
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 pairwise externally tangent circles with radii  $1,2,3,$  respectively. A circle passes through the centers of  $\Gamma_2$  and  $\Gamma_3$  and is externally tangent to  $\Gamma_1$  at a point  $P.$  Suppose  $A$  and  $B$  are the centers of  $\Gamma_2$  and  $\Gamma_3,$  respectively. What is the value of  $\frac{{PA}^2}{{PB}^2}?$ *Proposed by Kyle Lee*

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