Man goofball Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⠎⠀⠘⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣤⣶⡾⢿⡿⠟⠛⠉⠉⠉⠶⠄⢄⠀⢀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⢠⡂⠈⢀⡀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⡠⠴⠖⣦⡀⠀⡔⠁⡎⠍⡓⠀⠀⢀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠳⠟⠁⠀⠀⠀⠈⠉⠁⠋⡆⠐⠆
⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡐⢤⠀⠀⠀⠀⠀⡀⡀
⠄⣕⢦⣶⣤⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⢐⢠⡀⠀⠀⠀⢀⣀⠀⠀⠤⠀⢠⠀⠀⠀⠀⠈⠀⠉
⢽⣋⠸⠁⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠁⠀⠀⠀⠀⢀⠄⠂⣌⢋⡡⠀⠂⠀⠀⠌⡀
⠈⠈⣾⡿⠀⢰⠠⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢌⠐⠁⠀⠀⠈⠉⠀⠀⠀⠀⠄
⠁⠠⠹⠇⠀⠘⠁⠀⠀⠀⢀⠀⠂⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠻⢆⡄⢀⣠⣄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⣅
⠀⠐⠂⠀⠀⠀⢩⠾⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢲⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠻⠿⠟⠋

⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⢿⠿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡟⠉⠁⠈⠉⠙⣿⡁⠙⢻⣿⠋⢹⡿⠀⠀⠀⠀⠸⢿⣁⣀⣀⡀⠀⢸⣿⡁⠙⢻⡿⠉⢹⡟⠉⡀⠀⡈⠙⣿⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡛⠙⠃⠀⠙⠉⣿⣿⣶⠾⠃⢀⣾⣥⠀⠀⠀⠀⢠⣼⣟⠉⠛⠁⣠⣿⣿⣿⡶⠟⠁⢠⣿⠀⠸⠃⣸⡿⠀⢺⠂⠀⠀⠀⠀⠲⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣷⡀⣾⣿⣿⣇⣀⣠⣴⣿⣿⣿⡀⣠⣄⠀⣸⣿⣿⣿⣦⣄⣨⣿⣿⣀⣀⣤⣶⣿⣿⣧⣄⣴⣯⣀⣴⣿⣦⣀⡀⣀⣀⣾⣿⣿⣿⣿
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡟⢸⣿⣿⣿⣿⠿⠛⢻⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⣛⣵⣿⠀⣿⣿⣿⡿⢃⡴⠛⠋⣽⠻⣦⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣾⣿⣿⣿⡿⢸⣿⣿⡿⠱⣟⣷⣾⣆⣼⣿⠟⡄⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣷⣷⣌⡛⢿⣇⣷⣿⠛⢫⣾⠿⣿⡿⡇⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⢏⣤⣟⣰⠋⠉⠀⣸⠨⡎⠉⣿⡏⠀⠈⠁⠀⣾⠷⣴⠌⢿⣿⣿⣿⣿⣿⣿⠟⣩⣄⠌⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⢈⣿⣿⢁⣴⣿⣽⣿⣶⣇⠘⣿⣦⡠⣶⣄⣀⣟⣀⣈⣂⠺⠉⣍⠙⠛⠋⡡⣿⢞⣻⠀⠈⠩⣔⢂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢣⡟⣾⢫⣿⣿⣿⣿⣏⣉⣭⡾⠷⠋⠀⠀⠉⢉⣛⣛⠋⠻⣶⣤⣿⣸⣿⠏⣼⣟⣿⡇⣰⣿⣷⣿⣷⡸⠁
⠀⣦⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢃⣾⣻⣿⣿⠿⢟⣯⡶⢿⡅⢀⡀⠀⠀⣠⣤⣬⡍⠙⣿⣿⣿⣶⣾⣷⣿⣧⣿⡇⣹⡿⢳⣿⢳⣟⢌⣱⣇⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⢸⣟⣯⣬⡵⣞⣫⣥⣶⡿⠿⠛⠁⠐⠀⣛⣋⠉⢡⣶⡿⠟⢋⣙⣿⣿⣿⣿⠿⣷⣙⣃⣼⣧⡟⢣⣾⣿⣟⠿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢿⣇⣭⣵⠾⢛⣭⠄⣿⡴⠿⠃⠀⠀⠘⠛⠛⢡⣾⣿⣿⠿⣿⣿⣿⣿⢿⣛⣓⣼⣿⣿⣿⢦⣴⣿⣽⣿⢹⣇
⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣮⣥⣴⣾⣿⣿⡄⣿⣴⡿⠀⠀⠀⠻⠛⢳⣾⡿⠉⣁⢸⠟⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣔⣿⣿⡿⣿⣿
⠀⣿⣿⡀⠀⠀⠀⠀⠀⣄⢸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣟⣾⣦⡄⣸⣯⣴⣿⠋⣠⣾⣿⣄⠃⡸⣿⣿⡌⣿⣝⣻⣻⣿⣿⣾⣻⢏⠿⡿⢯⠟
⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣾⣿⣿⠁⢻⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣴⣿⣿⣿⡏⣾⣿⣿⣿⣿⡄⢡⢻⣿⡏⣉⠉⠁⠉⠋⠤⠤⠬⠋
⠀⣿⣿⣿⠀⠀⠀⠀⠱⠶⢿⡝⠃⠀⠀⢻⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣷⣌⠻⠤⠔⠁
⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠈⢿⣿⠏⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⠋
⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠄⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⡄
⠀⣿⣿⣿⣿⠀⠀⣄⠀⠠⣶⡷⠂⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⢠⣿⣿⠉⠉⠁⠀⠀⠀⠀⠆⠀⠰⠆
⠀⣿⣿⣿⣿⡆⠀⠈⠀⠀⠋⠀⠀⠀⠀⣸⣿⣿⣿⣿⣟⠈⠻⣿⣿⣿⣿⣿⠸⣿⣿⠟⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡐
⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠉⣠⠀⣿⣿⣿⣿⡇⢰⡿⠃⠀⠀⠀⢸⠀⠀⠀⠀⢦⠤⣀⠀⠊
⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⡇⢠⣿⡀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⢻⣧⠸⣿⣿⣿⡄⢂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣥⡀⠀⠈⣿⡆⠛⣿⣿⣷⡌⠀⠀⠀⠀⠀⢀⠀⠙⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⢟⠻⠿⢷⠀⠀⠈⠃⢨⣿⣿⣿⣿⣤⣀⠀⡰⠊⠀⠀⠀⠈⠃⠀⠀⠀⢠⠀⠀⢀⠀⠀⠁⠀⠀⠀⣼⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⡘⣇⠀⠀⠀⠀⠈⢿⣍⣟⣟⣋⠶⡁⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣿⣦⡀⠀⠀⠀⠀⠙⢿⡿⣿⣷⠐⢀⠀⠈⡀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⠀⣿⣿⣿⡇⣀⣀⠀⣀⣀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⢦⡹⣷⡡⢄⠀⠀⠘⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⡄⢄⢀⡄⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠙⠄⢱⣌⠳⡀⠀⠘⠂⠀⠀⠀⠐⢄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⡇⠤⠤⠀⠄⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣿⣷⣄⠠⠀⠀⠑⠠⠤⠄⠀⠑⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢃⠩⠛⣽⠃⡃⠀⠀⠀⠀⠀⠐⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⢊⡰⠆⠉⠆⡹⢄⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢐⡠⠀⠁⠐⠊⠛⠃⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠒⠀⠈⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⠿⠟⠻⠿⠿⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠐⠂⣤
⠀⠀⠀⠀⠀⢀⡄⢘⠅⡆⠁⠀⢀⣴⡿
⠀⠀⠀⠀⠀⡞⠀⠸⠀⣀⣤⣶⣿⡿⠃
⠀⠀⠀⠀⢰⣇⠀⠀⠀⠈⣿⠿⠋
⠀⠀⠀⠀⠀⠉⠙⢦⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠈⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⡙⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠐⠘⠉⠷⠀⠙⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣶⣤⡠⠔⠢⢼⣆⠀⡤⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠇⠐⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⢿⣿⣿⣷⡀⠀⠈⠀⢄⠠⠤⠀⠀⠀⢛⠿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣋⣴⣙⣩⣿⣿⣿⠀⠀⡄⠀⠙⢿⠀⠠⣀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⡀⠀⠀⠀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣧⡀⠁⠀⠀⠀⠑⠀⠀⡉⣛⠤⠀⠤⠤⢤⠤⠤⠤⢀⣀⠀⠀⠀⠈⠳⣶⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡏⣿⣿⣿⣷⣄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⠀⠀⠹⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡏⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣬⡀⠀⠀⠀⠀⠀⣀⣀⣤⡶⣶⣿⣿⣶⣶⣤⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡇⢹⣯⢻⣿⣿⣿⠋⠉⠉⠉⠛⠿⣷⣶⣶⣿⣿⣿⣿⣯⢣⣿⢿⣿⣛⣽⣿⣿⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡸⣿⡌⣿⣿⣿⣧⣄⣿⣿⠆⠀⣻⣿⣿⣿⣿⣿⣷⣷⣟⢁⠟⣋⣽⣶⣿⢿⣜⣏⣛⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢿⣿⢹⣿⣿⣿⡿⠛⠁⣀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠛⠛⠛⠛⠓⠓⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣎⢿⣧⡹⠼⣿⡇⠀⠸⠿⠿⠿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⠹⣿⣄⠹⣷⣤⣤⣤⣤⣤⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡎⢷⡘⢮⡷⡝⢿⣿⣿⣿⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡌⢿⣄⠻⣝⢦⡙⢿⡟⠇⢿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠀⠉⠃⠉⠀⠉

⣿⣿⣿⡿⣿⡿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠁⠀⢁⡈⠈⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠛⠛⢿⣿⣿⣿
⢿⣿⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣼⠁⠀⣀⣴⡾⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣿
⣻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣼⣿⣶⣿⣿⣿⣷⣶⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣿⣿⣿⣿⣿⣿⣿⠛⡻⣿⣿⣿⣿⣍⠁
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠹⣿⣿⣿⡿⠷⠦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢶⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠲⠶⠴⣿⣿⣿⡟⠛⠧
⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠉⢸⡿⢷⣤⣤⣤⣤⣴⣿⠟⠿⠥⣄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡀⠻⣿⢿⡏⠋⠁⠀⠀⢰⣿⣧⠀⠀⠀⢀⠔⠀⣠⣄⣠⠃
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⠉⡀⠉⠉⠻⣿⣦⡁⢸⣧⣴⣶⣶⣾⣶⣿⣿⣆⣰⣾⠁⠀⠀⢠⠟⡶⠒
⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠘⠃⢴⡇⠂⠄⢀⣿⣿⣿⣿⣿⠛⠉⠉⠉⠉⠻⠻⣿⣿⣇⡀⢶⠈⠉⠋
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠣⠅⠀⡀⠄⠀⠀⠉⠛⠋⠀⣶⣦⣤⣴⣾⡇⠉⠉⠈⠙⡶
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⡾⣙⠛⠛⠛⢉⣡⡄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣄⣀⡀⠀⢀⣼⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠟⢻⡍⠀⣠⣿⣿⣿⣿⣿⣟⣁⣠⣤⣤⣤⣤⣀⣀
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣴⣾⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠢⣈⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⣠⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⠟⠛⠙⠋⠉⠉⠙⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣿⣿⡟
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣾⣿⣧⣀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠿⠛⢛⣩⣷⣿⡿⠟⠛⢿⣿⣦⣄⡉⠉⠛⣿⣿⣧⡀
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢰⣾⣿⣿⡿⠋⣀⣀⣀⠀⠙⢿⣿⣿⣶⣄⣿⣿⣿⡷⠀⠀⠸
⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢋⣵⣿⣿⣿⡏⠀⠈⠋⠙⠟⠛⠀⠂⢹⣿⣟⣿⡈⠉⠋⠁⠀⠰⢧
⣿⣟⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⠀⠀⠀⢠⣧⣿
⣿⣿⣯⣑⣐⡤⠤⢤⡤⠄⠠⣤⠀⠀⢀⣀⣀⣀⡀⠀⣀⣛⣛⣛⣁⢀⣀⣀⣀⣀⣀⣀⢀⣀⠀⡙⣿⣿⡟⢀⣀⣴⣿⣾⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠐⠀⠀⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⠀⠀⠒⠀⣠⠴⣶⡿⠟⢛⣿⠳⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⣠⡞⡁⠉⠀⠀⢈⡿⣁⠈⠀⠘⡆⠠⠄⠀⠀⠐⠀⠈⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣿⣷⣿⡝⠛⠯⣍⡓⠢⣼⣃⡀⢤⠈⢰⠏⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣛⠇⠀⡀⠶⢶⣤⣨⣿⢶⣿⠿⣦⠶⢿⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣚⣩⠶⠂⠀⠐⢂⡸⠃⠀⣤⠀⠀⠀⠀⠀⠀⠾⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠴⠚⣉⣤⠄⠀⠀⠀⠀⠀⠤⢤⡀⠶⠤⠭⠂
⠀⠀⠀⠀⠀⠀⣶⣶⠶⠶⠖⠒⠒⠚⠋⢉⠠⢤⠖⠛⠉⠀⠀⠀⢀⣀⣴⣿⣿⣷⢿⣰⠛
⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⢀⡠⠄⠀⠈⠀⠀⠀⠈⠉⣵⡶⠛⣿⣏⣩⣿⢿⡿⠋⠚⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⠟⡻⣋⣭⣽⣶⡷⠞⢡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠛⣷⣿⠛⠉⠉⠉⠁⠀⢀⣼⣿⢢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠙⠻⣷⣤⣀⣀⣤⣶⡿⢻⡟⠈⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡰⠃⠀⠀⠀⠀⢩⣿⡿⠟⡁⠲⣿⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣐⠐⢠⠎⠀⠀⠀⠀⠀⢀⣿⣏⣰⣿⠟⢂⣟⣀⣀⡤⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⢷⡎⠀⠀⠀⠀⠀⠀⣼⣽⢃⡙⣿⡃⢺⣿⡟⠛⠋⠛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠘⠀⠀⠀⢸⣿⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⢠⠇⠀⠀⠀⠀⠀⢠⣿⢡⠆⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠏⠨⠀⠠⠴⠦⠀⠘⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣷⣦⣌⠢⣽
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣷⡎⠉⢛⠲⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣻⣿⠿⣿⣧⣽⠖⠀⣀⣁⡉⠁
⠀⠀⠀⠀⣠⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣧⣶⣾⣿⣿⣿⣿⣷⡘⢦
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡟⢋⣉⠻⣍⣹⣿⡿⢿⣛⡌⢷⡀
⠀⠀⢰⣿⣿⢾⣿⣿⣿⢿⡧⠻⠟⣿⣿⣦⣀⣙⠻⠿⠷⠈⠿⠛⢫⣀⡿⡿⣤⣷⡀
⠀⢰⣿⣿⣿⣿⣿⠏⣽⠘⣗⢀⠀⠈⠛⠟⠿⣯⣙⠳⠶⠀⠀⠀⠀⠉⢿⣿⣿⣟⣤
⠀⢸⣿⣿⣿⣿⣯⣿⣿⣾⣿⡏⠀⠀⠀⢀⠀⠀⣀⠀⠀⢀⣠⠂⠀⠀⠈⠻⣿⣿⣌⠳⣄
⢰⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣟⣤⣄⣐⣉⡔⠊⣁⣶⣶⠟⠁⠀⠀⠀⠀⠀⢿⣿⣿⣷⡝⢆
⠆⣾⣿⣿⣷⣿⣿⣿⣿⣿⣼⣷⣯⣤⣄⡀⠀⠰⡿⠟⣡⢀⣴⡼⢆⣻⣦⠀⠈⢿⣿⣧⣿⣤
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢛⠋⠉⣤⣷⠀⢤⠀⣶⠿⣧⣼⢋⠁⠀⠀⠀⠈⣿⢸⣷⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋⡿⠉⣙⣟⣷⣶⣿⣆⠈⠺⠿⣶⣿⡶⣶⣄⣼⡆⠀⠀⣿⠈⡏⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⡷⡤⢴⣿⣿⣿⣿⣿⣤⣤⣀⠀⠀⠙⠻⡿⠛⠇⠀⠀⠹⡤⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⠃⠈⢻⣿⣿⣿⣿⠏⠀⢙⡉⠂⢄⡀⠀⠀⠀⡀⠀⢠⠉⢀
⠀⠹⡟⣿⣿⣽⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣧⡶⣀⠀⠘⠁⠈⠐⠣⠀⠀⠀⠁⠀⠸⡆⠻⣷⣖⣶⢦⡤⢠⡀
⠀⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⣾⣿⣿⠁⣿⢀⣠⡤⣤⠰⡀⡀⠀⠀⠀⡏⢲⡍⠨⢬⣥⣤⣾⣛⣂
⠀⢿⣿⣿⠿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠞⢾⣿⣯⡿⠓⢿⣫⣩⣽⡷⠄⠁⢱⠀⠀⢰⡇⠀⠙⠀⠀⣄⠀⠈⠉⠙⠛⠁
⠀⣾⣿⣿⣿⣯⣿⣟⣛⣻⣿⣿⣿⣿⣷⡘⠧⣿⣤⣶⣾⠿⠛⠋⠀⠀⠀⢸⠀⠀⣸⣿⠀⠀⢤⡀⠸⡇
⠀⢬⢿⣿⣿⣿⣿⣿⣷⣷⣿⠿⣿⣿⣿⣷⣄⠻⣿⣿⣷⣤⣤⣤⣤⡄⠀⠀⣰⠃⢹⣿⠃⠀⢸⣷
⠐⣚⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡆⠈⠉⠻⠃⠀⠀⠃⢀⣿⡏⠀⠀⢸⣿⡁⠀⢰⠇
⠈⠭⠭⣴⡶⠾⠟⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢱⣶⢰⣏⣀⠀⢿⠛⠀⠀⢀⣾⣿⠀⣰⡏
⠀⠚⢹⣿⣻⣧⣶⣦⣤⣿⣿⡟⠛⠛⠿⠛⣉⣉⣵⣩⡅⢤⠀⠀⠀⠀⠀⠘⠓⢊⠑⠶⣄⣸⣿⠇⠀⡿
⠀⠀⣼⣿⣿⣯⣽⡟⣯⣼⣿⣳⣶⠒⠛⠉⠀⢀⣭⣍⠁⠁⠀⠀⠓⠀⠀⠀⢰⣾⠀⠀⠈⠙⠛⢯⡳⣤⡀⠠⢀
⠀⠀⠙⠛⠯⢳⣿⢾⣿⠋⢹⢻⣧⣤⡴⠶⢲⡿⠚⠋⣡⣦⣤⣶⠶⠀⠀⢀⣿⣿⠀⠀⠀⠀⢤⠈⣷⠀⠀⡄⠀⠈
⠀⠀⠀⠀⣰⡿⡧⣼⢹⡇⠀⠲⠶⠦⠔⠀⢀⠀⠠⣼⣿⣿⣿⣷⣾⣿⣿⣾⣿⡇⠀⠀⠀⠀⢸⠀⣿⠀⠀⠁
⠀⠀⠀⠈⠉⣠⣿⡿⢰⠧⣴⣤⣶⣶⣾⠿⠿⣿⣿⣟⣿⣿⣏⡕⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⡿⠀⡿
⠀⠀⠀⠀⠀⠙⢋⠑⠳⣾⣋⠩⠁⣌⢻⠎⣉⣽⣾⣿⣿⣧⣼⡧⣾⣿⣿⣿⡛⠀⠀⠀⠀⢰⠃⢰⠇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⣿⣷⠀⣼⣿⣾⠘⢻⠿⡿⢿⣻⡻⠇⠿⣿⣿⣿⠃⠀⢀⡀⢀⠎⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⡁⢿⠇⣿⣇⢈⠰⢶⡂⡿⡇⠀⢠⣿⡿⣃⡀⠀⢸⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠲⠖⠒⠒⠒⠲⡶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⢤⣾⡟⠛⢶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⠉⠹⡰⠁⠹⠀⠀⣣⣈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠂⠀⣀⣠⡁⠀⠀⠀⠀⠉⡏⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠳⠆⠀⠀⠀⠀⠘⠁⠙⠀⠀⠀⠀⢀⣴⠤⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠥⡴⠦⠤⢤⣤⡀⠀⣀⠀⠀⠀⠀⢠⣴⣁⠀⠀⠀⠀⠀⠠⣴⠓⠆⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠁⠀⠀⠈⠀⠀⠙⠋⠐⠶⠃⠤⣼⡁⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠔⠷⠛⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⢦⣠⠀⠀⠈⣻⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⢰⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢾⠛⠂⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⠋⠹⠇⠘⠉⠙⠛⠐⠶⢦⣀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣁⡒⠒⣠⣤⡤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣼⢷⡋⠈⠙⠋⠀⣀⠱⡄⠀⠀⣉⣡⢶⡆⠤⠄⣀⠀⠀⠀⠀⠀⢀⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⢠⠔⠊⠉⠉⠳⡛⠶⠋⠁⠀⢸⠃⣠⣀⡀⠀⠙⢒⣄⡀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠰⠿⠋⠈⠁⣽⠔⠋⠁⠈⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠉⠉⠒⢤⠀⠀⠀⠀⠀⠀⠀⡤⠐⠆⠀⢄⡀⠀⠀⠀⢹⣨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣴⠃⢀⡤⠂⠻⣛⡟⠐⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠳⣄⠀⠈⠁⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠉⠀⠀⢠⠋⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⡄⠀⠀⢀⡄⠀⠀⠀⣸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⣿⣶⣤⣛⣬⠝⠛⢳⡄⠀⠀⣸⠀⠀⠀⠀⠘⠋⠀⠀⠙⠊⠀⠀⠀⠹⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⠟⠃⠀⢠⡇⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠰⣇⣠⠃⠀⣠⡀⠀⠰⣟⣀⡠⠤⠶⠶⢻⡆
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠿⠀⠀⠈⠁⠀⠀⠓⠁⠀⢸⠶⠄⠠⠤⠤⢶⡞⠁
⠀⠀⠀⠀⠀⢸⡇⠀⢀⡤⠴⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠈⠢⢴⡃⠀⠀⠀⠀⠀⠀⠀⠒⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢀⣄⣸⠆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢧⣀⡠⠞⠓⠲⠤⠤⠶⠃⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⣤⣼⠏⠋
⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⣀⣀⣃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠙⣧⡀⣿⠃
⠀⠀⠀⠀⠀⢰⡇⠀⠘⣿⣿⠀⣹⣿⣿⣿⣦⣤⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⠛⠉
⠀⠀⠀⠀⠀⡿⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⣷⠀⢴⣿⣿⣿⠛⠉⠈⠉⠈⣙⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⣴⠶⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⠤⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡙⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⣮⠙⠻⢿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣷⣆⣄⣤⠌⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⢣⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⣿⣿⣿⡻⣿⣣⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠋⠀⣼⣿⣿⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠀⠀⣀⣤⠖⠚⣉⣉⡉⢻⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⠀⣾⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⡿⣤⣿⣿⣿⢩⣴⣿⣟⣁⣠⣴⣿⣿⣿⣮⡟⣿⣷⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⢷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠓⠉⢉⣙⣣⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⠉⠋⣿⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⢶⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⡿⣿⣥⣿⣦⣀⢸⡇
⠀⠀⠀⢰⣿⣿⡿⣻⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣬⣿⣿⣿⣿⣷⡈⣿⡿⠟⠋⠁⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠈⠻⣿⣿⣿⡿⢟⣫⣽⣭⡛⢿⣿⣿⣿⠿⠿⢿⣿⣿⣥⡟⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣯⣥⡀⠀⠘⠦⣿⠏⠙⠀⠀⠀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠘⣿⣿⡟
⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⡿⡿⣋⣉⠙⠿⠇⠀⠀⣤⡖⠀⠀⣀⣀⠀⢻⣿⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⢠⠬⣿⠟⣿⡟⣿⣷⣶⣼⠷⢾⣷⢾⠟⢿⠿⠃⠸⣏⡍⡦⠀⠀⠀⢀⣴⠋⡀
⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠘⠷⣾⡆⠘⠛⠉⠛⠛⣿⠀⠈⠛⠛⠋⠀⠀⠀⢱⡇⣾⠁⠀⠀⠀⢸⣿⠀⢸
⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣂⡀⠀⠀⣤⡟⠀⠀⠠⡄⠀⠀⠀⠈⣼⠁⡏⠀⠀⠀⠀⢜⡻⠂⠛
⠀⠀⠀⣻⣗⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠛⠁⣸⣧⣟⠀⠀⣠⠛⡄⠀⠀⢀⡷⠞
⠀⠀⢀⡿⠟⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣼⠁⠉⢻⣷⣾⣅⣀⡈⠀⠀⣼
