Occipital bone Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠦⠖⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣟⣷⣿⣧⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣋⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣟⣙⣫⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⣿⠧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠑
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠂⠀⠁⠂⣀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⡶⡡⠖⠉⠀⣀⠀⠀⠀⠀⠀⠀⠴⣄⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣾⣃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀
⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⡌⠿⣿⡷⢶⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠡
⠀⠀⠈⢿⣿⣿⣿⣿⣿⠋⣵⣦⡿⣷⣦⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⠄⠀⠀⠄
⠀⠀⠀⠸⣿⣿⣿⣿⢿⣿⣿⡿⡀⠉⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢶⠈⠉⠲⠀⠁
⠀⠀⠀⠀⠸⣮⣹⣋⣾⣿⣿⣧⡈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢃⠀⠤⠀⢀⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠹⠀⢶⣿⡱⠀⣦⣀
⠀⠀⠀⠀⠀⠸⣿⢹⣿⣿⣿⠟⠁⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠘⠂⢿⠀⡆⠀⠈⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⣆⣰⣾⣿⡷⠏⣤⠟
⠀⠀⠀⠀⠀⠀⢹⢸⣿⣿⣇⢐⣿⠇⢿⣿⡀⠀⠀⠀⣠⣴⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣼⣶⣄⠙⣿⣿⣿⣷⣾⠋
⠀⠀⠀⠀⠀⠀⢸⢺⣿⣿⣿⣿⣿⣻⣿⣿⡿⣤⣤⣤⣠⠿⠄⠐⠛⠀⠉⠀⠀⠀⣤⡄⠀⣀⠀⠀⣘⣿⣿⡟⢻⣷⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⡉⢿⣿⣿⣿⣿⣴⣇⣿⣗⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠀⡿⢦⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣷⠀⠛⠛⠛⠛⠿⢿⣿⣿⣿⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠻⣿⣿⣿⣿⣿⣟⢻⣿⣿⡁
⠀⠀⠀⠀⠀⠘⢯⠉⡄⠀⠀⠦⠰⣶⡿⠛⢻⣿⡇⣄⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣍⡁⠰⠛⠿⠻⠿⠿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠓⠻⢗⡴⢶⣶⡘⡀⢀⠈⠟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⢀⣀⣠⣤⣬⣿⢿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣾⣦⣤⢘⡌⠻⠈⠀⣀⣀⣀⣀⣀⣀⣴⣿⣿⣿⠃⠀⠀⣀⣾⡿⠟⠛⠷⡝⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠘⠀⢀⡀⣤⣀⣭⣽⣿⣽⣭⣿⣿⣿⣿⣶⣶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠃⠊⠙⠷⣤⣀⠃⢁⢘⡋⢨⠉⠛⢿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡖⢶⣆⣠⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⠿⢿⣷⠞⠛⡋⢠⣽⣿⡿⣿⠿⣧⣴⣗⣄⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡐⢟⠋⠛⠀⠀⠀⢀⣴⣶⣊⢀⡰⣿⣶⣴⢨⡌⠟⢷⡍⢿⠛⣧⣴⡶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠏⠉⠀⠀⢰⣶⠾⣷⣿⣿⣿⣛⡯⠁⣾⣿⣿⣾⣿⣶⣾⡃⣠⡄⠙⠈⠛⢋⣞⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠋⠀⠀⠀⠀⠰⢏⣿⣷⣻⣻⣯⣾⣧⣾⠀⣿⣿⣿⣟⣻⣿⡻⠛⠿⢃⡀⠀⠀⠁⠙⠸⠿⣆⡀
⠀⠀⠀⠀⠀⠀⡀⠿⠗⠀⠀⠀⠀⠀⣀⠚⣹⡛⣿⣿⣿⣿⣿⣖⠡⢺⣿⣿⣿⡟⠛⡿⠁⢿⢿⠂⠠⠀⠀⠀⠀⠠⠒⠸⠚⣦
⠀⠀⠀⠀⠀⠲⣿⠁⠀⢀⠀⠄⠤⣃⠿⣩⣉⡾⢻⣿⣿⣿⣿⡃⠀⢀⣽⣿⣿⣿⣷⠃⠐⠈⠙⠁⡓⠀⠄⠀⠀⠀⠀⠀⠀⠸⢷⡀
⠀⠀⠀⢠⣾⠋⢠⣀⣤⣠⣀⠀⠀⠹⡦⢻⣿⣴⣿⣿⣿⣿⡿⣁⠀⢠⣹⣿⣿⡿⢻⣿⡆⠀⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⢠⣶⢶⣷⣾⣿⣿⣿⡿⣶⣿⣾⣶⣿⠟⠿⠛⡛⣿⣥⡛⠘⣤⣤⠀⣌⠉⣀⡙⠃⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢻⡶
⠀⠐⡺⣧⣾⣿⣿⣿⣿⣿⣿⢇⣻⣭⡤⣥⣶⠿⣾⣿⠿⣿⠗⠀⣿⠉⠀⠆⠹⣿⣟⣿⣿⢦⣶⠴⣦⡴⢦⡰⢴⣦⣄⡀⠀⠀⠀⠀⠀⣧⣀
⠀⣶⣗⣿⣿⣿⣿⣿⣭⣾⣶⡿⠯⢯⡷⠈⠙⠉⠉⠉⠁⠀⢂⡰⠀⠀⠀⣀⣀⣈⣉⣁⣈⡈⠙⠀⠀⠀⠻⣴⠲⢄⠘⠻⡟⢤⣀⡀⠀⠈⢫
⢠⣾⣿⣿⣻⣿⣿⡿⠿⠋⠀⣀⣤⣶⣿⣷⣶⣷⢶⢶⣶⣞⣗⠤⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⣿⠷⢶⡤⡀⠀⡀⡉⠀⠀⠘⠊⢱⡿⣆⠀⢸
⢘⣿⣿⣿⣿⡿⣁⣼⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣾⠛⣑⢩⠄⠀⣤⠀⢿⣿⣿⣛⢠⠷⠟⠂⠠⠀⠀⠀⠁⠺⠃⠷⠃⠀⠀⠀⠀⠁⠀⢀⣿
⠛⢍⠽⠿⠉⣼⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣛⠓⠿⢙⠙⠠⠃⠀⢙⠆⢺⣿⣿⠌⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠐⠀⠀⠀⠘⡧
⠰⣿⣷⢲⣾⣿⣿⡿⣿⣿⣿⣿⣿⡟⡱⢪⠍⡑⡘⢧⡍⠈⠀⠀⠀⠀⢸⣿⣿⡂⠁⠀⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡄
⠘⢿⣷⣟⣿⣿⣏⣙⣻⣯⣿⣿⡍⣸⡧⣸⡇⣀⠉⡈⠃⠀⠀⠀⠀⠀⢸⣿⣏⡅⡀⠀⠀⠀⣶⡃⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠂
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠾⣛⠲⡎⣘⠛⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣁⠀⠀⠀⠀⠻⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠻⣿⡷⣽⢿⡯⢿⣷⣬⣻⢏⠀⠄⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠸⣿⣦⣬⣷⡿⢁⣂⡧⠤⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣷
⠀⠀⠀⠀⢹⣷⢸⡶⣢⢷⡧⡡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠻⠿⣧⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⢘⡿⣿⠀⠑⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠁
⠀⠀⠀⠀⠈⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠏
⠀⠀⠀⠀⠈⠻⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠄⠠⠀⠀⠀⠀⠀⠀⠀⠉⢋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣷⠶⠟⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠁⠀⠀⢀⣤⣠⣠⢠⣀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⣠⣠⣄⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣻⡏⠀⠀⠀⠠⣼⣿⠿⠿⣿⣿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣱⠾⠟⠛⣿⠀⠀⠀⢻⣿⣿⣶⡀⠀⠀⠀⠀⠀⣠⠟⣶⡟⠀⠀⢀⣤⣾⣿⣇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⣀⠀⠀⠙⢿⣿⣿⣤⣄⣀⣀⣴⠋⢠⠋⠀⢀⣴⣿⠿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣤⡀⠘⢻⣿⣿⣿⣻⠛⠁⠘⠁⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣣⡀⠀⡙⢿⣿⣉⠁⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⢾⠁⠀⠀⠀⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠐⣃⠀⠉⠀⠀⠀⠀⠀⡟⠁⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇⠀⢹⠆⠀⠀⠀⠀⠀⠀⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣬⣀⣀⠀⡀⢀⣀⣠⣠⠞⠁⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣾⣷⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⣉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣠⣄⣁⣰⣀⣤⣤⣠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣿⣿⣿⠁⣸⣿⣿⣿⡿⠙⠉⠰⢴⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠟⠻⣿⡟⠀⣿⣿⣿⣿⣀⢐⠂⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⣬⣾⣿⣿⣿⣿⣿⡟⠁⣠⣼⡟⠀⠀⢻⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⣋⠀⠀⡀⠀⠛⠿⠿⠿⢽⠎⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣸⣿⣿⣿⡿⠿⠛⠁⠀⠐⠊⢉⣉⠀⠀⠈⠀⠀⠈⣀⣀⣀⡀
⠀⠀⠀⢠⣿⣿⣿⣿⡄⠤⠒⢀⣤⣶⠿⠯⠍⢿⣷⡄⠀⣼⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⢠⣼⣿⣿⠟⣉⣤⣴⣿⣿⠃⠀⠀⠀⠀⠐⠹⣷⠀⣿⣿⣿⣿⠟⠋⠉⠉⠉⠙⠛⠢⢄
⠀⠀⣴⣿⢋⣵⣿⣿⣿⣿⠟⠈⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐
⠀⢠⣿⣥⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠈⠻⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡭⠚⠉⠋⠻⣏
⠀⠀⠀⠈⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⢹⣿⣿⠏⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠻⠟⡀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠐⠶⠶⠶⠆⠀⠀⠲⠀⠀⠴⠟⠃⠀⠀⣠⣷⠀⠎⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⡀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣴⣶⣿⠟⠉⡞⡜⣠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⣶⣆⠀⠚⠃⢹⣥⡀⠀⣀⣀⡀⠀⢀⣬⠁⣠⣶⣿⣿⠟⣿⣷⡀⠞⡜⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢠⡿⠟⠛⠂⡄⠀⠀⢹⣧⡀⠀⠀⠀⡀⡿⠁⠁⢸⣿⣧⡄⠀⣿⣿⠁⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⡀⠀⠸⣿⣿⠗⠒⠈⠁⢰⠀⠰⢾⡏⠀⠈⠃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⣿⠟⠀⠀⠀⠀⠀⡰⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣀⠜⠁⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡟⡻⡿⣿⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠚⠛⠐⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠲⠖⢶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣛⣯⣵⣶⣶⣶⣬⣿⣿⣿⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢊⣩⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠣⠰⠆⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⣨⣷⣿⣿⣿⣿⣿⣟⣱⣿⣯⡙⣿⣿⣿⣿⣿⣿⣿⣿⡅⠙⠄⠀⠷⣄
⠀⠀⠀⠀⠀⠀⣰⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡧⣹⣿⣿⣿⣿⣿⣿⣿⣿⡹⠼⣿⣷⠌⠭⡀
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⡿⠙⢻⡇⣿⣿⣿⣿⡿⠛⣻⡃⠩⠉⠡⠠⢀⠐⠀⠸⢆
⠀⠀⠀⢀⡾⢻⣿⣿⣿⣿⣿⢿⣿⠇⠸⢉⣛⠉⠀⠀⠉⠳⣿⣿⣿⡿⠀⢲⣟⠁⠀⠛⢷⣒⡀⣀⡀⠀⠀⡁
⠀⠀⠀⢈⢁⣿⣿⣿⣿⣿⣿⢿⣿⠂⣾⣿⣿⠄⠀⠀⠀⡀⣿⣿⣿⣿⣷⠘⣿⣰⠖⣶⣤⡀⠁⠀⠉⡄⠀⠈⠧
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⢿⢹⣿⠐⣟⣻⣯⣤⠂⠀⣼⡇⣿⣿⣿⣿⡿⠘⠛⢋⠾⠟⠁⠠⠀⢀⠀⢩⣇⠁⠀⠀⡀
⠀⠀⢠⢻⣿⣿⣿⣿⣿⣿⣿⠸⠟⠸⣿⣿⣷⣧⠶⠾⠛⡁⠘⣿⣿⣷⣦⣤⣤⣤⣤⣄⢠⣶⠶⣦⡙⠃⢉⣁⠀⠀⠁
⠀⠀⠘⢟⣿⣿⣿⣿⡿⣿⣋⣤⣶⠋⠻⠿⠿⠟⠚⠛⠁⠄⠀⣈⣉⣀⠉⠭⣭⣥⡐⣛⣚⣈⠛⠻⠻⠀⢸⣟⠁⢋⠁
⠀⠀⢦⠉⣽⣿⣿⣽⣿⡿⠛⣋⣭⣤⣶⠶⠾⠛⠉⠠⠿⠓⠀⣿⣿⣿⣿⣷⣶⣦⣤⣭⣙⠛⠻⣿⣿⣶⣾⡛⠃⡄
⠀⢀⢜⣶⣿⣿⣿⣿⣿⣿⣿⣟⡛⠋⠁⠀⠀⢀⣠⣶⣄⣀⠀⣿⣿⣿⣿⣿⢿⣿⠭⠉⢉⠛⠓⠂⠙⠿⣿⡉⠂⠸⣤
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣖⠂⢀⣠⣿⣿⠿⢿⡶⢰⣿⣿⣿⣿⣷⠶⠄⣴⣦⡦⠌⠃⠈⠉⢂⠘⠓⠒⠀⣹
⠀⠲⡙⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⢻⣿⣭⡙⠛⢁⡼⠻⣿⣿⣿⣍⣿⣗⣻⣶⡀⠀⠀⠀⠀⠓⠀⠀⠀⣞⠋
⠀⠀⠺⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⠋⡉⣡⣾⡽⠤⢦⠙⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠄⠛⠉
⠀⠀⠀⠀⠚⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠐⢀⣼⠟⠁⠀⠀⠀⠑⢎⢿⣿⣿⣿⣿⣛⡛⠒⠂⠀⠀⠀⠈⣾
⠀⠀⠀⠀⠀⠀⣝⣿⣿⣿⣿⣿⣿⡿⢃⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠂⠛⢹⣿⣿⣿⣿⣶⣦⣤⠤⠀⡔
⠀⠀⠀⠀⠀⠀⠈⢸⡿⢿⣿⣯⣤⣠⠀⢠⣿⡇⠀⠀⠀⠀⠶⠀⠀⠀⠀⢳⣾⣿⣿⣧⣬⣭⠥⢤⣠⡞
⠀⠀⠀⠀⠀⠀⠀⢱⠞⣾⣿⠏⣙⢷⣾⠏⡹⢷⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⢀⣿⣿⣿⣏⣑⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠈⣃⣠⡄⠁⠈⢷⡀⠀⠀⠀⠀⠀⠋⢀⣾⣿⣿⣿⠿⠛⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣾⣿⣶⣶⣴⣶⣁⣤⣟⠻⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣿⣏⣽⠻⠛⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣽⣿⣫⣿⣾⡿⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠴⠐⠂⡀⣠⠀⠤⣠⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡠⠐⡄⠵⠁⠃⣀⠀⠤⢰⡆⡆⠀⠸⢀⠀⡀⢐⡶⠁⠀⣀⠀⠀⠀⠉⢀⡄⡀⠤⣀⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠁⠉⠀⠀⠀⠂⣁⣀⣀⣈⣶⣤⣤⣿⡾⠻⣶⣷⣯⣤⣷⡐⠆⢈⢀⢢⣄⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠼⠐⢄⠀⠄⠀⣀⠀⢲⣿⣿⣿⣿⣿⡟⣿⣿⣟⣠⣿⣽⣿⡟⣿⣿⢿⣿⣿⣾⡁⣨⠃⡀
⠀⠀⠀⠀⠀⠀⠰⡂⠀⠀⠄⠙⢦⣲⣿⣿⣿⣿⣿⡟⢿⢻⡇⣿⣿⠟⠁⣿⣯⠿⡁⢨⡏⢊⢻⡿⢿⢿⣧⡏⡠⣀⠄
⠀⠀⠀⠀⢠⠜⠃⠀⠐⠀⢠⣼⣿⢿⣿⣿⡟⡏⣿⢡⠋⢼⣧⣿⣿⠀⢠⣿⣏⠈⢁⢸⠃⠀⢯⡇⠀⠘⠿⠘⣼⣃⣄
⠀⠀⣀⢰⠌⠀⠀⠀⣀⢲⣿⣿⣿⢸⣿⣿⡷⠇⢿⡏⡄⡾⣼⣿⡏⠀⢸⣿⠾⠀⢸⡟⠀⠀⠀⠆⠰⣾⠶⢾⡿⣷⡻⡂
⠀⠀⠑⠂⠀⠀⠀⢀⣸⣿⣯⢿⣿⣾⣿⣿⣵⠧⡜⠂⡿⣻⣿⣿⡇⠀⢸⡇⠀⡈⠀⠃⢀⠀⠀⠀⠀⢸⠀⠰⣴⡤⣵⡾⣤⣄
⠀⠀⠀⠀⠀⠠⠀⣿⣻⣿⣿⢸⣿⣿⣿⣿⣿⠶⢉⣨⡿⣿⢻⢃⠀⠀⢸⡇⠀⣽⠀⠀⠘⠇⠀⠄⠀⠘⡀⢰⡀⡆⣴⢠⡦⣹⡇
⠀⠀⠀⠀⢀⠃⣼⣿⣟⣿⠯⠒⢙⣋⣭⡵⠶⠟⠛⣋⣁⡁⠀⢎⡛⠚⢈⣡⣤⣴⠶⠾⠛⠛⠿⠻⠷⣶⢶⣬⡀⠇⢼⡋⠗⡈⢿
⠀⠀⠀⠀⢜⣿⣿⣿⡴⠟⠛⣋⡁⠈⠅⠒⠃⢈⣨⣤⣤⣤⣶⣷⣍⠙⠉⠀⢀⣠⡤⢤⢤⠤⡤⢀⡀⠀⠀⠉⠻⠶⡾⣬⡷⢟⠸⣧
⠀⠀⢠⡤⢤⡟⣋⣤⠖⠊⣁⣤⣴⢶⠆⣾⣿⡿⢿⠟⠟⠛⢹⠛⡏⣜⣴⢡⢈⠂⠁⠢⠛⠈⠀⠊⠅⢤⠘⢳⠄⣀⠀⠹⣧⣿⠂⠀⠑
⠀⠀⠸⡴⢾⢓⣡⣤⣤⣾⣸⠟⢁⡄⠑⠁⢀⠀⣾⠀⠂⠀⠀⠈⠁⡤⢰⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠳⠄⢀⡈⠀⡠⠛
⠀⠀⠘⠂⣧⣿⣿⣽⡟⡿⠋⢀⡧⠃⢀⠇⣈⡤⠋⣴⡀⠀⠀⠀⣷⡄⣿⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠂⠀⠛⢠⣽
⠖⠢⠀⡄⢹⣿⣿⡏⢁⢴⠊⡟⢃⣠⡴⠻⠋⠑⠂⠈⠐⠀⠀⡀⢿⣦⠘⢾⡷⠀⣀⠀⠀⠀⠀⡆⠀⠀⠀⠆⠐⢆⠈⠀⠀⠀⣰⠟
⠐⠓⠠⡤⠀⣿⡿⢡⣟⠘⣨⠴⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⡆⣑⣾⣇⠀⠀⠙⠀⢃⣨⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⡀⣠⡜⠁
⠀⢸⣇⠙⡑⠛⢇⣸⠶⣯⠙⡆⢠⣀⠀⠈⠀⣀⠐⡄⡈⣶⣿⠿⠿⠿⢴⡂⠀⠈⠻⡇⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁
⠀⠀⡤⠄⡨⠶⠽⣆⠞⢡⠶⡈⠄⠁⠄⠆⣀⠉⡼⣻⡿⠋⠁⠀⠀⠀⠀⠀⢻⣆⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠘⠢⠀⠀⠙⠋⡴⣿⣠⠋⣠⣶⡖⠢⣤⡀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⠀⠈⠃⡶⢄⠀⣤⢀⣀⠀⠀⠀⢻
⠀⠀⠀⠡⠀⠀⠀⠀⠉⣉⡉⠁⠘⢀⣞⠀⡈⠱⡏⠰⠃⠶⠒⠶⠰⠒⠲⠆⠲⠀⢂⣡⠀⠀⠀⢢⠀⣠⣬⠿⠛⣯⠐⡷
⠀⠀⠀⠘⠈⠀⢀⠐⠁⠈⣿⡿⣀⣦⠙⠢⢕⣤⡇⠀⠚⠃⠖⢰⠒⠒⠲⠐⠒⠂⢰⠙⣦⠀⣀⣀⡙⢿⡇⢰⣿⣿⣿
