
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
In a rhombus \(ABCD\), angle \(BAD\) is acute. A circle inscribed in this rhombus touches sides \(AB\) and \(CD\) at points \(M\) and \(N\) respectively. It intersects segment \(CM\) at point \(P\) and segment \(BN\) at point \(Q\). Find the ratio \(BQ\) to \(QN\) if \(CP: PM = 9:16\).

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