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data structures

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C. Count Binary Strings

time limit per test

2 secondsmemory limit per test

512 megabytesinput

standard inputoutput

standard outputYou are given an integer $$$n$$$. You have to calculate the number of binary (consisting of characters 0 and/or 1) strings $$$s$$$ meeting the following constraints.

For every pair of integers $$$(i, j)$$$ such that $$$1 \le i \le j \le n$$$, an integer $$$a_{i,j}$$$ is given. It imposes the following constraint on the string $$$s_i s_{i+1} s_{i+2} \dots s_j$$$:

- if $$$a_{i,j} = 1$$$, all characters in $$$s_i s_{i+1} s_{i+2} \dots s_j$$$ should be the same;
- if $$$a_{i,j} = 2$$$, there should be at least two different characters in $$$s_i s_{i+1} s_{i+2} \dots s_j$$$;
- if $$$a_{i,j} = 0$$$, there are no additional constraints on the string $$$s_i s_{i+1} s_{i+2} \dots s_j$$$.

Count the number of binary strings $$$s$$$ of length $$$n$$$ meeting the aforementioned constraints. Since the answer can be large, print it modulo $$$998244353$$$.

Input

The first line contains one integer $$$n$$$ ($$$2 \le n \le 100$$$).

Then $$$n$$$ lines follow. The $$$i$$$-th of them contains $$$n-i+1$$$ integers $$$a_{i,i}, a_{i,i+1}, a_{i,i+2}, \dots, a_{i,n}$$$ ($$$0 \le a_{i,j} \le 2$$$).

Output

Print one integer — the number of strings meeting the constraints, taken modulo $$$998244353$$$.

Examples

Input

3 1 0 2 1 0 1

Output

6

Input

3 1 1 2 1 0 1

Output

2

Input

3 1 2 1 1 0 1

Output

0

Input

3 2 0 2 0 1 1

Output

0

Note

In the first example, the strings meeting the constraints are 001, 010, 011, 100, 101, 110.

In the second example, the strings meeting the constraints are 001, 110.

Codeforces (c) Copyright 2010-2024 Mike Mirzayanov

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