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, or food.

• Players are represented by the characters 'C' for Cat and 'M' for 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, or down. They cannot jump over a wall or 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.