In the hidden temple in the mountains east of the Calm Lands, you can battle against any aeon that you have acquired by challenging Belgemine. Everytime that you win against Belgemine’s aeon, she will reward you with rare items. Defeat her Bahamut, and she will give you the key item Flower Scepter. Atfer fighting and defeating all of the aeons in the game, including the Magus Sisters, and performing the sending, Belgemine hands over the Moon Sigil, which removes the NO AP ability from Yuna’s legendary weapon.
To acquire the Cloudy Mirror, you must win the chocobo race under Remiem Temple. Winning the race is quite easy on the first try. To improve the item you win in this race, you need to use as many shortcuts as possible and avoid hitting any of the poles.