Allah Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⣄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⢿⣆⠰⠁⠉⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢻⡆⠀⠀⠀⠀⠹⡷⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⠋⠀⠀⢰⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣧⠈⠀⣤⠀⠤⠒⠁⠀⠀⠀⠀⠀⢀⣤⡴⠞⠋⠉⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⢿⠀⠀⢸⡀⠀⣀⡀⠐⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⣿⣿⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠳⣼⠀⠀⢸⡇⠀⠉⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⣄⢀⡀⠀⠀⠀⣾⡀⡰⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠹⣷⡄⠘⡇⠀⠊⠁⠀⠠⠒⠀⠀⠀⠀⢸⠀⢹⡈⣷⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻⠀⠀⡇⠙⠆⡇⢠⠶⠶⢦⣤⣄⣀⡀⠀⠀⠘⠗⠋⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠸⠶⠛⣦⣤⠇⢀⠞⣿⣼⣞⣛⣿⠾⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⡄⠠⠄⠈⠡⢞⣩⠞⠘⢿⣿⣭⡴⠞⠋⠁⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠸⣿⡇⠀⠀⢰⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⡴⠞⠋⡁⠀⢠⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⢿⣿⣷⡀⠀⠀⠀⠀⣿⣿⠀⠀⠸⣷⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⢸⣿⡏⠓⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣾⠏⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡟⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣷⡀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢀⣿⠁⢸⣿⣧⡀⠀⠀⠀⠀⣠⣾⠇⠹⣿⣿⣶⣤⣀⣠⣤⣾⠃⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣀⣼⡏⠀⠀⢿⣿⣿⣶⣶⣶⣾⣿⠟⠀⠀⠙⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠟⠀⠀⠀⠀⠙⠿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠃

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢀⣠⣶⠆⠀⠀⢹⡇⢸⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣽⡟⠀⠀⢀⣀⣠⣤⣴⠶⠾⠛⠛⠉⠀⠀⣀⣄⠘⣷⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⡾⠛⠉⣉⡄⢠⡄⢠⡄⠀⠀⠛⠻⡿⠉⠀⢿⡄⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣻⡿⣷⠀⠀⠀⠘⢻⣇⣸⣇⣸⡇⠀⠀⠀⢀⡀⠀⠀⢸⣇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⡿⣿⠶⠟⠀⡀⠀⠀⠀⠙⢋⡙⠋⠀⠀⠀⢠⡾⣷⡀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠀⠀⣠⡶⠟⣷⠀⠀⠀⠀⣼⢿⣆⠀⠀⠀⢸⡇⣘⣷⡀⠀⢻⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣀⠀⣿⠀⠀⠀⠀⣿⡀⢿⡄⠀⣀⠸⣧⠸⣿⠳⠄⠸⣧⠸⣷
⠀⠀⠀⠀⠀⠀⣠⡀⢠⡄⠀⣼⠏⣰⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⣇⢰⣟⠀⣿⡀⢹⣇⠀⠀⣿⡄⣿
⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⢰⡟⢠⣿⡏⢰⡟⠀⣸⡇⠀⢸⡇⠀⢿⡄⠉⠀⠘⣷⡀⢻⡆⠀⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠁⣾⠃⣾⢻⣇⢸⣇⠀⠙⠃⠀⣸⡇⢀⠘⢷⡄⠀⠀⠘⣿⠘⣿⠀⠈⣿⢸⣧
⠀⠀⠀⠀⠀⠀⢹⣿⣆⣸⡏⢠⡿⠈⣿⡀⢿⣄⣀⣀⣴⠟⢀⣿⣧⡈⠻⢶⣤⣴⠟⢰⡟⠀⠀⢻⣾⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣍⣉⣤⡿⠁⠀⠘⣷⣄⠈⠉⠉⣁⣤⡾⠃⠈⠻⢶⣤⣤⣤⣴⠟⠁⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⣀⣤⠴⠞⠃⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠄⢠⣄⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⢳⣧
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠃⠛⠃⠹⡘⢦⠀⠀⠀⠀⣸⣷⣄⠀⠐⠛⠉⡁⠀⡄⣆⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠶⠱⡶⠺⠦⠀⣘⠀⠀⠀⠀⠻⣿⣿⣧⠀⠀⠀⢹⠀⣧⣸⠀⠀⠶⠋⢁⣀⡀⠀⠀⠸⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣶⠶⠊⣽⣍⠁⠠⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠈⠛⠉⠁⠀⠀⠀⣼⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠲⠖⢢⠄⠀⢀⠁⠠⢉⠍⠚⠓⠂⠀⠀⠀⠀⢸⣿⡇⠀⠀⠄⠀⠀⠀⠀⠀⢠⣿⡍⠉⣙⣿⣿⡇
⠀⠀⠀⠀⢀⠀⠐⠳⣤⡤⣴⡿⣏⣙⣈⣷⣶⣋⠀⠀⠀⠀⠈⣿⡇⠀⠠⣤⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠈⠉⠀⢰⣉⠳⣤⣀⣀⡭⢧⣉⢉⡩⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⠻⠉⣀⣤⠄⢴⠄⠚
⠀⠀⠀⠀⡄⠀⠀⣀⡬⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⣾⣿⠟⠋⠁⠀⡴⠖⠋⠁⠀⠀⠀⠻
⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⣀⢀⣤⠀⠀⠀⠀⠀⠀⡼⣿⡄⠈⠉⠉⢙⠛⠛⠛⠛⠛⣛⣻⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⣤⢰⡇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣼⠁⢹⣧⠀⠸⣆⠉⠀⢀⣤⣶⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡾⢃⡀⠘⣿⣧⡀⠙⠀⢰⡿⠿⠛⠉⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣆⡀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠁⠹⠟⠀⠙⣿⣷⣄⢀⣯⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣶⣶⣶⣶⣿⣿⠟⠁⠀⠀⢳⠀⠀⠀⠘⢿⣿⣿⣿⣷⣤⣀⣀⣀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⣀⣁⣠⣤⣤⣄⣈⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠴⣚⠭⠵⢒⣒⣒⣒⣒⡒⠮⠭⣓⠦⣌⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡴⣫⠖⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣍⠲⣝⢦⣀⠀⠄
⠀⠀⠀⠀⠀⠀⡀⢀⡴⣫⢊⣴⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡑⣝⢦⡀⢀
⠀⠀⠀⠀⠀⠀⢀⡞⡜⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡏⠹⣿⣿⣿⣌⢣⢳⡀
⠀⠀⠀⠀⠈⠁⡞⡜⢰⣿⣿⣿⡿⡾⣷⣿⣿⣿⣿⡿⡟⣿⣿⣿⣿⣷⠀⣹⣿⣿⣿⡆⢣⢳
⠀⠀⠀⠀⠂⢸⢳⠃⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣧⣡⢴⣿⣿⡟⢿⣿⠀⣿⣿⣿⣿⣿⠘⡜⡇
⠀⠀⠀⠠⠀⣾⢸⢸⣿⣿⣿⢱⣿⣿⣿⡟⢹⣿⣿⡟⠘⣿⣿⣇⢈⣻⡇⢹⣿⣿⣿⣿⡇⡇⣷⠠⠄
⠀⠀⠀⠀⠀⢻⢸⢸⣿⣿⣷⣤⣿⣿⡟⣴⣸⣿⣿⣿⠀⣿⣿⣿⡀⢿⣧⢸⣿⣿⣿⣿⡇⡇⡿⢀
⠀⠀⠀⠀⠁⠸⡜⡄⣿⣿⣿⢻⣿⣿⢱⡇⣿⣿⣿⣿⡇⢸⣿⣿⣧⠸⣿⠀⣿⣿⣿⣿⢠⢣⡇
⠀⠀⠀⠈⠐⠀⢷⢳⡘⣿⣿⡆⢿⠏⣸⣇⠘⠿⠿⠟⣱⡄⠙⠛⠛⣸⣿⡇⢿⣿⣿⢃⡞⡾⠀⠂
⠀⠀⠀⠀⠀⠠⠈⢧⡳⡘⢿⣷⣄⣴⣿⣿⣦⣄⣤⣴⣿⣿⣶⣶⣾⣿⣿⡇⣸⡿⢃⢞⡼⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠳⣜⢦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡴⣣⠞⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⡲⢍⡛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⢛⡩⢖⣭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠮⣽⣒⣒⡦⠤⠤⠴⣒⣒⣯⠵⠚⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠈⠉⡉⠉⠉⡉⠉⠁⠄⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠀⠀⠀⣴⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⢠⡀⠀⢸⣧⠀⠀⠀⠀⠀⠀⡄⠀⠀⢻⣿⣿⣿⣿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣧⠀⠀⠀⠀⠒⠋⠀⣤⠀⠀⠀⢸⣇⠀⠀⣿⡆⠀⠀⠉⠳⣾⡇⠀⠀⠀⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣿⣶⡤⠀⠀⠀⠀⡎⣿⠀⠀⠀⢸⣿⡀⠀⢸⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡟⠀⠀⠀⠀⠀⠀⠈⠀⢿⣇⠀⠀⠜⠻⣿⣦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣤⣶⣦⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠀⠀⠀⠠⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⠀⠀⢀⡾⠛⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⣤⣴⡿⠁⠀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠸⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⡿⢻⣿⣿⣿⣿⣷⣦⣀⣀⣀⣼⡟⠁⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠙⠿⢿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⢀⣀⣀
⣿⣿⠿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣬⣭⣿⣿⣭⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣀⣤⡴⠟⠀⠀⠀⠀⠀⢸⣿⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠿⠛⠉⠁⠀⢠⣄⢀⣶⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢶⣿⣷⢀⣴⣞⠛⠉⣱⡆⢸⡇⢸⡆⠀⠀⠀⠙⡿⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠟⠁⠀⢸⣿⣷⡄⢻⣇⣨⣷⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠟⠛⠁⠈⠙⠋⠈⣍⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠀⣠⡀⠀⠀⠀⢸⣿⡀⠀⠀⢿⣿⣆⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⢀⣾⣿⡇⠀⣴⠀⠈⣿⡇⠀⣠⢸⣿⡏⠀⣰⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣀⡀⣼⣿⣿⠁⠠⣇⢀⠀⢻⣷⢠⡇⠈⣿⣿⡀⣿⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠛⢠⣿⢻⣿⠀⠀⠙⠋⠀⢸⣿⡌⠻⠏⠸⣿⣇⠈⠉⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⡏⢸⣧⠀⠀⠀⠀⢀⣾⣿⣧⡀⠀⠀⢸⣿⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⣿⡇⠸⣿⣦⣤⣤⣴⣿⡟⠙⣿⣿⣷⣾⣿⡿⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠹⢿⣿⣿⡿⠟⠁⠀⠈⠻⠿⠿⠟⠁⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠛⠋⠁



⣿⣿⠛⠿⢻⠿⡿⢻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⡀⠀⠠⡀
⠀⠀⢀⠔⠈⢀⣤⠾⠛⠋⠉⠉⠉⠙⠛⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠁⠀⠄⠉⠙⠳⣦⡈⠑⢄
⠀⠠⠂⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠈⢢
⠀⠁⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈
⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡇
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡇
⠀⠀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠉⠀⠀⠀⠀⠀⠀⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⠁
⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⣸⠄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠿⠃⠀⢀⠀⠀⣾⣧⠀⣿⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠀⠀⣿⣿⠀⣿⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⣿⡇⠀⠀⢸⣿⠀⢸⣿⡀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠘⢿⣶⣾⣿⣷⡀⣀⣿⣿⡀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⢿⣿⣿⠿⠁⠸⠇⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⢀⣠⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡤⠖⠀⠀⣠⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⢠⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢀⠴⠚⡁⠀⠀⣶⠀⠈⣷⠀⠀⠠⣄⡴⠀⠀⢻⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣼⣧⡤⠀⠀⠀⡔⣇⠀⠀⢸⣇⠀⢹⡇⠀⠀⠀⠃⠀⠀⠀⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠏⠀⠀⠀⠀⠀⠀⢿⣄⣀⠈⠻⠷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⣴⣿⣿⣿⣆⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⢹⣿⣿⣿⡿⠂⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣀⠀⠀⣠⠟⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠋⠀⠘⣿⣿⣿⣷⣄⠀⠀⠀⠀⣸⣿⣿⣷⡀⠀⠀⠀⢸⣿⡏⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣦⣤⣴⠟⣿⣿⣿⣷⣤⣀⣀⣼⡟⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠙⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⢠⠏

⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣇⢸⣏⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⣿⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢟⣹⣏⠹⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⣉⣿⡉⠷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠁⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠀⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢳⡾⢷⣞⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣶⠿⣶⡛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢴⡟⠱⡎⢹⣦⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⣾⠉⠶⠉⣷⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⢻⣇⠰⠇⣸⡟⢹⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠉⣿⡀⠶⢀⣿⠋⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡈⠛⠛⠙⢷⡾⠛⠛⠛⢁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠙⠛⠛⠻⣶⠟⠙⠛⠋⣀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⢦⣀⡴⠋⠙⢦⣀⡴⠋⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠳⣄⣠⠞⠉⠳⣄⣠⠞⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢀⡴⠛⢧⡀⢀⡴⠛⢦⡀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣠⠞⠻⣄⠀⣠⠟⠳⣄⠀⣠⠟⠁
⠀⠀⠀⠀⢠⣄⣄⣠⠀⡙⠏⣁⠀⢀⠙⠋⡀⠀⣀⠙⢋⡀⣤⣠⣠⡄⠀⠀⠀⠀⣤⣠⣀⡄⡈⠛⢁⡀⠀⡈⠛⢁⠀⢀⡈⠛⣁⢠⣀⣄⣤
⠀⠀⠀⠀⢸⣛⣾⡇⠧⠤⠘⢋⣀⣧⣤⣤⣼⣂⡉⢃⠢⡼⢺⣷⣛⡇⠀⠀⠀⠀⣟⣳⣿⡸⠤⠄⠛⣁⣸⣥⣤⣤⣇⣈⠛⠠⠤⠇⣿⣞⣻
⠀⠀⠀⠀⠘⠕⢦⡹⠇⣴⣿⠛⠉⠉⠀⠀⠉⠉⠛⢿⣦⡘⠏⡰⠊⡑⠀⠀⠀⠀⠉⠲⣌⠿⣠⣾⠟⠋⠉⠁⠀⠈⠉⠙⠻⣷⣄⠻⢡⠖⢙
⠀⠀⠀⠀⠈⣀⡀⢰⣿⠋⢀⠀⠀⠙⠀⠀⠀⢠⡀⠀⠙⢿⣦⢌⣀⠁⠀⠀⠀⠀⢁⣀⢠⣶⡟⠁⢀⠀⠀⠀⣀⠠⠄⢀⣀⠈⢻⣷⡤⣁⡌
⠀⠀⠀⠀⠀⢉⢠⡿⠁⠀⠈⠀⢀⠀⣀⠀⠀⣤⣤⠀⠀⠈⣿⡆⡙⠀⠀⠀⠀⠀⠈⡁⣼⠛⠀⠀⠠⡄⠈⠁⠀⠂⠀⣈⠁⣷⠀⠙⣷⢈⠃⠄
⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⢠⣤⠘⣷⢀⣴⠞⣛⠉⣀⠀⠀⣿⣏⠙⠁⠀⠀⠀⠀⠈⢹⣿⠇⠀⡐⠂⣠⣶⠀⢸⣷⠀⢿⡆⣿⠀⠸⣿⡌⠁
⠀⠀⠀⠀⠠⢤⢸⣏⠀⢠⠀⠁⢀⡼⣏⢋⡽⠿⠛⠛⠁⠀⣿⡯⣤⠄⠀⠀⠀⠀⠤⣌⣿⡆⠠⡧⢤⣻⡜⠲⠂⣿⠀⢘⣧⣿⠀⢠⣿⢠⠤
⠀⠀⠀⠀⠀⣤⡈⣿⡇⠹⠶⠞⠋⢠⡈⠛⠳⠂⠀⠀⠀⢰⣾⠃⣴⠈⠀⠀⠀⠀⢰⣅⢹⣷⡀⢧⣼⠋⢿⣤⣼⠛⣷⣶⠏⡟⢀⣼⡟⣠⡆
⠀⠀⠀⠀⠠⠀⢤⡘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⡠⠀⡄⠀⠀⠀⠀⠄⠠⣄⠛⢷⣄⠁⠀⠀⠉⠀⠀⠀⠀⠀⣠⣾⠟⢀⠄⢠
⠀⠀⠀⠀⢰⣅⣋⣴⢗⡙⠿⢷⣶⣤⣠⣤⣠⣴⡾⠿⢋⡐⢧⣝⣨⡔⠀⠀⠀⠀⣬⣘⣡⠾⣊⠻⠿⣶⣦⣠⣤⣤⣤⣶⠿⠟⣑⠲⣬⣓⣥
⠀⠀⠀⠀⢸⣾⡿⣇⢊⠉⠰⣦⠀⢯⠉⠉⡹⠁⢴⠆⠊⡹⣸⣟⣷⡇⠀⠀⠀⠀⣷⣿⣿⡸⡉⠀⠶⠄⠸⡍⠉⢩⠏⠠⠶⠘⢉⠇⣿⣻⣾
⠀⠀⠀⠀⠈⠁⠉⠈⢀⡴⢦⡀⠀⢀⡴⢦⡀⠀⢀⡴⢦⡀⠉⠈⠈⠉⠀⠀⠀⠀⠉⠈⠁⠁⣠⠷⣄⠀⠀⣠⠶⣄⠀⠀⣠⠶⣄⠈⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⢙⣶⣏⠀⠀⢹⣶⡋⠀⠀⣹⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠈⣳⣾⠁⠀⠈⣷⣞⠁⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡴⠋⠀⠙⢦⡴⠋⠀⠙⢦⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠞⠁⠈⠻⣤⠞⠁⠈⠳⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡶⢶⣶⠟⠻⣦⡶⢶⣄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣴⠶⣶⡾⠛⢷⣶⠶⣦⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⢾⡇⠈⡃⢸⣷⣸⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⡶⣿⠀⢛⠀⣿⣦⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠘⢧⡜⢣⣼⠃⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢻⣤⠛⣤⡟⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠾⠻⠟⠷⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡶⠟⠿⠻⢶⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣄⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣠⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣜⣷⣸⣇⣼⣣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣻⣄⣿⣰⣟⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢸⡇⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣿⡀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡠⠒⠈⠁⠉⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠈⠁⠒⢄
⠀⠀⠀⠀⡘⠀⢦⣄⢠⣤⣀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠠⣤⡀⢤⣄⡀⠀⢃
⠀⠠⠊⠁⠀⠀⠀⠈⠓⠹⠙⠷⣄⠀⠉⢢⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⡔⠉⠁⠀⠀⠀⠉⠎⠏⠻⢦⡀⠈⠑⠄
⣠⠃⠀⠀⠀⠀⠤⡄⠀⢲⠀⣷⠘⣇⠀⠀⢣⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⢤⠀⠀⣆⠸⡆⠹⡄⠀⠘
⠙⡄⠀⠀⠀⠀⠈⡅⠀⠸⣇⠈⣇⠹⠀⠀⡌⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⢣⠀⠀⠀⡀⠀⢌⠀⠀⢹⠀⠹⡄⠇⠀⢠
⠀⠑⢄⠀⠘⠤⠆⠳⠶⠋⠙⠷⠋⠃⣀⠜⠀⠀⢀⡤⠖⠊⠁⠈⠑⠲⢤⡀⠀⠀⠣⣀⠀⠳⠴⠈⠷⠶⠁⠳⠞⠝⢀⡠⠂
⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠉⢇⠀⠀⠀⠀⠀⠀⠀⡌⠁
⠀⠀⠀⠀⠀⠑⠤⣀⠀⣀⠠⠔⠁⠀⢀⣀⣰⠇⠀⠹⠶⣄⡀⣷⣦⣄⡀⠀⠸⣆⣀⡀⠀⠈⠲⠄⣀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡔⠉⠀⠀⠀⠈⡡⠒⠊⠱⡌⡏⠛⠿⣶⣄⠀⠀⠉⠢⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠑⠄⢀⠀⠃⡅⠀⢠⡈⠻⣦⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⡠⠊⠙⣺⠆⠀⠀⣿⡀⠘⣧⠀⢹⡆⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢠⠀⠀⢂⢻⠀⠀⠀⢸⡇⠀⠘⣧⢀⡇⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠈⣇⠀⣸⠸⣦⣀⣠⡞⠻⣦⣤⡞⡸⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠈⠛⠁⠀⠉⠋⠉⠀⠀⠈⠉⠜⣁⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⡤⠖⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡴⣦⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠀⠀⠀⠙⢻⣄⠁⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠼⠃⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠘⠀⣀⣠⠄⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠉⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡀⠀⠤⠂⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⡏⠀⠈⡇⠀⠀⠀⠐⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢸⣿⠙⠇
⠀⠀⠀⠀⠀⠀⠠⡄⠈⠳⣿⠀⠀⣧⠀⠉⣁⠤⠀⠀⠀⠀⠀⢀⠀⣄⢠⠀⠀⠀⢸⣄⡰⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢹⢦⡀⣿⠀⠈⠀⠀⠲⠀⠀⠀⠀⢸⡀⣹⣈⡇⠀⠀⠈⠛⠃⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣐⣻⠀⠸⡇⠑⣿⢠⣚⣛⣲⣶⣦⡤⠀⠀⠀⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠷⠞⣠⢞⡼⣿⣯⠼⢟⣻⡿⠖⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⣧⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⢳⣀⣁⣀⣬⠶⠋⠀⠙⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⢿⣿⠀⠀⢸⣄⣰
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠰⠆⠀⠛⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⢸⣿⡿⠆⠀⠀⠀⢸⣿⠀⠀⠈⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣼⠏⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡟⢰⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢸⡇⢸⣇⠀⠀⠀⠀⠀⠀⢠⡿⣿⣧⡀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢀⣿⠀⢸⣿⣦⡀⠀⠀⢀⣠⣿⠁⠻⣿⣿⣶⣤⣤⣶⡿⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣤⣾⠏⠀⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠈⠻⠿⣿⠿⠟⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣦⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠅⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⢀⣀⣤⡴⠾⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⢀⡀⣀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠋⠉⠀⠀⠀⠸⣯⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢆⢰⣤⡄⢿⠳⣄⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⣤⡴⠾⠛⠉⠁⣀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠙⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢠⠀⡈⠀⠀⡁⠈⠆⠹⡆⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⡆⠀⣧⠸⡆⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⣌⣛⠜⠷⠋⢙⡖⢀⣡⠇⠀⠀⠀⠀⠈⠻⣿⣿⣿⡀⠀⠀⠀⢸⡀⣸⣤⡇⠀⠀⠛⠉⢀⣀⣄⣀⡀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠙⣷⠾⠖⢋⣽⣍⠉⠀⠄⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠈⠛⠁⠈⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢰⣤⠖⣥⠀⠀⠈⠈⠀⡈⠟⡛⠴⠶⠶⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⣏⠉⠙⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠼⣍⢀⢀⠘⣴⣦⠶⠿⠀⢦⣤⣤⠖⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠰⣦⠄⠀⢈⠿⣏⠛⠋⠙⠳⠶⡞⢻⠿⠷⢦⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠈⠋⠀⠀⠀⢀⣠⣴⣿⣿⣿⡿⠟⠛⠉⠁⠀⣀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠳⢦⠈⠳⠶⠶⠶⠛⠻⢦⣤⡤⠆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠛⠉⠈⢃⣠⡴⠶⠛⠐⠟⡀⠉
⠀⠀⠀⠀⠰⣄⠀⠀⣀⣠⠖⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⣶⣿⣿⠟⠋⠀⠀⠀⠴⠛⠉⠁⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⡆⠀⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⡀⠀⣼⠀⠀⠀⠀⠀⠀⠰⢿⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⣧⠀⠀⣶⠀⠶⠆⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠛⢁⣿⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢸⣿⣆⠀⠉⣦⠀⠀⣠⣾⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢃⣤⡀⠀⢿⣿⣦⡀⠀⠀⢰⡿⠛⠉⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠟⠁⢀⠟⠛⠀⠈⢿⣿⣷⣤⡀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⡿⠟⠁⠀⠀⠘⡆⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⠀⠀⠀⠛⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⣀⣠⡴⠞⠀⠀⠀⠀⠀⢹⣿⡟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠞⠛⣫⠉⢠⡄⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠉⠀⠀⠀⣰⣿⠀⠸⣧⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⡀⠀⠀⠀⠀⠀⠀⣿⣀⣴⣿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡿⠿⠇⠀⠀⠀⠀⠀⠀⠈⠛⠋⡀⠉⠁⠀⠀⠀⠀⣼⣆⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠋⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⠐⣿⣿⣆⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⢹⣿⡿⣆⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⠃⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⣿⡏⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡿⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⢠⣿⠇⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⣼⣿⠀⠀⣿⣿⣆⠀⠀⠀⠀⢀⣴⣿⠇⢻⣿⣿⣦⣄⣀⣠⣴⣿⡟⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠸⣿⣷⣶⣾⣿⠇⠀⠀⠸⣿⣿⣿⣶⣶⣶⣿⣿⠏⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠙⠿⣿⠿⠋⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣾⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠏⠀⠀⠀⢀⣤⣶⡾⠋⠀⠀⠀⣤⡶⠂⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠃⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⣽⣿⣿⠃⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣏⠀⣼⣿⣿⠏⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣿⣿⣿⣶⠟⠀⠀⠘⣿⣿⣿⣆⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⠃⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣶⣿⣿⣿⠟⠛⢿⣿⣿⣿⡿⠟⠁⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⣀⣀⣴⡿⠃⠀⠀⠸⣿⠀⣿⡏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⣽⡿⠀⠀⠀⢀⣀⣠⣤⣴⡶⠾⠟⠛⠋⠁⠀⠀⣀⣀⠀⣿⡆⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⣴⠾⠛⠋⢉⡁⢠⡄⢠⣤⠀⠀⠘⠻⢷⣿⠉⠀⢸⣇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⢿⣶⣄⠀⠁⠀⠀⠾⣿⡇⢸⡇⢸⣿⠀⠀⠀⠀⠀⠁⠀⠀⠈⣿⡀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣟⣡⡿⠀⠀⠀⠀⠀⠘⠷⠿⠿⠾⠃⠀⠀⠀⢀⣼⣆⠀⠀⠀⢻⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡶⠿⠛⠉⠋⠉⢀⣤⣦⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⣾⡏⢻⣦⠀⠀⠸⣷⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⣿⡆⠀⠀⠀⠀⣿⠛⣷⡄⠀⠀⠀⢻⡇⢦⣿⣧⠀⠀⣿⡄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢁⣤⠀⣿⡇⠀⠀⠀⠀⣿⡆⢸⣧⠀⢠⡄⢸⣧⠈⣿⡍⠃⠀⢹⣇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣤⡄⢀⣦⠀⢠⣿⠁⣼⣿⠀⣿⠇⠀⠀⠀⠀⢸⡇⠈⣿⠀⣿⡇⠈⣿⡄⠸⣷⡀⠀⠈⣿⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣯⠀⣾⠇⣸⣿⡇⠀⣿⠀⢀⣿⠂⠀⢸⡗⠀⢻⣇⠈⠁⠀⠘⣷⡀⠹⣧⠀⠀⢿⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠉⢸⡿⠀⣿⢹⡇⠰⣿⠀⠈⠿⠂⠀⣸⡇⠀⠀⢻⣦⠀⠀⠀⠘⣿⠀⣿⡆⠀⢸⣷⢸⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⢀⣿⠃⢸⡟⠸⣿⠀⢿⣆⠀⠀⠀⣠⡿⠁⣸⣷⡀⠙⢷⣤⣀⣠⡿⠀⣿⠇⠀⠀⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣯⠛⠿⠋⣠⡿⠁⠀⢻⣧⠀⠙⠻⠿⠟⠋⢀⣴⡟⠙⢿⣦⣀⣉⠙⢋⣁⣼⠟⠀⠀⠀⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠿⠛⠁⠀⠀⠀⠙⠿⣶⣤⣤⣴⡶⠟⠋⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠘⠟

⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⠼⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⢄⡀
⠀⠀⠀⠀⡔⢉⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠈⠑⡄
⠀⢀⠤⠒⠀⠈⠙⢆⢿⢶⣄⠘⠐⢄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠄⠚⠀⠌⠳⢜⡿⢦⣄⠐⠂⢄
⢀⠆⠀⠀⠀⠀⡀⠀⢠⠀⡌⢳⡀⠀⢡⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡀⠀⣄⠠⡝⢧⠀⠀⢂
⠣⠀⠀⠀⠀⠀⡏⠀⠸⡆⠹⡄⡇⠀⢠⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⠃⠀⢸⡀⢳⡈⠇⠀⠠
⠀⠣⡀⠀⢢⡤⠰⠤⠞⠹⠶⠓⠀⡠⠂⠀⢀⣠⠴⠒⠉⠑⠲⢤⡀⠀⠀⠢⡀⠀⢦⡄⠳⡤⠎⠳⠴⠟⠀⡠⠃
⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠈⠙⡀⠀⠀⠀⠀⠀⠀⡌⠁
⠀⠀⠀⠀⠈⠢⠤⣀⣀⠤⠊⠀⠀⣀⣤⠇⠀⠘⠷⣤⡈⣷⣦⣄⠀⠀⠧⣄⡀⠀⠈⠢⠄⣀⡠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠊⠀⠀⠀⠀⡅⠊⠉⠱⠸⡍⠛⢿⣦⡀⠀⠈⠳⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣈⣀⠠⠀⢰⡄⠰⣦⠙⢷⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠐⠀⡀⣼⠁⠀⠈⣧⠀⠹⡄⠘⡇⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⡇⠀⢡⢹⡀⠀⢀⡿⣄⠀⢹⢠⠇⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠘⠦⠞⠈⠿⠶⠟⠀⠙⠿⢋⠎⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠤⠤⠤⠤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠞⠂⢸⣿⣿⣷⣦⣄⡀⠀⠀⣀⣤⣶⣾⡿⠿⠛⠛⣻⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⣠⣀⠀⠀⠹⠀⠀⠀⠙⠻⠿⣿⣿⣿⣶⣿⡿⠛⠉⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠿⣿⣷⣦⣄⠀⠀⡀⠀⣠⣶⣿⠟⠻⢿⣿⣿⣦⡀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡀⣀⣀⠀⠟⠀⠀⠉⠛⢿⣿⣦⡉⢠⣟⠉⠀⠀⠀⠀⠙⠻⣿⣿⣦⡀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⠛⠛⠃⠀⠀⢀⡄⠀⠀⣦⡈⠻⣿⣾⣿⣷⡄⠀⢴⣧⠄⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⣴⡄⠀⠀⠸⣿⡀⠀⢻⣏⠀⠈⠻⣏⠻⣷⣄⠈⠀⠀⠀⠀⠀⠘⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⠿⠁⠀⢰⢿⠀⠀⠀⠀⢻⡇⠀⠈⢿⣆⠀⠀⢻⡀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⡞⣾⠀⠰⠄⠀⢸⣿⡀⠀⠈⢿⡆⠀⢸⠃⠀⠀⠙⢿⣷⣄⠀⠦⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢿⡀⠀⢰⡇⢻⣦⣀⣀⣤⡿⠹⣷⣦⣤⣼⠇⠀⡾⠀⣀⣤⠴⠂⠙⢿⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠈⠻⠿⠿⠛⠀⠀⠈⠛⠛⠋⠀⡼⠑⠋⠉⠀⠀⠀⠀⠀⠹⣇⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢀⠀⠀⠶⣄⠛⠁⠀⢀⣾⣶⣿⣿⣷⣄⠀⠶⠀⠀⣀⡟⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⣰⠋⠀⡀⠀⠘⠀⢀⣴⡿⠟⠉⠀⠈⠙⢿⣿⣶⣾⣿⠟⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣀⠀⠀⣴⣿⠋⠀⠀⠀⢼⣶⠀⠀⠈⠉⠉⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢉⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠻⢿⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁


⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠤⠤⠤⠤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠞⠂⢸⣿⣿⣷⣦⣄⡀⠀⠀⣀⣤⣶⣾⡿⠿⠛⠛⣻⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⣠⣀⠀⠀⠹⠀⠀⠀⠙⠻⠿⣿⣿⣿⣶⣿⡿⠛⠉⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠿⣿⣷⣦⣄⠀⠀⡀⠀⣠⣶⣿⠟⠻⢿⣿⣿⣦⡀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡀⣀⣀⠀⠟⠀⠀⠉⠛⢿⣿⣦⡉⢠⣟⠉⠀⠀⠀⠀⠙⠻⣿⣿⣦⡀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⠛⠛⠃⠀⠀⢀⡄⠀⠀⣦⡈⠻⣿⣾⣿⣷⡄⠀⢴⣧⠄⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⣴⡄⠀⠀⠸⣿⡀⠀⢻⣏⠀⠈⠻⣏⠻⣷⣄⠈⠀⠀⠀⠀⠀⠘⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⠿⠁⠀⢰⢿⠀⠀⠀⠀⢻⡇⠀⠈⢿⣆⠀⠀⢻⡀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⡞⣾⠀⠰⠄⠀⢸⣿⡀⠀⠈⢿⡆⠀⢸⠃⠀⠀⠙⢿⣷⣄⠀⠦⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢿⡀⠀⢰⡇⢻⣦⣀⣀⣤⡿⠹⣷⣦⣤⣼⠇⠀⡾⠀⣀⣤⠴⠂⠙⢿⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠈⠻⠿⠿⠛⠀⠀⠈⠛⠛⠋⠀⡼⠑⠋⠉⠀⠀⠀⠀⠀⠹⣇⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢀⠀⠀⠶⣄⠛⠁⠀⢀⣾⣶⣿⣿⣷⣄⠀⠶⠀⠀⣀⡟⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⣰⠋⠀⡀⠀⠘⠀⢀⣴⡿⠟⠉⠀⠈⠙⢿⣿⣶⣾⣿⠟⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣀⠀⠀⣴⣿⠋⠀⠀⠀⢼⣶⠀⠀⠈⠉⠉⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢉⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠻⢿⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁


⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠂⠀⠀⡌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠆⢀⣠⡤⠶⢚⣛⣛⣻⣛⣛⣛⡒⠶⢤⣄⡀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡴⢞⣩⠴⢚⣉⣥⣤⣴⣶⣶⣦⣤⣬⣉⡓⠦⣍⡳⢦⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣠⡾⣫⠖⣉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣉⠲⣝⢷⣄⠰
⠀⠀⠀⠀⠀⠀⠀⡄⢀⡴⢫⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⡝⢦⡀⠠⢀
⠀⠀⠀⠀⠀⠄⠀⢀⡞⣱⢃⣼⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇⠹⣿⣿⣿⣧⡘⢎⢳⡀
⠀⠀⠀⠀⠐⠈⠀⡾⢰⢃⣾⣿⣿⣽⣷⣾⣷⣾⣿⣿⣿⣿⡿⡟⣿⣿⣿⣿⣿⣇⠀⠹⣿⣿⣿⣷⡘⡆⢷⠈⠁
⠀⠀⠀⠀⠡⠀⢸⢃⡇⣼⣿⣿⣿⣿⣷⣵⣿⣿⣿⣿⣿⡅⢃⣁⣿⣿⣿⢿⣿⣿⠀⣿⣿⣿⣿⣿⣧⢸⡘⡇⠠
⠀⠀⠀⠀⠀⠀⣿⢸⠀⣿⣿⣿⣿⢫⣿⣿⣿⣿⡿⣿⣿⣿⡿⠹⣿⣿⣿⠀⠻⣿⡄⢹⣿⣿⣿⣿⣿⠀⡇⣿
⠀⠀⠠⠀⠉⠀⣿⢸⢸⣿⣿⣿⡇⢛⣿⣿⣿⢏⡀⣿⣿⣿⣧⠀⢿⣿⣿⡇⢰⣿⡇⢸⣿⣿⣿⣿⣿⡇⡇⣿⠀⠉
⠀⠀⠀⠐⢠⠀⣿⢸⠀⣿⣿⣿⣷⢸⣿⣿⡏⣼⢡⣿⣿⣿⣿⡄⢸⣿⣿⣷⠀⢻⣷⠀⣿⣿⣿⣿⣿⠀⡇⣿⠀⠄
⠀⠀⠀⠀⠀⠀⢸⡌⡇⢻⣿⣿⣿⢸⣿⣿⢁⣿⢸⣿⣿⣿⣿⡇⠈⣿⣿⣿⣧⢸⣿⡀⢿⣿⣿⣿⡟⢸⢡⡇⠀⢀⠁
⠀⠀⠀⠀⠀⠃⠀⢷⡸⡄⢿⣿⣿⠘⢿⠏⢸⣿⡀⠻⠿⠿⠟⢰⣆⠈⠛⠛⠋⣼⣿⡇⢸⣿⣿⡿⢠⢇⡾⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠠⠈⢷⡹⣄⠻⣿⣧⣀⣠⣿⣿⣷⣄⣀⣠⣴⣿⣿⣷⣶⣶⣾⣿⣿⣧⢸⣿⠟⣠⢏⡾⠁⠤⠐
⠀⠀⠀⠀⠀⠀⠀⡀⡀⠻⣌⢦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⢃⡴⣡⠟⢀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⣍⠲⣌⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣡⠖⣩⠞⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⢦⣝⡒⠤⣍⣙⣛⠛⠛⠛⠛⣛⣋⣩⠤⢒⣫⡴⠛⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠉⠛⠲⠶⠭⣭⣭⣭⣭⠭⠶⠖⠛⠉⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠆⠀⠀⠤⠀⠀⠐⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠂⠄⡀
⠀⠀⠀⠀⣀⠔⠈⠀⣀⣴⡶⠿⠟⠛⠛⠛⠻⠿⢿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠛⠉⠉⠉⠙⠛⠷⣦⣄⠈⠑⢄
⠀⠀⢀⠞⠁⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣄⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠙⢷⡄⠀⠱⡄
⠀⠠⠁⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠘
⠀⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢻
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇
⠀⠀⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠀⠀⠀⠀⠀⠈⣿⡟⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⣸⠃
⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⡀⡀⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣧⡿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣿⠂⠀⠀⠀⠀⠀⣴⣶⡀⢰⣶⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⣶⡇⠀⠀⢿⣿⡇⠸⣿⣿⣦⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠃⠀⠀⢸⣿⡇⠀⣿⣿⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⢀⣾⣿⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⢊⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠻⣿⣶⡿⣿⣿⣧⣀⣠⣾⣿⣧⠀⢸⣿⣿⠀⢹⡇⠀⠀⠀⠀⠀⠀⢡⡞⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠋⠀⢸⠇⠀⠀⠀⠀⠀⣴⠋⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡄⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠞⠛⠉⠀⠀⢀⡀⠀⠘⣿⡈
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⢀⣤⠴⠶⢛⣯⢹⡆⢰⡄⠀⠀⠙⢷⠏⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢈⣡⣾⠿⠋⠈⣿⣷⣄⢸⣇⢈⣷⣸⡇⠀⠀⠀⠈⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣽⠟⠛⠀⠛⠛⠉⡛⠁⠀⠀⣤⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠀⢀⡄⠀⠀⠀⢰⣿⡀⠀⠸⣿⣷⡀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣰⣿⣿⠀⢰⠀⠀⣿⡇⠀⡄⣿⣷⠁⢠⠃⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣤⢀⣿⣿⡇⠀⣿⣠⠀⢹⣧⢸⣇⡸⣿⡇⢺⣤⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠁⢸⡟⣿⠃⠀⠈⠁⠀⢸⣿⡈⠉⠀⠹⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣾⠇⣿⣇⠀⠀⢀⣠⣿⢿⣷⣄⣀⣠⣿⡇⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣾⡿⠀⠹⣿⣿⣿⣿⣿⠃⠈⢿⣿⣿⣿⡟⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⠈⠉⠉⠀⠀⣠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⠟⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢀⣀⣠⣴⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠙⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠟⠛⠉⢀⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⠉⠁⠀⢀⣾⡇⠀⢸⣷⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⡞⣿⠃⠀⠈⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣠⣼⣿⣷⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠋⡀⠉⠋⠁⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⢷⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⢠⣿⡿⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⢀⣿⣿⡇⠀⠀⢹⣿⣿⣷⣄⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⢻⣿⣿⣿⣿⣶⣤⣴⣶⣿⣿⣿⠁⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣶⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⠏⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠛⠛⠋⠙⠛⠛⠛⠿⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⢺⣦⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⣠⡄⠀⢸⣷⡀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⡴⠛⠁⠀⠀⢸⣿⣷⡀⠀⠈⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⣠⣤⠀⠀⣀⣠⣴⠾⠛⡁⣀⠀⠀⠀⠀⠈⣿⡟⠃⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⣻⡿⠀⠉⠉⠁⠀⢸⣿⣇⣿⠂⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠘⠟⠛⠋⠀⠀⣀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⣴⠞⠋⠀⠀⠀⠀⠀⣀⡄⠀⠀⢰⣷⡀⠀⠰⣿⣧⠀⠈⣿⡇⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠸⣿⡇⠀⠀⢻⣿⠀⠀⣿⣇⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢠⡇⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣷⠀⠀⠈⣿⣇⠀⢹⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⢸⣇⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⡄⠀⠀⢹⣿⠀⢸⣿⡆⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠸⣿⣄⣰⣿⠸⣿⣇⡀⢀⣰⣿⢿⣿⣦⣤⣼⣿⠀⠈⣿⡇⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⢻⣿⣿⠏⠀⠻⣿⣿⣿⡿⠃⠈⠻⠿⠿⠟⠁⠀⠀⡿⠁⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣤⣄⣠⣤⣤⣤⣶⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⣠⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠹⢦⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠐⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⢹⣉⣩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡀⠙⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣯⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⣭⠇
⠀⠀⠀⠀⠀⠀⠀⢯⣍⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⠛⠁⠀⠈⠛⠶⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⠇
⠀⠀⠀⠀⠀⠀⠠⠞⡟⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢀⣠⢷⣄⡀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢳⣷⣷⠃⠀⠀⠀⠀⠀⠀⣠⣴⣶⡾⠿⠀⠀⢠⣾⠋⢁⣀⡈⠻⣷⠀⠀⠸⠿⢷⣶⣦⣀⠀⠀⠀⡞⠛⢛⢳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⡟⠁⠀⢠⣾⣿⡿⠟⢰⣿⣿⣿⠀⠿⢿⣿⣷⡀⠀⠈⢿⣿⣧⠀⠀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢰⠛⠛⠛⢳⠀⠀⣤⣶⣶⣍⠿⠀⠀⠀⠸⠿⡿⠀⠸⠿⣿⣿⣿⣿⠇⠀⣿⣿⠇⠀⠀⠈⠿⠿⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏⢀⣾⣿⢿⣿⣿⠀⠀⢀⣾⣿⡇⠀⢠⣾⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⠀⠀⣏⣉⣉⣹
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢣⣾⣿⠏⣾⣿⡇⠀⠀⣼⣿⡟⠀⢠⣿⣿⠇⠀⠀⣠⣤⣄⠀⣀⣀⡀⠀⢠⣿⣿⢃⣤⣤⡸⣿⣿⡇
⠀⠀⠀⠀⢀⣀⣘⡛⠿⣣⣿⣿⠃⢰⣿⣿⠀⠀⣸⣿⡿⠁⢀⣾⣿⡏⠀⢠⣾⣿⡿⠻⣿⣿⡟⠀⢀⣿⣿⡿⠟⣿⣿⡟⣿⣿⣿
⠀⠀⢠⣾⣿⠿⠿⢿⣿⣿⣿⣁⠀⣿⣿⡇⠀⣰⣿⣿⠁⠀⣼⣿⡟⠀⢀⣿⣿⠏⠀⣰⣿⡿⠁⢀⣼⣿⡟⠀⢰⣿⣿⠁⠘⣯⢭⡀
⠀⠀⢺⣿⣇⣀⣠⣾⣿⡟⠻⢿⣿⣿⣿⣤⣾⣿⣿⡇⣠⣾⣿⣿⠁⣠⣿⣿⣿⣀⣴⣿⣿⠃⣠⣾⣿⡿⠀⢀⣿⣿⠃⣠⣾⠋⠀⢹⡆
⠀⠀⠈⠻⠿⢿⠿⠟⠋⠀⠀⢀⣿⣿⠏⠉⠁⠻⠿⠿⠛⠙⠿⠿⠟⠋⠘⠿⠿⠟⠉⠿⠿⠿⠻⠿⠿⠁⠀⠀⠻⠿⠿⠋⠁⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣜⡻⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣴⣶⡿⠛⠁
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁
⠀⠀⢀⡴⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⡆⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠰⡆⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠟⣹⡟⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⢛⣏⠻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠾⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣠⣄⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣠⣄⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡾⢫⡝⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠫⡝⢷⣠⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⣧⡰⢀⣿⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⣿⡀⢆⣼⣋⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣌⠉⠉⣻⣏⠉⠉⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡉⠉⣹⣟⠉⠉⣡⣄
⠀⠀⠀⠀⠀⠀⣞⠁⠈⣳⣞⠁⠈⣷⣎⠁⢈⡷⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⣱⣾⠁⠈⣳⣞⠁⠈⣳
⠀⠀⠀⢀⣀⣀⡈⡳⠞⠁⢈⠳⠞⠁⠈⠳⣋⢄⣀⣀⡀⠀⠀⢀⣀⣀⡀⡙⠞⠁⠀⠑⠞⡁⠈⠳⢞⢁⣀⣀⡀
⠀⠀⠀⢸⡿⣿⡰⠄⣘⣡⣾⡤⢤⣧⣉⡃⠢⢃⡿⣿⠀⠀⠀⠀⣿⢿⡘⠄⢘⣩⣼⡤⢤⣵⣌⣃⠠⢆⣿⣿⡇
⠀⠀⠀⠈⠈⠆⣩⡾⠛⠁⢠⠀⠀⠀⠉⠛⢧⣉⠘⠁⠁⠀⠀⠈⠈⠳⣉⡴⠟⠉⠀⠀⠀⠀⠈⠙⢷⣍⠰⠁⠅
⠀⠀⠀⠀⠗⢰⡟⠀⠐⠀⠀⠀⠀⢀⣃⠀⠈⢿⡅⠳⠀⠀⠀⠀⠞⢠⡿⠁⠀⡂⠠⠐⠂⠁⠰⢢⡄⠻⣇⠺
⠀⠀⠀⠀⠊⣽⠀⠀⢠⡀⢳⣈⣤⢚⡋⣀⠀⢸⣷⠑⠀⠀⠀⠀⠘⣾⡇⠀⠄⢉⣤⠀⣶⠀⢿⢸⡇⠀⣷⠑
⠀⠀⠀⠀⠦⢽⡀⢠⠈⣀⠼⢯⣩⡟⠛⠉⠁⢸⡏⠴⣀⠀⠀⠀⢠⢽⡇⢸⠆⣾⣎⠒⢸⣇⠸⣿⡇⢠⡟⠤
⠀⠀⠀⠀⠓⠘⣷⡈⠋⠁⠐⠂⠉⠉⠀⠀⢠⣿⠁⠞⠀⠀⠀⠀⠻⠀⣿⣌⠻⠃⠙⠶⠋⠙⠛⠙⢁⣾⠃⠺
⠀⠀⠀⢠⡢⣃⡬⠻⣶⣤⣀⣀⣀⣀⣤⣶⠏⢤⡘⢀⠀⠀⠀⠀⣀⢞⡤⠹⣷⣤⣀⣀⣀⣀⣤⣴⠟⢥⣘⢄⡁
⠀⠀⠀⢸⣿⣿⡘⠁⠰⡄⢭⠉⠉⠏⠠⠆⠈⢆⣿⣿⡀⠀⠀⢀⣿⣿⡴⠉⠰⠤⠹⠉⠉⠭⠡⠆⠊⢓⣿⣿⡇
⠀⠀⠀⠈⠀⠀⢀⠜⠳⣄⣠⠞⠳⡄⣠⠞⢧⡀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡴⠳⣄⢀⡴⠳⣄⣠⠞⠳⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠳⣄⣠⠞⠳⣄⣠⠟⠣⣄⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⡼⠻⣄⣠⠞⠳⣄⣠⠞
⠀⠀⠀⠀⠀⠀⠀⣈⣡⣤⣤⡾⢧⣤⣤⣈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⡤⣤⡼⢷⣤⣤⣌⣁
⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⡏⢘⡂⣿⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣿⠀⡃⢹⣶⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣻⣮⣴⣋⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣙⣦⣵⣟⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠈⠁⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠈⠁⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡷⢼⡧⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⢾⡧⢾⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠷⠼⠀⠀⠀⠀⠀⢸⣿⡟⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠹⡄⠀⠀⣼⣿⡀⠀⠀⠀⣾⣿⣆⠀⠃⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠆⠀⣠⣾⡿⠀⠀⠀⠁⠀⠀⢸⣿⡇⠀⠀⠀⠸⣿⣯⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⢠⣾⠏⣼⠇⠀⣀⠀⣠⠄⠀⢸⣿⣇⠀⢀⡀⠀⢻⣿⡆⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢠⣿⠏⢰⣿⠀⠀⠉⠹⠃⠀⠀⢸⣿⣿⠀⠈⠉⠀⠘⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣾⠏⠀⢸⣿⡀⠀⠀⠀⠀⠀⣠⡿⢿⣿⣆⠀⠀⠀⠀⣿⣿⠀⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣀⣼⡟⠀⠀⠘⢿⣿⣦⣤⣤⣴⣾⠟⠁⠈⢻⣿⣷⣦⣤⣶⣿⠏⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁


