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You should redirect by mapping the pages appropriately.  The old sites seems like you are kind of abandoning it, so whatever rankings it currently has, will likely drop over time and the links will lose value.  As long as you are staying active with the new site, everyone wins - visitors get to an active site that is being updated, instead of an abandoned one.  You win because you are not going to have to worry about the old site or losing that traffic or losing link juice.  The search engines win because when they send traffic to the old URLs, the visitor ends up on a better site with fresh content.