Christian cross Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠃⠀⢠⣶⡄⠀⠘⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣽⣿⡟⠛⢻⣿⣧⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠂⣀⣀⣸⣿⣇⣀⣀⠐⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣿⣿⣿⡇⠀⢼⣿⣿⣯⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠒⠸⠿⠿⢿⣿⡿⠿⠿⠇⠒⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣽⣇⣀⣀⡀⠀⢀⣀⣀⣿⣭⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠴⠂⠀⠀⢸⣿⡇⠀⠀⠐⠦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣫⣿⣿⣿⡇⠀⢹⣿⣿⣯⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠠⠂⠀⢸⣿⡇⠀⠰⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣽⣿⡇⠀⢸⣿⣯⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⣛⣵⣿⣦⣙⠒⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡹⣿⣿⣿⠏⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⠛⢻⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⢹⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠳⣄⣀⣀⣀⡀⠀⣠⣿⡞⠀⢸⠘⣇⠀⠀⢀⣀⣠⣴⡾⠁⢳
⠀⠀⠀⠀⠀⣠⣾⢟⣼⣷⣶⣦⠤⠭⠭⠭⠽⠯⠞⠀⠘⠤⠬⠽⠭⠭⠥⣬⣿⣾⣷⣌⠳⣄
⠀⠀⠀⠀⠘⣧⣴⡻⣿⣿⣿⣿⣤⣤⣤⡤⠤⠤⣄⠀⢀⡤⢤⣤⣤⣤⣤⣿⣿⣿⣿⠿⢃⡼⠃
⠀⠀⠀⠀⠀⠀⠙⣿⠘⠫⣷⠿⠟⠛⠛⠛⢿⣿⢸⠀⢸⣾⡿⠛⠛⠛⠛⠛⢿⣮⡃⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠹⠴⠊⠁⠀⠀⠀⠀⠀⠀⣿⢸⠀⢸⢹⡇⠀⠀⠀⠀⠀⠀⠙⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣄⣬⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣳⣿⣿⣿⣆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣽⣥⣟⢿⣿⡿⢽⣤⣨⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣍⡠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⡦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣩⣾⣯⣿⣹⣿⣯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⣿⠧⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⢸⡿⣆⠺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢸⡑⠀⣄⣀⣀⣀⣀⣄⣤⣄⣀⣀⣀⣀⣀⣠⣆⣜⡷⢤⣦⣤⣀⣀⣀⡄⢠⣶⣷⡦⣾⡾⣋⢼⣇⡀⠀⠀⠀⢀⠠⣤⣀⢴⣮⡽⠋
⠸⣿⣿⢿⣯⣿⡿⣿⡿⣿⠛⡞⣛⣟⣛⠛⠛⠋⣁⣤⣶⡿⠗⣚⠃⠀⠀⢸⣭⣠⣯⡯⣴⢻⣌⠙⣿⣏⣙⢛⣿⣿⣯⡼⢻⣿⡿⣿⣷⣀⡄
⢸⠛⣛⠿⠿⠿⠾⠿⠿⠿⠶⠶⣮⣭⡉⠛⠛⠉⠙⢹⣙⡛⣖⠰⠦⠀⢀⣐⠿⠿⢿⣷⣘⣯⣠⣾⠿⠿⠿⠿⢭⣽⢿⡿⣾⠿⣿⠿⠟⠝⠁
⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⠀⠀⢹⣿⣞⣻⡇⠀⠀⠀⠀⠀⠀⠉⠀⠈⠺⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠆⣠⣿⠀⣯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠷⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠹⣿⣿⣽⠏⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠚
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡿⣫⣬⡈⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣯⣤⡠⣬⣾⣿⣿⣿⣮⣥⠀⢌⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡜⣿⣿⣿⣿⡿⣃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢹⣿⣿⣿⢃⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠎⠓⢤⣀⡀⠀⠀⠀⠀⢘⣿⡞⠛⠟⣿⡌⢿⠀⠀⠀⠀⠀⢀⣀⣴⠿⠇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡦⣹⣶⠚⣛⡛⠿⣛⣛⠛⢻⣧⣞⣂⠀⠀⣿⡈⠛⠛⠛⠿⢿⣛⣫⡥⠺⢧⡜⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⡡⣠⣾⣿⣿⣿⣿⣿⣷⣄⣉⠉⡉⢙⣉⠙⠛⠖⠈⠋⣘⣛⣛⣛⣋⣩⣴⣿⣿⣾⣯⡢⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣮⣛⢿⣿⣿⡿⠿⢿⣿⠿⠿⠿⣿⣿⣇⠀⣤⠀⢸⠿⠿⠿⠿⠿⣿⣿⣿⣿⡿⣫⣆⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠙⠋⢁⣬⠭⠿⠿⠿⢿⣗⠂⣿⣏⠀⢍⠀⢸⡿⠿⠿⠿⠿⢷⣮⣝⡻⠞⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣣⡾⠋⠀⠀⠀⠀⠀⠀⢸⣇⡟⠃⠀⠀⡴⣿⠀⠀⠀⠀⠀⠀⠈⠻⢿⣃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢃⡀⠂⠘⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡹⠃⠛⠀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡄⣀⠆⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡓⢯⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣧⢸⠀⢺⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⢇⠘⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣴⢸⣷⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣾⣿⡆⠀⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢱⣿⣿⢏⡄⢷⢹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⡟⠱⡻⣿⣿⣿⣷⢞⣃⣹⠟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⠷⣿⣎⠻⡿⠋⢀⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣆⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⣿⣟⢹⣟⠻⣿⠛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣵⣿⠃⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⣿⡄⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣼⣧⢿⣧⠀⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠈⠛⡿⠋⠀⣀⣴⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢦⣾⣿⣿⣦⡀⢀⠴⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⣀⣸⣿⣿⣾⣿⣿⣿⣿⡟⠁⠀⠹⢿⣿⣿⣻⣿⣇⣀⣀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣶⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⣤⣽⣿⣷⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠿⠿⠿⠶⠶⠶⠦⠤⠤⣀⣀⣀⡀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣶⣾⣿⡿⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣧⡙⢿⣿⣿⠀⠀⡼⠋⣼⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣿⠀⠜⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡩⡄⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⣿⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⣿⣿⣿⣿⣶⡧⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⡽⠿⠞⠋

⣿⢿⢿⠿⠿⡿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿

⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢠⣴⡇⣷⣢⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠽⢯⡯⡅⣭⣭⠿⠍⠀⠀⠲⠐⠒⠒⡗⠒⠒⠚⠂⠀⠒⠰⠒⢺⡷⠶⠆⠀⠀⠀⠀⠒⠒⢺⠓⠒⠒
⠀⠀⠀⠘⡇⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠗⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⢀⠀⡀⡀⡀⠀⢀⠀⠀⠀⢀⣀⣀⣀⣇⣀⣀⣀⠀⠀⠀⢀⣀⣸⣇⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⡏⠉⠉⠁⠀⠀⠀⠀⠉⢉⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⣀⢤⡇⣣⢄⣀⠀⠀⠀⠀⢀⣀⣸⣟⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣀⣸⣇⣀⡀
⠘⠛⠂⢉⡆⠎⡘⠚⠛⠀⠀⠀⠛⠛⢻⣾⠛⠛⠀⠀⠀⠀⠛⠛⢹⡏⠛⠛⠁⠀⠀⠋⠚⠛⢻⡟⠛⠃⠁
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠉⠀⣀⣀⠀⠉⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠃⠀⣰⣿⣿⣿⣿⣆⠀⠘⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⢀⣤⣶⣿⣿⣿⣿⣿⣿⣶⣤⡀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀⠀⠉⣿⣿⣿⣿⣿⣿⠉⠀⢀⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦⣄⡀⠀⠀⠙⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠋⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠉⠉⠉⠛⢿⣿⣦⣀⣀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣀⣀⣴⣿⡿⠛⠉⠉⠉⠙⠻⣿⣷⡄
⠀⠀⠀⠀⣠⣶⣿⡿⠁⠀⣴⣶⣶⣶⣄⠀⠙⠿⠿⠿⠿⠿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠿⠿⠿⠿⠿⠋⠀⣠⣶⣶⣶⣦⠀⠈⢿⣿⣶⣄
⠀⠀⠀⣼⣿⠟⠁⢀⣀⣼⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣧⣀⠀⠈⠻⣿⣧
⠀⠀⢰⣿⡏⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢹⣿⡇
⠀⠀⢸⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⡇
⠀⠀⠀⢿⣿⣄⠀⠉⠛⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠛⠉⠀⣠⣿⡿⠁
⠀⠀⠀⠈⠻⣿⣷⣦⠀⠈⠿⣿⣿⡿⠟⠀⢀⣤⣤⣤⣤⣤⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣤⣤⣤⣤⣤⡀⠀⠻⢿⣿⣿⠿⠃⠀⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⡀⠀⠀⠀⣀⣴⣿⡿⠛⠛⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠛⠛⢿⣿⣦⣀⠀⠀⠀⣀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⣉⣛⣛⣛⣋⣻⡀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃⠀⣛⡟⢛⣻⣟⣟⠁⠐⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠈⣿⣿⣿⣿⣿⣿⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⢀⣤⣶⣿⣿⣿⣿⣿⣿⣶⣤⡀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠉⢿⣿⣿⣿⣿⡿⠉⠀⢀⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣧⡀⠈⠛⠿⠿⠛⠁⢀⣼⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣤⣄⣀⣤⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⢒⠫⢝⡲⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⠐⠁⠀⠀⠙⡸⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⣦⣴⢷⣤⣾⡇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢩⠏⣼⣿⡿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣇⢻⣿⠒⣘⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⢒⠶⠤⢤⣾⣿⠻⡐⡜⠀⢿⡆⠘⠦⣈⣧⣤⠤⠖⠲⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠕⠁⠀⠙⢷⣷⣶⣬⣧⠁⠀⣀⠈⠀⠀⣯⣽⣶⣶⡿⠋⠀⠒⠝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⡀⠀⠀⠠⣾⣿⣿⣿⣿⢇⠰⢏⠳⠀⠔⠁⣴⣿⣿⣧⠄⠀⠀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⡂⠀⠾⢛⣛⣳⡈⠙⠃⠙⠂⠁⢀⣀⣾⣿⣭⣛⣛⠀⠀⢈⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠚⠉⠉⠀⠈⠻⣆⡈⢿⣿⣿⠉⢁⡹⠟⠁⠀⠉⠉⠓⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⢿⣿⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡘⠁⠈⠹⣿⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢻⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡀⠀⠀⠀⠈⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣇⠀⠀⠀⠀⣀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⠛⠻⣷⣶⣿⡶⠛⠠⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢮⣊⣛⣉⡨⠞⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⣖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⡜⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣿⣿⣯⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⡏⣿⣿⠙⢫⡖⢳⡀
⠀⠀⠀⢀⠀⠀⠀⢀⣤⠛⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡈⠓⠶⠿⣿⣿⡷⠘⣫⣼⣧⡢
⠀⠀⠠⠀⠀⠘⣶⡾⣿⠀⠁⠀⢀⣤⠖⠀⠒⠀⠀⠀⠂⠴⠆⣐⡶⠤⠀⠈⠤⣤⣾⡄⠞⠀⠂⠀⠀⠂⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎
⠀⠀⠀⢘⠿⠟⠛⠉⠙⡷⠖⠂⠘⠛⠻⠦⠀⠀⠀⠀⠉⠻⠇⠻⠿⠂⡄⠀⠀⠺⠿⠣⢟⠚⠄⠀⠀⠈⠈⠛⣡⡶⣴⡄⢹⣿⠃⣾⢿⣿⣟⠑⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠅⢸⡇⣿⣿⠀⣡⡜⠡
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⢈⡇⣿⣿⠀⣿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⣼⡇⣿⣿⠀⣿⣄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢾⠛⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣾⣿⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⡰⠟⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⡖⠇
⠀⠀⠀⠀⠀⠀⣠⠔⣺⣇⡹⡟⣀⣗⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⢆⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⡇
⠀⠀⠀⠀⢀⡄⠁⡶⠛⢩⣰⣇⡹⠙⢤⣈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠘⠾⢷⠈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡇
⠀⠀⠀⡇⠚⠩⡵⠒⣂⠖⣩⠋⡥⢤⡒⠿⠟⡛⠂⠀⠀⠀⣴⡠⣄⠀⠀⠀⠰⡆⠀⣈⡤⣵⣀⠀⠀⢖⣃⣀⣐⣶⣶⣀⣸⣇⣁⣀⣲⣶⣀⣒⡄
⠀⠀⠀⡏⠹⢿⣄⠛⣻⣦⡘⣷⣞⣻⡟⢳⡚⠟⠀⠀⠎⠀⣤⡽⣂⠀⣻⡄⠀⣠⣤⣀⡺⣥⡄⠠⠆⠈⢯⡉⠹⠿⠏⠉⢹⡏⠉⠉⠉⠿⢏⣉⢻
⠀⠀⠀⠃⢶⡛⠓⠈⠀⢱⠟⠛⣿⠉⠈⣸⠿⡳⠓⠀⠀⠀⠹⠦⠉⠀⠀⣤⡀⡇⠀⠨⠥⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠁⡀⠑⣤⡇⢀⡇⠈⣤⠚⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠑⢮⢰⣾⣷⢾⢥⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⡟⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⢺⣷⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡛⢠⣿⣿⡆⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣬⣵⡗⣬⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣸⠃⢠
⠀⠀⠀⠀⠀⠀⠀⢨⠦⠤⠿⠧⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⢿⡿⠿⡿⡿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣛⣻⣿⣿⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⢄⡀
⠀⠀⠀⠀⠀⠻⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢃⣀⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣶⣰⣷
⠀⠀⠀⠀⠀⠀⠈⠢⠹⢦⡀⠀⠀⠀⠀⠀⠀⠰⣿⠋⠛⠉⢻⠆⠀⠀⢀⡀⠀⠀⠰⣟⠉⡉⢉⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣀⡀⣀⣀⣀⣤⣠⣀
⠀⠀⠀⠀⠀⠀⣤⠀⣨⡾⠟⢦⡀⠀⠀⠀⠀⠀⢰⣼⡇⣶⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⡟⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⡿⠻⠿⣿⣿⣿⣿⣿⢿⣿⡻⣽⣿⣳
⠀⠀⠀⠀⠀⠰⣿⣿⣷⣅⣀⠈⠙⢦⡀⠀⠀⠀⠸⠋⠃⠛⠀⠀⠀⣠⠟⠛⢦⡄⠀⣀⣁⣥⣤⣦⠀⠀⠀⠀⠀⠀⣾⡁⢰⡆⠀⠈⠙⠙⢻⡟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢈⡙⠿⢸⡿⠀⠀⢈⣿⣦⡀⠐⠟⠾⠛⠛⠻⡧⠚⠁⢠⣆⠀⠈⠓⢿⠀⠀⠀⢰⠀⠀⠀⠀⠀⣻⢹⡇⠘⠃⠐⢛⡀⠀⢸⡃⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢹⣃⠀⠀⠀⠀⢰⣿⠙⡏⢀⣀⣀⣀⣀⣀⣀⣶⣆⣸⣧⣠⣶⣦⣼⣶⠶⠶⠶⠆⠀⠀⠀⣰⠃⠈⢠⠀⣤⢸⣿⠀⠀⢸⡇⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣮⠁⠀⠀⠀⠸⠏⠀⠀⠈⠉⠉⠉⠀⠈⡿⠉⠉⠁⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⢰⡏⣄⣠⢸⡇⣿⢸⣿⠀⠀⢸⡇⠜⠉⠥
⠀⠀⠀⠀⠀⢀⡀⣿⣿⢀⠀⠀⠀⢲⠦⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡶⡆⠀⠋⠁⣿⣿⢸⡇⠟⠈⠁⠀⠀⢸⡇⢠⣀
⠀⠀⠀⠀⠀⢸⠶⠿⡿⢾⡏⠀⠀⣸⢹⡇⡀⠀⠀⠀⣠⠀⠀⠂⠀⢰⣿⢿⣷⡄⠀⣀⠀⣰⡆⠀⡟⠉⠁⢲⠆⡿⠋⠈⠄⠀⢺⡇⠀⠀⢸⡇⢘⣿
⠀⠀⠀⠀⠀⠈⠒⠌⡳⠦⠀⠀⠀⣾⣸⠃⠇⠀⠀⠀⣿⠇⠀⠀⠀⠸⣿⣼⣟⠁⠀⡏⠀⢿⡇⠀⣷⣶⢻⣿⠀⢀⠀⡶⡇⠀⢸⠟⠀⠀⢸⡇⠰⠿
⠀⠀⠀⠀⠀⠀⠈⠛⠤⠀⠀⠀⠀⡀⢁⠀⠃⠀⠀⠀⠿⠆⢀⡀⣠⡖⠉⣍⠈⠓⡤⣡⡀⠈⠁⠀⠼⢿⣦⢻⡌⣿⠀⡯⠟⠀⠻⣏⠁⠀⠈⣇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡂⠀⠀⠀⢹⡯⠀⢀⣴⠒⠚⠛⠛⣻⡿⠷⠶⠐⠛⠒⠒⠓⣘⡿⠁⠀⠀⠀⢸⡏⢻⡄⢋⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢸⡟⠀⢸⣿⡆⠀⣀⡀⠀⠏⠀⢉⣽⠷⠷⣦⡀⠀⣇⠸⠥⠃⠀⢰⣧⣾⣇⣤⣿⠀⣄⠀⠐⠜⠿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠠⠚⠛⠓⠲⠂⠀⠘⠉⠀⢾⣿⡇⠀⠀⠀⠀⠃⠀⣭⢳⣿⣿⡌⣷⠀⣇⠀⠀⠀⠀⠸⣿⣿⡇⣿⣿⠀⠀⢠⣤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⣴⣿⣶⡷⠸⠿⠃⠀⠀⠀⠀⠱⣤⣿⡆⣭⣭⢸⡿⢶⣇⠀⡇⠀⠀⠀⠉⠙⠃⣿⣿⠀⠀⢸⣿⠄⠀⠀⠈⣯
