
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 $f(x)=\frac{ax+b}{{x}^{2}+1}$ is an odd function with a domain of $\left[-1,1\right]$, and $f\left(1\right)=1$.  
$(1)$ Find the values of $a$ and $b$, and determine the monotonicity of $f\left(x\right)$ on $\left[-1,1\right]$ (proof not required);  
$(2)$ Given $k$ is a real number, and the function $g\left(x\right)=x^{2}-2kx+4k+3$, if for any $x_{1}\in \left[-1,1\right]$, there exists $x_{2}\in \left(1,+\infty \right)$ such that $f(x_{1})=g(x_{2})$, find the range of values for $k$.

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