Chain Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⢀⣀⣀⣀⣠⣖⡭⠛⠚⠛⠯⣝⣢⣀⣀⣀⣀⣠⣚⡽⠒⠛⠛⠛⢯⣝⣦⡀⣀⢀⣴⡮⠛⠛⠛⠻⢭⣲⣄⠀⠀⢠⣞⠟⠛⠉⠉⢛⣽⣦⣀⣀⣀⡀
⠀⠐⠻⠿⠿⠿⠿⣿⢧⡀⠀⠀⢀⡸⢿⡯⠿⠿⠿⢿⡿⣇⣀⡀⠀⣀⡸⠠⣠⠘⠀⠻⡿⢇⡀⠀⠀⠀⣻⢿⡿⠿⠽⢿⡿⡆⠀⠀⢀⣘⠿⡿⠻⠿⠿⠯⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⠿⠽⠽⠖⠋⠀⠀⠀⠀⠀⠙⠒⠚⠻⠿⠿⠒⠛⠁⠀⠀⠀⠈⠓⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠙⠛⠛⠛⠛⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⢒⡭⣑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⠭⣐⢤⡀
⠀⠀⠀⠀⠀⠸⡘⣄⢈⣧⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⣄⢈⣧⠱
⠀⠀⠀⠀⠀⠀⠙⠮⣙⣢⡙⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣙⣢⡙⢄⣀⣀⠰⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⡑⡦⣝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡑⡦⠕⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡹⣄⡜⢆⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡹⣄⡔⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠭⠦⡑⢤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠭⠦⡑⢤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢢⡽⠲⡝⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢢⡽⠲⡝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡳⢄⡸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡳⢄⡸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⣠⣤⣀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠠⣿⠁⠀⠙⢷⣄⠈⠳⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠙⢧⡀⠀⣀⣹⣦⣄⣈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠈⠙⢿⣏⣉⣀⣉⡙⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⠻⠟⠉⣻⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣦⣤⣤⡾⢻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡏⠀⠀⠀⣾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣧⣤⣤⣠⡿⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡏⠈⠉⠉⠻⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠈⠻⠾⠿⣦⡀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠳⣶⡶⡶⠿⣧⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠘⣧⠀⠀⢹⣇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢹⣆⣀⣀⣿⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠟⠉⠉⠉⣹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣠⡾⠿⠟⠋⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣰⡿⣦⣤⣤⣴⠾⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣸⠇⢠⡿⠀⠀⠀⢀⡿⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣀⢀⠉⠛⠷⣾⡇⠀⠀⠀⣾⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣶⡛⠉⠉⣳⣶⣄⠀⠙⢷⣄⣼⠇⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠙⢿⡛⠛⢿⣇⠘⣿⣦⣤⣼⠏⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠹⣦⣌⠙⢿⣆⢹⣧⣄⣀⣀⣀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⣟⠙⠷⣤⣿⡄⣿⣍⣉⣿⠟⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣿⠀⠀⠈⠙⣧⢸⣏⣉⣁⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣻⣦⣤⣤⣀⣻⡄⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠟⠋⢀⣀⡈⠉⢻⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⡾⠛⠻⠿⠿⠛⢁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⠞⠉⠻⢶⣤⣤⡶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣴⣿⣥⡶⢶⣴⣟⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣯⣤⣤⣤⣄⠙⢷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⢀⢀⣠⣿⣦⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡟⠛⠁⢨⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢰⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡿⠛⠛⠛⠿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⢿⣦⣤⣄⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⣀⠉⠉⢹⣧⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⡟⠛⠛⠛⣿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣇⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣆⣀⣴⠟⠀⣸⡇
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠉⠉⠁⣀⣴⠟
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠶⠶⠞⠛⠁
⠀⠀⠇
⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠴⢦⣤⣤⣄⡀⠀⠀⢀⣀⡀⠀⠀⣀⣤⣤⣴⡶⠶⠶⢶⣤⣤⣀
⠀⠀⠀⣠⣤⣶⣶⣾⣿⠻⢷⣶⣾⣿⣭⣶⣶⣶⣿⣷⣶⣶⣾⣿⣿⣿⣿⠛⠿⠿⠿⠿⣿⣿⣿⣿⠿⢿⣿⣿⣶⣍⢳⡄
⠀⠀⣼⣟⣭⣷⣶⠶⢿⣶⠶⣶⢶⣦⣍⣿⣷⠀⠀⠀⣿⢋⣽⣶⣟⣿⣛⣿⣻⣿⣿⣶⡎⣿⣷⠀⠀⠀⠀⠀⢹⣿⣿⣷
⠀⠀⠘⢿⣿⣻⣿⣿⣿⣿⣿⣿⣜⢿⣿⣾⣯⣤⣶⣤⣿⣿⡿⢿⣽⣿⣿⣿⣿⣿⣟⢿⣿⣿⣷⣶⣶⣶⣶⣿⡿⣿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠙⠷⠾⣯⣭⣽⣯⣿⡿⠿⠖⠛⠉⠉⠉⠉⠉⠉⠙⠛⠿⠿⢿⣿⣯⣿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣒⣙⠇⠀⠀⠀⠀⠀⠀⠀⠀⢦⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⡦⣄⠀⠀⠀⢀⣠⢴⣖⡀
⠀⠀⠀⠈⢻⣿⣶⣤⣤⣶⡿⢷⣦⣄⣀⣀⣤⣾⣃⠙⢿⣾⣤⣶⣦⣄⠀⠀⣴⡶⠄⠀⠰⣶⣦⣤⣠⣄⠀⠀⢺⣝⣉⡉⠀⣤⣛⣻⣷⣶⣾⡿⠚⠉⠈⢻⡆
⠀⠀⠀⠀⠀⠈⠙⠓⢯⣭⡷⠰⣴⣶⣿⣟⡟⢿⣾⡷⣩⡿⠻⠯⣭⣛⡳⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⢷⣾⣿⣿⣭⣼⣥⣤⣽⣿⡿⣿⣿⢿⣿⡷⢄⣤⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⣈⣉⣿⡟⠋⠐⠶⠞⠋⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⣿⠃⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠘⠷⣵⡾⠽⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⣤⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣶⡿⠃⠀⠘⠉⠁

⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶
⠉⠉⣹⣿⣿⡿⠿⠿⠿⠿⣯⣿⣯⠉⠉⠉⠉⢩⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⡍⠉⠉⠉⠉⣽⣿⣿⠿⠿⠿⠿⢿⣽⣿⣏⠉
⠀⠀⢿⡿⣿⣇⣀⣀⣀⣘⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⣿⣀⣀⣀⣀⣿⣿⣿⡇⠀⠀⠀⠘⣿⢿⣿⣇⣀⣀⣀⣸⣿⣿⡿
⣶⣖⣮⣾⠟⠿⠻⠿⠿⠿⠛⢿⣷⣶⣶⣶⣶⣦⣵⠿⠻⠿⠿⠿⠿⠛⠿⣿⣶⣶⣶⣶⣶⣵⡿⠛⠿⠿⠿⠿⠟⠻⣿⣷⣶
⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣋⣉⠛⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⠛⠛⢷⣤⡴⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⡟⠀⠀⠀⢠⡿⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠊⠉⣦⣠⡴⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⣾⣋⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣋⣉⠝⠁⠀⣠⠞⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣵⠞⠋⢻⣤⣠⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣾⠁⠀⠀⢀⣽⠃⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠔⠉⠙⣦⡴⠟⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠔⠁⠀⣠⣾⣡⣴⠞⠋
⠀⠀⠀⠀⠀⠀⢀⡤⢚⣩⣤⠝⠁⠀⣠⠞⠉⠉⠉
⠀⠀⠀⠀⠀⡴⢋⡶⠋⠉⢿⣄⣤⢾⡇
⠀⠀⠀⠀⣼⢡⣏⣀⠀⠀⣠⡟⢁⡾⠁
⠀⠀⠀⠀⡿⠊⠀⢀⣧⠞⢋⣴⠟
⠀⠀⡠⠂⠀⢀⣴⣯⣤⡶⠋⠁
⣠⠊⠀⣀⡴⠋⠀⠉
⠻⣶⡞⠋
⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠾⠛⠛⠛⠛⠷⠶⣤⣴⡶⠾⠛⠛⠛⠛⠳⠶⣦⣤⠶⠾⠛⠛⠛⠛⠷⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣀⣤⣤⣴⣦⣤⣤⣄⠀⠙⢷⣤⣤⣤⣶⣤⣤⡾⠋⠀⣀⣤⣤⣴⣶⣤⣤⣄⡀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠉⠀⠀⠀⠀⢠⡿⠉⢿⡄⠈⣿⠀⠀⠀⠀⣿⠁⢠⡾⠉⢿⡀⠀⠀⠀⠀⠉⢻⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠘⢷⣄⡀⠀⠀⠀⠈⣿⠀⠘⣧⣄⣿⠀⠀⠀⠀⣿⣀⣼⠃⠀⣾⠁⠀⠀⠀⠀⣠⡾⠃⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠉⠛⠛⠛⠛⠛⠛⢷⣄⠀⠉⠛⠛⠛⠛⠛⠛⠉⠀⣠⡾⠛⠛⠛⠛⠛⠛⠉⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⢶⣤⣤⡴⠶⠾⠛⠛⠷⠶⢶⣤⣤⣴⠶⠾⠛⠛⠷⠶⢶⣤⣤⡴⠶⠾⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢹⣇⠀⢀⡿⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠉⠀⠈⠁⠀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠶⠶⠶⢦⣤⣤⡶⠶⣦⠀⠀⣴⠶⠶⣤⣤⡴⠶⠶⠶⠶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⣀⣠⣤⣤⣤⣤⣄⣀⠈⠙⣷⣤⡿⠀⠀⢿⣤⣴⠛⠁⡀⢠⢤⣤⠤⢤⣤⣄⡈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⡾⠋⠁⠀⠀⠀⢠⡟⠙⣷⡀⠘⣿⠀⠀⠀⠀⣾⠃⢀⣾⠃⢶⠆⠀⠀⠀⠉⠙⢿⠆⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣧⣀⠀⠀⠀⠀⠘⣧⠀⠸⣧⣀⣿⠀⠀⠀⠀⢿⣀⣼⠏⠀⣿⠃⠀⠀⠀⠀⣀⣾⠃⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠈⠙⠛⠛⠿⠿⠛⠛⢧⣄⠈⠙⠛⠛⠿⠿⠛⠛⠋⠁⣀⣼⠛⠛⠿⠿⠛⠛⠋⠁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢶⣤⣤⣤⣤⡴⠾⠛⠛⠶⢶⣤⣤⣤⣤⡴⠶⠟⠛⠷⢶⣤⣤⣤⣤⡴⠶⠟⠋

