
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
The function $f(x)= \frac{ax+b}{1+x^{2}}$ is an odd function defined on the domain $(-1,1)$, and $f\left( \frac{1}{2}\right)= \frac{2}{5}$.

$(1)$ Find the expression for $f(x)$ and determine the monotonicity of $f(x)$ on $(-1,1)$ (no proof required);

$(2)$ Solve the inequality: $f(t-1)+f(t) < 0$.

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