
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 a function $f(x) = x + \frac{t}{x}$ ($t > 0$), it exhibits the following properties: the function is decreasing on $(0, \sqrt{t}]$ and increasing on $[\sqrt{t}, +\infty)$.
(1) If $g(x + \frac{1}{x}) = x^2 + \frac{1}{x^2}$, find the explicit form of $g(x)$.
(2) Given the function $h(x) = \frac{4x^2 - 12x - 3}{2x + 1}$ ($x \in [0, 1]$), use the above properties to find the range of $h(x)$.

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