
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 $f(x)=(2m^{2}-6m+5)x^{m+1}$ is a power function and an even function.

$(1)$ Find the explicit formula of $f(x)$;

$(2)$ If the function $y=f(x)-2(a-1)x+1$ is monotonic on the interval $(2,3)$, find the range of the real number $a$.

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