A symmetrical convex grid polygon is a polygon such that:

• All its vertices have integer coordinates.
• All its internal angles are strictly smaller than ${180}^{\circ }$.
• It has both horizontal and vertical symmetry.

For example, the left polygon is a convex grid polygon which has neither horizontal nor vertical symmetry, while the right one is a valid symmetrical convex grid polygon with six vertices:

Define $A(N)$, the minimum area of a symmetrical convex grid polygon with $N$ vertices.

You are given $A(4)=1$, $A(8)=7$, $A(40)=1039$ and $A(100)=17473$.

Find $A(1000)$.