Norseman Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⡀⣠⣤⣤⣄⢀⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣷⣄⠀⠀⠀⢠⣾⣿⡇⣿⣿⣿⣷⢸⣿⣷⡄⢀⠀⠀⣠⣾⣿⣿⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⣿⣿⣿⣿⣮⢻⣿⣿⣿⡇⢱⡟⢻⡎⢸⣿⣯⣿⡟⣵⣿⣿⣿⣿⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⣿⣿⣿⣿⢿⣿⣧⣿⡇⢻⡆⢸⡟⢸⣿⣼⣿⡿⣿⣿⣿⣿⣿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⢿⣫⣾⣿⣿⣿⡇⣮⣿⣿⣵⢸⣿⣯⣿⣷⣝⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣯⣾⠟⣹⢡⣿⣿⣿⣿⡜⣏⠻⣷⣽⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣭⣙⡁⠰⠏⠸⣿⣿⣿⣿⠇⠹⠆⢈⣉⣭⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣨⣿⣿⣿⣶⣦⣤⣙⣋⣥⣴⣶⣿⣿⣿⣅⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠉⠉⠙⠛⠛⠿⠿⢿⣯⣽⡿⠿⠿⠛⠛⠋⠉⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣧⠀⠀⠀⢀⣈⡻⢶⡿⢟⣁⡀⠀⠀⠀⣼⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⣿⡄⣠⣾⡿⠛⠋⠀⠀⠙⠛⢿⣷⣄⢠⣿⣿⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢯⡄⢀⣤⣶⣶⣤⡀⢠⡽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣿⡇⣼⣿⣿⣿⣿⡇⢸⣿⣿⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢻⣿⣿⣷⣼⣿⣿⣿⣿⣧⣾⣿⣿⡟⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⣿⣿⣿⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁

⡿⢿⡿⢿⠛⠋⠀⣠⣽⡿⢛⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠙⠻⣿⣿⣯⣛⢻⣿⣷⣿⠋⠐⢛⣿⣿⡟
⣧⠈⣇⠐⡀⢀⣴⠏⠉⣠⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⣰⣆⢰⣦⣤⠃⠀⠀⠀⠀⠀⠀⠙⠻⣿⣧⣈⡿⣿⣧⡆⠀⠹⣦⡅
⠡⡾⡃⠀⢠⡟⠃⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⠇⣬⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣦⡘⣿⣷⣤⠄⠈⠙
⣤⠁⠈⣱⡟⢀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣧⢀⣤⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣌⢿⣿⣾⡶⢾
⠀⠀⢰⣿⢁⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣄⠈⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⢿⣿⡶⠞
⠳⢴⣟⢁⡞⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢞⠈⠀⠹⠶⠴⠟⣿⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡌⣿⣿⣴
⠂⣼⠋⠸⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⣋⣴⣶⣴⣖⢶⠨⣶⣸⡟⣧⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⣿⣷
⢲⡿⠀⢆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⢿⣿⠟⠛⣷⠲⣿⡃⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿
⢺⡇⢀⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⣵⣿⣟⣻⣾⣽⣧⡈⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢺⣿
⢻⡇⠸⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠜⠁⣠⣾⣿⠟⠉⠻⣿⣝⢷⣌⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣸⣿
⢽⡇⢀⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⠀⠀⣼⣿⠕⠨⣥⣦⣅⣽⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢙⣿
⣿⡿⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⡄⠀⢠⣿⣿⣷⣦⣿⣿⢿⣿⣿⠿⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿
⢿⡇⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⣿⣯⣠⣿⣿⡿⣿⣏⢱⠇⢸⣿⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⣻⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢸⣿⣿⣿⡿⠀⠤⠸⠿⠿⣿⣿⣿⠸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⣻⡇⢸⡇⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⡀⠀⣿⡟⠀⣿⣿⣿⣿⣿⣰⣦⠀⠀⠀⣿⣿⣿⡄⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⣿⡇⢨⡇⠀⠀⠀⠰⠾⠟⠻⣄⠊⠀⠁⠁⠀⣽⡇⢠⣿⣿⣿⣿⣿⠻⣏⠀⠀⠀⣿⣿⣿⡇⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⠀⣿
⣿⠃⠈⡇⠀⠀⠀⠀⢠⠄⠠⡇⠀⠀⠀⠀⠠⣿⠃⢸⣿⣿⣿⣿⣿⠳⡾⠀⠀⠀⠻⠋⣍⣽⣿⣿⣿⣿⣿⣷⠀⠁⠁⠂⠀⠀⣿⣀⣿
⣿⡄⢀⡗⠀⠀⣖⡀⠘⠀⢾⣿⠂⡇⠂⠆⢰⣿⠀⣿⣿⣿⣿⣿⣿⠐⡟⢀⠀⠀⣀⡠⠟⡻⢯⣿⡿⠿⠿⣿⣄⡀⠀⠀⠀⠀⣿⣜⣿
⣿⣿⣸⡗⠀⠠⠀⠉⠓⣶⠦⢉⡙⢧⣤⡄⢾⣿⠀⣿⣯⣿⣾⣿⡏⣨⣷⣿⡟⠉⣋⣑⣄⡀⠀⡄⠀⡠⠦⠤⠿⣿⣷⣄⠀⠀⣿⣿⣿
⡿⣿⡇⠱⣢⡴⢰⡾⠀⣿⣿⠿⢿⣷⠿⠇⢹⣿⢰⣿⣿⣿⣿⣿⣿⢻⣿⣿⡟⢿⣿⣿⣿⢁⣤⠖⠉⠉⠙⠶⣶⣬⣉⣭⣑⣀⣿⣿⣿
⣿⢿⣿⡀⠹⠟⣡⠃⢸⡟⠀⠀⠒⣿⠏⣣⣿⡿⢸⣿⡟⣿⣷⣿⣿⢸⣿⣿⡟⣻⣻⣿⣿⠀⠛⡠⠖⠄⠢⠔⣐⢉⡻⣿⣿⣟⡙⢿⠟
⠇⣼⣿⣏⣦⡾⣿⠀⣿⣗⠀⠀⣆⠐⣆⠈⣿⣇⣨⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇⠀⣿⣿⡇⠔⠋⠴⠆⢠⣆⢀⡈⠠⠁⡙⣿⣿⣧⢶⣾
⣸⣟⣛⡿⣧⣤⡼⠠⣿⣯⣠⣦⠈⢺⣼⡖⠻⠿⠿⠊⢹⣿⡿⢿⡿⢸⣿⠿⠀⣰⣿⣿⢁⡀⠀⢄⡰⣾⡟⠩⡴⠀⠻⡷⢹⢹⡿⢾⡟
⣟⣛⣿⣄⣀⣀⣀⣠⣶⣿⣿⣍⣹⣿⡟⣉⠀⢈⣏⣢⣿⣿⡀⠀⣴⢿⠄⠀⣏⣘⠟⠉⠀⠀⠉⠑⠶⢣⠄⠂⠐⢴⣐⣿⠀⢰⣥⠆⣰
⠿⣿⠆⣶⣶⣿⣿⣿⡻⣭⣿⠉⢼⣿⣿⣷⣶⣶⣾⢁⣿⣿⣿⠀⢽⣾⢩⣍⣽⣿⣀⢠⡄⠀⠀⠠⣀⣄⠒⠸⢟⣿⣾⣿⣧⣿⠃⣴⣿
⣴⣿⢿⠤⣿⣿⡟⠁⠈⣻⡷⣤⣦⡿⢿⣿⠿⢿⣿⣿⢿⣟⣙⣤⣶⣿⣼⣿⣿⣿⣿⣿⣷⣄⢴⣨⣉⠀⡸⢦⣼⢱⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣏⡦⠜⣃⠀⣶⣜⣿⣿⣿⣿⣷⣿⣿⡀⠈⢛⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣉⣻⣿⡏⠺⢽⣟⣇⡼⣺⣷⣿⢿⣿⣿⣿⣿⣿⢿
⣯⣭⣥⣤⣴⣿⠃⣰⣿⣿⣿⠿⢿⡿⢿⣿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠻⠟⠛⠿⠛⠛⠿⡿⠓⠚⠛⠛⠛⠛⠉⣙
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢜⠇
⣤⣬⡀
⠉⠘⠑⢄⡀
⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⢀⣀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢻⠙⡇⠉⠏⡏⣹⠂
⠀⠀⢠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⢈⠀⡷⠀⢸⡇⣼
⠀⠀⢀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⢉⠁⢀⢀⠀⢀⡀⠀⣼⣠⠊⠁⣬⡀⣁⢀⡘⣧⡟
⠀⠀⠈⠎⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⣴⠶⢦⢠⡷⢨⢨⣞⣿⠆⠸⠟⢳⣦⡜⠿⠿⠷⠾⠇⠿⡆
⠀⠀⠀⠈⠜⣷⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠆⠀⡄⡘⠋⠇⠀⢿⣄⣳⣯⣞⣛⡡⠚⠛⠓⠛⠂⠼⠧⣝⣿⡁⠀⢲⡏
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣅⡄⠀⠰⡀⣠⡄⢹⣿⣿⣭⣥⣄⣀⠀⠀⠠⣄⠀⠀⠉⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠈⢻⣦⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⢄⣽⣇⣡⢄⣌⢂⣼⣋⠉⢀⡀⢄⣉⠻⣦⠀⠈⠻⣦⡀⢢⡀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠦⢀⡀⠀⠀⠀⢠⣤⣄⠀⣳⣴⣿⣶⣯⣿⣾⠋⠙⠒⠿⣅⠀⣈⡑⣌⠄⠀⡀⠈⢳⡄⠹⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣨⣵⠿⠂⠹⠿⠻⠿⡛⠿⠛⠋⠐⢦⠙⠆⠒⠀⠀⠀⠀⢸⡿⣟⠁⢈⣸⡀⢼⣦⠀⢻⡄⠹⡌⢧
⠀⠀⠀⠀⠠⠚⢀⡶⠞⠓⢺⠟⢀⣤⡴⠀⠰⠂⠀⠙⢆⠑⠄⠀⠀⠀⠈⢀⠀⠀⠤⠶⠋⣰⣿⢷⠿⣿⣇⠀⠻⣦⠀⢿⠀⢻⡘⡇
⠀⠀⠒⠛⠛⠁⠈⠀⠀⠀⠠⠀⢿⠋⢠⠞⣡⠄⢃⢀⣼⡇⣸⣶⣾⣧⡼⠀⣀⣼⣁⣠⣵⣿⣿⠀⠀⢻⣿⠀⠀⠈⣧⠘⡇⣿⣷⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣁⠔⢁⣠⣨⣽⣟⣘⡋⠛⠿⠿⠁⣸⣿⣿⣿⣿⣿⡿⠃⢠⠆⢸⢹⡆⠀⠀⢹⡇⠁⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣿⣿⡿⠒⣫⠁⠀⠀⠈⠢⠜⢿⣿⣿⠈⠉⣉⣀⡴⠋⠀⠀⣸⣷⡇⠀⣾⠃⠀⣸⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⢰⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⡏⢷⣀⣀⣠⠀⢀⡔⠀⠉⣿⡇⠀⣿⠀⠀⢿⣿⡟⢇⠀⠙⠛⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡙⣿⣆⠀⠀⠀⠀⠀⠈⢻⣝⠢⢉⡀⠀⠀⢀⡴⠏⢠⡿⣇⠀⣿⡇⠀⣿⣿⣷⠀⠀⠀⢀⢠⡝⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣷⡀⠀⠀⠀⠀⠘⣿⡓⠀⢤⠤⠀⠀⣀⣴⢿⡇⠘⠀⢏⢷⠀⢻⣿⣿⣧⠰⡄⣈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣮⡢⡀⠀⠀⢻⣿⣯⣷⣶⠴⠟⢋⡔⣸⣇⡀⠳⠈⠈⢧⠀⢿⣿⣿⣧⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠢⡀⣾⠸⢤⡷⣄⠀⠀⠉⡴⢻⣿⡇⠀⠀⠀⠀⠳⡀⠁⠙⠫⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢉⠳⣬⡻⣄⠀⠈⠆⡙⠢⣄⡴⠾⣿⡇⠀⠀⠀⠈⢢⡈⠂⠀⢤⡈⠀⠀⠀⠠⢤⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠙⠦⣕⠦⣄⠱⣴⠟⣵⠿⡟⠻⣆⠀⠙⣶⣤⣿⣦⣀⠘⠋⠀⠀⠀⠀⣄⠈⠹⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⢅⡈⠙⠳⢾⣳⣄⠙⢦⡈⠢⡀⠙⠿⡟⠿⣿⣦⡀⠀⠀⠀⣀⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⢈⣉⣛⡯⠛⠢⢈⡂⠀⠈⠀⠀⢀⡈⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢍⠒⠲⠀⠘⢧⠀⠀⠀⠀⠘⠿⠾⠧⠀⠠⢀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡀⠀⠀⠀⠀⣠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⡐⠂⣄⣀⠔⡈⠀⣄⡐⡀⠀⠀⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠠⠀⣄⠠⠍⠚⠤⣯⠑⡒⢍⠢⡈⡀⢠⣶⠀⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⣀⠨⡐⢎⡱⢎⢃⠼⣠⣌⣁⡀⠘⣆⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠢⠛⢯⡝⣣⢊⡱⣅⠌⠻⣿⡆⠘⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢸⢟⠢⣄⠠⠀⢄⡀⠠⠀⠀⠂⢴⡟⠡⠜⣩⣷⡑⠞⢁⡠
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠠⢡⢢⣝⣧⢣⡈⠄⠛⠀⣀⣴⣦⣄⠂⠀⣎⡐⣦⣤⢒⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣉⡺⡼⣃⣂⣀⢠⠀⠚⠉⠉⠙⢿⠀⠀⣿⣿⢱⠞⣋⣱
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠉⠛⢃⣿⣿⡇⢀⠀⠀⠀⠀⠀⠀⡄⢆⣛⡛⡠⣛⣯⠰⠌
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⣗⢢⢀⠹⢎⡽⠃⠀⠀⠀⠀⠀⢀⡂⣸⡻⠬⣭⣥⠩⠸⠂⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢋⣩⠤⠸⠎⠀⠀⣄⢤⣴⣾⣟⣤⣛⣴⣦⡀⠘⢷⡒⢒⣉⠂⡠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⣡⡴⢀⠀⠒⣋⣹⡈⠀⠀⠀⠀⠈⠉⠀⣵⢄⠀⠁⠋⢈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠄⣀⠀⡀⠄⠀⠀⠀⡁⠔⠁⠀⡁⡿⢴⡇⡇⢻⠛⡓⢶⢒⡁⢄⡱⢶⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⣀⡰⠦⢴⡶⠋⠀⠂⣔⠿⣱⢆⡄⢱⢸⢺⢲⡇⡸⠧⢖⣣⡔⢊⣡⠼⡒⢀⡀⠈⣅⠒
⠀⠀⠀⠀⠠⠀⠀⠙⣦⣠⠞⠀⠀⢀⢨⡖⡠⢛⠨⠇⠻⢈⣍⣧⡼⠠⠋⢋⣡⣖⢡⡔⣧⢹⠈⢠⡀⠂⣀
⠀⠀⠀⠀⠀⡄⠈⠀⠈⠁⣴⢆⡛⣼⣫⡜⠱⢏⠜⣱⣆⠹⡼⣶⣋⢫⠘⡞⠉⣾⣶⡌⢦⣹⢘⠁⣉⠂⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⣤⢙⣾⡱⢎⣼⣿⣿⠐⣋⡭⢭⢾⠰⢠⡀⠉⠉⢁⡈⠔⢏⡖⠉⠶⠄
⠀⠀⠀⠀⠀⠀⢠⣾⢩⣀⡤⡈⠀⠉⠈⠄⣡⣾⣿⣿⣿⡌⠁⠛⠀⠀⠘⣧⢻⣿⣿⠿⠛⠈⠁⣸⣤⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠈⢿⢀⣈⣉⡁⡀⠀⠠⠈⢿⣿⣿⣿⣿⣇⠀⠐⠆⠀⢀⣧⠆⠋⠀⠀⠀⠂⠁⢹⣿⣆⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢈⠘⣉⡿⣀⣵⣿⡀⠁⠈⣿⣿⣿⣿⣿⡄⠀⠐⣤⠺⣶⢛⣂⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠃⠀⠤⠬⠭⠃⠀⠀⣿⣿⣿⣿⣿⣿⡔⠂⠀⣣⢸⡯⡽⠄⠀⠀⠀⠀⠀⠀⢬⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠤⠟⣚⡜⢁⠀⢀⡆⠹⣿⠿⠿⠿⠠⢄⠤⠁⠈⠛⠯⠤⡀⠀⠀⠀⠀⠈⠈⠵⠇
⠀⠀⠀⠀⡀⠄⠀⠀⠙⠗⠰⠛⣩⡔⡬⠂⣩⡗⠀⠈⠛⡗⠀⠶⠖⠢⡅⣆⡅⡀⠀⠠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣰⢎⠴⢇⠈⣡⡄⠀⢀⣷⢀⣤⣄⣠⣞⢮⡛⡎⣒⠂⠀⠀⠀⠀⢀⠃⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠛⢀⣨⠛⡡⢚⠀⢨⡇⠁⠘⣮⢱⡭⢣⢍⡔⠉⠻⠃⠶⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⣾⣄⣨⠁⠾⣛⡭⢾⣲⣢⠀⠈⣗⡫⢘⣥⡾⠓⠲⣿⡀⣤
⠀⠀⠀⠀⠀⠀⡀⠒⣉⠀⠀⠀⠀⠀⠁⠘⢻⠖⠠⣩⡬⣭⡄⠀⠸⠶⡘⣿⣆⠈⡀⠹⠤⠿⠐
⠀⠀⠀⠀⠀⠀⠀⠘⠡⣶⠈⣅⠀⣴⡔⠆⠁⢀⡐⠛⠄⠈⠟⠀⠀⠀⠦⠼⠛⢻⡇⠘⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠉⠁⢸⡮⡐⠛⠛⢖⠀⡘⢀⠁⠺⠦⣀⠀⠀⠈⠱⠀⠀⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⣄⡀⠀⣤⠾⡃⢉⡄⢿⡇⠋⠶⣌⡔⠙⢢⡜⣱⠗⡀⠘⠛⠲⣶⡖
⠀⠀⠀⠀⠄⠀⠀⠃⠀⢀⡠⠚⠀⠙⡄⢳⡖⠿⠁⢹⡈⣧⠰⡈⠁⠈⢶⠂⠘⠇⠐⣾⡆⠈⠓⠌⡵
⠀⠀⠀⠀⣀⠤⢀⣠⣶⡿⠁⢀⣴⡁⠀⠀⢰⠜⢛⡈⣵⠚⠄⢻⠀⠚⠶⠇⠀⡀⢆⠀⢠⡆⢹⠂⠁
⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠈⠛⣛⣄⠀⠙⠆⠉⣃⡀⠦⣘⣦⣤⣤⣤⣄⠊⠀⠀⠀⠙⠢⡈⠠⠄
⠀⠀⠀⠀⠐⠟⠀⠀⠊⠡⠀⠀⠀⢻⡀⠴⢆⠀⠘⠻⢦⡠⢎⢻⣿⡿⠿⠟⠂⠉⠢⡤⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠿⠶⠀⠎⠁⠈⠀⢃⠐⠀⠆⠑
⠀⠀⠀⠀⠀⠁⠀⠐⠊⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⢀⣴⣧⣴⣿⣿⠿⠿⣿⣿⣦⣼⣦⡀⠀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⡫⣢⣕⣽⣿⣿⣿⣿⣷⡀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⡀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣆⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢉⣿⣿⠿⡟⠿⠟⠻⠿⣿⣿⠿⠟⠻⠿⠻⠿⣿⣿⡉⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣥⠆⡀⢸⣿⣏⣴⣦⣄⣀⠀⠀⢠⡄⠀⠀⣀⣠⣴⣦⣹⣿⡇⢀⠠⣴⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠘⠄⢙⣾⣿⣿⡿⠿⣿⣿⣿⣶⠿⠿⣶⣿⣿⣿⡿⢿⣿⣿⣷⡋⠠⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣘⠛⣻⣿⣿⣿⠀⠀⠀⠈⠹⣻⠀⠀⣟⠃⠀⠀⠀⠈⣿⣿⣿⣟⠛⣃⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣙⣈⡟⢟⣿⣿⣶⡶⢀⣴⣏⣉⠀⠀⣉⣹⣦⡈⢶⣾⣿⣿⡻⢻⣁⣋⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣾⣷⢿⣿⣿⣿⣶⢟⡽⠿⠻⣿⣿⡟⠿⢯⣿⢾⣿⣿⣿⡿⣾⣷⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⠛⠛⣿⣿⢃⣿⣁⣴⡿⠿⠿⢿⣦⣀⣿⡈⢿⣿⠛⠛⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣾⣶⡿⣇⣾⣷⣿⢟⣿⣿⣷⣶⣶⣾⣿⣯⣻⣿⣾⣷⣰⢿⣶⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡅⠿⣯⡵⠃⢸⣿⣿⣿⣿⣿⡟⠹⡏⢻⣿⣿⣿⣿⣿⡇⠸⢮⣽⠿⢨⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠖⣌⣠⠹⣿⣿⣿⣿⣿⣿⢀⠄⢀⠈⣿⣿⣿⣿⣿⣿⠏⣄⣠⠲⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠄⢹⣿⣿⣿⣿⣿⣸⣷⣫⣴⣿⣿⣿⣿⣿⡏⠀⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠠⠆⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠙⣷⠙⣿⣿⠋⣾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠀⠁

⠀⠀⠀⠀⠀⠄⣠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡂⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠀⠠
⠀⠀⠀⠀⠀⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡘⡔⠵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠸⠃⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⢁⠘⡖⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡀
⠀⠀⠀⠀⢐⠘⡄⢢⡀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢑⠀⠁⠘⣄⠀⠀⠀⠀⠀⠀⠀⡜⠘⠃
⠀⠀⠀⠀⠘⣦⠘⢆⠹⡦⢀⣀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠂⢀⠈⢂⠀⠀⠀⢀⣠⠎⠀⢶⡇
⠀⠀⠀⠀⠀⠡⠑⢌⡗⢧⡀⠀⢨⣃⡂⠀⠤⠥⠤⠥⠄⠠⠤⣀⠠⠀⣀⣐⣀⣠⡀⠁⢨⡇⠈⠉⠁⠀⠀⡠⢡⠁
⠀⠀⠀⠀⠀⠀⠑⠤⣉⢆⠑⢤⠁⢠⠀⣴⠀⠀⠀⠀⠀⢠⢪⠃⡀⠀⠀⠀⠀⠀⠈⢉⡉⢣⠀⠀⠁⡰⢃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣧⠤⠤⡷⠯⠤⠤⠤⠤⠤⠤⠵⠶⠮⠄⠤⠀⠤⠄⢴⠬⣃⣸⡤⠤⠤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⢠⡷⣀⣰⠖⠂⠶⢄⣰⠀⠀⠀⣣⠅⠒⠒⠦⣀⡀⡀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡌⠁⠀⠰⠎⢻⣿⠗⠻⠖⠀⢰⠿⣿⠿⠉⠟⠉⠙⡗⠀⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢃⢰⢃⣴⡀⠀⢀⠐⠞⠀⠈⠁⠀⠘⡄⠈⠀⢀⠀⠀⢸⠈⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠯⠝⡞⠉⠀⠘⡶⠥⠤⢤⠔⠺⠀⠀⠨⡗⢠⣀⠠⢥⢾⠉⣄⠀⢀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡲⣞⠀⡴⡠⢃⠀⠀⠘⡼⠿⠣⡴⠞⠳⢶⠁⢀⠀⡸⣠⢋⠢⡀⢻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢸⡄⠙⡐⢀⢷⠀⣦⡤⠋⠀⠀⣀⣷⡀⠀⠀⠱⣅⡔⣡⠃⠈⡄⢹⢓⡒⣣
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠈⠇⡌⢀⠖⠉⠱⠒⢶⣬⣤⣤⣬⣭⣥⠶⡉⠑⡄⠑⡄⠀⠌⠥⡞⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠘⢰⠀⠀⠆⠀⠈⠀⢠⠊⠁⡄⠹⡐⢄⠑⢄⠈⠆⠈⡄⢸⡜⢀⠁⠰⡀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢁⢀⠘⡄⡄⠀⠀⠀⠀⠃⡄⠈⠀⠀⠐⡀⠀⢢⡆⢐⠀⡆⢸⠓⠘⡀⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⡸⠿⠀⠘⠃⡘⠄⠀⢰⡎⠀⠂⠀⠀⠀⠁⠀⠀⡇⡸⠐⢣⠛⠆⠀⠱⠀⢇⠧⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠁⠀⠀⠀⠀⠁⠀⠾⡼⡆⠀⠀⠀⠀⠀⠀⠀⢠⣧⠛⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠁⠀⠀⠀⢀⠔⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄⢀⡜⠁