⠀⠀⠀⠀⠈⣉⠀⠀⠀⣠⣿⣿⡏⢀⡀⠀⠈⡙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣌⣸⢿⠗⣿⢻⣷⣤⡀⠈⣿⣿⣿⡀⣄⠀⣀⡀⡀⢄⣀
⠀⠀⠀⠀⠀⢱⠄⠀⠘⠛⠛⠯⠤⢤⡂⠀⠀⡀⢸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡁⠀⣸⣿⣸⣷⡾⠿⠀⢻⡈⢁⠿⡬⣡⣇⢩⣍⣈⡀
⠀⠀⠀⠀⠀⠀⢫⡄⠀⠀⠀⠀⠀⠨⣷⣶⠉⠻⠈⣿⣿⣦⣄⣀⢀⣀⣀⣠⣴⠏⠀⡴⠞⠛⣧⣼⠋⠀⠠⠢⠄⡤⠤⠰⠤⠤⠰⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠦⠐⠸⣿⣿⣿⣿⣿⡯⢿⣉⢁⣠⢞⣁⠦⠒⠉⢁⢀⣀⡀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⣄⣠⣌⠍⠀⢀⢀⠈⠉⠉⡿⠋⣄⢀⣉⣈⣀⡉⣁⢀⣁⠅⣀⣀⣀⣤⣀⢄⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡉⠋⢿⡈⠌⢉⠉⠈⠀⠀⠀⠀⠀⠀⡰⢀⣄⠄⣠⢠⣤⢠⡄⡤⠜⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⠒⠾⠆⠒⠸⠷⣆⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣿⣤⣤⣥⣤⣥⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠁⡂⠀⠀⠆⠂⠆⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠈⠀⠀⢀⡀⣠⣁⣠⣴⠦⣴⣔⣡⡀⠄⠀⢀⡀⠁⠈⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⣀⣀⣸⣿⣿⣯⣿⢻⡏⠴⡏⡿⢹⢿⢿⣿⣯⣠⢁
⠀⠀⠀⠀⠀⠀⢠⠌⠀⠀⢠⣿⣾⣿⣿⡟⠁⠁⣿⣿⡇⢸⡟⠇⡄⡆⢨⠏⠙⢻⠣⣠⠂
⠀⠀⠀⠀⡀⠤⠀⠀⢀⢰⣿⡟⣿⣿⣿⠇⠀⡄⣫⣿⠁⢸⡻⠀⠀⠀⠀⠀⠠⠦⠦⡽⡶⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⣿⣹⣿⣟⠂⠀⣸⣿⠟⠀⢸⡆⠘⠀⠀⠀⠀⠀⠀⠐⠒⢚⢛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣧⣿⠃⠟⣛⣫⠭⠶⠟⡉⠁⠆⠂⠘⣀⣠⠤⠶⠶⠦⠤⣤⣄⠂⠀⠘⠉⢳
⠀⠀⠀⠀⠀⠀⣻⡿⠗⢊⠫⠉⠁⢐⣀⣠⣤⣤⣶⣶⠈⠁⠀⠄⠰⢠⠠⠀⠄⠀⣈⠙⠶⢴⢰⠈⠆
⠀⠀⠀⠀⢘⣁⠣⢀⢀⣠⣶⠟⠙⠙⠛⠉⠉⠀⠀⠐⠘⣩⢤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡈⠙⠄⢰
⠀⠀⠀⠀⢀⢐⣾⣯⣿⠞⠁⡌⠀⢀⣠⠐⢠⠀⠀⢀⡦⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡇
⠀⠀⠀⣁⠀⠘⣿⡿⠀⡞⢈⣥⠖⠋⠁⠁⠀⠀⢀⢀⣿⠈⠨⠠⠀⠀⠀⠂⠂⠀⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠐⡄⠂⠟⢇⡧⠖⠋⢀⢀⠀⠀⠀⠀⢰⣾⠼⠿⠄⠀⠀⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠚⣻⠂⡰⢂⠀⢈⠐⠀⠁⢠⠊⠁⠀⠀⠀⠈⢦⠀⠠⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠚⠓⠂⠞⡍⠉⠂⠴⠣⢀⡠⣀⠠⢀⠀⠀⠃⠀⠀⢁⠀⢀⣢⢠⡀⡍
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠸⣿⣠⠌⠰⢀⣽⠀⠒⠒⡐⠒⠂⠐⠂⠘⢣⢀⣀⡼⠿⢰⣾⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢃⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⡶⠋⢩⡍⣟⣦⠀⡿⠿⢢⠠⡤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢹⣤⠐⠄⣿⣤⣀⠀⣀⣀⣠⠜⢀⠤⢺⣹⠏⠀⠤⢈⡀⠁⠌⠡⠭⠅⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠆⠘⠿⠿⠿⠛⠛⢥⣴⣡⠀⠊⡉⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⡄⠀⠢⠀⠀⠁⠐⠐⢐⠀⢀⢒⢀⡐⡐⡘⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠒⠟⠰⠸⢦⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠁⠈
⣶⣶⣶⣶⠶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⣶⡶⡶⣶⢶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⢀⣤⣠⣤⣤⣤⢀⣠⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠡⠀⠉⠀⠀⠀⠀⠐⠈⠘⠃⢈⠀⠓⠻⠩⣟⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠁⠀⠀⠀⡶⡆⠀⠀⠀⠀⠀⠱⡤⠦⢤⠀⠀⠈⠰⢹⣯⠙⠈⠩⢟⣟⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠀⠀⠢⢤⣍⡚⠤⡠⣠⡶⠤⢾⢏⡀⣄⢀⡀⠙⠓⠮⡿⠿⢦
⠀⠀⠀⠀⠀⠀⣰⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⣷⡀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠣⠨⣑⡾⣇⣾⣸⣷⣤
⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⢿⣯⣿⣿⡆
⠀⠀⠀⠀⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣇
⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠘⠿⠁⣿⣅⠀⠀⠀⠀⠇⠀⠀⠀⠀⣀⡀⣂⣀⠠⠀⢬⢽⣿⣿⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣿⡯⠀⠂⠀⠀⠀⠘⠂⠉⠂⢸⡟⠶⠁⠀⠂⠀⠙⣿⢻⣿
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠜⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢲⡌⣣⡀⠀⠀⠀⠀⠠⣿⣷⡟
⠀⠀⠀⣧⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⣀⠀⡄⢸⣿⡏⠀⠀⢀⠀⠀⠀⠀⠘⠾⣾⡞⠁⠀⠀⡆⣠⠐⣿⡟⠁
⠀⠀⠀⠙⢻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣤⡉⠂⠇⠘⣿⣇⠀⠅⠁⠀⠀⡀⢄⡐⣂⣿⡗⠀⠀⢰⢴⣿⣼⣿⠇
⠀⠀⠀⠀⠀⠱⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡀⡁⢹⠂⢀⣏⡀⠀⠀⠀⡀⠀⠃⢠⣿⣿⣯⢝⣃⠀⠀⡯⢻⡟⠁
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢘⢅⠦⢛⡛⠁⠈⠉⠙⢷⡄⢂⡠⢶⣧⣷⣿⣷⣵⣟⢧⠑⠀⢸⣿
⠀⠀⠀⠀⠀⠠⡆⠀⠀⠠⡌⣀⣰⡐⠦⠗⠀⠀⠋⢈⡠⠞⠀⠀⠀⠀⠀⠀⠙⢢⡘⢿⣿⢿⣯⡹⠀⠈⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⢨⣷⡖⠚⢁⠀⢰⡆⢠⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡟⠂⡟⣄⣀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣤⣾⠏⠀⣼⣾⣿⡟⠉⢨⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠴⣿⣟⣷⠏⠀⢙⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠹⠆⢰⠙⠛⢛⠁⠀⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⡏⠁⠀⢠⠘⠹⡇
⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣟⣀⡁⢸⠀⠀⢸⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢲⡄⠒⠆⣿⣶⣀⣚⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠈⠁⠈⠛⢿⡄⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠞⡼⡿⠃⣄⣀⣿⠋⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠻⣿⣿⣶⣤⣤⠇⠠⠄⣪⠎⡐⠁⠀⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣤⣼⣿⣿⣿⣿⣷⣾⣯⣤⣴⣡⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⠙⢻⡟⠻⡟⠿⠛⡿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⣤⠀⠷⡇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⡄⠀⠀⢠⣀⣄⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⣶⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⣿⣿⣿⣿⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠤⠀⠀⠀⢠⡀⠀⠀⠀⣠⣤⣤⣄⡀⢀⡀⠀⠀⠻⠿⠿⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡾⠋⠀⠀⠀⠀⠀⠘⠛⠐⣦⣀⡈⠿⢿⣿⣄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⡀⢺⣿⣿⣿⣦⡈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣼⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠐⠂⠈⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧⠀⠀⠀⠀⠀⠰⠸⢷⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠂⣀⣠⠞⠉⠙⢤⠀⠀⠀⠀⠀⠐⣠⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢰⡀⢳⣄⠀⠀⣰⣶⡖⣸⣞⢉⣤⡀⠀⠀⠠⣥⡀⠀⠀⠰⠻⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣱⡬⠝⠤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠥⠀⣤⣾⣿⣿⣿⣵⡷⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢻⣐⣀⣀⣿⠁⠀⢹⣿⣿⣿⣿⣷⣿⣾⣿⡛⢛⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⣿⣿⣿⣿⣟⡛⠛⠝⣿⣿⣿⣿⣿⠛⠛⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⡿⠟⠋⠉⠉⠙⠿⣿⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢴⣶⡎⠿⣤⣶⣄⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡜⠛⠋⠙⢛⣈⡝⠀⣛⣿⡛⣛⢩⣿⣿⢿⣶⣆⢒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣻⠮⠉⠀⢀⣤⣤⣦⣿⣿⡮⣴⣼⣿⣿⣮⣿⣬⣿⢪⡅⢻⠟⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠋⠁⠀⡄⢠⡻⣷⣾⣫⣿⣿⣷⡶⣸⣿⣿⣿⡿⢿⡿⢻⡿⠅⠄⠀⠜⢻⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠎⠀⢀⠀⡌⠀⠒⣛⢺⣿⣿⣿⣿⣦⠩⢾⣿⣿⣿⢿⣿⠠⣾⡂⢈⠀⠀⠀⠀⠠⠌⡿⢦
⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⡈⠀⠤⣀⠗⣰⣱⣞⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣾⣋⠐⠙⠋⣐⡂⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⡶⡏⣠⣤⣦⣠⠄⡀⡸⠖⢻⣷⣿⣿⣿⣿⣟⢀⠐⠜⢻⣟⡿⢻⠿⠆⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⢀⣾⣛⣽⣿⣿⣿⡿⣻⢿⢿⠿⠿⣛⡟⣩⣽⣷⣏⢘⣶⡆⢰⠌⣆⣌⣁⣀⡀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⡸⣯⣿⣿⣿⣿⣿⣟⣭⣷⡚⣇⡿⠿⠿⠿⠟⠛⠁⠀⠁⠀⠈⠁⠛⠿⠿⠫⠿⠏⠙⠛⢹⠎⣛⣳⣦⣄⡀⠀⠀⠘⢷⣦
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠛⢉⣈⣧⣤⣤⣤⣤⣄⣠⡬⠊⠀⢀⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⠈⡋⠉⠅⠀⠹⠌⣶⣤⡀⠀⣿
⠀⠀⠀⢿⣿⣯⣿⣿⠟⣵⣤⣴⣾⣯⣿⣿⣿⣿⣿⣾⠿⣛⠱⢂⡀⠸⣿⣿⡿⢛⣿⡿⠉⢋⡈⡉⠣⣄⢣⣶⠄⠀⠀⠈⠙⠉⢀⡟
⠀⠀⠀⣿⣿⣿⠟⣵⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⡧⡙⠇⠀⠈⣣⠀⣿⣿⠗⡋⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣇
⠀⠀⠀⢸⣿⣇⣾⣿⣿⢿⣿⣿⣿⣿⠛⠻⢇⠽⡘⠣⣬⠈⠀⠀⠀⠀⣿⣿⡇⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⣿⣾⣿⣿⣯⣝⣿⣿⡿⠟⠻⠗⢸⠂⡌⠁⠓⠀⠀⠀⠀⠀⣿⣿⠄⠉⠀⠊⠰⡟⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠘⢿⣿⣿⢿⣿⣿⣿⣟⠇⢠⠓⠒⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⠆⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠈⠹⣿⡟⣻⣾⣽⠿⡇⠘⢁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⢻⣿⢿⡛⣿⣞⡛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠑⠍⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠏⠁⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢲⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣖⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠈⠙⣽⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠛⠀⠀⠀⠀⠀⠐⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⢠⣤⣤⣧⣤⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠇⣀⣴⣶⣧⠀⠀⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⣴⢋⣿⠃⠀⠀⢀⣾⡟⠛⠛⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠈⠁⢻⡀⠀⠈⠻⣿⣿⣆⠀⠀⠀⢀⡼⢃⡿⠁⠀⣰⣾⡿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢷⣆⠀⠸⣿⣿⣿⣿⡶⡞⠡⠞⠀⣠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⡀⠈⠻⣿⣿⠃⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⡞⠁⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠚⡃⠈⠁⠀⠀⠀⠀⡼⠉⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡀⢹⠂⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠦⠤⠀⠤⠴⠴⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠒⠲⢦⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠄⠠⠄⠀⠀⠀⠀⠀⠀⢀⣀⣉⣍⣉⣛⠛⠿⠿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⡿⠿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⡏⠁⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠂⠀⠠⢄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣧⣹⣿⣿⣿⣿⣿⣿⣷⣄
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡆
⠀⠀⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣽⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠉⠗⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢶⣤⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⣉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠀⠀⠀⠀⣴⣦⣶⣾⠆⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣽⡿⣶⡨⣿⣿⣿⣿⣿⣿⡗⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⣺⣿⡿⠟⠁⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⢻⣿⣿⣿⣧⣼⣿⣿⣿⡿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⠈⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠋⣴⣿⣿⣿⣅⠈⠈⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡿⡟⢆⠀⠀⠀⠀⠘⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡆⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠄⢠⣄⠪⣄⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠋⠉⠉⢻⣷⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠁⠀⠀⠀⠁⠸⣄⠀⠀⠀⢸⣿⣏⡂⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠆⠀⠰⠿⢿⣿⠗⠠⠔⢀⣠⣶⣾⣿⣶⣵⣿⣾⣿⡿⠛⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⢼⣿⡿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠄⠀⠀⠀⠀⠀⢸⣿⠛⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢀⡀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠐⠉⠀⠼⠿⠛⠛⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢀⡀⣀⣰⠤⠄⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣖⣀⣤⣾⣿⣷⣰⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⢹⣿⠉⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣛⢿⡵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣠⡾⠃⠀⠀⢻⡗⠀⠀⠹⠛⠉⣿⣿⠊⠛⣇⠀⠳⢠⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⣿⣋⣩⠭⠉⢠⡾⠟⠀⠀⠀⠀⢾⣷⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠋⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⡁⠈⠀⠉⠈⠙⠲⢄⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣿⣿⣿⡶⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠀⠀⠀⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠸⢆
