If $x + \sqrt{x^{2} - 1} + \frac{1}{x - \sqrt{x^{2} - 1}} = 20$, then find the value of $1000 \times \left( x^{2} + \sqrt{x^{4} - 1} + \frac{1}{x^{2} + \sqrt{x^{4} - 1}} \right)$.

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