Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. The radius of circle $C_1$ is in the form \(\sqrt{k}\). Find the value of \(k\).

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