
AutoToS prompt:

The task is a 5x5 mini crosswords. A state is a 2D array representing the current puzzle state, where the initial grid is all "None". Note that some of the possible answers are not exactly 5 character long. Given an input of possible answers to horizontal clues and vertical clues, how is the successor state function defined? Please first think step by step. Then provide the successor state function in Python code. 

The possible clues for each row and each column are given separately. Here is an example of possible horizontal and vertical clues:

horizontal_answers = [
        ["tasks", "goals", "plans", "agend", "chores", "works", "deeds", "items", "lists", "brief"],
        ["motor", "power", "drive", "diesel", "steam", "pumps", "crank", "gears", "turbn", "motor"],
        ["grand", "artsy", "showy", "ornate", "fancy", "vain", "proud", "vogue", "swank", "luxus"], 
        ["venue", "salle", "forum", "atria", "lobby", "parls", "court", "malls", "mall", "lobby"], 
        ["jeer", "scoff", "sleer", "deris", "sneer", "scorn", "derid", "gibes", "gibed", "flout"] 
]
vertical_answers = [
        ["amass", "stack", "hoard", "pile", "store", "heaps", "massy", "gathe", "lumps", "mound"],
        ["nilga", "goral", "eland", "lepus", "gazal", "kudu", "oryx", "gnu", "imps", "carb"],
        ["scheme", "design", "ettle", "nettle", "sting", "wiles", "plans", "ideas", "plots", "cocks"], 
        ["spout", "nosle", "snout", "mouth", "nostr", "ports", "inlet", "vents", "outlt", "beaks"], 
        ["drier", "arid", "sere", "parch", "dryer", "wring", "drear", "sear", "pall", "lack"]
]

where horizontal_answers is a list where element i is a list of possible answers to clue in row i, and vertical_answers is a list where element i is a list of possible answers to clue in column i.

Model response:
To generate the successor state function for a 5x5 mini crossword grid, you need to consider updating the state gradually by filling in words from the possible answers for horizontal and vertical clues. This function will return a list of valid successor states based on the next possible word for both horizontal and vertical clues.

When implementing this, you shall:
1. Define the initial empty 5x5 grid.
2. Iterate through each possible word for the horizontal clues and place them in the grid.
3. For each horizontal placement, iterate through each possible word for the vertical clues and place them in the grid.
4. Ensure that any new word placed must align and be valid with existing placements.

Here's the Python function:

```python
def successor_state(state, horizontal_answers, vertical_answers):
    import copy

    def is_valid_state(state):
        for i in range(5):
            horizontal_word = ''.join([state[i][j] if state[i][j] is not None else '*' for j in range(5)])
            vertical_word = ''.join([state[j][i] if state[j][i] is not None else '*' for j in range(5)])

            if horizontal_word.strip('*') and horizontal_word.strip('*') not in valid_horizontal_words[i]:
                return False
            if vertical_word.strip('*') and vertical_word.strip('*') not in valid_vertical_words[i]:
                return False
        return True

    def generate_successor_states(state, possible_answers, is_horizontal):
        successors = []
        for i in range(5):
            for word in possible_answers[i]:
                new_state = copy.deepcopy(state)
                if is_horizontal:
                    if len(word) <= 5:
                        for j in range(len(word)):
                            new_state[i][j] = word[j]
                else:
                    if len(word) <= 5:
                        for j in range(len(word)):
                            new_state[j][i] = word[j]
                
                if is_valid_state(new_state):
                    successors.append(new_state)
        return successors
    
    valid_horizontal_words = [
        set(horizontal_answers[i]) for i in range(5)
    ]
    valid_vertical_words = [
        set(vertical_answers[i]) for i in range(5)
    ]
              
    current_turn_horizontal = all(state[i][j] is None for i in range(5) for j in range(5))
    if current_turn_horizontal:
        return generate_successor_states(state, horizontal_answers, True)
    else:
        return generate_successor_states(state, vertical_answers, False)
```

### Explanation
- **`is_valid_state`**: This validates if the current state of the grid forms valid partial words horizontally and vertically.
- **`generate_successor_states`**: This generates successor states by attempting to place each possible word in the grid for either horizontal or vertical positions and returns valid new states.
- **Initial Check**: The function determines whether to place horizontal words first by checking if the state is entirely empty.

This function will proceed to fill the state with valid words step-by-step for both horizontal and vertical clues while ensuring intermediate states are valid.

AutoToS prompt:

Please provide a goal test function in Python ensuring that all cells are not None, all words horizontally are matching at least one of the horizontal answers, and all words vertically match at least one of the vertical answers. The function should receive a state, horizontal_answers, and vertical_answers as before and return True for a goal state and False otherwise.

Model response:
To implement a goal test function in Python, you need to ensure that:
1. All cells in the state are filled (i.e., not `None`).
2. Each horizontal word matches one of the horizontal answers.
3. Each vertical word matches one of the vertical answers.

Here is the function to achieve this:

```python
def is_goal_state(state, horizontal_answers, vertical_answers):
    def all_filled(state):
        for row in state:
            if any(cell is None for cell in row):
                return False
        return True

    def all_horizontal_words_valid(state, horizontal_answers):
        for i, row in enumerate(state):
            word = ''.join(row)
            if word not in horizontal_answers[i]:
                return False
        return True

    def all_vertical_words_valid(state, vertical_answers):
        for i in range(5):
            word = ''.join(state[j][i] for j in range(5))
            if word not in vertical_answers[i]:
                return False
        return True

    if not all_filled(state):
        return False
    if not all_horizontal_words_valid(state, horizontal_answers):
        return False
    if not all_vertical_words_valid(state, vertical_answers):
        return False
    
    return True
```

### Explanation
1. **`all_filled(state)`**: Checks that every cell in the 5x5 grid is filled (no `None` values).
2. **`all_horizontal_words_valid(state, horizontal_answers)`**: Checks that every horizontal word formed in the state matches one of the allowed horizontal answers.
3. **`all_vertical_words_valid(state, vertical_answers)`**: Checks that every vertical word formed in the state matches one of the allowed vertical answers.

The main function `is_goal_state` aggregates these checks, returning `True` only if all conditions are satisfied, otherwise `False`.

System message:
Goal Soundness Test Passed
Soundness test
Successor Iteration 1

AutoToS prompt:
Successor state has the same number of filled cells as the parent state.
Let's think step by step. First, think what you did wrong.
Then, think of in what ways successor state should be different from the parent state.
Then, provide the complete Python code for the revised successor function that returns a list of successor states.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.

