The positive integers $a, b, c$ satisfy the following conditions:

1. $\text{gcd}(a, b, c) = 1$
2. $\text{gcd}(a, b + c) > 1$
3. $\text{gcd}(b, c + a) > 1$
4. $\text{gcd}(c, a + b) > 1$

Determine the smallest possible value of $a + b + c$.

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