Rubber duck Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠿⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⣠⣤⣤⣴⣶⡾⠟⠻⣷⡀⠀⠀⠹⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣏⠉⠁⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠘⠻⢷⣶⣤⣄⣀⣀⣀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠉⠉⠛⠛⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⡾⠿⠛⠉⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢷⣶⣶⣶⣶⣶⣶⣶⡾⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⣀⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠠⢤⡟⠄⢠⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠰⠄⠀⡈⣠⣤⠀⠀⠀⠀⠀⠞⠉⠉⠙⢦
⠀⠀⠀⢰⠙⠺⠐⠀⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⡄⢧⠀⠀⠀⠰⠋⠐⣁⠉⠀⠀⠀⢶⢮⡑⡀⠈⠀⢸⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢨⠆⠀⠀⠀⠀⣰⣁⡀⢀⡰⠆⠀⠀⠀⠀⡴⠃⠀⠈⠊⠓⠀⠀⠀⠐⠁⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠛⠒⠒⠒⠒⠒⠉⡅
⠀⠀⣰⠃⠀⠀⠀⠀⠠⣤⣀⣈⣉⢴⠆⠁⠀⠀⠰⡷⠦⣤⠴⠆⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠎⠀⠈⣠⠔⠒⠒⠒⠒⠙⠀⢰⠃
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⢀⠆⠀⠀⠀⢷⢦⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢆⠀⠀⠁⠀⠀⠀⢀⡾⢗⡴⠃
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣀⣀⡸⣣⠎⠀⠀⠀⠀⠀⠁⠉⠭⢍⣁⣀⣀⣠⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣬⡭⠥⠖⠋
⠀⠀⠀⠉⠲⠤⣄⣀⣀⣠⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠚⠋⠛⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⢀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡽⠀⠀⠀⠀⠃⠘⣤⣠⡀⠀⢀⠀⠀⠋⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠡⠖⠒⠀⡀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠊⠀⠼⠀⠀⠘⢧⡙⠆⠐⠂⠀⠀⠀⢸⠤⢶⢴⡀⠀⣠⠖⢄⡀⠀⠈⠻⢵⣮⠕⢙⠁
⠀⠀⣾⠦⣄⣀⣀⡠⠴⠒⠀⠀⠀⠲⢤⠀⠀⠀⠀⠘⠦⠽⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡇⠀⣀⣬⡭⣀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⢠⡀⠀⠀⠀⣤⠤⠒⠉⠁⠀⠀⠀⠁⠀⠀⠀⡆⠀⠘⠒⠂⠐⠒⠀⠀⠀⢀⣀⡄⡴⠈⡇⠪⠥⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠱⣄⠀⠀⠻⣀⣀⣀⣀⡴⠁⢀⡀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠐⠋⠉⠉⠉⣠⡾⠃⠀⠈⢀⠁⣀⠀⢀⣠⠎⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠈⠑⠲⠤⠤⠬⠭⠤⠶⠒⠋⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣈⣒⠶⠾⠟⠋⠀⠀⠀⠀⠀⠑⠺⠭⣉⣀⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠈⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⢀⡀⠀⠀⣀⣴⡇⠀⠀⣾⣧⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠐⣏⠉⠛⠋⠉⠈⡀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠙⣆
⠀⠀⠀⠘⢷⣄⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠈⠛⠶⠶⠼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠈⠙⠳⠶⠤⣤⣤⣤⣤⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠉⠙⠛⠛⠛⠶⠶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠁⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠋⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⡴⠚⠋⠉⠛⢦⡈⠙⢦⡖⠛⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⣸⠥⢤⠤⠤⢤⠤⣷⠀⠀⠹⣦⠤⠤⠤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢻⠒⠚⠛⠛⠋⢛⡟⠀⠀⠀⢹⠛⠛⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣤⠴⠚⠀⢀⡤⠒⠚⠓⢦⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠙⠲⠦⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠴⢤⣀⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣠⣤⢭⣍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠤⠤⠤⢤⣄⣀⣀⡀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⣀⡤⠖⠚⠉⠉⠀⠀⠀⠀⠀⡠⠤⢄⡀⠀⠀⠈⠉⠉⠛⠲⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠸⢦⡀
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⡔⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⠇⠀⢠⡿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⠉⠙⠛⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠦⠤⠤⠤⠶⠖⠛⠉
⠀⠀⠀⠀⠀⢠⡞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⡾⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⣸⠃⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣤⣤⣤⣤⡤⠴⠞⢉⣠⡤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⣷⡲⠦⢤⣤⣤⣤⣤⠤⠶⠚⠋⠁⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠘⣟⡛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸⡟⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠶⠶⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠲⠶⠶⠖⠒⠒⠛⠛⠋⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠛⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢾⠗⠀⢸⠷⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠉⠉⠉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⠀⠀⢀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⡼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠉⠉⠉⠁
⠀⠀⠀⠀⡼⠁⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢰⠇⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⣸⠀⠀⠀⠀⠀⣈⠙⠒⠶⠦⠤⠴⠖⠚⣉⡤⠖⠦⣄⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣭⡟⠒⠒⠒⠒⠛⠉⠁⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢸⡀⠀⠀⠀⠀⢹⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠶⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⠟⠛⠻⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢣⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠻⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡼⠀⠀⠀⠀⠀⠀⠀⢸⣇⣾⣏⣳⡇⠀⠀⣿⠇
⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠓⠒⠒⠒⠲⠦⢤⣀⠀⠙⠿⠿⠟⠁⠀⢠⡿
⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡇⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠈⠻⢷⣤⣄⣀⣀⡤⠤⠖⢀⣀⡴⠞⠁⠀⠀⠀⠀⢠⡿⠁⠀⣀⣀⣤⠶⠶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⡿⠁⣠⡾⠋⠉⠁⠀⠀⠸⠷⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣥⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠓⣦
⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠉⣆⠀⠀⠈⣷
⠀⠀⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠋⠀⠀⠀⢀⡞⠀⠀⠀⣿⡄⠀⠀⠸
⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡇
⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⡿⠀⠀⢰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢠⡿
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠤⠤⠤⠖⠋⠀⢀⣴⡟⠁
⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋
⠀⠀⠈⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣶⡶⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠁⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠠⣾⣡⡀⠀⠀⣤⣤⡀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⢼⣛⢦⣽⠿⣷⡄⠻⠿⠃⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⢿⣟⠉⠉⠁⠀⢀⣠⣿⣿⣄⣀⣀⣤⣤⡤⠖⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠷⣶⣶⣶⡿⠿⠻⠿⠛⠛⠻⢶⣄⠀⠀⠀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⢠⣶⣒⣀⣀⣠⣿⡿⣿⠢⣤⠾⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠐⢦⣉⠉⠉⠉⢉⣁⣴⡿⠁⠀⣰⣿⠃⠀⣠⣴⡾⠟⠛⠛⠛⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠛⠛⠋⠁⠀⣠⣾⡿⠋⣠⣾⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠻⠷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⡿⠟⠋⠀⣰⣿⡟⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⢀⡘⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⠀⣠⣞⣩⡿⠛⣩⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠛⠿⠛⠁⣴⠿⠟⢛⣩⡴⠾⠛⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠙⠛⠛⠛⠶⢶⣾⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠉⠛⠛⠛⠓⢲⣦⣤⣤⣶⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣴⡿⠿⠟⠛⠛⠿⠿⠯⠻⢿⣿⣿⣿⣶⣶⣶⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⣀⣴⣾⣿⣵⠶⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠈⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠷⠟⢉⣿⠻⣿⣿⣷⣦⣤⣤⣤⣬⣭⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠹⣿⣧⡀⠀⠉⠉⠉⠉⠛⠛⠛⠉⠁⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠈⠛⢿⣿⣶⣶⣦⣤⣤⣤⣴⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠒⠒⠒⠒⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠫⠥⡀⠀⠀⠀⠀⣀⠤⠤⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠘⠂⠐⠒⠉⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⡩⠀⠀⠀⠀⠀⠀⢀⣶⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣀⣀⣀⣀⠀⢀⡀⠀⠀⠨⣈⣉⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⠑⠂⠠⢄⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡺⢋⠜⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣁⣐⠒⠀⠤⠄⠤⢀⣀⡱⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⣀⠀⠤⠀⠐⠈⠀⠁⠊⠔⠁⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢞⢿⣿⣿⣷⣶⣶⣾⠟⣁⠠⠄⠂⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠢⠝⣛⡛⠫⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⣀⠠⠔⢚⡄⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⡠⠔⠊⣠⢆⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠈⢀⡠⠔⢉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠁⣀⠔⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠈⠁⠀⡠⠔⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⢀⡒⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠀⠀⠀⠀⠀⠐⠒⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡟⠙⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⠖⠑⠀⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠉⠀⠟⠀⠃⠀⠀⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠋⣤⠄⡆⠀⠀⢀⠀⠓⠠⠀⠛⢱⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣉⠀⣳⠂⠃⠀⣄⠐⠁⣀⠙⢦⢻⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡆⠻⠟⡀⠛⠀⠀⠀⠉⠀⠀⢉⡶⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠴⢾⡷⠃⠀⡀⡀⠀⠃⣴⠐⠀⠁⠀⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡷⠁⠀⡤⠀⢠⡀⠁⠈⠀⠄⠁⠠⡦⣾⢧⡆⡘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡴⣧⣄⢠⣶⠄⠀⣿⢀⠀⠀⢀⠀⢤⡄⠀⠀⢾⠇⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⡈⠉⡌⠁⠀⠀⢀⠈⢞⠗⢁⣀⠀⡀⢀⡄⠀⠀⠀⠀⢆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠛⠋⠁⠀⠀⠆⠠⠶⡜⢀⡀⢰⠎⠛⠂⠉⠀⡀⠀⢀⠈⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⢸⡆⠁⠀⠀⡠⠞⠒⠘⠳⠶⠾⣀⡀⠘⠈⠀⠠⡌⠓⠢⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠀⠀⠂⢀⠊⣠⣾⠏⠶⢸⣦⠐⣮⣀⠈⣾⠆⡈⠠⠴⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠂⠠⠄⠲⣿⠰⠈⣥⣰⣀⡌⠁⠖⠉⠁⠀⠀⠻⠃⡐⢃⡄⡀⠂⡀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠡⣶⠂⢫⡶⠀⠙⣆⢀⠀⠟⡡⣴⡶⠁⣼⠇⣀⡻⠟⠱⠿⣯⣂⡐⠓⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⠆⠴⢫⡍⡑⠂⢈⡉⢈⡑⢤⣎⠴⠚⠓⣿⡐⠓⠻⠤⢴⠐⠃⢬⠍⠁⡳⠂⣠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣌⠁⠀⠀⠀⢰⣄⠀⠀⠘⣃⡘⠿⠇⠉⠀⠙⡃⣠⣦⠸⠇⠀⣦⡈⢳⠂⢐⠋⢢⣄⠈⣠⣭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠘⣷⠀⠈⡉⠁⢀⣘⣿⠇⠐⠚⡛⠃⠛⠂⠘⠤⠄⠀⠆⠛⣵⡎⢡⡾⢠⡍⣀⡀⠹⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⠀⠀⠀⣠⡌⠈⠡⠦⠀⣩⣄⠙⣠⣤⠀⠠⣦⣄⢢⣛⡤⠀⢀⣈⠀⠀⠀⢀⠈⠟⠣⠤⠄⠐⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⠇⣛⠋⠈⣿⠀⠀⠈⠉⠁⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢿⡛⠃⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠄⠈⣡⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⠞⠱⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠲⠀⠀⠀⠂⠂⠂⠂⠀⠀⠂⠀⠂⠐⠐⠐⠀⠀⠀⠀⠀⠀⠈⠀⠁⠃⠘

⢀⣤
⢸⢛⠀⠀⠀⠲⣀⠀⢀⠀⠀⠀⠀⠀⣐⠀⠀⢰⠆⠤⢦⣤⣤⣄⣀⣀
⢸⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⢀⣀⡆⠸⠃⠀⢀⠙⠚⡻⢯⡙⠲⠨⣧⠈⣠⡂⠀⠀⠀⠀⠀⢠⡆
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⣿⡇⠸⠇⠀⠀⡤⢤⠁⢸⡟⠙⠙⠿⣷⠊⠁⠀⠀⠀⠀⠀⣼⡁⢀⡇
⢸⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠁⡹⡇⢰⠀⠀⢀⠀⠀⠀⢿⢿⠀⠒⢰⡘⢷⠀⠀⠀⠀⣶⣷⡿⠀⣩⡇
⢸⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⣠⣤⣿⡇⢸⡇⠀⠷⠄⠀⠀⣾⣿⠷⡄⠠⣉⣻⣆⣤⡰⣆⣿⣿⠁⠀⣿⣇
