
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 an ellipse C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with two foci $F_1$ and $F_2$, and an eccentricity of $\frac{\sqrt{6}}{3}$. Points A and B are on the ellipse, with $F_1$ on segment AB, and the perimeter of $\triangle ABF_2$ equals $4\sqrt{3}$.

(1) Find the standard equation of ellipse C.
(2) For any point P on the circle O: $x^2 + y^2 = 4$, draw two tangents from P to ellipse C, intersecting circle O at points M and N. Find the maximum area of $\triangle PMN$.

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