
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 $f(x)=\frac{6x+b}{{x}^{2}+a}$ defined on $R$ is an odd function, and $f(1)=\frac{3}{2}$.  
$(1)$ Find the analytical expression of the function $f(x)$;  
$(2)$ If there exists $x\in \left[1,3\right]$ such that the inequality $|f\left(x\right)-m|\leqslant 1$ holds, find the range of real number $m$;  
$(3)$ If for all $n\in \left[0,1\right]$, and for all $t\in \left(0,+\infty \right)$, the inequality $f(t)+nf(\frac{t}{3})-s≤0$ holds, find the minimum value of the real number $s$.

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