Golden coins of varying quality come in finitely many (e.g., three) weights. You are told the weight amounts (e.g., 1 lb, 1 lb 1/2 oz, and 1 lb 1 oz); they differ by pairwise commensurable numbers in a given unit of weight.

Finitely many (e.g., four) sacks of golden coin are filled. Within any sack the coins are as abundant as you like and of uniform quality – which quality you aren't told.

From a golden lamp a genie appears. You are granted an exact reading on the combined weight of your choice of coins – just the one weighing – and exact results of any arithmetic calculations you propose.

Each successive sack is filled with which quality of coin?