⠀⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⠀⢄⠀⡹⣖⠘⠲⢄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⠂⠐⠲⣶⣀⣭⣴⡋⢁⣀⡈⠉⠓⠢⣄⡀
⠀⠀⠈⠳⣦⣍⣛⠷⠒⠂⣄⡈⠲⣤⠒⠶⣾⠉⡠⣤⡖⠀⣿⡏⢷⣸⣟⠠⠽⠖⢣⣿⠋⠉⢰⠦⡈⢻⣶⠠⢄⡀
⠀⠀⠀⠀⠀⠉⠻⣿⡷⠲⢡⣿⠢⣹⣿⣀⡿⠗⠃⣚⣗⣲⠽⢮⡤⠤⠤⠤⠒⠒⠫⢅⢭⣂⣧⡄⢦⣼⡿⠒⠒⣦⠕⠒⠐⠀⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢶⠷⠤⠰⠶⠶⠖⠛⠛⠁⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠺⣦⡉⠒⠢⢴⣁⣪⠍⣶⡖⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⣈⣁⣴⠟⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠙⠛⣛⣛⣭⣤⣶⣶⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⢰⣾⣿⣿⣿⠿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠈⠉⠉⣰⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠳⣿⣷⣄⠀⠀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣿⣿⣧⢸⣿⣿⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠋⠀⠈⠻⢿⣿⣿⡟⣼⣿⠟⠁
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣰⣿⣿⠏⠘⠉⠁
⠀⠀⠀⠀⣿⣿⡟⠉⠉⠉⠉⠉⠉⣭⣉⠻⣿⢟⣵⣾⣿⠟⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣶⡀⠀⠀⠀⢻⣿⣷⣾⣿⡿⠛⠁
⠀⠀⢠⣇⢿⣿⡗⣿⣿⡇⣀⡀⠀⠀⠉⣉⣭⣵⡖
⠀⢠⣿⣿⠈⠛⣽⣿⡿⣼⣿⣿⣿⣿⣿⣿⠿⠋
⢀⣾⣿⢇⣤⣴⣿⣿⠃
⣼⣿⣿⣿⣿⣿⣿⠇
⢿⣿⣿⣽⡎⣿⣿
⠀⠉⣿⣿⡃⣿⣿⡀
⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⢿⣿⡇⢻⣿⡇
⠀⠀⢸⣿⣇⣸⣿⡇
⠀⠀⠀⢻⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡶⠒⡊⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠚⣀⡜⢙⣁⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢧⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡎⢠⡯⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢩⠁⠆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣏⠳⠀⠀⠀⠀⠀⠀⠀⠌⢸⠅⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠠⠗⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡄⠘⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⣀⣁⠂⠀⠀⠀⠀⠀⠀⠀⢸⠹⡛⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠊⡠⠿⣿⡏⠐⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢓⡆⢡⣄⠀⠀⠀⡀⠀⠀⡀
⢤⠖⠒⠒⠒⡐⠛⠲⠆⠂⠃⢰⠏⣏⠠⠊⠁⠀⠀⢂⣴⣄⡀⠀⠀⠀⠀⠀⠠⣈⠳⢟⣠⠼⢂⡈⠔⠖⢲⣄⣳⠲⢤⡀
⠠⠴⢋⡝⠙⢻⡋⠶⠆⠀⡈⠁⠖⠁⠀⠀⠀⢀⠦⠶⠐⢆⡙⢢⠀⠀⠀⠀⠀⠀⠑⠢⠉⠐⠃⡁⠠⠰⣄⡉⠛⠲⠇⠴⠶
⠀⠀⠁⠀⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢲⠉⢅⢄⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠂⠈⠀⠀⠉⠒⠢⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣝⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠖⠠⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠀⢰⣿⡿⢿⣿⣆⠀⠀⢀⣴⣿⣿⣿⣿⣦⠀⠀⣿⣿⣿⣿⣷⣄⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⢸⣏⣿⣿⣹⣹⠃
⠀⠀⠀⠀⠀⣠⣿⠻⣿⡄⠀⠀⠀⢸⣿⣧⣾⡿⠏⠀⠀⣾⣿⡟⠉⠉⠙⠁⠀⠀⣿⣿⡏⠙⢿⣿⡇⠀⢸⣿⣿⣀⣀⡀⠀⠀⢸⣿⣿⣁⣀
⠀⠀⠀⠀⣰⣿⣽⢿⣽⣿⡄⠀⠀⢸⣿⡿⢿⣿⣦⡀⠀⣿⣿⡄⠀⠀⣀⡀⠀⠀⣿⣿⡇⢀⣼⣿⡇⠀⢸⣿⣿⠿⠿⠃⠀⠀⢸⣿⣿⠷⠾
⠀⠀⠀⣰⣿⣿⠏⠈⢿⣿⣿⡄⠀⢸⣿⣧⣴⣿⡿⠃⠀⠹⣿⣿⣿⣿⡿⠟⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⢸⣿⣿⣷⣶⣶⡆⠀⢸⣿⣿
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠁⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉
⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⣶⣶⠀⠀⠀⣶⣶⡆⠀⣶⣶⣶⣶⣶⣶⡆⠀⢰⢶⡶⡖⣶⢲⣶⠀⠀⣶⣶⠀⠀⣴⣶⠄⠀⣴⢶⣶
⠀⠀⢠⣿⣿⡿⠛⠻⣿⠿⠀⠀⣿⣿⡀⠀⢀⣿⣿⡇⠀⠛⠛⣿⣿⡟⠛⠃⠀⠘⠛⠛⣿⣿⠛⠛⠀⠀⣿⣿⣤⣾⣿⡟⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⢰⣦⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣍⠀⠀⠀⣿⣻⣿
⠀⠀⠘⣿⣿⣷⣤⣴⣿⡿⠀⠀⣿⣿⠇⠁⢹⣿⣿⡇⠀⣤⣤⣿⣿⣷⣤⡄⠀⣠⣶⣾⣿⣿⠀⠀⠀⠀⣿⣿⠀⠻⣿⣧⠀⠀⣽⣿⣽⣶⣶⣶⡆
⠀⠀⠀⠈⠛⠻⠿⠛⠋⠀⠀⠐⠛⠛⠀⠀⠘⠛⠛⠃⠀⠛⠻⠿⠿⠻⠿⠃⠀⠈⠛⠿⠛⠁⠀⠀⠀⠀⠟⠛⠀⠀⠿⠿⠀⠀⠻⠿⠿⠷⠿⠻⠃
⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢸⣿⣿⣧⡀⣠⣿⣿⣿⠀⢸⣿⣿⣷⡀⢸⣿⣿⠀⠀⣴⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣽⣟⣄⠀⠀⣴⢿⣿⣿⡿⣷⡀⠀⢸⣿⡿⢿⣿⣆
⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⠀⢸⣿⣿⣿⣷⣸⣿⣿⠀⢰⣿⣿⠉⠈⢹⣿⣧⠀⢸⣿⣿⣤⡿⣽⠀⢸⣟⣿⠉⠉⢻⣟⡇⠀⢸⣿⣷⣼⣿⡟
⠀⠀⢸⣿⣿⠙⢿⠟⢹⣿⣿⠀⢸⣿⣿⠹⣿⣿⣿⣿⠀⠘⣿⣿⣄⣀⣼⣿⡇⠀⢸⣿⣿⠯⠟⠋⠀⠸⣟⣷⣄⣾⣿⣿⡇⠀⢸⣿⣿⣿⣿⡄
⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⠀⠘⢿⣿⣿⠀⠀⠻⣿⣿⣿⣿⡿⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠻⣯⣿⣿⣿⣿⠄⠀⢸⣿⡏⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⠀⢠⡶⣶⢶⣶⢶⡶⡆⠀⢰⣶⣶⠀⠀⢰⣶⣶⠀⠰⣶⣶⡆⠀⣴⣶⣶⠀⠀⢶⣶⣦⠀⠀⠀⠀⠀⣴⡶⡶
⠀⠀⠀⠀⠀⠰⣿⣿⡏⠙⠋⠀⠈⠛⠻⣿⣿⡞⠛⠃⠀⢸⣿⣿⠀⠀⢸⣿⡏⠀⠀⢻⣿⣷⠀⣿⣿⡇⠀⠀⠈⣿⣟⡇⠀⠀⠀⢰⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⡀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠸⣿⣿⡄⢀⣿⣿⡇⠀⠀⠘⣿⣿⣼⣿⣿⠀⠀⠀⠀⠸⣟⣿⣤⣶⣤⣿⣻⠇
⠀⠀⠀⠀⢠⣶⣤⣤⣿⣿⡿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⢹⡷⣿⠿⣿⢿⡏
⠀⠀⠀⠀⠈⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠙⠻⠛⠋⠀⠀⠀⠀⠀⠈⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⣠⡀⠀⠀⣤⢤⡄⠀⣤⠤⡄⠀⢠⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣼⣿⡿⠀⠀⠹⣟⣷⣰⣿⣿⠃⠀⠐⠟⠟⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⠀⠹⣾⣿⡶⠃⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣄⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢀⣼⣿⣿⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠁⠈⢻⠟⠁⠀⠀⠀⢿⣿⠇⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⢀⠀⠀⠈⡀⠠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⠀⠀⠐⡎⢈⣀⠀⠠⠊⡐⠁⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠆⡄⢢⡚⡼⢻⠖⣆⣰⠀⠠⠀⠀⠈⡄⠀⠀⠄⠁⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠊⠀⠑⠀⠺⣉⠟⠈⢠⠔⠂⡈⠁⠀⠉⠉⠀⠐⠰⣖⣰⡒⡂⡠⠀⠁⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠠⠐⠈⠀⠀⠮⠍⡁⠶⢠⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠙⠫⡛⡇⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠰⠏⠘⠡⠀⠊⢀⡅⠀⠂⠀⠀⠀⠀⠀⠀⣈⢳⡴⡇⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⡹⡀⠉⡄⠀⣠⡔⠊⣙⡆⢀⣀⠀⠐⠀⠀⠈⠉⡈⢇⡵⢁⣴⢀⠀⠀⠀⠢⠈
⠀⠀⠀⢀⡀⠀⠀⠀⠢⣼⣇⠁⢤⠇⣤⠽⢹⡌⠒⢞⡵⣋⠄⠀⠀⠀⠀⡀⢉⡘⡀⠛⢹⣦⣞⣐⠀⠂⠀⠀⠀⠀⡀
⠀⢀⢀⡀⢈⠂⢠⠿⡂⠀⠋⣠⠀⢀⡼⠀⠂⡞⠀⣠⣴⣬⠄⠀⠀⠀⠀⠁⣨⠋⢉⣩⣄⢸⡭⣧⣤⠀⣠⠄⣀⠀⠀⡀⠐
⠀⠈⣫⠂⠋⠔⢙⣺⠇⠀⠐⠀⢉⠏⠀⡤⠤⠞⢁⢴⢦⡨⢀⢄⠐⢀⣠⣛⣁⡀⢹⢉⣉⣬⣽⣿⢿⣿⡟⢿⣿⠾⡛⠿⡿
⠀⠉⢤⡌⢁⡉⡍⢀⣠⠞⠈⢀⡤⠚⠒⢄⠀⠀⢿⠷⢦⡵⣂⣈⠫⠎⠡⡿⡄⠤⠄⠘⠤⠠⡤⠤⣤⣩⡿⢶⡶⠛⠿⠷⠶
⠀⠀⠈⠉⠈⠀⠈⠀⠀⠀⣠⠌⠀⠘⡻⠚⠂⠀⠀⠲⠀⢉⠁⠀⡠⠀⠐⢂⠉⢲⣤⡄⡐⣀⡭⠊⠁⡂⠊⣁⠴⠒⠒⠄
⠀⠀⠠⠤⠄⠂⠄⠒⠒⠉⡤⣄⠞⣃⡳⠏⠑⠢⠤⠆⢰⠈⠁⢁⣀⡀⠣⠦⠐⡈⠐⠅⠐⣋⠔⡶⠀⠈⣉⣁⣀⣀
⠀⠀⠐⠒⠒⠚⠉⠩⠁⠚⠀⠁⠠⢌⣀⠄⠖⠧⠠⡶⠈⠓⠚⠁⢀⠠⢉⡭⠄⣉⢹⡒⢋⠁⢀⠉⢛⡨⠠⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠉⠉⠀⠐⠃⠀⠊⠀⠀⠁⠁⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣶⡇⢸⣶⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣴⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣦⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⣷⡄
⠀⠀⠀⢀⣴⣿⣿⣿⡟⣰⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⠇⠸⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣆⢻⣿⣿⣿⣦⡀
⠀⠀⢠⠞⠛⠉⠉⠋⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠉⠉⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣴⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠆⠀⠘⣿⣧⡀⠀⠀⣠⣿⡟⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠻⠿⣿⡿⠿⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠁
⠀⠀⣀⣠⣤⣶⣷⣭⡛⠛⢛⣵⣾⣶⣤⣄⠀⣼⣿⣿⣿⣿⡆⢀⣤⣴⣾⣦⣝⠛⠛⣫⣶⣷⣦⣤⣀⡀
⢠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠻⣿⣿⣿⡿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋⣡⣤⣶⣶⣌⡉⢉⣴⣶⣦⣤⣉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡄⠀⠀⡀
⠀⠀⠀⣻⡇⠀⢀⣥⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢄⠀⢄
⠀⠀⣸⣽⣷⠊⠁⡸⠤⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠑⠦⠀⠶⣀⣇⡀
⠀⠀⠩⣼⣿⣷⣄⠀⡡⠒⢐⡩⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⢹⣀⣴⣆⠹⢎⡇
⠀⠀⠀⠙⣟⣦⢿⢮⣀⣞⠻⠆⣛⠉⠸⠏⣹⡶⠶⠄⠀⠠⠀⣤⠄⣤⣰⣣⢤⣤⡴⣿⠿⡃⠙⡦⢾⡅
⠀⠀⠀⠀⠘⣿⣏⣹⣏⡝⠀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠀⠠⠀⠙⠃⠉⡀⠿⠾⢷⡀⠈⢦⡸⠚⠁⢈⣙
⠀⠀⠀⠀⠀⠀⠸⠳⢄⡀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠤⣤⠞⠁⠀⠀⡌⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠟⠋⠙⠻⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡶⠿⠛⠉⠁⠀⣀⣤⣤⣀⠀⠈⠉⠛⠿⢶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⠉⠁⠀⢀⣀⣤⣴⠾⠛⠋⢹⡏⠙⠛⠷⣦⣤⣀⡀⠀⠈⠉⠙⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣶⠶⠟⠛⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠛⠻⠶⣶⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣀⣴⠾⠟⠛⠛⠻⠷⣦⣀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⢀⣼⠟⢡⡄⠘⠃⠘⠃⢠⠌⠻⣧⡀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢸⡇⠀⠀⣼⠇⠰⠆⢠⡶⠟⠻⢶⡄⠰⠆⠸⣧⠀⠀⢸⡇⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣷⠶⠶⣿⠀⠶⠀⣿⡁⠀⠀⢈⣿⠀⠶⠀⣿⠶⠶⣾⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣿⡀⠀⢻⣆⠰⠆⠈⠻⢶⡶⠿⠃⠰⠆⣰⡟⠀⢀⣿⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢸⣇⠀⠈⠻⣦⡘⠃⢠⡄⢠⡄⠘⢃⣴⠟⠁⠀⣸⡇⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢻⣆⠀⠀⠈⠛⠷⣦⣤⣤⣴⠾⠛⠁⠀⠀⣰⡟⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠻⣧⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣼⠟⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠈⠻⣦⣄⠀⠀⢸⡇⠀⠀⣠⣴⠟⠁⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠈⠙⢷⣤⣸⣇⣤⡾⠋⠀⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⠈⠛⠛⠁⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠉⠉⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⢀⣤⠤⠿⠿⠿⠤⠤⠤⠤⠿⠿⠿⠤⣤⡀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⢠⣤⣤⡄⣿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⣷⠠⣤⣤⡄⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⡄⣿⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⣿⠠⣤⣬⣤⣤⣿
⠀⠀⠀⠀⠀⣴⣿⣦⣤⣤⣤⣤⣤⣤⡀⣿⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⣿⠠⣤⣤⣤⣤⣤⣤⣴⣿⣦
⠀⠀⠀⠀⠀⠈⠛⠁⠀⣤⣤⣤⣤⣤⡀⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⢠⣤⣤⣤⣤⣤⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢠⣦⣤⠀⣿⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⣿⠀⣤⣤⡄⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠁⠀⠀⠹⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟⠀⠀⠀⠀⠻⣿⠟





⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⣀⠀⢀⠀⢀⡀⠀⣀⠀⢀⠀⢀⡀⠀⣀⠀⣀⠀⢀⡀⠀⡀⠀⠀⠀⠀⡀⢀⡀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⣠⠞⠁⢀⡤⠖⠒⠛⠓⡲⠦⣄⠀⠙⢦⡀
⠀⠀⠀⡼⠁⢀⡞⠁⠀⠀⣠⠖⠺⣧⠀⠈⢳⡀⠀⢷
⠀⠀⢸⠁⠀⡞⠀⠀⠀⠀⠹⣄⠀⠈⠁⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢻⠀⠀⣇⠀⠀⢀⠀⠀⠈⡇⠀⠀⠀⢠⠇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⣉⣩⠤⠤⠤⠤⠤⠤⣭⣉⡛⠲⢤
⠀⠀⠘⣆⠀⠙⣆⡀⠈⣳⠒⠛⠁⠀⢀⡴⠋⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢰⡋⠥⢴⣂⣀⡴⠚⠉⠙⡶⠚⢙⡆
⠀⠀⠀⠹⣦⡀⠀⠙⠲⠦⠤⠤⠶⠚⠉⢀⣠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⡉⠓⠲⠮⠥⠤⠤⠤⠴⠖⠚⠉⣀⣴
⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣀⣤⣤⣴⣶⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⡿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⡿⣷⣿⣭⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⣋⣉⣉⣉⣉⣉⣉⣙⠓⠶⢿⣽⣿⣷⣾⣿⣽⣿⣿⣿⣿⣿⣽⣿⣾⢿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢠⠞⣉⣩⠄⠀⣠⠖⠒⠲⣭⠽⠳⡄⠈⣿⣿⣾⣿⣯⣽⣿⣿⣿⣭⣿⣷⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡈⠓⠦⠬⣽⣉⣁⣀⣀⣬⠥⠴⠚⢁⣠⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⢿⣿⢿⣾⣿⣯⣿⣿⣿⣿⣭⣿⣿⣾⢿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣷⣯⣿⣟⣻⣿⣿⣿⣿⣿⣻⣿⣽⣿⢟⣿⣿⠷⠛⠉⠉⠉⠉⠛⠻⢿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⣻⣷⣿⣽⣿⣟⣿⣿⣿⣿⣛⣿⣯⣿⣾⣿⡟⠁⠀⣠⠴⠒⠒⠒⠦⣄⠀⠙⢿⣿⣫
⠀⣠⠴⠚⣋⣉⣭⣥⠤⠤⣭⣍⣉⡙⠒⠿⣿⣿⣿⣽⣿⣻⣿⣿⣿⣿⣛⣿⣿⣿⣾⡟⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠹⣯
⡞⠁⢴⠋⣥⣴⣀⣀⡴⠚⠛⢳⡶⠏⢳⡄⠈⣷⣯⣿⣟⣿⣿⣿⣿⣿⣿⣻⣿⣽⣿⡇⠀⣾⢀⣠⡖⠀⠀⢀⡤⢤⡀⠈⣇⠀⢹
⣿⣤⡈⠙⠲⠮⠭⠭⠤⠤⠤⠤⠶⠒⠋⣀⣴⣿⣿⣿⣿⣛⣿⣿⣿⣿⣟⣿⣯⣿⣿⡇⠀⢻⠈⠙⢧⣀⣠⠏⠀⢀⡷⠦⢹⠀⢸
⣯⣻⣿⣿⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⣿⡿⣿⣿⣾⣯⣿⣿⣛⣻⣛⣛⣿⣿⣭⣿⣿⣿⡄⠈⢧⡀⠀⠉⠁⠀⠀⠘⠁⢀⡟⠀⢸
⣿⣻⣿⣿⣽⣿⣿⣿⣛⣻⣿⣿⣿⣽⣷⡿⣫⡿⠙⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠋⠙⢿⣿⣄⠀⠙⢦⣀⡀⠀⠀⣀⣠⠞⠀⢀⡞
⠙⢿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⠈⠉⠉⠉⠉⠀⢀⡴⠋
⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣶⣶⠶⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠒⠒⠂⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⢀⣤⣄⢠⣤⡀⣠⣀⢀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠘⣯⠇⢹⢾⡐⣇⣸⠂⣸⠀⢸⣧⠜⣧⡤⠀⣏⠁⣿⣉⢸⢛⡆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⢀⣴⠟⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠻⣦⡀⠀⠀⠀⠀⠇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⢰⣿⠁⠀⠀⠀⠈⠻⣷⣄⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⣿⡆⠀⠀⠸⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠈⠻⣷⣜⠛⠁⠀⠀⠀⠀⢀⣴⠟⠀⠀⢠⠃⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣤⠈⠻⣧⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠎⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠈⠻⣦⡀⠀⠻⣦⡀⠙⠀⢀⣴⠟⠁⠀⠀⠀⠀⡌⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠈⠻⢶⡶⠌⠻⢶⡶⠟⠁⠀⠀⠀⠀⠀⠐⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⡀⠀⠀⢀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⡶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠾⠋⣁⣤⣤⡈⠙⠷⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠿⠛⠛⢉⣁⣤⣴⣾⣿⣿⣿⣿⣷⣦⣤⣈⡉⠛⠛⠿⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠙⠻⢿⡿⠟⠋⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠄⠈⠀⢀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⡀⠀⠀⠐⡀⢀⣠⠀⠀⠀⠰⡷⡄⠀⡀⢦⡀⠆⠠⠆⢈⣧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⡄⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠁⠀⠀⠘⠃⢤⠍⢧⣿⣧⠉⢻⡄⠀⠶⣿⡿⠈⢣⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠒⠲⢛⡃⠒⠠⣄⠀⣺⡿⠂⠘⢧⣼⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠘⢯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣳⣄⠈⠙⣿⣧⠱⣄⠈⣿⣀⠀⢠
⠀⠀⠀⠀⢀⣤⣾⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⢈⠁⠀⣀⡀⠈⠂⠂⠖⠈⠉⠑⢻⣿⣄⣿⡀⢹⡶⢠⡯
⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣪⣌⣑⣈⣃⡀⠀⠀⠀⠀⢀⣀⣴⣿⣿⡙⢷⣈⣿⣼⣧⠖
⠀⠀⠀⢹⣿⡿⠟⠁⠐⠿⢷⣮⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠀⢀⡽⠛⣶⡶⠾⣿⣿⣄⣿⣿⠁⢀⣿⣿⣿⠗⠂
⠀⠀⠀⢸⡟⠀⢀⠀⠀⠀⢠⣿⣷⡀⢸⣽⡿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡟⡄⡩⠴⡞⠋⡀⠀⡀⠀⠈⢹⣿⣧⠈⣿⣿⡟
⠀⠀⠀⠿⠶⠒⣿⣧⣾⣿⣿⣿⠿⠇⠜⠻⡄⠀⠿⣿⠟⢿⡿⠿⣿⠿⢛⠛⠻⠋⠊⠟⠀⠂⠒⢿⡐⠛⣺⡟⢿⢀⡏⠑
⠀⠀⠀⠀⠀⠀⢀⣿⣶⣦⣬⢀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠛⠚⠛⠛⠛⠉⠋⠉⠁⠀⠀⠀⢀⣸⣿⣿⣿⡧⠀⢻⠶
⠀⠀⠀⠀⠀⣀⣄⣤⠛⢉⣈⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣠⣄⣀⣶⣿⣿⣿⠿⠟⣿⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠛⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠈⠉⣉⢉⢛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣤⣄⣀⠉⠉⠙⠛⠛⠛⠋⠉⠉⣁⣀⣤⣤⣶⣶⣤⣤⣀⣈⠉⠉⠛⠛⠛⠛⠋⠉⢉⣀⣠⣤⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⣉⣀⣀⣀⣀⣉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡟⠁⣠⣾⣿⣿⡿⠟⠛⠉⠉⠉⠉⠛⠿⣿⣿⣿⣷⣄⠈⢻⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⡟⢀⣼⣿⣿⡿⠋⣀⣴⣾⣿⣿⣿⣿⣶⣤⣼⣿⣿⣿⣿⣷⡀⠹⣿⣿⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⡟⠀⣼⣿⣿⠏⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⠇⢰⣿⣿⡿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⣿⣿⣿⣿⣿⡆⠸⣿⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣿⣿⣿⣿⠀⢸⣿⣿⡇⠀⣿⣿⣇⠈⠻⣿⣿⡿⠋⢀⣴⣿⣿⡏⠈⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⢹⣿⣿⣿⡇⠘⣿⣿⣿⠀⢻⣿⣿⣷⣆⡈⠉⢀⣴⣿⣿⣿⣿⠃⢰⣿⣿⣿⠃⢰⣿⣿⣿⡏⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠈⣿⣿⣿⣧⠀⢻⣿⣿⣧⡀⠹⣿⣿⣿⣷⣶⣿⣿⣿⣿⡿⠃⣠⣿⣿⣿⡏⠀⣼⣿⣿⣿⠁⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠹⣿⣿⣿⣧⠀⠻⣿⣿⣿⣤⡀⠙⠻⠿⠿⠿⠟⠛⠁⣠⣴⣿⣿⣿⠟⠀⣼⣿⣿⣿⠇⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠹⣿⣿⣿⣷⣄⠘⠿⣿⣿⣿⣷⣶⣦⣤⣤⣶⣶⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿⠏⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡄⠘⢿⣿⣿⣿⣷⣄⡈⠙⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢁⣠⣾⣿⣿⣿⡿⠃⢠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡀⠙⢿⣿⣿⣿⣿⣶⣦⣤⣀⣈⣉⣉⣁⣀⣤⣴⣾⣿⣿⣿⣿⡿⠋⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⣀⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣦⣀⠙⠻⣿⣿⣿⣿⠟⠉⣀⣴⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣤⡈⠙⠋⢁⣤⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣴⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠀⠈⠁⢀⠶⠂⠀⠀⠀⠀⠀⠀⠀⢸⣫⡆⢐⢮⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠺⠂⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣤⣄⠁⠀⣠⡄⠀⠀⠀⠀⠀⣀⣤⣠⡋⠈⢩⣤⣶⣦⠄⠀⠀⠀⠀⠀⠐⠛⠢⣤⡤⠒⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠈⠛⠚⢈⡶⠭⢶⣁⠉⠀⠀⠀⠀⠀⠀⠀⣔⡶⠊⠉⠈⠐⡶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣠⡄⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢼⡝⠂⠙⠾⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣺⡃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣠⢀⡘⠓⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⠁⡀⠀⣤⠀⠀⠀⠀⠀⠀⠰⣖⡴⠈⠪⠞⢱⢰⣲⠂
⠀⠀⠀⠀⠀⠀⠐⠷⡵⠄⡦⢱⡎⣥⢀⡴⣠⡀⠀⠀⠀⠀⠀⠀⠬⣖⣭⡲⠅⠀⠀⠀⠀⠀⠀⠀⠀⣓⠊⢑⡊⢰⠉⠐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠇⠻⡂⠀⠘⠈⠀⠀⠀⠀⠀⡀⠶⠀⠔⠙⠂⠀⠶⠀⠀⠀⠀⠀⠀⣠⣖⢭⠃⣨⡁⢹⣱⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠑⠀⠠⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢔⡮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢦⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢋⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡡
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠇⡰⠲⠔⢦⢸⣷⠂⠀⠀⠀⠀⠀⠛⣨⣄⠀⠭⢀⣸⡀⠓⠀⠀⠀⠀⠀⠘⣂⣅⡀⠸⠇⣀⣦⡘⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠺⣁⠀⠀⣁⡶⠈⠁⠀⠀⠀⠀⠀⢈⡀⢈⡹⣿⢎⠀⣀⡁⠀⠀⠀⠀⠀⠀⡀⢈⣉⠾⠦⣉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⢿⡇⠳⢞⠦⠆⢰⠯⡄⠀⠀⠀⠀⠀⡤⢘⠇⢀⣖⠀⢸⠑⡴⠀⠀⠀⠀⠀⠠⠌⠋⠁⢸⡆⠉⠏⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡉

