
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 eccentricity $\frac{\sqrt{3}}{3}$, let $F_1$ and $F_2$ be the left and right foci of the ellipse $C$, respectively. A line $l$ passing through $F_2$ intersects the ellipse $C$ at points $A$ and $B$, and the perimeter of $\triangle F_1AB$ is $4\sqrt{3}$.
(I) Find the equation of ellipse $C$;
(II) If there exists a point $P$ on the ellipse $C$ such that the quadrilateral $OAPB$ is a parallelogram, find the equation of line $l$.

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