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠈⢳⡂⢄
⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡙⡗⠕⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⢠⢃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠘⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⡀
⠀⠀⠀⢸⠘⡄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠀⠁⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢈⡇
⠀⠀⠀⢸⣄⠘⠄⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠔⠁⡢⡆
⠀⠀⠀⠀⠋⠀⠀⠆⢀⠀⠀⠀⡰⠀⠤⠄⠀⠀⠀⣐⣒⣀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⢀⠳⡀⠐⠃⡨⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠓⠰⠿⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⣼⠁⣀⣢⣪⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡺⠀⢠⠀⢀⢐⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠁⢸⡇⠀⠀⣉⠉⣉⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⠈⠉⠑⠒⢫⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⡜⠑⠒⢉⠤⣤⣭⡁⢤⠀⠀⠀⣸⢥⣤⡤⢬⡵⠦⣰⡀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⣰⠀⡀⠀⠈⠂⠙⠟⠁⠈⠁⠀⠀⠉⠻⠟⠂⠁⠈⠀⡘⠀⠑⠄⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢬⣀⣇⠴⡗⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⠀⣀⠇⠈⢄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠁⢈⠁⠀⠀⠐⠀⢀⠀⠀⠀⡂⠐⠀⠀⠀⠈⡁⠸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⢁⠀⠀⠀⠀⡀⠒⠋⠑⠒⠉⠀⠐⠀⠀⠀⠀⢁⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢢⠀⢠⠀⡀⢀⠠⡶⠎⠁⠀⢀⡠⠛⢤⣀⠀⠀⢦⣄⠠⢄⠀⠀⡀⣎⠠⢬⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⢀⠀⡸⠋⠠⠙⠙⢙⡤⠤⣤⠤⠤⡤⠽⢅⠈⠢⠀⠣⡀⠀⠉⠖⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⡆⠀⠀⠀⠀⠁⠀⡠⠁⢀⠀⠙⡑⠂⠈⠂⠀⠁⠀⢁⠀⢁⠀⢰⠀⢠⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁⠘⡀⠁⢰⠀⠀⠀⠀⠀⢁⠔⠈⠀⠀⠘⠄⠀⠢⣅⠀⠀⢸⠀⠌⠓⢸⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠇⠀⠛⠁⠀⡀⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀⠀⠛⡜⠲⠀⠀⠁⢀⢆⠧⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⢞⡼⡀⠀⡀⠀⠀⠀⠀⠀⢀⣌⠘⠃⠀⠀⠀⠀⠀⠘⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⡠⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢷⣤⣀⠠⣤⣴⣿⣿⢸⠆⠰⡇⣿⣿⣦⣤⠄⣀⣤⡾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⢸⣿⣷⢸⣿⣿⣿⢠⡾⢷⡄⣿⣿⣿⡇⣾⣿⡇⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣾⣿⡿⣸⣿⣿⡟⢺⡵⢮⡗⢻⣿⣿⣇⢿⣿⣷⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣴⣿⣿⡿⡇⢈⢻⡟⡁⢸⢿⣿⣿⣦⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⡛⠿⠋⣼⠃⠸⠇⡸⠇⠘⣧⠙⠿⢛⣛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣹⣿⣶⣶⣤⣄⣈⣁⣠⣤⣶⣶⣿⣏⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠿⠿⠿⣿⣿⠿⠿⠿⠛⠛⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡟⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢻⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠛⠶⠿⠛⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⡀⠀⡠⣰⣿⣶⣤⣤⣶⣿⣆⢄⠀⢀⡾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⢰⡇⣿⣿⡉⣥⣬⢉⣿⣿⢸⡆⢸⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠉⢿⣿⣿⡿⠉⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣾⣷⣧⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢾⣿⣿⣿⠟⠛⠻⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣛⣿⣾⣿⣿⣿⣿⣿⣷⣀⣄⡘⣿⣽⣤⣤⣾⣿⣿⣿⣿⣿⣿⣷⢿⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⣿⣿⢿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣯⣾⣿⣿⣿⡿⣻⣽⣿⣿⣯⣯⣹⣿⣿⣟⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣬⣙⡿⣿⢿⣿⣿⣯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣿⣿⣿⣿⢟⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡿⣷⣿⣷⣿⣿⣿⣿⡇⠛⠛⡈⠻⣟⢷⣿⣿⣯⣳⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣽⣿⣿⣟⡟⣱⣿⣟⣿⣿⣿⣿⣿⣿⣿⣌⣹⣿⡿⣿⣿⣿⣿⠿⣯⣹⣼⣿⣿⣿⡇⠀⢾⠁⣀⣬⡳⣟⣿⣿⣿⣽⣄
⠀⠀⠀⠀⠀⢀⣾⣾⣿⣽⣻⢯⣾⣿⣿⣿⣇⡈⣻⣿⣿⡟⢿⣟⠿⣿⣿⣿⣿⣿⣿⣿⠛⣻⠟⣼⣿⣿⡇⠺⢫⣾⢿⣿⡿⣜⢾⢿⣿⣿⣾⣆
⠀⠀⠀⠀⠀⡾⢺⡿⣿⣷⢋⣬⣽⣿⣿⣿⣿⢿⣿⣿⣿⣿⣾⣉⣉⣹⣿⣿⣇⣿⣿⣯⢛⣿⣾⣿⣿⣿⣿⢐⡛⠉⠋⣨⠾⡉⠈⢿⣿⣽⣷⣿⡆
⠀⠀⠀⠀⣸⣽⣿⣹⣿⢧⣬⡛⠿⣹⠛⠊⣩⣿⣿⣿⡿⠿⣷⣎⣙⣿⣿⣽⣿⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠐⣠⠴⣾⡜⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢀⣿⣞⠗⣿⡟⣾⣿⡿⠆⠂⠠⢀⣿⣿⣿⣿⣦⠍⠹⠍⠻⡿⣿⣿⣟⢿⣿⢿⣿⣭⣍⣈⢿⡿⣿⣼⣿⣿⣣⠧⠀⠀⠀⣀⢸⣿⣿⣿⣿⣇
⠀⠀⠀⢸⡿⣽⣺⣿⠃⠉⠉⢀⡀⢤⣴⣤⣽⣿⣿⣿⡿⡀⠀⠀⡜⠀⡎⣿⣿⣯⣿⣧⣿⣮⣿⣿⣷⣇⣿⣏⣈⣁⡁⠀⣀⣶⡾⠛⠇⣿⣿⣿⣿⣿
⠀⠀⠀⢺⣷⡽⣿⣿⠠⠀⣠⠄⠂⠴⢿⣿⣿⢿⣿⣿⡟⠀⠀⠀⠀⠀⠇⣇⣿⣿⣿⣿⠟⠛⠛⣿⠃⢨⣿⣯⢰⣿⣿⣿⣩⠋⣀⣠⣄⢿⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣽⣿⣿⠘⠚⠀⠀⠀⢀⣠⡌⠙⣼⢿⣿⠄⠀⠀⠀⠀⠀⣀⣿⠋⠁⠀⠐⢤⣶⡊⢀⣴⣿⣿⣿⢈⣿⣿⣿⣿⣿⢿⣿⣿⢸⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣻⣿⢰⣿⣷⣶⣿⣿⣿⣿⣶⣤⣭⠟⣰⠀⠀⠀⠀⠚⠀⣿⢀⢀⣄⣀⣼⣿⣿⣿⣿⣿⣿⣿⣨⣩⣍⣹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⡿⢿⡏⢠⣯⠔⠀⣞⡀⣠⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣽⣿⡿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡸⣁⣌⣠⣶⣦⡼⢿⣷⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠘⣏⣿⣿⣿⣷⢿⣿⣿⣿⣿⢿⣿⣽⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣷⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⣸⣻⣿⣿⣯⡟
⠀⠀⠀⠀⠀⠹⣿⣿⣟⢾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣵⣷⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣾⣿⣿⣷⡳⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣽⣿⣿⣟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣯⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣛⣛⣿⢟⡵⣿⣟⣿⣿⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣽⢿⣿⣿⢿⣳⣝⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⣻⣵⣿⣿⣿⣿⢿⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣟⢿⣿⣿⣿⣽⣻⢽⣻⡿⣿⣿⣿⣿⣿⢿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣟⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣯⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣯⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣟⣿⣿⣿⣽⣿⣿⣟⣿⣟⣿⣿⣿⣿⣯⣿⣿⣽⣟⣻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⣴⠚⠛⠳⣦⣄⡀⠀⠀⠀⠀⠘⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡔⠊⠁⠀⠀⠀⣠⣾⣿⡧⡠⢤⣆⠹⡌⠻⣦⡀⠀⠀⠀⠀⠈⢰⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣄⠀⠀⠀⢀⣼⢻⡟⢸⡇⣃⡸⠟⠀⣹⣀⣈⣿⣦⣤⣤⣤⣤⣾⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡘⠿⠿⣿⡿⡿⠋⠉⠉⠛⠓⠦⠬⠞⠋⠁⠀⠀⠈⠃⢿⠛⠛⠛⢉⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⣼⣴⣞⠛⠛⠟⠻⡶⣤⢴⡾⠿⠉⠛⠛⠛⣾⣿⣛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣻⣾⠉⢙⠗⣶⣶⢤⠉⠩⡥⣴⣶⣟⢛⡉⢸⣿⢮⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⢞⣽⠏⠈⠉⠙⠙⠛⡓⢀⠀⠐⢥⡩⢍⠉⠁⠀⠻⣦⢳⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣫⢿⣀⡷⢶⣚⣻⣿⣁⡀⠀⣤⣀⣿⣿⣋⣭⣽⣒⢿⣧⡻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡯⠘⠉⢀⣾⢯⢞⣽⣿⣿⣿⣷⠿⠒⠛⠛⢿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠇⠀⠀⣾⢱⢫⣿⣟⣭⣴⣶⡶⠒⠙⠋⠫⢓⣶⣶⣶⣮⣽⣿⢻⡏⠑⣼⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⣀⣀⡟⣿⣿⣿⣿⡏⡽⠋⢉⡛⡓⢒⢚⠩⣼⡻⢿⣿⣿⣿⢸⣿⡤⢘⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠞⠻⣧⣿⣿⣿⣷⡇⣿⣱⣷⣿⣿⣷⣿⡙⣶⣿⣿⣿⣼⠀⠀⠈⠻⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⢻⣿⣿⡏⣿⣿⣿⡜⣿⣿⡇⣿⣿⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣸⣿⣷⣿⣿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⣿⢯⡿⢫⣿⡽⠻⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣽⡯⣋⡔⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀
⠀⠀⠀⢀⡠⢀⣠⡾⠁⠀⠀⠀⠀⠀⣠⡶⡿⠁⠀⠀⠉⢿⠷⣦⣀⠀⠀⠀⠀⠀⠀⠈⠀⢙⠳⣦⣀
⠀⢀⡴⢋⡤⠚⠁⠀⠀⠀⠀⠀⣠⣾⣿⢣⡧⣪⠵⣮⡑⡌⢷⡈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠑⣾⡟⣧
⠀⣿⠀⣾⡀⠀⠀⠀⠀⠀⢀⣼⢿⣿⠋⢸⣇⢇⠈⢿⡿⡀⠘⣧⠀⠈⢿⣧⡀⡀⠀⢀⣀⣀⣴⣿⡷⢸
⠀⣿⠈⣻⣿⣶⣶⣤⣤⣤⣾⠿⠶⠶⢶⣾⣬⣓⠭⠭⣪⡴⠖⠛⠛⠛⠛⢿⡛⣿⣿⡿⣿⣿⢿⠋⢀⡾
⠀⠘⢷⣄⠉⠛⠙⠿⠋⡟⠏⣀⣂⣀⣐⣀⡨⠉⠛⢚⠁⢐⣀⣠⣥⣤⣁⣀⣣⢹⣄⣁⣀⣈⣤⡾⠛
⠀⠀⠀⠉⠛⠓⠶⠶⢾⣴⣿⡍⠛⠋⠙⠏⠻⡿⢶⡦⢞⡿⠻⠇⠀⠋⠉⠋⢩⢿⣿⣯⡉⠁
⠀⠀⠀⠀⠀⠀⣠⡶⢻⣾⡏⠐⠓⣺⠶⣦⣦⣀⡈⡟⢋⣇⡤⣶⣶⡷⢟⡛⠋⠉⣿⣷⡙⢦⡀
⠀⠀⠀⠀⠀⡰⢋⣴⢿⣿⠃⠒⠾⠷⠟⢿⣟⢦⢉⠀⠀⠉⠼⠱⠿⠽⠖⠛⠖⠂⢿⣷⡙⣎⢳⡄
⠀⠀⠀⠀⢰⣣⣟⣵⣿⡏⢀⡀⢀⣀⣒⣚⣁⡔⠀⠸⠀⠀⠘⠲⣍⣭⣥⣤⣤⣀⡄⢻⣷⡘⣧⣿⡄
⠀⠀⠀⣠⣿⣿⣛⡵⢻⣧⣴⠿⢟⣛⣋⣛⣿⣇⣀⠀⠀⡴⢀⣴⣿⣿⣿⣒⣒⡲⢭⡛⢿⣿⣜⢿⣷
⢀⣴⣾⣏⠼⠋⠁⠀⣼⡼⣡⢞⣭⣿⣿⣿⣿⣶⣶⡷⠶⠒⠻⠿⢿⣷⣾⣿⣿⣿⣷⣿⣦⢻⣿⣿⣿⣤⡀
⣾⣼⢁⡇⠀⠀⠀⣾⠟⣼⢣⣿⣿⠟⢋⣉⣉⣉⣡⠴⠷⠶⠾⠶⢬⣉⣉⣉⣉⣙⠻⣿⡟⡞⡏⠻⣏⢻⣷⣤⡀
⠿⣿⣾⠀⠀⠀⠀⣿⢸⢧⣿⣿⣷⣾⡿⣿⡿⠯⣤⣄⣐⡒⣚⣫⠵⠛⣿⣿⣿⣿⣿⣿⣧⢹⣿⡀⠈⢻⠏⣿⣿
⠀⠿⠃⠠⣦⣤⣼⡇⣿⣾⢿⣿⣿⡿⣰⡟⡁⢀⣤⢉⡍⢉⡉⢠⠘⣷⣝⡻⠿⣿⣿⡏⣿⢸⡿⠷⠆⢸⣰⣿⣿
⠀⠀⠀⠀⠈⠛⠿⠞⠻⣿⣸⣿⣿⢳⣿⣾⡇⣾⣟⢸⣷⣼⢿⣾⣧⣿⡝⢿⣿⣿⣿⡇⡿⣾⠇⠀⠀⠈⠻⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⢸⣿⣿⣷⣿⢻⣾⣿⣿⡎⢿⣿⣿⣿⢸⣿⣿⣿⣷⡇⣾⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣾⣿⣿⣿⣿⣼⣿⣿⣿⣿⡀⣿⣿⣿⢸⣿⣿⣽⣿⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠏⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣦⡏⣿⢿⣿⡏⣽⠟⣩⣿⣧⠏⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣟⣼⢏⠞⢁⡴⢻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣿⡿⣣⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣡⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⢠⠁⠨⠀⠀⠀⠀⠀⠀⢀⣤⡂⠉⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢸
⠀⠀⠀⠀⠀⡎⠀⢰⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠉⠑⢄⠀⠀⠀⠀⠀⠀⢠⠊⠀⡆
⠀⠀⠀⠀⠀⡇⠀⢀⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⡔⠁⠀⠠⠁
⠀⠀⠀⠀⠀⣆⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡠⠊⠀⠀⠀⡌
⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⢀⠌
⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⢀⠎
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⡰⡃
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠇
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢰
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣤⣤⠴⠊⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡟⠉⠀⠉⡇⠉⠁⠐⠒⠂⠠⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⣳⠆⢀⠒⠓⠂⠠⠤⢀⡀⠀⠀⠀⠀⠉⠉⠐⠒⠠⠄⣀⡀⠀⠀⠀⠀⠃
⠀⢹⠀⢸⡆⠀⠙⢒⡛⣳⣤⣍⣑⠂⠦⠤⣀⡀⠀⠀⠀⠀⠀⠉⠐⠒⢺
⠀⣾⠀⠸⢿⠀⠀⠘⢷⣓⠋⠈⠉⠀⠁⠀⠀⠈⠉⠐⢦⠠⠄⢀⣀⡀⣸
⠀⡿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⡇
⠀⡇⠀⠈⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⢆⠈⢢⡀
⠀⡃⠀⡦⣠⣖⡐⠖⠳⡀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⢆⠀⢡
⠀⡏⠉⠳⠁⣀⣀⣉⠑⠪⡂⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡀⠀⠈⣤⠎
⢠⠁⠀⠘⢮⢯⠉⠡⣏⠑⠇⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠱⡀⠀⠈⢄
⡈⠀⠀⠀⡰⢿⣠⠔⠁⠀⠀⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⢆⠀⠈⠢⡀
⡇⠀⠀⢸⠀⠘⠃⠒⠒⠁⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠑⢄⠀⠈⠒⢤⡀
⢃⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠈⠒⢄
⠈⠑⠤⠌⢇⣀⣠⡤⠤⠤⠒⠉⠀⠘⠦⣄⡀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠑⢄
⠀⠀⠀⠀⠀⣠⠏⠀⠐⠤⠧⡀⠠⠤⠤⠤⠀⠠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡱
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⣌⡀⠀⠀⣀⡠⠤⠔⠒⠈⠉⠉⠐⠒⠤⠠⠤⡤⠤⠠⠄⠠⠤⠀⠀⠒⠒⠒⠒⠉

