K. Deck-Building Gametime limit per test1 secondmemory limit per test1024 megabytesinputstandard inputoutputstandard outputYou are playing a deck-building game with your friend. There are N cards, numbered from 1 to N. Card i has the value of A_i.You want to build two decks; one for you and one for your friend. A card cannot be inside both decks, and it is allowed to not use all N cards. It is also allowed for a deck to be empty, i.e. does not contain any cards.The power of a deck is represented as the bitwise XOR of the value of the cards in the deck. The power of an empty deck is 0.The game is balanced if both decks have the same power.Determine the number of ways to build two decks such that the game is balanced. Two ways are considered different if one of the decks contains at least one different card. Since the answer can be very large, calculate the answer modulo 998\,244\,353.InputThe first line consists of an integer N (2 \le N \le 100\,000).The following line consists of N integers A_i (1 \le A_i \le 100\,000).OutputOutput an integer representing the number of ways to build two decks such that the game is balanced. Output the answer modulo 998\,244\,353.ExamplesInput
4
16 12 4 8
Output
9
Input
4
1 2 4 8
Output
1
Input
2
1 1
Output
5
Input
6
1 1 1 2 2 2
Output
169
NoteExplanation for the sample input/output #1Denote S and T as the set of cards in your deck and your friend's deck, respectively. There are 9 ways to build the decks such that the game is balanced.  S = \{\} and T = \{\}. Both decks have the power of 0.  S = \{2, 3, 4\} and T = \{\}. Both decks have the power of 0.  S = \{\} and T = \{2, 3, 4\}. Both decks have the power of 0.  S = \{2, 4\} and T = \{3\}. Both decks have the power of 4.  S = \{3\} and T = \{2, 4\}. Both decks have the power of 4.  S = \{2, 3\} and T = \{4\}. Both decks have the power of 8.  S = \{4\} and T = \{2, 3\}. Both decks have the power of 8.  S = \{3, 4\} and T = \{2\}. Both decks have the power of 12.  S = \{2\} and T = \{3, 4\}. Both decks have the power of 12. Explanation for the sample input/output #2The only way to make the game balanced is to have both decks empty.Explanation for the sample input/output #3There are 5 ways to build the decks such that the game is balanced.  S = \{\} and T = \{\}. Both decks have the power of 0.  S = \{1, 2\} and T = \{\}. Both decks have the power of 0.  S = \{\} and T = \{1, 2\}. Both decks have the power of 0.  S = \{1\} and T = \{2\}. Both decks have the power of 1.  S = \{2\} and T = \{1\}. Both decks have the power of 1. 
4
16 12 4 8

9 
['divide and conquer', 'math', '*2500']