It's time for the first game of the global edition of the Squid Games.

The game is played in a tower with $$$n$$$ floors numbered from $$$1$$$ to $$$n$$$, each floor containing $$$k$$$ rooms numbered from $$$1$$$ to $$$k$$$. There are $$$m$$$ stairs; the $$$i$$$-th stair connects room $$$u_i$$$ on floor $$$f_i$$$ with room $$$v_i$$$ on floor $$$f_i + 1$$$.

There are $$$2\cdot10^9$$$ players invited to play the game. At the beginning, each of the $$$2 \cdot 10^9$$$ players must choose any room on the first floor. The game lasts for $$$n$$$ turns. In turn $$$t$$$ ($$$1 \le t \le n$$$), the following actions occur in order:

1. For every room $$$j$$$ on floor $$$t$$$, if there are more than $$$C_{t,j}$$$ players, random players are eliminated until exactly $$$C_{t,j}$$$ remain in that room.
2. If $$$t \lt n$$$, each surviving player in a room on floor $$$t$$$ must move to a room on floor $$$t+1$$$ using one of the available stairs connected to their current room. Players unable to move are eliminated. Moving down is not allowed.
3. If $$$t = n$$$, all surviving players are declared winners of the first game, and the game ends.

Player 456 wants to maximize the number of players who survive. For each turn $$$t$$$, compute the maximum number of players that can be alive at the end of that turn, assuming optimal choices for the initial distribution and movements. Note that for each $$$t$$$, these values are calculated independently.