Quail Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⠡⣶⡆⣻⣿⣿⠛⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠸⣿⣿⣿⣷⣬⣴⣿⣿⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣇⠀⠙⢿⣿⣿⣿⣿⡿⠿⠷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠷⣄⣀⣀⡼⠋⠁⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠙⠋⠀⢹⡄⠀⠀⢠⡀⠀⢠⡀⠀⢀⡀⠀⢀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡅⠰⠴⠂⠻⡄⠀⠀⠻⠄⠀⠻⠄⠈⠳⠀⠘⠦⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⠀⠀⠙⢦⣄⠀⠀⠀⢦⡀⠀⣤⠀⢠⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⡀⠉⠳⢦⣀⠈⠁⠀⠈⠁⠀⠉⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣷⠶⠾⢿⣶⣦⣤⣀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⣸⡏⠀⠀⠈⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⠁⢀⣀⣰⣿⡀⠀⠀⠀⠀⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⣁⣀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⣉⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣿⣿⣦⡘⠍⢿⣸⡷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⣛⠛⠛⠛⠛⠿⠿⣿⣿⡀⣸⣟⣴⠾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢋⣵⠞⣋⣉⣥⣀⠀⠀⠀⠀⠀⠙⢿⣿⣏⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⡵⢋⣤⠞⣋⣥⣤⣶⠤⠄⠀⠀⠀⠀⢸⣿⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠊⣱⠟⣡⡾⢟⣩⣵⣶⠶⠖⠀⠀⠀⢠⣾⣿⠃⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⢊⡵⢛⣡⡶⠿⢛⣩⣴⡶⠟⢋⣡⣾⡿⠛⠁⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀⠁⠖⣋⣥⠶⠞⣛⣩⣥⣶⠿⠟⠋⠁⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣦⣭⣤⣐⣛⣛⣋⣭⣭⣤⣤⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢏⡾⣿⠛⠛⠛⠛⠛⠻⠿⣿⣟⣛⣋⣉⡉
⠀⠀⠀⠀⠀⠀⢀⣼⢋⡼⣡⢏⡾⠁⠀⠀⠀⠀⢀⣀⣠⣤⠽⠿⢭⣍⢻⡓⠶⠄
⠀⠀⠀⠀⠀⢠⣾⣷⣏⣴⣧⠞⠁⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⢫⡶⠛⠉⠙⠛⠳
⠀⠀⠀⠀⠀⠘⠛⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣭⣤⣴⣦⣌⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠱⣀⡀⠀⡀⠀⠀⠈⠃⣸⠃⣀⣧⠀⠀⠀⠀⠀⠀⣠⣾⣡⣾⣿⠿⠛⠛⠛⠿⣷⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠙⣧⠀⡍⣿⣤⣀⣻⣿⣧⣬⣿⣾⣷⣶⣶⣶⣿⢯⣽⣿⣿⡃⠀⠀⠀⣀⠴⠋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠼⣿⠦⢹⣀⣷⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣧⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠤⡄⢀⠀⠀⠞⠀⢀⣤⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠄⢲⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇⠀⠠
⠀⠀⠀⠀⠀⠀⢱⠀⠸⡷⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⣟⠿⢿⣿⣿⣿⣿⠁⠘
⠀⠀⠀⠀⠀⢀⢀⣶⡄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⣫⣍⡈⠁⠉⠊⠉⠷⠯⢾⣿⠃⠀⡄⠀⠀⠀⢌⠀⣘⠃
⠀⠀⠀⠀⠀⢈⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠋⡭⠽⠏⠐⠃⠀⠁⠈⠀⠀⠬⢡⣾⣿⠟⠀⢀⠁⠀⡀⠀⠀⣐⠁⠂
⠀⠀⠀⠀⢄⢠⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⢉⣹⠭⠤⠂⠀⠀⠈⠀⡀⠂⠈⡀⠂⢉⣺⣿⠋⠀⠘⠣⠀⠄⣠⢀⡀⠋
⠀⠀⠀⠠⡈⣻⡷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡋⠰⠒⠀⠀⡀⠀⠀⠠⠀⠀⢀⠐⠀⢀⠤⢻⣿⡏⣿⣦⣄⣰⡄⣀⡈⠃⣈
⠀⠀⠀⠠⢈⢊⠷⣿⣿⣿⣿⣿⣿⣟⡿⢿⡛⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⢀⠈⡀⠀⣀⣼⣿⣿⣯⢻⣽⣿⡟⡃⠿⡿⠋⠁⢤⠈⡙⢀
⠀⠀⠀⠀⠙⢇⢨⣿⣿⣿⡟⣾⢿⣯⢻⣶⣿⡁⠐⠀⢠⡐⠀⡀⠀⠠⣀⣀⣠⣼⣾⣿⣿⣯⣿⣯⡠⡛⢻⣷⠆⢶⢁⡀⠀⡄⠀⠁⠌
⠀⠀⠀⠀⠀⠀⢸⣿⣟⡿⣿⣶⣿⣷⠟⠛⠋⠉⠉⠛⣶⡟⣿⠿⣶⣾⣿⡿⢉⣬⣿⣿⣿⣿⢿⣟⣿⣿⠂⣸⠙⢘⣺⣝⡲⡀⠒
⠀⠀⠀⠀⠀⠀⣸⣏⡹⣆⡽⠟⠉⠀⠀⣀⠀⢰⢦⣄⠙⣃⡸⡆⢹⣟⡉⠋⠛⠛⠻⣿⣿⣿⣟⣫⣿⣦⣏⡈⠓⠊⠙⠝⠷⢛⠂⠄⠂
⠀⠀⠀⠀⠀⢀⣿⣥⣾⠏⠀⠀⠀⠐⢭⣟⣭⠍⠒⠚⠁⠉⠀⠀⠈⡆⠈⠉⠒⠡⠤⠶⣻⣿⣏⣛⡛⣮⣿⣽⣷⣶⡥⠤⠥⠠⠄
⠀⠀⠀⠀⠀⠸⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠾⡄⠙⠂⠀⢀⣀⣴⠃⠋⠩⢉⢉⠛⡫⠋⠙⠋⣻⣷⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⢀⡀⠀⠙⠦⠁⠀⠒⠠⠘⠢⠤⠖⡡⠴⠁⠈⢣⠄⠋⠉⠀⠀⠀⠁⠉⣰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠓⠓⠒⢦⣤⣤⡤⠔⠀⠀⠀⠀⠀⣖⡀⠀⠀⠀⠘⢿⠿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠇
⣶⣶⣾⢷⣶⠶⣶⣶⣶⢶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣴⣶⣦⣶⣶⣿⣧⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠛⠛⠛⠢⣄⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⢸⣿⣿⣿⣟⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⣙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢛⣿⣼⣿⣿⣿⣿⡻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⢸⣿⣹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡻⣿⣿⣿⣿⣿⣿⠟⣸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣷⣿⣿⣿⣽⣶⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⣶⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣭⢿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡻⠚⠷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢟⣿⣿⣿⣿⢾⣿⣿⡿⣿⣿⣿⡷⣿⣟⠿⣿⣿⣿⠿⣿⣡⡋⣹⠛⣻⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣿⣿⢿⣿⣿⣙⢿⣿⣻⣯⣁⣹⣷⡬⢿⡛⣯⡽⠻⠿⣭⣝⣋⡻⠿⣭⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⡿⢻⣿⣿⣶⣿⡿⢿⣏⣉⣘⣷⣄⣹⣯⠙⣷⠾⣧⡌⠛⣷⠶⢾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⣿⣿⣿⣿⣿⡿⣿⣁⠻⣧⣄⣈⣽⣿⣀⠀⣈⣿⣤⣸⠯⣉⣻⢏⣿⣗⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢏⡼⠻⣫⣿⣿⣿⣿⣻⣿⣻⣿⣉⣹⣿⢿⣆⣈⣉⣿⣧⣤⣴⣛⣷⣾⡡⣿⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣿⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⠿⣾⢿⣿⣄⣉⣙⢿⣦⣀⣠⡜⠿⣶⣙⢿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⣠⡶⣫⡿⣻⣿⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿⣿⡛⢿⣦⣈⣭⠛⢷⣤⣤⣴⣯⡹⢿⣮⡽⠋
⠀⠀⠀⠀⠀⢠⡾⣫⢞⣵⢾⣿⡿⠟⠋⠙⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣯⣹⣿⡿⠿⠚⠉
⠀⠀⠀⠀⢰⣿⡽⢡⣾⣥⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⣿⡟⠋⠻⠿⠛⣿⡇
⠀⠀⠀⠀⠸⣿⣥⣿⡿⠋⠀⠀⢠⠒⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⣠⣼⣿⣷⣴⣶⣶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠻⢿⣿⣟⢿⣯⣅⡀⠙⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣴⡁⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⣾⡿⠿⣶⡀
⠀⠀⠀⠀⠀⣠⠼⣓⣲⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣄⣀⣤⣄⣀
⠀⠀⠀⠀⣰⣿⣿⣟⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⠀⠉⠳⣄
⠀⠀⠀⠘⠣⣿⣷⡷⢟⡿⠋⠉⠈⠉⠓⢦⣀⠀⠀⠀⠀⢰⣻⠄⠘⠃⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⢠⡟⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⣼⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡟⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⢹⡀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣧⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠘⣆⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠈⠉⠓⠒⠒⢒⡀⠀⠈⢧⡀⠀⠈⠳⣄⡀⠀⠈⠙⠒⠶⠤⠄⡀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡖⠒⣶⠚⠛⠉⠙⠲⢤⣀⡿⠀⠀⠀⠀⠉⠛⣶⠒⢲⠖⠚⠋⠙⠦⣤⣈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⠷⠶⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠿⠶⠞⠂


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⡴⠶⢟⣫⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⣟⣵⢾⡿⣟⢿⣶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡷⠭⡄⢠⣿⣼⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⡿⠛⠀⠈⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠛⠋⠉⠉⡉⠉⠻⣈⠙⢯⣄⡠⠈⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠾⠥⡌⠹⣅⣕⣁⡕⢂⣠⠀⠈⢫⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢊⡙⠉⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣴⡿⠃⠀⢀⣄⣠⡔⠀⣠⠆⠀⢀⢀⣀⣤⣴⣿⣥⡀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣾⣟⢁⡄⠀⣀⠉⠉⠓⠺⠷⣾⣶⣿⡿⢿⣿⣿⣿⣿⣀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⣿⣿⠛⠉⠛⢉⠁⠀⠀⠀⣀⣠⣿⣟⣛⣻⣿⣿⣾⣿⣿⢤⣤⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣵⣿⣿⣿⠛⠉⣉⠉⠀⠀⠀⢤⣾⣿⣭⣽⣿⣽⣿⣿⣿⡹⣭⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠶⠐⠀⠀⢀⣶⣾⣿⢷⣾⣯⣿⣯⣽⣿⡿⣽⣿⣽⡏
⠀⠀⠀⠀⠀⣠⢔⣃⣽⡟⣿⣿⡻⢯⣭⣭⠿⣭⠽⣛⣻⣿⣿⣿⣿⣾⣿⣟⣽⣿⣾⣿⣻⠏
⠀⢀⡠⠔⣯⠴⣾⣯⠛⠘⣻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⣴⠩⠜⠟⠋⣀⣸⡬⠶⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉⠛⣿⡉⠉⠙⣿⣿⠟⠉⠉⠁
⠈⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣳⡈⢿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠃⠻⣿⣾⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⡙⣦⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠛⠪⣟⣯⣙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠⠠⠄⠄⠄⠠⠠⠀⢄⠀⡀⠀⠀⠀⠀⡉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣞⣛⡛⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣡⣌⣉⠉⠙⠛⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢛⡿⠁⣠⣭⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢸⠇⠀⠷⠼⢃⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠞⣧⠀⠀⢸⡀⠀⠀⠀⢻⣅⣀⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣞⣯⣭⠷⠶⠖⠛⠃⠀⠀⠘⣧⠀⠀⢀⡼⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣿⡽⠞⣫⡽⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣫⠞⠉⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⡾⠃⢀⣀⠀⠀⢠⠴⠆⠀⠀⠀⣴⢠⡇⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡾⢁⡴⠋⠉⠀⢰⠇⠀⠀⠀⣠⡴⠋⣼⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣼⠁⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⠇⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣰⠏⢰⡏⠀⠀⠀⠀⣰⠆⠀⠀⣸⠇⠀⠀⣰⠏⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⢀⣼⠋⠀⣼⠁⠀⠀⠶⠚⠁⠀⠀⠚⠋⠀⣠⡾⠁⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⣠⠞⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⢉⡀⠀⠀⠀⢀⣴⠋
⠀⠀⣠⡾⠋⠀⢀⣤⠾⠻⢦⣀⣤⣤⠴⠶⠞⠛⠉⠀⢀⡏⢹⠆⢀⡴⠟⠁
⢀⡼⠋⣀⣤⠞⠋⠁⠀⠀⠀⠉⠻⠶⣤⣀⣀⣀⣀⣀⢸⣇⡿⠛⠋
⠻⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡿⠿⠶⠶⣄
⣶⡶⠚⠒⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣀⡤⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠛⣫⣉⣁⢈⣉⣀⠳⣄⠀⠀⠀⠀⠀⢠⡔⢆⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠔⡈⠉⠉⢙⣉⡁⣈⠉⠉⢢⡘⣆⠀⠀⠀⠀⣼⠁⢸⡆⠀⠀⠀⢀⠀⠐⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠋⠉⠐⠲⣔⡤⠀⠉⣲⢠⡀⢻⡌⢆⠀⠀⠀⡇⠐⣾⢡⠀⢰⢀⡖⡇⢰⢸⠀⠀⠀⠀⡧⠤⡀⡄⣶⡤⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡎⠀⠁⠸⠃⠸⠁⡌⣆⠀⠀⠳⡤⠻⠌⠣⠺⠾⠄⠷⠸⠘⠁⠀⠀⠀⠇⠴⠃⠆⠿⠀⢿⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠔⢆⠀⣤⣦⣽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⣄⡌⢦⣿⣦⡻⣿⣿⣟⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠀⠀⠈⠱⣞⠟⠋⢨⠛⣦⣩⣛⢦⡍⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠠⡀⠙⠈⠻⠖⠿⠃⠀⣁⡀⢁⠈⣳⣾⣷⣦⣝⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⣷⠟⠐⢴⣤⢀⣍⠲⢬⢦⣹⣭⣽⣝⡛⢶⡌⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡐⣠⠀⠀⠈⠀⠂⢤⣄⢀⣭⣃⣀⠰⣬⡅⠹⠏⢧⡀⠑⠠⣄⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⢀⠀⠀⠀⠀⠀⠶⣬⡙⠉⢀⣉⠻⠦⢰⡆⠂⠀⢀⠲⠤⠈⣷⣿⣦⡑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠱⣈⠑⠻⠃⠀⠈⣙⡆⡘⠦⠀⠀⠳⣄⢮⣻⣿⣦⡝⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠦⠀⠀⠀⠀⠀⠀⠹⣯⣝⣲⠶⢾⣿⠙⠊⠳⢆⠀⠢⣦⡌⢉⠻⣟⣻⠷⡮⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⡀⠀⠀⠀⠀⠈⠉⠛⠻⠄⢈⣁⣀⣀⡈⠙⠲⠬⣿⠆⢠⣌⠛⢛⣿⠐⠢⣍⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⠒⠦⠀⣀⡀⠀⠀⠀⠀⠉⠩⠥⢀⠲⣶⣤⡈⠠⠢⠽⠆⣬⡉⣝⣲⣘⣁⣌⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣤⢁⣀⠀⠙⠃⢀⣀⣀⠀⠀⢉⣠⡬⣍⣀⠿⠷⠞⠉⠉⠉⠉⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⡍⢽⡿⢻⠶⠶⣄⣤⠤⢤⣬⡥⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⣴⠃⠀⠀⠀⢹⠂⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢫⠞⠁⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⢋⡴⠃⠀⠀⠀⠀⠀⠀⢠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⢶⡏⡁⢻⡄⠀⠀⠀⠀⠀⢀⣠⠞⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠏⠱⠋⠱⢲⣶⠴⠚⠉⠉⠃⠀⠸⣖⣳⣭⣍⣥⠞⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉


⡿⠿⠿⡿⡿⠟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣛⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿

⠀⠀⢀⣤⠶⣛⡛⣿⣷⣦⡀
⠰⠮⣯⡭⠿⣿⣷⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠉⢀⣀⣼⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢸⣟⣡⡞⢻⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⢸⣿⠿⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⢸⢻⣄⠸⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⢠⣾⣌⢷⢦⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠘⠁⠀⢈⢂⠾⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠈⡀⡨⠱⣟⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠋⠍⠛⠛⠛⠻⠿⠿⠿⠿⠿⡿⠿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠈⠉⠻⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢢⡤⠴⠖⠚⠂⠉⠉⠉⠀⠁⠈⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⣼⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢤⣤⣺⣾⣆⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠹⢟⣋⢁⣄⣤⣬⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢛⡿⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⣦⣴⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⠛⠋⠉⠉⠁⠀⠀⠈⠉⠉⠙⠛⠿⣿⣿⣷⣦⣀⠀⣿⣿⡏⠀⠛⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣾⣿⣷⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠛⠋⠉⠉⠙⠛⠻⢿⣿⣿⣦⣄⣴⣿⣿⠟⠁⢀⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⣠⣿⣿⠟⠋⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠈⢙⣿⣿⣿⠟⠁⠀⠀⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣴⣿⣿⠟⠁⣀⣄⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⣾⣿⡟⠀⠀⢀⣾⣿⡿⠋⠁⠀⠀⠀⠀⢉⣿⣿⣿⠟⠁⠀⠀⢿⣿⣧⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⢸⣿⡿⠀⠀⢀⣿⣿⠏⠀⠀⢀⣤⣴⣤⣴⣿⣿⠟⠁⣀⣀⠀⠀⠈⣿⣿⡆⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⡟⠀⠀⢠⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⡇⠀⠀⢹⣿⡇⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣇⠀⠀⠸⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⡇⠀⠀⣸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⢸⣿⣧⠀⠀⠘⣿⣿⣄⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⣰⣿⡿⠀⠀⠀⣿⣿⠇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠈⢿⣿⣆⠀⠀⠘⢿⣿⣦⣄⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠁⠀⠀⣼⣿⡟⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠙⠿⣿⣿⣷⣶⣶⣿⣿⣿⠟⠋⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣴⣿⣿⠋⠀⠀⠀⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠀⠀⠀⠀⠈⠻⢿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣠⣤⣴⣿⣿⡿⠋⠁⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⣿⡿⡿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣿⣿⠇⢹⣿⡇
⣿⣿⢳⣺⣿⡇
⣿⣿⣷⣾⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⢛⢻⡇⠀⠀⠀⠀⠀⠀⡁
⠀⠀⣱⡄⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣭⣼⠃⠀⠀⠉⠀⠐⢠⡆
⠀⣥⣤⣿⣯⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⣟⠉⠀⠁⣰⡏⢉⣉⣛⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣷⣽⣖⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣴⣲⡤⣯⣟⣧⣶⣿⣿⣍⢥⣀⣨⡇
⠀⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣉⣉⣉⣁⢀⣼⣿⣿⣿⡟⠻⣏⢉⢛⠉⢷⣴⣾⡆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠟⠛⠀⠀⢀⠀⢶⣶⣷⣾⣿⣶⣼⣍⣉⡙⢿⣿⣿⣿⣟⣛⣿⣿⣿⣁⢠⡏⠉⡇
⠀⡿⣿⣿⣿⣿⡟⠘⣿⣾⣾⣿⣿⣶⣦⣾⣷⣝⣛⠛⠿⠿⣯⣾⣿⣿⣿⣷⣦⡉⠛⢛⣿⣿⡥⢜⠻⣯⡀⢈⡇
⠀⣿⣿⡿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⢦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⣿⣿⣾⣿⣿⡇
⠀⣿⣿⡧⣽⣿⡿⢾⣛⡉⠁⠉⠙⠛⠿⡿⣿⣿⣿⣥⠾⠒⠙⠻⢿⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡇
⠀⣿⣿⣶⣽⣯⣝⠻⡟⠁⠀⣀⠀⣰⡀⠘⣿⣿⣿⣿⣦⣽⣮⣵⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⡷⢛⣿⣿⣿⣟⣯⣥⣤⣶⣿⣿⣿⣷⣤⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⡭⣼⢼⣟⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⣤⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣯⠿⣯⣿⣾⣿⢿⣿⣿⣿⡿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⣮⣿⣿⣿⣿⠿⣿⣟⣿⣿⡇
⠀⡿⠻⣽⠿⠿⣿⡿⢿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢷⣾⣿⣿⣿⢿⡇
⣀⣈⣁⣉⣉⣉⣉⣉⣈⣋⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⢉⣥⠞⠋⠁⠉⠉⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⣷⣄⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⣀⣀⣀⣤⣤⡤⠶⠶⠶⠶⠶⠖⠒⠒⢀⣄⣀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⠙⠛⠷⠶⠶⣶⣿⠷⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢮⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⣠⣤⣤⣶⣶⣾⣷⣶⣶⣶⣶⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣴⠾⠟⠛⠉⠉⠁⠀⢸⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣦⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣶⠟⠋⠉⠀⠀⣿⡇⠀⠀⠀⠉⠙⠻⢷⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡇⢀⣴⠟⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣾⠏⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⡀⠀⠀⠀⠀⢀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣆⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣦⣄⣀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⣻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣿⣿⣯⣷⣦⣄⠀⠈⢿⣆⠀⢀⠀⠀⣀⣀⠤⣀⡠⣀⣠⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⣽⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠛⠟⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠋⠹⠏⠙⠛⠛⠛⠟⠛⠛⠛⠛⠉⠙⠏⠙⠫