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⣾⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡇
⠀⠀⢸⣷⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣾⡇
⠀⠀⠀⠻⣷⣄⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣩⡞⢉⡟
⠀⠀⠀⠀⢸⣟⢳⣄⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡾⢁⣴⡿⠁
⠀⠀⠀⠀⠀⠹⣿⡿⣷⣄⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡎⢱⠌⣩⠖⢋⣿⠇
⠀⠀⠀⠀⠀⠀⠘⢷⡙⣦⠰⣄⡈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣩⣿⣦⣜⣠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠹⠞⢿⡆⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⢀⠘⢿⠟⣑⡊⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣷⣴⣆⠹⣦⡀⠄⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⢦⡄⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢠⣾⣖⣦⠔⠋⢥⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣍⣘⠟⢷⣭⣧⠀⠀⠘⣧⠀⠀⠀⠀⠀⢀⣠⣽⣿⠉⠹⢿⣄⡀⠀⠀⠀⢠⡟⠀⠀⠸⣯⣉⣭⠴⠂⠃⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣽⣷⣌⢤⣭⡀⢷⠀⢸⡆⠀⠀⡀⢶⣿⣿⣿⡟⠀⠀⠈⢿⣧⠒⢄⠀⡜⠀⠀⠀⡞⢹⣿⣄⣀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠍⠙⢢⡈⠙⠓⠀⠀⢱⡄⠊⠰⠷⠏⢹⣿⡃⠀⠀⠀⠀⠈⢳⠆⠐⣴⡀⠃⠐⡇⠾⠻⣯⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠆⣻⠂⠙⢻⣄⠀⣠⠞⠁⡔⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠘⢳⣶⣿⣧⣤⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⠿⡟⠰⠎⣿⣾⠋⠀⠀⠈⣠⣤⣬⣝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⣷⠀⠀⢛⣽⠟⠋⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⣿⣐⢷⣷⣼⣟⣻⣦⣠⣶⣿⣿⣿⣿⣿⣄⣐⣂⣒⣀⣀⣀⣀⣀⣀⠀⣠⡾⠀⣤⣽⣿⠳⣤⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢧⣸⣋⠻⢿⣿⠛⣛⣻⣭⣽⣧⢷⣾⣿⡅⠠⠤⠌⠉⠉⢉⣍⠉⠛⢛⡛⠐⠋⢻⡙⠷⣄⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⣾⣏⣿⣶⣾⡗⢸⣿⣿⣏⣁⣡⣤⣶⣏⣀⣀⣀⣀⣀⣀⣀⠁⠀⠀⠀⠙⠀⠰⣌⢳⣄⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣽⣷⣿⣿⡿⢻⡟⠛⢋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠿⢿⣷⣶⣤⣤⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠋⣿⠀⣋⣁⣹⣿⣴⣶⣄⠸⠀⠐⣠⣾⣿⣿⠣⠤⢈⣻⡏⣭⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⢀⣿⣿⣿⣷⡼⣿⡛⢿⣇⠀⠚⢠⣿⣿⣿⢿⣶⣾⣿⣹⡇⢸⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⢸⣿⡍⠲⠿⠷⠎⡁⠀⠁⠀⠀⠈⠙⢦⣀⠺⠷⠞⠙⠉⢻⡀⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣸⡇⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢠⣿⣷⡶⠀⠀⠀⠀⣤⠀⠁⠀⠀⠘⢶⡀⠱⠄⠀⢠⣾⣿⠿⣧⢠⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⣿⣿⠇⣸⣿⣿⡇⣤⠀⢀⣾⡿⠒⠻⣶⣾⣷⠾⣷⡀⠀⠀⠘⡏⢁⣼⣿⠘⣀⠠⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⠷⢀⣿⣿⣿⣷⣿⡆⠉⠁⠀⠀⠴⠆⢨⡀⠀⠈⢁⣤⣀⣼⠃⣼⡿⠿⠋⣩⠶⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠘⣷⣌⡉⠛⠛⠛⠛⢀⣠⡶⣿⠳⢶⣶⣾⣿⡄⠠⣤⣈⡁⠀⢉⣁⣠⡶⠋⣙⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⢺⡁⣼⡿⣿⣿⡿⠛⠋⢩⡗⠀⠀⣀⠙⠉⠉⠀⠀⠺⣿⣿⣿⣿⣿⡿⠁⠐⢿⣦⡀⢳
⠀⠀⣀⣀⣤⣀⣦⣴⣶⣾⣿⣿⣿⣿⣿⠟⠉⢁⠂⠉⠀⣁⣤⠑⠀⠁⣀⠃⠀⠀⠀⠀⠀⠀⢻⡍⠛⠉⠉⠁⠀⠀⠀⠁⠻⣾⡖⢀⠠⣀⠀⣤⣤
⠀⠞⠛⡿⠁⣿⠈⢻⡍⣿⣿⣿⢿⣿⡇⠀⣇⠋⠀⠁⠶⠉⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣀⠀⢘⡆⠀⠀⠀⠈⣧⠈⢠⣩⠀⠄⣴⠛⠃⠀⡀⣀⢠
⠀⠲⠿⣁⠒⠶⣂⡀⠓⣤⣙⡿⢿⣶⡃⠀⢻⠂⠀⣴⠃⠀⢠⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠄⠀⠉⠀⠨⡅⠀⠀⠀⢰⡀⢨⠐⢇⠀⣄⠻⠄⠀⠀⠙⢩⡿
⢀⡀⢀⣩⠷⣄⠙⠛⢀⠝⠋⣵⡶⠾⢇⠀⢸⠆⠁⠙⢀⠀⢰⠶⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⢽⢃⡶⢀⣯⢛⣭⠁
⠈⠁⠬⠛⢠⡈⢓⣶⣤⠘⢷⣌⠻⢶⡞⢧⡘⠀⠀⠀⠛⠂⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⣄⠀⠀⠀⠀⠀⠁⠀⢀⠃⣾⡯⢳⢮⣵⠋
⠀⠐⠤⠂⠘⠂⠂⠀⡂⣠⣸⣿⢷⣾⣿⠄⠁⠀⠀⠀⡀⠀⠀⠀⠠⠂⠀⣀⠀⠀⠀⠀⣾⡗⣾⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡧⠄⠈⢀⣅⡠⢈⡧
⠀⠀⠒⠚⠁⡴⠐⠀⠙⠉⠛⠛⡬⢿⣷⡄⠀⠀⠀⠀⡇⠀⠀⠀⠐⡄⠀⠈⠁⠀⠀⠈⢁⣻⣯⠂⠀⠆⠀⠀⠀⠀⠀⠨⡀⠀⠀⢀⡀⠩⢓⡾⠂
⠀⠀⠀⠀⠠⠄⠖⢈⠶⢘⡆⢸⡁⣾⢻⠀⠀⠀⠃⠀⡇⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢸⣿⠿⠀⣀⠀⠀⢀⡄⠀⠀⠀⠁⠀⣖⠋⣧⠀⠈
⠀⠀⠀⠀⣄⣀⠄⠠⠂⠠⡇⠈⢛⠟⣸⡆⠀⠀⢸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠂⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠠⠟⠁
⠀⠀⠀⠀⠈⠁⠀⠀⣶⠀⡘⠂⠘⠃⣹⣧⡀⠀⠀⠃⡈⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠈⣇⠀⠀⠀⣿⣄⡏⣀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠘⣇⡀⠙⣿⣿⣦⡀⠀⠁⠀⠢⠈⠢⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⣷⢸⣿⢿⡍⠀⣘⠃⠲⣀⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡈⠻⢿⣷⣦⣤⡀⢀⣀⡀⠀⢀⠀⠀⠀⠀⡀⣀⠀⠀⠀⢸⣷⣿⢺⣷⠀⠈⢉⡒⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⠬⢉⣙⠛⠦⣄⠀⣀⠀⠳⠀⠀⢠⣷⠙⠀⠀⠀⠀⣿⡟⢺⡇⠀⠘⠰⢍⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⣌⠁⠈⠻⣆⠀⠀⣸⣝⢦⠋⠀⠀⠀⠀⠀⣼⣁⣜⣳⣤⠴⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠰⣿⠁⢠⣿⡿⠀⠀⠀⠀⠀⡀⠐⣿⣿⢛⡣⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⣯⡿⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⡿⠁⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⠆⣿⣿⣿⣿⢰⣶⣦⣀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣦⣄⡠⣠⣜⢿⣿⣿⠀⡿⢁⡈⢿⠀⣿⣿⡿⣣⢄⢄⣠⣴⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⣿⣿⣿⣿⣿⣷⢹⣿⣧⣿⣿⠀⣰⡟⢻⣆⠀⣿⣿⣼⣿⡏⣾⣿⣿⣿⣿⣿⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣍⡉⠛⠿⠿⢸⣿⡟⣿⣿⠀⣿⡀⢀⣿⠀⣿⣿⢻⣿⡇⠿⠟⠛⢉⣩⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣓⣂⣴⣿⡿⣽⣿⣿⢠⡘⢿⡾⢃⡄⣿⣿⣯⢿⣿⣦⣐⣚⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠽⣟⣯⣾⣿⣿⡏⢸⣷⣌⣡⣾⡇⢻⣿⣿⣷⣽⣻⡯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣮⣭⣙⡛⠿⠿⠁⣼⣿⡿⢿⣿⣧⠈⠿⠟⢛⣋⣭⣵⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣸⣿⣿⣿⣷⣶⣿⣿⣄⣠⣿⣿⣶⣾⣿⣿⣿⣇⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠙⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠀⠉⠙⠿⠿⠋⠉⠀⠀⠀⠀⢀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⣠⣤⣤⣤⣤⣤⣤⣄⠀⠀⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣍⠉⣤⣤⠉⣩⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⢻⣿⣿⣿⣿⡿⠿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢹⣿⣿⡏⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠠⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢣⠑⣄⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠂⠐⠳⠀⠀⠀⠀⠀⡰⢃⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠢⣱⢴⡈⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠂⢠⠀⠐⠂⠈⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢌⠣⡑⢄⠈⠠⠀⡄⠀⠀⠀⠀⠐⡨⡁⠀⠀⠀⠉⠉⢚⠙⠃⠀⠀⡠⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⠤⢸⡞⠀⠀⠀⠀⠀⠄⠒⠢⠀⠄⠀⠀⠠⠬⠥⡦⠠⠆⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⢸⠣⠔⠚⣀⣓⣠⡇⠀⠀⣰⣀⣀⢓⡠⣄⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⠑⠘⡿⠁⠏⠀⢸⠑⠿⠃⠀⠀⠈⠃⠀⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠢⡼⠔⠓⢤⣄⡀⢀⡠⡄⠀⠀⡇⡀⠀⠀⣠⠚⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡠⡵⠀⣄⠘⠀⠀⠨⣰⠧⣀⠴⠶⡝⠉⠀⢸⠀⠓⢄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠃⢑⡀⠸⡂⣴⠤⠃⠀⢀⡴⣄⠀⠈⢦⣠⢡⠃⠀⠈⡟⣀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠔⠉⠜⠛⠻⠥⢤⡤⠤⡝⠃⡉⠀⠑⠀⠀⠡⢞⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠈⠀⡀⢀⠔⠈⠀⠀⠈⠀⠀⠀⠀⢰⡄⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⣴⠂⢠⠄⡄⠀⢠⡐⠁⠀⠀⠀⠀⠀⠱⠀⠀⡄⢼⠁⠰⠀⡄⡄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠌⠈⠀⠀⠈⠀⢠⣼⡁⠀⠀⠀⠀⠀⠀⣠⠰⠂⠈⠀⠀⠀⠰⠙⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢀⠔⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⠀⣴⡇⠀⠀⠄⠀⠰⠚⡇⠀⢾⠇⠀⡞⠓⠄⠀⢠⠀⠀⣿⡌⢰⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠿⣠⠻⢣⠀⠀⠀⠐⠛⢧⠑⠀⠙⠃⠀⢃⡟⠓⠂⠀⠀⢀⠻⢇⡘⠇⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⡟⣤⣙⣡⣹⠀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⢣⣜⣡⡜⡇⠸⣶
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣧⡱⠄⣧⠙⣥⠝⢹⠀⠀⡀⠒⠶⠤⣤⡄⠀⠀⣤⡤⠴⠶⠂⣰⠀⢰⠜⢡⡌⠣⡇⢸⢉⠖⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡼⣿⡀⡇⡿⢈⢾⢆⠀⠀⣿⡦⣀⠠⠀⠁⠀⠀⠁⢠⠄⣀⢼⣿⠀⢀⢺⢇⡰⡇⡇⢸⡿⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣇⡌⡃⡟⣄⠻⢣⣸⠀⠐⠋⠁⠈⢻⡀⠀⠀⠀⠀⢰⠋⠁⠈⡻⠀⢸⢃⡸⢇⡜⡇⢘⣤⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣇⢇⣧⠛⣥⡙⣹⠀⢸⡇⠀⠀⠈⠁⠀⣰⡄⠀⠈⠀⠀⠀⣺⠂⢸⡜⢣⡜⢣⣇⠘⣿⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢬⢨⡙⢆⢋⣴⠈⠀⡮⠁⠐⠁⠠⠀⠀⠉⠉⣆⠠⠀⠈⠂⠈⣶⠈⠰⣎⢃⡷⣩⢨⠝⠁⢺
⠀⠀⠀⢀⠀⠀⡀⡆⢸⡇⠰⢡⠗⣬⡈⣙⢕⠁⠀⢡⠀⡀⠀⣀⠃⠀⠀⠀⠀⠘⡀⠀⡀⢠⡉⠀⠑⠜⣋⢠⣕⢷⡑⠀⢾⠀⡆⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠇⢸⡇⡰⠨⠾⠸⢳⠝⣷⠁⠀⢻⡇⠿⠦⣿⣦⡀⠀⠀⣠⣾⡧⠾⠇⢸⠓⠀⢲⡞⢵⡻⠸⠦⠑⢆⢸⠀⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡸⠟⡛⠇⣿⢀⢿⡀⠠⡦⡁⠀⠀⠈⠅⠈⠀⠐⠁⠸⠀⠀⠀⡀⣶⠀⣸⠇⠀⡇⠿⠛⠿⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢡⡌⢫⣤⢻⢸⣤⢃⠀⡰⣇⠀⠁⠄⠠⠈⠐⠀⠀⡀⠆⠈⠀⡴⣈⠀⣇⡜⠀⢏⣤⠛⣤⡙⢸
⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⡌⢡⣦⠋⣾⢸⡛⡆⠀⢻⣦⠀⠀⠠⡁⠀⢆⡀⠀⡄⠀⢀⢠⣾⠏⠀⡿⢣⠀⣧⠫⣶⠉⢴⠘⠀⡁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡧⡱⡷⣿⢸⣿⠀⠀⠂⠛⠀⠀⢃⠄⠀⠈⠓⠀⢃⠃⢀⠸⠋⠐⠀⢰⣿⠀⡇⡷⡀⢾⢂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢃⡘⠟⣔⢿⢸⢿⠀⠀⠀⢀⣈⠱⠈⠀⠠⠀⠀⠀⠈⠱⢈⣀⠀⠄⠀⠸⣿⠀⠟⣄⠿⣠⠻⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠸⢸⣥⢻⠀⠂⠘⠃⠀⠆⠀⠀⠈⠀⠀⠀⠶⠀⠐⠀⠀⠀⢃⣼⠀⢸⣿⠀⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢠⢸⠟⣼⠀⠅⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠈⠀⣌⢻⠀⢸⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢸⣾⠀⠀⠀⠀⠈⠢⣸⣄⠀⠀⠀⠀⣸⡠⠎⠀⠀⠀⠀⢸⣿⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡔⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢈⣤⣤⣤⣀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠂⠀⠂⠀⠀⠀⠤
⠀⠀⠀⠀⠀⣶⡀⠀⠹⢷⣿⡿⣿⠻⣿⣾⠟⠁⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢺⡶⠚⠋⠉⠙⠓⢶⡗
⠀⠀⠀⠀⠀⠀⠙⢶⠖⢺⣿⣾⣿⣷⣾⣿⠚⢴⠛⠁⠀⠀⠀⠀⠀⠀⠑⠀⢀⠛⠀⠀⢀⣀⡀⠀⠀⠛⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⣼⡍⠲⣿⡷⠊⣿⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠿⠽⢿⡇⠀⠀⢾⣶⡗⠀⠀⢸⡿⠯⢿
⠀⠀⠀⠀⠀⠀⠀⠙⢰⣷⣹⣮⣉⣹⣾⢱⣷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡏⣷⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠶⠦⣤⠤⠖⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠹⣿⡟⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⢀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⢿⣿⡆⠀⢈⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣀⢠⣴⣦⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢶⣿⡶⢿⡿⠀⠘⡇⠀⠀⠀⠶⠟⠆
⠀⠀⠀⠀⠀⠠⣿⢿⠿⣿⢿⢿⠿⣿⠷⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠙⠲⣿⠖⠊⠀⣐⠀⢿
⠀⠀⠀⠀⠀⠀⢿⣾⣶⡿⣾⡷⣶⠿⢺⢶⠟⣿⣦⣤⣄⣀⡀⠀⠀⠀⠙⠟⠀⣀⣀⣀⠀⠻⠃⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⢿⣾⣿⣾⣾⣾⣷⣿⣿⣟⡋⠉⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⢶⡀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠈⠀⠙⠻⠿⠦⠀⠀⠀⠀⡞⠀⠀⢠⠤⡄⠀⠀⢻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠘⠛⠃⠀⠀⣼⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⢀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠈⢻⣖⢦⣀
⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⢎⢳⣬⡑⢦⡀
⢀⡞⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⡆⠱⡌⠲⣝⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⣸⣇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠹⣤⡤⠿⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳
⡇⢸⡄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢻⠈⠲⢄⡈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸
⡇⠀⢧⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡇⠀⠀⠉⠒⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸
⢧⡀⠈⢳⡀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⠛⢍⡉⠉⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⢀⣇
⠸⡟⢆⠀⠹⣄⠘⣏⠓⠲⠤⣄⣀⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠀⠀⠈⠓⢤⡀⠘⡆⠀⠀⠀⢀⣀⡤⠔⡟⢁⡥⠚⡹
⠀⢳⡈⠳⣄⢻⡳⣽⣆⠀⠀⠀⢀⡿⠐⠒⢒⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣐⠛⠒⠂⠺⠤⠤⣀⣀⠀⠉⠢⣹⡏⠉⠉⠉⠁⢀⠌⠀⡌⢀⡜⢡
⠀⠀⠑⣄⠈⠙⢷⡈⠙⣆⠀⢀⡞⠛⢉⡩⠭⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣒⣒⠄⠈⠉⠉⠉⠙⠒⠒⠒⠲⠭⠽⢶⣿⣇⠀⠀⠀⡰⠃⠀⣀⠴⠋⢠⠏
⠀⠀⠀⠈⠑⠦⣄⣙⣄⡈⠳⡞⠀⠀⡏⢎⣹⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢭⠉⣴⢈⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣭⣦⠀⢹⠀⢀⠞⠀⢀⠼⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢾⣁⣀⣀⣈⣶⣶⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣢⣬⣵⣋⣀⣀⣀⣀⣀⣀⣀⣀⣘⣦⣭⠞⠀⢸⡆⢁⣀⣴⣁⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢨⠀⢀⣿⠀⠀⠀⠀⢀⠀⠈⡉⠁⠀⠉⣡⠏⠀⠀⠈⢇⢀⠇⠀⠀⢀⣀⠀⠈⢇⠉⠛⠛⠛⣏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⠀⠀⢸⣿⢧⣤⣔⣗⠛⠉⠓⠿⠳⢄⣰⡏⠀⠀⠀⠀⢸⡼⠂⠙⠋⠉⠛⠶⢤⡈⡀⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⡇⣿⠀⠉⠢⣭⡟⢻⣿⣿⣶⠦⢿⣧⠴⠒⠢⢤⣾⡛⢿⣿⣿⠋⣹⡿⡟⠛⠳⣿⢦⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠀⠀⣰⠃⢿⠀⠀⢠⠉⠫⡒⠻⠿⠯⠤⢖⣿⠀⠀⠀⢸⡏⠙⢻⡶⠶⠎⠊⠀⠹⡀⢀⡏⡆⠉⠲⢤⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⡆⢺⡟⢀⣼⣆⠀⢘⠀⠀⡠⠝⠖⠒⠈⠁⢹⠀⠀⠀⢸⣇⠐⠀⠈⠢⢄⠀⠀⠀⡇⣸⠁⠸⡄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠒⢧⣼⡥⠺⢹⠈⠳⣤⡔⠉⠀⠀⠐⡀⣠⠤⣾⠀⠀⠀⠀⡯⣄⡀⠀⠀⠀⠙⠢⣀⡷⠙⡆⠀⠘⢆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⠒⢯⠀⠃⢸⠀⠀⣿⠉⠉⠙⠉⢻⠏⣁⠔⣿⠀⠀⠀⢨⡗⢄⡹⣦⠴⠒⠒⡟⢹⡇⠀⣽⣄⠀⠈⢧⢸⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣙⠷⡞⡆⠀⣸⡆⣸⢣⠑⠄⡀⠀⠘⢧⣽⡿⠿⢦⣭⡤⠿⠿⢾⣶⠃⠀⠀⡸⠀⢸⣧⢠⠛⡌⢣⡀⠀⢻⣇⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠚⢿⡟⠒⢷⡄⣰⠁⢹⣇⣀⣳⡀⠀⢑⡴⠋⠀⠀⢀⠃⣻⡀⢄⠀⠀⠈⠳⣤⠞⠁⣠⠏⢸⡟⠀⢧⠀⠙⣶⣏⣭⣹⡆
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢻⡀⠘⣤⠃⠀⡎⢣⠀⢷⡿⠶⠋⢀⠀⠠⠔⣁⣴⢏⢷⣄⡀⠀⠀⠀⠺⢦⣴⣁⣠⣎⠀⠀⠈⡆⢰⣟⣠⡤⢤⣽⡄
⠀⠀⠀⠀⠀⠀⡎⠀⠀⣴⢸⡇⠀⣿⠀⡼⠀⠈⡷⠚⠉⢛⡶⠶⢶⢶⣛⣁⣀⣀⣀⣈⣉⣙⣛⠟⡟⠋⠙⠧⡀⠈⠳⡄⠀⢻⢸⠻⣅⣤⠚⢅⠐⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⢀⡇⠀⡏⢠⠃⠀⡞⠀⠀⣰⠋⠀⠀⢈⡹⠛⠛⠛⢿⣯⡉⠉⠻⢏⡉⠉⠳⡄⠀⠙⡄⠀⠹⡄⢸⣿⢠⡏⠘⣆⠈⡀⢸⡄
⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⢸⡇⢰⡇⢸⠀⢸⠁⠀⠀⡇⠀⠀⢠⠎⠀⠀⣰⠋⠜⢣⠉⠲⢄⠀⠱⡄⠀⠘⡄⠀⢘⡀⠀⢳⡌⡇⣾⠀⠀⠹⡄⢁⠀⣿
⠀⠀⠀⠀⠀⠀⠃⢠⠞⣇⢸⡇⡼⣇⠸⡀⢸⠀⠀⠀⡇⠀⢠⠇⠀⡰⢊⠁⠀⢩⠀⢳⡀⠀⠙⢦⡸⡄⠀⢱⠀⠀⡇⠀⣿⢳⡇⢿⠀⠀⠀⡇⢸⠀⢿
⠀⠀⠀⠀⠀⠀⣶⣿⠀⢸⡄⣷⠃⠸⣄⣷⠈⢧⠀⠀⢃⠀⣸⢠⠞⠀⡘⠀⠀⠘⡆⠀⢣⠀⠀⠀⠹⣧⠀⢸⠀⣸⠀⣰⣻⠀⠀⠘⣧⠀⠀⡇⢸⡼⢠⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼⠻⡏⠀⠀⠙⠋⢇⡜⢣⠀⠸⡀⣽⡎⠀⠀⡇⠀⠀⠀⠗⠀⠈⡄⠀⠀⠀⢸⠀⡎⢠⠛⣦⠋⠻⠀⠀⠀⠘⡆⢰⢿⣸⠧⣸⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡼⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⢣⣰⡇⣾⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⡇⠀⠀⢀⡇⡜⢷⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⠁⢧⠀⠀⡇⠀⠀⠀⢸⠀⠀⠁⠀⣠⠞⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢸⡀⠀⠀⠈⠀⠈⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢳⡀⠀⠃⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠀⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏

⠀⠀⠐⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂
⠀⠀⠀⠘⠮⠙⠛⠛⠛⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠀⠀⢉⠵⠂
⠀⠀⠀⢁⣀⠀⠀⠀⠀⠠⣄⣙⡟⢦⣀⡤⠤⠤⠴⠶⠶⠶⢦⣄⢀⣴⠒⠒⠒⠢⠤⠤⢤⣀⡄⣡⣎⣠⠄⠀⠀⠈⠉⣀⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠈⠁⢿⠉⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢶⠿⠿⠀⠀⠀⠐⠲⠇⠘⢶⡄⠀⢀⡀⠀⢀⡵⠂⠠⠖⠛⠂⠀⠘⢿⠿⡶⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠉⢿⠀⠀⠀⠋⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠙⣄⠀⢀⡿⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⢤⠂⢱⣼⣦⣤⣤⣴⣦⣄⣰⡀⠀⠀⢺⡿⠀⠀⢀⣆⣠⣴⣦⣤⣤⣴⣥⣮⣤⣿⢢⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⠾⣷⣠⠿⠟⣿⣷⣤⣀⣼⣇⣴⣤⣾⣿⠉⢻⡇⠾⠷⣖⠂⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉⠋⢩⡿⠿⠿⢿⡟⠻⠉⠹⣿⠟⠛⢻⡿⠿⠛⠻⢯⡙⠉⠁⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠖⡋⠀⠈⡏⠀⠀⠀⢀⣰⣉⣀⠀⠒⠶⢦⡀⠀⠀⠁⠀⠀⣠⡤⠤⠄⠀⣀⣀⡀⠀⠀⠂⢹⠀⠀⠙⠒
⠀⠀⠀⠀⠀⢰⢁⠀⠀⠇⠀⠀⠀⠋⣤⠞⠛⢛⠋⠈⢻⣷⣄⣀⢀⣐⣊⣁⣶⡶⠒⡛⠛⠳⢬⢻⠀⠀⠸⠀⠀⡈⡆
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢸⠁⡀⢠⣼⠇⠀⠘⣿⠿⣿⣿⣿⠿⣿⡿⠀⠰⢧⡄⢀⠀⣧⠀⡗⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡸⡖⠁⢸⣧⡁⠂⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣼⡇⠈⣆⡿⢸
⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠛⠀⢻⠀⢉⡿⣹⠟⠒⠶⠶⣿⣿⣿⣶⣶⡖⣾⡟⣿⡟⠀⣿⠃⠀⠑⡜⣦⣳⢀
⠀⠀⠀⠀⠀⠀⠙⠱⡀⠀⠀⠀⠀⠀⠀⠣⠘⣧⣾⠀⣿⡗⢱⠃⠿⠻⣎⢹⡅⠸⣷⣿⢃⡜⡿⠀⠀⠁⠛⢠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⠀⠀⠀⠳⣄⠹⣿⡀⣿⣷⡿⠷⣿⠶⢿⣾⣿⢀⣿⢿⣿⠄⠀⠀⣶⡀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡄⠀⠄⢀⠘⠳⡜⡓⢿⣏⠀⢀⠀⠀⠀⣹⡿⣾⢃⠞⡃⢠⠀⢠⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠀⠐⠀⠀⢹⡄⢃⣰⠀⢛⣠⢆⡌⢠⡏⢠⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠦⠍⢁⠀⠒⠉⣚⠻⢾⣣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠈⢹⠇⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⢀⣤⠀⠋

⠀⠀⡄
⠀⠀⠀⠀⠂⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡀⠒⠠⢔⡀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣇⡄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⢁⠠⠀⠤⠤⠤⠤⡀⠀⠄⠀⠤⢈⠀⡄⠀⢀
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⣾⣧⠃⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢸⡥⠂
⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠼⣛⡽⠆⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⢤⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⣀⡿⠛⠀⠈⠁⠀⠀⠀⣼⠶⠿⠿⣾⣧⡀⠀⢸⣦⠀⠷⠿⣏
⠀⠀⠀⠀⠀⠨⠉⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡀⠀⠙⢇⠼⣿⣿⡏⠁⠀⠀⠀⠉⠙⠂⠄⡀
⠀⠀⠀⠀⠀⠘⡁⠀⠀⡰⡏⠀⠀⠀⢀⣾⣋⠄⠀⠀⠀⢻⡉⠻⠙⠦⠀⠀⠀⠀⠀⠀⣀⠀⠑⢤⣀
⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⡷⠀⢸⠀⢸⢋⠴⢶⣦⣸⡄⠁⠀⢀⠀⠀⠉⠉⠢⣄⣠⣸⣯⣤⡈⣱⠋⡆
⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⡈⡄⣿⡇⢳⣄⣾⠉⢻⡀⠰⡀⠑⡇⠀⠀⠀⠀⠀⠈⠉⠉⢹⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢳⠘⢾⣿⠊⡦⠩⢅⣒⣟⠷⣶⣤⣹⡄⠀⠀⠀⠀⠀⠀⣠⣼⠁⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆⢀⡀⠀⠈⠈⠙⢻⠶⠛⠁⠙⣆⢰⠓⠂⠉⡛⠷⠤⠤⠔⠚⠉⣙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣡⣠⣀⣀⢹⡀⠠⠖⠂⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠣⣀⡴⠶⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠴⠒⠂⠀⠉⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠉⣠⠔⠊⣀⣀⣈⢉⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠞⣁⡴⢧⣄⠠⠶⠶⠤⠰⠶⣄⡀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡾⢁⠀⢀⣿⠷⠆⠀⠤⣭⠏⢴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠸⠋⠀⣀⣉⣫⡥⠩⣟⣓⡉⠙⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⠈⢸⠘⠀⡄⠀⢀⠐⢒⡂⣄⣓⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠸⠰⣇⠋⠰⠈⡀⠡⠍⠰⠸⢀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠁⡀⡀⠃⠘⠧⠙⢳⣤⣤⣤⣤⡾⢋⡹⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⡀⢃⡅⢰⡘⠦⠠⠀⡿⠁⠈⢻⠧⠠⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⣸⣿⣄⠉⠂⡴⠀⡇⠀⠀⠨⠗⠄⠞⢃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠻⣷⣾⠁⠀⠐⢦⡴⠂⠀⣸⣾⠛⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣾⣿⡉⢦⡀⠈⠳⢦⠀⠚⠃⢠⡴⣋⣠⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡟⢳⡄⠁⠀⠀⠈⠛⠾⠟⢩⡾⠋⢉⡡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠘⠻⣦⡀⠠⣀⣀⠀⠠⠏⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣄⡀⠀⠙⢦⡈⠙⠛⠃⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠢⡄⠁⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢰⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣈⣠⡿⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡘⢦⡀⢄⣠⡶⠒⠛⠙⢷⣚⠛⠛⠒⣋⡿⠗⠒⡲⢤⣆⢠⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢷⣄⠉⠲⢿⡦⠀⠀⠉⠛⠶⠂⠀⠀⠶⠚⠉⠀⠀⠐⣿⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡌⣻⣷⣄⠔⠀⠀⠀⠐⢦⡍⠀⠀⠀⠉⠀⢀⡀⠀⠀⠲⢄⠀⣿⣧⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣲⣿⣧⣤⣿⠁⢲⣦⠴⣿⢶⣄⠁⠀⠀⢰⠀⢸⢋⣴⡶⢦⡴⠀⣦⣙⣿⡯
⠀⠀⠀⠀⠀⠀⠀⢤⣬⢿⣿⠀⠋⢹⠀⡈⠃⠓⢿⠾⢻⡆⠀⠁⠀⠈⠱⠻⢬⠷⡺⠁⠀⠈⣤⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠶⢦⡞⣰⡷⠈⣲⠟⢤⡀⢠⠤⣄⣀⣠⣄⠠⠀⠙⢄⠰⣧⢫⡁⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⣿⠀⢹⢛⣿⢻⣶⣢⠈⠚⠲⡆⢠⢰⠄⠀⢴⣾⣿⠯⢻⡉⣿⡀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢛⡿⠀⢘⡇⠘⢸⣿⣅⣁⡀⠀⠙⣻⠃⢀⣀⣠⣝⠛⠋⠯⢴⣿⠏⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠂⠹⣦⣾⣁⣀⣼⡿⣿⠉⣙⣛⡛⢛⣛⣋⠙⣿⡏⠙⢻⣶⣶⠏⢩⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠲⢾⠉⣹⠏⡀⣻⡝⡆⡿⠿⢿⡿⠟⣿⠸⣿⣀⢴⣉⢉⡤⠒⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣠⡙⠉⣷⢻⡷⣄⣀⣁⣴⣿⢰⡇⢁⣀⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣴⡟⣾⡇⣛⣛⣛⣃⣿⢷⠷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢘⠬⣛⠭⠀⠭⢝⣫⡎⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢀⡤⠆⠀⠀⠲⠤⠹⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣁⡀⣀⣄⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⠛⠹⢧⡿⠅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣴⣭⣿⣼⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⣀⣣⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠴⢊⠀⢨⠀⠢⠄⣀⠔⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡆⣺⡀⠈⠢⣀⣀⣨⣦⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢡⡄⣏⠀⠀⠀⠀⠀⠁⠄⠑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠰⠆⠈⠉⠋⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⡀⠠⢁⢉⡖⢱⠪⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠤⠊
⠀⠀⡠⠖⡋⠹⠡⠀⠀⢈⡒⢧⡑⢤⣂⣈⡁⡅⡈⠈⠉⠀⡜
⢠⠞⠀⠀⠠⠌⣀⠀⡤⠁⠀⠀⠹⡄⠁⠀⠀⠀⠀⠀⣠⠎
⡎⠀⠐⠤⣛⠰⠄⡐⠀⠀⠀⠀⠀⡇⠠⠀⢈⡢⣤⢴⡵
⡇⠀⠀⠀⠀⠉⠰⣁⠀⠐⡀⠀⠸⡅⠀⠈⠁⠐⠗⡾⢇
⢧⠀⠁⠈⠐⠤⢼⠄⢀⠉⠉⠂⠀⠈⠀⢀⡠⢠⠰⢅⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀
⠘⣆⠀⠀⠀⠀⠐⡀⠀⠀⢉⡡⠞⠓⠠⡎⠀⠸⡐⠀⢡⣄⡓⠀⡀⠀⠀⠀⠀⠀⠀⠐⠍⡠⠼
⠀⠈⠳⠤⣤⣤⠤⠼⠶⠊⠁⠀⠀⠐⠒⠞⠀⠈⠱⡀⠈⠬⡑⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠰⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⠐⢀⠀⢠⠖⢁⣰⣖⣉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢚⡣⠂⠨⠥⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⡌⠂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⣸⠀⠈⠙⠃⠀⠐⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⣍⡆⠀⠀⠀⠠⠀⠀⠈⠀⡏⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠉⢸⠆⠀⠀⠀⢁⡦⠀⠀⢇⡏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡤⢿⡷⠐⣏⣠⣖⡒⠉⡁⠀⣠⠊⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⢠⡾⢋⠅⡼⢟⣡⣄⠄⣰⠃⡀⠀⡀⠉⠳⣄⠐⠁⠀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣮⡰⣇⠈⢧⠸⢊⣴⢋⡜⠁⣼⠃⢸⠀⠀⠈⠛⢆⠈⠢⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡲⣆⠹⣿⠇⢸⢠⢺⡎⢸⢀⢸⡎⢀⡟⠀⠈⠉⠒⠄⠀⠀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡉⠟⣸⠇⠠⠟⠈⢯⠃⣟⢸⠸⡆⠸⠁⠀⠀⢀⠀⠀⠀⠴⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠓⣀⠀⠈⣦⠀⠀⠀⠈⢧⠙⠮⠧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣤⣦⠀⠶⠆⠛⠛⠋⠀⣤⣤⡤⠀⠙⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢀⣄⠤⠠⠖⠓⠂

