
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 $C:\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1({a＞b＞0})$ with its right focus at $F({\sqrt{3},0})$, and the ratio of the length of the major axis to the length of the minor axis is $2$.<br/>$(1)$ Find the equation of the ellipse $C$;<br/>$(2)$ Let the line $l$ passing through the point $A\left(1,0\right)$ intersect the ellipse $C$ at two distinct points $M$ and $N$. If the point $B\left(0,1\right)$ lies on the circle with diameter $MN$, find the equation of the line $l$.

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