Annoying Reciprocals

Calculus Level 3

If \( x \) is a complex number satisfying \( { x }^{ 2 }+x+1=0 \), then for all positive integers \( n \): \[\sum _{k=1}^n \left(x^k+\frac 1{x^k} \right)^2 =n+a \left \lfloor \frac nb \right \rfloor \]

where \( a \) and \( b \) are positive integers. Find \( a+b \).

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