
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 for $x \in (0, +\infty)$.
(1) Find the expression for $f(x)$.
(2) If the function $g(x) = \log_a \left(a \sqrt{f(x)} - x \right)$ is monotonically decreasing on the interval $[2,4]$ ($a>0$ and $a\neq 1$), determine the range of the real number $a$.

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