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Well for the most part, with battles you want to abuse the hero-action button. For a round or two (or more, later in the game), and cross paths as often as possible to build up points. And then use the tri-attack. Early in the game I suggest having Zephyr at the top of your order list in the menu. That puts him in the middle during battle, behind the others. So you get a resonance point from his first hero action, and he can mow down enemies pretty effectively and the other two can finish off the damage he'd cause. Once you get the hang of it you can build better strategies later. But yeah. It took me a while to get the hang of the battle system, but once you come to terms with it, its fun. Most just hero action after hero action and then massive abuse of the tri-attacks
But if you have any specific questions I'll answer as best I can
I'm not sure what's got you with the world map. Other than everything's essentially "locked" and you have to open it up by placing hexes on paths to open up areas
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How do you perform a Smackdown attack?
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During a Hero action, knock an opponent into the air.
Proceed to jump and fire at them when you are over head, knocking them into the ground.