Input state: [[None, None, None, None, None], [None, None, None, None, None], [None, None, None, None, None], [None, None, None, None, None], [None, None, None, None, None]]
Example wrong successor state: [[None, None, None, None, None], [None, None, None, None, None], [None, None, None, None, None], [None, None, None, None, None], [None, None, None, None, None]]
Horizontal clues: [['tasks', 'goals', 'plans', 'agend', 'chores', 'works', 'deeds', 'items', 'lists', 'brief'], ['motor', 'power', 'drive', 'diesel', 'steam', 'pumps', 'crank', 'gears', 'turbn', 'motor'], ['grand', 'artsy', 'showy', 'ornate', 'fancy', 'vain', 'proud', 'vogue', 'swank', 'luxus'], ['venue', 'salle', 'forum', 'atria', 'lobby', 'parls', 'court', 'malls', 'mall', 'lobby'], ['jeer', 'scoff', 'sleer', 'deris', 'sneer', 'scorn', 'derid', 'gibes', 'gibed', 'flout']]
Vertical clues: [['amass', 'stack', 'hoard', 'pile', 'store', 'heaps', 'massy', 'gathe', 'lumps', 'mound'], ['nilga', 'goral', 'eland', 'lepus', 'gazal', 'kudu', 'oryx', 'gnu', 'imps', 'carb'], ['scheme', 'design', 'ettle', 'nettle', 'sting', 'wiles', 'plans', 'ideas', 'plots', 'cocks'], ['spout', 'nosle', 'snout', 'mouth', 'nostr', 'ports', 'inlet', 'vents', 'outlt', 'beaks'], ['drier', 'arid', 'sere', 'parch', 'dryer', 'wring', 'drear', 'sear', 'pall', 'lack']]

Model response:
To address the issues with the original approach, let's think step-by-step:

1. **Identify the Mistakes**:
   - The original function attempted to generate successor states directly without guaranteeing any new cells were filled. Therefore, it could return states identical to the input.
   - It did not handle the transition from placing horizontal words to vertical words clearly, which must happen sequentially.

2. **Determine the Required Changes**:
   - A successor state should differ from the parent state by having at least one additional word placed in the grid.
   - Generate valid successor states by filling the grid according to the current phase (placing horizontal or vertical words). Start with horizontal placement and switch to vertical once all rows are attempted.

3. **Revised Successor Function**:
   - The function must clearly track the filling process and ensure progression, creating new states with progressively filled cells.

Here is the revised successor function which ensures that the successor state has more filled cells than the parent state and handles the transitions properly:

```python
def successor_state(state, horizontal_answers, vertical_answers):
    import copy

    def is_valid_state(state):
        for i in range(5):
            horizontal_word = ''.join([state[i][j] if state[i][j] is not None else '*' for j in range(5)])
            vertical_word = ''.join([state[j][i] if state[j][i] is not None else '*' for j in range(5)])

            if horizontal_word.strip('*') and horizontal_word.strip('*') not in valid_horizontal_words[i]:
                return False
            if vertical_word.strip('*') and vertical_word.strip('*') not in valid_vertical_words[i]:
                return False
        return True

    def generate_horizontal_successors(state):
        successors = []
        for i in range(5):
            if all(state[i][j] is None for j in range(5)):  # Find the first completely empty row
                for word in horizontal_answers[i]:
                    if len(word) <= 5:
                        new_state = copy.deepcopy(state)
                        for j in range(len(word)):
                            new_state[i][j] = word[j]

                        if is_valid_state(new_state):
                            successors.append(new_state)
                return successors
        return successors

    def generate_vertical_successors(state):
        successors = []
        for i in range(5):
            if all(state[j][i] is None for j in range(5)):  # Find the first completely empty column
                for word in vertical_answers[i]:
                    if len(word) <= 5:
                        new_state = copy.deepcopy(state)
                        for j in range(len(word)):
                            new_state[j][i] = word[j]
                        
                        if is_valid_state(new_state):
                            successors.append(new_state)
                return successors
        return successors
    
    valid_horizontal_words = [
        set(horizontal_answers[i]) for i in range(5)
    ]
    valid_vertical_words = [
        set(vertical_answers[i]) for i in range(5)
    ]

    # Determine if the grid has completely empty rows or columns to decide mode
    horizontal_successors = generate_horizontal_successors(state)
    if horizontal_successors:
        return horizontal_successors
    
    vertical_successors = generate_vertical_successors(state)
    if vertical_successors:
        return vertical_successors
    
    return []

```

### Explanation in Detail
1. **`is_valid_state`**: Ensures that horizontal and vertical partial words formed in the grid so far are valid.
2. **`generate_horizontal_successors`**: Generates valid successor states by attempting to fill the first completely empty row with each possible horizontal word from the clues.
3. **`generate_vertical_successors`**: Generates valid successor states by attempting to fill the first completely empty column with each possible vertical word from the clues.
4. **Determine Mode**: The function first tries to generate successors by filling horizontal words. If this is not possible (all rows are filled), it generates successors by filling vertical words.
5. **Sequential Progression**: Ensures that once all horizontal placements are attempted, it will move to vertical placements.

This approach ensures a valid progression from the parent to successor states, and that each new state has more filled cells than its predecessor.

System message:
Successor States Soundness Test Passed
Completeness test
Successor Iteration 2

AutoToS prompt:
Successor function when run on the state [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]] failed to produce all successors.
Missing successors are: [[['a', 'g', 'e', 'n', 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]], [['d', 'e', 'e', 'd', 's'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]], [['i', 't', 'e', 'm', 's'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], ['a', 'r', 't', 's', 'y'], [None, None, 'l', None, None], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], ['s', 'a', 'l', 'l', 'e'], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], ['m', 'a', 'l', 'l', 's'], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], ['s', 'l', 'e', 'e', 'r']], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], ['s', 'n', 'e', 'e', 'r']], [['a', None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], ['a', None, 't', None, None], ['s', None, 'l', None, None], ['s', None, 'e', None, None]], [[None, 'g', 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, 'r', 't', None, None], [None, 'a', 'l', None, None], [None, 'l', 'e', None, None]], [[None, None, 'e', 'n', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 's', None], [None, None, 'l', 'l', None], [None, None, 'e', 'e', None]], [[None, None, 'e', 'm', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 'u', None], [None, None, 'l', 't', None], [None, None, 'e', 'h', None]], [[None, None, 'e', 'n', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 's', None], [None, None, 'l', 't', None], [None, None, 'e', 'r', None]], [[None, None, 'e', 'p', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 'r', None], [None, None, 'l', 't', None], [None, None, 'e', 's', None]], [[None, None, 'e', None, 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'i'], [None, None, 'l', None, 'e'], [None, None, 'e', None, 'r']], [[None, None, 'e', None, 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'y'], [None, None, 'l', None, 'e'], [None, None, 'e', None, 'r']], [[None, None, 'e', None, 'w'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'i'], [None, None, 'l', None, 'n'], [None, None, 'e', None, 'g']], [[None, None, 'e', None, 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'e'], [None, None, 'l', None, 'a'], [None, None, 'e', None, 'r']]]
First think step by step why the successor function failed to produce all successors of the state.
Then, fix the successor function.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.
Horizontal clues: [['tasks', 'goals', 'plans', 'agend', 'chores', 'works', 'deeds', 'items', 'lists', 'brief'], ['motor', 'power', 'drive', 'diesel', 'steam', 'pumps', 'crank', 'gears', 'turbn', 'motor'], ['grand', 'artsy', 'showy', 'ornate', 'fancy', 'vain', 'proud', 'vogue', 'swank', 'luxus'], ['venue', 'salle', 'forum', 'atria', 'lobby', 'parls', 'court', 'malls', 'mall', 'lobby'], ['jeer', 'scoff', 'sleer', 'deris', 'sneer', 'scorn', 'derid', 'gibes', 'gibed', 'flout']]
Vertical clues: [['amass', 'stack', 'hoard', 'pile', 'store', 'heaps', 'massy', 'gathe', 'lumps', 'mound'], ['nilga', 'goral', 'eland', 'lepus', 'gazal', 'kudu', 'oryx', 'gnu', 'imps', 'carb'], ['scheme', 'design', 'ettle', 'nettle', 'sting', 'wiles', 'plans', 'ideas', 'plots', 'cocks'], ['spout', 'nosle', 'snout', 'mouth', 'nostr', 'ports', 'inlet', 'vents', 'outlt', 'beaks'], ['drier', 'arid', 'sere', 'parch', 'dryer', 'wring', 'drear', 'sear', 'pall', 'lack']]

