
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 system $xOy$, it is known that circle $P$ intersects the $x$-axis at a segment length of $2\sqrt{2}$ and the $y$-axis at a segment length of $2\sqrt{3}$.

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

$(2)$ If the distance from point $P$ to the line $y=x$ is $\dfrac{\sqrt{2}}{2}$, find the equation of circle $P$.

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