Arranging Crystal Balls
Description
In the world of Compfestnesia, Pak Chanek discovers a secret underground dungeon. Inside it, there is a treasure chest that is surrounded by $n$ statues that are arranged in a circular manner. The statues are numbered from $0$ to $n-1$ with statue $i$ being to the left of statue $i+1$ and statue $n-1$ being to the left of statue $0$.
Pak Chanek observes that each statue is holding a crystal ball with an integer between $0$ and $m-1$ inclusive. Let's say the integer in the crystal ball of statue $i$ is $a_i$.
The dungeon provides instructions that every integer in the crystal balls must be $0$ in order to open the treasure chest. To achieve that, Pak Chanek is given an integer $k$, and he can do zero or more operations. In a single operation, Pak Chanek does the following:
- Choose exactly $k$ consecutive statues. In other words, choose the statues $p, (p+1) \bmod n, (p+2) \bmod n, (p+3) \bmod n, \ldots, (p+k-1) \bmod n$ for some chosen index $p$.
- Do one of the following:
- For all chosen statues, change their values of $a_i$ into $(a_i+1) \bmod m$.
- For all chosen statues, change their values of $a_i$ into $(a_i-1) \bmod m$.
Help Pak Chanek find the minimum possible number of operations to open the treasure chest.
The first line contains three integers $n$, $m$, and $k$ ($2 \leq n,m \leq 10^6$, $nm \leq 2 \cdot 10^6$, $1 \leq k < n$) — the number of statues, the bound of the integers in the crystal balls, and the number of statues that can be operated in a single operation.
The second line contains $n$ integers $a_0,a_1,\ldots,a_{n-1}$ ($0 \leq a_i < m$) — the integers in the crystal balls.
If it is possible to perform zero or more operations so that $a_0=a_1=\ldots=a_{n-1}=0$, output the minimum number of operations required. Otherwise, output $-1$.
Input
The first line contains three integers $n$, $m$, and $k$ ($2 \leq n,m \leq 10^6$, $nm \leq 2 \cdot 10^6$, $1 \leq k < n$) — the number of statues, the bound of the integers in the crystal balls, and the number of statues that can be operated in a single operation.
The second line contains $n$ integers $a_0,a_1,\ldots,a_{n-1}$ ($0 \leq a_i < m$) — the integers in the crystal balls.
Output
If it is possible to perform zero or more operations so that $a_0=a_1=\ldots=a_{n-1}=0$, output the minimum number of operations required. Otherwise, output $-1$.
5 9 3
8 1 4 5 0
4 4 2
1 0 0 0
5 5 2
1 0 0 0 0
7
-1
10
Note
In the first example, Pak Chanek can do the following operations:
- Do the $a_i := (a_i-1) \bmod m$ operation $3$ times for statues $1$, $2$, and $3$. Now $a=[8,7,1,2,0]$.
- Do the $a_i := (a_i-1) \bmod m$ operation $1$ time for statues $3$, $4$, and $0$. Now $a=[7,7,1,1,8]$.
- Do the $a_i := (a_i+1) \bmod m$ operation $2$ times for statues $4$, $0$, and $1$. Now $a=[0,0,1,1,1]$.
- Do the $a_i := (a_i-1) \bmod m$ operation $1$ time for statues $2$, $3$, and $4$. Now $a=[0,0,0,0,0]$.