Inspired Infinite Equation

Algebra Level 4

\[ 1= x^{2} - x^{3} + x^{4} -x^{5} +\ldots \]Solve for \(x\) in the equation above.

If the sum of all values of \(x\) can be represented in the form \(\dfrac{a+b\sqrt{c}}{d}\), such that \(a,b,c\) and \(d\) are integers and the fraction is in lowest form and \(d> 0\), find \(a+b+c+d\).

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