
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
A moving point $A$ is on the circle $C$: $(x-1)^{2}+y^{2}=1$, and a moving point $B$ is on the line $l:x+y-4=0$. The coordinates of the fixed point $P$ are $P(-2,2)$. The minimum value of $|PB|+|AB|$ is ______.

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