Hermione loves to play the following computer game, in which the player controls an unordered multiset of integers. Initially, the multiset is empty and the player's score is $$$0$$$. At any moment in the game, the multiset contains at most $$$k$$$ integers (not necessarily distinct). In one turn, the player can choose one of the following moves:

• Insert. Choose an integer $$$2$$$ or $$$4$$$ and insert it into the multiset. This move does not change the score, and it is only allowed if the size of the multiset before the move is strictly less than $$$k$$$.
• Merge. Choose an integer $$$x$$$ such that the multiset contains at least two copies of $$$x$$$. Remove two copies of $$$x$$$ from the multiset, and insert one copy of $$$2x$$$ into the multiset. This move adds the value $$$2x$$$ to the player's score.

The player can stop the game after any turn, at which point the player's score becomes final.

Hermione has been on vacation for the whole summer, and she hasn't played the game in a while. Today, she opened the game on her laptop and saw the leaderboard with the highest scores she's ever had: $$$h_1, h_2, \ldots, h_n$$$ in some order. Now she is curious how she managed to achieve those scores.

For each $$$h_i$$$, find any multiset of integers that Hermione could have at the end of the game if her final score was $$$h_i$$$, or determine that it is impossible to achieve such a score.