
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 $C: (x-2)^2+y^2=1$ and a point $P$ on the line $l: x+y+1=0$. If there exists a line $m$ passing through point $P$ intersecting circle $C$ at points $A$ and $B$, with point $A$ being the midpoint of $PB$, then the range of the x-coordinate $x_0$ of point $P$ is \_\_\_\_\_.

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