Don't push my buttons!

Jenny is the operator of a large garbage disposal machine. It is a fairly simple machine and constitutes of only three buttons: One green \color{#20A900} {\text{green}} button, one blue \color{#3D99F6} {\text{blue}} button and one red \color{#D61F06} {\text{red}} button.

Given that initially nn items are put in the machine, it will only destroy garbage in the three following ways:

  • The green \color{#20A900} {\text{green}} button, if pressed destroys 12\frac{1}{2} of the items,leaving n2\frac{n}{2} of the items, but it only works if the number of items nn is divisible by 2.2.
  • The blue \color{#3D99F6} {\text{blue}} button,if pressed destroys 23\frac{2}{3} of the items,leaving n3\frac{n}{3} items, but it only works if the number of items nn is divisible by 3.3.
  • The red \color{#D61F06} {\text{red}} button always works and destroys only 11 item leaving n1n-1 if pressed.

Since Jenny is a lazy operator she wants to minimize the number of times she presses the buttons? What is the minimum number of times Jenny has to press the buttons in order to completely destroy 466559466559 items?

As explicit examples for n=10n=10 items Jenny would have to do 44 button presses (red \color{#D61F06} {\text{red}} ,blue \color{#3D99F6} {\text{blue}} , blue \color{#3D99F6} {\text{blue}} ,red \color{#D61F06} {\text{red}} ) to minimize the number of presses as shown below.
101=910-1=9 \longrightarrow 93=39\diagup 3=3 \longrightarrow 33=13\diagup3=1 \longrightarrow 11=01-1=0

For n=6n=6 items the optimal solution is 33 presses (blue \color{#3D99F6} {\text{blue}} , green \color{#20A900} {\text{green}} ,red \color{#D61F06} {\text{red}} ) or (blue \color{#3D99F6} {\text{blue}} ,red \color{#D61F06} {\text{red}} ,red \color{#D61F06} {\text{red}} )
632211=06\diagup 3 \longrightarrow 2 \diagup 2 \longrightarrow 1 - 1=0 or 632111=06\diagup 3 \longrightarrow 2 - 1 \longrightarrow 1 - 1=0


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