
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. Point $A({2, \sqrt{2}})$ lies on the ellipse and satisfies $\overrightarrow{AF_{2}} \cdot \overrightarrow{F_{1}F_{2}}=0$.
(I) Find the standard equation of ellipse $C$;
(II) Determine if there exists a circle $x^{2}+y^{2}=r^{2}$ such that dynamic line $l$: $y=kx+m$ is tangent to it, where $l$ intersects ellipse $C$ at points $P$ and $Q$ with $OP \perp OQ$. If such a circle exists, find the value of $r$; otherwise, provide a reason.

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