Circular Dungeon

You are creating a level for a video game. The level consists of $$$n$$$ rooms placed in a circle. The rooms are numbered $$$1$$$ through $$$n$$$. Each room contains exactly one exit: completing the $$$j$$$-th room allows you to go the $$$(j+1)$$$-th room (and completing the $$$n$$$-th room allows you to go the $$$1$$$-st room).

You are given the description of the multiset of $$$n$$$ chests: the $$$i$$$-th chest has treasure value $$$c_i$$$.

Each chest can be of one of two types:

• regular chest — when a player enters a room with this chest, he grabs the treasure and proceeds to the next room;
• mimic chest — when a player enters a room with this chest, the chest eats him alive, and he loses.

The player starts in a random room with each room having an equal probability of being chosen. The players earnings is equal to the total value of treasure chests he'd collected before he lost.

You are allowed to choose the order the chests go into the rooms. For each $$$k$$$ from $$$1$$$ to $$$n$$$ place the chests into the rooms in such a way that:

• each room contains exactly one chest;
• exactly $$$k$$$ chests are mimics;
• the expected value of players earnings is minimum possible.

Please note that for each $$$k$$$ the placement is chosen independently.

It can be shown that it is in the form of $$$\frac{P}{Q}$$$ where $$$P$$$ and $$$Q$$$ are non-negative integers and $$$Q \ne 0$$$. Report the values of $$$P \cdot Q^{-1} \pmod {998244353}$$$.