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Galio AP Build
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Also it displays that you will have 280 AP with this build but i assure you it will be alot higher, they dont count his passive when they calculate.
Mercury Threads - Makes you less vulnerable to cc and it will make you even tankier to AP, + some ap to yourself again thanks to your passive.
Abyssal Scepter - A must have on all galio's, it gives you tankyness and about 100 AP + a nice aura to weaken your enemies.
Deathcap - Simply for the damage.
Void Staff - Same reason as deathcap, also makes you able to do damage to the tanky enemies.
Banshee's Veil - Great deal of tankyness and AP again thanks to passive and also it makes it harder for your enemies to interrupt your ultimate in teamfight because they first have to get through the shield.
Aegis - Nice aura to team and makes you generally tankier.