
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
On side \( BC \) of the rhombus \( ABCD \), a point \( M \) is chosen. Lines drawn through \( M \) perpendicular to the diagonals \( BD \) and \( AC \) intersect line \( AD \) at points \( P \) and \( Q \) respectively. It turned out that lines \( PB, QC \), and \( AM \) intersect at one point. What can the ratio \( BM : MC \) be?

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