Thorough Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢾⠁⠛⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⠦⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⠀⠐⠆⠀⠀⠈⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⢢⡀⠰⡄⠀⠀⢸⡇⠘⠳⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠁⠀⠘⡄⠀⠀⡇⠀⠀⡄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠸⣦⠀⠃⠀⠀⠁⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⠀⠀⠀⢸⣷⡟⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠋⠉⠛⠋⠛⠋⠀⠀⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣉⡛⠀⠀⢀⠀⠀⢀⣠⠴⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢓⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⢀⣀⣤⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠙⠷⡄⠀⠀⠐⠀⠀⣠⣴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠠⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⢻⣧⡀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⡆⢰⣿⠀⠀⠀⠀⠀⠀⠠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣿⣿⣆⠀⢀⣼⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⡄⣀⣤⣷⡘⡏⠀⠀⣀⠶⠀⠀⢼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠙⠳⠄⣸⡇⠀⠀⠀⣨⣥⣤⣄⣀⠀⠀⠘⣧⣿⠃⠀⢻⣏⣤⡾⠋⠀⠀⣴⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢻⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠛⠋⠉⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠈⣧⠀⠀⠈⢿⣿⣛⠛⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⠁⠀⠀⢠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⡀⠀⠀⠈⠳⡄⠀⠈⢻⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⣴⡿⠏
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡰⢦⡀⠀⠘⢦⡀⠰⣿⣿⣿⢻⣿⣿⣦⡤⢄⣀⡀⠀⢀⣾⣿⣿⣿⠏⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⡟⠰⢦⡀⠙⢦⡘⢿⣿⡎⠈⠉⠑⢹⣿⡏⣼⣿⣹⣿⡟⠉⠁⣠⠾⠁⠀⣀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⢸⡟⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣈⠛⡷⣤⡙⢶⣽⣧⡀⠠⠀⢸⡟⠙⣿⣧⣿⠋⠙⠿⠿⠿⠶⠿⠿⣛⠉⠁
⠀⠀⠀⠀⠀⠀⣼⡇⠘⢿⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣿⣿⣏⠙⣦⣿⣿⣿⣶⠀⠊⠀⡐⢀⣿⠃⠀⠀⠐⣧⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠹⣌⠻⣤⡀⠀⠀⠀⠀⠀⠀⢶⠙⠿⣿⣆⠀⠘⢿⣿⣿⣧⠀⢠⢠⣾⠏⠀⠀⠀⠀⠸⡆⠀⠀⠐⢿⡄
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠘⢧⡀⠉⠀⠀⠀⠀⠀⣀⡀⠣⠀⢿⣿⣡⠀⠈⢻⣿⣿⣷⣶⣿⡿⠊⠉⠙⠛⣼⣄⠁⠀⣤⠐⠸⡇
⠀⠀⠀⠀⠀⠀⡿⠷⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠙⠧⠓⡀⠀⢻⣿⠉⢹⣿⠀⠀⠀⠀⠊⠁⠛⠀⡀⠸⠀⠀⠁⠀⠀⡠⠖
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠘⠄⠈⣿⡄⢾⡇⠀⠀⠀⠀⠈⠄⠀⠀⠀⢸⡆⠰⠀⣤⠏⡠⠂
⠀⠀⠀⠀⠀⠀⢸⣷⣦⣤⣀⣀⣀⣀⣠⡤⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⢸⡇⠀⠀⢀⣤⡶⠲⠟⠒⠒⣠⣄⢀⣾⡵⠊
⠀⠀⠀⠀⠀⠀⠈⢯⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠂⠀⡆⠸⡇⣀⠴⠚⠁⠀⡤⠒⠋⠉⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠤⠴⠶⠶⠶⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠿⠋⠀⠀⠀⠊⠀⠀⠀⠀⣠⡼⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠉⠀⠀⠀⠁⠀⠀⠀⠀⢀⡾⠋⠑⢻⡟⠀⠙⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠃⠛⣓⡀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣴⠎⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣶⡾⠛⠃⢀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⣤⣀⡀⠀⠀⢀⣀⣤⣦⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠁⠀⠈⠹⣷⠐⠂⣾⠋⠁⠀⠉⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡇⠀⠀⣤⡀⠀⠁⢀⡀⠈⠀⢀⣤⠀⠀⢸⡧⢀
⠀⠀⠀⠀⠀⢀⢠⡶⠶⠽⢾⠈⠀⠀⠀⣰⡟⠙⢷⡀⣾⣷⢀⡾⠋⢻⣆⠀⠀⠀⠁⡶⠭⠶⢶⡄⡀
⠀⠀⠀⠀⠀⣼⠕⠀⠀⠀⠀⠀⠀⣿⣦⣸⡇⠀⠈⢻⣿⣿⡟⠁⠀⢸⣃⣴⣿⠀⠀⠀⠀⠀⠀⠪⣦
⠀⠀⠀⠀⠀⣿⡄⠀⠀⢿⡟⠺⠶⣼⣿⡿⣧⠠⣷⣄⣣⣜⣠⣾⠆⣾⢿⣿⣧⠶⠗⢻⡿⠀⠀⢠⣿
⠀⠀⠀⠀⠀⢘⠻⠂⠀⠸⢷⡄⠀⣠⣈⢣⢶⡍⢿⣿⢘⡃⣿⡿⢩⠖⡜⣁⣀⠀⢠⡾⠇⠀⠐⠛⡃
⠀⠀⠀⢀⣤⣬⡀⠀⠶⣶⣶⣿⣦⡌⡻⣿⣦⠱⠆⢭⠎⠱⡭⠐⠏⣴⣿⣟⢡⣴⣿⣶⣶⠶⠀⢀⣵⣄⡀
⠀⠀⢠⡿⠋⠁⠀⢀⣠⣬⡽⠿⠧⣐⠗⠄⡍⠾⠃⢶⣼⣧⡶⠘⠷⢩⠠⠺⣂⠼⠿⢯⣥⣄⡀⠀⠈⠙⢿⡄
⠀⠀⢎⡇⠀⠀⠰⣿⣉⠀⠀⠰⠶⣿⣿⡷⢰⠆⣉⣿⣿⣿⣿⣉⠰⠆⢾⣿⣿⠶⠆⠀⠀⣉⣿⠆⠀⠀⢘⡱
⠀⠀⠘⣷⣄⡀⠀⠈⠙⢛⣳⣶⡖⠩⡦⢒⣃⢶⡄⠾⢹⡏⠷⢠⡶⣘⠐⢴⠍⢲⣶⣞⡋⠋⠀⠀⢀⣠⣾⠃
⠀⠀⠀⠈⠛⢛⠁⠀⠶⠿⠿⣿⠟⢋⣵⣿⠟⡰⠆⣚⢆⡰⣓⠰⢆⠻⣿⣮⡙⠻⣿⠿⠿⠶⠀⠈⡛⠛⠁
⠀⠀⠀⠀⠀⢨⣴⠄⠀⢰⡾⠃⠀⠛⢉⡜⠲⣃⣾⣿⢨⡅⣿⣷⣘⠷⢣⡉⠛⠀⠘⢷⡆⠀⠠⣤⡅
⠀⠀⠀⠀⠀⣿⠃⠀⠀⣾⣧⢴⠾⢻⣿⣷⡟⠰⡿⠋⡜⢫⠙⢿⠆⢻⣾⣿⡟⠶⡦⣼⣷⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢻⡢⡀⠀⠀⠀⠀⠀⣿⠟⢩⡇⠀⢀⣼⣿⣿⣧⡀⠀⢸⡏⠻⣿⠀⠀⠀⠀⠀⢀⢔⡟
⠀⠀⠀⠀⠀⠈⠘⠷⠶⣲⢾⢀⠀⠀⠀⠹⣧⣠⡾⠁⢿⡿⠈⢷⣤⣼⠏⠀⠀⠀⡀⠷⣖⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⡇⠀⠀⠛⠁⠀⡀⠈⠁⢀⠀⠈⠛⠀⠀⢸⡷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⠀⢀⣠⡿⠠⠄⢿⣆⡀⠀⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠻⠚⠉⠁⠀⠀⠈⠉⠛⠟⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⢠⢟⡦⡀⠀⢀⣄⡨⠤⡡⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠒⠒⠳⣦⡠⡱⠃⠙⢮⣲⡟⠁⠀⠀⠈⢻⣤⣴⠶⠭⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⢤⣤⣤⡿⠀⠀⣀⠀⠨⡟⠁⠀⠀⠀⠙⣷⣠⡞⢳⣀⣼⣏⠀⢀⠀⠀⠱⡇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠁⠀⠀⠈⢳⣤⣸⣯⢽⢺⣇⠀⠀⠀⠀⢀⣾⡅⠉⠛⣽⣇⡿⢲⢏⣳⢀⣠⣟⠠⠌⠺⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⡗⢲⡾⣿⣷⡈⠀⢰⣿⡷⣤⢞⠦⡏⡿⢇⠀⡰⣽⣿⡁⠀⠈⣽⡟⠁⠀⠀⠀⢸⢸⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣭⠭⠭⠍⠝⠶⢶⣟⠃⠁⠸⡹⣞⡢⣈⣿⡙⣡⢎⡧⣥⢫⣿⣟⣞⡵⣏⣄⡤⣪⣿⠇⠀⠀⠀⠀⢠⡞⠋⠉⠑⠠⠍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢿⣷⡢⣄⡿⣧⣽⣺⢟⠇⢃⡞⡙⢎⢣⢫⡽⢫⢸⢽⣟⡯⣪⡚⣂⢹⠀⣠⣤⡾⣧⠤⡤⠀⠀⣾
⠀⠀⠀⢀⡀⠠⡶⠞⠛⢾⡀⠀⠀⠀⡤⠬⠮⣛⠵⣾⣏⡇⢮⣻⣼⢸⢰⣹⣌⠆⡾⡷⢨⣤⣋⢕⣩⠥⢶⢨⡜⣵⢏⠏⠀⠉⢶⣇⣀⡴⢿⣤⡀
⠀⠀⠀⠈⡄⡅⠀⠀⣀⣬⡿⣶⣶⣦⢤⢘⢡⣭⣍⣀⡂⢽⢲⣿⢚⣦⡻⣷⡟⣰⠥⣵⣊⢺⢡⣫⠔⡇⡍⠘⣼⣎⢮⣤⢔⣶⣿⡿⣟⣀⠀⠀⠑⡌⡰
⠀⠀⠀⠀⠰⣧⠀⠈⢳⠮⠁⠀⠑⣝⣗⣝⠆⢧⢰⣒⣭⣦⢣⡾⣉⣯⢵⡤⢼⣵⣿⣶⡎⡿⢺⣧⣿⠔⣡⣡⠮⠿⣿⣩⢏⠗⠉⠀⣻⣹⠇⠀⢀⡧
⠀⠀⠀⢀⣤⠿⠷⣾⣛⣷⣦⣴⣄⣸⡽⡟⣎⢪⡓⠽⡾⠿⠘⣧⣽⡿⣋⢼⣹⣬⣟⣻⣻⣗⣤⠶⢖⣍⣳⣯⣞⣵⣹⣿⣗⣰⣶⣦⡴⠿⠤⡴⢟⠁
⠀⢠⠖⡕⠁⠀⣠⠼⣏⠛⠻⢟⢤⣳⣍⢭⠤⠿⣝⢩⢟⣓⣾⢾⣑⡿⢿⡟⣾⢳⡻⣟⢧⣪⢽⣼⣯⢼⣦⠗⣛⣂⡸⠭⣄⠲⡝⠋⠀⠀⠀⠈⠑⠮⡢⢤⡀
⠀⠀⠑⣥⡀⠀⠙⢶⡃⣀⢀⡠⣿⣾⣏⠷⢾⣗⡎⡟⣼⣿⡿⣹⣿⢞⢳⣶⣿⣿⣶⡏⠾⣿⣳⣝⢷⠊⣵⡿⣖⣢⠌⣳⠮⠵⢊⠗⠀⠀⠀⠀⢠⢖⠝⠁
⠀⠀⠀⠈⢻⢶⡾⠛⠙⠻⣟⣓⣒⣭⢗⡩⠴⣢⣬⣙⠮⣿⡥⠿⢺⣭⣿⣿⣿⣿⣿⣿⣛⡧⣷⢚⣾⡲⣍⡛⠫⠔⣊⠥⣚⠼⣭⣵⣦⣄⣤⡾⠷⣧⡀
⠀⠀⢀⣔⡱⠋⠀⠀⠀⠀⢴⡫⢆⡰⢯⡐⠫⠽⣼⢟⣠⠷⣝⢫⢿⡶⣸⡿⣿⣿⠿⢧⡱⣿⠏⣾⣿⡟⣾⡨⢭⡿⢶⣹⡿⣿⠊⠁⠈⢨⠷⣄⠀⠈⢙⢄
⠀⠘⠓⠪⡒⢄⡀⠀⠀⠀⣠⣟⠲⠝⣒⡎⠭⣭⢴⠻⡕⢿⡟⣷⡻⢳⣵⣮⢧⠿⣼⡷⣞⢍⡷⣿⢭⣵⣃⣝⣶⠀⣓⣙⣯⡳⣵⣖⣤⣻⡗⠋⠀⠀⣜⠥⠃
⠀⠀⠀⠀⢀⣵⠞⠲⢶⠾⠻⠿⠛⢽⣻⣞⢟⣽⢻⠿⣙⠵⠶⠛⢽⣯⣬⣝⡛⣏⡗⣩⣾⣚⢻⡀⣶⠿⣖⢬⡣⡹⣾⣟⡯⠙⠛⠻⢿⣭⡿⢶⣶⠚⠁
⠀⠀⠀⢀⢺⠁⠀⢠⣏⣯⠀⢀⢴⣡⣋⣿⣖⡲⢋⢋⡔⣿⢿⣇⣾⣸⠿⣿⢟⡗⠘⢏⣻⣛⡮⢣⠻⡛⠭⠦⢳⠸⣝⢝⣝⢄⡀⢀⡲⢷⡀⠀⢹⡆
⠀⠀⠀⠎⠘⢄⠀⠀⠉⣹⣾⣿⠿⠝⠚⣳⡽⡟⡆⣺⢸⠼⣫⢃⡧⣩⢟⣒⡟⣼⢿⣮⠻⡥⣷⢯⣗⠨⢉⣛⣛⣃⡇⠓⠻⠿⠿⣾⡓⠋⠀⠀⢘⠘⡀
⠀⠀⠀⠀⠀⠉⠛⣶⠞⠉⢹⠷⣂⠀⣠⣵⢟⡜⡃⠿⢒⣋⢕⣩⠛⡃⢾⡾⠰⡹⣇⠇⡇⡟⣯⡳⢸⢼⡽⣖⣭⣲⡒⠚⠀⠀⠀⠈⡿⣤⡴⠦⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠘⠖⢻⡾⠛⠋⠁⣇⠉⡬⣫⡲⣿⣶⡇⣫⣞⣣⢣⡱⣌⡴⢱⢰⣵⡾⣟⢺⣾⠙⠪⢽⣷⠀⠀⠀⠀⠀⠸⠐
⠀⠀⠀⠀⠀⠀⢠⡐⠢⢄⣀⣀⡼⠃⠀⠀⠀⠀⢰⣿⡫⠞⠉⣹⠙⡽⣽⣿⢣⣛⢌⡡⢋⢄⡯⡛⠪⡽⣮⡆⠀⢠⣽⠷⠶⣔⣐⣒⣐⣓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡇⡇⠀⠀⠀⢀⣼⡟⡀⠀⢈⣿⣿⡞⠀⢱⣿⣸⠢⣵⠛⢾⣿⠇⠀⡈⢿⣿⡾⠯⡼⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡧⢐⡒⣽⠋⠁⢿⣑⠧⣾⢻⣟⣠⣀⠈⡿⠁⠀⠀⠀⠀⢹⣧⣗⣻⡏⠙⢧⡀⠀⠀⢀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⢆⠀⠀⠁⠀⣹⡟⠉⢧⡼⠋⢿⣄⠀⠀⠀⢀⢼⡃⠈⠉⠀⠀⣾⠛⠛⠃⠈⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠂⣐⠶⠟⠛⣧⡀⠀⠀⢀⣼⠯⡳⣄⢠⢊⠊⠻⢦⠤⠤⢊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢫⠒⡢⠛⠁⠀⠈⠺⣱⠁⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⡋⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢰⡶⠀⠀⢰⠆⠀⠀⠀⣤⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠈⠁⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠁⠤⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣀⡤⠚⣋⠈⠁⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠐⠋⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀
⠀⠀⢀⡞⢁⡠⠤⠴⠶⠶⠤⣤⣀⡘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⣬⡕⠉⠀⠀⠀⠀⠀⠀⠀⠍⠙⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⣿⠀⠀⠐⠂⠀⠲⠦⠤⠶⠦⠤⠄⠀⠈⠙⠳⣦⣀⠀⠰⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠒⠛⠀⢀⣀⠤⠀⠉⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠶⠛⠋⢀⠀⠀⣀⠄⠀⠉⠛⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠈⠳⣄⡀⠀⠀⠘⠛⠉⠉⠀⣀⠤⠚⠀⠠⠚⣁⣠⡴⠾⠛⠛⢩⠙⠻⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠻⠶⣤⣄⣀⣀⡀⣉⣀⣠⣤⠶⠶⠛⠋⠁⠀⠀⠀⠀⠈⡄⠀⠀⢻⡛⠻⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⣼⠂⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢠⡆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⡿⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⠀⣀⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠚⠳⣦⠄⢀⠀⠸⢀⡀⠄⠺⠄
⠀⠀⠀⡀⠀⠀⠀⠀⢀⠂⢀⠀⠀⠀⠉⠹⡤⠄⠀⠀⢀⡇⠠⢄⡴⠄
⠀⠀⠀⠀⠠⠤⠄⣀⣍⠀⠀⠐⠀⠤⠲⠸⣷⠖⣲⠶⠞⠀⢀⡠⠂⠀⠁⢢⠀⢀⠀⠀⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣁⢠⣄⣀⡤⠄⣿⣾⡟⣳⣮⣵⡞⢢⠁⠀⢠⡏⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠠⠀⠤⠀⠁⠘⢦⣈⣤⣿⣏⡝⢎⠡⣽⣿⣵⣽⣿⣯⢻⣤⣶⣿⣿⡀⠀⢱
⠀⠀⠀⠀⠀⠠⢦⡀⠀⣠⠦⡤⣿⠯⣑⣾⣳⣾⣿⣟⣿⣿⣿⣷⡹⠻⠻⠿⣿⣤⡟⠁
⠀⠀⠀⠀⠈⠤⠲⠿⠶⣶⠴⣎⠉⠻⢿⣿⣿⣿⣶⣹⣿⣿⣿⣿⣷⣴⡿⣿⣟⢫⣏⣓⣓
⠀⠀⠀⠐⢊⠀⢠⣶⢦⣼⣿⢮⣶⣴⣚⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡌⢉⡻⠟⡿⢭⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣖⣉⠄⢂⡀
⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡿⣿⣿⣿⣿⣿⣿⣷⡶⠆
⠀⠀⠀⠀⠀⠐⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡍⠙⠛⠻⡗⠿⣿⢽
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠾⠶
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡿⣿⣿⣿⣿⡟⣿⣛⣻⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢹⣿⣿⣿⣿⠀⠀⠈⠋⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣾⡟⠋⢿⣟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣜⣿⣿⠀⡘⡏⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣟⣿⣿⠁⡐⣤⣠⡀⢠⠇⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⢸⡏⣿⢿⣿⢀⣿⣿⣿⣿⡏⠀⢀⣇⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣇⡾⣹⣿⠘⢿⣿⠿⠋⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⡉⠘⡋⠀⣾⡿⠂⠀⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢃⠀⠀⢠⣾⡇⢰⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣅⣸⡇⠀⣆⠀⣸⣿⡿⢿⣬⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠫⠁⠀⣍⡐⣿⣿⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠙⢳⠀⠀⢀⠈⠛⡉⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⢿⣿⣾⣷⠆⠈⠄⠀⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠸⠿⢿⣿⣀⣠⣿⣾⣿⠿⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⡠⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣴⣿⡀⢀⡖⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣴⣾⡿⣽⣿⣿⣿⣿⣿⣿⡿⣷⡞⠫⡏⠉⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⣿⣇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⠥⠀⡀⠄⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⢸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠳⠶⠚⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠀⠠⡿⠿⠿⠟⠛⠋⠉⠉⠀⠀⢀⣀⣤⣴⡖⡀⠀⠀⠀⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⣤⣤⣶⣶⣿⣿⣿⣿⠟⢩⣗⠠⠀⢰⣷⠈⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⢠⣀⣿⠀⠀⠋⢀⣤⠀⣀⣍⠴⣍⡛⢿⣿⣿⡿⡟⢿⠿⢿⣿⠛⠍⠀⣼⣿⡀⠒⣸⣿⠀⣤⠆⢀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣼⡿⠞⠀⢈⠗⠲⣂⠄⢈⡅⠉⢁⡀⠆⣉⡩⠛⠉⠁⢀⠐⠂⡄⠑⠐⠭⣢⡀⡀⢨⠑⠣⠛⠻⢇⠁⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠊⠀⠀⠠⠖⠧⢀⡄⠀⠀⡀⠀⠁⠘⠋⠈⠁⠀⡀⠄⠂⠙⡀⢰⣷⠀⠠⡀⠀⠀⠈⠻⠗⡤⠀⠀⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⠃⠀⠀⢀⠂⠠⢸⣇⣾⠀⠈⠁⠀⠀⠀⢠⡆⠈⢀⠀⠂⢠⣇⣸⣿⡀⣀⣀⠄⢀⠅⠀⠁⢠⡀⠀⢀
⠀⠀⡆⠀⠐⠀⠀⠀⠤⣀⣀⠀⢀⣠⣤⡤⣬⣍⠀⠀⠴⠄⠀⣠⡄⣺⣇⢒⠶⠖⠠⡚⠛⠙⠉⠉⢀⠄⠊⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠁⠨⠑⠠⠄⠉⠀⢀⠀⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠐⠆⠨⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠂⠈⠘⢿⣿⣶⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡰⢠⢤⣄⡄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠙

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠛⠿⢑⣵⡞⡏⡿⠀⡇⢸⠏⢛⣦⣁⠙⠉⠙⠻⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⠟⠃⠀⠀⢰⣿⡍⣇⣏⡏⢸⣇⡟⢠⡿⠙⣿⡋⠀⠀⠀⠔⠃⠑⠋⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠛⠏⢉⣨⣿⡿⠋⠁⠀⠀⣀⠄⢸⣿⣧⣿⢹⣧⣿⡾⣠⡿⢡⣼⢟⣿⡦⠀⠀⠀⠀⡀⠀⠀⠉⠽⡿⣯⣌⠉⠟⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⢷⣶⣟⡇⡁⠀⠀⡀⠄⠀⠀⠀⢸⣿⣿⣯⣿⣿⠞⣴⢋⣰⢟⢁⣿⣿⡇⠀⠀⠀⠀⠛⠂⢄⠀⠀⠀⣡⣟⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠇⠁⠀⠄⠋⠀⠀⠀⠀⠑⢦⣀⣀⣠⣿⣿⡿⢿⣿⣡⣞⣵⢞⣱⣿⢿⣛⣡⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣄⠀⠁⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⡏⠈⣠⡖⠚⠂⠐⠒⠒⠒⠶⠶⣶⣤⣽⣿⣿⣮⣹⣿⣿⣿⣿⣷⣾⣭⣶⣿⣟⢛⣿⠀⣐⣀⣄⠀⠀⠀⠀⠐⣤⡀⠀⠐⠛⠂⢦⡈⠙⠻⣿⣿⣿
⣿⣿⣿⣗⠤⣺⣷⠄⠀⢀⡀⠂⠀⠀⠀⠀⠈⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⢱⠗⠃⢸⡟⢿⣿⣦⡀⠀⠀⠀⠀⠠⡀⠀⠀⠼⣷⡆⢔⣽⣿⣿
⣿⣿⠏⢀⡐⠀⠀⢀⡀⠊⠀⠀⠀⠀⢀⡄⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠠⣿⡇⢸⣿⢿⣿⣶⣄⠀⠀⠀⠘⠂⡀⠀⠀⠐⢀⠈⢿⣿
⠛⢻⣶⣿⣦⣠⣴⠅⠀⠀⠙⣶⣾⡆⠈⠾⠃⠹⣿⣿⣿⣿⣿⣿⣿⣟⣯⣿⣿⣿⠃⣿⠁⠀⡀⣸⣿⠁⣼⣿⣷⣤⡝⠿⢷⣄⠀⠀⠀⠈⢵⣤⡠⣾⣷⣾⠛
⠘⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⣘⠛⠃⠀⠀⠀⣠⡘⢿⣿⣿⣿⣿⣿⣾⣿⡿⣿⣿⣿⡗⠀⢴⣾⣿⡏⣰⣿⣿⣽⣿⡿⢿⣾⣿⣿⢦⣀⠀⠀⠈⠉⠀⠀⠀⠘
⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⢂⣁⣴⣶⣿⣿⣿⡸⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢰⣿⡟⠱⣿⣿⡿⡟⠋⠀⠀⠀⠀⢻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠠
⠀⠐⠈⠡⠀⠀⢀⣤⢞⣛⣭⣾⣿⣿⣿⣻⣿⣿⣿⣥⠀⠀⠀⠀⡟⣿⣿⣿⣿⣻⣿⣆⣂⣼⠟⣀⣴⣿⣿⣄⣠⣠⠦⣼⣷⣷⣾⣿⣏⣍⣉⣷⡀⠀⠈⠒
⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣯⢻⢾⣿⣿⣿⣿⠟⣋⣀⣹⣿
⠀⢰⢴⢻⡏⡉⠀⢀⢀⠀⠀⠀⠀⣿⣁⣀⠀⠉⠙⠛⠁⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⣠⣿⡿⢋⣻⠀⠀⠐
⠀⢸⡌⢉⣉⣩⣥⣤⣦⣤⡀⠀⢀⠻⣿⣿⡿⣦⡀⠀⠀⠀⠀⠀⠸⡿⣻⣿⡿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⡿⢯⣾⣿⣿⡇
⠀⠈⣷⣜⣩⣭⣽⣶⣶⣷⣶⣒⣂⣰⡬⠉⠋⠙⣿⡄⠀⠀⠀⠀⣠⣿⣿⣿⠇⢸⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⣠⣟⣡⣼⣿⣿⣿⣧⣱⣄⡀
⠀⢀⣿⣛⣿⡛⠀⠀⢉⣿⡿⠏⣻⣿⡇⠀⠀⠀⢿⣿⡄⠀⣴⣿⣿⣿⣿⣿⠇⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢋⣤⣹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⢿⣷⣯⣿⢿⠿⣿⣿⠿⣿⣻⣿⣿⣥⠀⠀⠀⠈⠧⠙⠀⠈⠙⠛⠛⢛⣯⣲⣿⣻⣿⣿⡏⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⡿⠿⠟⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠯⢿⣿⣷⣿⡿⠀⠀⠀⠀⠀⠈⠻⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⣲⣿⣿⠀⠁⠈⠉⠀⠀⠀⠀⠀⠂
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣿⡟⠀⢄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢞⠟⠁⠀⠐⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠
⢠⠀⠁⠀⠀⡀⠀⠀⠀⣤⡒⠂⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⢁⡄⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊⠀⠀⠈⢂⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⣠⣤⣴⣤⣄⣴⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢳⠀⠁⠁⠀⠀⠀⢀⠀⠀⠀⠈⠉⠙⠛⢿⣿⣿⣷⣶
⠓⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠄⠂⠀⠀⢹⡿⠃⠙⠉
⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⢄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠘⡁⠀⠂⠀⡐⢤⣄⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠀⠈⠉⣺⣿⣿⣠⣿⣶⣾⠆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣻⣿⠛⠛⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⣰⠁⠀⠶
⠀⠁⢰⡀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠤⠀⣲⠞⠛⠻⣇⠐⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⢠⠋⠁⠊⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠶⢦⣤⡄
⠀⠁⠀⠀⡀⠀⠀⠀⠠⠤⣀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠉⢘⡄⠀⢀⠀⠀⠀⣠
⠀⠀⠀⠆⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢹⣄⣸⣦⣶⣾⠋
⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠋⠀⠀⣠⣖⠀⠀⠀⠀⠀⠀⢈⣿⣿⡛⠉
⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⢄⡀⠀⠸⣿⣷⣤⡆⢠⣾⡿⠋⠄⠀⠀⠀⠐⠀⠈⠁⠀⠙⠦⣀⡒
⠁⠀⠀⠀⠀⠀⠀⢠⠀⠠⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣌⣙⠇⠀⡸⣿⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠔⠒⠊⠩⠛⠛⠛⡿⡿⢉⣤⢚⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⣼⣿⣷⣾⣷⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠛⠉⠉⠉⠉⠉⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣼⣷⣶⡶⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣿⡏⠀⠀⣠⢤⣀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⡋⠀⠀⠄⣈⣿⣏⣹⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢻⠁⠀⠘⠟⠘⠋⠻⣇⠦⡄⠀⠀⠆⡆⠀⣿⠇⠀⠀⠉⠉⢻⠓⠈⠋⠀⠙⣿⣧⡸⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⢸⣆⣀⡀⡀⠀⢶⣦⠀⠀⠀⠀⠀⣇⢠⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠈⠈⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣮⣿⣿⣧⣴⢾⡛⠋⠄⠀⠀⠀⠀⡯⠼⡇⠘⣿⣷⣦⣤⢢⡦⠀⠀⢙⣹⣷⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡜⢿⣿⣿⣿⣦⣿⠁⠀⠀⠀⠀⠀⢞⡔⢻⣶⣿⣿⣿⣿⣸⣥⣴⣾⠿⠿⠏⢱
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠻⣮⢻⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠰⢷⡈⢿⣿⣿⣿⣿⣿⣿⡏⠙⢷⣶⡄⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣆⠀⠻⣷⣹⣿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠘⣷⣈⠻⣿⣿⣿⣿⣿⣿⣷⣶⣿⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢻⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⡈⢿⣿⣿⣿⠿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢻⣧⣧⠀⠀⠀⠀⠀⣿⣦⠀⠀⠸⣿⣿⣌⢻⣿⡏⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢆⠀⠀⠘⠙⢻⡌⢿⡇⠀⣿

⠀⠀⠀⠀⠀⠀⢀⣴⡶⢲⡆
⠀⠀⠀⡀⠀⣠⡟⢠⡇⣼⣷⣤⣴⣤⣶⣶⣶⣖⣷⣿⣿⣿⣿⣶⣶⢶⡷⣾⣿⣾⣷⣾⣿⣷⣶⣿⣷⢿⣶⣶⣶⣶
⠀⠀⠀⣷⣿⣿⣷⣾⣿⣟⣿⣿⣿⣿⣟⣿⡻⢻⢿⡯⢿⠽⠝⠿⠿⡏⠛⠿⠻⠿⠟⠻⠿⠿⠿⠿⠻⠻⠻⠿⢿⡿
⠀⠀⠀⣈⠚⠛⠛⠿⢿⣿⣿⣿⠛⠋⠟⠏⠀⢠⣄⡀⢀⠀⠀⠀⢀⣀⡀⢰⣶⣶⠄⠀⣴⣶⡄⡀⢀⡴⢶⣤⢀⣠
⠀⠀⢸⣿⣿⣿⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⣭⣭⣭⣭⣯⣀⠀⣸⣿⡷⢾⠛⣿⠁⠀⣿⢡⡿⣽⣾⡇⠀⢸⢸⣿
⠀⣶⣿⣿⣿⠟⠂⣌⣯⣿⣿⣿⣣⣶⣆⠀⠀⣿⣿⣿⢛⣛⣛⢳⣿⣿⣿⣿⣶⣿⣦⣶⠛⠒⠛⠻⠿⣿⣗⣚⢸⡿
⠀⣿⡿⣿⣷⣀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣶⣿⣷⣾⣯⣿⣿⡟⢛⣏⢁⠘⠀⠀⠁⣀⣾⣻⣜⡛⣿⣟
⠀⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣽⣿⣯⣩⣿⣿⣿⣿⣿⣿⣿⣷⣡⢸⢆⣴⣇⣴⠶⣬⢹⣿⣿
⠀⣇⣐⣫⣿⣿⢿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣯⣉⣉⣻⣿⣾⣿⣿⣿⣿⡟⡛⣿⣿⣿⡿⣿⡉⣿⡅⣿⡀⢸⣿⣽⣿
⠀⡟⡻⣿⡿⠻⣯⣿⢿⣿⣿⣿⣿⣿⠿⣿⣼⣿⣯⣥⣿⣿⣽⣿⡟⠛⣹⣿⣛⣻⣿⣿⣯⣭⣿⣷⣿⣿⣾⠿⣿⣀
⠀⣿⣻⣿⣧⣲⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⠿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⢿⣿⣿⣿⣻⡦⢂⣹⣟⣿⣿⣶⣾⣿
⠀⢿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣯⡀⣀⣿⣿⣿⣿⣩⡍⣿⠉⠉⢉⠉⢻⣿⣿⣻⣾⣿⣿⡿⢿⣼⡿⣟⣛
⠀⢻⣄⢀⡈⢳⣿⣿⣿⣿⣿⣿⢇⠀⢹⣿⣿⣿⣛⣿⡿⣯⠿⣿⠿⣶⡢⡄⣶⣿⣿⣿⣴⡿⠷⢻⡿⠙⠀⠀⠈⠁
⠀⠁⠄⠂⠉⠉⣿⠿⣿⣿⢿⡟⣷⢶⡦⡽⠿⡋⡩⠀⠁⠈⠀⠀⣰⡿⣾⡷⢿⡾⢭⣻⣟⣀⣴⡄⠸⡀
⠀⢸⠂⠀⠀⢀⡷⠄⠉⠀⠈⠋⠉⠉⠐⠣⠖⠀⢽⣤⣀⠾⡿⢟⡾⣵⡟⠐⠿⣻⣄⠈⢻⣹⡟⠃⠀⠯⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠁⠀⠀⢠⡜⢆⠰⡀⢰⡜⣠⡷⡡⣠⠆⡤⠀⣠⠄⡉⡙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⠘⢦⣄⣰⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣞⣥⣾⣪⣖⠕⠁⡙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠠⣄⠀⠒⣬⣷⣾⡿⣿⣿⠿⣽⣿⣿⡿⣿⣭⡻⣿⡟⠿⣿⣿⣿⣿⣷⣖⣋⠥⣀⣌⠵⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡀⢄⣈⣿⡿⡿⣿⡏⢀⡟⠁⠀⢸⣿⣿⣄⣸⣿⣿⡌⣯⠀⠸⣯⠙⠿⣿⣿⣿⣯⣟⡥⢐⡌⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣌⣶⠟⠋⠀⠀⣿⠀⢸⢱⣷⣶⣿⣾⣿⣿⣿⣿⣿⣿⢸⡄⠀⣿⠀⠀⠐⠉⢛⠿⣿⣿⣯⣒⣁⣠⠔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡶⠯⠑⠀⠀⠀⠀⢿⡄⠸⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⠀⢠⡿⠀⠀⠀⠀⠀⠀⠁⠙⣻⠿⠿⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⢈⠀⠀⠀⠀⠀⠀⠘⣷⡀⠹⣎⠻⣿⣿⣿⣿⣿⡿⢟⡴⠃⢀⣼⠃⠀⠀⠀⠀⠀⢀⠴⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⢦⣀⠀⠀⠀⠀⠀⠈⢿⣤⡈⠛⠦⢭⣭⣭⠥⠖⠋⠀⣠⡾⠁⠀⠀⠀⣀⣴⠚⠓⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⠀⠀⡉⠻⢶⣤⣤⣀⣀⣠⣤⣶⡟⠭⢐⡢⣥⠒⢏⠉⢈⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠹⠛⠱⠒⠲⠾⠛⢿⠟⠛⠿⠒⠙⠋⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣀⣀⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢴⡻⣏⢹⡿⣿⢿⠉⢉⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣶⢲⣿⣿⣦⣄⠙⠋⢀⣼⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣾⣿⣿⣿⠛⠂⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠈⠻⢿⣿⣿⣿⡿⣿⣳⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⠙⣿⣿⣦⣤⣴⡀⣩⣽⡇⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁⠡⣾⣻⣿⣿⣿⣿⣿⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⢨⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⢀⣘⡀⠴⠟⠁⠑⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡈⣸⣿⡶⢶⣿⣷⠠⣄⡀⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠐⠃⠐⠂⠀⠀⠀⠀⠀⠀⠁⢻⣿⣿⣶⣶⣿⣿⣿⣿⢷⣄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⢻⠛⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡛⢻⡟⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣀⡈⡀⠀⠀⠀⠀⣀⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣾⣿⣟⣉⠉⠀⠉⢙⡖⠁⠀⢩⣿⣾⣿⡄⠢⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠰⣾⣦⣀⠀⠀⠀⣀⠛⠛⠛⠿⠻⠷⠦⠀⠀⠼⠦⠿⠿⠿⢿⣿⣆⢦⡀⠀⠀⠀⢘⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠦⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⠀⢠⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠻⠂⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⠀⠐⠃⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⡻⠃⠀⠠⠄⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠛⠛⠉⠁⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠁⢀⠀⠐⠠⠶⠖⠐⠛⠂⠒⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢠⡘⠷⣄⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠐⠀⠀⡠⠀⠀⠄⡶⠄⠷⠒⠀⠶⠂⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⡑⠷⣦⡄⠀⠄⢀⢀⣀⣀⣀⣀⣀⣨⡆
⠀⠀⠀⠀⠡⠀⠀⠀⣀⢀⠀⢀⣀⡀⣷⠦⠤⢀⣀⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⡀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠙⡄
⠀⠀⠀⠀⣄⠐⠈⠀⢀⣀⣀⢼⣶⣷⣿⣶⣄⢀⡉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣤⣀⠀⠀⠀⡀⠀⠀⠀⠉⠉⠉⢓
⠀⠀⠀⠀⠀⠠⠖⠒⠋⠉⣡⣈⠀⢈⣀⣻⣯⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠿⣮⣝⣓⣀⣀⣈⣁⠀⠀⠀⠐⠀⠈⡀
⠀⠀⠀⠀⠀⠀⡠⢤⢖⡙⠀⠀⠀⠿⠤⠀⠸⣿⠷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣈⣍⣃⣀⣈⣉⣉⠀⠀⠀⠠⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠐⠐⠂⠀⠈⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣧⡙⠛⣿⣄⣉⠉⠁⠀⠒⣊⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠈⢈⣉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⣿⡀⠀⣿⣿⡇⣧⡀
⠀⠀⠀⠀⠀⠀⣼⡏⢸⡀⢹⣧⠀⣿⣿⣧⢸⡷⠄
⠀⠀⠀⠀⠀⠈⡏⠀⣼⣧⠈⣿⡄⣿⣿⣿⠈⢿⣷⣄⠀⣤⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⠹⣿⣿⣿⣿⣀⠠⣙⠻⠀⣿⡟⣻⣴⣦⣧
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⣿⡄⢹⣿⣿⣿⡇⠀⠀⠀⠀⣿⢼⡽⣜⣃⠙⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⢻⣿⣿⣻⣶⣶⣶⣿⣿⣾⡆⣿⣭⣾⣻⣽⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣼⣿⣿⣿⢿⠿⠿⠛⠋⠷⣿⢿⣿⣿⣿⣿⣿⠄⠐⢠⣀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠻⣿⣿⣿⠟⠋⠁⣀⠠⠄⠀⣢⣴⣿⣾⣿⣏⠙⢿⣮⡓⢄⡻⣭⣑⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢽⡀⠀⣽⣿⣁⣤⠠⠊⠁⢀⣶⣿⣿⣿⡿⠏⡿⠿⣀⣠⡌⡿⣦⡉⠺⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠿⠏⣿⣟⢻⡀⠀⠀⠀⠹⠳⣦⡀⠀⠀⠀⡏⢸⡇⠀⠘⢿⣦⡈⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠈⣿⡄⣇⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠓⠘⠃⠀⠀⠈⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⣀⣴⣿⣧⣽⠀⠀⠀⠀⠀⠁⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠿⠛⠛⠛⠛⠛⠛⠳⢶⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣄⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣶⣶⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣛⣛⣛⣛⡳⢶⣶⣶⣶⣷⣷⣄⣠⣤⢴⠄
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠟⢁⣴⠾⢻⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⣇⣠⡞⠁⠀⢀⣁⣽⣿⣿⣿⠆⢹⡿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠿⠇⠸⠿⠦⠥⡀⠀⠘⠿⣿⣿⡿⠋⠀⡿⢁⣽⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣤⣿⣧⣼⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠑⣦⡀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠈⠒⢠⡿⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠠⠆⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⣶⡄⠈⠉⠁⠀⠈⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡾⢋⣀⣀⣀⠀⠀⢲⣾⣧⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠔⠀⠀⠀⠀⠀⠀⠐⠠⣀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⣴⣾⣿⣿⣿⣿⣆⠀⠹⣿⣶⡀⢀⣤⣤⣀⡀⠀⠀⠴⠞⢣⣹⣿⣿⣿⣷⣶⣶⣤⣀⠀⠁⠀⢢⣘⡄
⠀⠀⠀⠀⡁⢈⡁⣀⠀⠀⡰⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣷⣄⠈⠛⠛⠛⠃⠌⠉⠙⢷⣷⡜⢣⢃⡾⠛⣿⣿⣿⡿⣿⠿⡿⠷⣤⡠⠌⡩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣼⣾⡛⢾⠙⣿⣿⣿⣿⣿⠟⣿⠏⠙⣿⣾⣿⣿⣿⢿⣷⡀⠸⣿⡤⢣⡟⠀⠀⣿⣏⡇⠀⠀⠀⠀⠀⠈⣷⠘
⠀⠀⠀⠀⠀⠀⢠⢀⣼⠏⠙⠛⡄⠀⠘⠉⠀⠁⠀⠀⠋⠀⠠⣾⢿⡿⣉⣁⢻⡿⠀⣰⣟⣵⣟⣿⠀⢼⣿⣟⠁⠀⠀⠀⠀⠀⡠⡌
⠀⠀⠀⠀⠀⣴⢃⣾⣿⣤⠄⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⡏⢠⣿⣭⣿⡿⠀⢠⡿⣿⡿⣹⠟⢿⣿⣿⣏⠀⠀⠀⠀⠀⣼⠷
⠀⠀⠀⠀⢰⡧⣾⣏⣼⣿⣷⣾⣶⣥⡀⠀⠀⠠⠀⡀⠀⠠⠞⠀⠂⢹⣰⠎⠀⢀⣼⣻⡿⣓⡙⠀⢠⣙⣻⠿⢷⣆⣀⣀⣀⣿⠘⢄
⠀⠀⠀⢀⣧⣼⠻⡟⠉⠛⠣⠀⠉⠙⣿⡆⠀⠀⠀⠁⠀⠀⠀⢀⡤⠈⠁⠀⣰⡿⠿⢡⣾⣿⡗⠀⠀⠀⠀⠃⠨⣍⠛⠉⢸⣿⠀⠀⠑⡄
⠀⠀⠀⢸⠿⡿⠼⣿⣷⣤⣬⣂⣄⣼⣿⠟⣤⠠⣄⠀⠀⡠⠄⠈⠀⠀⢀⣴⢟⡤⢀⣾⡏⠛⢿⣦⡀⠂⠀⡀⠀⠀⠀⣰⡗⣿⢸⠀⠠⣈⠲⣄
⠀⠀⠀⡇⣠⠀⠀⠀⠙⠛⠛⠿⠟⠛⣡⣾⣏⣦⣈⠳⠈⠀⢀⣤⣴⠶⢋⡴⢋⡴⢋⠉⠀⠀⠀⠉⠻⠄⠀⣤⣤⣤⠾⡿⠇⠿⣠⠀⠀⠀⠀⠈⠣
⠀⠀⠀⣷⠹⠆⠀⠀⠀⠀⠀⢀⠀⣾⠟⢿⣿⡿⠋⠄⠀⣰⣿⠋⢀⣴⠟⣀⣼⣷⣶⡆⠀⠀⠢⡀⠀⠀⠀⠀⠘⠀⠀⠀⢠⠀⣿⡆⠀⠀⠈⠀⠀⠇
⠀⠀⠀⣿⣴⣿⣷⣶⣦⣴⣶⡟⢴⣿⣦⡴⠂⠤⠀⠀⣠⣿⣣⠾⠋⣹⣿⢿⣿⡿⣿⡷⣄⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣿⣿⣷⡄⠀⠀⠂⢸
⠀⠀⠀⣿⢿⣿⣿⣿⣿⡿⣳⣾⣷⠟⠉⠀⢀⡤⢄⣼⠟⠋⢉⣶⠶⠞⠁⠀⠉⠀⣨⣓⡝⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⢹⡄⠸⣿⠿⠟⣿⣄⣀⡼
⠀⠀⠀⢻⡄⠉⠛⠿⠿⠾⢛⡉⠀⠀⣀⡖⣋⠴⢎⣤⣀⡀⠊⠀⠀⠀⠁⠀⠀⠀⣼⣿⠇⢢⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠇⠀⢿⣿⣷⡿⢿⣿⠁
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠁⢀⣴⡶⠞⢫⠞⠁⣤⠜⠁⠀⠀⠐⠀⠀⢀⠀⠀⠀⠰⣿⣇⠀⠘⢫⢳⡄⢀⠀⠀⠈⠀⢀⡀⠘⣧⠀⠘⣿⣿⡤⠈⢻⣇
⠀⠀⠀⠀⠀⢻⣦⣤⣤⣴⣾⡏⠀⢠⡇⢠⡷⢴⡀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⢸⣿⣿⣿⢆⠀⠀⢠⠀⠀⠀⠀⠀⠈⠁⠀⠈⡄⠀⣿⠋⡷⣄⠘⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⢧⠀⢿⡄⣿⣷⣟⡟⡆⢠⡀⠐⣆⣠⢤⡄⡌⢻⡏⣿⣾⣿⠹⢿⣷⡄⠀⠀⠆⠀⠀⢀⠀⢀⣤⡇⠀⣿⠀⣴⣀⠀⢻⡅
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⠀⠘⣧⣈⣿⣿⡇⣸⢸⣇⣢⣭⣽⣦⡙⠞⣄⠙⣿⠃⠹⣦⣈⣻⣷⡄⠀⠀⣱⡦⣌⠀⠠⢟⡅⠀⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠛⠢⠀⠙⢧⠙⢯⠸⣿⡌⢻⡟⢿⣯⡻⣿⡄⠘⠀⠘⣧⡄⠹⣈⠛⢿⡇⠀⠀⠉⠘⠿⣋⠀⢈⠃⠀⠻⢀⢠⡛⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⢼⠀⢀⠀⠈⢷⡐⢶⢻⣇⠸⣿⢳⣟⣷⣞⣿⣇⠤⠀⠉⢻⣤⣇⣀⡾⠁⠀⠀⠀⠀⠀⢀⢀⡏⠀⢠⣶⠃⢸⡇⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣿⡿⣯⠁⠙⣦⢀⡀⢁⠀⠈⣧⠈⠉⢿⡀⠻⣦⣌⣭⠁⠈⣿⡄⠀⣀⡤⢬⣽⠏⠁⢀⣼⣿⣶⣄⠀⠀⠸⠃⠀⣾⡁⠀⢸⡇⠀⠰⡇
⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠈⢧⡁⠄⠀⠀⠈⠳⡀⠠⠿⣶⣻⣿⣿⣷⣾⣿⣿⠋⠁⠀⠀⢹⢷⡄⣸⣿⣉⣾⣿⠧⠀⣾⠀⠀⡟⠃⠀⢸⠅⠀⣺⠇
⠀⠀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠒⠀⠻⣦⡠⠀⠀⠀⠈⠁⠒⠲⠲⠤⢬⣤⡭⠶⠒⠲⡄⠀⠀⠈⢈⣿⡿⡯⢵⣾⣳⣄⣦⠉⠀⢸⠁⠀⠀⣸⠆⠰⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣇⣈⢁⣀⣀⣀⡀⠈⠻⣦⣙⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠺⢿⣧⣾⣿⣿⣿⣿⡏⠀⢀⡟⠀⠂⣰⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⢿⡀⠉⠉⠉⠀⠈⠛⠶⣤⣀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠹⣿⣿⣿⡿⠛⠀⠀⡸⠁⠀⢀⣧⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣴⣒⠛⠟⠉⠓⠀⣀⠼⠁⠐⠀⢸⢹⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠰⢄⠀⠀⠀⠄⠈⢉⠀⢀⠀⠠⡄⠘⠗⠀⠀⡸⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⠀⠀⠀⠁⠀⠤⢀⣀⠀⢀⠀⣀⠀⠰⢀⡀⠟⠁⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠑⠈⡙⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣶⣦⣀⡀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠉⢀⠈⠈⠁⠈⡉⠈⡉⠉⠁⠈⠉⠀⠙⠛⠋⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⣿⣿⣤⣤⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠉⠉⠉⠉⠀⠉⠉⠈⠉⠈⠁⠉⠉⠀⠉⠉⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⣄⡀⢠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣬⡛⣶⣂⠈⣵⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⡻⣿⣿⣾⣥⣾⣿⣿⣿⣿⡟⡟⣿⣿⣿⡇⣠⠂⠀⠀⠀⠀⠀⠸⠋⣽⣶⡄
⠀⠀⠀⠀⠀⠀⠀⡀⣠⣴⣶⣿⣿⣧⡘⠀⠙⡿⣿⣿⣿⣿⣿⠯⠡⢾⣷⠸⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⢮⣻⣿⣶⣦⣄
⠀⠀⠀⠀⠀⢠⠂⣼⣿⣿⣿⣿⣿⣿⣿⣦⡄⢹⣿⣽⣿⣿⣏⣿⡄⠸⡴⢌⣽⣽⣿⡇⡀⠀⠀⠀⠄⠀⠀⠀⠈⢿⣿⣿⣿⢸⡄
⠀⠀⠀⠀⠀⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⣿⣻⣿⡿⣶⣷⣸⣽⣾⣿⡟⠈⢻⡍⠀⠀⠀⠀⠀⠀⠀⡄⣡⠻⢯⣿⠀⣳
⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠛⢷⡿⠿⠿⠿⠿⠋⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠂⣼⠟⢰⡏
⠀⠀⠀⠀⠈⢿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣰⣦⣦⡈⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣼⣿⡿⣋⣿⣯⢇⡾⠁
⠀⠀⠀⠀⠀⠀⠓⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣌⡿⠋⠀⠈⠉⠉⠛⠛⠻⢿⣷⣶⣀⣌⣤⣭⣿⠿⠿⣻⣫⣾⣿⠏⠄⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⡇⠀⠀⠉⡠⢀⠠⠖⢢⣼⣿⣿⣿⣿⠟⠛⠛⠛⠿⠋⠉⠋⢿⣏⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠀⠀⠀⠙⠟⠋⠆⠀⠀⢹⣷⣀⠀⢏⠈⣄⣨⣴⣿⠟⠋⠀⣠⣞⣆⠀⠀⠀⠀⠀⠀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢹⣦⡀⠀⠀⠘⠄⠀⠈⠻⣿⣷⣾⣶⣿⠿⠋⠁⢀⡰⠚⠙⠿⠛⠀⠀⠀⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⢤⠀⡄⢀⣿⣿⣷⡀⠀⠀⠀⢀⡀⠀⠈⠛⠛⠋⠁⠐⣂⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢹⠉⢋⡀⣀⠀⠀⠀⠈⠐⠀⠄⠠⠶⠶⠖⠈⠀⠀⠀⠀⠀⠀⠀⢠⣠⣾⡏⢏⡷
⠀⠀⠀⠀⠈⠉⠀⠰⠀⠀⠀⣿⠀⠀⠉⢻⣦⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣄⡴⠀⣿⡿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⣏⠀⠀⢰⣿⠟⠛⠓⠻⣤⣀⣤⣄⡀⣠⣶⣢⡾⠟⠳⣿⣞⡿⠁⠸⠙⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠼⠂⠀⢈⣇⣀⢈⣶⣤⣿⣭⠿⣿⣿⣿⠿⡛⢂⡠⠿⢷⣿⡇⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠈⠙⢻⣯⣿⣉⣻⠞⠁⠉⢿⠋⢠⣸⠟⢷⣶⣺⢹⡆⠀⠀⣆⡀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⡻⠉⠁⠠⠠⢌⣤⡸⢁⢀⣾⠻⠻⣿⠇⠀⢰⡃⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⢠⡄⢾⣿⣷⣷⡞⠓⣴⣣⢼⣿⡁⠀⢠⡉⠓⣾⢿⠂⠀⠎⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣠⣆⣤⣾⡻⢿⣿⡦⣴⠟⠁⣠⣠⣿⡶⠺⡇⠀⠄⠀⠀⠈⠀⠀⠀⠀⣾⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⠟⠉⠉⡉⢍⣇⠹⣷⡎⠀⢀⣿⠿⠛⠻⢿⡷⠦⠀⣴⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢉⣄⣴⣿⡷⠿⣧⣼⣿⠁⣠⣾⡅⢠⣾⣿⣿⣿⢣⣾⡿⣃⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠿⢻⣿⣿⠟⢻⡿⠟⢽⣆⡀⠉⢷⡀⣿⣿⣿⡿⠛⡛⡕⠁⠀⠈⠀⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⣈⣋⣀⣀⠁⢀⣴⣿⠿⠟⠀⣹⣧⣿⡿⠉⣀⠒⢩⠷⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢋⣤⣾⣿⡁⢲⣿⣿⡿⠃⠀⣀⠮⠱⡐⣶⣾⣷⡀⡀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠙⠛⠛⠉⠀⠀⠈⠀⣠⣾⡿⠛⠉⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢐⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⣀⠘⠈⠁⠀⡀⠀⠀⢸⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡠⠤⠴⠧⠄⣀⡎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠞⡡⠔⠀⠘⣟⠀⢢⣍⠺⡁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣠⠔⢀⣁⢀⣉⣛⡑⠠⢀⠀⠐⡑⡼⠁⡷⢌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠒⠲⠀⠀⠐⠡⡊⠀⠀⠀⡄⡁⠉⠑⠢⣄⠀⢬⡃⢠⡧⠜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠒⠀⠀⠃⠀⠀⠀⠀⠀⢡⠡⠀⠀⠀⠀⠙⣦⡑⢏⠈⠈⠀⠀⠖⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢃⢣⡀⠄⠂⡜⠁⡟⢢⠑⡀⢀⡙⢢⠔⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣓⣋⣀⣐⡂⠤⣄⡀⠑⡑⢄⠜⢀⡜⠀⠀⢱⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠊⠉⠈⢆⠀⠉⠁⠂⢍⡓⠾⡛⢤⡾⣀⠀⠀⠀⡇⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠈⢌⣦⣠⡴⠒⠚⠳⢶⡈⠺⡏⠢⠍⣁⡉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⡄⠀⢀⡼⠓⠢⢍⡓⢤⡀⠀⢹⡑⢌⠢⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣘⡾⠖⠋⠀⠀⠀⠀⠈⠲⣝⢦⣸⠇⠀⠣⡘⡄⢀⡆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡋⠁⣀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢦⢹⡀⠀⠀⠐⠘⡄⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠙⢾⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⠀⡀⠐⡡⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠙⢦⡀⠀⣠⠞⠁⠀⠀⡰⠃⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠙⠺⣯⢋⠄⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣳⠴⠋⠀⠐⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢀⣾⣿⡟⠻⣿⣿⡄⠀⠀⠀⠀⢀⣀⣀⣸⣉⣉⣹⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⠛⢻⣿⣶⡀
⠀⠀⠀⢸⡿⠟⠛⠚⠛⢿⡇⠀⠀⠀⢸⣿⣿⣿⡿⠛⡛⠻⣿⣿⣿⣿⠀⠀⠀⠀⣀⣀⣸⣿⣇⣀⣀⠀⠀⠀⢰⣿⠿⠿⠇⠀⠸⠿⠿⣿⡄
⠀⠀⠀⠈⠀⠀⣀⣀⡀⠀⠁⠀⠀⠀⢸⣿⣿⣿⢠⣴⣿⣤⢸⣿⣿⣿⠀⠀⠀⠀⠿⠿⢿⣿⡿⠿⠿⠀⠀⠀⢸⣿⣀⣀⡀⠀⢀⣀⣀⣿⡇
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⢛⣠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠈⠻⣿⣿⠿⠃⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⡿⠿⠟⠋


