Filthy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡋⢀⣀⣀⣈⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣤⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠋⠀⣻⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡞
⠀⠀⠀⠀⠀⠀⠀⡄⠀⣤⡄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠍⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠆⠀⠀⣹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄
⠀⠀⠀⠀⢀⣄⣀⣡⠀⠻⠃⠀⠀⠀⢨⢿⣿⣿⣿⠿⠋⠀⠐⠁⠀⠀⠀⠀⠀⠀⠁⢰⣾⣶⣶⡆⠀⣴⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠟⢱
⠀⠀⠀⠀⠀⠻⢙⠛⠳⠃⠀⠀⢀⠀⢺⣾⠛⡱⣿⠀⠀⠾⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⡿⠛⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠘⢃⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠙⡏⠀⡇⣿⡖⠁⠦⠠⠄⠀⠀⠤⣤⡀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠐⠊⢈
⠀⠀⠀⠀⠀⠀⠀⣠⡄⣤⣤⣄⠀⠀⠈⠃⠘⠇⣿⡇⠀⠀⠀⠀⠐⠀⠶⣷⣮⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠦⠔⠉
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⡇⣿⣷⡀⠀⠐⠞⠛⣤⠴⠿⡿⢿⣿⠟⡟⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣄⠀⠀⠚
⠀⠀⢰⣿⣿⣎⢿⣿⠟⢻⣿⣿⣿⣿⣷⠀⠀⠁⢸⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⣿⡃⣧⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡿⠁⠀⣤
⠀⢦⣼⣿⣿⣿⢿⣣⣦⣿⢃⡿⢿⣿⣷⣀⡀⠀⢸⣿⠀⠠⠀⢰⣆⠄⠀⠀⠀⢺⣿⠾⠧⠀⣐⠂⠀⡀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠘⠉
⠀⠀⢿⣿⣿⣼⣿⣇⡜⢿⣏⠀⣼⣿⣿⣿⠛⠀⢸⣷⢈⣄⣤⠼⣿⣾⢷⠖⠿⣸⣟⣀⣶⣸⡟⡃⢀⠉⠀⠈⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢦
⠀⢀⣾⣿⣹⣿⣿⣿⣥⡜⣿⡂⠻⣾⣿⡏⠀⠀⢸⣯⣴⣶⣾⠿⠿⠿⢾⣿⣶⣿⣿⣿⣿⣿⣷⣿⣭⡀⠸⣷⣶⡦⣄⡀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣷⡀
⢈⣽⣿⣿⣿⣿⢿⡏⠈⠄⢾⣧⠞⣫⡿⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⡛⡗⠘⠟⣷⠀⢠⣤⣀⡀⠀⠀⠉⠉⠙⠳
⡽⣿⡿⣿⣿⣿⣴⣧⣤⡀⡞⠋⠐⣿⣿⡄⠀⡄⣹⣿⣦⠀⠀⠀⠀⠀⠀⠘⢁⣹⡏⢹⣿⣿⣿⡿⠿⠿⠀⠈⣷⢿⣶⡦⠈⡆⠸⣿⣿⣿⣿⣶⣦⡄
⠉⣿⡷⢿⢿⣡⡟⠈⣿⣿⡏⢀⢠⣿⣿⣧⢸⠂⣿⣿⣿⢀⡀⢰⣿⣠⡤⠀⢈⠝⠀⠀⠈⠉⡉⠁⠀⠀⠀⣀⠈⠀⠈⠑⠶⡇⠀⣿⡏⠻⢿⣿⣿⣷⠀⢲⡆
⠀⣿⣇⣾⢘⣧⠻⠟⢳⣿⣿⣾⡞⣿⣿⣿⠞⣠⣿⣿⠋⢸⣗⣿⣿⡿⣃⣄⠁⢀⡀⢓⠂⣀⠀⠀⠀⠀⠀⣿⣷⣄⣀⠀⠀⠀⠀⠹⢷⡶⠚⠛⠁⠉⠀⠸⠇
⢰⠿⣧⣿⠘⣇⠀⠀⠻⠛⣿⣿⣿⡍⣿⣿⣿⡿⢹⣿⢰⠟⢿⣿⣿⣿⣟⣿⠀⠈⠿⡏⠁⠈⠰⠄⠁⣀⣀⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠙⠀⣿⣿⡿⣿⣆⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣇⢸⣿⡿⢳⣿⣿⣿⡿⢛⡋⢀⣶⣟⣤⣴⣶⣶⣾⡿⠛⠛⠁⣿⣿⡿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀
⢀⢠⣿⣿⡇⠻⣿⣧⡀⠀⠠⠟⣿⠙⣿⣿⣿⣿⠻⣿⣷⣾⣿⣟⢿⡇⠉⠁⠀⣄⠀⠈⣿⡿⢹⣾⣿⣷⡆⠀⢹⣿⣿⣿⣿⣿⣿⣧⣵⣼⡽⢛⣿⣦⡀
⠘⣾⣿⣿⣿⣿⣿⣿⣦⠀⡾⠟⢾⢣⣿⠋⢈⣠⣴⣿⡿⠋⠘⠿⠸⠀⠀⠀⠐⣿⣶⣶⣟⣛⣿⣿⣿⣿⣿⣿⣤⣿⣟⡿⠿⠿⣿⣿⣿⣿⠟⠻⣋
⠀⡏⣿⠿⡿⢿⣿⣿⣿⣷⣄⠀⣿⣶⡟⠀⢈⣿⣿⣿⠁⠀⠀⠀⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣡⣾⣿⣷⣦⣀⣹⣿⣿⢀⡀⠉
⠐⡇⣿⡠⣧⠀⠉⢹⣿⣮⣿⣧⣹⡏⢻⣼⣿⣿⣻⣿⡄⠀⠀⠀⣟⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣶⣷⠂⠁⠐⠂
⠈⣱⣿⡈⠫⣖⠂⡎⠁⢻⣿⣿⢿⣣⣿⣿⣿⣿⢳⣿⡆⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣯⣉⠙⠛⠛⠉⠀⢀⡴⠄
⣼⣷⣿⠁⠀⠪⠀⠁⢀⣠⡟⢿⣿⣿⣷⣿⣿⣿⣿⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠉⠁⠀⠀⠀⠘⠻⠇
⢼⡟⡟⠀⠀⠀⠑⠀⠀⠉⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⢿⣿⣿⣿⣿⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣀⠐⠶⠖⠀⠀⣾⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠉⠉⠻⢿⣿⣿⣿⡿⠿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⢦⣤⠄
⠀⠘⠆⠋⠀⢤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢐⣄⠀⠿⠿⠿⠿⢧⡀⣀⣠⣤⠘⠀⠤⠾⣿⡶⣏⢀⡀⠃⠀⠀⠀⠀⠀⠒⠀⣀
⠀⠀⠀⠀⠀⠙⠛⠻⠿⠟⠛⠛⠉⢻⣿⣿⡿⡟⠉⠀⣴⣶⣄⣶⣶⣿⣿⣷⣤⣿⣦⣴⣶⣤⡞⣛⣿⣶⣾⢵⣶⠂⠀⠘⠷⣶⣦⡄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣠⡀⠀⣻⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⠅⠀⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⣀⣄⠀⣀⠤⠶⣶⣿⣿⠛⠻⠿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠁⣀⣰⣿⣶⣜⣛⠻⣿⣷⣄⠀⠀⠉⣝⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡀⠀⣿⣿⣿⡿⢿⣿⣶⣯⣀⠉⠳⣶⣶⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢂⣾⣿⣿⣿⡇⠀⠙⢿⣿⣿⣿⣿⡌⢷⣿⣷⣄⡀⠈⠛⠙⠛⢿⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣄⠀⠀⠙⣿⣭⣉⣀⣼⡅⠸⣿⣿⣦⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⡿⢸⣿⣯⣼⣿⢿⣿⣿⣿⣿⡤⠀⠀⠀⣛⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⣻⡇⠈⡏⢸⡇⡿⡈⠙⢁⣼⣿⣇⣤⣶⣿⠟⢛⣿⡿⣿⣿⠟⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣿⣿⣶⡇⢸⣿⠀⠙⠷⠟⢿⣿⣿⣿⣿⣿⣶⣾⣿⠁⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣼⣿⣿⡿⠃⣸⣿⣶⣿⣿⣶⣤⣿⡿⠿⣿⣿⣿⣿⣷⣄⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠁⠐⠛⠉⠙⠛⠛⣿⣿⠿⣿⣦⣀⣀⣼⡿⠿⢿⣿⣿⠏⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣇⡿⠃⠷⢸⣿⣿⣿⣿⣿⡶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢰⣴⣿⣿⣿⡎⠙⢠⠀⠀⠀⢸⣿⡟⢛⣽⣿⣷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣧⠀⠀⠀⠀⠈⣿⣿⣿⠉⠇⢀⣾⡇⠀⠠⠄⣸⣧⢾⣿⣿⣿⣧⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠙⠻⣯⣷⣶⡀⠀⢀⣿⣿⣿⣧⣾⣿⣿⡇⠀⠀⣺⣿⣿⠀⢨⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣧⣸⣿⣿⣿⣿⣼⣿⡽⣿⠀⢸⡿⣻⣿⠆⡾⣻⣿⣿⠈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣷⡿⣠⣏⣴⣿⣿⣆⠙⣿⣿⣿⠛⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢶⣶⣶⣶⣦⣼⣿⣿⡿⢶⣿⣿⣿⣿⣿⣿⠋⢿⣿⣿⣿⢹⣾⢋⣿⣿⣿⣿⣿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡁⢼⣿⣿⣿⣿⡄⠉⣿⣿⡿⣿⡇⠘⣾⣿⣿⡇⣼⠿⣿⣿⣿⣿⣏⢨⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣥⣾⣿⣿⣿⣿⢣⠐⣿⣿⡿⣿⠃⣷⣿⣿⣿⣷⣧⣴⣿⣿⣿⣿⣿⣼⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⣷⣼⣿⣿⣿⣿⣿⣞⣼⣿⣧⣷⠈⣼⣿⣿⣿⡟⠈⢸⣿⣿⣷⣿⣿⣟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠂⢸⣿⣿⣿⣞⣿⣿⣿⣿⠃⢰⣿⣿⣽⣿⣿⣿⣿⣿⣿⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⠀⠀⠘⢉⣿⣿⣿⣿⣿⣿⣿⠂⢸⣿⣿⣿⣹⣿⣿⣿⡟⠄⣼⣿⢿⣿⣿⣿⣿⣿⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠨⣿⣿⣿⣿⣿⣯⠀⠀⢿⣿⣿⣾⣟⣿⠟⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⡃⠈⡇
⠀⠀⠀⠀⠀⠰⠶⠿⠿⠿⠿⠶⠀⠀⠀⣀⠀⠠⡀⢸⣿⣿⣿⣿⣷⣶⣠⣸⣿⣿⣿⣿⠛⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠛⠻⢿⣿⡶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣸⣿⣿⣿⣿⣿⣿⣿⡿⠷⣾⣿⣶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣷⣶⣶⣿⡿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣦⣦⣤⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⣤⢠⡄⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠿⠸⠇⠛⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠤⠂⠒⠒⠾⠈⠛⠉⠁⠀⠀⠀⠰⠁⠀⠁
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢶⣀⣈⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡃⠀⠀⢀⠘⠀⢸⠀⠀⢀⡄⠁⠈⠁
⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠄⠐⠒⠓⠒⠒⠒⠒⠒⠛⠓⠒⠚⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⢧⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⡖⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⠀⣀⠀⠠⠤⢠⣾⠁⠀⠀⠀⡠⠊⠀⠀⠄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠋⠛⠛⠉⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⣰⠇⠀⠀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣤⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⢸⠀⠀⠀⠘⠷⠶⠤⠀⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠙⣾⣅⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⢸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⢈⡅⠰⡄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠒⠄⠀⠀⠈⠁⡀⠑⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠂⠀⠀⠀⠄⠁⠀⠸⣯⣭⣭⣭⣄⣀⣀⡀⠀⢀
⠀⠀⠀⠀⢸⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠏⣉⣁⡀⠀⢸⣶⣤⣄⣀⣀⣀⡀⠠
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣄⣤⣤⣤⢀⣸⠍⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠉⠉⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⠛⠋⠉⠉⠀⠀⠀⢸⣿⠛⠒⠀⠀⠀⠀⠀⣾⠋⠀⠄⠈⠀⠄
⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠤⠤⠤⡤⠤⠴⠄⠀⢸⣿⣄⣀⣀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠛⠛⠚⠃⠀⠀⢸⣿⠀⠀⠈⠉⠁⢀⣤⣿⠀⠀⠐⠓
⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⡀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠦⠤⠤⠤⠴⠿⠿⣿⢸⣿⣿⣿⣿⡏⢰
⠀⠀⠀⠀⠀⠉⠒⠶⠦⠐⢂⡀⣀⡀⠤⢤⣤⡤⠤⠤⢤⡤⠤⠤⢤⣿⣯⡤⠤⠤⠤⠤⠴⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠭⠭⠥⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢆⠀⠀⠀⠀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⢆⠸⠀⠀⠀⡼⢁⠄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠴⠚⠃⢐⣇⡀⠐⠁⠘⡄⡇⠀⠘⢡⠊⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠈⠐⠆⡀⠀⠃⠁⠀⣰⠃⠀⠀⠈⠢⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⡀⠠⠄⠐⢯
⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⠤⠤⣄⣀⡀⠸⡦⡀⠀⠀⠀⣯⡴⠅⠀⠀⢸⠉⠉⠉⠉⠉⠓⢦⠀⠀⡠⢐⠴⠂⠠
⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠉⠙⠛⠓⠦⠤⠄⠳⣀⠠⠤⠚⠄⠀⠀⡴⠒⠀⢸⠀⠀⠉⠀⠀⠀⠀⠃
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⠸⠀⠀⡾
⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⢄⠀⠀⠀⠀⠇
⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⣀⡰⠃⠀⠀⢸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠋⠀⠀⠀⠐⠉⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠈⠓⠤⢄⣀⣀⠀⠀⢀⣠⡃⣠⡤⠶⣎⢩⠃⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢰⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠘⡖⢦⡄⠈⠍⠀⣈⣉⡽⠀⠀⠀⠀⠀⠀⠀⢀⡉⠐⠂⠀⠈⠒⠏⢈⣩⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⣿⠋⡀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⡇⣏⡧⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠤⢀⢤⠛⡦⠲⠎⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣴⡒⠒⠀⠐⠂⠀⠀⠀⠀⣖⠄⠐⠀⢠⡎⠀⠀⡈⠃⠠⢄⠀⠀⠀⠀⣇⠀⢀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠉⣩⠆⠐⠀⠀⠀⠀⠀⠙⠀⠠⠖⠀⠰⠂⠀⠊⠀⢀⣘⣄⠤⠀⠤⠅⠀⠀⠉⠀⠈⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠓⠢⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣰⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣉⢑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠬⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣆⣠⠖⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣲⡼⣷⢿⣷⣟⠧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⡿⠁⣿⢸⡿⡄⠳⡌⠳⡙⢶⠦⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠠⣶⢋⡏⢰⡇⠀⢿⢸⡇⢻⠀⠹⡄⠹⣎⢻⣽⠉⠻⣿⣿⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠏⠉⢿⢹⡇⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠉⠀⠿⠀⠀⠹⡽⣝⣮⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⡽⣱⢛⡏⠀⠀⠘⣿⣷⣦⡀⠟⠀⠀⢀⣤⡀⠀⠀⠈⠁⢠⣄⣀⢀⣘⢲⡀⢳⡘⢎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢰⡇⢸⡇⠀⠰⡟⣿⣿⣿⠿⢀⠀⠀⠈⠀⠁⠀⠀⠀⠀⡾⠿⠋⣾⣟⣼⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡼⠀⠀⠀⠹⣯⣼⣇⠀⣼⡿⢷⡄⠀⠀⠀⢠⡖⠀⠉⠁⢰⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠻⡗⠀⠙⠶⣤⡤⠞⠁⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣤⣤⣤⣤⣤⣴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⡿⡇⠀⠈⣿⣶⠚⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠚⠉⠉⠁⠀⠈⠙⠒⠒⠛⠋⠁⠀⠤⣀⡉⠳⢤⡀
⠀⠀⠀⢿⣙⠲⠦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣼⣇⠀⠀⠠⠤⠤⣤⣤⣶⣷⣶⣄⠀⠀⠀⠀⠀⢠⣼⣷⡆⠀⠀⠙⢦⡀
⠀⠀⢀⣤⠟⠀⣴⣿⡿⢟⠡⣅⠠⣤⢄⡀⣀⢀⣙⣟⣷⣿⣿⣿⡟⠈⠻⠿⠿⣿⣤⣄⡀⠀⢀⡟⣿⡿⣵⠄⠀⠀⠀⢹⡆
⢀⣶⣿⢟⣿⣃⣟⠉⢰⡟⠛⠛⠛⠛⠛⠛⠉⠹⣯⣇⠘⠏⠟⠁⠀⠀⠀⠂⠀⣛⡛⠿⣿⠅⣠⣄⡈⠈⠁⢸⡆⢀⣴⠏
⠈⠉⠁⣾⡿⢻⣿⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡤⡤⠤⢶⡶⠀⠀⠀⠀⠉⠛⠑⠈⠀⠼⠿⠁⠀⠀⠈⣷⠟⣷
⠀⠀⠀⠈⠀⠻⠟⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠳⠶⣦⣴⣦⡀⠀⠀⠌⢀⡂⠄⠀⠀⢻⣾⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢐⠚⢿⠿⠿⡿⠱⠶⠀⠀⠀⠀⢘⠀⠀⠀⢻⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⢠⠀⣴⣤⠀⠀⠀⠀⠀⢀⣀⣀⣤⠾⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⢤⣤⣤⣤⣥⡤⠶⠖⠲⠚⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠈⠀⠀⠂⠀⠀⠀⠠⠀⢐⠀⠂⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠋⠀⢀⡲⠐⠀⠀⠳⡀⠀⢐⠀⢀⠀⡁⠈⠀⠀⠀⡠⠾⠆⠴⢱⠀⠀⠀⠀⠆⢀⣷⣐⠃⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠲⣤⠀⠀⢣⠀⠈⠀⢨⠀⠀⠀⢀⠔⠉⢠⠁⠄⢀⢤⡑⡀⠀⠀⠀⠘⢛⢉⡼⠆⠰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⡤⡎⣿⣿⡆⢀⢠⢀⠂⠤⠀⢆⠴⠠⡄⠉⠀⠀⠀⠈⠃⠈⠁⢀⠀⠀⠈⢉⢁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠲⠀⠀⠀⠀⣰⠐⡃⢸⣿⠃⣠⠈⡉⡻⠓⠞⠃⠛⠶⢁⠀⡠⢊⠂⣀⣀⡀⠀⠘⠀⠁⢀⣀⡄⠀⠄⠀⠒
⠀⠀⠀⠀⠀⢻⠋⢀⡄⠀⣤⠀⠀⠀⠀⡸⢯⣀⡇⢽⣿⢲⠆⣤⠃⠀⠠⠚⡀⣛⠚⣠⠗⠴⠛⠀⠹⣿⣇⠀⠀⠈⠀⠀⡄⠀⠀⠄
⠀⠀⠀⠀⠀⠘⠂⠀⢲⣷⠞⠀⠀⠀⠀⠦⠀⠙⣿⣾⣭⠏⢠⠁⡘⠁⣸⡔⢫⡔⠋⠀⠀⠀⠀⠀⠀⠻⢿⠥⢲⠀⠀⠀⡙⠀⠀⣦⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠈⣍⣹⠀⣂⠄⠚⠁⠀⣻⠞⣠⣈⢛⡾⠥⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⣴⡋⣿⠔⣸⣧⠀⠀⢈⠀⣯⣂
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣷⣼⣤⡀⠀⠀⠀⠀⠁⠀⣀⡤⠞⠁⠀⠀⣿⣿⣷⡄⢉⣽⠃⢺⡿⡦⠈⠋
⠀⠀⠀⠀⠀⢧⡼⠀⢸⠥⠴⠆⡈⠀⣤⠄⠀⠀⠀⠀⠀⠻⣿⣃⡌⢿⣧⢀⠀⠀⠀⠖⠁⠠⠈⡀⠀⠀⠘⡍⢿⠈⠁⠸⠤⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠘⢀⠀⠃⣅⠀⠠⠀⠀⠀⠀⠀⠀⢠⣿⢼⣯⣙⡇⠀⣰⠇⠀⠠⠀⠀⢀⣠⠂⠀⠠⠀⡇⠀⠀⠗⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⣤⡤⠀⠀⠀⠃⠀⠀⠀⠀⠘⡿⠂⠀⢉⠛⢾⢁⠀⠀⢐⣄⣴⡿⠀⠀⠀⠀⠦⢳⠀⠉⠀⢉⢉⡁⡅
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠛⠀⠈⡄⠀⠀⠀⠀⠀⣰⠚⠁⠀⠐⡍⠩⠙⡀⠐⠀⣾⢻⣿⡇⠀⢢⠄⠀⠀⠈⠀⠀⣀⣨⡀
⠀⠀⠀⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠞⢛⡈⠑⡶⠶⠙⡠⠸⠃⠀⠀⢱⣿⣯⣿⣥⡔⠲⠄⢠⡔⠀⠹⣿⡿⠃⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣇⢀⡾⠃⠀⢠⠑⡬⡄⠐⣊⠀⠠⠀⠀⠀⣏⠀⣹⡏⠻⢷⡶⣟⠁⢸⠀⠀⠈⠀⣠⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠘⣏⠁⣠⡀⠈⡻⠀⢼⡘⢛⡔⠡⡄⠀⠀⠀⣽⣿⣷⠀⠄⠀⢧⠀⢸⠖⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠈⠁⢀⠡⠀⠈⣄⠀⠀⠀⠐⠀⡼⠉⡋⢀⢀⠘⠂⠀⡾⢟⣿⠏⠙⢦⡄⠂⠀⡆⠀⠀⠀⠀⠚⠾⠏⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠉⠈⡅⠀⠀⠀⠈⠠⡄⠀⢀⢀⠐⠀⠃⠄⠀⠀⡴⡎⠊⢘⣫⠀⠀⠳⠆⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⢳⣋⣉⣀⣀⠾⡇⠀⠀⠀⢀⣼⡤⠀⠰⣿⠁⠀⠀⠀⠳⣴⢃⠆⡀⠀⠀⠘⠓⠦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡘⠀⠀⢰⠀⠀⠀⠳⠀⠀⠸⡋⠀⠀⢀⡼⢏⡀⠀⠀⠀⢘⢃⠀⠠⠀⠠⢀⣄⠀⠠⠀⠰
⠀⠀⠀⠀⠀⠀⡰⢄⠀⠠⠀⠀⠀⠉⠁⢀⠁⠀⠀⠈⠀⠆⢸⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣥⠀⠂⠀⠀⠖⠨⠁⠀⢀⡀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠒⠀⠒⠒⢈⠀⡛⠠⣄⠀⠀⠰⡏⠀⠀⠘⠀⢠⠃⠤⠰⠁⠄⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⣈⠀⠁⠀⠀⠀⠀⠈⠑⠢⣼⠁⠀⠀⠀⡠⠂⠔⠠⠃⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⡄⢀⢀⢀⢠⣀⡀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠘⠃⠈⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢁⡐⠒⠒⠀⠤⡀⠀⠀⠀⠀⢠⡄⣁⣀⣶⠖⠀⠀⠀⠀⠀⡀⢤⣤⡀⢠
