BALKAN-SL 2012 N2

Let the sequences $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ satisfy $a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$ . Let $c_n = a_n + b_n$ for all positive integers $n$ . Prove that there do not exist positive integers $k, r, m$ such that $c^2_r = c_kc_m$ .