A pack of cards contains cards with four identical cards of each value. The pack is shuffled and cards are dealt one at a time and placed in piles of equal value. If the card has the same value as any pile it is placed in that pile. If there is no pile of that value then it begins a new pile. When a pile has four cards of the same value it is removed.
Throughout the process the maximum number of non empty piles is recorded. Let be its expected value. You are given rounded to 8 decimal places.
Find . Give your answer rounded to 8 digits after the decimal point.
一副牌中有 种不同价值的牌，每种价值的牌都有 4 张，共计 张牌。我们将这副牌洗乱后，按序一张一张摸牌，并将摸到的牌放入与该牌价值相同的牌堆中。若不存在这样的牌堆，就把此牌视为一组新牌堆后继续摸牌。如果某牌堆有 4 张牌，则将此牌堆移除。
在摸牌过程中，我们记录下非空牌堆数的最大值，并记 为其期望。已知 四舍五入至小数点后 8 位后为 。
求 ，将你的答案四舍五入至小数点后 8 位。
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