Let $\Gamma_1$ be a circle with radius $\frac{5}{2}$. Points $A$, $B$, and $C$ are on $\Gamma_1$ such that $\overline{AB} = 3$ and $\overline{AC} = 5$. Let $\Gamma_2$ be a circle tangent to $AB$ and $BC$ at $Q$ and $R$, and also internally tangent to $\Gamma_1$ at $P$. $\Gamma_2$ intersects $AC$ at $X$ and $Y$. The area $[PXY]$ can be expressed as $\frac{a\sqrt{b}}{c}$. Find $a+b+c$.

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