Model response:
To understand why the successor function failed to produce all successors, we need to reconsider the scope and exact procedure of generating successors. The function must generate valid states by extending the current state through placing valid words in the next available row or column, one at a time. 

Here are the key steps to fix the function:
1. Ensure the function generates successors by filling one empty row or column at a time.
2. Validate that placing a word in a row or column doesn't invalidate the grid.
3. Differentiate between generating horizontal or vertical successors based on the current state.

### Revised Plan
1. Identify the first unfilled row for horizontal successors, and the first unfilled column for vertical successors.
2. Generate all possible states by filling this row or column with every possible word from the respective answers.
3. Each state must align with previously filled parts of the grid.

### Revised Successor Function
Here is the revised code with enhanced logic to ensure all valid successors are generated:

```python
def successor_state(state, horizontal_answers, vertical_answers):
    import copy

    def is_valid_state(state):
        for i in range(5):
            horizontal_word = ''.join([state[i][j] if state[i][j] is not None else '*' for j in range(5)])
            vertical_word = ''.join([state[j][i] if state[j][i] is not None else '*' for j in range(5)])

            if horizontal_word.strip('*') and horizontal_word.strip('*') not in valid_horizontal_words[i]:
                return False
            if vertical_word.strip('*') and vertical_word.strip('*') not in valid_vertical_words[i]:
                return False
        return True

    def generate_horizontal_successors(state):
        successors = []
        for i in range(5):
            if any(state[i][j] is None for j in range(5)):
                for word in horizontal_answers[i]:
                    if len(word) <= 5:
                        new_state = copy.deepcopy(state)
                        can_fit = True
                        for j in range(len(word)):
                            if state[i][j] is not None and state[i][j] != word[j]:
                                can_fit = False
                                break
                            new_state[i][j] = word[j]
                        if can_fit and is_valid_state(new_state):
                            successors.append(new_state)
                break
        return successors

    def generate_vertical_successors(state):
        successors = []
        for i in range(5):
            if any(state[j][i] is None for j in range(5)):
                for word in vertical_answers[i]:
                    if len(word) <= 5:
                        new_state = copy.deepcopy(state)
                        can_fit = True
                        for j in range(len(word)):
                            if state[j][i] is not None and state[j][i] != word[j]:
                                can_fit = False
                                break
                            new_state[j][i] = word[j]
                        if can_fit and is_valid_state(new_state):
                            successors.append(new_state)
                break
        return successors
    
    valid_horizontal_words = [
        set(horizontal_answers[i]) for i in range(5)
    ]
    valid_vertical_words = [
        set(vertical_answers[i]) for i in range(5)
    ]

    horizontal_successors = generate_horizontal_successors(state)
    if horizontal_successors:
        return horizontal_successors

    vertical_successors = generate_vertical_successors(state)
    if vertical_successors:
        return vertical_successors
    
    return []
```

### Explanation:
1. **`is_valid_state`**: Ensures partial words formed so far match the clues.
2. **`generate_horizontal_successors`**:
   - Finds the first row with any `None`.
   - Tries to place each possible word from the horizontal answers in that row.
   - Checks if the word aligns with any pre-filled cells and if the resulting state is valid.
3. **`generate_vertical_successors`**:
   - Similar logic to horizontal successors but for columns.
4. **Generation Control**:
   - Tries to generate horizontal successors first. If none are found (i.e., all rows are filled), then it tries vertical successors.

### Conclusion:
This revised function ensures that the successor states progressively fill unknown cells, thereby generating complete sets of valid successor states.

System message:
Successor States Soundness Test Passed
Completeness test
Successor Iteration 3

AutoToS prompt:
Successor function when run on the state [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]] failed to produce all successors.
Missing successors are: [[['a', 'g', 'e', 'n', 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]], [['d', 'e', 'e', 'd', 's'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]], [['i', 't', 'e', 'm', 's'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], ['a', 'r', 't', 's', 'y'], [None, None, 'l', None, None], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], ['s', 'a', 'l', 'l', 'e'], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], ['m', 'a', 'l', 'l', 's'], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], ['s', 'l', 'e', 'e', 'r']], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], ['s', 'n', 'e', 'e', 'r']], [['a', None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], ['a', None, 't', None, None], ['s', None, 'l', None, None], ['s', None, 'e', None, None]], [[None, 'g', 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, 'r', 't', None, None], [None, 'a', 'l', None, None], [None, 'l', 'e', None, None]], [[None, None, 'e', 'n', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 's', None], [None, None, 'l', 'l', None], [None, None, 'e', 'e', None]], [[None, None, 'e', 'm', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 'u', None], [None, None, 'l', 't', None], [None, None, 'e', 'h', None]], [[None, None, 'e', 'n', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 's', None], [None, None, 'l', 't', None], [None, None, 'e', 'r', None]], [[None, None, 'e', 'p', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 'r', None], [None, None, 'l', 't', None], [None, None, 'e', 's', None]], [[None, None, 'e', None, 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'i'], [None, None, 'l', None, 'e'], [None, None, 'e', None, 'r']], [[None, None, 'e', None, 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'y'], [None, None, 'l', None, 'e'], [None, None, 'e', None, 'r']], [[None, None, 'e', None, 'w'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'i'], [None, None, 'l', None, 'n'], [None, None, 'e', None, 'g']], [[None, None, 'e', None, 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'e'], [None, None, 'l', None, 'a'], [None, None, 'e', None, 'r']]]
First think step by step why the successor function failed to produce all successors of the state.
Then, fix the successor function.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.
Horizontal clues: [['tasks', 'goals', 'plans', 'agend', 'chores', 'works', 'deeds', 'items', 'lists', 'brief'], ['motor', 'power', 'drive', 'diesel', 'steam', 'pumps', 'crank', 'gears', 'turbn', 'motor'], ['grand', 'artsy', 'showy', 'ornate', 'fancy', 'vain', 'proud', 'vogue', 'swank', 'luxus'], ['venue', 'salle', 'forum', 'atria', 'lobby', 'parls', 'court', 'malls', 'mall', 'lobby'], ['jeer', 'scoff', 'sleer', 'deris', 'sneer', 'scorn', 'derid', 'gibes', 'gibed', 'flout']]
Vertical clues: [['amass', 'stack', 'hoard', 'pile', 'store', 'heaps', 'massy', 'gathe', 'lumps', 'mound'], ['nilga', 'goral', 'eland', 'lepus', 'gazal', 'kudu', 'oryx', 'gnu', 'imps', 'carb'], ['scheme', 'design', 'ettle', 'nettle', 'sting', 'wiles', 'plans', 'ideas', 'plots', 'cocks'], ['spout', 'nosle', 'snout', 'mouth', 'nostr', 'ports', 'inlet', 'vents', 'outlt', 'beaks'], ['drier', 'arid', 'sere', 'parch', 'dryer', 'wring', 'drear', 'sear', 'pall', 'lack']]

