Tigger Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠤⡄⡀⠀⠀⠀⠀⠀⢀⡀⣠⠤⠄⠀⠀⠠⠤⢀
⠀⠀⣀⠀⡀⣴⣲⠹⢿⡿⠍⠁⠀⠉⠂⠀⠁⠀⢘⠀⠡⠒⠒⠀⠂⠀⠏⡀⠁⠘⠠⠠⠠⢒⡀⠩⡟⣢⡀
⠀⠀⠀⡞⢻⢁⡏⢀⡐⣦⠻⣦⢼⠀⠀⠀⢠⠄⠇⠀⠀⠀⠀⠀⠀⠀⡀⠃⡀⠀⠀⠀⠀⢰⣿⡍⢠⣽⣿
⠀⠀⣀⣍⠉⣩⣴⡾⠛⣌⣷⢹⣏⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢻⣾⠓⣌⣹⡿⡇
⠀⠸⣧⡿⠃⠙⢿⣷⡆⣾⠟⣿⣿⠿⠂⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡰⣆⠀⠀⠀⠀⢺⡿⣾⠅⣘⣻⠃
⠀⠀⠙⢻⣤⡆⣴⣤⡟⠉⠿⣿⣿⣷⠆⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠰⣮⣛⠇⠀⠀⢀⠠⣸⣷⣿⠙⣿⡻
⠀⠀⠀⠈⠛⠛⠁⠉⠀⠀⠀⠉⢻⣿⣄⢠⠀⠀⠘⠎⠳⡀⡀⠄⠐⣦⣜⠻⠀⠀⠀⠀⠱⣽⣼⣿⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣠⠀⠀⠀⠖⣾⣿⣦⡀⢨⣟⢓⠠⠀⠀⠠⢽⣿⣿⣿⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⠀⠀⠹⣿⣻⣿⡿⠈⠉⠀⠀⠑⡀⠀⣀⢿⡻⢿⣽⠉⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠘⣿⣯⡽⠃⠀⠀⠀⠀⠀⠁⢌⣤⡽⢧⠈⢽⡿⣀⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⣠⡇⠈⣼⣬⠛⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠇⣹⠇⠀⠙⠠⣶⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡘⣿⢡⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠇⠀⠎⣿⡟⠀⠀⠀⠀⠈⠙⢟⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣻⣟⠓⠊⣠⣾⠅⠀⠀⠀⠀⢀⠊⣀⣤⣿⠯⢀⣀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⣶⣶⠿⡿⠿⠟⠛⠓

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⡟⢿⣏⣻⣈⣿⣱⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀⠹⠀⠿⠠⠆⣀⡄
⣿⣿⡟⠻⢦⣾⣿⣿⣿⣿⣷⣿⠟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⠈⠁⠀⠀⠀⠀⠈⢋⣀⣄
⣿⣿⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠠⣜⢿⡉⢻⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⢠⡶⢀⠼⣳⣄
⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⠉⠙⡻⣿⣧⡸⣇⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠏⡼⠃⠀⣠⣶⣶⠢⣄
⣿⣿⣿⣿⣿⣿⣿⣷⡿⠛⢁⣙⠻⣮⣻⣿⡿⢣⣿⢿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⣀⠀⢦⡀⠁⠠⠋⣠⠶⣷⣦⡈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣾⠞⡡⣍⠳⣮⣿⣿⣿⢛⣿⢒⡒⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡭⣭⠀⢤⡀⠀⠀⢛⣡⠶⡿⣌⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠶⣽⣾⣿⢩⢿⣿⢯⣿⣶⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⠋⠐⢆⠠⢀⡻⠀⠉⠑⣩⣿⣊
⣿⣿⣿⣿⣿⣿⣿⡿⢋⣶⠒⠷⣶⢿⣿⣿⣿⣿⠾⠿⢿⢷⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢁⢀⣀⣈⠀⠀⠀⠀⢀⠉⣁⣯⠙⢦⡀
⣿⣿⣿⣿⣿⣿⣿⣼⣟⣴⠆⠳⠮⣭⣿⣿⣿⣿⣶⣷⣶⣮⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠋⠋⠁⠁⠀⠀⠀⠀⠒⣊⣩⣏⡙⠦⠙
⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⠖⡀⣛⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠤⡿⣝⠷⡄⠀⡀
⣿⣿⣿⣿⣿⢿⣷⣿⣾⣿⠟⣃⡠⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡻⠷⣌⠀⠈⠀⠁⣀
⣿⣿⣿⣿⣸⢷⡏⣿⢿⣧⣾⣫⣤⣺⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠭⠛⠳⠈⠃⢀⠀⡆⢡⢸
⣿⣿⣿⣿⣿⣾⢱⣏⣾⣿⣿⣿⢻⣿⣿⡿⢻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠇⢱⠈
⣿⣿⣿⣿⣟⡽⣿⣿⡿⣿⣿⣿⢿⡻⣝⢯⡿⢿⣿⠽⠋⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢦⣄⠀⠀⠀⢂⠄⡠⢀⠀⠀⠀⠄⠀⠀⡐⠄
⣿⣿⣿⣿⣿⣟⣏⡻⡿⣿⢿⣿⡎⢷⡝⣾⣿⣾⣿⣛⣥⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠳⠤⠀⠀⠀⠈⡔⣃⡎⠀⠀⠀⠀⠠⠦⠄
⣿⣿⣿⣿⣿⣿⣿⠿⠟⣿⣾⣿⣿⣷⣿⣿⣿⡿⠯⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣂⡀⠀⠀⠀⠁⠀⠀⠈⠀⣠⣀
⣿⣿⣿⣿⣿⣿⣥⣶⣾⣿⣿⣿⣿⣿⣾⡿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⡿⠛⣁⣽⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠶⣄⡀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠶⣦⣤⣤⠶⢟⣛⠭⠭⢭⡭⠭⠭⣝⣛⠷⢦⣤⠾⠟⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠈⠀⡠⠼⠒⠛⠛⣻⡟⠛⠛⡓⠶⣄⠀⠀⢠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠉⣠⢀⡴⣪⠷⠿⠟⣫⡙⠻⠷⠮⡕⢦⡀⣀⠉⠛⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡟⢁⡞⡴⢃⠘⠀⣠⠖⡴⠿⣻⣿⠷⣄⠢⣄⠀⠃⡸⣆⠳⡄⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⡾⣸⡇⠿⠶⠜⣯⠀⢰⢞⣥⢌⢳⡄⠀⡿⠰⠶⡷⣸⠂⢿⣄⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⣰⡟⣼⢻⣿⣶⢶⡆⣌⠃⢸⠋⠼⢎⠻⡁⠈⡥⢰⠶⣶⣯⡟⡆⢻⣦⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡷⢀⣿⠇⣟⢸⠸⣌⡂⠁⢐⡟⠣⠀⠀⠀⠀⠚⣛⡀⠉⢚⡰⠃⡇⣿⠸⣿⡄⢲⡿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⡿⣿⠀⢿⡈⠳⠤⡤⢆⡼⠋⠀⠀⠈⠁⠀⠈⠹⢆⡢⢤⠤⠞⣡⡿⠀⣿⢿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿⡜⢮⣝⡷⣴⣟⠣⡄⢀⡀⠀⠀⠀⠀⢀⠀⡠⠞⣻⡦⡾⣫⡶⣡⣿⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠛⣿⣌⠛⣶⣿⡂⠓⠢⠀⠙⢦⠁⢡⠖⠉⠠⠒⡋⢘⣿⣶⠛⣡⣿⠛⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡇⠀⠀⠿⣗⠘⣿⣧⡀⠈⠉⠀⠀⣹⣃⠀⠀⠉⠀⢀⣾⡿⠁⣾⠿⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠘⢷⣄⠹⣿⣿⡇⢹⣽⣽⣿⣯⡏⢸⣿⣿⡟⣥⡿⠁⣠⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠙⢿⣶⣄⡈⠁⠘⣿⣿⣾⣿⣿⣿⣿⡷⣿⣿⠏⢘⣩⣴⡾⠋⠡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣿⡜⣿⣈⣛⣿⣿⣟⣁⣿⢫⣾⠋⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⠛⠛⠛⠛⠛⠛⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⠀⢀⣠⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⡤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣠⣷⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣟⠁⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣀⠠⣤⣤⡖⠒⢒⣶⣿⠀⠈⣡⠂⠀⠀⣀⣠⣤⣤⣶⣶⣶⠶⣖⣲⣤⣤⣤⣀⡀
⠀⠀⠀⡀⢴⣾⣿⠀⠀⢻⣿⣇⠀⠀⢿⣿⠷⠞⣁⡴⣺⣿⡿⢛⣵⡿⢋⣥⡶⠛⠋⠉⣉⣠⣤⣤⣬⣵⣦⣀
⠀⢀⣾⠀⠈⢻⡿⠛⠉⠀⠀⠀⠀⢀⡞⣡⠤⠐⢮⣿⣿⠋⢠⣾⠟⣱⣿⠟⠀⣠⣶⣿⠿⠛⠉⠉⢉⣉⣉⣙⠓⢄
⠀⣾⣿⣶⣴⠁⠀⠀⠀⠀⠀⠀⢠⢾⣾⣡⠞⠛⢦⣿⣿⢠⣿⠁⣸⣿⡏⠀⣴⣿⠋⠀⠀⢀⣤⣾⡿⠛⠉⠉⣉⠛⠷⣄
⠀⡋⠙⠛⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⡾⢡⣶⣿⠿⣾⡟⢸⡟⢠⣿⡿⠀⣼⣿⠃⢀⡴⢚⣿⢟⡥⢀⣤⣾⡿⠟⠛⢷⣿⡳⡄
⠀⣧⣀⣠⣾⣄⠀⠀⠀⠀⣀⣴⣿⣿⣧⣮⠔⣀⣒⣾⣧⣸⠇⣾⢻⡇⠀⣿⣿⠀⣿⣴⡋⠐⣋⣴⣫⡭⠁⠀⠀⠀⠀⠙⢿⣿⣤⡀
⠀⠘⢿⠿⠟⠁⢸⣿⣯⠁⢹⣿⣿⣿⣿⣷⠿⣿⠿⣿⣿⣿⡄⣿⢸⣇⡆⢿⣼⢞⣃⡀⠙⢻⣿⣯⣥⣴⣶⣶⣿⣿⣿⣶⣄⠈⠉⢻⡄
⠀⠀⠀⠈⠐⠒⠾⠿⠇⠀⠚⠋⠉⠛⠉⠇⠀⠰⠀⠀⠀⢻⣧⣹⣆⢿⣿⢏⡾⢫⠿⠶⠶⢛⡽⣻⡿⠋⠉⠀⠀⠀⠈⠉⠻⣦⣄⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣆⣰⣎⡀⣿⡀⡼⠛⠿⣿⣿⣿⣼⡇⡏⣠⠆⣰⢏⡾⢁⣤⣶⠾⠿⣶⣶⣦⡄⠀⠉⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠹⣿⣿⡇⢰⣧⣴⣿⡿⠛⢋⣁⣀⣀⣀⠀⠈⢻⣿⣦⣄⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣾⣿⣯⣽⠧⡴⣋⣉⣭⣉⠙⢷⣤⣌⡉⠙⢿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⡏⢸⡷⢂⠈⡁⠒⠶⣌⣛⡻⢦⡍⢿⡄⢸⡇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣿⡟⣀⠀⠋⡀⠉⣉⠳⠦⢭⠙⠆⠻⠾⣿⠾⣷⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠋⣙⣤⡈⡵⣩⣌⠁⠉⢀⠶⡀⠛⢠⠀⣦⣽⢷⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⢸⡏⣿⣇⡝⢯⠅⠀⠒⠀⣤⡁⣥⢈⡀⢩⣤⠞⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠳⠺⡇⢿⣿⡿⠺⣶⣲⣾⣶⣿⣷⣷⣍⣵⣿⣿⠿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠴⠸⠟⢹⡿⣿⣯⣭⣿⣿⣿⡿⣫⣿⠩⣿⡐⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⢻⠷⠟⠂⠙⠟⡸⠟⡫⠵⠾⠛⠋⣹⣿⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡼⠿⣶⣼⠚⠕⢂⣰⠲⣒⣤⣾⣿⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡈⠛⠷⠶⠶⢠⣾⡏⢹⣿⣿⣋⣏⡠⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢨⠛⠲⠄⠠⢾⠛⢿⣧⠞⠁⣏⠤⣤⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠀⣀⠀⢀⠃⠀⢰⡏⡶⠁⠀⠀⠈⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠟⢦⠟⠢⠖⠂⢸⠁⠀⡄⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠆⠠⠄⠆⠃⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡀⠀⢀⠀⠲⡄⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡐⣠⣇⠖⣰⡆⠈⠿⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠈⠞⠉⠗⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⣍⢀⣀⣀⡠⣤⢈⣤⡤⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⡌⢘⡱⢦⠘⢿⣿⡇⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢠⠈⡁⢂⠀⠎⠂⠟⠄⡍⢻⡦⣀⣀⠠⣔⡊⡉⠉⠉⠙⢉⠰⢆⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡇⠀⣠⠖⠜⠀⠜⣆⠀⢰⠀⠹⢎⠛⠂⠀⠙⠈⢦⠀⠆⢀⠀⠘⠇⢉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠁⠀⠤⢀⠦⠀⡟⠸⠀⠀⡌⣧⢰⡄⠀⢘⡄⡁⠘⡄⠁⠐⡘⠘⣮⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠖⠐⣴⣄⣤⣠⠜⣵⢃⠀⠀⡇⡉⠀⠇⢠⠀⠃⠘⡀⢙⠀⡀⠆⠂⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠤⢴⣿⣝⠂⡜⢁⣮⠆⠔⠘⠀⡆⡀⣽⠀⢀⠀⠀⠈⡀⠁⠀⣶⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠮⣐⠇⠊⢡⠔⠂⢠⠆⠀⢷⠃⠘⠘⠀⠀⠰⠁⢀⢣⣿⢊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣍⠀⠀⠃⠀⠐⠃⠀⠀⢸⡀⢃⠀⢀⣴⠞⢑⠃⡟⠸⣏⣡⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣾⣤⡐⢤⡃⠀⠀⠀⠸⠿⠟⠛⢿⣷⡌⠑⢺⡅⠀⠙⢧⣤⠗⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠮⠿⢻⡿⠉⠀⠀⠀⠀⠀⠀⢸⣟⠒⢀⣼⠀⠀⠀⠀⠙⠦⠾⣈⢃⡁⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡄⠀⢸⡷⠶⠄⠀⠆⠀⠀⠀⣼⡅⠀⢤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠂⡸⢿⡄⠀⠀⠀⣀⠀⠚⠀⠀⠀⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠦⢤⣠⠀⣀⠀⠀⡅⠐⠷⠴⢦⡄⠠⠾⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠙⠷⠈⠁⠭⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠂⠀⢁⡀⠀⢠⣶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠐⠋⠄⠀⠀⠀⢋⠀⠀⠀⠀⠁⢠⣤⠀⠈⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠻⣆⠀⠄⢘⡀⠀⢀⣀⠘⠀⠢⠀⠀⠀⠘⠀⢠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣘⣂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡆⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠃⢤⣬⣽⡟⢉⡀⠐⠂⠀⠀⠐⣠⡌⠉⠉⠙⠂⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠈⢛⣡⡄⠀⠀⠐⠛⠻⢿⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠀⠀⣀⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡉⠙⢿⠀⠀⣦⠀⠀⠉⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠛⠛⠧⠀⢀⠐⣶⣤⡴⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣤⣤⣤⡄⠀⣼⡷⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢟⠉⠀⠚⠉⢀⣼⣆⠐⡗
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⡀⠀⣴⣶⣄⠀⠈⠀⠒⡄⠉⠉⠁⠛
⠀⠀⠀⠀⠀⢀⡔⢿⠁⢺⡷⢼⢉⣠⣼⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⣠⣙⠿⡶⠂⠁⠀⣤⣤⡉⠋⠀⠀⠀⠀⠀⢰⣤⣀⠈⡃⠀⠀⠀⠀⢀⡄⣠⣤⣄
⠀⠀⠀⠀⣟⣉⣁⠀⠀⠀⢠⣤⣶⣤⠀⠀⠀⠀⠀⠀⣈⠉⠉⠀⠁⠀⠀⠀⠀⢠⢠⣿⡿⠟⠆
⠀⠀⠀⠀⢿⣿⣿⠆⠀⠀⢠⣤⠄⣨⡀⠀⠀⢀⣀⣄⠛⠓⠀⠀⠀⠀⠀⠀⢀⠂⠸⠋⠀⣠⣾⡄
⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠘⢁⣾⠃⡞⢩⠹⡏⠸⣿⣷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⠀⠠⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠤⠿⠆⣴⣤⣀⡀⠀⠀⡀⠀⢤⠄⠀⠈⠁⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⠋⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⠤⢤⣶⡾⠀⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢀⣀⣤⣤⣤⣄⣀⣀⡀⢀⣬⣍⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣡⠚⣟⢛⣿⣟⣓⢮⠳⡄⠀⣼⣾⣿⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡇⣻⠗⠞⠲⢈⣣⢀⡀⣏⢢⡐⢻⣗⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⢶⠉⡟⠱⠀⠘⢀⣩⢭⣴⡯⣠⢹⡄⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⢳⡟⠁⠀⠀⢰⣿⠑⠿⣁⠇⢸⡇⣿⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠀⠀⠀⠀⢀⠃⠐⠔⣀⣀⡸⠀⣿⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠧⠐⠄⢠⠴⠂⣀⡀⣼⠖⠁⡹⠇⢰⠛⠀⠀⣉⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣇⠀⠀⠒⢒⣾⠀⣨⡤⠶⠿⠀⠀⢐⣾⣷⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠉⠉⠓⣾⠓⠚⠃⠀⠀⠀⠀⠀⠛⢱⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠒⢤⠤⠲⠴⠖⠉⡳⢆⣤⣤⣠⡀⣶⣶⡄⡾⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠙⢦⣤⠀⠘⠻⠀⠻⣀⠀⠀⣠⡬⠟⠛⢀⣼⣵⠃⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢢⠀⠉⢿⡀⠀⡄⠀⡀⠉⠁⠀⠀⢀⣠⣾⠟⠁⡼⣹⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠣⡱⡀⠈⢷⠀⠀⠀⠀⠀⣀⣀⠤⢛⡿⠋⢀⠞⡰⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠹⡆⠀⠈⠳⡀⠀⠠⠊⠀⠀⣠⡿⠁⡰⠃⠘⠁⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠰⠄⠀⠀⠠⠀⠀⣰⠾⠛⠀⠈⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⡴⠊⠉⠙⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠉⠉⠲⡄
⠀⠀⠀⠀⣸⠃⠀⣀⣤⣤⣤⣀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢁⣠⣤⣤⣄⡀⢀⣿
⠀⠀⠀⢠⠇⠀⣤⣿⣿⣿⣿⣿⣷⡄⠐⢄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣒⣲⡲⢶⡶⠶⣂⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡜⠀⣴⣿⣿⣿⣿⣿⣯⡄⠈⣧
⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠳⠦⢶⡶⠉⢉⣉⣥⣭⣉⣉⣛⣛⣛⣋⣍⣩⣥⣍⣉⠛⠹⣷⠶⣾⠋⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸
⠀⠀⠀⠀⠣⠀⣿⣿⣿⣿⣿⣿⡟⠿⠷⠀⣠⡖⠁⣠⠞⠛⠋⣉⣁⣤⣬⣭⡄⣬⣭⣤⣌⣉⡉⠛⠻⣦⠀⠐⣦⡀⠸⠿⠟⣿⣿⣿⣿⣿⣿⡇⠠⠃
⠀⠀⠀⠀⠀⢱⣽⣿⣿⣿⡿⠟⠃⠀⡠⠈⠀⠀⠜⠁⣠⣾⡿⠿⠛⠉⢉⡉⢡⣉⣁⠈⠉⠹⠿⣿⣦⡀⠑⠄⠀⠉⠠⡀⠀⠙⠿⣿⣿⣿⣿⣵⠁
⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⠀⡠⠊⢀⡞⠁⠀⢠⠞⠉⢁⣠⡄⠀⠞⠋⢠⣊⠉⣙⠛⠆⢠⣤⣀⠉⠛⣦⠀⠀⠹⣆⠈⢢⡀⢸⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⢸⡿⠋⠁⣀⡾⠁⢠⡟⠀⠀⠐⠉⢠⡴⠋⠉⠠⠖⣶⠞⠀⠉⠀⠿⢷⡶⢦⠈⣉⠳⣄⠈⠳⠀⠀⠙⣦⠀⢹⣦⡈⠉⠻⢿
⠀⠀⠀⠀⠀⠀⢸⡦⠀⢰⡿⠁⢀⣿⡆⠀⠀⠀⠀⠈⠀⣾⠀⠀⣰⠏⢰⡃⡆⢰⡈⢈⢳⡀⠀⢹⡆⠈⠀⠀⠀⠀⠀⣿⣇⠀⠹⣿⠀⠰⡾
⠀⠀⠀⠀⠀⠀⠈⠋⢀⣿⠀⠀⢸⣿⡷⠀⠾⣶⣾⣷⡀⢿⡀⠀⠛⠃⡜⠈⠃⠀⠁⢀⡃⠃⠀⣸⠇⣰⣿⣶⣶⢦⠸⣿⣿⠀⡄⢻⣇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⢰⡇⠈⠈⣷⡀⢀⣠⣤⣤⣄⠈⠣⡀⠀⠀⠁⠀⠀⠀⠂⠀⠉⠀⠀⠀⢀⣤⣤⣤⣀⠀⣸⡏⠈⠀⣧⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⢸⣿⠀⣰⣿⡉⠻⣿⠹⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣽⡿⢹⡟⠋⣹⣷⡀⢸⣿⠀⢿⣿⡀
⠀⠀⠀⠀⠀⡘⠀⣿⣿⠀⠘⡏⠀⣿⠙⢿⣄⠘⠳⠤⠴⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡷⠤⠶⠋⢠⡾⠟⢹⡇⢸⡏⠀⣸⣿⡇⠘⡄
⠀⠀⠀⠀⢠⠀⢸⣿⣿⠀⠀⣇⠀⢻⡄⢤⡉⠛⠶⠆⠀⡸⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠻⡀⠀⠶⠞⠋⣠⠀⣼⠁⢰⡇⠀⢹⣿⣿⠀⢠
⠀⠀⠀⠀⠁⢠⣾⣿⡟⠀⠀⢿⢠⠘⠷⣄⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢉⠙⢁⣴⠏⣠⢸⠇⠀⠸⣿⣿⣦⠀⠁
⠀⠀⢠⠀⠀⢸⣿⣿⡇⠀⠀⣿⢸⡇⢀⠙⠖⠣⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠢⠓⠞⢁⠀⡿⢸⡆⠀⢀⣿⣿⣿⠀⠀⢠
⠀⢀⠃⠀⠀⢻⣿⣿⣧⠀⢠⣿⡆⢶⡸⣷⡄⠁⠀⡠⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠤⣤⠳⣴⣿⣰⠆⣾⡇⠀⢨⣿⣿⣟⠁⠀⠀⢃
⠀⣾⠄⠀⠀⠈⣿⣿⣿⠀⠀⣿⣿⠘⢷⠈⠁⢁⡞⢡⠆⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣄⠁⠀⣸⠟⢸⣿⡟⠀⣸⣿⣿⡏⠀⠀⠀⢼⡄
⠀⡟⡤⠀⠀⠀⢹⣿⣿⣆⠀⠹⣿⡄⠀⢠⣰⡟⠀⣡⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⠀⢀⡀⠘⣷⣰⠁⢀⣿⡿⠃⣀⣾⣿⣿⠁⠀⠀⠠⠘⡇
⠠⣾⡖⠀⠀⠀⠀⣿⣿⣿⣇⡀⠈⠻⣤⡀⠻⢧⣾⠟⠀⠆⠀⣿⡉⠉⠙⠛⠶⠖⠛⠉⠉⣹⣿⡏⠇⠘⢿⣦⢿⡃⣠⡾⠋⢀⣠⣿⣿⣿⡇⠀⠀⠀⠐⣼⡆
⠀⠸⣭⠆⠀⠀⠀⠐⠛⣿⣿⣿⣷⣄⠉⠻⣄⠘⠻⠷⢶⣤⣄⡈⠻⣿⣷⡄⠀⠀⣶⣿⡿⠋⣀⣤⣴⠶⠾⠛⢋⡼⠋⣡⣼⣿⣿⣿⡟⠓⠀⠀⠀⠀⢼⡽
⠀⠀⠉⣄⣄⠀⠀⠀⠀⠘⠛⠻⠛⣿⣷⣦⣌⣿⡶⢦⣤⣤⣀⣀⠀⠀⠙⠣⣴⡤⠛⠁⠀⢀⣀⣀⣤⣤⠶⣾⣯⣴⣾⣿⡟⠻⠟⠛⠀⠀⠀⠀⢀⣄⡌
⠀⠀⠀⠻⣿⣷⣀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣧⣤⡄⣄⠠⠠⠈⠀⠀⢀⣿⣅⠀⠀⠈⠠⠠⢀⡄⣤⣴⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⢀⣰⣿⡿
⠀⠀⠀⠀⠈⢻⣿⡀⣤⠀⠀⠀⡀⠀⠀⠉⠩⠿⢿⣷⣤⣤⣄⣀⣀⣠⠴⢿⣿⡿⠳⢦⣄⣀⣀⣤⣦⣴⣿⣿⡿⠉⣁⣼⠁⡀⠀⠀⢠⡄⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣶⣾⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⢡⣀⠀⠛⠀⣀⣼⠁⢹⣿⣿⣿⣿⣿⠁⣀⠘⠋⠁⠀⣿⣶⣶⣿⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⠿⣷⣦⠀⣴⣿⣦⣤⣤⣿⣿⣿⣿⣤⣾⣭⣯⣿⣯⣯⣼⣶⣾⣿⣿⣿⣿⣷⣶⣿⣷⣄⣠⣶⡿⢿⠏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠈⠀⣠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠋⠉⠋⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣍⣽⠻⣿⣿⣿⡿⠿⣟⣿⡿⢻⣎⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣯⠀⠁⠈⠀⠀⠛⠋⠈⢁⣸⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⢀⡀⠀⠀⣀⠀⡀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠲⠞⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⢠⡶⠟⠛⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣮⣀⠰⠤⢀⡈⠛⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠋⠱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠀⠂
⠀⠀⠀⠀⠀⣀⠀⢰⣶⠀⣰⣶⠖⠀⢀⡀⠀⠀⠀⠀⢰⣤⡤⠶
⠀⠀⠀⠀⠀⠀⠠⢼⣧⣦⣿⣏⣴⠀⣾⡿⠃⣴⢷⡀⠈⠋⠀⣰⣤⠀⢠⣤⡆⠀⠀⣐⣲⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣷⣿⣷⣶⡇⠹⠃⠀⠀⠐⠙⣿⣠⣈⣿⣿⣴⣆⢹⡏⣤⢸⠃⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡻⠇⠀⠀⠀⠀⠀⣀⣼⣿⠿⠿⠿⠿⣿⠷⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡧⠀⠀⠀⠀⠀⠈⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⡀⠀⠀⠀⠀⢉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⣥⣦⣲⣞⠁⠀⠀⠀⠶⢿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣌⠹⣯⣿⠀⠀⠀⠀⢰⡿⢟⣛⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢆⠃⢸⡄⠀⠀⢀⣿⠿⠿⢿⣧⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⢦⡀⠀⠀⡗⠻⣿⣄⠀⣏⠐⠒⠦⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠀⠀⣾⣟⣿⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣽⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣮⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⡔⢄⣀⠀⣤⣤⣄⡀⠀⠀⢀⡠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠻⡶⠾⠷⣢⡶⠖⢄⠰⠂⣡⡞⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣄⠀⢐⡘⣛⢐⢒⡂⠑⡄⠉⠁⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣿⠐⣆⠐⡯⢈⠄⣐⡄⡇⣴⠀⢠⡸⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⠈⡀⠘⠉⠀⠈⠸⠃⠉⠣⡼⡆⢸⡇⢸⠃⣹⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢠⢦⡀⠘⠀⠀⠀⡇⠀⢁⢬⡄⠁⢸⡇⠀⠀⠁⠾⠯⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣙⡇⠀⡀⢄⡀⠀⠀⡇⠐⠁⠀⡼⢃⡤⠶⣚⡿⠟⣛⣿⠿⠿⠷⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢽⡧⠙⢄⡨⠯⠀⡈⠞⢼⡯⡑⠂⢀⠠⣐⠻⠁⠂⢉⣁⣒⣒⣚⣛⣩⣭⣿⣟⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⡤⢤⡤⠴⡃⢠⡻⡌⠀⠀⠀⢀⡼⠀⠑⠀⣀⣁⠈⠤⠤⠤⣤⣌⣉⣁⣀⣀⣤⣤⣄⣀⡀⠭⣙⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡅⠀⠀⠈⠇⠀⠻⠀⢀⡀⠹⡄⠂⠀⠚⢉⣐⠒⠶⠶⠶⠂⠒⠒⠲⠶⠶⠦⠤⠭⢭⣉⣉⠙⠓⠦⣍⡲⢬⡑⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣤⣤⡤⣤⡤⠊⠀⠀⠁⢀⣄⠘⠢⢍⣉⣛⠲⠶⠶⣶⣶⡧⠖⠒⠒⠲⠶⠦⢤⣍⡙⠲⣤⡀⠙⢦⡙⣾
⢀⡴⠒⠒⠒⠲⣶⢤⣄⣀⣀⣠⡤⠴⣖⠀⠀⠉⠀⠀⢀⠀⠺⠿⣦⡙⠷⣶⣦⠀⠉⢹⠗⣴⡿⠋⠠⠒⠚⢛⣛⣛⣓⠲⠦⣍⡲⢌⠹⣆⠀⢳⢸
⣯⢀⡴⠀⠀⠀⠘⠀⠈⠳⠀⠙⢧⠀⠈⠁⢀⣠⠤⠶⢿⣷⣦⣄⠀⠈⠀⢸⠟⠀⢀⣏⠎⣀⣤⠤⣤⣤⡤⠭⠯⠅⠈⣉⡒⠊⠙⠦⡑⡜⣆⢸⣿
⠛⠚⢞⡀⠀⠀⠀⢀⣠⢤⣄⣀⣠⡤⠤⠒⠉⠀⠀⠀⠀⠀⠈⢹⠄⠀⣀⣎⣀⣠⣛⣋⠈⠁⠀⠀⠀⠀⣀⣤⡤⠴⡟⠉⠉⠉⠓⢆⢹⣷⣿⣾⠃
⠀⠀⠀⠀⠀⣀⢠⣤⣖⣒⡊⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢶⠚⠉⠉⠻⠅⠈⣉⣉⣉⣤⣤⡤⠤⠤⠤⠴⠒⠋⠄⣀⣀⣈⡙⠻⣿⣿⠟⠁
⠀⠀⠀⠀⢰⡷⣛⣿⡞⢣⣹⠄⠀⢀⡀⠀⠀⠀⠈⡙⠉⡍⠙⢺⠓⠒⣒⠛⠉⡉⠉⠀⣄⠀⢹⣦⠀⠳⡄⠹⣆⠀⠐⢤⣌⣉⣹⡷⠟⠁
⠀⠀⠀⠀⠀⠙⠃⠘⠛⠯⠓⠒⠉⠁⠀⠀⠀⠀⠀⠈⠛⠓⠶⢿⣤⣤⣿⣤⣤⣷⣤⠤⠿⠶⠾⠿⠗⠒⠛⠒⠛⠓⠚⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠤⢤⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇⢠⣦⠙⠶⠶⣒⢲⣧⣬⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠂⢀⣁⣸⢼⡇⠀⠀⠀⢡⠉⠁⠰⠛⡝⠛⡲⢄⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⢉⣽⣿⡦⢉⡌⠁⠀⢿⣦⣀⠲⠼⠏⠁⡈⠐⠙⢄⢠⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡅⣰⠛⡞⠉⠀⡿⢀⣾⠇⣼⠃⠰⠭⠻⣹⣦⠤⠀⠓⡛⠠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⡟⠀⠓⢸⠏⠀⣼⡃⢸⡟⠠⣿⠸⣤⣠⠄⠿⠟⠓⠀⠀⠙⢦⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣱⠀⠰⣷⠂⢰⣿⠀⢈⠀⣆⠙⢀⣤⠖⠁⣠⣴⣦⣤⣀⣀⣼⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⡖⠀⡟⠀⣹⣿⡀⢸⣧⠙⠷⣾⠁⣠⣾⣿⣿⣿⠀⣾⣿⡿⢧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⢸⡃⣰⣾⡀⠹⣿⣧⠘⠻⣄⠀⢹⣾⡟⠻⣿⣿⣿⣾⠀⠀⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠹⣇⠁⠀⢸⣧⠀⠈⠙⠛⠦⣽⠀⢸⣿⡇⠀⣼⡿⡿⠉⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄⠛⢾⣼⡄⠙⠓⠾⠶⣷⣦⣽⡆⢸⣿⡱⠄⠘⣇⣀⡀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⢦⡀⠀⠀⠙⣏⢿⡇⠸⣿⣦⣾⣿⡭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠙⠺⠦⠙⠀⠈⢿⣤⠀⠀⠊⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⡀⠀⠘⠛⠋⠹⠋


