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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