
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 $A(1,0)$, the moving point $C$ moves on the circle $\odot B: (x+1)^{2}+y^{2}=8$. The perpendicular bisector of segment $AC$ intersects $BC$ at point $D$.  
$(1)$ Find the equation of the locus $E$ of point $D$.  
$(2)$ Let $M$, $N$, and $P$ be three points on line $E$ such that $\overrightarrow{OM}+\overrightarrow{ON}+\overrightarrow{OP}=\overrightarrow{0}$, where $O$ is the origin. Find the range of $|MN|$.

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