Problem 1. Cowreography

USACO 2024 US Open Contest, Gold

The cows have formed a dance team, and Farmer John is their choreographer! The team's latest and greatest dance involves $N$ cows ($2 \le N \le 10^6$) standing in a line. Each move in the dance involves two cows, up to $K$ positions apart ($1 \le K < N$), gracefully jumping and landing in each other's position.

There are two types of cows in the line – Guernseys and Holsteins. As such, Farmer John has documented the dance as a sequence of

length-$N$ binary strings

, where a $0$ represents a Guernsey, a $1$ represents a Holstein, and the overall string represents how the cows are arranged in the line.

Unfortunately, Farmer Nhoj (who choreographs for a rival team) has sabotaged the dance and erased all but the first and last binary strings! With a big competition quickly approaching, Farmer John must waste no time in reconstructing the dance.

Given these two binary strings, help Farmer John find the minimum number of moves in the dance!

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $N$ and $K$.

The second line contains the first binary string.

The third line contains the last binary string.

It is guaranteed that both binary strings contain the same number of ones.

OUTPUT FORMAT (print output to the terminal / stdout):

The minimum number of moves in the dance.

SAMPLE INPUT:

4 1
0111
1110

SAMPLE OUTPUT:

3

One possible dance:

0111 -> 1011 -> 1101 -> 1110

SAMPLE INPUT:

5 2
11000
00011

SAMPLE OUTPUT:

3

One possible dance:

11000 -> 01100 -> 00110 -> 00011

SAMPLE INPUT:

5 4
11000
00011

SAMPLE OUTPUT:

2

One possible dance:

11000 -> 10010 -> 00011

SCORING: Inputs 4-5: $K=1$Inputs 6-7: Both strings have at most $8$ ones.Inputs 8-15: $N\le 5000$Inputs 16-23: No additional constraints.

Problem credits: Benjamin Qi