
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 the three lateral edges \( PA, PB, PC \) of a triangular prism \( P-ABC \) are mutually perpendicular, let the dihedral angles between the planes \( P-AB \) and \( BC, P-BC \) and \( CA, P-CA \) and \( AB \) be \( \alpha, \beta, \gamma \), respectively. Then, \(\frac{\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma}{\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma} = \) ______.

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