Natural science Emojis & Symbols

⠀⠀⠀⠀⠀⠠⣶⣶⣶⣴⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⣿⠛⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠚⠻⣿⣿⣿⣷⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⠿⠿⠿⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣤⣄⠀⠀⠿⠛⠛⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⠀⠀⢠⣤⣾⣿⣿⣿⡟⣷⣆⠀⠀⠀⠀⠀⠀⣠⣾⡻⣿⣿⣿⣷⣤⡀⠀⠀⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣿⣿⢿⢩⡄⠻⠸⢳⣿⣷⡀⢀⡀⠀⣼⡟⡟⠿⠟⢐⣿⣻⣿⣷⣦⣤⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⣿⡥⠠⠀⣥⠐⠀⢛⠘⠀⠱⣾⣷⡞⠃⡿⡜⠀⠀⢸⠇⠠⠨⣿⡿⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣿⣓⣈⣁⣙⣃⣀⣬⠴⠜⡀⢹⣿⣧⡧⠇⣃⣀⣀⣉⣈⣁⣈⣉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣟⣿⣿⣯⣹⣿⣿⢻⡳⡠⡑⢾⣿⢘⣽⣿⠿⢿⣷⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡽⢆⣋⠿⠿⢏⡛⠠⠍⠐⠄⢹⣿⠀⠘⢞⡄⢞⡉⠿⢿⣟⣹⢯⢿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣳⣌⡣⠆⢀⡀⠀⠙⠀⠀⠀⢸⡇⠀⠀⠘⡟⠀⠀⣤⠠⢿⣿⣶⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣶⣍⣐⡪⠀⣜⠁⠂⠀⠀⠀⠁⠀⠀⠰⣇⣻⣤⠱⣚⣪⡷⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠹⠟⠻⡿⠯⢿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⠽⢿⡿⠛⠿⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⣘⠑⠄⠀⢀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣰⣾⣿⣶⣷⣿⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢿⣿⣿⡿⠿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠀⣠⣼⣏⢛⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣀⣠⠤⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠀⠺⣿⣿⣿⣯⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠢⠈⠀⠀⠤⠈⠀⠀⠹⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣦⠄⠀⠀⠁⠀⣴⣿⣿⣶⣌⠻⣿⣿
⠀⠀⠀⠀⢠⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣿⠟⠋⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣤⢀⠂⢰⢿⣿⠄⠂⠀⠀⠤⡑⠀⠀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠲⢾⣽⢚⡵⠿⠃⠂⠀⠀⠀⠀⠀⠀⢲⠀⠀⠈⠙⢿⣿
⠙⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣦⣶⣤⣍⣭⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠙
⠀⠀⠙⠻⢤⣩⣈⡷⠠⠀⠀⠀⠄⠀⠦⠒⣾⡟⠋⢉⡭⠋⢀
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⢁⡠⢒⡡⠔⠀⠀⠀⠔⠉⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⡥⠖⠋⡴⡟⡴⠀⢀⡁⢄⠁
⠀⠀⠀⠀⣠⣴⣾⠿⠛⠁⠀⠀⣼⣷⣶⠁⠀⠹⣴⣘⡆
⣦⣤⣘⠛⠋⠉⠀⠀⠀⠀⠀⢸⣏⡹⠃⠀⠀⠀⠙⠦⣿⣦⣤⣀⡀
⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢸⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠾⣧⣬⠀⠁⠀⠀⠀⠘⠁⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢉⠠⠆⠐⠈⠉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⠈⠉⠁⠀⠀⡙⠂⠀⢈⡆⣼⢷⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡔⠋⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⢶⣄⠀⠒⠀⢤⠄⢀⡀⠉⠀⠂⠻⡇⢈⡏⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⠟⠠⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⣠⠶⠀⣄⠀⢀⠀⠀⠒⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠻⢦⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠾⠞⠂⠀⠿⠆⢘⡆⠀⠀⠀⢈⠄
⠀⠀⠀⠀⢀⡴⣠⣔⣋⣀⡀⡠⡤⣤⣦⣄⡀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠘⠇⠰⠿⣟⣿⣯⡙⠳⣦⡈⠙⣿⢶⡦⠠⣄⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⢠⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⠻⣿⣿⣆⢈⠛⢷⣄⠈⣇⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠭⠷⠖⣲⣞⣃⠤⣤⣄⠀⡀⠰⠎⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢹⣿⣿⣿⡀⡌⠻⡄⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⣠⣶⣿⣷⣄⠀⠀⠀⠀⢸⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢹⣿⣿⠟⠙⠢⢯⡇⠀⠀⠀⠀⠀⠈⢏⠻⣯⡛⠙⣷⠖⣦⠀⠀⠀⠀⠀⠀⡀⠀⠀⡾⢻⣿⣿⣿⣿⣆⢀⠀⠀⢸⡏⠉⡻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⣿⡄⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠻⡉⢻⠄⣿⡄⣳⣤⢀⠆⠀⠀⠀⠉⠀⢰⡇⠈⢿⣿⣿⣿⣿⠃⠀⠀⠀⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡄⠀⠙⢣⡄⠀⠂⠸⢣⡎⠀⢈⣠⢺⣶⣀⡤⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠐⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣶⠿⣿⣀⣇⣴⣿⢺⣤⡄⠀⠋⠂⠀⡉⢦⣀⣭⣽⡇⠀⠀⠀⣿⣿⣿⣿⠀⢀⡀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡿⠻⡌⣿⠁⠙⣿⣿⠋⠀⠀⣞⠀⠐⡀⠘⡆⠉⠃⣼⡇⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠘⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⢷⣄⠂⣯⡔⢻⣿⣿⣵⣠⡜⠻⠀⢀⣸⠁⣸⣆⠀⣿⡇⠀⠀⢰⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⡟⠉⠁⢸⣿⣿⢉⣿⡇⠀⠀⠠⠟⠠⠟⠉⠁⠻⠇⠀⠀⠘⢿⣯⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡇⣼⣿⣅⡀⡀⠀⣀⠀⠀⠀⠈⠁⠀⠀⣼⣿⣡⣾⣿⡇⠀⠈⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣺⣣⣿⣿⣿⡇⠰⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢙⣿⠿⢋⣤⡞⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢹⣿
⠀⠀⠀⠀⠀⠀⠋⠀⠀⢺⣿⢨⣿⣿⣿⣇⠀⡏⣠⣤⣤⠀⠠⡀⠀⣶⣶⠶⣿⣿⣶⡟⣏⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣸⣿
⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⡏⣼⣿⣾⣿⣿⡀⠘⣧⠶⠋⢀⣠⣅⣴⣻⣧⠰⠘⠛⢿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢁⣸⢃⣿⣿⣿⣿⣎⠑⠂⠀⣀⣴⣏⡟⠋⠉⠉⠻⣶⣄⣠⡔⠋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⢸⣿⣿⣿⣿⠿⠋⠀⣼⣿⡉⡟⠁⠀⠀⠀⠀⢸⣿⠋⠉⠙⠛⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣾⢀⣿⣿⣿⣿⡦⠤⠆⣼⡏⠙⣿⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠈⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣶⠇⣼⣿⣿⣿⣿⠁⠀⢴⠿⠁⢠⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⠳⣶⡖⠙⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢡⣿⣿⣿⣿⣿⠄⣼⣧⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⣿⣯⠐⠈⠙⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⡇⣼⣿⣿⣿⣿⠋⣾⠋⠙⢰⣾⠏⢀⠀⠀⠀⠀⢠⣅⣾⡿⠋⠒⠉⠉⠓⠛⠃⠀⠀⠀⠀⠀⡆⢀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢐⣿⢱⣿⣿⣿⣿⣏⣾⡇⠀⠀⠘⡟⠀⡘⢀⣤⡶⢠⡟⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣯⣿⡿⠹⣿⢿⡿⠉⠁⠀⠀⢰⡇⢰⣇⣼⡿⣡⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠳⣿⣿⠟⠁⠀⣿⣿⡇⠀⣤⡀⣠⠟⠃⣜⣿⣿⣶⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡿⠛⠛⠛⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠆⠀⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠢⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣤⣤⣀⡀⠀⣠⣤⣹⣿⣿⣦⠀⠀⠀⠀⠀⠈⠐⠶⠤⠀⣈⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠠⢼⣿⣿⣿⣿⣿⡿⠿⢿⣿⢿⣿⣶⣾⣀⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡉⠛⠉⠁⠀⠀⠉⠻⡎⢟⠻⣿⣿⡧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣄⡉⢓⠼⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⢱⡀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠉⠉⠑⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢄⡀⠀⡐⣀⣤⣴⣤⣄⠀⡂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣽⣿⣿⣿⣿⣿⣿⣶⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⣿⣿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⡺⡇⣀⠀⠀⠀⠀⡶⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣳⠄
⠀⠀⠀⠀⠀⣨⡆⡚⠁⠀⠀⠀⠀⢀⠘⣟⣯⠄⠀⠀⠀⢀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣀⠁⢈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⢀⣀⠄⠀⠀⡠⠀⠐⠂⢬⣴⡮⠅⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠚⠃⠀⠂⠀⠤
⠀⠀⠀⠀⣠⠴⢘⠇⠀⢀⠈⠁⠀⠀⠽⣞⢀⠤⠀⠀⠀⠠⣤⣤⣀⠀⠀⠀⠀⠀⡀⠀⣙⠒⠿⠀⠀⠀⠀⠐⠀⠀⡀⢀⠐⠋
⠀⠀⠀⡇⠈⠀⣮⡛⣀⢀⠤⠀⣀⠀⠀⢀⣥⠶⠀⠀⣀⣴⣿⣿⣽⣤⠶⠶⠦⠀⠹⣷⣿⣦⣀⣤⡤⠀⢀⡀⢀⠀⠙⠀⢀⡄
⠀⢀⠶⠀⠃⢷⠟⣯⣉⠀⠀⠁⢀⣴⣈⢉⣠⡀⠀⣰⣿⡿⠋⣼⣿⣤⣍⠉⢠⣴⠶⢮⣟⣾⢿⡋⠀⡰⠋⠀⠈⠀⠀⠀⡼⢠⠋⣀
⠀⠀⠀⢲⣶⠀⠀⠁⠈⠀⠀⢀⡀⠈⠻⠿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠞⠁⠀⠀⠉⠁⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠰⠈⢧⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠈⣷⡄⠀⢀⣴⠇⠀⠀⠒⡁
⠀⠀⠐⠁⠀⠀⠀⠀⠀⣾⠛⣛⣰⣄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢰⣜⠳⣤⠿⢋⣀⡀⠀⢺⣇⡄
⠀⠀⠠⡀⠀⢀⠀⠀⢠⣾⣿⢫⣾⢿⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⣷⡾⠃⠀⠀⠀⠸⣿⡧⣴⠞⠉⠁⠀⠀⠻⠅⠁⠀⠀⡥
⠀⠐⣄⢀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣶⠇⠃⠀⠀⣦⣀⠀⠒⠈⢆⢀⠁⢀⠀⢀⠀⠀⢹⣿⣿⣷⠀⠊⠀⠠⠘⢠⠀⠀⠠⠴⡀⠆
⠀⠀⠈⠓⠇⠀⠀⠀⠀⠈⠻⢿⣿⠋⠀⠀⠤⠀⠸⣿⣷⣄⠀⠀⠉⣡⠿⠄⠀⠀⠀⠸⣿⢛⣿⠒⠂⠀⠀⠘⠂⠀⠀⠀⠔⢆
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠸⢀⡀⢡⡀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀⠀⢸⠀⠀⢀⣀⠀⠀⠙⡾⠁⠀⠀⠀⠀⠀⠀⢠⣄⠀⡀⠤⠁⢀
⠀⠀⠀⠀⠈⢢⣾⣴⠀⠀⠀⠀⠀⠁⠐⠁⠀⠘⢦⠀⠀⠘⣿⣷⡀⠃⠀⣴⣿⠃⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡐⢲⣾⠇
⠀⠀⠀⠀⠀⢀⢸⣃⣤⠴⠖⠂⠀⠀⢀⡤⠻⣤⡀⠀⠀⠀⠘⣿⡇⢠⣼⡿⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠛⠁⠘⠋
⠀⠐⢲⣄⠀⠀⠉⠳⢤⡀⠀⢰⢀⠀⣾⡡⠶⡙⠷⠀⠀⠀⠀⢹⣿⡋⠁⠀⠀⠀⠀⠀⠀⠰⣾⣿⠆⠀⠁⢀⠀⠙⠁⠀⠀⠀⡄
⠀⠘⢹⠇⠀⠀⠀⠤⠀⠀⠀⠀⠀⡄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⣀⠀⠀⠀⣿⣿⠀⠀⠀⠚⠃⠂⠀⣈⣀⣀⣈
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣤⣜⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⠟⠫⣿⡟⠿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡰⠿⡿⠶⢤⣉⣿⣿⣾⣿⣿⣯⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⣰⠖⠻⣶⣶⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⣿⡿⡿⠀⠀⠴⢾⣷⣾⣿⣿⣿⣿⡀⢿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠁⣀⣤⣿⣿⣷⣦⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠛⢀⣾⣃⣼⠁⡠⠀⠀⣸⣿⣿⣿⣿⣿⣿⡉⢿⣧⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢠⣼⣿⠉⠻⢻⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣁⣨⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⡷⣠⢿⠀⢀⠀⠀⠀⠀⠀⢴⡆⠀⠐⠀⠀⢻⡿⠇⣠⣴⠿⠟⢷⣿⣆
⠀⠀⠀⠀⠀⢶⣴⣯⡹⠿⢿⣿⠿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⡀⠀⠀⢾⠀⠀⠀⠀⠀⠘⠃⠀⣠⣤⠞⠋⣠⠊⠰⠃⠀⠀⠀⢹⣿⠧⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡗⢀⡀⠘⢃⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⠀⠀⠀⠀⠀⣀⠠⠈⠿⠀⡰⠾⡵⠂⠘⡅⠀⠀⠀⠸⠁⢀⡲⠐⢂⡄
⠀⠀⠀⠀⠀⠀⠙⠿⡷⠶⠾⠛⣨⣽⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣷⣾⣿⠀⠀⠀⠀⠀⠈⠃⠲⠞⠋⠀⠀⡴⠛⠐⠀⢺⣗⠀⠁⠂⡈⠉⠀⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡧⠀⠁⠀⢉⣏⠈⠠⠀⠀⠦⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠉⠞⠂⠈⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡏⠙⠛⠛⠁⢹⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠠⠤⡛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣶⠏⠀⠀⠀⠀⠀⠀⠀⢠⡶⠆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢸⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⡏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⠀⠀⠀⠀⠀⠀⡿⠋⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠘⢻⡀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣀⡀⠀⠀⠀⠀⠀⡀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⠌⡧⠀⠀⠀⠀⠈⠉⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢳⠲⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⣘⣧⠀⠀⠀⠀⣈⢦⣔⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠠⡄⠁⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⢀⡾⠟⢿⡷⠂⠀⠐⠟⠛⢿⣤⡌⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⢽⣧⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠠⠏⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣾⣿⣿⣿⣯⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣉⣽⣿⣿⣿⣿⣿⣿⣿⣖⣄⣀⢴⢴⢶⣿⣾⣧⡶⢀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡿⠹⣿⣿⣿⣿⣿⣿⡏⢫⢣⡱⡉⣫⢿⣿⣿⣏⡄⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠉⠀⠀⢹⣿⢾⡿⣻⢉⣹⢺⣇⣷⣿⣿⣾⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⡰⢻⣾⣿⣿⣿⣿⣏⡏⣧⡿⣿⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣷⡾⣿⣿⠛⠛⠛⠛⠙⢻⣧⠸⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢿⣿⠀⠀⠀⠀⠀⠀⣾⡗⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣯⠀⠀⠀⠀⠀⢀⡾⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠅⠼⠇⠀⠀⠀⠀⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⣿⠖⠀⠀⠀⠀⣰⣖⠂⠠⡀
⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⣦⠜⠿⣿⡻⣿⠿⣿⡌⢠⡴⡟⣿⣿⣦⡴⡠⠖⢄
⠀⠼⠟⣻⣿⣿⣿⣤⣲⣄⠀⠀⡀⢀⣀⣦⣼⣿⣿⣯⣾⡿⣧⢼⣿⣿⣷⣞⢙⣿⣿⣿⣿⡖⢳⣖⣾⣦
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⡔⣻⠟⢿⣿⣿⣿⣯⣴⣿⣷⣷⣾⣾⣿⣿⣿⣿⣿⣿⢻⣿⣿⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣾⣿⡙⢿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣠⣸⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠄
⠀⢠⡄⠀⣠⣧⣶⣿⡟⠻⣿⣿⣗⣻⣼⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣋⡀
⠀⠐⠘⡞⠟⢻⣿⣿⣤⣬⡿⣽⣽⣗⡷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠛⣆⣨
⠀⠀⠀⠀⠀⠈⠀⠋⠋⠻⣛⣟⣿⣿⣿⣿⣏⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣩⣷⣿⡿⣿⣿⣿⣾⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢽⣿⠟⢿⣿⣿⢿⣛⡿⠛⣾⣿⣿⣿⣯⠯⠽⠝⠋⠉⠉⠉⠉⠀⠙⠋⠘⠋⠁⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⣿⣷⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣍⠁⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⢛⠂⠤⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣠⢰⡂⠀⣠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⠢⠀⣶⠠⠀⣼⡒⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣴⢐⡾⣀⣺⡽⢷⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣾⣟⡈⠩⣔⠠⠀⠀⢮⣧⠀⠀⠀⠀⠀⠘⢿⡻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡷⣿⣿⡷⢊⣉⣇⠀⠀⠈⠹⠆⡀⠀⠀⠀⠀⠀⢘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣯⠁⠌⠀⠁⠀⢡⠄⠀⠀⢀⣀⣷⣮⠴⣶⣾⠛⡿⠛⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠬⠟⠛⠛⢉⡿⠃⠀⠀⠀⠐⣀⣤⡤⠲⠞⠻⡋⠀⠀⠀⢸⣿⠢⠀⠈⠂
⠀⣤⣀⠀⢀⢀⣠⣤⠤⠤⡀⠀⢠⠤⠦⠤⢀⠤⢖⠆⠔⠲⠓⠋⠉⠁⠀⠀⠀⠈⠂⠀⡍⢱⠂⣹⣿
⡄⠛⠿⠯⠮⢵⢬⠳⠐⠦⡖⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⢀⡒⠄⠀⠀⠀⠀⠀⠀⠀⢈⣠⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣯⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣟⠛⣷⠀⠀⠄⠁⠀⠈⠀⠀⡀⠂⠀⠀⠀⠀⠀⠔⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣼⣻⣿⡋⠄⠀⠀⠠⢀⠠⢀⠀⠄⡁⠂⠈⠀⠀⢀⠀⠐⠊⠁⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠂⠐⠈⣉⣉⠛⠉⢁⠀⠀⠨⠕⠢⠀⠤⠔⠠⠄⠤⠆⠉⠒⠁⠀⠀⠀⣰⢶⠀⠀⠐⠀⠈⠑⠂⠤⢀⡀⣤⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⢞⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢷⡻⡧⣬⣤⣧⠌⣁⠂⢄⠨⢘⣿⡔⠟⡽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⡋⣰⣤⣀⣶⣿⣿⠴⠀⠉⠀⠀⣏⣻⣧⣤⣙⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠃⠐⣰⢃⣯⣠⣼⠛⠁⠀⠀⠀⠀⠀⢰⡯⣷⡄⠍⡁⠸⠅
⠀⠀⠀⠀⣠⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣴⠛⠓⣘⠛⣿⣡⣬⡿⠋⠀⠀⠀⠀⠀⠀⣠⠀⠀⠛⡳⢷⡆⠆
⠀⠀⠀⠀⠘⡀⠀⠀⠐⣚⠃⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡿⠿⡡⢠⣨⡁⣾⣷⣶⠏⠀⠀⠀⠠⠄⡐⠲⡌⢿⣧⠀⠀⣯⡈⡇
⠀⠀⠀⠀⠀⠈⣦⠀⠥⠏⠀⢀⡪⠁⡔⠊⠛⣛⠛⣻⢻⡓⠄⢲⣓⣾⡿⠷⠋⠉⠁⠀⠀⢀⠐⠁⠀⠀⠠⠗⢈⣾⠀⠀⠶⠁⠁
