
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 $C$ with its center at the origin, its left focus at $F_{1}(-\sqrt{3},0)$, and a point $M(\sqrt{3}, \frac{1}{2})$ on the ellipse.  
$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ A line $l$ passing through point $P(1,0)$ intersects the ellipse $C$ at two distinct points $A$ and $B$. If the area of $\triangle AOB$ (where $O$ is the origin) is $S= \frac{4}{5}$, find the equation of line $AB$.

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