Model response:
To understand why the previous successor function failed, let's carefully examine our logic:

1. **Current Problems**:
   - The previous function did not correctly identify and handle rows and columns that were partially filled.
   - The iteration and placement logic were not robust enough to cover all cases of partial fillings.

2. **Required Corrections**:
   - Identify both fully and partially filled rows/columns correctly.
   - Ensure the function correctly places words by matching partially filled cells.
   - Generate all valid successors by placing words in the next available row or column one at a time.

### Revised Successor Function
Here is the corrected and more comprehensive function:

```python
def successor_state(state, horizontal_answers, vertical_answers):
    import copy

    def is_valid_state(state):
        for i in range(5):
            # Check each horizontal word
            horizontal_word = ''.join([state[i][j] if state[i][j] is not None else '*' for j in range(5)])
            if '*' in horizontal_word:
                horizontal_word = horizontal_word.split('*')[0]
            if horizontal_word and horizontal_word not in valid_horizontal_words[i]:
                return False
            
            # Check each vertical word
            vertical_word = ''.join([state[j][i] if state[j][i] is not None else '*' for j in range(5)])
            if '*' in vertical_word:
                vertical_word = vertical_word.split('*')[0]
            if vertical_word and vertical_word not in valid_vertical_words[i]:
                return False
        return True

    def generate_successors(state, possible_answers, is_horizontal):
        successors = []
        for i in range(5):
            indices_to_fill = [None] * 5
            valid_row = True
            for j in range(5 if is_horizontal else 5):
                if is_horizontal:
                    if state[i][j] is not None:
                        indices_to_fill[j] = state[i][j]
                else:
                    if state[j][i] is not None:
                        indices_to_fill[j] = state[j][i]

            if all(ch is not None for ch in indices_to_fill):  # Skip fully filled rows/columns
                continue
            
            for word in possible_answers[i]:
                new_state = copy.deepcopy(state)
                can_fit = True
                for j, ch in enumerate(word):
                    if indices_to_fill[j] is not None and indices_to_fill[j] != ch:
                        can_fit = False
                        break
                    if is_horizontal:
                        new_state[i][j] = ch
                    else:
                        new_state[j][i] = ch
                
                if can_fit and is_valid_state(new_state):
                    successors.append(new_state)
            if successors:  # If we're able to generate at least one valid successor, stop and return
                return successors
        return []

    valid_horizontal_words = [
        set(horizontal_answers[i]) for i in range(5)
    ]
    valid_vertical_words = [
        set(vertical_answers[i]) for i in range(5)
    ]

    # Try to generate horizontal successors first, if no valid horizontal successors, then vertical.
    horizontal_successors = generate_successors(state, horizontal_answers, True)
    if horizontal_successors:
        return horizontal_successors

    vertical_successors = generate_successors(state, vertical_answers, False)
    if vertical_successors:
        return vertical_successors
    
    return []
```

### Explanation:
- **`is_valid_state(state)`**: Checks partial words horizontally and vertically to ensure they are valid sketched words.
    - Horizontal and vertical partial word checks include splitting by '*' to handle partially filled cells efficiently.
- **`generate_successors(state, possible_answers, is_horizontal)`**:
    - Determines whether to generate horizontal or vertical successors.
    - Iterates through each row/column and attempts to place possible words.
    - Ensures any new word placement aligns with the already placed characters.
    - Generates new states only if valid placements align with the current grid's state.
    - Stops and returns valid successors as soon as they are found, ensuring efficiency.

This function ensures that all possibilities are considered, and all valid successors are generated by properly extending the current state step-by-step.
Successor Iteration 4

AutoToS prompt:
The following exception was produced when testing the successor function: 
  File "<string>", line 41, in generate_successors

list index out of range. Please fix this exception in the successor function.
Model response:
It looks like the function may be attempting to access indices that are out of bounds due to incorrect logic handling both horizontal and vertical expansions separately. This mistake usually happens when the function tries to process partially filled rows or columns with mismatched lengths.

### Solution:
1. Ensure correct indexing within the bounds of the 5x5 grid.
2. Split and check each potential word placement to match the exact indices in the grid.
3. Fix any off-by-one errors by carefully managing the row and column limits.

Here is the revised function:

```python
def successor_state(state, horizontal_answers, vertical_answers):
    import copy

    def is_valid_state(state):
        for i in range(5):
            # Check each horizontal word
            horizontal_word = ''.join([state[i][j] if state[i][j] is not None else '*' for j in range(5)])
            if '*' in horizontal_word:
                horizontal_word = horizontal_word.split('*')[0]
            if horizontal_word and horizontal_word not in valid_horizontal_words[i]:
                return False
            
            # Check each vertical word
            vertical_word = ''.join([state[j][i] if state[j][i] is not None else '*' for j in range(5)])
            if '*' in vertical_word:
                vertical_word = vertical_word.split('*')[0]
            if vertical_word and vertical_word not in valid_vertical_words[i]:
                return False
        return True

    def generate_successors(state, possible_answers, is_horizontal):
        successors = []
        for i in range(5):
            if is_horizontal:
                indices_to_fill = [state[i][j] for j in range(5)]
            else:
                indices_to_fill = [state[j][i] for j in range(5)]

            if all(ch is not None for ch in indices_to_fill):  # Skip fully filled rows/columns
                continue
            
            for word in possible_answers[i]:
                new_state = copy.deepcopy(state)
                can_fit = True

                for j, ch in enumerate(word):
                    if (is_horizontal and state[i][j] is not None and state[i][j] != ch) or (not is_horizontal and state[j][i] is not None and state[j][i] != ch):
                        can_fit = False
                        break
                    if is_horizontal:
                        new_state[i][j] = ch
                    else:
                        new_state[j][i] = ch

                if can_fit and is_valid_state(new_state):
                    successors.append(new_state)

            # When an unfilled row/column is processed, attempt to return as soon as successors are found
            if successors:
                return successors

        return []

    valid_horizontal_words = [set(horizontal_answers[i]) for i in range(5)]
    valid_vertical_words = [set(vertical_answers[i]) for i in range(5)]

    # Check if we should generate horizontal or vertical successors based on the current fill status.
    horizontal_successors = generate_successors(state, horizontal_answers, True)
    if horizontal_successors:
        return horizontal_successors

    vertical_successors = generate_successors(state, vertical_answers, False)
    if vertical_successors:
        return vertical_successors

    return []
```

