
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 the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{3}}{2}$, and the line $l$: $x - y + 2 = 0$ is tangent to the circle $O$ centered at the origin with the semi-minor axis of the ellipse $C$ as its radius.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Is there a line that intersects the ellipse $C$ at points $A$, $B$, and intersects the $y$-axis at point $M(0,m)$, 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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