Let sequence $\{a_1,a_2,\dots \}$ with integer terms satisfy the following conditions: 1) $a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots$ ; 2) $2a_1=a_0+a_2-2$ ; 3) for arbitrary natural number $m$, there exist $m$ consecutive terms $a_k, a_{k-1}, \dots ,a_{k+m-1}$ among the sequence such that all such $m$ terms are perfect squares. Prove that all terms of the sequence $\{a_1,a_2,\dots \}$ are perfect squares.