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⣿⣟⣸⠇⠀⢀⡾⠀⠀⢰⠆⠀⣿⣶⡇⣿⣿⠠⣷⡾⢿⠚⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠛⢤⡠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣫⡇⠀⠀⠙⠉⠀⠀⠀⡈⠀⡇⠀⢸⣶⠀⣯⡻⠇⠉⠉⠁⢻⡕⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠐⠿⠀⠰⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⠀⠀⠀⠀⢀⢼⠟⢻⡇⠀⠘⠟⠀⢿⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡽⠧⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣶⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⢿⢦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⣸⠳⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⣀⣴⣄⣾⣶⡴⣿⣿⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⡀⢀⣸⣿⢹⣿⣿⣿⣾⡅⣿⣷⠿⢿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠓⠖⠒⠒⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⡟⣿⣿⣯⣿⠻⣿⣿⣷⣶⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣛⣷⣿⣿⣯⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢳⣿⣿⣿⡿⢷⣹⣿⣿⡙⢿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠥⠤⠾⣿⣿⣿⣟⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣼⣟⣿⣿⣿⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣃⣀⣉⣉⣉⣤⣤⣭⣭⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣏⣩⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠷⠶⠶⠶⠖⠒⠒⠒⢒⡒⣒⣒⠒⠒⢻⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⢷⣄⣿⠛⢻⣿⣷⡿⡿⠿
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⢉⣩⣯⣽⢿⠻⣿⣭⣭⣭⣭⣭⣥⣤⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠘⣿⣄⡿⠐⠋⠞
⠀⠀⠀⠀⠀⠀⠉⠉⣿⡶⠲⢶⡾⣡⣿⣦⠘⢿⣿⣶⠶⠶⠶⠦⠀⣾⡇⠀⢰⣿⡆⠀⣸⣿⡀⠀⣼⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣩⣿⡽⣣⣿⣿⣷⡈⢿⣯⣭⣍⣉⣉⣛⣿⡇⠀⢸⣿⣯⠀⣿⣿⡇⢸⣿⡗⠀⠀⢸⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢿⡟⣰⣿⡟⣉⠉⣷⡈⣿⣛⣲⣶⣒⣒⣿⡇⠀⢸⣿⣿⠀⣿⣿⡇⠘⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣴⣆⣰⣄⣿⣿⣾⣧⣿⡟⣿⣽⣷⣿⡇⣿⠤⠤⠤⠤⢤⣿⠇⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣧⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣸⣿⣿⣿⣯⠿⣿⣿⣿⣿⢹⣿⡇⣿⣿⡏⣿⠄⣿⢭⣭⠛⠋⢙⣿⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣯⣿⣿⣶⣾⣷⣿⣿⠧⣿⣾⣿⣇⣿⣿⢧⣿⠂⣿⣿⣿⣛⠛⢻⣿⠀⠀⠸⠿⠿⠀⠿⠿⠇⠀⠿⠟⠀⠀⢸⣇⠀⠀⠀⢸⡇
⠀⣉⣿⡿⠿⣬⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣷⣿⣒⡒⠶⠖⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣀⢀⣾⡟
⠐⠛⠻⠿⠿⠿⠿⣿⣿⣟⣛⡒⢒⡀⢠⣤⣴⣿⣿⣿⣶⣿⣿⣿⣿⣿⣾⣿⣿⣶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣾⣶⡶⠿⠶⠾
⣀⣀⣢⣤⡄⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠃⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⡀⠐⠒⠂⠒⠈⠉⠉⠉⠁
⠀⠀⠑⢀⠶⠊⠀⠀⠀⠀⠀⣤⣀⡁⡀⢀⠀⢤⣍⣀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣀⣨⠽⠵⠀⠀⠀⠀⠀⠲⠂⠠⠄⠤⠄⠀⠄⣀⡀⠀⠤⠰⠂⠉⠉⠉⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⠤⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠒⠒⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⣀⣀⣀⣀⣤⣄⣠⣤⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠿⠻⣿⠟⢻⠿⢿⠿⢻⢿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣷⣾⣶⣿⣶⣿⣷⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡄⢀⣀⣴⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠻⠟⠙⠁⠀⠀⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⢘⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣋⣤⣷⣄⣈⣉⣃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠙⣤⣠⠌⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠔⠀⠐⠧
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠈⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⡿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⡛⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠴⢆⣾⡿⣫⡌⢻⣦⡀⠀⠀⠀⠀⣸⣶⡙⠻⡇⠀⠀⢀⣴⣦⠀⢀⠀⠀⢰⣿⡀⢀⡀⢀⣴⣿⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠍⠀⣮⡀⣥⠙⢿⣶⡆⠀⠀⢸⣿⡇⠀⠅⢨⠀⢸⣿⣿⠀⠘⠓⠀⣿⣿⡇⠈⠉⠈⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠙⠃⠻⠁⠨⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠈⢩⡅⠀⠀⢀⣶⠀⡀⢠⣿⢺⣇
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢸⡜⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡃⠀⠀⠀⣿⣿⠀⠀⠀⢀⣾⡟⠀⠒⢸⣿⠀⠉⢸⣷⢸
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢱⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢸⠀⠀⠀⢸⣿⣿⠀⠀⢠⠀⢿⣿⡇⠀⠰⣆⢹⣿⠀⠀⠀⠈⢻⣏⠀⠀⢸⣿⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠁⠀⠀⠀⡆⠀⠸⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢘⠁⠈⠀⢸⣿⣿⠀⠀⠘⠓⢸⣿⡆⠀⠀⠀⠀⣿⠄⠀⠀⠀⢸⡏⠀⠀⢈⣿⠀⠀⡠⣿⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⡿⢿⡇⣸⠀⠀⠀⢀⣼⣿⡇⠈⠀⢘⠀⢸⣿⣿⡄⠀⠀⠀⠈⠏⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢠⡰⠆⣻⠀⡆⠁⢂⣸
⠀⠀⠀⠀⣄⠀⠀⠀⢀⡀⠐⠇⠀⠀⢹⠀⠀⠀⠀⠀⠉⠉⢰⠀⢠⠀⠉⠉⠁⠀⢀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠆⠂⢸⣿⢸⡇⠀⠹⠂⣿⡄⢈⠱⠂
⠀⢰⡆⡆⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢸⠀⠀⢘⢁⠰⠿⣦⡄⠀⠉⠙⠃
⠀⠀⠁⠀⣠⠞⣻⣄⠅⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠀⣿⠀⢘⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⣿⣿⠀⠀⠐⠚⠀⠐⠂
⠀⠀⢤⣴⠗⣫⠭⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣄⠀⠀⠀⠀⣿⠀⠠⠀⠀⠀⠀⠀⣸⣿⣷⠀⠂⠀⠀⠉⠛⠊
⠀⠀⠀⠀⠉⠓⠯⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠒⠿⠀⢠⡄⠀⠐⠀⠉⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⣯⣋⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣶⠞⠝⢶⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠅⠄⠀⠤⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠠⣏⢘⠃⢱⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠉⡈⠔⠃⠀⠀⠀⠙⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⢀⣿⣆⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣾⣏⣿⡀⠀⠉⢱⠀⣠⣤⣤⠠⣾⠿⢽⠿⣿⢿⣿⣷⠄⢠⣄⣀⠀⢰⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⢠⣿⣏⣿⣷⡀⠀⢸⠀⣿⣿⠉⠀⠀⢀⡔⢂⢐⡒⣄⠀⡇⢸⣿⡿⡄⢸⠀⣤⣤⠄⢠⣤⣤⠄⢸
⠀⠀⢰⡿⣟⣯⣿⣯⡁⠀⢸⠀⣿⠋⠰⠀⠀⣮⠛⠛⠛⠛⠋⠀⣧⢸⠟⠀⠇⢸⠀⣿⣃⠀⠘⣇⣋⡆⢸⠀⠀⢀⡆
⠀⠀⠀⣴⣿⣷⣿⢿⣷⡄⣸⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠂⢸⠀⡿⠀⠀⠠⡇⠀⠇⢸⠀⠀⣼⣿
⠀⢀⣾⣿⣿⣏⣿⣿⣿⡄⢿⠀⠋⣶⠐⣠⠖⣯⠀⠀⠀⠀⠀⢠⣿⡎⢠⠆⠀⢘⡀⠃⠀⠀⠀⠃⠀⠠⢼⠀⣸⣿⣿⣧
⣠⣿⣿⣿⣿⣧⣷⣿⣿⣿⣿⠀⠀⢼⠋⢠⠤⠯⠤⠤⠤⠤⠤⠴⣿⡿⠦⣄⣀⣸⣁⣉⣉⣙⣉⣿⣿⣿⣏⣰⣿⣿⣿⣿⣆
⠉⠉⠉⠉⠉⠱⢿⣿⣿⣿⣿⣶⣶⣸⣦⣭⣭⣭⣭⣭⣭⣭⣭⣭⡘⣧⣀⣻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠹⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡿⠿⠛⠉⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠃⠀⠓⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⡇⠀⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣀⠤⢤⡈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⡤⠚⠁⠀⠀⠘⠢⣄⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢉⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣠⡔⠋⠀⠀⢀⡴⣺⠿⢭⠿⢿⣦⡀⠀⠀⠙⣦⡈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⢹⡆⠀⠀⢠⣯⡟⠁⠀⢐⠀⠀⢹⣿⡀⠀⠠⡇⡿⠦⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⡇⠀⠀⠸⡿⡇⠉⠉⠙⠀⠉⣨⣿⠃⠀⢸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣩⢼⠀⡇⠀⠀⠀⠙⠮⣒⡶⠶⠶⣚⡿⠋⠀⠀⢸⢸⣯⡓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣉⠤⠒⠉⠁⢸⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢸⢸⠀⠈⠓⠦⣍⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⡥⠖⠋⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠀⠀⠀⣀⡀⠙⠳⢬⣙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⢺⣥⢴⡛⡇⠀⠀⣠⡶⠿⢷⣦⡀⢸⡄⣇⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⡿⣿⠀⣰⡿⠻⠛⢷⡄⠀⠈⢳⢦⣉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣧⠀⠐⣿⠀⠲⠆⣿⡇⢸⡇⣿⠀⠀⢠⢞⡿⠋⢹⠉⠻⣿⣦⠀⠀⠀⡇⣿⠀⢿⡇⢙⠃⣻⡗⠀⢀⡏⡟⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡀⠀⠙⠧⣴⣶⠿⠁⠸⡇⢸⠀⠀⡟⣸⠁⠀⢸⠀⠀⠘⣿⣧⠀⢠⡇⡏⠀⠈⠛⠾⠟⠋⠀⠀⣸⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢠⡇⣿⠤⠤⢼⠤⠤⠄⢿⣿⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢷⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⡇⡗⠀⠀⢸⠀⠀⠀⢸⣿⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡄⠀⠀⠀⠀⠀⠀⠀⣧⢸⡄⢸⡇⡧⠤⠤⢼⠤⠤⠤⢼⣿⠀⡿⢸⠇⠀⠀⠀⠀⠀⠀⢰⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⣴⠒⠒⡖⠛⠛⡆⣿⢸⡇⢸⡇⡇⠀⠀⠸⠀⠀⠀⢸⣿⠀⡇⢸⢰⠖⠒⠶⠒⠲⡆⣼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣹⡀⠀⠁⠀⠀⡇⢿⢸⡇⢸⡇⡇⠀⠀⢀⡀⠀⠀⢸⣿⢰⡇⢸⢸⠀⠀⠸⠀⢰⣷⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣏⠛⠛⠛⠛⠛⠁⢸⢸⡇⢸⡇⡏⠉⠉⠉⠉⠉⠉⢹⣿⢸⠃⢸⠈⠙⠛⠛⠛⠋⡟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡇⡇⠀⠀⠀⡀⠀⠀⢸⡏⢸⠀⢺⠀⠀⠀⠀⠀⢰⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⢿⣿⠿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡄⢸⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⡀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣶⡿⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠋⠉⢷⣄⠀⠀⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⡟⠁⠀⠀⠀⠙⢧⡀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡇⠀⠀⠀⠀⠀⠈⠗⠲⠶⠶⠦⢽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣄⠀⢰⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠔⡩⢾⣿⣦⣸⠀⠀⠀⠀⢸⣯⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠊⠀⠀⠈⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠷⠶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢣⠞⠁⠀⣀⣄⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢁⡴⠃⠀⠀⢸⡅⠀⢱⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⢀⣴⣟⣋⣠⡏⠀⠀⠀⠀⠘⢧⣀⠞⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠊⠉⠉⢦⠀⠀⠀⠀⢸⠀⠀⢀⠤⣄⠀⠈⠉⠉⢉⡙⠛⠒⠒⠶⠤⢤⣤⣙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡟⠛⢻⠛⠻⡀⠀⠀⠀⡜⠀⢠⠃⠀⢸⠄⠀⠀⡰⠉⢹⡆⠀⠀⠀⢀⡾⠋⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠁⠀⢸⠀⠀⡇⠀⠀⠀⡇⠠⠞⠒⠄⠾⡄⠀⡤⢇⣀⣼⡇⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡀⠀⠘⡇⠀⢸⠀⠀⡇⠀⠀⢸⠁⠈⠛⠓⠒⠒⠁⠀⠓⠂⠤⠤⠜⠀⣰⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠘⠷⠶⢦⡷⠤⠾⠤⣤⣇⣀⣀⣼⣀⣠⣤⣤⣤⣤⡤⠤⠤⠤⠤⣤⣤⣤⠋⠉⠉⠛⠻⢷⣄
⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠫⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢺⣯⣀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⣶⣦⣾⣋⠁⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⣴⡾⠿⠟⠛⠛⠛⠛⠻⠿⠿⠷⠶⠶⠶⠶⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⠶⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠐⠒⠂⠀⠰⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠦⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠇⠠⢸⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡗⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢸⠀⠠⠦⠴⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡇⡀⠠⠞⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⠀⣀⠀⠀⠀⢸⡇⠀⠀⣿⠡⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡇⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⣿⢏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⢀⣀⣸⣇⠀⠀⠹⠴

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⣿⡯⢿⣿⡽⣿⡏⣿⣏⣿⣟⣽⢏⣿⣍⣿⣿⣋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠀⠂⠓⠀⠀⠀⠠⢿⣿⣧⠿⣶⣯⡿⣇⣯⡟⣮⣯⣾⣞⣹⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⡇⠀⠀⠀⠀⠀⠀⠀⢈⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⣿⣽⣿⣽⣿⣾⣿⣷⡿⣾⣽⣟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡
⡇⠀⠀⠀⠀⠀⠀⠀⢈⡛⠛⠛⠓⢶⡀⠀⠀⠀⠀⠀⠀⢉⠽⠛⢿⢿⣾⣯⣿⣷⣶⢿⠟⠻⢌⠀⠀⠀⢰⣿⣶⠀⠀⠀⠀⠀⠀⠀⠤⡀⠠⠃
⡇⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠋⠩⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⡟⠙⠉⠀⠀⠀⠀⠠⠀⠀⠘⠛⠛
⡇⠀⠀⠀⠀⠀⠀⠀⠹⣶⠶⢶⣶⣶⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠈⠑⣦⡤⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢾⡀⠀⡀⠀⠘⠡⠔⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⠌⠩⣤⣶⡶⠟
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠀⠀⠢
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⡠⠌⠀⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⠖⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣒⢶⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢧⣛⡼⠥
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⠘⠋⢀⡟⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⡇
