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Equipment can be further improved through the use of black magic, by cursing it. Cursed equipment have a +20% boost to their base stats, but if you continue to strengthening it and it fails, you will incur a -10 drop in strengthening levels and lose the boost. Although you can acquire curse redirection dolls, these can currently only be found as rare drops in the Atrasia Castle Chaos Dungeon and higher, which requires being level 116+ to even enter. Because of this, you will not want to curse any equipment until you have already gotten it to a +30 strengthening level.
There is a Black Magic shop tucked away in a corner of the city in the Burning Desert, run by the Javir Brothers. You can walk up to their shop normally or use the map to fast-travel to them. They will handle everything related to cursed equipment.
The top option in the Javir Brothers' shop allows you to curse a piece of equipment in your inventory. You can chose any item, but it is only recommended to curse items that have already reached +30 strengthening.
The Javir Brothers' shop menu. Chose the top option to curse a piece of equipment, or the bottom option to strengthen a piece of equipment that has already been cursed.
Once you select a piece of equipment, you'll see the cost: 100,000 gold and 1,500 black magic crystals. Black magic crystals are usually acquired from Chaos Dungeon 4th floor monsters, but can also be acquired through the Cross Field token shop and as rewards from clearing bosses in the Distorted Boundary boss rush mode, etc.
You will be warned about the benefits and dangers of cursing equipment (boost to stats when cursed, but loss of levels and boosts if strengthening fails) and asked to confirm your selection. Choose the button on the bottom right to continue.
Cursing equipment is so dangerous, a cut scene will play where the Javir Brothers will warn you (again) and ask if you really, absolutely, positively, want to curse your piece of equipment. Choose the top (purple) option to confirm for the second (and final) time.
You have successfully cursed your equipment! It will now have a swirly purple mist in its icon whenever you see it in inventory or decks.
The info box for your equipment will now denote its cursed status in the name, and show a bar explaining it's cursed and the boost you get from it.
Once equipment is cursed, you will have to go back to the Javir Brothers' shop any time you want to strengthen it.
Strengthening equipment requires an appropriate number of high-tier strengthening stones of the matching element and/or type. For weapons, you can chose any combination of matching element and rainbow (all element) strengthening stones.
Remember that if this fails, your weapon will lose stats, so think carefully before doing this!
By using curse redirection dolls, failing a strengthening step will not undue the curse. Basically, you never want to curse past +30 without also using these dolls. However, the higher the level, the more dolls you need to spend to protect your equipment. Whether the strengthening succeeds or fails, you will lose the curse dolls, so spend them wisely!
Curse dolls are rare drops in higher level Chaos Dungeons. Sometimes these will drop as sealed dolls, and you will need to buy a special certificate in the shop (1,000,000 gold per certificate, limit 4 per day) to unseal them.
Additionally, you can create new dolls in the Javir Brothers' shops with the appropriate set of materials.