
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 $f(x)=x-\frac{4}{x}, x \in [1,2]$.

(1) Find the range of the function $f(x)$;

(2) Let $F(x)=x^2+\frac{16}{x^2}-2a\left(x-\frac{4}{x}\right)$, $x \in [1,2]$, $a \in \mathbb{R}$, find the minimum value $g(a)$ of the function $F(x)$;

(3) For $g(a)$ in (2), if the inequality $g(a) > -2a^2+at+4$ holds for any $a \in (-3,0)$, find the range of the real number $t$.

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