
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 are on the $x$-axis, the length of the major axis is $4$, and the point $(1, \frac{\sqrt{3}}{2})$ is on the ellipse $C$.
(1) Find the equation of the ellipse $C$;
(2) A line $l$ with a slope of $1$ passes through the right focus of the ellipse and intersects the ellipse at points $A$ and $B$. Find the length of the segment $|AB|$.

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