⢸⣿⣦⣄⠀⠀⠀⠀⢨⡄⠀⠀⠈⠙⢛⣾⡇⢸⡁⠀⢐⣶⡄⠀⢹⣿⠠⠆⠀⠉⢹⣿⣿⠷⣿⡿⡟⡇⠀⢼⣤
⢸⣿⡿⣿⠨⠀⠄⠀⣼⢇⠀⠀⠀⠀⢠⢿⡧⢸⠆⢠⠤⠹⠇⠀⣾⣿⠃⣤⡀⠀⢸⣯⣟⡿⣛⠀⠷⠁⣴⣿⣿
⢸⣿⠀⢈⡁⠀⠀⢠⣃⡘⡀⡀⠀⠀⠀⠈⣇⢸⣷⣟⣾⣶⢀⠀⢿⣿⣶⣿⣿⣿⣿⣏⣿⡇⠉⠀⢐⣻⣿⣽⣿
⢸⣿⣴⣬⣿⡀⠀⠾⠟⠽⢷⡇⠀⠀⠀⠀⣿⢸⡿⣿⣿⡁⠀⣲⢾⣯⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⡽⣿⣿⣿⣿
⢸⣿⣿⣿⡃⢀⣦⣀⣀⣀⣼⡇⡄⣤⣴⡴⡿⢸⡇⠐⣇⣟⠀⠉⣿⡏⠙⠛⠋⠛⢹⡟⠙⣸⠟⠛⠿⡿⣿⣿⣿
⢸⣿⡿⣛⣿⣿⣇⠀⠄⠀⢻⡇⠰⣮⠔⣯⡇⢸⡇⢀⣿⣿⠦⠀⢿⠓⠂⠀⠀⠀⢸⡆⠀⢸⠄⠀⠀⠀⠼⢿⣿
⢸⣿⣿⣥⣾⣿⠛⠀⠀⠈⢻⣿⠷⣴⣾⣿⡿⢸⣿⣗⣯⣉⣁⡤⠯⠥⠤⠤⠤⠄⠘⠗⠒⠒⠒⠛⠓⠀⠈⠉⠁
⠘⠿⠿⠿⠿⠛⠂⠒⠒⠒⠛⠛⠋⠉⠉⠉⠁⠀⠁

⠀⠀⠀⠀⢀⣤⣶⣆⣤⠶⠂⠀⠐⠂⡒⠀⢀⣠⠤⠆⠤⠤
⠀⠀⠀⠀⢸⣿⣿⣿⣦⣤⣪⡷⢾⢿⣥⣿⣦⠹⣿⣦⢀⠀⠃⠀⠰⢿⡀⡀⠄⠀⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣤⣤⣤⣀⣠
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣻⠿⣻⣧⣶⣿⣿⣿⣿⣷⠒⠀⠌⡉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠉⢹⣋⣙
⠀⠀⠀⠀⢸⣿⣿⣟⣣⡿⢿⣿⡏⠉⠐⠘⡟⠛⠟⠁⠉⠻⠷⡶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣿⣿⣻⢟⡿⣶⡆⣀⣉⡉⠉⠋⠠⠀⠀⠊⠒⡖⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣿⣿⣿⣯⣼⣦⣽⣿⣿⣢⠌⠀⢀⣄⠀⠀⢘⡁⢀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⢸⣿⣿⠀⢝⣯⣶⣬⣿⡿⣿⣶⣶⣿⣿⣿⣷⣌⣟⠻⢿⡀⠀⢠⣄⣄⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡀⢀⡅⠀⠻⠤⠨
⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⣿⣄⡸⠿⣿⠟⢻⡌⠛⠁⡈⢘⢿⠏⣄⠀⠈⠉⠩⠭⠉⠁⠀⠀⠀⠀⠀⠀⢀⡷⣆⢈⡉
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣻⣟⣽⣷⡤⠀⣎⣁⡀⣀⣄⠀⠀⠈⠊⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⣛⡘⡇
⠀⠀⠀⠀⢸⣿⣿⡛⣿⣿⣿⣿⣿⣼⠉⡡⣇⡤⢁⠀⠛⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢰⣠⣿⣿⣿⡆⢘⠿⠟⠾⢿⣧⣶⣶⣰⣖
⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⠟⢷⠓⡶⠏⠀⠃⢠⠄⠀⠀⠂⢀⣶⣦⣽⣷⣶⣶⣤⣤⣽⣿⣉⣀⠀⠁⣿⣀⠀⢴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠻⣿⡿⠿⠽⢶⣄⠈⠤⠀⣀⣶⣤⣯⣿⣦⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣦⣬⣿⣾⣼⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣠⣿⣿⣴⡖⠀⣠⠁⢀⣠⡉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡛⣿⣿⡛⠛⠛⢻⡟⠛⢛⣛⣛
⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣯⣄⣷⣷⣶⣿⣿⣿⣿⣿⣶⣾⣿⡿⠿⡿⢿⣟⣿⠟⠛⠻⣿⢿⣽⡿⠛⠛⠛⠛⠛⠛⢹⡟⢉⣉⣉⢉
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣉⣉⣙⣛⡛⣋⣀⣀⣀⣉⡉⢁⣀⢀⡇⠀⠉⠁⠀⠀⠀⣾⠠⣿⢣⡀⠀⠀⠐⠂⠀⢸⡧⠽⠛⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡯⠿⠋⠉⣻⡟⢻⢿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣠⡶⣦⣤⣤⣼⣧⣤⣤⣤⣤
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⣡⣴⡶⠟⠋⣉⣵⡦⠻⠿⠿⠛⠛⠉⠙⠻⢻⣿⣿⡿⣟⡛⢻⠛⣻⣿⣾⡛⣟⠛⠛⣿⣿⣿⣷⣿⣿⣭⣿
⢀⣀⣀⣀⣸⣿⣿⠿⢿⣿⣯⣥⣤⣴⣾⣿⣿⣿⣿⣯⣆⣀⣀⣠⣤⣴⣿⣿⣋⡙⠫⣝⣳⣿⡻⢿⠿⠁⢧⠈⠹⠿⠿⠏⠛⠛⡻⠿⠟
⠀⠉⠻⢿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠏⠒⠉⠓⠀⠁⠉⠉⠫⠖⡿⠿⠾⠟⠋⠈⠀⠐⠋⠉⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠈⠻⢿⣿⠋⠉⠛⠛⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠶⠓⠒⠒⠊⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠃⡀⡄⠀⠀⠀⠀⠀⠈⠓⠒⢦⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠀⠈⠀⢄⠃⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠁⢠⠡⠙⢆⠀⠀⠘⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠗⠀⠀⠀⠀⠹⡖⠼⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠀⢀⠀⠓⠒⡤⠂⠈⠀⠀⠹⢶⠼⡦⡴⠀⠀⠠⢂⠆
⠀⠀⠀⠀⠀⠀⠀⢦⠀⢰⡀⣠⢀⠀⢤⠀⣠⠀⠁⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡰⣂⣽⢦⡀⠀⣡⣶⡄⡀⢀⣠⣌⠀⠉⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠁⣃⣠⡈⠉⠁⠀⡐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡁⠀⠂⠂⣀⠙⠋⠙⣀⠈⠛⠋⠀⠠⠃⢦⡦⠤⡀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⡀⡁⠀⢀⢀⡄⠈⢀⡄⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢠⠂⠀⠈⢻⣷⣠⠀⠀⢀⡀⠀⠀⡄⠐⡌⢢⡃
⠀⠀⠀⠀⠀⠀⠘⠀⢀⢤⣳⠥⠬⠦⠥⠴⣫⢖⡄⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⢄⡣⡧⠬⠴⠌⣴⡥⣶⡀⠀⡸⠁⠀⠈
⠀⠀⠀⡀⠀⠀⠀⢀⠮⣗⠯⠭⠭⠭⠭⠽⠉⡷⢖⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠫⡳⢻⠭⠭⢽⣿⣿⡟⢔⡻⠄⠈
⠀⠀⠀⠀⠆⠀⠀⠶⡽⠃⣏⣉⣉⣋⣉⣙⠛⢭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠳⠝⠀⣬⣭⣭⣭⣭⣭⣇⠈⢪⠽
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⠉⢉⣩⢉⣤⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⠀⠓⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠤⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠄⠀⡄⠂⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠰⠟⠀⠨⠅⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠰⠠⠳⠜⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠅⠽⠈⠁⠐

⠀⠀⠀⡾⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁
⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠼⢿⣦
⠀⠀⠀⠉⠳⢤⣆⣰⠐⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣠⡄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠃⠓⠀⣸⠏
⠀⠀⠀⠀⠀⢀⠽⠋⡱⠛⣻⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⠿⣿⠛⠈
⠀⠀⠀⠀⠀⠀⠤⠀⢰⣲⣶⡦⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⡭⣞⠿⠁⠀⣇
⠀⠀⠀⠀⠀⠀⡠⠻⢋⡵⣿⢦⡀⠀⠀⠀⠠⠈⠰⠀⡠⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⢠⣿⢋⡞⢡⢤⡀⠀⠀⠀⠀⢰⠚⠳⡄
⠀⠀⠀⠀⠀⢺⣏⠙⣁⠤⠋⣲⣇⠀⠀⠀⠀⠈⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡼⡆⠀⣀⡇⠀⠀⠀⠀⢈⡀⣠⠁
⠀⠀⠀⠐⡻⢓⢊⡏⡉⠎⠛⠏⠁⠀⠀⠀⠘⡜⠁⠀⢢⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠶⢯⢯⣰⡿⣦⣶⣲⡥⠚⡇
⠀⠀⢀⣰⠧⠱⠪⢆⡀⠄⣍⢶⡅⠀⠀⠀⠀⠇⠀⠒⠀⠀⠀⠐⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⢰⢮⣋⢦⡏⢐⢢⣧⡐⠁
⠀⠀⡊⠃⢚⡐⠳⠈⢱⠤⠣⢭⣁⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⠈⠁⠈⠀⢄⠠⠀⠒⠀⠀⠈⠙⠤⠜⠛⡹⣝⡫⣿⢟⡷
⠀⠀⢫⠫⠇⡀⣋⠘⢈⠃⠇⡘⢁⠀⠀⠀⠀⠘⡀⠘⡖⠀⠄⡱⠀⠐⠋⠁⠀⢀⡄⠀⠀⠀⢀⣠⣄⡀⠘⠋⣿⣻⣛⣿⡆
⠀⠀⡞⠒⢂⠉⡅⢤⢈⠧⢸⠀⠃⠁⠀⠀⠀⠀⠑⢀⠓⢦⢐⡀⣢⠀⠀⠀⠉⠉⠀⠀⠁⠀⡟⠀⠀⣷⠖⠲⠟⣹⡳⠋⠁⡀
⠀⠐⠙⡧⠬⠂⡑⣄⡼⠐⠞⢀⠁⠀⠀⠀⠀⠀⠀⠠⠀⢈⠁⠉⢂⠀⠀⠀⠁⠁⢈⠂⠀⠀⢳⣀⠀⠘⠦⡤⢔⡵⣟⠁⡰
⠀⠀⠀⠇⢠⣠⣁⠎⡙⠃⡇⠃⠀⠀⠀⠀⠀⠀⠀⢁⠠⠈⠠⢄⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠚⠋⠀⠀⠉
⠀⠀⠀⠀⠈⠀⠁⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⠀⡀⢀⠀⠀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀
⣿⣿⢛⠿⢛⠛⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡖⠂⠈⢉⡀⣾⣤⣶⣿⡶⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣴⣿⠯⠍⠩⢤⢐⣮⡟⣉⣛⣿⣿⣿⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢀⡠⢿⣦⣌⠉⢻⡿⣦⠙⠣⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡀⡀⠀⢀⡠⠞⠉⣴⠏⠙⣿⣧⠈⠉⠉⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⡜⣁⠔⠋⣀⡴⠞⠁⠀⠀⢹⣿⡆⢧⠀⢠⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠗⠚⠉⠁⠀⠀⠀⠀⠀⢈⣿⡇⡘⠀⠀⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⣟⠿⣦⠀⠀⢀⣤⣤⣄⣀⣸⠻⣧⣷⡄⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡏⠀⠉⠁⠀⣸⡟⠙⠿⢓⠄⢀⣿⣿⣿⡆⢿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢿⣿⣻⡟⡇⢸⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡙⣧⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⣸⣿⣽⣷⠣⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣴⡘⢷⡈⢆⠒⠍⠍⢀⠀⠀⠀⠒⠉⠁⠘⣼⣿⠀⣸⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣆⠻⡎⠆⠀⠣⠔⠊⠉⠀⠀⠀⣠⡼⣿⢏⡇⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣭⡻⠆⠘⠀⠀⠀⠀⠀⢀⣶⣿⣵⢞⣁⣾⣷⡏⣿⢼⡟⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⣭⡛⠃⠀⠀⠸⣿⠀⠀⠸⣿⣿⡿⢫⡾⣿⣿⢣⠿⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣦⡍⠀⠀⢀⡀⠀⠀⠀⠀⠀⢻⣷⣾⢟⣼⣿⠃⠈⠸⢸⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣠⡶⠞⠉⠀⠀⠀⠀⠀⠀⢸⣯⣥⡾⠛⠁⠀⠀⢀⢸⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⡄⠀⠀⡀⠀⠀⠀⠀⣸⣿⣏⠁⠀⠀⡰⡠⡘⣸⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣘⠿⣷⣶⣶⣶⣿⣿⣧⢀⣷⢇⣇⡇⣷⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠉⠙⠛⠛⢸⡿⠸⣿⡇⠹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⣡⡇⠀⠀⠇⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠹⣷⣖⣀
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⠀⠀⠀⠀⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⣄
⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠈⠓⠤⠤⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⡄⠀⠩⠶⣄⠀⢺⣄⣀⠸
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠤⠒⠈⠉⠉⠉⠉⢳⣄⠀⠀⠀⠀⢀⡤⠤⠤⢀⡸⣆⠀⠀⠀⡈⠀⢀⡈
⠀⠀⠀⠀⢰⡀⠀⠇⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠈⡁⠀⠱⠀⠀⠙⡌⠛⠒⠉⠀⠦⠀⠁⠈
⠀⠀⠀⠀⠀⠉⠀⠀⢰⠃⢠⢆⡔⠚⠛⡶⠀⠀⠀⠀⠀⠀⢠⣆⠈⣂⣀⣀⡀⠀⢹⡄⠀⠈⠉⠑⢦⠠⡐⠆
⠀⠀⠀⢀⠀⠀⠀⠀⢾⡀⠘⠿⠷⠶⠶⢤⣄⢠⡾⠿⢦⣄⡏⢹⡆⢹⣉⡙⣿⠀⠈⡇⠀⠀⢀⠠⢤⠀⠀⣀⠁
⠀⠀⢬⣉⡇⡄⠀⠀⠈⡇⠀⠀⣠⠲⡆⠀⢹⣿⠀⠀⢀⣿⠗⠒⣷⢸⡟⠛⢁⡔⠲⠗⠦⡄⠀⠀⠤⠣⣌⠁
⠀⠀⠀⠘⢋⠙⠐⠂⠀⢹⡀⠀⠹⢦⣄⡠⠜⠉⠛⠒⠉⠸⠆⠀⠿⠸⠃⠀⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠻⠄⠀⠀⠀⢈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠘⠭⠁⠀⠀⠀⠀⠀⠐⠩⠗⢦
⠀⠀⠀⡆⠀⠀⠀⠀⡄⠀⠀⠉⠓⠲⠦⣤⣤⣤⣤⣤⢤⠤⢴⣖⣒⣺⡿⡏⠳⠤⠀⣠⠀⡀⠀⠀⠀⠐⠆⠀⠰
⠀⠀⠀⠳⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣉⠉⢉⡛⠉⠉⠉⠁⠀⠀⠙⠦⠀⠀⠻⠤⢧⡀
⠀⠀⠀⠀⠈⠑⠒⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⠒⠊⠑⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈



⣿⣿⠿⢿⢻⠿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠲⡄
⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⢪⠐⠇⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠛⣠⠞
⠀⠀⠀⠀⠀⠀⡞⠓⢤⡜⠀⠀⠀⠸⡆⠀⠈⡇⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢸⠃⠀⠘⠦⣄⡀⠀⠀⢳⠀⠀⢻⠀⡏⠀⠈⠉⠉⠙⠛⠲⣤⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠙⡆⠀⠈⣇⠀⠸⡆⢹⠀⣰⢲⠶⠤⣤⠟⠁⣀⡴⠋
⠀⠀⠀⠀⠀⡇⠀⠀⡠⠐⠢⠄⢸⣀⡠⠞⠉⠙⢧⡏⢠⡏⢛⡶⠞⢁⡴⠮⡉⠳⡀
⠀⠀⠀⠀⠀⢹⠀⠀⡇⠀⠀⠈⠄⢫⠀⢠⠖⠉⠁⠀⠈⠳⣏⡠⠴⠿⠓⠒⠛⠉⠉⠱⡄
⠀⠀⠀⠀⠀⠈⢧⠀⢇⠀⠀⠀⢸⠈⢆⡜⡔⠉⠉⠒⠢⠀⢸⠀⣀⠤⠴⠒⠒⠒⢄⠀⣧⡤⠖⠲⢤⣀
⠀⠀⠀⠀⠀⢀⣸⡄⠈⢦⠀⠀⣸⠀⠘⣤⠁⠀⠀⠀⢠⠀⢸⠀⠇⠀⠀⠀⠀⠀⠈⡄⢏⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠰⣅⠀⡱⠀⠀⠉⠉⠁⠀⠀⠈⡀⠀⠀⠀⠸⢀⡏⡀⢀⠀⠀⠀⠀⠀⠀⠇⢸⠀⡠⠖⢆⠐⡆⢳
⠀⠀⠀⠀⠀⢹⠂⠉⡑⠒⠒⠒⠒⠒⠠⠤⠼⠤⠤⠤⠧⠼⠽⠧⠼⠤⠤⠤⠤⠤⠤⠼⠤⠯⠔⢀⡸⠀⠇⠈⣇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠉⠉⠙⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⢸⠀⠸⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣧⠀⢸⢦⣠⠇
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠶⢻⡀⡸⠀⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡈⢧⣁⣀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠶⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣤⠶⠶⠞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢀⣀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠉⠉⠉⠛⠳⣦⣄⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⢀⡿⠃⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⣳⣶⢞⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⢀⣿⣷⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⢸⡇⠙⠶⠶⠖⠋⠀⠀⠀⠀⠀⠀⢠⣾⠛⠛⢿⡆⠀⣰⡾⠛⠛⢿⣦⡀⠀⢠⣿⣿⡄⠀⠀⣾⡟⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣼⠇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠻⢷⣶⣤⣄⠀⣿⡃⠀⠀⠀⣿⡇⢀⣾⠇⠸⣷⡀⠀⣿⣧⣤⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⣶⡀⠀⣹⡿⠀⢿⣧⣀⢀⣠⣿⠃⣼⡿⠛⠛⢿⣧⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠉⠛⠛⠋⠁⠀⠛⠀⠀⠀⠀⠛⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⢹⣧⣤⣄⡀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⢀⣾⠁⠀⠈⠙⣦⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣿⠃
⠀⢸⡻⠀⠀⠀⠀⣿⣀⡀⠀⠙⠻⢶⣦⣤⣤⣤⣤⡤⠤⠶⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠀⠀⠈⠻⡏
⠀⠈⠛⠶⢶⠶⠚⠉⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⠀⠀⢠⠏
⠀⠀⠀⢀⡆⣾⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⠶⠞⠉
⠀⠀⠀⠸⣧⡻⠀⠀⠀⠀⢀⣼⣥⣤⣤⣤⣤⣤⠶⠶⠞⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠳⠶⠶⣤⣤⣤⣤⣤⣤⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠙⠿⢶⣦⣴⠶⠛⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⡀⠀⡁⠀⠀⡀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠈⠀⠙⠨⠀⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⣤⣄⡄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⠍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣰⢂⡇⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠛⢇⠈⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣈⣆⠀⢠⠀⠁⠀⠠⡆⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡑⢀⣫⡽⢿⠶⠂⠀⠀⢀⡇⠐⠂⠚⠃⠂⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣛⡥⣀⡀⠈⠀⣤⠀⠀⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⢧⡀⠀⠙⠒⢶⠊⠐⠄⡀⠀⠂⠠⠤⣀⠀⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢠⠀⠀⠙⢦⡄⠀⠀⠑⢦⠀⠁⠀⠀⠀⠀⠸⠇⠀⠀⠀⢠⠋⠀⠀⠴⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠆⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠉⡄⠀⢀⠂⣀⡀⠀⠁⠀⠀⢀⡣⡌⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⠔⠁⠀⠀⠀⠐⢒⠐⠀⠈⣄⠀⠀⠀⠀⠈⠀⠃⣴⠀⡀⢀⠀⠓⠒⠀⠀⢀⠀⠁⠀⢲⣀
