Problem Statement

There are $N$ distinct points on a two-dimensional plane, numbered from $1$ to $N$. The coordinates of point $i$ are $(X_i, Y_i)$.

The distance between $(x_0, y_0)$ and $(x_1, y_1)$ is defined as $\max(|x_0 - x_1|, |y_0 - y_1|)$.

Find the number of routes that travel from point $1$ to point $N$ via some points and minimize the total distance traveled.
More precisely, find the number of sequences $p$ of positive integers of length $k$ satisfying $p_1 = 1$, $p_k = N$, $1 \leq p_i \leq N$ $(1 \leq i \leq k)$ and $p_i \neq p_{i+1}$ $(1 \leq i \leq k - 1)$ (added at 22:35) that minimize $\displaystyle\sum_{i = 1}^{k - 1} \max(|X_{p_{i + 1}} - X_{p_i}|, |Y_{p_{i + 1}} - Y_{p_i}|)$.
Since the answer can be very large, output it modulo $998244353$.

Input

$N$
$X_1$ $Y_1$
$X_2$ $Y_2$
$\vdots$
$X_N$ $Y_N$

Output

Output the number of ways of moving that minimize the total distance, modulo $998244353$.
Also, output a newline at the end.

Sample

3
0 0
0 1
1 2

2

5
0 6
1 5
3 2
5 5
3 0

4

10
-8 1
3 -5
-4 8
5 -4
-1 -6
6 0
-1 -4
2 -9
-6 7
10 -9

14

14
0 0
7 -7
11 -3
9 -1
7 1
7 -5
10 -2
8 0
4 4
5 1
5 3
7 -3
6 2
12 -4

334