Školjka

%V0 Three strictly increasing sequences $$a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots$$ of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$, the following conditions hold: (a) $c_{a_n}=b_n+1$; (b) $a_{n+1}>b_n$; (c) the number $c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n$ is even. Find $a_{2010}$, $b_{2010}$ and $c_{2010}$.