
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 its left and right foci being $F_1$ and $F_2$ respectively, and point $P(-1, \frac{2\sqrt{3}}{3})$ lies on the ellipse $C$, $|PF_2| = \frac{4\sqrt{3}}{3}$. A line $l$ passing through point $F_1$ intersects the ellipse $C$ at points $M$ and $N$.
$(1)$ Find the standard equation of the ellipse $C$ and its eccentricity;
$(2)$ If the area of $\triangle OMN$ is $\frac{12}{11}$, with $O$ being the origin, find the equation of line $l$.

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