Description

There are $n$ radioactive atoms that will undergo fission reactions for $k$ seconds. If at the beginning of the $t$-th second, atom $i$ is bombarded by $b\ (b>0)$ neutrons, it will release $a_i + b$ units of energy during the $t$-th second and will release $1$ neutron to each of the atoms numbered $x_{i,1}, x_{i,2}, \ldots, x_{i,a_i}$. Thus, at the beginning of the next second ($t+1$), the number of neutrons hitting each of $x_{i,1}, x_{i,2}, \dots, x_{i,a_i}$ will increase by $1$ (if $t=k$, meaning this is the last second, then neutrons will not be released by bombarded atoms). If an atom is not bombarded by any neutrons at the beginning of a second, it will not release energy or neutrons.

At the beginning of each second, atoms numbered $v_1, v_2, \ldots, v_m$ are bombarded by $1$ neutron. Therefore, from the beginning of the first second until the end of the $k$-th second, how much energy will each atom release?

Output the answer modulo $998244353$.

Input Format

The first line contains three integers $n, k, m$, representing the number of atoms, the duration of the reaction, and the number of atoms hit each second.

The second line contains $m$ integers $v_1, v_2, \dots, v_m$.

The next $n$ lines contain the information for each atom, formatted as follows:

The $i$-th line contains $a_i + 1$ integers, where the first number is $a_i$, followed by $a_i$ integers $x_{i,1}, x_{i,2}, \dots, x_{i,a_i}$.

Output Format

Output one line containing $n$ integers. The $i$-th number is the total energy released by atom $i$ from the start of the $1$-st second to the termination moment of the $k$-th second, with the result taken modulo $998244353$.

3 3 1
1
1 2
1 3
1 1

6 4 2

3 1000000000000000000 1
1
1 2
1 3
1 1

151723985 433897441 433897439

Hint

Explanation of sample #1:

• In the first second, atom $1$ is hit by $1$ neutron, releasing $1$ neutron to atom $2$, thus releasing $2$ units of energy.
• In the second second, atom $1$ is hit by $1$ neutron, releasing $1$ neutron to atom $2$, and releasing $2$ units of energy. Simultaneously, atom $2$ is hit by $1$ neutron, releasing $1$ neutron to atom $3$, and releasing $2$ units of energy.
• In the third second, atom $1$ is hit by $1$ neutron, releasing $2$ units of energy. Meanwhile, atom $2$ is hit by $1$ neutron, releasing $2$ units of energy, and atom $3$ is hit by $1$ neutron, releasing $2$ units of energy.

Thus, atom $1$ releases a total of $6$ units of energy, atom $2$ releases $4$ units of energy, and atom $3$ releases $2$ units of energy.

For all the tests, $1\le n,\sum a_i\le5\times10^5$, $1 \leq k \leq 10^{18}$, $0 \leq a_i \leq 5 \times 10^5$, $1\le m,v_i,x_{i,j}\le n$, and $v_1,v_2,\dots,v_m$ are distinct, while $x_{i,1},x_{i,2},\dots,x_{i,a_i}$ are also distinct.

This problem has a large input, please use faster I/O methods.