
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 $xoy$, the curve $y=x^{2}-8x+2$ intersects with the coordinate axes at points that all lie on a circle $C$.  
$(1)$ Find the equation of circle $C$;  
$(2)$ Let the center of circle $C$ be $C$, and the coordinates of point $D$ be $(2, \frac {1}{2})$. Determine a point $P$ on the line $x-y-6=0$ such that $|PC|+|PD|$ is minimized, and find the coordinates of point $P$ at this time.

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