
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 $p: (x+1)(x-5) \leqslant 0$; $q: 1-m \leqslant x \leqslant 1+m (m > 0)$.

$(1)$ If $p$ is a sufficient condition for $q$, find the range of real numbers $m$;

$(2)$ If $m=5$, and "$p \vee q$" is true, "$p \wedge q$" is false, find the range of real numbers $x$.

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