
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
Define a function $f(x)$ on domain $D$. If for any $x \in D$, there exists a constant $M > 0$ such that $|f(x)| \leqslant M$ holds, then $f(x)$ is called a bounded function on $D$, where $M$ is referred to as an upper bound of the function $f(x)$. Given the function $f(x) = 1 + a \cdot \left( \frac{1}{3} \right)^{x} + \left( \frac{1}{9} \right)^{x}$,

$(1)$ When $a = - \frac{1}{2}$, find the range of $f(x)$ on $(-\infty, 0)$ and determine whether $f(x)$ is a bounded function on $(-\infty, 0)$. Please explain your reasoning.

$(2)$ If the function $f(x)$ is a bounded function with an upper bound of $4$ on $[0, +\infty)$, find the range of the real number $a$.

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