
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
A function $f(x)$ defined on $D$ is said to be a bounded function on $D$ if there exists a constant $M \geqslant 0$ such that $|f(x)| \leqslant M$ holds for any $x \in D$, where $M$ is called an upper bound of the function $f(x)$. Given the function $f(x) = 1 + a(\frac{1}{2})^x + (\frac{1}{4})^x$, find the range of values of the real number $a$ if the function $f(x)$ is a bounded function on $[-2, 1]$ with an upper bound of 3.

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