
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 circle $C$: $x^{2}+(y-1)^{2}=5$, and line $l$: $mx-y+1-m=0$, where line $l$ intersects circle $C$ at points $A$ and $B$.

1. If $|AB|= \sqrt {17}$, find the angle of inclination of line $l$.
2. If point $P(1,1)$, and $2 \overrightarrow{AP}= \overrightarrow{PB}$, find the equation of line $l$.

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