
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 $l: x-y+2=0$ intersects the $x$-axis at point $A$, and point $P$ is on line $l$. There is exactly one point $B$ on the circle $C: (x-2)^{2}+y^{2}=2$ such that $AB \perp BP$. The set of possible values for the $x$-coordinate of point $P$ is __________.

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