
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 $f(x)$ with domain $D$, if there exist constants $m, M$ such that $m \leqslant f(x) \leqslant M$ holds for any $x \in D$, then the function $f(x)$ is called a bounded function on $D$, where $m$ is called the lower bound of the function $f(x)$, and $M$ is called the upper bound of the function $f(x)$. Especially, if ${{'}}={{'}}{{'}}$ holds, then $m$ is called the greatest lower bound of the function $f(x)$, and $M$ is called the least upper bound of the function $f(x)$.

$(1)$ Determine whether $f(x) = \sqrt{x+1}-\sqrt{x}, g(x) = 9^{x}-2\cdot 3^{x}$ are bounded functions? Explain your reasoning.

$(2)$ If the function $f(x) = 1+a\cdot 2^{x}+4^{x}, x \in (-\infty, 0)$ is a bounded function with $-3$ as its lower bound and $3$ as its upper bound, find the range of the real number $a$.

$(3)$ If the function $f(x) = \frac{1-a\cdot 2^{x}}{1+a\cdot 2^{x}}, (x \in [0,1], a > 0)$, $T(a)$ is the least upper bound of $f(x)$, find the range of $T(a)$.

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