1906M · Triangle Construction

Problem: You are given a regular $N$-sided polygon. Label one arbitrary side as side $1$, then label the next sides in clockwise order as side $2$, $3$, $\dots$, $N$. There are $A_i$ special points on side $i$. These points are positioned such that side $i$ is divided into $A_i + 1$ segments with equal length.

For instance, suppose that you have a regular $4$-sided polygon, i.e., a square. The following illustration shows how the special points are located within each side when $A = [3, 1, 4, 6]$. The uppermost side is labelled as side $1$.

You want to create as many non-degenerate triangles as possible while satisfying the following requirements. Each triangle consists of $3$ distinct special points (not necessarily from different sides) as its corners. Each special point can only become the corner of at most $1$ triangle. All triangles must not intersect with each other.

Determine the maximum number of non-degenerate triangles that you can create.

A triangle is non-degenerate if it has a positive area.

Input Format: The first line consists of an integer $N$ ($3 \leq N \leq 200\,000$).

The following line consists of $N$ integers $A_i$ ($1 \leq A_i \leq 2 \cdot 10^9$).

Output Format: Output a single integer representing the maximum number of non-degenerate triangles that you can create.

Note: Explanation for the sample input/output #1

One possible construction which achieves maximum number of non-degenerate triangles can be seen in the following illustration.

Explanation for the sample input/output #2

One possible construction which achieves maximum number of non-degenerate triangles can be seen in the following illustration.