The following modes can be accessed on the Title Screen under an additional "Extra" menu option.
Description
Effect
Collection
Beat the main game
New Game+
Beat the main game
Records
Beat the main game
Survival
Beat the main game
How to unlock Achilles (Survival Mode)
After you beat the game, play through and beat Survival mode (which consists of 117 battles). Upon beating Survival Mode, you unlock a screen that says "A New Party Member has been added".
To use Achilles, go to Survival Mode, and choose preset party. He will be at the bottom. Enjoy.
