Wrong Answer on test 233 (Hard Version)

This is the harder version of the problem. In this version, $$$1 \le n \le 2\cdot10^5$$$. You can hack this problem if you locked it. But you can hack the previous problem only if you locked both problems.

The problem is to finish $$$n$$$ one-choice-questions. Each of the questions contains $$$k$$$ options, and only one of them is correct. The answer to the $$$i$$$-th question is $$$h_{i}$$$, and if your answer of the question $$$i$$$ is $$$h_{i}$$$, you earn $$$1$$$ point, otherwise, you earn $$$0$$$ points for this question. The values $$$h_1, h_2, \dots, h_n$$$ are known to you in this problem.

However, you have a mistake in your program. It moves the answer clockwise! Consider all the $$$n$$$ answers are written in a circle. Due to the mistake in your program, they are shifted by one cyclically.

Formally, the mistake moves the answer for the question $$$i$$$ to the question $$$i \bmod n + 1$$$. So it moves the answer for the question $$$1$$$ to question $$$2$$$, the answer for the question $$$2$$$ to the question $$$3$$$, ..., the answer for the question $$$n$$$ to the question $$$1$$$.

We call all the $$$n$$$ answers together an answer suit. There are $$$k^n$$$ possible answer suits in total.

You're wondering, how many answer suits satisfy the following condition: after moving clockwise by $$$1$$$, the total number of points of the new answer suit is strictly larger than the number of points of the old one. You need to find the answer modulo $$$998\,244\,353$$$.

For example, if $$$n = 5$$$, and your answer suit is $$$a=[1,2,3,4,5]$$$, it will submitted as $$$a'=[5,1,2,3,4]$$$ because of a mistake. If the correct answer suit is $$$h=[5,2,2,3,4]$$$, the answer suit $$$a$$$ earns $$$1$$$ point and the answer suite $$$a'$$$ earns $$$4$$$ points. Since $$$4 > 1$$$, the answer suit $$$a=[1,2,3,4,5]$$$ should be counted.