### Explanation:
1. **`is_valid_state(state)`**: Checks the grid to ensure partial and completed words are valid based on the provided answers. It handles both horizontal and vertical words using consistent bounds.
2. **`generate_successors(state, possible_answers, is_horizontal)`**:
    - Determines whether to fill rows (horizontal) or columns (vertical).
    - Ensures all indices accessed are within bounds (0 to 4) for the 5x5 grid.
    - Manages potential word placements and checks if they align with partially filled states.
    - Ensures any new state is valid before adding to the successors list.

This function maintains better indexing logic and handles both row and column expansions appropriately, preventing the index out-of-range error.
Successor Iteration 5

AutoToS prompt:
The following exception was produced when testing the successor function: 
  File "<string>", line 37, in generate_successors

list index out of range. Please fix this exception in the successor function.
Model response:
To fix the "list index out of range" error, we need to carefully handle the bounds and ensure we don't access indices outside of the 5x5 grid. Let's refine the logic to ensure all operations are within valid indices.

### Detailed Steps:

1. **Ensure correct bounds checking**: Make sure we are accessing only within the valid 5x5 grid.
2. **Correctly handle partially filled rows/columns**: Ensure methods for checking and placing words align with the grid's state without exceeding boundaries.
3. **Refine logic to prevent off-by-one errors**.

Here's the improved function checking for bounds explicitly:

```python
def successor_state(state, horizontal_answers, vertical_answers):
    import copy
    
    def is_valid_state(state):
        for i in range(5):
            # Check each horizontal word
            horizontal_word = ''.join([state[i][j] if state[i][j] is not None else '*' for j in range(5)])
            if '*' in horizontal_word:
                fragments = horizontal_word.split('*')
                horizontal_word = fragments[0] if fragments[0] else (fragments[1] if len(fragments) > 1 else '')
            if horizontal_word and horizontal_word not in valid_horizontal_words[i]:
                return False
            
            # Check each vertical word
            vertical_word = ''.join([state[j][i] if state[j][i] is not None else '*' for j in range(5)])
            if '*' in vertical_word:
                fragments = vertical_word.split('*')
                vertical_word = fragments[0] if fragments[0] else (fragments[1] if len(fragments) > 1 else '')
            if vertical_word and vertical_word not in valid_vertical_words[i]:
                return False
        return True

    def generate_successors(state, possible_answers, is_horizontal):
        successors = []
        for i in range(5):
            indices_to_fill = []
            valid_row = True
            for j in range(5):
                if is_horizontal:
                    if state[i][j] is not None:
                        indices_to_fill.append(state[i][j])
                    else:
                        indices_to_fill.append(None)
                else:
                    if state[j][i] is not None:
                        indices_to_fill.append(state[j][i])
                    else:
                        indices_to_fill.append(None)

            if all(ch is not None for ch in indices_to_fill):  # Skip fully filled rows/columns
                continue
            
            for word in possible_answers[i]:
                new_state = copy.deepcopy(state)
                can_fit = True
                for j, ch in enumerate(word):
                    if indices_to_fill[j] is not None and indices_to_fill[j] != ch:
                        can_fit = False
                        break
                    if is_horizontal:
                        new_state[i][j] = ch
                    else:
                        new_state[j][i] = ch
                if can_fit and is_valid_state(new_state):
                    successors.append(new_state)
            
            if successors:
                return successors  # Return after first successful row/column
        
        return []

    valid_horizontal_words = [set(horizontal_answers[i]) for i in range(5)]
    valid_vertical_words = [set(vertical_answers[i]) for i in range(5)]

    # Try to generate horizontal successors first, if no valid horizontal successors, then vertical.
    horizontal_successors = generate_successors(state, horizontal_answers, True)
    if horizontal_successors:
        return horizontal_successors

    vertical_successors = generate_successors(state, vertical_answers, False)
    if vertical_successors:
        return vertical_successors

    return []
```

### Explanation:
- **Index Safety**: Managed indices through a bounded loop, ensuring any list access is within bounds (0 to 4).
- **Word Placement**: Added word placement logic that respects the filled state of each cell.
- **Validation Checks**: Included checks to ensure any modification stays within valid rows/columns, making sure the state update is safe.

This approach should prevent "list index out of range" errors by carefully controlling index access within the 5x5 limits.
Successor Iteration 6

AutoToS prompt:
The following exception was produced when testing the successor function: 
  File "<string>", line 47, in generate_successors

list index out of range. Please fix this exception in the successor function.
Model response:
The "list index out of range" error typically arises from trying to access or iterate beyond the bounds of the list. I will carefully review the logic to ensure we access indices properly within the 5x5 grid.

Let's refine the function to prevent such errors:

1. We need to confirm that for each word we try to place horizontally or vertically, it doesn't exceed the bounds of the 5x5 grid.
2. Fix any indexing and bounds checking errors to make sure we stay within the correct range.

Here is the revised function:

```python
def successor_state(state, horizontal_answers, vertical_answers):
    import copy

    def is_valid_state(state):
        for i in range(5):
            # Check each horizontal word
            horizontal_word = ''.join([state[i][j] if state[i][j] is not None else '*' for j in range(5)])
            if '*' in horizontal_word:
                fragments = horizontal_word.split('*')
                horizontal_word = fragments[0] if fragments[0] else (fragments[1] if len(fragments) > 1 else '')
            if horizontal_word and horizontal_word not in valid_horizontal_words[i]:
                return False
            
            # Check each vertical word
            vertical_word = ''.join([state[j][i] if state[j][i] is not None else '*' for j in range(5)])
            if '*' in vertical_word:
                fragments = vertical_word.split('*')
                vertical_word = fragments[0] if fragments[0] else (fragments[1] if len(fragments) > 1 else '')
            if vertical_word and vertical_word not in valid_vertical_words[i]:
                return False
        return True

    def generate_successors(state, possible_answers, is_horizontal):
        successors = []
        for i in range(5):
            if is_horizontal:
                current_line = [state[i][j] for j in range(5)]
            else:
                current_line = [state[j][i] for j in range(5)]

            if all(ch is not None for ch in current_line):  # Skip fully filled rows/columns
                continue

            for word in possible_answers[i]:
                if len(word) > 5:
                    continue
                new_state = copy.deepcopy(state)
                can_fit = True
                for j in range(5):
                    ch = word[j] if j < len(word) else None
                    if current_line[j] is not None and current_line[j] != ch:
                        can_fit = False
                        break
                    if is_horizontal and ch is not None:
                        new_state[i][j] = ch
                    elif not is_horizontal and ch is not None:
                        new_state[j][i] = ch
                if can_fit and is_valid_state(new_state):
                    successors.append(new_state)

            if successors:
                return successors  # Return successors after processing the first row/column with empty cells

        return successors

    valid_horizontal_words = [set(horizontal_answers[i]) for i in range(5)]
    valid_vertical_words = [set(vertical_answers[i]) for i in range(5)]

    # Try to generate horizontal successors first, if no valid horizontal successors, then vertical.
    horizontal_successors = generate_successors(state, horizontal_answers, True)
    if horizontal_successors:
        return horizontal_successors

    vertical_successors = generate_successors(state, vertical_answers, False)
    if vertical_successors:
        return vertical_successors

    return []

