
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
In the rectangular coordinate system $(xOy)$, the center of the ellipse $(C)$ is at the origin, and its foci $F\_1$ and $F\_2$ are on the $x$-axis with an eccentricity of $\frac{1}{2}$. A line $(L)$ passing through $F\_1$ intersects $(C)$ at points $A$ and $B$, and the perimeter of triangle $ABF\_2$ is $16$. Determine the equation of $(C)$.

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