Wedding cake Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣌⣁⢴⡶⢊⣡⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⣿⢛⡗⣺⠛⢙⣽⡏⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡏⣿⢬⠛⣾⣕⡸⣽⠧⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣌⠙⠋⠀⠀⢙⠚⢋⣰⣗⠠⠿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⡟⡟⡟⡟⡶⢾⢻⢳⢨⢹⢳⢲⡞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⡇⡇⡇⡇⣿⢾⢸⢸⢸⢸⢸⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢇⡇⣇⠗⠀⠓⣾⢸⢸⠸⠊⠘⢸⣇⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡿⡗⢸⣽⢀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠈⢁⣰⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⢰⢣⢩⢼⢶⣿⣿⣿⡿⡿⡿⡿⡿⡟⡟⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢸⢸⢸⢸⡼⠋⠹⣿⣿⡇⡇⡇⡇⡷⠑⡇⡇⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠾⠼⠜⠊⠀⠀⠀⠘⠻⣇⣇⠧⠋⠀⠀⠈⠣⠿⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠁⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠙⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⢤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⠦⣀⠄⠈⠉⠉⠉⠉⠀⠀⣀⠠⠂⠁⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠑⠒⠴⠂⠊⠈⠉⠀⢀⠀⢠⡉⣅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⢱⣍⣼⣿⣮⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡈⠓⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣐⣬⠵⠟⣇⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠓⠲⠤⠤⣭⣉⣉⣉⣉⣉⣉⣭⠤⠤⠖⠚⠭⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⣗⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣝⢻⣆⠂⡠⣤⠀⠀⠀⠀⠀⠀⠀⡀⢠⠞⠂⠀⠀⠦⣾⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣳⣿⡋⠤⢉⡵⡄⢠⠤⠆⠀⠋⠁⠁⠀⠀⠀⠁⣙⣧⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡟⢩⣽⠏⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣽⡿⠏⠙⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣬⣁⠀⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠀⢀⣀⣤⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠎⢉⠙⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠶⠶⠒⠒⠋⣉⠭⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠑⠆⠦⠄⡀⠀⡀⠀⢀⠀⣀⣠⠀⠤⠒⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡎⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠐⣮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠳⣄⠀⠈⠙⠒⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠄⠀⠀⠁⠀⠀⣴⢏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡉⠂⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣬⣉⣒⡒⠒⠲⠶⠶⠦⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⡏⠉⠉⢙⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣾⠥⠀⠀⠀⠀⠈⣓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡵⠋⠀⠀⠀⠀⠀⠀⠀⠈⠱⡌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠬⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⢬⣉⣉⡛⠛⠛⢛⣉⣉⡥⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣮⣩⡼⢿⣉⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⢿⠭⡏⣿⢹⡏⣿⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣇⡼⠛⣇⣿⣸⠓⡟⢓⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣄⣁⣀⣀⣠⣉⡥⠾⢷⡘⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⣏⣿⣿⣿⢹⢹⢹⢸⢸⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⡇⣗⣿⠟⣿⢸⢸⢸⢸⠸⢼⢸⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠁⣧⠟⠉⠀⠈⠛⠼⠼⠈⠀⠈⠺⠘⢡⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣜⡙⣨⡻⢶⣤⣴⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⡿⡯⢿⡏⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⣏⣿⡯⠋⠀⠈⠳⣿⣿⣿⣿⠟⠁⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⡈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⣁⣠
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠙⠛⠻⠿⠿⢷⣶⣶⣶⣶⣶⣶⣶⣶⡾⠿⠿⠿⠛⠋⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣰⠷⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣛⠂⣠⠀⠀⠠⣁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⠈⣉⣉⠲⠔⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣱⢆⠘⠩⣵⡙⠎⠃⠑⠾⣘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⣻⠘⠂⢉⡉⢀⣠⠾⠃⠀⣉⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠛⠛⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠉⠁⠀⠀⠁⠘⡆⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣢⣄⡐⠠⠤⢤⣀⣀⣬⠤⠄⠒⠾⣏⣁⡦⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠘⠧⠚⠦⠐⠶⠐⠛⠬⠑⠖⠁⠈⡀⠔⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⠀⠀⠒⠒⠲⠶⠶⠖⢒⣒⡀⠈⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠃⠀⠀⠀⠀⠠⣷⡶⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢿⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⢶⠋⢠⠀⣬⠀⡴⠻⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣯⣵⠾⠀⢾⣦⡄⠀⠐⠠⢤⠘⠁⠀⢀⠛⢳⠘⣿⢼⠟⠿⡟⢩⡷⠴⣖⡜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠈⠚⡀⢨⣿⣄⡀⣈⣀⡠⠀⠰⠎⠠⠁⢙⠻⡑⠦⣿⣿⢞⣛⣋⡡⠤⠖⠊⠁⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠴⠿⠋⠀⠀⠉⠉⠐⠒⠒⠒⠲⠆⠛⠺⡏⠉⠉⠉⠉⠀⠀⠀⣀⣠⠤⠂⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⡄⠀⠀⠀⠀⡲⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⠉⢸⡇⠀⠀⠀⠀⠉⠀⠈⠁⠰⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠽⣟⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠇⠲⢄⣀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠉⠀⡄⠀⠀⠀⠀⠸⡇⢠⠀⠀⢱⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠳⢄⣀⠈⡉⠓⠒⠶⠤⢤⠠⣤⡤⢤⡤⠴⠶⡶⢤⡀⣈⣁⣀⣤⣾⣶⡃⠛⢠⡄⡠⢀⡐⠋⠀⢀⡜⣯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠠⣉⠓⢷⣈⡀⣄⠀⠀⠚⡉⠧⠀⠐⢹⠆⣘⠈⣟⣋⡙⢻⣟⣋⣵⠊⡶⢥⢫⡷⠋⠀⣀⠴⣪⠞⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠨⠥⠹⠶⢆⣉⣡⡤⢤⣤⣤⣤⠼⢾⣏⣯⣛⡾⠋⠑⠿⠴⠛⢛⣉⢤⣲⣯⢶⣛⣥⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠽⣛⠒⠶⠶⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣠⣠⣭⣉⣉⣰⣦⣶⢶⣶⢟⣛⣻⡿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠠⠄⠀⠠⠌⢬⣉⣩⠍⠉⠋⠭⠩⠽⠦⠖⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⣀⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠛⣿⠐⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠒⠲⠦⠀⠀⠀⠰⠖⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠭⡭⢭⡉⣙⡒⣒⣰⠶⡟⠋⠉⣉⣉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⠀⠉⠁⠀⠀⠳⠋⠛⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠾⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠷⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠒⣲⡖⠒⠒⠒⠒⢒⡒⠒⠒⠒⣲⣶⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣰⡿⡟⣆⠀⠀⡰⢻⡿⡄⠀⣰⢻⢿⢧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⣺⠝⠁⠹⣌⠳⠞⣡⠃⠱⡙⠴⣣⠇⠈⠣⣽⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢍⣁⣉⣑⣒⠒⠒⠂⠠⠤⠤⠤⠤⠤⠤⠄⠒⠒⢒⣒⣊⣉⣀⡩⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢢⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡠⠚⠁⠈⢳⡀⠀⠀⠀⡠⠲⡀⠀⠀⠀⠀⣀⣀⡤⠊⠀⠈⢧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⠳⣄⡠⠎⠀⠀⢱⡀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⢾
⠀⠀⠀⠀⠀⣀⡠⠤⠤⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠎⠀⠀⠀⠀⠀⠀⠀⢀⡾⠤⢤⣀⡀
⠀⠀⠀⠀⡾⠦⠤⣄⣀⣀⣉⠉⠙⠛⠓⠒⠒⠒⠒⠲⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠛⠋⣉⣉⣀⣀⠤⠤⢿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⡭⣍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡩⡴⢆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡼⢱⡌⢧⠀⠀⠀⠀⠀⠀⢠⠞⣭⢣⠀⠀⠀⠀⠀⡸⢱⢻⠸⡄⠀⠀⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡴⢁⡏⢧⠈⢣⡀⠀⠀⠀⣠⠋⢠⢻⠀⢧⠀⠀⠀⣰⠁⡟⠈⡆⠳⡀⠀⠀⢸
⠀⠀⠀⠀⣯⣤⣤⣤⠴⠊⣠⠎⠀⠈⢆⠀⠑⢤⣠⠞⠁⢠⠏⠀⢧⠈⢧⡀⡴⠁⡸⠁⠀⠘⢆⠙⠲⠒⣻
⠀⠀⠀⠀⢟⠲⠤⠤⠖⠋⠁⠀⠀⠀⠈⠳⣄⡀⢀⣀⡴⠃⠀⠀⠈⠣⣀⠉⣀⠔⠁⠀⠀⠀⠀⠉⠒⠉⢹
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠒⠒⠛⠙⠦⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠞⠉⠛⠒⠒
⠻⠭⢽⣒⣒⣒⡲⠶⠤⠬⠭⠭⠭⠭⠭⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠭⠭⠭⠭⠭⠭⠤⠤⠶⢒⣒⣒⣒⣯⠭
⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠙⠁⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣾⣿⣷⣿⣿⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢠⣤⠀⠀⠀⢠⣤⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠶⠾⠛⠙⠻⠶⠾⠛⠙⠻⠶⠾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⡿⢶⣶⣶⡶⣶⠶⣶⣶⡶⡶⣶⢶⣶⣾⡷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣤⠀⠀⠀⠀⣠⡀⠀⠀⠀⢠⡄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢶⣤⣴⠾⠛⢷⣦⣴⡾⠟⠿⣶⣤⣶⠟⠻⢶⣤⣴⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠉⠻⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢠⣶⡄⠀⠀⠀⣴⣦⠀⠀⠀⢠⣶⡄⠀⠀⠀⣴⣦⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠶⠾⠛⠉⠻⠷⠶⠟⠋⠙⠻⠶⠾⠛⠉⠛⠷⠶⠟⠋⠙⠻⠶⠾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠈⠀⠂⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢺⠉⠉⠉⡸⣤⡼⡖⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢺⣅⠀⠈⠒⠲⠶⠳⣄⠠⠊⣠⣾⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠈⠪⣝⣒⣒⣒⣒⣒⣒⣊⡩⠔⠃⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠠⠀⠈⠁⠈⠄⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡷⠤⠲⢄⠀⠀⣀⡠⢀⠀⠀⠀⡰⠋⠉⢹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠗⢄⡠⢬⣉⠉⠀⠀⠀⠀⢈⣉⣀⣀⠔⣟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡇⠀⡑⢼⠓⠄⣀⣙⣦⣤⣖⣉⠩⠥⠚⡧⠔⡁⠄⡟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡇⠀⠂⠈⠳⢄⡠⠤⠞⠶⢧⣀⣀⣀⡰⠃⠈⠀⡀⣇⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⢀⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⣀⠔⠁⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡢⣄⠀⠀⠉⠓⠓⠴⠄⢠⣄⣠⠤⠤⠔⠒⠉⠀⠀⠀⡠⢞⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢝⡲⠤⢄⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⡠⠤⢒⡩⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠥⠭⠭⠭⠭⠭⠤⠐⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⡀⡆⢀⣰⢀⢀⢀⠀⡀⠀⠀⢀⡀⢀⠀⢰⢂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠱⠿⠘⠹⠡⡧⡇⠾⠞⠞⢸⠼⢰⠇⠀⢏⠈⢏⠼⠼⠆⠾⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠙⣦⡴⠋⠉⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠸⣿⡆⠀⢈⣀⡀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⢿⣿⠀⢈⣿⣿⣄⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⣼⣿⣀⠀⢛⣽⣿⣷⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠠⣞⡛⣿⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠀⠈⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⣿⣧⠀⢀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠶⠤⣭⣭⣬⣭⣭⣤⡤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢠⡶⠦⠀⠀⡴⠖⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⣸⠇⠘⠦⡼⠶⠷⢤⣤⠟⠟⠓⢻⠀⠀⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠁⠀⠀⠀⠈⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠛⠳⠶⠦⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⠛⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⠟⣓⠀⠀⠀⠀⠀⠈⠉⣉⠉⠉⠉⠉⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠙⠛⣄⠀⠀⠀⠀⠀⢼⣫⣅⠀⠀⠀⠀⠀⠀⢿⣱⡄⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠂⠀⠀⠀⠀⠀⠘⠿⣦⣤⣤⡶⠋⠉⠉⠹⢶⣦⣤⣤⡴⠂⠀⠙⠶⠖⠃⠀⠀⠀⠀⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠋⠀⠀⠀⠀⠀⠀⣀⣤⠟⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡈⠉⠛⠓⠒⠶⠦⠤⢤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠄⠀⠀⢤⣤⠤⠶⠒⠋⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⢹⠃⠈⢧⡀⠀⠀⠀⠀⠸⣧⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠦⠀⠀⠀⠀⠀⠀⠀⠀⢿⣵⣥⠀⠀⢀⣠⠞⡿
⠀⠀⣀⣤⠶⠖⠛⠋⠀⠀⠈⢿⣶⣤⣤⣤⠞⠁⠈⢶⣄⠀⠀⠀⠀⠀⠀⣀⣼⠟⠟⠲⣄⠀⠀⠀⠀⠀⣀⠄⠀⠈⠻⣿⠿⠃⠀⠇⠀⠀⠀⢀⣀
⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠙⠿⢷⣶⣶⣶⡿⠟⠁⠀⠀⠀⠈⠻⢶⣶⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠈⠛⠿⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⢶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⣶⠶⠶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢶⣶⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⠴⠶⠶⠚⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠒⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⢀⣤⡀⠀⣀⣦⡀⠀⣠⣄⠀⢀⣴⣄⠀⢀⣤⡀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⡟⠭⢝⣽⡳⠭⢟⣽⠫⠽⣻⣟⠯⠝⣿⡛⠭⢟⣽⡫⠭⢻⣿⢤⣀⣀
⠀⠀⠀⡠⢞⡭⠗⠒⠒⣏⠻⠭⠝⠋⠫⠭⠟⠙⠯⠭⠋⠙⠭⠽⠋⠻⠭⠝⠉⠫⠭⠟⣹⠛⠚⠾⠭⣛⢦
⠀⠀⠀⢳⣈⡓⠒⠦⠤⠬⠭⣓⣒⣖⣦⣤⣤⣤⣤⣤⣤⣤⡤⡤⣤⣤⣤⣐⣒⣒⣊⡩⡥⠤⠤⠖⣚⣡⠼
⠀⠀⠀⢸⡇⠉⠉⠑⠒⠒⠒⠢⠶⠤⠤⢥⢭⡭⡬⣬⢥⢭⡭⣭⣭⢥⢥⠥⠬⠤⠤⠒⠒⠚⠒⠉⠉⢸
⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⢠⡿⣦⣄⣀⣀⢴⣶⢄⣀⣀⣠⣶⣤⣀⣀⣠⣴⣦⣀⣀⣀⢴⣶⢄⣀⣀⡤⣶⡤⣀⣀⡠⣼⢿
⠀⠀⠀⢀⣷⣌⠓⠒⠚⣩⣌⡙⠒⠒⢋⣤⣉⠒⠒⢈⠡⣈⠐⠒⠚⣩⣌⡙⠒⠒⢋⣤⣉⠒⠒⢊⣡⣿
⠀⣠⠤⢺⡇⠛⠭⠹⠭⠛⠙⠫⠭⠭⠝⠉⠻⠭⠩⠽⠋⠛⠭⠭⠭⠛⠙⠫⠭⠭⠝⠉⠻⠭⠫⠽⠋⢸⠲⠤⡄
⡞⠡⣎⣙⢧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠜⢉⣳⠈
⢹⠲⠤⣨⣉⣙⠙⠛⠿⠧⠶⡲⣖⢶⣲⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣔⣒⣒⣲⣖⣶⠶⠭⠯⠛⠞⢘⣉⡥⠴⠲
⢸⠀⠀⠀⠀⠈⠉⠓⠓⠒⠖⠦⠴⠤⠤⠬⡠⢅⠭⡨⣄⠥⡨⣄⢅⡬⡠⠤⠤⠴⠤⠶⠒⠒⠓⠉⠉⠀⠀⠀⠈
⢸
⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣼
⢸⡿⢗⡦⠤⠤⢖⡾⠿⣲⠤⠤⢴⡺⠿⣗⠦⠤⠤⣖⠿⢷⡢⠤⠤⢴⡺⠿⣲⠤⠤⠤⣲⠿⢗⡦⠤⠤⢖⡿⣻
⢸⠻⣖⠬⠍⠩⢥⡺⢕⡠⠍⠉⠭⣶⠿⣢⠭⠉⠭⢔⡪⣷⠬⠍⠩⠥⣺⢵⡢⠭⠉⠭⢐⠿⣶⠬⠍⠩⢥⡺⢛
⢸⡀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠀⢨
⠀⠉⠑⠒⠦⠤⠤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠤⠔⠒⠉⠁
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠶⡲⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠀⠁⠈⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡆⢌⡙⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠦⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⠶⢌⠩⡘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠀⡌⢳⠀⠦⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⠤⠤⠤⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐

⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠄⣂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡀⡠⢀⣰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⢉
