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Attack Range is the range at which a hero can attack something. There are two main types of ranges: Melee, and Ranged.
Melee heroes are generally Strength or Agility heroes, though there are a few Intelligence heroes who are melee. Melee heroes must walk up to an enemy to attack them, and they cannot miss attacking up hills. Their bashes are magical damage, reduced by magic resistance, and they stack fully. Melee Cranium Bashers have a 15% chance of proccing. Melee attacks are affected by damage return.
Ranged heroes are generally Agility or Intelligence heroes, though as of recently, ranged strength heroes have been introduced (Huskar). Ranged heroes can attack from a range, varying between 140 and 770, and they have a 25% miss chance attacking up hills. Their bashes are physical damage, reduced by armor, add to lifesteal, and only one can trigger per attack. Ranged bashers have a 10% chance of proccing. Ranged attacks are not affected by damage return.
Goblin Techies has the greatest attack range of all range heroes, 650, only surpassed by Dwarven Sniper's Take Aim (550+220). Templar Assassin has the lowest at 140 (which eventually grows up to 360 with Templar Assassin's Psi Blades, exceptionally low for a ranged hero. The next lowest attack range is 330, given to Moon Rider.
600 is the most common, given to 20 out of the currently 46 ranged heroes in DotA.
Naturally, melee heroes all have a very short attack range. An attack range of 100 is the minimum and most common (given to two thirds of melee heroes), while 150 is the maximum, held solely by Doom Bringer. 125 and 128 are the only other two values used in attack range.

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