
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 the function $f(x)={\log_4}({{4^x}+1})-\frac{x}{2}$, $x\in R$.  
$(1)$ Determine whether $f(x)$ is an even or odd function on its domain. If it is, please provide a proof.  
$(2)$ If the function $h(x)=f(-x)-\frac{1}{2}{\log_2}({a⋅{2^{x-1}}+\frac{1}{{{2^{x+1}}}}+a})$ has only one zero on $R$, find the range of real numbers for $a$.

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