\[ {a_n} = \frac{\displaystyle \prod_{k=1}^{n-1} a_k}{\displaystyle \sum_{k=1}^{n-1} a_k} \]

If the sequence \(a_n\) satisfy the recurrence relations formula as described above with \(a_1 = 1\). Determine the value of \( \displaystyle \sum_{i=1}^7 \frac1{a_i} \).

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