AND, OR and square sum

### 题面描述 给定 $n$ 个非负整数 $a_1,\cdots,a_n$。 你可以进行如下操作：选择两个不同的下标 $i,j$ 满足 $1\leq i,j\leq n$，并将 $a_i\gets a_i\ \mathsf{AND}\ a_j,\ a_j\gets a_i\ \mathsf{OR}\ a_j$，**两个赋值同时进行**。AND 是按位与，OR 是按位或。 你可以进行任意次操作。求操作后所有数的平方和的最大值，即 $\max \sum a_i^2$。 ### 输入格式 第一行一个整数 $n\ (1\leq n\leq 2\times 10^5)$。 第二行 $n$ 个整数 $a_1,\cdots,a_n\ (0\leq a_i<2^{20})$。 ### 输出格式 输出一行一个整数表示答案，即操作后所有数的平方和的最大值。 Translated by Alex_Wei.

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?

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 < 2^{20} $ ).

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

1
123

15129

3
1 3 5

51

2
349525 699050

1099509530625

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 $ .