You are given (n) points on the plane, the coordinates of the (i)-th point are ((x_i, y_i)). No two points have the same coordinates. The distance between points (i) and (j) is defined as (d(i,j) = |x_i - x_j| + |y_i - y_j|). For each point, you have to choose a color, represented by an integer from (1) to (n). For every ordered triple of different points ((a,b,c)), the following constraints should be met: if (a), (b) and (c) have the same color, then (d(a,b) = d(a,c) = d(b,c)); if (a) and (b) have the same color, and the color of (c) is different from the color of (a), then (d(a,b) < d(a,c)) and (d(a,b) < d(b,c)). Calculate the number of different ways to choose the colors that meet these constraints. The first line contains one integer (n) ((2 \le n \le 100)) — the number of points. Then (n) lines follow. The (i)-th of them contains two integers (x_i) and (y_i) ((0 \le x_i, y_i \le 10^8)). No two points have the same coordinates (i. e. if (i \ne j), then either (x_i \ne x_j) or (y_i \ne y_j)). Print one integer — the number of ways to choose the colors for the points. Since it can be large, print it modulo (998244353). In the first test, the following ways to choose the colors are suitable: (1, 1, 1); (2, 2, 2); (3, 3, 3); (1, 2, 3); (1, 3, 2); (2, 1, 3); (2, 3, 1); (3, 1, 2); (3, 2, 1).