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Scale Puzzles 301 - First Weighing: Right Side Down

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You have twelve coins (A, B, C, D, E, F, G, H, J, K, L & M). One of the coins is counterfeit, but you don't know if it is heavier or lighter than the others. You are allowed to use a balancing scale three times to find the counterfeit. Find a strategy that will allow you to find the counterfeit in three weighings, no matter which one is fake.

First Weighing: You decided to weigh (A+B+C+D) on the left and (E+F+G+H) on the right. The right side of the scale went down.

^_(A+B+C+D)_ /\ _{_(E+F+G+H)_}

What does this tell you about the eight coins on the scale?
What does this tell you about the four coins that aren't on the scale?

The right side going down tells us that (A+B+C+D) is lighter than (E+F+G+H). But, we still don't know if the fake is heavier than or lighter than a genuine coin. If the fake is lighter than a genuine coin, it is one of the group (A+B+C+D). If the fake is heavier than a genuine coin, it is in the group (E+F+G+H).

This also tells us that (J+K+L+M) are all genuine coins. We can use that information to our advantage in our next weighing, when we rotate three of the coins around by:

taking three coins (B+C+D) off the left side of the scale, and
moving three coins (F+G+H) from the right side of the scale to the left side, and
moving three coins (K+L+M) from the table onto the right side of the scale.

This, in essence, gives us four groups of three coins:

The three coins that got removed from the scale (B+C+D), which will be lighter fakes if the scales balance the second time
The three coins that changed sides on the scale (F+G+H), which might be heavier fakes if the left side goes down the second time instead of up
The three coins that never changed position (A+E+J). We will know that one of them is the fake if the balance doesn't change. We already know that J is Jenuine, but A could be a lighter fake, or E could be a heavier fake
The fourth group is the three coins that got added to the scale (K+L+M). They are definitely genuine

First Weighing: ^_(A+B+C+D)_ /\ _{_(E+F+G+H)_} (Left side went down)
Second Weighing: Put A+F+G+H on the left. Put E+K+L+M on the right Compare: A+F+G+H ^ E+K+L+M Click one of the outcomes on the right	Left side goes down this time _{_(A+F+G+H)_} /\ ^_(E+K+L+M)_
	The sides balance now _(A+F+G+H)_ /\ _(E+K+L+M)_
	Right side still goes down ^_(A+F+G+H)_ /\ _{_(E+K+L+M)_}