
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 point P that is a moving point on the circle $C: x^2 + y^2 - 2x - 4y + 1 = 0$, the maximum distance from point P to a certain line $l$ is 6. If a tangent AB to circle C is drawn from any point A on line $l$, with the point of tangency at B, then the minimum value of AB is ______.

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