
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
A line passes through the right focus $F(\sqrt{2},0)$ of the ellipse $C$ and also passes through the endpoint of the minor axis. The slope of this line is $\frac{\pi}{4}$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Let $O$ be the origin. If point $A$ is on the line $y=2x$, and point $B$ is on the ellipse $C$, and $OA \perpendicular OB$, find the minimum length of segment $AB$.

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