
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 \( PABC \), if point \( P \)'s projection onto plane \( ABC \) is the circumcenter \( O \) of triangle \( ABC \), and the planes opposite to edges \( PA, PB, \) and \( PC \) have dihedral angles \(\alpha, \beta,\) and \(\gamma\) respectively, and the angle between \( PO \) and \( PA \) is \(\varphi\), then:

(I) \(\cos \angle BAC = \frac{\sin^2 \frac{\beta}{2} + \sin^2 \frac{\gamma}{2} - \sin^2 \frac{\alpha}{2}}{2 \sin \frac{\beta}{2} \cdot \sin \frac{\gamma}{2}} \) (and similar expressions hold for other angles);

(II) \(\frac{\sin \frac{\alpha}{2}}{\sin \angle BAC} = \frac{\sin \frac{\beta}{2}}{\sin \angle ABC} = \frac{\sin \frac{\gamma}{2}}{\sin \angle ACB} = \sin \varphi\).

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