⠀⠀⠀⠀⠀⠀⠈⢔⣻⡁⠀⠰⠁⣾⠂⠀⠀⠿⢷⢶⡾⣡⣼⣾⣏⠒⡀⠀⠀⠐⠀⠀⠀⠀⠈⠐⠀⢨⠔⠀⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠰⠀⣷⢀⡀⠀⠀⢀⠈⣿⣿⢟⠛⡉⣴⣧⣤⣀⡐⢀⠀⠀⠀⠀⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠉⠁⠀⠀⣰⣿⠁⠴⠇⢙⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣰⡟⠀⡁⠀⣈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠖⠓⠛⠦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⣿⠤⢚⠦⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠄⠐⢶⣖⣴⣌⣷⡀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣑⢂⠠⢈⣿⡀⠀⠀⠀⠀⠀⣀⣤⣾⣸⡽⣧⡀⠀⢨⡿⡿⢿⣿⡟⠀⠆⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡎⣴⢧⣺⢿⣦⣤⣤⡴⠿⠋⠁⣡⣿⣧⢸⡇⠀⠈⠀⠀⠀⢹⣿⠶⠀⢠⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠲⠶⣤⣀⠻⣿⣶⣌⡿⣶⣿⡿⣥⢀⣀⢰⣟⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⡈⣬⡟⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣦⣴⣴⣶⣧⣡⠀⣉⡛⢮⡝⠻⢿⣿⣿⣷⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⢀⣠⠠⡴⣿⣤⡔⣷⣿⣥⣤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡼⣿⠟⠁⠀⠈⠙⠻⣿⣿⠆⠈⡓⢦⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⡟⢻⣿⣿⡾⣿⠻⣿⣤⠶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠻⡀⠀⠀⠀⠀⠀⠈⠛⢧⣲⣍⡂⢈⠑⠦⢤⣀⣀⣀⣀⣄⣠⣤⡴⢘⣛⣤⣤⣰⢶⣋⣿⠟⠙⠷⣦⣤⣴⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⢳⣄⡀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣧⣀⣤⣤⣄⣨⠿⣿⣡⣿⢔⡠⠿⣿⢟⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⢻⡶⢤⣀⡀⠀⠀⠀⠀⣀⠉⠛⠿⠿⣷⣿⣽⣶⣷⣾⡶⠀⠀⠈⠿⣟⡁⠀⠀⠀⠆⠀⠓⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⣾⣿⣿⣿⣿⣿⣷⣿⠿⠿⠿⢿⣶⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣍⣾⢽⣿⣿⣛⠿⡿⠿⢻⣯⣡⣾⣟⣼⣭⡶⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⣿⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣹⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡹⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣩⣿⣿⡟⠛⣩⠉⣹⢫⡮⢹⣿⣿⣿⣿⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣿⣿⣿⣷⡝⢎⣹⣶⣿⣿⣿⣿⣟⣻⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣮⣿⣿⣿⣟⣭⡉⣶⠏⠉⣻⡿⣿⣽⣿⡶⣚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⣿⣿⣿⣹⣫⣿⣸⡕⢎⣶⣿⣴⣿⣿⣿⣿⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡿⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣮⣿⣯⣾⢿⣿⣿⣿⣷⣤⣴⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣻⣿⣿⣿⣿⣿⣥⣷⣾⣿⣿⣿⣿⣿⣿⡟⠿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⡿⣿⣿⣿⣿⢹⣿⣿⣿⣏⣾⣿⣿⡿⠵⠁⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣴⣾⣗⣽⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠷⢿⢿⡿⠿⠛⠿⠿⠦⢿⢿⣿⡟⠭⠿⢿⠷⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡔⠊⠀⠀⠀⠀⠀⠀⠇⢀⡀⠀⠀⠀⠀⠀⠀⢂⠀⢀⠀⠂
⠀⠀⠀⠀⢐⢀⠂⢐⠀⠀⠀⠀⠀⢉⡜⠐⣗⠀⠂⠀⠀⠀⠀⠐⡗⠀⠀⠀⠀⠀⠀⠀⠖⠁⢠⡄
⠀⠀⠀⠀⢀⡙⠀⠠⠂⠀⠀⠀⠀⠊⡇⢰⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡁
⠀⠀⠀⠀⠃⠀⠀⢀⠓⠀⠀⠀⠈⠀⢇⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠘⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⣀⠀⡔
⠀⠀⠀⠀⠌⠀⠐⢁⠁⠀⠀⠀⠀⠀⠀⢀⠨⠦⠐⠶⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⣬⡂⣡⣤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣀⣀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⡠⢊⣠⠞⢈⢡⠂
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠐⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠳⠿⠥⠒⢡⠻⡄
⠀⠀⠀⠠⠖⣋⣽⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠿⠛⠋⠙⠳⠴⠤⠄⠀⠀⠢⡀⠀⠄⣢⡮⠈⠤⠀⠤⠂⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠘⠋⠋⠀⠘⠀⠀⠀⠀⠀⠀⠄⠔⡒⣂⣀⠂⣠⠤⠤⠈⠁⠀⠀⠀⢀⢈⡔⡀⢲⠟⠁⢈⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣂⣀⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠁⠁⠀⠡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠸⠿⠿⡀⠀⠀⣶⡆⢀⡴⢺⣧⣭⡁⢳⣿⡏⠀⠀⠿⠀⠀⠀⠈⠛⠛⠉⠀⠙⠛⠋⠁⠈⠛⠒⠋⠀⠉⠛⠚⠃⠈⠓⠒⠂
⠀⣠⣤⢸⣶⣦⢀⡄⢸⣇⣨⣏⣈⣁⣈⣉⣁⡀⠀⠀⠀⠀⠰⠃⠘⣦⡞⠁⢱⡴⠁⠙⣦⡞⠀⢳⡞⠀⠙⣶⠋⠀⢷⠎⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠈⢫⡌⠛⠋⠀⠀⣼⣿⣿⣿⣾⣿⣷⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⢸⣷⠀⠀⣿⡇⠀⢸⣧⠀⠀⣿⡄⠀⣸⣇⠀⢀⢿⡄⠀⡀⢠⣤⣀⢀⡖
⠾⡿⠈⢱⣿⡆⠀⠘⣏⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠋⠈⠀⠘⡅⠙⢀⠋⠈⠀⠈⠀⠙⠀⠁⠈⠀⠈⠀⠙⠈⠁⠸⣿⣿⣿⣿
⠐⠀⠀⠀⠈⠀⠀⠀⢈⠀⠀⠀⣀⣀⣀⢈⣁⠀⠀⠀⠀⠀⢀⠀⠘⠃⠀⠘⠁⠠⡄⢠⣵⣤⣼⣤⣄⡖⠀⠀⠈⠀⠀⠀⠀⠀⠐⠒⡋⠀⠠⠛⠿⡿⠟
⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⢿⣧⠀⠀⠀⠀⠠⠤⡀⠀⠈⠃⠰⢤⣠⣾⡿⡿⠻⢿⣷⣼⣷⣴⢂⣀⠀⣴⣶⣧⡀⢠⡆⢠⡀⢰⢰⠂⠠⠰
⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣯⢿⣿⢾⠀⠀⠀⠀⠀⢀⡁⠀⠃⠀⢀⢰⣿⣿⣿⡆⣰⣿⣮⢛⢿⣿⡇⠀⠀⠹⠿⠟⠀⠘⠇⠈⠁⠾⠿⢷⣶
⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⠀⢀⠴⠶⠟⠫⡅⠀⢘⡷⠈⢭⣟⣻⣿⡟⢻⣿⣟⣶⣾⣿⡗⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠰⢶
⣭⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡆⠠⣀⡀⢀⣀⡴⠀⠈⠀⠐⠊⣻⣿⣿⠾⣿⣿⣿⣿⣥⡼⠧⠀⠀⠈⠀⠀⠀⠀⠁⠀⠐⡞⠆⠀⠀⣈
⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣀⡜⠃⢠⠾⠛⠛⠓⠄⠀⠉⣂⠀⠈⠁⡹⠿⠿⣿⣿⠿⡟⠫⠀⠀⠀⠐⡂⠐⠦⠶⠛⠛⠂⠀⠉⠓⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠈⢳⡄⠠⣀⠀⣀⣀⡤⠀⠀⠀⠂⠀⢀⡁⠀⠀⠘⠈⠀⠘⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠆⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠹⣄⠀⠀⣸⠇⠠⠞⠛⠋⠓⠦⠀⠂⠀⣴⡶⢀⠀⠀⠀⣰⣶⣶⡀⠀⠀⢐⣾⠿⣿⣿⠿⠟⠋⠉⠀⣀⠛⠛⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠟⠓⢧⡀⠐⢦⣄⣠⡤⠖⠀⠀⠀⠠⠀⠘⠁⠀⠂⠙⠿⠟⠀⠀⠀⠉⠉⣉⣉⣀⡀⠀⣀⡄⠀⠙⢻⡷⠸⣿⣿
⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⠀⠀⢸⡇⠰⠚⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠂⢰⣦⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠁⠀⠀⠺⠇⠀⠗⠀⣁
⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠀⣀⣈⠀⠐⠤⠄⠀⠀⠀⠚⠃⠀⠀⡤⠀⠀⣴⣶⠆⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⢀⣄⢀⠀⠄⠀⠐⠐⠂⠸⣿
⠀⠉⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠘⠛⠐⢿⣿⠲⠿⠀⣤⣤⡀⢘⣿⣿⣿⣇⣶⡆⠀⠉⠉⠁⢨⣿⣿⣿⣿⢶⠀⠀⠀⠰⡀⠀⠀⠑
⠀⠀⠂⠤⠄⣚⠳⣼⡑⢞⣿⠀⠀⠀⠀⠀⣀⠸⠷⠀⠛⣿⣇⠘⠿⠿⠇⠛⣿⠄⠘⠛⢀⡀⠹⣿⣿⣿⡿⡟⠀⠠⠀⠀⠈⠀⠀⠈⠀⠠⠴⠤⣄⠐
⠀⣴⡄⠀⠀⠀⢠⡿⢙⣫⡞⠀⢀⠀⠀⣿⣿⡗⠀⠀⣴⣿⣟⣤⣤⣄⡀⠀⢀⣰⢤⠀⠈⠁⣠⡈⠉⠛⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⢐⡀⠐⠀⠀⠇
⠄⢸⡇⠀⠀⢀⡘⣰⡾⣼⠇⠀⢈⠀⠿⠇⣽⣄⢸⣟⡸⠋⢿⠿⣯⡟⢤⠀⠀⢼⣻⠸⠀⠀⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠳⢽⠽⠃⠈⠉⠁⠘⠟⠀⣾⠁⠸⣿⣿⡀⠀⠁⠀⠙⠀⠀⠀⢀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠒⠀⠀⠀⢸⡇⠻
⠑⡀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⡆⠀⠀⢻⣿⣷⡀⣶⠦⣾⡿⠋⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠀⠺⠇
⠀⡆⠀⣉⠙⠛⠀⣀⠀⠀⢀⣾⢯⣿⣿⣷⣿⣧⠐⠈⢿⣿⣿⣟⢛⣁⡀⠀⠀⣴⡀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠸⣿⠇⠀⠄⣡⣴⣶⣷⣶⣤⡀⠆
⠘⡇⠈⠙⠃⢰⡆⠙⠀⠀⢸⣿⣿⣅⡸⢻⡿⣿⠠⠀⢠⣿⣏⣿⣽⡍⣿⣀⡀⠀⢀⠈⠀⠠⡆⠀⣾⣿⠆⢀⣤⡀⠠⠀⣼⣟⣹⣿⣿⣯⣽⣷⡌⠄
⠰⡇⠸⠿⠀⠘⠃⠘⣇⠀⠀⠛⢿⣭⣿⣿⣿⠟⠄⠀⢾⣿⣿⣿⣿⣟⠛⠻⠿⣀⡀⠀⣠⡀⠀⠀⣨⣇⠀⢀⣉⠀⠐⠰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐
⠀⣁⡠⠤⢄⣀⡤⢤⣍⠀⠀⠀⠀⠀⠙⠉⠈⠀⢠⣄⠀⣬⣉⡛⠻⠿⣿⠇⠸⣿⣿⡦⢈⣹⠿⠚⢿⡿⠃⠺⠿⠇⠈⠁⠻⣿⣽⣿⣿⣿⣵⡿⠁⠁
⣼⣿⠾⣧⣤⢩⡞⡯⢴⢩⠽⣖⠀⠀⠀⠀⢠⣤⠀⠙⠀⢀⡈⠉⠀⠀⠀⠀⠀⠈⠉⠐⠀⠉⠀⠀⣾⡆⠀⠘⠃⠸⠿⠀⠂⡈⠛⠛⡟⠛⠋⡀⠂⣠⣶
⡱⣿⣮⣗⡣⠒⠽⠣⠘⢜⣗⢾⠜⢀⣄⣤⣿⠃⠰⠇⢠⡟⠁⠘⠉⠉⢳⣤⠛⠉⠙⢲⡴⠛⠉⢳⣤⠞⠉⠙⠳⣴⠛⠋⠃⠀⠈⠀⠁⠈⠀⠀⠀⣯⠿
⢠⣦⣄⠀⠠⠶⠶⠆⠀⠀⠀⢠⣄⣼⣿⣿⣿⠀⠀⣄⠀⣤⠆⠀⠀⠀⣠⣏⡀⠀⢀⣼⣃⠀⠀⢀⣿⡀⠀⠀⣨⣇⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⢀⡀⠐⠀⠐⠀⣤⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣀⠀⠀⠘⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠛⣝⠷⢚⣳⡀
⠀⠀⠀⠖⠓⠀⠀⠀⠀⠀⠀⠃⠀⠀⠓⠡⢿⣧⡀⢀⠀⣰⠿⠿⠾⡟⠁⠀⠄⠀⠀⠀⠀⠋⠿⡿⠁
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⢶⠲⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠙⠀⠀⠀⠀⢸⠍⢠⠀⠀⠄⠀⠀⠸⡟⠘⡛⣷⠖
⠀⠀⠀⠠⠄⠈⠁⠀⠀⠀⠈⠐⠀⠆⠁⠰⠀⠀⠈⠁⠘⢦⣄⣿⠀⠀⠀⠶⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⢠⠀⠀⠀⠀⠐⠿⠋
⠀⠀⡀⠀⠀⠀⠚⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⢀⢠⠆⠐⠀⠀⡀
⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣼⡛⡷⣦⣀⠴⠂
⠀⠈⢀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠀⣿⣷⣷⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⣼⣥⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣄⠀⠀⠀⡀⠀⡠⠇
⠀⠂⠀⠀⠈⠳⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠾⠷⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⡄⠁⠀⠀⠀⢀⠀⠀⠐⢀⣹⣿⣿⣏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡀⢀⠀⡆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢬⡙⡆⠀⠀⡉⠻⠏⠁⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⢀⠀⢲⢃⠘
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⠀⠀⠁⠀⠀⠤⠀⠀⠀⠀⠉⣿⣿⣿⠿⢀⠈⠀⠈⠆⠀⠘⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⣤⠐⠀⠀⢠⡶⠂⠀⠀⠀⠀⠀⠀⢀⣐⠂⠀⠀⠀⠐⠀⢀⠁⣰⡽
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣠⠔⢚⠟⠁⠀⠀⠀⢠⣀⣤⠂⠀⠀⠃⠀⠀⠀⠀⠆⠐⠂⠀⠈⠀⢠⠞⠃
⠀⠀⠀⠀⢀⡞⠀⠀⠀⣴⣋⠤⠚⠁⠀⠀⠀⠀⠀⠘⡿⠿⠄⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⡎⢀⣠⣴⡴⠿⠓⠒⠒⠲⢤⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡝
⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⣰⣿⣽⣷⡀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣴⣾⠟⢿⣿⣿⡟⠉⠳⢦⣀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⡀⠀⠀⠀⠀⠰⣿⣇⠀⠀⣾⣿⣿⣦⠘⣿⣿⣧⡴⢤⣴⣿⡆⠀⢠⣿⠀⠀⠀⠀⠀⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⠀⠀⠀⠈⢻⣆⠀⠻⠿⢿⠿⣶⣿⣿⣿⣶⣾⣿⠿⠃⣠⡿⠁⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠻⣦⣼⣶⣾⣿⣿⣿⣿⣿⡋⣿⣖⣶⡾⠛⠀⠀⠀⠀⠀⠀⣼⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠧⣿⠇⣿⡇⠀⠀⠀⠀⠀⠀⣰⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣓⣿⣾⣿⠅⠀⠀⠀⠀⠀⢠⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣶⣤⣄⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠁⣿⣀⣀⣀⣤⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⠉⣿⣿⣿⣿⣿⣿⢿⣿⣯⣿⣥⣿⡟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣤⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⠇⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠇⠀⠀⢻⣿⡿⣿⡿⢿⣿⠿⣿⡟⠸⡆⣷⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡌⠙⠀⠺⢀⠀⠹⣷⣿⣷⣾⡷⣶⠋⠁⠸⠷⠸⠟⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡟⢁⠁⠀⠀⣽⣤⣈⣽⣿⣿⢿⢿⣅⠀⢀⡀⠀⠀⢸⡀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢋⢸⠗⠈⠀⠀⢼⣟⣂⣌⠹⠐⠀⣸⠠⣌⠀⡄⠁⠀⠀⠈⡴⠦⢈⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠑⠀⠂⠾⡶⠀⠀⢸⢋⡿⠿⣇⣰⣦⠙⣀⡇⢠⡇⠀⠆⠀⠀⠧⠱⡌⢀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠉⠀⠋⡀⠀⠀⠀⠀⢸⣔⣠⣤⣿⡇⠋⠰⡙⢀⠃⠐⢘⠁⠀⠀⠀⠀⡄⠀⡄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠐⢸⠆⠀⠀⢀⠀⠀⠀⣀⣿⡷⠹⣿⠡⡆⠰⠀⠈⠀⠀⡄⠀⠀⠀⠀⠁⠘⠀⠈⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⡁⢠⠄⠀⠛⠀⠀⢠⢿⣹⣶⣖⣿⣾⣿⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠂⡜⢡⠛⡀⠀⠀⠀⠀⠀⢻⣿⣿⠻⣿⢏⣿⣷⡶⡠⠉⣠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠈⢷⣦⡀
⠀⠀⠀⠀⠀⣠⣶⠛⠉⠀⢀⡠⠀⠌⢀⠀⠐⡦⠀⢀⣸⠴⢿⣧⡶⣹⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠃⠃⠀⠀⠈⠻⣷⣆⣀
⣀⣤⣤⠾⠛⠁⠁⠀⠀⠀⢘⠃⠀⠀⠀⠂⠀⡶⠊⠁⠀⡇⠈⣿⣾⣿⣿⣿⣿⣿⣿⣴⠆⠀⠀⠈⠀⠀⠀⠀⠘⠀⡤⢠⠀⠀⠀⠀⠀⠈⢯⠶⣤⡄⣀
⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠚⢿⣿⣿⣿⣿⣯⣭⣽⣿⠁⠀⠀⠀⠀⢂⠀⠀⠀⢀⠀⣺⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣾⠀⠀⠀⠀⣃⠀⢀⠞⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠀⠀⠀⢀⡀⠀⠈⠀⠠⠀⠈⠂⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢘⠀⠀⠊⠉⢠⡀⠊⠀⠀⠀⠀⠀⠈⢿⣿⠟⠉⠀⠀⠀⠀⠀⠡⡀⠀⠦⠀⠀⠀⠘⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡀⠰⡆⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢲⣰⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⢠
⠀⠀⠀⣰⠉⢳⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⠁⠤⠉⠠⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐
⠀⠀⠀⠺⠰⠉⠆⠀⠀⠀⠀⠀⣶⣏⣳⡖⠀⠀⠀⠀⠀⢀⣘⣁⠀⠀⠀⠀⠀⠀⠹⠺⠈⠁⠀⠀⠀⠀⠀⢨⡀⠗⣈⠂
⠀⠀⠀⠐⠿⠗⠃⠀⠀⠀⠀⠀⠁⠙⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠺⠏⠁⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢘⣏⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡉⣯⠩⠀⠀⠀⠀⠀⠀⡑⣇⣿⡋⠀⠀⠀⠀⠀⣾⠃⢷⡄⠀⠀⠀⠀⠀⢸⠦⠿⠀⠀⠀⠀⠀⠀⠠⠛⡈⠓⠆
⠀⠀⠀⢓⣿⡀⠂⠀⠀⠀⠀⠨⠃⣇⡿⢡⠀⠀⠀⠀⠀⢻⠀⣾⠃⠀⠀⠀⠀⠀⡅⠀⠀⣏⠀⠀⠀⠀⠀⢐⠀⠁⠘⡂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⢸⣁⡿⠀⠀⠀⠀⠀⠀⠹⣀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠴⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠰⣤⣤⡔⠢⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⢐⡀⠀⠀⠀⠀⠐⢀⡊⣒⠀⠀⠀⠀⠀⠀⠀⡤⢢⠄
⠀⠀⠀⢐⢠⢰⠂⠀⠀⠀⠀⠀⡜⠀⠂⠀⠀⠀⠀⢀⢀⡻⠻⡟⡀⠀⠀⠀⠀⠀⠈⠀⠃⠔⠀⠀⠀⠀⠀⢀⣄⣀⡀
⠀⠀⠈⢹⡸⢀⠁⠀⠀⠀⠀⠐⠢⢝⢡⠃⠀⠀⠀⠀⠀⢿⡘⣿⠀⠀⠀⠀⠀⠀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠘⢖⣲⠗
⠀⠀⠀⢻⣒⣚⡗⠂⠀⠀⠀⠀⢶⣌⣰⠦⠀⠀⠀⢀⠁⠪⡄⠼⠉⠀⠀⠀⠀⠀⠀⢇⡼⢀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡀⢀⠁⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⢀⣄⡄⠀⠀⠀⠀⠀⠀⠀⢨⣡
⠀⠀⠐⡘⠁⠋⠪⠀⠀⠀⠀⠀⢈⠇⣺⠀⠀⠀⠀⠀⠀⠒⠐⠀⠠⠀⠀⠀⠀⠀⠼⠃⠷⠀⠀⠀⠀⠀⠀⠀⠚⠳
⠀⠘⠒⠄⠀⠀⠔⠀⠀⠀⠀⠀⣰⠐⡰⠀⠀⠀⠀⠀⠬⠀⠀⠈⠤⠀⠀⠀⠀⠈⢦⣛⣠⠆⠀⠀⠀⠀⠀⠀⠂⠘⠂
⠀⠈⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⢤⠔⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠚⠘⠆⠀⠀⠀⠀⠀⠀⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⣶⣤⡴⡇⠀⠀⠀⠀⠀⠘⢡⠛⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡆⣖⠀⠀⠀⠀⠀⠠⠌⠀⠀⠠⠄
⠀⠀⠠⠄⣿⠅⠃⠀⠀⠀⠀⠀⢂⠸⠀⡂⠀⠀⠀⠄⠀⡌⠈⣄⠀⠀⠀⠀⠀⠀⣠⡞⠤⠀⠀⠀⠀⠀⠀⢴⡄⠀⣼
⠀⠀⠀⠶⠊⠛⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠑⠃⠉

⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠘⠿⠀⠙⠢⠀⠀⢀⠔⠋⠀⠿⠁⠀⠀⢿⡥⣷⡦⡀⠀⠀⢀⠠⣶⡇⣿⠃⠀⠀⣹⣟⣷⡦⠀⢀⠀⠀⣴⡾⣿⡏
⠀⠀⢰⣤⣀⣀⣀⣿⣀⣀⣀⣤⡆⠀⠀⠀⠈⢷⣍⡵⠾⢴⣦⠼⠾⣍⣼⠏⠀⠀⠀⠈⢿⣽⡅⠰⣼⣶⠆⣽⣽⡏
⠀⠀⠀⠔⠁⡀⢘⣿⠀⢀⠈⠢⠀⠀⠀⠀⠀⢰⣿⡅⣤⡾⡿⣦⡆⣿⣷⠀⠀⠀⠀⠀⣤⣿⣶⡐⢻⠛⢂⣶⣿⣄
⠀⠀⠀⠹⠄⠀⠀⠉⠀⠀⢠⠇⠀⠀⠀⠀⠀⠐⢿⣷⣶⠆⠀⢶⣶⣿⠃⠀⠀⠀⠀⠨⣏⢿⣽⠀⠀⠀⠸⣏⣟⣹
⠀⠀⠀⠰⠉⠁⠀⠀⠀⠉⠉⠆⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠈⠙⠆⠀⠀⠀⠀⠰⠉⠉⠁⠀⠀⠀⠀⠉⠉⠙⠆
⠀⢿⣿⣶⢄⡀⠀⠀⠀⢀⡠⣶⣿⡿⠀⠀⢦⡠⣶⠖⢤⡀⡀⣠⠔⢶⠦⣠⠆
⠀⠈⣿⡂⢀⡙⢆⣀⡴⢋⠀⢐⣿⠀⠀⠀⠈⢻⣷⣀⣠⢿⣿⢤⣀⣰⣿⠃⠀⠀⢠⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⢠
⠀⠀⢹⡗⠒⠶⠶⣿⠶⠶⠒⢾⡏⠀⠀⠀⠀⠀⣿⠁⡰⠾⠟⠶⠀⣹⠇⠀⠀⠀⠈⠹⠄⢀⠠⠀⢿⡿⠠⢄⡀⠰⠋
⠀⠀⢸⣇⡐⡊⠸⠿⠆⢁⢂⣸⡇⠀⠀⠀⠀⠀⠈⡧⠴⠂⠀⠳⠴⡏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣹⡇⠀⠈⠉
⠀⠀⠀⠙⠛⠛⠁⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢀⠀⠀⠀⠀⢀⠤⠀⢀⡀⠀⢸⡿⣶⣤⠀⠀⠀⠀⠀⢀⣔⡶⣿
⠀⣆⠘⠓⠀⠀⠐⣷⠂⠀⠀⠚⠁⣰⠄⠀⢇⠀⠀⠀⠑⢰⣧⠚⠁⠀⠈⣀⠇⠀⠀⢟⠙⠘⠈⠳⣀⣠⠞⠉⠓⠉⡟
⠀⠘⢧⠀⠴⠒⠁⣿⠈⠲⠦⠀⡼⠃⠀⠀⠀⡩⠋⠉⠉⣿⣿⠋⠉⠉⠩⠁⠀⠀⠀⠈⡶⠒⠶⠲⣿⡷⠶⠶⠲⣖⠁
⠀⠀⠀⢆⠀⠀⡰⠙⢆⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠉⡴⠉⠉⠳⡈⠁⡄⠀⠀⠀⠀⠸⣇⠀⠀⠠⢾⠷⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠓⠒⠀⠀⠀⠐⠒⠛⠁
⢀⡄⠠⣄⠀⠀⠀⠀⠀⠀⠀⣠⠄⢠⡀⠀⢰⣿⡷⠦⠀⠀⠀⠀⠠⠴⣿⣷⠀⠀⢰⣶⣶⣤⣀⠀⠀⠀⠀⣀⣤⣶⣶
⠈⢿⡀⠈⠑⠄⠀⠀⠀⣠⠞⠁⢀⡻⠁⠀⠘⣿⣁⠀⢀⠀⡀⠀⠀⢀⣹⡟⠀⠀⠀⢻⣿⡳⣯⣷⣄⣠⣮⣽⢚⣿⡟
⠀⠈⠅⠈⠀⠀⠀⣷⠐⠁⠀⠀⠈⠀⠀⠀⠀⢉⡟⠂⠈⢹⣿⠉⠀⠚⣏⠀⠀⠀⠀⠀⠻⣷⡜⠟⣿⡿⠻⢰⣿⠟
⠀⠀⠀⣰⠋⠀⠘⣿⠋⠀⠙⢄⠀⠀⠀⠀⠀⢸⣄⠀⣠⠍⠇⣄⡀⢀⣾⠀⠀⠀⠀⠀⠀⣿⣧⣼⠿⢿⣦⣾⣿
⠀⠀⠀⠣⣰⡀⠀⠉⠀⢀⡆⠌⠀⠀⠀⠀⠀⠀⠙⡟⠉⠀⠀⠈⠙⡛⠁⠀⠀⠀⠀⠀⠀⢈⠟⠛⠁⠈⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠟⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠹⠿⠿⠿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣰⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣤⣶⣶⡿⠿⠟
⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⢿⣛⣫⣭⣷⠶⠛⠁
⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠚⠛⠛⠛⠋⠉⠁
⠀⢰⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢠⣤⣤⡄⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡤⠟⠉⠉⠁⠓⠶⠶⠦⠤⠴⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠂
⠀⢸⣿⣿⣿⣿⡿⠋⠁⠀⠈⠉⠻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠂⢠⠆
⠀⠀⠹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠙⣿⣿⣷⣄⡀⠀⠀⣀⣴⣿⣿⡇⠀⠈⠋⠉⠉⠉⠉⠉⠉⠋⠙⠁
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠢⠤
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⡴⣾⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠺⡦⡈⣿⣿⣿⣟⡿⠋⣠⠖⠚⠛⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⠋⠉⠙⣷⠋⠀⠀⠀⠤⠄⠀⠒⠂⠊⠁⠀⠊⠉⠁⢦
⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⣿⣿⣆⣀⣤⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣽⣿⣿⣿⣿⣿⠛⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣤⣤⣽⣿⣿⣿⣯⣤⣤⣀⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢰⠊⢡⣿⣿⣿⣟⣛⣋⣉⣯⠉⢷⡀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⡿⢹⣿⠀⢻⣇⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢸⣿⣿⣿⣿⣟⣛⢚⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠧⠀⠀⠹⣿⣿⣿⣿⣿⡟⠁⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠙⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡴⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢾⠆
⠀⠀⠀⠀⠀⠀⣂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⣀⠀⠀⠙⠒⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠋⠙⠛⠳⢦⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣠⡾⠁⢀⡴⠞⠋⠉⠙⢻⡟⠺⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡤⢀⠀⠘⠀⠀⠀⢸⡿⠁⠀⠀⣼⠿⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⢿⣧⠤⠀⠁⠀⠀⢸⡇⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢡⣾⡇⠀⠀⠀⣿⣇⠀⢠⣿⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⣾⡏⠀⠀⠀⢀⣿⣿⠷⠾⠏⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣾⡏⠀⠀⠀⢰⣿⠣⣈⠀⢤⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢈⢷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⣿⠇⠀⠀⠀⠀⢸⡏⢰⡷⣯⠘⠀⠀⠀⠀⠀⣀⢀⡀⣼⣤⣢⣤⣠⣦⡆⣴⣿⣦⣤⣴⣶⣶⣿⡿⣿⠿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣣⣶⠀⠀⠀⣠⣴⡾⣿⣁⠀⠀⠀⠀⠀⢠⣵⠟⠛⠛⠻⢿⡇⢩⣿⣿⣿⠛⠟⠋⠉⠉⠁⠄⠀⠈⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⠀⣰⣿⡼⠂⠁⠀⠀⠀⠀⢀⣴⠿⣿⠿⠟⠛⠛⠛⠛⠙⠛⠉⠀⠀⠀⠀⠀⠀⣀⢀⣀⣨⣶⣿⣧⣾⣷
