
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 and one focus at $F(-2, 0)$, and the eccentricity $e = \frac{1}{2}$.  
(1) Find the equation of the ellipse $C$;  
(2) Let point $M(m, 0)$ be on the major axis of the ellipse $C$, and point $P$ be any point on the ellipse. When $|MP|$ is minimized, point $P$ is exactly 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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