
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 $f(x)=x^{-t^{2}+2t+3}$ is an even function $(t\in\mathbb{Z})$, and it is monotonically increasing on $x\in(0,+\infty)$.  
$(1)$ Find the expression for $f(x)$;  
$(2)$ If the function $g(x)=\log_{a}[a\sqrt{f(x)}-x]$ is monotonically decreasing on the interval $[2,4]$ $(a > 0$ and $a\neq 1)$, find the range of the real number $a$.

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