Consider the following variant of "The Chase" game. This game is played between players sitting around a circular table using two dice. It consists of 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 where 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 be the expected amount that the winner will receive. For example is approximately 96.544, and is 2.82491788e6 in scientific notation rounded to 9 significant digits.
Find , giving your answer in scientific notation rounded to 9 significant digits.
以下描述的是追逐游戏的一个变种， 位玩家围坐圆桌边，用两粒骰子来玩这个游戏。这个游戏共有 轮，每一轮结束后，有一名玩家将被淘汰，他需要向奖池里投入一定量的钱。存活到最后的人获胜，拿到奖池里的所有钱。
每一轮开始时，每个骰子将分别随机分配给一名玩家。每一轮由若干次操作组成。每次操作时，两名手持骰子的玩家掷出手里的骰子，如果掷出 1、2，这名玩家就将骰子传给他左边的玩家；掷出 5、6 则将骰子传给他右边的玩家；否则则把骰子留着。
如果一名玩家同时持有两粒骰子，该轮结束。这名玩家被淘汰，并且需要向奖池内投入 元，其中 是这一轮的操作数。如果某名玩家在某一轮开始时就拿到两颗骰子，那这名玩家即刻淘汰，并且一分钱也不用出。
记 为：有 名玩家时，胜者从奖池内得到的钱的期望值。已知：、（科学记数法，且保留 9 位有效数字）。
求出 的值，并将你的答案用科学记数法表示，保留 9 位有效数字。
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