
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 one focus of the ellipse $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ coincides with the focus of the parabola $y^{2}=4x$, and the chord length intercepted by the directrix of the parabola is $3$.

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

(2) If the line $l$ passing through the point $M(1,\frac{1}{2})$ intersects the ellipse $C$ at points $A$ and $B$, and point $M$ is the midpoint of chord $AB$, find the equation of line $l$.

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