```

### Explanation:
1. **`is_valid_state(state)`**: Checks whether horizontal and vertical words formed match the given clues, managing splitting by `'*'` properly.
2. **`generate_successors(state, possible_answers, is_horizontal)`**:
   - It identifies whether to work on rows (horizontal) or columns (vertical).
   - It constructs `current_line` from the grid and guarantees it won't exceed the bounds (`len(word) <= 5`).
   - Logic now carefully ensures word placement does not extend beyond the 5x5 range.
   - After processing the first row/column with empty cells, it returns the generated successors.

This refined approach should manage indices within the expected range of the grid and prevent "list index out of range" errors.
Successor Iteration 7

AutoToS prompt:
Successor state has the same number of filled cells as the parent state.
Let's think step by step. First, think what you did wrong.
Then, think of in what ways successor state should be different from the parent state.
Then, provide the complete Python code for the revised successor function that returns a list of successor states.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.

Input state: [[None, None, None, None, None], [None, None, None, None, None], ['l', 'u', 'x', 'u', 's'], ['l', 'o', 'b', 'b', 'y'], ['j', 'e', 'e', 'r', None]]
Example wrong successor state: [[None, None, None, None, None], [None, None, None, None, None], ['l', 'u', 'x', 'u', 's'], ['l', 'o', 'b', 'b', 'y'], ['j', 'e', 'e', 'r', None]]
Horizontal clues: [['tasks', 'goals', 'plans', 'agend', 'chores', 'works', 'deeds', 'items', 'lists', 'brief'], ['motor', 'power', 'drive', 'diesel', 'steam', 'pumps', 'crank', 'gears', 'turbn', 'motor'], ['grand', 'artsy', 'showy', 'ornate', 'fancy', 'vain', 'proud', 'vogue', 'swank', 'luxus'], ['venue', 'salle', 'forum', 'atria', 'lobby', 'parls', 'court', 'malls', 'mall', 'lobby'], ['jeer', 'scoff', 'sleer', 'deris', 'sneer', 'scorn', 'derid', 'gibes', 'gibed', 'flout']]
Vertical clues: [['amass', 'stack', 'hoard', 'pile', 'store', 'heaps', 'massy', 'gathe', 'lumps', 'mound'], ['nilga', 'goral', 'eland', 'lepus', 'gazal', 'kudu', 'oryx', 'gnu', 'imps', 'carb'], ['scheme', 'design', 'ettle', 'nettle', 'sting', 'wiles', 'plans', 'ideas', 'plots', 'cocks'], ['spout', 'nosle', 'snout', 'mouth', 'nostr', 'ports', 'inlet', 'vents', 'outlt', 'beaks'], ['drier', 'arid', 'sere', 'parch', 'dryer', 'wring', 'drear', 'sear', 'pall', 'lack']]

Model response:
To address the issue where the successor state has the same number of filled cells as the parent state, let's revise our approach. 

### Key Points to Fix:
1. **Incremental Filling**: The successor state should have at least one new cell filled compared to the parent state.
2. **Sequential Filling**: Fill cells row-wise starting with the first incomplete row and column-wise starting with the first incomplete column.
3. **Mixed Priority**: Prioritize horizontal filling first, then vertical.

### Steps:
1. **Identify the first empty row for horizontal filling**.
2. **Identify the first empty column for vertical filling**.
3. **Ensure incremental state updates to add at least one new word**.

### Revised Successor Function:

```python
def successor_state(state, horizontal_answers, vertical_answers):
    import copy
    
    def is_valid_state(state):
        for i in range(5):
            # Check each horizontal word
            horizontal_word = ''.join([state[i][j] if state[i][j] is not None else '*' for j in range(5)])
            if '*' in horizontal_word:
                fragments = horizontal_word.split('*')
                if fragments[0]:
                    horizontal_word = fragments[0]
                elif len(fragments) > 1:
                    horizontal_word = fragments[1]
                else:
                    horizontal_word = ''
            if horizontal_word and horizontal_word not in valid_horizontal_words[i]:
                return False
            
            # Check each vertical word
            vertical_word = ''.join([state[j][i] if state[j][i] is not None else '*' for j in range(5)])
            if '*' in vertical_word:
                fragments = vertical_word.split('*')
                if fragments[0]:
                    vertical_word = fragments[0]
                elif len(fragments) > 1:
                    vertical_word = fragments[1]
                else:
                    vertical_word = ''
            if vertical_word and vertical_word not in valid_vertical_words[i]:
                return False
        return True

    def generate_successors(state, possible_answers, is_horizontal):
        successors = []
        for i in range(5):
            if is_horizontal:
                current_line = [state[i][j] for j in range(5)]
            else:
                current_line = [state[j][i] for j in range(5)]

            if all(ch is not None for ch in current_line):  # Skip fully filled rows/columns
                continue

            for word in possible_answers[i]:
                if len(word) > 5:
                    continue
                new_state = copy.deepcopy(state)
                can_fit = True

                for j in range(5):
                    ch = word[j] if j < len(word) else None
                    if current_line[j] is not None and current_line[j] != ch:
                        can_fit = False
                        break
                    if is_horizontal and ch is not None:
                        new_state[i][j] = ch
                    elif not is_horizontal and ch is not None:
                        new_state[j][i] = ch
                if can_fit and is_valid_state(new_state):
                    successors.append(new_state)
                    break  # Ensure we only get one valid successor at a time
            if successors:
                return successors  # Return successors after processing the first row/column with empty cells

        return []

    valid_horizontal_words = [set(horizontal_answers[i]) for i in range(5)]
    valid_vertical_words = [set(vertical_answers[i]) for i in range(5)]

    # Try to generate horizontal successors first, if no valid horizontal successors, then vertical.
    horizontal_successors = generate_successors(state, horizontal_answers, True)
    if horizontal_successors:
        return horizontal_successors

    vertical_successors = generate_successors(state, vertical_answers, False)
    if vertical_successors:
        return vertical_successors

    return []
```

### Explanation:
1. **`is_valid_state(state)`**: Checks whether partial words formed horizontally and vertically are in the valid set.
   - Handles separating valid parts of the word via `'*'`.
2. **`generate_successors(state, possible_answers, is_horizontal)`**:
   - Determines whether to fill rows (horizontal) or columns (vertical).
   - Iterates through each row/column and attempts to place each possible word.
   - It ensures new states have at least one more cell filled by breaking after finding a valid word for the first row/column with `None` cells.
   - Manages access within bounds (0 to 4) for the 5x5 grid.
3. **Sequential Filling**: Ensures horizontal successors are attempted first. If none, vertical successors are attempted next.

