Knock on wood Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠟⣔⡿⠋⢲⡝⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢐⠀⠀⠀⠀⠀⠸⡀⠌⢁⣄⣨⣿⢻⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⢤⣹⣿⣛⣿⠟⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⢈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⢠⡖⠙⠳⣄⡤⣀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⣱⠃⣸⠇⠀⠈⠱⢤⣤⠴⠶⢤⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⡏⠀⠀⡏⠀⠒⠀⠀⠀⠉⠀⠤⠀⠛⣦⡤⣄⡃⠀⡛⠀⠀⠈⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣁⠀⢷⡀⠙⠀⠀⢀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⡀⣸⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⢆⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⡏⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣯⠀⠀⠘⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣟⠀⢰⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⠋⠀⠀⠃⠀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣻⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠠⠠⠀⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡧⠠⠀⡤⠅⠩⠉⠉⠍⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠨⠀⠀⠨⢡⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠄⠸⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⣿⢿⡿⠿⡿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣤⠀⢠⡄⠀⣤⠀⠄⣤⣄⢠⡄⢠⠀⢤⡄⢠⠀⠀⢦⡀⢰⠀⣤⠠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢠⣿⠀⢸⡇⢀⡟⢰⣰⠿⢰⣆⡙⠾⣷⡈⢷⣌⠻⡇⢸⡇⢸⡇⣿⢰⡇⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⣬⡟⢰⣾⣇⢸⠃⣾⢋⣾⡿⠿⠿⣦⡙⣷⡈⢿⣦⣴⠈⡇⢸⡇⡟⢸⠁⣾⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⢸⣿⡇⣸⢰⡇⣿⡏⣰⣾⣦⠹⣷⢸⣷⢸⡜⣿⠀⣿⢸⠁⡇⢸⠀⣿⣧⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢠⣿⡇⢹⣿⠃⣿⢸⢫⣿⡇⢾⣧⢹⠀⡿⢰⣿⠈⣇⢹⣧⡏⣼⢸⡇⣿⡇⢸⡿⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⣿⡿⢃⣼⣿⡀⡿⣼⢪⣿⡇⠘⣷⡏⣸⡗⣼⢳⢀⣿⡈⢿⡇⡿⢸⡇⣿⡇⢸⠃⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⠟⣡⠞⢋⣙⢿⡽⡌⣷⢻⣷⢰⣤⣠⣿⢸⡏⡞⣸⢹⣧⠈⢀⡇⣾⡇⣿⠇⣾⢰⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⡏⣰⠟⢿⡿⣿⡙⣌⢷⡽⣿⣿⣿⡯⣿⢑⣱⣿⡄⣿⡟⢸⡇⣿⡇⣹⠀⣿⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡇⡏⢰⠆⣿⢸⣷⢹⡜⣧⣽⠿⣋⣼⠏⣼⣿⣿⡇⢻⠃⣾⡇⣿⣇⢿⢀⣿⡜⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠈⢿⠹⡀⣆⡿⢸⣿⢸⣷⠟⢿⣿⣿⠃⢰⡿⣿⢿⣧⣼⠟⢫⣷⡈⢿⣿⠈⣿⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢸⡇⢻⣿⠇⣿⡟⣸⢋⣴⡌⢿⡿⡏⢺⣧⣽⢬⡟⡍⢀⣿⣿⣷⢸⣿⠆⣼⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢿⣀⣦⣼⡟⣰⡇⣼⣿⡷⢸⣇⣧⢸⣿⡏⣾⡾⢁⡾⠛⢿⡿⢸⣧⣾⡿⣳⠞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣧⠀⡈⢻⡟⣰⣟⣧⠸⣯⡏⢠⣿⣿⢸⣿⢳⣿⢣⣿⣷⣷⢸⣿⢸⣿⣿⣴⡏⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢻⣧⠘⣿⢡⣿⣿⣞⢷⣮⣴⣾⣿⡇⢸⡟⣾⠇⣿⡟⢱⣿⡈⣿⢸⣿⣿⣿⠇⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⡇⣼⡏⡈⣿⢧⢻⡟⣾⣿⠁⣿⡇⣿⢠⣿⠃⣾⠘⡇⣿⢸⣿⢸⣿⢸⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠸⡆⢻⠁⣿⢰⡇⢹⡇⢸⣿⣿⡟⡇⢻⣧⣿⠀⣿⡀⢿⣼⠃⣿⢸⣿⢸⣿⠈⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢻⠀⢿⠸⣾⡇⢸⡇⢸⡇⣸⣿⣿⣷⣿⣸⣷⢿⡆⢻⣷⣤⢀⣾⢏⠼⣿⡌⢻⣆⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡆⣿⡇⢸⡇⣾⠃⣿⣿⣿⣧⣯⣿⣿⣼⣿⣌⠻⠿⠛⣡⣾⣆⠹⣷⡄⢿⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⢸⡇⣿⣿⢸⡇⢹⣾⣿⣿⣿⢹⡟⣹⣿⣿⣇⠹⣿⡏⣼⣿⠃⡽⣧⠹⣿⡌⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣼⠁⣸⡇⣿⣿⢸⣇⢸⡏⣷⣿⡟⣼⡇⣿⣯⣿⠸⣧⢸⡇⣿⡿⢰⡇⢹⣧⠹⣇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠍⠀⣿⠀⡏⣿⡸⣿⡘⣿⣿⣿⣷⣿⡇⣿⣿⣿⠀⣿⠘⡇⢻⡇⢸⣿⠀⢿⡆⢻⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠰⣿⣿⡇⣿⣁⣿⣿⣿⣿⣿⡇⣿⣿⣿⣇⢿⡇⣿⢸⣿⡈⣿⣆⠈⣿⠘⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⡿⢰⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⠄⣿⡌⣯⢷⡈⢿⡆⢸⡄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⡇⣼⣿⣿⣿⣿⢸⢻⣿⣿⣿⣿⣿⢟⠻⣿⣿⣿⣿⡿⣷⡜⣎⢷⡈⣿⠈⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⡇⣿⡇⣿⣿⣿⢸⢸⡏⣿⢻⣿⡇⢾⡇⢹⣿⣿⣿⣿⣿⣷⠘⣎⢷⠸⡇⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⣿⡇⣻⡇⢻⣿⢻⣿⣸⣷⣿⣼⣿⣷⣸⣯⣾⣿⣿⣿⣿⢣⡜⣷⡈⢺⡇⣇⢧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠰⠿⠇⠘⠟⠸⠿⠺⠟⠇⠿⠻⠇⠿⠿⠿⠿⠿⠿⠃⠿⠇⠾⠷⠘⠿⠘⠇⠻⠸⠈⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠡⡀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢘⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠒⠤⡀⠀⠀⢸⡇⠀⠀⠀⠒⠒⠛⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢩⠞⠡⠄⠐⢿⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣺⠉⠀⡄⢸⣀⣀⣰⠔⠃⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠋⠁⠈⠈⠳⣦⣷⠔⠋⠩⢳⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣀⣴⣄⠀⠐⠄⡈⢷⠀⠀⣥⠃⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠠⠤⣤⢾⡁⠠⡏⠛⡖⠶⣚⡯⠖⠋⠩⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠞⠀⢶⠙⢲⠞⢋⠁⠀⠀⣀⣠⡎⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢸⢣⠈⠲⣬⠶⠖⠋⠁⠀⢹⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢧⣀⣄⣀⣤⠖⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐⠄⢉⡭⠟⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣀⣠⣄⣤⠊⠁⠀⠀⠀⠐⠠⣤⣀⡀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡦⣔⠀⠀⠀⠀⠈⠉⣩⠗⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⡄⠀⠀⢀⠞⢣⠠⠆⠤⠴⣤⠦⠀⠴⠠⠤⠤⢤⣤⠀⣾⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢦⡴⠃⠀⢸⠀⢀⡀⠀⡀⠀⠀⣑⣑⣒⠂⠤⠸⡇⣿⣿⣿⣿⢆⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣵⣀⣌⡁⣀⡀⣁⣀⣀⡀⣀⠈⢀⣄⣰⣿⣿⢹⣿⣯⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⣿⣿⣿⡇⠀⠀⠙⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⣿⠈⠉⠉⠉⢹⡏⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠄⢠⣄⣀⣀⡀⠀⠀⡇⠀⠀⠀⠀⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢯⣅⣤⣤⣄⣀⣀⡀⠈⠙⠳⣤⠿⠒⠒⠄⠀⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣦⣀⠉⠀⢀⣠⡴⠀⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠛⠋⠀⠀⣼⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⡆⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡄⠀⣿⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠈⠛⠃⠀⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠀⢀⠀⢀⠀⠀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⣀⠤⠴⠞⠛⠋⠙⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⢸⣴⠋⠀⣼⢆⣿⠀⡎⠈⡇⠀⡇⠈⠀⢸⣴⠃⠀⠀⠀⡏⠈⡇⠀⡷⣸⡇
⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠘⠀⠳⠀⠛⠈⠓⠀⠑⠴⠃⠀⠳⠔⠁⠘⠈⠳⠀⠀⠀⠳⠔⠁⠐⠃⠙⠂
⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢠⡄⠀⢠⣄⠀⠀⣤⠀⢀⣤⡤⣤⡀⠀⠀⣠⣤⢤⣄⠀⠀⢠⡤⣤⣤⡀
⠀⡞⠀⠀⠀⠀⠀⡀⡴⠒⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣧⠀⣼⢿⡄⢰⡏⠀⣾⠁⠀⠈⢻⡄⢰⡏⠀⠀⠘⣧⠀⢸⡇⠀⠈⣿
⠐⡇⠀⠀⠀⠀⠀⠀⢧⣘⠔⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⣠⡟⠘⣇⣼⠃⠀⣿⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⣸⠇
⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠸⣿⠃⠀⢿⡿⠀⠀⢻⣄⡀⣀⡾⠁⠘⣧⣀⢀⣰⠏⠀⢸⣇⣀⣠⡿
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠉⠋⠁⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠘⠳⢤⣄⣀⡀⠀⠀⣀⡤⠶⠚⠁⠀⠀⠀⡆⠀⢰⠒⠆⢰⡒⡦⠀⣖⠒⠀⢠⢆⠀⠐⡖⠂⢰⠀⣰⠒⡆⠀⣦⢰⠀⢰⡒⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⣀⠆⠸⠉⠳⠀⢯⣁⠀⠞⠚⠄⠀⠇⠀⠸⠀⠹⣠⠇⠀⠇⠻⠀⠠⣩⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡦⢄⣠⣧⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⢄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⠶⠶⠤⢼⠆⠈⠒⢇⣿⠀⠘⡄⠀⠀⠀⠀⣀⢀⡞⠀⠀⠀⠸⠦⡞⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⢀⣆⢀⣀⣦⠀⠀⠀⠀⠻⠀⣀⠹⡀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠈⠉⠈⢿⠀⠀⠰⣷⡀⠀⣿⡆⢳⣴⡆⠘⠤⣷⠦⠤⢷⠤⠤⣷⠤⠤⢷⠦⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⣄⠀⠀⢹⡇⠀⣿⠆⢠⣿⠀⠀⠀⠁⠀⠀⠋⠀⠀⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢀⠘⢇⠀⠘⠷⠀⠿⠀⠾⠁⢷⠀⠀⢀⣠⣤⡴⢴⠶⣤⡶⣿⠻⡟⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠚⢻⡅⠈⢷⡈⢷⠙⣷⣬⣷⠿⠞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠀⠀⠀⠀⢀⣀⡶⠒⠶⡞⢿⡳⢦⡀⢻⡆⢈⠃⠀⠀⢹⡇⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⠞⠁⠀⠙⠦⡀⠘⠆⠛⡀⠽⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠂⠀⠀⢸⠖⢤⣲⡄⠀⠀⠀⠉⠀⠀⠀⢿⡀⠀⠀⠀⠀⢸⡗⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⡇⠀⣏⢹⡄⠀⠂⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡇⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠠⠀⠀⢘⣿⣠⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣤⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⠓⠺⠷⠾⠶⣶⡀⠀⠀⡀⠀⣠⡟⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣴⠟⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠙⡆⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢺⣶⣛⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠀⠀⠀⠀⠀⠙⠉⢳⡀⠀⠘⣷⡀⠠⠀⠀⠀⣀⡀⠀⣠⣶⣴⠶⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣦⣀⣀⣦⣀⣀⣦⣀⢮⣍⢷⠀⠘⢧⡀⠀⣠⠞⢉⡽⠟⠁⠉⠁⠀⠀⢠⠖⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠞⠁⠀⠊⠀⠚⠁⠀⠀⠀⠀⠙⢦⡏⠛⠛⠓⠀⠀⠀⠀⡤⠰⠏⠀⠀⠀⢱⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣣⣄⣀⣦⣀⣀⣢⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠘⠁⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢆⣀
⠀⠀⠀⠀⢸⠉⠉⢹⠻⡆⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡆
⠀⠀⠀⠀⢸⠁⠀⡧⠀⢠⢰⣶⡄⠀⠀⢠⣄⢠⣄⠀⠀⢠⣤⠀⢀⣤⣄⠀⠀⢀⣠⣀⠀⣠⡀⠀⠀⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⢰⠁⠀⢸⠀⣿⡇⠀⠀⢸⡟⢸⣿⠙⢦⠀⣿⠀⣿⠁⠸⣦⠀⣾⠁⠈⢶⣸⣇⠀⠀⡿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠸⡀⣄⡜⠀⢸⣷⠀⠀⢾⡁⠀⣿⠀⠀⠑⣿⢸⣟⠀⠀⣿⣾⣿⠀⠀⠈⢹⡟⠀⠹⣆⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢸⠀⠒⠉⠊⢀⠀⢸⡇⠀⠀⠀⣷⠀⣿⠀⠀⠀⣿⡞⣿⠀⠀⣿⡏⣿⠀⠀⣸⣾⡇⠀⠀⢻⡀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣆⠄⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿⡄⣿⡀⠀⠀⣼⡇⠹⣆⣰⡿⠁⠙⢧⣠⠟⢸⣧⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠙⠁⠛⢁⠀⠀⠉⠁⠀⠀⣩⡀⢀⣀⠀⠀⢀⣠⣀⠀⣠⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠰⠦⣄⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠈⣿⣇⡞⠈⢷⡀⣼⠀⠸⣧⢹⡏⠑⣆⠀⠀⢸⡄
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣤⣖⠊⠉⠙⢦⣄⠀⠈⣿⠀⠀⢰⡄⠀⣿⣿⡇⠀⢸⣧⣿⠀⠀⣿⣾⣧⠀⠘⣷⡀⠘⡇
⠀⠀⠀⠨⠀⠀⠀⠀⡀⢠⠶⠶⣭⣳⣦⣀⠀⠙⠀⠀⠸⣇⠀⡜⢹⡀⣿⢻⡇⠀⢸⣿⣿⡀⠀⣹⣿⣿⠀⠀⢹⡇⠀⡇
⠀⠀⠀⡆⠀⠀⣀⡀⠁⠘⣦⡀⠀⠙⢷⡙⠗⠀⠀⠀⠀⢻⣦⡇⠀⣷⣿⠈⢿⡀⣸⡏⠘⢧⣠⡟⠠⣿⣤⠤⠞⠁⠀⡇
⠀⠀⠀⠑⠊⢻⠀⠀⠀⠀⠈⠛⠦⠤⠤⠇⠀⠀⠀⠀⠀⠀⢻⠇⠀⠸⠇⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠇
⠀⠀⠀⠀⠀⣹⠖⠒⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠤⠤⠤⠒⠒⠒⠒⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢼⣀⣀⣠⣠⠤⠤⠤⠴⠒⠒⠒⠒⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁
⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠏
⠀⠙⢿⣿⣿⠿⠛⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢀⣨⣵⡔⠀⠀⠀⠀⠀⠀⠀⣡⠄⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⣿⡿⣿⣧⠆⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣿⡿⠀⠀⠀⡤⠀⣘⠀⠀⠀⠀⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠁⠢⡀⢰⠇⢰⠁⡄⠀⠀⠀⠀⠹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢺⠀⡜⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣱⠆⠀⠀⢀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀⢀⠃⠀⠀⠀⠀⠀⠀⠈⢡⡾⢳⣾⣿⣿⣶⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⣤⠿⣳⣿⣿⣿⣿⣿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢰⣿⣿⣿⣿⠿⣿⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⠟⠋⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢠⠄⠀⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⢀⡰⠎⠀⢠⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠰⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⡄⢀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠐⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⡶⠂⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢿⣶⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⣼⡿⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣅⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀
⡀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣐⠒⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣶⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠦⣀⠀⢀⣼⡇⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡏⢈⠳⣌⣙⡛⢁⣿⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠘⣧⠀⢀⡾⠉⠉⠓⢦⡀
⠀⠀⢸⠿⡦⣤⡿⠛⠙⠛⠶⢺⡟⠀⠀⠈⠉⠳⡌⣟⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⣠⠤⠤⡾⠀⠀⠀⠀⠀⠀⠻⠛⡿⠀⠀⠀⠀⠀⣷
⠀⠀⣏⠀⠱⡄⢰⡀⠀⠀⡀⢸⡀⠄⠀⠀⠀⠀⠘⢿⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⢸⡇⠀⠀⠈⠛⠹⠶⠛⠻⣾⡇⠀⠀⠀⢀⣤⡀⠀⠀⢰⣷⠀⢻⡄⢀⣀⠀⠸⡇⠰⣄⡀⠀⠀⢦⡀⠀⠀⢲⣄⠀⠀⠀⣦⡀⠀⠀⣹
⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠘⣿⣧⠀⠀⢸⣿⠂⠀⢻⣾⡟⠀⠀⠛⢛⣿⠟⠛⠛⢛⡿⠛⠛⢛⣿⠟⠛⠛⣻⡿⠛⠛⠛
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡀⠀⢸⣿⡁⠀⠀⠀⠈⠁⠀⠀⠀⠋⠀⠀⠀⠈⠁⠀⠀⠀⠋
⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⣿⡆⠀⢸⣿⠀⢠⣿⠏⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣠⣤⣦⣤⣤⣄
⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠱⠂⠀⠹⠀⠀⠀⠻⠛⠀⠘⠛⠀⠘⠿⠀⠈⣧⠀⠀⠀⣠⣴⣶⣾⡟⠛⢿⡟⠻⣿⡏⠙⣷⡀⢻⡄⢻⣧
⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⡀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠾⠛⢿⣄⠀⠀⢻⣄⠈⢿⡄⠙⣿⣆⢘⣷⣄⣿⡄⢻⣇
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣿⣄⡀⠀⠈⢻⣆⠀⠀⠻⣧⠈⠉⢀⢸⣿⠋⠉⠁⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈⠂⠀⠀⠀⠀⠀⢃⡰⢶⣞⠉⠉⠉⠻⣄⠙⢷⡈⠙⢶⣄⠀⠻⠇⠀⣀⠀⠀⢁⠈⢸⣿⠀⠀⠀⢀⠐⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠈⠙⢦⣀⠀⠈⠳⠈⠛⠀⣠⢹⣿⡰⠀⠀⠀⠀⠠⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⡀⠀⠀⠀⠀⢸⣁⣀⠠⣄⠀⠀⠀⠉⠃⠀⠠⠔⠀⠉⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠄⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢿⡉⠉⢳⣼⣿⣀⠀⠄⠁⠆⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠸⡇⠀⠀⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⡏⠀⠀⠻⣦⡟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⢈⣀⣸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠘⣧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢻⣄⠂⠀⠀⢀⣀⢹⣿⣾⡷⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⢧⡀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠐⣊⣥⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠻⣦⣤⣵⣤⣀⣤⣄⣀⣀⠀⠀⠈⠀⠀⠀⢀⡠⢌⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢿⣆⠀⠀⣀⠔⠈⣁⣴⠟⠁⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣼⣤⣴⡾⠋⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠉⠁⢠⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢀⣤⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣠⡶⣤⡾⣧
