
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 proposition $P: \exists x\in R$, $ax^{2}+2x-1=0$ is a false proposition. Let the set of real numbers $a$ be $A$, and let the set $B=\{x\left|\right.3m \lt x \lt m+2\}$. If _____, find the range of real number $m$. Choose one of the following three conditions: ① if "$x\in A$" is a necessary but not sufficient condition for "$x\in B$"; ② "$x\in B$" is a sufficient condition for "$x\in \complement _{R}A$"; ③ $B$⋂$\complement _{R}A=\varnothing $, and complete the blank line in this question according to your choice, then solve the problem.

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