
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 a straight line \( l \) intersect each side (or its extension) of the closed planar polygonal chain \( ABCD \) at points \( P, E, Q, F \) respectively. Then
\[
\frac{AP}{PB} \cdot \frac{BE}{EC} \cdot \frac{CQ}{QD} \cdot \frac{DF}{FA} = 1.
\]

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