⠀⢀⣀⣼⠁⠀⠀⠀⠀⠀⠀⣿⣀⣤⣄⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠶⠶⠾⠛⠁
⢰⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⣧⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⡜⠻⣶⡖⠳⢦⠟
⢿⠀⣤⡀⠀⠀⠰⣄⠀⠀⠠⣄⠀⠐⣤⠛⠛⢷⡀⠀⠘⣧⡀⠀⠀⠀⠀⣀⣾⠉⠛⣷⡾⠟⠉⠙⠋⠀⠀⠀⠀⠀⣠⠴⠤⣄
⠘⠓⠚⣿⠶⠒⠒⢻⡷⠶⠖⢚⡷⠒⢚⣿⠒⠚⠃⠀⠀⠈⠻⣤⡀⠀⣿⣍⣀⣀⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⢳⡄
⠀⠀⠛⠁⠀⠀⠐⠋⠀⠀⠀⠋⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡇⠈⠉⠉⠛⠃⠀⠀⠀⠀⠀⢀⡴⠒⠺⠃⠀⠀⠀⠀⠀⣷⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀⡀⠀⠠⣄⠀⠀⠀⠰⣄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⢾⣿⠶⠶⠞⣿⠖⠒⠒⣺⡷⠒⠛

⢠⣶⣶⣶⣶⣴⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡀
⢸⣿⢋⡙⠉⠋⠀⢻⣟⡣⠟⣛⢻⠋⠟⡦⣿⣿⣿⠀⠀⠀⠀⠀⣤⡤⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡄⠇
⢸⡟⢮⡡⢇⣤⡾⢸⣿⣧⡟⠀⠀⡀⠀⠐⠙⢿⣿⠀⠀⠀⠀⡶⣾⣇⠙⠛⣰⣦⣤⡀⠀⠀⠀⢰⢸⣿⡿⣿⠙⡟⢃⢂
⢸⣿⠟⡁⠘⠛⣹⢠⣿⣷⠘⢠⡆⠀⠀⠀⠀⢻⡿⠀⠀⢀⣀⡴⠙⠛⠇⢠⡿⠁⠁⡀⠀⠀⠀⢸⣾⣿⣤⣿⠀⣞⠸⡘
⢸⣯⠀⡀⠐⣠⢋⢾⡟⢇⡅⠀⠁⠀⠀⡄⠀⣼⣇⠀⠀⠀⠉⠑⣀⣂⣴⣂⣀⣰⣴⠓⣀⠀⠀⠈⡌⢻⣿⣯⣛⣯⠀⡇
⢘⡿⠤⣿⠆⡧⠀⠛⡁⢚⠂⡀⢀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠈⠹⠻⣿⠿⢿⠻⠁⠂⠀⠀⠀⠁⢎⣻⡏⡽⡟⢇⠐
⢈⣷⠐⣿⣷⣅⣾⢾⣶⣸⣢⣼⠂⠀⠀⢈⠀⣿⣿⠀⠀⠀⠀⠀⠀⣰⡔⠵⡀⠀⠃⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈
⢨⣿⣏⣿⣿⠦⢰⣿⣵⣳⢟⣻⠐⡖⠂⣷⠀⣾⣿⠀⠀⠀⠀⣀⣶⣭⣳⠆⣐⣤⡶⡻⠂⠀⣀⠀⠀⠀⠀⠀⠀⣀
⢸⣿⣿⣿⡏⠘⣻⡿⠞⠽⠎⣥⢀⡁⡺⣭⠀⣸⣯⠀⠀⠀⢀⣿⣿⡿⣿⣻⣿⢿⣦⡉⠀⠀⠐⠁⠛⠐⠀⠛⠃⠒⠙⠙
⢸⣿⣿⣿⣿⣏⡅⠀⠀⡄⠒⢿⣹⠆⣝⢱⠀⢹⣿⠀⠀⠀⠀⠈⠉⠙⠟⠛⠙⠋⠉
⢸⣿⣿⣿⣛⠿⠓⠀⢲⣶⣿⣗⣿⠀⢠⡈⠀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀⠤⢠⣠⣄⣤⣶⣾⣦
⢸⣿⣿⣿⣍⣛⣹⣿⣻⣿⣿⣇⣇⠐⢘⣤⣠⣾⣿⠀⠀⢠⡖⠐⠀⠊⠉⠀⠀⠈⠀⠀⠐⠹⣿⣷⣾⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣄⣀⣿⣿⣿⣿⠀⠀⢨⣄⠀⠈⠺⠀⠀⠈⠈⠀⠀⢈⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⡆
⠀⠀⠀⠈⠈⠈⠁⠀⠀⠀⠀⠈⣉⣀⠀⠀⠀⠀⠁⠀⠀⢀⠬⠤⠀⠤⠄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠄⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠉⠀⠀⠀⠀⠀⠐⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠻⠏⠿⠿⠿⠿⠿⠟⠿⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣠⣴⡿⠿⠛⠉⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⢀⣤⣾⠄⠀⠀⠀⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠄⠀⠀⠀⠴⠟⠋⠀⠀⢀⣠⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣄⣀⣠⣤⣶⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⡿⠿⠟⠛⣷⣤⣀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠀⠈⡭⠖⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠟⠁⠀⠻⢿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠓⠦⠄⠤⠌⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⡈⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⣀⣤⠄⢀⣀⣠⣤⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠡⠙⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⢀⠉⠼⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢠⣤⠄⠀⠔⠀⣤⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⢀⡀⠀⠀⣸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣦⣤⣤⣄⣀⣀⣠⠤⠠⠤⠉⠀⠀⠀⠀⠀⢹⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣉⠀⠀⠀⠀⠠⠖⠂⣀⣴⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠿⠋⡙⠟⠉⠀⠀⢀⣤⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⠿⠛⠉⠀⠐⠀⠀⠀⣀⣤⣶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣄⡀⠀⠀⠉⠛⠿⠋⠁⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢨⢸⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣴⡆⠶⠆⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⡇⠀⠀⠀⠒⠀⠾⠋⠀⠀⠶⠀⠀⠙⠷⠴⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⠀⠐⣤⠂⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣶⠩⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣰⡂⠀⠀⠙⠛⠀⠒⠘⠟⡋⠀⡀⣰⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⣻⣗⠀⠀⠂⠀⣠⣿⠀⠀⠰⠄⠀⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠻⠟⠀⠀⠀⢤⣿⣿⣣⡤⢀⠀⠘⢻⠗⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄⠀⠀⠆⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠈⠡⠄⠀⠀⠀⠿⠀⠀⠀⠰⠞⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣶⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣀⣠⣄⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⢾⡿⣿⠿⣿⢛⣛⣛⣟⣛⣻⡿⢿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⣯⡔⠑⠪⢑⢁⡿⠨⠙⢻⣽⣻⣏⣿⣷⣿⣯⣛⡻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⡿⠟⠀⣨⣿⡚⠪⠪⠁⢹⡖⠒⠂⡆⠿⠁⠉⢿⣿⣿⣽⢿⣿⣝⣿⣿⣆⡀
⠀⠀⠀⠀⠀⢰⣿⠟⡻⢷⣄⣴⡾⢿⠷⢿⡠⠒⠪⢹⡆⠒⠊⢹⠎⠑⣶⣆⢛⡾⣽⣹⣿⡿⣿⣾⢿⣿⣦
⠀⠀⠀⠀⢰⣿⢳⢎⢆⣴⠏⠳⣀⡠⠐⠈⢇⠀⢀⣫⣀⣂⣠⣭⣐⢢⣟⣽⢿⣦⣼⠛⣿⣿⣿⣞⣷⡻⣿⣷
⠀⠀⠀⣠⣿⣿⣵⣤⣪⣓⡂⡼⠉⠗⡦⠀⣠⢖⣿⠿⢺⡟⠾⠃⣼⡋⢲⣓⣿⣇⣽⣶⢻⣿⣿⣞⣿⣿⣽⣿⣧
⠀⠀⠠⣿⣿⣟⣀⣾⣋⣉⣻⣟⣓⣛⣲⣾⣷⣋⣁⣄⣭⣇⣈⣝⣷⣙⣶⣟⣷⣟⣆⣻⣀⣻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢸⣿⣿⡟⡿⣩⠙⣭⡿⣹⣿⣟⣽⣿⣿⡿⣽⣿⡿⣽⣿⡏⢻⡽⣿⣿⡏⡏⣿⢹⢫⣿⡿⣻⣿⡿⣿⡽⣿⣧
⠀⠀⣿⣿⣿⠏⡕⢿⣸⣿⠇⣿⣿⠘⣿⣿⢏⡕⢿⣿⡇⣿⣿⣿⢸⡇⣿⣿⣻⡇⣿⢸⢸⣿⡇⣿⣿⣇⢻⡇⣿⣿
⠀⠀⢸⣿⣿⣼⣿⣼⣼⣿⣿⣬⣿⣷⣽⣿⣼⣿⣼⣿⣿⣽⣿⣿⣼⣧⣿⣿⣯⣷⣿⣼⣮⣿⣿⣮⣿⣿⣼⣵⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⣿⣿⣿⣔⣹⣷⣿⣿⡝⣦⠾⢯⣿⣿⣿⣿⣽⣿⢿⣿⣿⣿⣿⣿⣿⡿⢣⣾⣛⣽⣿⣿⣷⢷⣿⣿⣿⣿⡟
⠀⠀⠀⠙⣿⣿⣿⣟⣿⣿⣿⣿⣮⡿⣿⣬⣟⡛⢿⣿⣿⣿⣿⣿⣿⡿⣟⣭⣽⢋⣼⣿⣿⣿⣿⣏⡾⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠈⢹⣿⣿⣾⢯⡻⣿⣿⣿⣾⣽⡿⢯⣿⣦⣩⣏⣉⣿⣿⣾⡿⣿⣿⣿⣿⣿⡿⣯⢞⣯⣿⣯⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢻⣿⣯⣧⡹⢮⣛⢿⣿⣿⣿⣷⣿⣽⣛⣿⣷⣿⣭⣿⣾⣿⣿⣿⣿⣯⢞⣿⢿⣻⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣽⣷⣾⣟⢿⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⡳⣽⣺⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣿⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⢛⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠛⠛⠋⠉⠠⣤⣀⣼⢃⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⠀⣀⣀⣤⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⣿⡿⠉⠻⣿⣿⠟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣻⣿⣿⣿⣿⣿
⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢸⣿⠁⢀⡠⣄⠓⢲⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠿⢃⡤⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⣿⡏⠀⠈⠿⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣣⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣾⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢠⣿⣥⣄⢀⣾⣿⢟⣿⣿⣿⣿⣿⡿⠿⢟⡟⠈⡱⡜⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⢋⡵
⠉⠛⠿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⢸⣿⠀⢀⣿⡟⠉⣩⠿⠋⠁⠋⠁⡀⣸⠟⡯⠞⣰⣯⣷⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⢋⡴⠉⣠
⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣷⣦⣄⣀⣸⡟⠀⣼⡟⠀⠀⠀⠀⠄⡀⠀⠀⠔⠁⣈⣤⣾⣿⣿⠿⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⠋⢠⡾⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣷⠞⠛⠁⠀⠀⠀⠠⠊⠐⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣩⣇⣴⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⢀⡴⠊⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣃⡀⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠄⠀⢀⡴⠛⠀⠈⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⣞⣃⣀⣀⣀
⣀⣀⣤⣄⣀⣀⣀⡀⢠⡞⠁⠀⢠⠔⠀⠀⠐⠁⠀⠀⢀⣠⣾⢛⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠁⡴⢣⠆⠀⠄⠀⢀⠂⢠⣚⣟⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠉⠁⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣯⢀⡌⣰⡥⢀⠎⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠓⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣴⠁⡨⢋⣾⣥⡏⠀⡔⢀⣷⣮⣼⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠛⠉⠀⠈⠑⠦
⣿⣿⣿⣿⣿⣿⣿⣷⡟⢀⠄⠞⣼⢋⣟⣰⣼⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣠⡿⠄⣼⡗⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣶⣶⣤⣤⣤⣤⣀⣠⣤⣤⣤⣶⣶⣶⣤⡶⠞⠛⠉
⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣵⣿⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣷⣄⣤⣀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣀⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠠⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠳⢄⠀⠀⠌⠀⠘⢷⡀⠀⡀⠈⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⡀⠄⠀⠀⠙⠎⠀⠀⠀⠈⠷⠈⠀⠀⠀⠀⣿⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠂⠀⣾⠇
⠀⠀⠀⠀⠀⠈⢉⠀⠀⠄⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⢤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢶⠶⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠹⡄⠀⠀⠀⠸⣆⠀⠀⠀⡼⠁⠀⢀⣠⣀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠞⠃⠀⠂⠀⠀⠿⠆⠀⠀⠁⠀⢀⡀⠀⠋⠀⠀⠀⠀⠀⣰⡿⠋⠉⢀⣀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢀⡀⠀⠀⠀⣿⠛⠁⠀⠀⠀⢿⠁⠀⠀⠀⢠⡀⠀⠈⠃⠀⠀⠀⢀⡀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢃⣀⣀⣠⣿⠀⠀⢤⣤⣶⠾⠆⠀⠛⠛⠻⠃⢤⣀⡀⢀⣀⣴⡿⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣶⠟⠋⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠛⠉⠉⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠁⠒⠘⠷
⠀⠀⠀⠀⠀⠀⠘⠚⠛⠉⠉⠀⣀⣴⠾⠛⠀⠀⢀⠀⠀⣰⡄⠀⠀⠀⣶⠄⡀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠴⡆⠀⢠⣾⠏⠀⠚⠄⢰⣿⠀⠀⠐⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠚⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣯⣽⣿⣯⣽⣯⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞⠙⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣠⢦⡀⠀⠀⠀⢸⡉⠁⡠⠄⠀⠐⠂⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⢀⡟⠓⢦⡀⠀⢸⡹⠙⢦⡀⠀⢀⡾⠁⠀⠹⣦⣀⡴⣋⣳⠀⠁⠀⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⣾⠈⠙⢦⣼⠃⠀⠀⠙⠶⠟⠁⠀⠀⠙⢳⣾⣀⣄⠀⠀⡹⠿⠊⠀⢈⠀⠀⠀⠀⠀⠀⡄⠸⣀
⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣸⠉⡹⠷⣶⠃⠀⡠⢄⣀⡤⣄⠀⠀⣠⠤⣄⠼⠢⣀⡎⠀⠀⠀⠱⠞⢀⣀⡀⡰⠹⡀⢀⠀⠀⠀⠀⢠⠀⢘⡄
⠫⣵⠀⠀⠀⡰⠋⢉⡁⠀⠀⠈⡉⣀⠀⠀⠑⠖⠁⠀⣀⡀⠀⠑⢴⠁⣀⠠⠀⠀⠈⠁⠆⠐⡄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⡀⠸⡀
⠀⠀⡠⠏⢉⢁⡊⠁⠀⠀⠀⠀⠒⠀⠐⠠⡀⠀⠉⠁⠀⠈⠢⣀⠀⠀⠁⠀⠀⠐⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠚⠀⠀⠀⠀⠁⡡⠤⡹⡄
⠀⠀⢳⠀⢡⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠈⢆⠀⠐⠀⠀⠀⠈⠉⠙⢄⠀⠀⠀⢆⠀⠀⠊⢰⠂⠀⠀⠒⠒⠀⢀⠅⠀⢁⠘
⠀⠀⠀⢆⠈⡇⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⠈⠀⠀⢰⡄⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡄⠀⠀⠃⠀⠀⠀⠀⢀⠀⠂⠀⠠⠈
⠀⠀⠀⠸⡄⠁⠀⠀⠀⠀⠀⠀⠀⠠⠮⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠀⡇⠀⠀⠓⣄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠀⢘⠀⠀⣀⢀⣀⣠⠖
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⠀⣀⠀⠀⠀⠑⠈⠆⠀⠀⠸⠓⢄⠀⠀⠀⢰⠐⢄⣀⠠⠠⠋⠉⠈⠀⠀⡏⢩⣤
⣠⢄⡀⠀⠘⡆⡄⠀⠀⠀⠰⠝⠀⠀⠀⠰⠁⠀⠀⠀⠀⠈⠀⠑⡀⠀⢀⠄⠐⠤⠘⠐⠀⢀⡑⠈⠉⠀⢀⣄⠀⠀⣤⠤⠴⠾⠿⠾⠀⠀⠯⠗
⠙⠈⠁⠀⠀⠇⢰⠀⠀⢀⡀⠀⠀⠀⠀⡈⢀⡤⡠⠄⣀⠄⠀⠀⠈⠀⠁⠀⠀⢀⠶⠤⠴⠋⠙⠚⠛⣲⠧⣤⣙⡊⢹
⠀⠀⠀⠀⠀⢸⠈⡄⢀⡄⠡⣀⠤⠔⠋⠀⣀⡀⢀⣠⢄⣀⣸⠉⠲⠒⠉⢉⣉⡉⠀⠀⠀⣰⠦⣄⣀⣿⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠘⠦⣀⠤⠴⣄⣠⣤⡤⠤⣴⡷⠟⣿⡄⠀⠈⢹⡷⣄⡀⢠⡟⠉⠙⠶⣄⣼⠃⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠉⠀⠀⢹⡇⠀⠀⢸⠇⠀⠙⠟⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠗⠀⠀⢀⡀⠈⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠢⣿⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡶⣤⣄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣼⠃⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⣿⣿⡾⠿⠷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠁⣿⢸⠉⠙⠛⠲⢤⡀
⠀⠀⠀⠀⣀⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⣀⡟⠀⢢⣤⠞⠸⠶⠶⣦⣀⡾⠀⢀⣤⣄⣀⡀
⠀⠀⣠⠞⠁⠀⠀⠀⢹⠀⠀⠀⣀⣤⣄⠀⠀⢹⠈⢳⡀⠀⡏⠉⠉⢡⠀⡾⠁⠀⠀⠀⠀⠀⠉⠁⠀⢸⠃⠀⠈⠉⢳⡆
⠀⠀⢻⡄⢰⣒⡄⠀⣾⠀⠀⡟⠁⠀⠘⢧⡀⠘⡇⠀⠻⡄⣇⠀⠀⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⡟
⠀⠀⠀⢷⠀⠉⢀⣴⠏⠀⢀⡇⠾⠓⢦⡈⢳⡄⣧⠀⠀⠹⣿⠋⠀⢸⢸⡄⢀⡶⠲⠶⣤⣀⡀⠀⢠⡇⠀⢀⣈⡿
⠀⠀⠀⢸⡶⠛⠛⠛⠳⡶⠾⢧⣄⣀⡤⠽⠂⣹⣿⠀⠀⠀⠙⠀⠐⢺⠀⢳⡘⠷⣤⠀⠀⠈⢹⠆⣸⠦⠀⠀⣼⠁
⠀⠀⠀⢸⣧⠀⢀⣠⣄⣀⠀⠀⠈⢧⢠⣄⡀⠁⠘⢆⠀⣠⠀⠀⠀⢸⡀⠀⠉⠹⠉⠀⣴⢤⡾⠀⣿⣀⠀⣰⠃
⠀⠀⠀⠈⡗⠀⢸⠀⠀⠈⠳⡀⠀⢸⢹⠈⠛⢦⣀⣸⠁⢸⣦⠆⠀⢸⠋⠛⠛⣿⠀⢸⠃⠀⠀⢀⣠⣍⡙⠋
⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⣰⠁⠀⣸⢸⡄⠀⣸⣉⣁⣠⡿⠹⠶⠶⠚⠀⠀⠀⠛⠒⠛⠀⠀⠀⡟⠃⠈⣷
⠀⠀⠀⠀⢻⠀⠘⠓⠚⠋⠁⠀⣠⡧⠼⠃⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠁
⠀⠀⠀⠀⢸⣄⣴⣤⣤⣤⡷⠞⠋
⠀⠀⠀⠀⠀⠈⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣻⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢳⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢳⡀⠀⠀⠀⢰⣇⠀⠀⠀⠀⡴⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢳⡀⠀⠀⣼⢻⠀⠀⢠⡞⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢳⡀⢀⡏⠸⡇⣰⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢳⣼⠇⠀⡿⠃⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠀⠀⢿⡆⠀⠀⢻⠀⠀⠀⠀⠀⣸⠁⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡉⠛⠓⠶⢾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠟⠚⢉⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠶⠶⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢲⡶⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⣴⠾⡿⠀⠀⣠⡀⠀⠀⣿⠷⣦⣄⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠀⠀⢸⡇⣠⡾⠛⣷⡄⠀⣿⠀⠀⠉⠙⠻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠈⠻⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⠀⠀⠈⣿⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⢹⣷⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠘⣿⣄⠀⠀⠈⣿⡆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠁⠀⠀⠀⠘⣿⣆⠀⠀⢹⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⣿⡇⠀⢸⣿⠀⠀⢀⣴⡿⠿⢿⣦⣠⣄⡀⠀⠘⠿⠿⠿⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠀⠈⠀⠀⠀⠁⠀⠀⠸⠿⠀⠀⠀⠻⠋⠙⣿⣶⣶⣦⣤⣤⣀⡀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣴⡶⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⡿⠿⢿⣶⣦⠀⢀⣤⣿⠋⠀⠀⠹⠋⠀⠀⠙⠿⢿⠿⣷⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠉⠀⣀⣤⣴⣤⣄⣀⣀⡀⠀⠀⣼⡟⠉⠉⢉⣩⣿⠿⠋⠁⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠁⠀⠀⠀⠀⠀⢀⣀⣿⡏⠉⠀⠀⣼⡟⠉⠁⠉⠛⠛⠛⠛⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⢀⣠⣴⡾⠿⠛⠙⠻⠿⠀⠀⠉⠁⠀⢀⣀⣀⣤⣤⣶⣶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠐⠿⠿⠿⢷⣶⣶⣶⣶⣶⣶⣶⠾⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⠿⠿⠛⢋⣉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⢿⣿⠟⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⢀⣀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣶⣤⡄⠀⠀⠀⠀⠈⠻⢷⠿⣷⣶⡀⠀⢀⣴⡄⠀⢀⣼⡟⠋⠁⠀⠀⠀⠀⠀⠀⢴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⠻⠿⠿⠋⠀⢠⣤⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢠⣄⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⡇⠀⠘⣿⡄⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⣿⡇⠀⠀⢻⣧⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣟⣛⡻⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣸⡍⠉⠁⠀⠸⡇
