Problem Statement

Takahashi has decided to wander around his house.
During his walk, he will roam between $N$ points called Point $1$, Point $2$, $\\dots$, Point $N$, where Point $1$ is his house.
There are $M$ pairs of points connected by roads; let $(a_i, b_i)$ be the $i$-th of these pairs. There are $p_{i, d}$ roads with a length of $d$ $(1 \\leq d \\leq T)$ kilometers connecting Point $a_i$ and Point $b_i$.

Takahashi wants to know the number of $T$ kilometers courses that begin and end at his house. Here, a $T$ kilometers course is defined as follows.

• An alternating sequence with points and roads $v_0 = 1, e_0, v_1, \\dots,e_{k-1}, v_k = 1$ such that $e_i$ $(0 \\leq i \\leq k-1)$ connects $v_i$ and $v_{i+1}$, and the total length of $e_i$ is $T$ kilometers.

Help Takahashi by finding the number of such courses modulo $998244353$. Two courses are considered different when they are different as sequences.

Constraints

• $2 \\leq N \\leq 10$
• $1 \\leq M \\leq \\min \\left(10, \\frac{N(N-1)}{2} \\right)$
• $1 \\leq T \\leq 4 \\times 10^4$
• $1 \\leq a_i \\lt b_i \\leq N$
• $(a_i, b_i) \\neq (a_j, b_j)$ if $i \\neq j$.
• $0 \\leq p_{i,j} \\lt 998244353$

Input

Input is given from Standard Input in the following format:

$N$ $M$ $T$ $a_1$ $b_1$ $p_{1,1}$ $p_{1,2}$ $\\ldots$ $p_{1,T}$ $\\vdots$ $a_M$ $b_M$ $p_{M,1}$ $p_{M,2}$ $\\ldots$ $p_{M,T}$

Output

Print the number of desirable courses, modulo $998244353$.

Sample Input 1

3 2 2
1 2
1 0
1 3
2 0

Sample Output 1

5

Around his house, there are:

• one $1$-kilometer road connecting Point $1$ and Point $2$, and
• two $1$-kilometer roads connecting Point $1$ and Point $3$.

We have the following five desirable courses:

• $1 \\times 1 = 1$ course that goes Point $1$ $\\to$ Point $2$ $\\to$ Point $1$, and
• $2 \\times 2 = 4$ courses that goes Point $1$ $\\to$ Point $3$ $\\to$ Point $1$.

Sample Input 2

3 3 4
1 2
3 0 0 0
1 3
0 1 0 0
2 3
2 0 0 0

Sample Output 2

130

Around his house, there are:

• three $1$-kilometer roads connecting Point $1$ and Point $2$,
• one $2$-kilometer road connecting Point $1$ and Point $3$, and
• two $1$-kilometer roads connecting Point $2$ and Point $3$.

The desirable courses can be classified into the following categories, according to the points visited:

• Point $1$ $\\to$ Point $2$ $\\to$ Point $1$ $\\to$ Point $2$ $\\to$ Point $1$,
• Point $1$ $\\to$ Point $2$ $\\to$ Point $3$ $\\to$ Point $1$,
• Point $1$ $\\to$ Point $2$ $\\to$ Point $3$ $\\to$ Point $2$ $\\to$ Point $1$,
• Point $1$ $\\to$ Point $3$ $\\to$ Point $1$, and
• Point $1$ $\\to$ Point $3$ $\\to$ Point $2$ $\\to$ Point $1$.

We have $81$, $6$, $36$, $1$, and $6$ course(s) of these categories, respectively.

Sample Input 3

2 1 5
1 2
31415 92653 58979 32384 62643

Sample Output 3

844557977