Perspiration Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢟⢿⣿⣿⡿⣠⠖⠾⣷⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡿⣡⢀⣿⡆⠙⢿⡇⢧⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠁⠿⠂⡍⠀⠀⠀⠑⠮⣵⣦⡀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠐⡆⠀⠛⠀⣸⢻⡄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠐⠒⠃⠀⠀⢠⠷⢋⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣦⡐⠛⠛⠷⠆⢀⣴⣿⣿⠟⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣳⣶⣶⣾⣿⣿⣿⣏⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣇⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠛⠛⠛⠛⠛⠛⠷⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣧⣤⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠙⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣰⠋⠉⠛⠓⠶⠶⠦⣤⣤⣬⣙⣛⣛⣛⣛⣂⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡏⠀⠀⢀⠀⢤⣄⣈⠙⠯⡷⠀⠿⠀⣖⣲⠶⠆⠀⠀⠀⠀⠀⠀⠀⢠⡏⢹⡰⠃⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣴⠟⠉⢻⠁⠀⠀⠈⣭⣭⣭⣿⡿⠶⠀⠀⠀⠀⠠⢴⣶⣶⣿⡿⢿⣯⣥⠀⠀⣸⠀⠀⣧⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⣿⣀⣘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠄⠀⠀⣏⠀⠀⣿⠀⡰⢚⠋⠻
⠀⠀⠀⠀⠀⠀⠈⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⠶⠛⠚⠙⠏⠀⣠
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⡿⠟⢿⣄⣠⣄⢀⣼⠗⠒⣲⠟⠻⢯⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠓⠒⠛⠋⠁⠀⠀⠀⠀⠉⠛⠛⠿⠶⠶⠚⠉⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⣀⡀⠀⠀⢀⣸⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡦
⢀⡤⣤⣯⠹⠶⠞⠋⠁⢘⣇⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡼⠓⢦⡀⠀⢀⣀
⣹⡶⠄⠀⠀⠀⣀⣤⠶⠛⢻⣆⣀⣤⡴⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡼⣧⡀⠀⠉⠲⠋⢽⠦
⠛⢲⡾⢃⡶⠛⠉⠀⠀⠀⠀⠉⠉⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠳⣤⠾⠁⠀⠙⠳⢤⡀⠀⠀⠴
⠀⠈⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⡶⠤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⢀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣠⣀⣀⣈⣉⣉⣉⣉⡉⠉⠙⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢰⡏⠈⠉⠉⠉⠉⠉⠙⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠉⠉⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠚⠋⠉⢹⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠭⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠈⠈⢳⡀
⠀⣀⡤⠤⣤⣀⠀⠀⠀⠉⠓⠦⣄⠈⣇
⡞⠁⠒⠒⠂⠉⠓⢦⡀⠀⠀⠀⠈⢧⣿
⢧⡀⠀⠀⠀⣀⡀⠀⠙⣆⠀⠀⠀⠸⡏⠀⠀⠀⢀⣠⣀⣠⣤⣤⣀⡀
⠀⠉⠙⠋⠉⠉⠉⠙⠲⣌⣧⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⢠⡏⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠙⣆
⠀⠀⣠⢚⣛⠛⠒⠶⢤⡀⠀⠀⠀⠀⠀⢸⡇⣠⠴⠛⠉⠀⠈⠳⣤⣀⠀⠀⣹
⠀⠘⣇⠁⠀⢀⣤⠴⠦⢭⣷⡄⠀⠀⠀⣼⠛⠃⠀⠈⠓⠚⡁⠀⠀⢉⠛⢶⠏
⠀⠀⠈⠓⠛⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠹⣦⠀⠀⠀⠉⠉⠀⠀⠘⠬⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠶⠦⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣟⠻⢦⣀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠉⠁⠀⠉⠙⠛⠛⠛⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⡛⠶⠦⢤⣤⣀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠉⣻⠋⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠘⣧⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⡀⣠⡏⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢸⠿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡿⣋⡴⠳⠦⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⣀⠀⠀⠀⠈⢷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⢀⣤⣄⣉⣙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢲⡏⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠓⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠍⣿⣿⣿⣿⠟⢿⣇⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⣡⣶⣌⠻⢿⠃⣾⡆⢻⣷⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣸⣿⣿⣿⣷⡆⢸⣿⣿⠄⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠟⢛⠻⣿⣿⣿⣿⣦⣉⣩⣴⣦⣬⢹⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⠛⣋⣼⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣤⣤⡙⢿⣧⡘⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⢸⣿⣿⠆⣿⣿⣿⣿⣿⠃⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⢾⠟⣡⣾⡟⠛⢛⣋⣤⣾⣿⣿⢃⣶⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⢸⡟⣛⣉⣉⣉⣉⣉⣉⣉⣙⡛⢿⡇⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠙⠃⠠⠀⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠠⢼⡇⠀⠀⡀⠉⠓⠦⣤⣀⡀⣀⣀⣀⣀⣤⡤⠖⠒⠈⡍⢩⠁⡏⠈⠉⠛⠦⣄
⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢀⠀⡀⠀⣴⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠸⣦⣇⠀⠀⠀⠀⠀⠙⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣯⢰⡇⠈⡉⠰⠾⠛⠶⠃⠤⠀⠀⢉⣥⠖⠀⠐⠀⢠⠈⠁⣴⣏⣿⠢⣄⣉⠀⠀⠀⠀⠈⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⣟⠁⠀⠀⠀⢀⡀⣤⠀⠀⠀⠁⢀⣠⠄⠀⠀⠀⡜⠁⠋⣸⣏⣿
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣤⣤⣶⡌⣿⣤⣦⣄⠀⠈⠁⢠⠁⣀⠀⠀⣀⠁⠒⠀⠀⠀⠁⡀⠀⣹⣯⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢈⣀⣤⣴⣿⣿⣿⣷⢾⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⣠⣴⣿⣿⣿⣿⣅⣷⣤⣤⣀⠀⠐
⠀⠀⠀⢠⣶⣶⣿⣿⡏⢹⣿⣿⠏⣿⣼⣿⠙⣿⣿⣿⣿⣷⣄⠀⠀⠂⠚⠃⣀⣴⣿⣿⣿⣏⣿⣿⣿⣾⣿⣻⣿⣿⣷⣶⣄
⠀⠀⠀⢸⣿⣿⣿⠟⠁⢸⡿⠇⠀⣹⣿⠶⡖⢋⡍⠛⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣹⣿⣿⣿⠟⣿⣿⣿⠟⣻⣿⣿⢻⣿⣿⣷⣤⣀
⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠠⠈⣿⡄⢀⠀⠀⠀⠉⠉⠙⠛⠉⣿⣿⣿⠋⠉⢹⠿⠧⠈⣴⣿⡈⠁⠉⠻⢿⡟⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣉⣠⡶⣿⠀⠀⠀⠀⠀⠈⠛⠛⣛⡆⠙⠛⢉⣈⡁⣤⣀⣤⣾⣿⣻⣷⡘⠛⢠⡄⡈⠁⠠⢿⣿⣿⣿⣿
⠀⠀⠀⢀⣀⣠⣤⣶⣾⠿⠛⠛⠛⣧⣿⡟⠛⠿⠿⢿⣿⣾⣟⡿⠿⠟⠛⠟⠛⠛⠈⣡⣬⠀⣾⡿⠯⡤⠄⠀⠄⠉⠀⠀⠐⠉⠉⠙⠿
⠀⠀⠀⠺⡿⠿⣿⣬⣤⣤⣀⣀⣀⣼⣼⣿⣠⣤⣤⣤⣤⣄⡉⠀⠀⣤⣴⣾⣷⣿⠿⠿⠷⠶⢿⣿⡿⠿⡶⠶⣾⣶⡆⢰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠛⠛⠛⣿⣯⡉⠈⠉⠙⠁⠈⠋⣻⣿⣿⣯⣅⣩⣭⣶⣶⠛⠓⢾⣾⡇⠀⠀⣀⣉⢀⣀⣌⣉⣤⣴⣿⣿
⠀⠀⠀⠿⠛⠒⠿⠿⠿⢿⣿⣷⣤⣀⣿⢿⡧⠤⣴⠒⣿⣿⣿⡿⠋⠉⠭⠉⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠋⠁⠀⠈
⠀⠀⠀⣤⣤⣤⣤⣴⣶⣶⣿⣿⠿⣿⣿⣿⣧⣤⣴⣦⣽⣿⣿⣿⣶⠾⠿⢿⣿⣿⣿⣿⣿⡿⣿⢻⣿⣟⣛⣿⡿⣿⣻⣿⣿⣿⣦⣤⣴
⠀⠀⠀⣿⣿⣍⣉⣉⣉⣁⡀⠀⠀⢀⣿⣿⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⠀⣤⣤⣸⣿⡏⠀⠀⢀⠀⠈⢀⣤⣀⣸⣿⣿
⠀⠀⠀⠈⠉⠛⠛⠻⠿⡿⢿⣿⣶⣾⣾⣿⣞⡛⠛⠛⠒⠤⣤⠄⢿⡛⠛⠛⠻⠿⠿⣿⣿⣿⣾⣿⣿⣏⣥⣴⣶⣶⣶⣿⣿⡿⠛⠹⠿
⠀⠀⠀⣀⣀⣀⣦⣤⣶⣶⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣶⣿⣿⡿⠙⣿⣿⣏⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠈⣿⣿⣿⣥⣬⣤⣉⣠⣄⣀⡀⣿⡟⠉⠃⣀⣠⣬⣿⣿⣿⠿⠟⠿⡿⣿⣿⣯⣥⣤⢾⣿⣯⢀⣙⠃⠀⡀⠀⢀⣀⣠⣤⣾⣿⣿
⠀⠀⠀⠀⠈⠉⢋⣻⣟⣻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣄⣁⣶⣦⣤⠤⠿⠿⣿⣿⣾⣿⣿⣿⣿⣿⣿⢿⣿⠿⠟⠛⠉⠁⠈⠋
⠀⠀⢀⡀⠀⢀⣬⣉⣉⣀⣤⣤⣶⣶⣿⣿⣿⣷⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠛⣿⣿⠉⣉⡉⣤⠔⠂⠀⠀⠉
⠀⠀⢻⣷⣿⣿⣟⣛⣛⠉⠉⠉⣉⣹⣽⣿⠻⠿⢿⣶⣾⣶⣦⣤⣤⣤⡀⣀⣈⣽⣿⣿⣿⣶⣿⣯⣸⣿⣿⣦⣶⣦⣤⣴⣤⣤⣀
⠀⠀⠻⠻⠿⠿⢿⣿⣿⣿⣶⣶⣥⣼⣿⣷⣶⣶⣿⣿⣿⠿⠿⠛⢿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢛⡛⣉⣉⣍⣉⣤⣄⡁⢘⡛
⠀⠀⠀⠀⠀⠺⠆⢉⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⡀⠀⣀⠀⠀⠉⠘⠓⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣷⡼⠃
⠀⠀⡀⣀⣤⣴⣾⣿⣿⣿⡟⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣍⣀⣄⣤⣴⣦⣾⣿⣿⠿⠟⠉⠁
⠀⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣏⡙⠛⠿⠿⣿⣿⣶⡷⣤⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣀⣀⣄
⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣯⡿⠿⠛⣛⠋⣉⣉⣉⣀⣀⣀⣀⣀⣉
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣯⣽⣿⣾⣿⣾⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⠻⠋
⠀⠀⣼⣟⣹⣟⣿⣿⣿⣿⣿⣿⣷⣶⣾⡿⣿⣙⣿⣿⣿⣿⣿⢿⣿⠿⣟⠿⣾⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⣼⢟⣿⡛⣿⡇
⠀⠀⢸⣾⢻⣟⣻⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⣛⡛⣿⣿⣿⢛⣜⣿⣽⣿⣿⣷⣿⣧⣿⡹⣿⡄
⠀⠀⣌⢙⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣧⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⢟⣱⣿⣿⣿⣹⣿⣿⣇⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠰⣿⢎⣽⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⠻⣙⣿⣿⡟⣿⠏⣼⣿⣿⣿⣿⣿⣿⡿⣿⠁
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⢻⣿⣿⣿⣿⣾⣿⣟⣿⡟⣼⢿⣛⣿⣽⣿⣿⢻⡌⣀
⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣧⣴⣤⣼⣿⣿⡟⢿⣄⢻⡅⢥⡄
⢀⣄⡸⣻⣿⣿⣿⣿⠿⣿⢿⣧⠿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣿⣯⠿⢛⣿⣿⣿⣿⣿⣿⣹⣯⠙⠿⠋⢳⠈⢻⡀
⠀⠏⣰⠻⣿⣿⣿⣿⣷⡞⢿⣏⣦⣹⣿⣷⠆⣽⣿⣿⠟⣿⣿⣿⣿⡟⣿⡍⢁⠁⣠⢿⠛⡿⢿⣿⣿⡟⣧⢈⣿⡞⢀⣤⢿⡆
⠀⣿⠛⢀⣿⣿⣿⣿⣿⣿⣿⣽⣯⣬⣿⣽⣾⣿⡿⢿⣿⢿⣧⣿⡄⢻⣿⣤⣟⠠⣧⡼⢧⡰⣿⣿⣿⣿⠋⣼⡟⣿⣏⠛⠛⠁
⠀⠈⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⡎⣩⡟⠻⣿⡴⢾⢋⣶⣉⣱⣦⠰⢞⠶⣿⡿⢻⣿⣿⣿⣾⣿⣿⣿⣟
⡀⢠⣼⣿⠿⣛⣿⣿⣥⣴⢾⣿⡿⣾⣿⣿⣿⠷⣯⣿⢿⣴⠿⠛⢀⠶⣤⢿⡛⡉⠁⣴⣛⣏⣹⣶⣿⣿⣿⣿⣏⣙⡿⣏⡋
⠙⣻⠧⣄⣳⢮⣿⣿⣿⣧⣺⠿⣿⢫⣿⣿⣿⡊⣿⣿⣿⣬⡟⢳⣘⡖⢻⣴⠏⢠⠦⢈⣿⣿⣿⣿⢿⡤⣥⡼⢿⡬⠃⣬
⠀⣽⣶⣟⣱⠾⣿⠛⣿⠟⢿⡶⢿⣉⠿⣿⣿⡍⠿⣶⣿⠷⣍⠉⣁⡽⢮⣥⡾⠃⣠⣾⣿⣿⠟⣿⣿⣵⣽⠶⠏⡤⠺⠋
⠀⣼⣽⣿⣿⣿⣾⣷⣽⣶⢾⡿⠖⣉⣿⣾⣿⣿⣤⡈⠉⠶⢈⡀⠠⠶⠂⣉⣷⣿⣿⣿⣿⣿⣷⣿⠇⣻⡁⠋⣶⠄
⢄⡉⠙⠛⢿⡿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣧⣁⣰⡏
⠀⠛⠂⠳⡞⠒⠇⠈⢉⡍⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⢙⣫⣿⠿⣿⠗
⠀⢲⣦⠆⠐⠀⢛⠁⢠⠏⠸⣇⠹⣿⣿⣿⣿⣿⣿⣿⠿⢿⣉⡙⠫⠤⠒⠀⠀⠈⠛⠋⠀⠈⠛⠋
⠀⠀⠀⠀⠘⠟⢀⡇⢠⡶⠛⣃⠀⠸⣿⠘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⠁⡀⠀⢀⣤⠿⠛
⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠈⠉⠉⠐⠀⢀⠄⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡇⠀⠀⠀⠄⢈⠁⠄⠀⠀⡝⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡽⠁⠘⠦⣀⣀⣤⣊⡑⠤⠤⠞⠀⠀⠹⣍⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⡔⠉⠀⠀⠀⠈⠐⡄⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣄
⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀
⡠⠸⢄⠘⠀⠸⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⠀⠸⠀⡙⢀⠼⠠
⡰⠂⠤⠀⠀⠀⠃⠤⠀⠀⠀⠏⢇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⠏⢃⠀⠀⠠⠄⠃⠀⠀⠀⠄⠲
⠑⠒⠒⠄⠔⠓⠤⣀⠀⠀⠉⣇⠈⠢⡀⠀⠀⣿⣿⣿⣿⣏⣁⣉⣁⣉⣿⣿⣿⣿⡇⠀⠀⡠⠊⠀⡜⠁⠀⣀⡠⠔⠓⠄⠔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠈⢢⠀⠈⠓⢦⡥⣄⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⡤⢥⠶⠋⠀⢀⠎⠀⢰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠑⠤⡄⠀⠙⠢⡀⣿⣿⣿⣿⣿⣿⣿⡇⡠⠒⠁⠀⡤⠴⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠙⠢⠄⣀⠈⠙⢍⡀⠉⠁⢈⠝⠋⢀⣀⠤⠒⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠭⢦⡾⠕⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢍⡉⠙⠃⠒⠐⠒⠒⠒⠲⠊⠳⠒⠒⠒⠒⠐⠂⠛⠈⣉⠥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠒⠒⠀⢤⠤⠤⠤⠄⠤⠤⠤⢤⠄⠐⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣤⣤⣤⡼⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣦
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠰⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢰⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⣠⠀⠀⠘⠻⠿⠃⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢸⡏⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣴⠟⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⢿⡄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢿⣆⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠋⠁⠀⠀⠀⠀⢀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢿⣶⣶⣾⠇⣀⡀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⠿⣷⣦⣤⣤⣤⣾⣿⣿⣿⣷⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠁⢴⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢠⣌⠙⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⡄⠹⣿⣿⣿⠟⢋⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⣿⣿⠃⣼⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⡁⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡀⢿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⡷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢠⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠛⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠛⠛⠛⠁

⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⡄⠀⠀⣠⣀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⣠⣾⣿⣿⣿⣶⣿⣿⣿⣷⣄⣀⣼⣿⣿⣦⣴⣦⡀
⠀⠀⣠⠶⠚⣿⣿⣿⠉⠙⠳⢦⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⡼⢃⡴⠛⠛⠛⠛⠛⠳⣄⠈⢳⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢰⡇⡾⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢸⡇⣷⠀⢀⠀⢀⠀⣀⣀⣀⡿⢀⡇⠀⠀⢠⣿⡿⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⢉⢁⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⣀⣤⣤⣀
⠀⢳⡌⢻⣋⣿⣋⣿⣉⣿⣹⣇⡼⠃⠀⠀⢸⣿⠃⢸⣼⠆⠀⠀⢠⣆⡄⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠉⠳⣄⢈⣷
⠀⠀⠙⠻⢯⣄⣉⣰⠿⢤⡴⢿⠀⠀⠀⠀⢸⣿⠀⠀⠁⣀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣠⠶⠖⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⢀⣶⡞⣧⡉⠙⣷⠀⠁⢸⠀⠀⠀⠀⢸⡏⠀⠀⠀⠉⠛⠶⣄⡀⢠⣄⠀⣠⡴⠞⠉⠁⠀⠀⠀⠀⣄⡶⠸⣿⣿⡀
⠀⠀⠈⠛⠁⠈⠙⣆⠉⠀⣀⣾⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⠟⠛⢦⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠙⣧⢠⡟⠁⢹⡇⠀⠀⠀⠐⠦⣤⣄⡀⠀⠀⠀⠀⢤⣶⣖⣛⡀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢻⣄⣀⣤⣤⠴⢿⠈⢧⡀⢸⡇⢀⠀⡀⠀⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣧⡴⠞⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠘⣧⠀⠙⢻⡆⠘⠟⠁⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⣧⠀⠀⠀⠀⠈⢿⣝⣷
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⣿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠙⢦⣀⡇⠀⣆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠈⠁⠀⣿
⠀⠀⠀⠀⣠⢴⡞⠁⠀⠘⢧⡀⠀⠀⢀⠀⠉⠹⡄⠈⠁⠀⠐⠛⠳⠤⠖⠲⠴⠞⠲⠞⠉⠛⠉⠛⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠸⣧⠞⠀⠀⠀⠀⠀⠳⣄⠀⢻⡾⠧⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠐⣶⠆⠀⠀⠀⠀⠀⠀⠀⣉⣤⠶⠋⠁⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣄⡀⠀⠈⠉⠛⣶⠶⢦⣤⡤⠤⠶⢶⠚⠋⠁⠀⠀⠀⡀⠀⠈⢷⠀⠀⠀⢶⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠻⠚⢻⣿⣿⡷⠦⡾⠀⠀⠀⠀⢀⠀⠻⠟⠀⠈⣇⠀⠀⠀⠻⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⣿⣿⣧⠀⠸⣦⠶⡄⣠⠀⢸⡆⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠙⠁⠀⠈⣿⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣶⣤⣬⣥⣄⣀⣀⣀⣤⣤⣤⣴⣾⡇⢿⠶⠚⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⠇⣸⠶⠶⠶⠚⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣏⣠⡀⠀⠀⢀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠷⠶⠒⢛⣁⣿⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⠏⣠⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠿⠛⠁⣼⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣴⣤⣤⣤⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠶⠀⢸⠇⠀⠀⠀⠀⢸⡇⣐⠖⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣩⣿⡇⣼⣿⣿⣿⣿⣇⢸⣿⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠃⠸⡄⠀⠀⠀⠀⣠⠇⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣾⣇⢻⣿⣿⣿⣿⠟⣸⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡻⠿⠿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⡤⣶⡞⠁⠀⠈⠁⠀⠘⣳⡶⢤⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣛⡉⢡⣾⣿⣿⣿⣿⣷⡌⢙⡛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡞⠁⠀⢽⣿⡀⠀⠀⠀⠀⢀⣿⡇⠀⠈⢳⠀⠀⠀⠀⢸⣿⣿⣿⣿⢱⢚⣛⠃⠘⢿⣿⣿⣿⣿⡿⠀⢸⣿⣷⡜⣿⣿⣿⣿
⠀⠀⠀⢸⡇⠀⠀⣽⣿⣿⣶⣤⣴⣶⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⠅⠀⠈⠙⠛⠛⠋⠁⠀⢸⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⢸⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢰⡲⡄⠀⠀⠀⢀⡠⠖⠉⠉⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢤⠀⠀⠀⠀⢀⠔⠊⠉⠉⠑⢄