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

⠀⠀⠀⠀⠀⠸⣌⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢄
⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠋
⠀⠀⠀⠀⠀⢠⡿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⡓
⠀⠀⠀⠀⠀⢸⣍⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⢀⠀⠙⠛⠳⣦⡄⠀⠀⠀⠀⠀⢸⣶⡞
⠀⠀⠀⠀⠀⢩⣭⣭⠀⠀⠀⠀⠀⢀⣀⣤⣿⠀⠀⣠⠶⠋⠳⢦⡄⠀⢸⣿⣤⣀⠀⠀⠀⡸⠿⢧
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣰⣿⠟⠉⢠⣶⣶⠿⢰⣿⣷⡈⠿⣶⣦⠀⠉⢻⣷⡄⠀⢳⣶⡞
⠀⠀⠀⠀⠀⣎⣉⣉⡆⢀⣴⣶⣮⠛⠀⠀⣸⡿⠇⢸⣿⠿⠿⠿⠂⠻⠿⠀⠀⠀⣿⡇⠀⣸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⢠⣾⡿⣿⡿⠀⠀⣾⣿⠁⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⢣⣤⣴⠃
⠀⠀⠀⠀⣀⠿⠿⣣⣿⠟⢸⣿⠇⠀⣼⣿⠃⢀⣿⡿⠀⣠⣾⡿⢷⣾⡿⠀⣰⣿⡿⢾⣿⡿⣿⣿
⠀⠀⣴⡿⠿⠿⣿⣿⣏⣀⣿⡿⠀⣰⣿⠏⢀⣾⣿⠁⣰⣿⡟⠀⣼⣿⠃⢠⣿⡟⠀⣾⣿⠁⣹⡿⣄
⠀⠈⢿⣷⣤⣾⡿⠋⠙⣿⣿⣷⠾⣿⣿⣤⢿⣿⣧⡴⢿⣿⣧⣾⣿⣧⣴⣿⡿⠀⢸⣿⣧⡴⠏⠀⣸⠆
⠀⠀⠀⠉⠉⠁⠀⣀⣀⢿⡿⣤⣤⣬⣿⣥⣤⣭⣭⣤⣤⣭⣁⣀⣉⣉⣈⣉⠁⠀⢀⣉⣉⣀⣤⡾⠋
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣶⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠁
⠀⠀⠜⠛⠛⠛⠿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⣤⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡞⠀⠀⠀⣀⣴⡶⠋⠀⠀⢀⣴⠖⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡟⠀⠀⢠⣾⣿⡟⠁⠀⠀⢀⣾⡏⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⢉⣾⣿⡟⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣄⣼⣿⣿⣠⡄⠀⠸⣿⣿⣷⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⠉⠀⠀⠀⣿⣿⣿⣷⣤⣤⣾⣿⣿⠇⠀⢀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣤⣴⣾⣿⡿⠿⣿⣿⣿⣿⠿⠃⣠⣴⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⢉⣉⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⠛⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣉⣀⣀⣀⣀⣉⣷⣤⡀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⡖⠛⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⡟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠙⠲⣄⠀⣠⡴⠋⠉⠳⢤⡀⠀⠀⠀⢹⡟⢻⡟
⠀⠀⠀⠀⠀⠀⠀⢰⠶⠷⠶⠷⢶⠄⠀⡆⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠉⠛⠛⠛⠛⠋⠙⠳⣄⣴⠾⠷⠾⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣤⣤⡼⠀⢸⣇⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣤⣤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣤⣞⠋⠈⣳⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡴⠋⠀⠉⠉⠉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢀⣀⣀⣤⣤⡤⠤⠤⠤⣤⣤⣜⣧⣤⡶⢖⣛⢛⣍⢉⣭⢙⣛⠛⣲⠶⢦⣤⣟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡄⠀⣨⠷⠖⣛⡉⢡⡤⡄⢠⢶⡄⣴⢶⠀⡤⣬⣤⢼⣻⢸⣽⢸⣽⢸⣽⣿⣿⣿⣿⣷⣿⡆⡍⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣀⣀⣤⡤⢴⣼⢧⡶⠆⢤⢸⡇⣿⢸⠀⡇⢸⢸⡇⣿⢸⠀⡇⢸⣿⢸⣿⢸⢿⠸⠿⠸⠿⠿⢿⢿⣿⣿⣿⡇⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠛⣩⡿⣢⠿⠛⠓⠒⠛⠲⣝⢸⡇⣿⢸⠀⡇⢸⢸⡇⣿⢸⠀⡇⢸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⡿⠃⠀⠀⠀⠀⠀⠀⠈⢿⡁⠛⠘⠀⠁⠈⠀⠁⠉⠈⠀⠁⠈⢿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⢠⣦⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣀⣤⣤⣤⡤⠤⠴⠶⠶⠶⠒⠒⠛⠛⠛⠉⠉⣉⣉⣉⣀⣀⣠⣤⣤⣤⣞⣳⣭⡉⠙⠓⠂
⠀⠀⠀⠀⠀⠠⠶⠶⢻⣧⠴⠶⠒⠒⠒⠒⠶⠤⣴⡇⣀⣀⣤⣤⡤⣤⠤⠴⢶⠶⠒⠒⠚⠛⠛⠛⠉⠉⠉⠉⠉⠀⢀⡴⠋⠉⠉⠉⠉⠙⢻⡗⠂
⠀⠀⠀⠀⠀⠐⠒⠚⠛⣷⠒⠘⠋⠈⠉⠘⠃⠒⢸⠡⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⣾⠤⠤⠶⠶⠒⠒⣒⣛⣛⣛⣓⠺⠶⠧⢤⣄⣀⣤⣿⠤⠤⠼⠷⠶⠖⠒⣶⠲⠶⠤⢤⣄⣀⣀⢸⡇⠀⠀⣀⣀⣀⡀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⣿⠚⣛⠛⠋⠉⠉⠉⠉⢡⡌⠉⠉⠛⠓⠲⢶⠶⠶⠶⠒⠚⠛⣛⠛⣉⡉⣿⠀⠀⠀⠀⠀⠀⠉⠹⣟⣛⢉⣍⢩⣍⢩⡉⣛⢺⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣿⢰⣶⠀⠀⣾⡇⢸⡇⠀⢠⣦⡄⠀⢸⡅⣾⡇⢸⣿⢸⢿⡇⣿⣇⣻⣀⣀⣀⣀⣠⣤⣤⠤⠿⠭⠶⠶⠖⠒⣲⠲⠿⢬⠁
⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⣿⠀⠀⣿⡇⢸⡇⠀⢸⣿⡇⠀⢸⠶⢷⠷⠛⢛⡛⠛⣻⣍⠉⢩⣍⠉⢡⣦⠀⢰⣶⠀⢰⣿⡄⢰⣿⡆⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⣿⠀⠀⣿⡇⢸⡇⠀⢸⣿⡇⠀⢸⡁⣿⡇⠀⣿⣿⠀⣿⣿⠀⣽⣿⠀⢸⣿⠀⢸⣿⠀⢸⣿⡇⢸⣿⡇⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠀⠀⠁⠁⠈⠁⠀⠈⠈⠀⠀⠀⠀⠁⠁⠀⠉⠁⠀⠉⠉⠀⠈⠉⠀⠈⠈⠀⠈⠈⠀⠈⠉⠀⠈⠉⠁⠈⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⠀⠰⡄
⠀⠀⠀⠀⠀⠀⢄⠂⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠂⢸⣇
⠀⠀⠀⠀⠸⢀⡀⠤⠘⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⣿⠀⠀⠼⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢂⡠⠔⠈⡟⢧⡀
⠀⠀⠀⣤⣤⣥⣤⣤⣤⣧⣌⢻⡄⠀⠀⠀⠀⠀⢀⡠⠊⠁⡀⢹⠀⣶⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣤⣤⣤⣧⣬⢿
⠀⠀⠈⠳⣤⣀⣠⡴⠟⢿⠁⠀⢿⠀⠀⠀⠀⠀⡅⠀⠀⣾⡇⢸⡇⢹⡇⠀⢿⡄⠀⠀⠀⠀⠀⠀⠳⢦⣄⣤⠶⠛⣿⠁
⠀⠀⠀⠀⡰⠛⢁⡶⣆⣸⠐⠆⢸⠀⠀⠀⠀⠐⡇⠓⢸⢹⠁⢸⡇⢸⣇⠀⠘⣷⠀⠀⠀⠀⠀⠣⠁⡴⠛⠀⣠⣁⣹⡘⠆
⢀⠜⠀⢸⠁⠴⣸⣄⡈⢻⡿⣆⠀⠀⠀⠀⠀⠀⢧⣠⡟⠸⣷⣼⣷⠇⠻⣶⣶⠏⠀⠀⠀⠀⠀⡄⡸⠸⣷⠶⠻⣿⢻⡿⣦
⡆⠱⢁⣧⡶⠛⡉⠝⠛⣸⠁⡘⠀⠀⠀⠀⠀⠀⠀⠉⠤⠒⠀⢹⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡇⢀⡅⠠⠤⢀⣸⠀⡘
⠙⠾⠛⢳⣀⣡⡴⠶⠛⣃⡼⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⢿⣋⣤⠶⠞⢋⣱⡾⠁
⢸⡀⢀⣴⠟⠿⠶⠶⠟⠋⠀⠃⠉⠛⠻⠷⠶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⠟⠛⠉⢅⣨⡴⠛⠿⠶⠿⠛⠉⢷⡀
⠀⠙⠛⣡⡼⠖⣠⡴⠿⠛⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣭⡴⣦⣀⡴⣤⠞⠷⠾
⠀⠀⢧⣨⣵⠟⠁⣀⠐⢂⠘⠋⠈⠙⠛⠻⠷⠶⠶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠿⠛⠛⠋⠉⠁⠀⠀⠈⡉⠤⠄⠐⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢠⡇⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⡞⣧⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠹⣿⣈⡟⠷⣦⣀
⠀⠀⢀⣠⠴⠚⠛⠋⠉⠛⠛⠛⠶⢶⣤⣈⣻⠷⣾⣙⡿⢶⣌⡙⠷⣄
⠀⣴⠋⠀⣠⠴⠛⠋⠉⠉⠛⠛⠷⢶⣬⣉⣻⠷⣾⣙⡿⣦⣉⠻⢦⣌⠻⣆
⣼⠇⠀⣰⠃⠀⠀⣔⠁⠀⠀⢀⠔⢶⠈⠉⣻⠷⣾⣙⡷⣦⡙⠳⣄⡙⢧⡘⣧
⣿⠀⠀⣿⡀⠀⢀⠏⢀⣴⠞⠻⣦⠘⡶⣬⣽⣳⣾⣙⣷⣬⡛⢦⡌⢳⡌⢷⡸⡇
⢿⡆⠀⠹⣷⡀⠈⠛⠋⠀⢠⣤⡄⠈⡿⠶⠋⠤⣿⣙⡗⠈⠻⣦⠹⣆⢻⡌⡇⡇
⠘⣿⣄⠀⠈⠻⢷⣦⣤⣤⣤⣤⡴⠞⠙⡷⢶⣿⢻⠛⠟⠖⠀⣈⢳⡘⣆⢧⢣⠃
⠀⠈⠻⣷⣤⣀⡀⠀⠀⠀⠀⠲⣀⣤⢖⡁⠈⡥⠊⠀⣾⠀⣇⢸⡄⠻⡜⣾⠏
⠀⠀⠀⠀⠉⠛⠛⠿⠿⠟⠛⠛⠉⡔⠁⢢⡞⠀⠰⡼⢻⣤⠿⠦⠇⢷⠱⣜⠢⠤⠄⢀⣀⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠤⣄⣠⠺⣦⡴⠶⢛⡚⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠻⠶⠈⠻⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡟⢅
⠀⠀⠀⠀⠀⢀⡆⠀⢰⡇⢰⡆⢸⠁⠀⢸⣿⣿⣿⣿⠟⣽⠹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠘⠘⠌⠆
⠀⠀⠀⠀⣴⣩⣇⠀⣸⣇⢀⡇⢸⠀⠀⢸⣿⣿⣿⡏⢸⣿⣧⢹⣿⣿⣿⠀⠀⠀⠀⠀⠪⣈⡲⠆
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠈⠉⠀⠈⠀⠀⢸⣿⣿⣿⡅⢻⢷⣾⢜⣿⣿⣿⠀⠀⠀⠀⠈⠒⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣟⢿⣟⢳⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣴⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⣾⣷⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠛⢻⣿⣿⡄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣘⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⡖⣿⣿⡀⢸⣿⠟⡿⠖⢀⣠⣱⣾⣧⠀⠀⠀⠀⠀⠀⠀⣿⠟⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠐
⠀⢠⣠⣦⣄⡀⠀⠀⠘⣷⣿⣿⣿⣿⣿⣦⣿⣤⢿⣿⣿⣁⠀⠙⠃⢀⠀⠀⣀⡀⢀⣀⣿⠀⢸⣿⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⠂⠤⠄
⠀⢰⣿⣿⣿⣿⣶⣄⣼⣿⣿⣿⣿⣯⣵⢸⣿⣏⣸⠟⣿⡛⢩⣤⠀⠠⣤⠺⣿⢿⣿⢡⣿⣷⣼⣷⣴⣶⣶⣦⢼⣿⣿⠀⠀⠢⠴⠶⠖⣶⡤⠄⠈⣿⡇⠀⠀⠠
⠀⢸⣿⣿⣿⣿⣿⣾⡿⠻⢿⣿⡟⣿⣿⢸⣿⣩⣙⣇⣹⠙⣿⣿⣤⣤⠏⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⡇⠀⢲⣾⡿⣿⣿⣗⠀⢰⣿⠇⣀⣤⣶
⠀⣼⣿⣿⣿⣿⣧⣽⣥⣼⣼⣿⢹⣿⣿⣿⡇⢉⠙⠛⠏⣿⣽⣀⣠⡆⡀⢀⣴⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⡀
⠀⣹⣿⣽⣿⣿⣿⣻⣻⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣝⣣⣬⣌⣀⣉⠛⠿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣹⣇⣿⣶⣿⢿⣿⣿⣿⢿⣿⣿⡇⠙⣿⣿⣍⢻⣿⡿⠛⣿⣿⠛⢿⡗⣷⡇⢤⢠⠰⠿⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢽⣿⠏⠈⠿⠾⠿⣿⣻⣃⣻⣿⣿⡟⠿⢿⡿⡿⣿⡷⠶⠿⣿⠾⠟⣁⣀⡛⢚⣛⠒⢄⢺⣿⣟⣿⠻⠛⠃⠻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡇
⠀⣿⣿⣰⣦⣠⡄⣤⠈⠀⠉⠹⠀⣿⡟⠛⢻⡿⠿⣿⣿⠟⠿⠿⢿⠀⡿⢿⡇⢤⣽⠇⡜⣿⣿⣿⣿⣿⣏⡿⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⡇
⠀⠃⢁⣼⣿⣛⡇⠙⠀⠀⢛⡠⠈⠻⣿⡿⣿⣿⡟⠟⢻⠛⢙⣶⡎⣤⢰⡎⠀⢸⣿⠀⢾⡧⡇⡞⣿⣶⣼⣗⣶⢶⡷⣾⣷⣾⢲⡤⢤⣯⣿⣯⣬⣥⣭⣿⣿⣿⡄
⠀⠀⠀⠉⠉⠉⢑⡒⣈⣧⠈⠑⠀⠰⠏⠏⠙⠉⠁⠀⠈⠀⠀⠉⠣⠿⠈⠃⠈⠉⠉⠛⠒⠚⠂⢀⠈⠉⢻⡏⣿⠶⢶⡷⠦⠦⠴⠦⠤⡴⠷⠴⣶⣷⣿⣿⣟⡚⠁
⠀⠀⠀⠀⠀⠀⠐⠛⣛⢿⣿⣿⣿⣛⣓⣀⣀⣆⠐⢒⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⡀⣈⠙⡛⠛⠛⠛⠋⠋⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⢿⣿⣿⣿⣿⡿⣿⢿⣾⣋⣠⣤⣤⣤⣴⣤⣶⣶⣶⣦⣤⣶⣶⣶⣶⣤⣤⣀⣴⣦⣶⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠛⠋⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣤⣤⣴⣤⣤⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣨⣤⣙⣶⣴⣾⣿⣿⣿⣿⣯⣿⣯⠙⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⡇⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣅⢸⣿⠀⠀⠀⣠⣾⣿⢟⡿⠟⣿⣿⣿⣿⡿⠿⢿⣿⣿⣼⣿⢏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⣠⣿⣿⣷⣿⡵⠿⠟⠉⠻⣅⣉⣴⢸⣿⣿⢹⠏⣾⣿⣿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡀⢇⠀⢠⣧⠉⠉⠻⣄⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⢎⣼⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⢀⣠⡴⣿⣿⣼⣄⣾⣿⣷⣄⠀⠘⢷⣦⣄⡀⠀⠀⠀⣀⣀⣤⢻⣿⣿⠿⠛⠿⢟⣽⣿⣿⣿⡿
⠀⠀⠀⠸⡀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣆⡀⠘⢻⣿⣿⣶⣾⣿⡿⣱⡿⠋⠁⠀⠀⢀⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠈⠙⠿⠿⠿⣿⣿⣦⣀⠙⡻⢿⣿⣿⡟⠁⠀⢀⣀⣤⣶⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢿⣿⣶⣤⣄⣀⣀⣀⠀⣀⠀⠈⣉⣫⡽⠟⠛⠛⠛⠿⠷⣶⣾⠿⠿⠿⣿⣿⣿⣿⣿⣁⣀
⠀⠀⠀⢀⣤⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⡏⠀⠀⠀⣀⣠⣴⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣿⣿⣿⡿⠿⡿⠦⣀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡟⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠂
⠀⠀⠀⠉⠁⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡷⣄⢀⣠⣴⣾⣿⢿⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⢠⠿⠛⠛⠻⠿⠿⡿⠿⠿⠛⠁⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣿⡿⠋⠁⠛⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣤⣀⣠⣄⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣛⣿⣿⣿⣿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⠻⣿⣷⣟⣽⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣟⣾⡇⣷⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣁⠀⠀⠀⠀⠰⠘⠃⠀⠀⠀⠀⠀⢰⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⠛⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣤⣤⣶⣾⣿⣾⣿⣿⣿⣧⣠⠀⣶⣾⣿⣶⣄⣄⡴⢀⣶⣿⡟⢿⠟⡿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⡽⠇⠁
⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠙⣿⡿⣋⣺⣿⣿⣿⣿⣿⣿⣟⣿⣷⣴⠿⠿⠿⠟⣯⣿⣧⣿⣯⣭⣤⣤⣾⣧⣬⣿⣬⣽⣿⣿⣿⣿⣏⣯⣩⢸⣦⣤⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣸⣿⣧⢀⣤⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠗
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠛⠙⠉⠛⠛⠛⠛⠋⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠐⠒⢦⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⢼⠁⠀⡀⠀⠀⢀⡠⠂⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⢰⠀⢷⠀⠀⢀⡎⠀⠒⢋⡡⠔⠂⠓⠀⠀⢠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡅⠀⠀⡆⡼⣠⡏⠀⠀⠈⠀⢀⡀⠁⡀⣀⡀⠀⠀⠀⠢⠳⢄⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠛⣿⡦⠀⢰⠂⢸⡀⠀⠊⣳⡀⠈⠴⢆⠤⣀⠀⠈⠉⠀⡣⠠⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⠀⢹⠀⢳⡷⡡⠈⢆⡸⢷⡀⢣⣿⢣⡀⠐⠒⠲⡄⢡⠀⠀⡆⠘⣀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣠⣿⣿⠻⡀⠀⢿⠹⣆⠈⢷⣤⣥⡈⣿⣶⡙⣶⣗⠀⠈⠉⠀⠀⠁⠀⠈⠙⢢⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⠈⢙⣿⣶⠀⠀⠈⠓⠿⣦⡈⢯⡛⠭⠛⠻⢿⣾⠾⢷⣼⣠⣇⡀⣀⣀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⡴⣻⠁⢻⣦⣄⠀⠀⠀⠀⠙⠮⣧⣶⣤⣦⢀⣉⣁⣂⢀⠈⣭⣽⣿⡏⠀⠀⠸⠀⠀⢸⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣽⣿⣾⠀⠀⣿⣯⣿⢦⣕⣒⣄⠀⠉⠻⢷⣄⡀⠢⢤⣄⠈⢈⠈⢉⣁⣠⣴⣿⠄⠀⢀⡿⠁⠉⠉⢩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣾⢿⣿⡀⠀⠘⣿⣿⡇⠈⣁⣽⢷⣤⣤⣤⣈⠉⠛⠦⣄⣀⠀⢏⣿⣿⣿⣻⠋⠀⠀⠊⠁⠀⠀⠉⠀⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢈⣿⠏⢸⡿⣇⠀⠀⠈⠻⣗⣀⣿⣿⣤⡀⠈⠙⡟⠛⠒⠲⡾⣯⣟⣿⡿⠿⠛⠁⠀⠀⡀⠀⠀⠀⣸⡦⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⡟⣏⡔⠀⣾⣿⣷⣄⣀⠀⠈⠻⣟⣋⣹⣛⠻⠶⠾⠿⠤⠧⠿⠽⠿⠞⠛⠛⠁⠀⠀⠈⣥⠀⠀⢰⣧⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡏⠙⢻⣷⣔⡝⠹⣻⠛⡿⣿⣿⣶⣄⡉⠻⢯⣍⠒⠤⣤⣤⣀⣀⠀⠀⠀⣀⣀⣀⢴⣶⣶⠏⣱⠚⠚⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠡⡇⠀⠈⢻⣿⣿⣦⣻⣔⣏⡿⣿⣿⣿⣿⣿⣷⠎⡇⠀⢹⣶⣤⢏⣿⣿⣿⣿⣿⡷⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣿⣶⣶⡀⠻⡏⠉⠍⣹⢿⣏⣿⣿⣿⣿⣿⢿⠀⣷⠀⠠⣸⣏⣾⡿⠿⠛⠋⡉⠈⠁⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣾⡄⠀⠴⠊⣿⣿⣿⡟⣵⣷⣿⣆⠸⣤⡶⠛⣩⡶⠶⠖⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⣿⣷⠙⡆⠀⢠⣿⣿⣟⢟⣻⣧⡿⠿⠤⢟⣒⡉⠁⠀⢀⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⡟⣼⣶⣤⣘⣿⣿⡶⠚⠉⠀⠀⢠⣶⣟⣿⣿⣶⣾⣿⠯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⠈⢏⠀⢀⡀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠠⠶⢶⣶⡶⠶⠿⣿⣿⡿⡿⠿⠟⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⣀⣸⡷⣶⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠀⠀⣀⣀⣠⣾⣿⠿⠋⣽⠏⠙⣿⠟⠻⠿⣿⣿⣿⣧⣽⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠛⠛⠛⠋⠉⠁⠀⠀⣼⠏⠁⡸⠃⠀⠀⠀⠀⠀⠀⠈⢻⡍⢿⠸⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⣧⢻⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣿⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡼⠛⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⢳⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⡇⣿⢸⠇⠔⣽⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡓⢄⠀⢹⣿⠋⣀⢸⣿⠏⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⢟⣫⠴⠶⠶⢷⣍⠈⢷⣀⢡⣶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⢀⡾⢁⣤⡤⡒⡠⣟⢦⡀⠈⡎⠹⣏⡀
⠀⠀⠀⠀⢠⣤⢞⡵⢇⣧⡘⢩⣿⠀⡇⣁⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠘⢿⠁⠀⠀⣌⠃⢺⠏⡸⢁⠿⢺⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠈⣵⢤⣊⡉⠉⣀⡴⢃⡎⢠⡟⢀⠊⡣⠀⡀
⠀⠀⠀⠀⠀⠐⡇⠀⠀⣹⡟⠛⠉⣉⣤⠟⠉⢉⡼⣡⢾⣿⡄
⠀⠀⠀⢀⣴⣣⡱⣄⠀⠈⠉⢉⠟⠀⠀⢀⣀⡿⢰⢃⣿⣿⣧
⠀⠀⠀⠘⠛⠿⠿⠾⠵⠒⣂⠓⣒⣦⠥⠤⠤⠅⠇⢠⣿⣿⡏⡠⢤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣧⠙⠿⣻⣾⠿⠿⠶⢌⣻⡟⠀⢴⣿⡿⠁
⠀⢀⣠⣤⠴⠶⠤⠤⣄⠁⢿⣯⠤⡼⠯⣤⣀⠀⠀⠀⠈⠉⢹⠚⠋
⠀⠀⠙⣿⣿⣿⣷⣦⣠⢠⡟⠁⠀⣠⠦⠤⢌⣙⣶⡖⠦⣄⠀⠙⢦⡀⠀⢀⠴⣲⣿⡇
⠀⠀⠀⠀⠙⠿⢿⣿⣿⣽⣧⠖⡿⢁⡴⣾⡛⣝⢯⣿⣄⠈⢳⡤⢤⡷⠐⢃⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡠⠋⣸⡇⢸⡇⣯⡾⣿⢿⣿⣾⠏⡟⠀⠀⣿⠈⢿⠀⢼⡿⠟⠁⠀⠀⠀⠀⢀⡴⢦⢤⢄⡀
⠀⠀⠀⠀⠀⠀⢰⠃⠀⢿⢳⣴⡗⣸⡿⠵⢋⣼⢋⡴⣧⣀⡴⡿⠀⢸⡄⣀⡀⢤⣀⡀⣀⣀⠤⠾⢦⣖⣚⣪⠅
⠀⠀⠀⠀⠀⠀⢸⣆⠀⢸⡀⠈⢷⣀⣈⠿⠉⣰⠟⢀⡿⠀⣰⠃⠀⣼⢁⣴⣶⣿⡿⣱⠯⣧⠤⠴⢶⣤⣬⢯⠶⠆⠠⢄⣠⣤
⠀⠀⢀⣤⣖⣈⣜⣧⠀⠀⠙⢦⣤⠬⠤⠶⠛⢁⣠⠞⣡⠞⠁⣠⣾⡟⠾⠿⣟⡋⢰⣿⡶⢯⡼⣀⡾⢕⣢⠌⣷⢠⣶⣿⠟⠃
⠀⠀⠀⠙⠻⢿⣿⣮⠳⡀⠀⠘⠳⣶⣶⣶⡾⠿⠗⠋⠁⠀⢀⣿⣿⣤⡤⣄⠈⠙⢦⡉⠑⠺⡛⠓⠺⡉⢀⣛⣁⡉⠛⠲⢄⡀
⠀⠀⠀⠀⠀⣠⠞⣀⣤⣴⡀⠒⣏⠀⠀⠙⢻⡷⠶⠴⠶⢒⣽⡜⣿⣿⣿⣿⣿⣦⣄⠹⣆⠈⢿⣿⣶⣜⢎⢿⣿⣿⣿⣷⣦⣽⣄
⠀⠀⠀⠀⣴⣡⣾⣿⣿⠟⡇⣾⣾⠲⢤⣤⡿⠀⠀⢠⢱⣿⡿⠁⠈⠻⢿⣿⣿⣿⣿⣷⣼⣇⠀⠙⠻⠿⠿⠄⠉⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠐⠛⠛⠛⠉⠀⠀⠱⡿⠋⠀⠀⠀⠀⠀⠀⠘⡿⠋⠁⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡵⠚⡛⠲⣯⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⢀⣠⠷⠶⢾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠠⠄⠀⠤⠄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⣄⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣨⠁⠠⠞⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠋⢠⣀⡀⠙⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢻⡀⠀⡏⣿⣧⠀⠀⣼⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡄⠀⠓⢰⠇⣿⠸⠀⠚⠁⢀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡗⠀⣠⠉⢀⡏⠀⠀⣀⠀⢸⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣀⡼⠟⠋⠁⣠⠞⠀⠱⣄⠈⠳⢾⣧⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⣠⡴⠛⠁⠀⠀⠀⠈⠳⣦⣀⠈⠙⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠶⠛⠉⠰⡀⠀⠀⠀⠀⠀⠀⣴⠙⠳⠶⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⡀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠛⠛⠒⠒⠀⠀⠨⣽⣷⣶⣶⠶⠶⠾⠿⠛⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⠈⠉⠙⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⠛⠻⠶⢶⣤⣤⣄⣀⣀⣀⣤⠤⢴⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⢤⣤⣤⣴⡶⠾⠟⠛⠉⠉⠀⠀⠀⠈⠙⠛⠳⠶⠶⠶⠶⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⣠⠀⢸⠀⢠⡄⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⡀⢸⡇⠀⡦⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⡄⠸⠆⠘⠀⠸⠀⠾⠁⣴⠃⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣦⠈⠧⠘⠃⠘⠇⠰⠇⢠⠞⠀⣠⠟
⠀⠀⠀⠀⠳⢦⣄⠈⠳⠄⢁⣠⠤⠶⠶⠶⠦⣄⡈⠁⠴⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⠀⠹⠆⠈⠀⢀⣀⣀⣀⣀⡀⠀⠉⠠⠞⠁⣀
⠀⠀⠀⠀⠀⣄⡈⠛⠀⣴⠋⣡⡴⠖⠓⠀⠀⠀⠙⢦⡀⠘⠋⣁⣤⠶⠂⠀⠀⠀⠀⠀⠀⢠⣀⠙⠃⠀⣠⡶⠛⣉⣭⣽⣿⣿⣷⣄⠀⠀⠛⢁⣠⡴⠖
⠀⠀⣀⣀⣀⣀⡉⠁⣼⠁⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣉⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⠁⠀⢰⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⢉⣁⡀
⠀⠀⠀⠀⣉⣉⡁⠀⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣉⣉⣀⣀⡀⠀⠀⠀⠀⢉⣉⣉⠀⠀⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢉⣉⣁⣀⡀
⠀⠀⢀⣀⠉⠉⠁⠀⣷⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣸⠃⠀⠉⠉⣉⣉⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⢻⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡿⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠈⠉⠙⠳⠦⣤⣼⣧⡴⠖⠛⠉⠉⠉⠙⠳⠦⣴⣯⣤⡴⠞⠋⠉⠉⠀⠀⠀⠀⠉⠉⠓⠦⢤⣠⡈⠛⠋⠁⠀⠀⠀⠀⠈⠉⠛⢁⣤⡤⠴⠚⠉⠉
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠉⠙⠶⠤⣤⡤⠶⠛⠉⠉⠉⠙⠳⠦⣤⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⠤⣤⣤⠤⠒⠉⠉⠉⠉⠒⠦⢤⣤⠤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠶⠛⠉⠀⠈⠙⠶⠤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⠤⠖⠋⠁⠀⠉⠲⠤⠤⠖⠉








