
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 an ellipse $C$ with its center at the origin $O$, one endpoint of the major axis is $(0,2)$, and the quadrilateral formed by the endpoints of the minor axis and the foci is a square. The line $l$ intersects the $y$-axis at point $P(0, m)$ and intersects the ellipse $C$ at two distinct points $A$ and $B$, with $\overrightarrow{AP}=2\overrightarrow{PB}$.

$(1)$ Find the equation of the ellipse;

$(2)$ Determine the range of values for $m$.

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