Problem #1: Near the line

For a complex number \(c\), we define the shift of \(c\), denoted \(f(c)\) as the average between \(c\) and \(|c|\).

Then, define \(g(c)\) as the limit of the sequence \(c, f(c), f(f(c)), \dots\) if it exists.

Now, define \(G(N) = \sum_{k=1}^N g((k/N)e^{i(k/N)})\).

Find \(\lfloor G(10^{10}) \rfloor\)