
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 $(x^{2}+y^{2}=8)$ with a point $P_{0}(-1,2)$ inside it, $AB$ is a chord passing through point $P_{0}$ with an angle of inclination $\alpha$.

(1) When $\alpha=135^{\circ}$, find the length of the chord $AB$.
(2) When the chord $AB$ is bisected by point $P_{0}$, circle $M$ 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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