HMMT 2022 U12

A unit square $A B C D$ and a circle $\Gamma$ have the following property: if $P$ is a point in the plane not contained in the interior of $\Gamma$, then $\min (\angle A P B, \angle B P C, \angle C P D, \angle D P A) \leq 60^{\circ}$. The minimum possible area of $\Gamma$ can be expressed as $\frac{a \pi}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.