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

(Ⅰ) Determine whether there exists a real number $m$ such that $P = S$. If it exists, find the value of $m$; otherwise, explain why not.

(Ⅱ) Determine whether there exists a real number $m$ such that $x \in P$ is a necessary but not sufficient condition for $x \in S$. If it exists, state the range of $m$; otherwise, explain why not.

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