
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 the ellipse $W$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, the short axis of the ellipse is $2$, and the ellipse passes through point $P(1, \frac{\sqrt{3}}{2})$,
1) Find the equation of the ellipse;
2) Line $l$ intersects with ellipse $W$ at points $A$ and $B$. Does there exist a point $C$ on ellipse $W$ such that quadrilateral $AOBC$ is a rectangle? If it exists, find the area of rectangle $AOBC$. If not, explain the reason.

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