
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]


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Given the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ with its left focus $F_1$ at coordinates $(-\sqrt{3}, 0)$, and $F_2$ as its right focus. Let point $M$ be a point on the ellipse $C$, and the perimeter of $\triangle MF_1F_2$ equals $4 + 2\sqrt{3}$.

$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Through the fixed point $P(0, 2)$, draw line $l$ intersecting the ellipse $C$ at two distinct points $A$ and $B$, and $OA \perpendicular OB$ (where $O$ is the origin). Find the equation of line $l$.

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