
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 maximum distance from a point on the ellipse $C$ to the right focus is $3$.

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

$(2)$ A line $l$ with an existing slope intersects the ellipse $C$ at points $A$ and $B$, and the circle with $AB$ as its diameter passes through the origin. Find the range of values for the $y$-intercept of the line.

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