# Themed Challenge 2 (Manga/Anime): The Exhaustive Battle

Pikachu and a wild Rattata have been battling for a long time. They have been fighting for so long that their accuracy is haywire, and their health is one on-target hit away from being a K.O. Any attack now, however powerful or weak, if on target, could defeat the opponent.

At this stage of battle, Rattata has $$2$$ attacks. He can use Tackle, which has a $$\frac{1}{3}$$ chance of being on target. He can also use Hyper Fang, but only after $$2$$ moves have occurred (i.e. Rattata moves, then Pikachu moves, then Rattata can use it or save for a later move). This attack has a $$\frac{2}{3}$$ accuracy rate.

Pikachu, on the other hand has $$3$$ attacks. One is Quick Attack, which has a $$\frac{1}{3}$$ probability of being on target. Another attack is ThunderShock, which has a $$\frac{1}{2}$$ probability of being on target, but can only be used after $$2$$ moves. Finally, Pikachu can use ShockWave, a move with Infinite Accuracy, which, mathematically speaking has an accuracy rate of $$1$$. However, it can only be used after $$4$$ moves.

In this battle, the highest probability of Rattata winning and defeating Pikachu is given by $$\frac{M}{N}$$ , where $$M,N$$ are co-prime integers. Find $$M+N$$.

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