
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 Cartesian coordinate system $(xOy)$, the ellipse $C$ has its center at the origin, with the foci $F_1$ and $F_2$ located on the x-axis and an eccentricity of $\frac{\sqrt{2}}{2}$. A line $l$ passing through point $F_1$ intersects the ellipse $C$ at points $A$ and $B$, such that the perimeter of triangle $\triangle ABF_2$ is $16$. Find the standard equation of the ellipse $C$.

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