⢀⣈⠁⠀⠀⠀⡞⠀⡠⣄⣀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠁⠀⠀⡰⠁⠀⡤⣀⠀⠀⠀⢣
⠓⠁⠀⠀⠀⢀⢧⡞⠡⠄⠈⠍⢹⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢠⡧⣤⠊⠔⠀⠩⠙⡆⠀⢣
⠀⠀⠀⠀⡰⠃⠀⡇⠀⠁⠈⠀⢸⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢸⠀⠈⠀⠁⠀⣏⡇⠘⡄
⠀⠀⠀⡴⠁⠀⠀⣧⡀⠰⠆⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢸⢄⠐⠾⢂⡰⠁⠀⠀⡇
⠀⠀⠈⣳⣄⢤⣼⣧⡿⣗⣛⢿⢢⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣧⠔⡶⠾⣾⡏⠉⢙⠳⡒⠤⢞⡀
⠀⠀⡜⠈⠉⢺⠄⡧⣧⣎⣱⣼⢰⡃⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠉⢳⢸⠉⠗⠒⢺⠀⡇⠀⠀⠹
⠀⡼⠀⠀⠀⡼⠸⡅⢸⠇⠈⡏⢀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⠇⢸⠀⡆⠀⢸⠀⡇
⢠⠃⠀⠀⢠⠃⠀⢣⠘⡄⢀⡇⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡼⠀⠀⠀⠘⠀⠈⡆⡇⠀⢸⢸
⠘⡄⠀⠀⢀⡆⠀⠀⠓⢳⡸⠘⠁⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⢣⠀⠀⠀⣴⠀⠀⠓⡇⠀⣾⠊⠀⠀⡇
⠀⠈⠒⠒⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⠒⠊⢸⠀⠀⠀⣇⠀⡟⠀⠀⢀⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢹⠀⡇⠀⠀⠸⠀⠠
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠛⠛⠛⠛⠛⠛⠷⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣰⠟⢿⣀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣤⣀
⠀⠀⠀⠀⢀⡾⠃⠀⠈⠻⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⣾⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⢸⡏⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⢸⡇⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠷⣦
⠀⠀⠀⠀⣿⡀⠀⡴⠋⠓⠲⠤⠤⣤⣄⣀⣀⡐⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠈⢷⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡼⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣾⠃⠀⠀⠀⢀⣀⠀⠙⢿⣒⡆⠀⡇⠀⡤⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⡄⡎⠀⠀⠀⢀⣿⠁
⢀⣴⠶⠶⡟⠀⠀⠐⠒⠒⠿⠿⣷⣦⣍⠁⠀⠀⠀⢉⣉⣡⣤⣤⣤⣶⣦⠄⠀⠀⢰⡇⠀⢳⠀⠀⠀⠀⣼⠇
⢼⡇⠠⢤⠇⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠭⣍⡉⠁⠀⢸⠀⠀⢸⡆⠀⣀⠴⠿⣦⡀
⠘⠷⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣼⠧⠜⣴⠒⠀⢸⡇
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢠⣤⣤⣴⠟
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠁
⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁
⠀⢀⣠⣴⣛⣳⣄⠉⠛⠳⠶⣤⣤⡄⠀⠀⠀⠀⢠⣀⣀⣀⣠⣤⡴⠶⠚⠛⠙⠛⠳⢦⣀
⠀⢿⢻⡀⣿⠄⠉⣷⠀⠀⠀⠀⠘⢳⣶⣶⠦⠤⠟⠉⢉⣡⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠈⠻⣏⣡⡤⠶⠛⣦⣀⣀⣤⠾⠋⠀⠘⠦⠴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⣿⠶⠶⣤⣄⢀⣠⣤⠶⢯⣀
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠙⡏⠁⠀⠀⢀⣉⡇
⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⢘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⢶⣀⡶⠶⠋
⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⠴⠶⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣬⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⣾⡁⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠋⠉⠁⠀⠻⣤⣤⣤⡤⠴⠶⠚⠛⠉⠁⠀⠀⠈⠛⠶⠶⠶⠛⣷⠒⠋⠁⠻⣄
⣴⠟⠛⠶⣤⡀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣴⣄⠀⠙⢷
⠘⢷⡄⠀⠈⢻⠋⠻⣷⡀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠘⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⠈⠛⠶⠶
⠀⠀⠻⣄⠀⠀⣠⡴⠞⢳⣄⠀⢠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣤⣤⣤⡤⠶⠛
⣀⣀⣀⣈⣛⣛⣉⣀⣀⣀⣙⣷⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠑⢄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⡒⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⣴⠒⠀⠀⠀⢤⠀⡄⠻⢷⠴⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠁⠈⠁
⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⡀⢴⣂⠘⠠⠂⠊⠀⠀⠀⠀⠀⠈⠲⠤⣀⣀⡠⠔⠁⠠⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠄⠀⠋⠙⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠰⢀⡠⠂⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⢀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠄⠈⠁⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡚⠀⠀⢦⠀⠑⠀⠸
⠀⢀⠀⠀⠀⠁⡀⠀⠶⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠘⠐⢋⣰
⠀⠋⠐⠊⠕⠈⠈⠁⠰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⠒⡄⠀⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡖⠢⠤⠀⢰⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠁⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⣄⠀⢀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠰⠂⠀⠀⠀⠀⣀⣠⢒⡨⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠛⠖⠁⠪⡀⠀⠀⠀⡀⠐⠀⠀⣄⠂⢤
⠀⠀⠀⠀⠢⠀⠔⠁⠀⠅⢠⠐⣄⠀⠈⠄⠐⠂⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⢸⠁⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠄⠀⢐⡴⠃⠈⢄⣀⡄⢀⠈⠢⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠛⠅⠠⡤⠐⠅⠁⠀⢹⠀⠈⡂⠀⠀⠪⠂⡀⠄⠒⠈⠋⠑⠠⡀⡀⠀⠰⠘
⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⠖⠀⠀⣂⡖⠁⠀⠀⠀⠀⠈⡱⠊⣀⠠⠐⠈⣄⠀⠀⠘⡤⠂⠈⢉⡐⢖⡒⠀⠲⡁⢢⢄
⠀⠀⠀⠀⠀⠁⠴⡆⢄⣀⢀⣀⠴⠁⣠⠆⠀⠀⠀⠀⠐⠀⢶⡁⠀⠀⠀⡈⠄⡀⠀⠂⠑⡀⠀⠀⠀⠀⢱⠂⠉⠓⢄⠈⠂
⠀⠀⠀⠀⠀⠀⠔⠁⠈⠃⠘⠁⠈⠈⡠⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠁⠘⢃⠀⠰⠀⢰⠀⠀⠀⢀⠸⠀⠀⠀⠈⠑⠈
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⡄⠁⠀⠀⠀⠀⠀⣜⡇⠠⣀⡀⠀⠈⠀⢀⣠⡦⠖⠂⠈⠉⠉⠉
⠀⠀⠀⠀⠈⠀⠖⠂⠀⠀⠀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠉⠢⠖⠘⢱⢫⢉⡅⣌⠄
⠀⠀⠀⠀⠀⠐⠂⠒⠀⠠⠤⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⠾⠿⠿⠿⠿⠿⠿⠶⠶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣶⣦⣀⠀⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠁⠘⢿⣆
⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠋⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢻⡆
⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿
⠀⣠⣴⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢀⣿
⢰⡿⠁⠃⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠶⣏⠀⠀⠀⢸⡟
⣿⡇⠀⠀⠀⠀⠀⠘⣿⠦⣤⣤⣠⣤⣤⣤⣤⣤⣠⣤⣤⣭⣭⣥⣤⣤⣤⡤⠤⠶⠶⠶⠛⠛⠛⠉⠉⠁⠀⠀⠈⢷⡀⢠⡿⠁
⣿⡆⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⢻⣇⠀⠀⠀⠀⠀⢀⠀⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠸⣿⡀⠀⠀⠀⠀⢸⠀⣼⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢸⡇⠀⣤⠖⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⢻⣧⠀⠀⠀⠀⠈⢰⡏⠀⢿⡄⠀⠀⠀⣀⣀⣀⣀⣀⣀⡈⠙⠻⠶⠿⠀⠘⠁⠀⠙⠛⣁⣤⣶⣾⣿⣯⣤⣤⠀⠀⠀⢹⣧⣤⣤⣄
⠀⢈⣿⡆⠀⠀⠀⠀⣿⠀⠀⠸⣇⠀⠀⠤⣤⡿⠿⠿⢿⣿⣿⣿⡿⠶⠂⠀⠀⠀⠀⠀⠛⠻⠿⠯⣥⣤⣤⡤⠄⠀⠀⠀⠈⡏⠁⠉⠻
⣰⡟⠉⢙⠳⢦⡀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠠⠶⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⠁⢀
⢻⣇⠀⠈⢙⡷⠛⠄⠿⣆⣀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟
⠈⠻⣶⣤⣀⣥⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠈⠉⠉⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡟⠛⠷⢶⣤⣄⣀⣀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢠⣀⣀⣤⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⢧⣀⣀⣤⣤⣀⣼⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠙⠸⡆⠀⠀⠀⠀⠀⠀⢻⠀⣇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢸⡆⡏⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠈⣿⡇⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢻⣷⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⣀⣽⡟⠛⠛⢿⣶⣦⣄⡀⠀⠀⠀⢀⣀⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡞⠋⠉⣤⠿⣦⣤⣠⣤⣤⣽⣷⣤⣀⣰⡟⢻⣷⡟⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠶⠟⠛⠁⠹⣦⣄⣀⣀⣀⣨⣿⣀⣀⣀⡉⠉⢩⡟⠀⣸⡏⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡶⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠉⠉⠉⠛⠛⠛⠀⠀⣿⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠶⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣠⣤⣶⠶⠾⠿⠿⠿⠿⠿⠿⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢻⣤⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠈⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠘⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠞⠀⠀⠉⠉⠉⠛⠓⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⡟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠶⣶⣤⠀⣷⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⢠⡄⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⢿⠃⠀⠀⠲⠾⠿⠷⣦⣌⠉⠀⠉⠀⢛⣋⣉⣤⣤⣤⣤⡤⠀⠀⢸⠇⠸⡇⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠤⡼⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠛⠿⠯⢭⡉⠁⠀⣾⠀⠀⣿⠀⣀⠴⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣠⠿⠔⠳⠂⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⢀⣤⣤⣤⣄⣀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⣤⡶⠛⢉⣠⣤⣤⡈⠙⠳⣦⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠛⠁
⠙⠛⠛⢻⡙⣧⡟⠉⠀⠀⢈⡿⢦⣀⠀⠀⠉⠛⠶⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⢀⣀⣤⡤⠖⠛⠉⠉⠉⠉⠛⠓⠶⣤⣀
⠀⠀⠀⠘⢧⣽⣷⡾⠛⢳⡞⠁⠀⠉⠛⠲⠶⠦⠶⠶⠞⠛⠻⠧⠤⢦⣤⡼⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠈⠉⣩⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠟⠛⠛⠛⠛⠲⠶⣤⣿⣄⢀⡀⠰⣌⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⠶⠶⠚⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⣼⠛⣦⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⠀⠀⠀⣀⣀⣤⣤⡤⠶⠖⠒⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠷⣄⠀⠀⠀⠀⠀⠀⠀⠉⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⠈⠛⢦⣄⡀⠀⠀⠀⢀⡟⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠳⠶⣤⣼⡇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠶⠶⠶⠶⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣶⣶⣶⣤⣤⣤⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣛⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣇⡤⠒⠺⠿⠋⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⢠⣄⠠⡄⠈⣙⡂⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡅⠀⠠⡆⠀⠄⢠⠃⠀⠀⠀⢨⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠒⡄⡴⠀⠉⡆⠚⠀⢀⡠⠔⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢘⢀⠋⠉⠲⡁⢀⡀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⣄⡸⡇⠀⠸⠀⠀⢀⣱⢘⣡⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠐⠒⠴⠶⠈⠁⢀⡀⠣⡀⢈⣼⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠶⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⢀⠆⠀⠨⠄⠀⠀⠃⠈⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⢀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢈⠗⠠⢀⣀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢄⡴⠁⠀⠀⣀⣤⣤⣤⡄⣀⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣧⠊⠀⠀⢰⡋⠖⠄⠀⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣻⠀⠠⠀⣀⢀⡔⠁⠀⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠀⢸⣇⣿⡷⠇⠰⣀⣀⡴⠁
⠀⠀⠀⠀⠀⢠⢤⡜⢱⠋⠀⢸⠻⠏⢠
⠀⠀⠀⠀⠀⢜⠀⠀⡭⠒⠀⢸⡞⣀⣸⡤⢤⣀⠀⢀⡀
⠀⠀⡔⡆⠀⣘⠝⡏⠀⠀⠀⢸⢡⡟⡞⠀⠀⠎⣸⠁⣸⠤⡃
⡤⠤⠅⠁⡀⡍⠀⠀⡠⢄⠀⡘⢸⢡⠁⠀⠀⠠⢸⠉⠯⠃⠃
⠀⠠⣁⠎⠀⠀⠰⠎⠀⢸⣦⡇⠀⡇⠀⡀⠀⠆⡌
⠀⠀⠀⠀⢠⠤⣴⠀⠐⠻⠷⣀⠀⡱⢀⠇⢸⢠⠃
⠀⠀⠀⠀⡼⡄⠙⠀⢠⠂⠀⡜⠉⠁⠹⣄⡼⠈
⠀⠀⠀⠀⢏⠱⠒⢶⣁⠀⢀⣃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣤⡶⠛⠛⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣰⡟⠈⢿⡀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣟⠀⠀⣰⠗⠀⠙⢷⡶⠛⠁⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⣿⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢻⣄⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⣿⡁⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡿⠀⠀⠀⠈⢷⣤⣀⠀⠀⣀⣤⡾⠁⠀⠀⠈⣷⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⣠⣴⣿⡉⠛⠛⢉⣿⣤⣄⣀⡀⠀⠉⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠟⠛⠉⠁⣀⠈⠻⠶⠶⠟⠁⠀⠈⠉⠛⠻⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⢼⡏⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣀⡀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠛⢷⣄⠀⠈⠁⠀⠀⠀⠀⠀⠘⣷⡀⠀⢀⣴⠾⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⣾⠃⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠀⠀⠈⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⡀⠉⣸⣿⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠳⣤⣾⠟⣷⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠸⠀⠀⠀⠀⢹⡟⠁⢸⡀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣙⣶⠟⠁⠘⠶⠀⠀⠀⣰⡞⠀⠀⢸⣧⡀⠀⠀⢀⡾⠣⠊⠀⠀⠠⡲⠁⠀⠀⠰⠁⡀⠀⢀⡤⠖⠁
⠀⠀⠀⠀⠀⢀⣞⡽⠁⠀⠀⠀⠀⡀⠀⠀⣹⣇⠀⠀⣸⣿⠵⡀⡺⠋⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⣀⣠⠾⠋⠀⠀⣀
⠀⠀⠀⠀⣡⡿⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠿⣿⠀⢸⣿⣷⡖⡳⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⣀⠴⠚⠁
⢀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠀⢸⣿⣷⣿⡅⢀⣔⠀⠀⢀⠀⠀⠀⢀⡤⠾⠁⠀⠀⡶⠚⠁
⠈⠐⢼⠗⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⢀⣿⣿⠿⠋⠠⠜⠁⠀⠀⠔⠀⣠⠶⠃⠀⠀⠀⢀⣠⡤⢂
⠀⠀⠀⠀⢸⡄⡀⠀⠀⠀⠀⠀⢀⣁⣠⣴⠟⠀⡾⠟⠁⠀⠀⠀⠀⠀⠀⣠⠂⠈⠀⠀⠀⠀⡰⠖⠋⠁⠂⠁⠀⠀⣀⢀⠄⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠘⡇⠀⠀⠁⠐⠀⣠⢾⣹⣿⣅⢲⣤⡞⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⣀⠀⠊⠀⡀⠀⠀⠀⠀⣀⠰⠋⠀⠀⠀⠠⠐⠂⠈
⠀⠀⠀⠀⠀⠣⣀⢀⠀⢀⠔⠁⣾⣿⡝⣿⣿⣷⡀⠈⠀⢀⡤⠞⠋⢀⡀⠀⠐⠀⠉⠁⠀⣀⠔⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣟⠀⠁⠀⢰⣿⣿⣏⣹⣿⣿⢲⣦⣶⡉⠀⣀⠐⠋⡠⠀⠁⠀⠀⠀⠀⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣸⣏⡛⠓⠲⢦⣿⣿⣿⣼⠋⣰⡿⢑⠙⣷⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⣞⣿⡌⠳⢦⡀⣿⣿⣿⣽⣿⣦⠟⣱⠋⠁⣿⣶⡻⠂
⠀⠀⠀⢠⣿⡟⢻⠋⠠⣾⠿⣿⣿⣷⣯⣬⣥⣶⣿⣄⢀⡿⠉
⠀⠀⢠⣿⡟⢀⣿⣦⣏⣡⣰⣅⣽⣿⣿⣿⡻⠉⠀⠋⣽⠃
⠤⢤⣿⣟⢀⣿⣿⣶⣤⡴⢏⢍⣥⣼⣿⡁⠀⢀⣠⡾⠁
⠀⣼⣿⢿⡟⣿⣭⣟⣃⣭⡿⠟⠛⠉⠙⠛⠛⠛⠁
⢠⣿⣧⠞⣰⣿⣿⠿⡛⠉
⠘⣿⣇⡴⠛⠉⠉
⣶⡾⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣽⣷⣶⣶⣶⣦⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⣠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣴⠀⡰⢁⠞⠁⠀⠀⠐⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡟⠀⠀⠀⠀⣠⣿⣿⣿⠇⣜⠵⠋⠀⠀⣀⢀⡷⣫⡤
⠀⠀⠀⢠⣶⣷⠀⣀⠀⠀⠀⠀⠀⠀⢀⢆⡿⣱⢇⢠⣾⣴⣿⣵⣿⣃⠴⠀⠀⠀⢀⠔⣱⣯⡾⠋⣀⡀
⠀⠀⠀⠸⣿⣧⠌⢿⣿⡀⠀⠈⠐⠂⠞⠼⣰⢧⣿⢿⣾⣿⣿⣿⣿⡟⠀⠀⢀⠔⣹⣿⣿⣫⣠⣴⠏⠁
⠀⠀⠀⠀⠈⣿⣶⣿⣿⣿⠀⡄⠀⠁⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⢟⢁⣤⠖⣡⣿⣿⣋⣉⢼⣯⠖⠁
⠀⠀⠀⠀⠀⣿⡟⠉⠙⠻⠀⢸⣶⣾⡷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢃⣴⣾⠟⣻⣿⢿⡥⣚⡡⠄
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣿⣿⣿⡗⠀⢸⣿⣿⣿⣿⣿⡝⣻⣴⣿⢟⣷⣿⣿⡷⡿⠚⠉⣀⣤⡦
⠀⠀⠀⠀⢏⠀⠀⣴⠳⠀⠀⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣾⣿⣿⢿⣷⣿⣟⡩⠟⠁⢀⣴⣛⣫⣥⠒
⠀⠀⠀⠀⠘⣠⣼⡉⠀⠀⣠⣿⣿⣿⣷⡀⠙⣿⡿⣻⣿⣿⣿⣿⣿⠿⠎⠉⣠⣴⣾⣿⣿⣿⠿⣩⡄
⠀⠀⠀⠀⠙⠃⣿⠦⣄⣠⣹⣿⢿⠻⣿⣿⠀⢘⣴⣿⡿⢷⣾⠋⢛⣠⣶⣿⣿⣿⡿⢛⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⣿⠋⣻⣏⣻⡿⠤⣬⣟⣾⣇⠀⢄⣍⡻⣿⣿⡾⢿⣿⣿⣿⣿⣯⣶⠉⠁⠀⢤⣆
⠀⢀⣠⡴⠶⢾⣿⢤⣁⠊⣿⣇⠀⣿⣿⣿⣿⡄⠹⣟⣳⠘⣿⣟⣯⣿⣿⣿⣿⣭⣼⡿⠛⠛⠋⣀
⢠⡿⠁⠀⠀⢸⣿⠀⠉⢻⡿⢋⣳⣯⣿⣟⡻⠇⣼⣿⢑⡆⣿⣿⣿⣿⣟⣿⣿⣿⣿⠴⠔⠞⠁⠁
⣿⠁⠀⠀⠀⣸⣿⠀⠀⠀⢻⣿⣿⣿⡿⣿⡵⢞⣩⣌⣿⠇⣿⣽⡿⠟⠃⠈
⢿⡀⠀⠀⠀⠛⠈⠀⠀⠀⢸⣿⣿⣿⡿⣾⣿⣷⣽⣿⣥⠾⠋⠈
⠘⢷⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⠿⠋⠉
⠀⠀⠙⠷⣶⣤⣤⣴⠾⠋⠀⠈⠉
⣿⡯⣟⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠲⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣧
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢀⡞⠁⠉⠳⣄⡈⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠘⢦
⠀⠀⣀⡴⠛⠉⠀⠀⠀⠀⢉⡟⠳⣄⠀⠀⠀⠙⠦⣄⠀⠈⠙⠒⠲⠶⠒⠋⣹⠁⠀⠀⢈⡷⣖⠋⠉⠉⠉⠙⠓⠦⣄
⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠈⢳⡄⠀⠀⠀⠈⠓⢦⡀⢠⡖⠀⠀⢰⠇⠀⠀⡴⠋⠀⠈⠳⡄⠀⠀⠀⠀⠀⠈⠳⡀
