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

(Ⅰ) Find the equation of the ellipse C;

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

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