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠉⠉⠓⠒⠂⠀⠉⠉⣁⣀⡄⣴⠋
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠢⠴⠤⣠⣃⡼⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⢧⣙⣄⡤⠦⠔⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠⢄⠀⠙⡙⠀⠀⠈⠢⠈⢄⠠⠄⡠⠀⠖⠁⠀⠠⠋⠋⠀⣠⣴⠖⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⢀⣣⣴⣠⣄⣠⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⣄⠀⢠⡀⠘⠛⠁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⡀⠛⢛⣦⠀⠀⠹⡏⠀⠀⣴⡛⠛⠀⠀⢿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠍⠁⢺⣿⠓⠦⠴⠳⡖⠙⣿⡷⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠙⢧⠈⠀⠀⠀⣽⠋⠷⠤⡴⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡷⢶⣴⡐⠈⠀⠀⠀⢸⠀⠃⠀⠀⠠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠋⡟⢁⢈⡀⠀⠀⢘⡄⠩⣄⡐⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡶⡻⠁⡬⠥⢤⣀⣘⡡⠀⡾⢵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣱⢄⡀⠀⠀⠒⣉⢵⡱⠷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠉⢑⠚⡛⡋⠁⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠤⠴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⠀⣤⣤⠄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠮⡥⠉⢡⡽⠷⡜⠁⣤⢾⣷⡈⠟⡃⣸⡷⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⢈⣧⣶⢶⡰⣟⣽⢳⣧⢻⣿⢯⣶⣷⣾⡽⣿⡿⠈⠽⠛⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⣏⡬⢿⣿⠛⠓⠈⠀⠀⢀⣀⣠⣤⣀⣉⡉⠁⠻⣿⣃⣴⣿⣼⠏⢤⡀
⠀⠀⠀⠀⠀⠀⣼⣃⡏⢰⣿⣶⠊⠁⠀⢀⣤⢞⡟⢋⣹⠯⣌⣿⣷⣿⣂⣦⠀⠐⡍⢻⣟⠁⢃⠷⡀
⠀⠀⠀⠀⠀⢸⡩⢮⡕⠆⣻⠁⠀⣠⡾⡹⢿⣭⡽⣶⣿⣿⡶⠟⡛⠈⠀⠁⠀⠀⠐⠀⠘⣿⣯⠚⡧
⠀⠀⠀⠀⠀⢺⣶⢄⡟⢈⣿⠀⠀⣿⣟⣷⢭⣍⡁⠻⣿⣿⣷⣋⢽⣀⡀⠀⢀⡔⠀⠈⠀⢸⡷⢾⡶
⠀⠀⠀⠀⠀⢨⡹⠾⣷⡈⠿⣦⡴⣿⡋⠀⠠⣄⠴⠷⣌⠳⣤⡘⢦⣈⠙⠓⠋⠀⠀⠃⠀⠐⡷⢶⡁
⠀⠀⠀⠀⠀⠀⢻⡄⠾⣛⠛⣫⣽⡏⠂⢀⣲⡃⠀⠀⠀⠙⠚⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⣼⡯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠨⢙⠘⠛⠛⠊⠁⠀⣤⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣸⡷⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠆⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡞⣩⣹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣻⢯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⠆⠳⣿⣵⡸⠏⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣤⣴⣴⢶⣾⣾⠉⢿⣿⣣⣾⣾⣿⣿⣿⣿⣿⡶⢦⢤⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢠⡤⣦⢦⣼⠿⢛⡻⠟⣷⣀⣪⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣟⡷⢼⠯⠇⢀⣦⣄
⠀⠀⠈⢀⣤⣶⠿⠿⠳⣷⣿⠝⠓⠉⠉⢁⡴⢻⡟⣿⣿⡿⠟⠋⠉⡁⠀⠠⠠⠠⠠⠡⠉⠛⠷⣤⣶⢿⣮⣛⣉⣡⡄
⠀⢀⠶⣿⠯⣍⢿⣿⠟⠉⠀⠀⠀⠀⢠⣯⣷⣮⣛⡿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⡃⠛⠋⢩⣿⡆
⠀⣼⠲⠞⠧⠙⣼⡏⠀⠀⠀⠀⠀⠀⣾⡌⢻⣿⣿⣇⡀⠀⠀⢀⣀⣠⣤⡔⢲⣖⣶⣶⣶⡄⣀⠀⠀⠈⡿⣿⣿⡷⣿⢛
⠀⣿⠛⡿⢷⡞⢿⠃⠀⠀⠀⠀⠀⠀⢺⣵⡆⢥⡄⣿⣷⠶⢋⡽⠛⢛⣯⠒⣩⡭⣬⠉⣷⠀⣾⣗⣄⠀⣷⣮⣁⠸⢿⡞
⠀⢹⣺⢳⣿⢏⡐⣜⡦⠀⠀⠀⠀⠀⠈⢷⣶⣗⣲⢿⡶⢃⣴⣲⣮⣅⣀⡼⠿⠿⠯⢿⣯⠈⢀⣡⣥⣿⣭⣻⣭⣧⡍⣠
⠀⠈⢿⣿⣥⢾⣿⢩⢮⢀⣷⣄⡀⠀⠀⠀⠉⠻⠿⢮⢡⣼⠽⠿⠛⠉⠁⠀⠀⠀⠀⢀⣨⣿⢮⡽⢿⣿⣆⣫⣭⠟⢠⠇
⠀⠀⠀⠻⣿⢛⣏⡘⢫⡠⣽⡶⢶⣝⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣯⠿⣉⡧⠔⠘⣇⢻⡟⠋⣰⠃
⠀⠀⠀⠀⠈⠙⡯⣿⣌⣱⣮⠿⣑⡛⢈⡳⢠⣾⢻⣿⠱⡴⢇⣾⣉⣻⣻⣿⣿⣿⣿⣿⣷⣯⣼⢀⡜⣿⣶⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠍⡙⠓⠾⢿⣿⣌⣡⣿⡴⢟⣙⡃⢀⣾⣟⣻⣿⣓⡿⣿⣿⣿⣿⠿⠞⢹⠌⢻⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⢀⡉⠉⠁⠙⠛⠻⠿⠛⠿⠟⠟⠇⠃⠛⠉⢉⡀⠄⠒⡧⢀⣼⡃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⢜⡑⢿⢐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠃⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⢦⣤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢀⠶⣆⣰⣯⡿⠟⢻⣽⣿⣦⣅⠀⢀⡠⢲⣭⣵⣟⣻⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢽⣿⢿⣁⣀⣀⣀⣀⣩⣥⡿⣸⡆⢣⣵⣟⣻⣾⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣻⣖⣎⣝⢣⠮⣝⣿⣿⡆⡾⣷⡸⣃⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⣉⡛⣩⣟⡿⠛⠉⣿⣿⢚⡷⠟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⣀⣼⣫⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⢶⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣀⣀⣼⠃⠀⣄⠠⠤⢶⢀⡉⠀⠀⢀⣦⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⣿⠀⢰⣿⠀⠀⠀⠋⠈⠁⠈⡤⠟⢻⣿⡇⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡀⡙⣄⢼⣏⡀⠀⠀⠠⠑⢲⡁⠠⠾⠾⢿⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠸⡇⠀⠘⣿⣦⣤⡀⠀⠨⠦⠀⠄⠀⠀⠉⠙⠁⠉⡏⣶⠂⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⣭⣇⠀⢷⡀⠀⠘⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣤⣄⠀⠀⠀⢀⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⢸⣿⣤⣀⠹⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢀⣠⡾⠟⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠄⠀⠈⠛⠷⣝⣟⠻⣷⣟⣷⣤⡀⠀⠂⠁⠀⠀⢀⣠⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠈⠛⠿⣿⣿⣏⣉⠀⠀⠀⢀⣠⣴⣿⣿⠋⠀⢀⣤⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠑⢦⣀⡀⠈⠛⢿⣿⣷⣤⣀⣀⣀⣴⡿⢁⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠁⣽⣦⣤⣤⠛⣩⣿⡿⠿⠿⠛⠂⠉⡉⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠙⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠚⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣻⣿⡿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⢋⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⡃⣽⣯⣵⣶⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠶⣏⡁⠀⡙⢿⣿⣿⣿⣿⠟⣡⣤⣤⡉⠛⣿⣹⣿⣿⣿⡟⣩⣟⣛⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣗⣦⣹⡆⠀⠓⠈⡿⠛⠃⠀⣼⣿⠹⠸⡇⠀⢙⣿⣿⡿⠋⣰⠟⣿⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠻⢿⣿⣤⡾⢆⠙⣦⡀⠘⢻⣿⠀⢈⠁⣠⣾⣿⠛⢁⣴⣷⣼⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⡥⡝⢿⡳⢦⣪⣝⣿⣶⣌⣀⠀⠸⢿⣿⡟⢡⣶⣿⣿⡟⡻⢿⠟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡾⣛⣫⠟⠻⣿⣿⡟⠙⣭⠉⠳⣷⣞⠻⣿⣼⣿⣿⠏⠀⠀⠀⠐⢫⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠘⠋⣴⣿⡞⣡⣿⡇⢸⣿⣱⣦⡿⠟⣷⣿⣿⣵⡦⠀⠀⠀⠀⠀⠀⠀⠚⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⢿⣵⣿⠑⣶⣿⣿⣥⣤⣄⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠹⠏⠀⣏⣽⡟⠁⠉⠨⠙⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠠⠀⠀⠄⠀⢋⣿⠁⠀⢰⡀⢰⣿⣿⠀⠀⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⡀⠀⢸⣟⠀⣀⣸⣧⢸⡿⢻⡗⠀⠈⠸⠿⣿⣤⣂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⣿⡞⢹⣿⣿⣿⠇⢸⣧⠀⠀⠀⠀⠛⠊⢿⣧⣦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣷⠠⣿⣿⣿⡃⡘⣿⠀⠀⠀⠀⠀⠀⠘⠻⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠛⠊⠻⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡆⠀⣿⣦⡙⠇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⢬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⣿⣿⣿⣦⠀⢼⠀⠀⣀⠀⣀⣤⠀⠀⠀⠀⠙⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣁⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⢿⣿⣿⣿⠀⠀⣄⠲⣿⣦⠷⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁⠁⠀⠀⠀⠀⠀⠤⢦⣹⣷⣴⣾⣿⠿⠉⢀⣠⣿⠀⢸⡏⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⡰⠀⠉⠀⠀⠀⠈⠛⡋⣽⣿⣷⣤⣤⣾⠙⣿⣤⠀⣿⡄⠀⠈⢷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⣠⣴⣯⡗⣏⣿⣷⠡⢙⣿⣶⣾⣿⡶⣿⣿⣆⡀⠈⣷⣞⡁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠛⠒⠲⣤⣤⣶⡻⣾⣏⣽⡿⢿⣿⣿⣾⣿⣿⣤⡙⣷⣿⣷⣿⣤⡌⢿⣿⣿⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⠉⠙⢻⢾⣿⣿⣿⣧⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠒⠂⠘⠛⠛⠛⠛⠛⠛⠓⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢠⡞⠁⠐⠂⠬⢍⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⠭⠄⠂⠈⠙⣦
⠀⠀⠀⢸⡖⠚⢋⣉⣓⡒⠦⢝⡮⠷⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠴⠭⣞⠥⠖⣚⣋⣙⠛⠒⣿
⠀⠀⠀⠈⢷⡉⣉⠀⠨⢭⣍⣻⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣼⣋⣭⠍⠀⣈⡉⣸⠇
⠀⠀⠀⠀⠘⢧⡠⠤⠖⠒⠒⠒⠛⠦⣄⠀⠀⠙⢦⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠞⠒⠒⠒⠒⠦⠄⣰⠏
⠀⠀⠀⠀⠀⠈⢷⣐⠊⠉⢉⣉⣉⡙⠛⠛⠒⠦⣄⡳⣐⣢⡴⣁⡤⠖⠛⠛⠛⣉⣉⣉⠉⠉⢒⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠧⣤⣤⠬⣭⣭⣭⠯⠭⢽⣏⢘⣿⠭⠭⣯⣭⣭⡭⢤⣤⡬⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣁⠤⠄⠐⣶⠤⠒⠊⡝⣿⢾⡿⡑⠒⠤⢴⡒⠀⠤⢄⣉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⣉⠤⠒⢈⡏⠀⠀⣠⢾⢀⡏⡘⡇⠹⢆⡀⠀⠙⣏⠐⠢⢌⠓⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠤⠂⡩⠳⡴⢺⠁⠀⣸⢀⢃⢹⠸⠀⢻⠲⡴⠫⡁⠢⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⢠⠊⠀⡜⠀⠇⠀⢰⠇⠈⠃⠈⣿⠀⠈⡄⠑⡀⠈⢦⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠀⡘⠀⠀⡆⢀⡞⠀⠀⠀⠀⠸⣆⠀⠇⠀⢸⠀⢈⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠶⠚⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⠭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣾⠕⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⡋⢉⢿⣿⣦⢠⠀⢀⡀⣰⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣎⣽⣿⣷⣶⣦⠿⣿⡀⠻⠟⠋⠙⣿⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣿⣿⣫⣾⣿⡍⣝⡷⡄⠀⠀⠀⣜⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣸⣯⣼⣿⣿⠟⠍⢿⣯⣷⣷⠢⣄⣤⣧⣄⣤⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⡿⢿⣿⣿⡓⣤⣼⣟⣿⣅⣤⣿⣩⣽⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠳⣄⠲⣿⢿⣿⣦⢬⣿⣍⡻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠠⢞⣿⡿⠀⠀⠀⠉⣍⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡊⠽⣷⣦⣤⣾⣿⣿⣷⣬⣶⣆⣺⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢉⡟⠛⢹⡿⣏⠀⠈⠙⣿⣿⣿⣿⣿⣯⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣤⣾⣤⠶⠛⠁⠨⡟⢦⣄⣈⣻⣿⣿⣿⡇⢼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡴⠦⣄⠱⣌⣠⣿⠿⢿⡟⢿⣿⡙⢿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠒⠰⠿⠷⠿⠟⢤⣤⣶⣼⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢶⡻⠮⣅⣈⣀⣠⣤⣴⣶⡾⠛⠿⠅⠉⣛⣿⣿⡟⠀⠀⠀⠀⠸⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠈⠉⠁⠀⠀⠀⠀⠉⠙⢻⣷⣦⣌⣿⣿⣿⡇⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣬⣽⣶⣶⣀⣐⣨⣤⣶⣿⣿⡿⠋⣿⣿⡟⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢹⣿⣿⣿⠟⠀⢰⣿⡿⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣀⣀⣿⡿⠁⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠋⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢧⡀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣀⣀⠀⠀⢀⡼⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢈⣿⠞⠋⢉⠀⠀⢀⣠⣤⣨⣅⣤⣄⡀⠀⠀⡉⠙⠳⣿⡁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⣀⣴⠟⢁⣴⢿⣷⠖⠛⠉⢞⣡⢭⡭⣌⡳⠉⠛⠲⣾⡿⣦⡈⠻⣦⡀⠀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⢋⡴⢻⡇⠀⠀⢀⣘⠛⠛⣩⠤⠤⢍⠛⠛⣃⡀⠀⠀⢸⡟⢦⡙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠞⠉⣰⣿⣇⡀⠀⠊⢁⡠⠖⠒⠒⠒⠒⠲⢄⡈⠓⠀⢀⣸⣿⣆⠉⠳⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢀⡄⠉⣽⡏⠻⡄⠀⠺⠀⠀⣠⠔⠢⣄⠀⠀⠗⠀⢠⠟⢹⣯⠉⢠⡀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣾⢇⣾⠁⠀⠗⢢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠲⠀⠈⣷⡼⣷⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢸⡏⠈⠙⠲⠄⠀⠙⢺⣿⣏⣀⠀⠀⠀⠀⣀⣹⣿⡗⠋⠀⠠⠖⠋⠁⢹⡇⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢸⣇⣠⡤⡠⢞⣣⣤⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠤⢤⣜⡳⢄⢤⣄⣸⡇⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣠⣴⡟⣇⠀⢰⣿⣦⡻⢄⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢟⣴⣿⡆⠀⣘⢻⣦⣄⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣬⣧⠻⠆⢸⣿⣏⠻⢦⡀⠀⢠⡄⠀⠀⢠⡄⠀⢀⡴⠟⣹⣿⡗⠰⢟⣼⣥⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣯⣍⡛⠛⣻⣿⣓⠦⠀⠀⠈⠻⣤⣤⠟⠁⠀⠀⠴⣚⣿⣟⠛⢛⣩⣽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠹⣿⣿⠿⠳⠶⡶⢤⣴⣦⡴⢶⠶⠞⠻⣿⣿⠏⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣀⣿⣿⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⣿⣿⣀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣆⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡏⠖⡀⠀⠀⠀⠀⢀⠲⢹⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣟⣉⣉⣉⣙⣩⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⡤⠠⣤⠠⣤⠀⠀⣠⠔⠐⢤⠀⠀⣤⠀⠠⡄⠠⣤⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⣿⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⢻⡄⠀⠀⢻⠃⠀⣿⠀⠈⠀⠀⣿⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠐⠛⠀⠀⠙⠒⠄⠚⠀⠀⠛⠒⠒⠃⠐⠛⠀⠀⠛

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠢⢰⠀⠀⠀⢰⢀⠠⢠⠀⠀⡄⠀⠀⠀⢀⡄⡄⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡑⠄⡗⠆⠀⠀⢠⡧⠰⠃⠠⢊⢆⡀⡴⠈⣼⠊⠀⢀⠠⢀⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠘⣇⢰⡄⠀⠸⣣⡶⠊⢐⠍⢉⠌⢀⡼⠁⣐⠈⢤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠂⠘⣈⡟⣲⠞⣋⣶⠿⠗⠚⠃⠰⢋⡀⠄⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⣿⠛⣡⡾⠡⠁⣀⠠⠴⡒⢠⠋⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⣤⠞⡇⣰⡟⢀⣠⠊⠀⠀⠀⢈⠃⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⢀⣧⡟⠀⡾⠁⣀⣀⣀⣠⠾⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⢞⠟⠁⡸⣳⢋⠉⠉⠉⠀⠈⢹⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⢞⡵⠋⢀⡤⠗⡵⠁⠀⠀⢀⡴⠁⢈⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠀⣠⠎⠀⢀⠀⠀⢀⠄⡘⠁⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠌⡈⠀⠀⡆⠀⣀⡡⠊⠀⠠⠊⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡾⠀⠀⠀⣦⡞⣡⠴⠊⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡀⢼⣿⠏⠁⠀⠀⠀⢀⠂⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠈⠀⣸⣿⠀⠀⠀⣠⣀⠅⠂⡐⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡀⠐⠈⡄⡿⠀⡠⠊⠍⠀⢀⡔⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠁⠀⡠⡾⠀⣧⠊⢀⠌⢀⡴⠋⠠⠈⡀⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⠎⠀⡇⢠⠇⠀⣈⡔⡱⠁⠠⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠃⠀⠈⢠⠀⡄⡴⡋⢰⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⡀⠸⠇⢡⡗⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡴⠶⠶⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⡰⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣤⠒⠋⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠬⠳⡀
⠀⠀⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⣠⡇⠀⠲⣄⠀⠀⣠⡟⣡⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣇⢠⣀⠀⣀⣹⢦⡀⠻⣄⣴⠟⡇⠀⠀⠹⡳⡜⡝⡴⢹⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣿⢀⣙⠻⡦⣹⣾⣌⡦⠾⠴⠴⠣⠶⣦⣄⣸⢠⡷⢁⠟⣿⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢹⡀⠉⢳⣝⣜⢿⠏⠀⠀⠀⠠⠊⠀⠈⢷⡇⢸⡇⡜⢸⣿⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢱⠛⠍⠉⢻⣾⡀⠀⠀⠀⠀⠀⠀⣀⣼⣇⣸⣷⠋⠉⠉⣹⠉⢹⡄⠀⡇⠀⢀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁⠀⠉⠑⠚⠉⠉⠉⠉⠀⠀⠈⠁⠈⠓⠒⠚⠉⠀⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⡀⡀⡀⠀⠀⣀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⣼⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⢠⣿⡿⠿⠛⠛⠿⢃⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠈⣿⠿⠟⢋⣉⣤⣤⣶⣾⣿⣷⡀⢺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⠇⠹⣿⣿⣿⣿⣿⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣈⣉⣙⢋⣠⣄⡙⠛⠻⠿⠟⠀⠹⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣦⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣧⠀⣉⣉⣉⣙⣛⠁⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⠛⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡛⣛⣛⡛⢛⣻⣿⣿⠿⠿⢿⡿⠿⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣾⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⣿⣿⣿⣿⣮⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⡵⣿⣿⣯⣽⣷⣾⣿⣿⣷⡤⣤⣤⣤⣤⣤⣤⣰⣦⣄⣀⣀⣀⣀⣀⣠⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⣷⠼⣿⣿⣿⡿⠿⠿⣽⠿⠿⠛⠛⠻⠿⢿⡿⠟⠛⠛⠛⠛⠛⠻⠿⠿⠟⠛⠛⠛⠉
⠀⠀⠀⢸⡿⣿⣿⣏⣿⣷⣿⢧⣿⣿⣿⡃⠀⠀⠀⠸⣿⡄
⠀⠀⠀⢾⡇⣿⣿⢹⡼⣿⣿⣿⣿⣾⣿⡅⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠸⣿⣿⢿⢸⣿⢻⡻⡿⣿⣿⣿⣧⠀⠀⠀⢠⣿⠃
⠀⠀⠀⢰⣿⡏⡿⣸⣿⢸⣷⣝⢾⣿⣿⡷⠀⠀⢀⣾⠏
⠀⠀⠀⢸⣿⢸⣇⢻⣿⣇⢻⣿⣷⣝⡾⣟⣷⢰⡿⠋
⠀⠀⠀⠈⢿⡸⣿⡌⢿⣿⣷⣭⣻⣯⣟⠿⣿⣿⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠁⠹⢿⣦⡙⠿⣯⣟⡻⠿⠿⠷⠊⠉⠙⠛⠛⠛⠛⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⡀⠀⠠⠀⠄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣽⠇⠀⠀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⡀⠀⠀⠀⠦⠹⠴⠋
⠀⠀⠀⠀⢷⡶⠃⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠻⡆⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠖⠀⠀⠀⠀⣴⡀⠀⠀⠓⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠉⠉⠤⠄⠈⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠟⡆⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⡰⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠐⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⢄⣶⣿⣿⣤⣰⣿⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣟⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣿⣿⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠃⠀⠀⣿⠁⠘⢠⠄⣴⡢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣤⣴⣄⣺⣣⣖⣷⡀⠈⠀⢴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠀⠄⠀⠈⠉⠑⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢰⣆⣠⣤⢦⡤⠀⠀⠀⢷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢂⡀⠚⠳⢾⠋⠀⠀⠀⠀⢸⣿⠎⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⢺⡘⠇⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣀⣀⣀⣠⠐⠉⠙⣿⣷⡄⠀⠀⡞⠧⣀⠘⢦⠀⠀⠬⠚⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠁⡹⠀⠀⠈⠀⢻⣷⠀⢀⠀⠀⠈⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠳⠢⠆⢁⠀⢺⡟⢠⡏⠀⠀⠀⠀⣡⠋⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢠⠆⡂⠀⠐⠲⠶⣫⢠⠏⠀⣠⡴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠁⠀⢀⠀⠀⠀⠠⠄⠀⠀⡰⠋⠀⠀⠀⡠⠂⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⡔⢀⢇⠀⠀⠀⠘⢀⠎⠀⠀⠀⠀⠀⠀⠀⢈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠝⠒⠀⠈⠁⠸⠃⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠂⠁⠀⠠⢤⣾⣞⡁⠀⢀⣀⡤⠔⠀⢀⡴⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⠛⠛⢅⠀⠀⠀⠀⠀⠒⠋⠉⠙⠛⠛⠯⠥⠤⣶⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⡀⠀⠀⠀⠀⠁⢠⠟⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠄⠀⠀⠀⠀⠀⢀⠘⠋⣉⡝⠁⠤⣤⡤⡄⢤⠖⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠂⠈⠙⠐⠊⠁⠀⠀⠀⠀⣀⠴⠀⣤⠖⠃⢰⣴⣶⣤⠴⡾⣷⠒⠈⣭⣤⠂⠀⠀⢀⠤⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⣀⡠⠶⣿⠉⠉⣹⠛⠙⣿⠦⣀⣉⣳⠀⠀⠉⠁⠀⠀⠈
⠀⠀⠀⠈⡀⠀⠀⠀⠀⢀⡤⠤⠚⠉⠉⠉⠉⠀⢀⠈⠁⠀⠀⢸⠳⠀⣻⠦⣼⣇⠀⠈⢉⡟⠛⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⢨⣿⣧⠀⣿⣤⣈⡏⠁⠀⣼⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠀⠀⠀⠀⠾⠿⠿⠀⠈⠛⠻⠿⠶⠚⠙⠓⠀⠈⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣽⣯⣿⢷⡄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⢀⣀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣿⣿⡏⢘⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣿⣿⣧⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⡿⢿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣾⣿⣿⣿⡆⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠾⡿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠙⠿⡿⠟⠁⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠒⡗⠂⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣀⢄⣀⣀⣒⡲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠉⢓⣠⣿⣿⠟⠛⠋⠙⠛⠻⣿⡷⠀⠀⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠤⢒⣋⡽⡫⢀⡠⠔⠒⠂⣼⠶⣦⡅⠀⠀⡧⠈⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠢⠤⣴⢫⡏⣜⡵⠋⠀⠀⠠⢉⡠⠶⢶⡇⠀⠀⣟⡀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⡠⠞⣱⣟⣼⠟⠁⠀⠀⠀⢠⡞⠀⠀⠀⠁⠀⠀⣟⣁⡤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⣤⡶⠚⠻⠉⠉⠀⠀⠀⠀⠀⢀⣞⣀⠀⠀⠀⡀⠀⠀⣿⣏⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠾⣯⡶⢶⣶⣤⣀⠀⠄⠀⢀⣠⡞⣿⣿⠛⠀⢸⡇⠀⠀⣿⣿⣢⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢡⡏⢠⡟⠉⠛⠻⠛⢿⣸⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢁⡿⠡⢠⠇⠀⠀⡈⠛⢧⠀⠀⠀⠀⢸⡇⠀⠀⢈⡡⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⢠⢣⢺⡇⠇⠹⢀⠸⠀⢈⠀⠈⠳⣄⠈⠉⢹⡇⠀⠀⠈⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⡠⠋⢠⠞⣷⠀⠐⠸⡀⠀⠸⣆⠀⠘⣿⠓⠦⢴⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢉⢉⡤⠞⠁⠀⠘⣦⣦⣀⣷⣄⠀⠈⠻⢾⡟⠀⠀⢸⠇⠀⠀⢠⣀⡀⠤⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡠⠖⠋⠉⠉⣁⡽⢷⡄⠀⠈⢻⡄⠀⠈⠀⠀⠀⢸⠉⣿⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡀⠀⠀⢀⠀⣶⣤⡟⢀⠆⣼⡇⠀⠀⡀⠀⠀⢸⠀⣯⣦⡀⠀⠀⠀⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡉⠂⠀⠃⠀⠁⠀⡠⢻⢴⣅⢠⣾⣿⠁⠀⠀⡇⠀⠀⢸⣾⠓⠊⢹⣦⣄⣉⠀⠀⠒⠊⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠒⠚⢻⡫⡔⠂⡍⠉⠉⠙⢧⡀⢰⠀⠀⠀⢸⡇⠀⠀⠀⣿⣄⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣠⡴⣻⠗⠶⣶⣃⣀⣄⣀⠀⠉⣿⠀⠀⠀⢸⣄⣤⣀⡀⢸⣯⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠤⠄⠒⠚⣋⣵⣾⣿⣤⠞⠉⠀⡀⢀⡽⠗⢂⣼⠀⠀⠀⢸⠃⠔⠚⢿⣿⣿⡿⣶⣭⣙⠓⠒⠒⠀⠺⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣤⣤⣶⡶⠿⠛⠉⡿⣿⠟⠀⠞⠁⠙⣿⢸⣄⣴⣿⠀⣀⡀⢸⡇⢄⣠⡆⡟⠁⠉⡆⠉⠛⠻⠿⠿⠿⠿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⠁⠀⠀⠀⠀⡼⠁⢹⣟⡀⠙⠦⠤⡞⠶⣽⣛⣟⠉⠉⠉⠙⣟⣛⡭⣖⢳⡤⠴⢧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⢻⣷⣤⣄⣀⣨⣵⢟⠽⠛⠛⠶⠾⠛⠹⣅⣶⣿⡿⠁⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣀⣴⣡⠤⠤⠤⠤⠼⠧⠬⠭⠿⢿⠿⠊⠀⠀⠀⠀⠸⠀⠀⡟⠉⣽⠁⠀⠀⠀⠀⠀⠈⢓⣦⠤⢤⣤⣄⣀⡀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣠⢔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢀⣉⣤⠴⠂⠀⡏⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠈⠉⠙⣷⡞⠁
⠀⠀⠀⠀⠀⠀⡸⠁⠀⢀⡾⠣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠷⠞⠒⣻⠛⠉⠉⢹⢢⣄⠀⡇⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⣾⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠁⠀⡠⠊⠈⠀⠀⠐⠚⠉⠾⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠈⠓⠒⠾⢿⢀⠀⠀⣀⣀⣠⠤⠴⠒⠚⠉⢹⠋⠁⠀⠀⠀⠀⢙⡂⠀⢰⡃⠀⠀⡀⡀⢇⠀⠀⢀⣀⣠⠤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⢦⠀⠀⠣⢹⣸⡶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⠓⠷⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠

