
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 prism \( A B C - A_{1} B_{1} C_{1} \), \( A B \) is perpendicular to the lateral face \( B B_{1} C_{1} C \). Point \( E \) lies on edge \( C C_{1} \) such that \( E \ne C \) and \( E \ne C_{1} \). Given that \( E A \perp E B_1 \), \( A B = \sqrt{2} \), \( B B_1 = 2 \), \( B C = 1 \), and \( \angle B C C_1 = \frac{\pi}{3} \), find the tangent of the dihedral angle between the planes \( A-E B_1 \) and \( A_1 \).

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