⠀⠀⠀⠀⠀⠀⠄⠀⣀⠄⠀⠐⠀⢀⠀⠄⠀⠀⠀⠀⢹⡂⠀⠁⠀⠀⠀⠇⠈⠛⡇⠈⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⣚⣚⢉⣠⡜⠂⠀⠀⠀⠀⢁⠀⠀⠀⠄⠀⠁⠀⠀⠘⠑⢀⣘⡃⣄⣠⠀⠁⠈⠉⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⣉⡡⡬⠐⠙⡀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⢈⡿⢄
⠀⠀⠀⠀⠀⠀⠈⡄⠐⡗⠉⡉⠉⠀⣁⡤⠔⠒⠒⠒⠛⢣⡀⠀⠀⠀⠐⡖⣦⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡤⠚⠉⠀⠀⠀⣤⠀⠀⠀⠉⠉⢀⠤⠀⠘⠃⠿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠁⠀⠀⠀⢀⠄⢰⠁⠋⢳⡀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠐⢶⠆⠛⣶⠑⠂⠀⠀⠀⢘⠁
⠀⠀⠀⠀⠀⠋⡀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠇⠐⠑⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰⠓⠾⡏⠐⠂⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠠⡀⠒⠹⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠢⡜⠁⠀⠀⠀⠀⠱⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠠⠀⢄⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠠⠠⠦⠤⠶⢄⣀⠀⠀⠀⠀⠀⡠⢰⠈⠓⠀⠀⠀⠀⠀⠀⠀⠈⠀⠗⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⡀⢈⣉⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣛⠿⠿⠿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠙⠉⠉⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣤⣶⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠁⠀⠈⠉⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⢮⡻⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⣌⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢤⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣤⠴⠖⠛⠉⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢤⣄⣀⠀⠙⠷⣍⣉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣀⣀⣙⣻⣿⣷⣶⣦⣭⣿⣶⣤⣤⣀⣀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠉⠁⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣽⣿⣿⣿⣿⣿⣿⡇
⠠⣌⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⢾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⠦⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠡⠶⠶⢂⣤⠄⢀⠀⠀⢀⣠⠀⣰⠾⠋⠐⡡⢋⣽⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣍⣛⠾⢿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣦⣤⣼⡿⠖⠋⠁⠀⠀⠈⡠⢞⡿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣶⣦⣭⣙⡛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠔⡡⢞⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⢖⣩⢞⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣋⢴⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣷⢄⡀⠀⠀⠀⢀⡴⠶⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⢦⣤⠴⡞⠁⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⡇⠂⠉⠀⠈⠧⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣠⠀⠀⠠⠤⠤⠤⠀⠀⠉⠉⢈⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠀⠀⠀⠀⠀⠠⠀⠀⢀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠒⠋⠙⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⢀⣀⣠⣤⣤⣤⠶⠶⠶⠶⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣤⣖⣚⠋⠉⣉⣈⣀⣠⣤⣤⣤⣤⣤⠾⠧⠤⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠾⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣬⣤⣷⣶⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⢀⡀⠤⢐⣀⣤⡴⠶⠛⢻⡿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣊⡭⠶⢟⣋⠁⠀⠀⠀⠀⠀⢡⡿⠟⠋⠁
⠀⠀⠀⠀⠀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⡤⣒⡭⠖⠋⠁⠲⠶⣶⠪⠉⠀⢀⣰⣄
⠀⠀⣠⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠷⣪⣵⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠙⢿⣦⡀⠀⠀⢀⣀⣀⣀⡤⠔⣡⣾⣏⣴⣿⡋⢦⣉
⠀⠀⠀⠀⠉⠻⢿⣶⣦⣭⣭⣷⣾⣿⣿⢿⠟⢻⣿⢷⣤⣀⣼⠆⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠸⢦⠀⠹⣧⡀⣍⣁⣰⠀⠀⠀⠀⠀⠈⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣆⠈⠻⢶⣭⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⡉⢀⡀⠨⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⠋⠀⠀⠘⣿⣮⣧⣀⠀⠐⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡿⢹⡿⠈⣷⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢚⣫⠸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡇



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣿⣭⣭⣯⣬⣭⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠓⠦⠤⠖⠋⠙⢦⣀⠆⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣄⠈⠀⠀⠀⠀⠀⠀⠀⢀⡔⠩⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠙⠲⣦⡀⠀⠀⠀⣰⠏⠀⠀⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⣠⠞⢳⡝⠲⣤⣼⠃⠀⠀⠀⠠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠚⠒⠂⠘⠁⠀⠀⠙⠛⠛⢻⡀⠀⠀⠀⠀⠈⠛⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⢠⡀⠀⢀⡴⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡤⠤⠤⣤⣤⣼⠳⣖⠋⠀⠀⠀⢠⠞⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⠀⠀⠀⠈⢹⡟⠛⠳⣤⣀⠀⠘⣷⠶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠈⠛⣏⠁⠰⣞⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠃⠀⠀⠀⢰⣏⠀⠀⠀⠀⣀⠀⣾⣄⠀⠈⠙⠚⠦
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣤⡞⢻⣄⠀⡼⢻⣉⠛⠀⠸⣄⣠⡄
⠀⠀⠀⠀⣀⠀⣰⠏⠉⣿⢣⠾⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣰⠀⡜⣼⠛⠡⠚⣻⡗⣾⡓⠀⠀⠑
⠀⠀⠀⢀⠋⠉⠁⠀⢠⣿⡆⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⣼⡉⠀⠀⠀⡙⠀⠈⠙⢆⠀⠤⣄⣀
⠀⠀⣀⣈⣣⡀⠀⠀⣺⣿⣥⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠝⠀⠀⠙⣇⣀⢄⠁⠀⠀⠀⠀⢻⡤⠞⠁
⢀⡴⠋⠀⠉⠻⢦⡾⢻⣯⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠒⢲⡶⠚⠑⡟
⠈⠻⣄⠀⠀⢠⡟⢁⡾⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡤⠀⠀⠀⠐
⠀⣠⠟⠀⣴⢋⡴⠛⡁⢺⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡑⢾⣷⠂⢈⡁⠐
⠰⡏⠀⠀⣻⣸⡇⠈⠀⢈⡇⠀⣠⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⡀⣀⡹⡄⠈⠂⠈⠃
⠀⠙⢦⠄⠹⠏⠀⣀⣀⠀⣇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⠼⠀⠉⠻⣏⠛⢻⣄⡀
⠀⠀⠀⢀⡠⠖⠚⠁⠀⣬⠁⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⣀⠇⠀⠀⠛⠛⢧
⠀⠀⠐⢋⠀⠀⠀⠀⢀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⠀⠀⠀⢀⣻⠆⠀⠀⢀⠠⠬⣄
⠀⣟⠉⠁⠐⣀⠀⢀⣼⠀⠀⣠⣎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡁⠀⠀⠈⠉⠛⡞⢻⣶⡶⠋⠀⠀⣠⡿
⢀⡞⠁⠀⠀⠈⣳⣮⡟⠀⠘⡇⠈⠀⢀⡴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄⠀⠀⠀⣸⠋⠈⢷⣄⠀⠘⣯
⠈⠳⠳⣄⡀⠛⠙⠏⠀⠀⣠⠇⠀⢀⣜⣳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⠜⡌⠀⠀⠀⠛⢦⠄⠚⠃
⠀⠀⠐⠈⣷⡐⠠⣦⣤⣞⢙⡆⠀⠀⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢦⣤⠴⢦⣄⣼⠃
⠀⠀⠀⠀⠀⣀⡀⠀⡼⠀⠈⠑⠀⠀⠚⠉⢻⣄⠀⢀⣀⡀
⠀⠶⡤⠀⢀⡟⠈⠉⠁⠀⠀⠀⢀⡴⠒⠢⠼⠋⢱⠎⠀⠘⢧
⠀⠈⠐⠐⡺⣁⣀⡀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⣨⠁⠀⠀⠀⠉⢱⠆
⠀⠀⠀⣼⠁⠀⠀⠉⠋⠰⢾⣅⠀⠀⠈⠑⢦⣠⣿⣀⣀⣀⣠⣤⣞⣀⢀⣠⡀⣦⠢
⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⣇⡞⠀⠀⠀⠀⠰⠿⣾⡋⠉⠉⠁⠀⠈⠙⡆
⠀⢀⠀⠀⠀⡟⠀⠀⠀⠛⠻⠤⣄⠀⠀⠈⠀⠀⠸⡇⠀⠀⠀⠀⢀⣼⠁
⠀⠈⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⢳⡀⢀⣴⠟⠉⠁
⠀⠘⠀⠀⠀⠀⠀⠀⠘⠦⣤⠖⠢⢬⡷⠛⠓⢦⡴⠋⠉⠛⠁
⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠈⠊⢂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠄⠀⠠⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠢
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢡⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⣂⠁⡀⠀⠀⠑⠀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠐⢄⣀⡠⠂⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠄⠐⠂⢄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠐⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠀⠀⢀⢬⠀⠀⠀⠀⠀⠣⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⡁⠀⠀⠤⠀⠊⠀⡠⠀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠈⠀⠀⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠠⠔⠈⠀⠀⠀⠄⠌⡐⢄⠀⠠⡀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⢀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠔⠀⠀⠀⠀⠀⠀⠂⠀⠄⠠⠡⡀⠈⠐⠒⠈
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠐⠎⠀⠀⠑⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠤⠂⠀⠀⠀⢄⠀⠀⠀⠀⠀⠠⠀⠀⢉⠤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠋⠀⢀⠂⠀⠀⠀⠈⠀⠀⠱⠐⡀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⢢⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠊⠀⠐⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⢀⠤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢶⣄⢀⣤⢦⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠱⠆⡀⠈⣧⡀⠀⢀⣄⢀⠀⠄⠻⡿⣶⣇⠇⢭⠇⣳⡾⡽⠃⢠⠀⠀⣇⠀⠈⢠⡎⠀⣀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠄⣠⠘⠓⠨⡀⠀⠈⢃⠤⣍⠁⠀⣴⣷⡿⠀⣲⠠⣷⣶⡄⠀⢉⡅⠠⠈⠀⠀⣈⠓⠚⠀⣄⠄⠈⠉
⠀⠀⠀⠀⠀⡔⠓⠚⠂⠈⠉⠀⢹⣶⣾⣇⠀⠀⠘⠲⠀⢇⠀⣀⣋⠂⠶⡵⠆⢊⣓⣀⢀⡇⠰⠚⠁⠀⠀⣽⣶⣿⠁⡈⠉⠁⠘⠒⠓⣂
⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⢀⡄⠉⣀⣁⣉⣁⣙⣲⣦⣁⣈⣳⣉⣽⣷⣾⣹⣷⣼⣏⣹⣋⣀⣡⣴⣖⣁⣈⣁⣉⡈⡉⣀⡀⠀⠀⠀⠀⠀⠠
⡤⠠⠀⠀⠀⠠⠀⠄⠀⠀⠀⠤⠀⠠⠀⠀⠄⠠⠀⠀⠀⠄⠀⠠⠠⠠⠄⠀⠀⠀⠠⠠⠄⠄⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⠄⠤⠀⠀⠀⠄⠀⠄⠀⠠
⡇
⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣈⡄⢢⢠⢢⢇⢀⢠⢎⢀⡀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠈⠀⠀⠈⠀⠀⠏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⡟⣠⢀⡄⣠⢀⡄⣠⢀⠄⡄⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠘⠐⠃⠘⣰⠃⠊⡼⠁⠊⠘⢱⠇
⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠰⠃⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⡇
⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠒⠖⠒⠒⠒⠲⠲⢒⣶⠶⠒⠒⣶⠒⣶⢶⣶⢲⡶⣶⠖⢲⡒⠒⠲⣶⡒⠶⠲⠒⠒⠒⠂⠖⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠑⡂⣀⣀⠀⠀⠀⠊⣹⣾⣷⠟⠈⠁⢀⢠⠁⡾⠙⢛⠻⠁⣝⢯⣄⠹⡛⠋⠹⡄⢁⢀⠈⠁⠙⣷⣿⣮⠑⠀⠄⠀⣀⣀⡀⡆
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠘⠜⠦⢉⣁⡘⠃⠀⠀⠙⠠⠞⠁⠀⣬⣿⡦⢰⡢⢠⣾⣥⠆⠈⠑⠠⠜⠀⠀⠀⠻⢀⡉⠱⠏⠈⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣀⠰⠊⠀⡿⠀⠀⠘⡆⠀⠹⠁⣠⢦⣾⠃⡼⠄⡈⣷⣧⣦⡀⠉⠂⠐⠟⠀⠀⠺⡆⠉⠂⣠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠈⠁⠐⠀⠀⠁⠀⠀⠀⠛⠉⠸⣵⠗⠰⢷⡅⠈⠙⠃⠀⠀⠀⠀⠀⠐⠀⠉⠐⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠋⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⠙⢦⠙⠦⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⠞⣠⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠿⣄⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⡤⠖⠒⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⠏⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡖⠲⣤⠀⢠⣞⠛⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠋⠀⠀⢸⡟⠛⢧⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣴⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣤⡆⠀⠰⣿⣿⣿⣿⣿⣶⣶⣶⣶⠂⠠⣤⣤⣄⣀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄⡀⣥⠄⢿⣃⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡈⢿⣿⣉⣁⣸⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⡋⠃⣥⡀⢈⣭⠀⠀⡀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣟⠳⣤⠜⠀⠸⣿⠀⠠⣤⣴⣿⣿⡟⠋⠁⠈⣨⣿⣿⡿⣾⣿⡆⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣵⠀⠀⠀⠀⢀⣿⠀⢠⠜⠋⠉⠻⡟⠀⣠⣴⣿⠛⣿⡟⠻⢿⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⣸⣟⡳⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣶⣟⠻⠟⠁⠀⠁⢀⡈⣏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠻⠇⠀⠀⣤⠀⠀⠀⢨⣿⠀⠀⣀⣀⣠⣤⣿⡟⠿⣏⡗⠀⣀⠀⠀⠛⢹⡀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣤⣤⡀⠀⠀⠀⢺⣿⠀⠈⣉⣽⣿⠋⠉⣡⣾⣿⣧⡅⠉⡀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠋⠛⠋⠂⠀⠀⣿⣿⠇⠈⣋⣘⡉⠠⠀⠙⣻⠉⠈⠀⠚⠅⢤⠇⠀⣷⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣥⡜⠂⠀⠀⠀⠀⠀⣿⠘
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠐⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣴⣾⣿⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣉⣹⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠉⠉⢩⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠘⠿⠿⢏⡙⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢻⣿⢄⠀⢠⣾⣿⣿⣿⠋⢁⣤⣤⣾⣿⣿⣿⣿⣷⣶⣔⠄⣤⣦⣶⣿⣮⡑⠦⣼⢇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣌⣿⡇⠃⠘⠛⠛⠛⠛⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣀⣿⣼
