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Problem 9: A Relative of Thue-Morse Word

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Let $\mathbf{c} = (c_0,c_1,c_2,\ldots)$ be the least increasing sequence of positive integers starting with 1 and satisfying the condition $$(*)\hspace{1cm} n\in\mathbf{C} \Leftrightarrow n/2\notin\mathbf{C},$$ where $\mathbf{C}$ is the set of elements in the sequence $\mathbf{c}$. The first elements of the sequence $\mathbf{c}$ are $$1,3,4,5,7,9,11,12,13,15,16,17,19,20,21,23,25,27,28,29,\ldots$$ Observe both that all odd integers are in the sequence and that gaps between two consecutive elements are either $1$ or $2$.

What is the relation between the sequence $\mathbf{c}$ and the infinite Thue-Morse word $\mathbf{t}$?