
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 $C$ centered at the origin, one focus at $F(-2,0)$, and the ratio of the major axis to the minor axis as $2: \sqrt{3}$,
$(1)$ Find the equation of ellipse $C$;
$(2)$ Let point $M(m,0)$ be on the major axis of ellipse $C$, and point $P$ be any point on the ellipse. When $|\overrightarrow{MP}|$ is minimized, point $P$ coincides with the right vertex of the ellipse. Find the range of the real number $m$.

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