⠀⠀⠀⠀⠀⣠⡾⠛⠋⠠⠤⠀⠀⠀⠙⡆⠀⠀⠀⢳⡤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣼⠋⠬⠁⣀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡀⠀⠳⣒⠠⠤⠄⣀⠉⠉⠙⠛⠓⠲⠶⠦⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⢀⡀
⠀⠀⠀⢸⡯⠀⠀⠀⠀⠀⠉⠀⠩⢄⠘⠁⠀⠀⠇⠀⠀⠁⠉⠂⠀⠀⠀⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡽⠛⠉⢹⡶⢦⡀
⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠈⠃⠒⠒⠓⠲⠦⠴⢦⣤⡴⠒⠒⠒⠂⠠⠀⠐⠂⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⠔⣀⡸⠃⠀⠹⡆
⠀⠀⠀⠀⢻⣖⡢⠤⠤⠀⢀⠀⢀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢠⡟⡰⣡⠋⡤⢀⠀⠀⠀⢹
⠀⠀⠀⠀⠈⢿⡟⠿⠛⡒⡒⠤⠀⣠⣀⣀⠀⠀⠈⠁⠐⠀⠀⠀⢀⠈⠀⠒⠀⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡞⢀⣷⠁⠰⠀⠎⣠⠃⠀⢸
⠀⠀⠀⠀⠀⠈⠳⠶⢤⣤⣀⡈⠁⠒⠀⠀⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠈⠉⠉⠉⠉⠉⠉⢹⠇⢸⢸⡀⡕⠸⠐⡇⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠦⢤⣤⣀⣀⡀⠉⠐⠒⠠⠄⢀⣀⣀⠀⠀⣀⣀⣀⠀⠀⠀⠤⠤⢄⣀⣸⡀⡎⡸⠁⣇⠀⡆⡇⢸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠳⠶⢤⣄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣄⢸⡇⠁⠀⢰⡈⣰⠃⣧⡜⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠠⣇⠐⡈⢄⠉⢁⢜⡟⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠲⠤⣤⣀⣀⠹⣦⣈⠀⠤⢐⣩⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿
⣿⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠀⠀⠀⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏
⣿⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣿⠀⠀⠀⢾⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢻⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠂⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣹⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡿⠛⠿⠿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣋⡭⢷⡄
⠀⠀⠀⠀⠀⢀⣤⡴⠶⠒⠲⠼⣆⠀⠀⠀⣧
⠀⠀⠀⢀⡼⠋⠀⠀⠉⠀⠀⠀⠈⡆⠀⠀⠘⣟⣓⠒⠒⠒⠶⠦⢤⣤⣀⣀⣀
⠀⠀⠀⣾⠁⠉⠀⠀⠀⠀⠐⡀⠸⠁⠀⠀⠀⠀⠂⠠⠉⠀⠀⠀⠀⠀⠤⠄⠉⠉⠉⠉⠙⠛⠛⠛⡒⣒⡶⠖⢦⣤⡀
⠀⠀⠀⢿⡀⠠⠄⠀⠐⠒⠀⠤⠤⠦⠄⣤⣀⡠⠤⠠⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡠⢀⡸⠁⠙⣦
⠀⠀⠀⠸⣗⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢏⢎⠌⡠⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠻⡏⠛⠩⠙⠐⡀⠤⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠠⠄⢀⣀⣀⢀⣀⣀⣠⡏⡰⡏⠠⠀⠂⡔⡀⢠⡇
⠀⠀⠀⠀⠀⠙⠛⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇⣧⢻⠀⠘⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠦⠤⣤⣄⣀⠀⠒⠂⠀⠀⠐⠒⠀⠀⠀⢈⡁⠒⢺⡀⠀⠁⠸⢂⠇⡇⠇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⣤⣀⡀⠀⠀⠀⠀⠈⠉⢘⡇⠀⢀⠑⠊⡠⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠤⣄⣀⡀⢳⣌⡀⠀⠐⣊⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠂⠀⠍⠀⠂⠀⠈⠀⡀⠀⠄⢤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠥⢈⡑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠿⠷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢠⢦⣀⠀⣀⠤⠔⠒⠉⠀⢀⣠⣴⡶⡆⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠙⠻⠶⣶⣤⣀⠠⠜⠋⠉⠀⣀⣤⠀⢰⠟⠛⠉⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡇⢀⣤⡴⠞⠛⠉⢩⠀⢸⡇⠀⠀⠀⠀⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠅⡂⠀⠠⢀⣾⠀⢸⣅⠀⢀⡐⠁⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢢⡇⠀⡤⠂⠸⠀⢼⡗⠀⠝⠔⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⢸⠌⠇⠀⠁⠀⠀⠀⣸⠁⠁⠎⠀⠸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠅⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠁⣧⠀⠎⠀⠀⠀⣾⡆⠀⠀⠀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⡇⢸⣧⡷⠤⠦⠠⠆⠀⠛⠂⠒⠂⢈⠥⢤⡄⢸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⡂⢘⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡄⢠⣤⣤⣤⣤⣄⡄⠀⡴⠴⠶⠶⠾⣐⡾⠇⢸⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠁⢀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⢸⠁⣦⡀⠀⠀⢀⠀⣷⠀⠠⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡀⢈⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⢸⢸⡆⠀⠀⠆⢸⠀⣿⠀⠀⠀⡐⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠈⡂⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸⠇⠰⠃⠄⠘⠀⣿⣈⠀⢬⠂⡀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠨⡀⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⢸⣤⠈⠀⢄⠀⠀⠀⣿⠴⠺⠣⠈⠁⠀⠀⢼⡗⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢾⣿⡲⢀⣠⣴⡇⠀⣿⠁⡐⠀⠂⡆⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠓⣂⡇⠀⣿⠀⠀⢀⡀⡇⠀⠀⣋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢰⢨⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⠼⠾⠦⣬⣀⠀⠀⡀⠀⠁⠀⠛⠳⠖⣀⠤⡇⠀⠀⢵⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠕⠄⠰⢸⠀⠀⡀⢸⠀⠀⠀⠀⡀⠤⡐⠪⢉⢁⣈⠀⠂⠀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠉⠁⠀⢀⢿
⠀⠀⠀⠀⠀⠀⡀⠁⠰⢸⠀⠀⠂⢸⣀⣠⠴⠊⠀⠁⠀⠀⠀⠀⠈⣁⢀⡀⠀⠒⠐⠀⠀⠈⠉⠉⠒⣢⣄⣀⠀⠀⠠⢸⣽⠀⢀⠐⠊⠉
⠀⠀⠀⠀⠀⢀⠈⡀⢸⣸⠀⢐⣩⡼⠿⠟⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠤⠄⠈⠉⠛⠲⢍⣸⣿⠠⣀⣀⠠⢤⣆
⠀⠀⠀⠀⠀⢈⣡⡤⠚⠛⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠈⠄⢐⡀
⠀⠀⠀⠴⠚⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⡏⠀⠀⠀⠠⠊⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⢀⣠⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢄⡠⠐⠐⣢⠔⠈⠑⡄⠀⢤⡀⠀⠀⠀⠀⠀⢀⣺⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⠎⠀⣠⠞⠁⠀⣠⠊⠀⠀⡄⢳⠀⠀⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠐⠋⠀⠀⠉⠀⠀⠚⠁⠀⢀⡴⣡⠞⢈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡇⠀⠀⠀⡼⠀⠀⣸⡟⠁⠀⠫⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠁⠀⢰⡇⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠤⣄⡀⠀⡏⠙⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣈⡉⠁⠐⠠⡀⠀⠀⠀⠀⠀⠀⣀⡼⠃⡀⠀⠀⠀⠀⠀⡆⠈⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣦⡀⠘⣆⢀⡴⠛⠛⠉⠉⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⡿⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠉⠀⠀⠉⠉⠉⠉⠉⠁⢀⠀⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠳⠶⠶⢶⡆⠀⠀⠀⢠⡶⢶⣶⣶⣶⣶⣶⣦⠀⡴⠒⢦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡄⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢙⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠀⠈




⣿⣿⣿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣾⣶⣶⣶⣿⣶⣿⣵⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢰⡶⠶⠶⠶⠶⢦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢸⠃⠀⠀⢰⠶⠶⠶⠶⠖⠒⠒⠒⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⠀⠀⠀⣿
⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⡆
⠀⢸⡀⠀⠀⢸⠀⠀⠀⠀⣟⡽⠯⠭⠿⠿⠛⠳⠙⡇⠀⢸⣏⡭⠭⠭⠭⠽⠭⢍⣶⠀⠀⢸⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⢰⣧⡄⠀⠀⠀⠀⠀⢸⣽⠀⠀⢸⠇⠀⠀⢸⡇
⠀⢸⡅⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡏⡇⠀⠸⣿⡇⠀⠀⠀⠀⠀⢸⣼⠀⠀⢸⡀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⣻⠇⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⠇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡀⠀⠀⠀⡇⡧⠀⠀⠀⠀⠀⠀⣷⡇⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⣿⠂⠀⠀⠀⠀⠀⢸⣾⠀⠀⠘⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢻⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣾⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣿⡆⠀⠐⣿⡁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢼⠀⠀⠀⠀⠀⠀⣿⡁⠀⢰⣿⡇⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⡇⠀⠀⢀⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢷⢸⠀⠀⠀⠀⠀⠀⣿⠇⠀⠈⣧⠆⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⡇⠀⠀⠘⡇
⠀⢸⡇⠀⠀⠘⡇⢀⢤⡀⢸⣜⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⣻⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠘⡇⠀⠀⠀⡇⢼⠀⠇⠸⠯⠭⠿⠿⠷⠿⠟⠿⠆⠀⠀⠫⠿⠶⠶⠾⠶⠿⠿⠿⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⡜⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⠳⠤⠞⢀⣄⣀⢀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⢤⣄⣤⣄⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⢀⠁⡆⢸⢹⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠋⠛⠉⢻⢸⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⠈⠒⠁⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⠀⣧
⠀⠈⡇⠀⠀⠸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠎⡄⠀⠀⡇⠀⠀⠀⡿
⠀⢸⡇⠀⠀⢐⡇⠀⠀⠀⠸⢾⠭⠭⠿⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⠭⠭⠭⠭⢶⠇⠀⠀⣗⠀⠀⠀⣟
⠀⢸⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣟
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢰⣖⣒⣒⣶⣶⣶⢶⣶⣶⣶⣾⣷⡾⣶⠶⠶⠶⠶⢶⣶⠀⠀⠀⡗⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⡇⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠘⡇⠀⠀⢀⡇⠀⠀⠀⠀⡿⣿⠾⣿⣿⣿⣿⡿⠿⠾⠶⠿⠷⢾⣓⠺⠶⣖⠚⠛⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⣠⣧⣤⣤⡤⣷⠶⠦⡶⢦⣴⠴⠦⣤⣶⠶⣦⣶⣶⣶⣶⡶⢶⣶⡦⠶⠶⢶⠿⢿⠿⠗⠒⠧⠀⢤⡤⠷

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠸⣿⠇⠀⣰⠆⢸⠀⠀⣶⡶⣶⡄⣶⡆⠀⣴⣶⠶⠀⣰⣶⣶⡀⣰⣶⠶⠂⣶⡶⠶⠀⠀⢸
⠀⢸⠀⠀⠀⣠⢾⣿⡟⠛⠁⢠⣾⠀⠀⣿⡟⠛⠁⣿⣧⣤⣿⣿⣭⣠⣿⠷⢿⣧⣬⣽⣿⠆⣿⣿⣅⠀⠀⢸
⠀⢸⠀⠀⠰⠏⢸⣿⡇⠀⠀⢸⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⠀⠀⣿⣧⣾⡿⢸⣿⣾⣿⡇⣼⡿⣿⡆⢰⣿⠿⠇⢸⣿⣼⣿⠃⠀⠀⢸
⠀⢸⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⠀⠀⠿⠏⠻⠿⠸⠿⠟⠿⠇⠙⠿⠿⠃⠘⠿⠿⠇⠸⠿⠙⠿⠆⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠘⠋⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⣛⣛⣛⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⠤⠤⠤⠤⠶⠶⠋⠀⠀⠀⠉⠁⠠⣧⣤⡤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠉⠉⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣴⣁⡀
⠀⠀⠀⠀⢀⣤⣶⣿⣧⣀⣀⣠⣾⣿⡶⠶⠶⠶⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢰⡿⠉⠀⠙⣿⡟⢛⣿⠟⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⠁⠀⠀⠀⢸⣿⣾⠃⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⠀⠀⠀⠀⢘⣿⣿⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢿⢧⠀⠀⠀⣸⡟⣿⣆⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠋
⠀⠀⠀⠀⠻⣷⣤⣴⡟⠁⠈⢿⣦⣤⣾⠏⣀⣀⣀⣀⣠⣤⡤⠤⠤⠶⠖⠒⠒⠚⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠑⢬⢦⡀⠀⠀⠈⠐⠀⡢⠄⡀⠀⣀⣐⣠⡒⠚⡁⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠐⢂⣠⠄⠀⠱⢤⠂⠄⠄⠠⠄⡤⠚⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠢⠖⠉⠀⠀⠀⠈⠙⢅⠐⠂
⠀⠀⠀⠀⠰⠒⡊⠐⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠄⠄⠀⠀⠀⠀⠁⠂
⠀⠀⠀⣠⢈⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠤⠀⠀⠀⡖⢲⠙⠉⠀⠀⣸⢙⠀⠀⢐⡄⠐⠈⠉⠀⠀⠉⠻⡄⠂⠀⠀⠀⠀⠉⢲⣄
⠀⢀⡀⠴⠈⠠⠄⡀⠀⠁⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠻⡄⠄⠀⠨⠃⠀⠀⢀⣀⣀⣀⠸⠃⠆⠀⠀⠀⠀⠀⣘⢃
⠀⠀⠱⠰⠃⠀⢦⣤⠀⠀⠀⡦⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀⠀⠈⠇⠀⠀⠀⠀⠂⣀⣀⣀⣀⠤⣄⢠⠀⠀⠀⠀⠀⠀⢣⡞
⠀⠐⢢⠠⡀⠀⠀⡄⠇⠀⠀⠋⠀⠀⠀⡇⢸⠀⠀⣛⡀⠀⠀⢫⠀⠀⢸⢆⠀⠀⠀⠀⠀⢸⡉⡍⠀⠀⠀⠀⠈⡛⢢⠶⠀⠀⠀⢀⠜⣡
⠀⠀⠰⠀⣉⠂⠀⠇⢠⠀⠀⠀⠀⠈⠉⠳⣴⠀⠀⠀⠀⠀⠀⠈⠆⠀⠨⡄⢂⠀⠀⠀⡀⠈⠛⣓⠂⡀⠀⠀⠀⡔⠀⠀⠀⠀⡀⢈⢨⡀
⠀⠀⢄⠀⠸⠃⠀⢰⠸⠀⠀⢀⣄⡀⠀⠀⠘⠀⠀⠠⢤⢄⠀⠀⠀⡀⠀⡙⣄⢂⡀⠀⣤⡀⠀⠀⠀⠀⠀⢀⣄⠡⡀⠄⠀⠠⠁⠀⢳⡆
⠀⠀⠐⣠⠁⠀⠀⢸⣤⡄⠀⠀⠿⠆⠀⠀⢀⡀⠀⠀⢨⡈⣄⠀⢀⠗⠀⠁⠈⠉⠉⠉⠑⠚⠒⠒⠒⠂⠀⠀⠈⠉⠉⠛⠈⠀⠀⠀⢐⠜
⠀⠀⠀⠋⣄⠀⠀⢸⣬⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠂
⠀⠀⠉⠣⠈⠢⢀⠀⠁⠁⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡠⠊⠈⠀⠈⠀⡠⠆
⠀⠀⠀⠀⠁⠉⠀⠉⢣⢲⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡠⢀⠐⠈⠹⠲⡂⢤⠄⢄⡀⠀⢀⠠⠀⢀⠀⠁⠒⠲⠅⠀⠒⠢⠈⠈⠑
⠀⠀⠀⠀⠀⠀⠂⠀⢀⠑⠁⠒⠒⠀⡂⠜⠠⢤⠑⠀⠲⡤⠀⠀⠀⡀⠠⠉⠈⢃⠬⡀⡀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠸⠃⠉⠐⠐⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡞⠉⠉⠉⡟⠉⠉⠙⡆⠀⠀⡇⠀⢰⠖⠒⠒⢲⠒⠒⠒⢲⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡟⠉⠉⠉⡏⠉⠉⠉⡇⠀⠀⡇⠀⢸⠒⠒⠒⢺⠒⠒⠒⢺⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⣀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠛⢻⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⣧⣤⣤⣤⣧⣤⣤⣤⡇⠀⠀⡇⣶⣼⣀⣀⣀⣸⣀⣀⣀⣸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠁⠀⠀⢸⠁⠀⠀⢹⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡗⠒⠒⠒⡗⠒⠒⠒⡇⠀⠀⡇⠀⢸⠦⠤⠤⢼⠤⠤⠤⢼⠀⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠧⠤⠤⠤⠧⠤⠤⠤⠇⠀⠀⡇⠀⢸⣄⣀⣀⣸⣀⣀⣀⣸⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣽⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⣿⠛⣿⠀⠀⣴⣦⡀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠰⣿⣅⠻⣧⠀⠀⣿⣰⡏⠀⣼⠃⣸⡿⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣦⣄⡈⠻⣷⣿⣇⠀⣿⡟⠀⣼⣧⠞⠋⠀⠀⠀⠀⠀⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⣄⣉⡻⣶⣄⡙⠟⠀⠉⠀⣈⣃⣀⣀⣀⡀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠙⠛⠛⢿⣟⠀⢀⣰⡟⠉⠙⣿⡍⠉⢻⣆⠀⣀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣤⠶⠞⣻⠟⢰⣟⠙⢿⣆⠀⠈⢿⡄⠀⠙⠛⠙⠻⢿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠻⣧⣤⠞⠁⠀⠈⣿⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠁⠀⠀⠀⠀⢹⡇⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⢷⣆⠀⣀⣤⡴⠾⠛⠋⠁⣀⣹⣷⣶⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠋⠁⢀⣠⡴⠞⠛⠉⠁⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣠⠶⠶⣦⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⣾⠁⠀⠀⠈⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠹⣧⣀⣀⣴⠏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠁⠀⠀⠉⠉⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀⣀⣀⣀⣀⢀⡀⡀⠀⠀⠀⢺⣆⣿⣼⣿⣧⣸⣇⣾⣿⣼⣿⣴⣿⢿⡗
⠀⠀⠀⠀⠀⣾⠛⣷⢹⣤⣿⠹⢀⡰⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⣯⡿⢿⣿⣿⣻⠃⠀⠀⠀⠈⠿⠏⠿⠟⠟⠙⠛⠋⠛⠻⠋⠃⠿⡭⠁
