
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 $f(x)=\frac{ax+b}{1+x^2}$ defined on $\mathbb{R}$ is an odd function.  
$(1)$ Find the value of the real number $b$;  
$(2)$ When $a \gt 0$, determine the monotonicity of the function $f(x)$ on the interval $(1, +\infty)$;  
$(3)$ When $a=1$, let $g(x)=mx^{2}-2x+2-m$. If for any $x_{1}\in [1,3]$, there exists $x_{2}\in [0,1]$ such that $f(x_1)+\frac{1}{2}=g(x_2)$ holds, find the range of values for $m$.

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