Let $S_1$ be a circle of radius 1 and center $(-1, 3).$ Let $S_2$ be a circle of radius r and with  center at A =(5, -7). Points $P_1,Q_1$ lie on $S_1$ and $P_2,Q_2$ on $S_2$ such that $P_1{P}_2$ is tangent to both $S_1$  and $S_2.$  Also $Q_1{Q}_2$ is tangent to both $S_1$ and $S_2.$ The line joining mid-points of $P_1{P}_2$ and $Q_1{Q}_2$ meets X-axis at B. If $AB=\sqrt{53},$ then the value of $r^2$ is