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, \(\langle a^{-1}\rangle=\langle a^2\rangle\), and \(2< a^2 < 3\). Find the value of \(a^{12}-144a^{-1}\).

(第十五届AIME1997 第9题)