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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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