
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 the ellipse $C$ is at the origin, its foci lie on the $x$-axis, the focal distance is $2$, and the eccentricity is $\frac{1}{2}$.
(1) Find the equation of the ellipse $C$;
(2) Let line $l$ pass through point $M(0,1)$, and intersect with ellipse $C$ at points $A$ and $B$. If $|AB|=\frac{3\sqrt{5}}{2}$, find the equation of line $l$.

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