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Masteries can help strengthen your players, or give them extra defensive or special capabilities. Click on your name at top of game screen to access adding Masteries. Search online for MCOC Mastery Guides (Ronin Nupe used to have a good one, stopped supporting it after a while but still a very good beginner guide if it’s still out there).
** Basically some good starting Mastery Nodes are Precision/Cruelty (increases Critical Hit Rate and Damage, really adds a punch to your attack), BlockProficiency (defense, lowers the amount of damage you take if you are hit while blocking, good especially if newer player), and under Proficiencies add Parry (lookup how to use), Limber (if opponent stuns you, reduces how long you are stunned for), and Dexterity (that one only needs to be Level 1). Those are very entry-level, the Mastery Guides will go much further in-depth.
And Node Buffs are extras that the opponent has been given for particular nodes they are on, they either strengthen your opponent, or cause bad things to happen to you during the fight. And some node buffs are linked to other defensive nodes as well.
FYI, an Orange symbol there means the node is giving itself the boost (not linked from any other node). And on maps, can see if a node has a purple or orange border around the defender Profile Pic on node.
And Dup'd Hero means that you have pulled him from a crystal a 2nd time (or “Awakened” him using an Awakening Gem, triangle symbol on the Inventory Item) , and he has some additional Abilities unlocked then (see top of that hero's Profile screen).
Class-specific cores are sometimes randomly on Daily Class Cat Quests, but rare. But don't bother with the 6 different Class Masteries anyways.
“Uncollected” means having beaten the Collector in Act-5 Chap-2 at least one time thru.
And as you get more mastery points you can apply, just follow some Mastery guides or screenshots of suggested mastery setups.