
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 $F_{1},F_{2},|F_{1}F_{2}|=2 \sqrt{2}$, and points $A$ and $B$ on the ellipse. $F_{1}$ lies on the line segment $AB$, and the perimeter of $\triangle ABF_{2}$ equals $4 \sqrt{3}$.
(1) Find the standard equation of ellipse $C$.
(2) Draw two tangent lines $PM$ and $PN$ of ellipse $C$ passing through any point $P$ on the circle $O: x^{2}+y^{2}=4$, intersecting with 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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