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⣿⣿⣿⣧⣀⠀⠀⢠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⣄⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⢀⠠⡈⢻⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡶⠀⠀⠀⠙⠷⣶⣿⣷⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⡆⠀⢀⠈⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣾⣹⡇⣀⣾⢞⣿⠏⠀⠀⠀⠀⠀⢸⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠉⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣤⣤⣤⣤⡀⠈⠻⠃⠀⢻⣿⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠸⣿⣿⣿⡇⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⡇⠀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⢻⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⠀⠠⣤⣄⣤⣴⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⣿⣿⣿⣿⣷⣄⠀⠀⠘⣿⣿⡇⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢸⡿⡏⠀⣿⠇⠘⣿⡟⢿⣄⠀⠀⢿⣿⡇⠀⠀⣀⣿⣿⣄⡄⡠⠂⠀⠀⠀⠉⠙⠓⠂
⠀⠀⢀⠐⢄⣀⠀⢨⣣⣠⣀⠱⣤⡘⢦⠱⡄⣸⢹⡇⣼⠃⠀⣰⠏⠀⠀⠉⢀⣠⣾⣿⣷⣾⣿⣿⣿⣿⣿⣟⣡⢎⣴⡠⠀⢀⡠
⠀⠀⠀⠐⢦⡽⣷⣤⣿⣿⣿⣷⣽⣿⣌⣧⣳⣿⣼⣵⣷⣄⣴⣃⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⣴⣫⣿⡔
⠀⣈⣀⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣄⣀⡀
⠈⠉⠉⠉⠉⠉⠉⢉⠁⣀⠀⢀⣀⣐⣒⣒⣒⣒⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣀⣀⣀⣀⣀⣀⣣⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠁⣀⣀⣀⣀⣀⡈⠉⠓⠦⣄⠀⠀⢣⡷⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⠴⠚⠉⢀⣀⣀⣀⡀⠉⠑⠢⣄⠈⢱⣶⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢠⠞⠁⣠⠔⠊⠉⢀⣀⣀⠈⠉⠲⢤⣀⠽⢇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢰⠃⢀⡞⠁⣠⠖⠉⠁⣀⠈⠉⠓⣤⡞⠱⣄⠈⢧⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠇⠀⡎⠀⡼⠁⣠⠞⠉⠀⠉⣱⢾⠁⢱⡀⠸⡆⠘⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢰⠁⠀⠇⠀⡇⠀⡞⠩⠟⡅⠈⡇⠀⡇⠀⡇⠀⣧⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡀⠘⡆⠀⣇⠀⢧⡀⠑⠲⠚⠁⣰⠃⢀⠇⢀⡇⠀⡇⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢧⠀⠱⡀⠘⢦⡀⠙⠲⠤⠴⠚⠁⣠⠎⠀⡼⠀⣰⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢣⡀⠙⢦⡀⠉⠒⠦⠤⠤⠖⠊⠁⡠⠞⠀⡰⠃⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⢦⣀⠉⠓⠲⠤⠤⠤⠴⠒⠋⢀⡠⠚⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠢⣄⠈⠙⠲⠦⠤⠤⠤⠤⠖⠚⠉⢀⡤⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠙⠒⠶⠤⠤⠤⠤⠤⠴⠖⠋⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣭⣩⣭⡭⠭⠭⠭⣭⣭⣭⣭⣭⣭⣍⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣾⡶⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⣜⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣘⠦⢏⡀⠀⠰⠂⠀⠀⠉⠀⢀⠀⠤⠤⣴⠆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠁⠀⣏⠙⢳⣴⡯⢐⡱⠀⠀⠀⠀⠀⠑⠳⠀⠀⢀⣿⣿⠿⠋⠢
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠶⠚⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⣼⣿⠁⠀⠀⢧⡄
⠀⠀⢺⣿⡀⠀⠀⠀⠀⣠⣄⣢⣜⢿⣛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣶⡶⠂⡐⠀⠀⠀⠀⠀⠀⠀⠙⠝⠻⠟⠀⠀⣼⣿⠇
⠀⠀⠀⣿⣧⠀⠰⠒⠚⠁⠈⠙⠛⠛⠪⣻⣗⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠌
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣽⠆⠀⠀⢹⣆⠀⠀⠀⢀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣂⡦⣄⡠⠿⣏⠀⠀⠀⠈⢿⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⢀⣿⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠠⠜⠃⠀⠀⢀⣀⣠⣀⣀⣠⠠⢐⡈⠀⣿⣤⡼⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣰⣿⣄⣠⣾⡿⠀⠀⠈⠁⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠋⠀⠀⠀⠀⠀⠀⡆⠄⠈⢠⡖⢦⠈⠈⠁
⠀⠀⠀⠀⢠⣾⠓⠛⠻⠿⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠃⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠈⠁⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣸⠛⠚⠙⠁
⠐⠿⠿⣤⠋⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠘⠀⠀⠀⢹⣏⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣄⣠⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⢾⣟⣳⣢⣴⣦⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢠⣤⣶⣆⠀⠀⣨⡿⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠉⠛⠛⠻⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⠋
⠀⠀⠀⠀⠀⠯⠀⠹⣿⡀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡜⠃
⠀⠀⠀⠀⠀⠀⣥⣀⠷⡇⠀⠀⠀⠀⠀⠀⠀⠀⣘⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢡⣦⡌
⠀⠀⠀⠀⠀⠀⢼⡿⠆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⠜⠢⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣿⣷⡶⣿
⠀⠀⠀⠀⠀⠁⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠈⠿⠿⢗⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣶⠆⠀⠀⠀⣴⣿⣿⡿⠋⠹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣶⡦⠀⠿⠇
⠀⠀⣠⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢋⠀⠀⠉⠁
⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷
⠀⠀⣬⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠘⢿⣿⠆⠀⠀⠀⠀⠀⠀⠈⣙⡅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠙⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⢹⣞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⡄⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠸⣷⡔⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⠀⠀⢶⡀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⢹
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⢠⣝⣿⣟⢷⣦⣄⡀⠸⣷⡀⠀⠀⢻⣇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸
⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⢸⣿⡇⠈⠻⢯⣟⡿⣿⣶⣥⣤⣤⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢸
⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠇⡾
⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⣿⣮⡝⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟⡼⠃
⠀⠈⢳⡝⣦⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⣿⢻⡇⠀⠀⠀⡀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣤⠾⣿⣿⣿⣿⣿⠞⠁
⠀⠀⠀⠙⢮⡙⢷⣤⣀⠀⢸⣿⣷⡄⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠹⣷⣤⡀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠙⠷⣬⣽⣿⣾⣿⣿⣿⣶⣤⣤⣾⣿⣿⣿⣄⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⣴⠘⣿⣿⣆⠀⠀⠀⠀⠀⣀⣬⣝⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⡇⠀⢀⣼⡟⠀⢹⣿⣿⣄⣀⣠⣴⣾⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣶⣿⡿⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡎⣿⡻⣷⣹⡟⢋⣀⣀⣴⡿⣵⣿⡿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣞⣛⣿⣿⣿⣿⣾⣟⣿⣾⣿⣵⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣯⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣾⣿⣟⣿⣿⣿⣛⣛⡿⣿⡟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣾⣿⣿⡿⠟⠛⠿⣮⠻⣿⣿⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣤⣤⣤⡴⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⣦⣤⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⢡⣤⣤⣤⡜⣿⣿⣿⢫⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⣴⠯⣭⣿⣶⣤⣤
⠀⠀⠀⠀⠈⠶⣤⣠⣤⣤⣤⡴⠶⠚⠋⠁⠀⠀⠀⠀⣰⣿⣿⣿⡟⢱⣿⣿⣿⣀⠀⠀⣀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠖⠛⠉
⢠⣆⠀⠀⠀⢠⡌⠉⢩⣥⠀⠀⣤⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠃⢸⣿⣿⣿⣿⣿⣟⣿⠋⣥⡄⠀⣤⠀⣤⠀⢠⡄⢠⣤⠀⣤⠀⣤⣤⣤⡄⣤⣤⣤⠀⣴⡄
⢸⡿⣧⡀⠀⣿⣿⠀⠸⣇⠀⠀⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⣯⣴⠀⣾⣿⣿⣿⣿⣿⣿⠃⠀⣿⡇⠀⣿⠀⣿⠀⢸⡇⢸⣿⣆⣿⠀⠉⣿⠉⠁⣿⠋⠉⠀⣿⢿⣆
⢸⡇⣼⠇⢸⡇⢸⡇⠀⣿⠀⠀⣿⠀⠀⢰⢿⣿⢿⣿⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⡇⢀⣿⠀⣿⠀⢸⡇⢸⡏⢻⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿⠀⣿
⢸⣿⡏⠀⣿⠀⠈⣿⠀⣿⣸⡆⣿⠀⠀⣰⣷⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢸⣷⣾⣿⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⣿⠀⠀⣿⣶⡆⠀⣿⣼⠇
⢸⡏⢿⡄⢻⡆⢰⡟⠀⣿⣿⣷⡿⠀⢀⣿⣿⡏⣿⣿⣿⡿⠃⠀⢻⡿⠽⠿⠛⠋⠀⠀⠀⠀⢸⡏⠀⣿⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⣿⠀⠀⣿⠉⠀⠀⣿⢿⡄
⢸⡇⣼⠇⠘⣷⣼⠁⠀⣿⡏⣿⡇⠀⢸⣿⡟⣹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⣸⡇⢸⣧⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿⠸⣧
⢸⣿⠋⠀⠀⢻⡏⠀⠀⣿⠃⢸⡇⣠⡿⠋⣼⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣿⠀⢿⣿⢿⡇⢸⣿⠀⣿⠀⠀⣿⠄⠀⣿⣷⣶⠀⣿⠀⢻

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠐⠒⠒⠒⢒⡽⠿⠻⠟⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠉⠀⣠⣆⣿⣿⣽⣿⣾⡇⢢⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄
⢠⡤⠖⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⡟⠁⠈⣷⣾⣷⣿⠆⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⢀⠆⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠉⠙⠋⠀⠀⠀⠀⠈⠈⠙⠚⠀⠘⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠤⠀⣀⠀⣐⠛⢖⣒⣓⣚⣓⣒⣲⡦⠴⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⣄⣀⡀⠀⠀⠴⠟⠙⠫⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣠⣀⡀⠀⠶⠟⡥⠉⠩⡍⠉⠉⠉⢉⣅⣴
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣅⠠⠚⠛⠙⣀⣀⣀⣠⣇⣀⣠⣇⣀⣀⣀⣀⣀⣸⣀⣠⣄⣀⣀⣀
⠀⠀⡀⠀⢀⣀⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⢀⣤⠗⠁⠀⠀⠀⠈⠻⣷⡄⠀⢀⠐⠂⢀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣴
⡼⡏⠀⠀⠀⠀⢀⠀⠀⠘⢿⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣝⠊⢂⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣚⡋
⣧⡏⠀⣨⣷⠊⠣⠀⠀⠀⣸⠀⠀⠀⠀⠀⠁⢣⠀⡴⠶⠶⠦⡄⢸⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠻⢤⣧⣴⡁⠀⠀⠀⣱⣤⣧⠾⠋
⠘⣿⣄⠁⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠛⠈⢆⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉
⠀⠈⠛⠿⣶⣶⣶⠷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠰⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣕⣀⣀⣠⡔⠁⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠠⡴⠒⠶⠤⠤⢤⣤⡄⣀⢠⣤⣄⣠⣤⣄⣤⣠⣾⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠛⠋⠁
⠀⣐⣠⡠⠤⠶⠶⣏⡿⠿⠿⠛⠛⠛⠻⠿⠛⠛⠛⠛⠛⠩⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠉⠁⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⡀⠀⠀⠀⠀⣾⣿⣬⣹⣿⣟⡒⢾⣶⣶⣾⡿⠚⣛⣟⣉⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⡿⠀⠀⣷⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⢾⣬⣌⣉⣤⣴⡇⠈⣉⠀⠀⣀⡀⠀⢠⠀⠀⠀⠀⠈⠋⠳⠶⠖⣺⣿⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠿⠟⠛⠻⠿⠿⣿⣿⣦⣄⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢋⣴⣿⣿⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⢀⣠⣶⣾⣿⣿⣿⣷⣶⣄⡀⢀⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⣠⣿⣿⠟⠉⠁⠀⠈⠉⠻⣋⣵⠿⠋⠛⠛⣥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⢰⣿⡿⠁⠀⢀⣤⡶⢄⣤⠾⠋⢵⣿⡆⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⢾⣯⠰⣛⡥⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠸⣿⣷⡀⠀⠈⠻⠿⠟⠁⠀⢀⣾⣿⠇⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠹⣿⣷⣤⣀⠀⠀⠀⣀⣤⣾⣿⠏⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣶⣦⣤⣤⣤⣴⣶⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉





⣿⣿⣛⣛⣛⣛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⣤⣀⡀⢀⠄⠀⠀⠈⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⢻⣿⣿⣤⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠙⠢⣄⠀⠐⢦⣄⠀⠀⠀⠦⣤⣀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢗⣀⡐⠤⣤⡀⠰⠄⠀⠇⠁⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠛⠷⠎⡕⢦⣀⠀⠑⠶⠚⠈⠀⠙⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⠄⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⡀⠈⡀⠀⠀⢠⡀⠈⠑⠄⠈⠛⢤⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠇⠀⢰⣰⡄⠸⡇⢠⡄⢀⠀⠀⠀⠦⠀⠀⢠⣄⠻⣇⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⡄⠀⣞⣇⢀⠆⠘⠇⠀⠀⠀⠀⠠⠤⠎⣼⠏⠀⣿⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣌⢿⣿⣎⣇⢀⣠⠤⠴⢶⣾⣿⣿⠿⡏⠉⣘⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣾⠀⠀⠀⠈⢣⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉

⠀⢠⡶⣤⣶⡆⠂⢠⣢⡄⠀⠀⠀⠀⠆⠀⠀⣢⣄⢀⡄⢤⣐⡄⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⢸⣇⣨⣿⡟⠀⠎⠰⡦⠐⠀⠅⡈⡏⡥⠀⡟⢿⡋⢹⣶⢂⠱⠂⠘⠇⠀⠀⠀⣠⡀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⣀
⠀⢸⣿⣿⣿⣟⡉⣻⣏⠗⠚⣸⢀⡐⣧⡐⠀⡍⢹⣽⣿⢻⣘⣿⢀⠐⠂⠀⠀⣶⣿⣯⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⢸⣿⣿⣿⠁⢀⣏⡙⡂⠀⢸⠦⠤⡿⠇⠂⡁⢸⠤⡿⣻⣿⡏⣀⠀⠀⠀⢰⣿⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⢸⣿⣿⣯⠀⠚⠛⠡⠅⠀⢸⡆⢲⣷⡿⡄⠁⣾⣿⣿⣿⡿⠃⠉⠁⠀⠀⣼⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣴
⠀⣼⣿⢸⣿⢠⣶⣶⡟⡃⠀⢰⠀⢽⣇⡁⡀⢰⢻⡟⠉⢿⣱⠀⠀⠀⠀⠀⡿⠀⡿⠀⢰⣶⣶⠶⠄⢀⣀⢠⡇
⠘⠛⠛⠛⠛⠛⠛⠛⠓⠓⠒⠛⠓⠚⠛⠓⠓⠛⠛⠓⠓⠛⠓⠒⠚⠒⠂⠀⠃⠀⠑⠀⠈⠀⠀⠒⠚⠛⠛⠛⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀
⠀⡄⠀⣴⣦⣀⣀⡀⢀⡠⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⣤
⠀⠃⢸⣿⣿⣿⡯⢍⡩⠉⠋⠰⠱⠀⠀⡄⠐⡀⢰⢥⠂⠀⠀⠀⣸⡇⠀⠀⢠⣿⣿⡼⠀⠀⠀⠀⠀⠀⣠⡄⠈⠛
⠀⠁⢸⣿⣿⠀⠀⠉⠍⠠⠱⢾⣷⠤⠀⠄⢠⢀⣴⣶⣦⣤⣤⣴⣿⠷⠄⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡄
⠀⠀⢺⣿⣿⡃⠈⠈⠷⡬⢀⣄⠖⠀⠀⠀⢫⢘⣿⠿⣿⣿⣿⡟⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠸⡇⢸⡇⠀⢰⡷⠿⡿⠉⠀⣴⣤⡀⠀⢹⡏⠀⠀⢸⠀⡉⠀⠀⠀⠀⢸⡇⢿⠀⣸⡷⢶⣶⠶⠄⠀⠀⠠⢴⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠖⠒⠒⠒⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢉⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢠⣾⣿⣟⣻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣥⣾⣿⣿⡾⠧⠀⢾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⢠⣿⣿⠿⢿⣿⣣⣤⣴⣾⣷⡴⣿⣿⡿⣿⣿⣿⡋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⡄
⠀⠀⠀⠀⠀⣿⣴⣿⣿⣿⣿⣿⣿⣿⠿⠿⠀⠉⣿⣔⣟⣻⣙⡁⠀⠀⠀⠀⢰⡔⠀⠀⠀⠀⠀⠀⣿⠀⢿
⠀⠀⠀⠀⢸⣿⡿⣥⣿⣿⡟⢁⠈⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠻⣄⡀⠀⠀⢀⡤⠋⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠙⠉⠉⢹⣿⣿⣧⣽⣿⣿⣛⣿⠈⢻⣿⣿⣿⣿⣿⡿⠗⠀⠀⠘⠿⣿⣿⠿⠃⠀⠀⠈⡇
⠀⠀⠀⠀⢸⣶⣶⣿⣶⡤⣸⡿⢿⣿⣦⡀⠀⢸⣿⠀⣾⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡏⠛⠻⠿⠷⣿⣿⣖⡉⠻⠷⠀⢸⡟⢸⡇⢀⣀⣀⣀⣀⣀⣤⣶⣶⣾⣿⣿⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠘⢿⡿⣿⣿⣷⠆⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⡟⠉⠙⠿⠗⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⢻⣿⡿⠋⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⢸⠀⠀⠀⠀⠀⠀⢸⣿⡟⣻⡿⠋⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣴⣄⣠⠀⠀⢸⣿⡇⠘⣇⠀⠀⠀⠀⠀⣀⠀⡏⣿⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⣼⣿⣷⣤⣿⣤⣾⣿⣷⣾⣿⣿⣷⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠰⡄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠉⠓⠢⣄⡵⠶⡉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡤⠴⠒⠒⠲⣤⣔⠋⠙⢦⡈⠑⠠⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣐⡠⠐⠒⠐⠒⠤⣧⡖⠁⠀⠀⠀⢀⡴⠋⠈⢧⡀⠀⠙⢆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠲⢄⠀⣠⠏⠀⠀⠀⠈⣇⠀⠀⠈⢣⠀⠀⡄
⠀⠀⠀⠀⠀⡰⠱⣿⣿⡿⠟⠉⠁⠀⠄⡀⠀⣀⣀⣀⠀⠀⠰⡃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢧⠀⠇
⠀⠀⠀⠀⠘⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠡⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡜
⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡷⠄⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸⠁
⠀⠀⠀⢸⠀⠀⠀⣠⣦⣄⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⠟⠁⠀⠀⡇⠀⠀⠀⣸⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠧⠄⠀⠠⠧⠄⠒⠉
⠀⠀⠀⠀⠠⡀⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠑⠄⠈⠿⠿⠟⠛⠀⠀⠀⠀⠀⣀⣤⣶⣶⣄⠌
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠤⠤⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠾⠿⠛⠛⠛⠛⠿⠷⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣤⣀⣀⣀⡀⠀⠀⠀⠀⠈⢻⣆
⠀⢰⡟⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠿⠛⠛⠛⢛⣛⣛⣛⣻⣿⣿⣯⡉⠙⠻⢿⣿⣷⣤⠀⠀⠀⠀⢻⣆
⢀⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⢁⣀⡤⠖⠚⠋⣩⡉⠙⠻⡹⡿⡟⣿⣧⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⠈⣿
⣸⡇⠀⠀⠀⠀⠀⠀⣠⠋⠛⣿⠿⠛⠋⠁⠀⠀⢲⡟⣻⣿⠀⠀⠀⣤⠔⠮⣿⡀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⢸
⣿⠃⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⠎⢰⣷⣤⣿⣿⣭⣭⣶⡦⠽⠶⠆⠈⣙⣦⠀⠘⢿⡟⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⢠⡏⠀⣼⣿⣿⣿⣿⣿⣿⣷⡅⣩⣴⣤⣝⣛⡿⡇
⣿⡄⠀⠀⠀⢠⡿⣤⢆⣀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⠟⣣⠌⠉⠛⠋⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢹⡇⠀⠀⠀⣸⡇⢦⠜⣥⢆⣠⡀⠈⢷⣄⠀⠻⣿⣿⣿⠟⣱⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⣿⡀⠀⠀⣿⠷⠺⡤⢢⠼⣁⢝⢁⡀⠙⠷⢤⣈⠛⠏⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠹⣧⠀⠀⢻⠘⡖⢧⠴⠤⢃⣈⢋⣹⡃⡀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠹⣧⡀⢸⡏⠳⢢⣀⠦⠎⣅⡨⣁⡙⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠙⣷⡄⠇⠶⠊⣅⡰⣤⠈⣄⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⠶⣶⣶⣦⣤⣤⣠⣾⠋
⠀⠀⠀⠀⠈⠻⣦⡈⠒⠉⣄⠴⠴⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⢤⣀⣀⠀⠀⠀⠀⠀⠒⢦⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠀⠀⠀⠈⠙⠛⠁⠀⠈⠠⣄⣀⣙⣛⣶⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣾⣷⣶⣤⣬⣷⣶⣶⠾⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠱⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣛⠻⢶⣄⡀⠀⠀⢠⡇⢀⢀⠀⠀⠀⠀⢤⡄⢐⢺⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣀⠭⠀⠉⠽⠿⢠⣄⡉⠳⣦⡀⠀⢳⡼⢯⠸⣀⡤⢰⢀⡇⠸⢸⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡦⣄⡀⠀⡢⠁⠉⡀⣈⠛⠆⠀⠀⠀⠀⠉⠀⠈⠉⠃⠀⠘⠀⠀⠀⢸⡧⡀⡈⣀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢀⡐⠃⠀⠦⠉⠏⣄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠠⢀⠇⡟⢹⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⢠⠚⢠⣚⣑⠦⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖⠀⣦⠰⡇⢿⣝⣶⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢺⡇⠈⠲⢻⣎⢿⣧⣿⣟⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣧⣼⠇⣶⢲⣮⣟⢿⣿⣿⣮⢿⢠⣞⠿⣝⡻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⡀⠐⣼⡅⢻⣿⠁⠹⡿⣄⣴⡿⠳⣶⡮⢷⠾⢭⣙⣶⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡖⠋⢀⡁⠀⣼⠿⡾⣾⡗⣠⣻⣿⡒⠦⡉⣷⠢⣌⠻⢮⣻⣦⠛⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠋⢀⣌⠱⡄⣇⠀⣿⡮⣝⣿⣿⠃⠁⠀⠘⠢⠻⣦⡙⠒⠉⢿⡉⢳⠄⣷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⣦⠸⠁⡀⡀⠘⡧⢹⡏⠹⣿⠯⡀⠻⡝⠂⠂⠐⢾⣦⠈⠛⣦⣍⡃⡳⣌⠛⣾⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⠀⠁⠸⠴⣧⢸⡿⣧⡈⣦⣽⣶⣿⡢⣌⡒⠄⠈⣳⠀⣘⣿⡳⣿⢿⣶⡘⣷⣌⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠰⡆⠀⠁⢸⡷⢬⡟⢶⣵⣱⣎⢲⣮⠭⡀⠱⣿⡇⠈⠉⠻⣶⣶⣽⢷⢿⣈⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠁⠀⠀⠀⢀⠄⠀⢦⡙⢀⡙⠉⢿⣟⢣⣀⡹⣤⣼⡗⠄⠟⢾⡟⠫⣥⣌⠀⠉⠦⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠀⠀⠀⠀⠀⡀⣷⡜⣿⡿⢦⣙⠪⡈⠓⠙⢈⠉⢸⠶⣏⡹⣄⠈⣻⣇⡄⡰⡀⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠶⠄⠒⠉⠘⢿⣾⡃⢺⡏⢣⣀⠀⠰⢦⡑⢤⣓⢦⠙⣤⣸⣟⠋⠙⣽⣿⡈⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣤⠦⠄⠚⠀⢧⡀⢳⣄⠻⣷⣤⣶⣍⡙⢿⣿⣓⡟⢡⣉⣳⣄⣨⣿⡃⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠠⣄⢀⣙⡰⣌⠓⢸⣦⣉⠛⣿⠦⢉⣻⣿⠪⢯⡉⠉⠛⣿⣧⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣴⣄⢻⡟⠛⢧⠀⡤⡈⢲⣽⠶⠾⠫⠉⠛⠒⠁⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣤⣼⣿⣳⠶⠾⠋
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⣬⣯⠾⠟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣚⣯⡿⠋⢠⡎⢡⠶⡯⠊⠙⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠟⠋⠉⠉⠉⠍⣩⢽⡠⠞⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⠿⠿⠿⠿⠿⠿⠿⢷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣿⣿⣿⣶⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣠⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⣯⡉⠙⠛⠿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠘⣿⡄
⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⠟⠉⠀⠀⣀⣤⡤⠶⠖⠛⠛⠻⢿⡟⢿⣿⡿⣿⣿⣧⠀⠀⠀⠈⢻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠹⣿⡀
⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⣠⡴⠞⠋⠉⠀⢀⣠⣼⣽⣦⠀⠀⠙⠈⠃⠙⠈⠻⣿⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣧
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠉⢹⠿⠛⠋⠁⠀⠀⡀⠀⠀⠹⣏⣨⣿⣷⡀⠀⠀⠀⣴⡖⠙⠢⢿⣄⠀⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣾⣦⣄⣻⣟⣋⣩⣭⣑⣶⣤⣽⣷⠶⠀⠀⠉⡳⣄⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠂⠒⠂⠤⠄⠲⠿⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠺⣿⣿⣿⣿⣶⠾⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⣇⠀⠀⠀⠀⠀⢸⡟⠷⠞⣄⡤⠀⠀⠀⠀⠀⣿⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢸⣿⠀⠀⠀⠀⠀⣿⡇⢰⣤⠾⣄⣴⡀⣠⠀⠀⠹⣷⡀⠀⠘⢿⣿⣿⣿⣿⠟⣡⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠈⣿⡆⠀⠀⠀⠀⣿⣧⡰⣆⣀⢭⣀⡟⠁⡷⠞⠀⠙⠻⣦⣄⡀⠙⠿⣿⠋⢠⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠹⣿⡀⠀⠀⠀⣿⢠⡀⣩⠁⢈⡁⢸⠞⠱⣤⠾⣄⡔⠀⠉⠛⠒⠄⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⢻⣷⡀⠀⠀⣿⣀⡟⠉⠶⠊⠛⠋⢦⡴⠰⣀⠜⣄⣠⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⢻⣷⡀⠀⢹⡏⠹⠞⣇⢀⠰⠴⠂⣄⣠⢩⢀⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠹⣿⣄⠈⡇⢤⣤⠉⠉⠀⣄⢀⠈⡁⢈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⣶⣤⣤⣤⣤⣤⣀⣀⡀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⢥⣀⠷⠃⠘⡋⠀⠙⠛⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡈⠁⠀⠷⠔⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠒⢶⣦⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠖⠂⠀⠉⠉⣉⠀⠀⠙⠳⠶⢶⣠⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣦⣤⣀⣐⣦⣄⣀⠀⠀⠤⣤⣀⣀⣈⣙⣿⣿⣶⡾⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣶⣶⣦⡄
⠀⠀⠀⠈⠋⠁⢈⣿⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⣯⣵⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠖⠛⠛⠲⢤⡀
⠀⠀⠀⠀⣴⢿⣿⣻⣿⣿⣿⣿⠿⠿⢦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⠁⢠⣄⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠉⢿⣿⣿⣛⡵⠟⠁⠀⠀⠀⠈⠙⠶⡀⠀⠀⠀⠀⠻⢼⠃⠈⠁⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⡾⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡏⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣧⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣆⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠈⠛⠲⠦⠤⠤⠄⠀⠀⠹⣆⠀⠀⠈⠳⣄⠀⠀⠀⠙⠲⠦⠤⠤⠤⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣦⠤⢤⡴⠶⠒⠛⠦⣄⣀⠈⣧⠀⠀⠀⠈⠙⠲⢦⡤⢤⣤⠤⠶⠞⠳⣄⣀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣧⣤⣾⡄⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣠⣤⣼⣧⣤⣿⡀⠀⠀⠀⠀⠉⠉⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⢀⠐⢀⠘⢀⠐⠀⠀⠃⡀⠃⡀⠀⢀⠐⢀⠘⢀⠐⠀⠀⠂⡀⠃⡀⠂
⣀⠐⢀⠐⢀⠐⠀⡀⠂⡀⠂⡀⠀⢀⠐⢀⠐⢀⠐⠀⡀⠂⡀⠂⡀⠂
⢀⠐⢀⠐⢀⠐⠀⠀⠂⡀⠂⡀⠒⢀⠐⢀⠐⢀⠐⠀⠀⠂⡀⠂⡀⠂
⠀⠠⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠄
⠉⠠⠈⠠⠈⠠⠀⠀⠄⠁⠄⠁⠤⠉⠠⠈⠠⠈⠠⠀⠀⠄⠁⠄⠁⠄
⠉⠠⠈⠠⠈⠠⠈⠁⠄⠁⠄⠁⠀⠉⠠⠈⠠⠈⠠⠈⠁⠄⠁⠄⠁⡤⢤
⠀⢀⠈⢠⠈⢀⠀⠂⡄⠁⡄⠀⡀⠈⢀⠈⢀⠈⢀⠀⠀⡀⠁⡄⢁⣧⣸⡀
⠐⢀⠐⢀⠐⢀⠀⠀⡀⠂⡀⠂⡀⠐⢀⠐⢀⠐⢀⠀⠀⡀⠂⡴⢋⡤⠤⡈⢣
⠒⢀⠐⢀⠐⢀⠀⠂⡀⠂⡀⠂⡀⠐⢀⠐⢀⠐⢀⠀⠂⡀⠂⣧⠺⣄⣠⠇⣸
⠀⠀⠐⠀⠐⠀⠀⠀⠀⠂⠀⠂⠀⠐⠀⠐⠀⠐⠀⠀⠀⠀⠂⣸⠓⢤⡤⠚⢧⠀⠀⠀⠀⢀⣤⣤⡀
⠠⠈⠠⠈⠠⠀⠀⠀⠀⠄⠁⠄⠀⠠⢀⡶⠚⠲⡆⠀⠄⠁⢄⡇⠄⡟⢹⠀⠘⡄⠀⠀⠀⢸⣇⣀⡇
⠤⠈⠠⠈⠠⠈⠠⠄⠁⠄⠁⠄⠉⠤⢘⡷⠚⠠⡏⠀⠄⠁⣼⠁⢴⠃⠀⡆⠀⢧⠀⠀⠀⢸⡗⠒⡇
⠠⠈⠠⠈⠠⠈⠀⠀⠁⠄⠁⠄⠉⠠⠘⣟⠉⢠⡏⢠⣤⢁⡇⠁⡎⠐⠂⢸⡀⠘⡄⣤⡄⢸⡇⡇⡇
⣀⠀⢀⠐⢀⠐⠀⡀⠂⡀⠂⡀⠂⢀⠘⣏⡉⢠⡗⠀⣿⣿⡀⢸⠁⠈⡁⠀⣇⠀⢣⣹⠁⢸⡇⡇⡇
⣀⠐⢀⠐⢀⠐⠀⡀⠂⡀⠂⡀⠒⢀⠸⣧⣴⢀⡗⠀⣸⠇⣀⡏⡀⠂⠄⠀⠸⡀⠘⡇⠀⢸⡇⡇⡇
⢀⠐⠀⠐⢀⠐⠀⠀⠂⡀⠂⡀⠀⢀⠸⣧⠄⢀⡗⠀⡼⠀⣸⢦⣀⡶⢦⣀⡤⢇⠀⢳⠀⢸⡇⡇⡇
⠀⠠⠀⠠⠀⠠⠀⠀⠄⠀⠄⠀⠀⠀⠰⡇⠶⠈⡧⢸⠃⢀⡏⠄⠀⠓⠚⠁⠀⠸⡀⠈⡆⢸⡇⡇⡇
⠉⠠⠈⠠⠈⠠⠀⠁⠄⠁⠄⠁⠄⠉⢰⡗⠢⠈⡧⣞⠀⣼⠁⠄⠁⠄⡁⠀⠀⠀⢣⠀⢹⢸⡗⠓⡇
⠈⠠⠈⠠⠈⠠⠀⠁⠄⠁⠄⠁⠄⠈⢰⡟⠓⠈⡇⢸⠛⠅⠁⠄⠁⠠⠄⠀⠀⠀⠈⠓⡏⠈⢇⢠⠃
⠉⠀⠈⢀⠈⠀⠀⠀⡀⠃⡀⠁⠀⠈⠀⠙⢋⠛⢁⠜⠀⡀⠃⡄⠁⡀⠀⠀⠀⠀⠀⠀⠓⠀⠈⠃
⠒⢀⠐⢀⠐⢀⠀⠂⡀⠂⡀⠂⡀⠐⢀⠉⢉⠙⠻⠟⠳⠶⠲⡴⠚⠿⠛⠛⠛⠛⠋⠉⠉⠉⠉⠈⠉
⠒⢀⠐⢀⠐⢀⠀⠂⡀⠂⡀⠂⣀⠐⢀⠐⢀⠐⢀⡀⠂⡀⠂⡀⠂⣀
⠀⠀⠐⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠰⠀⠶⢎⡶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠶⠶
⠤⠀⠠⠈⠠⠈⠀⠄⠁⠄⠁⠄⠀⠠⠈⠠⠈⠠⠈⠀⠄⠁⠄⠉⠄⠉
⠠⠈⠠⠈⠠⠈⠀⠄⠁⠄⠁⠄⠉⠠⠈⠠⠈⠠⠈⠀⠄⠁⠄⠁⠄⠁
⢤⠈⠠⠈⢠⠈⠀⠀⠁⡄⠁⡄⠀⠠⠈⠠⠈⢠⠈⠀⠀⠁⡄⠁⡄

⠀⠀⠀⢀⡴⠒⡿⠛⠛⠒⢤⡀⠀⠀⠀⠀⠀⠀⣠⠶⠒⡶⠒⠒⠦⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉⠑⠲⣄
⠀⠀⢰⠋⢀⠰⢆⠀⠀⠀⠀⢷⡀⠀⠀⠀⢀⠞⢩⠀⠎⠀⠀⠀⠀⢘⣆⠀⠀⠀⠀⢠⠋⠀⠀⠀⢠⡄⠰⡆⠹⡄⠀⠀⠀⢠⠞⠀⢀⡠⠤⠤⢄⠀⠈⢧
⠀⠀⣼⠰⠁⠀⠀⢹⠉⠉⠙⢮⡇⠀⠀⠀⢸⡄⠀⠻⠤⠀⠠⠤⠊⠁⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣒⠄⠀⡇⠀⠀⠀⢸⠀⢠⠏⢰⡿⣷⠀⢳⠀⢸⡀
⠀⠀⢹⠀⡱⠀⠀⢎⠀⠀⠀⢸⠃⠀⠀⠀⠘⣎⠒⢦⢀⣀⠀⢀⣀⠤⡾⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠸⡄⠈⢧⡘⠿⠟⣠⠞⠀⡼
⠀⠀⠈⠳⢧⣀⠀⢀⣳⣲⠾⠃⠀⠀⠀⠀⠀⠈⠳⢾⣦⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠘⠦⣄⡉⠉⢉⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⣠⠤⢲⡖⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⢒⡖⠤⣄
⠀⠀⢠⠞⣡⠖⠋⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⡴⢻⡂⠀⠀⠀⠀⢸⡷⡄⠀⠀⠀⠀⢠⠞⢁⡤⠚⢅⠉⠲⡉⢧⠀⠀⠀⢀⣠⡔⠒⠛⠉⠉⠉⠉⠓⢦⣄
⠀⠀⡞⢸⠇⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⢸⠁⢸⠀⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⡯⠰⠁⠑⢄⠀⠱⡀⠘⡋⡇⠀⣰⢋⡹⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢳
⠀⠀⣇⠘⣆⠀⠀⠀⠀⢠⡞⢡⠇⠀⠀⠀⠸⡄⠈⠷⣄⣀⣀⣠⠞⠀⡞⠀⠀⠀⠀⢧⠄⠧⠀⠀⠢⠀⠘⣀⢷⠇⠀⠹⣭⣻⠚⠒⠒⠒⠒⠒⠒⠛⣻⡿⠏
⠀⠀⠘⢦⡉⠻⠲⠶⠖⢋⡤⠋⠀⠀⠀⠀⠀⠙⢤⡀⠀⠈⠁⠀⣠⠞⠁⠀⠀⠀⠀⠈⢧⣤⠋⠢⡤⠷⢒⣡⠏⠀⠀⠀⠀⠉⠓⠶⠤⠤⠤⠤⠖⠚⠉
⠀⠀⠀⠀⠉⠙⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠓⠚⠉

