If $$\alpha = \lim_{x \to 0^+} \left(\frac{e^{\sqrt{\tan x}} - e^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}}\right)$$ and $$\beta = \lim_{x \to 0} (1 + \sin x)^{\frac{1}{2}\cot x}$$ are the roots of the quadratic equation $$ax^2 + bx - \sqrt{e} = 0$$, then $$12\log_e(a + b)$$ is equal to _____