⠀⠀⠀⠀⠀⠀⠀⠐⢲⠚⠛⠛⠛⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠛⠃⣠⠦⠃⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢨
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢦⣀⣀⣴⣆⣿⠄⠀⠀⠀⠀⠀⠀⢀⠆⣴⣂⠀⡀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠈⠈⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⢸⠛⠁⠀⠉⠁⠠⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣮⣏⡓
⠀⠀⠀⠀⠀⠀⢀⡣⢀⣠⠀⠀⠀⢀⣈⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡭⣭⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠑⠋⠛⠛⠛⠙⠋⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣐⣸⡉⠄⠱⠂
⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠃⠍⠙⠁⠂
⠀⠀⠀⠀⠀⠘⠻⠶⢦⣤⣤⣤⣤⣤⠴⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢄⣀⠀⠀⢀⣀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠱⡴⠁⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠤⠵⠦⠤⠤⠤⠾⣥⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠑⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⡠⠒⠲⡆⠀⠀⢀⡤⢤⡀⠀⠀⣠⠔⠪⣧⠀⠀⠈⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠙⠋⠀⠀⠀⠱⢄⡠⠎⠀⠀⠙⠒⢊⠁⢠⡀⠸⡆⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡀⠀⠀⢀⡀⠀⡐⢀⣀⣀⣀⣄⣀⣀⣀⡀⠀⠣⠔⢱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠥⠶⠒⠚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠉⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⢠⣯⠔⠒⢄⡀⠀⠀⢀⣀⠔⠦⡀⠀⠀⠀⢠⡾⠒⠒⢤⡀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢾⠁⠀⠀⠀⠙⠢⠤⠛⠉⠀⠀⠙⢄⠀⠀⣸⠁⠀⠀⠀⠙⠒⠚⠋⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠳⡀⠀⡟⠀⠀⠛⠀⠀⠃⠀⠀⠀⠈⠁⠀⢀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣉⣁⣀⡤⠤⠤⠤⠤⠼⠧⠤⠤⠦⠤⠤⣤⣀⣀⣤⠀⠀⠀⠀⣤⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⠶⠤⣄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⢀⡤⠒⠒⢷⡄⠀⠀⠀⠀⢀⠤⠤⣠⡀⠀⠀⠀⢀⡴⠒⠲⡜⣦⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠙⠳⠤⠔⠋⠀⠀⠀⠀⢇⠀⠀⠀⡰⠋⠀⠀⠙⠻⠦⡠⠔⠋⠀⠀⠀⠘⡾⡆⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠁⠀⠐⠂⠀⠀⠀⠀⠒⠀⠀⠶⠀⠀⡧⠇⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠀⣀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠁⠀⠀⡄⠀⠙⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠔⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢠⡐⠉⢶⢡⠗⣩⣄⢵⡜⡶⢉⣤⠐⠤⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡰⡑⢏⡁⣤⣍⡘⢻⡇⢸⠈⠥⣼⠆⡿⢛⣭⢶⣀⢤⣿⠠⣵⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣁⣼⠋⠻⣭⣧⢀⣹⠞⣧⡝⠒⠖⡥⢊⡛⠮⣤⠼⣷⢟⡁⠝⡿⢈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢮⠂⣽⣷⣢⡑⠈⢀⣲⠟⠉⡝⠻⡖⡙⢯⠤⣀⣁⡔⢧⢙⣄⣾⢦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠉⢈⠛⡙⣧⣞⣉⣋⠴⠖⠳⠼⠧⠱⣗⡬⠟⠢⠤⡤⣱⡟⠊⠁⠀⢸⠃⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⣷⠘⢦⣸⣏⠉⠀⠀⠀⠀⠀⠀⠀⢜⡩⠤⠶⡀⢢⡀⠀⠀⠀⠀⠀⡼⠇⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠱⠶⠋⣤⠙⠢⣌⠂⠐⠶⠶⠆⡦⠘⣧⢰⣤⠻⠔⡗⠂⠀⠀⣈⠔⢁⡸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⡲⠶⡏⠛⣧⢡⣀⠀⠍⠩⠵⢒⣩⠟⢺⠷⢚⣋⠔⢰⠋⠖⠄⠈⠀⢀⢸⣧⣛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢳⡦⣴⠉⢄⠙⠒⠁⠈⠉⠁⠉⠁⠀⡘⣉⠀⠋⠈⠢⡈⠓⠈⠁⠀⠈⠁⣸⠛⣏⣬⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣬⡏⣁⡌⠛⡲⡿⢥⣀⣀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⣚⣁⡠⣤⡴⠞⡅⠙⢇⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠼⢧⣹⡇⢤⡂⣀⡞⢉⣩⢉⢿⢋⠻⡋⠹⢏⡻⡝⢉⢴⢁⣲⠠⣙⡐⡴⠬⠟⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠉⠈⠑⠘⠛⠢⠛⠤⠟⠢⠟⠣⠝⠣⠜⠛⠢⠛⠒⠛⠑⠋⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢙⡇⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⡏⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⣈⢃⠠⢀⠀⠀⠖⣩⠗⠤⣍⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⣄⡴⠂⢤⠀⠀⢀⡀⢀⡜⠁⣐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⢎⠨⣡⣙⡆⠑⠤⣀⡭⠲⢸⣶⠖⠚⠂⣹⣅⠒⠒⠒⠒⠒⣲⢦⡄⢙⠄⢸⠛⣆⠋⠁⡀⠔⠋⢠⣏⠜⣢⠐⢄⢀
⠀⠀⠀⠀⠀⢀⠐⡡⢝⡬⠟⠒⠉⠈⡗⠀⣰⡎⣺⡖⠲⡉⣡⠚⢰⡃⢤⣉⣁⠠⠔⢊⠴⡋⣳⠤⠉⠃⢹⡧⡘⢗⠀⢂⠠⠂⡍⠂⠛⢌⣾⡃⢔⡆
⠀⠀⠀⠀⠀⢰⡔⠺⠋⠀⠀⠀⠀⠀⣇⠀⠀⣀⢁⡤⠊⠀⠈⢦⡁⡇⡘⠑⠒⠂⠉⠁⠈⠰⠋⠠⠋⠉⠁⠤⣠⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢁⠿⠆⣥
⠀⠀⠀⠀⠀⠀⣧⢱⠋⡱⠐⣀⠠⠀⠈⠛⠶⣬⣅⣀⡀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠲⠊⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠁⠀⢀⡠⠠⡄⣙⡐⣼⡇
⠀⠀⠀⠀⠀⠀⡇⠈⠓⠳⠬⢇⠰⡏⣙⠆⣠⡄⠀⡉⡍⠛⡛⠓⠒⠒⠒⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⣙⠋⢉⠁⡠⣀⠐⢤⢉⢺⠠⣱⣑⡶⠚⠋
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⠚⠣⠽⣄⣲⡅⢬⡇⢌⡖⢈⢤⢈⠢⣈⢢⡌⠢⠌⡁⡎⢌⣲⠠⣳⠠⣽⡔⣬⠌⠾⠂⠛⠑⠉
⠀⠀⠀⠀⢀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠚⠑⠀⠛⠤⠟⠣⠝⠣⠽⠓⠘⠛⠒⠋⠒⠋⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀
⠀⠀⠀⡠⠃⠐⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣰⠃⠀⠑
⠀⠀⠀⢂⠀⢠⡌⢦⡑⠀⢠⡀⢠⡈⠠⣉⡑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡮⠤⡄⣎⠢⠄⠐⢂⡵⠃⣄⢀⡎
⠀⠀⠀⢎⢰⡞⣧⠀⠈⠀⠤⢒⣏⠴⠶⠉⣽⠃⣁⠤⣀⡀⠀⠀⠀⠀⠀⡰⠒⣋⣠⡆⢠⠀⠀⠀⠀⠀⢀⣀⠠⣠⣏⢠⣁⠖⢟⡆⡖⠤⠀⠁⠀⡰⣯⠀⠃
⠀⠀⠀⠀⠈⢩⡇⠈⠀⠂⠈⡍⢪⡷⠌⠘⢫⡴⠫⡗⠤⣈⡉⠉⢀⡰⠆⠘⠏⠑⢤⣹⠂⢹⢍⡉⠉⢉⣀⠠⠖⠁⡴⢄⡉⠴⢪⠰⣆⠲⡄⠀⠁
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢄⢁⠨⠃⠲⡅⠸⣁⢺⣒⡀⠒⣊⡡⠔⠣⣇⠀⢠⣼⢃⠉⡸⢷⠬⣑⠒⠤⠀⣂⠠⡆⣸⠇⢡⠗⠀⣉⢃⡄⠀⠀⠠⡆
⠀⠀⠀⠀⠀⢈⡚⡁⡄⠄⢀⠈⠀⠀⠀⠀⠈⠦⣀⠄⠀⠀⠀⠀⠀⠰⢡⠟⠁⡤⠒⢏⠘⠳⠈⡄⠀⠈⠉⠀⠀⠀⠈⢀⡠⠆⠀⠀⠀⠈⠀⢤⠊⢃⣼⡄
⠀⠀⠀⠀⠀⠀⠳⠼⡅⡎⢠⡶⠐⡀⢤⣀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣦⠐⠀⠀⠀⠀⠀⠑⠲⠀⠀⠀⠀⠀⠀⢀⠀⢨⣁⠠⣠⠒⢎⡜⣃⢢⡯⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠳⠬⢇⣲⡇⢬⠊⣱⠆⡁⡄⠂⡤⠐⣀⠀⣄⠐⢀⡔⢂⡀⠂⡠⠒⣤⠒⢤⢊⠲⠠⣙⠠⣹⣐⣾⠤⠿⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠓⠚⠓⠜⠳⠬⠷⢌⠾⣈⡾⢌⡾⠇⡹⠦⣱⠷⠬⠾⠬⠟⠲⠛⠓⠋⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟



⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠀⠈⠉⠉⠉⠉⠉⠉⠉⠐⠒⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡠⠊⠉⠹⠄⠀⠀⡰⠊⠢⣀⠀⣀⠔⠉⠑⢻⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠑⠀⠊⠀⠠⠀⠀⠀⠀⠀⠰⠀⠈⠆⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢤⠀⠀⠀⣠⣀⢠⠀⠀⠤⠠⠰⠆⠤⠀⠀⠀⠀⢑⡈⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⡀
⠀⠀⠀⠀⠀⢠⠔⠂⠁⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⣅⠀⠀⠀⢀⣞⠔⠀⠒⢄⠀⠀⠀⣠⡠⠒⠢⡀⠀⠀⠀⢰⠟⠂⠐⠠⡀⠀⠀⣠⠾⠁
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠘⠇⠀⢀⠀⠀⠑⠂⠚⠋⠀⠀⠀⠐⢄⠀⢀⡌⠀⠀⠀⠀⠈⠐⠊⠉⢠
⠀⠀⠀⠀⠀⢠⠐⢄⠀⢰⠀⠀⠘⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⢀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠉⣁⣀⠀⠀⠤⠤⠤⠤⠿⠤⠤⠤⠦⠤⠈⣤⣀⣀⣦⠀⠀⠀⠀⢴⠀⠀⠆
⠀⠀⠀⡀⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠂⠤
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⢳⣄⠀⠀⠀⠀⢀⠤⠤⠐⣦⠀⠀⠀⠀⠀⢀⠄⢀⢀⠀⠀⠀⠀⠀⡠⠠⠤⡐⣄⠀⠀⠀⠀⠈⡂
⠀⠀⠀⢰⠘⠷⣀⡠⠔⠁⠀⠀⠀⠰⠀⠀⠀⠀⡠⠁⠀⠀⠩⢷⠄⢀⠠⠊⠀⠀⠀⠘⠸⡆⠀⠀⠀⢠⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⠚⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⢴⠄⠀⢀⣷⠀⠀⠀⢾
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡄
⠀⠀⠀⠀⠁⠀⠐⠗⠀⠀⣀⠀⠐⠗⠀⠀⠒⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠌⠀⠃
⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⣥⣠⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡞⢁⡴⠶⠿⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⢸⡅⠀⠀⠈⠉⠁⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠙⠳⠦⣤⣄⣀⣻⣤⡤⠶⠋⠉⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢈⠍⠉⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠊⠉⠈⠲⢄⡀⠀⠀⢀⣀⠔⠋⠈⠂⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠻⢦⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣴⠟⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁⠀⠀⠀⠉⠙⠓⠶⠶⠤⠤⠤⠤⠶⠶⠖⠛⠉⠀⠀⠀⠈⣗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣢⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠉⠀⠀⠀⡮⠂⠀⠀⠰⣲⠀⠀⠀⠸⠕⠀⠉⠁⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠐⠠⠠⠠⠒⠑⢄⡀⠀⠀⠀⠀⠀⢀⠤⠰⢀⢀⣀⠤⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠏⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡁⠀⠀⠀⠀⠈⠙⠳⠶⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠖⠛⠉⠀⠀⠀⠀⠸⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠳⡀⣦⣄⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣠⡤⠶⠠⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢤⠀⠀⠀⠉⠉⠃⠜⠛⠒⠒⠒⠲⠍⠖⠒⠒⠛⠛⠂⠊⠉⠀⢠⡄⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠊⠈⠢⢄⡀⠀⣀⡤⠛⣄⠀⠀⠀⠀⠀⠀⢀⠴⠁⢄⢀⠀⣀⠄⠊⠈⠲⠤⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠑⠒⠐⠐⠒⠋⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣀⠈⠛⠲⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠚⠉⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣄⣀⣈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⣀⣀⣠⡤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⣶⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠒⢿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⡒⠒⠒⠒⠾⠧⠀⠀⠀⠀⠒⠒⠒⢒⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡴⢤⠽⢏⢉⣝⡛⣛⣛⠿⣿⠛⠉⢉⣡⣀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠸⠇⠈⠉⠀⠀⠀⠹⠞⠙⠋⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢼⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠤⠤⢭⣭⠭⠭⣍⠭⣭⠭⠩⠭⡭⢭⣭⣭⣤⠤⠒⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⡟⣄⠀⠀⠀⢀⣴⣷⡄⠀⠀⢰⣿⣿⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣀⢞⡿⠻⡘⢦⡀⣠⠎⡸⢏⠻⡀⢠⢃⡏⢺⡸⣄⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠹⠶⠋⠀⠀⠱⢤⣉⣡⠞⠁⠈⢧⡙⣡⠞⠀⠀⠙⠲⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠢⠤⠤⣤⣉⣉⣉⣉⣒⣒⡒⠒⠒⠒⠒⠒⠒⣒⣒⣒⣉⣉⣉⣉⣤⠤⠤⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢲⡄⠀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⣠⠞⠁⠈⢳⡀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢷⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠒⠋⠀⠀⠀⠀⠀⠻⡄⠀⣠⠞⠁⠀⢣⠀⠀⠀⢀⠞⠉⠉⠀⠀⠀⠀⠈⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⢦⡀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠋⠉⠉⠉⠓⠦⠤⢤⣄⣀⣀⣀⣀⣀⣀⢀⣀⡀⠀⣈⡉⢁⣀⣀⣀⣀⣀⣠⡤⠤⠴⠋⠉⠉⠙⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠒⠒⠒⠦⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀⣀⣁⣉⣁⣉⣁⣀⡤⡤⠤⠤⠤⠤⠤⠤⡄⠒⠒⠒⠊⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⢖⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⢀⠖⣯⢻⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡏⣸⡄⢳⡀⠀⠀⠀⠀⠀⠀⣰⠃⣶⠘⡆⠀⠀⠀⠀⠀⡜⢰⠚⡆⢳⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠏⣰⠃⢣⠀⠹⣄⠀⠀⠀⢀⡼⠁⢰⠙⡇⠹⡄⠀⠀⠀⡼⠁⡎⠀⢻⡈⢧⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢄⣀⣠⠴⠞⢁⡴⠃⠀⠈⢧⠀⠈⠳⣄⡴⠋⠀⣠⠇⠀⢻⡄⠙⣄⢀⡾⠁⡸⠁⠀⠈⠳⣄⠙⠒⠛⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠲⠤⠤⠤⠒⠉⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⣀⡴⠃⠀⠀⠀⠹⣄⠈⠉⣀⡼⠁⠀⠀⠀⠀⠈⠑⠒⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⡠⠤⠤⠔⠺⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠒⠢⠤⠤⢄⡀
⠀⠀⠀⠀⢸⣓⡒⠤⣤⣤⢄⣈⣉⣙⣒⣒⣲⡒⠒⠂⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠄⠒⠒⠒⣒⣒⣋⣉⣉⣁⣤⢤⣤⠤⢒⣚⡇
⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠒⠰⠤⠭⠭⠭⠭⠭⠭⠍⢀⣀⣀⣀⣀⣀⣐⣒⣒⣒⣒⣂⣀⣀⣀⣀⣀⡀⠩⠭⠭⠭⠭⠭⠭⠤⠔⠒⠒⠒⠊⠉⠉


⣷⣷⣿⣿⣶⣷⣾⣷⣷⣿⣶⣶⣿⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣾⣿⣿⣶⣶⣿⣾⣿⣷⣿⣿⣿⣾⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣻⣟⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣦⡰⣄⡀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⢦⣀⣰⡿⠿⠛⠋⣠⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⣶⡇⢀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠄⠀⢀⡤⠔⢉⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡄⠀⠀⠀⠐⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠹⣇⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⣒⠄⣀⠀⣤⣼⣾⠿⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⢠⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠿⠿⠋⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠟⠋⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⣿⢡⣄⣀⣀⣀⣤⣶⠾⠿⠿⠛⠓⠆⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠙⠃⠻⠿⠿⠿⢟⣁⣀⣠⠔⠂⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠭⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⣠⣴⠒⠛⢿⡿⠋⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠰⠶⠒⠛⠁⠐⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⢀⣤⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢹⠵⠋⢉
⠀⠀⢨⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠃⠐⠃⠀⠚⠀⠘⠃⠀⠘⠀⠀⠀⢀
⠀⠀⢠⣄⣤⢤⡀⢀⣤⣤⣶⣾⡿⠿⠿⡟⢻⣿⣿⡑⠀⠠⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⣆⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⡄⣀⣷⣻⠇
⠀⠀⠀⠛⢿⣼⡇⢸⣿⣿⣪⣁⠰⠒⠊⠈⠉⠿⠿⡦⠴⠆⠀⠁⠚⠳⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠛⠉⠙⠷⠞⠂⠲⠞⠉⠙⠋⠀⢰⡏⣿⠇
⠀⠀⠀⠀⠀⠈⠁⠈⠽⠽⢿⣿⣷⣄⣙⡂⠀⠀⠀⠀⠀⠀⠠⠔⣈⣤⣾⣿⠿⢃⡀⠀⠀⠀⠀⠀⣿⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠶⣤⣉⡙⠛⠀⠀⠀⠀⠐⠂⠀⠈⢙⣿⡿⢋⣡⣶⡿⠁⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢤⣄⡀⠀⠀⠤⢊⣡⡶⣿⢯⡿⠃⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣲⣿⠍⢴⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⢀⠔⢩⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠉⠀⠀⠉⠱⣇⠀⢘⣦⠔⣁⠧⠴⠤⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠖⠚⠀⢀⠀⠀⠀⠀⢨⠀⠀⠀⠉⠳⣄⠀⠀⠀⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⢀⡤⠀⠄⠂⠀⠉⢀⠀⠀⢸⠀⠐⠀⠀⠀⡒⠀⠀⠀⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢆⠀⠀⠠⠂⠀⠀⢀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠂⠀⠀⠀⠀⣀⣠⢀⠀⠉⠂⠀⠂⠃⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠂⠀⠀⠈⠉⠁⠀⠀⠀⠀⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣦⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄⠀⠀⠀⠀⠒⠊⠁⠀⠀⠀⠀⠂⠱⠀⠀⠀⢠⠞⠢⣴⣴⣿⡷⣶⣿⣿⣿⣿⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⡧⠂⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠠⡄⠀⠀⠀⢘⠃⢠⠈⠿⣿⣷⣿⣿⣿⣿⣿⠀⡞⠁⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣻⣄⠀⠀⠀⠀⠀⠀⠀⠚⠛⠂⠀⠀⠁⠈⠀⠁⠀⠛⢦⣶⣿⣿⣿⣿⣿⣿⡏⠸⣆⠀⢀⠂⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠙⠻⢯⣝⢳⣷⣴⣶⣶⣶⣶⣤⣤⣤⣜⣿⣧⣠⣤⣿⣿⣷⣿⣿⡿⠟⠛⠻⡿⠛⠒⢄⠈⠑⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠋⠁⠁⠈⠑⣄⣀⡀⠈⠟⢛⠻⠿⠧⡽⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠄⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠈⠑⢀⠀⠀⠘⢄⡀⠀⣀⠔⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣤⣤⠤⠞⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⠀⠀⠀⠀⠀⣠⠤⠼⣤⠏⠘⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠘⡄⠀⢈⠑⠢⠀⡠⡊⠀⠀⠀⠙⣴⣰⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠨⡙⣢⠵⠄⠀⠑⣀⠞⡄⠀⠀⠀⢀⠁⡘⣧⠤⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢼⠁⢈⡴⠆⠰⢇⠀⠐⠀⠘⢀⢤⡒⠂⢼⣿⡙⡷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠙⡿⠾⠷⠿⡄⠉⠈⢫⣌⣁⣨⣥⣾⡀⠀⠁⣼⠇⠉⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠫⣁⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠈⠓⠂⠉⠓⠚⠛⠓⠊⠀⠀⠀⠊⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠱⡀⠀⡼⠛⠉⠙⢳⠀⠀⢠⡶⠛⠉⠙⢏⠁⢠⠞⠉⠙⣦⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠿⣖⡲⡇⢰⣇⡀⠀⠀⡈⠀⠀⢾⡀⠀⠀⠀⣸⠀⢸⣤⣤⠶⣿⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⡇⢸⠟⠋⠉⠉⠙⠤⠤⠞⠙⠛⠋⠩⢽⠀⢸⠒⠚⠉⣻⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⡁⠂⣴⢁⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣀⣤⣴⣿⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡀⠀⠀⠑⠶⠞⠋⠉⠉⠉⠀⠀⠒⠒⠂⠈⠉⢻⣻⡄⠘⢯⠟⠛⠉⠉⣾⡟⠻⢶⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠫⠻⢕⣿⣿⠦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⣧⣀⣸⣀⣠⡴⠟⣻⢿⣧⠀⠀⠛⢦
