
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  $\triangle ABC$ ,  $\angle B = \angle C = 30^{\circ}$  and  $BC=2\sqrt 3$ .  $P, Q$  lie on segments  $\overline{AB}, \overline{AC}$  such that  $AP=1$  and  $AQ=\sqrt 2$ . Let  $D$  be the foot of the altitude from  $A$  to  $BC$ . We fold  $\triangle ABC$  along line  $AD$  in three dimensions such that the dihedral angle between planes  $ADB$  and  $ADC$  equals  $60$  degrees. Under this transformation, compute the length  $PQ$ .

*(Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 8)*

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