Let \(R\) be a subset of \(\mathbb N \times \mathbb N\) defined as follows:

\(R= \{ (a,b) \in \mathbb N \times \mathbb N: b \) is the lowest natural number with \(\gcd(b+1, 2b+1)=a\}\).

For each \((a,b)\), let \(A\) denote \((a+b)\). What is sum of all possible values of \(A\)?

\[\] **Notations:**

\(\mathbb N \) denotes the set of natural numbers.

\(\gcd(\cdot) \) denotes the greatest common divisor function.

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