The equation $a\sin x+\cos 2x=2a-7$ possesses a solution if 
 

$\begin{array}{l}a\sin x+1-2{\sin }^{2}x=2a-7\\ 2{\sin }^{2}x-a\sin x+(2a-8)=0\\ \sin x=\frac{a\pm \sqrt{a^2-4\times 2(2a-8)}}{2\times 2}\\ \sin x=\frac{a\pm \sqrt{a^2-16a+64}}{4}=\frac{a\pm (a-8)}{4}\end{array}$

$\sin x=2$ or $\frac{a-4}{2}$

$\sin x=2>1$ no solution

$\begin{array}{l}-1\le \sin x\le 1\\ -1\le \frac{a-4}{2}\le 1\\ -2\le a-4\le 2\\ 4-2\le a\le 4+2\\ 2\le a\le 6\end{array}$