Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions:

1. Each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are not relatively prime.
2. All other pairs of the five integers are relatively prime.

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