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Geometry Level 5

Let \(ABC\) be an acute angled triangle. Let \(A,B,C\) be its angles. For \(n=1,2,3\) , we define

\[\large x_{n}=2^{n-3} \left(\displaystyle\sum_{cyc \{A,B,C\}}\cos^n A \right) + \prod_{cyc\{A,B,C\}} cos A\]

Find the minimum value of \(\large\displaystyle\sum_{n=1}^{3} x_n\)

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