
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, the circle $C$ passes through the points $A(4,0)$ and $B(2,2)$, and the center of the circle $C$ lies on the line $x+y-2=0$.  
$(1)$ Find the equation of the circle $C$.  
$(2)$ If point $D$ is any point on the circle, and the coordinates of fixed point $E$ are $(5,0)$, find the equation of the locus of the midpoint $M$ of the line segment $DE$.

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