
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
In the plane $xOy$, the ratio of the distance from the moving point $P$ to the point $F(\sqrt{2}, 0)$ and its distance to the line $x = 2\sqrt{2}$ is $\frac{\sqrt{2}}{2}$.
(1) Find the trajectory equation of the moving point $P$.
(2) Let $O$ be the origin. If point $A$ is on the line $y = 2$, and point $B$ is on the ellipse $C$, and $OA \perp OB$, find the minimum length of the line segment $AB$.

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