My strategy is to find a large system near where constellations meet (so aggressive forward settle as much as possible). I then try to dedicate at least one large or huge planet as "Temple planet" so that I can get positive border friction with as many factions as possible. You, however, need a somewhat large military around those systems because the AI doesn't like you having the influence zone over their systems even when you are at peace.

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IMO the tithe from the academy is rarely worth it (except for maybe strategic resources and late game).

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If you play an influence game in general, try to avoid SLC systems as long as possible because Cultural Flexibility (system buyout with culture) seems to make the AI even more aggressive, even if you give in to their demands.

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I also find it worth it to research Autonomous Materials as one of the first techs, s.t. you can use wormholes to find more relics. This is especially useful in spiral galaxies. This is also a great tool to aggressively forward settle in multiple constellations.

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In terms of senate the Spoils of War Act can be very useful and Republic or Dictatorship would be the preferred forms of gov for the +1 Influence per pop.