
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
Circles \(\mathcal{C}_{1}\) and \(\mathcal{C}_{2}\) have radii 3 and 7 respectively. The circles intersect at distinct points \(A\) and \(B\). A point \(P\) outside \(\mathcal{C}_{2}\) lies on the line determined by \(A\) and \(B\) at a distance of 5 from the center of \(\mathcal{C}_{1}\). Point \(Q\) is chosen on \(\mathcal{C}_{2}\) so that \(P Q\) is tangent to \(\mathcal{C}_{2}\) at \(Q\). Find the length of the segment \(P Q\).

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