KöMaL 2010 A516

In each of five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ there is initially one coin. There are two types of operation allowed: Type 1: Choose a nonempty box $B_j$ with 1$\le j \le$4. Remove one coin from $B_j$ and add two coins to $B_{j+1}$. Type 2: Choose a nonempty box $B_k$ with 1$\le k \le$3. Remove one coin from $B_k$ and exchange the contents of (possibly empty) boxes $B_{k+1}$ and $B_{k+2}$. Prove that for every integer $0$\le n \le$2^{2010}$ there is a finite sequence of such operations that results in boxes $B_1$, $B_2$, $B_3$, $B_4$ being empty and box $B_5$ containing exactly $n$ coins.