HMMT 2022 G6

Let $A B C D$ be a rectangle inscribed in circle $\Gamma$, and let $P$ be a point on minor arc $A B$ of $\Gamma$. Suppose that $P A \cdot P B=2, P C \cdot P D=18$, and $P B \cdot P C=9$. The area of rectangle $A B C D$ can be expressed as $\frac{a \sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers and $b$ is a squarefree positive integer. Compute $100 a+10 b+c$.