
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 $F_1(-1,0)$ and $F_2(1,0)$ being the left and right foci of the ellipse, respectively. A line $l$ passes through point $F_2(1,0)$ and intersects the ellipse $C$ at points $A$ and $B$. The perimeter of $\triangle ABF_1$ is $4\sqrt{3}$.  
$(I)$ Find the equation of the ellipse $C$;  
$(II)$ If $OA \perp OB$, find the equation of the line $l$.

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