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡛⢝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠊⣏⣸⢀⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠉⣹⡍⢹⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⠭⠤⠼⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⡐⢀⡠⠾⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣴⣿⣷⣦⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢫⠟⠱⢽⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⠔⠀⠀⠀⠄⠂⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢟⠅⠉⠁⠀⠀⠀⠀⠀⠊⠁⡝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢟⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢋⠅⠁⣂⣀⠀⠀⢀⢀⣠⣄⡀⠐⠀⠈⠀⠀⣨⣙⢝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⠕⠑⠂⠀⠀⠀⠀⠀⣠⣶⡫⠟⢧⣄⠀⠀⠀⠀⠀⠀⠀⠐⢙⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠕⠋⠀⠀⠀⠀⠀⠰⣤⡾⠫⠊⠀⠀⠑⢝⢷⣄⠀⠀⠀⠒⠀⠀⠀⠙⢿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣷⠁⠀⠀⠈⡀⠀⠉⣩⡾⡡⠊⠀⠀⠀⠀⠀⠂⠑⢝⢷⣅⠀⠀⠀⠀⠀⠀⠀⠗⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠄⠀⠀⣴⣶⡤⠘⠛⢺⡧⠭⠥⠍⠁⠀⠀⠀⠀⢽⡿⠛⠁⠀⣢⣖⠀⠐⠂⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢹⣽⣟⠀⠀⢸⣟⠀⢠⡤⠤⠬⠥⣄⠀⣿⡇⠀⠀⢘⣿⣿⣃⠀⠁⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠠⠴⢺⣿⡷⠀⠀⢹⡷⠀⢸⡅⢠⡄⡂⡇⠀⣾⡇⠀⠀⠰⣿⣿⠂⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⠂⣀⣸⣿⣟⠀⠀⢸⣗⠀⠘⡄⢸⡇⡇⡇⢀⣻⡇⠀⠀⢺⣿⣿⡂⠐⠆⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠄⠀⠠⣾⣿⣯⠀⢀⢸⡯⠀⠠⠀⠸⠇⡈⡧⠄⣽⡇⠀⠠⢴⣿⣿⡇⠀⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡂⠀⠀⠛⠛⠛⠂⠀⢸⡷⠀⢰⡇⠐⠐⠆⡇⠀⣿⡇⠀⠀⠘⠉⠛⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣁⡀⠈⡉⠉⣉⣁⣹⣹⡶⠈⠉⢉⠩⠿⢽⣿⣏⣏⣀⣁⣉⣀⣉⣉⣀⣁⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠁
⠀⠀⠀⠀⠠⢒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⠀⠀⢠⣦⣀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠀⠁⢸⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀
⠀⠀⠀⢄⠀⠀⠀⠀⠀⢠⣀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⢀⡠⠤⠀⠈⠀⠀⠀⠀⠁
⠀⡌⣤⣌⣢⣀⣤⣤⣤⠈⠀⠀⠀⣿⣿⢿⣿⣿⠇⠀⠀⢸⣿⣿⡟⠛⠛⠃⠀⠀⠀⠀⠀⢁⠀⠀⠂⢤⡀
⣰⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣼⣿⡿⠀⠀⠀⢸⣿⣿⣅⢀⡐⠀⠊⠅⠀⠀⠀⢠⡀⠀⠀⠀⠙⠆
⡇⠀⠸⢿⡿⠻⢿⠟⠉⠀⠀⠀⠁⠆⡂⠈⠻⠃⠀⠀⠀⠸⠿⠿⠹⠀⠃⣠⠀⢀⣀⠀⠀⢀⣿⡀⠀⠀⢠⣢⣀⣀⣀⣀⣀⣀⣀
⠐⡄⠀⠀⠀⠀⠀⠀⢄⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⠂⠀⠠⡀⢁⣀⢾⣿⠿⠿⢿⣿⠛⠻⢟⡛⠉⢉⠛⠛⠻⣿⣿⣿
⠀⠘⡄⠀⠀⡀⠀⠀⠈⢂⠀⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠟⠀⠀⠀⠀⠿⣿⡟⠋⠀⢠⣿⣏⣠⠤⠖⠒⠂⠀⠀⠀⠀⠀⠁⠉
⠀⠀⣁⠄⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣤⣶⣾⢿⡿⠛⡚⠛⠛⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢸⠀⠀⠈⠀⣤⠖⠂⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⢀⡀⠀⢀⠄⠀⠰⢤⠀⠔⠃⠀⠀⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢀⡀⠀⠤⣬⠝⢫⣴⠊⢉⢰⠀⠴⠒⠚⠍⣤⠄⠀⠝⠀⠀⠠⠄⠀⠀⢤⣦
⠀⠀⠀⠀⠀⠀⢀⡾⠃⡤⠄⠀⠀⠐⠂⠠⠉⠨⠀⠀⠀⠀⢬⠴⠀⠄⠘⠈⠏⢨⡁⠀⠐⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢀⣴⠶⠷⠐⠀⡖⠒⣆⠴⠀⠉⠊⠚⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢜⢢⡄⠀⠀⠀⠀⠀⣿⠷⡄
⠀⠀⠀⠰⠏⠀⡤⢤⡀⠀⣁⡴⢉⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⢀⠀⠘⠐⣀⣀⠀⠄⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡿⠀⠀⣐⠀⠀⠂⠨⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠩⡄⢍⣷⡁⢾⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠃⣰⠋⠹⢏⣤⣒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠅⣸⠀⢀⣌⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢋⠅⢠⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⡁⡟⢴⡄⠛⠅
⠀⠀⠀⢀⣀⠀⡀⢠⡞⢶⣇⣴⣵⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢸⠻⡎⢷⠂
⠀⠀⠀⠀⡀⠀⣠⡏⣵⣙⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠣⣾⠀⠀⣸
⠀⠀⠀⠀⠳⣴⣿⣶⣝⣿⣿⣿⠿⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠁⠀⢀⠀⠀⠐⠀⣠⡊⠈⠓⠚⠃
⠀⠀⠀⠀⣠⣿⣿⣾⡿⢿⠟⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢺⡄⠰⢆⠀⢠⡜⠃
⠀⠀⢀⡄⣙⡿⠛⣩⣛⠿⠆⠀⠲⡀⠑⡀⠀⢀⡀⢠⠀⡀⠀⠀⠀⠠⡀⠉⠡⣁⠈⠁⢞⡁⠠
⠀⠀⠐⡻⠋⠐⡹⣵⠿⠇⠀⠀⠀⣅⣤⢈⣀⠄⠉⢀⡀⣀⡀⣀⡘⠀⠀⢳⡀⠈⠽⠯⡁⠁⢦⠰⡿⠈
⠀⠀⠈⣄⢨⠙⣠⣿⣠⡆⠀⠀⠀⠸⠓⠸⠋⣢⢀⡁⡀⠉⠁⠀⠈⠉⠒⠈⢤⠠⠀⠈⢳⡤⠈⠇⠄
⠀⢀⣀⡀⠠⣼⠋⢉⣉⡀⠀⠀⠀⠀⠀⣰⣞⡻⠯⠉⠀⠒⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⡄⠀⠁
⠀⠈⠄⣠⠹⡏⡇⢸⢶⡧⠀⢀⠀⢰⡷⡿⢿⡅⠢⠀⠀⠀⠀⠀⣤⠀⠠⢠⡂⡀⠀⠀⠀⠀⢀⢠⡶⢀
⠀⠀⢠⠃⠀⣱⣿⣶⣻⣿⣄⡏⣄⢾⣿⠇⠈⡠⢀⠀⠀⠀⠀⡀⠀⠁⡰⠁⠈⠀⠂⠀⠀⠀⠠⠎⠀⠀⠀⣠
⠀⠀⠰⠆⠀⠒⠒⢒⣾⣩⡏⠀⠿⠓⠋⢀⠈⠀⠈⠁⠀⢀⠈⣀⠀⡗⠁⡈⠁⡀⢀⣨⣀⣶⡅⠡⠆⣤⠊⣉⠅
⠀⠀⠀⠓⠀⡀⢀⣿⣰⢿⠫⡆⡀⣑⡴⠈⢠⡄⢤⡥⠀⢬⣤⡴⢸⣧⡀⠀⢒⢻⣄⣠⡹⣽⠾⣱⢦⣼⣇⡀
⠀⠀⡃⠘⠃⣤⢤⡈⠋⠸⡟⠗⠃⢘⢣⣌⣿⣉⢳⣶⣖⣆⡽⡃⠾⠀⣝⠶⣯⣤⣼⠚⠏⢹⡳⣴⣿⣿⡘⠳
⠀⠀⢃⣐⢤⡍⠀⠀⠀⠐⡇⡦⠐⠺⠺⠙⠟⠹⠛⠉⢉⠁⠀⠉⠉⠋⠛⠷⢭⣦⡿⠀⠀⠀⠀⠎⢼⠋⣃
⠀⠄⠀⠿⠛⢿⠀⠀⠀⢀⣹⣄⠀⠀⠈⢀⠈⠤⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⢀⣌⠈⣰⠓⢽⠂
⠀⡂⠀⢠⠐⡎⢃⠀⣰⣾⡛⠋⠃⠀⠀⠀⠁⠀⠤⠀⠠⢀⠀⠀⠁⠀⠀⠀⠛⠿⣴⣆⡄⠟⠛⣇⣹⠉⠫⠁
⠀⠀⠀⠤⠄⠼⠈⣦⠋⠨⡅⠀⡄⠀⠀⠀⣰⣳⢘⡞⢓⣞⢻⡦⠄⠀⠀⡀⠘⡄⠸⣿⡗⣼⠋⢀⣛⠠
⠀⡆⠀⠀⠻⠶⠓⠚⢂⣠⢦⠀⠤⡐⠉⣡⠺⠷⢭⠞⠛⠟⠫⡔⠻⢊⣿⣲⣞⣹⣅⣾⡱⢦⢮⣟⠉
⠀⡄⠀⠀⢀⠠⡌⠉⣂⣋⢈⠀⡘⠛⠀⠩⠋⢙⠷⠉⠀⠀⠠⠷⠷⠟⠶⠿⡯⡏⠉⢶⠆⠀⠈
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠛⣋⣉⣡⣦⣴⠍⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⡇⠈⠣⡈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣯⣯⣽⣿⣿⣾⣷⠄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣷⣌⠳⣄
⠀⠀⠀⠀⠀⠀⣰⡿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣷⡌⠳⣄
⠀⠀⠀⠀⢀⣼⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣦⡙⠦⡀
⠀⠀⠀⣠⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢦⡀
⠀⠀⣼⡯⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠻⣆⠀⠀⣀⣀⠄
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⠛⠋⠉⠉⠀⠀⠀⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⡄⣻⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣿⠂⠀⣀⣠⣾⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣶⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⢹⣿⣿⣿⣿⣿⣧⠀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⡿⠿⠿⠿⠿⠷⠤⠤
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇⣾⠁
⠀⠀⠀⠈⠉⠛⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⠃
⠀⠀⠀⠀⣀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠆⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠙⠛⠛⠶⠶⢶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠷⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⡀
⠀⠀⠀⠈⠃⠂⠀⡀⠀⢀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡢⠀⠀⢂⣤⠔⠛⠛⠁
⠀⠀⠀⠀⢒⣌⡳⣾⣄⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⡄
⠀⠀⠀⠀⠈⢉⣿⣿⣷⣮⣶⠁⠂⠀⠀⠈⠀⠀⠀⠀⠀⠐⠊⠀⠑⠀⣀⠄⠀⠀⠀⠀⠀⠈⡠⣦⡀⠶⡤⠄
⠀⠀⠀⠀⠀⠰⢻⠛⢳⡦⣿⢀⡀⢈⢤⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠘⠉⠀⠀⠀⠀⠀⠐⠚⠁⣄⡁⠀⢀
⠀⠀⠀⠀⠀⠀⢈⣹⣭⣽⣆⠄⠀⠀⠎⠠⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠂⠀⠀⠀⠀⠀⣠⣤⠀⠰⠷⠅⠊⠐
⠀⠀⠀⠀⠐⠀⠀⠙⢿⣧⣟⡊⠀⢲⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠩⣶⠥⠀⢉⣯⠐⡀⠀⠀⠀⠀⠐⢠⢁⡔⠀⠀⠀⠀⠐⠀⠀⠈⠀⡼⢂⠐⠀⠀⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠈⠽⠃⠐⠀⠀⡆⠀⠀⠈⠀⠉⢀⠀⠀⠀⠀⠀⢀⡀⠌⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠑⣤⣤⡤⠤⠤⠂⠁⠀⠄⡘⣀⡃⢼⠆⠈⠑⢏⠁⠂⠠⡬⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄⠀⠀⠀⠀⠓⠒⠀⢀⡠⠀⠋⢰⣋⣠⣉⡁⠘⢄⠠⣀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⢀⠔⠊⠁⠀⠀⠀⠁⡠⢉⡃⢴⡄⠈⢃⠀⠑⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⣛⡚⢛⢹⡇⠀⠀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠲⡿⠥⣈⠛⠏⠩⠀⢠⠎⠀⠀⣤⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠷⢶⠒⠘⠆⠘⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡤⠤⠤⠔⢛⣂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠆⠀⢱⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠘⢠⡟⠲⣄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡤⠀⠀⢹⡀⠀⠻⠇⢠⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡤⠀⠀⠸⡆⠀⠐⡀⠀⠂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⡇⠀⠀⢸⣥⠀⠀⣿⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠠⠤⠀⠠⠄⠶⠂⠾⢀⠇⠀⠀⠸⡇⠀⠀⠈⡷⠀⠀⢻⠀⠠⠄⠺⠇⠠⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⡀⢀⣰⡆⠀⠀⡠⠟⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡡⠁⠀⠀⢠⠀⢆⠀⠀⠀⠀⣀⡄⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠀⡁⠂⠔⠁⠁⡂⠀⠂⠉⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⢢⠀⠀⠀⠀⠀⠀⠊⠀⠀⠆⠀⠀⠀⠄⠘⠀⠈⠡⠀⠋⠀⡀⠀⠀⠄⠄
⠀⠄⠀⠀⠀⣠⠀⠀⠀⣀⠈⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠚⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⣤
⠀⢀⠀⠀⠀⠐⠁⠀⠈⣿⠀⢠⡄⠀⠈⠁⠁⠒⠈⠉⠃⠄⡈⠡⠀⠠⠤⠀⠊⠑⠦⠀⠀⠀⢠⣄⠀⢸⡏⠀⠀⠑
⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣀⡀⠶⢶⠀⠀⠀⠀⠈⠀⠐⠀⠁⠀⠁⠀⠀⢰⠶⠆⣀⡀⠹
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠇
⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠒⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠔⢉⡤⠒⢶⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⡗⠒⠦⡈⠒⢄
⠀⠀⠀⠀⢀⠞⠁⣴⢋⠔⡩⣻⡆⠀⠀⠣⣀⣀⠀⠀⠀⣀⣠⠜⡀⢈⣰⣟⢌⠢⡘⢦⠀⠑⡄
⠀⠀⠀⡠⠁⣤⠚⡧⡎⡼⢨⢯⢟⡷⣦⣄⣠⠬⠷⠀⠚⠤⢄⣠⣴⢾⡻⡝⡅⢣⣹⢼⡓⣦⠈⢆
⠀⠀⡰⢁⡴⢋⡾⣱⢫⢗⡿⣼⣨⣊⡶⠋⠀⠀⠀⠀⠀⠀⠀⠙⢶⣐⣅⣷⢿⡺⡝⣎⢳⡜⢲⡈⢆
⠀⠠⢁⠞⣠⠟⡴⢣⡯⠞⠚⢧⡇⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣸⠼⠉⠓⠽⣌⢢⡙⣄⠱⡌⠄
⣐⡡⠮⠴⠷⠚⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠙⠚⠳⠽⢮⣢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣥⣀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣘⣿⣁⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣾⡟⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⡛⣷⠶⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠹⣇⠿⢤⣸⠓⠒⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣤⡀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⣿⣿⡧⠀⠀⠈⢳⣴⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⠲⠷⠶⢤⣤⡔⣂⣀⣀⡐⠲⣶⣾⣦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⢸⠀⠀⢰⣿⠀⠯⢭⣭⡭⠭⢽⡿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠿⢿⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡟⡇⠀⢸⣿⠀⠘⡈⠳⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡯⡇⠀⢸⣿⠀⠀⣏⡙⠛
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⢤⣇⣶⣾⠀⠀⠀⠀⢸⠀⠀⠘⠓⠀⠀⢸⣿⠇⠀⢸⣻⠂⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣴⡏⣇⣾⢻⣿⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠸⣼⣟⡁⣿⡞⠁⢰⠀⠀⠀⠀⠘⠒⠒⠂⠠⠤⠤⠤⠤⠤⠀⣀⡀⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣴⣿⣿⣦⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣤⣀⣠⣾⣿⣷⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠙⢿⡿⠋⠉⠉⢹⣿⣿⣿⣿⣿⣿⡿⠉⠉⠙⠿⣿⠟⠉⠉⠉⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⠀⠀⠀⠸⣷⣄⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⢀⠀⠀⠀⠹⣿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠸⣿⣦⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⡄⠠⢤⠀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠙⠻⣇⠀⠀⠀⠘⠄⡄
⠀⠀⠀⠀⠀⡴⠦⠀⠲⠤⢘⢦⠀⢐⠠⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣄⠀⠀⠁⠢⢄⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠛⠈⠁⠐⠀⠦⣤⣤⣀⣀⡘⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠀⠀⠄⠒⠀⠀⠈⠃⠐
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠈⠀⢀⢀⣀⠈⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠄⠀⢸⠞⠉⢆⠀⠀⠀⢀⠉⠀⢀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⡀⢀⠈⠙⠖⡋⠘⠂⠂⠀⠚⢂⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠄⠉⠐⡀⠚⡅⠛⠐⠧⡤⠀⢶⠀⠆⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠸⠖⠌⠀⠸⠧⡇⠀⡀⠠⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢀⠷⡀⡁⠞⠀⠀⠀⠀⢀⠀⠆⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠐⡄⠀⡐⠂⠄⠈⠙⢱⠾⠶⣁⠀⠀⠒⣈⠦⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠘⠛⠲⣄⢸⡆⢠⡄⣵⢀⡔⠋⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⠈⠀⠀⠀⠀⠄⢸⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠀⠠⠀⠀⠀⠀⢀⠊⠘⡁⡀⠀⠀⢀⣸⠀⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠸⠷⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣠⠞⠀⠀⠀⢱⠈⠑⠔⠀⡆⢥⡀⠀⠀⠀⠻⣄⠀⠀⡾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠙⢄⠁⠀⠀⠀⠄⠀⠀⠀⠀⠁⢀⢧⠀⠀⠀⠀⠈⠀⠀⠙⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠟⠴⡆⣿⢳⡄⠧⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⢀⣬⠀⠀⠀⢿⠠⠃⢰⠀⢀⠀⠀⠀⠐⠶⠄⠀⠠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣁⠛⢢⡀⠀⠸⣧⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣍⡄⠀⠳⠀⠀⡿⣇⢰⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣧⡟⢰⡗⠀⣰⠇⣇⣿⠀⢀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠁⠀⠀⠈⠀⢀⣤⡠⠞⡩⠀⡞⠀⡐⠛⣠⣿⠟⠀⠸⣁⠉⣠⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠂⠂⠀⠴⠦⡈⠀⠄⠀⠊⢀⣼⠁⠐⣇⠀⢼⡸⣟⡄⠁⠀⠀⢨⣄⠠⠀⠀⢤⡀
⠀⡀⠀⠀⠀⠀⠀⠘⣃⡀⠀⢄⡑⠆⠀⠑⠲⡆⠰⠶⠋⠉⠃⢰⡿⠄⠀⠓⠿⠄⠒⠀⡴⠕⠘⠧⠀⣀⠀⠀⠀⢀⡀
⠀⠅⠀⠀⠀⡠⠄⠀⠈⠀⠘⠉⣔⠊⡇⠀⠀⠱⠀⢢⡀⠁⠸⠸⠫⢶⡀⢢⣴⠲⠶⢏⡁⠀⠈⠦⠘⠋⡈⢰⠐⠊
⠀⠂⠀⠀⠀⠀⠤⠀⠀⠠⠀⠀⠈⡠⠇⣘⠂⢀⡲⡦⢄⢠⡴⢐⠁⢺⠀⣠⠀⠀⠐⢀⠱⠀⠁⠀⠆⠀⠐⢣⣄⠀⡁
⠀⠀⠀⠀⠀⠀⠘⡂⠀⠘⠑⠄⠘⣠⠠⣀⡼⠏⠀⠀⠀⡈⢤⢀⣀⠈⠻⠑⡀⢠⠀⢀⠠⠃⠂⠉⠤⡷⠄⠀⠀⠈⠉
⠀⠃⠀⠀⠀⠈⠒⠓⠖⣠⡄⠐⠀⡠⠏⠈⠀⠚⠀⠀⠀⠈⠠⠠⠈⡼⠀⢈⠲⢈⡕⠠⠆⠠⡄⠹⠏⠀⠉⠁⠠⠐⠐⠒
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠃⠀⠘⠂⠀⠀⠀⠀⠀⠚⠒⠈⠁⠀⠁⠀⠉
⠀⠛

⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠰⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣆⠀⠀⠀⠀⢠⣾⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⣠⣾⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠿⣿⣿⣶⣤⣀⣀⣲⡄⢻⣿⣿⣿⣿⣷⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠁⠸⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠄⠀⢹⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⡟⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠄⣿⣿⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠃⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢔⢶⣪⣧⣵⣥⣭⣏⣜⡰⠢⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣐⣤⣲⡽⠞⠛⠉⣽⣿⣿⣿⡟⣿⣿⡉⠙⠳⢭⣗⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡋⠁⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠈⠙⢿⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠨⡑⠦⣄⣀⠈⢿⣿⣿⣿⣿⣟⣹⠃⠀⢀⣀⡤⢖⡫⠒⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠴⢂⢌⣉⡛⠛⠿⠿⠿⠿⠗⠒⢋⡩⡵⢮⠋⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠜⠙⠑⢰⠐⠺⡙⠌⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠶⠶⠲⠒⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⡠⠔⠋⠉⠀⠀⢀⣀⣀⣀⣀⠀⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⢷⣶⣶⣶⣶⡯⠶⢤⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢷⣶⣶⣶⣶⡶⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠼⣿⣿⣿⡿⠁⠀⠀⠁⠀⠀⠈⡄⠀⠀⠀⠆⠀⠀⠈⠀⠀⠌⣿⣿⣿⣿⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠠⠦⠤⠐⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⠀⠀⠄⢀⣀⠄⠀⠀⠀⠀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⢀⣀⣀⣀⠀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠈⠙⠢


⡿⠿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⡀⠀⢀⢰⡄⠀⠀⠀⠀⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠘⡀⠀⠀⠀⠁⠦⠽⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠇⠀⠀⢠⣤⣀⣀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⢀⣿⢣⣦⡀⠀⠀⠀⠀⡃⠀⠀⣕⠦⣙⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⣠⣎⣿⡄
⠀⠀⠀⠀⠀⣸⡇⣿⠍⠁⠀⠀⠀⠀⠇⠀⢀⣸⠓⠈⠛⠛⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠉⢸⣧
⠀⠀⠀⠀⠀⣿⠃⠂⠀⠀⠀⠀⣀⣠⣟⡉⣾⠿⡀⠁⠤⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢈⡜⠉⣿⣿⡇⠀⠀⢀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⣿⡟⣿⣿⣿⡄⠄⠼⢶⣿⣿⣿⣿⣫⠛⠻⠆⠀⠀⡀⡀⢸⡟
⠀⠀⠀⠀⠀⠈⣿⡌⣴⣦⡄⠀⠉⡇⣿⣿⡷⠦⠶⠠⢜⣿⣿⣿⣿⣿⣦⣴⠊⣱⣛⢾⣆⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣷⡽⣿⣷⡆⠀⡇⣸⣿⣿⢳⡲⡶⢶⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⢟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⣿⣯⣤⡇⢿⡿⠁⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣶⣿⡿⣳⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣮⣻⠆⠈⠏⠠⢸⣿⡀⠀⣿⣿⣿⣿⣿⡿⣟⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣥⣄⣘⣻⣷⣈⣿⣟⣻⣯⣷⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⢄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣟⣻⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠋⠧⠽⠈⢪⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢵⣿⠀⠀⣿⠀⢿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣌⣛⣀⣀⣛⣃⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⡏⠉⠉⠉⠉⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠇⢰⣯⣿⣦⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢒⡭⠚⣿⡀⠘⠾⠽⠋⠀⣿⡓⢯⣒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡮⠕⠊⠁⠀⠀⣹⣧⣦⡀⢀⣴⣄⢿⡇⠀⠈⠑⠺⢵⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣪⡵⠚⠁⠀⢀⡀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣸⡇⠀⠀⢀⡀⠀⠈⠓⠮⣕⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢹⡅⠀⡴⢻⣿⣯⠳⡄⠀⡏⣿⣿⡇⢸⣿⣿⢹⡇⢠⠞⣹⣾⡝⢦⡀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⡇⢸⢁⣀⠉⢁⡀⢸⠀⢧⡟⠉⢃⡘⠉⠛⣿⡇⡇⢀⡈⠉⣀⡀⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⢸⣾⣿⣧⣿⣽⢸⠀⣼⠇⣴⠋⡏⢦⠀⣿⡇⡇⣿⣿⣴⣯⣷⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢸⣷⣿⣦⣾⣿⣼⠀⢿⠠⡇⢰⡇⢸⡆⣾⡇⡇⣷⣗⣽⣻⣆⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣺⡇⢸⣿⣿⡟⣿⣽⢻⢰⣿⡶⡇⢸⡇⢸⢷⣿⡧⡇⣯⣯⣿⢽⡏⡇⢰⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠸⠽⠿⠥⠿⠯⠼⠀⡛⣶⡇⢸⡇⢸⡆⣻⡇⠧⠿⠿⠭⠿⠯⠇⠀⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⣤⣧⣤⡤⣤⣤⣤⣤⣤⣴⣷⣿⣿⣿⣿⣿⣷⣾⣧⣤⣤⣤⣤⣤⢤⣤⣴⡧⢧⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣀⣠⣄⡾⣿⠷⢍⣵⣤⣠⡀⠔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢫⣭⠾⣵⠞⠳⣿⠶⢼⡋⢡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⡶⢸⠦⣰⡤⡞⢧⠞⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⠉⢣⣧⡟⢉⢀⡞
⠀⠀⠀⠀⠀⠱⡴⢬⠒⠄⢀⠀⠀⠀⠀⠀⠀⠈⣹⢗⡾⢩⢗⢋⡞⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⡉⣈⠄
⠀⠀⢀⣤⣦⡟⠳⠟⣣⣾⣳⣌⣰⣖⢀⣤⡀⡀⣣⠼⠷⡅⠈⠾⠀⠀⠀⣀⠀⠂⣤⣤⣠⣤⡟⣫⣽⠋
⠀⢶⠛⠿⠛⠢⠟⠳⢷⣯⠿⠤⣥⣿⡽⠬⣿⡭⣽⠃⢰⡯⣏⢘⣍⡑⢻⠉⠛⠉⣷⠖⢉⣀⡼⠋⣱⢮⣤⡀
⠀⠀⠡⢔⡟⢫⡶⢤⣾⣻⠟⡛⠳⠶⠶⢵⠁⣦⡞⠀⣮⠵⠀⣸⣭⣟⣿⣿⣭⡸⣽⠛⢟⢋⣛⡺⣿⣧⣿⣿⣦
⠀⠀⠀⣾⡿⠿⠖⣘⣣⣭⠤⠷⠶⠶⠾⠿⢩⡅⣈⡉⣠⢻⠬⠥⣤⣘⣇⠄⢀⣵⢶⢷⠏⠳⠋⢹⡳⣡⡶⠋⠁
⠀⠀⣁⣠⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢛⡤⣭⡅⡄⠂⠀⠀⠀⠀⠉⠙⠓⠦⣬⣈⠳⠾⡳⢾⡴⠋
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣼⠀⢆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⡀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠂⡦⡰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡇⣼⠃⠀⣨⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡞⠛⠁⢰⠶⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢡⡾⠷⢶⣄⠬⣬⡤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡰⠖⢀⣸⢹⠀⢠⡖⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢿⣹⠞⢛⡿⡟⢶⡶⢻⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠞⣤⢾⠚⠙⢦⡞⠀⣼⠶⢧⣼⡄⣧
⠀⠀⠀⠀⠀⢠⠖⠋⣾⢁⡴⠶⠖⣋⡀⣠⡵⣄⣠⡾⢇⢸
⠀⠀⠀⠀⠀⠈⠲⠚⠉⠣⡡⢶⣴⠋⠛⣓⠀⢀⠘⣄⡐⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠱⢟⣩⡯⠗⠛⠉⠑⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠟⠉

⠀⠀⠀⢐⣂⠀⠀⠀⡶⡄⠀⠀⠀⠀⠠⠾⠆⠀⠀⠀⠀⠠⣶⠀⢶⠄⠀⠀⠀⣀⣴⣯⣬⣷⣄⡀⠀⠀⠀⠀⠀⡶⠀⠀⢴⡄
⠀⠀⠘⠁⠈⠃⠀⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⠛⣧⠀⠀⠀⢸⡇⢀⣿⠀⠀⠀⡏⠀⠈⡏⠀⠈⡇⠀⠀⠀⠂⠀⠀⠆⠀⠀⣿
⠀⠀⡶⠘⠃⢶⢰⡆⠛⠀⣶⠀⠀⠀⠀⠀⠀⢀⣤⣿⠄⠀⢻⠁⢸⡏⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟
⠀⠀⠓⠒⠒⠚⠘⠓⠒⠒⠛⠀⠀⠀⠀⠀⠀⠚⠛⠛⠂⠀⠘⠀⠈⠃⠀⠀⠀⠇⠄⠀⠇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢀⡀⠀⡀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⠁⠒⢹⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⡀⠀⠀⢀⣀⠀⣀
⠀⠀⢀⣿⣷⣼⣧⣿⣿⡄⠀⠀⢀⣾⣦⠀⠙⠃⠀⠀⠈⠓⢒⡋⠄⠘⠛⠀⠀⠀⣀⠀⠰⣌⣿⡾⢿⣿⠀⠀⠘⠚⢠⣿⣄
⠀⠀⠈⠁⠘⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⢘⡇⠐⠀⠀⠀⠀⠈⣿⠀⠀⠨⢾⣿⠀⠙⠀⠀⠄⠀⢿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⣿⣦⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠰⠿⠶⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⢀⣇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⣀⣬⣤⣶⣿⣿⣿⣿⣷⣶⣬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠏⢠⡇⠘⠇⠀⠘
⠀⠀⠀⠀⣼⣻⣽⣾⣿⣯⣿⣿⣿⣿⣿⣿⣻⣿⣽⣿⡀⠀⠀⠀⠀⢀⣤⣤⣅⣀⡃⠀⠠⠄⢸⡇⠀⠄⠀⠀⣀⣠⣤⣤⡀
⠀⠀⣠⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠿⠟⠓⠀⠀⠀⠈⠙⠛⠻⠿⣿⣷⣷⣶⣼⣧⣴⣶⣾⣿⡿⠿⠟⠋⠉
⠀⠀⠀⠀⠉⠛⠋⠙⠛⠉⠀⠀⠉⠁⠈⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠛⠋⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡽⢷⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡽⠋⠀⠀⠙⣯⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡿⠿⣼⣇⠀⠀⠀⠀⣸⣯⠷⢯⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣯⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠺⣻⡄⠀⠀⠀⠀⢠⣿⠕⠀⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠀⢀⢔⡶⣿⢴⠀⠀⠀⠀⠀⠀⠀⢸⢻⡀⠀⠀⢠⡏⡇⠀⠀⠀⠀⠀⠀⠀⡮⣿⢶⡢⡀
⠀⠀⠀⠀⠀⠀⣷⡏⠀⢺⡰⡀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⢸⡇⡆⠀⠀⠀⠀⠀⠀⢀⢆⡟⠀⢹⣾
⠀⡠⣲⡿⠿⠾⠷⢿⠀⠀⠙⠿⣿⣴⣶⣒⣲⣖⣚⣼⠀⠀⠀⠀⣧⣓⣲⣖⣢⣶⣤⣷⠿⠋⠀⠀⡿⠾⠷⠿⢧⣖⢄
⣔⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿
⠹⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞
⠀⠈⠪⢿⠶⣶⡶⣿⠀⢀⣴⢾⠿⠿⠍⠉⠙⠉⢩⢹⠀⠀⠀⠀⡏⡍⠉⠙⠉⠩⠿⠛⣷⣦⡀⠀⣿⢶⣶⠶⡿⠕⠁
⠀⠀⠀⠀⠀⠀⠟⣧⠀⣼⠙⠁⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⡊⡧⠀⣼⠻
⠀⠀⠀⠀⠀⠀⠈⠺⠿⠿⠽⠀⠀⠀⠀⠀⠀⠀⢸⢸⡆⠀⠀⢠⡇⡇⠀⠀⠀⠀⠀⠀⠀⠯⠿⠿⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⢸⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠⣺⠏⠀⠀⠀⠀⠹⣷⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠙⠳⠶⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢽⡦⣤⢶⡖⠀⠀⠀⠀⢳⡶⣤⡴⡯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣿⡇⠀⠀⠀⠀⢸⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢧⡀⠀⠀⢀⡾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⢦⡴⡻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡰⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⡸⠁⣰⠁⡔⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⡇⢸⠀⢸⡤⡄
⠀⠀⠀⠀⠀⠀⢀⡜⢹⡇⡰⠁⠘⠇⣸⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⠁⡼⡀⡜⠀⡇
⠀⠀⠀⠀⠀⢀⡞⢀⡞⠀⡇⠀⡸⢠⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡏⢰⠁⠀⡇⢠⠷
⠀⠀⠀⠀⠀⣸⠀⢀⠇⡜⠀⠰⡅⣼⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣸⠁⡤⠀⢰⠁⣿
⠀⠀⠀⠀⢀⡇⠀⢸⠀⡶⠀⢀⣧⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢧⠀⠀⠀⠀⠀⢠⠃⠀⢠⣏⡞⠀⢀⡆⢠⠋
⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠑⠺⠽⠇⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠀⠀⠀⠀⣠⠋⠀⠀⠿⠾⠤⢄⡎⢀⡞⠀⣸
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠸⡀⠀⠀⡇⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢀⡇
⠀⠀⠀⢰⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⠔⠉⢀⡞⠀⠀⢰⠃⠀⢠⣧⠠⠾⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⣾⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⣀⣸⡀⠀⡠⠊⠀⠀⡼⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⢠⠆⠀⠀⠀⣾
⠀⠀⠀⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠑⠾⡁⠀⢀⡜⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠⡏
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⢀⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⠁
⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣻⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣟⣿⣻⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⠀⣤⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣽⠀⢿⡤⠠⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡼⠹⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢰⠁⠀⢰⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡰⠃⠀⠀⠀⣦⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⣀⠃⣀⣀⡀⠀⠀⡆⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠁⠀⠈⠉⠀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡀⠁⠀⠒⠂⠀⠀⠋⠀⠀⢳⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣠⠁⢀⣀⣀⣀⡀⢀⣀⡀⠀⠀⣦⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣰⠁⠀⠈⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⡄⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⠸⠇⠀⠲⠖⠒⠐⠒⠒⠒⠲⠶⠶⠃⠀⠰⠆⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣠⡷⠀⠤⠤⠴⠦⠤⠤⠦⠤⠤⠴⠆⠠⠶⠶⠾⣀⣸⠧⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣀⠠⢠⡄⢀⣀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡋⠉⠉⠉⠀⠀⠀⠈⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⣉⣉⣉⣉⣀⣁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠀⠀⣾⠁⠀⠀⠀⣨⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⡀⠀⢀⡀⢁⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠤⠐⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⣠⠶⠋⠉⡄⠑⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⣠⡴⠋⠀⠀⠀⢀⡇⠀⠈⠛⢤⡀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠊⠉⠀⠀⠀⢀⡴⠞⠻⡄⠀⠀⠀⠙⢦⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢀⡰⣿⠛⠋⠉⠉⢻⠶⢆⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠠⠔⠉⠀⠀⠀⠀⠀⢀⠏⠀⠉⠀⠀⠀⢰⡆⠀⠀⢳⠀⠀⠀⠈⠣⡄⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⢠⠀⠀⠀⠟⠀⠀⠀⡇⠀⠀⡌⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⡠⠀⠀⠀⣴⠚⠻⢆⡰⠏⠀⠀⢸⠶⡀⠀⠀⠃⠀⠀⠇⠀⠀⢀⣴⣿⠀⡆⠀⠀⣀⡀⠙⢆⡀⠀⠀⠀⠀⠙⣤
⠀⠀⠀⠀⠀⠘⣦⣠⠴⡟⠀⠀⠀⠀⠛⠠⠖⠋⠁⠀⠀⠀⠈⠀⠹⣄⠀⢸⡆⣨⠀⢀⡴⠋⠀⠀⠀⠹⢀⠘⠋⣁⠀⠀⠙⢤⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠳⣤⣧⣇⡴⠋⠀⠀⠀⢀⡄⠀⠈⠁⠀⠉⠀⠀⠀⠀⠙⣤⡀⠀⠀⡰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⡆⠙⠋⠙⠑⡆⠀⠀⠈⠛⠀⠀⠀⠀⠙⠛⠛⠤⠦⠄⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⠙⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀⣀⡀⢀⣀⣀⣀⠀⠀⢸⡏⠀⠀⠈⠉⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⡾⠉⠉⠉⠉⠉⠉⢹⠋⠉⠉⠉⠉⠉⠉⢹⡇⠀⢸⠃⠀⠀⠀⠀⡇⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢨⡇⠀⢸⡀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡄⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⣠⡄⣤⠀⣿⠀⢀⠀⡀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⠷⠃⠀⣯⠐⢯⣤⠃⠀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠳⠆⠀⠀⠘⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀⣀⣀⣀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈
⠀⠀⠀⠐⣷⣄⣸⡇⠀⢠⣦⣀⡀⠀⠒⠒⠒⠒⠒⠂⠘⣿⠀⠐⠒⠒⠒⠒⠒⠛⠶⠖⠀⠀⣠⣄⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⠀⠤⠀⠉⠁⠉⠉⠋⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠉⠉⠛⠋⠀⠀⠠⠶⠦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠒⠛⠉⢹⡆⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣤⣤⣤⡤⠀⣤⠉⠑⠒⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣤⡀⣴⠊⢳⠀⠀⠉⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⣤⢀⣈⣄⠀⣼⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠛⠋⢀⡴⠞⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢰⡏⠀⢀⡀⠴⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠓⠄⠺⢆⡀⡨⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠘⠛⡆⠀⠀⡶⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠘⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣇⣀⣀⠀⠀⠀⠀⠤⠤⠼⠀⠸⠤⠀⡄
⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⣿⡿⠿⠿⠀⠀⠀⠀⠄⠀⢀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⡘⡛⣃⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⡘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣿⣛⣛⣛⣛⣛⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠶⣤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢷⠃⡏⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⢢⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢦⠀⠁⣤⠀⠀⣤⣠⠄⠀⠰⠬⠀⠀⠄⠀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠉⢻⠄⡟⠀⠀⠀⡄⠀⠀⠀⠀⠠⡄⢛⠂⠘⠴⢆⠙⠆⡀⠀⠁⠀⠀⠀⠇⠀⠀⠘⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡀⠀⠀⠀⠀⠘⠀⡒⠙⠂⠀⠀⠀⠄⠠⡙⠛⠀⠀⠀⠀⠀⠀⠐⡀⠀⠰⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠀⠆⠡⠀⠀⡄⠘⠀⠄⡀⠀⠻⣨⠆⠀⠴⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠁⢀⡄⠀⠀⢂⠀⠀⠀⠁⠂⡀⠀⢈⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣘⡣⠀⠠⣅⠀⠒⠀⠂⢰⡫⠀⠀⢈⡅⠌⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⢂⠈⠁⢀⠀⠑⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠈⠀⠈⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⠀⠄⠀⠂⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣩⠀⠀⡄⠀⡀⠀⠞⠀⠀⠰⠀⠀⠁⠀⠂⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠒⠀⠀⠰⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠠⠠⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⢀⠀⠀⠀⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠀⠶⠚⠚⠀⡂⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠀⡘⠁⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⣤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⠁⠀⠀⠀⠀⠂⠀⠀⠐⠰⠄⠀⠀⠀⡀⠄⠀⠀⠈⠀⠀⢀⡠⠁⠂⠀⠀⠀⢰⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠀⡠⠀⠀⠀⠀⠂⠠⠈⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠙⠃⡄
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢄⡀⠀⡐⠀⠀⠄⣀⠴⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⡠⠀⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡘⠀⢀⠈⠘⠧⣄⠀⠘⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⢀⡽⣀⠀⠀⠀⠀⣠⡴⠂⣀⠄⠁⠀⢩⠄⢀⡀⠁⠄⠀⠈⠰⣄⡈⢷⡄⠀⠀⠀⣰⠠⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢠⣦⠀⠀⠀⣿⠠⡈⢿⣄⡀⠤⠀⠀⠀⠁⠀⠀⠀⠒⣤⠋⠁⠨⣇⠀⠀⠠⠔⠋⠀⠈⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢃⠀⠀⢹⡀⠈⠀⣈⡗⢂⠀⠀⠀⠀⠀⠀⠠⡀⠒⠖⠀⢸⠇⢀⣴⠎⠁⠀⠀⠀⣀
⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡶⠀⠀⠃⢀⣀⡬⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠑⣀⡔⠋⣁⠐⠀⠀⠀⠀⠀⠀⠁
⠀⡘⠀⠀⠀⠰⠁⠀⠀⠀⠀⡘⠀⠀⠁⢐⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢚⠣⠀⠴⠫⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠒⠂⠀⠀⠀⠂⠀⢀⠀⣀⠀⠀⠂⠢⠀⠀⠀⠠⠀⢀⠀⠀⠀⢀⠀⠀⠈⠀⣀
⠀⢨⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠀⠂⡃⠀⠀⠀⠀⠀⢀⡅⠂⡱⠀⠄⠀⠀⠐⠑⠀⠀⢀⡀⠈⠑
⠀⢉⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠲⠁⠈⠀⠠⠉⢀⠀⠀⠸⠀⠀⠀⠁
⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⠉⠛⠻⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠙⢿⣿⣿⣿⣿⣆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡆⠀⠀⠀⢻⣿⣿⣿⣿⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⡇⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣈⣿⣿⡟⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠉⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠛⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⢉⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠑⠤⠤⡀⠀⢀⡀⠤⠤⠀⠀⠀⠀⠠⠤⠤⢀⡀
⠀⠀⠐⠀⠀⠀⠀⢾⡀⢰⣶⣶⣶⣶⣦⣼⣿⣧⣴⣶⣶⣶⣶⡆⠀⡧⠀⠀⠀⠀⠂⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣿⣿⣶⣶⣶⠶⢶⣶⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡈⠉⠉⠉⠁⠀⢸⣿⡇⠀⠈⠉⠉⠉⠁⣼⣷⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⣠⠜⠀⠀⠘⢿⣿⣿⣿⣧⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡄⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣾⡿⠛⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣟⠋⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⢢⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⠓⠀⠀⠀⠀⠀⣾⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⠀⠀⠀⠀⣴⣿⠇
⠀⢸⣿⣿⣿⡿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠒⡗⠂⠀⠀⠀⠀⠀⣤⢤⣤⣀⣸⠀⣏⣀⣠⣤⣤⠀⠀⠀⠀⢸⣷⣀⠈⠀⠀⠀⠀⢸⠏
⠀⢸⣿⣿⣿⣶⣲⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠷⠤⠄⠀⢲⠀⡔⠂⠤⠤⠾⠀⠀⠀⠀⠈⠉⠻⣦
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇⠀⠀⠀⠀⠀⠁⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⣠⣆⡀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠒⣀⣤⣀⣀⣀
⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⣠⡹⣄⠺⠉⡿⢿⣶⣦⡀⠀⠀⠀⠀⠀⠀⣐⣾⠟⠛⠻⣿⡃⠀⠀⠀⠀⠀⠀⠘⣀⡀⠉⠉⠙⢿⣿⡆
⠀⠀⠀⠈⣭⢭⣭⢭⣭⢭⣭⠁⠀⠀⠀⠀⠀⠘⠻⠮⠳⣤⡁⠸⠛⣿⠀⠀⠀⠀⠀⢀⣠⣭⣽⣶⣶⣦⣽⣓⠂⠀⠀⠀⠀⠀⠀⠻⣷⡄⡰⣝⣦⠁
⠀⠀⠀⠀⠿⠶⠿⠶⠿⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠉⠓⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⢠⡀⣦⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢻⢓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⣶⢒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣃
⠀⠀⠀⠀⣤⡄⣹⣿⣏⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⡬⠋⠻⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡿⢉⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣼⣿⣧⡄
⠀⠀⠀⢤⣾⣧⡌⠉⢡⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⣟⡽⣇⡿⣝⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢉⡤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠁
⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠚⠛⠀⠀⠀⠀⠀⠀⠀⠿⠮⠽⠭⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⣰⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣀⢰⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⢀⣤⡀⠀⠀⠀⠀⠀⢀⣶⣶⣦⡤
⠀⠀⠀⠀⠙⠻⠿⠿⠗⠙⠀⠀⠀⠀⠀⠀⠀⠀⣠⣍⣣⡹⢣⣞⣋⡀⠀⠀⠀⠀⠀⠀⠀⡇⣸⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡄⢀⣠⡀
⠀⠀⠀⠀⠀⣼⣿⡇⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⢈⣽⡬⣷⡯⣬⡍⠁⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠿⠷⠰⡇⠀⠀⠀⠀⠀⠀⠀⠙⠟⠋⠲⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⡇⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⣀⠴⢂⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣄
⠀⠀⠀⠀⢰⣮⣤⣾⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣿⣏⣤⡟⠁⠀⠀⠀⠀⠀⠀⢀⣤⠶⣿⣛⠿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉⡉⢹⢆⡜
⠀⠀⠀⠀⢸⣿⣿⣥⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢿⣿⣭⡶⢛⣩⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠁⢿⡄⠁
⠀⠀⠀⠀⠸⢿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠈⠙⠲⢍⡥⠒⠉⠀⠀⠀⠀⠀⠀⠀⢰⣷⣍⠻⡶⢋⣴⣧
⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⣀⣁⣀⣈⣠⣏⠠⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⣠⠖⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⢀⣀⣤⣄⣀⠈⠀⠤⠀⡀⠾⠅⣀⣀⣉⣴⣒⣀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡒⠀⢀⣠⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠋⠉⠙⠉⠉⠁⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠒⠛⠓⠂⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⢀⣀⣀⣀⣀⢀⣀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⣤⣿⣧⡄
⠀⠀⠀⠀⢰⣶⣾⣷⣶⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠸⠿⣿⡿⠿⠂⠀⠀⠸⢿⣿⠿⠀⠀⠰⠶⠶⣿⡷⠶⠶⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠸⠿⣿⡿⠿⠇⠀⠀⠿⢿⣿⠿⠗⠀⠀⠀⠀⣿⡇⠀⠀⢀⣀⣀⣸⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠶⢾⣿⠶⠶⠀⠀⠀⠀⣿⡇⠀⠀⠀⠺⠿⢿⣿⠿⠿⠇⠀⠰⣦⣿⣇⠀⠀⠈⠉⠉⢹⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠿⠏⠛⠀⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⡀⢻⡿⢀⡀⠀⠀⠀⡀⢻⣿⢁⡀
⠀⠀⠀⢿⡿⢿⡿⢿⡿⠀⠀⢾⡿⢿⣿⢿⡿⠀⠀⠀⠐⣦⣴⠆⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢶⣤⣼⣏⣤⣶⠂⠀⠀⠀⠠⡄⠀⠀⠈
⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠛⠉⣰⣧⠉⠛⠂⠠⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠐⠋⠙⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⠤⠄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⠶⢧⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣤⡀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⠶⣴⣿⣷⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠋⠛⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠀⠀⢸⣶⣞⢬⣷⣶⣾⠠⠀⡐⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢿⣡⠄⡀⢠⣼⣏⣷⣝⣿⣤⠀⡀⢤⣹⢿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠁⠀⠉⢲⣾⠟⣿⡟⢻⣶⠊⠁⠀⣽⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢻⡇⠀⠀⠘⡇⠄⠈⠀⢄⠏⠀⠀⠀⡿⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⠂⠃⠀⠀⠀⠘⡂⢈⠀⡊⠀⠀⠀⠀⠃⢊⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⢁⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠋⠁⠀⠀⡀⢀⠀⢈⢋⡃⠀⠀⢀⡀⠀⠀⠉⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡄⠀⠀⠀⠀⠿⡀⡄⢬⡁⣩⢄⡄⣸⠃⠀⠀⠀⠠⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢜⡴⠛⠁⠀⠠⣤⡐⠂⠄⠮⢮⣟⣧⠾⢆⡠⠚⣠⣤⠀⠀⠙⠳⣜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠁⢀⡘⠃⠚⠉⡄⢶⡀⡀⢴⡶⣿⡷⣶⢄⡀⣱⠖⡌⠙⠀⠋⣀⠁⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⢲⠏⠀⠐⢶⣆⠀⠘⣧⠀⢠⠴⠻⠳⢄⠀⢸⡟⠄⢀⣶⠦⠀⠈⢶⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣀⠀⠈⠋⠁⣀⣩⣱⢂⡗⢰⡀⣰⢐⣇⢳⡩⣁⡀⠙⠁⠀⢀⣨⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠋⡩⠛⠻⢂⣶⣀⣷⡏⠘⠋⠚⠷⠟⠎⠛⠈⣷⣆⣘⣆⠚⠛⠫⡈⠣⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠂⠂⠓⠁⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠙⠂⠂⠒⠈⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢰⠀⠀⣤⠀⢀⠀⠀⢰⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠀⠁⠒⠣⠀⠓⠀⠉⠘⠁⠘⠚⠀⠀⠈⠉⢨⠀⠃⠀⠈⠁⠓⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⣐⣂⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⠿⠿⠿⠿⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣲⠀⠒⠒⠒⠒⠒⢲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣤⡶⠶⣦⣤⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠉⠀⠀⠄⡀⠈⠉⠛⠳⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠰⢞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣠⡴⠶⣤⣀⡀⠀⠀⠀⢀⠤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠈⠀⠀⠀⣠⠶⠛⠋⠁⠀⣀⠀⠉⠛⠳⡦⠀⡀⠀⢠⠀⠀⢠⣦⣀⡀
⠀⠀⠀⠀⢴⠖⠛⠉⢁⠀⠀⠉⠀⠈⡇⠀⠀⠀⢠⠊⠀⠁⠀⠀⡀⠀⡄⠀⠁⠀⠈⠀⠀⠘⠀⠈⠉⡟
⠀⠀⠀⠀⠸⡀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⢰⠀⠀⡆⠀⠀⠀⡄⠀⠐⠀⠂⠀⢰⠀⡇⠀⠇⠀⠰⠂⢠⠁⠀⠀⠀⡇
⠀⠀⠂⠀⠀⢇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⠇⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⠀⠀⠀⢸⣀⣀⣀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠒⠒⠒⠲⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣲⠶⠖⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠤⢤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢰⣶⡆⠀⠉⢁⣬⠴⠒⠒⠒⠦⡄⠀⢀⣀⡜⢧⣀⡀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡧
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡰⠋⠀⠀⠀⠀⠀⠀⢹⠀⠉⠙⢦⡴⠉⠉⠀⢰⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⡀⠀⠀⢸⠁⠀⠀⡀
⠀⢰⣶⣶⣶⣾⣿⣷⠆⣰⣶⡆⠀⠀⠀⢀⣠⠞⠀⠀⠀⠘⠃⠀⠀⠀⢸⠀⠀⠀⢀⣉⠀⣁⠀⠀⠀⢰⡇⠀⠐⠾⡿⠖⠒⢾⠖⠒⠾⡿⠷
⠀⢈⣛⡛⠛⢻⣿⡏⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠞⠛⢛⡿⠀⠙⠙⠛⠷⢾⡇⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠈
⡜⠁⠀⠈⠑⢸⣿⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢠⢶⢦⠀⠀⠘⢿⠟⠂
⠣⣀⣠⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡴⡮⢸⣿⣿⡇⢫⡦⡀⠀⠀⠀⠀⠰⠛⠒⠒⠚⠲⠆⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢀⣀⣊⣋⡀⠈⢿⣿⠃⢀⣘⣙⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢸⡿⣿⠀⣀⣀⡀⡀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠸⡿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠⠥⠄⠀⢸⣿⣿⣿⣿⣹⢿⣿⣹⣿⣿⣿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡭⡍⠉⠀⣿⣿⡄⠉⢉⢭⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢸⣷⣿⠈⠉⠉⠉⠉⢀⣀
⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⡖⠀⠀⠀⠀⠘⠫⣦⣴⣿⣿⡇⣔⡯⠋⠀⣀⠀⠀⡀⡀⠀⠀⢀⠀⡀⣀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⢸
⠀⢀⠀⠀⠀⠿⣿⣿⣿⣟⠇⠀⠀⣀⠀⠀⠀⠈⢸⣿⣿⡇⠁⠀⠀⠁⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠈⠀⢸⣿⣿⠀⠀⠀⠯⠭⢭⢨⡭⠭
⠀⣿⣿⣿⣧⣯⣿⣿⣿⣭⣯⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠀⢸⡿⢿⠀⠀⠀⠀⠀⢸⢸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⠿⠿⠟⠀⠀⢤⠤⢼⠤⠤⡄⠀⠀⠀⡤⠴⠃⠳⢤⡀⠈⠉⠉⠀⠀⠀⠀⠀⢸⢸
⠀⢿⡿⠿⠯⠯⣿⣿⣿⣩⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣤⠄⠈⢩⡆⡏⠉⠀⠀⣠⠤⣀⠀⣀⣀⣀⡘⠛
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠂⢶⠁⠀⠀⠀⠀⠀⠀⠠⡧⠄⠀⠰⠧⠇⠀⠀⠈⢧⡐⡋⠳⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠐⠰⠀⢸⠀⢼⠀⠖⠐⠀⠀⠀⠀⠀⠛⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠋⠀⢹⣿⣿⠁
⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠒⠒⠒⠚⠀⠙⠚⠒⠒⢢⠁⠀⠀⠀⠀⠀⢰⣢⠂⠀⠀⠀⠀⢰⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶
⠀⠀⠀⠠⣄⢾⣿⢄⣠⠀⠀⠀⠀⠀⠀⠀⠜⠒⠒⠖⢲⠀⢔⠲⠒⠒⠪⠄⠀⠀⢀⡀⣻⡾⣿⢰⣟⠀⡀⠀⠘⠛⠛⠛⢻⣿⣿⠛⠛⠛⠛
⠀⢠⣤⣶⣬⣼⣿⣬⣵⣦⣤⠀⠠⣤⣿⣤⠀⠐⢘⠀⢸⠀⢸⠀⠓⠐⠀⠀⠠⠀⠸⠼⢿⣟⣿⢻⡿⠧⠇⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠈⠉⠛⣭⢹⣿⢩⣝⠋⠉⣤⣤⣤⣿⣤⣤⡄⠀⠀⢸⠀⢻⠀⠀⠀⠺⠇⠸⠖⠀⠀⠛⢹⣿⠈⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠈⠀⢻⣿⠃⠈⠀⠀⠀⠀⣤⣿⡀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠂⠲⠆⢰⠖⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⢸⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⡶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣀⣀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣤⡀⠈⠁⢀⣤⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤
⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠻⣿⠀⠀⣿⠟⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⣿⡇⠀⢀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠀⠀⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⡀⠀⢸
⣿⡇⠀⢸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡇⠀⢸
⣿⡇⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⢀⠀⠀⡀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢸
⢿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣴⣿⠀⠀⣿⣦⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼
⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠿⠋⠀⠀⠙⠿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠞⠓⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣾⡿⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣄⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣮⣼⣷⣌⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠋⠀⠉⡿⣿⣿⣿⣷⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠁⠀⠀⠀⠀⠡⡇⠙⢿⣿⣿⣞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⡄⠀⠀⠀⠀⠀⠸⡀⠀⠘⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠒⢦⣌⣿⣿⣿⣿⣼⣇⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣯⡙⠛⠿⢿⠦⢭⣷⣦⣀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡾⢻⣿⡄⠀⠀⠀⢀⣤⣴⣦⣶⣤⣄⣉⠓⠠⠀⢇⠀⠘⣿⣿⡿⠳⣄⣀⣈⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⣿⠿⣿⡉⠉⠉⣫⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⢿⡄⢠⣿⣿⣿⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄⠀⠹⡷⠶⠛⠡⠚⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⣳⣾⣿⡿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡞⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⣿⣿⡯⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠘⠟⠀⠀⠀⠘⣤⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠈⣿⣿⣿⢇⣤⣄⡀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠔⢒⣋⠀⣀⡀⠀⠀⣀⠀⠀⠀⣴⣿⡿⣿⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠡⠦⠤⠭⢳⡆⠹⣿⣷⣾⣟⠟⢠⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡄⠀⣰⣿⣿⣷⡏⢀⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡦⢇⠹⣛⠻⠻⡟⠁⣸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢎⠃⢸⣷⡁⠀⣰⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠁⠀⣆⣴⣿⣿⣷⠀⢹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⣷⣷⣿⣿⣿⡿⠟⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⢠⠀⠘⢻⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡿⠀⠙⠻⢫⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⣿⣿⠇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢛⣰⣿⡇⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠟⠀⠎⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠉⠃⠀⠀⠀⠀⠀⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡄
⠀⠀⠀⠘⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠄⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⢺⠆⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢤⡐⠁⡰⠀⠀⢰⠖⣶⠟⢷⡞⡙⡆⠀⣴⣷⣄⠀⢰⡞⢳⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⣶⠀⠘⣶⠞⠃⠀⠀⠀⠀⠀⠈⢼⠄⠾⡧⠈⠀⣸⣇⠈⣿⣄⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢇⡶⠄⠁⠀⠀⠀⠀⠀⠀⠈⢀⣌⠀⠀⠀⠀⡾⠟⠉⡀⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠘⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⣁⠉⠀⠀⠀⠀⠀⠀⠘⠄⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡶⢳⡄⠀⠀⠀⣸⠦⠀⠀⠀⢠⡎⢁⠀⠀⠾⡶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠘⢠⠉⠀⠀⣰⢶⡄⠁⠀⠘⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠰⣧⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠆⠂⠾⠖⠒⠀⠀⠀⠀⠀⢁⠉⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⢀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠦⠖⠀⠀⠀⠐⠲⠲⢀⠀⠀⣀⠈⠀⠀⠀⠀⠈⠓⠀⠀⠀⣨⠀⠀⠘⠛⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠛⠁⠀⠀⠙⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢱⣤⣿⣇⣼⣀⡔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡄⠀⠀⢤⣹⣿⣿⣿⢥⣬⣽⣿⣾⣥⠤⠀⠀⢰⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣟⠃⠐⠺⢿⣯⣝⣻⠾⠿⣋⣠⣿⠷⠶⠄⢉⡿⢯⡁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣌⣷⡶⠀⠈⠀⠈⠀⠀⠚⢙⣽⣿⢿⣯⣽⣿⣿⠻⢦⠀⠀⠀⠀⠀⠀⠐⣲⣮⠤
⠀⠀⠀⠀⠀⠀⠀⠰⠏⠙⠀⠀⠀⠀⠀⠀⠂⠈⠹⠾⡛⠹⠛⠿⠙⠀⠀⡠⠊⠀⠀⠀⠀⠈⠁⠙⠆⢀⡀⢀⡀
⠀⠀⠀⣈⣣⣞⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⡄⠘⣴⣾⣷⡄⢀⠁⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢿⡯⡤
⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠎⢈⣿⣆⣾⣧⣿⣔⠍⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⣿⣿⣿⣿⣿⣿⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⡀
⠀⠀⠀⠀⣑⣾⡴⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⠧
⠀⠀⠀⠀⠀⠘⠇⠀⡀⠀⠀⢀⠔⠒⡤⡴⣾⣿⣝⣛⣛⣻⣏⣿⣿⣿⠟⢿⠒⢀⠔⣦⡀⢠⡀⡤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢽⣾⣃⠀⠛⠶⠶⣤⣤⣬⢟⣩⡍⣿⣍⣽⣡⠰⠄⠄⢦⡽⠚⠛⠁⠶⢿⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠙⣿⣽⠉⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇
⠀⠀⠀⠀⢠⣄⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣭⠁
⠀⠀⠀⠀⠈⢁⡙⠛⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡛
⠀⠀⣤⠐⠻⠿⣿⣷⡦⢤⣤⠀⠀⠀⠀⠀⠀⣀⠠⢤⣴⣶⣿⣿⣿⡇⠀⠉⠀⠀⠀⠀⢀⣁⣤⠄
⠀⢀⣼⣴⣶⡄⣀⠈⠻⣏⡀⠀⢀⡀⠂⠉⢁⣤⣖⣋⣩⠄⠀⣿⣿⡇⠸⠀⠀⠐⢤⣾⣟⠋⣽⠷⠉
⠀⠈⠉⣹⣵⣄⠄⣭⢀⡈⢉⣰⣾⣶⣶⣿⣿⣿⣿⣿⡟⠀⠀⣼⣯⡄⢰⡄⠓⠒⠛⣋⡉⣿⠨⠟⢲⣮⣿
⠀⠀⡽⢿⣿⣿⣿⣿⠁⢠⣆⣿⣿⣿⢸⣿⣿⡇⢶⣿⡇⠀⣤⢸⡿⠇⢸⠀⢸⠀⣾⣇⣸⣧⣤⣌⣺⣿⣿⣀
⠀⢶⣷⣾⣿⣿⣿⢋⣰⣿⣿⡿⠟⣻⣿⡿⣿⣿⣿⣿⣧⠀⣉⣼⣿⣷⡘⠶⠌⣴⣝⢿⣿⣟⣟⣿⣿⣿⣿⣿⡇
⠀⣭⣭⣭⣥⣿⣿⡿⠏⠉⠉⠈⠛⠛⠂⠒⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠃⠀⠀⢀⡠⠴⡚⠍⠉⠉⠙⠉⠉⠉⠩⠓⠦⢄⡀⠀⠀⠘⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⣿⠆⢀⣞⡍⢊⠀⠐⠀⠀⣀⢤⡀⠠⠀⠁⠂⠀⠑⢩⣳⡀⠰⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⣿⣶⠝⡩⠂⠀⠀⠀⠀⢸⣿⣾⠇⠀⠀⠀⠀⠀⠀⠐⢍⠻⣶⣿⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢿⣧⠸⣯⠈⠀⠀⠀⠀⠀⠀⢹⣾⠛⠋⠙⢠⣄⠀⠀⠀⠀⠀⠁⣽⠇⣼⡿⢠
⠀⠀⠀⠀⠀⠀⠀⢤⢻⣾⣿⣦⣘⣧⡀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣠⣤⣈⠀⠀⠀⠀⢀⣼⢃⣴⣿⣿⡟⡤
⠀⠀⠀⠀⠀⠀⠀⡰⣿⣾⣿⣿⣿⣿⣿⣦⣄⢀⡠⣴⣿⣿⣿⡿⠿⠚⠋⠈⠶⣄⣴⣿⣿⣿⣿⣿⣿⣾⢂
⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣇⣽⡾⠿⠶⢶⠀⣀⣴⣾⡀⠈⠻⣿⣿⣿⣿⣿⣿⡥
⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣩⣍⡉⠛⢋⠈⣿⣷⣄⣀⣸⣿⣿⡿⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠿⣟⣻⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⢶⡟⠀⠜⢳⣿⣯⡭⢬⣿⣟⣻⠿⣇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠟⠋⣿⣿⣿⣿⠺⣿⣿⣿⣿⠉⠁⠄⠀⣹⣿⣿⣶⣤⣼⠿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣿⣿⡻⣿⣆⢹⣿⣿⣿⡀⡀⠀⢰⣿⣿⠿⠅⠤⢱⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣿⣿⣷⣌⠙⠄⢿⣿⣛⡛⣿⡛⣻⣿⣿⣧⡄⢀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣮⣸⣿⣿⣿⠿⢶⣿⡿⠋⣰⣿⣿⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡟⣠⣾⠁⢸⣾⣿⣿⣿⣧⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡌⠿⣻⣏⣉⣟⠿⢥⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡯⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⠋⢏⠏⠻⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⡇⣿⠀⢸⡇⠙⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣧⣿⣤⣬⣇⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣤⣄⠀⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣲⣯⡇⠸⣿⣿⠇⢨⣷⣢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣐⠮⠒⠉⠀⡈⣇⣄⠀⢀⣄⠈⡇⠈⠑⠪⢗⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣨⠗⠊⠀⣀⡀⠀⠀⡇⣿⣿⠀⢸⣿⠇⡇⠀⠀⢀⡀⠈⠓⢯⣐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣥⠀⡴⠻⣿⡏⠢⡀⡇⣿⣿⡀⣸⣿⡇⡇⣠⠊⣷⣯⠓⡄⠀⡏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⣇⣤⠀⣤⡀⡇⡇⡏⢉⣥⣍⠉⠁⡇⡇⣠⣄⢠⣄⠀⠀⡃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⣿⣿⣿⣻⡇⡇⣇⡇⡞⢸⠈⡆⢠⡇⡇⣿⣿⣟⣟⠇⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⠀⣿⣿⣿⣿⡟⣇⣽⣇⡇⢸⠀⣇⣼⣇⡅⣿⣿⣿⣿⢲⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠀⣿⡿⠛⣿⣇⠇⡏⡏⡇⢸⠀⡇⠉⡏⣇⣿⣟⠻⣿⡰⠈⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡧⣼⣤⣤⣤⣤⣤⣤⣤⣧⣷⣷⣾⣶⣷⣦⣧⣤⣤⣤⣤⣤⣤⣤⡤⢧⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⠾⢇⡈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣁⠶⢾⣧⣿⠶⢆⣹⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⡻⣷⡎⣿⡏⢶⡾⣻⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡝⣷⢛⣬⠛⢯⣜⢳⡟⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢙⣽⣶⣶⣍⠁⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢹⣍⠻⠏⣵⠃⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠌⣾⣧⠸⡿⢂⠸⡿⠀⣿⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⣀⠀⠀⠀⠀⢀⠠⡑⢜⢎⣾⣿⠀⣶⢌⢱⡆⢸⣿⣎⢔⡀⡡⠀⡀⠀⠀⠀⢀⣀⣤⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣏⡙⠛⠿⢶⣦⣤⣦⣅⣪⣿⣿⣿⣿⣇⣿⣜⢾⣧⣾⣿⣿⣿⣮⣔⣡⣶⣤⣶⠾⠟⠛⣉⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠏⣠⣏⣻⡛⣲⣦⡤⢈⣉⠉⠛⠛⠻⣿⣿⣿⡐⣿⡷⢸⣿⣿⡿⠛⠛⠋⢉⣉⠠⣤⣶⡞⣻⣏⣷⡈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡶⡙⣾⣏⠻⠏⣛⡆⡦⣙⡋⣴⣾⣟⣷⣦⣴⡿⢎⡻⢿⣤⣦⣿⣿⣶⡬⣛⡡⡆⣌⡛⠿⢟⡿⣾⡱⣶⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⢹⡷⠺⣟⢿⡿⠙⠃⠛⠃⣘⣁⣡⣧⣶⣶⣶⣹⣷⣶⣶⣶⣥⣁⣘⡁⠛⠓⠛⢹⣿⢟⣿⠳⣾⠃⣰⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⡿⠚⣉⣠⣤⠶⣶⣿⡿⢿⣿⣿⣿⣿⣟⣿⡉⣹⣾⣿⣿⣿⣿⣿⠿⢿⣟⡶⢶⣤⣌⡙⠺⣇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠟⠁⠁⠀⠀⠀⡠⢞⣿⣿⣿⣿⡏⢼⣿⣿⡜⢻⣿⣿⣿⢟⠧⡀⠀⠀⠀⠈⠉⠻⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⡩⢪⢟⣿⠁⣰⣟⣿⡄⢸⣿⡯⠮⡑⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣿⠀⡽⢫⠾⣇⢸⣿⠣⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠸⣿⡎⣼⡿⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⣷⣿⡤⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⣟⣋⣙⣿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢿⡿⢻⡫⣿⠇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⡾⢡⡟⣧⡙⣧⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠠⡿⠌⣁⡉⠴⡧⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⠃⢷⣿⣷⡆⠻⣦⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣹⣿⣄⣸⡿⣣⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⣧⣿⡟⣿⡟⣿⣧⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⣥⣬⣾⣧⣤⣝⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣦⡉⠺⣧⣿⠎⢁⣼⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣙⣧⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠋⠙⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠉⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢀⣀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⢻⣇⢈⣵⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣖⢛⣢⣾⣿⠄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋⠀⠉⠙⠁⠛⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠹⢿⠟⠉⠙⣄
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀
⠀⠀⢀⡁⠀⠀⠛⠐⡄⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢀⣤⡴⠄
⠀⠀⠀⠀⠀⠀⠖⢠⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠹⣆
⠀⠀⠀⣴⣾⣿⣟⣻⣍⣯⣽⣿⣿⣿⡿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣖⣲⣿⡄
