\[ a \left [ \frac { \left ( \sqrt{2 + \sqrt{4+ \sqrt x }} \right )^b }{b} - \frac {c \left ( \sqrt{2 + \sqrt{4+ \sqrt x }} \right )^d }{d} + \frac {e \left ( \sqrt{2 + \sqrt{4+ \sqrt x }} \right )^f }{f} \right ] \]

Ignoring the arbitrary constant, the antiderivative of function \( \sqrt{2 + \sqrt{4+ \sqrt x }} \) can be written as the expression given above for positive integers \(a,b,c,d,e,f\) with \( \text{gcd}(c,d) = \text{gcd}(e,f) = 1 \).

What is the value of \(a+b+c+d+e+f\)?

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