F. Serval and Bonus Problemtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputGetting closer and closer to a mathematician, Serval becomes a university student on math major in Japari University. On the Calculus class, his teacher taught him how to calculate the expected length of a random subsegment of a given segment. Then he left a bonus problem as homework, with the award of a garage kit from IOI. The bonus is to extend this problem to the general case as follows.You are given a segment with length l. We randomly choose n segments by choosing two points (maybe with non-integer coordinates) from the given segment equiprobably and the interval between the two points forms a segment. You are given the number of random segments n, and another integer k. The 2n endpoints of the chosen segments split the segment into (2n+1) intervals. Your task is to calculate the expected total length of those intervals that are covered by at least k segments of the n random segments.You should find the answer modulo 998244353.InputFirst line contains three space-separated positive integers n, k and l (1\leq k \leq n \leq 2000, 1\leq l\leq 10^9).OutputOutput one integer — the expected total length of all the intervals covered by at least k segments of the n random segments modulo 998244353.Formally, let M = 998244353. It can be shown that the answer can be expressed as an irreducible fraction \frac{p}{q}, where p and q are integers and q \not \equiv 0 \pmod{M}. Output the integer equal to p \cdot q^{-1} \bmod M. In other words, output such an integer x that 0 \le x < M and x \cdot q \equiv p \pmod{M}.ExamplesInput
1 1 1
Output
332748118
Input
6 2 1
Output
760234711
Input
7 5 3
Output
223383352
Input
97 31 9984524
Output
267137618
NoteIn the first example, the expected total length is \int_0^1 \int_0^1 |x-y| \,\mathrm{d}x\,\mathrm{d}y = {1\over 3}, and 3^{-1} modulo 998244353 is 332748118.
1 1 1

332748118 
['combinatorics', 'dp', 'math', 'probabilities', '*2600']