⠀⠀⠀⠀⠀⠙⠛⠃⠘⠁⠉⠈⠘⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠠⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠔⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⡄⣠⣤⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣤⣠⡤⢰⠆⠀⠀⠀⠀⠀⠀⣰⡟⢻⣾⣿⣶⢿⣿⣏⣲⡆⠿⠀⡄⠀⠀⠀⠀⣴⢶⡄⣿⢸⡇⣾⠋⢸⣇⣿⢹⠇
⠀⠀⠀⠀⢾⣻⣺⣿⣾⠟⠿⢿⣼⠯⠀⠀⠀⠀⠀⠀⠠⡘⠷⠾⠛⠛⠉⠘⠃⠙⠋⠑⠃⡠⠃⠀⠀⠀⠀⠻⠾⠃⠙⠛⠁⠙⠛⠘⠁⠻⠐
⠀⠀⠀⠀⠘⠛⠉⠉⠉⠀⠀⢀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠀⠀⠐⠒⠂⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠾⠷⠞⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣡⣠⡄⠀⠀⠀⠀⣠⡦⣠⡶⣦⣴⠶⣦⣴⠆⢠⡄
⠀⠀⢀⣤⡤⣶⣖⡦⢰⣿⡄⣿⣛⠀⣧⣸⣿⡇⠀⠀⠀⠀⠀⠀⣀⢀⣤⣤⣾⢻⣾⣷⢿⣿⡃⠀⠀⠀⠀⢿⣤⠿⠦⠞⠛⠶⠟⠿⢶⡛⢠⠃
⠀⠀⢻⣄⣤⢹⡟⠷⠼⠗⠿⠰⠾⠛⠋⠈⠋⠂⠀⠀⠀⠀⠀⢺⣿⣿⠷⠾⠉⠛⠁⠁⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠔⠁
⠀⠀⠀⠉⠁⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁⡀⠀⠀⠀⢀⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣠⡶⠆⣶
⠀⠀⠀⠐⣿⣻⢾⣏⣿⣿⡾⣷⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠲⣴⢶⣶⢃⣶⣆⣶⢸⣾⡇⠀⠀⠀⠀⠀⠠⢀⣤⣄⣾⠛⣿⣿⠞⠉⣛⣷⣏
⠀⠀⠀⠀⠹⠃⠀⠉⠉⠈⠁⠉⠛⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⠷⠾⠛⠿⠟⠹⠯⠁⢲⠀⠀⠀⠀⠀⢿⣁⣽⠻⠶⠋⠛⠀⠀⠀⡈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠠⠤⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠙⠁⠀⠂⢀⢤⡄⠰⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢊⡈⠁⢨⡉⠉⠉⢠⡈⠁⢁⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠈⡇⠀⣸⡀⠀⠀⣸⡀⠀⣸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠠⠏⠉⢻⠋⠉⠉⢻⠋⠉⢻⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢠⡆⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢀⡗⠒⢻⠗⠒⠒⢻⠗⠒⢻⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢘⡇⠦⢾⡆⠶⠤⢾⠦⠤⢾⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠸⣇⣀⣸⣀⣀⣀⣸⣀⣀⣸⣷⣶⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⠁⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣠⡶⠛⠛⢶⣄⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⢻⡆⠀⠀⠀⣿⠁⠀⠀⠙⣧⠐⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢾⣇⠀⠀⠀⢈⡇⠀⠀⢰⣏⣤⠶⠶⢶⡏⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢹⡄⠀⠀⢸⡷⠖⠛⠋⠉⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡄⠀⠀⠸⡇⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡀⠀⠀⣷⣀⣀⡼⣿⠶⢦⣤⣤⣤⣴⠋⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⠶⠞⠃⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠲⠶⠶⠚⠛⠋⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠟⢿⡀⣶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢸⡇⣿⠀⠀⠀⠀⠉⠙⢷⡄
⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠰⣄⣠⡾⠇⠹⠶⣶⣶⣧⡀⢀⣾⠁⠀⢀⣤⣀⣀
⠀⢀⣶⠟⠉⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⣆⠀⠀⠀⡟⠛⠛⠛⢻⡆⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠉⠛⠿⠃⠀⠀⢸⡏⠈⠉⠙⠛⠶⢦
⣴⡿⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣤⡴⠞⠻⣦⡀⠀⠀⢸⡇⠀⠹⣧⡀⠀⣇⠀⠀⠀⠀⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣰
⠹⣷⠀⠰⣟⢻⡆⠀⠰⣿⠁⠀⠀⣿⠁⠀⠀⠀⠘⢿⣄⠀⠀⣷⠀⠀⠘⣷⡀⣿⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢠⡟
⠀⠹⣇⠀⠙⠋⠀⠀⣠⡿⠀⠀⠀⣿⠀⡿⠷⣤⡀⠀⠻⣆⠀⢻⡀⠀⠀⠈⢷⣿⡶⠟⠀⠀⣿⢸⡇⠀⠀⣰⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠐⢲⡿
⠀⠀⢿⣀⣀⣤⣴⣾⣏⣀⠀⠀⢰⡇⠀⡇⠀⠀⠙⢦⡀⠙⣧⣸⡇⠀⠀⠀⠈⢿⡇⠀⢀⣀⣿⠈⣷⠀⢰⡟⠀⠀⠈⠉⠛⠳⢶⣤⠀⠀⣾⠀⠀⠀⠐⢶⣾⠃
⠀⠀⢸⡟⠋⠉⠁⠀⠉⠉⢿⠟⠛⠳⢶⣇⣠⡤⠶⠚⠋⠀⣸⢿⡇⠀⠀⠀⠀⠈⠁⠀⠈⠉⢻⠀⠘⢷⣜⠛⢷⣦⠀⠀⠀⠀⠀⣿⠀⢠⡿⠶⠆⠀⠀⣼⠃
⠀⠀⢸⣇⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢧⠀⣤⡀⠀⠀⠁⠈⢻⡀⠀⠀⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠉⠛⡟⠁⠀⢠⣤⣤⣰⡟⠀⣾⡇⠀⠀⠀⣸⠇
⠀⠀⠸⣿⣉⠀⠰⡏⠉⠉⠛⠶⣄⠀⠀⠀⠸⡇⣿⠛⢷⣄⡀⠀⠀⢹⡆⠀⣿⡀⢀⠀⠀⠀⣿⢶⣦⣤⣤⣤⡅⠀⠀⣾⠁⠈⠉⠁⠀⠉⠛⠳⢶⣴⡟
⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠘⡆⠀⠀⠀⡇⣿⠀⠀⠈⢻⣦⣠⡞⠀⠀⣿⣷⠋⠁⠀⠀⣿⠀⠀⠀⠉⣽⠃⠀⢰⡏⠀⠀⠀⠀⣀⣤⠶⢶⣄
⠀⠀⠀⣿⠀⠀⢰⡇⠀⠀⠀⠀⣰⠇⠀⠀⢰⡇⢿⡀⠀⠀⣼⡇⢉⣀⣀⣠⡿⠸⣧⣶⠶⠿⠛⠀⠀⠀⠀⠛⠛⠛⠿⠃⠀⠀⠀⢰⡟⠳⠀⠀⣹⠄
⠀⠀⠀⢸⡀⠀⠸⠧⠤⠤⠴⠞⠃⠀⠀⢀⣾⣀⣸⡇⠀⠀⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⡶⠟
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠀⢰⣤⡾⠛⠉⠉
⠀⠀⠀⢘⣇⣀⣼⣷⣤⣤⣤⣼⡷⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠶⠒⠒⠛⠉⠉⠉⢙⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠴⠖⠒⠚⠉⠉⠉⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣞⡋⠉⠉⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠟⠋⠉⣛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⠆⠒⠲⢄⠈⣿⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠘⡰⢁⠀⠀⠐⢎⠃⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀⣠⠤⠾⠷⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠁⡇⠀⠩⠂⠸⡀⡄⣿⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠚⠉⠉⠀⠀⠀⠤⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⢹⣧⠰⡀⡘⠤⠀⢀⡴⡰⢡⣿⠃⠀⠤⠤⢀⣀⣤⠤⠖⠚⠉⠉⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢻
⠀⠀⠀⠀⠀⣀⠬⢿⣦⠑⢌⡛⠊⠉⠐⣵⣿⣣⡤⠖⠚⠋⠉⣁⣠⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⣤⣶⣿⠿⠿⠷⣿⣷⣶⣤⣧⣶⣿⣿⡿⠿⠛⠻⣿⣿⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⠀⠐⠚⠙
⢀⣾⠻⣽⠿⠯⠍⠑⠢⠙⢿⣯⠅⣰⡿⡧⠂⠉⠭⢭⡃⠈⢻⣳⣀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣼⠃⠐⠁⠊⠙⠆⢀⣀⠀⠈⣿⣦⡿⠈⢀⡠⠒⠉⢲⡱⠀⡀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⣀⣤⠞⠃
⣿⠀⠀⢆⠑⠀⠀⡸⠀⡠⠀⣿⣿⣇⢀⣀⠁⠈⠑⡠⠃⠀⡇⢸⢿⠀⠀⠀⠠⠤⠖⠀⠀⠀⡀⠀⢈⣀⡤⠴⠖⠛⠉⠁
⢻⣇⠀⠄⠀⡠⠞⠁⡰⠃⣰⣻⠛⣿⡈⢦⡀⠀⠤⠔⠀⠠⠁⡿⡟⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠉
⠀⠻⣧⡀⠀⠤⠔⠊⣁⣾⣿⣧⠴⠻⣷⣔⢌⣉⠀⠀⠀⢴⣼⣿⣃⠀⢀⣠⠴⠒⠋⠉
⠀⠀⠈⠛⠿⠶⠾⠿⠿⠛⠉⠀⠀⠀⠈⠛⠷⣶⣤⣶⣶⣿⠿⠟⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠉⠙⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢁⡾⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⢂⡾⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⠀⠠⠀⡼⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠠⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠁⡼⢁⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⡠⠁⢀⠆⡼⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠠⠃⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠔⠁⠀⠈⡼⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⡠⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠊⠀⠀⠀⡼⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠀⠀⡔⠁⠀⠀⠀⠀⢠⠂⢀⠀⠀⠀⢠⠃⠀⡄⠀⣼⢃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠔⠀⢀⠎⠀⠀⠀⠀⠀⢠⠏⢀⠊⠀⠀⢠⠃⠀⠈⠀⣼⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠌⠀⡰⠃⢀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡰⠁⢀⠄⠀⣼⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢠⠂⢀⠜⠀⢀⠊⡠⠁⠀⢀⡼⠀⡘⠀⠀⡰⠁⠀⠎⠀⣼⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡴⠁⣠⠋⠀⡠⠃⠔⠀⠀⢀⡞⠁⡰⠁⠀⡼⠁⠀⡐⠀⣼⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⠞⢀⡞⠁⢀⠔⢁⠈⠀⠀⢠⠏⠀⡐⠁⠀⣼⠁⠀⠐⠀⣼⠃⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣀⠐⠁⡰⠋⠀⠠⠂⣠⠃⠀⠀⢠⠏⠀⡔⠀⠀⡰⠃⠀⠀⠀⣼⠃⣰⠃
⠀⠀⠀⠀⠀⠀⢠⠞⢁⠜⠁⣠⠞⠁⢀⡔⠁⠀⠁⠀⠀⢠⠏⠀⠈⠀⠀⢰⠃⠀⠀⠀⣼⢃⣰⠃
⠀⠀⠀⠀⢀⡴⠋⡰⠁⢀⠞⡡⠀⢠⠎⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⠇⠀⠀⠀⣰⠃⣰⠏
⠀⠀⠀⣠⠞⠁⠊⢀⡔⠁⢠⠃⣰⠃⠀⢀⠀⠀⠀⢀⡼⠁⠀⢠⠂⠀⡾⠀⠀⠀⣰⠇⢰⠏
⠀⢀⡼⠃⠀⢀⡴⠋⠀⢠⠇⠠⠃⠀⠀⠌⠀⠀⣠⠞⠀⠀⡐⠁⠀⣸⠁⠀⠀⣰⠏⢠⠏
⣠⠟⠀⠀⢀⡞⠀⠀⢀⡞⠀⠀⠀⢠⠎⠀⢀⡴⠋⠀⠀⢠⠃⠀⣰⠃⠀⠀⣰⠏⢠⡏
⣿⠛⠛⠛⢻⠛⠛⠛⢻⠛⠛⠛⠛⠛⠛⠛⠛⡟⠛⠛⠛⢻⠛⠛⢿⠛⠛⠛⣿⢠⡟
⣿⠀⠀⠀⠰⡆⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣷⡟
⣿⣤⣤⣴⣴⣷⣴⣶⣶⣶⣴⣶⣦⣤⣴⣴⣦⣦⣴⣷⣤⣴⣶⣤⣾⣤⣶⣶⣟
⣟⣿⣻⣻⣟⣟⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣛⣻⣟⣿

⠀⠀⠀⠀⠀⠀⠀⣄⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢦⠛⠻⠟⠻⠟⢺⠛⡶⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣲⣤⣿⣶⣿⣦⣴⢀⢀
⠀⠀⠀⠀⢨⣘⣡⣄⣃⣄⣀⣄⡷⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢠⠄⠀⠁⠀⣼⣸
⠀⠀⠀⠀⠀⡼⠟⢻⡟⠛⠇⠈⠁⠀⠀⢀⣢⣤⣾⣦⣾⣧⣤⣀⢀⡀⠀⠀⢱⣾⢶⣶⢷⣾⠒⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⢹⠀⠉⠈⠠⠺⡇⡇⠀⠀⠈⠀⠈⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣾⢶⣶⢷⣾⠲⠆⠁⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣀⣰⣄⣴⡆⢀⡀⠀⠀⠀⠀⠈⠁⠈⠏⠀⠉⠀⠀⠀⠀⠀⠀⡀⠀⣤⣀⣴
⠀⠀⠀⠤⠚⢋⡏⢩⡏⠉⡏⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠟⠻⡟⠛⠻⢛⣰⢰
⠀⠀⠀⠰⠆⣸⣤⣬⣏⣠⣷⣾⠀⠀⠀⠀⡀⠀⣤⡀⣠⠀⠀⠀⠀⠀⠀⠁⣠⣈⣃⣄⣠⣾⣂⡸⠴
⠀⠀⠀⠀⠀⠛⠉⡿⠉⠙⠂⠀⠀⠀⠀⡄⠻⢻⠛⠛⠛⠓⢛⠂⡆⠀⠀⠀⠸⠟⢹⡟⠙⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡄⣸⣇⣸⡇⣴⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢻⡿⠙⠿⠈⠁
⠀⠀⠀⠀⠀⠀⣀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣰⣷⣴⣿⣀⣤
⠀⠀⢀⠀⡷⢶⡿⢿⡿⡶⡶⣤⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠖⠈⠋⠟⣉⠙⡏⣷⠤
⠀⠀⠀⠀⣁⢸⠁⢸⣇⢰⡇⡜⠀⠀⠀⠀⠰⣶⢾⡷⢾⡷⠴⡆⢤⠀⠀⠀⠛⢼⣤⣷⣤⣴⡇⠷⠄
⠀⠀⠀⠀⢹⠿⢻⡿⠿⣯⠙⡃⠀⠀⠀⠀⠆⠀⠀⠁⢘⠀⠚⡇⡞⠀⠀⠀⠀⠋⠡⠸⠃⠈⠃
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠾⣿⠾⢷⡚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢥⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡔
⠀⠀⠀⠀⠀⠀⡀⠀⠈⣄⠉⢄⣞⡑⠂⠀⠁⠀⢀⣆⣦⠀⠀⠀⠀⣠⡿⢫
⠀⠀⠀⢀⠠⠈⢆⢠⡁⠈⠄⠈⢿⣿⠀⠀⠑⠢⠾⠋⠀⠳⢤⣤⠾⠋⠀⡎
⠀⠀⠀⠀⠀⠀⠢⠀⠉⠋⠀⠀⢸⣿⢤⠄⢈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⣀⣤⢶⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡠⠔⣙⡂⠢⠙⠈⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣲⡌⠀⢠⠁⠈⡄⠀⠘⡀⠀⠐⢄⠠⢄⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠀⡎⠀⠀⠇⠀⠀⢈⠀⠀⠀⠀⠀⠀⠈⠂⠘⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠤⠤⣔⣚⠉⠀⠀⢸⠃⠀⠀⢇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠉⠙⠒
⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠻⢿⣷⡀⠀⡇⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣫⣴⣶⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠿⠋⠀⠀⠀⢹⡝⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⡾⠓⠢⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠈⢆⣀⠔⠂⡉⠄⠀⠐⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡡⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉
⠀⢠⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⢀⣠⠞⠋
⠀⠸⣿⣦⡄⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠈⠻⣝⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⢀⣠⣤⣶⣻⣥⣴⣀
⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⣀⡠⠔⠒⠒⡤⠄⠈⠉⢦⡀⠀⢤⡀⠀⠀⠀⠀⠀⠀⡀⠀⣺⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠏⠀⠀⣠⠞⠀⠀⠀⡤⠊⠀⠀⠀⠹⡄⠀⠀⠀⠀⠠⠶⠛⠋⠉
⠀⠀⠀⠀⠀⣠⠊⠀⣰⠃⠀⠀⠼⠁⠀⠀⣠⠞⠀⠀⠀⢠⡇⢠⣧⡤⢄
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣴⣋⡼⠋⢀⡤⠀⡧
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡞⠀⠀⢀⣾⡏⠀⠀⢘⠒⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⣾⠀⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⠀⠀⠀⠙⠛⠲⠤⢀⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⠛⠶⣄⡘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠙⠓⠦⣄
⠀⢀⣤⣶⣤⣀⡀⠉⠉⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⣠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠒⠄
⢸⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠘⣇⠀⣠⡶⠟⠛⠛⠋⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⠞⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠸
⢸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠉⠛⠉⠛⠉⠃⠀⢀⠤⢤⣀
⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⣦⠘
⢸⣿⡿⢋⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿
⠘⠋⠘⠛⠛⠛⠛⣷⠀⠀⠀⠀⠀⠟⠛⠛⠛⠛⠛⠻⠿⠿⣿⡆⡔⠋⠉⠙⣦⡿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣷⡀⠀⢀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠛⠛⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠸⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⣿⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢸⡇⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⡿⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠸⡇⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡈⠉⣀⣄⡀⠉⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡈⠁⠀⠀⠈⠁⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣩⠞⠁⠀⢹⡎⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣴⣾⣷⡆⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠾⠥⠤⠴⣤⠚⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣛⣛⣛⠛⣡⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢠⠇⠀⣠⠿⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⣸⣿⠟⣀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⣶⡖⠛⠁⣠⠞⠁⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣯⠙⢫⣭⣶⠟⣡⣾⣿⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠉⠓⢿⣲⡞⠁⠀⣠⠞⠉⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣼⣨⣛⣡⣾⣿⡿⣫⣶⣦
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⠳⢤⡞⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣼⣼⣿⠽⣝⡿⣫⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⠣⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢲⣿⣿⡛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠚⠁⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣡⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⢦⣄⣀⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⡽⣯⢷⠸⣿⣿⡜⠎⢸⠀⣿⣿⣿⣿⡟⡿⢸⠃⢎⣾⣟⡾⢣⡿⢋⡾⣿⣿⠟⣠⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⣯⢧⠘⣿⣧⠀⠀⠀⢹⢿⠉⡟⠃⠇⠇⠀⣜⡟⡼⢡⡟⣱⢟⣾⠟⣡⣾⡿⢟
⠻⣿⡿⣿⡿⣝⢿⠙⠃⠀⠈⢎⠂⠘⡿⡄⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠞⠈⠠⢋⡞⡵⠋⢀⣼⡿⣫⡴⢟
⣿⣮⡙⢮⡙⢮⡣⡁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠊⠀⠴⠋⣨⠞⡫⠖⡩
⡳⢿⣻⢦⡈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠕⠈⢐⣬⣶
⡻⠶⢍⡓⠯⣳⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡽⣞⣿
⢭⣓⠢⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⣷⣄⣀⣠⡾⢹⣦⣤⣴⢾⡏⠀⠀⠀⠀⠀⠀⢀⡠⣖⡯⠷⢛⣋⡽
⣟⣶⠤⢄⣀⠀⠀⠀⠀⢤⣀⣀⣀⣀⣿⡌⠻⠟⠁⠀⠛⠋⠁⣿⣷⣤⣤⣤⣴⠖⠋⠑⠉⠁⠀⣈⣠⣤⢶
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠐⡉⠉⡉⠙⡏⠁⠀⢧⠃⠐⡆⠐⡆⠈⡉⠡⡶⠋⣄⠀⠀⠀⠀⠈⠙⠒⣚⣋⣩
⣀⣀⣀⡀⠀⠀⠀⣀⣀⣠⡇⠀⠃⠠⡇⠀⡄⢸⡇⠀⠀⢠⠇⢐⠧⢤⡇⢀⣇⣀⣀⠀⠀⠀⠉⠋⠉⠛⣓
⣿⠿⠿⠟⠉⠉⠉⠉⠙⠛⢿⠀⠸⠀⠀⠀⣄⠀⡇⢸⡀⢸⡄⠘⠀⢸⠃⣼⠛⠉⠉⠉⠀⠀⠀⠒⠲⠶⠶
⣖⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡧⢴⣖⣛⠒⠛⠉⠉⠉⠉⠉⠈⢉⣉⡙⢿⣇⠀⠀⠀⠀⠀⠀⣐⣢⡤⢭⣭
⣵⣖⣂⠄⠀⠀⠀⠀⡠⠴⠛⠛⠙⠻⣿⢀⣠⣤⡀⢀⣤⣤⡀⣿⡟⠛⠛⠛⠳⢤⣀⠀⠀⠀⠀⠈⠉⠙⠛
⡿⢟⠂⠀⣀⠤⡂⠀⠀⠀⠀⠀⠀⢀⣿⠞⠉⠛⢷⣸⠟⠉⠛⠾⣇⠀⠀⠀⠀⠀⠀⠹⣲⣤⣿⣷⣶⣮⣽