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢲⣦⡀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡶⡏⢀⠀⠻⣤⣾⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠊⠉⢻⣏⠁⠈⢣⢦⣀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣆⠳⠦⠙⣧⠀⠀⠣⣤⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠙⢲⡄⠀⠈⢳⣄⠀⠈⠢⣄⠀⠙⣄⠀⣠⠴⢦⡤⠴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡆⢣⣬⢻⡁⠀⠈⢿⣄⠀⠀⡘⢧⠀⠈⠻⣏⢢⡟⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢰⣿⡄⠀⠀⢻⡆⠀⠈⠻⣆⠀⠀⠈⢦⣤⠄⠹⣾⡇⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻⢿⣿⡄⢠⠀⢳⡄⠀⠀⠘⣷⠀⠀⠈⢻⡀⠀⠈⢿⡃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠰⢿⣿⣿⡌⠆⠈⠱⡄⡀⠀⠘⣷⡄⠀⠀⠹⣆⠀⠈⢿⡄⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⠰⣿⣿⣿⡙⠀⡈⠻⡵⡀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠈⢿⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠻⢾⣿⣴⣾⣷⡙⣏⠀⠹⣆⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠑⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⣿⣿⣿⣿⣷⢁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣺⣿⡿⠟⠛⢻⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠀⠀⠀⠠⣶⢻⣿⡆⠂⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠠⣠⠄⠒⠈⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡷⢀⠀⠀⠀⠛⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠤⠼⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⠥⠮⢿⣿⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢻⡟⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⠀⠓⢺⣿⣿⣷⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⣠⢟⣤⣿⣿⠿⢹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠷⢉⣿⣿⣷⡻⣦⠀⣀⡀⠀⢂⣠⠖⠱⠋⠛⠉⠁⠀⢾⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣮⣿⣇⠀⠀⢀⡸⢟⣻⣿⣿⡜⣷⣽⣿⠶⠋⠁⠀⠀⠀⠀⢀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠻⠿⣶⣴⡶⠋⢀⣼⣿⡟⠉⣿⠿⠋⠁⠀⠀⢀⡠⠀⠀⠠⠋⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⡀⣤⡾⠋⣠⡴⡃⣾⢃⠂⠀⠀⣀⣀⣠⡔⢚⡛⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣦⣼⠿⠟⠉⠀⠠⠐⣼⣾⣿⠀⠓⠖⠛⠋⠓⢀⡠⠖⠛⠉⢉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠆⠀⠀⠀⠄⠂⠀⠀⠀⠠⣿⠀⠈⠉⠉⠠⢞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡀⠀⠀⠀⣴⢃⡼⣄⢳⡄⠀⠀⠀⣠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣙⠦⡀⣸⢃⠎⠀⠈⢆⢻⡀⡠⠞⣡⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⡇⡏⠈⠁⢸⣇⠀⢠⣄⣤⠀⢀⣿⠀⠉⠈⡇⣿⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣛⡲⢦⣸⡀⠀⠀⠀⠸⠀⠀⣿⣧⣿⡇⠀⠘⠀⠀⠀⠀⣽⣤⠶⣚⡋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⠉⠀⠈⣷⢄⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⢀⢴⡏⠀⠈⢹⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⠀⠀⠀⠘⢆⠁⠀⣠⣤⣴⣾⣧⣿⣶⣤⣤⡀⠀⢁⠞⠀⠀⠀⠸⢸⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⢧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⢏⣀⣀⣀⡀
⠀⠀⠐⢾⡛⢉⣩⡭⠭⠍⠉⠙⢦⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⠞⠉⠉⠭⠭⣭⡉⠙⣻⠖
⠀⠀⠀⠀⠙⢦⡙⢆⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠏⢸⣿⣿⣿⣿⣿⣿⠈⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢀⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣉⡀⢸⣿⣿⣧⣿⣿⣿⠀⣈⣙⣿⣿⣿⣿⣿⣷⡀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣳⣤⣉⡁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣉⣡⣴⣋
⠀⠀⠀⠀⢀⣴⣞⡉⠴⣄⠀⠁⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠁⢀⡴⠌⣙⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀⡀⠀⠀⠀⠀⠈⠉⠻⠿⠟⠛⠿⠿⠿⠿⠉⠁⠀⠀⠀⠀⣀⣠⠴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠚⠋⠉⠁

⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

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

⠀⠀⠀⠀⠀⠀⠀⠀⠌⠄⠀⠀⠀⠄⠀⠀⠀⠀⠠⠌⠄⠀⡂⠀⠁⠀⠠⢐⠀⠠⠀⠈

⠀⠀⠀⠀⡺⢂⡀⠀⠀⠀⠀⢠⡄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⢠⣿⣦
⠀⢀⠀⠀⠉⠈⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠈⠳⣤⣤⣴⠶⠊⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠉⢻⡆
⠀⠈⠷⣤⣤⣤⠶⠊⠀⠈⠳⠶⠶⠞⠋⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣦⣤⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⢠⡄⣠⣤⣀⡀⠀⠀⠠⠶⣶⣦⠄⠀⠀⠀⠴⠶⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⣴⠚⠉⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⣴⠟⢡⠀⠀⠀⠀⢀⠀⠀⢐⡞⠙⠃⠀⠀⠀⠀⠀⢴⣦
⠀⠀⠈⢷⣄⣀⡀⠀⠀⢻⣧⣀⣀⣀⠀⠀⠘⣿⣄⣀⣀⡀⠀⠀⠙⠶⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⠀⠐⢊⡶⠀⠀⠀⣤⣀⣉⡼⠃
⠀⠀⠀⠀⠀⣀⣴⠿⠁⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠸⠋⣹⠆⠀⠀⢈⠏⠛⠉⠀⠠⣄⣠⠞
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠘⠊⠃⠀⠀⠀⠀⠀⠉⡶⣦⠀⠁⢿⣀⡀⠀⠀⠀⠸⢦⣀⠄⠀⠀⠀⠀⠀⠀⠀⣶⠀⠓
⠀⠀⠀⠀⠀⡖⠋⠋⠀⡀⠀⠀⠀⠀⣀⠄⠀⠀⢰⡏⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠖⢲⡆
⠀⠀⠳⠶⠾⠃⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠈⠛⠓⠃⠀⠀⠀⠻⠷⠶⠶⠀⠀⠀⠀⠀⠐⠦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣄⡀⠀⣆⠀⠀⠀⠘⢃⣭⡄⠀⠀⠀⠀⠀⣾⣁⡀
⠀⠀⠀⢀⣰⠟⠳⠿⠂⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠘⠁⠀⢻⠀⠀⢀⠀⠈⠀⡃⠀⢰⡄⠀⠀⢙⣹⠟
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠰⣤⣤⠞⠁⠀⠀⣦⣄⣀⣤⡤⠏⠀⠀⢸⣤⣤⠾⠁⠀⠀⠙⠛⠛⠋⣥⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⡆⠀⠀⡴⠿⠿⣷⡄⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢘⠁⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⡿⠁⠀⢸⣷⣤⣶⠟⠀⠀⠀⢰⣀⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⢿⣧⠀⠀⠀⣿⡆⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡈⠙⠀⠀⢿⣿⡆⠀⢸⣿⡆⠀⠀⣿⣿⡀⠀⢸⣿⡀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠈⣿⡀⠀⠀⢸⣧⢀⠀⢸⣿⢠⡄⠀⣿⡆⠀⠀⢸⣿⠇⡀⠈⣿⡆⢠⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢛⣻⠇⠀⢠⠾⠃⠀⠀⢸⣿⠘⠤⠈⣿⠀⠙⡆⢸⡇⠀⠀⠈⣿⠀⠹⡀⣿⡛⠈⠶⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠚⠋⠅⠀⠀⠸⡀⠀⠀⠀⠀⣿⣆⣀⠀⣿⡇⠀⠀⢸⣧⠘⠞⠀⣿⠀⠀⠃⢻⡇⠀⣶⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠷⠂⠀⠀⠃⠀⢀⠀⠿⢿⣿⡿⢿⣿⣷⣤⣤⣸⣿⣀⠀⠀⣿⡀⣰⡗⢸⡇⠀⢿⡇⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠤⣦⡀⡴⠆⠀⠀⢀⣾⠿⣧⠀⢸⣇⠀⠀⣸⣯⣽⣿⢿⣿⣿⡿⠋⢹⣷⢻⡇⢸⡇⠀⢸⣷⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠀⠈⢳⠃⢀⠀⠀⣞⠁⠀⠙⢿⣿⣿⣿⠿⢿⣿⠉⠀⠀⣿⡇⠅⠀⢸⡏⢸⡁⢸⣿⠀⣸⣿⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢀⣠⠖⠋⠀⣀⣿⣦⣄⡀⠀⠈⣿⠀⠀⠘⣿⠀⠀⠀⢻⡇⢣⣀⣼⣧⠸⣷⣬⣿⣴⠏⠘⢿⣷⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⣠⠖⠋⢁⣤⣶⡿⠟⠛⠛⠻⢿⠀⠀⣿⠀⢀⣀⣿⣀⣀⡀⢸⣿⠀⠙⠉⣿⠀⠙⠛⣿⡏⠀⡰⣄⠉⢡⣄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠀⣀⣴⡿⣋⠁⢀⣤⣀⠀⠀⠀⠀⣀⡿⠀⣿⠿⣿⠿⠿⠿⣿⣿⣿⣷⣶⣿⣶⣦⡄⢻⡇⠀⠓⠋⠀⠀⠘⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡿⠟⠁⠰⢿⡷⠈⠻⣏⣠⣴⣶⡿⠟⢁⠀⠻⣷⣦⣄⠀⠀⣨⣿⣤⣾⣿⣿⡟⠋⠀⢸⡇⠀⠀⢼⣧⡀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⣤⠞⠁⢀⣤⣶⡿⠿⠛⢋⣥⡀⠐⢿⣷⠀⠀⣈⣿⣿⣿⠿⣿⡏⠁⠀⣿⠁⠀⠀⢸⡇⢸⠀⠘⣿⡁⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠘⠉⠀⣀⣴⠿⠋⠁⢀⣠⡴⠋⠉⣀⠀⠀⣁⣴⣾⠿⠋⠹⣿⡇⣿⠁⠀⠀⠟⠀⣤⡄⠸⠃⠘⠒⠀⢻⣷⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣴⣾⢿⠏⠀⡐⠊⠉⢁⣠⠴⠛⢁⣠⣾⠿⠋⠀⣦⡴⡴⠀⣡⡿⠀⢀⣠⡄⠀⠶⠃⢀⡀⠀⢠⣄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣾⠀⠈⠍⠀⠞⠉⠀⣀⣴⡿⠛⢁⡠⠆⢀⣠⣤⣶⡿⠋⠀⢠⠟⠻⣷⣤⣀⣤⣾⣧⣤⡞⢿⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡘⠖⣃⣠⣴⣾⠟⢉⡤⠖⠉⣠⣶⡿⠟⠋⠁⠰⣀⠀⣦⡀⠀⠘⠻⠿⠛⠀⠙⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠟⡿⠀⠀⠈⠀⢠⣾⡟⢡⣄⢀⠖⠀⠀⣀⣤⣾⣿⣶⣤⠀⠀⣇⠀⠺⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⣴⡿⢻⠀⠀⠈⠿⢀⣴⣾⠟⠋⠉⠉⠉⠛⠀⠀⢈⡤⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⡿⠋⠀⢸⣄⢀⣀⣴⡿⠋⠀⣀⡤⠀⠀⠀⠀⠠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⣸⠆⣤⠀⠸⣌⠣⢄
⠀⠀⠀⠀⠀⢉⢀⠘⠣⠀⠈⠣⡈⠳⢄
⠀⠀⠀⠀⠀⣼⠇⣤⠀⢷⡄⠀⠈⠣⡀⠱⣄
⠀⠀⠀⠀⡠⢿⠀⠈⣧⠀⠹⣦⡐⠀⠙⢦⠈⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⣼⡆⠀⠸⡆⠀⠙⠳⣄⠀⠄⣧⠈⢳⣄⡐⡐⢰⠄
⠀⠀⢀⠂⢰⣿⣿⠀⠀⠸⡄⢂⡀⢹⡄⠀⠘⣧⠀⠹⣇⡇
⠀⠀⡀⠀⣿⡿⣿⣧⠀⠀⠘⣈⠀⠘⢷⠀⠀⠘⣧⠀⠉⠻⣄⡄
⠀⠀⠁⠀⣿⡇⢿⣿⡇⠀⠀⠽⣆⠀⠈⢧⡀⠀⠈⠀⠀⢀⠙⣷⡄
⠀⠀⠤⠀⢎⢳⣬⡿⣷⠀⠀⠀⣽⠄⢀⠈⠃⠀⠀⠘⠀⠀⡄⠘⠙⢦
⠀⠀⠀⠄⠀⢎⣧⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⣘⠆
⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⡆
⠀⠀⠀⠀⢀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠘⣄⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢁
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢹⣿
⠀⠀⠀⠀⠀⠀⠘⣆⠀⢠⠉⢿⣿⣷⡵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡰⣿⡇⠘⣥⡈⢙⣿⡿⢿⣷⣄⡀⠀⠀⠀⠈⠀⠀⠀⠈⠒⠈⠁⠀⠀⠀⠠⣶⣄⠁
⠀⠀⠀⠀⠀⣼⣷⡿⠃⢐⣈⠡⣼⣿⣤⣦⠞⣿⣿⣯⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠁
⠀⠀⠀⠀⣸⢻⠅⣿⢂⡈⠙⠃⠈⠙⣿⡿⠻⢋⠋⠛⢷⣤⡠⠂⠀⠀⢳⣤⣤⡀⠀⠀⠀⠚⠁
⠀⠀⠀⢀⢿⢀⣤⣽⣀⣀⣀⣤⣧⠖⣩⠂⠴⢏⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠢⡈⠉⠉⠉⠁⠀⠉⠠⠚⠁⠀⠐⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠹⣿⣿⠷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛
⣿⣿⠛⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠛⠂⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣤⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⡀⡀⠄⢰⠀⠒⠒⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⡂⠀⠀⣥⡴⠶⠾⠖⢶⢊⠉⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⢸⠘⣒⣛⣛⢚⡛⢓⡓⠛⡀⣸⡿⣿⣿⣏⠇⠀⣀⣠⣦⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠨⢭⣭⠄⡜⢲⠃⣈⡡⢙⠘⠨⢖⣀⠚⠓⣛⠛⡿⣶⡅⠀⢹⣷⣶⡏⠀⡄⠀⠀⠀⠀⠀⠀⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⡂⣾⠗⠀⠧⢸⣿⣿⢿⣦⠰⠇⣾⡿⠈⢸⣿⠀⢸⣿⡇⠀⢠⣷⢹⣿⣶⡆⠀⠀⠀⢀⡀⠀⡆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠘⠁⠛⢛⠀⠃⢸⣿⡿⣘⣻⡀⠀⢙⢁⠀⠘⠛⡀⡘⢿⡂⠀⢸⣿⢸⢿⣿⠃⠐⡀⠀⠀⢀⡀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠠⠘⡀⠐⠀⣴⠀⣾⣿⠀⠀⢸⣿⢀⣿⣿⠂⠀⡼⣳⠀⢰⣿⠀⢡⣿⠁⠀⢸⣿⠀⢣⡝⠀⡶⠃⢘⣋⠉⢱⡦⣤⢚⣦⠃
⠀⠀⠀⠀⠀⠀⠃⣴⠤⣀⠀⠙⢾⡽⢭⠀⠘⢸⣿⢸⣏⣉⠀⠀⣿⣽⠀⢸⡍⠀⢸⣿⠀⠀⠈⣿⠀⢸⡇⣿⣀⠀⡿⣯⡁⢸⠁⠀⡌⢟⢹⣤
⠀⠀⠀⠀⠀⣀⣠⡇⠐⢉⣤⣞⣩⣷⠟⣀⣠⡘⢋⠀⠋⠉⠀⠀⠉⠈⠀⠈⢁⠀⠘⣭⠄⣤⣦⣧⣀⣈⣡⣥⣬⣤⣤⣭⣥⡤⠤⢴⡅⢊⡤⠁⠂
⠀⠀⠀⠀⠀⠀⠉⢉⣙⢏⣠⣼⣟⣏⢈⡉⠲⠂⠀⠠⠄⠀⠀⡀⠀⢀⣤⣤⣤⣄⣀⣧⠀⣿⣿⣿⣿⠿⠿⠿⠿⡿⠟⠻⠛⠶⠈⠉⠉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⢀⡁⠉⠁⠉⠉⢀⠀⠁⠹⣆⣧⢴⢶⣦⣎⠤⠔⠒⠂⠐⠀⠀⠀⠰⠒⢒⣒⠉⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⠈⡑⣲⠒⠚⠳⠖⠋⠉⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢛⣿⠟⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢸⣅⢒⡀⠀⠀⠀⣿⣈⣉⣁⣿⠀⠀⠀⣿⣈⣋⣁⣽⠆⠀⠀⣿⣀⣀⣀⣹⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠩⡍⠁⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠐⢷⡄⠀⠀⠀⠀⢙⡿⠟⠀⠀⠀⠀⠙⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⣤⠾⠁⠀⠀⠀⠀⢠⡼⠇⠀⠀⠀⠀⢸⣇⣀⡀⠀⠀⠀⠸⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡳⢄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣯⡟⠀⠀⢀⠀⠀⣤⣿⡽⠂⠀⣀⠀⢰⣤⣮⡿⠀⠀⣀⠀⢰⣤⣮⡷⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⣀⣼⠇⠀⠀⠀⠀⢺⣅⣠⡟⠀⠀⠀⠀⣿⣀⣼⠏⠀⠀⠀⠀⣿⣀⣼⠏⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⡄⠀⠀⠀⠀⢸⣯⡁⠀⠀⠀⠀⠀⣿⣍⠀⠀⠀⠀⠀⢸⡇⢛⣀⠀⠀⠀⠀⣿⠀⣀⡀
⠀⠀⠀⠀⠀⠠⣆⠀⢘⣿⠇⠀⠀⠀⠀⢸⣇⢀⡀⠀⠀⠀⠀⣿⡀⣀⠀⠀⠀⢀⣸⣷⣋⡿⠀⠀⠀⣀⣿⣞⣻⠗
⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢠⠤⡄
⠀⠀⠀⠀⠀⠀⢀⡄⠉⠐⣦⠀⠀⠀⠀⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣧⢘⡇⠀⠀⠀⣠⠀⢰⣿⣦
⠀⠀⠀⠀⠀⠀⠘⠷⠶⠾⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣧⣤⡿⠀⠀⠀⢠⣬⣥⣼⠇⠀⠀⠀⠻⣦⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡖⠂⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢠⣄⢰⡇⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⣴⢲⣆
⠀⠀⠀⠀⠀⠀⢾⣃⠀⣈⣻⠆⠀⠀⠀⠀⢿⣍⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⣿⣿⢙⡧⠀⠀⠀⣀⠀⠙⢛⡿
⠀⠀⠀⠀⠀⠀⠈⠙⣛⣭⠁⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠙⠶⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⢉⡙⢩⡍⢿⣿⣿⣟⠏⠈⠀⠀⣠⣿⣝⠻⢍⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣄⣲⠄⠀⠀⠀⠀⣀⣶⣶⣦⣤⣤⣄⡴⠼⣯⠿⠄⠰⠓⠀⠛⠛⠛⠑⠐⣿⠟⠻⢿⣿⣿⣿⣟⠛⣀⡈⢙⠻⠟⠛⣿⣿
⣉⣉⣙⣁⡀⠀⠀⠀⠀⠉⠻⠿⣿⣿⠛⠿⣿⣶⡂⠀⠀⢠⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⣀⢀⣀⣀⣐⣻⣿
⠈⠉⠉⠉⠉⠋⠉⠉⠀⠀⠈⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⢿⣿⣿⣿⠆⠀⠀⠁⠀⠀⠈⠀⠐⠂⠚⠛⠛⠋⠉⠉⢉⣉⣉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⡀⠀⢸⣿⠿⣿⣿⣿⣿⣿⢻⣿⣀⣀⣀⣴⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡽⠂⢀⡴⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠺⣇⣄⣈⠳⣿⠟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠠⣀⠹⣦⠀⠈⠝⠆⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡼⢧⣉⠁⠈⢧⡀⠀⠻⡄⠀⠐⣄⠀⠀⢀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡀⡀⡹⡆⠀⠀⢿⡀⠀⠈⠳⡀⠈⠢⣴⠋⣼⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣧⠈⠁⠘⣆⠀⠈⢷⡆⠀⠀⠹⣄⣀⠸⣿⡏⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢺⣿⣧⠀⡀⠹⣄⠀⠀⠹⣦⠀⠀⠹⡏⠀⠈⢷⠖⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠠⣿⡟⣧⢻⠀⠙⣄⠀⠀⠹⣦⡀⠀⠘⢦⡄⠀⢳⡀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⢸⣿⣿⣇⠂⢀⠹⣶⡄⠀⠘⣷⠄⠀⠈⠀⠀⠈⢿⡆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⢺⣷⣶⣿⣎⠻⠀⠙⣧⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠑⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣿⣿⣿⣿⡄⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠶⠿⠛⠉⠉⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠠⠸⠼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣐⡒⠁⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣆⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠔⣚⣛⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⠈⠥⢭⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡟⠸⣿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⠀⠈⡟⢻⣿⣿⡵⣦⠀⠀⠀⠐⠄⢀⡴⠋⠀⠚⠛⠁⠸⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣿⠀⠀⠀⣰⢿⣿⣿⣞⢷⣴⣧⣤⠖⠁⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡺⣿⣯⣤⣤⠟⠁⣸⣿⠟⢻⣿⠟⠋⠀⠀⠀⡀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⢠⡼⠛⣁⡔⢉⣿⠁⠀⠀⢀⣀⣀⡤⣔⠋⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢤⡿⠟⠋⠀⠀⠀⢲⣾⣿⠈⠶⠞⡛⠓⢀⡠⠞⠒⠁⠍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣁⡀⠀⠀⠀⠀⠀⠀⢸⡿⢀⠀⠀⠀⠲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⠿⠛⡿⠻⡿⢛⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠒⠒⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⢀⡀⠒⠂⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⢀⣴⡆⢀⣴⠋⠁⣤⣬⣹⣿⣄⡄⢠⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢇⠀⠍⢱⣬⡵⢾⠟⠛⢿⣿⣿⣇⢻⣾⠟⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠆⢈⠈⢭⠓⠂⠀⠴⣿⣿⣴⣿⢛⣥⢾⣿⡆⠀⠒⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣦⣄⡀⢠⣿⡆⠀⠀⠀⠀⣂⡈⣓⠺⣧⣀⣠⣦⣬⣭⣍⣽⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣷⣤⣄⡀⠀⠈⢳⣽⣿⣤⣤⣬⣉⣛⠻⢿⣿⣿⣿⠟⠁⠀⠀⣠⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⢉⡙⠻⣿⣧⡀⠀⣀⣙⡛⢿⣿⣿⣿⣋⣿⣿⣿⣿⣵⣶⣿⠛⣿⣟⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢿⣿⠿⣿⢿⣿⡿⠋⢉⣿⣿⢿⣿⣿⣿⣿⣿⡏⠉⠻⣼⡁⠉⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣈⣀⡡⠉⠉⠀⠀⠀⠀⠛⠋⠁⠈⠿⣿⣷⣷⡀⠀⠘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣉⠀⠄⠀⠀⠀⠀⠀⢀⣠⣾⢿⣦⣾⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡻⣿⣦⠀⠀⠀⠀⢐⢋⣶⣿⣺⣿⣿⠶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠹⡟⠛⠉⡻⣧⣀⣀⠈⢨⣿⡍⣿⡿⢿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣶⣬⣛⣶⣬⣿⣿⣿⣦⠈⣿⡏⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠿⢁⠋⠀⢱⣤⠇⠀⠀⣀⣴⡆⠀⠀⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⠃⠀⢀⣾⣿⡿⠿⢤⣾⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡔⠀⣴⣿⣿⣿⠋⢸⣠⡿⠟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣾⣿⣿⣿⢀⣁⣸⣿⣧⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡹⠿⣿⠿⣿⣿⣿⣿⡦⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠠⣴⣿⡷⢶⣮⠭⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣧⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣼⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢻⡀⠀⠸⣇⠀⠀⣼⠇⠀⢠⡏⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠈⣷⠀⠘⣇⠀⠀⣿⠀⠀⣿⠀⠀⣸⠃⢀⡾⠁⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠘⣧⠀⠻⠀⠀⠛⠀⠀⠋⠀⠀⠟⠀⣼⠁⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠦⠤⣤⣀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⡀⡄⡆⢰⢠⢀⠀⠉⠙⠶⣄⠀⠀⠀⠀⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠰⢦⣄⡀⠀⠀⠀⠉⠳⠀⠀⣠⠞⠁⢀⣦⠈⠇⠃⠃⠁⠈⠘⠘⠸⠑⣴⡄⠈⠳⣄⠀⠀⠞⠉⠀⠀⠀⣀⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⠀⠀⢠⡞⠁⢀⡤⠋⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠉⢤⡀⠈⢳⡄⠀⠀⣤⠶⠛⠋⠁
⠀⠀⠀⠈⠉⠛⠛⠲⠶⢤⣤⡀⠀⢠⠏⠀⡈⠛⠀⠀⠀⣠⡢⠓⢉⣁⣈⡉⠚⢔⣄⠀⠀⠀⠛⢁⠀⢹⡄⠀⢀⣤⡤⠶⠖⠛⠛⠉⠁
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡟⠀⠠⠭⠁⠀⠀⢬⠊⣠⠞⠋⠉⠉⠙⠳⣌⠑⡥⠀⠀⠈⠭⠄⠀⢻⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠀⢸⠃⠠⠭⠍⠀⠀⠸⠇⢰⡏⠀⠀⠀⠀⠀⠀⢹⠆⠸⠆⠀⠀⠩⠭⠄⠘⡆⠀⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⠀⢸⡄⠐⣒⣂⠀⠀⢰⡆⠘⣇⠀⠀⠀⠀⠀⠀⣸⠇⢰⡆⠀⠀⣐⣒⠂⢠⡇⠀⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣧⠀⠐⣒⡀⠀⠀⢚⢄⡙⢦⣄⣀⣀⣠⡶⠋⣠⡓⠀⠀⢀⣒⠂⠀⣼⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⢀⣀⣤⣤⠶⠶⠚⠛⠁⠀⠘⣆⠀⢑⣤⠀⠀⠀⠙⠕⡤⣈⡁⢉⣁⢤⠪⠋⠀⠀⠀⣤⡂⠀⣰⠃⠀⠈⠛⠓⠶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠀⠀⠈⢧⡀⠈⠓⣄⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⣠⠚⠁⢀⡼⠃⠀⠀⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠰⠞⠋⠁⠀⠀⠀⣀⡴⠀⠀⠙⢦⡀⠈⠟⢀⡆⡄⡄⡀⢀⢠⢠⢰⡠⠻⠁⢀⡴⠋⠀⠀⢦⣀⠀⠀⠀⠈⠙⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠁⠃⠇⠸⠘⠈⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠉⠛⠒⠶⠤⠤⠶⠒⠛⠉⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⢠⡟⠀⣴⠀⠀⣤⠀⠀⣤⠀⠀⣦⠀⢻⡄⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⢀⡿⠀⢠⡏⠀⠀⣿⠀⠀⣿⠀⠀⢹⡄⠀⢿⡀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣼⠁⠀⢰⡟⠀⠀⢻⡆⠀⠘⣧⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡟⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢻⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣀⡴⢚⣉⣁⡀⠁⠉⠉⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠦⠴⠃⣠⠞⢁⡞⠋⠀⢀⡔⢋⠉⠀⠠⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠈⢣⠀⢸⡆⠀⠀⡸⢠⡎⠉⠢⠀⠈⢣⡄⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠁⠀⠀⠀⠀⣠⠏⢀⣾⡇⢆⡇⢡⠘⢌⢂⣰⢱⢀⠀⡥⢆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣼⡏⢀⡞⠁⠘⢦⠱⣀⠀⣠⣤⣥⡖⣚⣐⣡⠇⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⡎⠀⠀⠀⠈⢆⣀⣉⠢⡀⠤⠒⠋⠉⠁⢀⡞⠁⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠒⠢⢄⠀⠀⠀⣀⣀⣸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠳⣌⠢⢄⡉⠐⢲⠞⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⠙⠄⠀⠀⠉⢻⡇⠀⠀⠀⠀⠛⣀⡠⠤⠶⠶⠤⢄⡠⢀⣨⠷⠖⠒⠛⠉⠀⢰⠖⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠠⡀⠀⠈⡄⠀⢀⡼⢷⡀⠀⢠⡖⠛⠓⠤⣀⢀⣠⣤⡀⠈⠋⢁⣀⡀⠀⣿⠛⠒⠋⠀⢨
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠙⠢⣰⠀⣸⠁⡜⠁⠀⠀⣙⡷⣾⠁⠀⠀⠀⠈⠉⠁⠀⠀⠑⠢⠞⣹⠟⠊⠁⠀⢀⡶⢤⠎
⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⢂⡇⣸⢁⡠⠖⠋⢁⡞⢉⣷⣤⣄⣀⣀⡀⠀⠀⣀⣀⡤⢞⣁⡀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠉⠉⠁⠒⠠⠈⠟⣁⣁⣀⣀⡼⠉⡴⠋⠁⢡⠉⠉⠉⠉⠛⠿⣟⡶⠾⠛⠉⠉⠑⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⣀⣤⢸⠀⠀⣠⠟⣠⠊⠀⣀⠤⢼⡴⠂⠤⠤⢤⣀⠀⠙⢄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠈⣇⢀⠔⢡⠼⠁⠀⡼⠁⠀⠈⠀⠀⠀⠀⢠⠟⠀⠀⠈⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⡴⠃⠀⢀⡼⢁⣠⠔⠋⠀⠀⠀⣠⠏⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠈⣓⣒⣀⡛⠻⢭⡐⠢⢄⣀⣠⠔⠛⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣠⠖⠉⠀⠀⠈⠙⢦⠀⠈⠢⡀⠈⠳⡒⠤⠄⠀⠀⠈⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠊⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠘⡄⠀⢳⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣸⠷⠦⠤⠤⠤⠶⠒⠒⠂⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣀⣤⠤⠒⠊⠁⠀⠱⡄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣰⡋⢈⣉⡁⢀⠀⢈⣉⡁⢨⣷⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢻⡇⠀⠀⠀⣼⠀⠀⠀⠀⢸⠜⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠱⡀⠀⠀⢀⡀⠀⠀⢠⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣠⠔⠛⢢⣄⣀⣀⣠⡖⠛⠢⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⡁⠀⠈⠙⢦⠀⠀⡴⠋⠀⠀⢈⣂⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠒⠤⣀⢣⡜⣀⠄⠒⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢠⠏⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠹⡄⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠠⢄⡀⠀⠀⠀⣰⣋⣠⠧⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⣄⣙⣆⠀⠀⠀⢀⡠⠄⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡯⣄⡀⠀⠙⡆⣠⠞⠉⠁⠀⠀⠀⠀⠉⠒⠢⠤⠤⠤⠤⠔⠒⠉⠀⠀⠀⠀⠈⠉⠳⣄⢰⠋⠀⢀⣠⢽⡄
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⢣⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡜⠁⠀⠀⠙⣆
⠀⠀⠀⠀⢠⡇⡎⡜⢠⡆⢸⠀⠒⠒⠒⠒⠒⠒⠒⠚⠛⠓⠒⠒⠤⣀⠤⠒⠒⠚⠛⠛⠓⠒⠒⠒⠒⠒⠒⠂⠀⡇⢰⡄⢣⢱⢸⡄
⠀⠀⠀⠀⠘⢷⠀⠃⡏⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠒⠲⣄⡤⠔⠤⢄⡀⠀⠀⠀⠀⠀⠀⠘⡼⢱⠘⢀⡼⠃
⠀⠀⠀⠀⠀⠈⠛⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠙⠲⢄⣀⠀⠀⠀⠀⠀⠈⠒⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣆⣀⣤⡤⠴⠚⠁