⠀⢰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣤⣾⣿⣷⣆
⠀⢸⡇⢰⣦⠀⢀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⡿
⠀⢸⡇⠸⡿⠀⠸⡯⠀⠸⡧⠀⣿⠀⠀⠀⠀⠀⠀⣹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣿⣿⣷⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠁
⠀⠸⠗⠚⠛⠒⠚⠛⠒⠚⠓⠚⠿⠀⠀⠀⠀⠀⠀⢾⣻⠇⠀⠀⠀⠀⠀⠀⠀⠠⣟⣿⣿⣤⣿⣿⣷⠀⠀⠀⠀⠀⠀⠉⠛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⢠⣾⣿⣿⣷⣶⢰⣶⣿⣿⣿⣧⠀⠀⠀⠀⣴⣶⡟⢛⠒⣦⠀⠀⠀⠀⠀⠀⠎⠉⣠⡴⣭⡑
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠃⠀⠀⠀⠻⠿⣿⣿⡏⠀⣿⣿⡿⠿⠋⠀⠀⠀⠀⣿⣿⣌⠻⠋⣼⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡼⠃
⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢅⡀⠀⠀⢀⡠
⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠠⠤⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⠟⢷⡜⠀⠀⠀⠀⠀⠀⣿⣿⠋⣴⡌⢻
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⢱⠀⠀⠀⠀⠀⠀⣿⣿⣧⣍⣥⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⠻⠿⢿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⣀⣀⡤⠚⠀⠀⠀⠀⠀⠀⠙⢻⡿⢿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⣠⣴⣦⣤⣤⣤⣤⣾⣻⣶⣤⣤⣤⣤⣴⣦⡄
⠀⠀⠀⠘⣿⡀⠀⣀⣀⣉⣩⣭⣭⣩⣉⣉⣀⡀⠀⢀⣿
⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⢹⣿⠉⣿⠉⠉⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠾⠿⠿⠾⠿⠿⠷⢶⣤⡀
⠀⠀⣠⣤⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⣀
⠀⢸⣿⠉⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠛⠷⣦⣀
⠀⢸⣿⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠈⠛⠛⠛⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠷⢶⣄⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠶⠶⠶⠾⠟⠋⠀⠀⠀⠀⠙⣷⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⠿⠿⠿⠿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⢶⣶⣶⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠹⠘⠃⠛⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢙⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⢛⠚⡛⢛⢛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠹⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠛⠙⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠻⣧⠀⠀⠀⢀⣠⣤⣄⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⠟⠀⢀⣰⠟⠋⠈⢻⣷⠟⠋⠻⣧⠀⠀⠀⣴⠟⠉⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⣄⠀⠀⠀⠀⢀⣴⢟⣡⣤⣤⣴⣿⡁⠀⢀⣴⣯⡀⠀⢸⣏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣿⡄⠀⣠⡾⠛⠻⠟⣉⣀⡀⠀⠉⠻⠿⠛⠿⣯⣿⡆⠀⠻⣦⣄⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⢠⣴⢿⣤⢶⣤⣿⠋⠉⠻⣦⣀⣴⠶⢶⣄⠹⠿⢷⣤⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣸⣿⣿⠁⣾⡇⠀⠈⠻⣦⡀⠀⠈⠻⣧⡀⠀⠙⢷⣄⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢿⣅⣿⠆⢈⣿⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢷⣄⠉⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠈⣿⠁⢰⣿⠁⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢷⣤⣿
⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⢀⣿⡀⠀⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠺⠋⠻⢶⣤⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠁⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⡈⠻⣦⡀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⣋⣿⣦⣌⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⠟⠋⣾⠋⠉⠉⠉⠛⠻⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠙⠻⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠈⠻⣦⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠟⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⣈⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⣿⠀⢶⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠠⣴⣿⣾⡂⠀⠀⠁⠀⠀⠀⠀⢸⡇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠏⢀⣬⡄⠹⢿⡟⠁⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢶⣄⣹⡏⠀⠈⠛⠛⢀⣤⡄⢇⣀⠀⠀⠀⠀⢴⣿⣿⣿⣦⠀⠀⠀⠀⡄⠈⠀⠀⠀⠈⠀⡿⢃⣤⡀⣀⠀⠀⠀⡴⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠚⣿⣭⣻⡷⠖⣰⢃⣼⣶⣷⡄⠀⢀⠀⢒⠂⢻⣿⣿⣿⡿⠃⠀⠀⠈⠀⢀⣄⠈⠶⠄⠈⠁⣘⠛⠙⠿⠁⠞⣿⠁⠀⢠⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠸⠈⣿⠿⠏⠁⢀⡉⢸⣿⣿⣿⡿⢀⡃⠀⣀⠀⠀⠙⠙⢿⡠⡆⠀⠐⠛⠐⣤⡈⠑⠀⢀⡢⢠⡟⢰⣇⣠⠄⠀⢤⣀⡙⢾⠀⠀⢙⡃
⠀⠀⠀⠀⣠⠟⢠⣶⣷⠀⠐⣶⡿⣧⡤⠉⠛⠉⣠⠘⢧⣀⠁⠰⠐⠆⠀⠸⠗⢀⣠⡄⠀⠀⠀⠉⠀⣀⠀⠀⠛⠠⢾⣯⣿⣾⣿⣷⡄⠀⣠⣦⡀⠀⣛⠄
⠀⠀⠀⠀⠿⠀⣿⣿⢏⡀⢿⣿⣿⣿⡄⠂⠀⢾⠃⠀⢀⠙⠀⣠⣤⡀⢀⣀⠙⠉⠉⠀⠀⠀⠀⠀⠀⠙⠿⠲⠄⣰⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⠏⠀⠀⠃
⠀⠀⠀⠀⠀⢸⣿⠏⠸⡇⢸⣿⣿⠋⠀⢄⠀⠀⠀⣀⠸⣇⠀⢨⣽⣷⣿⣿⣷⡀⠀⡙⠁⠀⠀⠀⠄⢠⢀⠀⢀⡘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡆⠄
⠀⠀⠀⢀⡀⠘⣃⠀⣤⣆⠀⠈⡀⠀⣀⢚⡄⠧⠻⠌⠀⠉⠀⠸⣿⣿⣿⣿⣿⠇⠀⠁⠀⠀⣄⠀⠀⠀⠀⠀⠘⠁⢈⠻⣿⣿⣿⣿⠿⠃⢰⠀⠈⠀⠀⠹
⠀⠀⠀⠈⣇⠈⣿⣷⣈⢻⣿⣠⣤⡀⠈⣉⣁⡀⢀⠘⠓⠚⠛⠀⠙⠻⠛⠛⠁⢰⣤⣴⡆⠀⣬⡃⠀⠀⠀⠀⣰⣿⣿⠄⠈⠛⠋⢉⡴⠖⠘⢄⠀⠄⠀⠈⣤
⠀⠀⠀⠀⠙⠁⠛⠛⠛⠷⣿⣏⡉⢀⡾⢹⡉⢻⡝⠷⠀⢀⢈⠁⢀⣠⣬⠑⠒⢈⡀⠈⣠⣄⠈⠛⠶⠀⢀⡀⠈⠙⠏⠀⣰⣷⠀⠙⢁⣤⣤⣠⣤⡄⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⣷⠀⣾⣿⣿⡿⠘⢧⣈⢁⣼⣯⠑⣀⣤⣾⣄⠘⣿⣿⠃⠀⠘⠿⢾⣿⠏⠀⠞⠀⢠⠞⣿⣄⠀⣠⣶⣿⣿⠀⠀⢻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⢻⠃⠀⢀⣉⣉⠀⠉⢸⡏⢸⡄⢻⡄⣈⠁⣠⣾⣿⣿⣿⣷⡆⢸⠀⠣⡈⠛⠛⠁⠀⠻⣿⣿⠟⠀⠀⠄⠙⢿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⣄⠀⠘⣿⣿⡟⠀⠀⠈⠳⢦⡤⠟⢠⡟⣼⣿⣿⣿⣿⣿⣿⠃⠰⡄⠀⠉⢠⣼⣿⡆⠀⠀⣠⠀⠞⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣬⠓⣄⡀⠀⠁⠘⠀⠀⠐⠀⢠⠀⠋⠀⠈⠻⣿⣿⡿⠿⠋⠀⣠⣴⣾⣆⠙⠿⠟⠁⠚⠋⢩⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢯⣽⡶⠖⠀⠀⠀⡀⣼⡼⠆⠀⠀⠀⠀⣠⣶⣾⡂⠈⢳⣿⣿⠏⠰⣾⣿⣿⠀⣾⣿⡄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢶⡌⢳⠀⠀⢰⣶⣶⡆⠀⢉⠛⠋⠃⠀⠀⢈⡀⠠⣄⠈⠋⠋⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⡄⠀⠀⠀⠀⠻⠏⠀⠀⠁⠈⢰⢄⠴⠀⢸⠀⠀⠀⢰⠇⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⢠⣶⣥⣤⡀⠀⠀⢌⠡⠇⠀⠀⠀⠀⠈⠳⠄⠀⢠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢿⣿⣿⣶⣤⡀⠀⠀⣴⣿⣿⣿⣷⡆⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⣤⣤⡈⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠷



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

⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡆⣾⣿⡗⣰⣿⡆⣠⣄
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣥⣿⣿⣧⣿⣿⣱⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠉⠉⠀⠀⠙⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢋⣴⣾⣿⣿⣿⣿⣷⣶⣄⣈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡟⠠⣿⣿⣿⠛⠛⢛⣿⣿⣿⣿⣿⣿⣃⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠁⢸⣿⣿⣧⣴⣾⡿⠛⠉⠉⠁⠀⠀⠀⠀⠀⣀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡀⢸⡿⣻⣿⣿⠏⠀⠀⣠⣤⣶⣶⣶⠾⠿⠿⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢞⣵⣿⡿⠃⠀⢀⣾⣿⣿⣿⣿⣿⣤⣤⣶⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⢵⣛⣭⣭⣅⣀⠀⢾⣿⣿⣿⣿⣷⣿⣯⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢟⣵⣳⣿⣿⣿⣿⣟⠻⠿⣦⡹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡁⢤⣤⣾⣯⣷⣽⣿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢗⠶⠟⠃⣯⢹⣿⣿⠛⠡⣤⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣹⣛⢷⣼⠟⣾⢏⣼⣿⣿⠳⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠷⠮⣽⣷⣿⢃⣾⣽⣷⡾⢳⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣽⣿⡿⣋⡼⣾⣿⣿⢿⡵⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣵⣿⣿⣿⡉⣿⣿⡿⣿⡿⣿⢿⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣟⣿⣿⣷⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣻⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠹⢜⡈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠈⠁⢒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⡜⢩⣦⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠰⢧⣤⣿⣇⠀⡎⢀⠜⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⠀⠀⢠⠃⠈⢿⣿⣿⠀⠳⠧⠤⠊⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡎⠀⡠⠤⣉⠀⠀⠀⠀⢀⣴⠋⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡸⠀⢘⢀⠔⠉⡇⠀⠀⣼⣿⣿⣷⠯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣰⠃⠀⠈⠓⠒⠊⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠈⠑⠒⠁⠀⣸⠀⠈⠑⠒⠉⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⠉⠁⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⠒⠒⠒⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣤⣤⣤⣶⣾⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣷⣶⣤⣤⣤⡤⠆⠘⢆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⠀⠀⠄
⠀⠀⠀⠙⣾⣷⣿⣽⣯⣤⣶⠶⠋⠐⠀⠀⠀⠀⠀⠀⢨⣿⣿⡅⢀⣀⣠⣤⣤⣄⠀⠀⠀⠀⣼⢟⣽⣿⣿⣭⣥⡴⠆
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⡶⠦⡄⠢⠀⠀⠀⢸⣿⣿⡿⠛⠋⠉⠉⠹⠿⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⡿⢯⣝⡓⠆⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣟⠲⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠙⠳⠈⠉⡠⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠌⠙⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⣶⠶⢶⣤⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠒⣀⣀⣀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⣹⣷⡀⠀⠀⠀⠀⠀⠀⡀⣠⠊⠁⠀⠀⠀⠀⠀⡈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣾⡇⡀⠶⠿⠿⠿⠿⠷⠄⠀⠀⠀⠴⠁⡇⠀⠁⠀⠀⠀⠀⠀⢰⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⣤⢠⡄⣤⢠⡄⡌⠀⠀⠀⠄⠀⢣⡀⠂⠀⢀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣿⠇⠀⠉⠈⠁⠿⠈⠁⠀⠀⠀⠀⠀⠀⠀⢙⠒⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⢀⣴⣤⣤⣤⣴⠶⠖⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣂⣀⣀⡀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣤⣤⣤⣤⣤⠆
⠀⠀⠀⠾⣳⣿⣷⣶⣶⣿⣛⠁⠀⠀⠄⠀⠈⠀⠀⠀⠀⢠⣿⠟⠛⠛⢿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣟⣧⣤⣄⣀
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣟⡓⠶⠄⢀⠀⠀⠀⠀⠀⢸⡿⢠⣴⣤⠘⣿⡇⠀⠀⠀⠀⠀⠀⠙⣻⡻⣿⣿⣿⡿⢯⣍⡛⠂
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡻⢮⣍⡓⠂⠀⠀⠀⠀⠀⠀⣼⡇⠀⠘⠁⠀⣿⡇⠀⠀⠀⠀⣤⣶⠟⠋⠀⠘⠻⢯⡙⠷⣌⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⠁⠀⠁⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠌⠉⠉⠉⠪⢢⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠌⠀⠀⠀⠀⠀⠡⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠸⢁⠀⡠⡄⠀⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠂⣠⠀⢤⠲⡏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠂⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢶⠸⠖⠒⠒⠒⠒⠒⠒⠶⠞⠀⠀⠀⠈⣥⡛⡟⡛
⠀⠀⠀⠀⠀⠀⠰⡟⢢⠤⢄⠀⠛⢤⡀⣀⣀⣀⣀⡠⠂⠀⢸⠁⠀⠀⠈⡇⡗⠲⠶⢶⢠⣤⣤⣄⣀⣀⣤⣶⠘⠓⠒⠒⠒⣤⠾⠿⣏⡛
⠀⠀⠀⠀⠀⠀⠀⠘⢇⣀⠜⠁⠂⣤⡀⢁⡅⠀⢠⡀⠀⢀⠸⠶⠶⠶⠶⠇⡇⠀⠀⣾⡌⣿⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⢠⠊⠀⠀⠀⠐⣴⠁⠀⢀⣀⣀⡀⠀⠘⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⣈⠉⠈⣅
⠀⠀⠀⢠⠀⠀⠚⠁⠀⠀⠈⠉⠈⠙⠉⠂⠐⠂⠂⠀⠓⠐⠓⠀⠀⠀⠀⠘⠃⠛⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⡀⢸⡇⠀⠀⢹⠄
⠀⠀⠀⠀⢹⠟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⢻⡟⠛⡟⠛⣟⠛⣿⠛⣻⠛⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡄⢀⡖⠓⠒⠒⠚⢲
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⢸⠀⡇⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⢁⠀⡀⠀⠀⠈⢀⠀⣲
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⡇⠀⢸⠀⡇⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⡀⢸⠉⢰⡖⠘⣿⠒⣶⠂⡒⡎⢳⠀⢸⡇⠀⣧⠀⣸⣠⣧⣤⣷⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠄⠀⠠⠦⠀⠀⠠⠄⠀⠶⠇
⠀⠀⠀⠀⠀⡃⠨⠄⠸⡇⠀⠷⠀⢿⠀⠅⠇⠸⠀⠸⢧⡤⠿⠠⠼⡿⠓⠂⠛⢻⡿⠶⠤⣤⠠⠤⠤⠤⠤⠠⠄⠐⠒
⠀⠀⠀⠀⠀⠁⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡃⠀⠀⠀⠀⠀⠀⠀⣺⢴⣶⣶⠿⢶⣿⣷⣶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠐⠦⠾⠤⠤⠴⠴⣏⠀⠀⠉⠉⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠴⠤⠶⠖⠚⠋⠉

⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⡏⢻⡆⢠⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠸⣿⠀⢠⡀⠀⢿⡄⠹⣾⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣾⠋⠒⠀⠈⠀⠘⣄⣼⡇⠀⠸⣟⠢⠈⠻⠛⣿⡇⠀⠈⠷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠹⠛⠛⠖⠀⠀⠀⠀⠀⠀⠉⠀⣀⣠⣤⣤⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⣿⣿⣿⡿⠋⠈⠁⠈⢷⣄⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⢽⣿⡿⠟⠁⡄⠀⢠⡆⢠⠊⣩⣷⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢀⣤⡶⠀⡤⠀⣾⣿⠈⡆⠈⠀⢷⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠤⡊⢹⠟⣠⠃⠀⣺⣿⣿⣤⣷⠀⢦⢸⡄⢹⡄
⠀⠀⠀⠀⢀⡀⠀⠀⢚⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⢠⣾⠟⠀⡷⣹⣿⣿⣿⣿⡄⢸⣸⡇⢸⡇
⠀⠀⠀⠀⠈⠁⣴⠋⠛⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⡟⠳⠈⠀⠉⠜⠻⣦⠀⢤⣿⣿⣿⠿⣿⡇⠈⠁⠟⢸⡇
⠀⠀⠀⠀⢐⣦⠀⠀⠀⠀⣿⣯⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣄⠀⢠⠄⣼⣿⢀⣾⡿⣽⣿⠀⣿⡇⠀⠀⠈⠁
⠀⠀⠀⠀⠹⣷⠀⠀⠀⢠⣿⣿⣿⣿⣷⡄⢀⠀⢢⣀⡀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠈⠳⡙⠀⠀⢀⢉⣰⣾⣿⡟⣿⡇⠀⠛
⠀⠀⠀⠀⠀⢿⠀⠀⠀⣼⣿⣿⡿⠛⠃⠘⢋⡷⠀⠀⠀⠀⠀⠀⠀⣴⠖⠀⢠⠀⠀⠀⠀⠀⠀⠈⠉⠟⢻⣾⣼⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣿⡦⠼⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠋⢰⠁⠀⠈⠀⠀⡀⠀⠀⠐⢲⠚⡿⠗⢫⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⣼⣿⣿⢿⡿⠿⠿⣠⣀⣀⣴⣾⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⡧⠀⠈⢳⣄⠘⠇⠛⠙⠛⣡⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⠻⠁⠄⠸⡷⠶⠟⣳⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣳⣶⢞⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢀⠀⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡟⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡉⢁⢀⣀⣠⡄⠀⢴⣼⣿⠟⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⢒⡾⣿⣿⠆⣴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣬⣽⣿⣿⣶⣯⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠉⠀⠀⠈⢢⡀⠀⠀⠀⢀⠀⠀⠀⠀⣧⠀⠀⢀⣰⣷⣿⣿⣿⣧⣼⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠰⡄⠀⠀⠹⣷⣄⢳⣄⡀⠀⠀⠈⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⢿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢻⡀⠀⢀⣚⣿⣷⡟⢷⣿⡆⢾⣿⣿⠞⣿⣿⣻⣿⣿⣏⣄⠀⠀⠠⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣇⠀⠀⢻⣶⣿⠉⣿⠟⢷⠈⣿⡇⢸⣿⡏⣸⣿⣿⡟⠻⢿⣿⣿⠀⣤⡀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⣿⡯⠀⢹⠀⣿⣿⣿⡗⠞⠉⢻⣿⣿⠿⢷⣦⡐⠻⣿⠄⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠁⠀⢸⠀⢻⣿⡟⠀⠀⠰⣼⠟⠁⠀⠀⠈⠿⣿⣷⠀⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠿⠁⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠈⠛⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⡷⠒⢱⣶⣶⡤⡀⠐⠲⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⠁⢀⣴⣾⣿⠿⣿⣿⣦⠘⠺⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⡿⣿⣿⡟⠀⠁⣾⠿⠟⡿⠂⣸⣿⠀⠀⢀⣈⡛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠐⠃⣿⣿⡁⠀⢠⣏⠀⣚⡟⠋⠏⠟⠀⠀⣼⡟⠀⣠⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣿⡟⠀⠀⢻⣿⣇⠀⠸⣿⣶⣿⠁⠀⠀⠀⠀⠀⢿⡄⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⠹⣿⣯⡜⣴⠀⢻⣿⣄⡀⢿⡿⠁⠀⠀⠀⠀⠀⠒⢺⡃⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡀⠻⠉⠁⠉⠃⠘⣿⣿⣿⣧⣤⣸⣶⣶⠀⠀⠀⠡⠌⠁⢨⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⡀⢀⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⢀⣠⣴⡿⠿⠿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢄⠀⠀⠀⣼⣿⣿⣿⣿⣧⡀⠀⣀⣤⣴⣶⠾⡟⠋⠘⠛⠁⣰⣟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⣿⣿⣿⣿⣿⣿⣇⠉⠁⣴⣿⣶⣶⣾⢷⡶⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣼⣿⣿⣿⣿⡿⢫⣾⣿⣿⣿⡿⣿⣷⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠁⣾⣿⣿⣿⣿⣧⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⢹⣿⣿⣿⣿⡏⠀⣼⣿⣿⣿⣿⣿⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠁⢸⣿⣿⣿⡏⢢⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⣾⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⣿⣿⣏⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⣿⣿⣿⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣰⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢻⡿⡀⠉⢫⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢸⣷⢷⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣸⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠿⠿⠿⠿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⢀⣿⡀⠃⠀⠀⠀⢠⣄⣸⡇
⠀⠀⠀⠀⠀⡐⠁⠆⠀⠀⠀⢹⡆⠀⠀⠀⠁⠀⢸⡿⠈⠁⠀⢀⣴⠛⣁⣻⣽⣷⣶⡄⢠
⠀⠀⠀⠈⠀⢻⠿⠿⠶⢦⣄⣀⠀⠀⠐⠀⠀⠀⢸⡆⠀⠀⣰⡟⣡⠞⠋⠉⠉⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣢⡀⠀⠀⠀⣿⠇⠀⣸⠏⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣽⡇⠀⠀⣿⡀⢰⣿⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⠃⢸⡟⣤⠀⣠⡿⠓⠊⠀⠀⠀⠀⠈⠀⣦⣀⠀⠀⠀⠀⠀⢀⣴⡶⢂
⠀⠀⠀⠀⠀⠀⢠⠀⢠⣿⣿⠀⠀⣾⣿⣿⣿⠟⠉⠀⠀⣀⣀⣀⣤⣴⣦⢧⣿⣷⣄⣤⣴⣶⣿⡿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⢇⣃⠀⠀⣹⣾⣿⡟⠀⠀⢀⣴⣿⣛⣛⣛⣿⣫⡿⠿⠛⠉⠉⠁⠀⠀⢀⣸⣿⣠⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⣆⣾⣿⣿⡋⠁⠀⢀⣿⣿⣿⠛⠛⠉⠉⠁⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⡿⣷⣦⠀⡀
⠀⠀⠀⢠⠀⢠⣶⣿⣿⣿⣿⣟⣿⣿⣿⡖⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢀⣴⠿⠟⠛⠻⠽⣿⣿⣭⣅⣙⡿⠿
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠟⢹⣿⣷⣾⠄⠀⢘⣿⣿⣿⡿⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⡏⡿⠛⠿⢿⣿⠆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣏⣿⣿⣿⠦⠀⢈⣿⡟⣿⢀⠀⠀⠀⢠⡏⠀⠀⢀⣀⡀⠀⢹⣿
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣿⣿⣿⣾⠟⠋⢰⣥⠦⠤⠀⢀⣾⠀⣠⣴⡿⠉⠛⢦⡘⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⠟⠛⠛⢿⣿⠒⣼⣿⠃⠀⠀⢠⣿⣿⣿⡿⠎⠁⠀⠀⠀⢿⡿⠀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣻⣿⣷⡟⡁⠀⠀⠀⠀⠉⠙⣿⣅⠠⡀⠀⠈⢻⣿⣿⠋⣠⡄⠀⠀⠀⢨⣷
⠀⠀⠀⢹⣿⣿⣿⣿⣿⡏⣠⡿⠀⣸⠄⠀⠀⠀⠠⠏⠈⠙⠋⠒⠀⠈⢻⡿⣅⠘⣖⣀⣠⣶⣿⡟
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣾⠃⡏⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⣷⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣀⢻⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢇⠀⠀⠉⠉⠘⠛⢻⣿⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣶⣾⣀⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣀⣾⠇⠈⢳⡄⠀⠀⠀⠐⢞⣿⡇
⠀⠀⠀⠀⣤⠹⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⢈⣵⠂⠀⠀⠱⡀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠐⠒⠈⠋⠀⠀⠀⢸⡿⠀⠀⠀⠀⡏⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⠀⠀⠀⠀⠀⠆⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣯⣾⣧⡄⠀⠀⠀⠀⣠⡟⠀⠘⠆⠀⠀⠀⢀⣰⣼⡿⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣤⡄⠀⣰⡟⠀⠀⠀⠀⠁⠀⠀⢁⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣭⣵⣿⣿⠀⠀⠀⠀⠀⢠⢄⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⣿⣿⣄⡀⢀⡀⠀⣠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣿⣵⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⣠⣤⣤⣤⣤⣄⣀⣀⣰⣾⠛⣷⣀⣀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⢸⡟⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠈⣿
⠀⠀⠈⢿⣤⣤⣤⡶⠶⠶⠾⢿⡟⠛⢿⡿⠶⠶⠶⣦⣤⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⣦⡀
⢀⣀⣀⣀⣀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⣿⠋⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠛⠻⠶⢶⣤⣀
⣿⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠳⣦
⣿⡀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠛⠛⠛⠛⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠻⠶⣦⡀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠙⠳⢶⣦⣤⣤⣤⣤⣤⣤⣤⣴⠶⠟⠁⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⣀⣀⣀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠉⠉⠉⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡶⢶⡶⢶⠶⣶⠶⡶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠟⠷⣦⡘⠟⠘⠇⠸⠃⠻⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠘⣿⡶⠰⡆⢸⠆⢶⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⣀⣴⢿⣤⢠⡄⢠⡄⣤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⡁⢈⡁⢈⠁⣉⠀⣉⠀⠀⠀⣠⡶⠛⠷⣦
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣾⣧⣜⡁⠛⠀⠛⠀⠀⠀⣿⡀⠀⠀⣿⠇
⠀⠀⠀⠀⣼⠏⠁⠀⠀⠈⠹⣧⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠷⣶⡾⠋⠀⠀⠀⠀⠉⠻⣿⠿⠿⢶⣄⠀⠈⠻⠶⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣀⣀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⣠⡿⠀⠀⢠⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠙⠻⠷⠾⠟⠋⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⢿⣄⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢾⣿⣥⣀⡀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⣶⣾⠋⠉⠙⢷⣄⠀⠀⠀⠀⠈⠻⠶⢶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠈⠻⣦⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣄⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠙⢷⣄⠀⠀⠀⣠⡶⠶⠶⠟⠋⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠙⢷⣶⠟⠋⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠙⢶⣤⣤⣤⣤⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠦⠀⠀⢀⠀⠀⠀⠀⠀⠀⢾⠄⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠀⠰⠉⠀⠆⠁⣸⠆⠀⠀⠀⠰⢶⣶⣶⣦⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠁⢀⠀⣴
⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡄⠀⠈⢫⢿⣿
⠀⠀⠀⠀⠀⣶⣾⣿⣶⡆⠀⢀⠀⠀⠀⠀⢙⣿⣿⣿⠋⠀⢀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⢸⡀⠀⠀⠀⠀⢸⣾⠃⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⢀⡞⠀⠀⠀⠀⠻⠿⠛⠿⠋⠀⡼⠀⠀⠀⠀⠸⠛⠁⠀⠀⢀⣾⠃⠀⠀⠀⠀⡾⠙⢷⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠾⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠉⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣄⣦⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⣄⣀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠦⡀⠀⠠⠀⣠⠏⠀⠀⣀⣤⣤⣤⡀⢹⣿⣿
⠀⠀⠀⠀⠀⢠⡞⢻⣏⠀⠀⢰⠀⠀⠀⢸⣿⣇⠆⣽⣿⠀⢰⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⢠⣧⢴⠀⢀⣶⣾⣿⣬⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠟⠃⠀⢀⡞⠀⠀⠀⠀⠉⠉⠉⠉⠉⣠⡿⠀⠀⠀⠀⠐⠤⠤⠤⠂⢀⣾⡟⣿⣆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠤⠴⠞⠋⠀⠀⠀⠀⠐⠀⢀⣠⣤⡶⢿⡉⢸⣿⣿⣷⣦⣤⣤⣤⣶⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠚⣴⣾⣿⣿⠟⠉⢀⣀⣀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠀⣰⣾⢿⣷⡆⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⠀⠀⠀⠠⡇⠀⠀⣶⣶⣶⡄⠀⠈⣿⣿⡏⠀⠀⢸⣿⡿⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠺⣷⣴⣿⠋⠀⢰⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⢰⡇⠀⡄⠀⠀⢹⡟⣿⠀⠀⢀⣿⣿⡇⠀⠀⣶⣿⣷⡄⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢆⡘⡔⠀⠀⠈⠀⠉⠀⣠⣾⣿⣿⣷⢄⠀⠿⠿⠿⠃⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⠀⠏⠁⣤⠾⣷⢶⠶⡶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⢷⣦⠀⣿⡿⠿⠿⠀⢸⡿⠿⢿⣦⠀⣴⡾⠛⠻⠇⠀⣴⡾⠛⢿⣦⡀⢰⣿⣧⡀⢸⣷⠀⠀⣴⣿⣧⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣾⠟⠀⣿⡿⠿⠗⠀⢹⣿⣶⣾⠋⠀⠙⠿⢷⣶⣤⢸⣿⠀⠀⠀⣿⡇⢸⣿⠹⣷⣼⣿⠀⢰⣿⣉⣿⣆⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣿⣷⣴⡦⠀⢺⡿⠀⠻⣷⡄⢶⣦⣤⡾⠟⠀⠻⢷⣤⣾⠟⠁⠸⣿⠀⠘⢿⡿⠀⣿⠏⠉⠹⣿⠄⢿⣿⣦⣶⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠀⣀⡀⠀⢀⣀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⢀⣀⣀⡀⠀⣀⡀⠀⠀⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣿⣏⠀⠻⣷⣠⣿⠋⢀⣾⡟⠉⠉⠻⠀⣿⣟⣉⣉⠁⠈⢻⣿⠁⠀⣿⣿⣦⠘⣿⠀⢸⣿⣋⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠙⣿⡿⠀⠀⢹⣿⠃⠀⠘⣿⡄⠘⢿⣿⠀⣿⡟⠛⠋⠀⠀⢸⣿⠀⠀⣿⡇⢻⣿⣿⠀⢸⣿⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠛⠋⠀⠀⠘⠛⠀⠀⠀⠙⠻⠿⠿⠛⠀⠛⠛⠛⠛⠂⠘⠟⠻⠓⠀⠛⠃⠀⠙⠛⠀⠘⠻⠛⠛⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣷⣤⣤⣶⡶⢶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⣿⠀⠈⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⣤⣾⠋⠀⣾⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢹⣿⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠐⢿⡅⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠀⣿⠀⣿⠀⠀⠀⠀⢸⣿⠀⢀⡿⠐⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⣿⡶⠟⠁⠀⠀⠈⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠀⣀⠀⠀⠀⠀⣿⠀⣿⠛⠛⠛⠛⠿⠟⠀⠀⠀⠀⠀⢰⡄⠀⠈⠙⠻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁⠀⣠⡾⣿⡟⠀⠀⠀⠀⢿⣄⠙⢷⡀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢸⡇⠀⠀⢀⣴⠄⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠁⠀⠀⠀⣰⠟⢠⡟⠀⠀⠀⠀⠀⠀⠹⣧⠀⠻⣦⠀⠀⠀⠀⠙⠷⠄⣀⣬⣄⡀⠰⠟⠁⠀⠀⠀⠀⠈⠙⠷⣦⣀
⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⡆⢀⣀⣀⣀⠀⣼⠏⠉⠉⠻⣆⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⡀
⠀⠀⠀⠀⢾⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣿⡀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣿⡇⠈⠛⠛⠋⠀⢿⣄⠀⠀⣰⡟⠀⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻
⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⢷⣄⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢀⣴⠄⠙⠛⠛⠋⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⣀⣤⡶⠟⠛⠛⠛⠛⠛⠛⢻⣦⠀⠀⠀⠀⠀⠈⠛⠷⣦⣽⣷⣶⡄⠀⣼⡇⠀⠀⣿⡇⠀⠀⠰⠟⠁⠀⠀⢸⡇⠀⠀⠙⠷⠆⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠘⢿⣥⣤⡾⠛⠻⣿⠛⠻⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠈⠁⠀⠀⢀⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠛⠷⢦⣄
⠀⠀⠀⣨⣿⣀⣀⣀⣀⣀⣀⣀⣠⣿⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠙⣷
⠀⢀⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠈⣿⡟⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣤⣄⠀⠀⢹⡇
⠀⣿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⢸⣷⠀⠀⣀⣤⣤⡾⠛⠛⠛⠛⠳⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⢸⡏⠀⢹⡇⠀⢸⣇
⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⣼⡏⠁⠀⠀⢀⣤⣤⣄⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⡇⠀⢸⡇⠀⢸⡇⠀⢸⡟⠻⣄
⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⣶⡶⢿⡅⢀⣿⠶⣶⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡾⠛⢷⡄⠀⣿⠀⢸⡇⢠⡿⠁⠀⠘⣷⠀⢸⡇⢠⣿
⠀⣿⠀⠀⠀⠀⢰⡟⠉⠙⢷⡄⠀⠀⠀⢸⣿⠀⣿⡇⠀⣿⡄⠈⠙⠛⠁⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣇⣀⣼⡇⠀⣿⠀⢸⡇⢸⡇⠀⠀⠀⣹⡇⢸⡇⢸⣿
⠀⣿⠀⠀⠀⣤⣼⣧⣤⣤⣾⣥⡄⠀⠀⢸⣿⠀⣿⡇⠀⣿⡄⠛⠛⠛⠛⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡟⠋⣹⡇⠀⣿⠀⢸⡇⠘⣷⡀⠀⢠⡿⠁⢸⡇⠘⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⣿⡁⠈⠛⠛⠋⠀⣿⠁⠀⣿⠀⠀⠀⠀⠀⣀⢸⡿⠟⢹⡇⠀⠻⣦⣼⡇⠀⢸⡇⠀⢸⡇⠀⢸⣧⣼⠟
⠀⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⠀⣿⡇⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⠀⠀⠀⢀⡾⠋⢸⣷⠶⢻⡇⠀⠀⠈⢹⡇⠀⢸⣇⠀⣸⡇⠀⢸⡏
⠀⣿⠀⠀⠀⠀⠛⠛⠁⠘⠛⠛⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠟⠁⠀⠸⠿⣶⠾⠇⠀⠀⠀⢸⡇⠀⠈⠛⠛⠋⠀⠀⣸⡇
⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠀⣿⡇⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠾⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⣠⡿
⠀⢻⣦⣤⣤⣤⣶⣤⣤⣶⣤⣤⣤⣤⣤⣾⡏⠀⠘⢷⣦⣦⣤⣤⣤⣤⣤⣤⣤⣤⣾⠋⠀⠀⠀⠀⠀⠀⣤⣾⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣦⣶⠞⠋
⣿⣿⣿⡿⠿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣴⠿⠛⠙⠳⣤⣤⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣠⠎⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣠⠖⠂⠀⢀⡜⠀⢀⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠒⡆⠀⠘⢧⡶⠟⠉⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠘⠗⠁⠀⠀⠀⠀⠀⣤⠀⠊⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡿⠀⠀⠀⠀⠀⠀⢊⡨⠐⠒⢦⠀⠀⠀⠀⠀⠀⡰⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠴⢒⢿⣄⠘⡄⠀⣠⣾⣾⣿⣷⣤⢔⢾⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣉⡉⠀⠀⣼⣿⣿⣿⣟⢯⠎⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⣿⣿⣷⣶⣿⣿⣿⣿⣿⠟⡧⠰⠯⢄⡀⠀⢀⣠⠤⢠⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡽⢣⠞⠑⣧⠀⠀⠙⢾⡋⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⣿⣿⣿⣿⣿⠟⣵⣋⡤⠶⠛⠳⣤⣀⣠⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⢢⡼⠋⠉⠉⡹⢫⣞⡠⢿⣶⡒⠒⠒⠀⠩⢩⡤⢀⣀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⠞⠀⠀⢀⡞⡵⠁⠙⡄⠀⠉⠻⣷⣄⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣧⠀⠹⣆⡠⣴⣫⠞⠀⠀⠀⡇⠀⢀⣤⣾⣿⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⢀⣠⠴⠛⠉⠁⠀⠀⠀⠀⠈⠳⣤⠏⡐⢸⡃⠀⠀⢀⡞⠀⠀⢻⣿⣿⠟⠙⠋
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⢔⣵⠏⠁⠀⢠⠏⠀⢠⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡱⠋⠁⠀⠀⡴⠃⠀⠀⠈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⠞⠁⠀⠀⢀⣼⠁⠀⠀⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡵⠃⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡤⠤⢤⣤⣏⠟⠀⠀⠀⠀⠰⠿⠛⠛⡟⠉⠉⣿⡏⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠇⠀⠀⠀⢀⠔⡁⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢻⠰⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠈⠢⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣼⠀⢷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⡠⠊⡀⠀⠈⠢⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡿⠀⠸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠴⠚⢁⡠⠀⣈⠔⠊⠀⡤⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢒⣉⡥⢔⡉⠄⣀⠔⣉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠤⣚⣩⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠄⠀⠀⠘⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⣤⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⡜⢹⣧⠀⢀⡴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⢀⡤⢦⡄⢤⡄⢠⠄⢸⡇
⠀⠀⠀⣿⠀⠀⣼⠉⠉⢻⣆⢸⣇⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣧⣀⣀⠈⢷⡜⠀⢸⡇
⠀⠀⠀⣿⠀⠈⠉⠁⠀⠉⠉⠁⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁⠉⠉⠁⠀⡼⠀⠀⢸⡇
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣞⡅⠈⢀⠘⡄⠀⢀⡠⠔⢂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣈⣿⢹⣷⣤⣿⠀⢱⡠⠋⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠓⠢⡀⢸⣿⣿⠘⡌⣿⡟⡟⠀⢸⠁⢀⡎
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠠⠀⠀⠀⠈⠓⢻⣿⠀⣇⠈⣿⣣⠴⢾⡶⢤⣳
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣿⣷⠂⠉⢋⣬⡝⠿⣆⢹⡆⡿⠁⢀⡎⡠
⠀⠀⠀⠀⠀⠰⣗⠀⢠⣀⡀⠀⠀⠹⣄⠴⣿⣿⣿⣧⡈⢿⣿⠁⠀⣸⢀⡇
⠀⠀⠀⠀⠀⠀⠼⢷⣆⠙⡅⠀⠀⠀⠉⣴⣁⡙⢻⣿⣿⡌⢿⣆⠀⣿⡼⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢈⣷⣄⣙⠳⣌⡢⣀⢴⢯⣀⣤⢬⠿⠿⣿⣿⣸⣿⠐⣯⠃⢀⣤⣾⣿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠾⠛⠠⢈⡛⢿⣿⣯⣿⣄⡀⠹⣄⢆⢤⡈⠻⣿⣿⢸⡟⢠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡻⣾⢁⣍⠙⢦⡙⣿⣆⠻⣷⡄⠙⣿⣠⠟⣁⣀⡤⢤⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣄⢳⣈⣟⡓⠦⣍⡻⢿⣦⠘⢿⣦⣿⣯⣾⣿⣿⡿⠢⢵⡹⣿⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⡴⠾⠟⠛⠛⠛⠛⠛⠛⠛⢷⣄⡉⠻⣿⣿⣿⡿⠁⠀⠈⠳⣿⡇⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢺⣷⢴⠿⠿⠂⢀⠄⡔⢻⡄⠀⠀⠀⠀⠉⠻⣦⡈⠻⣿⡀⠀⠀⠀⣀⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡄⠀⠀⣮⠀⣿⠈⢳⡀⠀⠀⢄⠀⠀⠈⠛⢷⣾⣿⠶⠾⠛⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢘⡇⢸⡄⢀⣿⠃⠙⣦⡁⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢆⢸⣇⠈⢷⣸⣿⡀⠀⠙⠁⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣷⢿⣳⡞⢿⣿⣧⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⡍⢙⣳⡿⡀⠀⠹⣿⣧⠀⠀⠀⠀⠘⡟⢧⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⡏⠉⣿⢄⣈⡐⢀⣏⠻⣷⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠗⠈⣹⣆⠀⠉⠙⠿⣷⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡡⠞⣉⣤⡿⡷⢤⡄⠤⠔⠛⠛⠛⠻⠶⢦⣤⣤⣄⣀⣤⣤⣤⣤⣴⣶⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠑⣩⡴⠿⠟⢩⠞⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣡⠚⠁⠀⠀⠀⣾⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠹⣄⠘⠆⣠⡀⠀⠀⠀⢀⣴⣾⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠉⠀⠀⠀⣰⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⣁⡉⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⢉⡠⠖⠋⠁⠉⠓⠦⣌⠙⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⣁⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡆⢸⡇
⠀⠀⠀⠀⠀⠲⣶⠶⠶⠾⠿⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠾⠿⠶⠶⢶⡶⠂
⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⣤⠀⢀⡤⡄⢀⡤⡄⢠⠀⡄⣠⢤⡀⠀⣤⠀⠠⣤⠄⣤⠤⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⢰⠿⢧⠸⣄⡄⠸⣄⡄⢻⣠⠇⣿⠻⡅⣸⠿⣇⠀⣿⠀⣿⣁⡀⠘⣧⡀
⠀⠀⠀⠀⠀⢀⣼⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣄
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⣿⠉⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⡏⢹⡟⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⣡⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣌⠙⠲⣄⡀⠀⠀⠀⠀⣀⠴⠚⢉⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⡉⠓⠦⠖⠋⣡⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⡆⢸⠯⡀⠀⣙⡤⠀⠀⠀⠀⠠⣄⣀⣤⣀⣀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣇⣿⣿⣤⢸⣞⠆⠆⣹⡇⠀⠀⠀⠀⠀⠹⣟⡻⢾⣿⣷⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⢀⣴⣌⣿⣟⣛⣩⠟⣾⡤⠖⠒⠓⠯⣌⠀⢀⠀⣼⣣⣿⣽⣿⣿⠄⠀⢐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣧⣿⣼⡿⢿⣿⡻⣶⣶⡗⠚⠂⡆⢀⠲⢶⣿⣿⣿⢿⣽⣿⣇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡠⠀⣠⠤⣶⣿⣿⣿⣆⣀⣽⠛⠉⠉⢯⠹⠁⠓⡲⠖⠏⢴⠞⢃⡼⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡶⠟⠲⢭⡿⠋⣹⣿⣿⠂⠀⢠⣶⣾⣶⣦⣤⣄⣨⡤⠤⠴⣿⡀⣱⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡞⣿⣔⢙⣀⣨⣼⠛⠋⣹⣷⠖⠉⠁⠈⠹⣏⠻⣿⡇⠀⠀⠀⣀⡼⣿⣿⠟⠂⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⣃⢹⣍⣿⣭⣤⠟⠢⢀⣿⡅⠀⢸⡇⠀⠀⢹⣆⣯⣤⡴⠶⢾⡁⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣯⢘⣿⡇⠤⠄⠀⣀⠞⢿⡧⠹⣿⡇⠀⠀⢸⠟⠉⠀⠀⠀⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠃⠀⠀⠀⢀⣾⢇⠘⣏⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣴⡿⣸⠄⠙⣤⣤⠀⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢁⡾⠀⢷⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢀⡸⠻⣴⡾⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠳⠒⠶⣾⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠡⠀⠉⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡄⢀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⣱⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢠⠀⠹⡆⠀⠀⢀⣾⡀⠀⠀⠀⠀⠀⠀⢈⣷⡀⠀⠸⣆⠀⠀⢿⠇⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣧⣄⠀⠀⢀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⣿⡀⠀⣸⠇⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⣀⠀⢈⣻⣙⣿⣿⣿⣷⡾⠂⠉⠀⠁⠀⠀⠀⠀⠈⠳⡀⠀⠈⠻⣆⢸⡄⢀⡟⠀⠀⠀⢦⡀⠉⠈
⠀⢀⠤⢶⣶⣿⣶⣯⣥⠋⠉⠹⣿⡟⠀⠸⠛⢶⣄⡀⢀⠀⠀⠀⠀⠈⣇⠀⠀⠈⢻⣷⡿⠀⢤⡀⠀⠈⠁⢈⡙⠓⠢⡉
⠀⠁⠀⠀⠀⠁⠈⠀⠈⡀⠀⠀⡿⠀⠀⢠⣤⡀⠻⣷⣾⡀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠙⠁⠀⠀⠙⡗⠀⠀⠀⠙⢣⡄⢸⣇⢀⣸
⠈⣍⠀⠀⠀⠀⠀⣠⣶⣷⠀⠀⠃⢀⡤⠤⠄⠘⠶⠬⠻⡧⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠀⠀⢢⡈⠁⢻⣶⣿⣤⡄
⠀⠈⠀⠀⠀⡴⠺⠟⠛⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡅⠀⣠⣶⣤⠀⠀⠀⣧⡄⠀⠀⠘⠀⠀⠀⡸⠀⠀⢴⣿⣿⣿⣧⣄⢀⠀⠀⢀⡤⠀⢀
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⠏⠁⠆⠀⠀⢻⡇⠄⠀⠀⠀⠀⠠⠁⠀⠀⠴⠿⣿⣿⣿⣿⠟⠉⠉⠁⠀⠾⢄
⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⡔⠀⠀⠀⢰⡀⠀⠈⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠈⠁⠀⢠
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣸⠃⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⣾⠏⠁⠀⠀⢀⣠⠤⠄⠬⠀⠀⠻⣏⠉⠲⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⣾⠀⠀⢁⠀⠀⠀⠀⢸⣿⣤⣶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠹⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠄⠠⡈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠸⠆⠀⠀⠀⣿⣟⠥⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠈⠠⡀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣿⠀⠀⡾⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢹⣦⠘⢆⠀⠀⠀⠀⠀⠀⠀⠸⣆⠘⣿⠀⠰⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣔⡀⠙⢷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣿⡀⠁⠀⠀⠁⠀⠠⠐⠒⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⣀⡴⠚⠑⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠂⠀⠀⡀⠁⠀⠀⠀⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠉⠑⢂⣠⠇⠀⠀⢰⠀⠀⠀⠀⠆⠁⠀⠀⠀⡀⠀⠀⠀⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠓⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⡀⠀⢀⠀⠈⠆⠀⢀⣦⣤⡤⠤⣤⣄⣀⡀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠊⡑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡄⠀⠀⠀⣠⠀⠋⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠁⠀⠊⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣈⠐⣴
⠀⢀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠻⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢉⡀⠄⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⣠⠄⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠈⠃⠀⣠⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⡀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆




⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⡇⠀⠀⢸⣿⣉⣿⣋⣹⣯⣯⣍⣿⣯⣇⣹⣋⣷⣋⣿⡿⣏⣼⣹⣯⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⢟⣛⣛⣛⣛⣛⠿⣿⣿⣿⡇⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣋⣵⣾⣿⣿⠿⠿⢿⣿⣿⣶⣍⠛⣵⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢏⣼⣿⡿⣋⣵⣶⣿⣿⣷⣶⣍⠛⣵⣷⡜⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣾⣿⡟⣼⣿⡿⣛⣭⣭⣝⠛⣵⣿⡜⣿⣿⡌⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⢸⣿⣿⢰⣿⣿⢟⣵⣷⢹⣿⣿⢸⣿⣧⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣸⣿⣿⢸⣿⣿⡸⣿⣿⣿⣿⡿⣸⣿⡿⢸⣿⡟⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⢻⣿⣧⠻⣿⣷⣮⣝⣛⣭⣾⣿⡿⣣⣿⣿⢣⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⠻⣿⣷⣬⣛⠿⢿⣿⠿⢟⣫⣴⣿⡿⣣⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣷⣬⡻⢿⣿⣿⣿⣾⣿⣿⣿⠟⣫⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣭⣭⣭⣵⣶⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡄⠀⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⡸⠶⠄⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄⠀⠀⠀⠈⠀⠀⠨⣀⠀⠀⢄⠀⢸⠀⢸⠃⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠟⠂⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⡠⢿⠇⠀⢀⣴⠉⠀⠀⠀⠀⠠⢀⠘⢆⢸⡀⠈⠀⠀⠴⠁⠀⠀⣼⠁⢀⡤
⠀⠀⠀⠀⠀⠀⠐⠂⠰⢶⣆⠠⣼⣿⣿⡗⠈⠀⠐⠿⠛⠀⠀⠀⠀⠀⠀⠀⠡⣄⠱⠇⠀⠀⠄⠀⠀⠀⢠⠏⣠⠚⠀⢀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠄⠻⣿⣿⣦⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠈⢦⠀⠀⡘⠀⠀⠀⠀⣼⡞⢁⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⣅⠀⢀⠀⠂⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠆⠐⠉⠉⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠿⠇⠀⠻⣄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠃⠀⠀⠀⡀⠀⠘⠂⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣷⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⢀⣄⣼⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⣶⣶⣿⣿⡹⣅⠀⠀⠀⡀⠚⠉⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⣣⣾⣿⣿⣿⣿⣿⣶⠶⠤⠀⣴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⣠⣼⣿⠃⢿⣿⣿⣷⡟⠁⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠋⠉⠈⠈⠀⠁⣼⡿⠘⠛⢷⣖⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠙⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣇⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠄⠀⢀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣉⣿⣻⣿⣿⣭⣍⣿⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣯⣭⣽⣿⣯⣭⣽⣽⣯⣯⣽⣽

