Weird Recurrence, Weirder Sum

Algebra Level 2

Let \(a_1 = 2,\) \(a_2 = 3,\) \(a_3 = 7,\) and \[a_n = 1 + a_1 \times a_2 \times \cdots \times a_{n-1}\] for \(n\ge 4.\) Find \(\displaystyle \sum_{n=1}^\infty \dfrac 1{a_n}.\)

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