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I also had a better build:
Bloodthirster x2
Zephyr x2
Last Whisper
And a Blade Of The Ruined King
You can change a Bloodthirster to a Black Cleaver if enemies have much armor
With this build can also get around 4k2 with The Culling. But with Baron buff and 2 elixirs you can even reach over 4k6
And 30% CDR (40% with Black Cleaver)
And 51% lifesteal (33% with Black Cleaver)
Also 20% movement speed increased
With these items you can get more advantage in mid game and enough survival in later even without armor and magic resistance
If your allies have buffs that increase your damage/attack speed, that will be awesome!
hurricane
zephyr
ghostblade
last whisper
black cleaver
and last the The Bloodthirster.
according to my calculations it should yield a max damage without armor being involved around 4200. It also holds a nice amount of cd reduction and armor pen, so it would make it hard for the opponents to build against it.
Because it cost around 16.7k I would recommend either breaking off and kill the monsters or waiting to finish the ghostblade til last.