
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 \( P \) is a point on the edge \( AB \) of the cube \( ABCD-A_1 B_1 C_1 D_1 \), and the angle between the line \( A_1 B \) and the plane \( B_1 C P \) is \( 60^\circ \). Find the tangent value of the dihedral angle \( A_1 - B_1 P - C \).

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