
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 $l$: $mx-y+1-m=0$.
(1) Determine the relationship between line $l$ and circle $C$;
(2) If point $P(1,1)$ intersects chord $AB$ such that $\frac{AP}{PB}=\frac{1}{2}$, find the equation of line $l$ in this case.

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