
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 $\triangle ABC$ with vertex $A(0,1)$, the equation of the line containing the median $CD$ on side $AB$ is $2x-2y-1=0$, and the equation of the line containing the altitude $BH$ on side $AC$ is $y=0$.  
$(1)$ Find the coordinates of vertices $B$ and $C$ of $\triangle ABC$;  
$(2)$ If circle $M$ passes through three distinct points $A$, $B$, $P(m,0)$, and a line with slope $1$ is tangent to circle $M$ at point $P$, find the equation of circle $M$.

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