⣫⣤⣖⣯⠖⠋⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠘⡏⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢀⠀⠈⢙⢿⣿⣿⣿⣿
⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠐⢝⢮⣲⣍⡻⣿⣿⣿
⠴⣊⠔⣠⢔⡤⣠⠖⡡⠀⣠⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⢰⡵⣄⡙⢿⡿⣿⣶⣝⡿
⣚⣵⣿⡿⣫⡾⣣⢞⢄⣴⣷⡃⠀⠀⡜⡆⠀⠀⠀⢀⠀⠀⠀⣀⢹⣷⡀⠀⠠⡈⢻⣮⡿⣦⡙⢮⣿⣿⣿
⣿⡿⣫⡼⢋⡼⣳⣯⣾⣿⡿⢁⠆⣼⢷⠇⢀⢰⢰⢸⢠⠀⢸⣸⡈⣿⣷⣱⡱⡹⣽⣿⣿⣿⣿⣮⡹⣿⣿
⣿⣿⢟⣴⣿⣿⣿⣿⣿⣿⢃⡞⣰⣿⣿⠠⣟⣼⣼⣿⢸⡄⡀⣇⣧⢸⣿⣷⣷⡽⣽⣿⣿⣿⣿⣿⣿⣮⡻

⡽⣿⣿⣧⡻⣎⢿⣿⡞⣟⣿⢿⣧⢿⣿⣿⣿⡇⢿⣿⢿⣿⣿⠈⣿⠘⣿⡿⣿⢸⣿⣿⡿⡼⣼⣿⡟⣼⣿⠏⢠⡟⡱⣿⢏⡾⣱
⣧⡈⠻⣏⢷⡙⣆⠻⣿⡜⠜⡏⠿⡜⣿⡘⡏⠃⠘⠙⠈⠁⠸⠀⠏⠀⡟⠃⠉⠘⠸⢹⠇⢡⢇⡿⠁⣹⡟⢠⠏⢀⡼⢣⢎⣾⡿
⣯⡻⢄⠘⢧⠹⡌⢧⠙⡕⠀⠀⠀⠀⠀⠡⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠘⠀⠀⠟⢀⠏⢀⡞⠁⠁⡴⢋⡴
⣿⣿⣦⠁⠀⠑⠈⠀⠀⠉⡩⠄⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠞⢑⣿⠿
⣲⠍⠫⠳⢄⠀⠈⢲⣄⠀⠀⠑⠀⠀⣷⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢠⣄⠠⠤⠀⠂⢛⣉⣍⡉⢠⡴⠶⡟⢸⣿⠾
⠉⠓⠂⠀⠀⠰⠄⠀⠹⣿⣦⡀⠀⠀⢹⣿⠀⠀⢼⣿⡄⣰⠧⠤⠴⠤⢘⣊⣩⣤⡤⠶⣶⡞⢹⡷⠋⢁⡀⣙⣿⠀⢸⠇⣿⣇⠔
⣿⣢⠄⡀⠀⠀⠀⠀⠀⠙⢿⡷⡈⢷⣄⡙⠀⣷⡀⠉⢡⡴⠒⠛⠛⢾⠏⠁⠀⢻⡇⠀⠸⡇⢸⠃⢠⣏⣉⡭⣿⠀⣼⢠⣿⠵⢾
⣯⡉⠁⠀⠀⠐⠀⠀⠀⢰⣘⣿⣷⠈⢿⡝⢦⣿⠻⣦⢸⡇⠀⠶⠀⣼⠀⢸⠀⢸⡇⠀⠀⠀⢸⠀⢸⣏⡅⠀⣿⠴⣏⠸⣿⡈⠄
⠿⢧⣶⣶⣶⣖⠚⠛⠉⣉⣉⣀⣀⣀⣠⣿⣆⠙⠃⠀⢸⡇⠀⣤⡈⢻⠀⠘⠀⠈⡇⠀⣇⠀⢸⣆⠀⠙⢁⣼⣇⣀⡼⢀⣿⢂⠉
⣅⠐⠉⠍⠉⠛⠻⢶⣠⣉⠙⠻⢿⣿⡛⠛⠛⠂⠀⠀⢸⡇⠀⠉⢀⡄⠀⣸⣇⣀⣅⣤⢿⠴⠚⣋⡓⠒⣋⣥⣬⣭⣴⣿⠋⠄⠄
⠧⠄⠀⠀⠀⠀⠀⠀⢀⣨⣵⣶⠄⠈⣹⡷⠄⠀⠀⠀⢸⣧⡤⠶⠛⠛⣋⣉⣉⣉⣥⣴⣤⡶⠟⢫⠹⠛⣛⣯⠵⠘⡈⠄⠀⠀⢤
⠓⠀⠀⠀⠀⢀⠤⡶⠿⠟⠛⠡⠴⠿⠿⠶⣶⠆⠀⠰⠶⠶⠶⠶⠾⣿⠿⠿⠛⠛⠉⠀⠉⠀⠀⠀⠂⠡⠰⡘⡊⠥⠀⠀⠀⠀⣀
⣥⣄⡀⠀⠀⠀⠂⠁⠀⠀⠀⣰⣶⡶⢂⣴⢋⣠⣴⡆⠀⢀⡀⢶⣤⣈⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⡀⠐⠲⣾
⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠏⣰⣿⠿⠛⠉⢸⣿⣠⡿⠳⣼⣿⠈⠉⠛⠻⢦⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣌
⢕⡤⠒⡡⠂⠀⠀⡀⢀⠾⠋⠀⠚⠉⠀⠀⢸⡿⢸⣿⡟⠁⠀⠘⠿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢤⡪
⠋⡠⣪⠎⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⠏⠀⠀⠀⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⡈⢦⡱⣄⢻
⣫⣾⢗⢔⠄⡰⣠⠂⢀⠄⡄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠠⡀⡀⠀⢡⠀⠱⣀⠠⡑⣄⠳⣜⢷
⡿⣵⣿⣯⡾⣵⠃⢠⣾⡟⡠⣠⠇⣾⣴⣧⠀⡰⠀⡀⢠⢰⡄⢠⡄⠀⡄⠀⡆⠀⢸⣷⣇⢀⢷⣼⡄⡈⣧⠀⢻⣷⡱⡜⢧⡙⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⠀⠀⠀⢀⣠⠖⠛⠛⠛⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠀⠀⠈⠙⢳⣄⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡿⢦⠀⠀⠀⠀⠀⠀⠈⠻⢯⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣝⢦⡀
⠀⠀⠀⠀⠀⢀⣤⠶⠶⠛⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣝⢦⡀
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣤⣬⣿⣿⡦⣄
⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣽⣦⡀⠀⠀⠀⣠⣾⡿⠛⠋⠀⠀⠒⢍⢷⠈⠳⣄⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢠⣾⡿⠋⠉⠉⠉⠉⠻⣟⢦⡀⢰⣿⠃⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⣀⣙⣶⣄⣀
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠠⡿⡏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⢹⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⢷⡫⠝⠛⠚⠚⢯⡳⣄
⠀⣠⠖⠚⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡐⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡄⠀⠀⠀⠀⣠⣾⡿⡵⠋⠀⠀⠀⠀⠀⠀⢻⢹⡄
⣸⠃⠀⠀⠀⠈⢳⣄⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡜⣆⠀⠀⠀⠀⠀⠀⢀⣾⣏⣙⣿⣿⣶⣶⣶⡿⠟⣿⢹⠁⠀⠁⠀⠀⠀⠀⠀⢸⣸⠇
⣿⠀⠀⠀⠀⠀⠀⠹⣆⢷⠀⠀⢹⡆⠀⠀⠀⠀⠀⣀⣤⡤⠶⢤⣽⣮⣷⣦⣤⣤⣴⣾⣿⡿⠿⠛⠛⠻⣿⡍⠷⣄⠠⣿⣸⡀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⢿⡀⠀⠀⠀⠀⠀⠀⠘⢿⣷⠶⠾⠃⠀⠀⠀⣴⣾⠟⠋⠉⠉⠙⠙⠓⢍⢿⡉⠛⣹⣿⠋⠀⠀⠀⠀⠀⢸⣿⠀⠈⠳⢼⣯⣣⣀⠀⠀⠀⠀⢀⣾⡟⠳⣄
⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣯⣧⠀⣿⣧⠀⠀⠀⠀⠀⠀⣸⣿⢀⣀⣤⣤⣽⣿⣿⣷⣶⣶⣾⣿⣿⣾⢿⠿⢿⣶⣄
⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡐⣧⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠲⣼⣿⣆⡀⠀⠀⢀⣴⣿⣟⡭⠗⠚⠓⠒⠒⠦⡹⣆⢠⣾⠟⠉⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣜⢧⡀⠀⣀⡀⠀⠀⢀⡴⣳⣧⠴⣾⣿⠿⢿⣿⠿⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢻⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣝⣲⣦⣴⣖⣾⣭⡾⣫⠗⠋⠉⠉⠉⠓⢬⡳⣼⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡽⣄⠀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣠⣴⣟⣟⣛⣛⠿⣿⣯⠀⣾⢱⠃⠀⠀⠀⠀⠀⠀⠀⣷⢹⣧⢳⡀⠀⠀⠀⠀⢀⠀⣀⢞⡾⠉⠛⠿⢿⣿⣿⡿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⠙⢮⢿⣷⣿⡸⣆⠀⠀⠀⠀⠀⠀⠀⣼⣼⠃⠙⢯⣷⣶⣶⣶⣶⣿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠙⢿⣼⡧⣄⣀⣀⣀⡤⣞⡽⠃⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢿⣇⠀⠀⠀⠀⠀⠈⡸⣿⠃⠀⠀⠈⠛⠲⠶⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣤⣤⣤⣴⣾⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⠭⠭⠿⠿⠿⠿⠦⠴⠶⠶⠶⠶⠶⠶⠶⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⠗⠛⠛⠛⠛⠒⠒⠒⠒⠒⣒⣶⣶⠶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠤⠄⠒⠒⠈⠉⠉⣀⠀⢸⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢠⠒⠒⠂⠉⠉⠉⢸⠀⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠉⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢠⠖⠒⡆⠠⠖⠒⢲⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠈⠀⠀⢸⠀⢸⡇⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣇⣸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡛⢻⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣇⠈⠒⠒⠃⠠⢄⣀⣸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠉⠁⠒⠒⠤⠄⣀⣀⠀⢸⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉
⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋
⠀⢈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣆⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁
⠀⠀⠙⢿⣝⢷⣄⠀⠀⠀⠀⠀⠀⠀⠛⠙⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⠖⢀⣀
⠀⠀⠀⠀⠈⠳⢍⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⣀⣤⢶⣿⣿⣿⣶⣿⣿⠿⠞
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡤⠴⢤⣤⠖⠉⠀⠈⣹⠖⠁⠀⠀⠈⣣⠄⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠤⠒⣋⣽⣾⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⡞⠁⠀⠀⣠⡞⠁⠀⠀⠀⣠⠞⠁⠀⠀⢀⠈⢳⡄⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⣠⠟⠀⠀⠀⡼⠋⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣼⡀⢸⠇⣀⣀
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⠀⠀⠀⢀⠼⢃⡴⠛⠉⠀⡈⢳⡄
⠀⠀⠀⠀⠀⢺⡀⠀⠀⠂⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢻⣦⠞⠀⠀⢰⠏⣠⠌⠃
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢰⣿⡏⠀⠀⠀⣨⠉
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠀⠀⠀⠀⣿⠀⠛⠳⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⡟⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠙⠳⣦⣀⠀⠀⡧⠀⠀⠀⠉⠙⠒⠤
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠉⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠲⠧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠓⠦⣄
⠀⢀⣴⣶⣶⣤⣄⣀⠀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠐⠀⢀
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠹⡄⠀⠀⣠⣶⠾⠿⠛⠛⠛⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣿⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠓⠈⠩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠿⠇⠀⠀⠀⢀⠀⠀⠀⢀⡇
⢸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠙⠦⣀⠈⡇
⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⢸⣿⣿⢏⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⣿⠀⡇
⠘⠛⠁⠛⠛⠛⠛⠛⢿⡇⠀⠀⠀⠀⠀⢀⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠀⡰⠛⠉⠉⠻⣦⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣇⠀⠀⠀⠀⣿⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠻⣦⣤⣤⡾⠋⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡇⢀⡀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠇⠈⠇⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠒⠊⢻⠀⠀⠀⠀⠈⠑⠦⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⢀⡤⠴⠖⠚⠉⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⢀⡏⠀⡄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⢸⡇⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⣠⢿⠇⠀⠀⠉⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⢷⡀
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢳⡀
⠀⠀⠀⠀⠀⠈⠁⢤⣤⣤⣽⣿⣦⣀⣀⣠⣶⣿⡿⢿⣦⣀⠀⠀⢀⣹⣿⣿⣿⣶⣶⣶⣶⠶⠟⠉⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠁⠀⠀⠉⠉⠛⠛⠋⠉⠛⢛⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢠⠂⠀⠁⠀⠲⠤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠓⠂⠀⠉⠓⠒⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠐⣦⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⢹⡇⠈⢿⣷⣄⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢀⣿⠇⠀⠈⣿⣿⣧⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣾⣿⡀⠀⣠⣿⣿⣿⠀⢻⣿⣆⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣿⣿⣷⡀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣿⣟⠉⠙⠻⡄⠀⠉⣯⣭⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠁⠀⠀⠀⠈⣇⠀⠀⡇⠀⢀⣿⡛⠛⠀⠈⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠒⢻⡇⠀⠀⡟⠀⢀⡇⠀⣸⠁⠉⣳⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢸⡇⠀⢰⠇⠀⣸⢁⣰⣿⣠⠞⠁⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢸⣧⣤⣿⣦⣶⣿⣿⡿⠛⠁⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠉⠉⠁⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢳⡀⠀⠈⠇⠀⠀⢠⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⡇⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢸⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠁⠀⠀⠣⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡴⠞⠉⠀⠀⠀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣞⠍⠉⠉⢺⠕⠈⠑⠔⠀⠀⠀⠀⠀⠀⠘⡶⠒⠓⠲⣄
⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠘⠒⠒⢦⡀
⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠃⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣣
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡄⠀⠀⠀⠀⡀⢀⡥⠤⠲⣤⡞⠉⠀⠀⠉⡖⠉⠁⢸⣿
⠀⠀⠀⢸⡄⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣖⠉⠉⠉⠲⣇⠀⠀⢿⣇⠀⣶⣷⣦⣿⠀⠀⢸⣿
⠀⠀⢀⣠⠷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⡀⠀⠀⠀⠙⠀⠀⠸⣿⡄⡇⢰⡛⠉⢇⠀⠸⣿⡇
⠀⠀⢾⠁⢀⣠⠤⠖⠒⢤⣄⡀⠀⡖⠉⠈⠣⡀⢇⠀⠀⠀⠀⠀⠀⠀⢻⣧⠘⠺⠟⠀⠘⡄⠀⣿⡷⣄
⠀⠀⡸⠓⠉⠀⣤⡄⠀⠀⣿⣿⡄⣷⠀⢠⡀⠙⢼⡀⠀⠀⢦⡀⠀⠀⠘⣿⣄⣀⣀⣤⣾⠳⣶⣿⠇⠘⡆
⠀⠀⠹⡀⠀⠀⠛⠃⠀⠚⠛⠿⣤⣿⠀⠸⠷⠀⠈⢣⠀⠀⠸⣿⡦⣠⣤⣿⣿⠿⠛⠋⠁⠸⣀⣸⣦⠀⡇
⠀⠀⠀⢱⡀⠀⠀⢠⣤⡄⠀⠀⢹⣿⣆⠀⢀⣀⡀⠘⢶⣾⡿⠿⠇⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⠁⣸⠃
⠀⠀⠀⣼⢣⠀⠀⠀⠿⠃⠀⠀⣼⣿⡟⠀⣸⢿⣷⣶⡿⠃⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⢀⣀⣠⠞⠁
⠀⠀⠀⡏⠀⢧⠀⠀⠀⣀⣤⣾⣿⢿⣿⡿⠟⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅
⠀⠀⠀⢷⠀⠈⠶⢶⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠃
⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠓⢤⣤⡤⠊⠉⠁⠀⠠⠀⠀⠀⠀⠠⣔⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠉⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⣠⠴⠊

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢆⡁⢨⡀⢀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠲⠁⠐⠂⠐⣌⠷⣈⡀⠀⠒⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣾⡗⡏
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠠⢀⠀⠳⠀⠀⢃⣧⣶⡄⠈⠢⣀⠀⠀⢀⣿⡾⠻⡄⠀⠀⠀⢀⣴⡿⠋⢸⠁
⠀⠀⠀⠀⢀⠔⠢⠀⡃⡐⢦⡁⠀⢲⠀⠀⢹⣿⣇⠀⠀⠀⠉⠒⠛⠋⠀⠀⠙⠦⠤⠶⠟⠋⠀⠀⡾
⠀⠀⠀⠀⠀⠐⢄⠀⠤⠈⠈⡁⠛⠀⠀⠀⢸⣿⣿⠀⠀⠁⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⣠⣴⣧⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠰⠀⠀⠀⠀⠀⢠⠸⣏⠣⡈⡢⠌⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠒⠛⠉⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⡉⠈⠢⠦⣄⠂⠀⡠⠀⢄⠄⠶⠂⣀⠠⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣧⠀⠀⢠⠀⠀⠀⡆⠀⠀⠘⣄⠀⠀⠘⢮⠀⠢⠄⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⡼⠀⠀⠀⡇⠀⠀⠀⠈⠦⠀⠀⠀⢄⠀⠀⠀⠀⠑⡄⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠀⠀⢰⢇⠀⠀⠀⡇⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠀⠀⠙⠲⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢛⣋⣥⣦⣤⣄⡀⠀⠀⢸⡀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠿⣿⡆⠀⠀⣧⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣀⣠⣤⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⡿⠟⠛⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠺⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣫⠟⠉⠀⠀⠀⠀⠀⢿⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⢠⣾⠿⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠈⠱⣄⠀⠀⠀⣀⠤⠔⠒⠀⢉⣉⣀⣀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠈⣢⠔⠋⢀⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣴⣶⣶⣶⡶⠶⠶⠶⠿⠶⠾⠿⢿⠿⠿⠿⠿⠿⠟⠿⠿⠻⠛⠛⠛⠻⢿⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⡀⠀⠀⣀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠘⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠻⡇⠀⢸⣿⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣯⢿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣧⠀⠠⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣆⢺⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢹⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣻⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣛⢽⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⠀⠀⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡹⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢾⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⠀⠀⢻⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠤⢢⡄⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⢀⠀⣿⣸⣧⠀⣀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠐⠀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠑⢿⣆⡬⢤⣤⣥⣭⠂⢀⠀⢰⠀⠀⢸⡇⠀⠀⠰⠂⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣽⠀⢰⣿⣠⡸⣷⣀⣀⣀⣩⣠⣾⠀⠀⢸⣗⣒⣒⣒⣒⡷⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⢻⠰⠞⠛⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣻⠀⠓⠤⠛⢠⣦⣶⣶⣶⡶⣶⣶⠀⠀⢶⣶⠶⠶⠶⠶⡿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⢠⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢨⠀⠀⢸⡗⠀⠀⠀⠀⡃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠂⠄⠀⠀⠸⣿⠀⠀⠀⢀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⡇⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⣵⡂⣀⠀⠀⠀⣿⠀⠀⠀⢀⠀⢸⠀⠀⢸⡇⠀⠀⠀⡄⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠘⣿⡇⠀⠀⠀⠀⣿⣄⣀⣀⣉⣀⣸⠀⠀⢸⣷⣀⡀⠀⠀⣤⡇⠀⠀⠀⡄⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡃⠤⠀⠀⠀⠉⠀⠈⠉⠉⠉⠉⠀⠀⠈⠉⠈⠉⠉⠉⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⡀⢿⣇⣀⢀⠀⡀⡀⢀⠀⠀⡀⣀⣀⡀⢀⠀⠀⡀⣀⣀⢀⠀⠀⠴⠖⢠⡇⠀⠀⣀⣘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣭⡭⠱⠛⠛⠛⠛⠛⠻⠛⠿⠛⠿⠿⠽⠧⠽⠬⡧⢤⠶⠶⠶⠾⠶⠶⠶⠿⠾⠷⠶⠤⠭⠬⠴⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠉⡍⠉⠁⠛⢛⠙⢋⠜⢋⠗⠠⠖⠠⠖⡰⠆⡰⠆⠶⠲⠆⠴⢢⠦⢠⠄⡤⢀⢄⠀⢠⠤⠴⢠⠄⠶⠰⠖⠰⠂⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡔⣠⢂⠐⡀⠐⡡⢊⠥⢊⠠⢊⠄⢊⠤⢊⠔⡢⠀⡠⢚⠔⡡⠒⡡⢊⠔⡨⠊⢔⠤⠂⠀⡡⢊⠔⡡⠂⠀⡠⣚
