Strong Arm Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠻⢶⣤⣠⣤⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠻⠟⠁⡀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠙⢿⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠲⣾⠀⠀⠀⠀⠀⠙⣦⠎⠀⠀⠀⠀⢙⡛⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡰⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠃⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣷⠀⠀⠀⠀⠀⠀⢳⣀⣤⠄⠀⠀⣸⡄⠀⠀⠀⠀⢸⡆⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢹⡆⣠⠆⠀⠀⠀⠸⣿⠁⠀⠀⠀⢸⡟⠃⠀⠀⠰⣾⡗⠀⠠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣻⣯⠄⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠃⠀⢠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⡀⠀⣻⣀⣀⣀⣀⠀⢸⡇⠀⠀⠀⢸⠃⠀⠀⠀⢰⡏⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢽⠟⠋⠉⠀⠀⢈⡭⠉⠉⠛⢳⡄⠀⣿⠦⠤⠀⣠⣯⣉⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠀⠀⠀⠈⣷⠶⠋⠙⠒⠚⠉⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⣼⠇⡀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠘⢦⣐⣤⣄⣠⣤⣤⡤⠶⠛⣡⣾⣥⣄⣤⡴⠶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠿⢿⣿⣥⣄⣀⣤⣤⠾⠛⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠁⠀⠀⠀⠻⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣠⣶⠞⠛⠉⠉⠉⠉⠉⠙⠛⠷⢶⣤⣀⡤⠶⠶⠛⠿⠶⠶⣤⣄⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠙⠢⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠶⠶⠶⠶⣶⣶⣶⣶⣶⠶⣿⣶⣤⣶⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⡀⠀⠀⠀⠀⠠⢤⣤⣤⣤⣤⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⣿⣿⣙⠿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣿⣿⣿⣿⣷⣦⣍⠛⠿⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣭⣿⣷⠈⠽⣿⣿⣿⣿⣿⣿⣶⣦⣬⡙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣬⣍⣛⠻⣿⣿⣿⣿⠘⠋⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣭⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⣿⠏⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⣽⣶⣶⣶⣤⣤⣙⡻⠿⢿⣿⣿⣿⣿⠋⣵⣶⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠛⠃⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⢹⣿⣿⣿⣿⣿⣿⣿⣟⣛⠀⣶⣴⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⠃⣴⡆⢲⡀
⠀⠀⠀⠀⠀⠀⣰⣷⢸⣿⣿⣿⣿⣿⠃⣼⣿⠁⣿⡇
⠀⠀⠀⠀⠀⢰⣿⣿⢸⣿⣿⣿⣿⠏⣼⣿⣿⠀⣿⡇
⠀⠀⠀⠀⢀⣿⣿⣿⣾⣿⣿⣿⡟⣸⣿⣿⡻⠂⢿⣷⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣆⠈⣿⣿⣆⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣤⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣇⢸⠏⣿⣇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢰⣿⣿⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⠿⠟⣋⣀⣀⡀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣴⣾⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⠟⣱⣿⣿⣿⣿⡏⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⣠⣾⣿⣿⣿⣿⣿⠿⠿⠿⠛
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣾⣿⣿⣿⣿⡟⠀⣾⣷⣦⣄⣈⠉⠛⠛⠛⠛⠛⣉⣁⣤⣶⣿⣿⣿⣷⣦⣤⣤⣤⡤⠀⣠⣶
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⣿⠋⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⣁⣤⣾⣿⣿
⠀⠠⣄⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⡿⠛⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⢹⣷⣦⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠴⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡛⠿⠿⢿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⢉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢉⠉⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢠⣠⡄⠈⡄⠀⡁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠁⠀⡎⢀⣤⣠⠀⠑⢄
⠀⠀⠀⠀⠀⢀⡔⠁⠀⢠⡾⠁⠙⠒⠃⠓⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠓⠚⠁⠹⣦⠀⠀⠱⣄
⠀⠀⠀⢀⡴⠋⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠈⠳⣄
⠀⠀⢠⠎⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠘⢦
⠀⢠⡏⠀⠀⠀⢀⣿⣿⣥⠤⠦⠠⠤⣤⣤⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣦⣤⡤⠤⠀⠦⢤⣹⣿⣆⠀⠀⠀⠈⣧
⠀⡾⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⢀⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣅⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⠀⢸⡆
⠀⣇⠀⠀⠐⠙⠛⠁⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠙⠋⠐⠀⠀⢠⡇
⠀⠈⠳⣦⣄⣀⠀⠀⠀⠛⠛⣛⣛⣻⣿⣶⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣾⣟⣛⣛⡛⠛⠃⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⠃⠀⠃⡀⠀⠀⠀⠀⠉⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⣿⣀⣜⢁⡼⠁⣴⠃⢀⡀⣀⡀⠀⠀⠀⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣼⡃⢀⣾⣿⣿⣧⣠⠄⢀⣈⡁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⣿⣿⣿⣥⣼⣿⣿⣿⣿⣿⣷⡤⠂⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠛⠛⠉⢻⣿⣿⣗⠃⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡹⡇⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡇⣟⠀⠀⠀⠀⢠⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠁⠀⠀⠀⠀⠈⠀⠈⢧
⠀⠀⠀⢀⣠⣴⣾⠿⠟⠓⠂⠀⠀⠀⠀⠉⠳⣦⣄⠀⠀⢀⠐⠄⠓⣀⣀⣀⣀⣀⡀⠀⠀⠐⠌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⣧
⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⠶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⢿⣷⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⠀⠀⣿⡀⠀⠀⠀⠀⠀⢀⣿⣿⣦⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢰⣾⣿⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣴⣿⣿⣿⣿⣿⣿⡿⢿⣷⡤⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢐⣿⣿⣷⣲⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⡐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢤⣿⣿⣿⣿⡿⣿⣿⠿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠲⠾⠛⣫⡿⠟⠛⠋⠀⠀⡸⣿⣿⣿⣿⣞⣿⣿⣷⣰⡟
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡟⠉⢁⣿⣿⣿⣿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⢀⢀⣠⣿⣷⣤⣼⣿⣛⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠀⠀⠪⢹⣿⣿⠃⠀⠀⠀⠈⠀⠀⠉⠛⠿⣿⣿⣿⣷⣶⣤⣀⣠⣤⣾⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠉⠋⢰⠟⠟⠁⠀⠀⠀⠀⠠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠛⠛⠿⠿⠿⠿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⣀⠰⡄⢳⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠐⠞⠐⢭⣠⣇⣼⡴⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣀⠼⠋⠙⠓⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⣠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢀⡜⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠠⠋⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣠⣤⠶⠶⢤⣤⣀⠀⢀⣠⡴⠖⠚⠉⠉⠒⠂
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠟⠉⠀⠉⠉⠉⠀⠉⠛⠯⠟⠉
⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠔⠚⡉⠁
⠀⠀⠀⠀⠀⢺⣇⠣⠀⠀⠀⢀⣠⣤⠞⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠹⢦⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⢶⣶⣶⣦⠤⠀⠀⠀⠀⠈⠉⠙⠛⠓⠂⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⣤⣒⡲⠤⣤⣤⣀⣤⣤⣶⡶⠶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣄
⠀⠀⠀⠀⠀⡞⠀⡰⠐⠂⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠂⠒⠰⡀⠘⡆
⠀⠀⠀⠀⡜⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠸⡄
⠀⠀⠀⢰⠃⠀⠀⢹⣠⠤⠤⠤⣤⡴⠚⠋⠛⠚⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠚⠋⠛⠲⣤⡠⠤⠤⢤⣸⠁⠀⠀⢳
⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⢄⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇
⠀⠀⠀⠿⣅⣀⣀⣀⠀⠈⠉⠁⠀⣠⡀⠈⠳⣄⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⡴⠋⠀⣠⡀⠀⠉⠉⠀⢀⣀⣀⣀⣽⠇
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠖⠒⠋⠁⠙⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠞⠁⠉⠓⠒⠶⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣿⣿⡻⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣿⣿⣶⣬⣙⠻⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣽⣿⣄⠸⠿⠿⣿⣿⣿⣿⣶⣤⡝⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣍⡻⣿⣿⡇⢉⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢿⣒⣂⠛⠛⠿⣿⣿⣿⣿⣿⣿⡯⢟⣰⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣍⡛⢻⠟⠀⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣟⡀⢤⣭⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⣿⣿⣿⣿⣿⢃⣾⠁⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⣿⣿⣿⣿⢃⣾⣿⢠⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣸⣿⣿⣿⠇⣾⣟⠿⠘⣿⡄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣷⡄⠹⣿⣦⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣦⣄⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡄⠿⢹⣷⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⠟⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣨⣵⣶⣾⣿⣿⣷⡄
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⠟⣹⣿⣿⣿⡟⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⣾⣿⣿⣿⣿⡿⠿⠿⠃
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣤⣾⣿⣿⣿⡿⢁⣾⣷⣦⣄⣉⡉⠛⠛⠛⣉⣩⣤⣶⣿⣿⣿⣶⣦⣤⣤⠄⢀⣴⣾⠆
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⠟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣉⣥⣶⣿⣿⠟
