
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 the line $l: x-y+1=0$. If there exist two points $M$ and $N$ on the circle $C: \left(x-2\right)^{2}+\left(y-1\right)^{2}=1$ such that $\angle MPN=60^{\circ}$, determine the range of the abscissa of point $P$.

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