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⣿⣷⣄⣤⡄⢠⣶⣶⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣹
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡸⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣀⣵
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠊⠱⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⡿⠟⡋⠁⠀⢀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠟⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⠶⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡆⣿⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡋⠋⠛⠛⠛⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣿⣿⣿⣿⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⣿⣿⣿⡿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⡉
⠀⠀⠀⠀⠀⢠⣾⣷⣆⠸⡆⠀⡿⠁⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡏⣿⣿⡏⠀⣷⠀⡇⣀⣤⣤⣶⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣿⣶⣧⡘⣧⡀
⠀⠀⠀⠀⠀⠸⣻⣿⡟⢀⡇⠀⣿⢻⣿⣟⣻⣄⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡘⣷⡄⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡶
⠀⠀⠀⠀⠀⠀⠈⠁⣠⠞⠀⠀⣿⣸⣗⣷⣹⣽⣿⣿⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡘⣿⣆⠀⠀⠀⠀⠀⠀⠀⣇⢠⡇
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡘⣿⡄⠀⠀⠀⠀⠀⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠙⠛⠛⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡘⣷⠀⠀⠀⠀⠀⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⡘⡇⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠓⠒⠶⠶⠶⢤⣤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠤⠤⠶⠶⠶⠟⠛⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⣾⠂⠀⠈⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⡏⠀⠀⠀⢃⠀⠀⠁
⠀⠀⢀⣀⣤⣤⣤⣤⣴⣾⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣴⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠚⢃⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠸⠟⠿⠟⣿⣿⠿⠿⠿⠟⠻⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⣀⣾⠁⠀⠀⠀⠈⠛⠿⢿⠿⠿⣿⠟⠛⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠠⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⢁⠀⠛⠋⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢀⡀⠀⠀⠀⠀⣿⣟⠀⠐
⠀⠀⠀⠀⠠⣤⣤⣼⣶⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡄⠀⠀⠀⠀⠀⣒⣶⠶⠶⢶⣖⠀⢠⠀⠀⢸⡏⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠴⠶⠾⣿⣿⣿⣿⣶⣤⣤⣶⣴⣧⣤⠾⠯⠍⠀⠀⠀⠀⠀⠀⠀⡪⠭⠭⠭⠭⠵⠀⠈⢧⡀⠀⠻⠶⠂⠀⠀⠹⣿⣄⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⣽⣯⡭⠭⠳⠶⠞⠓⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠋⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⣠⣼⠽⠿⠦⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠛⡍⠀⢠⠭⠁⠀⢐⢢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠬⠶⡄⠀⠈⠚⠁⣀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣞⠢⣄⣀⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠉⠉⠈⠊⠓⢾⣆⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣖⠽⠃⠀⠀⠀⠀⠀⢀⡀⠁⠀⠙⣆⠀⠀⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡞⠁⠀⡤⣶⠻⠀⠀⡔⠁⠈⡆⢠⠀⠸⡄⠀⠀⠀⠸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠳⡀⠀⠳⡇⢀⣀⣼⠔⠂⠀⡇⡾⡁⠀⠽⠦⠄⣀⣠⣤⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠹⣄⣀⠷⢚⡄⠸⢄⣀⡴⠁⡿⣜⡂⠤⢀⡀⢡⠊⣄⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢴⡋⣁⣤⢴⠟⠲⣄⣀⣀⣀⢧⠀⠉⠙⠲⢆⡽⠤⢿⣤⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡝⠋⠉⢸⠀⢰⡏⢁⡄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠇⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠘⠁⠀⠀⠈⠳⡈⢣⣾⠆⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢰⢹
⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣶⣄⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠏⡄
⠀⠀⠀⠀⠀⣠⠞⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢳⠈⡓⠀⠀⠀⣸⠀⠀⠀⠀⠀⢓⢡
⠀⠀⠀⢀⡞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣇⠀⢀⡌⢠⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠘⣸⠀⢀⠀⣄
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣾⢀⡏⣀⣀⣀⣐⣳⠐⢯⠍⣰⠖⠀⢱⠀⠈⢟⣉⡓⢲⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣛⣳⣜⠯⣝⣓⠠⠤⠬⣓⡦⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⠴⠋⠀⠀⢈⡉⣉⠍⢥⡀⠀⠀⠀⡤⠶⠋⠛⠒⢲⡄
⠀⠀⠀⠀⠈⠙⠲⢤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⡅⠙⠄⠀⣀⠅⠀⠀⠘⠓⠒⠢⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠐⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠚⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⢿⣦⡀⠀⠀⠀⠀⣠⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⣿⡇⠀⠀⠀⠀⠻⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⠾⠟⣀⣤⣶⣶⣶⣶⣤⣶⠿⠛⠻⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⣶⠾⠿⠿⢷⣦⣼⡟⠉⠀⠀⠀⠈⠙⠁⠀⠀⠀⠈⠛⠋⠉⠛⣿⣆⠀⠀⣠⣴⣶⣄
⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⣿⠁⠈⣿⡇
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⣦⡛⠿⠿⠟⠁
⢠⣶⣶⡄⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣤⣄⡀
⠹⢷⡾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣦
⠀⢠⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣰⣿⣧⣀⣤⣦⣤⣤⣴⣆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣼⣷⣤⣤⣤⣶⣦⣤⣀⣀⣠⣴⣶⣤⣠⣴⣾⣯
⣿⡇⠈⠙⢹⣿⠉⠉⠉⠻⣷⣶⣶⡿⠛⢷⣶⣦⣴⣶⠿⠋⠀⠉⠉⠉⠀⠈⠉⠛⠛⠋⠁⢿⡏⠉⠉⢈⣿
⠹⣷⣄⣀⣸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣧⣀⣠⣾⠟
⠀⠈⠙⣿⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡟⣿⡏⠁
⠀⠀⠀⣿⡜⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣿⠃
⠀⠀⠀⣿⡇⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢰⣿
⠀⠀⠀⢹⣧⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢸⣿
⠀⠀⠀⢸⣿⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⣸⡏
⠀⠀⠀⠸⣿⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠿⣶⣾⣿⣅⣀⠀⠀⠀⣿⣧⣴⣶⠿⠿⠿⣶⣤⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣤⣤⣴⣿⣧⣶⠶⢶⣶⣤⣍⣙⠛⠻⠿⠟⢛⣉⣥⣴⡶⠿⢶⣦⣝⣿⣆
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⠋⣀⣀⠀⠀⠈⠉⠛⢻⣿⠿⠿⠟⠛⠉⣴⡶⠷⣶⡌⠻⣿⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢼⣟⣻⡇⠀⠀⠀⠰⣿⣿⡷⠀⠀⠀⠸⣿⡀⠀⣸⡿⠀⠈⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠈⠙⠋⠁⠀⠀⠀⠀⠉⠉⠁⠀⣀⣀⠀⠙⠻⠿⠛⢁⣀⠀⢸
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⡀⠀⢸⣿⣻⡷⠀⠀⠀⢰⣿⣿⡇⢸
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣠⣤⣄⠀⢺⣏⣀⣿⡇⠀⠀⠉⠋⢁⣤⣤⣄⠀⠉⠉⠀⢸
⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣏⣉⣿⡇⠈⠙⠛⠋⠀⠀⠀⠀⠀⢾⣏⢈⣿⡇⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣌⠛⠛⠋⠀⣀⣠⣤⣶⣶⠶⣶⣶⣤⣈⠛⠛⠋⠀⢀⣴⡿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣿⣿⣿⣿⣛⣋⣉⣀⣀⣀⣀⣀⣉⣛⣿⣿⣶⣿⣛⣋⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠄⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠈⠈⠁⠐⣄⠀⢰⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠂⢀⠀⣀⣀⣴⣀⠀⠀⠀⠈⠁⠀⠉⢴⣿⣇⣸⡿⠲⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣵⣿⣿⣿⣿⣷⡶⠒⠂⠉⢷⡀⠀⠉⠛⠉⠛⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠠⡄⢙⠤⠤⠀⢼⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⡅⠈⠂
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢀⡀⠀⢃⡄⢠⣿⡄⠀⠀⠈⣻⢿⣿⣿⡿⠟⠓⠦⣄⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⣤⣀⠔⠁⠀⠈⠈⢇⠀⠀⢀⢘⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡄⠀⠀⠀⡌⢦⣴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠉⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣽⣯⣮⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⣠⠞⠛⠛⠛⠛⠷⠶⣦⣄⣀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣄⠀⠀⠈⢻⣆
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⠀⢿⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣼⡇⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢻⡆⠀⢀⣠⣶⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⣿⡿
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⢠⣆⠀⠀⣾⣿⡆⠀⠘⠁⠀⠀⠀⢀⣤⡶⠚⠂⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠈⠷⠀⠈⢿⣷⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠂⠐⢿⣷⣾⣿⣿⣿⣿⣀⣤⣶⡷⠀⠀⢠⣄⡀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣬⣭⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠉⠻⣦⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣶⣤⣤⣤⣤⡤⢀⣠⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠉⣡⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣨⡆⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠻⣷⡀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⢻⣷⡀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢋⣿⡏⠀⠀⢻⣷⡄⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⣸⣿⠁⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⢠⣿⡏⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣀⣴⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⣸⣿⣧⣴⣶⣶⣿⠆⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠛⠉⠀⠀⠀⢠⣶⣦⠀⠀⠀⣰⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⢸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⠀⠐⠿⢛⠻⠓⣛⣠⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⡿⠃⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠁⠒⠆⢀⡀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣠⣾⣿⣿⣿⣿⣿⣷⣆⠀⠀⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⢫⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠉⣡⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠲⠂⠀⠛⣿⠟⠋⠍⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣄⢀⡿⣷⣄⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠓⠈⠻⡿⠀⣠⡌⠉⣼⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠀⢀⠀⡈⠁⠀⠀⠀⠞⠻⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⣛⠛⠹⢄⠀⠀⠀⠆⢹⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠊⠁⣤⣶⣶⣦⣤⣴⣟⠀⢹⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⠀⠀⣛⢷⡄⠀⠀⠀⠐⢶⡜⠷
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠿⣿⠀⠀⢀⠙⣿⣿⣿⣯⢽⠿⠁⠛⠟⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣦⡀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡯⠁⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⣿⡿⠁⠈⠓⠆⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠉⠉⠻⣀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠖⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠈⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⡏⣿⠟⢹⣿⣿⡿⡻⡝⣯⢻⣝⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⢶⣿⠉⣽⣿⠋⣙⡟⠛⣿⡛⠛⠛⠛⢛⡟⠻⡛⢛⠛⣿⡟⠛⣛⠙⢫⣭⠍⢹⣷⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠸⠁⠀⢹⠃⠀⢫⠁⠀⡭⠀⠀⡁⠀⠸⡏⠀⢱⡆⠀⢻⠇⠀⢻⠁⠀⢻⠀⠀⠻⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿
⡇
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠐
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀
⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠰⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠘⠅⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡁
⡇⠀⡀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⡇⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠐⡀⠀⠀⠀⠀⠀⠀⠀⠆
⡇⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⣧⣀⣀⣄⣀⣄⣀⣀⣤⣀⣠⣄⣤⣤⣄⣠⣤⣄⣤⣄⣀⣀⣀⣀⣀⣠⣄⣤⣀⣀⣄⣤⣀⣀⣀⣠⣄⣄⣀⣀⣀⣀⣀

⠀⠀⠀⠀⡠⠒⠉⠉⠉⠈⠐⠒⢲⡀
⠀⠀⠀⡜⠀⠀⠘⠃⠀⠀⠀⠀⠀⠱⡄
⠀⠀⢠⠟⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⢀⢏⠻⢀⡭⠓⠒⢺⠖⠄⠀⠀⠀⠀⡅