⠀⠀⠀⠀⣦⣙⠻⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠋⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣁⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠁⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠠⣄⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠖⠁⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠚⠁⠒⠂⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⣀⣀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⣀⠴⠒⠉⠉⠀⠀⠀⠀⠀⠀⢸⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠤⠖⠒⠢⠤⣀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣀⠀⠀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠑⠒⠋⠉⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠶⠒⠋⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢹⣟⢹⡖⢲⣤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⠉⣿⠛⢸⡿⣆⠀⠉⠑⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⡇⢸⣿⠀⢸⠁⠀⢄⣀⠀⠀⠀⠀⣝⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢤⣿⣇⢸⣿⠀⣸⡀⠀⣼⣿⠓⠂⣀⣠⠇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣆⠋⠁⢠⣀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠏⠀⠉⠉⠙⠋⠉⠉⠀⠘⢧⠀⠀⠙⢷⣤⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠙⢷⣄⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠙⢷⡀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠙⢦⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣹⣷⣄⢻⣧⡀⠀⠈⠲⣤⡀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠉⠉⠉⠉⠉⠛⢿⣿⡿⣷⣿⣿⣶⣄⠀⠈⠳⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⢶⣤⡉⠉⠛⠻⠶⣤⣤⣀⠀⢠⡄⣠
⠀⠀⣀⣤⡴⠶⠶⠿⠿⠿⠓⢶⣾⠟⣡⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⠛⠷⠄⠀⠀⠀⠉⠉⠀⢀⣼⠏
⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣼⣿⠟⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣷⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣿⣿⡋⠀⠀⢀⣀⣠⡴⠾⠛⠉⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⡿⢟⡽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠰⠟⠛⠛⠛⠛⠛⠉⠉⠁⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⡿⠿⠛⣉⠔⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠐⠒⠚⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠢⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠘⣄⠀⠀⠀⠀⠀⠠⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠔⠁⢀⡠⠤⠒⠢⢄
⠀⠀⠀⠐⠢⢤⣀⣀⣀⠀⠀⢸⠀⢡⠘⡆⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⣀⠀⡠⠔⠂⠀⠀⠀⠀⠀⡐⠁⠀⢀⡔⠊⠉⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢸⠀⠸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⣀⡀⠀⠐⠻⡄
⠀⠀⠀⠀⠀⠀⠂⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠂⠀⠀⠰⡄⠀⠀⠀⠑⠦⠤⠤⠤⠤⠤⠤⠒⠒⢄⠈⠂
⠀⠀⠀⠀⠀⠐⠠⠤⠤⠤⠤⠤⠶⠤⠄⠒⠁⠀⠀⠀⠉⠒⠀⠤⠀⠀⠄⠐⠂⠐⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢨⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢠⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠜
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⡞⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠀⡇
⠀⠀⡜⠀⠀⠀⢀⡠⠖⠒⠒⠂⠂⠉⠀⠀⠀⠉⠀⠀⢰⠀⠀⠹⡀⡠⠚⠉⠁⠀⠀⢠⡀⠀⠀⠀⢰⠇⠀⠀⡆⠀⣀⣀⣀⣠⠴⠒⠃
⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠁⠀⠀⢀⠀⠠⠊⠳⣄⠀⠀⢸⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⢀⡀
⠀⠀⢀⠀⢀⣀⡀⠀⠀⠈⢉⣀⠀⠀⢀⡈⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⣀⡠⠂⠈⡆⠀⠀⠀⠘⡆⠀⠀⠀⠀⠠⠤⠤⠂⡠⢄⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠱⡀⠀⠀⠀⠸⣄⣀⠀⠤⠐⠚⠉⠁⠀⠀⠀⠘⠀⠀⠀⠀⠙⠊⠁⠒⠒⠒⠒⠒⠉⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⠶⠚⢶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡄⠀⢀⠀⠀⠀⠀⠉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡼⠀⢀⡏⠀⢀⡟⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢰⠇⠀⣸⠁⠀⣼⠁⠀⢀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠾⣦⣤⣿⣀⣀⣿⡀⠀⠈⠛⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣠⡼⠋⠉⠉⣿⠛⠛⠁⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢸⡇⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣸⡇⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠻⢿⣷⣦⣀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⣀⣀⠀⠀⣿⣇⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⢻⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⣴⡾⠿⠛⠛⠛⠛⠛⠛⠿⢿⣦⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⣀⣀⣠⠔⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⣠⣾⣷⣄⡀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣀⣴⠟⠉⠀⣈⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠞⠋⠀⠀⠴⠚⠋⠉⠀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉⠁⠀⠀⠈⠛⠷⠶⣶⣶⠾⠿⠟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⠀⠀⠀⠐⠒⠒⠲⠶⠶⠶⠶⠶⠶⠄⠀⠀⠀⢀⣠⣴⡿⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠉⠛⠻⠿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣭⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⡀⠀⠀⢠⣾⣿⣿⡄⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣴⣿⣿⣿⣿⣷⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠙⣿⣿⣷⣾⡿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⣤⣤⣿⣿⣿⣿⣿⣿⠋⠰⠷⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡙⠻⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣌⡙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣼⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⡿⠿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠉⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠉⠙⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⢄⠀⣄⢘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⣤⠄⣜⠂⢨⠇⡼⢈⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢠⣤⠤⣤⣈⣓⣻⣾⠿⠋⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⢀⠔⠋⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠘⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⣤⣴⠶⠶⠶⣤⣀⠀⠀⢀⣤⠴⠚⠋⠉⠉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⢰⡯⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠀⠈⠉⠉⠒⠈⠙⠺⡿⠚⠁
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠠⣿⠰⣄⠀⠀⠀⠀⠀⣀⣤⠞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠘⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠈⠛⠲⠦⠤⠤⠤⠄⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⢯⣍⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢬⣭⣽⣿⡶⠾⠿⠛⠛⠋⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀



⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠈⡉⠢
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠱⡀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⢄⡀
⠀⢀⠊⠀⠀⠀⠀⠀⠀⢄⠀⠀⠰⣀⢬⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠁
⠀⢸⠀⠀⠀⠀⠠⡀⠀⠀⢢⠀⠀⡇⢠⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⡞⠀⠀⡀⠀⠀⠈⠢⡀⠀⣅⡠⠓⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⢠⠃⠀⠀⠈⠳⢄⠀⠠⠦⠊⠁⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⡜⠀⠀⠀⠀⠈⠂⢀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠃⠀⠀⠀⠀⢀⡴⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⢠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒
⠀⢇⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣠⠄⠒⠉
⠀⠈⢆⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢀⡇
⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠤⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⠀⠀⠀⠀⣀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠑⠢⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠒⠀⠀⢤⣀⠀⠈⠙⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡀⠀⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠰⠶⠀⢹⣖⡀⠀⠀⠀⠀⠀⠉⠛⠳⢦⡀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠶⠦⣄⡀⠀⠀⠀⣧⣴⠿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⣰⠟⠁⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠰⠋⠉⠙⠛⠛⠷⢦⣄⣀⠀⠀⠀⠀⣰⠞⠉⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣧⠔⠉⠓⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⠙⣁⣠⡤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⣸⣿⢻⡇⠀⠀⠀⠀⠀⢠⡿⠁⣿⠘⡇
⠀⠀⠀⠀⠀⢠⣿⠃⠘⡇⠀⠀⠀⠀⢠⡟⠁⢸⡇⠀⣧
⠀⠀⠀⠀⠀⣾⡏⠀⠀⠇⠀⠀⠀⢠⡟⠀⠀⢸⡇⠀⣿⡄⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⠶⠶⠶⠶⠶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⣸⡟⠀⠀⠸⠀⠀⠀⠀⣾⠁⠀⠙⢶⣷⡀⠘⢿⣦⣀⣠⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠿⣶⣦⣄
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠹⣧⠀⡀⠻⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢹⣤⡇⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣼⣿⡿
⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣠⣾⣿⣿⣿⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠶⠖⠛⠉⠉⠉⠙⢿⣿⣦
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢀⡆⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡴⠋⠀⠀⠀⠀⣰⡿⠛⠿⣶⣤⣄⣀⣀⡀⣀⣀⣀⣤⡶⠟⠋⠀⠀⠤⠤⢤⣴⣶⣶⣾⡿⠟⢿⣿
⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠀⠀⣸⣿
⠀⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⢀⣴⣿⡟
⠀⠈⢿⡌⠛⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⣤⣤⣤⣤⣶⣶⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⡢⠀⢈⡆⠠⡀⠄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣀⠀⣀⣼⣥⣤⣊⣠⣚⣡⣋⠼⠃
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢠⡟⠻⠦⣤⣴⣇⣈⣩⣭⠟
⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⡀⣰⡟⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⣷⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣤⣤⣄⡀⠀⠀⠀⠀⣀⣤
⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠇⢸⡇⢀⣀⣤⣤⣤⣄⣀⡀⠀⠀⣠⡾⠛⠉⠁⠀⠀⠈⠉⠛⠿⣶⣤⣶⠿⠋⠁
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠋⠉⠀⠀⠀⠈⠉⠛⢷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇
⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⣸⡏⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⢀⣀⣤⣴⠾⠋⠀⠻⣆
⠙⠷⠶⣶⣶⣶⡶⠶⠒⠒⠀⠀⠉⠙⠛⠛⠛⠉⠉⠻⢦⣄⡀⠀⠹⣧
⠀⠀⠀⠀⠈⠉⠛⠻⠶⣶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣴⡿⠿⢷⣶⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠀⢸⣷⠀⠀⠈⠙⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠐⣄⠀⠀⢈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡈⠳⣦⣄⣷⡈⠻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⢭⣙⢻⣄⠈⠙⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣍⠛⠻⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠛⠛⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⣀⣤⡤⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣸⣏⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⢰⡏⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⠶⠛⠛⠙⠛⠶⣤⡀⢀⣴⠞⠛⠉⠛⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⣤⣤⣤⣤⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠋⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⣤⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢠⣀⣦⡄⢸⠀⡆⠸⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡞⣹⠛⠧⠼⠤⠿⠾⠛
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⠀⡟
⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣀⣀⣀⣀⡤
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠘⣇⣀⡤⠤⠤⢤⣄⣠⡶⠟⠉⠁⠀⠀⠀⠀⠈⠙⠛⠋⠉
⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠉⠛⠦⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣀⣀⣤⡴⠾⠥⠀⠳⣄⡀
⠀⠀⠀⠈⢧⣤⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠐⣄⠀⠈⠙⣶⣄
⠀⠀⠀⠀⠀⠉⠒⠦⢤⣄⣀⣤⡔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣆⠀⠀⠘⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⢶⣶⠶⠶⠾⠟⠛⠉⠁⠈⢿⣦⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢹⣿⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠁⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⢲⡀⢣⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⢆⢠⠄⠀⢸⡄⠀⢣⠀⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢰⠚⠉⠀⠀⢸⡃⣀⠌⢀⣾⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣀⣿⣶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⢀⣷⣶⠿⠿⠿⠿⠿⠿⠛⠉⠉⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣶⣾⡿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣦⣄
⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾
⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠈⠻⠿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣴⡶⠿⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣡⣁⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡞⢠⠗⢚⣳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣀⡞⣠⢋⡴⠁⠉⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣄⣶⣿⣻⣦⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠉⠻⢿⣿⣿⣿⣿⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣏⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠂⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡯⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⠀⠀⠀⠀⠀⢻
⠀⠀⠰⠚⠋⠉⠉⠉⠉⠉⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⡠⠴⠒⠒⠒⠲⢦⣄⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⠀⠀⠀⠀⠀⠀⠀⠈⠑⣾⣿⣧⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⠧⠀⠀⠀⠀⠀⣠⠀⠀⠙⢿⣿⣿⠀⠀⣀⣀⣤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣶⣄⣀⣀⣀⣺⣿⣂⠀⠀⠀⠀⠉⠃⠀⠻⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣀⣤⠀⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⢿⣏⣛⣀⣈⣉⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠥⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠁⣠⠋⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠶⠚⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠊⠉⣉⣑⣒⣶
⠀⢀⣠⠄⠖⠁⠈⡁⠀⢀⣀⣲⣤⣄⣤⡀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠉⠉⠉⠉⠉⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣧⣤⡤⠴⠶⠖⠲⣶⣶⣦⡤⠤⠤⠤⠼⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠒⠚⠓⠶⠒⠒⠒⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠭⢖⡒⠤⣄⡀
⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⣄⣀⣀⣽⣄⣀⣿⣀⣀⣀⣠⣤⣴⠶⠶⠶
⠶⠶⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⣴⠋⠉⠉⡽⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣸⠁⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣧⠴⣦⣴⡿⠶⠟⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⡶⠞⠋⠁⠀⢠⡿⠁
⠀⣀⣀⣤⣶⠾⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠉⢁⡾⠁⠀⠀⠀⢠⣿⠃
⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄
⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣤⣀
⠀⠈⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠙⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡄⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠦⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡉⠀⢀⠄⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⡠⠘⠐⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⢳⡀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠠⠊⣀⣀⡤⠖⠚⠙⠓⠦⡀
⠀⠀⠀⠀⠒⠲⢤⣀⣀⣀⡀⠀⠀⢸⠀⠰⡀⢳⡀⠀⠀⠀⠀⡈⠀⠀⠁⠀⠀⠀⢀⡀⢀⡠⠒⠉⠉⠀⠀⠀⠀⠀⡔⠁⠀⠀⢠⠞⠉⠉⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠑⠄⣸⠀⠀⡇⠈⠇⠀⠀⠀⢀⠁⠀⠀⢠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠞⢧
⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⣀⣀⣠⠄⠀⠀⢠⡀⠀⠀⠀⠘⢦⣀⣀⣀⣀⣀⣀⣉⣁⡠⠤⢄⠈⠳
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠒⠒⠀⢀⣀⣀⣀⣁⠠⠂⠀⠀⠈⠠⠤⣀⣀⠀⠀⠈⠀⣀⡀⠀⢀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⣆⠽
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠠⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢠⣄⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡠⠊
⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⢀⡎⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⠶⠛⠉⠁⠀⠀⠀⠀⣼⠀⠀⠆
⠀⠀⢀⠞⠀⠀⠀⠀⠀⢀⣠⣄⣀⢀⡀⠴⠒⠒⠒⠒⠢⠀⠀⠀⡌⠀⠀⢧⠀⠀⣀⠤⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤
⠀⠀⡞⠀⠀⠀⠀⠰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆⡜⠁⠀⠀⠀⠀⠀⢈⣦⠀⠀⠀⠀⢰⡇⠀⠀⠘⣀⡠⠶⠒⠒⠒⠋⠁
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄⠀⠀⠀⢷⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⢀⣀⠔⠀⠚⠁⡈⠓⡄⠀⠀⠸⡇⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠐⠠⠄⢀⣀⣄⣀⡀⠀⠀⠀⣀⠠⠀⠀⠀⢄⠘⠆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠁⠀⢱⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠒⠂⠖⠊⣁⠴⢦⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢧⣀⣀⣀⠤⠐⠒⠋⠉⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠈⠒⠉⠑⠒⠒⠒⠒⠒⠋⠁⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣟⣻⣿⣝⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣭⣿⣿⣭⣭⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣷⣶⣿⣿⣏⢻⣿⣿⣿⣿⣿⣿⣿⣟⠉⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣌⣿⣾⣿⣿⠿⣋⣥⣴⣿⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣹⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⢿⣟⣉⣿⣇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣹⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⣤⣄⣄⣘⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣠⡴⡄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡿⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣽⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⣹⡇⡿⣼⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⠿⡇⢁⣉⣤⣿⣿⣿⣿⣿⠛⣿⡿⠟⠛⣿⣿⣿⣤⣿⣿⡇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⣿⣿⣿⠠⣿⠃⠀⢠⣿⣿⣿⣿⡿⡟
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀⠀⣿⣿⣿⣦⣿⣄⢀⣼⣿⣿⣿⣿⣷⣆⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠿⠿⠿⠝⠃⠀⠸⣿⣿⣿⣿⣟⣟⣿⣟⣻⣿⣿⢿⣿⣿⣶⡄
⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣏⣿⣿⣿⣶⣾⡿⠿⠿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⢣⠈⠙⠻⢿⣿⡿⣧⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄⠀⣸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⠾⠋⠀⠀⠀⠀⠀⠀⢿⡇⠈⠛⠳⠦⣄⣀
⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠀⠀⠻⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠴⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠓⠶⠤⠬⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠙⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⢲⣾⣠⣤⡀⡀⠀⠀⢀⢀⣷⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⣾⣿⠿⠿⣷⡧⣀⢹⣿⣟⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣼⣿⠃⠀⠀⠀⠙⠺⠾⠛⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠅⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢫⠋⠀⠀⠀⠀⠀⣰⡟⢹⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣴⡿⢡⠃⠀⠀⠀⠀⠀⢰⠏⢀⣾⡿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⡣⡞⠀⠀⠀⠀⠀⠀⠀⢀⢼⣿⠇⣀⣤⡶⠞⠛⠛⠋⠙⠛⠛⠛⠿⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣞⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⡶⣶⠶⠟⡛⠛⠛⠛⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⣿⣟⠂⠨⠂⠄⠀⡀⠀⠀⣰⡿⠃⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤
⠀⠀⠀⠀⠀⢸⣿⣿⣄⠁⠉⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄
⠀⠀⠀⠀⠀⠘⣿⣿⡻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⢀⠀⠀⠀⡀⠀⠄⠄⠀⠀⠀⡆⢢
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣎⠑⠀⠀⠀⠀⠀⠀⡀⢠⣤⡴⠠⢀⠀⠀⠈⠢⣙⠻⣿⣶⣶⣤⣤⣤⣤⣴⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣮⣥⣀⣀⣀⣀⣠⣥⣿⣾⡁⢋⠠⢀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠛⠿⣿⣿⣿⠿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⢿⣿⣿⣿⣦⣠⠁⠎⡠⢀⠄⡀⠀⠀⡀⠀⠀⣀⠛⠋⠉⡀⠘⣪⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣾⣤⣥⣮⣊⣄⣨⠊⠀⠊⡠⢪⢊⢈⣐⣵⡿⠅⠈⠳⣄⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⢿⣿⡿⠿⠿⠿⠟⢻⣿⡇⠄⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡌⡄⠀⠀⠈⢻⣦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠆⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿⣛⣿⡷⠾⣿⣍⣽⣿⢦⣶⡶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⣻⡿⠉⠛⠋⠉⠀⠀⣀⡉⠉⠀⠀⠛⠷⣾⠿⣦⣴⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣶⡾⠛⠛⠉⠀⠀⠀⠀⢀⣶⡿⠛⠻⣷⣦⠀⠀⠀⠀⠀⠸⢿⣬⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣥⡿⠃⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⣀⡀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠉⠙⣷⣶⣄
⠀⠀⠀⠀⣤⣶⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣿⣷⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣽⡇
⠀⠀⠀⣼⣏⣽⠃⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠉⣿⡏⠁⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⣧⣀
⠀⠀⠀⣹⡟⠋⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠛⠛⠛⠻⣿⡟⠛⠛⠛⠻⣿⠟⠂⠀⠀⠀⠀⠀⠀⢸⡟⢻⡆
⠀⠀⣾⢿⡧⠀⠀⠀⠀⠀⢠⠀⣀⣀⠀⣠⡀⡄⢠⠌⣛⡃⠀⢀⡀⣤⢀⠄⡠⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇
⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⢸⠀⣽⡇⠀⠱⡀⡇⢸⠀⡂⠇⠀⠐⡄⠀⢸⠄⡧⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦
⠀⠀⣸⣯⡀⠀⠀⠀⠀⠀⠈⠀⠉⠁⣀⣝⣥⣥⣼⣦⣝⣥⣤⣜⣥⣤⣬⡂⠙⠀⠓⠀⠀⠀⠀⠀⢸⣇⣽⠆
⠀⠀⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⢤⡤⣽⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⠄⠀⠀⠀⠀⠀⠀⠈⣿⠋
⠀⠀⠻⣿⡏⠀⠀⠀⠀⠀⡀⠀⢼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⡀⠀⠀⠀⢀⣾⢿⡇
⠀⠀⠀⣼⣷⣦⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠸⣷⣾⠃
⠀⠀⠀⠹⣧⣽⡆⠀⠀⠀⢻⣿⣿⡶⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠰⣾⣿⣿⠃⠀⠀⣤⣶⡏
⠀⠀⠀⠀⠉⠛⣧⣤⡀⠀⠈⠿⠛⢿⣦⣀⠀⠀⠀⢸⣿⣿⡀⠀⠀⢀⣠⣶⠟⠻⠏⠀⠀⣾⣏⣽⠇
⠀⠀⠀⠀⠀⠀⢿⣍⣿⡆⠀⠀⠀⠀⠙⠻⣿⣷⣶⣿⣿⣿⣷⣶⣿⡿⠟⠁⠀⠀⢀⣤⣴⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣶⣶⣤⡀⠀⠀⠀⠉⠛⠻⢿⠿⠛⠋⠁⠀⠀⢀⠀⣸⣿⣩⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣾⣿⣤⣴⣦⣤⡀⠀⣀⣤⣤⣀⢀⣶⡟⢻⣿⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢦⣿⠟⠻⣿⣥⣽⡿⠋⠛⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣯⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄
⣿⣿⣿⣿⣿⣿⡿⣫⣴⣾⣿⣻⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣻⣿⣿⣿⣆
⣿⣿⣿⣿⣿⢯⣴⣿⣿⣿⢟⣿⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢻⣿⣿⣟⣿⡋
⣿⣿⣿⣿⣿⠸⣿⣯⡭⣥⣞⣹⣶⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣭⠭⠕⠒⠻⠒⠊⠈⠙⠃
⣿⣿⣿⣿⠇⢠⣾⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣾⡏
⣿⣿⣿⡟⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⠇
⣿⣿⡿⠁⢰⣿⣿⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⠀⢸⣿⣿
⣿⡿⠁⣄⢸⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣻⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⡄⢸⣿⣟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀
⣿⢁⣾⣿⣿⣿⣿⣿⡿⣿⡿⠛⣽⣿⣿⣿⣿⣿⣿⣀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣾⣿⣿⣿⣿⣿⡟⠀⠀⢀⣴⣶⣿⣿⣶⣶⣆⡀⠀⠩⣿⣿⣿⣿⣷⣄
⡟⢸⣿⣿⣿⣿⣿⣷⣀⣵⡆⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣿⣿⣿⣿⣿⣿⠀⢾⣿⣿⣿⣿⣿⣷⣠⣴⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣿⣿⣿⣿⡆
⣿⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣭⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⣿⣿⣿⣿⣿⢠⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣭⣟⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⣿⣿⣿⣿⣷
⣿⣧⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢉⡍⠁⣤⡹⣿⣿⣿⣿⣿⠈⣦⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢩⡍⢀⣄⠻⣿⣿⣿⡟
⣿⣿⣭⣛⣛⣛⡛⠹⠛⣋⠡⠾⣿⣿⣿⣿⣿⡿⢿⣺⣆⢻⣆⢿⣿⣮⣉⣻⣿⣿⠀⠉⠉⠛⢛⡛⠋⠟⠛⡋⠡⠿⣿⣿⣿⣿⣿⡿⠷⠛⠊⣿⣌⣿⣷⡌⠙⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡿⠟⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⣷⡟⢳⡶⠴⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢳⡟⢃⣿⠷⠀⣿⣷⡌⠉⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⣿⠁⢸⣿⠀⢸⡿⠉⠃⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣼⣿⠀⣿⣿⠀⢸⡇⠀⠀⢳⣤⣄⠀⠀⠀⠀⢻⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⣿⡀⣿⣿⠀⣸⣧⠀⠀⣾⣿⡉⠉⢠⣄⣤⠞⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡻⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣶⡔⠋⠁⢀⣀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⠏⠉⠙⠛⠛⠛⠛⠛⠛⠁⠈⠻⣆⠀⠀⠙⢿⣦⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠙⢿⣷⡀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢸⣧⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣦⣤⣄⣸⣿⣧⡀⢻⣷⣄⠀⠀⠈⠛⢶⡄⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠁⠈⠉⠉⠛⠻⢿⣿⡿⣿⣷⣿⣿⣷⣤⡀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣬⡉⠙⠛⠻⠿⣷⣤⣄⣀⠀⠀⠀⢀⣀⡀⢘
⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣾⡿⢣⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠊⠙⠿⣷⣤⡀⠀⠀⠈⠉⠛⠿⠷⠀⠀⠉⣴⡿
⠀⣀⣤⠶⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⣿⡿⢋⣴⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⣀⣀⣀⣀⣠⣼⠏
⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠂⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⣿⣿⡏⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⣫⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣼⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠁⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⡡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠃⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠈⡉⠰⡀
⠀⠀⠀⠀⣠⠆⠈⠁⠀⠀⠀⠀⠱⡄⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⣤⣄⣀⡀
⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⡀⠀⠀⠘⡄⢈⡓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⠇⠀⠀⠀⠀⡀⠀⠀⠘⢄⠀⠀⠹⠀⡝⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⢸⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⢢⠀⡿⡀⠁⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⡇⠀⠀⠐⢤⡀⠀⠀⢸⡀⣠⠗⠊⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⢸⠁⠀⠀⠀⠀⢍⣃⡀⠈⠉⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⡎⠀⠀⠀⠀⠀⠀⢀⡔⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠁⠀⠀⢀⣀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠸⡀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢡⡀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢳⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡞
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢤⡀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢄⣀⠀⠀⠀⠀⢀⡤⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣰⣦⣶⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣽⣽⣽⣽⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣯⣭⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠛⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⣤⡶⠟⠛⠀⠘⢦⣀⣤⣰⣦⡀⠀⠀⣀⠀⢀⡿⣓⠂⣖⡂⠒⠀⢀⡀⠀⠀⣠⣶⣠⢄⣠⠎⠀⠘⠛⠷⣦⣄
⠀⠀⠙⠁⠀⠀⠀⠀⠀⣶⣿⣾⣿⡿⢶⠊⠙⠳⢴⡿⠻⣯⣌⠿⠋⠀⣶⠶⠛⠙⢲⠾⣿⣿⣼⣿⡆⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣭⠆⠀⠈⠀⢀⡄⠀⠀⠒⠈⠋⠀⠔⠀⠀⠀⣄⠀⠀⠀⠀⢮⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⣻⡷⡀⠀⠀⠀⠈⠁⠀⠀⠀⡰⢿⡁⠀⠀⠀⢰⣿⠃
⠀⠐⠈⠲⠂⠀⣀⣠⣴⣿⣄⢻⡇⠛⠦⣞⠋⠙⠻⠦⠤⡄⠁⣡⡤⠤⠞⠁⠈⢛⡦⠞⠓⡿⢃⣼⣷⣤⣀⡀⠀⠒⠀⠒⠄
