
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  $ABC$  be a triangle with circumcircle  $\omega$ . Let the bisector of  $\angle ABC$  meet segment  $AC$  at  $D$  and circle  $\omega$  at  $M\ne B$ . The circumcircle of  $\triangle BDC$  meets line  $AB$  at  $E\ne B$ , and  $CE$  meets  $\omega$  at  $P\ne C$ . The bisector of  $\angle PMC$  meets segment  $AC$  at  $Q\ne C$ . Given that  $PQ = MC$ , determine the degree measure of  $\angle ABC$ .

*Ray Li.*

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