The printed guarantees
Open ×10 costs 4,000 gems. The chests carry pity floors: an S-Grade is guaranteed within 27 or 49 draws depending on the chest. The planner turns those guarantees into worst-case and expected costs for a specific gem budget.
Duplicates feed the blacksmith
Six slots feed one ATK/HP pool, and upgrade levels: not quality: are the long grind. Sort By Quality and feed spares to the Blacksmith.