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠗⢠⣾⣿⣿⡄⠀⠈⠠⠖⢥⣤⣬⠑⠦⠈⠀⠀⣾⣿⣿⣦⠐⠿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠘⣿⣿⣿⣷⠀⠀⠀⠈⠒⠀⠐⠂⠀⠀⢀⢸⣿⣿⣿⡟⠀⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠁⠸⣶⠦⠔⠈⠒⠊⠐⠤⢶⡿⠀⠙⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁

⣿⣿⠟⠿⠛⠿⡿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡄⢸⢱⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢤⠀⡀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⣴⣷⡀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠻⢿⡶⠀⣀⣴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂
⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠺⠀⢹⣿⠿⠑⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣱⣄
⠀⠀⠻⣿⣦⣄⣀⣀⡄⠀⠀⠀⠢⣄⣀⠀⡀⠀⠀⣀⣾⢻⡄⠓⠄⠀⠀⠀⠠⡔⠀⠀⠀⠀⠀⡉⠛⠳⠄
⠀⠀⠀⠈⣿⡏⠉⡉⠘⢧⣀⠀⠈⠁⢀⠀⠀⠀⢠⣿⡿⠈⣷⠀⠀⠀⠰⣤⣴⡇⠀⠀⠀⠀⣼⡷
⠀⠀⠀⠀⢹⣿⣴⣽⣤⡀⢿⣀⡀⠀⣼⠀⡔⢀⣾⣿⠇⠀⢻⡷⣄⠀⠀⠈⢻⡇⠀⣠⣴⣾⡿⠁⢀⡄
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣌⣻⣧⣰⣿⣾⣾⣿⣿⠟⠀⠀⠘⣷⡽⣿⣗⣀⣠⠓⠛⠋⣭⠗⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠘⠻⢮⣍⠉⠁⠀⣴⡈⣡⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠛⠉⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⡤⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⢤⣀
⠀⠀⢠⠎⢰⡎⡂⠀⡙⠆⠀⠀⣀⠤⣄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣠⠤⢄⡀⠀⠀⠰⢋⠀⢱⢰⡆⠱⡀
⠀⠀⢰⣀⠈⠧⢰⠠⠼⡾⠶⠺⣆⠐⠋⡟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⡙⠒⡲⠛⠒⠶⢷⠧⠀⡎⠼⠃⡀⡆
⠀⠰⠾⣯⣀⠂⢈⠄⠑⢱⠆⠀⠀⢙⡆⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡜⠠⡏⠀⠀⠀⠐⡎⠊⠠⢁⡐⣀⣼⠷⠆
⠀⠀⠀⠘⠌⠷⣮⡏⠦⠞⠀⠀⠀⠀⡄⠀⠹⡀⠐⠀⠀⠀⠀⢠⣶⣶⣶⣄⠀⠀⠀⠈⠀⢄⡞⠀⢠⠀⠀⠀⠀⠀⠳⠤⢱⡕⠼⠡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠁⣦⡀⠀⠀⠀⣹⢀⡀⣀⢹⠀⠀⠀⠀⣴⠊⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠘⠁⠒⣢⠀⠘⠂⠀⠀⡟⠀⣠⠖⠚⠃⢀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⠀⠀⠀⠱⠗⠐⠛⣂⠞⠓⠾⠏⠀⠀⠠⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠉⠃⠀⠀⠀⠐⠉⠁⠀⡀⠠⠊⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢀⠀⠀⠀⣴⠀⠀⠀⣀⡀⡕⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡳⣽⠲⢚⣥⣝⠒⢿⠟⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢸⠅⢤⠀⢏⢁⠜⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⠁⠒⣺⠒⠊⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠀⠛⠒⠂⠛⠐⠒⠚⠃⠤⠄⣀⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⢣⠀⠀⠀⢀⣀⡀⠤⠤⠤⠤⠄⠀⣀⡀⠀⠀⠀⡜⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⣒⠮⠑⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠞⣀⠀⠀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢠⠀⠀⠀⠀⢀⠠⡜⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢼⢷⣤⡀⠀⡞⢰⠇⠋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠂⢀⠞⠉⠀⣰⠁⢨⠀⡃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠠⢀⣶⠊⠀⠀⠀⠈⠻⠩
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⣠⡴⢻⡟⡄
⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠰⠉⠀⢰⣾⡇⡆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⢀⡴⠃⢀⡤⢨⡿⡑⡠⠊⠁⠀⠀⠀⠉⠢⣤⠂⠉⠀⠀⠈⠁⠐⠠⡀
⠀⠀⠀⠀⠀⣼⢤⣀⠀⠀⢠⠞⣱⣿⡾⠁⡍⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⢠⠉⠀⠀⢘⣆⠭⢛⣡⠋⢀⡤⢧⡀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣦
⠀⠀⠀⠀⠘⢷⣄⡈⠉⠒⠛⣫⣴⠖⢋⣀⣈⡛⢿⣿⣿⣿⡿⢿⣛⣉⣤⣽⡇
⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣤⣤⣭⡭⠶⠛⠉⠁⠀⠀⠀⠉⠁⠛⠒⠒⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢟⠿⣥⣄⣀⣀⣠⣤⣴⣶⣶⣿⣿⣶⣶⣶⣦⣶⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⠛⠿⢛⣿⠿⠿⠟⣛⣛⠻⠿⣿⣿⡿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠈⠑⠦⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡷⢋⠙⠉⠀⢀⡬⠁⠈⠋⠽⡛⡲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡿⢁⣠⠤⠂⠀⠈⠑⠒⠀⠀⠀⠐⠬⣩⣽⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣧⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⣿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⡄⡘⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟⢱⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣠⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣣⠋⠀⠀⠄⠀⠀⠀⠀⠀⢰⠀⠀⢠⠀⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⢋⡆⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⢧⣿⣇⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⡿⢾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⡴⢾⣿⡿⠿⠛⣉⢁⣼⣵⣴⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⡗⣸⣿⣿⣿⣿⣿⣿⣿⣷⢀⡀⠀⢀⣼⠃⣽⢋⠇⠀⠰⡏⣸⣿⡟⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣸⢿⣿⡟⠸⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⡿⣿⣿⣟⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠜⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣷⣿⣷⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡎⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⢹⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣎⢿⣿⣿⣿⡿⠿⣿⣟⠻⣿⣧⣶⡄⠀⠀⠀⠀⢾⢟⣫⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠈⠸⢿⣿⣿⣿⣿⣹⢿⣿⣿⣿⣿⣿⡛⠛⠋⠥⣤⡤⣀⠀⠰⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡝⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣁⡛⠋⠠⠬⠐⣣⡐⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀⢘⣓⣀⡈⠱⠶⠀⠶⢿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⢈⡀⡱⢿⣿⣿⣿⣇⣻⣿⡿⣝⣿⡿⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠛⠋⠉⠛⠻⠿⠿⠿⠶⠖⠀⢀⣀⣈⣤⣥⣴⣼⣿⣿⡿⢿⡾⡿⢽⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⢿⣿⣿⡷⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠠⣈⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠐⢄⠈⠳⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣠⣶⣦⣈⣷⡶⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢀⣴⣶⣶⡾⠟⠉⠛⠛⠉⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠅⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡎⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡞⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣶⣶⣤⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡜⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠿⠛⠛⠉⠉⠉⠛⠻⢿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠁⠀⠀⠀⠀⢀⠄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣼⢸⡏⠀⠀⢀⣤⣶⣾⣿⣿⣷⣦⣀⡴⠛⠁⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢰⡏⣼⣃⣀⣴⠟⠋⠉⠀⠀⠀⠀⠉⠙⠳⠦⢤⣀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⡾⠁⠋⢉⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠤⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠀⠀⠀⠀⠀⠀⠀⡀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠶⠶⠶⠚⠛⠉⠁⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣮⣙⠛⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠂⣠⠊⠁⠀⠀⠀⢦⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠛⠉⣁⣴⠞⠁⠀⠀⠀⠀⠐⡌⢷⡀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣟⣻⣿⣿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠟⠋⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠔⢂⣤⣄⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⣾⣟⡛⢾⣩⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡀⠿⣿⠿⣣⣧⡤⠒⠒⣶⣶⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⣾⢟⣿⣿⣿⣶⡶⠶⢯⣟⡿⠋⢢
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢽⣿⣿⠛⣛⣻⡿⠁⣢⣛⣦⡈⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⡇⢹⣿⡇⣩⣿⣧⡆⢀⡿⡙⣿⢷⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⢸⣿⠛⡿⢿⡉⠉⠻⣿⣷⣿⣟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⠄⢳⣮⣿⣷⠞⣋⣾⣯⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡟⡿⣾⣿⣿⣟⣉⠬⠟⢫⣟⢶⡂⠲⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡈⠛⢻⡿⢿⡆⠀⢸⠙⣾⡧⠤⠽
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠠⠀⣀⣸⣧⣾⢹⠐⠙⡇⠈⢷⠖⠂⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⢀⡞⠁⠀⠀⠉⢿⣄⣠⡙⣦⣬⡿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣾⡤⠤⣄⠀⠀⠀⠻⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⡀⠸⡄⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢉⣿⠀⢸⣆⣆⣰⡅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⢻⠀⠀⠻⡇⠀⠀⠠⣠⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⣷⣤⠀⠁⠲⠄⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡐⠀⠀⢘⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⠿⠶⠖⠊⠈⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⡀⠀⠀⢠⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⢿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣯⣷⢾⣿⣿⡿⣾⣽⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠄⠀⠀⠀⡀⣀⣠⣓⡿⠋⠀⠀⣪⣕⠀⠀⠙⢿⣚⣄⣀⣀⠀⠀⠀⠠⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⣶⡶⠟⠛⡛⠛⠛⠿⣦⢔⡽⢞⡳⢯⡢⣴⠿⠛⠋⢛⠛⠻⢶⣴⡗⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢿⣷⣦⠰⠸⢰⢀⢯⠞⢀⣨⣅⡀⠳⡽⡀⡂⡇⠆⣴⣾⡿⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠠⢈⡛⠻⢆⣤⠼⠮⣿⠞⢋⣉⣉⡙⠳⣿⠵⠧⣬⡰⠟⢛⡁⠄⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢾⣿⠀⢒⠒⣁⡟⡞⣭⣿⢋⡾⣯⠖⠲⣽⢷⡝⣿⣭⢳⢻⣈⡐⡒⠀⣿⡷⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⣵⡿⠋⠻⣧⡤⣔⣬⣧⡿⢛⡚⠈⠑⠩⠋⠙⠍⠚⠁⢓⣛⢿⣼⣵⣢⢤⣼⠟⠙⢿⣎⢦
⠀⠀⠀⠀⠀⠀⠀⣜⣽⣟⠀⠀⣔⡵⠚⢁⡾⢁⡞⣭⡽⡀⣪⣶⣆⢰⣶⣅⢀⡯⣭⢳⡈⢷⡈⠓⢮⣢⠀⠀⣻⣏⣧
