
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 center of ellipse $C$ is at the origin, and its foci $F_1$ and $F_2$ are on the $x$-axis, with an eccentricity of $\frac{\sqrt{2}}{2}$. A line passing through $F_1$ intersects $C$ at points $A$ and $B$, and the perimeter of $\Delta E$ is $16$. Then, the equation of $C$ is.

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