
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 $O: x^{2}+y^{2}-4x-2y=0$, a line $l$ passes through the point $P(0,2)$. <br/>$(1)$ If the chord intercepted by line $l$ on circle $O$ has a length of $2$, find the equation of line $l$; <br/>$(2)$ If the ratio of the arc lengths of the minor arc and the major arc intercepted by line $l$ on circle $O$ is $3:1$, find the equation of line $l$.

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