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠟⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠛⠛⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠛⠁⢀⠀⣠⣶⣮⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⢀⡀⠀⠙⠿⠛⢾⣿⠁⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠡⣏⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣾⣿⣧⣼⡿⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣿⣿⡿⢟⣄⠻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣐⣿⣿⣿⡏⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⣿⣿⠿⠿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⢴⣿⣾⡐⣄⠈⣿⣄⠀⠀⠀⠀⢀⠀⣰⢟⣵⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣏⣧⡀⠙⢷⣶⡖⠒⠃⠰⠋⣻⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠻⠿⠟⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢴⣶⠖⠒⠶⠂⠴⠷⠺⡿⢿⣿⣷⠰⠶⠾⠷⢶⠶⣶⣶⣶⣦⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠖⠀⠀⠐⠁⠂⠊⠀⠀⠈⠂⠀⠛⠁⢠⡀⠈⠕⠫⠃⠞⡿⣿⣿⡿⡿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠚⠁⠀⠀⠀⠀⠀⡆⡆⠀⠀⠀⠀⠀⠀⡈⢁⢆⡤⣬⣀⡀⠀⠈⠒⠁⡾⢹⣏⠙⠗⠸⠉⠻⠷⠿⠛⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡮⠃⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠀⠄⠀⠄⠀⠀⡀⠀⠚⢇⡨⠠⠁⠀⠀⢠⠀⣀⣸⣼⠂⠀⠀⠀⠀⠀⠰⠀⣏⠻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣴⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠃⠘⢻⣦⣅⠘⠛⠈⠟⠛⠋⠛⠛⠓⡼⣬⢧⠤⢶⡀⣤⢀⡙⢉⡧⢘⠈⢳⡄
⠀⠀⠀⠀⣀⡾⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⣿⣄⠀⠱⠀⠘⢀⠀⠀⠀⠐⠜⠀⠀⠈⠀⠰⠎⣷⠿⣷⣷⣅⣾⣿⣶⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠘⣟⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢪⠰⣾⣏⣹⣿⣿⣷
⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢈⣗⠶⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣿⣿⣿⠂
⠀⢀⠶⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢀⢀⣡⠴⣿⣆⠀⠀⠀⠁⠀⢰⡓⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⢀⠀⠀⠈⢿⣿⣿⣿⣧
⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠠⡀⠀⠀⠀⠀⠀⠈⠛⠇⠸⣿⣦⠀⠀⠀⠀⠀⠘⠇⢀⠀⢀⡀⠀⠀⢀⡠⠒⣄⣤⠀⠄⠐⢆⣾⣿⣿⣿
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⡉⠐⠂⠀⠀⠐⠀⠠⠒⠀⠋⠤⢄⣼⠗⣽⡍⠀⠀⠂⠀⠰⠘⣿⡾⣯
⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠓⡆⢸⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⣤⠿⣧⠈⠇⠀⠁⠀⠀⠀⠀⣿⣧⣹
⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠉⠛⠀⢸⣿⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢩⣏⣻⢽⡆⠀⠀⠈⠀⠀⠀⠈⣽⣿⣿
⢸⡄⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⡄⠀⣷⠀⢸⣿⣿⠀⠀⡀⢀⡀⠀⠀⠂⠀⠀⠀⠟⢬⣧⣈⠀⠀⠀⠀⡀⢀⣠⠀⢻⣿⠇
⠙⠓⢻⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢩⣀⡈⠰⠀⢩⠀⢸⣿⣟⠀⠦⡑⠀⠀⠀⠀⠀⢀⢀⠰⠇⣾⣿⣇⠃⠀⠀⢰⣔⣼⣧⣜⣟⣿⠃
⠀⠀⠘⠶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡑⠉⠏⠹⠜⢶⡄⠀⢸⡿⠈⠀⠀⠀⠀⠀⠀⠀⣖⢀⢚⣹⢿⠿⡟⢿⡾⠀⠀⠠⡏⢻⣿⣿⠞⠁
⠀⠀⠀⠀⠙⡧⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣥⠶⠇⠒⣀⣻⠁⠀⠨⠶⠄⣀⡀⠀⡀⠆⠀⠀⢠⡀⣿⣾⠿⠿⢈⣣⡴⠣⠀⠀⢪⣳⣿⠁
⠀⠀⠀⠀⢰⢟⠀⠀⠚⠀⠀⠄⠀⠀⠀⠀⠀⡀⠈⠂⢁⡹⢄⡛⠛⣉⡙⠀⠀⠀⠀⠀⠘⢷⡂⠱⣠⠀⢿⣿⡾⣝⣿⣿⠿⣬⣖⠻⡀⠈⠀⠐⣿⣇
⠀⠀⠀⠀⣶⠆⠐⠀⠀⣤⡈⠀⣠⢤⡐⣄⢘⠃⠀⠀⠘⠁⠈⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⡀⠻⣿⣿⡿⣿⣝⢬⣏⠠⠌⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠸⣧⠀⠀⠀⢺⣅⣮⡭⡜⠇⠐⠀⣀⡀⢀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣸⣿⡧⢀⣿⡎⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠹⣆⠀⠀⣿⣯⠛⠀⣄⢠⣄⣦⣿⣇⠸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⢂⠀⡧⣿⣴⢖⣤⠀⢀⣠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠹⣦⢿⡟⠀⠀⢰⣿⡿⣿⣿⡁⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠿⣿⣮⣽⣿⠁⠀⠀⣹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢁⠈⣷⠀⠀⡞⠛⠓⠛⣏⠀⠀⠰⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠀⠀⠘⠋⠉⠀⠀⠈⡅⠚⠹⣷
⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠢⠃⠀⠀⠠⡟⠀⠀⠰⣧⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡋⢠⢴⠀⢸⣄⣄⠀⢠⠇⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠘⠇⢀⣀⣀⠿⠿⣷⣦⣼⡇⠀⠀⠀⢿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣿⣼⡇⠈⠈⠀⢸⣿⡿⠒⠶⣦⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠙⠻⣷⠀⠀⠀⠈⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣱⠿⡿⠋⠰⠄⣀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠈⢿⣿⣿⣿⣦⣄⣀⣸⡏⢀⠤⠔⣊⡼⢁⡈⠁⠀⢠⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⢹⣿⣿⣿⠿⣿⣿⣴⡴⠶⠟⠉⢀⡎⢀⣠⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⢻⠛⠙⠛⠋⠱⢏⠱⠟⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⡅⠀⣿⣾⡁⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⡀⠀⠈⠉⠁⠁⠀⠀⠀⠀⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⡤⢤⠤⣼⡶⢮⠟⠛⠁
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠔⠀⠀⠂⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⢀⠒⠐⠁⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⢡⡀⢠⠀⢇⠀⠈⠌⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿
⠀⠀⠀⠀⠊⠀⠁⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣿⣿⡿
⠀⠀⠀⠃⡠⠁⠀⢈⡀⢀⠸⠀⠀⠀⣄⠀⠀⠠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠀⠀⠀⠁⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠰⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡤⢀⢞⡁⠢⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠀⠀⡘⡀⠄⠀⠀⠀⠀⠀⠠⠔⠈⠀⠀⠨⠁⠐⠀⠈⠊⠀⡠⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠠⡄⠀⠐⠀⠀⠈⠀⢦⠀⠀⠀⠤⠈⠀⠈⠂⠄⠐⠁⢀⠈⢰⡶⢀⠂⢂⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡣⠿⠿⠿⠓⠀⠂⠉⠛⠛⠛⠛⠋⠁
⠀⠀⠀⣃⢀⡀⡀⢀⠀⠀⣀⢀⢠⠐⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡏⡇⠈⠈⠁⢘⠀⣠⠠⠰⠖⠈⠇
⠀⠀⠀⢫⡗⢓⢊⠉⡁⡀⡡⠠⠄⠓⡈
⠀⠀⠀⠀⠋⠈⠉⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠁
⠀⠀⠀⠀⡄⠀⠀⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠄⠀⠀⠀⠂⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⠿⢿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠉⢻⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡅⣈⡙⠟⠋⠀⠀⠀⠀⠀⣶⣶⣾⣯⡀⠀⠀⠀⠀⢘⣲⣦⣤⡄⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡸⡘⠿⡿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠊⠁⠀⠀⠀⠀⠘⠭⠿⠿⠟⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣬⣉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣠⠏⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠘⠿⠿⠛⠛⠛⠉⠛⠛⠿⣿⣿⣷⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⣀⣀⣀⡤⠴⠚⠉

⣙⣦⡀
⠉⢻⡄⠑⠢⣄⡀
⠀⠀⣷⠀⠀⠀⠹⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⢰⢇⡘⠢⠏⠉⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠶⢤⣀
⢀⡟⠈⠃⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦
⢸⡇⠀⠀⠀⠀⠀⣠⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⣠⡤⠶⠶⣦⡀⠀⠀⠈⢳
⠈⢧⠀⠀⠀⠀⣼⠃⠀⠈⢷⡀⠀⠀⠀⢀⣠⡴⠟⠋⠀⠀⠀⠀⣴⠞⠉⠉⠉⠉⢻⡆⠀⠀⠀⠈⠳⣤⣀⣀⣠⡴⠞⠁⠀⠀⠀⢀⡞⠁⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠈⠳⣄⣀⣰⠇⠀⠀⢀⣠⢿⣄⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⣀⡾⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⣼
⠀⠀⠀⠈⠉⠹⡄⠀⣴⠋⠁⠀⠹⣏⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠓⠲⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠛⠋⠁⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠹⣄⡇⠀⠀⠀⠀⠙⣶⠛⠉⠉⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠘⢧⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠋⢧⡀⠀⠀⠀⠀⠈⢷⡚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠋⠛⢳⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠳⡄⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⠟⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠙⢦⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣤⠞⠁⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣈⠳⡄⠀⠀⠀⠀⠹⣦⠀⠀⢀⣀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠙⢦⡀⠀⠀⠀⠘⣷⠋⠉⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠈⠳⣄⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⣰⠛⢦⣀⣀⠀⠹⣧⡀⠀⢀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠻⣄⣀⣉⣙⣷⣤⣼⣷⠾⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠶⠾⠿⠿⠿⠟⠛⠛⠉⠉⢻⣄⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡻⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠠⡀⢸⣷⠀⠀⠀⣄⣰⡀
⠀⠀⠀⠀⠀⠀⢄⠐⠲⠿⠛⠂⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠀⠀⠀⣹⣆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠈⠻⡉⠀⠀⠀⠀⠴⠦⠀⠀⠡⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⣄⠀⠀⠀⠀⠢⠘⠀⠀⠣⠤⠤⠀⠀⠀⣀⣑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠈⠇⠀⡀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⠻⠔⢠⠂⠈⠲⠀⠠⡀⡅⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠁⡀⠀⠀⠀⠀⠀⢠⣏⣠⣇⡀⠂⠠⡙⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠂⣱⡄⠁⡄⠀⠀⠀⠀⠀⠈⠉⠒⢤⣇⡆⠀⠀⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠙⢤⡀⠀⠙⢿⡆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠄⠀⡀⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠉⠐⠂⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠕⡂⠤⠀⣀⠀⠀⠀⠀⠀⠀⠴⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠐⢶⡿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠱⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⡀⠀⠀⣤⡀⠀⠀⠀⣀⣀⠀⠀⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠈⠈⣰⣟⢇⣶⣾⡟⠀⠀⠀⠀⢁⣿⠀⢠⡆⢰⣟⢧⣦⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠔⢹⣏⡜⠳⠛⠧⠂⠀⠀⠀⠈⣿⠀⢸⣧⢾⣏⠼⡿⠹⠏⠳⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠂⠀⠐⠄⠀⠐⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠐⠐⠐⠐⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣦⠀⢰⣷⣶⣦⠀⠀⠀⠀⠀⢰⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣆⠀⣸⣷⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠂⠸⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⡉⠉
