
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 right square prism \(A B C D-A_{1} B_{1} C_{1} D_{1}\) with a base \(A B C D\) that is a unit square, if the dihedral angle \(A_{1}-B D-C_{1}\) is \(\frac{\pi}{3}\), find the length of \(A A_{1}\).

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