⠀⠀⠀⠀⠀⡞⠁⠀⢰⣿⣟⠊⡠⠤⣌⠉⠑⠒⠒⠲⠶⠦⠤⠤⠤⠤⠶⠶⠖⠛⠛⠛⠉⣉⡠⠴⠊⠕⢋⡟⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⣇⠀⠀⠈⠻⣙⡏⠀⠀⣸⡄⠀⠈⠓⠀⠀⠀⠀⠀⠤⠤⠤⠐⠒⠒⠊⢉⣽⠒⠲⣤⣠⠶⠋⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠘⢮⡢⡀⠀⠈⠙⠒⢶⠟⠛⠲⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡈⠉⠁⢰⣷⡀⠀⢀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠺⢕⣠⢄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠁⠀⠀⠀⠙⠲⢛⣛⣭⣔⣮⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⣲⣤⣄⣤⡤⠤⠤⠤⠤⠤⠀⢴⣀⣀⣀⣤⡬⠽⠟⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡏⢀⣶⣿⣿⣷⣶⣾⣿⣷⡄⠀⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⢶⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⠀⠈⠻⣿⣿⣿⣿⢿⣿⣄⢀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⢀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⠻⠟⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠉⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠹⣿⣿⣿⣿⡿⠛⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠻⣿⣿⣿⣦⣿⣿⣿⡇
⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⠁
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣈⣻⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⣹⣿⠂
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⢛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣀⣠⣤⣶⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⢿⡿⠛⢿⣷⢶⣾⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢀⣿⣌⠈⠃⠀⠘⠋⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⠟⠛⢿⢻⣿⣿⡿⣿⣿⡏⠉⣇⠀⠀⠀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡏⠁⡟⠀⠀⠀⠘⢸⣧⣿⡇⠃⠁⣇⠀⠘⢷⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠋⠉⣿⠀⢰⡇⠀⠀⠀⠀⡾⣼⣿⢇⣀⣤⡝⣆⣴⢿⣿⣿⣿⣄⠀⠀⠀⠀⢀⣴⠿⣤⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⢀⣿⡇⣼⣧⣤⣤⣤⣾⣾⡿⠋⠈⠉⠙⣿⢙⣵⣿⣯⣽⣿⠛⣷⡀⠀⠀⢾⡅⢾⣯⣞⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠑⠃⠀⠀⣾⠛⠛⢻⣿⣿⢿⠛⠿⣍⠀⠀⠀⠀⠀⣿⡿⠋⠉⢹⣿⡟⡄⢸⣧⡀⠀⠈⠻⣦⠙⢉⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣠⣾⠋⠀⠀⣸⣿⢿⠈⠳⠀⠀⠀⠀⢀⣀⣀⡁⠀⠀⣠⣿⣿⣿⣵⣿⠙⢿⣄⡀⣰⠟⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⣼⠟⠁⠀⠀⢺⣟⢿⡘⣇⠀⡀⠀⢤⣄⣀⣈⣩⣤⣴⣾⣿⠿⣿⣿⣿⡇⣠⡾⣻⣿⠋⠀⢸⣿
⠀⠀⠀⠀⠀⢀⣠⡿⠁⢀⡾⠃⠀⠀⠀⣠⣾⣿⣿⣷⣌⣦⡙⢶⣤⣄⣉⣭⣿⣿⡿⠛⠁⠀⠈⢻⣿⣾⣯⣾⣿⠃⠀⢀⣿⡿
⠀⠀⠀⠀⢰⡟⠁⠀⠀⠈⣷⠀⠀⠀⣼⡟⠁⡠⠾⣛⣿⣿⣿⣿⢿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠘⠿⣿⣿⣏⠙⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠈⢿⣶⣄⠚⢁⣸⠆⠀⢰⣿⣄⠀⠐⠛⠉⠴⠾⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣡⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠈⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠉⠛⠿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠈⠻⣿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢷⡀⠀⣀⣀⠈⠙⢿⣿⣿⣿⣍⣉⣉⣁⣀⣀⣀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⣴⣶⣿⣿⣋⣵⡿⡿⣿⠋⠙⠻⢶⣾⡟⢿⣿⣿⣿⣿⣿⣿⣯⡵⠂⠀⣸⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣾⣾⣷⣦⡀⢤⣈⣿⣧⠘⣿⣿⣿⣿⣿⣿⠋⠀⠀⢠⣿⣿
⠀⠀⠀⠀⢿⣿⠟⠁⠀⠀⠀⠈⠹⣿⣿⣿⣷⣿⣿⣿⣷⣼⣿⣿⣿⣿⡃⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⡟⠙⣿⠀⠘⢉⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⣿⠀⠺⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣟⣿⣇⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⣿⣿⠿⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣁⣉⣭⣭⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠞⣆⣨⣇⠉⢅⠆⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠔⠂⠀⢳⡒⣶⣄⠀⣐⠇⢆⡄⠕⠉⢐⣾⣎⡦⠚⠘⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⣇⠆⠀⢸⣿⡇⠃⢘⣯⠦⣫⡴⠔⠀⠀⠙⢯⣟⠿⣾⣞⡇
⠀⠀⠀⠀⠀⠀⠀⢌⣀⣿⠀⡰⠈⡀⠀⣜⠙⣧⣄⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠉⠡⢹⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⠿⠷⠾⠷⠛⠑⠲⠿⠻⠿⣿⠀⠈⠀⠀⠀⠀⠀⠀⡈⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡦⠀⠀⠀⠀⠀⠀⢰⠄⠀⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣾⠉⠐⢢⠄⠀⢠⡴⢸⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀⠀⠀⢀⠀⠀⠤⠌⢸⣿⡟⠀⡄⠙⡳⠴⡊⠋⢈⣼⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠺⡀⠀⠎⣿⠃⢹⢀⠀⠀⠀⠀⢀⠈⡟⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡈⢸⡇⠀⠀⠀⢸⡀⠋⢻⡏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⢁⠉⠀⠄⠀⢠⠀⠁⠐⣾⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠌⠋⠀⠀⠀⠀⠀⠀⠀⠀⣻⣮⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⡀⠀⢠⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢇⠀⠀⠀⠀⠀⠀⢨⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠐⠀⠀⡂⠈⠈⢀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠧⢀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⡴⣶⠦⠶⠶⣴⠖⠶⠿⠶⣶⡶⢶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡟⠛
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣭⣿⣿⣽⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠨⠿⠿⠿⠿⠿⠏⠈⠙⠷⠋⠻⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⣒⡶⠖⢒⣒⡒⠲⢖⣒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⠶⠤⠈⠩⠴⠶⠾⠥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⣠⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠘⠦⠬⠥⠤⢽⣿⠿⠦⠽⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡄⠀⠀⠀⢠⡶⠒⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⣃⣒⡊⣁⣛⣉⣙⣋⣓⣊⠀⠀⠀⠀⠀⢠⣿⣻⠛⠋⠛⠁⠀⠀⠀⣶⣶⣶⣼⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠾⠿⠿⠿⠿⠿⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣿⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠉⠉⠉⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠞⣡⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣄⢳⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣟⡁⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡼⣦⣙⢷⣶⣤⣴⣶⠾⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣷⣤⣤⣤⣴⢟⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣙⣿⣿⣿⣿⡟⠋⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠤⣉⡳⢿⣿⣿⣿⣿⣵⠏⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⠞⢋⣠⣤⡶⠚⠋⠉⠉⠉⠉⠉⠉⠙⠓⠦⣝⣻⣿⣿⣿⣿⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣷⡞⢛⣽⠟⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠟⢻⡇⠀⠀⠞⠛⠛⣻⣿⡶⠂⠀⠀⣤⣴⡶⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⡙⢿⣿⠀⠀⠐⠚⠛⠛⣹⠟⠁⠀⠀⢸⣿⣿⣦⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢻⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠁⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡌⠁⠈⠁⢀⠀⠀⠀⣀⣤⢿⣤⣄⣀⣀⣽⡆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡇⠀⣿⡄⠀⠘⣩⣶⣾⣿⣿⣿⣿⣿⣿⣦⣜⣿⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠘⠿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣿⣿⣿⡇⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⣿⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢸⣿⡿⡟⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠀⠹⣷⣇⠀⠈⠻⣿⣶⣧⣴⣶⣶⣾⣵⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⢟⣫⣾⣧⠀⠀⠈⠛⢶⣄⠀⢹⣿⣿⣿⣿⠏⣿⡿⠁⣿⡿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋⢁⣰⣯⡷⠿⢿⣦⡀⠀⠀⠀⠉⠳⢾⣿⣿⣿⣿⡼⠛⠁⠀⣿⠛⠷⢭⣙⠻⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠹⣿⣦⣄⡀⠀⠀⠀⢻⣿⣿⠏⠀⠀⣠⡼⠃⠀⠀⢀⡀⠉⠒⠈⠙⠻⣷⣦⣀
⠀⠀⠀⠀⠀⣠⣶⡿⢛⡵⠊⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠿⠿⠿⠿⠿⠿⠿⠶⠦⠴⠿⠿⠶⠖⠛⠉⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⢠⣾⣿⣯⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⡴⣲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢠⡀⠀⠀⢀⠙⢿⣧
⠀⠀⢀⣾⡿⢀⣾⠋⠙⠲⢦⡀⠀⠀⠀⠀⠀⢠⠀⠀⣿⣷⡀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠘⣧⠀⣰⣿⠀⠘⣿⣇
⠀⠀⣼⣿⠃⣼⠇⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢸⡆⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⣿⣤⣿⡏⡀⠀⠘⢿⣧⡀
⠀⢰⣿⡟⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢸⣿⡄⠸⣿⣿⣇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡄⠸⣿⡇⣿⣿⣿⢣⠇⠀⡄⠈⢿⣷⡀
⠀⣼⣿⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣦⣿⣷⡀⣿⣿⣿⡄⠠⣯⣭⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣿⣿⣼⣿⣿⡿⠀⢠⡇⡀⠈⢿⣿⡀
⠀⣿⣿⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⣆⢹⣿⣿⣷⣼⣿⣿⣷⡀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⣠⣿⠃⣧⠀⢸⣿⣷
⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⣀⣿⣧⣼⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣤⣤⣶⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠿⣿⣿⣿⠟⠉⠙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⣠⣿⣿⠏⠀⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠐⢿⡿⠃⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠟⠋⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠋⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣩⣭⣽⣿⣿⣿⣷⣦⡀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⢀⣠⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠟⠙⠻⣿⣿⣦⡀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⣠⣾⣿⣿⣿⣄⣤⣴⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡇⠙⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡇⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⠿⠟⠋⠉
⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⠿⠟⠋⠉
⠀⠀⢀⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⡿⠟⠋⠉
⠀⢀⣾⣿⡿⠻⣿⣿⣦⡀⢀⣠⣤⣶⣿⣿⣿⡿⠟⠋⠉
⢀⣾⣿⡿⠁⠀⣈⣿⣿⣿⣿⣿⡿⠟⠛⠉
⣼⣿⣿⣵⣶⣿⣿⡿⠿⠛⠉
⠻⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⣀⠄⡤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠐⡠⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⢂⡄
⠀⠀⠀⠀⡔⣕⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⠁⠀⠀⢀⠄⠀⠀⠉⠱⣀⢀
⠀⠀⠀⠀⠁⣍⡑⠒⠂⠀⠤⠔⠒⢂⡀⠀⠀⠀⠀⣁⠁⠈⠀⠘⠀⠀⠆⠀⠑⣉⡔⠀⠀⠘⣦⡠⢄⣠⠴⠋⠀⠀⠀⣄
⠀⠀⠀⠀⢸⡡⣄⣀⣀⣀⡭⠭⣉⣋⠌⢳⠀⠀⢀⢘⠫⠀⠀⠀⠀⠐⢒⣋⠭⠚⠈⠀⠀⠀⢹⣼⣨⣭⣧⣠⢄⣠⢄⡜
⠀⠀⠀⠈⠒⠭⠤⠀⣀⣀⣐⡚⠩⠅⠂⠁⠀⠀⠈⠒⠤⢄⣀⣀⣀⣀⣀⡠⠤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠘⣏⣭⣭⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠀⠠⠒⠀⢲⡀⠀⠀⠀⠀⠀⠀⢀⠤⣢⣄⠀⢀
⠀⠀⠀⠄⠓⠤⢤⣄⣀⣀⣀⣀⡀⠼⠖⠵⠀⠀⠀⠀⠀⠀⠌⠀⠈⠈⠀⠀⠀⢸⠃⠀⠀⠀⢀⡠⠒⡁⠀⠂⠂⠢⢊⡩⠈⡀
⠀⠀⠄⠰⢤⣤⣀⣀⣀⡀⢀⣀⣀⣠⣴⠆⠀⠀⠀⣠⣖⠀⠠⠀⠀⡀⠀⢄⡰⠃⠀⠀⢀⠴⠈⠐⢠⠀⠁⠐⡁⣀⠄⠀⠄
⠀⠀⢃⠀⠀⡉⠈⠉⠙⠋⠉⠉⢉⡨⠋⠀⢀⠀⠀⠈⢍⠳⣇⠁⠐⣀⡴⠋⠀⠀⠀⠈⠪⢧⡄⠘⠊⠐⠀⠀⡂
⠀⠀⠀⠑⠊⣀⣤⡅⢐⡒⠒⠚⠉⠀⠀⠀⠁⠀⠀⠀⠀⠐⢹⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⠉⠉⠄⡀
⠀⠀⠀⠀⠀⠳⠄⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢱⠀⠀⠀⠀⠐⠉⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢀⡀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠎⡅⠤⠤⠄⠒⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠈⣀⣤⠞⡫⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢻⣆⢀⣼⢢⡇
⠀⠀⠀⠀⢀⠀⠈⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠟⠋⠁⠈⠱⡘⢄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠻⡿⠃⢈⠇
⠀⠀⠀⡠⢰⢄⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⠳⡄⠀⠀⠀⢠⠚⠀⠀⡏⣽⠶⠾⢸
⠀⠀⠈⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣳⠷⠁⠀⠀⠀⠀⠁⠚⠂⠀⠐⠒⠊⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣿⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠿⠟⠛⠛⠛⠉⢉⣉⣽⡿⠿⠿⣿⣭⣭⣭⡙⠛⠛⠛⠿⠷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠁⠀⢀⣀⡀⠀⢰⡿⠟⠛⣿⠋⠀⠀⠀⠈⢻⡇⠙⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⢰⡟⠛⠻⣶⣾⠇⠀⠘⣿⡀⠀⠀⠀⠀⣸⡟⠀⠿⠷⠿⠿⠿⠿⠷⢶⣦⣄⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⢀⡀⠙⢿⣦⣤⣤⣾⠟⠁⠀⠀⣴⡾⢶⣦⣄⠀⠀⠈⣻⡇⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠀⠀⢰⣿⡋⠁⠀⠀⢸⡟⢻⣧⠀⠀⢩⣿⠶⣶⣄⠀⠀⢻⣆⠀⠈⠙⠻⠶⠾⠟⠁⠀⢀⣴⣿⠁
⠀⠀⠀⠀⠀⣿⡋⠛⢷⣦⣄⣀⠀⠀⠙⠻⢷⣶⣤⣾⠇⠀⢿⣆⣀⣼⡇⠀⠈⠻⢷⣤⣼⡿⠀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠋⣻⡇
⠀⠀⠀⠀⠀⢹⣷⡀⠀⠈⠙⠛⠿⠷⣶⣦⣤⣤⣀⣀⣀⣀⡈⠙⠛⠋⠀⠀⠀⢀⣀⣈⣁⣀⣤⣤⣶⡶⠾⠟⠋⠉⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⣿⡋⠻⢷⣤⣀⡀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣴⠿⠋⣿⡇
⠀⠀⠀⠀⠀⢻⣧⡀⠀⠈⠙⠻⠷⣶⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⡾⠿⠛⠉⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⣿⡟⠻⣶⣤⣀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⢀⣠⣴⡾⠛⣿⡆
⠀⠀⠀⠀⠀⢻⣧⡀⠀⠉⠛⠻⠷⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⡶⠿⠛⠋⠁⠀⣠⣿⠃
⠀⠀⠀⠀⠀⣾⠟⠿⣦⣄⡀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⣀⣤⡾⠟⢿⡆
⠀⠀⠀⠀⠀⢻⣧⠀⠈⠙⠛⠿⢶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⠾⠟⠛⠉⠀⢠⣿⠃
⠀⠀⠀⠀⠀⣾⠟⢿⣦⣄⡀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⠿⠷⠶⠶⠶⠾⠿⠿⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⣀⣤⣾⠟⢿⡆
⠀⠀⠀⢀⣤⣿⣇⠀⠈⠙⠻⠷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⡶⠿⠛⠉⠀⢀⣿⣧⣄
⠀⣠⡾⠟⠉⣼⡟⢷⣤⣀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⢀⣠⣴⠟⣿⡌⠙⠿⣦⡀
⣼⡟⠀⠀⠀⢿⣇⠀⠉⠛⠿⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠾⠟⠋⠁⢀⣿⠇⠀⠀⠘⣿