⠀⠀⠀⠀⠀⡀⠤⠔⠊⠁⠁⠈⠉⠉⠙⠃⠚⠒⠛⠒⠓⠶⠃⠶⠃⠶⠅⠾⠤⡾⠶⠷⠞⡴⠿⠶⠷⠶⠶⠳⠗⠶⠒⠷⠲⠧⠾⠖⠺⠚⠛⠙⠢⣄
⠀⠀⠀⠀⠛⠩⠂⠦⢤⠄⣤⣠⡄⣠⣄⣠⣀⣀⣀⣀⣀⡀⢀⡀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡄
⠀⠀⠀⠀⢸⠑⠡⠒⠤⠢⠔⡡⢒⠤⢂⠤⠊⠄⠈⠠⢊⠡⢋⠝⣉⠝⡩⢋⠉⠉⠝⡉⢝⠉⠋⠝⡋⠋⠙⠋⠜⠁⠟⡡⢋⠉⡩⢉⠍⡩⢋⠍⡩⠉⢋⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠉⠀⠁⠐⠁⠊⠀⠀⠔⠁⠈⠀⠊⠐⠁⠐⠀⠂⠐⠀⠊⠁⠂⠠⠊⠐⠀⠂⠀⠀⠈⠀⠁⠈⠀⠁⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⣀⠀⣠⣤⣀⣈⠱⠟⣀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠟⢀⣴⠞⣛⣛⣻⡾⣋⣿⣆⠙⠛⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠃⢿⡏⠹⣯⡾⢋⣵⡟⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣿⠀⢸⡇⠀⠙⣿⢛⡉⣻⡄⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢃⣴⠏⣀⠟⠁⠀⠀⠙⣋⣵⠟⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⡾⠋⠀⣠⡶⠾⠛⠉⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠂⠉⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠈⢸⠀⠀⠀⠀⡇⠀⠀⠀⢐⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⠀⠀⠀⢈⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⠀⠘⡇⠀⠀⠀⡿⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠸⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡟⠀⠀⠀⠀⡇⠀⠀⠀⣷⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⠀⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠈⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⢀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠄⢸⡇⠀⠀⠀⢸
⠀⢸⠀⠀⠀⢀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠸⠀⠀⠀⣿⠀⠀⠀⠀⡇⠆⠀⠀⣟⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⡇⠄⠀⠃⡏⠐⠀⠀⢸⠀⠀⠀⠀⡏⡀⠀⠀⠀⡇⠀⠀⠇⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⡇⠀⠀⠁⡇⠀⠀⠀⢸⠀⠀⠀⠀⣷⠀⠀⠀⠀⡇⠀⠀⡄⣿⠸⠀⠀⢸⡇⠀⠀⠀⢸
⠀⣿⠀⠀⠀⢀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠸⡇⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠈⡇⠄⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢠⡇⠀⠀⠀⢸
⠀⢻⠀⠀⠀⠰⡇⠇⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡿⠀⠀⠀⠀⡇⠀⠀⠀⡿⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⢸⡇⠀⠀⠀⢸
⢠⢸⠀⠀⠀⠸⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠐⢸⡇⢀⠀⠀⢸
⢸⢸⠀⠀⠀⢰⡇⠀⠀⠀⡇⠘⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡧⠀⠀⠰⢐⡇⠀⠀⠀⢸
⠨⣼⠀⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⡏⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠐⣻⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⢀⠨⡇⠀⠀⠀⢸
⠘⢸⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⠀⠀⢰⡇⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠚⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢖⡒⠒⠢⠶⠟⠁⢀⠀⡀⠀⠹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡏⠐⠀⠀⠂⡀⢤⣸⡄⢇⠀⠀⠉⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡿⠀⠃⡀⢠⣀⣀⠀⠀⢩⣻⣦⠣⣄⣀⣐⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢀⠉⠳⡀⢉⣿⣐⣿⣦⣿⣽⣿⣿⣿⣿⣿⣿⣿⠿⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⢠⣀⢀⠈⠃⠹⣿⣿⣿⣿⣿⡏⠛⡀⢉⣇⠂⠂⠀⠐⠂⢸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠈⠙⠻⣷⣤⣶⣾⣿⣿⣿⣿⣿⠿⣀⠬⠭⠿⠘⡀⡧⠀⡤⠾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⢀⠀⠀⠄⣀⣝⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣳⣶⣾⣿⣶⣦⡴⠀⠁⠈⠈⠙⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡽⠛⣦⡙⢿⣿⣿⣿⢳⣝⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢦⣄⡀⢀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣼⣿⣿⣦⣝⡻⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⡟⠒⠀⠀⠈⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢀⣞⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣀⠀⠀⠀⠀⠀⣄⠈⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⡌⣁⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⡿⠅⠀⠈⠁⠀⠀⠀⣶⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣌⠉⠸⠛⠈⠀⠻⣿⣿⣿⣿⣧⠄⠀⠈⡀⠀⠀⠀⢺⣿⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣽⣿⣶⣤⣄⣠⡶⠉⠙⢿⣿⣷⣆⠀⠀⠀⠀⠀⣨⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣶⣤⣤⣄⠙⣿⣿⣟⠀⠀⢠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡅⠈⠿⢿⣿⡷⣶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣽⡇⡀⣿⣆⠀⠉⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⣿⣿⠃⠀⠘⣇⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⣿⣿⠀⢿⣷⠀⠐⠀⢹⠀⠀⠀⠝⠿⣿⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣿⣿⡿⠀⠸⡿⠀⠀⠀⠀⢄⣀⣤⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡿⠿⣖⣢⣀⠀⠠⠐⣺⣿⣿⠿⠿⠿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠠⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠹⢂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠂⠀⡔⠉⠃⠀⠀⠀⠀⡔⠀⠢⡀⠀⠀⠀⠀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠘⣄⠀⠰⡀⠀⠀⡀⠸⢄⡀⣅⢀⠠⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠔⠒⢬⣦⡀⠈⢲⠉⠀⢀⡠⢋⠉⠀⠀⣀⡠⠔⠫⠀⠀⠀⡄⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⢶⡀⠀⠑⠀⠈⠀⠀⠀⡏⠀⢀⣠⢤⠶⡀⠀⡇⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠚⢓⡄⠀⢠⡄⠀⠀⠀⡘⠀⠀⣧⠬⢆⣇⠱⠠⠇⠃⣀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠁⠀⡆⠈⠉⠂⠀⡐⠀⠀⠀⠦⣀⠂⢀⡗⠀⠒⠀⠠⠄⡀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣱⠀⠀⠀⡰⠁⠀⠀⠀⢠⠀⡇⡇⡇⠀⠀⠀⠀⠀⠑⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⢬⡀⠀⣠⠁⠀⠀⠀⠀⠀⠂⡇⡇⠠⣀⡀⠀⠀⠄⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡀⠉⠦⡷⠳⠀⠀⣀⠀⠀⣠⠁⡇⠀⠀⡀⣉⠐⠐⠋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣨⣂⣀⣥⠀⡄⢠⣋⠀⠥⠐⠂⠃⠀⠀⡇⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⠁⠀⡋⠀⡇⢹⡀⢙⠸⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢤⡤⠆⠀⠰⠀⢇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢈⢁⠀⡠⡄⢃⢸⣆⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡸⠀⠒⣔⣁⡨⡅⠃⡅⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢳⢀⢠⠤⠵⢶⠒⢰⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠥⠓⡶⣖⣐⠿⣉⠁⠀⠀⠀⠀⠀⠀⢸⠸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⢿⡡⣿⡾⡾⠿⠭⢟⡉⠁⢠⡈⠁⠚⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠁⠀⢀⠀⠀⠀⠀⠚⠒⠲⠽⠧⠼⠦⠤⠴⠞⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠉⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠠⠤⠄⢀⠀⣀⡀⠀⠀⠀⠀⠔⠒⠔⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠒⠒⠖⠶⠶⠤⠤⢤⣀⣉⣀⣁⠀⠐⠒⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⣠⠆⠠⠀⣂⠀⠀⠀⢀⣈⡉⠉⠙⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢻⠀⠀⠀⠀⠸⣿⠈⠉⠀⢻⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢸⠀⠠⠂⠀⠀⣿⠀⠀⠀⢸⠀⠀⠅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠀⠀⢸⡇⠀⠔⠀⠀⣿⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣄⢄⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡼⢻⢸⡇⠀⠀⠀⡄⣿⠀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠷⢮⠞⠿⢳⣶⣦⠇⣿⣤⢄⣀⣨⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⡇⠁⠁⣄⣀⡀⠀⠀⠉⠈⠉⠛⠚⠏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢸⡇⠀⠀⡏⢈⣐⡉⠓⠶⠤⢤⣤⣀⡄⠀⠀⡇⢠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⢰⡂⢸⡇⠄⠉⠡⠒⠂⢤⠈⠃⠀⠀⠃⠘⠀⢤
⠠⣄⣀⠀⠀⠀⠄⠀⢸⠀⣹⡇⠀⢸⣇⠸⠷⢦⣤⣀⣀⠀⢸⠀⠀⠀⠠⠀⠸⠀⠹⠂
⠀⠀⠈⠙⠓⠢⢤⣄⣸⠀⣿⡇⠀⠈⠉⠛⠻⠶⢶⣤⣭⣉⣛⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠑⠲⡢⠤⢄⣀⠀⠀⠀⠀⠈⠉⠙⠃⠀⣸⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠕⠃⠒⠦⠤⣄⣀⣘⣻⢀⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠂⠜⠖⡲⢄⣀⡀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠫⠲⣶⢤⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣦⡀⠀⠀⠀⢀⣤⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⣀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣦⣠⣴⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠒⠀⢲⡄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠉⠉⢻⣿⣿⣿⠿⠛⠛⢻⣿⡿⠋⠉⠀⠈⣿⠛⠉⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⢻⡇⠀⠀⠀⢰⠃⠀⠀⠀⣀⣀⣀⣼⣿⡆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠻⠀⠀⠀⢸⠀⠀⠀⢰⣿⡿⠿⠿⢿⣧⠀⠀⠀⢸
⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣦⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢿⠀⠀⠀⠸⣿⣤⣤⣴⡿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢰⠁⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁⠀⠈⣦⡀
⠀⠀⠀⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢷⡀⠀⠀⠀⠘⢦⣀⣀⣠⣦⣤⣤⣴⣿⣧⣄⣠⣴⣿⡿⠆
⠀⠀⠀⠀⠹⡇⠀⠀⠀⢰⣦⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⣷⡀⠀⣀⣀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠋⠀⠀⡇⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠘⠤⢤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢻⡀⠀⣀⣀⣠⣤⣴⡁⠀⠀⠀⠀⠀⣸⣿⣿⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠟⠉⠙⢿⣿⣿⣿⣿⣿⡿⠁⠀⠉⠙⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁



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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡟⠻⠿⠧⢽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⠀⢹⡛⡟⠛⠿⠿⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣉⣿⣿⣿⣿⣷⣤⣦⣠⣄⠹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣇⣈⣿⢻⣿⣇⣿⡦⠼⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣧⣀⠹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡛⣿⡿⣿⣇⢿⣿⣸⡿⠿⣿⣿⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡟⠁⠀⠀⠉⠋⠀⣀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣾⣷⣬⣿⣾⡿⠋⠀⠀⠈⠙⠁⢀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡉⢉⣿⣿⣿⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣌⠉⣹⣿⢹⡇⠀⠀⠀⠀⢀⣾⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠟⠻⣿⣿⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣷⡾⠛⢎⠻⣶⣤⣤⣶⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣭⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣭⣺⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡿⢻⠁⠘⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣿⣿⣿⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢇⣿⣿⣿⡇⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⢸⠟⢻⡿⠀⠀⠀⠀⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋
⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠨⣟⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⢳⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⣠⡴⠋⠁
⠀⠀⠈⠻⣝⠳⣄⠀⠀⠀⠀⠀⠀⠈⠁⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⢀⣠⣤⠶⣚⣀⣤
⠀⠀⠀⠀⠈⠑⠌⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⣀⠀⠀⢤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣠⠴⣞⣯⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠑⣶⠞⠁⠀⢀⣠⠛⠁⠀⠀⢀⡱⠂⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠐⢊⢡⣴⡿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢀⡾⠁⠀⠀⣠⠟⠁⠀⠀⢀⡴⠋⠀⠀⠀⢠⠀⢻⡄⠀⠀⠀⠀⠤⠒⠋⠉
⠀⠀⠀⠀⠀⢀⡤⠂⠀⢠⠞⠀⠀⠀⠼⠃⠀⠀⠀⣠⠞⠀⠀⠀⠀⢀⡿⢀⣼⡥⠤⢤⡀
⠀⠀⠀⠀⠀⣾⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⠋⣰⠟⠁⢀⡤⠀⢿
⠀⠀⠀⠀⠀⢻⡄⠀⠀⡀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⣽⡿⠁⠀⠀⠺⠤⠚⠁
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⣸⣿⠇⠀⠀⢀⡾⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠛⠲⢦⣤⣀
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢤⣄⡀⠀⠀⢈⡀⠉⠙⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠈⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣄⡀
⠀⠀⣀⣤⣄⣀⠀⠉⠉⠒⠂⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠐⠤
⢠⣾⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠈⠱⣄⠀⠀⠀⣠⣤⣴⡶⠶⠶⠛⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⡀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⢸
⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠰⠎⠉⠩⠭⠭⠭⠭⠭⠭⠭⠭⠿⠃⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠉⠳⢆⡀⢸
⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⢸⣿⡿⢋⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠀⡇⢸
⠘⠋⠀⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⡸⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⢀⡘⠋⠉⠉⢳⣄⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⢸⡿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⣦⣤⡴⠟⠁⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⠀⢠⡄⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠀⠘⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠐⣩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⣍⠂⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠐⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠆⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠘⣽⠛⠛⠛⢯⠁⠈⡽⠛⠛⠚⣮⠃⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⡀⠀⡇⠀⠀⠀⣿⢀⣀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠠⠝⠒⠒⠒⠊⠄⠠⠑⠒⠲⠒⠣⠯⠬⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠐⣭⠉⠉⠉⢭⠃⠘⡭⠩⠈⠍⣍⠂⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠰⣝⣀⣀⣀⣚⡄⢠⣓⣀⣀⣀⣫⠄⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠀⣠⠴⣒⣯⣽⣓⢦⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠲⣄⡱⠋⠀⠀⠀⠈⢳⣁⡴⠚⠉⠙⠲⣄⠀⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠘⡆⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢾⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠒⠒⠲⠍⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡥⠼⠂⠀⢀⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣰⢫⠄⠀⠀⠀⣸⠃⠀⠈⣷⠀⣤⠶⠶⣦⠀⢠⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠿⢦⣤⡾⠃⢸⡇⠀⢸⠇⢀⣿⠟⢙⡷⠀⣠⣤⣤⡀⠀⢸⠇⠀⠀⢠⠞⢀
