
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
Choose one of the following three conditions to complete the question below and solve it:  
① The circle passes through the point $C(3,4)$;  
② The center of the circle lies on the line $x+y-2=0$;  
③ The chord cut by the circle on the $y$-axis has a length of $8$ and the coordinates of the center $M$ are integers.  
Given that the circle $M$ passes through the points $A(-1,2)$ and $B(6,3)$ and ____.  
$(1)$ Find the equation of circle $M$;  
$(2)$ Find the equation of the line containing the chord with midpoint $(2,1)$.

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