
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  $AB$  be a diameter of a semicircle  $\Gamma$ . Two circles,  $\omega_1$  and  $\omega_2$ , externally tangent to each other and internally tangent to  $\Gamma$ , are tangent to the line  $AB$  at  $P$  and  $Q$ , respectively, and to semicircular arc  $AB$  at  $C$  and  $D$ , respectively, with  $AP<AQ$ . Suppose  $F$  lies on  $\Gamma$  such that  $ \angle FQB = \angle CQA $  and that  $ \angle ABF = 80^\circ $ . Find  $ \angle PDQ $  in degrees.

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