Score : $600$ points

Problem Statement

Given is an integer sequence of length $N$: $A=(A_1,A_2,\cdots,A_N)$.

You can repeat the operation below any number of times.

• Choose an integer $i$ ($1 \leq i \leq N$) and add $-1, 2, -1$ to $A_{i-1},A_i,A_{i+1}$, respectively. Here, $A_0$ stands for $A_N$, and $A_{N+1}$ stands for $A_1$.

Determine whether it is possible to make every element of $A$ $0$. If it is possible, find the minimum number of operations needed.

Constraints

• $3 \leq N \leq 200000$
• $-100 \leq A_i \leq 100$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\cdots$ $A_N$

Output

If it is impossible to make every element of $A$ $0$, print -1. If it is possible to do so, print the minimum number of operations needed.

4
3 0 -1 -2

5

We can achieve the objective in five operations, as follows.

• Do the operation with $i=2$. Now we have $A=(2,2,-2,-2)$.
• Do the operation with $i=3$. Now we have $A=(2,1,0,-3)$.
• Do the operation with $i=3$. Now we have $A=(2,0,2,-4)$.
• Do the operation with $i=4$. Now we have $A=(1,0,1,-2)$.
• Do the operation with $i=4$. Now we have $A=(0,0,0,0)$.

3
1 0 -2

-1

4
1 -1 1 -1

-1

10
-28 -3 90 -90 77 49 -31 48 -28 -84

962