
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 equilateral triangle \( \triangle ABC \), there are points \( D, E, F \) on sides \( BC, CA, AB \), respectively, dividing each side into a ratio of \( 3:(n-3) \) with \( n > 6 \). The segments \( AD, BE, CF \) intersect to form triangle \( \triangle PQR \) (where \( BE \) intersects \( AD \) at \( P \), and intersects \( CF \) at \( Q \)). When the area of \( \triangle PQR \) is \(\frac{4}{49}\) of the area of \( \triangle ABC \), find the value of \( n \).

(1992 Japan Mathematical Olympiad Preliminary Problem)

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