
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Circle $C_1$ has its center at $O$, which lies on circle $C_2$. Points $X$ and $Y$ are the intersections of circles $C_1$ and $C_2$. Another point $Z$, lying on circle $C_2$ and outside circle $C_1$, has distances $XZ = 11$, $OZ = 13$, and $YZ = 5$. Determine the radius of circle $C_1$.

Present the answer in LaTex format: \boxed{Your answer}
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