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 $1\le a,b,c,d\le m$ 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 $m=100$?