For all $t>0$, 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 achieved?
$$
\begin{aligned}
S & =r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2} \\
& =\left(r_{1}+r_{2}+r_{3}+r_{4}\right)^{2}-2\left(r_{1} r_{2}+r_{1} r_{3}+r_{1} r_{4}+r_{2} r_{3}+r_{2} r_{4}+r_{3} r_{4}\right) \\
& =\left(-\frac{1-10 / t)}{1 / t}\right)^{2}-2 \frac{-2}{1 / t} \\
& =t^{2}-16 t+100
\end{aligned}
$$

A study of this quadratic shows that the minimum value of $|S|$ is 36 and is achieved at $t=8$.