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

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

(2) Does there exist a real number $m$ such that "x belongs to P" is a necessary and sufficient condition for "x belongs to 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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