The n-queens puzzle is the problem of placing n queens on an n x n chessboard such that no two queens attack each other.

Given an integer n, return the number of distinct solutions to the n-queens puzzle.

Examples

Example 1

Input:  n = 4
Output: 2

There are two distinct solutions to the 4-queens puzzle as shown.

Example 2

Input:  n = 1
Output: 1

Constraints

• 1 <= n <= 9