⠀⠀⠀⠀⠀⣀⡠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢤⡔⡶⢶⣶⠶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⡀
⠀⠀⢀⡴⠋⡹⠛⠚⡇⠙⠗⢆⠀⠀⠀⠀⠀⣀⡠⠤⠤⠶⠶⠤⢤⣀⠀⠀⠀⠀⠀⠀⢽⡖⡆⢽⠀⣇⣹⢨⡇⠀⠀⠀⠀⠀⣠⠾⡍⠽⢀⣁⠺⢋⢷⡄
⠀⠀⡼⠀⢠⠁⠀⠀⠀⠀⠈⡈⡆⠀⠀⠀⡾⠅⢀⡀⠁⠀⠐⠀⣀⠨⠽⡄⠀⠀⠀⠀⠸⣿⡇⢸⠐⡇⡇⣼⠁⠀⠀⠀⠀⢰⡏⠳⡙⠍⢢⣢⢙⡁⠧⢻
⠀⠀⢧⠀⢸⠀⠀⠀⢸⠀⠀⣷⠇⠀⠀⠀⣇⠀⠀⠀⠀⠈⠈⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠻⣿⣼⣾⣶⡷⠃⠀⠀⠀⠀⠀⢸⡲⢱⣅⢫⡠⣒⢕⣘⠵⣼
⠀⠀⠘⢦⡘⡄⠀⠀⠈⠀⣠⠟⠀⠀⠀⠀⠈⠒⠤⢤⣤⣤⣤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⡤⡔⢃⣴⣈⠚⣲⠏
⠀⠀⠀⠀⠉⠛⠲⠤⠗⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣋⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠥⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠙⣾⠁⣠⠞
⠀⠂⠀⡴⠚⠉⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⣠⠤⢴⡋⠁⠀⡠⠶⢦⡉⢢⡀⠀⠀⠀⠀⠀⠀⠀⣷⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⣾⣿⠁
⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠙⢶⡤⣬⣆⠀⢧⣱⡶⠃⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡡⠤⠤⢬⡙⢦
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⣖⣶⣷⡶⠼⣦⡴⠚⢉⡷⠀⣿⠀⠀⠀⠀⠀⠀⣰⡃⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⡏⡐⠁⡔⠉⠑⠀⠘⡈⡇
⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢸⣯⣏⣿⣏⣹⣛⠏⠉⠉⠀⢠⠃⠀⠀⠀⠀⠀⠀⣯⣉⣉⣩⡇⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠈⡅⢐⠀⢀⢃⡇
⠀⠀⠀⠈⠓⠢⠤⠤⠶⠒⠋⠀⠀⠀⠀⠈⠓⠳⠼⠭⠿⠿⠒⠚⠋⠓⠋⠀⠀⠀⠀⠀⠀⠀⢯⠁⠀⢨⠇⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⠢⠬⠅⠒⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠆⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠂⠀⣀⢠⠀⠀⠄⠀⠀⣤⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⣿⣿⣿⣶⣌⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠷⠄⡀⢉⠘⣀⡆⠐⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣻⣄⠀⠹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠐⠁⢂⠀⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⠀⠙⣻⣿⣻⣇⠀⠀⠀⠀⠀⠰⢸⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠈⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⢀⡽⢿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠂⣈⡁⢀⣀⠀⢀⠀⢀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⠷⠟⢡⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠁⠀⠀⢱⠈⠀⠀⡅⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⠀⢨⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⣿⡆⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣿⣧⣄⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡄
⠀⠀⠀⠀⠀⠀⢳⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣯⣶⣯⣽⣻⣿⡷⠿⡄⠀⠀⠀⠀⠀⣠⣄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠙⠐⠀⢀⠀⠀⠀⡴⠀⣀⣀⡀⠀⠀⢿⡿⠉⠛⠋⠛⠛⠿⢿⡟⠛⠛⠛⠛⣙⠁⠨⢆⠀⠀⠀⣠⣞⣩⣟⣋⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢠⡆⢀⡔⢲⡋⠉⢤⡈⠻⣄⣠⠋⠀⠀⢀⣀⣀⣀⣤⣸⣇⣀⣀⣤⣤⣤⣜⣥⣼⣦⣼⣿⣿⣿⢿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠴⠋⠳⢾⣇⡠⠘⣷⠀⣀⣧⣤⡴⠿⠛⠛⠋⠉⠉⠉⠿⡉⠉⠉⢙⣿⣿⣿⣿⣿⣿⣿⠿⣳⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣵⠶⠩⠀⠀⢻⡄⠂⠘⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣤⡴⠟⠉⠉⠉⠉⠉⠉⠀⠉⠁⠙⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⢠⣂⡁⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠰⡋⠗⠀⠀⡐⠠⢐⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⠟⣣⡤⠄⠐⠞⠻⠶⡤
⣿⣿⣿⣿⣿⣿⣿⡷⠶⠶⠒⠶⠖⠒⠿⠿⠲⠖
⣿⣿⣿⣿⡟⠓⠲⠆⠀⠀⠀⠀⣿⣷⠀⠀⡀
⣿⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⣼⣿⣯⠀⡴⠁
⣿⣿⣿⣿⠥⢴⣤⠀⠀⠀⠀⢿⣿⣿⣿⠃
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠁⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣀⣠⣾⠿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⢀⡤⢾⣿⠿⣿⣿⣿⣿⡃⠀⠀⡀⡧⢚⡀
⣿⣿⣿⣿⢀⠀⣀⢀⡀⡀⢀⣿⣹⣷⣿⣿⣿⣽⣠⢀⣀⣄⣄⣠⣫⣀⠞⠁⠀⠀⠀⣈⣽⣷⣤⣴⣿⣿⣤⣤⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣀⣀⡀⠀⠙⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣼⣿⣶⣿⣭⣍⣿⡆⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⢿⡗⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡶⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⣿⠀⠙⢷⣄⠀⠀⣠⡾⠋⠀⣿⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠉⠋⠀⠀⠀⠀⠀⠀⢙⣷⣾⡋⠀⠀⠀⠀⠀⠀⠙⠉⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠛⢷⣄⡀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⣼⣿⣭⡀⡄⠀⠀⢠⢀⣭⣿⣧⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠘⠿⠿⣷⠀⠀⠀⠀⣾⠿⠟⠃⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠃⠀⠀⠀⠀⠀⣠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠂⠀⠀⠀⠀⠀⠘⠷⡾⠃⠀⠀⠀⠀⠀⠐⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣶⣶⣶⣶⣶⣴⣶⣶⣴⣶⣦⣶⣶⣶⣶⣶⣴⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⣤⣶⣶⣤⡀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠿⣼⡿⠞⠿⠧⠀⠭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠧⠀⠀⠁⠃⠀⠀⢴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⠀⠀⠀⠀⡐⠄⢀⡀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⡄⠀⠁⠀⣼⣿⣾⡟⣛⠶⣅⣀⡀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢲⡆⠀⣷⡄⢿⣇⠸⠇⡘⣿⢻⣶⣦⣄⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡐⠶⣁⠀⣅⣰⡌⣱⣦⣽⡿⢲⣤⣿⣿⣭⠰⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠀⣾⡄⢸⣦⣿⣙⣿⣾⣿⣿⣿⣾⣿⣍⠿⠂⢢⣘⢿⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣠⠄⣈⡙⡋⠉⣯⡀⣹⣿⣿⣿⠓⢙⠛⠛⣾⠻⣏⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠟⠁⠀⢲⠛⣿⡀⠙⢏⠛⠃⠈⠋⢰⣿⠀⠹⣟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠂⠀⡿⠂⠈⠇⠙⣝⠦⠀⠁⠈⠀⢰⡿⣿⡳⠄⠈⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢈⠀⢠⠐⠟⠀⠈⠷⡄⠰⣶⢞⠻⠍⢾⡷⠀⠂⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠟⠙⢿⠀⠚⢠⣄⣢⠀⢀⢶⣶⠾⠃⠠⢀⠠⠯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⠑⠂⠀⠟⠉⠐⠂⠂⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣀⠔⡔⡀⣤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣘⣣⣽⣿⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣯⣭⣵⣶⣶⣦⡙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢁⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣽⣿⣽⣿⣽⡿⢡⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣵⣿⣿⣾⣿⣿⣿⣿⣿⣧⣿⣽⢿⣾⢯⣾⠏⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⣯⡿⣵⡿⣡⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣵⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⡾⢋⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣫⣴⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠟⣡⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣾⣿⣿⠿⠛⢁⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢺⣿⠋⠁⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣶⣶⣿⣿⢶⣦⡤⠴⠶⠶⡾⣥⠤⠲⠶⠶⠄⠐⠶⠶⠶⠦⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠷⠆⠠⣤⣘⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡄⠀⠶⠶⢶⠶⠶⠶⠶⠶⠶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣠⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣾⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⣿⡏⠀⠀⠀⠀⣀⠀⢀⡀⣀⢠⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣤⠀⢸⣿⣧⣀⡀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣾⡟⠀⠀⠀⢸⣿⣿⣿⣿⠄⠉⢉⣩⣍⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿
⠀⠀⢀⠀⠀⠀⢹⣿⠿⢿⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⡿⣿⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⠿⠿⠿⠿⠿⣿⣿⣿
⠀⠐⡛⠛⠂⢰⡟⠁⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢸⢇⡟⡇⠇⠀⠀⠀⠀⠀⠀⠉⠁⣒⢉⡀⠀⠐⠚⠂⣿⣿⣿
⠀⠉⠉⣀⣠⣸⣧⣤⣤⣤⣴⣷⣧⣴⣴⣦⣶⣴⣼⣾⣷⣷⣷⣶⣦⣀⣀⠀⣀⡀⣀⢀⢀⣀⢀⠉⣀⣤⣿⣿⣿
⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣼⢟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠠⢐⠀⠐⠐⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣷⣟⡧⣿⣟⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠒⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠿⢿⣵⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠥⣔⠘⢓⡂⣬⣧⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢎⣰⡬⠱⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠉⠇⢲⣿⣥⣻⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡯⢃⢨⢟⣜⣿⡿⠏⠄⣤⠓⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆⠒⠈⠈⠞⡣⢀⡸⢏⣻⡖⠿⡮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣤⠀⡈⠲⣭⢨⡲⢤⡆⠉⢭⣑⣯⡙⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠃⢀⡤⣚⣋⠐⢤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠐⢁⠀⠙⠿⠻⡌⢔⠹⣯⣙⣶⣴⣆
⠀⠀⠀⠀⠀⠀⠀⢤⠐⠂⠚⠭⠟⡽⣆⣀⣤⡈⠢⠀⠀⠀⠀⠀⠀⣄⠀⠄⠀⠀⠀⠈⠙⣶⠞⠓⣊⣥⣛⢾⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠘⣦⠄⠠⠀⣼⠏⠀⣠⠀⠀⠈⣆⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣭⣼⡃⣙⣻⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⡺⠍⠈⠲⠏⠖⢀⡀⣸⡆⠀⠀⠀⠀⠀⠙⢤⠀⡀⠀⠀⠀⠀⠘⠒⣷⡮⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⠛⡦⢀⠀⢰⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠠⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠈⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣨⡴⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⢀⡀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠞⢲⠳⠀⢀⠜⡄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⢀⢀⢸⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠃⠐⠈⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⡋⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⢀⣤⣄⡈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢠⡶⢻⣀⣨⠏⠛⢯⣙⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠒⠒⣿⠀⠀⠀⡟⠀⠀⠉⠁⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠁⣀⣤⠤⠀⠀⠀⠀⢸⡄⠀⠀⣇⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⡴⠀⠠⠞⠉⠀⠀⠀⠀⣴⠀⣸⣄⠀⠀⠙⢦⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡴⠋⠀⠀⠀⠀⠀⡀⠀⣠⠞⠁⢠⡟⠈⠙⠓⠲⠲⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣠⠞⠃⠀⣠⠞⠁⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⠆⠘⠃⠀⠀⡀⠁⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢠⡟⠀⠞⠁⠀⠠⠶⠖⠛⣁⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⢀⣠⡤⠞⢓⡖⠒⣶⣶⠒⠚⠛⠋⠉⠀⠀⠀⢀⡶⣦⠀⠀⠀⠀⢀⣴⠋
⢠⠞⠋⣁⡤⠞⢋⣤⠞⠁⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⡇⣹⠀⣀⣠⠴⠋⠁
⢹⡶⠞⠉⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⢂⡇⡿⠋⠉
⣿⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⣧⣤⣄
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡄⠈⠻⣿⣿⣿⣿⠟⠁⢠⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠛⠉⠁⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠈⠉⠛⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⢀⣤⣶⠟⢛⣛⠋⠾⠦⣥⡢⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣤⡾⡒⠒⢲⣶⡖⣒⢤⣀⣀
⠀⠀⣤⠖⣾⣻⣿⣿⡾⡿⢻⠿⣷⣦⠀⠛⢷⣶⡤⠶⢤⣭⢦⣞⡛⣫⣀⡭⣉⣉⣛⢱⡾⡔⢻⡿⢧⣯⣿⡶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠿⣿⡈⢙⣻⡟⠀⢈⠗⠲⣒⠊⣝⣾⣋⣈⠂⣁⡈⠁⣴⡚⠋⠀⠩⠿⣍⣉⣽⣷⡜⣾⢿⣷⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠿⠟⠀⠐⠁⠂⡀⣡⡖⠉⠚⠙⠁⢈⡉⢠⡭⢍⣶⣞⠩⣤⠴⣤⣽⣷⣏⢨⣫⣰⣷⣾⣧⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠠⡄⠀⠀⠀⢳⠎⡉⠝⢒⠀⠚⢿⠯⣵⡙⣽⣗⣬⢿⣴⣶⡶⠾⣿⠂⣭⡟⢟⣿⣕⡈⣗⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠒⠐⠌⢀⢨⣉⡀⠒⢻⡾⣵⣾⣿⣭⣭⣷⣿⣟⣻⣷⣾⣷⣖⡿⣷⡄⣹⡿⣿⢿⣷⡍⣋⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⡐⢦⣴⢀⠀⠀⠛⠩⣿⣵⠟⣷⣎⣽⠛⠻⠜⠀⢈⡈⠛⠛⠐⢿⣽⢶⣷⢻⣿⣿⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⠘⣂⢜⠃⣀⢉⡀⠀⢀⠂⠈⠉⢤⠒⠚⢂⡥⠐⠋⣀⠌⡠⠄⠐⡥⠒⡻⠛⠛⢿⣿⣯⣳⣍⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⡰⠆⢈⠕⢉⠋⠹⣻⣢⢕⠦⠌⠀⢩⠉⠁⠀⠀⢘⡁⠀⢀⠄⢀⠤⠚⠁⠤⠠⣄⡿⠛⡛⠙⠿⠟
⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠣⠻⠃⢁⡈⠝⠳⠋⠆⣙⠉⠲⠂⠂⠀⠈⠩⠶⠀⠀⡤⠤⠀⠀⠿⠋⠀⠀⡐
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡄⠴⠬⠙⠄⠙⣛⣳⡤⡐⠲⠦⠉⣀⡢⠀⠀⠀⠀⡴⠀⡴⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡶⠊⠀⠈⠡⠄⠤⠄⠀⢲⣤⠔⠒⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⢠⣼⡁⡀⠀⢀⡀⠀⠀⠀⢀⡬⠁⠀⠀⠀⠀⡀⠁⢀⣤⠀⠀⠐⡈⠊⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢄⠒⠯⠔⠊⠀⠀⠀⠀⠀⠀⠀⣀⡴⠸⣅⡤⠤⠦⠒⠀⠠⠀⠘⠁⠀⠴⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣴⠼⠟⡿⠖⠉⠀⠀⠀⠀⠀⠄

⣰⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⢶⢶⣾⡷⡶⡶⣶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚
⠈⠙⢿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡼⠞⡏⣿⡇⢹⡿⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠈⠻⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⣇⣇⣿⣿⣸⣇⣿⣿⣷⠀⠀⠀⠀⠀⡠⠊
⠀⠠⣄⡄⡄⢠⠙⠻⣿⣦⣄⡀⠀⠀⢸⣿⣰⣿⢿⣿⣽⣿⣿⣿⣿⣸⣿⠀⠀⢀⠐⠉
⠘⢛⣮⣯⡗⠁⠀⠀⠀⠙⠿⣷⣶⣀⠀⠻⣿⣿⣰⡿⣿⡜⡇⡆⣿⣿⠟⣠⠀⠑
⢀⡸⣷⣟⠉⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣆⡈⠙⠿⣷⣿⣷⡷⠟⢋⡥⠊
⠀⠀⣤⣿⣿⣿⣷⣶⣶⣿⣿⣦⠀⠀⠉⣿⣻⣶⣄⠀⠛⠋⣀⠔⠉⡠⠀⠀⣤⣶⣦⣤⣤⣴⣢⣄⣀⡀
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡏⣼⠀⠀⠀⠈⠻⢿⣿⡷⣧⣸⡁⠔⠈⠀⠀⠐⠻⢿⣷⣎⡛⠿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢹⢻⡏⠁⠈⠻⣿⡿⣄⠀⢀⣀⡤⢀⡽⠻⣿⣟⣿⣶⣄⠀⠀⠀⠀⠀⠈⣿⣿⠿⣿⣾⣿⣿⣷
⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⢸⠁⢸⠀⠈⢉⠴⡃⠀⠀⠈⠙⠿⡿⠛⠷⣦⡀⠀⠀⠀⠈⠀⠀⠀⠹⡿⣿⣿
⠀⠀⠀⠀⠜⠐⠃⠀⠀⠀⠆⠀⢰⣡⠞⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢄⠈⢻⣦⠀⠀⠀⠀⠀⣀⣀⣀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢡⡀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠉⢻⣿⣧⠀⠀⠀⠀⠙⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⢷⣿⣦⡀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⢻⡷⣀⠀⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠈⢿⣿⣿⣷⣾⣿⣿⣷⡀⠀⢀⣴⡶⠖⠂⣸⣿⣠⣾⣷⣦⡀
⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡷⠁⠀⠀⠀⠉⠉⠻⣿⣿⣿⣮⡢⡀
⢀⣠⣶⠿⣆⡶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣮⡢
⠙⠃⢘⣻⣿⣿⣛⡇⠀⠀⠀⠀⠀⠀⠀⠒⠻⠛⢉⢋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣦
⠀⠀⠀⠀⠀⢰⠚⠃⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿⡟⠿⣿⠃⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⣇⣀⡳⣢⡯⠴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠰⣿⣭⣱⠼⠃⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠁⠀⠟⢋⡤⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⡀⠀⠀⠀⠀⠐⠂⠀⠂
⠀⠀⠀⠀⢠⣤⣞⣸⣃⡟⠁⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠁
⠀⠀⠀⠀⠘⢍⠙⡻⠟⠀⠀⠀⠀⠈⠖⠀⠀⠀⠀⠀⠀⠀⠉⣉⣡⣿⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⢀⠞⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⡀⠀⢐⣲⡀⠁
⠀⠀⠀⣤⣀⠀⢻⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠄⠔⢀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠚⢛⡟⠂⠈
⠀⠀⠀⠘⠋⠻⢿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣶⣿⣷⡿⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠁
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣆⣻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠉⠛⠉⢙⣿⣧⣶⣷⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⣶⣿⠆
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⡏⠀⠉⠑⠀⠀⡐⢀⠀⠀⠀⠀⠀⠀⢸⠃⠛⢻⠁
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⢠⣦⠆
⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠈⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠁
⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⠆⢠⠲⠷⠄⠀⠀⠁
⠀⠀⠀⠀⢠⣲⡿⠋⠀⠀⢠⡽⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠹⠀⠀⠀⠀⠀⠛⠶⡌⠒⠆
⠀⠀⠀⠦⣈⣋⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠲⠄⠀⠤⣄⣄
⠀⠀⠀⠀⠨⣿⣄⣀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⡏⠀⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⢸⣿⣿⠿⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠶⠖⠠⠄
⠀⠀⠀⠀⠀⠘⠻⣿⣆⡴⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣇⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠡
⠀⠀⠀⠀⠀⠀⠘⢪⣏⣁⡄⠉⠒⠐⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⡦⠈⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⣫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣤⣴⠟⠁⠀⢦⣼⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⢠⣤⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⣷⠀⠀⠻⣦⡿⣾⠛⢛⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠾⠋⠀⠀⣿⠀⠀⠀⠀⠀⣽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⣀⣤⠶⠞⠛⠳⢶⣄⠀⠀⠈⠳⣤⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⡴⢟⣩⣤⡴⠆⠀⠀⠀⠙⣧⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢠⡾⢋⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣴⠋⠀⠈⡁⠀⣠⠄⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡾⠁⠀⢠⡾⠁⣸⠏⠀⢠⡏⠀⣾⠁⢀⡿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡿⢁⡤⢠⡟⠀⠀⠉⠀⢀⣾⠃⡾⠃⢠⡾⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣼⠁⣾⠁⠈⠀⠀⠀⠀⠠⠾⠁⠀⢀⣴⠟⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢱⡏⠀⠿⠀⣠⣴⠞⠃⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⡇⠀⠀⠀⠉⠀⠀⢀⣠⡴⠞⢋⡶⢶⣄⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠘⠛⠲⠶⠶⢶⣚⣛⣉⣡⣤⣤⠆⢿⡘⣯⠁
⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⣠⡶⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠘⢷⡸⣧
⠀⠀⢀⣠⡴⠞⠋⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠙⠛⠛⠛⠦
⠀⠀⠘⠻⠶⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠉⠉⠉⠉⠉⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⣿⣷⣤⡴⢦⣤⣄⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢠⣿⣿⡟⣉⠤⠒⣒⡛⠻⠻⠿⣷⠀⠈⢻⣿⡄⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⠂⠙⠋⠉⠀⢀⡸⣽⠇⣀⣰⣆⠘⢤⠀⢸⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠇⠀⠀⠀⢰⠀⣿⣿⣿⣿⣷⡁⣤⣴⣻⣇⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡾⢃⢆⡀⠀⢻⡀⠻⣿⣿⡿⣫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡿⢲⠈⠈⠘⡀⡙⠢⠌⠛⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠷⠩⡋⡉⢂⠐⠡⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢐⡡⠐⠀⠤⠄⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⣶⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣄⣀⣈⡉⢀⣀⣀⣤⣭⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠚⠉⠁⠒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢠⣚⡦⠄⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠉⠠⠖⠊⠉⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠐⠒⠋⠉⠉⠙⢲⡄⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡀⠀⠀⠠⢀⠀⠀⢀⠎⡔⢀⠄⢀⠂⢠⡞⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⡄⠊⡠⠀⠀⠀⠀⠀⣠⠂⠜⢠⠂⡰⠃⣰⠏⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⠔⢉⠔⠀⠀⠀⠀⠀⠀⠀⠠⠎⡰⢁⠜⢁⡾⠁⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⠔⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡠⢋⡴⠋⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡠⠋⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⣠⠞⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⢀⣠⡴⡟⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣯⠀⣴⠛⠉⠁⢰⠁⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣿⠀⠀⠀⠈⢧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⡀⠀⠀⠀⠈⣇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣇⠀⠀⠀⠀⠸⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠹⡄⠀⠀⠀⠀⢣⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡼⠇⠹⣤⣀⣀⣀⣠⣦⣹⣦⡤⠄⣠⣤⡤⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠶⠶⠶⠶⠖⠒⠒⠒⠒⠂⠈⠉⠐⠂⠉⠛⠀⠀⠒⠊⠘
⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠾⠟⠛⠛⠛⠛⠻⠷⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣠⣄⣀⣀⡀⠀⠀⠀⠀⠈⢻⣆
⠀⢰⡟⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠿⠛⠛⠛⢛⣛⣛⣛⣻⣿⣿⣯⡉⠙⠻⢿⣿⣷⣄⠀⠀⠀⠀⢻⡆
⢀⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⢁⣠⡤⠖⠚⠉⣩⡉⠉⠻⡹⡿⡟⣿⣧⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⠈⣿
⣸⡇⠀⠀⠀⠀⠀⠀⣠⠉⠛⣿⠿⠛⠉⠁⠀⠀⢲⣟⣿⣯⠀⠀⠀⣤⠔⠢⣿⡀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⢸
⣿⠁⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⠎⢰⣷⣤⣿⣟⣭⣭⣶⡦⠽⠶⠂⠀⣙⣦⠀⠘⢿⡟⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⢠⡏⠀⣾⣿⣿⣿⣿⣿⣿⣷⡅⣩⣤⣤⣝⣛⡟⡇
⣿⡀⠀⠀⠀⢠⡿⡤⢆⡀⠀⠀⠀⢸⡇⠀⢹⣿⣿⣿⣿⣿⣿⢟⣡⠌⠉⠛⠋⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢹⡇⠀⠀⠀⢸⡇⢦⠜⣥⢆⣠⠀⠈⢷⣄⠀⠻⣿⣿⣿⠟⣵⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⣿⡀⠀⠀⣿⠷⠺⡤⠢⠼⣁⢝⢁⡀⠙⠷⢤⣈⠛⠏⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠸⣧⠀⠀⢻⠘⠖⢦⠔⠦⢁⣨⢉⣙⡃⡀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠹⣧⡀⢸⡏⠳⢂⣀⠦⠎⢅⡌⣁⡉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠙⢷⡄⠃⠶⠊⣅⠰⣤⠀⣤⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠳⠶⢶⣶⣦⣤⣤⣠⡾⠋
⠀⠀⠀⠀⠈⠻⣦⡀⠓⠁⣄⠴⠔⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤⣀⣀⠀⠀⠀⠀⠀⠒⢦⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠀⠀⡀⠈⠙⠋⠁⠀⠀⠠⣄⣀⣙⣛⣶⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⢾⣷⣶⣤⣬⣷⣶⡶⠾⠟⠛⠉⠁

⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠛⠋⠉⣻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⣾⣵⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡷⠚⠛⠓⢦⣄
⠀⠀⠀⢠⣞⣿⣿⣿⣿⣿⡿⠟⠛⠦⣄⡀⠀⠀⠀⠀⠀⣠⣏⠀⣴⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠈⡿⠿⠯⠖⠋⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠙⢮⡀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⣸⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢿⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢸⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢧⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠓⠦⣤⣀⣀⡀⠀⠀⢳⡀⠀⠈⠳⣄⠀⠀⠈⠳⢤⣄⣀⣀⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⣤⡤⠴⠖⢦⣀⠀⠙⣆⠀⠀⠈⠛⠶⢤⣤⣄⣠⣤⠤⠶⢤⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣇⣠⣿⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⢀⣀⣼⣇⣠⣿⠀⠀⠀⠀⠉⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣇⣴⣦⡞⢹⣷⣶⠛⢷⣴⣷⣶⠞⣿⠛⠻⢿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣥⠾⠛⣿⣧⠾⠙⢿⣤⡼⠏⢻⡿⠀⢿⡶⠻⣿⡤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠄⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢴⡿⠟⠀⠀⠀⠀⠒⢄⢸⣇⣾⡯⠴⠒⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⣒⡲⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠙⢟⠙⢦⡂⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⠩⠄⢢⠠⠈⠉⢷⣆⣄⠀⠀⠁⣾⣿⣿⣶⠀⠀⠀⠸⣷⡄⠈⠑⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠀⠀⠀⡈⣀⣤⣴⣾⣾⠉⣿⡄⠀⠀⠛⠻⠿⠃⠀⠀⣸⣷⡘⢿⣶⠞⠁⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠁⠀⠀⢀⣼⡿⠋⣀⠄⠠⠈⠐⣹⣷⣀⡢⠀⠀⡀⠀⢀⣤⣯⣿⣷⠀⢉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⣿⣿⣠⠎⠀⠀⠀⣰⣷⡿⠋⠉⣵⣿⡏⡀⢰⠟⠁⠀⠈⣿⣿⣧⣀⣄⠀⠀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⡁⠀⠀⢠⣾⢻⣿⣟⠂⣸⣿⣿⣿⣧⠂⠀⠀⠀⣠⣿⠛⢿⣿⣧⣤⣴⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣶⣾⣾⣯⣿⡿⠾⠿⣿⣿⣯⡹⣿⣽⣀⣠⣴⣿⠃⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡓⠀⠀⢨⡦⠀⠀⠀⠀⣷⠀⠀⡀⣠⣬⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠤⣤⡀⠁⠀⠻⠿⢅⠀⣠⣄⠀⠀⢑⡃⠀⠼⢺⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣴⣿⣥⣿⣷⠟⠙⠃⠖⠀⠱⠀⢸⠉⠀⡄⠀⢈⣥⣄⣄⣰⢿⢷
⠀⠀⢀⡠⠔⠀⢄⡀⡀⠀⢠⣳⣶⣾⣿⣿⣿⣿⡿⢿⣿⣄⣤⠼⡧⠶⠓⠷⢾⣥⡚⠊⠆⠙⠿⠛⠛⠉⠈⠀⠀⡀
⠀⠀⠈⠙⠋⠀⠀⢻⣄⣐⠚⠋⠉⠁⠉⠀⠘⠀⠔⢉⠖⠉⠛⠂⠈⠚⠀⠀⣀⣹⠣⠄⠀⠀⠀⠀⠀⡀⠀⠀⠚⠂
⠀⠀⠀⠤⠴⠒⠒⠨⠾⠛⠿⠆⠀⠤⠀⢠⠴⠖⢀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢠⣾⣿⣿⣧⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠻⠁⠿⠻⠿⡿⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠷⠦⠄⣄⣀⣘⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣭⣤⣤⣤⣤⣤⣤⣬⣭⣷⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠁⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⢛⣛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣼⣿⡄⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⠀⠀⠈⠋⠁⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣄⡤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⣿⣿⣿⣿⣿⣧⡀⠀⠀⢦⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠤⢀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⢜⣏⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠸⡿⣿⣻⣿⣯⣩⣭⣥⣤⣀⣀⣸⣧⣀⣾⣿⣿⣷⠀⠀⠀⣠⣤⣤⣌⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣆⠀⠀⢉⣷⣾⣿⣆⡉⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⣿⣿⣿⣿⣿⣶⣤⣤⣯⣍⣠⡀⢹⣿⣿⣿
⠀⠀⠀⠀⠈⡿⢿⡿⣿⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢡⠘⠙⠈⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⣠⣀⣜⣴⣾⣠⠀⠀⠀⠺⡁⢹⠃⠙⢷⠍⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⣠⣦⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⣀⣉⣯⡀⣠⣞⣰⣦⣤⢀⡄⢠⣠⣦⣠⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⡠⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣶⣶⣼⣿⣼⣷⣴⣦⣴⣤⣴⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠲⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢾⣍⠀⠀⠲⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡿⠁⣀⣤⡀⠈⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣸⡇⠀⢿⣤⡿⠸⣇⣈⣹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠶⠋⢸⠛⣇⠀⠀⠀⢀⡾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢛⣋⣥⡤⠶⠶⢦⣤⣀⣿⠀⠘⢧⣄⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣤⠞⠋⠁⣀⣀⣀⣀⣀⡀⠈⠁⢀⠀⠀⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⡴⠋⢀⣤⠶⠛⠉⠉⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⠟⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⠏⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢱⡏⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣾⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠁⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣿⠛⠶⠶⠦⠤⠤⠶⠶⠚⠋⠁⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⣀⣴⠞⠉⠀⠀⣀⣴⠿⢦⣄⣀⠀⠀⠀⢀⡞⢹⡇⠀⢀⣠⡴⠞⠉
⠀⠀⢀⣤⠴⠛⠉⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⠉⠙⠛⠒⢺⡇⣿⠛⠛⠉⠁
⠀⠀⠘⠓⠶⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠛⠛⠛⠦

⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠔⠁⠀⢀⠀⠉⢢⡀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⠔⣅⠀⠀⠀⡨⠢⡀⠀⠀⢀⣤⣻⣿⣿⣿⢟⣤⡀
⠀⢸⠀⠀⠀⠀⠀⠠⠄⡃⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡌⠀⢸⡄⠀⢠⡇⠀⠁⠀⠀⢸⣿⣧⣿⣿⣿⣼⣿⡇
⠀⠸⡀⢀⠀⠂⠀⠀⣠⠃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠰⡀⡸⠁⠀⠀⢇⢀⠆⠀⠀⠸⣿⢏⣿⣿⣿⡹⣿⠇
⠀⠀⠈⠲⠤⠠⠤⠞⠁⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⠣⠄⠀⠠⠜⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠗⠁

⠀⠀⢀⠄⠒⠀⠒⠠⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⡀⠀⠀⠀⠀⠠⢔⣂⣀⠐⠢⢄⠀⠀⠀⠀⣠⡴⠶⠿⣶⣦⣄
⠀⢰⠁⠀⠀⠀⠀⠀⠽⠆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣻⡄⠀⠀⣐⡝⠁⠀⠀⠙⡄⠈⠂⠀⠀⣰⢫⣾⣿⣿⣮⢻⣿⡆
⠀⢀⠀⠀⠀⠀⠀⠀⡅⡂⠀⠀⢸⣿⣿⣿⣿⣿⣿⢾⡇⠀⠀⠌⡇⠀⠀⠀⠀⠸⣄⠘⠀⠀⢿⢸⣿⣿⣿⣿⣧⢻⡿
⠀⠈⠢⡀⠀⠀⠀⣀⠜⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⢜⣦⡀⠀⠀⠀⡨⠁⠀⠀⠈⢷⡻⣿⣿⣿⣿⡷
⠀⠀⠀⠈⠁⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠂⠂⠁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠋⠁
⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⢀⢔⣿⣷⡆⠀⠉⠦⡀⠀⠀⢀⣴⠋⠉⢽⣿⣷⣤⡀⠀⠀⢀⠎⠁⠀⠐⠆⣌⠱⡀⠀⠀⢀⣾⣿⣿⣯⣹⠻⣷⡀
⠀⠈⠀⠙⠉⠀⠠⣴⣆⠁⠀⠀⢸⣿⣶⣾⣿⣿⠋⠹⡇⠀⠀⡂⠀⠀⢀⣄⠀⠀⠀⢠⠀⠀⣾⣿⣿⣿⠻⣿⣿⣿⣧
⠀⠸⣤⣶⣄⠀⠘⠻⢋⠇⠀⠀⠸⡟⠙⠿⣿⣯⣄⣴⠇⠀⠀⠡⡀⠀⠀⠉⠀⠀⢀⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠈⠛⠏⠤⠤⠖⠁⠀⠀⠀⠀⠈⠲⠾⠿⠛⠋⠁⠀⠀⠀⠀⠑⠢⠀⣀⠠⠔⠂⠀⠀⠀⠀⠘⠻⢿⣿⡿⠟⠋

⠀⠀⠀⠀⣀⠄⢤⡄⠤⠀⠀⠀⠀⠀⠀⣀⣤⡀⢤⣤⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⢠⠊⠀⢀⣤⠙⢶⠀⠀⠀⠀⢠⣾⣿⡿⢛⣷⡙⠀⠀⠀⠀⠀⡠⠊⠀⢠⠂⢘⠀⠀⠀⠀⠀⣠⣾⣿⣟⣿⣿
⠀⠀⣇⠀⣴⠟⠁⠀⡔⠀⠀⠀⠀⠻⣿⠋⣡⣾⣿⡟⠀⠀⠀⠀⠎⠀⣀⠔⠁⡠⠊⠀⠀⠀⠀⣼⣿⢿⣿⣿⡿⠃
⠀⠀⡉⢳⣄⣀⠤⠊⠀⠀⠀⠀⠈⣷⣝⠻⣿⠿⠋⠀⠀⠀⠀⠀⠳⣈⢀⡠⠊⠁⠀⠀⠀⠀⠀⠹⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣄⠀⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣾⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⢿⣿⠿⠿⢿⠋⠉⠉⠙⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡏⠀⠀⠈⡷
⠀⠀⢀⣠⣄⣠⣤⣶⣶⣾⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⢰⣛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠀⠀⠀⠘⠁
⡴⠐⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠙⢿⣿⣷⣶⣦⣤⣤⣄⡀
⠁⠀⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠉⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡏⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣖⣋⣭⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠫⢭⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠿⢿⡿⠿⠛⠋⢁⣀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⣋⡥⠴⣒⣒⣶⣿⡷⠦⠤⠴⠋⣇⣠⠴⢀⠾⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡾⢷⣾⠟⢛⠛⠛⠁⠀⡀⠀⠀⠀⠈⢁⠒⠛⠙⠎⠳⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣾⣿⣿⡿⠇⠀⠀⠀⠉⠉⠁⢀⣀⠈⠑⠟⠺⢆⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⡿⢋⣁⣠⠀⣀⣀⡀⠀⠀⣗⢦⣠⡘⣏⡙⠲⢄⠤⢬⣳⣴⠜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠘⢛⡋⠉⠋⠀⠠⡟⣿⠇⠀⠀⠘⠂⠀⠙⠮⢽⣦⡀⠙⠋⠻⣦⡀⢼⠷⡄
⠀⠀⠀⠀⠀⠀⣠⠞⣀⠀⣴⡟⣠⡄⡆⢦⡇⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠦⢄⡀⠉⠳⣄⣹⡄
⠀⠀⠀⠀⠀⣰⠃⣰⠟⠀⣼⠏⢸⠀⣇⠘⠇⡆⠸⣆⡀⠀⠀⠀⢀⡤⠀⢀⡀⠀⠀⠀⠀⠉⠳⢤⡹⠉⠹⡄
⠀⠀⠀⠀⣰⠇⠈⣡⡇⢀⠀⠀⠀⠀⠻⠿⠀⢿⡦⣟⠹⡄⠀⡰⠋⡴⡷⡿⠁⠀⠀⠀⢸⣧⠀⠀⠡⣄⠀⢹⣄
⠀⠀⠀⢠⡟⣠⡴⠉⣇⣼⠀⠀⡀⠀⠀⢠⡀⠈⠃⠸⡄⢠⠞⠀⡜⠀⠀⠀⠀⠀⠀⠀⠘⡟⠶⡆⠀⠈⠁⠸⣼⣦⡀
⠀⠀⠀⣼⡿⠉⠀⠀⠛⠸⡆⢠⡇⠀⠀⠘⠇⡀⠀⠀⣰⠃⢠⢸⠁⡇⠀⠀⠀⠀⠀⠀⠀⠉⢧⠉⢧⠀⠀⣄⠀⢧⣵⣄
⠀⠀⠀⡏⠀⡀⢰⠀⠀⠀⣇⢸⡇⠀⠀⣄⢀⡇⠀⢰⠃⠀⢸⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣰⠀⠸⣤⠈⠛⣎⢦
⠀⠀⠀⡇⠀⢷⣸⠀⠀⠀⠹⠋⠀⠀⠀⢸⡾⠀⢀⡏⠀⣴⡎⠁⠀⢷⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠉⠀⠀⠙⣷⣆⠘⠺
⠀⠀⠀⡇⠀⠈⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⣿⡇⠀⠀⠸⡄⢀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣀⣴⣶⣦⢽⣦⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠙⣾⣿⣿⣝⢧⣹⣿⣷⣜⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣬⣻⠿⢻⣮⣻⣿⣿⣿⡌⢿⢾⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣬⣙⣷⣾⣿⣷⣝⣧⡙⢻⢀⣼⠙⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⣽⣿⣿⣢⣭⣶⣬⣽⣾⣟⣼⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⢋⣩⣀⣀⡈⠙⣧⣸⡷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠙⣿⣿⣿⣎⣿⣿⠀⠘⣿⢲⡿⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡖⠀⠙⣿⣿⣿⠟⠁⠀⣠⣿⣿⣷⣿⣶⣍⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣍⠁⢰⠟⠃⢀⣴⣿⣿⡿⠿⠛⠋⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠊⠉⠉⢻⣿⣾⡇⣀⠐⠝⢋⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣾⡶⠿⡻⢿⣿⣿⣿⣾⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⡿⣿⣿⣿⣦⡀⡤⠙⠛⠛⡿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⣼⣻⣦⠿⣏⣷⣾⣿⡿⢿⣿⣿⣶⡜⠀⡿⢠⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢘⣻⣿⣿⣿⣿⣿⠟⣫⣶⣿⢿⣿⣿⣿⣄⠀⠄⢠⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠃⢳⡜⣿⣿⣿⣿⣿⣿⣿⣋⣵⣿⣿⣿⣿⣿⣦⣄⢈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⢻⡿⢮⡻⢮⡛⢿⣿⠿⢿⣯⣤⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⢈⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠁⠀⠈⠉⠉⠛⠲⢦⠀⠀⢀⡴⠋⠀⢙⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣠⡴⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣀⡴⠞⠋⠉⠉⠉⠉⠉⠛⠂⣠⡾⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣶⠀⠀⣠⠞⠋⠀⣀⡴⠋⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢠⡟⠀⠀⠀⢀⣤⠖⠛⠋⢠⣾⡁⠀⣠⡾⠋⢠⡀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣾⠀⠀⠀⢠⡟⠁⠀⠀⢠⡟⣀⣹⠾⠋⠀⠀⠈⣧⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣏⠀⠀⠶⢾⡇⠀⠀⠰⣿⠛⠉⠀⢰⡶⠶⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢿⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢀⡟⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠘⣧⠀⠀⠀⠈⠛⠶⣤⣤⡤⠶⠋⠀⠀⠀⢀⡾⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⣀⣤⠶⠛⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⣀⣀⣀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣶⡆⠈⠟⠁⠉⢠⣶⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⣴⡶⠀⠀⢀⣴⣿⣿⣿⣿⢿⣿⣿⣿⣾⣿⣀⠙⠛⢻⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡔⠿⠟⣀⡨⠀⠰⠀⢙⣿⣿⣿⠛⠃⠈⠁⠀⠉⢿⣿⣿⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣧⣼⣿⠟⣳⣀⠠⠀⣸⣿⣿⣿⡀⠀⠀⠀⠀⢀⡈⠁⠉⢻⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⢤⡈⢹⣿⣿⣿⣧⠄⠀⠀⠀⣈⣡⣶⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣾⣿⠀⣭⠛⢻⣷⡶⠓⣠⣴⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢿⣿⣋⣤⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⣿⣿⠿⣿⣽⠿⢛⣭⠿⠛⠛⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⠁⠸⠏⠋⠀⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠋⠘⣉⠁⠀⠈⠀⣠⣴⠀⠀⣠⡴⠿⠿⠿⠟⠛⢿⡟⠛⠋⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣾⣿⣿⣦⣴⡅⡀⠀⣵⣾⣿⣿⣷⠟⠋⡰⠀⢶⠒⢄⠉⣁⣴⡔⠂⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣳⡶⠋⠀⠀⡀⢇⢘⣀⢻⣷⡼⣶⣿⣇⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⢦⣀⣴⣿⣿⣿⠈⢻⢿⣿⣿⣿⣿⣿⠟⣄⠉⢠⣥⣜⠛⠿⠏⣨⣿⣶⣿⡿⠛⠉⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡿⠟⠀⠀⠸⡄⠈⠉⠉⠉⠀⢠⣴⣶⣿⣿⣿⣿⣶⣖⠛⢛⠿⠏⠀⢀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠋⣀⣀⡀⠐⠲⢿⣶⠀⠀⠀⠀⠀⡝⠻⣿⣿⣿⣿⣿⡿⢰⣿⣷⠆⠐⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⣿⣷⣾⣿⣿⣿⣤⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠺⢿⡏⢠⣼⣅⣤⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡛⢻⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣷⣤⣄⣾⣿⣿⣿⣷⣾⣾⠇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⡿⠋⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⠠⠄⠛⠃⠀⢠⣿⣿⣿⣿⣿⣦⠀⠀⢀⠀⣀⣠⣴⠀⠀⡠⠀⠀⠀⣀⣄⣁⣄⠀⠀⠬⠛⢿⠛⠟⠓⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⣠⣬⡙⠿⠻⢿⠟⠿⠓⠀⠛⠛⠛⣿⣷⣶⡟⠁⠀⠈⢿⠿⢿⣏⣉⡘⠃⠀⣀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣾⣿⣿⣶⠋⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣷⣦⠀⡀⢠⡔⠀⠀⠈⠈⠀⡸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⣿⣿⡀⠈⢠⡰⠤⠄⢀⣤⣞⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠺⠿⣿⣷⣶⣀⡤⠀⣀⣀⣾⣿⣿⡿⠛⠁⣀⣄⠱⣦⣄⣴⣿⠿⣛⣉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢰⣶⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⣾⣾⣿⣿⢟⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠻⢿⣿⣇⠛⠙⠻⠇⠻⡷⠿⠿⠻⣿⣧⡾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠁⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⣀⣤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠠⠂⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⢤⣝⢀⠀⠈⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠺⣭⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠤⠀⢰⡏⠀⠀⠀⢀⣀⣰⣁⣿⡇⠀⠁⠃
⠀⠀⠁⠀⠐⠁⠀⢠⡟⠋⣿⠹⢿⠃
⠀⠀⠀⣶⣶⣶⠀⠘⠀⠀⠈⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⡀⠈⣹⠟⠈⠀⣀⠰⠾⢂⣀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢾⠟⢲⠅⠀⠀⣴⡏⢀⠑⠚⠏⠀⢀⣴⠂
⠀⢙⣿⣾⣟⠀⣩⣥⣥⣙⢦⠀⣰⣾⠇⠀⡀⠀⠀⠀⠀⠀⢀
⠀⡿⣿⣯⡍⠀⠿⠾⣿⡹⢇⡱⣿⡏⢀⡨⠁⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⣷⣿⣿⣿⠿⡟⠀⠁⣿⣰⣇⣼⣾⠟⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⢀⠀⠀⠠⢛
⣼⣿⣿⣿⣿⣿⣤⣤⣀⠈⠻⣿⣿⢏⣠⣤⣴⡿⠋⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣿⣯⡷⢾⣧⣿⣤⣤
⣹⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣾⣿⣿⣭⣿⠁⠀⢠⣤⠸⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣾⣿⣿⣿⣿⣿⡟⢿⣷
⢻⣿⣿⣿⣿⣿⠋⣀⣠⣼⣿⣽⣿⣿⣿⣿⠀⡀⠘⠃⠂⠀⠀⠘⠛⢾⣿⣆⠀⠀⠀⡐⢶⣼⣷⣶⣿⣿⣿⣽⣿⣿⡇⡿⣿
⢺⣿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣿⣷⣾⠁⢠⣶⣁⠀⠀⢀⡀⢸⣿⡏⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣟⣟⢃
⢸⣿⣿⣿⣿⣿⣿⢟⣭⣯⣽⣿⣿⣿⣿⢟⠁⢀⠀⣸⣤⣤⠀⠻⠟⢹⣿⣧⣄⣠⣈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷
⢸⣿⢿⣻⣿⣛⢿⣷⣿⣿⣿⣿⣍⢿⠃⣀⣼⣿⣷⣿⣿⣻⡖⠂⠀⠴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⠋⣌
⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣾⣫⣿⢿⣿⣿⣿⢿⡇⠀⠀⠀⠈⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣼⣛⣿⣿⣿⣿⣷⡀⠀⠀⠠⠁⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧⣿⣽⣿⣿⣿⣿⡿⣿⣿⢟⢿⣀⣂⠸⣤⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⣿⣿⣿⣯⢿⣷⣿⢿⣿⣡⣶⣮⠙⣿⣷⣿⡿⣿⣿⣿⡿⣿⣿⡏⠙⠿⠟⢿⣿⣿⣿⣿⣿
⠘⢟⢉⢉⣩⣭⣿⣿⣿⣿⣿⡟⠺⠟⣷⣿⣶⣻⣿⣿⡧⠾⠧⢺⡇⢻⣿⣿⣷⣿⣿⣿⣿⣿⡿⣿⣿⣾⣿⣿⣿⡿⢻⠿⡿
⢠⣵⣿⣿⣿⡟⠙⢻⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣦⣧⣀⣨⣿⡰⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠟⣭⣄⣛⣿⣤⣤⣀
⢸⣿⣿⣿⣿⣧⣼⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠁⡻⣿⣀⡉⠙⠛⠉⣡⠀⠈⣩⣤⡉⠀⠀⠉⠁⣯⣽⣿⣿
⠀⣻⡿⠜⢃⠙⢿⣿⠿⢻⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠐⣻⣿⡟⢻⣤⣴⣶⠏⠶⠆⡀⠹⠛⢀⣨⣤⣤⣤⣍⣽⣿
⠀⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀⣿⣿⣷⣾⣿⣿⣿⢿⡿⠿⠏⣶⠶⠀⠀⠰⠛⠿⣦⣶⣾⡷⣾⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⡿⡻⠟⣋⠛⠛⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢻⣶⣤⡀⠀⠀⠀⣤⢾⠐⢹⣧⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⣾⠟⣷⣴⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠻⢿⣇⠀⢶⣄⠀⡀⢘⣽⣿⣿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣧⣤⠶⠿⠻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⡸⠂⠼⣇⣰⣿⣿⣿⣤⣾⣿⣿⣯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⠀⣻⡏⡻⢿⣿⣷⣿⣽⠸⠟⢿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣅⣀⣀⣈⣁⣀⣀⣀⣀⣀⣈⣉⣀⣉⣈⣈⣡⣄⣤⣀⣈
⣿⣿⡟⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⠾⠿⠿⠿⠿⠿⠿⢶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣶⣶⣶⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠋⣉⣉⣉⣙⣛⣿⣿⣿⣦⠉⠉⠻⢿⣿⣿⣶⡄⠀⠀⠀⠀⢻⣧
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡟⠁⣀⣠⠴⠖⠛⠋⣉⡉⠙⢿⡙⣟⢿⢻⣿⡆⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⠀⢿⡆
⢸⡿⠀⠀⠀⠀⠀⠀⠀⢀⠜⠽⢿⣿⡿⠷⠛⠉⠀⠀⠠⣶⠿⣿⡗⠀⠀⠁⢀⣈⠤⢸⡇⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿
⣾⡇⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠈⠁⠀⠀⠀⡰⠀⣤⡀⢹⣾⠿⠛⠠⢤⣀⣻⣧⣄⠈⠙⢦⡀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⡾⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⠟⠡⠉⠀⠠⢴⣶⣿⡄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡷⠸⣿⣿⣷⣶⠶⢥⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⡇⠀⠀⠀⠀⣸⡟⠖⢡⡤⠀⠀⠀⠀⢻⡆⠀⠘⣿⣿⣿⣿⣿⣿⡿⣫⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣷⠀⠀⠀⠀⣿⠁⠳⠞⢧⠾⣤⢤⠀⡈⢿⣆⡀⠘⢿⣿⣿⣿⢋⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⢿⡆⠀⠀⠀⣿⠳⠻⡤⠺⠴⢧⡠⢋⢀⡀⠉⠛⠦⣄⡉⠻⠁⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠘⣿⡄⠀⠀⣿⠸⡖⢧⡠⢦⠜⡀⡈⡁⣹⠃⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠘⣷⡄⠀⢻⠛⠧⢎⠀⣰⣤⢉⠁⣙⠁⡙⠃⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠘⢿⣄⠘⡆⣄⡜⠛⠀⣀⢀⢙⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⠿⣶⣤⣤⣤⣤⣀⣀⡀⠀⢀⣼⠟
⠀⠀⠀⠀⠈⠻⣧⡁⠀⣤⠟⠃⠈⡅⡈⠋⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠿⣦⣀⠀⠙⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⡀⠲⠶⠤⠀⠀⢤⣀⣀⡀⠈⣹⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣤⣀⡀⠀⢀⡀⠀⠈⠁⡀⠀⠀⠐⠦⣤⣤⣨⣭⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣿⣶⣶⣾⣿⣿⠷⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣠⣀⠀⠀⣠⠖⠑⢢⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣼⡿⠟⠓⡾⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⣶⣾⣿⢋⡄⠈⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡼⣿⣿⣿⡷⣿⡅⠒⣹⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣊⣜⣡⣿⣿⠇⣿⣶⣶⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣼⡋⠛⠏⣿⣿⣧⡙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠊⣩⣿⣽⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡸⠃⠀⠀⠀⢁⣴⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡌⠠⡄⣠⠄⠀⠀⢀⡔⠀⠈⠁⢠⣴⣶⣿⣿⣿⣿⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠟⠛⠛⠛⠒⠀⠆⢀⣀⡀⣀⠀⠀⠀⠈⠛⠟⡟⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢉⣉⣉⣉⡀⠀⡀⢄⣀⣀⠀⠁⠀⢀⠀⣠⢠⣄⠿⣣⢈⣙⣛⣻⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠟⠛⠉⠉⢉⣉⣅⣈⡁⠀⡀⡀⢀⠀⠄⠙⠦⢌⣽⡉⠒⠠⣀⡨⣿⡥⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠁⢀⡀⠉⠋⠁⠀⠉⠀⠀⡄⠀⠒⠀⠦⢀⠈⠉⣉⠀⠿⢩⣀⠙⠻⠶⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣿⣿⣶⢟⡃⠀⠀⠀⠀⠘⠋⠀⢄⠀⠀⠀⢄⣉⡛⢷⣤⡶⠶⠻⢿⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⣾⠗⠛⠿⢿⠇⠀⠀⢀⠠⡦⢄⠀⠀⣤⡀⠃⣴⠢⣬⣷⡦⣬⡥⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡈⢶⣾⣻⣿⣿⣷⣾⣿⣿⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠹⠿⠛⠛⠋⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠤⠤⢞⡓⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠘⠣⣿⠦⠠⣄⠀⠈⠙⣬⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠆⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡭⡉⠉⠉⣩⣍⡩⠋
⠀⢀⣤⣤⣤⣀⠀⠀⢀⣤⣬⣬⢉⠀⠀⠀⠠⠊
⠀⢿⣿⠿⠛⠛⠳⣶⣿⡿⡿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⢀⡠⣾⠜⣿⣿⣿⣿⣿⣿⣿⡳⠬⠂
⠀⠀⠀⡀⡀⠴⢦⡾⣿⣿⣿⣿⣿⣆⠻⠃⣤⡿⣏⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⡊⠀⠁⠀⡀⢈⠀⠉⠉⠻⣿⣿⠀⠳⢛⠿⢿⠀⣿⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠻⠷⣄⠀⣠⣄⣀⡀
⣧⠀⠀⠀⡀⠨⡧⠂⠀⠀⠀⠀⠘⠋⢈⡓⡋⠀⠈⢉⣘⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣠⠀⠺⠛⣒⢦⡀
⡇⠀⢠⠈⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠯⠘⠂⠀⠀⠆⠰⠆⠀⡔⠄⠀⢈⠁⠀⠄⠀⠀⠀⠀⠀⣠⡛⠿⣿⡧⠠⣤⡘⢿⣷⡀
⡇⠂⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⡛⣶⣄⠀⡘⠆⠀⠀⢀⡊⠁⠀⢀⣠⣦⠀⠀⠈⠉⠑⠦⣄⠀⠈⠛⢻⣆⠻⣷
⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣞⣻⣷⣿⣿⣀⠀⠣⠀⠀⠀⣿⣷⣆⠀⠐⠁⠈⠢⡀⠀⠀⠀⠈⠀⠀⢀⣘⢁⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣞⣿⣷⣶⠿⢷⣬⣤⣤⡀⠀⡙⠁⢙⢶⣆⣀⠀⠀⠱⣆⠀⠀⠀⡀⠀⠀⠘⠀⠊⡓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⣯⣽⣿⣿⣿⢿⣟⣳⣚⣂⠀⠀⠀⠨⠿⠿⢢⣤⠠⢻⡆⠀⣐⠕⠀⠒⠃⠑⠀⢬⢞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡉⢿⢻⣿⣿⣥⡼⣛⠛⠻⢉⣻⡝⠛⠒⠒⠃⠁⠰⠈⠣⡜⠊⡠⡄⠤⠉⠀⠀⢀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣬⣉⡁⣉⣩⣯⡝⣉⣻⢤⠅⢤⣄⡀⠈⠀⠀⠀⡄⣶⠙⠠⠀⠀⠀⠣⠈⠀⠁⠀⠘⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠪⠉⠀⢶⡶⠟⠓⠂⠀⠲⠨⢷⣄⠀⠀⠀⡄⠀⡡⠀⠱⢲⣐⡂⠀⠀⠀⠀⡀⢀⡿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠎⠀⠀⠛⣿⡿⠁⠀⠀⠈⠠⡄⠀⠓⠀⠛⢀⢐⣈⣋⡀⠀⠀⠀⠀⠁⢼⠿⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣢⠈⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠈⣆⡠⡈⠀⠀⠈⠳⣛⣃⠀⠀⠀⠤⢀⠠⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⠀⠲⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣦⣆⣀⣀⣽⣿⡄⢀⠀⠀⢠⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⣿⢃⣤⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠸⣟⣾⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣄⠈⠛⢿⠋⢀⠀⠁⠀⢀⢀⡜⠡⣼⣅⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣝⣷⡀⠄⣮⠀⠀⣤⡮⢠⠛⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠷⣾⠀⢀⠘⡏⠹⠈⠙⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣿⡿⣾⠀⠀⡆⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⡄⠀⢧⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡀⠉⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠉⠉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠚⠉⠉⠀⠛⠓⠒⠒⠲⠟⠀⣠⣶⡶⠶⠶⣦⡀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣀⠠⢄⣀⣒⠂⠠⠄⡀⠘⠀⡀⠠⠬⠙⣦⠀⠈⣷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡠⠔⣃⠤⠚⠀⠈⠉⠘⠢⢄⡀⢘⣁⠤⠒⣀⠀⢿⡀⠀⠻⣆⡀⠀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡌⠀⠈⢀⡠⠖⠚⠃⢉⠁⠂⠉⠀⠀⠀⠀⠀⠀⣱⢌⡷⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠀⣠⠞⠁⢀⡠⡤⠤⠤⣈⠓⢄⠀⠀⠀⠀⣠⢋⡤⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡻⠜⠁⢠⠖⢉⣠⣤⣀⠀⠈⠑⢦⢣⡀⠀⢰⣧⣏⣤⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⢦⢠⡏⢠⣿⣿⣿⠋⠀⠀⠀⠈⣿⣠⠀⠀⣿⣿⣿⣦⡀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⣿⡇⠘⣿⣿⣿⣤⡀⠀⠀⠀⣧⢻⡤⠒⠻⠯⣿⡿⠃⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠟⣷⡀⠈⠉⠛⠉⠀⠀⠀⣰⡣⠈⠄⡀⠀⠀⠀⠁⢦⣼⡿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠛⠛⢛⡷⣼⠿⣦⣄⣀⣀⣀⠴⠛⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⢷⢿⣷⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣘⠃⢸⠀⠀⠀⠉⡹⠋⡛⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢹⡇⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠀⡞⣠⠀⠀⠀⠁⠀⠹⣤⡀⠙⠶⠤⢤⣐⣶⣖⣊⣲⣤⠟⠀⣸⡇⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⡀⠰⡉⢆⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠛⠉⠉⠉⠉⠁⠀⢀⡴⡣⠀⡌⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠸⣇⠀⠘⢎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡔⢁⣾⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⡀⠘⣦⠀⠀⠑⣌⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠼⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⣋⣙⠷⢾⢷⣤⣤⣈⠳⡌⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣸⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⡥⠶⠛⠉⠀⠀⠀⠀⠀⠀⠉⠳⡌⢢⠈⠳⣤⡀⠀⠀⠀⠀⣰⠏⣰⠟⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⡑⢦⡈⠙⠛⠲⠚⠛⢁⡴⠋⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠙⠢⣌⠑⠒⠒⠒⠈⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠐⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⢠⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⡸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣤⡾⠋⣠⠈⠁⠘⢀⠀⠐⠉⠀⠀⠀⠐⠉⠛⠁⢀⡾⠁
⠀⠀⢀⣠⡴⠾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣈⣭⠿⠋⠀⠀⠠⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠘⢿⣤⣀⣉⢉⣫⣤⣀⣀⣀⣀⣠⣤⣤⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⢰⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠏
⠀⠀⠀⠀⠈⢹⣿⣟⠁⠀⠈⣿⣍⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⠃⠀⢀⣀⣤⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠻⣧⡀⠀⠀⠀⡄⠉⢶⡀⠀⠀⠀⠀⠀⣀⡀⠀⠋⠉⢉⣠⣤⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣼⣇⠀⠘⣧⣤⣤⣴⡶⠟⠛⠛⠛⠛⠻⣏⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡄⠀⢿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠆⠈⠻⢦⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣧⠀⠘⢧⣄⡀⠀⠀⠀⠀⣾⣉⣉⣀⣀⣀⠀⢀⣀⣀⠉⠉⠉⠁⢙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠋⠉⠉⠀⠀⠀⠀⠉⠙⠛⠳⠦⠤⠬⠭⢭⣉⠉⠻⣦⠀⠙⢿⣟⠻⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⣶⣆⠀⠰⣦⣤⣄⣀⠀⠀⠀⢀⣽⠆⠀⠻⣦⡀⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠈⠻⣯⡙⠻⠿⠟⠛⠁⠀⠀⠀⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠛⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠟⡿⠿⠿⠻⠿⠏⠙⡿⢿⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣛⣛⣛⣛⣛⣿⣟⣛⣛⣛⣛⣟⣛⣛⣛⣛⣛⣻

