Let ABCD be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows:

A(a, 0), B(0, b), C(−c, 0), D(0, −d), where 1 ≤ a, b, c, d ≤ m and a, b, c, d, m are integers.

It can be shown that for m = 4 there are exactly 256 valid ways to construct ABCD. Of these 256 quadrilaterals, 42 of them strictly contain a square number of lattice points.

How many quadrilaterals ABCD strictly contain a square number of lattice points for m = 100?