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I travel a lot so I switch servers quite a bit. I usually use NA, OCE, Garena and less frequently, EUW.
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For Guinsoo's, you get it to convert the crit chance of it to on-hit, so it applies on dagger, E and R as well. He builds Collector as last item as well, so he does 120 bonus physical damage on-hit. Lich Bane does very little damage in comparison to Nashors due to its 2.5 second cooldown which is extremely high during a rotation, and you're bound to get more damage off Nashor's instead of Lich Bane due to its 100 AP and on-hit on every dagger, E and R.
Lich Bane also only does damage to the target you shunpo to, like what Kuro_Shiro said. I tested it out a minute ago on Practice Tool. I won't comment anything on how well Luden's work with Kat as I have not tried it yet, but going Mythic (NH or Riftmaker) first item does not deal as much damage as rushing BoTRK or otherwise Nashor's.