Cat and Mouse II

A game is played by a cat and a mouse named Cat and Mouse.

The environment is represented by a grid of size rows x cols, where each element is a wall, floor, player (Cat, Mouse), or food.

• Players are represented by the characters 'C' (Cat) and 'M' (Mouse)
• Floors are represented by the character '.' and can be walked on
• Walls are represented by the character '#' and cannot be walked on
• Food is represented by the character 'F' and can be walked on
• There is only one of each character 'C', 'M', and 'F' in grid

Mouse and Cat play according to the following rules:

• Mouse moves first, then they take turns to move
• During each turn, Cat and Mouse can jump in one of the four directions (left, right, up, down). They cannot jump over the wall nor outside of the grid
• catJump and mouseJump are the maximum lengths Cat and Mouse can jump at a time, respectively. Cat and Mouse can jump less than the maximum length
• Staying in the same position is allowed
• Mouse can jump over Cat

The game can end in 4 ways:

• If Cat occupies the same position as Mouse, Cat wins
• If Cat reaches the food first, Cat wins
• If Mouse reaches the food first, Mouse wins
• If Mouse cannot get to the food within 1000 turns, Cat wins

Given a rows x cols matrix grid and two integers catJump and mouseJump, return true if Mouse can win the game if both Cat and Mouse play optimally, otherwise return false.