⣼⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠙⠲⣽⡦⠶⢯⡀⣀⣼⠁⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢹
⡇⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠉⡏⠉⠓⠲⠦⣤⣀⣸⡆⠀⠀⠀⠀⠀⠈
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠺⣅⠀⣧⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢰
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠋⠉⠉⠉⠛⠛⠒⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠉⠛⠲⠦⠤⠤⠤⠴⠺⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⢷⢀⡤⠄⠀⠀⠀⠀⠀⠀⠤⢤⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⢁⡘⠛⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⣠⠖⠛⢲⡘⣿⡷⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⠁⠀⠀⠀⣷⡇⠀⣤⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠘⢦⣀⣀⠀⠉⠀⠐⢯⠇⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⣧⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⡟⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠦⣤⣀⣀⣀⣀⣀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠦⣄⠀⠈⠹⡯⢭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⢤⣀⠀⠈⠙⠲⣄⣻⣼⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⢻⡓⠶⢦⣠⡽⢿⡀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢠⢳⡀⠀⠀⠀⢸⡇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⢹⠀⠀⠀⠀⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⠤⠤⠤⠼⠤⠤⠾⠤⠤⠤⠴⠧⠤⠼⠿⠆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀
⠀⠀⠀⠀⢠⠏⠓⡾⠁⠀⢹⡀⠀⠀⠀⡀⠀⠀⢀⡤⠶⠒⠲⣤⡀
⣠⣤⢠⣄⡾⠀⣰⡁⢸⣡⠀⣧⠀⢀⡴⠁⢀⡔⠁⠀⠀⠀⠀⠈⢷
⠷⢫⢩⣄⠀⠀⠀⠈⠳⡄⣔⣿⡆⢉⡤⠲⢾⢀⡀⠀⠀⠀⠀⠀⣸
⠀⠈⠁⠘⣆⢀⡴⠲⢤⡝⢮⠁⢹⡏⠀⠀⢨⡿⠟⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡀
⠀⠀⠀⠀⠈⠉⠀⠀⣀⣙⣭⡇⠈⠳⢤⡤⣾⠦⣄⣀⣀⣀⣴⣏⣀⣠⣤⡤⠤⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⡖⠲⢤⡀⡀⣄
⠀⠀⠀⠀⠀⠀⢰⠋⣭⣾⢭⠳⠀⡀⠀⠔⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢳⡀⣸⠳⠷⡏
⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣤⡄⠰⠵⠀⡠⢆⠀⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣏⣹⣿⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⠀⠙⠋⠀⣍⡆⠀⠀⠙⠲⢤⣀⠐⠠⠄⣄⢀⡀⣀⣠⠟⠀⠀⣠⢴⡁⠈⣯⡞⡫⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢨⣱⢀⠴⡀⢀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡖⠚⠒⢻⣹⠁⠀⠀⠀⠙⠀⣉⣽⣩⠧⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠶⡇⠀⠀⠈⠛⠁⠧⠇⠀⠀⠀⣀⣀⣀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣄⡷⠴⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡛⠤⣿⡀⠀⠀⠀⠀⢀⡤⠶⡎⠙⠍⠳⡄⠀⠀⢸⣄⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁⣹⠲⢲⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣌⣐⠈⠉⠒⠒⠾⠭⠄⣀⣙⣦⣶⣚⣁⣀⠀⠈⣧⠿⠍⠉⠉⢧⠀⠀⣠⠞⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠙⠓⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⢠⠀⠙⣶⣿⠀⢤⡀⠀⣸⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠉⠉⠓⠲⠤⢤⣀⡾⠀⠀⠀⠘⠀⠀⠳⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧⡠⠤⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⡄⠀⢸⡀⠀⠀⠀⢀⣀⣀⣀⣤⡤⢤⡖⣲⠋⠉⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠻⡀⠈⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⡜⢠⠇⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣎⠓⠀⢱⡀⠀⠀⠀⣀⣀⣀⣠⣎⣠⣋⣀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⢥⡄⣀⡜⠁⠀⠀⠀⠀⠊⠉⣿⣉⣉⡥⠤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣶⣶⡒⠒⠒⠚⠹⣏⠉⠉⠉⠉⢉⡩⠕⠛⠧⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⠛⠛⠻⣝⢦⣀⣀⡼⡆⠀⠀⠀⡯⣀⣀⣠⠴⣒⡿⠿⢍⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⣒⣛⣛⣩⠬⠭⠖⠊⠁⠀⠀⠀⠑⠲⠤⣉⣉⡓⠒⠶⠤⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣒⣒⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⠃⠀⠀⠀⠀⠀⠀⠀⠠⣖⡒⠒⠚⠋⠉⢉⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠽⠂⢰⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠰⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡖⢀⡈⠢⡀
⠀⠀⠀⠀⠀⣠⠴⠒⠉⠉⡉⠉⠓⠲⠞⢀⣄⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡜⠙⢆⠙⢤⠴⠒⠋⠉⣉⠉⠑⠲⢤⡀
⠀⠀⠀⡴⠋⣠⠔⠊⠉⠉⠉⠉⠙⢲⠀⠈⠢⡳⣄⠑⢄⠀⠀⠀⠀⠀⠀⢀⠇⢰⠃⢀⡴⠃⡀⢰⡊⠉⠁⠈⠉⠉⠒⢤⡈⠳⡄
⠀⢀⠎⢀⠞⠁⠀⠀⠀⠀⠀⠀⢠⠇⡰⠙⣄⠘⡌⠢⡈⠳⡄⠀⠀⠀⠀⡜⠀⡏⢀⡎⢠⠎⠉⢦⠑⣄⠀⠀⠀⠀⠀⠀⠙⢆⠈⢆
⠀⡜⠀⡎⠀⠀⠀⠀⠀⠀⢀⡴⢁⠜⠁⠀⠘⡆⠸⡀⠘⢦⡈⠢⣠⡤⣴⠁⡸⠀⡸⠀⡎⠀⠀⠀⠱⡌⢢⡀⠀⠀⠀⠀⠀⠈⡆⠘⡄
⠀⡇⢸⠀⠀⠀⠀⠀⠀⣞⢁⡔⠁⠀⠀⠀⠀⠃⠀⡇⠀⠀⠑⢄⠙⠇⠃⢠⠃⠀⡇⠀⣇⣀⡠⠤⠤⠞⠂⠙⡆⠀⠀⠀⠀⠀⢸⠀⡇
⠀⡇⠈⡆⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡇⢀⠇⠀⠀⠀⠈⠳⡄⠀⠘⠋⣉⣁⠀⣤⠤⠴⠒⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⡌⠀⡇
⠀⠸⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡜⠀⠀⠀⠀⠀⠀⠳⡄⡞⠉⠁⠘⡄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡼
⠀⠀⠙⢄⠘⠢⣀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣵⢣⡀⠀⠀⠘⢆⠈⠢⣄⠀⠀⠀⠀⠀⠀⢀⡠⠚⢀⠜⠁
⠀⠀⠀⠀⠑⠦⣈⠉⠒⠒⠒⠒⠊⢁⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⠦⣀⠉⠒⠒⠒⠒⠊⢁⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣴⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⢿⣿⣿⠏⠀⠀⠈⠃⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠛⠁⠀⠀⠀⠀⣾⠇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⢻⣿⣿⣿⣿⣷⣦⣀
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⣙⠿⣿⣿⣿⣿⣿⣷⣄
⠀⢨⣿⠃⢠⣤⣀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣮⣝⠻⢿⣿⣿⣿⣦
⠀⢸⡏⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠛⢿⣦⡙⣿⣿⣿⡄
⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣿⣷⠸⣿⣿⣇
⠀⢸⠁⠐⠐⠛⠋⠙⠛⠻⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢹⣿⡇⢻⣿⣿⡀
⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢿⣿⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠲⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⣀⡴⠛⠉⡠⠒⢁⡞⣠⡴⠶⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢧⠞⠁⠀⣠⠊⠀⣀⠾⠋⠁⢀⣀⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡼⢁⣀⣴⣯⣤⡴⠛⠉⠉⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⡄⠀⣀⡤⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢠⡎⠈⠑⠦⡀⠀⠀⢰⣦⣟⠀⢸⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⡠⠚⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠛⠀⢀⡾⠁⢈⡷⠋⠀⠀⠀⢠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠛⠓⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⣤⠒⠒⣲⠄⠀⠀⣼⡀⢀⡜⠁⠀⠀⠀⠀⠻⢋⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⠀⢻⠀⠀⠀⠠⣔⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠹⠴⠛⠁⠀⠀⠀⡿⠌⢸⡇⠀⠀⠀⠀⣠⠞⠋⠉⠉⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡁⠀⠘⣇⠀⠀⠀⠈⠛⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠞⠹⡇⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⡆⠀⠀⣴⠉⠀⠙⣆⠀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡤⢿⠀⠀⠀⢿⣤⡤⠶⠶⠶⠟⢋⡇⠀⠀⠀⡄⠀⠀⢹⡙⠛⢻⠏⠀⢸⡍⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⢇⠀⠀⠘⢧⡀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢳⡄⠀⠀⠉⠈⠁⠀⠀⢸⣇⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠷⣤⣘⡄⠀⠀⠀⠙⠛⠚⠛⠁⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⠶⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣴⠾⠋⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣰⣿⣿⣷⣭⣿⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢠⣇⡘⠻⣿⣿⡏⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢰⠋⢙⡷⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠁⠉⠙⠻⠿⠀⠀⠀⢠⡾⠋⠀⠀⠀⢠⡞⠙⠓⢾⠁⢀⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣛⡭⠿⠿⢷⣶⡦⠤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢸⡆⠂⢴⡿⠤⡎⣀⡾⠁
⠀⠀⠀⣠⠶⠚⠛⠛⠉⢁⡴⠚⠁⠀⠀⠀⠀⠸⣤⣤⣬⠿⣤⣀⠀⠀⠀⠀⢀⣠⡴⣏⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⢀⡀⠠⠘⠁
⠀⠀⣾⠁⡀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠈⡼⠉⠛⠛⠛⠛⠋⣹⣦⢘⣷⠀⠀⠀⠀⢀⣤⠞⣡⡴⠋
⠀⠀⠉⢶⠁⢀⡀⢀⣀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡇⠀⢀⠀⠀⠀⠀⠌⢿⠟⠙⠻⢶⣶⣾⡯⠶⠚⠁
⠀⠀⠀⠈⠛⣿⡀⣸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⠀⠀⣹⠗⢐⣶⠀⡄⢸⡇
⠀⠀⠀⠀⠀⠈⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣘⠀⠙⠃⠀⠾⠏⢀⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣦⠀⠀⠀⠀⠀⢀⣿⣤⣀⣀⠈⠛⠛⠶⠶⠶⠶⠖⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣼⠏⡰⠃⢸⣇⠀⠀⠀⠀⢸⡗⠀⠉⠉⠙⠛⠓⠶⠤⣤⣄⣀⣀⣼⠁
⠀⠀⠀⠀⣠⡿⢡⠞⣀⣴⣏⣹⡄⠀⢀⣴⠞⢻⣇⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⣹⠏
⠀⠀⢀⣾⠋⡴⠃⠰⣿⡟⢿⡀⢻⣄⣼⣿⣄⠀⢙⡷⣴⣆⡀⠀⠀⠈⠙⠛⣺⡧
⠀⣰⠟⣡⠞⢦⢀⣴⠋⠀⠈⢷⣀⣩⡼⠋⠙⠛⠉⠀⠀⠈⠙⠳⠶⣤⡶⠿⣿⣟⡳⠶⣤⡀
⣼⣋⣴⣁⣠⡾⠟⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠆⢀⡿
⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣥⣼⣿⣶⣶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠧⣄⣀⣀⣙⠉⠁⠀⠀⡔⠋⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⣶⣦⣭⣤⣭⣉⣉⣛⡓⠶⠶⡤⢿⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣙⣛⣛⣿⣷⣿⣇⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣷⣼⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣠⣤⡤⠄⠀⠀⠻⢿⣿⣻⠟⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⣀⣴⣿⣶⣦⣄⡹⣿⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⡿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⡻⣿⣿⣿⣿⡿⠋⠁⠀⢀⣟⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠉⣠⣿⣿⣦⡙⡿⠋⠀⠀⠀⣠⣾⣿⡄⠀⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⠻⡄⠀⠀⠀⠉⠙⣿⡅⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣦⡁⠀⠀⠀⠀⠀⠙⣧⣄⠀⠀⠀⠀⠈⠳⣦⣰⣤⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⢷⣤⣄⣴⡏⠹⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠉⠛⢿⣿⣷⡄⠀⠀⠀⣀⠠⠄⠐⠚⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣾⣥⣿⠛⠃⠀⠈⠁⠀⠀⠀⠀⡠⠂⢻⣿⢿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠶⠶⣶⣦⣤⣀⠀⠀⠀⠀⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⢳⣸⣆⢥⣤⣤⣤⣄⣀
⠀⠀⠀⠀⢀⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⢟⣷⢖⡠⣾⠿⣯⠁⠀⢰⡆⠀⠀⠀⠀⢀⠜⠁⠀⠀⢀⣴⡿⣻⣿⣾⠓⠂⠘⠉⠛⡷⣦⡀
⠀⠀⠀⣴⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢞⣴⣟⢿⣆⠀⢹⡆⠀⢸⣧⠀⢀⠀⡰⠃⠀⠀⠀⢠⣿⠋⠀⠘⣿⢿⡄⠀⠀⠀⠀⠈⠪⡻⣦
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣯⢻⣆⠈⡷⠀⠸⠿⠀⣁⠎⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠹⣎⢿⡀⠀⠀⠀⠀⠀⠀⠽⣧
⠀⢠⣿⡁⠀⠀⠀⠀⠀⠀⣠⣶⢂⣥⠀⠀⠀⠙⢿⣿⡿⣿⠄⢕⣦⡤⠀⠀⠁⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢿⣼⣧⣀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⢸⣿⠄⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⠀⠀⠀⠀⠙⠛⣋⣛⣶⣿⣿⣷⣤⣴⣿⣷⣶⣤⣤⣄⢸⣧⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠂⠀⠛⢻⡇⠀⠀⠀⠀⠀⢰⣿⠀⢻⣿⣿⣿⡿⠿⣿⡿⠿⠟⠉⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡇
⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⣀⠀⣀⣤⣿⠇⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠁
⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⣀⣀⣠⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⣤⣤⣤⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠄⠀⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣠⣿⡟⠀⠀⢸⡧⠉⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠁⡞⠁⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠠⠀⠐⠒⡄⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡿⠃⠀⢄⡀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣂⡔⠁⠐⠤⠸⠁⠀⠀⠀⠈⠈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡤⠀⠀⠀⢠⠖⡉⠂⠠⠀⠀⢀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠄⡽⠀⡀⠀⢀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠉⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⠀⠀⠀⠙⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢀⠒⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢄⡀⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠚⠣⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣄⠐⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠃
⠀⠀⠀⠀⡄⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⢀⡀
⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠈⠉

⣟⣿⣿⣻⣛⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣻⣿⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠆⠠⠀⠀⠃⠀⠑⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⢰⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢀⠀⠐⠂⡰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠐⠚⠁⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠳⠤⢄⠀⢀⠠⠊⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡋⠀⠀⠀⢐⠂⠀⠀⠀⠀⠸⠶⢖⠒⠲⠦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠠⠀⠐⠒⠦⢤⠁⠀⠀⠀⢀⡀⠀⠀⠀⢀⡃⡀⠀⠐⠄⢀⡀⢈⡒⣄⠀⠀⠀⠀⠀⠀⠀⠈⠒⣕⢄
⠀⠠⢂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠙⠢⡀⠀⣠⡁⠀⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠔⠃⣄⡄⠀⠉⠒⠤⢁⣴⠋⠀⠀⠀⠀⠐⠄⢀⠀⠀⠀⠀⠀⡄
⢰⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠹⠂⠀⠀⠘⠭⢶⠏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⡈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠊⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠱⡐⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠈⠃⡀⠀⠀⠀⠀⠀⢀⠀⡢⠂
⠀⠀⠈⠪⠐⠄⢀⠀⠀⠀⡀⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠒⠉
⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣭⣿⣯⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⣆⠀⠀⢀
⠀⠀⠀⠀⠀⢤⠀⡀⠷⠀⠀⠾⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⣰⡸⣿⠼⠸⣿⣴⣠⡾
⠀⠀⠀⣀⠀⠈⢁⣠⣴⣶⣶⣦⣤⡈⠡⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠤⢤⣄⣢⠟⣫⣴⣶⣶⣶⣶⣤⡙⠷⣣⣴⠆
⠀⠀⠀⠈⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣽⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⢵⡂⣀
⠀⠀⠒⠂⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠋⠁⠀⠀⠀⠀⠀⠒⠛⠿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣛⠉
⠀⢀⣀⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠠⠤⠄⠀⠀⠀⠀⠀⠀⣠⣭⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣦⣤
⠀⠀⠀⠐⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠨⢶⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⣟⠂
⠀⠀⠀⠊⠁⠀⡈⠙⠛⠿⠿⠟⠛⢁⡀⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⡳⢦⣝⠻⠿⠿⠿⠿⢛⣡⣜⠯⠙⠓⠂
