
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 proposition: "$\exists x\geqslant 2$, the inequality $x^{2}-x-m\leqslant 0$ is a false proposition. Find the set of real numbers $B$ for the value of $m$; If the solution set of the inequality $(x-a)(x-a-1) \lt 0$ is $A$, and if $x\in A$ is a sufficient but not necessary condition for $x\in B$, find the range of real numbers for $a$.

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