
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 that the center of ellipse $C$ is at the origin, with the axes of symmetry being the coordinate axes, and passing through points $A(0,2)$ and $B(\frac{1}{2},\sqrt{3})$.
$(1)$ Find the standard equation of ellipse $C$.
$(2)$ Find the coordinates of the points where the line $\sqrt{3}y-2x-2=0$ intersects with ellipse $C$.

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