⣿⣿⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡿⠻⢳⣤⡿⠟⠟⠻⠛⠛⠛⠛⠻⠛⠛⠻⠛⠛⠛⠛⠛⠻⠻⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠨⠀⣤⠭⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠂⠀⠀⠀⠀⠀⠀⢄
⠀⢠⠄⠀⠀⠀⠤⠊⠉⠑⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⡁⠀⠐⠠⠤⠐⠀⣀⠤⠐⠀⠈
⠈⠘⠀⠀⠀⠀⠢⣀⣀⡠⠓⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠉⠐⢖⢒⠒⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠀⠀⠀⠀⠀⠀⠀⡀⠂⠄⠀⠀⠀⠰⡇⠀⠀⠀⠀⠘⡇⠀⠀⢀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠄⠀⠀⠀⠀⠙⠲⠤⣤⣀⣸⣠⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⠄⠀⢀⠄⠀⠐⠀⠈⠓⠎⣀⡀⠀⠈⠀⣠⣈⠀⠀⡀
⠀⠀⠀⢀⠀⠆⠀⠐⠂⠀⠀⠀⠈⠀⠀⠉⠀⡀⠂⠀⠀⠘⠁⢄⣰⠁⠀⠉⠒⠤⡁⠂⠤⣀⡀
⠀⠀⠀⠀⠀⠐⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠤⠂⠀⠀⠀⠀⠀⠀⢀⣀⡠⢄⡀⠉⠒⠂⠍⣙⠒⠲⠤⠤⣄
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠆⠀⢀⡀⠤⠀⠒⠂⠁⠀⠀⡀⠈⠁⠒⠤⡀⠀⠈⠉⠐⠂⠊
⠀⠀⠀⠀⠀⠘⠊⠁⡀⢀⠀⢀⡠⠄⠠⠤⠄⠁⠀⠀⡅⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠂⡆
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⢢⠒⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠔⠀⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠑⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡔⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⠉⡀
⠀⡤⠄⠤⠒⢪⠡⠉⡠⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁
⠀⠀⠂⣀⡹⠁⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡰⠀⠂⠰⠀⠀⠰⠀⠚⠀⠂⠰⠀⠆⠰⢀⣆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⣧⣶⣴⣶⣾⣴⣧⣼⣶⣷⣴⢳⣿⣶⣧⣾⢿⣦⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⠀⢀⣰⣆⣴⣶⣶⣤⣶⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⡄
⠸⠚⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢰⠀⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠋⠁⠀⣤⣿⣿⣿⣿⣿⣿⠃
⢸⢰⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣷⣷⣶⣶⣤⣤⣶⣶⣦⣤⣶⣿⣿⣿⣿⣿⣿⣧⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⠏
⢸⣾⣿⣿⣿⣿⣿⣾⣿⡛⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⢛⣛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠉⠸⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠫⠙⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠉⠁⠛⣻⣿⣿⣿⣿⡇
⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⢰⠟⠁⢹⣿⣿⣿⣿⣿⣯⢁⣀⣤⣤⣴⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠚⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠀⢀⣀⣀⣾⣿⣿⣿⢳⣢⣀⢀⡀⠀⠀⠀⢸⠀⠀⠀⠒⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⢿⣿⡇
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣷⠀⠀⠀⢸⠂⠀⠀⢠⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢀⣿⡇
⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢶⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⡇
⢰⣮⣿⣏⣿⡛⠟⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣶⣄⡈⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡇
⠈⣿⣿⣿⣿⣷⣄⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡌⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⣽⣿⣿⣿⣿
⢸⣿⣿⣜⣿⣿⣿⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⡆
⢸⣿⣿⣿⣿⣿⣿⣷⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃⡘⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⡄⢽⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠤⠤⠤
⠀⢸⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠘⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⢸⣿⡿⣿⣿⣿⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢴⣾⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⢠⡾⣿⣷⣿⣿⣿⣿⠃⢻⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⣿⣯⠻⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣯⠥⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢹⣷⢹⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⠛⠉⠀⠀⢸⣭⣭⣥⣤⡤⡔⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⢻⢺⣇⣿⣿⣶⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣅⡙⠿⢿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⡿⠿⠟⣿⣿⣿⣿⠏
⠀⠀⠀⡠⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠸⠁⠃⠀⠈⠀⠀⠀⠀⠀⠙⠻⣿⣿⣟⣿⣹⣭⣭⣬⣶⣿⣿⡿⠟⠛⠁
⠀⢀⣾⣶⣤⣿⡿⠋⠁⠀⠙⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠻⠛⠛⠙⠛⠛⠻⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢤⣀⠀⠀⠀⠀⢸⡀⣼⡟⠀⠀⢀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠙⡷⣄⠀⠀⠘⣿⡿⠁⠀⠀⣀⣀⡀⠹⡓⠠⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠘⢷⡄⠀⣿⠁⠀⠀⠈⠤⠠⠬⣽⡇⠀⢁⠋⣠⡄⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⣿⣄⡇⠀⢠⡴⠶⠦⣀⠀⠀⢉⣢⣬⡞⠁⡘⠱⠢⢄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠑⢦⣼⣿⣧⠀⠀⠀⠀⠀⠈⢿⡛⢿⠁⠈⠙⡶⠀⢀⣤⣺⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣏⠑⠢⣄⠈⠛⣿⠀⠀⠀⠀⠀⠀⠈⡇⠈⠀⠀⢀⡼⠊⢉⡠⠼⠷⣦⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣳⣄⠘⣧⠀⠀⠀⡀⢀⠀⡇⠀⠀⡴⠋⠀⠀⣀⠤⠒⠒⠙⡇⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠃⠀⢄⠉⠉⠉⠓⣿⠖⠒⠉⠀⢴⠀⣠⠠⠊⡀⠀⡠⠚⠁⠀⠀⠀⡴⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢣⡀⣠⠚⢁⠀⠀⠀⠀⡇⢀⣿⣴⣞⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠀⠀⡟⣱⠀⠿⠃⠀⠀⣸⢀⣼⡿⠋⢁⠤⢒⠂⠙⠓⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠛⠆⢧⢾⠁⠈⡴⠁⠀⠀⢠⣧⡾⠋⢀⣴⠿⠿⠒⠒⠒⠒⠻⡄⠁⠑
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠁⠀⠀⣬⣍⠉⠒⢤⣴⣿⢠⣿⡕⠁⠀⣠⠿⠟⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⡇
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠦⣝⣿⣼⣿⢀⣼⣾⣇⠠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠻⡇
⠀⠀⠀⠀⠀⠈⠣⢤⣤⣤⣤⣀⠀⠀⠀⠀⣠⡄⠀⣘⣿⣿⣟⡁⠈⠹⣏⠉⠀⠀⢀⣀⣄⣂⠩⠶⠬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡶⠤⠤⠤⣴⣾⠡⠃⠈⢻⢫⡀⠀⠸⣯⣴⡶⢿⣗⡲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠞⠋⢧⠀⠀⠀⠀⠀⠱⠀⠀⡿⢿⡁⠀⠀⠈⠑⠨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠁⠀⠀⣚⣦⠀⠀⠀⠀⠀⠀⠀⣆⠀⠉⠙⠲⠦⢤⣄⣈⣊⣣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢀⡤⠖⠋⠁⠀⠙⠢⣄⡀⢰⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢤⣀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡄⢄⡀⠀⠐⢄⠀⢆⠀⠀⠀⠀⠀⠀⡄⠀⢦⠰⢄⠰⠀⡆⠆⡀⠄⠠⡀⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣄⣀⡀
⠀⠀⡠⣾⣷⡀⠀⠀⢠⣼⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠐⢋⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡿⠓⣀⠀⠀⣬⣇
⠀⠀⢈⣼⣷⣄⣿⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⣽⡇⠀⣼⣿⠀⡀
⠀⠀⣼⣿⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣽⣅⠀⠀⠀⠀⠀⠀⠀⣼⣻⣿⡇⠉⠉⠉⠉⠉⠉⠁
⠀⠸⢧⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠖⣩⣿⣿⠀⠀⠀⠀⠀⠀⠐⢣⣿⣿⣿⡄
⠀⠠⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠏⣿⡿⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠿
⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠻⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠃⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⡀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡀⢠⣾⣦⣄⣀⡠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣯⣀⣀⡀⠀⠀⠀⠀⡐
⡇⢸⣿⣇⣉⡉⠉⠉⣿⡋⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⡍⠻⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⠐⠚⠛⠓⢺⡟⠛⠓⠒
⠥⠬⢿⠿⠿⣿⣆⢰⠃⢣⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠹⠆⠇⣠⣶⣇⠐⣶⣶⣶⡶⠀⠀⠀⢸⣀⣻⣿⣿⣿⣆⠀⡏⢱
⠀⢀⠎⢂⠀⠈⢿⡏⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢸⠘⣧⠀⢸⣍⠛⠛⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠠⢣⠀⠈⠻⣿⠁⠀⡇
⠀⠀⠀⠈⠀⠀⠼⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠂⠘⠿⠗⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠐⠁⠀⠂⠀⢠⠇⠀⠀⢸⡄
⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠐⠀⠂⠀⠀⠀⠐⠐⠀⠀⠀⠀⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣶⠲⠲⣂⡦⠽⠍⠚
⠀⠀⠀⠠⢀⠀⣀⡀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⠇⡀
⠀⠀⠀⠘⠂⠚⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠈⠂⠈⠉⠉⠁⠉⠉⠁⠢
⠀⠀⠀⠀⢀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⣀⡀⠀⠤⠀⣀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡔⠈⠁⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢄
⠀⠀⠀⠀⡆⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⢀⠠⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠑⠐⠐
⠀⠀⠀⠀⡇⠠⠄⠀⢰⠀⠀⠀⠀⠀⠀⠀⣇⢂⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡖⢉⡁⠀⠃⠀⣲⡀⠀⠀⠀⠀⠀⠀⢄⡐⠠⠂⠀⠤
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠁⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⠇⠀⠀⠀⠀⠐⡀⢁⣰⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠂⠀⠀⠀⠀⡄⢸⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠄⠠⠀⠐⠈⡄⠀⠈
⠀⠀⠀⠀⢄⣀⠀⠈⣀⢖⠁⠀⠀⢀⠎⢁⢀⠁⣐⡂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡭⠀⠀⢈⠐⠀⠀⢀⡀
⠀⠀⠀⠀⠘⠈⠙⠉⠀⠀⠀⠀⣠⠧⠆⠀⠀⠀⠀⠀⠀⠑⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⢀⠁⢀⠀⣀⢀⡀⣀⠠⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠈⠃⠁⠁⠀⠁⠈⠀⠀⡨⠋⠀⠀⠀⠀⠀⠀⠋⠀⠀⠁⠁⠀⠀⠀⠀⠔⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠚⠌⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡀⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠒⠀⢁⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⡟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣶⣶⣿⣿⣿⣿⡻⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣯⣷⣾⢿⣟⣿⣿⣿⣿⣷⣬⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡿⠛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣹⠿⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠧⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣭⡽⠋⠁⠁⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣫⡤⠖⠋⢀⣀⣤⣄⣀⣀⣤⣴⣶⣻⡿⣘⠃⠀⠀⠀⣠⣴⠋
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⠈⠿⠀⠈⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠈⠛⠛⣹⣿⣿⣿⣿⣿⢿⡿⣿⡟⣿⣿⣿⣿⡉⡀⠀⠀⠀⡀⡺⠗⠋
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣛⣁⣀⣀⣀⠀⠀⠭⠭⣭⣴⠖⢦⣀⡘⠁
⠀⠀⠀⠀⠀⣾⡟⠋⠉⠘⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⣶⣿⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢐⣂⣀⡀⢠⣤⢩⣿⢿⣿⣿⣿⣿⣿⣿⠿⠛⡇⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠶⠷⠶⠿⢟⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠤⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡲⠒⠤⣀⢤⡀⠀⠀⠑⠺⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡰⠃⠀⠈⣷⣀⠀⠀⠀⡏⠉⢩⠟⡖⠚⠒⠓⡞⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢅⠀⢀⠞⠀⠈⠑⠢⣤⣣⠔⠁⢀⣇⠀⠀⢀⣇⠀⠀⠀⠀⢀⣀⠿⠭⠭⠭⠭⠭⠽⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠈⡇⠀⡠⠊⠀⠉⠉⠁⠀⠳⡄⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⣀⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣡⣤⣤⣀⣀⣠⣔⣹⣄⣸⣇⣀⣇⣀⣀⣀⡀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⢰⡋⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⡆
⠀⠀⠀⠀⠀⠹⣭⠭⠭⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⣭⠏
⠀⠀⠀⠀⠀⠀⣷⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡞⢳⠀⢰⠋⡆⠀⡖⢳⠀⢠⠊⣦⠀⢰⠑⡄⠀⡎⢲⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⡇⠀⠇⢸⡀⢸⠀⣿⠀⠀⠀⡇⠀⡇⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⣿⠀⠀⠀⡇⠀⡇⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⣿⠀⠰⠀⡇⠀⡇⣸⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢹⠋⢱⠒⡖⠒⣾⡻⡆⢲⡶⢶⠒⢶⠶⡖⠋⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⠀⠀⠀⠀⡃⠀⡇⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⢸⠀⡇⠀⣾⠁⡇⢸⣷⣾⠀⢸⠀⡇⠀⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠦⣿⠀⠰⠚⡇⠀⡇⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣧⠀⣇⢸⠀⠸⡖⡇⠀⢿⢀⠇⠸⣇⣿⠀⢸⢀⡇⠀⣇⣸⠀⢸⡀⡇⠀⣇⢸⠃⠸⡀⡿⠀⢸⢀⠇⠀⣇⣻⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢩⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠻⠴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠠⢾⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⢀⠀⠀⢳⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⢀⡾⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢳⠀⠘⡇⠀⠀⢠⠏⠀⠀⡄⠀⠀⠀⠀⣦⠀⠀⠾⠀⣶⣶⣖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡌⢧⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠧⠀⣤⣾⣿⣿⡯⢄⣴⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⠃⠀⠦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠈⢻⡏⠀⠐⠋⠉⠁⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⠠⠤⣄⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣾⢃⡤⠄⠀⠙⠀⢸⣶⣄⣀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠸⠇⠊⠀⠀⠉⠀⠀⠀⠈⠉⠈⠁
⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡄⠀⣀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⣤⣶⣿⣿⣾⣿⡄⠀⠀⠀⠠⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢛⢹⡿⢿⣿⢻⠇⣠⣄⣀⠀⠀⠲⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣶⠞⢻⣟⣹⠿⠿⠋⠉⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠋⠀⠀⢻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠁⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣀⠀⠤⠀⠀⠄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠰⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⣠⠎⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢀⣠⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡟⢻⣤⣿⠃⠀⠜⠁⠀⠘⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣴⣿⠋
⠀⠀⠀⢀⣄⠀⠀⠀⠛⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⢳⡶⣤⡀
⠀⠀⢸⡏⠀⢰⣤⣤⣤⣼⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠘⢷⣴⣅⠀⠉⢀⣿⡇⢸⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⢹⡉⠀⠀⢸⡿⠃⣼⡇⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⣠⠟⠀⠀⣠⣶⡿⠟⠋⠁
⠀⠀⢀⣤⣤⣴⡾⠃⣿⠃⢀⣿⣿⠀⠀⣿⡆⠀⠀⣷⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⣿⡇⠀⠀⣤⣄⠀⠀⢀⣼⡿⠋
⠀⠀⣾⠃⠀⠀⠀⠀⢻⡆⠘⣿⣿⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⣿⠀⡀⠀⢀⣿⠃⠀⣼⣿⠃
⠀⠀⠘⣷⡄⠀⠀⠀⠀⢿⡄⠹⣿⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⣼⡟⢡⡀⠀⠀⢰⣿⠀⠙⢿⠛⠁⠀⣰⣿⡏
⠀⠀⠀⢿⣄⣴⠟⠻⡆⠈⠻⣦⡘⣧⠀⢿⡄⠀⢰⣿⠀⠀⠀⢰⣿⡁⣿⡇⠀⠀⣾⡟⢠⣤⡿⠁⠀⣴⣿⠿⣶⡀
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣼⡟⠛⢻⡇⠸⣧⠀⢸⣿⠀⠀⠀⢿⡏⠀⢹⡇⠀⢸⣿⣧⣼⠏⠀⣠⣾⠿⠋⠀⣿⡇
⠀⠀⠀⠀⣿⣷⣶⡟⠀⣼⡟⠀⠀⡈⢿⣄⣿⡄⠈⣿⡆⠀⠀⠸⣧⠀⠈⢿⣄⣈⡿⠀⠀⣠⣾⠟⠁⠀⠀⠀⢻⣇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⢀⣾⠃⠀⠉⠻⣧⡀⢿⣷⠀⠀⠀⣿⡆⠀⠀⠉⠉⣡⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⢆⣀⠀⠀⣿⣇⣾⢃⣼⠃⠀⢀⣼⡿⢾⣿⣧⡀⠀⢻⣧⠀⠀⢀⣼⡿⠉⠀⠀⠀⠀⠀⠀⠀⣄⣠⡟
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⣿⠃⣾⠃⠀⢰⣿⡟⠀⠀⠀⠹⣷⡀⠘⣿⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⢀⠀⢰⡿⠋
⠀⠀⠀⠀⣼⡏⠀⠀⠀⠈⠿⣾⡏⠀⣠⣿⠏⠀⠀⣴⠇⠀⢙⣷⣴⠟⠉⠀⠀⠀⠀⣤⠀⠀⠀⢀⣼⣷⠟⠁
⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠻⠿⠿⠟⠁⠀⠀⠸⣧⣤⣶⠿⢛⠁⠀⠀⠀⠲⣤⡴⠏⠀⠀⢀⣼⠟
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⣾⡇⠀⠒⠶⣿⠋
⠀⠀⠀⠀⠀⠛⣷⠀⠀⣶⠀⠀⠀⠀⠙⢷⣿⣀⣤⠀⣴⣶⡶⠏⠙⢷⣴⡾⠿⠋⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠻⣶⠾⠋⠀⠀⠀⠀⠀⢀⣈⣽⣿⣏⡉⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠞⠛⠋⣉⣉⡛⠿⣷⣤⡀⠀⠀⠈⠙⠳⠶⠖⠋⠁
⠀⠀⠀⠀⢀⣤⢶⣄⢠⣿⣏⠀⠀⠀⠀⢦⣀⣿⠀⣉⣀⣀⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠙⠷⠾⠏⢸⣿⠛⢷⣤⡀⠀⠀⠉⠀⢾⣋⣉⡽⠇⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠻⣷⣤⡀⠀⠀⠉⠹⠶⠾⣿⣻⡿⠀⢀⠀⠙⠻⣿⣶⣤⡀⠀⠀⠀⠀⣧⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⠀⠀⠀⠀⠉⠻⢷⣤⣀⠀⠀⠀⠚⢡⡴⣶⣏⣿⠆⠀⠀⠉⠙⢿⣷⣦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡶⠛⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀⠀⠀⠀⠀⠉⠁⣀⣀⣠⣤⡶⠛⠉⢻⡆⠀⢀⣀⡀⠀⠀⠀⠀⢀⣴⠖⢶⡀
⠀⠀⠀⠀⢠⡾⠛⢶⣄⣠⡿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣤⣤⣶⡿⠿⠛⠉⠀⠀⠀⠀⢸⠃⡴⠉⠉⢻⣆⠀⠀⠀⠈⠃⠴⠾⠃
⠀⠀⣠⣤⣤⣿⣄⣀⣿⡷⠶⣦⡈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠴⢤⣾⡇⠀⢀⣀⣀⡀
⠀⠀⠘⢧⡽⠉⠉⠙⣿⠀⠀⢸⡿⠀⠈⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣴⠟⠉⠉⠙⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠛⠃⠀⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠋⠁⠀⠀⠀⠀⠀⢿⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⣾⠀⠀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣿⡶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⣠⣄⠀⣿⡀⠀⠀⠀⡠⢺⠆⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⡀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⢀⡀⠙⠋⢀⡘⠇⡴⠒⢲⠀⣿⣀⠤⢄⡀⠀⡠⠤⠤⡀⠰⡹⡄⡘⣷⡀⢠⣤⠀⢾⡷⢀⢃⢀⠅
⠀⠀⠀⠀⠀⠀⠈⠰⣔⠶⡁⠈⢲⠃⢀⡠⠾⠟⠀⠠⠀⢱⣼⠀⠀⠀⠘⠆⠼⢃⠀⣿⡇⣀⡄⠀⠀⠀⣸⣸⠏⠀⠄
⠀⠀⠀⠀⠘⠇⠀⠐⢸⠛⠛⠢⣬⣧⡎⠀⠀⡜⠂⠀⠀⣼⣿⣄⠀⠀⠚⡄⠀⠈⣇⣿⢏⡼⣥⠋⠉⣴⣵⠏⠤⠂
⠀⠀⠀⠀⠀⢀⠤⠊⠉⡕⠂⠉⡝⠁⡣⠀⢀⡅⠀⠠⠊⢸⣿⠈⠂⠀⠀⣄⠀⠐⠁⠙⣟⠀⠀⠀⠀⠋⠉⢱⠀⢠⣄
⠀⠀⠀⠀⠀⠈⠙⠻⢿⢷⡖⠈⠁⠀⠀⠀⠘⡀⠀⡀⠀⣿⣿⠇⠀⡀⢀⠄⠀⠀⠁⠐⠈⠑⣤⣄⠀⠀⠐⠣⡀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣊⣴⢿⠀⠀⠘⣤⣀⠀⢐⠁⠈⠁⠀⢸⣿⠀⠀⠀⠀⠱⠀⢀⠤⡈⠀⠀⡸⠋⠀⠠⣀⣠⣃⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠛⠉⠁⡜⠳⢄⣘⠀⢨⡀⠀⠐⠂⠀⠀⢾⣿⠆⠀⠄⢢⠀⠀⡇⠀⢘⡀⠔⠙⡄⢦⣀⣨⣭⠟⢩⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠄⡀⠘⠀⠐⠄⠀⢸⠀⠀⠀⣸⣿⡄⡀⠀⡼⠀⠀⠆⠀⠈⠀⡤⠐⡧⠚⠫⡒⠲⡀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⢻⣿⠋⠀⠀⠘⢄⠀⠀⠀⠁⠈⠀⠀⣅⡰⢦⠍⣉⣀⡀⣠⠄⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡆⠀⠀⢀⠀⢀⠎⠀⠒⠄⡴⣿⣿⡷⠤⠐⠂⠈⣆⠀⣀⡃⠀⠘⠑⠋⢷⣾⡿⠟⣻⣿⣋⣀⣀⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠐⠀⠉⠉⣄⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣀⡋⠉⠀⠝⠈⣤⣅⠤⠈⠁⠉⢫⡉⠉⠐⠶⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⠔⠉⠈⠑⢆⠀⠐⢻⣿⠓⠀⢠⠊⠀⠈⠱⢄⣀⡤⠻⠛⠢⢄⣸⠷⣦⣘⣄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⠀⠁⠀⠀⠈⡄⠀⢸⣏⠀⠀⠆⠀⠀⠀⠀⢠⣉⠀⠀⣇⢀⣠⡦⣙⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⣄⠀⠀⢀⡜⠘⣄⠀⠀⣠⠤⠄⠒⠁⠀⠉⠍⢩⡅⠳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⣴⣦⡄⠀⠀⢠⡄⠢⠙⠃⡐⠁⣂⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡄⣠⠀⡐⠋⡁⠀⠀⠀⠀⠐⠛⣟⢫⠙⢀⠀⠲⠶⠒⠗⣠⣞⠟⠩⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⢿⣽⣿⡄⢀⣷⡶⠖⠂⠀⢀⣔⠁⡀⠀⠀⠃⢀⣤⡖⢠⣿⣉⣀⡠⢤⠐⠂⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠳⠿⣿⣟⠟⠠⠉⠃⢀⣀⡶⠾⠶⠬⠄⠸⠂⠺⣾⣏⡁⠙⠩⠤⠀⣀⣀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡍⢉⢿⣿⣁⠀⣬⡭⠛⠀⢪⡦⠀⠂⡐⢉⡀⣬⡤⠄⠀⣐⡇⣀⡀⠀⠈⠑⢤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠘⠀⡿⠉⠳⠀⠀⠙⠿⠆⢻⡄⠀⠀⠠⣧⠀⢀⡴⠚⢗⠀⢃⣾⠳⠿⢷⣩⣌⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡎⠀⠇⠐⢀⣄⠀⠀⠐⠒⢻⣄⢀⠀⣀⠀⠀⡌⠀⠔⠸⠆⠈⠉⠙⠳⠞⠄⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⡈⠃⣤⡄⠠⠂⠉⠀⠀⡄⠀⣴⠿⠿⠋⢶⡀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠴⢰⣦⡄⢹⣶⣦⡄⠀⠠⣴⣆⣰⣧⠀⣿⠆⠂⠀⢠⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⠀⠀⠀⠀⠉⣻⣿⣿⣿⠉⠂⠀⠐⠃⠐⢻⡄
⠀⠀⠀⠀⠀⠀⠰⡗⠀⠘⢡⣤⣷⡶⠀⠠⠈⠈⣱⣌⠁⠰⣆⠀⠐⠶⠠⠬
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠃⠀⠀⣼⣯⠘⠇⠀⠀⠀⠀⠀⢈⡀⠄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⢋⠁⣉⣣⡞⢀⡏⣿⣷⣌
⠀⠀⠀⠀⠀⠀⠈⠀⣤⠐⠋⢱⣭⢚⠍⠈⡇⠘⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠿⠀⠈⠸⣿⣇
⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣘⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠒⢢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠄⠛⠀⠈⠁⢑⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠌⠀⠀⠀⠀⠀⠀⢊⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠀⠀⠀⠀⠀⠀⠀⠘⣰⠀⢀⡤⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡧⠒⢠⡌⢤⡒⠮⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠀⠀⠀⠀⠀⠐⢢⠀⠀⢀⠀⢀⢀⠀⠀⣤⡴⠶⠆⠗⠒⠒⠒⠒⠒⠒⠒⠶⠁⠀⠀⠀⢩⠖⡿⢛⠇
⠀⠀⠀⠀⠀⠀⠰⡟⢂⠤⢤⡀⠛⢆⡀⣀⣀⣀⣀⡠⠞⠀⢐⠁⠀⠀⠀⡇⢸⠺⠶⢶⠄⣦⣤⣤⣠⣤⣠⣶⡂⠛⠚⠒⠒⢢⠜⠿⣿⡉⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢇⢀⠜⠁⠂⢼⡄⠈⣄⣀⣨⡄⠀⢀⡨⠶⠶⠶⠶⠧⣼⠀⠀⢼⣧⢻⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⢨⠀⠎⠀⠀⠀⠈⢤⠋⠀⠀⠀⡀⠀⠀⠈⢂⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣀⠀⡆⠇⠀⠀⠀⠀⠀⢸⠀⠎⠁⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣰⠉⠉⢹
⠀⠀⠀⠀⠂⠀⠚⠃⠀⠀⠈⠉⠉⠉⠉⠉⠐⡆⠀⠀⠚⠀⠓⠀⠀⠀⠀⠐⠃⠘⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠠⡿⠀⠀⠈⡇
⠀⠀⠀⠀⢹⠛⢛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⣟⠛⢛⠛⠛⠛⠛⠛⢛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠀⡔⠒⠂⠀⠀⠓⡄
⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢸⠀⠀⠁⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠀⠁⠈⠉⠈⠉⠁⢉⡆
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢸⠀⠀⠀⠆⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡇⠈⢹⠙⠻⠉⡿⠉⣷⡆⢲⠂⠉⡇⠀⢻⠀⠸⠀⣀⣤⣦⣤⣼⣆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠰⠠⠀⠶⠀⠤⠠⠤⠀⠤⠜
⠀⠀⠀⠀⠀⡇⠀⡸⠘⡶⠀⠇⠀⢿⠇⢸⠀⢠⡇⠀⡾⡰⠴⠤⠮⣿⠧⠄⠘⠫⡿⠶⠶⢤⠄⠀⠀⠀⠀⠠⠤⠤⠀⠊
⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠉⠉⠉⠉⠉⠉⢹⣥⣶⣾⣿⣶⣿⣷⣶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠜⠆⠾⡠⠆⠴⠤⢿⠀⠀⠉⠉⠉⠉⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⡤⠴⠒⠒⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣛⣛⣁⣛⣻⣟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣯⣭⣽⣭⣿⣿⣿⣽⣍⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠾⢛⠋⠉⠁⠀⠀⠙⠻⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠋⠀⢀⠈⠀⠀⠀⠀⠀⢲⡛⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⠿⠋⢤⡄⠀⢠⣹⠀⠀⠀⠀⠀⢀⠀⠓⢀⣤⣨⣤⣽⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣼⠟⠋⠁⠀⠀⠀⠀⠀⠈⠈⠢⣤⣀⣀⣀⡁⠀⠀⠀⣼⣷⢦⣤⣽⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣰⢾⡇⢀⡴⠶⢤⠴⠊⢉⡳⢶⡀⠀⠀⠀⠀⠉⡇⠀⠀⢠⣿⣥⡾⠿⣿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⣰⢏⠳⣼⠋⠀⠀⠀⢠⠓⠒⠤⠸⡄⠀⠠⢀⡤⠞⠁⠀⠀⢻⣿⣿⣶⣠⣤⣾⣿⣿⣇⠀⠑⠢⠄
⠀⠀⠀⢀⣿⠃⡰⠋⠀⠀⠀⠔⠁⠀⢀⡀⠘⢷⣄⠀⣹⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣴⣦⡴⠂
⠀⠀⠀⢸⡏⢸⡇⠀⠐⠇⠀⠀⠀⡴⠛⠉⣱⠀⢿⡀⣿⠁⢀⣀⠀⠀⠀⠈⠻⣯⣿⣿⣿⣿⣿⣿⣷⣞⢸⣟⠁
⠀⠀⠀⣿⣯⣼⡅⢀⠈⠀⣷⡀⠀⢡⠼⡋⠉⢠⡌⢷⣿⣆⣸⣿⡇⠀⠀⠀⠐⠚⠿⣿⣿⣿⣿⣿⡿⠛⣫⣼⣷⡄
⠀⠀⠀⢹⣿⡌⣷⠀⠀⠀⠎⠉⠢⣌⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇⣦⣦⣆⡀⢀⣴⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⡗
⠀⠀⠀⠘⣿⣇⣿⠀⠀⢈⠄⠀⠀⠙⢯⡽⣶⣟⡷⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠹⣿⣿⣷⣶⣅⠤⠐⠌⠀⠀⢀⣠⣿⠛⠶⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⣿⣿⣦⣟⣷⣶⣶⣤⣴⠟⢩⣎⢷⡈⢶⣾⣿⣿⣿⣿⣟⣻⣿⣿⣿⣤⣤⣤⣭⣭⣿⣷⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣿⣹⡆⠈⠳⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡃⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠙⠻⢿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣯⠙⣿⣿⡟⢩⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠈⢿⣄⣾⣿⢰⠤⣿⣿⣿⣿⣿⣿⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣾⣿⣿⣏⠘⢰⡟⣿⣿⣿⣿⣿⣿⣶⣶⡄⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡅⣋⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡏⣿⣿⣿⣿⣿⣿⣿⣬⣧⡌⢿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠛⠛⠋⡞⢻⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣻⣍⣿⢿⣿⣿⣿⣯⣿⡿⣿⣿⣦⣝⢿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⣀⣤⣴⣿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⢟⣵⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣽⣦⡙⣿⣾⣿⡀
⠀⠀⠀⠀⠀⢀⣴⡵⠋⠙⠿⠯⠩⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⣿⣟⣿⣌⢻⣿⣧
⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠁⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣯⢻⣆⠙⣿⡄
⠀⠀⠀⢀⠀⢂⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⡦⣿⣿⣿⣿⣿⣿⢿⣭⣾⣿⣧⣙⡉⠛⠛⢿⣿⣿⣿⣿⣧⠹⣯⣽⣧⢨⠃
⠀⠀⠀⣿⠁⠀⠠⣄⠀⣰⣆⣿⣿⣿⣿⡿⡆⢠⣿⣿⣿⣿⣿⣿⢰⣿⡿⢻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠈⠻⣿⣿⣿⣎⠙⣿⣻⣎⢣⡀
⠀⠀⠀⠉⣄⢀⣠⣿⣾⣿⣿⣿⣿⣿⠁⠀⠀⣸⣿⣿⣿⡿⢿⣴⣿⣿⣿⣾⣧⠀⢸⣿⣿⠟⠉⢻⡏⠉⠷⣤⡙⢷⣦⡘⢿⣿⣿⣧⢿⣷⢻⡎⠁
⠀⠀⡘⠀⠀⠀⠀⠙⣿⢟⣿⣿⣿⣿⡇⠀⢀⣿⣿⣿⣟⣿⣾⠉⣿⣿⠙⠻⣿⣆⠘⠉⠁⠀⠀⣸⠇⠀⠀⠩⠹⣶⣿⣷⣄⠈⢻⣽⠀⢿⣧⠻⠸⡆
⠀⠀⡇⠀⡄⠀⠀⣾⣿⠘⣿⣿⣿⣿⣿⣆⠸⣿⣿⣿⣿⣿⣿⡇⣻⣥⣴⣦⣼⡷⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⢘⣿⣿⣿⡄⠙⢿⣷⢈⣿⣆⠀⠛⡀
⠀⠀⡇⠀⢿⡠⣶⣿⣿⣇⣿⣿⣿⣿⡇⠹⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡘⢯⣿⣿⡄⢨⣿⡆⢹⣿⡗⠀⠁
⠀⠀⠷⠦⡼⣿⣽⡿⠛⢻⣿⣿⣿⣿⣇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⡆⠀⠀⢠⣠⣯⣀⠠⡂⠀⣀⣴⣍⣶⣿⣿⡇⠀⣻⡷⠈⠉
⠀⠀⠀⠀⠀⠁⠉⠁⠀⠻⠿⢿⣿⣿⡍⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡷⢨⣿⣿⣾⣳⣿⣿⣿⣭⣾⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠻⠿⠗⠘⠻⣿⣿⡿⠋⠉⠀⠉⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢿⣶⢸⣿⠛⠛⠀⣿⠟⢿⣦⢰⣿⡛⠷⠀⣴⡿⠛⢷⣆⢸⣿⣆⠠⣿⠀⢠⣿⣿⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠟⠋⢸⣿⠛⠛⠀⣿⠿⣿⡋⢈⡛⠛⣿⡜⣿⣄⠀⣸⡿⢸⣿⠹⣿⣿⢀⣾⣧⣼⣷⠀⣿⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠛⠛⠛⠀⠛⠀⠙⠃⠈⠛⠛⠋⠀⠈⠛⠛⠋⠀⠘⠋⠀⠘⠛⠈⠋⠀⠀⠛⠃⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣄⢠⣄⠀⢀⣄⠀⣀⣤⣀⠀⢠⡀⢀⣀⣤⣄⢀⣄⡀⠀⣠⠀⣠⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣼⣿⠈⢻⣶⡿⠁⣾⠋⢉⣙⡁⢸⡇⢸⣿⣭⣭⢸⣿⣿⣄⣿⠁⣿⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⣿⠀⠀⣿⡇⠀⠻⣧⣬⣿⠇⢸⡇⢸⣿⣤⣤⠸⣿⠈⢻⣿⠇⣿⣧⣤⡄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⢾⣏⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⡉⠈⠙⣿⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠁⠀⠀⠈⢻⣦⣴⣿⣦⣄⣤⣴⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣘⣿⣀⣀⣀⣀⣀⣈⡉⠀⠀⢸⡟⠁⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣦⣄⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆⢀⣄⠀⠀⠻⣷⡀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢰⡿⠁⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⡄⠀⠸⣷⠀⠹⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣤⣀⣼⣿⣶⠾⠟⠛⠃⣼⡇⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⡇⢰⣿⠀⠶⣿⣧⣤⣤⡄
⠀⠀⠀⢠⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⢠⣿⡷⠟⠁⠀⠀⠀⠈⠉⠛⠿⣶⣶
⠀⢀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠛⢷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠿⠋⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤
⠀⢨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣸⣷⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛⠛⠛⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⣿
⠀⠛⢿⡄⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣄⡀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⡷⠀⠀⣸⣿
⠀⠀⢾⢿⣄⠀⠀⠀⠀⢸⣿⣀⣀⣽⡏⠙⢿⣆⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣼⡿⠛⠛⠛⠛⠛⢻⡟⢻⣿⠋⠀⠀⣴⡿⠛
⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠙⢷⣄⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣶⣶⣾⣷⣾⣿⣶⣶⡄⠉
⠀⠀⠀⠀⠀⠀⣹⡿⣶⠀⣾⠇⠘⣿⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⣰⡿⢿⣦⣀⣀⣀⣿⣀⣀⣀⣀⡀⢀⣿⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⢿⡇⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢨⣿⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣀⣀⣀⣾⣟⡉⠉⢉⣉⣩⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⣿⡇⠀⠀⠀⢀⣾⠿⣷⡀⠀⠀⠀⠀⣿⡇
⠀⠀⣰⡿⠛⠛⠛⠛⢛⣿⡀⠘⠛⠛⠛⠛⠛⠛⠙⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⣧⣀⣼⡇⠀⠀⠀⠀⣿⡇
⠛⠛⠛⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣀⣉⣉⣁⣀⣀⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠷⠿⠿⣷⡿⠶⠶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣶⣶⣶⣶⣶⣿⣧⣴⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⣿⡇⠀⠀⠀⠀⠾⠿⠿⠆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠻⢷⣤⣄⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣿⠉⠉⠉⠁⠀⣻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠙⠀⣿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⣿⡇
⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠉⠉⣿⣶⣄⣀⣀⣀⣀⣀⣤⣾⠟⠛⠃⠀⠀⢿⣷⣤⣴⡄⠠⣤⠀⢴⣴⣤⣴⣤⣤⣿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣉⣿⣛⣛⣿⣋⣉⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢤⢎⠒⢠⠀⠀⠀⠀⠀⢠⡀⠄⠀⠀⠀⠀⢁⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠐⠀⠀⠀⠐⠖⠀⠀⠀⠈⠃⠘⠤⠈⠀⠀⠀⠀⢈⡁⣀⣠⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⡈
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠈

