Xiao Chang and Xiao Peng want to play a game.

The game is played on a directed acyclic graph (DAG) with $$$2^n$$$ nodes (labeled $$$0$$$ to $$$2^n-1$$$) and $$$m$$$ edges. There are $$$k$$$ pieces placed on the graph, with the $$$i$$$-th piece initially located at node $$$a_i$$$. The two players take turns, with Xiao Chang moving first. On each turn, a player can freely choose any piece and move it one step along any outgoing edge from its current node. If a player cannot make any move on their turn (i.e., all pieces are on nodes with no outgoing edges), that player loses the game. Note: Multiple pieces can occupy the same node.

Naturally, Xiao Chang and Xiao Peng are perfectly rational: once the initial positions of the pieces are set, they both already know who will win or lose the game. So, they decide to manipulate the initial positions $$$a_i$$$! They observe that each $$$a_i$$$ can be uniquely represented as an $$$n$$$-bit binary number. If they write down the binary representations of all $$$k$$$ pieces row by row, they will form a $$$k \times n$$$ binary matrix, where the value at the $$$i$$$-th row and $$$j$$$-th column indicates whether the $$$j$$$-th bit (from least to most significant, with the least significant bit being the 1st bit) of $$$a_i$$$ is 0 or 1.

They assign a position number $$$(i-1) \times n+j$$$ to each cell $$$(i,j)$$$ in the matrix. Starting from position 1, they take turns deciding whether each cell should be 0 or 1, with Xiao Chang going first, followed by Xiao Peng, until the entire matrix is filled (i.e., Xiao Chang decides the first position, Xiao Peng the second, Xiao Chang the third, and so on). In this way, they "scheme against each other" to determine all $$$a_i$$$.

Afterward, Xiao Chang makes the first move in the DAG-based game described earlier. However, the game is still too simple for them, so they want to ask you: Under these rules, who will be the ultimate winner?