⠀⠀⠀⠀⠀⣸⣿⣿⡆⢠⣿⣿⣷⠀⢠⣿⣿⡆⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⣼⣿⣿⡇⠀⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⡇⢘⣿⣿⣿⠀⢘⣿⣿⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣻⣿⣿⣇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠘⣿⣿⣿⠀⠘⣿⣿⣇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢹⣿⣿⣿⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠘⢿⡿⠇⠀⠻⣿⡿⠁⠀⠻⠿⠟⠀⠹⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠀⠘⣿⣿⠟⠀⠻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢰⠈⠈⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠃⠐⠂⠀⠙⠿⠟⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⠘⠿⠟⠃⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠙⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⠀⣀⡄⠀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠁⠀⠀⠀⠀⢈⣭⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣶⠹⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣷⠀⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⡠⣾⣿⣿⣿⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡈⠛⢿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⣀⠙⢿⣿⣿⡟⣿⡽⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⡽⣿⣟⢿⣿⣿⡇⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣝⢿⣿⣿⣿⣿⣿⣷⣄⠙⢛⣬⠻⣿⣮⣝⣃⡸⠿⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣭⣿⣿⣿⣭⣽⣾⠿⠋⠀⠀⠉⠙⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠔⠀⠐⠒⠉⠉⠐⠊⠉⠐⠢⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠠⡀
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⠀⠀⠠⠤⠔⠈⠁⠒⠂⠈⠀⠉⠀⠉⠑⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⢀⡁⠀⠀⠀⢀⡠⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⣀⠀⠀⠀⠀⠀⢠
⠀⢀⠆⠁⢡⠄⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⢀⡀⠜⠐⠄
⠀⠈⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘
⡰⠁⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠣
⡇⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠤⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔
⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⡀
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⢀⡀⠀⠀⠀⠀⠀⡀⠀⣀⠄⠀⠀⠀⢀⡀⠤⠐⠈

⠀⠀⠀⢀⡀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢀⡀
⠀⠀⡔⠁⣀⠀⠀⠀⡀⠑⢄⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈⢢
⠀⢸⠀⠈⠀⠁⠀⠈⠈⠀⠈⡄⠀⠀⠀⡎⠀⠘⠗⠀⠀⠺⠃⠀⢱⠀⠀⠀⢠⠁⠀⠑⠁⠀⠈⠒⠁⠀⡇
⠀⢸⠀⠀⢄⡀⠀⢀⠄⠀⢀⠃⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠘⡀⠀⢀⠄⠒⠢⣄⠀⠀⡇
⠀⠀⠣⡀⠀⠈⠉⠀⠀⡠⠊⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠈⠁⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠠⠤⠤⠤⠤⠄⠤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⠁⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣶⣦⡀⠀⠀⠀⣴⡏⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢘⡿⣬⡻⣯⡇⠀⠀⣴⡟⡃⢴⣮⣜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡜⣻⣟⣽⡄⠀⠀⠀⠀⠙⠺⠿⠏⠁⠀⠀⠀⠉⠛⠛⠿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠸⡿⢫⠙⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠾⢦⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⢀⠀⠀⠀⣀⣤⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡺
⠀⠀⠀⠀⠾⢿⢿⢿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠁
⠀⠀⠀⠀⠐⢋⣟⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢯⣛⠷⡙⠛⢦⡀⠀⠈⠙⠂⠀⠀⠀⠈⢶⡀
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢀⡀⠉⠢⣄⠀⠀⠘⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀
⡅⠀⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⢹⡗⢦⣀⠀⠈⠻⣦⣀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄
⠃⠀⠀⠀⢀⡴⠻⢻⣽⠍⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣀⠈⠻⢷⣄⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀
⠘⠀⠀⠀⣌⣥⣿⡾⡿⠿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣤⣀⠝⢿⣤⡘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠂⠀⠀⠘⠮⡻⠁⣷⣧⣾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣄⡙⢿⣦⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠉⠈⠛⠋⠙⠛⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡞⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡿⣯⠾⣿⡇⠀⠀⠀⣠⢶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠒⣷⣢⡿⠃⠀⠀⣾⣿⣌⣏⡞⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠁⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⣬⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠴⠋⠉⠀⣰⠿⠿⢿⡿⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠔⢁⣤⣶⣶⢌⠻⢷⣊⠈⠙⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⢠⢣⠀⡀⢀⡠⢂⣉⣤⣾⠻⣿⣿⣿⣿⣿⣿⣦⡝⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣇⣾⣿⣵⢋⠴⠋⠈⢿⡟⠆⠸⣿⣿⣿⣿⣿⣿⡛⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⡈⣱⠊⠀⣀⠀⢈⡇⠀⠀⢹⣿⣿⣿⣿⣿⡇⠏⢹⡀
⠀⠀⠀⢀⣠⡴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣋⠘⠛⠁⠙⠛⠛⠛⠀⣤⣾⣿⣿⣿⣿⣯⠇⢸⣽⠃
⠀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⣶⠶⠆⠀⠂⠀⠀⠉⠻⣿⣿⣟⣿⡏⠀⣾⡏
⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢻⣿⣄⣀⣀⡀⢀⣀⠀⠀⠀⠙⣿⣿⣿⡶⠁
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡔⠀⠀⠀⠀⠀⠀⠀⠻⢦⡽⡿⠋⠉⠁⠀⠀⠀⢀⠀⠀⠙⣷⠟⠁
⢈⠻⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠈⡀⡺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⢀⡀
⠈⠻⣦⣬⡉⠛⠛⠛⠛⠛⠛⠛⢉⠉⡤⠀⣾⠀⠈⠀⠀⠀⠀⠀⠀⠸⣤⢀⠀⡀⠀⠀⠀⠀⢶⠀⠀⠀⠈⣧⣄⡠⠒⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⣷⠄
⠀⠀⠈⠙⠲⠼⠄⠀⢀⣃⠈⠈⠄⠐⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢄⣠⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣄⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠦⣤⣤⡄⠀⠀⡀⠀⠀⠀⠙⢦⣤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⠀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠉⠉⠁⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠄⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠄⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡠⠤⠔⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠉⠉⠀⠉⠀⠀⠂⠀⢀⠈⠐⠀⢀⠀⠈⠐⠀⣆⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠙⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⡈⠀⠄⠀⠀⠑⠠⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠘⢆⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢐⠀⠀⠀⠐
⠀⠀⠀⠀⠠⠁⠀⠀⢀⠀⠀⠈⠳⣄⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢤⠤
⠀⠀⠀⠠⠁⠀⠀⠠⠂⠀⠀⡠⠁⠀⠙⠢⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⣀⣀⡀⠠⢴⣢⠄⠴⠀⠡⡀
⠀⢀⠔⠁⠀⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠤⠀⠒⠀⠉⠀⠀⠒⠠⠀⠈
⠐⠡⣤⠐⡈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢃⠀⠀⠀⠀⡀
⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡁⠐⠀⠈
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠉⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠄⠂⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⠉⠙⢍⠉⠉⠉⢹⠋⠉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡩⠝⢃⣀⠤⠔⠂⢉⣩⣽⠇
⠀⠀⠀⠀⠀⠀⡰⠋⠑⢆⠀⢀⣠⡾⠷⠞⠛⠛⠛⠋⠉⠉⠉⠉⢉⣉⡁⠤⠤⠐⣒⣢⡶⠶⢖⣂⠤⠔⠒⠉⣀⣠⡤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠲⠥⠤⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⢉⣡⣤⣶⠖⣻⣭⠭⠔⠒⠉⢁⣀⣤⠴⠶⠯⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⠁⣀⣤⠴⠚⣋⠭⠕⠂⠘⢇⣀⣠⣤⡶⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠙⢇⠀⣀⡤⠖⠋⡡⠔⠊⠉⢢⠀⢀⣠⣴⠟⠋⠁⠀⡽⠀⠀⢀⣠⣴⠖⠒⠒⠒⠤⣀⠀⠈⠢⡀
⠀⠀⠀⣀⠔⠒⠉⠁⢣⠀⠀⣀⣠⠜⠋⣁⡠⣤⠊⠙⢢⠀⢀⣰⠛⠹⢧⣤⣤⠤⠶⠊⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠘⢆
⠀⢀⠞⢧⠀⠀⠀⢀⣴⡷⠟⠋⠀⡔⠹⠂⠀⠿⠶⠶⠾⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠴⠶⠶⠛⠀⠀⠀⠘⡄
⢠⠃⠀⠀⢑⣤⣶⢿⠏⠀⠀⠀⠀⠣⢤⣤⣤⠤⠤⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠘⠒⠚⠛⠉⠉⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣤⡄⠒⠒⠒⠠⢄⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠧⠤⠤⣄⡑⡄⠀⠀⠀⢨⠈⠁⠀⠀⠀⠀⠀⢹⡆⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⣪⠍⠉⠀⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠛⠁⠀⠀⠀⠀⠙⠲⠦⠤⠤⠴⠖⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢇⣀⣀⣀⣀⣀⣠⠴⠋⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢀⠤⢒⣪⣽⣶⣿⣿⠿⣿⣿⣷⣄⠀⠀⠀⢀⣤⡖⠒⠒⠒⠒⠠⣄⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠘⢧⣑⠤⢄⣀⣀⣀⡠⠤⢐⣪⣵⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⠀⣇⠁⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⣶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡒⠤⣀⣀⠀⠀⠀⠀⣀⡠⢄⣪⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣶⣶⣿⣿⣶⣶⠿⠟⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠕⠊⠁⠀⠀⠀⠀⠀⠀⠈⠐⢴⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡤⠴⠖⠒⠀⠭⠍⠉⣉⠀⠀⠐⠛⠿⢿⣿⣿⣿⣷⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠀⠒⠂⠤⢀⠀⠈⠐⠢⢄⡀⠀⠈⠙⠻⢿⡆⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠈⠑⠠⡀⠀⠀⠉⠂⢄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠘⣎⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⣀⠑⠠⢀⠀⠀⠈⠢⢄⠀⠹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠰⠀⠈⢦⡀⠀⠁⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠍⢒⠤⣀⠀⠁⠢⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⠃⠀⠀⠀⡹⢦⣀⠀⠀⠈⠑⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⢲⠦⢦⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠠⠂⠀⠀⠀⡰⠁⠀⠈⠓⢆⠤⢀⠀⠀⠀⠈⠐⠀⠠⠄⣀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠴⢪⢄⡠⠎⡄⠻⣿⣿⣿⣿
⣿⣿⣿⡿⠁⠀⠀⢀⠔⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠐⠀⠭⢀⡒⠀⠀⠀⠠⠀⠀⠀⠀⠐⠉⣉⢠⣄⠒⠋⢉⣀⠠⠴⣀⠀⠉⠛⠛⠋
⣿⠿⠋⠀⢀⠠⠂⠁⠀⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠉⠀⠉⠁⠀⠈⠉⠁⠉⠉⠐⠠⠀⠀⠀⢀⡀⣤⠔⠊
⣿⣿⣷⡮⠥⠄⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⣆⡀⠀⠀⢀⣠⣾
⣿⣿⣿⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠹⣿⣿⣿⣿⣿⣿
⣿⣿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣤⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀⠀⠀⠤⠀⠀⠤⠀⠀⢀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠪⢒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠐⠒⠶⣶⣄⡀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠉⠻⢷⡂⠤⠒⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⢼⡇
⢀⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢸⣦⡀⠀⠀⠀⠀⢀⡠⠛⠲⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⡈
⡟⠁⠀⠀⠘⡄⠀⠠⠄⠀⠀⠀⠀⣻⠀⠈⠳⡀⠀⢠⠔⠁⠀⠀⠀⠀⠙⢄⢀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠁
⢹⡄⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢳⣤⣶⣷⣦⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁
⠀⢹⡄⢰⣄⢤⣼⡄⠀⠀⠀⠀⠀⣸⣽⡄⠀⣀⢀⠆⡶⣄⡀⢐⡤⠀⣀⣀⣨⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠃⠀⠀⠀⠀⠀⣰⠄