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣛⣝⣛⣩⣋⣻⣙⣹⣟⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠀⠀⠁⡀⠀⠠⡔⠚
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠇⡇⠀⠀⠐⠀⠓⢠⣦⢡⠁⠤
⢀⡐⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠆⢃⡀⠀⠀⠈⠀⠀⠀⠀⣀⡠⢀
⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⣀⣠⣤⠴⠂⠋⠁⠱⠈⡈⠀⠁⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠈⢀
⠻⢛⣀⠀⠀⠀⢀⡀⣤⣠⣼⠀⠨⢛⢏⣇⠇⢋⠄⢁⠄⠀⡐⠀⠇⡀⠀⠒⠄⠀⠀⡸⡄⡀⣐⡀⠦⠒⠊⠉⠁⠀⠈
⠀⠛⠿⠸⣿⡄⠿⠓⠉⢡⡃⠀⠀⠀⠀⠀⢠⠉⣸⢸⡃⢰⡇⢰⠀⠳⣔⡄⢤⠄⠂⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠊⠀⢢⣆⠀⣄⣧⠼⠇⠿⠁⠤⠷⠦⠷⠙⠊⠉
⠀⠀⠀⠀⠀⠀⠠⡄⢀⣄⢸⢠⣽⠼⠈⠁
⠀⠀⠀⠀⢀⠰⣴⡴⠮⠿⠛⠋
⠀⠀⠀⠀⠈⠁
⠀⢀⠀⠀⠠
⣶⣶⠿⣶⠶⠶⣶⢶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⣿⣿⣿⣟⢁⣠⣿⣿⣿⣄
⣿⣿⠛⠛⠛⠛⠁⠈⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈
⣿⣿⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠜⣩⢞⣉⠽⡆
⣿⡟⠀⡞⠉⡶⠋⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢾⠁⣿⢁⠈⢹
⣿⡿⠀⡙⠛⠛⢓⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣷⡊⠃⢹⠐⣢
⣇⡁⠀⣧⣨⣇⣸⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠰⢿⠇⢙⠓⠜⠛⠸⢇⣠⠴
⣿⣿⠀⡴⠀⠖⢶⠀⢰⣶⡆⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⣹⡆⣰⡁⠈⠛⢃⡀⢠⡘⠃
⣿⣿⠀⠳⣄⣠⠞⠀⢸⣿⡇⠀⠀⠀⠀⠈⠆⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠙⣧⢻⠃⢹⣷⠳⢶⣜⠃⣆⠁⠙⠃
⣿⠁⠀⡴⠀⠞⢳⠀⠀⠙⠃⠀⠀⠀⢀⡾⠶⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢿⡀⠻⡿⢶⢀⠈⡃⠀⠉⠁⠉
⣿⡄⠀⠙⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠄⠀⠀⠀⠀⠀⠀⣰⠋⠣⠈⠙⠠⠀⠀⢋⠘⠻⠄
⣿⣿⠀⡍⠉⠉⠉⠀⢰⣶⡄⠀⠀⠀⠀⠈⠁⠠⣿⠁⣶⠆⢠⣴⣿⡎⠒⠀⠀⠀⢀⣤⠀⢰⡏⢀
⡏⡟⠀⡟⠚⠓⠒⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠙⣫⣿⡞⢑⡀⠔⠚⠜⠀⠀⠰⠏
⣿⡄⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⡟⣡⠀⣿⡁⠀⠠⠞⠃
⣿⣷⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠟⠓⠚⠀⠀⠀⠀⠀⠤
⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠦⠟⠈⠛⠋⡻⠧⠂⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠍⣍⠃⣀⡔⠈⣿⠇⡀⣠⠘⠁⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠆⠘⢠⠆⠋⣴⠀⡁⠄⠠⠚⡓⠄⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⣀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢀⢀⠄⠀⠥⠒⠆⠃⡹⠘⠑⠒⠤⢂⠈⣁⠀⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡀⠙⠀⠀⠀⠀⠙⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠠⢾⡇⠐⡚⣹⣼⢁⠀⡤⣬⠁⢈⣭⡑⠀⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢉⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠿⠋⠁⠀⠀⠃⠈⠆⠠⠀⠊⠴⠦⠄⠐⢀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⠈⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠘⡭⢔⠶⡒⢆⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⣀⣘⢁⠀⠀⠀⢢⡀⣑⠀⠀⠀⠀⠈⠛⠲⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⢀⠁⠒⢮⡒⠄⠒⠄⠠
⠀⠀⠀⠀⠀⠀⠨⠴⣋⠙⠭⠥⣬⡪⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠶⢒⡺⠳⠆⠈⠧⢤⡀⠅
⠀⠀⠀⠀⠀⠆⠀⣈⡥⣌⠥⠀⣒⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡃⠦⠥⠤⢁⠢⣀⣈⠳⠙⠂
⠀⠀⠀⠀⠘⠀⠈⠩⠤⢂⣩⠅⠨⡱⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢒⡈⠑⠀⠠⠐⢀⠙⢧⡰⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠬⠥⠲⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠲⣜⠀⡆⠀⠐⠀⠁⠈⢡
⠀⠀⠀⠀⢀⣀⠀⠀⢰⠒⠤⢀⡲⠐⠊⠤⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠮⣝⣂⣨⣦⡐⢈⠁⠀⠀⣄⢀⠎
⠀⠀⠀⠀⣸⡇⡆⠀⠁⢀⣴⢁⠔⣨⠍⢩⠾⠶⠊⠉⠙⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣳⣾⠋⢉⣠⣯⠕⠦⠈⠰⢌⠏
⠀⠀⠀⠀⠀⣿⢀⠄⣠⡁⠁⠀⠀⠀⠰⢋⠛⠀⢄⣉⡒⠉⠉⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⣿⡳⡄⢈⡽⠏⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠸⣡⠶⠞⠁⠀⠀⠀⠀⠀⢈⠀⠀⡀⠠⠁⠀⠏⣠⣾⠋⠁⠀⠀⠀⠀⠐⠻⠃⠀⠀⠀⠐⠛⡿⢧⢁⡈⠀⢦⡀⠸⠂
⠀⠀⠀⠀⠀⠀⢳⣒⠑⠂⠀⠀⠀⠑⠴⣜⠃⢠⠀⢢⡐⣤⠀⢿⣧⢠⠀⠀⠀⠀⠘⠉⠀⠀⠀⢀⡐⢿⡷⡷⠘⠉⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢸⠗⠂⠀⠀⠀⡀⠀⢠⠈⠀⠀⠀⠠⠉⠁⠲⠀⠙⠟⠁⠀⠀⠀⠀⠀⣰⣖⠚⣯⡷⠈⠰⠃⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠖⠀⠒⢐⣄⡑⠀⠢⠠⡐⠂⠀⠐⡶⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠓⠉⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢢⣄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠘⠄⣀⠀⠀⠀⠀⠀⠀⠈⠀⡌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⢻⡶⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⠄⢙⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠉⠀⠀⣠⡔⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠈⠣⠀⣀⣠⣴⢠⣶⠀⡘⠇⣾⠘⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣈⢻⠇⠘⣶⠈⠿⡿⠀⠻⡄⠳⠀⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⣛⠳⠸⠇⠰⡇⠹⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠙⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣶⡠⣦⡉⠓⠀⠀⠀⠾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠈⣿⠃⢿⠙⠂⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⠈⠀⠀⠀⠀⠀⡬⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⠀⠀⡨⢀⡼⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠧⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣉⠁⡘⠃⠈⢠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣗⠚⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣕⣚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣴⠶⠒⠲⠦⣄⡀⠀⠀⢀⣀⣀⠀⢀⣀⣀⡀⠀⠀⢀⣠⡴⠖⠒⢷⣆
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢨⠁⣠⠾⠛⠛⠛⢛⡙⠛⠛⠛⢷⣤⡈⣱⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠈⣇⢸⣆⣠⠖⠋⠈⠰⣲⡶⠶⠶⠛⠙⠻⠶⠶⣄⠠⣁⠙⠣⣄⣴⡞⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢮⠛⡫⠀⣾⠃⠐⢦⣤⣀⣰⡏⢸⣆⣀⣤⡬⠂⠈⣿⠀⢺⡛⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠘⠁⠚⢡⡶⠶⢆⡈⠏⣩⡴⢰⣍⡹⢃⣲⠶⢶⣎⠓⠢⠙⠃
⠀⠀⠀⠀⠀⠀⢀⠔⢡⠃⢰⣄⠉⣠⠶⣶⡅⢰⡃⠁⠘⠘⡇⢠⣴⡶⣤⡈⣠⡆⡀⢤⠈⣄
⠀⠀⠀⠀⠀⢰⡏⢰⡏⣿⠈⢿⣷⡄⠑⢲⢟⠌⠇⠀⠀⠀⠃⣿⣖⠊⢡⣾⡿⠁⣷⠘⣧⠈⡆
⠀⠀⠀⠀⠀⣸⠇⣾⡇⢿⣆⠀⠙⠷⡄⠀⢸⢆⠀⠀⠀⠀⡀⠻⠈⢠⠾⠋⠀⢀⣿⠀⣿⡄⣇
⠀⠀⠀⠀⢠⡏⢸⣿⡇⠘⣿⣷⣦⡶⠶⠄⠁⡌⠀⠀⠀⠀⢱⡀⠠⠤⣤⣶⣶⣾⠃⠀⣿⡇⠸⡇
⠀⠀⠀⠀⠸⡇⠀⣿⣷⡄⠚⠻⣿⡁⡀⠀⢀⠻⠷⣶⣴⡶⠟⠀⠀⠀⠀⣿⠟⠃⠀⣴⣿⠇⢰⡟
⠀⠀⠀⠀⠀⠙⢦⠹⣿⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⢘⡇⠀⠉⢀⡀⢸⠏⠀⢠⠄⠀⣿⡟⣰⠟
⠀⠀⠀⠀⠀⠀⠈⠣⠘⢧⡀⢳⣶⠒⠙⠷⣦⡶⠔⠚⠛⠢⢤⣤⡶⠋⠀⣀⣼⠃⡸⠋⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⠀⠷⠄⢠⡐⠊⠀⠀⠀⠈⠀⢈⡄⠀⠊⠁⢀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠹⢶⣤⣤⣤⣤⡴⠟⠀⠀⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠏⠀⠁