⠀⠀⠀⠀⠀⠘⠁⠈⢰⠆⠀⢰⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠋⠏⣿⠶⡖⣿⠙⠌⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢀⠇⡇⠀⡠⠴⡄⢀⢠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⡇⠘⠾⠀⠘⠊⠀⠀⠙⠛⢤⠀⠙⢆⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢇⠓⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣙⠚⠁⠀⠀⠀⢀⡠⠔⠒⠉⠑⠒⡶⠀⠀⠀⠀⢨⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡗⡒⠉⠁⢀⣠⣢⠀⠀⡸⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠜⢢⠀⠀⣞⠀⠈⡇⢀⢧⡆⠀⣀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⡜⠀⠀⢿⢀⡴⠃⣸⠁⠹⠏⣅⢸⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠙⠊⠐⠉⠀⠀⠉⢀⣴⣿⡄⠀⢀⠡⢞⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠢⣤⣀⣠⣤⣾⡟⠋⢹⠇⠀⢀⣤⠞⢛⣣⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠈⢿⡿⠛⡟⣀⡴⠋⠀⡴⠋⣟⠉⢹⠈⢩⠃⠁⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢯⠟⠒⡄⠈⠓⠤⣍⣛⣉⣁⣠⠤⠞⠁⢀⣸⠤⠞⠀⢇⣠⠆⢠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢇⡔⠰⡇⠀⠀⠀⢀⣀⡴⠋⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠑⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠍⠊⠳⣌⡑⠒⣒⣳⠎⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⣏⣀⣀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢫⡉⠉⠉⠁⠀⠀⠀⠀⠈⣹⠃⠀⠀⠀⣠⠦⠤⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢨⢗⢳⠀⣀⡤⡀⠀⢀⠀⠀⠉⠒⣦⠀⡠⢷⡒⢦⢀⠀⡝⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣾⠀⠙⠊⠁⠀⠈⠉⠻⣄⠀⢀⡴⣃⡰⢋⡵⠟⠳⢮⣎⡠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠾⢛⣛⡿⢝⡢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠲⣬⠖⠁
⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠈⠑⠪⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢾⣿⠿⠟⣍⣿⠛⠛⣻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠔⠉⠀⣒⣊⠉⢉⠈⣁⡻⣿⢿⣿⠇⠀⢀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠄⠀⠘⠂⠀⢀⠈⠀⠰⠁⠀⠉⠀⡀⠨⠟⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⡀⡈⢄⠀⠀⣰⢂⠀⠀⠠⢀⠛⠊⠀⠠⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⠁⠀⠈⠣⢀⡈⠁⡀⠀⠄⠁⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢈⠃⠀⠋⠁⠠⡀⠀⡄⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣂⣀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⣀⣀⣀⢀⣀⠂⢾⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠘⣿⢳⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠏⠈⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⡀⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⣠⣤⡴⠶⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠞⠋⠉⠀⢀⣠⡄⠀⠀⠀⠠⣌⡙⠷⣄
⠀⠀⠀⠀⠀⠀⠠⢾⣿⠁⠀⠀⠀⠀⠿⢋⠀⡠⠀⢠⡀⠘⠷⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢰⢃⣞⢣⠀⢰⢻⣦⣂⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠆⠀⢰⠄⠀⣸⣿⡏⢸⠀⢸⠘⣿⣿⣄⠀⠀⠸⣦⠔⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢠⢿⠀⠈⢳⡀⠀⡞⠀⢾⣆⠀⢰⠋⠀⢰⠀⣿⣞⣽
⠀⠀⠀⠀⠀⠀⡞⠡⡙⠦⠘⠋⠀⠈⠣⣭⠉⢀⡀⠈⢧⡉⠵⡋⠀⢸⠃⢹⣍⡇⠀⠀⡤⠖⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⡿⢹⣆⠀⠀⣿⠀⠉⠁⡀⢨⣤⣄⣀⠙⠲⣄⠀⠘⠃⠘⣿⠁⠀⠀⢀⡩⠴⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣀⢰⣿⠀⠀⣰⠃⠀⠀⠀⠈⠙⠒⢚⠀⠀⠀⠀⣿⠀⢀⡴⠋⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠘⠛⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣸⡏⣰⠋⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⣧⡴⠖⢒⣒⣀⠒⠲⢤⣸⠀⣴⡟⢰⠃⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠹⣧⡀⠘⢁⠀⠀⠀⠀⠀⠀⠀⢀⣬⠾⠋⠀⢸⡀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣠⣝⡓⠲⣿⣦⣤⣤⣤⣤⡶⠋⠀⠀⠀⢀⡀⢉⢾⣤⣀⣀⠴⠋⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡜⡴⠓⣋⡤⣾⣧⢭⣽⣍⡗⢦⠀⠀⡰⣇⢙⢿⣁⡏⠀⠠⠔⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣿⠆⠀⠀⠸⢞⡟⣿⠓⠄⠋⠀⣯⡳⣄⣷⢈⠏⠀⠈⢳⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠉⠁⢀⣀⣀⣠⡞⢁⣿⠀⠀⠀⠐⢿⡵⢉⡹⣌⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠠⣇⡀⣀⡴⠋⠀⠀⢸⣤⣾⣿⣧⣥⣤⣄⣸⡇⢾⠀⠈⢡⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢠⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⣤⡶⠶⠶⣶⡌⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢿⣙⢿⡃⢻⡇⠀⠀⣿⠁⣼⣋⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠭⠤⠖⠒⠚⠃⠀⠀⠀⠀⠈⠈⠷⠤⠤⣈⡓⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣍⡉⠉⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿⣷⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⢿⣿⣿⣖⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⢀⣨⣿⣿⣿⣿⠀⠈⠉⢛⡿⠿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢏⣿⣿⠟⣸⣿⣧⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢻⣿⣿⣿⣿⡟⠁⣰⡿⣻⡿⣯⠉⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠠⢻⣿⣿⣿⢻⣿⣴⡿⢱⡟⣧⠸⡆⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣅⢨⣿⣿⣹⣾⣻⡿⠁⣿⠀⢹⡄⢻⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠿⠭⣿⣿⣿⣿⡿⠁⠠⣿⠀⠈⢧⣤⠧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠃⢠⣿⣿⣿⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⢈⣿⣿⣿⡅⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠉⠉⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣰⣯⡴⠞⠛⠉⠉⢁⡀⠀⠀⠈⠉⡙⠻⢦⣄
⠀⠀⠀⠀⣠⣶⡏⠁⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⠀⠙⢷⡄⠉⢷⡄
⠀⠀⠀⠈⢡⣿⠃⠀⠀⠀⠀⠀⢁⠖⣠⡎⠀⠈⡷⣄⢈⠁⠀⠀⠹⡄
⠀⠀⠀⠀⢸⣿⡆⠀⠀⡀⠀⠀⣾⣼⡇⢰⠀⠀⠃⣿⣦⡆⠀⠀⠀⢳⠀⢀
⠀⠀⠀⠀⢻⣿⠑⠀⢠⡿⠀⢠⡞⠿⠃⣸⠀⠀⡇⠙⠟⣻⠄⠀⢀⢸⡾⢉⣱
⠀⠀⠀⢀⡼⢿⡀⠀⣞⡇⠀⢀⠱⣄⣠⠏⠀⠸⡽⢄⣠⢏⡀⠀⣿⠘⣷⣧⡜
⠀⠀⠀⣿⢀⢦⡙⠀⠉⠀⡄⠈⣓⠶⠀⢠⠄⠀⠙⢦⡐⠫⠀⠀⢹⠀⢻⣤⡇⠀⠀⠔⢊⠡⠤⠤
⠀⠀⠀⠹⣞⠈⣷⡀⠀⢸⡇⠀⠈⠀⡀⠰⠶⠦⣄⡀⠙⠢⡄⠀⠈⠀⠘⣿⠀⠀⠀⢀⡤⠖⠚⠓⠒⠄
⠀⠀⠀⠀⠈⠳⣌⣁⡀⡎⡇⠀⠀⣼⠁⠀⠀⠀⠀⠈⠉⠐⣁⠀⠀⠀⢠⣿⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠓⠃⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⣼⡇⣰⠋⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠸⣇⡤⠖⠒⣒⣒⡂⠠⠤⣄⣿⢀⣼⠟⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠘⢷⣄⠀⠲⠉⠀⠀⠀⠈⠉⠁⠀⠀⢈⣩⡾⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣙⡳⠦⢬⣶⣤⣀⣀⣀⣀⣠⡴⠋⠁⠀⠀⠀⠈⠓⡶⣀⠀⠀⣀⡴⠋⢀⠄
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⡟⣹⠤⠖⢉⣹⡏⢛⣛⡟⢻⢤⡀⠀⠀⢰⡊⠲⣾⠞⡽⠉⠉⣁⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠈⠁⠀⢺⡭⡿⣿⢯⡛⢵⠃⢠⠳⡀⠀⣏⠙⡖⠈⠓⣧⡀⠀⠄⠒
⠀⠀⠀⠀⠀⣰⣯⠶⠶⠋⠀⠀⠀⠈⣸⠇⣼⠀⠀⠀⠀⣼⡿⣸⠶⢷⠸⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⡼⠁⠀⠀⣀⡴⠲⠒⢻⠁⣰⣿⣇⠀⠀⠀⠈⣿⠁⡴⠛⠧⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠳⠤⠤⠞⠁⡀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣾⣄⣧⣀⣀⣼⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⢶⡶⠒⠒⢲⣶⠈⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⢿⣉⠿⡅⢸⡇⠀⠀⢸⡿⢠⢾⣉⡧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⡥⠤⠖⠒⠛⠃⠀⠀⠀⠀⠀⠀⠘⠦⠤⢤⣈⠛⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣤⡾⠟⠉⠉⠉⠉⠻⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣾⠏⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣼⡏⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⢠⣄⠀⠹⣷⣄⣠⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢠⣿⠁⠀⠀⠈⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣴⡟⠀⢀⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢻⣦⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⣿⡀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⣠⡿⠀⠀⠀⠀⠈⠿⣦⣀⠀⠀⠀⠀⣀⣴⡿⠁⠀⠀⠀⠘⣿⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢀⣀⣿⡟⠛⠿⠿⠛⢻⣿⣀⡀⠀⠀⠀⠘⠛⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⡶⠿⠛⠋⠙⢷⣤⣄⣠⣤⡾⠋⠙⠛⠿⢶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠋⠉⠉⠀⠀⠀⢀⣴⠄⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠻⢷⣄⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⣤⣶⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⡶⠃⠀⣿⠇⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢸⣿⠀⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣤⣤⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⣀⣀⣤⣤⣾⣿⣷⣤⣴⣶⣶⣿⣯⣀⡀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣷⣦⣤⣾⠟⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⡀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⠀⠀⠀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣶⡿⠋
⠀⠀⠈⠙⠷⣦⣤⣀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠰⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⣀⡀
⠀⠈⠉⠉⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠉⠛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠈⠙⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣶⣦⣤⣤⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⡁⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠉⠙⠛⠛⠛⢻⣿⡿⠛⠉⠉⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠈⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⢈⡷⣄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡀⠀⠀⠀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡤⠒⠁⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣠⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠛⠢⡺⣯⡆⠀⠀⠀⢰⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⠀⢠⠉⠈⣀⠙⣷⡀⠀⠀⢸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠐⢹⠀⣿⠀⢸⣿⡑⢾⡇⠀⠀⠀⣿⠃⠀⠐⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⡟⠉⠛⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢿⣀⣠⠄⠐⢸⣧⣄⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⠛⢛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠨⠀⠀⢸⡟⠉⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡿⠛⠛⠒⠚⠀⠀⠘⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠰⠁⠀⠀⢠⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡶⣛⠶⠥⠾⠶⢦⡴⡟⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣥⣼⣿⣿⡿⣿⣿⣷
⠀⢸⢹⢳⡒⣿⣥⣄⡴⠛⣁⠀⠀⠄⠤⠊⠀⡇⠀⡇⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠁⣸⣿⡇
⠀⢸⣏⠀⠀⢸⡟⣡⠂⢩⠃⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⣿⡟⠁⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠘⠋⠀⠀⢸⢸⡇⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠰⠀⠀⠀⠀⠀⠛⠀⠀⠸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠿⢯⣑⣠⡄⠀⠀⠀⠀⠀⠀⠀⢸⡧⠓⢒⡲⠀⠀⠀⠀⠀⠀⣽⣿⣿⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⣣⣀⣀⣀⠀⣀⠔⠊⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠉⠙⡂⠈⠉⠀⠑⠢⢔⡻⢶⣊⡉⡉⠐⢢⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⢒⠧⣀⠀⠀⡔⠙⠶⣤⠤⢤⠖⠋⠉⠀⠙⢀⡸⠀⠨⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⢊⡀⠈⠳⣀⠄⠀⡠⢸⡞⠑⠒⠤⢄⣠⠂⠉⠀⢁⡾⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠔⠡⡃⠈⡆⠀⢸⢇⠜⢠⠃⠘⡀⢀⡠⠔⠉⣱⠒⡲⠯⢤⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠙⡄⠀⡩⠖⠓⠤⣼⣸⠑⠶⢤⠀⣵⠁⠀⠐⢹⠝⠛⠄⠀⢨⠀⠈⣱
⠀⠀⠀⠀⢰⣶⡦⠤⠤⠤⠤⠤⢼⠤⠤⣤⠾⢯⠤⠤⣤⠮⠤⠟⡄⢠⠀⣿⢸⠀⠀⡸⢿⠤⠤⣼⠤⢬⢷⡭⠤⠤⠤⠤⠤⠤⠤⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠣⠀⠈⣦⠴⠥⠤⡀⣠⠜⣎⢻⡇⡾⠀⠀⡇⢸⣀⡴⠃⠀⡌⢀⠕⡖⠢⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠴⢒⢽⣀⣀⡠⢾⠁⠀⢈⡟⢿⣧⠤⠤⠷⡊⢇⠀⠀⠀⣷⠃⠀⢸⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢸⠀⠀⠀⢀⡉⠶⠮⠒⠁⣻⣄⠀⡖⠙⢝⡗⠄⠠⡇⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢄⠀⡠⠋⠀⠀⡠⠐⡾⠁⠈⠑⢧⡀⠀⡗⠒⠊⠀⢀⠞⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠀⠀⢠⢊⡠⠊⠀⡠⠔⠒⠙⢏⠉⢳⠲⡖⠊⠁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⢋⠁⠀⡠⠊⠀⠀⠀⠀⠈⠢⢄⣣⠼⠀⠀⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢦⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠛⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⡤⠀⠀⠀⠀⠀⠀⠛⠋⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢻⣿⣿⣿⣿⣿⣿⣤⣶⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣿⠇⠀⠀⠀⢀⣶⣄⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⣾⣿⣿⣿⡿⠙⢿⣿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡄⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣿⣿⣧⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⣤⣀⡀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⠛⠿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⡏⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⣸⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⣼⣿⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃⢀⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣴⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠋⠀⠀⠀⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢆⠠⢠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠁⢀⡤⢄⡈⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡞⠁⠀⠀⠈⢦⠀⠉⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⡇⠀⠀⠀⠀⡼⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢂⡠⠈⠲⠤⠤⠚⠁⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠞⠀⢀⠄⢠⠀⠹⠀⠀⠀⠀⠀⢐⠤⠲⢤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⢰⣅⠀⠀⠀⢳⠀⣠⣤⣉⠐⢋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡯⣧⠀⠀⠈⡿⢥⣀⠉⠳⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣯⠚⣏⠛⠿⠤⢤⠞⣈⠀⡌⢳⡀⢸⡇⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣆⠊⠢⠤⠤⠼⣄⣚⡀⠘⣼⢃⡜⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠒⢲⠆⠐⠀⠀⢈⠉⢯⠉⣄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢆⣀⣀⣀⣀⣀⣀⣀⣀⣀⢴⡃⠀⠀⡀⠀⢠⡄⠘⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢤⡀⠀⠀⠀⣀⣀⣀⠤⢽⡀⠀⠑⢄⡀⠅⡠⠃⢱⠀⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠓⠒⠛⠛⠉⠉⠀⠀⠤⠀⠀⠙⡄⠀⠀⠉⡾⠀⠀⠈⡆⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠥⠤⠤⠤⠤⠤⠤⢤⣤⠤⠤⠤⠾⠤⠒⠊⠀⠀⠀⠀⠳⠬⣿⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠈⠐⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠂⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠀⡜⣀⣀⠠⠤⠤⠤⢉⣀⠀⠀⠠⠒⠒⠒⠒⠁⠈
⠀⠀⣀⡀⠀⠀⠀⠠⢀⠀⡇⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠈⢦⠀⠀⠀⠀⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀
⠀⠀⠀⠀⢸⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⠉⠒⠤⠤⠂
⠀⠀⠀⠀⠸⠄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⢀⣀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠰⣅
⠀⠈⠀⠑⢄⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠱⣄
⠀⠀⠀⠀⢀⠕⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠓⢒⠃
⠀⠀⠠⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣀⠠⠐⠁
⠀⢠⠁⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢄⠁
⠀⠒⠈⠀⠁⠂⠤⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠘⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠛⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⣓⠶⠒⠒⠒⠒⡛⢍⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠔⠂⠉⠀⠀⠡⡀⠀⡎⠀⠀⠑⠢⠄⢘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠁⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⡏⢹⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡷⢹⡀⠀⢰⠁⠈⡆⠀⢀⡠⡎⡟⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢓⢤⡀⠀⠀⡇⡄⢧⡀⡸⠀⠀⢇⡰⢉⠊⡜⠁⠀⢀⡤⡚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⢌⠓⢤⣧⣘⠤⢽⣇⣠⣄⣸⡧⠼⣀⣇⡤⠚⡡⢊⠞
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠙⣄⠑⢄⣈⠗⠊⠉⠀⠀⠀⠀⠉⠑⠺⣁⡠⠊⣠⠋⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢖⢍⠒⠒⢺⢦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡴⡟⣉⠭⠉⣙⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢌⣉⣇⡎⠀⠐⣋⣭⡛⢦⡀⢀⡴⢛⣭⣙⠂⠀⢱⣸⣠⠞⠉⠁
⠀⠀⠀⢀⣀⣀⡠⠤⠔⠒⠒⠊⠉⣹⠀⠀⠸⢯⣿⠟⠀⡇⢸⠀⠿⣿⡿⠇⠀⠀⣏⠉⠑⠒⠒⠢⠤⢤⣀⣀⡀
