
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 \( \triangle ABC \), the sides opposite the angles \( \angle A \), \( \angle B \), and \( \angle C \) have lengths \( a \), \( b \), and \( c \) respectively. Given \( c=2 \) and \( \angle C = \frac{\pi}{3} \):

(1) Find the maximum value of the perimeter of \( \triangle ABC \).

(2) When \( 2 \sin 2A + \sin(2B + C) = \sin C \), find the area of \( \triangle ABC \).

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