
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 ellipse A, B satisfies: passing through the right focus of ellipse C, $F(\sqrt{2}, 0)$, and the line passing through the endpoint of the minor axis has an inclination angle of $\frac{\pi}{4}$.
(1) Find the equation of ellipse C;
(2) Let O be the origin. If point A is on the line $y=2$, point B is on ellipse C, and $OA \perp OB$, find the minimum length of segment AB.

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