
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 $p: \exists x \in [-1,2]$, $x^{2}-2x-m \gt 0$ is a false proposition.  
$(1)$ Find the set of real numbers for $m$, denoted as $B$;  
$(2)$ Let the solution set of the inequality $(x-2a)(x-(a+1)) \leqslant 0$ 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}
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