
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 \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of line segment \( BM \). Given that \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \).

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