
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 the set $A=\{x|6+5x-x^{2} \gt 0\}$ and the set $B=\{x\left|\right.\left[x-\left(1-a\right)\right]\left[x-\left(1+a\right)\right] \gt 0\}$, where $a \gt 0$.  
$(1)$ If $a=2$, find $A\cap (\complement _{R}B)$;  
$(2)$ Let proposition $p:x\in A$, proposition $q:x\in B$. If ¬$p$ is a sufficient but not necessary condition for $q$, find the range of real numbers for $a$.

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