
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 line $l$: $y=kx+\sqrt{3}$ intersects the $y$-axis at a point which is one of the foci of the ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{m}=1$ ($m > 0$).

(1) Find the equation of the ellipse $C$;

(2) If the line $l$ intersects the ellipse $C$ at points $A$ and $B$, is there a value of $k$ such that the circle with diameter $AB$ exactly passes through the origin $O$? If such a $k$ exists, find its value; if not, explain why.

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