
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 triangle \( \triangle ABC \), \( \angle B = \angle C = 30^\circ \) and \( BC = 2\sqrt{3} \). Points \( P \) and \( Q \) lie on segments \( AB \) and \( AC \) respectively, with \( AP = 1 \) and \( AQ = \sqrt{2} \). Line \( AD \) is perpendicular to \( BC \) at \( D \). When \( \triangle ABC \) is folded along \( AD \), the dihedral angle between plane \( ADB \) and plane \( ADC \) is \( 60^\circ \). Find the length of segment \( PQ \).

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