A15399 | Arranging Crystal Balls

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:

1. 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 $p$ .
2. 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 $-1$ .