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠐⠂⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⣰⠋⣷⡀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣠⣤⡀⣀⣀⠀⣀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⢰⠗⠒⠻⣧⠀⣾⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⡇⠀⠁⠈⢷⣀⠏⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠠⠿⠄⠀⠰⠿⠷⠙⠿⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠿⠷⠙⠿⠶⠒⠀⢈⡟⠀⠀⢸⣿
⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡿
⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠦⠤⠤⠄⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⢤⣤⣤⡴⠞⠛⢛⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⢀⣤⡉⢳⡄⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢠⡄⠀⠀⠻⠾⠇⣸⠟⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡟⠀⠀⠀⠸⣇⠀⠀⠀⠀⢠⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠋⠉⠉⠉⠉⠉⠻⠶⣤⣤⣄⣙⠷⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⣠⡶⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠛⠛⠉⠀⢀⣤⣤⡄⠀⣶⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⢀⠀⠀⢠⣶⣛⡿⠂⠀⠰⠿⠶⠛⠀⢀⡿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣴⣟⣿⠟⠀⠈⡉⠉⠀⠀⢀⣀⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠉⠀⢠⣾⣯⡿⠁⠀⣾⣯⠿⠃⠀⢀⣼⠃⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠉⠙⠳⠶⣤⣀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⢀⣀⣉⡙⠛⠳⠶⠶⠶⠤⠤⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⣠⡴⠞⠉⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⠉⠙⠳⢦⣤⣀⡀⠀⢠⡞⣷⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⣴⣚⣉⣀⣤⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⢠⡟⠰⠶⠶⠶⠛⠛⠉
⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⠻⠶⠶⢶⡄
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠤⠀⠠⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⠔⠀⢠⣤⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢡⠏⠆⠀⠀⠁⠀⠠⣾⣛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⣠⠏⠘⡇⠀⠈⠒⣶⡾⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠿⠋⡡⠔⠁⠀⡴⠛⢦⡀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠿⠛⠋⢁⣀⠤⠀⠀⠀⠊⠀⠀⠈⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢟⡩⠔⠂⣀⣤⠶⠋⠉⣀⠄⢀⡠⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⣩⠖⠉⣀⡴⠞⠉⠀⣠⡴⠋⣡⠔⢋⡴⠊⠀⡰⠁⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⠞⢁⣠⠞⠉⢀⣠⠶⠋⢁⣠⠞⢁⣴⠋⠀⣠⠞⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⣠⡾⢃⡴⠋⠀⠠⠚⠉⢀⣤⠖⠋⣁⣴⠟⣉⠀⠐⠁⠀⠀⠀⠀⣀⣾⡇
⠀⠀⠀⠀⠀⣠⣾⢟⣤⠾⠋⠀⠈⠀⠀⢀⣠⠴⠚⠉⢀⣠⠞⣋⠴⠊⠁⠀⠊⠀⠀⠀⢀⣼⣿⡿
⠀⠀⢀⣤⡾⣋⣴⡿⠛⠀⠀⠀⠀⠀⠊⠁⢀⣀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⣾⣿⡟⠁
⠀⣴⠟⠑⠊⢙⣿⣾⣉⣀⣤⣤⣶⣯⣤⣶⣿⣿⡛⠋⢀⠀⣀⠀⠀⠀⢀⣠⣴⣶⣾⣿⡿⠋
⠀⠻⠾⠷⠛⠋⠉⠉⠉⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⠛⠛⠛⣿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⠻⢿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠉⠉⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠟⠛⠋⠉⠉⠉⠛⣿⡟⠛⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣯⣥⣤⣤⣤⣤⣤⣤⣀⣀⢠⣿⡏⠀⠀⠀⠈⢙⣿⣷⣤⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠉⠉⠉⠁⠀⠈⠉⠉⠙⢛⣿⣿⣿⣦⣀⠀⣠⣶⡿⠋⠀⠙⢿⣦⡀
⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠿⠋⠁⠈⠙⢻⣿⣿⣋⠀⠀⠀⠀⠀⣹⣿⣆
⠀⠀⠀⣼⡿⠁⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⠋⠉⠀⠀⠀⠀⢀⣴⣿⠋⠙⢿⣷⣄⠀⣴⣿⠟⠿⣿⣧
⠀⠀⣸⡿⠁⠀⢀⣴⣾⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠉⢿⣿⡿⠁⠀⠀⠈⢿⣧
⠀⢠⣿⠃⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠘⣿⡆
⠀⣸⣟⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⢿⣧⠀⠀⠀⢹⣧
⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢿⣧⠀⠀⢸⣿
⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠈⣿⣧⠀⢸⣿
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠸⣿⡄⣸⡟
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⣿⣇⣿⠃
⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠈⠻⣷⣄⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⣡⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣼⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⡀
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠅⠰⠄⠛⢠⡭⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣾⠟⢥⠆⠂⢈⡁⡀⠂⢐⣿⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢿⡧⠃⣾⡷⠀⣰⣾⣅⢷⠟⢐⡉⡶⠀⠘⡈⣄⢾⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⠀⢰⣿⠉⠱⢖⠆⠈⣡⠆⠌⠁⢤⠐⠂⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣠⡞⢷⡆⠀⢱⠸⣿⠀⣴⠇⠐⣖⠙⠁⠖⠂⢀⠡⠀⠐⠂⢁⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣿⣄⠈⠱⡀⢸⡄⣻⠀⣿⣯⢉⠴⡖⢈⣥⡈⠄⠂⠂⣩⠄⢄⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠸⣆⠀⣳⠘⡇⢻⡄⠏⣗⢸⡏⠄⠀⠀⠀⠉⠠⠈⠁⢒⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠘⣆⠈⢷⡇⠞⡇⠰⢻⡄⣐⠉⡀⠆⠐⠀⠀⠩⠀⣰⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢧⠠⢿⣼⡇⠀⢛⢷⡛⡁⠹⠖⢃⣃⠸⠀⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⢧⠐⣿⣿⠀⠀⢫⣻⣵⣀⣛⠘⠄⢶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡙⡙⢇⠀⠀⠑⣝⠻⠶⠿⣛⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⠀⠀⠀⢸⠎⢳⡕⠈⢧⡀⠀⢸⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⣧⠈⠀⠳⣄⣸⠀⢰
⠀⠀⠀⢸⠀⠀⠀⢸⠃⠀⠙⢧⠀⠀⠘⣿⡀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⠀⠙⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⠒⢲⠖⠤⢤⣀
⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠁⠀⢿⡄⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⡀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠈⠛⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⢨⢀⢱⡀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠐⢄⠀⡠⠊⠀⠀⠀⠊⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡆⣾⢸⣿⡄⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⡞⠀⠀⠀⠀⢀⡠⠋⠀⢀⠐⠁⠀⠀⣀⠤⠀⠀⠈⠁⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠃⢻⢸⣿⣷⡀⠀⠀⠀⠀⠘⡄⠀⠀⢸⠀⠀⢀⠠⠐⠁⠀⠀⠀⠁⠀⣠⠔⠉⠀⠀⠀⠀⣠⡴⠖⠛⠉⠉⠉⠉⢱⠂⢄⡀⠀⢸
⠀⠀⠀⢠⠀⠀⠀⠀⡇⢰⠀⠀⢸⢸⣿⣿⣷⡄⠀⠀⠀⠀⠙⣷⡖⠉⠐⢍⠀⠀⠀⠀⡀⠀⠀⠀⡔⠣⡀⠀⠀⠀⣰⠚⠁⢣⠀⠀⠀⠀⡀⠔⠉⠓⢄⠈⠢⣜
⠀⠀⠀⢸⠀⠀⠀⠀⡇⠈⠀⠀⢸⣿⡟⠛⠱⢿⣄⠀⠀⠀⠀⠘⢦⠀⠀⠈⢷⡀⠠⠊⠀⠀⠀⡌⠀⠀⠈⠢⣀⠞⠁⠀⠀⣼⣿⣶⣶⡎⠀⠀⠀⠀⠀⠙⣾⣿⡄
⠀⠀⠀⢸⠀⠀⠀⠀⢧⠀⠀⠀⣸⣿⠗⣄⠀⠀⠹⣦⠀⠀⠀⠀⠈⢣⠀⠀⠀⣿⠁⠀⠀⠀⡐⠀⠀⠀⠀⠀⡍⠀⠀⢀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣹⣿⣿
⠀⠀⠀⢨⠀⣀⣀⢀⣸⠀⠀⡎⠀⠀⠈⠛⢷⣄⠀⠈⢣⡀⠀⠀⠀⠀⢣⡀⠀⢀⡇⣀⣀⣠⡁⠀⠀⠀⠀⢰⣿⡦⠂⠁⠀⠉⠻⠟⠋⠁⠢⠄⡀⢀⠄⠊⠈⠉⠛
⠀⠀⠀⡼⠋⠀⠁⠀⢹⣅⠤⠽⠢⣀⠀⠀⠀⠙⠛⠤⠀⠑⡀⠀⠀⠀⠀⠱⣄⡘⠋⢹⠀⠀⠈⠑⢄⠀⠀⠸⣿⠃⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠙⠲⠤⣀⡀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠚⢄⠀⠀⠣⡀⠀⠀⠀⠆⠀⠀⡿⣀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀
⠀⠀⠸⡀⠀⠀⠀⠀⠀⡝⠷⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢦⠀⠀⠀⠀⠀⢻⡲⣄⠉⠒⠰⠖⡆⠀⣀⣤⠈⠑⢲⣾⣿⣿⣷⣄⠀⠀⠄⠚⠢⢀⠀⠀⠀⡸
⠀⠀⠀⠑⢖⠀⠀⢀⣀⣧⠀⠱⣌⠒⣤⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢱⠘⡆⠀⠀⡰⠉⠉⠁⠀⢳⡀⠀⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢱⣶⡶⠁
⠀⠀⠀⠀⠀⠉⠈⠉⠁⠈⠲⠄⢼⠵⠗⠉⠀⠠⠄⢀⣀⣀⠀⠀⠀⠤⠾⣦⠀⣤⣐⣼⠦⠿⠀⠊⠁⠀⠉⠁⠈⠉⠙⠶⡉⠉⠉⠙⢆⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠀⠀⠠⠀⠀⠒⠒⠊⠀⠉⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠷⠤⠤⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠂⣀⠐⠀⠀⠀⠀⠈⣰⠀⠙⠁
⠀⠀⠀⠀⠀⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠖⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠈⠄⠀⠀⠀⡀⠀⠀⠀⠀⠐⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠤⡀⠀⠀⠀⠈⠀⠀⠂⡀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠄⠀⠀⠈⠣⡐⠠⠄⠀⠀⠀⡀⠉⢁⡀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠀⠀⠀⠀⡈⠀⡀⠀⠁⠀⢀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠠⠀⠠⠤⢀⠀⠄⠠⠄⠁⠀⠀⠤⠀⠀⠄⡀⠀⠀⠢⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡀⠁⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠢⠁⠀⠠⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠂⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⣀⠀⠀⠐⠀⠀⠲⠠⠐⣄⢀⡀⠉⣠⣤⠄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠉⠀⣀⡠⠌⠁⠒⠀⠤⠄⢀⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⣦⠜⡓⠚⠃⠁⠀⠴⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⠝⣢⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⠚⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡐⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠔⠁⠀⠀⠀⠀⠀⠀⣠⠀⠈⠿⠵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⡠⠤⠄⠒⠀⡀⠈⡁⠤⠲⠀⠒⠀⣒⡉⡁⠤⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣥⣥⣬⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣗⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣿⣿⣷⣶⣿⣿⣷⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠋⣉⣉⣤⣤⣤⣤⣴⣶⣶⣶⣤⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣵⣷⣾⣿⣿⣿⣿⠏⠙⠿⡿⠯⣽⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠋⠖⠘⠟⠁⠶⣤⡀⠀⠀⣀⠉⢻⢻⣟⣛⣺⣶⣦⡄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣟⣆⡀⠐⠀⠀⠀⠁⠀⡀⠀⢈⣀⣁⣈⠀⠀⠀⢈⣹⣿⠀⢀⣾⠟⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣑⡿⡇⠀⠀⠀⠠⣄⠀⠀⠈⣠⠦⠊⡙⠻⠿⠿⢥⣣⣤⣾⡏⢀⣿⣇⡀
⠀⠀⠀⠀⠀⣀⢿⠛⠛⠛⠛⠛⠛⠛⠀⠁⠀⠀⠉⠀⠀⣸⠑⡘⠀⠣⠀⠐⡠⢀⣀⣈⣍⣀⣨⣹⣿⣷
⠀⠀⣴⡮⠛⠛⠚⠒⠂⠀⠀⠀⠋⠀⠀⠀⠐⠂⢀⣬⣹⣧⣁⠀⠀⠀⠈⠉⣤⣯⠊⠉⠛⠛⠛⠛⠛⠁
⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣦⡄⠀⣤⢼⣿⣿⢆
⠀⠘⢟⣦⣀⣀⣀⡠⠀⣾⣿⣿⣿⣿⣿⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⡄⠉⠀⠙⠿⣿⡣⡀
⠀⠀⠀⠉⠛⢟⣄⠀⢉⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⢻⣿⡀
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠚⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣄⣁⡴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠓⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⣿⣧⠻⠷⠀⠀⠀⠀⠀⠀⣤⡆⣠⣿⣿⡬⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢬⣛⣿⣷⣬⣿⡆⠀⠀⠀⠀⠀⠀⠀⠯⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣀⣤⠀⠀⠀⠀⠀⠘⢽⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢨⣟⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⣀⣠⣡⣄⣽⡛⢋⣴⡆⠀⠐⢦⠙⢋⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⡶⢀⢾⡁⣯⡉⣤⡄⣿⣿⣷⠦⠀⠒⠓⠉⠀⠀⠀⠀⢀⣠⣠⡶⢞⠀⢀⢼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢂⣼⣤⣿⠶⣷⣞⡿⢃⣾⣿⣿⣿⡱⣆⠁⠀⠀⠀⠀⠀⠒⠚⣻⠍⢠⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⡿⣿⣧⣾⣿⣾⣯⡽⢓⣿⣿⢿⡟⢧⡑⠂⠀⠀⠀⠀⠀⢀⣶⡼⢣⣾⣿⣿⡷⡘⢟⣉⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠲⢿⣤⣿⣽⣯⣾⣿⣏⢴⣿⣿⣿⣾⡉⠆⣠⢀⠠⣶⣾⣶⣾⣿⣿⣷⣿⣿⣿⣿⠓⣬⣿⣿⠃⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣠⠆⠁⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⣯⠀⠃⣘⢟⣻⣿⣿⣿⣿⣿⣿⢟⣱⣿⣿⡿⢮⠑⢸⠟⠁⠀⠀⠀⠀⢁⠈⡄
⠀⠀⠀⠀⠀⢀⣼⣿⡷⠞⠋⠉⠀⠀⠀⠀⢸⡏⠉⠛⠛⠻⠋⠀⡀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣻⡶⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠰⠚⣷⣀⡀⠀⢀⣴⣾⣿⣾⣿⣿⣿⣿⡿⢻⡍⠛⠿⠟⠛⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠲⠿⠛⠁⠀⠉⠉⠁⠀⠀⢀⣸⣇⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠒⠒⠒⠶⠢⠀⠺⢧⡐⣶⡦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⡻⠿⣛⡃⢘⣛⠿⢟⣽⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢫⣮⣻⣧⣼⣟⣵⡝⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣉⣉⢈⣉⣿⣏⣹⣿⣉⣉⣉⣉⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡜⢟⣽⡟⢻⣯⡻⢣⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣟⣵⣶⣭⡅⢸⣭⣶⣮⣻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣙⢿⣿⣿⣿⣿⣿⡻⣿⡿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢶⡊⣉⡠⠤⠤⠤⠤⠤⣈⡙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢘⣧⠀⠀⠀⠀⠀⠀⣀⣉⡲⣄⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡀⠀⠰⠭⠤⠳⡀⠀⢀⠔⢋⠙⠒⠮⣝⠣⣌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⡇⠀⠀⢀⠴⠒⢳⡀⠘⣆⢀⡀⠀⠃⠈⢣⠈⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡎⢹⡀⠀⠙⠒⠒⠉⢳⠀⣿⠀⡀⠁⡀⢰⠁⣾⠁⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠚⠒⠓⠒⠓⠒⠒⠒⠒⠒⠒⠓⠛⠒⠒⠒⠒⠚⠒⠛⠒⠓⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⡒⠒⠒⠒⢲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠒⠒⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⡴⠛⠢⡀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠠⠤⠄⠀⠀⠀⠤⠤⠀⠀⠘⠳⡀⠀⡜⡭⠒⠍⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠠⠤⠄⠀⡇⠈⣇⠀⠀⠀⠙⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠽⡄⠈⠓⠢⠤⠤⠤⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⠉⠉⠉⠉⠀⠀⠈⠲⢤⣀⡀⠀⠀⡷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⢧⣶⡀⢀⡤⠂⠀⢠⣼⠀⡏⠙⠢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠼⠦⠾⠷⠧⠾⠤⠤⠤⠌⠀⠟⠓

