
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 an isosceles triangle \(ABC\) with \(AB = AC\), points \(P\) and \(Q\) are taken on the sides \(AB\) and \(BC\), respectively, such that \(P\) is the midpoint of side \(AB\) and the angles \(\angle PQB\) and \(\angle AQC\) are equal. Let \(M\) be the foot of the altitude of triangle \(BPQ\) drawn from vertex \(P\). Find the ratio of the lengths of segments \(CQ\) to \(QM\). If the answer is a fractional number, it should be written as a decimal with a dot, for example, "0.15".

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