⠀⠀⠀⠀⠀⢳⡹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠻⠄⣼⠃⠀⣸⠇⣼⠋⠀⢠⡟⢀⡿⠀⠀⠀⠘⣆⠈
⠀⠀⠀⠀⠀⠀⠙⠲⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⠶⠖⣿⠁⠘⠃⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢠⣷⡂⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣤⡾⠃⠀⠙⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠈⠀⠀⠘⢦⡀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠙⠲⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠉⠉⠋⠉⢬⢧⠀⠀⠀⠀⠀⢀⢿⡀⠀⠀⠀⠀⠀⢠⠏⠀⠘⠀⠀⠈⢳⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠲⣤⣴⣺⠽⠂⠙⠦⢤⣤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⢧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠉⠳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢠⡟⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣿⣠⠴⠖⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⢉⣁⣤⣭⣍⣉⣛⠷⣾⡛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠞⠀⠀⠀⠸⣇⣠⠀⠀⠒⠉⠁⠀⠈⠳⣄
⠀⠀⠀⠀⢀⣴⢋⡀⣠⠾⠋⠉⠀⠀⠀⠉⠉⠻⣦⠙⢷⡔⠒⠒⠒⠂⠀⠠⠤⠤⠄⠀⠀⠒⠒⠒⠂⠀⠈⠛⣲⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⢀⡞⠁⢀⡽⠋⠀⠀⠀⠀⣀⣀⡀⠀⠀⠘⣷⡀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠄⢿⡀
⠀⠀⠀⢸⠇⢠⡏⠀⠀⠀⠀⠀⡜⠀⠀⠘⡆⠀⠀⠘⣷⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠒⠒⠒⠛⠉⠉⠉⠉⠉⠀⠀⠀⢀⣠⡞⠁
⠀⠀⠀⢸⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⡇⠘⣷⠒⠋⠋⠉⠉⠉⣉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢠⡴⠶⠒⠒⠊⠉⠉⠉⠉⠉⠉⠛⢶⡀
⠀⠀⢀⡼⠃⡟⠀⠀⠀⠀⠀⣀⠤⠦⣄⠀⠸⡆⠀⠀⠀⣿⠀⠹⣟⠉⠉⠉⠉⣉⣁⣤⡀⠀⠀⠀⠀⠀⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⣾⠁⢸⠇⠀⠀⠀⢀⠞⠁⠀⠀⠘⡄⠀⣧⠀⠀⠀⢹⡄⠀⣿⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡀⣸⡄⠀⠀⢀⠏⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⢸⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡏⢹⡇⠀⠀⢸⠀⠀⡄⠀⠀⢠⠃⠀⡟⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⠛⣿
⠀⠀⢸⡇⠘⡇⠀⠀⢸⡀⠀⠧⠤⠖⠃⠀⢀⠇⠀⠀⠀⢸⡇⣸⠇⠀⠄⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠒⠒⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⡿
⠀⠀⠀⣧⠠⣷⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣼⠁⣿⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠴⠖⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⢤⠤⢴⡇
⠀⠀⠀⢹⡆⠸⣇⠀⠀⢱⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢰⠏⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⣉⣉⣀⣀⣤⣤⠤⠴⠶⠒⣾
⠀⠀⠀⠈⣷⠠⢻⡄⠀⠀⠳⠤⠒⠋⠁⠀⠀⠀⠀⣠⠏⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠉⠉⠉⠉⠉⢉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢹⡄⠀⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣤⠞⠓⠒⠒⠒⠒⠒⠒⠛⠛⠉⠉⣉⣉⣡⣤⠤⠄⠒⠈⠉⠉⠉⠉⢉⣀⣠⡤⠶⠖⠚⠛⠛⠁
⠀⠀⠀⠀⠈⢷⣀⠀⡏⠛⠶⢤⣤⣄⣠⣤⠶⣋⡴⠿⠥⠤⠤⠤⠀⠀⠒⠒⠒⠛⠛⠋⠉⠉⢉⣀⣀⣤⣤⡤⠶⠶⠖⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣴⣤⣤⣤⣶⣚⣉⣀⣀⣀⣀⣀⣤⣤⣤⡤⠴⠶⠶⠖⠚⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣦⣽⣛⣛⣛⣛⣛⣀⡰⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⢃⣶⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⢀⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣝⡻⠿⣿⣿⣿⣿⠿⢋⣼⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⢠⣤⣶⢶⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⢋⣴⣶⣤⠸⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⡿⣱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⢣⣾⣿⣿⣿⣷⣬⣛⠿⠿⠋⠀⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⡿⣱⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣘⠻⠿⠿⣋⣤⣤⣉⣛⣛⣋⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠉⠙⠶⣦⣤⣀⣀⠤⠄⠀⠁⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⡉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠏⠉⠁⠀⠀⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⢶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠉⣇⡄⠀⠀⠘⡏⠀⢰⡄⠀⠀⠀⠐⣶⣖⠀⠀⢸⡄⠀⣀⡀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢀⠓⠂⠀⠐⣿⠀⠸⡇⠀⠀⠀⡸⠉⢾⡆⠀⠀⢿⡀⠘⠉⢻⡖⢠⡄⢰⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⣇⠀⠀⠀⢈⡤⠀⠀⠀⣥⠀⠀⣷⠀⠀⡰⠓⠒⠘⢦⠀⠀⠘⣧⠀⠀⢠⠇⠀⠀⢸⡇⠘⣶⠆⢤⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢌⣲⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⢿⡇⠒⠃⠀⠀⠀⠼⠠⡀⠀⣿⠀⠀⠀⠆⠀⠀⢸⡇⠀⣿⠀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠑⠀⠀⠀⣿⠀⣿⠀⠈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢠⣤⢈⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠛⠄⡤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣦⣀⣦⣼⣧⣶⣶⣾⣿⣿⣿⡟⢀⡀⢳⣤⣤⣤⣄⣶⣾⣿⠇⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣀⣽⣿⣿⣿⣿⣿⣿⣿⢏⣷⣦⣼⣿⣿⣿⣿⣿⣷⣶⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣿⡿⣿⣿⣯⣵⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠈⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠟⠋
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⣧⣼⣿⠀⠆⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⣾⡟⢟⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡀⢀⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⣤⣀⡻⣧⠀⢂⣀⣀⣠⣄⣀⣀⣠⡤⠴⠖⠛⠉⠉⠉⠉⠛⠛⠳⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢲⠁⠀⠀⠀⠀⠂⠾⠀⢠⠀⠀⢠⡆⠀⡄⠀⠀⠀⢀⠀⠀⠀⢀⡇
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠠⠇⠀⠀⡌⠀⠀⠃⠀⢠⠃⠀⢀⡀⠉⠀⠀⠀⢸⠀⠀⠀⠄
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢖⠂⠀⠀⠀⠀⠀⠀⠰⠀⠁⢠⠇⠀⢀⠀⠀⡟⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠆⠀⠀⡏⠀⠐⡆⠀⡜⢀⠚⡡⢫⡄⠀⠀⠀⠀⢸⠀⠀⠀⢹
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⠁⢀⣨⡇⠀⡌⠊⡾⢰⣯⡴⢂⡄⠀⠀⡼⠀⠀⠀⠈
⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⡀⢀⡌⠄⣺⣿⢠⢰⢧⣧⢣⣿⣿⠀⣠⡎⢀⣼⡧
⠁⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⠀⢾⠁⣾⣰⡽⢻⠃⢀⣾⡿⣿⣾⣿⣷⣷⠆⣴⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠆⣲⡟⣿⢡⣿⣤⣾⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⡅⠀⠈
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⡴⠁⡀⣥⣿⣿⣿⣿⡿⢸⣿⣿⣿⠋⢠⣿⢫⡛⠁⡀⠁
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠁⢦⣀⣀⠀⠀⠀⢀⠂⠀⠀⠁⡄⢸⠋⠨⠄⡾⠀⣹⢏⡿⠻⠈⢠⣮⣾⡽⠀⠀⣼⡠⡿⠀⡄⠠⠄
⠀⠀⠀⠀⠀⢹⠀⠀⡀⠀⢰⢿⡟⠟⠀⠀⢘⠀⠀⠀⢰⠁⠙⠐⠁⠀⠁⢸⡟⣾⠹⢶⣡⣾⣿⣧⣦⡁⣰⠳⣡⠇
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠈⣇⢽⠘⠀⢰⣾⠀⠀⠀⠸⠀⢀⠀⠀⠀⠀⣼⣧⡏⠄⣿⣿⣿⣿⣿⣿⣷⠋⣰⠋⢀⡴⢳⣄⣴
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠐⠔⠈⣤⣦⣄⠁⠘⣆⠀⠀⢀⠀⢠⡠⡀⢳⡀⣿⡜⣿⣹⣿⣿⣿⣿⣿⣿⠇⢸⢇⢀⠈⠀⣼⠟⠃
⠀⠀⠀⡄⠀⠀⠀⠙⢦⡀⠀⠁⣼⣿⢇⣰⣀⢹⠀⠀⢸⠀⢼⣿⣧⣿⡇⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠀⠲⠘⡆⠀⠀⠈⠁⠀⠀⠀⢠
⢀⠀⠀⣄⠀⠀⠀⠀⠀⠙⢶⡖⠻⢷⣿⡇⢸⣷⢣⠀⠀⣇⣤⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡄⠘⠀⢀⠀⠈⠀⠆⠀⠀⢠
⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢉⠀⠆⠈⠈⠀⢠⣹⡿⣻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡇⠀⠀⠀⠀⠠⡾⠐⠀⠀⢫⠄
⣤⣼⢤⣤⣴⣦⣤⣶⣴⣤⣤⣦⣴⣿⣶⣿⣾⣦⣴⣦⣶⣧⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣤⣤⣴⣴⣤⣶⣦⣴⣶⣴⣴
⣿⣷⣼⣷⣭⣿⣥⣵⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⠀⠀⡠
⠀⢀⠔⠂⠁⠀⠃⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠈⠛⡀⢴⠉⣆⣠⠓⠀⠂⠀⠀⠀⠀⠀⠀⡰⠂⡉⠠⢀⣀⡀⠉⢀⢀⣀
⠀⢀⠀⠐⠀⡖⠀⡆⢐⡇⠆⠋⡠⡞⠀⠀⠀⢀⠴⡁⣢⠛⠛⠟⠛⢡⡁⢣⠃⠀⠀⠀⡴⠧⠠⢌⣾⠉⠛⡅⠉⠉⣈⡄
⠀⠈⠄⠀⢰⠃⢊⢡⣀⣠⣠⣌⠣⡭⠀⢀⠀⢸⢀⠂⣉⣀⣀⣤⣬⣥⠼⢬⠅⠀⠀⢠⢣⡀⠤⠒⣿⠜⣦⠽⡔⠒⠎⠰
⠀⠀⠀⠀⠈⠋⠉⠉⠉⠀⠀⠀⠤⣱⠀⠀⠐⠈⠙⠁⠋⠁⠸⠉⠃⠈⠀⠱⠆⠀⠀⠀⠁⢹⠀⠔⠁⠸⠈⠃⠀⡞
⠀⠀⠀⠀⡄⠀⢀⠀⡀⠀⠀⠆⠂⠁⠀⠀⠀⠀⠀⠆⠁⢢⣀⡀⠬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣓⠶⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⠄⢤⠦⡀⠢⠚⢤⣀⣎⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣀⣴⣄⣀⡀⠀⠀⠀⠠⢀⡠⠀⠤⠠⠄⡀⠈⠣
⠀⠀⡀⢀⣗⡀⠀⠀⢀⡭⠀⠁⡈⠀⡀⠀⠀⠀⢀⠀⠀⣰⠀⠀⠸⠛⠏⠐⠀⠀⠀⢁⠀⢨⠃⠀⠀⠀⠀⠨⣤⢤⠄⡄⡆
⠀⠀⢣⠀⣈⠛⡱⠶⠟⠷⡂⠂⠋⣏⠁⠀⠀⠀⡌⠀⠀⠘⣶⣦⣤⣀⣴⠊⠀⠀⠀⠈⡀⠈⡔⠀⠈⡤⠉⡶⠀⠀⠰⠧⠁
⠀⠀⠉⡽⠀⢂⠁⣀⠚⢄⢥⢴⡖⠊⠀⠀⠀⢠⠊⠠⠀⠾⠐⠐⡡⡜⠀⠀⠀⠀⠀⠀⠀⢀⣇⡵⠦⠕⠶⠞⠚⣾⠛⠋
⠀⠰⠒⠲⣏⣠⠥⢤⠊⠁⠻⠉⠀⠀⠀⠀⠀⢸⢐⣈⣄⣀⡄⢄⡣⠍⠀⠀⠀⠀⠀⠀⣩⡰⡀⠀⡀⠀⡀⣠⠞⠃⠐⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠂⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠔⠄⠀⠀⠀⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠐⠀⠀⠀⠡⡀⠀⠀⠀⠐⠢⡀⣤⠴⢤⠃⠀⠘⡠⣔⡀⠀⠀⠀⠀⢀⠀⠀⠤⠠⢀⠐⡢
⠀⠀⠀⠳⢄⣠⣔⣀⣀⡀⣀⣤⠴⠃⠀⠀⠀⠀⠐⠨⡀⡤⠊⠀⠉⢙⠀⠀⢸⠄⠀⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⢈⠄⢤
⠀⠀⠀⠀⢀⡀⠀⠙⠻⠋⠩⣄⢀⡀⣀⡀⠀⠐⠂⢰⡁⣀⠀⠂⡆⢘⠡⡅⢃⡇⠀⠀⠀⠠⠀⠀⣸⠀⠠⠆⠴⠀⣀⣘
⠀⠀⠀⣀⣀⡗⢦⠀⠤⡒⢒⡍⠃⠁⣧⠀⠀⠀⡠⠰⣓⣈⣤⡄⠤⠬⡥⠲⡖⠑⠀⠀⠀⠄⣀⠀⠲⠚⠓⠉⠋⠉⠋⠉
⠀⢠⠏⠽⠀⣺⡍⢀⣈⠣⠌⠦⠶⠲⢎⠀⠀⠀⠱⠀⡈⠹⢄⣠⠖⢞⠁⡀⡓⠀⠀⠀⠀⠀⠀⠒⡀⣀⠀⠀⠄⠂⢀
⠀⠹⢶⠒⢋⡁⢉⣀⢀⠀⡠⡀⣀⢀⡠⠃⠀⠀⠀⠘⠀⠘⠀⠀⠆⠀⠀⠁⠈⠃⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣀⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠔⡀⠀⠀⣠⠐⠉⠁⣀⢀⡀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⡈⠀⠀⠀⠄⠀⠀⠤⠀⠈⠘⠉⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⡹⠝⠂⠀⣁⢢
⠀⠀⠀⠀⠀⠀⢊⡁⠀⢀⡤⠊⣀⠔⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⣁⠆⠀⠀⢀⣠⠈⡀⠐⠀⠁
⠀⠀⠀⡠⣀⡀⠧⢈⡡⠝⢤⡖⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠤⠀⠀⠁⣤⣒⠋⠉⠈⢉⡠⠀⠀⣈
⠀⠀⠰⠀⠟⠡⠞⠂⠱⡤⠄⣡⠀⠈⠀⠀⠀⠀⠀⠀⠂⡡⠄⠀⠀⠀⠀⠀⠐⠀⠀⣠⡄⠂⠐⠛⠉⠀⢊⣧⠾⠚⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡯⢃⡤⣀⠔⠊⠁⠀⠀⠀⣠⠴⠦⠀⠁⠀⢀⣠⠔⣞⡭⠦⡉⠂⠀⠀⡢⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠉⠁⠴⠉⠊⠀⢠⡀⠠⠂⠐⣈⡤⠀⢀⣠⠴⠚⡽⠂⠉⡄⠀⠀⠀⠀⠀⠊⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢬⢻⢰⠀⠐⢾⢤⠀⠀⠄⠒⠉⠀⣠⠴⢊⣡⣴⠾⠋⠁⡠⠀⠰⠂⠄⠀⠀⠀⠀⡀⠀⠀⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⠲⠶⠾⠇⢠⣄⢀⡷⠀⠀⠀⢀⣨⣔⡾⠋⠉⠀⠀⡀⠀⠂⠀⠀⠀⠀⠔⠀⠀⢀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣚⣉⡡⣶⡀⢸⣏⠊⠉⡟⠗⢀⣠⡶⣻⠟⠉⠀⠠⠤⠒⠉
⠀⠀⠀⠀⠀⢠⠏⠘⠁⠤⠀⠚⣗⣀⣈⣼⣭⢔⣚⠁⠡⢊⡡⠀⠀⠀⠀⡚⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠸⠀⠞⢛⡀⢈⡫⠀⢸⢉⣯⣗⠉⢰⡜⠖⢿⠄⠖⠀⢀⡠⠆
⠀⠀⠀⠀⠀⠀⢀⠚⢰⣧⣈⠳⡀⢻⡿⠉⡡⠠⢸⢍⡦⠀⠁⠸⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠑⠲⠞⠭⠴⠖⠛⠁⠀⡳⢄⣆⠚⣥⣰⠟⡾⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣈⡥⠒⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠮⠄⠱⠾⠛⠈⠀⠀⠀⠀⠀⠠⠀⠈⠀⢀⡡⠖⠋⠁⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣪⣐⡀⣠⠴⠚⠁⠀⠀⣀⢴⠂⠀⠊⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆⠀⠀⠀⣠⠶⠛⢲⢄⣏⠁⠘⠉⠀⢀⣀⡴⣶⠯⠞⠋⠀⠐⠀⠀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⠀⠀⠌⠙⢂⠼⢉⡱⡍⣦⣀⢀⣀⡭⣦⡉⠄⠀⠀⠀⡀⠀⢀⠠⠊⠁
⠀⠀⠀⠀⠐⠲⡆⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡃⣡⢾⢄⡷⣿⢿⡿⠋⠀⠀⢻⡄⠀⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⣻⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣦⣹⣖⣫⡴⠂⡈⠁⠀⠀⠀⢈⣇
⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠖⠉⠙⢅⣠⠷⠤⢤⡀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⠀⠣⠀⢀⡄⡀⠀⠐⢺⡇⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⠋⠈⠃⠀⢠⠀⢸⣇⡼⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⠀⢸⣤⣼⣿⠟⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠁⠉⢸⠁⠀⠀⠀⣠⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⠀⠀⠀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠲⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣽⣀⣀⣀⣀⣀⣀⣀⣤⠷⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠄⠀⠀⠀⠀⣀⣀⠀⠀⠸⣀⣠⣤⣤⣀⡀⠀⢰⢲⠶⠶⢖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⢆⠔⡪⠓⠋⠉⠙⢣⡤⡊⠁⡠⡲⠟⠻⢷⣌⡀⡄⠀⠈
⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⢀⢄⡲⠖⠛⠓⠣⢄⠎⠀⠀⢀⡋⢹⠉⠑⣘⡘⡘⠀⢀⡀⠀⠹⡁⢃⠀⠸⠘
⠀⠀⠀⠀⠈⠻⡢⣀⢀⢜⠕⠁⡠⡤⣤⡤⡦⡀⡏⠟⠋⠙⣽⠌⠆⠀⢁⠁⠇⠀⠘⠈⡆⠀⠁⠘⡄⠀⡎
⠀⠀⠀⠀⣀⣤⡼⠶⠿⢮⣄⠀⡆⡅⠀⠀⠘⡵⢰⠸⠀⠀⠈⢆⠸⠀⠘⡀⡇⠀⢸⠀⢐⠠⠝⠆⢰⠀⠰⢃⣀⣀⣀⣤⠤⠄
⠀⠀⢰⠪⠋⠁⠀⢀⠀⠀⠙⡵⡇⡇⠀⣀⠀⠱⣹⡄⡆⠀⣀⠀⠳⣇⠀⢡⢡⠀⠀⡘⡄⡀⠀⠘⠀⣣⠼⡸⡒⠉⠁
⠀⠀⠈⠆⢇⠀⠘⡀⢈⠀⠀⡀⡅⠃⠀⢸⠄⠀⠱⠃⢡⠀⠸⠳⡀⠈⠀⠈⡄⡄⠀⠰⡈⡐⠀⢠⠀⢡⠀⠘⢐
⠀⠀⠀⠸⠘⡀⠀⠓⠃⠀⠰⣡⡇⠀⠀⠨⠼⠀⠀⠣⠈⡆⠀⠇⠈⢆⠀⠀⢱⠈⢆⡀⠈⠁⢀⠆⣸⡈⠒⠐⣸⠇
⠀⠀⠀⠀⢃⢣⠀⠀⡤⢄⠀⠀⢣⠀⠀⠀⣀⠠⡀⠀⢣⢡⠀⠐⣹⢆⠳⡀⠰⠚⠢⢌⣀⣈⠡⠊⠀⠀⢀⠜⡝
⠀⠀⠀⠀⠘⠘⠄⠀⢃⢀⠇⠀⢸⠸⠀⠀⡁⢆⠡⡀⠤⠇⠦⠀⠃⡆⠓⠐⠒⠊⠁⠀⢀⠟⠒⠀⠐⣊⠥⠊
⠀⠀⠀⠀⠀⠇⢃⠀⠈⠁⠀⡠⢃⢰⣀⡀⠇⡀⠂⠐⠒⠒⠀⠉⠉⠁⠀⠀⠀⢰⠂⠒⡩⠊⠑⢄⠹⡀
⠀⠀⠀⠀⠀⢸⠘⠄⠒⢂⡉⡔⠻⠤⠄⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠏⠁⠀⠀⠀⠀⠑⢼⡄
⠀⠀⠀⠀⠀⠀⠉⠉⡹⢁⠔⠉⠢⡑⠤⣀⠀⠀⠀⣀⠤⠊⠢⠄⣀⣀⠠⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡣⠂⠀⠀⠀⠈⠒⠤⠌⠉⠡⠔⠚⡄⠀⠖⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢴⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠊⡀⠀⢤⡤⠤⠤⣤⡤⠤⢄⠆⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣀⢀⣸⣀⣀⡀⣸⣀⡀⣸⡄⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠛⠛⢻⠛⠛⠛⢻⠛⠛⢻⠁⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠄⠀⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠟⠛⢿⠟⠛⠛⢿⠟⠛⢻⠂⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⢸⠃⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡶⠶⢾⡶⠶⠶⢾⡶⠶⢾⠁⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡀⠀⣸⡀⠀⠀⣸⡀⠀⣸⢡⢶⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣹⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⢀⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⠾⠒⢈⣡⣄⣒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡅⠀⢀⠀⠤⢥⡛⠻⠟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣀⠀⠀⡉⠀⠀⠀⢱⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡸⢁⡔⠀⡇⠀⠀⢀⠜⢀⠔⠒⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠆⢢⡆⠀⠀⠚⠓⡇⠀⢠⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡺⠇⠘⠀⠤⠖⠧⠔⠧⣀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡁⢤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣶⠆⠀⡜⢂⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠐⣸⠀⠀⡇⠀⠑⠤⣀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⡑⠦⣀⠎⠀⠀⠹⠀⠀⡇⠀⠀⠀⠀⢈⠏⣉⡀⢀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⠱⣄⠎⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⣆⠬⠄⠴⢀⠏⠱⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠈⠣⣄⠀⠀⠀⠀⠀⠃⠀⠀⠐⡄⡤⣇⠀⢤⣀⠸⠤⢃⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠡⡀⠀⠙⠂⠤⣀⣸⠆⠀⠀⠀⠀⢠⢦⡀⠠⠄⠔⠀⡛⡀⢀⣓⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⢀⠀⡠⠚⠁⢹⠟⠲⡄⠀⠀⢣⣀⠨⣗⡂⠀⠀⠻⠡⠊⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢠⠂⢀⠄⠸⣀⠙⡀⡠⢒⡭⠀⡦⡈⠢⣤⠃⠈⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡴⠁⣠⠃⠀⠀⢘⠀⠷⠀⡜⠑⠀⠃⠀⢄⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣀⠎⠀⡸⠁⠀⠀⠀⢀⠑⢠⠎⠐⠁⠀⠀⠀⠀⠱⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡄⠈⠓⠰⠃⠀⠀⠀⠀⠒⠘⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢈⠑⠠⢄⣀⠬⢀⡀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⠄⠄⠊⠀⠀⠀⠀⠍⠑⠒⠒⠚⠳⢤⠄⢠⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⠠⣀⠀⠀⠀⣀⠜⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠘⠛⠋⡓⠒⠚⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠴⢍⡁⢀⣀⡀⠀⠀⠀⠀⠀⠑⡔⢁⣀⡀⢀⣷⠀⠀⢀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⢶⡺⠟⠻⠋⢁⣀⣥⣨⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠟⠋⠁⠈⠑⠂⣁⣠⣴⣾⣿⡿⠟⠉⢰⠾⠋⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⡁⡨⠤⢒⣀⠤⠀⠀⣒⣿⣿⣿⠿⢋⡡⢀⠀⠀⠁⠴⣋⣥⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣭⠒⠉⠀⠀⠀⠒⠉⠉⠀⠀⣠⣿⣿⠟⡋⢁⡴⠟⣋⠤⠀⡀⢀⣴⣿⢿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠭⡪⣝⢦⣤⠴⠚⠁⣠⣴⣶⡿⠟⢉⠴⠋⠀⠀⠔⠋⣀⣴⣿⣿⠟⠋⠒⠉⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢈⠈⠳⣿⡄⢀⣤⡾⡛⠭⣀⡤⠖⠋⠁⣠⣶⣴⣿⠿⣛⣽⣯⠔⣊⣤⠞⣋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢃⣀⣐⣈⣄⣤⣽⠿⣿⣟⠉⠐⠉⣀⡴⢂⣴⣾⢿⣛⣭⠶⠛⠙⠉⡴⠋⠟⣁⣼⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡜⢋⠉⡅⣠⡿⠉⠒⠄⠈⢻⣷⡶⢋⣵⠾⠛⠩⠒⡹⠋⢀⠔⠂⣠⣴⣊⣴⡿⠛⠵⢛⣆
