
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.
(1) Find the trajectory equation of the circle center P;
(2) 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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