⠀⠀⠀⠀⠀⠀⠀⠐⠂⠙⠂
⠀⠀⠀⠀⠀⠀⠒⠒⠻⠼⡁
⠀⠀⠀⠀⠀⠘⡂⢀⡄⠈⠉⠁
⠀⠀⠀⠀⠀⠘⠛⠻⠤⠄⢶⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠠⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⡨⠆
⠀⠀⠀⡀⠀⣠⠂⠀⠶⢷⡝⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢀⣀⠴⠶⢤
⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣵⣄⣤⣶
⠀⠀⠀⠶⠠⣰⠆⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡁⠀⢄
⠀⠀⠀⠀⠔⠁⠀⠒⣀⠀⠶⣀⡀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠁⢀
⠀⠐⠧⠄⡤⠀⠀⢀⠙⣀⡀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣘⢀⠤⠚⠐⠾
⠀⠀⠀⠀⠸⠇⠂⠉⡀⢀⡉⠹⠄⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠐⠪⣭⣋⠠⣤⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣤⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣶⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡎⢇⠀⠀⢒⠄
⠀⠀⠀⡠⠔⠒⠒⢢⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠡⡀⠀⠀⢀⠜⠀⠈⢆⠀⠀⠀⣀⠤⣀
⠀⠀⠀⡱⠀⠀⠀⠘⢲⠶⡏⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠂⠀⠀⢊⠀⠐⢳⠂⠁⠐⠲⡛⠀⠰⠁⠀⠈
⠀⠀⠈⢳⠀⠀⠀⠀⠀⡆⢣⣴⠃⠀⡄⠀⡠⠀⠀⠀⠀⠀⠀⠀⢐⠇⠀⠀⠀⠐⣷⣖⡁⠀⠀⠀⠀⣑⣤⠀⠁⠒⠈
⠀⢠⠤⠶⠀⠈⠁⠈⠄⠉⠁⠈⡆⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⡡⠤⠀⠄⣀⣀⡤⠋⠀⠀⠀⠀⠀⠈⠢⢄⡀
⠀⢇⠀⢀⡁⠀⠀⠀⠐⠀⠀⢐⡍⡂⠀⠐⠀⣊⠀⠀⠀⠀⠀⢊⠤⠀⠀⠀⠀⢈⠽⠋⠙⠀⠂⠠⠐⠀⠛⢍
⠀⢰⠁⠀⠀⠐⠀⠁⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠒⠀⠀⠀⠒⠁⠀⠀⠠⠀⢌⢑⡶⠀⠀⠀⠀⠀⠀⠀⠴⣖⠉
⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠄⠀⠠⠤⠤⢤⠤⠀⠀⠤⠵⠂
⢹⠀⠀⠒⠒⠒⠁⠀⠈⠉⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠤⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠚⠀⠀⠀⡇⠀⡀⢀
⠀⠀⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠃⠀⠀⠀⢀⠀⠀⣀⣀⠤⠀⢀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠁
⠀⢧⣀⠄⠀⠀⢣⣀⠄⠀⠀⠀⠳⠤⠂⠘⠦⠄⠀⠘⠦⠔⠀⠈⠳⠤⠤⠂⠈⠳⠤⠤⠔⠊⠀⠳⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⢹⢁⣿⡈⡏⢸⠅⢫⡀⢼⢙⣾⠭⠀⣷⣼⣀⣿⡈⡏⢹⡏⡯⢸⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠈⠀⠁⠁⠈⠁⠉⠀⠈⠉⠈⠀⠀⠁⠀⠉⠀⠁⠁⠈⠀⠉⠉⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠖⠒⠶⠦⢰⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠶⠶⢾⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣤⣶⣶⠸⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢛⡛⠿⠇⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣦⣶⣾⡆⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⢠⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣶⣶⣬⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣙⣋⣥⣤⢹⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⡇

⠀⠀⠀⢹⣄⢲⡲⢡⡇⠀⢠⠘⣴⠢⢼⣠⠂⠀⢧⡙⣏⡙⢣⣸⠀⠀⡇⢸⢨⡇⢸⠀⠀⣣⢰⡌⡴⠀⠀⢂⣱⠞⠲⣀⡇
⠀⠀⠀⡄⡟⢦⣽⡎⠃⠀⠘⢳⣌⣵⣎⠹⡀⠀⡏⢻⣿⣽⡏⢱⡀⠀⡃⠸⣼⠃⢰⠀⠀⢹⠦⠵⠈⠀⠀⡤⣻⣤⣼⡏
⠀⠀⠀⢷⡘⣤⣭⢃⡆⠀⢸⣴⠿⢍⠙⢶⠁⠀⡷⣬⡰⣇⡠⡞⠀⠀⣳⠄⠐⢀⣿⠀⢸⣈⣤⡬⢠⠃⠀⡀⢏⡉⣍⠁
⠀⠀⠀⡎⣷⡌⢢⠙⡇⠀⢃⠈⣷⠺⢷⢈⠂⠀⢁⠹⡟⡟⠇⢸⠀⠀⢱⠀⢸⣿⡏⠀⠀⡋⢸⣴⡏⠀⠀⢱⢾⡀⢠⢖⠁
⠀⠀⠀⡇⢣⡙⠚⡇⡀⠀⠘⢣⣌⣱⣄⠹⠀⠀⡞⢶⣩⣷⡞⠣⠀⠀⣼⣆⢰⡌⠣⠀⠀⢇⢰⠬⡼⠀⠀⠇⡟⠿⠟⠃⡄
⠀⠀⠀⡸⣦⡹⣙⢰⠇⠀⡆⢸⠀⠀⢸⢀⠀⠀⢢⡈⢻⣅⠳⡼⠀⠀⠁⢿⣿⡁⠶⠀⠰⢹⡶⢷⠑⡀⠀⢦⣀⠊⢑⣠⠇
⠀⠀⠀⡁⣟⢳⣾⡆⡇⠀⣧⢸⠀⠀⠸⠉⠀⠀⡄⣹⠷⠾⡗⢛⠀⠠⡀⠈⢸⠃⠀⠀⢰⡘⡤⠤⢠⠂⠀⠐⣽⣶⣾⡏
⠀⠀⠀⢻⡈⠦⠤⢠⡇⠀⡇⢸⠀⠀⢠⠀⠀⠀⡼⣦⡑⢢⡬⢻⠀⠀⡇⠀⢸⠀⠈⠀⠀⠉⠸⢶⣯⠀⠀⡆⠡⠔⠣⠁⡀
⠀⠀⠀⡊⠿⣧⣵⡌⡇⠀⢇⠸⣄⠀⢼⣠⠀⠀⢧⡘⣏⡉⢳⣸⠀⠀⡇⣰⢸⠀⢀⠀⠀⣠⠰⡞⠇⠀⠀⠈⢓⣀⣠⡿⡁
⠀⠀⠀⢃⡹⣌⣙⢁⡇⠀⡜⢳⣬⣹⣄⢹⠀⠀⡏⢻⣷⣼⡏⢹⡄⠀⡇⠙⣼⠀⣿⠀⢠⢹⠇⠺⠈⠀⠀⡄⠏⣻⡋⠇⡄
⠀⠀⠀⡜⣷⡝⢮⡙⡇⠀⢰⡈⠳⢤⠙⣾⠀⠀⡵⣬⡰⣧⡠⢞⠀⠀⠷⠀⡰⢠⡿⠀⠘⣌⠆⡤⢀⠆⠀⠑⣬⠁⢨⡤⠃
⠀⠀⠀⡇⢣⠹⠿⡇⡗⠀⢆⠘⣗⠚⢷⣨⠀⠀⢄⠹⣟⡟⠇⣸⠀⠀⢳⠀⢿⣿⠃⠀⠀⡍⢸⢷⣯⠀⠀⠀⡼⢿⠾⡇⡄
⠀⠀⠀⢸⣆⢳⡒⢱⡇⠀⠜⠳⣌⣱⣄⠹⡀⠀⡞⢷⣽⣷⡞⠃⠀⠀⣞⡄⢠⠈⢆⠀⠀⣡⢰⡆⡴⠀⠀⢂⣰⠊⠳⣀
⠀⠀⠀⡄⣟⢦⣼⡎⠃⠀⠀⠀⠀⠀⠈⠀⠁⠀⢳⡀⢫⣅⠱⡴⠀⢸⠈⣿⢾⠃⣸⠀⠀⢹⠆⠿⠀⠀⠀⡤⣹⣤⣴⡟
⠀⠀⠀⢳⡈⣤⢬⢁⡇⠀⢀⠀⠀⠀⢸⠈⠀⠀⣄⣽⠷⠾⡗⢻⠀⢸⠀⢾⢸⠀⢼⠀⢸⣌⠄⠸⢠⠃⠀⡀⢋⡝⢍
⠀⠀⠀⡎⣷⡌⢲⠙⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜⣦⣀⣢⡼⢻⠀⢸⡀⢸⢸⠀⢸⠀⠀⡉⢸⣶⡏⠀⠀⢙⢾⡀⣰⣟⠁
⠀⠀⠀⣇⢣⡙⠛⡇⡀⠀⢠⡈⢶⠂⠤⣀⠄⠀⢧⡘⣏⡉⣓⣸⠀⢸⠁⢸⢸⠀⣌⠀⠀⢧⠸⠈⡬⠀⠀⠇⠟⠻⠛⠃⡆
⠀⠀⠀⡼⣦⡹⣝⢰⡇⠀⠰⢓⣤⣹⣦⠸⡄⠀⡏⢻⣷⣼⡇⢹⡄⢸⡇⢸⣸⠀⣹⠀⠀⢹⡶⢿⠁⠀⠀⢦⣄⠈⢙⣴⠃
⠀⠀⠀⡁⢏⢳⣾⡆⡇⠀⢘⣌⠙⢤⡙⢶⠁⠀⡽⣬⡐⣧⡨⢟⠀⠸⣷⠂⡤⢠⡟⠀⠸⡈⠄⠤⢰⠆⠀⠐⣽⣶⣾⡏
⠀⠀⠀⢹⡈⠢⠴⢠⡇⠀⢨⠈⣿⠒⢿⣀⠆⠀⢇⠹⣟⡟⠇⣸⠀⠀⢱⠀⢿⣿⡇⠀⠀⡉⠸⢷⣯⠀⠀⣇⠡⠜⠡⢀⡄
⠀⠀⠀⡆⠻⢧⣵⡌⡇⠀⠘⠱⣤⣙⣆⠙⡀⠀⡞⢷⣭⣷⡞⠣⠀⠀⣾⣄⠸⠄⢳⠀⠀⣠⠰⡞⢏⠀⠀⠈⢻⣠⣰⡋⡀
⠀⠀⠀⢇⡹⣌⣉⢁⡇⠀⠀⠀⠀⠀⢀⣀⡁⠀⢳⣈⢺⢄⠱⡾⠀⠀⡏⢿⣿⠀⣴⠀⢠⢹⡗⠛⠈⠀⠀⡀⠋⣹⣋⠇⡄
⠀⠀⠀⡞⣷⣝⢶⡙⡇⠀⢀⣴⠀⠀⢸⠉⠀⠀⣄⣽⠿⠾⡗⣙⠀⠀⡆⢸⠇⠀⢫⠀⠘⣌⠆⡴⢀⠆⠀⠐⣮⡁⠨⡴⠃
⠀⠀⢀⣇⣃⠙⢿⡇⣇⠀⡈⢸⡀⠀⢀⢀⡀⢀⡞⣶⣁⣲⣜⣫⡀⠀⣇⣸⣰⣀⣸⡀⠀⣍⣸⣷⣟⠀⠀⣀⣾⣿⣿⣇⣄
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⠀⢰⣿⠀⠀⣴⣦⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⣾⣿⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⢿⣿⣿⣿⠀⢸⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡀⠀⢠⣿⣿⣿⣀⣸⣿⣄⣀⣿⣿⣀⣸⣿⣇⣀⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠿⠿⣿⣿⠿⢿⣿⡿⠿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠀⢸⣿⠀⠀⣿⡿⠀⠘⠿⠃⠀⠘⠋⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠉⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡿⠛⠉⢀⣠⣴⣾⣿⣦⣄⡀⠉⠛⠿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠈⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡾⠟⠁⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠻⢷⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⠈⠙⠻⢿⡿⠟⠋⠁⣀⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣄⣠⣴⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⢷⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣟⣘⡾⠾⣼⣼⣰⢢⢄⠀⠀⠀⠀⠀⠈⡈⠁⠀⠄⠀⠠⠀⠁⠀⠐⢀
⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⢀⠈⠻⡇⢠⣴⠀⠈⠉⠛⠺⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠶⠷⠶⢤⣄⣑⡆⠙⠂⠕⠚⣣⡰⠟⣠⠋⠀⠀⠀⠀⠰⠁⠀⠄⠀⠀⡀⡐⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠉⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠐⢠⡄⠀⠀⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠐⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⢸⠃⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠜⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠘⠆
⠀⠀⠀⠀⠀⢈⠀⠀⠀⡄⠓⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡈
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⡐⠒⠅⠀⠀⡒⠲⡁⠈⠀⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠴⠦⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠜⠒⠶⠶⠶⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠶⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⠟⠏⠛⠋⠙⠈⠛⠙⠻⡟⠻⡲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠏⠛⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡹⣦
⠀⠀⠀⠀⠀⠀⣼⠏⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⣴⠅⠀⠀⠀⠀⣠⣴⡾⡟⠟⠻⠹⠿⠟⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡄
⠀⠀⠀⠀⢰⡯⠀⠀⠀⣰⡾⡛⠏⠁⠀⠀⠀⠀⠀⠀⠘⠱⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⣿⡐⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣇
⠀⠀⠀⠀⣿⠀⠀⢸⣿⠉⢀⣠⣶⠾⠿⠟⡿⣶⣤⠀⠀⠀⠀⠀⠀⠸⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⣿⣄⠀⢸⣿⢠⡾⢋⣠⣤⣄⡀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠸⣷⠀⢸⡏⣿⣷⣿⣿⣯⢙⣿⡅⠀⠀⢸⣷⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠹⣦⠀⢿⣿⣿⣿⣿⣿⡏⣿⡇⠀⠠⢸⡟⠀⠀⠀⠀⠠⢵⣿⠁⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠻⣿⣿⣿⡿⠾⠋⢀⣀⣶⡟⠁⠀⠀⠀⢀⣲⣿⠿⣦⣤⣤⣤⠤⠶⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣙⠛⠿⠶⠿⠿⠛⠉⠀⠀⠀⢀⣰⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣴⣤⣀⣠⣠⣤⣠⣵⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠈⣁⠤⠒⠈⠁⠀⡀⠕⠊⠩⠻⡒⠶⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠄⠁⣀⠀⠀⠊⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠒⠐⠀⠈⠙⠢⣅⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠘⠻⠶⠖⠉⣐⡠⢤⡤⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠱⡄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣹⢋⠜⠛⠓⠤⠠⣤⣤⣤⣀⠙⡄⠀⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠓⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢺⠀⠀⢸⠀⢠⠢⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⢕⠞⢢⣀⡇⠀⢀⡤⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠂⡁⠀⡐⣿⢠⡇⠷⢟⠓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣰⠎⡀⡀⠀⠀⣢⣻⠀⠉⠼⣗⡏⢲⠀⢆⡸⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠛⢧⠈⠙⢻⣯⢿⠉⠁⠀⠀⠈⠁⢸⢳⠀⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣻⠯⣴⢀⠈⣿⠛⠆⠀⠐⠄⠀⠠⠘⠸⢶⡞
⠀⠀⠀⠀⠀⠀⢀⣠⣸⠀⣰⢒⡠⠹⢅⠞⣸⡀⠀⢀⡘⣫⣹⣣⢰⠘⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠓⠓⠒⠚⠛⠃⠀⠞⠛⠙⠊⠉⠘⠛⠉⠛⠀⠀⠂⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢤⡔⠲⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠙⣧⠀⠘⡆⠀⠙⣄⡑⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠙⢿⡄⠀⢿⡄⠀⢿⡄⢠⣿⠿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢶⣄⠀⠣⠄⠀⣀⣤⣾⣩⠿⠿⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣧⣤⣾⣷⣾⠟⠉⠉⠀⠀⠈⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⢹⠿⢿⡟⠇⠀⠐⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⣶⣄⣀⣀⣴⣦⣄⣴⢶⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣷⡀⢰⣦⠉⠌⠓⣈⠙⠛⠃⢘⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⣦⣿⣧⣤⡀⢉⠲⠄⢀⡞⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡔⣻⣼⢷⣦⣙⣻⢿⣷⣤⣾⡇⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⡇⢙⠣⠉⠻⣿⣉⠡⡙⢻⣿⣇⠸⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠦⣹⣶⣤⣉⣻⣿⣿⣤⡏⢹⣷⣿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⡿⠸⣿⡋⠛⠿⠿⣭⣼⡷⣾⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣄⢂⣤⣉⡙⢲⣴⣿⠟⠋⡙⣯⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠸⢷⣧⣠⣽⠋⠉⠉⠳⠁⡠⢺⣋⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣁⣛⣽⣿⣿⡿⡷⣺⣶⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣛⡀⠤⠿⠿⠟⣹⣿⠙⠙⢿⣆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠰⣦⠀⠀⢺⣇⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠛⣤⢘⣇⡤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣳⣆⠀⡐⡿⠁
⠀⠀⠀⠀⠰⠾⠾⠒⡒⣇⣪⡓⠓⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣄⣟⢆⢸⢃⣤⡼⠞⠃
⠀⠀⠀⠀⠀⠀⣐⡶⠋⢐⣇⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⣈⣨⡭⠝⡿⡗⠤⣶⣴⡄
⠀⠀⠀⠀⠀⠐⠋⠂⠀⢐⡗⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠉⠁⣺⡇⠈⢶⣀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠈⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠜⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠
⠀⠀⠀⠀⠀⠀⢬⢵⠷⣛⠀⠀⠀⠀⠀⠀⠀⠠⡄⣆⡄⠀⠀⠀⠀⠀⠀⣀⡇⠀⢠⡥⠀⣀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⡺⡿⡖⠀⠀⠀⠀⢀⠀⠀⡘⠣⣀⣀⡔⠓⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠼⠉⠉⠉⢉⡽⠯⣉⠉⠋⠉⠙
⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠰⡆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠋⢀⡆⠘⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣥⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠑⠀⢨⣇⠀⠈
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠘⠉⡴⢰⡈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲

