Is \(12^2\) special?

Algebra Level 4

\[\begin{cases} \{a^{-1}\}=\{a^2\} \\ 2<a^2<3 \end{cases}\]

Suppose that \(a\) is positive and satisfies the above conditions, find the value of \(a^{12}-144a^{-1}\).

Notation: \( \{ \cdot \} \) denotes the fractional part function.

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