
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 the vertex A (0, 1) of $\triangle ABC$, 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, and P $(m, 0)$, and a line with a slope of 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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