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

⠀⠀⠀⠀⡴⠊⠉⠐⠄
⠀⠀⠀⢸⠁⢰⠓⠀⠘⡀
⠀⠀⠀⢸⡄⢸⡛⠻⣄⡇
⠀⠀⠀⢸⢷⡀⠙⢲⡜⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⡀⠙⣋⠭⢻⠈⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠘⢦⡇⢠⡟⠀⣿⠠⡀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠈⢧⠀⢳⣞⣩⠷⠾⠶⠤⣄⡀⠀⠀⠈⠦⢠⠀⡼⠁⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠓⠓⠛⠛⢲⣄⠙⢖⠒⠠⢤⠈⢀⡄⠀⠐⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢖⠚⢳⡁⠸⡇⠈⠁⠺⠼⠁⢀⠴⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠳⣄⠙⠢⣤⡀⠄⠀⠈⠚⠥⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠵⠃⠀⡴⣤⢀⡀⢸⡆⠈⢧⣬⠵⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠀⠰⠂⠘⠀⠓⠧⠤⣵⣄⠙⢮⣌⣠⣤⣤⣈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡴⠀⢠⡀⠀⠀⠀⠈⠓⠲⠶⡴⢶⡞⣹⢧⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⣻⠁⣸⠃⢸⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣧⢐⣛⣄⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡙⢧⣀⠙⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣦⣬⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣤⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠈⢻⣿⣆⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠙⣿⣷⡀⣾⡿⠋⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠉⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⢠⡾⠋⠉⠛⢷⡄⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠈⢿⣷⡄⠀⠀⠘⢷⣤⣤⣤⠾⠁⠀⢀⣀⣀⣀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠉⠀⠀⠀⣴⠟⠉⠉⠙⢷⡄⠀⠀⠘⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣧⡀⠀⠀⠀⢻⣿⣤⣀⠀⠀⠀⠀⠀⠘⠷⣦⣤⡶⠟⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⢿⣷⡄⣀⣤⣾⣿⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠋⠁⠀⢨⣿⣿⡿⠟⠋⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠻⣦⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⣶⠛⠶⣦⣄⠀⠀⠀⢻⣿⣆⠀⠀⠀⢀⣠⣾⣿⣿⣦⠀⠀⠀⠀⠀⠛⢶⣤⣴⠾⠃⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⢻⣆⠀⠀⠙⠳⣦⡀⠀⠹⣿⣦⣤⣾⣿⠿⠛⠁⠙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀⠙⢷⡀⠀⠀⠈⢻⣄⠀⢿⣿⡟⠋⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣠⡶⠶⠶⣦⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠙⢷⣄⡀⠀⢹⡆⠘⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢰⡏⠀⠀⠀⠈⣿⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠈⠛⠳⠾⠃⠀⠘⢿⣿⣦⣄⡀⠀⠀⢸⣿⡇⠀⠀⠘⣧⡀⠀⠀⣠⡟⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣾⣿⠇⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⠿⠛⠛⠉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⡀⠈⠉⠉⠙⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠛⢉⣁⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠋⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⡇⢀⣴⠞⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣀⣻⣿⡥⠼⠇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣟⣛⣿⣯⡭⠿⠷⠶⣾⡾⠋⣩⡼⠁⣰⣶⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⣿⣿⣿⢟⣿⡽⢿⡯⠿⢿⡿⠿⢟⣻⡿⠶⠾⠾⡿⠛⢹⣿⠃⢀⣯⣶⡶⠆⠀⣸⠁⢠⣿⠁⠀⣿⡇⠀⣾
⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⣯⠀⠀⣹⣿⡿⣧⡟⠁⣴⣾⠟⢋⡉⠙⢿⠟⢁⣴⣶⠀⢰⡇⠀⣿⡟⠀⣸⠋⢠⣶⠆⠀⡏⠀⣸⣿⣧⣀⣉⣠⣤⡏
⠀⠀⠀⠀⢀⡾⠁⢠⣶⣶⣶⣿⠞⣋⣉⠉⠹⣏⠀⢠⡴⠁⠰⠿⢃⣠⡏⠀⣿⣿⠇⠀⣾⠀⠐⠿⠃⢀⣿⣄⣈⣭⣤⣼⠷⠶⢟⣛⣿⠉⠉⠉
⠀⠀⠀⠀⠈⢷⣄⡈⠙⠛⢿⣿⠿⠛⡋⠀⢸⡏⠀⣸⣇⠀⠰⠾⠿⢻⣷⣄⣬⣥⠄⢠⣿⠷⠶⠖⠛⠛⠋⠉⠉⠀⠀⢀⡀⢀⣿⣿⠇
⠀⠀⠀⠀⣠⣀⣈⣙⡷⠀⢸⠃⠠⠿⠇⠀⣾⣃⣀⣿⣿⣷⡶⠶⠾⢿⣍⣛⣋⣡⡴⠏⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⡿⠁⣼⣿⡏
⠀⠀⠀⢰⣏⣉⣉⣉⣠⡴⠟⠳⠶⠶⢞⣛⡛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⡿⠁⣰⣿⡟
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣈⠙⠿⣿⣿⣿⣿⣿⠟⠋⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣀⠉⠛⠉⠀⠀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⣶⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠄⢠⣶⣤⣤⣤⣤⣶⣿⣿⣶⣤⣄⣤⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣌⡛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣴⣶⡆⠀⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⣿⣿⣿⣿⡟⠛⠃⠘⠛⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠌⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠳⣦⣄⠀⣿⠀⣠⣴⠞⠃⠀⠀⠀⣰⢤⡹⣽⢃⣤⡆⠈⠀⠀⠀⠀⠀⠀⠶⠀⠰⠇
⠀⠀⠀⠀⠀⠰⣶⠆⠀⠀⠀⠀⠀⠀⠂⠀⢉⣓⢾⣞⡉⠀⢐⠀⠀⠀⠀⢈⣛⣻⣿⣟⣋⡁⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠠⡷
⠀⠀⢂⠨⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⣤⠾⠋⠁⣾⠀⠙⠻⢦⡀⠀⠀⠀⠻⠞⢱⣻⡜⠾⠇⠀⠀⠀⠀⠀⠀⢐⡴⠀⠰⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠤⣤⢠⣬⣼⣥⡤⢤⡤⠀⠀⠀⠀⠠⠄⠈⠀⠉⠀⠄⠠⠀⠀⠀⢀⡀⢀⢀⠀⠄⠀⢀⠀⠀⠀⠀⢀⠀⠀⠻⡦⠀⣴⠦⠄
⠀⠀⠘⢣⣿⣯⣝⣿⣿⡜⠁⠀⠀⠀⠀⠂⠀⠀⣤⡀⠀⠐⠀⠀⠀⠀⠀⠀⢀⡉⠀⢉⡤⠀⠀⠀⠀⠀⠘⠶⠖⠀⠀⠀⠉
⠀⠀⢰⣮⢟⣿⣽⣿⡿⢴⡆⠀⠀⠀⠀⠂⠀⠈⠉⠀⠀⠐⠀⠀⠀⠀⢠⡄⠡⢔⠀⠤⠀⢠⠀⠀⠀⠀⠀⠀⣰⠄⢀⡀⠘⠛⢧
⠀⠀⠉⠉⠀⠡⣭⡍⠀⠉⠉⠀⠀⠀⠀⠈⠁⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠃⠭⠘⢠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡄⢓⣿⢿⢠⡄⣠⡄
⠀⠀⠀⠀⠠⣀⣀⣀⠀⠀⠀⠀⠀⠀⢬⠑⢣⣄⣻⣠⡄⠪⡥⠀⠀⠀⠀⠀⠠⠠⠡⠀⠀⠀⠀⠀⠀⠀⢩⢻⡶⣍⣹⣩⡶⣟⣯⠁
⠀⠀⠀⠀⠠⠘⠛⠣⠀⠀⠀⠀⠀⠀⠴⢄⡠⠜⢻⠳⢆⡠⠦⠀⠀⠀⠀⠀⠀⠡⠡⠌⠀⠀⠀⠀⠀⠀⠰⣰⣥⠞⣿⠳⣮⣇⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠡⠆⡀⢭⢀⠰⠮⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠫⠆⡹⣿⢾⠱⠏⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠍
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣰⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳
⠀⠰⣶⢫⠧⣩⠉⣏⠴⡜⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠢⣤⣾⢾⣆⣤⠴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢙⡛⢴⢽⣿⡯⡦⢚⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⣽⣻⣯⣿⠋⠀⠀⠀⠀⠀⡀
⠀⢀⣮⣱⡄⠾⠭⠗⣀⢟⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣫⠷⢾⣹⡷⢿⣑⣄
⠀⠀⠉⠉⠁⢏⣒⡟⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣹⠁
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⡤⠀⣀⣀⠀⡀⣤⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢻⡀⣀⠄⢻⡟⡼⠀⠀⢠⢸⠁⠘⠁⢀⠀⠀⠐⠌⠀⣷⠀⡏⢆⠰⡅⣿⠘⠇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢣⢈⡎⣧⠇⠀⢸⣧⠀⡐⠋⠀⡘⠀⠀⢠⠎⠀⠀⠀⠘⣀⣷⡟⡁⠁⠀⢠⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠀⢹⠀⠀⢸⣿⠀⠁⠀⠀⣧⡤⠰⠋⠀⠀⠀⠀⠀⠀⣷⠰⠀⠀⠀⠀⢹⠆⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣆⠀⢸⣿⠠⠈⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣿⠻⠀⠀⠀⠀⠘⣋⣠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢽⠀⠀⡀⠀⠀⠀⢸⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⡷⣴⠒⠛⠁⠀⠀⠀⠀⢀⣠⣶⣶⣄⣸⣠⡾⣶⣦⡀⢰⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⠇⠀⠀⠀⠀⢀⣠⡶⢿⣿⣿⣿⣿⣿⡛⣅⠉⠙⠃⢸⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⡇⠀⠂⢀⣠⡶⠟⠉⠀⠀⠙⠻⣿⣿⣿⡇⢸⣿⣦⡀⠘⡇⢏⣀⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠉⠷⣿⣿⣇⣤⠞⠋⠅⠈⠀⢀⠀⠀⢠⠠⡄⡙⢿⡟⢸⣿⣿⣿⣷⣧⡇⠸⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠌⢀⣠⢼⣿⠛⡇⠀⢀⡶⠒⠲⠓⠐⠁⠀⠀⡗⣱⢾⣏⡼⣿⣿⣿⣿⣿⣿⠦⣀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢉⣠⠖⠋⠡⢼⡿⠀⢃⡴⠻⣤⠀⠀⢠⠀⠀⠀⠀⠀⢹⣾⡇⠃⠀⢙⣿⡟⣿⢶⣟⣿⡄⠆
⠀⠀⠀⠀⠀⠀⣤⡴⢞⠋⠀⠠⡄⡀⢸⣷⠓⡟⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⣿⡿⢀⣴⣿⣿⣿⡟⢸⣿⡟⡃⡄
⠀⠀⠀⠀⠀⠀⠈⠀⢻⠀⠂⠐⠛⠙⢾⣿⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡇⢾⣿⣿⣿⡏⡧⢸⡽⡆⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⢸⣿⠇⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⡟⢹⣿⣿⣿⡇⣷⢸⠇⣣⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠆⠀⠀⠀⠀⢠⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⣠⣾⣿⣷⡺⢿⣿⣿⣧⡏⠘⡇⣏⢸
⠀⠀⣤⣄⠴⣶⣼⣿⣾⠀⠀⠀⠂⡀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠐⢀⣾⣿⣿⣿⡟⡁⣾⣿⣿⢿⡇⣠⠃⣿⢸⣦
⠀⢸⣿⣿⡿⣿⣟⣿⣿⠀⠀⠀⠀⠁⢸⣿⣿⣅⠀⠀⠀⠀⠀⡄⠀⣰⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⢸⡗⣿⠀⡏⢸⡷
⠀⢸⣿⠿⠷⡯⠿⡿⠿⠀⠀⠀⠀⠀⢸⣿⣇⠋⠀⠀⠀⠀⠀⠁⠚⠻⡿⡻⠛⠟⢿⣧⠊⠟⠿⠿⢸⣷⡿⠟⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⡅⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠀⠀⠘⠛⠁⠀⠀⠀⢸⠋⠁⠀⠀⢰⡇
⠀⢀⠀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣄⡤⠼⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠚⡇
⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⣀⡀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢃⡈⢉⠈⠀⠀⢄⠀⢱⣆⣐⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢢⣌⠀⠀⠀⢋⣁⠼⠒⡉⠤⠖⠨⡑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢁⠀⠀⢰⡈⠠⡴⠂⠉⢀⠀⠀⠈⠙⠳⡤⠇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⢠⣔⣢⡄⢈⠁⢩⠀⠀⠀⢆⠀⠅
⠀⠀⠀⠀⠀⠔⠀⡀⠀⠀⠀⠀⠀⣄⠀⠠⠀⠀⠐⠁⠈⡀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠂⠐⠒⠒⠂⠒⠀⠂
⠀⠀⠀⠀⠐⠠⠀⢀⠀⠀⣀⡔⠀⢈⠂⠀⠀⡠⠈⠁⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠈⣀⠉⠀⣈⠈⠹⠀⡴⠁⡂⠀⠀⠀⢀⡀⠘⡄
⠀⠀⠀⠀⠀⢣⡈⠀⠀⠉⡀⡀⠉⠀⠉⠀⠆⢊⠐⠋⡂⠀⠈⠒⠀⠅⠀⠀⢀⠄⠀⠤⡀
⠀⠀⠀⠀⠀⡀⢱⢔⣄⣀⣀⡀⠀⠀⠀⠀⢃⠀⠀⠈⠃⠀⠰⠆⠀⠀⠀⣰⠃⠀⠀⠀⢈⡄⡀
⠀⠀⠀⠾⠀⠀⠤⠱⠤⡄⢼⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠍⠀⠀⡰⢃⠊⢐⡀⠠⠁⠂⡍⠂⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⢋⣖⠀⠀⠀⠀⠀⠰⠨⠬⠉⣂⢝⠡⡄⠸⣄⡐⠭⠔⢀⣀⠂⠀⢀⠀⠀⠄
⠀⠀⠀⣠⢀⠅⠄⠉⠈⠗⡧⡅⠀⠀⠀⠀⢀⣣⠁⠀⠤⢄⢀⢑⣷⡴⠤⠈⠀⢠⠤⠤⢆⡠⠶⠤⠘
⠀⠀⠰⠧⠀⠀⠑⠂⠈⠠⠁⣈⣄⠀⠀⠀⠀⠉⡄⠀⠀⠊⡀⠀⠈⠀⠀⠐⣠⢦⡤⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠂⠀⠀⠀⠀⠀⢨⡉⣩⣉⡅⠀⠠⡀⠀⠁⠀⠉⣕⣾⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠿⠄⠇⠀⠀⢀⢠⠀⠀⠀⠀⠴⠁

