
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 the line $l: (2k+1)x + (k-1)y - (4k-1) = 0$ ($k \in \mathbb{R}$) intersects with the circle $C: x^2 + y^2 - 4x - 2y + 1 = 0$ at points $A$ and $B$.
1. Find the equation of line $l$ when $|AB|$ is minimized, and calculate the value of $|AB|$ at this time;
2. Find the equation of the tangent line to circle $C$ that passes through point $P(4, 4)$.

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