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠙⠉⠐⠈⠀⠀⠀⠀⠀⠑⡤⠡⠾⠤⠤⠥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠈⠉⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠊⠉⠀⠈⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠓⣦⡤⠀⢰⡆⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⣀⠠⠤⢤⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⣲⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡀⡜⢸⠄⠀⠀⢀⠀⠀⠉⠉⠉⠀⠒⠒⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⠉⠉⠀⡙⢣⡀
⠀⠀⠀⠀⠀⠀⠀⣃⠄⠣⠈⡀⡀⢀⢄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⡀
⠀⠀⠀⠐⡀⠀⠀⡆⠀⠀⠀⠑⠊⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢆
⠀⠀⠀⠀⠠⠀⢰⠆⠀⠐⠲⠦⠄⠀⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢣⡀
⠠⡀⠀⠀⠀⢀⡟⠀⢠⠤⢤⣄⠀⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⢘⠀⠀⠘⠒⠀⠊⠀⠆⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⢸⢀⠀⢰⠒⠒⢦⠀⢻⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⠉⠀⠀⢘⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⢱⠀⠠⡀⠀⠍⠀⠔⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⢠⣼⣁⡃⠀⠄⠌⠰⢆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠑⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣎⣀⣀⣀⣀⣀⣈⣐⣒⣛⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠈⠉⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠀⠉⠛⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢀⡀⠀⠄⣀⣀⣀⣉⣉⠉⣉⠉⠀⠀⠐⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣼⣷⣾⣿⣿⣿⣿⣿⣿⢿⣿⣶⣼⣷⣷⣤⠑⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣾⠟⣻⠻⠛⢹⣧⠀⠈⠀⠈⡟⠛⢻⣿⣟⢿⣷⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⢸⣷⣨⠿⠿⡷⠀⣾⣏⡀⢉⣀⠛⠠⠄⣹⣿⣿⣿⣷⣦⣈⢦⡀
⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⠻⣿⡞⠳⠶⣶⣿⣿⣶⣿⣾⣿⣶⡦⠘⠠⠀⣈⡛⢿⣿⣷⡙⡆⠀⠀⡀⠀⠀⠐⠆
⠀⠀⠀⠘⠳⠤⠬⣤⣄⡀⠀⠀⠀⠑⢄⣀⡀⠀⠀⠠⣿⣇⠀⠀⣼⣿⠿⢿⣭⣿⣿⣿⣅⣶⣶⠶⠉⣿⠞⣿⣿⡇⢰⠀⢀⠀⢸⡄⣤⢄⡸⠃
⠀⠀⠀⠄⠠⠤⣀⣌⢈⣠⣆⡀⠀⠀⠀⠈⠛⠶⢄⣀⣈⠻⣷⣾⡿⣉⣴⣾⣿⠿⣿⣿⣿⣉⠄⠀⠀⣭⣽⣿⣿⠅⢸⠀⢸⣄⣸⣇⡼⢋⠀⠀⣂
⠀⠀⠀⠀⠀⠀⣀⠤⠉⠉⣀⡁⠀⠐⠳⠶⠖⠒⠒⢿⣥⣶⣿⢛⣿⣿⣿⣿⣿⣧⣄⠈⠛⠿⣷⣶⣾⣿⣿⣿⡿⠀⢸⡇⠀⠀⠉⢙⡿⢋⣙⡚⠁
⠀⠀⡠⠞⠙⠦⣤⣠⣄⠀⠀⠀⣀⣴⠊⠒⠠⠖⠢⠀⠉⠀⣮⣸⣿⣿⣿⡿⢿⣿⣿⣧⠀⠀⠀⠁⠀⠐⠋⢸⣗⣄⡀⡷⠲⠶⢶⣼⣅⣺⣿⢿⣆
⠀⡏⠀⠀⡀⠀⠈⠁⠘⣦⠀⠀⢁⡀⣦⡉⠂⠀⠀⠀⠀⠀⣇⣻⣿⣿⣿⡏⠈⢻⣿⡄⢰⣾⣷⡶⠀⢠⣾⣧⠉⠿⡇⣿⠴⠤⢮⠤⣄⣭⣩⡄
⠀⣗⣀⣤⡽⢿⡶⡿⠿⢏⠁⠾⠾⠷⠶⠛⠂⠀⠀⠀⠀⠀⠘⠛⠁⠙⠛⠀⠀⠸⣿⣧⡳⣈⠙⢻⣿⣿⣿⡟⠀⠀⡇⢹⡀⠀⠀⠀⠀⠀⠀⠄
⠀⣨⢉⣠⣶⣏⣠⡂⠐⣠⣶⣶⣶⢶⡆⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠹⣿⣻⣦⣤⣴⣿⣿⣿⣷⡄⠀⡇⠈⣇⠀⠀⠄⠀⠀⠀⠐
⠀⠈⠟⠉⢿⣿⣥⡟⢠⣾⠍⠀⠒⠈⣿⣆⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠿⣿⣿⣿⣿⣿⣷⣼⣷⣦⠘⣦
⠀⠀⠀⠋⠙⠙⠏⠀⢹⣿⠇⢀⠀⠀⣿⣿⡄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢟⡦⣿⣿⣿⣿⣿⣿⡿⠇⢰⣌⡳⣄
⠀⠀⠀⠀⣄⠀⠀⠀⣿⣿⠀⠀⣠⣶⣿⣿⡄⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣯⣹⣿⣿⣿⣷⣳⣾⣿⣷⡈⢦⡀
⠀⢀⠴⠘⠉⠀⠠⢠⣿⣿⢤⣾⣿⣾⣿⣿⣷⡌⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣆⣀⡀⠀⠀⠀⠹⡜⡿⣿⣿⣟⣻⣿⣿⣿⣿⣦⠙⢦⡀
⠀⠀⠀⠀⣀⠀⠀⣼⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠀⠀⠹⡴⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣢⣝⢆
⠀⢀⡇⢂⣨⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣿⣿⣿⣿⣿⣿⣿⣷⠽⣿⣿⣷
⠀⠈⢿⠟⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢱⢹⣿⣿⣿⣿⣿⣎⢻⡦⠙⢻⣿
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠈⠉⠨⠮⠛⢿⣧⣿⣷⡄⠀⡄⠀⠀⣀⣀⣀⡀⠤⠤⢤⠀⠀⠀⠂⠀⠀⠀⠀⠀⢣⠹⣿⣿⣿⠛⣿⣧⡽⣯⣆⢻
⠀⠀⠀⠀⠀⣰⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠈⢻⣿⡻⣿⠀⠓⠀⠁⠁⠀⢻⡅⠀⠀⢨⡳⠀⠀⠀⠀⠀⠀⠀⠠⠈⢇⠻⣿⣿⣷⣿⣿⣷⡉⢿⣷
⠀⠀⠀⢸⢰⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣻⡇⠰⡀⠀⠀⠀⠀⠻⣄⠤⠤⠥⠴⠆⠀⢠⣄⠂⢠⣄⠀⠈⢆⠹⣻⣿⣿⣿⣿⣿⣀⠹
⠀⠀⢀⠄⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⡘⢿⣿⡀⠄⡀⠀⠀⠀⠠⣄⣀⣀⣀⢀⡀⠀⢸⣿⣧⡙⢿⠀⠀⠈⢆⠹⣿⣿⣿⢿⣿⣿⡀
⠀⠀⠐⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠻⣧⠀⣰⠀⠀⠀⠀⠉⠙⢻⣿⠾⠛⠓⠚⠻⠿⣷⠞⠙⠶⠂⠈⠀⢻⣿⣿⣿⢿⣿⠛
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⠙⡦⠼⠷⠶⠋⠉⠉⠡⠀⠀⠀⢠⣄⣛⡶⠂⣀⢈⡟⠀⠀⣄⢻⣿⣿⡾⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠁⠙⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠒⠚⠈⠀⠀⠀⠙⡄⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠛⠀⠀⠀⠀⠀⠀⣤⢀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠸⣦⠀⢀⠀⠀⠨⠀⠀⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠐⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠐⠉⢴⠖⣀⣤⠤⠀⡊⠃⠀⠀⠠⠤⠀⠀⠀⠸⠃⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⡀⠒⠐⠀⠀⠉⡀⠂⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠲⠂⠀⠈⡁⠀⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠛⠉⠁⠀⠀⢀⠀⠈⠉⠛⠻⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠀⠀⠀⠐⡿⠇⠀⠀⠟⠏⠀⠀⢠⣦⠀⠀⠙⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⢿⠇⠀⠀⣀⣤⣤⣴⣦⣤⣤⣀⡀⠁⠀⠀⠀⠀⠈⢻⣿⣄⣠⣤⣶⡆
⠀⠀⠀⠀⠀⠀⣴⣿⠏⢀⣠⡀⠀⠀⣠⡾⠛⠉⠉⠀⠀⠀⠀⠈⠉⠻⣷⣄⣠⣤⣶⡶⠿⠛⠛⠉⠁⠸⣿
⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠙⠀⣠⡾⠋⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠾⠛⠋⢉⣀⡀⣤⡀⣶⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⢰⣟⣁⣀⣤⣴⡶⠾⠟⠛⠉⠁⢠⣤⣤⡐⣇⢾⡏⠿⠸⣧⣼⡇⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⢸⣿⣁⣀⣤⣴⡶⠿⠟⠛⠉⢁⡀⢠⡴⣦⢸⡇⠀⠀⣷⣼⣇⢻⡜⣷⠀⣤⢻⡄⢿⡀⠀⠀⠀⢹⣧
⠀⣤⣤⣶⠿⠟⠛⠉⣉⢠⡀⣤⠰⠾⣟⠉⣷⣸⣷⠘⢿⡇⠀⠀⢹⡇⢻⡎⣷⠙⠶⠛⠈⢁⣀⣠⣤⣶⠶⠿⠛
⠀⢻⡇⠀⠀⠀⠀⢿⡋⠙⣷⢻⡆⠀⢿⡄⢹⡏⢹⡆⠘⣷⠀⠀⠀⠛⠈⣁⣠⣤⣶⠶⠿⠛⠋⢉⣿⡇
⠀⠈⣿⡀⠀⠀⠀⠸⣿⠃⢹⡎⣷⣠⡘⡧⠀⠛⠈⠁⣀⣀⣤⣴⡶⠿⠛⠛⠉⣽⠇⠀⠀⠀⠀⢸⣿⠃
⠀⠀⢹⣧⠀⠀⠀⠀⢿⠄⠈⠛⠈⣉⣠⣤⣴⡶⠿⠛⠛⠉⠁⠀⠀⠀⠀⣠⣾⠋⠀⣀⡀⠀⢠⣿⡏
⠀⠀⠈⣿⡄⠀⠀⣀⣤⣴⡶⠾⠟⠛⠻⢿⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠟⠁⠀⠈⠛⠂⣠⣿⡟
⠀⠀⠀⠸⠿⠟⠛⠻⣿⣧⡀⠀⠀⠀⠀⢀⠈⠙⠛⠳⠶⠶⠟⠛⠋⠁⠀⢠⣶⠄⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠻⠇⠀⠀⣠⣤⠀⠀⢐⣶⠄⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣧⣶⣤⠀⡀⠀⠈⠁⠀⠀⠀⢀⣀⣤⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣷⣶⣿⠿⠿⠟⠛⠉


⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣻⣿⣻⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠐⠐⠲⠤⣴⣤⣤⣴⣶⣦⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣈⣤⠀⠠⣰⡿⢶⡄⢉⠿⠿⠻⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⣿⡟⢀⣾⢿⣿⠇⠀⢻⣿⠿⣋⠈⢷⠂⠀⠈⠙⠷⣶⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⢏⡜⠋⡩⣼⢿⣿⣿⠀⠀⠀⠻⢿⡇⠀⣽⡗⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣸⡟⣼⠀⢰⣿⠙⠷⣾⢯⣷⢛⣿⣿⣿⠕⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠛⢿⣧⣴⣿⣘⢿⣿⣿⣿⣯⣾⠛⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠸⡇⠹⡏⢻⣿⣭⣥⣴⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣿⣿⣿⣿⣿⣇⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⢿⣿⣿⣿⣷⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⣿⡿⡟⠛⠸⣿⣿⣿⡷⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠁⠀⠀⠀⠘⡀⢎⢻⣿⣿⣿⠑⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠁⢢⢧⢿⣿⠟⠃⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣦⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠶⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣰⡀⣿⣿⣿⠃⢠⣿⠀⠀⢰⡆⠀⠀⣿⣿⣿⣿⠀⣾⣦⡀⠀⣸⣿⣀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡟⣿⡍⣿⢛⣿⠀⢸⣿⡇⠀⢸⣧⠀⠀⣿⣟⣿⡿⢸⡟⠙⡟⣆⠈⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢺⣇⢳⠸⣧⣿⣼⣿⠆⢸⣿⡇⠀⢸⣿⠀⢠⣟⢸⣿⡃⣼⠇⠘⢡⣿⡄⣻⣷⡄⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠹⣯⣿⣿⡇⣼⣿⠁⠀⣿⡟⠀⣿⣧⡿⣿⣧⣿⠀⢀⣾⣿⠇⣾⡟⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢿⣿⣦⡘⢿⢿⣿⠀⣿⣿⡇⢰⣿⡇⣰⣿⡟⣹⢿⣿⡇⣠⣾⣿⡟⣴⣿⣷⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠀⢻⣿⣷⣘⣿⣿⠁⣿⣿⠀⢸⣿⡇⣿⣿⣷⠃⣼⣿⣴⣿⡿⢋⣾⣿⠃⠀⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⣿⣿⣿⣿⠀⣿⣿⠀⣿⣿⠡⣿⣿⢃⣶⣿⣿⣿⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⣿⣿⣿⣿⠀⣿⣿⠀⣿⣿⢘⣿⣿⣿⢢⣿⣿⡿⢿⠟⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣄⣀⣠⣽⣿⣿⣿⣿⠀⣿⣿⣾⣿⡏⢸⣿⣿⣷⣿⣿⣿⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣾⠉⠀⢠⣿⣿⣿⣿⣿⡁⢀⠈⣻⣿⡋⢁⠀⣹⣿⣿⣿⣿⣿⣿⣟⠛⢳⠾⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠙⠛⠶⠾⠟⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠾⠿⠿⠯⠉⠉⠩⢿⣧⣀⣤⣴⠷⠄⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠘⢆⠀⠀⠀⢠⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡰⠋⠀⡠⢇⢸⠀⠀⣰⢃⠄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠉⠀⠵⠆⠈⠀⠈⡈⡀⠀⢁⠊⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠤⠀⠀⠀⣀⠇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⡠⢀⡂⠈⠁⠆⠧⠀⠠⠤⠤⡀⠀⠀⠀⠀⣀
⠀⢀⡤⠊⠉⠀⠀⠀⠀⠀⠉⠙⢖⠒⠸⠿⢦⣄⣀⣉⢭⠁⢀⣀⣆⠀⠀⢠⡤⠀⢸⠀⠀⠂
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⢋⠇⠀⢸
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠮⡀⠁⠀⠀⡇
⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢀⣀⠜⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠋⠀⠀⠀⠈⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠓⠢⠤⢀⣀⣀⣀⡴⢠⠴⠖⠿⣁⠃⠀⢀⣠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢜⠉⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣛⣶⣈⠄⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢤⢀⡀⣀⡀⣉⡀⣤⠮⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⠋⢠⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠞⠉
⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⡀⠀⠀⠀⢋⡁⠚⠆⠙⢉⠃⠐⢢⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠒⠠⠄⠀⠀⠀⠀⠀⠘⡎⠀⢀⠐⢉⡠⠀⡆⠀⠈⠇⠀⠀⠀⡟⠀⠐⠠⠄⠀⠁⡀
⠀⠀⠀⠀⠀⠠⠄⣈⡁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡄⢠⢀⠄⠉⠛⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠂⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢖⣀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⣻⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽

⠀⢰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠔
⠀⢸⣿⣿⣄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣼⣿⣿⣿⣷⣦⣤⣤⣾⡇
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⣠⠴⠒⠒⠛⠓⠲⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⢀⣿⠀⣀⠤⠄⠀⠀⠒⠺⡇
⠈⣹⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⡆⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠁⠁
⢰⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠳
⠘⣿⢿⣿⠅⠀⠴⠒⠒⠀⠀⠐⠁
⠀⠀⠀⠹⡄⠐⠲⠆⠂⠀⠀⠂⠃⠐⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⢉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢿⠙⢿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⢉⣙⣛⠃⠀⢘⢋⡉⡉⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠁⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠛⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⣁⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠮⠏⠁⠛⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⢠⣆⡀⢠⠖⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢹⡿⠃⠄⠀⡀⠿⠺⠞⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⠏⠉⠃⠈⢂⣸⠠⢠⠀⠐⢎⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡦⢦⣀⢻⠉⢹⠔⠀⠀⠘⠆⠀⠀⠀⢀⢴⡌⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠅⠠⠀⠘⠀⠀⢀⠀⠀⢀⠀⠀⡀⠀⡃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⡀⠀⠤⠖⠀⠀⣀⣀⡄⠀⠁⠐⠈⡀⠅⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢮⣄⣡⡠⠤⠶⣾⣿⣿⣿⣿⣷⣆⠀⠀⡀⠱⡐⣄⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠠⣤⣴⢞⡀⢙⢹⣷⠑⢶⣀⠀⣀⡤⠤⣬⣉⣉⡉⣰⠂⠋⠈⡅⠘⣄⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠒⢊⠁⠀⠀⠀⠑⠏⢻⣮⠈⢿⡇⠀⣸⢿⡁⠀⠀⣹⠍⠛⠛⠉⠲⠞⠀⠠⠀⠘⢂⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣠⡴⠊⠐⠆⠀⠀⠀⠀⠀⠉⡅⠀⠀⠁⣴⠀⠀⠀⠀⣴⠁⠀⠠⠆⠀⠀⠀⡀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠋⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢷⠀⠙⡄⠀⠀⣷⠀⠀⠀⠀⠀⢀⡈⠀⠀⠁⣀⢜⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠙⣦⡳⠄⣀⣀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠐⠂⠀⠀⠙⡯⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠴⠟⠋⢁⡐⠋⠻⠦⣄⡣⣿⠄⠀⢀⣀⣀⡀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠴⠉⠀⠀⠉⠀⠈⠁⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠄⡀⢤⡀⢀⣠⠀⠀⣀⣀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠓⠈⠀⠀⠀⣠⣶⣄⣈⠙⠻⣿⣿⣿⣿⠿⣿⠋⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⡖⢈⣥⣶⣿⣿⡆⠀⠰⣿⣭⣭⣿⡤⠀⣎⡉⠉⠉⠺⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⢹⣿⠿⠀⣸⣿⣋⣿⣿⠃⠀⠀⠙⣿⣟⣉⣶⠀⠀⣯⠁⠀⠀⠀⠈⠛⠿⢷⣾⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢃⡴⠉⠁⣴⢢⣿⡹⠿⣻⣿⠀⠀⣀⣀⡘⠛⣿⡂⠀⢰⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠇⣼⠇⠀⣲⣿⡏⠈⠓⢺⣿⢫⣾⠛⣻⣿⣿⣿⣷⠙⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠏⠀⠐⡛⢻⣿⣦⣤⣿⣇⠘⠿⢿⣿⣿⣿⣟⣡⣾⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠀⠀⠀⠁⠈⣿⡎⠻⣏⠛⢻⣿⣟⣛⣉⣉⣭⣿⣿⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⠛⠀⠀⠙⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⣿⣿⣿⣿⣿⣿⣿⣧⣤⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣾⣿⣿⣿⣿⡿⢻⠟⠁⠈⠹⡟⣿⢧⣝⠳⣭⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣟⡽⠁⠀⠀⠀⠀⠀⢋⠈⢂⠙⢷⣾⣯⣿⣏⠛⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣯⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⡘⣿⣿⣿⢿⠄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣳⣼⣿⣯⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣲⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡍⢀⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢨⣿⣿⣿⡏⠀⠀⣀⠖⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣏⠁⢀⡀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣷⠀⣾⡿⢻⣿⡇⠀⢸⣿⡇⠀⠀⠀⣿⡀⠀⠀⢸⣿⣿⣿⣿⡿⢀⣿⣿⣶⣄⠀⠀⣿⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠫⣿⠀⢿⡗⢘⣿⡇⠀⣸⣿⣷⠀⠀⢰⣿⡇⠀⠀⢸⣿⢏⣿⣿⠇⢸⡟⠁⢹⡏⣷⡀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡌⠳⡀⢿⣇⣿⣧⢸⣿⡇⠀⣿⣿⣻⠀⠀⠀⣿⡏⠀⠀⣼⡇⣸⣿⣿⠀⣿⡗⠀⠜⢠⣿⣷⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠄⠁⠀⢻⣿⡸⣿⣿⡧⠀⣿⣿⡇⠀⠀⣸⣿⠃⠀⣾⣿⣧⣿⢿⣿⡄⣿⠀⠀⠀⣼⣿⣿⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣄⠀⠠⣿⡟⢻⣿⡇⢰⣿⡿⣷⠀⢠⣿⡿⠀⠀⣿⣿⡿⢳⢿⣿⣿⡇⠀⣠⣾⣿⣿⠇⣼⢯⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⣿⣿⣷⡆⠸⣏⣻⣿⡇⢸⣿⣧⡏⠀⣼⣿⣿⠈⣿⣿⡟⣡⠏⢘⣿⣿⢡⣾⣿⣿⠿⣣⣾⣫⡟⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⢿⣿⣿⣦⣽⢻⣿⡇⢸⣿⣿⠀⠀⣾⣿⡟⢀⣿⣿⣷⠏⠀⣾⣿⣯⣾⣿⢛⣠⣴⣟⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠻⣿⣿⣿⣿⣿⠃⢸⣿⣿⡆⢠⣿⣿⠇⣸⣿⣿⣇⡜⡁⣼⣿⣿⣿⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⠆⢸⣿⣿⡇⢸⣿⣿⠆⢸⣿⣿⣿⡇⣱⣿⣿⣿⡿⠿⠟⠋⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⣹⣿⣿⣿⣿⣿⠂⢠⣿⣿⣷⣼⣿⡿⠀⢸⣿⣿⣿⣴⣿⣿⣿⣿⣍⣀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡰⠂⠛⠋⢙⣻⣿⣿⣿⣿⣾⣿⠋⠀⠀⠙⢻⣿⡿⠛⠁⠀⠘⠻⣿⣿⣿⣿⣷⣿⣿⣿⣟⠛⠛⠶⡾⠇