⠀⠀⠀⠀⠀⣦⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢤⣠⠄⣀⡴⠒⠂⠠⠤⣀
⠀⠀⠳⣬⣐⠿⠥⠀⣈⠀⢤⡤⠤⡶⢈⢡⡄⠐⣶⢲⣹⣏⣩⠧⢎⣿⠂⠀⡄⡙⣶⡤⢀
⠀⠀⠀⠀⠉⠻⣷⠒⢤⡗⢌⣿⠠⠿⠊⣸⠗⣴⠿⠤⠤⠄⠐⠒⠋⠭⢤⣾⣄⢢⡿⠃⠒⡦⢐⣂⣀⣀⡢
⠀⠀⠀⠀⠀⠀⠈⠑⠛⠓⠒⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢌⡑⠚⠤⠖⢰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠶⠯⠤⠀⠁

⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⣠⠖⠋⠀⠀⣀⣄⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⢿⣿⣿⣿⣶⡄
⠀⠀⡼⠁⢠⡖⢲⡀⢧⣬⠇⡴⠶⡄⢹⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠛⣿⣧⣀⣼⡿⠛⢿⣿⡄
⠀⢸⠃⠀⠈⠓⠚⠀⠀⠀⠀⠓⠖⠃⢸⠃⠀⠀⠀⣀⣴⠂⠀⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣦⣾⣿⠇⠀⠀⠀⠀⣠⠆
⠀⢸⠀⢰⡏⢹⡄⠀⠀⠀⠀⣀⣀⣠⠞⠀⢀⡤⢞⡟⠁⠀⠀⣿⣿⡏⠉⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⣀⣴⡿⠁
⠀⠘⣇⠀⠉⠉⠀⠀⡴⠋⠉⠉⠁⠀⢀⡴⠋⣰⠋⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠋⠉⠉⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠈⠳⣄⡀⠀⠀⣧⠀⠀⠀⠀⢾⡉⣳⠞⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠰⢾⣧⠟⠁
⠀⠀⠀⠀⠀⠉⠙⠒⠛⠀⡴⠋⠉⢳⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠀⢠⣾⣿⣷⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⣶⣶⣤⣤⣤⣤⣤⣤⣶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣶⣶
⠀⠀⠀⠀⢿⡇⣷⣶⣾⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⢸⡿
⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⢿⣿⣿⣿⡿⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠈⢻⣿⡿⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠙⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣝⠿⣿⣿⣿⣿⣿⣿⠿⣫⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣮⣛⠿⠿⣛⣵⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⣰⠏⠀⡀⠀⠈⢳
⠀⠀⠀⠀⠀⡟⠀⢨⡟⢃⠀⠀⣇
⠀⠀⠀⠀⠸⣧⠀⢸⡷⢾⣾⡀⢸
⠀⠀⠀⠀⠀⣿⡄⠈⢷⣄⡈⠻⣾
⠀⠀⠀⠀⠀⡟⢻⣄⠀⠀⣹⣦⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢣⠀⠸⣟⢉⡭⢹⡇⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠨
⠀⠀⠀⠀⠀⠈⢦⡀⣸⠃⠀⣺⠃⠀⣿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠑⡀⠀⠀⠀⠀⠀⢰⠏⡐
⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠸⡟⢀⣰⣿⡄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢠⣟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠹⣷⣁⣬⠿⠶⠟⠷⠶⢦⣤⡀⠀⠀⠀⠀⠈⠧⢠⡆⠀⡾⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠈⠛⠶⠖⣾⠞⠶⠶⣤⡀⠙⢶⡶⠤⠤⢄⡀⠀⠁⠀⡀⠀⠀⠴⠞⠛⠋⠉⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⣠⣎⡔⠋⢻⡄⠀⢱⠄⣀⡎⢹⠆⠀⠘⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣥⣉⠨⣇⠀⠘⢯⡀⠊⠀⠈⠙⠋⠀⠀⢸⠋⡿⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢧⡀⠈⠛⠦⣄⡀⢀⡀⠀⠆⠈⠛⠧⣄⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣾⣁⡵⠀⠀⣀⣀⠀⠀⢠⠊⢳⡄⠀⢹⠨⠝⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⠠⡇⢸⠋⠒⢇⠀⠙⣇⣰⡸⢹⠛⠛⠙⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣒⣶⡶⠶⠛⠃⠀⠛⠁⠀⠁⠀⠉⠓⠲⠶⠾⢧⣀⠈⠻⢶⣦⣴⣮⡶⢦⣄⡀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⣤⡀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⣶⠞⢋⣿⣆⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠸⣷⡏⠀⣹⡆⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡀⠀⠀⠀⢸⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠀⢠⡿⠅⢀⡟⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠐⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⣧⠐⣋⣹⡄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠻⣏⠉⠉⠙⣦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣦⡉⠛⢷⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡿⣶⣾⡇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢡⣸⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠉⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢹⣿⠁⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶⣄⢸⣿⢀⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠋⠀⠀⠀⣀⠀⠀⠀⠀⠰⣿⡀⠀⠀⣠⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣄⠀⠀⢸⣿⠀⠀⠀⢴⡆⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⣿⡇⠀⣸⣿⠁⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣨⣿⣆⡀⠀⣿⡇⠀⠀⠀⢠⣤⠀⠀⠀⢸⣿⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣸⣿⣴⡾⠟⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠋⠙⠿⣷⣿⣧⠀⠀⠀⠀⠀⣤⡀⠀⢸⣿⠀⠀⣰⡶⠀⠀⠀⢀⣠⣶⡿⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⢿⣦⣄⡀⠀⠙⣿⣄⢸⣿⠀⣰⡿⠁⣀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠉⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠉⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣜⣿⣾⣿⣴⣿⣷⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⢀⣀⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⠶⠀⠰⠆⠀⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠛⠃⠈⠉⠉⢉⣩⣿⢿⣿⣿⣿⡿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⢁⣾⠟⣿⡿⣿⡄⠉⠛⢿⣶⣄⡀⠀⠀⠀⠀⢀⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣶⣶⣶⡿⠟⠉⠀⠀⢠⣾⠏⠀⣿⡇⠘⢿⣆⠀⠀⠈⠙⠻⢷⣶⣶⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣴⣾⠟⣿⡿⠁⠀⠀⠀⠀⠘⠋⠀⠀⣿⡇⠀⠈⠉⣀⠀⠀⠀⠀⠀⢹⣿⠿⣷⣦⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⣿⡏⠀⢠⣿⠃⠀⠀⠀⠀⠸⠃⠀⠀⠀⣿⡇⠀⠀⠈⠛⠀⠀⠀⠀⠀⢸⣿⡄⠀⠉⢿⣏⠉⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⡾⠟⠀⠀⢸⣿⠀⠀⠀⠀⠰⠆⠀⠀⠀⣰⣿⣧⡀⠀⠀⠀⠺⠇⠀⠀⠀⠀⣿⡇⠀⠀⠈⠛⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣿⡇⠈⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣿⣇⡀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⢋⣉⣩⣭⣥⣤⣤⣤⣤⣭⣭⣍⣉⡛⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠛⠿⢿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⢸⣿⣿⣷⣶⣬⣍⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢙⣉⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢇⡸⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⣾⣷⡌⣿⣿⣿⣿⡿⢀⣭⣭⣭⡍⣩⣭⣭⣭⣭⡍⢭⣭⣭⣥⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣹⣏⢸⣿⣿⣿⣿⢁⣛⣛⣛⣛⠀⣛⣛⣛⣛⣛⣃⠘⣛⣛⣛⣃⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⠸⣿⣿⣿⡇⣼⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣧⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⡛⣛⣣⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⡿⢟⣩⣴⣶⣦⣍⠻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⠿⠿⠿⠟⠘⠿⠿⠿⠿⢿⣿⡶⢛⣩⣶⣿⠿⣋⣭⡙⠿⣿⣶⣍⡛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠻⣿⣿⣿⡆⢿⣿⣿⡿⢋⣥⣾⣿⠟⣋⣴⣾⣿⣿⣿⣷⣦⣙⠻⣿⣶⣬⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣷⡘⣿⣿⠀⣿⣿⢋⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣍⣃⡙⢻⡆⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⡆⢿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⡟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢻⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⡟⣡⣿⡟⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⣛⣛⣛⡀⠹⣿⣦⡙⠿⣿⣿⣿⠿⢋⣴⡿⠋⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠛⢿⣷⣬⣉⣥⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠉⠛⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠀⠀⠀⣀⣴⣿⣶⣶⣶⣶⣶⢰⡿⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠀⠀⣾⣿⣟⣿⣿⣿⣿⣯⣾⣿⡄⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠷⡾⣿⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⠻⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⡇⠸⣿⠛⠛⢻⣿⣿⣿⣿⣿⣯⣻⡿⠀⢀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢻⣿⢰⡇⠀⢻⣇⠀⠀⠙⢿⣿⣿⣿⣿⡿⠁⠘⠻⣿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⢀⣠⡿⠋⠀⢸⣿⠺⠃⠀⣀⣿⡄⠀⠀⠀⠉⠛⠿⠟⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⣿⣶⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠙⢿⣿⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⠀⠀⢻⣿⡄⠀⠀⠈⠉⠉⠘⡁⠋⠘⢙⢉⠈⢋⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠚⠀⠀⠀⠀⠀⢠⠂⠀⠠⢤⣤⣄⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣠⣠⠄⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣻⡟⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⣿⢧⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣠⠆⠀⠀⠀⡴⢛⡇⠀⠀⠀⣾⠀⡦⠂⠑⣾⢏⡞
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠁⠀⠀⠀⠊⢁⣼⠃⠀⠀⠀⣿⣶⠀⢠⣾⣿⣾⡅
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣩⣇⠀⠀⠀⠐⠀⣴⡟⠁⠀⠀⠀⠀⠙⠿⣇⠘⣿⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⢉⣿⣿⠄⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠡⡀⠹⣿⣳⠀⠀⠀⠀⠀⣠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣴⣿⣿⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠱⣴⡉⠉⠓⠀⠀⠴⢯⣠⣤⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣠⣾⣿⠿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢘⣿⣷⣶⣦⣤⣤⣀⣀⣨⡍⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⣾⣦⣾⡟⢻⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠀⠈⢻⣿⣿⣏⠀⠀⠈⢿⣄⣆
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠙⠻⠃⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⠓⣄⠀⡴⠻⣿⣧⡀
⠀⠀⠀⠀⢸⠀⣾⠟⢤⡀⠀⠀⠀⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣬⢻⣿⡀⠹⣿⡇
⠀⠀⠀⠀⠀⢿⠋⠀⣼⣷⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣇⠀⠛
⠀⠀⠀⠀⣀⠀⠀⣼⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠘⣯⡈⠢⡀
⠀⠀⠀⢸⣟⠀⣰⣿⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡿⣿⣦⠈
⠀⠀⠀⡿⢿⣿⡿⢿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⢯⠉⠀⠈⢻⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡲⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢰⠏⠂
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠂⠀⠀⠠⠒⠀⠚⠀⠀⢀⡠⠀⠂⠃⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⢸⡆⣰⣧⣆⠒
⣴⣧⣿⣇⡆⠀⠀⠘⣯⠐⠂⠀⠀⢲⡄⠀⠀⠂⠻⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢰⢷⠀⢠⣾⢷⣭⠌⠛⢶⠒⢠⡆
⣻⡟⢫⣿⣧⣷⣏⡠⠀⠔⠐⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡔⢯⡀⢘⡁⢂⠪⡦⠴⡘⢦⣝⢳
⠚⣯⡶⢹⣿⣿⠋⠂⠀⠂⠐⡈⢺⣶⣆⣤⣀⣤⢀⠀⠀⠀⢀⡀⣤⢀⣠⡊⠃⣴⠞⠃⢜⡺⠤⠉⠱⣿⠀⠈⠀⠙⢿
⢈⠁⠖⢸⣉⡤⠎⣠⠆⠀⠀⠐⠲⣿⣿⡿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠈⠈⠺⣂⠘⣯⡤⠒⣀⡠⣜⡀⠀⠈⠶⢾
⠸⠄⢠⣟⡉⠠⠠⠶⢀⣰⢿⠑⢠⣙⣯⢓⣹⣭⠟⣋⡁⠀⠀⠀⠙⢿⠀⠈⠙⠳⣎⢢⡟⠒⠉⠈⠣⠌⡂⠀⠠⡀
⠀⠀⢠⡏⠠⢰⠂⢠⠊⠙⢄⣯⡀⠀⠉⠁⠀⢄⣪⣷⣻⣓⣦⣴⡶⠂⠀⡀⠁⣌⠚⢿⣗⠄⣇⠀⢰⡆⠈⠲⡀⡀⠑
⠀⡰⠋⠀⢀⡬⣟⠀⠀⠀⢣⣀⠧⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠉⡋⠀⠀⠉⠐⠉⠄⠀⠁⢿⡁⣀
⠀⠀⠀⠈⠀⠀⠀⠙⠊⠀⠊⠁⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⠘⠉⠀⠄⠐⠤⠦⠀⠽⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣟

⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⢀⣨⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣲⣫⠄
⠀⠀⠀⠀⢠⣄⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣍⣩⢧⠶⠂
⠀⠀⠀⠀⣄⣈⣱⣆⣠⠤⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣤⣩⣿⣦⣶⠛
⠀⠀⠀⢠⡌⠉⢡⣎⣁⣀⡄⠀⣠⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣉⣤⣾⣥⡼⠳⠶⠾⠃⢀⠀⣠⠄
⠀⠀⠀⠀⠓⠚⣯⠉⢉⢁⣠⠀⠋⡀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⠀⡞⠀⠀⠀⢰⠃⠁⢺⣤⠿⢶⣧⡾⢿⣶⠶⠾⠋⠊⠀⠟
⠀⠦⠐⠶⡴⠾⢛⡓⠟⠉⠁⠀⠀⣧⠀⡀⢀⡀⠀⠈⢧⡄⠠⢠⠆⠀⠐⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⣟⣫⡿⠶⢟⡿⠶⠶⠂
⠀⠀⠶⣦⣀⣴⠞⠛⣶⣄⣼⠻⠇⠘⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣳⣫⣿⣭⣿⣟⣻⡷⢶⡾⠇
⠀⢀⣤⣬⣡⣂⣴⣾⣇⠈⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣋⣽⠶⠖⠺⢛⣿⣷⣤⣤⣶⠄
⠀⠉⠉⠉⠙⡉⠉⣹⡋⠛⠋⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠛⠶⠼⠿⠿⣿⣧⣤⣤⡤⠆
⢀⡝⠛⢿⣍⣠⠾⠻⡿⢷⠆⣦⠆
⠈⣿⠏⣉⣈⠈⣀⢾⣧⣀⠀⡾⠟
⠀⠀⠀⠀⠿⠋⠏⠈⠙⠛⠋⠁
⠀⠄

⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠉⠉⠉⠉⠙⠻⢶⡀⠀⠀⠀⠀⠀⢠⡶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠷⣦⠀⠀⠀⠀⠀⠀⣴⠞⠛⠉⠉⠉⠉⠉⠉⠉⠙⢻⢶⡄⠀⠀⠀⠀⠀⢀⡶⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⢀⣴⠄⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⢠⣦⡀
⣤⣤⣤⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣤⣤⣤⣤⣤⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣤⣤⣤⣤⣤⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣤⣤
⠀⠀⠀⠁

⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⡏⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⢹
⠀⠀⠉⠒⠒⠒⠒⠉⠀⠀⠈⠓⠒⠒⠒⠊⠀⠀⠀⠑⠒⠒⠒⠚⠁⠀⠀⠉⠒⠒⠒⠒⠁⠀⠀⠈⠓⠒⠒⠒⠊⠀⠀⠀⠑⠒⠒⠒⠚⠁⠀⠀⠙⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢀⣿⠃⠠⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⢸⣿⠀⠀⡄⢉⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⢸⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠤⠤⣀⣀⣀
⠀⠀⢸⣿⠀⢺⣟⠲⢀⣤⡀⠀⠨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⣐⠒⠶⠄⠀⠀⠀⠀⠀⢰⡿⠒⠒⠓⡲⣌⢳⡤⣍⡒⠦⣄
⠀⠀⠀⣿⡄⢸⣿⣭⡀⠉⠄⡀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠰⠙⠂⠀⠀⠀⠀⠀⠀⣸⡧⠤⠤⠤⠵⣌⠛⠅⡀⣉⠳⡌⢣
⠀⠀⠀⠸⣷⡘⣯⡿⣯⣒⠄⢁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠓⠛⠛⠂⠉⠉⠉⠒⠮⡟⢠⣷⢿
⠀⠀⠀⠀⠹⣧⠈⠿⣲⣿⣛⣦⣄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇⢸
⠀⠀⠀⠀⠀⠹⣷⡀⠙⢹⣿⣿⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⣇⣸⢱⡌
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠙⣿⣿⣿⡿⣅⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣇⡞⠀⠏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠙⠙⠙⠻⠿⢿⡶⠻⠿⣖⡶⠴⠶⠓⠛⠀⣀⣴⠾⠋⠀⠀⢀⡤⠶⠚⠛⣉⣉⣉⣉⣉⡝⠛⠚⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠟⠋⠁⠀⠀⣠⠞⠁⣠⡴⠞⠋⠉⠉⠉⠉⠉⠉⠛⠶⣤⡈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣰⠃⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⢻⡆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⡁⠀⢀⡠⠤⠶⠖⠲⠶⠾⢧⣀⠀⠀⠀⣿⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣧⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣼⠇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣄⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⢀⡼⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣼⡇⠀⠀⠀⠀⠀⢀⠀⠠⠖⠚⠉⢀⣠⠔⠋⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣇⡇⠀⠀⠀⠀⠀⢸⠶⠶⠖⠒⠋⠉⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣠⠤⠖⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠰⣦⣄⣀⣠⣤⣤⡀⠀⠀⣰⠿⣶⣎⣠⣀⠀⠀⣠⡀⠙⣠⣄⡀⣀⠀⠀⣾⠛⠛⣛⢿⣦⣠⣴⡾⠛⢳⡄⠀⢸
⠀⠀⠀⠈⠙⠻⣭⠧⢸⣿⣿⣿⢿⣷⣾⠟⠿⢿⡛⠀⠀⠀⠀⠀⠉⡩⣿⣶⠾⣯⣽⣤⣿⣿⠿⣿⣿⣶⣠⣼⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠷⣦⣼⠿⠉⠀⠉⠁⠀⢠⠧⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠈⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠆⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⢉⣀⣀⣀⣀⡉⠙⠛⠛⢿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠻⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣾⣶⣶⣶⣤⣤⣤⣄⡀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠉⠉⠙⠛⠛⠿⠿⢿⣿⣷⣦⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠛⠉⡀⠀⣀⣀⡉⠉⠛⠛⠻⢿⣿⣦⣀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣶⣤⡀⠙⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣤⠀⠈⠻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣶⣤⣤⡀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⠋⠀⠀⠀⠉⠙⠛⠛⠻⣿⣷⡄⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⣤⠙⢿⣷⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣸⡿⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⣴⣿⠃⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⣸⡿⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⣸⣿⠁⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⣿⡿⠀⠀⠀⣸⣿⡿
⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⣿⣿⠇
⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⣿⡿
⠀⠀⢀⣿⣿⠁⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣿⣿⠇
⠀⠀⣼⣿⡟⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢰⣿⡇⠀⠀⠀⢰⣿⡏
⠀⠀⣿⣿⠇⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⣸⡿⠁⠀⠀⢀⣿⣿⠇
⠀⣸⣿⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢸⣿⡏
⠀⣿⣿⠃⠀⠀⢀⣤⣿⣿⣤⣀⣀⠀⢸⣿⠀⠀⠀⢠⣿⣿⠃
⣰⣿⣿⣠⣴⡿⠿⠛⠉⠉⠉⠹⠿⢿⣿⣇⠀⠀⠀⣸⣿⡏
⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡀⢀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⣾⣿⡟
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋
⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁
⠀⢸⣿⣷⣤⣶⣶⡄⠀⣰⣿⡿⠃
⠀⠘⣿⣿⠟⠀⠙⣿⣿⣿⡟⠁
⠀⠀⣿⣿⠀⠀⢠⣿⣿⠋
⠀⠀⣿⣿⣀⣴⣿⡿⠃
⠀⠀⢹⣿⣿⣿⠟⠁

⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⢠⡀⠀⣶⡀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆
⣸⡆⢀⠀⠰⠇⠀⠀⣿⡀⠀⠀⣧⠀⠈⠀⠀⡀⠀⠸⠇⠀⠈⠀⠀⢸⡃⠀⠸⠃⠀⠀⠀⣧⠀⠀⠀⠀⠘⡾⠀⠀⠀⠀⣷⠀⠀⠀⣞⡄
⠈⠀⣿⡄⠀⠀⣄⠀⠛⠁⠀⠀⠙⠀⠀⠀⢸⣧⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⢰⢆⠀⠋⠀⠀⢠⠀⠀⠀⠀⢰⣣⠀⠉⠀⠀⠀⠙⠁
⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⢠⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣏⡇⠀⠀⠀⠈⠋⠀⠀⠀⠀⢠⡄⠀⠀⢸⣂
⢠⡀⠀⠀⠀⠀⠉⠀⠀⠀⢠⡶⠀⠀⠀⣄⠀⠀⣰⡀⠀⠘⠃⠀⠀⡀⠀⢸⡇⠀⠀⠀⣀⠀⠀⠀⠙⠁⠀⠀⠀⠀⣄⠀⢸⡃⠀⢠⡗⠀⠀⠘⠋
⢸⡇⠀⠠⣿⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⣽⠆⠀⠸⠃⠀⠀⠀⠀⢀⣿⠀⠘⠃⠀⠀⠀⣯⠀⠀⠀⠀⠀⡀⠀⠀⠸⣼⠀⠈⠁⠀⠀⠁⠀⠀⠠⣆
⠀⠀⠀⠀⠉⠀⠀⢸⣥⠀⠀⠸⡇⠀⠀⠈⠀⢀⠀⣀⡠⠤⢤⠤⠼⣿⣄⣀⣀⠀⠀⠀⠉⠀⠀⠀⠀⠸⣵⠀⠀⠀⠈⠀⠀⠀⢀⡄⠀⠀⡀⠈⠉
⠀⠀⢠⡄⠀⣴⡀⠀⠁⠀⠀⣀⡤⠴⢒⠾⣛⠽⣋⠤⣒⠭⠓⠊⠁⠈⢣⢰⠂⣉⠉⠙⠓⠶⢶⣤⣀⣤⣉⣀⠀⠀⠀⠀⠀⠀⢸⣱⡀⢠⢛
⠀⠀⠰⡿⠀⠙⠀⠀⣠⣴⠛⠁⣠⡾⢑⡩⣰⠝⣊⠥⠖⠊⠁⠀⠀⠀⠃⣂⢓⠤⣐⠆⠄⠀⠀⠀⠙⠳⢤⡩⢉⡓⠲⠤⣄⡀⠈⠛⠀⠈⠋
⢰⡄⠀⠀⠀⣠⠔⠭⠭⠁⢀⠞⡩⢒⠕⣊⠔⣊⠥⢂⠁⠀⠀⠀⠀⠀⠀⢹⡀⡡⠔⡒⠨⠥⠀⠀⠀⠈⠁⠙⢷⠊⠕⡂⠄⠉⠲⢄⠀⠀⠀⣷
⠸⠇⠀⣠⢊⠑⠨⠍⠀⢀⠏⡨⢊⠵⣊⠕⡪⢔⣊⠁⠀⠀⠀⠀⠀⠀⠀⠈⡅⡊⠥⠐⠂⠉⠥⠀⠀⠀⠀⠈⠀⠳⣭⢔⡂⠥⢀⠀⠙⢦⠀⠉
⠀⢀⠮⣄⢄⡀⠁⠀⠀⣼⣡⢞⡥⡪⣕⠯⣒⠭⣒⠠⠀⠀⠀⠀⠀⠀⠀⠀⢧⡰⠌⢁⡂⠤⠀⠁⠀⠀⠀⠀⠀⢂⠹⡯⢒⠩⠅⣀⠀⠠⡱⡄
⢠⡯⠾⠼⠯⠤⣖⡠⠀⣏⠔⡡⡪⢋⣥⣊⣴⣪⣤⣊⣡⣀⡠⢀⠀⠀⠐⠀⠨⢒⡩⢔⡂⠬⣐⣒⠲⠶⢀⡀⠀⠀⣂⠹⡥⢂⡡⣔⡨⠥⠵⣽⡄
⠛⠀⠀⠀⠀⠀⠀⠈⢰⣧⠿⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢾⣴⣄⠀⢰⠄⣪⡵⠾⠛⠒⠒⠒⠒⠒⠒⠒⠾⠿⢆⣷⠟⠋⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⠑⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⡎⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢿⡀⠀⢠⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡉⠛⣁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⣀⡠⠀⡠⢄⡠⢂⠤⢀⠤⢀⠤⢀⠤⢀⠄⡠⢀⡄⣠⢄⡀⢀⠀⢀⢀⡀⣀⠀⡀⢀⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠘⠁⠚⠁⠒⠁⠂⠡⠊⠵⠊⠔⠥⠊⠕⣩⠔⡡⢊⠔⡡⢊⠕⢋⠔⡡⢊⠕⡫⢒⠕⡢⢖⠕⡡⢒⡴⣢⠔⡡⢂⠔⡠⣠⢄⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠛⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠤⠠⠀⠀⠠⡄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⢀⢠⢰⡼⠚⠉⠉⠙⠢⡀⠐⢥⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⢠⢐⡨⠞⠁⠀⠀⠀⠀⠀⠀⠹⡀⠠⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⢐⣨⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠘⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⢲⠎⠀⡜⣵⣯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢔⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢁⢀⠄⡘⡀⠀⢸⣯⣦⠀⠙⡆⠀⠀⠀⠀⢀⡤⠞⢁⢠⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠄⢎⡥⠞⠉⢷⠲⣀⠈⢻⡮⠁⠀⢸⠀⢀⡠⠖⠁⢤⣰⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠌⣨⠞⠁⠀⠀⠀⠈⢣⣙⢷⣼⠇⠀⠀⣿⢏⢡⠠⣐⠼⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠴⠞⠚⠓⠤⢄⡀⠀⠀⠀⠀⢈⠽⠁⠀⡀⣼⣼⡼⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⣰⢰⣶⡶⣦⣙⣦⣀⡤⠞⠁⢀⠠⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⡀⠄⣪⡷⡻⡳⠤⠬⠿⠛⠋⢁⢀⡀⡜⣊⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠄⢒⡥⠞⠙⠧⣇⣡⢱⢠⢰⠰⠐⣊⡬⠵⠊⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⢠⠰⣨⠔⠉⠀⠀⠀⠀⠀⠉⠙⠓⡶⣾⡛⡏
⠀⠀⠀⠀⠀⡴⠁⠀⢀⡜⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠻⡲⠃
⠀⠀⠀⠀⢰⠁⠀⢀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⡐⣣⠞⠁
⠀⠀⠀⠀⡇⠀⠀⠼⠐⣇⠀⠀⠀⠀⠀⠀⣠⠖⠉⡀⢆⡴⠊⠁
⠀⠀⠀⠀⢳⠒⣒⡂⠀⠈⠢⢤⣀⡤⠖⠊⠁⡠⢘⡴⠋
⠀⠀⠀⠀⠀⠳⣄⠒⡤⢀⡀⡀⠀⠀⢠⠐⣢⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠥⡮⢝⣁⠥⠖⠋⡁⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⢀⡀⢀⡀⢀⡀⢀⡀⢀⡤⢀⠀⡀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠂⠵⢊⡴⢊⠴⢊⡴⢊⠔⣣⠔⣉⠔⢋⠴⣊⠕⣫⠔⣩⠖⣩⠔⣩⠖⣩⠞⣡⠖⡡⠞⣡⢖⡥⢊⠔⣫⠞⡵⣫⠞⡵⢋⠞⡵⣣⠞⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠀⠁⠀⠁⠀⠁⠀⠁⠀⠀⠀⠁⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣤⣶⣤⣀⠀⠛⠁⠀⠛⡅⠀⠀⠀⠛⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⣀⣤⣤⣴⡶⠶⣦⣤⣤⠗⠀⢠⣤⣌⢳⡿⠉⠹⣣⠀⠀⠄⠀⠀⢀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠩⣿⣿⣿⣉⠻⣿⡗⠀⣾⣿⡷⠀⠤⠀⠀⠀⠙⠗⠀⣤⣤⠀⣰⠆⠰⠴⠛⠁⠀⣤
⠀⠀⠀⠀⠀⢀⣀⡀⣤⡤⠾⣿⣿⡆⢘⡿⠛⠛⠛⠾⠶⠶⠿⠟⠁⠀⠀⠰⠀⠀⠀⠀⠀⢀⣀⡀⠉⠆⠀⢀⣀⠀⠀⠀⠘
⠀⠀⠀⢠⡾⢋⡀⢿⣿⣷⡾⠟⠻⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⣀
⠀⠀⠀⢹⣷⣿⡿⢀⠟⠁
⠀⠀⠀⣈⣴⣿⣿⡁
⠀⠀⣾⠟⢿⡿⠋⣷
⠀⠀⠹⣶⣿⣦⣠⣿⠃
⠀⠀⠘⠂⢹⡿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠾⠷⣶⣦⡀
⠀⠀⠀⢀⣷⣿⣯⣅⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠛⠉⣿⣿⣷⠀⣴⣯⣝⣗⠄⣀⣤⣀⠀⢠⣭⠴⢶⣤⢤⡀⣀
⠀⠀⠀⠻⣏⠙⠻⢿⣘⣾⢧⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⠻⣿⣿⣷⣄⣸⡿⣿⣖⣛⣫⣿⠉⣳⣾⣯⠙⢏⣾⠃⠀⠀⠀⠈⠉⠑⠚⠝⢶⡄
⠀⠀⠀⠘⠮⣓⠤⣼⣿⣿⢓⣮⣷⡷⠗⣲⣥⣾⠋⣿⣿⣿⢀⡨⣻⠟⠋⠁⠀⠀⠀⠀⠈⠻⠧⣿⣿⣿⢆⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠻⣿⣿⡷⠐⣿⣧⣿⣷⡿⠖⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⣛⣤⣔⣮⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢀⣠⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣀⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡽⣤⠚⠉⢣⢀⡄⠀⠀⠀⠀⠀⠀⢠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣤⣠⡷⠒⠁⣪⠎⠀⠀⠀⠀⠀⠀⠙⢻⢷⣆⡉⣷⣄⣂⡶⠐⠒⠒⠒⠤⣤⣄⣀
⠀⠀⠀⣀⣠⣤⣐⢒⣒⠶⣴⠤⠤⣀⣀⠀⢠⡞⠁⣴⣋⣈⣀⡠⠐⠈⡀⠀⠀⠀⠀⠀⠀⣎⣔⣺⣀⠣⢧⣽⣠⣿⣡⡾⢿⡏⠉⡓⡌⣿⣷⢽⡓⠒⣠⣤⡀
⠀⠀⠀⠀⠉⠉⠉⣿⠻⡅⢸⢳⠶⣤⣍⢲⢾⠇⣾⠉⡋⢩⠟⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠙⠛⠓⠶⠤⣍⣩⣤⣾⣽⣶⣾⣿⣿⣧⣼⠋⣽⠟⠻⠷
⠀⠀⠐⠒⠾⠷⠶⠿⠦⢬⣭⣧⡀⠛⠛⣚⣋⣉⣠⠴⠟⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃


⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣶⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣙⣿⣡⠄⢄⢸⣇⡤⠀⠀⠀⡆⣶⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠑⢄⠀⢹⡏⢀⡴⠃⢀⡿⠯⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡆⠀⢰⠈⠀⣤⠑⢼⡷⢋⣄⠀⠈⡆⠀⢰⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢤⣠⠀⡄⠠⠤⠷⢇⠀⢸⠀⠘⣿⠇⣸⣇⠸⡿⠀⠀⡇⢀⡼⠯⠤⠄
⠀⠀⠀⠀⠀⠀⢘⣛⣷⣇⢀⣀⣀⣠⣵⣾⠀⠀⡠⠊⢸⡏⠳⣄⠀⢀⣷⣋⣀⣀⣀⡀⠀⢀⡔⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠁⠀⠀⠀⠀⠀⡱⢎⠀⢠⢾⣧⡀⢈⡵⢏⠀⠀⠀⣀⠀⠐⠚⡻⣟⠻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣄⠈⠳⣄⠺⠿⠗⣠⠞⠁⢀⡝⢇⢸⣇⡽⠫⡀⠀⠑⢄⠻⠿⠛⣠⠊⠀⡨
⠀⠀⠀⠀⠀⠰⠶⠾⡷⠶⠾⡷⠶⢾⠷⠶⢶⡷⠶⢶⣿⣿⠶⠶⣾⠷⠶⢾⡷⠶⢾⡶⠶⢾⡶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⡊⠀⡠⠋⣤⣶⣦⠑⢄⠀⢉⡶⣋⢸⡇⢱⣞⠁⢀⡴⠋⢴⣾⡦⠙⢦⡀⠙
⠀⠀⠀⠀⢀⣀⣦⣽⣮⡤⠀⠀⠉⠀⠀⢀⡵⢋⡀⠈⢻⡷⠃⠀⡱⢎⠀⠀⠀⠀⠀⢀⡀⠉
⠀⠀⠀⠀⠀⠀⠋⠜⠁⠀⠈⠉⠉⢉⡽⣿⠀⠀⠙⢦⣸⣇⡠⠊⠀⠀⡟⢏⠉⠉⠉⠀⢹⢿⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⡿⡏⠀⢹⠀⠀⣾⡆⢹⡏⢰⣿⡄⠀⡇⠀⣷⢖⠒⠂⠘⠀⠋⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠇⠀⠘⡀⠀⠙⣡⢾⡗⣄⠛⠀⠀⠇⠀⠸⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣲⣾⠁⢠⠞⠁⣸⣇⠀⠑⢄⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠿⠸⠀⠀⠀⠚⢹⡇⠑⠐⢋⣿⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠉⠉⠉⠉⠉⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣠⡾⠛⠉⠉⠻⣦⠀⠀⣀⡀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣼⡟⠀⠀⠀⠀⠀⢹⡆⠀⠉⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣤⡿⠿⠛⠻⢿⣦⡀⢠⡟⠀⠀⠚⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⡴⠛⠁⠀⠀⠀⠀⠀⠙⣷⡾⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢠⡶⠋⠃⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣿⠀⠀⠀⣴⠿⠿⠛⠁⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣼⢷⣤⣤⣤⣤⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⣼⠷⣦⣼⠏⠉⠁⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⡟⠀⠀⠀⠀⣸⠇⠀⠈⢻⣆⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢸⣇⠀⠀⠀⠀⢹⡇⠀⠀⠛⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣴⢿⣄⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣰⣏⣀⣽⣷⣤⣤⣴⡏⠀⠀⠲⠋⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡴⠟⠉⠉⠉⠉⠙⢿⣿⠁⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠋⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡾⠛⠃⣠⡶⢿⡾⠟⠁⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠘⠇⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠀⠀⠀⠀⣼⣏⡛⣿⠶⠶⠶⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⡴⢋⣿⠁⠀⠀⠀⣼⠇⠈⠙⣯⡀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⡼⠁⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠈⢻⣤⠟
⠀⠀⠀⠀⠀⢸⠁⠀⠸⣷⡀⠀⠀⠀⠈⣷⠀⠀⢲⣌⡯
⠀⠀⠀⠀⠀⡏⠀⠀⢀⡿⢷⣦⣀⣀⣀⣿⠀⠀⢠⣿⡟
⠀⠀⠀⠀⢰⡇⠀⠀⣸⡧⠶⠟⠿⠿⣿⡟⠀⠀⠉⢸⡇
⠀⠀⠀⠀⠈⣷⡴⠛⠁⠀⠀⠀⠀⢀⣿⠁⠀⠀⢀⡿
⠀⠀⠀⠀⣠⠏⣠⠖⠦⣠⣴⣶⣶⡿⠁⠀⠀⠀⣼⠃
⠀⠀⠀⣴⠃⠸⡇⢀⣾⠋⠀⠉⠁⠀⠀⠀⢀⡼⠁
⠀⠀⣼⠃⠀⠀⢀⣾⠻⣦⣤⣀⣀⣀⣤⠖⠋
⠀⢰⡏⠀⠀⠀⣾⠃⠀⢠⡏⠉⢉⡾⠁
⠀⢸⠃⠀⠀⢸⡇⠀⢀⡞⠀⢨⡟⠁
⠀⢸⡇⠀⠀⠈⠳⠶⠋⢠⣴⠏
⠀⠘⢿⣄⡀⠀⠀⣰⣾⠟⠁
⠀⠀⠈⠛⠻⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠛⠀⠀⠒⠈⠁⠀⠈⠀⠀⠸⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠒⠒⠒⠲⠶⠶⠢⠲⠄⠀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣤⣤⠤⢤⣄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠠⠤⡤⠤⣤⣤⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠤⠤⣤⣤⠄⠀⠀⢀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠀⠀⠀⢀⣀⢀⣶⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣷⣤⣀⣈⠙⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢍⣀⣄⣀⣈⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢡⣄⣀⣀⣄⣄⣀⣀⡀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠠⡀⣀⣠⣀⣠⣄⣀⣀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠉⠉⠉⠉⠉⠁⠁⠙⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠚⠂⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠐⠲⠶⠶⠶⠖⠀⠀⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠠⣾⣿⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⣤⣤⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠯⠙⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡿⢋⣡⣤⣤⣤⡀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢁⠀⡀⢀⠀⣀⣀⠀⡀⢻⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⣷⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠀⠘⠒⠛⠛⠛⠓⠒⠀⠀⠀⢉⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠞⣶⠀⠘⠒⠓⠚⠋⠘⠛⠀⠀⢸⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡿⠟⠛⠉⠁⠀⠈⠙⠳⠦⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⠿⠏⠁⠈⢻⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠛⠛⠿⠋⠉⠻⣿⣿⣿⡿⠟⠻⣿⣿⠿⠛⠁⠀⠀⠀⠘⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣋⣀⣠⣤⡼⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⢿⠿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠙⣆⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⢣⡀⠀⢀⣴⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣦⡀⠀⠀⠈⢀⣿⡁⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠙⣿⣿⣦⠀⠀⢸⣿⣿⣷⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢻⣿⣷⣄⠘⠿⣿⣿⣿⣿⣿⣦⣤⣤⠞⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⠻⣿⣷⣄⡀⠈⠙⠛⠿⠿⠏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⣀⠀⠀⠀⠈⠉⠓⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣶⣦⣤⣤⠄⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠓⠶⢾⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠙⠛⠛⠷⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⢿⢻⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣟⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢁⣤⣤⣤⣄⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣤⣤⣤⣄⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⡙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣁⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣉⠛⠻⢿⣿⣿⠿⠛⢋⣡⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⡉⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡼
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⠿⠟⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻
⠀⠀⠀⠀⠀⠀⣿⣿⣇⣴⣶⣦⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢷
⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣸
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⠿⠟⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡾
⠀⠀⠀⠀⠀⠀⣿⣿⣇⣴⣶⣦⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡾
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⠿⠟⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣇⣴⣶⣦⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⢼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠉⠋⣿⣿⣿⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣿⡇⠀⠀⢸⣿⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠙⢿⡇⠀⠀⢸⡿⠋⠀⢘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⢻⣷⡄⠀⠈⠃⠀⠀⠘⠁⠀⢠⣾⠟⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⣿⠟⠉⠻⣿⡄⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⢠⣿⠟⠉⠻⣿⠀⣀⣤⣤⣀
⠀⠀⠀⠀⣼⡿⠋⠙⠻⢿⣿⣧⠀⠀⠹⣿⡄⠀⠀⠀⠈⢿⣧⡀⠀⠀⢀⣼⡿⠁⠀⠀⠀⢠⣿⠏⠀⠀⣼⣿⡿⠟⠋⠙⢿⣧
⠀⠀⠀⠀⠹⣷⣤⡀⠀⠀⠈⠛⠦⠀⠀⠹⣷⡄⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠏⠀⠀⠴⠛⠁⠀⠀⢀⣤⣾⠏
⠀⠀⠀⠀⠀⠈⣹⣿⣷⣦⣄⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⢀⣠⣾⠏⠀⠀⠀⠀⠀⣠⣴⣾⣿⣏⠁
⠀⠀⠀⠀⠀⣼⡿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣤⣿⡇⠀⠀⢸⣿⣤⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⣇
⠀⠀⠀⠀⠀⠸⣿⣦⣤⣤⣤⣶⣶⣶⣶⣶⣦⣀⠀⠀⠈⠙⠻⡇⠀⠀⢸⠟⠋⠁⠀⠀⣀⣴⣶⣶⣶⣶⣶⣤⣤⣤⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⠿⣶⣶⣶⣶⣶⡿⠛⠉⠀⠀⢀⣠⡆⠀⠀⢰⣄⡀⠀⠀⠉⠛⢿⣶⣶⣶⣶⣶⠿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⢺⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⣿⡇⠀⠀⢸⣿⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡗
⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠟⠀⠀⠀⠀⠀⢰⣿⠟⠋⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠙⠻⣿⡄⠀⠀⠀⠀⠀⠻⢿⣿⣿⡟⠁
⠀⠀⠀⠀⢠⣾⠿⠋⠁⠀⠀⣀⡄⠀⠀⢠⣿⠇⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠸⣿⡄⠀⠀⢤⣀⠀⠀⠈⠙⠿⣷⡄
⠀⠀⠀⠀⢿⣇⡀⢀⣠⣴⣿⡿⠀⠀⢠⣿⠏⠀⠀⠀⠀⣴⡿⠃⠀⠀⠘⢿⣦⠀⠀⠀⠀⠹⣿⡄⠀⠀⢿⣷⣦⣄⡀⢀⣼⡿
⠀⠀⠀⠀⠈⠛⠿⠿⠛⠉⣿⣇⠀⣠⣿⠏⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠹⣿⣄⠀⣨⣿⠉⠛⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠀⠀⠀⣰⣿⠋⠀⠀⡀⠀⠀⢀⠀⠀⠹⣿⣆⠀⠀⠀⠙⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡁⠀⢀⣼⡇⠀⠀⢸⣧⡀⠀⢘⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⡿⣿⡇⠀⠀⢸⣿⢿⣶⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣾⠿⠁

⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠠⢟⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡄
⠀⡼⣤⢻⡏⠱⣾⣶⣶⢩⣶⠞⢢⣤⣀⠺⢢⢀⡀⢸⠥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⣾⣷⣯⡷⠶⢠⡄⣀⣀⠬⡀⣀⣰⢸⡇⢰⡀
⠀⠾⣋⠴⡏⠲⣿⣿⠗⢏⠋⢘⡋⣻⠕⣇⠟⣿⠷⠞⣎⣘⡆⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⣼⢆⣸⡳⡔⢊⡅⣳⡓⣜⢸⣿⡿⡟⠠⡧⠽⣏⡙⢧⣽⣇
⠀⣟⠹⣤⣑⡾⣱⠹⡆⣵⡤⠾⡀⢟⡅⢻⢆⣋⣄⣨⢥⡭⠌⣿⣾⢶⡎⠿⣶⣶⡟⣿⣿⢴⡾⢇⢺⡃⢩⡁⣨⣿⡙⢤⣽⣷⠔⡳⢈⠲⣯⠤⣾⣦⡽
⠀⠑⠀⣾⡹⢆⣯⢀⣿⣇⡓⢛⣀⣋⢱⢾⣯⠔⡰⢣⡜⢻⠳⢼⢷⣸⠗⢂⠸⣿⢏⢹⢇⢚⣝⠺⣿⢳⡏⢻⠀⢇⠑⣤⡈⡛⠁⢤⠘⡄⢼⡐⡗⡸⢧
⠈⠀⠈⠻⠁⠴⣏⠹⠬⠘⠁⠛⠋⡔⢦⠟⣃⠹⢤⠎⠱⡌⢳⠦⡛⠇⠨⠤⠀⠘⠉⠋⠑⠃⠔⠬⠿⠞⠳⣼⡓⠢⣥⠆⡸⠙⡆⣨⠀⢰⢀⠛⡥⢁⠲
⠀⠀⢀⡆⠠⡆⡀⠓⢌⡁⠀⠀⠀⠀⠀⠀⠋⠁⠉⠓⠀⠘⠛⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢷⠀⢦⠈⠈⢶⢌⠛⡛⠒⠀⠀⠀⠀⠈⠉⠂⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠉⠉⠐⠈⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠴⠖⠛⠋⠙⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠋⠉⠉⠀⠀⠀⠀⣀⠠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣁⣠⡤⣄⣀⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⡖⠚⠻⣛⠉⠙⠋⠉⠉⠓⠒⠲⠤⠤⣤⣀⣀⠠⠤⠒⣿
⠀⠀⠀⠀⠀⠀⢀⡠⢾⡉⠀⠀⠉⠢⣄⡀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠳⣄⠀⠀⠀⠀⠉⠓⢦⡀⠙⠒⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠙⠒⠢⠄⠀⠀⠀⠶⣄⡀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠳⢤⣄⡀⠀⠉⠉⠉⣹⠃
⠀⠀⠀⠀⢀⠔⠉⠻⣄⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⠀⠀⠀⠀⠀⠉⠑⠒⠒⡾⠃
⠀⠀⠀⠀⢘⠀⠀⠀⠈⠱⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⣀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢰⠓⢆⠀⠀⠀⠀⠉⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣶⠞⠁
⠀⠀⠀⠀⡴⢦⣈⠑⢢⡀⠀⢀⠀⠀⠉⢉⣻⢧⣄⢀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⡤⠤⣬⠷⠤⣉⣭⠽⠓⠋⠉⠉⠁⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠉⠀⠈⠙⠛⠿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠛⠉⠛⠻⢿⣷⣄⠀⠀⠀⠀⢀⣠⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⢰⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡗⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠋⣡⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣤⣤⣴⣿⡿⠋⠀⣠⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣀⠀⢻⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⣠⣾⡦⠀⠈⠉⠛⠛⠛⠋⠁⠀⠀⠚⠛⠛⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣽⣿⡇
⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠋⠉⠉⣿⣿⠋⠛⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⢀⣠⣴⣾⣿⣷⡆
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⢀⣴⣾⣿⠿⠛⢩⣾⣿⣃⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⢻⣿⣯⣄⡀⠀⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠉⠛⠿⣿⣷⣦⣿⣿⡆
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⡀⠀⣠⣾⡿⠋⠀⠀⣠⣤⠀⠀⠀⠀⠀⠉⠛⠿⠛⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⡿⠋⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⣿⣿⠀⣀⣄⡀
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠉⠋⢁⣾⣿⠃
⠀⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⢀⣾⣿⠃
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⠃
⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⢙⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⣾⣿⠃⠙⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠙⠻⢿⣷⣦⡀⠀⠀⠀⢸⣿⡇
⠀⢠⣿⡟⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡀⠀⠀⠈⣿⣧
⠀⣼⣿⠃⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿
⢀⣿⡿⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⢸⣿⠇⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠘⣿⣇
⢿⡿⠀⠀⠀⠀⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⠀⠀⠀⢿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠁⠀⠀⠀⠀⠈⠙⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠋⠁⠀⠀⠀⣀⡴⠖⠳⢦⣀⠀⠀⠀⠈⠙⠳⠶⢤⣀⣀
⠀⠀⣀⣀⣠⣤⠴⠶⠛⠛⠉⠁⠀⠀⠀⣀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠉⠛⠳⢦⣄⣀⠀⠀⠀⠈⠉⠛⠛⠶⠦⣤⣄⣀⣀
⣾⠋⠉⠉⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠉⠙
⣿⠀⠀⠀⣤⠶⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⣤
⣿⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⢸⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠
⠸⡇⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸
⠀⣿⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣿
⠀⢸⡆⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢰⡇
⠀⠀⢿⡀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⡿
⠀⠀⠘⣧⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣼⠃
⠀⠀⠀⠹⣆⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣰⠏
⠀⠀⠀⠀⢹⡄⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⢠⡏
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠛⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠳⣤⡀⢀⣤⠞⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠛⠛⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⢀⣴⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣶⣶⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⣀⣜⣿⣟⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣖⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢦⣿⡴⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡂
⠀⠀⠀⣽⢿⣿⡈⢹⠋⣸⣿⢿⡅⠀⠀⠀⢻⣿⣶⣨⣿⣅⣰⣿⣿⠂⠀⠀⠀⢳⣾⣄⠀⣿⢀⣠⣷⡚⠀⠀⠀⠈⡿⣷⣆⢉⣏⢁⣶⡿⡟
⠀⠀⠀⠰⣏⣁⣹⣿⣿⣁⣉⡷⠀⠀⠀⠀⠀⣈⣙⣿⣿⣿⣏⣉⠀⠀⠀⠀⠀⣍⠀⣈⣽⣿⣯⡁⢀⣉⠀⠀⠀⠀⠀⣍⣈⣿⣩⣏⣀⡅
⠀⠀⠀⠽⣿⣿⢁⣼⣄⢸⣿⡾⠅⠀⠀⠀⡼⣿⠿⢙⣿⣍⠹⢿⢿⠄⠀⠀⠀⠼⢿⠉⠀⣿⠀⢙⡿⠧⠀⠀⠀⠠⢿⡟⠏⣀⣇⡀⠿⣷⠧
⠀⠀⠀⠀⠁⠈⠩⢾⠮⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⣿⠓⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠆

⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⢤⡀⣦⠙⢾⠞⢡⣆⣠⠄⠀⠀⠀⣴⡆⢿⠶⣿⡴⣿⢀⣦⠄⠀⠀⠠⣤⣴⠀⢙⣿⠋⢀⣆⣤⠀⠀⠀⠠⣤⣄⠀⣸⣿⡁⢀⣤⣤
⠀⠀⠀⢠⠽⠛⣦⠞⢦⡞⠻⢥⠀⠀⠀⠀⣉⡽⠿⢦⣿⡤⠿⢯⣉⠁⠀⠀⠀⠹⠛⣷⢯⣿⡽⡾⠛⠏⠀⠀⠀⠀⠙⠛⡿⣬⣯⡼⢿⠛⠁
⠀⠀⠀⠘⣳⣶⠛⢶⠞⢳⣶⡚⠀⠀⠀⠀⣍⣹⣶⠚⣿⠙⣶⣟⣉⡀⠀⠀⠀⣰⣤⡿⣟⣿⣻⢷⣴⣆⠀⠀⠀⠀⣠⣼⢷⢛⣟⢳⢿⣤⡄
⠀⠀⠀⠉⠁⠟⡤⢾⠶⡌⠇⠉⠁⠀⠀⠀⠛⠃⣾⠞⣿⠳⢿⠈⠟⠁⠀⠀⠈⠛⠛⠀⣠⣿⣄⠀⠋⠛⠀⠀⠀⠈⠉⠉⠀⢩⣿⡁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠛⠛⠉⣤⣄⠀⠛⠋⠉⠙⠛⠻⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⢀⣤⡤⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠈⠻⠀⠀⣀⠉⣛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠠⣶⡶⠁⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠙⠃⠉⠃⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣽⣾⡗⠀⣴⡆⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠉⠉⠀⠀⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⡄⢠⣾⠗⠀⠘⠟⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣽⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡛⠁⠰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⡿⠦⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⠿⠟⠛⠀⠀⢰
⠀⠀⠀⠀⠀⢠⣋⠟⠃⠀⠀⠀⠀⣀⣤⣤⣀⣠⣤⣠⣶⣤⣶⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠉⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠻⠋⠁⠀⠀⠀⠀⠀⠉⢀⡜
⠀⠀⠀⠀⠸⡏⠹⣇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠏⠙⠋⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢿⡄⠈⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠻⠟⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠈⢷⣀⡀⠂⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣄⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣶⣟⣷⣾⣿⣶⣦⡄⣄⡀⣤⣤⣦⣤⣵⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⣶⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠸⠿⠿⠿⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠘⠿⠾⠇⠀⠀⠀⠈⠿⣿⠟

⠀⠀⠀⠀⢐⡄⣠⡀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⢤⣔⣦⢤⠀⠀⠀⣴⣶⣶⣦⡀⠀⠀⣠⠶⡶⠶⡄
⠀⠀⠀⠀⢙⡃⣙⡋⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⣾⣿⣿⠀⠀⠀⢸⣹⣿⣿⢻⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⢽⠤⡧⠤⡇
⠀⠀⠀⠀⠈⠃⠙⠁⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠑⠽⠽⠃⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠘⠦⠧⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣚⡓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠖⠉⠁⠉⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢐⡁⠀⠀⢀⡜⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡽⠒⢚⣁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⣢⠊⠁⣠⠞⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡴⢫⠔⠉⠈⠳⢶⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢃⠀⠀⠀⠀⠀⠀⣇⠮⡀⣀⡠⠔⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣀⠀⠈⠳⡀⠘⠀⡠⠔⠒⡲⠊⢁⣴⣉⣥⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠀⠀⠀⢮⣀⢠⡞⢀⡴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢙⡤⠈⠀⠀⡙⠋⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠔⠁⠀⡠⢤⡈⠱⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⢋⡷⠒⠊⢀⡠⠋⠀⢄⠀⠙⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⡤⠊⢀⡴⠋⠉⠒⠉⠁⠀⢰⠀⠈⢧⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠂⣏⣠⡔⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣨⡀⠀⢀⡴⡱⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣠⣓⣒⡩⠜⠁
⠀⠀⠀⠀⠀⠀⡠⢔⣩⠎⢀⢴⠁
⠀⠀⠀⠀⠀⡜⡴⠁⠈⢊⠇⡼
⠀⠀⠀⠀⠀⡄⠓⠤⠒⣁⠜⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣴⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⠶⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣙⠻⠀⠟⢋⡽⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢰⣋⠳⠿⠸⠷⠊⣹⢀⣤⡀
⠀⠀⠀⠀⣀⠀⡏⢹⢀⣠⡖⠮⣷⠀⣾⠵⢲⣄⡀⡏⢹⠀⣀⠀⠀⠀⠀⢠⠶⢤⡇⢸⢠⠬⣕⣦⢠⣖⡫⢤⢸⡀⣧⡴⢦
⠀⠀⠀⠘⢏⡙⠃⢸⣸⠀⡟⠲⣤⠀⣠⠖⢻⠀⣇⡇⠘⢋⡹⠇⠀⠀⠀⣈⡷⠄⠀⢿⡞⢠⣄⡉⠈⣉⣤⠘⣯⠇⠀⠴⣏⡀
⠀⠀⠀⣴⠚⣡⠴⣦⡉⠀⢧⣄⣸⠀⣇⣠⠼⠀⠉⣴⠦⣌⠓⣦⠀⠀⠘⠥⠴⣚⡽⠆⠀⠸⡤⠿⠘⢧⡼⠀⠀⠾⣝⡲⠤⠽
⠀⠀⠀⠈⠉⠱⣏⣡⠖⠛⠲⣌⡙⠀⢋⣡⠖⠛⠲⣌⣹⠎⠉⠁⠀⠀⣞⠳⣄⢧⡀⢾⣉⡗⠀⠀⠀⠀⠐⣏⣹⠆⣠⢇⡴⢛⡆
⠀⠀⠀⣠⢤⣘⢧⣉⠓⠶⠚⣉⣤⠀⣤⣉⠓⠶⠚⣉⡼⣃⡤⣄⠀⢀⡬⠷⠈⠛⣋⠀⢩⣤⠀⠀⠀⠀⢠⣬⠁⢈⠓⠋⠰⠯⣄
⠀⠀⠀⠙⢦⣌⠙⢋⣡⠀⡟⠁⢸⠀⡇⠈⢹⠀⣌⡙⠋⢡⡴⠋⠀⠘⠲⢲⠂⡾⡽⠀⠷⠼⠃⡴⠶⡄⠻⠤⠇⠸⡽⡆⢲⠲⠚
⠀⠀⠀⠰⣏⡤⡆⢸⢹⣀⡷⠞⣉⠀⣉⠓⢾⣀⡏⡇⢰⢦⣹⠆⠀⠀⠀⠻⠴⢃⣙⣒⣲⠂⣤⠙⠞⢡⡄⢲⣒⣚⣁⠳⡼⠃
⠀⠀⠀⠀⠀⠀⢧⡼⠀⠉⢓⣫⠽⠀⠿⣝⡚⠉⠀⢧⡼⠀⠀⠀⠀⠀⠀⠀⠀⢯⣥⣤⠀⡈⢯⡳⠴⣫⠏⠀⢠⣬⣭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡤⢶⠀⡶⢤⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡼⠳⣄⣳⢰⣃⡴⠳⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡶⢼⠉⣧⠶⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣙⠾⠹⢚⣩⠇
⠀⠀⠀⠀⠀⢠⡶⣴⢋⣇⣠⢽⣶⠀⣶⡯⣅⣸⡝⣧⢶⡄⠀⠀⠀⠀⠀⠀⡴⣄⢠⣄⠰⣟⠻⢿⢰⠿⢚⡽⠀⣀⢠⡤⡄
⠀⠀⠀⠀⢠⡴⢛⣀⡸⡿⣸⠓⢮⠀⡵⠚⣇⢻⠇⣀⡛⢦⡄⠀⠀⠀⠀⡴⠇⣿⣿⢸⣀⡬⠟⣂⢀⡚⠯⣄⣸⢹⣿⡆⠷⣄
⠀⠀⠀⠀⣀⠉⣹⣛⣋⣤⡻⣄⢸⠀⡇⣠⠞⣠⣘⣛⣯⠉⣁⠀⠀⢀⡴⢋⣤⣬⠋⠘⣩⡖⠋⢹⢸⡏⠓⣮⡙⠈⢫⣤⣌⡓⣦
⠀⠀⠀⣘⣏⢳⣧⡹⣄⣀⣙⣮⡻⠀⢟⣡⣊⣀⣠⠏⣼⡾⣹⣃⠀⠀⠉⢹⣋⡵⣶⢰⠦⣝⠲⢾⢸⠷⢚⡥⢶⢰⡶⢍⣳⡍⠁
⠀⠀⠘⠯⡭⢠⣭⢄⡬⠭⠭⣭⠅⠀⡨⣭⠭⠭⢥⡠⣭⡄⢭⠽⠃⠀⠀⣠⢤⣀⣿⢸⣀⡤⠟⣂⢀⡚⠯⣄⣸⢸⣇⣠⣥
⠀⠀⠀⠸⠥⠋⣯⠾⣤⣠⢞⡵⢻⠀⡟⢮⡳⣄⣤⠷⣽⠙⢮⠇⠀⢠⡶⢼⣶⣬⠋⠘⣭⡖⠋⢹⢸⡍⠓⣮⡝⠈⢫⣴⣾⠖⡦
⠀⠀⠀⠀⢠⣖⠾⠿⢖⡀⢿⠀⣸⠀⣇⠀⡿⢀⡲⡾⠷⢲⡄⠀⠀⠀⠉⣗⡈⣱⣶⢰⠦⣝⡲⢾⢸⠷⢚⡥⢶⢰⣶⡍⣐⡏⠁
⠀⠀⠀⠀⠀⢩⣟⣤⢸⡿⣼⣋⣥⠀⣬⣙⣧⢿⡇⣤⣻⡍⠀⠀⠀⠀⠀⠉⣇⣹⠻⠼⢀⡤⠟⣂⢀⡘⠧⣄⠸⣼⢿⣇⡏⠁
⠀⠀⠀⠀⠀⠀⠉⠘⠷⠇⢠⢟⣫⠀⣝⡻⡄⠸⠶⠃⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⢒⡿⠟⠘⠻⣷⡋⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠶⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠛⠞⠛⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⡠⠀⠀⠑⠀⠀⠀⠘⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⢀⡈⠐⠀⠀⠀⡀⠀⠀⢀⡁⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⠀⢰⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⠢⢄⠀⠁⠂⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⠀⠀⠠⠊⠀⠀⠀⠀⠑⠀⠒⠀⡀⠀⠀⠐⠆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠘⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠀⠀⠑⢒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⡀⠀⠸⠀⠀⠀⣀⠰⠀⠈⠀⠀⠀⡇⠀⠀⠀⠙⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠱⡁⠉⠀⠀⠀⠀⠁⠀⠐⠀⠀⡀⢠⠀⢀⡉⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⢀⢀⠀⠀⠀⣀⠀⢀⠀⠣⡈⠓⠢⠤⠤⠤⢐⡙⠀⠒⠉⢀⡌⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠈⠰⠁⠀⠀⠈⠁⠀⠂⠒⠚⡉⠀⡑⢲⠒⠍⠀⠀⠀⠀⠀⢙⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠠⡀⢈⠢⠄⣀⣀⣰⠀⠀⠀⠈⠣⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠌⢰⡀⢀⣀⡀⠀⣄⡀⠠⡄⠈⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠁⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⠀⠀⠢⡀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠃⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣀⣀⣀⣴⡾⠿⠿⠿⢷⣢⣄⣀⣀⣀⣰⣚⠽⠛⠛⠛⢿⣳⣄⡀⠀⣠⣶⠿⠛⠛⠿⣖⣤⡀⠀⢀⣴⠿⠛⠛⠛⣿⣦⣀⣀⣀
⠀⠘⠿⠿⠿⠿⣿⢧⣀⣀⣀⣀⠿⡿⠿⠿⠿⢿⡿⣄⣀⣀⣀⡸⢿⡿⠻⠿⢿⣿⣀⠀⠀⣀⡿⣿⠿⠿⢿⣿⣆⠀⢀⣀⡿⣿⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠚⠋⠁⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣤⣤⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣶⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠿⢿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢠⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣷⣶⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠻⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣠⣤⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣶⣦⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⡿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠻⠷⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣤⣶⣼⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣶⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠟⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠿⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⣶⣶⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣷⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠿⠿⠟⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠙⠻⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣠⣶⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⠋⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣉⡛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠿⠟⠋⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣈⡛⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⣿⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠛⣋⣈⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿⣥⣌⡙⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣼⣿⣿⡽⢿⣦⠀⠀⠀⠀⣼⡟⠻⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠉⣤⣾⣿⠿⣶⣾⠟⣿⣿⣷⡈⣱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣿⠟⢱⣿⣿⣇⠹⢟⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣬⣽⡿⣯⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣾⡗⠛⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠶⢾⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡏⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈






⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠟⠻⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⣿⢟⠿⢻⣿⣿⡿⠿⠿⣿⣿