If the inequality ${\sin }^{2}x+a\cos x+a^2\ge 1+\cos x$ holds for any $x\in \mathbf{\text{R}}$,then find the range of values for negative $\mathbf{\text{a}}$.

Given $A\equiv (0,2)$ and two points $\mathbf{\text{B}}$ and $\mathbf{\text{C}}$on the parabola $y^2=x+4$ such that $AB\perp BC$, determine the range for the y-coordinate of point $\mathbf{\text{C}}$.

If $lo{g}_4(x+2y)+lo{g}_4(x-2y)=1$, then the minimum value of $\left|x\right|-\left|y\right|=$.