| Spongebob wants to go out trick-or-treating on Halloween, but has to work a graveyard shift at the Krusty Krab. Luckily, his ghostly fry cook friend, the hash-slinging slasher, is here to cover. | |
| At the start of the shift, there are \(N\) patties on the grill (numbered from \(1\) to \(N\)), the \(i\)th of which weighs \(A_i\) grams. For each order that comes in, the slasher removes from the grill a non-empty sequence of patties at contiguous indices \(i..j\) (patties \(i..j\) must all be on the grill at that moment). The *deliciousness* of the order is defined as \((A_i + ... + A_j)\), modulo \(M\) because too much meat can be overpowering! Note that if a patty is removed, a future order's range cannot span across that empty spot. Also, there may be patties left over at the end. | |
| As proof of his hard work at the end of the shift, the slasher will compute a hash by taking the [bitwise XOR](https://en.wikipedia.org/wiki/Bitwise_operation) of the deliciousnesses of all the orders. How many distinct hashes are possible across all valid sequences of \(0\) or more orders? | |
| # Constraints | |
| \(1 \leq T \leq 125\) | |
| \(0 \leq A_i \leq 1{,}000{,}000{,}000\) | |
| \(1 \leq N \leq 9{,}000\) | |
| \(1 \leq M \leq 5{,}000\) | |
| The sum of \(N + M\) across all test cases is at most \(200{,}000\). | |
| # Input Format | |
| Input begins with an integer \(T\), the number of test cases. Each case begins with a line containing the two integers \(N\) and \(M\). Then, there is a single line containing \(N\) integers, \(A_1, ..., A_N\). | |
| # Output Format | |
| For the \(i\)th test case, print "`Case #i:` " followed by the number of different possible hashes. | |
| # Sample Explanation | |
| In the first sample case, there are \(3\) patties with \(A = [2, 2, 0]\) and \(M = 10\). The possible valid order sequences (patties removed) and hashes are: | |
| - No patties removed \(\to\) hash \(0\) | |
| - Patties \(1..1\) \(\to\) hash \(2\) | |
| - Patties \(2..2\) \(\to\) hash \(2\) | |
| - Patties \(3..3\) \(\to\) hash \(0\) | |
| - Patties \(1..2\) \(\to\) hash \(4\) | |
| - Patties \(2..3\) \(\to\) hash \(2\) | |
| - Patties \(1..3\) \(\to\) hash \(4\) | |
| - Patties \(1..1\) and patties \(2..2\) \(\to\) hash \(0\) | |
| - Patties \(1..1\) and patties \(3..3\) \(\to\) hash \(2\) | |
| - Patties \(2..2\) and patties \(3..3\) \(\to\) hash \(2\) | |
| - Patties \(1..1\) and patties \(2..3\) \(\to\) hash \(0\) | |
| - Patties \(1..2\) and patties \(3..3\) \(\to\) hash \(4\) | |
| - Patties \(1..1\), patties \(2..2\), and patties \(3..3\) \(\to\) hash \(0\) | |
| There are \(3\) distinct hashes the slasher can make: \(0\), \(2\), and \(4\). | |
| In the second case, the slasher can make hashes \(0\) (for example, with no orders) and \(1\) (for example, with \(5\) orders, each including one of the patties). | |