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