⢿⣆⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠾⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⢀⣾
⠈⠻⣷⣄⡀⠀⠀⠀⠉⠛⠷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠟⠋⠁⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠈⠙⠻⢷⣦⣤⣀⡀⠀⠀⠈⠉⠙⠛⠻⠿⠷⠶⣶⣶⣶⣶⣦⣤⣤⣶⣶⣶⣶⡶⠶⠿⠿⠛⠛⠉⠉⠀⠀⠀⣀⣠⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⡶⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠤⡀⠀⠀⠀⠀⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⢦⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⡤⠠⡤⠤⠞⠀⠀⠀⠀⠀⡆⢀⡰⠿⠿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡤⡴⡁⠂⠀⠀⠀⢀⠀⢠⠀⠀⠂⠀⠀⠸⠎⠄⠀⠀⠀⠙⠂⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠤⢒⠠⠄⠁⢀⡀⠁⠈⡀⠀⠒⢰⠦⠂⢄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠈⠐⠂⡀⠁⠂⠀⠀⠀⠀⠀⡜⢦⣜⣾⠀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠀⠀⠂⡀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⣷⣄⠀⠀⠀⠀⠀⠁⠂⠂⠄⠀⠑⣀⠀⢨⣻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠿⢿⡛⠛⢿⣿⣿⣿⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣘⡒⠾⡉⠹⢿⣿⣶⡯⣶⣦⢠⡄⢠⣿⣠⣶⣿⣟⡒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣰⢆⡰⠚⠋⠀⠀⠉⠋⡀⣿⣿⢸⡇⢸⣿⣿⣿⠟⠿⠍⠀⠁⢀⣀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣴⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣬⣷⠆⠀⠀⠀⠀⠀⠉⠉⠉⠇⡎⠁⠠⠈⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠉⠻⠯⠫⣿⣿⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⠆⠀⠀⠀⠠⠠⣌⣀⣀⠀⢁⣀⠀⣀⠀⠀⠠⠀⠀⠀⠨⠤⠤⠤⠄⠤⠤⠤⠤⠤⣤⣾⣿⠃⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡯⢀⣄⢀⠀⢠⠀⣿⣿⣿⣤⣾⣽⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⣤⣀⣉⣿⣿⣦⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠧⣬⠃⡬⠞⠁⠀⣿⣿⣿⣿⣏⣿⣿⡷⢦⣠⡞⠀⠂⢠⣤⠀⠀⠀⠌⡿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣁⠈⣹⣿⣿⠿⠻⢿⢿⠇⠀⠈⠀⠀⠒⠉⠁⠈⠀⠀⠀⢐⣟⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠚⠛⠒⠒⠒⠚⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⢆⣢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠲⣦⣿⣷⣯⣿⣻⡧⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣿⢹⣿⣿⣿⢟⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁⠈⢻⢿⠵⢫⡿⢛⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⠟⠛⠛⠛⠿⢿⣷⣦⣀⣴⣾⣿⡿⠿⠿⢿⣷⣶⣤⣀⠀⢸⡇⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿⣻⣍⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣼⣧
⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡆
⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣃⡀⠀⢀⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣛⣘⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⣤⣤⣤⣤⣶⣾⡿⠟⠋⠻⢿⣷⣶⣶⣶⣶⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢲⡲⠄⣠⡀⠀⠀⡀⠀⠀⡀⠀⣼⡗⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡠⠴⠀⠈⠿⣶⣯⣇⠀⢀⣄⣫⣜⣁⣸⣙⣧⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠄⢹⣿⠀⠛⠛⢋⠉⠉⠀⠈⡉⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢀⠈⠁⠂⠁⢸⣿⠀⠀⠀⠀⠈⡄⠀⠂⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠜⢸⠀⠀⢀⠸⣿⡆⠀⠀⠀⣸⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠧⠄⠠⠠⠠⠄⣿⢠⠀⠀⠀⡏⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⢹⡏⠀⠀⠀⡟⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢹⠀⢸⡆⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠁⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠⡎⢱⠶⠀⠀⠀⠀⠀⢀⡤⠄⡁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⢱⠖⠀⠀⠀⠀⠀⢀⡤⠀⣀⠀⠀⠀⠉⠉
⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠚⠒⠉⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠚⠂⠉⠀⠴⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠈⠉
⢢⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠀⡀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡖⣠⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠀⡀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢴
⠧⠀⠀⠀⠀⠀⠀⠀⠀⢐⡈⠛⣕⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⢐⠈⡟⣕⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠟⠳⠇⠀⠀⠀⠀⠀⠀⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠟⠻⠇⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠻⠓⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡆⢰⡏⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠆⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⢄⠀⠁⢰⢢⠆⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠈⠀⠀⠀⠀⢂⠈⠇⢰⣦⠆⠀⠀⠀⠛⠉
⢀⣴⠤⠶⣤⡴⠖⠛⣛⠋⠙⡟⠉⠁⠀⠀⠀⠀⠉⠀⠎⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣴⡦⠶⣤⡴⠖⠛⣛⠋⠛⠟⠉⠁⠀⠀⠀⠀⠉⠀⠎⠀⠀⠀⠀⠀⠒
⣀⣄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠈⠷⠚⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠑⠈⠶⠊⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡌⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⡎⢱⠶⠀⠀⠀⠀⠀⢀⡤⠐⣀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢱⠶⠀⠀⠀⠀⠀⢀⡤⠀⡂⠀⠀⠀⠉⠉⠁
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠚⠂⠉⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠎⠀⠀⢀⡀⡀⠀⠀⠘⠛⠛⠋⠀⢶⣤⠆⢘⠂⠀⣀⠀⠀⠀⠀⠠⣀⣘⣯⣷⣾⠉⠁⠀⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣵⠄⠐⠒⠠⠁⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠈⠀⠀⠀⠈⠋
⣲⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠀⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠚⠚⠛⠛⢐⠀⠀⠀⠀⠀⠀⠀⣤⢀⣦⡀⣄⠀⠀⢠⣤⣀⢠⠀⠀⠀⠀⠂⠀⢀⣶⣾
⠧⠀⠀⠀⠀⠀⠀⠀⠀⢐⠈⠻⣕⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠾⠶⣶⠶⠄⡀⠀⠀⠀⠀⠒⠈⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠏⠳⠇⠀⠀⠀⠀⠀⠀⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠏⠻⠇⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠲⠓⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡆⢰⡏⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡆⢐⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⢀⠀⠃⢰⣢⠆⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⢂⠀⠃⢰⠦⠆⠀⠀⠀⠛⠉
⣀⣠⡤⠶⣤⡴⠖⠛⣛⠋⠙⠟⠉⠁⠀⠀⠀⠀⠉⠀⠎⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣠⠤⠶⣤⡴⠖⠛⡛⠋⠙⠟⠉⠁⠀⠀⠀⠀⠉⠀⠎⠀⠀⠀⠀⠀⠂
⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠨⠷⠚⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠑⠨⠴⠚⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠄⠀⠀⠀⠀⠀⢠⡖⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⡤⠊⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡴⢤⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡦⣼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⢾⣯⣉⣉⣤⡷⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⣠⠤⣀⣀⣨⡷⣈⠑⢴⣇⣀⣀⠤⣄⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⣀⡇⠀⠀⠀⠀⣷⣈⠑⢺⠀⠀⠀⠀⠸⣀⡀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠸⠭⢅⡀⠀⠀⠀⠀⠘⠢⠴⠋⠀⠀⠀⠀⢀⡨⠭⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢣⡀⠀⠀⠀⣇⢀⣀⠤⣄⠀⠀⠀⠀⣠⠤⣀⡀⣸⠀⠀⠀⣀⡜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⣧⠀⠈⠓⠦⣄⣈⡉⠀⠀⠈⠳⠤⠤⠞⠁⠀⠀⢉⣁⣠⠴⠚⠁⠀⣼⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣟⠲⠚⢲⡀⠀⠈⠉⠉⠛⠒⠒⠒⠒⠒⠒⠛⠉⠉⠀⠀⢀⡖⠓⠖⣻⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢻⡀⠀⠀⢳⣀⡴⠛⣦⠀⠀⢀⡤⢤⡀⠀⠀⣴⠛⢦⣀⡞⠀⠀⢀⡟⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠳⣄⡀⠀⠈⠀⠀⠘⠦⠤⠞⠀⠀⠳⠤⠴⠃⠀⠀⠁⠀⢀⣠⠞⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⢸⠙⢦⡀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠚⠋⠀⠀⠀⠀⢀⡴⠋⡇
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⡠⠖⠋⠀⠀⣇
⠀⠀⠀⠀⠀⠀⣿⣄⠀⣀⣀⠀⠈⠙⠓⠲⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠖⠚⠋⠁⠀⣀⣀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⢿⠈⠉⠁⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠈⠉⠁⡿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⢧⠀⢠⠞⠛⢦⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⡴⠛⠳⡀⠀⡼⠁⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈⠙⠉⠀⠀⠘⣇⠀⠀⣠⠟⠀⠀⠻⣄⠀⠀⣸⠃⠀⠀⠉⠋⠁⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⠑⢦⠀⠀⠀⢀⡜⢳⡀⠀⠀⠀⣠⠛⣆⠀⠘⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⡀⠀⢧⣀⡼⠀⠀⠀⢘⣦⡴⠃⠀⠀⠀⢇⣀⠞⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⣀⠀⠀⠀⠀⢰⣿⣻⡄⠀⠀⣿⢶⢻⠀⠀⠀⢸⢯⣽⠂
⠀⠀⠀⠀⠀⠀⢀⣤⣠⠁⠀⠈⡹⠀⠀⠀⠀⠀⣗⠔⡇⠀⠀⢸⠢⣽⠀⠀⠀⢸⠢⣼
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⣦⡔⠁⠀⠀⠀⠀⠀⣹⣭⣽⠤⠤⠼⠵⠾⠤⣤⣄⣏⡒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢗⢦⡀⠀⠀⠀⠀⠀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣀⠀⠀⢀⣄⠀⠀⢀⡀⠀⢠⡄⠀⠑⣳⠀⠀⠀⢠⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠚⠉⠛⠒⠋⠈⠙⠛⠉⠙⠛⠁⠘⠒⢺⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⣧⡰⠖⢦⣀⣀⣀⡠⠶⢦⣀⣀⣀⡴⢤⢸⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⣁⣨⠏⠉⠑⠶⠊⠀⠀⠀⠳⠤⠞⠁⠀⠹⢤⠼⠋⠻⣏⠁⣹⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢦⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠳⣌⣀⣠⠶⣄⣀⣠⠟⢦⣀⣀⡴⠋⠣⢤⣤⠞⠳⣀⣀⣠⠦⣀⣠⠞⠙⠦⣄⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢤⡤⣤⣤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣟⣽⠭⠯⠼⠤⠷⠞⠒⠓⠚⠒⠓⠒⠒⠚⠒⠛⠺⠷⠯⠽⣭⣟⣳⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣀⡀⣀⡤⢤⣀⢀⡤⠶⢦⣀⣠⠤⢤⡀⣠⠤⢤⡀⠀⣀⣈⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠒⠋⠀⠀⠙⠁⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠙⠁⠀⠀⠙⠋⠁⠀⠙⢧⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⣀⣀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣄⣀⣀⣀⠀⣀⣀⣠⠞⠀⡼⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⣿⣿⡯⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⡇⢹⣿⣿⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣨⣿⣿⣿⣿⠿⠟⠋⣹⣟⠿⢿⣿⣿⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⣿⡿⠁⠀⠀⠘⢿⡿⠃⠀⢹⣿⣷⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⣿⣿⣿⠏⠀⠀⠀⣀⠀⢠⡄⠀⠀⢸⣿⣿⣇⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⣻⣿⠉⣿⠀⠀⣶⢠⣤⠁⠀⠈⣁⣤⣸⣿⣿⡿⡵⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⣷⣾⣿⡄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠀⢻⣧⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⢹⣿⡿⡓⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣸⣿⣿⠀⠈⣿⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⡿⠁⠛⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠘⠃⠹⣾⣡⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⠃⠀⠀⠰⠯⠽⠇⠀⠀⠀⠀⠀⠀⠀⠀⢂⣺⣿⡁⠀⠀⠈⣿⡃⡀⠙⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣣⡅⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃⠀⠘⠂⣟⡹⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⠀⢠⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠁⠀⠀⠀⣰⣿⠀⠇⢴
⠀⠀⠀⠀⠀⠀⠈⠋⠀⣰⣿⠀⠀⠀⠀⢶⡦⢼⠃⠀⠀⣠⣶⣿⣿⣿⡟⠋⠀⣀⣶⠤⠾⣿⣁⠘⠀⠀⠀⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡻⡆⠀⠀⣸⠇⠀⢾⠀⣠⣶⣿⣿⣿⡿⠟⠀⠀⣿⠇⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠲⠆⠀⠀⠀⠾⠇⣴⡄⠀⣿⠂⠀⠙⠇⣿⣿⣿⣿⠋⠀⢀⣐⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣽⣄⣟⡀⠀⠀⠀⢹⠟⣿⠋⠀⠀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢠⠄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠻⣤⡿⠃⠀⠀⠀⠀⠀⠉⠀⠠⣶⠃
⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠘⠇⠀⠀⣤⡄⠀⠀⠀⠀⣲⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣹⣦⠄⡀⠀⢀⠈⠄⠡⠌⠤⠀⡜⠀⠀⠰⠆⠀⠀⠀⠁
⠀⢰⡄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠈⠙⠻⠾⣏⣷⣤⠴⢚⣚⡛⠒⢓⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡶⠒⠈⢉⣭⠉⠉⠑⠂⠄⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠛⠀⠀⣹⣿⣷⡀⠀⠀⢀⠰⣄⠀⠀⡠⠊⠀⡀⠀⠀⠁⠀⠀⠀⠲⠀⠀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡄⠦⡡⠢⢈⡵⣤⠀⠀⠀⠋⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⡷⠀⣀⣴⡿⠇⠸⠑⠆⢠⣄⣀⠀⠀⢠⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⢸⣿⣿⣯⠀⠀⠳⡌⠀⡱⠀⠀⠀⠀⠀⢿⣮⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢨⣿⣿⣏⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣌⣀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢀
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠀⣀⠀⡠⢸⣆⠀⠀⠀⠀⠀⠀⠀⠱⠾⣿⠃⠀⣤⠀⠀⠀⠀⠀⠀⢀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠉⠐⠘⠧⠌⠀⠀⠀⠀⠀⠀⠀⢸⠟⠃⢾⠀⠀⠀⠀⠀⠀⠘⠃⠈⠁⠈⠄
⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢿⣿⡆⠀⠀⠐⠺⣌⡃⠀⠀⠁⠄⢀⡀⠀⠀⣙⣡⠞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢾⣿⡄⠀⠀⠘⣂⡬⡇⠘⠃⠀⠀⢀⡄⠀⠀⢹⣉⢠⠇⠀⡀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⣿⡅⠰⠦⠀⣀⠻⣇⠀⠀⠀⠀⠈⠁⠀⠓⢀⡷⣎⢺⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠰⣿⣇⠀⠀⠀⠏⠱⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⡟⣡⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣏⠀⠀⠀⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠶⢦⢩⡄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣿⣿⠀⠀⠀⢋⡀⠿⡀⢶⠀⠀⠛⠀⠀⠀⠁⠀⢛⠓⢯⠸⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⣛⢻⠀⠛⠀⢠⣽⡶⡇⠀⠀⠀⠀⠀⠠⡆⠀⠀⠘⡟⢲⡄⢡⠶⠀⡦⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠘⡆⠀⠀⢀⠁⡹⢧⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⢹⣋⡾
⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠃⠀⢠⡀⣷⢡⣻⡀⠀⠙⠀⠀⠀⠀⠈⠁⠀⠀⠽⠒⢤⡇⠀⠰⠄⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⣀⣀⣸⡀⠀⠀⣘⢁⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⣸⡇
⣿⣿⣿⣿⢿⣿⣿⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣌⣤⣬⣥⣤⣯⣤⣽⣮⣤⣿⣡⣯⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⢤⣀⣀⣀⠀⠀⡞⠙⢦⣀⣤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠹⡄⠈⠙⠛⠻⣄⣸⠃⠀⠀⠀⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣀⣠⢤⡀
⠀⠀⠀⠀⣰⠒⠲⣏⠉⠀⠄⠀⠘⠲⣤⣠⡤⠞⠁⠀⠀⠀⠀⠀⣠⠖⠋⢳⠀⠀⠀⢠⡏⢀⣠⡇
⠀⠀⠀⠀⠘⢦⣤⡼⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠖⠋⠀⠀⡀⠀⠉⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠈⠣⡀⠀⠂⠀⣠⠤⢤⡀⠀⠀⠀⠀⣰⠒⠲⢤⡀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢼⣁⣀⡼⠁⠀⠀⠀⠀⠈⠲⠤⠼⠃⠈⠀⢀⣀⠀⠀⠈⠀⠀⠈⣀⡴⠋