⠀⠀⢻⣿⣷⡈⢿⣿⡆⠀⠀⠀⣠⡿⠻⠃⠚⠉⠉⠈⠐⠋⠻⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠟⢋⣡⣤⣶⠄⠀⢀⣠⡴⠞⠋
⠀⠀⠀⢿⣿⣷⡈⢻⣿⡄⠈⠈⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠉⠣⠛⡄⠀⠀⠀⠀⠀⠘⣧⣴⣾⣿⣿⣿⡿⠷⠛⠉
⠀⠀⠀⠈⢿⣿⣷⣼⠛⡇⢀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⢸⡉⠀⠀⠀⠠⡙⣶⣤⡀⠀⠀⠀⣀⣀⡀⣠⡴⠔⠋⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⢢⣼⠿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣿⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣠⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡱⠂⠈⠛⢿⣿⣤⣌⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢆⠀⠙⠻⣿⣧⣄⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⢀⢂⢀
⠀⠀⠀⠀⢺⢆⡀⠀⠀⠀⠀⣀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠢⣍⣿⣷⣏⣍⣇⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⠔⢱⡿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠀⠉⠁⠄⠀⠀⡀⠀⡀⠀⠀⢀⠃⣔⣼⢕⠌⠀⠈⡉⠑⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⡀⠄⠂⠈⠀⡠⢤⡤⢓⠶⡰⢆⠆⠀⠀⠪⢨⠛⢩⠢⡀⢘⣈⣿⠗⢁⠔⡷⠊⡝⢆⠀⠀⠀⠀⠈⠁⠒⠤⢀
⠀⠀⠀⢀⠄⠈⠀⡈⠀⠀⠀⠈⠈⠓⠶⡟⣓⠘⠀⠀⠀⠀⠸⡃⢀⠾⢍⠃⢀⡇⠀⠈⠢⠥⡀⠴⠻⠿⡍⠗⠀⠀⠀⠀⠀⠰⢉⠢⣄
⠀⢀⠔⠁⢀⠄⠒⠉⡒⢄⠀⠀⠀⠀⠀⠒⠁⠃⠀⠀⠀⠀⠀⠀⠀⢈⢏⠀⡾⠯⠵⠀⠀⠀⠈⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢂⠈⢳⡀
⢀⠂⠀⠀⢅⠀⡎⡣⢄⣈⣀⣔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣬⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠙⠢⠬⠞⠃⠙⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⡰⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣤⣤⣀⣀⣀⣠⣴⡿⠁
⡇⠀⠀⠀⡠⢆⡠⢄⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⢪⣇⣀⡀⠀⠀⠀⢀⣾⡻⠙⠁⢀⠁⡠⠾⠙⠛⠙⠻⠿⠛⠁
⡇⠀⢰⢉⠤⡢⡝⠆⡖⠦⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠳⠀⠀⠕⣹⡿⠛⠟⠷⣆⠀⠀⢯⡀⠀⣴⡡⡀⡑
⢻⡄⠀⠑⠢⠤⠾⠒⢁⡠⣄⠀⣀⢔⡽⠧⠕⠀⠀⠀⢀⠁⠃⠀⠨⢪⠏⠀⠀⠀⠀⠀⡆⠀⠠⠷⣌⠓⠈
⠀⠻⣦⡀⠀⠀⠀⡜⠁⢰⠦⣹⡆⡄⠀⠜⠀⠀⠀⠀⡈⠀⠀⢔⢔⣝⠀⠀⠀⠀⢀⡔⣀⣤⠄⠀⠈⢢
⠀⠀⠙⣷⣤⡀⠀⠣⠤⢩⡭⠾⠉⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠⣠⡞⠀⠉⠁⠐⢊⠙⣎⣉⠉⠉⠀⠀⠈⣧
⠀⠀⠀⠀⠊⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢄⡞⠀⠀⠀⠠⠐⠀⡠⠧⠀⠀⠀⠀⠀⣠⣯⠄
⠀⠀⠀⠀⠀⠀⠘⠋⠽⣿⣶⣤⣄⣀⠂⠄⠄⠀⠌⠀⠠⣢⠟⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⣠⣴⠟⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣷⣶⡞⠀⢀⣰⡋⠀⠀⠤⣉⠃⠀⠀⠀⠀⢀⣠⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠁⣀⣴⠿⠛⣿⣽⢶⣶⣶⣶⣶⢾⡽⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⠞⠀⠠⡈⠁⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣋⣴⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠘⢿⣷⣄⠀⠀⣴⣾⣿⠿⠿⠿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠻⣿⣦⣼⣿⠋⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⣰⠟⠁⠀⠈⢻⡄⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠻⣿⣆⠀⠀⠀⢻⡄⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠹⣿⣧⠀⠀⠀⠛⠶⠶⠾⠋⠀⢀⣤⠶⠶⣤⡀⠀⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠈⢿⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⢿⣀⠀⠀⣀⡿⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⠀⠀⠀⣘⣿⣷⣦⡀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠋⠙⣿⣷⣴⣾⡿⠟⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⢶⣄⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠋⠁⠀⠀⠀⢿⣿⡛⠁⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢹⡇⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠁⢀⣤⣤⣀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠈⣻⣿⣤⡀⠀⠀⠀⠀⠹⣧⡀⠀⣀⣼⠃⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠸⣇⠈⠉⠻⢦⣀⠀⠈⢻⣿⡄⢀⣠⣴⣾⡿⠟⢿⣿⡄⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠹⣦⡀⠀⠀⠙⢷⡀⠀⢿⣿⣿⠿⠛⠁⠀⠀⠀⢿⣿⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠈⠻⣦⡀⠀⠀⢻⣆⠸⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣰⠟⠁⠀⠉⢻⡄⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠈⠙⠷⣦⣤⡿⠀⠙⣿⣧⣄⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿⡄⠀⠀⠀⣸⡇⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣼⣿⠁⠀⠀⠈⠛⠶⠶⠶⠋⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠟⠛⠛⠋⠉


⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⡶⢶⣶⠶⠶⠶⠶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣒⠒⠒⠒⠒⠒⠒⠲⠶⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠚⠧⣬⣉⣉⡉⠙⠛⠛⠛⠛⣉⣉⣉⡽⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡿⢶⡶⢦⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⡶⢾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡧⢸⡇⢸⠀⣿⠀⡇⢸⠁⣿⠀⡇⢸⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠾⢷⣼⣤⣿⣤⣧⣼⣥⣿⣤⣷⡾⠷⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢾⣷⣶⣶⣶⣶⣶⣶⣾⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣈⣉⠉⠉⠉⠉⣉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢉⣛⠛⠛⠛⠛⠛⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠀⠀⠀⠀⠀⣠⡾⡛⢳⡄
⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣤⣀⣀⠀⠀⠀⣿⠀⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠀⣿⡀⠀⠀⢠⡾⠻⢌⢁⣴⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⢈⡇⠯⠝⣧⠀⠀⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⡆⠀⣿⡀⠀⣨⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣶⣶⡾⠋⠀⠀⠙⠻⠿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⠿⠛⠁⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠫⠭⢻⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣾⣶⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡄⢠⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣁⣤⣾⣿⣿⣿⣿⣿⣦⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⡆⠀⢀⣀⣥⣄⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⣀⠀⣙⡂⠀⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠿⠿⠿⡿⠿⠿⠿⠛⠋⠙⠛⠃⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣡⣤⣠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣾⣿⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢋⣿⣾⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⢫⠿⢀⡦⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡛⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⣄⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⣷⣾⣷⣦⣤⣤⡄⠀⠀⠀⠀⡐⠀⠀⠰⣿⣿⣷⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠈⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠰⡇⠀⢚⣾⣿⣿⣿⢻⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣧⣄⣀⣼⣿⣏⣉⣁⣀⠀⠀⠀⠀⠀⢀⢀⣾⣿⣿⣿⣇⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠿⢷⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⠓⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡜⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣿⣯⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⢟⣿⣿⣗⡼⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⣼⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⢸⣿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⡇⢸⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⠘⢿⣿⣿⠉⣿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢿⣿⣿⣿⣿⣿⣿⣿⣉⡉⠉⠙⢻⣿⣷⠘⣿⣿⣶⣿⡆⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠿⣿⣿⣿⣿⣿⣿⡇⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠃⠙⠻⠛⠛⢻⡀⠈⠙⠻⠿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⢠⣶⡆⢰⡦⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⣰⣶⠀⣲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⣼⡟⠠⢏⡀⢠⣿⠁⣠⠖⣠⠄⢀⡤⠀⢰⣿⠀⠀⠀⢀⣿⠃⠴⣋⠀⢠⣤⠀⣠⠖⣠⠄⢠⡀⢠⡆⠀⣿⠇⢠⡆⣠⡄
⠀⠀⠀⠀⠀⣿⠁⠀⢸⡇⢸⡇⠀⠙⢸⣏⠀⢿⡅⠀⢻⡟⠀⠀⠀⢸⡟⠀⠀⣿⠃⢿⡁⠀⠙⢸⣏⠀⢿⡁⢻⡁⢸⣿⠀⠈⠛⣸⡇
⠀⠀⠀⠀⠀⠛⠀⠴⠋⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠛⠠⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⢀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⣀⣠⣘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⠀⣴⣿⣿⣿⣿⣷⡄⣰⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⠇⢻⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠋⠁⠀⠈⠙⠻⠿⠟⠋⠀⠀⠙⠻⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠉⠁⢄⠀⠀⣴⣾⣿⣿⣶⡄⠀⣠⣶⣿⣿⣿⣦⠀⢀⣴⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠆⠊⠀⠀⠙⠻⠿⠿⠟⠁⠀⠈⠻⠿⠿⠿⠋⠀⠀⠛⠿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⡀⠀⣠⣶⣶⣷⣶⣄⠀⢀⣴⣶⣿⣷⣦⡀⠀⣠⣶⣾⣷⣶⡄⠀⢠⣴⣾⣿⣶⣤⠀⢀⣤⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠠⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠃⠈⠻⣿⣿⣿⣿⠟⠀⠙⢿⣿⣿⣿⡿⠃⠈⠻⣿⣿⣿⣿⠏⠀⠹⢿⣿⣿⣿⡿⠁⠘⢿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠄⠠⠀⠄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⠻⣿⠟⠉⠁⠀⠀⠀⠉⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢀⡴⠖⠓⢶⡄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠈⠳⢦⣤⠾⠃⠀⣠⣤⣤⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠹⡆⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠹⣄⣀⣀⣴⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⣆⠀⠀⠀⠀⠀⠀⣈⣧⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠙⣦⠀⢀⣠⡴⠛⠉⠻⣧⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠙⢷⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⡀⠀⠀⠀⠀⠘⢿⣏⠁⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣸⠇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠰⣿⣿⣷⣦⡀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⣹⣧⡀⠀⠀⠀⠀⠈⠛⠒⠚⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠙⣿⣿⣿⣿⣧⡀⠀⠀⢹⣧⣀⣠⠶⠛⠉⠈⣷⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠻⢿⣿⣿⣿⡆⠀⠘⣿⠋⠁⠀⠀⠀⠀⢻⠀⠀⠀⠀⡾⠉⠀⠈⢻⡄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠉⠛⠿⠋⠀⠀⠘⠷⣄⡀⠀⠀⠀⣾⠀⠀⠀⠀⢷⣀⠀⢀⣼⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⡿⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣄⣀⣀⣀⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣾⢿⠿⢶⣦⣄
⠀⠀⢩⠙⢆⠐⠈⠙⢿⣦⡀
