
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
Inside circle $O: x^{2}+y^{2}=8$, there is a point $P(-1,2)$. Chord $AB$ passes through point $P$ and has an inclination angle of $\alpha$.

$(1)$ When $\alpha=135^{\circ}$, find the length of $AB$.

$(2)$ When chord $AB$ is bisected by point $P$, write the equation of line $AB$.

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