Sum Over Subsets

Description:

Input:

The first line contains one integer $$$m$$$ ($$$1 \le m \le 10^5$$$): the number of different values in the multiset $$$S$$$.

Each of the next $$$m$$$ lines contains two integers $$$a_i$$$, $$$freq_i$$$ ($$$1 \le a_i \le 10^5, 1 \le freq_i \le 10^9$$$). Element $$$a_i$$$ appears in the multiset $$$S$$$ $$$freq_i$$$ times. All $$$a_i$$$ are different.

Output:

Print the required sum, modulo $$$998\,244\,353$$$.

Sample Input:

2
1 1
2 1

Sample Output:

9

Sample Input:

4
1 1
2 1
3 1
6 1

Sample Output:

1207

Sample Input:

1
1 5

Sample Output:

560

Note:

A multiset is a set where elements are allowed to coincide. $$$|X|$$$ is the cardinality of a set $$$X$$$, the number of elements in it.

$$$A \subset B$$$: Set $$$A$$$ is a subset of a set $$$B$$$.

In the first example $$$B=\{1\}, A=\{1,2\}$$$ and $$$B=\{2\}, A=\{1, 2\}$$$ have a product equal to $$$1\cdot3 + 2\cdot3=9$$$. Other pairs of $$$A$$$ and $$$B$$$ don't satisfy the given constraints.