
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 \(ABCD\) be a regular tetrahedron with each edge length \(a\), let \(E\) be a point inside the face \(ABC\) and \(F\) inside face \(BCD\). Define \(s_1\) as the sum of the distances from \(E\) to the planes \(ABD\), \(ACD\), and \(BCD\), and \(s_2\) as the sum of the distances from \(F\) to the planes \(ABC\), \(ABD\), and \(ACD\). Similarly, define \(S_1\) as the sum of the distances from \(E\) to the edges \(AB\), \(BC\), and \(CA\), and \(S_2\) as the sum of the distances from \(F\) to the edges \(BC\), \(CD\), and \(DB\). Determine the value of \(\frac{s_1 + s_2}{S_1 + S_2}\).

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