⠀⠀⠀⠀⠀⠀⢸⠁⠄⠀⠀⠉⠀⢈⡴⠚⠛⠢⡈⠀⠀⠀⠠⠀⡠⠚⠉⠉⠙⠓⠒⠒⡖⠛⠁⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⡀⠠⠀⠀⠁⣠⠞⠀⠀⠀⠀⠈⠳⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡤⠚⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⢴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡴⠋⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡞⠁⠀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣤⣤⡖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡦⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠖⠚⠋⣽⣷⡄⠉⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡌⠢⣄⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠶⠶⣶⡶⠶⣶⡶⠶⠶⠶⠶⠶⢶⣶⣶⣶⢾⠃⠀⣿⡄⠈⠓⠦⣄⣀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡜⠁⠀⠀⠀⠹⣄⠀⠀⠀⣠⠏⠀⠀⠈⣿⠀⠀⠸⡟⣆⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣬⣷⣤⣾⣥⣤⣤⣤⣤⡇⠀⠀⠀⢳⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠈⠁⠀⠒⠀⠀⠀⠈⠂⢄⠈⠈⢹⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠘⠁⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠀⣤⠰⠂⠸⠆⠿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠿⠿⠚⠛⠛⠛⠋⢹⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⢡⠊⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠻⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡿⠿⡿⠻⠿⠿⠿⡿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣷⣾⣷⣶⣿⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⢴⠟⠻⠿⣿⣶⣤⢸⡞⣇⠀⠀⠀⠉⠁⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠒⠴⣄⣀⢙⣿⣿⣻⠏⠀⢀⡀⠄⠈⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡤⠔⢁⣠⣶⣿⣿⣿⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣆⣠⠀⢀⣀⠀⠀⠀⠀⠀⣤⣩⣰⣾⣿⣿⣿⠿⢋⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⠻⢷⣿⣿⣷⡄⢀⣀⣴⣿⣿⣿⣿⣿⡏⣠⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣻⣤⣤⣄⣉⠉⠃⠸⠿⢓⣿⡿⠟⢋⣤⣾⣿⣿⡿⠛⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⢟⡟⡹⢿⣿⣇⣰⣦⣤⣤⣷⣿⣿⣿⣿⠿⢃⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣀⠙⠚⠚⢿⣯⣻⣿⢿⣿⣿⡿⠟⣡⣵⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣮⠽⣽⣷⠶⣤⣤⣀⠈⠙⠛⣛⣉⣠⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣛⣽⡗⣭⣿⣶⣾⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢭⣇⣻⣟⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⡄⠀⠀⠀⠀⠀⣠⠞⢳⡀⠀⠀⠀⠀⠀⢀⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⣹⠀⠀⠀⠀⠸⡅⠀⢀⡇⠀⠀⠀⠀⢰⠋⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢶⡖⠃⠀⠀⠀⠀⢀⡹⠶⠋⠀⠀⠀⠀⠀⠀⢳⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣛⠋⠽⢷⠀⠀⠀⠀⢸⡛⠛⠻⡇⠀⠀⠀⠀⢸⠿⠶⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠙⠒⢿⠀⠀⠀⠀⢸⠉⠙⠺⡇⠀⠀⠀⠀⢸⠓⠢⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢹⠀⠀⠀⠀⢸⣄⠀⠀⡇⠀⠀⠀⠀⢸⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠙⠒⢾⠀⠀⠀⠀⢸⠀⠙⠲⡇⠀⠀⠀⠀⢸⠑⠢⢤⡇
⠀⠀⠀⠀⣠⠤⢤⣀⠀⠀⢀⣀⣨⣶⣄⣀⣸⣀⣤⣤⣀⢸⠦⣄⣤⠧⣤⣀⠀⠀⣸⣄⡀⢀⡇⢀⣀⣀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⡏⠀⠀⠈⢳⡞⠉⠀⠀⠉⢳⡟⠉⠀⠀⠀⠈⢿⡔⠁⠀⠀⠀⠈⢷⠋⠁⠈⠙⢿⡟⠉⠀⠀⠉⣷⠋⠁⠉⢳
⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⢀⡟
⠀⠀⠀⠀⢸⡏⠙⠒⠒⠓⠦⠤⣀⡠⠤⠲⢤⣀⡀⣀⣀⠤⠲⣄⠀⠀⠀⢀⡴⢆⡀⠀⢀⡼⠣⣄⣀⡠⠃⠙⠒⠒⠚⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣀⠀⠠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡴⠋⠉⠙⠺⣇⡴⠋⠉⠁⠉⠉⢢⣀⡴⠖⠛⠛⠙⠛⢦⠀⣠⠖⠋⠉⠉⠙⠳⣄⡤⠖⠚⠒⠶⣄⣠⠖⠚⠛⠛⠲⣄⡷⠖⠒⠲
⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠁
⠈⢳⡦⠤⢀⡠⢤⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡖⠦⣀⣀⡠⢴
⠀⢸⠄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠙⠒⠒⠒⠒⠉⠁⠈⠓⠦⢤⣤⠴⠚⠉⠙⠲⠦⠴⠞⠁⠈⠉⠓⠊⠉⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⡜⠁⠉⠈⠓⢦⣠⠶⠚⠙⠋⠒⢦⣠⠴⠚⠋⠉⠉⠙⢢⡀⡴⠚⠉⠉⠉⠉⠳⣄⡴⠊⠋⠓⠶⡄⢠⠖⠛⠛⠛⠓⣦⡴⠚⠙⠛
⢇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈⠁
⠈⠛⠲⠦⠤⠤⠦⣀⡀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⠞⠳⢄⣠⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠠⡇⠀⠂⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠲⢄⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣀⠖⠉⠀⠀⠀⠀⠢⡀⠔⠁
⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⣀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢀⠄⠀⠀⢠⡶⠮⢿⣿⡆
⠀⠀⠀⠀⠀⠈⠓⠀⠤⠀⠀⠀⠀⠀⠀⢂⣠⡀⢠⡞⠓⠀⠀⠀⠄⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠒⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠉⠀⠀⠀⠀⠉⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⢄⡀⠀⠀⠀⠀⠀⠀⡄⣀⡀⠀⠐⠀⠒⠂⠚⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠐⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡬⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠨⠛⠀⠢⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⠀⠈⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠘⡁⠀⣠⡴⠞⣉⡅⠞⣹⣷⠂⠁⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⠀⠀⠀⠀⠐⣦⡴⠚⣡⡒⣽⠏⠉⣰⡿⢉⡔⠁⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢀⠔⣠⠞⢋⣳⡄⢊⡷⠜⣠⡾⠋⣠⠞⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⢞⢁⢠⡌⢠⠟⢀⣴⡾⢋⠠⠞⠋⠀⠀⣀⠴⠀⠀⠀⠴⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣡⡆⣰⢿⢊⣩⣶⠟⠝⣁⠀⠀⠀⠀⢀⣾⣿⡇⠀⠄⠁⠀⡤⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣇⢅⣘⣥⣶⠟⠋⣠⣶⡾⠟⠀⠀⠀⢠⣾⣿⣿⡇⠀⠀⢢⢠⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢀⠀⠀⠀⠀⡄⠊⠀⠀⠀⠈⠛⠿⢟⢉⣠⣴⣾⣿⣿⣂⡀⣀⢀⣴⣿⣿⣿⠟⠀⠀⠀⠼⢿⡧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠠⡸⣷⠀⠑⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠈⠙⠻⠿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⢰⣶⣞⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠤⡆⠀⠀⠀⠠⢤⣤⣼⣿⣛⡋⡉⢥⣔⣤⡀⣠⣴⠖⠒⣢⣴⡦⠤⠤⣆⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣶⣶⣿⣀⣠⣤⣶⣿⣿⣿⣿⣿⣵⣦⣬⣭⣿⣙⣿⣷⣦⣥⣬⣽⣿⣿⣿⣯⣤⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⢴⣦⣶⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣴⠀⢲⣿⣿⢿⡺⡿⠻⠟⠙⠛⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣰⣦⣻⡿⢿⠃⠹⠂⠀⠉⠉⠀⠀⠀⣠⠖⠋⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡱⠒⠤⢤⣀⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠶⢤⣻⡇⠀⠀⢀⠀⠱⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣤⡐⢄⠉⡇⠀⠀⠀⠀⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⢠⠇⡉⠉⠽⡟⢤⣀⢂⣴⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣞⡋⠀⠀⠀⠀⣷⢲⣻⢷⣩⠏⡩⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⢈⡵⠦⢤⣠⠜⠉⣾⣱⡺⠃⣰⡴⠃⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡏⠀⠀⠀⠀⠉⠐⠃⠛⠁⡰⢫⣽⠃⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢇⠀⢓⠦⠤⢄⣀⣀⣀⡼⠁⠈⠀⡆⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠁⠀⠀⢀⡜⢸⠀⠀⠀⠀⢠⣿⣀⡠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠘⣆⢸⠀⠀⠀⠀⠸⣿⡠⠤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⣷⠄⠀⠀⠀⠘⣼⡀⠀⠀⢀⡀⢻⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠛⠋⠀⠀⠀⠀⠀⢻⣦⡀⠀⢀⡴⣿⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠑⠴⠋⡠⠃⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣸⢔⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢉⣾⣷⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠛⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⣿⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⡿⠛⠛⠉⠉⢠⣶⣿⣷⣦⠀⢠⣄⠙⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡟⠉⡉⠙⠿⠇⢸⣿⣟⠀⣿⣿⣿⣿⣿⡇⢘⣿⠀⠙⠛⠛⠛⠛⠻⠿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣇⠘⢿⣷⣦⣴⣿⣿⣿⡄⠙⢿⣿⣿⠟⠁⣼⣿⣶⡶⠞⠻⢿⣿⣶⣶⣤⡈⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠀⣀⣹⣿⣿⡟⠋⠙⢿⣶⣤⣤⠤⠴⠾⣿⣿⣿⠀⢰⣶⣤⡈⠙⠻⠟⠃⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣀⠙⠻⢿⣿⣿⣿⣧⡈⠛⠿⢿⣿⠇⢸⣧⠀⣿⣿⡏⢠⣶⣤⡈⠻⣿⡧⠈⣿⣿⣿⣷⣶⣶⣾⣿⠿⠛⢁⡀
⠀⠀⠀⠀⠀⢿⣿⣦⣄⣈⠙⠛⠿⠿⣷⣶⣤⣤⣴⣿⣿⣦⡈⠉⣀⣼⣿⣿⣿⣦⣤⣤⣴⣿⣿⡿⠿⠟⠛⢉⣀⣤⣾⣿⠇
⠀⠀⠀⠀⠀⣀⠙⠿⣿⣿⣿⣷⣶⣤⣤⣄⣈⣉⣉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⢉⣉⣉⣀⣤⣤⣴⣶⣿⣿⣿⡿⠟⢁⡀
⠀⠀⠀⠀⠀⢿⣷⣦⣄⡉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⣁⣤⣶⣿⠇
⠀⠀⠀⠀⠀⡀⠙⢿⣿⣿⣿⣶⣦⣤⣄⣈⣉⡉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⣉⣉⣀⣤⣤⣶⣾⣿⣿⣿⠟⠁⡀
⠀⠀⠀⠀⠀⢿⣷⣤⣈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢉⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣶⣤⣄⣈⣉⠉⠙⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠋⢉⣉⣀⣤⣴⣶⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⣿⣶⣄⡉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⣁⣤⣾⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣤⣈⣉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⢉⣉⣠⣤⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣿⣦⣄⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢁⣤⣾⡇
⠀⠀⠀⣀⠀⠘⠿⣿⣿⣷⣦⣤⣄⣉⡉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⣉⣁⣤⣤⣶⣿⣿⡿⠟⠀⢀⡀
⢀⣴⣾⣿⠀⣶⣤⣈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠿⠛⢉⣠⣴⡆⢸⣿⣷⣄
⣾⣿⣿⣿⣆⠘⢿⣿⣿⣶⣤⣄⣉⡉⠙⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠉⣉⣁⣤⣴⣾⣿⣿⠟⢀⣾⣿⣿⣿
⠙⢿⣿⣿⣿⣷⣄⡉⠛⠿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⣁⣴⣿⣿⣿⣿⠟
⠀⠀⠙⠻⢿⣿⣿⣿⣷⣦⣤⣈⡉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⣁⣠⣤⣶⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⡿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⢧⡴⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⢿⡿⢋⣛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⣶⣶⣿⣿⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣡⣿⣿⣿⣿⣌⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⣛⢻⣿⣿⢿⣿⡿⢠⡙⢿⡿⢡⣦⢻⣿⣿⣿⠿⠿⢟⣴⣷⣌⣛⣡⣶⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠏⣶⣭⣴⣿⡌⢿⠟⣤⣝⣣⣿⣿⣦⣴⣿⣿⠘⣿⣿⢱⣿⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠿⠟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⣛⣛⣛⣛⣛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⣛⣛⣛⣛⣛⣁⣀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⡍⢿⣿⣿⠏⣼⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣩⣍⢻⣿⡿⣡⡌⣿⣿⣿⣿⣿⣿⣿⡿⢋⡻⣿⣿⣿⣿⠏⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣌⣛⣫⣼⣿⣿⣧⣍⣩⣴⣌⢻⣿⣿⡿⢸⣿⣿⣦⣭⣶⣿⡇⢻⣿⣿⣿⣿⣿⣿⢱⣿⣷⣬⣥⣦⢹⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⢿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣜⠿⢋⢹⣿⡿⡏⢸⣿⣿⣿⣿⣿⣷⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⠾⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠻⣤⠇⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⡀⠀⠀⡀⠀⢌⡀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡀⢘⢧⠀⠃⡐⠢⡀⣋⡤⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠉⠀⠨⠌⢿⡉⣿⠓⠦⢯⣠⠶⠛⠀⠒⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠉⢙⠀⠀⠀⡀⡀⠒⠢⢿⡚⣓⣬⣤⣑⣜⠁⠈⣠⡄⠒⠀⠄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠁⠀⠀⠀⢀⠸⡄⣨⠖⢉⣉⠁⡱⠚⠋⢉⡄⢩⠈⠙⣆⠛⠤⡈⡑⠀⢂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠱⠁⠀⠀⠀⠀⠀⠀⣀⠽⠫⠀⠐⠒⢲⣷⠪⢆⡀⠈⠊⠄⠂⠘⠷⣄⠑⡠⠔⠉⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡿⡰⡄⠀⠀⠀⠀⠀⣼⠇⠑⠀⠈⢁⢀⣀⣼⣷⡒⢤⠀⣠⠀⠖⣠⣐⡾⡁⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡨⡷⢄⡀⠀⠀⠈⠺⠛⠛⠋⠉⠉⠀⠉⠀⠈⠙⠓⠚⠛⠉⠉⠁⠈⠀⠀⠀⡀⠀⠰⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣙⠂⠫⡉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⣁⣀⠀⢂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡼⡀⢢⢘⡇⢠⡖⣄⠀⢀⡠⠤⢄⣀⡀⠀⠀⠀⡩⠭⢁⣀⣠⡀⠀⢠⠎⢀⠞⢲⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠡⠑⠈⢃⠓⠞⠀⠈⠛⠁⠊⠉⠢⠉⠹⣄⠀⡜⠠⡰⠋⠙⣦⠀⠀⠁⠀⡼⠀⠀⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⢠⠀⠌⢧⠀⠀⠀⠀⠀⠀⡐⠀⢦⠀⠀⠉⠀⡐⠀⢀⠀⣨⢷⡄⠀⣼⠃⠀⠘⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⠘⠀⠁⠀⠑⢠⠀⠀⠈⠀⡇⠁⠀⢢⠀⢀⠄⢀⠐⢸⠰⠇⠀⢹⣰⡇⢐⠰⠆⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⡀⣣⡠⠄⠀⣿⠀⠀⠀⠀⡟⠀⠀⠠⢣⡇⠀⠐⠀⡘⡀⠰⡀⠀⡿⢁⠒⣀⠠⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡑⠄⠉⠒⠚⠹⣂⡈⢀⠂⢳⠀⠂⠀⡼⣇⠀⠐⠀⠀⠱⠀⣀⡴⠁⠈⠋⡀⠄⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣀⠠⠢⠀⢀⠉⠉⠉⠛⠢⠐⠀⠀⠈⠛⠀⠉⠀⠈⠈⠁⠀⢤⢀⣁⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⡀⣀⠂⠘⠔⠀⠄⢠⠀⠀⠀⠂⠠⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣧⣀⣠⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠙⠛⠛⠉⠀⠈⢻⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡏⠀⠀⣀⣀⣀⣤⡀⢀⣸⠋⠉⠻⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠈⠳⠤⠞⠉⠉⠁⠀⠉⠉⠁⠀⠀⣰⣿⠃⠉⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣄⠀⠀⢀⣠⣶⣶⣴⣶⣾⠿⣷⣶⣶⡿⠟⠁⠀⠀⠀⣀⣙⣿⣦
⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠁⠙⠻⠿⠿⠿⠛⠉⠉⠉⠀⠀⣀⣀⣀⣤⣤⣴⣶⣾⠿⠿⠟⠛⢹⣿⡀
⠀⠀⠀⢠⣶⣿⠟⠋⠁⠀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⡾⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣤⣤⣤⣾⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠈⠉⠉⠉⣿⡟
⠀⠀⠀⢸⣿⣴⡿⠿⠿⢿⣶⣄⡀⠀⠀⣠⣴⡾⠟⠛⠛⠛⠻⠿⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣧
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⠿⠿⠟⠛⠋⠁
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⡾⠿⠿⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⢀⢀⣄⡀⣠⣄⠀⣠⡄⣠⣀⣤⣠⣤⡀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢨⡐⠀⠀⠀⠀⠸⠿⠏⠟⠧⠿⠞⠀⠧⠴⠛⠿⠤⠸⠁⠷⠋⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠘⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⡆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢑⠀⠀⠀⠹⣾⡇⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠸⡇
⠀⠀⠀⠀⢆⠀⢠⡀⠀⠀⠀⡶⠃⠀⠀⠀⠀⠁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⢤⠀⠀⠄⠀⡏⠀⣷⠄⢠⡆⣄
⠀⠀⠀⠀⠤⡖⠦⠄⠠⠤⣻⡆⠀⠀⠀⢀⠠⠀⠟⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣈⠦⢤⣧⡖⡧⡄⣿⢄⡜⡇⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⢄⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠁⠀⠈⠛⡆⡇⣿⡿⣿⢳⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠑⠀⠀⢀⡀⠀⠀⠠⡼⠄⠀⠀⠀⠀⠀⠀⠄⠸⣧⢹⠇⠿⢞⣴⠃
⠀⠀⠀⠀⠀⠀⠒⠲⢶⡶⠼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠉⠀⠁⠀⠀⠀⠀⠀⢠⣤⡤⠶⠀⠦⠷⠮⠀⢀⣀⠬⠅⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⡿⠿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣶⠄⠀⢰⡶⠀⠀⣶⠆⠀⠀⠀⠉⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣤⣴⣾⠟⠋⠀⠀⠸⠗⠀⠀⠿⠃⠀⠸⠟⠀⠀⠻⠇⠀⠐⠿⠀⠀⠺⠇⠀⠈⠻⣿⣦⣄
