AMC 8 2020 Problem 22

When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$$$7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26$$When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$$$N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1$$ $\textbf{(A) }73 \textbf{(B) }74 \textbf{(C) }75 \textbf{(D) }82 \textbf{(E) }83$