Problem Statement

You are given positive integers $N$, $M$, and $K$, and a sequence of $M$ non-negative integers $A = (A_{0}, A_{1}, \dots, A_{M-1})$. Here, $2^{N - 1} \leq K < 2^{N}$ holds.

In the input, $K$ is given as an $N$-digit number in binary notation, while the other integers are given in decimal notation.

Additionally, $A$ is not given directly in the input. Instead, for each $i = 0, 1, \dots, M - 1$, you are given a sequence of $L_i$ integers $X_{i} = (X_{i,0}, X_{i,1}, \dots, X_{i,L_{i}-1})$ such that $A_{i} = \sum_{j=0}^{L_{i}-1} 2^{X_{i,j}}$. Here, $0 \leq X_{i,0} < X_{i,1} < \cdots < X_{i,L_{i}-1} < N$ holds.

Find the inversion number, modulo $998244353$, of the sequence $B = (B_{0}, B_{1}, \dots, B_{MK-1})$ defined as follows.

• For any integer $a$ such that $0 \leq a < K$ and any integer $b$ such that $0 \leq b < M$, the following holds:
• $B_{aM+b}$ is equal to the remainder when $\operatorname{popcount}(a \operatorname{AND} A_{b})$ is divided by $2$.

What is $\operatorname{AND}$? The bitwise $\operatorname{AND}$ of integers $A$ and $B$, denoted as $A \operatorname{AND} B$, is defined as follows: In the binary representation of $A \operatorname{AND} B$, the digit at the $2^k$ ($k \geq 0$) place is $1$ if and only if the digits at the $2^k$ place in the binary representations of both $A$ and $B$ are $1$; otherwise, it is $0$. For example, $3 \operatorname{AND} 5 = 1$ (in binary: $011 \operatorname{AND} 101 = 001$). Generally, the bitwise $\operatorname{AND}$ of $k$ integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1 \operatorname{AND} p_2) \operatorname{AND} p_3) \operatorname{AND} \dots \operatorname{AND} p_k)$, and it can be proved that this is independent of the order of $p_1, p_2, p_3, \dots, p_k$.

What is $\operatorname{popcount}$? For a non-negative integer $x$, $\operatorname{popcount}(x)$ is the number of $1$s in the binary representation of $x$. More precisely, for a non-negative integer $x$ such that $\displaystyle x = \sum_{i=0}^{\infty} b_i 2^i\ (b_i \in {0, 1})$, it holds that $\displaystyle \operatorname{popcount}(x) = \sum_{i=0}^{\infty} b_i$. For example, $13$ is 1101 in binary, so $\operatorname{popcount}(13) = 3$.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq M \leq 2 \times 10^5$
• $2^{N-1} \leq K < 2^{N}$
• $0 \leq L_{i} \leq N$
• $\sum L_{i} \leq 2 \times 10^5$
• $0 \leq X_{i,0} < X_{i,1} < \cdots < X_{i,L_{i}-1} < N$
• All input values are integers.
• $K$ is given in binary notation.
• All numbers except $K$ are given in decimal notation.

Input

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

N M
K
L_{0} X_{0,0} X_{0,1} \cdots X_{0,L_{0}-1}
L_{1} X_{1,0} X_{1,1} \cdots X_{1,L_{1}-1}
\vdots
L_{M-1} X_{M-1,0} X_{M-1,1} \cdots X_{M-1,L_{M-1}-1}

Output

Print the answer.