
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 square \(ABCD\) with side length \(a\), vertex \(A\) lies in plane \(\beta\), and the other vertices are on the same side of plane \(\beta\). The distances from points \(B\) and \(D\) to plane \(\beta\) are 1 and 2, respectively. If the dihedral angle between plane \(ABCD\) and plane \(\beta\) is 30 degrees, then \(a =\)?

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