⠀⠀⠀⠀⠀⢠⠀⠀⣠⣶⣿⣿⣿⣿⣯⡻⢄⡿⢁⡸⠧⢴⠄⠀⠀⢠⣿⣿⣟⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⠶⠄⢀⡀
⠀⠀⠀⠀⠀⠉⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡾⢯⡘⠃⠀⠀⠀⠀⠀⢨⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⢿⠿⠻⣿⣿⣿⡿⣟⣉⣁⡐⣶⡶⣶⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣭⣿⣿⡙⠛⠁⢠⣗⠫⠽⠖⠀⠀⠀⢺⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠙⢛⣿⣿⣿⣯⠿⣯⠳⡄
⠀⠀⠤⠀⠀⣾⣿⣿⣿⣿⡟⣋⣭⠏⣡⣶⣾⠄⣩⠜⠂⠀⠀⠀⠀⠘⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠃⠀⠙⠛⠻⠷⠋
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⡙⢻⣿⡆⠙⠉⢱⠎⣿⣼⡦⠄⠀⢀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣠⠦⣄⡀⠀⠀⢹⡆⠀⠀⠀⠄
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣤⡙⠋⢷⣿⡷⣷⣿⢈⣀⢀⣼⣿⠋⠀⠄⢀⣦⠔⠂⠀⠀⠀⢀⣼⠁⢀⣤⣤⡾⠛⠀⠀⠘⣀⠀⢸⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣯⣴⠿⠛⠉⠉⠙⣿⠳⠷⠀⣴⡟⠁⠀⠀⠀⠀⢠⣾⣿⣴⣿⣿⣍⠁⠀⠀⠀⠀⠘⢿⣼⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡟⢻⣧⣼⣶⡟⠁⠀⢀⡀⠀⠀⠈⠓⠓⢾⡟⠁⠀⠀⠀⠀⠀⢳⣿⣿⣿⡟⠠⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⡄
⠀⠀⠀⠀⢹⣿⣿⣿⣍⣽⣿⡿⣿⠫⠘⣷⠀⠀⢸⠇⠀⠀⠀⠀⠀⢈⡇⠰⣀⣠⡀⠀⠀⡈⢿⣿⣿⣇⠀⢠⡅⠀⠀⠀⠐⠀⠀⣿
⠀⠀⠀⠀⠈⣿⡟⣿⣿⢿⢏⡝⠀⣴⠾⡏⠁⠀⢻⠀⠀⠀⠀⠀⠀⠾⠁⠀⠈⠁⠈⠉⠀⠀⠙⣗⣿⣿⡄⠘⠷⢎⢀⡀⠀⣸⣶⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣴⣾⡏⣾⠁⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣿⣿⢷⣦⡹⢯⣿⣻⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣿⡇⣿⠀⠀⠧⠀⡆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⠀⠀⢷⠻⠀⠀⠙⠛⠟⠛⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣯⡏⢰⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣶⣼⡧⡴⠞⢠⣼⣾⣷⣄⠀⠀⣤⣤⣴⣶⣦⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣹⣿⣧⡟⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠉⠉⠁⠀⡀⠚⡟⠁⠉⠛⡇⠈⠉⠉⢿⡥⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠴⠀⠹⣿⣷⣿⣿⣷⣿⣾⡙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⢠⡶⠿⠀⢠⣿⠶⢡⣄⠀⢰⣄⠀⢶⣾⠟⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣟⣻⡿⠆⠀⠀⠀⠀⠀⢀⠤⠤⣾⠇⠃⠀⠀⠀⣤⣷⡀⠀⢀⣠⡀⢸⣧⡈⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⠀⠀⢀⠟⠀⠀⠀⠀⢠⡿⠏⠀⠈⠀⠙⠁⠸⣇⠉⠉⠉⣹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⡻⠁⠢⠀⠀⠀⠀⠀⠀⠈⠂⠀⢠⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣯⣿⡤⣿⣦⣀⡀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣠⣞⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⡄⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤⣤⣤⡤⠲⡎⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣗⣻⣣⡀⠣⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢠⡾⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣏⣌⢀⡛⡉⠓⠀⢀⡄⠀⠀⠀⢰⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⡐⢦⠀⡄⠀⢀⣠⣶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⢸⡿⣍⠻⢍⠛⠙⢶⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⠶⠫⠛⡩⠛⡩⢿⡇
⠀⠸⣷⠄⣱⢖⣳⡾⡷⢶⣌⣛⡲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣛⣡⣖⣾⢷⣞⡲⣎⠠⣾⠃
⠀⠀⠹⡦⢹⣯⠖⣉⣩⣄⣀⡛⢿⣾⣍⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡹⣷⡿⢛⣀⣢⣍⣉⠲⣹⡇⢶⠏
⠀⠀⠀⠹⡆⢻⢸⡏⢦⡤⠄⠬⠍⣓⠻⣾⡿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⣋⠭⠤⠠⣦⠴⢹⣇⡟⢰⠏
⠀⠀⠀⠀⠹⡆⢯⠳⣄⡙⠢⣄⡀⠀⠘⡒⢝⡷⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣻⢾⡩⢒⠋⠀⠀⣠⠔⢋⣠⠞⡼⢰⠏
⠀⠀⠀⠀⠀⢿⡘⣷⣄⢉⠑⠒⠭⠍⣐⠄⣄⡈⠋⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣋⠿⠙⢁⢠⠠⡂⠭⠭⠒⢊⡉⣠⡾⢀⡟
⠀⠀⠀⠀⠀⢸⡆⢹⣎⠓⠒⠊⠣⣐⠀⣈⡝⠛⢶⣤⠉⢟⣳⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢾⡿⠋⣤⠶⠛⢪⣁⢐⣂⠜⠑⠒⠚⣱⡇⢰⡇
⠀⠀⠀⠀⠀⠈⡯⢸⡿⠟⢒⣁⡀⠀⠀⠀⠘⢤⣂⡈⡙⠾⣷⣿⣿⣧⠀⠀⠀⠀⠀⢀⣼⣟⣿⣾⠷⢋⢁⣐⡴⠃⠀⠈⠀⣀⣉⡒⠻⣿⡇⢽
⠀⠀⠀⠀⠀⠀⢿⡘⡇⣾⠉⠡⡉⡌⠉⡃⠒⡤⢀⡀⠱⣤⠈⡻⣧⣻⣧⢠⣾⣷⠄⣾⣟⣼⢟⠁⣤⠊⢀⠀⢤⠒⢚⠉⢡⢉⠈⠉⣷⢸⢂⡿
⠀⠀⠀⠀⠀⠀⠈⠳⣽⠮⡝⠛⠛⠓⠚⠓⠚⠓⠒⠯⢤⣖⣁⠈⠈⠳⣿⣇⣿⣿⣸⣿⠞⠁⠁⣈⣴⡤⠽⠒⠚⠓⠚⠓⠚⠛⠛⣫⠵⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⣶⣤⣐⣐⡈⢒⡄⢀⡠⢄⠉⠛⠶⣤⣹⣿⡏⣻⣿⣏⣤⠶⠛⠉⠠⠄⡀⢠⡒⢀⣂⣀⣠⡶⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⡻⢫⡟⢻⢳⡞⡿⠳⢶⣿⣿⡫⣿⣿⣿⣯⢝⣿⣿⡶⠞⣿⢳⠾⡟⢻⡝⠟⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠐⠁⠃⢲⡇⢸⠀⡹⢀⠀⠚⢻⣿⣿⡏⣿⣟⢹⣿⣿⡟⠃⠀⡀⣏⠀⡇⢸⡖⠸⠘⠂⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠒⢀⠀⢩⡇⠈⡆⠳⢌⢀⡀⠈⠀⢿⠃⣿⣿⠘⡿⠀⠁⢀⠀⡡⠞⢰⠁⢸⡍⢀⡀⠒⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⠐⢇⡇⠐⢳⠀⠹⡄⠸⠤⣁⠀⠄⡜⠀⣿⣿⠀⢣⠠⠀⣈⠤⠇⣠⠏⢀⡞⠂⢘⡸⠂⣨⠗
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠶⣐⡀⠩⠷⡀⠈⠢⣄⠘⠦⡾⠀⠀⣿⣿⠀⠈⢷⠴⠁⣠⠔⠁⢀⠾⠍⠀⣂⠷⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⣄⠂⢔⢄⠘⢩⠦⣀⣈⣉⡟⠁⠀⠀⢻⡇⠀⠀⠈⢻⣉⣁⣠⠴⡍⠃⡠⡣⠓⣠⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣘⡒⡓⠤⣈⣁⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣈⣀⠤⠚⢒⣃⢀⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣌⡬⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢥⣡⠏⠉⠉⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⡀⠀⠀⣀⡀⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠰⡂⠀⠐⠚⡠⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⡈⠀⠀⠠⠀⠀⣀⣱⠮⢀⠀⠓⠘⠓⠃⠀⠀⡀⢌⡃⠠⡆
⠀⠀⠀⠓⠓⠀⠀⠀⠠⠀⠢⢨⠀⠀⠀⢀⠀⠉⠀⠐⠸⡁⢐⠀⠁⠊⠀⠀⠀⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠄⠌⠀⢁⠂⠃⢸⡓⠃⠰⣀⠇
⠀⠀⠘⠀⠇⠀⠐⢀⠶⠀⠀⠈⠀⠀⢀⠀⠀⠀⠄⢀⠠⡤⠀⠠⠀⠀⠄⠀⡆⠀⢴⠀⢉⠀⠐⠀⠀⠀⠀⢄⠠⠁⠈⠄⠠⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠉⠔⡀⠢⠀⠂⠀⠀⠄⠀⠠⠀⠀⠄⠈⠈⣠⣀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⢀⣀⠀⢀⠠⡁⢈⠆⢄⠀⠤⣄⠀⠀⢔⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⢴⠆⡄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠦⠘⠊⠁⠀⠀⠀⠀⠀⠀⠁⠀⣱⠀⠀⠀⠀⢀⠈⠠⠄⠈⢷⠀⠢⡤⠃
⠀⠀⠀⢾⡐⠁⠙⢀⡇⠀⠀⠖⠂⡀⠀⢐⡛⠆⠐⠶⣆⣶⡄⠀⠮⠆⠀⠌⣀⢂⡂⠀⠔⠢⠀⠸⠭⠭⠿⠿⠽⠇⠀⠌⠉⠈⢀⠅⠀⠀⠀⠐
⠀⠀⡀⠀⣅⡋⣀⠁⠁⠁⠀⠈⠀⠀⣲⢢⡀⠀⠀⠧⣍⠴⠀⠠⣄⠄⠑⠀⢻⠀⠈⠀⠀⠀⠀⠀⠉⠈⠚⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉
⠀⢀⠤⢌⠉⢠⠈⠉⠀⠠⠀⠀⠀⠀⠠⣉⠀⢻⠒⡄⢠⠀⠈⣡⢀⡐⠂⠠⠃⠀⠀⠀⠄⠀⡄⠁⠀⣤⠀⣴⡶⢿⠂⡀⢠⡀⠄⡀⠈⠉⣂⠈
⠀⡇⣀⠈⠀⠈⡀⠀⠀⠈⠁⠀⠀⣦⡄⠉⡀⠈⠒⠁⠀⠈⠃⠀⠀⠁⠤⠆⠀⠠⠀⠂⠀⠂⡄⠀⠘⠓⠛⣂⡅⠂⠀⠀⢀⢌⠉⢋⡀⠁⠄⠈
⠀⠑⠤⠜⠀⠀⠀⡀⠐⠀⠀⢀⠤⠤⠤⠀⠫⠀⠀⠀⠀⠰⢆⠀⢀⠀⠘⠂⠀⠜⣁⠀⢿⠆⠠⣛⠃⠰⠊⠁⣀⠋⡁⢀⣠⡃⣀⢀⠋⠤⢰⡄⠆
⠀⠀⣰⡆⠀⢀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠀⠠⡦⠀⠐⠀⠀⠀⢠⠀⠁⡠⠒⠢⡀⢀⠠⠢⡀⠀⠀⠀⠤⠀⣿⠀⠶⠀⡀⠀⣀⢄
⠀⠀⡷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠃⠄⠀⠀⠀⠈⠀⠒⠉⠀⠀⠀⠀⠱⠄⢤⠃⠘⢈⠌⠀⡡⠀⢀⠀⠀⢰⠀⠀⠀⢠⠀⠀⠃
⠀⢀⡀⣠⠀⠀⠄⠄⠂⠀⠠⠴⡏⠲⢀⠔⠁⠐⠄⠀⢠⠒⢸⣿⠷⠾⡀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠐⠀⠂⠀⢀⡞⠀⠀⡀⠀⠀⠀⠰⠁
⠀⠑⠸⠈⠈⠆⠀⠀⠀⠠⠀⠀⠇⠀⠈⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢟⡠⠀⠂⡄⠀⢬⣽⠀⣄⡠⠨⠀⡀⠐⢀⣁⠀⠂⠀⠂⠀⠁⡀
⠀⠒⢤⠈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠈⠀⠄⠀⢨⠾⡀⠀⠀⠀⡀⠀⠀⢸⠀⠐⠉⠁⠀⠀⣉⢀⠀⢈⡉⣍⠀⠂⠘⠄⡃⠉⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠈⠀⠐⠊⠀⠀⠂⠐⠀⠀⠀⠈⠀⠀⠑⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⡿⡇
⠀⠀⠀⢰⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡶⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣴⣇
⠀⠀⠀⠀⠙⣿⣿⣿⣽⣿⣶⣤⣀⢀⠰⠃⠘⠆⠀⢀⣤⣶⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢻⣙⣻⣿⣿⣿⣿⣿⣷⡇⠡⠄⠸⣾⣿⣿⣿⣿⣿⣟⣋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠉⣵⣿⣿⣿⡽⣛⢵⡂⠙⠋⢐⡪⣝⢯⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠈⠷⠀⠀⡀⢀⣰⡄⠀⣸⢿⣿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢰⢹⣿⣾⣿⣿⡏⣆⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⠂⠁⣠⣴⣿⠿⠋⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠶⠭⠭⠷⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⣵⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠸⢿⠿⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢢⢂
⠀⠀⠀⠈⠖⡼⢀⣃⡤⢠⡄⣀⠀⢄⡘⠃⠘⠃⣀⠀⢀⠠⡄⢰⣔⣇⢧⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢳⡧⠀⡄
⠀⠀⠀⠀⠘⡍⣑⡒⠽⡶⠳⢺⢧⣦⡍⠁⠈⢉⣴⡤⣟⠺⢳⡮⠔⡚⡉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠮⢷⠢⠌
⠀⠀⠀⠀⠀⠀⠨⠦⠎⠑⠘⠋⢋⣹⡄⠀⠀⢠⣏⠙⠋⠋⠚⠹⠴⠏⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣮⡯⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⣴⣤⣤⣤⣶⣦⣉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⡀⠀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⣛⡻⠿⠿⠿⢛⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠀⢠⣀⣼⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢩⣙⣛⣛⣋⡭⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢻⣛⣩⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⣀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡗⠲⡀⠈⡀⢀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇⠀⠻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄⣀⣀⣠⣔⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⣤⣴⣿⣿⠃⠀⠀⠀⠈⢰⡇⣄⣀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣠⣤⠴⢿⣈⣿⡟⠉⠻⢿⡄⠀⠀⠠⡆⢸⣷⣿⠉⠈⣿⣿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠻⣯⠑⢛⡁⠂⠈⢻⣿⣷⠀⠀⡀⠀⢀⣤⡀⣷⣿⠿⣛⡆⣸⡿⢿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⢿⣿⣧⢠⣿⠀⠀⡿⠀⢉⡟⠀⠏⣸⣿⣄⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢠⠡⠹⠆⠀⠀⠈⠀⠀⠘⢻⣿⣾⣧⠀⠰⢷⡄⠸⠷⠠⠂⢁⣾⣯⡿⠛⠿⠿⠿⢇⣤⣄
⠀⠀⠀⠀⠀⣀⣸⣤⣤⣄⡀⣠⠂⠀⠘⢿⣭⢿⣿⣷⠀⠀⣿⠀⠀⠀⠀⠀⠘⡿⠛⠁⠀⠀⠂⢀⣿⠹⠙⢷⡀
⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠁⠈⣀⠉⢿⣾⣿⣿⡀⠀⡽⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐⢀⡴⠻⠁⠀⠀⠀⠁
⠀⠀⠀⠘⢷⠶⠦⠤⠂⠄⠀⠀⢂⣄⡈⠻⣾⣿⣿⣿⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠪
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣌⣻⣿⣿⣧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡈⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠰⣶⣬⣁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣷⢆⠄⠀⠀⠀⠀⠀⡐⢋
⠀⠀⠀⠀⠀⠀⠈⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣤⣄⠀⠀⢠⣦⣰⣾⣴⢂⡤⠶⠦
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣄⣀⣤⣀⣀⣀⣤⡶⠶⠖⠛⠛⠻⣿⣿⣧⣄⡀⠉⠉⠻⣿⠿⢶⣄
⠀⠀⠀⠀⠀⠀⣾⠙⠛⠻⠿⠿⠿⠿⠛⠉⠀⠐⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⡄⠀⣶⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠸⣧⡈⠀⠀⠀⠀⠤⠐⠂⠀⠀⠀⠀⢀⣤⡴⢊⡿⢳⣼⡟⣹⠟⡀⢹⡄⠀⢼⣁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⣦⣤⣠⠤⠄⠀⠀⢀⣍⣸⡿⠋⢀⣬⣿⣷⡿⠟⡃⢸⣷⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠟⠉⠀⠀⣶⠋⠋⠀⣠⡾⠃⢸⡟⣆⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠇⠀⠀⠀⡰⠃⠀⠀⡄⠉⠀⠀⣿⠀⠘⡄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣼⠁⠀⠀⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠭⢽⡟⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⡿⠒⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣍⣉⣿⣿⡟⣙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣝⣿⣿⣏⣉⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣄⣀⣈⣫⡉⠉⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠉⣙⣉⣀⣀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡁⠀⠀⠀⠉⢢⣀⠀⠙⢦⡀⠀⠀⠀⠀⣠⠞⠁⢀⣤⠊⠀⠀⡀⣀⣹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡋⠉⠉⠉⠁⠀⠈⠑⠢⣀⠹⣦⣿⣧⡞⢁⠠⠒⠉⠀⠀⠀⠉⠉⠉⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢶⣶⣶⣶⣦⣶⣦⣤⣄⣁⣾⡏⣿⣮⣤⣤⣤⣶⣴⣶⣶⣶⠶⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠋⠉⢉⡠⠀⠀⠀⠀⣿⣧⣿⡇⠀⠀⠀⠲⡉⠉⠉⠑⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⠠⠞⠀⣀⠔⠁⠀⠉⣿⡏⠀⠀⠀⢄⡀⠀⠦⠀⠈⠉⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠊⡀⠀⠲⠒⣨⠁⢀⠊⠐⡰⣿⡷⡀⠈⢄⠀⢩⡐⠲⠂⠀⡈⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⡀⡰⠁⠰⠃⢀⠂⠀⠔⠀⣿⡇⠐⣀⠀⢂⠀⠳⠀⠱⡀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣴⣆⣠⡃⣠⠎⠀⠀⠀⠀⠀⠈⢦⡀⣣⣠⣷⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢠⣄⠴⣶⢲⡆⢤⣴⣤⠄⣰⢲⡶⢂⣤⠄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠻⣧⣄⠀⠹⠀⣽⡇⠰⠉⠀⣠⣾⠋⠀⠀⠀⠀⢠⣾⠏⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⢻⣿⡄⠁⠏⢀⠁⣰⡿⠉⠀⠀⠀⠀⠀⠀⢊⣽⣿⣯⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠈⡿⠞⠀⠈⠻⣏⠀⠀⠀⠀⠀⣴⣶⣶⡽⢻⡿⡿⢻⣹⣾⣶⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⣤⣄⡘⢿⣿⠇⠀⢇⠃⠀⣿⣿⠋⣀⣴⡄⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⡠⠀⠀⢠⠤⠾⠟⠋⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⢧⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡶⢶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢶⣦⠆⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡾⣏⡙⣃⢈⡄⠀⠀⠤⠄⢀⠀⠀⡀⢠⣶⡄⢀⡀⠀⠀⡠⠤⠀⠀⠀⢈⣠⣼⣧⣄⡃
⠀⠀⠀⠀⠀⠀⣨⣭⣿⣿⣟⣯⡭⡄⠀⠀⠀⠀⠀⠈⣓⢉⣿⣿⣿⡉⢉⠀⠀⠀⠀⠀⠀⠀⣈⣉⣿⣿⣋⣀⡀
⠀⠀⠀⠀⠀⢀⠃⣾⣿⣿⣿⡇⢇⠰⠀⠀⠈⠙⠱⠋⠁⠞⠈⣿⠉⠣⠀⠙⠟⠛⠁⠀⠀⠎⠀⣿⣿⣿⣿⠁⠰⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣇⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣿⣿⣿⣿⡷⠦
⠀⠀⠀⠀⠀⢸⠁⣿⣿⣿⣿⡟⢸⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⠇⠹⣷⣿⣍⣠⠊⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠉⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠐⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⠲⣄⠀⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⢇⡀⡀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡇⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠰⠋⠀⠀⢶⠀⠆⡟⠠⣅
⠹⣀⠠⠀⠀⣀⠀⠀⠐⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠞⣽⡷
⠀⠈⠓⠂⡀⠈⢷⣌⣁⠠⠋⠀⠀⠀⠀⠀⢀⣶⠶⠀⡀⢸⡇⢀⡴⠀⠀⠀⠀⠀⠀⡴⠁⠀⢜⣡⠟
⠀⠀⠀⠀⠈⠐⠂⠂⠀⠀⠀⠀⢤⠀⢀⡴⢻⡅⠀⠀⠀⠊⠓⣯⣤⣤⣶⡶⠄⠠⠀⠙⠒⠂⠈
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡀⠑⣤⣤⣾⠇⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⢠⣾⡆⣿⠀⠀⣀⣴⢾⣁⣠⡿⠀⠀⠑
⠀⠀⠀⠋⡀⢽⡳⢄⠀⠿⠁⠀⠉⠉⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠞⣿⠃⠀⠀⠠⠀⠀⠀⣀
⠀⠀⠴⣤⠆⠀⠹⡌⡇⠀⠀⠀⠀⠀⠀⠀⣀⢤⢠⡀⠀⢴⡔⠁⢠⣾⠃⠀⠀⠨⡀⠨⣀⡾⠁
⠀⠀⠀⠈⠐⠆⠐⠛⠀⠀⠀⣴⣆⠀⠀⠀⠙⠳⢶⣦⠊⠈⣿⡀⢀⡼⠀⠀⠀⠀⠁⠂⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡲⣯⠀⠀⠀⠀⠀⠈⠓⠶⠶⠂⠈⠉⠀⠀⢀⡶⠀⣠⠄
⠀⠀⠀⠀⠀⣤⠄⠀⠀⢤⡀⠈⠙⣦⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣼⠁⠀⡁⡀
⠀⠀⠀⠀⠀⠉⠳⢿⣀⠀⠙⠆⠶⠟⠀⠀⠘⠑⢤⠀⠀⢣⠀⠀⠀⣤⠖⠋⢀⢠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠃⢄⠘⡆⠀⠀⠈⢆⠄⢈⡼⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢇⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠆⠉

⠀⠀⠀⠀⠀⠀⠀⢩⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⣄⡀
⠀⠀⠀⠀⠠⠒⠀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢤⣤⣤⢜⡤⣤⣤⠤
⠀⠀⠀⠀⠇⠀⠀⣿⣻⣇⠀⠀⠀⠀⠀⠀⠀⢀⣏⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠜⠛⠛⠁⠀⠀⠀⠀⠀⠀⠰⢻⠓
⠀⠀⠀⠀⠈⠂⠨⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢠⡴⠚⠓⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡚⣓⠄
⠀⠀⠀⠀⠀⣧⣠⢤⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁
⠀⠀⠀⠀⣿⣾⠛⠚⢫⣾⠆⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⡀⡀⣠⠀⣠⣤⢤⡄⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂⠀⡀
⠀⠀⠀⠀⡿⣶⣭⣾⣿⡷⣿⠇⠀⠀⠀⠀⠀⠀⢸⡯⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠻⡀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⢀⣿
⠀⠀⠀⠉⠲⠋⠉⠉⠙⠚⠉⠁⠀⠀⠀⠀⠀⠀⢮⣽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣿⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢀⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢴⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢛⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣦⣴⡂⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣼⡃⠀⠀⠀⠀⠀⠀⠀⠬⠚⢏⠉⣏⣻⠁⠀⠀⠀⠀⠀⠔⢠⣿⣿⡌⠢⠀⠀⠀⠀⠀⠀⠀⣷⣏⣿⡇