⡀⠻⣣⠔⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⢠⠃
⠑⠉⠁⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠤⠤⠠⠤⣄⡀
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢄⡀
⠀⠀⠀⢸⠀⠀⢀⢂⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣀
⠀⠀⠀⠘⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠲⠤⠤⣤⣴
⠀⠀⠀⢸⠰⡁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢤⡠⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⣴⡿⠉
⠀⠀⠀⠈⢢⡁⠄⡀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡉⠛⠀⢠⠒⡀⠀⠀⢁⠞⠉
⠀⠀⠀⠀⠀⠙⢦⡀⡀⠈⠠⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠩⠐⠁⠀⢁⡀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠲⠶⣦⣌⡀⠈⠂⠁⠀⠀⡄⠀⠄⠀⠀⢀⡀⠀⠸⢃⠀⠀⢀⡥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠺⠶⢴⣆⣀⣀⣀⠄⠀⠀⣽⣦⡬⠼⠗⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⢿⠛⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢼⣿⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣽⣿⢿⣽⣾⣿⣿⣿⡗
⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠒⠚⢻⣿⣾⢛⢛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠞⣀⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣧⢀⣀⢠⣤⣤⣴⣶⣶⣤⣄⠀⢨⡿⢯⡽⠉⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢠⣀⢰⣤⣄⣎⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣷⣿⣿⣿⡻⠆⠀⠀⠢
⠀⠀⠀⠀⠀⠀⢀⣯⣿⣿⣧⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧⣖⢳⣶⢢⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡯⣯⠻⢿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣆⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠠⣹⣿⣿⣿⣿⣿⣿⠿⣿⣇⠜⣐⣲⣦⠐⢿⣿⣿⣿⣿⣿⣶⢟⣻⣿⣿⣿⣿⣿⣿⣤⡙⢿⡧
⠀⠀⠀⠀⠀⠐⠹⣿⣿⣿⣿⣿⣧⣾⣭⣿⣟⠝⠇⢱⠇⣿⠿⣆⡿⢫⣶⠉⢛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⠗
⠀⠀⠀⠀⠀⠘⣴⢿⣿⣿⣿⣿⣿⣟⣿⡋⣩⣶⣿⡄⡘⠻⡄⣹⣗⠉⠁⠀⠀⠉⠙⠛⣿⣿⣏⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠘⢿⡟⢿⣿⣿⣿⣶⣡⣾⣷⣦⡀⣽⡶⠂⠠⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⡏⠛⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠘⣉⢙⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⣈⣻⣿⣃
⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠈⣿⣿⣿⣿⡿⣿⣏⣛⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⣿⣿⣽⣿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢽⠉⠦⠙⠻⢿⡿⢉⣝⡟⢿⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⢫⡄⡹⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⠀⠉⠒⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⡑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⢦⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣸⠃⠀⠀⠀⠀⠈⠳⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡉⠛⢛⡥⠶⣤⣴⠞⠻⣗⠶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢠⣇⣰⣯⣄⣰⠏⢿⠀⠀⢹⡄⣇⠀⢹⡆⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⡾⠋⠀⢠⡟⠙⣦⠸⡇⠀⠀⣧⢻⠀⠀⣿⠉⡟⠙⣇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡾⠀⢀⣾⡇⠀⠀⠘⣧⠀⠘⣇⢿⠀⠀⢸⣾⡇⠀⠸⡇⣷⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢸⣭⡷⠀⡾⠁⠀⣼⡿⠷⠀⠀⠀⢹⡄⠀⢹⡜⡇⠀⠀⢿⣷⠀⠀⢻⣿⡀⠈⣇
⢀⣀⣀⡀⠀⠀⠀⠀⠀⡼⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⢷⠀⠀⢷⣿⠀⠀⠘⣿⡄⠀⠘⣿⡇⠀⢿⡀⠀⣠⠤⢤
⢸⡅⠈⠙⠳⣄⠀⠀⡼⠁⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠘⣿⡇⠀⠀⠹⡇⠀⠀⠘⠇⠀⢸⣷⣸⡇
⠀⠙⣦⠀⠀⠈⢧⣸⠃⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠓⠶⠞
⠀⠀⠸⡇⠀⠀⠘⠿⢦⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⣷⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣴⠞⠛⠉⠉⠙⠓⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠻⠦⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡀⢠⡇⠀⠀⠀⠀⠀⠀⠙⠶⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣼⠛⠻⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⣄⣀⣀⣤⢶⡀⠀⠀⠈⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣧⣤
⣸⠃⠀⠀⠀⠀⠉⠛⠶⢤⣀⡀⣰⠟⠁⠀⠀⠀⠉⠁⠀⠀⠙⠳⠶⠶⠚⠉⠻⡄⢀⣀⣤⠴⠖⠛⠉⠁⠀⠘⣇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠙⠋⠀⠀⠈⠉⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡖⠁⠀⢠⢤⠀⠀⠀⠀⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢚⠀⠀⡜⢀⣤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣭⠃⠀⠙⠋⢁⠀⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠸⣦⣀⡼⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡞⠀⠀⠀⡀⠀⡉⠑⠒⡄⢀⠀⠀⠀⡰⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢒⠏⠻⠤⠕⠀⢾⣿⣿⣿⢖⠝⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣴⣶⣤⣴⣿⣿⣿⡷⡍⣴⡁⠀⠀⠀⣀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣯⢟⠞⠱⡄⠈⠓⣖⠉⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡻⢿⣿⣿⣿⡿⡣⢯⠶⠚⠙⠦⣤⡼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣹⠊⠀⢀⢞⠞⡔⠛⠿⣬⡉⠁⠀⡶⠒⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠰⣇⢀⡰⡻⠃⠀⢸⠀⢀⣨⣿⡆⣸⠃
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠙⢦⠌⡐⣟⠀⠀⢠⠃⠀⢻⣿⡿⠙⠋
⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸⣢⠾⠁⠀⡰⠁⠠⣾⣿⡟
⠀⠀⠀⠀⢠⡏⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠜⠁⠀⢀⠜⠀⠀⣤⣾⣿
⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡡⠋⠀⠀⢠⣾⣷⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠘⣇⠀⢸⠀⠀⠀⠀⢀⠤⠒⠤⣾⠝⠀⠀⠀⠐⠛⠛⢻⠉⠙⣯⠉⠙⣇
⠀⠀⠀⠀⠀⠈⠂⠀⠂⠀⢀⠔⠡⢄⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠈⡀⠀⡇⢇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⠄⠀⠑⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢣⢀⡇⠘⡄⢸
⠀⠀⠀⠀⠀⠠⠴⠚⣉⠄⢀⠅⠀⡠⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⢣⢸
⠀⠀⠀⠀⠀⠀⠠⠭⠒⠭⡀⢄⣪⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠈⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠤⠤⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠐⢤⡀⠀⠐⣶⣄⣠⠴⠚⢉⡉⠉⠉⠉⠉⠐⠒⠂⠤⢄⣀⡀⠀⣈⣉⠲⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⢀⣼⠃⠠⡀⠀⠀⠀⠉⠑⠂⠤⣀⠀⠀⢀⣠⡤⠽⠻⠅⠀⡠⠚⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣱⢾⡉⠀⠀⠀⠀⢀⠞⠀⢄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠏⠀⣇⠀⠀⠀⠀⠀⢀⡠⠔⠚⠫⠤⣀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣸⡿⠀⠀⡇⠙⢳⠄⣠⠖⠉⣠⠴⠒⠛⠒⢄⠱⡄⠀⣀⣠⡤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⡇⠀⠀⢡⠀⢀⣴⠁⠀⣸⠅⠐⠀⠄⠀⠦⢱⠱⠊⠉⠐⢄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣀⠈⠀⠀⠘⣀⡏⣼⠀⠠⡟⠀⠠⣦⠀⠁⠺⠏⢸⠀⠀⠀⠀⠑⠢⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⠀⠀⡠⠒⠛⠀⢿⠘⠀⠱⡆⠀⠀⠀⣒⡀⠐⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣿⢠⠎⠀⠀⠀⠀⠈⢧⡀⠀⠙⣤⠄⠈⠉⢴⡶⠖⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⡠⠛⣱⠋⠷⠒⠁⢰⣾⣝⢶⠤⣄⡀⠙⠦⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠋⠁⠀⣀⠤⠔⠲⠎⢀⡼⠁⠠⠤⢴⢶⡆⠸⣒⠇⠑⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡎⢀⡴⠋⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⡎⡼⠷⠚⢩⠋⠀⠀⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢁⠊⠀⠀⠓⣄⠀⡴⠃⠀⠀⠀⠀⠰⢰⠃⠀⠀⢈⡶⢎⠋⠹⡙⠿⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢀⡿⢅⠀⠀⢆⠀⠀⢸⢸⠰⡖⠈⠁⠀⠈⢧⠀⠈⢮⠢⣴⢄⢱⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠎⠀⠈⠑⢄⠈⠢⡄⠈⠄⣿⠀⠀⢄⠀⠀⠀⢣⠀⠀⠑⢎⠑⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠁⠀⠀⠈⢳⡀⡀⠀⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡜⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠦⢄⣰⣀⣀⡀⠀⠀⠀⠉⠁⣀⣀⣀⣀⣤⠤⠖⠚⠃⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠽⠉⠉⠉⢙⠫⢄⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠀⠀⠀⡞⠁⢯⠁⠀⠀⣀⡴⠃⠀⠀⢣⠀⠀⠀⠸⣄⢀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠛⢦⠀⠑⢶⡏⠀⠀⠀⠀⠀⢈⣣⠀⡠⠂⢉⣸⡇⠀⠀⠀⠀⠀⠀⠈⢙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠸⣅⠀⠀⣠⡝⠒⠡⠤⠤⠔⠋⠉⠉⢉⠀⢸⣀⠔⠋⠈⠳⢤⡶⠒⣄⠘⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⠄⠀⠙⢧⠀⠈⢱⡀⠀⠀⣦⠀⠀⠀⠀⠘⠁⠀⠋⠱⣄⠀⢢⠎⢀⣠⠋⢀⡭⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠈⠓⠤⠞⠃⠀⠈⠃⠀⠀⠂⢀⠀⠀⠆⠀⠀⢹⡴⠛⣳⠏⠀⣀⣃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠷⠤⠤⠒⠒⠉⠉⠉⠉⠉⠙⠒⠒⠒⠊⠀⠀⠡⠤⠤⠤⠉⢉⡁⠼⠋

⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢣⡀⣆⠀⣴⡿⠛⢣⠴⢶⡄⢀⣠⠂⡔⠻⠳⡀⢠⡀⠀⣠⡶⢠⠛⠻⣶⡄⠀⡆⢰
⢄⠀⠀⠀⠀⠈⠋⠙⠧⣽⣆⡘⠦⣴⣾⡄⠀⣿⢯⣥⡾⢅⡀⢈⣹⣦⡹⣷⡇⠀⣼⣦⣄⠌⡀⣈⣠⠿⠋⠙⠁⠀⠀⠀⢀
⠘⠧⠰⣄⠀⠀⠀⠀⠀⠈⢻⣥⣀⠙⠟⠁⢀⠀⠀⣰⠆⠈⢩⠉⠀⢴⠀⠀⠀⠀⠻⠟⢃⣀⣿⠋⠀⠀⠀⠀⠀⢀⣴⠂⠞
⠀⠀⠀⢘⣷⡄⠠⢴⣟⡀⠀⠈⠉⢉⣜⣛⣛⣿⠟⠁⠀⠀⠿⠇⠀⠈⠻⣿⣟⠛⢳⣌⠉⠉⠁⠀⢘⣵⠤⠀⣰⣿⡁
⠀⠀⠀⠈⠁⢀⣠⠜⠻⣟⠻⢦⡔⣱⠏⣹⣋⣄⣈⡙⠭⠃⠂⠐⠭⠝⣃⣀⣍⣿⡉⣯⠱⣠⡶⢛⣿⠛⠦⣄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⠁⠯⠴⠿⠭⠭⠭⠭⠤⠤⠯⠥⠤⠭⠭⠭⠭⠽⠷⠼⠇⠽⠶⠟⠁











⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣖⢲⡖⢶⣖⣒⣒⣒⣒⠒⢲⢲⠒⠒⣒⣒⣒⣒⣶⠖⣲⢒⡶⣤⡀
⠀⠀⠀⢀⡀⠀⠙⠲⣤⣿⣵⠾⠛⢄⣻⣈⣿⣙⠉⢩⡔⢒⠄⠡⢠⣢⣌⠉⠙⣩⣏⣰⢏⠜⠷⣮⣽⣦⡔⠋⠀⢀⡀
⠀⠀⠀⢨⣿⠏⠀⠒⢟⡅⠀⢀⣀⣀⡙⢥⣭⣽⣷⣦⠀⠀⢰⡶⠀⠀⢀⣴⣿⣭⣭⡝⢁⣀⡀⠀⠈⡻⠓⠂⠙⢿⡍
⢀⡴⠡⠟⠁⠀⠀⠀⠀⠀⣠⣟⠉⢡⣴⣆⠀⡀⠀⠀⠗⠀⣈⣃⣀⠰⠏⠀⠉⠁⢀⣴⣍⠉⢹⣆⡀⠀⠀⠀⠀⠀⠙⠆⢲
⠘⠁⠀⠀⠀⠀⣄⣠⣶⢻⡝⠈⡵⠛⠿⡟⠀⢸⢿⣜⠻⣏⠁⠈⢑⡾⢓⣲⣟⠀⠘⡿⠟⠲⡈⠙⡝⢲⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢐⠃⠸⠀⠘⢿⣦⣤⠃⠶⠋⠀⠈⠃⠈⢦⣆⠌⠠⠋⠁⠘⠶⠖⢣⣠⣾⠟⠀⠹⠈⢃⠈
⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠞
⣿⣿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡔⠉⠙⡆
⠀⠀⠀⠀⠀⠠⠠⢄⠠⠀⠀⠀⡞⠀⠀⠈⠉⢑⠀⠀⠀⠒⠤⡤⠂⢱⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠢⡀⠀⠁⠀⢄⣼⡠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣨⠔⠁⠀⢰
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢐⣿⡭⠏⠉⠉⠁⡆⢈⠁⠢⡀⠀⠀⢠⠃⠀⠀⠀⠈⠢⢤
⠀⠀⠀⠀⠀⠙⠂⠐⠀⡠⠊⠀⢹⡄⠀⢠⠒⢁⠜⠀⠀⠈⠲⣤⣦⣀⣤⡤⠐⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⡛⣆⢀⡱⠘⠤⣤⢄⣤⠶⠊⢻⡁⠀⠈⠉⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠑⠚⠉⠳⣀⡰⠀⠈⢦⠀⠀⠀⡴⠋⠀⠀⠀⢸⢱⠀⠀⢀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠑⢄⠀⠑⠒⠢⡀⠀⡼⠀⢇⠀⡯⠑⡄⠈⢦⡀
⠀⠀⠀⠀⠀⡰⠋⠉⠆⠀⡠⠄⠠⡀⠀⠀⠈⢄⠙⢦⡀⠀⠙⠚⠁⠀⠀⡉⠁⠀⠘⢆⠀⢋⠢⡀
⠀⠀⠀⠁⠉⠃⠀⠀⠘⢸⠁⠀⠀⠑⠤⠄⡀⠄⠁⢀⠈⠓⢤⡀⠀⠀⠀⠑⢀⠀⠀⠈⢧⠈⢆⠘⢆
⠀⡀⢀⡱⡀⠀⠀⠀⢰⣼⠀⠀⠀⠀⠀⢀⡔⠀⠀⡀⠀⠂⠄⡉⠳⢤⡀⠀⢀⠀⠢⡀⠈⢧⠘⡀⠀⢣
⠘⡀⠀⠀⠙⠢⡀⠀⠈⡇⠀⢀⡠⠔⠊⠁⠀⠀⠀⠈⠢⡀⠀⠀⠑⠢⢝⠦⣀⢢⠀⠈⢄⠘⡆⢡⠀⠀⠃
⠀⢱⡄⠀⠀⠀⠈⠑⣶⢿⣖⠛⠀⠐⠒⠒⠤⡀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠈⠪⡳⢧⡀⠀⠡⣽⠸⠀⠀⠘
⠀⡎⠀⠀⠀⣀⡤⠖⠉⠉⠧⠀⠀⠀⠀⠀⣠⠞⠐⠢⠤⣀⡀⠀⠑⢄⡀⠀⠀⠈⠢⡑⡄⠀⢹⠀⠀⠀⡆
⠀⠉⠉⠉⡩⠊⠀⠀⠀⠀⠠⠀⠀⠀⠀⢻⠀⢄⡀⡀⠀⠀⠉⠉⠒⠲⠬⢦⣄⡀⠀⠘⢌⠂⠀⢆⠀⠰
⠀⠀⠀⠀⠀⠀⡠⠶⡄⠀⠀⠆⠀⠀⠠⠜⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⢤⡀⠑⡅⠸⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠐⠀⠀⠀⠠⠄⠈⣙⣾⣄⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠉⠉⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⢀⡀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣹⡆⠊⠤⠶⠂⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣈⠀⠛⠋⣱⠉⢶⡄⠀⢀⣠⡿⣀⡿⢛⣷⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⣌⠷⢺⣷⠀⠄⠀⠀⠀⠀⠀⠈⠻⢿⠳⠀⠀⠋⠀⣀⡤⠖⠚⠷⢶⢿⣏⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣝⢐⡛⠀⠀⡠⠔⠈⠉⠉⠒⣄⠀⠉⠀⠀⣠⠞⠁⠀⠀⠀⠀⠺⡾⠛⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣄⠀⠿⠀⠼⠁⠀⠀⠀⠀⠀⠸⡆⢠⣠⡞⣁⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠤⠭⣶⣤⠞⠀⠀⠀⠃⢀⠀⢸⡇⣼⠏⣿⣿⠏⣀⣀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠤⠤⣤⡤⠤⣀⡀⢠⠟⠛⠒⠁⠼⠋⠀⠀⠀⠀⠀⠁⠀⡾⣰⢃⣤⠾⠛⠉⠉⠉⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⡴⠋⠀⠀⠀⠀⠹⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⡼⣱⡿⠋⠀⠀⠀⠘⡀⠀⠀⠀⠀⠈⢳⠄
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢸⠁⠀⠀⠀⢠⠀⠀⠀⠈⢳⡀⠀⠀⢀⣤⠀⠀⠈⣠⣯⡾⢋⠀⡀⠀⠀⠀⠐⡤⢤⡶⣦⣀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⣄⠀⣸⣄⠀⠀⠀⠀⠀⠀⡠⠀⠀⢳⠀⠀⢸⣿⠇⣠⣾⡿⠋⠸⣿⡿⠃⠀⠀⠀⢠⣶⠾⢧⣽⠋⣹⣿⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣈⡃⠉⠾⠶⠆⠀⠀⠤⠚⠁⠀⠀⢸⠀⠀⠘⣡⣾⠿⠻⡉⠻⠁⠀⢄⠀⠀⠀⠀⢀⢻⣿⣋⣏⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣼⠁⠻⡆⠀⠀⠀⠀⢀⣄⠀⠀⡸⢀⣴⡾⠋⠁⠀⠰⢻⠤⣍⠁⠀⢡⡄⠀⠀⠀⠜⠿⠋⠙⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⠔⠂⠐⠢⡀⢸⣿⡗⣰⣷⡿⣋⣤⠴⠶⠒⠒⠾⢄⠀⠀⠀⢐⡷⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⢹⠘⢛⡿⣫⡾⠛⠉⠀⠀⢀⢀⠀⠀⠀⠘⣂⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡞⣴⢋⡀⠀⠀⠀⠀⠀⠻⠛⠿⠲⠗⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣼⢃⡀⢀⣀⢀⡀⠀⠀⠀⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠲⠦⣌⣷⡿⣦⣄⠀⠀⠀⠀⠀⣾⣿⠙⠿⠃⠛⠓⠿⣥⣖⡀⠁⠀⣀⠀⠔⠁
⠀⠀⠀⠀⣀⣛⣻⢿⠏⠋⠀⠀⠈⠻⣄⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠃⣿⣭⡄
