
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 \( B = \frac{\pi}{4} \), angle \( C = \frac{5 \pi}{12} \), and \( AC = 2 \sqrt{6} \). \( D \) is the midpoint of \( AC \). If a line segment \( PQ \) of length 3 (with point \( P \) to the left of point \( Q \)) slides along line \( BC \), then what is the minimum value of \( AP + DQ \)?

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