
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
Let the parabola C be defined by ${x}^{2}=py$ ($p\gt 0$) with focus F. Point M lies on C, and $|MF|=3$. If the circle with MF as its diameter passes through the point $(\sqrt{2},0)$, determine the equation of C.

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