⠀⠀⠀⠀⢼⣿⡙⠃⠀⠀⠀⠀⠀⠀⠹⡼⣷⣿⣿⠁⢸⣞⡤⠀⠀⠀⠀⠈⣿⠶
⠀⠀⠀⠀⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣿⡟⠀⠀⠁⠁⠀⠀⠀⠀⢀⡾⠷
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠒⠀⠀⠀⠀⠀⡇⢸⣿⠃⣀⣠⣤⠀⠀⠀⠀⠀⢠⡿⠆
⠀⠀⠀⠀⠀⠈⠒⠤⠀⠀⠠⠀⠀⠀⢰⠃⣿⡏⠾⣿⡿⠃⠀⠀⠀⠠⠀⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡏⣰⣿⠔⠒⠓⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠⠂⢀⡞⣰⠟⡁⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠞⣰⠋⠀⠁⠰⠊⣁⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⠜⠁⠀⠀⠀⠀⠀⠉⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡙⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⣾⣏⠈⠉⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣿⠿⢟⣛⣿⣮⣛⠻⠿⠟⣿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣤⣾⣿⡟⢡⡾⣫⡿⣿⣿⢿⡿⣿⡿⣯⣻⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⣳⣤⣽⣿⡿⢱⣿⣿⡟⣰⢏⣿⣹⣗⣿⢷⡙⢿⣮⣛⠶⣿⣤⡀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣩⣽⣿⣿⣿⠃⣼⡿⣹⡇⣿⣿⢿⠙⣿⣽⣆⣿⢦⡙⢿⣷⣮⠻⣿⣷⡶⣆⠀⠘⠻⠶⠶⠾⠟
⠀⠀⠀⠀⠀⣠⣤⠖⠛⠉⢱⠞⢻⣿⡽⢟⣡⠞⣽⣿⣿⡇⣿⣿⣿⡆⣿⣿⢿⣧⣾⠻⣆⠹⣎⢷⠘⣯⠻⣟⠛⠛⢶⡄⠀⣤⠶⠖⠛⠻⢦⣄
⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠘⠻⣯⣭⠶⣻⣿⠾⢋⡾⣡⣿⣿⣏⣼⠇⣿⣿⡘⣏⢹⣧⣹⠌⣿⢿⡆⢿⡛⠋⠀⣴⣟⣁⠀⠙⠛⠛⠛⠛⠛⠋
⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⣰⣟⣫⡾⢿⡏⣿⣿⡟⣰⡏⣿⣇⢿⡿⣿⣿⠀⣿⣌⢳⣼⣿⡆⠀⠀⠀⠉⢻⡆⠀⠰⣾⣿⠆
⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⢻⡆⠀⢾⣿⣧⠾⠷⠞⠻⣷⢿⣟⣱⢿⡇⢿⣿⣎⢻⣿⢿⡶⣷⠾⠀⠉⠋⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣀⣀⣀⣹⣾⠟⠋⠀⣿⣼⠁⣿⢻⡇⠘⠷⠛⢀⣀⣠⣤⡴⠶⠛⢉⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠈⠛⠓⠒⠛⠛⠛⣋⣉⣩⣉⠛⠦⣤⣤⣤⣬⣅⡀⢙⣋⣡⡤⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣤⠴⠟⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣿⣿⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⠑⠦⡀
⠀⠀⠀⠀⠀⠀⣠⠂⠀⠉⠳⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⣤⣤⣤⣶⣦⣤⣼⣿⣿⣟⣛⠛⣳⣤⣴
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⣠⣤⣴⣶⣾⠿⠿⠛⠛⢛⠛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣏⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠟⠋⠁⠀⠀⢀⣀⡜⠘⣋⣙⣛⡿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⣑⣀⣀⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣉⣭⣿⣿⣿⣿⣿⣧⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⣴⣿⣿⣶⣷⡟⣿⣥⣭⣉⣉⣙⣟⣿⣛⡛⠋⢢⡙⢿⣿⡿⠟⠛⠁⢸⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣻⣯⣿⣿⢰⡿⠟⠉⠉⠉⠙⠛⠛⠿⢿⣿⣿⠧⠈⠁⠀⠀⠀⣀⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢾⣿⣿⣿⣿⣿⡇⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠉⠉⠉⠛⠿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣆⠀⡀⠹⡉⠛⠿⢷⣦⣤⣤⣤⣴⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⡿⢰⡇⣾⣵⢠⢡⡄⠀⠟⠀⣿⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢻⣶⣌⣏⣾⣷⣿⣿⣾⣿⣽⣦⣧⣤⡁⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢿⣾⣿⣿⣿⣿⠿⠚⣙⠋⢈⢻⢿⣏⣉⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠙⠛⠛⢟⣿⣿⣿⣿⠿⣿⣿⡿⣿⣧⣽⣷⣦⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⡷⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣦⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠈⢻⠾⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠟⠁⣼⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣷⣾⣿⡧⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⡿⠛⠋⠉⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢹⣦⣦⣀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣟⣛⢫⡉⠂⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⣾⣿⣿⠘⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⣰⣿⣿⢿⡄⠀⠘⢷⣄⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠃⣴⣿⣿⡏⢸⣧⠀⠀⠘⣿⣆⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⡇⢸⣿⠀⠀⠀⢹⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⠇⠸⢿⠄⠀⢀⡤⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⣶⣿⠉⢹⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠀⠘⣿
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠆⠀⠀⢻⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣿⣿⠀⠀⠀⠈⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠀⠀⠀⠀⠻⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⢤⡀⢀⡠⠐⠒⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣇⠀⣠⠀⠀⠀⢠⣤⠀⠀⠀⠔⠀⠉⠁⠢
⠀⠀⠓⠃⠀⠀⢰⠁⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⢁⣀⡀⠀⠉⠉⠀⠀⢀⡀⠈⠀⠀⢘⢀⡄⠀⠀⠀⠀⡔⠁⠉
⠀⠀⠀⠀⢠⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢀⠀⠀⠀⠀⢀⡜⠢⡠⠚⣄⠀⠀⢀⡴⠥⠄⠠⠤⢄⡀
⠀⠀⠀⠀⣈⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠥⢀⠀⠀⠀⠀⠀⠃⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠒⠈⠀⠀⠀⠀⠒⠎⠀⠀⣠⣒⡒⠦⠔⠒⠢⠤⡄⠀⠈⠑⢄⠒
⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡇⠰⡞⡵⣶⡦⠀⠀⣬⣭⣁⠁⠀⠀⠀⣸
⠀⡏⢉⠉⡉⠉⡉⢉⠉⣀⣀⡄⠬⠤⠄⢀⡀⠀⠀⠁⠀⠇⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⢹⢶⠀⢏⡻⡇⠀⢸⡸⣿⠁⢃⠒⣄⡴⠃⡖
⠀⡇⠀⠀⢀⢸⡀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠛⢭⠭⠍⠀⡥⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⢸⠈⢆⠀⢰⡥⠤⣴⠉⠁⢀⡌⢴⠉⠀⠆⡗
⠀⡇⢀⠀⡰⠊⠀⠀⠤⠤⢤⣄⠀⠐⠒⣠⢄⡀⢣⡠⠀⡇⠰⠐⠒⠐⠂⢲⠒⠂⠀⠀⢸⠀⠨⠷⡦⢭⣨⡥⡴⢚⠉⡹⢋⢧⣁⣤⣷
⠉⠀⠉⠀⡇⠀⠀⠀⠉⠐⣚⣧⠀⠀⢴⣻⡄⢃⡿⢤⠀⡇⢀⠀⠀⠀⠀⢸⠀⢀⠀⠖⠁⠁⠐⡞⠀⢰⠀⠅⢀⢸⠊⠐⠁⠈⡇⡐⡒
⣀⡄⠉⠁⠳⢦⠃⡵⠇⠀⣌⢫⠇⠀⣩⡛⡣⢸⠃⠜⠀⠃⢸⠀⠀⠀⠀⢸⠀⠈⠀⠐⠂⠈⠀⠳⠄⣨⡀⢘⢇⣀⠉⠒⠒⠘⣧⠆⢇
⢀⣠⣤⠦⠄⠠⠛⢦⠀⠀⠈⡵⢦⣤⣷⠍⠀⠌⠀⣁⠀⡁⢸⠀⠀⠀⠀⢸⡔⠁⠠⠔⠀⣰⢯⢻⡄⢙⡷⢡⠞⡀⠈⠑⠒⠒⠙⡀⢸
⠈⡆⠀⡬⢤⢰⣪⠞⠓⢤⣀⠘⠂⢐⣋⣤⠞⢋⡡⠜⡇⡀⢸⠀⠀⠀⠀⢸⠐⠤⢄⣀⣄⣀⡄⠀⣠⢈⣱⡁⠀⣣⣀⣀⣀⠠⠔⡇⠸
⣀⣿⠀⠇⠲⡼⡅⠀⠀⡇⠉⢛⠫⠚⡩⠔⠊⠁⠀⠰⠁⡀⠗⠠⠦⠠⠄⣸⠄⠂⠌⠣⠸⠄⢤⠔⡹⡏⢸⠁⠀⠀⢳⠀⠀⠀⠀⣭⢉
⠤⠘⠀⡇⡘⢰⣀⡀⠠⢼⢠⠋⡝⠁⠀⠀⠀⢀⠠⠺⣒⣊⡉⠔⠓⡄⠢⠔⠢⡀⠈⡉⠤⠶⣭⠊⠀⡆⢸⠀⠀⠀⢸⡇⠀⠀⠀⢸⠄
⠒⠉⠀⢹⠁⠈⠲⠒⠛⢋⡇⠸⠀⢀⡠⠔⢊⡡⢐⠩⠐⠉⡠⠔⠊⢳⠴⠒⠛⠩⠀⠀⠀⠀⣿⠈⠀⠉⢸⠀⠀⠀⡞⠹⡄⠀⠀⢸
⠀⠶⠄⠱⢤⣀⡀⠑⠢⢄⡑⠦⣮⣅⠴⠮⡔⠊⣀⠤⠂⠁⠀⠀⠀⠸⡄⢀⣀⠀⠀⠀⡐⠀⢸⠠⠶⠄⠸⣽⣖⣱⠇⠀⢳⡁⢮⡼⢀
⠀⠀⠀⡐⠁⠀⠈⠉⢲⠚⠻⣄⣠⣏⠀⠂⠓⢎⠀⠀⢠⠊⣿⠽⠂⣄⢗⠝⠁⠀⠀⠠⠀⠀⣴⠀⢀⣄⢰⠃⠄⢇⠀⠒⣂⢇⡀⣇⠈
⠀⠀⠀⡇⠀⠀⠀⢰⡟⠈⠀⠘⢟⠋⠀⠀⠀⠈⠄⠀⠠⢡⠀⢀⡶⣼⠚⡤⠖⠤⠒⣦⣐⠊⡿⠴⣥⠤⠥⠤⠤⢬⡖⠛⠚⠀⠀⢈⡆
⠀⠀⠀⡇⠀⠀⠀⠘⢹⠀⠀⠀⠘⡣⡀⠀⠀⠀⠈⠀⠘⢯⣉⡜⠂⠁⠀⡛⢄⢠⡾⠜⣹⡦⣤⡔⠉⠉⠈⠈⠁⠉⠓⠀⠈⠉⠓⠒⣁
⠀⠤⡄⡇⠀⠀⠀⡇⠈⡆⠀⠀⠀⡗⠓⢄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⠜⠉⠛⠉⠙⢎⠀⠀⢙⢆⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⡧⠤⠤⢤⡇⠀⢸⡄⠀⠠⡇⠀⠈⢢⡀⣀⠀⠈⡄⠀⠀⠀⠀⠎⠀⡘⠀⠀⠈⡈⢣⠀⠀⠁⢻⠉⠙
⠀⠀⠀⢹⣁⣹⣉⡀⠀⠀⠹⠤⠽⡥⢄⠀⠋⣕⢄⠱⡀⠘⣄⣀⠤⢸⠀⢠⠀⠀⠀⠀⠰⠈⡆⢀⠈⠈⡆⠀⠀⠀⠀⠀⠐⠓⠶⠳
⠀⠤⠤⢾⣀⠀⠁⣈⣧⠒⢷⢄⣀⣀⠠⡧⠡⣑⣀⡤⠋⠀⠉⠀⠀⢸⠀⠘⠄⠀⠀⠀⠀⠀⢡⣀⣙⣀⣱⣄⡀
⣀⣀⣀⣀⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣐⣲⣆⣀⣀⣀⣀⣀⣘⣙⣁⣘⣒⣒⣔⣒⣒⣚⣒⣒⣚⣃⣀⣀⣈⣁⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠞⠋⠉⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⢁⠤⠴⠶⠶⠶⣄⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⢦⠀⠀⠀⠰⣧⡿⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⢰⡶⠶⠤⣤⣤⣶⠟⠁⠀⣠⣦⠀⠀⠀⠙⠲⠦⠤⢤⡀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠳⢤⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣠⠴⠃⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠛⠓⠲⠶⠤⢤⣤⣤⣤⣤⣶⣾⠉⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣸⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣠⣤⡴⠶⠞⠛⠛⠋⠉⢠⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣾⣯⣾⠟⣿⠏⠀⢹⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⣿⣠⡴⠶⠟⠛⠛⠋⠉⠉⠉⠀⠀⠿⠟⠋⣡⠄⠟⠀⠀⢸⡼⡇⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⡾⢠⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⣤⣤⣤⣀⡀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⡠⢀⡠⠔⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⣽⡟⣋⠈⠛⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠽⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⢀⡼⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣿⠀⠀⢦⡀⠈⠻⢦⣄⣀⢢⣀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠋⠁⠀⢀⣴⠟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠘⢷⣄⡀⠙⠶⣤⣄⣈⠉⠙⠙⠳⢿⣦⣄⣀⣀⡀⠀⠀⣀⣠⣤⡶⠿⠛⠉⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⡙⠻⠷⣦⣤⣌⣙⡛⠳⠶⠦⠤⠄⠈⠉⠙⠛⠛⠛⠛⠉⠁⢀⣀⣤⡴⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠁⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠌⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠛⠛⠓⢯⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠃⠀⠀⠘⠛⠋⠀⣀⣀⣤⣴⣤⣀⠀⠀⢺⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⢁⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣜⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠂⠀⠲⣿⠹⢷⣴⣶⣶⣾⣶⣶⣾⣿⣶⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⣶⣦⣤⣶⣿⣿⢶⣤⣀⣤⣤⣄⠀⠀⠀⠀⠀⡿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠷
⠀⠀⣿⣿⣿⠋⣿⣿⣼⣿⣿⣿⣿⠿⠁⠀⠀⠀⠐⡇⠀⠀⠈⠉⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠈⠁⠀⠀⠀⣿⣿⡏⠉
⠀⠀⣿⡿⣿⠀⣟⣻⣾⢿⡿⣻⡏⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⣿⡿⣿⡟⠀⠀⠀⠀⠀⣿⠏
⠀⠀⣿⡇⣿⠀⠀⠀⠀⠉⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠛⠟⣿⣿⣿⡿⣧⣾⣿⣿⣿⣶⠀⠀⢰⡇
⠀⢰⣿⠀⣳⢤⡴⠦⠶⠶⠶⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⡀
⠀⢸⣿⠀⣿⣶⢶⡄⣮⣠⠀⣿⠀⠀⠀⠀⠸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⢼⣿⣿⣿⡇
⠀⠈⢿⣄⡏⣿⠄⡇⢸⣿⡇⣾⡇⠀⢰⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠀⢸⣿⣿⣿⣷⡀
⠀⢰⡇⣿⡇⠗⠘⠃⣾⢿⠃⣿⣧⢀⣿⣯⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠈⣻⣿⡿⠃
⠀⢸⡇⣿⡇⡎⣠⢄⢀⠀⠀⣿⣿⣿⣿⣤⣿⡀⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧⠀⠀⠀⣼⣿⣿⣿⣿⣷⣴⣾⣧⠉
⠀⠠⣶⣿⡇⠟⣿⡏⠰⠷⠆⣿⣿⣿⣿⡿⢿⣻⣿⣠⣄⣶⣄⠂⠀⠀⠀⠈⠁⠈⠃⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢻⣿⡇⠇⠈⠀⠁⠀⣀⣿⣿⡿⠿⠗⠾⢻⣿⣿⣿⠏⢏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣾⢿⣷⣄⣠⣴⣶⠛⠋⠀⢀⣴⡾⢓⣵⡿⠟⢹⠏⠂⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠉⢹⣿⣿⣿⣿⠁⢹⠇
⠀⠀⠘⣻⣿⣯⢿⣿⣿⡟⠛⢛⣷⡿⣶⠉⣽⠁⠀⡼⠘⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢹⣿⡏
⠀⠀⠀⣼⣿⣿⡿⣿⡟⢷⡄⢸⡏⠀⠙⣧⡏⢐⠀⣰⣂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⢸⣿⠃
⠀⠀⠀⠽⣟⣭⡞⠋⠀⠀⢳⣸⠇⠀⠀⠈⢿⣵⣾⣿⢷⣿⠿⡐⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⣿⠁⠀⣾⡿
⠀⠠⠖⠚⠉⠀⠀⠀⢀⣠⠄⠋⠀⠀⠀⠀⠀⠙⣿⢃⡴⠿⣀⡸⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠘⠿⣿⡄⠀⣿⣇⠄
⠀⠀⢀⣀⣠⣴⠒⢚⠉⠀⡀⢈⡄⠀⠀⠀⠀⠀⠈⢿⣿⣶⠏⢀⠀⠀⠀⠑⠀⠀⠀⠠⠉⠀⠀⠀⠉⠛⢠⡿⠋
⠀⠀⠈⠛⠛⠻⠿⣿⣆⠀⢳⡈⢧⡀⠀⠀⠀⠀⠀⣈⡿⣿⣷⣏⡀⠀⠀⠀⠐⠂⠀⠂⠄⠀⢀⡄⢲⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠺⢿⣿⣦⠀⢻⣌⢷⡄⠀⢀⣤⠾⠛⠀⠠⣄⠋⠀⠀⠀⡴⠀⠀⣤⠦⡤⡄⠈⢀⣿⡿⠛⠂
⠀⠀⠀⠀⠀⠀⠴⠛⣿⣿⣷⣀⣹⣦⣿⠾⠓⠂⢀⡀⢠⣠⠆⠉⠝⡀⠖⠂⢠⠀⠄⠀⠠⣍⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣽⠛⠛⢻⡿⠟⠉⠀⠀⠲⠀⠀⢠⠊⠀⠁⠘⠀⠂⢐⠀⢄⠉⠐⢚⣿⣿⣿⣿⡿⠀⠐⢀
⠀⠀⠀⠀⠀⢠⢀⣤⣶⣾⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⡦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡨⠄⠰⠆⠀⣰⠶⠊⢈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠁⡄⠀⠈⠀⠀⠂⠒⣀⢔⣩⠀⡠⢨⠹⡣⢱⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣒⡿⠀⡠⠞⢄⣶⠔⠈⢄⢀⡟⢐⠿⠛⠀⡠⠁⠈⢄⠘⠞⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢇⡚⣣⠼⢀⠄⣀⣉⠀⢙⡀⠰⠉⡠⠮⣬⠊⢣⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠢⡤⠐⠉⠀⠳⡋⡑⠀⠜⠐⡏⢁⣀⠘⠃⢀⠨⡀⢠⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⣠⣉⡧⢠⠠⡈⢁⣠⠀⠀⠤⠶⠃⡸⡟⠀⠀⢈⠂⠀⠀⠀⣀⠀⠆⢹⡆⠀⣠⠒⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠐⠀⢀⠁⠀⡺⠼⠀⠀⠀⠀⠘⠾⠁⠀⡠⢥⡂⠪⠄⠀⠒⠁⢀⠈⠀⡌⠉⠲⠒⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠂⠂⠑⠐⠀⠀⠀⠀⠀⠀⠀⠠⠘⠢⣴⣃⠀⠀⢀⠀⠀⠈⠰⡇⠈⠁⠡⠴
⠀⠀⠀⠀⠀⠀⢇⡀⠀⢇⠀⠢⠘⡈⠌⠄⢆⠀⠀⠀⢀⠤⢄⢀⠣⢤⡅⠀⠙⠀⠀⠘⠁⠀⢾⠐⠡⠄⡀
⠀⠀⠀⠀⠀⢀⢀⠈⣶⠀⠑⡄⢂⠀⢀⠀⡀⠀⡄⢢⣈⡑⠃⡘⠀⠀⠀⠴⢀⡀⢠⠒⡀⢰⠀⠁⠐⠒⠁
⠀⠀⠀⠀⠀⠣⠴⠀⠘⡇⠀⠰⣄⠁⠀⠀⠀⢘⠈⠀⠁⠘⠈⠀⠀⢀⠡⠂⠁⡌⠈⠀⠈⠁
⠀⠀⠀⠀⡠⡄⠀⠴⠀⢡⣢⠂⠈⠤⠔⠒⡀⢈⠀⠀⠀⠀⢀⠀⡐⠁⠀⠀⠀⡀⢰
⠀⠀⠀⠀⠈⠀⠀⠔⢒⠀⠀⠱⡀⠀⠀⠀⠈⠀⠀⣊⠄⠊⠀⠣⠃⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠈⠒⢄⠀⡠⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠂⠀⠀⠀⠀⠀⡀
⠀⠰⠁⠈⣟⠓⡐⡍⠐⠀⠀⢀⣖⠢⣠⠄⠒⠒⠐⢤⡄⠀⠃⡈⠡⠤⠔⠈⣒⠤⠄⠀⡀⢀⠐⠡⣞⠂⠀⠀⠠⡄⢤
⠀⠀⠀⠆⠀⠤⠄⠒⢲⠀⠙⠒⠒⠒⠀⠀⠥⢄⡤⠚⠢⣤⣄⢤⣤⠬⣁⣈⣈⠀⡄⢐⢽⡾⣬⡄⡴⠮⠵⢨⢈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⡌⢐⣻⡕⠂⠋⠉⠐⠒⠋⠐⠚⠠⠟⠉⠀⠈⠲⢄⡠⠀⠈⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠟⠉⣻⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠦⠖⠦⡀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣷⠀⠀⡷⡇⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠀⠀⠀⠀⡯⡀⠄⣐⡑⡘⢦⡀⠀⠀⠀⠀⡼⢡⣤⡌⣧⠀⡇⡇⠀⠀⠀⠀⠀⣿⣿⣶⣄⣠⢤⣼⡟