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⢄⠙⢄⠀⢄⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠑⠀⠑⠀⠀⠠⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠐⢀⠀⠀⠘⢟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣈⡂⡀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠉⠳⣄⡀⠀⠀⠀⢀⣤⠚⢿⣮⡿⣍⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⡈⠉⠉⠉⠉⠉⣠⣼⣿⣿⣿⣮⡳⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠈⠀⡀⢀⠔⢂⡴⠣⣼⣿⣿⣿⣿⣿⣿⣾⣶⡟⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠰⢀⠔⣀⣴⠏⠀⠠⣈⠺⣿⣿⣟⡿⣿⣵⣿⣾⣷⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡀⠒⣠⡾⣿⠿⡃⠀⠀⠀⠾⣫⠛⡥⣾⣿⣿⣿⣿⣿⣯⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢀⣠⣾⣿⡇⢻⣮⡏⠀⠀⢀⡴⠐⢋⣴⣽⣿⣿⣿⣿⣿⣻⣿⣿⣿⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣁⣴⣾⣯⠙⢻⣧⢸⣿⠖⣠⡴⠋⣠⣾⣿⣿⣿⣿⣿⡿⣷⣶⣝⣾⣿⣿⣷⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠘⠈⠻⣿⣶⣂⡙⢿⣷⠞⢛⣥⣾⣿⣿⣿⣿⣿⣟⣿⣾⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠄⠈⢿⣿⣶⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡀⢿⣿⣿⣿⡿⣿⣿⣯⣻⣿⣿⣿⣿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣧⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣮⣽⣿⣿⣿⣿⣿⣿⣟⣾⣯⡻⣿⣿⢾⡹⣿⣿⣿⣿⣿⣻⣿⣿⣽⣮⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡫⡻⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣝⢾⣻⣿⣯⣿⣿⣿⣿⣻⣮⣷⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⡿⣿⣽⣟⣻⣿⣷⣿⣿⣿⣿⢿⣿⣿⣿⣷⡟⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⣿⣿⣿⣧⡻⣯⡻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣗⢿⣿⣿⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣻⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠄⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⡀⠀⠀⢈⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢅⡀⠀⠀⠉⠙⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⠴⠖⠒⠂⣠⣄⠀⠀⠀⠀⢀⡤⠶⠒⠒⠲⢤⡀⠀⠀⠀⢀⣠⠤⢤⡤⡤⢄⡀⠀⠀⠀⠀⣠⠴⠒⢒⡲⠦⣄⠀⠀⠀⠀⠀⢀⠤⠆⣶⠦⣄
⠀⠀⢰⣁⡤⠴⠚⠉⠁⢀⡇⠀⠀⢠⡗⠒⠒⠒⠒⠒⢒⣿⠀⠀⠀⣿⡀⠀⠘⡐⠀⠀⢹⡆⠀⠀⠀⡷⣦⣝⣃⣀⣠⡴⡇⠀⠀⠀⠠⣟⠓⢲⡟⠒⣨⡇
⠀⠀⠘⢝⡓⠲⠶⠖⢚⡻⠁⠀⠀⠘⢏⡓⠒⠒⠒⠚⣉⠜⠀⠀⠀⠳⢌⣙⣛⣓⣒⡨⠝⠀⠀⠀⠀⠳⣄⣀⣀⣀⠤⠔⠃⠀⠀⠀⠀⠳⠅⠛⠻⠿⠋
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⣀⣀⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⢀⢜⢧⠀⠀⠀⠀⠀⠀⡠⢆⠀⠲⡄⠀⠀⠀⠀⠀⡴⡡⠒⠒⠪⠣⠀⠀⠀⠀⠀⠀⢠⢮⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠕⠁⢸⠇⠀⠀⠀⠀⢸⡀⡨⢶⠁⢸⠀⠀⠀⠀⠀⡗⡀⠀⠀⢀⢆⠂⠀⠀⠀⠀⠀⡇⠀⠙⠦⣄⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡊⠁⢀⣠⠎⠀⠀⠀⠀⠀⠀⠙⠤⠤⠗⠁⠀⠀⠀⠀⠀⠙⠾⣦⣐⡭⠊⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⣀⡠⠄⡀⠈⠉⠉⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢤⡀
⠀⠀⠀⢀⣴⣊⠁⠀⠀⢈⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⢀⢿⠀⠀⠀⢀⡠⠒⠉⠀⠈⠃⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⣠⡚⠁⠀⠀⢀⠇
⠀⢀⡔⠁⠀⠈⢳⢀⡴⠋⠀⠀⠀⣠⠊⠙⢵⣄⢀⠴⣫⠞⠀⠀⣰⣟⡳⣦⠀⠀⡠⠂⠀⠀⢠⠞⠁⠀⠀⠀⣠⢞⠕⠁⠀⠀⠀⣠⠊⠀⠑⢤⡀⡠⠊
⠀⢸⠀⠀⢀⣠⠞⠋⠀⠀⠀⢀⡜⠁⠀⢀⡠⢟⡵⠊⠁⠀⠀⣜⣿⣺⣽⣽⡧⠚⠁⠀⠀⢠⣇⡀⣀⡀⢔⡫⠚⠁⠀⠀⠀⠀⢰⡃⠀⠀⢀⠔⠋
⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠸⢦⣤⠖⠉⠊⠁⠀⠀⠀⠀⠀⠙⠾⠷⠛⠉⠀⠀⠀⠀⠀⠀⠓⠁⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⡤⣔⣒⣒⠢⣄
⠀⠀⠀⡠⠴⠤⠶⠤⠶⠦⣄⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠂⣎⠀⠀⠀⢱⠈⡇⠀⠀⠀⠀⢠⢶⣒⡉⠉⣏⢉⣀⢶⡄
⠀⠀⠀⠳⠤⠄⣀⣀⣠⠶⠞⠀⠀⠀⠀⢯⠲⠤⣤⣤⠤⢶⠃⠀⠀⠀⠀⠘⡧⠬⣭⣃⡭⠥⢊⠇⠀⠀⠀⠀⠘⠦⠤⠤⠥⢤⣤⣤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠐⠚⠁⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⠿⠛⠛⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠛⠛⠿⢿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀
⠀⠀⠀⢠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄
⠀⠀⢠⣿⣿⠋⠀⠀⠀⣴⣶⣦⣴⣶⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣦⣴⣶⣦⠀⠀⠀⠙⣿⣿⡄
⠀⢀⣾⣿⠃⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠘⣿⣷⡀
⠀⣼⣿⡏⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⢹⣿⣷
⢰⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿
⣸⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣤⣤⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣤⣴⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠿⠛⠛⠿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⡿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⡀
⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⣾⣿⠆⠀⠀⠀⠰⣿⣷⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⣼⡟⠀⠀⠀⠀⠀⢰⣿⡷⠀⠀⠀⠀⢾⣿⡆⠀⠀⠀⠀⠀⢻⣷
⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢹⣷⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢻⣧⠀⠀⠀⠰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠆⠀⠀⠀⣼⡟⠀⠘⣿⡄⠀⠀⠀⠀⠀⣠⣶⠿⠟⠛⠛⠛⠿⣶⣄⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠘⣿⡄⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⢰⣿⠁⠀⠀⢹⣷⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⣼⡟
⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠙⠻⠷⢶⣶⡶⠾⠟⠋⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠻⣷⡀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⢀⣾⡟
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣀⣀⣀⠀⣀⣀⣀⣤⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⣀⣀⡀⢀⣀⣀⣠⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⣿⣟⣟⡟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣷⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⠔⠋⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡰⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠁⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⠁⠀⠀⠁⠢⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠶⠲⠤⣀⠸⣄⣀⣀⣀⣀⣀⡀⠤⡏⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠂⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢄⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠠⠀⣀⣑⣅⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠻⡀⠀⠀⠀⠀⠀⢠⠏⠀⢀⣠⠴⠒⠋⣉⠝⢓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⣠⠤⠖⠁⠀⢠⠏⠀⢀⠄⢊⠤⠊⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠫⢍⡉⠈⠉⠒⢄⡀⠀⠀⠀⠀⢸⠤⠎⣀⣐⡁⠀⠄⢊⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠁⠂⠌⡑⠢⢄⡀⠙⢆⠀⠀⠀⠀⠑⠒⠶⠶⠶⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠐⠠⠀⢀⡀⢁⣢⡬⠑⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⣠⠆⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣷⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣾⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣰⠟⠉⠻⣆⠀⣰⠟⠉⠻⣆⠀⣰⠟⠉⠻⣆⠀⣰⠟⠉⠻⣆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢿⣀⠀⣀⡟⠀⢿⣀⠀⣀⡟⠀⢻⣀⠀⣀⡟⠀⢻⣀⠀⣀⡟⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣰⠟⠉⠻⣆⠀⣰⠟⠉⠻⣆⠀⣰⠟⠉⠻⣆⠀⣰⠟⠉⠻⣆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣀⠀⣠⡟⠀⢿⣀⠀⣀⡟⠀⢿⣀⠀⣠⡿⠀⢻⣀⠀⣀⡟⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣰⠟⠉⠻⣆⠀⣰⠟⠛⠻⣆⠀⣰⠟⠉⠻⣆⠀⠤⠤⠤⠤⠤⠤⠄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢿⣀⠀⣀⡟⠀⢻⣀⠀⣀⡟⠀⢻⣀⠀⣀⡟⠀⣠⣤⣤⣤⣤⣤⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠋⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⡶⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣦⠀⠀⠀⠀⠀⠙⠓⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠈⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡅⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⢻⣄⠀⠀⠢⣤⣀⣀⠛⠻⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⣸⠐⡇⠀⠀⠀⠀⡠⠊
⠀⠀⠈⠛⢷⣆⡀⠀⣿⣿⣷⣦⣌⠙⠿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⢩⡄⠀⡠⠊⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⢿⣦⣌⣿⣯⣉⠂⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣻⣾⣄⡀⠀⠀⠀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⢏⢿⣿⡉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣱⡏⢸⣙⣦⣺⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣼⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⡤⣛⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠶⠔⠒⠋⠀⢀⣰⠃⠀⠀⠀⠰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⡀⠀⠀⣠⣾⡇
⣼⣤⢀⡀⢀⡀⡀⠀⠀⠐⠊⠁⠀⠀⠀⣀⣤⠾⠛⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠁⠲⠦⠴⣱⣶⣿⣿⡿⠃
⢿⣿⣌⠉⠀⠤⠀⠀⠀⢀⣀⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣐⡒⠖⣻⡯⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⢹⠽⠟⠋
⠀⠉⠛⠿⠶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣀⣀
⠀⠀⠀⢀⣶⠾⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⢟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁
⠀⠀⠀⡀⠀⠤⠶⠂⠐⠀⠀⠀⠀⠀⠀⠀⠘⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⣾⣴⠎
⠀⠀⢸⣧⡤⡴⠒⡶⡀⠀⡄⠀⠀⠀⠀⠀⠀⢻⡨⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣯⣧⡬⡤⠤⠿⡄⢧⠀⠀⠀⠀⠀⠀⠈⣷⣿⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⣿⡅⣿⠀⠀⠀⠀⠀⠀⢸⣯⣭⠉⠁⠀⠀⠀⠀⠐⠐
⠀⠀⠘⣷⣟⠀⡀⢀⠀⣤⣐⠀⠀⠀⠀⠀⠀⠀⠸⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⢿⠀⠀⠀⠀⠀⠀⠸⣿⣿⠄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠹⣿⡴⡿⠿⡣⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⠷⣾⡄⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠙⢶⠖⠛⠛⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣰⣾⣿⠇⠀⠀⠀⠀⠀⠀⠈⢿⡽⠄⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣀⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠷⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠊⠉⠁⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⡔⠋⠉⠑⡄⢰⠋⠀⠀⠹⡄⠀⠀⠈⠀⢹⡀
⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⢠⠞⠉⠉⢳⡈⢧⡀⢀⣲⠃⠈⠣⣄⣤⠞⠁⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⡰⠋⢀⡠⠤⣄⠸⣄⡠⣀⡼⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣠⠎
⠀⠀⠀⡜⠁⢀⡏⠀⠀⠀⢳⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⣸⠁⠀⠀⠧⣀⣀⠵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⣀⡀
⠀⠀⡇⠀⢀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⡇⠰⡏⠀⠀⠀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠹⡀⠑⠦⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠉⠲⣄⠀⠀⠀⢸
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⢤⣤⠼⠀⠀⠀⡜
⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢂⣀⣠⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠉⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⣠⡴⢦⡀⠀⢉⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢰⡏⠀⠀⠛⠀⠀⠽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡈⣴⠀⠀⠀⠈⠀⠀⠚⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠎⠀⠀⠀⡄⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⣧⠀⠀⡰⠀⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠓⠊⠁⠀⠀⠀⣂⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠄⣀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⣉⠤⠁⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠁⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡂⠀⣀⠈⠀⢀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢢⠡⠰⡉⠀⠀⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠞⠁⠀⡠⠑⣁⢈⠀⠀⠄⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⢁⠀⡀⠀⠀⠀⠀⠤⠀⠀⠑⠀⠈⡀⠀⠀⠙⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡤⣢⣑⠀⠀⢈⠂⠀⢀⠄⠈⠂⠔⠀⠀⠀⠀⡀⠀⠈⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢹⣛⠛⠛⠉⠵⠖⢚⣠⣠⣀⣀⣀⠀⠐⠀⠈⠀⠐⠄⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⣀⠘⠋⠓⠀⠀⠀⠀⠀⠀⠀⢿⡟⠁⢹⡾⢶⣶⣤⣤⣤⣵⣈⠄⢸⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡜⣃⠒⠆⢀⠠⠄⠀⠀⣀⠈⠁⠀⣠⣠⠜⠁⣤⠾⡋⠉⠛⠉⡾⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠈⢩⢿⣿⠄⠀⠀⠀⠀⠈⠀⠸⠟⠒⠸⢷⣶⣶⣬⣭⣤⠀⠀⠀⠠⠠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡯⠀⠈⠉⠐⠒⠃⠤⠴⣶⣮⡀⠴⢂⣁⡠⠜⠉⡉⠈⠉⠙⠺⠻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢄⣀⣿⣂⠀⠀⠀⠀⠀⠀⠀⢀⡿⠏⠀⠚⢋⣻⡶⠶⢿⡧⠤⠤⠄⣊⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠐⠀⡈⠉⢐⠂⠀⠄⠀⠀⡀⣨⣴⠀⠐⠂⣉⣀⠀⠒⡉⠀⠀⠀⢉⠉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠷⡦⣦⣤⡑⣀⠀⠀⠀⠀⠀⠉⠁⠘⠛⠁⠤⠴⢋⣁⠀⠀⠀⠰⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠃⠻⠙⡛⠷⠶⡆⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢰⠌⠀⠀⠀⠀⠀⢀⠁⠀⠈⠉⠍⠉⠀⠟⠁⠰⣀⡈⠀⣀⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠈⠀⠀⠀⢈⠀⢈⢤⠀⠀⠀⠈⡠⠐⠂⢄⠀⠀⠀⠀⠀⠢⠈⣤⠈⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡠⠀⠀⠀⠀⠈⠀⠀⠘⣇⠂⠀⠀⠀⠀⠀⡘⠈⠀⣄⠈⡠⠀⠀⢤⡆⡠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡠⠊⠀⠀⠟⣴⣽⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡊⠀⠀⠀⣤⠉⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠢⢹⣿⣟⣀⣀⣀
