Consider the two$-$player $($Max player and Min Player$)$ zero$-$sum game. The game begins with a set of eleven coins. On any player$$s move, the player must split the set of coins into two subsets. Max$-$player is not allowed to split a set$/$subset of coins into two equal sizes. If it is max player turn and all the sets$/$subsets of coins have either one or two coins, max player will lose the game. Propose an appropriate utility function for playing the game.

$($a$)$Apply the minimax algorithm for finding the best action for the max player at the root.

$($b$)$Apply the minimax algorithm with alpha$-$beta pruning for finding the best action for the max player at the root.