CF1612E.Messages

Monocarp is a tutor of a group of $n$ students. He communicates with them using a conference in a popular messenger.

Today was a busy day for Monocarp — he was asked to forward a lot of posts and announcements to his group, that's why he had to write a very large number of messages in the conference. Monocarp knows the students in the group he is tutoring quite well, so he understands which message should each student read: Monocarp wants the student $i$ to read the message $m_i$.

Of course, no one's going to read all the messages in the conference. That's why Monocarp decided to pin some of them. Monocarp can pin any number of messages, and if he wants anyone to read some message, he should pin it — otherwise it will definitely be skipped by everyone.

Unfortunately, even if a message is pinned, some students may skip it anyway. For each student $i$, Monocarp knows that they will read at most $k_i$ messages. Suppose Monocarp pins $t$ messages; if $t \le k_i$, then the $i$-th student will read all the pinned messages; but if $t \gt k_i$, the $i$-th student will choose exactly $k_i$ random pinned messages (all possible subsets of pinned messages of size $k_i$ are equiprobable) and read only the chosen messages.

Monocarp wants to maximize the expected number of students that read their respective messages (i.e. the number of such indices $i$ that student $i$ reads the message $m_i$). Help him to choose how many (and which) messages should he pin!

Monocarp 是一个由 $n$ 名学生组成的小组的导师。他通过一款流行即时通讯软件中的群聊与学生们交流。

今天对 Monocarp 来说十分忙碌——他被要求向该群转发大量帖子和通知，因此他在群聊中发送了数量庞大的消息。Monocarp 对自己所指导的学生非常了解，因此他清楚每条消息应由哪位学生阅读：Monocarp 希望第 $i$ 位学生阅读第 $m_i$ 条消息。

当然，没有人会阅读群聊中的所有消息。因此，Monocarp 决定将其中一部分消息置顶（即“pin”）。Monocarp 可以置顶任意数量的消息；若希望某位学生阅读某条消息，则他必须将该消息置顶——否则该消息必然被所有人忽略。

不幸的是，即使一条消息被置顶，某些学生仍可能跳过它。对于每位学生 $i$，Monocarp 知道他们最多只会阅读 $k_i$ 条消息。假设 Monocarp 共置顶了 $t$ 条消息：若 $t \le k_i$，则第 $i$ 位学生将阅读全部 $t$ 条置顶消息；但若 $t > k_i$，则第 $i$ 位学生将从所有置顶消息中随机选择恰好 $k_i$ 条（所有大小为 $k_i$ 的置顶消息子集等概率被选中），并仅阅读所选中的消息。

Monocarp 希望最大化期望意义上阅读到各自指定消息的学生人数（即满足“第 $i$ 位学生阅读了消息 $m_i$”这一条件的下标 $i$ 的个数的期望值）。请帮助他确定：应置顶多少条消息？具体是哪几条？

The first line contains one integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of students in the conference.

Then $n$ lines follow. The $i$-th line contains two integers $m_i$ and $k_i$ ($1 \le m_i \le 2 \cdot 10^5$; $1 \le k_i \le 20$) — the index of the message which Monocarp wants the $i$-th student to read and the maximum number of messages the $i$-th student will read, respectively.

第一行包含一个整数 $n$（$1 \le n \le 2 \cdot 10^5$）—— 表示会议中学生的数量。

接下来是 $n$ 行。第 $i$ 行包含两个整数 $m_i$ 和 $k_i$（$1 \le m_i \le 2 \cdot 10^5$；$1 \le k_i \le 20$）—— 分别表示 Monocarp 希望第 $i$ 个学生阅读的消息编号，以及第 $i$ 个学生最多会阅读的消息数量。

In the first line, print one integer $t$ ($1 \le t \le 2 \cdot 10^5$) — the number of messages Monocarp should pin. In the second line, print $t$ distinct integers $c_1$, $c_2$, ..., $c_t$ ($1 \le c_i \le 2 \cdot 10^5$) — the indices of the messages Monocarp should pin. The messages can be listed in any order.

If there are multiple answers, print any of them.

第一行输出一个整数 $t$（$1 \le t \le 2 \cdot 10^5$）—— Monocarp 应当置顶的消息数量。
第二行输出 $t$ 个互不相同的整数 $c_1$, $c_2$, ..., $c_t$（$1 \le c_i \le 2 \cdot 10^5$）—— Monocarp 应当置顶的消息的索引。这些消息可以以任意顺序列出。

若存在多个合法答案，输出其中任意一个即可。

Let's consider the examples from the statement.

1. In the first example, Monocarp pins the messages $5$ and $10$.

   • if the first student reads the message $5$, the second student reads the messages $5$ and $10$, and the third student reads the messages $5$ and $10$, the number of students which have read their respective messages will be $2$;
   • if the first student reads the message $10$, the second student reads the messages $5$ and $10$, and the third student reads the messages $5$ and $10$, the number of students which have read their respective messages will be $3$.

   So, the expected number of students which will read their respective messages is $\frac{5}{2}$.

2. In the second example, Monocarp pins the message $10$.

   • if the first student reads the message $10$, the second student reads the message $10$, and the third student reads the message $10$, the number of students which have read their respective messages will be $2$.

   So, the expected number of students which will read their respective messages is $2$. If Monocarp had pinned both messages $5$ and $10$, the expected number of students which read their respective messages would have been $2$ as well.

3. In the third example, the expected number of students which will read their respective messages is $\frac{8}{3}$.

4. In the fourth example, the expected number of students which will read their respective messages is $2$.

我们来考虑题目陈述中的例子。

1. 在第一个例子中，Monocarp 将消息 $5$ 和 $10$ 固定（置顶）。

   • 若第一位学生阅读了消息 $5$，第二位学生阅读了消息 $5$ 和 $10$，第三位学生也阅读了消息 $5$ 和 $10$，则已阅读各自对应消息的学生人数为 $2$；
   • 若第一位学生阅读了消息 $10$，第二位学生阅读了消息 $5$ 和 $10$，第三位学生也阅读了消息 $5$ 和 $10$，则已阅读各自对应消息的学生人数为 $3$。

   因此，阅读了各自对应消息的学生人数的期望值为 $\frac{5}{2}$。

2. 在第二个例子中，Monocarp 将消息 $10$ 固定（置顶）。

   • 若第一位学生阅读了消息 $10$，第二位学生阅读了消息 $10$，第三位学生也阅读了消息 $10$，则已阅读各自对应消息的学生人数为 $2$。

   因此，阅读了各自对应消息的学生人数的期望值为 $2$。若 Monocarp 同时固定（置顶）了消息 $5$ 和 $10$，则阅读了各自对应消息的学生人数的期望值同样为 $2$。

3. 在第三个例子中，阅读了各自对应消息的学生人数的期望值为 $\frac{8}{3}$。

4. 在第四个例子中，阅读了各自对应消息的学生人数的期望值为 $2$。