This revised function aims to ensure incremental progression in each successor state, incrementally filling the grid correctly and ensuring valid placements.
Successor Iteration 8

AutoToS prompt:
Successor state has the same number of filled cells as the parent state.
Let's think step by step. First, think what you did wrong.
Then, think of in what ways successor state should be different from the parent state.
Then, provide the complete Python code for the revised successor function that returns a list of successor states.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.

Input state: [[None, None, None, None, None], [None, None, None, None, None], ['g', 'r', 'a', 'n', 'd'], ['v', 'e', 'n', 'u', 'e'], ['j', 'e', 'e', 'r', None]]
Example wrong successor state: [[None, None, None, None, None], [None, None, None, None, None], ['g', 'r', 'a', 'n', 'd'], ['v', 'e', 'n', 'u', 'e'], ['j', 'e', 'e', 'r', None]]
Horizontal clues: [['tasks', 'goals', 'plans', 'agend', 'chores', 'works', 'deeds', 'items', 'lists', 'brief'], ['motor', 'power', 'drive', 'diesel', 'steam', 'pumps', 'crank', 'gears', 'turbn', 'motor'], ['grand', 'artsy', 'showy', 'ornate', 'fancy', 'vain', 'proud', 'vogue', 'swank', 'luxus'], ['venue', 'salle', 'forum', 'atria', 'lobby', 'parls', 'court', 'malls', 'mall', 'lobby'], ['jeer', 'scoff', 'sleer', 'deris', 'sneer', 'scorn', 'derid', 'gibes', 'gibed', 'flout']]
Vertical clues: [['amass', 'stack', 'hoard', 'pile', 'store', 'heaps', 'massy', 'gathe', 'lumps', 'mound'], ['nilga', 'goral', 'eland', 'lepus', 'gazal', 'kudu', 'oryx', 'gnu', 'imps', 'carb'], ['scheme', 'design', 'ettle', 'nettle', 'sting', 'wiles', 'plans', 'ideas', 'plots', 'cocks'], ['spout', 'nosle', 'snout', 'mouth', 'nostr', 'ports', 'inlet', 'vents', 'outlt', 'beaks'], ['drier', 'arid', 'sere', 'parch', 'dryer', 'wring', 'drear', 'sear', 'pall', 'lack']]

Model response:
First, let's analyze why our previous implementations did not generate any successors with a different number of filled cells compared to the parent state:
1. **Mistake Analysis**: The previous logic might be prioritizing certain conditions, hence not covering all possible valid placements. We need to revise how we handle rows and columns to ensure at least one new entry per successor state.
2. **Criteria for Successor State**: A successor should have more filled cells than its parent state and should be selected according to the correct word clues (horizontal or vertical).

### Revised Plan
1. **Identify the first row or column with empty cells** and attempt to fill it.
2. **Generate all valid states from the initial state by filling in additional words**.
3. **Each successor state must have at least one additional filled cell** compared to the parent state.
4. **Ensure that word placements do not violate the valid grid state**.

Here's the corrected function:

```python
def successor_state(state, horizontal_answers, vertical_answers):
    import copy
    
    def is_valid_state(state):
        for i in range(5):
            horizontal_word = ''.join([state[i][j] if state[i][j] is not None else '*' for j in range(5)])
            vertical_word = ''.join([state[j][i] if state[j][i] is not None else '*' for j in range(5)])

            if '*' in horizontal_word:
                partials = horizontal_word.split('*')
                if not all(part == '' or part in valid_horizontal_words[i] for part in partials):
                    return False
            elif horizontal_word and horizontal_word not in valid_horizontal_words[i]:
                return False

            if '*' in vertical_word:
                partials = vertical_word.split('*')
                if not all(part == '' or part in valid_vertical_words[i] for part in partials):
                    return False
            elif vertical_word and vertical_word not in valid_vertical_words[i]:
                return False
        return True

    def generate_successors(state, possible_answers, is_horizontal):
        successors = []
        for i in range(5):
            if is_horizontal:
                current_line = [state[i][j] for j in range(5)]
            else:
                current_line = [state[j][i] for j in range(5)]

            if all(ch is not None for ch in current_line):  # Skip fully filled rows/columns
                continue

            for word in possible_answers[i]:
                if len(word) > 5:
                    continue

                new_state = copy.deepcopy(state)
                can_fit = True

                for j in range(5):
                    ch = word[j] if j < len(word) else None
                    if current_line[j] is not None and current_line[j] != ch:
                        can_fit = False
                        break
                    if is_horizontal and ch is not None:
                        new_state[i][j] = ch
                    elif not is_horizontal and ch is not None:
                        new_state[j][i] = ch

                if can_fit and is_valid_state(new_state):
                    successors.append(new_state)
            
            if successors:
                return successors  # Stop after generating the first set of valid successors

        return []

    valid_horizontal_words = [set(horizontal_answers[i]) for i in range(5)]
    valid_vertical_words = [set(vertical_answers[i]) for i in range(5)]

    # First try to generate horizontal successors
    row_successors = generate_successors(state, horizontal_answers, True)
    if row_successors:
        return row_successors

    # Then try to generate vertical successors
    col_successors = generate_successors(state, vertical_answers, False)
    if col_successors:
        return col_successors

    return []
```

### Explanation:
1. **`is_valid_state(state)`**:
   - Checks for valid partial words both horizontally and vertically.
   - Splits words by `'*'` to manage and check partial phrases.
2. **`generate_successors(state, possible_answers, is_horizontal)`**:
   - Handles both horizontal and vertical filling.
   - Tries placing each word in the specified row/column.
   - Ensures that successors contain at least one additional filled cell compared to the parent state.
   - Only one row or column is processed to ensure incremental state filling.

This approach ensures that each successor correctly extends the state by filling in new words while maintaining validity constraints.

