An array $$$h$$$ of length $$$l$$$ is a hilly array if $$$h_i = \min(i, \, l - i + 1)$$$ for every $$$1 \le i \le l$$$. That is, it rises $$$1, 2, 3, \ldots$$$ up to a peak of $$$\left\lceil l/2 \right\rceil$$$ in the middle and then mirrors back down. For example, the hilly array of length $$$5$$$ is $$$[1, 2, 3, 2, 1]$$$ and the one of length $$$6$$$ is $$$[1, 2, 3, 3, 2, 1]$$$. Each length gives exactly one hilly array.

Consider $$$n$$$ intervals, where the $$$i$$$-th interval is $$$[a_i, b_i]$$$. An interval contains a value $$$v$$$ if $$$a_i \le v \le b_i$$$; if $$$a_i \gt b_i$$$, the interval is empty and contains nothing.

A hilly array of length $$$x$$$ fits a sequence of intervals if we can find a subsequence of $$$x$$$ intervals such that each element of the hilly array, in order, comes from the corresponding interval in the subsequence. Formally, a hilly array $$$h$$$ of length $$$x$$$ fits a sequence of $$$n$$$ intervals if there are positions $$$1 \le i_1 \lt i_2 \lt \cdots \lt i_x \le n$$$ such that interval $$$i_j$$$ contains $$$h_j$$$ for every $$$1 \le j \le x$$$. Let $$$f$$$ be the length of the longest hilly array that fits, or $$$0$$$ if none does.

For example, take the $$$5$$$ intervals $$$[1,1], [2,1], [2,2], [1,1], [1,2]$$$. The longest hilly array that fits is $$$[1,2,1]$$$, placed at intervals $$$1$$$, $$$3$$$, and $$$5$$$, which contain $$$1$$$, $$$2$$$, and $$$1$$$ respectively. So $$$f = 3$$$ here.

You are given two integers $$$n$$$ and $$$k$$$. Compute the sum of $$$f$$$ over all $$$k^{2n}$$$ ways to form the sequence of intervals, where each $$$a_i$$$ and each $$$b_i$$$ independently takes a value from $$$1$$$ to $$$k$$$. Since the answer can be large, output it modulo $$$998244353$$$.