⠀⠠⢀
⠀⠐⠄⠀⡀
⠀⠀⠀⠛⠛
⠀⠀⠛⠻⠆
⠀⠟⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⣦⡶⠦⠀⠀⠀⠀⢀⣴⠂⢸⣏⠀⣾⠋⢉⣥⣶⣶⣄
⠀⠈⠀⣰⡟⠛⣿⣏⠀⠻⠦⠀⠉⠀⠉⢙⣦⡴⢖⣤⡬⠉⡒⠦⣄
⠀⠀⠀⠈⠓⠀⠈⠉⠁⠀⠀⠀⠀⢀⡴⡿⠃⡼⣫⠄⢀⣀⣈⠚⠋⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠁⣼⠋⡰⠚⠉⢉⣉⡙⢦⣤⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢻⣼⠃⣴⢀⡤⠞⠉⠉⠙⠃⣅⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⢷⡏⣿⢸⡇⠀⢹⠎⢀⣤⠶⠶⠶⣦⡘⠁⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣄⢸⡇⣼⠁⡜⠱⠂⠉⢀⣀⣀⣀⡀⠙⠖⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠮⣈⣃⣙⡴⠁⠀⣀⣴⡿⠟⠛⠛⠿⣤⣄⣤⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡓⡚⡃⡸⠀⢀⡾⠟⠉⣀⡤⠶⢦⣤⡈⡏⣉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠡⢸⡇⢰⠏⠀⢠⡾⠉⠀⣀⣀⣈⠻⣿⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠃⢸⠀⠁⠀⣠⢋⡀⠠⠞⠉⠭⣉⣑⠢⠤⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠤⡴⠮⠥⠆⠀⢠⡤⠔⣠⠟⠚⣓⣒⠊⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠔⣠⠶⠞⠁⣳⡇⠀⢡⡾⠋⠀⠒⠢⣤⢠⢬⣖⣠⠤⠠⢔⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡼⣡⠆⣠⡾⡛⡅⠀⣿⠁⣠⠀⠀⣧⡸⢋⡴⣰⣲⠀⡲⢤⡵⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠉⠀⠜⠀⡁⢸⣇⠀⢀⡿⠶⣿⠰⠏⠀⡶⢉⠷⠸⠧⠙⣺⠆⠱⠄⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⣛⡛⠃⠀⣇⢦⣙⣧⡘⣇⢠⡌⠀⠀⠜⢠⢃⡆⡔⣀⡜⠐⠐⣶⠆⠒⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢭⠀⢀⠱⣬⣧⣍⣙⠻⣜⢦⣹⡆⢀⡃⢸⢹⡇⢿⡀⠀⡀⢀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⠀⠈⠰⢤⣍⢳⠹⣄⡙⢳⣌⣳⣌⠳⣝⣦⡡⡄⠃⢻⡤⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠆⠀⠀⠀⠘⡄⠓⢤⠑⢖⠠⠭⠙⠓⠊⢭⡥⡔⠐⠶⠒⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡶⢠⡀⠊⣩⠭⠙⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢧⡀⠑⢛⣉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠲⢄⡢⣭⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠃⠰⠞⠛⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⢀⡄⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠤⠼⠨⠇⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣀⠁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠢⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠉⠁⠹⠖⠂⢤⡀⠀⠀⠀⠀⠀⠀⠠⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⢦⣀⡤⠞⠉⠱⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠰⡎⢀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠖⠳⠎⠀⢠⠈⠑⠲⠦⢤⣤⡤⢀⠀⢤⡤⢤⡄⠀⠐⣠⡿⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠐⣇⡠⡏⠀⠀⢸⠃⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡜⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤⣤⡤⠾⣴⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠉⠁⠀⢀⡜⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠙⢧⡀⠀⠒⠒⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠉⢰
⡀⠀⠀⣠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡐⠰⠆
⠀⠀⢠⡼⠆⠀⠀⡀⠀⠀⠀⢀⡀⠂⠀⠠⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠄⢠⡇⠀⠀⠈⠉⠀⠀⢀
⠈⠁⢠⡜⡀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠲⠲⢤⣌⣉⠒⠠⣄⣀⡙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⣀⠠⠤⠒⣊⣤⠖⠁⠙⢦⣀⠀⠀⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠤⣌⣉⠓⠚⠛⡿⠛⠛⠛⣿⠟⠛⠛⠉⢀⣀⣀⣀⠠⠤⠔⠒⠊⠉⠀⠀⠀⠀⠀⠈⠁⠘⠧⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡙⠀⠈⠙⠉⠁⠴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠒⠒⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⢀⣀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠏⠀⢰⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠲⠶⠶⠶⠺⠟⠋⠁⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⢀⡁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⢨
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣀⡀⠀⠹⣄⣀⣤⣤⣶⠶⠶⣤⣤⣀⣠⠞⠋⠀⡠⢄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠸⠁⢙⡴⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣌⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣡⡴⠋⠀⠀⠀⠀⣠⣴⣦⣄⠀⣠⣤⣤⣄⠀⠀⠈⠳⣆⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡼⠋⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⢳⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⢀⣴⣿⣿⡿⣤⣼⣿⣿⣿⡋⢻⣿⣿⣿⣦⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⡛⣻⣿⣿⣿⣿⠿⢛⣿⣿⣿⣿⣿⣦⡀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠲⠒⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡤⣬⣽⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⣾⣷⣾⣷⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣤⣤⣀⣀⡀⠀⠀⡇⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠚⠋⠉⡠⠁⠀⠀⢨⠋⠙⠛⠛⠛⠻⡉⠉⠀⠩⡈⠉⠙⠲⢤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠑⢄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠑⣄
⠀⠀⠀⢀⣠⡤⠶⠖⠒⠚⠛⠛⠛⠛⠓⠒⠚⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠘⠶⠤⠶⠦⠤⠤⠤⢤⣤⣤⣀⣀
⠀⠀⡼⢫⢝⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠛⢦⡀
⠀⠈⢧⣸⢎⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⢘⠀⢹⡄
⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣇⣈⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣠⡴⠗⠒⠲⣄⡀⠀⢀⣠⡤⠤⣼⣀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣣⡘⣟⣘⠿⡈⢳⣠⠏⣥⣦⢠⣤⠉⢧⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⣿⣿⣷⠉⢸⡏⠰⢠⣶⣶⣦⠶⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠻⠟⢋⣠⠏⠹⣄⠘⢿⣿⡿⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣬⣍⣛⣋⠁⠀⠀⠈⠳⠶⠶⠶⠋⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⢿⣿⣿⣿⡄⠀⠈⠉⢻⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⠾⠛⢻⣧
⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⢻⣿⡆⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠉⢻⣿⣿⣿⣿⣆⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣶⣶⠾⠿⢷⣤⣀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡀⠀⢀⣸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣄⠀⠀⢠⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣾⣿⣿⣿⣿⣤⣶⡶⠶⠾⠿⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣶⣶⣦⣄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⡿⠋⠁⠈⠙⠛⠛⠛⠛⠻⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠀⡠⠦⠄⠀⠀⠀⠀⠚⠙⠀⠀⣿⠛⠛⠂
⠀⠀⠀⠚⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠰⢶⡶⠆⠀⠀⠀⣰⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⣀⣀⡴⠚⠉⠛⣿⣿⣦⣄⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠟⠉⠀⠈⠙⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⣾⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣠⣿⣿⡿⠀⠉⠛⡟⠀⠀⠀⠠⠴⠒⠒⠻⠿⣿⣿⣿⡿⠿⣶⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠃⢀⣼⣿⣿⡟⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠸⣿⣿⣿⣆
⠀⠀⠀⢀⣀⣀⣀⣴⣫⢄⣀⣀⣀⣼⣿⣟⣉⣀⣀⣀⡀⠀⠹⣦⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⡀⠈⠉⠛⠛⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠲⣄⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡀⠀⢨⡛⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⢄⠀⣧⡉⣷⣾⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⠀⠘⢳⡆⠀⣤⣿⡇⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆⢸⣸⣧⡰⡶⣏⣳⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⣄⣦⢀⡈⢦⣀⣨⣯⣿⣯⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣻⡄⡤⣀⣻⢿⣷⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⡟⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⢈⣷⣠⣿⣴⣤⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣀⠀⣤⣦⣤⣠⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⢦⡈⡷⡄⣾⡰⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠙⡷⣧⠘⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢤⡴⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣛⢦⣠⣾⣮⣭⣭⣚⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⡾⠛⣹⡟⣿⠙⡟⢟⠻⢿⣶⣦⣍⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠴⠊⠀⢀⡠⣄⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣴⣿⣏⣠⠞⢁⢣⠃⡧⠇⡈⠇⢀⣿⠻⣷⣍⠺⣾⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢯⡀⠀⠋⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⢾⡟⣿⡞⢏⡡⠞⠻⠽⠀⠈⠳⠃⠈⠋⣄⠈⠻⣳⣜⢷⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡏⣾⢡⣹⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠙⣿⠻⣧⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⢠⡏⢸⡇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⠁⠀⡴⣂⠀⢹⡀⢻⡄⡇⠀⠀⠀⠀⠀⠀⡴⠉⠙⣦⠤⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⢨⠱⣾⠇⢠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⢸⡇⢸⢷⢸⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⢸⠀⣿⡀⠸⣌⡸⠀⠀⠀⢄⠀⠀⠀⢀⡆⠀⠀⠀⣠⡇⢠⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠘⣠⡿⡏⠲⣀⠀⠀⠀⠀⠈⠓⠶⠖⠋⠀⠀⣠⠞⠁⡇⢸⣿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣦⣿⠁⡇⠀⠀⠉⠒⠤⣤⣀⣀⣀⢠⡤⠲⣿⡁⠀⠀⠘⠮⣷⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡏⠀⠀⠀⢀⠴⠊⣿⠀⢀⡠⠴⠳⣄⣷⠙⠦⣀⠀⠀⠈⡶⢻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡇⣿⡀⠀⠈⠳⠦⣼⣿⡦⠋⠀⠀⠀⠀⢹⡏⠓⠚⠉⠀⠀⡇⢀⠱⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⡇⡇⠀⠀⠀⢀⠇⡇⢥⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⡦⠈⢆⡃⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢸⣵⠃⠇⠀⠀⢀⡌⢸⠀⠸⡄⠀⠀⠀⠀⡟⢨⢣⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡜⢀⠇⠀⠀⠹⣀⣤⣄⣼⠁⢸⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⡜⠀⠀⠀⠀⡟⠟⠻⠿⠆⢸⡀⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⡄⢰⢠⣏⠟⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠰⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠆⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠶⡄⣄⠇⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠖⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠚⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⣠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠈⠤⠵⠮⢤⣶⠦⢤⣀⠀⠀⠀⠀⣀⣀⣀⣀⠾⠤⣀⣁⣠⡇⠀⠀⠀⠀⠀⠀⣀⣧⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠈⠉⠉⠉⠉⠀⠈⠙⠿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠼⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⢤⣿⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣸⠀⢀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠉⠓⠀⠈⠀⠀⠙⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢠⠃⢀⣘⡰⠋⢀⡤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠛⠶⣤⡀⠀⠀⠾⢀⠞⠉⣠⣖⣩⡤⠔⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡀⠠⠤⠤⠀⠤⠄⣀⡀⠀⠀⣼⠗⠚⠁⠀⠀⠀⠀⠀⠁⠈⠻⣦
⠀⠀⠀⠀⢀⡼⠃⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠁⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⢀⡞⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠉⠉⠉⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⣾⠔⠁⠀⠀⠀⠠⠶⠚⠋⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⡤⢤⣀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢤⡀⠙⢦⡀⠀⠀⠀⠀⢠⡞⠀⣠⢞⡀⠀⠈⠳⡄⠀⠀⠀⠀⠀⣧
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⣙⠦⣀⠙⠦⠀⠀⠀⠛⢀⠞⣵⣿⣿⣠⠀⠀⢹⠀⠀⠀⠀⠀⢸⡛⣦
⠀⠀⡏⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⣿⣿⣤⡌⠳⡄⠀⠀⠀⠐⣏⠀⠹⢿⣿⠟⠀⠀⣼⢠⡤⠀⠀⠀⢸⡇⣹
⣰⠒⣇⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⠿⠾⠟⠀⢠⠇⠀⠀⠀⠀⠈⠠⣄⡤⠴⠒⠛⠁⠀⠀⠀⠀⠀⠀⢸⣧⠟
⣯⠀⢿⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠘⢦⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠈⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠚⠉⠉⠉⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣀⣀⡀⠀⠀⠀⠶⠶⠿⢷⡀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠁⠀⠀⢀⣀⣀⣀⣀⣀⣳⡀⠀⠀⠀⣰⠆⠈⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢰⣥⠤⠴⠒⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⠠⠼⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠉⠻⣦⠀⠀⠀⠀⣀⣨⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠉⠛⠲⠦⣤⣄⣀⣀⣀⣀⣀⣠⣤⡴⡖⠚⢉⠔⢀⡏⠀⢀⣘⣧⠀⠀⠛⠉⢀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠙⠻⣬⣉⠉⠩⠙⡇⠀⠈⠲⠋⠀⣸⠁⡶⠋⠁⠙⠶⠶⠖⠒⠲⡄⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⣤⣤⣤⣤⠤⠿⠚⠀⠀⠀⠀⠠⡴⠶⠶⢶⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⢈⣉⣡⠟⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡷⠀⣭⣻⣿⣿⣿⡟⠚⠛⢛⣻⣧⣤⡿⠋⠉⠉⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠉⠚⠙⠟⠿⠧⢄⣴⠟⡉⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢻⡾⠃⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⣤⣠⡴⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⣤⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣀⣀⣙⣶⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠒⢶⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⣸⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠉⠁⠀⠈⣿⠹⣆⡀⠀⠀⠀⣀⣀⣤⠞⠋⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⡞⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠈⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⠙⢦⣤⣄⣠⣤⡴⠶⠋⠁⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠂⠀⠀⠀⢸⣷⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆⠀⠀⠀⠀⠀⠀⠠⣄⣀⡤⠶⠚⠛⠛⠋⠙⠛⠛⠻⠦⣤⣀⠀⣼⠋⢀⣠⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢠⣀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠉⠻⣧⡀⠀⠈⢉⡴
⠀⠀⠀⠀⠀⠀⠀⣶⡄⠸⣿⡄⠀⠙⣦⠞⠁⣀⣠⣤⣤⣴⠆⠀⠀⠀⠀⠀⠉⠻⣟⠲⠄⠀⠈⠻⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠙⠁⠀⣴⣧⠶⠛⣩⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠸⣇
⠀⠀⢀⣠⡤⠶⠦⣤⡀⠉⠛⠶⣼⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡤⠤⠦⣤⣀
⠀⢠⡟⠁⠀⠀⣀⣀⣽⣦⣄⣀⡟⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣀⡀⠀⠀⠙⢷
⠀⣿⠀⣠⡴⠚⠉⠉⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠳⢦⡀⢸
⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⢀⣀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠙⢿
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠷⢦⣄⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠶⣤⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⡄⠀⠈⠛⢯⣙⠛⠛⠛⠉⢉⣥⠞⠉⠀⠀⣼⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⣴⠏⠀⠀⢻⡄⠀⠀⠀⠙⠷⣄⣀⡴⠋⠁⠀⠀⢀⡾⠉⢳⡄⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡾⠃⠀⠀⠀⠀⠻⣄⠀⠀⠀⢀⡼⢿⡄⠀⠀⠀⢀⡾⠁⠀⠀⢻⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠹⣆⠀⣠⣿⣁⣀⣽⣦⡀⢠⡞⠁⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠁⠈⠉⠉⠀⠈⠳⠟⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢀⣀⣀⣠⣤⣤⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣴⠞⠋⠉⠉⠀⠀⢀⣠⢴⡏⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⣷
⠻⣤⣀⣀⠀⠀⣰⠟⠁⢰⡇⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⢾⣏⣠⡴⠋
⠀⠀⠉⠉⠙⠛⠻⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠈⠉⠛⠒⠲⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⣠⣶⣿⣷⣾⣿⣷⣾⣿⣿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠋⠀⠀⠀⠀⠉⠀⠉⠈⠉⢿⠁⣾⣿⣿⣿⣿⣿⠟⢿⣿⡿⣿⣿⣿⣿⣿⣏⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⣿⣿⣿⡿⠋⠀⠀⠀⠈⠁⠈⠋⠈⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢨⢭⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡀⡀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡼⠲⡌⠃⠀⠀⠀⣠⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣲⡏⠁⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢄⢴⣔⡀
⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠣⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠐⠨⢀⠀⢀⠀⠀⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠃⠁⠈⠘⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠐⠄⠀⢁⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⢁⠀⠁⠀⠀⠢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠪⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠐⠢⢀⠅⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⢁⠘⠑⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⢁⡰⠓⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠈⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠂⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠐⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠶⠲⠶⠤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠔⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠘⠀⠸⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢦⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠣⠄⡀⢠⠀⠀⠀⠀⠀⠰⠖⠒⠙⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠤⣤⡤⠄⠁⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠐⠀⠁⠠⠀⢰⣆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠳⡀⠀⠀⠀⢀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⢀⠀⡘⠁⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢀⠍⠀⠀⠀⠉⠁⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠀⠒⠒⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠛⠉⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠀⠉⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠦⠀⠀⢀⡶⠠⠤⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⢸⡤⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠑⠚⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠋⠙⠳⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡫⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡠⠂⠀⠀⡠⠀⣠⢾⣀⣞⡴⠋⠙⣆⡄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⢀⣠⣴⠯⠖⠋⠀⠛⠋⠁⠀⠀⠀⠘⣷⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠁⠰⠦⢄⡀⠀⠀⠀⠀⢀⡴⠒⠃⢻⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣶⣶⡍⢠⡇⠀⠈⢡⣶⡆⠀⢸⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠬⠭⠀⡼⠀⠀⠀⠘⣛⠥⠀⢸⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡀⣀⡤⠤⠤⢤⡄⡄⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣤⣤⣤⣤⡤⢤⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢴⠞⢻⠁⠸⠀⠀⠀⠈⡏⢳⡦⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠁⠀⠀⡜⠀⠘⢆⠀⠀⠀⠀⣸⠁⠈⡇⠀⠀⢉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠱⡀⠀⠀⢹⠀⣀⡈⠳⡄⣠⠞⢁⣀⣸⠃⠀⠀⡎⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠑⠀⠀⠈⠉⠘⠧⣄⣸⣁⣀⡼⠀⠁⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢩⠩⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠈⣀⠠⠤⠤⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⡟⠠⣄⡀⠀⢀⠀⡇⠀⠀⠀⢀⡃⠀⠈⡇⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡤⠞⣷
⠀⠀⠀⠀⠀⠀⢹⠶⠦⣭⣙⣻⠀⠑⠦⠤⠴⠊⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢰⣏⣩⣥⠴⣾⠋
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⣀⣸⣏⡄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⠛⠳⠶⣤⣶⣶⠶⠶⢤⣤⣧⣧⣤⣴⣶⣚⣋⡉⠁⠀⠀⠀⠀⢸⣀⣀⣀⣀
⠛⠛⠛⠛⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⡿⢿⠉⠉⠉⠉⠉⠉⠛⢿⡂⠀⠀⠀⣀⡼⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⣿⣤⣀⣀⠀⠀⣐⡮⣷⣿⣶⠒⠛⠉

⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣾⣟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠘⠻⠷⠶⠶⢺⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣠⡤⣄⠀⠀⢀⣴⠄⢀⣴⠆⠀⢠⣾⡟⢀⣤⠀⠀⣠⡀⢀⣾⣏⡼⠁
⠀⢀⡴⠶⠶⣤⡀⠀⠀⢀⣿⡏⣴⣿⡁⠘⡇⢀⣾⡟⠀⣼⡏⠀⣰⣿⠏⢠⣿⠏⠀⣾⡟⠀⣼⡿⠋⠀⡀
⢰⡏⠀⠀⠀⠸⡷⠀⠀⣼⡿⠀⣿⡏⠛⣾⠗⠻⣿⣀⣴⣿⠣⢶⣿⠷⠖⢾⡏⣠⣼⣿⢀⣼⡿⠀⠀⡼⠁
⠸⣇⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠛⠚⠋⠀⠀⠉⠉⣼⡏⢠⣿⠋⣇⠀⠈⠛⠉⠀⠛⠋⠸⣧⣠⠞⠁
⠀⠈⠛⠶⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⣴⠋⠉⢉⣿⠟⠲⢾⡟⠋⣿
⢰⣇⠀⡴⣿⡄⣰⢻⣷⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠸⣧⣠⡏⠀⠀⠀⠀⣼⣄⡀
⢸⣿⣸⠃⣿⣷⠇⢸⣿⠀⢀⣤⡄⣤⡄⠀⠀⢠⢤⠀⠀⠀⠀⣨⠉⠀⣤⠀⠀⢠⣿⣿⡟
⢸⣿⡏⠀⣿⡿⠀⢸⣿⢠⣿⣣⠇⠙⡿⢤⣤⠈⢻⠦⣤⡄⠀⣿⠀⢠⣿⡆⢠⠈⣿⣿⣿⠆
⢸⣿⡁⠀⣿⡇⠀⢸⣿⢸⣿⠁⠀⡰⠁⣼⡿⢀⠏⢠⣿⠇⣸⣿⣄⡞⣿⡷⠋⠘⠻⣿⡏⣤⣀⣤⣦⣄⣀
⠈⠛⠁⠀⠀⠀⠀⠀⠋⠀⠙⠛⠋⠀⠀⣿⣧⠞⠀⢸⣿⡴⠁⢙⣋⢤⣿⡇⠀⠰⣿⠿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⣠⠖⠚⢻⣦⣀⣀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠀⠀⠀⢰⡋⢀⣼⡿⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⡉
⠀⠸⠥⠂⣠⡿⠋⠉⣸⡷⠀⣰⡿⢡⠇⠀⠀⠀⠀⣀⣀⠀⠀⢙⣛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⢀⣀⣼⡟⠁⠶⣾⡋⠀⢰⣿⡵⢃⣴⡾⢛⡆⠸⣿⡀⠁⠀⢿⣄⠉⠀⢠⣴⠞⢳⠀⢀⣴⠶⣾⡏
⡼⠉⢸⣿⠙⣦⠀⠈⣿⢀⣿⠋⠀⣼⡟⠓⠋⢡⢞⣘⣿⣆⡠⣋⣻⣷⣠⣿⠟⠚⠋⣰⣿⠃⣸⡟⢀⡖
⢧⡀⠀⠓⠀⠟⠀⢠⣿⢹⡇⢀⡼⠙⠧⡤⠖⠋⠯⡥⠿⠃⠸⢭⠼⠟⠀⠻⢦⠤⠚⠙⠷⠖⠻⣧⠞
⠈⠛⢶⣤⣤⣤⣾⠟⠃⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⢤⡀⠀⢀⣠⡤⠤⠤⠶⠦⠤⠤⣤⣀⠀⠀⣠⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⢹⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠞⠁⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢀⡞⠀⠀⡇⠀⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⡰⠉⢳⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠰⡆⠀⠀⠀⠀⠉⣻⠋⠁⠀⠀⠀⠀⣶⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠱⢤⣀⣀⣠⠴⠋⠳⢄⣀⣀⣀⣠⠏⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣤⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⠤⣤⣤⢬⡓⠦⢤⣀⣀⠀⠀⠀⠀⣀⣀⡤⠔⠋⠑⠀⣾⡄⠛⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⣤⣄⠸⣿⠁⠉⠉⠐⠃⣶⣶⠈⠋⢩⣽⡍⢁⡌⠀⣾⣻⡆⠀⣾⡇⣴⢲⣄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣼⡏⠙⢿⡀⣿⠀⠀⠀⠀⠀⣿⣿⠀⣰⡿⠛⠁⢸⣧⠀⣿⣿⣧⠀⣿⡇⡿⡟⣻⣇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⡁⢀⣼⠁⣯⣤⣤⡤⠀⠀⢿⣿⣴⣿⠃⢤⠄⢸⣿⠀⣿⠀⢿⡀⣿⡇⣿⡇⠀⢹⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣏⣿⣿⣷⡀⣿⡇⠈⠁⠠⠄⣹⣏⣿⡇⢀⠀⠀⢸⣼⠀⣿⡄⢸⡇⣿⡇⢙⡃⠁⢨⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠐⣿⡋⠉⢻⣇⣿⡅⠀⢰⠄⠀⢨⣿⠙⣧⠀⠀⠀⠸⣿⠀⡿⡇⢸⡇⢸⡇⣿⠇⠀⢠⣧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿⡇⠀⣸⡏⣿⣇⠀⠀⠀⠤⢸⣿⠀⢩⣧⡀⣀⠘⣿⠂⣿⡇⠀⣿⣿⠃⣿⡇⠠⢸⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠂⠘⠿⠾⠋⢠⠘⠛⠟⠛⠛⢀⠸⠟⠀⠀⠙⠿⠃⠀⠹⠃⠻⠁⠁⠘⠟⠀⠳⠶⠶⠟⠁⠖⣇

⠀⢀⣤⣶⢿⣿⣿⡷⣦⣄⠀⠀⣀⣤⣴⣶⣶⠿⠿⠿⠿⠿⢷⣶⣶⣤⣄⡀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣷⣦⣀
⢠⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠟⠋⠉⢀⣠⣤⠴⠶⠆⠀⠀⠀⠈⠉⠛⠻⢿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣯⣿⣧
⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣾⣿⣿⣿⣿⣿⡟⠁⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⢿⣿⣿⣿⣿⣿⣿⣿
⠹⣿⣿⣿⣿⣿⡏⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⢻⣿⣿⣿⡿⣿⡿
⠀⠙⠿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡾⠟⠁
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣴⡿⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⢷⣦⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⢀⣴⠶⣛⡓⢶⣤⡀⠀⠻⣿⣾⡿⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣰⡟⠁⣾⣿⣿⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢿⡟⠁⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠰⣤⡾⢧⣴⠆⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠿⢷⣶⣦⣤⣤⣤⣀⣀⣀⣤⣤⣤⣶⣾⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⠀⠀⠀⢦⢹⣧⠀⠀⠀⠀⠀⢀⣿⢧⡇⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⣿⡇⠀⠀⠀⠸⡆⢿⣇⠀⠀⠀⠀⣾⡏⣼⠁⠀⠀⠀⢸⣿⠘⣿⡄
⠀⣠⣶⣿⣟⣻⡿⣿⣇⠀⢿⡇⠀⠀⠀⠀⢿⠘⣿⡄⠀⠀⢰⡿⢠⡏⠀⠀⠀⠀⣿⡇⢀⣽⡿⣿⣻⣿⢷⣦⡀
⣰⡿⣵⣿⣿⣿⣿⣯⢻⣧⠸⣿⡀⠀⠀⠀⠸⡇⢹⣷⠀⠀⣿⠇⣼⠁⠀⠀⠀⢠⣿⢡⣿⢯⣾⣿⣿⣿⣷⣽⣿
⣿⣷⣿⣿⣿⣿⣿⣿⣧⣿⡇⢻⣧⠀⠀⠀⢀⡿⢸⣿⠀⠀⣿⡄⣿⠀⠀⠀⢀⣾⠏⣾⣏⣿⣿⣿⣿⣿⣯⣿⣿
⢹⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣷⣔⠚⣛⣡⣿⠃⠀⠀⠹⣷⣌⡛⠒⣢⣾⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿
⠈⢿⣞⢿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣙⣿⣿⡟⠋⠁⠀⠀⠀⠀⠈⠛⢻⣿⣿⣁⡀⠀⣿⡾⣿⣿⣿⣿⣿⣟⣾⠟
⠀⠈⠻⣿⣽⣿⣿⣿⣵⣿⠿⠿⠿⠛⠛⠻⠿⠿⠿⠷⠶⠶⠾⠿⠿⠿⠟⠛⠛⠿⠿⢿⣷⣯⣿⣿⣿⣽⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣴⣶⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⣀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⣹⡇
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣦⡿
⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⢸⣿⣿⣿⣿⣿⣿⠉⠉⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⢲⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠐⠿⠶⠒⠒⠒⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⣦⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠉⢉⣉⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⡶⠛⠛⢿⣟⠳⠄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠾⠇⠀⠀⡄⠀⠀⢄⡀⠀⠀⠀⠀⠤⠤⠖⠋⢀⡾⠋⠀⠀⠀⠀⠘⢷⡀⠀⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢀⠀⡀⢻⡀⢳⣄⠉⠑⠒⠂⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠘⣇⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠓⠭⣶⣷⠟⠉⠛⠶⢦⣤⡤⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⢾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢸⡆⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢶⣦⡀⠀⢸⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡇⠀⠿⠁⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠃⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⣶⣇⠀⡗⠀⠀⠀⠀⢿⣆⠀⠀⣠⠀⠀⠀⢠⣀⠀⢀⣿⠇⠀⠀⠀⠀⣾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠈⠙⢧⡇⠀⠀⠀⠀⢸⣿⣷⠛⠁⠀⠀⠀⠀⠉⠻⣿⡿⠀⠀⠀⠀⠀⣿⠀⣸⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠋⠁⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡾⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣴⡋⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⣧⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⡼⠀⡟⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣍⣁⣸⠀⠀⠀⠀⠀⢀⣴⣾⣋⣉⣠⣤⣤⣤⣤⣤⣬⣿⠀⠀⠀⠀⠀⠀⠁⠰⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⠀⠀⠀⠀⠀⣿⣿⣿⠿⠛⠛⢿⣟⣛⣻⣿⣿⣿⠀⠀⠀⠀⠘⣇⣀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢿⡿⠛⣒⣚⣻⣭⣭⣭⣤⣤⣬⣿⠀⠀⠀⠀⠀⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠸⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉

⣾⡆⠀⠀⣰⡿⠀⠀⣿⠀⠀⣴⡿⠻⣶⡄⠀⣠⡾⠿⢷⣦⠀⠀⠀⢰⣷⠀⠀⠀⠀⣾⡟⠿⣶⣄⠀⠀⢰⣧⠀⠀⠀⠀⢸⣿⠛⠿⠿
⣿⡇⢀⣼⠟⠁⠀⢸⣿⠀⢸⣿⠀⠀⠈⠋⠀⣿⡇⠀⠀⠛⠁⠀⠀⣾⣿⡆⠀⠀⠀⣿⡇⠀⠈⢻⣧⠀⢸⣿⠀⠀⠀⠀⢸⣿
⣿⣷⡿⠋⠀⠀⠀⢸⣿⠀⠈⢿⣆⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⢠⣿⢿⣷⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⢸⡟
⣿⡟⢿⣦⡀⠀⠀⢸⣿⠀⠀⠈⢻⣷⡀⠀⠀⠀⠹⣿⣄⠀⠀⠀⢸⣿⠈⣿⡆⠀⠀⣿⡇⣀⣴⡿⠃⠀⢸⣿⠀⠀⠀⠀⣸⣧⣤⣤⣤
⣿⡇⠀⠹⣷⡀⠀⢸⣿⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠘⣿⣆⠀⠀⣿⡏⠀⢹⣷⠀⠀⣿⡟⠛⠿⣷⡄⠀⢸⣿⠀⠀⠀⠀⢹⣿⠉⠉⠉
⣿⡇⠀⠀⢹⣷⠀⢸⣿⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠘⣿⡆⢰⣿⠷⠶⢾⣿⡄⠀⣿⡇⠀⠀⠸⣿⡄⢸⣿⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⣿⡇⢸⣿⠀⢶⣆⠀⠀⣸⣿⢰⣶⡀⠀⢀⣿⡇⣼⣿⠀⠀⠀⣿⣇⠀⣿⡇⠀⢀⣼⡿⠁⢸⣿⠀⠀⠀⠀⢺⣿
⢿⠇⠀⠀⠀⢹⡿⠘⣿⠀⠈⠻⣷⣶⠿⠃⠀⠙⢿⣶⡿⠋⠀⣿⡇⠀⠀⠀⢹⣿⠐⣿⣷⣶⠿⠋⠀⠀⠸⣿⣶⣶⡶⠆⠻⠿⠶⠶⠶
⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⢀⣤⣤⣄
⢰⠏⠀⠀⢻⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤⡄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡞⠁⠀⠹⡆⠀⢠⡟⠀⠀⢹⡆⠀⠀⠀⢀
⣿⠀⠀⠀⢸⠇⠈⣿⣿⠦⠤⠤⠤⠤⠤⠴⢿⣿⣿⠧⠶⠶⠶⠶⠶⠶⢾⣿⣿⠆⢸⡇⠀⠀⢠⡇⠀⢸⠃⠀⠀⢸⡇⠀⠀⣠⣿⣄⡀
⣿⠀⠀⢀⡟⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢸⡇⠀⠀⣼⠃⠀⢸⠀⠀⠀⡾⠁⠀⠀⠈⢿⠁
⣿⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠃⠀⠀⢸⠀⢀⡼⠃
⣿⣠⠟⠀⠀⠀⣀⣤⣤⢤⡄⠀⢀⡤⢦⡀⠀⢀⡴⣦⠀⠀⢀⣤⠤⣤⡀⣤⠀⠀⢸⣇⡼⢣⠶⣄⠀⣼⣠⡞⠁⠀⠀⢀⣤⠤⣤⡀
⣿⠋⠀⠀⠀⣰⠋⠀⢷⡀⢻⡀⣼⠁⢸⡇⠀⠸⣇⢸⠀⢀⡟⠁⠀⠀⢹⣿⠀⠀⢸⡟⠁⢻⡀⣿⠀⣿⠏⠀⠀⠀⢰⡏⠀⠀⣀⡿
⣿⠀⠀⠀⢀⣿⠀⠀⠈⠛⣾⠶⠋⠀⠘⡇⠀⠀⠙⣿⠶⢾⡇⠀⠀⠀⢸⣿⠀⠀⣼⡇⠀⠀⢙⡿⠶⢿⠀⠀⠀⠀⣾⡶⠚⠛⠉⠀⢠
⠙⢦⣤⣤⠞⠉⢷⣤⣤⠾⠃⠀⠀⠀⠀⠻⣤⣤⡴⠃⠀⠀⠻⣤⣤⣤⠟⠻⣤⡼⠋⠷⣤⣤⡾⠁⠀⠘⢧⣤⣤⠾⠉⠳⣤⣤⣤⡴⠋
⢀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⣀⣀
⣿⡇⠀⠀⣿⡇⢸⣿⠀⠀⠀⢼⡇⠀⢠⡿⠋⠙⣷⠀⠀⣰⡟⠉⠻⣧⠀⠀⠀⣿⣇⠀⠀⢠⣿⠉⠙⢷⡄⠀⣿⡇⠀⠀⠀⣿⡏⠉⠉
⣿⡇⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⡇⠀⣿⠃⠀⠀⣿⡇⢀⣿⠁⠀⠀⣿⠀⠀⢰⣿⣿⡀⠀⢸⣿⠀⠀⢈⣿⠀⣿⡇⠀⠀⠀⣿⡇
⣿⣇⣀⣀⣿⡇⢸⣿⠀⠀⠀⢸⡇⢰⣿⠀⠀⠀⠉⠀⢸⡟⠀⠀⠀⠉⠀⠀⣸⡏⢹⡇⠀⢸⣿⠀⣠⣾⠏⠀⣿⡇⠀⠀⠀⣿⣇⣀⣀
⣿⡏⠉⠉⣿⡇⢸⣿⠀⠀⠀⢸⡇⢸⣿⠀⠛⠛⢻⡇⢸⡇⠀⠛⠛⣿⡆⠀⣿⠇⠸⣿⠀⢸⣿⠛⠻⣷⣄⠀⣿⡇⠀⠀⠀⣿⡏⠉⠉
⣿⡇⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⡇⠈⣿⠀⠀⠀⢸⡇⢸⣷⠀⠀⠀⣿⡇⢸⣿⠛⠛⣿⡇⢸⣿⠀⠀⠈⣿⠄⣿⡇⠀⠀⠀⣿⡇
⣿⡇⠀⠀⣿⡇⠀⢿⣆⡀⣀⣾⠇⠀⢻⣇⠀⢀⣾⠇⠀⢿⣄⠀⢠⣿⠃⣼⡇⠀⠀⢸⣧⢸⣿⠀⣀⣼⠟⠀⣿⣇⣀⣀⡀⣿⣇⣀⣀
⣽⣥⣤⣤⣽⣥⣤⣤⣽⣿⣯⣭⣤⣤⣤⣽⣿⣿⣯⣤⣤⣬⣿⣿⣿⣥⣤⣿⣥⣤⣤⣬⣿⣬⣿⣿⣯⣥⣤⣤⣽⣿⣿⣿⣭⣼⣭⣭⣭
⣿⣿⣩⣽⣩⣍⣹⣍⣽⣏⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡾⠿⠻⠿⠷⢶⡟⡛⠻⠏⣉⡛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡾⠛⢛⣙⣃⣘⣿⣿⣿⣷⣦⣽⣷⣦⣼⣿⣦⡘⢿⠾⠟⠛⠻⠿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣶⠾⠟⣛⣫⣴⣾⡿⢟⣛⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣶⣿⣿⣥⣶⡶⣈⣽⠇
⠀⠀⠀⠀⠀⠀⠸⣧⣄⡲⠿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣛⠲⢦⣬⣽⣶⣿⣿⣿⠟⠽⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣷⣌⠻⠶⢶⣄⡀
⠀⠀⠀⠀⠀⢰⡟⢡⣾⣿⣿⣿⣿⣿⡟⠋⢠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⡦⠀⢹⣿
⠀⠀⠀⠀⠀⣿⢡⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢉⠍⣩⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣰⡟
⠀⠀⠀⠀⢸⣏⠈⣸⣿⣿⡿⣱⣿⣿⣿⣿⣿⣯⣾⣫⣾⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠙⣷
⠀⠀⠀⠀⠀⢻⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣍⠱⣾⡏
⠀⠀⠀⠀⠀⢸⣇⢻⡟⣹⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣭⣿⣿⣿⣿⣿⣿⡏⢩⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠻⣦⡄⣿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡹⡾⠿⢿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠛⠻⡯⠙⢿⡿⢉⠀⢸⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠹⡏⠈⢿⠉⠛⠻⠿⠿⠯⠭⠽⠟⠛⠛⠛⠭⠙⠛⠀⢀⠤⠀⠘⠧⠈⡠⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢣⡀⠀⠀⠁⠐⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⠀⠀⢄⢰⢃⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠱⡐⢄⡃⠀⠀⠀⠾⠛⠿⠿⠟⠋⠀⠀⠀⠘⠛⠛⠿⠏⠹⠂⠀⠀⠀⠈⢸⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣭⡚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢣⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢂⣈⠠⢤⣄⠀⠀⠀⠀⠀⠀⡸⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢈⣆⠀⠀⠀⠀⠀⠀⣾⣷⣦⣤⣤⣶⣿⣿⡄⠀⠀⠀⠀⠴⣁⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢋⠖⠁⠈⢆⠀⠀⠀⠀⢀⣿⣿⠛⠻⠟⠙⣿⠿⠇⠀⠀⡠⠊⠀⠀⠙⢎⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢡⠃⠀⠀⠀⠀⠀⢄⡀⠀⠸⠏⣿⣦⣤⣤⣴⣿⠇⣀⣠⠚⠀⠀⠀⠀⠀⠈⢃⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠃⠀⠀⠀⠀⠀⠀⠀⡍⠓⠶⠠⣂⠉⠉⠩⠍⢐⡜⠻⠀⠀⡆⠀⠀⠀⠀⠀⠈⡀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢠⠀⠀⠀⠀⠀⡄⠀⠀⠇⠀⠀⠠⡀⠉⠀⠐⠂⠁⡠⠃⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡰⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⡞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠲⣤⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠴⢶⢋⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡼⢿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⠀⠀⢸⡇⠳⠭⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣠⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⡤⢤⢤⢄⣀⡀⠀⠀⢀⡄⠒⠲⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣰⣶⣴⣶⢠⣇⣰⢀⠞⡴⡱⢰⢲⢸⠇⡏⠇⠀⡐⠁⡑⢀⡏⠀⠁⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⢠⣘⠿⠞⠛⣻⡟⡇⠈⢿⠃⠈⠀⢰⣠⡾⡟⡆⠀⢠⠀⡄⢁⡀⢠⢸⡇⠰⠑⠁⣴⢀
⠀⠀⠀⠀⠀⢠⣼⠋⠀⠀⢀⠓⠁⠃⠀⢠⢀⠀⣠⣼⣿⢿⡏⣴⢃⠆⡰⢁⣾⠁⣿⣿⣣⢀⠀⢰⣿⣿⣶⡤⠆⡈⣳⡀⣤⡄
⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠈⠰⣀⢰⡆⣾⣸⣶⣿⢿⡿⣿⣿⣧⣿⢰⣇⣾⣧⣾⣫⡿⡡⠁⡀⢸⣿⡟⢨⡄⠸⢇⡓⠁⣸⠁
⠀⠀⠀⠀⠀⠐⢻⣀⠀⡀⢲⡄⢸⣇⣿⣻⠿⢯⣷⣾⣦⣿⠘⣿⣿⢿⣿⣿⣿⣿⡇⣼⡶⠀⢰⣾⣿⡇⠡⢵⣶⠈⢰⣶⣧⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣄⢿⡄⣿⣸⡿⠋⠁⠀⠘⢿⣏⣉⠁⠀⠙⠻⠾⢿⣿⣿⣿⣇⣭⠤⠀⠼⠛⠹⣿⣅⢠⡍⠀⣀⣱⠉⠁
⠀⠀⠀⠀⠘⢠⡙⣤⠿⠚⠛⣯⡀⠀⠀⠀⠀⠘⠉⠭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡂⠀⠀⠀⠀⠙⢿⣿⣿⠾⢶⠋
⠀⠀⠀⠀⠀⠐⠮⠁⠀⠀⠀⠙⡇⠀⠀⠀⠀⠽⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⠃⠁⠀⠀⠀⠀⠀⠙⢿⣶⡂
⠀⠀⠀⠀⠀⡀⠸⣄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠀⠀⣀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠑⠢⠬⠱⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠰⠆⠀⠀⠀⠀⠀⠀⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣞⠀⣤⡙⢷⣤⡤⠤⢴⣏⣙⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡄⠀⠐⢘⡆⣯⠁⠀⠈⠘⠁⠛⠉⣩⠿⠿⠷⢤⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣶⡴⢿⣿⣇⠻⠀⠴⠀⢠⣿⡀⠀⣄⢰⡓⠎⠱⠩⡶⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢉⣴⣯⠿⠋⢰⡞⢀⣤⠌⣹⠟⣒⠦⣤⢥⡀⠈⠀⢤⣸⣀⠳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣽⠋⣜⣁⣿⠄⢀⣾⠀⣼⡏⢰⡏⢀⠉⠓⢉⣿⣿⡟⠁⠀⢲⡀⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⣤⠇⠀⠈⣸⡃⠀⣺⡇⠀⠟⠀⠸⣇⠘⠷⢚⠉⠛⠀⠘⠀⠀⠀⠙⣧⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⢠⠘⣹⠁⠰⣿⡇⠀⣤⡸⣦⡈⣰⠟⠡⢴⣴⣿⣿⡶⢲⣶⣶⣅⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⢯⡟⠀⣏⡀⢰⣿⣧⠈⣿⡇⠙⢫⣏⢠⣼⣿⣿⣿⣿⢃⡾⠿⢿⢛⠞⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣏⠘⡇⠰⢿⡅⠀⢻⣿⣄⠁⢻⣤⠀⣷⣿⠋⠛⣿⣿⣿⣿⠃⠀⢸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠘⣧⠀⢀⠸⣷⡄⠀⠁⠙⠛⢶⣿⠀⣿⡿⡄⠀⢿⡿⠿⠠⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⡄⠙⠻⣾⣇⠘⠙⠻⠷⢾⣷⣤⣿⠂⣿⣧⠱⢐⠘⢇⡀⡄⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠛⢦⡀⢀⠀⠈⢳⡜⣿⡀⠹⣿⣶⣾⣿⡭⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡍⠳⢶⡈⠓⠁⠙⣷⣀⠀⠠⠈⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣄⠀⠉⠛⠒⠛⠳⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡀⠠⠄⠀⠀⠀⠄⢀⡀⣠⣤⣤⣤⣀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣀⣀⡀⠀⠀⠠⠄⠀⠈⣻⣿⣿⣿⣿⣷⡀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠤⠀⠀⠁⠐⠀⠄⡀⠈⠉⠛⠿⣿⣧⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⠁⠂⢀⠀⠀⠙⠢⢀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠈⢫⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠄⡈⠂⢄⠀⠀⠁⠢⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠆⠀⠳⡀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈⠐⡠⣀⠀⠑⠌⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠀⡘⠀⠀⠀⡘⠲⣄⠀⠀⠈⠀⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠿⠒⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⢀⠌⠀⠀⢀⠔⠀⠀⠀⠙⠣⠀⠠⠀⠀⠀⠈⠁⠂⠀⠀⠤⠀⣠⠤⠤⠀⢒⣫⠶⠆⠚⡀⠘⢿⣿⣿⣿
⣿⣿⠟⠁⠀⢀⠔⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠌⠡⠄⠀⠀⠬⠀⠠⠔⡒⠛⠀⠀⠐⠂⠈⡰⠀⠀⠈
⣿⣥⣤⡐⣈⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡉⢃⠀⠀⠀⠀⢀⣴
⣿⣿⡿⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣿⣶⣶⣾⣿⣿
⣿⣿⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣼⣷⣶⣦⣀⣴⣾⣧⣴⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠀⠀⠀⠀⢠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣷⡀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⡴⠋⠉⠙⠋⠉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⢀⡞⠋⢋⣛⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣟⡋⠉⠳⣄⠀⠀⢀⠾⢛⡳⣤⣀⡤⠗⠊⠀⠀⠀⠚⢹
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣿⣿⣿⠀⠀⠀⠀⣾⠀⣴⢋⠁⢠⡉⡟⡏⠁⠀⠀⠀⣿⠉⢩⣯⣸⣿⠄⠀⣿⠀⠀⠀⠀⣾⡷⢻⡥⠖⠒⣖⡂⠀⠠⣴⠞
⢀⡴⠖⠛⠉⠁⢙⣦⣀⣀⣼⣁⣰⣦⡈⠓⠀⠀⠀⠹⡄⠻⣌⣛⣻⠵⢡⡇⠀⠀⠀⠀⢹⣄⠈⠛⠟⠋⠀⣠⡾⠁⠀⠀⠀⠉⣀⡼⠃⠀⠀⠈⠙⠛⠛⠉
⢺⣄⣀⠀⠀⠀⠀⠀⣀⠤⢶⠋⣿⠠⠇⣀⠀⠀⠀⠀⠙⠶⣶⣽⣶⢶⠎⠀⠀⠀⠀⠀⠀⠻⠯⣀⡠⠶⠋⠚⠁⠀⠀⠀⠈⣏⠉
⠀⢿⡁⠀⡀⠀⠐⢛⡇⠀⠈⠳⠤⠴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠉⠙⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠉⠛⠧⣤⠴⠞⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⡒⠒⠈⠛⠛⠿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⠀⠀⠈⠉⠉⠛⠒⠒⠒⠚⠛⠛⠳⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡤⠀⠀⠀⠈⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⢀⣀⣀⣤⠶⢾⣿⠿⠿⣿⣶⣤⡀
⠀⢣⡀⣩⡧⣤⠎⠀⠀⠀⠀⡈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⣠⠞⢉⡳⢤⣀⣠⣤⣤⣤⠤⢤⣀⡤⠾⡉⠙⢦⣀
⠀⡿⣹⣿⢁⡎⠀⠀⣶⣀⣀⢹⡀⠹⣷⢹⠓⢦⡄⠀⢰⠉⠀⢀⣞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣸⣀⠀⠀⢱
⠀⡇⢹⠇⡜⠙⠉⠉⠀⠁⠉⠉⠱⡀⢹⡈⡇⡴⠁⢠⠊⠀⡤⠊⢿⠁⢀⠔⠉⠓⠒⠒⠒⠚⠲⡄⠀⡟⠻⣦⠀⠈⢢
⠀⣇⢼⡴⠁⠚⠙⠀⠀⠀⠚⠙⠂⠳⣤⣷⡏⠀⠀⢹⠄⢠⠏⠀⢸⣤⡎⠀⣀⡀⠀⠀⢀⣀⠀⠸⣤⣇⡴⠛⣦⠀⢞
⣠⠊⠀⣇⡀⠘⢗⡒⠾⠖⠒⡾⠀⣤⡏⠸⡄⠀⠀⠸⡄⢸⠀⠀⢧⡀⠀⠀⣁⣁⠠⠄⢈⣈⠀⠀⠀⡸⠀⠀⣳⠀⡞
⢧⠀⡴⣏⠓⠒⠬⡓⠦⠖⡞⠚⠉⣁⢧⠀⡽⠀⠀⠀⠳⠞⠀⠀⠀⣿⡷⣄⣘⢯⣉⣉⣡⣯⠤⢿⣿⠃⠀⠀⠙⠚⠁
⠀⠹⡀⠈⡷⠀⠀⢻⠛⣻⠀⠀⠸⡅⢀⣟⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⡄⠈⢱⡬⢥⠏⠀⢰⠊⢹
⢀⡴⢳⡀⡇⠀⠀⠘⡦⡇⠀⠀⠀⣇⡞⠈⢧⠀⠀⠀⠀⠀⠀⢀⡜⢇⢠⠇⠀⠀⣇⣸⠀⠀⠸⣆⡎⢣
⠛⠠⠤⠽⣇⠀⠀⣸⣁⣱⡀⠀⣠⣏⣀⣀⣨⠇⠀⠀⠀⠀⠀⠛⠲⠼⢿⡀⠀⢀⣇⣹⡀⠀⣠⣏⣠⡤⠗
⣤⣤⣤⣤⣬⣵⣾⣥⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣮⣤⣤⣭⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢋⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢸⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣷⠤⠖⠳⠤⢤⡴⢦⣄⠀⠀⢠⠏⠘⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠖⡻⠃⠀⠀⠀⠀⠀⠈⠀⠈⠙⢶⠋⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⢠⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⣜⣀⠀⠀⠀⠀⠀⢀⣀⡀⣠⠔⠋⠁⠑⠂⠀⠀⠀⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡰⠛⢦⠀⠈⡍⢀⡤⠒⠢⠖⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠲⣶⢾⠃⠰⠃⠀⠸⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠇⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⡄⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢤⣀⠀⠀⠃⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀⠮⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠠⠤⣀⣀⣀⣀⣠⡀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠤⠔⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⢟⠶⣄⡀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠐⣺⠏⢠⡞⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠚⢷⡦⠽⠷⠶⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⠴⠒⠒⠋⠁⠀⢸⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣤⣇⣀⣀⣀⡀⠀⠀⠉⠉⠙⠛⠓⠒⠒⠂⠈⠉⠳⣄⠀⠀⠀⠀⠀⢸⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠘⣇⠀⠀⠀⢀⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠃⠀⢀⡟⠀⠀⢀⡞⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⣠⡴⠋⠀⠀⣠⣟⠀⠀⠀⢰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⣠⠞⠁⠈⢧⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠴⠞⠁⠀⢀⣀⣤⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢰⡶⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⢿⣷⡀⢀⠤⠒⠒⠊⡹
⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⢇⠀⢀⣠⡞⠁
⠀⠀⠀⠀⢰⣁⣤⣴⣶⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠛⠻⣿⣶⣶⣤⣈⣏⠁⠀⠙⠢⣀⡀
⠀⠀⠀⡠⠒⣿⣿⣿⠿⠟⠛⠉⠘⠛⠉⠁⣸⣿⠟⠁⣼⣿⡿⠟⠋⠀⠀⠀⠀⠉⠙⣿⣿⠂⢍⠒⠒⠒⠉
⠀⠀⢰⠁⠀⣿⣿⠀⠀⠀⠀⠴⣶⣤⣄⠀⠋⠀⠀⠀⠉⠀⠀⣠⣤⣶⠦⠀⠀⠀⠀⣿⣿⠀⠈⡆
⠀⠀⠘⡄⠀⠘⠋⠀⠀⠀⠀⠉⠻⠟⠉⠑⠀⠀⢸⠁⠀⠀⠊⠉⠻⠟⠉⠀⠀⠀⠀⠙⠃⠀⢠⠃
⠀⠀⠀⠈⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠉
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠄⠀⠀⢸⡴⠊⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠠⠀⠀⠀⢀⡞⠀⠈⠀⠀⢸⡀⠀⠀⠠⡀⠀⠀⠀⡦⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣶⣾⣤⢤⣾⣄⣀⠀⠀⠀⣸⠁⠀⠀⠀⠱⡄⠀⠀⠇⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⡋⠰⠇⠞⠚⠙⠉⡙⢷⣎⡁⠀⠀⠢⢄⠀⢈⡠⠄⠒⠒⠢⢄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⣠⢞⣽⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⡦⡀⠠⠀⢠⠋⢠⠀⠀⠂⠀⠀⢱⠀⠤
⠀⠀⠀⠀⠀⠀⠺⢣⢊⡇⠶⠀⠀⠈⠀⠀⠄⠀⠈⠁⠀⠠⠦⡿⣿⣜⡄⠀⢸⡀⢫⡀⠀⢀⡀⠀⡸⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⠋⠹⠘⠅⠀⠀⣹⣦⢍⣉⣉⠤⠚⢁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡈⠓⠤⣄⣀⡠⠴⠊⢀⠴⠃⠀⠀⠀⠀⠀⣀⠜⠃⠀⡀⠀⢄⠀⠈⠑
⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⣉⡶⠢⠤⠤⠤⠒⠊⢷⠤⣀⠀⢀⠀⡠⠊⠁⠀⠀⠀⡇⠀⠸⡄
⠀⡴⠉⡩⢅⠈⣇⠀⠀⢤⣴⠊⢁⡇⠀⠀⠀⠀⠀⠀⠘⡄⠀⢹⠿⠋⠀⠀⠀⠀⠀⣈⠥⢄
⠀⢣⣼⣁⣄⡇⠈⡇⠀⠈⠉⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣹⠚⠁⠀⠀⠀⠀⢀⠤⢺⠔⠈⠚⡍⢱
⠀⢮⡤⢽⣉⡜⠉⣀⠀⠀⠀⠀⣀⣀⠀⡇⠠⠤⠄⡀⠀⣤⣄⣀⣀⡀⠀⠀⠘⠤⣌⠪⣶⡷⠙⡇
⠀⠀⠀⠰⣒⣾⣾⠽⠒⠊⠉⠉⠀⢠⠶⠃⠀⠀⠀⠳⠦⠀⠀⠀⠀⠈⠉⠁⠒⠦⢼⣷⣋⠑⠊⠁
⠀⠀⢀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⡄
⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠐⠁⠐⠱⢠⠂⠀⠀⠀⠠⠀⠐⠀⠆⠀⠤⠆⠈⠇⢣⠂⠧⠀⠀⠖⠀⠀⠀⠴⠠⠀⠀⠏⠁⠞⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⠀⠁⠀⡆⠈⠥⠃⠀⠀⠀⠀⠀⠀⠀⢠⠑⡖⠀⠀⢀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡰⢠⠀⠉⠁⠀⠁⢴⢀⠈⠁⠐⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⡀⠁⠀⠓⠀⠀⠀⠄⠈⠀⠀⠀⠀⠂⠀⠁⠀⡨⠀⠀⠀⠀⢠⣀⠃⠀⠌⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠈⢄⡀⠀⠀⢀⠀⢀⠰⡀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠈⢀⠎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠁⠈⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⣠⠀⠁
⠀⠀⠀⠀⣀⠀⢀⠀⠀⠱⠄⠦⠀⡀⠀⠈⠀⢀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢁⡀⠀⠀⠀⠀⠀⡁⠐
⠀⠀⠀⠀⠀⠃⠡⢄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⡀⠀⠀⠀⠊⠀⠐⠂
⠀⠀⠀⡀⠀⡄⠀⠈⠀⠄⠀⠀⠀⠀⢨⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠁⠀⠓⠀⢀⡐⠀⠀⠁⠄⠄
⠀⠀⠀⠁⠐⠁⠀⠀⠀⠠⠀⠀⡀⠀⠘⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠐⡩⠂⡀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣀⠀⠀⠤⠂⠀⣂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⣂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣄⠠⡀⠀⠀⠀⣼⠵⢲⠄⠸⢤⡤⠶⠤⠤⠋
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠄⡀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠆⠠⢤⠖⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⢄⠈⠂⠄⡀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⡿⠀⢠⡞⢛⡿⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⣼⠀⠉⢀⣴⠞⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣠⡤⠖⠚⠉⠉⠉⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠿⢠⠟⠁⣀⣴⣯⣤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉
⠀⠀⠀⠀⠀⠀⣠⣴⣾⠋⠀⠀⡴⠋⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢿⠷⣄
⠀⠀⠀⢀⣴⠟⠉⠀⠘⣦⣶⣿⠶⠟⠋⠉⠉⠉⠉⠉⠉⠉⠛⠷⢦⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣷⣀⣿⣄⠘⢷⡀
⠀⠀⢠⡟⠁⠀⠀⢀⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⢸⣿⡀⢸⣷⡀⣿⠙⠋⠹⣧⡈⢷⡀
⠀⢀⡟⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢠⠟⢙⣧⠿⠉⢿⠟⠀⠀⠀⠘⣷⠀⢷
⠀⣿⠁⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠷⠾⠛⠉⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠈⣧
⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿⡄
⠀⣯⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠛⣛⠛⠃⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⣿⠀⠀⠀⢿⡀⠀⠀⠀⠀⣴⡾⠟⠛⠛⠻⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⢠⡾⠋⠙⠳⢦⡀⠀⠀⠀⠀⠈⠛⢾⣧⣀
⠀⢿⡄⠀⠀⣸⡇⠀⠀⠀⠀⠀⣠⡴⠛⠉⠳⣤⡉⠛⢿⣦⡀⠀⠀⠀⢀⣾⠏⢀⡴⠋⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⣿⠙⣷
⠀⠸⣧⢀⡼⠋⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⣈⡙⢦⡀⠉⠛⠀⠀⠀⠸⠃⣰⢟⣴⣿⣇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣿⠀⣿
⣠⡾⢻⠏⠁⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⢀⣼⣿⣯⣀⡙⠃⠀⠀⠀⠀⠀⠸⡇⢸⣿⣹⣿⣻⠀⠀⠀⢰⣇⣤⡴⠀⠀⠀⠀⣿⡀⠋
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠘⢿⣿⣿⣿⠃⢸⡇⠀⠀⠀⠀⠀⠙⣦⣙⣻⣿⡧⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⢠⡿⠁
⢹⣆⠀⣿⠀⠀⠀⠀⠐⠶⠤⠿⠦⠤⠤⢤⣤⣭⣭⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠙⠳⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠒⠒⠛⠒⠲⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣨⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠛⠂⠀⠀⠀⠉⠉⠉⠉⠛⢿⡄⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢀⣀⣀⣀⣤⣄⣄⢀⣀⣀⣥⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠰⠷⠖⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⡶⠟⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⣿⡍⠀⠀⠀⢀⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠉⠛⠻⣷⠶⠶⠶⠶⠶⣶⣾⣿⠿⠋⠉⠀⠀⢹⡿⣆⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣀⣠⣄⢀⡀⠀⠀⠀⢸⣧⣻⡄⠀⠰⠞⠛⠛⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⠷⣤⣶⣮⣿⣿⠯⠽⠀⠀⣴⠟⠉⠙⣧⣤⣤⣤⣼⠛⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠒⠚⠃⠀⠀⠀⢀⣀⣀⣀⣨⣗⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠉⠓⠶⣤⣄⣀⣀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠺⠟⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣈⣉⣩⣽⣿⣿⣯⣽⣿⡷⠒⠲⠖⠛⢉⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⠉⠉⠉⠉⠁⠀⠀⢠⡾⠋⠉⠉⠉⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣿⠟⠀⠀⠀⠀⠀⢠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣃⣀⡀⠀⠀⠀⢠⡿⠋⠉⠉⢿⡀⠀⠀⠀⠀⠀⢸⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠉⠉⠉⠛⠛⠓⠿⣷⠀⠀⠀⠈⢳⣄⣀⠀⢀⣠⠛⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣿⠛⠛⠛⠁⠀⠀⢀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠟⠛⣿⠶⠶⠶⠾⠃⠀⠀⠀⠀⠀⠹⠷⠞⠻⢿⡟⠛⢻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠉⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣻⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢺⠜⠛⠈⠉⠐⠾⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠘⢛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⢛⣭⣭⣥⣤⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣫⠄⣰⠋⣠⠶⠶⢤⡬⢭⣙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢁⠾⠯⡾⢣⠞⠚⠋⠩⠶⢬⡁⣄⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢀⠈⡀⠀⡡⠟⣡⠤⡴⠛⠶⣄⡈⢩⠅⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡶⢰⠇⡧⡄⢃⡾⣡⣤⣤⠶⠶⣄⣛⠘⡋⠱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠠⠟⡴⢃⠄⢋⡼⣡⣤⣄⣴⡶⣔⡮⡑⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠤⠚⠈⠋⢚⣁⡀⣠⣅⣀⢈⡙⠢⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣉⠀⠀⠀⠰⢊⣉⣉⣭⣅⠠⢍⡛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠉⠀⠀⣀⣀⣂⠰⣬⣁⣀⠨⣍⠉⠙⡳⡝⣆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠠⠖⠈⣶⣶⢏⡻⣆⠡⣍⠶⣎⠳⠴⡂⠤⠄⠜⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⢠⠈⢉⢀⢠⣄⠨⡛⣌⢇⠺⠓⠒⠒⣈⣉⡤⠖⠻⠿⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣂⡀⠀⠡⠖⠀⠋⣈⠁⠀⠈⠸⠀⠙⡒⠋⠹⠋⡴⢆⣴⢞⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣭⣅⣠⣷⠀⠈⠴⠁⠀⠂⠼⠀⠐⠀⠁⠀⠒⠋⠚⣙⠋⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⡌⠘⠓⣾⢈⠁⠀⠀⢀⣉⠘⠂⠈⠍⣀⡀⠶⢚⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠁⠠⣈⣥⣾⣅⠏⠉⠀⠀⠀⢀⠔⠀⠈⠁⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠞⢁⠀⠈⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠤⠒⠉⡀⠀⠀⠀⠀⠀⠀⠠⠼⠄⢀⠠⡀⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠖⢁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⡒⡀⠀⣦⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠷⠚

