
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 $\triangle ABC$, $\angle ACB = \frac{2\pi}{3}$, and the sides opposite to angles A, B, and C are a, b, and c respectively.
(1) If a, b, and c form an arithmetic sequence with a common difference of 2, find the value of c.
(2) If the area of the circumcircle of $\triangle ABC$ is $\pi$, find the maximum value of the perimeter of $\triangle ABC$.

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