⠀⠀⠀⠰⠶⠶⠶⡷⣿⣿⢎⢾⢾⡱⠆⢾⡇⢸⢸⡱⣏⠀⢈⣙⢾⡷⣋⡁⠀⣹⢌⡇⡇⢸⡷⠰⢎⡷⡷⡱⣿⣿⢾⠷⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⢻⣻⣯⠀⠀⠫⡳⢤⡈⢷⡈⢧⣙⣲⠁⢜⢿⠏⠸⡿⡫⠈⣟⢛⡼⢁⡾⢁⡤⢞⠝⠀⠀⣽⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡻⣷⣄⣴⡟⠓⠫⢟⡟⣷⣭⢥⢀⡤⣔⣆⣠⢦⢤⡀⡬⣭⣾⢿⡛⠝⠚⢻⣦⣠⣾⢟⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢼⣿⠀⠬⠬⡉⣧⢧⣛⣿⣜⢧⣝⠦⠵⣛⡼⣣⣿⣛⡼⣼⢉⠩⠥⠀⣿⡧⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠐⢈⣥⣴⠎⡛⢲⢖⣿⢦⣌⣉⣉⣡⡴⣿⠲⡖⢛⠱⣦⣬⡁⠂⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣼⡿⠟⠰⢸⠸⠈⣞⢦⠈⢙⡋⠁⡰⣳⠁⠆⡆⠆⠻⢿⣷⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⠿⠷⣦⣤⣡⣠⣤⣶⠟⠪⣳⢮⡵⣞⠕⠻⣶⣤⣤⣨⣤⣴⠾⠿⡧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠂⠀⠀⠀⠉⠉⠙⡍⣷⣄⠀⠀⢝⡫⠀⠀⣠⣾⢽⠋⠉⠉⠀⠀⠀⠐⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣟⡿⣾⣿⣿⣷⢿⣻⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠃⠈⠉⡉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠰⡀⠘⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⢄⠀⢡⠀⢱⠀⣷⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢳⣤⠤⠖⠯⡤⠞⠖⠛⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠰⠋⣿⡇
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⣤⣶⠶⠶⠶⠶⣦⣄⣀⣤⣾⠿⠛⠉⠀⠀⠀⠤⠛⠉⠁⠀⠘⣿⡄
⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣀⣀⣀⣀⣤⡤⠔⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠈⠻⣿⣷⣤⣤⣀⣀⣀⣀⣤⣤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⠀⠀⠀⠀⠈⢻⣷⣦⣤⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠙⢿⣷⣤⣀⡀⠀⠀⣹⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠆⠤⠀⠀⠀⠆⠀⠄

⠆⠀⠈⠀⠈⣀⠁⠀⢠⡄⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⢀⣀⠀⠀⣀⠀⠀⠁⠈
⣀⠀⠀⠀⣴⡄⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣶
⠉⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠦⡀⠀⡀⠀⠀⠠⡀⢀⡴⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⠀⣄⢹⡏⢀⠀⠀⢀⠀⡿⠀⠖⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢤⣶⠿⠶⣿⣾⡁⡮⢀⠀⢸⢀⣿⣿⡶⠿⣶⣧⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠲⣄⠀⠁⡄⣀⡀⠀⢠⣴⣼⣷⣄⣿⣯⢿⣷⣾⣧⣼⡿⢹⣿⣁⣼⣯⣤⣤⠀⠀⣀⣠⠀⠀⣀⠴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠐⡂⢀⣼⡦⢰⡆⠀⠁⢠⠈⠛⣯⢙⣿⡿⢃⣼⡿⣿⠿⡿⣷⣈⠿⣿⡏⣹⠟⢁⡄⠀⠀⠁⣤⠤⣿⣀⠀⣀⠘
⠀⠀⠀⠀⠂⢦⡄⠁⠀⠛⢿⣥⣤⢢⣤⣀⠀⣀⠀⠀⣿⣾⢋⣴⠏⠁⠀⠀⠀⠁⠀⠙⣷⡘⢿⣟⢰⠀⣀⠀⢀⣤⣤⣤⣤⡝⠛⡀⠀⢠⣤⠐
⠀⠀⠀⠀⠀⣼⢧⣤⡾⠉⠉⠛⠣⠀⣥⡀⠹⢯⡟⢡⠿⣷⢿⣿⠀⠀⢀⠀⠀⠀⠐⠀⣸⡿⢹⡿⣮⠑⣿⠻⠁⣠⡍⠰⠛⠋⠈⠳⢦⣌⣿⡄
⠀⢀⠿⠀⢾⣛⠟⠉⣶⠀⠀⡈⠱⣦⣹⡿⠃⠀⢿⠀⢐⡾⠀⣿⡇⠀⣤⡀⠀⣰⠄⢀⣿⠇⠨⣷⠀⠰⠏⠀⡿⢫⣼⠆⢀⠀⠀⣼⡆⠙⢿⡻⠦⠾⠂
⠀⠀⠓⣾⠗⢡⠴⢿⣏⣶⣌⣻⣄⣸⡿⠃⠂⠀⣷⣄⡀⠄⠄⢸⣇⡀⠙⠛⠛⠛⠀⣠⡿⠀⠄⢀⣀⣴⡆⠀⠀⢻⣿⣀⣼⣷⣲⣏⠿⠶⢄⠙⢶⡖
⠀⢀⣴⣄⣀⡘⢒⣶⣿⢿⢟⡉⠙⣿⣇⠘⣦⣀⣭⠙⢑⠶⠒⠀⠭⠻⣶⡆⣠⣴⡾⠯⠅⠐⠶⣊⠛⢡⣇⣰⡆⢀⣽⡏⠛⣻⡿⠿⣶⣖⣈⢀⣤⣴⡀
⢠⠉⢻⡴⣯⣿⠟⢯⣀⡑⣺⣿⣶⣿⣟⣷⣬⡛⠞⠛⡉⠠⡈⢂⡠⢈⣻⣷⢿⣿⣁⠀⠤⠋⠀⠉⡛⢣⣟⣣⣶⣏⣿⣷⣿⣟⠓⣀⡉⠻⢿⣿⣶⠞⠛
⠛⢸⣿⡿⠋⠁⠒⣶⣼⣿⣿⡿⠿⠿⢿⣶⣿⣿⣇⠀⣿⠀⢠⣼⣧⣿⣿⠏⠘⣿⣿⣶⣧⣦⠀⣾⠟⠀⣿⣻⣷⡾⠿⠿⠿⣿⣿⣧⣴⠆⠈⠙⣿⣿⣽
⡿⠿⠉⣀⣴⡾⢿⣿⣿⣿⣿⡀⢰⠛⠂⠙⠛⣿⠿⣆⡈⠣⠀⠉⢻⡏⢀⠀⠀⢨⠈⡿⠋⠀⠠⠂⣠⠿⣿⡻⠏⠀⠛⣷⠀⣸⣿⣿⣿⣿⠿⣦⣄⠀⠙
⠃⣀⡴⠋⠀⠀⠀⠀⠉⠙⠻⢷⣄⠓⠲⠾⠿⣿⠄⠈⠿⠷⢦⣄⠸⢇⣺⣧⣴⣿⣀⠛⢀⣴⡶⠿⠇⠀⣿⡿⠷⠦⠒⢁⣼⠿⠋⠉⠀⠀⠀⠀⠙⠳⣆
⣾⠏⠀⠀⠀⠀⠀⠀⠈⠑⠲⣄⡉⠂⠀⠀⠀⠹⣦⡐⣧⡄⠀⠙⠷⡈⠻⣯⣿⡟⠋⡴⠋⠁⢠⣤⡄⣠⡿⠁⠀⠀⠐⢊⣡⡴⠂⠁⠀⠀⠀⠀⠠⠀⠙
⣿⠀⠀⠀⣠⡶⠟⠛⠋⠷⣦⡈⠻⢷⣤⣤⡄⠀⣿⣉⣤⡄⢀⠀⠀⠉⠀⠘⠟⠀⠘⠀⠀⠀⠀⣤⣌⣿⡇⢀⣤⣤⡶⠟⢁⣤⠶⠞⠛⠛⠶⣤⡀
⣇⠀⢀⣼⡏⠀⠀⠀⠀⠀⠈⣻⣦⣄⣨⢻⣧⠀⠁⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠁⢰⡿⣏⣀⣴⣿⡇⠀⠀⠀⠀⠀⠘⣷⣄
⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣹⣿⣯⣍⠈⠁⠀⠿⠶⠖⠲⠞⢿⡆⠀⠀⠀⠀⠀⠀⢠⡿⠗⠚⠒⠲⠾⠇⠈⠁⢉⣵⣿⣿⡇⠀⠀⠀⠁⠀⢀⣾⣿⡿
⣿⣄⠐⠏⠻⣤⢀⣀⡀⣠⡼⠟⠉⠉⠉⠀⠀⠀⣴⣀⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⣢⠀⠀⠀⠈⠉⠉⠙⠷⣤⠀⣀⡀⢤⠞⢫⠟⢀
⣿⠉⠀⠀⠁⠐⠊⠉⠉⠁⠀⠀⠀⢀⡀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠙⠀⠈⠀⠀⠁
⢿⣶⣾⠷⣦⠀⠀⠀⠀⣀⣀⠀⠈⠛⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠙⠁⠀⣀⣀⠀⠀⠀⠀⢰⡶⢿⣶
⣿⣿⠁⠀⠉⠀⢀⣴⣿⣿⣿⣤⣴⣶⠤⣀⢀⣀⡀⠀⠀⠀⠀⠀⠠⢠⡤⠀⠀⠤⡤⠘⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⢴⣦⣤⣿⣿⣿⣶⣄⠀⠘⠃⠀⣻
⣿⣿⣧⣀⣀⣴⣿⣿⠋⡙⠉⠉⣉⣁⣤⣤⡄⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠁⣤⣤⣄⣉⡉⠉⠉⠙⢿⣿⣦⣀⣀⣴⣿
⣿⣿⣿⣿⣿⣿⠿⠃⠈⢙⠛⠟⠛⡉⠉⠀⠀⠙⡆⠀⠀⠀⠀⢀⡀⠀⢀⣘⢓⣀⠀⠀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⣛⠛⠟⠛⠁⠀⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠿⣶⠿⢨⣇⠀⣆⠘⣧⠀⠀⠀⠀⠈⠓⠈⠈⠉⠉⠉⠁⠘⠋⠀⠀⠀⠀⢀⡏⢠⡆⢨⡧⠸⣷⡾⠋⠀⠀⠀⠀⣾⣿⣿⣿
⣸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠁⠘⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠛⢩⣿⣿⣿⣷⡀⠀⠀⠀⠀⠂⠀⢤⡿⠿⢶⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣶⠾⢿⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋
⠀⡿⢸⣿⢿⣿⣿⣄⠀⠀⠀⠈⠀⠸⠀⠀⠀⠹⣿⣿⠋⣂⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡏⠣
⣤⠄⣸⡟⢸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠂⢹⣷⣮⡅⠀⠀⠀⠀⠀⠀⠀⣥⣾⣟⠀⠿⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⢸⣿
⢏⣀⣿⣇⢸⣿⣿⣿⢿⣿⣷⣄⢠⣤⣦⣶⠀⠀⠋⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⢿⣟⠀⡷⠀⠃⠀⣤⣦⣤⣤⣀⣴⣿⣿⢿⣿⢠⣿⢸⣿⣦
⠜⢻⣿⣿⠀⣿⣿⠏⠀⠋⣽⣿⣿⣿⠋⠉⠀⠁⡄⢱⣧⠐⠁⠀⠀⡄⠀⠀⠀⠀⠀⡆⠀⠀⠻⢰⣧⠀⠀⠈⠉⢩⠟⣿⣿⣯⣍⠉⠘⢿⣼⠛⣼⣿⣏
⠀⢿⣿⡏⢻⡧⢤⣤⣤⠸⣿⣿⢻⣿⣦⣁⣀⡀⢰⣿⣿⠇⠀⠀⠀⣻⠀⠀⠀⠀⣼⡀⠀⠀⠸⣧⣿⣧⠀⣀⣀⣠⣿⡟⢿⣿⡇⢰⣤⣤⠼⡾⢻⢻⣿
⠀⢸⣿⡀⠀⢹⣦⣿⡍⢶⡟⢡⡞⠹⣏⣉⠉⠁⢸⣿⠛⣷⠻⣦⡀⠉⠀⠀⠀⠀⠈⠁⣠⠾⣳⠟⢿⡇⠀⠉⣉⣉⡟⢳⣆⠙⣿⢴⣽⢷⣾⠁⠀⣼⡿
⡀⠀⢹⠇⠀⢾⣿⢶⡋⣾⣗⣾⡇⠀⢸⡟⠀⠀⣼⡿⠀⢘⣷⡄⢁⣀⣤⠀⠀⣀⣠⡌⢁⣶⣋⠀⢸⣿⠀⠀⠘⣟⠀⠸⣿⢷⣿⢿⣯⢿⣿⠁⠰⠘⠇
⠃⠀⠈⠀⠀⠸⣿⣸⣿⣾⡿⠟⣥⡜⠋⠉⠀⠐⡙⠷⠀⠛⠛⢿⠀⣿⡟⢷⣴⢟⣿⠁⣾⠟⠉⠓⠈⠻⠇⠀⠈⠛⠁⣤⠿⠿⣯⣾⡏⣼⡿⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣿⣿⢸⡏⣷⡤⣤⣄⡀⣀⠠⠊⠁⠀⣿⠃⠀⡀⠃⠘⢇⡀⠀⠰⠟⠀⢸⠀⠘⣿⣦⠀⠁⠄⢀⣀⣠⣴⠆⣴⣿⡟⠻⣿⡄⡄⢸⠁
⠀⠀⠀⢀⠀⢀⣸⡏⢸⠁⣿⣆⢸⣿⣧⠈⣀⠀⣰⡿⠛⢣⣤⣀⠀⠀⠈⣡⣀⠃⠀⠀⣀⣠⣄⠛⢼⣷⠀⣄⠁⣸⣿⣿⠰⣿⢽⡇⢸⣿⡅⠀⠈
⠀⠀⠀⠘⠂⣼⣿⠁⠘⣆⠹⣧⣿⠁⢿⡆⢹⣇⠙⢀⡄⠀⠀⠀⠀⠀⠂⠉⠋⠀⠀⠀⠃⠀⠀⠀⣀⠙⢡⣿⠀⣿⠁⢿⣷⣹⠟⠀⡌⢹⣿⡆⢠
⠀⠀⠀⠀⣸⡟⡇⠃⠀⠈⢠⣿⡏⠀⢸⡇⢸⣿⣧⠈⠀⠰⠀⣴⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⣰⣿⡇⠀⡿⠀⠈⢿⣯⠀⠂⠀⡴⢙⣷⡈
⠀⠀⠀⠴⣟⠃⡘⠀⠀⢀⣿⠀⠀⠀⠈⠀⠸⣏⠹⡇⠀⠀⠀⣿⠀⠘⠻⣶⣶⡿⠃⠀⣶⡄⠈⠀⢠⠟⠉⣳⠀⠐⠀⠀⠀⢹⣆⠀⠀⠡⠊⡼⢷
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⣠⠀⢠⠄⠁⠀⠀⠄⠀⠀⢘⣿⣷⣄⠀⠀⠀⠀⠀⣰⣿⣟⠀⠀⠘⠁⠀⠈⠀⣰⠆⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠉⠱⠦⠀⠒⠀⡄⠀⠀⣠⡿⠀⠹⣿⣶⣄⣀⣴⣾⠟⠀⢹⣦⠀⠀⠀⡄⠘⠂⠠⠶⠉⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠿⠧⠀⠀⠘⠿⠃⠈⠟⣯⠀⠀⠸⠿⠇⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣤⠀⠀⠀⣀⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⣀
⠀⠀⠀⠀⠩⢡⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠨⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠿⠛⣫⣭⡁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠀⠀⠀⢀⣾⣿⠏⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⠸⣶⡄⠀⠀⣂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡏⠀⠀⢻⣿⣦⡀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇⠀⢀⣼⣿⣿⣿⢄⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣥⡤⣼⣿⣿⣿⡿⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⢱⣓⣾⣿⣿⣿⣀⠶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣤⣭⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⣿⣿⣿⣿⡿⡛⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⡀⠀⠀⣾⢲⡞⣺⢣⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⡿⢸⡇⠀⣼⣿⣽⡶⠉⠚⢿⣿⣟⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢺⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣽⣿⡟⠀⠀⠀⠘⣿⣿⣽⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣸⣿⡟⠀⠀⠀⠀⠀⠘⣿⣿⣫⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣽⡟⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡽⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣷⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢤
⠀⠀⠀⠀⢠⠃⠀⡖⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠐⡆⠀⢣
⠀⠀⠀⢀⠏⠀⠘⡇⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⡟⠀⠈⢇
⠀⠀⠀⣼⠀⠀⠀⡷⠖⠚⠒⠒⠶⠋⠁⠈⠉⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠉⠀⠉⠳⠖⠒⠒⠒⠶⡇⠀⠀⢸⡄
⠀⠀⢠⣿⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠐⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠁⠀⠠⢤⣤⡤⠄⠁⠀⠀⣼⣿
⠀⠀⠀⠛⠶⠤⣤⣆⡀⠀⠀⣀⡴⢧⣀⠈⢳⣶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣤⡉⣀⣠⢫⣄⣀⠀⢀⣰⣶⣴⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠻⣿⣿⣧⡀
⠀⠀⠀⢠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄
⠀⠀⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦
⠀⢠⣿⣿⣿⣿⣿⣷⣤⣶⣶⣦⣴⣿⣿⣿⣿⣿⣿⣦⡀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣤⣶⣶⣦⣸⣿⣿⣿⣿⣿⡆
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢺⣿⣄⣀⡀⠁⠀⠀⠀⢈⣧⣄⣀⣀⡀⠀⠀⠀⠀⠀⠈⢀⣀⣤⣿⡗⠀⠀⠀⡀
⠀⠀⠀⠀⣠⠀⠀⠀⠀⣰⠏⠀⠈⠉⠛⠃⠀⠀⠀⠸⡇⠉⣍⡉⠁⠀⠄⠀⠀⠀⠘⠛⠉⠁⠀⠹⣆⠀⠀⠈⠀⣄
⠀⠀⠀⣸⠃⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣶⣴⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⢆
⠀⠀⢸⢧⣀⠀⠀⣛⠛⠁⠀⠀⢤⡄⠀⠀⢀⣠⡴⠺⠉⢿⠦⠤⠒⠀⠐⠀⢄⡀⠀⠀⢠⡤⠀⠀⠈⠛⡛⠀⠀⣀⡼⣇
⠀⠀⠈⢻⣭⣀⡈⠉⢷⡀⣀⣀⡀⠳⣆⠀⠀⠈⠉⠀⠀⡀⠣⠄⠀⠀⠀⠈⠀⠀⠀⢀⠈⢀⣀⣀⢀⡾⠁⢀⣀⣭⡟⠁
⠀⠀⠀⠀⠙⣷⣦⣀⠀⠉⣋⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣛⣻⣿⣙⠉⠀⣀⣴⣾⠋
⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⢿⣿⡟⠁⠙⣷⣄⣀⡀⢀⣀⣠⣄⣀⡀⢀⣀⣤⣾⠋⠀⠀⠈⡿⠿⠿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠠⣟⣛⠟⢩⠝⠉⠀⣆⠉⠉⠙⠋⠙⣺⠄⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⢹⣭⠇⠀⡷⠖⠋⠀⡙⠛⠦⡀⠀⠀⣭⡋⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣾⠀⠀⣷⣄⣀⠀⣧⣀⣀⠀⠀⠀⣷⣥⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠈⣏⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿⠛⠓⠀⠟⠒⠚⠃⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⢹⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢧⠀⠆⠀⠀⠀⠀⠀⠀⡼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣯⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣄
⣿⣿⣿⣿⣿⡿⢫⣶⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣽⣿⣿⣿⣦
⣿⣿⣿⣿⡟⣴⣿⣿⣿⢏⣽⠿⠟⠟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡿⢫⣿⠿⠯⠿⣥⡀
