
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 $O: x^2 + y^2 = 8$ with a point $P_0(-1, 2)$ inside it, and $AB$ is a chord passing through point $P_0$ with an inclination angle of $\alpha$.
(1) When $\alpha = 135^\circ$, find the length of chord $AB$;
(2) When chord $AB$ is bisected by $P_0$, find the equation of line $AB$.

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