
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 $p: \left|1- \frac{x-1}{3}\right| \geq 2$, and $q: x^2 - 2x + 1 - m^2 \geq 0$ ($m > 0$). If the negation of $p$, $\neg p$, is a necessary but not sufficient condition for the negation of $q$, $\neg q$, find the range of real numbers $m$.

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