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Algebra Level 5

Let \(\Lambda\) be a real number that can be represented in the nested radical form as

\[\Lambda = \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \cdots}}} \]

If the closed form of \(\Lambda\) is

\[\Lambda = \displaystyle {\frac {{\sqrt[{3}]{a+b{\sqrt {c}}}}+{\sqrt[{3}]{a-b{\sqrt {c}}}}}{d}}\]

where \(a, b, c\) and \(d\) are positive integers with \(c\) square-free. Find the smallest possible value of \(a+b+c+d\).

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