
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 the rhombus \(ABCD\), point \(M\) is chosen on side \(BC\). Perpendiculars are drawn from \(M\) to the diagonals \(BD\) and \(AC\), intersecting line \(AD\) at points \(P\) and \(Q\) respectively. It is known that lines \(PB\), \(QC\), and \(AM\) intersect at a single point. What is the ratio \(\frac{BM}{MC}\)?

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