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All the King's Wine
For me just wanna say even though havent read other commwnts. Just want to explain that perhaps the solution above is correct.
The issue is the Exact bottled that was poison. I only thoughtof statistics. Its like he's right above , number the bottles were the 10 servants drinks. Who ever dies after 3 weeks look into their data. Look into the same numbers they died then vary the numbers to the surviving victims. Then remove the identical numbers where the survivors have also drinked(it means it doesnt have poison). Until you get that One numbered bottle that is only been drinked by the dead servants.
You would really get the exact bottle although thats a lot of wine haaacontext