
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 $1$.

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

$(2)$ Determine whether there exists a line $I: y=kx+m$ ($k\in \mathbb{R}$) that intersects ellipse $C$ at points $A$ and $B$, such that $|\overrightarrow{OA} + \overrightarrow{2OB}| = |\overrightarrow{OA} - \overrightarrow{2OB}|$ holds. If such a line exists, find the range of values for the real number $M$. If it does not exist, explain why.

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