
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
For any \( t > 0 \), consider the sum \( S = r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r_{4}^{2} \), where \( r_{1}, r_{2}, r_{3}, r_{4} \) are the roots of the polynomial

\[ P_{t}[X] = \frac{1}{t} X^{4} + \left(1 - \frac{10}{t}\right) X^{3} - 2 X^{2} + \sqrt[3]{2 t} X + \arctan (t) \]

What is the minimum value of \( |S| \) and for which \( t \) is it reached?

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