⠀⠀⠀⠈⠉⠉⠓⠒⠢⠤⠤⢄⣀⣹⠀⠀⠀⠀⠀⠀⡰⠁⠈⢆⠀⠀⠀⠀⠀⠀⣏⣀⡠⠤⠤⠔⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⡏⢇⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⡸⢹⣉⡑⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣫⣍⣀⣒⣉⣼⠞⢣⡀⠀⠀⠈⠩⠭⠭⠍⠁⠀⠀⢀⡔⠳⡧⠤⠤⣑⠵⢄⡀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⠊⡠⠊⢉⡶⢄⣀⠀⠀⠀⠀⣀⡠⢴⡉⠑⢄⠙⣄⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⢊⡤⠚⢹⠉⡖⢺⡏⠙⠋⢹⣗⠒⡍⢻⠓⢤⡑⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡬⠚⠁⠀⢀⡜⡠⣁⠎⢱⠀⠀⡎⠈⢳⠘⢸⠀⠀⠈⠓⢥⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣼⡸⠊⠁⠀⠸⡀⢀⠇⠀⠈⣇⢾⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⣇⡸⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠑⡄⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢸⠀⣀
⠀⠀⠀⠀⠰⠋⠁⠀⠀⠐⢀⠆⢙⠓⠋⡅⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⠀⠀⠠⠂⠛⠲⡌⣀⣀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠂⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠁⠀⠅⠘⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠈⠈⡀⠐⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠰⠠⠏⣀
⠀⠋⠠⠔⡄⠈⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠐⠂⡀⠈⠈
⠀⠀⠀⠊⠀⠀⠀⠀⡈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡶⢤⠤⠀⣴⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠚⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡠⡀⠀⡀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠘⠀⠀⠀⢀⠀⢠⠉⠤⠒⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⡀⡟⠑⠫⢙⠤⡀⠀⠀⠄⠈⠀⠰⡀⠰⡆
⠀⠀⠀⠈⠂⠐⠁⡀⡔⢐⠊⠆⠀⠈⢄⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠋
⠀⠀⠀⠀⡀⠀⢠⡖⠀⠀⢳⠔⠂⠀⡄⠑⢄⠐⠂⢀⡀⠀⠀⢀⢀⣴⡀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠈⢙⠈⠀⠀⣀⠋⠀⠀⠀⠀⠌⡀⠐⠀⢀⢀⠀⠀⠱⡔⢀⠈⠀⢀⣀⡀⠀⡀
⠀⠀⠀⠀⠁⣀⣄⠀⠀⠂⠀⢀⠞⠁⡀⠀⠀⠀⠀⠌⣀⠖⠀⠀⠀⢂⠀⠀⢨⠠⡀⠀⠀⠁⠐⢀⠀⠤⡉⠂⠄
⠀⠀⠀⠀⠀⢀⠙⠘⢻⡼⠛⠉⠸⠙⠀⠀⠀⠀⠠⠠⠉⠃⠀⠀⠐⠂⠶⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠢⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⢠⡶⢀⠀⠀⠀⠀⠀⠀⢡⣠⠤⠐⠂⠒⠒⠈
⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠂⡅⠀⠀⠀⠀⠀⠘⠷⠰⠹⢱⢂⢀⢀⠋⠉
⠀⠀⠀⠀⠀⠒⠐⠀⠠⠄⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠠⡐⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢨⢠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣧⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢨⠁⠩⡤⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠣⣀⡀⠀⡼⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢚⡰⠾
⠀⠀⠀⠀⠀⠀⠀⠰⢻⠋⠹⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⢡⠋⣉⣉⢉⡉⠁⡀
⣦⡖⠊⠂⠀⠀⢠⠂⠐⠀⠐⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⠀⠀⠀⠈⡏⠁⡏⠀⡇
⢿⢷⡇⡞⠁⠒⠁⠀⣇⠀⠀⠀⠀⢠⣇⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣯⠔⡖⠒⠊⠈⣮⡄⠀⠀⠘⣯⠉⠙⠃
⠈⠉⠉⠓⢤⣀⣀⠾⡏⡄⠀⠀⡼⣁⠀⡹⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠒⠓⠤⣀⠔⠁⣿⠀⠀⠀⠻⢷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣆⠀⠈⠛⠙⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡄⠄⠐⠊⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⠓⠒⠀⠒⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡞⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣀⠔⠈⠀⠑⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠞⠫⡀⠀⠀⠀⠉⠉⠉⠑⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡴⠋⠈⠢⣀⠀⠀⠀⠀⠀⠈⠑⣤⣠⣀⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡾⠁⠀⠀⠈⠲⢄⣀⡠⠤⠶⣄⣏⠠⢛⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡜⠀⠀⠀⠀⠈⠉⠒⠒⠲⠦⠤⠴⣧⡇⢠⢿
⠀⠀⠀⠀⢀⡼⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠇⠀⠀⢀⠀⠀⠀⠀⡼⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣮⠏
⠀⣀⣀⣾⣉⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠝⠀⠀⠀⠸⣏⠳⠆⢰⣙⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠏
⠑⠪⢿⣍⣉⠙⡷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡢⢄⣨⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠢⡄⠀⠘⡅⠉⠑⢦⡀⠀⠀⠀⢰⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠈⣦⢠⠃⠀⠀⠀⢱⠀⠀⣠⠋⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠊⠃⠒⠢⣄⠀⡞⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⡷⠚⠁⠀⠀⠀⠀⠃⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠈⠻⡄⠀⠀⢀⣀⣀⣤⡧⠀⠀⣤⣠⣎⡀⠀⠀⠀⠀⠀⠀⣇⡠⠔⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡖⠙⠉⠉⠉⠙⠲⢤⡇⠀⠀⠀⠀⢀⡽⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣀⠀⡼⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠙⢄⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧⡀⠀⠀⠀⢀⡜⠀⠀⠀⠀⣀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡴⠚⠉⠉⠉⠉⠉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⣠⠔⠒⠂⠙⢒⠶⣐⡻⠭⠝⠛⠙⠋⠩⠿⢶⣢⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣖⡶⠿⠋⠙⠋⠙⠋⠽⢜⣣⢄⡀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⢀⠞⠀⠀⢀⣀⡰⡩⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣄⠤⠤⠤⣤⢞⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡝⢦⣒⣫⢄⣀⡀⡄
⠀⠀⠀⠀⠘⠒⠒⢴⡃⢸⢱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢻⡠⠤⢤⡆⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⡅⠘⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⣀⡼⠧⣄⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣬⠀⠀⠀⠇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⠴⠀⣠⠞
⠀⠀⠀⠀⢠⠎⠁⠀⠀⠘⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⠃⠀⠀⠀⠸⡼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠓⠒⢏⡀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠘⢮⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡵⠃⠀⠀⠀⠀⠀⠙⢞⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⠃⠀⠀⠀⠉⠑⠒⠒⡆
⠀⠀⠀⠘⠒⠂⠁⠒⠢⣄⡀⠀⣻⠪⢗⣤⠤⠄⡠⢤⣤⣒⠾⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠷⣲⣤⠤⡤⠀⠤⣔⡪⠕⣻⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠉⠙⡆⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢀⠧⠤⣄⣀⣀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣀⣠⠔⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⣄⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠳⠊⠁⢀⠜⠁⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠙⡶⠤⠤⠤⠤⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢇⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣄⣀⣴⠋⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠉⠀⠀⠈⢳⢤⣀⠀⠀⢀⣈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡠⠚⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⡎⠈⠉⠙⠛⠀⠲⠂⠚⠋⠉⠉⢇⠀⠀⠀⠀⠀⢀⡎⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣠⢼⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⡞⠀⢸⠀⠀⠀⠀⠀⢀⠜⢆⠀⠀⠀⠀⡇⠈⠢⣀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢦⠀⠀⠀⢠⠋⠀⠈⠓⢤⣀⣀⣣⠀⠀⠀⠉⠑⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⣶⣾⣿⣾⣶⣿⢶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⠾⣶⠷⡶⢶⢶⣶⣿⠷⠶⠾⠷⡾⠷⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⡇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠄⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣄⡄⠀⠀⢀⠀⠀⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⠠⠄⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣤⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⠤⠀⠀⠀⠀⠄⠂⠒⠂⠤⢄⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠔⠂⠉⠉⠡⠂⢀⠑⢄⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠐⠁⠈⠁⠙⡁⠀⠉⠀⠀⠀⠀⠀⠈⡄⠀⠑⠈⠎⠐
⠀⠀⠀⠀⠀⠀⠀⢨⢀⣀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠒⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⢠⠤⠠⠤⠌⡀⢰⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠀⠀⠀⠀⠀⠐⡔⠈⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⢁⣀⣤⣤⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢸⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠞⠉⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢀⠚⡄⠀⠀⠀⠀⠀⡆⢠⠀⠔⠁⠌⠀⢀⡇⠇⠀⠀⣠⠴⠖⠒⠢⣄⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢁⠀⢀⠠⠠⡀⡇⠈⠁⠀⢂⠀⢀⠞⠁⠸⠀⠚⠁⢸⠆⠀⠀⠈⠳⡄⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠈⠀⡊⠀⠀⠸⡇⠀⠀⠀⠀⢄⡎⠀⠀⣴⡆⢠⠀⠘⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⢆⠀⣷⠀⠀⠀⠀⠰⠀⠀⠀⠀⡟⠀⠀⢰⠁⢱⠈⡐⡇⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠑⠾⠀⠀⠀⠠⠘⠓⠒⠁⠀⠇⠀⠀⣿⠀⠀⡆⢁⠓⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⡜⠀⠀⠀⠀⠀⠀⠘⡆⢸⠀⠀⠀⡏⠀⠀⠛⡀⣀⣁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠀⢸⠀⠀⢰⡇⠀⠀⠀⠑⠈⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⣁⣠⠇⠀⠀⠘⡄⠀⠘⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢧⡀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⠓⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⢀⣀⣠⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⡀⢀⣀⠀⠐⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠀⠀⣂⠀⠈⡉⠀⡽⠈⠉⠉⠐⠒⠚⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡇⢀⡀⠙⡄⠀⠥⢰⢁⡄⠀⠀⠀⠙⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠁⠈⠉⠀⠀⣯⠉⡀⠀⠐⠊⠀⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡃⠀⢠⠃⠀⠈⢣⠀⢠⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠤⠀⢀⠯⠇⠀⠸⡀⠀⠀⣀⣇⢉⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠔⠶⠄⠶⠄⠈⠀⢀⡀⠳⡀⠀⢁⡏⠉⠛⠒⠒⠒⠒⠒⠒⠒⠲⠶⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠶⠶⢶⠀⠀⣎⡁⠀⠈⠉⠕⠃⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⠀⠄⠀⡀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠠⡁⠀⠐⠄⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠉⠑⠒⠀⠤⠤⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⣤⠄⠀⠀⢀⡠⡴⠆⠀⣲⡆⠀⢀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⠗⠁⠀⠀⠀⠊⠰⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⠀⠀⡄⠀⠀⣀⣠⠂⠀⢠⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⠘⣂⢰⣷⣮⡏⢠⡃⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢡⠞⠀⠀⣿⣿⠟⢸⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠙⠠⠎⡀⠀⠀⣇⡏⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡀⠰⠋⠁⠀⠀⣿⢠⡄⢷⠚⠲⢦⠤⣤⠴⣄⠄⣄
⠀⠀⠀⠀⠀⠠⡋⠇⠀⡸⡔⠀⠀⠀⠀⠀⠀⠀⡘⢁⠅⠀⠀⠎⠀⣀⢠⣄⠐⣅
⠀⠀⠀⠀⠀⠦⣋⠁⡈⠹⠀⠀⢀⡴⠀⠀⢠⠀⠁⡜⠀⠀⠀⢀⠃⠋⠉⠤⠀⠉
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠄⠀⠀⠀⢠⡜⠀⢰⠀⠀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⡏⠀⠠⢷⣧⠁⠀⠘⡆⢠⠃⢀⠃⡜
⠀⠀⠀⠀⠀⢀⠀⡀⣸⡆⠸⡇⠀⠀⠆⠈⢹⠶⠞⠂⡞⠀⢸⠒⠃
⠀⠀⠀⠀⠀⠀⠘⠀⠧⠥⠀⠀⠠⠊⠀⠀⠘⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠄⠀⠠⠤⠤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠉⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣘⣿⡀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣄⡀⠀⠀⢠⣿⣥⣤⣤⣶⠾⠛⠁
⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠉⠛⠻⠃⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡈⠉⠉⠁
⠀⠀⠀⠀⠰⣿⠉⠈⠙⢿⣦⣀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣤⣶⠿⠛⠻⣷⡀
⠀⠀⠀⠀⠀⠙⠿⠷⠿⠟⠛⠉⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⠻⢷⣦⣤⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡄⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⣀⠀⠀⠀⠀⣀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⢷⡀⢣⠘⢷⡄⠀⠐⣄⠀⠀⠸⡇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣤⢀⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⢀⣤
⠀⠀⠀⠀⢀⠙⣆⠳⡌⢿⣦⡀⠈⡆⠀⠀⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣰⣿⣷⠈⠂⠀⠀⢰⠏⠀⠀⠀⠀⠀⢠⠤⠀⡀⢀⣴⠏⢠⡿⠁⠀⠀⢀
⠀⡄⠀⠀⠈⢣⠘⠀⣿⠸⣿⣅⠆⠀⠀⠀⠹⣄⠀⠀⠀⢸⡟⠀⠀⠀⢹⢈⡟⢢⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣀⡰⠁⣾⠁⣠⠎⡄⠀⢀⣴⣿
⠀⠀⡀⠠⠀⠀⠑⣸⠋⢤⣟⠉⠺⣦⠀⠀⠀⠈⠃⠀⠀⠈⣷⠄⠀⠀⢨⢘⣿⡆⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠋⠀⢰⡏⠰⠈⠆⠀⣴⡿⠛⢡
⠀⠀⢀⣤⠀⠀⠀⠀⢇⠈⢹⣆⠐⠟⢷⣀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⢸⠸⣇⠴⢀⡾⢃⠀⠀⠀⠀⠀⠴⠁⡄⢠⡟⠀⣧⡘⠀⣸⠏⣀⣴⣿
⠀⠠⡀⠀⠈⠳⣌⣠⣿⡀⠀⠻⡄⣇⠈⢳⡄⠀⠀⠀⠀⠀⣸⡇⠀⠀⢠⡾⣰⠏⢸⣼⡧⠀⢠⠜⠀⡀⠐⠀⠀⣰⣿⠇⢀⣿⠁⢠⡏⣰⢋⣼⠇
⠀⠀⠈⠂⠐⠢⠤⢁⡍⢻⡄⠠⡘⡀⡆⠀⠗⠀⠀⠀⠀⠀⢹⣧⠠⠀⠀⡆⢹⠀⣧⣿⠁⣄⠄⠀⢰⣰⠃⠀⢠⣿⠏⠀⣼⠋⠠⢋⡴⣧⠟⠁
⠀⠀⠀⠠⡄⠀⡀⠀⠘⡄⣿⣄⡐⢰⡷⢄⠐⠀⠀⠀⠀⠀⣿⡁⠀⡄⢠⠃⣼⠀⢻⡅⠀⣇⠀⢀⡟⠀⣠⠀⢫⠇⣰⣟⣡⠎⢠⠋⠘⠁⠀⠔
⠢⣄⠀⠀⠘⣂⠈⢦⠀⠘⢬⡛⢿⡄⠳⣈⡇⠀⠀⢀⠀⠀⢈⣿⣶⠑⠈⢾⢃⣴⠿⢃⣼⠟⢀⡾⠃⢾⠁⡠⠋⣠⡷⠋⡸⢀⡄⠀⠀⠐⠂⣄⡴
⠀⠘⡃⠈⠀⠹⢆⡀⠂⣀⠀⠉⣿⠃⠀⠠⡉⢦⡀⠀⠀⠠⣿⣿⣇⡀⠐⢹⣿⡇⠀⣼⠁⢠⡿⢀⡶⠃⠈⢠⡾⠋⠀⠀⢠⡟⠁⡰⣂⣤⠞⡝⢣
⠀⠀⠈⠲⣐⠀⠀⠙⠂⠙⢦⡄⣿⣇⠀⠀⠈⠜⠽⣢⠐⠀⠈⠙⣿⣿⣦⠈⠹⣿⡀⣿⣴⡿⠁⡀⠁⢲⣶⠟⠃⠀⠀⣠⢞⣠⠀⣴⢋⡤⣞⡶⠆
⠀⠈⢢⣁⢨⣇⡀⠀⠐⢦⠘⣯⠈⠻⣷⣦⣄⣀⠠⣽⣀⠁⠀⠀⣼⣿⣿⡇⠀⣿⡇⢹⣿⠁⠰⣁⣠⣾⠃⠀⠀⢀⠞⣁⣾⡥⢾⣿⣩⡶⠋⡠⠖
⠀⠀⠈⣏⠉⠙⢿⣅⠀⠈⢣⠙⠲⣤⣀⠈⠻⢿⣷⡌⢻⣷⢇⣴⣿⣿⡿⢠⣾⣯⣎⣾⡿⠀⠀⣿⠉⡠⠀⠀⡀⢀⣴⣾⣏⣴⠾⠛⢁⡀⢈⣤⣾
⠀⠶⢤⣈⡛⠛⢆⠨⠳⣄⠘⠳⣄⠀⢸⡇⠀⠘⣿⡿⢸⡟⢻⣿⡟⣿⢡⣿⣿⣿⣿⣯⣤⣾⣤⡿⠀⠃⡠⢂⣤⡾⣻⡿⠏⠁⣤⠔⢃⣼⣾⠷⠉
⣄⡀⠀⠀⠙⠶⠾⣟⢿⣉⢓⣦⠸⣄⣀⣙⣶⣴⣿⠁⣾⣷⠙⣿⣇⢿⣿⣿⣿⣿⣿⣿⡟⣛⣉⣡⡾⠋⣰⣿⠍⣰⡟⠀⠴⣉⣤⣿⡿⢋⡁
⠀⡉⠑⠄⠀⠀⠀⠘⢦⣙⣿⣿⣷⣤⣄⠀⢹⣿⣿⣧⡘⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⢋⣭⣭⣥⣾⣯⣥⠾⠫⢤⣾⠿⠿⢛⡿⠛⠋⠁⠀⡠
⢦⣄⡀⣈⠁⢀⡀⣄⠀⠀⠀⠙⠛⠛⣿⣇⢿⣿⣿⣿⣿⣿⠟⠋⠩⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣟⣁⢠⣶⣶⣟⣡⡶⠾⠋⣀⣠⡔⣂⣻⣷
⣀⡀⡀⠀⠈⠁⠉⠛⠷⠶⠶⠾⢶⣄⡹⢿⣬⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣤⣼⡿⢿⣛⣿⣭⣭⠿
⣋⡉⠁⠂⠉⠛⠶⠶⠶⣦⣤⣤⠶⣿⣿⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣟⣿⣭⡿⣧⠿⠶⠟⣉⣉⡄⠐⠂
⠙⠾⠶⣿⣁⠴⠶⣤⣤⠴⢦⣄⣀⣬⣿⣿⣿⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡛⢛⣉⣹⣧⣤⣴⡚⢛⣷⣾⣧⣄⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⣏⠉⠛⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⠟⣿⠟⣛⣛⠋⣻⣿⣟⢉⣉⠙⠿⠿⢛⣁
⠠⠀⠄⠤⣀⣀⣀⡀⣀⣽⠷⣦⣌⣛⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠛⢿⣛⠛⡿⠿⢿⠿⠿⣟⣛⣛⣋⣩⣍
⢀⡤⠌⠀⠉⠛⠑⣉⠉⢡⣶⣄⠉⢉⣼⣿⣿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⢻⣬⡟⠲⢦⣉⠛⠛⠳⠾⠓⠚⠛⠛⠓⠢⠀⠄
⣀⣀⣤⢶⠾⠟⠾⠛⠳⠿⡥⠈⠟⠿⣿⣼⣟⣿⣿⣿⣿⣿⣿⣶⣤⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣮⡛⠿⣦⣉⡉⠛⡢⣤⢀⣀⡀
⠋⠉⠀⠀⠀⠀⠀⠀⠐⠀⣁⢀⣶⠶⢿⢠⣼⡿⣿⡏⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣌⠉⠻⣿⣿⣷⣌⡙⢻⣶⣤⣠⡀⠀⠈⠛⠿⠛⢣⣶
⠀⠀⠀⠐⠉⠁⠀⢀⡄⣴⠏⠉⣁⠔⣤⢶⣿⣴⣟⣸⡏⣿⣏⣿⠏⢘⣯⢿⣿⡟⢧⡻⣿⣿⣧⣀⠈⠙⠳⠦⡷⢶⡍⢻⣯⡉⠤⡀⠠⢀⡈⠀⠈
⠀⠀⠀⠀⣠⡤⠴⠏⠀⠀⠔⠉⠰⠞⣁⣼⡏⢁⣼⠏⢀⣾⣏⣿⠀⢻⣇⢨⣿⢃⠸⡋⣿⣯⡛⢻⣷⣄⡀⠀⠀⠁⠙⠦⠹⣿⣤⡈⠉⠠⠀⣀
⠀⣄⡲⠿⠋⠀⠀⠀⠀⣠⢴⠿⣋⣾⣿⠛⠐⡏⠁⠀⣿⢹⣿⠃⠀⠸⣯⠹⣷⡘⢆⠙⣋⢿⠿⣿⣿⣿⣿⣌⡲⣦⡀⠑⠢⣝⠻⡿⢶⣶⣄⡀
⠚⠁⠀⠀⠀⠀⠀⠀⢈⣡⣁⣼⣿⠏⠉⠂⡔⢁⠀⢀⣾⣦⣿⠀⠀⢰⣿⡇⠈⢻⡎⡆⠹⡈⠀⠈⣿⣿⡏⢿⣷⣜⢷⡄⠀⠻⣄⠘⠊⢻⣏⠻⣆
⠀⠀⠀⠀⠀⠀⣰⠟⠋⢉⡟⢻⡉⠠⠀⡠⠃⠀⠀⢸⣿⡟⠁⠀⠀⠀⣿⠇⣀⣾⡃⢨⡀⠁⠀⠀⢻⣎⡳⣤⣙⠿⢷⡍⠻⣦⠀⠙⣦⡈⠉⢷⡌
⠀⠀⠀⠀⡀⢨⣿⢀⣸⡏⢀⣭⠋⠀⣼⠀⠀⠀⠠⢾⣿⡅⠀⠀⠀⠈⢻⣤⡙⡯⣿⠘⣧⡀⠀⠘⠙⢈⠙⢦⠈⢇⠀⠰⡆⣹⢷⣄⡘⠷⣄⡀⠈
⠀⠀⢠⣤⣤⣿⠁⢾⣩⣾⠟⠁⢀⢾⡏⠀⢀⣦⣴⢾⣿⠟⠀⠀⠀⠀⠀⣿⠟⠁⣼⠃⠈⣿⣦⡀⠀⠈⠀⠈⠂⡀⠐⠄⠀⠘⠷⣄⡙⠳⡀⠹⣧
⠀⢀⣿⠁⠀⠀⢈⣾⠛⠋⠀⢀⡟⢸⡇⢠⢻⠉⠀⢀⣿⠀⠀⠀⠀⠀⠀⣹⣷⡄⡈⠃⠄⠈⠚⣿⡆⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢌⠻⣆⡡⠀⠻
⡰⠟⠋⠀⠀⢠⣞⠅⠀⡀⢠⠊⢠⡇⠀⠸⠀⣧⠀⢸⠇⠀⠀⢀⡠⠖⠈⢿⡟⢧⣁⠀⣴⡀⠐⠿⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠌⣻⢷⣄
⠁⠀⠀⠁⠀⠼⠃⠀⢠⡸⢀⡤⢿⠀⣠⣴⠾⠛⠀⡞⠀⠀⢀⡏⠀⠀⠀⣟⢧⡘⠏⠐⠨⣿⣄⠀⢦⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙
⠀⠈⠀⠀⠃⠀⠀⢤⠋⢠⢸⡴⠃⢸⡏⠀⠀⠀⠸⠇⠀⠠⠊⠁⣄⠀⠀⢹⠆⠣⠈⢠⡄⠀⢻⡆⡘⠌⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠈⠛
⢠⠄⠈⠀⡀⠀⠰⠃⠀⢸⣼⠃⠀⢼⡇⠀⠀⠠⠀⠀⠠⠀⠀⢠⡁⠀⠀⠹⢦⡀⢣⠈⠉⢦⢼⡇⣶⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠁⠀⠀⠀⠀⠀⠀⣠⠞⣱⠏⢀⠔⡘⠀⠀⢠⠀⠀⠀⣰⠀⠀⣸⡇⠀⠀⠀⠀⡅⠈⢄⠀⠸⡈⢿⠻⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⡞⠁⡼⠃⠐⠁⠖⠁⠀⠀⠃⠀⠀⡀⢙⠂⢀⢳⠈⠇⠀⠀⠸⡃⠀⠀⢠⡀⢸⡇⢣⠀⡄⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣜⠷⠈⢰⠀⢿⠀⠀⠀⠑⠀⠀⠀⠇⠸⢷⡠⡋⠡⡀⠁⠀⠀⠀⠀⠈⠋⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⠀⠙⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣾⡄⠻⡀⠀⠀⠀⠀⠀⡃⠰⠒⠂⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠈⢣⡀⠀⠸⡄⠀⠀⡇⠀⣼⠀⠀⣠⠋⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠈⠢⡀⠑⡄⢀⠹⠀⡀⡇⢃⢧⠂⡔⠁⢀⡴⠋⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠈⡀⠈⠀⠁⠁⠀⡇⠈⢀⢎⢔⠔⢉⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⢄⣀⠅⠈⠀⠀⢀⠠⠀⠀⠄⠑⠀⠠⢒⠡⠀⡀⠤⠒⠋
⠀⠀⠀⠀⠀⠙⠛⠒⠒⠦⠤⠤⠀⢁⡀⠀⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⢤⣶⠤⠒⠖⠋⠑⠒⠚⠉⠓⠒
⢠⡤⠤⠤⠴⠒⠋⠉⠐⠒⠤⣀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⡄⠤⠤⠤⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠒⠃⠈⠉⠀⠉⠀⠀⣀⡠⠔⠃⠴⠉⠀⠀⣀⠠⠔⠒⠀⠈
⠀⠀⡀⢀⡀⢀⠀⢉⠐⣂⡀⢀⣁⣀⣀⣐⣻⣯⡍⠉⠉⢉⣩⠋⠀⠀⡀⠉⠀⠈⡳⠨⡖⠢⡀⠄⡀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⢞⡤⠿⠼⠞⠲⢥⣀⣀⣀⡬⢤⣄⣈⣩⣄⣯⣄
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠶⠪⠉⠑⣚⢋⠩⠍⠀⠒⠒⠒⠒⠒⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠉⠉⠉⠁⠒⢤⡤⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣠⣤⣶⠶⠶⠶⠶⣶⣤⡈⢿⡆⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⡁⠤⠤⢀⡀⠀⠀⠈⢻⣌⠻⠦⡀⠀⠀⠑⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⣁⡀⠀⠀⠤⣄⠀⠈⢆⠂⠀⡆⠆⠈⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⣦⠀⠀⠶⠀⠀⠀⠘⢤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⡤⠄⠊⠅⠀⠀⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡀⠉⠀⠀⣀⠔⠋⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣩⠀⠀⠀⣒⠖⠂⠁⠀⠸⠉⠉⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠀⠀⠊⢠⠀⠀⠀⠀⡧⠀⠀⠠⢄⠀⠈⠢⡀
⡔⠀⠚⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠀⠀⠀⠀⠀⠀⠁⠂⠀⡞⠀⠀⠀⠀⠀⠑⢄⠀⠑⡀
⠁⠀⢠⣿⠉⠐⠀⠠⢀⡀⠀⠀⢀⠄⠈⠑⢪⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⠐⡀
⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠈⠉⠀⠀⡠⠔⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠐⠂⠁⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⡰⠉⠆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠒⢄⠀⠀⣀⠤⠊⠀⠀⠈⢆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢏⠁⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠎⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣿⠃⠀⣱⠆⠀⠀⠀⠀⠀⠀⢂⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣶⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠁⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢏⡴⡚⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢖⠻⠊
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣤⠀⠠⠄⡠⡀⠤⠀⢠⡄⠀⠀⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣛⡱⠄⠀⣣⡀⢶⠖⠈⠰⠈⠐⢶⠆⣠⣃⠀⠴⡙⡳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⢸⠶⠀⣋⡑⠴⡴⠞⠒⠒⢶⡤⠄⣉⣃⠰⢾⡊⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠈⢾⠷⡠⠴⣂⠁⡀⣄⠸⠦⡹⡁⠼⣠⡄⡀⢉⡲⠠⡰⢾⠎⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣉⠀⠚⠃⡰⠊⢠⡐⠈⠀⢨⠿⠂⠀⢻⠃⠂⠲⠯⠀⠈⠀⣠⠈⠲⡐⠛⠂⢈⣁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠈⠀⠢⠀⠟⠋⢦⠋⡻⠞⠁⠀⡰⠀⠀⢈⠀⠀⠐⡀⠀⠙⠾⡋⢳⠎⠋⠃⠀⠂⠘⠡⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡆⢀⡆⠀⠓⣴⣄⠀⠈⠉⠍⡃⠤⣀⡼⣆⡠⠄⡋⠉⠉⠀⢀⣴⡔⠃⠀⣇⠀⣶⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⡄⡚⠉⣙⠁⡄⠈⠛⠠⢇⣠⡇⠄⡀⢡⡉⡀⠀⣦⣀⠧⠘⠋⠀⡄⢙⡩⠑⡂⣤⡄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⣉⠣⣆⠨⣟⣛⠀⢀⢀⢁⡨⠶⣐⢬⣄⣘⣀⣬⢴⡐⠮⣀⢁⢠⠀⢘⣛⠉⡄⠧⣋⡁⠀⡀⡆
⠀⠀⠀⠀⠀⠀⠐⠐⡗⡯⠙⠧⢘⡯⢚⡅⣸⠀⢂⢿⠨⡅⠫⠉⣤⡼⠿⡿⣀⡹⠥⢐⡤⢸⢋⡀⢸⠀⢑⡿⣟⠠⠟⠯⡖⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⣶⣛⡃⠰⡽⠾⡄⠋⠈⢂⣹⣉⡝⠙⠅⢰⠁⠼⠙⣍⣹⣀⠂⠈⠁⡼⠻⡥⠀⢋⣳⡆⣀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡥⢍⡤⢻⡤⠜⠂⡤⡠⠶⠀⠃⠀⠇⠀⡠⡎⡤⡀⠀⠇⠀⠂⠰⠶⡤⡄⠚⠦⣴⠃⣌⠅⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⢀⠣⠀⠛⢘⠑⢄⠀⠠⠴⠇⠀⠀⠬⠅⠀⠀⡷⠠⠄⢀⠔⢹⠘⠃⠠⢓⠀⠙⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢀⡉⠀⠓⣖⠏⢵⠾⠑⠂⢄⠈⠀⠀⢰⠄⠀⠈⢀⠀⠒⠙⢶⠍⢑⡖⠃⠀⣃⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠈⢻⠖⣨⠒⢄⢐⠰⠐⠀⣾⠃⢀⠞⢆⠀⢻⡇⠔⠰⢀⢀⡔⢨⡐⢾⠋⠐⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢠⠛⠋⢡⠘⠣⠄⢄⠁⣼⠁⠘⠂⢹⡌⡁⠀⠤⠛⢠⡉⠛⠣⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣚⠉⠠⢷⡆⢉⡉⠛⢐⠘⠋⣉⠁⣶⠧⠈⢙⡧⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠮⠁⠀⠃⣤⠉⠓⠠⣈⡠⠐⠉⢡⡄⠃⠀⠩⠞⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⣰⣿⡆⠀⠀⣰⣿⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⠀⠀⢰⣿⣆
⢠⣿⡿⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⢿⣿
⣾⣿⠃⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠘⣿
⢿⣿⡄⠀⠀⣿⣿⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⢠⣿
⠈⢿⣿⡄⠀⠘⢿⣿⡄⠀⠀⠀⢸⣿⣿⣿⠟⠻⠿⠛⠻⣿⣿⣿⠛⠻⡿⠛⠻⣿⣿⣿⡀⠀⠀⠀⠀⢠⣿⡿⠁⠀⢠⣿⡿
⠀⢸⣿⡇⠀⠀⣸⣿⡇⠀⠀⠀⢸⣿⣿⣿⣦⣀⣀⣀⣴⣿⣿⣿⣦⣀⣀⣠⣾⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢸⣿⡇
⢀⣿⣿⠁⠀⢠⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠈⣿⣿
⠈⣿⣿⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⠛⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⣿⣿
⠀⢹⣿⣧⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⣀⣤⣶⣶⣶⣄⣀⣽⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⣼⣿⠏
⠀⠀⢻⣿⡆⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⢰⣿⡟
⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠈⠉⠉⠉⠁⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡟⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠁⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠻⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡏⠉⢻⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡿⢷⣄⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠛⠛⠻⠷⣾⣟⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢙⣷⡾⠟⠛⠋⠛⠛⠿⣶⣄
⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⢠⣿⠛⣷⡌⠻⣧⡀⠀⠀⠀⠀⣸⡏⠀⣰⡿⠋⠻⣆⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠘⣿⡀⠈⠻⣦⡀⣀⣠⣿⠁⢠⣿⠁⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⡿⠋⠀⠀⠀⠀⣿⡇⠀⠀⠈⢻⣾⣿⡇⢀⣸⣯⣤⣤⣴⠶⠾⠿⠇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢿⣿⠟⠛⠙⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣤⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣶⣶⡶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣉⣹⣉⣉⣉⡭⠿⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⢠⠖⣚⣯⡥⠴⢒⣛⣿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⣠⠤⠭⠓⠀⠀⠀⢀⣉⠥⠴⠒⣋⡩⢤⡾⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣀⣠⠤⠖⠒⣉⣡⠤⠖⠊⠉⢀⡴⠋⡠⢯⠉⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡴⡇⢸⣡⠤⠖⠚⠉⠁⠀⠀⠀⢀⡴⠋⡠⠊⠀⠈⢏⢱
⠀⠀⠀⣠⠔⠚⠉⠉⠉⠉⠉⠓⠢⣄⡴⢋⡰⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠊⠀⠀⢀⡤⠾⡌⣏⠉⠉⠉⠓⠲⢄⡀
⠀⢠⠞⠁⣠⠔⣞⠉⣿⠉⢹⠶⡴⠋⡰⣎⠀⡇⢸⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⠀⢀⡴⠋⢀⣤⢳⠸⣽⡏⠙⡶⢤⡀⠙⢦
⢰⠃⢀⣞⠁⠀⠸⡄⢿⢠⡿⠊⡠⠺⡀⠘⣆⡇⢸⠀⠀⣀⣤⣴⠋⣠⠞⠁⠀⠀⢀⡞⠀⣰⠋⠀⠳⣇⢻⠀⡜⠁⠀⠙⣄⠀⢣
⡏⠀⣼⠀⠙⠲⢄⣳⠼⣏⣠⣾⡶⠯⠷⠒⠛⠋⠉⠉⢹⠁⡶⣌⡟⠁⠀⠀⠀⠀⢸⠀⢰⠋⠙⠒⠦⣽⡼⢿⡤⠔⠚⠉⢹⡄⠈
⡇⠀⣯⠉⠉⢉⡩⢷⣤⡿⣤⠤⠤⣤⣀⣀⣀⣀⣀⣀⣈⣷⢿⣜⢧⡄⠀⠀⠀⠀⢸⠀⢸⠒⠒⢋⡩⢿⢦⣼⠻⢍⡉⠉⢹⡇
⢹⡀⠘⣤⠔⠋⢠⠏⣧⠳⡈⠙⠢⣤⠇⠀⡞⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠘⡆⠈⢧⠚⠉⢠⠏⢸⠘⡄⠀⠈⠓⡞⠀⣸
⠀⠳⡀⠈⠢⣄⡞⠀⢸⠀⢹⣤⠔⠁⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⠳⢤⡃⠀⡾⠀⢹⣀⡠⠊⠀⡰⠃
⠀⠀⠈⠢⣄⡀⠉⠉⠛⠋⠉⢀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠉⠙⠛⠚⠋⠁⣀⡤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠠⠀⠀⠀⡄⢠⢿⣷⡀⠁⠀⠁⠀⠐⡀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣤⡀⠀⠂⠀⠀⠀⠿⠸⢿⣇⠀⠀⠀⠀⢀⡴⡫⣷⠈⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠹⡌⢎⢳⣤⠐⠀⢀⢸⠁⠀⣆⢹⡀⠀⠀⣰⠞⢱⢣⡟⠀⠀⠀⠈⡐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⠀⠐⠀⠀⠀⠸⡇⠈⢄⠟⣷⡆⠀⠇⠀⠀⡇⣼⣷⠀⡴⠃⢀⣏⢎⡟⠀⠀⠀⠈⠀⣀⠈⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⢿⡳⣴⣄⣀⠀⠐⣇⠀⠘⣞⣰⣽⣿⠤⠤⠶⠻⠿⠿⡿⣅⢀⡞⡿⣜⡟⠀⢁⣠⠤⠚⣡⢾⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⢮⠪⣝⣗⢶⣾⣀⠴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⠔⠋⠁⣠⢞⣵⡏⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠄⠐⠠⡀⠀⠘⢧⠀⠙⠪⣪⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣠⣞⢕⣿⡏⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠘⢧⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣀⡄⣠⡤⠤⠶⢦⣴⡶⠂
⠀⠀⠙⠻⣿⠯⣟⣻⢏⣓⠿⣖⣻⠿⠀⠀⢀⡤⢴⠶⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⣀⣀⠀⠀⠀⠹⡅⠀⠀⣀⠤⢒⣭⠾⠛
⠀⠀⠀⠠⠉⠓⢦⡉⠙⠛⠿⢶⡿⡀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣟⢷⡀⠀⠀⢀⡴⠟⣥⣀⣀⣴⣉⣀⠀⠀⠀⠀⠸⣦⠛⣠⣴⠞⠋⠀⠄
⠀⠀⠤⠄⠀⡀⠀⠉⠳⢤⡀⣼⠁⠀⠀⠀⠀⠁⠀⣻⣿⣿⣇⣽⣿⣷⠀⠀⣼⠻⠓⠛⠹⣿⣿⡟⣨⡗⠠⠀⠀⠀⢻⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⣿⡄⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠠⠀⠸⡅⠉⠙⠓⠶⠆⣠⣀⠀⠁
⠀⣀⣠⣶⠶⠞⠋⠭⣭⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣧⣤⣤⣶⣶⣶⣶⣴⠿⠖
⠀⠉⠿⠏⠩⢉⣀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣵⣶⠖⠚⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠉⠑⣲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠢⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠀⠀⢐⣰⢿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⢀⠈⠲⢦⡀⠀⠀⠐
⠀⠀⠀⠀⠀⣠⡴⣏⡬⠗⠋⠉⠙⡄⠈⠀⠀⠀⠀⠀⠀⠀⠲⠔⠓⠒⠦⠤⠚⠛⠛⠒⠒⠀⠂⠀⠀⠀⠀⠀⢠⣿⣷⣿⣟⣾⣥⣂⣽⣷⣄⠂
⠀⠀⠀⣠⣾⣗⣋⡥⢤⠤⠴⠖⠶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣄⣀⠋⣤⢊⠂⢀⠀⠀⠀⠀⢀⡴⠉⢻⡀⠁⠀⠉⡁⠉⠙⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣼⠫⡫⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠠⠐⠀⠀⠀⢀⣴⢿⡢⡀⠀⠩⡄⠈⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⢀⠄⠊⠀⢀⣼⡋⣠⠞⠁⢀⣹⣷⣦⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⣀⡤⠞⢿⠷⣴⣝⢮⡲⢄⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⠞⢁⣠⢖⠋⠘⣯⢋⣻⠋⠛⢳⣶⣦⣤⣤⠤⠴⢶⣶⣿⠋⠄⠀⢸⡄⠌⠉⠻⠿⢦⣕⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠵⠞⡁⠀⠀⠀⢸⢇⢈⠇⠀⣰⠋⢸⡏⢿⠀⠀⠀⡜⠘⠿⣆⣌⠄⠈⣏⠂⠀⠀⠀⠀⠁⡉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⢀⠀⠀⢸⡏⡌⠀⡞⠀⠀⠈⣿⢸⠈⠀⢰⠁⠀⢠⠘⢿⡌⠄⢧⠀⠀⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣾⡝⡰⠃⠀⠀⠀⠀⠘⣷⢠⢀⡇⠀⠀⠀⠀⠀⠹⣤⣬⠀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡀⠀⠐⠀⠀⠀⠀⢻⣼⡘⠃⠀⠂⠀⠠⠈⡄⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠂⠈⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠘⣦⡀⢀⣠⣴⠾⠋⠉⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣳⣆⠀⠙⠻⣉⠀⠀⢰⡖⠛⠒⠲⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠙⠢⡀⠀⠈⠳⣄⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠀⢠⡇⠀⠀⠀⠀⡘⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡄⢺⡄⠀⠀⠁⠄⠀⠀⠀⠀⣤⢺⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⣽⣾⣧⣤⣠⣌⠀⠀⠀⠀⢸⠿⣀⡀⠀⠀⠀⠀⠠⠧⠼⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣯⣇⡽⢏⡡⢄⠉⢩⣇⠀⠀⢠⡟⠀⠀⠙⠂⠀⠰⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠉⠂⣀⠳⣋⠾⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠇⠀⠀⣿⠀⠀⣄⡉⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣄⡄⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⣀⡀⠰⢒⣰⡏⠀⢰⣿⠁⠀⠈⢦⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠉⠉⠙⠄⠀⠈⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏⠉⠁⠠⠿⠟⠀⢠⡄⠈⢧⡦⢀⣤⣶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠼⢉⣧⠁⠀⠀⢀
⠀⠀⠀⠀⠀⢠⡄⠀⠀⡰⡞⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⡅⢀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢹⣙⣤⣤⠴⠋
⠀⠀⠀⠀⣠⠼⠿⠀⠚⠇⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⢸⡙⣏⢌⣽⡀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⢸⣿⡗⠓⢺⠁
⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣤⣿⣿⣦⡇⠀⠀⠀⡰⠟⠲⠋⠀⠀⠀⠀⠀⠻⠀⡇⠀⢠⡆
⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀⢧⣠⠴⠚⢋⣩⣿⣿⣿⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠃⠀⣼⠁
⠀⠀⠀⠈⠛⠓⠒⠒⠛⠳⠤⠤⠤⠤⠤⠤⠜⢹⣻⡧⠤⠤⠀⠈⠛⢛⡿⡾⠋⠉⠓⠦⢄⡀⢀⣀⣠⡥⠾⠷⣦⢰⡦⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢤⣀⠀⠈⠉⠉⠉⠉⣉⡽⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠒⠒⠚⠛⠢⡀⢀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡶⣶⣤⣄
⠀⠀⠀⠀⠀⢶⡦⢀⣀⣀⡀⠀⠀⠀⠠⠴⠋⢈⡉⣻⣷⠀⠀⠀⠀⠈⠿⢇⣤⣤⣤⠀⠀⠀⠄⠐⠋⢀⡤⣈⣿⠇
⠀⠀⠀⠀⠀⠀⣰⠿⠛⠙⢻⣦⠀⣀⠀⠐⢒⡿⣾⡋⢀⣀⡀⣀⣀⣀⣠⣟⣋⡉⢉⡟⠀⡒⠒⢢⣉⣹⠻⣯⣄⠀⠠⡄
⠀⠀⠀⣠⠠⠾⠿⠞⢉⡽⠛⠀⢴⣶⡆⢀⣿⣿⡇⣷⣬⣿⣿⢹⣹⣿⣛⠛⠉⠴⠛⢁⣴⣄⢀⠺⠛⠛⠒⠿⡈⠁⠀⢸
⠀⠀⣠⠴⠶⢿⣃⣴⣶⠀⣶⡷⠶⢭⣿⡿⠿⠛⠛⢿⡻⢿⣧⣿⣿⣿⣿⣴⣿⡇⠐⢿⣿⠿⠏⢀⡤⠶⠶⢶⡌⣷⠶⡅
⠀⢰⠃⠀⠀⠀⠙⡆⠀⢹⢇⠁⣠⡾⠋⣇⢀⣀⠀⣡⣿⠀⡞⠀⠀⠀⠀⠘⡆⠀⢄⠀⣀⠶⠛⢀⣇⡤⢖⣀⣾⠇
⠀⠸⡀⠀⠀⠀⢀⠇⠀⠘⢧⠚⠡⠶⣶⡿⣧⣴⡿⠟⠃⠀⠹⣄⠀⠀⠀⠜⠁⠀⠈⣂⣁⡚⢹⡿⠛⣶⣿⡟⢉⣀⣀⣀⡀
⠀⠀⠉⠲⠶⠂⠀⠀⢠⡴⠚⠛⠓⢶⡏⠀⠸⣿⣓⠶⡚⠋⠛⠻⢯⡉⠀⠀⠀⠴⠋⠉⠈⠉⠻⡅⠀⠸⣿⡽⠻⣅⠀⠈⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠹⡆⠀⠙⣏⠀⠈⠂⠀⠀⠀⣧⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⡄⠀⢹⡇⠀⠀⠁⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠃⠀⠀⠀⠀⠀⢀⠏⠀⠀⠸⣆⠀⠀⠀⠀⠀⡸⠁⠀⠀⠣⡀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⠄⠊⠀⠀⠀⠀⠑⠤⣤⡤⠴⠋⠀⠀⠀⠀⠈⠳⠦⠤⠖⠈⠀⠀⠀⠀⠀⠈⠒⠖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡤⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠑⠀⠤⠄⢀⣀⠀⢀⣀⣀⣀⡀⢠⠄⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠾⠀⠐⣶⡶⠍⠈⠈⢭⣤⣤⣤⠀⡜⢸⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠣⠜⠒⠃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣉⡳⠶⣄⣀⡴⢠⠶⢖⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣏⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠁⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠁⣉⠤⠚⣄⡤⠤⠀⠀⠀⠀⠀⠀⢀⣀⠀⡼⣉⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠈⠑⢆⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠀⣀⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⠃⠒⠒⢲⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣅⡇⠀⠀⠀⠘⣄⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠚⠁⠀⠀⠀⠀⠓⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠠⠴⠖⢛⢙⡒⠒⢤⣀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⡼⢠⠖⢈⡀⠤⠄⡉⠲⣈⠧⠀⠀⠀⠀⠀⠡⠞⠀⠀⠀⠀⢠⡈⠁⡀⣔⣀⡠⡄⠀⠀⠠⡑⣄⠠⠁
⣸⢰⠁⡰⠥⢈⣩⡉⣾⡆⠘⢼⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⢀⠁⠀⠈⢡⠄⠐⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠠
⣷⢸⠀⢇⠡⡘⠻⣦⡉⠁⠾⣿⠀⠀⠀⠀⠀⡴⠟⢀⠀⡀⠀⢀⠠⡁⢾⣀⠄⣐⢨⣬⡀⠀⠊⠀⠀⠀⠀⣀⣀
⠘⣆⠣⣈⠒⠬⠥⠖⢋⡼⣲⠄⠀⠀⠀⠀⠂⠂⠀⠡⣴⠀⠀⠀⠈⠁⠀⠀⠘⠎⠂⠈⠁⠀⠠⠤⠄⠀⠀⠁⠒⠉⠁⠀⠑
⠰⠿⣵⣮⣉⣀⣂⠐⢝⣮⠃⠀⠀⠀⠀⠀⠙⠛⠤⡀⠈⢀⢀⠀⠀⠀⠠⢀⡀⠀⠀⠀⢠
⠀⠀⢸⡋⠿⠟⠻⠭⠍⠙⠧⠖⠀⠀⠀⠀⠀⠈⠂⠀⠠⣤⠀⠀⠀⠙⠇⠶⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠼⠂⠀⡇⠀⡰⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠑⠄⠀⠄⠂⡁⠔⠀⠀⠲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠒⠇⠀⠀⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠉⠉⠉⠉⠉⠓⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⠠⠄⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⣾⠀⠸⠀⢀⢆⣠⠀⢀⣤⣠⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠰⡀⠀⠀⡷⢀⠛⣄⡌⣶⠁⠾⠀⣄⢆⢠⡆⠈⠟⣧⠎⠘⡃⠱⢟
⠀⠀⠀⠀⠀⢠⠐⢀⡠⠶⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠘⠀⠘⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢰
