
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 $C: x^2 + y^2 - 4x + 2y + 1 = 0$, and a line $l: y = kx - 1$.
1. What is the value of $k$ when the line $l$ passes through the center of the circle?
2. Is there a line $l$ that intersects circle $C$ at points $A$ and $B$, and the area of $\triangle ABC$ is 2? If it exists, find the equation of line $l$; if not, explain why.

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