Binomial theorem shall not save you!

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