
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 $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a > b > 0)$ passing through point $A(1,\frac{3}{2})$, the quadrilateral formed by the four vertices of the ellipse has an area of $4\sqrt{3}$.

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

(2) Determine if there exist two distinct points $E$ and $F$ on the ellipse $C$ such that: $\quad$(i) the slopes of lines $AE$ and $AF$ are negative reciprocals of each other; (ii) the midpoint of segment $EF$ lies on the $y$-axis. If such points exist, find the length of the chord formed by the intersection of the angle bisector of $\angle EAF$ with the ellipse. If not, explain why.

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