⠀⠀⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣦⢴⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⡿⠟⣛⢻⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⡿⣋⣤⢺⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢙⢿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⡿⢯⢧⣿⢿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⡟⢿⣿⣧⡹⣿⢻⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡹⣯⡄⢷⢸⣦⢻⣿⣿⣿⢸⣿⣿⣿⣿⢟⡓⡴⣿⣽⡜⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣎⠻⣿⣷⡙⣎⢿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣻⣭⣷⡌⣿⠦⣆⣙⣸⣿⣿⣿⢸⣿⣿⣿⡿⢅⣿⣴⢁⡽⣿⣾⣿⣿⡇
⠀⠀⣿⣿⣿⢿⣿⣿⣽⣷⣮⡻⣷⣿⣎⢻⣿⣿⣿⡇⣿⣿⣿⣿⡜⣿⡜⢧⣼⡇⣿⢻⣿⣿⣿⣿⢸⣿⣿⡿⠱⢟⣿⢡⣯⣾⡽⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣧⣾⣿⡟⣉⡹⠷⢿⣿⣿⣧⢹⣿⣿⡇⣿⣿⣿⣿⠹⣮⡻⣦⠹⣇⣿⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⢻⣶⣟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣟⣾⣿⢻⣾⡏⣿⢸⣿⡇⣿⣿⡇⣿⣿⣿⣿⣆⢿⡷⠻⣿⣿⣿⣮⣿⣿⣿⣿⢸⣿⣿⣿⣭⣩⣾⣭⠿⣶⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣏⣛⣼⣿⣿⣾⣿⣿⣿⣧⢿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠷⢌⠲⣬⢽⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣻⣿⣿⢟⣷⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣻⣿⡟⣿⣿⡘⠿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣷⣼⣿⣬⣼⣿⣶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠃
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠃⢹⣛⡛⠛⠛⢛⡿⡛⠿⡻⣿⣿⣿⣿⣿⡇⢸⡭⢛⠋
⠀⠀⣿⣿⣿⣿⣿⣿⠟⠛⠟⠉⠉⠟⠙⠻⠟⠛⠹⠀⠸⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⠀⠀⠃⠈
⠀⠀⣿⣿⢿⢿⣟⡏⢀⠀⠀⡐
⠀⠀⡿⠉⠈
⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣷⣄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⣼⣿⣿⣿⣟⣃⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠀⢀⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣄⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣒⣒⣒⣒⣒⡂⢰⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⠂⢸⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣦⣼⡄⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠩⠭⠭⠤⠤⠬⢭⣭⣭⣉⣛⣛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⣦⣭⣭⣙⣛⡛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠒⠒⠒⠛⠋⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠀⠠

