
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
Given a circle N passes through points A(3, 1) and B(-1, 3), and its center is on the line $3x-y-2=0$.
(Ⅰ) Find the equation of circle N;
(Ⅱ) Find the equation of the circle symmetric to circle N with respect to the line $x-y+3=0$;
(Ⅲ) If point D is any point on circle N, and point C(3, 0), find the equation of the trajectory of the midpoint M of segment CD.

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