⣿⣿⣿⣿⡇⢹⣿⡟⣷⣷⣭⣽⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⠟⠋⠉⠉⠉⠁
⣿⣿⣿⡿⠀⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠁⢠⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠀⣤⣿⡟
⣿⣿⠉⠀⣸⣿⣇⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⠀⣿⣿⣇⡀
⣿⠃⣼⣷⣿⣿⣿⣿⣿⣿⡿⢟⣿⣿⣿⣿⣿⡟⠉⠑⢚⣿⣿⣿⣿⣿⣿⣿⠀⢀⣾⣷⣿⣿⣿⣿⠀⠀⠀⣠⣤⣤⣦⣤⣄⡀⠀⠒⣻⣿⣿⣷⣦⡀
⡟⢸⣿⣿⣿⣿⣿⡏⣘⡭⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣽⣻⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣍⣀⡄⢀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣮⣽⣻⣿⣿⣿⡆
⣷⠘⣿⣿⣿⣿⣿⣿⣿⣷⣦⣝⣛⣻⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⢀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣤⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷
⣿⣧⣈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣋⠉⢁⡘⢿⣿⣿⣿⣿⠘⣆⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⠉⣁⡹⣿⣿⣿⡿
⣿⣾⣍⣛⣛⣛⠋⠟⢛⡩⠼⣿⣿⣿⣿⣿⡿⢿⣶⣘⣧⡸⣿⣦⣙⣻⣿⣿⠀⠙⠝⠛⣛⡛⠹⠟⣛⠩⠾⣿⣿⣿⣿⣿⡿⠷⠶⢹⣦⣻⣿⣮⠙⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡶⠿⠿⠿⠿⠿⠿⠿⠷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠰⠿⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠛⠛⠿⣦⣄⠀⠀⠀⣠⣴⣿⠿⠿⠿⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⣼⡟⠋⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠻⣆⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣤⣤⠶⣾⣿⠋⠀⠀⠈⠻⠶⣤⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠋⠀⠀⠀⠀⠠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣀⣀⣀⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣶⣦⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣶⣾⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡤⠄⠤⠠⠠⠀⠤⠀⠤⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⠀⠀⠀⠀⣆⠀⢃⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⡇⠀⠀⠀⡏⠳⣄⠐⠠⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡈⠀⠀⠀⠀⠹⣔⠀⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠖⡇⢀⡄⠀⠀⠀⠘⢤⡈⠠⠛⣆⠀⠀⠀⠀⣠⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡏⢸⢣⡀⣀⠀⠀⠀⠱⣄⠀⠊⣧⠀⠀⠌⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠬⠸⡇⠀⠑⠹⠌⣟⢦⡀⠀⢹⢀⠉⢸⠀⠘⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠲⡀⠳⣄⠀⠀⢀⡞⠐⣷⣠⠏⠸⠁⢸⠁⢠⡾⡀
⠀⠀⠀⠀⠸⡗⢄⡀⠀⠀⢱⠄⢠⠈⠢⡈⢢⣀⡞⢡⠀⠿⠃⡐⠀⠂⡜⠀⢸⠀⣇
⠀⠀⠀⠀⠀⠑⠄⠈⢦⠀⠀⠱⡀⠡⡀⠀⢄⠙⠀⠀⠄⡀⠊⠀⠀⡴⠀⣠⠏⠀⢙⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠐⠢⠤⣝⠀⠀⠀⠀⡄⠁⠀⠈⣤⠀⠀⠞⠀⠀⠄⣰⠷⠞⠁⡀⠀⣼⠀⠀⡰⠋⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢓⣙⢲⡀⠀⢸⠀⡆⠀⠙⠀⠀⠀⠀⢈⠀⡡⠔⠀⡰⠁⣰⠃⠀⠠⠐⣉⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢆⠹⡄⡟⢠⠀⠀⠀⠀⠀⠀⠀⠰⠎⠀⠀⡎⢀⡖⠁⡀⢀⡴⢉⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠆⠟⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡞⣰⢾⢃⡮⠐⢺
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠘⡀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⠀⡿⢫⢸⣼⡱⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠠⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢁⠉⡄⢁⠁⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠒⠀⢃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠕⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠤⠀⠀⢀⣀⡀⠀⠤⠊⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠈⠐⠨⠀⠄⠂⠐⠠⠀⠀⠀⠀⠠⠀⠄⠠⠚⠑⠀⠀⠐⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣠⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠒⠁⣈⣭⣥⣦⣄⠈⠙⠛⠍⠛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡷⠀⣠⣿⣿⣿⣿⣿⣿⡏⢁⠀⠀⣠⣀⠻⣶⣴⣦⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠗⠈⠻⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⢈⣿⣿⡿⣿⣿⣿⣷⣿⣦⡄⠉⢠⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⠀⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠓⠙⠛⠋⠛⢿⣯⣤⣾⣷⣿⣻⡯⣉⣽⣿⣿⣾⣿⣄⠀⠸⣿⣿⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠻⣿⣀⠀⣀⣀⣄⣿⣿⡟⠉⠛⠿⠛⠀⢸⣿⣿⣿⣿⣿⣿⡥⢄⠈⠉⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⣴⣿⣿⣤⠙⢿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠈⠿⢿⣿⡿⠛⠿⣷⣿⣷⣶⣼⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣏⠡⠈⠉⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⣹⣿⣿⡿⢇⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣸⣿⣇⡘⠛⠒⠀⠀⠀⠀⠀⠀⠈⣩⣶⡄⠘⠻⢿⡇⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⡏⢹⣿⣿⣏⣽⣿⣿⣿⣟⠃⢀⡄⢀⡄⠀⠀⠀⠀⠀⢹⡿⢧⢠⣰⡞⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠅⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠌⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠻⣿⡟⣽⣿⡍⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠷⠆⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢁⣤⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣿⣤⣄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠀⠀⣀⡖⣛⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠻⠈⢡⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⢢⣯⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠋⠻⣿⣿⠋⠉⠉⠙⠇⠀⠀⠀⠀⠀⢠⡄
⠀⢰⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡁⢀⣰⡾⠋⠘⠿⠉⠿⢗⣠⣦⣠⣤⣼⣿⠇
⠀⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⠓⡂⢀⡀⠘⠀⡘⠻⠿⢟⡛⠉⠋
⠀⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢽⡟⠛⣉⣥⠌⠑⠘⣿⣋⠈⠛⠷⠾⡇⠀⠀⠀⠀⢠⢠
⠀⠘⣿⣿⣿⣿⣿⣿⣿⡌⣇⠀⠀⠀⠀⠀⠀⣴⣾⡟⣤⣰⣮⠡⠀⠀⠀⠠⣾⡇⠀⠀⠀⠀⠀⣄⣀⣄⣀⣬⣀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⢸⣶⣄⣼⣿⣦⣰⣿⣿⡇⠛⢷⣿⣦⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⣷⠛⠋⠉⠉⢹⣿⡇⠐⠂
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡟⣶⣶⢿⣿⣿⣥⠤⡴⢞⠻⢿⣀⣀⣀⣰⣿⣔⣤⣤⡴⠆⠀⢠⣤⡄
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⣿⣄⣾⡓⢺⣇⠀⢉⢻⣿⣿⠿⠟⠉⠋⠀⠀⢀⠼⠟⠷⠆
⠀⠀⠀⠀⠀⠈⠻⢿⡿⠋⢿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⢠⣴⣿⣫⡿⠁⡺⢍⡙⣊⠸⠃⠀⢀⠀⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡟⣿⡀⠰⠀⠀⠉⢁⣤⣆⠴⠋⠀⠠⠞⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠿⣿⣿⣿⣦⣤⣤⣤⣴⣿⣿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣼⠃⠀⠀⠀⠻⣿⣿⣿⣿⠟⠉⠁⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⠀⠀⠀⠀⠀⠙⣿⣿⣿⣤⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠸⣿⡀⣀⠀⠀⣄⣸⠏⣿⡿⣿⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢻⣿⠇⠀⠀⢹⠋⠀⢻⠀⠻⣷⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⢸⣿⡀⠀⠀⣼⡇⠀⢸⣄⠀⠘⢷⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠈⠻⢷⡶⢾⠟⠀⠀⠀⢿⣤⠀⠈⢻⣄⠀⣠⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⣄⣠⣤⠰⣦⡀⠀⠀⠀⠀⠙⠷⠾⢿⣿⠛⠃⠈⠁⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⣦⡀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣦⡀⣤⣀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⢸⣿⣷⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠑⠀⠄⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣾⣶⣠⣴⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣿⣿⡋⡀⠀⠀⠀⠀⠀⣾⢻⡏⠀⢠⣿⣿⣿⠿⣿⣿⡇⠀⠀⠂⠀⠀⢀⣠⠔⠀⡠⠬⠓⠓⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣇⣽⡽⣧⠀⠀⠀⠀⠀⡇⠀⡇⠠⢫⣿⣿⠦⠚⣻⡿⠃⠀⠀⠰⠉⠁⠀⠀⠀⠘⠁⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⣿⡧⢶⠟⠀⠀⠀⠀⠀⢻⠈⣧⡶⣿⣏⢀⡀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠀⠀⣿⠀⠁⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣿⢿⣾⡄⠀⠀⠀⠀⠀⠘⣇⠸⣀⣿⢿⡷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠀⢰⣇⢰⡄⠟⣠⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣽⡾⠙⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣘⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢻⣤⣿⠠⠈⡇⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢷⡟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⢿⣿⢻⣷⠀⠀⠀⠀⠀⠀⠀⠋⠀⣲⡆⠸⣿⣏⠀⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣄⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⢻⣏⠀⠈⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣸⣿⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠘⣿⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣯⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡹⡇⠀⣇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡜⢿⠁⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠻⣦⣹⡇
⠀⠀⠀⠀⠀⠀⠀⣰⢢⣦⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠿⣿⢹⠋⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⣠⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠂⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘

⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣙⡓⠀⠀⡀⢠⣶⠇⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠰⣿⣾⣗⠀⠀⣼⣿⣶⠟⣃⣀⡀⢿⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠀⠀⢹⣿⣿⣮⣿⣿⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⠛⠛⠛⠛⠻⠿⠿⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⣄⣤⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿⡶⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠘⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡿⠿⠿⠿⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣰⣦⡿⣹⡟⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⣦⠔⠛⠿⠿⠾⠋⠀⠀⠀⠀⠺⢧⣀⡀⢀⡾⠁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⣠⣾⡷⢷⣄⠀⠀⠀⠀⢀⣤⣠⡤⠞⠉⢀⠁⠱⠀⠒⠂⣴⡄⠀⠀⠀⠀⠀⢈⠿⠋⠀⠰⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⢀⡼⠋⠁⠀⠀⠙⢻⣦⠀⠦⣉⢱⡏⣀⠄⠀⠋⠐⣠⡄⠀⢀⣿⣷⠀⠀⠀⢀⣴⣯⠖⠀⠀⠀⢀⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⣀⡀⠂⢀⡀⠀⣻⣿⠀⣼⣿⣿⡀⡤⠚⠥⠋⠀⠘⢀⣼⠠⣻⣿⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠋⣠⠆⠠⠀⠀⠀⠀⠀⠀⢦⣼⣿⠞⠁⠀⠔⠉⠀⠠⣾⣿⣷⣿⣿⣿⣇⠀⠀⠀⠀⡠⠌⢉⣠⡾⠿⠋⠁