⠀⣀⣴⣶⣿⠿⠛⠉⠀⠀⣀⠀⠀⢀⡀⠀⠀⣀⠀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⠀⣀⡀⠀⠀⣀⠀⠙⢿⣷⣤⡀
⣼⣿⠛⠁⠀⠀⠀⠀⠀⠈⠛⠁⠀⠘⠋⠀⠀⠛⠁⠀⠘⠋⠀⠀⠙⠃⠀⠈⠛⠀⠀⠙⠃⠀⠈⠛⠀⠀⠀⠈⠛⢿⣷⣤⡀
⣿⣧⠀⠀⠀⠀⢠⣤⠀⢀⣤⡀⠀⢠⣄⠀⠀⣤⡄⠀⢠⣤⠀⠀⣤⡄⠀⢠⣤⠀⠀⣠⡄⠀⢀⣤⠀⠀⣤⡄⠀⠀⠈⢻⣷
⠹⣿⣦⡀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣠⣿
⠀⠈⠻⢿⣶⣤⣄⣀⠀⠠⣶⠄⠀⢰⡦⠀⠀⣶⠄⠀⢰⡶⠀⠀⣶⠆⠀⠰⣶⠀⠀⢶⡆⠀⠠⣶⠀⠀⠀⠀⠀⣠⣶⡿⠏
⠀⠀⠀⠀⠉⠙⠛⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⣀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣦⣤⣤⣤⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠟⠿⠿⣿⣿⣿⡿⠟⠛⠋⠉⠉⠉⠉⠙⠛⠛⠋⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠤⠠⠄⠀⠀⢀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⢈⣂⣥⣾⠛⠀⠀⠁⠒⢯⡄⠀⠑⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠘⠿⠿⡋⠁⠙⠷⠶⠶⢼⣿⣶⠆⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⡅⡾⠀⢤⣤⠀⢀⣀⢰⣡⠄⠀⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠉⠙⠁⠛⠃⢸⠞⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣥⣮⣾⣶⣦⣄⡎⠀⠀⠀⠀⠤⠦⢤⡀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡌⠢⠼⠷⠨⢟⠃⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡠⠤⢖⠊⠉⡀⠛⢷⣿⣴⣶⣤⢞⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠃⠀⠀⠀⢂⠐⣿⣦⣤⣸⣿⣟⡁⠋⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠘⣦⣄⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⡵⠀⢰⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢣⡀
⠀⠀⠀⠀⢰⠁⣴⣿⣿⡾⠁⠀⢸⡇⠀⠀⢸⣤⡄⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⣾⡆⠘⠉⢷
⠀⠀⠀⠀⠈⠀⣾⣿⠟⠁⠀⠀⣾⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡄⠀⢈⢃
⠀⠀⠀⠀⠀⣠⣿⠇⣀⣄⡀⠀⡾⣿⣧⠀⠀⡧⣦⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⣼⣿⣿⡧⠤⠍⠨⣅
⠀⠀⠀⠀⣰⣿⣿⣦⣶⡤⠭⣿⢹⣿⣿⣧⠰⣇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠸⢏⡙⠡⡦⣄⠀⠀⣼⡄
⠀⠀⠀⠀⠹⢿⣿⠿⠂⠀⠀⠀⢹⣿⡿⣿⣷⣍⣤⡄⡀⠀⠌⠋⠁⠀⠀⠀⣀⣠⠍⠑⡄⠈⡇⢠⣿⡇
⠀⠀⠀⠀⠀⣾⢋⡔⠀⠀⣴⠞⠒⠼⠿⠟⢿⣿⣄⣾⣿⣶⣿⣶⣶⣶⣶⣿⡭⣥⡤⠜⢉⣸⣧⠞⠉
⠀⠀⠀⠀⠀⣿⣾⣴⢀⣾⣅⡀⠀⠀⠀⠀⠳⣦⣹⣛⢻⣿⣿⣿⣿⣿⣻⣽⣿⡙⢿⣯⡴⠉
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠓⢦⣠⣀⠠⣾⣿⣯⣤⣥⣼⣿⣿⣿⣿⣿⠟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠛⠿⢿⣿⣿⣿⣿⡿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⠤⠔⣶⣆⠀⠀⡀⠀⠄⠀⠀⠀⠀⠉⠂⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡤⡀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠌⠐⠄⡀⠀⠄⡀⠢⡀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠠⠉⠀⠀⢄⡀⠀⠀⠀⢙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠅⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠉⠒⠤⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠫⠒⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠄⠁⠀⠀⠀⠀⠈⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠅⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡠⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⡄
⠀⠀⠀⣠⠀⠀⠀⠈⢢⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⢴⡾⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠉⠒⠠⣄⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⡤⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⡂⠄⠒⠉
⠀⠀⠀⠙⢄⡀⠁⠂⠄⡀⠀⠀⠀⠀⠉⠉⠩⠉⠉⠉⠉⠉⢉⣉⣙⠛⣛⠛⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠈⠀⢀⣠⠴⠎⠁
⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠉⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⣀⡤⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠉⠐⠂⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⢡⠰⠞⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⣀⠀⠀⠈⠁⠀⠒⠂⠀⠉⠀⠀⢀⣀⢠⡤⠖⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⠤⠦⣐⢶⡒⠊⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠚⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠓⠲⢤⡈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢼⢱⡀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣄⣀⠀⠈⢣⠀⠀⣹⠦⢿
⠀⠀⠀⠀⠀⠀⠀⡼⠚⠀⠀⠀⢀⡀⠸⣁⣀⡀⠀⠀⠀⠀⠀⠙⠢⠤⠴⠦⣄⢰⠃⠀⠀⣀⣈⣉⡿⠥⠖⠋⠁⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⢯⣈⡉⠁⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⢻⠈⡏⠉⠉⠉⠀⠀⠀⠀⣀⣠⢴⡟⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡞⣿⠇⠀⠀⡠⠤⠭⢷⣶⣶⣤⣤⣤⣤⣤⣤⡤⠤⢤⠤⠬⠿⠥⠤⠶⢖⣒⣒⣋⡭⠵⠚⠉⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠸⡗⣇⠀⠀⠀⠓⠤⣄⡀⠀⠀⠉⢉⡽⠯⡍⠉⠉⠉⠉⣹⠿⢯⠉⠉⠉⠉⠉⠀⠀⢀⣀⡤⠴⠋⢸⡇
⠀⠀⠀⠀⠀⠀⣰⢳⡼⠃⠀⠀⠀⣠⠖⠛⠒⠒⠒⣾⠦⠤⠭⠤⠤⠤⠤⠥⠤⠬⣥⠤⠶⠖⠒⠚⠋⠉⠀⠀⠀⣠⠞⣧
⠀⠀⠀⠀⠀⠀⢻⠸⣀⠀⠀⢀⣀⢹⣄⣀⣀⠀⠘⠛⠁⠀⠀⠰⣤⡴⠆⠀⠀⠘⠛⠁⠀⠀⠀⢀⣀⣠⣤⡶⠞⠁⢀⡏
⠀⠀⠀⠀⠀⠀⣼⢳⣌⣳⠀⢳⠬⠭⣤⣀⣈⣉⣉⣉⣛⣛⣛⣒⣒⣒⣒⣒⣒⣒⣛⣿⠿⠽⠟⠛⠛⠉⠁⠀⣀⣴⠏⢧
⠀⠀⠀⠀⠀⠀⢧⠀⠙⢿⡀⣾⠤⢤⣀⣀⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⣖⣺⠭⠟⠉⠀⡼
⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠉⠉⠙⠓⠒⠒⠻⠿⠯⠭⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠓⠚⠛⠉⠉⠁⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠶⠤⢤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡤⠤⠴⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⠇⡀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡝⠀⠀⠙⢦⡀⠀⠀⠀⢀⡴⠋⠀⠂⠀⠈⠀⠐⠃⠀⠳⢄⠀⠀⠀⣠⡴⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠞⠀⡀⠀⠀⠀⠙⢦⣀⡴⠋⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠚⠁⠁⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠉⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠋⠐⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⣴⡋⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⣀⣀⠀⣀⡀⡀⢀⣀⢀⣀⡀⠀⢀⣐⣤⣤⠤⢤⣤⣤⣽⡦
⠀⠀⠀⠀⠉⠙⠛⠷⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠃⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈





⣤⡴⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣽⣷⣮⣾⣶⣬⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⣿⣿⣿⣿
⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⣿⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⡇⠀⠀⠄⢀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠂⠀⠀⠀⠀⠀⠀⡀⠈⠀⠁⠀⠐⠂⢀⠀⠀⠀⠠⠀⠁⠀⠀⢀⣿⣿⣿
⣿⣿⣿⣶⠈⠀⠀⠐⠠⠤⠀⠈⠀⠀⠀⠀⠀⠀⠐⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⣿⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣿⣿⣿
⣿⣿⡿⠁⠀⠀⠈⠀⠐⠀⠤⠠⠄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠤⠀⠤⠀⠒⠀⠀⠁⠀⠀⠘⣿⣿⣿
⣿⣿⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⡇⠀⠀⠄⢀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠐⠀⢀⠀⠀⢀⠠⠈⠁⠀⠀⢀⣿⣿⣿
⣿⣿⣿⣶⠈⠀⠀⠐⠠⠠⠐⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⡿⡻⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⡻
⣇⢄⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢀⠄
⣿⣷⣤⣀⠀⠀⠁⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠉⠀⠀⢀⣠⣵⣾
⣿⣿⣿⣿⣿⣿⣶⣶⣤⣦⣀⣀⣠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡤⢳⠀⠀⠀⠀⣀⣀⢄⡀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠢⠤⡀⢧⠜⠀⠀⢠⠊⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢻⡴⢧⠛⠁⠀⠠⠀⠀⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠘⣆⠀⢀⣈⣦⢾⠀⠀⠀⠀⢀⠔⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠁⣈⡀⠈⢳⢀⠟⠀⠀⠉⢾⣄⠂⣀⠔⡡⠐⡆⠀⡠⠟⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠸⠿⠇⠀⢀⡏⠀⢠⣦⣤⣄⢷⠒⠋⢁⠄⠂⠀⠾⡁⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡞⠉⠑⣦⣤⣠⣀⣤⠞⣷⠀⠀⡻⠋⠁⠀⠑⢶⡇⠀⠀⠀⠀⠈⢦⠀⠀⠀⡦⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⣮⣀⣹⣶⡞⠁⠈⢙⡋⠀⠑⡼⠷⡞⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣸⠶⠖⠛⠁⠀⠀⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠉⠉⠉⠀⠈⠛⠛⣩⠼⠬⣝⠳⠶⢇⡀⡀⠀⠀⠀⣀⣰⠟⠛⣻⠏⣯⠁⠀⠀⠀⠀⠀⠄⢊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡆⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢀⣀⣉⣉⣉⣉⣉⡉⠀⠀⠀⠈⠩⠡⠀⠀⠀⣀⡀⣠⠞⡏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢳⣠⠖⢦⡀⣠⡀⢠⠤⡄⠀⠀⣠⣄⠀⠀⢀⣤⢴⣄⠐⣦⢠⡴⣤⣤⡴⠛⠉⢹⡇⠀⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇⠀⠀⠘⠁⣇⡏⠀⢸⠀⠀⡏⠙⠿⠟⠋⠀⠀⠈⠉⠁⣾⠀⠀⠉⠀⠀⠀⢠⡆⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣄⡀⠸⠃⠀⠀⠶⠀⠉⠀⠀⢸⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⣀⠤⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⣿⣿⣒⡶⢤⣦⣤⡤⣠⣀⠈⡁⢨⣧⠀⢀⠀⠀⢀⣀⣠⡤⠤⠤⠤⠴⠄⠒⠒⠈⣁⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠽⢯⣽⣷⣾⣿⣷⣶⣶⣶⣤⡤⠬⠭⠤⠤⠐⠒⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣷⣯⣗⣒⠤⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢀⣀⣤⢤⠤⠄⠠⠐⠀⠂⣉⡤⠜⡀
⠀⠀⠀⠀⠀⠀⡀⠄⠐⠂⠈⢹⠀⠀⠉⠛⠛⠿⠟⣻⡶⢶⣿⢯⣭⣭⣭⢭⡭⠵⠦⠴⠦⢤⣤⣤⣤⠤⠤⠥⠄⠒⠒⠈⠉⠀⠀⠀⡏⠁⠐⠂⠤⢀
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⢸⣧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣼⡇⠁⠀⠀⠀⠀⠈⠒⡄
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠸⣧⣕⡪⢅⣒⠤⢤⣀⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠤⠀⠒⠈⢑⣠⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣾⣿⣥⣶⣦⣭⣭⣄⣁⣒⣒⣒⣒⣀⣀⣒⣀⣀⣒⣂⣀⣤⣅⣴⣴⣦⣿⣶⢾⡻⠟⠋⠀⠀⠀⠀⠀⠀⣠⡴⠃
⠀⠀⠀⠀⠀⠉⠛⠲⠦⣄⣀⡀⠀⠀⠆⢠⠈⠉⠙⠋⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠙⠀⠈⠀⠀⠀⠀⠀⣀⣠⠤⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣶⣄⣨⠍⠑⢒⡐⠲⠤⠤⠤⠤⠤⠀⠀⠀⠀⣠⠄⠠⠤⠄⠤⠤⠈⠃⠐⠊⠭⢉⣄⡴⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠒⠙⠒⠒⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠒⠒⠚⠉⠑⠒⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡄⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⠀⠀⢠⡏⠀⠀⢀⣴⡇⠀⠀⢰⡆⠀⠀⠰⢣⠀⠀⠈⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⠂⠠⠜⠛⠀⠀⠚⠛⡃⠠⠀⠉⠁⠀⠀⠀⠉⠀⠀⠰⡀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠤⢤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢀⢷⡤⢴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⡀⠰⣕⣀⠀⣠⢠⣀⠀⠀⡠⢰⠄⠀⠀⢀⣇⠀⠀⠀⣸⠦⢠⣬⣾⣿⡇⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡇⡈⡿⠋⠙⠓⠻⣷⣷⡶⠧⠈⠷⣶⡾⠿⠋⠙⠷⠿⠋⠙⠉⢰⣿⡟⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡇⠀⡇⠁⠀⠠⠀⠈⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⡆⢸⣿⡇⢷⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡅⠀⠃⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⢿⠿⢳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⢠⠀⠀⠀⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⢺⡿⣦⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢰⠀⢀⣾⠄⠀⡀
⠀⠀⠀⠀⢀⠠⠂⠩⠐⠴⠄⢸⣷⣿⠸⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⢸⡇⢸⣴⣮⣿⠀⠀⠀⣑⠴⡄⡀
⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠸⣯⣹⢰⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢃⠸⣿⣭⣿⠀⠀⠐⠁⠁⠈⠈⢂
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠈⠢⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠆⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠂⠀⠁⠀⠐⠒⠂⠀⢤⣤⣀⣀⣀⣀⣀⣀⠤⠤⠤⠶⠂⠈⠀⠐⠂⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠉⠛⠻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣱⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡏⠻⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⠠⠤⠄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡉⠉⠀⢀⣀⣀⣤⡤⠽⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⠋⠡⣤⢶⣛⢉⣡⣴⣻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠉⠉⡟⠋⠑⠨⠴⡀
⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⢣⣀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⠙⠒⠲⠤⣤⣀
⠀⠀⠀⡠⠖⠋⠁⠠⢔⣮⡅⠻⢼⣽⣿⣿⡶⠦⣤⣠⣇⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣇⣭⣵⠦⣄⠀⠈⠙⠲⢄
⠀⠀⠈⠀⠀⢠⣴⣿⣛⣻⣥⡴⠶⠛⠛⢛⣻⣿⣷⡿⣯⣿⣓⣲⡦⢤⣼⣀⣀⠀⠀⡷⠟⠛⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠦⣀⠈⠉⠉⠙⠳⠀⠀⢠⣶⣿⣯⠭⠶⠶⢒⣻⠿⣋⣉⣽⠿⠛⠷⠿⠿⠽⠇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠈⠙⠒⠦⢤⣄⣀⣀⠀⠉⠁⠀⠀⠀⠀⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠒⠒⠲⠶⠶⠶⠤⠤⠶⠶⠶⠒⠒⠒⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⡿⠛⠛⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣿⣁⠀⣫⣵⣶⣶⣦⣄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣉⣿⡿⠛⠿⣿⣿⣿⣿⣷⣼⣇⢘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⡀⠀⠀⠀⠀⠀⠀⢀⠀⢯⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠘⣹⣿⣿⡆⢠⣿⣿⣏⠑⢸⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣵⠃⠀⠀⠹⠛⠁⠈⠙⠳⠀⠀⠈⢧⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣰⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢠⢴⣿⣿⣷⣄⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠶⠶⠄⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠙⢆⣀⣀⣀⣀⣠⠞⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡈⠓⢼⣿⣿⡿⠟⢉⡤⠊⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⡇⠀⠀⠉⢻⣧⣖⠋⠀⠀⢸⣿⣿⡳⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣠⢴⣶⣿⣿⣿⣿⣻⣿⣿⣿⣧⠀⠀⣴⣿⣯⣻⣷⠀⠀⣼⣿⣿⣿⡽⣿⣿⣷⣶⢤⣀
⠀⠀⠀⢠⣯⣽⣾⣿⣿⣿⣿⣿⠳⣿⣿⣿⣿⣿⠀⡼⠋⠻⣿⣿⣿⣷⡀⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⣾⣝⣆
⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⣿⡟⣛⣒⣙⣻⣿⣿⡟⠁⠀⢀⣿⣿⡟⠉⢹⣿⣿⣿⣋⣐⣛⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠼⣾⣿⣿⣿⣿⣿⣿⡝⣿⡜⣿⣿⣿⣿⣿⣷⡀⠀⣼⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⡽⠄
⠀⠀⠀⠤⢌⣉⠛⠻⠿⣿⣿⣿⣿⡜⣷⣾⣿⣿⣿⣷⣀⣿⣿⣿⡇⣰⣿⣿⣿⣿⣶⣏⣾⣧⡿⠿⠛⢋⣉⡠⠤
⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣬⣉⠛⠿⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠘⣉⣥⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣌⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣡⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣉⠻⢿⡿⠟⣋⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡦⠢⠀⠀⡰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠗⠀⢀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠂⢀⠠⠔⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠠⠤⠀⠀⠒⠈⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠲⠒⠒⠖⠙⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢠⢊⠈⠉⢂⢠⠚⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡄⢄⠀⢀⡠⠤⠦⠼⢤⡆⢉⡉⢢⣈⠢⢈⣳⠀⠀⠒⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⠀⠘⠚⠢⠷⠅⠭⡀⠀⠀⡀⠃⠀⠀⠀⠺⠕⠚⠓⠀⣀⠠⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠁⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠒⠄⡀⠀⠀⠐⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠔⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠈⠑⠂⠒⠠⠤⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠤⠄⠀⠐⠒⠊⠁⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠈⠑⠦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠉⠉⠉⠁


