
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 $p$ be the proposition that the equation $x^{2}-2ax+2a^{2}-a-6=0$ has real roots for $x$, and let $q$ be the proposition that $m-1\leq a\leq m+3$.  
$(1)$ If the proposition $\neg p$ is true, find the range of real numbers for $a$.  
$(2)$ If $p$ is a necessary but not sufficient condition for $q$, find the range of real numbers for $m$.

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