For all x > 0, let y₁(x), y₂(x), and y₃(x) be the functions satisfying dy₁/dx − (sin x)²y₁ = 0, y₁(1) = 5; dy₂/dx − (cos x)²y₂ = 0, y₂(1) = 1/3; and dy₃/dx − ((2 − x³)/x³)y₃ = 0, y₃(1) = 3/(5e), respectively. Then lim[x→0⁺] (y₁(x)y₂(x)y₃(x) + 2x)/(e³ˣ sin x) is equal to ______________.