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⣿⣧⣼⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡾⢻⡏⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣼⣇⣀⣻⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⡿⠿⢿⣿⡿⠿⠿⢿⣿⣿⠿⢿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠿⠛⠋⠉⠉⠀⠀⣀⣠⣤⣴⠟⠛⢻⣦⣶⠟⠉⠙⣷⣤⣀⣀⠀⠈⠉⠉⠙⠛⠿⢿⣿⣶⣄
⠀⠀⠀⠀⠀⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⢼⣟⠉⠉⠁⢻⣆⣀⣴⠏⢻⣄⣀⣠⡿⠉⠉⠙⣻⡦⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧
⠀⠀⠀⠀⠀⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⠶⠿⠿⠷⠶⠶⠿⠿⠿⠶⠾⠛⠛⠉⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠛⠿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣶⣾⣿⡿⠿⠛⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⠀⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣦⣄⣀⣀⣠⣴⣿⡿⠻⣿⣷⣤⣀⣀⣀⣤⣾⣿⣿⣷⣄⣀⣀⣴⣿⡟⢿⣿⣦⣀⣀⣀⣠⣴⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠙⠻⠿⠿⠿⠿⠟⠋⠀⠀⠈⠛⠿⠿⠿⠿⠿⠛⠁⠈⠛⠿⢿⠿⠟⠋⠀⠀⠙⠻⠿⠿⠿⠿⠛⠉⣿⣿
⠀⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄
⠀⠀⣴⣿⡿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢿⣿⣆
⠀⢸⣿⣏⠀⠀⠙⠿⣿⣿⣶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣿⣿⠟⠋⠀⠀⣻⣿⡇
⠀⠀⠻⣿⣷⣤⣀⡀⠀⠈⠙⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠁⠀⢀⣀⣤⣾⣿⠏
⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣀⣤⣤⣴⣶⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠨⣧⣴⡦⢠⣢⠁⡲⢀⠀⢀⣀⡠⢐⡆⠈⢻⠐⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠉⡓⠃⠁⠀⢈⣯⡓⢿⣍⣠⠎⣽⠀⠀⢠⠃⠀⢸⣼⡇⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡉⣁⣤⡔⠀⠈⠻⣟⡷⢶⢏⣠⣷⢌⣜⢎⡀⢀⣻⠸⣣⣬⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢫⠞⠉⠀⠀⠀⠀⠀⠀⠙⣽⣾⣿⣿⠿⠷⠿⠶⠛⠑⠒⠛⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠡⠆⠀⠀⠀⠀⠀⠀⠴⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠓⠠⢤⠄⠀⢠⡄⠂⣷⣿⡟⠀⠤⣀⠀⠀⡀⠀⣠⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⣠⠿⡄⠀⡸⢧⣀⣿⢿⠧⠀⠀⠟⠀⠰⠁⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠑⡌⠁⠀⠉⠉⠀⠀⠘⠿⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⠀⡃⢰⠀⠀⠀⠀⢀⡄⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢀⠁⡞⠀⠀⠀⠀⠈⣇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣿⡀⢠⠃⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⡤⠀⠀⢀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢠⠀⠀⠠⠀⠰⠿⢙⣯⠄⠀⠀⠀⠀⠀⠀⢧⠸⣿⡴⡇⠀⠀⠀⠀⠀⠠⣽⣧⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠈⠐⠴⣀⠁⠸⢆⡀⠏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠃⠀⠀⠀⠀⠀⠰⡿⢀⡰⠇⠉⣀⠮⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠗⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠠⠻⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡁⠀⠀⠀⡀⠀⠀⠄⠀⣀⠀⠀⡁⢈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠥⠄⠀⠀⠤⠀⠒⠬⠂⡀⠀⠀⠅⠲⠈⠅⠀⠀⠠⠤⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⡀⠀⠀⠀⠀⡠⠊⠱⡀⠀⠀⠀⠀⠀⡠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⡇⠀⠀⠀⠀⢇⠀⣠⠁⠀⠀⠀⠀⠸⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡵⢌⡀⠀⠀⠀⢰⠶⠛⢦⡄⠀⠀⠀⠀⣤⣉⠬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣉⡇⠀⠀⠀⢸⠓⠂⠤⡆⠀⠀⠀⠀⠓⠤⠄⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⢸⣄⠀⠀⡇⠀⠀⠀⠀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠒⠠⠆⠀⠀⠀⢸⠀⠉⠒⡇⠀⠀⠀⠀⡉⠑⠂⢼
⠀⠀⠀⠀⢀⣀⠠⠤⠄⠒⠓⠂⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠘⠒⠂⠤⠤⢀⣀
⠀⢠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⢠⠁
⡇⠀⠀⢀⢾⠀⠀⠀⠀⠀⠀⢀⠠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢄⡀⣀⣀⠀⠀⠀⠀⠀⡠⢤⡀
⠣⢀⡠⠎⠀⠢⣀⠀⢀⣀⠔⠁⠀⠀⠀⠑⠤⠄⠀⠠⠤⠔⠊⠀⠀⠀⠀⠀⠈⠑⠂⠀⠒⠊⠀⠀⠈⠒⡖
⠀⠀⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠄⣀⡀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀⠀⠀⠀⠀⡇
⠀⠀⡟⠉⠀⠀⠀⠀⠉⠒⠒⠀⠀⠐⠒⠊⠁⠀⠀⠀⢀⡀⠀⠉⠐⠒⠒⠒⠈⠁⠀⠀⠀⠈⠑⠒⠒⠒⡇
⠀⠀⡇⠤⠔⠒⠀⠀⠢⢀⣀⡀⣀⣀⣀⠠⠔⠂⠉⠁⠀⠀⠉⠒⠤⠤⠤⠤⠐⠒⠀⠉⠁⠒⠤⠤⠤⠒⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢧
⠀⢊⡀⢠⠊⠉⠐⡔⠒⠂⡤⠐⠠⡤⠐⠠⡤⠀⠤⡠⠄⠠⡤⠀⢤⠤⠤⣄⠤⠠⢤⠒⠒⢴⠉⠈⢱⣀⡠
⠀⠀⠈⠉⠓⠐⠚⠢⠄⠴⠧⢀⠠⠦⢀⠠⠦⣀⡠⠧⢀⡨⠆⣀⠹⣀⣀⠼⢀⣀⠼⠤⠤⠞⠑⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀
⠀⣰⠃⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⡇⠀⠀⢸⡃⠉⠙⠲⠦⠤⣄⣀⣤⡤⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀
⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠲⠦⠴⠖⠒⠒⠒⠲⠤⣤⣀⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠙⠒⠦⠤⠤⠴⠒⠛⠙⠛⠓⠒⠒⠶⢤⣈⡇
⢰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡏⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⡇⠀⠀⠀⣿⠑⠲⠖⠚⠉⠓⠲⠤⣄⣠⠤⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⣇⠀⠀⠀⢸⠛⠒⠒⠓⠢⠤⣤⣤⠤⣄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠓⠒⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠙⡄⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠦⠤⣤⣤⠤⠤⠤⣀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠒⠲⠦⣄⣀⣀⣀⣀⡤⠤⠶⠦⢤⣀⢀⣀⠤⠖⠲⠶⠖⢺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⡈⠉⠛⠻⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣦⣤⣀⠉⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡈⠙⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⠾⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⡶⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⡶⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣶⣶⠶⠿⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⠶⠿⠟⠛⠛⠉⠉⠉⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⡶⠾⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⠶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⠶⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣶⠶⠾⠿⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣀⣀⣀⣤⣤⣴⣶⡶⠾⠿⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠻⢦⣤⡾⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠙⠻⣦⣄⣹⠟⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣤⡀⣻⡟⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣄⡀⣸⡷⢶⣄
⠀⠀⠀⠀⠀⣠⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⡀⠀⢻⣆
⠀⢀⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢹⣇⠀⣸⡿
⠀⠿⣷⠶⠶⣶⣶⣶⡶⠶⠶⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠻⣷⠀⠀⢸⡟⠛⠛⠁
⠀⠀⢻⡀⠀⣀⠀⠀⡀⠀⠂⠀⣀⠀⠀⠂⠀⢀⠀⠛⠀⠰⠈⠁⠀⡄⠀⠘⠀⠀⠀⡉⠀⢿⠀⠀⣸⡇
⠀⠀⢸⡇⡀⠁⢀⣀⣁⠀⠀⠐⠀⠀⢀⡤⠶⢬⣄⠀⠀⠄⠀⣠⡴⠶⠷⣶⣤⡀⠀⠁⣀⣸⠀⠀⣿⡇
⠀⠀⢸⣧⠶⠚⠉⠁⠈⠑⢄⡐⣀⡴⠁⠀⠀⠀⠉⠳⢦⣤⡾⠃⠀⠀⠀⠀⠙⠻⠿⠿⠿⣿⠀⠀⣿⡇
⠀⠀⢸⣏⣴⠖⠛⠓⠶⢤⣀⡉⠉⢀⣠⠶⠶⠶⣤⣀⡀⠀⠀⠀⣠⠶⠶⠶⢶⣄⣀⣠⣴⣿⡇⠀⣿⡇
⠀⠀⢸⣿⠁⠈⠁⠤⠀⠀⠈⠉⢛⠉⣡⠀⠆⠀⠀⡉⠙⠛⠛⢋⡁⠰⠀⢀⠀⠙⠛⠛⠉⣿⡇⠀⣿⠃
⠀⢀⣼⣿⣀⠐⠂⠀⠀⠂⠀⠀⡉⠀⣤⠀⠀⢀⡀⢠⡄⠀⠃⠀⠀⠀⡄⠀⠀⣄⠀⠚⣠⣿⠇⠀⣿
⢰⡟⠛⠛⠻⠿⢿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣠⣤⣴⣾⣿⡿⠀⠀⣿
⠀⠛⠶⣤⣤⣤⣀⣀⣀⣈⣉⣉⣉⣉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⠶⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡞⠉⠙⠀⠂⠰⣀⠀⠁⠀⠀⠀⠀⠀⠀⠤⣤⣀⣀⢀⡀⠀⠀⠈⠀⡭⠅⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⠀⡄⣀⣠⠔⡅⠀⠙⡦⠀⠀⡀⢀⠀⠀⠀⠀⠈⠉⠙⠻⠓⠓⢲⠺⡟⠻⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⢁⠸⣆⢡⡰⡿⣂⠒⢴⠃⠀⢐⣤⠙⠛⠁⠀⠆⠀⣠⡀⠀⠈⠀⠀⢈⡀⠀⢀⢠⣀⡰⢶⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣌⡀⣽⠋⠘⣧⡉⣀⣼⣦⣀⣸⣿⣧⣤⣄⣤⢠⡀⡄⠀⢢⣦⣠⣤⣿⣥⣼⠷⢶⡿⣿⡿⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢈⠍⠉⠀⠚⠋⠉⠉⠁⠀⠀⠈⠁⠈⠉⠉⠁⠚⠁⢙⣤⠸⠐⠉⠁⠀⠀⠀⠀⠈⢀⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣙⣿⠀⠀⠀⠀⠀⠀⢰⠀⣿⣿⣆⢴⢤⠀⠀⠀⠀⢋⣫⠷⢂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠖⠀⠒⠉⠋⠈⠁⣻⠆⠋⠛⠁⠀⠀⡀⠈⢣⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡂⠀⠀⣳⣶⣦⠀⠀⠀⢀⣸⣷⣶⠦⠀⠒⡴⠆⠐⣛⣿⠀⣽⡷⢎⠀⠀⠀⠀⠀⠀⢰⠃⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⢀⣀⣹⣋⣀⡉⠉⢁⣁⣀⣾⠇⠈⠉⠁⣀⣉⠉⣡⡀⣤⡖⡋⠀⠀⠀⠀⣠⣴⡇⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣽⣧⣀⡀⠀⠀⢀⣨⣯⡟⠀⣀⠉⠁⣀⣀⣠⡾⠃⢾⡛⣦⣤⣤⣶⣿⣿⣿⡗⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⡞⠋⠈⠉⠉⠉⠉⣉⣼⠟⠉⠉⣉⣠⠬⠤⠄⠀⢸⣿⣡⢗⠚⠙⠛⠻⠿⠗⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠘⠚⣻⡧⢤⣤⣄⣀⣀⣀⣈⣿⠀⠀⢀⣠⣀⣤⣶⡿⠇⠀⠋⣡⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢾⡿⣶⣎⢅⣀⡀⠉⠉⠉⠉⠉⠁⠈⠀⠀⠀⢀⣀⣨⣷⢶⣞⡋⠛⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣿⣦⣤⣁⣀⣉⣉⣉⣉⣈⣉⣉⢉⣁⣤⣤⣤⣿⠾⣈⡁⣠⣿⡖⠃⢘⣶⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⢹⡏⠛⠻⠛⠿⠶⠾⢃⠔⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠤⠄⠐⠒⠋⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢳⡄⠀⠀⢠⠏⢻⡄⠀⠀⢠⡟⠹⡄⠀⠀⢠⡞⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⠇⠀⠀⢸⡄⢠⡇⠀⠀⢸⡄⢠⡇⠀⠀⠸⡇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⢙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣧⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣬⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⣄⣀⣀⣠⡴⠶⢤⣀⣀⣠⡤⢤⣄⣀⣀⡤⠶⢦⣄⣀⣀⣠⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⣄⣈⣁⣀⡤⠤⢤⣉⠉⣁⣤⣤⣈⠉⣉⡤⠤⢤⣀⣉⣁⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢀⡀
⠀⡆⠀⠀⠀⠐⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠐⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠀⠀⢀⣈⠁
⠀⠐⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⢠⢤⠌⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠐⠄⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠄⠒⠀⠀⠄⠐⡀
⠀⠀⠀⠀⠀⠀⠈⠓⠠⠄⣀⠀⢀⣠⠣⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠐⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠠⠀⠙⠲⠄⠀⠤⠀⠀⠀⡀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠡⣀⠀⠀⠀⠀⠁⠒⠢⢰⠶⡫⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠠⠄⠀⠂
⠀⠈⠑⠀⠀⠀⠀⠀⠁⠐⠊⠀⠀⠀⠀⠀⠀⠀⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⡀⡀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠶⢠⣂⡡⢀⡂⠤⠒⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⡄⠀⠈⠉⠉⠀⢀⠠⠄⡆⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠈⡁⢈⠥⠐⠒⠀⠀⠀⠀⠀⠀⠓⠄
⠀⢆⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠃⡀⠀⠈⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⡌⠀⢀⡀⠀⢤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣔⣦⡀⠀⠀⠈⠀⢠⠀⠀⠉⠱⠂⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠉⠛⠋⠹⣿⣇⠀⠀⠀⠀⠀⠘⠇⠀⠱⠂⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠟⠁⠚⠀⠈⠓⠀⣿⡀⠀⠀⠀⢠⡀⣴⠶⣦⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⡇⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⠀⠻⣤⣶⣤⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡿⣁⡀⠀⡀⢀⠙⡄⠈⠀⠀⠀⠀⣀⡈⠉⠉⠁⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⢈⢹⣯⢹⡉⡙⠋⣴⣷⠄⢀⣄⠀⠈⢯⡅⠀⡄⠀⠀⠀⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⣿⣏⠈⣿⢷⣄⠸⣿⠀⠀⠀⣹⠷⠸⡇⣰⠤⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢠⡙⡅⢹⡀⠻⠎⢹⠃⣿⠁⠀⢠⡜⢷⡄⠈⠓⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⢀⡇⠘⣷⡀⠠⡀⠃⠠⠀⡐⢨⡇⢸⡇⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠀⢨⠉⠈⣧⣴⠀⣡⣌⠐⠂⠈⣾⠏⢨⡟⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⡈⠁⡈⢰⣦⠉⠻⣔⡹⠋⠃⢴⠁⣿⠀⢸⡇⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡇⢠⡀⢈⡁⠀⣀⣯⣧⡸⠀⠈⢠⡿⠠⢾⣧⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠋⠿⠁⠀⠀⠀⢬⠙⣱⠇⠠⠈⣸⠇⠀⣹⡏⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠏⠀⠀⠀⠀⠀⠉⠛⠀⠃⠀⡼⢷⠜⠛⠃⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠀⣮⡐⣀⣀⠀⠀⢠⠀⠀⡀⠺⠇⠀⠀⠀⠀⠀⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠠⠡⠈⠀⠤⠉⠀⠀⡄⠀⢸⡁⠚⠀⠠⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡀⣘⠀⠀⠀⠀⠀⣤⠀⠀⠈⢶⠃⠀⠀⢋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡇⠹⠷⠶⠀⠰⢾⠀⢀⠀⠀⠇⠀⠀⠀⠘⡁⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡁⠃⠀⢠⠙⣠⡇⠀⡄⢁⡝⠀⠀⠀⠀⠀⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠌⢁⡀⠀⠀⢰⡏⠀⣠⠀⡿⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡈⢛⠀⠀⠀⣄⠈⠁⠀⠀⠴⠀⢾⠁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⣟⠀⠀⡘⠀⠘⣆⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠘
⠀⠀⠀⠀⠀⠀⠐⢰⠂⠀⠀⢸⢀⡿⠀⠐⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⡄⢸⠀⠌⠐⠀⡰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⠃⡼⠏⠀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢀⡦⠀⡇⠀⣠⠋⢀⣇⠘⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢸⠀⠐⡏⠈⠀⠀⣼⢈⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⡏⠀⣸⠀⠀⠀⢷⣼⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⡦⠄⠀⠀⠐
⠀⠀⠀⡼⠀⠀⢿⠓⠺⠶⡾⠃⠀⠀⠀⡴⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⢠⡏⢈⡁⣼⠂⣴⠊⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠈⠀⠀⠘⣄
⠀⠀⢨⡵⠻⠃⣿⠞⠁⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠂⢀⠘⠂⠀⠀⠀⠀⡜⠓⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢈⢓⡀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡰⡟⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠘⡪⣄⠀⠀⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⣬⠷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⡀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⣶⣿⢶⣄⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣶
⠀⠀⣼⠋⠛⣤⢈⣻⣦⣤⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡰⢆⣋⣿⡶⠖⠙⡇
⠀⣰⠅⢀⡅⠀⠙⠻⠳⣄⣉⣓⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⡄⢶⣖⠚⣙⡤⠛⠏⠉⠁⠀⠀⠰⣶⠘⡆
⣰⣇⠐⢹⠇⠀⠀⣀⣀⠨⠛⠛⠹⠛⠦⢤⡠⠭⣿⣿⣛⣉⠉⣿⠧⢤⠴⡄⠻⠋⠉⠉⠀⠀⠄⠁⣤⡀⢀⣠⠶⠛⠁
⠈⠉⠓⠲⠶⣦⣈⣿⠁⠀⠀⠀⠀⠀⠆⡀⠄⢂⠀⡀⠀⠀⠀⠸⢇⠀⠠⡀⠂⠂⢐⣤⣀⣀⡠⠶⠟⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠲⠶⠶⠷⢆⣀⣀⣀⣠⣅⣠⣄⣀⣄⣈⠴⠶⠶⠆⠐⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣆
⠀⠀⠀⠀⣠⡶⠀⠈⠁⠀⠀⠀⠘⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡤⠓⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠰⢾⡟⣠⡄⠀⠀⣤⣶⣤⡀⠀⠈⠻⠧⠤⠄⠀⢠⡇⠀⡇⠠⠠⡄⠀⣠⣤⣤⠄⠀⠀⣼⡇⢸⡇⠘⠁⠀⠀⠀⠀⠀⠾⠅⠀⠀⣰⣿⣿⣷
