
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 that the function $f(x)=\frac{ax+b}{1+{x}^{2}}$ is defined on $(-1,1)$, $f(-x)=-f(x)$ always holds, and $f(\frac{1}{2})=\frac{2}{5}$. Determine:<br/>$(1)$ the analytical expression of the function $f(x)$;<br/>$(2)$ if $f(x)$ is an increasing function defined on $(-1,1)$, solve the inequality $f(x-1)+f(x) \lt 0$.

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