⠀⠀⠀⠀⠀⠹⣧⣠⣀⣀⣼⣿⣿⣿⣿⢿⣿⣿⣷⣴⣷⣶⣾⣿⣿⣦⣾⣀⣾⣿⣿⣟⠛⠛⠛⠛⢛⣿⣿⣀⠀⣄⣡⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡄⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⡀⢹⠀⠀⠀⠀⣰⠋⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡞⠁⠀⣠⠎⢣⠈⡆⠀⠀⣰⠃⠰⠃⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⠀⠀⠀⠡⠠⠚⠁⠀⠈⡆⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⠓⢄⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⠀⠀⡐⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠈⡌⠂⠐⠈⠁⠒⠒⢦⡀⠀⠀⢀⡀⠀⠄
⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢏⠓⠒⠼⢮⣶⣤⣀⣀⡈⠁⡄⠀⠀⢀⣠⣃⠀⠀⠀⢠⣤⡀⠀⣿⠀⠀⠃⠒
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠉⠈⠀⠀⠈⠂⠀⠀⠀⠈⠀⠀⡰⢋⡇⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠱⡼⠀⠀⢰⠇
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠑⢄⡰⠃⠀⠀⣸
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠄⢀⡴⠁⠀⠀⠀⡏
⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡀⠴⠊⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣀⣠⠄⠒⠋⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠐⠃⠚⣉⠁⠈⠀⠠⠀⠠⠤⠤⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠗⠢⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡾⣂⣡⠂⠒⠒⠒⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⣀⣀⣈⣉⠭⣜⠄⣦⠿⢢⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢽⡿⠡⢄⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⡧⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡧⠄⠀⢤⠉⣶⠒⢲⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠞
⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠂⠀⠚⠁⠀⠉⠀⢰⣊⡐⠐⠁⠃⣰⠋⠉⠀⠨⠇⣀⣤⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠉⠙⠂⣠⡤⠀⠀⠀⠀⠀⠀⠈⠇⠈⢀⡠⠊⠸⣌⠇⢈⡵⠀⠀⠀⢳⠀⠀⠀⠀⡟⠀⠈⠉⠒⠀⠀⠀⠦⣄
⠀⠀⠀⠀⠀⠠⣔⡒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢷⠒⢢⠀⢀⠔⠑⠛⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠲⠄⠠⢤⡤⠄⠀⠀⠀⠀⠀⠈⠳⠾⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠶⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠰⠖⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠒⠲⠤⠤⠤⠤⠤⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠂⠂⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣦⣤⣀⠀⠀⠀⣠⣀⣦⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠋⠀⣀⣀⡉⢙⠛⠙⣿⢿⣋⡑⠐⠒⠻⢿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣤⣮⣭⣿⣿⣿⣦⣠⣿⣿⣿⢭⣽⣦⣀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠊⢉⠙⢿⣾⡟⣻⣿⢹⣿⣯⣿⠿⠏⣰⠋⢈⣦
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠛⢦⣤⣬⡶⣂⠉⠁⠃⡠⠙⠀⠩⣭⣥⣶⣾⣯⣉
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠐⠛⢻⡿⣿⢿⣤⡶⠶⣷⠾⢾⣿⠟⢻⠋⠛⠃
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠠⣿⣿⣶⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠹⢿⢿⣿⣿⡻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣋⣼⠇⡇⠸⡀⠈⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠿⣿⣿⡿⠉⠀⠂⠀⠁⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡅⠀⠀⠀⠀⠀⠀⠟⣿⣿⠀⢀⠆⠀⠀⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣴⡟⢶⣄⠀⠀⢿⣿⠀⣿⠀⠀⠀⢴⠀⢸⣿⡙⠀⠀⣠⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠐⣿⡿⠌⢿⡇⢰⣿⣿⣿⣿⡇⠀⠀⣿⡇⣽⣿⣷⣆⣾⠏⢀⣾⠳
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠘⠋⣿⣦⠘⣷⣿⣿⣿⣿⣿⡵⠀⢠⢿⡇⣿⣿⣿⣿⣿⡴⣻⡏
⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⣿⣿⠐⠀⠏⣿⣿⣿⣯⠋⣿⡏⣰⣿⢷
⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⠀⠀⠀⣀⣿⣿⣿⢿⢿⣿⣿⣿⣆⠀⡾⣿⣿⣿⠏⢸⣿⣿⣿⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣤⣀⡄⢻⣿⣿⣼⡘⣿⣿⣿⣿⣴⣷⣿⣿⣿⣄⡾⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⣿⣿⣿⣿⣿⣧⣽⡏⣿⣿⣿⣿⣿⢸⣿⣾⣷⣿⣿⣿⠙
⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀⠀⠀⢴⣿⣟⢻⣿⣿⣿⡇⢿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡟⣿⣦
⠀⠀⠀⣰⣿⣶⣤⣄⢠⣰⣿⣷⠀⠘⠛⠛⠛⠻⠿⣿⣷⣾⣿⣿⣿⣶⣿⣿⠿⠟⠛⠛⠛⠛⠒⠂
⠀⠀⠀⠙⠛⠛⠓⠀⠚⠛⠛⠛⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣄⠀⠘⠲⠶⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠁⠀⣸⠇⠀⠀⢠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⠴⠚⠁⠀⠀⠀⣿⠀⠠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢰⡇⠀⠀⠀⠀⠀⡞⠁⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠈⠳⣄⣀⣷⠀⠀⠀⡀⠀⠙⠶⠊⠙⠋⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⢢⣼⣠⠃⠀⠀⠀⠀⠈⠀⠀⠀⣠⠏⠳⡄⠀⠀⠀⠀⠀⣮⠽⠃⢿⣳⢰⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢯⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠿⡗⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣅⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠒⠂⠀⠀⠀⣿⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⣠⠖⠚⠢⠤⠤⠤⠤⠤⠤⠤⠔⠒⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢀⡖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠖⠒⠒⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀⠸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠄⠒⠒⠒⠚⠛⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⢤⣤⠤⠴⠚⠋⠷⣤⣤⠴⠓⠦⢤⡤⠴⠛⠓⠒⠒⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣏⠉⣿⠀⠀⠀⠀⠀⢠⠞⠛⢣
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠘⠦⡤⠞⠀⢀⠀⠀⠀⠀⢀⡞⢳⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠸⠤⣿⡀⣧⠞⠀⠀⠀⠀⠀⠀⠀⢰⣛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡿⠀⠀⠀⠈⠳⣬⢶⣯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡄
⠀⠀⢀⣀⠤⠴⠒⠒⠲⢤⣀⢠⡴⠶⠶⠒⠒⠒⠲⡆⠀⢸⡇⠀⠀⢹⠀⠀⣠⠴⠚⠋⠉⠛⢲⡴⠏⠉⠉⢳⢘⡋⠉⠉⠉⢹⠀⢠⡟⠃⠀⠀⠸⠃
⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠘⣾⡄⠀⠀⠀⠀⠀⠀⢳⠀⡸⠀⠀⠀⡸⢀⣾⠃⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠸⡇⢸⡇⠀⠀⠀⣤
⢸⡆⠀⠀⠀⠀⣤⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣤⡴⠞⠀⡧⠀⠀⠀⣧⡞⠡⠀⠀⠀⠀⠀⡞⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃⡄⠀⢀⡇
⡎⢃⠀⠀⠀⠀⠀⠀⠀⣤⣾⡁⣶⠀⠀⠀⠀⢸⣇⣀⣸⠃⠀⠀⠀⣿⠀⢀⡄⠀⠀⠀⠀⠙⠒⠙⡇⠀⠀⠀⣠⢤⡀⠀⠀⠀⣧⠸⣧⣗⣲⡾⠁
⣇⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠈⠉⠉⠏⠀⠀⠀⠀⠉⠁⠈⠉⡂⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡆⡼⡏⠉⣳
⠸⡄⠻⠀⠀⠀⠘⠛⠀⠀⠀⢀⡿⠀⣀⣀⣀⣀⣀⣀⣰⠀⠀⠀⠀⠀⠀⠀⢈⠙⠢⢄⣀⣀⣀⡠⡴⠦⠀⠐⣧⠀⠀⠒⠈⠉⡟⠙⠧⠼⠛⠁
⠀⣧⠰⢇⠀⠀⠀⠀⠀⢀⠰⣻⠁⠉⠉⣉⣉⣉⣠⣿⠏⠁⠈⠉⠉⠉⠉⣩⡟⠶⣤⣀⣀⣀⣀⡴⠶⠶⠶⠶⠋⠳⠶⠚⠛⠛⠁
⠀⠸⣆⡀⠉⠉⠀⠉⣉⣠⠞⠛⠛⠛⠋⠉⠉⠉⠉⠘⠓⠛⠛⠛⠉⠉⠙⠉⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⣤⡴⠶⠶⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡖⠳⢶⠋⠉⠀⠀⣠⠞⠋⣠⣤⠴⠦⣤⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠙⣆⠀⠀⠀⠀⠘⠁⢠⠞⠁⠀⠀⠀⠀⠉⢣⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡦⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣶⡆⠀⠀⠀⡇⠈⡷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠉⠁⠀⠀⢀⠇⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠤⠤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠐⠀⠀⠀⠀⠤⠀⠴⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⣠⡴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡄⢠⡾⠉⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡷⠚⠁⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢸⠀⠀⢀⡔⠛⠒⠦⣄⡀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠑⠀⠀⢀⡀⠀⢰⡂⠀⠀⠀⠀⠀⣼⠀⢠⠏⠀⠀⠀⠀⠀⠙⢦⡀⠀⢠⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣄⠀⠀⠀⢰⡏⢹⠀⠀⠁⠀⣀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡘⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⡿⢹⡆⠀⠀⣿⠀⢸⠀⠀⠀⣼⠉⢳⡀⠀⡿⠀⢸⠀⠀⠀⠀⣀⣀⣀⣠⣤⣼⣿⠀⠀⠀⠘⠛⠛⠓⠶⢦⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⡇⢸⡇⠀⠀⡟⠀⢸⠀⠀⢠⡏⡠⠚⠧⡼⠃⠀⢸⢉⡏⠉⠉⠉⠉⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠠⣤⡤⠞⠁
⠀⠀⠀⠀⠀⢾⣋⠉⠀⠀⠀⠀⠀⠀⠈⠁⠟⢸⡇⠀⢰⡇⠀⠈⢇⣀⡼⠀⠓⠙⠳⠖⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣤⣤⠟⠳⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠉⠛⢶⡄⠀⠀⠀⣠⠤⠤⠦⠀⣇⠀⢸⠓⠢⡀⠈⠉⠀⡤⠤⠀⢀⡴⠀⠀⠸⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠀⣀⣠⠤⠴⠏⠀⠀⠀⠙⠲⠂⢤⠀⢿⣤⠞⠓⠋⠀⠀⠀⠀⠓⠒⠒⠒⠲⡆⠀⡖⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⣄
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣄⠀⠀⠀⠛⢶⡄⠀⠈
⠀⠀⠀⠀⠀⠛⠦⠤⠤⠤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠾⠤⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠶⠶⠦⠤⣤⣤⣀⣀⣀⠀⠀⠀⠀⢀⣠⡿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣀⣀⣀⡈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⢀⢤⡀⠀⠀⠀⠀⢀⣀⠺⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠎⠀⠀⠀⠀⣇⣈⡇⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠰⠭⣆⢼⠝⠀⠀⠀⠀⠀⣔⡦⠀⠀⠀⠀⠀⠀⠁⠀⠾⠆⠀⠀⠸⢤⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡤⠤⢤⠀⠀⠀⣀⣀⣀⡀⣀⣤⣤⠀⡤⠤⠤⡄⠀⣠⠖⠂⢰
⢀⡤⠒⠉⠁⠀⠉⠳⣼⠉⠉⠁⠀⠈⡅⢰⠁⠀⢸⢀⡴⠋⠀⠀⠀⠛⠁⠀⠈⠶⠂⠀⠀⢸⠀⡏⠀⠀⠈
⢸⠀⠀⠀⣤⠀⠀⠀⣿⠀⠀⠀⢠⡤⠇⣸⠀⠀⢸⡞⠁⠀⠀⠀⡔⠋⡀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡆
⡆⠆⠀⠀⠀⠀⠐⠺⢤⡆⠀⠀⠠⠧⢤⡇⠀⠀⢸⡄⠤⡀⠀⠀⠈⠉⠙⠀⠀⠰⠲⠀⠀⠀⡆⢳⡧⢞⠁
⢣⠸⠀⠀⠀⠦⠀⠀⢈⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠧⢀⠀⠀⠀⣠⡄⠀⣤⠀⠡⠠⢐⠇⣏⡤⠞
⠘⡄⢆⣀⠀⢀⡀⠄⣾⣉⣉⣉⣉⣩⣿⠓⠒⠒⣂⣀⡸⠳⢤⣀⣈⡠⠶⠶⠶⠊⠲⠒⠒⠊
⠀⠑⠦⠤⣤⠤⠴⠊⠉⠉⠁⠀⠉⠀⠈⠉⠉⠁⠀⠉⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⢤⡴⠚⠉⢉⡽⠛⣉⣁⣉⡉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠙⡆⠀⠀⠀⠋⡰⠋⠁⠀⠀⠉⢂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠻⠃⠀⠇⠀⠀⠀⠀⡅⠀⠐⠿⠀⠀⢸⠀⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⠄⠠⠄⠊⠀⠀⠀⠀⠀⠙⢤⣀⠀⢀⡠⠊⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠈⢳⡀⠀⢀⡴⠒⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠔⠒⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡴⠁⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡦⠂⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠁⠀⠀⠀⠆⠀⠀⡀⠀⠀⢀⡁⠀⠀⣈⣀⠀⠀⠀⠀⠀⣰⠏⠀⡀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠆⠀⠀⠀⢀⠄⠀⠀⠀⢨⡀⣠⠌⠀⠉⠁⢠⣀⠀⣠⠁⠀⠃⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠀⠀⢀⠞⡇⠀⠁⢀⡀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠈⢆⡏⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⠁⡇⠀⢸⠀⡷⠀⠀⠏⢹⡄⡸⠀⣧⣤⣤⡤⠤⠤⠤⠼⠷⠀⠀⠉⠉⠙⢓⡆
⠀⠀⠀⢠⡶⠒⠒⠒⠒⠒⠒⠛⡆⡇⠀⣿⠀⢡⠀⡸⠰⠡⢍⡇⠀⠰⠊⠀⠀⠀⠀⠀⠀⠣⣀⣀⡦⣌⡽⠋
⠀⠀⠀⠀⠉⠓⢦⠀⠀⢀⠤⠤⠁⢇⠀⡗⢢⠀⠉⠁⠤⠤⠤⠆⠀⠰⠀⡄⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⢠⠔⠒⠊⠀⠀⠈⠁⠐⡆⠸⠖⠉⠁⠀⠀⠀⠉⠉⠉⠳⠀⠋⠉⠁⠀⠀⠀⠀⠀⠈⢧⣀⡀⠀⠀⢠⣖⠒⠲
⠀⠀⠀⠸⠤⠀⣀⣀⣀⣀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⣼⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠺⡖⣤⣤⡄⠈⣣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢙⡄⣈⣼⣷⣼⣺⣇⠝⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢘⣷⣿⣿⣿⣿⣿⣿⣧⡀⠀⠸⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣶⣿⠿⣿⠏⠁⣽⠻⣿⣿⣿⣷⡀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⠀⠀⠀⠠⠀⠀⠀⠩⠛⠋⠙⠃⠀⠸⣷⠀⠀⠀⠀⡐⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣠⠊⣽⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠌⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠉⢀⢠⠆⢈⣀⡀⢈⠛⠚⣡⠞⢹⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⢀⠀⠊⢀⠞⠀⣀⠾⠚⠛⠿⡖⢆⡀⢀⡀⣠⣶⣿⣿⣿⣷⣿⣧⣤⠏⠀⠈⢧⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠛⠀⢰⠇⠀⠀⣀⠀⢻⣤⣧⣤⣿⠟⠉⠋⠛⠿⣿⣿⡏⠀⠀⠀⠀⠀⣿⡼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡆⢀⣀⠀⠀⣴⢺⣿⣿⠋⠡⠔⠁⣀⠀⢉⣿⡇⠀⠀⠀⠀⠈⣿⠃⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢡⡈⠉⠀⢀⣿⡿⡟⣧⣜⠲⣤⡄⢨⣤⢞⣿⡇⠀⠀⠀⠀⠀⢷⣄⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣠⡴⠋⠑⠒⠊⠘⢷⣿⠟⠻⠦⢠⡶⠛⡫⠀⣿⣄⠀⠀⠀⠀⠀⡞⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⡷⢶⡆⠙⠻⡆⠀⠀⠀⣠⣿⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠁⠈⠀⠀⠀⣦⠐⠆⠀⠀⠀⠀⠠⠤⡄⢀⠠⠁⠀⢹⡆⠘⣿⠀⠀⢰⢿⡏⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⢄⠀⠀⠀⢸⠀⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡆⠀⠈⣿⡖⣿⡶⠏⢁⣿⣷⣦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣳⣿⣡⣠⠚⠃⢸⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⣼⡿⠁⠀⠀⠒⠀⠀⠀⠀⠀⢸⣿⣏⡀⣒⣟⡥⣮⡼⠁⠀⠀⢿⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣼⠀⠀⠈⠁⢀⡤⠊⠀⠀⠀⠀⢠⠄⡈⣙⣿⡇⠉⢩⡄⣾⠂⠀⠀⠀⠀⠹⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠰⣯⠾⢻⡀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠐⠋⠃⣾⣷⣥⣠⣀⠀⠀⠀⣼⠏⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⡂⢠⣏⣽⡆⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⣲⣶⣿⣿⣿⣷⣬⣷⣄⡀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⢲⡏⠁⠀⠈⠙⠛⠳⢾⡧⣤⣀⣄⠀⠀⢘⣷⣤⣷⣶⣿⡿⠿⠛⠉⠉⠋⣿⣿⣧⣟⣻⡿⣿⣫⣾⡇
⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠈⣶⣽⢦⣤⣀⣀⣄⠀⠀⠀⠐⠭⡛⢟⣋⣩⣥⠭⠉⠋⠁⢠⣤⣤⣾⣔⣤⡿⣿⣷⣿⣿⣷⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠚⠒⡅⠀⠛⠈⠉⠀⠀⠀⠈⠀⠀⠈⠙⠦⣤⣄⣀⠈⠛⠿⣿⡶⠶⠖⠛⠛⠛⠉⠙⠉⠉⠉⠀⠐⠛⠛⠛⠛⣏⣉⣉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⣾⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠿⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡀⠀⠀⠀⠀⢰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⢋⣽⠟⠋⠉⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⠿⢿⣭⡙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠈⢿⣆⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⠀⢰⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣀⡀⠀⠀⠀⠀⢀⣠⣴⠟⣿⠏⣿⠀⢸⡿⣿⠻⣦⣄⡀⠀⠀⠀⢀⣠⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⣍⠛⠛⠛⣿⣿⡏⠉⢀⣾⠏⠀⣿⠀⢸⡇⠹⣧⠀⢩⣿⣿⠛⠛⢛⣹⡿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠙⠿⣦⣄⣈⣛⣡⣴⠟⠁⠀⠀⣿⠀⢸⡇⠀⠙⢷⣤⣙⣋⣠⣴⠟⠋⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣸⡏⠀⠈⣿⠀⠀⠀⠉⠉⠉⠉⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⣀⣴⠏⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⣀⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⣀⣠⣤⠤⠤⠶⠤⢤⣤⣌⣉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⢷⠀⠀⠀⠀⠀⡜⠀⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣆⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⠇⠀⠀⠀⢀⣼⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣷⣤⣤⣤⣾⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣀⣀⣀⠀⣀⡀⣀⣀⢀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⣤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠅⠀⠉⠙⠋⣤⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢴⣶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣟⡋⡈⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⠿⠟⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣤⣦⠥⠀⠦⠀⢀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⣀⣤⠶⠞⣻⣿⠿⠟⣙⡷⠬⢥⣤⣴⣶⣻⣿⣿⣿⣿⣿⠁
