
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 $f(x)$ is an odd function defined on the set of real numbers $R$. When $x \geq 0$, $f(x) = \frac{-x}{x+1}$.  
$(1)$ Determine the monotonicity of the function $f(x)$ on $\left[0,+\infty \right)$ and provide a proof.  
$(2)$ Find the analytical expression of the function $f(x)$ when $x < 0$.  
$(3)$ If the inequality $f(k-t^{2})+f(2t-2t^{2}-3) \lt 0$ holds for any $t \in \left(-1,1\right)$, determine the range of real numbers for $k$.

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