
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, given that the center of circle $M$ lies on the line $y=-2x$ and is tangent to the line $x+y-1=0$ at the point $P(2,-1)$.  
$(Ⅰ)$ Find the equation of circle $M$.  
$(Ⅱ)$ If point $A(3,0)$ and point $B$ lies on the circle, find the minimum value of $|AB|$.

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