
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 $f(x)=\frac{{x+b}}{{a{x^2}+1}}$ defined on the domain $(-1,1)$ is an odd function, and $f(\frac{1}{2})=\frac{2}{5}$.  
$(1)$ Find the values of real numbers $a$ and $b$;  
$(2)$ Determine the monotonicity of the function $f(x)$ on $(-1,1)$ and provide a proof;  
$(3)$ Solve the inequality related to $t$: $f(t+1)+f(2t) \lt 0$.

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