
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 coordinates, the focus is on the x-axis, and the eccentricity is equal to $\frac{\sqrt{2}}{2}$. One of its vertices is exactly the focus of the parabola $x^2=4y$. What is the standard equation of the ellipse C?

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