
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
Convex quadrilateral \(ABCD\) has right angles \(\angle A\) and \(\angle C\) and is such that \(AB = BC\) and \(AD = CD\). The diagonals \(AC\) and \(BD\) intersect at point \(M\). Points \(P\) and \(Q\) lie on the circumcircle of triangle \(AMB\) and segment \(CD\), respectively, such that points \(P\), \(M\), and \(Q\) are collinear. Suppose that \(\angle ABC = 160^\circ\) and \(\angle QMC = 40^\circ\). Find \(MP \cdot MQ\), given that \(MC = 6\).

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