
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
Let the ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0) \) pass through the point \( (0, \sqrt{3}) \), with eccentricity \( \frac{1}{2} \). The line \( l \) passes through the right focus \( F \) of the ellipse \( C \) and intersects the ellipse \( C \) at points \( A \) and \( B \). The projections of \( A \), \( F \), and \( B \) on the line \( x=4 \) are denoted as \( D \), \( K \), and \( E \) respectively.

1. Find the equation of the ellipse \( C \).
2. Connect \( AE \) and \( BD \). When the angle of inclination of line \( l \) changes, do lines \( AE \) and \( BD \) intersect at a fixed point? If so, find the coordinates of the fixed point and provide a proof; if not, explain why.

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