
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 triangle $ABC$ with $\angle B = \frac{\pi}{3}$,  
(I) if $AB=8\sqrt{3}$ and $AC=12$, find the area of $\triangle ABC$;  
(II) if $AB=4$ and $\vec{BM} = \vec{MN} = \vec{NC}$ with $AN=2\sqrt{3}BM$, find the length of $AM$.

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