The set of values of a for which the equation ${\mathbf{Sin}}^4\mathbf{+}{\mathbf{Cos}}^4\mathbf{x}\mathbf{=}\mathbf{a}$ has a solution is .

${\sin }^{4}x+{\cos }^{4}x=a$

$\Rightarrow {\text{\hspace{0.17em}\hspace{0.17em}1-2\hspace{0.17em}sin}}^{\text{2}}{\text{\hspace{0.17em}\hspace{0.17em}x\hspace{0.17em}\hspace{0.17em}cos}}^{\text{2}}\text{\hspace{0.17em}x\hspace{0.17em}=\hspace{0.17em}a}$

$\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}1-}\frac{\text{1}}{\text{2}}{\text{\hspace{0.17em}sin}}^{\text{2}}\text{\hspace{0.17em}\hspace{0.17em}2x\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}a}$

$0\le {\sin }^{2}2\text{x\hspace{0.17em}}\le 1\Rightarrow 0\le \frac{1}{2}{\sin }^2\text{\hspace{0.17em}2x\hspace{0.17em}}\le \frac{1}{2}$

$\frac{1}{2}\le 1-\frac{1}{2}{\text{sin}}^{\text{2}}\text{\hspace{0.17em}2x\hspace{0.17em}}\le \text{\hspace{0.17em}1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\therefore \text{\hspace{0.17em}\hspace{0.17em}a\hspace{0.17em}}\in \left[\frac{1}{2},1\right]$