⠀⠀⣠⢞⣫⡤⠶⠚⢋⣽⠶⠋⢑⣶⣒⠒⠒⠐⠀⠈⠙⠛⠿⢿⣿⣀
⠀⡔⣱⢟⣩⡴⣲⣶⣿⠀⢀⣴⣫⢡⠟⠉⣉⡤⢀⢠⡶⡞⢻⣥⡤⠼⢿⢶⣄⡀
⢠⣽⡟⣛⠻⢿⣿⣏⠁⠀⠛⣿⣿⣡⠴⠐⠻⠾⡣⠜⠁⠁⠈⠊⡠⣿⢾⠟⠨⢙⣳⣄
⢀⣿⡿⢡⡾⢻⣷⣧⣈⣆⣀⣛⢿⣇⣴⡷⣊⣜⠻⢦⡄⠀⠉⠀⢤⣀⣡⣷⠢⣾⣟⣜⣷⡀
⢸⡿⣿⢈⣹⠁⣼⢣⢋⡌⢉⠙⢻⣿⣿⣿⣥⢿⣲⣄⡉⠃⠚⠤⣆⡙⡛⢳⠷⢾⣟⣫⣽⣷
⢸⡏⢱⣿⢏⡤⢻⡏⣼⠃⡺⣤⢾⡶⣏⣻⣿⣿⣷⣾⣉⢀⣢⡤⠀⠀⢡⣏⣽⣾⣿⣿⣿⡏
⢸⣿⢸⡞⠏⠀⡸⢧⡽⣶⠛⣿⣾⣿⢿⠏⠀⣹⡿⣦⣭⠙⢻⣿⣋⣉⣽⣿⣿⣿⣽⣿⣿⠃
⠈⠻⣾⡏⣷⣿⠁⠂⣠⣏⡼⠟⢩⡏⡾⠀⣰⢻⠇⠈⢱⢛⣬⣬⣴⡟⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠹⣟⠣⡛⢧⡲⡛⠉⠀⠀⠀⣿⠃⠀⢹⢿⠀⠀⢸⣾⠁⢿⣿⠀⠀⢻⣿⣯⡿⢭⢇⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠑⠀⣤⣉⠄⠀⠀⠀⠐⡇⠀⠀⢸⡿⠀⠀⢸⡇⠀⢸⣧⠀⠀⠀⢻⣿⣿⣿⣾⣿⣿⣿⣿⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡏⠀⢀⣎⡇⠀⣸⠘⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⢿⣿⣿⣿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡀⠀⠀⠀⣿⠀⠈⣷⠇⠀⠘⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⠀⠀⠙⠃⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡄⠀⢸⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⡇⡀⠀⠀⠀⡄⠈⠷⣀⣌⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠒⠒⠒⠂⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⠠⠀⠀⣑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⠂⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡇⣼⠀⣀⡀⠰⠀⠀⠀⠀⠀⠀⠀⣴⠁⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠈⡆⠀⠀⠀⠀⠒⠐⠁⢈⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⢀⡈⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡈⠆⡀⠈⠂⠔⠁⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣧⡤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢛⡿⠄⠤⠢⣔⠀⠈⠉⠑⠓⠉⠺⠄⡛⠃⠚⠀⠀⠂⠀⠂⠀⣀⡀
⠀⠀⠀⠀⠀⣠⠀⡀⡀⠀⠈⠀⡀⠀⠀⠀⠀⠰⠒⠂⠀⠀⠀⠀⠀⠀⠢⠐⠇⣀⠐⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⢄⠓⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠍⠀⠠⣀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠁⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡤⠀⣅⠠⠂⠒
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢲⠒⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠛⠻⣟⠛⠟⠿⠾⠛⣻⣏⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⣿⣿⣿⠋⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣿⠃⠀⠀⠀⠀⠛⠿⢿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣴⣶⠀⠀⠀⠀⣺⣶⠮⣿⣇⠀⡀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠁⠉⡤⠀⠀⢡⠉⠈⠀⠀⣻⣿⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⠀⠀⠀⠀⢀⠶⠀⣺⣿⣿⣿⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣦⣀⠀⠀⠀⠈⠉⠀⠾⢿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣿⣿⢻⠈⣿⠿⣶⣶⣾⠁⠀⠀⠈⡿⢿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⣾⡿⠁⠉⢹⡆⠀⣿⠀⠈⠛⢻⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⠃⠀⠀⠸⢿⣄⢿⠀⠀⢠⠎⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⠀⠀⠀⠀⠨⣿⠀⢀⢋⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢁⡿⠀⠀⢀⠎⠀⢈⣿⣗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄
⠀⠀⠀⠀⠐⡄⠈⣄⢀⠜⠈⠇⠀⠀⠏⠀⡠⢋⣿⡟⢦⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠠⠞⠓⠀⠛⠁⠀⠀⡀⠀⠀⠀⢸⣡⠟⠀⢿⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⢃⠀⠀⠀⠀⠹⠀⠀⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠈⠉⠒⠴⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠀⠀⠀⡤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠐⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡴⠀⠀⣀⠤⡶⠦⠒⠾⠯⠖⠊⠙⠺⠥⢖⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⡔⢦⣀⣀⠀⠈⢿⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⡇⣠⢖⣋⡉⢹⠀⠀⠙⠒⠒⠛⠻⣦⢸⣆⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⡷⠟⠁⢰⠆⠀⠀⠀⠀⠀⠰⠆⠀⠀⠘⣏⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⡇⠀⠀⠀⠀⣀⠀⠀⣀⣀⠀⡀⠀⠀⠀⣿⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣧⠀⠀⢀⠀⠀⢉⣉⠭⠤⠤⡜⢄⠀⢸⣏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡄⠀⡌⡠⠞⠓⡒⡚⠫⣉⢹⡜⡄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢣⣇⠔⠚⡄⠈⠀⢸⠉⠁⢇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠘⠦⠤⠚⢀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠲⠦⠴⠖⠚⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⣿⡍⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣼⡿⠉⠉⠒⠦⣄⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠸⡕⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣻⠃⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠘⠮⣑⠤⣄⣀⣀⢀⣀⣀⠴⣊⠵⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠺⠯⠭⠗⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣤⠶⠖⠲⢶⡤⠶⠚⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠿⡉⠙⠏⠀⠀⢀⡴⠋⣠⠴⠒⠛⠓⠦⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣀⠀⠀⢹⡄⠀⠀⠀⠈⠀⡜⠁⠀⠀⡀⠀⠀⠈⡆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠿⠂⠀⢰⠃⠀⠀⠀⠀⠀⡇⠀⠀⠸⠿⠂⠀⠀⣸⠀⠟⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉⠁⠂⠀⢀⠀⢀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⡄⠀⣠⠞⠋⠁⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢿⠀⢀⠔⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡴⠃⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠈⠳⣬⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⣤⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡆⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⡀⠀⠀⠈⠀⠀⠀⠳⠀⠀⠀⠀⠃⠀⠀⡠⠛⠲⢤⣀⠀⠀⠀⠀⢀⡞⠁⢀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⠀⡀⠀⢰⠂⠀⠀⠀⠀⢀⡇⠀⡞⠁⠀⠀⠀⠈⠓⢄⠀⠀⡾⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣤⡀⠀⠀⢠⡞⢹⡀⠀⠁⢀⣀⠀⠀⢸⡇⢠⠁⠀⠀⠀⠀⠀⠀⠀⠳⣠⡇⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⠇⣷⠀⠀⣸⠀⢸⠇⠀⠀⡞⠉⣧⠀⢸⠃⢸⡀⢀⣀⣀⣀⣀⣀⣠⣤⡿⡇⠀⠀⠈⠙⠛⠓⠶⢦⡀
⠀⣀⣤⣤⣤⣤⣤⣤⣤⣭⣿⡄⣿⠀⠀⣿⠀⢸⡆⠀⢰⢃⠖⢙⠶⡏⠀⠸⣉⡏⠉⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠰⢦⡶⠚
⠘⢏⣀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢿⠀⠀⡏⠀⠀⢧⣠⠞⠈⠉⠙⠒⡇⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡴⠞⠙⠒⠛⠁
⠀⠀⠈⠑⣦⠀⠀⠀⡴⠒⠒⠃⢸⡀⢀⣏⣙⠄⠀⠀⢠⡖⠲⠤⠴⠇⠀⠀⠓⠒⣦⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⣠⠤⠖⠚⠋⠀⠀⠀⠙⠛⠓⡆⠘⠷⠞⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⣳⠀⢟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⣴⠒⠚⠲⢤
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⣳⠀⢀⣸
⠈⠉⠉⠙⠛⠓⠒⠲⠶⠤⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡤⠟⠀⠀⠈⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⠤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣛⣓⣲⣶⣶⣤⣤⣤⣶⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣋⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⠴⠶⠒⢶⣤⠶⠚⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠓⠚⢯⡉⠛⠀⠀⢀⡴⠋⣠⠴⠚⠛⠓⠲⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⡀⠀⠀⣧⠀⠀⠀⠀⠀⡜⠁⠀⢀⣀⠀⠀⠈⡆⠸⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠋⠀⢀⡏⠀⠀⠀⠀⠀⣇⠀⠀⠘⠟⠁⠀⠀⡇⠀⠋⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣳⣄⣀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⣀⠴⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⠀⠀⠄⠀⠄⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣠⡤⠤⢤⣄
⠀⠀⠀⠀⠀⡟⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⢷⠀⣠⠞⠁⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢷⡀⢴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣸⠞⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⠀⠀⠀⠀⠸⠁⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⣜⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⡇⠀⣠⠊⠀⠉⠓⠦⣀⠀⠀⠀⡞⠀⠀⠂⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⢶⡀⠈⠇⠀⠀⠀⠀⢀⡇⠀⠃⠀⠀⠀⠀⠀⠈⠳⡀⢸⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⢳⡀⠀⢀⡏⠀⡇⠀⠀⣰⠶⣄⠀⢸⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡀⠀⠀⠶⣯⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⢸⡇⠀⢸⠇⠀⡇⠀⠀⡏⣠⢿⣄⡜⠀⢸⠶⡖⠒⠂⠊⠛⠋⠉⠋⣹⠁⠀⠀⠀⢀⣀⣀⣽⠇
⢰⡞⠋⠉⠉⠉⠉⠉⠉⠉⢻⢸⡇⠀⢸⠀⠀⢧⠀⡸⠁⠓⠲⢬⡇⠀⠘⡛⠁⠀⠀⠀⠀⠀⠀⠀⠈⣤⣀⣨⠥⣤⣼⠟
⠀⠙⠓⢦⡀⠀⠀⢀⣠⠤⠼⠀⡇⠀⣸⠒⡄⠈⠉⠁⡤⢤⣀⣀⡇⠀⠈⠧⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⢀⣠⡤⠴⠃⠀⠀⠈⠒⠶⢤⠀⢷⣤⠟⠛⠁⠀⠀⠀⠓⠒⠒⠒⢢⠀⡶⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⣠⠤⠴⢤⣄
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⠀⠀⠙⢲⠄⠀⢨
⠈⠓⠒⠒⠲⠶⠦⠤⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡼⠂⠀⠀⠛⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠛⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⣀⣤⡌⠁⠀⠀⠀⠙⠿⠟⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠉⠉⠙⠒⢀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠒⢦⣤⣴⣿⣟⣿⣶⣄⣀⠒⠀⡈⠉⠰⠶⠶⠶⢶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣰⣶⠞⠛⠋⠁⠀⠀⠈⠛⠻⣶⣾⣁⣠⠤⠄⠀⠀⠾⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣦⣄⣀⡀⠀⠉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠙⠛⠛⢷⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣦⡀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⣴⡿⢧⡀⠀⠀⠤⣤⣤⣤⣤⠤⠀⠀⠀⣰⣿⣧⣀⡈⢻⡦⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣿⣄⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⢿⣷⣄⠙⢷⡀⠀⠛⠛⠛⠃⢰⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠙⠻⣷⣄⠙⣆⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣧⣤⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠷⣶⡶⠾⠿⢿⣿⣷⣶⣶⣶⣿⢿⣿⣿⣿⠿⠿⠿⠿⣶⣶⣶⣶⢶⣶⣶⡦⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠙⣷⣄⠀⠀⠀⠘⠿⠿⢆⣰⣾⠋⠀⢿⣄⡀⠀⠀⠀⠻⠿⠿⢆⣸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠛⢷⣶⣤⣤⡤⣴⣾⡟⠁⠀⠀⠀⠙⠻⣶⣶⣤⣀⣴⣶⠞⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢰⡾⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⡄
⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠈⢧⣤⣀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣤⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⣦⣤⣴⣶⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⡴⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠀⠀⢀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠾⠿⠿⠟⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⡋⠉⠉⠉⠉⢻⠟⠛⠛⠛⠛⢻⡿⠛⠋⢻⣿⠏⠉⠉⠉⠻⣏⡉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣿⠋⣽⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠇⢰⡿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡿⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠃⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⢀⣧⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣧
⠀⠀⠀⠀⠀⢰⡎⠀⢠⡇⠀⠀⠀⠀⣿⣀⣠⣶⠆⠀⠀⠀⠛⠀⢸⣿⠀⠀⠀⠀⢴⣶⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⠃⠀⠸⠟⠀⠀⠀⠀⣿⠛⣿⡄⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⣴⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣆⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠉⣷⡄⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠠⠤⠤⠤⣾⣿⠀⠀⠀⢿⡇
⠀⣀⡄⠛⣻⡿⠛⠀⣠⣶⣾⣷⡀⠀⠀⠀⠀⠉⣷⡄⠀⠀⢸⣿⠁⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⠀⠀⠀⢸⣇
⠀⠻⣧⡾⠋⠀⢀⣼⠟⠉⢀⣼⣿⠀⠀⠀⠀⠀⠈⢻⡆⠀⢸⣿⠀⢀⣠⣴⣿⣤⣤⣤⣤⣤⣤⣤⣴⠛⠋⠉⠉⢹⣿⠀⠀⠀⠈⣿⡗⠐⠒⠀⡀
⠀⣼⡏⠁⠀⣠⡾⠃⠀⣰⡿⠙⢿⣦⡀⣀⣄⣀⠴⠿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⣉⣭⡿⠿⠷⠾⠿⠷⢿⣿⠀⠀⠀⠀⣿⡷⠲⠷⠾⢧⣀
⣶⠉⠀⢀⣴⠏⠁⢀⡸⠋⠀⠀⣼⠟⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⡿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣿⡷⠰⢦⣀⣈⡛
⠙⢶⣤⣿⠇⠀⢀⣜⠃⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⣶⣶⣿⣿⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⡷⠀⠀⠉⠻⢿⡄
⠀⠀⣿⠇⠈⣴⡾⠋⠀⣠⣾⣿⣶⣶⣶⣶⣶⣶⠾⠛⠛⠛⢻⡟⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⠾⠾⣿⣧⠀⠑⣶⣀⣀⡟
⠀⠀⣿⣆⡀⢛⣷⣦⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠱⣦⣼⠿⠟⠁
⠀⠀⠀⠛⠻⣿⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⡀⠈⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⠁
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⢿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣿⣦⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⡿⠿⠿⢿⣿⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡏⠉⠉⠉⢻⡟⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⡏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣆⠀⣠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⡄⠻⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠉⠉⠋⠉⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠉⠹⠿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠿⠋⠁⠀⢀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⣤⣄⠀⠀⠀⠈⠉⠙⠿⠷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⣀⣀⡰⠶⠞⠉⠁⠀⠀⣀⣠⣶⠿⣧⡀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣨⡛⠛⢷⣶⣀⣀⣀⠀⠀⠀⠈⠉⠉⠛⣶⣶⣦
⠀⠠⣶⣶⣄⣉⣉⣀⣀⣀⡤⢶⣶⠾⠛⠋⠀⠀⠛⠛⠲⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠉⠛⠛⠛⠲⠶⠶⠶⠶⠶⠈⠛⠋
⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣦⣤⣤⣤⣤⣴⣶⣶
⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⢢⠀⠀⠀⠋⠀⠀⠈⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠸⠗⡆⠀⠀⠀⠘⠋⡀⠀⠀⢒⠴⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⢀⡦⡀⠞⠁⠀⠠
⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠈⠀⠘⠓⠀⠐⢂⠒⠀⠀⠁⠀⠀⠀⣔⠀⡀⠀⡈⠘⠷⠆⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠲⠀⠀⠀⠈⠂⠁⢀⣀⣀⠈⠑⠎⠀⠀⠀⠀⠀⣠⣤⡤
⠀⠀⠀⠀⠘⠁⠀⠘⠃⠂⠀⠉⢠⠄⢐⣶⣶⣿⣿⡿⣿⠟⢁⡃⠄⢠⣀⠀⠀⠀⠰⠈⢁
⠀⠀⠐⠠⠀⠀⠂⠀⠀⠀⠀⢢⣄⠀⣐⣿⣿⠏⢿⡏⠁⡀⡆⣿⡾⢿⣿⣿⣦⡀⠀⠀⠉⠀⠀⠄
⠀⠀⠐⠦⢀⣤⡀⠂⠀⠀⣴⣿⣿⣏⣫⣿⢉⣠⣽⣥⣥⣤⣥⣀⠀⢀⡿⣿⣿⣿⣆⠀⠀⠀⠐⠀⠀⠀⠀⡀
⠀⠀⠘⣦⡞⠻⠍⠒⡦⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡃⢤⡚⢿⣝⣆⣈⡤⠒⠈⠁⠁⢀⢰⠿⠃⠂
⠀⠈⠀⠘⠓⠀⢉⣴⣾⣿⠿⠿⢿⣿⣿⣿⣿⣿⣯⣽⣛⣻⣙⢉⣻⣿⣿⣦⡼⣝⣯⣤⣠⠄⠀⢀⠀⠈⠁⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⣟⣿⠛⠯⠒⡚⠙⠋⠀⢀⠀⠀⠉⢻⣅⡿⠷⢾⡟⣖⣛⡄⡃⡸⠣⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠌⠉⠉⠀⠀⠀⠀⠁⠀⠘⢿⣇⣠⣴⡭⣤⣬⡤⡥
⠀⠈⠀⠠⠄⠀⡀⠀⠀⢠⡄⠂⠀⠀⠰⢦⣄⠀⠀⠤⠋⡀⡀⠠⠈⠀⠀⠀⠠⠖⢩⣍⡅⠉⢟⡜⠀⠁⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⡠⢞⣻⣯⡕⠠⠄⠀⠠⠀⠀⠀⢀⡀⢄⠀⠀⠂⠛⠙⠚⢯
⠀⠠⠀⠀⠀⠀⠀⡀⠀⢤⠔⠠⣧⡤⣄⢀⡉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠉⠉⠥⠌⢃⡠⠄⠀⠀⠀⠀⠉
⠀⠀⠀⠑⠀⠔⢠⠴⣛⣻⠄⠒⠀⠀⠀⠀⠀⠐⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠤⠈⠉⠙⠓⠖⠐⠀⢷⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡿⢻⣿⣿⣿⣿⢶⣾⡿⠍⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠺⣭⣙⣧⠘⠟⢟⠿⣙⡇⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣣⣾⣿⣞⣶⣿⣿⣦⣤⡈⠙⠛⠀⢠⣾⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⠈⢹⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣻⠿⡿⠍⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⡀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡀⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⣿⣿⣿⣿⠾⣿⣿⣿⣄⠀⠈⠙⠻⠿⣿⣿⣿⡏⢿⣧⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠉⠀⠀⢹⣿⣿⣷⣶⣶⣶⡄⣠⣶⣾⡇⢾⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠋⠈⢡⣾⠟⣿⣿⡿⣿⣿⣿⣟⣾⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣷⡟⢛⣿⣿⣷⣿⣿⡿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⢸⠏⠛⢻⣿⡟⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⢤⣼⡤⡰⠛⠁⠀⠀⠁⠠⣀⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠰⠁⢸⣿⣿⢶⣶⣿⣷⠁⣄⣀⡀⠀⠀⠀⠀⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⣿⣿⣿⣲⣬⣭⣾⣿⣶⣿⣿⡷⣄⠀⠀⠀⠘⣏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⣿⣧⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠅⠀⣼⣿⣿⣿⣿⣿⣿⡿⡽⠋⠁⠰⠏⡌⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠋⡙⢛⣿⠋⠉⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⣿⣷⠎⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣭⣙⠃⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⡏⣀⣠⡤⢀⣈⣀⠀⠁⠀⢀⣀⡀⠬⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣋⡉⠉⠀⠈⠉⢙⡖⠋⠀⠀⣬⡿⣷⣿⣿⣆⠘⡄⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢤⣵⣦⡀⠀⠀⠀⠀⠀⢸⠿⠿⢿⣿⣿⣷⣶⣤⣀⠉⠉⢈⣧⣄⣿⣿⣿⣿⣿⣿⣆⢰⣄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣀⠀⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠙⠻⣧⣙⠂⠀⠀⠀⢀⣰⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠀⠚⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⠏⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⣠⡀⠀⠀⠀⠀⠀⣠⣤⡀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠟⠀⠀⠀⠀⠀⠷⠞⢁⣌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣲⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣤⠴⣄⠀⠀⢺⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡝⣉⡀⠀⠀⠀⠀⠘⣧⣼⠇⠀⠀⠀⢸⡇⠀⢸⠂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠈⢯⡽⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠛⠋⠀⠀⠸⡽⠇⠀⠀⠀⢠⡖⢦
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢿⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣤⣀⠙⠋⠀⠀⠀⠀⣀
⠀⠀⣠⠶⠋⠉⠉⠉⠉⠉⠉⠛⢷⡀⠀⡴⠒⠛⢻⡴⠛⠉⠉⠙⢦⡀⠀⣠⠶⠛⠉⠉⠛⠶⣄⠀⢠⡞⠁⠀⠀⠀⠀⠉⢳⡀⢀⡴⠚⠉⠉⠙⠛⠲⢦
⠀⡴⠃⠀⠴⠀⠀⣀⣀⡀⠀⠀⢸⡇⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⣴⠋⢠⠄⠀⠀⠀⠀⠈⢳⡟⠀⠀⠖⠀⠀⠀⠀⠀⣷⡏⠀⠀⠀⠀⢀⣠⣤⡾
⣸⠁⠀⠀⠀⣴⠛⠉⠉⢻⣴⣆⣼⠁⠀⡇⠠⡄⠀⢠⣤⠀⠀⢀⣤⢿⡇⠀⠈⠀⣀⣀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣠⠤⠴⠒⠙⣆⣀⣀⣀⡀⠈⠛⠓⠦⣄⠀⣴⢲
⡏⠀⠀⠀⢸⠃⠀⠀⠀⠈⠁⠈⠁⠀⠀⣷⠀⠀⠀⢸⣸⠀⠀⠀⢷⣸⠀⠀⠀⢰⠁⢸⠀⠀⠀⠀⣿⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⢳⡀⠀⠀⠈⣧⠈⠋
⣧⠀⠀⠀⢸⡀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠉⠀⠀⠀⠀⢻⡄⠀⠀⠈⠓⠋⠀⠀⠀⢀⣧⣤⣤⣀⠀⠈⠛⠛⢶⣠⡶⠶⢤⣄⣤⠞⠐⣦⠀⠀⣻
⢹⡀⠀⠀⠈⢳⡀⠀⠛⢶⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠶⠤⠷⠀⠀⠀⠀⢵⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡿
⠈⢳⡀⠀⠀⠀⠉⠛⠒⠋⠀⠀⠀⠀⣿⢻⠉⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡏⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁
⠀⠀⠙⠳⣦⠀⠀⠀⠀⢀⣤⡄⠀⠀⣿⢸⣀⣀⠀⠀⣿⠙⣦⠀⠀⠀⢀⣸⠟⣆⢠⣶⠀⣤⡇⠀⣇⡾⣇⠀⠀⠀⢠⡶⢾⠁⠀⠀⠀⠀⠀⠀⣤⡀⡿
⠀⠀⠀⠀⢸⠀⣀⠀⡀⢸⠀⣷⡆⣴⣿⠈⠉⢹⡇⠀⣿⠀⢹⠀⣤⡄⣿⠉⠀⣿⢸⡏⠷⠿⠇⠀⠀⠀⢻⢰⢷⡀⢸⠁⠘⣇⢸⢳⡄⣾⢧⠀⡏⠛⠁
⠀⠀⠀⠀⢸⢸⡟⠞⢻⢸⡆⠀⣿⢿⠀⠀⠀⠘⣿⣧⣿⠀⢿⠀⡇⠙⠋⠀⠀⢹⢸⠇⠀⠀⠀⠀⠀⠀⠈⠛⠘⡇⣸⠀⠀⠹⣾⠈⠷⠏⠸⣦⡿
⠀⠀⠀⠀⠘⠿⠀⠀⠘⠿⠀⠀⠹⣾⠀⠀⠀⠀⠀⢸⣿⠀⢸⣄⡇⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠛

⣿⣿⣷⣄⠀⠀⠉⢳⡸⣦⠿⠒⠲⣇⣴⠨⢙⡢⡽⢿⣯⣾⣿⣿⣯⣿⣷⢙⣿⣯⣭⡝⠃⠀⠀⣿⠏⢀⡇⠈⢻⡇⠈⠻⠇⠹⠏⣿⡱⣶⢠⡿⠃⠉⢾⡟⠀⠘⡛
⣿⣿⣿⣿⣿⣶⣾⣉⠀⡸⠿⠃⠠⠨⠧⣤⠀⠹⣷⣰⣿⢟⣹⣯⣷⣞⣿⡾⢟⢋⡞⣅⢠⠆⠶⢨⡶⠀⠀⡄⣈⠁⠀⠀⠀⠀⠀⠈⢿⡟⠁⠀⣄⠀⠹⠀⠀⡸⣧
⣿⣿⣿⣿⣿⣿⠿⢿⢟⡙⢣⣤⡄⠀⠀⠈⠃⠰⠃⢉⡙⣮⣭⣴⣿⣿⣁⣽⣹⠻⢳⢏⡝⠀⠀⢰⣷⡶⣦⣰⠈⠁⠀⠀⠀⠀⠀⠀⠹⡅⠀⢀⡈⠀⣶⠘⠋⢠⣬
⣿⣿⣿⠿⠛⠿⢷⡶⣿⣿⠊⠿⠗⡿⠒⠀⠀⠁⠀⣹⠷⠷⣄⠈⠛⠃⠈⠁⠛⠛⠛⠀⠀⠀⣿⠙⠛⣿⡟⡻⢃⠀⠀⠗⣠⠀⠀⠶⠄⢀⠀⠁⠀⣀⣼⣧⠖⠟⠋
⣿⣿⣷⣶⣴⣾⡏⢀⣿⣧⣼⣿⠾⣀⣠⡔⠀⢀⣤⣭⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠄⠀⢀⡤⠀⡿⠸⠅⠉⣀⠉⠀⠀⠘⢄⡀⡄⠀⠀⠀⠈⢹⡏⠡⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣷⣿⣿⣭⣥⣴⣮⠿⢖⡛⣲⡴⠶⠆⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠈⠛⠁⣾⣆⡀⣈⠑⣿⠁⠀⠁⠀⠙⠀⠀⠀⠈⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⡎⠋⢉⠀⠀⠀⠈⠉⣇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣴⢠⣿⠐⡟⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣸⡇⣴⣠⡄⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣤⣄⢼⢯⣛⣠⣿⣻⣻⡄⠀⠀⠀⠋⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡟⢀⡀⠀⣄⠀⠀⠀⠀⠀⣠⠤⠶⠶⠄⠀⠀⠀⠀⠀⠰⠆⠀⠀⠘⣫⣤⢣⡟⢟⡾⢭⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⡀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣄⠁⠁⠀⠀⡿⠈⡀⣼⠁⠀⢠⠄⠀⠀⠀⢠⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠋⣚⡃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡸⢦⠀⠈⠐⠀⠀⣀⠘⢛⠁⠏⠀⠀⠀⠀⠀⠀⠀⢈⣻⣯⠗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⣵⠿⢛⡉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠈⢣⡆⠀⠀⠀⣀⠂⠀⠀⠀⠀⠈⠃⣁⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣴⠟⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠁⠻⠂⢀⠀⠀⣠⡄⠀⠀⠁⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢄⡱⣛⠅⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⣽⡏⠀⠀⠀⠀⠀⢸⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣸⡿⠃⢰⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣎⡿⢿⣦⡶⠖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠋⢀⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣦⣀⠀⢀⠈⠀⠀⠀⠀⠀⠠⠿⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠿⣍⠉⠂⣧⡤⠠⠄⠀⠀⠰⡻⣧⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⡗⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢲⣏⣷⣤⣄⠌⣯⢵⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⡻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⢠⣴⣔⡷⠸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢹⣾⠃⣴⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⢀⡎⢁⣼⢦⡶⠀⠈⠻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⣉⠉⠁⣇⣿⠛⢣⣾⣆⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠡⣈⣛⣿⠿⢼⣷⣿⡟⠋⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⣪⠗⠛⢫⡌⢴⣾⡾⢥
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⣀⣈⠧⠻⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠚⢭⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠛⠛⠻⢿⣟⠛⠛⠛⠳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢁⣤⡶⠾⠷⢦⣄⠙⣷⡀⠀⠀⠀⠹⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢠⡾⠁⠀⠀⠀⠀⣽⣧⠘⣷⡀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⠁⠀⠀⠀⠀⣼⠇⠸⣧⠸⣧⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⡀⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠰⣿⠀⠀⢸⡇⣿⠄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⠁⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣨⡿⠶⣶⣶⣶⣶⣿⣷⣾⡟⢰⡟⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢰⡟⢋⣭⣿⣿⣿⣿⡟⢀⡿⠁⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⢡⣶⣿⠿⠋⣠⡿⠁⠀⠀⠀⣰⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⣾⣧⣤⣴⣾⣯⣤⣤⣤⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣼⣇⣿⣿⡇⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣀⣀⣴⠟⠋⠉⠙⢿⣿⣇⣠⡴⠟⠉⠉⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠉⠉⠀⢀⣠⣤⣀⠀⠈⠉⠉⠀⢀⣠⣤⣀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⣤⡴⠟⠉⠈⠉⠻⣦⣤⣤⡾⠋⠉⠈⠙⢷⣤⡄





⡿⠿⠿⠿⠿⠻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⠎⠘⠀⠀⠀⠀⢀⠀⠀⡠⠒⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⡤⠁⠘⢄⠀⢇⠀⢀⡠⠤⠀⠺⣄⢀⡜
⠀⠀⠀⠀⠀⠊⠁⠀⠈⠉⠁⠉⠑⠊⢢⡀⢄⢡⠀⠑⢟⣀⣈⣙⡷⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢂⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠈⠒⠁⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠉⠀⠀⠀⢀⠴⠊⠁⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡶⠀⠁
⣠⣤⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢋⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣿⠗
⠉⠁⠀⠀⢠⠏⠀⠀⣀⠀⡀⠀⠀⠀⠈⡄⠣⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡸⠱⡀⠀⠀⢀⢼⠔⢒⢼⡀
⠀⠀⠀⠀⣼⠀⠀⠈⠢⠥⠃⠀⠀⠀⠀⢱⠀⠑⣄⠀⠀⠀⠀⠀⠈⡡⠊⠀⠀⠀⠈⠒⠀⠆⢠⢇⣸⣭⠕
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠈⠀⠄⣀⡠⠤⠚⠀⠀⠀⠀⠀⠀⣀⡤⢦⣳⠄⠠⠛⠈
⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢠⠦⠔⠃⠀⠀⠤⠴⠒⠒⠂⠀⠐⠲⢄⣀⠤⠖⠊⠁
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⡶⣷⠄
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡞⠚⠫⢉⡉⠉⠁⠀⠀⠀⠀⠒⠂⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠀⠀⠖⠐⢂⡨⠟⠁
⠈⠑⠂⠠⢐⢉⣽⢮⠂⠀⢸⠁⠀⢈⡩⠟⠁⠒⡲⠛⢻⡟⣁
⠀⠀⠀⠀⠀⠀⠫⠭⠤⣠⡈⡗⠊⠁⠀⠀⠀⠘⠒⠚⠙⠛⠁⠘

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣛⣛⣛⣛⣛⡻⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡜⣿⣿⣿⣶⣶⣶⣶⣾⣿⡜⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⣫⣵⣦⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢟⣥⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣱⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⡇⠿⠟⠼⠿⠿⠿⠿⣿⠿⠿⠿⢟⣼⣿⣿⣿⣿⢻⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⢸⡇⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⣿⣿⡇⡇⠀⠀⣠⣄⠀⠀⣠⣄⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣿⢹⣇⡇⣀⣉⡁⠈⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⢋⣅⣻⣾⣿⡇⣭⡻⣿⣿⣦⡀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣦⣭⣝⣛⣛⣯⣭⣾⡿⠿⠋⠀⠒⠋⠙⠒⠒⠋⠙⠒⠒⠋⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄⣄⣤⡄⣤⢀⡤⠀⣤⠀⢠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠂⠘⠙⠓⠋⠋⠓⠛⠛⠂⠘⠁⠚⠛⠁⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣱⠈⠀⠀⠈⠀⠀⠀⠀⢐⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣄⠤⡀⠀⠀⠀⠄⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⢿⣝⣷⣄⣩⣷⣦⣶⣋⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠀⠀⡉⣻⣿⣾⣿⠟⠉⢀⠮⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⡈⠀⠟⢛⣿⣿⠛⠙⢦⣄⠰⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣤⡤⠈⠀⠛⠻⠿⠾⠛⠛⠁⠔⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⣽⢃⣴⡞⣟⣸⣿⣷⡾⡿⠟⠿⣿⣿⣕⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⣀⢾⣿⣇⠿⣿⣿⣿⣄⣀⡀⢀⡞⢻⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣩⣾⣿⠰⣶⣿⣿⣿⣄⡿⠿⣘⣽⠛⠟⢿⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣿⣿⣿⣶⡏⢹⣿⣿⣿⣷⣷⣻⡃⢰⣶⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢻⣿⣿⣿⣏⣇⡘⠃⣻⣿⣿⣿⡟⢿⣦⡙⣿⠂⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠀⢀⠘⣿⣿⣿⣿⢿⣷⣶⣿⠙⠛⠿⣷⣤⣼⣿⢧⣴⡦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣡⢀⡼⠇⠀⣾⣷⣏⣼⣇⣹⠛⠻⣶⡎⠈⢻⠃⢰⣾⡿⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢠⠇⠸⠁⠀⠀⣸⡿⣿⣿⣿⣾⡾⣷⡄⢱⣶⣾⣆⣤⣿⣦⢀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⡇⠀⢸⡇⠀⢃⡴⠈⠉⠻⡅⣤⣿⣿⣧⢻⡷⡏⢿⠈⢛⠭⡞⣿⢾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⢧⣄⡟⢠⢀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣯⣻⣇⢿⡇⢦⢹⡟⣿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢂⠀⠀⠐⠀⣿⣟⠐⠀⠁⠈⠀⣸⣟⣸⡿⢛⣿⣿⣷⣿⡘⣏⠻⢾⡇⣿⣧⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠳⣵⣀⣷⣾⣿⡀⠈⡛⠀⡖⠉⠹⣿⡏⣽⣿⣿⢿⣾⣿⣿⣶⣷⣧⣸⣧⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠈⣿⡁⠈⢿⣷⣼⣧⣤⣷⣠⡖⢿⣷⠺⠶⢿⣆⣿⠻⡿⣟⣿⣼⣯⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠘⢿⣦⣌⠛⠛⢻⣿⣿⣿⣅⡾⡨⣴⣧⣼⣿⣿⣞⣡⣈⣍⠻⣿⣿⣦⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠿⣿⣶⣼⡿⣅⡙⠻⣿⣿⣿⣿⣿⣷⣽⣿⡉⢡⠈⡆⢿⡏⠛⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⣇⣘⡛⢿⢸⠇⣹⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣼⣌⣿⣹⣖⣯⣭⣄⣼⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠙⠓⠢⣄⡀
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠚⠛⠛⠛⠒⠲⢤⣀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢘⣄⣠⠤⠤⠤⠤⠤⠤⣀⡀⡰⠋⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⡤⠚⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠙⢲⡞⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠁⣠⠴⠦⢄⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠉⠓⠒⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠠⠄⣿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠲⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠼⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠞⠁⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣊⠤⠚⠁⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡔⠁⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⣀⠴⠊⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠢⡀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⠉⠁⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡺⠀⢠⠊⠁⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠉⠁⠀⠘⠢⠆⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠢⢄⣀⠀⠀⠀⠀⠀⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣠⣾⣦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⠏⢿⢏⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠁⠉⢹⠱⡇
