You are given an array $$$a_1, a_2, \ldots, a_n$$$, where $$$n \leq 8$$$ and $$$a_1 + a_2 + \cdots + a_n \leq 100$$$.
Construct a simple polygon$$$^{\text{∗}}$$$ with at most $$$333$$$ vertices that has exactly $$$$$$\frac{(a_1 + a_2 + \cdots + a_n)!}{a_1! a_2! \cdots a_n!}$$$$$$ different triangulations$$$^{\text{†}}$$$. It can be proven that such a polygon always exists.
$$$^{\text{∗}}$$$A simple polygon is a polygon that does not intersect itself and has no holes. In other words, no two non-consecutive edges can have common points, and consecutive edges must have exactly one common point — the vertex between them. Consecutive edges may be collinear.
$$$^{\text{†}}$$$A triangulation of a polygon with $$$m$$$ vertices is a set of $$$m-3$$$ diagonals that intersect only at vertices. A diagonal is a segment between two vertices which lies inside the polygon and has exactly two common points with the polygon sides — the vertices it connects.
The first line of each test contains a single integer $$$n$$$ ($$$2 \leq n \leq 8$$$) — the number of elements in the array.
The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \leq a_i \leq 100$$$) — the elements of the array.
It is guaranteed that $$$a_1 + a_2 + \cdots + a_n$$$ does not exceed $$$100$$$.
In the first line, output a single integer $$$m$$$ ($$$3 \leq m \leq 333$$$) — the number of vertices in the polygon.
In the $$$i$$$-th of the following $$$m$$$ lines, output two integers $$$x_i$$$, $$$y_i$$$ ($$$-10^6 \leq x_i, y_i \leq 10^6$$$) — the coordinates of the $$$i$$$-th vertex of the polygon.
The polygon must be simple. The vertices may be given in either clockwise or counterclockwise order.
31 1 2
8 0 0 2 1 4 2 6 1 3 5 4 7 0 5 1 2
24 1
5 -2 -2 -3 1 0 3 3 1 2 -2
In the first test, the required polygon has to have $$$\tfrac{4!}{1! 1! 2!} = 12$$$ triangulations. The following are all the triangulations of the example polygon.
In the second test, the required polygon has to have $$$\tfrac{5!}{4! 1!} = 5$$$ triangulations.