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠀⠂⣰⠠⠄⠀⠰⠀⠠⡀⠂⡀⢀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⠐⡶⣦⣤⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣾⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣾⣿⣤⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣴
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⡿⠛⠻⡿⢿⠟⡛⠿⠟⠿⡿⢿⠿⠃⠻⠛⠛⢻⣿⣿⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠂⠤⢁⠀⠸⢃⡀⠻⠇⣠⠀⢼⡇⠟⡁⠠⠆⣩⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣷⣿⣿⡆⣺⣿⣿⣷⣾⣿⣿⣿⠷⠟⠛⠒⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⠁⡔⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣀⠀⠀⠀⠈⠋⠀⠈⠁⠀⠀⠺⡞⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠟⠉⠉⠛⠉⠻⢿⡟⠛⠁⡠⠐⠀⠀⠀⠙⢢⣤⡀⠀⢰⣾⣷⡆⠀⠀⠱⣄⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣠⠴⠶⠿⠸⣧⠀⠀⠙⠛⠁⠀⣼⣇⠙⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡔⠃⠀⠀⣠⡟⡠⠀⠀⠀⣐⡿⠒⠀⣤⠀⢀⠘⠉⠻⣧⣄⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣏⠀⠀⢠⢺⣿⠀⢰⣾⣿⡄⠀⠀⠀⢰⡟⠻⠷⣤⣴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠷⠷⠟⠃⠙⠛⠛⠊⠿⣧⣤⡶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣶⣼⣿⣷⣤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣷⠞⠉⠀⠀⠀⠀⣀⣀⣀⣀⡀⣠⣿⣾⣿⡟⠛⠋
⠀⠀⠀⠀⠀⠐⠛⠛⠉⠉⠁⠀⠀⣤⡤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢸⣿⠟⣽⣿⠋⡙⠻⢿⣿⣿⣿⣿⡅⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢃⣼⠟⠁⠀⠁⠀⢸⣿⣀⣀⣽⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⣿⠁⠈⣿⠀⢀⡘⣶⣸⣸⣿⠉⠁⠀⢀⢸⣠⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣼⣷⣾⣷⣧⣾⣶⣿⣦⣤⣿⣿⣿⣿⣿⣴⣶⣦⣿⣾⣯⣿⡏
⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠐⢢⣤⡀
⠀⠀⠀⠀⠀⣀⡀⢀⡀⣀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠒⠃⠐⠋⠃⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣸⣾⣷⡷⣻⣷⣺⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣶⢞⣿⣿⣿⣿⣷⣀⠀⠀⣀⠶⠠⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣶⣤⣠⣤⣤⣤⣀⣿⣿⣛⣿⣿⣿⣿⣿⣷⣦⣅⠉
⠀⠀⠋⠀⠙⢻⣿⣿⡏⣿⣿⣿⣷⣾⣿⣿⡯⢻⣿⡿⣷⣿⣿⣿⣏⠿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⡁
⠀⠀⠀⠀⠐⠛⠻⣿⣗⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠚⠋⢘⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣟⣀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠤⠀⡐⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠠⡷⠈
⠀⠀⠀⡀⠀⠀⠀⠀⠽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠘⢿⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣄
⠀⠀⠀⠘⣧⡀⠀⢀⠐⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠁⠀⠩⠓⢾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠉⠛⢿⣷⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠽⢿⣿⢿⡿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠂⠛⠛⠛⠛⠻⠛⠻⠛⠉⠠⣨⡁⠉⢹⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠉⡙⠻⡿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠗⠀⣘⣸⣿⠻⣿⢽⣿⣿⣿⣿⣷⣤⣀⣴⣶⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡹⡷⠃⠘⠉⡈⠿⠟⢡⣿⠓⠟⢿⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢉⡸⠃⠀⠋⠑⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠠⠀⠚⠉⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡔⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⡠⡤⠖⠒⠀⠆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠶⠒⠂⠆⢄
⠀⠀⠀⣠⢪⠚⢄⠀⢀⢀⣄⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠜⢇⠀⢀⡀⣇⠀⠐⠄
⠀⠀⢰⢡⠇⠀⢀⠯⠀⠀⠀⠙⣴⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡎⠀⢀⠷⡁⠀⡇⠈⣦⠋⠄
⠀⠀⡎⡜⠀⠀⠂⠀⣱⠊⠑⡎⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠀⠀⠌⠀⢱⠞⠙⢶⠁⡄
⠀⠀⡇⡧⠀⢰⠆⠠⡇⠀⠀⢸⠀⢧⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠠⠆⠀⣯⠀⠀⢸⠄⡇⠰
⠀⠀⡇⢃⠀⠀⠀⠀⡁⢄⡤⠦⠀⠈⠛⠒⠀⢀⠴⡶⠀⠀⠀⠀⢡⢰⠀⠀⠄⠀⢸⢤⡠⠿⠀⠅⠈⠀⠀⠀⣀⣈⢻
⠀⠀⠸⡘⡄⠀⠘⣔⠀⠀⡀⡠⠂⡠⠀⠀⠀⠀⠙⠋⡀⠀⠀⠀⠸⡌⡆⠀⠈⣶⠁⠀⠇⠀⠂⠀⠀⠀⠀⠐⠶⡽⠀⡖
⠀⠀⠀⠱⡑⢄⠎⠀⠁⠈⠋⠀⡀⠁⠀⠀⠀⠀⠀⢭⣺⠀⠀⠀⠀⠙⣜⢆⡔⠀⠈⠀⡏⠀⡠⠁⠀⠀⠀⠀⠳⠶⠄⠉
⠀⠀⠀⠀⠈⠚⠷⠦⠄⠦⠂⠈⠀⠀⢞⣟⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠷⠦⠤⠠⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠘⠀⠀⠄⠀⠐⠒⠀⠄⠊⠘⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠃⠂⠀⠄⠁⠀⡅⠊⠈⠂⠁⠂⠄⠂⠐⠁

⠀⠀⠀⠀⡀⠀⢤⣀⠀⢀⣄⢀⠀⡀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣀⡀⢀⣠⠀⠀⠀⣠⢀⣠⠀⢀

⠀⠀⠀⠀⠀⠀⣀⣠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⣀⡀
⠀⠀⠀⢀⠔⡽⡋⠀⠀⠀⠄⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡫⡏⠀⠀⠀⡇⠈⠢⡀
⠀⠀⢀⠎⡎⠀⠑⡄⠂⠉⠃⠢⣀⠧⡠⠶⡂⠀⠀⠀⠀⠀⠀⠀⠀⡌⡜⠁⠈⣤⠂⠉⡏⠢⣀⠵⡀
⠀⠀⡜⢸⠀⠀⠌⠈⢢⠄⠤⡠⠠⠀⠉⠛⠁⠀⢠⣒⣆⠀⠀⠀⢸⢰⠁⠀⡐⠀⢢⡤⠧⣤⠛⠀⠁⠀⠀⠠⠤⡄
⠀⠀⡇⣇⠀⢀⡀⢀⡎⠀⠀⢡⣀⣄⡄⠀⠀⠀⠀⠉⣉⣀⠀⠀⡄⢘⠀⢀⡁⠀⣼⠀⠀⢀⣀⡀⠀⠀⢐⣛⡆⢁⡀
⠀⠀⡇⡏⠀⠈⠁⠀⢣⠀⢀⡎⢀⠀⠐⠀⠀⠀⠀⠘⠵⠯⠀⠀⢃⢨⠀⠀⡅⠀⢹⡀⢀⣌⠀⠀⠀⠀⢠⣩⣁⠸
⠀⠀⢱⢸⠀⠀⠡⡠⠂⠈⠀⢀⢦⢀⠁⢭⣭⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠐⣀⠎⠈⡏⠀⡶⠀⠄⠀⠀⠈⠁
⠀⠀⠈⢦⢣⡀⡰⠉⠂⠠⠖⠁⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡱⡀⡠⠋⠂⠀⡗⠊⢀⠍
⠀⠀⠀⠀⠑⢽⣤⣀⠀⢀⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⣧⣀⠀⠀⣇⠔⠃
⠀⠠⠀⠀⠀⠠⢀⣈⠉⠠⡠⠠⢀⠀⠀⠄⠀⢀⠀⠀⠀⠀⢠⠀⠀⡄⣀⠀⢀⠌⠉⠅⡠⠠⡀⠄⠀⠀⡀⢀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠉⠁⠢⠄⠀⠀⠚⠀⠀⠀⠀⠁⠚⠂⠀⠆⠀⠀⠀⠀⠀⠉⠙⠀⠜⠀⠀⠚⠂⠘⠘⠀⠀⠐⠂⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣞⣾⢁⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⠀⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢻⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⡠⡏⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⠃⠀⠃⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⡐⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⡀⠉⠉⠈⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣆⢆⡀⢠⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡆⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⡄⠈⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡈⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠃⠀⠀⠀⠀⠐⠦⠄⠆
⠀⠀⠀⣀⣠⡀⠀⣀⡀⠈⠉⢀⡠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⡀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠴⠶
⡄⠀⠀⠈⠁⠁⠀⠁⠀⠀⠐⠊⠀⠀⠀⠀⠀⢤⣀⢀⠐⠲⠶⠀⠀⠀⠈⠉⠉⠉⠁⠀⣤⠚⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡦
⠀⠉⠀⡀⣀⣀⠰⠶⠃⠀⠀⡀⠀⠀⠤⢠⡤⠀⠉⠉⠀⠀⠀⠠⠷⠷⠀⠀⠀⠀⢀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠘⠉⠀⣠⠤⡠⠤⠀⠄
⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣤⣤⣾⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣛⣩⣙⠻⢦
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⢄⡀⠀⡀⠀⠀⠂⡘⠀⣦⠀⠀⠀⠀⢀⣴⣫⣶⡿⠟⠛⠻⢷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢱⠀⡜⢷⣤⣿⣿⣶⣿⣿⣾⣾⣿⣿⣿⣿⣿⠀⠀⢀⠄⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⢙⣋⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠀⡄
⠀⠀⠀⠀⠀⠄⠐⠆⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⡿⣟⠻⣿⣿⣿⠇⠘
⠀⠀⠀⠀⢄⢸⣆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⠿⠉⠷⠠⠀⠀⠀⠘⢳⣾⡏⠀⠀⠀⠀⠐⢀⢌
⠀⠀⠀⡀⠘⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠿⡏⣈⠉⠀⠠⠀⢀⠀⠀⡐⠢⣿⠏⠀⢄⠀⠀⡀⠀⠒
⠀⠀⠀⠈⣾⢔⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠌⠀⠀⠀⠀⠀⠀⡀⠀⢁⡔⣾⣿⣦⣀⣠⡀⡀⠃⢁
⠀⠀⠀⢈⡸⣾⣿⣿⣿⣿⣿⡿⢟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⡿⣼⡿⠋⠻⠟⠁⠠⠀⠃⡀
⠀⠀⠀⠈⠀⣿⣿⣿⣽⣛⣟⣽⣿⢧⣤⣀⣦⣆⣄⣀⣢⣤⠴⢻⣿⣿⣿⣿⣶⣅⠹⢇⣛⣴⢄⠀⡀
⠀⠀⠀⠀⠀⣿⢓⣏⣿⠟⠋⠁⠀⠀⣀⠈⠷⢹⡍⣿⠟⠧⠾⢿⣿⣿⣟⢿⣿⣕⡀⠂⠁⢘⢎⡵⠄
⠀⠀⠀⠀⢠⣯⣴⠏⠀⠀⠐⢻⣦⠌⠂⠙⠈⠀⠀⢹⠀⠒⠠⠄⠨⣿⣿⣿⣟⡿⣿⣶⣴⡤⠌
⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡈⢁⠀⠠⠴⡋⠉⠉⠈⡉⢋⠉⢉⡽⠶⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠄⠈⠂⠀⢁⡄⣉⣉⠓⠂⣀⠀⠊⠀⠀⡀⠀⠀⠀⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠢⠆⠉⠁⠀⠀⠈⠉⠉⠋
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠐⠒⠒⠂⠀⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢐⡨⠴⠒⠛⠋⠛⠛⠛⠛⠓⠶⢦⢤⣈⠉⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣈⠴⠚⢉⣠⠤⠒⠘⠉⠉⢉⠉⠉⠙⠒⠒⠦⢤⣉⠙⠓⠦⢄⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣡⠞⣁⠴⠚⠉⠡⡄⡄⠀⠀⠀⠘⠈⠀⠀⠀⠀⢠⠤⠀⠉⣳⢆⠀⠙⢦⡈⢢
⠀⠀⠀⠀⠀⠀⠀⡰⢡⠎⣡⠞⠁⠀⠀⠀⠀⢀⡠⠤⠖⢒⠐⠒⠰⠒⠠⠤⣀⡀⢀⡾⠃⠀⠙⣆⠀⠻⡄⠱⡀
⠀⠀⠀⠀⠀⠀⡄⣰⠃⡴⠁⠀⠳⠒⢀⡤⠊⠁⠹⡐⢈⠀⠂⠀⠄⢈⠄⣾⢁⠞⡽⢧⡈⠙⠀⠈⢣⠀⠘⣆⠠
⠀⠀⠀⠀⠀⢀⢰⠇⡰⠃⠀⠀⠀⡰⠃⠀⡈⠀⠀⠘⣄⠴⠂⠀⠒⠤⣸⡳⢫⠞⢂⡡⠘⣆⠀⠀⠈⢧⠀⠘⡄⢆
⠀⠀⠀⠀⠀⠆⡟⢀⠇⠀⠀⠀⢠⠁⠀⠀⠐⠀⡠⠊⠀⢀⠔⠁⠀⠀⠔⡱⠳⡈⠀⠐⣉⠄⢆⠠⣦⠌⢧⠀⠰⠸⡄
⠀⠀⠀⠀⠀⠐⡇⢸⠀⠘⠙⠂⡸⠤⠀⣀⡀⢠⢁⢔⠔⠁⠀⠀⡀⢊⡜⠁⠀⠘⣆⢥⣖⡾⡛⡆⠀⠀⠈⡆⠀⡆⠇
⠀⠀⠀⠀⠀⢘⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⡀⠟⠀⠁⠀⠀⡌⣔⣲⠘⡆⠀⠀⠀⢱⠀⠁⠘⠊⢱⠀⠀⠀⢣⠀⠁⢸
⠀⠀⠀⠀⠀⠸⡇⠘⡄⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠱⢤⡠⠞⠀⠀⠀⠀⣜⠀⠀⡔⠔⣸⠰⠦⢀⡙⠀⠀⠂
⠀⠀⠀⠀⠀⢃⢣⠀⢣⠄⠈⠑⠀⢃⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⢔⠅⠀⠰⣧⡁⢐⣤⣶⠏⠀⠀⠀⡇⠀⠃⡆
⠀⠀⠀⠀⠀⠀⠌⣧⠈⢇⠀⠀⠀⠀⠑⢄⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡶⠋⠀⠀⠀⡼⠀⠠⢐
⠀⠀⠀⠀⠀⠀⠈⠜⣦⠈⢣⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⠀⡤⡠⠠⡠⠤⠤⠄⠀⠀⠀⠠⠤⠤⠀⡴⠁⠠⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠌⢦⠀⠱⣄⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠒⠂⠒⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠀⢠⠲⣢⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢰⣻⡼⠀⠀⢀⠀⠊⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣑⠢⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⡀⠤⠂⢁⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢅⣐⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⣂⣁⡤⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠂⠤⠤⠐⠒⠒⠊⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⢀⠔⠀⡡⠂⠀⠐⠀⠀⡠⠂⢀⠀⠐⠁⠠⠊⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⠥⠤⢜⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠍⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣐⠄⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡾⠙⣿⢿⡿⠂⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣒⢫⣭⡁⠙⠠⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠞⣳⡿⢿⡾⢿⣿⡻⢓⡂⠀⡽⡄⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣽⡞⢩⣚⣿⠿⡛⢋⣳⠞⣵⣡⠆⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣫⣽⣯⡟⣿⢾⢟⠚⠖⠷⠦⢹⣭⠏⢀⡦⣴⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣿⣿⣿⣏⡰⡿⣞⣯⣬⡿⠂⣤⣠⡞⣘⣦⣴⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣻⣞⣽⣽⣝⡻⠷⠳⠿⣷⣷⣾⣯⣽⣿⣗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⣫⠷⢞⣛⡙⣻⣉⣿⣦⣿⣛⣷⠯⣜⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣼⣫⣼⣯⣽⣯⣤⣿⣻⣿⣟⡛⢿⡼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠋⠉⠑⠚⠧⠴⠶⠿⣽⣥⣴⣺⡶⠭⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡃⢀⡃⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠑⠉⠉⠉⠉

⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⢠⠞⢹⡠⠤⠒⠒⠛⠢⡀⠀⠀⠀⣴⣷⢛⣫⣭⣷⣶⣶⣄⠀⠀⠀⣰⡏⠀⠀⢀⡠⢄⡙⢆⠀⠀⢀⠸⣿⣿⡿⢟⣛⣛⢦⡀
⠀⠀⣇⠀⠀⠳⣄⣀⣀⣀⣀⣱⠀⠀⠸⣿⣿⣧⡻⠿⠿⠿⠿⠿⠄⠀⢸⠁⠈⠒⠊⠁⠀⠀⠘⡎⡆⠀⣿⣷⣮⣵⣾⣿⣿⣿⣧⢳
⠀⠀⡏⠢⡀⠀⢸⠁⠀⢸⠀⢹⠀⠀⢳⣝⢿⣿⡿⣸⣿⣿⢸⣿⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸
⠀⠀⠹⣄⠈⡳⠃⠀⢀⠇⡠⠃⠀⠀⠈⢿⣷⡎⣱⣿⣿⡟⣾⠟⠀⠀⠀⠳⡊⠉⠉⠑⠢⠴⢊⠜⠀⠀⠙⣩⣶⣶⣮⣝⣛⣛⡵⠃
⠀⠀⠀⠈⠙⠢⠤⠤⠚⠋⠀⠀⠀⠀⠀⠀⠉⠚⠿⠿⠛⠈⠁⠀⠀⠀⠀⠀⠈⠓⠢⠤⠔⠚⠁⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠋
⠀⠀⠀⢀⣠⡤⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠲⢶⣦
⠀⠀⠰⠛⠋⣀⣀⡠⠛⠛⠃⠀⠀⠀⠀⣤⣶⡿⠿⠿⣷⣦⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⢷⣌⠀⠀⠀⠀⣠⣾⣿⣿⠷⣦⡙⠃
⠀⠀⡀⠀⢠⣿⣿⣧⠀⠀⢠⣀⠀⠀⠸⣿⣿⠃⠀⠀⢻⣿⣿⠻⠄⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠙⠀⠀⠀⣼⣿⣿⠟⢁⣼⣿⣿⠂
⠀⠀⣿⡇⠀⠉⠟⠁⠀⠘⣿⡿⠀⠀⠀⢸⣿⣷⣦⣶⣿⣿⡅⠀⠀⠀⠀⣄⠀⠠⡾⠋⠀⠀⠀⠀⠀⠀⠸⣿⣿⠋⣠⣾⣿⣿⠃
⠀⠀⠈⠁⠀⣠⣦⣄⡠⠀⠉⠁⠀⠀⠀⠸⣿⡿⠛⠻⢿⣿⡷⠂⠀⠀⠀⠉⠳⢦⣀⠀⠠⠀⠀⠀⠀⠀⢰⣌⡛⠿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠠⢾⠃⠀⠀⢀⣨⡴⡄⠀⠀⢀⡤⣸⣿⣿⣿⠟⣓⣀
⠀⠀⠄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣿⣿⣿⣿⣿⣿⡀⠀⢠⠁⡞⠀⠀⣰⠋⠀⠀⠈⡄⠀⣾⢃⣾⣿⡿⣱⣾⣿⣿⣧
⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⠀⠀⢸⡞⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢁⣾⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣷⣿⣿⣿⣿⣿⠟⠀⠀⠀⠢⡀⠀⠀⠳⣀⠀⢀⠜⠀⠀⠙⣿⣿⣿⣧⠻⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠋⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠙⠒⠁⠀⠀⠀⠀⠈⠙⠻⠿⠷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷
⠀⢴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⢁⡠⠤⠤⢄⡀⠀⠀⠀⢀⣿⣧⢀⣀⣤⣀⡀
⠀⠀⠀⢻⡧⡀⠀⠀⠀⢀⠀⢀⠀⠀⠛⣄⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠰⠁⡰⠋⠰⠄⠀⠀⠙⢆⠀⠀⣾⡟⣵⣿⣹⣿⣿⣿⣦
⠀⠀⠀⠀⠣⠈⠒⠂⠉⢠⠂⠈⠃⢂⡀⠹⣷⣦⣶⣾⣿⢽⣷⣶⣶⡦⠀⢠⠀⡅⠀⠒⠈⠁⠀⠀⢸⠀⠀⣿⢹⣿⣯⣽⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠡⠀⢀⣀⣐⠠⠤⠤⠐⠁⠀⠹⢿⣿⣿⣿⣽⡟⠶⠚⠛⠀⠀⢂⣣⡀⠀⠀⠀⠀⢀⡜⠀⠀⠸⡏⢿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠒⠤⠤⠐⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋
⡶⣶⢶⢶⠶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⣶⢶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡦⢠⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⢩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⣿⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠈⢿⡿⠀⠀⢀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣸⡿⠀⠀⠀⠀⠀⠈⠀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⣶⣶⣶⣶⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⢿⣿⠃⠀⠀⣠⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣷⣶⠆
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠈⠛⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡿⠋
⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀
⠀⠀⠀⢠⣿⠁⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣦⡀
⠀⠀⠀⢸⣿⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠉⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⡿⠟
⠀⠀⠀⠀⢿⣧⣾⠏⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋⠁⠀⠀⠀⢀⣴⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⢈⣿⡿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⢰⣿⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⡿⠃
⠀⠀⠀⠈⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⢛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣿⣷⣶⣶⢶⣶⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣠⣶⣿⣷⣶⣄⡀⢠⣶⣰⣶⣤⣀⣀
⠙⠿⠋⠁⠀⠈⠙⣿⣿⣿⣛⡿⢿⣿⣏⣤
⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⣿⡛⣿⡮⠙⣥⣤
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⣿⣿⣷⡈⢅⣶⣿⣷⠘⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠁⣈⢧⠛⢍⠤⢿⣜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠙⠷⡀⠒⠓⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡲⠣⡄⠈⠐⠃⠀⡐⠀⠀⠈⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⡉⠋⠹⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠄⢤⣤⣄⠁⠊⢁⡀⠛⠁⠀⠐⠄⠂⠅⢀⡀⠀⠀⠠⠾⢆⡉⠛⠃⢩⣿⣮⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣖⣽⣷⠀⠂⡉⠀⠀⠐⣧⣄⠀⢠⢞⠓⢀⡀⠀⠀⠀⠙⠦⠀⠀⠀⠺⠇⢨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣳⣾⣿⣛⣷⡆⠤⠈⠂⠀⠀⢿⠟⢷⣀⡀⠀⠀⠉⣆⠀⠀⠀⠀⠀⠀⠐⢢⠑⠮⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⣿⡿⢿⣿⣭⣽⣿⣿⣗⣢⡀⠂⠀⠛⡻⢿⣦⡀⠀⠘⣷⡀⠀⠠⢆⢀⣀⡠⣄⡀⠆⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢟⣿⣿⣿⣿⠝⢳⣾⣷⣦⠤⡤⢤⣭⣍⣩⡹⠟⢀⠸⡇⠠⡶⠅⠀⠀⠠⠀⠈⠙⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣛⠹⠻⠞⠻⠯⡶⠤⠬⣴⠚⠆⠈⠀⠀⢠⠀⠀⠀⣘⢂⡤⠊⠂⠉⠀⡄⠀⠀⠔⠚⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣢⠶⠞⣛⣋⣲⣟⠛⠁⠟⢳⣦⠀⠀⠀⠀⠃⡛⠄⠈⢤⣂⢠⠈⠀⠀⠀⠀⠁⢨⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣹⠋⠁⣤⣦⣤⠁⠈⠙⠷⠀⠀⠐⣇⠀⢮⡀⣠⢸⡔⠂⠀⠀⠀⠀⠆⠐⡯⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡢⠋⠁⠀⠀⠀⣿⠏⠀⠀⠀⠀⠈⠣⡀⠀⠠⠀⠐⠶⣒⣶⡇⠀⠀⠀⠀⠀⠩⢵⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠞⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⣿⣞⣦⡀⡀⠀⠈⢶⣇⠀⠀⠀⠀⠂⢸⣟⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣿⣿⣦⡔⠀⠀⠜⣾⡏⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟⠛⠛⠁⠀⡀⣾⣯⢜⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⠻⣿⣿⠿⠋⠀⠄⠀⠁⠀⢰⠞⣹⢛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣇⢤⡄⠙⠀⡄⠀⠀⢀⡴⠲⣧⣉⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⢻⣆⣰⠋⠀⣴⡿⢏⠹⢴⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣈⣿⣦⣤⠀⠒⡆⠀⢰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣿⠀⠈⡅⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠿

⠀⠀⠀⠀⠠⣄
⠀⠀⣀⣈⠒⠘⡆⡀⠀⠀⠀⠀⠀⠀⢰⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠘⠛⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⢀⣸⡟⢛⣿⡦⠀⠀⠀⠀⠀⠠⠤⣴⢿⣿⢷⠀⠀⠀⠀⠀⢠⠞⣵⣺⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⢀⣈⡷⡀⣟⠑⢂⣀⠀⠀⠀⠀⠀⠀⠀⣻⡿⠸⠀⠀⠀⠀⣰⡿⠞⣻⠿⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠰⡇⠾⢱⠈⠼⢺⡧⡷⢐⠀⠀⠀⣠⡀⢰⠟⣦⠀⠀⠀⠀⠼⠁⢠⣞⣡⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢏⢢⡄⠀⠀⠙⠚⠁⠀⠀⠀⠙⠚⠉⠀⠀⠀⠻⠣⠞⠀⠈⠃⠀⠀⠀⠀⠘⠋⠉
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⣤⠴⠶⠶⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⣴⣦
⠀⠀⠀⠀⠀⠴⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡗⠤⠀⠀⠀⠘⠏⣿⣿⡀⠀⠀⠀⠀⣠⣴⣶⣿⣶⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⣼⢿⡆⠀⠀⠉⠙⠳⣤⣶⡶⠿⠟⠉⠀⠀⠀⠀⠀⠀⣸⠋⠘⡧⠀⠀⠀⣼⠇⠻⣿⡿⠿⢿⣧⡀
⠀⠀⠀⠀⠀⠀⢿⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠐⠇⠀⠀⠀⠀⠀⠐⢿⣄⢀⣨⣃⠃
⠀⠀⠀⠀⠀⠘⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⢐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣻⣄⠀⠁⠀⠀⠀⠀⠀⣴⠄⠀⠀⠁⠀⠀⠀⢾⡆⠰⡆⠀⠀⢰⠄⣴⡆
⠀⠀⠀⣰⣿⣿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⠿⣿⣿⠀⠀⠀⠀⠀⠐⢶⣿⡶⠶⠶⠖⠀⠀⠀⠈⠁⠀⢷⣶⣤⠏⠀⠉⠁
⠀⠀⠀⠃⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⡆⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡏
⠀⠀⢀⡼⠋⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣷⠀⠀⠀⠀⠀⣸⠏⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢿
⠀⠀⠮⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠹⡀⠀⠀⠀⠀⣟⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⡀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⢀⣤⣿⣷⣦⡀⠀⠀⣀⠀⠀⠀⢀⣿⣀⣴⣡⠀⠀⠀⠀⢀⡤⠖⢒⣶⣿⡀⡄
⠀⠀⠀⠀⢀⣸⣇⡜⠁⠀⠀⠀⠀⠀⢾⠛⣿⣿⣏⠳⠄⠐⠋⠀⠀⠐⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣋⠛
⠀⠀⠀⠀⢸⣿⡏⠉⠉⠁⣄⠀⠀⠀⠈⠉⠻⡟⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⣄⠀⠀⠀⠀⠀⠀⣠⠟⠋⠉⡽⠃
⠀⠀⠰⠲⠾⠿⠿⠷⠆⠀⠈⠀⠀⠀⠀⡀⠀⠁⢹⡀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢼⠀⠀⠀⠀⢠⡞⠁⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢞⣻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠂⠀⢀⢈⣁⡲⢉⡃
⠀⠀⠀⠀⠀⠀⠀⠦⢨⣶⣿⣿⣿⡃⠀⠀⠌⢘⡿⢿⠏⢡⡆⣠⣾⣿⢾⠤⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⢸⡻⣃⢀⣠⢛⣿⠗⠉⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠙⠛⠋⠉⣳⣿⣕⣺⣿⣿⠀⠀⠀⠀⠀⠀⣤⣤⣦⣴⢦⣀⡀⣄⠀⢄
⠀⠀⠀⠀⠀⣠⣼⣿⣤⡄⠀⢰⣿⣿⣟⡷⢞⡷⡄⠀⠀⢀⡶⣽⡏⠭⢼⡉⡔⢿⣿⢑⣧⡀
⠀⠀⠀⠀⠀⠀⢼⣿⣟⣿⣧⣬⣝⣿⣯⡳⠚⠸⣄⡀⢘⣾⢰⠮⠹⠓⠚⠠⢰⣶⡼⠿⠆
⠀⠀⠀⠀⠀⠀⠨⡉⠉⢻⣷⠀⠀⢻⠏⠁⢈⢿⡁⢈⠾⠃⠀⠀⢀⣤⣀⣈⣉⣀⠁⣀⡀⢀⣀⢀⡄
⠀⢀⣈⣭⠽⢤⣠⠀⠀⠰⡿⡆⢀⡆⠀⠐⠃⠀⠀⠀⢡⣴⠀⢠⣞⣠⣞⠚⠉⠉⢉⣙⣋⣭⢿⣿⣿⣷⢤
⠀⣈⣩⣭⢛⡃⢉⢰⡦⣤⣌⣿⠟⠁⠀⠀⠀⠀⠀⠀⠐⠒⣶⣶⣿⣷⣾⣥⣵⣶⣾⣷⣿⣿⣿⣿⣿⣿⡶
⢠⣿⣿⡿⢆⡀⠸⣿⣏⣿⡟⠃⠀⠀⢀⡀⠀⠀⠠⡬⠶⠖⠒⠊⠉⠘⠻⠿⠿⠁⠀⠀⠑⡒⢛⢟⣛⣿⡷
⠀⢿⣿⣽⣿⣿⣞⢘⠋⠁⠀⠀⠀⠀⠉⠀⠲⠆⣨⡉⠒⣤⠀⣀⡘⠁⣠⣴⣏⣶⣾⡿⢶⠺⣯⡁⡛⢻⠇
⠀⠈⠛⠿⣿⣾⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣓⠠⡿⣻⣧⣿⣿⣿⠿⢿⣿⡟⣸⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠲⢂⣀⣤⣠⣂⣏⢀⠸⢿⣟⠷⠿⡿⠿⠿⠶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠸⣷⣋⡴⠫⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣐⣾⡿⣻⢧⣙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠉⠿⠼⠛⢋⡩⡦⠳⢖⡦⡿⠹⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣉⣨⡽⠿⢂⣀⡀⠙⠀⢀⠐⠜⢃⡊⣷⣕⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣠⡾⣴⡟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣷⣂⣁⣿⠄⣸⢋⡧⠄⡀⠋⡀⡀⣘⡚⠙⢛⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⡀⣰⡶⣿⡽⡚⣷⠟⣠⠞⠴⣌⡜⣁⡤⢓⣶⣹⢦⠀⡅⠀⡀⠀⢀⣠⣤⢦⡶⣾⣿⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⡀⠈⡤⣭⡥⢋⡾⣫⡨⠞⣻⢃⣞⡏⡐⣷⣾⣿⣷⣿⣭⡏⢸⣿⣀⣟⡹⢿⠾⡿⠿⣷⣞⡻⠋⠈
⠀⠀⠀⠀⠀⠈⠀⡧⢿⣿⣡⣞⣻⢯⣶⣡⣻⣿⣿⣿⡻⣝⣶⣿⡺⠋⣴⣿⣿⣜⣿⣥⣽⡿⠛⣛⢛⣭⣷⡄
⠀⠀⠀⠀⠀⠀⢠⠷⠚⡿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠀⣰⣟⣿⣿⣿⣿⣧⣴⣾⣧⣿⡿⠭⠟
⠀⠀⠀⠀⠀⡆⠀⠀⠑⠀⠀⠠⠈⠙⠛⣿⣿⣿⣬⣍⣁⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⢿⣿⢭⠁⠀⠀⠠⠖⠀⠀⠆⠀⠀⢠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⡴⠿⠧⣤⣍⡙⠛⠛⢋⢹⣿⣿⣾⣿⣷⣹⠿⣌⠛⠀⠀⠠⠆⢀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠁⠀⠉⠉⠉⠉⠓⠊⠡⠶⣖⣾⣿⣿⣇⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣴⣶⣿⣿⠿⠿⠿⠿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣦⡛⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣋⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⣤⣄⡀⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢸⣿⣿⣿⣦⣭⣭⣥⣶⡄⠀⠀⠀⠀⢠⣶⣿⠿⣿⣿⣷⣦⣍⠻⣷⣬⡻⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠸⣿⣿⣿⣿⣿⣿⠟⣛⣩⡄⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣸⣿⣿⣌⢻⣷⡜⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢿⣿⣿⣿⡏⣴⣿⣿⠿⠟⡀⠀⠀⠀⢸⣿⣏⣾⡟⣻⣿⣿⣿⠟⣸⡿⢃⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠘⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⡀⠀⠀⢸⣿⡿⢿⣿⣿⠻⣿⡇⣾⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠘⠿⠟⠿⠿⠟⠓⠿⠃⠻⠇⠀⠘⠿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⢠⣦⣴⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⠶⢦⣤⣤⣤⣤⣤⣦⣦
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠶⠙⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣾⣿⣷⣌⠻⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣁⣘⠻⣿⣿⣿⢃⡴⠞⣛⡛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇⣾⣵⣿⣿⣿⣿⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣿⣿⣿⣿⣿⣛⣛⣛⣿⣿⣇⢻⣿⣧⠻⣿⣿⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣷⣌⡛⠿⠿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁⠀⠈⠻⣿⣿⣿⣿⣶⣶⣶⣶⣶⡄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⠀⢠⠈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠈⣧⠀⡀⠀⠀⢠⡶⠋⠁⠀⠀⢐⣶⡿⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣦⣤⣿⣼⣧⣤⣤⣿⣤⣤⣤⣤⣤⣌⠁⢈⣿⠿⠶⠆

⠀⠀⣀⠀⢀⠀⡀⠀⠀⠀⠀⢀
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣻⣵⠶⡗⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⣤⢾⣿⣿⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⣤⣶⣶⣿⣿⣿⠚⣏⡟⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⢞⣉⣴⡆⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⢿⠟⡍⡤⣾⣵⣿⣿⣇⣠⣴⣶⣾⣿⡿⡿⢿⣿⣿⣿⣻⣻⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⣠⣴⣾⢿⣿⢿⣿⣻⠿⠾⠟⢀⣴⣽⣶⣾⣯⣿⣻⡿⣯⣿⣷⣟⢿⣻⣿⣽⡿⠽⠯⣟⡿⠞⠻⢦⡄
⠀⠀⣀⡴⡞⠛⣿⣟⣿⣿⣮⣾⢿⣿⣿⣶⣶⣿⣿⣿⢏⣼⣝⣿⣿⣿⣿⣿⣧⣟⠛⡫⠎⠁⠀⠂⠄⡒⡎⢧⡚⡴
⠀⠉⠉⠉⠉⠋⠇⠸⡯⣿⣿⣿⠾⠻⠯⢟⣿⡧⣿⣿⣿⣾⣿⡿⠀⢿⡷⢷⣿⣍⣩⢃⠀⠌⠀⠈⠀⠀⠀⣦⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠲⣡⣐⣿⣼⣿⡿⠿⠿⠛⠋⠁⠄⠀⠲⠿⣷⡊⢈⢈⢀⢁⣀⠤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠼⠉⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⠿⢷⣤⠤⠤⠄⠀⠀⠀⠀⠀⣀⠤⣄⣀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡦⣤⣤⡤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠰⠖⠛⠉⠉⠉⠉⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠈⢷⡀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢀⡻⣔⠛⠉⠉⠉⠉⠛⠶⣄⡀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢠⠞⠁⠙⣦⠀⠠⣤⣄⡀⠀⠈⠳⡄⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡏⠀⠀⣀⠨⣷⣇⠀⠈⠹⣆⠀⠀⢹⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⡏⠀⢨⡝⢹⡆⠀⢹⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠸⣇⠀⠀⠻⣄⠀⠉⠉⠀⣠⠟⠀⠀⣸⠇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠙⣆⠀⠀⠈⠛⠲⠶⠛⠉⠀⠀⣰⠏⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠳⢦⣄⣀⠀⠀⣀⣀⡴⠞⠁⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠁