Starflower Serenade (Hard)

The king of the winter kingdom decided to gather the most beautiful flowers and make a garland for the queen, who loves flowers. The flowers she likes are special. They bloom only after a magic seed is planted and starlight falls on it. There are $N$ kinds of flowers that can bloom from a magic seed, and the flower of the $i$-th kind blooms with probability $p_{i}$. Whether a seed blooms does not depend on other seeds. To make the garland, at least $M_{i}$ flowers of each kind $i$ are needed. Suppose flowers bloom from seeds one at a time until the garland is complete. What is the expected number of seeds needed?

The first line contains an integer $N$ ($1 \leq N \leq 6$), the number of kinds of flowers. Each of the next $N$ lines contains two integers: $M_{i}$ ($1 \leq M_{i} \leq 10$), the number of flowers of kind $i$ that are needed, and $a_{i}$, the weight of the blooming probability. $0 < a_{i} \leq 100,000$, and the probability $p_{i}$ that a flower of kind $i$ blooms is $a_{i} / \left(\sum_{i=1}^{N} a_{i}\right)$.

Print the expected number of seeds needed to make the garland. Under the given conditions, the value is always a positive rational number. Write it as $P/Q$, where $P$ and $Q$ are coprime positive integers. Print the unique integer $R$ with $0 \leq R < 998244353$ such that $R \times Q \equiv P \mod 998244353$.