⠀⠀⠀⠀⠀⠈⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⡐⠓⡘⠀⠛⠛⠀⠀⠀⠀⠀⠀⢀⠈⠈⠁⠡⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⡶⣶⠶⠶⠶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⢶⣶⣶⣶⠶⠶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠚⠸⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⣸⢼⣷⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠿⢽⣿⠿⠧⠄⢠⣴⣶⣤⣄⠀⠀⠀⠀⠀⠀⢠⡀⡄⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⣰⠚⠉⠁⠀⠀⣠⡀⠉⠻⣿⣿⣦⣄⠀⠀⠀⢡⠈⠉⠁⠀⢰⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⡿⠀⠀⠀⠀⠀⢹⣿⣆⣀⠀⠀⠉⠉⠉⠀⠀⢸⠀⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡍⠛⢤⡀⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀⠈⠐⢄⡠⠎⠀⠀⠀⠀⠲⣦⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠃⠀⠀⠘⢷⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⣦⢀⠜⢁⣀⣀⣀⣀⣀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠛⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⢀⠀⠀⠘⠛⢿⣿⣿⡾⣿⣿⣿⣿⣿⡟⠉⠉⢉⠂⠤⡀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⠙⠲⣄⠀⠈⣿⢫⣄⠈⠛⠿⠟⠋⠀⠀⠀⠸⠾⠃⠴⢮⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⢙⣶⠞⠁⠀⠉⠁⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣠⡴⠏⠁⠀⠀⠀⠀⠚⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣥⣄⡀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠂⠀⠀⠀⠀⣠⣴⡿⣿⣿
⠀⠀⠀⣤⣾⣠⣄⣀⡀⠀⠀⠰⠶⠀⠀⣠⡾⠋⠀⠀⠀⠙⠳⣼⣇⣀⣠⣤⣜⣀⣴⣶⣤⣤⣰⣟⣭⣿⣇⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣾⣞⠛⠃
⠀⠀⠀⠈⠻⣿⡟⠛⠋⠀⣠⡶⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠲⡤⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠈⠻⣦⣤⣾⡿⠂⠀⠀⠀⢀⠏⠀⠀⠀⠀⠁⠀⠀⠀⠇⠀⢻⣿⣿⣿⣿⡿⠁⠀⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠈⠉⠉⠁

⠀⠀⢠⣤⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣤⠄⠀⠀⠀⠀⠀⠀⡠⣴⡄⠀⠀⠀⢀⡤⣶⡀
⠀⠀⢸⣟⣛⣤⡀⠀⠀⠀⠑⠀⢠⣤⠀⢀⣀⡀⣴⠂⠐⠁⠀⠀⣿⡀⠀⢠⣤⣤⡴⠟⠀⣿⠁⠀⠀⢀⡎⢡⣟⣷
⠀⠀⠈⣿⣿⣿⣷⡀⠀⠀⠀⢾⣤⡄⠀⠀⠀⠀⣈⣿⠆⠀⠀⠀⢈⣵⣦⢸⣿⡟⢇⣰⢛⡅⠀⠀⢠⣦⠉⢣⢿⣮⡀
⠀⠀⠀⢧⠈⠛⢿⣿⣄⣤⣤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠙⠃⠀⠐⠋⠀⢠⣤⣆⣸⣿⣾⡆⢻⣷
⠀⠀⠀⠈⠷⠂⣤⣼⣃⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣄⠀⢦⢒⣮⣬⣿⣿⡿⣿⠃⣼⡿
⠀⠀⠀⠀⠀⣴⣷⣿⠿⠏⠻⣿⣿⣷⣶⡄⣀⠀⠀⠀⠀⠀⠸⠋⠈⠀⠀⣀⠓⠀⠈⠷⣤⣿⣿⣿⣿⠻⢷⡄⠈⠁
⠀⠀⠀⠀⠰⠟⠛⠁⠐⠮⢍⣈⣻⣿⣯⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠼⠿⣷⣌⣛⣋⣩⡄
⠀⠀⠀⠀⠀⠀⠀⣉⢸⠇⠀⠈⠉⠀⢉⣉⡁⠀⠀⠀⠠⠤⠄⠦⡤⠄⠀⠀⠀⢀⣀⡀⠘⠛⠛⠛⠋
⠀⠀⠀⠢⢥⡈⠄⡀⠀⠀⠀⣠⣤⡀⠘⣷⣀⠹⣦⡀⠀⠀⠀⠀⠀⠀⢀⣐⠯⢚⣿⠃⢰⣿⣦⠀⠀⠀⠀⠀⠀⡠⠏
⠀⠀⠀⠀⠈⠙⠋⠀⠀⢀⣚⡛⢿⡇⠀⠘⣷⡆⠭⢉⡣⡀⢀⠐⢁⣜⠩⠅⣠⣾⡃⠀⢸⣿⢿⣧⠀⠀⠀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡆⠀⠚⠚⠎⠒⠒⢽⣠⣿⡷⠆⠁⠰⢷⠗⠁⠀⣽⣏⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠛⠁⠀⠘⣳⠄⠀⣀⠘⡄⣿⠃⠀⡀⠄⣜⠇⠀⠀⣻⠋⣿⣿⣇⠀⠀⠰⢀⡀
⠀⠀⠀⠸⣿⣤⡀⠀⣼⡉⠃⠀⠀⡀⢂⠀⠀⠘⣣⣼⡁⠀⠈⠁⠐⢘⣤⡸⠋⠀⠀⢸⣿⢀⣿⣿⡧⠀⠀⠆⢀⡀
⠀⠀⠀⠀⠘⢿⡿⡆⠘⢿⣟⡀⠀⠀⡞⠀⠀⢀⣿⠈⠀⠀⠀⠀⠀⠈⢳⣧⠀⠀⠐⢸⣿⣸⡿⠋⠀⠀⠀⠹⢿⣷
⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠘⠻⠓⠂⠀⠀⡠⠜⠁⠀⠀⠀⣤⣦⠀⠀⠀⠈⠲⣄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠈⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠠⡀⠀⠘⠿⣿⣷⣤⣄⡀⠀⠀⢠⣾⣶⢆⣀⠀⠀⠀⠀⣤⣌⣿⣿⠃
⠀⠀⠀⢀⣾⣿⣿⣿⣷⡀⡀⣠⣴⣿⣿⣿⡿⠃⠀⠀⠀⠘⠠⠤⠤⣄⠉⢻⡇⠀⠀⣟⠹⣟⣿⣷⣄⣄⣾⣿⣿⡿⣽⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠀⠈⠛⣧⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⣿⣾⣿⣿⡏⠻⠋⢿⣿⣾⣿⡄
⠀⠀⠀⠀⠈⠿⠟⠋⠀⠁⠸⣿⣿⣿⠟⠀⠀⢸⢯⣴⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠛⠿⣿⠟⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠋⠁⠀⠀⠀⠘⣶⣿⣱⣛⣤⣤⡠⢶⢾⡿⢚⡟⠀⢠⡄⠀⠀⠰⡄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⢳⣄⣠⣾⡿⠆⠀⢀⣞⣷⡆⠀⠿⣮⣿⡻⢿⣿⠻⣿⣿⣿⡏⠀⢀⣾⣿⣿⡷
⠀⠀⠀⠀⢀⣠⠙⠟⠉⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠘⠁⠈⠙⠛⠋⢀⣴⣿⣿⠛⣭⡖⠀⣠⠀⠀⠀⢀⣤⣤
⠀⠀⠀⢆⠟⠁⠀⢀⣤⡀⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡿⢋⣵⢇⣿⠃⠀⡠⠀⠀⠀⡉⢸⣿⡇
⠀⠀⠀⠀⢀⠠⢶⣿⣯⠟⠀⠀⠀⢰⣎⠙⣿⣿⣿⣷⣦⠀⠀⠀⢠⣾⡿⢛⠡⣾⢻⣿⢸⡏⠀⠴⠁⠀⠀⠀⠆⣀⢹⡇
⠀⠀⠀⠀⣿⣿⡿⠟⠁⠀⠀⠀⢀⣴⣿⣷⣿⣏⣿⣿⣿⠀⠀⠀⣿⠟⣡⣿⣥⡿⣽⣿⠿⠁⠀⠀⠀⠀⠀⠀⠛⠻⢿⡇
⠀⠀⠀⠀⠈⠉⠀⢀⣀⠀⣀⣤⣾⣟⡟⠿⣍⠙⠛⠻⠋⠀⠀⠀⠛⠚⠛⠛⠋⢁⣬⠵⣶⣿⢤⣄⣀⡀⠀⠀⠀⠈⠿⠃
⠀⠀⠀⢀⠀⡘⠻⣿⣽⣿⣿⣿⣿⣿⣧⡁⠘⠃⠀⠀⣀⡀⠐⢂⣶⣶⣶⠶⠀⢀⣤⡬⣽⡻⣷⣤⣿⣙⢻⠷⣆⣄
⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣴⣯⣤⡽⢿⡏⠉⠁⠀⢠⠾⠟⠛⠶⠈⢏⣉⣩⣴⣾⣶⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠾⠟⠋⠀⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠿⠃⠉⠛⠛⠉⠁⠉⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠉⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⣿⠿⠛⣷⣶⣿⡛⠛⠻⠿⠛⠻⠿⢿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠆⠀⠀⠀⠀⠀⠈⠻⣿⣿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣷⣦⣄⡀⠀⠀⠀⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠠⣦⣿⣿⣿⠿⠿⠻⠶⣄⡀⠉⠀⠀⠀⠘⠷⠤⠀⠀⠀⠀⠉⢀⣠⠶⠟⠿⢿⣿⣿⣿⣴⠔⠊⠘⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣶⣶⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣶⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⣿⡟⢁⠉⣻⣷⡄⠀⠢⠀⠀⠄⠀⣠⣾⣟⠁⡈⢻⣿⣿⡿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⣟⠁⠀⠉⡳⣾⣿⣧⣼⣧⣼⣆⠀⣡⣌⠀⣰⣧⣼⣧⣼⣿⣿⡿⠉⠀⠈⣳⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣳⣊⣿⣟⣿⣻⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⣹⡽⣑⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⢿⣿⣿⣿⡿⢻⣿⣿⡟⣿⣿⣿⣿⡿⣿⣿⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣽⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣯⣿⣿⣿⡁
⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣇⣨⣳⣿⣿⣿⣿⣿⡞⠀⣿⣿⠀⢷⣿⣿⣿⣿⣿⣞⣅⣸⡫⠁
⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⣻⣿⣏⣹⣿⡿⢳⣿⠇⠀⢿⡿⠀⠸⣿⡞⢿⣿⣍⣹⣿⣟⡼⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣿⣏⣿⣻⣟⡽⠀⠀⢸⡇⠀⠀⢯⣿⣿⣿⣹⣿⣤⡟⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠿⠟⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠻⠿⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣾
⣿
⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣴⡄⣿⢿⣿⣿⡿⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⣡⣤⣤⣤⣿⡟⣿⣷⣿⡃⠋⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⣿⡏⢩⢉⣷⡿⣿⣿⡏⢹⠉⠁⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⣿⠂⣾⣽⣿⣿⣦⡙⣷⣬⠶⡞⠛
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣾⡟⣸⣿⣿⣿⣿⢻⣷⢸⣿⣧⢷
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⢳⣿⣿⣿⣿⡟⣾⣿⠈⣿⣟⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣿⣿⣿⣿⡻⠟⣽⣿⣿⣀⣿⠏⠼⠀⢠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣶⡶⣶⣦⣴⣀⣀⠀⠀⢀⡭⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⣿⣿⣿⣿⣿⣿⣴⣿⡿⢻⣿⣉⡿⠈⠉⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠙⠻⠿⣿⣿⣿⣿⣛⣩⣿⡟⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⣠⣿⣿⣿⣿⣿⡿⣫⣾⣍⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣟⣻⣭⣾⣿⣿⣿⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠋⣵⣿⣿⣷⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣏⣁⣠⣿⣿⣿⣿⡀⢐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠢⡤⠤⡤⠤⣠⣀⣀⡀⣸⣴⣿⢹⣷⣼⠄⣀⡀⣠⡠⠤⡠⢦⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠀⠘⣦⣽⣿⣿⣻⣭⡍⡏⠈⡏⢩⢙⡹⠻⠙⣣⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⢀⣠⠄⠀⠰⠁
⠀⢂⡛⣯⠛⠘⠫⠝⣿⡛⠅⠣⠀⠓⠘⣛⣚⠠⠛⠈⠋⡏⣻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⢠⠀⢀⠀⠁
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠉⠈⠂⠀⠈⠁⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⣀⠠⣤⠀⠀⠀⣤⢄⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢰⠁⠀⠀⠀⠉⡀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠴⠀⠂⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⠢⠀⠀⠠⠊⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠄⢀⠀⠀⠀⠐⠂⠀⠉⠀⠁⠂⠺⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡤⠀⢄⡀⠸
⠀⠀⠂⢬⠓⠉⠀⠄⠐⢒⡏⠈⠈⠀⢟⠒⠀⠀⠀⠑⢫⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠻⠈⠀⠠⠟⢃⡀
⠀⠀⠀⠀⢣⡖⠢⡍⠙⡻⢃⠀⠀⠀⠚⢛⠁⡥⠒⣦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠠⠃⣀⠘⠄⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠈⣰⡇⠀⠀⠀⢳⡈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡀⠀⠀⠀⣸⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⢣⡀⠀⢀⠃⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠠⠀⠀⠀⠀⠀⢄⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠁⠇⠀⠀⠀⠀⠀⠰⠈⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢠⢠⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠤⠀⠀⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠆⠉⡞⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣀⠓⠀⢃⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠄⠀⠣⡀⠀⣠⠀⠀⠀⡀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⢈⠀⠀⠀⠉⠀⠉⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠁⠀⠎⠁⢱⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠒⠀⠐⢤⠊⠁⠀⠉⠓⠴⡀⠀⠒⣁⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠁⢴⡃⠀⠀⠀⢘⡆⠌⢀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡸⠃⠆⠀⠀⠀⠀⠀⢆⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⢈⠁⠐⡀
⠀⠀⠀⠀⠀⠀⠜⠁⢂⠘⡀⠀⠀⠀⠘⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⠀⠀⠀⠀⠀⢲⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⡤⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠼⠀⠀⠀⠀⠀⠀⠀⡆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⢀⠠⠁⠀⠐

⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⢸⣿⣿⣹⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣏⣿⣿⡇
⠀⠀⠀⢸⣿⣯⣿⣿⣿⣿⣿⣟⠉⣿⣬⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣥⣿⠉⣻⣿⣿⣿⣿⣿⣽⣿⡇
⠀⠀⠀⠀⠹⣿⣽⣿⣯⣀⣸⣿⣿⣿⠛⢤⣩⣿⣙⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣋⣿⣍⡤⠟⣿⣿⣿⣇⣀⣽⣿⣯⣿⠏
⠀⠀⠀⠀⠀⠈⢿⣿⣟⣿⣏⠉⠉⠙⠛⢿⠋⠉⠉⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠉⠙⡿⠛⠋⠉⠉⣹⣿⣻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡗⠒⠒⠦⠤⠧⡀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⢀⠼⠥⠴⠒⠒⢺⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⡀⠀⠀⠀⠀⠀⢈⣓⣤⡀⠀⠀⠙⠻⣦⡀⠀⢀⡀⠀⢀⣴⠿⠋⠀⠀⣀⣤⣚⡁⠀⠀⠀⠀⠀⢀⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠉⠉⠉⠉⠉⠁⠈⠀⠈⠙⠲⢄⡀⠈⠻⣄⣾⣷⣠⠟⠁⢀⡠⠖⠋⠁⠀⠀⠈⠉⠉⠉⠉⠉⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣼⣿⣿⣧⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠁⣀⣀⣀⡽⠓⠊⠉⠉⢉⡠⢿⣿⣿⡿⢄⡉⠉⠉⠓⠚⢯⣀⣀⣀⠈⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠚⠉⢉⣉⡤⠚⠁⠀⠀⣠⠔⠋⣠⢻⣿⣿⡿⣄⠙⠢⣀⠀⠀⠀⠓⢤⣉⠉⠙⠓⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣁⠤⠖⠚⠉⡼⠁⠀⣀⣠⠞⠁⢀⡴⢁⠞⣿⣿⠱⡌⢦⡀⠈⠳⣄⣀⠀⠈⢧⠉⠓⠲⠤⣨⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢀⣠⠖⠋⡹⠋⠉⣰⠃⠀⢠⠞⢀⠎⠸⣿⣿⠆⠱⡀⠳⡄⠀⠘⣆⠉⠙⢏⠙⠲⢄⡀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⡼⠁⠀⣰⠃⠀⣰⠃⠀⡞⡐⠀⣿⣿⠀⢂⢳⠀⠘⣆⠀⠘⡆⠀⠈⢧⠀⠀⢹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡶⣾⡁⠀⣠⠃⠀⡰⠁⠀⣼⠄⠀⠀⢿⡿⠀⠀⠠⣧⠀⠈⢦⠀⠘⣄⠀⢈⣷⢶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣽⣿⣯⣿⣦⣼⡁⠀⣼⠋⠀⠀⠀⠘⠃⠀⠀⠀⠙⣧⡀⢈⣧⣴⣿⣽⣿⣯⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣯⣿⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣿⣽⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⣹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⣀⣈⡑⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠔⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⣀⠔⢳⡀⠀⣤⡄
⠀⠀⠀⠀⠀⠈⢣⡄⠈⡻⢦⡄⢀⣞⣿⠗⠂⠀⠀⠀⠀⠠⢄⠀⡥⢄⠀⠀⠀⠀⠀⢀⣀⣀⢀⢾⣿⠶⠊⠀⠄⠈⠣⠘⠛⠁
⠀⠀⠀⠀⠠⣠⡾⠁⠀⢷⣿⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⠀⠀⠀⠉⡀⠈⠀⠀⢀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠉⣳⣶⣦⡀⠀⠂⠁⡤⠂⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄⠀⠀⠀⠎
⠀⢀⣀⣄⠊⠁⠀⠀⠀⠀⡰⠋⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤
⠀⢿⣿⣟⣀⣤⣤⡄⣄⠠⠄⠂⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣬⣿⣿⣿⣿⡇⠀⠀⢀⣀⣀⣀⠀⠀⠀⠠⠄
⠀⠙⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡤⠂⢤⠂
⠀⠀⠸⢟⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⣿⣿⣿⣿⠀⠀⠀⠔⠋⠉⠀⢠⠏
⠀⣘⣱⣿⣿⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡶⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢉⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠹⣻⣿⣿⣯⢹⣿⡿⠿⠶⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠘⣰⣄⠀⣦⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢻⡟⣿⡿⣷⡙⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠛⠉⠉
⠀⠀⠀⠀⠸⢿⠃⣼⠃⢀⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢁⢟⣿⠏⠁⠀⠀⠀⢤⣤⣴⣿⠁⢀⣽⣿⡟⠉⡇⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⢸⣧⣿⢴⣾⣿⠿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⣾⡟⠀⠀⠀⣈⡁⠚⠻⣿⣿⣧⣈⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⣠⣾⣿⡷⠈⠁⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠈⠗⠛⠉⠉⠉⠉⠛⠀⠀⠀⢀⡀
⠀⠀⠀⠸⡟⠿⠋⠀⠀⡀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⢂⡄⣁⢴⡀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠐
⠀⢀⡰⠊⠀⠀⠀⠀⠀⠦⣽⢐⣷⣿⣶⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⡶⣾⣦⡀
⠀⡜⠁⠀⠀⠀⡀⠁⢦⢄⣮⢹⢻⣧⣿⡿⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣇⣀⣼⣿
⠀⠀⠀⠀⠀⣤⡍⢲⣬⢴⣿⣿⣾⣿⠟⠁⠀⠀⠀⣠⡻⡯⢯⣘⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣿⠟
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⣿⣿⡿⠃⠀⠀⠀⠀⢈⡿⡻⢀⡨⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠀⣀⣀
⠀⠀⠠⠀⠒⠉⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣾⣀⣨⠄⣗⠽⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢈⠳⣿⡇⠀⠈⠁
⠀⠀⠀⣠⣴⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣻⠆⢡⣷⠾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠠⣶⣾⣿⡿⠃⠀⠀⠀⡀⣀⡀
⠀⠀⡀⠀⠰⣿⡿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢨⣧⡟⠃⣡⡿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⡶⢬⣜⣽⣿⣿
⠀⠈⠁⠀⠀⢙⡿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠸⣷⣾⡜⢲⣧⣽⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣐⣛⢿⣬⣭⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢩⣿⡀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣟⣟⣽⣇⣺⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣯⣟⣿⢷⡷⣾⣿⠏
⠀⠀⠀⠀⠁⢈⣷⠾⠿⢻⣿⣄⡀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⡿⠿⠿⠟⠁⠀⠀⠀⠀⢂⡀⠀⢸⣿⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣾⣿⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡚⡠⠠⠠⡄⠀⠄⠀⠉⠀⠒⠂⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⡀⡀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠘⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣠⠚⠀⠀⡀⠠⡄⠀⠁⠈⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠈⠁⠉⠉⠀⠑⠤⠂⠀⠀⠀⠀⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠁⠈⠀⠀⠀⠀⠀⠀⠉⠀⢹⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡂⠄⠀⠀⠀⠀⢐⣓⣒⣛⣒⣒⣳⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⢄⠀⢀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⡩⠘⠁⠄⡒⠀⠀⠂⠀⠀⠀⠄⠂⠀⠀⠂⠠⠀⠀⠀⠀⠺⠂⠀⠀⠀⠈⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠛⡄⠛⠂
⠀⠀⠀⢠⢅⣠⠀⠀⠀⠀⠛⢠⡶⣆⠀⢶⠆⠀⡴⠶⣦⠀⣴⠀⠀⢀⡴⠲⣦⠀⣠⠖⠶⠄⠐⣆⠀⣴⠀⠀⠈⠢⠤⠚
⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⢸⡷⣯⠀⢸⠀⢸⡇⠀⢸⠄⣿⠀⠀⢸⡇⠀⣹⠀⡿⠀⠰⡤⠀⠘⣾⠃⠀⠀⠠⠀⠠⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠧⠞⠀⠾⠆⠈⠷⠶⠛⠀⠻⠶⠶⠈⠷⠶⠛⠀⠻⠶⠾⠃⠀⠰⠏⠀⠀⠀⠁⡀⠁⡀⠈⠁
⠀⠀⠀⠀⠐⠠⡁⠀⡀⠈⠀⠠⠂⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠐⢀⢀⡘⠀⠃
⠀⠀⠀⠀⠀⢀⠄⠀⢀⠀⠀⢼⠒⠂⠀⠈⡈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠠⢄⠀⠎⠀⠀⠄⠠
⠀⠀⠀⠀⠈⠠⢀⠀⠂⠀⡀⠀⢀⠔⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⢤⠀⠀⠀⠀⠰⠂⠀⠀⠀⡘⠀⠈⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠒⠆⢠⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠦⠀⠀⠀⢶⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠖⠀⠆⢀⡄⠐⡠⠀⡄⠀⣐⠀⠠⡄⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡈⠉⠐⠀⠀⠰⠒⠀⠀⠀⠰⡐⠒⠀⠰⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⣄⠀⠀⠀⠴⠂⠨⢦⡀⠀⠀⠀⠀⠁⠁⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡄⠀⠀⡤⠒⣆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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



⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⠀⡀⠈⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠃⠙⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠋⠉⠀⡠⡠⡀⠉⢛⡻⢿⣿⣿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡄⠀⠀⢨⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣟⡧⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣾⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⣿⣿⢯⣿⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠈⠁⠀⠀⠀⠃⢀⢿⣿⣟⣽⡟⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣽⣯⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣽⣯⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣯⡿⠿⠿⠋⠀⠀⠀⠀⠒⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣟⣿⣿⣿⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⠿⣿⣏⣿⣿⡇⠀⠀⠀⠀⠈⠛⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣯⣾⡅⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣿⣿⢿⣿⠃⢴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣸⣿⡿⠿⣷⣽⣿⣟⡀⠈⠛⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣾⡆⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⢀⣤⡀
⠀⡀⠀⠀⠀⢠⡄⠙⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢰⡄⠀⠀⠀⢠⠀⠀⠀⢂⡀⢀⠄⠀⠀⠈⠋⠁⠐⠀⠐⠀⠀⢸⠀⠀⢠⣼⣀⠘⠛⡟⠁⠀⠀⢠
⠀⢽⡀⠀⠀⢈⣿⣾⢂⠘⣆⠀⠀⠀⠀⠀⣴⣾⣦⣿⠀⣤⣷⣶⢀⡞⠀⣧⠊⠄⠀⢠⡎⠀⠀⢧⡠⢀⠀⠀⠀⠀⠀⠀⣸⠀⢠⣼⢿⣯⡀⣸⠁⠂⠀⢠⠇
⠀⠈⣷⣀⠁⠈⣽⣭⣿⣴⠟⣆⣠⡀⣤⣀⡛⠿⣿⣿⠋⢻⣯⣵⣿⠀⠀⢹⡇⠀⢤⣿⠃⢀⣠⣾⣷⢀⡀⢠⡀⡀⠀⢀⣿⠎⠱⣿⣻⣿⣴⡟⠀⠀⠀⡿⡠
⢤⣀⣸⡏⠳⣄⠨⣿⣾⣿⡆⠘⣯⣹⣿⢿⣷⣶⣟⢻⣆⠰⢿⣿⣿⣧⡀⠀⢳⡁⠈⣣⢾⣿⣿⣏⠉⢿⢙⣿⡿⣷⣶⣞⠛⠀⠰⣿⣻⣿⣻⣿⡄⠀⠸⢡⣥
⣦⣽⣶⡉⠀⠘⠣⡼⢿⣟⣷⢡⣿⣯⡽⢿⣿⣟⣿⣶⡉⠻⣾⣿⣿⡋⢿⣆⠸⠀⠰⣧⣾⣿⡯⠀⠀⣌⣯⣹⢿⣯⡿⣿⣴⡄⠐⢿⣿⣿⣿⣽⣧⡎⠼⠉⣽
⣿⠃⠘⠓⠦⣴⣦⠈⠛⣿⡿⠛⢿⣿⠇⡘⠉⠿⣿⢿⡿⠀⠈⣿⣿⣅⠸⣿⠦⢀⠰⣿⡷⠇⠀⠀⠁⣿⣿⣿⡈⠋⠿⢻⣿⣿⢈⣿⣿⣆⢽⣷⢿⡥⣠⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣷⣿⣿⣿⣼⣯⡟⣻⠶⣄⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣽⣶⣄⣀⣀⣠⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢾⣻⣿⣿⣿⣾⣾⡏⠁⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⠀⠀⢄⠠⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣀⣀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠸⣿⡿⣿⣿⣷⣦⣤⣴⣶⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠿⣇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⡿⣿⣿⡾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠙⠿⣶⣼⣿⣧⡀⠈⠟⠿⣿⣿⣿⣿⠿⢃⡾⠚⠟⠉⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠷⠶⠤⠶⠏⠁⠀⠈⠀⠠⠀⠈⠀⠀⠀⠀⠉⠀⣀⣿⣿⣿⣿⢏⣴⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣠⠀⣤⣄⣀⣤⣤⣤⣤⣴⣾⣿⣿⣿⣏⢸⣯⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣾⣿⣿⣿⣿⣏⢸⣿⡿⣯⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣦⡙⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠰⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢠⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢨⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣺⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⣿⣿⡿⠯⠁⠀⠉⠻⣿⣿⣷⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢽⣿⣿⣿⣿⡧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣱⣿⣿⠏⠀⠀⠊⠀⠀⠀⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣘⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⡙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣵⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⣿⣿⣿⣷⣦⣤⣴⣶⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣀⣠⣴⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⣠⣴⣾⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⠀⠀⡀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠰⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⢠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠁⠀⠀⡀⡀⠀⠠⠀⠁⢀⠀⢤⡀⠀⡀⡀
⠀⠀⢀⢀⠀⡀⡀⠀⠊⠀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠐⠂⠈⠁⠀⠀⠀⠀⠀⠄⠨⣄⣄⠀⠠⠀⠀⠀⠙⠀⠂⠂
⠀⠀⠈⠂⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠨⡹⢚⠃⠈⠉⠀⠀⡆⠀⠀⠀⠀⠈⠍⠍⠉⠉⠀⠀⠀⠀⠀⠀⠠
⠀⡀⢄⠋⡀⠓⠀⠠⠀⠀⠀⢕⡁⠰⠠⠈⡁⠐⡂⡀⠀⠠⠁⠀⠀⠀⠀⠀⢀⡀⣠⡄⡦⠀⢀⠀⢀⠀⠂
⠠⠀⠀⠀⠁⠀⠀⠈⠀⠀⠢⠈⢁⠈⠈⠀⠀⠀⠀⠀⠐⠂⠀⡀⠀⡀⠀⢀⡘⠁⡡⠄⣀⠀⠀⠄⠉⣄⠁
⠀⠁⡈⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠠⠀⠋⠀⠈⠁⠉⠀⠀⠀⠀⣶⠀⠀⠀⠂⠓
⠀⠰⡅⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠠⠀⠀⠁⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⢀⠈⠀⠀⠀⠐⢐⠷⠶⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠂
⠈⠁⠂⠈⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⠕⠀⠠⠀⠂⠅⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠄⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⠀⠀⢀⠀⡀⠀⣠⣿⣄⠀⢀⠀⠀⠀⠀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠻⣄⠈⢲⣷⣶⠿⠿⠿⣿⣿⡶⠁⣠⠶⠁⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠂⠉⠉⠁⠉⢉⣛⣿⣿⣶⣤⣄⣀⡈⢓⣿⣿⣿⣧⣰⣤⣿⣿⣿⣛⣁⣤⡴⠶⠟⠻⠿⠍⣉⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠁⣀⣤⣎⣁⣈⣹⣿⠟⠙⢻⣿⣿⡿⣿⣿⡿⠋⠻⠍⠲⣤⣤⣤⠷⠦⢄⡀
⠀⠀⠀⠀⠀⠈⠀⠀⢀⠔⠉⠀⠀⡤⢻⠛⠀⠀⠀⠀⢸⣿⣟⣛⣿⣿⣷⡀⠀⠀⠀⠈⢦⠉⠲⣄⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⣟⠀⠀⠀⡠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠢⣀⠀⠀⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡤⠖⠊⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⡄⠈⠛⠶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⠻⣿⣿⣿⡿⠛⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⢀⢴⠞⠛⡲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⢙⠄⢰⡋⡈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⠛⠷⡦⡀
⠀⠀⠀⠀⠀⢠⢡⠃⠀⢎⣉⡜⢀⣠⣤⢄⡀⠀⠀⠀⠀⢰⠁⠀⠈⠉⠀⠀⠉⠁⠀⠈⡆⠀⠀⠀⠀⣀⡠⣤⣀⡀⢃⣉⡕⠀⠘⡜⡄
⠀⠀⠀⠀⠀⠆⠈⣦⣤⣤⣴⡶⣽⣿⠛⣦⡾⠷⣦⡀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢀⣴⠾⢷⣴⠛⣿⣯⢶⣦⣤⣤⣜⠀⢡
⠀⠀⠀⠀⠀⣶⣟⠻⣷⣌⣻⣧⣬⣿⣦⣹⣷⡀⣷⡾⢦⡀⠑⡄⠀⠀⠀⠀⠀⠀⢠⠊⢀⡴⢷⡿⢀⣾⣏⣴⣿⣥⣾⣟⣡⡾⠟⣻⣶
⠀⠀⠀⠀⠀⣿⠛⢟⡩⡉⠂⠀⠂⠂⠀⠉⠍⠛⠻⢷⣸⡟⣆⠈⢢⠀⠀⠀⢀⡔⠁⣴⢻⣇⡾⠟⢛⠩⠉⠀⠀⠀⠀⠐⢉⢍⡿⠛⣿
⠀⠀⠀⠀⠀⠹⣻⡟⠛⠢⣀⢦⣤⣄⣀⠀⡀⠀⠀⠂⠌⠓⢼⣆⠀⠱⠀⠀⠎⠀⣸⡧⠋⠡⠒⠀⠀⠀⠄⣀⣠⣤⡔⣡⠖⠛⢻⣟⠏
⠀⠀⠀⠀⠀⠀⠹⣛⡿⠟⠛⢦⢠⣘⡉⣹⠲⢶⡄⠀⠀⠀⠀⠙⣆⠀⡇⠸⠀⣰⠋⠀⠀⠀⠀⣠⡶⠞⣏⢉⣃⡄⡴⠛⠻⣿⣛⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡶⠚⢛⢠⠀⠉⠛⢤⡔⠛⢣⡄⠀⠀⠀⠸⣤⡿⢿⣴⠃⠀⠀⠀⣠⡞⠛⢦⡴⠛⠉⠀⡄⡛⠓⢶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⠶⠏⢘⠀⠀⠀⡐⢝⢶⠋⠙⣦⠂⣀⣤⣿⡿⢿⣿⣤⣀⠀⣵⠋⢙⡶⡫⠂⠀⠀⠀⡃⡿⠶⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⠾⣄⠁⠤⠠⡌⠢⠠⢑⡫⢴⣯⢿⣿⣿⡯⢿⣿⣿⡿⣽⡦⢗⡋⠌⢔⢠⠀⠄⠈⡰⠷⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣧⣠⣿⢉⣶⡒⣲⣾⣩⣿⢷⣿⣷⣿⣿⢿⣧⣼⡿⣿⣿⣾⣿⡾⣿⣉⣷⣖⣲⣶⡉⣷⣄⣼⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢻⠟⡶⠛⠉⣩⣵⣾⣿⣿⣿⢏⠇⠟⣿⣿⢱⠹⡹⣿⣿⣿⣷⣮⣍⠉⠛⢾⠻⡟⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡜⠁⣰⣿⣿⣿⣿⣿⠿⢡⠏⢸⠀⢸⠇⠀⡇⠱⡌⢿⣿⣿⣿⣿⣿⣆⠈⢧⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠁⠀⠙⠿⠿⠿⢛⣡⠔⠋⡠⠃⠀⠀⠀⠀⠙⣄⠙⠦⣌⡛⠿⠿⠿⠃⠀⠈⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣀⡠⠴⠒⠒⠋⢉⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⡉⠙⠒⠒⠦⢄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠘⠋⡀⠤⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠤⣀⠙⠃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡔⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⢂⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠸⠤⠌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠡⡠⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⡄⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠐⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠂⡀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢠⣿⣿⠄⣿⣧⠀⠀⠀⢀⣤⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⣤⣴⣾⣿⡏⢼⣿⣿⣷⣶⣦⣼⣿⡏⡸
⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠇⢎⢻⣝⣿⣿⣿⡿⠘⣽⢀⣀⣀⣀⣤⣴⣶⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠉⠉⠓⠿⢿⣿⣿⣷⣽⣿⣿⣯⣈⣿⠟⠋⠀⠀⠀⠙⠻⢿⣿⡟⢁⣤⡉⡩⠟⣛⣭⡿⠿⠛⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣏⣉⣉⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣽⡯⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣴⣶⣶⣶⡶⠾⠿⠷⠶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠉⠀⠀⠀⠀⠈⠙⠋⠉⠉⠁⠀⠀⢠⣤⣀⡰⣜⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⡄⣔⠀⠀⠀⠀⠀⠄⠀⠀⢡⠇⠃⢀⣽⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢘⣓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣤⣽⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⣿⡠⢀⣖⠀⠀⠀⣠⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢻⣿⣿⣿⣤⣤⣾⣽⣾⣴⣤⣾⣿⣿⣿⣿⣿⢋⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣛⡫⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⡀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠃⠦⠉⠀⠰⠟⠛⠋⠁⠀⠀⠀⠀⠄⣀⠂⠀⠰⠊⠀⢀⢀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⠀⠁⠰⠰⠀⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠰⠀⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠁⠀⠞⠀⠀⠀⠓⡀⠀⠀⠀⠀⠐⢺⡆⠀⠀⠀⠀⠚⠻⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠄⠂⠀⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⠀⠀⠀⡀⡀⠀⢠⠆⠀⠀⠘⠀⠀⠄⢀⡀⢄⠀⠀⠀⠀⠀⠀⣴⣃
⠀⠀⣠⠀⠀⠀⢀⣀⠈⠋⠀⠀⡄⠀⠀⠀⢬⡻⣄⠔⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢶⡇⡘⣏
⠀⠀⠃⠀⣀⠈⠐⠀⠀⢀⠀⠉⠀⠀⠀⡐⢉⠥⠂⠠⠆⠀⠀⠀⣠⣀⠀⠀⠀⠀⠆⠀⢠⠀⢈⣩⣵⣵⠇
⠀⣸⠀⠀⠛⠀⠀⣀⠐⠈⠀⠄⠀⠀⢀⡁⠊⢀⡛⡀⠠⠆⠀⠘⠑⠀⠀⠀⡦⠀⠀⠀⠈⠀⠀⠀⠹⡏
⠀⣼⠀⠀⠀⠐⠀⠁⢀⡀⢀⠀⠐⠂⢣⡞⠄⣼⠃⢹⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠀⠀⢠⡄⠄⡀
⠀⠦⠀⡿⠇⠀⠀⠀⡀⠀⠌⠁⠀⢇⠁⢀⡾⡟⢎⡎⠀⠀⣀⡰⠄⠊⡷⠀⠀⠀⠈⠀⠂⠀⠀⠀⠰⢠⠁
⠀⣴⡔⢳⣦⣀⠀⠀⠐⠤⠀⠀⢀⠈⠣⡾⠈⠠⠌⠀⠀⠁⠻⢯⠆⠀⠁⠈⢀⣀⠈⠓⠒⣂⣸⡧⠄
⠀⠀⠀⠀⢋⠁⠀⢀⡀⠤⢒⡖⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠏⠀⢰⠋⠁⠀⡇
⠲⣄⢠⡞⠀⠀⠀⢀⠖⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⢠⠆⠀⠀⣀⣤⣼⡤⠴⠊⠀⠼⣆
⢰⡿⣆⡔⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⢈⡀⠀⠀⠀⠀⠁⡆⠀⠀⠀⣇⠀⣸⠇⠀⢀⠀⠀⠛
⠀⠙⡏⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⡒⠁⠀⠀⠀⡄⡀⠖⠀⠀⣰⠚⢻⡍⠁⠀⢰⣶⠧⠶⣄
⠀⢀⡄⠀⡀⠐⢀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠠⠂⠀⠨⠄⠀⠀⢑⠈⠀⢀⣻⠤⠞⠁⡀⠈⠉⡇⠀⠀⢉⠃
⠀⠘⠏⠀⠱⡀⠀⠀⠖⠀⠀⠒⠚⠉⠉⠉⠁⠀⣀⣀⣴⣶⡶⠆⠀⢀⣯⠟⠀⠀⠢⢏⠀⠀⠑⠧⡐⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢠⠞⢻⣭⠭⢽⡓⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣚⡯⠭⣭⡟⠳⡄
⠀⠀⠀⠀⣿⠀⠻⠿⠁⣀⣈⠑⢄⠙⠻⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠿⠋⡠⠊⣁⣀⠀⠿⠟⠀⣿
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠻⠟⠁⠀⠑⣄⠈⠢⡑⢄⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠀⠀⡠⢊⠔⠁⣠⠊⠀⠈⠻⠟⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠹⡄⠀⣤⣤⠀⠀⠔⠋⠀⢱⣄⠈⢆⡑⣄⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣠⢊⡴⠁⣠⡎⠁⠉⠢⡀⠀⣤⣤⠀⢀⠎
⠀⠀⠀⠀⠀⠀⡇⠀⠛⠛⠀⡌⠀⣀⣠⠃⠈⠳⣜⠛⢌⠳⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⠜⡡⠛⣣⠞⢁⠘⣄⣀⠀⢡⠀⠙⠛⠀⢸
⠀⠀⠀⠀⠀⠀⢧⠀⠀⣴⣤⡅⠘⠿⠟⠀⠋⠀⠈⢳⣄⠱⣝⢄⠀⠀⠱⡀⠀⠀⠀⠀⢀⠎⠀⠀⡠⣋⠎⢠⡞⠁⠀⠙⠀⠹⠿⠃⢈⣤⣦⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠈⠢⣄⠙⠋⠁⢀⣤⣤⠀⠀⠀⡰⢅⣈⠳⣌⠁⢧⡀⠀⠱⠀⢀⡀⠀⠎⠀⢀⡼⠛⣩⠞⣁⡈⠆⠀⠀⠀⣠⣤⡀⠸⠙⠋⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⡘⡈⠛⠛⡀⠀⠀⡇⠀⠋⠙⠪⣑⠦⣙⢦⡀⣨⣿⣿⣅⢀⡴⣋⠴⣊⠔⠊⠙⠁⢸⠀⠀⢀⠛⠛⢃⢂⣀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠇⠀⡀⠀⠡⣾⣿⠧⣀⣉⠉⣁⡲⣵⣬⣓⢿⣿⢻⡟⣿⡿⣞⣥⣮⣴⣊⡉⣉⣀⠸⣿⣷⡌⠀⢀⠀⠸⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣄⡈⠄⠀⢣⣠⠤⠷⢒⠒⡉⠩⣿⣭⣍⣛⣿⣿⣿⣿⣛⣩⣭⡿⠍⢉⠐⡒⠾⠤⣤⡜⠀⠠⢃⣠⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡩⠟⠋⠁⣀⣠⣤⣈⡑⠾⢦⠶⢾⣿⣻⣿⣿⣿⣿⣿⣿⡷⢶⡴⠷⠊⣁⣤⣄⣀⠈⠙⠻⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠊⠍⣙⡽⢃⠠⢀⣶⢶⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⡶⢦⡀⠄⡘⢯⣋⠩⠙⢦⡀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⣺⢟⣪⠴⠛⠉⢀⣠⣶⣫⣽⠿⣻⡿⠃⣸⣿⣿⣇⠈⢿⣯⠻⣼⣝⡶⣄⡀⠉⠛⠦⣕⡻⣗⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠙⢉⠁⠀⠀⡀⠘⠋⢉⠔⠁⡼⢻⠁⣠⢿⣿⣿⡿⣆⠈⡛⢷⠈⠢⡉⠙⠓⢄⠀⠁⠈⡉⠋⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠔⠁⠀⢀⢸⣿⠀⣐⠁⢠⠟⠄⡄⢠⠇⢸⣽⣯⡇⠸⡄⢡⠠⠻⡄⠈⣂⠀⣿⡇⠀⠀⠈⠢⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⠁⢸⡏⣸⡿⠀⣟⡜⡐⢀⡎⠀⢸⢾⡷⡇⠀⠱⡀⢂⢳⣳⠀⢿⣧⢹⡇⠈⠄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣄⠂⠀⠈⢰⡿⠁⠀⠟⡰⠁⡸⠀⠀⠘⣿⣿⠇⠀⠀⢧⠈⢆⠻⠀⠈⢿⡆⠁⠀⠘⣀⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢦⡀⠀⠆⠀⠀⠀⡰⢁⣶⠇⠀⠀⠀⠛⠛⠀⠀⠀⠘⣶⡈⢆⠀⠀⠀⠐⠀⢀⡴⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠳⠴⡀⠀⣀⣠⡷⢋⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡙⢾⣄⣀⠀⢀⡦⠞⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠙⠊⠉⠉⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠉⠉⠑⠋⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⢠⣄⣠⡀⠂⣴⢄⣺⡫⢛⠀⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡀⠒⢀⣿⡟⠛⠏⣅⠨⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠂⠡⠃⠘⠾⠀⠃⣀⢌⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠓⠄
⠀⠀⠀⠀⠀⠆⠄⠠⠀⡄⡌⠀⠂⠀⢀⡨⠁⠧⠀⠠⠀⠀⠀⠀⢀⠀⠀⠠⡀⠀⣠⣘⠻⠀⠀⠠
⠀⠀⠀⠀⠀⢷⢈⠂⠠⠈⠀⠀⣤⠐⠌⠱⠆⠀⠀⠀⠀⢄⠀⠀⠈⡠⠈⠃⡅⢷⢠⠈⡄⢁⠢⠂
⠀⠀⠀⠀⠀⡉⠴⢆⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⣄⠈⢷⣄⠀⠘⣅⢎⠛⣿⡠⠖⣖⠂⠜⠁
⠀⠀⠀⠀⢠⠀⠄⡐⢦⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠁⠰⡌⠢⣚⠀⡈⠺⢷⡞⠁⠠⠀⠠⠂
⠀⠀⠀⠀⠘⠀⠀⠀⠂⠬⠀⠀⠀⠀⠈⢶⡤⢀⡀⠀⠀⢀⡈⠠⠉⢤⡈⠀⡄⠛⣐⣈⣘⣐⣆⠴⠚⠁
⠀⠀⠀⠂⠀⠉⠀⠀⠀⠚⠀⠀⠄⠀⠀⠀⠱⠏⠲⠂⠀⠀⠘⠄⠘⡄⠀⠨⢠⣌⠀⣈
⠀⠀⠀⠀⠀⢈⠀⠂⡤⠀⡀⠀⠀⠢⠄⠀⢀⠀⠀⠄⠢⢄⠀⢀⠀⠔⢰⠀⠉⠍⠀⡈⠀⢛⠉⠟⠒⠂
⠀⠀⠀⡀⢀⡄⠐⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠐⠤⡈⣀⠏⠀⡀⠀⠀⠒⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⢂⡐⠙⠀⢁⠀⡀⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⠁⡋⠀⠄⠄⢀⠀⠠⢠
⠀⠀⠄⠀⢀⠈⠡⠀⢀⡜⢁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠄⠈⠈⠀⠀⡀⠀⠀⠀⢀⡀⠤⠆⠐
⠀⠀⠠⠀⠠⠐⡂⠐⢀⡰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠃
⠀⠀⠀⠘⠀⠀⠁⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠆
⠀⠀⠀⠀⠀⡀⠁⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠⠀⠀⠀⠠⠈⠁
⠀⠀⠀⠀⢀⠀⠰⠂⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠁⠀⡆⠀⠀⠄⠁
⠀⠀⠀⠐⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢀⠀⠀⠀⡀⠄⠐⠀⢈
⠀⠀⠀⠠⠀⠀⠀⢸⠀⢇⢀⡀⠠⠔⠒⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠹⣄⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠠⢈⠀⠀⠀⠡⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠁⠘⠀⠀⡀⠑⢀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠀⢀⣴⣤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣸⣭⡾⠾⢉⡀⣼⠚⠻⡘⡿⣿⡟⢹⣖⠠⣄⣠⣴⣿⡿⢿⠛⢋⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⠍⣠⣧⣄⣠⣴⠁⢠⠀⣙⢦⣥⣼⡟⢿⣿⠿⢛⣯⡵⣿⢠⡀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⣋⡽⠻⣦⣾⣯⣿⢛⣫⣻⣟⣁⣥⣖⣿⢿⣶⡟⣿⣀⣯⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠞⠁⠀⠀⠀⠀⠀⢀⠜⡏⠀⠀⢀⣟⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡰⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠳⢤⡀⠼⢾⣦⡤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠐⠒⡖⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠁⠀⠀⠉⠉⠉⠑⠦
⠀⠀⠀⠀⠄⠒⠂⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⢰⡶⠀⠀⠀⠀⠀⡠⢶⡆⠀⠀⠀⣠⡶⣆
⠀⠀⠀⠀⢿⣯⣷⣄⠀⠀⠀⢁⡈⠒⠀⠴⠄⠰⣇⡈⠀⠀⠘⢧⠀⠰⣶⣶⡜⠉⡼⠃⠀⠀⡰⢀⢀⡻⠆
⠀⠀⠀⠀⢸⡛⡿⣿⣆⠀⢀⠈⠓⠂⠀⠀⠀⠀⠛⠁⠀⠀⠀⠰⣿⣴⡿⡍⠃⣩⠞⠀⡀⡘⢓⣄⣏⢻⣦
⠀⠀⠀⠀⠈⠳⠅⢈⣻⠏⣻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⣌⣛⣷⣾⣿⡿⢀⣿
⠀⠀⠀⠀⠀⠀⢠⣮⣿⠿⠛⢿⣷⣶⣦⣀⡀⠀⠀⠀⠈⢹⠋⠉⠀⠀⡒⠀⠙⣦⣴⣿⣿⣿⠻⣮⠁⠋⠁
⠀⠀⠀⠀⠀⠀⠟⠛⠁⠐⠦⢄⣩⡿⣿⣼⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠜⢿⣦⣉⣉⣭⠆
⠀⠀⠀⠀⠀⢀⠀⠀⠦⡘⠀⠀⠀⠀⢰⡤⠤⡀⠀⠐⠒⠚⠒⠒⠀⠀⣠⢤⣤⠀⣉⡉⠉⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠲⡶⠂⠀⢀⣴⣿⡇⠈⢿⣄⠚⠶⢄⠀⠀⠠⢀⡠⠶⠒⣸⡏⠀⣿⣿⡄⠀⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣼⣧⡀⠀⢿⣦⠁⠈⢃⢀⣶⣌⠀⢀⣼⡷⠀⠀⣿⢻⣿⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⡿⠿⠃⠀⢐⡦⠀⠀⠘⠙⡿⠀⠀⠀⢰⠆⠀⢘⡇⢻⣿⣇⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠿⣦⣄⠀⢌⢉⠀⠀⠀⢂⠀⠀⢻⣴⡣⠀⠈⠀⠀⣠⣜⠋⠀⠀⣽⡇⣾⣿⡏⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠙⠟⠃⠈⠸⣄⣀⠀⠊⠀⠀⡼⠇⠀⠀⠀⠀⠀⠐⠿⡀⠀⢀⣾⡥⠛⠉⠀⠀⠀⠙⠛⠂
⠀⠀⠀⠀⠀⢴⣷⣤⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⢠⣾⣦⣀⠀⠀⠈⠂⢀⣄⣀⠀⠀⠀⠀⠀⢀⣤⣶⡄
