
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 sets $P=\{x|x^{2}-8x-20\leqslant 0\}$, $S=\{x||x-1|\leqslant m\}$.

$(1)$ If $(P\cup S) \subseteq P$, find the range of the real number $m$;

$(2)$ Is there a real number $m$ such that "$x \in P$" is a necessary and sufficient condition for "$x \in S$"? If it exists, find the range of $m$; if not, explain why.

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