
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 $O$, its foci on the $x$-axis, and its eccentricity as $\frac{1}{2}$, the distance from the right focus to the right vertex is $l$.

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

$(2)$ Does there exist a line $l: y=kx+m$ $(k \in \mathbb{R})$ that intersects ellipse $C$ at points $A$ and $B$, such that $|\vec{OA} + 2\vec{OB}| = |\vec{OA} - 2\vec{OB}|$ holds? If it exists, find the range of the real number $m$; if not, explain why.

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