If the shortest distance between the lines \(\frac{x+2}{2} = \frac{y + 3}{3} = \frac{z-5}{4}\) and \(\frac{x-3}{1} = \frac{y - 2}{-3} = \frac{z+4}{2}\) is \(\frac{38}{3\sqrt{5}}k\) and  \(\int_0^k [x^2]dx = α - \sqrt{α}\), where [x] denotes the greatest integer function, then 6α³ is equal to ______