For all positive integers \(n\), the sequence \(\{a_n\}\) and \(S_n\) satisfy:

\[\large S_n=\sum_{k=1}^n a_k \\ \large a_n+S_n=2n\]

Find the value of \(\displaystyle \lim_{n \to \infty} a_n\).

*This problem is a part of <Inconsequences!> series.*

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