
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 that the solution set of the inequality $2ax^{2}+4x+b\leqslant 0$ with respect to $x$ is $\{x|x=-\frac{1}{a}\}$ and $a \gt b$, then $ab=$______, and the minimum value of $\frac{2a+{b}^{3}}{2-{b}^{2}}$ is ______.

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