Bettter BackLink structure does not help to rank?
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Without looking in depth, first guess is that it definitely could be some keywords were suppressed due to artificially created backlinks - links are still heavily weighted in Google.
It could also be age of domain and on page optimization, but the fact that you said the site ranks nowhere and not just below the other indicates possible penalty.
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On page factors are also OK , it can be due to over optimization of keyword through link building (this can be true for first case but still I need some concrete stance and what can be done to remove the same)
but second case is not at all understood.