
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 the tetrahedron \(ABCD\), \(AD > AB\), \(AD \perp AB\), \(AD \perp AC\), and \(\angle BAC = \frac{\pi}{3}\). Let the areas of triangles \(ADB\), \(ADC\), \(ABC\), and \(BCD\) be denoted by \(S_1\), \(S_2\), \(S_3\), and \(S_4\), respectively. Given that \(S_1 + S_2 = S_3 + S_4\), find \(\frac{S_3}{S_1} + \frac{S_3}{S_2}\).

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