Paragraph: Let $p,q$ be integers and let $\alpha,\beta$ be the roots of the equation $x^2 - x - 1 = 0$, where $\alpha \neq \beta$. For $n=0,1,2,\ldots$, let $a_n = p\alpha^n + q\beta^n$. FACT: If $a$ and $b$ are rational numbers and $a+b\sqrt{5}=0$, then $a=0=b$.

Question: $a_{12} =$