⠀⠔⠀⢠⠉⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀
⡆⣧⠀⣾⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⣇⠔⢩⠆
⠇⠀⠀⠈⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠀⠰⠊⣰⠁
⠈⡄⠀⠀⠀⠉⠛⢠⠄⠷⠀⠀⠀⠀⠀⣤⣀⠀⠉⠂⠀⣀⣀⡀⣀⠀⠀⠀⠀⠐⠧⣀⠙⠀⠀⠀⠀⠠⢀
⠀⠘⠦⣀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠈⣆⠘⠃⠀⠉⠀⠠⠋⠀⢓⠀⠀⠀⠀⠀⢸⠀⠉⠦⠀⢠⠒⠂
⠀⠀⠀⠈⠃⠘⣇⠀⠀⠀⠀⠀⠀⢀⠰⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⡦⠀⠀⠈⠳⠀⢀⡠⠋
⠀⠀⠀⠀⢸⠀⠹⡄⠀⠀⠀⠀⢀⡹⠖⠒⠉⠉⠉⠁⠒⠦⣀⠀⠀⠀⠰⡁⠀⠀⠀⡔⠁⢠⡎
⠀⠀⠀⠀⠀⣧⠀⢳⠀⠀⣀⠔⠉⠀⣀⠀⠀⠀⢀⣀⠀⠀⠈⠲⡀⠀⠀⠘⠀⢠⠞⠀⢠⠟
⠀⠀⠀⠀⠀⢀⣄⣠⣧⡴⠁⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⢀⡄⠀⠱⠀⠀⢤⣠⠋⠀⢀⠎
⠀⠀⠀⠀⠀⢯⠁⠀⢸⠁⠀⠀⠀⠀⠈⠀⠀⢀⣀⠤⠚⢩⡆⠀⠀⣛⡀⣸⠃⠀⢠⠎
⠀⠀⠀⠀⠀⠘⣇⠀⡆⠀⠀⠀⣿⡏⠉⠉⠉⢁⣀⣤⣶⡿⠁⠀⢀⠀⢙⠟⢦⣴⠋
⠀⠀⠀⠀⠀⠀⢹⡀⢡⠀⠀⠀⢻⣿⣶⣾⣿⢋⣉⢀⡼⠃⠀⠀⢨⡟⠁⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⢧⠈⣄⠀⠀⠀⠙⢿⠁⢀⢠⡤⠊⠀⠀⠀⣠⠏⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠓⠶⠶⠶⠶⠖⠋⠉⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠡⡤⠒⠒⠒⠒⠀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠰⠀⠀⠈⠉⠉⠙⠓⠒⠲⠤⡶⠤⠼⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠠⠤⠤⠄⢾
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠉⠻
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠣⣀⣀⡠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡖⠒⠲⠶⠖⠊⠻⣤⣤⣀⣀⣀⣀⣀⣠⠤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⠀⠀⠀⠀⠁⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠧⠴⡇⠀⠀⠀⠀⢀⠞⠀⢊
⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⡇⠀⠀⠤⠒⠁⠀⠀⣸
⠀⠰⢂⠀⠀⢋⠀⠀⠀⠀⣀⠦⠚⠣⠁⠀⠀⠀⠀⠆⢨
⠀⠉⠂⠠⣭⣭⡥⠍⠁⠎⠑⠀⠀⠀⠨⡆⠠⠄⠚⠠⠙
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠺⠟⠓⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⢀⠈⠉⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⣙⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢰⣷⣦⣀⣾⣧⣴⠀⢹⡀⢀⣀⣀⣤⣤⡶⢶⣶⣶⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⣠⣾⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⣿⣿⣿⡟⠿⠿⠙⠧⠀⠿⠿⠛⠉⢁⣀⣤⣤⣤⣤⡤⠤⠈⠙⠛⠛⠷⠶⠤⠄⠀⠠⠏⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⣿⣿⣿⡟⠢⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⣉⣡⣴⣾⣶⣦⣍⡁⠚⠻⢿⣦⣤⣀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠖⠀⣾
⠀⠀⠀⠀⠀⠀⣀⣤⣾⡄⠈⠻⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣶⣶⣿⣿⣿⠋⠀⠀⠈⠙⢿⣦⡌⠉⠉⠉⠙⠛⠃⠀⠀⠀⠀⣤⣿⡯⠀⣼⡏
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⠄⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣠⣴⣿⠋⠉⠉⠛⠛⢁⡀⢧⡇⢷⣀⣈⣛⡿⢷⣤⣤⣤⣤⡀⠀⠀⢠⣴⣶⣾⠁⢀⣼⠇
⠀⠀⠀⢀⣴⣾⡿⠛⣿⣟⠉⠀⢻⣿⣿⠿⠍⠁⠀⠀⠀⣩⣥⡿⠃⠀⣠⣶⣿⣿⠟⠛⢻⣧⠀⠉⠉⠙⠻⣷⠦⣤⡀⠙⠻⣦⡀⠉⠛⠿⠤⠄⢺⠟
⠀⠀⠴⢿⣿⣿⣤⠀⠙⢿⡀⠀⢠⡿⠃⠀⣀⣤⠆⠀⣾⣿⠋⠀⠀⠀⠉⣀⣴⣶⡆⠀⠀⠛⠀⠀⢀⠀⢤⣄⠀⢈⡁⠀⠰⣌⠳⠀⠀⠐⢦⣴⣿⣆
⠀⠀⠈⠙⣻⣿⣷⡦⠀⠈⠳⠀⠀⢀⣴⣾⡿⠃⠀⣠⣿⠇⠀⠀⠀⠀⠚⠛⠀⠸⠿⣿⡇⣦⠀⠀⠀⣙⢶⣾⣆⠀⠉⢷⡀⠹⣦⡀⠀⠀⣄⠈⠻⣿⡆
⠀⠀⢠⣾⣿⣿⣿⣶⣦⣤⣤⠄⣠⣿⣿⠟⠁⠀⢸⡿⠁⠀⢀⣀⠀⣤⡀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠘⣦⡁⢹⠀⠀⠀⠛⣴⣼⠿⠆⠀⠘⣦⡀⠘⣷⡀
⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⢃⣴⣿⡿⠁⠀⠀⠀⠈⢠⡟⢠⣿⣿⣶⡿⢧⠀⠀⠀⠀⠻⣿⠋⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠙⠋⠀⠀⢠⡀⠈⢿⡆⢹⣧
⠀⠈⠉⢉⣽⣿⣿⠟⠁⣰⣿⣿⡿⠁⠀⣠⣿⡀⢠⣿⣿⠀⠉⠉⣉⣁⣄⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡆⠀⢠⣶⣿⣷⡶⠖⠘⣌⢷⡀⢸⣿⠀⢻⣧
⠀⠀⢠⣿⣿⣿⡟⠀⠸⠋⣽⣿⡗⠀⣼⣿⣿⡇⢸⣿⡿⠀⠘⠻⣿⣿⣿⠿⣧⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣟⣋⡜⢁⡤⠀⠸⣾⡇⢸⡿⠀⠀⢻⣧⡀
⠀⢀⣿⣿⣿⣿⣷⠀⠀⢰⣿⣏⡀⠰⣿⣿⡿⠀⢸⣿⠃⠀⢲⣄⠙⠶⠶⠾⢿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⢀⣴⣃⣤⠄⠀⣿⠃⢸⡇⠀⠀⠈⣿⡿
⠀⣼⣿⢿⣿⣿⣿⡆⠀⠟⢿⣿⢁⠀⠸⣿⡇⠀⢸⣿⠀⠀⠀⠈⠛⠒⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠁⠀⠉⠁⠀⢀⣸⡟⠀⣿⠀⠀⠀⠀⢿⣿
⠘⠋⣡⣿⣿⣿⣿⣷⠀⠀⠘⢹⣿⠀⠀⢹⣿⠀⢸⣿⣆⡀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠈⢙⣻⡇⠀⢹⡆⠀⠀⠀⣾⣿
⠀⣰⡿⢿⣿⣿⣿⡿⠀⠀⠀⣾⣿⡇⠀⠀⠻⣇⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠠⡀⠙⢿⣿⡇⠀⢸⡇⠀⠀⠀⣿⣿
⠀⠁⠀⣼⣿⣿⣿⡇⠀⠀⣼⣿⣿⡇⠀⠀⣤⣍⣷⣤⣀⣀⠀⠲⣦⣄⣀⣠⡞⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢻⡄⠈⠛⠀⣷⣾⣷⡀⢠⣀⣿⡟
⠀⠀⢰⣿⣿⣿⡿⠀⠀⠰⠟⢩⣿⡆⠀⠀⠙⠿⠟⠋⢉⠉⢉⣀⣨⣿⠟⠋⠀⠀⠀⣼⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⣿⡄⠀⠀⣿⠀⠀⡀⠈⣿⡟⠁
⠀⠀⣾⣿⣿⣿⠃⠀⣸⡇⠀⠘⢻⡧⣀⠀⠀⠀⠀⠀⢙⣿⡿⠿⠿⠃⣀⠀⠀⢀⣠⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⣬⣷⠀⣼⡿⢀⣼⣿⡿⠋
⠀⢰⣿⣿⣿⣏⣤⣾⣿⠇⠀⠀⢸⣷⣿⡧⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠴⠾⠿⠿⣿⣿⡏⠡⣄⡤⠖⠒⠶⠾⠿⠷⠒⢀⣱⠀⠀⣹⣿⡆⣿⠃⣾⣿⡟⠁
⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠛⠁⣹⢿⣀⡄⠀⠚⢿⣿⠂⠀⠀⠀⠀⠴⠶⣶⣾⣿⣛⠀⠘⠻⠿⣿⣶⠀⠀⢠⡾⠛⠁⠀⠾⠋⠉⠀⣿⣾⣿⠟
⠀⠈⣿⡿⠿⠛⢻⣿⣧⣶⣶⣿⣷⠀⠀⠞⠋⠀⠀⠀⣠⣿⠓⠀⠀⠀⠠⠶⠤⢤⣄⣉⣉⠉⠀⠀⠀⠀⠙⢶⡗⠊⠀⠀⠀⣠⡤⠴⠋⢠⣿⣿⠏
⠀⠀⠈⠀⠀⠀⠈⣿⣿⣿⣿⣿⣧⡔⠀⣠⣶⠋⠀⠾⣫⡵⣦⣤⠀⣶⡀⠠⣄⢀⣀⣉⡉⠙⠛⠛⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⢠⣶⠝⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡿⢻⣿⣿⣿⢠⣾⣿⠇⠀⠀⠀⠀⠀⠉⠁⠀⣿⣷⣷⣤⣤⣥⣤⠄⠀⠀⠀⢀⣠⣴⣿⣶⣶⣤⣀⣀⣼⠟⠁⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡟⢹⣿⡿⣫⣶⣧⠀⠀⠀⠀⠀⠀⢀⡀⣾⣿⣿⣿⣿⣿⠿⣧⣤⣶⣾⣿⡿⠛⠛⠛⢻⠏⣿⠋⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠊⠙⢟⣵⣿⣶⣦⣤⣤⣀⠈⢴⣿⣿⣿⣿⣿⣿⡀⢻⣿⡿⠋⠻⠇⠀⠀⣸⠋⣸⣿⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣶⣆⢻⣿⣿⣿⣿⣿⣷⣤⣧⣀⣀⣀⣤⣶⣶⣷⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⢿⣿⣿⣿⣿⡄⣿⡏⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠯⢼⣛⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⠸⣿⣬⡉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣆⠁⠈⠙⠻⢶⣶⣤⣤⣀⣀⣀⣀⣠⣤⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠜⢀⠥⠦⣖⠲⣶⣗⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣬⣜⡳⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡞⠁⠀⠀⠀⣋⢿⣍⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠁⠀⠀⢂⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⢀⣀⠤⠬⠴⠥⣀⠀⡤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⡀⠀⢀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠉⠀⠀⢀⠀⠄⠀⢛⣧⡸⢣⡆
⠀⠀⠀⠀⠀⠀⡠⠚⠉⠄⠀⠩⣉⠛⢯⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠸⠀⠀⢀⠂⠁⠀⠀⢀⠗⢲⢦⠲⡬⠂
⠀⠀⠀⠀⠀⡼⠁⡄⠀⠈⠀⠀⠈⠄⠀⣆⡠⠖⡲⠋⠁⠀⡇⠀⠈⠃⠌⠉⠒⢤⡁⠀⠀⠀⠄⠂⠀⢱⠊⠳⠇
⠀⠀⠀⠀⠀⡇⠀⢂⠀⠀⠈⡀⢀⡴⠊⠁⢠⠋⠀⠀⠀⠀⡅⠀⠀⠀⠈⠢⠀⠀⠈⠢⡔⠉⠀⠀⡠⠃⠀⢠⠃
⠀⠀⠀⠀⠀⠹⡈⢧⣊⡂⢀⡼⠋⠀⠀⠰⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠡⠀⠀⠀⠙⣄⠤⠊⠁⠀⣠⠎
⠀⠀⠀⠀⢀⠐⢻⡿⣏⣹⠎⠀⠀⠀⢀⡅⢲⣦⡀⠀⠀⠀⠃⠀⠀⠀⣠⢲⣦⣅⠀⠀⠀⠈⣦⠠⠒⠊⠁
⠀⠀⠀⠀⠀⠉⠁⠶⢀⠏⠀⠀⠀⠀⡌⢿⣿⡿⠃⠀⠀⠀⠆⠀⠀⠀⢿⣿⣿⠯⠄⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣦⣄⣥⣴⣦⡀⠀⠀⠄⠀⠁⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠟⠛⠉⠉⡹⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠊⠀⠀⡠⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡩⠲⣄⡀⠱⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠎⣀⡴⠺⣙⡆⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣘⡒⠠⠼⠥⠷⠖⣢⣤⣤⣐⣂⣤⣐⣂⣀⠭⠬⠽⠒⣊⣩⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⠉⠙⠒⠒⠒⠦⠤⠤⠤⠤⠤⠤⠒⠒⠒⠊⠉⠉⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠦⠤⠤⠤⠤⠤⠤⠤⠤⠒⠒⠒⠒⠚⠉⠉

⣿⣿⢛⢿⢛⠟⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⠿⠟⠑⠿⢻⣟⣿⠿⠿⠻⣿⣿⣷⢹⠻⠷⣋⣛⣦⣍⠂⣦⣽⢽⣽⣿⣹⣷⠾⡿⠈⣿⣞⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⣿⣿⣾⣿⣷⣯⠻⠿⠿⡃⣊⠙⣁⢙⠿⢷⣻⢸⣿⢷⡍⢾⣿⣿
⡿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢿⣋⣿⣴⣶⣿⣷⣧⡫⠛⣿⣧⣟⣥⡘⠻⢆⣿⣿⣷⣤⢿⣟
⠃⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣹⣻⣿⣽⣿⣯⢯⡄⣏⢥⣽⣿⣽⣷⣄⡂⠼⣯⣾⣆⠸
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣸⢻⣿⣿⣿⠿⢞⣧⣿⣿⣾⣿⢿⣷⡄⡑⢶⣌⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣼⣿⣯⢛⣷⣾⣯⢻⠿⣿⣿⣷⣿⡁⢿⣿⣿⣻⡏⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢸⣿⣟⣿⣣⣾⣿⣯⣟⡻⢿⣷⣿⣿⣻⣿⢸⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣏⣿⠀⢠⣾⣿⢍⣹⣿⡯⣿⣏⡛⣿⣷⣿⣝⡧⣿⣿⣾⣿⣟⢾⢩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡿⢏⣸⣯⣿⣶⣽⣝⡿⣦⠉⣿⣿⢹⣿⡏⣹⢿⡯⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠚⠟⠂⠀⠀⠈⢉⣿⣼⣿⣿⣿⣿⣿⣿⣾⠂⣼⣙⡷⠿⡅⣼⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠹⣿⣿⣿⣿⣼⣯⣿⠷⡟⠿⣧⠘⠁⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠟⢿⡂⣷⣿⣂⠐⣡⡷⣿⡷⣰⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠐⣻⣶⣿⣾⢧⡭⣿⠃⣴⣿⣷⢌⡹⣿⣿⣷⣉⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣩⣾⣿⣿⣛⣦⣹⣷⢻⢸⣷⣿⡟⢸⣷⣼⣿⣿⣿⣷⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢟⣩⡿⣿⣿⣁⠉⣌⣼⣿⢑⣿⣾⠉⠉⣿⣽⣽⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣿⣿⣿⣯⣼⣿⢿⠟⣀⡉⠀⡶⢛⣫⣴⠺⣷⣙⣿⠸⠿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠂⣸⠾⠷⠚⠋⠈⡱⠈⡇⡀⠁⠠⢤⠠⡎⣿⣛⠁⣛⣙⡇⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⢟⡓⢀⡴⣾⢿⢋⣠⣤⣬⣟⠹⠇⠐⣒⢠⢀⡀⠿⣿⡇⢿⢟⣽⢢⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠛⣽⡤⢺⡟⣶⢿⣿⣿⣿⣼⠷⣷⠈⠋⣿⣋⣿⣯⣸⡷⠀⢰⣾⣿⢨⣭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡕⣌⠳⡞⠾⢟⣽⡛⣿⣿⣏⣶⣻⣴⢳⣿⣽⠟⣿⣶⠰⠃⣼⣷⡿⠘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣣⣽⣦⠘⠙⣼⣿⣿⣏⠙⠣⣿⣿⡹⣅⣿⣿⠇⠘⢱⡏⠉⣧⣷⣷⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠆⠈⣙⣥⣾⡧⣔⢿⣿⡇⠀⡁⣶⣶⣧⡌⣤⣤⣿⣽⡇⣻⣟⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣈⡠⣤⣤⡝⠶⢿⢿⡟⢋⣵⣤⣿⣿⣿⣫⡆⣿⣿⣿⠷⢻⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣿⣿⡈⣥⡄⠒⠨⠸⢾⣿⣟⢷⣻⣽⣿⡏⣣⠘⣟⣭⣆⣾⡷⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⠿⣿⣿⣶⣼⣷⣀⡙⠿⠟⠿⢳⡏⠓⠛⣃⠉⠠⠹⣿⡃⣿⣽⣿
⣾⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⣖⠿⡿⢿⣿⡻⠟⠇⠄⠈⣉⠀⣁⢈⣭⣋⡄⢰⣷⢹⣇⣿⣾⣿
⣯⣿⣿⣇⣿⣳⣦⣀⠀⠀⣀⣄⣶⣿⣮⣍⡹⢶⣖⠦⣀⡀⠄⠒⢻⣿⣷⣦⣿⣿⡷⢯⣹⢷⣿⣿⣿⣽⣿⣿⣿
⠉⢩⠉⠁⣭⢥⣿⣿⣿⣿⡿⣛⣿⣿⣿⠿⠟⣋⣵⣮⣿⣿⣿⣟⣲⣼⣽⢻⣿⣷⣅⣿⣿⡆⢽⢿⠛⡇⣹⣿⣿
⠸⠠⠃⣶⡿⣸⣻⣿⣿⣿⠿⠿⣛⣭⣶⢾⣿⣿⣿⣿⡿⣿⣿⣿⣟⣿⣏⣹⣿⣯⣼⣷⣿⡷⣻⣿⣇⢩⣿⣿⣿
⢢⣾⠴⡦⠀⢿⠛⠛⠙⢑⢢⣿⣿⣽⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣷⣮⡹⣋⡇⣿⣿⣿
⣶⣒⣸⣷⡘⣯⡇⣿⣵⣿⣿⣿⣿⣷⢺⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠻⢿⠿⠿⣿⡿⢿⣿⣿⡿⣷⣞⢟⡳⢿⣻
⣿⣶⣶⣿⣷⢿⣿⢸⣿⣿⣿⣿⣿⣿⣇⣛⠻⣿⣿⣿⣿⣿⣷⣿⣿⡗⣿⣿⣿⣿⣿⠷⠞⠿⢶⣯⣭⠭⣷⠻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢛⢩⠀⣆⣤⣄⠤⣚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡉⢦⡋⠳⠐⣭⠂⠩⣼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣸⡅⠾⡽⢇⢶⣭⡩⠴⣧⠿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⡰⠛⠒⠾⣟⣾⣲⡶⠥⠛⠉⠊
⠀⠀⠀⠀⠀⠀⠀⣸⡫⣲⡵⣿⣶⣚⡷⣿⡀
⠀⠀⠀⠀⠀⠀⢰⠉⡟⢁⠞⢋⡟⠛⠻⣷⣿⡄
⠀⠀⠀⠀⠀⠀⢸⢆⡇⣰⡾⢋⣤⡾⢷⢶⣶⣵⣄
⠀⠀⠀⠀⠀⠀⣸⢸⣧⠏⣼⠋⣀⡴⠋⣭⣥⠴⠛⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢠⠏⣾⣳⢋⡏⣴⡟⣡⡞⢩⢛⣿⠯⠽⠿⠿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⢸⠣⢸⣏⣸⢰⢹⢁⡏⡴⢫⣾⠃⡶⢖⣒⣛⣛⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣴⠏
⠀⠀⠀⠀⠀⢸⠁⡧⠻⣻⠀⢸⣾⢰⣇⣱⡏⣼⠃⣿⠏⢩⡖⣿⣿⢷⣟⠳⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣩⡄
⠀⠀⠀⠀⠀⡜⠘⡄⢹⣿⡄⣾⣿⢻⣾⣿⣾⢻⣿⡟⡜⣼⡿⣿⠅⡼⢡⢯⢯⡭⠽⢦⡀⠀⠀⠀⠀⠀⠀⠫⣁⣽⢄⡀
⠀⠀⠀⠀⢰⡅⠀⢡⠏⠀⠉⢻⣿⣌⣷⣯⣿⢸⡝⣿⣱⣿⣷⠟⢀⡗⢃⣦⡦⠑⡦⠽⣿⣄⠀⠀⠀⠀⠀⠀⠈⠣⣼⢉⣦
⠀⢶⡫⠍⠙⠁⠀⡎⠀⠀⠀⠀⠈⠛⠉⠙⢿⣷⣧⢿⡏⡟⣽⠀⣿⣰⠊⣼⠱⡇⣧⡟⣩⠿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢷⣋
⠀⠚⠧⠤⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣼⡐⢿⣜⣆⠀⠘⠮⣿⣜⣃⣼⣿⣷⣴⠟⢿⡆⠀⠀⠀⠀⠀⠀⢸⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠾⢻⢹⡿⠝⠛⠦⣬⣑⠾⣦⣭⠿⣭⣉⡻⣦⣤⠛⢦⣀⠀⠀⠀⣠⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣖⠉⠊⢳⣎⣒⡲⠚⣲⡽⠷⠋⠁⠀⠀⠙⠛⠦⢽⠳⣦⣏⠛⣶⡞⢫⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠳⠤⣌⣉⡨⠱⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢭⡉⠛⣛⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⡑⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢣⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠤⠤⠬⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⣤⠀⢤⡤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⢾⠻⡰⣄⠙⣷⡘⣇⠸⣷⢰⡘⣧⢸⠏⢹⣶⡖⡶⢶⡶⢢⣿⡿⢂⠞⣱⡿⠓⡔⠒⣲⣼⠏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠻⡌⢧⠀⠁⢱⠘⣧⢸⡇⢹⢸⡯⠘⣇⡟⡞⠀⡼⡼⣿⠁⡿⢁⡏⣼⡄⠈⣰⠟⣰⠏⢀⠄⢿⡟⠀⡚⡉⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⣄⣇⣸⠀⣇⠈⡄⣼⡘⡇⣿⠀⡇⠀⠙⢹⠁⠀⠁⣇⢻⠀⣧⡈⡇⢸⡃⢰⠃⠀⡇⢠⡿⠀⢀⠙⢸⡙⠻⠄⢃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⠋⣆⡸⢀⡏⠀⢀⢹⡇⢳⠸⣇⠈⢠⠃⠘⠀⢰⡀⡿⣸⢶⠎⢳⡘⢦⠓⠈⠀⡀⠇⠈⣇⠰⢸⣇⠚⢈⡀⠀⢦⡁
⠀⠀⠀⠀⠀⠀⠀⢸⣈⣹⡗⠌⣧⢸⠇⢰⠘⣎⢣⠈⠀⢻⣤⡏⠄⣰⡀⣸⡁⢡⠏⢸⡇⠈⣷⢸⣧⠀⠀⣧⢰⣈⠜⠢⣸⠞⠒⠌⡉⠢
⠀⠀⠀⠀⠀⠀⠀⣬⠙⣧⣙⡦⠘⢾⡄⡿⠀⡘⠦⣀⠀⢸⡯⢁⣸⣟⣼⣳⢣⠏⠀⢸⠁⠀⢻⡰⢿⡄⠀⠘⢦⠌⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢠⡿⢷⡇⠺⣌⡳⠎⠙⢣⣤⣍⣛⠚⢍⣺⠃⠀⠈⠉⠉⠙⠂⠤⣀⠀⠁⢀⣸⡙⠇⠳⠄⠀⠈⡄
⠀⠀⠀⠀⠀⠀⣾⣧⡴⠀⡠⠊⠀⠀⠀⠀⣞⡋⠙⠳⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⡙⠒⣤⡠⣄⠀⠀⢰
⠀⠀⠀⣠⢤⡖⣧⠜⠀⠐⠀⠀⠀⠀⢀⠴⣌⡻⢷⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠃⠑⠢⣤⠀⠀⠑⢄
⠀⠀⠀⠉⠉⠉⢹⡄⠀⠀⠀⠀⠀⠀⠈⢗⠢⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠠⡀⠀⠀⠀⠙⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠘⡦⠀⠑⢢⠀⠀⠀⠀⠀⠱⣄⠀⠓⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠑⢳⠀⠀⠀⠡⠆⡾
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉

⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠱⡀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡄⠀⠆
⠀⠀⠀⠀⠙⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠎⠀⠀⠀⠀⠀⠑⠢⣶⣧⠀⠀⠀⠀⠀⠀⣢⣿⡤⡇⠀⠀⠀⠀⠀⢀⣿⠗⠊
⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⢼⣿⡀
⠀⠀⠀⠀⠸⠃⠻⠀⠀⠀⠀⠀⠐⢢⣼⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠸⡜⡇
⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠎⠘⠄⠀⠀⠀⠀⠀⠰⠀⠡⠀⠀⠀⠀⠀⠀⠐⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⣠⣀⠔⠁⠀⠀⠘⡀⠀⠀⢀⠄
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠳⢄⣶⣀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠈⠙⢿⣶⠄⠀⠀⠺⣿⠁⠀⠀⠀⠀⠀⠳⣿⡴⠊
⠀⠀⠀⠓⢲⣾⠆⠒⠈⠀⠀⠀⠀⢿⡏⠑⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣰⢿⡟⠀⠀⠀⠀⠀⠟⢻⠀⠀⠀⠀⠀⠀⠀⡟⢹⠀⠀⠀⠘⠃⣇⠀⠀⠀⠀⠀⢸⠋⣷
⠀⠀⠀⠀⠚⠟⠃⠀⠀⠀⠀⠀⠸⠈⠛⠐⠆⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠈⠉⠒⠋⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠈⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⡀⠀⠀⣀⠦⢄⣀⡤⠚⢦⣀⣀⠀⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⢠⣄⢀⡰⡩⠝⠫⢅⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣅⢨⠀⠁⢐⠀⢀⣤⣶⣧⠀⠀⠀⡔⠈⠉⠀⢂⢹⢢⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣄⠀⠈⠢⣸⣿⡿⠃⠀⢙⠆⢃⢠⣾⣿⣿⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠤⠬⠷⢄⣀⣰⠖⠶⠭⣶⡚⠍⡆⠉⠛⠋⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢨⠟⠒⠲⣦⠠⠭⢝⡲⠦⡷⠒⠚⠀⠀⣠⣾⡥⠤⠄⠶⠒⠒⡖⠊⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡘⠉⠉⢩⠃⠈⠐⢶⠊⠅⣥⠤⢤⠖⠊⠁⠈⣅⡀⠤⠐⠲⠾⣄⣦⣽⡪⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣃⠀⣠⠃⠀⠀⠀⠀⠉⠉⠀⠀⢸⠀⠀⢀⣠⠯⠀⢀⡠⠤⠤⠤⠿⠦⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠔⠊⠁⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠅⠀⠀⣔⠀⠉⠉⠁⠒⢤⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠘⠒⢒⠒⠊⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⢤⣀⣀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠉⠉⠙⠒⠒⢲⠞⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⡰⢹⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡰⠁⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡰⠁⠀⠸⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠧⠤⠀⣴⠁⠀⠀⠀⣄⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣀⣙⠲⡄⠀⢠⣅⣀⣀⣈⣓⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉

⡿⠿⠿⠿⠿⢻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⡀⢀⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣍⣉⠁⠀⠀⠀⠀⠀⠀⠀⠉⣉⣭⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⡄
⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢀⡼⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⢠⡟⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⢠⡟⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⣿⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⢀⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⣼⣿⣄⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠈⠉⠉⠛⠲⢦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠤⣀⡉⠉⠓⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠲⣤⠤⠤⣌⡑⠢⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⣼
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣤⣤⠀⠀⠀⠛⠄⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡆⢠⡟
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣾⠁
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⡟⠰⣀⠔⢋⣤⣶⣿⣷⣶⣶⣶⣶⣦⣤⡄⢸⡄⠀⠀⠀⠀⢘⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢷⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⢳⡈⢳⡀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣉⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⠉⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠻⡇⠙⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⠏⠠⠒⠈⠀⠀⠀⣿⣏⠀⠔⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢁⠀⣀⠀⡠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠂⠀⠀⠀⡐⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢄⢈⢀⡁⠀⠨⣒⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣉⣁⣧⡀⠀⠈⠘⠢⠄⡀⠀⠀⠀⠀⠐⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⡗⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠉⠀⠀⠀⠃⠀⠀⠀⠀⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠄⠀⠐⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⡄⠀⠲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠁⠠⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡠⠔

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢷
⠀⠀⠀⠀⠀⠀⠳⠦⠤⣀⡀⠀⠀⠠⠤⢒⣛⣛⣛⣹⣍⣝⣃⣐⡤⢤⢄⣤⠶⠋⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠞⠋⣁⣠⣄⢈⣍⡈⣁⣈⡛⠶⡝⢦⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⢚⡄⠀⠀⠈⢩⠈⠀⢈⣁⢩⡌⠉⠍⡀⠀⠈⠇⠀⢀⡆
⠀⠀⠀⠀⠀⢀⡼⠂⣠⠄⠀⡿⣷⠀⠻⠦⡀⡄⠘⣆⡇⠠⡱⠖⠐⡆⣰⠄⡾⠄⡈⠷⡀
⠀⠀⠀⠀⣰⠋⠀⣼⠋⢠⡿⣠⠟⠢⣴⠶⠦⡀⠀⢀⠀⠀⠀⠂⠀⡠⢤⡄⠴⡴⡎⢳⡟⡄
⠀⠀⠀⣰⠃⠀⠸⡇⠀⣾⠃⠹⣿⠥⠜⠤⠠⣿⡶⠇⠀⠀⠀⠀⢺⡤⠤⣧⠁⡿⢹⡈⣧⢸⡄
⠀⠀⢠⠇⠀⠀⠘⡿⠀⡏⡠⣄⣀⣓⣽⠾⢠⡉⠀⠳⡄⠀⠴⠀⡀⠀⣀⠀⠀⠀⡀⢠⡿⠀⠉⣄
⠀⠀⣺⠀⠀⠀⠖⣀⠀⠼⠄⠔⠀⠈⢠⠀⢌⠳⠀⢀⢇⡑⠀⠂⠀⠀⠞⣠⠀⡍⢹⠀⡇⠀⠀⠣
⠀⠀⠿⢴⡃⠀⣿⣇⢀⣀⠀⠀⣶⣦⢝⠳⢌⠳⣄⠘⣤⡉⠀⢀⠀⢠⠞⣡⡶⢠⡎⡰⣷⣀⣀⢰⡇
⠀⠀⠀⢻⡇⠚⠉⢙⣦⡉⠀⠀⢻⣧⣌⠛⠦⠀⠀⠀⠈⢣⡤⠊⠀⠀⠀⠜⠁⠘⣡⡿⠮⠁⠿⠌⠁
⠀⠀⠀⠙⠷⡀⠀⠀⠈⠙⠲⡀⠻⣜⢿⠐⣿⠀⢀⣬⠳⠾⣷⢶⡞⢻⣿⣧⡴⠿⣛⣠⡖⠚⠙
⠀⠀⠀⠀⠐⠻⣶⣶⣤⡀⠀⠴⢦⡹⣿⣧⡙⢠⣸⣿⣿⣿⣿⣿⡇⣜⠉⠀⠀⢸⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣆⡀⠈⠙⠻⣿⡏⣼⣿⣿⣿⣿⣿⡏⣿⠿⠀⢠⢴⠾⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣤⡀⢿⣿⣾⢻⣿⣿⣿⡿⢻⠁⢀⣤⡾⣎⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠹⣷⣤⣭⣵⣦⣄⣧⣴⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠈⠙⠛⠛⠛⢉⣾⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⣶⣶⡿⠃

⠀⠀⢀⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠓⠒⠒⠒⠒⢲⣤⣤⡤⠤⡄
⢠⠞⢉⠀⠀⠀⠘⡆⠀⢀⣀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⡠⠒⣦⠖⠋⠉⠣⠀⠈⠆⣻
⡷⠀⠀⠀⠀⠀⠀⡿⢺⠆⣸⠖⠉⡀⠔⠲⢄⠀⠀⢀⣔⡡⠚⣡⠒⠒⠦⡀⣠⠞⠉⠉⠑⣦⠀⠀⠀⠀⢀⣀⠤⢄⣀
⠓⠦⣀⡀⠀⢠⠞⠁⣸⢀⣡⠔⠋⠀⠀⠀⠈⢧⣰⠟⠉⠀⠀⢧⠀⠀⠀⣼⣇⠀⠀⠀⠀⢸⣂⡠⠒⠉⠁⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠙⠒⠉⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠘⢆⠀⠀⠉⢹⢾⣇⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣧⡠⣤⣤⣀⡴⠚⠑⠶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⠤⣄⣀⡀⠀⠀⠚⠯⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠴⡒⠶⠈⠀⠀⠀⠚⡗⠀⣐⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠈⠉⠁⠒⠚⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡈⠳⠤⠤⠖⢒⣏⣠⠤⠖⠛⠉⠁⠀⠙⠲⣤⣄⣀⠀⢠⠐⠤⠒⢀⣠⠞⠣⠤⠄⣤⢤⡸⠏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⡉⣁⡀⠤⠐⠚⠉⠁⠀⠀⠀⠀⢀⣀⣤⠒⠋⠀⠀⠉⠉⠉⠉⠛⠉⠉⠁⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉⠣⣨⠛⠒⣦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⡖⠊⠉⠻⣝⠛⠒⠛⠉⠉⠈⢧⡀⠀⠀⠀⠀⠈⠢⢴⡁⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡠⢤⣀⣀⣠⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣦⠄⠑⢄⠀⠹⣆⡤⢤⡤⣄⠀⠀⠙⢦⡷⣄⠀⠀⠀⠀⠈⠙⢲⠆⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢲⠈⠀⣸⠁⣰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣠⣾⢛⡁⡠⢎⠀⠈⠳⢤⡀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⣀⡼⠀⠀⡟⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢰⠃⢀⡿⣞⠀⠀⠙⠦⣀⠀⠈⠑⠢⠤⣀⡀⠈⠙⢒⣤⠀⠀⣀⡤⠶⢾⢉⣸⠋⠉⠉⠯⠵⠒⠚⠙⠢⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃⠀⠹⣄⠈⠳⢤⡀⠀⠀⠉⠒⠤⣄⣀⠀⠈⠉⠁⣺⠸⡴⠛⠂⢸⢃⡏⣁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠉⠓⠦⣄⡀⠀⠀⠈⠉⠑⠒⢤⡏⠸⠤⠤⠒⢊⡜⢉⣁⡾⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣤⣀⡀⠉⠙⠒⠶⠤⢄⣀⣘⣓⣲⠋⠉⠻⠖⠚⠉⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⣿⣿⣿⣶⣦⡤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣀⣀⡈⠉⠉⢻⠀⠀⠀⠀⠀⢠⠟⠛⠛⠉⢉⣉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠤⣀⠀⢸⠀⠉⡟⠉⠉⠀⠀⠀⠀⠀⠈⠓⡟⠉⢹⠋⠁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠁⠓⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠛⠚⠉⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠤⠤⠤⠤⠤⠴⠚⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠤⠤⠤⠤⠶⠶⠚

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠉⠭⠉⣉⠙⠛⠻⢿⣿⣿⣿⡿⠿⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠈⣻⠟⠉⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠈⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣠⣤⣤⠄⠀⠹⣇⠂⠄⡀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠑⠠⢀⠀⠀⠛⢷⣦⣄⠀⠀⠈⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣾⣿⡿⠃⠀⠀⠀⠘⢧⡀⠀⠉⠐⠠⢄⡀⠀⠀⠀⠀⠀⠁⠒⠤⢀⡀⠉⠒⠠⢀⣉⢻⡆⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⡿⠁⠀⠀⡀⠀⠀⠀⠝⠶⡀⢀⠀⠀⠀⠉⠐⠂⠠⠀⢀⡀⠀⠀⠀⢉⣁⡠⠴⠞⢉⡅⠀⠀⠈⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡿⠁⠀⠀⠔⠀⠀⣠⠎⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⡀⡀⠀⠀⠰⠟⠉⢠⡴⢖⣡⡍⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠇⠀⡠⠊⠀⡀⠔⠁⢀⠠⠊⠀⠀⠀⠀⠀⠀⠠⠀⠉⠒⠆⠀⠠⠤⠤⠤⠤⠀⠶⢂⡈⠉⠁⢀⣉⡱⠂⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠉⢻⣷⡖⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⠀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⢀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠻⣿⣿⣿
⡿⠿⠛⠉⠀⠀⢀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣄⠀⠀⠈⠙⠛
⣷⣤⣄⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠠⡀⠀⠀⠀⠀⢀⠀⠀⠠⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⡡⠋⢁⠄⠒⣀⠀⢀⡈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡄⠀⠀⠀⠐⠀⣴⠃⠠⠊⠀⠀⠀⠀⢳⣦⠈⠻⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⠀⠀⠀⠀⠀⢰⡇⠀⠇⠀⠀⠀⠀⠀⠀⠃⠐⡄⠘⢿⣿⣿⡿⠁⠀⠀⣠⣽⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠎⠀⠀⡠⠶⠾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⣸⣆⠈⢛⠉⠉⢤⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠇⠀⣌⠀⠀⠀⠀⠀⠀⣠⠄⠠⠒⠀⠐⠑⠊⣰⡏⠙⡄⠈⠅⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⠀⠰⠀⠉⠂⠄⣀⡄⠀⡏⠀⠀⠀⠀⠀⠀⣴⣉⢈⠀⠴⠐⠚⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠘⠀⠆⠀⣀⠀⢀⠋⠁⠀⡑⠋⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⡀⠀⠀⡀⢰⠀⠀⢠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⠰⢀⣀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⠀⠃⡄⠀⢆⡡⠄⠀⠀⠀⠀⠀⡀⠤⠰⠎⡡⠊⠱⠤⢅⢤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡤⠃⠀⠋⡴⢂⡀⠀⢀⡴⠊⠀⢀⡈⠖⠁⠀⠀⠀⠈⡄⠡⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠀⠀⠀⠀⠈⠧⠐⠉⠀⠀⠀⠈⠁⡜⠀⠀⠀⠀⢀⡰⠄⠃⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠏⣉⣁⢐⠀⢄⡄⠒⠈⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠑⠢⠀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠓⠰⠦⠤⠤⠀⠀⠐⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠐⠂⠤⢄⡀⠀⠀⢀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠔⠈⠒⠀⠀⠀⠤⠤⠤⢀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣀⣀⣀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⡱⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢾⣱⠀⠀⠛⠻⠀⠀⢠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⡇⠇⠀⠀⠀⠀⡀⠀⠉⠐⠀⢜⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠛⠊⣆⠀⠀⠀⠈⠂⡄⠀⠀⡘⣸⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣠⣤⡀⣭⠆⢀⣀⡠⠴⠃⠀⠀⡠⠃⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⡴⠖⢣⢴⠶⠋⠉⠛⣡⠘⠖⠛⠲⢦⡀⠃⢀⠞⠁⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡸⠁⢺⠁⡴⣱⢏⣥⢾⡷⢂⣼⠁⡔⠀⠠⢤⣄⠉⢳⡈⠀⢀⠾⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⡻⢧⣻⣂⠘⣧⠉⢰⢻⢡⠎⢀⢾⠃⢠⠇⠀⣠⣀⠈⠢⠀⠙⠂⠀⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣝⣻⠀⣻⠟⠀⡏⢸⢹⡇⣼⢠⢸⡏⠀⣾⠰⠀⠀⠀⠉⠂⠀⠀⣄⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⠖⠄⠛⢠⡟⠀⠸⠃⠈⣯⠀⣧⢸⡘⢧⠀⠃⠀⠀⠀⠐⠀⠠⠀⠺⠿⠃⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⣀⡠⣶⠉⠀⠀⠀⠛⣀⡀⠀⢸⠄⠀⠀⠀⠈⠧⠘⠢⠷⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐
⣠⣴⣶⡄⠰⠶⠈⠛⠛⠁⠀⣤⣤⡤⠀⠈⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⢀⣄⠤⠠⠖⠓⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠿⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣤⣤⣤⣄⡈⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣤⣤⣤⣤⣀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣸⣷⡄⠀⠀⠀⠀⢸⣧⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⠛⠟⠃⠀⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⣠⡼⠏⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⢀⣠⡾⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠈⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠶⠶⠶⠊




