
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
An ellipse \( C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) \((a > b > 0)\) passes through the point \((0, \sqrt{3})\) and has an eccentricity of \(\frac{1}{2}\). A line \( l \) passes through the right focus \( F \) of the ellipse \( C \) and intersects the ellipse at points \( A \) and \( B \). Points \( A \), \( F \), and \( B \) are projected onto the line \( x = 4 \) as \( D \), \( K \), and \( E \) respectively.
1. Find the equation of the ellipse \( C \).
2. When the line \( l \) changes its angle of inclination, determine if line segments \( A E \) and \( B D \) intersect at a fixed point. If they do, find the coordinates of this fixed point and provide a proof. If not, explain why.

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