
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 propositions $p: |m-2| < 1$ and $q$: the equation $x^{2}-2\sqrt{2}x+m=0$ has real roots.

(Ⅰ) If the proposition "$p \wedge (\neg q)$" is true, find the range of the real number $m$.

(Ⅱ) Given the proposition $r: a-2 < m < a+1$, if $p$ is a sufficient but not necessary condition for $r$, find the range of the real number $a$.

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