
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 $f(x)=\frac{{4}^{x}+1}{{4}^{x}+a}$ is an odd function.<br/>$(1)$ Solve the inequality $f(x)>\frac{5}{3}$;<br/>$(2)$ Let $g(x)=\log_{2}\frac{x}{2}\cdot\log_{2}\frac{x}{4}+m$. If for any $x_{1}\in \left[2,8\right]$, there exists $x_{2}\in \left(0,1\right]$, such that $g(x_{1})=f(x_{2})$ holds, find the range of real number $m$.

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