RMO Practice Problems - Number Theory #3

Let $(p_{i}, q_{i}, r_{i})$ be the ordered triple of primes that satisfy the below condition

$p_{i}|q_{i} ^{r_{i}} +1 \quad \quad \quad q_{i}|r_{i} ^{p_{i}} +1 \quad \quad \quad r_{i}|p_{i} ^{q_{i}}+ 1$

Find the value of $\sum_{i=1}^{n} p_{i} + q_{i} + r_{i}$ where $n$ denotes the number of solutions.

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