
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
Three circles \(\Gamma_{1}, \Gamma_{2}, \Gamma_{3}\) are pairwise externally tangent, with \(\Gamma_{1}\), the smallest, having a radius of 1, and \(\Gamma_{3}\), the largest, having a radius of 25. Let \(A\) be the point of tangency of \(\Gamma_{1}\) and \(\Gamma_{2}\), \(B\) be the point of tangency of \(\Gamma_{2}\) and \(\Gamma_{3}\), and \(C\) be the point of tangency of \(\Gamma_{1}\) and \(\Gamma_{3}\). Suppose that triangle \(ABC\) has a circumradius of 1 as well. The radius of \(\Gamma_{2}\) can then be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find the value of the product \(pq\).

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