
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 regular quadrilateral prism \( ABCD-A_1B_1C_1D_1 \) with a base edge length \( a \) and a lateral edge length \( b \), point \( P \) is a moving point on the diagonal \( AC_1 \). Let the angle between \( PC \) and plane \( ABCD \) be \( \alpha \) and the angle between \( PD \) and plane \( CDD_1C_1 \) be \( \beta \). Find the value of \( \tan \alpha \cdot \tan \beta \).

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