
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 that point $P$ is a moving point on 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 point $A$ is taken arbitrarily on line $l$ to form a tangent line $AB$ to circle $C$, with $B$ being the point of tangency, then the minimum value of $AB$ is _______.

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