
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 an ellipse $C$ with its center at the origin and its major axis along the $y$-axis. The distance from the foci to the endpoints of the minor axis is $\sqrt{2}$ times the length of the semi-focal distance. The shortest distance from the foci to a point on the ellipse is $\sqrt{2}-1$. Find the equation of the ellipse $C$.

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