Suppose that you have N labeled coins pinched in one stack in your fingertips (your palm is above your fingers and your palm is facing down, so that you can drop as many coins as needed from the bottom of the stack) and you have a table which only allows for two other stacks (call these L for left and R for right). You can freely choose L or R for each coin, but the coin dropping order is fixed. You can also freely pick up coins from either stack (the reverse of a drop).
For general N, what is the minimum pickups needed to reverse the stack order? I'm curious about the total number of coins picked up and about the total number of pickup events, but am officially asking only for the latter.
As an example, I believe the minimum for N=7 is 5. We start holding 1234567. Drop 247 in L and 1356 in R; pick up 1356 from R. Drop 16 in L and 35 in R; pick up 24716 from L. Drop 76 in L and 241 in R; pick up 35241 from R. Drop 54 in L and 321 in R; pick up 321 from R. Drop 321 in L; pick up 7654321 from L.