Circle $w$ with center $O$ meets circle $\Gamma$ at points $X$ and $Y$, and the center $O$ is on circle $\Gamma$. A point $Z$ on $\Gamma$ lies outside circle $w$ such that the distances are $XZ = 11$, $OZ = 15$, and $YZ = 13$. If the radius of circle $w$ is $r$, find $r^2$.

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