F. Colorful Polygontime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou 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$$$ diagonals that intersect only at vertices. A diagonal is a segment between two vertices which lies inside the polygon and has exactly two common points with the polygon sides — the vertices it connects.InputThe first line of each test contains a single integer $$$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$$$.OutputIn the first line, output a single integer $$$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$$$-th vertex of the polygon.The polygon must be simple. The vertices may be given in either clockwise or counterclockwise order.ExamplesInput31 1 2Output8
0 0
2 1
4 2
6 1
3 5
4 7
0 5
1 2
Input24 1Output5
-2 -2
-3 1
0 3
3 1
2 -2
NoteIn the first test, the required polygon has to have $$$\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.