The three roots of the cubic $30x^3 - 50x^2 + 22x - 1$ are distinct real numbers between $0$ and $1$. For every nonnegative integer $n$, let $s_n$ be the sum of the $n$th powers of these three roots. What is the value of the infinite series \[ s_0 + s_1 + s_2 + s_3 + \dots \] ?

Let's think step by step and output the final answer within \boxed{}.
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