⠀⠀⠀⢀⣰⠛⠉⠙⢦⡄⠀⠀⠀⠀⠀⠙⢶⣄⠉⣡⠽⠊⠀⠀⠀⠀⠀⡇⠈⢭⠀⣿⠀⣱⡇⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠙⠓⠴⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⣟⠓⠛⠚⣿⣤⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⠉⠉⠛⠋
⠀⠀⣠⠔⢒⡒⠒⢤⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢶⣶⣤⣤⡄
⠀⢸⡇⡈⠀⠒⡃⠴⠃⠀⢸⠀⠀⠀⠀⣀⠴⢊⠅⣢⢟⠎⠳⡄⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡛⢑⡀⢃⢲⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢐⡧⢀⠋⠾⠓⡁⠔⣨⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣻⠈⠀⢸⢸
⠀⠸⡇⠀⠀⠀⠀⢠⣄⣀⡟⠀⠀⠀⠀⡷⣄⠑⣖⣡⠴⠋⠀⠀⠀⠀⠀⢀⣠⠔⣒⣿⡟⠉⢳⡆⠀⠀⠀⠀⠀⣷⣽⢀⠀⠀⣿
⠀⠀⠈⠒⠦⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣳⣄⣤⣤⣤⡶⠟⠁⠀⠀⠀⠀⠀⠙⠾⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⢠⣴⡳⣷⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢴⠞⠓⡆
⠀⠀⠘⣎⢳⠘⠉⠸⣿⡀⠀⠀⠀⢀⡤⠴⠉⠀⠀⠀⠀⠈⢓⣢⠀⠀⠀⠀⡶⣯⣶⡯⠕⠒⠢⣄⠀⠀⠀⠀⠀⠀⡨⠷⠾⣄
⠀⠀⢴⡚⠂⠀⠀⠀⢻⠃⠀⠀⠀⢸⡕⠀⢄⡀⠀⢀⣠⠒⢁⣼⠀⠀⢀⡴⢇⣻⠀⠀⣀⣴⣾⣿⠆⠀⠀⠀⡴⠋⠀⠀⠀⠀⠳⡀
⠀⠀⠀⢹⠓⢄⠀⠀⠈⢳⡀⠀⠀⠸⣧⣄⠀⠈⢙⠁⣀⣴⠟⠋⠀⠀⣸⣻⣦⣤⣴⡞⣣⠟⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢼⣀⣈⣆⣀⣤⠾⠁⠀⠀⠀⠀⠙⠻⢶⣶⣶⠞⠁⠀⠀⠀⠀⠈⠙⠾⣽⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⢀⢀⡞
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣠⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠘⣇⣀⡼⣏⠀⠀⠀⠀⢰⠛⢦⣀⣀⡀⠀⠀⠀⣠⢶⡄⠀⠀⢀⡤⠖⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⢀⡿⠛⣧⣞⣲⣦
⠀⠀⠀⠀⠀⡿⠙⡇⠈⠀⠀⠀⠀⠈⣧⡾⠀⠈⠉⠉⠉⠉⢧⢸⠇⠀⠀⠙⡟⠒⠶⠶⠶⠒⡿⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⡄
⠀⠀⠀⢠⣎⠀⠀⢙⡆⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⢻⠢⠤⠤⠴⢺⡇⠀⠀⠀⠀⠀⢸⡏⠈⠉⠉⠁⠹⡇
⠀⠀⠀⢸⣇⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠈⠳⠤⢤⡤⠞⠀⠀⠀⠀⠀⠀⢫⠢⠤⠤⠤⠤⠒⡇
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⣩⣭⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⣿⡿⠟⠛⠉⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠃⠘⠿⡅⠀⠀⠀⣼⡏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣊⣁⣤⠴⠞⠋⠉⠛⣶⣾⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠉⢀⣿⣿⡿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⣠⣼⣿⣿⡇⠀⣀⣤⣾⣯⣿⣿⣿⡿⠿⠿⠿⣿⣷⣶⣤⣄⣀⠀⢀⣀⣤⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⡋⡠⢾⣿⣿⣿⣽⣦⣾⣿⠿⣛⣫⣭⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣴⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⡖⠋⠋⠀⠀⠉⢈⣟⣿⣿⠏⠀⢸⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠏⣀⡀⠀⠀⠀⠀⢠⢿⣿⣿⠰⢤⣚⡶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢼⡇⠀⠈⠀⠀⠀⠀⠂⠁⢊⣸⣿⣮⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⢀⣠⣤⣤⠶⢒⣒⡀⠀⠳⣄⣀⢀⡀⠀⠀⠀⠀⠉⠛⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠐⠒⠒⠶⠶⣶⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠿⠦⣝⡒⠶⠦⠤⣤⣝⡻⠿⣦⣔⣦⣀⣐⣦⣕⡺⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⣠⡄⠙⣲⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⠒⠂⠈⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠒⠚⠩⠭⠵⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡤⠤⠶⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠦⠤⡀⢀⠀⠀⠈⠑⢄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⣞⣠⠶⡤⢽⣷⠶⣀⠂⣰⠀⠀⠀⠀⠳⠀⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡏⠀⣠⣿⣧⣾⣽⣶⣵⣶⡟⣿⠛⠋⠈⡗⢀⡼⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⢧⠘⣧⡽⣿⣿⠿⢊⡙⠁⣾⣧⣤⣭⠒⢺⠋⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣀⠀⠸⣄⠱⠌⠶⠒⠥⠞⣯⡟⢂⣈⠤⠉⠀⢠⠃⠀⢰⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢏⡛⠶⢤⣁⡛⠠⠤⣀⡀⠀⣬⠋⠁⠤⣤⣤⣦⣄⣀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠅⠙⢯⡑⠀⢙⣥⠖⠋⠉⠀⠀⠀⠀⠈⠉⣿⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⡀⠀⠈⢿⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⣧⠀⠀⠀⠳⣌⢢⡀⠀⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⣦⠀⠀⠈⠻⢷⣤⣀⣠⣤⣶⣿⣇⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡿⠷⠖⣶⣾⣿⣿⡝⡁⠛⢽⣿⡟⠛⠛⠉⠁⠀⠀⠀⢀⡴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢰⠟⠋⠀⠀⠀⣸⣿⠞⠙⢿⣿⠃⠀⠀⠹⠷⠀⠀⠀⠀⠀⣀⡴⠋⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄⠀⠀⠀⠀⠀⠀⢀⣹⣿⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠰⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡈⠓⠦⢤⣄⠀⠀⠸⢻⡇⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⣴⠟⠁⣠⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢃⠀⠀⠁⠛⢦⡀⠸⠟⢆⠀⠀⠀⢹⣀⠀⠀⠀⠀⢸⡏⠀⣰⠃⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠑⢀⠀⠀⠀⠹⢆⠀⠈⠂⠀⠀⠀⡾⡇⠀⠀⠀⢸⡇⢰⠉⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⢇⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢏⡁⠀⠀⠀⠈⢳⠇⠠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠄⢸⠀⠀⠀⠈⡿⠀⠀⠀⠀⢸⡇⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡀⢠⡄⠀⢘⡇⠀⠀⠀⠀⢸⡇⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠦⣴⣦⣵⣤⠟⠀⠀⠀⠀⠀⢸⣇⠌⠁⠀⠀⠀⠀⢀⣀⣤⣤⠾⠛⠛⢉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⢀⠀⡀⢉⠏⣇⠀⠀⠀⠀⠀⠀⡘⠋⠀⠀⠀⢀⣠⠴⠛⠋⠁⢀⠀⠋⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠁⠔⠠⠋⠘⢁⣉⣀⡈⠉⠓⣧⠀⠀⠀⠀⠀⣀⢧⡤⠀⣠⣴⠟⠁⠀⠀⠀⡔⠊⠀⡠⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠉⠐⠚⠓⠛⠛⠋⠉⠉⠀⠀⠀⠘⡀⠀⠀⠀⠘⣏⡀⢀⣴⠟⠀⠀⠀⢀⣤⠋⠀⠀⡐⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢠⣸⣆⡜⠁⠀⠀⠀⣴⠎⠁⠀⠀⢀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡘⡟⢨⠇⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡀⣿⠀⢸⡄⢀⣰⠋⠀⠀⠀⠠⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⡝⠀⠀⣧⣾⢁⡀⠀⠀⠀⣠⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⢀⡴⠋⠉⠓⠒⠒⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣹⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠿⣿⣾⡿⠟⠉⠁⠀⠀⠉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠿⠿⠛⠛⠛⠉⠉⠉⠙⠛⠛⠿⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠋⠉⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠲⢦⣍⠻⣿⣆
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠙⢿⣷⣦⣤⣤⣶⣶⣄⣴⣾⡿⠛⠁⠀⢀⣠⡴⠾⠛⠋⠉⠉⠉⠉⠉⠉⠛⢶⣄⠀⠀⠙⣧⡙⣿⣇
⠀⠀⢀⣴⣿⡿⠿⢿⣿⣦⡀⠈⠉⠉⠉⢉⣽⣿⠟⠁⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠘⣷⠀⠀⠸⡇⢻⣿
⠀⠀⣾⡿⠁⠀⠀⠀⠈⢿⣷⠀⠀⢀⣴⣿⠟⠁⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⣄⣼⠃⠀⠀⢸⡇⠀⢠⡇⣸⣿
⠀⠀⣿⣇⠀⠀⠀⠀⠀⣸⣿⠀⣠⣿⡟⠁⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢦⣄⠀⢻⣅⠀⠀⠀⢸⠇⠀⣼⢣⣿⡏
⠀⠀⠹⣿⣶⣄⣀⣠⣴⣿⠏⣰⣿⠏⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⡴⠞⠛⠳⣆⠀⣿⠶⠋⠻⠆⠉⠀⠀⢠⡟⠀⠀⢁⣾⡿⠁⣀⣤⣶⣦⣤⣀
⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⢠⣿⡏⣰⠆⠀⣿⠀⠀⠀⣠⠤⠀⢸⡇⠀⠀⠀⣽⠀⠿⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢠⣾⡿⢁⣾⡿⠛⠉⠉⠛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢠⡏⠀⢸⡇⠀⠀⠘⠷⠤⠶⣄⠙⠶⠶⠞⠃⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣴⣿⠟⠀⢸⣿⠁⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⠀⠈⣷⠀⠀⠀⠀⢤⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⢀⣴⣿⠿⢿⣷⣄⠸⣿⣆⡀⠀⠀⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡈⢷⡀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠁⠀⣠⣴⣿⠟⠁⠀⠀⠙⣿⣧⠙⠿⣿⣷⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡌⠳⣤⣀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠟⠛⠉⠁⠀⢀⣠⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣬⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣷⣶⣶⣶⣿⣿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣶⣿⠿⣿⣦⣄⣀⣀⣀⣀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠟⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠕⠆⠦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⡏⢡⡀⠑
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠉⠒⠉⠉⡠⠂⠀⠀⢀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢐⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⢠⠂⠂⠒⠀⠀⠀⠀⠀⢤⢀⡗⠈⠢⠤⠄⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠉⢀⠽⡽⠋⠒⠂⠀⠀⠀⠀⠤⠐⡇⠀⠀⠀⠀⠀⠀⠈⠆⢄⠀⢄⢀⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠲⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⢠⡃⠀⢱⠂
⠀⠀⠀⠀⢬⠇⠀⠀⠀⠀⠀⢸⠌⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡶⢫⣜⠒⠊
⠀⠀⠀⠀⠂⠲⢁⠀⠀⠀⠀⠇⠀⠠⠒⠎⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡰⠡⠃⠀⠉⡆
⠀⠀⠀⠀⠀⠃⠈⡄⠲⠀⠴⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠰⠖⡐⠠⠁⠀⠀⠠⠁
⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠜⡰⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠏⡐⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⡠⠄⠒⠀⠈⠀⠈⢸⠄⠀⡠⠂⢸
⠀⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠀⢀⠼⠠⣀⣀⡀⠀⢠⡁⠊⠆⠀⠠⠀⠀⣉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠕⡀⠀⠚⠀⠀⠀⠈⡇
⠀⠀⠀⢀⠤⠒⣋⣀⣤⣆⣁⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⠖⠛⠉⠁⠀⢀⠌⠁⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⢤⠔⢦
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢸⡤⡄
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⡠⠤⠖⠒⠊⠉⠉⠉⠀⠀⠀⢀⠜⠓⣺⣷⡅⠀⠀⠠⠋⡰⠁
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣀⠈⢁⡴⠁⠀⠀⠀⣺⠃
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠬⠽⢷⠉⠀⠀⠀⠀⣉⠝
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⣀⡤⠀⠉⢁⡠⠀⠀⡠⢴⠃⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠘⣄⠘⠂⠀⠀⠀⢈⡡⠂⠀⠈⣩⠄⠠⠤⡶⠁⠀⠀⢀⠴⠊
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠙⠒⠠⠤⠀⠷⠤⠤⠾⠏⠠⠤⠒⠊⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢲⠈⡗⠒⠠⣤⢤⠤⠤⠔⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠃⣠⠃⠀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⠤⠤⠔⢁⣠⠤⠊⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⣀⡤⠤⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣄⠀⠀⢤⠀⠀⡀⠀⢀⠀⠷⠀⠀⠀⡄⠀⡦⠀⢀⠀⢰⡄⠀⢀⡀⠀⠀⠀⢆⠀⠀⢀⡀⠀⠀⡀
⠚⢰⡀⠈⠀⠀⠇⠀⠸⠆⠀⠀⡆⠀⠁⠀⠀⠀⠘⠀⠈⠀⣄⠘⠇⠀⠀⠀⠋⠀⣤⠈⠃⠀⠐⠖
⠀⠈⠁⠀⢸⠀⠀⢠⡀⠀⠀⠀⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣹⠀⠀⠀⠛⠀⠀⠀⢠⠀⠀⣆
⢲⠀⠀⡄⠀⠀⠀⠐⠃⠀⠀⣆⠀⢲⠀⠈⠀⠀⡄⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠴⠀⠗⠀⠸⠀⠀⢈
⠈⠀⠀⠋⠀⢰⡄⠀⢰⠀⠀⠁⠀⠀⡀⢀⡀⠠⣧⣀⡀⠀⠀⠛⠀⠀⠀⢲⠀⠀⠘⠀⠀⠀⡀⠀⡀⠘
⠀⢀⡆⠠⡆⠀⠀⣀⠤⢐⣠⠻⢛⠥⠢⠐⠀⠀⢈⡄⠂⠌⠉⠙⠒⠶⣤⡬⠀⣀⠀⠀⠀⠸⡥⠀⣥
⢀⠀⠋⠀⣠⣔⠚⠉⡴⡫⠡⢊⠩⡂⠈⠀⠀⠀⠀⠐⠈⠂⠉⠀⠀⠀⠈⠙⠲⡠⠈⠁⠢⢄⡀⠀⢠⡄
⠸⠀⡠⠊⠀⠁⠀⡜⢘⠨⡂⢑⢅⠂⠁⠀⠀⠀⠀⠀⠁⠁⡀⠀⠈⡀⠀⠀⠀⠘⢦⠀⠀⡀⠈⠢⡈⠁
⢀⢎⣢⠴⠀⠀⢰⢐⠕⠨⠞⠁⡀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⡀⢀⢀⠀⠀⠀⠈⣧⢐⠀⠀⡄⢝⡄
⠟⠉⠀⠈⠉⠀⣬⡵⠖⠓⠉⠈⠉⠉⠈⠙⠓⢦⣤⡀⢠⢠⣤⠴⠿⠓⠂⠡⠠⠤⠾⢜⡵⠚⠉⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣗⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣉⣁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠉⠀⠀⠁⠀⠂⠀⠀⠂⠔⠄⠠⠢⠠⠈⢄⠁⠌⠀⠨⠢⢀⠕⢔⢄⢀⢀⠀⡠⡂⡠⢀⢄⠔⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⣤⣶⣾⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠿⠿⠿⠿⣿⡿⠋⠁⠀⠀⠀⠀⠉⠻⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣄⠀⠀⠀⠀⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠿⠟⢛⣛⣛⣛⠛⠿⢿⣿⣷⣤⣉⠻⣿⣿⡄
⠀⠀⠀⢀⣠⣤⣤⣄⡈⠙⠿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠿⠛⣉⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⣿⣿⣦⠹⣿⣿⡀
⠀⢀⣴⣿⠟⠛⠛⠻⣿⣦⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢛⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋⣍⠻⣿⣿⣧⠸⣿⣿⡆⣿⣿⡇
⠀⢸⣿⠁⠀⠀⠀⠀⠸⣿⡇⠀⢠⣾⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣆⢋⣴⣿⣿⣿⠀⣿⣿⢁⣿⣿⠃
⠀⠸⣿⣆⠀⠀⠀⠀⣼⣿⠃⣰⣿⣿⣿⣿⠟⣰⣿⣿⣿⣿⣿⡿⠿⠻⠿⣿⣇⢰⠎⠙⠿⣆⣻⣿⣿⡟⢰⣿⣇⣼⣿⡟⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠙⠿⣿⣶⣶⣿⠟⠃⣰⣿⣿⣿⣿⡏⣰⣿⣿⣿⣿⣿⡟⢰⣿⣿⣷⠘⣿⢰⣾⣿⣷⣿⣿⣿⡿⢡⣿⣿⣿⣿⡿⠁⣴⣿⠟⠋⠙⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢠⣿⣿⣿⠡⣶⣿⣧⡘⠿⡿⠟⣰⣿⣾⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⠟⠀⢸⣿⡇⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢰⣿⣿⠸⣿⣿⣿⣶⣶⣶⠌⣿⣶⣶⣿⣿⣿⣿⣿⣿⡿⠛⣡⣾⣿⣿⣿⣿⣯⡀⠀⠘⣿⣧⡀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⢸⣿⣿⣆⠹⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⡿⠛⣉⣴⣿⣿⣿⣿⡿⠋⠉⠻⣿⣦⠀⠈⠻⢿⣶⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⢻⣿⣿⣷⣬⣙⠛⠿⠿⠿⠿⠟⠛⣋⣩⣤⣶⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣦⣴⣶⣿⣷⣄⡀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠙⠻⠿⣿⣿⣿⡿⠿⠛⠁

⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣀⡀⠀⠀⠠⠈⣂⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⢩⣅⠀⠀⠀⢈⣭⡑
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣽⣿⣿⠛⢻⣭⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠿⢀⠔⠐⠊⣹⢃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣿⣿⣿⡶⢛⣉⣭⣉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠂⠁⠀⠐⠈⢀⢀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣦⣴⣾⣿⣿⣶⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠂⠀⡠⡄⠄⠂⢉⡂⢠⠅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠠⡮⢆⣠⣘⠝⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⣭⣭⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠀⠘⠄⡀
⠀⠀⠀⠀⠀⠀⣼⡎⠑⠢⢄⣲⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠐⠀⠀⡒⠁⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⢸⡏⣠⠞⠉⠉⠉⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⡀⠀⢀⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠒⡄⠀⠀⢀⣄⠀⠀⠀⢀
⠀⠀⠀⢤⣐⡀⠚⠝⣿⣿⣿⣟⠋⠀⠉⣉⣭⡏⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⡀⠒⠂⠁⠀⠁⡀⠄⠁
⠀⠀⠀⠀⠀⠀⠃⠤⠌⣉⠉⠛⠛⠛⠉⠙⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣁⠤⠒⠒⢒⣒⠒⠂⠄⠀⠀⠀⠀⠀⢤⠐⠒⢀⡀⠁⠀⠐⠲⢤⣀⡀⢀
⠀⠀⠀⠀⢀⠔⠂⠉⠑⠂⠈⠄⣐⠉⠭⠄⠀⠬⢍⠀⠀⠀⠀⠀⠠⠦⠐⠉⠀⠀⠀⠀⠀⣀⠠⠌⠋⢈⠥⡀
⠀⠀⣠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠢⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⡆⠪⠑⢄
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠠⠊⠄⠀⠑⡄⢱⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣⠀⠣⡀
⠀⠀⠀⠀⢄⡀⢀⡀⢀⠀⠀⡀⠠⠤⠊⠀⠀⠀⠀⠀⢠⠀⠀⠂⢠⠃⠀⠀⣃⣀⡀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢆⠀⢡
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠨⠑⢄⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠸⠀⠀⠀⠘⡀⠀⡃
⠀⠀⠀⠀⣴⢀⠀⠃⠃⠀⠈⠁⠀⠒⠂⠊⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⢀⠎⠀⠀⠑⠄⠀⠇⠀⠀⠀⠀⡇⢀⠁
⠀⠀⠀⠀⠀⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⠠⠄⠀⢠⠀⠀⠀⠀⢀⡠⠃⠠⠃⠀⠀⠀⠀⠈⠀⠉⠉⠉⠁⠂⠌⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠁⠀⠀⠀⠈⠒⠒⠐⠊⠀⠀⡰⠁⠀⠀⠀⢀⠠⠤⢤⠠⠄⠀⠠⡀
⠀⠀⠀⠀⠀⡠⠄⠀⠤⡀⠋⡰⠸⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⢀⢜⠀⠀⠀⠀⠀⠇⠖⠆⠘⠒⡒⠠⠌⡇
⠀⠀⠀⠀⠌⢀⠄⠀⢄⠈⡖⠀⠀⠂⠁⠀⡟⢠⢀⠤⠀⢄⠀⠀⠣⣓⠱⠁⠀⠀⠀⠀⢀⠇⠀⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢂⠘⠄⠄⢸⠀⠇⠀⠀⢀⠃⢀⠋⠑⠁⠔⠀⠆⠱⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⡄⠊⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠈⠃⠵⢄⠐⠓⠊⢀⡀⠔⠁⠀⠀⠢⣀⠑⠌⣲⡶⠵⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠇⠦⠤⠀⠤⠄⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡵⣞⡭⢞⣡⡶⢖⣠⣤⠄⢀
⠀⠀⠀⠀⠀⡀⠀⠤⠈⣧⣈⣘⣏⣻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢐⣮⣕⣺⠭⠶⣚⡭⡭⠶⢛⣻⣿⣷⡄
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠻⠿⠿⣿⣥⣭⣻⣶⣶⣴⣶⣾⣯⣴⣿⣿⣦⡻⡭⠦⣕⢲⣏⣿⣽⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣭⣽⢿⠿⠿⢿⠟⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣋⣈⣉⠈⠉⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣭⣉⣉⣀⣀⣀⣰⣿⣥⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣷⣦⣬⣭⣍⣛⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣿⣿⣿⣿⣿⣝⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⡟⣿⡹⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣧⣻⣮⡿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠢⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡈⠁⢄⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠁⡀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⢀⠀⠀⠀⢂
⠸⠀⠁⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⠌⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠂⠈⠁⠀⠀⠀⡀⠀⠈⠁⢒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠁⠀⠈⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠂⢁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠂⠀⠀⠂⠀⠀⠀⠩⢤⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⡒⠋⠁⠀⢀⣀⣀⣤⡀⢴⠖⠂⠈⠉⠁⠀⠀⠑⠩⢒⡅⠀⢈⢉⣉⣠⡖⣶⠦⠖⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠰⠉⡄⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠉⠀⠀⠡⠀⢾⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊
⢠⠆⠀⠀⠀⠀⠀⠀⢀⠁⠀⡁⠀⠀⡄⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞
⡆⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠋⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡛⢸⡄
⠑⠀⠂⠀⠀⠀⠀⠀⢻⣀⠈⠁⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⣀⠀⡐⠧⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠸⠀⠻⠀⠠⠀⢀⠀⠀⢀⣀⡀⡈⠀⠠⠆⠈⢰⠊⡰⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⠀⠀⠠⢦⣴⣀⣀⣤⡉⣀⠇⠀⢰⡄⠀⠀⠴⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣆⠤⠀⡠⠀⠀⡀⠁⠉⠁⠛⠈⢹⠀⢀⡄⠈⠢⠀⠀⢃⠡⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣀⠌⠈⡳⢀⠀⠰⠒⣆⢀⡀⠐⣄⣀⠐⣄⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⣩⣥⠰⠢⣭⠁⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠠⠦⡀⠀⠠⢀⡀⢠⠈⣄⡀⣀⣐⠆⣀⣈⡂⢀⡀⣀⠀⡀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡘⠘⠈⠉⠉⠁⠀⢈⣉⠀⠂⠈⠁⠈⠇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠄⢀⡤⠄⠄⠀⠀⠈⠊⠀⠁⢤⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠄⡤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠒⠒⠒⠀⠓⠐⠀⠀⠀⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⣀⡠⠤⣐⣒⡭⠤⠤⠴⠶⠶⠦⠤⠤⣄⣀⠀⡀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⢀⠤⣶⠭⠒⠊⠉⠀⢀⡀⠀⠀⠀⠀⢤⠀⢀⣀⡀⠈⠑⠮⡆⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⡀⠁⠀⢀⡔⠁⠊⢀⠐⣀⠠⢂⢊⠴⠀⡆⢀⠀⡖⢸⠀⠀⡥⠔⠁⠀⢀⡇⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⡜⠀⠀⠀⡎⠀⠀⠀⢐⣲⠄⡀⢰⡘⠂⣰⠃⣂⡠⠦⠸⠘⠜⠀⠀⢀⠔⠉⠀⠀⢠⠂⠀⢸
⠀⠀⠀⠀⢰⡇⠀⠀⠸⡇⠀⠀⠘⠐⠒⠂⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡠⠄⠊⠁⠀⠀⣀⠔⠁⠀⠀⢸
⠀⠀⠀⠀⢸⠱⡀⠀⠀⠈⠑⠢⠤⠤⣀⣀⣀⣀⣀⡀⠤⠤⠤⠒⠒⠉⠁⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠸⡄⠙⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠉⠁⠐⠒⠒⠂⠤⠤⠤⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠓⠦⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⣝⠻⠖⣲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠞⠀⠀⠀⣠⢾⡷⡾⢀⡜⠙⣆⠀⠉⢫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣃⠀⢀⡠⢞⣁⠀⠀⠷⠋⢀⣀⠘⢆⣦⢀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⡹⡴⠋⠀⠈⠈⠁⠀⠀⠀⠈⠈⠀⠈⠙⡶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡇⠀⠀⠀⢠⠄⠀⠀⠀⠰⡦⠀⠀⠀⡟⠱⡄⠀⢀⠜⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣇⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡷⠚⢁⡴⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠢⣤⠔⠀⠀⠀⢀⡴⠓⠋⠻⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣤⣀⣀⣀⣀⣠⣤⡖⠻⡇⠀⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢙⠲⠋⠀⠳⠋⢘⡶⡹⠓⠒⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢸⡀⠀⡠⠔⠶⣯⢾⣁⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠙⠲⣇⠀⠀⣸⠘⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡏⠓⢤⣀⠜⡽⠋⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⣄⣀⠀⠐⣏⠙⠒⢢⠀⠀⠀⠀⠀⣿⣷⡶⣫⣾⣶⣶⣶⣶⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠰⢧⡐⢥⡄⣰⠇⢀⣤⢖⠒⠒⢿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡀⠀⢀⣠⠇⠀⠉⠁⡴⠋⠘⠦⣳⢄⠀⠹⡟⢻⠛⠉⠛⢿⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠱⠤⠞⠁⠀⢀⡤⢊⠀⠀⢀⠀⠉⠻⣷⣤⠇⢸⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣔⠋⠀⠀⠀⡸⡏⠉⢹⡀⠀⠀⣸⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠴⣗⣋⣉⣹⠇⠀⠀⠿⢝⣛⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠉⠉⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣄⠀⠀⢀⢤⡀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠫⡀⠀⠀⠉⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢀⣠⠴⠞⠉⢀⣀⡤⠔⢙⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⣏⣠⣴⡾⠟⠛⠛⠉⠉⠙⢶⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡙⢦⡰⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠈⠇⠀⠘⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠤⠴⠶⠶⠶⠶⠶⠦⢤⣄⣀⣤⢀⡴⠋⠀⠀⠀⠀⡼⠋⣹
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠖⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠉⣡⠿⠿⡦⠀⠀⠀⠠⠊⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠞⠁⠀⣰⠃⠀⠀⠀⠀⠠⢺⡏
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠒⢳⣆⡠⠞⠁⠀⠀⠀⠀⠰⠒⣻⠃
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠘⠛⠉⣹⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⠀⠀⠀⣠⠂⠀⢀⣠⠴⣾⠁⠀⠀⠀⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠳⣄⠀⠀⠀⠀⠀⠉⠉⣡⠀⠐⠒⠒⠋⡁⠀⠀⢀⣠⡞⠁⠀⠀⠀⠀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠤⠖⠋⠀⠀⣀⡤⠴⠋⠀⠉⣉⣩⠞⠀⠀⠀⠀⠀⡰⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠙⠛⠒⠒⠒⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠐⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⠤⣤⣤⠐⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡿⠀⠀⠉⢹⡏⡽⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠉⠀⣠⠇⠀⠀⠀⡼⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣁⡀⢀⣀⣄⡼⠃⠀⢀⣀⡤⠃⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠀⠀⢠⠖⠚⠉⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠤⠤⠴⠖⠒⠋

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⣦⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢶⠖⠄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠑⣤⠁⠀⠀⡖⠊⠑⣄⠀⠀⢠⡄⠋⠀⠎⡂⣠⠭⢄⣮⣀⠇⠀⠀⠀⠀⡠⠀⠄⠡⠜⠀⠻⠀⣀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⠞⠉⠀⠀⠄⠀⠀⠀⠈⡄⠠⠀⢠⠊⢰⠉⢑⠁⠀⠀⡇⢀⡴⢦⢀⡥⠀⠁⠀⢀⢀⠔⠲⡘⠃⠈⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⡈⠒⠊⢀⣄⠓⠤⠞⢀⣘⡧⠴⠈⢂⠘⠀⠉⠁⡾⠦⠤⠃⠯⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⣀⣴⣬⣄⣀⠀⠈⠀⠶⠀⢧⡴⢰⣻⢤⠋⠀⠙⡆⠰⡃⠀⡄⠐⠉⠳⡀⠀⠐⠂⣤⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⠳⣤⡀⠀⠀⠉⠁⠀⠀⠙⢷⡈⠃⠀⠀⠴⠄⢠⣧⠘⠦⣤⠔⡁⠀⣉⢈⡐⣆⠀⢀⡆⠸⠆⠀⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣤⠀⠀⣀⡀⠘⠛⠦⠸⣭⢰⡏⠀⠉⠋⠙⠋⠀⣆⠀⣤⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠱⣀⢤⠀⠀⠀⠀⠀⠀⡘⢧⠀⠀⠉⠀⠀⠁⠞⠈⠐⠋⣀⢄⠈⠳⠤⠴⠃⠘⠗⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⠈⠀⢑⠀⠀⠠⠴⠋⠀⠀⢡⠀⠀⠀⠀⠀⠆⠀⠋⠇⠈⠚⠁⢀⡀⠀⠻⠃⠀⠐⠗⠀⠀⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⣜⠀⠀⠀⠀⣀⣀⠀⠲⢀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢀⡄⠙⠃⠀⠀⣤⠀⢠⠔⠒⠀⠒⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠸⣄⡀⠀⡞⠁⠀⠉⠀⠀⢃⡠⠒⠛⠛⠢⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⢠⠾⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠙⢇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢘⡦⠒⠈⠀⠒⠤⠔⠊⠉⠉⠒⢄⣠⠔⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠒⡀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠉⠑⢆⠀⠀⠀⢠⠄⠐⠀⠀⠀⠀⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠰⠃⠄⠀⠠⠀⠀⠈⠀⠤⠀⠀⠀⠀⠘⣦⡓⠆⢀⣔⠄⣰⠃⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠔⠋
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠶⠶⠶⠚⠉⠣⢄⣀⡀⠿⢤⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠁⠀⠀⠈⠀⠈⠁⠈⠉⠀⠀⠐⠈⠁⠈⠁⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⣀⣀⣠⠴⠋⠀⠂⠩⠭⠒⠊
⠀⠀⠆⠄⠢⠈⠀⠆⠠⠄⠀⠀⠠⠠⠄⠠⠀⠠⠀⠀⠄⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠠⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⣼⠄⣦⠃⠀⠴⠐⠂⠄⠀⠀⠀⠀⠀⠀⠆⠀⠉⠀⠀⠀⡁⠆⠣
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⡥⠄⠀⠀⠀⠀⠀⠀⠀⠈⠶⢽⠽⠿⠙⠁⠀⠀⠀⠁⠀⠐⠁⠂⠀⠀⠀⠚⠂⠁⠀⠁⠀⠐⠒⠷⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣂⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