
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 the medians of the triangle \(ABC\) meet at \(M\). Let \(D\) and \(E\) be different points on the line \(BC\) such that \(D C = C E = A B\), and let \(P\) and \(Q\) be points on the segments \(BD\) and \(BE\), respectively, such that \(2BP = PD\) and \(2BQ = QE\). Determine \(\angle PMQ\).

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