Coin Towers

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 \(B_{k+2}\).

Warm up problem: Using only operations of Type 1, what’s the most coins you can have in total across all 6 boxes?

Main problem: Using operations of either type, what’s the most coins you can have in total across all 6 boxes?