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?

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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 $ .