Sum of Intersection Sizes

You are given $N$ sets $S_1, S_2, \cdots, S_N$.

For each $k$, choose $k$ of the sets among $S_1, S_2, \cdots, S_N$ and take the size of their intersection. $a_k$ is the sum of these intersection sizes over all ${N \choose k}$ ways to choose the sets. In formula form:

$a_{k} := \sum_{\substack{\tau \subseteq \{1, 2, \cdots, N\} \\ |\tau|=k}} \left|\bigcap_{i \in \tau} S_{i}\right|.$

For every $k=1, 2, \cdots, N$, find $a_{k}$ modulo $998\,244\,353$ $(=119 \times 2^{23}+1)$. $998\,244\,353$ is prime.

The first line contains an integer $N$. ($1\leq N\leq 1\,000\,000$)

The next $N$ lines describe the sets. The $i$-th of these lines contains space-separated integers in the following format.

• The first number is $|S_i|$. ($|S_i|\ge 1$)
• The next $|S_i|$ numbers are the elements of $S_i$, $S_{i,1}, S_{i,2}, \cdots, S_{i,|S_i|}$, given in ascending order. ($1\le S_{i,j}\le 10^9$)

The sum of all set sizes does not exceed $1\,000\,000$. That is, $\sum_{1\leq i\leq N}|S_{i}|\leq 1\,000\,000$.

Print $N$ lines. On line $k$, print $a_{k}$ modulo $998\,244\,353$ for each $1 \leq k \leq N$.