CF2181J.Jinx or Jackpot

Jack is in his favourite casino and has $1000$ dollars. The casino has literally nothing but a single slot machine. Jack knows the history of this casino. Once upon a time, the future owner of the casino was walking and suddenly saw an array of $n$ integer choices $p_{1}, \dots, p_{n}$ each from $0$ to $100$. He picked an index $i$ ($1 \leq i \leq n$) uniformly at random and thought that it was a good idea to create a casino in which there is only one slot machine with jackpot probability of $\frac{p_i}{100}$. And he created it.

Jack knows the array of choices $p_{1}, \dots, p_{n}$ that suddenly appeared to the owner during the walk, but he does not know which $i$ the owner picked. However, the chosen index $i$ is fixed forever; the slot machine always uses the same $p_i$ as explained below.

On the slot machine, Jack can bet $x$ dollars, where $x$ is a non-negative integer, and pull the lever. Then:

1. With probability $\frac{p_i}{100}$ it will be a jackpot, and the slot machine returns $2x$ dollars to him, so he gains $x$ dollars.
2. With probability $1 - \frac{p_i}{100}$ it will be a jinx, and the slot machine returns nothing to him, so he loses $x$ dollars.

Even if Jack bets $0$ dollars, he will understand whether it was a jinx or a jackpot.

Also, the slot machine is not very durable, so Jack can play at most $k$ rounds on it.

Find the maximum expected profit Jack can achieve by an optimal strategy. Here a profit is defined as the final amount of money Jack has minus his initial $1000$ dollars.

Of course, Jack can't make a bet that is more than his current balance.

杰克正在他最喜欢的赌场里，身上有 1000 美元。这家赌场里只有一台老虎机。杰克了解这家赌场的历史：很久以前，这家赌场未来的老板在路上散步时，突然看到了一个由 $n$ 个整数 $p_{1}, \dots, p_{n}$ 组成的数组，其中每个 $p_j$ 均在 $0$ 到 $100$ 之间（含端点）。他从 $1$ 到 $n$ 中均匀随机地选取了一个下标 $i$（即 $1 \leq i \leq n$），并认为以 $\frac{p_i}{100}$ 为中奖概率设置一台老虎机是个不错的主意——于是他就照此建起了这家赌场。

杰克知道老板散步时偶然看到的那个数组 $p_{1}, \dots, p_{n}$，但他并不知道老板当时究竟选中了哪一个下标 $i$。然而，这个被选中的下标 $i$ 是永久固定的；如前所述，这台老虎机始终使用同一个 $p_i$。

在该老虎机上，杰克每次可以下注 $x$ 美元，其中 $x$ 是一个非负整数，然后拉动拉杆。此时：

1. 以概率 $\frac{p_i}{100}$ 触发“头奖”（jackpot），老虎机返还 $2x$ 美元，因此杰克净赚 $x$ 美元；
2. 以概率 $1 - \frac{p_i}{100}$ 触发“厄运”（jinx），老虎机不返还任何钱，因此杰克净亏 $x$ 美元。

即使杰克下注 $0$ 美元，他仍能获知本次结果是“厄运”还是“头奖”。

此外，这台老虎机不够耐用，杰克最多只能玩 $k$ 轮。

请计算杰克通过最优策略所能获得的最大期望收益。此处“收益”定义为：杰克最终拥有的钱数减去其初始资金 $1000$ 美元。

当然，杰克不能下注超过其当前余额的金额。

The first line contains two integers $n$ and $k$ ($1 \leq n \leq 100\,000$; $1 \leq k \leq 30$) — the number of choices and the limit on the number of rounds. The second line contains $n$ integers $p_{1}, \dots, p_{n}$ ($0 \le p_i \le 100$) — the choices.

第一行包含两个整数 $n$ 和 $k$（$1 \leq n \leq 100\,000$；$1 \leq k \leq 30$）—— 分别表示选项数量和轮次上限。
第二行包含 $n$ 个整数 $p_{1}, \dots, p_{n}$（$0 \le p_i \le 100$）—— 表示各个选项。

Output a single real number — the expected profit Jack can achieve by an optimal strategy. Your answer will be considered correct if its absolute or relative error is at most $10^{-4}$.

输出一个实数——Jack 通过最优策略所能获得的期望收益。若你的答案的绝对误差或相对误差不超过 $10^{-4}$，则视为正确。