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠾⣿⣿⡿⢿⣿⡿⣛⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⢸⣿⣿⣿⣇⠺⣺⡏⠀⠀⠘⣧⣏⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⣀⣀⣀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⡇⠀⠻⣿⠟⠁⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣦⡀⠤⢀⣰⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠟⠻⣿⠟⣉⣿⣿⡏⢸⡿⢹⣏⢩⣽⠋⢻⣿⣿⢻⠛⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⢸⡇⣿⠀⣿⣿⣿⡇⣸⢃⣿⣿⢸⣿⠀⡌⣿⣿⢸⢰⡌⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⡾⢱⣿⠀⣿⣿⣿⠇⡏⢸⣿⣏⢸⣿⠀⣇⢹⣿⢸⠀⣿⠈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣐⢿⣿⠐⢟⣿⣿⠀⠁⣾⣿⡿⢸⣿⠀⣿⠸⣿⢸⠀⣿⡄⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⡎⢿⢸⣿⣿⣿⠀⡆⢻⣿⡇⢸⣿⠀⣿⡆⣿⢸⡆⣿⡇⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢻⡇⣸⢸⣿⣿⣿⠀⣿⢸⣿⡇⢸⣿⡆⢿⣇⢻⡸⡇⣿⢁⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣈⣼⣿⢸⣿⣿⣿⢠⣿⠈⣿⡇⢸⣿⡇⣼⣿⠸⢡⣇⣁⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⢸⣿⣿⣿⢸⣿⡄⣿⣧⢸⣿⡇⢿⣿⣧⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣤⣶⣿⣯⣸⣿⣧⣘⣯⣈⣿⣦⣼⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⡴⢰⣧⣴⣷⠛⡆⠀⡔⣼⢠⠇⣠⣿⢀⢷⣰⠎⣻⠛⠃⢯⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁⠈⠀⠈⠈⠛⠁⠈⠋⠙⠃⠈⠁⠘⠉⠀⠋⠀⠉⠀⠐⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⣀⣄⠀⠀⡄⢀⣄⡀⡀⣠⠤⠀⢀⣄⠀⣠⡀⣤⡄⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⢯⡼⠁⠸⠋⢿⢹⡴⠛⢷⡶⠀⠼⠹⠖⠿⠸⣯⡁⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣥⣿⡉⣿⠉⣼⣷⢒⡀⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣾⣟⣷⣿⢿⣏⣽⣾⣗⣶⣒⡂⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠯⢿⣿⠄⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⡶⠛⠑⢢⡀⠀⠀⢀⠔⠀⢀⠀⠀⠀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣇⣾⣷⣼⣧⣤⠄⢸⣿⡷⢀⡆⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⠿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣶⡇⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⠿⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⡝⠀⣀⡀⠀⣀⠀⠐⣁⣤⡾⠛⣾⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣲⣄⣼⠋⠁⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⣿⡿⠙⠀⠀⠉⠋⠁⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣀⠀⢀⣀⣴⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡀⠀⠀⠀⠀⠀⣈⣙⣿⣟⣉⣉⣻⣿⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠛⠋⠋⠉⠁⠉⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠙⠋⠀⠀⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢃⠀⠀⠀⢸⠇⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠈⠀⢀⣀⣀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠀⠀⢀⣤⡶⠞⠛⠛⠛⠛⠛⠿⢶⣤⡀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠄⠀⠈⢻⣆⠀⠠⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣠⣤⣈⣻⣶⣦⣴⡶⠟⠋⠁⠀⠀⠀⠀⣀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠛⠉⠁⠀⠈⠉⠉⠉⠛⠻⡟⠻⠿⢿⣿⣟⡏⠉⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⡀⠀⢻⣤⣀⣹⣿⠟⠁⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠁⠀⠀⠑⠦⠄⠀⠀⠀⠀⠀⠀⠠⠞⠛⢿⣦
⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣿
⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡟⠃
⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠙⠷⢶⣤⣤⣤⣤⣤⣴⣶⡶⠶⠿⠿⠛⠛⢹⡇⠀⢀⡴⠞⠛⠛⠿⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡟⠀⢀⣠⣤⣀⡀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡇⠀⠀⠀⠉⠉⠛⠛⠋⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡼⠋⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⡏⢳⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣠⠟⠁⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⢸⡇⠀⢻⡄⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣴⠋⠀⠀⠀⣿⠈⠙⠳⠶⣦⣤⣤⣤⣤⣤⣤⡶⠶⠟⠛⠉⠀⠀⠀⠀⢸⡇⠀⠀⠹⣆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⡞⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧⠈⢷⡀
⠀⠀⠀⠀⣠⠴⠲⣄⡾⠃⣰⠏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠈⢧⡀⠿⠒⠛⠛⠲⢤⡀
⠀⠀⢀⡾⠁⠀⠀⠈⠀⠼⣅⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠙
⠀⠀⣿⢀⡀⠀⠀⠀⠀⣠⢈⡷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠸⣆⠀⣄⠀⠀⠀⠀⣀⣻
⠀⠀⠈⠙⢧⢶⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠸⡆⠈⣷⠀⣠⣤⡟⠉
⠀⠀⠀⠀⠀⠘⠓⢿⣤⠷⣴⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢷⡴⠛⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⢉⡟⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡟⠛⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣀⣤⠴⠶⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⣿⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠶⠶⠶⠛⠛⠛⠛⠙⠛⠛⠛⠛⠻⠶⢦⣤⡀
⠀⠀⠐⢿⣥⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠛⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⣀⣠⡌⠁⠀⠀⠀⠙⠿⠿⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⠀⠈⠙⠛⠛⠦⢀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣄⣀⣰⣿⢿⣷⣦⡀⠀⠤⠀⠈⠛⠣⣤⣤⡤⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⣤⣤⠿⠿⠟⠋⠉⠉⠛⠿⢿⣤⣤⠛⠀⣀⣀⠀⠀⢀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣇⣀⡀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡻⢶⣶⣦⣄⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣤⡀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠿⣦⡀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⣴⡟⢳⣄⠀⠀⠐⠶⠶⠶⠶⠶⠤⠀⠀⠀⣴⣿⣿⣦⣄⣈⠿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠰⣿⣧⡀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⠻⣷⣦⡈⠻⣦⠀⠈⠉⠉⠉⠁⣸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠈⠻⢿⣦⡈⢳⡀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣦⣴⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠹⣧⡀⠀⠀⠀⠘⢿⣿⣷⠉⣴⣿⠋⠨⣿⡀⠀⠀⠀⠈⢻⣿⣿⣆⠀⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠙⠻⣦⣤⣄⣀⣄⣠⣤⣿⠟⠁⠀⠀⠈⠻⢧⣤⣤⣄⡀⣠⣤⣴⠿⠋⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣹⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢀⣀
⠀⠀⠀⠀⠀⠀⣴⠛⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠙⠛⢷⣄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡗⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠘⠿⠷⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠀⣠⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠷⠶⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣴⡶⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣶⣶⡾⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠛⠛⠻⣿⣿⠶⠶⠶⠶⢶⣶⣶⡶⠶⣶⣾⠟⠻⠿⠻⣿⡿⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⢸⣿⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠉⢱⡆⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣹⡟⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⠃⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢻⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣰⠟⠀⢠⠎⠀⠀⠀⢰⣿⠂⠀⣀⡀⠀⠀⠀⢿⡆⠀⣾⡏⠀⠀⠀⠀⢀⣀⠀⣹⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠰⣿⠀⠀⣾⡆⠀⠀⠀⠈⣿⣦⣾⠛⠁⠀⠀⠀⠀⠀⣀⣿⠃⠀⠀⠀⠀⠙⣿⡷⠿⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⡀⠁⠀⠀⠙⠁⠀⠀⠀⠈⠛⠉⢿⣇⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⢀⣸⣧⣤⡴⠿⠷⠀⣀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠺⣿⡇⠀⠀⠀⣿⡇
⠀⣠⡖⠉⣹⡿⠋⠁⣠⣾⣿⠿⢧⡄⠀⠀⠀⠀⠀⢿⣧⠀⠀⠸⣿⡇⠀⠀⠀⠠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⡇⠀⠀⠀⢸⣧⡀
⠀⠙⣷⡾⠋⠀⠀⣴⡟⠁⠀⣠⣾⣟⠀⠀⠀⠀⠀⠀⢹⣧⣀⣠⣿⡇⢀⣠⣴⣾⣷⣶⣶⣶⣶⣶⣶⣤⣶⡛⠋⠁⠀⠀⣿⣿⠀⠀⠀⠀⣿⡏⠈⠉⠁⣀
⠀⣾⡏⠀⠀⢠⣾⠋⠀⢀⡾⠏⠙⢿⣦⣀⣤⣤⣤⠼⠛⠿⠿⠛⠛⠛⠋⠁⠈⠉⠀⠈⠁⠀⠀⣠⣤⡿⠿⠟⠻⠿⠿⠿⣿⣿⠀⠀⠀⠀⣿⡿⠛⠛⠿⠿⣤⡀
⣾⠉⠀⢀⣰⡿⠁⠀⣀⡟⠁⠀⢠⣾⠟⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣿⡿⠐⠶⣄⣀⣙⠁
⠉⢷⣦⣿⡟⠀⠀⣠⡛⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣶⣶⣶⣿⣿⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣿⣷⠀⠀⠉⠙⠿⣧
⠀⠀⣿⡇⠀⢡⣶⠟⠁⢀⣴⣿⣷⣶⣶⣶⣶⣶⣶⡶⠟⠛⠛⠛⣿⠋⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠶⣶⢾⣿⣧⠀⠁⣤⣄⠀⣸
⠀⠀⣿⣇⠀⠘⢿⣤⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠃⣤⣈⡿⠿⠋
⠀⠀⠈⠻⠿⣶⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢀⠀⠒⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⠛
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠘⠃⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⡄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⢷⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⡿
⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣶⣾⣿⣿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⠿⢿⣿⠿⠛⠛⠛⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⠀⢀⣼⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⠸⠿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡾⠛⠛⠒⠛⢻⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠁⠀⠀⠘⠛⢿⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠛⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣀⡀⠀⠀⠀⠀⠛⠛⢻⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠟⠋⠀⠁⠀⢀⣤⣾⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⡿⠿⣧⣤⡄⠀⠀⠀⠀⠀⠈⠙⠻⠿⠷⣤⣤⣄
⠀⠀⣠⣄⡀⠰⠿⠋⠉⠁⠀⣀⣠⣤⣴⡿⠿⠋⠉⢻⣷⣀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠗⠀⠈⠉⠙⠿⠿⠷⣦⣤⣤⣤⣠⣤⣀⡘⠿⠿
⣀⣀⣹⣿⣿⣶⣶⣶⣶⣶⣾⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣁⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠋⠉⠿⠟⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣾⣿⣦⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠒⠒⢦⣤⣀⢠⠰⠖⠾⠴⣶⡾⣿⡿⣶⣶⠼⠖⠶⢠⣀⣠⣤⠖⠛⠲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⣤⣅⣈⡷⠄⢀⣠⣾⡟⣋⣩⣷⣾⣷⣯⣉⡻⣿⣤⣀⠀⠴⣋⣀⣥⡶⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⠍⠅⢀⣀⡡⠤⢞⣋⣩⣴⠇⢷⣬⣍⣛⠦⠤⣁⣀⠀⠭⢽⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠉⢀⣠⢿⡅⠠⠞⣫⡷⣲⣾⢁⣸⣶⡲⣯⡛⠆⠀⣽⢧⣄⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⠛⣡⡿⠀⢠⡼⠀⣠⢛⡋⠀⠈⣛⢣⡄⠸⣤⠀⠸⣷⡙⢷⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢛⡇⠶⠿⡄⢠⡏⠈⠀⠀⠀⠈⠈⣧⠀⠼⠷⠆⣟⠃⠀⢻⣧⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢸⣿⢸⡟⢸⣷⢴⣶⢦⡈⢻⣇⠀⠀⠀⢠⣿⠇⣠⠶⣶⢴⣿⠘⣿⢸⣿⠠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣼⣿⢸⡇⢆⡉⠳⡍⠹⡧⢸⣿⠂⠀⠀⢺⣿⠠⣿⠉⡵⠋⣁⠆⣿⢸⣿⣄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⢸⡄⠈⣛⡃⠀⠀⡗⣸⠃⠀⠀⠀⠀⢻⡐⡇⠀⠀⣙⡋⠀⣿⢸⣿⡿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣿⣿⢼⣷⣶⣬⡶⢆⠔⠀⡌⠀⠀⠀⠀⠀⠈⡅⠀⢔⠶⣮⣼⣶⣿⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣼⣿⠧⣸⣧⠟⠀⠀⠄⠀⠀⠀⠀⠀⠐⠆⠀⠈⢧⣿⡰⢻⣿⣼⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣝⢛⠿⣿⠀⠀⠀⣮⡀⠀⣸⡀⠀⣠⡆⠀⠀⢸⡿⣟⣛⣿⣿⡧⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠿⣟⣿⣿⣧⣬⣂⡐⠢⠈⠻⢶⡀⣰⠾⠋⠠⠚⣁⡪⣦⣽⣿⣟⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⣻⡉⠙⠳⠋⡭⠛⢃⠀⠀⣨⡀⠀⢈⠛⠩⡍⠳⠋⠈⣙⡟⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⢤⣠⡀⠘⣿⣦⣤⠾⣿⣿⢦⣴⣿⡟⠀⢠⣠⠀⠀⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⡄⠀⠀⠀⢹⣿⣾⣉⠀⠀⠀⠈⠀⠀⠀⢈⣹⣾⣿⠁⠀⠀⠀⡄⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿⡄⠀⠀⠈⠉⢿⣿⣶⣦⣤⣠⣠⣤⣶⣿⡿⠏⠉⠀⠀⢀⡼⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠛⠻⠟⢿⣿⣿⠟⠿⠛⠃⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠤⢤⣤⣤⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠶⠶⠤⠞⠁⢠⣀⣠⣴⣾⣷⠆⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣴⣶⣿⣶⠶⣀⣤⠆⠉⠛⠛⠩⣥⣼⣿⡿⠷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢀⡴⣋⠄⠈⠉⠁⣼⠋⣴⣾⣶⣄⣠⣤⡀⠉⠁⠘⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⡞⣼⠃⣤⣤⣄⢸⡏⣼⣿⣿⣿⣿⣿⣿⠏⠰⣄⣛⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣀⡀⠀⠀⢷⡹⣧⣿⣿⣿⡌⠃⠟⠛⠛⠛⠛⠛⠻⣷⡄⠀⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠈⠉⣙⣷⡄⠈⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⡄⠈⢳⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⣴⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣇⠀⠘⢿⣦⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠈⣩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠹⡆⠀⢲⣝⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣀⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠋⠉⠙⣿⣿⣷⣖⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⡏⢀⣼⡇⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⣡⣶⣶⣦⡖⠀⢻⣟⣛⡺⣆⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⣾⣿⣷⡀⠰⠞⠉⠉⣉⣉⣙⡛⠲⢦⣄⠠⠶⣸⡇⠴⠻⠿⣟⣤⠀⠀⠀⠀⣿⣏⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣄⢻⣿⣿⡧⠀⠀⢠⣤⡽⣿⣿⠛⠲⠀⠙⠆⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⣿⢃⣉⣁⠀⠀⠀⠈⠉⠐⠛⠛⠉⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⡢
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⢹⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⡿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣤⠤⠀⠾⠛⠀⢄⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡟⢸⣿⢡⣴⣟⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣇⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣜⣿⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⢿⣿⡷⠖⠀⠀⠀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡻⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠶⠶⠾⠿⠛⠀⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣷⡙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠙⠓⠶⠶⣤⣤⡤⠶⠶⠞⠛⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠁⠀⠀⠀⣀⣀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⣠⡴⠞⠛⠉⣉⠛⢷⡄⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣼⠏⠀⠀⢀⡈⠋⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⢸⡿⠶⠶⠞⠋⠀⣰⢿⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠘⢷⡶⠶⠶⠶⠞⠋⠀⠙⢷⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡙⠛⠶⠛⣋⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠙⢷⣤⡾⠋⢙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠙⢷⣴⠟⠙⠳⣦⡟⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠈⠃⠀⣀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡀⠀⢀⣴⠟⠁⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠈⠻⡶⠟⠁⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⠶⠶⠞⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣠⠖⠛⠛⠛⠓⢦⡀
⠀⠀⢀⣠⠤⠤⠤⢤⣀⠀⠀⠀⠀⣴⡛⠓⠈⠳⢦⣀⣰⠃⠀⣠⠔⠒⢤⠀⠙⡆
⢠⡞⠉⠀⣀⣀⠀⠀⠈⢧⣠⠶⠋⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⢧⣀⣠⠆⠀⠀⡿
⣿⠀⢰⠋⠀⠈⢱⡀⠀⠈⠁⢀⣀⣀⠀⠀⠀⠀⠀⠲⢿⣿⣶⡄⠀⠀⠀⣀⠼⠁
⢻⡀⠀⠳⠤⠤⠞⠁⠀⠀⣴⣿⠿⠋⠀⠀⠀⠀⠀⢀⡤⠦⣄⠀⠀⠀⠈⠛⠶⣤
⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢦⣄⠀⠀⠀⠀⢸⣋⣦⠈⢧⠀⠀⠀⠀⠀⠺⣄
⠀⠀⠀⢉⡟⠁⠀⠀⠀⠀⠀⡏⠀⣤⣿⣧⠀⠀⠀⠸⣿⣿⠃⣸⠀⠀⠀⠀⠀⠐⢿
⠀⠀⠀⠈⣿⡁⠀⠀⠀⠀⠀⢧⠀⠻⣿⣿⠀⡠⠤⠀⠘⠳⠖⠛⠋⠀⠀⠀⠀⠀⣈⡷⠶⠶⠚⠓⢦⣀⡀
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠴⠞⠓⠤⠞⢁⣤⠒⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠠⣼⠀⠀⢀⡤⢄⠀⠀⢹
⠀⠀⠀⠈⢧⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣠⡶⠋⡧⠀⢫⣀⡸⠀⢀⡼
⠀⠀⠀⠀⠀⠳⠶⢦⡀⠀⣀⠤⠄⠠⢀⠀⠀⠀⠉⢻⠀⣄⠀⠀⠀⠠⣀⣼⢋⣠⠞⠁⠀⠀⠀⠀⢺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠁⠀⠀⠀⠀⠱⣄⡀⠐⠾⠟⢀⡀⠄⠀⠀⠈⢻⠉⠁⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠄⠀⢀⡀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢠⠎⠀⢀⠔⠃⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⡆⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢰⠇⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⢽⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣮⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠈⢀⠀⠀⠀⠀⠸⠗⠀⠀⠀⢀⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠑⠤⣀⡀⠀⢀⣀⡠⠔⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠂⢸⡿⠛⢳⡖⠂⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠠⠄⠘⠷⢤⡀⢀⡼⠇⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡀⢀⣽⣏⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⣤⢴⠧⠞⠁⠈⠳⠤⠴⣤⣠⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⡄⢄⡀⠤⠤⠄⠀⠀⢀⠤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⠛⢛⣯⡟⠃⠢⡈⢠⣞⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡆⢀⠒⣨⠨⠼⠁⢰⢀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢐⠋⠈⡆⠈⠐⢰⠚⠸⡘⡇⢸⡂⣸⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢨⣄⠀⠀⠀⠀⠀⠀⢀⣃⠃⢸⡇⠁⠘⢱⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣛⡄⢀⠀⡀⠀⢸⠁⠂⠀⠼⢃⠤⢖⣾⠿⣛⡿⠿⠷⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣽⠆⡃⡀⠿⠀⡀⠛⢞⠭⠒⠀⢄⠚⠙⠀⢉⣐⣶⣮⣭⠭⠭⠭⢽⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⢶⠒⢶⠚⠅⠲⡓⠀⠀⢀⡤⠏⠀⠀⠀⠀⠀⠒⠒⠲⠶⠤⠤⣤⣤⣶⣦⠤⣔⠪⣝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⣀⠀⠈⠀⠈⢀⡴⠀⠹⠀⠀⢀⡰⠭⢭⣉⣉⣁⣈⣩⣉⣉⣉⣉⣐⡒⠲⠤⣌⠙⠲⢍⢢⣹
⠀⣠⠤⠤⢤⣄⣀⠀⠀⠀⣀⣈⡉⠉⠉⠙⠉⠀⠀⣤⣤⡻⣦⣬⡙⠒⠲⣭⢩⣿⠟⣁⠤⠤⠤⠬⠭⣝⡒⢌⠙⢦⠀⢣⢹
⣎⢀⡤⠀⠀⠙⠈⠹⡅⠙⢯⠈⠑⠀⢀⡀⠤⢾⣦⣄⠈⠉⠀⢹⠇⠀⣸⡑⢉⣁⣀⣀⣀⣰⡿⠉⠑⠂⠉⠳⢌⢪⢷⢈⣾
⠛⠻⢂⠀⠀⠀⠀⠤⢤⣀⣈⡤⠄⠊⠁⠀⠀⠀⠀⠉⡇⢀⣀⣏⣀⣔⣋⡈⠉⠀⠀⠀⢀⣤⠤⢰⠏⠉⠉⠒⣄⢻⣾⣾⠏
⠀⠀⠀⠀⣠⣤⡶⢦⢍⡁⠀⠀⠀⠀⠀⢀⣀⣤⣄⡲⡆⠉⠁⢙⣀⣀⣭⠥⠤⣤⠤⢒⠒⡒⠚⠁⠠⢄⣀⣉⡛⣿⡿⠃
⠀⠀⠀⠀⢯⡥⢿⣋⡴⢃⡀⠤⠀⠀⠀⠀⠈⣠⡆⢉⡏⠉⢩⠁⢠⡀⠀⣦⠀⣿⣦⣈⣳⣈⣧⣀⣑⣶⡤⠼⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣃⠀⠀⢀⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣭⣧⣀⣹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣓⣾⣿⣿⣽⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡿⣏⣹⣏⢻⠾⣿⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢉⠧⠉⣿⣿⡿⣿⡦⣉⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡌⢷⣆⢮⣿⢱⣿⣵⣯⣇⢭⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣾⣽⢾⣿⠏⣽⣿⣿⡟⣽⣛⣿⡦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢸⣏⣾⢏⣼⣻⣞⣴⣷⣫⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⡸⣿⣿⣻⢿⣿⣿⣿⡿⠿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠟⢉⣷⣽⣿⣿⣟⣻⣾⣷⣶⣶⣶⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣧⣹⠿⣯⣭⣭⣈⢿⡿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣿⣟⣻⣿⣿⣷⣤⣿⣿⣿⣶⡃⠻⣿⣿⢟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣼⣿⣿⣿⣿⣿⢷⣝⠻⢿⡿⣧⣇⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣹⣿⣯⡟⢿⣿⣷⣽⣾⡽⢿⣦⣿⢸⣿⣿⣿⣷⣴⡄⣠⢀⡄⠀⣀⣀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿⣷⡟⠿⣶⣮⢷⢸⣿⣿⣽⣿⢹⣿⠿⢾⣷⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡈⢿⣼⣿⡿⠟⠋⢷⡹⡄⢹⡸⣿⣿⣿⡿⢿⣷⡌⣿⠀⠀⠀⠘⠛⠛⠛
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠏⠀⠀⣰⡎⢿⡇⣧⢈⡷⠋⠉⣿⣯⣉⣙⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⡿⠃⠀⠀⣾⣿⣿⣾⣧⢸⠘⠁⠀⠀⣷⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⠀⠘⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢀⣀⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄⡀⣸⣿⣇⣤⣶⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠛⠉⠉⠛⠛⠉⠀⢿⣿⣿⡿⠋⠀⠀⠀⠀⠙⢉⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣠⣤⣶⣦⣄⠀⣼⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⠁⢀⣴⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢿⣿⡆⠙⢿⣿⣿⣿⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⣿⣿⣿⣿⠆⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠻⠁⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢸⣿⣿⠏⢰⡟⠋⠉⠛⣿⣧⠀⣰⠂⠀⠀⢀⢠⣿⡿⠃⢀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⢀⡴⠛⠙⠻⡿⠿⠋⠀⠀⠀⠘⠛⠿⠛⣍⢉⠉⢻⣦⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠎⠐⢶⠗⠀⡄⠀⠀⠀⠉⠁⠀⠀⠢⠀⠙⠋⠀⣰⠋⠀⠀⢸⣿⣿⣤⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⠀⠀⠀⠂⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠀⣾⣿⠇⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠙⠿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡿⣠⡤⠶⢦⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡔⠋⠙⢢⡙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼⠷⢦⡀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠁⢠⠖⠚⠷⣇⠀⠀⠀⠀⠀⠀⢧⣀⢠⡀⠀⠀⢠⡄⠠⠜⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠸⣆⡀⣀⣿⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠄⠀⠀⠐⡟⠈⠓⠒⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡉⠉⢸⡆⠀⠀⠀⠂⠀⣴⠖⢻⠛⠛⢻⠓⠓⠶⠶⢶⣦⠀⠀⠀⠰⡿⠓⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣧⠔⠀⠀⠀⢰⡿⡖⠺⡗⠛⢻⣋⣙⣟⣓⣿⣿⠀⠀⠀⣸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⣦⠀⠀⠀⠘⢧⡵⠖⠛⢛⡩⢭⣭⡉⠉⠉⠁⠀⠀⢢⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⡀⠀⠀⡆⢩⣀⠀⠀⠀⠀⠁⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⠀⠀⣼⣿⣿⣿⣄⣀⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⣿⣿⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣴⣤⣄⠀⠀⠀⠀⠀⠀⠼⡿⠋⠻⠿⣿⣿⣧⠀⠀⠀⣀⠀⢀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⣿⣿⠀⠀⠀⣱⣘⣙⣋⣜⠛⣟⣋⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠀⠀⠀⠁⠉⠁⠉⠉⠁⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⠀⢹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣁⡐⢀⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⢿⣏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠋⠛⣭⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣀⡀⢨⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⢀⣴⣀⣀⡄⠀⠀⠻⢿⠁⠀⠀⠙⠿⠏
⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⡀⠛⠀⠀⠀⢶⣾⣿⡶⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣄⠈⠀⠀⠓
⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣄⡀⠀⠉⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡀
⠀⠀⠀⢸⣿⣟⣿⠹⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣹⡇⡁⢁⠄
⠀⠀⠀⠀⢻⣿⣴⣦⣿⣿⣿⣿⣿⣿⡟⢩⣿⢿⢿⡿⢱⣻⣹⣷⡧⣤⡈⠻⣿⣿⡿⢿⣿⠿⠿⢿⣷⡇
⠀⠀⠀⠀⠸⣯⣭⣤⣤⣬⣿⣿⣿⣻⠖⢬⠛⠛⠷⠉⠀⠻⠏⠫⡱⢏⣴⣤⣬⣥⣤⣿⣷⣶⣦⣤⣴⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⣤⠀⠀⠈⠀⠈⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢦⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⢦⣀⠀⢀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⠖⠉⠁⠉⣧
⠀⠀⠀⠀⠀⠙⢶⡄⠙⠛⠋⢀⡴⠋⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⡔⠉⠱⡤⠶⠦⡄
⠀⠀⠀⠀⠀⠀⣨⡗⠀⠀⠀⢾⡅⠀⠀⠀⣤⡙⠳⣤⣤⡶⢋⣴⠆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡄⠀⢀⣀⣇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠾⠋⣠⡴⠶⣦⡀⠛⠦⠀⠀⠈⢻⡆⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡆⠀⠀⢀⡀⢠⡦⠴⠚⠁⠀⢯⡀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠙⠃⠀⠀⠀⣠⡾⢁⣀⣀⡈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠖⠁⠳⡈⡇⠀⠀⠀⠀⠀⠉⣹⣠⢦⠀⠀⡶⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣴⠟⠁⠉⠻⢦⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇⠀⠀⠀⠀⠀⠀⠟⠁⠈⠓⠚⠁

⠀⠀⠀⠀⢀⡤⠒⠓⠒⠒⠲⢄⠀⠀⠀⠀⠠⠤⠤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠓⠒⠒⠲⣄⠀⠀⠀⠀⠠⠤⠤⠤⠤⣤⣀
⠀⠀⠀⢀⠎⠀⠀⢀⠶⡀⠀⠀⢳⡀⠀⠀⠸⢭⠭⠭⠭⠓⠚⠳⡄⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢠⠶⡀⠀⠈⢳⠀⠀⠀⠺⣭⠭⠭⠍⠓⠚⢳⡀
⠀⠀⠀⡞⢀⡤⠶⡁⠀⢈⠲⢤⡀⡇⠀⠀⡠⡏⠀⣵⠀⠰⡎⠀⣷⡄⠀⠀⠀⠀⠀⠀⡏⢀⡤⢞⡁⠀⢉⠢⢤⡀⡇⠀⠀⡠⡇⠈⣶⠀⢐⡎⠀⣧⡄
⠀⠀⠀⢷⡚⡆⠰⠏⡀⠹⠆⢸⢳⡎⠀⠀⠑⢧⠀⠀⠐⠀⠁⠀⡿⠊⠀⠀⠀⠀⠀⠀⣷⣻⡄⠰⠇⡀⠹⠂⣸⢳⡇⠀⠀⠓⣇⠀⠉⠰⠀⠁⢀⡷⠂
⠀⠀⠀⠘⣌⠹⣄⣠⡂⢄⡰⠋⣡⠃⠀⠀⠀⠈⢦⣘⠅⠨⣂⠞⠁⠀⠀⠀⠀⠀⠀⠀⠸⡌⠱⣄⠀⠀⢀⡴⠋⣱⠃⠀⠀⠀⠘⢦⡀⠒⠂⣠⠞
⠀⠀⠀⠀⠈⠢⣤⠟⣉⠻⣄⠔⠁⠀⠀⠀⠀⠀⢀⡞⢫⡝⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣠⠻⣉⠻⣠⠞⠁⠀⠀⠀⠀⠀⢠⡟⢫⠟⣷⡀
⠀⠀⠀⠀⠀⢠⡟⠀⠛⠐⢻⡄⠀⠀⠀⠀⠀⢀⡎⠈⠁⠘⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡛⠀⠓⠐⢳⡀⠀⠀⠀⠀⠀⢠⠏⠘⠁⠘⠀⢳⡀
⠀⠀⠀⠀⢀⡏⠘⡄⠀⠠⠃⢳⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠑⡀⠀⠠⠁⢡⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⣼⠀⠀⠘⡤⠁⠀⠈⡇⠀⠀⠀⢠⡇⡆⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⡼⢠⠀⠐⡴⠁⠀⠘⡇⠀⠀⠀⢰⠇⡆⠀⠀⠀⠀⢰⢸⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⠁⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⢸⠀⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠨⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠆⠀⠀⠀⠀⢸⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⢋⡡⠀⡀⣀⢒⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣉⠔⢦⡅⠐⠲⠀⡒⢦⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢣⢄⡼⠟⢩⡤⠺⢯⡸⠿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡿⠀⠀⠀⠺⠯⢀⠹⠗⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⠀⠀⠐⢰⡘⠠⠈⠈⠍⠀⠂⡤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢇⠠⠀⠀⢉⠊⠱⣶⠶⣴⡟⠡⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠘⠄⣶⡘⠮⢁⡀⡇⠀⠘⡇⢈⡕⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢺⣏⠻⣦⡄⠀⠱⣤⠄⠀⣴⡞⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢿⣏⠣⠀⠙⠲⣀⣉⣠⢞⣥⡄⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⠬⣧⡀⠀⠀⠈⠋⣰⠋⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠟⢂⠀⠙⢦⡘⠓⠆⠀⠀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠒⠠⠁⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠒⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠰⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣡⠇⠐⠀⣀



