
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 Cartesian coordinate plane $(xOy)$, a circle $P$ intercepts a line segment of length $2 \sqrt {2}$ on the $x$-axis and a line segment of length $2 \sqrt {3}$ on the $y$-axis.
(I) Find the trajectory equation of the center of the circle $P$;
(II) If the distance from point $P$ to the line $y=x$ is $\frac { \sqrt {2}}{2}$, find the equation of circle $P$.

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