
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 $f(x)=ax^{2}-2ax+2+a$ $(a < 0)$, if $f(x)$ has a maximum value of $1$ in the interval $[2,3]$.  
$(1)$ Find the value of $a$;  
$(2)$ If $g(x)=f(x)-mx$ is monotonic in the interval $[2,4]$, find the range of values for $m$.

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