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠛⡒⢦⣥⣴⠶⠂⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠃⠀⣀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣬⡽⠧⣤⠾⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣊⣭⣽⣿⡷⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣊⣵⣾⣿⣿⠋⢹⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⣼⣿⣿⣿⣟⠁⠀⠈⠦⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⢟⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣬⣭⣥⣦⣄⣀⠀⠀⠀⠀⡇⢠⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⢸⠃⣿⣿⣿⣿⣿⣿⡆⢀⡤⠒⣿⣿⣷⣶⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣻⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣼⣿⣿⠿⢿⠿⠻⣿⣿⣿⣿⣿⣿⣿⠃⠾⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⣿⣿⣿⣷
⠀⠀⣠⠚⠛⢿⡁⠀⠀⠀⠀⠈⠙⠻⣿⡿⠟⢁⣠⣴⣾⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⡻⠿⣿
⠀⢰⠁⢀⣀⣼⣿⠀⠀⠀⠀⠀⠀⢠⡿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⣷
⠀⠀⠓⠤⢴⠿⠏⠀⠀⠀⠀⠀⠀⣾⣧⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠳⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⠁⣠⣤⠈⠙⣿⣿⠟⠁⠠⠶⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠒⠂⠀⠀⠈⡭⢖⡥⠄⠀⠻⢿⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⠻⣿⣷⣄⠀⠀⣤⠋⠀⣠⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢧⣘⢋⣷⣦⠘⢶⣾⣙⣯⣽⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠁⠀⢸⣿⠫⠛⣋⢠⣿⣇⡀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⢡⣀⠰⣒⠒⠶⡖⠂⠀⢸⣿⣿⣿⣿⣿⣿⣟⠉⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠤⠤⠤⠄⠒⠁⠀⠀⠀⢈⢹⣆⠘⣧⠀⠸⢄⠀⣸⠟⢫⣿⣿⠿⠋⠙⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠁⠀⠀⡄
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢣⠁⠈⠳⠀⣀⣋⣑⣰⣿⣿⡀⠀⠀⠀⢱⠀⠀⢰⠁⡄⠃⡀⡃⠀⡘⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢣⡀⠀⠀⠶⣤⣶⣾⣿⣿⣷⡄⠀⠀⠀⢣⠀⢸⢸⡁⢹⠗⢲⢀⠃⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢳⡀⠀⠀⠀⠀⣼⡿⠻⠿⣿⣆⠀⠀⠈⠆⠺⣝⠀⢸⠀⣾⠜⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠙⠒⠒⠛⠛⠛⠁⠀⣸⣿⣿⣧⡀⢀⡇⠀⢹⠀⢸⠀⣾⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠰⠊⠘⢿⢟⣿⠿⠿⣷⠠⣾⠀⣸⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⢳⠀⢻⡄⣿⠀⣿⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠰⣾⠁⣿⡆⣿⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣀⣴⣯⣴⣿⡗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠘⠛⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣾⣿⣻⣶⣤⣀⣠⡴⠞⣋⣭⣴⠶⠶⣭⣓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣽⠋⣠⣾⣿⣿⡷⠛⠋⠛⠻⢷⣕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠋⠀⠀⠀⠀⠀⠹⣾⣿⣿⡿⣿⣿⣾⡿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠙⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⢀⣀⠀⠀⠀⠻⣼⠏⣒⡢⠉⠡⣒⡈⣿⠃⢀⣀⣀⣀⢀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢻⣄⡉⠁⠀⠀⠉⣁⢉⣿⣿⡿⠋⠁⠉⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠐
⠀⠀⠀⠀⠁⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣿⡱⣦⣶⣤⡾⢩⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⡐⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠳⡌⢉⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣌⡻⢋⣴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣤⠾⠵⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⢹⣿⣿⣿⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠋⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠟⠛⠉⣹⣿⣿⣿⠏⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⣦⡀⣷⡀⠀⢠⣿⣿⣿⡟⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣶⠀⣴⠀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣧⢠⣿⣿⣿⣿⠁⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣼⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠏⣠⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣶⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢯⠄⠸⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠃⠈⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣶⣦⣶⣶⣄⠀⢀⣴⣶⣦⣴⣾⣿⣿⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⣴⣶⣶⡀⠀⠀⣰⣶⣶⣄⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠃⠀⣰⣿⠏⢻⣿⡇⠀⠀⣿⣿⠋⣿⣷⡀⠀⢻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠛⠉⠛⠀⠀⣿⣏⠎⢡⣽⡇⠀⠀⣿⣥⠉⢏⣿⡇⠀⠘⠋⠙⠻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⠠⠂⠀⠀⠀⢻⣿⣇⣾⡿⠀⠀⠀⠸⣿⣆⣿⣿⠃⠀⠀⠀⢢⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⣥⠀⠀⠀⠀⠀⠉⠛⠋⣀⣤⣴⣤⣄⡉⠛⠋⠁⠀⠀⠀⠀⢀⡅⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⢣⠈⠀⢀⣶⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢀⣶⡆⠀⠈⢠⠃⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡁⠀⠘⣿⣿⡄⠀⠘⢿⣿⣿⣿⣿⠟⠀⠀⣼⣿⡇⠀⠀⣁⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠹⣿⡿⣶⣄⣀⡈⠉⠉⢀⣀⣴⣾⣿⡿⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠙⢿⣮⡻⢿⣿⣿⣿⣿⠿⣫⣾⠟⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡈⠻⣿⣶⣦⣤⣤⣶⣿⡿⠃⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠈⠙⠛⠛⠛⠛⠉⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠦⠤⠤⠶⠖⠋





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⣀⣤⣖⣰⣤⢤⠤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠛⠲⣄⠀⣷⢸⡇⢰⣶⣶⣿⣶⣶⣆⣀
⠀⠀⠀⠛⠛⠛⠛⠿⠴⠿⢿⣿⣿⣿⣶⣦⣄⠀⠀⠀⢠⣤⡀⢰⣾⣿⣷⣶⣤⠀⠀⠀⠀⢀⣾⢸⣏⣷⢘⠀⢸⣾⡗⣼⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣾⣷⡀⠀⡂⢀⣴⡶⣶⣤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣜⣓⢄⠀⣿⣿⢛⣉⣉⣠⣹⡁⣿⣿⣋⣲⣹⣧⣥⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣂⢃⠈⢿⣿⣿⣯⣿⡉⠀⠀⠀⠉⠙⠛⠛⠻⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢟⠹⣿⣿⣍⢦⣼⣿⣿⣿⠿⢟⠛⠶⠦⣄⣄⣀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢈⠀⣿⣿⣷⢸⣿⣿⣿⣿⣶⣾⣿⣷⣶⣤⣄⡁⠘⠻⠟⠂⢀⣃⡄⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠈⠀⣽⣿⣟⣼⣿⣿⣿⣿⣿⡿⣃⡿⣿⣿⣿⣿⣷⡄⠀⢸⣿⢽⣿⣿⠿⣷⣌⡆
⠀⠀⠠⠄⠀⣀⠀⠀⠀⠀⢿⣿⣿⣧⣠⣿⣿⣿⣻⣿⠀⠿⠿⠿⢇⣿⣿⣿⡒⠬⣿⣻⣷⡀⣹⣿⣻⣿⢇⠑⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠒⠦⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⠘⣿⣛⡃⣛⣉⡴⢿⣦⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⣿⣿⡏⡀⣿⣿⣧⣿⠻⡿⠿⠛⠁⠀⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⠇⠀⣿⢛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢼⣿⣷⠾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⣣⠀⠁⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⣶⡍⢁⣶⣤⠀⠻⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣞⠘⠛⢁⣈⠻⠛⠀⠀⠀⠙⣿⠁⠀⠈⠏⠀⠀⠀⢀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⢰⣇⠂⡀⠀⠀⠋⣀⠀⠙⣏⡠⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢚⠄⠀⠒⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣿⠟⢀⠅⠀⠀⠀⡉⠀⠀⢻⣿⣆⠀⢀⡠⠔⠂⠉⠄⠀⠀⠀⠀⠥⠀⡀⠀⠘⣿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⢀⡂⠀⠄⠀⠀⠁⠀⢀⣾⣿⣇⠀⠈⠀⠙⠀⠸⠃⠀⠀⡀⠀⢄⢀⠀⠀⠀⠉⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⣴⡖⣌⣤⡀⣄⡀⠀⠈⠉⢀⠀⠈⠀⠱⠄⠀⠀⠀⠉⠀⠀⢉⠀⠄⡀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⣶⣶⣤⣶⣾⣿⢯⣾⣿⠋⣰⣿⢿⣴⣤⡄⠀⠙⠆⠁⠈⠀⠀⠀⠀⠀⠀⠀⠘⠦⢐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠿⠿⠿⠛⠛⠉⢠⣾⣿⠇⠉⠉⢀⣾⡿⣟⣷⡆⡀⢀⣀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠒⠈⠀⢸⣿⣍⠁⠀⣠⣾⣿⣿⣿⡟⠁⣸⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠄⠀⢰⣿⣿⣿⣿⠏⢠⣾⣿⡆⠘⣿⡇⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠃⠀⣿⣿⣿⣿⡟⠀⡆⢻⣿⣧⠀⣿⡇⠀⠀⠀⠀⠀⠐⡄⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠀⣿⣷⠀⠀⠀⠀⣠⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⠀⢹⠟⠀⠀⠐⠆⠁⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠠⡀⠘⣿⡄⠀⠀⣠⣶⣶⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⡱⠀⠹⠏⣧⣾⡿⠋⠁⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠁⠘⣛⣛⣛⠛⠛⢯⣭⣴⣾⡄⠀⠀⠀⡙⠋⣀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠀⠀⢤⣿⣿⣿⢀⠀⠈⠻⠿⣿⣿⡄⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠙⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣾⣷⡀⢹⡿⠁⢴⡶⢀⠀⢾⠇⠈⣉⡉⠓⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⣉⠀⠀⣡⣶⡄⠀⠈⠀⣤⠀⠈⠿⠟⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣿⣿⣿⣿⣷⠀⠀⠀⢤⣤⣴⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⡀⠀⢠⣾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⡄⡇⣰⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣼⣿⣿⠃
⢀⣤⣤⣤⣾⡟⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠸⣿⣿⠧⠤⣄
⠳⡿⣿⣿⣿⣇⠈⠉⣁⡐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⣤⣶⠀⠀⠀⡀
⠀⠈⠑⢻⣿⡟⠀⢀⡀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣀⠀⣿⣿⣿⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠹⠁⠀⠠⠤⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⣿⣿⣶⣤⣤⣤⠁
⠀⠀⠀⠀⠀⠣⣀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠝⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣤⣤⣤⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡦⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠒⠒⠀⠀⠰⠆⠀⠛⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠰⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣎⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢊⠀⠀⠈⡄⢀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠘⢄⠀⠄⠀⠠⠄⠀⡀⠀⠀⠀⠀⢀⡄⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠃⠀⠀⠈⠁⠀⠀⢨⠗⢄⠘⠁⢠⡄⠀⡆⠀⠀⠰⢣⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢠⠈⠤⠀⠀⠀⠂⠀⢀⠴⠳⠀⠀⠀⢱⢀⡄⠀⢈⠈⢱⡌⡆⠀⠖⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⡀
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢠⠀⠀⠀⠋⠴⠀⢀⡄⠀⢨⠀⠀⠀⠉⠀⠁⠀⠀⠀⢰⠁⡇⠂⠀⠀⠠⠀⢰⠆⠴⠀⠀⠀⠃⠀⠐
⠀⠀⠀⠀⠀⡌⠀⠃⠀⠀⠀⠀⢠⠇⡀⠀⠀⠈⠁⠀⢸⠅⠀⠀⠀⠀⠀⠀⡀⠀⠘⠰⠿⣇⠀⠉⠈⠀⢨⣀⠀⠀⠛⠆⠁⠀⡁
⠀⠀⠀⠀⢠⠃⣆⠀⠀⠀⠀⢀⡇⣀⠀⠀⠘⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡏⢠⠄⢨⠀⣸⠀⠀⣀⠀⠠
⠀⠀⠀⠀⠈⣢⣀⠀⠁⠀⠀⡾⠠⠀⠀⠀⠋⠁⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠗⠂⠀⠂⠀⠁⠀⠀⠘⠀⡀
⢀⠀⠀⠢⣴⠉⠉⠛⠻⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡜⠢⠀⠀⠀⠀⠀⠀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠒⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠑⠒⠤⢄⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠈⠁⠒⠀⠤⠤⠤⠄⠀⠀⠀⠀⢱⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠈⠁⠀⠛⠁⠴⠂⡀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠰⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠐⠂⠁
⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣈⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣴⣿⣶⣦⠠⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠘⡛⠃⣐⣿⣿⣿⡟⠶⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠗⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣄⠀⠀⢿⣿⣿⣿⣿⣇⢰⡀⠀⢀⣤⠼⠧⠦⣿⣷⣴⣶⡦⠐
⠀⠀⠀⠀⢀⠄⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⠀⠀⢘⣿⣿⣿⣿⣗⣼⣅⠀⠙⣿⣷⣀⣀⡹⣿⣿⣥⣤⣼⡌⣀
⠀⠀⢠⣖⠁⣴⢆⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠙⣿⣿⣿⣿⣿⣿⣫⣴⣿⣿⣿⣿⣿⣋⣀⡀⣹⣿⠋⡏⢹⣿⣿⣻⡉⡍⢉⡭⣤⠬⠂
⠀⢠⣾⣟⣓⣈⣤⣴⠄⠛⠋⠋⠉⠁⠒⠒⠂⠙⠓⠿⠿⠿⠿⠓⠿⠿⠟⠻⠿⠟⠛⠿⠿⠿⠿⠛⠛⠀⠂⡛⠛⠛⠿⠅⠰⠈⠉⠉⠈⠁⠈⠀⠀⠀⠠
⠀⠀⠀⠈⠉⠁⠄⠈⡉⠈⡀⠀⠀⠀⣀⠐⢀⠀⠐⢚⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠐
⠀⠀⠠⣌⡃⢿⡌⣉⡂⢱⣝⠲⡏⣙⢳⣴⠈⠍⣴⡍⠊⣇⠥⡂⠀⢀⠶⠞⠦⢼⠀⡆⠠⢈⡤⠸⢵⢂⣶⢁⡔⠂⢲⡐⠆⠐⠈⠄⠰⡤⢆
⠀⠀⢀⣿⢙⡆⢸⢫⠁⠔⢿⠇⠘⡸⢻⠛⣤⠗⣾⢈⡝⣺⢶⣧⢡⠙⠤⢠⣐⣦⣾⡅⣶⠻⣍⡆⠞⣀⣽⡆⠶⡏⠋⡍⡉⠁⣀⡛⠃⣷⡤⠖⠣
⠀⠀⢈⣿⢸⣇⣷⡫⠜⢻⣴⡏⠰⣉⢾⣀⣻⡀⢾⠀⠼⢋⢸⡇⡛⠷⢬⡘⢼⡡⡤⢀⣅⠔⢻⡳⢦⡥⠺⢛⠿⠁⠦⣯⡌⠷⡋⠁⠀⣾⣁⣀⠁
⠀⠀⠀⣿⢸⣏⢸⡇⣿⠛⢸⣇⢘⣟⢾⡑⢽⡆⣿⠳⣿⣯⢱⣎⢵⠻⠎⢣⠿⣵⣄⢸⡇⢠⣬⠁⠀⣿⡶⣻⢧⡀⣶⡰⢶⠶⡀⢆⡿⣡⠈⠒
⠀⠀⠉⣿⠘⠷⢸⡦⠯⣀⢸⡇⣀⣉⣻⡓⢺⠇⢹⡘⣿⠣⢸⡅⢒⡼⢦⡡⠾⢼⡇⢸⡏⣈⣭⠗⠀⣿⠒⡯⢿⡁⣽⡵⢿⣷⡅⣬⡝⣡⢌⡉
⠀⠁⠈⠉⠛⠒⠺⠧⠤⠤⢼⣅⣈⣁⣽⡉⠉⠁⢻⠃⠒⠒⢾⠔⠠⠜⢵⣧⠀⠼⣆⢡⡦⠩⡍⠁⠙⣿⠃⠑⢻⡓⣿⡴⣿⣶⠄⢼⡅⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠒⠶⠶⠼⠧⢤⣀⣘⡇⠀⠀⠀⢰⡎⠁⠈⠂⠐⣿⠂⠰⢼⠳⣿⠤⠟⣀⣀⣾⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠷⠤⣤⣄⣀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢘⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠓⠛⠒⠤⠤⠀⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠫⠙⠉⠉⠀⠀⠀⠈⠀⠀⠀⠈⠙⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠛⠳⠁⠀⠀⠀⠀⡀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣶⣃⣀⠀⠀⣀⣠⣎⠀⠀⠀⠀⡱⠄⠀⠤⡀⠀⠠⡀⠀⠉⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠟⠈⢛⠟⠣⡴⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠒⠂⡐⢀⣘⠈⠙⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣻⠀⢀⡎⠀⠀⢸⡇⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠢⡮⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⣉⣄⠘⣇⡀⠰⣾⣿⣦⣀⣳⣄⢈⣿⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠅⠀⢈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣶⣶⣅⣄⣀⣀⡀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣮⡻⣿⡿⠻⣯⣽⣟⣻⣿⣿⣿⣿⠁⠀⠀⠀⠀⣛⣿⣿⣿⣿⣯⣭⣭⣿⡿⣛⣴⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠹⣿⣿⣿⣮⣹⣿⠿⢿⣿⣿⣿⡖⢤⠴⠒⠢⣼⡟⣻⣿⠿⢿⣯⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠋⠙⡿⢿⣿⣿⣆⠀⠹⠉⣿⡇⠀⠀⠀⢠⠉⢠⡿⠁⣠⣾⣿⡿⣋⣵⠟⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⡿⢷⣝⠳⡈⠻⣦⠀⢀⣿⡇⠀⠀⠀⢸⡀⠘⠁⣴⣿⠟⣡⠞⡛⠛⠀⠀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⠿⣷⣬⡑⠮⡂⠈⠷⢸⣿⡇⠀⠀⠀⠈⡇⢀⡼⠋⠠⣪⠥⠬⣡⣤⣤⣤⣤⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠁⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⢀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣾⣭⣭⣁⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠀⠀⡀⢰⣿⣿⣿⣿⣄⠋⠁⡤⠄⠊⠃⠄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠠⣀⣀⡀⠀⠻⠟⠉⠉⠙⢿⣷⣿⣤⣄⢀⣀⣄⣡⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢾⣥⣤⣤⣤⣤⣤⡽⠿⠿⠧⣤⣤⣤⡄⠈⠉⠀⠉⡙⢻⣿⣤⣬⣭⣽⣿⣶⣶⣶⣾⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣟⣹⢛⣟⢋⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠋⡄
⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⣀⣀⣀⣽⣄⡀
⠀⠀⠀⠀⠀⢀⡼⠿⠟⠛⠛⠉⠉⠉⠙⠛⣻⣥⡶⠦⠀⠀⠀⢀⣀⣀⣤⣤⣤
⠀⠀⠀⣠⡶⠋⠀⣀⣀⣀⡠⣤⣴⡖⢺⣿⣿⣧⢤⠀⠶⠚⠛⠉⠉⠁⠀⠀⢸
⠀⠀⠘⠛⠛⠉⢩⠏⠘⠛⠀⠈⡟⣷⠞⠁⢸⠀⣎⡄⠀⠀⠀⢀⣠⡄⠀⠀⢸
⠀⠀⠀⠀⠀⢠⣯⠤⢦⣀⡀⢰⣿⡿⢤⣤⢾⡄⢻⡄⣀⠴⠚⠋⢹⠃⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⣾⡿⢳⣄⠀⠈⡇⠀⠙⢦⣀⣀⣀⡟⠀⠀⢠⠃⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢲⣴⡟⠁⠈⣿⠣⡴⠞⠀⠀⠀⠀⠉⣹⣿⣿⣶⣾⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠸⣇⠀⠲⣦⡀⠀⠀⡰⠋⢑⠞⠉⠁⠀⠀⠀⠀⠙⢿⣧⡔⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢻⣦⡼⠁⠀⡏⠀⠀⠀⠀⠀⠀⢀⣤⣾⣭⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠘⣿⡁⠀⠀⣇⠀⠀⠀⠀⣠⣾⣿⠷⣫⠑⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢹⢇⣿⠶⣿⣿⣶⣴⣾⣿⡫⠕⡡⣐⡵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢸⢸⣡⡶⡟⢝⢝⡿⢿⣭⢖⠔⣊⠤⢚⣽⣷⡀
⠀⣠⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⡟⢸⡻⡌⣌⢇⠡⣻⣶⣿⣷⡫⣷⢯⣣⢿⡻⣧
⠀⠙⡟⠳⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇⣿⠝⣎⢂⣦⣷⡿⠟⠉⠘⢿⣕⢿⣷⠱⣒⣽⣇
⠀⠀⢧⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⢿⠧⡇⣿⣮⣶⠿⠛⠉⠀⠀⠀⠀⠈⢻⣯⢾⣿⡻⣝⣿⡄
⠀⠀⠈⠉⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡹⡀⡟⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢻⣾⣷⡽⣧
⠀⠀⣦⡀⠈⣇⠀⠀⣀⣼⠀⠀⠀⠀⠀⢻⣗⡝⣽⣿⢷⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣛⡭⢔⣿⡆
⠀⠀⠘⢏⢳⢸⡄⢀⡍⢙⡖⠀⠀⠀⠀⠀⠹⣮⣪⢿⢸⠟⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⡛⠽⠧⠀⣀
⠀⠀⠀⣼⠈⣦⣧⢾⠀⣯⠀⠀⠀⠀⠀⠀⠀⣬⠇⣸⢸⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣋⣠⣴⣶⣿⠿⡃
⠀⢳⡲⡈⣇⣹⣻⡼⡰⠃⠀⠀⠀⠀⠀⠀⠀⠙⠦⡇⡎⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠋⠉⠀⣿⣿
⠀⢠⣷⣉⢫⣹⡟⡗⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠾⣄⠧⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠳⣆⠀⢀⣠⢄⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠘⢤⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣽⠞⠛⠉⠉⣩⡿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣀⣤⣿⣃
⠀⠀⠀⠀⠀⠀⢛⣷⠀⡀⢠⠀⢀⠀⠀⣀⣐⣊⣼⣟⡀⢀⣀⣀⡀⠀⣼⠁⣠⡞⣠⠟⠀⠀⠀⠀⢀⣠⠿⠂⠶⠖⠠⢶⡟
⣀⣀⣀⣴⠛⠦⡼⠿⠜⢳⠿⣤⣯⠀⣸⣿⣿⣿⣿⣿⣿⣟⣁⠈⠉⣉⣉⡴⠛⠫⠿⢾⣥⣀⣰⣛⣉⣁⣀⣀⣠⣄⣠⣼⣥⣾⣥

⠀⠀⢀⡤⠒⠉⠉⠉⠑⠢⣄
⠀⡴⠁⠀⠀⠀⠀⢀⠀⠀⠈⢣
⡜⠀⠀⢐⠞⠓⣤⠖⠛⢦⠀⠀⠇
⡇⠐⡀⢸⠀⠀⡻⢣⡤⠃⠀⠀⢸
⢣⠀⠳⣄⡙⠋⠀⠀⢀⡠⠆⠀⡘
⠈⢦⡀⠀⠉⠉⠉⠉⠀⠀⠀⡠⠃
⠀⠀⠙⠢⢤⣀⣀⣀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⠈⠲⣄
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠸⡀⠑⠂⡀⣀⡀⠀⢠⣾⠀⢸⡀⠀⠀⡐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠻⣷⡀⠈
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠑⢂⢟⣟⠻⣥⡎⠋⠁⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⡔⡡⠔⠀⠙⠛⠤
⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠈⠲⢤⡄⠀⢠⠢⠎⠨⠧⠘⣣⣤⣤⣶⠤⠞⠀⢡⠤⠶⢋⡟⠉⠑⢮⡞⣄
⠀⠀⠀⠀⣴⡉⢷⠂⠈⠉⠇⠀⠀⠀⠀⠉⠒⢾⠃⢀⡤⢀⡞⠁⠀⠀⠙⡆⠀⢀⣘⡄⠑⡏⠀⠀⠀⢈⡗⠏
⠀⠀⠀⢠⠧⡑⠢⣉⡋⠀⡀⠀⠀⠀⠀⠀⠀⣿⠀⢈⢤⠈⠳⣤⣤⣤⠞⠁⠰⢉⠋⣑⣄⡙⢤⣀⣠⣾⠓⠞
⠀⠀⠀⠈⡗⢬⡖⠚⣁⡤⠟⠂⠲⡀⠀⠀⠀⠸⡀⠠⡠⠙⣀⠄⠂⠰⠀⠀⠈⢮⡀⡐⠈⣏⠉⠀⠘⠘⠉
⠀⠀⠀⠀⠈⠉⣽⠫⠀⠈⠀⣀⡤⠇⢀⡀⣀⡀⢱⣀⠀⠀⠈⢁⣰⠃⠀⠀⠀⢠⠓⣄⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠙⢦⡴⠶⣏⠉⢣⡀⢸⠘⢣⠠⠖⠈⢹⣿⡂⠻⠏⢉⠿⢙⣳⡏⣠⠋⠑⠖⠃
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⢦⡀⠙⢾⣄⠻⣀⠆⣆⣈⠀⠅⠀⠀⠀⠀⢨⢀⠟⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠈⠉⡹⠃⠈⠁⠐⢎⠳⡊⣉⠟⠁⠈
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠃⠀⠀⠀⠀⠸⡆⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢻⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠘⠇⠀⠘⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠦⠴⠇⣀⡀⡀⠀⠄⠠⠤⠖⠀⡂⢉⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢰⠂⠐⠄⠀⠀⠀⠀⠀⠘⢃⣀⠇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠁⠀⢈⠖⠒⠲⠶⢶⡆⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠶⡾⠇⠀⠀⠀⢸⣧⣤⣀⣼
⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⣄⠀⠀⠀⠀⠀⠀⣇⣀⡧⠀⠀⠀⠀⢀⡗⢀⡇⠀⠀⠀⠐⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠈⠙⠖⠐⠒⠾⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠉⠓⠒⠲⠒⠖⠋⠁⠀⠀⠙⣧
⠀⠀⠀⠀⠀⢧⠀⠤⠤⢀⣀⠲⠶⠖⠒⢲⠒⠒⠒⠂⡁⠀⠀⠀⡓⠒⠒⠒⠲⠦⢤⣄⣀⡀⠀⠀⠀⠀⣀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠟⠛⠂⠁⠀⠀⠀⠈⢦⠀⠀⠀⠀⢀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⣀⣤⣤⣤⣤⡤⠤⢤⡀⣧⠀⠀⣠⣯⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡴⢾⣿⠿⠟⠛⠋⠉⠀⠀⠀⢼⠁⠙⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⣼⣠⣄⣘⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⣠⣤⢤⡤⠀⣠⣀⣻⡏⢭⣽⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⣠⣶⣿⣷⣶⡶⠾⢧⠀⠀⠀⠀⣶⠾⠟⠛⠋⠉⠉⠁⠀⢹⣟⣿⣿⣿⣿⡼⠀⠀⠀⠀⡀⠙⠄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⡇⠀⠀⠀⢐⡆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠰⠶⠖⠓⠖
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣠⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠘⣿⣿⣿⡟⠻⠿⣿⣿⢽⣿⣿⣿⣷⡄⣀⡤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷
⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠈⣫⣾⣿⣿⣿⠿⠋⠁⠀⢀⡄⢀⣤⣤⣤⣄⠀⠀⠀⠀⣀⣀⠨⡀
⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⢸⣿⣿⣿⠟⠉⠀⠀⢀⣰⡾⢰⣿⣿⣿⣿⣿⡷⠀⠀⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⢿⠟⠀⠀⠀⠘⣿⡿⠃⠀⠀⠀⠀⢼⣿⠀⠘⣿⣿⣿⣿⡿⠆⠀⢠⣾⣿⣿⣯⣉
⠀⠀⠀⠀⠀⠸⡖⢦⡀⠀⢰⡿⠁⠀⠀⢠⣶⠾⠋⠁⠀⠀⠈⠉⠉⠡⠀⠀⢀⣼⣿⡏⣭⠋⠘⡄
⠀⠀⠀⠀⠀⠀⣧⡠⢷⠀⠀⠁⠀⠀⠀⢸⡏⢠⣴⣆⢢⣀⠀⢀⣴⣤⣄⠀⠘⠋⠉⢸⠿⣧⣰⠃
⠀⠀⠀⠀⠀⠠⠶⠚⠋⠁⠀⠀⢤⣦⣶⣮⣴⣿⠛⠿⠀⢻⣿⣿⠿⠛⠋⠀⠀⢀⠠⠜⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣆⠀⠀⢸⡟⠛⠲⠒⠀⠁⣈⣤⣤⣴⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⡄⠀⢸⣷⣶⣶⣶⣿⣿⣿⣯⣥⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠸⠛⠛⠏⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠳⠦⠤⣀⣠⣤⠴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⢛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⣀⣀⠀⡀
⠀⠀⢠⣤⣤⣤⣦⣤⡤⢶⠆⠀⠀⢩⣥⣭⢄⣈⠀⠀⠀⠀⠀⠺⠀⢉⡉⠹⣦⠀⠀⠉⠙⠻⣿⣿⣶⡀
⠀⣠⣿⣿⣿⣿⣿⣿⣄⡞⢡⡀⠀⠈⠙⢿⣾⣿⣦⣄⠀⠀⠀⠀⠀⠈⠁⠸⠤⠂⠀⠀⢳⣌⠉⠉⠙⢿⡄
⠀⣿⣿⣿⣿⡿⣿⠿⡟⠃⣀⣠⠠⠀⠀⠀⠈⠟⠋⠀⠀⠀⢀⣤⣶⣿⣷⣄⣼⣧⡿⠂⠉⢁⠀⠀⠀⠘⠁
⠀⢿⣿⢿⣿⣻⢻⠿⣿⠿⣿⡧⠌⠷⠤⠀⠄⠤⠂⠀⠤⠀⠀⢙⣿⣿⣾⣿⣿⣿⣧⣶⣾⣿⣿⣿⣿⣿⡇
⠀⣾⣿⣧⣮⣭⢠⣰⣿⣦⣴⢃⣀⣉⠋⢽⣶⡄⠈⠁⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⣹⢿⣿⠇
⠀⠻⣿⣿⣿⣿⡶⣿⣿⣿⡇⢸⣿⣿⣶⣾⣷⠄⡀⣝⣿⡿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠐
⠀⠐⣿⡿⠾⠿⣷⣿⣿⣿⣧⣠⣭⣭⣡⣬⣽⣷⣩⣿⣟⡀⣿⣶⣶⣶⣤⡼⠛⢻⣧⣸⣿⣿⣿⣿⠏
⠀⠀⢤⠀⡀⢀⠀⣿⡏⡽⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⡛⠳⡤⠜⠛⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠚⢿⣿⣅⣸⣇⠉⢿⡿⢿⣿⣿⣿⣿⡏⣦⠻⠿⠿⠟⠿⠙⠿⢯⣙⡶⠶⠟⡓⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠺⠷⠦⣘⣻⠿⣿⠿⠿⠗⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐

⠀⠀⠀⠀⠀⠀⠐⠤⡀⠀⠀⢀⠠⠤⢄⣀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣀⣀⣀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢹⣂⣴⣷⢖⡟⡛⠾⣙⠲⣖⣶⡿⣛⣾⠿⠦⣼⡿⠤⠙⠋⠁⠀⠀⠈⠀⢐⡴⠒⢂⣀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣶⡿⣫⡾⠁⣾⣀⣱⡀⢉⢴⣬⡉⠉⢁⡠⠖⠒⠒⠢⠦⣤⣀⣀⣀⣀⡠⠔⠃⣴⣾⠭⠟⢉⡀
⠀⠀⠀⠀⠀⠀⢀⡛⢿⣉⠈⢦⠟⠉⠛⣧⡄⠀⠀⢱⠞⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠳⠶⠤⢤⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠠⣈⢳⡆⢀⣶⡄⠈⢇⢀⡐⠁⠀⣠⣴⣿⣭⣭⣍⡙⠲⠦⣤⣤⠴⢾⠳⢓⡯⢷⣿⣭⣴⠏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣷⣿⣿⣻⣦⠔⠋⢀⡴⠚⣉⣥⣶⣿⢣⢒⠈⢦⣤⣤⣤⣦⣵⣾⠭⠴⣬⣿⣿⡿⢰⠁
⠀⠀⠀⠀⠀⠀⠈⠿⣝⡲⣄⣝⣮⣉⣉⣁⣤⠴⣋⡴⠛⠁⠀⠀⢦⠔⢩⡵⡀⠙⠉⠉⠉⢉⣵⣞⣿⣿⣿⣿⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣙⣻⠷⠶⣶⣿⣿⣿⡶⠟⠩⠴⠶⢤⣤⣴⣮⣂⣢⠜⣜⢄⠀⣀⣤⢟⡵⣫⡹⣾⣿⠏⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣱⡟⠓⠆⢶⢦⣤⣄⣀⣀⣠⣴⠯⠖⢚⣽⣟⡿⢯⠭⣟⣾⣷⣟⡹⣴⣿⣛⣹⡿⠟⠁⣠⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣟⠿⡔⠦⡀⠈⠲⢬⢉⠉⢀⣠⣔⢾⣟⢋⣭⣽⡆⣠⣾⠟⣳⠿⢛⣋⣉⣭⣥⣤⣴⣾⣻⡟⣮⣻
⠀⠀⠀⠀⠀⠀⠀⠦⠀⢵⠀⠀⣶⣴⡞⡉⠙⠛⠳⣤⡉⢿⠔⠈⠋⠙⡿⢡⣾⡷⠚⠋⠛⠙⠒⣶⣾⢟⡗⠟⣻⡿⢹
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠸⡄⠀⡎⢿⣿⣶⣿⣷⡾⣿⣿⣦⢫⠜⢠⢠⠀⣾⠿⡿⡓⣣⢚⡼⠞⣋⣭⢽⢿⠟⡻⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡇⠀⡟⠆⡛⠛⠾⢾⣞⠽⠫⣿⡟⣰⢣⠃⣰⠏⣿⡿⡇⣰⠟⣠⡞⢻⡶⣶⣮⣔⣁⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢧⠐⣷⠤⠬⢭⣿⣦⢹⣞⡳⣼⣇⣧⡏⣰⡿⣥⢈⣶⡿⠋⣰⡿⣫⢚⠷⣾⡿⡋⢙⡿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡘⢦⣜⣿⣅⣚⣟⣛⣻⡵⠜⣛⡉⠥⠬⢬⣩⡟⣛⣋⣠⡶⠋⣈⣠⣍⡛⠰⠴⡞⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⠻⡙⠛⣻⠿⠶⣦⣽⣤⡶⠐⠒⠚⣼⣷⣯⣬⣥⡶⣟⠉⢺⣼⣯⣿⠻⢛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢞⣳⣦⣬⣮⢤⢤⣄⠈⣟⠻⠰⡥⣀⢉⣽⠟⡞⠹⢹⢟⢺⡳⢦⣭⡾⣹⣷⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⠀⣶⣿⡄⣺⣶⠋⢀⠜⡠⠋⠨⠃⠜⡡⠁⠎⠼⠀⠛⣧⠀⡧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠼⠾⢱⣿⢁⣄⣰⣛⡋⢳⡒⣂⣋⢉⣲⣤⣾⡂⠈⠩⡆⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣮⣿⠁⣵⣬⡙⠮⣝⢀⣀⣉⡉⠍⠚⠉⣻⠽⠇⣨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⣈⣁⣟⠀⠀⢈⡼⢉⠔⣡⠄⡠⠈⣳⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠭⣴⠶⢋⡠⠋⡴⢁⡞⡰⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠚⠁⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠖⠒⠈⠀⠒⠛⢃⡀⠀⠀⠀⠀⣠⣴⠶⠔⣇
⠀⠀⠀⢀⡶⠦⣦⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡀⠀⠠⠄⠤⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⢀⠘⡄⢀
⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣇⣃⣀⣧⣤⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠖⠚⠛⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠡⠉⠠⠀⠀⠀⠀⠀⠀⣄⠀⠘⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠘⣃⣤⣠⣤⣄⠀⠐⠲⠎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣤
⠀⠀⠀⢀⠀⠀⠠⠠⠀⠀⠀⠀⠀⣴⢀⠖⠉⢀⠀⠀⢡⠈⠉⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡤⣄⠀⠀⠀⠀⠈⠉⠋⠉
⠀⠀⠈⠈⠀⢀⠀⠀⠀⢀⡀⠀⠀⣸⢛⡁⠶⠈⠀⠰⠆⢤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣤⠟
⠀⠀⣤⠴⠂⠛⣡⡴⠦⣭⡁⢰⣿⠇⠀⢹⡄⠀⣠⠀⣴⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠠⣾⠏⠀⠀⠘⠿⠟⠻⣄⣀⣀⣈⠇⠀⠀⠀⡄⠀⠀⠀⢰⡞⢁⡴⢲⡄⠀⠀⠀⠺⠖⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣤⠴⢮⣻⣼⡏
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⠃⠀⠀⠈⠁⠑⠀⣀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠀⠐⠀⠒
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡂⠄⠨⠀⠠⠅⠸⠀⠀⠓⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣭⣭⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠵
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⢠⠀⠐⠀⠀⠀⣀⣠⡤⠠⠀⠀⠀⠆⠰⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠶⠂⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡚⡒⠀⠀⠀⡀⠉⠵⠆⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠈⠁⣈⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠐⠆⠀⢨⣍⡥⠀⠁⠀⢁⠀⠒⠀⠄⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⢀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡤⠤⠀⠈⢙⠉⠀⠋⠀⠀⠤⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠻⠚⠴⢇
⠀⠀⠀⠀⠐⠶⠲⠶⠶⠶⠒⠂⠀⠀⠀⠰⠞⠀⠀⠀⠀⠴⠦⠄⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⢀
⠃⠓⠢⣤⠀⢰⣶⣀⠀⠉⢉⣴⠒⠀⠀⠀⠀⠀⡀⠂⡔⠀⠀⠀⠀⠀⠀⢀⠀⡔⢄⠀⠀⢤⣀⣀⠀⠀⠀⠀⢠⠾⠿⠋⠀⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠁⠀⠀⠀⠀⠰⠶⠀⠐⠈⠔⠂⠀⠀⠀⢀⡀⠹⠋⠠⠄⠀⠀⠀⠀⢠⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⣀⠀⠈⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢞⣥⠞⠉⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⣿⠁⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⢻⡄⠀⠀⢀⡟⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠙⠶⠶⠛⠀⡏⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢳⡀⠀⠀⠀⠀⠃⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠀⢣⠀⠀⡀⢠⣦⠀⠀⡖⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⣼⡇⡠⢷⢂⡍⠉⠓⠛⢾⠤⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⡼⠉⣿⠀⠀⠁⠀⠀⠀⠀⠸⢦⠀⠀⠈⢣
⠀⠀⠀⠀⠀⢀⣾⠃⠀⢲⡆⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⠈⢷⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠙⠢⣄⣸⢀⣀⡀⠀⠀⠀⠀⢸⠀⠀⢳⡀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⡴⠛⠉⠈⣍⡳⡄⠀⠀⢸⠀⠀⠈⢷⡀⠀⣠⢾⣀⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣾⣀⣠⡤⢜⡶⠳⠀⠀⠀⢸⠀⠀⠀⠈⢷⡞⠻⡄⠀⢠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠙⠛⡶⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡸⢻⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡇⢸⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⡇⢸⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠁⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡾⠀⢸⡇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⡇⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⠀⠀⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⢱⡇⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡎⠀⢸⣇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠃⠀⠀⣽⠋⠉⠉⠁⠈⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢠⡄⠀⢀⠀⠀⠀⠻⠖⠚⠛⠛⠒⠒⠋⠀⢀⡴⢇⣤⢤⡤⢴⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢾⣀⣀⡠⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣟⡿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⢻⣿⣿⣿⣸⣿⢿⣿⢣⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣥⣶⣾⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠈⢿⡿⠿⠇⡿⠋⠉⠙⠻⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⣛⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠈⣿⣶⣿⠿⠿⢛⣛⣭⣭⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠶⢦⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⠀⠀⠀⠀⢹⣆⣀⣀⣤⣄⣀⣴⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣦⣤⣤⣽⣿⣿⣶⣤⣤⣴⣿⣿⡿⠿⠛⣿⡟⠛⢉⣠⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣦⣤⣾⣿⢹⠉⢹⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⡁⠀⠀⠀⠀⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⠃⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⡿⠀⠀⠀⠀⢸⡇⣿⣿⣯⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡆⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⣀⣤⣄⣀⡀
⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣾⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣄⠙⣿⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠙⡎⠉⢹⠀⠀⠀⠀⠀⠀⠀⣠⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠄⠀⠀⠀⠀⠀⣓⣀⢾⣤⣶⣥⣴⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢢⠀⠀⠠⢾⠟⠁⡿⣆⣮⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⡄⠀⢀⡀⠰⠁⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡠⠐⠁⠀⠀⠀⠀⣿⣿⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⣽⢟⣽⣹⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠟⠻⣿⡇⢍⡙⣯⢤⣴⣾⡯⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⡴⣠⢿⡿⣌⠈⠉⠉⢀⡟⡽⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⢱⢯⠟⠀⠀⠀⠀⠀⡾⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⠏⠀⠀⠀⠀⢠⠶⠿⢻⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡻⣕⢄⠀⠀⠀⠀⣃⠀⠄⢀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⡾⢂⠀⠀⠿⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⢀⡈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠄⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡛⠋⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠞⠶⠟⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⠂⡀⠀⢀⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡿⠘⡇⠘⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢰⣿⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣤⣾⡏⢀⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣸⣿⣿⣿⣿⣿⣧⠋⠀⠘⢶⡄⣟⣘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢽⣿⣿⣿⣿⣿⣿⡀⠀⠀⠁⠞⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⡀⢀⡀
⠀⠀⠀⠀⣀⠀⠀⠈⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠄⠂⠀⠂⠀⠀⠔⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⠛⠻⠟⢿⠂
⠀⠀⠀⠀⠀⠀⢀⠤⠐⠁⠀⠀⠀⠀⠀⡀⠀⠈⣀⣠⣤⠤⡤⠀⠐⠀⠲⠛⢻⣿⣿⣿⣾⣿⠷⠶⠶⢦⡀⠠⠶⢖⠉⣀⡟⠁
⠀⠀⠀⠁⠢⢄⡋⠀⠀⠀⠀⠀⠀⠀⢉⡽⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣶⣦⡀⢰⡆⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⡟⢻⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣇⠘⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢹⣿⣿⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠘⣿⣿⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⣿⣿⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠘⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⡇⠀⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⠇⠀⣹⠿⢿⣙⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣴⣦⣶⣶⣶⣶⣶⣶⣦⣶⠾⠿⠿⠛⠁⠀⣠⣤⣶⣿⣿⣾⡿⠃⣠⠾⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢋⣴⣿⣿⣿⢿⠟⠛⠁⢉⣁⡀⣀⣀⣀⣀⣤⣤⠝⠛⠟⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠐⠋⠈⠉⠁⠁⣠⠄⣠⣾⣿⢿⡿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠘⠡⠞⠱⠋⠁⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠠⠒⠒⠢⢄⠀⠖⡙
⠀⠀⠙⠖⠢⣄⠀⠀⢻⠸⡠⠀⢀⣤⣀
⠀⠀⠀⢨⠀⠈⠳⡀⠀⣯⣁⠻⣄⣀⠀⠀⢀⣀⣠⣤⣤⡶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣀
⠀⠇⠀⠈⠉⠐⠲⠽⠦⡏⠙⡿⣛⠭⣛⡛⠋⠉⠀⠀⠤⠄⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⢘⢀⣀⡠⠒⠒⠢⠔⢻⠹⡈⠢⡉⠙⠛⠫⠍⡑⠒⠢⢄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠈⠁⠀⠑⠉⢉⡇⠀⠈⡄⠁⠀⠈⠒⢄⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⣠⡞⠀⢰⠀⡇⢀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⣷⡀
⠀⠀⠀⢠⡞⠁⠀⠀⠃⠀⡅⠈⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⢷⡀
⠀⠀⣠⠏⠀⠀⠀⡀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⣧
⠀⢰⠏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠰⡀⢻
⢀⡟⠀⠀⠀⠀⠠⠈⠀⠀⡇⠀⠀⠐⡀⠀⠀⠀⡀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡆⠀⢇⢸
⢸⠃⠠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠱⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⢰⠀⢸⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢧⠀⠀⠀⠱⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠸⡄⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⣆⠀⠀⠀⠱⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡇⠀⡿
⣿⠀⢀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠘⣆⠀⠀⠀⠱⡀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣇⠀⡇⣸⠇
⢻⡀⠀⡀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠘⣆⠀⠀⠀⠈⢆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⣷⡟
⠘⣇⠀⠱⠀⠀⠄⠀⠀⠀⠀⠘⢄⠀⠈⢦⠀⠀⠀⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣷⡟
⠀⢹⡄⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⣇⣼⠏
⠀⠀⢻⣄⠀⠱⡀⠀⠀⡀⠀⠑⡀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠘⣄⡴⢿⣟⡢⠄
⠀⠀⠀⠙⢦⡀⠙⣄⠀⠈⠢⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣆⣠⠾⠋⠅⠂⠑⠀⢄⢀
⠀⠀⠀⠀⠀⠙⢦⣌⠳⢄⠀⠙⢤⡀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢳⣠⡴⠟⠁⡄⡄⠲⢤⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣭⣒⣀⡉⠢⢄⡈⠑⠲⠤⣄⣉⣐⣒⣤⣤⡴⠾⠛⠉⠉⠃⠠⠘⢀⡆⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠁⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢀⣴⠆⣀⣤⣤⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢛⣵⠟⠉⠀⠈⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣾⣿⠇⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⢿⡆⠀⠀⠀⠀⢰⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⡀⢀⣠⡞⠀⣿⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣇⠀⠙⡛⠛⠉⠀⠀⠋⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠹⡆⠀⠐⠀⠀⠀⢠⠀⠀⡆⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⢸⣇⠀⠁⢠⡶⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣸⡆⠀⢳⣰⡀⠲⠲⢾⣭⣤⣄⡸⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣰⢿⣧⡰⠋⠷⠤⠞⠀⠀⠀⠀⠈⣟⠛⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢰⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⢶⣤⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠻⢦⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠙⠓⣾⣧⠶⠶⢦⣀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢻⡄⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⢠⠞⠉⠀⠀⠰⢦⣌⢵⠀⠀⠀⠀⣷⠀⠀⠀⠀⢿⡄⠀⣠⡴⠻⣤⣤⠶⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣰⣇⡀⢀⣤⢤⣙⣷⠸⠦⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⢿⡴⠋⢻⡄⠀⠀⣴⣾⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⢦⣬⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣠⣤⡀⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣰⢏⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⡿⠸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢸⡄⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣸⠀⠀⣿⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢰⠃⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⣶⢸⡇⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡏⠀⠀⣾⠃⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⣮⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢸⠃⠀⢰⡟⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢿⠀⠀⠘⢿⣶⠤⠤⠤⠖⠈⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢀⡀⠀⠀⠀⣾⠃⠀⣀⣀⣀⡀⠀⠀⢀⡹⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣒⣦⠀⠀⣸⠇⠀⠀⠀⠙⠛⠛⠉⠉⠉⠉⠛⠛⠉⠀⠀⡀⣴⠞⣤⣴⢦⣶⠖⢾⡤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠔⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠛
⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠿⢿⢿⣿⡿⠿⢿⡿⠿⣿⠿
⣿⣶⣷⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠁⠀⠀⠸⠇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⢦⣄⠀⠀⣼⠀⠀⠀⠈⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢤⣠⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⣿⣵⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢂⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣀⠀⠀⠀⢀⠐⠢⣄⠀⠠⠈⠠⣄⠀⠀⠀⠊⠁⣾⡏⠀⠀⠀⠀⢠⡄⢀⣀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣈⡀⠀⠈⠙⠀⠀⠀⠙⠃⡀⠀⠀⣸⡿⠂⠀⠴⠃⠀⠹⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠂⠂⠉⠉⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠠⠶⣎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠉⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣼⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠉⠉⠁⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⢶⣯⡴⠇⠀⠀⠤⠤⣿⣟⣠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠒⢄⡈⠳⣌⢀⠠⣼⠓⠀⢀⣿⠃⠀⠀⠀⠘⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢳⡄⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠈⠓⠀⠀⢹⣿⡟⠛⠛⠛⠓⠒⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⡀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣷⣿⠿⠆⠀⠀⠀⣴⣿⡷⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣿⡟⠁⠀⠈⡄⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣾⣿⠇⠀⠀⠀⠹⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣼⣟⣀⣀⡀⠄
⠀⠀⠀⢀⣴⣾⣿⣿⡿⠶⠶⢦⡀⠀⢰
⠀⠀⠀⠈⠛⠿⠟⢿⡏⠹⠁⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⣈⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡠⠔⢺⡙⢧⣴⡇⢙⡦⡀⠰⢊⡩⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣦⣤⣄⢰⣀
⠀⠀⠀⢠⡎⠀⠀⠈⠙⠲⣿⠟⡉⡀⠙⢲⡎⠀⠀⠈⠀⠀⠀⡀⠤⠀⠀⢀⠜⢡⠤⠈⢻⣿⣿⣿⣿⣄⠀⠀⠀⢄⣀
⠀⠀⠀⢸⣇⠀⠱⣦⣠⣶⠹⢄⣛⡤⠀⠺⠷⢤⡀⠀⣀⡀⠀⠀⠀⠀⠀⡀⢠⣤⠀⣤⠀⣬⠉⣭⣽⣛⠃
⠀⠀⠀⠘⣿⣄⡤⢿⣧⣿⣷⡿⣵⡛⠁⣀⣔⡿⣿⣿⣿⠀⠀⠀⠀⡀⠀⢀⣈⣉⣀⣉⣀⣿⣶⣿⣿⣿⣿⣦⣀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⢻⣿⣾⠟⠓⢦⡀⠀⠈⠁⠈⠉⠹⠛⢛⡏⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠘⢿⣿⣷⣦⣀⡉⠓⠶⠚⠓⢛⣷⣰⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣶⣤⣴⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣾⠿⠾⣫⠀⠀⢠⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢰⣿⣿⡋⠀⠀⢠⡾⠀⢀⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣾⡇⠀⣸
⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⠟⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⣂⣂⣀⣀⣐⠒⠒⠒⢒⠒⠒⠂⣀⣐⠐⠒⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠤⣄⣀⣐⣂⣒⣂⣀⣐⣒⣒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣀⣤⠤⢶⠶⣶⣶⠾⠿⠒⠀⠐⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠀⠀⠉⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⣶⣿⣿⠿⢛⣂⣶⣾⣽⢋⠙⣿⣻⣿⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣻⠫⢛⢛⢛⣛⣛⠛⣉⠛⠋⣩⣿⣿⣯⣙⡛⠛⡙⠉⡛⢛⡛⠛⢋⣙⠻⣛⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣛⣯⣯⠯⠿⣿⣞⡿⠯⠀⠀⠭⣿⣿⠿⣤⡉⠳⢾⣶⣌⡳⣮⣝⠳⢾⣍⡓⠶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣷⠶⢟⣛⣿⣿⡾⣞⣿⣷⠃⣶⠀⠑⣠⣿⣿⣿⡶⡇⢿⣷⣮⠛⢿⣶⣭⣙⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣥⡶⣿⠿⣫⣿⣿⣿⣿⣿⡋⣸⢣⠊⢹⡟⢯⢻⣿⣷⡏⣯⣿⣿⣿⣄⠛⢿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣋⣵⣾⣿⣿⣿⣿⢿⡏⠙⢠⣿⡏⢠⡿⠋⠀⢻⣏⡉⠤⢝⣿⣷⡹⣿⣷⢷⣞⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⠟⢹⡿⡏⠦⡸⣟⣼⣿⡇⣾⡇⢀⣧⠰⣿⣟⣇⡚⣿⣿⣧⠩⢻⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⣠⡎⡛⣑⡁⣞⣿⣿⣿⣿⠀⣿⢳⣾⣿⡀⣿⣿⣷⡍⣽⣿⣿⣧⠀⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣮⡿⠧⢭⣬⣷⣿⣿⣿⡿⢻⣤⣿⣿⣿⣿⣿⣿⣿⣿⢧⣳⣿⣿⣿⡆⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠭⢛⡦⡬⣾⣷⣿⣿⠟⠁⢷⣿⣿⣿⣿⣿⡟⣿⣿⣿⣻⣦⣽⢿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠐⠙⠛⠝⠻⠋⠀⠀⠀⠉⠀⠀⠁⠁⠁⠀⠙⠿⠟⠛⠘⠛⠙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡃⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⠀⠐⠃⡀
⠀⢀⣀⠀⠐⣬⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠐⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⣠⠏⠀⠀⠀⢀⠴⠀⠀⣁⠈⠀⠀⠀⢀⠀⠀⡴⠋
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⢀⣀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⢀⡈⢉⠀⠀⠉⡠⠆⠀⠃
⠀⠀⠀⣀⠀⠀⠀⠋⠀⠈⠛⠂⠴⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠃⠀⠀⡄
⠀⠀⠂⠌⠀⠀⣀⢀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⡒⠤⠀⠀⢸⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⡀⢀⡠⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠄⠄⠀⠀⠀⠀⡀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⠀⢀⠌⠀⠠⠒⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠱⠄⠀⠣⡀⠐⠁⠿⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠐⠀⣀⠀⡀⠀⠀⠀⠀⠋⠒⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⡴⠒⠀⠀⠀⠀⠀⠐⡌⠓⠀⠃⠀⠀⠀⠀⠀⠁⠂⠀⠀⠐⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⡄⠐⠀⣠⠞
⠀⠀⠤⠀⠀⠓⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣴⢟⣉⡁⢀⠀⡀⠀⠀⠀⠀⣉⡡⠤⠔⠋
⠀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠷⠀⠀⠀⢀⡤⠀⡆⠀⠔⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠒⠋⠁⠐⢀⣡⣌⠢⣀⡤⠟
⠀⠉⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠋⠉⠉⠈⠉⠈⡧⣈⠂
⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⣠⡄⠀⠹⡀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⡰⠃⠀⠀⠀⠄⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⡀⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠢⢌⠙⢤⠀⠀⠀⠀⠀⠁⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⢇⠀⠀⠀⠀⢤⣀⠀⠀⠀⣠⠂⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠒⠒⠀⠀⠀⠀⠀⠐⢄⠀⠈⠀⠀⠀⠀⠀⢀⠀⡸⠁⠀⠀⠀⠀⠈⠗⢠⡎⠀⠀⠀⡀⠀⠀⠀⠀⠈⡦⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡤⠒⢀⠀⠄⢀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢀⠤⠀⠀⠀⠀⠃⠀⠀⠠⠀⠸
⠀⠀⠀⠀⠁⠊⠀⠀⠀⠈⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠐⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⢀⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡈⠉⢁⠀⠏⢠⣋⣀⣀⣀⣤⣶⠊
⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⠛⢻⣕⣶⣢⣧⣶⣷⣶⣶⣿⣿⡿⠋⠉⠉⠁⡀
⠀⢨⡉⠓⠒⣲⣌⣀⣔⣄⣁⠀⡄⠀⠀⠀⣠⣬⢤⣍⣛⣃⣬⣽⣷⣿⣿⣻⣩⣙⣻⣟⣿⡿⠁⠀⠀⠈⠉
⠀⠈⣯⣫⣭⡶⠟⢓⣺⡟⣏⠀⠀⠒⠶⠾⠋⠉⠳⣬⣭⣯⡭⢴⣶⣶⣾⣿⣽⡯⢷⡿⠋⠀⠀⢀⠄⡘
⠀⠀⢛⣫⣟⣽⣟⣑⢎⣽⣮⢳⠂⠤⠤⠤⠦⣤⠀⠀⢨⣿⣿⣿⣿⢷⣶⣾⣽⣿⣋⣤⣄⣤⣶⣅⡠⠓
⠀⠀⠀⠉⠉⢙⣻⣖⣻⣯⢺⢆⠀⠀⠀⢀⣀⣌⡇⠀⢸⣿⡟⠁⣤⡀⠨⣙⣫⣽⣷⢮⡉⠍⠛⢦⣄
⠀⢰⣄⣛⣲⣤⣤⣷⠳⣗⡷⡘⣟⡝⣏⣽⣛⢟⠇⠀⠈⠛⠛⠮⣭⣤⣭⡶⣶⣄⠁⠠⠉⣰⠀⢸⡯⠛⠴⢤
⠀⠈⢮⡋⢓⡙⠘⢻⣧⠈⣿⢿⣓⣋⣩⢥⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⢷⣿⣧⡴⠮⣄⣀⣀⣤⣱⡖
⠀⢀⣄⠈⠀⠈⢻⠧⣿⣷⣌⡳⠭⡭⢶⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣿⣛⣻⡵⣋⣝⣪⣉⡉⠙⠛⣃
⠀⠘⢧⣄⠀⢊⣍⣙⣾⣿⣿⣻⣿⣿⣿⣿⣾⣶⠊⠁⠀⠀⠀⠀⠠⢀⣧⣬⣉⣫⣽⣿⣶⣿⠿⠛⠛⠿⠥⠞⠉⣀
⠀⠀⠦⠜⢉⠀⣿⣟⡭⣿⣿⡿⠿⣿⣿⣍⣿⣿⣿⣷⣦⣤⡄⣠⠀⣿⠿⠷⣷⡞⣶⣾⢷⡾⣒⣚⡳⢄⡀⠀⢸⠃
⠀⠀⠉⠙⣶⣿⣻⢏⠞⠛⢙⠯⣁⣲⣬⢭⣿⣻⡿⠛⠛⠋⠀⣿⠂⠙⡟⠷⣿⣟⠿⠿⣿⡿⠭⢍⢻⣤⣥⠄⠁
⠀⠀⠀⣨⠝⠋⠁⠀⠀⠊⠉⠳⠖⢶⣽⣷⣽⣽⡇⠀⠀⠀⠀⣾⣄⠀⣇⣤⣼⡳⣭⣽⠏⠀⠁⠀⣧⣨⣥⠛⠛⠁
⠀⠀⠀⠹⠰⠄⠈⠀⡄⠐⠀⠀⢀⣁⣉⠿⠿⠾⣧⣀⣀⣠⣴⣿⣥⣾⠿⠋⠉⠑⠦⠜⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣸⣤⣤⣤⣤⣠⣤⣤⣤⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⣴⣶⣶⡤⣴⣶⣶⡤⢧
⠀⠀⠀⠀⠀⢧⠹⠿⠿⠋⠈⠻⠿⠿⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠻⠿⠿⠁⠹⠿⠿⢃⡜
⠀⠀⠀⠀⠀⠈⢳⠀⢀⣤⣤⣄⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⣀⣀⣀⡀⢠⡎
⠀⠀⠀⠀⠀⠀⠈⢷⣼⣈⣏⣸⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣁⣤⣄⣱⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡼⣶⣶⣶⡶⢲⣶⣶⣶⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⣾⣿⣿⣦⣾⣿⣿⡖⢳
⠀⠀⠀⠀⠀⢳⡈⠛⠋⠁⠀⠉⠛⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠛⠿⠿⠁⠹⠿⠟⢣⠎
⠀⠀⠀⠀⠀⠀⢱⠀⣀⡴⠲⣄⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠐⢓⠓⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⣥⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⢀⡀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⢙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢿⣿⣿⣿⣶⡄⠀⠀⠀⠛⣿⣦⡀⠀⠀⠀⣠⠶⢋⣠⣀⣀⣤⣦⡲⣤⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠙⣯⣿⣿⣿⣯⣶⣶⡂⠀⠀⠉⢿⣿⣆⣠⠞⡁⠀⠊⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠿⠟⠡⣦⣭⣽⣷⣶⣽⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⢹⣾⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⢿⣿⡿⢻⣭⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⢿⡿⠁⠀⠀⠀⠈⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣿⣯⣴⠿⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⢀⣠⣦⡹⣟⣿⣿⡟⠀⠀⠀⠀⠀⠰⠰⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠀⠀⠀⠀⢀⡴⠏⢁⣠⠴⠶⠶⠖⠒⠶⠶⢤⣤⣀⠀⢠⣻⣿⣿⣿⣾⡿⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠷⠄⢻⡀⠀⠀⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠙⠶⣝⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠺
⠀⠀⠀⠀⠀⠀⠸⠿⠷⠟⠃⠀⠀⠹⠇⠀⠀⠀⠆⣀⣴⡾⠿⠟⠛⠓⢦⠀⠀⠀⠈⢻⡿⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⣿⣦⡀⠀⠛⣩⣴⣠⣤⣤⣄⣄⡦⠀⠀⢀⠏⠰⠋⠀⣿⠀⠀⠀⠀⠰⠶⠰⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⢿⣿⡀⠀⠿⣿⡏⠸⢿⣻⠗⠀⠀⠀⠀⠀⢤⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢮⡽⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⢀⡄⠀⠾⠐⣰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠒⢶⣤⡀⠀⠀⠀⠀⠀⡜⢁⣴⡤⣂⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⠀⢀⣤⡄⣠⠟⠈⠢⠀⠀⠀⠀⠀⠉⠙⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣴⣞⣀⣤⣤⣤⣤⣡⠀⠄⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣍⣄⣠⣴⣾⣿⡿⠃⠀⠀⠀⢰⣼⡏⠘⣿⠿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣯⣿⡿⡍⠸⠛⠿⠁⠀⠀⠀⠀⣿⡏⠀⠀⠸⣇⠉⠙⢿⣿⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠿⣿⣿⡿⢀⣀⠠⢂⡀⠀⠀⠀⠀⣸⣿⡏⠀⠀⣴⠋⠀⠀⠀⠈⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣤⣴⡿⠁⠀⠀⢠⣷⣿⣿⣧⡴⠟⠁⢐⣮⣶⣶⣶⣵⣝⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣅⠀⠀⢀⣠⣿⡿⠿⠛⠁⠀⠀⠀⢨⣿⣿⠉⠉⠛⠛⠙⠛⠛⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⡿⠛⠁⢀⣤⡀⠠⡀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡿⠟⠛⠉⠉⣿⣿⡟⢉⡄⠀⣰⠏⠀⠙⢶⣿⣦⢳⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠉⠁⠀⠀⠀⠀⠀⢹⣿⣷⣿⣧⣼⠃⠀⠀⠀⠀⠙⢿⣿⡷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡿⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡔⣍⣁⣀⣀⣈⣹⡵⣄⣤⣤⣤⣤⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢸⣿⣿⣿⡟⣳⣾⣤⣠⡈⢀⠀⠀⣀⣱⣧⣼⢻⣿⣿⣿⡆⠈⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⢠⡟⠀⣹⢟⣣⣹⣏⣘⣿⣯⡉⢹⡄⣿⣿⣿⣧⠀⠀⠀⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⢤⣿⣿⣿⣿⡶⣟⣽⣿⡿⠟⢛⡛⠿⢿⣿⣿⣿⣇⣿⣿⣿⣿⡄⠠⢀⣯⣉⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸⣿⣿⣿⡇⢠⣿⡟⠁⠐⠀⢸⠃⠀⠂⠉⢿⣷⡌⢸⣿⣿⣿⡇⠀⠀⠀⠸⡗⢻⣿⣷⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠀⠛⠀⠀⢸⡀⠀⠀⠚⠈⢿⣷⢸⣿⣿⣿⡇⠀⠀⠀⢀⠟⣫⡿⠿⢿⡃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠱⢤⠤⢼⣿⣿⣿⡿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣾⣿⣿⣿⡷⢤⡤⣤⣼⡼⡿⠁⠰⣟⣙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇⢻⣿⡄⠿⠀⡀⠀⠀⢀⡀⠚⣰⣿⠇⣿⣿⣿⣿⠀⠀⠀⢸⣩⠼⠃⠠⣖⠻⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⡄⠻⣿⣶⣄⣁⠐⠂⣀⣥⣾⣿⠋⣸⣿⣿⣿⠇⠀⠀⠀⡘⠁⠀⣴⠲⣎⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠹⠅⠛⠻⢿⡿⠿⠿⣛⡫⣹⠙⠛⠛⠛⠛⠉⠉⢉⡉⠁⣴⠶⡜⠓⢾⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣾⣽⢷⡞⣷⢠⡶⣦⢠⢶⡄⣿⢹⡆⠹⢦⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠼⣿⡁⢸⣧⠟⠸⣠⡇⢙⣿⣇⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠚⠋⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠃⠏⢇⢠⠖⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢸⠇⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⡞⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠈⠀⠀⠈⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠨⠀⡄⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⠀⠂⠀⢀⠀⢰⠀⢠⠀⠀⣼⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⠶⠤⢤⣀⣀⣤⠤⠴⠖⠂⠐⠋⠱⣩⣿⣦⠀⠰⠸⡄⠀⠃⠀⠋⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⡤⠞⠀⢇⠿⢿⡁⠀⠘⠁⠙⢆⠀⠀⢰⣾⡧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴⠖⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢤⣀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠈⠩⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠔⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠾⠃⠀⠘⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠈⠃⠀⠀⠀⠐⠀⠀⠀⢀⢈⡓⢦⡀⠀⠀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠁⠀⣀⠀⠀⠀⣣⠀⠀⠐⠶⠖⠀⣂⠀⢘⠳⠲⣄⠙⢄⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⢠⠎⠀⡼⠋⠀⠁⠀⠀⣞⡀⠀⠉⠛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠰⡇⠀⠀⠀⠀⠼⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠹⡆⡞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠊⠀⠀⠀⠀⠀⢐⡅⣠⣠⠆⠀⠀⠈⢇⠀⠀⠀⠀⠀⣀⠴⠻⠈⠃⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⡄⠈⠁⠀⠀⠈⠀⡄⠈⠑⠦⠶⠖⠋⡅⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠠⠀⠂⠀⣠⢀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠴⠂⠀⠀⠀⢀⡄⠀⠐⠣⠋⢀⣾⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠄⠀⠀⠀⠈⠁⠀⡴⠀⢀⠎⠀⠀⣠⠟⠀⠀⠀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠔⠂⠸⠤⠀⠈⠰⠖⠋⠁⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢊⠀⠀⠀⣠⠎

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣼⠁⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⠀⣸⠃⠠⡏⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣧⠀⠀⠀⠀⠀⢰⡏⠀⢸⡇⠀⠀⢰⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡄⠀⠀⠀⠀⢸⠇⠀⢸⡇⠀⢰⡟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⢇⠀⠀⠀⠀⣾⠀⠀⢸⡇⠀⣾⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⡟⠀⠀⢸⡇⣰⠏⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⣤⠀⠈⣧⠀⠀⠀⡇⠀⠀⢸⣇⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⠀⢹⠀⠀⠻⡄⠀⠀⡇⠀⠀⢸⣿⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢳⣄⡦⠀⠀⣿⠀⠀⡇⠀⠀⢸⣿⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠹⣿⡀⠀⠸⣇⠀⣷⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠘⢷⡄⠀⢸⣆⣿⠀⠀⣸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠈⢷⡀⠀⢿⣿⠀⠀⣿⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢿⡀⠈⣿⠀⠀⣿⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠘⢧⡀⢹⡄⠀⣿⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⣧⠈⢃⠀⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣄⠘⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢻⡄⠸⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢃⠀⢺⡅⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣱⡄⠀⢸⡇⠸⣇⣀⡼⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣧⣤⡜⠳⣤⠟⠉⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠋⠁⣠⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠶⠛⠋⠁⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⣠⠤⠚⠋⠁⠀⠀⠀⢠⠀⠀⠀⠘⡀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⠖⠈⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⢀⡾⠃⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠘⠃⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⢦
⠀⠀⣠⠟⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠃⠀⠀⠈⢧
⠀⢰⡏⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠈⣵⠀⠀⠀⠀⠀⠈⣷
⢠⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⢸⡇
⡜⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⢹
⣷⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀
⠈⢷⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⣸
