
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 $C$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$, a line $l$ with a slope of $\alpha$ passes through the left focus $F(-\sqrt{3},0)$ intersects the ellipse $C$ at points $A$ and $B$. When the slope of the line $l$ is $\frac{π}{6}$, $|AF|=7|BF|$.
$(1)$ Find the standard equation of the ellipse $C$;
$(2)$ Let point $O$ be the origin, find the maximum area of triangle $\triangle OAB$; and find the equation of the line $l$ at this point.

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