⠀⢀⣾⣿⠿⢅⣩⣯⣬⠑⠮⠋⣵⣅⠲⣛⣿⣷⡀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢸⣿
⠀⣾⣿⣟⠂⣾⣿⣟⣅⡀⠀⣀⣙⢿⣿⡩⣽⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⣿⣿⣿⣅⣻⣿⣟⠓⠁⠀⠻⠻⣡⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⣿⣿⣿⣿⣿⡅⠀⠀⣤⣤⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠘⣿⣿⣿⣿⣿⡆⡴⠿⠿⣦⣰⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣠⡀⢀⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠐⠀⠂⠀⠀⡀⠁⠀⠀⠀⠀⠚
⠀⠀⠈⢿⣿⣿⣿⣿⡀⢺⡿⢁⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠲⠤⠄⣠⡈⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠲⢄⠀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠆⠒⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢰⣶⣾⣿⣶⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⠆⠀⠠⣶⣶⣶⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢠⢰⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡇
⠀⠀⢀⡀⠀⠨⠀⠀⠀⣀⠀⠀⢀⠤⠤⠇⠧⠤⢄⠀⠀⢀⣄⣀⣸⣇⣀⣠⡀⠀⠀⠄⠀⡀⢇⠀⠠⠀⠀⠀⢀⣠⣄⣀⣿⣒⣀⣤⡀
⠀⠀⠀⠀⠀⠰⢐⠂⠀⠀⠀⠀⠈⠒⠓⡆⡖⠒⠚⠀⠀⠈⠋⠉⢹⡏⠉⠙⠁⠀⠈⠈⠀⠀⡆⠀⠀⠁⠀⠀⠈⠙⠋⠉⣿⠭⠉⠛⠁
⠀⠀⠀⠀⠀⢰⠐⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢀⡀⢸⡏⠀⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢀⣰⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢹⠋⠙⠃⠀⠀⠙⠛⢻⡟⠛⠋⠀⠀⠈⠋⠙⡏⠉⠃⠀⠘⠛⣿⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣄⣀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠈⢻⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢰⠀⠀⠀⠀⠀⣠⣼⣧⣤⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⢉⣀⡀⠀⣀⣘⠍⣀⣀
⠀⠘⠛⢻⠛⠛⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣀⣸⣿⣀⡀⠀⠀⠀⠀⢀⢂⠀⠀⠰⠂⠄⠀⠰⠂⠀⠒⠂⡃⠒⠚
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠛⢻⡟⠛⠃⠀⠀⠀⠛⢟⣿⠿⠛⠀⠀⠀⠋⠫⠨⠉⠃⠀⠀⡆⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⢸⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠇⠞⠀⠀⠀⠀⠸⠾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡉⢲⡤⠈⠛⢷⡄⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠻⢿⣷⠞⣁⣀⣌⣸⡟⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢅⣠⡶⡸⠁⠀⢩⣽⠟⠁⠀⢀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣥⠔⠁⠀⢠⡜⠃⠶⠾⢀⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣠⣄⡀⠀⠀⢀⡞⠀⠀⣀⡼⠃⠀⠀⣤⠟⠀⠀⢠⣤⠞⠛⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠈⠙⣷⡀⠙⣦⡴⠏⠀⢀⡾⠋⠀⠀⠀⣾⡏⠀⠀⠠⡾⡅⢀⠈⣸⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣰⡟⠀⠀⢀⡾⠁⠐⠀⢀⣴⠟⠀⠀⢀⡼⠃⠘⠛⢠⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⢸⣿⠟⠀⢰⣶⡞⠁⠀⡄⣀⡾⠋⠀⠀⠀⣾⠃⠀⠀⢀⣾⣿⣀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢈⣿⠁⠀⠀⣰⡟⠀⠀⠀⢐⡿⠁⠀⠀⠀⣰⠇⢀⠄⢠⣿⣿⢻⠏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⠁⠀⢀⣰⠏⠀⠀⠀⣰⡿⠁⠀⠀⢀⡼⠋⠀⣜⢣⣿⢻⣿⣷⠄⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣤⣿⠃⠀⠀⠈⠃⠀⠀⢀⣾⡟⠁⠀⠀⣼⣿⠇⠀⠀⠜⣾⣿⣺⣿⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡿⠃⠀⠀⠀⠀⠀⣄⠀⠘⠛⠀⠀⠀⢜⣿⠃⢠⣦⡤⣰⡟⣉⢸⣧⣼⣠⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠏⣰⣿⣿⣿⣿⣿⡈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣿⣿⣿⣿⣿⣧⠁⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠉⠛⠿⢧⣿⡄⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⣆⠀⠀⠀⠈⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣿⣿⠁⠛⠰⡄⠀⡀⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠀⠀⡀⠈⠑⣝⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣿⠮⠀⠀⠀⠁⣠⢨⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⢢⣿⣿⡟⠓⠆⠀⠀⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡿⣿⡟⢁⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣨⣿⡿⢯⣤⡍⠁⢀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⣿⣿⡇⠘⢷⡋⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⣪⣶⣿⣿⡷⠐⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⢠⡾⢇⠈⠚⠿⢧⡿⣮⣳⣄⡀⠀⢠⠂⠀⠀⠀⠀⢠⣶⢎⣿⣿⢿⡟⢹⠃⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠉⠊⠙⢶⣁⠌⣠⣤⡌⣴⡟⣱⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠱⡀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣾⣟⣼⣿⣿⣿⣍⠙⢦⣤⠤⠀⠤⠿⣿⣶⡙⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⢛⠄⠀⠀⢤⣀⠀⠀⠀⠀⡉⠻⢿⣯⠉⠻⣿⣿⢀⠀⠙⢷⣶⣶⡿⠿⠛⠵⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⡀⠀⠀⢈⡿⢢⣤⡀⣀⠀⠀⠀⠀⠀⢈⢻⣇⠩⢆⡀⠙⠿⣦⡄⠀⣠⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⡒⠤⠤⣬⣭⡀⠉⠉⣿⠿⢶⣤⣴⠂⢸⣿⣴⡥⢈⡁⠀⠈⠛⢾⣿⣏⢀⣴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠈⠁⠶⣄⠀⠒⠤⠈⠙⠁⢸⣿⠛⠃⠀⠀⠉⡀⠀⠀⠈⠉⠉⡵⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠀⠀⠀⠰⠄⠘⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣦⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⠉⠀⢀⠈⠀⡁⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⡀⠄⠈⠀⠀⢠⡄⠀⡌⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠶⠀⢀⣠⣅⠀⠀⠀⠀⢀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠀⡀⠩⢄⣩⠥⣔⠃⠀⢠⣀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢡⠀⠈⢠⡖⣤⢰⡆⣁⢀⡄⢀⠠⠂⡄⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢐⣤⡑⠀⢰⣇⠃⠀⠅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⠋⠈⣴⠀⡦⠈⢡⠞⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⣀⠶⠀⠄⠚⠠⡶⢶⡉⠁⠀⢂⡢⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠁⠈⠓⠒⢤⠸⡏⢠⠈⣧⢠⠖⠉⠁⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠐⠀⠀⠀⠢⠀⡆⠀⠀⠀⠉⣠⠀⠤⠁⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠐⠀⠀⠀⠀⠀⠈⢇⡀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡀⠀⠀⠀⢀⣠⠸⠟⠸⠁⠀⠀⠀⢀⠀⠀⠈⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⣀⢞⠁⠀⠀⢠⠁⠈⠀⠀⠀⢳⠀⠀⠀⠈⠳⡄⠀⣟⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⡂⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣤⠀⣀⣠⡀⢸⡀⠀⠀⠀⠀⢰⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠃⢸⠃⠈⠙⡏⢙⠁⡃⠀⠀⠀⠀⠀⣄⡀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⢻⣄⠀⠀⢿⣨⡀⡇⠀⠂⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡇⠈⢦⠀⠈⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡚⡽⠢⡶⠀⣸⠀⣛⡇⠀⢀⣠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⣀⢄⡴⢛⠃⡜⠀⡰⠃⣰⣿⠇⡀⢏⠉⢁⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠃⠡⠀⠴⠦⠀⢀⠀⠐⢁⣨⡃⠐⡇⠀⣿⢸⣣⠀⠀⠁⠸⣇⠠⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠐⢃⠀⠐⣤⠔⢂⠁⠛⣪⠀⠖⠋⠉⠁⢰⡿⠄⠈⠚⠃⠒⠂⡰⠓⠘⠣⠀⡄⠀⠀⠀⣠
⠀⠀⠀⠀⠠⠆⠀⠀⠀⠈⠁⢾⠉⡇⠀⡀⠂⠐⠤⠄⠘⢘⠛⣿⠀⠶⠖⠚⠻⣅⠀⠈⠳⠉⢡⠄⡂⠉⠁
⠀⠀⠀⠀⠀⠀⡂⠐⠀⡀⠀⢠⠜⠁⡌⣡⡬⠿⢲⠲⠖⡰⠀⢻⣰⠖⠀⠀⠀⠄⡆⠀⣄⢂⡀⠈⠶⠤⣠⠐
⠀⠀⠀⠀⠀⢄⢡⡀⠀⠀⢃⠀⢲⡤⠴⢋⡔⠀⠀⠨⠐⠤⠤⠀⠉⢅⠀⢆⠀⢂⠙⠁⣤⡀⢋⣀⠀⠀⡡⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠘⠇⠀⠠⠪⠁⠀⠈⠄⠀⠀⡄⠈⠂⠀⢋⠀⣎⠁⡴⠉⠉⠀⠦⠈⠁⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣦⣤⣶⣶⣶⣶⣶⣶
⣿⡿⡉⠿⣯⡿⢽⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡄⢠⠀⠀⠘⣃⠶⣤⠃⡀⠐⢰⣤⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡀⠑⠘⣷⢂⡮⠀⢸⠀⡜⠒⢾⠁⠁⠈⢣⡾⠇⠘⡀⢠⠴⣄⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⡼⢧⡄⢄⠰⡉⢋⢣⢰⠈⡄⡇⠀⠘⠀⠀⠀⠀⠀⠀⠈⠔⡘⢳⠎
⠀⠀⠀⠀⠀⠀⠠⣴⣝⠀⡀⠺⢚⠓⡈⢣⠐⡀⠀⠀⠀⠅⠃⡀⠀⠀⠀⠀⠀⠀⠀⢈⠈⠔⡡⠂⡠⠂⣭⡹⠋⠄
⠀⠀⠀⠀⠀⠲⠑⣾⣙⣛⠢⣀⠠⡑⢌⠢⡁⠸⡘⠀⡀⡀⠆⠅⠂⠀⠄⠀⠀⠀⠀⠀⠊⢀⠐⣈⠔⡀⠀⢁⠄⠂⢀⠄
⠀⠀⠀⢀⣀⡈⣓⠬⣍⠢⢥⡀⠑⠬⡀⡑⢁⠤⠙⠁⠀⠀⣤⡀⠀⠀⠀⠈⠀⢠⡀⢈⠔⠡⢊⠁⠀⠄⠊⠁⠐⠙⠓⢻
⠀⠀⢀⡨⣭⣹⡭⠉⡒⠀⢀⡈⢁⡤⠈⠙⠖⠀⠀⠀⡠⠊⠀⢙⣄⣀⣤⣤⣴⡶⢶⣥⠀⣀⠀⠀⠀⠀⠄⠂⠠⠄⠃⠘
⠀⢀⡒⠀⠌⠥⠐⠂⠀⠉⠀⠀⣁⠚⣷⣏⠠⠄⢀⠖⣡⣤⣤⣤⡀⠷⣤⣌⣉⠛⣦⡙⢿⣦⣌⣁⠳⠀⠀⠐⠀⠤⠁⠀⠐⠀⡀
⠀⠀⣹⡋⣉⣀⣒⡒⠀⠀⠄⠀⣀⢼⣿⡧⡀⣺⠟⣿⣇⠀⡈⢻⣷⣀⡈⠙⣏⠓⠈⣥⣄⠙⢾⣿⣷⣌⠃⠀⠒⠀⠉⠀⢸⢋⣇⡀
⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢠⣽⣧⠞⣹⣷⣿⣧⣴⣿⣟⣓⠃⣴⣿⣯⣉⠉⢿⣷⡀⠈⡿⠷⠀⠙⢿⣝⣦⣄⡀⠀⠀⠀⠀⢁⣀⣐
⠀⡈⢉⠁⢈⠁⠠⠠⠄⢤⢌⣼⠋⠘⣰⣿⡋⣡⣼⠿⠿⣿⢦⠉⣷⠾⢿⣶⣆⠀⢩⣦⢷⣄⡀⠀⠠⣈⡩⢤⣥⠖⠀⠀⠀⢠⣄⡀
⠀⢘⣶⣦⠀⠀⠀⢀⣉⣻⣿⣷⡏⠼⣓⣸⢿⡁⢼⡄⢾⡽⣯⣷⢏⡀⠀⢀⡉⠻⢆⡀⢌⡿⠻⠀⣤⠘⢷⣈⣡⣦⢀⠀⠀⢢⠨⢏⡀
⠀⠸⠝⢋⡀⠀⠭⠀⠀⠳⣶⣶⣶⠦⡌⠙⣾⢷⣜⡂⠈⣟⠉⠛⣄⠙⣆⠈⢳⡆⠀⢱⢳⣵⣄⡀⠣⣿⡾⠉⠁⠀⡈⠀⠀⠈⠁
⠀⠈⠁⠤⠀⡂⠠⣀⠀⢐⠁⢈⡿⠿⡿⣜⣿⡟⢿⣧⠀⣿⢣⠀⠘⠓⠤⠳⣼⣷⣷⡖⢸⡁⢨⠛⡻⢳⣶⠀⣦⣶⠀⠀⠀⡤⢔⠠⠄
⠀⠀⠘⢛⣹⡆⠄⢀⠨⣀⠅⢼⡖⣉⠁⣸⣿⣷⣼⣿⣷⣀⣆⢡⣄⣀⣀⣀⣮⣰⣿⣙⣦⣽⣤⣄⣣⣤⡄⠀⣫⡀⠀⠢⢄⠙⠏⠁
⠀⠀⠀⠈⠈⣁⠤⡥⢢⡔⠙⣈⣧⡄⣸⣿⣠⣀⣈⡟⣋⣿⢟⣷⠛⣧⣤⣄⣀⣁⣀⠬⠉⠻⢿⣿⣤⣭⣬⣻⡿⠿⢆⢀⠀⠌⠂⠁
⢠⡾⠁⠀⢙⠥⠰⠶⠄⢩⣾⡟⠣⡙⡿⠋⠙⠿⣶⣿⣿⣿⡟⡁⢸⣀⣴⣾⣽⠟⢋⡹⠗⠂⣤⣉⠛⠿⢿⣿⣿⣦⣌⢺⣧⡀⢦⣄
⠈⣉⡀⠸⣷⣶⣿⣄⣾⣿⠣⠦⢀⠙⠀⠀⠀⠀⣿⢷⣿⣿⣿⡇⢸⠃⡾⠟⢡⡞⢩⠀⠀⠀⠈⣿⣿⣿⣷⣖⣼⣿⣿⣦⣘⢷⣾⣷
⢰⠶⣿⣷⣿⣿⣿⣷⣿⡿⢿⠄⠈⠁⠀⢄⡀⠂⢻⣈⣹⣿⣿⡇⠘⠘⢸⠃⣾⡇⢸⡆⡷⠀⠀⠘⠛⣿⣿⣯⣿⣻⢿⡿⢄⣬⣿⠋
⣌⣳⣾⣻⣿⡿⠻⠿⣿⣷⣯⠂⠀⣦⠀⠈⠁⠈⢛⡟⣹⣿⣿⡇⠴⠀⡇⢸⣿⠃⢸⡇⡽⠀⠀⠀⢀⠉⢻⡿⣿⣇⢸⣿⣤⣌⣳⣤
⢾⣿⣿⣇⣤⣤⠄⠀⣿⣿⣿⣟⢛⣿⠃⢀⣈⣿⣺⣿⣿⣟⣿⠆⣾⣀⠁⣸⠓⢶⠨⡇⠻⡀⠀⠀⠀⠱⣦⢣⣸⣿⣾⣍⡏⠙⠻⣶⣀
⠀⣿⡿⣿⠃⠀⠀⠀⠀⢯⣵⢾⡏⣿⡆⢫⡭⣷⢿⣏⣿⣿⣏⣼⠟⣁⡚⠛⠐⢆⢰⣿⡄⢁⡗⠀⠀⠀⠈⠃⣻⣿⡏⢻⡧⢘⣽⣿⣿
⠀⣿⣟⢟⡛⣷⣤⡤⠀⣤⣍⣿⣿⡿⠿⠟⠛⢿⠿⢿⠿⢿⠿⠿⠾⣷⣾⣷⣶⣌⡻⠿⠿⠦⠱⠜⣦⣀⣀⣀⣻⣯⣰⣿⣁⣿⣿⣿⣿
⠀⠹⣿⣶⣾⠅⢛⡓⠐⠋⠉⠉⠋⠁⠀⢰⣠⡤⡤⢂⣀⣅⠑⠀⣒⢀⠀⢝⢹⣿⠿⠿⡟⠿⠻⠶⢦⣬⡭⢭⣽⠛⠙⣿⣿⠶⣏⡻⠋
⠀⠀⠈⠙⠻⠷⣉⣀⡠⢶⣶⡍⠿⠷⠟⣿⠏⠀⢀⢈⠱⣯⣤⡄⠉⠈⢃⣤⠈⠀⠠⠀⠂⡀⠀⠆⠠⠀⠀⠀⠘⠛⠷⠞⣹⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠘⣽⡿⢀⡤⡤⢤⠤⠀⠀⢀⢦⢻⡧⣤⡤⣾⡧⣰⣶⣤⣟⢛⡒⢶⡶⠄⠀⠒⢺⣖⣀⣤⡶⠶⣒⠀⣚⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⣷⣿⣥⡛⣿⣿⠶⢞⣷⢋⣿⣃⣤⣽⣿⣽⣿⣉⣉⣼⣗⣴⠀⡙⢀⠀⣙⡷⢿⣿⣷⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣯⡧⣿⠧⠾⢛⣰⡷⣿⢯⣿⣻⣿⣿⣿⣿⣿⣿⣿⣭⣏⣩⣠⣾⢿⣿⠻⡷⠚⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣬⣭⣭⣤⣤⣤⣿⣿⣿⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠻⣿⡿⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣾⣿⣷⣶⣿⡇⠀⢸⣿⣧⣀⣄⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠉⠉⠛⠛⠛⠛⠛⠛⣿⡇⠀⢸⣿⣿⡟⠉⠉⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⣶⣦⣴⣶⣶⣿⣿⣶⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣀⣀⣀⣀⣸⣿⣿⡇⠀⣿⣇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⡇⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⡇⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣀⣿⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡤⠖⠒⠒⠦⣄⠀⠀⠀⡠⠔⠒⠒⠂⢄⡀⠀⠀⢀⠤⠀⠒⠀⠢⡄⠀⠀⠀⠀⠀⠂⠐⠂⠄⠀⠀⠀⣠⣶⠿⠿⠿⢶⣤
⠀⠀⣺⠀⠀⠀⠀⠀⠈⠃⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⡎⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⠙⠃
⠀⠀⠈⠲⠤⣄⣀⡀⠀⠀⠀⠀⠑⠤⠄⣀⡀⠀⠀⠀⠀⠘⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⡄⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠲⣀⠀⠀⠀⠀⣰⠃⠀⠘⢄⠀⠀⠀⠀⢀⡄⠀⠀⠢⡀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⡀⠀⠀⣀⣼⠇⠀⠀⣠⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠈⠙⠋⠛⠉⠁⠀⠀⠀⠀⠉⠁⠒⠈⠁⠀⠀⠀⠀⠈⠙⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⢸⣿⣿⡏⠉⠉⠙⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠈⢿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⢀⡴⠖⠛⠒⠶⣄⠀⠀⠀⡠⠔⠂⠀⠒⠤⡀⠀⠀⣠⣾⠿⠿⠿⢷⣦⡀⠀⢠⣶⡿⠿⠿⠿⣶⣄⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣆
⠀⠀⣿⠀⠀⠀⠀⠀⠘⠃⠀⠰⠁⠀⠀⠀⠀⠀⠑⠀⢰⣿⡁⠀⠀⠀⠀⠛⠃⠀⣿⣏⠀⠀⠀⠀⠈⠛⠀⣿⣿⣏⠀⠀⠀⠙⠟⠛⠀⢰⣶⣿⣦⣀⣀⣠⣿⣿⡿
⠀⠀⠘⠳⢦⣤⣀⣀⠀⠀⠀⠀⠑⠠⠄⣀⡀⠀⠀⠀⠀⠻⢿⣶⣦⣤⣄⡀⠀⠀⠘⠿⣷⣶⣤⣄⣀⠀⠀⠘⢿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⠀⠀⠈⠙⢆⠀⠀⠀⠀⠀⠉⠈⠛⢿⣆⠀⠀⠀⠀⠈⠉⠙⢻⣷⠀⠀⠀⠉⠉⠙⠛⢿⣿⣿⡀
⠀⠀⢳⣄⠀⠀⠀⢀⣸⠇⠀⠘⢄⠀⠀⠀⠀⢀⡌⠀⠘⣿⣦⣀⣀⢀⣠⣾⡟⠀⢻⣧⣄⣀⢀⣀⣴⣿⠀⢻⣿⣷⣤⣀⣠⣴⣿⣿
⠀⠀⠀⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠈⠙⠛⠿⠟⠛⠉⠀⠀⠀⠉⠛⠛⠟⠛⠋⠁⠀⠀⠙⠿⠿⣿⡿⠿⠟⠁
⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⠀⣰⣾⣿⣿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠻⢶⣦
⠀⣿⣿⣿⠀⠀⠀⠘⠛⠛⠁⣿⣿⡁⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠃
⠀⠘⢿⣿⣿⣿⣶⣶⣤⡀⠀⠹⢿⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣤⣤⣀
⠀⠀⠀⡈⠉⠙⠛⠿⣿⣿⡄⠀⠀⠈⠉⠙⠛⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡆
⠀⢻⣿⣿⣤⣤⣤⣴⣿⣿⠋⢿⣿⣦⣀⣀⣠⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠀⠀⣀⣼⠇
⠀⠀⠙⠻⠿⠿⠿⠿⠟⠁⠀⠈⠙⠻⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁

⠀⠀⣴⣾⣿⣿⣿⣿⣶⡄⠀⢀⡴⢢⣑⣊⠑⠠⣀⠀⠀⠠⠀⢈⡒⣪⠒⠠⡀⠀⣠⣶⣿⣿⣿⣿⣶⣄⠀⢀⣴⣾⣿⣿⣿⣷⣦⡀
⠀⢸⣿⡟⠁⠀⠈⠙⠿⠿⠀⣮⣐⠁⠀⠀⠀⠱⠜⠂⢠⢣⠊⠀⠀⠀⠀⠅⠔⢠⣿⣿⠉⠀⠀⠉⠻⠿⠇⣼⣿⡏⠁⠀⠈⠙⠿⠿
⠀⠘⣿⣿⣶⣦⣤⣤⣀⠀⠀⠹⣌⡲⡠⢤⣀⡀⠀⠀⠀⢆⠐⢀⠤⡄⣀⠀⠀⠀⢻⣿⣷⣶⣤⣤⣀⠀⠀⠘⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠉⠙⠛⠻⢿⣿⣷⠀⠀⠀⠀⠈⠑⠓⢮⢱⡀⠀⠀⠈⠁⠈⠊⠐⡈⢂⠀⠀⠈⠙⠛⠛⠿⣿⣿⡄⠀⠀⠉⠙⠛⠻⢿⣿⣧
⠀⢰⣿⣧⣀⡀⣀⣠⣿⣿⠂⢔⢖⡀⠀⠀⢀⡀⣣⠇⠐⠒⠄⢀⠀⢀⡀⠃⠣⠰⣶⣶⣄⣀⢀⣀⣼⣿⡇⢶⣿⣦⣀⠀⢀⣠⣿⣿
⠀⠀⠙⠿⣿⣿⣿⣿⠿⠋⠀⠈⠑⠯⠓⠟⠪⠵⠉⠀⠀⠀⠢⠔⢊⡒⠴⠀⠀⠀⠘⠻⢿⣿⣿⣿⠿⠛⠀⠀⠛⠿⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣆
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣦⡀
⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇

⢬⠤⢤⡄⠠⡄⠀⠀⠀⣠⠀⠠⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣉⣀⣑⣃⣀⣉⠓⢒⣋⣉⣀⣀⣀⣀⡀
⣿⡛⠿⢿⣿⣿⣦⣼⣿⠟⢲⣾⣻⠉
⢿⣿⣶⣄⠈⠉⢛⣿⣿⣴⠿⠏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⡟⠉⢻⣿⣇⣀⠉⠹⠟⠁⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗
⠿⣦⠀⣿⣿⣿⡃⠀⠀⠀⠘⡧⢸⠀⠀⠀⠤⠴⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⡷⢻⣶⣿⣿⣿⣿⡆⠆⡄⠈⠀⢼⠀⠀⠀⠀⠀⠈⠉⢉⣀⠀⠀⠀⠀⢠⠄⠀⢀⢁⣀⣀⣈⣀⠀⠿⠀⠠
⣻⡛⣋⣿⣿⣿⣯⡉⠒⣇⣘⡀⢺⣶⡦⣤⣤⠀⡀⠀⡼⣿⣧⠀⠀⢀⣰⡂⣀⣠⠚⢫⡿⠿⠟⢶⣾⡍⣷
⢤⣤⣤⡍⣭⢿⣹⠟⠀⣐⡿⣇⣾⡠⠟⣾⢱⠿⢾⡶⠿⢿⣖⢈⡁⠛⠘⠿⡿⢷⠆⠈⠀⠀⠈⠉⣍⣍⣩
⠈⣜⠛⣬⢱⡋⢕⠁⢞⠲⠁⣼⠩⣾⣶⡷⢺⢷⢶⣷⣤⣭⣬⣿⣿⠿⣧⣶⡶⠀⠀⠀⠀⠞⢨⣹⣿⣾⣿
⢀⠀⣀⢿⡄⣷⣸⠁⠖⣀⢇⣽⡴⠌⣩⣤⣩⣨⣼⣦⣿⢹⡿⣭⣿⣺⣿⠃⠃⣀⠀⠐⢬⣷⡿⢿⣿⠿⣼
⠷⣤⣿⣶⣼⡿⠿⠶⠖⠒⠂⠀⠐⠒⠺⠿⣿⣿⠿⠛⠛⠁⠀⢋⣇⣉⣍⠙⠚⠤⠶⠧⠾⠭⠭⠯⠭⠠⣿
⠂⠀⠂⠀⠒⠂⠀⠒⠀⠒⠂⠐⠐⠂⠂⠐⠚⠃
⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢗⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠵⠃⠀⠀⠀⠀⢰⡶⠻⠀⠀⠀⠀⠃
⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⡀⠀⠘⠓⠦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠘⡗⠹⡀⠀⠀⠀⠁⠐⠀⠀⢀⠂⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠑⠎⠁⠀⠀⠀⠀⠀⣠⠖⠛⠋⠉⠘⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠤⢤⡏⠁⠀⠀⠈⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⣼⠁⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠀⢿⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠐⠤⣄⡀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣸⠤⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠢⣄⣀⣀⢤⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠀⠈⡄⠀⠀⠈⠓⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠟⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⣀⠔⠁⢰⠢⣤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡄⠀⠀⠈⠑⠎⠁⠀⠀⠀⠰⡦⠆⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠚⠁
⠀⠀⠀⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠃
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣾⣵⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣚⣼⣴⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠍⡹⣿⣿⡻⢩⡖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡻⡇⣸⠺⠷⣅⢳⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠛⣟⣛⠻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⣠⣤⢿⡟⣻⣎⣰⠟⣤⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣴⢡⣾⡾⠠⢤⠠⢄⣽⣿⣟⣻⢧⣼⣟⣯⣽⣋⣆⣠⢤⠸⣾⠷⡅⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣐⣿⣼⣖⡒⣦⣬⣦⣛⣩⡾⢛⢌⣧⡀⡋⣻⣿⣟⣡⣾⣮⣀⣀⣐⣁⢿⣤⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡝⣿⣯⠀⠒⢜⣿⣿⡿⢿⡏⡲⣿⡍⠛⢿⣃⣿⣏⣴⣿⣿⠟⠉⠹⣿⣾⢯⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢍⠛⡻⠘⠧⠙⠻⣑⣬⣙⣛⠻⠿⠈⣥⠾⢭⠛⠣⠃⠰⡿⠷⢨⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣾⣿⣿⡔⠂⣾⡏⡾⠛⠃⠀⠀⠀⠉⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡘⣿⢅⠉⣿⡏⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⡿⣯⣴⣯⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠿⣼⡟⢻⠀⣾⣿⡼⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣉⣽⣾⣾⣿⣿⡷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢳⡏⡇⣼⢹⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠇⣧⣟⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⡿⢿⣀⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠵⠍⣷⣸⠋⠰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡝⣧⠀⢹⢩⠀⣌⣩⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡔⠷⢁⡤⣿⢾⣆⠈⢻⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠴⢾⣿⡾⣷⣭⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⣷⡟⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠁



⣿⣿⡿⠻⠟⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢰⣶
⠀⢀⣤⣤⣼⣧⣤⣤⠀⢠⣤⣤⣿⣤⣤⡄⠀⣤⣤⣼⣧⣤⣤⠀⢠⣤⣵⣿⣮⣤⡄⠀⣤⣤⣼⣧⣤⣤⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⢹⡏⠉⠉⠀⠀⠀⠊⣿⠑⠀⠀⠀⠈⠀⢸⡇⠀⠈
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⢻⡇⢀⠀⠀⠀⣀⠸⣽⠂⡀⠀⠀⢀⡀⠃⠘⢀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⡏
⠀⠀⠻⠛⢻⡟⠻⠗⠀⠘⠯⠛⣿⠻⠽⠃⠀⠣⠌⠅⠈⠡⠌⠀⠘⠛⠛⣿⠛⠛⠃⠀⠺⠛⠛⡟⠛⠟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢠⢿⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠠⠙⠄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣸⣇⡀
⠀⠀⠴⠦⣾⡷⠶⠆⠀⠰⢊⠲⣿⢖⡑⠆⠀⠶⠶⣾⣷⠶⠶⠀⠰⠦⣾⣿⣿⠴⠆⠀⠶⢾⣿⣿⡷⠶⠄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠈⢹⡏⠑
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠔⠁⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⣀⣸⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣀⡀⠀⠀⣀⣀⣿⣇⣀⡀
⠀⠀⠀⠛⢻⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⠟⠃⠀⠈⠛⠛⣿⡟⠛⠋
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⢈⠈⠀⠀⠀⠀⢀⣀⣿⣇⣀⡀⠀⠀⡀⠀⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢻⡇⢀⡀
⠀⠀⠐⠂⠠⠠⠀⠚⠀⠀⠘⠒⢦⡴⠒⠂⠀⠘⠛⠉⣿⠉⠛⠃⠀⠁⠉⠁⠈⠁⠂⠀⠀⠈⠛⠛⢻⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠀⠀⠀⡀⢸⡇⣀⠀⠀⠀⠀⢛⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠉⠁⠀⠈⠉⢹⡏⠉⠀⠀⠈⠉⠉⠉⠁⠀⠘⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⠗⠂⠀⠐⠇⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠉⠁
⠀⠀⢠⡄⠀⠀⠀⢀⣰⣇⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠶⠂
⠀⠘⢻⡟⠋⠀⠀⠀⢸⡇⠀⠀⠘⠀⠀⢀⢀⣇⢀⠀⠀⠀⣀⣸⣷⣀⠀⠀⠀⠀⡖⡀⠀⠐⠀⠠⠐⠃⠘⠒⠐⠒⠃
⠀⠀⠸⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⡟⠉⠀⠀⠀⠙⢹⡏⠙⠁⠀⠀⠉⠃⠋⠁⠀⢠⢠⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠉⠁
⠀⠀⠀⠀⣿
⠀⠀⠀⢠⡷⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣧⣶⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡂⢔⣷⡀⠀⠀⠀⣤⠤⢠⡆⠀⠀⡀⢀⡀⠀⣠⡀⠠
⠀⠀⣀⠸⠇⢿⡏⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢠⢨⡗⠻⣦⠀⠀⠺⣦⣮⡤⠀⢸⠒⢺⣯⠉⣽⡇⣹
⢀⣼⠻⢷⣶⡾⠟⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢱⣅⡀⠀⠀⢀⣾⡟⠁⠐⠀⠀⠀⠀⡶⠋⠁⠀⠾⠗⠛⠙⠒⠹⣿⣷
⣿⣿⣷⣾⣿⣷⡆⠐⠯⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣾⣿⣿⣇⣐⡴⠀⠀⠀⠀⠀⣴⡀⠀⢸⠂⡎⠀⣀⠔⠁⢸
⣿⣿⣿⣿⣿⣿⡆⠈⠓⠀⠀⠀⠀⠀⠀⠀⡠⠀⢤⢀⢀⠀⣠⣄⣤⡀⠀⡀⢀⣿⣿⢣⡄⠀⢀⡆⣴⣦⢻⡁⠀⠆⠰⠅⠀⢡⠎⠠⡸
⣿⡿⢿⣿⠿⣿⣷⣤⣠⣠⡖⠀⠀⠀⡄⠸⣧⣼⣋⠸⡂⠂⠀⠀⠀⠀⠘⢾⣿⣿⣿⡿⠯⠠⣼⡧⣬⣽⣿⡧⠈⠀⠈⣆⢀⠓⠈⣽⣿
⢀⡄⢈⣱⢀⣉⣟⣛⣩⡘⠛⠋⠒⡈⠀⠁⣉⣿⠋⠯⠁⠀⠀⠀⠀⠀⣀⢸⣿⣉⣹⡇⡀⢀⢻⣿⣿⣿⣿⠃⠀⢨⣆⠹⠏⡀⠀⢘⣿
⣎⠿⢌⡋⢈⡛⠿⠟⢛⡛⠆⠠⠙⠁⠀⣼⣿⡇⠀⠀⠀⠰⣀⠀⠀⠀⠀⢾⣿⣿⣿⣷⣱⡀⣎⣿⣿⣿⣿⡆⠄⠈⠀⣤⣿⣧⣿⣿⣿
⣍⠓⠛⠉⠛⢉⣥⣐⡓⡀⠐⢤⣤⠌⢰⣿⣿⣷⠀⢿⣄⠀⠈⢿⣆⠀⠀⢸⣿⣿⡿⢏⠉⢷⠘⣿⣿⡿⣿⣷⠀⢠⠁⢸⢻⢿⣹⣿⣿
⠍⠉⠀⠀⠁⠀⠀⠉⠩⠀⠈⣶⠉⠀⣾⣿⣿⣿⠀⢌⣿⣦⣤⣿⣿⣀⣷⣹⣿⣿⡇⠘⣦⡸⣇⢸⣿⣿⣿⣿⡀⢠⡶⢸⠈⣸⣿⣿⣿
⠠⣌⡉⠉⠽⠆⠀⠤⠤⠀⠀⠨⠍⠐⣿⡉⣿⣿⠛⠺⣿⣿⣏⣄⣀⣿⣿⣿⣿⣿⡧⠀⢻⡅⢿⡆⡟⠈⠹⠟⠳⣾⣦⣾⣷⣿⣿⣿⣿
⠛⠛⠓⠖⠦⠐⠒⠒⠓⠂⠀⠀⠀⣀⣿⣿⣿⣿⣷⣄⠘⣫⢿⣿⣿⣿⠿⢯⢽⣿⣇⣠⡟⣧⠀⢷⡇⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢁⣀⣀⡤⣾⣿⣿⣿⣿⣷⣶⣶⣿⡛⣿⣿⣿⣯⠙⠢⣄⣡⣮⣤⣄⣀⡀⠀⠀⢿⣿⡟⠃⠘⣇⠈⡅⠀⠘⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⡇⠛⠀⠀⡏⠁⠀⢻⣿⣿⣿⣿⣶⣾⣿⣷⠀⠀⢹⣆⢻⡀⠈⠙⠛⠛⠿⠿⠿⠿⣿⣿⣿
⠛⠿⠿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣇⠀⠀⠀⠄⠀⠀⠈⠻⠛⠻⠿⢿⣿⣿⣟⠶⠦⡈⢿⣦⣃⢠⠀⠀⢀⠐⠀⠒⠟⣻⣿⣿
⣡⡈⠘⣿⠇⣼⣿⣿⣿⣿⡿⡿⠛⠛⠉⠁⠙⠇⢀⣾⡀⡴⠇⢠⣷⡤⠄⠀⠀⠈⠉⠛⠧⠴⠛⠊⠉⠛⠿⡁⠆⠰⡜⠀⠉⠩⢉⠵⣶
⣽⣷⡖⣸⣎⠉⠐⢾⣿⣿⣅⣷⣄⠀⠠⣚⣓⣺⣿⣿⣿⠹⠿⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣰⣾⣏⣠⣄⣼⠛⠈⠛
⣿⣯⣷⣿⣿⣦⣴⣦⣍⡛⠛⠛⠛⠛⠒⠢⠉⠡⠚⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⣄⣀⣠⣄⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⣿⣿⡿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡿⣻⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠂⣽⡗⢙⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠈⠀⢸⣷⣼⣧⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⢲⡒⠲⢾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⢸⣿⣿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣁⠀⣸⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢉⡀⠀⠸⠟⣿⡿⠳⡧⢈⡟⣱⠟⢿⣿⣶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠰⢜⠛⠒⠀⡷⣤⡴⢡⣦⡌⠻⣟⡛⠓⠒⠨⠁⠤⣁⣊⣉⣈⡉⡙⠫⠿⠿⠗⠶⣲⣶⣦⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⡃⢀⠀⢛⣺⣷⢀⣘⠿⣏⣀⠘⠛⠁⠀⠈⢤⣀⣊⣍⣑⣂⣀⣀⣈⣉⡉⡉⠁⠤⠄⠠⠀⣀⣀⣉⣑⣀⢀⠀⠀⠀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⡆⢐⡁⠉⠂⣻⡇⠐⣿⡃⢠⣶⡖⠀⠐⢻⡏⠉⣩⣿⣯⡍⠉⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠒⠳⠀⠘⡷⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣯⡟⢺⢞⠀⠀⣿⡄⠀⣿⠇⣾⣿⡯⠀⡀⢺⡄⣼⣿⣿⣿⣿⠂⠀⢱⠀⢀⣴⣾⣿⣆⠀⠀⢠⠀⢠⣴⡀⠀⠀⢀⣾⣏⡇
⠀⠀⠀⠀⠀⠀⢤⣶⠰⠿⣿⠿⡷⢸⠟⣁⠿⠿⢆⠀⣿⠀⢿⠿⣧⠀⠀⢸⡃⢻⢿⣿⡟⢿⠀⠀⢸⡄⢸⣿⣿⣿⣿⡄⠀⢺⡉⣿⣿⠇⠈⣀⢀⠛⣿⡥
⠀⠀⠀⠀⠰⠤⠀⠁⠉⠂⠀⠀⠀⠸⠿⠥⠺⠿⠟⠂⣠⡤⠤⠂⠀⠀⠀⢸⡧⠍⠀⢀⡀⢸⣿⠂⠸⠇⠿⠿⠿⠛⠟⠃⠀⠀⣗⠛⠛⠂⠨⠉⠈⠀⡁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠈⠛⠀⠀⠀⠀⠀⠁⠀⠀⠛⠖⠒⠒⠀⠤⠀⠙⠁
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣻⣻⢿⣻⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠩⠞⠀⠀⠠⠀⠔⠲⠀⠠⠠⣤⢀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠛⠛⠻⡖⠉⠃⠈⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠠⣴⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠲⠶⣶⠶⣿⡷⢶⡶⠶⠆
⠀⠿⠿⠿⣿⣿⡿⠿⠿⠿⠀⠀⠀⣀⣈⣳⣴⣋⣀⡀⠀⠀⢼⣶⣶⣾⣿⣿⣶⣶⣾⡇⠀⠀⠀⠀⠙⣿⡟⠁
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⣿⠛⠛⠃⠀⠀⠈⠁⠀⠈⣿⡏⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡀⠘⢿⢧⣟⡟⠀⡀⠀⠀⠀⢀⣤⢾⣿⡷⣤⡀⠀⠀⠀⠀⠀⠀⠿⣿⠿
⠀⠀⣤⣤⣤⣾⣿⣤⣤⣤⡄⠀⣀⢿⣲⠾⠇⠿⢶⣺⢇⡀⢠⣤⣾⣥⣼⣿⣧⣬⣷⣤⡄⠀⣤⣿⣦⣤⣿⣤⣤⣿⣤
⠀⠀⠉⠉⠉⢻⣿⠉⠉⠉⠁⠀⠈⣾⠽⠲⡆⣶⠺⢽⡎⠁⠈⠛⢻⡛⢻⣿⡟⢛⡟⠛⠁⠀⠈⠻⠃⢰⣿⡆⠈⠟⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠁⠀⢨⡇⣿⠀⠀⠁⠀⠀⠀⠀⠙⢻⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠠⠿⣎⡷⠧⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⡠⠤⠴⣾⣿⡦⠤⠤⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠴⠊⡏⠈⡏⠲⠄⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⠶⡄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⢀⣀⣀⣀⣸⣿⣄⣀⣀⣀⠀⠀⠀⠀⠐⠯⣷⣦⣤⣄⣀⣸⣴⣇⣀⣠⣤⣴⣾⠽
⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢠⣼⣸⣏⣸⣧⣿⣼⣸⣿⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣙⠈⣛⣿⣿⣿⠿⠋
⠀⠀⣄⡀⢻⣿⠃⣀⡄⠀⠀⠀⠀⠉⠁⠀⢹⣿⠁⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⢰⡏⠉⠉
⠀⠀⣿⡿⢿⣿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⢀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠂⠀⠐⠐⠀⠀⠂⠀⠀⠀⠐⠀⠀⠿⠶⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠛⠉⠀⠀⠀⣿⣿⡻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⣿⣧⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⣿⣭⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⢻⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣸⡷⠤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣴⣿⡷⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣅⡀⠻⣏⣽⣧⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣶⣦⣿⣿⣿⣿⣿⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢫⣽⣿⣿⣿⣿⣿⣿⠟⠁⢠⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⣁⣄⣻⣿⡀
⠀⠀⠀⠀⠀⢸⣿⡷⠶⠶⠄⣞⣉⣛⣉⣉⣉⣉⣀⣀⣈⣻⣿⣿⣿⣷⣾⣿⣿⡿⠟⠁⣠⣾⣿⣷⢶⣶⠶⣶⣶⠾⣿⡿⢿⣿⡿⢿⣶⠀⣸⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣦⣶⡿⠿⣿⣿⢛⣿⣿⣿⡛⣉⣿⣿⣽⣿⣿⣿⣿⣿⡿⠋⣠⣶⣿⣿⣟⣭⣿⣷⣿⣿⣿⣤⣾⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⡄⠈⢀⠐⠁⠸⠟⡿⠛⣙⡿⣛⡿⢻⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣍⣽⣿⣿⣿⣿⣿⡿⠿⠿⠿⠻⢿⣿⡿⠟⢛⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢾⣿⣧⡐⢁⣠⣾⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣽⣿⠻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣯⣴⣿⣿⣩⣿⣿⣿⣽⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣍⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣁⠠⠾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⠅⠀⠲⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⠖⠈⠀⣾⣿⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣠⣤⡶⠛⠻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⢤⣹⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⠀⠀⠀⣄⢹⣿⣭⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣄⠸⣿⡄⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⣈⠲⣿⡟⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠙⣦⣿⣟⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠘⢿⣽⡿⠞⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠰⢮⣹⣷⣴⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣤⡙⣿⣄⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠙⣿⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠰⠦⣿⡄⣸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣄⡀⢹⡏⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣉⠙⢻⣧⢼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢈⠑⢺⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠙⠓⢾⣿⣀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⢀⡀⢸⣿⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢤⠀⠈⢙⣿⣇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣄⡀⠀⠀⠈⢀⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣶⣶⣾⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣭⣭⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣴⠟⠁⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣊⣾⠟⠁⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣊⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡑⢄
⠀⠀⠀⠀⠀⠀⠀⡠⢪⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⡠⢪⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⡠⢊⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡕⢄
⠀⡠⣊⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⡜⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⢣⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠑⢌⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡡⠊
⠀⠀⠀⠑⢌⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠑⢍⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠑⢜⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⠻⣦⡀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⠻⣦⡀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⣛⣛⠥⠊

⣾⡿⡿⠿⡿⠿⢿⣿
⠻⠿⠿⠟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿


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