$\overrightarrow{a}$  and  $\overrightarrow{b}$  are non-collinear vectors such that  $\left|\overrightarrow{a}\right|=2,\overrightarrow{a}.\overrightarrow{b}=1$  and the angle between  $\overrightarrow{a}$  and  $\overrightarrow{b}$  is  $\frac{\pi }{3}$ . If $\overrightarrow{r}=x\overrightarrow{a}+y\overrightarrow{b}+z(\overrightarrow{a}\times \overrightarrow{b})$  satisfying  $\overrightarrow{r}.\overrightarrow{a}=2,\overrightarrow{r}.\overrightarrow{b}=8,$ $(\overrightarrow{r}+5\overrightarrow{a}-7\overrightarrow{b}).(\overrightarrow{a}\times \overrightarrow{b})=4\sqrt{3}$  then which of the following is true