${\text{Number of integral values of a for which f(x)=log(log}}_{\frac{1}{3}}\left(lo{g}_7\right(sinx+a\left)\right)) be defined for every real value of x is$

$$\begin{gathered} {\text{log}}_{\frac{1}{3}}\left(lo{g}_7\right(sinx+a\left)\right)>0 \\ \Rightarrow 0<lo{g}_7(sinx+a)<1 \end{gathered}$$

$$\begin{gathered} \Rightarrow 1<sinx+a<7 \\ \Rightarrow 1-sinx<a<7-sinx \forall x\in R \\ \therefore 2<a<6 as minimum of sinx =-1 and maximum of sinx =1 \end{gathered}$$