Sum of Intersection Sizes

You are given $N$ sets $S_1, S_2, \cdots, S_N$. Each $S_i$ is the set of all integers from $l_i$ to $r_i$, inclusive.

For each $k$, choose $k$ of the sets $S_1, S_2, \cdots, S_N$ and take the size of their intersection. $a_k$ is the sum of these intersection sizes over all ${N \choose k}$ ways to make the choice. 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 998244353. 998244353 is a prime.

The first line contains an integer $N$. ($1 \leq N \leq 300000$)

Each of the next $N$ lines contains two integers $l_i$ and $r_i$, separated by a space, which describe $S_i$. ($1 \leq l_i \leq r_i \leq 998244352$)

Print $N$ lines. For every $1 \leq k \leq N$, print $a_k$ modulo 998244353 on the $k$-th line.