
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 circle $C\_1$ has its center at the origin $O$ and is tangent to line $l\_1$: $x - 2y + 3\sqrt{5} = 0$, let point $A$ be a moving point on the circle and $AM$ be perpendicular to the $x$-axis at point $M$. Furthermore, moving point $N$ satisfies $\overrightarrow{ON} = \frac{2}{3}\overrightarrow{OA} + (\frac{2\sqrt{2}}{3} - \frac{2}{3})\overrightarrow{OM}$. Determine the equation of the curve traced by moving point $N$, denoted as curve $C$.
(I) Find the equation of ellipse $C$;
(II) If line $l$ intersects with ellipse $C$ at two distinct points $A$ and $B$, where $\overrightarrow{OA} \perp \overrightarrow{OB}$ ($O$ is the origin), determine the range of the length of line segment $AB$.

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