The sequence ${x}_{n}$ is defined by $x_1=\dfrac{1}{2},~~~ x_{k+1} = x_{k}^2 + x_k.$ If $S=\dfrac{1}{{x}_{1}+1}+\dfrac{1}{{x}_{2}+1} + \dfrac{1}{{x}_{3}+1} +\cdots + \dfrac{1}{{x}_{100}+1}$ then find the value of $\left\lfloor S\right\rfloor$.

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