Given a nonnegative real number x, let \langle x\rangle denote the fractional part of x; that is, \langle x\rangle=x-\lfloor x\rfloor, where \lfloor x\rfloor denotes the greatest integer less than or equal to x. Suppose that a is positive, \left\langle a^{-1}\right\rangle=\left\langle a^{2}\right\rangle, and 2<a^{2}<3. Find the value of a^{12}-144 a^{-1}.