
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 functions $f(x)=1+a(\frac{1}{2})^{x}+(\frac{1}{4})^{x}$ and $g(x)=\log_{\frac{1}{2}}\frac{1-ax}{x-1}$.<br/>$(1)$ If $g(x)$ is an odd function, find the value of the real number $a$;<br/>$(2)$ Under the condition in $(1)$, when $x\in \left[-3,2\right]$, the function $y=f(x)+m$ has a zero, find the range of the real number $m$;<br/>$(3)$ A function $f(x)$ defined on $D$ is considered a bounded function on $D$ if for any $x\in D$, there exists a constant $M\geqslant 0$ such that $|f(x)|\leqslant M$ holds true. The constant $M$ is called an upper bound of the function $f(x)$. If the function $f(x)$ is a bounded function on $\left[0,+\infty \right)$ with $5$ as the upper bound, find the range of the real number $a$.

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