The Chase II (noch nicht übersetzt)

Consider the following variant of "The Chase" game. This game is played between $n$ players sitting around a circular table using two dice. It consists of $n-1$ rounds, and at the end of each round one player is eliminated and has to pay a certain amount of money into a pot. The last player remaining is the winner and receives the entire contents of the pot.

At the beginning of a round, each die is given to a randomly selected player. A round then consists of a number of turns.

During each turn, each of the two players with a die rolls it. If a player rolls a 1 or a 2, the die is passed to the neighbour on the left; if the player rolls a 5 or a 6, the die is passed to the neighbour on the right; otherwise, the player keeps the die for the next turn.

The round ends when one player has both dice at the beginning of a turn. At which point that player is immediately eliminated and has to pay $s^2$ where $s$ is the number of completed turns in this round. Note that if both dice happen to be handed to the same player at the beginning of a round, then no turns are completed, so the player is eliminated without having to pay any money into the pot.

Let $G(n)$ be the expected amount that the winner will receive. For example $G(5)$ is approximately 96.544, and $G(50)$ is 2.82491788e6 in scientific notation rounded to 9 significant digits.

Find $G(500)$, giving your answer in scientific notation rounded to 9 significant digits.