Koxia and Sequence

1000ms

262144K

Description:

Mari has three integers $$$n$$$, $$$x$$$, and $$$y$$$.

Call an array $$$a$$$ of $$$n$$$ non-negative integers good if it satisfies the following conditions:

$$$a_1+a_2+\ldots+a_n=x$$$, and
$$$a_1 \, | \, a_2 \, | \, \ldots \, | \, a_n=y$$$, where $$$|$$$ denotes the bitwise OR operation.

The score of a good array is the value of $$$a_1 \oplus a_2 \oplus \ldots \oplus a_n$$$, where $$$\oplus$$$ denotes the bitwise XOR operation.

Koxia wants you to find the total bitwise XOR of the scores of all good arrays. If there are no good arrays, output $$$0$$$ instead.

Input:

The first line of input contains three integers $$$n$$$, $$$x$$$ and $$$y$$$ ($$$1 \leq n < 2^{40}$$$, $$$0 \leq x < 2^{60}$$$, $$$0 \leq y < 2^{20}$$$).

Output:

Output a single integer — the total bitwise XOR of the scores of all good arrays.

Sample Input:

3 5 3

Sample Output:

Sample Input:

100 0 100

Sample Output:

Sample Input:

79877974817 749875791743978 982783

Sample Output:

Note:

In the first test case, there are $$$12$$$ good arrays totally as follows.

$$$[0,2,3]$$$, $$$[0,3,2]$$$, $$$[2,0,3]$$$, $$$[2,3,0]$$$, $$$[3,0,2]$$$ and $$$[3,2,0]$$$ — the score is $$$0 \oplus 2 \oplus 3 = 1$$$;
$$$[1, 2, 2]$$$, $$$[2, 1, 2]$$$ and $$$[2, 2, 1]$$$ — the score is $$$1 \oplus 2 \oplus 2 = 1$$$;
$$$[1, 1, 3]$$$, $$$[1, 3, 1]$$$ and $$$[3, 1, 1]$$$ — the score is $$$1 \oplus 1 \oplus 3 = 3$$$.

Therefore, the total bitwise xor of the scores is $$$\underbrace{1 \oplus \ldots \oplus 1}_{9\text{ times}} \oplus 3 \oplus 3 \oplus 3 = 2$$$.

In the second test case, there are no good sequences. The output should be $$$0$$$.

Informação

Provedor Codeforces

Código CF1770F