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣆⠀⠀⣀⣤⣶⣾⣷⠀⠀⠀⠀⣈⣋⣙⠛⢲⣄⠀⢹⡞⣷⢿⣦⡀⢀⣴⣟⣾⣿⡆
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠋⠀⠘⠶⡌⣻⣿⣿⣿⣿⣿⣷⣿⡁
⠀⠀⠀⠀⠀⢹⣿⣿⣿⢿⡟⣿⣿⣿⣿⠉⠀⣀⠀⠀⠈⠂⠂⠀⠐⠀⠀⠀⠀⠸⣷⣽⣿⡃⠛⠘⣿⣷⣿⠅
⠀⠀⠀⠀⠀⠀⠙⠋⠁⠈⠀⠻⣿⡿⠋⠀⠀⡿⢴⣦⣀⠀⠀⠀⢀⣀⢠⣤⠀⠈⠙⠛⠛⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣄⠀⠀⢠⣄⠀⢸⡟⣻⣷⡶⣶⣬⣿⣿⡭⠃⠀⣾⣤⣄⠈⠃⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡸⣷⠛⠋⠉⠀⢰⣿⣿⣆⠀⠉⠙⠃⠹⠃⠹⠿⠿⠃⣠⣴⣿⢟⣻⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠠⠰⠋⠀⢀⣀⠀⠀⠀⠸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣸⡿⢛⣵⣸⡏⠀⢈⠀⠀⠐⠩⣿⡆
⠀⠀⠀⠀⠀⢀⣀⢶⣿⡽⠃⠀⠀⠠⣮⡙⣿⣿⣿⣶⠀⠀⠀⣴⣿⢛⡱⣾⢻⡯⡿⠁⠰⠁⠀⠀⠐⣀⣹⡇
⠀⠀⠀⠀⠀⠸⣿⠟⠋⠀⠀⠀⢀⣴⣿⣿⣿⣼⢿⣿⠀⠀⢸⡟⣡⣾⣷⡟⠿⠟⢃⡀⠀⠀⠀⠀⠈⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣿⣿⡎⠝⢆⠈⠈⠁⠀⢀⠀⣉⣉⣁⠀⠐⠒⠹⣿⣅⡒⣦⢤⣄⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢲⣶⣿⣿⣿⣿⢿⣿⣿⣆⠀⢀⡤⢐⣲⣤⡞⠛⠛⠉⣠⣶⣿⡉⢻⠟⠟⣛⣢⣩⣵⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⣽⠿⠛⢠⣾⡿⠟⠁⠈⠁⠀⠀⠀⠳⣶⣶⣶⡶⠏⠙⠛⠛⠛⠙⠛⠆

⠀⠀⢀⣤⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣤⡀
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⠉⠓⠠⢠⣽⣷⣤⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣾⣯⡄⠀⠚⠉⢙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣏⣀⡀⠉⠈⠘⣶⣾⣿⣟⣿⣏⣦⣄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⣶⣹⣿⣻⣿⣷⣶⠋⠁⠉⢀⣀⣹⣿⣿⣿⡿
⠀⠀⠘⣿⣿⣛⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⣛⣿⣿⠁
⠀⠀⠀⠘⣿⣿⣷⣶⣿⣿⣿⣧⠄⡀⣸⣧⡀⠁⠈⠙⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠋⠁⠈⢀⣼⣇⡀⠠⣼⢿⣿⣿⣶⣾⣿⣿⠃
⠀⠀⠀⠀⠹⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠉⠻⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⠏
⠀⠀⠀⠀⠀⢻⣿⣿⣶⡶⠶⠶⠶⠶⠶⠖⠒⠒⠿⠷⢦⣀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣠⠟⠁⠀⠀⣀⡴⠾⠟⠒⠒⠶⠶⠶⠶⠶⠶⢶⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠘⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠹⣦⣶⣶⣴⠏⠀⣠⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣶⣶⣶⣾⣿⣷⣶⣶⣦⣤⣤⣤⣤⣄⣈⣑⣢⣽⣿⣿⣯⣴⣊⣁⣠⣤⣤⣤⣤⣴⣶⣶⣾⣿⣷⣶⣶⣶⣶⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣩⣻⡟⠛⠛⠛⠻⢏⣛⣛⣛⣛⣛⠻⠿⡙⣿⡇⢸⣿⢋⠿⠟⣛⣛⡛⣛⣛⡩⠟⠛⠛⠛⢻⣟⣍⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⠤⠄⠀⠀⠀⢐⠋⠀⠀⠀⠀⢀⡠⢘⢿⣷⣾⡿⡃⢄⡀⠀⠀⠀⠀⠹⡂⠀⠀⠀⠤⠤⢿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠾⣯⣿⠁⢀⣀⠤⠀⠀⠐⠀⠀⠀⠀⣠⠖⠉⡰⢨⠀⣿⣿⠀⡍⢆⠉⠲⣄⠀⠀⠀⠀⠂⠀⠀⠄⣀⠀⠈⣿⣽⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠋⠉⠀⣀⣀⠤⣠⡀⢄⣴⠋⠁⢀⡞⢠⠃⠀⢻⡿⠀⠘⡌⢳⡀⠈⠑⣦⡠⢀⣄⠤⣀⣀⠀⠈⠙⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⡄⣠⠴⠛⠉⢀⡴⠁⠴⢛⡟⠀⣠⠏⢠⠃⠀⠀⢸⡇⠀⠀⠘⡄⠹⣄⠀⢻⡟⠢⠈⢦⡀⠉⠛⠦⣄⢠⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⡇⠀⠀⣰⠋⠀⣼⡖⠊⠀⡴⠃⢰⠃⠀⠀⠀⢸⡇⠀⠀⠀⠘⡄⠘⢦⠀⠙⢲⣧⠀⠹⣆⠀⠀⠸⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⣷⣖⣾⡁⠀⢠⡏⠀⢀⡼⠁⢠⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡄⠈⢧⠀⠀⢻⡄⠀⢈⣷⣲⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⢿⣷⣞⣿⣣⢠⣞⡁⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣈⣳⡀⣜⣿⣳⣾⡿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠿⠏⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠹⠿⠎⠁





⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⠶⠶⠶⣶⡶⠶⡶⢶⡶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⡇⠀⢸⠀⡏⢸⠀⠹⠇⡇⠈⠇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣦⣤⣾⡿⠀⠸⢀⣼⣆⠀⠀⠄⠀⠀⠻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⢃⣠⣤⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣧⣴⣦⠙⠁⠀⠰⠷⣄⠐⠢⢴⣷⠄⠙⣵⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢛⠉⢙⠿⣿⠀⠻⣄⢶⣤⡮⠽⠗⡀⠂⠀⠠⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠂⠸⣷⡈⠒⠢⢻⣼⠁⠀⣤⡼⠟⢋⡆⠀⠫⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⢻⣿⣤⣤⣬⣵⣶⠖⠙⠁⣀⣿⡇⣰⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⢣⠿⠛⠛⠛⠁⠘⠛⠻⠿⠻⢯⠘⠻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣾⠇⠀⠀⠈⠃⠐⡒⣳⢒⡘⠀⠀⠀⠀⠗⣤⢾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡉⣩⣶⠶⣦⣄⠘⠣⠀⠸⡿⠋⣠⣴⢾⣿⣄⢁⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠋⠁⢻⡇⠙⣎⣿⠀⠀⠀⢷⣿⡿⠋⠀⣿⣿⠎⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠉⢿⢆⠹⡏⠀⠚⠇⠀⢿⡏⢀⣴⣿⡿⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢹⠇⣸⣋⣙⡃⠀⡀⣸⡄⠀⠈⠀⣺⡟⣉⠰⠒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣴⣿⠉⢿⣿⣷⣄⠈⠁⣠⣶⣿⣿⠛⣿⢷⣄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⣻⣷⣮⣿⣿⣿⣷⣿⣿⣿⠟⣁⡤⢿⣿⠟⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡄⠛⠁⠉⠉⠻⡿⠟⡿⢿⡿⠛⠛⠓⢿⡗⣴⡃⠁

⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡠⠀⠠⠄⠀⠀⢀⣀⡀
⠀⠀⠀⢩⠄⡠⠀⠄⠛⠀⠀⠄⠀⠀⠀⢀⣾⣯⣿⣿⣦⣄⡀⠹⢿⠀⢤⠿⠇⡈⢀⣬⣣⡀⠐⠀⢿⡏⢹⡆
⠀⠀⠀⠀⠤⠂⠰⡄⠣⢦⣁⡘⠀⠀⠀⢸⣿⣿⣿⣿⡷⠿⣻⣶⣄⡀⠠⡿⠀⠐⣉⠷⠙⠋⣄⡀⠀⢿⣾⠏
⠀⠀⠀⠀⠀⠀⠴⣩⣛⡎⣰⠯⠄⠀⢠⠀⠙⠻⣷⣿⣴⣿⣶⣿⣿⣿⡆⠀⠀⣀⣻⣄⢀⡞⢸⡇⡀
⠀⠀⠀⠀⠀⠀⣋⡑⡵⠿⣟⠷⢀⠆⠘⡷⢾⠇⠒⠤⣌⠙⠻⠿⠿⠟⠁⠀⠀⢸⠉⠀⣍⣩⣹⡇
⠀⠀⠀⠀⢀⠀⠀⣈⢻⡶⠿⣎⠄⠀⠀⠈⠓⠲⠦⠼⠜⠃⠀⠀⠀⠀⠀⠀⠀⠈⣶⠞⢃⣤⡸⣇⠀⠀⢀⣤⡄
⠀⠀⠀⢤⣈⠣⡀⠈⠀⠀⠀⠀⠀⢀⣤⣦⡿⡿⣶⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⢋⣻⣿⣧⣾⡆⠀⢸⣻⡇
⠀⠀⠀⢥⣈⢷⣝⠀⠐⢶⡄⠀⠀⠸⣏⣤⣦⣤⣼⠇⠀⠀⠀⠸⡇⡞⠀⠀⠀⠀⠀⢠⠹⣿⣿⣽⣧⠀⠸⠿⠁
⠀⠀⠀⠀⠉⠛⠽⡷⣭⣶⡀⣤⣀⡀⠛⢟⣫⡀⣶⣤⡄⠀⠀⠀⢱⣀⠀⠀⠀⠀⠀⠀⠡⡹⢿⣿⠇
⠀⠀⠀⢰⡞⡹⢛⣿⢟⡞⢿⣿⣿⣤⠀⢿⡋⢃⣹⣿⣏⠀⠀⠐⠀⠙⡄⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠘⢡⠏⡠⢍⢲⣷⢹⣿⣾⣇⠈⠛⠚⠁⠉⠁⠀⠀⠀⠀⢠⠁⣿⠇⠀⠀⠀⠀⠀⣀⣀⣠
⠀⠀⠀⣤⣶⡀⢼⡴⣮⠿⢺⠯⠛⠛⠃⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⢠⠀⠀⠂⠀⠀⠀⢀⣾⣿⢿⣿⡦⠠
⠀⠀⠀⢻⡌⡇⢠⡴⠯⠆⠈⠀⣀⢠⡀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠛⡇⠀⠀⠀⠀⠀⠀⠙⣿⣹⣿⣗
⠀⠀⠀⢸⣿⠇⠸⠇⠀⠀⢠⠃⠉⠁⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠳⢀⡀⠀⠀⠀⠀⠀⢨⠁⠛⠟⠄
⠀⠀⠀⠀⢀⣠⠤⢄⠀⠀⢸⠀⠀⠀⠀⠋⡤⠀⠈⡀⠀⢀⠐⠊⠁⠀⠃⠘⡌⠀⠀⠀⠀⠀⠀⠝⢘
⠀⠀⠀⠀⡞⠀⠠⠄⠑⠄⠀⠃⠀⠀⠀⠀⢖⠱⠀⢁⠀⠀⠀⠀⠀⠁⠀⠈⠡⠀⢠⣾⣟⠇⠀⠀⠁
⠀⠀⠀⢀⣅⠀⠠⠄⠀⣸⢀⠴⠷⠶⠚⢲⡄⠀⠲⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⡤
⠀⠀⠀⠀⠀⠀⠀⣤⢠⠇⠸⣿⣰⣶⣦⣤⠁⠀⠀⠀⠈⠀⠀⠀⠤⠐⢦⡀⠘⠲⢤⣶⠧⠬⡟⢷⡚⠁
⠀⠀⠀⠀⡆⠀⠀⠂⡜⠀⣀⡀⣀⡉⠁⠀⠀⢠⣄⡠⣴⣠⢴⡶⣠⣄⡀⠉⠛⠋⠉⡡⠶⠋⠺⣿⠷⠄⢀
⠀⠀⠀⢠⡇⠐⠀⠠⠃⢸⣽⠿⠿⢷⢶⡀⢰⠾⣿⣼⣻⣧⣾⣧⣿⣈⡗⠀⢀⡔⠉⣠⠴⢃⠰⠂⠀⣴⣿⡆
⠀⠀⠀⠈⠳⠴⠔⠁⠀⠘⢻⡟⢳⣾⢻⡁⠸⣧⡼⠻⢿⣏⠳⣼⡏⠋⠁⢠⠊⠀⢺⠁⠀⠈⣤⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠉⣩⢿⠆⠀⠀⠀⠚⠓⠉⠉⠛⠁⠀⠀⠀⠘⠂⡁⢾⠀⣀⠀⠀⠀⠀⠻⠟⠁
⠀⠀⠀⢀⣴⡜⠿⣿⣀⠀⢀⣄⡰⡶⣾⢛⣇⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠌
⠀⠀⠀⢸⣟⠤⣢⠙⠋⠀⠀⠉⣾⠻⣏⣿⢿⡟⠀⠉⠈⡙⠶⠤⢤⣤⣤⠀⠀⣠⠀⠀⠜⠀⠤
⠀⠀⠀⠀⠘⠺⠛⣁⢾⣿⣀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠐⠀⠲⠒⠻⠛⢉⠁⠘⠛⠚⠖⠀⠻⠖
⠀⠀⠀⠀⠀⠀⠛⠛⠂⠙⠋⠀⠀⠀⠀⠀⠀⠲⠞⠁⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠰⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢠⠠⢶⣔⡖⣶⣤⢤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣰⡜⢣⡌⢰⠂⠴⠟⠋⠉⢈⠑⠱⠈⠙⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⣶⠻⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠖⠀⢀⣤⣀⠀⠀⠂⠉⠁⢀⡀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣰⠄⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢾⣿⣿⣧⠀⠀⠉⠠⠆⢰⣶⡞
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠐⠀⠐⠂⠀⠀⠀⠀⠀⢠⠀⠀⠈⠛⣿⣿⡦⠾⠀⠐⠀⠀⣬⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢺⠶⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠈⢠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣴⣶⣶⡆
⠀⠀⣧⣠⣏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⡔⠀⠀⠀⠀⠀⠳⡀
⠀⠀⢻⣟⠇⠀⠀⠀⣨⡁⠀⠁⠘⠀⠀⠀⠀⠀⠀⣎⣿⣿⣿⡟⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢀
⠀⠀⠀⠹⠃⠀⠀⢈⡧⠀⠀⠀⠘⢢⢤⣶⣾⡆⢸⣿⢿⠿⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠈⠠⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠓⠀⠀⠀⠄⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢃⠀⠀⡈⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⣶⠆⠀⠀⠀⢢⡤⠄⠀⠠⠤⢶⣛⠀⡀⠀⡄⠀⠐⠀⠀⠁⠈⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣴⠀⠀⠀⠘⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠠⣶⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠔⢻⠙⠋⠀⠀⠀⠀⠀⢀⣰⣿⣿⡻⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡓⠦⡄
⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣲⡌⠀⠀⠀⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣏⠁⠀⠠⣴⡇⠀⠀⠀⠈⢻⣿⣿⣿⡿⣿⡛⢻⣰⠀⠀⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠛⣛⣷⣤⢀⡀⠀⠀⠉⠃⠀⠀⠠⠋⠀⠀⠀⠀⠀⣈⣿⣿⣿⣾⠛⣿⡵⠯⠀⠀⠀⢀⣤⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⢺⣿⣿⣷⡄⠀⠀⠀⠀⣠⠀⠀⠀⠀⣠⠶⣾⣍⣼⠞⢿⣿⣿⡙⣿⣖⠁⢀⠤⢾⠛⣶⠀⢸⣄⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃⠀⠀⢠⣴⠃⠀⠀⠤⠊⠀⠘⠆⠂⠀⠀⠀⠉⠛⠿⠣⠿⣧⠈⠓⠀⡄⢰⣀⠈⣥⡀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠉⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠈⡀⣈⠁⠀⢀⡄⠀⠀⠀⠣⠀⠛
⠀⠀⠀⠀⠀⡄⠀⠀⢀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣤⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢀⣿⣶⠞⠋⠀⠀⠀⠀⠀⠀⣲⣂⠀⠀⠀⠀⠤⠀⠀⠀⢀⣿⠟⣳⢄⢀⣠⠂⠀⠀⠀⠀⠀⢠⣼⠃
⠀⠀⠀⠀⠋⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⡿⠋⢶⠀⠀⠀⠀⠠⣤⣶⡟⣤⢨⡥⣿⣼⠁⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣦⡄⠀⠀⠀⢸⡿⣦⢫⡸⡷⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣳⣿⣽⠀⠀⠀⠀⢻⡏⡌⠋⣷⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡏⠀⠀⠀⠀⠈⠓⢋⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠄⠀⠙⠙⠀⣠⡶
⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉
⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡧⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣻⡔⠀⠀⠀⠀⠀⠚⠿⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠁⡝⣿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠑⠂⠁
⠀⠀⠀⠀⢺⣷⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠛⠓
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣶⡄⠀⠀⢠⣶⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣷⣤⣤⣾⠿⠛⠁⠀⠀⠀⠀⠀⠠⠔⠒⢾⡀⠀⠠⠆
⠀⠀⠀⢠⡖⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠘⠤⣤⣤⠤⠏⠀⠀⠀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣶⠃
⠀⠀⢀⠑⢒⣿⣿⡇⠚⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⡏⡟⠉⡁
⠀⠀⠀⠉⢁⣿⣿⡏⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣥⣤⣯⣤⣼⣤⡄⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⢀⣀⠇⠓⠛⠃
⠀⠀⠀⠀⠉⢸⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠰⠡⠀⠀⠐⠋⠉⡛⢻
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⢰⠀⢸⡇⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸
⠀⠀⠀⠠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⢰⣿⣿⣿⠿⠛⣛⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠸
⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠾⠛⠁⠀⠀⠀⠀⠀⠀⣤⣤⡌⠀⣤⣤⡤
⣀⣀⣀⢀⡀⣀⡀⣀⣀⢀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⡀⣀⢀⣀⣀
⣿⣿⠟⠻⡿⢛⠿⢿⠿⣻⡿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡄⢠⡤⢶⣶⣤⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⡁⠀⣻⡿⠁⠀⠀⠀⠀⢀⣤⣤⣤⣄⣠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣖⠂⢀⣐⡐⢲⣶⣶⣆⢠⠀⢦⣤⡀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠔⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠛⣠⠹⡟⠁⠀⠁⠈⢹⣿⣶⣸⣿⣣⣤⠄⢀⣴⣿⡖⠀⣀⣀⠘⠋⠳⠡⠀⠀⠀⠰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⠀⠁⠀⣤⡀⢰⠃⠈⠩⣿⢳⣾⣉⣴⡿⢻⡟⢶⣴⣿⣿⡕⡌⣣⢰⣶⣶⢤⣄⠤⣤⣤⡄⠐⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⠀⠀⢈⣁⠘⠒⠁⠚⣁⣠⣶⣿⣭⣄⣧⣼⣤⣧⠯⣿⣿⡽⢦⢧⠿⣿⣒⠚⠒⠛⠛⠁⠒⠀⠀⠀⠀⢈⣉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠛⠒⠒⠒⠚⠀⠁⢩⣈⣀⠀⠀⠀⠈⠉⢠⡄⠀⠉⢀⣤⣴⣶⣴⣶⣶⡶⣶⠶⣛⣾⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠖⠊⠀⠁⠀⠀⠀⠐⠨⢭⣭⣵⣧⣴⣶⣦⣶⣿⡿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠤⠤⣀⣳⣶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠸⠃⠀⠀⠀⡒⠒⠪⠿⠯⠙⠻⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢮⣽⠂⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠋⠐⠀⠀⠀⠄⠀⠀⢸⣥⠀⠀⠀⢸⠿⠀⠀⢸⣟⠀⠀⣾⡏⠀⠀⠀⢶⣶⡤⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⣘⠀⠀⠀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⢶⣴⠶⣶⣾⣶⣆⣦⣾⣿⣢⣦⣿⣶⣶⣶⣶⣿⣿⣷⠶⠶⠶⠛⠛⠓⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣾⣅⣀⠠⣤⣜⣓⡒⠶⠊⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠸⠓⠀⠉⠱⠿⣿⣿⣿⣤⣶⣾⣿⣿⣿⣿⣿⠻⣿⡿⠿⠿⠛⣛⣙⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⣿⡿⠟⣻⣿⣿⣶⡶⠛⢳⣶⢶⣿⣿⣬⡿⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠋⠙⠛⠀⠉⠭⠀⠀⠁⠀⠈⠛⠛⠛⠋⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⢿⣿⣷⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⡉
⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠐⠛⠛
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠲⠒⠀⠈⠙⠒⠛⠻⣻⣿⣦⣤⣤⣤⣤⣤⣴⣦⣶⣦⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣬⣭⣭⣤⣭⣦⣤⡶⠾⠳⠖⠃⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⡽⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠟⣋⣴⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠶⠾⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢲⡙⢷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣷⣻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣎⣧⡀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣻⡗⢂⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⡿⣿⣿⣿⣿⣿⠄⢠⠀⠀⠀⣿⣿⠆⠀⠀⠀⣴⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠾⢿⣿⣿⣿⠁⠸⠀⠀⢠⣿⠏⠀⠀⠀⠀⣿⡷⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣛⣓⣿⣿⡿⠋⠀⢰⣦⣤⣿⢿⡄⠀⠀⠀⣸⣿⡿⢿⣿⢿⣋⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠘⠋⠙⣁⠲⡀⠀⠀⠈⠁⠉⠀⢾⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢛⣿⣿⣿⠟⠀⠀⠀⠀⣀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣤⣀⠀⠈⢉⣿⣿⢦⡤⢀⠀⠈⠁⠀⠀⠀⠀⠀⣠⠖⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡍⠋⡉⠓⠶⣿⣿⣿⣇⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣬⣧⣼⣦⣄⣈⣿⣿⣿⣀⡈⠁⠀⠀⠀⠀⠀⠀⣾⡿⠀⢠⣤⣾⣿⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣰⣵⣄⣉⣹⣿⣿⣿⡿⠿⣷⣶⣦⡀⠀⠀⠀⣰⠿⠃⠀⠈⠛⠉⢻⣿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠰⣎⡹⠟⡻⠅⠀⠈⠉⠻⣿⣿⣾⣿⣿⣿⡧⠀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠉⣲⣏⡀⠊⠀⠸⠒⢀⣀⠄⠾⠛⢻⣿⣿⣿⡇⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢠⢾⡿⠒⡟⠛⢃⡈⡰⣇⣤⠀⠐⠀⠠⠒⢙⣿⣿⣿⣧⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⠉⢦⡔⠐⠂⠀⢀⠈⣷⡊⣽⠁⣤⣤⡆⠰⢤⠽⣿⣿⣿⣿⡿⠀⠀⠀⠀⣠⠞⢃⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⢶⢦⠘⢢⣾⠃⡴⣿⢿⣷⣿⠀⡟⣹⣿⣦⣣⣴⣿⣧⣾⣿⡇⠀⠀⠴⠊⠙⠛⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠁⠀⠘⢓⣦⣿⢿⣏⡛⠘⣿⡇⢠⡟⣽⣿⣿⣿⣿⣿⣿⣿⢿⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢩⣿⠁⠀⣾⣿⣿⠈⡿⢁⣮⡁⠻⣿⣿⠿⢿⣿⣿⣿⣦⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸⢤⢏⣿⣿⢿⣮⣷⣿⣃⣭⣴⢇⣤⣤⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢀⡀⡯⣟⣹⠖⣿⣛⣫⣿⣿⣿⠻⣿⣿⣿⣿⢿⡷⣿⠏⠀⢸⣠⠛
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢁⣄⢹⣾⣟⣷⣾⣹⣿⣼⣯⣝⡧⣽⣽⣿⣿⣿⣿⣿⣻⣦⠙⢠⡶⠙⣇
⠀⠀⠀⠀⢸⣷⣷⡀⠀⠀⠀⢘⣿⣿⣽⣏⣿⣿⣿⢵⣷⣴⣿⣧⡜⠉⣀⠿⣟⠀⡟⣿⣆⡈⢹⣶⡾⠛⠀⡀
⠀⠀⠀⠀⣼⣿⣿⣇⡀⠀⠐⠎⣉⣉⣾⣣⣿⣿⣿⣿⣧⣿⣶⠍⠀⠀⠐⣾⣿⣦⣜⠊⣁⣼⠟⠻⣶⠒⢮⡴⠒⠂
⠀⠀⠀⢸⢿⣿⡇⠹⣇⠀⡀⠶⢉⣭⢛⣛⣿⠛⢿⠿⡅⠍⢹⣶⣆⣤⣾⢿⡟⣿⣿⢁⣼⢿⠀⢠⡽⠢⢌⡀
⠀⠀⠀⠈⢿⡿⠾⣦⣿⡄⠀⠀⠘⣙⣭⣝⣧⣶⣤⣴⠟⠒⣾⣗⢿⣿⣿⣿⣗⣾⣿⣿⣿⡉⣰⠟⡤
⠀⠀⠀⠠⣿⡿⢓⣾⢿⣿⣾⣷⣿⡉⣁⣼⣿⣿⢛⠙⠢⠐⠿⠟⢿⣧⣽⣿⣿⣻⣿⣿⣿⡷⠋
⠀⠀⠀⢘⡿⣤⣻⡿⠉⢻⣽⠯⣍⣻⡿⢛⣿⠻⠿⣷⣆⠩⠶⣾⣿⣷⣿⣿⣿⣿⣩⣏⣛⡧
⠀⠀⠰⠞⠛⠚⠋⠀⠠⣟⣿⣓⣿⡿⠋⠉⠀⠀⠀⠀⠉⠿⠏⠤⠬⡿⠛⢿⢿⣟⠛⠿⡇
⠀⠀⠀⠀⠀⠀⣶⣄⠀⠻⣮⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⠈⠙⠆⠀⠈⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣆⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣆⣀⣤⣶⣾⡿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠉⠉⠀⢿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⣼⣷⠦⠀⠐⠃⡜⠓⠖⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠄⢨⡀⠟⠆⠈⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠠⡀⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢬⡅
⠀⠀⠀⠀⠀⠀⡃⠂⠁⠐⠟⠀⠀⠀⠐⢦⠀⠈⠂⠀⢀⠀⠀⠀⡄⠀⢄⡰⣄⠉⠲⡘⠀⣠⠁
