103725: [Atcoder]ABC372 F - Teleporting Takahashi 2

Problem Statement

There is a simple directed graph $G$ with $N$ vertices and $N+M$ edges. The vertices are numbered $1$ to $N$, and the edges are numbered $1$ to $N+M$.

Edge $i$ goes from vertex $i$ to vertex $i+1$. (Here, vertex $N+1$ is considered as vertex $1$.)
Edge $N+i$ $(1 \leq i \leq M)$ goes from vertex $X_i$ to vertex $Y_i$.

Takahashi is at vertex $1$. At each vertex, he can move to any vertex to which there is an outgoing edge from the current vertex.

Compute the number of ways he can move exactly $K$ times.

That is, find the number of integer sequences $(v_0, v_1, \dots, v_K)$ of length $K+1$ satisfying all of the following three conditions:

• $1 \leq v_i \leq N$ for $i = 0, 1, \dots, K$.
• $v_0 = 1$.
• There is a directed edge from vertex $v_{i-1}$ to vertex $v_i$ for $i = 1, 2, \ldots, K$.

Since this number can be very large, print it modulo $998244353$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $0 \leq M \leq 50$
• $1 \leq K \leq 2 \times 10^5$
• $1 \leq X_i, Y_i \leq N$, $X_i \neq Y_i$
• All of the $N+M$ directed edges are distinct.
• All input values are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$ $K$
$X_1$ $Y_1$
$X_2$ $Y_2$
$\vdots$
$X_M$ $Y_M$

Output

Print the count modulo $998244353$.

Sample Input 1

6 2 5
1 4
2 5

Sample Output 1

5

The above figure represents the graph $G$. There are five ways for Takahashi to move:

• Vertex $1 \to$ Vertex $2 \to$ Vertex $3 \to$ Vertex $4 \to$ Vertex $5 \to$ Vertex $6$
• Vertex $1 \to$ Vertex $2 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $2$
• Vertex $1 \to$ Vertex $2 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $4$
• Vertex $1 \to$ Vertex $4 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $2$
• Vertex $1 \to$ Vertex $4 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $4$

Sample Input 2

10 0 200000

Sample Output 2

1

Sample Input 3

199 10 1326
122 39
142 49
164 119
197 127
188 145
69 80
6 120
24 160
18 154
185 27

Sample Output 3

451022766