⠀⠀⠀⣀⣀
⠀⠀⢠⣿⣿⣷⢦⣄⡀
⠀⠀⠀⣿⡟⠻⢷⣬⡙⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⣿⡇⠀⠀⠈⠙⠷⣯⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠈⠻⣿⣷⣄⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠹⣿⡙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⡄⠈⠻⣿⣿⡆
⠀⠀⠀⢹⡇⠀⠀⠀⠀⢰⣄⠀⢸⣷⡀⢘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣸⡇⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⢿⡄⠀⠀⠀⢀⣿⣆⠈⢿⣧⠀⠻⣿⣿⣷⣤⣀⣀⣀⣀⣤⣸⣿⣠⡟⠁⠀⢳⡀⠀⢹⣿
⠀⠀⠀⠀⠀⢻⣦⣄⢠⣼⣿⣿⡄⢈⣿⣆⠀⢹⣿⣿⣏⠻⣿⣿⠿⢿⣿⣿⣿⠿⠿⠷⣾⣷⣤⣀⣿⡀
⠀⠀⠀⠀⠀⠀⢯⡻⢿⣿⣿⣿⣿⣿⣿⢏⠀⠈⠀⠙⠿⢷⡈⢻⣶⡿⠶⡷⠶⢦⣴⣶⣤⣄⣀⢉⡉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠹⢤⡈⠙⠛⠿⢿⣿⣄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠈⠉⠛⠛⢦⣠⡄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣓⣶⠶⠿⠿⠿⢛⣷⣶⠖⠀⠀⠀⠀⠀⣸⡟⠋⠙⠋⠉⠂⠈⠲⣤⣄⣀⠀⠉⠛⢦⣀⠙⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠴⢾⣿⣥⡄⠀⠒⢛⣿⣿⡏⠀⠀⠀⠀⠀⠀⢤⣽⣿⢠⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⠀⢻⣧⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⣠⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⣀⣀⠈⢉⡀⢿⣶⣄⡀⢠⡀⠀⠀⠀⠈⢻⣿⣆⠀⠛⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡧⠀⠾⢻⡟⠁⠀⠀⠀⢀⣴⣿⠃⠒⠺⢿⣿⣿⡿⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠐⣿⡟⡄
⠀⠀⠀⠀⠀⠀⠋⣿⠟⠁⣠⣾⣿⠋⠀⣠⠄⠀⣼⣿⡋⠀⠀⠀⢀⡈⠀⠀⠀⢻⣿⣿⠉⠛⢿⣷⣄⠀⠀⠈⢹⣧⠀⠀⠀⠀⠀⢻⠀⢰
⠀⠀⠀⠀⠀⢀⣼⠏⠀⡰⢫⡿⠁⣠⣾⡏⠀⢸⣿⣿⡇⠀⠀⠀⠘⣿⡆⠀⠀⠘⣿⣿⡄⢀⣈⣿⣿⣷⣦⣀⣼⣧⣤⣄⣀⣠⡆⢰⠀⣾
⠀⠀⠀⠀⠀⢸⣿⠇⠀⢠⡿⠀⠘⣫⣿⡇⠀⢸⡛⣿⣧⠀⠀⠀⠀⠙⠃⠀⠀⠀⠘⢿⣿⣾⣿⣿⣿⣿⠋⠙⠻⣿⠄⠀⠉⠉⢳⣿⣤⡆
⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⠁⠀⠐⣿⣿⡀⠀⠘⠃⠙⠻⣿⣦⣤⣤⣀⣀⠉⠙⠓⠶⠾⠿⠿⠿⠿⠿⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠻⣃⡇
⠀⠀⠀⠀⠀⠋⠁⠀⢸⣿⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠠⠄⠚⠛⠛⣛⠒⠒⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⡀⠠⣄⡀⠠⣀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⣠⡄⢸⣿⠀⠀⠛⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⣤⡀⠀⢠⠈⠹⠟⠳⠄⠀⠉⠳⣤⣈⠙⠦⠈⠳⠆⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠘⢻⣇⢸⣿⡆⠀⢀⣿⣧⡖⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣷⡀⢧⠀⠀⡀⠀⠀⠀⡀⠈⠻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡎⣿⣷⠀⠸⣿⡟⠀⣰⢃⣾⣿⣿⣿⣿⣿⣿⣏⠙⢿⡎⠙⣿⣿⡇⢸⠁⠀⣀⠀⠻⣆⠀⠟⠀⢰⣶⣦⣄⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠈⠙⣇⢹⣿⡀⠀⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡆⠈⣿⠀⠈⠙⣿⡐⣦⠀⠙⢦⠀⠙⠀⠀⠀⠘⠛⠻⣿⣿⣷⣶⡶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⡇⠀⠋⠀⣿⣿⣿⣿⣿⠿⠋⠙⣿⠁⣿⠇⠀⠈⠳⡀⠀⠛⣷⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠰⠃⢀⠟⠀⠀⠀⠀⠣⢄⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⢸⣿⣿⣿⡧⠖⢨⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⠛⠛⠻⣦⣄⡀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢪⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠘⣿⡟⠁⠀⠀⣼⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⡀⠐⠋⠉⠛⠿⣷⣶⣶⣾⣿⣧⣄⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢰⡄⠀⢿⣦⠀⠀⢰⣿⣆⢈⡄⠀⢰⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢉⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢿⡀⠘⣿⣧⣤⡟⠀⠉⠉⡹⢁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠘⢷⡀⠈⠻⠿⣿⣷⣶⣾⡤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⡀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠋⠁

⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣯⣷⣆⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢀⡄⣉⣋⣷⣿⢿⣤⣼⡟⠋⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡦⣻⣉⡁⠛⠙⠿⣿⢾⡿⠿⠙⠣⡀⡀⢠⡾⢋⣽⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣭⣙⡏⢢⣴⣾⣿⣿⡿⠁⠀⣳⣌⠛⣾⢣⡿⠉⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠈⠈⠛⠛⠿⣟⠿⠟⠛⢉⡟⠿⢢⣀⠊⢡⡿⣼⠇⠈⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⢀⣠⠄⠰⠗⠀⠀⣤⡼⢶⣶⠖⢼⣿⡰⢸⢹⠀⢸⡇⣿⢀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⡄⠰⣥⡀⠀⠀⠀⠖⠴⣷⣶⣾⣿⣿⣜⡄⠀⢸⡇⣿⣸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠂⢹⢀⠉⠀⣀⣠⢞⣿⠶⣶⣶⣾⣿⣷⠁⡇⢀⡘⣷⢹⣇⠿⣫⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣈⡱⣾⣹⡟⣁⣤⡞⠻⠛⠟⠿⠏⠀⠇⠐⣷⣿⡌⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠠⠖⠉⢰⢿⣿⣿⣿⠌⠟⠁⠈⡀⢤⣁⠀⠘⠀⠀⠀⠀⣿⢿⣿⣜⣫⠎
⠀⠀⠀⠀⠀⠀⢀⢔⠚⠉⠆⠀⠀⠀⠀⣠⢧⠸⠋⠁⠀⠀⠀⠀⢠⠈⠛⠷⠀⠀⢀⣆⢰⣿⣼⣷⡈⠁
⠀⠀⠀⠀⠀⢠⣡⠁⠀⠀⠀⠀⠀⠀⢰⡟⢬⠠⠤⠀⠀⠀⠀⠀⠀⠇⠀⢀⡄⠀⢸⣿⣿⣿⣼⢸⠁
⠀⠀⠀⠀⠀⠸⠁⠁⠀⠀⠀⠀⠀⠀⣗⡠⠄⠈⠈⠀⠀⠀⠀⠀⠀⠠⠀⣸⣧⠀⢸⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣦⣜⣹⢿⡿⠃⡉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠉⠙⠉⠀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⢀⣀⠀⠀⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⣾⡆⣀⠉⠉⠛⠻⣷⡞⠻⡆
⠀⠀⢀⡴⣞⣿⣿⣿⣿⣿⠉⣀⣀⡰⣀⡀⣿⣿⠀⠗
⢀⡀⢫⣴⣋⡵⠶⠋⢁⣠⣤⣿⣿⠧⠈⣙⣿⣿⡇
⢊⣤⣪⡿⠋⠀⢀⣼⡿⣿⣿⠿⠁⠀⠲⠘⠿⠿⣧⡀⠀⠃
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⣄
⣸⣿⠏⠉⠉⠁⠐⠒⠙⣿⢷⣄⠀⠀⠀⠀⠀⠀⢶⡀⠀⠙⠿⣿⣿⣤⡀
⣛⣿⣄⢠⣤⣤⣴⠶⢶⣿⣮⣿⡂⠀⠀⠠⠐⠇⠘⢿⣶⣄⣀⡀⠈⠛⠿⢧⣤⣀
⣿⣿⡿⣨⣿⠀⢀⠀⠀⠀⠙⠿⣷⡀⠀⠀⠀⠀⠀⠀⣼⣹⡏⣿⡗⢶⣶⡈⣿⣿⣷⣦⣤⣀
⠀⠈⠹⣿⣿⡇⠉⠈⠉⠐⢄⠀⠈⢻⣄⠀⠀⠀⠀⠀⣿⣿⠀⢹⠀⠺⣿⣷⣮⠉⠉⠈⠒⠽⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠐⢠⣀⣿⡳⣤⣤⣤⣤⣿⣿⣘⣸⣿⡒⠚⠛⠛⣺⠟⠁⠐⠀⢈⠛⢿⣿⣯⣁⡀⠀⢠⣤⣤⣤⣤⢤⣤⣴⣾⣿⣧⡀
⠀⠀⠀⠘⡞⣿⡇⠀⠀⠀⠀⠀⠀⠙⠛⠿⢶⣴⣗⣉⣿⣿⣿⠿⣏⠻⠻⠟⢻⣏⡛⠛⠛⠛⠋⠁⠀⢉⡉⠉⠉⠛⠻⡍⢉⢻⡍⠀⠁⠀⠉⠉⠙⠻⣧⡀
⠀⠀⠀⠀⣷⠘⣿⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠛⠟⠒⠒⠲⠦⠬⠿⠶⠶⣶⠶⠶⠚⠛⠛⠛⠉⠛⠛⠚⠺⠷⠀⠀⡀⠀⠀⢀⣀⡈⠻⣦⠴⢠
⠀⠀⠀⠀⠘⠇⠈⢳⣄⣈⣀⢀⠀⢀⢆⡀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣄⠀⠀⠀⠀⢀⣾⠨⢯⢤⣄⠤⠀⠩⠄⣹⣿⡇
⠀⠀⠀⠀⠀⠸⠟⠉⠉⠊⠛⠛⠷⢶⡂⠷⠛⠿⡛⠒⠿⠶⢿⣶⡾⠶⣾⣷⣿⣿⣶⣶⣶⣶⣖⣒⠒⠒⠶⣒⣆⠻⠾⢿⣶⡶⠾⠗⢒⣀⠰⠠⢠⣤⣤⣤⠙⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠀⠀⠘⠋⠀⠀⠀⠛⠃⠀⠀⠘⠛⠀⠀⠀⠉⠉⠒⠛⠒⠒⢲⣿⣿⡃⣀⡈⡛⠛⠀⠀⠘⠿⠃⠀⠀⠻⠿⠁⠀⠈⠉⠛⠛⠁⣿⡁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣛⡛⠛⠓⠛⠻⠛⠛⠛⠛⣛⡛⠛⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⢿⠿⠟⠛⠛⠛⠛⠛⠿⠿⠿⠿⠷⢾⣷⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠘

⠀⠀⠀⠀⠬⠀⠀⠀⣀⡀
⠀⠀⢠⣄⣾⣦⡀⣠⠤⡹⠻⣆
⠀⢀⣠⣼⣿⢿⠄⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠖⠓⠘⠛⣿⢰⡄⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢈⣧⣿⣞⣋⠉⠈⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠾⢻⡙⣿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⡃⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸
⠁⠀⠸⢳⣿⢳⡦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⢙⢹⣧⡄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠒⢰⣾⣖
⠀⣀⣴⣶⣿⠻⠇⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⠯⠅⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢈⣽⡟⠙⠢
⠛⠁⢘⠹⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠰⢸⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠘⠀⠉⠨⣿⣿⣤⡀
⠀⠀⡝⢂⣿⠄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⢴⣾⣟⠇⠀⠙⠆⠀⠀⡠⠖⡀⣠⣤⠀⠀⠀⠀⢀⣤⢤⣾⡶⠀⠈
⠀⠀⡄⠘⣿⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⢹⡏⠃⠀⠀⠀⠀⠀⠀⣀⡀⣹⡿⠲⢄⡀⠀⠁⠀⢘⣽⡿⠖⡤
⠀⠰⠈⢤⣿⠃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇⠃⠀⠀⠀⠀⠀⠘⠉⠉⢿⣿⣤⡀⠀⠀⠀⠀⢀⢸⡇⠂
⠀⠐⠀⢸⣿⡂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡇⠤⠀⠀⠀⠀⠀⠀⡠⠀⢺⡏⠀⠈⠀⠀⠀⠀⡀⢼⡇
⠀⠁⠈⣺⢿⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⢾⡟⡉⠂⠀⠀⠀⠀⠁⣸⡇⠰
⢸⣳⣄⢼⣾⡧⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⣭⣧⡀⠂⠀⠀⠀⠀⠀⠀⠀⢊⡇⠀⠀⠀⠀⠀⠠⠁⣼⡇⢀⡄
⣻⣀⣈⣻⣻⣃⣉⣹⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣏⣈⣿⣗⣉⣵⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣸⣉⣻⣟⣉⣇
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠠⠠⠤⠀⢀
⠀⢀⣰⣶⣟⠲⢶⢖⠳⣘⣫⣶⠿⠿⠞⠛⢓⢹⠄⠂⡈⣡⢰⣺⣦⣷⣠⡄⠒⠚⠚⠇⡴⡂⢄⣃⣀⡀⢀
⠀⣠⡼⢖⣀⣤⣯⣥⠴⠂⣿⠃⠀⣀⠀⣄⣾⢻⠀⣀⠝⠨⠧⢸⡽⣏⠉⠳⢄⣄⣤⢤⡀⣤⣴⣛⣃⢼⣌⡙⣤
⢀⠏⡀⠈⠁⢈⢇⣘⣆⢀⡼⡆⠒⠉⢣⣀⢯⡘⡆⠀⠠⠤⠀⠈⢃⡷⠲⠆⠀⠀⠐⠊⠏⠁⠀⠉⠹⡀⠀⢈⣛⠁
⠊⢿⡾⠠⣿⣾⣹⡽⣝⢧⣀⣈⣀⠀⣉⡠⠀⠘⣇⠀⠐⢠⣯⠴⣟⢲⣄⢀⣐⣆⡀⢀⠶⠂⠈⠛⠂⠀⠃⣀⠉
⠀⢀⠇⠀⠀⡞⡩⠙⣶⡾⠟⠛⠻⡛⠻⣟⡣⢾⣿⣭⣶⣿⠋⠛⠉⣠⣟⠦⢌⠿⣻⣿⣾⠛⡯⡀⠠⠂⠀⠀⢀
⠀⢨⠔⠊⡰⣠⠇⢡⣿⠷⣶⡶⠀⠹⢷⡟⠙⠷⢼⣮⢻⣻⣧⠄⠞⢻⣯⣿⣃⣼⠇⠀⠈⠛⣧⣼⡁⠀⠀⠀⢨⢰⡀
⠰⠖⠶⠂⠤⠀⠀⠈⣩⡗⠈⠀⢴⡄⠀⡓⢊⠒⠶⠶⣾⡽⡈⠋⣄⣀⡏⣩⣷⡋⠀⢀⣀⣀⢸⣿⣖⡆⢸⣄⠱⣣⠂
⠀⠰⠀⠀⢘⠄⠀⠀⠀⠱⢤⣀⠤⡄⣰⡀⠙⢂⠀⠀⢪⡄⠇⡠⡤⠀⣤⢸⡂⠛⠃⢀⠁⢉⠾⣟⣿⠇⠀⠈⣄⠉
⠀⠃⣈⠉⠩⠓⠒⢲⡞⢁⠘⢿⣟⣻⠷⠖⢶⣿⣦⣐⣿⣧⣤⣀⢀⡼⠗⠛⠗⠢⡄⠛⠋⠜⠃⣁⣀⡀⠀⣀⡄⢀
⠀⢶⠉⡚⡶⠞⠶⡖⡽⠎⠃⠼⢸⣵⠴⠚⠛⠛⠿⢏⡙⠛⡛⠛⣻⠆⠀⠀⠀⠀⠙⢶⣵⣧⣼⡿⣭⣙⢷⣄⣀⠆⣃
⠀⠈⢻⠉⠀⠀⠃⠞⠀⠀⠀⠀⢈⢻⣾⣷⠄⠺⣿⡆⣿⡇⠉⣉⣿⠆⠂⠰⠶⠈⠐⢀⣏⣽⠻⠋⣹⡛⠿⠻⡆⠀⣹
⠀⠀⢩⠁⠀⠀⠀⠀⣀⡀⡵⠖⣶⠞⣇⠀⢀⣀⠀⢀⡋⠀⠘⠛⠣⡄⠷⠀⠀⠀⠀⠊⠉⠈⡆⣀⣭⣩⠀⠜⠀⠙⠃
⠀⠀⢨⡇⠤⢖⠨⠱⢠⠀⠀⠀⠀⠀⣨⣁⠠⣠⠀⡈⠀⠀⠀⠀⣰⣬⠀⠂⢀⠀⠀⠀⠀⠀⣙⢏⣁⠤⠀⠰
⠀⠀⣚⠦⣿⡴⣴⠂⠟⠀⠸⠁⠀⠀⣸⣭⠓⠻⢺⣾⡃⢤⢠⢤⣼⣿⣲⢶⠃⠘⠿⠗⢀⣶⠳⠾⠲⣄⠀⠒⠿⢻
⠀⠀⠀⡞⢿⣠⣤⠠⠒⠓⡆⠈⠀⣜⢟⣁⡈⠚⠓⢋⣷⠻⠇⡆⠨⡿⢿⠀⠀⠀⠀⠀⠀⣹⠈⠁⠰⠟⠛⡖⣺⣿⢧
⠀⠀⠀⢀⠈⠎⠘⢲⣥⠀⢈⢶⣺⢶⠊⠛⡛⡁⠸⡈⢻⠀⠀⠓⠼⠀⢸⡏⣡⠶⠄⠤⣸⠛⠟⢀⡄⠀⠀⢰⠿⠛⠈
⠀⠀⠀⠉⢠⡔⠓⠂⣶⠒⡡⡶⠀⠛⠀⠀⢀⡁⠀⢉⢠⠀⠀⠚⠻⢂⡞⠼⢿⡷⢶⣟⡹⢠⡀⠀⠀⠊⡠⠀⠆⠄
⠀⠀⠀⠀⠠⠱⡤⠀⠄⠀⡄⠀⠉⣥⠀⡠⢬⣁⣀⡠⠈⠀⠀⠤⠀⠈⠊⠉⠸⠉⠓⠛⠀⠀⠘⠃⠄⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⣤⣤⠄⣀⣠⣤⠴⣶⣶⠶⡒⠒⠒⠶⠆⠀⣀⣤⣴⣶⣿⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡷⢖⠛⡩⢶⠃⠿⢟⣫⣽⣶⣶⡜⢟⡴⣛⠽⠻⣟⣭⢾⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⢏⣴⠇⣠⡴⢖⣒⣛⠉⢷⡁⠀⣹⠗⡞⠀⠁⠠⣾⣿⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢻⣿⡾⢸⢏⡶⢋⣽⣧⠶⠿⢃⡤⢂⣤⣦⣄⠀⢀⣶⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠈⡏⡼⠋⠘⣠⣴⣯⣵⣦⣤⣬⡽⠞⠫⠵⢦⣌⣧⠸⣿⢡⣟⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡷⣰⠃⠀⣸⣿⢻⣤⡥⣋⠕⡊⡲⣶⣦⡴⢣⡿⠀⠀⠃⣸⣿⣆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠄⠁⠀⢰⠋⠀⢀⣤⣬⣤⠄⣽⣇⡴⢂⠀⣯⣜⠏⡀⠀⣿⣿⣹⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⡇⠀⠀⠀⠸⢦⣤⠄⠀⠀⢒⣭⢿⣯⡶⣻⠏⠋⣙⣴⠃⠠⢿⣿⠉⡟⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⣻⡿⠃⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣶⠿⠟⢁⠄⠀⠸⣿⣾⠃⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠙⢿⡁⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣲⣶⠶⠞⢉⡀⠂⠆⢻⡏⡇⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⠳⣤⣀⣙⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⡿⣾⢁⡀⣴⣃⣾⠀⣾⣿⢸⡿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣤⣼⣷⢾⠋⠀⠈⣇⣿⡆⢻⡇⣿⢋⡼⣻⣫⡏⠀⣿⡟⣾⠃⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢏⠁⠙⠅⠈⠀⠀⢀⣸⢇⡏⣸⣿⣷⠏⣸⣩⠟⢀⣾⣟⣽⣫⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣼⢡⣟⡾⠁⢠⡟⣡⣼⠿⣫⣾⢁⣥⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⡼⣡⠟⣋⣤⣴⠿⣛⣩⣴⠟⠫⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⣠⠟⣛⡵⣟⣯⣷⠾⠟⢛⣉⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡡⠂⣴⢃⡾⣫⠾⠛⡉⠀⢀⣶⢛⡍⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢠⢺⢇⡞⡽⢁⡞⠋⠀⢸⠟⣙⠋⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡌⢸⢸⠣⠇⠈⢧⣀⣀⣾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠁⢸⢸⡄⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⣆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠈⠆

⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠘⠀⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣋⣥⣶⣿⠿⠛⣿⡟⢿⣿⡇
⠀⠀⠈⠀⠉⠛⣆⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠿⢛⣉⢩⡴⣯⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠰⢶⢦⣬⡓⠈⠀⢘⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⣁⠈⠉⠉⡿⢻⡟⣱⣿⣿⣿⡿⣿⡿⢩⡶⠾⢿⣧⣭⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⠤⠬⢥⡿⠍⠠⢺⡄⠘⣿⣿⣿⡿⠟⠋⢁⣴⣿⠿⠻⢷⡆⠁⠸⣿⢾⣿⣽⣶⣿⠟⢡⡿⣶⠿⠶⣯⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢨⠙⠻⠥⢽⣆⠀⠈⠀⠀⢻⡉⠙⠃⠀⢀⣼⣿⣯⠘⠁⣻⠁⠄⠀⣨⣿⣿⣿⡟⠁⢀⡿⠓⢻⣾⣛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢨⣿⣿⡿⣿⣿⠠⢵⣌⣃⠈⢷⣄⠀⠀⠙⢿⣿⡏⠀⢀⡙⠀⢀⣼⣿⣿⠿⠋⠀⣰⡿⠤⢯⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢰⣶⠶⠶⠋⢿⣿⣿⢦⡉⠤⠘⢻⣷⣤⣀⡘⠏⠁⠀⠈⣧⣴⣾⣿⡿⡏⢀⣤⣾⣟⣻⣿⣾⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣷⡶⠞⣻⣐⣹⣶⣏⣜⣶⣹⣶⣿⡿⣿⣿⣶⣄⡀⢀⡉⣽⣿⣿⡄⢹⣿⣿⣷⣿⡛⠀⠊⠙⠋⠐⣻⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⠛⣙⣋⡽⠖⢋⢿⣷⣿⡟⠁⢩⣷⡀⠈⠻⣷⣼⡨⠻⣿⣧⣼⣿⡿⣴⡏⠀⠀⠀⠀⠀⠘⠁⡿⢻⣿⣿⣿⣿⣿⡇
⠀⠀⢀⠈⠃⠈⠿⠏⢡⣴⢾⣥⠟⢀⣿⡧⠀⣸⣿⠟⣤⣀⣹⣿⡿⣦⣾⣿⣿⡉⣠⣬⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣼⠿⣿⣿⣿⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠛⢫⡿⣥⡾⢟⣿⣷⡀⢙⣿⣷⡽⠟⠋⠁⠀⣿⣿⣿⣝⣿⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠿⣿⡇
⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠰⢿⣷⣿⣿⠀⢘⣿⢿⢿⣿⣶⣶⣷⣦⣽⣾⣿⣿⡿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠉⢁⡾⠇⠀⢸⠋⣸⣿⠟⠀⠙⠀⢟⠙⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠟⣿⡟⠀⠀⠈⡆⠸⢀⣿⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣶⣿⡇⠀⠀⠠⣷⡀⢸⣿⣿⣿⣀⠀⢀⠂⠴⣶⣄⡀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⡀⡄⣿⣇⢸⣿⡟⣿⣿⠃⠀⠀⠀⠹⡛⣿⢧⡴⢃⠆
⠀⠀⢠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⠀⠘⣿⣀⡶⢹⣿⣿⣿⢸⡿⠁⢻⣿⠀⠀⠀⠀⠀⠀⠹⠁⡼⢻⣾⡡⢊⡤
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡀⢰⣿⣿⡁⠘⣿⣿⣿⣿⣇⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⠈⠀⢉⣿⢿⣿⣷
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣋⡗⠈⣿⣿⣿⣿⣏⢠⠂⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢈⣿⢻⣥⡀
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠛⠞⠢⠹⣿⣿⣿⡇⡞⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⡃⢛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⣇⠀⠀⣰⣄⠙⢿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⡸⠗
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⣿⠀⠀⢹⣿⣿⣆⡉⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢔⣿
⠀⠀⠀⠀⢸⣿⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠾⠠⢘⡿⠀⠀⣿⣿⣿⣿⣷⡀⠀⠨⡇⠀⠀⢀⡀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠉⢴⡂
⠀⠀⠀⠀⣼⣿⠟⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣇⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⢉⠠⣦⣼⣿⡄⣯⣤⣿⡄
⠀⠀⠀⢸⣿⠿⠶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⣿⡄⠀⢸⣿⣿⣸⡿⣿⠀⠀⢨⡗⠈⠻⣿⢧⡏⠑⢯⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢼⠸⣿⣿⡿⣿⡿⣿⠿⠃⠀⢀⣠⣾⣷⠀⠀⢯⡚⠀⠀⠈⢻⣧
⠀⠀⠀⠀⢿⡀⠀⠀⠀⡀⠀⠚⠀⠀⠀⠀⠀⠈⠣⠟⢀⣻⣿⣿⣧⣀⡄⢠⣿⠃⢿⣿⢀⡀⢸⣿⡀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⢸⣧⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢊⣭⡾⣿⣯⡙⠛⠹⣿⣯⣀⡘⣿⣿⣇⣸⣦⣽⣤⡀⠀⠀⢻⣧⣀⡴⠃
⠀⠀⠀⠀⠸⢿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣷⣯⢑⣿⣼⢿⣷⣎⠑⣸⣿⣿⡟⢻⣿⣿⣛⣿⡿⠿⣷⠄⠀⣠⣿⣯⣤⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠓⠲⠶⢖⣒⣶⣿⡌⣷⣿⡏⣤⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢉⣸⣿⣿⣿⣿⣿⣶⣦⠘⣿⣿⣿⣿⣍⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⠏⠙⠻⠿⣎⣹⣿⣿⣿⣷⣦⣈⠹⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡄⢀⠀⢸⢏⣹⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠸⠀⠁⠈⠀⠀⠀⠀⠡⠀⠀⠀⠀⢠⠃⠀⠄
⠀⠀⠀⢠⣄⣀⠀⢪⣩⣆⢠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⡇⠰⠀⠀⠀⠐⠀⡔⠀⠁⠀⡁⠀⠆
⠀⣀⠐⢦⡀⠙⣷⣘⢳⠏⢰⢀⠏⠰⠀⠀⠀⠀⠀⡔⠡⡐⢤⠤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⠀⠀⠈
⠀⠸⣆⠀⠙⠄⠈⣿⢼⠇⠰⣎⡴⠓⠄⢀⢔⣤⡞⠀⢀⠇⠈⡆⢠⠔⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠈⠑⢄⢸⣆⢻⣶⡆⠐⣽⠁⠀⠀⢈⠘⣿⡇⠀⢨⡆⠀⢿⢀⡟⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⢦⠻⡈⣻⢃⢣⡏⠀⠀⠀⠀⠄⠻⣷⡄⠸⣆⣀⡞⣼⢁⠇⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⡄⠀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠣⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠣⠼⡿⠿⠋⠜⡿⠑⠁⠀⠀⠀⠀⠀⠀⠈⠯⣤⠀⢳⠆⠀⢰⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠘⠀⠀⠀⠀⠀⠀⡴⠒⠂⣴⠶⢄⡀⢳⣼⣇⣼⣿⠷⠚⠃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠑⣄⡀⠈⠷⣇⡙⢶⣍⢯⠉⣤⣤⡶⠿⢛⡋⡳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⠀⠹⢦⣦⡭⣷⣶⠽⠷⠦⣤⠤⡤⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣉⣀⣀⣑⣀⣐⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⣟⡛⢻⡟⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀⠀⣀⣀⣤⣄⣄⣀⣀⣀
⠀⠀⠀⢀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⢉⣉⡀⢀⣈⣉⣉⠛⠛⣉⣤⣤⠀⢩⣍⣙⠛⠶⣦⣄
⠀⠀⠀⣾⠃⢀⣀⡈⠉⠛⠛⠶⢶⣦⡴⠞⠋⠁⠀⢀⡖⠋⠉⠀⣴⠏⣉⠉⠉⣽⣛⣋⣴⠖⡲⠀⠀⢹⡇⠀⠀⠙⢿⣦⣀⣤⣴⠶⠶⠟⠛⣷⡄
⠀⠀⠀⢿⠀⠈⠉⠻⢿⣶⣤⣀⡀⠀⢠⣀⠀⠀⠀⠊⢀⣠⣶⠟⠀⣿⡁⠀⠀⣼⠀⠘⠃⠞⠇⠀⢀⡌⠳⢦⣴⣇⣀⠙⠏⣀⣤⣶⣿⡷⠂⣿⠁
⠀⠀⠀⠸⡇⠀⠀⢰⣿⣿⣿⣿⡏⠀⠠⣬⠳⣄⠀⠙⠛⣩⡿⠷⢿⡟⠃⠀⣈⣵⣀⣀⢀⣐⣶⣾⣋⣭⡙⣾⡷⣄⠸⡆⢀⡉⡻⠟⠉⠀⣸⡏
⠀⠀⠀⠀⢻⡀⠀⠀⣙⣿⡿⠁⠀⠀⣀⠩⣤⣌⠳⡝⣿⣏⣼⣿⣶⣿⡟⠛⣫⡽⠿⠭⠿⠶⠶⢿⡟⠻⠟⢛⣿⢈⡣⠶⣏⠤⠈⠃⠀⣴⡟
⠀⠀⠀⠀⠀⠙⣦⡀⢉⣿⠍⠉⢛⣒⡭⢤⣀⠙⢷⣝⣮⠻⠿⢿⣶⣤⠁⠘⢁⣠⣤⣤⣤⣤⣤⣤⣉⠃⠐⠫⢖⣩⣴⢺⡅⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⣸⠷⣾⣯⠥⢤⣘⣛⠃⠐⠾⣿⠶⢭⣝⡓⠀⠴⢤⡀⠀⠀⢻⣿⣭⣤⣤⣤⣤⣼⡟⠀⠀⠪⢭⣤⣖⣲⣶⢒⡂⢺⣷⡀
⠀⠀⠀⠀⠀⢠⠏⣞⠟⢻⡧⣤⣙⣛⣉⣻⠷⠒⠒⠦⢬⣽⣓⠂⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⠟⠉⠀⠀⠀⠀⢁⣉⣀⠡⢽⡭⠁⢺⣿⠃
⠀⠀⠀⠀⠀⣿⡈⠐⠀⣸⡧⢬⣭⣥⠤⢽⣯⠭⠭⢍⣛⣒⡦⠀⠀⠀⠀⠀⣀⣀⣠⣤⣼⠿⢶⣦⣤⣤⡀⠀⣐⠭⣈⠉⢹⣇⠀⠺⣿⡆
⠀⠀⠀⠀⢠⣯⠔⡢⡀⣸⣿⡀⢻⣿⡀⠘⣿⠀⠀⠄⠶⣖⣻⡅⢀⣤⣶⣋⣉⣁⣠⣤⣤⣤⣤⣀⡤⢍⣻⡾⠋⣑⣈⠁⢸⣇⠀⠙⣮⣷
⠀⠀⠀⠀⣼⠡⡪⢞⠁⣼⣿⡆⠸⣿⡇⠀⠻⣦⣄⠀⠉⢻⣿⣿⣿⣿⠀⠀⣿⣄⣧⣦⣦⣤⣤⣼⠀⠀⣿⠇⠀⣹⠃⠀⣿⡗⠀⠀⠩⣻⣧
⠀⠀⠀⠀⣿⢠⢖⡡⠀⣾⣿⠃⢐⣿⣿⣦⡀⠘⢧⠀⠀⠈⢿⣿⠯⣿⡄⠀⡟⡿⡟⡟⣿⢻⣿⢿⡀⢸⣧⣤⠾⠃⠀⢀⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢿⡶⣫⠖⢀⣿⡇⠀⠈⠸⣿⣿⣷⡀⠈⡇⠀⠀⠼⣿⣢⠙⣿⣄⣿⠉⠧⡇⠟⠈⠸⣸⣷⡏⣼⠁⠀⢀⣦⣿⠿⠃⠀⡀⠀⣜⣿⡇
⠀⠀⠀⠀⢸⡆⠁⠀⢠⣿⡆⣸⡟⡀⠉⢻⣿⡇⠀⢻⡆⠀⢸⣿⣿⣁⢻⣿⣿⠀⠀⠀⠀⠀⢠⢿⡏⣷⠿⢿⣿⣿⡏⠈⠀⠀⢱⣱⣦⣼⣿⠃
⠀⠀⠀⠀⠈⣿⣴⡄⡄⠻⡿⣿⣿⡅⠀⠘⠛⣗⠀⠈⢳⣄⠸⣿⣿⣯⣾⡇⠸⣦⣤⣤⣤⣴⠏⢸⣇⣿⠟⠀⠀⠈⠀⠀⠀⣠⣣⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠹⣿⣧⣷⢠⢠⠡⢻⣦⠀⠀⠀⠀⠀⠀⣈⠛⣶⣾⣿⣿⠏⢧⣀⣉⣥⣤⣭⣏⣀⣸⢻⣿⡆⠀⠀⠀⢠⣀⣰⡟⢳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢾⣷⣤⣿⣬⢠⢰⠀⢠⢃⢅⣧⣉⢹⡵⠛⣯⣷⣶⠶⠶⣦⣤⣤⡤⣬⣠⢟⡀⢦⢦⢰⣸⢻⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣧⣦⣧⣮⡿⠿⠿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣼⢸⡞⡄⢾⣸⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠋⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡟⠻⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⢠⡀⣄⠀⠀⠀⠀⢀⣠⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣦⣼⣶⣴⣾⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠰⣀⠀⣀⣤⣤⣤⣭⣵⣲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⣀⣠⢶⡿⠋⠉⣠⡀⠨⣍⡻⠁⠀⠋⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⢀⡰⣳⢫⣪⠖⠁⠀⠂⣋⡬⣽⠀⠀⠦⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⣠⠔⣱⣣⡿⠁⠀⠀⠀⡼⠁⠀⠘⠀⠀⣟⣀⡤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⡶⠚⠋⠋⠁⠀⠀⠀⠀⣸⣅⡀⠀⢠⠀⠀⣿⣎⢼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢻⡿⣫⠟⣶⣤⣀⣀⡤⢾⠙⠛⠉⠀⣿⠀⠀⣿⢿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⣼⠃⣀⠉⠁⠱⢾⡀⠀⠀⠀⢽⠀⠀⠹⢻⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡝⢜⡏⠁⠇⡀⠄⠁⠀⠳⣄⠐⠂⣾⠀⠀⠘⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢀⠜⢈⠜⣇⠀⠀⠇⠀⠸⣄⠀⣿⠓⠤⣮⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠡⠖⠁⠀⠘⣶⣤⣽⣦⡀⠈⠳⣿⠀⠀⣿⠀⠀⢠⣄⡀⠰⠖⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠒⠚⠻⡄⠀⢸⡆⠀⠹⠀⠀⢸⠈⣇⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⢀⣽⡛⠁⢀⣾⠇⠀⠀⠀⠀⢸⣰⠋⢣⣀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠤⠤⠰⡎⡠⠉⠍⠛⠙⢆⠀⠀⠀⠀⢸⡇⠀⠀⣿⡉⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⡷⠶⣶⣡⣄⣀⠀⠙⡇⠀⠀⢸⡄⡄⡀⢸⣯⡢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⣊⣩⣶⣿⣷⠚⢀⣀⣀⡜⠛⢠⡇⠀⠀⢸⠁⠘⡻⣿⡿⡿⢶⣭⣙⣂⣀⣈⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡶⠶⠾⠛⠋⠁⢸⢻⣇⠀⢇⠀⢘⣘⢦⣾⣧⠤⠤⢼⣬⣴⢇⣟⠀⢨⠀⠈⠉⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡠⠁⠀⢻⣦⡀⠁⠙⢪⡽⣾⠧⣤⣤⠴⢶⢚⣽⣼⠃⠉⠳⣄⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣠⣞⣀⣀⠀⠸⠯⡙⠻⢿⣿⠝⠀⠀⠀⠈⠀⢘⠟⢻⠇⠀⠀⠀⠀⠙⢢⣄⣀⣀⣀⡀⠈⠒⡀
⠀⠀⠀⠀⠀⢠⠃⠀⠀⣠⢶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠀⡀⣀⣤⠔⠀⣸⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠈⠉⠳⣶⠁
⠀⠀⠀⠀⠀⠌⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠚⠋⡏⠉⠁⣸⣲⣄⣿⡀⠄⠊⠁⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠸⣀⢀⣼⠃⠀⠀⠀⠀⠀⠀⢀⡠⠴⡶⠟⠋⠀⠀⢐⠀⠀⡈⠈⠁⠈⢻⠁⠀⠀⠀⠀⢀⣀⣠⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠶⠆⠒⠒⠚⠉⠉⠀⠀⣸⠁⠀⠀⠀⠀⠐⡂⠀⢇⠀⠀⡀⠄⣆⡤⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠂⠸⠦⠦⠴⠞⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⣾⣷⣶⣊⣽⣷⣄⡀⠀⠀⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣖⣛⣏⢻⣷⣿⣿⣷⣀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⠙⡛⣿⣷⣝⠿⣿⣿⣿⣿⣉⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⢁⣿⣟⣾⣝⡺⢿⣿⣿⣿⣍⠻⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⠁⠟⣛⣿⣷⣬⣓⣬⣿⡆⠘⡄⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡄⠀⢀⡄⠀⣴⣿⠿⣃⠀⠀⢸⡿⢿⣿⣿⣿⣿⡛⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⣟⣠⣾⡟⠁⢹⣁⣴⣨⣩⣿⣯⣿⣿⣿⣿⣿⣬⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⠋⠁⡴⠞⣿⣿⠋⠛⠿⢻⣿⣿⣿⣿⣯⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣿⣲⣶⣷⣿⣿⣿⣄⠀⣶⡇⣿⣿⣿⣿⢿⣿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⣸⣿⣿⣿⣿⣿⣿⠟⡦⠀⢻⣿⣿⣿⣿⢈⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⢠⣾⣿⢿⣿⡿⣿⡿⠀⣆⡀⢺⣿⣿⣿⣿⠞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠈⢻⠾⠛⠀⠘⠛⠐⢣⠱⣿⠻⢸⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⡌⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣩⣥⠈⣉⣹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠾⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠅⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣹⣾⡿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⣀⣀⣤⣦⣤⣤⡄⢉⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⢹⠃⠀⠀⠐⣰⠟
⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠈⠻⣦⡀⠠⣀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⠟⢻⡅⠀⢀⠀⠔⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣧⣤⡿⢢⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣧⠀⠀⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣤⣴⣶⡟⠉⠹⣿⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢊⣉⣩⡍⠙⠛⠻⣆⠸⣿⣧⣼⣿⣗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢁⡒⠛⠛⠿⢿⣧⢠⣆⣿⡛⣯⡭⠏⠛⠉⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡈⣽⣵⣁⡃⢀⣸⣿⣦⡿⣿⣿⣷⣶⣶⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣿⣿⣿⣿⢿⢺⡇⣿⣯⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣸⣘⣿⣿⣿⣿⣿⣿⣿⣿⠻⠀⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠈⢁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡟⡕⡻⢻⣿⣿⣿⣿⠙⢿⣍⠢⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⢏⠌⠀⠉⣟⢅⠙⣆⠻⣢⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡎⣀⡀⢀⣬⣺⣶⣌⣷⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⣶⣾⣿⣿⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⡿⣿⢸⣿⡟⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣷⣿⣽⡿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠹⡇⣿⣿⢹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢿⣿⠀⢘⡇⠻⠋⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣾⡇⠄⢸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⠇⠀⢸⡄⡄⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⢸⠀⠀⠈⡇⡇⠀⣿⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣷⡇⠀⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠘⣿⠀⠘⣷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢹⠀⠀⢸⡼⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣃⣀⣀⣀⣀⣀⣈⣀⣀⣀⣓⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠉⠁⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠴⠀⠀⠀⢿⣦⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠿⠦⣀⠀⠀⠒⠂⠀⠀⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡴⢤⠀⠀⠀⠐⠀⠀⠀⣀⣤⣦⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⡀⠀⠀⠀⠀⢠⣿⠋⠀⠈⠀⠈⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠀⢿⡿⠋⠠⡀⢠⣾⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠄⠀⠀⢸⡏⠱⠀⠘⠲⣤⣀⠀⣀⣱⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⠨⡄⠄⢀⣼⡄⠀⠀⠀⠀⢈⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⢾⣥⠀⠈⣶⣿⠿⢿⣿⣶⣶⣿⡟⠉⣤⡤⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣻⡬⠃⠀⣿⠏⢀⣀⣉⠛⠿⠿⢏⠀⢸⣦⠄⠀⠀⣤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘⠻⣆⣸⣿⡟⠷⠲⠶⠂⠀⠸⠋⠀⠉⠀⠿⠧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡻⢆⣈⠿⣿⣷⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢞⠉⠀⢼⡧⠙⠻⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠀⠀⠘⢓⢶⣤⣀⠉⣃⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣤⠁⠀⢸⣄⠄⠀⠊⠙⢫⣿⡟⢳⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠻⠀⠀⠀⠀⠘⢋⣽⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠁⢠⡇⢢⡀⠀⠀⠀⠸⠉⠉⠒⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠰⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠾⠁⣾⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠈⠓⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠀⠰⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠄⠀⠀⠴⣤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠞⠘⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⣤⣾⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⠡⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠠⢤⠀⠀⠀⠀⣧⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣾⣿⡏⠁⠀⠀⢀⠀⠀⠀⣀⣀⣠⣶⣤⣤⣤⣤⠞⠁⠀⠀⠀⢠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⡀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠒⣀⠀⣶⣾⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⡟⠛⣻⣟⣛⣛⣛⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣭⣭⣽⣿⣯⣯⣼⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣔⣂⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣣⣾⣀⣴⡅
⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⢀⠆
⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⡻⣿⣿⣿⣿⣿⡶⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢟⣽⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⡎⡻⣿⣿⣿⣶⣶⠶⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢿⣿⣿⣿⠿⣿⣿⣷⢸⣿⣿⣿⣷⡝⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⠏⠀⣸⣿⡿⣼⣿⣿⣿⣛⣋⣈⢻⡿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡷⢠⣿⣿⠿⣛⣿⣿⣿⣿⣿⣿⣧⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⢀⣤⣤⢄⣀⣈⠉⠁⣰⣾⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣄⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣄
⠀⠀⢀⣠⣴⣾⣿⡟⢫⣍⣙⣿⢿⣿⣿⣷⡼⠛⣩⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣟⣿⣍⠛⢻⣿⢿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⢠⣾⣿⡟⠉⠀⣼⣿⣿⣿⠿⢿⣷⣾⣭⣭⣥⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣮⢻⡆⠊⠙⠀⠀⡿⠸⢹⣿⣿⡿⣿⣿⢻⣿⣿⣿
⢻⣿⣯⠁⣠⣾⣿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⠀⠀⠀⠜⠁⠤⠟⣿⣿⡇⣿⡏⣼⣿⣿⣿
⡞⢡⡟⢠⡛⢻⣿⠆⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⢹⡿⣿⢠⡿⢹⡏⠁
⠻⠞⠀⠀⢹⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠩⣿⣿⣿⣎⠻⡼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠃⠑⠝⠦⠓⠈⠷⠄
⠀⠀⠀⠀⢸⡋⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣶⡶⢦
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣷⣯⣿⣿⠿⢿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣧⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣀⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⡏⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⡗⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⡏⢸⠹⠿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠒⠯⣅⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢆⣴⠀⢙⣧⢸⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⠟⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⡀⠀⠀⠛⠛⣡⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣛⣿⣆⠼⠉⠛⠓⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⣠⣴⣶⠀⠀⠀⠀⠀⠀⣧⣻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡾⠿⣿⠏⠀⠀⠀⠀⢤⣤⣽⡅⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡟⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣷⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠐⢹⣿⡿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⡎⠀⠁⠀⠀⠀⠀⠙⢷⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢼⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⣊⣽⣿⣿⡿⠦⣤⡄
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⣧⠿⣯⣿⡿⢷⢿⣿⣿⣯⠽⢿⡅
⠀⠀⠀⣠⠴⢟⣫⣴⡮⠀⣈⡁⠠⠿⠀⠉⢀⡄⠈⢿⢍⣰⣾⠆⠀⠞⠃⢬⣶
⠀⠀⠀⠐⠛⠋⠉⠉⠉⠀⠉⠁⠀⠀⠀⠘⠻⠡⠀⣼⠿⣴⣶⣶⡀⣾⡦⣰⣿⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣻⣥⣬⣭⣶⣀⣽⣧⣼⠋⠀⠀⢢⣤⠀⠠⢤⣄⣀⡀⠀⠀⡠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⣋⠩⣞⣿⣿⣋⣩⣽⣿⣿⣯⢿⡄⡈⠈⣿⡇⠀⠀⠀⠀⠉⠉⡻⠃⠀⠤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠛⠃⠚⠒⠋⠙⠛⠃⠙⠷⠛⠛⠛⠿⠛⠛⠓⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠐⠂⠤⢤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠈⠳⠙⠲⡔⠒⠛⠛⠛⠓⠲⠶⠒⠂⠈⠉⠉⠁⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡀⠀⠀⡷
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢒⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢰⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⢰⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠐⠮⣄⣀⣀⣀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠙⢾⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣾⠀⣉⠀⠀⣀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⡏⠛⣿⣧⡀⠈⠂⠶⠶⣦⣌⡉⠉⠉⠘⠛⠉⢻⠀⣼⣟⠷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠣⣼⣿⣿⣿⣶⣤⠶⡚⠅⠀⠉⠓⠀⠀⠀⠀⣻⡿⠿⢿⡤⢰⡀⠉⠁
⠀⠀⠀⠀⠀⠀⣰⣣⡴⠀⠀⠀⠀⠀⡰⠁⠀⠠⠖⠛⠛⠓⢦⣄⡛⢶⣤⠄⣉⣀⡀⠉⠀⠀⣠⠖⠓⠢⢄⢀⠀⠠⠣⣄⢠⠚⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⢁⡞⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠁⠐⠋⠐⠒⠀⠼⡆⠴⠤⡀⠀⠀⠀⢀⡜⠁⠀⠈⠑⠀⣁
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢢⡀⠀⣠⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢠⡤⠀⠀⠀⠀⠀⠀⢀⠀⠉⣤⡄
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡉⠂⢿⡀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣠⠂⢸
⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠿⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄⡀⠀⠀⠀⠀⣀⣠⠜⠀⢸
⠀⠀⠀⠀⠀⢸⢘⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣰⠃⢰⣿⣿⣿⣿⣷⣦⣀⣭⠙⣿⠻⠋⠙⠿⠛⠯⠉⢯⠭⠭⠓⢿⡟⠙⠋⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢿⠀⢸⣿⡟⠋⠛⢿⣿⣿⣿⣿⣿⠂⠀⣰⠷⣄⠤⠤⠤⠤⣆⠴⡟⠁⠀⠀⢀⡾⢦
⠀⠀⠀⠀⠀⠀⠀⢧⢠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢸⣿⢧⡀⠀⠀⠉⠙⢿⣿⣯⣴⣿⠁⠀⠀⡤⠀⠀⠀⠀⣠⠃⢀⣄⡠⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠈⠳⣄⠀⠀⠀⠀⠀⠀⠙⡆⠘⢻⡄⠱⠾⡤⠲⢀⠀⣿⣿⣿⣿⣀⣀⠢⠀⠀⠀⡠⠀⠄⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠠⢄⠀⢹⡄⠈⢷⣤⣄⣀⣰⣯⠤⠄⠴⠤⠦⣤⡏⣠⠀⢠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠺⣅⠀⠀⠉⠉⠉⠀⠤⠄⠀⠀⠀⠀⠀⠈⠁⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠞⣧⠚⡧⠊⠑⠸⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⣴⣆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⢠⣶⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣴⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣫⡖⣦⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⠵⣯⣼⣿⣿⢸⣧⡴⢞⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡔⢻⣶⣿⣿⡿⣟⠈⢷⠄⠙⠃⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣼⢋⣴⣉⣿⣿⠿⠋⣵⠄⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣴⠟⣵⣧⣼⡿⢋⡅⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣿⠀⠀⠀⠀⢀⡾⢫⣾⣿⠟⠉⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡇⠀⠀⠀⣠⡟⣱⣿⡿⣁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⣰⡟⣼⡿⢫⡄⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⢸⣟⢱⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣧⠀⢰⡟⣼⠟⣥⡌⠀⠀⠀⠀⠀⣠⣾⡿⠿⣿⣿⣿⠿⣿⣿⣶⣄⠀⠀⣿⣿⣋⣣⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡻⣤⣿⢱⠏⣦⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⢸⣿⣿⠀⠈⢻⣿⣿⣦⣼⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡴⠛⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⣦⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠛⡟⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢈⣿⣶⡄⠈⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣟⢛⡿⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⣷⣦⣤⣟⢷⣭⡻⢶⣦⣤⣤⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡋⣻⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⢻⣿⣿⣷⣬⡛⢶⣭⣛⠿⣍⣽⣿⠛⣿⠟⣿⠟⣿⣏⣹⣧⠼⣛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆⠈⠙⠻⠶⣮⣭⣽⣛⣓⣛⣛⣫⣭⣭⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣶⣴⠶⠤⠤⣤⣄⡀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠋⢀⡀⣹⣆⠰⡶⠶⢶⡿⢿⣿⣿⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⡧⠸⣿⣿⡿⢃⣠⠾⠿⠛⣷⣶⣍⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠋⢰⣟⠁⢠⡞⠋⣡⡾⣶⣵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⡿⠙⣿⡿⠛⢉⣤⡿⢿⣟⠿⠷⡦⠙⢰⣿⣿⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣁⠀⠹⣏⠀⣾⠋⣠⣿⡛⣿⣍⡽⣦⣀⣿⣿⣯⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢟⣿⣿⠟⠀⣴⡟⠁⢸⠇⣰⠿⡟⢧⣙⣻⣷⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⣿⡏⠀⣼⣿⡿⠁⠈⢠⣟⠀⠹⣆⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⠁⢹⣿⣷⡏⠀⠘⠛⠛⠛⣿⣶⠋⠩⠭⣉⠛⠻⣯⣝⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⡀⡈⠟⣿⣇⣶⢀⣠⠄⠀⠹⣿⣦⡀⠵⠒⡨⠀⢈⡿⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⣿⣷⣷⡀⠙⣿⣿⣿⣿⣴⣦⣤⠌⠻⣿⣿⣾⠿⠿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣏⠛⢦⣌⠛⢿⣿⣿⡟⠁⠀⠀⢠⣀⣀⣀⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠓⢄⠙⢿⡇⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠂⢀⢈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⡀⠀⠉⠸⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⡄⠉⠀⣠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠠⣄⡈⣩⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠀⢠⠞⣷⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⡇⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠙⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠑⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣠⡤⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⣠⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟⣰⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⡀⠀⣀⡼⠀⠀⠀⠀⠀⠀⣴⢋⣭⣦
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠋⠉⠀⠀⠀⠀⠀⠀⠀⡽⣾⡿⠟
⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠁⣀⠴⠛⠉⢉⡷⠶⠤⣄⡑⠢⣄⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⡼⢠⠟
⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠉⡀⠠⢔⡪⠉⠀⠓⠄⠀⠈⠙⠺⢷⣄⠀⠈⠱⡀⠁⠀⠀⡼⢡⡯⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠆⢀⡀⠀⠉⢶⡿⠁⠀⠀⠸⢀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⡼⢁⣾⠆
⠀⠀⠀⠀⠀⠀⠀⢸⡟⡀⠙⠻⠃⠀⠀⠀⠀⢀⠞⢀⠹⡶⠀⢀⠀⢸⢇⣢⣠⡀⡴⢁⡾⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⣷⣷⡀⠠⠀⠀⠀⠀⠀⠞⣰⠏⠀⠠⠂⢸⣿⣿⣿⣿⣿⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⣿⠟⠁⠀⣠⣾⡟⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡛⠢⢅⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢁⣴⣟⠀⠐⠈⢉⠀⠁⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⢀⣼⣿⣄⣀⣠⠀⠀⠀⣠⣿⡿⢿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⡿⠀⠀⢀⡼⡿⠉⢀⠈⠉⠈⠻⣿⣿⣿⣦⡀⠀⠀⠀⡀⣤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢟⣿⣿⣿⡿⣁⢀⣴⡟⠀⠀⠀⠀⢀⣀⠀⠀⠋⢻⣿⡏⠁⠀⠀⣀⣿⣳⡃⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣾⣿⣿⠏⠅⣩⣿⣿⡇⠀⠀⣀⠀⠀⠈⠟⠀⠀⣸⣿⡇⠀⠀⠀⠈⣯⣭⡀⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣏⡵⡀⢸⣿⣿⣯⠁⡆⠘⠃⠠⠀⠀⠓⠀⣄⣿⠉⠃⠀⠀⠀⢓⢻⣿⠃⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⠁⠐⢠⣿⣿⣿⣿⣷⠀⡠⠐⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢿⣈⣸⠇⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣰⣿⣿⣿⣿⣿⣿⠀⠰⠄⣀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⢩⣵⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠶⢿⡿⢿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠓⠀⠉