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢒⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠊⠐⠁⠀⠀⡀⠀⡀⠀⣠⣦⡏⣺⠇
⠀⠀⠀⠀⠐⠒⠠⠄⠠⠀⣀⡀⠀⠀⠀⡀⠊⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠔⠉⠑⠺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢉⠚⠒⠀⡀⠀⠀⠈⠀⠊⠀⠁⣀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠄⠠⡨⠁⠄⠆⢈⡋⣛⣆⠠⢄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡙⠚⠀⠋⠃⠘⠚⠁⣮⡕⠀⠀⠀⠀⠉⠉⠒⠂⠂⠤⠤⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠉⠐⠣⠀⠋⠙⠃⠉⢀⠂⠄⠀⠒⠢⠢⠀⠤⢄⣀⠀⠈⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⠤⠀⠀⠈⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠢
⠀⠀⣀⠠⠤⢄⡀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡅
⢠⠊⠁⠀⠀⠀⠈⠢⡀⠀⠇⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠙⣄⠸⡀⠀⠀⠀⠀⢀⠁
⢱⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠱⡀⠀⡀⠀⡘
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠈⣆⠙⢦⠀⣠⠃
⠀⠀⠈⠓⠤⣄⣀⠀⠀⠀⣀⡸⠤⠇⠉⠁⠑⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡽⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠙⢂⠶⠴⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠸⠁⠋⠈⣋⣈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠐⢣⠴⢄⣀⣀⢸⠃
⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠚⠀⠒⠁⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⢀⡔⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠦⣤⢤⢀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⠛⢦⠀⠀⢱
⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⣀⡄⢶⠒⡎⠒⠤⡖⠲⠤⠞⠉⠁⠀⠀⠀⡱⠀⠀⢉⠆
⠀⠀⠀⠀⠀⠀⠐⢄⣀⣸⡌⢳⡙⣦⡓⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⡏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡹⡿⣌⢲⠦⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⠈⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠃⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡤⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠥⢦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠈⣷⠈⠳⡀⢠⠖⣉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⡇⠸⡷⡄⠈⠛⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠃⠀⣷⠙⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⠀⠀⠑⣦⣄⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⡟⡝⡏⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡗⠚⠾⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⣳⣀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣲⡤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣁⠐⢤⣶⣶⡶⢶⣦⣤⡄⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣿⣿⣿⣿⣌⣿⣿⣿⣷⣏⣱⣶⣷⡝⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡲⠉⣿⣿⣿⢿⡿⣿⣿⣅⡉⠻⣿⣿⡟⣠⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠅⠀⣿⣫⡁⠄⠣⣈⣁⠼⡟⠑⣬⣍⣹⣟⣥⠏⠀⢻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠖⠉⠙⢻⣀⢀⠄⠀⣄⠀⠀⢰⣿⣿⡿⠀⠀⠀⠠⡸⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠄⠀⠀⠁⠈⠀⠀⠀⠀⠀⠘⢿⣿⠁⠀⠈⠳⢿⡁⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣇⣤⡿⠀⠀⣀⡀⢠⡀⠉⠀⠄⠈⡟⠀⣨⣄⢠⣄⢙⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⠀⠀⢐⣿⣟⡇⡀⣄⢶⣄⣄⠐⠂⠈⠑⡌⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⡿⣷⣦⡜⠀⣿⣧⣸⣿⣿⠟⠉⠀⢤⣦⣶⣜⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⢻⡿⠀⣰⣏⢻⣧⠈⣿⣇⠸⣾⡜⠻⢛⢛⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢿⣇⢸⠋⢸⠊⢿⣿⣿⣿⡦⠙⡏⢀⠞⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢷⡜⣦⢸⢀⣸⣿⡄⢰⣧⣹⠤⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⢠⣾⠿⠛⠊⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢠⣠⣀⣤⣠⣤⣴⣶⣤⣷⣶⣶⣿⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⠷⠚⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢿⣿⣿⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⣾⣿⣿⣿⣿⣿⣿⣏⠙⣿⣿⡆⠀⠙⠋⣀⣀⠀⠈⠉⠉⠁⠀⠰⣾⣿⣿⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠁⠀⠘⠿⠍⠉⢹⣷⠸⣿⣇⠀⠀⢰⣿⣿⡆⡄⣷⣄⣄⣠⣀⣿⣿⣿⣿⣷⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣠⣤⣤⣴⣿⣶⣶⣄⠀⠀⠈⣿⣷⣻⣿⠀⠀⢸⣿⣿⣷⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⣶⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠉⡿⡇⠀⣿⣻⣿⣿⣿⣿⣿⣧⠀⠀⢿⣿⢺⣿⠀⠀⢸⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠼⠿⠒⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⠀⣼⣻⣿⣿⣿⡿⠟⣹⠀⠀⢻⣿⣿⡟⠀⠀⠀⠹⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠂⠀⠈⠁⠀⠀⠀⠈⠉⠛⠟⠿⠟⢠⡏⠀⣀⡺⣾⡿⠃⠀⠀⢠⣴⡏⣼⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠣⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⣘⣿⣿⣿⣿⡿⠁⠀⠉⢿⣿⣿⣿⣿⣿⣿
⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣤⣴⡶⠾⠿⠿⠶⠌⠻⣷⣼⣟⠀⠈⠘⣿⣿⣿⢿⣿
⣦⣝⡳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⡀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⢀⣤⡂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠒⠻⠿⠃⠀⢛
⣿⣿⣿⣿⣶⣮⡛⠶⣤⣄⡀⠀⠀⢠⣠⠮⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡿⠏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⣿⣿⣿⣿⣿⣿⣿⣦⡌⠉⠻⣷⣦⡄⣀⠀⠀⠀⠀⣠⡴⠚
⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⣸⣿⣿⣿⠢⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣽⣿⣿⣿⣿⡿⠁⠀⠀⠀⣸⣿⣿⣿⠟⢏
⣿⣿⣿⣿⣟⠃⠀⠀⢀⣴⣿⣿⡿⢧⣟⢋⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠈
⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣷⣾⢋⡱⣦⣂⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⣶⡶⠶⠿⠾⣿⡧⣽⠷⢶⣶⣤
⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣿⣵⣶⠿⣩⢿⠂⡑⣢⣦⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⣻⣿⣭⣾⣃⢩⣾⡔⠗⢋⣧⠃⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠤⠆⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣤⣷⣤⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡘⢾⣿⣟⣟⣗⠁⢧⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠯⠿⠪⢻⣾⡤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠐⠀⠀⠐⠀⣉⣀⡽⠧⡀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⠀⠈⠤⠒⢻⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⢀⠊⠀⠈⠀⠀⠀⡄⠀⣀⠥⠐⢤⡮⡏⢩⡿
⠀⠀⠀⠀⠀⠒⠒⠒⠲⠶⠦⠤⠤⢄⣀⣀⠀⠀⠀⢀⡀⠊⡀⠁⠀⠁⠀⠀⠀⠀⠀⡀⠀⢀⠜⠉⢑⠺⠿⠁⠀⠀⠠⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠈⠉⡑⠓⠾⠵⡌⠀⠀⠀⠀⠉⠀⠊⠀⠘⠀⠀⠁⠀⠘⠁⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⣀⠈⠀⢋⢠⠀⣀⠴⠂⡂⣠⠀⢺⠷⢶⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠤⠀⠐⠉⢀⡨⢌⡽⠉⡦⡥⣡⣇⣷⣤⠖⡃⠉⠉⠛⠑⠂⠤⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⠀⠀⢤⡔⣀⡐⠀⣄⡀⣄⢡⠤⠀⠛⠃⠀⡀⠀⡀⠀⠀⠀⠀⠤⠄⢉⣉⠓⠒⠒⠦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠐⠀⠀⠀⠓⠀⠁⠩⠀⠀⠀⠩⠐⠊⠁⣚⠁⠋⠓⠘⠲⠲⠤⠀⠀⠄⠉⠉⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠛⠚⠲⠐⠀⠀⠀⠐⠀⠀⠐⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡞⠁⢸⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢮⠙⢾⡀⠄⢸⡌⢧⡀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣷⠉⠙⠛⠞⣆⡛⠹⡀⠻⢮⣙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⢀⡤⠄⠀⡠⠤⠤⠭⠀⠉⠓⢶⣀⠀⠀⠉⠛⠳⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣞⡔⠉⠀⠀⠁⠀⠀⠀⠀⢀⠔⠊⠁⠄⠒⠉⢉⣀⣠⣤⣤⣤⣍⡛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⣈⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢦⣀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⡾⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣦
⠀⠀⠀⠀⠀⢸⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠸⡇
⠀⠀⠀⠀⠀⢨⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣿
⠀⠀⠀⠀⠀⡞⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣿
⠀⠀⠀⠀⢸⣧⠀⠀⡇⠀⠀⠀⠂⠉⠉⠉⠉⠁⠀⠀⠀⠀⣀⠀⣀⡾⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⢄⢀⡀⣿
⠀⠀⠀⠀⠸⡏⠀⠀⡇⠀⠀⠀⢀⡤⠤⠤⠤⠴⠒⠒⠺⡧⠉⠉⠉⠛⠁⠀⠠⣄⣠⡴⢻⣿⣿⡶⣤⣀⠈⣿⠁⢻
⠀⠀⠀⠀⠸⡇⠀⠀⡇⠀⠀⠰⠋⠀⠀⠐⠒⠒⠂⠐⠀⠁⠀⠀⣄⢢⠀⠀⠀⡠⠏⠙⠛⠿⠿⠷⠤⠬⠗⢸⠀⡏
⢀⡴⠚⠛⠛⠷⣶⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⠀⠀⡎⠀⠈⠒⠶⠤⠴⠄⠀⠀⠀⣼⣼⣁⣀⣀
⡏⣠⠴⠦⣤⡀⢸⠛⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣴⠞⠁⠀⠀⠘⠁⠀⠀⠀⠁⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠉⠉⠉⠛
⠁⠁⠀⠀⣌⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⡏⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⡛⠋⠛⠆
⠀⢦⠀⠀⠀⢹⡄⡆⠀⣰⠃⠀⠀⢀⡴⠋⠀⠀⠙⠀⠉⠙⠃⠀⠀⠀⠀⣴⠶⠶⠄⣿⠀⠀⠀⠀⠀⣄⠀⠘⡏⠁⠀⢀⠔
⠀⠈⠳⣴⠀⢀⡇⡇⢰⠇⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠂⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⡇⠠⠞⠁⠠⠒⠁
⠀⠀⠀⠘⣇⠈⠀⢷⣾⠀⠀⠀⠀⠀⢀⣴⠾⠿⡏⢩⠓⢦⡀⠤⣄⣀⡀⠀⠀⠀⠀⣼⠀⠀⠀⢸⣼⠃⠀⡀
⠀⠀⠀⠀⠙⢦⣤⠼⣿⠀⠀⠀⢀⡴⠋⠀⠀⢀⣨⣽⡀⠀⢹⡶⠤⢀⣈⣁⠀⠀⠀⡟⠀⠀⠀⣾⡿⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⡴⠋⠀⠀⡴⠚⠉⠁⠀⠈⠁⠀⠃⠀⠜⠻⣿⣧⠀⣸⠁⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡾⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣼⡯⢢⠃⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⣠⠋⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠙⢧⣀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣶⣶⣶⣞⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣉⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣽⣿⣿⣿⣿⣯⣽⣿

⢀⢀⠴⠀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠠⣀⠀⣀⠀⣦⠀⡀⡀⠀⢀⡀⠀⡀⠀⠀⠄⠀⢠⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠀⠄⠀⣀⠀⡀⠀⠒⠀⠀⢀⠀⠠⠀⠉⠀⡏⠜⠀⡀⢀⡘⠃⠀⠠⠀⠰⣲⣀⠘⠁⠀⡀⠀⠬⠀⠈⠉⢩⡀⠓
⠀⠀⠀⠀⠀⠉⠈⠡⠀⠀⠀⠄⠀⢰⡀⡇⠀⠀⠀⠀⠀⠀⠈⠀⣄⢀⢴⡄⠀⢻⣮⡄⠀⠠⠃⠀⠐⠀⠆⠐⠀⠆⢠
⠘⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣇⢲⢀⠀⠃⠸⠗⠀⠀⢀⣾⣿⣼⡃⠀⠐⠗⠀⢠⡄⠀⠀⠀⠂⠀⠈
⠀⢀⠀⠀⠀⣀⠀⠛⠀⠀⠰⠂⠀⠀⡡⠁⠀⠀⠈⠀⠀⠀⠀⠾⢁⢂⠈⠀⠚⠃⠀⠀⡀⠀⠀⠀⢸⠇⠸⠐⠻
⠀⠈⠓⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⣄⠁⠈⠀⢻⡄⠃⠀⢠⠀⠁⠈⠀⠆⢀⡐⡆⠀⡆⠁⠀⢠⠀⠀⠠⠀⠀⠄⠀⠀⠀⡄
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣜⠄⡇
⠨⠇⠀⠀⢰⠼⡀⠀⠀⠀⡀⠂⠀⠀⠄⠠⠀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠗⢀⡀⡈⠺⠷
⠀⠆⠄⢀⡈⠉⠁⠀⠀⠀⠁⠀⠠⡦⠀⠁⡄⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠈⠁⠉⠀⠑⠀⠀⠀⠀⠂
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠶⠆⠀⠰⠂⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠆⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠐⠂
⠀⠀⠀⠐⠀⠀⢀⡄⠀⠀⠐⢆⠀⠀⢀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⢀⠂⠀⠃⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠠⠀⠀⢀⠒⠀⣀⢀⡀⠋⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⣀⡀⠒⠀⢄⡂⢀⡀⠀⡅⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠄⠛⠁⡁⠀⡈⠁⠈⠁⠲⠂⠀⠀⠄⠀⠀⠐⠆⠀⠀⠀⠠⠀⠀⠈⠁⠉⠀⠉⠉⠀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠁⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀
⠀⣾⣿⠁⠀⠀⠀⠀⠀⠰⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠆⠀⠀⠀⠀⠀⠈⣿⣷
⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣈⣙⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⣁⡀⠀⠀⠀⠀⠀⠀⢸⣿
⣾⣿⠀⠀⠀⠀⠀⠀⣾⡿⠿⠿⠿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠿⠿⠿⢿⣷⠀⠀⠀⠀⠀⠀⣿
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⡀⣠⣶⣶⣄⢀⣠⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠙⠿⣿⡿⠋⠙⢿⣿⠿⠋⠛⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣏⠀⠀⢈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠛⠛⠿⢿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⠿⠟⠛⠛⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⡿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠆⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠰⣿⣿⣷⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣦⣄⡀⠀⠈⠛⠋⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠉⠛⠁⠀⢀⣠⣴⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠋⠉⠢⡀⠀⠀⠀⠀⢀⡴⠚⠉⠓⣆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡌⠀⡀⠀⠀⠹⡄⠀⠀⣠⠋⠀⠀⠀⠀⢸⠀⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⡇⠀⠀⠀⢀⠀⢳⠀⢰⠃⠀⡀⠀⠂⠀⡸⠀⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢱⠀⠀⠐⠀⠀⢸⡄⡇⠀⠂⠀⠀⠀⢠⠃⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⠀⠀⣿⠀⠀⢀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡤⡌⢣⠀⠀⠀⠀⡇⠀⠀⠀⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⢦⠀⠀⢸⢁⠇⠈⢢⠉⠀⢸⡇⠐⠈⠀⢀⡞⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠃⠓⠒⠛⠘⠛⠉⠉⠑⠒⠚⣄⠀⠀⢀⠌⠉⠙⠋⠉⠉⠳⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢀⡴⠉⠀⠀⢀⣀⠠⠀⠀⠀⡀⠀⡠⠄⠒⠢⢄⠀⠈⢦⡴⠃⠀⠀⠀⠐⡀⠀⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠈⠙⡄
⠀⠀⠀⡾⠀⢠⠦⡤⠤⡤⠵⡀⠀⠀⠁⠞⠒⠲⠰⠒⠘⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠁⠀⠀⠀⠀⠀⠓⠈⠒⠲
⠀⠀⠀⣧⠀⢇⣀⣐⣚⣁⣀⡇⠀⠀⠀⢦⠐⠒⠒⠒⢒⠇⠀⠀⠀⡀⠀⠂⢀⡎⠀⠂⠀⠀⣀⣀⣸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠤
⠀⠀⠀⢧⡀⠈⠢⣤⣠⡤⠊⠀⠀⠀⠀⠈⠒⠤⠤⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⢹⡀⠀⢀⡠⠄⠀⠀⠀⠀⠀⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠉⠁⢀⡞⠉⠙⠓⠲⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⢀⡤⠛⠋⠉⢀⣀⣤⣀⣄⠼⠉⠛⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⡤
⠀⠀⠀⡼⠀⢀⡔⠋⠉⠳⠤⢤⡄⢀⠀⠀⡤⡶⢋⠀⠀⡏⢹⠀⡆⣷⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢠⠇⠀⡼⠀⠀⠀⠀⠀⠈⠳⠋⠋⠛⢡⠃⡎⣼⠀⡇⢸⠀⡇⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠋⠁
