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)