⠀⠀⠀⠀⠀⠀⣴⡟⡩⢿⣷⡈⢮⣼⣟⣉⣀⣴⣆⣤⣥⣽⡟⢋⣴⡤⠔⠂⢀⣠⠾⢳⢂⣴⣿⠟⣫⠕⣀⠔⢒⣫⡽
⠀⠀⠀⠀⠀⣾⡝⠀⠁⠚⢻⣷⣿⠿⡫⢁⣾⣿⣿⠿⣅⠹⣽⣏⠀⠀⠐⠊⢁⣴⣴⡷⣟⠋⠠⢊⣵⣞⣥⡾⠟⠉
⠀⠀⠀⠀⠀⣿⠀⢴⣤⣾⣿⢫⠀⠀⠀⣼⣿⣟⠁⠀⠀⠀⠿⣿⡴⢖⣫⡵⠟⣋⠵⢞⣩⣠⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠻⠶⠿⠿⠿⢿⡀⠁⠜⣾⣟⣁⣀⣀⣤⣷⣒⣀⣿⠶⠛⠉⣚⠉⢀⣚⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣠⣿⣿⣟⠛⠟⠛⠻⡯⡹⣿⡤⠒⠋⠁⣺⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠙⠳⢮⣐⠄⠀⣿⣇⣤⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠒⠚⠛⢦⣴⠋⠉⣉⠉⢣⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣏⠀⠀⠒⢒⣷⡿⠖⠒⢶⡶⠒⠋⠉⠉⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠋⠀⠀⠀⠈⣧⠀⠀⢹⣇⠙⠂⠄⠀⠁⠀⣀⣠⣤⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠁⠀⠈⢧⡀⠈⠛⠛⣱⣿⡀⠀⠀⠻⣷⣦⣄⡀⠀⠩⣛⡿⠋⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠒⠚⢁⣿⣷⡀⠀⠀⠸⣿⡇⠀⠀⠈⣿⡇⠈⣻⣿⣿⠿⠒⠰⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢉⣳⡀⠀⠀⠀⢹⣿⣇⠀⠀⠀⢻⣿⠀⣀⣠⣾⣧⣶⣶⣿⣿⣍⠀⠺⢿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣄⠀⠀⣈⣿⣿⠀⣀⣀⣈⣿⣶⣶⠿⠿⠟⠉⠙⠻⣿⡏⠀⠀⠁⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠙⠛⠶⢿⣿⣿⣦⣬⣽⡿⠿⡟⠁⠀⠀⠀⠀⠀⠀⠈⠙⠀⢴⣬⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣦⠀⠀⠀⠈⠙⠺⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣈⣣⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⢳⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣨⡿⡇⢹⠀⠀⠀⠀⠀⡀⢄⠠⡈⢧⣸⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣤⡀⠃⠀⠀⠰⢄⡱⢤⡙⣬⣷⣽⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠶⠒⠀⣈⡤⠎⠉⠂⠀⠳⣤⣄⣙⠳⢿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣿⣾⡿⠛⢿⣿⣟⣛⠓⠶⢤⣹⣿⠿⠿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⣿⠀⠀⣿⡇⠈⠉⣿⠓⠒⠀⣈⡥⢀⡾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣶⠀⣿⡇⡔⠀⣿⠀⢀⣾⡟⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢿⣿⠀⡿⣼⠁⠀⣿⠀⢸⣿⣴⡖⠀⡾⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢸⣿⡀⢱⡇⠀⢀⣿⠀⠈⢻⡿⠁⠘⢁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡇⣾⡇⠀⠈⡥⠀⠀⡾⠁⠀⡰⣈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠁⢻⠇⠀⢀⡇⠀⠀⠀⠀⠘⣿⣃⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢸⡿⠀⣠⠎⠀⠸⡇⠀⠀⠀⠀⣰⣿⠇⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⡿⠀⠀⠀⠁⠀⠀⠀⠈⢡⣯⣾⢱⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⢃⡄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢡⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣱⠟⣡⠟⣼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡉⠀⣡⣾⣣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⣶⣤⣄⣀⣀⣤⡤⠶⠿⠖⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⣫⣿⠿⣻⣿⡿⢿⣿⣿⣿⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠟⣋⣭⠶⣛⡭⢖⣫⡭⠖⠠⣚⡵⢚⣵⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⢛⣩⡵⠾⣛⣭⠶⣛⣵⣾⣿⣡⡤⠶⢛⣭⣾⣿⡿⠛⠁⠠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⢛⡩⠔⢚⣩⡵⢖⣋⡥⠖⣛⢉⣈⣩⡭⠶⢒⣻⣽⡿⢿⠵⠂⣀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⢿⠋⠁⠠⢚⣡⡴⢞⣫⡴⢞⡫⢕⣚⣍⣙⠋⢉⣀⣘⣿⣿⡟⠉⢀⣠⣴⡿⢟⣫⣵⣿
⠀⠀⠀⣀⣤⣶⣿⠟⢋⣡⠞⠁⢀⣴⠾⣋⣵⠞⠫⢕⣪⡵⢚⣭⣬⡭⠔⣀⣼⣿⠟⣛⣭⡴⠾⣛⣭⠶⢞⣿⣿⠟⠁
⣤⣶⣿⣿⣯⣿⣋⣴⡋⢀⣠⢞⣭⠶⡛⠉⣠⣴⣾⠟⣫⠾⣋⣥⣶⣶⣿⣿⠿⠛⠛⣋⡥⠖⠛⣩⣴⣾⣿⠟⠁
⣿⣿⠽⢟⣛⣛⣿⣿⣿⡿⠿⠿⢿⣿⣶⣿⣿⣿⣥⣾⣵⣏⣩⣾⠿⠛⠉⠀⢀⣤⡾⠋⢀⡴⢋⣵⡿⠋
⣿⣟⢉⣀⣤⣶⠶⠶⠶⣾⣭⣤⣌⣉⣉⣉⣉⣙⡛⠻⠯⣻⣿⣁⣤⣤⣤⣴⠿⠋⠀⠰⢋⣴⡿⠋
⣿⡿⢿⣿⠁⠠⣶⣾⡷⢖⡺⠭⠙⠛⠛⠋⣙⠛⠻⢷⣦⣸⣿⠟⠛⠉⠁⢀⡠⠔⣠⣾⡿⠋
⣿⣷⢤⣉⠳⢤⣈⠻⣷⡙⢏⠩⣤⠤⠤⢤⣌⠉⠀⢀⠀⢹⣿⡤⠴⠒⢉⣡⣴⣿⠟⠉
⣿⣿⠶⣭⣛⡢⠍⣓⠦⣝⡳⠥⢄⡀⠉⠉⠵⠋⠴⣋⡴⢻⣿⣠⡴⠞⣫⣿⠟⠁
⢿⣿⣿⣶⣿⣿⣿⣮⣽⣶⣍⣙⣓⠶⠶⠶⠶⠶⢛⣁⡤⢼⣿⢁⣴⣾⠟⠁
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣷⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣷⣦⡀⣠⣴⣶⣶⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⠁⠀⠀⠹⣿⡿⠉⠀⠀⠀⠉⢻⣷⣶⡿⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⡀⠈⣿⠁⠀⠀⠑⠺⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢀⣠⣤⣤⣴⣶⠶⣾⠿⠟⠛⠉⠉⢻⡄⣿⠀⠀⠀⠀⠀⢸⣿⣅⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⠍⠀⠀⠀⠀⠀⠸⣤⣁⣀⡤⠖⣻⡇⣿⠀⠀⠀⠀⠀⣸⡟⠛⠛⢛⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⣿⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⢸⡏⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠙⠛⣿⣏⡀⠀⠀⢀⣾⠇⠀⠀⠀⣼⠇⠀⠀⠀⠀⢀⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣆⠀⠀⠀⣠⡾⠃⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠔⠉⠀⠸⣿⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠿⠏⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠈⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢷⣄⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡏⠓⠀⠠⠴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢶⣦⣤⠤⠖⠊⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣶⣤⣤⣤⣾⣀⣀⣀⠀⠀⠀⠀⣀⣀⣠⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⠋⠋⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢰⠶⡆⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣠⠏⠀⠹⡄⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣠⠴⠞⠋⢀⡴⠋⠀⠀⠀⠙⢦⡀⠙⠳⠦⣠⡀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⢠⣴⣦⣄⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⣿⡅⢀⣿⠞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠈⠛⠛⠁⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠀⣟⣹⣯⣹⡾⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸⡇⠀⣟⢻⣟⢣⡿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢸⡇⠀⣿⠋⠉⣿⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⣿⠀⠀⣿⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⡶⡿⢏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣶⠋⣙⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣋⡙⢲⡀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⡇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠆⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠻⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡼⠃⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣼


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠛⠿⠿⠿⠻⡿⢫⡝⠿⢿⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣛⣛⣛⣻⣻⣿⣿⣛⣛⣛⣛⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠛⠛⠛⠷⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠀⠀⠈⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⢀⣾⠋⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣼⡇⠀⠀⣤⣤⣀⣀
⠀⠀⢠⣿⣷⣾⣿⣿⣿⣿⣿⣷⡀⠀⣴⣶⣶⣿⣿⣿⣿⡇⠀⢠⣶⣶⣾⡇⠀⢠⣶⣶⣶⡗⠀⠀⣀⣴⣾⣿⣿⣿⣷⣆⠀⣿⣦⣄⣸⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢠⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⠃⢀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢉⣿⣿⣿⣿⠃
⠀⢠⣿⣿⣿⣿⠋⠀⣽⣿⣿⠋⠀⣼⣿⣿⡿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⡇⠀⣿⣿⣿⡏⢠⣿⣿⣿⣿⡿⠛⠉⠛⠁⠀⠀⠀⠀⣸⣿⣿⣿⣷⡀
⠀⢸⣿⣿⣿⣿⣴⣿⣿⠟⠁⠀⢰⣿⣿⣿⢣⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣧⢸⣿⣿⣿⢣⣿⣿⣿⣿⠋⠀⠀⠀⢀⣀⣀⡄⠀⢰⣿⣿⣿⠇⠘⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⢀⣿⣿⣿⡟⢸⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣼⣿⣿⡿⣾⣿⣿⣿⠃⠀⣴⣾⣿⣿⣿⣿⡇⠀⣾⣿⣿⡏⠀⠀⢹
⠀⣿⣿⣿⣿⠋⠉⢉⣿⣿⣿⣿⣿⣿⣿⣃⣼⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡏⠀⢠⡿⠿⢿⣿⣿⡿⠀⠘⠛⠻⠟⠀⠀⠀⣸
⢀⣿⣿⣿⣿⣷⣶⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⡟⢿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣧⠀⠀⠀⣠⣾⣿⡿⠁⠀⠀⢀⠀⠀⠀⠀⢀⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿⣿⣿⣿⠟⢛⣿⣿⣿⣿⣇⣿⣿⣿⡇⠘⣿⣿⣿⣿⡿⠀⢿⣿⣿⣿⣿⣶⣿⣿⣿⠟⠁⢠⣶⣿⣿⣿⡆⠀⣠⡾⠃
⠾⠿⠿⠿⠟⠋⠁⢸⡇⣾⡿⠿⠿⠟⠀⠸⠿⠟⠛⠛⢹⣿⣿⣿⠃⠀⠸⠿⠟⠛⠁⠀⠈⠻⣿⣿⣿⣿⠿⠛⠁⠀⠀⠻⣿⣿⡿⢛⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢀⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⣀⣀⣤⣴⠾⢻⡿⠃⠀⠀⠀⠀⠀⠀⣴⣿⣶⠶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⣴⣟⣀⣀⣀⠀⠀⠀⠀⣼⣯⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢻⡿⠃⠀⠀⠀⢉⣩⡽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⢀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⣤⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠾⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⠀⠀⠄⡀
⠀⠀⠀⠀⢀⠄⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⡠⠴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⡠⠋⠀⠀⢸⠀⡀⠀⠀⠀⢨⠀⡀⢀⡠⠆⠀⠀⠀⠀⠀⠀⠈⠢⣄
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡞⠙⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⢘⡶⠒⠒⣏⠀⠀⠀⢀⡴⠉⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡤⠤⣀⣀⣀⣀⣀⡀
⢸⠀⠀⠀⠈⠒⠒⠚⠉⠀⠀⠀⠀⣧
⠘⢄⣀⣀⣀⣀⡀⠀⠀⠤⠤⠤⣼⠀⠐⢄
⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⡀
⠀⢑⠦⠄⠀⠐⠒⠈⠀⠈⠐⠾⡅⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁
⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠢⢄⡀⠀⠀⠤⠖⠒⠒⡎
⠀⠀⠀⠀⠈⠒⠠⠤⢠⡀⠠⢀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⠶⠒⠛⠓⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉
⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋
⠀⢈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⡆⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁
⠀⠀⠙⢿⣝⢷⣄⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⢀⣀
⠀⠀⠀⠀⠈⠳⢍⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⣀⣤⢶⣿⣯⣷⣶⣿⣿⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠰⢤⣤⠖⠉⠀⠈⣹⠖⠁⠀⠀⠈⢣⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⡀⠄⠒⡉⣽⣾⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⡞⠁⠀⠀⣠⠞⠁⠀⠀⠀⣠⠞⠁⠀⠀⢀⠈⢳⡄⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣠⠏⠀⠀⠀⡼⠋⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⣼⡀⢸⠃⣀⡀
⠀⠀⠀⠀⠀⢰⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠼⢃⡴⠛⠉⠀⡈⢳⡄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢻⣤⠞⠀⠀⢰⠏⣠⠌⠁
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢰⣿⡏⠀⠀⠀⣨⠉
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠀⠀⠀⠀⣿⠀⠙⠲⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠟⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠙⠳⣦⣀⠀⠀⠁⠀⠀⠀⠉⠙⠒⠤
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠰⠧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠓⠦⣄
⠀⢀⣴⣶⣶⣤⣄⣀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠐⠀⢀
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠹⡄⠀⠀⣠⡶⠶⠟⠛⠛⠛⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣿⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠳⠈⠉⠩⠭⠭⠭⠭⠭⠭⠭⠭⠿⠃⠀⠀⠀⢀⠀⠀⠀⠀⡇
⢸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠙⠦⣀⠀⡆
⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⢸⣿⣿⠋⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⣿⢀⡄
⠘⠋⠁⠚⠛⠛⠛⠛⢿⡇⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠀⡰⠛⠉⠙⠳⣆⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⠀⠀⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠻⣦⣤⣤⡾⠋⣾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡇⢀⡀⠀⠀⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠇⠈⠇⠀⠀⠹

⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⢿⡘⣿⣿⡏⢏⠺⡁⢿⣿⣿⣿⣿⢸⢣⡏⡸⣽⣿⣿⠟⣼⠟⣴⣿⣿⡿⢋⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠌⢯⢳⠈⢿⣿⠀⠀⠁⢸⣿⠻⢸⡏⡟⡘⠀⢱⡿⣱⠋⡼⢫⡾⣵⡿⢋⣴⣿⡿
⡻⣿⡿⣿⡻⣝⢿⠛⠃⠀⠈⠣⠣⠈⢿⡇⠀⠀⠘⠛⠀⠀⠀⠀⠁⠀⡱⠁⠁⠞⡵⢫⠞⠁⣴⣿⢟⣵⠾
⣿⣮⣙⢮⡙⢮⡳⡕⠀⠀⠀⠀⠀⠀⠘⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠑⠁⡠⠞⢁⡵⢛⠵⢊
⡳⢿⣻⢧⣈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠩⠒⠁⣢⣵
⡻⠶⣍⡓⠯⣓⢤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⣯⢷⣻
⢭⣓⡢⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⣀⣼⠻⣷⣤⣤⠞⣿⠀⠀⠀⠀⠀⠀⠀⣠⢴⣺⡽⠞⣛⣭
⣟⣲⠬⢅⣀⠀⠀⠀⠀⠤⣄⣀⣀⣀⣼⡏⠛⠿⠋⠀⠙⠛⠁⢸⣿⣦⣤⣤⣤⠶⠊⠑⠈⠉⠀⢀⣁⣤⡴
⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⡎⠉⡉⠙⣿⠁⠀⢱⡆⠀⢶⠀⡶⠀⠉⠉⣴⠛⢁⠀⠀⠀⠀⠀⠩⠓⣒⣛⣋
⣀⣀⣀⡀⠀⠀⠀⣀⣀⣠⣧⠀⠋⠀⢿⠀⡄⠈⡇⠀⠈⠀⡇⠀⡷⠤⣿⠀⣼⣀⣀⡀⠀⠀⠈⠉⠉⠙⣛
⣿⠿⠿⠟⠋⠉⠉⠉⠙⠛⢿⡀⠸⠀⢸⠀⣠⠀⡇⠀⡆⠀⣇⠀⠃⢀⡗⢰⠟⠋⠉⠉⠁⠀⠀⠐⠲⠶⠶
⣶⣂⡀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢴⣖⣚⠒⠛⠋⠉⠉⠉⠉⠉⢉⣉⣉⠻⣿⡀⠀⠀⠀⠀⠀⣀⣲⣤⠬⣭
⣷⣖⣂⠄⠀⠀⠀⠀⢠⠴⠛⠛⠛⠛⣿⡇⣠⣤⡄⠀⣠⣤⣀⢺⣿⠛⠛⠛⠻⠦⣄⡀⠀⠀⠀⠀⠉⡉⠛
⡿⢟⠒⠉⣀⠤⡂⠀⠀⠀⠀⠀⠀⠀⣿⠞⠋⠙⢿⣸⡟⠉⠙⠻⣷⠀⠀⠀⠀⠀⠀⠩⢗⣮⣼⣿⣶⣾⣭
⣫⣥⣶⣯⠗⠋⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⣿⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⡀⠀⠉⡻⣿⣿⣿⣿
⣿⠟⢋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⢆⠀⠀⠀⠀⠀⡀⠪⡳⣕⣮⣙⢿⣿⣿
⠴⣪⠔⣠⢔⡥⣂⠔⡡⠂⢠⢢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⢐⣝⢦⣈⠺⣿⢿⣷⣮⣻
⣚⣵⣿⡿⣫⡾⣫⢞⡕⣴⣷⡧⠀⠀⣼⡸⠀⠀⠀⢠⠀⠀⠀⢠⠸⣿⡄⠀⡐⣄⠙⣷⣝⣷⣌⢳⣿⣿⣿
⣿⣿⣯⡾⢋⡴⣻⣿⣾⣿⣿⢃⡎⣰⣧⡇⢀⢠⢠⢸⡀⡀⠀⡎⡆⢻⣿⡜⣜⣜⢮⣿⣿⣿⣿⣷⣝⢿⣿
⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⢃⡾⣰⣿⣿⣇⣺⢸⢸⣼⡇⡇⡄⣿⢻⡈⣿⣿⣾⣞⣯⣿⣿⣿⣿⣿⣿⣷⣽