Inspired Infinite Equation

Algebra Level 3

1=x2x3+x4x5+ 1= x^{2} - x^{3} + x^{4} -x^{5} +\ldots Solve for xx in the equation above.

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

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