It is given a positive real number $a$ such that: 
$$ \left\{ \frac{1}{a} \right\} = \{ a^2 \} $$
$$ 2 < a^2 < 3 $$
Find the value of $a^{12} - \frac{144}{a}$.

Let's think step by step and output the final answer within \boxed{}.
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