
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
Let  $\gamma_1, \gamma_2, \gamma_3$  be three circles with radii  $3, 4, 9,$  respectively, such that  $\gamma_1$  and  $\gamma_2$  are externally tangent at  $C,$  and  $\gamma_3$  is internally tangent to  $\gamma_1$  and  $\gamma_2$  at  $A$  and  $B,$  respectively. Suppose the tangents to  $\gamma_3$  at  $A$  and  $B$  intersect at  $X.$  The line through  $X$  and  $C$  intersect  $\gamma_3$  at two points,  $P$  and  $Q.$  Compute the length of  $PQ.$ *Proposed by Kyle Lee*

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