
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 with center $C$ that passes through points $(1,1)$ and $(2,-2)$, and the center $C$ is on the line $l: x-y+1=0$,  
$(1)$ find the equation of circle $C$;  
$(2)$ a line passing through $A(1,0)$ intersects circle $C$ at points $E$ and $F$, find the equation of the trajectory of the midpoint $M$ of chord $EF$.

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