
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 $f(x)=a-\frac{2^x}{{{2^x}+1}}$ is an odd function with a domain of $R$.  
$(1)$ Find the value of the real number $a$ and determine the monotonicity of the function $f(x)$.  
$(2)$ When $x\in \left[3,9\right]$, if the inequality $f({\log_3}^2x)+f(2-m{\log_3}x)≥0$ always holds, find the range of the real number $m$.

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