
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 point $O$ is the coordinate origin, the eccentricity of the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $\frac{\sqrt{3}}{2}$, and the point $(\sqrt{3}, \frac{1}{2})$ is on the ellipse $C$. The line $l$ passes through the point $(1,1)$, intersecting the ellipse $C$ at points $A$ and $B$.
(I) Find the equation of the ellipse $C$;
(II) Determine whether there exists a point $P$ on the ellipse $C$ such that $\overrightarrow{OA} + \overrightarrow{OB} = \overrightarrow{OP}$. If such a point exists, find the equation of the line $l$; if not, explain the reason.

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