Kugane, in one of the onsen.
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If you have the Centerio Hunt stuff unlocked you can beat up on hunt marked baddies for points and that will easily get you to 340. (though it might take a bit of time.)
I'm so hoping for no dungeon or trial for the MSQ's last part, but I am expecting one all the same. Either way excited or not I'm waiting for mid April to hit it.
I don't think Mendacity tomes would be too hard to farm up at this point, actually. Probably better to focus on them with a combination of running Ridoranna to collect that i360 gear. I'd recommend finding some groups to help clear Deltascape and Sigmascape normal as well so you can get the mendacity weapons (need to clear O8N four times for the crystalloid needed for the trade item). You should also be getting a bit of Genesis tomes too to trade for solid i390 pieces. When you're geared up enough for it, I'd recommend tackling Orbonne (so you can start turning in coins for Genesis gear upgrade items) as well as Alphascape normal. It'll take a bit of time and effort if you want to take it that far, but once you get to the i380+ range you should have no problems on gear until a good few levels into the next expansion. Even with just the Mendacity/Lighthouse gear you should be able to handily get through the end of the MSQ up to the end of the content coming out next week. Don't forget the dungeon drops to tide you over, as well! Between all that, there should be no reason to bother with the i330/340 gear at this point, especially when you factor first time bonuses and such.
Ridoranna? (Google..) Raid, not in my duty finder. All I have is Rabanastre and Deletascape, and I haven't tried them, or any other raids yet.
I've only been playing about 35 days now, and I've pretty much done only MSQ stuff, and a few extra dungeons occasionally when I had to gear up more to continue the MSQ.. I'll be completely lost when all that ends.
I'm excited for the Beast tribes alliance. They're great. I wonder if we'll get to feed a fish to a Snake or Cirina.