Binomial theorem shall not save you!

Algebra Level 5

For \(i = 1, 2, 3.\), define

\[\large S_{(i, n)} = \frac{\displaystyle\sum_{k=0}^{\lfloor\frac{n-1}{2}\rfloor}(-1)^k\binom{n}{2k+1}x^{2k+1}_i}{\displaystyle\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}(-1)^k\binom{n}{2k}x^{2k}_i}\]

If \(\forall n \in \mathbb{N}, S_{(1, n)}+S_{(2, n)}+S_{(3, n)} = S_{(1, n)}S_{(2, n)}S_{(3, n)}\), and \(x_1 = 1\:; x_2 = 2,\) find \(x_3\) up to three decimal places

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