
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 $f(x)=\frac{{3}^{x}+a}{{3}^{x}+b}$ and $g\left(x\right)=f\left(x\right)(3^{x}+1)+3^{-x}$.<br/>$(1)$ When $a=5$ and $b=-3$, find the value of $x$ that satisfies $f\left(x\right)=3^{x}$;<br/>$(2)$ When $a=-1$ and $b=1$, if for any $x\in R$ and $x\neq 0$, the inequality $g\left(2x\right)\geqslant m\cdot g\left(x\right)-10$ always holds, find the maximum value of the real number $m$.

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