
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
In the rectangular coordinate system $xOy$, a circle $C_1: x^2+y^2=r^2$ ($r>0$) is tangent to the line $l_0: y=x+2\sqrt{2}$. Point $A$ is a moving point on circle $C_1$, $AN$ is perpendicular to the $x$-axis at point $N$, and the moving point satisfies $\overrightarrow{OM}+\overrightarrow{AM}=\overrightarrow{ON}$, where $M$ is another moving point. Let the trajectory of point $M$ be curve $C$.

(I) Find the equation of curve $C$;
(II) Let $P$ and $Q$ be two moving points on curve $C$, and $T$ be the midpoint of line segment $PQ$. The slopes of $OP$ and $OQ$ are $k_1$ and $k_2$, respectively, and $k_1k_2=-\frac{1}{4}$. Find the range of $|OT|$.

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