
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  $\omega$  be a semicircle and  $AB$  its diameter.  $\omega_1$  and  $\omega_2$  are two different circles, both tangent to  $\omega$  and to  $AB$ , and  $\omega_1$  is also tangent to  $\omega_2$ .  Let  $P,Q$  be the tangent points of  $\omega_1$  and  $\omega_2$  to  $AB$  respectively, and  $P$  is between  $A$  and  $Q$ . Let  $C$  be the tangent point of  $\omega_1$  and  $\omega$ . Find  $\tan\angle ACQ$ .

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