⠀⢀⡞⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇⣿⡀⡇⢸⣠⢧⡎⢣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⡤⢤⠀⠀⠀⠀⢀⡖
⣞⠉⠄⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠿⠛⣿⡟⠻⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢻⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠷⢾⣿⠶⠶⠶⣿⡷⠶⠶⢾⣿⠶⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣶⣶⣾⣿⣶⣶⣶⣿⣷⣶⣶⣾⣿⣶⣶⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣤⣼⣿⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠋⠉⠉⠙⠛⠿⠛⠋⠉⠉⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣦⣤⣴⣾⣿⣿⣷⣦⣤⣴⣶⣿⣿⣷⣶⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠟⠛⠛⠻⠿⢿⠿⠟⠛⠛⠛⠿⢿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⡀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠶⠶⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠋⠁
⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠤⣄⣀⠀⠀⠹⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢈⡙⠿⣯⣉⠉⠛⢉⡉⠟⠛⠻⠿⣿⣶⣄⣽⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠖⠉⠉⠉⠁⠐
⠀⠀⠈⠻⡄⠀⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⣈⣹⣿⠇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣾⣁⡀⢤⠤⣤⣤⣀⣀⢀
⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⠙⠛⠿⠿⠟⠛⠉⠁⣴⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢙⡛⠻⠶⣶⣟⢿⡛⢿⣿⣷⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡈⣧⠀⠀⠀⠙⠷⣤⣉⠀⠀⠀⠀⣠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣀⣀⠀⣀⣠⣴⣾⠿⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢷⣄⠀⠀⠀⠀⠉⠛⠶⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠫⠾⠿⡿⠿⠋⠁⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠻⣿⣿⣷⣶⡦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠙⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⡶⠟⠛⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢠⣾⠉⠀⠀⠀⠀⢸⡟⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡇⠀⠀⠀⠀⠀⠈⣧⠈⠙⠳⢦⣄⡀⠀⠀⠀⣰⡿⣤⣉⣁⣀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠈⣋⠙⣿⡷⠶⠟⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣠⡶⠏⣡⠀⠀⢹⣇⠀⠀⠀⢹⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠁⠁⣊⡡⠀⠀⠀⠻⣄⠀⠀⣘⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠛⠉⠳⠞⠋⠙⠰⣶⣶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠙⢷⡀⠀⠀⣠⡶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⢀⢀⣀⠀⠀⢹⡇⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⢸⡇⢠⡾⠋⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠐⠁⠀⠀⠙⠃⠼⢿⣡⣿⢃⣀⠤⠤⠄⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⣿⢀⡤⠴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⣿⡀⠀⢠⡔⠒⠐⠒⠀⢀⠀⠉⠉⠉⡠⠆⠒⠂⢠⠀⠀⠀⠟⠋⠉⠀⠀⠀⠁⠀⠙⠋⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡒⠂⠁⠹⣧⠀⢨⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⢃⠀⠀⠀⠀⠀⢱⡀⠀⢰⡏⠀⠀⠀⢠⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⠉⠛⠦⢤⣿⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣻⠀⠸⡄⠀⠀⠀⠀⢨⡅⠀⠈⢃⠀⠀⠀⠠⠆⠀⢠⡎⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⣄⠀⠀⠀⣀⠔⠁⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠘⢇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⠿⠿⠿⠻⡿⠟⣽⠛⢿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣻⣟⣛⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣴⣶⣶⣾⣶⣶⣶⣶⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠉⠛⠿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠟⠉⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⠀⠀⣠⣾⣿⠿⠿⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⣰⣾⣿⠋⠀⠀⠀⠀⣼⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀
⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⣼⣿⢿⣿⣦⣀⣀⣀⣴⣿⠏⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇
⠀⠀⠀⢠⣿⣿⠏⠀⠀⢸⣿⠇⠀⠙⠛⠿⠿⠟⠛⠁⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⢀⣾⣿⡏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⣼⣿⣿⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣴⣶⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠸⣿⣿⠉⢻⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⣿⣿⡇⠀⠉⠛⠿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧
⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⢻⣿⡆
⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⣸⣿
⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠃⠀⢰⣿⡿
⠀⠀⠀⠀⠈⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠋⠁⠀⣀⡀⠀⠀⠘⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣶⣤⣄⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣠⣴⣶⣾⠿⠟⠉⠁⠀⠀⠀⠀⠻⢇⣀⡀⠀⠀⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⣤⣴⣶⣶⣶⣶⣄⡀⠈⠛⠁⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠻⠿⢿⣿⡿⠿⠛⠛⠛⠛⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡇⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣾⣿⣿⠿⠛⠉⠀⠀⠀⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣾⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⡿⠿⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣶⣿⣿⡿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣤⡄⠀⠀⠀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠸⠿⠇⠀⢿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠿⠿⢿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣦⣤⣤⣤⣶⣶⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠟⠉⠉⠉⠉⠉⠉⠻⣿⣿⡀⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⣠⣶⣿⠟⠁⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⢠⣶⣾⡶⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠻⣿⣿⣤⣄⣀⣠⣴⣾⣿⠿⠛⠁⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣶⣶⣦⣶⣶⣶⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⣿⣿⡿⠉⠀⠀⣹⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠁⠀⠀⠀⢿⣿⣿⣦⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⢠⣄⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠘⣿⣷⠀⠀⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⣿⣿⣧⠀⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⢰⣿⣿⣿⡄⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⣠⣿⡿⠻⣿⣿⣦⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⣠⣼⣿⡿⠁⠀⠈⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡖⠉⠀⣠⢊⣠⡀⠀⢬⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡤⣠⣾⣿⢟⡽⡟⣿⣿⡻⣦
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⢣⠎⢳⡹⡄⠀⠹⡜⡆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢱⣿⣿⡏⡞⣵⡜⣎⣿⣿⣎⢧
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⢸⠀⠀⡇⡇⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇⡇⣿⣿⢸⣸⣿⣿⢸
⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⢳⡀⠀⠘⣎⢷⣴⣿⣧⣴⡶⣷⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡜⣿⣿⣧⢳⡙⣿⣶⣿⣿⡷⣯
⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠓⢤⡀⠈⠢⣉⠛⢛⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠻⢿⣷⣝⠶⣿⣿⡆⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠇⣋⠚⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡸⠇⣃⠛⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⠿⢿⣦⣀⠀⠀⢸⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣽⡞⣧⣼⢝⣷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣿⣿⣿⣿⡷⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣿⣿⡿⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣻⣿⡟⢡⢶⣿⡷⢀⣤⠁⣌⡀⠀⠀⠐⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣽⢿⣿⣷⣸⣾⣦⡹⡐⢲⠈⠲⣜⠻⡾⡀⠀⢂⠆⠠⠄⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣻⡿⠋⠈⣄⠀⠳⣌⢢⠘⠄⠉⠛⠦⣝⡀⠠⡀⠲⠀⡄⡀⠱⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⣇⣠⣤⡈⢻⡐⢄⠀⠀⠀⠈⢙⣆⣳⡀⠢⠙⠈⢀⡀⡁
⠀⠀⠀⠀⠀⠁⣀⣢⣤⡀⠀⢀⡀⠀⠀⠀⣿⠈⠻⠿⣿⡃⣈⣉⠲⡆⠀⠐⡀⠁⠌⠛⡗⠈⠂⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⢸⠷⠿⠷⠄⠚⠀⠀⣾⣿⣿⣿⣽⣦⣼⡄⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠐⢿⣿⣿⣿⡟⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠉⡙⣿⣿⠏⠈⠀⠠⣤⣄⣠
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣄⠘⠃⠀⠀⡀⠤⠀⠄⠀⢀⠀⠀⠹⠋⠀⣿⣷⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⡛⠁⠀⠀⠊⠀⠀⠀⠤⠐⠈⠀⡈⠀⠀⠐⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⣿⣶⣧⡀⠀⠀⠀⠀⢀⠀⢀⠤⠁⠀⠀⢠⢸⣿⢋⣠⡔⡀
⠀⠀⠀⠀⠀⠀⠘⠓⠉⠁⠀⢠⡈⠁⣂⡄⣀⠀⠀⠀⠈⠀⠀⠀⢠⡶⢾⣾⡇⣡⢾⣭⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢠⡟⣽⠀⣿⡃⠀⣂⠒⠒⠄⠀⣀⣠⣿⣿⣾⣿⢿⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃⣸⢫⡇⠠⣿⣷⡤⡠⠚⢛⣿⣿⣿⣯⣵⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠾⠲⢴⣿⣿⡇⢀⡟⠀⠀⢠⠟⢹⡗⠁⣀⣼⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣈⡿⠀⠼⠀⠀⡰⠃⠀⢞⣠⣾⣿⣿⡿⠟⠀⠀⠙⢿⣿⣯⠉⠷⣍⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⣠⠃⢀⠜⠁⢀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠙⠃⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠂⠒⠻⠛⢿⡛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⣠⡀
⠀⠀⠀⣠⠀⠀⠀⣿⡇⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⡇⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠇⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠈⣦⡀⠀⠻⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⡇⠀⠀⢸⣿⣧⠀⢠⡄⠀⠀⠀⠀⠀⠀⠸⣆⡀⠋⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⢿⣟⣦⣀⣀⠀⠀⠀⠛⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢧⡘⠟⠀⠀⠀⢸⣿⠇⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢻⢹⣦⡆⣀⡀⡈⠁
⠀⠀⠀⠀⠀⢹⣷⣼⣋⡧⢶⢶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠈⠉⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣯⢥⣿⡳
⠀⠀⠀⠀⠀⠀⠛⠞⠯⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⡾⠛⠛⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣄⡀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⠀⣼⣿⠇⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⢼⣿⢦⣹⠀⠀⡇
⠀⠀⠀⠀⠘⡆⠀⢻⡟⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣠⣿⠀⢼
⠀⠀⠀⠀⠀⣷⣀⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡇⠀⠀⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠈⠻⠿⠋⠀⡞
⠀⠀⠀⠀⠀⠙⢯⣙⠦⣄⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣤⣎⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣀⡀⣀⣄⡀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠼⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢧⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠑⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄
⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠟⠉⠀⠀⢀⣴⡿⠋⣁⣤⣤⣈⠙⢿⣆
⠀⠀⣰⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⡿⠃⠀⠀⠀⢀⣾⠏⣠⣾⠛⠉⠉⠻⣷⡀⠻⣧
⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⣾⠏⢰⣿⣧⠀⠀⠀⠀⠘⣿⡀⢻⣧
⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢸⣿⠀⣾⡏⣿⡆⠀⠀⠀⠀⢻⣇⠈⣿⡀
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⣿⠁⢹⡇⠀⠀⠀⠀⢸⣿⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣇⠀⣿⡀⣸⡇⠀⣀⡀⠀⢸⣿⠀⣿⡇
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠸⣿⠀⢿⣷⡿⠿⠛⠛⠛⠿⢿⣧⣠⣿
⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⢿⣇⠘⣿⡄⠀⠀⠀⠀⠀⠀⠉⢻⣯
⠀⠀⠙⢷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣄⠀⠀⠀⠈⢿⣆⠘⢿⣦⣄⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣷⣄⣀⠀⠀⠻⣷⣤⣈⢻⣧⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⢻⣿⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⢀⣴⡶⠿⢶⣦⣀⠀⠀⣠⣴⡾⠿⣶⣤⡀⠀⠀⣠⣶⠾⠷⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣠
⠘⠿⠿⠛⠉⣀⣀⣀⠉⠛⠿⠿⠛⢁⣀⣀⣀⠙⠻⠿⠿⠋⢁⣀⣀⣿⠇⠀⠀⠀⠀⣀⣀⣀⡀⠸⣿⠿⠛
⢠⣤⣤⣴⡿⠛⠉⠛⢿⣦⣤⣤⣶⠿⠛⠉⠛⢷⣦⣤⣤⣾⠟⠋⠙⠻⣷⣤⣤⣤⡾⠟⠋⠙⠻⣶⣿⣧⣴
⠀⠉⠉⠁⢀⣠⣤⣄⡀⠈⠉⠉⠁⢀⣠⣤⣄⡀⠈⠉⠉⠀⣀⣤⣤⣀⠀⠉⠉⠉⠀⣀⣤⣤⣀⠀⠉⠉⠁
⢠⣦⣤⣶⠟⠋⠉⠙⠻⣷⣤⣤⣾⠟⠉⠉⠙⠿⣶⣤⣴⡾⠟⠉⠉⠛⢿⣦⣤⣴⡿⠋⠉⠉⠻⢷⣦⣤⣾
⠀⠈⠁⢀⣴⡶⠿⢶⣦⣀⠈⠁⣀⣴⡶⠿⣶⣤⡀⠉⠀⣠⣶⠾⠷⣶⣄⡀⠉⢀⣠⣶⠿⢷⣦⣄⠈⠁⣠
⠘⠿⠿⠛⠉⠀⠀⠀⠉⠛⠿⠿⠛⠁⠀⠀⠀⠙⠻⠿⠟⠋⠁⠀⠀⠈⠙⠻⠿⠟⠋⠀⠀⠀⠈⠛⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡆⠀⠀⠐⠀⠀⠀⣤⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⠤⣦⠶⣶⠒⠂⠖⠹⣧⣾⣅⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣑⣯⢟⣳⠚⣿⣤⣤⣿⡯⡚⠑⣝⡉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⢦⡔⣾⣿⣿⣾⣭⠟⣼⣿⢩⣿⠀⡶⠀⠿⠋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⢀⡿⢿⣿⣿⣿⣿⣿⣾⣯⣟⣽⡆⢸⣿⣤⣿⡔⠱⡖⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣧⠶⣶⣿⣿⢸⣿⣿⣿⣿⣿⣿⣳⣼⣿⣷⣿⣿⣀⣮⣷⣾⣯⣉⣟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⢻⡭⣤⣿⣿⣿⣿⠿⠞⢿⣿⣿⣾⣽⣿⣿⡴⣿⣿⣿⣿⣯⣻⣿⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣵⣿⣾⢣⣝⣻⣿⣿⣾⣿⣿⣷⣶⣷⣿⣿⣿⠿⣿⢿⡿⣿⣿⡄⢛⡽⣷⣯⣿⠟⠡⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⣴⣿⣪⡽⣿⣿⣿⣿⣿⣷⣾⣯⣶⣿⣿⣿⣿⣿⣿⣿⠏⠉⢺⣿⡟⠛⣶⣽⢏⢸⡖⠀⡇
⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣽⣍⢤⣿⣿⣿⣿⣩⣽⣿⣽⢿⣿⣿⣿⣷⣿⣇⣭⡭⢻⣿⡟⡽⢻⣾⣿⣿⡤⠛⠉⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⢫⠛⢻⣤⣿⣿⣟⡻⢑⢟⣻⣿⢴⣿⣿⣿⣿⣿⣿⠧⣽⣶⣿⣯⣾⣿⠛⢿⣠⡌⢀⡤⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣴⣯⣭⣿⣿⣻⣾⡿⣯⣟⣛⠻⢿⡿⣿⠉⢁⣠⣼⣿⣾⣿⠿⡝⠛⣹⣷⣷⣾⠘⣏⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠓⠛⠉⠁⠈⠉⠹⠿⣿⣿⣿⣿⣧⣴⠶⠀⠀⢿⡶⣼⣿⣧⣼⢃⠰⣄⣸⠾⣏⠹⣮⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⣿⢿⣧⡴⠠⣾⠓⡾⣈⣿⢵⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠊⠛⠛⠃⠙⡑⣿⠟⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣦⣄⠒⠄⠀⠀⠀⠀⢤⣶⣴⣮⣄⠀⠀⣼⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠈⠻⠛⠛⠋⠀⠈⢰⣶⣄⠀⠈⠻⢿⡿⠋⠀⠸⠋⠉⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠳⠖⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣤⣒⣤⡀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣽⣿⣿⣿⠟⠅⠀⠀⠀⠀⠀⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠢⠄⠛⠛⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⣤⣴⣿⣶⣦⣿⣾⢿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⠁⡆⠀⠀⠀⠀⠁⠈⠁⠀⠙⠃⠒⠤⣄⣀⠀⠀⠰⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣾⣧⣤⣄⣀⣤⣀⣀⠀⢀⣶⣀⣀⣠⣬⣿⣷⣶⣆⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠉⠉⠉⠉⠉⠙⠻⠿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⣿⠿⠿⢿⣿⣿⠟⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⣾⣿⣿⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⣀⣄⣠⣶⣶⣿⣿⣿⠟⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⣀⣀⣠⣶⣿⣿⣯⣭⣠⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⢿⡿⢻⣿⡿⠃⢠⣾⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⢿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⡭⠉⠀⣠⡿⢃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠰⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠂⠀⠀⠀⠀⠀⠸⡀⠀⠀⠠⣀⢤⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⠃⠀⠀⢀⡐⠁⣀⠤⠒⠘⡞⠁⠀⠀⣇⣀⠆⠀⠀⢀⡠⠒⠉⠀⠀⢱
⠀⠀⠀⠠⢦⠀⠀⠀⠀⢇⡀⣀⠔⢊⠩⠿⡏⠀⠠⢤⠄⠑⠦⡤⠊⡩⠋⣱⠄⠒⠉⠀⠀⠀⠀⣠⠃⡴⠞⠁⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠚⣉⠀⠀⠓⠾⣄⠀⠸⢤⣋⡠⠴⠳⣆⣀⣀⡀⢤⣐⡨⠔⠚⠣⡀⠀⠀⠀⠀⠘⠤⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠁⠀⠀⠀⠀⠙⢆⠀⢀⡁⢀⣴⡆⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠾⠂⢎⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀⠀⠀⠰⡰⠀⠀⠀⢀⠤⣄⢀
⠀⠀⠀⠀⣠⢬⡗⣎⣠⠤⠤⠊⠃⠀⠉⠐⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣐⠀⠙⢶⣄⢀⠀⠀⠀⠀⢈⠂⠠⠄
⠀⠀⠀⠀⢯⣉⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⣀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⡋⠻⠷⠿⠫⢥⢾⣕⣒⣒⢒⠒⢒⢒⣒⣒⣾⠡⠮⠢
⠀⠀⣀⠀⢈⣿⡿⡟⡆⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⢸⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⡎⠘⢷⣺⡿⠁⡅⠈⠄⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠈⠁⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤
⠀⠀⠀⠀⠠⡇⠀⢀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠐⠚⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣽⠲⢋
⠀⠀⠀⠀⠀⢀⡀⠀⣆⠀⠈⠻⠮⣄⠀⠀⠀⠀⠀⠁⠢⠠⠄⠀⠄⠤⠊⠉⠒⠦⠤⣤⠤⣤⠀⡤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠁
⠀⠀⠀⠀⢀⣀⢴⠉⠁⠀⠀⢀⠇⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠁⠠⢀⠔⠁⢰⠾⡀⠀⠀⠀⠳⢤⣴⣀⡑⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠈⠒⠒⠁⠀⠑⠤⠀⠀⠠⠴⠶⠯⢝⡛⠲⠶⣤⣄⣈⠉⢒⠲⠦⠖⢒⡒⠢⢠⠴⠓⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣚⠛⠭⠿⠿⠶⢦⡤⠄⠄⠴⠁⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠒⠀⠐⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠐⠒⠒⠾⢗⣒⣒⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠽⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⣿⣿⣿⣿⣿⣿⣿⣿⠱⡿⣟⠛⢻⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢻⡿⣻⣾⣿⣿⣿⣿⠘⠟⠩⡻⢿⢸⡟⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠫⠞⠏⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⣗⡭⠿⣿⡽⣿⢰⡶⠲⠰⣞⢸⣅⣿⣿⣿⡿⣿⣿⠿⡃⠘⠅⣒⣲⠂⠉⠹⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⢲⣚⣋⣥⣴⡴⣟⡷⠧⢨⣤⣤⣥⣽⢘⠸⣧⣿⣭⣿⢿⣿⠗⠡⠾⠿⣿⣿⣿⣶⢀⢸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠐⠀⡀⣨⡬⣟⣿⣽⣷⣾⣿⣿⣗⢈⣧⡈⢋⡻⢜⠮⣿⣿⣷⣿⣿⡟⠠⡀⠀⡀⢹⠿⣿⣝⡼⣦⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡁⠃⠀⡀⠀⠀⠀⠀⠸⠀⠉⢿⣏⣿⣾⣿⣿⣿⣿⢸⣉⣛⠝⣧⣼⣹⣿⣿⣿⣾⣾⡁⡀⠾⠛⠛⠀⠁⠀⡙⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⢀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠘⣦⠙⣿⢿⣿⣿⣿⣿⠸⣛⣿⡚⣏⢸⣾⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⣄⣀⢠⡄⠛⠂⣹⡍⢻⣿⣿⣿⠸⡿⠀⢐⡏⠸⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣎⣏⠋⠀⠀⠁⠀⠠⠀⣸⣿⣿⣿⠘⠛⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣻⡏⠀⡾⡀⢀⣀⠀⠊⠉⠉⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⢽⠗⣿⡛⢽⠁⢈⡇⠇⠀⢷⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣀⠀⠀⠠⠐⠀⠀⠈⠠⠀⠈⡀⠄⠀⠀⠘⢿⠖⠀⠘⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿
⢰⣶⢻⡦⠁⠀⠀⢀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠯⣿
⢨⡏⣿⠁⠤⠴⢂⠍⡀⠈⠁⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠠⠚
⠀⣡⠁⢦⠦⠆⡈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢺
⠰⡽⠣⠋⠒⣀⣀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠉⢙
⠀⠶⠄⢁⢀⡂⠁⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⠜
⢠⣤⣈⡷⢀⠉⠁⢒⡀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠚⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢸⡆⠀⠀⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡾⠀⠀⠀⣿⡟⣥⣾⣿⣿⣿⣷⣶⣦⣭⣝⣛⣛⣛⣫⡅⣾⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣇⠀⠀⠀⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠉⠀⠀⠀⠀⢿⣿⣿⣿⣩⣽⣿⣿⣿⣿⣯⣭⣻⣿⣿⣿⠀⠀⠀⠀⣤⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢟⣛⣛⣛⣛⡻⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠺⢡⣖⣰⡌⠗⣼⣿⣿⠟⠀⠀⠀⠀⢀⡟⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠓⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡌⢿⡿⣡⣿⣿⠟⠋⠀⠀⠀⠀⠀⠘⡇⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⡰⣶⣶⣶⣶⣶⣶⣶⡶⢠⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣌⡻⠿⠿⠿⠿⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠲⠋⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣳⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⣶⠀⠀⠀⠀⢠⣖⡶⠰⠾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⠈⠉⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠿⠃⠀⠀⠀⠀⢰⣟⣷⠀⠀⠀⢰⡏⠉⢷⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠟⠸⣟⡆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠳⠶⠋⠀⠠⣏⡇⠀⠀⠀⢠⠤⡄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⣯⠇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡈⠛⠃⠀⠀⠀⣀
⠀⠀⢀⡴⠛⠉⠉⠉⠉⠉⠙⢳⡄⢀⡴⠚⠛⣷⠞⠉⠉⠙⢦⡀⢀⡴⠛⠉⠉⠛⠶⣄⠀⡴⠋⠀⠀⠀⠀⠙⢦⠀⡠⠞⠉⠉⠉⠛⠲⡄
⠀⣰⠋⠀⠸⠁⣀⣤⣤⡀⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⠀⡖⠀⠀⠀⠀⠈⢿⠃⠀⠘⠁⠀⠀⠀⢈⡿⠁⠀⠀⠀⢠⣤⡴⠃
⢠⡏⠀⠀⢀⡾⠉⠀⠀⢷⠷⣼⠃⠀⡇⠐⠀⢀⠶⡄⠀⠠⡞⢹⠀⠀⠀⡠⢤⠀⠀⠀⠘⡆⠀⠀⠀⡔⠒⠋⠉⠱⠤⠤⠤⣤⡀⠉⠉⠲⣄⢸⣿
⢸⡇⠀⠀⢸⠁⠀⠀⣤⣀⣀⣀⣀⣀⣧⣀⠀⠀⠷⠃⠀⠀⠹⣿⠀⠀⠀⣇⡸⠀⠀⠀⢀⡇⠀⠀⠀⠧⣀⣀⡀⠀⡀⠀⠀⠀⣹⠀⠀⠀⢹⡄
⠘⡇⠀⠀⠘⣇⣀⣸⣧⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⠀⠀⠀⡼⠻⢿⣶⣄⠀⠀⠀⠹⡏⠉⠉⠛⠋⠁⠨⠃⠀⢸⣧
⠀⣿⡄⠀⠀⠈⠻⠭⡽⠇⠀⠀⠀⣴⢶⠚⠀⠀⢠⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⡞⢳⠀⠀⠉⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⢸⡄
⢰⠃⠙⠶⣄⠀⠀⠀⠀⢠⣄⠀⠀⣿⢸⠀⠀⠀⢸⡟⣆⠀⠀⠀⠀⣶⢦⠀⣤⠀⣸⡇⢸⣠⣦⠀⠀⠀⢀⣤⡔⠁⠀⠀⠀⠀⠀⣀⢸⡇⠀⢸⡇
⡿⠀⠀⠀⢸⠀⡀⠀⠀⣏⢹⣤⢠⣼⠈⠛⢻⠀⢸⠀⢸⠀⣠⡀⡟⠁⢸⢰⡟⠷⠿⠇⠀⠉⠸⣄⣶⡀⢸⠀⢻⡀⡶⡄⣴⢦⠐⡟⠛⠀⠀⠘⡇
⡇⠀⠀⠀⢸⣸⠟⠞⣇⣿⠀⢸⢸⠀⠀⠀⠸⢾⣼⠀⢸⠀⡏⠛⠁⠀⢸⣆⡇⠀⠀⠀⠀⠀⠀⠛⠁⣇⣼⠀⠀⢿⡇⠳⠟⠸⣦⠇⢀⠀⠀⠀⡇
⡇⠀⠀⠀⠈⠟⠀⠀⠙⠃⠀⠘⠿⠀⠀⠀⠀⠀⣿⠄⠸⣦⠇⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⣇
⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡀⠀⣿
⣿⠀⠀⠀⠂⠀⠀⠘⠆⠀⠀⠐⠄⠀⣰⣆⢀⡀⠰⠄⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠈⠃⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻
⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⢸⠞⣧⠀⠀⠀⣴⠀⢠⣦⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀⣿⣀⠞⣧⠀⠀⣤⠀⡄⠀⣴⠀⣀⢀⣄⠀⠀⢸
⢻⠀⠀⠀⠀⠀⡼⢻⠀⠀⢀⣄⠀⢰⠃⠀⠀⣿⠀⠀⢰⡏⡇⡾⠸⡄⠀⢸⠇⢳⡀⣷⣴⠀⢸⠉⠛⠀⢸⠀⠀⢸⠶⠻⠞⢹⡆⠉⢸⢹⡄⠀⢸
⢸⠀⢠⣤⠀⠀⡇⢸⠀⠀⣸⢹⡄⢸⠀⠀⠀⢸⠀⠀⢸⠀⠙⠁⠀⡇⠦⢸⠀⠈⠻⠁⢹⡆⢸⠀⠀⠀⢸⡆⠀⢸⠀⠀⠀⢸⡇⠀⢸⠈⣇⠀⢸
⠘⢷⠏⢸⠀⠀⡇⠈⡇⠀⡁⠀⡇⢸⠀⠀⠀⢸⡄⠀⢸⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠸⡇⠀⢸⠀⠹⣦⡞
⠀⠀⠀⠀⡆⠀⡇⠀⣧⠀⡇⠀⠹⠟⠀⠀⠀⠈⡇⠀⢸⠀⠀⠀⠀⣷⠀⣿⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⠀⣇⠀⢸
⠀⠀⠀⠀⢷⣸⠁⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡾⠀⠀⠀⠀⣿⠀⡻⠀⠀⠀⠀⠸⡇⣸⠀⠀⠀⠀⢻⡼⠁⠀⠀⠀⠀⠘⢦⡟
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠃⠀⠀⠀⠀⠸⣆⡇⠀⠀⠀⠀⠀⣷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠲⡀
⠀⠀⢀⡠⠤⢄⡀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠗
⢀⠞⠁⠀⠀⠀⠈⠢⡀⠀⢸⠀⠀⠀⠀⠄⠀⠤
⠄⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⡄⠀⠀⠀⠀⠀⡄
⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⣄⠀⠀⠀⢰
⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢢⡀⢀⡇
⠀⠀⠀⠑⠦⢤⣀⡀⠀⢀⣀⣘⠦⠎⠈⠁⠘⠒⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠉⢉⡹⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠒⡲⠖⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⡀⠀⠐⠠⠄⠐⠁⠋⠀⢉⠈⡉⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠦⠄⠃⡴⠔⣁⢀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠂⠐⠊⠀⠀⠀⠀⢢⡇
⠀⠀⠀⠀⠀⠀⡴⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⢱⢦⢤⠄⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠞⠳⡀⠀⠘⡀
⠀⠀⠀⠀⠀⠈⢳⠆⠀⠀⠀⡀⣄⠲⡒⢲⠰⠶⡖⠒⠲⠖⠉⠁⠀⠀⠀⢸⠄⠀⢈⡅
⠀⠀⠀⠀⠀⠀⠈⢆⣀⣀⣅⠳⣌⢧⣙⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢯⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡙⣿⣮⠓⡾⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠎⠁⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡠⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠶⣄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢻⡄⠙⢄⢀⡔⢉⡉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠇⣣⠈⣿⢆⠀⠉⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠋⠀⢸⠈⠣⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠅⠀⠈⢲⡤⡤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠠⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣰⠃⠀⠀⠄⣠⠀⠀⠀⠀⠀⠀⠀⢀⢊⡞⠀⡴⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠁⡼⢀⢈⡔⠁⠀⠀⠀⠀⠀⠀⠀⠌⣼⢰⣼⠁⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢄⠀⠀⠀⣀⣠⣤⣤⣄⣀⣀⣀⣠⢡⣁⣰⣠⣸⣀⣆⠜⠀⠀⠁⠀⠀⠀⠀⠀⢸⣦⡟⣿⡇⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠸⡆⢀⢧⢮⣴⡿⢛⣫⣥⣤⣤⣤⡙⠻⢟⠛⠻⡟⠛⢿⡿⣿⠾⣶⢿⡶⢦⡤⣤⢶⣤⣤⣾⢿⢷⣿⢱⢃⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣸⣿⠟⢁⣴⢟⣿⣿⣿⣿⣿⣿⣷⣄⠡⡀⠘⣆⠀⢣⡈⢣⠸⡄⠱⡄⠳⡜⢦⣹⣿⣿⢸⢀⡇⡞⢸⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢤⠀⡀⠀⢀⢸⣯⡟⠁⣴⢟⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣷⠸⣄⠘⡆⠈⣧⠈⡇⢹⣤⣿⣀⣿⣼⣿⣿⡇⡎⢸⠇⠁⡌⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢣⡈⠀⢷⢰⢰⣾⡟⠀⣼⠏⢾⡟⣼⡿⣫⣿⣿⣿⣿⣿⣿⣿⡇⢿⡄⣿⡆⣿⡄⣹⣸⣿⣿⣿⡟⣻⡿⣏⠃⡇⢸⠀⡄⠃⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢷⢰⡈⡘⠿⣿⡇⢰⡟⠀⣿⠀⣿⢁⣿⢯⣿⢟⣿⣿⣿⣿⡇⣸⣿⣿⡇⢸⣧⣿⣿⣿⡟⣸⢃⢹⢣⣿⠀⠃⡟⢰⣽⢀⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠈⠈⠃⢡⠀⣿⡄⢸⠃⠀⡇⠀⢻⠀⣿⠸⣿⢸⡟⣿⣿⣿⠇⣿⡟⣿⡏⣿⢿⣿⣿⣿⢡⡇⢀⠊⣼⣿⢀⠀⡇⣼⠀⣸⠀⠹
⠀⠀⠀⠀⢠⢰⡰⠀⣦⠀⠀⠁⢀⡀⢸⠀⢹⣷⠈⠀⠀⠁⠀⠈⢀⣈⣀⣉⣬⣥⣾⣿⡟⣼⣿⣿⣿⣷⣿⣿⣿⢇⣯⢸⡀⡜⠀⣧⡇⣿⠀⡇⣿⢀⡇⠀⣷
⠀⠀⠀⠀⠘⣆⣧⠇⢻⣤⠰⠼⠟⠃⠀⠀⢊⣿⣷⣶⡾⢿⣛⣿⣿⣿⡿⢿⣿⣿⣿⢿⣿⣿⣿⣿⠿⠟⠛⠋⢻⠘⡟⢸⣥⡧⢠⡏⡇⣿⠀⡇⠿⢸⡆⡇⢸
⠀⠀⠀⠀⠀⠉⠁⠃⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⣩⣴⣾⠟⣫⣥⣶⣶⣿⣿⣿⣿⣵⣿⣿⠿⠟⠁⠀⠄⠀⠀⠚⠀⢀⠛⠋⣴⣼⡃⣷⣷⡄⣇⣰⡾⠙⢂⣽
⠀⢀⡀⠀⠁⢀⣄⠀⠀⠒⠀⣠⣾⣿⣿⣿⢫⣾⣿⣿⡟⣾⣿⣿⣿⣿⣯⣿⣿⠿⠿⣛⠁⠁⢈⣤⣄⠉⠀⠀⠠⠀⠀⣀⠀⠀⠉⠁⠛⠉⠁⠀⠀⠀⠀⢾⣿
⠀⣼⣷⣶⣾⣿⠿⢿⣿⣿⣿⣿⣿⢡⣦⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣦⣤⣤⣶⣶⡾⠟⣻⣿⣶⣤⣤⣤⣴⣾⣿⣷⣌⣀⣀⣤⣴⣾⢿⣦⣀⣠⣼⣿
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢾⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠾⠿⠿⠿⠿⠿⠿⠿⠶⠿⠿⠿⢿⣿⣿
⣿⣿⣿⠿⣿⠿⢿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣮⣶⣷⣴⣧⣾⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠓⠋⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢯⣬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⢰⣿⠗⡘⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⢤⡀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣶⢉⣄⠀⠀⠀⠀⠿⡼⠃⠀⠀⠸⣇⠀⡿⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠘⠾⠁⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⠉⠀⠀⠘⠛⠁⠀⠀⠀⣟⡷
⠀⠀⠀⣀⡤⠴⠶⠶⠶⠤⣤⡀⠀⢀⣠⣤⣄⣤⠤⠤⣄⠀⠀⠙⣫⠤⠤⢤⣀⠀⠀⣠⠖⠛⠛⠛⠶⣄⠀⠀⣠⠤⠤⠤⣄⣀
⠀⢀⡞⠁⡠⠀⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⠋⠀⠀⠀⠈⢳⣤⠞⢁⠀⠀⠀⠈⠳⣴⠇⠀⡤⠀⠀⠀⠘⣦⠞⠁⠀⠀⠀⣀⣸⠇
⢀⡟⠀⠀⢀⡴⠛⠉⢳⣤⣀⡇⠀⡇⢠⠀⢀⣤⠀⠀⣀⠼⡏⠀⠁⢀⣀⠀⠀⠀⢹⠀⠀⠀⢀⡤⠤⠴⢻⡀⠀⣀⠀⠘⠻⠧⣄⠀⡴
⢸⠃⠀⠀⣼⠁⠀⠀⠈⠁⠉⠀⠀⣇⠀⠀⠸⣼⠀⠀⠸⣦⡇⠀⠀⣎⢸⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⠈⠉⠉⠉⢣⠀⠀⠈⣧⠉
⢸⡆⠀⠀⢻⡀⠀⣾⡉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠈⠋⠀⠀⠀⡼⢶⣦⣄⠀⠈⠉⠑⡶⠚⠒⠶⠖⠃⠰⠀⠀⢸⡄
⠀⣧⠀⠀⠀⠻⢯⣹⡿⠀⠀⠀⢠⣤⠼⠀⠀⢀⡀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⡼⣧⠀⠉⠉⠁⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⣠⠏⣧
⢰⠋⠳⢤⡀⠀⠀⠀⢀⣀⠀⠀⢸⢸⠀⠀⠀⢸⢷⡀⠀⠀⠀⢰⢤⠀⣀⠀⢸⡇⢿⣠⡄⠀⠀⠀⣠⣤⠂⠀⠀⠀⠀⠀⡀⢸⠁⠀⢻
⡾⠀⠀⠀⣿⢀⠀⠀⢸⢹⣦⢠⣼⠘⠛⣷⠀⢸⠀⣷⢀⡄⢸⠋⠸⡇⡟⠷⠿⠃⠈⠉⢹⢰⣆⢰⡇⠸⡄⡴⡆⢴⣆⢸⡟⠛⠀⠀⢸
⡇⠀⠀⠀⢸⣾⠓⢻⣸⠀⢸⣿⡀⠀⠀⠸⢷⣾⠀⣏⢸⠙⠋⠀⠀⣇⡇⠀⠀⠀⠀⠀⠈⠋⢸⣸⠇⠀⢹⣧⠹⠟⠸⣶⠇⢀⠀⠀⢸
⡇⠀⠀⠀⠘⡃⠀⠈⠛⠀⠈⠿⠃⠀⠀⠀⢸⣿⠀⢻⡼⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠀⠸
⣿⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠃⠀⡼⢧⢠⡀⠑⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⢀⡀⢀⣀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠉
⣿⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡇⠘⠋⣇⠀⠀⢰⢧⢀⡿⡄⠀⠀⡶⣆⠀⡀⠀⠀⢸⢷⡜⠹⡆⠀⢸⣀⣦⣀⣿⠀⠆⢰⣆
⢹⠀⠀⠀⠀⢰⠃⡇⠀⢠⡶⡄⠀⡇⠀⠀⣿⠀⠀⣾⢸⣼⠀⣧⠀⢰⠇⠹⣤⠟⢿⠀⣸⠀⠀⠀⡇⠀⢸⠛⠉⠉⢸⡇⠀⢸⢸
⢸⣀⡞⢣⠀⢸⠀⣇⠀⢸⠀⡇⢀⠁⠀⠀⢹⠀⠀⣿⠀⠀⠀⢸⠈⢸⠀⠀⠉⠀⢸⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⡇⠀⢸⠘⣇⢀
⠀⠉⠀⢸⠀⢸⠀⢸⠀⢸⠀⢻⣼⠀⠀⠀⢸⠀⠀⡏⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⡃⠀⠀⠀⢻⠀⢸⠀⠀⠀⠘⡇⠀⣸⠀⠈⠛
⠀⠀⠀⠸⡆⣼⠀⠘⢧⠎⠀⠀⠁⠀⠀⠀⠘⡇⠀⡇⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸⣀⡟⠀⠀⠀⠀⢳⣀⡇
⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⠁⠀⠀⠀⠸⡆⣸⠀⠀⠀⠀⢸⣤⠇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⢂⣠⣤⣶⣶⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠁⣠⣾⡿⠛⣋⡉⠀⠀⠀⠀⠀⠈⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡀⠈⠛⠁⢠⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣡⣤⣴⣤⣤⣀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣴⣿⣿⣿⠋⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⠛⠃⠘⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣉⣭⣭⣭⣭⣭⣭⣤⣤⣄⠀⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣴⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⢀⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏⢠⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