
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 $f(x)=x^{-2m^{2}+m+3}$ ($m\in\mathbb{Z}$) is an even function and is increasing on the interval $(0, +\infty)$.

(1) Find the value of $m$ and determine the explicit formula of $f(x)$;  
(2) If $g(x)=\log_{a}[f(x)-ax]$ ($a>0$ and $a\neq1$), is there a real number $a$ such that the maximum value of $g(x)$ on the interval $[2,3]$ is 2? If it exists, find the value of $a$; if not, explain why.

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