
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 of the coordinate, its focus is on the $x$-axis, and the maximum distance from a point on the ellipse $C$ to the focus is $3$, with an eccentricity of $e=\frac{1}{2}$.
(I) Find the standard equation of the ellipse $C$;
(II) If a line $l$ passing through the left focus $F\_1$ with a slope angle of $\frac{\pi}{4}$ intersects the ellipse at points $A$ and $B$, find the value of $|AB|$.

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