# Count Complete Tic-Tac-Toe Game Sequences Two players play on a standard `3 x 3` board. `X` moves first and the players alternate. A game ends immediately after a move creates three equal marks in a row, column, or diagonal, or after the ninth move if nobody has won. Different move orders count as different game sequences even when they produce the same final board. Implement `count_completions(board, next_player)` to return the number of distinct legal move sequences that start at the supplied position and continue until the game ends. ## Function Contract ```python def count_completions(board: tuple[str, ...], next_player: str) -> int: ... ``` - `board` contains exactly nine strings in row-major order. Each entry is `"X"`, `"O"`, or `"."`. - `next_player` is either `"X"` or `"O"`. - The input is guaranteed to be reachable by alternating legal play from the empty board, and `next_player` is consistent with the mark counts. - If `board` is already terminal, return `1`: the empty continuation is the one completed sequence from that state. - Otherwise, every legal choice of the next empty cell starts a distinct continuation. ## Constraints - The board is always `3 x 3`. - A winning position is terminal even if empty cells remain. - Your implementation must not count any moves made after the first win. - Return an integer. ## Examples ```text board = ("X", "X", ".", "O", "O", ".", ".", ".", ".") next_player = "X" ``` The result includes the continuation where `X` wins immediately in the top-right cell, plus every other legal first move and all terminal continuations below it.