
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 $P=\left\{ x\mid {{x}^{2}}-3x+2\leq 0 \right\}$, $S=\left\{ x\mid 1-m\leq x\leq 1+m \right\}$.

$(1)$ Is there a real number $m$ such that $x\in P$ is a necessary and sufficient condition for $x\in S$? If so, find the range of $m$.

$(2)$ Is there a real number $m$ such that $x\in P$ is a necessary condition for $x\in S$? If so, find the range of $m$.

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