Video solution:

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Written solution:

To see which numbers we can make by inverting it, let's make an inverting machine.

This would take $N$ and yield either $2N$ if $2N$ is even (which it always is) or $\frac{N-1}{3}$ if $\frac{N-1}{3}$ is an odd integer. Note that $\frac{N-1}{3}$ is an integer only if $N \equiv 1 \mod 3.$ Also, if $N$ is even, then $N-1$ is odd. That would mean $\frac{N-1}{3}$ would be odd. Therefore, our inverter machine yields $2N$ and also $\frac{N-1}{3}$ if $N \equiv 1 \mod 3$ and $N$ is even.

Now, we must see what the inverting machine can yield after $6$ moves:

$1)$ We can only get $2.$

$2)$ From $2,$ we can only get $4.$

$3)$ From $4,$ we can get $1$ and $8.$

$4)$ From $1,$ we can get only $2$; from $8,$ we can only get $16.$

$5)$ From $2,$ we can get only $4$; from $16,$ we can get $5$ and $32.$

$6)$ From $4,$ we can get $1$ and $8,$; $5,$ we can get $10$; from $32$ we can get $64.$

Move $6$ can yield $1, 8, 10,$ and $64,$ and their sum is $83.$

Thus, the correct answer is E.