Thank you for providing the detailed information and requirements. I'll think through this step-by-step and then generate the two functions as requested.

Step 1: Understanding the processed_global_state

The processed_global_state contains four main components:

1. available_move_actions: A dictionary of available move directions for each agent.
2. enemy_info: A nested dictionary containing information about enemies for each agent.
3. ally_info: A nested dictionary containing information about allies for each agent.
4. own_info: A dictionary containing information about each agent itself.

These components provide crucial information for decision-making, such as:
- Agent and enemy positions
- Health status of agents and enemies
- Visibility of enemies and allies
- Attack availability
- Cooldown status

Step 2: Considering task combinations

The optimal combination of tasks will depend on the current battle state. Here are some scenarios to consider:

1. All agents attack: This could be effective when enemies are within range and the team has a health advantage.
2. Some agents attack, others move: This could be useful for focus firing on specific enemies while repositioning other agents for better angles or to protect low-health allies.
3. All agents move: This might be necessary when retreating from a disadvantageous position or when trying to group up for a coordinated attack.

The planning function should consider these scenarios and choose the best combination based on the current state.

Now, let's create the two functions with these considerations in mind:

```python
import numpy as np

def planning_function(processed_global_state):
    available_move_actions, enemy_info, ally_info, own_info = processed_global_state
    llm_tasks = {}
    
    # Calculate team health and enemy health
    team_health = sum([info[0][0] for info in own_info.values()])
    enemy_health = sum([enemy[5][0] for agent_enemies in enemy_info.values() for enemy in agent_enemies.values()])
    
    # Identify the weakest enemy
    weakest_enemy = min([enemy for agent_enemies in enemy_info.values() for enemy in agent_enemies.values()], key=lambda x: x[5][0])
    weakest_enemy_pos = (weakest_enemy[6][0], weakest_enemy[7][0])
    
    for agent_id, agent_info in own_info.items():
        if not agent_info[4]:  # If agent is dead
            llm_tasks[agent_id] = 'none'
            continue
        
        agent_pos = (agent_info[1][0], agent_info[2][0])
        agent_health = agent_info[0][0]
        
        # Check if any enemy is in attack range
        enemies_in_range = [enemy for enemy in enemy_info[agent_id].values() if enemy[0][0] == 1]
        
        if enemies_in_range:
            # If there are enemies in range and agent health is good, attack
            if agent_health > 0.3:
                llm_tasks[agent_id] = 'attack'
            else:
                # If agent health is low, try to move away from enemies
                move_direction = get_safe_direction(agent_pos, enemies_in_range, available_move_actions[agent_id])
                llm_tasks[agent_id] = 'move' if move_direction else 'attack'
        else:
            # If no enemies in range, move towards the weakest enemy
            move_direction = get_move_towards(agent_pos, weakest_enemy_pos, available_move_actions[agent_id])
            llm_tasks[agent_id] = 'move' if move_direction else 'attack'
    
    return llm_tasks

def get_safe_direction(agent_pos, enemies, available_moves):
    enemy_positions = [(enemy[6][0], enemy[7][0]) for enemy in enemies]
    directions = {'North': (0, 1), 'South': (0, -1), 'East': (1, 0), 'West': (-1, 0)}
    
    max_distance = -1
    best_direction = None
    
    for move in available_moves:
        new_pos = (agent_pos[0] + directions[move][0], agent_pos[1] + directions[move][1])
        min_distance = min(np.linalg.norm(np.array(new_pos) - np.array(enemy_pos)) for enemy_pos in enemy_positions)
        
        if min_distance > max_distance:
            max_distance = min_distance
            best_direction = move
    
    return best_direction

def get_move_towards(agent_pos, target_pos, available_moves):
    directions = {'North': (0, 1), 'South': (0, -1), 'East': (1, 0), 'West': (-1, 0)}
    
    min_distance = float('inf')
    best_direction = None
    
    for move in available_moves:
        new_pos = (agent_pos[0] + directions[move][0], agent_pos[1] + directions[move][1])
        distance = np.linalg.norm(np.array(new_pos) - np.array(target_pos))
        
        if distance < min_distance:
            min_distance = distance
            best_direction = move
    
    return best_direction

def compute_reward(processed_state, llm_tasks, tasks):
    reward = {}
    available_move_actions, enemy_info, ally_info, own_info = processed_state
    
    for agent_id, task in tasks.items():
        if own_info[agent_id][4]:  # If agent is alive
            if task == llm_tasks[agent_id]:
                reward[agent_id] = 0.1  # Small positive reward for following LLM task
            elif task == 'stop':
                reward[agent_id] = -0.05  # Small penalty for stopping
            else:
                reward[agent_id] = 0  # No reward for not following LLM task
        else:
            reward[agent_id] = -0.1  # Small penalty for being dead
        
        # Additional rewards based on outcomes
        if task == 'attack':
            # Check if any enemy was damaged or killed
            for enemy in enemy_info[agent_id].values():
                if enemy[5][0] < 1:  # If enemy health is not full
                    reward[agent_id] += 0.2  # Reward for damaging enemy
                if enemy[5][0] == 0:  # If enemy is killed
                    reward[agent_id] += 1.0  # Larger reward for killing enemy
        
        elif task == 'move':
            # Reward for moving to a better position (e.g., closer to allies, further from enemies)
            agent_pos = (own_info[agent_id][1][0], own_info[agent_id][2][0])
            ally_distances = [np.linalg.norm(np.array(agent_pos) - np.array((ally[6][0], ally[7][0]))) for ally in ally_info[agent_id].values()]
            enemy_distances = [np.linalg.norm(np.array(agent_pos) - np.array((enemy[6][0], enemy[7][0]))) for enemy in enemy_info[agent_id].values()]
            
            if min(ally_distances) < 3 and min(enemy_distances) > 5:
                reward[agent_id] += 0.1  # Reward for good positioning
    
    return reward
```

These functions aim to enhance multi-agent reinforcement learning by promoting coordination and strategic decision-making:

1. The `planning_function` considers the overall battle state, including team and enemy health, to determine the best action for each agent. It prioritizes attacking when enemies are in range and the agent's health is good, while encouraging low-health agents to retreat. When no enemies are in range, it directs agents towards the weakest enemy for a coordinated attack.

2. The `compute_reward` function provides rewards that align with the LLM tasks and encourage beneficial outcomes. It rewards agents for following the LLM tasks, damaging or killing enemies, and achieving good positioning. It also includes small penalties for stopping or being dead to discourage these states.

These functions work together to promote effective team coordination, enemy reduction, strategic positioning, and damage mitigation, ultimately aiming to win the battle within the given constraints.