
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 that points \( D \) and \( E \) are on sides \( AB \) and \( AC \) of \( \triangle ABC \) respectively, with \( BE \) intersecting \( CD \) at point \( F \), let \( S_{\text{quadrilateral } EDD F} = S_1 \), \( S_{\triangle BDF} = S_2 \), \( S_{\triangle BCF} = S_3 \), and \( S_{\triangle CEF} = S_4 \). Determine the relationship between \( S_1 S_3 \) and \( S_2 S_4 \).

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