Successfully complete the game. You can now start a "New Game +" under the Hard or Very Hard difficulty setting. Although the dungeons will be more difficult and you will lose all your buildings and research, you will still have the characters, items and medals earned during your original session.
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When you complete the story line. You are offered a New Game +. You can choose to go to Hard or Very Hard mode. In these modes the dungeons are much harder. This allows you to start from the beginning, but your characters items and medals are carried over, but you must research everything again.