
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
Let \(ABCD\) be a rectangle, and let \(E\) and \(F\) be points on segment \(AB\) such that \(AE = EF = FB\). If \(CE\) intersects the line \(AD\) at \(P\), and \(PF\) intersects \(BC\) at \(Q\), determine the ratio of \(BQ\) to \(CQ\).

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