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

⠀⠀⠀⠀⠀⡠
⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠂⢀
⠀⠀⢸⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠒⣐⡴⠖⠒⠛⠉⠉⠉⠉⠉⠙⠒⠃⠉⠛⠀⢈⣉⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠈⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡎⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⠈⠉⠒⠦⣤⣤⣀⣀⡇⠀⠀⠀⠀⠘⡆⠀⢀⡠⠐⠀⣀⡀⠀⠀⣵⡀⢸⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠢⠤⣀⠀⠀⠉⢻⠀⠀⠀⠀⠀⢁⡴⠋⠠⠒⠉⠁⠀⠀⠀⠀⡷⠾⡷⠾⠛⠛⠶⠖⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠙⠢⣼⠂⠀⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠋⠀⠘⠀⠀⠀⢀⣀⣀⣀⡄⠤⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠠⢘⡻⣆⠀⠀⢀⡟⠁⠀⠀⠀⢀⣂⣠⣾⡇⠀⠀⠤⢤⣀⠠⣾⠉⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠒⢩⢴⣿⣦⣴⣾⣇⠀⠀⡀⢀⣾⡏⠉⠹⠋⠀⠀⠀⢠⣨⠛⠦⡉⠀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠛⠀⠀⠠⡙⢯⠩⠁⢿⣿⣿⡇⡡⠖⠀⠀⠀⢨⣿⡀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣅⠀⠀⠀⠀⠀⣀⣀⠒⣾⠟⠧⠤⠤⠴⠒⠒⠢⠸⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⣤⠀⡆⠀⠀⠀⠀⡀⠀⢧⠈⠰⠂⢀⡼⠻⠷⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠟⠑⠁⠀⠀⠀⢾⠉⠻⡧⢀⡀⠴⣯⠔⠀⠛⡿⡖⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠏⠀⠀⠀⠀⠸⡃⠐⢹⣉⠷⠐⠛⠒⠒⠟⠁⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡉⣆⠑⠢⠤⠴⠞⠉⠙⠁⠀⡸⠁⠀⢀⣠⠤⠤⠄⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣝⣒⡶⠖⣂⣀⣤⠶⠚⡅⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢏⠉⠁⠀⠙⢦⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠤⣤⣤⡤⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠶⠖⠒⠀⠀⠈⠒⠀⠐⠲⠒
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣳⣤⡤⠾⠷⢄⠀⠠⡀
⠀⠀⠀⣀⠤⠂⠀⠀⠀⠀⠀⢀⣼⠿⠁⠀⢀⣀⠀⠐⢤⡀⠀⠀⠀⡜⠁⠀⠈⠑⢦⠀⠀⠀⠀⠐
⠀⠐⢯⣀⠀⠀⠀⠀⠀⢀⣠⠞⣇⣀⠀⠀⠀⠈⠓⠦⠤⡙⠂⠀⠀⡄⠀⠀⠀⠀⠀⡑⠠⠤⠤⠐
⠀⠀⠀⠈⠙⠓⠒⠖⠛⠉⠀⠙⠋⠉⠉⠙⢶⡄⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⢰⣆⠠⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠉⣽⣦⡀⠀⠀⢀⣀⣤⣤⣀⣠⣯⣈⡆⠀⠈⢆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡤⠂⠀⠀⠤⠒⠛⣉⡿⠷⠾⠛⠉⠙⠛⠉⠻⣏⡁⠀⠈⢢⠀⢆
⠀⠀⠀⢠⣠⠊⠀⠀⠀⠀⠀⢣⣀⡀⠀⠀⣠⠊⢁⡤⠖⠒⠀⠀⠀⠤⠄⠀⠉⢙⠢⡀⠺⡀⠘⡄
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠹⠟⠡⡏⠁⠐⠁⠀⠀⠀⠀⠀⠈⠳⡀⠘⠁⠀⢣⠘⡄⠘⢆⢣
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠈⢿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠑⢶⣒⣏⠹⠷⢦⣀⠀⠀⠀⣰⣀⠀⢀⣠⠆⠀⠀⠀⠈⣇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡴⢻⡏⠙⠒⠒⠋⠁⠀⠀⢀⣀⡤⠍⠉⠉⠉⠉⠛⠿⣅⠀⠀⣆⠀⠀⢸⡄
⠀⠀⣧⠀⠀⠀⠀⠀⢠⠞⢠⡟⠣⡀⠀⠀⠤⠤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠁⢠⡇⠀⠀⠀⡇
⠀⠀⠸⣾⠀⠀⠀⠀⠏⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⡛⡇⠀⠀⠀⡷
⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⠀⣠⣴⣦⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠹⣄⠀⠀⠀⢸⡄⠀⣀⣤⣾⠶⠶⠛⠉⠉⠉⠉⠉⠀⠀⠀⠈⠲⣤⡄⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠛⠖⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠄⢂⡒⠒⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡹⡀⠀⠀⠐⢻⣷⣤⠄⠤⣾⡀⠺⠗⠀⣿⠛⠻⢶⣄⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣃⣷⠀⠀⢰⣾⣿⣿⢦⣤⢰⡷⠒⠀⠀⣿⡀⢀⣰⠏⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣀⢹⡏⠀⣠⠀⠀⢈⣽⣿⢷⣿⣿⣿⣄⣙⢯⢿⡟⠋⠁⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠠⣿⣼⠁⠊⠉⢻⠀⢸⣿⠁⢸⣿⡻⣿⣿⣷⣖⣺⠴⢤⡀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣰⠀⠠⠞⣄⢾⡇⠀⣾⠴⢿⣋⢻⣿⣿⣷⡋⠀⢹⡄⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⣿⢸⣿⣾⠟⠚⣦⡿⣼⠿⣿⣿⣷⠀⠸⣧⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⡟⡇⠀⠀⠀⢿⣿⣿⣇⠀⠀⢅⣉⠀⢿⡟⣿⣿⢧⢀⣿⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢢⣼⣥⠩⠀⠀⢀⣄⣽⣿⠉⢴⡀⡾⣿⡷⢸⡟⣿⣿⡎⢸⡿⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣸⣿⡟⢲⠶⠀⠸⣿⠟⢻⣆⠈⢷⡇⠉⠁⢸⡟⣛⡷⠇⣾⡇⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢹⡿⣿⠙⣶⣶⡶⣦⡀⢸⡿⢀⣘⠁⣼⣶⣾⣟⡿⠤⣼⡿⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⢹⣾⠿⠿⠿⢛⣶⣿⡟⡛⣟⣦⣿⣿⣿⣿⣿⠆⣿⠂⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢏⣿⠟⣶⣾⡟⣿⡟⡻⢿⣷⣾⣿⣯⢿⣻⠟⢷⣠⠏⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣬⣿⡆⠿⠉⣧⠸⣄⠁⠀⣿⣿⣿⣷⠁⣿⣿⣿⠁⠀⠇⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⣧⣤⣼⣿⣧⢻⣄⣴⣟⣸⣿⣯⡆⣻⠿⠿⠀⠀⠃⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⣍⡿⠃⣿⠈⢿⡜⠻⠿⠀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠁⠀⠀⠁⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⣒⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣴⣿⣿⣿⣶⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⣿⣿⣿⣿⣿⣿⣷⣝⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢿⣿⣿⣿⣿⣿⣿⣷⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠇⣻⣿⣿⣿⣿⣿⣿⣿⣧⡻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡴⠋⠉⠀⠉⠒⠤⠤⠄⠈⠛⠳⣦⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⡀⠘⣧⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡾⠃⢀⠤⠐⠀⢀⣠⡾⠟⢻⠀⠘⡀⠸⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⡔⣥⣤⡤⡄⠀⠉⠀⠛⠃⠀⠀⠰⡀⢻⡜⣇⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⡳⢿⡟⠔⡟⢀⠀⠀⠀⠀⠀⠀⢀⠱⠀⢿⣌⣓⠦⠤⠖⣫⡔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣾⠁⠀⠓⢀⠠⠀⠀⠀⢠⠊⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡝⣦⡀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⢤⣶⣿⣿⣿⡿⠟⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⡟⠛⣛⢫⣴⣋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣙⣓⠻⣇⠀⠀⠀⠀⢀⠤⠊⠀⠀⣷⣿⣿⣷⣦⣭⣭⣙⣓⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣓⣉⣿⣿⠄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠙⠒⠮⠭⣝⣒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢱⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⣼⡟⠻⣿⣿⣿⡿⠋⠀⠀⢀⠀⠀⢈⠓⢦⠤⣼⢣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣵⣶⣿⣿⣿⣿⡿⢀⣀⣀⠤⠒⡡⠾⣷⣶⡿⠛⠁⣆⠀⠀⢀⠀⠙⢆⢄⠁⡤⠃⠈⢳⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡱⢿⣿⣿⣅⣼⣿⠇⠒⠀⠒⠒⠉⠀⠀⠀⠀⠀⠀⡆⢰⠑⠄⠀⠹⡢⣀⠆⠉⠀⠀⠀⠀⠻⣎⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡵⠋⢀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠣⡈⠢⣠⢖⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢾⠋⡠⡾⢁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠢⠤⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣝⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡾⡱⠊⠉⠌⣀⢠⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣿⢸⡃
⠀⠀⠀⠀⠀⠀⢠⢞⡼⠁⠀⠀⠀⠀⠘⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠗⠒⠒⠒⠂⠈⢻⡎⡇
⠀⠀⠀⠀⠀⢠⢏⡾⠁⠀⠀⠀⠀⢀⡴⢋⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣼⢱⠃
⠀⠀⠀⣠⠴⢋⡞⠁⠀⠀⠀⠀⣠⢞⣴⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣰⢇⡏
⠀⠀⢰⡇⡞⠉⠁⠀⠤⢀⠀⢰⢱⡋⡟⣼⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣴⢏⡞
⠀⠀⠀⣇⢧⠀⠀⠀⠀⠀⠉⠢⣦⢻⡇⣿⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣠⣾⣿⡜⡆
⠀⠀⠀⠘⣎⢧⡀⠀⠀⠀⠀⠀⠈⢷⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠁⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⢻⡀
⠀⠀⠀⠀⠘⢧⡻⣄⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠣⣀⠀⢀⣶⣿⣿⣿⣿⣿⣿⡎⢧
⠀⠀⠀⠀⠀⠀⠙⢮⡳⣦⣀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣭⡓⠶⣤⣀⣸⣣⡀⠀⢀⠐⣼⢋⣓⠶⠶⠤⠤⣤⣤⡤⠶⠶⣒⣫⣭⡻⢤⡀⠀⠀⢀⣴⠛⠿⠿⠿⠿⠿⣛⣛⣛⣙⣛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠭⣍⣛⣉⣛⣛⣚⡵⠋⠈⠉⠉⠙⠛⠒⠒⠚⠉⠉⠉⠀⠀⠙⠦⢭⣓⣒⣛⡭⠞⠉⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣯⣽⣤⣤⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠁⠉⠉⠁⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⣀⠀⢀⣀⡀⢹⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⠿⠀⠉⠈⠁⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣯⣠⣶⣄⣠⣿⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣍⣩⣽⣿⣿⡏⠀⠑⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⣦⡀⢀⡀⠀⠉⠓⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣍⣉⣠⡾⠟⠋⠀⠙⠛⡀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣩⣛⠛⢙⣋⠋⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠑⠈⠀⠉⠙⠇⡺⠋⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣦⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⣦⣀⢀⣀⣀⣴⣿⣿⠿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⢀⡠⠞⠁⠈⠙⠛⠿⠻⠋⠉⠀⠀⢹⣿⡿⢷⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠙⢧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣁⣀⠀⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⡗⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⣶⠦⠘⠷⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣂⠀⠐⣄⠀⠀⠀⢀⠀⠀⣀⣴⠋⡀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠿⠷⠶⠿⠟⣿⣿⠿⢟⣿⣿⣿⡌⠗⢀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠾⠛⠉⣡⡶⣿⣿⡿⢻⠃⢸⠏⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⠀⠀⠀⠀⠀⠈⠉⣴⢿⣿⠃⡞⣠⠃⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⢀⡼⠃⣾⢃⡜⠛⠋⠀⢀⡾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⢀⡾⠁⠸⠃⠈⠟⡁⠀⠀⣸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⢠⠀⣰⠀⠊⠀⠀⠁⠀⠀⠐⠿⣷⣴⠏⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⠀⠀⢀⡟⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠟⠁⠀⠀⠀⠀⣼⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠰⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣈⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢶⣦⣀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣤⣄
⠀⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣢⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⡿⠓
⠀⠀⠀⠈⠛⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣿⣶⣦⣀⠤⣀⠀⢻⣿⣿⣯⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⢴⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣶⣬⣭⡍⠻⣿⣿⣿⣷⠒⠒⢂⣀⣽⣤⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣄⠀⠙⢿⣿⡇⠀⢀⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⣿⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⣡⣾⣿⣿⠏⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣫⣷⣾⣿⣷⣿⡉⢀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⠿⠿⣿⣿⣷⣶⣶⣶⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠄⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣫⠀⠀⠀⢸⣿⢿⣟⣵⣶⣉⣻⣿⣿⣿⣿⣷⣄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⣠⣤⣄⣸⠟⠼⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠆⠀⠈⠻⠿⠃⠀⠐⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣶⣶⣤⣀⠀⠀⣀⠀⣀⣀⡀⠀⠀⠀⣠⣴⣶⡿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣈⣻⣿⠿⠟⠻⠉⠉⢉⡉⠙⡛⠿⠿⣿⣿⣏⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⡁⠖⠹⡛⠘⠋⢁⡀⠈⠉⢉⣁⠈⠿⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠿⠀⠀⢀⡈⣀⡀⠀⠋⢀⣈⣁⠀⠀⠙⠻⢿⡃
⠀⠀⠀⠀⠀⠀⣴⠤⠄⢤⢴⡆⢸⡟⠂⠀⣘⣥⣤⣽⡃⠀⠀⢘⣿⣬⣉⣦⡀⠀⢿⣷⢀
⠀⠀⠀⠀⠀⠀⢸⣶⠀⡄⣸⡁⣾⣇⠀⠀⠀⠀⠀⠛⠗⠀⠀⠈⢿⠁⠀⠀⠀⠀⢸⣿⢻⡿⢛⠒⢿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣶⢾⠇⠉⠸⣿⣇⠀⠀⢀⡈⠁⠀⠀⠀⢀⠈⡉⠀⠀⠀⢘⣾⡿⢸⣥⠤⠠⣼⠂
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣄⣉⣠⠹⣿⣦⣤⣾⣷⣆⢻⣿⣶⣿⠁⣷⣄⣀⣰⣾⠏⢁⣿⣿⣦⣶⡏
⠀⠀⠀⠀⠀⠀⣸⠛⠁⢹⣿⢿⣿⣀⣈⠙⠿⣿⣿⣿⣿⣻⣿⣿⣾⣿⣿⣿⡿⠋⠰⣾⣷⢹⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠾⠟⠈⠉⠁⠈⠁⠀⠈⠛⢿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠙⠛⠻⣿⣯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠷⣄⠠⡆⠀⠀⠈⠍⠉⠁⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠭⠒⣪⠟⠒⢺⣀⣈⠛⣣⡰⠆⠖⠂⠋⣃⣈⣴⠏⡨⣤⡆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⡁⠜⠉⠓⠉⠀⢀⡆⠀⠀⠀⠀⠁⢀⡽⠌⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⣠⣀⣀⡀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⢀⠀⠀⠀⠀⠀⣀⡴⠞⣋⣭⠥⠤⠀⠀⠤⠬⢭⣝⡛⠶⣤⡀⠀⠀⠀⠀⠀⠀⣼⣷⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣶⣶⣶⡿⠋⢴⣿⣿⣿⣿⣿⣧⣰⣿⣿⣿⣿⣿⣷⠌⠻⣿⣶⣶⣶⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡝⢿⣿⣿⣿⠏⣠⠞⣿⠿⣛⣩⣭⣽⣿⣿⣯⣭⣭⣙⡻⢿⡛⢦⡈⢿⣿⣿⣿⠿⣣⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠄⠈⠉⠁⠚⠁⠘⠁⠾⠛⠋⣉⣭⣿⣿⣯⣍⡙⠛⠻⡆⠙⠀⠙⠈⠉⠉⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⣠⡖⠀⣼⡀⡀⠀⢀⡴⠋⠉⢀⣿⣿⡄⠈⠉⢶⣄⠀⢀⠀⣷⠀⠐⣦⠀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡟⣠⡆⢻⣷⣿⢿⡀⠀⢰⡆⠞⢋⡏⢻⠀⡄⠀⠀⣼⢿⣶⣿⢨⣧⡈⢷⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡟⢰⣿⠇⣹⠷⣬⣬⣑⡀⠘⢷⣉⠘⠃⣈⡼⠃⡀⣊⣥⣮⠾⣏⡀⢿⣷⡈⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣽⣿⣿⠃⣿⢿⡆⣟⢷⡽⠢⠉⣿⣿⠛⠙⠙⠸⠙⠛⢿⣿⡍⠔⠯⣾⢟⡄⣿⣿⡇⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⢡⣿⣿⣿⠘⣿⠈⡿⣿⣶⣴⣞⠃⠈⢙⠉⠡⠀⠀⠌⠉⠚⠉⠀⢻⣶⣴⣿⡾⡏⢸⡇⢸⣿⣿⡿⡆⢸
⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⣿⣿⣿⡈⣿⣆⣻⣿⣿⢯⣘⢦⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⣡⢏⣬⣿⣿⣷⣃⣿⡇⣼⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣼⣿⣷⣾⣿⣷⣌⡳⠅⠰⣄⡈⠢⠜⢀⣠⠆⠠⢖⣫⣴⣿⣿⣶⣿⣿⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣶⢻⣿⣶⣭⣒⠀⠙⠿⣶⣶⠿⠋⠀⢐⣯⣵⣾⣟⣷⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣾⣿⣯⣼⣿⣿⢶⣬⣤⣤⣿⣿⣤⣤⣤⡶⣾⣿⣿⣬⣿⣷⣶⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠙⣿⣿⣿⣿⣿⣿⣿⡀⣸⣦⣶⣶⣶⣶⣤⣧⢀⣿⣿⣿⣿⣿⣿⣿⠟⠙⠉⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠃⠈⢻⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠛⠁⣹⠇⠀⢀⣴⣣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣣⢄⡴⢃⣾⣴⠿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⠈⠉⠀⢴⣿⣿⣿⣿⣿⠋⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣙⡷⠲⠶⠺⣟⣼⣿⣿⡿⠃⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠿⢛⣿⣿⡟⠻⢟⣿⡟⠁⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣩⠟⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⣿⣿⣿⡿⠋⠔⠋⠠⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣼⡷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡃⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣻⡏⢻⡾⠿⣒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⡀⣿⣹⣿⣰⡿⢆⢰⡆⢀⣀⠀⢀⣄⣷⢟⣼⣡⣿⠀⠀⣤⣀⢻⣇⣙⣗⣶⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡟⢏⣿⡇⣿⣿⣿⣵⣇⢹⠃⠸⡿⢿⡿⢿⠃⣡⣞⣿⢿⣰⣷⡝⣿⣾⣿⣿⢦⣸⠎⣿⢿⣶⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⢿⡇⡟⣿⣇⣼⣿⡿⠙⠇⣿⣿⢻⡞⣧⣄⠀⠸⡿⢸⡧⠱⢟⡉⣸⣿⡿⣿⣿⣿⣟⠻⣦⡙⠿⣽⣼⣿⢻⠿⠟⠓
⠀⠀⠀⠰⠶⠶⠛⠡⠾⠦⠽⠿⣿⣿⠿⠷⠦⠀⠋⠿⠾⠿⠿⠿⠾⠷⠿⠦⠶⠿⠍⠁⠀⠉⠛⠉⠉⠁⠘⠓⠶⠿⠧⠬⠷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⣉⣉⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣞⣿⣿⡷⣶⣄
⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⣠⣿⡿⠻⣿⣷⣿⣯⣽⣿
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢰⣿⡟⠀⠀⢹⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠘⣿⠁⠀⢀⣿⣿⣿⡿⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣶⣿⣏⡻⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢯⡀⢩⣽⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠙⠾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠀⠀⠀⠀⠀⢿⣿⡇⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠈⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣽⣿⡀⠀⠀⠀⠙⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⣶⠄⠀⠀⠀⠀⠈⠓⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⢤⣄⡤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⡤⢂⣌⡫⠥⠄⠠⠶⠖⠲⠶⠄⠀⠒⠰⠚⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠶⡀⠀⢁⢐⣒⣠⣶⣠⣴⡾⠶⠟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⡄⠀⡄
⠀⠀⠀⠀⠀⣰⣧⣾⣄⣴⣵⣿⣤⣼⣿⡿⠿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠦⣀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠛⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⢀⡂⠀⠀⠀⢀⠄⠀⠀⢀⣀⣼⣷⣶⣶⣶⣿⣶⣶⣶⢤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡠⠀⠈⠐⠀⠀⣠⣾⣋⣽⠟⠛⢛⣩⠋⣩⣉⡙⢻⣿⣻⣿⡾⣿⣶⣆⡀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠒⡠⣀⣄⣴⣿⢿⣛⣏⣵⣿⣼⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⢻⣿⣷⣞⢪⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠂⢀⠤⢠⣀⣜⣻⣯⣟⣋⣵⣾⣿⣿⣿⣿⡿⠟⢟⣿⢟⣹⣿⣻⣛⠛⢿⣿⣿⣯⣾⣿⣯⣛⣿⣿⣯⣝⡠⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⣨⢫⡜⠛⣼⣿⢿⢿⣾⣿⢿⠇⡐⢷⣼⣷⣿⣿⣿⡛⠉⠉⠒⡿⠉⠻⣿⣿⣿⣿⣻⡯⣩⣿⣿⣍⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⣹⡼⢈⣼⠟⠋⠁⢸⣿⣿⣾⣤⣽⣿⣿⣿⣿⣿⣿⡅⠀⠀⢠⣷⡶⠀⠹⡟⠿⣿⣿⣿⣿⣧⣭⣹⣷⠉⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠸⠃⢒⣴⡟⣠⣾⡃⠀⠀⠀⣼⣿⣿⢙⣽⣿⣿⣿⣿⣿⣿⣿⣿⡗⠶⣿⣿⣗⠂⢤⡇⠀⠙⠻⣿⣿⣿⣿⣿⠞⠁⢀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⣀⠀⠔⢛⣯⣼⣿⣿⡇⠀⢀⠀⢸⣿⣿⣿⣠⣽⣿⡿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠗⠀⣸⠇⠀⢉⣫⣿⣿⣿⣿⣟⠛⢋⠁
⠀⠀⠀⠀⠀⠀⡀⠀⣤⣼⣿⣿⣿⣼⣯⣆⡨⠆⠈⣿⣿⣟⡋⢽⣿⣷⣭⣙⣻⣿⣿⣿⣿⣿⡿⠂⣬⡟⠀⠀⣠⣿⡿⣫⠿⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠻⠿⣿⣿⣿⣿⣓⠅⢀⠈⠿⣿⣿⣾⣿⣹⣿⣿⣿⣿⣿⣿⢿⢫⣴⣾⢟⣁⣴⣾⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠙⠻⣿⣾⣧⣄⣀⡈⠻⢿⣿⣿⣷⣄⣯⣿⣾⣾⣶⣿⣿⣷⣞⡯⢿⠛⠛⠀⠉⠀⠀⣤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⢶⠶⠶⢷⣿⣿⣿⣿⠿⣿⡿⣿⣿⡛⢯⠄⠒⠉⠁⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠠



