
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 a function $f(x)=\frac{{1-{3^x}}}{{1+{3^x}}}$.<br/>$(1)$ Determine the parity of the function $f(x)$;<br/>$(2)$ Solve the inequality $f(4^{x}+8)+f(-3\cdot 2^{x+1}) \lt 0$;<br/>$(3)$ Let $g(x)=4^{x}+2^{x+1}+m$. If for all $x_{1}\in R$, there exists $x_{2}\in \left[0,1\right]$ such that $f(x_{1})\leqslant g(x_{2})$, find the range of real number $m$.

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