CF1368D.AND, OR and square sum

Gottfried learned about binary number representation. He then came up with this task and presented it to you.

You are given a collection of $n$ non-negative integers $a_1, \ldots, a_n$. You are allowed to perform the following operation: choose two distinct indices $1 \leq i, j \leq n$. If before the operation $a_i = x$, $a_j = y$, then after the operation $a_i = x~\mathsf{AND}~y$, $a_j = x~\mathsf{OR}~y$, where $\mathsf{AND}$ and $\mathsf{OR}$ are bitwise AND and OR respectively (refer to the Notes section for formal description). The operation may be performed any number of times (possibly zero).

After all operations are done, compute $\sum_{i=1}^n a_i^2$ — the sum of squares of all $a_i$. What is the largest sum of squares you can achieve?

戈特弗里德学习了二进制数表示法，随后提出了这个题目并交给你解决。

给定一个包含 $n$ 个非负整数 $a_1, \ldots, a_n$ 的集合。你被允许执行如下操作：选择两个不同的下标 $1 \leq i, j \leq n$。若操作前 $a_i = x$、$a_j = y$，则操作后 $a_i = x~\mathsf{AND}~y$、$a_j = x~\mathsf{OR}~y$，其中 $\mathsf{AND}$ 和 $\mathsf{OR}$ 分别表示按位与和按位或运算（形式化定义参见“注释”部分）。该操作可执行任意多次（包括零次）。

在所有操作完成后，计算 $\sum_{i=1}^n a_i^2$ —— 即所有 $a_i$ 的平方之和。你能达到的最大平方和是多少？

The first line contains a single integer $n$ ($1 \leq n \leq 2 \cdot 10^5$).

The second line contains $n$ integers $a_1, \ldots, a_n$ ($0 \leq a_i \lt 2^{20}$).

第一行包含一个整数 $n$（$1 \leq n \leq 2 \cdot 10^5$）。

第二行包含 $n$ 个整数 $a_1, \ldots, a_n$（$0 \leq a_i \lt 2^{20}$）。

Print a single integer — the largest possible sum of squares that can be achieved after several (possibly zero) operations.

输出一个整数——经过若干次（可能为零次）操作后所能达到的最大平方和。

In the first sample no operation can be made, thus the answer is $123^2$.

In the second sample we can obtain the collection $1, 1, 7$, and $1^2 + 1^2 + 7^2 = 51$.

If $x$ and $y$ are represented in binary with equal number of bits (possibly with leading zeros), then each bit of $x~\mathsf{AND}~y$ is set to $1$ if and only if both corresponding bits of $x$ and $y$ are set to $1$. Similarly, each bit of $x~\mathsf{OR}~y$ is set to $1$ if and only if at least one of the corresponding bits of $x$ and $y$ are set to $1$. For example, $x = 3$ and $y = 5$ are represented as $011_2$ and $101_2$ (highest bit first). Then, $x~\mathsf{AND}~y = 001_2 = 1$, and $x~\mathsf{OR}~y = 111_2 = 7$.

在第一个样例中，无法进行任何操作，因此答案为 $123^2$。

在第二个样例中，我们可以得到集合 $1, 1, 7$，且 $1^2 + 1^2 + 7^2 = 51$。

若 $x$ 和 $y$ 以二进制表示，且位数相同（允许高位补零），则 $x~\mathsf{AND}~y$ 的每一位为 $1$ 当且仅当 $x$ 和 $y$ 在该位上均为 $1$；类似地，$x~\mathsf{OR}~y$ 的每一位为 $1$ 当且仅当 $x$ 和 $y$ 在该位上至少有一个为 $1$。例如，$x = 3$ 和 $y = 5$ 分别表示为 $011_2$ 和 $101_2$（高位在前）。于是，$x~\mathsf{AND}~y = 001_2 = 1$，而 $x~\mathsf{OR}~y = 111_2 = 7$。