⠀⡀⠀⡠⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⢀
⠀⢀⠁⠈⠉⠑⠀⠀⠀⠀⠀⣦⠂⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠠⠄⡈⡀⠤⠀⠀⠀⠀⠂⠂
⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠽⠚⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠉⠑⠂⠀⠀⠀⠠⠀⠀⢀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⣀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠈⠀⠀⠄⠠⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠠⠉⠉⠁⠀⠀⠀⠐⡄⢠
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠢⡈⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠠⠈⠃
⣤⣤⢦⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣷⣤⣤⣤⣤⣤⣤⣶⣤⣤⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣲⣴⣖⡂⣖⣶⢴⣶⢠⡖⢶⣰⣦⣰⢀⡶⡄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠘⠓⠂⠋⠓⠐⠞⠋⠓⠚⠙⠃⠛⠘⠛⠓⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⡖⢦⡖⣴⢒⡂⡆⣶⣒⢰⣆⣶⣶⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘⠀⠙⠚⠃⠃⠛⠒⠙⠈⠋⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⣏⢙⣦⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⣈⣋⠉⠟⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣡⡤⠤⠤⠤⠤⢀⡉⠻⣆⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⢸⠃⢸⠃⠀⠀⠀⠀⠀⠀⢰⡆⢸⠆⡷⢠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠊⠁⠀⠀⠀⠘⣆⠘⢦⣀⣀⣀⣀⣀⣠⠼⠁⣼⠋⠁⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣤⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣶⠛⠛⠛⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⢤⡀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡀⠀⠹⣯⡽⠙⢦⡀⠙⢦⡀⠀⠀⠀⠀⠸⠷⠟⠛⢛⣻⣻⣟⠁⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠶⢠⡆⣶⠀⠀⠙⢦⡀⠙⠂⠀⠀⠀⢠⠾⣆⣀⣿⣉⣉⣉⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣤⣈⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣟⠀⣹⠄⠀⢀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠿⣆⠰⠾⠷⠶⢶⡄⠀⠈⠁⠀⠀⠀⠈⠉⣿⠀⠀⣞⣹⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⢉⣭⣭⣭⣿⣭⣭⣽⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⠤⠤⠥⠤⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠒⠒⣶⡗⣶⠋⠀⠀⠀⠀⠀⠀⢀⡆⣿⠀⠀⠘⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢤⣤⣬⣭⣭⡽⠏⠉⠁⠶⣄⠀⠀⠀⠀⣠⡞⠃⣿⣒⠒⠒⢒⣒⣒⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠛⠃⠀⠀⠈⠉⠁⠁⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣂⣴⠖⠘⠄
⠀⠀⠀⠀⠀⠙⠀⢀⣴⠈⠉⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠰⡔⠓⠾⣯⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢦⠀⠈⠀⠀⠀⣄⠀⠀⠀⢻⡆⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⢻⡄⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠠⢀⡿⠋⢁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⠀⠀⠙⣧⠀⠀⠸⣧⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢷⣄⡀⠀⠀⢻⠃⠀⣾⣦⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠹⣆⠀⠀⣿⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢻⣿⡅⠀⠀⠀⣀⣾⣿⢿⣿⣿⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⣻⣆⢀⡟⠀⠘⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠋⠁⠁⠲⢶⣿⣿⣿⢿⠿⠏⠛⢳⣼⣿⣿⣿⠛⠓⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢿⣾⡇⠀⠀⢴⣁⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡂⠀⠹⣿⣿⠁⠀⠀⠿⠋⠉⠉⠁⠀⠀⠀⢀⣄
⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢶⢀⠀⠁⠀⠀⠀⢰⡏⠉⠀⠁⠀⠀⠀⠀⠀⠀⢀⣼⡗⠀⠉⠀⠀⠘⠻⡆⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠠⠔⠒⠲⠶⢶⡄⠀⢀⡈⠀⠡⠄⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⣾⣿⡟⢀⡴⠞⠓⢀⠀⠀⠃⠀⠀⢿⣿⣷⣤⢤⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⠴⠞⠀⠀⠉⠉⠙⠠⠀⠀⠂⠀⠈⠉⠉⠀⠙⠂⠄
⠀⢀⣴⠂⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢶⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠺⣿⣿⡆⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠘⣁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣠⣤⣝⣿⣷⣶⣶⣦⣇⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣡⣿⣿⣿⣿⣿⣻⢻⣿⣿⣿⣯⠀⠀⠀⠀⠀⠙⠉⠙⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⠃⢋⡹⣽⠿⠿⣿⣿⣏⠻⡿⠀⢀⣤⠓⣀⢀⠀⠀⠀⠲⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⢀⣨⣁⣼⣦⢰⣿⠛⣿⣿⣷⣾⣿⣿⠛⠟⠻⢷⣦⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣿⣿⡿⠁⠈⢿⣷⣼⣟⠉⠉⠟⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠉⠁⠉⠀⠀⠀⠘⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢩⣿⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⡅⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⢀⡄⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠠⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡤⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠞⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⢀⠀⠀⡰⠎⠀⠀⠈⠀⠀⠀⠀⠈⠀⠰⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠠⠀⡀⠀⠀⢡
⠀⠀⠀⠀⠈⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡈⣦⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣳⡿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠒
⠀⠀⠀⡘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⣀⣤⣤⠴⠒⠀⠈⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠺⠛⠍⠛⠒⠂⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛





⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⠛⠙⠛⠛⠶⣤⣀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠛⠛⠛⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢠⡶⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣆⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⡟⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠻⣏⠀⠀⠀⠹⣆⣀⣀⣀⣀⣀⠀⣿⢸⡇⠀⢀⣀⡼⠁⠀⠀⠀⠹⣆⡀⢀⣠⣤⣄⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠈⠉⠉⠉⠉⠙⣷⣿⢸⡇⣼⠟⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠉⠛⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠸⠿⢸⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⡾⠁⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣾⣿⠫⠀⡇⣵⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⢻⣶⣶⣧⣰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠻⢇⣿⣿⢸⡆⢿⣜⢡⠃⠀⠦⠀⠀⠀⠀⠀⠀⠀⠱⣥⠝⣝⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡌⣿⢏⣭⣄⣚⣁⣋⡶⠚⡉⠃⠀⠢⠤⠀⠀⠀⢐⣿⣴⣿⣮⣿⣿⡆⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠻⢿⡅⣿⣸⡿⢿⣿⣍⡭⣶⣶⠻⠚⠋⠐⡂⠀⠐⠶⢶⣶⢶⡶⠷⣏⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣶⡿⣟⡙⣇⠀⢉⣼⠾⠛⠉⢻⡋⡛⠒⠶⠦⡤⠴⢁⡤⠖⠀⣰⢿⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⣠⣴⣧⡙⢻⢂⠕⠳⣩⣾⠿⠁⠀⠀⣠⣠⣧⣀⡀⠀⠁⠐⠀⠀⢃⣠⣾⠁⠺⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠋⡠⣀⠉⠽⣠⣦⡾⠟⡷⠆⢀⠔⠚⠛⢿⣿⠻⣿⣿⡗⠋⠉⠉⠁⠈⢿⡦⣦⣹⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⣿⡤⠤⡤⠶⠲⣏⡁⠀⢰⡟⠛⠀⢀⡀⠀⠀⠹⡄⠉⡷⠃⠀⠀⠀⣀⠴⠛⢟⣛⠋⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀⢹⣧⣿⠶⠶⠞⠉⠀⢀⣿⣧⢀⣀⢸⡄⠀⠀⠀⣿⣈⣡⡶⠶⠛⠻⢧⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠺⣤⢀⣯⣗⠠⠤⠀⢠⣠⠞⠻⣿⡆⠻⣿⡇⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠙⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠃⠀⠀⠀⠀⠀⣸⡿⠀⢹⢙⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⡇⣯⠈⠺⢁⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛⣫⡜⢃⡄⠈⠛⠋⠱⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠟⠀⡈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢀⡄⢠⣴⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡘⠛⠀⠃⣾⢏⡀⢽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡄⢀⢠⡇⣼⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⢸⣮⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣁⣀⣀⣀⣀⣀⠈⠙⠒⣤⡀⢸⣿⣿⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⠉⢉⡿⢿⣦⡀⢨⡟⠻⣛⡻⠗⢾⡤⣤⣬⣍⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⡢⢹⣷⣈⣧⡀⠈⣇⠀⠂⢣⠘⢯⣄⣠⠀⠉⣶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠃⠀⠀⠙⠻⣯⣿⢀⠻⡄⠀⣸⠀⣼⣯⠇⠈⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⢠⡇⣠⠏⢠⠋⣀⣀⡴⢣⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣠⢀⣼⣶⢤⣄⣀⠀⠀⠀⠀⠀⣿⠛⢹⡴⠋⠀⠋⠀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢼⣿⢌⢹⡇⠈⡟⠟⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⡈⠈⣿⡇⠸⡔⢀⣠⠴⠶⣦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣖⡁⣰⠙⠻⣿⠋⠉⠉⠙⠳⣿⣴⣫⡵⢖⣀⣤⣿⡇⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡼⡉⠣⡀⠀⠀⢀⣸⠿⠷⢶⣿⣷⣞⡛⢿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠐⡶⠀⢀⣠⣤⣤⣤⣿⠀⠡⢀⡼⠉⢀⠔⠂⠀⠀⠀⣼⠃⠠⢦⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢣⡀⢸⠇⣰⣧⣀⠀⠀⠀⢰⡙⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⡆⠉⠻⢿⣷⡄⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠉⠛⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣳⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠉⠙⠛⠳⠦⣭⣭⣙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠙⠆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠟⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⠀⠸⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡾⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⠀⠀⣿⣷⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣍⠑⠈⠻⠉⠳⣆⡘⣿⣧⡀⢸⣿⡇⠀⢀⣤⢶⢾⣾⣶⠶⣲⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡉⠻⢏⠛⠓⠀⠀⠀⠀⠀⠁⠸⣿⣇⣾⣿⣠⣴⠟⠙⢁⡼⠛⠁⣸⣿⠿⠟⠛⣯⡿⠛⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠘⠏⠳⣄⠀⠳⡄⠀⠀⠀⠀⠀⠰⡀⢼⣿⣿⣿⡿⠃⠀⠀⠞⠁⢀⣴⠟⠁⠀⣠⣿⠋⠈⠛⠷⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⠀⠈⠀⠙⠢⡀⠙⣆⠀⠡⣀⠀⢀⣽⣿⡿⠁⠛⠀⠀⠀⠈⣰⡴⠛⠁⠀⠀⠰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠘⢆⠘⣧⡀⢿⣿⡤⣿⡿⠁⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠑⢬⣳⣾⣿⣷⣿⡇⠙⣀⣾⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡾⢹⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⢦⡀⠀⠀⠀⠀⠻⡟⢿⣿⣿⣳⢐⣿⡏⣿⣯⡦⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣀⣤⠶⠚⡋⣁⢠⠶⠿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣙⢦⡐⠀⠀⢀⢏⣀⣿⣿⣿⠹⣿⣼⣿⡫⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣶⢶⣿⠟⢉⢤⠒⠛⠉⠀⠀⠀⠀⠨⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣳⣿⣆⣸⡟⣬⠈⢻⢫⣿⣿⣿⣿⣿⠁⠎⠀⠀⠀⠀⡤⠃⢀⡞⣫⠖⠉⣠⡶⠟⠁⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⢿
⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣤⣾⣿⣿⣯⡥⠍⠙⣻⠿⣿⣼⣿⣧⣸⣼⣿⣿⣿⣿⣿⠎⢀⡄⢀⣠⣬⣴⣿⠋⠊⠁⣠⡾⠋⠀⠀⠀⠀⢀⠤⠚⣹⠟⠁⠀⠀⣿⣼
⠀⠀⠀⣴⣿⠀⠀⠁⣀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠀⠀⠉⢻⣿⣿⣼⡟⢯⠻⣿⣿⣴⣿⣾⣿⡟⠋⠁⠠⠊⣠⡾⠋⢀⠀⠀⠀⢀⠀⠀⢀⢼⡇⠀⠀⠀⠀⣿⠟
⠀⢀⣾⣯⣠⠤⠶⠒⠂⠀⠐⠒⠐⠲⠦⠤⢤⣀⣀⠄⣠⣤⣀⣈⠻⣿⣿⣾⡇⣿⣿⣿⡟⣹⣻⡾⠤⠴⠦⢾⣥⡴⣾⡵⠂⠀⠀⠀⠀⠀⢂⡾
⠀⢸⣯⡷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠾⠿⠶⣮⣽⡻⢿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣷⠶⠋⠁⠀⠈⠛⠿⣯⣴⣾⣶⣁⡴⠛⣡⡞⠁
⢠⣿⣭⣀⣤⣤⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⢿⣿⣿⣾⣿⢿⣿⣿⣿⢿⣿⣿⡶⠿⣿⠛⠛⠉⠑⠲⣌⣻⣟⣩⣤⠴⠟⠋
⣿⢁⡾⠉⠛⠯⣤⣤⣄⣀⠠⠤⠄⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⣫⠟⢻⡿⣿⣿⣿⠿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠈⠛⠣⢄
⡟⠋⠀⠀⠀⠀⠀⠙⠿⠗⣒⣒⣢⣴⣶⡦⠤⣶⣮⣷⣿⣿⣿⣿⣿⠃⠀⠎⠀⡏⠙⠻⠃⠀⠉⠛⠿⣟⡳⣄⠀⠀⠀⠀⠲⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠛⢙⣷⡶⢾⡿⢿⠿⠟⣻⣿⣿⡏⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡳⡄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡊⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠠⠞⢡⠞⡱⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⢆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⢀⠜⠁⣀⠴⢻⡿⠛⠲⢶⣄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⢆⠀⠠⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠁⠀⢈⡀⠔⠈⠀⠀⠀⠀⠈⠉⠘⠛⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢆⠐⣆⠅
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠸⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣿⠈⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⡀⠀⠀⠀⠀⢠⡆⢰⠀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢄⠀⠱⣜⣧⣥⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠲⠦⡀⢸⣇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢦⣤⣄⣑⣦⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⣤⣤⣤⠤⠤⢤⠀⠀⠀⠀⠀⣄⣀⣀⣀⡀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⢰⠃⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀⠤⠤⣄⠀⠀⠀⠀⠀⠀⡇⠀⢾⣉⣤⣀⣉⡷⠀⢸⠀⠀⠀⠀⢰⠃⠉⠉⠁⠀⠀⠉⠉⠈⢧
⠀⠀⠀⠀⣠⠤⠤⡀⢠⠇⢸⠀⠀⠀⠀⠀⢀⣀⣿⣀⡴⣠⣶⣠⣼⣤⡀⠀⠀⠀⠀⡇⠠⠒⠛⠛⠛⠉⠀⢤⢸⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⠀⠀⠀⠀⡼
⠀⠀⠀⠋⡁⠀⢀⡙⠚⠁⢸⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠛⢈⡠⠋⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⠀⢸⡗⠒⠒⠚⠒⠒⠒⢾⠁
⠀⠀⠀⢀⡉⠉⠉⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣄⣀⣸⠀⠀⠀⠀⠀⠀⣧⠀⢠⠏⠳⡄⠀⠼
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄
⠀⠀⢘⣙⣉⣋⣉⣋⣙⣋⡛⢳⠀⠀⠀⠀⣠⠤⠤⡄⠀⠀⠀⢠⡴⠲⣆⠀⠀⠀⠀⠀⠘⣶⢴⣦⣶⣴⣶⠟⠀⠀⠀⠀⠀⠀⢤⢸⡀⠀⠀⠀⢸⡡⡄
⠀⠀⢸⡏⡿⡏⡿⠿⢿⢹⢻⢸⠀⠀⠀⣼⢹⠀⠀⣷⠤⠤⠤⣼⠁⠀⡏⣷⡄⠀⠀⠀⠀⣷⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠉⣇
⠀⠀⢸⡯⡏⡇⡏⢹⢹⢸⢸⢸⠀⠀⠀⢹⣼⡄⠀⡟⠉⠉⠉⢹⡂⠀⣧⡏⠀⠀⠀⠀⠀⣿⠽⣏⢱⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡶⡀⠀⢸
⠀⠀⢸⣙⣓⣓⣓⣚⣒⣚⣚⣸⠀⠀⠀⠀⠈⠓⠚⠃⠀⠀⠀⠘⠓⠒⠋⠀⠀⠀⠀⠀⠀⢸⠖⠉⠉⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⣧⠀⢸
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠀⠘⠀⠉