System message:
Successor States Soundness Test Passed
Completeness test
Successor Iteration 9

AutoToS prompt:
Successor function when run on the state [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]] failed to produce all successors.
Missing successors are: [[['a', 'g', 'e', 'n', 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]], [['d', 'e', 'e', 'd', 's'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]], [['i', 't', 'e', 'm', 's'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], ['a', 'r', 't', 's', 'y'], [None, None, 'l', None, None], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], ['s', 'a', 'l', 'l', 'e'], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], ['m', 'a', 'l', 'l', 's'], [None, None, 'e', None, None]], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], ['s', 'l', 'e', 'e', 'r']], [[None, None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, None], [None, None, 'l', None, None], ['s', 'n', 'e', 'e', 'r']], [['a', None, 'e', None, None], ['m', 'o', 't', 'o', 'r'], ['a', None, 't', None, None], ['s', None, 'l', None, None], ['s', None, 'e', None, None]], [[None, 'g', 'e', None, None], ['m', 'o', 't', 'o', 'r'], [None, 'r', 't', None, None], [None, 'a', 'l', None, None], [None, 'l', 'e', None, None]], [[None, None, 'e', 'n', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 's', None], [None, None, 'l', 'l', None], [None, None, 'e', 'e', None]], [[None, None, 'e', 'm', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 'u', None], [None, None, 'l', 't', None], [None, None, 'e', 'h', None]], [[None, None, 'e', 'n', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 's', None], [None, None, 'l', 't', None], [None, None, 'e', 'r', None]], [[None, None, 'e', 'p', None], ['m', 'o', 't', 'o', 'r'], [None, None, 't', 'r', None], [None, None, 'l', 't', None], [None, None, 'e', 's', None]], [[None, None, 'e', None, 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'i'], [None, None, 'l', None, 'e'], [None, None, 'e', None, 'r']], [[None, None, 'e', None, 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'y'], [None, None, 'l', None, 'e'], [None, None, 'e', None, 'r']], [[None, None, 'e', None, 'w'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'i'], [None, None, 'l', None, 'n'], [None, None, 'e', None, 'g']], [[None, None, 'e', None, 'd'], ['m', 'o', 't', 'o', 'r'], [None, None, 't', None, 'e'], [None, None, 'l', None, 'a'], [None, None, 'e', None, 'r']]]
First think step by step why the successor function failed to produce all successors of the state.
Then, fix the successor function.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.
Horizontal clues: [['tasks', 'goals', 'plans', 'agend', 'chores', 'works', 'deeds', 'items', 'lists', 'brief'], ['motor', 'power', 'drive', 'diesel', 'steam', 'pumps', 'crank', 'gears', 'turbn', 'motor'], ['grand', 'artsy', 'showy', 'ornate', 'fancy', 'vain', 'proud', 'vogue', 'swank', 'luxus'], ['venue', 'salle', 'forum', 'atria', 'lobby', 'parls', 'court', 'malls', 'mall', 'lobby'], ['jeer', 'scoff', 'sleer', 'deris', 'sneer', 'scorn', 'derid', 'gibes', 'gibed', 'flout']]
Vertical clues: [['amass', 'stack', 'hoard', 'pile', 'store', 'heaps', 'massy', 'gathe', 'lumps', 'mound'], ['nilga', 'goral', 'eland', 'lepus', 'gazal', 'kudu', 'oryx', 'gnu', 'imps', 'carb'], ['scheme', 'design', 'ettle', 'nettle', 'sting', 'wiles', 'plans', 'ideas', 'plots', 'cocks'], ['spout', 'nosle', 'snout', 'mouth', 'nostr', 'ports', 'inlet', 'vents', 'outlt', 'beaks'], ['drier', 'arid', 'sere', 'parch', 'dryer', 'wring', 'drear', 'sear', 'pall', 'lack']]

Model response:
We need to ensure that the successor function properly fills in additional cells compared to the parent state, and to produce all valid successors. Here's a detailed step-by-step approach to fixing the function:

1. **Reasons for Failure**: 
   - The previous function did not sufficiently ensure that new cells were being filled in each successor.
   - It didn't fully cover all possible word placements for both horizontal and vertical fillings.

2. **Successor State Requirements**:
   - Each successor state should introduce at least one new word in an incomplete row or column.
   - Ensure careful iteration over the grid so that all possible valid placements are considered.

### Revised Plan:

1. **Identify the incomplete rows and columns**.
2. **Generate all possible successor states for each incomplete row/column**.
3. **Ensure each successor state has at least one more cell filled compared to the parent state**.

### Improved Function:

```python
def successor_state(state, horizontal_answers, vertical_answers):
    import copy

    def is_valid_state(state):
        for i in range(5):
            # Check each horizontal word
            horizontal_word = ''.join([state[i][j] if state[i][j] is not None else '*' for j in range(5)])
            if '*' in horizontal_word:
                partials = horizontal_word.split('*')
                if not all(part == '' or part in valid_horizontal_words[i] for part in partials):
                    return False
            elif horizontal_word and horizontal_word not in valid_horizontal_words[i]:
                return False
            
            # Check each vertical word
            vertical_word = ''.join([state[j][i] if state[j][i] is not None else '*' for j in range(5)])
            if '*' in vertical_word:
                partials = vertical_word.split('*')
                if not all(part == '' or part in valid_vertical_words[i] for part in partials):
                    return False
            elif vertical_word and vertical_word not in valid_vertical_words[i]:
                return False
        return True

    def generate_successors(state, possible_answers, is_horizontal):
        successors = []
        for i in range(5):
            if is_horizontal:
                target_line = [state[i][j] for j in range(5)]
            else:
                target_line = [state[j][i] for j in range(5)]

            if all(ch is not None for ch in target_line):  # Skip fully filled rows/columns
                continue

            for word in possible_answers[i]:
                if len(word) > 5:
                    continue

                new_state = copy.deepcopy(state)
                can_fit = True
                filled = False

                for j in range(len(word)):
                    ch = word[j]
                    if target_line[j] is not None and target_line[j] != ch:
                        can_fit = False
                        break
                    if is_horizontal:
                        new_state[i][j] = ch
                    else:
                        new_state[j][i] = ch
                    filled = filled or (target_line[j] is None and ch is not None)
                
                if can_fit and filled and is_valid_state(new_state):
                    successors.append(new_state)
            
            if successors:
                return successors  # Stop after generating the first set of valid successors

        return []

    valid_horizontal_words = [set(horizontal_answers[i]) for i in range(5)]
    valid_vertical_words = [set(vertical_answers[i]) for i in range(5)]

    # First try to generate horizontal successors
    row_successors = generate_successors(state, horizontal_answers, True)
    if row_successors:
        return row_successors

    # Then try to generate vertical successors
    col_successors = generate_successors(state, vertical_answers, False)
    if col_successors:
        return col_successors

    return []
```

### Explanation:
1. **`is_valid_state(state)`**: Ensures that partially or fully filled rows/columns match the provided clues.
2. **`generate_successors(state, possible_answers, is_horizontal)`**:
   - Iterates through possible words and tries placing each word in the specified row/column.
   - Checks if the word matches the pre-filled cells and if at least one new cell is filled.
   - Adds successors only if they are valid.
3. **Validation**: Ensures that each newly generated state does not conflict with the existing state and adheres to the constraints posed by the clues.

This refined function will generate valid successors by ensuring incremental updates, each with additional filled cells, covering all potential placements for both horizontal and vertical clues.

System message:
Successor States Soundness Test Passed
Completeness test

System message:
Maximum iterations reached for completeness test. Process stopped.