⠀⠘⢷⡀⠀⠀⠀⠀⠈⢄⡀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠶⠶⠶⠶⢾⣦⡴⢶⣤⡶⣤⡾⠛⠛⠛⠛⠛⠛⠛⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠁⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⠙⠳⡤⠶⠠⠀⡀⠀⠀⢀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠠⣼⣿⣓⠦⣄⡀⠀⢀⣀⡀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠳⣄⠉⠉⠀⠀⠉⠻⠋⠉⠩⠋⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠴⣚⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣾⠷⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⣥⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠳⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠁⠀⠀⠀⠀⠁⢐⡟⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠈⠛⠻⢯⣍⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⣻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⠀⢾⡛⠉⠋⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⢰⣴⣮⡀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡼⠃⢿⠹⣿⡏⠇⠙⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣠⢤⡖⠒⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠉⠁⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⠀⡤⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠲⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡏⢸⡄⠀⠀⠀⠀⢀⣄⠀⠀⠀⢘⡆⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠀⠀⠙⠶⣤⣤⠶⠋⠹⣄⣀⣠⡾⠁⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣴⣿⣿⢋⠀⠀⠀⢴⠂⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢹⣿⠿⠿⣀⠀⠐⢦⠇⠀⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣎⣤⣤⢰⡆⠺⣏⡀
⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣧⡬⢾⠶⠆⢸⣿⢻⣷⣄⡀
⠀⠀⠀⠀⣄⣈⡆⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣯⣤⣴⣏⣿⠎⢟⢿⣷⣦⣀⠀⣀
⠀⠀⠀⠀⠀⠈⢣⣄⣢⠀⠀⢀⣴⡿⢋⣹⡟⣹⠻⡿⢿⣿⣍⢵⡿⣷⢀⣩⣿⣿⣿⠷⠟
⠀⠀⠀⠀⠀⠀⠀⢝⣽⡽⣦⣯⡥⣠⣭⣤⡘⠻⣿⣤⢟⠛⠛⠄⢼⢯⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠿⢷⣼⣟⣳⠽⠟⢹⣿⣦⠙⢿⠖⣺⣟⠒⡠⠃⠀⠙⢷⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⢹⣿⡁⠀⢣⠀⠻⣿⣷⠐⠠⠉⢄⠌⣿⠃⠀⠀⢨⡇⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢸⣷⠽⢴⠆⠀⠀⣽⣿⡽⠤⠐⠁⠀⣀⡠⣶⠀⠠⠼⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⡏⠀⠈⠁⠑⠠⠁⠉⠂⠀⢠⡶⡎⡑⠂⡼⠛⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣌⡀⠴⠀⠈⠀⠀⡁⠀⡐⠀⢀⠀⠀⠀⡃⢈⡄⠀⠰⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⣸⣄⣀⢠⣴⡟⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡿⠉⠀⠈⢿⣿⣯⠁⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢘⠁⠀⠀⠀⠈⢛⡓⠛⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠈⠀⠀⠀⠀⠸⡿⡮⠜⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⢸⣤⠠⠜⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠍⡁⡀⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣦⡀⠈⠀⢢⠁⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⡆⠀⠴⢦⣤⣤⣄⣀⡀⠀⣀⡤⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⠰⢤⡀⠀⠀⢀⣈⣉⠉⠻⢿⣦⠷⠿⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣕⣄⠴⠴⠶⠆⠀⠈⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠛⠛⠛⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠘⠛⠲⠶⠤⣄⠀⠀⢀⠤⠴⠶⠛⠓⠀⢾⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠘⣿⠀⡤⢤⣤⠤⡤⠄⠀⠀⠀⠤⣤⡤⣤⢤⠀⣼⡏⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡟⠀⠀⠘⠻⠟⠃⠀⣿⠀⠀⠀⠛⠛⠓⠀⠀⠻⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢡⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠲⢤⣄⣀⣀⣀⡤⠶⠚⠁⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⠀⠀⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⣨⡄⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠇⠀⠀⢸⣿⣿⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⡄⠀⠀⠉⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠃⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡀⠀⠀⣀⠀⠀⣄⣾⣿⣇⣀⣀⡀⣰⣿⣿⡇⠀⠀⠀⠀⠀⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣠⣾⣿⣧⣼⣿⣟⣻⠀⠀⠀⠀⣀⣈⣽⣷⣤⣀⣄⣠⣶⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠛⠋⠙⠣⣤⣿⠟⠁⠀⠉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⠞⠛⠛⢇⠹⠇⠰⠛⢛⠛⠢⣻⡿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡙⢿⣿⣦⣴⢾⣟⢿⡇⢀⣾⣿⢽⣶⡠⢿⡿⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⢻⣜⣿⠏⠋⠀⣉⣴⣶⠒⢸⡃⠚⠉⠀⠘⡇⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠞⠯⡅⠀⢀⣾⣿⣿⣿⣷⣾⣧⢀⣀⠀⢀⡧⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣠⣷⣰⣿⣿⣿⠿⠿⣿⠿⢋⣠⠄⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⠸⢩⣿⣿⣛⣛⠿⠋⠀⠀⣰⠟⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣿⣿⣿⣧⣄⠉⠛⠉⠁⠀⢀⣤⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⢻⣽⣿⣿⣿⣿⣷⣄⣀⣀⣠⣾⢏⡼⠋⣛⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡠⠚⠹⣾⢟⡿⣫⠿⡟⣿⣿⡴⠋⢀⡔⠁⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⣈⡀⠋⢐⣯⠚⣁⠞⠁⠀⣰⠛⣦⠤⣀⣤⣾⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⣿⣿⣿⡷⠊⠀⠈⠁⠀⠰⠈⠀⣀⠚⢿⣿⣿⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢻⡅⠀⠀⠀⠀⠀⠀⠀⠜⠁⣀⣾⢯⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣶⣶⣶⠶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢚⡛⠛⠻⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢀⠙⠁⠰⠶⠀⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠛⠁⣴⠄⠠⡶⠀⠛⠙⠒⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡦⠀⢠⡄⠀⠀⠶⠆⠀⣶⡾⠉⠉⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣤⣤⣬⠵⠖⣊⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣶⣶⡾⠿⢟⠛⠉⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠙⠳⡄⠀⠀⠀⠈⠁⠈⠉⠳⡄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢰⠀⠀⠀⠀⠀⢹⡀⠀⡆⠀⠀⠀⠀⠀⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⣘⡇⠀⠀⠀⢸⡄⠀⣆⠀⠀⠈⡇⠀⡁⢀⢠⣇⠀⠀⡌⠀⠀⡄
⠀⠀⠀⠀⠀⢉⠭⣭⡣⡀⠀⠀⠀⡇⠈⠉⡁⠀⠀⡇⡎⠀⠀⢀⡉⠐⠀⡇⠀⢠⡠⢒⠲⣆
⠀⠀⠀⠀⡇⡀⠀⢜⣷⠈⠀⠀⠀⡇⠐⠄⠃⠀⠀⠷⠰⠀⠀⠀⣁⠋⠀⢧⠀⢸⡀⠁⠀⠘⠂
⠀⠀⠀⠀⣇⢃⢀⣶⣿⠀⣠⠀⠀⡇⠀⣀⠿⢀⣤⣶⣶⣶⣶⣄⠛⠤⢤⣼⠃⢤⡁⠀⠀⠈
⠀⠀⠀⠀⠘⣦⢈⣝⣇⠎⢀⠤⠤⢿⣏⠀⠀⣾⣿⣤⣿⣿⣿⣿⣧⠀⢀⠤⢦⣄⢳⠀⡀
⠀⠀⢀⣀⣀⣈⣷⠌⠁⡀⠀⠀⢀⠀⠈⢳⡜⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⡀⠘⠂⠀⣧⣤⣄⡀
⠀⢪⣍⠉⡻⠍⣿⠀⠀⣇⠀⠀⠃⠀⣀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠃⠀⡀⠀⠁⠀⠀⠀⢷⢲⣟⣷
⡇⣿⣿⣷⠁⣾⣿⣷⠶⡿⣄⠀⠀⠀⠘⣿⡆⠌⠛⠿⣿⣯⠽⠪⢴⣾⠇⠀⠀⠀⡤⣤⠌⠙⠿⠹⠶⠶⢦
⢻⡽⣿⣿⡇⣿⣿⣿⠀⣹⣜⢷⣄⠀⠀⠈⠻⣶⣤⣤⣀⣀⣤⣴⡟⠁⠀⠀⡀⠀⢀⣥⣭⡉⢂⣴⣿⣿⠆
⠀⠹⣎⢿⣷⢹⣿⣿⡆⣿⣿⣷⣌⠛⠶⣤⡀⠈⠻⢭⣉⣉⡍⠉⠀⠠⠒⢰⠇⡀⣼⣿⣿⣿⢰⣿⣿⠏
⠀⠀⠘⢧⢻⠘⣿⣿⡏⣿⣿⣿⡟⢷⣄⠈⠻⣦⣀⠀⠀⠀⢀⡠⠀⠀⢠⣧⠀⡇⣿⣿⣿⡏⣸⣿⡟
⠀⠀⠀⠈⣧⠀⣿⣿⡇⣿⣿⣿⡇⣼⣿⣦⡀⠀⠉⠙⠛⠛⠉⠀⠀⣠⣾⣿⠂⡇⣿⣿⣿⠃⣿⣿⠡
⠀⠀⠀⠀⢸⠐⢿⡿⠀⢹⣿⣿⣿⢿⣿⣿⣿⠳⣶⣶⣶⣿⠒⠲⣿⣿⣿⣿⠀⣿⣿⣿⡿⢠⣿⡟⡄
⠀⠀⠀⠀⠈⠷⡤⠴⡆⣾⣿⣿⢧⢸⣿⣿⣿⡀⣿⣿⣿⣿⡄⠈⣿⣿⣿⣿⠀⣿⣿⣿⡇⢈⣻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣛⣛⣼⢸⣿⣿⣿⠇⢹⣿⣿⣿⣿⠀⢿⣿⣿⡟⢸⠛⠿⠿⠡⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠹⣦⡛⠟⣋⠐⢦⡙⣿⣿⠟⠶⣌⡛⠛⠃⠘⠛⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢨⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠊⠀⠀⠀⠀⠀⠀⠙⠙⠂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠋⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠀⠐⣂⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡖⠀⠀⠀⠀⠀⣼⡆⠀⠀⠰⡀⠁⠋⠀⢹⡵⠋⠀⠰⢀⠀⢀⡁⢀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡭⠁⠀⠃⠀⠈⠓⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠊⠰⠄⠀⠀⠀⠀⠊⣭⢤⡼⢃⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠸⣦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠛⠒⠒⢿⡣⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⡾⠛⠓⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡄⢸⣷⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⣀⡏⠉⠁⠐⠘⠀⠀⠀⣤⠠⠞⠃⠰⠆⢠⣤⣌⡶⡚⠀⡆⠀⢩⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣼⠃⠁⠀⢰⠆⡴⠃⠰⠦⠐⠤⠄⠀⠃⠀⣀⣀⣤⢅⡆⠇⢴⡀⢈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⠗⠀⢠⡖⠐⠺⢦⡀⡦⠤⢿⣷⣶⣶⣦⣭⣥⣴⣾⣿⣷⣆⣀⠘⣿⣯⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⣦⣒⠲⣽⣮⣻⣯⣙⣭⢙⠉⢻⣭⣿⣤⣯⣛⣛⡛⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠉⠀⣯⣽⠙⣈⣁⢉⡞⠂⠨⣻⣿⡃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢊⠉⠹⠻⡇⠳⢷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠉⡛⣻⡆⢀⠤⡆⢄⢟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⢸⠀⣘⡇⠸⡀⣴⣿⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣼⣿⡏⠗⢀⣀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠃⠿⠿⠁⠶⠌⠁⠀⠙⠋⢹⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠰⠀⣀⠀⠀⠉⠙⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⣲⣿⣠⣄⢲⣴⣤⣤⣦⣬⣿⣟⡾⣠⣤⣀⣰⣽⣿⣵⣆⣀⠀⠠⠤⠶⡤⣦⣤⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠤⠤⠬⠿⠿⠶⠌⠐⠂⠒⠚⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠢⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⠁⠀⢱
⠀⠀⠀⠀⡐⠉⠈⠑⠃⠀⠀⠐⠀⠐⠀⠘⠂⠢
⠀⠀⠀⠀⢃⠀⠀⠄⠀⡀⠀⠀⠁⠀⠐⠀⠀⠸
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠈⠣⣀⠠⠀⠀⠀⢀⠀⡐⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠖⠉⠀⠀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⡀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⡀⠆⠈⠀⢠⠀⡠⠊⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢀⢀⠠⡙⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠇⠀⠘⢨⠖⠁⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⢆⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⠢⠀⠀⡄⠂⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⠈⠈⢀⠕⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠁⢺⠁
⠀⠀⠀⠀⠀⠀⡀⠔⠃⠀⢀⣀⠀⠀⠀⢀⣀⠀⢀⠇⠈⡡⢊⠒⠤⢀⡀⠀⠀⠀⢀⡀⣤⠒⠈⠀⢀⡀⢸
⠀⠀⠀⢀⡴⠊⠀⠀⠑⠊⠁⠀⠑⠤⠤⠃⠈⠁⠁⢀⠜⠀⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠆⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸
⠀⠀⠀⢇⠀⠀⠀⠀⢰⠀⠀⠀⠢⠂⠀⠀⠐⠀⠐⡀⠈⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠹⠂⠀⠀⠈⡄⠐⠀⠀⠀⠐⠀⠀⠀⠀⠉⠒⠂⠉⡇⠀⠀⠀⠀⠈⠠⠀⠀⠂⠆⠀⠀⠀⠰
⠀⠀⠀⢠⠁⠀⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠐⠂⣡⠙⠢⠀⠠⠊⠀⠀⠐⠀⡆
⠀⠀⠀⠀⠂⠄⠠⠄⠀⠀⢣⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠱⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⣀⣀⠀⠀⠀⠀⠀⢀⣀⠠⠠⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠉⠀⠈⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⡀⠀⠀⠀⠘⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⠅⣕⣀⣠⣔⠧⢨⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠂⠀⠀⠀⠘⠃⠉

⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠾⠦⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣤⣄⡀
⠶⠆⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡌⠑⣀⣀⣀⠀⣴⣶⣿⡿⠙⣻⡏⣬⣿⡾⣿⣷⣶⣄⣀⣀⣀⣀
⣠⣶⣾⣿⣿⣿⣿⣿⣿⡿⠛⠻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡀⠀⢢⡀⠀⠀⠀⠈⢿⣆⠘⣿⣿⣿⠉⠙⣻⣤⢾⡋⠀⠈⣿⣄⢀⣿⣍⣿⣟⣻⣿⣿⡇
⡿⠿⠋⣉⠁⠀⠀⠀⠀⠁⡄⠀⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⢿⣄⠀⠡⠀⠀⠀⠀⠀⢻⣆⠘⡏⢻⣶⡴⢿⣥⣿⣿⣷⠾⢿⠏⢙⣿⣡⣽⣛⣿⡟⢿⡇
⣀⣠⣴⣀⣨⣿⣶⡄⠀⢠⣄⠀⠀⠀⠀⠘⢯⡻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⢻⡆⠸⣟⠛⣷⣾⢻⣤⣀⣿⡶⢿⡗⢻⠋⢙⣏⠉⣻⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⢀⣤⠀⠀⠀⠀⢈⣿⣿⣿⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢿⡄⠛⠻⣟⣻⣼⡿⣿⣻⣷⣾⡿⠻⡿⠻⡿⠛⣿⣷⢼⣿
⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡁⠀⠀⠀⠀⠚⠁⠈⠛⣉⠙⠃⠈⠁⠀⠀⠀⠀⠉⠻⣷⡀⠀⠀⠸⣇⢸⣦⡿⣿⡉⢷⣾⠛⣿⣸⣷⣴⣿⡶⣿⣆⣰⣇⠈⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⢄⠝⠁⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⠈⠛⣧⣈⣷⡞⠛⣦⣿⣿⣿⡇⢹⣧⡈⣋⠈⢻⠛⢿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⡇⠸⣷⠿⣿⡇⢹⣿⠋⣿⣿⣿⣿⣾⣿⣿⡿⣷⢾⣷⣬
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⢀⣀⠀⠀⠀⠀⠟⢀⣽⣦⣤⣷⡞⣻⣾⣿⣿⣿⣿⣿⣿⣿⣷⣹⣦⣙⡋
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⣀⣀⣰⢿⡿⠛⠿⠟⠙⠿⠋⠻⠿⠛⣿⠀⠀⠀⢈⡁⣉⣧⣤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣻⣝⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⣿⣬⡙⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢼
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠘⢿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡈⠀⠀⠈⠁⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠘⢿⣿⡆⠀⠀⠄⢐⣿⡇⠀⠄⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠈⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⣿⡆⠀⢠⡜⣿⡄⠀⠀⠈⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡟⠋⠾⠿⡀⠀⠈⠉⠙⠻⢷⡀⠀⠀⠀⢰⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⠟⣡⣾⠁⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡄⠀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⢿⣿⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣤⡕⠤⠿⠋⠀⠀⠀⠠⠤⠤⠤⡿⢿⣿⣿⣿
⠀⢸⣿⣿⡿⣷⠀⢹⡍⠀⠀⠀⠀⠀⠀⠀⣄⠙⣆⠀⠀⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⡀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⣈⢻⣿
⠀⢸⣿⣿⣿⠿⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣿⣦⣤⣀⣿⠟⢋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠛⣩⣴⠞⠁⠀⠻⣷⡀⠀⢀⠀⠀⡀⠀⠹⢿⣿
⠀⢸⣿⣿⠟⣯⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠈⠛⠤⣤⣄⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⣾⠿⠟⠈⠉⠁⠀⠀⠀⠀⣿⡇⠀⠸⢇⠀⠻⣦⡔⣿⡟
⠀⠈⣿⣿⣠⡝⣾⣿⣯⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠙⢷⣤⣤⣤⡴⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠘⢷⡄⢹⡇⣼⢧
⠀⢸⣿⣿⠟⠁⠸⣟⣷⣙⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠻⠿⠟⠀⡿⢩⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠀⠀⠀⠀⠈⢳⡞⠳⢿⣿⢧
⣠⡿⠋⠀⠀⡀⠀⠻⢜⣿⣼⣱⠂⠀⠀⠁⠀⠀⣾⣿⣿⠆⠀⠲⣤⡀⠀⣀⡀⠀⠀⠀⣴⣿⠀⡅⠸⣿⠀⠀⣤⡠⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢠⣿⡿⣷
⠇⠀⠀⠀⠀⢷⣀⠀⠈⢻⣯⠀⠀⠈⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠈⠛⣻⣿⣿⠀⠀⢀⣿⠈⠀⢠⠀⠻⡆⠀⠸⣿⣾⢿⣧⠀⢀⠀⠀⠀⠀⠀⢀⡿⢶⣿⣿⣿⡟
⡃⠀⠀⠀⠀⢸⣿⣿⣦⣜⠿⣆⡄⠀⡐⠀⠀⠀⠀⠈⠻⡇⠀⣰⢠⣿⡟⠁⠙⣷⣶⡾⠇⠀⠀⠘⡄⢰⡆⠀⢀⡹⡿⠈⢿⣶⡾⠀⠀⠀⠀⠀⡼⢁⡼⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠰⣼⣦⠀⠙⠀⠸⣧⣄⠁⠀⣠⣦⡄⠀⠀⠁⠀⢿⢸⣿⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⡀⠀⠻⣦⠀⠙⣆⠀⠀⠘⠀⢀⠀⠀⠀⣦⠐⢚⣿⣿⣿⣿⠯
⢣⡀⢰⣦⠀⠀⢹⣿⣷⣀⠆⠀⠻⠞⢤⠘⠻⠟⠙⠀⠀⠀⠀⢸⡞⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⢋⣰⣏⣼⣯⣿⣿⣿
⠘⣧⠀⠻⣆⠀⠀⠹⠏⣿⣿⡡⡀⠀⠀⢾⣦⡀⠀⠀⠀⠈⠀⢸⣧⢉⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠀⠀⢀⡜⡠⢥⣿⢶⡗⣶⢿⣿⣿
⠀⢹⣧⡀⠙⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢩⣿⡟⠶⣄⡀⠘⠘⣿⡘⢌⠁⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣄⣀⠄⢀⣴⠁⠀⢱⣀⣦⣸⣷⣶⠶⣦⣽
⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣀⣀⣿⣿⣶⣥⡟⢶⣦⣿⡄⣀⠀⠀⠀⣸⠏⣀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣤⣛⣡⡴⣿⠺⡟⢙⡉⣅⣹⣦⣧⣸
⠀⠀⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠛⢿⣿⣿⣮⣽⣯⣴⡛⠛⠿⢦⣿⣁⣤⣦⣤⣀⣀⣰⠀⠀⡀⢀⣀⡔⣩⣿⣿⣝⣿⣿⣿⡷⢾⡟⢿⠛⠿⢿⡟
⠀⠀⣿⣿⣿⡎⠙⠻⠷⣶⣤⠀⠀⠀⠀⠀⠀⠀⣀⣤⠜⣿⣽⣏⣻⡿⢻⣿⡿⢷⣭⣭⣴⣄⣢⣉⣭⣝⠛⠛⣧⠜⣦⠀⡉⠛⠹⣯⣻⣿⣿⣿⣿⡧⣾⣧⣾⣾⣿
⠀⠀⣿⣿⣿⣿⡀⢠⡄⠈⠉⠁⠂⠀⠒⢀⡛⣉⣛⣭⠾⣙⣿⡛⢿⣟⢿⡿⣷⣿⣭⣼⣟⣹⡿⣿⡿⣿⣾⣏⣸⠇⣼⡇⢻⣳⣤⣙⣿⣿⣿⣿⣿⣷⣼⣇⣸⣍⣻
⠀⠀⣿⣿⣞⣿⡟⢿⠀⠀⠀⠀⠀⣀⣶⢟⣵⣟⣽⣾⣿⠻⣎⢻⣌⢿⡜⣯⢼⣋⣿⣉⣿⣉⣿⣿⡟⢋⣋⠛⠛⢷⣏⡶⠾⢃⣿⢾⠣⣦⣌⠙⠻⣿⣿⣯⣽⡋⢛
⠀⣰⡿⣿⣹⣿⠁⠀⠀⠀⢀⣴⣿⣿⣶⣿⣷⣿⡟⣿⣿⣿⣿⣷⣽⣖⣯⠽⣞⡟⢻⠏⣯⣼⡧⣼⣅⣿⠻⣷⣧⡴⠿⢷⣶⣾⣛⡾⢷⡿⣼⣷⣦⣽⣿⣿⣿⣿⢿
⢰⣿⣿⢿⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⠟⠉⠀⠈⠻⢿⣏⣿⣋⣿⣿⣿⣿⣿⣿⣿⣶⣟⣋⣼⣤⣾⣄⣽⣿⣿⣿⡇⢀⠀⠀⠙⠻⢷⣾⣓⣾⣏⣿⣿⢿⡝⠻⣿⣾
⣿⣿⣿⣿⣿⣃⣐⣿⡿⣳⣿⣽⠿⣿⡀⠀⠀⠀⠀⠀⠉⠻⣧⣿⣨⣿⣿⣿⣍⣿⢹⣿⣿⣩⣀⣹⣿⣿⣿⣿⣟⣧⠈⣷⣤⡀⠀⠀⠿⣿⣷⣾⣹⣏⡿⣽⣧⡘⣿
⣿⣯⣿⣿⣿⡿⣿⣿⣿⣿⣿⣽⣟⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣧⣾⢤⡿⣼⢿⣿⣿⣟⣛⣸⣿⣿⣿⣿⣿⣀⣸⣿⣿⣷⣶⣤⣀⠈⠛⠿⣿⣿⣾⣻⣿⣾
⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣿⢺⡷⣿⣿⣿⣟⣹⣿⣿⡿⠋⠉⠉⠛⠻⢿⣿⣿⣿⣿⣶⣤⡈⣿⢿⣿⣿⣿
⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⡀⠀⠉⠻⠿⣿⣿⣏⠿⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⠿⣿⡶⠟⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣶⣶⣤⣼⣿⡿⠋⠀⠀⠀⢀⣠⡾⠃⠀⠰⣿⣿⣶⣶⣶⣶⣶⣶⣈⣿⣷⠀⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢿⣼⣧⣽⣿⣿⣶⣶⣷⣾⡿⠏⠀⠀⠀⠀⠈⠉⠁⠀⠀⢸⣿⣿⣿⣿⡿⠃⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣠⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⢀⣠⣤⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠻⠁⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣀⢀⣀⡈⠁⠉⠀⠀⠀⠀⠀⠂⠄⠀⣰⡖⠛⠣⣸⣿⠟⢀⡴⢦⠀⠤⠄⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡞⣿⡏⢻⠇⠁⠀⣶⣶⣶⣤⣤⣀⣀⡀⠀⠀⢈⣤⣿⣵⣶⣿⣧⠀⠐⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⢶⣆⢶⣮⠀⢀⣿⣿⡛⢻⣿⣿⠿⣍⡳⢤⣼⣯⣏⣽⣭⠽⢿⡄⣀⣶⡎⠀⠒⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠤⢀⣸⣛⣨⣏⢠⣭⠁⢸⣿⣿⣄⣸⣀⠠⠰⣿⣾⣿⡏⢁⣻⣿⣶⣷⡰⠛⣉⡄⠀⠀⠘⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠈⣿⠄⣿⠆⣿⠂⣸⣿⣿⣂⣻⣫⣶⡴⢿⣿⡜⣤⣼⡽⢿⣆⢉⡴⠟⠛⠁⡴⠀⠳⡄⢹⣿⣿⠀⢀⡀
⠀⠀⠀⣴⡦⣶⣗⢾⡀⢘⠀⠘⠁⠀⠀⠀⠀⠀⠰⡇⢠⡅⠀⠀⠉⠈⠉⠈⠉⠛⠭⠉⠃⠉⠀⠒⣀⠙⠋⣠⣴⡆⢰⡇⠀⠀⢧⠀⣿⣿⡄⠿⠽⠶⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣷⠀⠇⠀⠀⠀⡀⣄⠂⢴⣲⣶⣶⣝⢳⡄⠀⠙⢷⡾⠋⢁⡄⠘⡇⠀⠀⡟⢀⡿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣿⣇⣄⠀⠀⠀⠀⠘⢀⣿⠿⠛⠋⢹⣙⣫⣽⣿⠿⠿⣿⣏⢿⠂⢲⣤⣀⣴⠟⢁⠀⢳⣀⣰⠃⣸⠁⢀⣀⣀⣀⡠⠤⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢺⣿⡇⠀⢸⣷⣾⠢⢤⣤⠴⠞⠛⢣⡤⣲⣿⡏⠀⣦⢹⣿⣾⣧⣤⣤⣉⣉⣴⠟⣤⠈⠻⠤⠞⠃⠀⣠⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣩⣿⣿⡶⣿⣿⣿⣶⣶⣶⣶⣿⣿⡗⣚⡽⣿⡇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣻⣉⣾⢏⣷⣄⣀⣀⣠⣴⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠥⢾⣿⢿⣿⣿⣿⡿⠟⠉⠉⠉⠙⠛⠱⣺⠿⣿⣭⣼⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠾⠯⠟⠋⠋⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢕⣯⠷⣫⣖⢋⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉

⠀⠀⠀⡠⠖⠒⠒⠤⡀⢠⢊⠝
⠀⠀⠚⢲⠤⣄⠀⠀⠘⡇⣆⠤⠀⢀⣀⡀
⠀⠀⠀⢸⠀⠈⠳⡄⠀⢱⣞⠒⢮⡁⠀⠁⠀⠀⢀⣀⣀⣤⣤⡴⠶⠶⠶⠶⠶⠶⢦⣤⣤⣀⡀
⠀⡎⠉⠉⠑⠒⠤⠬⢦⣸⠙⠛⣿⠭⣛⠛⠛⠛⠉⠉⠁⠌⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⣄
⠀⢰⠂⠀⢀⠤⠠⠤⡀⠼⡖⢎⠢⣍⠒⠛⠶⠖⡒⠒⠠⢄⡀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠳⣄⡀
⠀⠀⠂⠉⠻⠔⠒⢲⠀⠀⢡⠘⡀⠀⠑⢄⠀⠀⠀⠈⠂⠄⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⣠⠏⠀⡆⢸⠀⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⣧⡀
⠀⠀⠀⠀⣠⠞⠉⠀⢀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⢷⡀
⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠘⣷
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸
⠀⣸⠁⠀⠀⠀⠀⡀⡄⠀⢸⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠈⡀⠀⢣
⢠⡇⠀⠀⠀⠀⠀⠁⡇⠀⠈⡄⠀⠀⠈⡄⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢰⠀⠸⡄
⣸⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⢣⠀⠀⠀⠘⡄⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡆⠀⣧
⣿⠀⠈⠀⠀⠀⠀⠀⠘⡀⠀⠈⢧⠀⠀⠀⠘⡄⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢇⠀⠀⡇⠀⣿
⣿⠀⢀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⣆⠀⠀⠀⠘⡄⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡆⠀⣿⠀⣿
⣿⠀⠘⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⣆⠀⠀⠀⠈⢆⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⢹⠀⢹⢰⡏
⢹⡄⠀⠄⠀⠀⠀⠀⠁⠀⠀⠑⡀⠀⠘⢆⠀⠀⠀⠀⠡⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡇⣼⡿
⠈⣷⠀⠈⠄⠀⠐⡀⠀⠁⠀⠀⠘⢄⠀⠈⢧⡀⠀⠀⠀⠈⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⡆⠀⣷⡿⠁
⠀⠸⣇⠀⠘⡄⠀⠐⠀⠀⠀⢄⠀⠀⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣇⣴⠏
⠀⠀⠹⣦⠀⠘⢆⠀⠈⠂⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⢀⠀⠀⠘⣦⡼⠻⡧⣑⠤
⠀⠀⠀⠘⢷⣄⠈⢣⡀⠀⠑⢄⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣆⣤⠞⠋⠨⠂⠙⠂⠡⣀
⠀⠀⠀⠀⠀⠙⠳⣤⡙⠦⡀⠀⠑⢄⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⣧⣤⡶⠛⠡⡀⡀⠒⢦⡀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⣕⣂⠠⠑⠦⣄⠈⠓⠦⢤⣄⣈⠐⠲⢦⣤⣤⠶⠿⠛⠋⠉⠆⠀⡖⢂⢆⠀⠀⢀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣛⣚⣿⣿⣶⣶⣾⣛⣛⣛⣋⣉⣀⣀⣀⣀⣊⣈⣂⣙⣋⣕⣀⣁⣀⣁⣀⣀⣀⣀
⣿⣿⠋⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⠶⣦⣤⣤⣜⣿⣷⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢿⣷⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⣥⣄⡀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⢻⣿⡄⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⠀⢸⣯⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⣿⠇⠀⣠⣴⣾⠿⠿⣿⣿⣿⣿⣟⡋⢴⠛⠉⠁⣀⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⡇⠀⠀⠀⠀⢿⣇⠀⠀⣰⣿⣟⣠⣾⣿⣽⣾⣿⣿⣶⣿⣿⣿⣿⣿⣶⣄⠀⣾⠿⣦⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣀⣼⡿⣠⠴⠶⢤⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣭⣛⠝⠻⣿⣿⣿⣿⣷⣍⠀⢈⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠻⠟⣻⣦⣤⣴⣤⣶⣴⣶⣶⣾⣿⣿⡿⠟⢻⡅⠀⠀⠀⠀⢸⡇⠙⣿⣦⣿⣿⣿⣿⣿⣿⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣴⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿⢿⡟⠉⠀⠀⠈⣷⠀⠀⣀⣠⣾⣷⣶⣾⣿⣿⣿⡿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⣾⣿⣿⣿⣿⠉⠉⠀⠀⢸⣧⣀⣠⣤⣴⣿⣿⣿⣿⡿⠟⠛⠻⣿⣿⣿⣿⡇⣽⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⢀⣀⣰⣿⣿⣿⣿⣿⡿⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢻⣟⠋⣩⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⣿⣿⢿⣇⣀⠀⠀⠀⣸⡇⠀⠉⠉⣿⠀⠀⠀⠀⠀⣿⣶⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⢻⡇⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⣿⣿⣿⣌⣛⣛⣛⠻⣷⣿⣧⣄⣀⣀⣿⡀⠀⣀⣠⣾⣿⡿⠟⣡⣾⣿⣿⣿⣿⣿⣿⡿⣼⠃⣀⡤⣾⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣛⣯⣶⣯⢞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣎⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣿⠿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⢟⣭⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣿⣷⣷⡻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣽⣿⣷⣶⣶⣾⣿⣿⣿⡻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⢿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡛⢒⣷⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣮⣻⣟⣙⣻⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣭⣽⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⠿⢿⣿⣽⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣟⠿⣿⣿⣿⡿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠾⠿⠿⠿⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡠⢶⠿⠿⠆⠶⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡖⠂⠰⠛⠈⠛⢛⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢤⣀⣠⣄⣠⣄⢀⣀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡧⠀⢀⣠⣴⣾⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠛⠛⠛⠋⠈⠉⠙⠛⠋⠙⠉⠛⠻⠆⣤⡄⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠛⣫⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠉⠠⣦⠀⠀⠀⢀⣴⡄⠀⠀⠀⠘⠿⠂⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡍⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋
⢠⣤⣶⣤⣀⣴⡶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⣀⠀⠀⠀⠀⠀⠀⣯
⣈⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠶⠿⢷⣤⣤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⢀⣿⡆
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⢸⣿⣭⣶⠶⠮⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⢀⣼⡟
⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣛⣛⠛⠷⢤⣴⣶⠆⠀⠀⠀⣀⣤⡄⠺⠛⠉⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣉⣉⣭⣤⣶⣶⣶⣧⣤⣴⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟
⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⠛⠛⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⢠⣤⣿⠟⠃
⠀⠀⠀⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣯⠤⠆⠀⠀⠀⠀⣰⣿⠉⠉
⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⣦⣤⡾⡿⠇
⠀⠀⠀⠉⣿⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠛⠉⠉⠀⠉
⠀⠀⠀⠀⠘⠋⠉⢉⣴⣶⣤⣄⣀⠀⠀⠀⠐⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙⠛⠂⠰⢄⣴⠟⠉⠀⠀⠀⠀⢀⣀⢀⣀⢠⣴⣶⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⣀⣀⣴⠟⠻⢮⡉⠛⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡄⠀⢼⣿⠋⠁⠀⠀⠀⠙⢦⡈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠹⣦⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⣷⡀⠀⠀⠀⠀⠀⠀⠈⢷⡌⠻⣦⣄⣀⣀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⣷⡀⠀⠀⢀⣤⣤⡴⢾⣃⣄⡀⠨⣙⣻⣿⣶⣾⠿⠟⠻⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡈⠻⣄⡀⠘⠛⠛⠚⠉⠉⠉⠉⠓⠶⣯⣭⣛⠾⢯⡛⠋⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠈⠻⣿⣓⣀⣛⣛⣿⣽⠿⠷⠀⠀⠀⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣚⠉⣥⡶⢦⡄⢶⣤⣤⣤⣉⣉⣉⣉⣋⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠀⠀⠀⠈⠻⣆⡙⢦⡀⠀⠀⠉⠉⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠘⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠢⡄⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠡⢄⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢆⣀⠀⠀⣀⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠸⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢁⠀⣀⡉⠀⠉⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠢⢀⡀⠔
⠀⠀⠑⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⠰⢀
⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡀⠀⠀⢀⠁⠰⠁⡐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠆⠀⠴
⠁⡀⠔⡂⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣠⠀⠀⠀⡈
⠀⠂⠂⠈⠐⠐⠀⣟⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡀⠹⠀⠁⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⣰⠃⢱⢄⡀⠠⠀⠀⠀⠀⠀⠠⠀⠀⣠⢴⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⢨⡀⠑⠀⠠⣇⢨⠂⠀⠊⠁⠡⢸⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠄⢼⣀⡧⠤⠤⠤⠀⠀⠀⠶⠥⠤⣆⣸⡖⠒⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠅⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⡁⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡤⢠⣀⢀⠀⣀⣠⠤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠰⠀⠀⠀⠆⠀⠀⠂⠐⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡆⢰⡆⠀⠀⠀⢠⡜⢳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⡀⠀⠀⠀⣷⡄⠀⠀⠀⠀⢸⠇⢸⠀⠀⠀⠀⠀⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢞⡀⡐⡁⠀⠀⠀⣿⣿⣦⡀⠀⠀⢸⠀⠛⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⣿⡇⠙⢿⣷⣤⣬⣀⣀⣴⣿⣄⡀⢀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠈⠙⠛⠛⠋⠁⠙⠛⠿⠛⠛⣿⡟⠀⢠⡤⠤⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢷⣶⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠿⢧⣀⢘⡇⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⣦⣞⠁⢀⣀⣀⡀⢈⡿⢻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢉⣀⣠⡾⠷⠤⢤⣄⣾⣻⢧⡀⠀⢠⡇⠀⠸⣧⣼⠟⣭⠿⠛⠛⠙⣿⡇⠀⢿⣀⡀
⠀⠀⣀⣀⡄⠀⠀⠀⠉⠛⣿⡟⠁⠀⣀⣀⠀⠈⠙⣿⠅⠙⣆⣸⠀⠀⠀⢹⣿⣄⢹⣤⡤⠀⠀⡿⡷⠒⠺⣄
⠀⠀⣽⣀⡡⠀⠀⠀⠀⠀⣿⣿⢴⡾⠛⠛⠳⡄⠀⠘⣷⣀⠈⠛⠀⢰⡄⠀⢻⡉⠉⠉⠀⣀⡼⣡⣿⣤⣤⡟⠛⠛⠓
⠀⠀⠀⠀⠁⠀⠀⠀⣠⣾⣿⣿⣿⡀⠀⠀⠀⣻⠀⠀⣿⣿⣦⡀⠀⡾⢷⣀⣈⣿⠶⠶⠟⠋⣼⠁⠈
⠀⠀⠀⢀⣠⡤⠶⠿⠿⠿⣿⣿⣿⡳⢦⡤⠔⠃⠀⣰⣿⣿⡌⢳⡶⠇⠈⠛⠉⠁⢀⢠⣀⣀⣘⣦⡀⠀⠀⠠⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠿⢿⣿⣀⣀⣀⣠⣴⣿⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠜⠳⡘⢦⡀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠊⢁⡴⠃⠀⠀⠟⠛⠛⠋⠉⠀⠉⢿⡄⠀⠀⣠⠖⠋⠉⠓⠦⡆⠀⠙⠦⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣷⢀⡾⠁⣠⠖⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡞⠃⠐⠆⠀⠀⠀⠀⢹⡾⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠲⡼⠀⠀⠀⠀⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢛⣿⣿⣿⣾⡆⢠⣄⣀⡀⣀⠀⣀⣀⣤⣤⣑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⣁⣩⣿⣿⣿⣾⣭⣯⣽⡿⠛⠋⠉⠉⡙⠛⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠈⢉⡛⣿⣿⣿⡟⠀⠀⠀⠀⠘⠀⣾⠀⠀⢀⡀⠶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢹⠛⠟⠋⠰⠀⠀⠀⠀⣦⣤⠿⣧⡀⣸⣿⣴⣶⣬⡙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⡉⠳⣦⡀⠀⠀⢸⡇⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣯⣭⣦⣽⡯⠙⢙⣿⣿⣿⣿⣜⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣀⣌⠻⢦⣄⢸⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢱⣦⣶⣿⣿⣿⣿⣿⣶⣶⣶⣝⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⣿⣿⣰⣤⣤⡉⣻⠁⢀⡴⠖⠉⠀⠀⠀⠀⠀⠉⠉⠛⢿⣿⣛⢿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣿⣿⣿⣿⠿⠧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡿⢛⣵⣦⠀⠀⠀⠀⠀⠀⢀⠀⡤⠤⠄⠲⠒⠿⠿⠿⣿⣿⡟⠋⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⢋⠀⣾⣿⡿⠀⠀⡀⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⣿⣿⡿⠻⣿⣿⣿⠿⣿⢟⣾⣿⣿⣿⡷⠀⠀⠀⠀⠀⣴⡇⠀⣠⣤⣤⣀⣀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣼⣡⣾⡿⠋⠀⠀⠀⠙⣿⣾⣧⣾⣿⣿⠟⠉⠀⠀⠀⢀⣤⣾⠋⣴⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣧⡘⠀⠀⠊⠁
⠀⠀⠀⠀⠀⣠⣼⡿⠛⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢾⡿⠀⠀⢿⣿⣿⡿⢿⡿⠛⠀⠀⣰⣿⣿⣿⡇⢡⠀⠀⣤⠀⢰⡀
⠀⠀⠀⠀⢠⣘⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⢀⠀⣀⣤⡶⠇⠀⠀⠈⠁⠹⠄⠈⠁⠀⠀⣰⣿⣿⠉⠉⠁⢸⠀⠀⢹⣗⠻⡷⠆
⠀⠀⠀⠰⣋⣘⠇⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⡀⢠⡟⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋⠃⣼⠧⣤⠞
⠀⠀⠀⢸⣿⡿⢾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⢠⡿⠋⠀⡤⠚⠻⢿⡋⠀⢤⣴⡦⢾⡟⠀⠀⠀⠀⢀⢀⢠⡄⡀
⠀⠀⠀⠘⠿⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢰⣿⣷⡀⢸⣿⣿⣯⣴⠶⠟⠃⠁⢀⣨⣤⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡄⠉⠈⢿⣿⣿⣿⢁⡡⠐⢶⣿⣿⣻⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⠿⣇⠀⠀⠘⠋⠛⠛⠋⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢶⣤⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⣿⡟⠿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣀⣘⣻⣿⡀⠀⠀⠀⠀⠑⠀⠃⠰⠀⠀⠆⠄⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠛⠉⠝⣶⣿⣿⡿⣿⠇⠙⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣶⠀⠴⣶⣶⣏⠉⠛⠛⠶⣦⣤⣤⣀⣀⣀⣀⣤⠤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⣉⣙⣻⣶⣿⣿⣿⠟⣀⠀⠀⣀⣸⠛⠋⠁⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⣾⣿⡿⠷⣼⠿⣿⢿⣏⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⣀⣤⠴⠒⠉⢉⣤⠴⠛⠉⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⢻⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡍⠣⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡜⠻⢦⡤⣄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠦⡀⢮⡉⢉⠙⠓⠚⠛⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠢⠐⠂⠀⠒⠄⠐

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠔⠈⠉⠉⠐⠤⠐⠒⠒⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠁⠀⠀⠀⠀⠀⠀⢉⡉⠈⠩⠀⡂⠤⣀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠐⠄
⠀⠀⠀⠀⠀⢀⠄⠁⠀⠎⠀⠀⠠⠄⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃⠐⠂⠐⠉⡂⠀⠐⠁⠑⢠⡤⠄⠚⢄⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⠀⡀⠀⠀⠀⠓⠄⠀⠠⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⢈⣶⣶⣧⣰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡰
⠀⠀⠀⠀⢀⠣⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋⠀⠀⠀⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠈⢡
⠀⠀⠀⠀⠈⢤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠢⡀⠀⠀⠀⠈⠐⠤⠤⠤⠐⠁⠀⠀⢀⠄⠊⠀⠀⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⣀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢠⡀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠢⠐⠉⠢⢀⠀⠀⠐⠀⠤⠤⠄⠐⠈⠀⠀⢀⠔⠁⠀⡠⠁⠀⡠⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠢⡀⠀⠈⠀⠂⠀⠄⠀⠀⠠⠄⠐⠈⠀⢀⠤⠊⠀⡀⠀⠀⠀⠀⠀⡄⠘⣀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⣀⠀⠀⠄⢀⣀⡀⠀⠀⠀⣀⠀⠄⠐⠀⡀⠄⠂⠀⠀⠀⢠⡄⢆⠀⠀⠀⠘⢁⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠀⠠⠤⠤⠄⠀⠒⠒⠉⠁⠀⠀⠀⡀⠄⠊⠁⠀⠄⠂⠀⠀⠀⠉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⣠⡀⢀⠈⠐⢀⡀⢀⠀⠂⠁⠀⣀⠀⠠⠐⠈⠀⠀⠀⠀⠀⠈⢀⠠⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠸⠁⠉⢸⠀⠀⢻⣁⣹⠟⠀⠀⡗⠉⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠒⠈⣠⣔⣤⠈⠐⠂⠠⠀⠏⡄⠀⠀⠀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠀⠀⠀⠰⠀⠀⠀⠀⠀⡠⠚
⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⣠⢖⣤⠀⠀⠀⠀⠀⢸⠀⠒⠁
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠙⠦⠚⠀⠀⠀⠀⠀⠘⠀⠁
⠀⠀⢀⡴⢀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠘⠀⠀⠀⣀⠤⣤⠀⠀⠀⠀⠀⠀⡀
⢀⠔⠀⠈⠐⠌⡐⠤⣀⣸⠧⠤⢤⣤⣤⡀⢀⣀⣀⢙⣭⣝⣀⡀⠀⣰⢀⣄⡤⠠⢄
⠃⣤⠀⠀⠀⠀⠈⠠⠄⠁⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃
⠀⠠⠞⢠⡄⠀⡀⠀⠀⠀⡀
⠀⠀⠐⠛⠒⠁⠀⠚⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣰⠀⢀⣀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠒⠒⠒⠒⢺⠁⠀⠀⡶⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠀⠀⠀⠀⣬⠀⠀⠀⢧⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠆⠁⠀⠉⠢⡀⠀⠀⣨⠀⠀⠀⠀⡇⠀⠀⠀⢠⠄⠀⠀⠀⢌⠀⠀⢀⡐⠉⠀⠈⠁
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠀⠀⠀⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡄⢠⠀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⣟⣿⢧⢤⡤⢀⡓⣼⠀⡇⠀⣀⠀⠈⣟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⢀⣤⣦⣄
⠀⠀⠀⢠⣤⣿⣷⣏⠉⠉⠁⠈⢡⠘⡀⠇⠸⠿⢧⣤⣯⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠦⠴⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠓
⠀⠀⠀⠈⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣟⢛⣛⣠⢻⡤⣾⣀⣠⣤⠀⢠⡀⠀⠀⢸⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠤⣄⡨⢻⣦⣀⡻⣿⣿⣿⣩⣀⠀⠀⠈⠉⠛⠿⠷⣨⡄⠀⠸⣿
⠀⠠⣀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡶⡀⠲⢧⡁⢘⣧⣁⡨⣿⣿⣋⣰⣶⣤⣀⠀⠄⠀⠀⠀⠀⠀⠀⠛⢀⡀
⠀⠀⠈⠁⠙⠣⠴⢤⣤⣀⡀⣾⠟⠀⠀⢛⢧⠈⣼⣿⣧⣿⡇⣯⣉⠻⣿⣿⣿⣷⣶⣤⣀⡴⠒⠋⠉⣠⡅
⠀⠀⠀⠀⠀⠐⢠⣶⣿⣿⡆⣿⢸⡆⠀⣼⣕⠤⣿⣿⣯⣿⣷⣿⢿⣆⠬⠿⡿⣿⣏⠛⢿⣿⣿⣿⠉⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠹⡟⢀⠀⣿⣾⣷⣿⣷⡴⡔⣖⢧⣘⢿⣇⡀⡏⣿⣿⡆⠈⣿⣿⣯⠀⣿⣷⡶
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣆⠻⠎⣸⣿⣿⣿⣿⣿⣷⣾⣮⣷⠙⡓⠆⠉⢃⡸⢿⡇⡀⣽⣿⣿⠠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠘⣷⣶⣹⣁⣼⣟⣿⣠⣭⣼⣿⣿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⠀⢻⣿⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠋⠉⠛⠉⠉⠀⠉⠻⣿⣿⣿⣿⣿⠀⠓⠘⣿⣿⣿⠉⠉⠀⠈⠻⠿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⣠⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠾⠿⠿⠛

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣸⣯⣍⣭⣭⣩⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣶⣄⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠒⠒⣦⣀⠀⢀⡖⠉⠉⠉⠙⢿⡻⣦⡀
⠀⠀⠀⠀⡴⠋⣠⠴⠒⠒⠒⠒⠶⢤⣄⠀⠀⠀⠀⠀⠀⢀⠇⠈⠉⠉⠀⠀⠀⠀⠀⠀⡹⣆⠻⡟⢾⡛⠙⠛⠓⢦⣀⡀
⠀⠀⠀⡼⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⣆⠹⡄⠹⡄⠀⠀⠀⣏⠙⠳⢦⡀
⠀⠀⢸⣷⠷⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢳⠀⠀⠀⠀⠀⢻⠀⠀⠀⠹⣦
⠀⠀⣿⠃⠀⠀⣿⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣇⠸⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠳⣄
⠀⢸⠙⢄⣀⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠱⡃⠆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⣶⠏⠀⠘⣿⠀⠀⡄⠀⠀⠀⠀⢰⠀⠀⠀⢀⠀⢹⢷⡀
⠀⡎⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⣸⠁⠠⠀⠀⢀⣎⠀⠀⠀⠀⢳⠘⠀⢻⡀
⠀⣧⠀⠀⠀⣿⣿⣿⣿⣿⡿⢿⣿⠀⠀⠀⠀⠀⡄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢰⡏⢠⣧⠤⠤⣤⠞⠉⠉⠳⣤⠤⠀⢸⠀⠃⠈⡇
⠀⢸⡀⠀⠀⠹⣿⣿⣿⣿⣧⣾⢃⣄⣄⡀⢀⡄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⣼⣡⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢸⠀⠀⠀⡇
⠀⠈⢷⡀⠀⠀⡈⠛⠿⠟⠋⠁⠉⣿⠿⢣⣾⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢸⠀⢀⢠⡇
⠀⠀⠘⣷⢄⡪⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡞⠠⠃⡾
⠀⠀⠀⠈⢷⡍⠒⠦⠤⠤⠴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠊⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠈⣠⡴⠋⢀⣤⠤⢤⣄⡀⠀⠀⠀⠀⣀⣠⡤⠚⠀⠀⢀⣠⠞
