
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 the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with a focal length of $2c$, and points $P(c, 1)$ and $Q(\frac{2\sqrt{3}}{3}, -\frac{2\sqrt{3}}{3})$ both on the ellipse $C$.

1. Find the equation of the ellipse $C$.
2. If the left focus of the ellipse $C$ is $F$, and the line $l$ with the same slope as line $PF$ intersects ellipse $C$ at points $A$ and $B$, find the equation of line $l$ when the area of $\triangle AOB$ is at its maximum.

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