
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 that the center of ellipse $C$ is at the origin, the foci are on the $x$-axis, the length of the major axis is $2\sqrt{3}$, and the eccentricity is $\frac{\sqrt{6}}{3}$.  
$(Ⅰ)$ Find the standard equation of ellipse $C$.  
$(Ⅱ)$ If the line $l$ passing through the point $(1,0)$ intersects ellipse $C$ at points $A$ and $B$, and the circle with diameter $AB$ passes through the origin $O$, find the equation of line $l$.

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