
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 $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ passing through the point $A( \sqrt {3}, \frac {1}{2})$, with an eccentricity of $\frac{ \sqrt {3}}{2}$, and $F\_{1}$, $F\_{2}$ being its left and right foci, respectively.

(1) Find the standard equation of ellipse $E$;

(2) Determine if there exists a circle centered at the origin such that any tangent line to this circle intersects ellipse $E$ at two points $P$ and $Q$ with $\overrightarrow{OP} \perp \overrightarrow{OQ}$. If such a circle exists, find its equation and the maximum value of $|PQ|$. If not, explain why.

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