Consider an infinite sequence of positive integers which satisfies the property that the greatest common divisor of any two consecutive terms is strictly greater than the preceding term. Let \(N\) be the smallest possible value of the \(2014^{\text{th}}\) term of this sequence. Find the last three digits of \(2N.\)

**Details and assumptions**

- In other words, if \(\{a_i\}_{i=1}^{\infty}\) is the sequence, we have
\[\gcd (a_i, a_{i+1}) > a_{i-1} \quad \forall \ i \in \mathbb{N_{> 1}},\]
and you have to find the smallest possible value of \(a_{2014}.\)

- This problem is not original.

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