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠐⣿⣶⣶⣶⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠐⣿⣿⣿⣿⡿⠇⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⣤⣤⣤
⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⢀⣴⣾⣿⣧⣀⣠⣤⡤⠖⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⢀⣶⡿⣿⡿⢿⣶⣄⠀⠀⠀⠋⢻⣿
⠀⠀⠀⣿⠋⠉⠀⠀⠀⢠⣴⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢠⣿⣿⣹⣦⣀⢀⣽⣿⠶⠀⠀⠀⠘⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢤⣾⣿⣿⣿⠿⢿⣯⠙⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠉⠛⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢸
⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣿⣆⡀⠀⠀⠀⠀⢻⣿⣿⣽⡿⢻⣿⣿⣿⡆⠀⠀⠀⣼
⠀⠀⠈⣿⡿⣿⣶⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣀⣴⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣾⣿⣿⣿⣿⣷⠀⠀⣸⣿
⠀⠀⢠⣿⡇⠀⣹⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠷⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⢸⣿⡇⠀⠋⠀⠀⠀⠀⣸⣿⣿⡿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣾⣿
⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⣿⣿⣿⣷⡄⠀⠀⠀⣸⣿⠛⢿⣿⣿⣿⠀⠀⠀⠀⣴⣿
⠀⠀⢀⣿⡀⠀⠀⠀⠀⠠⠿⣻⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⢿⣿⣦⢸⣿⣿⣿⠀⠀⠀⠸⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⣿⣿⣿⡁⠀⠀⠀⠀⣾⣿⣿⣧⣿⣿⣿⢷⡄⠀⠀⠀⠹
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠐⣿⢻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⢹⣆
⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠁⣾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠂⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⠈⢧⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠛⠀⠀⠈⠉⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠃⠀⣾⣿⡿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⢳⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⡾⠁⠀⢠⣿⠟⠀⣸⣿⡟⠛⢻⣿⡧⠀⠀⠀⠀⠷
⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⡀⠀⠀⠀⠀⠴⣿⣿⣿⠷⠶⢾⣿⡷⠀⠀⠀⢀⣤
⠀⠀⢸⣿⣀⣀⣠⣤⣤⣤⣤⣤⠤⣿⣿⡯⠷⠖⠀⠐⠒⠒⠒⠒⢒⣿⣷⠀⠀⠀⠀⠸⣿⣿⣦⠤⠤⠀⠀⠀⠀⠛⠀⠀⠀⠀⠘⠀⠀⠀⠚⠛⠛
⠀⠀⠘⠿⠧⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣞⣛⣻⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢾⣟⣛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡯⠤⠼⣧⣿⣿⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣿⣿⣼⠧⠤⢼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡾⠛⠛⠯⣵⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⣿⠵⠛⠛⢷⣶⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠦⠤⠤⠬⢦⡀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢀⡴⠥⠤⠤⠶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⣀⣀⣀⣀⣙⣦⣀⠀⠙⢷⣄⠀⠄⢠⠀⣠⡾⠋⠀⣀⡴⣋⣀⣀⣀⣀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠙⢧⣾⣿⣼⠋⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣈⣼⡟⢹⣷⣥⣄⣤⣤⣶⣶⣶⣶⣶⡶⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠃⠉⢋⠭⠉⠉⠉⡈⢿⣿⣼⣿⢁⠈⠉⠉⠤⣚⠉⠉⠛⠳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠒⠀⠀⡠⠎⠀⢀⡤⠊⡀⠋⣿⣿⠉⢀⠁⠀⡀⠀⠘⢤⡀⠀⠐⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠔⠂⢀⣄⣠⠤⠲⠋⠀⠐⢀⠂⣿⣿⡀⠀⠣⠀⠈⢓⠠⢄⡠⠀⠀⠂⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⠊⠀⡌⠀⣴⠃⢀⠎⢀⢂⠎⢸⣿⠐⠔⠀⠱⡀⠈⢦⠀⠱⡀⠀⢦⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣞⠀⣠⠁⢀⠎⠀⡠⠂⠀⠘⠏⠀⠈⢦⠀⠐⡄⠀⣃⠀⣱⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣾⣤⡾⠁⠀⠀⠀⠀⠀⠀⠈⠳⣤⣼⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⣀⣨⣿⡻⢶⣶⣶⣶⣛⣛⣛⣛⡛⠓⠶⣶⣶⣶⣶⡿⣿⠍⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⢠⣍⡻⣦⠙⣿⡟⣙⠛⢲⠦⣍⣳⣤⢦⡉⠙⠿⣿⣟⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢦⡀⠀⢿⣧⢸⡇⡌⠳⠮⠿⠛⢶⣮⠉⠯⢄⣳⣠⣄⠈⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣧⡙⢦⣀⡿⠋⣴⣞⡥⢀⡀⡀⠻⢅⢄⡀⠤⡻⣿⡏⠓⠟⣈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⠃⣿⠹⣦⡟⠁⢨⢟⣉⣀⣀⣍⣠⣤⣤⣀⣿⡄⠙⡿⡇⠂⣼⣁⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⢸⠀⡏⢀⢸⡇⣿⢛⡩⠤⣀⣹⢿⣿⠻⠿⣿⣿⡛⠀⠁⠁⠀⣿⣿⢿⡃
⠀⠀⠀⠀⠀⠀⠀⣸⢳⣇⡜⣆⢿⢀⣼⡇⣧⣏⣴⣿⠽⠇⢀⠙⠻⠞⠿⣿⣿⡀⡀⠀⠀⣿⡶⠞⣷
⠀⠀⠀⠀⠀⠀⠀⣏⣼⠟⠀⠘⣾⡿⣺⢇⢸⡇⣭⣭⡓⢦⡀⠓⠒⠒⠀⠈⢻⣇⣸⠀⠀⠘⢷⢖⠏
⠀⠀⠀⠀⠀⠀⠀⡟⢹⠀⡄⢠⣿⣼⠁⣸⡈⢧⡙⠶⠳⠧⠙⠂⢀⡤⢊⠥⠊⠈⠛⠀⠀⠀⢠⣟
⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⠘⣾⣻⡏⡷⠴⣷⣬⣙⣛⣃⡒⢶⣦⣾⣴⠁⠀⠀⠘⣷⣶⠶⡶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢁⢨⡿⣆⢻⡏⣷⣷⡄⢟⠉⠛⠿⢿⣿⣮⢻⣿⣯⠭⡐⠀⠀⡈⠙⢧⠞⢋⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠙⢶⣿⡼⣿⠘⢦⠀⡆⡠⢚⠛⠻⣷⣿⣿⡧⠂⠀⠀⠀⣠⣞⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠙⢿⣿⣧⢠⡁⢰⣴⢣⠀⡼⠋⠈⠀⢹⡷⠶⠾⠿⠛⢻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠢⣄⠀⠙⢿⣧⣷⡈⣿⢸⡀⣇⠀⠀⠀⠘⠹⠶⠤⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠑⢌⠢⣄⠙⢿⣷⠘⡎⢧⡸⡄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡱⡌⠳⡌⢻⡇⢹⡌⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠦⠴⠃⢹⡀⠈⢆⢷⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⢀⣼⠃⠀⠈⠈⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⡙⢆⣀⠀⠀⠀⠀⠠⣴⣾⣿⢛⡛⣿⣧⣦⠄⠀⠀⠀⠀⣀⡰⢋⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠙⣦⣝⡇⠀⠀⠤⠴⢛⣤⠴⠞⠻⠦⣤⡙⠦⠤⠀⠀⢸⣫⣴⠋⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⠁⡿⠋⣭⣥⣄⠲⣚⡛⢷⣾⣿⣿⣷⡾⢛⣓⠖⣠⣬⣍⠙⢿⠈⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠞⠃⠀⠈⣤⠘⣛⡛⢿⣦⣤⣄⣤⣤⣴⡿⢛⣛⠃⣤⠁⠀⠘⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⡳⣾⣶⣦⣛⡁⣨⡉⠉⢻⡟⠉⢉⣅⣈⣛⣴⣶⣷⢞⢠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢸⡿⢂⢉⡙⠛⠷⣽⡁⠢⠀⠀⠔⢈⣯⠾⠛⢋⡉⡐⢿⡇⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⡆⢿⠇⠛⢩⣭⡑⠘⣶⡄⣴⣦⣰⣦⢠⣶⠂⢊⣭⡍⠛⠸⡿⢰⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⡷⡄⢤⡐⠦⠽⠇⡀⢉⠁⣨⡻⢟⣅⠈⡉⢀⠸⠭⠴⢂⡴⢰⢾⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡆⠻⣬⣛⠳⣶⠟⢱⡇⠀⠋⠀⠀⠉⠀⢸⡎⠻⣶⠾⣛⣥⠟⢰⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⣾⡧⣄⠀⣈⡓⢂⢀⣼⠗⠀⠀⠤⠤⠀⠀⠺⣧⡀⡐⠚⣁⠀⣠⢼⣷⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠈⠀⣿⢦⠛⢡⣿⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠣⣿⡈⠛⡴⣿⠀⠁⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣀⠀⠀⠙⠀⠙⢸⣏⢸⡏⠀⠀⠀⣀⣀⡀⠀⠀⢹⡇⣹⠇⠊⠀⠃⠀⠀⣀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣝⠻⣿⡳⠶⠶⣦⠀⠏⡴⠛⣠⡶⠟⠁⠈⠻⢶⣄⠛⢦⠻⠀⣴⠶⠶⢞⣿⠟⣫⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠳⢬⡙⠀⠀⠸⠇⠈⣠⠞⣩⣤⣦⠀⠀⣤⣤⣍⠳⣄⠁⠸⠇⠀⠀⢋⡥⠞⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⡄⠒⢬⣁⠀⢀⡀⠀⢠⣾⣿⣿⠟⠀⠀⠻⣿⣿⣷⡄⠀⢀⡀⠀⣈⡥⠒⢠⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢹⣧⣤⡘⣮⣉⡉⢀⠀⠀⠙⣉⣡⡤⠄⠠⢤⣌⣙⠋⠀⠀⡀⢉⣉⣵⢃⣤⣼⡏⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⣇⣨⡙⣿⡼⣷⡌⣫⡭⠶⠶⠒⠒⠶⠶⣭⣝⢡⣾⢧⣿⢋⣁⣸⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⣿⡇⣸⡇⣿⢃⣁⢀⣀⣙⣿⣿⣋⣀⡀⣈⡘⣿⢸⡇⢸⣿⣿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢧⢙⠃⠸⠈⣿⠠⣷⣶⣶⣶⣶⣾⠄⣿⠁⠃⠘⡋⡼⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⢧⣴⣌⢰⣦⣌⠻⠿⣡⣴⡆⣡⣤⡼⠻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠜⠿⣿⠻⠿⣿⠟⠣⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠋



⣶⣶⣶⣶⣶⡾⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣿⡷⢷⠾⠶⠾⠿⢾⣷⠶⢶⠷⠶⠶⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠙⢄⠤⡄⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⡿⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠁⠀⠄⠐⣷⠀⠀⢸⠀⡠⠚⠉⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡇⢸⡴⠁⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠀⠀⣠⡉⠀⢀⠞⠈⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠀⠀⠄⠂⠼⣃⠼⢄
⠀⢀⠀⠏⠀⠀⠈⢀⡐⠄⠠⢌⡡⡀⠣⣀⣀⠥⠿⠈⠁⠄⠀⠀⠀⠀⣖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣤⠎
⠰⡁⠁⠠⠄⠠⠀⢌⠃⠈⢁⡀⠨⠿⠆⠈⠠⢶⡂⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠄⠒⠠⠀⡄⡎⠀⢠⡃⠶⠴⢋⡀⠀⠀⢹⡱⠄⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢠⠎⡀
⠀⠀⠁⠀⠀⠁⢀⠁⠀⠀⠑⠒⠊⠁⢱⠀⠀⠈⣧⠀⠁⠠⠀⠀⠀⠀⠀⠈⡆⠀⢀⢀⣠⠋⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⡀⢀⣤⡀⠀⡆⠀⠀⠈⢄⠀⠀⠠⠛⠐⠂⠀⠀⢱⢀⠇⠀⠈⠘⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢌⡗⠁⠑⡄⠸⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⠀⠀⠀⠀⠉⠒⠂⠤⣀⣀⡄⠀⠈⠚⠃⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠤⠀⠤⠄⠀⠤⠀⠤⠤⠀⠀⠲⠂⢀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⠃⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣥⡆⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣤⣶⠚⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠒⠒⠋⠁⠐⠛⠀⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣶⣄⣈⢻⣿⣿⣿⣦⡲⣤⡀⠢⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣛⣿⣿⣿⣷⡝⣿⣦⡙⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣍⣻⣿⣿⣿⣿⡟⢻⣿⣿⡿⡜⣿⣿⣼⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠻⣿⣷⡀⠹⣿⡟⠈⢹⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣤⣤⣬⣉⣙⠛⠿⣿⣷⡀⢈⠻⣧⢰⡙⣷⣤⣸⣿⡿⢟⡻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⢿⣿⡿⣿⡿⢿⣦⣄⣭⣿⣮⣿⣾⡿⢿⣛⣿⣿⣿⡞⡫⠽⣮⣿⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⣀⣭⣿⣿⣧⣤⣄⡸⣯⣽⡿⠟⣫⠵⠞⣛⠉⣥⠄⠁⢾⣇⣿⢜⣿⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⢿⠟⠻⠿⣿⣿⣟⡭⢖⠫⠅⣐⡫⠭⢔⣒⣺⣯⣽⣿⣿⣿⣿⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⣻⣿⣶⣿⣿⡿⢿⣿⣩⠵⣚⣭⣶⣾⠟⢛⠋⢷⣌⠀⠀⠀⠹⡏⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢉⣴⡿⣻⣿⣿⣏⣴⣟⣭⣶⠟⣩⣴⣿⣿⠘⢿⡿⢹⡿⡿⢶⣿⣟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣱⣿⡟⣿⣿⣿⣟⣁⠠⡝⣿⢿⣿⡿⢧⡀⠀⠀⠀⠀⠘⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡿⣫⣾⣿⣿⣿⣟⠿⢃⣇⡁⠸⣏⠋⠀⠙⠄⠀⠠⢦⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⢻⣿⣿⣿⣿⣟⠻⠁⣾⣟⣿⣿⣷⡀⠀⠀⠶⣒⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⢿⣿⣿⣿⣿⣆⣧⢙⣬⣽⣿⣿⣿⣶⣄⠀⠉⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡈⠓⢌⠻⣿⣿⣿⣾⣎⢿⣿⣿⣿⠿⣟⠩⢛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣑⠄⠀⠈⢻⣿⣿⣿⣯⣻⣿⣿⣷⣿⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣽⣿⣿⣿⣷⣽⣿⣟⠿⠿⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⠷⡹⣝⢿⣿⣿⣌⣒⠲⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠿⣿⠇⠀⠀⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⢛⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢻⡭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠇⠀⠀⠉⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠿⢻⠟⠀⠉⠋⠀⠀⠀⠀⠸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⢻⣿⣿⣿
⣿⣿⣿⢿⡿⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠛⠁⠀⠀⠀⠀⠀⠀⠹⠿⣿
⣿⣿⢥⠟⠓⠠⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣏⠛⠓⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢳⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠹⠟⠛⠻⠛⠉
⠘⠂
⡌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠉⠛⢻⣿⣷⣿⢿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡿⣛⣵⣴⣦⣀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⠿⠟⣷⣿⣿⣿⣿⣦⣀⡀⠀⠀⢽⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⠅⠒⠄⣶⣍⡻⢿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⣿⣿⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢼⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⠀⠀⠐⡟⢡⡿⣻⣿⣿⣿⣿⣿⠶⣄⣀⡲⢿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠆⠀⢘⢆⡾⣱⣿⣿⣿⠻⣿⢿⠑⠈⠹⣙⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣏⡿⠃⡭⡸⣿⣿⡿⠋⠙⣿⡝⠧⡄⣰⡀⣻⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣧⡞⣹⡇⢠⡶⠐⠻⣿⣿⣧⣶⢹⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣹⣿⣿⣿⠟⠁⢹⣿⠜⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡟⠉⠀⠘⣸⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⠯⠀⠀⠀⠀⠿⠼⠻⠿⠿⠿⡿⠿⠿⠿⡿⡿⠿⠇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠻⣷⣶⣤⣄⡀⠀⠀⠀⣀⣠⣴⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠻⡏⠙⢿⣆⢀⡴⠋⠁⠀⠈⠉⠛⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⡇⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠛⠛⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣤⡴⠾⠋⠉⠉⠻⢦⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⣄
⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠙⠛⠿⠶⣤⡀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⣀⣀⡀⠀⣀⣴⣿⠃⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⠦⣤⣄
⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢻⣿⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢠⣿⣧⣄
⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠁⠀⡹
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣤⣤⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⠀⢹⣿⡏⠁⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⡿⣿⣷⣿⣿⣿⣄⡀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⡿⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⢸
⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⢰⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⣿
⢹⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⣧⡈⠙⢿⣿⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣴⠿⠻⠿⠁⠀⠀⣿
⠈⢿⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣍⣷⠶⠀⠀⠀⠀⢠⣿
⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣿⣿⣿⡌⠻⣿⡙⠛⠛⠿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢿⡟⣿⡉⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠒⠒⠓⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠓⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠏⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡉⠃⠀⠉⠿⠁⢀⡞⠉⠉⣹⡅⠛⢐⣧⡿⢋⡿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⠀⠀⡀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⢀⣀⣀⡀⠀⣠⣤⡀⠀⣠⠶⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣤⣙⠀⢸⡆⠘⡇⠀⣾⠋⠻⠆⢠⡟⠉⠳⠀⢸⣏⣉⡁⠘⣧⣤⡁⠀⠻⠶⣦
⠀⠀⠀⠀⠀⠀⠀⢶⣌⣽⠀⢸⣇⣀⡇⠀⢿⡀⣠⡄⠘⣇⠀⣠⠀⢸⣏⣉⡀⠰⣦⣨⡟⠀⠷⣤⠾
⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠀⠀⠈⠉⠁⠀⠈⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡍⠉⠀⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣀⣀⣀⣠⣤⣀⣀⣀⣀⣠⣤⣠⣤⣠⣤⣤⣄⣠⣤⣤⣤⣤⣠⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⢀⡾⠁⢀⠐⠀⠀⠀⠀⠠⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⡹⢿⡀⠀⢠⠀⣤⠤⢤⢤⣄⣲⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣼⠟⠈⠁⠀⠀⣾⡶⠄⠋⣰⡆⢫⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣏⠀⠀⠀⢀⡤⠚⢳⡀⠀⣼⣧⣀⣿⣿⣿⣶⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣷⠀⡠⠴⠏⠀⠀⠈⠛⠀⠚⢛⣿⣿⣿⣿⣿⣇⡨⡇⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⢸⣿⡸⠿⣦⣄⠢⣀⡀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠀⣿⣭⣥⣤⣄⡀⠀⠀⠡⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠱⠥⣿⡃⠀⣸⣿⣿⣿⠧⢤⣀⣀⣽⣿⢻⢛⣿⣿⡿⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⢠⠞⢁⡜⠁⣿⣧⣿⣿⣿⡿⢧⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠟⢻⣿⣿⠣⠀⣀⡀⠁⣠⣾⣧⣴⣿⣿⣿⣭⡏⢠⠾⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠠⡜⢳⣿⣧⠔⠋⣠⣾⣿⣿⣿⡇⢸⣯⣹⣮⣽⣿⢷⣿⣿⠇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⠒⢶⣄⠀⠀⠀⠈⢸⣿⣿⡶⠋⠀⣿⠿⣿⠛⣦⣼⣿⣿⣿⣿⣿⣾⣿⣿⠀⠈⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠷⣶⡄⠻⠀⠀⠀⡠⢺⣿⣏⠉⠑⠀⠛⠓⠚⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⠇⠐⠠⠁⡞⢡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⣿⠀⠀⠀⠜⣴⢿⣿⣥⣤⣀⠀⠀⠀⠈⠉⠙⠶⢿⣿⣿⠻⣿⣿⠏⠀⠐⠁⢀⡇⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠰⢃⠀⠀⡊⡘⠉⠘⡏⠈⢻⣿⠏⠀⠀⠒⠐⠛⢾⣿⣿⣏⠃⠈⡋⠠⢊⡼⣴⡿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⣄⠀⠀⠀⠀⠀⠀⠃⠀⢸⣿⡌⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠁⠀⢉⡵⠛⢛⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⡀⠉⢉⣷⣴⣦⣂⡀⠀⣠⣿⡿⠋⠀⠀⠒⠦⣄⣀⣈⣿⣿⣷⠀⣠⣶⡿⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⠀⠹⠿⢿⣆⣼⣿⡟⠁⠀⠀⣠⠉⠀⠀⠉⢉⣛⣻⣿⣻⠿⢿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⣿⣿⣯⣥⣀⠙⠉⠀⠀⢤⢀⣀⣈⣿⣿⠋⠑⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠹⠒⢦⠀⢀⠞⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣲⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣷⡼⣿⣿⣿⣿⡿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠰⣿⣿⣿⣿⠷⠛⣉⢉⣝⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠿⠻⣭⣤⣶⣾⣿⣿⣿⣿⣶⣭⣗⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣤⣀⠻⣿⣿⡟⠁⠀⠀⠘⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣷⣙⢿⣿⣄⠀⠴⠾⠿⠿⢿⡿⠟⣽⡿⣿⣿⣿⣿⣿⣿⣇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣷⣍⠿⡿⠟⠀⠀⠀⣠⠖⠛⠷⡶⠤⠾⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢛⣿⣿⣿⣿⣿⣅⣀⡄⣤⣾⣼⡇⢾⣧⠐⣷⣦⣐⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢿⣿⣿⡿⢿⣿⣿⣿⣧⣿⣿⣷⡻⣮⢻⣷⣬⡻⢿⣿⣿⡿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠽⢿⣿⣿⣝⣿⣿⣿⣿⣧⠻⣷⣝⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⣈⣿⣿⣬⣿⣿⢿⣷⣌⡻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠿⣿⣿⣿⣿⣿⣷⣭⣛⠿⠷⢿⣿⣿⣿⣿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⢛⣿⣿⣿⡿⠿⣛⣻⣿⣿⣿⣭⣥⣴⣶⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣾⣿⣿⣇⣿⠸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢿⡟⠟⠻⣿⣿⣿⣧⣻⣿⣿⣿⣬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣬⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⣷⣾⣿⣥⣼⣿⣿⡟⠉⣃⡀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⢟⣧⣾⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⣆⠘⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⢠⣿⣿⡟⢻⢸⣿⡆⠹⣿⡿⣿⢻⣿⠿⣿⡇⢹⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢼⣿⣿⣇⢸⡄⡿⠃⢸⣿⠓⣿⠸⣿⠇⣿⣇⢸⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⢸⡇⠁⢸⡇⠀⠀⠸⣿⠀⣿⡇⠉⠀⠉⠁⢸⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⣇⠀⠀⠀⠸⠀⣿⣷⠀⠀⠀⠀⢸⣿⠀⣿⣿⡄⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⢸⣿⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣛⣛⣉⣛⣻⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣹⣯⣭⣽⣿⣿⣿⣍⣩⣭⣿