International Mathematical Olympiad 2010 Problem 5

In each of six boxes $B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{6}$ there is initially one coin. There are two types of operation allowed:

Type 1: Choose a nonempty box $B_{j}$ with $1 \leq j \leq 5$. Remove one coin from $B_{j}$ and add two coins to $B_{j+1}$.

Type 2: Choose a nonempty box $B_{k}$ with $1 \leq k \leq 4$. Remove one coin from $B_{k}$ and exchange the contents of (possibly empty) boxes $B_{k+1}$ and $B_{k+2}$.

Determine whether there is a finite sequence of such operations that results in boxes $B_{1}, B_{2}, B_{3}, B_{4}, B_{5}$ being empty and box $B_{6}$ containing exactly $2010^{2010^{2010}}$ coins. (Note that $a^{b^{c}} = a^{(b^{c})}$.)