⠀⠁⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠑⠠⠀⠀⠀⠠⢀⠤⠷⠤⠒⠤⣀⣀⣤⠾⠋⢀⡔⠉⠀⡀⠀⠀⠉⠙⠛⠿⠭⠭⠀⠀⠠⣊⡵⠛⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣀⣀⣀⣀⣠⣤⡴⠶⠛⠉⠀⠀⠀⡞⠀⣠⠿⠶⠦⢤⣔⣂⣀⡠⠤⠴⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⠲⢤⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠻⠲⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠖⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠈⡆⠀⢄⠠⡀⠀⠀⠀⠀⠀⠀⣠⠖⡩⠐⢀⠠⠐⠂⡉⠄⢂⠀⠀⠀⠀⠀⠰⠘⡄
⠀⠀⠀⠀⠀⢠⠇⡀⠠⡑⠨⢑⡒⠤⢤⣀⡔⠁⠀⠔⠈⣀⠄⠂⠁⡀⠔⠁⠀⠀⠀⠀⠘⡄⠀⠘⡄
⠀⠀⠀⠀⢀⠎⠃⠃⠀⠈⠒⠠⢀⣉⠐⠂⠨⢙⠲⢤⡈⠀⢀⠔⠈⠀⠀⠀⠀⠀⠁⠀⠀⠰⡀⠀⠸⡄
⠀⠀⠀⢀⠎⠌⡘⠠⠀⠀⠀⠀⠐⠠⢈⠁⠒⠄⡉⠢⡙⢎⢀⠔⠀⡠⠀⠀⠀⠀⠀⡀⠀⠀⢃⠀⠀⢣
⠀⠀⠀⡎⡜⠠⠁⠂⠀⡄⠀⠀⠀⠐⢄⠉⠂⠀⠘⡄⠱⡈⣇⠀⠊⢀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠇⢸
⠀⠀⡸⠰⢀⠃⡈⢀⠀⠁⠀⠀⠀⠑⠄⠑⠀⠀⠀⠰⠀⠃⢹⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡸
⠀⠀⡇⠇⡄⠀⠁⢸⠀⡀⢀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢲⠃
⠀⠀⡇⠀⠃⠀⠀⠀⠆⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⢃⠟⡇⠀⠒⠀⠀⠀⠊⠀⢀⠔⢁⠀⡰⠃
⠀⠀⢇⢇⠘⠄⠀⠀⠈⠢⡑⠀⠀⠀⠀⠀⠀⠂⡶⡡⢃⠞⣰⠃⠀⠤⠀⠀⠀⠀⠀⠀⢐⡡⠚
⠀⠀⠘⢎⠢⡈⠐⠤⣀⠀⠠⠄⠀⠀⠀⠀⠀⢀⠔⡡⠋⣰⣏⠀⠀⠀⠀⠀⣀⣀⠤⠒⠉
⠀⠀⠀⠈⠳⢌⡐⠢⠄⢉⣀⠐⠂⠀⠀⠀⣀⡤⢺⣿⡴⠋⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠶⢶⣖⣒⣲⣒⣚⣻⣿⡿⢿⣏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓


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

⠦⡜⢲⠀⠀⠀⢀⡀⠀⠀⠠⡄⠈⠣⡀⠘⡄⠈⠀⠓⠊⠁⡀⠠⠀⠀⠠⡀⣀⠀⠀⣼⢗⠞⢕⠟⠁⠹⣼⠀⢁⣀⢧
⠀⠁⠀⠐⢂⣀⠀⠀⠀⠰⠆⠀⠀⡀⠈⠢⡘⠀⠀⠀⠖⢨⠅⠖⠈⠁⠠⡄⠰⠶⠸⢈⣇⡸⡘⠀⠀⠁⠀⡄⠀⠁⠠
⠀⠈⢿⣿⣿⣿⠁⠀⠀⠀⠄⠁⠀⠐⠃⠀⠈⠻⠆⠀⠁⠄⠐⠀⠀⠀⠀⠁⠀⠀⠀⠙⠖⠊⠃⠀⠀⣄⡤⢀⡁⠀⡄
⠀⠀⠀⢿⣿⠃⠀⢀⣧⠀⠀⠀⢀⠠⠀⠀⠐⣀⠀⠀⠁⠀⠀⠀⢴⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣸⢋⡉⢛⣓⣓⠿
⠀⠀⢴⡎⠁⠀⠀⠘⠋⣀⣀⣀⣀⣀⡀⠃⠀⡄⠀⠀⠀⠀⠸⢄⠀⠐⠃⠀⠤⠀⣶⠄⢬⠿⢆⡹⠧⠿⠧⣬⣩⡕⠿
⠎⠀⠀⣔⢕⣚⠶⡀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠴⠚⢡⡆⢰⡇⢠⡀⠀⠀⠀⠐⠛⠀⠀⠀⠛⢋⠈⠛
⠀⠄⠀⢋⢜⠈⡎⡆⠘⣿⣿⣿⣿⣿⡇⠨⠂⠉⣁⡂⠀⠀⢠⠄⠀⢠⡆⢀⠈⢁⠘⠃⠀⠀⠁⠀⠰⠆⠈⠱⠓⣴⡔
⢀⡂⠛⢀⢋⠸⠳⡁⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠀⠰⡇⠈⠁⢺⠇⠙⠀⣤⠀⠀⢀⠀⣠⡀⠀⠀⠀⠁
⠘⢌⣭⠫⡆⣤⠞⠘⠀⠋⠉⠉⠙⠋⣁⠀⠀⢄⡄⠀⠀⠀⠀⠀⠀⠸⠀⡄⠀⣶⠀⠛⠀⣀⠀⠀⠈⣥⠀⠀⢂⠀⠳
⠀⠀⠁⠀⠈⢀⠀⢠⣆⡃⢠⡄⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⡀⣠⢠⠄⠀⠁⠀⠀⠀⡤⠐⠀⠆⠀⠀⠀⠀⠀⠈⠐⠂
⠀⠀⠀⢰⣆⣠⣶⣿⣿⣿⣿⣿⣾⣥⠆⠀⠀⣶⠄⠀⠀⠤⠯⠑⡖⢙⡖⠀⠀⢀⠀⠀⡀⠈⠌⠠
⠀⠀⠔⠀⣹⣿⣿⣿⣷⡿⣿⣿⣿⣿⢀⡁⠀⠘⠀⠀⠐⠑⡎⣹⣉⠍⠅⢠⣦⠀⠀⠀⠀⠀⠀⠰⠂⢱⠄⡀
⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⠃⠀⠀⠀⠀⠀⠈⠨⠁⠘⠀⢀⡀⣀⡀⠠⠆⠘⢄⠐⠀⠀⠘⠒⢉⡁⠄⡀
⡀⠀⠀⣀⠀⠈⠛⠿⠿⠿⠿⠿⠛⠀⠀⣤⡀⣤⠶⣆⠀⠀⠂⠀⢀⡮⠳⠔⠛⠺⠀⠀⠀⢀⠀⠀⣀⠀⢨⠰⠈⠁
⠀⠀⠄⠉⠀⠀⠰⠀⠀⠀⡄⠀⡆⠀⠸⡯⣚⣇⣸⢮⢞⣠⣆⠀⢀⢸⢧⢢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡀
⠀⠀⢻⣿⣿⡿⠀⢀⡀⠀⢀⠀⠀⠀⠶⠀⠛⡘⠓⠀⠈⠓⠀⠀⠈⢗⢆⡩⡆⢀⡀⠀⠁⠀⠒⠀⡀⠀⠀⠀⢠⡀⠡
⠂⠀⠀⢻⡿⠙⠛⠟⠀⠀⠸⢷⣄⠀⠀⠙⠏⠁⠀⠛⠀⠀⠀⠀⠛⠈⠈⠉⠠⣿⡿⣠⣾⡧⠀⠀⡀⠀⠀⠀⠄⠃⠐
⢰⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡀⠀⠀⢀⡀⠀⠠⠆⠀⠜⠃⠀⣠⣄⢟⣽⣿⣧⠀⣤⠈⠁⠂⠐⡀
⡿⠧⠠⠠⠸⠂⠀⠀⠀⠀⠀⠀⠀⡌⢡⡀⠀⠀⢈⠁⠠⠁⠜⠂⢤⠀⡄⢿⢿⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡂⢐⡀⠀⠀⢀⠀⠀⠦⠀⠀⠶⠂⠀⠆⠰⠈⠀⠁⠠⠁⠀⠀⠀⠀⡠⠁⠈⠿⣿⣭⡄⠀⢠⡄
⣁⡽⢼⣠⠀⠛⠃⠰⠀⠄⠀⠀⠒⠀⢈⢃⠚⠀⢠⡀⠄⠀⠠⠁⡠⢁⠠⠀⠀⡠⢉⠀⠀⠈⡃⠀⢀⠀⢀⠀⢠⠀⡳
⡼⢤⠼⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⠁⠀⡘⢃⠀⡷⠀⣀⢀⡀⠀⠈⠏⠀⠀⠐⠈⠀⡴⠾⠍⡻⢏⢩⡢⠀⣀⡍⢨
⠀⠁⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡔⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⡏⡆⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⢀⠎⡜⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⢶⡶⣄⡀⠀⠀⠀⠹⣦⠀⠀⠀⡇⠸⡆⠀⠀⠀⠀⢠⡞⢸⠀⣰⠃⡜⠀⠀⣴⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠲⢄⡀⠀⠹⣷⡀⠀⢠⠀⢻⡀⠀⠀⢠⡏⠀⡇⢰⠃⡾⠁⢀⡞⠁⣺
⣀⣀⣀⣀⠀⠀⠀⠀⠀⠘⢦⡀⠙⢦⣄⠹⣷⡄⢸⠀⠀⢳⡀⢀⡞⠀⢸⢠⠃⣼⠃⢠⠎⠀⠀⡇
⠉⠉⠉⠉⠛⠛⠷⢦⣤⣀⠀⠙⠢⡀⠈⠳⣭⣿⡄⡆⠀⠀⠱⡜⠀⠀⣸⠆⢰⡏⢠⠏⠀⠀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣤⣈⠠⠀⠈⢻⣟⣷⡄⠀⢀⠃⠀⢠⡿⠀⡏⢠⠏⠀⠀⢠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣓⢷⣄⠀⠙⡌⢳⡀⠘⠀⠀⢸⡇⢸⢠⡟⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡝⢷⡄⠘⠀⠹⡆⠀⠀⠈⠑⣼⡏⠀⠀⠀⡸⠀⠀⠀⣀⠤⠴⠒⠉⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⢦⠹⠄⠁⠀⠀⡀⠀⠸⠁⠀⠀⡰⠁⣠⠶⠋⠁⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢷⡘⡄⠀⠀⠉⢠⡄⠀⠀⣸⡵⠋⢀⣀⡤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠹⡡⠀⠀⠀⢨⠀⢀⡾⠋⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢱⢃⠀⣠⡌⢀⡟⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⡏⠻⠃⠹⠇⢹⡀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠑⢄⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠳⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣴⢇⣤⣼⣶⣄⢶⣀⡴⠖⠚⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣛⣉⣉⣀⣿⣁⣽⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⣳⣤⠄⢀⣠⣴⡾⠛⠚⠛⢶⣶⣤⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢕⢶⣿⡿⠟⢻⠿⠒⠣⠀⠀⠀⢉⠙⢿⣤⡄⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⠋⠀⠃⠁⠀⠀⣀⣄⡀⠀⠀⠽⣿⠿⣾⣷⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡿⠖⠀⠀⠀⠔⠉⠀⠀⠈⠁⠀⠀⠈⠀⠈⢻⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⡿⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣫⣷⡁⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠉⠛⠦⣌⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢫⡻⢿⣿⣷⡷⣄⠙⠚⠳⢤⣤⠒⠅⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠛⠚⠛⠛⠋⠙⠿⠿⠿⠿⠛⠓⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠒⠒⠊⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⠀⠀⠀⢀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⢷⣦⡀⣿⣦⣤⡾⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡋⠀⠀⠀⠀⠛⣿⠿⠋⢡⣴⠾⣯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠳⣦⡏⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣄⡀⠀⠀⠀⠙⠦⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣶⣤⣤⣀⣈⠉⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠋⠹⢟⣿⣿⡿⠛⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠏⡀⠀⠀⠟⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣴⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⣤⣤⡄⠐⠀⠈⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠈⠛⠻⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠛⠉⠠⡠⢂⣶⣶⣦⡀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣥⣄⠈⠀⠀⠀⢸⣿⣿⣿⣇⣴⣶⣦⡀⠀⠘⢄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⣿⣿⣿⣧⠀⠀⠀⠀⠻⣿⣿⠏⢻⣿⣿⣿⡄⠀⠈⡍⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠁⢸⣿⣿⡿⢫⡀⠀⠠⠂⠀⢀⡀⠀⠀⢹⠿⠿⠃⠀⠀⠠⢀⠀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠉⠉⠀⣾⣿⠀⠀⢀⣾⣿⣿⣦⠀⠘⠀⠀⢠⣾⣿⣿⣯⡀⢠
⠀⠀⠀⠀⠀⠀⠰⠋⠉⠁⠀⣀⣤⣤⣀⠠⣾⣿⣿⠀⠀⠈⣿⣿⣿⣿⠁⠀⣦⠀⠈⢿⣿⣿⣿⣷⢨
⠀⠀⠀⠀⠄⠰⠀⠹⢄⠀⣾⣿⣿⣿⣿⠆⣿⠿⠃⠀⡴⠀⠹⢿⡿⠟⠀⠈⣿⣷⣄⠀⠙⠻⠟⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣟⠿⣿⠟⠋⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠒⣄⣰⣾⣿⣦⣀⣀⡀⠀⣠⣤⣤⡒⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠙⠟⠁⠀⢼⣿⣿⣿⣷⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⡈⠀⠀⠸⣿⣿⣿⡿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠈⠙⠋⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠗⣤⣤⣄⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡸⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⣿⣿⠘⠻⠛⠁⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣿⣿⠀⠀⣠⣤⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢻⠇⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⣾⡀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠁⢄⢠⣿⣿⡦⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⣶⣾⣿⣿⡧⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠙⠿⣿⣇⡀⠰⢴⠆⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⢶⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⢇⡠⠼⠿⢿⣧⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠛⢷⣶⣶⡆⠾⡦⡶⠎⡿⠛⠁⣀⢄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠛⠇⠴⣦⡧⢌⠧⠔⠊⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡘⢳⣬⡴⠛⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣔⣴⣿⠁⡴⢫⠀⡝⠄⣹⡦⢣⠄⠀⠀⠀⠀⠀⠀⠠⢠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⢖⡆⡉⠂⠀⣀⣀⣀⡀⠈⡋⠸⣛⣢⡇⣘⢼⡀⣤⣽⠁⠬⠄⠀⠀⠐⣶⣶⡅⠄⠁⠀⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⢿⣿⣏⠲⠺⠀⢠⠏⣻⢥⣿⣿⣷⣴⣿⢦⠀⠈⢳⠊⠱⡻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠦⣑⣀⣞⣠⠟⣛⠛⠛⢛⡉⠙⣄⣱⣤⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣛⢿⣿⣆⣀⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣺⢽⣶⣽⣺⢽⣿⣿⣿⣿⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣾⣽⣿⣷⣮⣻⣿⣿⡿⠁⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣴⠊⠻⠋⠛⠉⠙⠛⠻⢿⡃⠲⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⢞⡀⠀⠀⠀⠀⠀⠀⠀⠩⡂⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣼⡿⠀⠀⠀⠀⠀⠀⠀⣾⣷⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢈⡏⠁⣠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠁⠃⣨⠙⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠶⠀⠸⠇⢀⡾⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⡱⣦⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠧⠀⠀⠋⠠⠋⠔⠁⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠃⠀⠀⠀⢠⣄⣂⣀⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⡄⠇⠀⠀⠁⠀⠀⠀⠉⠙⠛⠻⠷⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠒⠶⠲⢀⣤⠄⣠⠴⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⠟⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠰⠀⠐⠂⠘⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠙⢻⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠆⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠞⠢⠄⠀⠤⠊⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄⡀⣀⡀⠠⢀⠐⠃
⠀⠀⠀⠀⠀⠀⡬⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣷⡾⠷⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⢛⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠀⡄⠐⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⢤⡀⢤⠈⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟
⠀⣠⠀⠀⠀⠀⠈⠀⣀⠒⣀⠀⠂⡐⢂⣤⡄⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⣀⠀⡀⣀⡠⠆
⠘⢋⡙⠉⠉⠁⡀⠀⠻⠿⣃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠚⠃⠰⠂⠉⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠼⢇⡨⠍⠀⠀⢀⠄⠀⠛⠀⠀⠠⠀⠐⢢⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡃⠐⢀⠀⠀⡁⢠⡤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠛⠂⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡼⠃⠀⠚⠃⠀⠀⠀⠁⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤
⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠁⠀⠀⢠⡀⢃⡰⠐⠶⠀⠑⠲⢄⣠⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡰⠚⠎⠀⣠⣀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀
⠀⠀⠐⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠉⠀⠀⡰⠀⠀⠀⠀⠠⠤⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀
⣀⣀⣀⣀⣈⣉⣉⣁⣀⣘⣢⣄⣀⣀⣐⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡶⣻⣽⣏⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠥⡀⠀⠀⠀⠀⠀⠈⡠⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢘⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠑⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢄
⠀⠀⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢂⣤⠀⠀⢀⡘⢀⡂⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆
⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠈⠀⠈⢠⡀⢠⡄⠁⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠔
⠀⠀⠉⠂⠀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠌⠀⠀⠀⠈⠀⡀⡁⠀⠀⠀⠀⠲⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠈⢂⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⣀⠀⠄⡔⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⣆⠀⠀⠀⠀⠐⠀⠰⠄⠀⡠⠂
⠃⠀⠀⠬⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⡅⠀⠀⠀⠐⠂
⠀⣊⡀⠐⠤⣀⡀⢀⡧⢀⡀⠀⠀⠀⢃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⠀⠐⡧⠀⠀⠐⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠄⠀⠐⠠⡸⠢⡀⠀⠀⢀⠀⠠⠀⠀⠠⠒⠂⠀⠀⠀⣀⠞⣄⠀⠀⠄⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠁⠀⡏⢲⠤⢈⠈⠠⢀⠀⡀⠀⢁⡠⠴⡞⢻⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⢸⠀⠀⠀⠈⠟⠐⠏⡀⠠⠀⠀⡇⢸⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠚⠛⡧⢼⠒⠒⠒⠒⠲⠂⠒⠘⠃⠒⠘⡧⢼⡏⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠨⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣄⣀⣀⣀⣀⣀⢀⣀⡠⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠠⠀⣩⠁⠉⠀⢹⠀⠀⠰⠄⠤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣴⣦⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣴⣶⣴⣤⣤⣤⣤⣤⣤⣴⣦⣴⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠚⠛⠲⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠚⠀⠀⠀⠀⠀⠀⣀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠖⠒⠲⢦⡀⠀⠀⠀⣀⡤⠤⣄⡀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢰⡏⢠⣞⣻⢮⠇⠀⢀⡞⣡⣤⡀⠈⢳⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡟⠀⠀⠘⣇⡀⢉⣡⠏⠀⠀⢸⡰⣿⣼⡿⠀⢸⡇⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡿⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠳⣌⣉⣀⡴⠋⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡇⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣀⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣼⡇⠀⠀⣠⡄⠀⠀⠉⠛⠛⠋⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠗⣦⢤⣀⣀⣀⣠⣤⢶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⣧⡛⢦⣠⡁⠀⣇⣠⢜⡿⠁⠀⠀⠀⠀⠀⢠⣤⣤⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠧⣄⠀⠁⢸⣀⡴⠋⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⢌⣙⣛⡋⠁⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⣀⣀⣠⣤⠶⢿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣟⠛⠉⠉⠁⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡟⠻⣦⣄⣀⣀⣠⣴⠟⢙⡷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠈⠳⠶⣤⣌⣉⣉⣉⣠⡴⠛⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣼⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⡴⠇⠀⠚⠖
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡶⣄⡞
⠀⠀⠀⠘⠛⢈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣤⡿⠀⠁
⠀⠀⠀⠀⠀⠀⢧⣶⣶⡆⠀⠀⠀⣠⣬⣗⣀⠀⠀⠀⠂⠀⠤⠄⠀⠀⠀⣀⠤⠒⠉⣿⣿⠏⢹⢳⡀
⠀⠀⠀⠀⠙⠒⠒⠛⠛⠃⢀⡠⠀⣸⣄⣉⣈⣷⣟⣒⣆⢀⠄⢂⣴⢖⠋⠁⠒⢴⢺⡿⠁⢀⡗⠂⣷⠤⠴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠈⢫⣿⣀⠤⢼⡇⣿⠋⠸⡯⠿⠽⢾⣿⣿⡖⠒⠚⢁⣄⣀⣿⠃⣠⠚⠁⢸⣿⠀⢠⣼⣿⣦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣹⣾⣿⣧⣶⣤⣧⣤⣴⣁⣇⡈⣇⠀⠀⠀⠀⢸⢃⣴⡏⡀⠀⢃⣿⡄⢠⡏⠉⢹⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠂⣼⢏⣿⢻⢹⣿⠛⢿⠻⠿⡿⡿⡿⣿⣿⣷⣶⣾⣾⣼⣿⣇⣜⣺⣗⣽⣾⣗⣤⣼⡷⢲⣶⡀
⠀⠀⠀⢀⣀⣀⣀⣀⣀⠈⢷⡞⠈⠋⣿⣀⣌⣧⣖⣞⣎⣆⣆⠇⡇⡇⡏⢹⢸⠉⠏⡟⢻⢻⣿⣿⣿⣿⣿⣟⣽⣿⣏⣀⣀⡀
⠀⠀⠀⠀⠉⠛⠿⠟⠿⣿⣿⡟⠒⠒⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣻⡳⠹⡽⣞⣦⣇⣧⢸⣸⡏⠁⠀⢺⠓⠻⡖⠊⠽⠷⠭⠄⠉
⠀⠀⠀⠀⠀⠀⠉⢁⣐⣶⣶⣤⣤⣤⡤⠤⠤⠤⠄⠀⠈⠉⠉⠛⠛⠛⠿⠿⣾⣬⣜⣌⣎⢋⢷⡈⠀⠘⠀⠀⠱⠄
⠀⠀⠀⠀⢀⣤⣽⣿⣿⣿⣿⣿⣷⣶⣶⣾⣶⣶⣶⣶⣦⠷⠶⠶⠦⠤⣤⣤⣤⣤⣤⣉⣉⣛⣛⣻⣦⠇⢀⣀⣀⣠
⠀⠀⠀⠀⠐⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣖⣙⣂⣂⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢻⣷⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠘⣿⡄⠀⠀⠀⢀⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣦⡀⠀⠀⠸⣿⡄⣿⡇⠀⠀⢠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⠀⣿⡇⢸⣿⠀⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡝⢿⣦⣿⡇⢸⣿⣴⣿⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⢻⣿⡇⢸⣿⡿⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡈⣿⡇⣾⡟⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢸⣷⡿⢡⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢻⡇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢸⣿⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢸⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣏⢠⣶⣸⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⢹⣿⣿⣿⣿⣟⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⣩⠋⣠⣾⣿⣿⢹⣏⣟⣿⣿⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⢋⡵⠊⢁⡜⡱⢻⡏⢻⠈⡟⡜⣿⢯⠳⡈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢉⡴⠊⣴⣠⢏⡜⢁⠇⡿⢸⡆⡇⢹⡘⣇⢳⠙⢦⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⣠⠞⠁⣠⠞⢡⠞⠀⡎⢰⡇⠀⡇⢹⠈⣧⢹⡎⢳⡾⢻⡄⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⣰⠋⢀⡞⠁⢠⠏⠀⡸⠀⢸⠇⠀⡟⠋⡆⢹⡍⠘⡄⢧⠀⠹⡄⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢠⠃⢠⡞⠀⢀⡏⠀⢠⠇⠀⣼⠀⠘⡇⠀⡇⠸⡇⠀⢱⠘⣇⠀⢹⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⢸⠀⢸⠁⠀⢸⠀⠀⣾⠀⠀⣧⠀⠀⡇⠀⢷⠀⡇⠀⢸⠀⢸⡄⢸⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢸⠀⢹⠀⠀⢸⠀⠀⢿⠀⠀⣿⠀⠀⣧⠀⣸⠀⡇⠀⢸⠀⢸⡇⢸⠀⢀⣿⡇
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⠚⣆⠘⣇⠀⠸⡄⠀⢻⡀⠀⢹⠀⠀⡟⠀⡇⠀⡟⠀⢸⠀⣼⢡⡞⣂⣼⡿
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠘⣧⠙⢶⠲⠻⡄⢀⣣⠀⢸⡇⠀⡇⢀⡇⢸⣇⢀⠷⣴⢃⡾⢀⣾⡟⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣈⠳⣌⠓⢤⡙⢦⠉⣆⠈⣇⢰⡇⡸⢡⡟⢀⡟⡰⣣⠞⣡⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣮⣷⣤⣙⢮⣳⣌⣆⡟⣼⢠⢃⠎⣠⣿⣾⣟⣥⣾⠟⠁
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⡟⠋⠁
