
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 line $l$ passing through point $P(1, 2)$ and intersecting the circle $x^2+y^2=6$ at points $A$ and $B$.

(1) If point $P$ is exactly the midpoint of segment $AB$, find the equation of line $l$;  
(2) If $|AB|=2\sqrt{5}$, find the equation of line $l$.

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