
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 $ax^{2}+bx+1 \gt 0$ is $\left\{{x|-\frac{1}{2}＜x＜1}\right\}$, if the inequality $bx^{2}-mx-2a\geqslant 0$ holds for all $x\in \left[4,+\infty \right)$, then the maximum value of the real number $m$ is ____.

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