Let $(a_n)_{n=1}^\infty$ be a real sequence such that \[a_n = (n-1)a_1 + (n-2)a_2 + \dots + 2a_{n-2} + a_{n-1}\] for every $n \geq 3$. If $a_{2011} = 2011$ and $a_{2012} = 2012$, find the value of $a_{2013}$.

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