
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
The function $f(x)$ is defined on the domain $D$. If for any $x \in D$, there exists a constant $M \geqslant 0$ such that $|f(x)| \leqslant M$ holds, then $f(x)$ is called a bounded function on $D$, where $M$ is called an upper bound of the function $f(x)$. The functions $f(x) = 1 + a(\frac{1}{2})^x + (\frac{1}{4})^x$ and $g(x) = \log_{\frac{1}{2}} \frac{1 - ax}{x - 1}$ are given.

1. Find the value of the real number $a$ if the function $g(x)$ is an odd function.
2. Under the condition of question 1, find the set of all upper bounds of the function $g(x)$ on the interval $[\frac{5}{3}, 3]$.
3. If the function $f(x)$ is a bounded function with an upper bound of 3 on the interval $[0, +\infty)$, find the range of values for the real number $a$.

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