"""
Tic Tac Toe Player
"""

import math

X = "X"
O = "O"
EMPTY = None


def initial_state():
    """
    Returns starting state of the board.
    """
    return [[EMPTY, EMPTY, EMPTY],
            [EMPTY, EMPTY, EMPTY],
            [EMPTY, EMPTY, EMPTY]]


def player(board):
    """
    Returns player who has the next turn on a board.
    """
    x_count = sum(row.count(X) for row in board)
    o_count = sum(row.count(O) for row in board)
    return X if x_count == o_count else O


def actions(board):
    """
    Returns set of all possible actions (i, j) available on the board.
    """
    return {(i, j) for i in range(3) for j in range(3) if board[i][j] == EMPTY}


def result(board, action):
    """
    Returns the board that results from making move (i, j) on the board.
    """
    if board[action[0]][action[1]] is not EMPTY:
        raise Exception("Invalid action")
    
    new_board = [row[:] for row in board]
    new_board[action[0]][action[1]] = player(board)
    return new_board


def winner(board):
    """
    Returns the winner of the game, if there is one.
    """
    for line in board:  # Check rows
        if line[0] == line[1] == line[2] and line[0] is not None:
            return line[0]
    for col in range(3):  # Check columns
        if board[0][col] == board[1][col] == board[2][col] and board[0][col] is not None:
            return board[0][col]
    if board[0][0] == board[1][1] == board[2][2] and board[0][0] is not None:  # Check diagonal
        return board[0][0]
    if board[0][2] == board[1][1] == board[2][0] and board[0][2] is not None:  # Check anti-diagonal
        return board[0][2]
    return None


def terminal(board):
    """
    Returns True if game is over, False otherwise.
    """
    return winner(board) is not None or all(cell is not EMPTY for row in board for cell in row)


def utility(board):
    """
    Returns 1 if X has won the game, -1 if O has won, 0 otherwise.
    """
    win = winner(board)
    if win == X:
        return 1
    elif win == O:
        return -1
    else:
        return 0


def minimax(board):
    """
    Returns the optimal action for the current player on the board.
    """
    if terminal(board):
        return None

    turn = player(board)

    def max_value(board):
        if terminal(board):
            return utility(board), None
        v = -math.inf
        best_move = None
        for action in actions(board):
            val, _ = min_value(result(board, action))
            if val > v:
                v = val
                best_move = action
        return v, best_move

    def min_value(board):
        if terminal(board):
            return utility(board), None
        v = math.inf
        best_move = None
        for action in actions(board):
            val, _ = max_value(result(board, action))
            if val < v:
                v = val
                best_move = action
        return v, best_move

    if turn == X:
        _, move = max_value(board)
    else:
        _, move = min_value(board)

    return move