⠀⠀⠀⠀⠀⠀⢓⣠⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⣄⠀⢳⣄⠀⠰⡀⢀⢳⣮⢠⠈⠄⢀⠔
⠀⠀⠀⠀⠀⠈⠀⠀⠈⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠘⡁⣄⠩⠃⠀⡘⢶⢦⡽⠇⠀⠅⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠃⠀⠀⠀⠐⢦⡄⠀⠀⠀⠀⠈⠀⡍⠣⠀⠠⡀⠓⢂⣐⠒⣀⣠⠶⠂
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠂⠀⠄⠀⠀⠀⠱⠋⠤⠀⠀⠀⠢⡂⡂⠀⠰⢀⣈⠈⡉
⠀⠀⠀⠀⠀⠀⢁⠐⠠⠄⢀⠀⠀⠠⠄⠀⢀⠀⠀⠁⠄⠀⠀⠀⠈⢠⡁⠁⠉⠀⠉⠈⠋⠙⠓⠂
⠀⠀⠀⠀⠀⠠⠀⠁⡀⢀⠀⠀⠁⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⢀⠌⠀⠀⠀⡈⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⢠⠈⠁⠈⡄⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⢀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠃⠈
⠀⠀⠀⠀⣀⠁⠈⠀⠄⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀
⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⢅⠁⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⠡⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⡀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢱⡀⠀⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⡰⠃
⠀⠀⠹⣿⣿⣦⣀⠀⠀⡰⢆⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠙⠻⣁⡐⠙⠆⠓⠈⣇⣀⣀⣼⣇⣀⣈⠧⠀⠠⠐⠀⠀⠀⠀⠀⢠⠃⠀⠀⠁⠀⢀⠀⠤⠀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠙⢛⢠⣴⠖⠋⠉⠙⠄⠈⠀⡄⠀⠁⠒⠂⠀⠀⠀⠀⠀⢸⠄⠀⠀⠈⠉⠁⠀⠀⠀⡀⣠⣀⠈⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⠀⠀⠀⣀⢤⡆⠀⠅⠂⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠽⠍⠙⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠒⠒⠂⠀⠀⠐⠂⢄⡀
⠀⠀⠀⣠⣴⣶⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠉⠉⢓⡐⠒⠂⠀⠀⠈⠉⠻⣦
⠀⡴⣾⠿⠿⡶⡤⣄⠈⠀⣖⢲⡖⢲⣴⣤⣄⠀⠀⠀⢀⠀⠴⢶⠋⢻⡀⠀⠀⢨⣿⠁⠀⠀⠀⠀⠀⢀⣻⠄
⠀⠀⣙⠻⡱⣓⡗⠀⠘⢼⠃⣜⠣⡞⠥⠗⢽⣳⠄⠀⠀⠀⠀⠘⢷⠮⢶⣦⣤⣸⣷⣤⣶⣷⣶⣶⡶⠿⠛
⠀⠀⠈⠋⠀⠈⠈⠱⠤⢥⠔⢤⠞⣥⠖⠀⠲⠟⠁⠀⠀⠀⠀⠀⠘⠀⣀⡿⠙⢇⠈⠉⠉⠉⠉⠙⠶⠶⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣴⠠⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠡⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⢀⠀⣰⡓⣐⣄⣶⣷⣰⣄⠀⠙⠻⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠘⠳⣄⡀⠀⠀⠀⠀⠈
⠀⢀⡏⣴⣆⡀⣿⣿⣍⣽⣿⠏⡷⢧⣀⡄⢋⠁⠦⠀⠀⠀⠀⠀⣀⣤⣠⣴⣤⠴⠊⢉⣿⢟⡶⣺⢿⣷⣾⠉⠈
⠀⠘⢦⡟⠁⠀⠉⡿⠛⠁⠉⠙⠃⠀⠺⢣⣄⠱⠀⡁⠀⠀⣠⣾⣿⣿⡇⣼⡿⢿⢸⢧⡿⣻⠟⢯⡞⠙⡾
⠀⠀⠀⠑⢰⣶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢠⠄⠀⠈⠙⠛⠉⠉⠁⠁⠀⠀⠉⠉⠉⠑⡄⠀⠀⠠⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂

⠀⢲⡖⡤⢄⠀⡠⡄⢲⠆⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⢠⣦⣄⠀⠀⠀⠀⠳⠿⠆⠀⠀⣄⣄⠀⠀⠀⣤⡤⠀⠀⠀⡀⠀⠀⡀
⠀⠈⢷⣠⠤⣯⢠⣠⡿⠀⠀⢸⡅⠁⢀⠀⢰⠇⠀⠀⠀⣬⢿⠀⠀⠠⡂⠀⠀⢧⠀⠀⠯⣀⠀⣄⢀⣈⠕⠀⠀⢇⣀⣤⢀⡠⠃
⠀⠀⠀⠧⠤⠣⠤⠇⠀⠀⠀⢸⣇⠀⣏⡴⠁⠀⠀⠠⣄⠀⢿⠂⠀⡠⠉⠑⢤⡆⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠀⢠⡀⠈⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠉
⠀⠠⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣄⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠠⡀⠀⢰⠀⠀⠀⠀⢈⡇⠀⢠⠀⠀⠀⠀⢠⠄⠀⢵⢈⢢⣠⡴⡉⢸⠀⠐⠀⢀⡀⠀⠉⡃⠈⢫⣏⢐⣤⣔⢸⡮⠁
⠀⠀⢰⣾⢿⠿⢻⣷⡅⠀⠀⢀⠈⠀⣵⠀⠈⠓⠒⡟⠲⠊⠀⠀⠀⡏⠀⠻⡀⢸⡆⠀⠀⠰⠾⢷⠒⠦⠀⠀⠈⠫⣙⠟⣏⡉⠃
⠀⠀⠀⢹⠀⠀⠐⢟⠁⠀⠀⠈⠂⠀⠁⠀⠀⠓⠂⠀⠒⠃⠀⠀⠀⢹⠉⠀⠉⣋⠀⠀⠀⠀⠛⠈⠚⠛⠀⠀⠀⠛⠂⠀⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣿⠀⠀⠀⠀⠠⣇⠀⠀⣇⠦⠀⠀⠀⢦⠁⠐⢀⡀⠀⠉⡔⠀⡀⠀⠀⠀⠄⢁⠀⠀⠀⢀⡮⣧⡄⠀⠀⢀⠀⠀⠀⠠⣍⣅
⠀⠀⠉⠲⡄⡷⡠⠚⠁⠀⠀⠻⣄⠠⠀⠀⠀⡱⠛⠹⠏⠓⢆⠀⠀⠀⢀⣀⣤⢀⠴⠀⠀⠀⢆⣤⣿⢧⡀⠀⠀⠻⣧⣤⣴⡉⣥⡀
⠀⠀⠀⠒⠉⠀⠁⠒⠀⠀⠀⠀⠀⠚⠀⠀⠀⠐⠶⠀⠀⠶⠀⠀⠀⠀⠀⠐⠈⠘⠒⠀⠀⠀⠘⠛⠛⠻⠃⠀⠀⠀⠙⠿⠟⠲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢰⠂⢀⠀⠀⢀⡀⢲⠀⡶⠢⠀⠀⠀⠀⠀⠲⡰⠂⠀⣀⠀⠲⣐⠂⠀⣿⢵⣆⠀⣤⡖⠀⢀⡑⢲⡀⠀⢲⣀⡢⠄⠀⠀⢤⣂⣰
⠀⠘⢄⣅⠄⡦⢀⣅⡜⠀⠃⠀⢀⡀⠀⢀⠋⠀⠛⠤⠢⡯⠐⢼⠋⠀⠀⢹⠒⢲⡿⡼⠁⠀⢨⠁⠈⠕⠀⠈⢿⡙⠓⢶⠒⠋⣹⠋
⠀⠀⠸⠶⠎⠋⠶⢾⠀⠀⠀⠀⠸⡁⠐⠁⠀⠀⠀⠠⠀⠀⠨⠀⠀⠀⠀⠹⠦⠂⢠⠆⠀⠀⠘⢎⣑⣮⠀⠀⠀⡗⡀⠀⢀⡰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡋⣀⠀⠉⠛⢒⣲⣷⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠦⠤⠤⠉⣻⣅⢹⣿⣿⣿⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢⡐⠂⠈⠻⠿⠟⣫⣽⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠂⣿⣿⣆⠀⠀⢀⡤⠀⡰⢊⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡅⠡⢷⣾⡟⠋⠀⠀⠳⠌⢹⣂⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠓⠒⠒⠚⠿⠇⢀⣴⣧⡀⠀⢹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⣤⣤⢤⣄⡀⠀⠀⠙⡿⠋⣀⣴⣾⣿⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣛⢉⣿⣿⡿⠁⣐⣶⠋⣴⣾⣿⣿⣿⣇⣀⣠⣶⠏⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡻⠖⠛⠁⠁⠘⠿⠿⣦⡙⢿⣿⣿⣿⣿⠿⠿⠋⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⢀⣀⠀⣀⣀⡀⢻⣿⣶⣝⡻⢿⣤⣴⢦⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣾⣿⣿⣿⣁⢿⣟⢻⣿⣿⣿⣿⣿⠃⠨⢯⣤⡅⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠋⠽⠛⠁⠈⠙⡿⢿⣿⣻⣽⣶⣿⡿⠁⡸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⣠⡼⢀⣘⣿⡿⠿⠿⣿⢧⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢻⡽⠿⠇⢸⣿⡋⠀⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⢀⣴⡾⠃⣼⣿⠃⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⠙⡿⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⣺⡁⠀⠀⢠⠏⠀⠀⠀⣠⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⡞⠀⠀⠀⣼⣿⡿⠛⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⣠⣅⠀⠀⠀⠹⣿⣿⣿⣿⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣤⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠛⠛⠛⠛⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣤⣤⣀⢠⣿⣵⣯⣼⡿⣷⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣾⣿⡟⠉⠙⢻⡟⠿⣿⠟⣱⡏⢹⣿⡤⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⡟⠀⠀⠉⢧⡀⢀⣀⣙⣷⣶⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢿⣿⣿⣿⡀⠀⠀⠀⣷⣸⣿⣿⣿⣷⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠿⣿⣿⣿⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⠁⢀⣴⣿⡟⣿⣿⣭⣟⡻⠋⠻⢿⣿⡏⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣿⡟⠉⠀⣀⣴⠿⠋⠀⡇⣿⠹⣿⣿⡇⠀⣠⣾⣿⠃⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣠⡾⠋⢉⣿⡟⠿⣿⣶⡾⠋⠁⠀⢀⣼⣵⠏⣰⣿⣿⣷⣾⣿⡿⣿⣦⠀⠀⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⡾⠋⠀⢠⣿⠟⠁⢀⣾⣿⡿⠀⣀⣴⣿⠟⠁⣴⡿⢹⡿⣡⣾⠟⠀⠈⣿⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⣴⣿⠟⠁⢀⣼⣿⡟⠁⣠⣾⡟⣡⣿⡟⠻⣿⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣠⣾⡿⠁⠀⣠⣿⠟⣻⣿⣿⣿⣫⣼⡿⠋⠀⠀⡇⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠉⠙⢿⣿⣀⣠⣾⠟⠁⣰⣿⠟⣿⣿⣿⠟⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣄⠀⠀⠈⣿⣿⣇⣤⣾⠟⣡⣾⣿⠟⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠈⠛⢧⣦⣤⣿⣿⣿⣿⣥⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⣸⣿⡿⣋⣽⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣀⣤⣾⣿⣷⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣥⣤⣶⣾⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⠀⢠⣤⣥⣾⣧⣬⣤⡄⠠⣤⠀⡀⠀⢰⠀⠀⠀⡡⠤⠁⠀⠀⠐
⠀⠀⠳⡀⠀⠀⠀⢀⣴⠋⢳⣆⠀⠀⠀⠀⡠⠊⠀⠀⠀⠊⢱⡄⠀⠀⠀⠀⠀⠀⣀⠘⢺⠿⢣⡜⠿⡖⠃⣀⠀⠀⠀⠀⠀⠢⡠⣆⠀⠀⢀⣠
⠀⠀⠀⠀⡘⠂⠂⣈⢻⡄⣼⢋⡀⠀⠐⡂⠀⠀⠀⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠛⡿⠋⠉⡄⡀
⠀⠀⠀⠀⠀⠈⠁⠉⢁⠸⢀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠑⣄⡳⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡈⠀⠀⠀⠀⢀⢁⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⢄⣀⠴⢊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠠⠆⡀⠑⢦⢠⠖⠁⠰⠄⠀⠀⠀⠀⠀⡀⢡⠌⠉⠁⡎⠀⠀⠀⠀⠂⠉⠉⠀⠀⠀⠙⣗⢸⡏⠁⠀⠀⠀⠉⠐⠂⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⢀⡡⢬⡇⠀⣿⠤⣅⠀⠀⠀⠀⠀⠀⠈⠴⠈⠹⠁⠶⠒⠁⠀⠀⠈⠂⠀⠀⠀⡀⠀⣯⢸⡃⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⢇⠈⠳⢤⠛⠈⢃⠴⠋⣀⠆⠀⠀⠀⠀⢠⠀⠛⠂⠛⠀⢤⠀⠀⠀⠀⠀⠀⢹⣏⢇⣀⡘⣏⣀⣠⢯⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠙⠂⠂⠀⡉⠀⠒⠒⠁⠀⠀⠀⠀⠀⠄⠈⠓⠐⠘⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠋⢹⡛⢟⠻⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠉⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠂⠒⠲⠤⠐⣄⣧⣥⡔⠡⠤⠒⠂⠀⠀⠀⠀⠀⡆⠀⠀⡥⠀⢀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠈⠀⠀⠀⠀⣶⡐⠙⡆
⠀⠀⠀⠀⠀⠀⠐⢻⠂⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⠔⠛⠢⣈⠆⠀⠀⠀⠀⢀⠘⣇⠀⠀⢰⡤⣴⠀⠀⢀⡟⢀⠀⠀⠀⠀⠀⣹⣇⢀⡇
⠀⠀⠀⠀⠀⠀⡠⡖⠋⠙⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠦⠤⠏⠀⠀⠀⠀⠀⠀⠀⢼⡄⢀⣻⣿⣿⢒⡀⣼⠁⠀⠀⠀⠀⠀⠀⣿⣷⣾
⠀⠀⠀⠀⠀⠌⠐⢇⠀⢀⠟⠀⠣⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠸⠐⣄⠀⠀⠀⠀⡌⠸⣇⠀⠙⢿⠿⠋⢀⣿⠞⡄⠀⠀⠀⠀⠀⢸⠳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢀⢴⡄⠈⠀⠀⠀⠀⠐⢀⠘⡦⠀⠈⠀⠠⡞⢀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠑⠂⡀⠀⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠰⠆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡽⠤⡧⠬⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠗⢏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠤⠐⠂⢆⡇⠀⠀⠀⠀⢤⡀⠀⠒⠒⠀⢀⡤
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠹⣦⡾⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠴⠖⠢⠔⢣⠤⠀⠀⠀⢰⠒⠇⢹⠋⠉⠉⠸⠐⡆
⠀⠀⠀⢀⠄⠁⠀⠀⠁⠀⠀⠊⢃⠀⠀⠀⠀⠀⠀⡆⡠⠊⠤⠆⠤⠄⠢⡀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠆⠀⠀⠀⠀⣄⠀⢠⡕⠀⣠
⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⢰⠂⠀⠀⠀⢠⠀⡀⠀⠹⠁⠨⠖⠀⠐⠲⠀⠹⠅⠀⣀⢠⠀⠀⠀⠀⠃⡀⠀⢈⠀⠀⠀⠀⠀⠢⡀⢦⠂⠐⡄⢀⠔
⠀⠀⠀⠀⠀⠀⢙⠒⠶⠚⡘⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣶⣦⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⣶⠂⠀⠀⠀⠀⣀⡴⣶⡆⠀⠀⢀⣤⢶⣄⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣽⣷⣄⠀⠀⠀⢀⡀⠓⠂⠰⠖⠐⢯⣄⠁⠀⠀⠻⣄⠀⣶⣶⣶⠉⣠⡟⠀⠀⢀⠮⣔⢫⣿⠄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣼⠛⢿⣿⣆⠀⢀⠈⠛⠂⠀⠀⠀⠀⠛⠋⠀⠀⠀⠐⢷⣧⣾⢯⠙⢘⡵⠃⢀⣀⠺⣧⣼⡟⢿⣆⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⢯⠄⢘⣿⢏⣻⡃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⢿⣷⣿⣿⡟⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣾⣿⠿⠻⢿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠹⡟⠉⠀⢀⡐⠎⡉⢳⣄⣾⣿⣿⡟⢿⣌⠑⠋⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠿⠛⠁⠰⠮⣁⣙⣿⣿⣴⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠾⠿⣷⣍⣛⣋⣭⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠔⢀⠀⠤⡸⠀⠀⠀⠀⢠⡤⣀⡀⠀⠀⠦⠶⠲⠴⠂⠀⢀⣀⣤⡤⠈⢙⠛⠋⠁⠀⠀⠀⠀⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢲⢴⣦⠄⠀⠀⣤⣶⣆⠈⢿⣴⠊⣶⣄⠀⡀⢀⡀⢀⢴⠍⠂⣿⠇⢸⣿⣧⠀⠀⠀⠀⢀⡔⠋⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣽⣿⡀⠀⣿⣆⣀⠒⢳⠀⢬⣔⡃⠈⣁⣺⡋⠀⢸⡿⣿⣧⠀⠀⠀⠉⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠉⠁⠀⠀⢾⣿⣿⣿⣿⠇⠀⢠⡎⠀⠀⠈⡛⣿⡏⠀⢀⠀⢭⠀⠀⢽⠗⣿⣿⡄⠀⢀⠄⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠐⣶⣄⠀⠀⣞⠛⠉⠁⡀⠄⠀⠈⢻⣦⣞⠁⠘⠋⠀⠸⣉⡾⠃⠀⡐⣿⢀⣿⣿⡏⠀⠸⠀⠁⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⢿⡷⡄⠸⣿⣒⠀⠀⡸⠀⠀⢠⡟⠉⠀⠀⠀⠀⠘⢿⡇⠀⠀⢫⣿⣸⡿⠋⠀⠀⠘⢿⣷⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣠⣤⡀⠀⠀⠀⠙⠛⠒⠀⠀⠄⠋⠀⠀⠀⣠⣦⡀⠀⠀⠐⠦⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢹⣿⣿⣦⡀⠀⠀⠀⣀⣠⣤⣦⠀⠈⠀⠀⠛⠿⠿⠶⣤⣀⠀⠀⣿⣿⣣⣤⡀⠀⠀⣰⣾⣽⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣤⣴⣾⣿⣿⣿⠋⠀⠀⠀⠀⠈⠛⠓⠢⠀⢿⠂⠀⢿⡔⠻⣿⣿⣶⣾⣿⣿⣯⣿⠃⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⣀⣀⠀⡠⠄⠀⠀⠀⠀⣼⣹⣿⣿⠿⡿⢻⣿⣳⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠙⢿⡿⠟⠹⠃⣿⣿⣿⡿⠀⠀⣲⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⡟⠀⠀⠈⠻⠿⠋⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠋⠀⠀⠀⢹⣼⣟⣗⣂⣀⡠⢤⣾⡇⣿⠃⢀⡄⠀⠀⢤⡀⡀⠤⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢶⣄⣴⡾⠷⠀⢀⣜⣷⡆⠈⠿⢿⡏⢻⣿⢻⣿⣿⣿⠁⢀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣀⡤⠚⠛⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠈⠁⠈⠉⠉⠁⣰⣿⡿⡋⣵⡇⠀⡦⠀⠀⢀⢴⣶⡄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠘⠋⠀⢀⣰⣦⡄⠀⠀⠈⣻⣿⣿⣿⣄⣀⠀⠀⠀⠀⣠⣶⣶⢟⣩⣶⢷⡿⠁⢀⠂⠄⠀⠠⠸⢿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠰⣾⣿⣿⠋⠁⠀⠀⠀⣿⣆⣿⣿⣿⣿⡆⠀⠀⣼⡿⠋⣤⣻⣯⣿⣽⠇⠀⠉⠀⠀⠀⢨⣶⣾⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠙⠟⠋⠀⠀⠀⢀⣠⣿⣿⣿⡛⠷⠿⡿⠃⠀⠀⠿⠵⠾⠿⠟⢙⣛⣩⣶⣀⡀⠀⠀⠀⠀⢙⣿⠇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⡔⠶⣿⣿⣾⣿⣿⣿⣧⠈⠱⠄⠀⢀⡀⠠⠐⣶⣦⣤⡄⠀⣉⣀⣿⣿⣮⣽⣟⠲⣶⢤⣀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣿⣿⣷⡄⢀⣶⣮⣭⠷⡿⠉⠉⠀⣠⠾⠿⢷⠈⢟⣙⣩⣵⣷⣶⣿⣟⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠉⠠⠿⠟⠋⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠁⠉⠉⠉⠁⠈⠉⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠀⠀⡠⠋⠊⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡠⠋
⡰⠋⠀⠀⠠⠔⣒⠒⠐⠄⠀⠀⠀⠀⠀⠀⠰⠀⢀⢾⣷⠶⠋⠀⠀⠀⡰⠋⠀⠀⢠⠔⣒⠒⠀⢄⠀⠀⠀⠀⠀⠀⠈⠀⢀⢾⣿⠶⠋
⠀⠀⠀⠀⠁⠠⣴⠄⠀⠀⠑⢄⠀⠀⠀⠀⠠⡠⢄⡀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠐⠁⠀⣴⠄⠀⠀⠁⠀⠀⠀⠀⠀⠘⡠⢄⡀⠀⠀⠀⣀⠴
⠀⠀⠀⠀⡄⠀⠹⡄⠀⢀⠀⠈⠆⠀⠀⠀⠀⠈⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⡄⠀⢀⠀⠈⠆⠀⠀⠀⠀⠈⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠐⡄⠀⠹⣷⣾⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠙⢷⣾⡇⠀⠈
⡀⠀⠀⠀⠀⠈⠳⣄⣀⠐⢾⡰⣼⠀⠀⢀⠤⠒⠒⠉⠉⠉⠁⠀⠐⠤⡀⠀⠀⠀⠀⠀⠲⣄⣀⠐⢾⡴⡱⠀⠀⢀⠠⠂⠒⠉⠉⠉⠉⠀⠐
⠈⠢⠀⠀⠀⠀⠀⠀⠉⠛⠓⠋⠀⠀⠀⠁⠀⠀⢀⡀⠂⠄⠄⠤⢀⠀⠈⠲⡄⠀⠀⠀⠀⠀⠉⠛⠓⠋⠀⠀⢠⠃⠀⡀⢀⡀⠂⠤⠄⠤⢀
⠦⡀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣯⠉⠁⠀⠀⠀⠀⠘⠢⣀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣯⠁⠁
⠲⡘⡂⠐⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠈⠲⣌⢆⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠳⣄⡀
⣿⠟⠛⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠉⠓⠶⠶⣶⣿⠟⠟⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠉⠓⠷⠶
⢀⣀⢌⣳⠋⠀⠀⠀⢀⠀⠠⠤⣤⣀⡀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⢐⣳⠋⠀⠀⠀⢀⠀⠤⠤⢤⣀⡀⠀⠀⠀⢀
⠭⠞⠋⠁⠀⢀⠔⠊⠀⠀⠀⣀⡀⠍⠻⣦⠀⠀⠀⠉⠐⠢⠤⠬⠬⠬⠭⠖⠋⠁⠀⢀⠄⠊⠀⠀⠀⣀⡀⠍⠻⣦⠀⠀⠀⠈⠐⠢⠤⠬⠤
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⣹⠏⠂⢈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢹⠇⠆⠈⡻
⠀⠀⠀⢠⠁⠀⢀⠀⠀⠀⡠⠋⠋⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢀⠀⠀⠀⡠⠫⠊⠀⢰⠏
⠀⠀⠀⠀⠄⢀⢾⣿⠶⠋⠀⠀⢀⡰⠃⠀⠀⡠⠒⠀⠀⠒⢄⡀⠀⠀⠀⠀⠀⢰⠀⢀⢾⣷⠶⠋⠀⠀⠀⡰⠋⠀⠀⡠⠒⠀⠀⠒⠄⡀
⠀⠀⠀⠰⡴⢄⡀⠀⠀⢀⣀⠴⠊⠀⠀⠀⢰⠁⠀⣴⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⡰⢌⡀⠁⠀⢀⣀⠴⠊⠀⠀⠀⠀⠁⠀⣴⠆⠀⠀⠐⢄
⠀⠀⠀⠀⠈⠁⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠈⡄⠀⠸⡄⠀⢠⠀⠈⢂⠀⠀⠀⠀⠈⠁⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠹⡄⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠙⣿⣶⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠹⣷⣾⡆
⠀⠀⢀⠄⠐⠚⠉⠁⠀⠉⠁⠒⠤⡀⠀⠀⠀⠀⠈⠳⣄⣀⢐⣿⡴⡧⠀⠀⢀⠤⠒⠚⠉⠁⠀⠁⠀⠀⠤⡀⠀⠀⠀⠀⠈⠳⣄⣀⢈⣮⡴
⠀⢠⠃⠀⠀⢀⡂⠒⠠⠄⠄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠉⠙⠓⠉⠀⠀⢀⠃⠀⠀⢀⡐⠒⠢⠄⠤⠀⠀⠈⠒⡀⠀⠀⠀⠀⠀⠉⠙⠓⠉
⠀⡆⠀⠀⠐⣿⠁⠁⠀⠀⠀⠀⠙⠦⣀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣯⠈⠁⠀⠀⠀⠀⠙⠦⣀⠈⠦
⠀⠠⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠈⠰⣌⢆⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠳⣌⢦⠀⣃
⠀⠀⠱⡀⠀⠀⠀⠀⠉⠓⠳⠶⢶⣿⠟⠛⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠉⠛⠳⠶⢶⣿⠟⠓⢀⣸
⣤⡀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢔⡣⠃⠀⠀⠀⣀⠤⠀⠤⣄⣤⡀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⢀⢀⢰⡳⠋⠀⠀⠀⣀⠤⠀⠤
⡀⢛⡧⠀⠀⠀⠈⠒⠲⠤⠬⠬⠭⠽⠞⠋⠀⠀⢀⠔⠊⠀⠀⠀⣀⡀⠌⢛⣦⠀⠀⠀⠈⠐⠲⠤⠤⠬⠬⠽⠚⠋⠀⠀⢀⠔⠊⠀⠀⠀⣀
⠃⢈⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⣹⣇⠂⢈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣸
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡠⠊⠉⠉⠰⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⢁⠦
⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡿⠧⠀⠀⠀⠀⠀⠀⠀⣀⡤⢚⢠⡀⠀⠀⠀⠀⠀⢸⠀⠐⠈
⠀⠱⢆⡀⣀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠈⠁⠀⠀⠀⠀⠈⠲⣀⣀⣀⠔


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠓⠀⠀⠀⠀⠀⠀⢀⠔⢁⣀⣈⠲⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣤⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠐⠉⠘⠁⠀⡁⠀⠀⠀⠀⠀⠐⠒⠓⠒⠂
⠀⠟⠃⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⣼⣤⠀⠀⠀⠀⠀⠀⠀⠓⠄⣀⡠⠐⠁


⠀⠀⣠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠴
