Check my answer: Shortest sequence from $1$ to $2^{2018}$, where each step either doubles or squares the previous value.

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My previous question was about the meaning of the problem. But now, I present a solution. Can MSE verify my answer?

Given a number, you can perform two operations on it: double it or turn it into its square. Your initial number is one. How many operations at least do you need to turn it into $2^{2018}?$

Is my solution correct?

$$\begin{align}2^{2018}\longrightarrow 2^{1009}\longrightarrow 2^{1008}\longrightarrow 2^{504}\longrightarrow 2^{252}\longrightarrow 2^{126}\longrightarrow 2^{63}\longrightarrow 2^{62}\longrightarrow 2^{31}\longrightarrow 2^{30}\longrightarrow 2^{15}\longrightarrow 2^{14}\longrightarrow 2^{7}\longrightarrow 2^{6}\longrightarrow 2^{3}\longrightarrow 2^{2}\longrightarrow 2^{1}\longrightarrow 1\end{align}$$

I find answer is $17$. Is it correct?