
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  $\omega_1$  and  $\omega_2$  be circles with radii  $3$  and  $12$  and externally tangent at point  $P$ . Let a common external tangent intersect  $\omega_1$ ,  $\omega_2$  at  $S$ ,  $T$  respectively and the common internal tangent at point  $Q$ . Define  $X$  to be the point on  $\overrightarrow{QP}$  such that  $QX=10$ . If  $XS$ ,  $XT$  intersect  $\omega_1$ ,  $\omega_2$  a second time at  $A$  and  $B$ , determine  $\tan\angle APB$ .
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*2015 CCA Math Bonanza Individual Round #15*

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