
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ whose center is at the origin $O$ and its axes of symmetry are along the coordinate axes. The top vertex of the ellipse and its two foci form an equilateral triangle with side length $2$.

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

$(2)$ If a line $l$ with slope $k$ passes through the point $M(4,0)$ and intersects the ellipse $C$ at points $A$ and $B$, and $\overrightarrow{OA} \cdot \overrightarrow{OB} > \frac{1}{2}$, find the range of values for $k$.

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