
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
For a function $f(x)$ defined on the domain $D$, if for every $x \in D$ there exists a constant $M > 0$ such that $|f(x)| \leqslant M$, then $f(x)$ is called a bounded function on $D$, and $M$ is referred to as an upper bound of the function $f(x)$. Given the function $f(x) = 1 + a \cdot \left( \frac{1}{2} \right)^x + \left( \frac{1}{4} \right)^x$,
(1) When $a = 1$, find the range of $f(x)$ on $(-\infty, 0)$ and determine whether $f(x)$ is a bounded function on $(-\infty, 0)$, with an explanation;
(2) If the function $f(x)$ is a bounded function with an upper bound of $3$ on $[0, +\infty)$, find the range of the real number $a$.

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