A936.Moortal Cowmbat--Gold

Bessie has been playing the popular fighting game Moortal Cowmbat for a long
time now. However, the game developers have recently rolled out an update that
is forcing Bessie to change her play style.
The game uses $M$ buttons labeled by the first $M$ lowercase letters ($1 \leq M \leq 26$). Bessie's favorite combo of moves in the game is a length-$N$
string $S$ of button presses ($1 \leq N \leq 10^5$). However, due to the most
recent update, every combo must now be made from a series of "streaks", where
a streak is defined as a series of the same button pressed at least $K$ times
in a row ($1 \leq K \leq N$). Bessie wants to modify her favorite combo to
produce a new combo of the same length $N$, but made from streaks of button
presses to satisfy the change in rules.
It takes $a_{ij}$ days for Bessie to train herself to press button $j$ instead
of button $i$ at any specific location in her combo (i.e. it costs $a_{ij}$ to
change a single specific letter in $S$ from $i$ to $j$). Note that it might
take less time to switch from using button $i$ to an intermediate button $k$
and then from button $k$ to button $j$ rather than from $i$ to $j$ directly
(or more generally, there may be a path of changes starting with $i$ and
ending with $j$ that gives the best overall cost for switching from button $i$
ultimately to button $j$).
Help Bessie determine the smallest possible number of days she needs to create
a combo that supports the new requirements.

• Test cases 2-4 satisfy $N\le 1000, K\le 50.$
• Test cases 5-8 satisfy $N\le 30,000, K\le 50.$

The first line of input contains $N$, $M$, and $K$. The second line contains
$S$, and the final $M$ lines contain an $M\times M$ matrix of values $a_{ij}$,
where $a_{ij}$ is an integer in the range $0 \ldots 1000$ and $a_{ii} = 0$ for
all $i$.

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