For all $x>0$, let $y_1(x), y_2(x)$, and $y_3(x)$ be the functions satisfying $$ \begin{array}{ll} \frac{d y_1}{d x}-(\sin x)^2 y_1=0, & y_1(1)=5 \\ \frac{d y_2}{d x}-(\cos x)^2 y_2=0, & y_2(1)=\frac{1}{3} \\ \frac{d y_3}{d x}-\left(\frac{2-x^3}{x^3}\right) y_3=0, & y_3(1)=\frac{3}{5 e} \end{array} $$ respectively. Then $\lim _{x \rightarrow 0^{+}} \frac{y_1(x) y_2(x) y_3(x)+2 x}{e^{3 x} \sin x}$ is equal to $\_\_\_\_$ i