Count N-Queens Solutions

Return the number of ways to place n queens on an n-by-n board with no shared row, column, or diagonal. Count distinct board arrangements; rotations/reflections count separately. n is an integer from 0 to 9. By convention n=0 has one empty arrangement. Return only the count, not board layouts. Examples (JSON; null means Python None): Input: {"n": 4} Output: 2 Input: {"n": 1} Output: 1

def count_queens(n: int) -> int:
    """Count boards by reserving columns and both diagonal families."""
    pass