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡴⣦⣤⡤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⢹⣿⢮⡄⠘⡿⠃⠸⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠻⠃⡄⢸⠋⠗⠹⠻⠅⠘⠻⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⡤⢤⢤⡶⡖⢶⠖⠶⣖⡖⣶⡟⣏⢻⣣⣸⠰⣇⢻⡇⢼⣉⣡⡜⠦⢢⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⡏⠓⣷⢋⡴⢃⡎⡼⢁⠈⠀⣼⠹⡇⣉⠃⢿⠀⢸⠏⡀⠈⢳⠹⣦⣼⣿⣆⣋⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠃⢀⢁⡇⠉⠈⠉⡾⠀⢡⡞⢰⠀⣿⠀⠁⠻⠀⠘⠀⡛⠀⠡⠀⠈⢧⡙⣄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠇⡿⠸⡇⣆⡇⢠⠟⡆⣿⠇⠀⣴⡯⠀⡄⠀⢠⠀⠀⡇⡀⢣⠈⡀⢀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣰⢹⡆⢷⣯⣗⣼⢠⡇⠋⣸⢸⡝⠃⣼⠃⢠⠸⡄⣴⡁⠃⠀⠀⠷⠠⢹⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⢻⣄⢻⡄⣿⣈⣽⠜⠛⠷⢽⣼⡦⣰⣯⣀⣸⠤⣤⣾⣯⢄⠀⠸⢐⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣏⡀⢿⡞⣿⣾⣟⡺⠀⠀⠀⠀⠀⠉⠙⠓⠛⠛⠈⠉⠁⠐⡶⠀⠀⣸⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢠⢿⣍⣛⢎⡷⣿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠀⠛⡄
⠀⠀⠀⠀⠀⠀⢸⣻⣀⡔⣻⠾⠉⡝⣿⣿⣝⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠈⠙⠚⠻⡍⠈⠏⠀⠘⢏⡙⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡠⠕⠚⢹⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⣠⠤⣄⣈⣳⣄⣂⣠⣤⣄⡀⠀⠀⣀⣀⠀⠀⢀⣀⣀⣤⡤⠤⣬⡝⠒⠓⠋⠑⠒⠂⠀⠀⠈
⠰⢀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠓⠚⠷⠟⠛⠑⠂⠀⠘⠊⠈⠈⠉⠉⠉⠉⠉⠉⠉
⠠
⠀⠀⠀⠀⠐⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠲⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⡌⠈⠉⠉⠒⠂⠀⠐⠊⠉⠉⠉⠀⠚⠦⣄
⠀⠀⠀⠀⠀⣀⡴⠊⠉⠉⠉⠉⠉⠙⠓⢶⣄⡀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡴⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠂⠉⠉
⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⣿⣷⣦⠄⠄⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⢀⡞⠹⣧⠀⠀⠀⠀⢸⠀⠀⠀⠐⡐⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⣼⠀⠀⠙⢧⣀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠒⠲⡴⠶⠤⠛⠒⠒⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⢻⠀⠀⠀⠀⠀⠉⠂⠀⠻⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢠⡞
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⢔⠁⢸
⠀⠀⠀⠱⠲⠤⢤⣄⣀⣀⣠⡴⠈⠢⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠒⠀⢀⠀⢠⡏
⠀⠀⠰⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠳⣤⣀⣀⣀⣀⣀⠴⠃⢄⠀⠀⢀⣣⠀⠑⢄
⠀⠀⠇⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢣⠀⢈⣿⡠⠀⡸
⠀⠀⠰⡀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⡸⠃⠀⠀⠻⣄
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠉⢙⠟⠒⠒⠒⠒⠚⠓⠉⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠏
⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠋⠁⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠋⠀⠈⠀⠒⠒⠒⠲⠒⠒⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣠⣤⡶⠟⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠂⡄⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠉⠛⠒⠒⠲⠖⠒⠚⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠚⠛⠂⠀⠀⠀⣼⠓⠒⠒⠳⠄⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⣀⠀⠀⠀⠀⠐⣧⡤⠄⡀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠷⠤⠴⠖⠚⠁⠠⣤⣄⣲⡤⠤⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠉⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡴⠊⠀⠀⢀⣠⠖⠁⣠⠖⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⢀⡤⠴⠋⠀⣀⡤⠖⠋⠙⠀⣀⠤⢒⠋⢀⠀⠎⠁⠀⡴⠋
⠀⠀⠀⠀⢀⡤⠚⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⣀⠤⠖⠊⠁⠀⠗⠒⠉⠁⠀⠀⠘⠂⠉⠀⠀⠛⠊⠀⢰⠃⣠⠎
⠀⠀⠀⠚⠁⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠚⠉⠀⠀⠒⠶⣤⣀⠀⠀⢠⡀⠀⠀⢀⡤⠚⠲⣦⣄⠀⢀⡏⠜⠁
⠀⠀⢉⡩⠶⠂⠀⠀⠒⠒⠒⠉⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⡀⠀⠀⣟⣿⣶⣾⣃⣠⠴⣫⢒⣴⣶⡇⠈⠀⡼⠁
⠀⠀⠀⠉⠒⠀⣠⠴⠂⠀⣀⠴⠒⠒⣆⠀⢸⠀⠀⠀⠀⠀⠀⠐⢳⣤⣀⣩⠽⠡⠤⢽⣄⠀⠠⣿⣿⣿⠃⠀⠀⡇
⠀⠀⠀⠴⠶⠯⠥⠤⢴⠞⠁⠠⣄⠀⢙⡠⠞⠃⠀⠀⠀⠒⠒⠲⠦⠬⠥⠤⣆⠀⠀⠀⣿⠳⠶⠚⠋⠁⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠊⢸⠆⠈⠀⢀⣀⣤⡶⠃⠀⠢⣄⡀⠀⠀⠀⠈⠓⠚⠋⠁⠈⠐⠢⡀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠀⠀⠀⢿⣿⣿⣷⣄⡐⠀⠀⠉⠓⠢⠤⣄⣀⣀⡀⠀⠀⠀⣀⣤⠴⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢨⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢠⠟⠻⡿⠛⠛⠻⢿⣿⣿⣷⣶⣦⣤⣄⣧⣠⡤⠤⠞⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠳⣄⠘⠢⣀⠀⠀⠹⡄⠈⠻⣿⣿⣿⣿⣿⠃⣀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠣⣄⠈⠑⠦⢤⣀⣀⠀⠈⠻⣿⣿⣿⠇⠈⣳⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣑⠲⠤⣤⣀⣈⣉⣭⠭⢖⣫⠵⠚⠛⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⠤⠤⠤⠖⠋⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣀⣀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡞⢅⡹⠿⠅⡸⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡻⠋⠗⠀⠸⠛⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣠⡅⡘⣿⡄⢢⢂⡾⣫⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⡤⡤⢠⢰⠖⢠⡎⡹⢡⠟⢡⢈⡎⣿⠀⠙⠒⠷⣤⣜⠉⠉⣸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠺⣙⡑⠯⡧⢠⡘⡎⢸⠸⡜⢠⢇⡟⠀⣾⠈⣷⢻⡃⠀⡉⠓⠈⡙⢔⣻⠁⢸⢄
⠀⠀⠀⠀⣀⣀⡀⣄⣫⣅⡿⠥⢬⡑⣿⣤⣧⣇⡃⠀⢡⠜⣼⡇⢰⡏⢸⡾⡀⣕⠠⡈⠀⠐⠒⠶⣤⣔⣡⡏⡈⠲⠄
⠀⠰⠿⠛⠛⠛⠓⠛⠫⣕⣶⣂⠞⠻⣆⣸⣔⣄⣽⣦⣄⡠⢦⣝⣶⣵⣼⣧⣵⢸⣇⣵⣄⣠⣀⡀⣀⣀⠈⢈⢁⡎⣓⣋⢠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠊⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣶⣶⣶⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠂⠀⢻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠌⠛⢿⣿⣷⣦⣀⠀⠀⢀⣠⣴⣶⣿⡿⠛⠁⠢⡀⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠀⢠⠂⠀⠀⠀⠈⠛⢿⣿⠃⠀⠙⣿⡿⠟⠁⠀⠀⠀⠀⠈⠄⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠄⢻⣿⣷⠀⠆⠀⠀⠀⡀⢀⡄⠀⠀⠁⠀⠀⣬⠀⠐⠀⣤⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠈⠉⠉⠀⠘⢄⠀⠀⠐⠠⠤⠀⠀⡄⠀⠀⠹⠀⠀⠀⠄⠀⠀⠀⢀⠌⠀⢸⣿⣿⡿⠛⠒⢄
⠀⠀⠀⠀⠀⠀⠈⠒⠒⣶⠀⠀⠀⠀⠈⠑⠤⠀⣀⡀⠠⠺⣀⠀⠀⢀⠔⠤⣀⠀⣀⣀⠴⠊⠀⠀⠘⠛⠛⡀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⡠⠄⠀⠠⠀⠀⠀⠉⢉⢡⠄⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⡠⢋⠀⠤⠀⢄⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠰⡃⠀⠀⢀⠀⡀⠈⠂⠤⠀⠀⠈⠀⠀⠈⠀⠠⡆⠀⠅⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠉⠀⠀⠀⠑⠄⡀⡀⣀⠄⠒⠒⠠⠄⠀⠐⠀⢠⡇⠀⡴⠚⠁
⠀⠀⠀⠀⠀⢀⣠⠤⣄⢀⡀⡀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠓⠐⠊⠀⢀⡆⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢠⠎⠀⠀⠊⠀⠀⠀⡃⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢀⠤⠤⢞⠁⠀⠈⠳⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⠀⠀⣀⣀⣀⠤⠐⠒⠊⠉⠀⠀⠘⡤⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢭⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠈⠁⠂⠐⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⠀⠀⠀⡠⠐⠒⠀⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠊⣄⡘⠃⣨⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠳⢴⣾⠃⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠄⠁⠈⠉⠁⠀⠉⠀⠄
⠀⠀⠀⠀⠀⣀⠤⠐⠁⠀⠀⣠⣀⡀⣮⡷⠀⠀⢧⣐⡠⡁⢠⣠⣤⡀⢀⣤⣔⡜⡁⠀⢈⠀⠀⠀⠀⠀⠀⢀⡘⠋⠑⠠
⠀⠀⠀⠀⠀⡇⠚⣷⣆⢠⢤⡌⣋⣵⣨⡻⡞⢠⣿⣅⡁⢳⡄⠉⠱⣤⣤⡍⠹⢼⡾⠀⣼⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⣀⣨⣟⡎⢛⣵⠳⢶⣿⣀⣢⠎⠈⠏⠀⠀⣹⠓⣶⣔⣛⣲⡶⠞⠛⠲⠏⠐⠀⢀⡀⠀⠀⡜⠃⡁⠐⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣬⣭⣤⣠⣡⠞⠉⠀⢀⡏⠠⠀⡏⠀⢌⢠⠉⣡⣈⠠⢀⠁⡸⣦⢴⡆⢣⣾⣥⣤⡶⠁⠀⡄⠄⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠶⠤⢒⣘⠾⢈⠀⢀⠔⠀⠠⠋⡨⠢⢼⡀⠀⠾⡐⠀⠈⡓⣶⣦⣶⠖⠋⡂⠀⠠⠀⠲⠀⠄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠒⠈⠉⠉⠉⠉⠉⠉⠃⡠⢀⡄⠀⠠⠔⢤⡒⢠⡔⠀⠀⡠⠼⢋⣀⣀⣒⣐⣿⠀⠀⠀⠀⣇⠢⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢼⠀⠀⠀⠀⠀⢁⠴⠀⢀⠔⠁⢀⠀⠀⠀⠈⠉⡻⠀⠀⢀⢎⠀⢠⠄⡏
⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠠⠆⢈⡁⠀⡠⠆⠀⠀⣤⢀⢜⠠⠄⠀⠂⠀⠀⢀⡜⠀⠀⢠⠀⠀⠁⡈⠔⢃
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⠀⡀⢀⡇⠀⢀⠔⡡⢘⣹
⠀⢀⣀⣀⡀⠀⣀⠀⢤⣤⣄⣀⣀⢀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⣠⢚⠝⡩⠞⢡⠟
⠀⣀⠄⠀⣀⣀⠀⠀⣤⣤⡤⠠⠌⢀⣈⣛⣿⠦⣤⡀⠌⠀⢀⡴⡄⠲⠤⣤⣤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⢤⣡⣪⠤⠊
⢰⠕⠊⠉⠁⠀⠀⠀⠀⠀⠁⠉⠃⠈⠀⠒⠛⠶⢢⣥⣦⡙⠖⠇⠤⠑⠂⠀⠈⠀⠉⠉⠉⠁⣀⠀⠀⠠⠔⢁⠀⠈⠾⠈⠁
⢨⣮⣐⣒⣂⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣄⣀⡀⠀⠀⢫⣀⠤⡔⠁
⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣙⣛⣒⣒⣂⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⢉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠀⠒⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢲⠀⠀⠀⠘⠓⠒⠒⠒⠽⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢟⣠⠞⠀⠀⣀⣀⡤⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⣀⡤⠖⠋⠉⠀⠀⢀⣠⠤⢤⣀⠀⠈⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡡⠖⠒⠒⠲⣤⣠⡞⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠃⢼⠀⠀⠀⠀⣿⣯⣿⢀⣀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠈⠳⣄⣀⣀⣀⡡⢾⡿⠿⠃⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠒⠲⠲⠞⠁⠀⠀⠀⠀⠀⠀⠒⠒⣲⠶⠦⣄⣀⣀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⣅⠀⡿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢶⣏⠁⠀⣽⠈⠙⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢰⣇⠀⢻⡒⢲⡦⠤⠤⠤⣤⡤⠤⠶⠶⢲⡿⡉⠁⠀⢸⠈⠱⠦⠟⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠘⣿⡀⠀⠑⠞⠙⣄⠀⠀⡎⠳⡀⠀⠀⣸⠀⠙⠦⡤⠃⠀⠀⣀⣀⣀⡖⠒⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣏⢧⠀⠀⠀⠀⠈⠙⠋⠀⠀⠈⠓⠚⠹⣆⣀⣀⣀⣠⠖⠋⢉⣁⡈⢃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠸⡌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⡨⠋⠀⠀⠀⡇⠀⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢳⡈⢦⠀⠀⠀⠀⠀⢀⡎⠱⣴⢋⡤⢄⠀⠀⡜⠑⠢⢄⢹⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠳⡄⠀⠀⠀⠈⠢⣄⢳⡞⠀⠀⠱⣀⠀⠀⢀⡤⠟⠁⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⢢⡀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⠢⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠐⢄⠀⠀⢸⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣦⠀⣨⣀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⢰⢋⡤⢤⠠⡶⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⠐⡆⡔⠐⢬⠀⠀⠀⠀⠉⠓⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠈⠙⠀⠺⠤⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠀⠀⠀⠀⣀⣀⡀⠀⣞⠛⠻⠿⡿⠓⠀⠲⡀⠀⠀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢁⠀⠤⠉⠉⠐⠑⠂⠀⠑⣄⠀⢀⠏⠔⠛⠒⠒⠒⠒⢒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠓⠀⠀⠐⠂⠈⠉⠁⠀⠈⠁⠀⠈⠉⠉⠉⠑⠒⠤⢄⡀⠽⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⠂⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠒⢄⣀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣀
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⠔⠂⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⡀⠤⣲⡅⠀⠀⠆⡠⡀⢀⣀⠀⠒⠀⠀⠘⡄⡄⢀⠏⠈⠀⠑⠒⠊⠢⣀⣀⡀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⢸⠀⠀⠀⢠⠒⠀⠠⠄⢒⣦⠁⠀⢀⠸⣿⡇⠀⣊⡸⠤⡏⠉⠂⠉⡯⣗⠄⠈⢆⣫⣬⡀⠐⠄⠀⠀⣴⣤⡀⠑⠐⠤⡦⢄⠀⡸
⠀⠀⠀⠀⠨⠀⠀⠀⠧⡀⠀⠀⡀⢿⣿⠃⠈⠁⠁⢁⡇⠀⡇⠐⠂⡬⢀⢎⣰⣓⢨⠀⠀⠀⢹⠿⡟⠠⠂⠆⠀⠹⢿⠃⠔⠀⠀⡤⠚⠉
⠀⠀⠀⣀⠀⡆⠀⠀⠀⠘⢄⠀⠈⠉⠀⠀⠁⠀⣀⠜⠀⡐⠉⠈⠉⠀⠀⠀⠀⠈⠀⠑⡀⠀⠀⢢⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡔⠀⠀⠀⢀⢀⡀
⠀⠀⣾⠉⠉⠝⠲⠤⠴⠒⠋⠑⠀⠤⠤⠄⠐⣎⠀⠀⠀⠆⠀⠀⠀⠠⠀⠠⠠⠀⠀⠀⡅⠀⠀⠀⠈⢂⡤⣀⡀⣀⣀⠠⠤⣊⠀⠀⠀⡠⠉⠁⢹⡆
⠀⠀⠉⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠑⠤⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠐⢨⠁⠈⠀⠁⠀⠀⢃⠀⠀⠉⢃⠠⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡧⢀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢀⡜⣁⣀⡀⣇⠀⠈⠀⠀⠀⠀⠀⠀⠸⠄⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠈⠉⠢⡵⠆⠤⠤⠤⠆⠂⠉⠛⠀⠀⠀⠈⠒⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠠⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⢲⠤⢀⡀⠀⠀⠀⠀⢸⠀⡸⠀⠀⠀⢨⠀⣀⠐⠠⠄⡠⢀⡀⡣⠀⠀⢈⠆⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣉⠀⠀⠐⠋⠀⠡⢀⠤⢀⡄⠀⠀⢸⠃⠁⠈⢄⠀⢑⠒⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡐⢡⠀⠀⢠⠀⢀⠐⠒⠄⠀⠀⠀⢀⢸⠀⡀⠀⡔⢠⠁⢣⡀⠀⠀⢸⠀⠀⠀⠀⠶⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡐⠀⠘⠀⠀⢈⠀⢅⠀⣠⠃⡰⢆⢀⠂⠑⡄⢉⡠⠅⡔⢆⣘⠁⡠⠃⢸⠀⠀⠀⢠⠀⢡⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡆⠀⠀⡐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡁⠀⢶⠀⠀⠘⠁⠀⣠⠜⢳⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢣⠀⠀⢰
⠀⠀⠀⠀⠀⠰⣄⡁⠙⠢⡀⠀⢘⠉⠉⠓⢠⡛⢅⡀⠈⢣⣄⡌⠑⢄⣨⠤⠚⠀⣀⡘⠈⡉⢖⡠⠶⠀⠀⢴⠀⠀⢀⠔⠋⢉⣩⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⢀⠄⠚⠉⣙⣻⣟⣋⣁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢿⣶⡦⢠⠶⠓⠊⠙⠛⠉⢩⣯⣉⠉⠉⠉⠓⠂⠁⣤⠀⣤⣴⠞⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⣇⠈⠻⠧⠀⠀⠀⣀⣒⣒⣒⠺⡗⣀⣀⣒⠢⠄⠀⠀⢠⡴⠟⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⠀⠀⠀⠛⠛⠉⣩⡭⢼⣿⣿⡿⢭⣍⠛⠛⠓⠀⠀⠀⢛⣠⠖⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠨⡿⠀⢠⣴⠊⠰⢛⣉⣥⣤⢰⣧⢠⣤⣉⡛⠆⠙⣦⠀⠀⢻⣿⢀⡓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⣠⠆⢀⠁⠀⠀⢋⣩⡍⢁⣼⣯⣀⠉⣍⡙⠂⠀⠉⡀⠠⣈⠉⣞⠁⢈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢡⡾⠁⢴⣿⠀⡀⢠⠈⠁⣘⠉⠀⠀⠉⢛⡀⠙⠀⠀⠠⣿⡆⠈⢣⡘⢧⡀⠊⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⡟⢱⢸⣆⣸⣌⣛⠂⠀⠀⢫⠂⠀⠀⠁⢬⠁⠀⠐⢛⣠⡅⣰⠈⣇⢳⡌⢣⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣿⣷⠘⢸⡏⣼⠋⠻⣘⣱⣄⠀⠀⠀⠀⠀⠀⢠⣌⣉⠝⠙⢏⢹⠂⡟⣾⣷⠀⡄⢲⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣿⠃⠀⢨⡀⢻⡄⠈⠈⠹⠿⠀⠀⠀⠀⠀⠀⠺⠏⠀⠊⢠⡟⢨⠀⠀⢻⣿⠀⡇⠀⠐⡈⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢻⣷⡀⣤⣷⣤⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣤⣾⣤⢂⣼⠿⠀⡇⠀⠀⢃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠈⠻⣿⣏⠑⢀⡹⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⠀⠉⣿⣿⠟⠀⢀⠆⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡀⠹⠿⢶⣬⣻⡴⢄⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⢀⠶⣿⣠⣾⠟⠋⣀⠞⠉⠀⠀⠀⢸⠃⠸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠂⠀⠀⠀⠘⠷⢤⡌⠀⠛⢶⡄⠈⣴⠾⠃⢈⣤⠶⠛⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢄⠀⠀⠀⠹⣶⢂⡀⠀⠀⠐⡗⠀⠀⢀⡐⣲⠏⢠⣤⡴⠂⠀⠀⣠⡄⠀⠀⠀⡈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⢷⡀⠀⠀⠘⢿⣶⡶⠶⠟⠛⠗⠶⢶⣾⠋⠠⠟⠁⠀⠀⠀⢠⣿⠁⠀⠀⠘⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠙⣦⡀⠀⠘⣧⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠸⠀⠀⣰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣦⠀⠀⠈⣷⠀⠀⠀⠉⢰⡂⠒⠊⠉⢁⣠⠖⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⣧⠀⠀⠸⣧⡀⠀⠀⠈⠛⢦⠀⣴⠟⠁⠀⠀⢀⠔⢠⡾⠋⠀⠀⠀⠀⠀⣾⠋⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⠀⠈⢧⡀⠀⠈⠳⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢁⡴⠋⠀⠀⠀⠀⠀⠀⣼⣇⡴⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠳⡀⠀⠀⠙⢷⡀⠀⠀⠀⣤⠞⣋⣡⠴⠋⠀⠀⠀⠀⢀⡞⠀⡰⠉⠋⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠙⣆⠀⠀⠈⠙⢦⡀⠐⠁⡼⠋⠀⠀⠀⠀⠀⠀⣠⡟⠀⢠⠁⠘⢋⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢦⡀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢷⠀⠘⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢠⡄⢠⠞⠉⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠂⢀⡀⠀⠉⠳⢤⡀⠀⠈⠀⠀⠀⠀⣀⣤⣶⠿⠋⠀⠀⡠⠞⠀⢀⣀⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⠀⠐⠂⠀⠀⠀⠀⢸⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡓⠢⣤⠀⠀⠈⠀⢀⣀⡤⠂⠀⠀⠀⠀⠀⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣟⣶⣶⡾⢿⢆⣴⠋⠁⠀⠀⠀⠀⠀⠀⠰⠛⠛⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠳⡀⠀⠋⣄⣀⣠⢴⢟⣡⠄⠀⠀⠀⠀⠀⠀⢰⠇⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⢸⠉⠀⢼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡐⠀⠀⠀⠤⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⠁⠀⠀⠀⠀⠀⠀⡂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢈⠀⠀⠀⠀⠀⠉⢢⡀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢀⠇⠐⠂⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠀⠀⠀⠀⠀⠀⠈⠐⠐⠃⠀⠀⠢⢄⠀⢠⠁⡌⠀⠀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⢀⠖⣠⣤⣄⣹⣦⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⡀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠠⣄⠘⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⣀⣤⡀⣸⡀⠀⢀⡄⠀⠀⣤⠀⠀⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠑⠶⠮⢤⣶⣾⣿⠜⠚⣥⡶⠟

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀
⠀⠀⣀⣤⣶⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⠟⠛⠲⠦
⠀⠈⠁⠀⠀⣾⠉⠙⠛⠻⠿⢿⣿⣿⣿⣶⣦⣤⣀⣀⣶⣇⣀⣤⣶⣾⣿⣿⡿⠿⠛⠛⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣏⢙⣿⡟⠛⠿⠿⣿⡿⣿⠿⠿⣿⠛⣿⡷⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣼⠃⠹⣆⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠤⠴⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠒⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⠤⠦⠤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠁⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⢲⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣀⣾⣀⣀⣀⣤⣤⣤⣯⣤⡤⠤⠤⣤⠴⠷⠶⠶⠶⠒⣶⠚⠛
⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⣿⠻⣄⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⣀⣀⣀⣠⣿⣀⣀⣀⣀⡀⢸⡇
⠀⠀⠀⢸⡀⠀⣿⡀⠘⣧⣤⡤⠶⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠒⠢
⠀⠀⠀⠈⢧⡀⠈⠙⠋⠉
⠀⠀⠀⠀⠀⠉⠛⠒⠂⠁