
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 circle $C_{1}:x^{2}+y^{2}=9$, point $A$ is a moving point on circle $C_{1}$, $AN\bot x$ axis at point $N$, and the moving point $M$ satisfies $\overrightarrow{OM}+2\overrightarrow{AM}=\left( 2\sqrt{2}-2 \right)\overrightarrow{ON}$. Let the trajectory of the moving point $M$ be curve $C$.

$(1)$ Find the equation of the trajectory curve $C$ of the moving point $M$;

$(2)$ If the line $l$ intersects curve $C$ at two different points $P$ and $Q$, and the circle with diameter $PQ$ passes through the origin $O$, find the range of the length of segment $PQ$.

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