Stone Pile Game

You are given n piles of stones, where the i-th pile contains a_i stones. Let \( x \) be the average of the sequence \( a_1, a_2, \cdots, a_n \), i.e., \( x = \frac{a_1+a_2+\cdots+a_n}{n} \). You are also given an integer k such that \( k \le x \). Two players, F and L, play the following game alternately (F moves first):

• Select two piles \( i \) and \( j \) such that \( a_i < x < a_j \). If no such pair of piles exists, then the current player loses and the opponent wins.
• Remove exactly \( k \) stones from pile \( j \) (which satisfies \( a_j > x \)) and add them to pile \( i \) (which satisfies \( a_i < x \)).

Both players play optimally. The game stops when a player is unable to make a move. In addition, if the game continues indefinitely, output Draw.

Your task is to determine the result of each game. For each game, output:

• Draw if the game will continue forever.
• F if player F (the first mover) wins when the game terminates.
• L otherwise.

Game Analysis

Since moving \( k \) stones from one pile to another keeps the total number of stones (and thus the average \( x \)) invariant, the only possible moves are those that transfer surplus from piles having more than \( x \) to those having fewer than \( x \). A move is only valid if:

• There exists a pile with \( a_i x \).
• After a sequence of moves, a pile will reach \( x \) only if the difference \( |a_i - x| \) is divisible by \( k \). Otherwise, the pile will never reach the balanced state, and the game will continue indefinitely.

If \( x \) is not an integer (i.e. \( \sum_{i=1}^{n} a_i \) is not divisible by \( n \)), then some piles will always be strictly less than or greater than \( x \), and the game will never terminate (i.e. the outcome is Draw).

Assume now that \( x \) is an integer and that for every pile, \( |a_i - x| \) is divisible by \( k \). Let the number of moves needed to bring all piles to \( x \) be given by:

[ m = \sum_{a_i < x} \frac{x - a_i}{k} = \sum_{a_j > x} \frac{a_j - x}{k}. ]

The players alternate moves. If no move is possible at the start, then player F loses. Otherwise, if \( m \) is odd, then player F makes the last move and wins; if \( m \) is even, then player L wins.

inputFormat

The first line contains a single integer T (1 ≤ T ≤ 10⁴), the number of games.

Each game is described as follows:

• The first line contains two integers n and k (2 ≤ n ≤ 10⁵ and k is a positive integer with \( k \le x \)).
• The second line contains n integers: a1, a2, ..., an (1 ≤ a_i ≤ 10⁹).

The total sum of n over all test cases does not exceed 10⁵.

outputFormat

For each game, output a single line containing one of the following:

• Draw — if the game continues indefinitely.
• F — if the first player wins.
• L — if the second player wins.

sample

3
2 1
1 3
2 1
1 5
3 2
1 2 3

F
L
Draw

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