⠀⠀⠀⠀⡀⠀⠁⠀⠈⠙⠓⠒⠚⠶⠒⢦⣄⣀⣠⡞⣂⣀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⢀⣔⡓⣠⣾⠋⠙⠍⠉⠃⢐⣒⣂⣤⣤⣄⡀
⠀⢀⠀⢲⠿⠆⠀⠀⠀⠀⠀⠉⠉⠄⠀⠀⠀⠙⠿⠟⠉⠙⠛⠳⣤⣀⡴⠏⠁⠀⠀⠀⢀⠀⣦⣈⣱⢟⣿⠷⠶⠶⠏⠛⠻⠏⠙⠒⠒⠀⠐⠶⠤⠤
⡰⠁⣰⠴⢢⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢈⣛⡶⠤⠤⠦⠴⣤⣡⡤⢞⡵⠟⠡⠄⠀⠒⠒⠒⠒⠒⠒⠊⠈⠤⢠⠤⠦⠖⠒⠒⠂
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠶⠦⠈⣣⡼⠻⠒⠒⠶⠖⠂⢈⣉⣉⠁⠀⠀⡀⠠⠠⠄⠀⠴
⠀⠀⠲⠶⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠛⢀⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⠶⣤⣤⣤⣤⡄⠀⢴⡦⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠲⠿⠷⢆⡀⠻⠝⢀⣠⣠⣤⡴⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⡚⠉⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢛⡭⠖⠉⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣋⠴⢚⡩⠀⠀⠀⠀⠀⣀⣤⣶⠾⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠐⣈⡤⠚⠁⠀⠊⢁⡤⢖⡫⠝⠈⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⣀⡴⠚⠁⠀⠀⣀⡴⣞⠵⠚⠁⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⢀⣠⠴⣋⡵⠊⣀⠴⠚⢁⠀⠀⣀⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣤⡀⠈⢧⡀
⠀⠀⠀⠀⣠⠴⢋⠵⠊⠁⠔⠊⠁⠔⠊⣡⣴⣿⠟⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⡟⠀⠀⠈⣿⡄⠈⠳⣄
⠀⠀⠀⢯⡁⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣷⣤⣤⠾⣻⠇⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠳⣄⠀⠀⠀⣀⣴⡿⠟⠉⠀⠀⠀⠀⠀⣠⠤⠖⠶⢤⡤⠶⠦⢤⡴⢛⡟⢿⣞⣋⠁⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠈⠳⣤⢞⠕⠋⠀⠀⠀⠀⠀⠀⠀⣼⠓⠒⠲⠤⢬⣷⠀⠀⣿⠀⢰⡇⠀⣿⠀⣩⡶⠶⠶⣦⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣀⣠⡤⢾⡷⣴⠒⢻⣷⣴⣷⣞⡃⣼⠁⠈⢢⣤⠿⡆⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣰⣿⠑⣄⠈⢳⡄⢸⠁⣧⡴⠋⢻⠀⠈⣪⡽⠿⠿⣟⣉⣡⢴⡧⠶⠶⣤⡀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⢹⡹⣄⡈⠳⣸⡇⢘⣧⡘⢷⣤⡾⠀⢰⡟⠶⠴⠤⣼⣧⠀⣸⡀⠀⠀⢈⣷⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠈⠳⣬⣽⡿⠋⣠⢾⠈⠙⣍⢳⣠⡶⠾⣿⣛⣛⣫⠵⠋⠀⠻⣍⣛⣛⣻⠇⠀⠀⢀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠹⣌⠓⠦⣬⣾⠹⣗⢦⡈⠻⣇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠉⡿⠲⠚⠉⠹⡎⠳⣽⣤⡾⠂⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⢸⡘⢦⣄⣀⣼⠇⠀⠀⠀⠀⠀⢠⠏⠀⠀⢹⡆⠀⠀⠀⢀⢴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠙⠲⠶⠾⠋⠀⢀⣀⣀⠀⠀⢺⣧⡤⠴⣾⠇⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⡴⠉⢹⠉⢳⡄⠀⠉⠛⠉⠁⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⢰⣧⡀⣸⣀⣾⠇⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠻⠿⠯⠽⠋⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢀⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

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




⣿⢿⠿⡿⡿⢟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣟⣻⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⡇⠇⠀⠀⢀⣄⠀⠀⠀⣀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⡇⠃⠲⣤⣴⠟⠻⠀⠀⠀⠘⣷⣶⡛⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⡇⠀⡴⠋⠛⠷⠀⠀⠀⠀⢰⠟⠙⢿⣷⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⢰⡇⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⣠⡴⠟⠛⠋⠉⠉⠉⠛⠻⣿⡷⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣤⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠃⠀⠀⠀⠘⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠘⠟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⡀⢀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢀⠎⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⣸⣥⣊⣤⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣴⠿⠁⠀⠀⠀⠀⠀⠀⠃⡨⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡠⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠉⢀⡀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠄⠀⠀⣄⠁⠀⢀⡠⠊
⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⢀⠊⠂⢤⠲⣄⠀⠀⠀⠰⠖⠉
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠁⠀⠀⢀⠤⣈⡀⠀⠀⢀⣀⣠⠤⠤⠤⠤⢤⡀
⠀⠀⠀⡜⠀⠀⣠⠖⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠈⠀⡠⠊⠁⠀⣀⡉⢩⣍⡉⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢀⠔⠊⠀⠀⣰⠏⠀⠀⠀⠐⢽⡄⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⡠⠂⠁⠀⠀⠀⠀⠹⢄⡀⠀⢀⣀⠜⠁⠀⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡄⠀⠀⢠⡇
⠀⠀⠀⠀⢻⢆⠀⠀⠀⢀⠔⠁⡠⠊⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠑⢕⡤⠖⢁⠔⡚⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⢀⡠⠊⡠⢖⢣⠌⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣾⠝⠁
⠀⠀⢀⠔⠉⠀⠊⠀⠠⠋⡰⠯⣓⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⣚⠵⠊
⢀⠔⡃⠔⠁⠀⠀⡰⠁⠌⠀⠀⠀⠉⠑⠂⠭⠍⣑⡒⠒⠒⠶⠴⠒⠒⣒⣒⠭⠭⠐⠂⠉
⠀⠉⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠁
⠀⣁⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠠⣤⣀⡀⠀⢀⠀⢀⣼⢶⡚⠳⠀⢀⣀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠾⢶⣦⣽⠀⠈⠉⠙⠁⠂⠉⠁⢰⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠠⠐⢢⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠰⠀⠀⣸⢳⡖⠀⠀⠉⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⢠⠃⠀⠀⡿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⣀⡤⠚⠁⠉⠙⠛⠁⠀⠀⢀⡎⠀⢠⠴⠃⠀⠉⠓⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢠⡀⠓⠋⠟⠀⠀⠠⣀⠄⠀⠀⣸⠃⠀⠀⠀⠰⠋⠘
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠠⢟⡧⣀⣀⣀⠰⡄⠀⠀⠀⠀⠒⣼⠶⠴⠚⠈⣳⢄⡀⡄⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠨⠃⠀⠀⠀⠀⡿⠇⠀⠀⠱⣈⣹⠋⠈⠙⠄⢰⣶⣄⣀⣅⠀⣀⣠⣴⣧⠀⠀⠈⠁
⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⠀⠀⠐⠤⣄⢶⡾⠛⠛⠀⠀⠀⠈⣟⠉⠉⢹⣛⣛⣯⡅⠀⢢⠁⣠⠄
⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣌⢻⣋⠙⠻⠿⢿⡄⠂⠈⠚⠁⠀⠀⡀
⠀⠀⠀⠤⠤⠂⠀⠀⠀⡀⠐⢢⣄⠀⣀⠀⠀⠀⠹⣀⠀⢠⠞⣿⢽⣯⣷⡄⠀⠀⢉⡄⠀⢠⣄⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠹⠀⠈⠀⠒⠒⢂⠀⠉⠁⠈⠀⠀⠀⡳⠳⠾⠛⠋⠀⠉⠀⠈⠀⠀⠈⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⠀⠀⠀⠀⢀⡴⠀⡠⣇⠀⠀⠁⠀⠀⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⡔⠊⠀⠀⡀
⠀⠀⢀⠠⠂⠀⠀⠀⠹⠿⠄⠀⠀⠀⠀⠀⠀⠜⠀⠀⡁⠁⠀⠀⠀⣠⡴⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⠁⠀⠠⡀
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠔⠉⠀⢠⣀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣠⡷⠾⠛⢹⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢇⡿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀
⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠊⠁⠀⢀⡴⠚⠉⠉⠉⠉⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠉⠉⠛⠦⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⠛⠻⣿⣷⡄⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢀⡞⠛⢿⣿⣦⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣿⣤⣴⣿⣿⡷⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⢸⣧⣤⣾⣿⣿⡆⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢹⣄⣀⡤⠿⠿⠿⠿⠿⠧⠤⢤⣀⣜⠀⠀⠀⢡⣀⣤⠤⠼⠿⠿⠿⠿⠿⢤⣄⣀⡟
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠀⣀⣀⣀⣀⠀⠈⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣔⠋⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⠤⠤⠴⠶⠒⠒⠒⠒⠶⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠊⠉⠀⠀⠀⣀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⢉⡤⠤⠤⠄⡀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⢰⠋⠀⠀⠀⠀⠈⡆⠀⠀⢯⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠔⠉⠀⠀⠀⠑⠄⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⣀⣀⣀⠀⠀⠘⣄⠀⠀⠀⠀⢠⠇⠀⠀⠀⠓⠄⣀⣀⠤⠊⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⠞⠁⠀⠈⠙⢆⠀⠈⠑⠒⠒⠊⠁⢀⡀⠀⠤⠤⠤⠤⠤⢄⡀⠀⠀⠀⠘⢄⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢀⡞⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠔⠒⠤⣄⡙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠙⠆⡀⠀⢀⡠⠃⢀⠔⠁⠀⢡⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠉⢦⠀⢀⠔⠋⠉⠉⠑⢆⠀⠸
⠀⠀⠀⠀⢸⠁⠀⠀⢀⣀⣀⠀⠁⠀⠀⢠⠃⠀⠀⠀⠀⠠⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡎⠀⠀⠀⠀⠀⢈⠀⢰
⠀⠀⠀⠀⢫⠀⠀⠔⠁⠀⠀⠉⢦⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⢸⠆⡄⠀⠀⠀⢀⠏⠀⠀⠘⢄⡀⠀⠀⡠⠊⢀⡜
⠀⠀⠀⠀⠸⡀⢸⠀⠀⠀⠀⠀⢸⠇⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠑⠂⡐⢄⠀⠀⠤⠜⠀⠧⢀⣀⣀⡌⠀⢀⠠⢄⣀⣉⣉⣉⡠⠔⠊
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢀⣠⠞⠀⠀⡠⠀⣉⡁⠒⠠⠤⣀⣀⠀⢀⡼⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠉⠙⠓⠒⠢⠤⡉⠁⠀⢀⡀⠤⠒⠒⠒⠒⠂⢄⡀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣈⠐⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠳⢄⡀⠀⠀⠀⠀⣀⣠⠤⠊⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡈⠉⠐⠒⠀⠤⠤⠤⠜⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠤⠄⣀⣀⣀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠒⠚⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠤⠄⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⡋⠃⠉⠘⡁⠉⠁⠀⠒⠒⢲⠊⠀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⠉⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠆⡸⠠⠤⠧⠤⠶⠒⠒



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⡴⢶⡶⣶⠶⠟⢿⡟⢻⡟⠻⣿⠻⡿⢳⡶⢶⡶⠦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣦⡀
⣿⠀⠸⠇⣿⠀⠀⢸⡇⢸⡇⠀⣿⠀⠁⠸⡇⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷
⣿⠀⠀⠀⢸⠀⠀⢸⠇⠀⠇⠀⡇⠀⠀⠀⠁⠀⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇
⣿⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠋⣛⠻⣿⣿
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢰⣶⡌⡧⢹⣿⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣈⠉⢁⣡⣿⣿⡇
⢸⡇⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣷⣶⣶⣾⣾⣾⣾⣷⣶⣷⣶⣶⣶⣶⣷⣶⣿⣶⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠟⠛⠻⠿⠿⢿⡅
⠀⣿⢾⣿⡷⠶⣿⣿⢾⣿⣽⣿⣿⣿⣽⣯⣿⣿⣿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠄⠀⠀⠀⠀⣼
⠰⡇⢸⣿⠀⠀⢽⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⣿
⢰⡇⢸⣿⠀⠀⢾⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⢀⡟
⢨⡟⠚⠛⠓⠒⡛⠛⠚⠛⠛⠛⠛⠛⠛⡛⠛⠛⠛⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠀⢸⡇
⠸⣧⣀⣀⣀⣠⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡀⢸⡇
