
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
Let the function be $y=ax^{2}+bx$.  
$(1)$ If $x=-1$, $1\leqslant y\leqslant 2$, and if $x=1$, $2\leqslant y\leqslant 4$. Find the set of all possible values of $4a-2b$;  
$(2)$ If $a=2$, $b=-1$, and for $x>\frac{1}{2}$, the inequality $y\geqslant 2mx-m-8$ always holds, find the range of real numbers for $m$.

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