# Still not as hard as Dark Souls

The video game Schmark Schmouls is an excellent video game with the latest graphics and cutting-edge gameplay and will be a surefire hit when it releases sometime in the near future (Soonâ„˘). Also, Schmark Schmouls has no relation to Dark Souls... please don't sue.

In the game, the player starts at checkpoint $$A$$, and must pass through the checkpoints $$B$$, $$C$$, and $$D$$ in order. After passing checkpoint $$D$$, the player wins the game. To pass each checkpoint (except for the starting checkpoint $$A$$), the player must defeat a boss.

Each time a player attempts to defeat a boss, the following outcomes could occur:

• $$\frac{1}{3}$$ probability: The player defeats the boss and passes the checkpoint.
• $$\frac{1}{3}$$ probability: The player loses to the boss and gets sent back to any of the previous checkpoints with equal probability. The boss at that checkpoint will not re-spawn, but all bosses in succeeding checkpoints will re-spawn.
• $$\frac{1}{3}$$ probability: The player loses to the boss, throws the controller across the room, turns the game off, cries in the corner, and never plays Schmark Schmouls again.

The probability that a player wins the game of Schmark Schmouls can be expressed as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are positive co-prime integers. Find $$a+b$$.

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