HMMT 2022 G4

Parallel lines $\ell_{1}, \ell_{2}, \ell_{3}, \ell_{4}$ are evenly spaced in the plane, in that order. Square $A B C D$ has the property that $A$ lies on $\ell_{1}$ and $C$ lies on $\ell_{4}$. Let $P$ be a uniformly random point in the interior of $A B C D$ and let $Q$ be a uniformly random point on the perimeter of $A B C D$. Given that the probability that $P$ lies between $\ell_{2}$ and $\ell_{3}$ is $\frac{53}{100}$, the probability that $Q$ lies between $\ell_{2}$ and $\ell_{3}$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.