Item synthesis is a long side quest, but can get you many items (including the best keyblade) for free. I recommend doing this one first as you can gain many levels and earn yourself many good items. To activate the item synthesis side quest head to traverse town and go into the accessory shop. Use the green trinity on the bottom level and you will be able to climb the new ladder to the item synthesis shop. Talk to the moogle by the furnace to pick an item to synthesise. The items on the left are the ones you will make and the items on the right are the special mix items you need. After you create all the items listed new ones will be made available to synthesise. The last item is Sora's best weapon the Ultima Keyblade. I will list the synthesis items that you need for each item and where they can be found. Mix what you can when you first get here, but you will most likely need more items.
Cottage - Bright Shard x2, Blaze Shard x3
Elixer - Blaze Shard x2, Frost Shard x2, Thunder Shard x2, Mythril Shard x4
Energy Bangle - Power Shard x2, Frost Shard x1
Power Chain - Spirit Shard x2, Power Shard x2
Guard Earring - Spirit Shard x2, Thunder Shard x1
Dark Ring - Lucid Shard x2, Bright Shard x1, Mythril Shard x2
Bright Shard - White Mushroom (Common)
Blaze Shard - White Mushroom (Common)
Frost Shard - Blue Rhapsody (Common), White Mushroom (Common)
Thunder Shard - Yellow Opera (Common)
Mythril Shard -
Power Shard -
Spirit Shard -
Bright Gem -
Blaze Gem -
Frost Gem -
Thunder Gem -
Mythril Gem -
Power Gem -
Spirit Gem - Air Soldier (Common)
Mythril -
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