
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the points $(\sqrt{3}, \frac{1}{2})$ and $(1, \frac{\sqrt{3}}{2})$, and point $A$ is the bottom vertex of the ellipse.  
$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ Two lines $l_1$ and $l_2$ passing through point $A$ and perpendicular to each other intersect with the line $y=x$ at points $E$ and $F$ respectively, given that $OE=OF$, find the slope of the line $l_1$.

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