. ....⠹⣷⡀⣰⡿⠁⢀⣾⠟⠛⢿⣶⡀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⣿⡟⠁⠀⢸⣿⠀⠀⢀⣿⡇⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⢷⣶⡾⠟⠀⠈⠻⣷⣶⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⡞⠛⠿⠦⠀⢸⣿⠀⠀⣿⣿⠀⣠⣾⠿⠿⣶⡀⢸⣿⠀⣠⣾⠟⠀⠀
⠀⠀⠀⠘⠻⠷⣶⣤⠀⢸⣿⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⢿⣧⠀⠀⠀
⠀⠀⠀⠲⣧⣤⣼⠿⠀⠸⢿⣦⣴⡿⠃⠀⠹⢿⣤⣴⡿⠁⢸⣿⠁⠈⢿⣷⡀⠀

. ....⠹⣷⡀⣰⡿⠁⢀⣾⠟⠛⢿⣶⡀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⣿⡟⠁⠀⢸⣿⠀⠀⢀⣿⡇⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⢷⣶⡾⠟⠀⠈⠻⣷⣶⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⡞⠛⠿⠦⠀⢸⣿⠀⠀⣿⣿⠀⣠⣾⠿⠿⣶⡀⢸⣿⠀⣠⣾⠟⠀⠀
⠀⠀⠀⠘⠻⠷⣶⣤⠀⢸⣿⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⢿⣧⠀⠀⠀
⠀⠀⠀⠲⣧⣤⣼⠿⠀⠸⢿⣦⣴⡿⠃⠀⠹⢿⣤⣴⡿⠁⢸⣿⠁⠈⢿⣷⡀⠀

. ....⠹⣷⡀⣰⡿⠁⢀⣾⠟⠛⢿⣶⡀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⣿⡟⠁⠀⢸⣿⠀⠀⢀⣿⡇⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⢷⣶⡾⠟⠀⠈⠻⣷⣶⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⡞⠛⠿⠦⠀⢸⣿⠀⠀⣿⣿⠀⣠⣾⠿⠿⣶⡀⢸⣿⠀⣠⣾⠟⠀⠀
⠀⠀⠀⠘⠻⠷⣶⣤⠀⢸⣿⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⢿⣧⠀⠀⠀
⠀⠀⠀⠲⣧⣤⣼⠿⠀⠸⢿⣦⣴⡿⠃⠀⠹⢿⣤⣴⡿⠁⢸⣿⠁⠈⢿⣷⡀⠀

. ....⠹⣷⡀⣰⡿⠁⢀⣾⠟⠛⢿⣶⡀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⣿⡟⠁⠀⢸⣿⠀⠀⢀⣿⡇⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⢷⣶⡾⠟⠀⠈⠻⣷⣶⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⡞⠛⠿⠦⠀⢸⣿⠀⠀⣿⣿⠀⣠⣾⠿⠿⣶⡀⢸⣿⠀⣠⣾⠟⠀⠀
⠀⠀⠀⠘⠻⠷⣶⣤⠀⢸⣿⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⢿⣧⠀⠀⠀
⠀⠀⠀⠲⣧⣤⣼⠿⠀⠸⢿⣦⣴⡿⠃⠀⠹⢿⣤⣴⡿⠁⢸⣿⠁⠈⢿⣷⡀⠀

. ....⠹⣷⡀⣰⡿⠁⢀⣾⠟⠛⢿⣶⡀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⣿⡟⠁⠀⢸⣿⠀⠀⢀⣿⡇⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⢷⣶⡾⠟⠀⠈⠻⣷⣶⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⡞⠛⠿⠦⠀⢸⣿⠀⠀⣿⣿⠀⣠⣾⠿⠿⣶⡀⢸⣿⠀⣠⣾⠟⠀⠀
⠀⠀⠀⠘⠻⠷⣶⣤⠀⢸⣿⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⢿⣧⠀⠀⠀
⠀⠀⠀⠲⣧⣤⣼⠿⠀⠸⢿⣦⣴⡿⠃⠀⠹⢿⣤⣴⡿⠁⢸⣿⠁⠈⢿⣷⡀⠀

. ....⠹⣷⡀⣰⡿⠁⢀⣾⠟⠛⢿⣶⡀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⣿⡟⠁⠀⢸⣿⠀⠀⢀⣿⡇⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⢷⣶⡾⠟⠀⠈⠻⣷⣶⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⡞⠛⠿⠦⠀⢸⣿⠀⠀⣿⣿⠀⣠⣾⠿⠿⣶⡀⢸⣿⠀⣠⣾⠟⠀⠀
⠀⠀⠀⠘⠻⠷⣶⣤⠀⢸⣿⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⢿⣧⠀⠀⠀
⠀⠀⠀⠲⣧⣤⣼⠿⠀⠸⢿⣦⣴⡿⠃⠀⠹⢿⣤⣴⡿⠁⢸⣿⠁⠈⢿⣷⡀⠀