The sequence 1, 2, 4, 5, 10, 11, 22, 23, 46, 47, $\dots$ is formed as follows:

$\bullet$ Start with the number 1.

$\bullet$ Add one to get 2.

$\bullet$ Double that to get 4.

$\bullet$ Add one to get 5.

$\bullet$ Double that to get 10, and so on.

We repeat the steps of "add one" and "double that", alternating between them.

The 100th term will be of the form $3 \cdot 2^k - 1.$  Compute $k.$

Let's think step by step and output the final answer within \boxed{}.
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