13:I[87,["736","static/chunks/89fc4f0e-9e200ae6a0166583.js","401","static/chunks/363642f4-32903b9154b7835a.js","153","static/chunks/153-58eddeb032cc4608.js","131","static/chunks/131-05ce00cfd760e12b.js","87","static/chunks/87-b76b60eef573119a.js","286","static/chunks/app/problem/%5Bid%5D/page-45cf3875257fc32e.js"],"Header"] 14:I[1742,["736","static/chunks/89fc4f0e-9e200ae6a0166583.js","401","static/chunks/363642f4-32903b9154b7835a.js","153","static/chunks/153-58eddeb032cc4608.js","131","static/chunks/131-05ce00cfd760e12b.js","87","static/chunks/87-b76b60eef573119a.js","286","static/chunks/app/problem/%5Bid%5D/page-45cf3875257fc32e.js"],"ProblemView"] 15:T628,

You are given a positive integer $$$k$$$ and an array $$$a_1, a_2, \ldots, a_n$$$ of non-negative distinct integers not smaller than $$$k$$$ and not greater than $$$2^c-1$$$.

In each of the next $$$k$$$ seconds, one element is chosen randomly equiprobably out of all $$$n$$$ elements and decreased by $$$1$$$.

For each integer $$$x$$$, $$$0 \leq x \leq 2^c - 1$$$, you need to find the probability that in the end the bitwise XOR of all elements of the array is equal to $$$x$$$.

Each of these values can be represented as an irreducible fraction $$$\frac{p}{q}$$$, and you need to find the value of $$$p \cdot q^{-1}$$$ modulo $$$998\,244\,353$$$.

Input

The first line of input contains three integers $$$n, k, c$$$ ($$$1 \leq n \leq (2^c - k)$$$, $$$1 \leq k \leq 16$$$, $$$1 \leq c \leq 16$$$).

The second line contains $$$n$$$ distinct integers $$$a_1, a_2, \ldots, a_n$$$ ($$$k \leq a_i \leq 2^c-1$$$).

Bitwise Magic

Examples