
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
Given a rhombus  $ABCD$ . A point  $M$  is chosen on its side  $BC$ . The lines, which pass through  $M$  and are perpendicular to  $BD$  and  $AC$ , meet line  $AD$  in points  $P$  and  $Q$  respectively. Suppose that the lines  $PB,QC,AM$  have a common point. Find all possible values of a ratio  $\frac{BM}{MC}$ .
*S. Berlov, F. Petrov, A. Akopyan*

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