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⢧⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣽⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣯⣤⣤⣤⣤⣤⣵⣦⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠦⠐⢒⣂⠠⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡐⠂⠀⠠⠀⠆⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠸⠶⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠀⠀⣤⣤⣄⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣽⣿⡆⠀⡀⠀⠂⠀⠍⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⠈⠁⠀⠀⠀⠔
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠛
⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⡆⠀⠀⠀⠀⠀⢀
⠀⠀⢦⣼⡇⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⢧⠁⠀⠀⠀⠂⠀⠀⠀⡀⠀⣀⣀⣀⠀⣤⣼⣿⡟⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠛⠛⠛⠰⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢇⠀⠀⠀⠸⠃⠀⠀⠀⠀⣶⡄⠀⠀⠈⠀⠀⠂⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠘⠀⠀⠀⠀⠀⠀⢠⣤⡤⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⢿⣿⣦⡯⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢚⡿⣻⣿⣈⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠋⠀⠀⣀⡄⠀⠀⠀⠀⢺⣷⣿⣬⢿⡷⠦⠀⠆
⠀⠀⠀⠀⠀⠞⠲⣄⡀⠀⠀⠀⠀⠛⠀⠀⠀⠜⠁⠀⠀⠀⠀⠘⢿⣿⣯⣄⢻⡆⠬⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠨⠿⢾⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠤⢔⡉⢡⠾⣿⣷⣴⢻⣇⠀⠀⠀⠀⣠⡄⠀⠀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠀⠀⠀⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠤⡙⢿⡀⠀⠐⠟⠑⠀⠀⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠈⠁⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢠⠀⠈⠁⠰⠀⢀⡀⠰⠤
⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡞⠂⠛⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⢀⡗⣴⠚⠉⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠠⠀⠀⠀⢠⠿⠋⠠⠀⢀
⠀⠀⠀⠈⠉⠹⢷⠀⠀⠀⠀⠀⠀⠀⠰⠃⠰⠄⠀⠀⠀⠀⢄⠾⠁⡄⢀⠀⣅⡘⠀⠀⠀⠀⠀⠀⠀⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠸⠷⣦⢡⡜⠃⢻⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣾⣦⠀⠀⠀⠀⢹⡄⡄⠀⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡃⠀⠀⠀⠀⠀⠁⠀⣀

⠀⠈⠉⠉⠛⠓⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠓⠢⢀⠀⠀⠀⠀⡄⠀⠀⣀⡀⠀⠀⠀⡔⠀⠀⠀⠠⠐⠛⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠐⢦⣾⣿⣿⣿⣿⣦⡤⠂⣠⠀⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣾⣾⣿⣿⣿⣿⣿⣿⣿⡛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢿⣯⣿⡿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⣿⣿⣿⣿⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⣿⣿⣿⣿⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢠⣿⣼⣾⣇⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠆⢠⣷⢠⣾⡇⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡠⢿⠟⠉⣿⣿⣧⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⢸⣻⣾⣿⡏⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⣶⣦⣴⣶⡶⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣁⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⡞⠙⣿⣶⡄⣿⡀⢰⢸⡏⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣿⣿⡿⢄
⠀⠀⠀⠀⢀⠴⠛⢀⣴⣿⣿⣿⣿⣧⣼⣦⣷⡄⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⠀⣿⣿⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠹⣿⣿⣿⣿⣿⡟⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠰⢷⣤⣿⣿⡿⡆⠁
⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠈⠛⠋⠉⠁⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⡈⠻⠿⠟⢠⠇
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡆⢰⠀⡄⠀⣿⠀⠀⠀⢣⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠸⣦⣷⣶⣿⣶⣷⣴⠇⣸⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⢤⣤⣄⣀⣀⠀⠈⠦⠠⣄⡀⠙⢦⡟⣋⣙⣿⣿⣿⣿⠶⠃⣀⡴⠰⠀⠉⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⡠⠐⠋⠁⠁⠀⠈⠉⠛⠛⠲⠤⠌⣹⣶⢤⣀⣀⣉⣗⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣀⣠⣤⣖⣋⡩⠭⠛⠛⠒⠀⠐⠀⠂⠈⠑⠢⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠀⠐⠲⠖⠚⠛⢲⠟⠽⠿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠿⠶⠤⣤⠄⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⣠⢴⠏⠀⠄⠀⣼⡇⢠⣿⣿⣿⡌⢿⠁⣙⢿⡉⢹⣦⣀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⡼⢀⡄⠀⠘⠁⣭⣿⣿⣿⣿⣿⣦⡀⠘⠆⢁⢹⡇⠈⠑⠦⡀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠊⠀⠀⠀⣀⣿⡀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⢜⣶⣇⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠘⣿⡾⢻⡏⢿⣿⣷⣿⣿⣿⣿⣿⠟⢹⣦⣿⣿⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠙⢿⣿⣿⣿⠟⠋⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⣶⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⢸⠇⣻⠿⢿⡁⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠿⢛⠿⠦⠄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠎⠀⠘⠶⣿⣡⣚⡷⠊⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠿⠗⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⣤⡶⣾⣿⣿⠶⣦⣄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠎⠹⠃⣤⡄⠀⢀⣤⡀⠟⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⣘⠀⡀⠀⢼⠆⠀⡀⢸⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡿⣴⢿⣦⢸⢠⣾⠷⡜⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡊⠺⣦⠀⠀⠀⢾⡷⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠀⣆⠘⡟⢰⢈⡿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠈⠀⠐⠀⠁⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢹⠀⠹⠾⠞⠁⢸⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⣶⣆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡄⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⢹⣦⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠸⣤⣭⣦⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠋⠉⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠤⠾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣝⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⣾⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣡⣄⣤⣤⣤⡀⠉⠁⠀⢳⠀⠀⢀⣶⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡧⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠏⠁⠈⠛⢻⣂⡀⠀⠲⣷⠀⢼⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡆⠀⠀⢀⣀⡈⠻⣗⠀⠀⠜⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⢶⠶⠞⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⣀⢀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠿⣿⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠟⠉⠀⠑⣄⠀⢀⢀⣠⣠⣄⢤⡈⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠺⡟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⣽⣿⠟⢻⠉⠛⠯⠟⠳⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⣴⡾⠤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣟⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢻⡿⣿⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣥⣿⣴⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣃⣼⣡⣟⣡⣾⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⢻⠟⠉⢹⣯⡿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢰⣠⠃⣠⠔⠉⢈⣿⣟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⡷⠊⠁⣠⠖⠉⠀⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⣾⡏⠀⢰⣃⠴⠊⠀⢀⣤⣾⠟⠋
⠀⠀⠀⢸⣿⣿⣿⠲⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⡜⠀⠀⢀⣴⣿⣟⣁⣤⣤⣶⣤⣄⡀
⠀⠀⠀⠀⠻⣿⣯⣹⣿⣙⣲⣿⡿⣿⣶⣦⣄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢰⠁⢀⣴⣿⣿⡿⠋⢁⡠⠝⠛⠿⣟⣷⡀
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣭⣿⠯⠭⣤⡽⠛⠿⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣇⣴⣿⢟⣟⡉⠤⠒⣉⠠⠔⠒⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣤⣤⡤⠽⠶⢤⣀⠈⠉⠛⠿⣶⣤⡀⢴⣶⣼⣥⡾⠟⣋⠔⠋⠉⠙⠚⠛⢳⣤⠒⠚⠉⢉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⡤⠤⠒⠚⠉⠉⠒⠒⠒⠚⣿⣿⣿⣿⣿⣴⠮⠥⠤⠤⠤⠴⠶⣉⡁⠀⠉⠑⠲⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⣤⣤⣤⣴⣶⣾⣿⠟⣋⠼⣿⣿⣿⣿⡒⠦⣄⣀⠀⠀⠀⠉⠑⠲⣤⣴⡿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⡿⠿⠿⣥⠖⠊⠁⡰⠁⡟⡿⣿⡮⢓⢤⡀⠉⠑⠲⠤⣄⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⠊⢩⡇⠀⢠⠃⠀⣇⠹⡘⣿⣄⠈⠉⠓⠦⣄⣠⣾⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡎⠸⣶⠛⡄⠀⢸⠀⢳⣹⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣇⠳⡈⢦⠀⢃⠀⢿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢶⣷⣼⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⠀⣀⢀⣀⣀⢀⣀⣀
⣿⣟⣟⣿⣟⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣟⣛⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠄⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⢨⠽⠂⠀⠀⠀⠀⡝⠐⠀⠀⣠⠁
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠁⠈⠂⡀⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⡀⠀⠀⠠⠨⠈⠀⠀⠀⠀⠀⡀⠡⠮⠠⠀⡀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⣄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠧⠈⠀⠐⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⢸⡀⠊⠀⠂⢸⠀⠀⢄⠠⠀⠀⠀⠀⢠
⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠈⠶⠠⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠐⠀⢈⠠⠠⠘⠈⠀⠀⠀⠀⠀⠈⠂⠀⡆
⠀⠰⠰⠀⠀⠴⡆⣠⣄⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠯⠭⢭⠭⠭⠭⠀⠀⠀⠀⠚⠁⡈⠑⠂⠀⠀⠀⠠⠁⠄
⠀⠀⠀⠀⠀⢆⣘⡠⠄⠀⠀⠐⠁⠐⠀⠀⠀⠈⠀⠀⠈⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠀⠄⠅⠐⠀⢀
⠀⡰⠒⠢⠀⠀⠀⠀⠀⠀⢀⡦⡄⠀⣶⠄⢠⡴⢦⡀⢰⠀⠀⢀⡴⢶⡄⠀⡴⠶⡄⠀⣦⠀⣴⠀⠀⠀⠀⠠⠁⠀⠈⡆⠒
⠀⠀⢀⠀⠆⠆⠸⠆⠃⠀⢸⡷⣇⠀⣿⠀⣿⠀⢸⡇⢸⠀⠀⢸⡇⠀⣿⢸⡇⠀⣦⠄⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠘⠷⠋⠐⠛⠂⠘⠶⠚⠁⠘⠓⠒⠈⠳⠞⠃⠈⠳⠶⠋⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡠⠀⠀⠀⠀⠀⠀⣼⣃⣘⠿⠃⡋⠀⠀⠀⢀⠁⠀⠂
⠀⠘⠃⠀⠁⠀⠸⠂⠀⠀⠀⠐⠇⠀⠀⠀⢄⠁⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠀⡀⡄⠀⠀⢀⡠⠔⠀⣠⠀⠀⠀⠁⠀⠀⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠐⠅⠀⠀⠉⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠃⠂⠇⠁⠑⠀⠀⠐⡂⠈⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠨⠀⠀⠀⠀⢀⢀⠂⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠄⡀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⡐⠀⠃⡀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢀⠠⠀⠀⠀⢈⠐⠁⠀⠸⣀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⡀⠀⠂⠀⠄⡐⠀⢂
⠀⠀⠀⠀⠀⡴⠁⠀⠠⠄⠀⠀⠀⠈⠸⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠁⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠹⠃⠀⡐⠄⠈⠁⠀⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢠⣶⣤⠀⠀⠀⢰⠄⡀⠀⠀⠀⠀⣀⢄⡄⠀⣶⠂⠀⠀⠀⢠⡔⣶⠆⠀⠀⡠⣾⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣾⣷⡀⠀⠀⣤⣈⡃⠐⠛⢘⣣⡌⠀⠀⠙⣦⡀⣿⣿⡍⡤⡟⠀⠀⣜⠦⣹⣿
⠀⠀⠀⠀⠀⠀⠀⢿⢘⠻⣷⣄⣤⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠂⠉⠞⠁⢤⣤⣹⣶⣧⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣰⣾⣥⣾⣤⣀⣀⠀⠀⠀⠀⠀⠠⣤⠤⠄⠀⣀⠠⢦⠐⢺⣶⣽⣿⡟⠏⡼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⢩⣄⠙⢿⣿⣏⣶⣄⠀⠀⠀⠁⠀⠀⠘⢦⣄⣶⣿⡻⠿⠟⡊⠙
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣰⠂⠉⠉⠉⣉⣉⠁⠀⠠⢤⠤⢤⠄⠀⠀⣀⣀⠘⠻⠿⠛⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢧⣴⡔⠁⠀⣴⣶⠀⢻⣦⠽⣦⡀⠀⠀⡀⢀⣴⠿⢹⡟⢐⣿⣦⠀⠀⠀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣽⣆⠀⢿⣮⡑⢪⣆⢠⣴⡑⢈⣰⣿⠀⢸⡿⣿⣇⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⠿⠟⠀⠰⣇⠄⣀⠹⣿⡗⠀⣤⣸⡆⠀⣸⠗⣿⣿⡀⠀⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⡀⢾⣉⡀⠠⢘⡀⠀⠙⣶⡟⠀⠉⠀⢼⣦⠋⠀⠀⣿⢠⣿⡿⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠃⠈⠛⢧⡤⣘⠁⢀⡼⠃⠀⠀⣀⠀⠀⠛⢆⠀⠸⠿⠚⠋⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣦⠀⠀⠀⠀⠀⣀⣁⠀⠀⠀⠸⣿⣧⣄⣀⠀⠁⣰⣦⣄⡀⠀⠀⢀⣤⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷⣀⣠⣶⣿⣿⡟⠃⠀⠀⠀⠲⠶⢤⡙⣷⠀⢸⣋⢿⣿⣦⣤⣿⣿⣟⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⢈⣀⡀⡀⠀⠁⠀⠀⣹⣲⣿⣿⢿⢿⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠟⠙⠃⣿⣿⣿⠇⠀⣶⣦⣄⠀⠀⠀⠀⠀⢀⡀⠀⠸⢿⣿⠇⠀⠀⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⡈⠋⠁⣀⠀⢸⡾⣿⣷⣤⣄⣲⣿⣛⠏⠀⣶⣀⡀⠶⠠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠾⠛⠋⠀⣼⣿⣧⠈⠙⠻⠋⠿⠩⢿⡿⠇⣀⣼⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢤⡾⠉⠈⣀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣀⣰⡿⢟⡗⣿⠁⢘⠃⠀⠐⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠃⠀⠀⢰⣍⣻⣿⣷⣦⠀⠀⢠⣾⢟⡹⣾⣻⣯⡏⠠⠎⠈⠀⢐⣄⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠋⠀⠀⠀⣰⣾⣿⡿⣿⣿⡿⠀⠀⢿⣫⣾⡷⠟⠿⣛⣧⠀⠀⠀⠀⠈⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣶⣶⣾⣿⣿⣿⡙⠳⠄⠀⢀⠀⠤⣤⣤⣤⠄⢈⣉⣿⣿⣯⣳⡶⢦⡤⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⣿⡿⣿⣿⡆⢠⣾⣭⠽⠾⠉⠉⠀⡴⠟⠻⢌⣛⣍⣭⣿⣶⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠿⠛⠁⠀⠀⠀⠀⠀⠉⠛⠿⠿⠋⠀⠉⠉⠁⠀⠉⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣼⣷⡀⠄⠀⠀⠀⣀⠀⣠⣾⣦⠀⠀⠀⢀⢤⠀⣿⡦⠀⠀⠀⠀⠀⢠⣄⠀⣠⣦⠀⠀⠀⠀⢸
⠀⠀⢀⡷⠊⣻⣿⡟⢣⡀⠀⣿⣷⡟⣿⣿⠀⠀⠀⠀⠀⠘⡏⡛⣠⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠀⠀⢀⡴
⠀⠀⢠⡤⢾⣿⣿⡛⢿⡆⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⣿⣷⠡⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠈⢿⠂
⠀⠀⠸⢡⣾⢻⣿⢉⡶⠇⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⡿⠃⠈⠁⠀⠀⢸⣿⣿⣗⣿⣿⣿⠀⠀⠀⣽⠄
⠀⠀⢸⠴⢋⣼⣟⠓⣥⠀⠀⠈⢿⣿⣿⠇⠀⠀⠀⠀⠐⢹⡋⠐⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⡆
⠀⠀⠈⢠⠴⣿⡑⢤⡄⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠁⠀⠀⠀⠈⡇
⠀⠀⠀⢁⣴⣟⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣰⣢⣤⣤⣾⠛⠂⠀⠀⠀⢸⡟⣿⡄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣿⣬⣿⠇⠀⠀⠐⠀⠀⢸⣿⣿⡇⠀⠀⣼⣿⣿⡀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⣶⣶⣠⣄⠀⠀⠀⠘⣿⣿⠖⣿⠂⠀⡀⠅⠀⠀⢸⣿⣿⠁⠀⠀⣿⣿⣿⠇⠀⠀⢀⣬⡏⢀⡾
⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠘⠳⠸⣆⠱⠱⠈⠀⠀⠀⠈⠻⠃⠀⠀⠀⠘⠿⠋⠀⠀⠀⠨⠙⢷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⢀⠀⠀⣤⣤⣤⢀⣠⣒⣢⣼⠶⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⢳⡀⠀⠀⣴⣆⠀⠀⠀⠀⠀⢠⡴⠊⠻⠋⠛⢰⠋⠘⢀⠁⠈⠉⠉⠉⠛⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠐⠀⠀⠀⠀⣠⠆⠀⠐⢲⡟⣧⠀⠀⠈⢶⣌⡀⡀⣀⡤⠏⠁⠀⠀⠀⣀⠘⠀⠀⡈⠀⠀⢀⣠⣰⣾⡿⠋
⠀⠀⠀⠄⠀⠀⣤⣰⣶⣶⣶⣶⣿⠙⠛⢿⣷⡶⠿⠦⠞⢇⣀⡀⠴⠿⠻⠋⠀⠀⠀⠀⠉⠙⠙⠛⠀⠀⣀⣄⣠⣤⣼⣷⣷⣾⡿⠾⠿⠿⠿⠟⠉
⠀⠀⠀⣀⣿⣶⠿⠛⣫⣾⣿⣿⣀⡀⠀⠀⣿⠶⠦⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⠆⠴⠶⣦⣌⡁⠀⠀⠙⠉⠹⡏
⠀⠀⠸⣾⣿⣁⠀⢠⣿⠟⡉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠯⠽⢜⣦⣅⣀⠁⢈⣷⣄⠀⠀⠁
⠀⠀⠀⠙⠉⠛⠛⠛⠁⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣶⣿⣿⣿⣿⣤⣔⣛⣙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣐⡆⠀⠀⠈⠙⠇⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠽⠿⠿⠿⠿⠛⠁
⠀⢀⠀⣠⣤⠀⠀⢀⠀⡄⠈⠈⠁⠀⠀⠀⢀⡄⠀⠀⢠⣔⠻⣿
⠀⠀⠁⠋⠉⠁⠉⠛⢋⠀⠀⠀⠀⠠⠀⠉⠘⠁⠀⡹⠊⣿⡒⠻⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠄⠀⣠⠼⢃⡀⠀⢲⣤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠒⠚⠋⠀⠀⠁⠀⢭⣿⣴⡆⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣄⣦⡖⣷⡀⣠⠇⢀
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠆⣀⣤⢴⣦⠾⢷⠷⠾⢿⢛⡗⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⣀⠀⠘⣺⠯⣿⣽⠟⣈⡿⠿⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠐⠋⠟⠢⠜⠆⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠛⠛⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⣶⣶⢤⡀⠀⠀⠀⠀⣀⢰⣶⣦⠀⠀⢴⡆⢠⣀⠀⠀⠀⠀⠀⢀⣠⡄⣶⡄⠀⣶⣶⠤⣀⠀⠀⠀⠀⠀⢀⡠⢴⣶
⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠘⣧⠻⠿⣐⡄⢀⠀⣰⡸⠷⢣⡟⠀⠀⠹⡖⠋⠑⠐⢄⠀⢀⠴⠀⠋⠛⢾
⠀⠘⢷⡤⠄⠀⣾⡶⠀⠠⣴⠟⠁⠀⠀⠀⠘⢳⡖⠋⠙⣿⣟⠋⠓⣶⡟⠁⠀⠀⠀⠣⡤⠤⡤⠤⣷⡯⠤⡤⠤⡤⠃
⠀⢠⡇⠀⠀⢀⢻⠓⠀⠀⠀⣧⠀⠀⠀⠀⠀⢿⣷⠘⠺⠹⠹⠚⢁⣿⡏⠀⠀⠀⠀⢸⠁⠀⠀⢀⣽⢇⠀⠀⠀⢸⡄
⠀⠀⠻⢶⠶⠃⠀⠀⠳⢴⠶⠋⠀⠀⠀⠀⠀⠉⡿⠿⠛⠀⠈⠻⠿⣏⠁⠀⠀⠀⠀⠘⠶⣤⣤⠈⠘⠀⠠⣤⡤⠞
⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠰⣶⣶⡄⡀⠀⠀⠀⠀⠀⣴⣶⣶⠀⠀⢦⡀⠄⡶⠤⡀⠀⠀⡠⠴⣖⠤⣠⠆⠀⠸⣿⡟⢭⡗⣄⠀⢀⡔⢻⡽⣻⣿
⠀⢹⣇⠀⡉⢢⠀⢀⠞⡁⠀⣿⠇⠀⠀⠀⠹⣷⣄⢀⣈⣿⣯⣀⡀⣰⣿⠃⠀⠀⠀⠹⣿⣆⢹⡿⣿⡿⣿⢉⣼⡿⠃
⠀⠈⣿⠖⠶⠶⢾⡷⠶⠖⢶⡟⠀⠀⠀⠀⠀⢹⠋⠁⠤⢾⢧⠄⠉⢻⠃⠀⠀⠀⠀⠀⢨⣿⡀⢠⣿⣧⠀⣸⣿
⠀⠀⣿⡀⠀⠀⠻⠇⠐⠀⣠⡇⠀⠀⠀⠀⠀⠈⢣⣀⡠⠀⠠⣄⣠⠏⠀⠀⠀⠀⠀⠀⠈⢿⣿⣯⠀⠨⣿⣿⠏
⠀⠀⠈⠛⠛⠊⠀⠀⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