시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
3 초 512 MB 0 0 0 0.000%

문제

The Merkle–Hellman Knapsack Cryptosystem was one of the earliest public key cryptosystems invented by Ralph Merkle and Martin Hellman in 1978. Here is its description

Alice chooses n positive integers {a1, . . . , an} such that each ai > Σi−1j=1aj, a positive integer q which is greater than the sum of all ai, and a positive integer r which is coprime with q. These n + 2 integers are Alice’s private key.

Then Alice calculates bi = (ai · r) mod q. These n integers are Alice’s public key.

Knowing her public key, Bob can transmit a message of n bits to Alice. To do that he calculates s, the sum of bi with indices i such that his message has bit 1 in i-th position. This value s is the encrypted message.

Note that an eavesdropper Eve, who knows the encrypted message and the public key, has to solve a (presumably hard) instance of the knapsack problem to find the original message. Meanwhile, after receiving s, Alice can calculate the original message in linear time; we leave it to you as an exercise.

In this problem you deal with the implementation of the Merkle–Hellman Knapsack Cryptosystem in which Alice chose q = 264, for obvious performance reasons, and published this information. Since everyone knows her q, she asks Bob to send her the calculated value s taken modulo 264 for simplicity of communication.

You are to break this implementation. Given the public key and an encrypted message, restore the original message.

입력

The first line contains one integer n (1 ≤ n ≤ 64).

Each of the next n lines contains one integer bi (1 ≤ bi < 264).

The last line contains one integer s mod q — the encrypted message s taken modulo q (0 ≤ s mod q < 264).

The given sequence bi is a valid public key in the described implementation, and the given value s mod q is a valid encrypted message.

출력

Output exactly n bits (0 or 1 digits) — the original message.

예제 입력

5
10
20
50
140
420
440

예제 출력

01001

힌트