Lucas Numbers?

Algebra Level 4

\[\sqrt[8]{2207-\dfrac{1}{2207-\dfrac{1}{2207-\cdots}}}\]

The express above can be expressed as \(\dfrac{a+b\sqrt{c}}{d}\), where \(a,b,c,d\in \Bbb{Z}\) with \( \gcd(a, b, d) = 1 \) and \( c\) is a square free integer. Find the value of \(a + b + c + d\).

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