Let $ABCD$ be a rectangle inscribed in circle $\Gamma$, and let $P$ be a point on the minor arc $AB$ of $\Gamma$. Suppose that $PA \cdot PB = 2$, $PC \cdot PD = 18$, and $PB \cdot PC = 9$. The area of rectangle $ABCD$ 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 $100a + 10b + c$.

Let's think step by step and output the final answer within \boxed{}.
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