
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
$$
S(t)=\left\{\begin{array}{l}
2\left(1-t+t^{2}-t^{3}\right)\left(0<t<\frac{1}{2}\right), \\
\frac{1}{2}\left(t+\frac{1}{t}\right)\left(t \geqslant \frac{1}{2}\right) .
\end{array}\right.
$$

Determine the intervals of monotonicity for \( S(t) \), and provide a proof.

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