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠀⠀⡘⠸⠀⠀⣠⠤⠤⣤⣔⠁⠀⠛⠀⠀⠙⣆⣀⣀⡐⣼⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠛⠳⢤⡀⠀⢠⢁⠐⣠⡮⠤⠖⠊⠀⠈⢳⠀⠓⠁⢠⠞⠁⠙⠦⣽⣆⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⠒⢢⠹⣤⡸⢀⠀⣏⠀⠀⠀⠀⠀⣾⠆⡇⠀⠀⡇⠀⠀⠀⢸⡇⢸⣸⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡧⠒⡆⠀⠉⠛⡘⠀⠹⡀⠀⠀⠀⠀⠙⡱⠀⠀⠀⢣⣀⠀⠀⠈⢁⠜⠳⣟⡮⠀⠀⠤⠤⢄⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢀⣺⡄⠀⠀⢀⠷⠀⠀⠙⠲⠤⠤⠴⠊⠀⠀⢠⠤⢶⠈⠛⠒⠚⠁⢠⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠄⠀⠀⡎⠐⠀⠀⠀⠃⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡀⢄⠖⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠁⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣎⠔⠂⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠺⠀⢀⡠⠒⠉⠲⡀⠀⠀⠀⠀⠶⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠘⠖⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠉⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠋⠀⠀⠀⠀⠀⠘⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⠀⠀⠀⠀⠸⡀⠉⠉⠒⠒⠶⠦⣤⣤⣀⣤⣄⠠⠤⡔⠲⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠆⠀⢀⠀⠀⠀⠙⢄⠀⢀⠔⠚⠳⣧⡰⠗⢦⡀⠀⣠⠃⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠋⠀⢀⠃⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠸⠁⠀⠈⡧⠚⡅⠀⠀⠀⠈⣇⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⠦⢄⣀⡘⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢱⠀⠀⠀⠀⣸⡀⠀⠀⠀⡵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢈⠃⠀⠀⢠⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠀⠀⡗⠒⠈⠉⠁⢇⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢱⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡥⠥⠤⠤⢤⣤⣤⣤⣤⣤⣤⠤⢤⣵⡞⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢵⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠘⣧⠐⠀⠀⠀⠀⠈⠐⠂⢄⣄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠁⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠨⢷⡀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣇⠀⢹⡄⢣⠀⠀⣠⠞⠁⠀⠀⠂⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣤⣤⣄⣀⣀⠐⢾⠀⠀⣿⡼⣤⠞⠁⠀⠀⠀⣨⠀⠀⠀⠀⠀⠙⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠉⢉⣙⡛⠛⣺⠛⢿⡾⣿⡛⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⠁⠂⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⠘⠳⡧⠝⠁⠀⠀⠀⠀⢹⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢹⣦⣄⣀⡀⠀⣀⣀⣠⣶⣯⣀⣀⣀⠀⠀⠈⢁⠀⠀⣀⡀⢀⣀⣀⠀⠈⣿⣴⣢⠤⠀⠀⠤⠤⣀⣶⠿⡁
⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⠟⠛⠛⠛⠟⠛⠛⠏⠛⠁⠐⠒⠐⠀⡂⠀⠘⠷⢏⡞⠉⠉⡙⢃⠙⠺⠿⠟⠛⠿⠟⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢰⠹⡀⠀⣀⢤⡄⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠀⠣⡟⠀⠳⠞⠀⠀⠳⠞⣄⡇⠙⢦⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⡟⠓⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠤⠼⠆⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⢀⡐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠴⢤⣀⠤⠔⠚⠁⠀⡀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⠀⠀⠀⡴⠛⢲⡀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠈⡇⠀⢰⣧⠀⢀⡇⢀⡸⢏⣀⠤⠶⡄⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠟⡰⢁⡄⠘⢧⡤⠞⠀⣼⠀⠀⠋⢴⠄⡿⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠉⠀⠁⠀⠀⠀⢀⣠⣾⣿⡆⠀⠀⠐⠚⣅⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠑⠲⣤⣤⣴⣶⣿⠟⠁⢨⠇⠀⢀⣤⠖⣉⣓⣒⡢⢄⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠹⣿⠟⠫⡇⣠⡴⠋⠀⣴⠋⢸⠉⠉⡇⠀⡸⠃⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣯⠝⠓⡆⠀⠙⠲⢬⣉⣉⣉⣁⣀⡤⠞⠁⠀⣀⡧⠴⠃⠘⣅⣰⠆⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢎⡠⠀⠇⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠣⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣐⠪⠔⠲⣄⠑⠦⠤⠼⣯⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⢹⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢖⠒⠒⠶⠖⠋⠉⠉⠉⠉⠑⢶⡇⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⣒⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⢸⡩⣉⠉⠉⠓⣯⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠙⢬⣧⠴⠊⠙⠤⠤⣖⠀⠀⠀⢀⡽⠀⣰⢭⣳⣤⣇⠄⡠⠃⣱⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣴⡓⢺⡀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⠴⣋⠳⢞⡡⠚⠉⠀⠉⠛⠒⠊⠁
⠀⠀⠀⠀⠀⠀⢾⠛⠒⠒⠭⢵⣒⠭⣚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠈⠙⠪⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣒⣒⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣤⡀⠀⠀⣄⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣦⠀⢹⣷⣄⠀⣿⣿⣿⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡈⢻⣿⣿⣿⡆⣾⣿⣿⣴⣿⣿⡏⠀⢀⣾⡇⠀⠀⣠⣾⣿⠋⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⡿⠀⣀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠙⢿⣶⣶⣾⣿⣶⣤⣀⠀⠹⣿⣿⣿⢿⣫⣽⣶⣶⣶⣾⣭⣟⠿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⣿⣷⣾⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⡿⠟⠉⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣭⣿⣿⣏⣾⣿⣭⣥⣭⣝⣿⣿⡿⠿⠻⠿⣿⣧⢻⣿⣿⣷⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⠿⠿⣿⣿⣿⢹⣿⣿⣿⣟⣧⣿⣿⣿⣾⣿⣿⣻⣾⣿⡟⣿⣿⣛⣉⣀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣯⡉⠉⠙⠛⠛⠋
⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠟⠉⢁⣴⣾⣿⣿⣯⡻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣾⡿⠟⠛⢋⣿⣿⣶⣽⣛⣿⠿⠿⣿⣛⣭⣾⣿⣿⣿⣿⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⣠⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃⢸⣿⣿⠏⣿⣿⣿⠏⢿⣿⡆⠀⠀⠻⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠸⣿⠃⠀⣿⣿⣿⡄⠀⠉⠛⠶⠄⠀⠀⠀⠀⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠹⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣴⣾⠛⠃⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢪⣾⣿⣿⡁⠀⡄⢉⠸⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣿⣿⣿⣧⠀⠁⡠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢿⣿⣿⡟⢀⣴⡅⠰⢐⠖⠁⣹⣖⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠑⢾⣻⣧⠀⡚⠚⠉⡽⢌⣁⣁⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠛⠟⠀⠈⠲⠊⠀⠀⠀⠉⠀⠀⠀⠀⠐⠒⠒⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⠁⠀⠀⡄⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢩⠀⠀⠀⠀⠄⠀⢢⣄⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⠠⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠁⠠⠄⠀⠃⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠀⠀⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⠁⠔⠚⠈⠈⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠧⣀⠀⢆⣀⡠⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠖⠒⠚⢱⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠹⠉⡗⠒⠂⠀⠀⠀⠀⠀⠀⠀⠤⣴⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄⡀⡌⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠚⠁⠘⠧⢔⡒⠠⠤⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡀⠀⢠⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣁⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⡚⣃⡭⠳⡤⠋⡆⢹⠀⠀⠒⡖⠆⠀⢀⡠⠊⠓⢄⡠⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⢐⡇⠑⢄⠀⠀⡇⠀⢣⠀⢀⠀⠀⢠⠁⠀⢥⠂⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸⠁⠹⣯⡧⢠⠇⠀⢸⡆⠉⣀⠀⣸⡀⣀⠈⠘⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢒⠞⠀⠀⢀⣙⣞⣡⣴⠲⣻⠙⣯⣽⣼⢛⠋⣫⢀⣉⢠⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⠀⠀⠙⠛⠿⠿⠿⣯⠴⣷⣾⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⠿⠿⠿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠍⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠴⠶⠶⠶⠶⠦⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠖⠒⠒⠲⠦⣄⡀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠰⠯⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠘⠓⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⣧⠖⠛⠛⣆⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⣼⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠁⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢀⡇⠀⠀⠀⢀⡼⠂⠀⠀⠀⠀⠀⠀⣸⠃⠀⠚⠂⠈⠙⠂⢀⡀⠀⠤⠞⠁⠀⢴⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⢸⡇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠙⠦⠶⡦⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡒⠒⠆⠀⠀⠀⠸⡄⠀⠀⠀⠀⣻⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⡇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣤⣤⠀⠀⠀⣄⠀⠀⣰⠋⠈⠉⠓⠲⢤⣄⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢻⠀⠀⠀⠀⠈⠳⡆⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⡟
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠙⠶⠤⣄⠀⠀⢠⡏⠁⠀⢀⣤⣄⡀⢈⣧⣤⠤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣼⠃
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⢠⡞⠀⢨⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⣰⠃⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢸⡄⠐⠋⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⢸⡇⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣞⠁⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠚⠻⠶⠶⠶⠶⠖⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡛⠉⠉⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⣿⣿⣿⣿⣿⡙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⢿⡿⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣀⣀⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠽⠛⠉⣿⠁⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣹⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⢿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡤⠀⡀⠐⡈⡉⠤⠄⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠭⢑⡑⠧⠘⢀⠞⠁⠀⢀⣤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣁⣀⡀⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⢡⠔⠂⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠔⠊⢉⡀⠀⢀⠠⠐⢈⡁⠄⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢔⢞⡉⢠⠖⢁⠤⠊⠁⠠⠐⢀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠁⠀⢀⠷⢅⡐⠀⢸⡇⠴⠊⠙⠒⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠨⠑⠀⠀⠀⠁⠤⡴⠙⠂⠉⠉⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠘⠀⡇⠀⠀⠀⡀⣈⠀⢀⣀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠠⡀⡃⠀⢀⡾⠉⠉⢀⠎⠀⠈⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡤⣾⡿⡀⠨⠃⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠰⠉⣴⠋⠀⠙⠢⠄⠊⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⠉⠈⠉⢀⡄⢀⠄⠀⠐⠒⠃⠀⠘⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⢀⠄⠂⠉⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡘⢷⣤⣀⠌⠀⠀⠒⠂⢤⠆
⠀⠀⠀⠀⠀⠀⢠⠤⣀⡱⡆⠹⣉⠒⠠⠤⠄⣤⠜
⠀⠀⠀⠀⠀⠀⠎⢠⠏⢩⢧⡀⠘⠝⠁⣎⠉⢽⠋
⠀⠀⠀⠀⠀⣸⢦⣴⣀⠔⠋⠳⠄⢈⠀⠈⠁⠁
⠀⠀⠀⠀⣴⠉⢈⡾⠍⡲⡶⢶⣤⢷⣄
⠀⠀⢀⣤⣴⣿⣷⣷⣶⣿⣶⣚⣋⣅⡀⢙⣿⣿⣷⣶⣶⣤⣤
⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣶⣶⣶⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠉⢻⡇⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠂⠸⠃⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠧⡀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⡀⠈⠁⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠀⠂⠐⠢⡑⠄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠄⠠⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠠⠊⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠂⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠴⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠀⡀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⠂⠄⢀⠀⠀⠀⢀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⠀⢀⠀⠈⠁⠈⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢈⠦⢀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⡀⠀⣀⣠⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣿⣿⣿⣿⣿⠛⠻⠋⠐⢄⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⠀⠀⠙⣿⣿⠏⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⠀⠀⠀⠀⠌⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⢀⣀⠄⢆⠁⠀⣀⠠⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠘⡀
⢻⡀⢲⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⢡
⠀⢳⡈⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠆
⠀⠀⠻⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢼⣄⣀⡤⠒
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠛⣋⣤⠖⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣯⣭⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠁⠀⢸⠃⠀⠀⡜⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⠀⠀⠀⠀⢀⣾⠃⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⡀⠀⠐⣄⠀⠀⠀⠀⣼⡇⠀⠀⣠⠟⣿⡀⠀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠸⣷⣦⣤⡾⠋⠙⠒⠛⠁⠀⠈⠛⢲⡞⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⡀⢀⣿⠀⠀⢀⣠⣤⠴⠦⣤⣄⡀⠀⠈⠳⣤⣤⣤⣤⣤⡶⠞⠁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⠀⠀⣻⠉⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⢠⡟⠀⠀⠠⠚⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠘⣧⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⢀⡿⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢸⠟⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣧⡀⠀⠉⠙⠲⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢶⣾⣁⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢀⣈⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⢀⣰⡏⠁⠈⠉⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠠⠶⠒⠀⠀⠀⠰⠯⣄⠀⠀⠈⠛⠶⣤⣀⣀⣀⣀⣤⠶⠋⠁⠀⠀⢰⠟⠉⠃⠀⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡿⠀⢀⣀⡀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⣾⠷⠛⠉⠉⠙⣇⣀⡀⠀⢀⡤⠶⢤⡴⠚⠲⢦⣘⣇⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠈⠉⠉⢻⣤⠏⠀⠀⠀⠁⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡏⠀⠀⢠⠀⠀⠀⠘⣦⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠸⠁⠀⠀⣿⠀⠀⠀⠀⠈⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣞⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣴⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣶⣆⡀⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⣿⣷⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣯⡻⠇⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠸⣿⣾⣿⣻⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣥⣤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢀⣈⡿⠟⠀⠀⠉⠿⢿⡿⢿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢿⡿⢯⣛⣿⡟⠻⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⡟⣦⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢸⣿⣯⣵⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣽⣿⡃⠀⠉⠓⠒⠒⠒⡖⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⣠⣿⣿⠟⠛⢿⣿⣧⡀⠀⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣻⣿⣿⠿⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠛⢻⢇⡀⠻⣿⡇⠀⠀⠀⣽⣿⠃⣀⡿⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠰⡆⠀⢿⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠃⠀⠙⠟⠀⠀⠘⠟⠁⠸⠧⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⠀⠛⠛⠂⠀⠐⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣦⠀⠀⠀⠀⢀⣴⣞⣛⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣯⣿⡇⠀⠀⠀⢸⣿⣷⣿⣧⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⢾⣿⣿⠀⠉⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⡾⠟⠀⠀⠀⠀⠈⠻⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⡇⣠⣬⡛⠁⠠⠀⠈⣿⣿⣶⣂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡇⢬⡽⠃⠠⠀⠀⠀⢻⣿⣏⣻⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣿⣟⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣞⣷⣹⣖⣦⣄⠀⠀⠐⣿⣿⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⣾⣿⣿⣿⣿⣿⡷⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⢗⣽⣹⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡿⢿⣿⢿⢿⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠶⣿⣽⠷⠋⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡾⠿⠿⢿⡿⠿⢾⡵⠿⢿⢶⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠎⠀⠀⠉⠁⠈⠓⠒⠓⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⢲⡀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣵⡾⣿⣶⣄⢷⠀⣰⠇⣠⠞⠙⢦⡲⣄⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠫⢎⡥⠖⢺⡽⡏⠁⢧⠀⠀⠀⠀⠀⢩⡄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠈⢠⣞⣉⡤⣇⠀⠀⣶⠀⢤⣀⡀⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢀⡟⠉⠈⢉⡁⠀⠀⠀⠀⣀⠈⠀⠀⠛⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⡴⢿⠉⠀⠀⠀⠘⠋⠀⢀⡀⠈⠙⠀⠀⠀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡞⠁⠈⠙⢲⡀⣦⠀⢀⡤⣤⡤⢤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⠁⠀⠀⠀⠈⢳⡀⠀⠈⠓⠚⠓⠚⠁⠸⢆⡞⠀⠀⠀⡐⡆
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠙⢲⡤⣤⡄⠀⠀⠤⢶⠋⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠙⡾⠲⢤⣀⡤⢼⡁⠀⠀⠀⠀⠀⢈⡧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⡼⠁⠀⠀⢻⣀⣠⡽⣆⠀⠀⢀⡴⣫⣀⣀⡟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡞⠁⠀⠀⠀⠈⠁⠀⠓⠮⣷⣤⡮⠴⠏⠀⠉⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠁⠀⣇⣀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⣀⠀⠀⣸⣿⠀⠀⡀⠀⠀⠀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠸⣿⣇⠀⢿⣤⡄⣿⣿⠀⢰⣇⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠈⢳⣔⣦⣿⣿⣦⣼⣿⣿⣿⣿⣦⣿⣿⣧⣿⡿⠁⢀⡄⠀⠀⠀⢀⣠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⣻⣿⣿⡿⠿⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣇⣴⢟⡆⢀⣤⣶⡿⠟⠁
⠀⠀⠀⠠⣤⣤⣀⣀⡀⠀⠀⠐⢦⣔⡶⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣦⣤⣀⡈⣻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣇⣤⡶⠚⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⣀⣤⣤⣤⣶⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣶⡶⠦
⠀⠠⠴⠶⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⣽⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠿⠿⢿⣿⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⢶⣤⣀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠿⢛⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣏⡉
⠀⠀⠀⠀⠀⢀⣴⠿⠛⠋⠉⠀⠀⣨⡿⠛⢹⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣯⡉⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⡿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡟⠮⠻⢦⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢻⡟⠀⣿⣿⢿⣿⡟⠻⣿⣿⢿⣻⣆⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠚⠀⢠⣿⡟⠈⢻⡇⠀⢻⣿⠀⠁⠹⠄⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠃⠀⢸⣿⡆⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠁⠀⠀⠀⠀⠀⠀⠐⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⡷⠀⠀⠀⠀⡈⠄⠴⠦⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⣁⣤
⠀⠀⠀⠀⠀⠀⠀⢠⣇⢀⡐⠡⠄⠒⡸⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡘⠓⢄⠀⠠⠀⣠⠞⠀⠀⠐⢄⠀⠀⠀⠈⣽⠃⠀⠀⠀⠀⢀⢿
⠀⠀⠀⠠⠂⠀⠀⡼⠁⠑⡇⣇⢀⠎⠀⠈⢃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠨⡇⡄⠜⠁⠀⠀⠀⠀⠀⠇⠀⢀⡼⠁⠙⡄⠰⣠⠔⠁⠀⠂⠀⠈⠂
⠀⠀⠐⠀⠀⠀⠼⠁⠀⠀⢴⡤⠃⠀⠀⠀⡄⢦⠀⠀⠀⠃⠀⠀⠀⠀⠀⠾⠀⠀⠀⢐⣧⠟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠁⠀⠀⡇⣴⠛⡄⠀⠀⡈
⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠎⠑⠒⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠒⠁⠀⠀⠀⠀⠀⢀⠜⠀⡀⠀⢀⣰⠃⠁⠈⠁⠀⠀⠀⡀⠔⠁
⠀⠀⠀⢀⣂⣤⣠⣤⣖⡉⠠⠤⠤⠤⠤⠤⠠⠀⠀⠊⠀⠀⠀⠀⠲⠴⠶⢒⣚⡉⠀⠀⠤⠤⠤⠀⠀⠒⠐⠊⡀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠀⠈⠈⠁






⣷⣶⡶⠶⢾⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⢆⠈⡄⢠⢠⡄⡌⢀⠆⡰⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⠈⢆⠘⡄⢣⠸⡄⣧⢸⢸⡇⡇⣼⢠⠇⡜⢠⠃⡰⠁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢄⠑⣄⢳⣘⣮⠷⠛⠋⠋⠉⠉⠙⠙⠛⠾⣵⣃⡞⣠⠊⡠⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡑⢦⡑⣮⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣵⢎⡤⢊⡠⠔
⠀⠀⠀⠀⠀⠀⠀⣈⠒⠤⣙⣲⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣖⣋⠤⠒⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢉⡒⢮⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡵⢒⡉
⠀⠀⠀⠀⠀⠐⠒⠢⠤⣭⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⠤⠔⠒⠂
⠀⠀⠀⠀⠀⠉⠉⣐⣒⣒⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣒⣒⣊⠉⠉
⠀⠀⠀⠀⠀⣀⠀⠬⠭⠭⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠭⠭⠥⠀⣀
⠀⠀⠀⠀⠀⠤⠀⠒⢒⣛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣛⡒⠒⠀⠤
⠀⠀⠀⠀⠀⠀⠒⢈⣉⠥⢞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡳⠬⣉⡁⠂
⠀⠀⠀⠀⠀⠀⠈⢁⠤⠒⣩⠼⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠧⣍⠓⠤⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⡠⠊⡱⢛⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡓⢌⠑⢄⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠊⡰⢋⡼⢹⢟⡶⣦⣄⣄⣀⣀⣠⣠⣴⢶⡻⡏⢧⡙⢄⠑⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢀⠎⢠⠃⡜⢰⠃⡟⢸⢸⡇⡇⢻⠘⡆⢣⠘⡄⠱⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠎⠀⠁⡈⠈⠃⢁⠈⠆⠱⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠘⠀⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠈⠁⠒⠤⡀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡐⠀⢃⠈⢄⠀⠀⢀⠀⠀⠀⠀⠀⠢⠈⡆⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢧⠀⢄⡙⠀⠁⠀⢊⡑⠀⠒⢄⠀⣀⠀⡇⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠇⠀⠀⠀⠈⠇⠀⠀⠨⠔⡁⠙⡄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⢀⡀⢶⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡠⠜⠁
⠀⠀⠀⠀⠀⣤⠤⠄⡀⠀⠤⠀⠒⠂⠤⣀⠀⢠⠁⡆⠈⠀⠀⠀⠀⠀⠑⠤⣄⣐⣒⢀⠀⠀⡀⠠⢎⠑⢄
⠀⠀⠀⠀⢰⣈⠀⠠⢍⡕⠀⠀⠀⣀⠀⠀⢧⠈⠀⠁⢀⠀⠀⠀⠀⡔⠁⠀⠠⡉⠉⢓⡦⠼⠒⠦⠞⠀⢀⣀⣄⠀⠀⠀⠀⠀⢀⡠⠤⠒⡄
⠀⠀⠀⠀⠀⠐⠚⠙⠊⢸⠀⡠⠂⠃⠈⢠⡎⢃⡲⠂⠈⡄⠀⠀⡜⠀⠀⠀⠤⠘⠒⠫⣄⠀⢀⠀⢀⡆⢀⠀⢈⡳⠤⠄⠒⠉⠁⠀⠀⠀⡧⠒⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⢆⠀⢨⠉⠀⠇⠀⠀⠀⠁⠀⠁⠀⠀⠠⡀⠀⡇⠀⠀⠀⠀⠀⢠⠎⠀⠀⣄⣇⡨⠷⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠬⢣⣤⠵⠄⠀⢀⠖⠢⣄⠞⢀⡀⠀⠀⠈⠙⠳⣄⡀⠀⠀⠀⢘⣤⠦⠛⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡉⠐⠒⡲⡂⢉⡁⠀⠀⠈⠁⠁⠆⡁⢧⠀⠉⠉⠉⡉⠩⠀⡂⠥⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠬⠅⢈⠠⣄⠀⠀⠰⠤⢀⣀⡀⠷⣬⠷⠄⠐⠀⠅⢀⡠⣅⢠⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡘⢄⡀⠀⠀⢀⡀⢀⡇⠀⠀⢀⠀⠄⠂⠀⠀⠀⠀⠈⠢⣘⠦⣈⡫⡶⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⢅⠀⠀⠉⠉⠀⣉⠥⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡳⣬⡷⣽⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠍⢁⠔⠒⢺⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣙⢹⠀⢀⠀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⢃⠀⠀⠀⠘⠐⠠⡀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠓⠒⠂⠛⠂⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣠⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠻⢿⣄⡀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠉⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠞⠉⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢦⠀⠀⠀⠀⠀⡤⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⣷
⠀⠀⠀⣠⣤⣤⣤⡄⠀⠀⠀⢿⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢿⠲⢤⣀⣀⣀⣀⣀⣤⡶⠀⠀⢀⡇⠀⠀⠀⠀⠐⠛⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠓⠤⣀⣉⣉⣩⠵⠋⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠈⠀⠀⢀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢦⡀⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⢰⠂
⠀⠀⠀⠈⠳⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⢄⠀⢀⡎⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠐⠉⢠⠃
⠀⠀⠀⠀⠀⠈⢢⡈⠑⠒⠤⣄⡀⠀⠀⠀⢸⠀⠀⠀⢀⣀⡵⠮⠤⠤⠤⠤⣤⣀⠣⠠⠐⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡀⠤⢄⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⠉⠒⠒⢚⡠⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⢀⠇⠀⠀⡀⠤⠒⠈⠁⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠓⠉⠁⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠖⠃⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢴⣁
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠉⠒⠒⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⡠⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠢⣀
⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠑⠤⡀
⠀⠀⢀⠤⠚⠁⠀⠀⣀⣀⣀⠀⠤⣄⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣀⡀⠀⠀⠀⠀⠀⠑⠢⣀
⠤⠮⠅⠒⠒⠉⠉⠁⠀⠀⠀⠀⡰⠃⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠈⢇⠀⠈⠉⠁⠒⠂⠤⠤⣄⣑⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡀⠤⠬⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⢡⢄⡀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠕⠒⠉⠀⠀⠀⢠⠃⠀⠈⠉⠓⣒⠢⠤⠤⠔⠒⣊⠉⠀⠀⠀⢸⠀⠀⠉⠒⠤⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡠⠊⠁⢱⠀⠀⢀⠎⠀⠱⡄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣠⠊⠀⠀⠀⠈⡆⢀⠎⠀⠀⠀⠘⢆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠞⠁⠀⠀⠀⠀⠀⢣⠎⠀⠀⠀⠀⠀⠈⢣⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⢀⣤⡶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢿⡟⠁⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⣸⡏⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⣀⠀⠻⣧⣀⣀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⣾⠋⠀⠀⠘⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣼⠏⠀⣰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠙⣷⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⣿⡀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡿⠀⠀⠀⠀⠘⢻⣦⣄⣀⣀⣠⣴⡿⠃⠀⠀⠀⢨⡿⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⣠⣤⣾⣯⠉⠉⠉⠉⣽⣷⣤⣄⣀⡀⠈⠁⠀⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠞⠛⠋⠉⠀⡀⠙⠻⠶⠶⠟⠋⠀⠀⠉⠙⠛⠿⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠿⢶⣄⠀⠀⠻⠃⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⣠⣴⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣿⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢀⣾⠋⠀⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠘⣿⡀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣿⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⣿⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠷⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠾⠿⠛⠛⠛⠻⠿⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⣿⣆⠀⣈⠛⢷⣄⡀⠀⠀⠀⠀⠀⠘⣷
⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠋⠁⠀⠀⠀⠻⠋⢸⣿⡆⠀⠉⠻⢶⣤⣀⠀⠀⠀⢹⣇
⢸⡏⠀⠀⣀⣀⣀⣠⣤⣴⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠈⠙⠻⠶⢶⣶⣿
⣿⠟⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⣀⠀⠀⠀⠠⡶⠾⠟⠛⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⠷⠂⠀⠀⠀⠀⣴⡄⠀⢸
⣿⠀⠀⢰⣿⡇⠀⠀⠀⢠⡶⠾⠛⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠿⠶⣶⠀⠀⠀⠀⢸⣟⣿⠀⢸
⢻⡇⠀⠹⣿⠟⠀⠀⠀⠸⣧⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣀⣠⡿⠀⠀⠀⠀⠈⠙⠋⠀⣿
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⢰⡆⠀⠀⣶⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⣤⡄⠀⠀⢠⡿
⠀⢻⡆⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⣾⠃
⠀⠈⢿⡄⠀⠀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣼⠏
⠀⠀⠈⢿⣄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⠾⠟⠻⣶⠟⠻⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠈⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣾⣿⣿⣿⣿⡿⠿⣿⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣿⣿⣿⢿⣿⡿⢻⣿⡇⠒⠈⣿⣿⠿⠿⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⡎⣭⢡⣿⣿⡇⠿⠗⣸⣿⣿⠇⣼⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣛⡛⢿⣿⣿⣀⣾⣿⣿⣿⣶⣿⣿⣿⡧⣾⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢣⣾⣿⣿⡿⠿⣿⣿⣿⠟⣋⣙⣻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠈⠙⢻⣿⣗⣿⣿⣿⣿⣷⠗⣸⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣷⣬⣥⣾⣿⣿⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣶⣿⣷⣦⡉⠉⠁⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠀⠀⠀⠻⠿⠋⠉⠛⠿⠏⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣿⣿⣿⣿⣿⡿⢆⡀⠀⠀⠀⠀⢠⣿⣿⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⣿⣯⡡⣿⣿⣦⣀⣀⣀⣛⡻⢃⡞
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⣿⣿⣿⣿⣿⠇⠈⠻⠿⠿⠿⠿⠿⠇⠼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡛⠿⠿⢿⡿⠿⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣶⣶⣶⣆⣀⣀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⣶⡄⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠘⣷⡀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⢰⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠶⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠈⠉⠓⢦⡀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠤⢤⣀⠀⠀⠀⠀⠀⠀⣀⡽⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢯⣀⠀⠀⠀⠀⠀⠀⣀⡤⠤
⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠚⠁⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠚⠁⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠚⠁
⠀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠
⠀⠀⠀⠀⠙⠲⢤⣀⣀⡤⠔⠋⠀⠀⠀⠀⠙⠲⢤⣀⣀⡤⠖⠋⠀⠀⠀⠀⠙⠢⢤⣀⣀⡤⠖⠋
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠈⠉⠓⢦⣀⣀⣀⣀⡤⠚⠉⠁⠈⠉⠓⢦⣀⣀⣀⣀⡴⠚⠉⠁⠈⠉⠓⢤⣀⣀⣀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