If you don't know what something on the chart means, check the legend on the lower left.
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The Monsters below are fought in Quest Mode, which is accessed via the Battles icon. There are a total of 8 areas. To clear an area, you must defeat each Monster you encounter, including the boss, without being defeated yourself. If you lose a battle, you must start over from the beginning of that area. Monsters encountered in Quest Mode have their own specified Power value, they don't necessarily use the Power value shown in the evolution guide. The power of each opponent is listed below that specific Monster. At the end of each area, you will unlock a background upon clearing it for the first time. The list of backgrounds and which area unlocks each can be found on the manual page.