Ranking at depth 3
Model's reasoning and response:
To rank the states, we need to evaluate how many goal constraints each state satisfies. The goal constraints are:

1. Row 1:
   - tile_1-1: white
   - tile_1-2: black
   - tile_1-3: white

2. Row 2:
   - tile_2-1: black
   - tile_2-2: white
   - tile_2-3: black

Let's evaluate each state:

### State 9
- tile_2-1 is painted black (satisfies goal constraint for tile_2-1).
- No other tiles are painted according to the goal.

**Constraints satisfied: 1**

### State 10
- No tiles are painted according to the goal.

**Constraints satisfied: 0**

### State 11
- No tiles are painted according to the goal.

**Constraints satisfied: 0**

### State 12
- No tiles are painted according to the goal.

**Constraints satisfied: 0**

### State 13
- tile_2-2 is painted white (satisfies goal constraint for tile_2-2).

**Constraints satisfied: 1**

### State 14
- No tiles are painted according to the goal.

**Constraints satisfied: 0**

### Ranking
Based on the number of constraints satisfied, the ranking from best to worst is:

1. State 9 (1 constraint satisfied)
2. State 13 (1 constraint satisfied)
3. State 10 (0 constraints satisfied)
4. State 11 (0 constraints satisfied)
5. State 12 (0 constraints satisfied)
6. State 14 (0 constraints satisfied)

Since States 10, 11, 12, and 14 satisfy no constraints, they are ranked equally in terms of constraints but are listed in the order they were evaluated.

```ranking
9, 13, 10, 11, 12, 14
```

Final ranking:
Rank 1: State ID 9
State Description: (tile_0-1: clear, tile_0-2: clear, tile_0-3: clear, tile_1-1: occupied by robot1 holding color black, tile_1-2: clear, tile_1-3: clear, tile_2-1: painted black, tile_2-2: clear, tile_2-3: clear, robot1: on tile_1-1 holding color black, color_white: available, color_black: available)

Rank 2: State ID 13
State Description: Tile tile_0-1 is clear. Tile tile_0-2 is clear. Tile tile_0-3 is clear. Tile tile_1-1 is clear. Tile tile_1-2 is now clear. Tile tile_1-3 is clear. Tile tile_2-1 is clear. Tile tile_2-2 is painted white. Tile tile_2-3 is clear. Robot1 is on tile_1-2, holding color white. Colors white and black are available for switching.

Rank 3: State ID 10
State Description: (tile_0-1: clear, tile_0-2: clear, tile_0-3: clear, tile_1-1: clear, tile_1-2: clear, tile_1-3: clear, tile_2-1: occupied by robot1 holding color black, tile_2-2: clear, tile_2-3: clear, robot1: on tile_2-1 holding color black, color_white: available, color_black: available)

Rank 4: State ID 11
State Description: (tile_0-1: clear, tile_0-2: clear, tile_0-3: clear, tile_1-1: clear, tile_1-2: occupied by robot1 holding color black, tile_1-3: clear, tile_2-1: clear, tile_2-2: clear, tile_2-3: clear, robot1: on tile_1-2 holding color black, color_white: available, color_black: available)

Rank 5: State ID 12
State Description: Tile tile_0-3 is now clear. Robot1 is on tile_1-3, holding color white. Tile_0-1, tile_0-2, and tile_0-3 are clear. Tiles in row 1 (tile_1-1, tile_1-2) and row 2 (tile_2-1, tile_2-2, tile_2-3) remain clear. Colors white and black are available for switching.

Rank 6: State ID 14
State Description: Tile tile_0-1 is clear. Tile tile_0-2 is clear. Tile tile_0-3 is clear. Tile tile_1-1 is clear. Tile tile_1-2 is now clear. Tile tile_1-3 is clear. Tile tile_2-1 is clear. Robot1 is on tile_2-2, holding color white. Tile tile_2-3 is clear. Colors white and black are available for switching.

