
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 point P_0(-1, 2) inside a circle defined by x^2 + y^2 = 8, let AB be a chord that passes through P_0 and has an angle of inclination α.
(1) When α = 135°, determine the length of chord AB;
(2) When chord AB is bisected by P_0, there exists a circle M that passes through point C(3, 0) and is tangent to line AB at point P_0. Find the standard equation of circle M.

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