
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 function $f(x)=ax^{2}-2ax+2+b$ ($a\neq 0$), it has a maximum value of $5$ and a minimum value of $2$ in the interval $[2,3]$.
$(1)$ Find the values of $a$ and $b$;
$(2)$ If $b < 1$, $g(x)=f(x)-(2^{m})\cdot x$ is monotonic in the interval $[2,4]$, find the range of values for $m$.

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