
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 $p:\exists x\in \left[-1,2\right]$, $x^{2}-2x-m \gt 0$ is a false proposition.<br/>$(1)$ Find the set of real numbers for $m$, denoted as $B$;<br/>$(2)$ Let the solution set of the inequality $\left(x-2a\right)\left[x-\left(a+1\right)\right]\leqslant 0$ be $A$. If $x\in B$ is a necessary but not sufficient condition for $x\in A$, find the range of real number $a$.

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