_{This problem is a programming version of Problem 252 from projecteuler.net}

Given a set of points on a plane, we define a convex hole as a strictly convex polygon such that:

1. the vertices are from the set of given points;
2. the polygon is convex;
3. none of the elements of the given set lie in the interior of the polygon. Non-vertex elements from the given set may lie on the boundary;
4. no three consecutive vertices are colinear.

As an example, the image below shows a set of twenty points and a few such convex holes. The convex hole shown as a red heptagon has an area equal to square units, which is the highest possible area for a convex hole on the given set of points.

For our example, we used the first points , for , produced with the pseudo-random number generator (given and ):

-
i.e. , −, − −, …

Given integers , and , generate ordered pairs with the pseudo-random sequence with parameters and . If a pair appears more than once, remove all but one occurrence. What is the maximum area of a convex hole with vertices from this set of points? How many different convex holes are there?