In one move, Milica can replace the whole string with $$$\texttt{AA} \ldots \texttt{A}$$$. In her second move, she can replace a prefix of length $$$k$$$ with $$$\texttt{BB} \ldots \texttt{B}$$$. The process takes no more than $$$2$$$ operations. The question remains — when can we do better?

In the hint section, we showed that the minimum number of operations is $$$0$$$, $$$1$$$, or $$$2$$$. We have $$$3$$$ cases:

1. No operations are needed if $$$s$$$ already contains $$$k$$$ characters $$$\texttt{B}$$$.

2. Else, we can use brute force to check if changing some prefix leads to $$$s$$$ having $$$k$$$ $$$\texttt{B}$$$s. If we find such a prefix, we print it as the answer, and use only one operation. There are $$$O(n)$$$ possibilities. Implementing them takes $$$O(n)$$$ or $$$O(n^2)$$$ time.

3. Else, we use the two operations described in the hint section.

A fun fact is that only one operation is enough. Can you prove it?

Try a greedy approach. That is, split each $$$a_i$$$ only as many times as necessary (and try to create almost equal parts).

We will iterate over the array from right to left. Then, as described in the hint section, we will split the current $$$a_i$$$ and create almost equal parts. For example, $$$5$$$ split into three parts forms the subarray $$$[1,2,2]$$$. Splitting $$$8$$$ into four parts forms the subarray $$$[2,2,2,2]$$$. Notice that the subarrays must be sorted. Because we want to perform as few splits as possible, the rightmost endpoint value should be as high as possible (as long as it is lower than or equal to the leftmost endpoint of the splitting of $$$a_{i+1}$$$ if it exists).

When we iterate over the array, it is enough to set the current $$$a_i$$$ to the leftmost endpoint of the splitting (the smallest current value). It will help to calculate the optimal splitting of $$$a_{i-1}$$$. For the current $$$a_i$$$, we want to find the least $$$k$$$ such that we can split $$$a_{i-1}$$$ into $$$k$$$ parts so the rightmost endpoint is less than or equal to $$$a_i$$$. More formally, we want $$$\lceil \frac{a_{i-1}}{k} \rceil \leq a_i$$$ to hold. Afterwards, we set $$$a_{i-1}$$$ to $$$\lfloor \frac{a_{i-1}}{k} \rfloor$$$ and continue with our algorithm. The simplest way to find the desired $$$k$$$ is to apply the following formula:

The answer is the sum over all $$$k-1$$$.

There are $$$q \leq 2\cdot 10^5$$$ queries: given an index $$$i$$$ and an integer $$$x>1$$$, set $$$a_i := \lceil \frac{a_i}{x} \rceil$$$. Modify the array and print the answer to the original problem. Please note that the queries stack.

The solution does not exist only for "small" $$$k$$$ or when $$$n+m-k$$$ is an odd integer. Try to find a construction otherwise.

Read the hint for the condition of the solution's existence. We present a single construction that solves the problem for each valid $$$k$$$.

One can verify that the pattern holds in general.

How would you write a checker? That is, how would you check that a valid walk of length $$$k+1$$$ exists in the given grid (with all the edges colored beforehand)?

If $$$a_i > b_i$$$, swap them. Imagine the pairs $$$(a_i,b_i)$$$ as intervals. How can we visualize the problem?

The pair $$$(a_i,b_i)$$$ represents some interval, and $$$|a_i - b_i|$$$ is its length. Let us try to maximize the sum of the intervals' lengths. We present three cases of what a swap does to two arbitrary intervals.

Notice how the sum of the lengths increases only in the first case. We want the distance between the first interval's right endpoint and the second interval's left endpoint to be as large as possible. So we choose integers $$$i$$$ and $$$j$$$ such that $$$i \neq j$$$ and $$$a_j - b_i$$$ is maximized. We add twice the value, if it is positive, to the original absolute beauty. If the value is $$$0$$$ or negative, we simply do nothing.

To quickly find the maximum of $$$a_j - b_i$$$ over all $$$i$$$ and $$$j$$$, we find the maximum of $$$a_1,a_2,\ldots a_n$$$ and the minimum of $$$b_1,b_2, \ldots b_n$$$. Subtracting the two extremum values produces the desired result.

How to handle point updates? (change a single value in either $$$a$$$ or $$$b$$$, and print the answer after each query.)

Try a greedy approach. That is, split each $$$a_i$$$ only as many times as necessary (and try to create almost equal parts).