Define a recurrence relation with starting value $x_0$ such that $-1 < x_0 < 1$ and

$x_{n+1}=\sqrt{\frac{1+x_n}{2}}.$

What is the value of

$\cos\left(\frac{\sqrt{1-x_0^2}}{\prod\limits_{n=1}^{\infty}x_n}\right)=\,\,?$

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