⠀⠀⠀⢀⡔⠲⣖⣒⠒⡛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⣤⣤⣀⣀
⠀⠀⢀⡞⣄⢘⠀⠉⠀⢀⠀⣳⠀⠀⠀⠀⠀⠀⣠⠴⠚⢋⣉⣥⠽⠷⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠶⠶⠶⠶⠶⣶⡞
⠀⠀⠀⢳⠮⠯⣥⠤⠭⡥⢾⣁⡀⠀⠀⠀⠀⠘⣧⠶⠯⣭⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣳⡟⢻⢿⣁⣄⡀⠀⠀⠀⠀⠀⣼⣡⣄⣀⣀⣀⠀⠀⢸⡆
⠀⠀⠀⢸⠀⡇⠀⡇⠀⡇⢸⠓⢸⠄⠀⠀⠀⠀⣉⣲⣦⡤⣤⣍⣙⡇⠀⠀⠀⠀⠀⣿⢹⡏⣿⣘⡿⣾⣿⡍⣿⡆⠀⠀⠀⠀⡇⢠⣤⡀⣀⡀⣀⣀⢸⡇
⠀⠀⠀⢸⠀⡇⠀⡇⠀⡇⢸⢀⢸⠀⠀⠀⠀⠀⠻⣤⡤⠶⠶⢾⣯⣅⠀⠀⠀⠀⠀⠻⠷⠶⠋⠈⠁⠘⠷⠶⠟⠁⠀⠀⠀⠀⣧⠈⠀⠉⠉⠁⠉⠉⢸⡇
⠀⠀⣴⠉⣷⣇⣀⣇⣀⣣⡿⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣤⣭⣍⣉⣉⣉⡽
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣏⣹⠆⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⣀⣀⣀⣀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⢠⣠⣿⢷⠿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢨⢷⡀⠀⠀⠀⠀⠀⠀⠸⣟⡻⢡⢎⣕⣶⣖⡊⡳⡄⢸⢿⠀⠀⠀⠀⡤⠒⠋⣚⣗⡛⢛⠦⡀⠀⠀⠀⠀⠀⠀⡾⢯⠼⡇⢰⠯⡏⢷
⠀⠀⠀⠀⠀⠤⠿⣤⡿⠀⠀⠀⠀⠀⠀⠀⢸⣧⡇⣾⠙⣿⠍⣻⣿⣿⣾⢸⠀⠀⠀⢸⡁⠀⠀⠀⠉⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⡇⣟⡇⡇⢸⢸⣿⢸
⠀⠀⠀⠀⠀⠀⣞⠸⠦⠤⡄⠀⠀⠀⠀⠀⢸⣿⠳⡙⢙⢶⡋⡡⣪⠏⢹⣿⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣇⣿⣷⡇⢸⣼⣿⡼
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⠓⠚⠛⠋⠁⠀⠸⠿⠀⠀⠀⠀⠀⠙⠦⡤⠶⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃⠈⠻⠛⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⢀⣿⠀⠀⠀⠀⠀⠀⡠⠔⠒⢆⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠕⠋⠉⠝⢆⠀⠀⠀⠀⠀⡼⢋⠻⡄⠀⣽⠀⠀⠀⠀⢠⣯⣄⠀⠀⠀⣭⠆
⠀⠀⠀⢀⡴⠟⢦⣀⠀⠀⠀⠀⠘⢿⡀⠒⢒⡮⠗⠀⠀⠀⢸⠁⢻⠃⣷⠀⣿⠀⠀⠀⠀⠸⠿⣿⣿⣿⡿⠛
⠀⠀⠀⠙⠦⠤⠦⠋⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠀⠀⢸⡒⠚⠒⣿⣠⣿⣄⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⠉⠉⠛
⠀⢠⠖⡩⠍⡑⢢⠤⠤⠄⠀⠀⠀⠀⢀⡤⠔⡢⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣖⣒⣺⢷⣶⣶
⠀⢸⠐⠀⠁⠀⢸⠀⢸⠀⠀⠀⢠⠊⠡⢢⡟⠟⠈⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣤⠠⠄⠄⡏
⠀⢸⠀⠀⠀⠀⢈⠀⢸⠀⠀⠀⢸⡀⠢⡥⢐⣨⠖⠁⠀⠀⠀⠀⣀⣠⣧⡤⠤⣄⠀⠀⠀⠀⢸⣿⠀⠀⢡⡇
⠀⠈⠲⠤⠤⠔⠋⠉⠉⠀⠀⠀⠀⠘⠓⠛⠃⠀⠀⠀⠀⠀⠀⣿⢄⣈⣩⣥⡴⠟⠀⠀⠀⠀⠘⠿⠤⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⡶⣏⢿⣿⣷⡀⠀⠀⠀⠀⠀⢀⡠⠔⠓⠦⣄⠀⠀⠀⠀⠀⢀⡠⣴⡶⠀⠀⠀⠀⠀⠀⠀⠰⡛⢻⠂
⠀⠀⣸⡜⠂⠀⠹⣧⠀⠀⠀⢠⡒⠁⠀⠀⠀⠀⡈⢽⡂⠀⠀⢸⢳⠟⠛⠉⢉⣷⡀⠀⠀⠀⣀⠔⠋⠙⠢⡀
⠀⠀⠓⡤⡀⠀⠀⠣⡀⠀⠀⢸⡀⠀⠂⠠⠀⠈⣠⡾⠁⠀⣸⣍⡚⢀⣠⢞⡿⠛⠁⠀⠀⢸⠁⠀⠀⠀⠀⢱
⠀⠀⢀⡇⠈⡄⣀⣠⠗⠀⠀⠈⠙⠷⣤⣤⣤⠟⠁⠀⠀⠀⠓⠿⣿⡯⠟⠉⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠊⠉⠁
⠀⠀⠀⠀⡤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⢷⣴⡟⠅⠀⠀⠀⢾⠙⡦⠤⠤⢤⡤⣤⠚⡆⠀⠀⣴⣊⡉⠉⢉⣲⡄⠀⠀⠀⠀⡼⠉⠻⠯⠿⡆
⠀⠀⠀⣠⠞⠀⢆⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠘⡶⠃⠀⠀⢸⣌⡉⠉⣉⢼⠁⠀⠀⠀⠀⣧⠀⠀⠀⠀⣴
⠀⠀⠀⣏⠉⠀⢸⡆⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠚⠁⠀⠀⠀⠀⢧⣍⣉⣠⠎⠀⠀⠀⠀⠀⡯⢀⣀⣀⡀⢽
⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡓⢦⣄⠀⠀⠀⠀⢻⣀⣾⡟⠀⠀⠀⠴⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠻⡳⣄⠀⠀⠸⣿⡟⠀⠀⢀⣠⠤⢤⣀⢳⠉⠲⡀⣠⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠁⠙⣷⡀⢠⡟⠀⠀⠐⠁⠀⠒⠒⠦⣿⡀⠀⠋⡉⣠⢄⣠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠹⣿⣼⡇⠀⢠⡴⠾⠶⣄⠀⠀⠀⢙⣢⣤⠟⠁⡠⠙⠶⠤⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠈⠲⣄⣹⣿⡇⠀⠀⠀⠀⠀⠀⠹⡞⠻⣏⠁⠈⠑⣴⠁⠀⣠⣖⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣯⠒⠢⢄⡈⠙⠻⣷⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⠀⠀⠀⣿⠔⠋⢉⣩⡽⣦⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠻⣦⡀⠹⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠞⠀⠀⠊⢀⡠⠤⠜⢷⠱⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⢦⠠⡈⠙⠓⠛⠳⢤⣿⣄⠤⠴⠊⢉⠇⢸⠀⢀⠜⠁⠀⠀⡠⠊⠁⠀⠀⠀⣸⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢢⡀⠀⠀⡠⠋⠁⠀⠀⠀⢸⠀⢸⡇⢂⡴⢂⡤⠊⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⣤⢊⠀⣴⡄⠀⠀⠀⡏⠀⣾⣿⡿⠒⠉⣀⣀⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢠⡀⢂⠀⣰⠃⠿⢀⡝⠀⠀⠀⣸⢁⣼⠟⠋⣠⣖⡯⠄⣂⣀⣈⣹⡙⠳⠢⡀
⠀⠀⠀⠀⠀⢀⡀⠤⠤⠬⢧⣀⡈⠙⠘⢯⡏⠀⣰⡎⠔⠀⠀⢠⣧⡾⠁⣠⡼⠟⠛⠉⠉⠉⠉⠉⠉⢳⠀⠀⢡
⠀⠀⡠⠒⠉⠀⠀⠀⠀⠶⠦⠀⠉⠑⢶⣽⡇⢸⣿⠎⠀⠀⣠⠟⠛⠋⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢸
⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⢦⣽⣷⣿⣟⣠⣾⣾⣷⡅⠀⠒⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠻
⠓⠤⣤⣤⣤⣤⣀⠀⠀⠀⠀⢠⡤⠀⢀⣻⣿⣿⣟⡀⠀⠙⣏⠁⠀⠀⠀⣀⣠⣒⡨⠭⠷⠮⣆
⠀⠀⠀⠀⠀⠀⠉⠙⣶⢤⣤⣤⣤⣶⡏⡜⠁⠙⢟⢯⠀⠀⠘⣯⣥⣶⣞⣻⡷⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡴⠛⢹⡀⠀⠀⠀⠈⠀⢣⠀⠀⢻⣿⡇⠒⠢⠌⠙⠢⢍⠳⡄
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠊⠀⠀⠨⢧⡀⠀⠀⠀⠀⠀⠀⠀⣸⠈⠙⠓⠦⢤⣀⣀⠀⠑⢼⢦⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⠴⠚⠉⠙⢦⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠈⠣⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠛⠶⢼⡀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣄⡀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢃⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢃⣾⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⣶⣶⣾⣿⣿⣿⣼⣿⣿⣿⢃⣴⣶⣶⣶⣶⣶⣾⡟
⠀⠀⠀⠀⣴⣎⠳⡄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⢃⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⡴⢋⣶⡆
⠀⠀⠀⠀⠸⣿⣧⠹⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡼⢡⣿⡿
⠀⠀⠀⠀⠀⢹⡇⣶⣦⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣠⣶⡆⣿⠁
⠀⠀⠀⠀⠀⠀⢿⣌⢻⣿⣦⣀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢀⣠⣾⣿⢋⣼⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣧⡙⢿⣿⣿⣷⣶⣤⣀⠀⠀⠈⠙⠿⠟⠋⠀⠀⢀⣠⣴⣶⣿⣿⣿⠟⣡⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣎⠻⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⡿⢋⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣬⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠁⠀⠀⠙⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣤⣤⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⡿⠿⢟⣛⣛⣛⣛⣛⣓⣚⠿⢿⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣛⡭⠶⠚⠋⠉⠉⠀⠀⠀⠀⠉⠉⠙⠓⠶⢭⣛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡿⢛⡵⠚⠉⠀⠀⣀⠀⢄⢐⡋⠀⠀⣴⡆⡴⠆⢀⡀⠀⠉⠓⢮⡻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢋⡴⠋⠀⠀⢐⣮⠆⠛⠑⠈⠀⠉⠀⠀⠀⠀⠙⠀⠧⢠⢂⢄⡀⠀⠙⢦⡹⣿⡀
⠀⠀⠀⠀⠀⢀⣾⡟⣱⠋⠀⠀⡆⢙⠚⠀⠀⠀⣀⣠⠤⠶⠶⠶⠦⠤⣄⡀⠀⠀⠉⠃⢚⣀⡀⠀⠙⣎⢿⣷⡆
⠀⠀⠀⠀⠀⢸⡟⡼⠃⠀⠐⣁⡌⠁⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠛⡀⠆⠀⠘⢧⢻⡃
⠀⠀⠀⢀⡴⡿⣸⠃⠀⠐⢒⠂⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠰⠚⠂⠀⠘⣇⢿⣦⡀
⠀⠀⠀⠘⣧⢣⡏⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢹⡸⣿⠃
⠀⠀⠀⢠⣿⢸⠃⠀⠀⣶⡄⠀⣠⡶⢦⠀⣠⡤⣦⠀⣦⠀⣰⢀⣤⠴⣄⠀⢀⣶⡀⠀⠦⣤⠦⢰⡶⠶⠆⠘⡇⣿⡄
⠀⠀⠀⣿⣿⢸⠀⠀⣸⣏⣷⠀⣉⠀⠀⠀⣿⠀⠀⠀⣧⠀⢹⠈⡷⢶⡟⠀⣼⣯⣧⠀⠀⣿⠀⢸⠷⠆⠀⠀⡇⣿⣿
⠀⠀⠀⠈⡿⢸⡄⠀⠋⠉⠙⠇⠙⠷⠟⠀⠙⠣⠟⠀⠙⠶⠛⠘⠇⠈⠷⠰⠏⠉⠙⠆⠀⠟⠀⠸⠷⠶⠆⢠⡇⣿⠃
⠀⠀⠀⢠⣯⡌⣇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣸⢱⣿⡄
⠀⠀⠀⠈⠻⢷⢹⡄⠀⠠⡤⠇⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣤⣷⡄⠀⢠⣟⣾⠿⠃
⠀⠀⠀⠀⠀⢰⣧⢻⡄⠀⠰⠋⣤⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠖⠀⠀⠀⠘⠙⠆⠀⢠⣟⣿⣏
⠀⠀⠀⠀⠀⠸⢿⣷⡹⣄⠀⠈⠉⡵⣠⣀⠀⠀⠉⠙⠒⠶⠶⠶⠶⠒⠋⠉⠀⠀⠀⡀⣍⠼⠃⠀⣰⢟⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣌⠳⣄⠀⠈⠑⠡⠃⢲⠀⣄⠀⠀⠀⠀⣀⠀⡀⢄⣀⠰⡻⠕⠀⠀⣠⠞⣡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⣷⣮⡳⢤⣀⠀⠈⠉⠰⠞⠸⠟⠀⠀⣸⠇⠵⠀⠉⠀⠀⣀⡤⢞⣥⠾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣭⣓⠶⢤⣄⣀⣀⠀⠀⠀⠀⣀⣀⣠⡤⠶⣛⣭⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⣷⣶⣮⣭⣭⣭⣭⣭⣭⣭⣶⣾⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠛⠛⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣤⣴⠶⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠉⠉⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⣆⠀⠀⠀⠈⡄⠀⡀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⠒⠒⠾⢶⣜⣿⡀⠀⠀⠀⡇⢸⡇⠠⠀⢀⣴⠟⢀⣀⠀⣰⣾⣷
⠀⠀⠀⠀⠀⠀⢀⣼⠃⣀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡿⠀⠀⢠⠁⡼⢀⠆⣠⣿⣿⡶⢿⣯⣾⣿⣿⣛⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣧⠀⣀⣞⡰⢁⢾⣾⣿⢟⣫⣾⣿⣿⣿⣟⣿⣭⣍
⠀⠀⠀⠀⠀⠀⢿⡿⠿⠷⠖⠀⠀⠀⠒⢲⢶⣶⣶⡸⣿⢻⠟⢁⣾⡿⣋⣴⠟⣉⣽⣿⣿⣿⣿⣿⣿⣅⡀⠀⡀⠀⠠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⣷⣿⣿⢟⡯⠑⠈⣀⣶⣿⣿⣿⣿⣿⣿⣿⠃⠀⣀⣠⣴⠿⣿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠈⢹⣷⠞⢿⣿⣿⣿⣿⣿⣿⢿⣋⣙⡓⠊⠉⠀⠀⠀⠁⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠋⣀⠀⠀⠀⢰⠀⣸⡿⠇⠈⣿⣿⣿⣶⣶⣾⡻⠟⠛⠻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠃⢸⣷⡀⠀⠀⡆⠀⣿⡇⠀⠀⠘⢿⣿⡿⠏⠒⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠘⠄⣢⠀⠀⠀⣰⡿⠁⠀⠀⢮⠛⣻⠀⣰⠁⣸⣿⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⣰⡟⠀⠀⠀⠀⡈⠀⢻⣷⡏⣰⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣰⠟⠀⠀⣠⣿⠏⠀⠀⠸⡿⠁⣫⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢁⠔⡰⠋⠀⠀⠀⠀⠀⠹⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣉⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠊⠻⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣙⠊⠙⠯⠻⢦⡘⣿⣧⡀⣾⣿⠀⠀⢀⣴⣶⣶⣶⠶⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡟⠛⠿⠛⠒⠀⠀⠀⠀⠈⠘⣿⣇⣿⣟⣠⡴⠋⠂⣼⠛⠁⣾⡿⠿⠛⣻⡿⠟⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⠙⠙⢦⠀⠱⣄⠀⠀⠀⠀⠐⣄⢸⣿⣿⣿⠟⠁⠀⠘⠁⣠⡾⠏⠀⢀⣼⠋⠀⠙⠿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠑⢄⠘⢆⠀⢀⡄⠀⣼⣿⡟⠐⠁⠀⠀⠀⣰⠾⠋⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠓⡈⢧⡈⣿⣷⢻⡿⣀⠀⡀⠀⠀⡼⠃⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣇⠀⣴⢿⣤⢜⠁⢀⠀⠀⠀⠀⠜⠁⡿⠀⠀⠀⠀⠀⣀⣤⠴⠖⢚⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠢⡀⠀⠀⠀⠙⠌⢻⣿⣶⢴⣿⣵⣿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣇⣠⣤⣤⠶⢛⡩⡴⠔⠊⠉⠉⠩⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢾⣦⡀⣶⣎⡓⢮⡿⣻⣮⣿⣿⡗⢀⠂⠀⠀⠀⢀⠄⢀⡾⣫⠝⢋⣠⠾⠁⠀⠀⠀⠀⣀⣤⣶⣷⣿⣆
⠀⠀⠀⠀⠀⣀⣠⣤⣄⣠⣴⣶⣾⣢⡛⠛⢿⣿⣏⣸⣷⣘⣷⣿⣿⣿⣿⣥⠂⣀⠀⢀⣠⣱⡴⠫⠚⢁⣴⠟⠁⠀⠀⠀⢀⡴⢞⡿⠋⠁⠙⣯⣿
⠀⠀⢀⣴⡏⠀⠉⠁⠈⠉⠉⠉⠉⠁⠀⠀⠈⠀⠈⠛⣿⣿⣎⡿⢿⢿⣿⣷⣼⣷⣾⡿⠟⠉⡡⠀⣰⠞⢁⠀⠀⠀⠀⠀⠁⠠⡎⠀⠀⠀⢀⣿⠗
⠀⢠⣿⣅⡤⠴⠒⠂⠀⠐⠒⠐⠲⠤⠤⣀⣀⢄⣤⣄⣀⡙⣿⣿⣾⡏⣿⣿⡿⣫⢿⡾⠦⠲⠤⢮⣥⣶⣵⠎⡠⢂⠀⢀⡔⣰⠃
⠀⣼⡽⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠛⠷⢾⣽⣻⢿⣿⣿⣿⣿⣿⣱⣿⡿⠞⣉⣁⠀⡈⠛⠿⣶⣾⣗⡥⠞⣉⡼⠃
⣰⠿⣦⣴⣶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣻⣿⣿⣟⣿⣿⣿⡛⠏⠉⠁⠀⠙⠲⣼⣯⠥⠖⠚⠋
⣿⡴⠃⠀⠉⠚⢾⣶⡖⠒⡂⢀⡀⠀⠀⢀⡀⢤⣴⣿⣿⣿⡷⠋⣹⢻⣿⣿⣿⠛⠿⣿⣿⣦⡀⠀⠀⠀⠈⣉⠂
⠋⠀⠀⠀⠀⠀⠀⠉⢋⣶⣶⡶⣿⣛⣫⣽⣿⣿⣿⣿⣿⣿⠃⠀⠁⠘⠁⠀⠉⠀⠀⠀⠉⠳⢝⢦⡀⠀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⡐⠋⠉⠀⠀⠚⣋⠉⠋⠉⢁⡴⢻⡿⣻⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠑⣟⣄⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠐⠉⣐⡵⣿⠶⢤⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⡄⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠊⠀⢊⠡⠄⠁⠀⠀⠈⠙⠊⠳⠦⡀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠐⡈
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠇⠀⠀⠀⠠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⢸⡰⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⡆⢰⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠐⢄⠘⣌⣧⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠄⠈⣧⠘⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣦⣤⡵⠾⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠚⢉⠖⠒⠙⠻⠿⠛⠉⠩⠍⠙⢷⣿⡿⠃⠀⢲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⣀⣀⣤⣀⠀⠘⠋⠀⡔⠀⠠⠤⣶⠃⣀⣤⠄⠉⣿⠖⣠⣤⡞⠉⠋⣑⣀⡀
⠀⠀⠀⠀⠀⠀⠔⠚⢁⣤⠋⠁⣤⡄⠙⢧⣤⣀⡐⠛⠉⣀⡴⢰⣦⡀⠀⢀⡼⠿⠛⠉⣁⣀⣀⣠⣼⠛⠿⡓⢢
⠀⠀⠀⠀⢀⣼⢫⣞⠉⣵⠀⡴⠋⠀⣀⡤⢞⡉⠉⠛⠻⠁⢠⡎⠉⠻⡄⡾⠀⣸⠞⠛⢉⡉⣯⡁⠁⣠⣤⣼⣿⣷⡄
⠀⠀⠀⠀⡸⢁⡟⠁⡠⠋⢠⠇⠀⡞⢡⠶⠿⠄⠀⠀⠦⡀⣿⠀⠙⠀⡞⠁⠀⣿⠀⣤⡌⠙⢹⡇⠀⣿⠀⡏⠉⣦⡘⠆
⠀⠀⠀⠀⣷⠁⠻⣀⡀⢀⠊⠀⢘⣇⠀⠀⠐⠲⣾⠀⢠⣿⠃⠀⠀⠞⠁⣄⠀⢿⣶⣯⣤⣤⣾⣷⣠⣧⢀⡿⠀⠈⢻⣾⡆
⠀⠀⠀⠀⠈⠀⠈⢶⠖⠤⠦⠤⠄⠉⠛⠷⠦⠀⢸⣆⢀⡋⣀⣷⣄⣠⡴⠟⠷⣾⣏⠀⠉⠉⠻⠟⠉⣡⠞⢀⣤⣴⣿⡯⠳
⠀⠀⠀⠀⠀⠀⠁⠒⠠⠄⠀⠠⠶⠗⣒⠖⠠⣴⡾⠛⠉⠀⠁⠊⠉⠉⠀⠀⠀⢠⡿⠛⠻⣷⣶⡴⠾⣿⠀⠘⡉⢳⣾⣷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠨⠿⠞⠉⣀⣠⡴⠚⠉⠒⠲⢤⣴⠧⣤⣼⣷⡶⠚⠿⠿⠁⠀⣽⣿⢟⣃⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡞⠉⠁⠀⠀⢀⠀⢀⠀⠙⣦⡀⠈⠁⣀⣀⡰⠶⠞⠻⠿⣿⡛⠉⣩⣿⣿⣋⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠷⠀⠀⠠⣾⠛⡀⠀⠳⠄⠈⢻⡀⢀⣽⢟⠳⣴⡶⠏⣠⡉⢣⡀⠈⠉⣰⡠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠀⠉⠂⢈⣉⣣⣬⣍⣉⢁⡀⠓⢇⠀⠀⠈⣳⡄⠉⠛⠓⠉⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣼⣟⣛⣿⣿⣿⠿⢿⣷⣿⠷⣶⠾⣧⣶⣶⣧⣾⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢿⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿⡿⠿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠉⠙⠻⣿⣶⣲⣮⣵⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁




⣶⢶⢶⢷⣶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶

⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣝⣄⡠⣤⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⢸⣽⠀⠀⠀⠀⠀⣠⠔⠊⠉⠈⠳⢄
⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⢰⡉⠀⠀⠓⠵⣄⠀⠀⠀⠀⢀⡞⡁⢈⢳⠀⢸⢸⠀⠀⠀⠀⠀⣿⣶⣄⣀⣀⣀⣴
⠀⠀⣠⠞⠛⠳⣄⠀⠀⠀⠀⠀⠘⠺⣄⠛⢁⡵⠚⠁⠀⠀⠀⢸⠀⡑⠀⢸⡄⢸⡎⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⠟⠉
⠀⠀⠙⠶⠤⠖⠊⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢸⡛⠒⠒⢺⣥⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠉⡙⠛⠛⠛
⣠⠖⢒⡒⠒⢤⣀⡠⡄⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⠶⢦⠤⢤
⡧⠀⠀⠈⠁⢸⠀⠀⡇⠀⠀⠀⢀⠔⢊⣥⣢⢟⠎⢱⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⡉⢡⠈⡅⡏
⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢟⠠⡀⠘⢉⠄⢂⡽⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⢀⡇
⢧⡀⠀⠀⢀⡴⠤⠔⠃⠀⠀⠀⠸⢦⣄⣯⡴⠚⠁⠀⠀⠀⠀⠀⣤⠖⠪⠿⠿⠀⢙⡇⠀⠀⠀⠀⠘⡇⠀⠂⠈⢸⠇
⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠵⢤⠶⠖⠛⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠰⡻⣎⢻⠿⢿⣦⠀⠀⠀⠀⠀⠀⡠⠔⠚⠁⠑⠲⣀⡀⠀⠀⠀⠀⣠⢴⣺⠽⣃⣀⠀⠀⠀⠀⠀⠀⠀⢫⣉⡼
⢀⣵⡈⠀⠀⠸⣿⠄⠀⠀⠀⡶⢉⠀⠀⠀⠀⠀⡀⠔⢿⠀⠀⠀⢸⠙⡟⠋⠉⠀⣨⡷⡀⠀⠀⠀⢀⠔⠋⠉⠉⠢⡀
⠈⠲⡄⠀⠀⠀⠹⡄⠀⠀⠀⣯⡀⠈⠐⠠⠄⠈⢀⣴⠟⠀⠀⣸⣍⡙⠁⣠⡴⣻⠿⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱
⠀⢸⠀⠙⡀⠀⣀⡼⠀⠀⠀⠉⠻⢶⣤⢠⡠⣲⠞⠁⠀⠀⠀⠓⢿⣟⣟⡵⠚⠁⠀⠀⠀⠀⠀⠰⡅⠀⠀⠀⠀⠀⢠
⠀⠈⠛⠓⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠋
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠐⣇⣀⣼⡏⠀⠀⠀⠀⡴⢢⣀⡀⠀⠀⠀⠀⢀⢤⠀⠀⠀⢀⡤⠤⠤⠤⠤⣀⠀⠀⠀⠀⠀⢈⡟⢻⣧⡶⣦⡀
⠀⠀⠀⣹⠛⡇⠁⠀⠀⠀⠀⢳⣴⠏⠉⠉⠉⠋⠉⣧⢸⠁⠀⠀⢻⠒⠶⠶⠶⠶⣾⠃⠀⠀⠀⠀⡸⠀⠀⠀⠉⠉⢇
⠀⢀⡎⠁⠀⢱⡄⠀⠀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠈⡗⠤⠤⠤⠚⡏⠀⠀⠀⠀⠀⡗⠉⠉⠉⠉⠉⣿
⠀⢸⣇⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠞⠁⠀⠀⠀⠀⠀⢇⠂⠀⠀⠀⠐⣺

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢰⣶⣶⣶⠀⠀⠀⠀⣀⣀⠀⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠐⣿⣿⠂⠀⠀⠀⠀⠀⠸⣿⡿⠀⢸⣿⣿⣿⠀⠀⠀⠐⣿⣿⠂⣿⣿⣿⣿⠀⠀⢠⣤⣄
⠀⠀⢠⡄⠰⣦⣀⣾⣷⡀⠀⠀⢠⡄⠀⠀⠀⣰⣷⡆⠀⠀⠀⣠⣶⣶⣶⣦⣄⣠⡦⠀⠀⠀⠀⢰⣶⣶⣦⣤⣤⡀⠀⠀⠀⠘⠿⣟⡀
⠀⣀⣼⣄⣀⣈⣉⢻⣿⡇⠀⢀⣸⣇⣤⣴⣾⣻⣿⣿⠀⠀⠀⣿⣿⣿⣿⡇⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⠉⠉⠁⠀⠀⠀⠀⢀⣿⣿⡀
⠀⢸⣿⣿⣿⣿⠇⢸⣿⠃⠀⠀⣿⣿⣿⣿⡿⢸⣿⠏⠀⠀⠀⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣠⣴⣟⣿⣿⡇⠄⢠
⠀⠠⡄⠈⠁⠀⠀⢸⣿⠀⠀⠀⣤⠀⡍⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠀⠀⢸
⠀⠀⠑⠊⠀⠀⠀⢸⣿⠀⠀⠀⠈⠒⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⡆⡀⠀⠀⠓⣼
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠙⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠃⠀⠀⠀⠀⠀⠀⠉⠀⠃⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠾⠿⠷⠂⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⢀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣛⣋⡉⠉⠳⣄⠀⠀⠀⠀⣰⣶⣿⣯⣤⡶⣀
⠀⠀⠀⠹⣦⡀⢠⣤⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⡏⠉⠉⠁⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⢸⣿⡏⠉⢉⢉⣿⣦⡀⢿⠃⠀⠀⠀⣿⣿⡀⠀⠀⣠⠾
⠀⠀⠀⠀⠘⢷⣝⢿⠟⣙⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠻⡿⠃⠀⠀⠀⠀⠀⠸⣿⣇⠀⠠⡚⣿⣿⠇⢸⠀⠀⠀⠀⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠈⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠿⠶⠶⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢸⣿⠀⠰⡅⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡟⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⡏⣿⠀⠀⠀⠀⠀⢸⣿⠀⢐⡅⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡇⠀⠘⠿⠂
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⣀⠀⠀⠀⢸⣿⣿⡇⠿⠀⠀⠀⠀⠀⠻⠇⠀⠐⠶⠶⠶⠶⠾⠀⠀⠀⠀⠘⠿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⢀⣴⣶⣄⣠⣶⣶⣀⠀⠀⠀⠀⠀⣰⣶⣦⣠⣶⢶⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣙⣟⣽⣷⣛⣏⣹⣦⠁⠀⠀⢀⣈⣻⣋⣻⣟⣞⣙⣧
⠀⠀⠀⠀⠀⠀⣸⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠻⠿⠿⠏⠫⠄⠀⢀⣿⣿⣿⣿⠿⠿⠿⠏⠋⠄
⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢳⣦⣔⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣉⣡⣀⣀⡀⠀⢸⣿⣿⣿⣿⣀⣈⣁⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣋⣉⠉⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⠿⠿⠿⠿⠿⠇⠀⠀⠁⣿⣿⣿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⢀⣀⣙⡛⠛⢻⣷⠀⠀⠀⠀⠀⠀⢀⣙⣛⡛⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⠀⠈⠻⠇⡀⠀⠀⠀⠀⣼⣿⣿⣧⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠈⠉⣄⡀⠐⢶⠒⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠐⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠴⠆⠁⠉⠢⣴⠀⠀⠀⠁⠀⠀⠀⢀⠾⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⡤⠤⠬⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢀⡠⠴⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⡄⠀⠀⠀⠀⠀⠀⢸⢀⡤⠀⠀⠀⠀⠐⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⠀⠀⡼⠉⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠀⠚⠛⠂⠀⡜⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⣼⢢⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⠁⠀⡠⠀⠀⠀⠀⠀⠠⡄⠀⠀⣀⡀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣴⣁⠀⠀⠀⠀⠀⠀⠯⠝⠯⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡶⠋⠀⠁⠚⠦⣤⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠠⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠋⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠐⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢐⠃⠔⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢶⠡⠀⢀⣠⡇⠀⠀⢀⣀⡀⢀⠀⠀⢀⣀⣀⢄⣀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠳⠌⣾⠟⣠⣶⣿⣿⣿⣿⡷⠈⠠⢨⣿⣿⣿⣿⠿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⡀⣴⣄⡏⠀⣿⣿⠿⢿⠇⠀⢠⠀⠀⢨⣿⣿⣏⠍⠄⠈⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣇⠨⣚⡁⠀⠹⣽⣉⠠⠀⠀⠊⢀⣰⣼⣮⠙⠿⠀⠶⢋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠘⠓⢦⠓⠄⠀⣀⣤⣤⡄⠀⠈⢸⠟⣿⣿⣡⣶⣾⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣷⣷⣝⣶⠾⢯⡉⠀⠇⠀⠰⣿⣤⡿⢿⢻⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⣿⣇⠘⣿⠄⠀⠀⠀⠑⢳⡗⠋⠉⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢿⣦⡙⠀⠀⠀⠀⠠⠀⠀⠆⠀⠛⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣯⠁⠨⣁⠂⡀⢂⢈⠀⢀⢀⠙⣻⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠻⠀⠀⠀⠈⠉⠉⠸⠘⢹⢙⡘⢟⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢄⠁⠀⠀⠠⠀⠀⠀⠀⠚⢁⣰⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⠤⠤⠴⠶⠟⠋


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

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠐⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣧
⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⣿⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣶⣶⡄
⠀⠀⠀⠀⠀⠉⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⠇
⠀⠉⠉⠓⠂⠤⢤⡈⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⣶⠀⠀⠀⠀⢠⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀
⠀⠀⢸⡟⠀⠀⠀⣠⣟⡼⠃⠀⠀⠀⠀⠀⠠⠐⣀⠠⡔⠁⠀⡰⢿⠀⠀⠀⠀⠀⠀⠀⣾⣛⣿⣿⣿⠟⠋⠁
⠀⠀⣾⠇⠀⠀⢀⣯⠟⠁⠀⠀⠀⠀⠀⠀⠁⠈⢁⠾⠀⢀⡜⠁⢸⠀⠀⠀⠀⠀⠀⠀⠉⠋⢻⣏
⠀⠀⣿⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠉⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⣾⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⡄⠀⠀⠀⣀
⠀⠀⠀⠀⢰⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡼⠁⠀⣠⡾⠁
⠀⠀⠀⠀⣼⣿⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣼⠃⢀⢼⠟
⠀⠀⠀⠐⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⣴⠃⠀⠘⠋
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣟⣿⣿