设函数$f\left(x\right)=\cos \left(x+\frac{2}{3}\pi \right)+2{\cos }^2\frac{x}{2},x\in R$.
(I)求$f\left(x\right)$的值域;
(II)记$∆ABC$的内角$A,B,C$的对边长分别为$a,b,c$，若$f\left(B\right)=1,b=1,c=\sqrt{3}$,求$a$的值.