
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
Let $A$ and $B$ be two moving points on the lines $y= \frac {2 \sqrt {5}}{5}x$ and $y=- \frac {2 \sqrt {5}}{5}x$ respectively, and $|\overrightarrow{AB}|= \sqrt {20}$. The moving point $P$ satisfies $\overrightarrow{OP}= \overrightarrow{OA}+ \overrightarrow{OB}$. Denote the trajectory of the moving point $P$ as $C$.  
$(1)$ Find the equation of curve $C$;  
$(2)$ If the coordinates of point $D$ are $(0,16)$, $M$ and $N$ are two moving points on curve $C$, and $\overrightarrow{DM}=\lambda \overrightarrow{DN}$, find the range of the real number $\lambda$;  
$(3)$ $M$ and $N$ are any two points on curve $C$, and the line $MN$ is not perpendicular to the $y$-axis. The perpendicular bisector of segment $MN$, line $l$, intersects the $y$-axis at point $E(0,y_0)$. Find the range of $y_0$.

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