
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 a triangular prism \( P-ABC \) where the three lateral edges \( PA, PB, \) and \( PC \) are mutually perpendicular, and the dihedral angles \( P-AB-C, P-BC-A, \) and \( P-CA-B \) are denoted as \( \alpha, \beta, \gamma \) respectively, find the value of \(\frac{\sin^{2} \alpha + \sin^{2} \beta + \sin^{2} \gamma}{\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma}\).

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