
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 defined on $(-1,1)$, and $f\left( \frac{1}{2} \right)=\frac{2}{5}$.

$(1)$ Determine the expression of the function $f(x)$;

$(2)$ Determine the monotonicity of the function $f(x)$ when $x \in (-1,1)$, and solve the inequality $f(2x-1)+f(x) < 0$

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