If $24 \int _ { 0 } ^ { \frac { \pi } { 4 } } \left( \sin \left| 4 x - \frac { \pi } { 12 } \right| + [ 2 \sin x ] \right) \mathrm { d } x = 2 \pi + \alpha$, where $[ \cdot ]$ denotes the greatest integer function, then $\alpha$ is equal to $\_\_\_\_$ .