⠀⢿⣍⠹⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⠟⢉⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠙⢳⣌⠛⢄⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠘⠀⢸⡇
⠀⠀⠀⠀⠘⢷⡌⠃⠀⠀⠀⠀⠀⡰⢃⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠁⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠄⠀⠀⠀⠀⢀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠞⠀⠀⠀⠀⠀⠈⢣⡘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣽
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠄⡤⠀⠀⠀⠀⠀⠀⠀⠀⡌⠄
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠲⣶⠖⠒⢲⣶⠺
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡆⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⡅⠀⣸⣿⢀
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣶⣟⣛⣿⣿⣻
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠏⠛⠏⠛⡟⠙⠙⠛⠋⠏⠻⠛⠻⠛⢹⠙⠛⠛
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⣿⠋⡄⣴⣦⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣌⠣⡙⠋⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣶⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡇⡀⠀⠀⠀⠀⠀⡇⠀⡀⠀⣰⠀⢀⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣿⣾⣷⣶⣷⣶⣿⣶⣾⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠛⠛⢿⣿⡿⠻⠿⣿⠿⠿⢿⣿⡿⢿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡀⠀⠀⠀⢸⠆

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠿⠋⠉⠉⠛⠟⠛⠛⠛⠻⡿⠿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠢⠤⠀⠤⠴⠚⠒⠢⠤⠴⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⡀⠀⣀⠜⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠤⠐⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⢀⣀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣿⣿⣿⠋⠁⠘⣤⡀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⣀⠜⠙⢿⣿
⣿⣿⡇⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠱⡀⠀⠀⠈⣿
⣿⡿⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢻
⡿⠁⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⡇⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢀⡤⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠛⠋⠉⠁⠈⠉⠉⠙⠛⠿⣿⣿⠿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⠀⠀⢮⡁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡀⠁⠀⢀⣀⡀⠀⠀⠹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠀⠀⣰⣿⡿⠁⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠙⠻⣦⠀⠀⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠇⠀⠀⣿⡟⠀⠀⠀⠀⠀⠱⢂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠴⢺⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢁⡠⢆⣩⠁⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠠⢀⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡙⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣬⠃⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢿⣿⣿⣿
⣿⣿⣿⠟⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠘⢿⣿⣿
⡏⠉⠁⠀⣀⣼⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣄⠀⠀⠉⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠉⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⢀⡤⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⢰⠃⠀⠀⠀⡸⠀⠀⠘⠄⠀⠀⠀⡏⠀⠀⠠⠂⠒⠉⠐⢄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⢂⠀⣀⠸⠁⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠃⠀⠀⠀⠀⣸⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠏⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠻⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢰⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⡠⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠎⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢠⠞⠓⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢻⡀⠀⣠⠇⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠃⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠈⠠⠒⠉⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠶⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣾⣁⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⣠⢾⠟⡥⢒⡽⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⣼⡱⢃⣔⡵⢋⠔⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⣰⣿⣷⣮⡩⠂⢮⣦⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠑⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⢀⠔⠃
⠐⠾⠿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠹⡷⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠁⠛⢧⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠤⠤⠤⠤⠴⠒⠉
⡶⣶⢶⢶⠶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⡶⣶⢶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠿⢿⣿⣿⣿⡿⠿⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⡀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⢈⣿⣿⣿⡁⠀⢀⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⡿⠀⠀⠀⢿⣿⡿⠀⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠿⠟⠛⠻⠿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠋⣠⣴⣶⣶⣶⣦⣄⠙⢿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⢀⣴⠟⠋⠉⠉⠉⠙⠻⣦⡀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛



⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢐⠅⠀⠀⠀⢰⣶⣄⠀⠀⠀⢠⣶⣦
⠀⠀⠀⢐⣾⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠛⠋
⠀⠀⠀⢺⣿
⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⡠⠔⠒⠒⠲⢦⣄⡀
⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠈⠝⣻⣦⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢶⣦⡤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠦⡀⠀⠀⠀⠀⠀⠠⠂⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠢⡀⠀⠀⡐⠁⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠑⢄⠀⠈⠒⠈⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡑⣄⠀⠀⠀⠁⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⡘⢬⠆⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠑⠦⡀⠀⠈⢢⠢⣱⠸⠂⠦⠤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠈⠢⡑⠘⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠓⠢⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢡⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠚⠁⠀⠀⠀⢀⣌⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢁⡆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⢒⠇⠀⠀⠀⠀⢜⠀⠀⠀⠀⠐⢶⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠀⠁⠘⠉⠀⠀⠠⣄⠼⣏⢠⡀⠀⠓⡄⠀⠀⢄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠁⠐⢄⠀⠀⡀⠀⢡⡄⠀⠚⠂⠒⢺⠽⡇⠀⠃⠀⡔⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⢄⠈⠳⡤⠵⠦⣄⠀⠀⠘⠇⠉⢙⠦⢤⣤⡤⢊⡴⠏⢤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠑⠀⠠⡄⠒⠀⠀⠀⢹⠂⠀⠉⠓⠔⠛⠋⠭⠀⠀⠴⠽⣶⠠⠈⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠛⠄⠀⠀⡠⠋⠘⡿⢀⣾⣄⠀⢠⠆⡀⢱⠄⠀⢀⡠⠆⢤⣬⡧⣀⠀⢈⡦⡀
⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠐⠒⠒⠒⠚⠛⠁⠀⠰⢄⣄⢠⡞⢀⣠⠔⣳⣏⠀⠈⠑⠢⣄⣸⡾⢀⡔⠋⢀⣀⣼⡥⠔⢋⣠⠏⣀⣼
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⢼⡦⠤⠤⠴⠎⣥⣤⣤⣤⠀⠀⠀⠑⠚⢻⡧⣤⣭⠥⠞⠓⠶⣋⣉⣉⡥⠞
⠀⠀⠀⠢⠴⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣄⣀⡀⠀⠀⠀⠀⠀⢤⣀⡀⢀⠠⠀⠐⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⡊⠐⠀⠀⡀⠀⠉⠉⠛⠒⠲⠤⠄⢟⢿⣷⣄⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⡰⡡⠊⠀⠀⠀⠈⠀⠀⠀⣳⠀⠀⠀⠀⠀⣸⠀⠛⢿⣷⣀⠖⣫⣭⣿⣦⡄⠲⡀
⠀⠀⠀⠀⠀⠌⠄⠀⠐⠈⠀⠑⣄⠒⠶⠖⠁⠀⠀⠀⠀⠐⠀⠀⣀⢯⢻⣿⣼⣿⣿⣿⣿⡟⠀⠈⢆
⠀⠀⠀⠀⠈⠈⠀⠀⠐⢄⣀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⡀⠹⣌⠈⡛⠻⢿⣿⠟⠋⠈⠀⠀⠈⠄
⠀⠀⠀⠀⡄⠀⠀⠀⡠⠀⠀⢄⠀⠀⠀⠀⠀⠀⢠⢞⣭⣽⣿⣿⡖⠛⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢹⠀⠀⠀⢄⠀⠀⢠⠇⠀⠀⠀⠀⠀⢻⣼⣿⣿⣿⣿⠏⠀⠡⠀⠀⢄⣀⣠⠂⠠⡀
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠫⠂⠀⠀⢢⣤⣾⣅⠀⠀⠀⠘⡆⠀⡄
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣠⠴⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⢀⠴⠁⡐
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠈⠾⠀⠀⠀⠀⠘⡆⠀⠀⢠⠔⣒⣲⣦⣄⠀⠀⠈⢿⣿⣿⣿⠁⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⠢⣀⡀⡀⡠⠈⠀⠀⠀⣏⣾⣿⣿⣿⣿⠇⠀⠀⠈⢿⣿⣿⣷⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠉⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠋⠀⠀⠀⠀⢘⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠠⠄⠊⠁⠘⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠁⡤⠴⠦⠀⠀⠀⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⢀⣴⣾⣿⣿⣦⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⠀⠀⡀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠟⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢹⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠘⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡷⠀⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠟⠁⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣡⠄⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣾⣿⣿⣿⣿⡆⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⡤⠀⠴⣂⡴⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣏⠀⠀⠀⠀⣠⡴⠚⠉⠀⢀⣀⣀⣠⠴⣂⡴⠚⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⢦⣀⣠⡞⠁⠀⠀⢀⣀⣬⡭⠷⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡈⠓⠒⠒⠒⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⣀⠴⠋⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠦⠤⣄⣀
⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠟⢦
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠈⢧
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡤⢋⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣤⣀⣀⣀⣤⠤⠖⠚⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠰⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠿⠿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠉⠉⠛⠻⣿⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠛⣷⣦⠀⠀⠀⠀⢸⡇⠀⡔⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡗⢺⣿⣿⠀⠀⠀⠀⢸⡇⠈⣿⣛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⣼⠇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡾⠁
⠀⠀⣀⠤⠀⢀⡶⠂⠀⠈⢳⣄⠀⠀⠀⠉⠑⠒⠒⠒⠒⠒⠈⠉⠉⠉⢉⣾⠏⠈⠻⣧⣀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠰⠟⠁⠀⠀⣀⡀
⠀⢸⠁⠀⢀⡎⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠈⠙⠻⠶⣦⣤⣤⣤⣴⠶⡇⠀⠀⡀⠀⠀⠀⠈⡇⠉⠲⡄
⠀⠈⠙⠒⠋⠀⠀⠀⠀⠀⢰⠀⠀⠀⠙⠛⠶⠶⠦⠤⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠁⠀⠀⠀⠀⠘⠤⠤⠃
⠀⠀⠀⠀⠀⡠⠂⢸⠀⠀⠘⢄⠀⣀⠆
⠀⠀⠀⠀⠀⠳⠴⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠷⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