
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 that the function $f\left(x\right)=x^{-2m^{2}+m+3}$ (where $m\in Z$) is an even function, and $f\left(3\right) \lt f\left(5\right)$.

(1) Find the value of $m$ and determine the analytical expression of $f\left(x\right)$.

(2) If $g\left(x\right)=\log _{a}\left[f\left(x\right)-2x\right]$ (where $a \gt 0$ and $a\neq 1$), find the range of $g\left(x\right)$ on $\left(2,3\right]$.

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