⠀⠀⠀⠀⠠⣿⣧⣤⣀⡐⢡⡔⡁⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠃⠀⠀⠀⡀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠀⡀⢀⠴⠶⠟⢁⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣮⣼⣿⣦⣤⣀⠀⠀⠀⠀⣤⣿⡷⠀⠀⣀⠀⠂⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡷⢾⣷⢀⣴⣦⣀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢢⣌⠙⠛⠁⠀⠀⠹⣷⣄⣤⣿⡿⠋⠀⠀⢰⣿⣆⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⣿⣿⠿⠛⠋⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠢⣤⡀⠀⠀⠈⠉⠻⠿⠦⢶⣤⣶⣾⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠐⠠⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⣀⣠⣄⠖⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠈⠻⠟⠑⣷⣄⣀⠀⠀⠈⠓⢶⣤⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠐⠋⠟⠋⠀⠀⠀⠀⠀⣀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠛⢦⣦⢀⣄⠙⠷⣦⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠻⣿⡷⢿⣿⣿⣿⣿⣿⣿⠛⠋⠋⠁⠙⠁⠀⠀⠀⢀⣴⣧⣠⡴⠂⠀⠀⣠⠞⠋⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢁⣾⣿⣿⠿⠛⠛⠛⠀⠀⠀⠀⠀⠔⠀⢀⠾⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⠿⠟⠋⠉⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⣀⣸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣤⣤⣹⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁⠀⢰⡄⠀⠀⠀⠀⣤⣤⡈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⣿⣷⣤⣤⣶⣶⣿⣿⣇⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠈⢻⣿⣿⠟⠛⠻⢿⣿⣿⡿⢿⣿⡿⠏
⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⣸⣿⡿
⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⡿⠿⠿⠿⠿⠿⠶⠦
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⠿⠛⠉⠁
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣠⣶⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣷⣄
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⡀
⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠟
⠈⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⣿⣿⡿⠿⠆
⠀⠹⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣷⣦
⠀⠀⠈⠙⠿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⡿⠟⠛⠉⠉⠉⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡱⠀⢀⠀⠀⠀⠉⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠀⣼⠁⢀⡜⠀⢰⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠷⢿⣄⣾⠀⢠⠃⢀⠄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠦⠼⠳⠶⢶⣶⡇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠔⠒⠒⠲⠶⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⡀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠙⡄⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣴⣶⠿⠿⠿⠛⠿⠿⢷⣦⣄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⡀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠛⠿⢿⣶⣶⣶⣶⡶⠶⠟⠋⠀⠀⠀⠀⣀⣀⣀⣠⣤⡤⠆⠀⠀⠀⠀⠘⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⢀⣀⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠋⣿⠏⠛⢿⣶⣤⣀⣀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⣿⠇⠛⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠸⠉⠙⢁⡀⠀⠀⠀⠀⠀⠀⠀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⣇⠸⢠⠀⠙⡇⠀⠀⠀⠀⠀⠀⢠⠿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⣀⡑⢄⠈⢠⣿⢰⡀⠰⡟⠀⠀⣀⣀⣀⢊⠀⡿⢇
⠀⠀⠀⠀⠀⢀⣴⡟⠀⢠⡶⠲⠞⣁⠤⢝⡌⢰⡼⠷⠼⣷⠀⠓⠤⠚⠛⣻⣿⣧⡱⣤⣼
⠀⠀⠀⠀⢠⢿⠟⣄⠀⡛⠀⠀⢈⠀⣠⠖⠚⢛⣡⣤⣤⡽⠿⠖⠒⣶⡄⠘⢿⣿⣿⠈⠁⠀⣠⠄
⠀⠀⠀⠀⠘⠈⠠⢾⢀⡄⣠⠒⣀⠼⠃⢠⣴⠟⠉⡀⢂⠁⠂⠀⢳⡌⢻⡀⢸⣿⣿⣦⡀⠾⠋
⠀⠀⠀⠀⠀⣄⣵⣶⣉⡔⠀⢺⡇⠀⢠⣾⢁⣠⠼⠈⢛⠟⢻⣷⣼⡿⣀⠙⣦⣿⣧⣹⣧⡤⠐⢢
⠀⠀⠀⠀⠀⢻⣏⠻⢋⠡⠆⠚⢃⣴⣼⡿⠯⠅⢂⣽⢶⣦⣦⣿⣿⣹⣿⢀⣧⢹⡟⠀⠀⢠⠖⣶
⠀⠀⠀⠀⠀⢀⣿⡇⠙⢀⣠⣤⣿⠿⡟⢀⡔⠚⢋⣊⣤⢀⡹⣿⣿⣿⣿⣿⠿⣿⡿⠃⠀⣾⠀⠋
⠀⠀⠀⠀⠀⣼⢟⡁⠀⢈⡿⠋⣫⣰⣷⣼⣷⠷⠿⡍⠂⣸⡇⠘⣿⣿⡿⣡⣾⣿⠁⠀⠀⡻
⠀⠀⠀⠀⠀⢇⡀⠈⢠⣿⠀⠀⣿⣿⣟⠻⠟⠶⣶⠿⣿⣿⣧⣼⣿⣿⠛⣿⠛⢿⣶⠉⣿⡇
⠀⠀⠀⠀⢰⡆⠀⠃⣼⣿⡧⢴⣿⣏⣹⣷⣦⣤⣿⣾⣿⣿⣿⢟⡿⠁⠀⣸⠀⠈⠻⠂⠻⠇
⠀⠀⠀⠀⠈⢹⣷⣠⡿⠁⠀⠠⠿⠯⠿⣽⣿⣯⣭⣶⣾⡭⠶⠋⣀⣤⣶⣟⣠⣤⡾⢄⣠
⠀⠀⠀⠀⠀⠀⢉⡼⠗⠶⠾⠛⠂⠀⠀⠙⠛⣿⠿⡿⠴⠶⠾⠿⢿⣿⡿⠿⠿⠃⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⣿⠀⣀⡤⣀⠐⠀⠀⢸⡇⠀⠁⢀⣀⡐⠀⠀⠀⠤⠀⠀⠀⠀⠀⣠⡴⠉
⠀⠀⠀⠀⠀⠀⣹⠀⠉⠚⠁⠀⢀⣡⠌⠁⠀⢠⣾⡿⠿⠿⠶⠷⠌⢀⡀⡠⠴⣴⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣿⣼⠙⢦⡀⠀⠠⠃⠀⠀⠀⢼⣭⠁⠀⡴⠦⠤⠖⢻⡝⠃⠀⣏⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠓⣴⣦⠤⢤⢠⠖⠋⠀⠉⢲⡇⠀⠀⠀⠘⠁⠀⣰⡇⠀⠐⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⣼⡟⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⢠⡏⢀⣿⡇⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣱⠘⠿⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⢀⠀⠀⢠⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⣠
⠀⠀⠈⠀⠀⠘⣇⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊
⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⢀⡄⢀⡴⠇⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢰⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣤⣶⡾⠟⡴⠋⠀⠐⠒⠲⠶⠶⠶⠶⠶⠶⠼⠿⠦⠀⠀⢻⡷⣄⠈⢳⣦⣄⠀⠀⣀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠳⠈⠑⠄⠉⢻⣷⣄⠈⠻⣦⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⢀⠔⠢⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠻⢷⣤⡄⠉⠓⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⣡⡈⠂⣀⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠶⣿⠀⠀⠀⠉⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠰⠖⠂⠀⢀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠓⠀⠒⠢⠤⠄⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣤⠤⠶⠖⠲⠛⢛⠣⠆⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢠⡄⠀⠀⠀⢤⡄⣠⠀⢤⠲⣶⣦⣬⣷⣤⣀⠉⠉⠉⠉⠀⢀⣤⢶⣶⣶⣶⠒⡶⢸⠇⡘⢠⡆
⠀⠀⠀⠀⣼⣿⠀⠀⠀⠘⣷⣿⠀⠀⠀⠘⠻⠛⠀⢹⡿⠃⠀⠀⢠⣿⣿⠁⠀⠙⠛⠁⠀⠀⢹⠆⠁⣾⠀⠀⠀⠀⢠⡀
⠀⠀⠀⢸⣿⣿⠀⠀⡄⠀⢹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⠶⢾⣿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣼⣧
⠀⠀⠀⣸⣿⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⣷⣤⡀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⢀⣴⣿⣿⠃
⠀⠀⢠⣿⣿⡟⠋⣩⡗⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⠇⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⣀⡐⠿⠿⣿⣿⣾
⠀⠀⠈⣿⣿⠁⠀⣿⡇⠀⠀⡀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠘⣷⠀⠀⢸⣿⡟⢠⡀
⠀⠀⠀⠈⠻⡆⠀⢻⡇⠀⢸⡇⠀⠁⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⣷⠀⠀⣿⠀⠀⢸⣿⡅⢸⣷⡄
⠀⠀⠀⠀⠀⠁⠀⢸⡇⠀⢸⡇⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⣿⠀⠀⣿⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢸⡇⠀⣠⣴⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡀⢸⣿⠀⠀⣿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⣀⣠⠀⠀⢿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⢿⠀⠀⠟⠀⠀⠀⢸⡟⠋⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⡿⣿⢿⠋⢁⡤⠴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡤⠀⣧⠄⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠒⠲⠶⠶⠿⣀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠂⢀⣀⡀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⠄⠂⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀
⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣰⠁⠀⠀⢸
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠀⠀⠀⡀
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⢧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠉⠉⠓⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⣷
⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⡀⣿
⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⣸⠀⡏
⠀⠀⠀⠀⠀⠈⠛⠳⠶⣶⣤⠴⠶⠶⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠈⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣄⣀⣀⡀⠀⠀⠀⠙⠳⢦⣄⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠶⠶⢬⣝⣓⢚⣡⠴⠁
⣿⣷⠿⠿⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣾⣿⣾⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⣅⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢛⣿⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣮⠛⢙⣿⡘⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠚⣿⣿⣿⣿⣯⡿⡃⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣄⠀⡀
⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠛⠓⢊⣁⣸⣿⣿⣿⡿⠦⣛⣀⠤⠤⠄⢀⠀⣀⣀⣄⣀⣀⠴⠂⠀⠀⢀⣠⣿⣰⣶⣤⣀⣩⢉⣿⡟⣰⠁
⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⢃⣶⣦⣈⠀⢀⣠⡐⠚⠋⠉⠀⠀⣀⣷⣶⣶⣤⣬⣿⠟⠛⠉⠉⠉⠉⠉⠙⠛⠛⠁
⠀⠀⠀⠀⣿⡿⠀⠀⢀⣴⣟⠀⠀⠀⠀⣸⢷⣿⣿⣿⢏⡦⢸⣿⣷⣦⣀⣤⣴⣾⠿⠛⠛⠛⠋⠉
⠀⠀⠀⠀⢹⣿⠀⠀⠾⠋⠹⣷⣤⠀⣼⣵⣾⣶⣿⣿⣘⣴⡟⠋⠉⠁⠀⠈⠉
⠀⠀⠀⠀⣼⡿⡀⠀⠀⠀⠀⣿⡿⠋⢀⣽⡛⠉⠀⢀⣿⠟
⠀⠀⠀⠀⢿⣿⣦⡀⠀⠠⠊⠁⠀⢀⣾⡿⣶⣶⡤⠞⠁
⠀⠀⠀⠀⠘⣿⣿⣿⣆⠀⠐⠀⠀⣾⠟⠀⢹⡏⠸
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣤⣦⣤⠾⠛⠋⠀⣸⡷⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠋⠠⠖⠋⢁⣰⡏
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣄⣤⡜⠋⢓⣾⣷⠁
⠀⠀⠀⢀⠞⠓⠳⠆⡾⣽⣍⣽⣭⣵⠿⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