For all x > 0, let y₁(x), y₂(x), and y₃(x) be the functions satisfying

$ \frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, \quad y_1(1) = 5, $

$ \frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, \quad y_2(1) = \frac{1}{3}, $

$ \frac{dy_3}{dx} - \frac{(2-x^3)}{x^3} y_3 = 0, \quad y_3(1) = \frac{3}{5e}, $

respectively. Then

$ \lim\limits_{x \to 0^+} \frac{y_1(x)y_2(x)y_3(x) + 2x}{e^{3x} \sin x} $

is equal to __________________.