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠡⠾⠋⠟⠋⠉⠙⠛⠉⠛⠷⠶⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⠾⠿⠿⠿⠿⠿⠿⠿⠷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣦⣄
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⢀⣤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⡀
⠀⢀⣴⡟⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⣷⡀
⠀⣾⡏⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢹⣷
⢀⣿⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣿
⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿
⠀⣿⡇⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢀⣿
⠀⢻⣇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡿
⠀⠘⣿⠀⣸⡟⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⣧⠀⣿⠇
⠀⢀⣹⣧⣿⡇⠀⢠⣴⣶⠿⠛⠛⠛⠻⠿⢶⠆⠀⠀⠀⠀⠀⣶⠿⠟⠛⠛⠛⠿⢷⣦⣄⠀⠘⣿⣰⣿⡀
⣰⡿⠛⣿⣿⠀⠀⠈⠁⠀⣀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠈⠁⠀⠀⣿⣿⠛⢿
⣿⡇⠀⠹⠟⠀⠀⠀⠀⠈⠛⢹⣿⣿⡏⠛⠁⠀⠀⠀⠀⠀⠀⠀⠛⠹⣿⣿⡿⠙⠃⠀⠀⠀⠀⠹⠟⠀⢸
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⣿
⠈⣿⣆⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣰⡿
⠀⠘⠿⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠿⠁
⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣶⡾⠿⠿⠿⠿⢷⣶⣦⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠙⠻⢷⣶⣶⣶⣤⣶⣶⡿⠟⠉⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⣀⣀⣀⣀⣀⣀⣀⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡟⢲⣾⣾⣶⡄⣶⣶⣶⡆⣶⣶⣶⣶⢶⣦⢰⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡏⠉⠁⢸⣿⣁⣿⢷⣿⡇⣿⣧⠁⢠⣿⠇⠸⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢸⣿⢿⡇⠀⣿⠶⢿⡇⠀⣼⠇⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣠⡇⢀⣿⣼⣿⣶⣿⣿⣿⣷⣾⣯⣀⡀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠮⠉⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠤⠭⠭⠍⠁⠀⠀⠀⠀⠀⠀⠁⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡀⢀⣠⣀⡀⠀⠀⠀⠀⣠⣶⡆⢄⠲⡄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠘⣴⠀⠿⣿⠏⣆⠀⢀⡀⠙⠛⠁⢸⡇⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⣄⣀⣀⣠⠿⠚⠊⠳⣤⣤⣴⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢹⣿⠿⠛⠁⣴⣰⣆⡄⠈⠻⠿⠿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⠀⠀⠀⡠⠀⠘⠋⠙⠃⠀⠀⣤⡀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⢼⣧⡤⢤⣤⠄⠠⠤⠶⣿⠃⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⣿⣶⠀⠀⠈⠀⠀⢸⣿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢏⠻⠶⢶⢶⢶⠞⠋⠌⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠆⠉⠐⠀⠐⠐⠈⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠐⠂⠐⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠐⠲⢄⡀⠸⡇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⡠⠒⠉⠁⣨⣇⣀⣀
⠀⠀⠀⠀⡠⠒⠂⠁⠈⠓⠀⠀⢀⣉⣦⣻⣍⣉⣉⣉⡉⠀⠀⠀⠀⠤⠤⠴⠧⠤⠤⢎⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡤⠤⠤⣴⠿⠛⠂⠒⠒⠒⠒⠾⢷⡖⠒⠒⠲⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣇⡀
⠀⣠⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⠔⠈⠀⠁⠢⡀⠀⡠⠀⠀⠠⡀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲
⡜⠁⠀⠀⠀⠀⢠⠔⠒⠓⠢⣴⠃⠀⠀⠀⡎⠀⠀⠀⠀⠀⠰⡎⠀⠀⠀⠀⠘⡄⠀⠀⠀⢳⡎⠁⠀⠉⠢⡀
⢑⠄⠀⠀⠀⢠⠁⠀⠠⡀⢀⡇⠀⠀⠀⠀⡇⠀⠀⣚⣧⠀⠀⡇⣽⡇⠀⠀⠀⡇⠀⠀⠀⠈⠳⢤⡔⠀⠀⢡⠀⠀⠀⠀⠙
⣾⠀⠀⠀⠀⠘⡀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠱⡀⠀⠛⣋⡠⠴⠧⠽⢥⣀⡀⡰⠁⠀⠀⠀⠀⠀⠀⠙⢦⣀⠆
⠘⠦⣀⣠⠀⠀⢱⠞⠁⠀⠀⢀⡀⠀⠀⠤⢤⡵⠊⠉⠀⣠⠂⠀⠐⡄⠀⠉⠳⣤⠤⠄⠤⠤⢄⣀⡀⠀⠙⢦⠀⠀⡦⠐⠊
⠀⠀⠀⠸⡀⣰⠋⠀⡠⠔⠈⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠘⠠⠤⠤⠃⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠓⠦⡈⢳⡔⠃
⠀⠀⠀⠀⣼⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣳⡀
⠀⠀⠀⢰⡇⡜⠀⠀⠀⠰⠻⡅⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⣸⢠⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠃⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⢻⡍⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠒⠋⠉⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⡟⠢⢤⣀⣀⡀⠀⠀⢀⣀⣠⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⢩⠉⣉⣹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠒⠀⠉⢀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠏⣉⣁⠼⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠰⢟⠓⠶⣄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⡞⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣄⡀⠘⡆⠡⠈⠲⣄⣠⣤⣄⣀⡼⠁⠀⠀⣇⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣇⠀⠀⣤⢤⣿⠴⠂⢸⡶⣦⠀⠀⡊⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡆⠀⠉⢉⠿⠤⠤⢿⡀⠀⡴⠒⠒⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠜⠀⠀⡴⠋⠀⠀⠀⠀⠉⠙⠲⣄⡼⠁

⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦
⠀⠀⢺⣿⣿⣟⣠⣶⢿⢷⣦⣄⠀⠀⠀⠐⢚⣹⣿⢿⣿⡿⣿⣧⣭⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡠⠄⢾⡗
⠀⠀⢸⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧⠀⠀⠐⠉⢉⡀⠘⢹⡎⠛⡂⠀⠉⠀⠀⠀⠀⢿⣿⣿⣿⣿⢇⣾⣿⣿⣶⣄
⠀⠀⢸⣿⣿⣿⣿⠉⠉⠉⢹⣿⣿⠀⠀⠀⠀⠈⠳⠀⣘⣏⠀⠙⠀⠀⠀⠀⠀⠀⢸⡿⠿⢿⡷⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠲⠆⠁⠀⠁⠀⠐⠇⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠹⣿⣿⣿⣿⠟
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⢀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠤⠄⠈⢃⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠿⢉⡋⣻⣤⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⡗⢦⡤⠞⢹
⠀⠀⢸⠀⣾⣻⣟⣻⡆⢶⠤⠀⠀⠀⠀⠀⠀⠀⡽⣧⣌⣡⣿⡛⠉⠛⠃⠀⠀⠀⠀⠀⠀⢀⡁⠀⢀⣴⡆
⠀⠀⠘⠀⠹⣿⣿⡯⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠟⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢷⡸⣿⠅⠢⢤
⠀⠀⢨⡀⢰⣿⣿⣷⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣌⣛⢃⣁⣴⠆
⠀⠀⠀⡇⢸⣿⢹⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢎⢱⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣉⡉⢽⣿⠟⠁
⠀⠀⢴⣷⣼⡿⢸⣿⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⣸⡥⢤⣿⠬⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠙⢦⣤⡶⠒⡆
⠀⠀⠘⠛⢹⡇⠈⣏⠙⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⡱⣎⠀⠱⡑⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠀⠈⢢⡇⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡒⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⢀⣭⣄⣤⣄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠚⢻⠙⠸⠻⠻⠻⢻⠳
⠀⠐⣦⣤⣽⣿⣟⣿⣿⣤⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⢀⣿⣟⣿⣿⢿⠀⠀⠀⠀⠀⣀⡘⢱⠀⠉⠸⠈⠈⢱
⢜⣱⣿⣿⣿⣿⣿⣿⣿⣿⣧⣱⠄⠀⠀⠀⠉⠓⢤⣴⣿⣿⣿⣿⣿⣾⣿⡂⠀⠀⠀⠉⠀⢸⣀⡤⣿⣧⣄⣼
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠉⢻⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⣭⣗⣶⢶⡖⣶⣪⡅
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠋⢳⣿⣿⣷⠀⠀⠀⠀⠀⠀⢾⣵⣿⣽⣿⣯⣿⠇
⠀⠀⠘⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠉⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠀⠀⠳⠿⣾⣽⡮⡷⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣶⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢯⡻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⢛⣋⣭⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⡛⣛⣉⣭⡴⠾⣛⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣉⣭⣽⠿⠟⢛⣻⣿⠟⣫⡿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠙⠳⠶⣾⣟⣋⣭⣶⣟⣭⡾⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠀⠠⠀⠠⠀⠀⠀⡠⠔⠒⡒⠢⣄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡔⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠘⠀⠀⠄⠀⠐⢤⡀⠀⠀⠀⣠⠖⡀⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⢐⣄⢦⡀⠀⠀⠤⠀⠀⣨⠂⣯⢺⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⡀⡄⡔⡣⠟⠰⠊⢂⣤⣭⣤⣈⠁⢛⡒⠃⠀⠐⢄⢸⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠈⠉⠁⠀⠀⠀⢠⣿⣿⣿⣇⣹⣦⠀⢀⡀⠀⠀⠀⢸⠏⠉⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠹⡇⠀⠀⠀⢀⣀⡤⠊⠘⣿⣿⣿⣿⣿⡿⠀⠀⠈⣉⠲⠀⢸⢰⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠑⢷⡀⢀⢞⣭⣄⣀⡀⠀⠈⠛⠿⠟⠋⢀⣠⣴⣾⠋⠌⢠⠗⢫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠳⣄⢆⠈⢻⣿⣿⣿⣿⡗⠚⠛⢻⣿⣿⡿⠁⠀⢠⠯⠴⢏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠲⠜⢢⡀⠀⠑⠝⠿⠛⠷⡶⢶⣿⣿⢏⠃⠀⣠⠋⠀⠀⠀⠀⠉⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠄⣦⠝⢦⡀⠈⢈⠂⡀⠈⠀⣈⢟⠆⠀⡴⠁⠀⢀⡦⠄⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⡀⣀⠠⠄⠀⠠⠀⡈⢢⡀⠑⠤⠥⠤⠴⠋⠀⡼⠁⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠑⢄
⠀⠀⠀⠀⠀⢺⡄⠀⠀⠀⠀⠀⠀⢀⠜⡡⠊⡠⠢⠙⠦⣤⣤⣤⣤⠤⠞⠁⢤⣤⠤⠒⠃⠉⠈⠉⠉⠀⠀⠀⠈⠉⠛⠢⠧
⠀⠀⠀⠀⡴⢋⡔⠀⠀⠀⠀⠀⠀⢁⠜⠀⠀⠁⠀⠀⠀⠀⢀⢂⣐⡖⠂⠁
⠀⠀⠀⠘⢧⡎⡴⢀⡀⡀⠀⠀⠀⠀⠀⠀⡀⢐⠠⢠⠰⢰⠬⠆⠈
⠀⠀⠀⠀⠘⠤⠴⡏⠀⣼⠃⠇⣰⠀⡀⡗⠀⡾⠬⠜⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠤⠖⠙⠶⠞⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠿⠛⠛⠿⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣤⣤⣄⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣶⠾⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠷⣶⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⣶⠶⠶⠶⢤⣤⣤⣤⣶⠶⠶⣿⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠹⣟⠛⠛⠒⠶⠖⠛⠛⠛⣻⡟⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠙⠷⢦⣤⣤⣤⣤⡴⠞⠋⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣤⣄⣀⣀⣀⣀⣠⣤⣶⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠉⠛⠛⠛⠛⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠶⠾⠛⠛⠛⠷⢶⣦⣤⣤⣄⣀⣀⣀⣤⣤⣤⣶⠾⠟⠛⠛⠷⠶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣶⣤⣀
⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀
⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢹⣷⣴⣦⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣴⣶⣾
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣶⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠊⣠⠊⠀⠀⠀⠆⠀⠀⠀⠐⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡰⠁⠀⠀⠀⡘⠀⡆⠀⠀⠁⢰⠈⢆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠃⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠴⡷⠆
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡠⠥⠐⠒⠒⠒⠒⢂⠐⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⣠⠤⠴⠒⣈⢔⠴⠀⡈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠔⠉⢀⣀⣠⠴⢊⡴⡱⡡⢡⠁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⠤⠄⠀⠀⠒⢸⢠⠃⡔⠊⢉⣀⠠⠤⠊⡜⢱⠁⠘⠀⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⡘⠀⠚⠉⢀⣀⠀⢠⠃⡟⠀⡆⠀⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢋⠏⠀⡼⢀⠇⢀⠇⢀⠃⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠢⡄⠀⠂⠀⠚⠠⠊⡀⠎⣠⠎⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠠⣤⣀⡐⠀⠨⠔⢚⣡⣾⠿⠃
⠀⠀⠀⠀⠀⠀⢠⡶⠐⣆⣀⣀⡀⡆⢀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠘⢧⣠⠏⢸⢸⠇⡇⢻⡸⠇⡇⡿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣀⢰⣆⢀⡴⢋⣠⣴⣦⣤⡀⠉⠓⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣈⡙⡞⢁⣎⣿⣿⣿⣿⣿⣦⡐⣆⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠓⠂⢄⡀⠀⢸⢛⣡⣾⣿⣿⠁⣼⢿⣿⣿⣿⣿⠟⠁⠑⢜⢿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⡟⠀⣠⣦⣄⠱⣄⢸⣾⣿⣿⣿⠇⣰⣿⣾⣶⣴⣽⡇⠀⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⢀⣼⣿⡇⣼⣿⣿⣯⠀⠈⠻⠿⠏⢞⣡⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠰⣿⠁⡀
⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⡿⢋⣁⡤⠀⠀⠐⠋⠉⢉⠀⠉⠉⠛⢻⣿⠙⣆⠀⠀⠀⣸⣟⢀⣧
⠀⠀⠀⢸⣿⣿⣿⣾⣿⣭⣾⡿⠋⠀⣀⠀⠀⠈⢉⣠⣴⠶⠾⠛⠛⠛⠿⣿⡄⠀⠀⠈⠁
⠀⠀⠀⠀⢻⣿⣿⡾⢯⣿⠟⠠⣤⡶⣉⡴⣡⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⠀⣿⣿⠇⠀⢿⣴⣾⢏⡞⣩⡾⠋⠀⢤⡀⠀⣂⣀⣠⣄⡀
⠀⠀⠀⠀⠀⣿⡏⠀⠀⢸⣿⣏⡞⣼⡟⠁⢀⠀⣀⣿⢀⣬⣿⣿⣾⣿⠞⢷⣶⣦⡀
⠀⠀⠀⠀⣠⡟⠀⠀⠀⠸⣿⣾⣿⣿⣾⣿⣤⣾⣿⡟⠸⣿⣿⣿⣿⠏⠀⣶⣿⡿⠃
⠀⠀⠀⠀⣸⡇⠆⠀⠀⠀⠻⢟⣿⣯⣿⣿⣿⣿⣿⣷⠀⠘⠛⢋⣁⠀⢸⣿⠉⢁⢀
⠀⠀⠀⠀⠛⠁⠛⠁⠀⠀⠘⣿⣿⣷⣿⠿⢋⣤⣤⣄⡀⠀⢀⣨⣥⣄⠀⠉⣁⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣾⣿⣷⣄⠛⠛⠄⣿⣷⣿⡿⠽⠚⢙⣋⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠁⣿⠀⠀⢸⣿⣿⣿⣶⠂⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⡟⠀⠀⢂⠻⠟⠿⣿⣷⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢸⣿⡀⠀⠈⠃⠑⠢⠀⢀⡉⢉⣉⣥
⠀⠀⠀⠀⠀⠀⠸⠤⠚⠛⠻⠿⣿⣿⣿⣷⣬⣿⣿⣦⣄⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠤⠖⠒⠊
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠿⠛⠛⠛⢻⣿⣿⣿⣿⠉⠙⠛⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣎⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠐⣇⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠹⣷⣦⣾⣿⣿⠿⣷⡀
⠀⠀⠀⠀⠀⣠⠾⠋⠉⠉⠙⣿⣷⣿⣿⣿⡿⢡⣶⣿⡿⠿⠶⣦⣄
⠀⠀⠀⠀⠉⠰⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⣼⠟⠹⠇⠀⠀⠈⠙⡧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢄⣆⣿⣿⣿⣿⣴⠏⣀⠃⠘⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⠀⠀⢿⠺⣿⠀⠐⠐⠚⠓⢶⣄⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠘⠀⢀⠈⠠⣏⠀⠀⠀⠀⠀⢨⣿⣾⣷⣾⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠶⠖⢰⣿⡀⠀⠀⢠⣶⣾⣿⣿⢟⣵⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠲⢶⣾⣿⣿⣷⢄⣼⡛⠻⢿⣷⣋⠥⡋⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢻⣿⣿⠻⣿⡏⢻⡏⢁⡏⠃⠀⠆⣿⣿⣄⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣥⡀⢰⣧⣠⣿⣦⣾⣷⠺⡇⠲⣠⠞⠉⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠼⣿⡖⢿⡿⠙⢿⠋⠻⠇⠈⣋⠀⣸⣅⣰⣷⢼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⠈⣅⠀⣴⣄⢰⣦⣰⣷⡤⣿⣗⢾⡟⠙⠟⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣖⢿⡟⠻⣿⠙⠿⠁⠉⠀⠀⠀⠀⣦⣀⣿⡤⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⣤⣀⣶⠀⠀⠀⠀⠀⠰⣿⠟⢿⠋⠻⠁⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⢤⢖⡀⢨⡀⢰⣆⣸⣷⢾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢼⣷⠺⣿⠙⠟⠈⡃⢀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠟⠿⠉⠻⡰⢛⠀⢉⠀⣠⡀⣶⣤⣿⠾⡏⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⢠⣸⣦⡘⣦⣼⣷⢾⣷⢻⡏⠻⠃⢙⡀⣸⡘⣿⣿⣿⣿⣇
⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢉⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⠣⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⣦⣽⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⣿⣿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣧⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢨⣿⣿⡶⠹⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⣿⡇⢹⣿⣷⣤⠀⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣷⣶⠜⠃⠸⣿⣿⣿⣦⠀⠈⠁⠖⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⢋⣵⠟⠉⠉⠀⣠⣼⣿⣿⣿⣿⣷⣦⣀⠈⢀⠀⢉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠟⠋⠁⣠⠟⣁⣀⡈⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠛⢦⡛⠱⣤⡀
⠀⠀⠀⠀⠀⣰⡾⣽⠋⢀⣴⣿⢋⡤⢁⣴⣿⣿⣷⣿⣿⣿⣿⣿⣿⢻⠻⠻⢿⡎⠻⣿⣿⣶⡄⠀⠙⢢⠈⠽⣦
⠀⠀⠀⠀⣾⣿⣿⣷⣴⡿⣫⣿⣟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣯⡟⢿⠘⠃⠁⠀⠑⠀⠹⠉⠻⣧⠀⠀⠀⠠⠀⢎⣷⡀
⠀⠀⠀⣼⣿⣿⣿⣷⣿⣶⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⡇⣗⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠈⡿⣷
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⢹⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣆⢰⣆⣄⣰⣤⣼⡏⠀⠀⠀⠠⠀⠸⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠄⠘⠀⣿⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢁⣤⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠻⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢠⠑⣀⣿⣿⣿⣿⣿⣻⡟⠻⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠟⣿⣹⡿⣿⡙⠉⠀⠀⢀⡀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠉⠉⡤⠄⠈⢇⣘⣿⠅⠂⢛⠀⠀⡙⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠁⠀⢰⡘⢃⠉⠀⠀⠍⠀⡰⣀⠂⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⢀⠆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠀⠈⠉⠉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠓⠠⠄⠀⠤⠦⠖⠀⠠⠀⠠⠄⠂⠄⠀⠿⠀⠆⠀⠄⠰⠀⠂⠀⠀⠀⠰⠐

⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠲⣤⡀
⠀⠀⠀⡼⡩⠖⠛⠓⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡇⠀⢻⠈⢇⡟⢲⡀
⠀⠀⢀⣟⡄⠙⢨⡇⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡁⠀⠀⠀⡼⠀⠜⠩⢧
⠀⠀⠈⠸⡀⢳⠌⠁⠀⠀⠀⢠⣀⡙⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠀⠢⡁⠀⣀⡴⠊⣧⡀
⠀⠀⠀⠀⠉⣻⠀⠀⠀⠀⠀⠀⡉⠛⠗⠱⣄⠀⢀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠉⢀⡤⣸⠁
⠀⠀⠀⠀⠀⡇⠀⠜⡸⠆⠓⠀⠀⡀⠀⠀⠋⢉⠉⠁⠫⠤⣀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⣨⠂
⠀⠀⠀⠀⢰⡇⠀⠶⠇⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⢄⠊⠀⠀⡰⠀⠀⠀⠙
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣇⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢄⠀⠀⠀⢀⠞⠀⢀⠂⠠⠤⢀⡠⢞⠀⠀⠘⣤⠋
⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢤⡀⠋⠂⢀⡡⠔⠋⠉⠁⠀⠈⠙⣶⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁⠀⠀⠀⠀⠀⠀⠂⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣦⡀⠀⠀⢠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠆⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣀⣀⣀⠀⢠⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢀⣠⣽⣭⠽⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣀⣀⣠⠞⠷⢴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⠃⠀⠀⠀⢿⡄⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠀⠀⣀⣀⣨⣟⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣻⠾⠉⠓⠒⠚⠂⠈⠿⠒⠊⠉