E. Expected Damage
time limit per test
2 seconds
memory limit per test
256 megabytes
standard input
standard output

You are playing a computer game. In this game, you have to fight $$$n$$$ monsters.

To defend from monsters, you need a shield. Each shield has two parameters: its current durability $$$a$$$ and its defence rating $$$b$$$. Each monster has only one parameter: its strength $$$d$$$.

When you fight a monster with strength $$$d$$$ while having a shield with current durability $$$a$$$ and defence $$$b$$$, there are three possible outcomes:

  • if $$$a = 0$$$, then you receive $$$d$$$ damage;
  • if $$$a > 0$$$ and $$$d \ge b$$$, you receive no damage, but the current durability of the shield decreases by $$$1$$$;
  • if $$$a > 0$$$ and $$$d < b$$$, nothing happens.

The $$$i$$$-th monster has strength $$$d_i$$$, and you will fight each of the monsters exactly once, in some random order (all $$$n!$$$ orders are equiprobable). You have to consider $$$m$$$ different shields, the $$$i$$$-th shield has initial durability $$$a_i$$$ and defence rating $$$b_i$$$. For each shield, calculate the expected amount of damage you will receive if you take this shield and fight the given $$$n$$$ monsters in random order.


The first line contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 2 \cdot 10^5$$$) — the number of monsters and the number of shields, respectively.

The second line contains $$$n$$$ integers $$$d_1$$$, $$$d_2$$$, ..., $$$d_n$$$ ($$$1 \le d_i \le 10^9$$$), where $$$d_i$$$ is the strength of the $$$i$$$-th monster.

Then $$$m$$$ lines follow, the $$$i$$$-th of them contains two integers $$$a_i$$$ and $$$b_i$$$ ($$$1 \le a_i \le n$$$; $$$1 \le b_i \le 10^9$$$) — the description of the $$$i$$$-th shield.


Print $$$m$$$ integers, where the $$$i$$$-th integer represents the expected damage you receive with the $$$i$$$-th shield as follows: it can be proven that, for each shield, the expected damage is an irreducible fraction $$$\dfrac{x}{y}$$$, where $$$y$$$ is coprime with $$$998244353$$$. You have to print the value of $$$x \cdot y^{-1} \bmod 998244353$$$, where $$$y^{-1}$$$ is the inverse element for $$$y$$$ ($$$y \cdot y^{-1} \bmod 998244353 = 1$$$).

3 2
1 3 1
2 1
1 2
3 3
4 2 6
3 1
1 2
2 3