⠀⢀⣐⣒⡂⠀⠀⠀⠀⠀⠀⣀⣒⣛⣲⣄⣀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠻⡿⠋⣠⣦⠀⠈⠁⠈⣁⣠⣴⠞⠀⠀⠀⠀⡰⠤⠀⠀⠻⠟⠻⠿
⠰⣾⣿⡏⠀⠀⠀⠀⠀⠀⠛⣛⣛⣛⣛⣻⣿⣷⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠙⠛⠀⠀⠀⠀⣤⣄
⠀⠉⢈⠛⠛⠈⢀⣴⣶⣶⣶⣶⡄⠈⠉⠉⠛⠛⠻⠿⠿⣧⡀⠀⠀⠀⠀⣀⠀⣼⠷⠀⠀⠀⠀⠀⣼⣿⣿⣄⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠠⠤
⠀⠀⢙⡁⡀⠀⠈⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠿⠿⠟⠀⠀⠦⠀⠀⠀⠀⠀⠀⡄⠀⠸⠀⢠⠄⠀⢠⠀⠀⢀⣤⣤⠁⢀⣀⡀⠀⠀⠀⠀⠀⠂⠘⠃
⠀⠀⠀⠀⣠⣧⠀⠀⠀⣴⡄⠀⠀⠀⠀⠐⠀⠂⠠⠀⠀⠀⠀⡇⠶⣶⣾⣿⣿⠿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠈⠃⠀⠀⠀⠀⠂
⠀⠀⠀⠐⢿⣿⡆⣄⢠⣿⣿⠄⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠁⠀⠀⠐⠒⠚⠛⠁⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀
⠀⠀⣠⠂⠈⣿⡇⠉⢸⡿⠁⠲⠀⠀⠀⢀⣿⢠⠀⢀⡀⠀⠀⠀⠀⣠⣤⡀⣴⣿⣷⣄⠠⣄⠀⠀⠀⠸⠇⠀⠁⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠌⠃⠀⢈⡣⡀⠀⠀⠀⠀⠛⠙⠘⣤⣿⣿⣦⠀⢀⣴⣿⣿⣿⣦⡈⠉⠙⠓⠌⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⣧⠀⠀⠀⠀⢠⡇⠀⡆⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢈⡀⠀⠀⣈⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠘⠃⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⢀⣶⣶⣶⣶⣶⣆⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠩⠭⠭⠭⠭⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⡄⡀⠘⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⣠⣶
⠀⠤⠤⠄⠀⢀⣶⡖⠀⠉⠉⠁⠐⠒⠒⠀⣀⣀⣀⠀⠀⠛⠉⠁⠀⠁⠀⠀⠀⠀⠀⢀⣄⡉⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢾⣿⣿⢠⡀
⠀⣠⣤⣤⣭⣭⣭⣭⣬⣭⣭⣉⣭⣭⣭⣬⣭⣭⣭⣥⣤⣤⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣼⣷⣤⣤⣤⣤⣤⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣤⣀⣉⣉⣉⣛⣛⣓⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉
⣿⣻⣛⣛⣟⣟⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣧⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠐⠲⢤⡀⠀⠀⠀⠀⢀⠴⡛⠋⠻⣾⢁⠀⣱⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⠈⣆⠀⠀⢀⠈⡑⠄⢀⠔⡁⠀⢁⡀⠀⠘⣾⣠⡿⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠢⣅⠻⢢⣹⣄⠀⠀⠳⣀⡸⢃⡈⠠⠀⠀⠀⠈⠁⠘⢧⣤⡀⠈⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠂⠈⡟⠡⠤⢂⣱⡀⢀⡞⢠⠀⢠⠀⠠⠆⢀⣡⠤⠅⣘⣷⡶⢶⣊⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣷⣄⣣⣐⣲⡞⠙⠤⠤⡝⠄⠤⠀⠠⠄⢰⡏⠙⠢⠰⢈⣟⠻⠦⠗⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣁⣀⡠⠭⠙⠛⠛⠉⢳⣄⡀⣐⣡⠷⠤⢶⡶⠶⠿⢧⣀⠀⡠⢞⣝⠀⣄⠀⠀⠀⠀⢈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠉⠣⢄⡀⠄⠀⠀⠀⠀⢈⣉⡉⠁⠀⠀⠀⠀⠈⠉⠀⠉⠉⠛⠉⠉⠋⠁⡀⠀⠐⠚⠍⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠙⣦⠀⠀⣨⠟⠛⠙⠛⠳⠄⠀⠀⢠⣶⠟⠛⠙⠛⣟⠉⠁⣠⠞⠉⠙⢳⡄⠀⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⣶⢤⣄⣸⠀⢀⣿⠀⠀⠀⠀⢰⠀⠀⢸⣾⠁⠀⠀⠀⠀⢸⡄⠀⣏⠀⣀⣠⣼⡇⠐⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠙⠒⢸⠀⢸⣿⣖⡒⠶⠤⢾⡀⠀⠘⣼⣦⣤⣤⣤⡴⢾⡇⠀⡟⣛⣫⡭⢾⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣀⠀⠀⢸⠀⢸⠃⠀⠀⠀⠈⠀⠙⠒⠛⠁⠀⠀⠐⠒⠒⢺⡇⠀⡏⠁⠀⠀⢠⡇⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠧⣌⠑⢲⠟⢠⢼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣧⡤⣴⣶⣿⣇⠠⢈⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣾⣇⠀⠀⠀⠈⠦⠴⠟⠋⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠚⠛⠋⠭⢽⡇⠀⠹⣯⠽⠚⠋⠉⠙⣾⡟⠺⢽⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡵⣪⢵⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠚⠛⠛⢻⣟⠄⠀⢸⠀⠀⣀⣤⣾⠿⣿⣶⡀⠈⠈⡙⣦
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠁⠈⣴⣿⣿⡳⠌⠉⠓⠲⠤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣿⣷⣶⠿⠾⠛⠋⣁⡤⢞⡵⢻⡇⠀⠀⠀⠜⣧
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠸⣿⡻⣦⣡⠖⠋⢱⡄⠀⠀⢀⣀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣀⡤⠤⠒⠋⠁⠐⠉⣠⡞⠀⠀⠀⢀⣄⣿
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠙⢧⣸⡁⢀⠀⣠⣿⠀⠀⠀⠈⠁⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠁⣠⣾⠉⠉⢷⢦⡴⠞⠉⠀⠀⠀⡠⣫⣾⠇
⠀⠀⠀⠀⠀⠀⠈⠳⡕⢄⡀⠀⠀⠈⠙⠒⢶⣾⠛⠓⠶⠤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣯⠈⠉⠉⠁⣺⣶⠀⠀⠀⢀⣭⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢍⡲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠈⠳⣦⡶⠾⠟⣛⡡⢴⣺⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⢽⣒⣦⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⣒⣺⣭⠷⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠯⠭⠭⠭⢭⣍⣉⣉⣉⣭⣭⣭⡭⠭⠤⠴⠶⠖⠚⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄⣀⣄⡀⣂⠣⠀⠀⠀⠀⠈⠉⠐⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⣀⡠⢴⡷⠻⣧⣀⣼⠟⠙⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠫⢄
⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡔⠉⠀⠈⢊⠓⠺⢧⣼⣖⡿⠛⡧⢦⠱⣀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⡇⣐⠂⠀⣀⣀⣀⠀⠀⠀⠉⠛⠓⠚⠀⠈⠑⠂⢀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⡷⠀⠉⠉⠀⠀⠀⠉⠒⠒⠐⠠⠀⠀⣀⡀⠀⠀⠀⠈⢲⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠂⠀⢠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⢄⡷⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠢⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠦⠶⡴⠖⢾⣀⣨⠔⠲⠐⠆⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡰⠁⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠳⣤⣼⣯⣹⠆⠈⢀⠠⠀⠑⠴⠤⠤⠤⠤⠜⠀⢀⠔⠁⠀⢀⢌⡟
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠑⢤⡀⠀⠀⠘⠈⢈⠅⠘⠒⠀⠂⠀⠒⠚⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠉⠓⠶⢤⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡠⠄⠒⠉⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠤⠤⠔⠒⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⢛⣛⣿⣿⣿⣿⣿⣿⣟⠒⠒⠙⣿⣿⣿⣿⣿⣿⣿⠿⠭⢙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⠤⣄⣸⣿⣿⣿⣿⣿⡏⠉⠒⠒⣿⣿⣿⣿⣿⣿⣿⠓⠢⠤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠲⢄⣀⢸⣿⣿⣿⣿⣿⡏⠙⠲⠤⣿⣿⣿⣿⣿⣿⣿⠓⠦⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⡹⠿⠿⠿⠿⠿⠧⠤⠤⠄⠺⠻⠿⠿⠿⠿⠿⠤⢤⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿
⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿
⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡟⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡇⠀⠀⠀⡰⠃⣇⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠓⠒⠒⠓⢄⡀⠀⠀⠀⣠⠞⠁⠉⠢⣀
⣿⣶⣴⠚⠁⠀⠈⠲⠤⣄⣠⠤⠚⠁⠀⠀⠀⠀⠀⠈⠑⠒⠒⠐⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣠⠴⠒⠊⠉⠉⠒⠦⣄⡀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠂⠉⠉⠉⠉⠑⠒⠤⣀⡀⠀⠀⣀⡠⠴⠊⠉⠉⠉⠒⢤⣀⠀⠀⣀⣸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⣀⠤⠤⢄⣀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠀⢸⣿
⣿⣿⣿⠤⠔⠒⠉⠉⠉⠉⠓⢤⣀⣀⣀⣀⣀⣀⡠⠤⠒⠉⠁⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠔⠒⠂⠉⠉⠉⠉⠉⠉⠓⠤⠤⠔⠊⢹⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡿⠒⠲⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡖⠉⠉⠻
⣿⣿⣀⠀⢠⠋⠀⠈⢱⠖⠋⠑⢢⠔⠒⠒⣤⠒⠒⠲⡤⠒⠒⢦⡤⠒⠒⢦⠔⠒⢢⠤⠒⠢⣤⠔⠒⠢⡔⠊⠉⠲⡏⠀⠀⠈⣆⣠⣴
⣿⣿⣿⣿⣿⣷⣤⣤⣾⣤⣀⣀⣼⣀⡀⣀⣧⣀⠀⣨⣆⡀⢀⣸⣆⡀⢀⣣⣀⠀⣸⡀⠀⢀⣸⡀⠀⣀⣾⣀⣀⣤⣷⣶⣶⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡯⠈⢍⠉⣩⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⢹⠍⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⣒⠄⠀⠴⡂⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠉⠀⠀⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣄⠀⠀⠀⣈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠅⠀⠀⠙⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀⢀⠀⠠⠞⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠄⠀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠘⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠐⡀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⣿⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡄⠔⠻⠀⠙⣿⠃⠀⠀⠐⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⡀⠀⠉⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⡀⠀⠀⠀⠀⠀⠀⠉
⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡀⠀⠀⡀⢀⠈
⠀⠈⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢈
⠀⠀⢸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠌
⠀⢀⠈⣄⠀⠀⡀⢀⠀⠀⠀⠀⠀⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢧⠀⠀⡐
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⢆⠐
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣀⠘⣎
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⢠⠊⡤⠗⠠⢠⠓⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⡆⢀⠀⠆⣠⡸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⣄⠐⠲⢡⡏⢀⠊⢀⠀⢀⡇⠀⠀⠀⠀⣾⢦⡈⡟⠀⠀⠀⠀⡇⠄⡀⠀⢹⢀⢵⠿⡔⠟⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣶⣮⣤⣤⣄⡀⠸⢄⣀⣚⠀⠀⡸⠡⠄⠁⠈⠀⣸⠀⢀⠀⠀⢀⣗⠼⣁⣥⠀⠅⠂⢠⡗⢎⣀⠂⠸⡔⠻⣤⠛⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⢿⣿⣿⣿⡄⠀⠀⠉⠛⠶⠥⠴⠵⣤⡂⠁⡟⠠⠈⠂⡀⢡⡇⠛⡎⣸⠀⢠⢁⠚⡆⢁⠝⣆⠀⡇⠬⠘⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠈⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠓⠛⠓⠊⠁⠛⠷⠛⠐⠓⠛⠁⠁⠉⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠠⣤⠭⠔⠒⠒⠈⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⡀⣀⣀⣤⣤⠶⠶⠒⣚⣿⡿⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠏⠉⠁⢰⡾⠻⢍⣁⣴⣾⣯⣿⣶⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢹⠉⠁⠒⠲⢷⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⢣⣀⡀⠀⢀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡏⠉⠓⠲⠦⣤⣄⡀
⠀⠀⠀⢀⠴⠚⠉⠀⡠⣖⣊⡁⠺⣽⣽⣿⣿⣷⢶⠤⣀⣸⡇⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⣃⣉⣑⡦⣄⠀⠀⠉⠓⠦⡀
⠀⠀⠐⠁⠀⠀⣠⣴⣾⢟⣛⣣⣤⡶⠿⠟⠛⠻⣷⣶⣾⣿⣿⣗⣖⡶⡤⢤⣸⣇⠀⠀⠀⠀⣯⠿⠚⠒⠁⠃⠀⠀⠀⠀⠈
⠀⠀⠈⢦⡀⠀⠛⠛⠛⠻⣍⠀⠀⢀⣤⣶⣞⣛⣩⣭⣤⠶⣿⣿⠟⢛⣻⣾⣿⣿⣿⣿⣷⣾⡷⠀⠀⠀⣠⠔⠀⢀⡠⣄⡴⠃
⠀⠀⠀⠀⠉⠓⠦⣄⣀⡀⠀⠀⠀⠈⠛⠛⢉⡁⠀⣀⠀⠔⠁⠉⠙⠉⠈⠁⠀⠀⢀⣀⣈⡩⠤⠔⣚⡩⣤⣶⣮⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠦⠤⢤⣤⣤⣭⡍⢉⡉⠉⠉⣉⡉⠉⡉⠉⠉⠑⠂⠐⠿⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠀⠂⠖⠲⢰⣤⢀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⡜
⠀⠀⠀⠀⡓⠲⠶⠤⠭⠽⠤⠵⢾⠀⠀⠀⠀⠀⠀⠤⠸⠖⠒⠺⠗⠒⢲⠧⢄⠀⠀⠀⠀⠀⠀⠀⣨⡀
⠀⠀⢴⣋⣙⣛⡒⠒⠒⣚⣚⣛⣋⣠⡄⠀⠀⠀⡇⡉⢛⠛⠒⠒⠒⠒⠒⠒⠚⠃
⠀⠀⢰⠀⣈⠉⢉⠙⠉⡋⢉⢉⠉⢰⡇⠀⠀⠀⡏⠀⠉⠠⠃⠀⠐⠋⠀⠁⢸⠂
⠀⠀⠈⣂⣀⢀⠀⠀⡈⠁⢈⢉⣀⣸⠇⠀⠀⠀⠸⢄⣒⣒⡒⠀⠒⢐⣒⣒⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⢀⠀⢀
⠀⠀⠀⢀⣬⠧⠶⢚⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣸⣀⣠⡧⣠⠄⣼⠤⣼⣀⡀
⠀⠀⠀⠰⢔⡺⠯⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠡⢈⣀⣈⣀⣀⣁⣈⣀⣈⣀⣡⠤⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⠀⠀⠀⠀⠀⠀⠀⡇⠢⢄⡠⠀⣠⠀⠔⢠⡔⠲⠢⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡆⠀⠀⠀⠀⠀⠑⠥⠬⠥⢴⣦⣤⣴⠦⠤⠤⠶⠒⠛⠁
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡆⠀⠀⢀⡌⠀⠀⠨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⣇⣁⠈⠀⠂⠈⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣭⣍⣉⣉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⡤⣐⢒⠒⢢⠀⠀⠀⠀⢠⠖⠐⠢⠤⢤⣤⣤⠸⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⠖⢾⠟⡓⡲⠦⠄
⠀⠀⠀⠏⠀⠒⠉⠘⠃⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠛⠓⠾⠶⠶⠲⠶⠿⠁
⠀⠀⠀⡶⣤⡤⠤⢾⠀⠀⠀⠀⢸⣿⠂⡠⣠⠄⠤⠀⢤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠛⠓⠚⠒⣒⣒⣛⣛⠁
⠀⠀⠀⣄⣀⣀⣀⣠⠀⠀⠀⠀⢸⣏⠒⠀⠀⠀⠀⠀⣈⣼⠀⠀⠀⠐⠠⠤⠤⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠭⠍⣩⡅⠭⠀⠀⠀⠀⠘⢧⣌⣁⣐⠒⠒⠒⢀⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢱⣾⣶⣷⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠂⡻⣀⢸⣭⠀⠱⣈⠂⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠄⠒⢶⡆⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢸⢤⡤⠙⠋⠙⠹⣼⠿⣍⠦⢼⠔⢢⡀⠀⠀⠀⢠⡔⠒⠚⡁⣾⠀⠀⢸⡇⣴⡏⡟⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⢤⣰⡄⠀⠀⠀⠀⢠⣿⣷⣬⣙⠺⣠⣶⣥⡀⢠⠊⢰⡇⢀⢠⠁⡌⠀⠀⢸⡇⣿⠃⠁⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⡀⢈⣠⣴⡞⠢⠐⠀⠘⠻⣯⣟⠿⣿⣷⡛⣷⢁⡀⣯⡇⢂⡴⠰⢁⠀⢀⢜⡃⢿⣳⣄⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠣⠙⣯⣿⣷⣾⣿⣿⣶⣿⣧⣵⣯⣴⠶⠫⢤⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢗⣿⣽⡿⡿⠷⠿⠋⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠘⣿⣏⠰⠠⠀⠀⠘⠃⠀⠀⠀⠠⠅⠀⠀⠀⠛⠀⠀⠀⠠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⡟⣿⠃⠒⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⡴⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡏⢱⠂⣤⣤⠀⠀⢠⡤⠆⢺⢀⣿⣧⡟⠀⠀⠈⠓⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⣇⠸⠀⠿⠳⡀⢀⡸⠳⡄⠘⢸⣿⡟⡇⠀⠀⠀⠠⠁⠈⢤⠀⢠⢐⠒⠨⠤⡆⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡇⠉⢀⠄⠀⠀⠉⠉⠀⠠⠀⢇⡿⣿⢁⠄⠀⠀⠀⠀⠀⢰⠛⠀⠀⢸⠀⠅⠀⠛⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠉⠀⣿⠀⣷⠀⠀⠂⠀⠀⡆⢸⡟⣹⣿⡆⠀⠀⡄⠀⠠⠃⠀⠀⠀⠀⠀⠀⠠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⠆⠀⡏⢰⡇⠀⠀⠀⠀⠀⣷⠀⠇⢯⣿⡇⠃⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢠⡴⠀⠾⠋⠉⠉⠉⠉⠉⠛⠀⠺⢼⡟⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⡏⠷⢴⠓⣟⣠⢦⢀⡶⣀⡶⢦⠀⣼⡇⢀⣇⠀⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⠁⠀⠀⠀⡏⠁⠀⠉⠀⠀⣷⠀⠀⢸⣧⣾⠿⡀⠠⠀⢣⡗⠀⠄⠀⠸⠀⠤⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢰⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣏⠀⠀⠀⢻⣿⠀⠉⠲⣄⢾⡀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠧⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠃⠀⠈⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠹⡆⠀⠀⢹⠀⠇⠀⠀⠀⠀⠀⠀⠀⠰⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⣄⠀⠀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠈⠀⠀⢀⢸⠀⠀⠀⢈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣀⡠⢶⣋⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣘⣦⣀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠖⠢⠤⠤⠄⣀⠤⢀⠤⠤⠤⠴⠒⠒⠉⠉⠉
⣀⣀⣀⢀⡀⣀⣀⣀⢀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣀⣀⡀⠀⣤⣴⡒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠋⠁⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⣇⣴⠷⢀⣀⢀⣠⡆⣠⣤⣵⣶⣿⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢲⡊⠁⢄⡄⠈⡕⠂⠐⠙⠉⠉⠉⠉⠀⠋⣹⡇⢀⠃
⠀⠀⠀⠀⠀⠘⢦⣤⣀⡀⣀⣀⠀⠀⠀⠁⠀⠀⢠⠀⢀⠀⣠⡇⠀⠀⠀⠀⠀⢠⠟⠁⡈
⠀⠀⠀⠀⠀⠀⢀⠉⠁⢉⠿⠛⠛⠐⠒⠺⠿⢋⡛⠛⠚⠃⠩⢥⡆⠀⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢦⣀⣈⣸⡗⠚⠒⠦⠔⠳⠾⠏⣻⠷⠶⠾⣃⣼⡇⠀⠀⢀⡎⠀⡰
⠀⠀⠀⠀⠀⠀⢀⠙⠻⣙⡿⠾⣿⣶⡶⣾⣭⡼⢿⡿⢭⠭⠟⣻⣆⣠⣾⣿⡇⡰⠁
⠀⠀⠀⠀⠀⠀⠘⠿⣶⠛⣧⣤⠀⢀⣤⣤⣤⣼⣿⠓⣒⠊⠿⢺⡟⠻⠿⠯⠖⠁
⠀⠀⠀⠀⠀⠀⠻⠶⢴⣿⠣⢄⠀⣤⣤⠤⠭⢵⣿⠴⣶⡶⠮⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠶⣶⣦⣭⣭⣥⣤⣤⣬⣀⣉⡁⣠⣤⣤⣖⣫⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠓⠒⠛⠛⠟⠛⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