
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: "For all $x\in \left[1,2\right]$, the inequality $x^{2}-2mx-3m^{2} \lt 0$ always holds" is a true proposition.  
$(1)$ Find the set $A$ of real numbers for which the proposition is true.  
$(2)$ Let the solution set of the inequality $x^{2}-2ax+a^{2}-1 \lt 0$ be $B$. If $x\in A$ is a necessary but not sufficient condition for $x\in B$, find the range of real numbers for $a$.

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