Let $ABC$ be a triangle such that $AB=\sqrt{10}$, $BC=4$, and $CA=3\sqrt{2}$. Circle $\omega$ has diameter $BC$, with center at $O$. Extend the altitude from $A$ to $BC$ to hit $\omega$ at $P$ and $P'$, where $AP < AP'$. Suppose line $P'O$ intersects $AC$ at $X$. Given that $PX$ can be expressed as $m\sqrt{n}-\sqrt{p}$, where $n$ and $p$ are squarefree, find $m+n+p$.

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