
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \frac{\sqrt{3}}{2}$, and the length of the line segment intercepted by the ellipse from a line passing through the focus and perpendicular to the $x$-axis is $\sqrt{2}$.

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

$(2)$ A line $l$ with slope $\frac{1}{2}$ intersects the ellipse $C$ at two distinct points $A$ and $B$. If $\angle AOB$ is obtuse, find the range of the $y$-intercept of line $l$.

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