
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 $C$ with its center at the origin and its foci on the $x$-axis, and its eccentricity is $\frac{1}{2}$. A line passing through the left focus $F$ of the ellipse $C$ and having an inclination angle of $60^{\circ}$ is tangent to the circle $x^{2}+y^{2}=\frac{b^{2}}{a^{2}}$.

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

$(2)$ If the line $l: y=kx+m$ intersects the ellipse $C$ at points $M$ and $N$ (where $M,N$ are the left and right vertices, respectively), and the circle with diameter $MN$ exactly passes through the right vertex $A$ of the ellipse $C$. Determine whether the line $l$ passes through a fixed point. If so, find the coordinates of that point; if not, explain why.

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