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠃⠀⢀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⢀⠀⢰⡄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣦⠀⠀⠀⠚⠁⡄⠀⠀⢿⡄⠈⣿⠀⠀⠙⢶⠇⠀⠀⢿⣿⣿⣷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠛⠀⠀⠀⠆⣿⠀⠀⣼⣧⡀⣸⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠹⡷⠆⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣴⣶⣄⠀⠀⠀⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠘⣿⣿⣿⣿⣆⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⢹⣿⣿⣿⡅⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢩⣭⣿⣷⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠘⣷⣾⣶⡆⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⠀⠀⣰⠿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⡾⠃⠀⣿⣿⣿⣧⡀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⣿⣿⠃⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠋⠀⠀⠀⠸⣿⣿⣿⣿⣦⣄⣀⣠⣿⣿⣿⣿⣦⣀⠀⠀⣿⠏⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠋⠀⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀⡀⢀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣾⣃⡠⢿⡼⡀⠲⢄⠀⢚⣧⡚⢧⣱⢰⣶⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⣱⣁⣾⣁⣧⣲⠶⢷⡖⠈⠒⣺⡿⠿⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠴⠶⡏⣲⠟⢱⢟⡒⢥⡾⢠⢄⣵⠇⠻⠿⢓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⣤⠾⠷⠒⡴⠛⠕⠋⠐⠢⠖⠢⠴⠾⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⣿⣯⡏⡱⣆⡠⢄⡰⠒⢒⠨⠝⣪⣭⢻⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣇⡏⣇⢀⣾⣶⡟⠛⢰⣧⣀⣀⢩⠟⠁⠉⢺⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⢸⣸⡽⢿⠬⠭⣉⠉⢻⣯⢳⡇⣴⣻⢿⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡲⣋⠴⢷⢺⣐⠤⣀⣭⡂⠙⠿⣶⣏⠪⡃⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣷⠏⢳⣧⢣⠉⠑⢦⡙⡜⢆⠀⣈⣾⣟⢯⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢶⢤⣞⡁⠑⠷⡴⢒⠝⢮⣈⡶⣋⠍⢮⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⣏⣉⣀⣶⣒⠜⠲⣿⡖⠳⡷⣿⠀⠀⡱⡝⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⠛⢩⣽⢯⠉⢠⣏⠀⠘⡇⣾⠖⡯⠝⡛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣞⡹⣶⠛⠐⣿⣥⣤⡸⡄⠀⡇⠇⠘⢹⠠⣤⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⠀⢸⠀⢀⡀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⡇⠀⡀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠸⡆⠸⠄⠼⠀⡾⠁⣤⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣧⠀⠇⠠⠇⢰⠇⢠⡄⠀⣰⠆
⠀⠀⠀⠀⢦⣄⠀⠳⡄⠙⢀⣠⡤⠤⠤⣤⣀⠘⠁⡴⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠘⢦⠈⠁⠀⠀⣀⣀⡀⠀⠀⠋⢠⠞⠁⡀
⠀⠀⠀⠀⢀⣈⠙⠂⢠⡞⢋⣤⠶⠖⠀⠀⠉⠳⣄⠀⠞⢋⣠⡤⠆⠀⠀⠀⠀⠀⠀⣀⡙⠓⠀⢀⣴⠞⣋⣉⣽⣿⣷⣦⡀⠀⠰⠋⣁⣤⠴
⠀⠀⣀⣀⣀⣉⠁⢰⠏⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢋⣁⡀⠀⠀⠀⠀⢀⣀⣀⣈⡙⠀⢀⡾⢡⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠙⣉⣀
⠀⠀⠀⠀⣈⣉⠀⢸⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢉⣉⣁⣀⡀⠀⠀⠀⠀⣉⣉⡁⠀⢸⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢈⣉⣉⣀⣀
⠀⠀⣀⣀⣀⠀⠀⢸⡆⠀⠀⢀⣀⣀⣀⣀⠀⠀⢀⡟⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠘⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⠇⠀⠀⠉⣉⣉⡁
⠀⠀⠀⠀⠉⠛⠲⠶⠿⠶⠛⠉⠁⠀⠀⠉⠛⠲⠾⠷⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠠⠴⠖⠋⠁
⠀⠀⠀⠀⢤⣤⣀⠀⠀⠀⠀⣀⣤⠤⣤⣄⡀⠀⠀⠀⢀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⢀⣀⣤⣤⣄⡀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠉⠛⠓⠚⠋⠁⠀⠀⠀⠈⠙⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁⠀⠀⠈⠙⠛⠓⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠲⠦⣄⡀⢀⣠⠴⠖⠶⣤⣀⢀⣀⡴⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣀⡀⣀⣀⠀⠢⢀⣀⠀⢀⣠⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢂⢠⢊⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡇⢸⠃⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢢⣼⡡⠞⡰⠖⢂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⢹⡀⡜⠁⠀⠘⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⠤⢤⣀⠀⡁⠀⠈⠁⡇⠀⠀⠀⢳⠀⣀⠔⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⢒⡶⠋⠀⠀⠀⠀⠈⠉⠢⡀⠈⠳⣇⠀⢰⠂⠀⠀⠀⢀⡮⠞⢋⠨⠵⠆
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠄⠀⠤⠤⣄⡀⠀⣠⠊⠁⠀⡤⠒⠋⢐⣲⡤⠤⠤⠄⡀⠀⠀⠘⡆⠀⠘⡧⣌⡙⠓⠒⠚⠉⠀⠚⢥
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⠁⠀⠀⠸⠀⢠⠊⠀⢀⠤⠠⢤⠀⠘⠦⡀⠀⠸⠀⠀⠘⣄⣜⠉⠉⠉⠉⠓⢦⡀⠇
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠉⠒⠦⣄⣨⢧⠀⠀⠀⠰⠀⣼⢆⠠⣡⠖⠀⠒⣿⠀⠀⣸⠀⠀⡇⠀⠀⢁⠈⠉⠲⣄⠀⠀⠀⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⣀⠀⠀⣠⠖⠋⠁⠘⡆⠀⠀⠀⠱⠝⣄⠙⠢⠤⠖⠊⣛⡤⠊⡌⢀⡴⠁⠀⠀⢸⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⢑⠞⠀⠀⠀⠀⠀⢳⠀⠀⠀⢠⠆⠈⠢⡀⠀⠠⠀⠉⠀⠀⣧⠊⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠊⠹⡀⢀⡴⠊⠉⠉⠙⢆⠀⠀⡎⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠲⢄⣀⡀⠀⠀⣳⠋⠀⠀⠀⠀⠀⠀⠑⢦⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠴⢊⠝⠋⠁⠀⠀⠀⠀⠀⢠⡏⠁⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⠭⣧⠤⢤⡀⠀⠀⠀⡴⠿⠶⣤⡈⠉⠀⠒⠠⠶⠶⠒⠒⠉⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⣠⡞⢀⡤⠂⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠙⢖⠤⢀⣀⣀⣀⠠⣤⡴⠋⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠉⠙⠳⢦⡀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠒⠒⠒⠒⠚⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠒⠚⢻⠤⠤⣤⡤⠤⠴⠶⠒⠈⠿⠄⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⣀⣀⣘⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠐⠢⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⠶⠋⠉⠀⡴⠛⡏⠉⠓⠲⡖⠒⠒⣶⣏⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢀⠞⠁⠀⢰⡄⠀⠀⠁⠀⠀⣿⠀⠉⠛⠲⠦⠤⣤⣤⠤⠴⠚⠁⠀⠀⠀⠙⠲⢤⣤⣀⣀⣀⠤⠤⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠊⠀⠀⠀⢸⠛⠦⣄⡀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⠀⠉⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⡠⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣁⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣽⣿⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣾⣿⡇⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠸⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠹⣿⠃⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣿⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣿⣿⣿⣿⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⣿⣿⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣿⣦⣿⣿⣿⣿⣧⣿⣷⣶⡆⠀⠀⠀⠀⠀⢀⣀⡀⠀⣿⣿⣿⣿⣿⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣿⣿⣿⡟⣷⡄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣿⣿⣿⠁⠙⢧⡄⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣿⣿⣿⣿⡆⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢧⣼⣿⣿⣿⣧⡼⠿
⠀⠀⠀⠀⠀⢰⣯⣄⣀⣰⣿⣿⣿⣧⢀⣠⣼⡆⠀⠀⣀⣤⠴⣿⣍⣀⢰⣿⣿⣿⣿⣇⣀⣩⣿⠦⣤⣄⠀⠀⠀⢠⣤⣄⡀⣾⣿⣿⣿⣇⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠈⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠁⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⡟⢿⣇
⠀⠀⠀⠀⠀⠀⢀⣰⠿⠀⣿⣿⣿⡁⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡹⣿⣿⣿⣿⢏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣥⣿⣿⣿⣿⣧⣼⡿⠄
⠀⠀⠀⠀⡴⢶⣯⣅⣀⣠⣿⣿⣿⣇⣀⣨⣽⣶⢦⠄⠀⠀⠀⣤⣤⣌⣷⣿⣿⣿⣿⣾⣡⣤⣤⠀⠀⠀⠀⠀⠀⣶⣶⣤⣀⣿⣿⣿⣿⣯⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⣶⠶⢶⠶⡶⣶⠶

⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣬⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣧⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⣾⣿⡀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡄⠀⣿⣿⣧⠀⣿⣿⣧⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢿⣿⠟⠀⢿⡿⠛⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠃⢸⣿⠀⠀⢸⣿⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⡀⠘⣿⡀⠃⠸⣿⠀⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⣿⠀⠘⠀⣿⡇⠀⠀⣿⠀⠁⠀⣿⡀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⡿⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⢠⣷⠀⣿⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣆⠀⢀⡀⠀⣿⣠⣤⠀⣿⡇⠀⠀⣿⣤⣿⡄⣿⡇⢠⣿⣷⡄⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣴⡿⢀⡼⠃⠀⣼⣿⣿⠀⣿⣇⠀⠀⣿⠉⣿⡇⣿⡇⠀⢿⣿⠿⠀⣷⡄⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⣠⡾⠃⠀⣴⡿⠉⣿⠀⣿⣿⣤⡄⣿⠀⣿⡇⢻⡇⠀⢸⣿⡄⠀⠉⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⡾⠟⠁⠀⢠⡿⣧⣰⡟⠀⣿⣿⠈⠀⣿⠀⣿⡇⢸⡇⠀⠀⣿⣇⠀⡀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⡰⠀⠀⠉⠙⡏⠀⠀⠀⢀⣿⢃⣿⣿⣷⠀⣿⣿⠀⠀⣿⠀⣿⣧⢸⣿⠀⠀⢸⣿⠀⢿⡆⠀⣿⡇
⠀⠀⠀⠀⠀⣾⣅⠀⣶⡆⠀⢀⠀⠀⠀⣸⡏⠈⣿⣿⣿⡆⢸⣿⠀⠀⢹⣾⠿⣿⣾⣿⠀⠀⢀⣿⠃⢸⡇⠀⣿⣿⠀⢠⣿⣆
⠀⠀⠀⠀⣶⠉⠻⣷⡀⠀⠀⠘⣧⣤⣼⡿⠁⠀⣿⡇⢻⣧⢾⣿⣶⣾⣿⠟⠀⠘⢿⣿⣶⣿⣿⡟⠀⠀⠃⠀⢸⣿⠀⠸⣿⣿⠷
⠀⠀⠀⣴⣿⠀⡀⠈⠀⢀⣴⡄⠘⠛⠛⠁⠀⠀⣿⡇⠀⠉⢻⣿⠛⠋⣵⣷⠀⠀⠸⣿⠘⠛⠉⠀⢀⡃⠀⠀⢸⣿⠀⠀⢻⣿⡀
⠀⠀⠀⣿⣿⣿⠇⠀⠀⠺⠿⠇⠀⠀⠀⠀⠿⠀⣿⡇⠀⠀⢸⣿⠀⠀⣿⣿⠀⠀⠰⣿⠀⠀⠀⠀⠉⢳⡆⠀⢸⣿⠁⠀⠀⢻⣷⡀
⠀⠀⢰⡏⠉⠁⠀⠚⠀⠀⠀⣀⣀⣀⣤⣄⠀⠀⣿⣇⠀⠀⢸⣿⠀⠀⣿⣿⠀⠀⠠⣿⠆⠀⠀⠀⠙⠛⠁⠀⠈⣿⢄⡄⠀⠀⣿⣧
⠀⠰⡆⠀⠀⠀⠀⢀⣠⣾⣿⣿⢿⣿⣿⣿⣷⣄⣿⣿⠀⠀⠸⠿⠀⠀⢻⣿⠀⣀⣤⣶⣾⣿⡁⠀⠀⣠⣾⣇⣀⣹⣾⣷⣄⣀⣸⣿
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠀⠀⠹⣿⣿⣿⣿⣿⣿⢶⣶⣶⣶⣶⣶⣾⣿⣿⣿⠿⣿⡙⢿⣿⣾⣿⠟⠻⣿⠿⢋⡁⠹⣿⣿⡿⠏
⠀⢸⣷⣤⣾⡿⠟⠋⣀⣀⣀⣀⡀⠀⢻⣿⣿⠀⢸⣿⠈⠉⠉⣟⠉⠉⢰⣿⡀⠀⠀⣿⡇⠀⠉⠉⠀⠀⢀⡀⠀⣿⡇⠀⠀⣤⣄
⠀⠀⡽⠛⠉⠀⠀⠐⠿⠿⠿⣿⡇⠀⠀⡉⠁⠀⢸⣿⠀⠀⠈⣿⡀⠀⠀⣿⡇⠀⠀⢻⣤⣶⣶⣶⣤⣄⡈⠁⠀⢿⣷⠀⠺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠋⠀⣴⣿⣿⡇⠀⢸⣿⠀⠀⢀⣿⡇⠀⠀⣿⡇⠀⠀⠘⡿⠿⣿⣿⠿⣿⣿⣿⣶⣾⣋⣀⠀⠈⠉
⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠹⣿⡄⢸⣿⡀⠻⠉⣿⡇⠀⠀⣿⡏⠰⠏⢸⣷⠀⠀⠀⠀⠀⠈⢉⣹⠿⢿⣿⣿⣷⣶⣶⡆
⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣀⠘⣿⣦⣙⡁⠀⠀⣿⡇⠀⠀⣿⠇⠀⠀⢸⣿⢀⣤⣤⣴⣾⣿⣿⠿⢸⣿⠉⠋⠉⠛⠋
⠀⠘⠃⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⣿⣧⠈⠻⣿⣿⣿⣴⣿⣧⣶⣶⣶⣶⣿⣿⢻⣿⠘⠟⠛⠉⠉⠁⠀⠀⢸⣿⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠉⢻⣿⣤⣬⣿⠇⠀⣿⡏⠉⠈⣿⡏⠈⠉⣭⡉⠉⠀⢸⣿⠀⠀⢠⣴⣶⣿⣷⠀⢸⣿⠀⠀⢹⣿
⠀⠸⣦⣀⣤⣶⣿⠿⠋⠀⢸⡇⠀⠻⢿⡿⠟⠀⠀⢹⡇⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠘⣿⠀⠀⠉⢁⣠⡾⠏⠀⢸⣿⠀⠀⠀⣿
⠀⠀⠙⠿⠿⠛⢁⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⠀⣿⡇⠀⠀⣽⡇⢠⣴⣼⣿⣇⣤⣤⣈⡉⠀⠀⠀⢸⣿⠀⠀⠀⠋
⠀⠀⠀⢀⠀⢰⣿⣿⣶⢾⣿⣿⡦⠀⠀⢾⣿⣶⣾⡿⠁⠛⠋⣿⡇⢠⠀⢹⣿⣿⡟⠙⣿⡟⠛⠿⠿⢿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠈⢧⠀⢹⡟⠁⠀⠈⠉⣠⡄⠀⠀⠻⣿⡏⠀⣀⠀⠀⣿⡇⢸⡷⣼⡿⢻⣿⣦⣿⡇⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⡶⠛⠁⠀⡀⠀⠀⣹⡇⠀⢿⡇⠀⣿⡇⠀⠀⣿⡇⠀⠉⠛⠿⣧⣀⣤⣴⣶⣿⠿⠛⢹⣿⡇
⠀⠀⠀⠀⠀⠀⡆⠀⠉⠁⠀⢠⣤⡾⠟⢀⣴⡿⠁⠀⢸⡇⠀⣿⡇⠀⠀⣿⡇⠀⠀⣠⣶⣿⣿⣿⣿⠉⠁⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠈⣀⣴⣿⠟⠁⠀⠀⣸⠇⠀⣿⡇⠀⠀⠋⢀⣴⣿⡿⢻⣧⢿⣯⣿⡆⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣶⣶⣿⡿⠋⠁⠀⠀⠀⠘⠃⠀⠀⣿⠃⠀⣠⣾⣿⠟⠁⠀⢸⣿⠀⠻⠿⠁⠀⢀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠋⠀⠀⠀⣾⣿⣦⠀⠀⠀⠀⢁⣴⣿⡿⠋⠀⣦⣄⣀⣼⡿⠀⢀⡀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⢀⣠⣶⣿⠟⠁⠀⣀⠀⠻⣿⣿⠟⠁⢴⠿⣿⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⢀⣠⣶⣿⡿⠋⠁⣠⡴⠛⣉⠀⠀⠈⢿⡆⠀⠀⠀⠻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣶⣿⣿⡿⢿⡏⠀⠀⠛⠉⣄⣼⣶⠀⠀⣠⣿⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠉⠀⢸⡇⠀⠀⠀⠀⠈⠋⠁⣤⣾⣿⠏⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠈⣿⣦⣤⣀⣀⣤⣶⣿⣿⠟⠁⣀⡾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠿⣿⣿⣿⠿⠟⠋⢀⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⡿⠿⠿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⡀⢀⣀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣁⣀⡅⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⣅⣀⣸⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠘⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⠀⠀⠀⢀⡠⠣⣀⠀⠀⠀⢠⠧⠤⢤⠀⠀⠀⠀⠀⠀⠀⢈⣛⣛⡀⠀⠀⠀⣀⣴⣷⣤⣀⠀⠀⠀⢈⣛⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠊⠀⠔⠊⠁⠀⠀⠀⠉⠐⠄⠈⡆⠀⢺⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠩⠩⠉⠉⠉⠉⠍⠭⠩⠦⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣂⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠢⡀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢟⣭⣭⣭⣭⣍⠿⣿⣿⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠉⢻⣿⡇⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⡀⠀⡁⠀⡇⠀⡀⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇⣿⣿⡇⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⡿⠛⢲⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠉⠙⠁⠈⠉⠀⠀⠟⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡼⠀⣦⣿⣿⣿⣿⣿⣦⣤⣜⣃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⡴⣦⡀⠀⡀⠀⣀⣼⡿⠁⠀⢀⣠⡾⢫⢿⢿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⣋⣾⣿⣿⣋⣠⠶⠋⣿⣧⣴⣾⠿⠋⢀⡀⣴⠁⣸⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠁⣀⡈⠩⢉⡛⢏⣩⣤⣤⣼⣿⣿⠿⣫⣵⢀⣀⣽⣷⠿⠚⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠚⣻⣿⣿⣿⣷⣤⣹⣿⣯⣳⡾⣾⣿⣿⣿⡿⠛⠋⢀⣀⣤⣤⣠⣤⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⣠⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣻⣶⣶⣶⣤⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠿⢿⣿⣿⣯⣍⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢁⡀⣾⣴⣷⣼⣿⣿⣿⣋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⢙⣉⣄⣿⣾⣿⣾⣿⣿⣿⣿⡿⠿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠻⠟⠛⠿⠟⠛⠉⠈

⠀⠀⠀⠀⠀⠀⡠⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢤
⠀⠀⠀⠀⠀⠀⣇⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⠃
⠀⠀⠀⠀⠀⠀⡟⠛⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⢻
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣚⣁⣈⣑⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⣶⠿⠶⠿⣶⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢰⠿⠶⠾⢶
⠀⠀⠀⠀⠀⠙⡖⠒⡖⠃⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠘⢲⠒⢲⠋
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡏⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠈⡟⢛⠛⡛⠛⡛⠛⡛⢛⠛⢛⠛⣛⢻⡇⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⢠⣤⣷⣤⣷⣤⠀⢀⣀⣇⣸⣀⣿⣀⣇⣀⣇⣸⣀⣸⣀⣿⣸⣇⡀⠀⣤⣼⣤⣼⣤⡄
⠀⠀⠀⠀⠈⡧⠭⠭⠥⣽⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢻⠬⠥⠭⢼⠃
⠀⠀⠀⠀⠀⡇⢸⢹⠀⣿⠀⡇⠀⡴⠚⠓⢦⠀⡴⠒⠓⢢⡀⣠⠒⠓⠲⡄⢸⠀⢸⠀⡏⡇⢸
⠀⠀⠀⠀⡖⠓⠚⠚⠓⠛⠒⠛⠚⠓⠒⠒⠚⠓⠓⠒⠒⠒⠓⠓⠒⠒⠒⠓⠚⠒⠚⠒⠓⠓⠚⢲
⠀⠀⠀⠀⡇⢀⠴⠚⠉⠓⢦⡀⠀⢠⠖⠊⠙⠒⢄⠀⠀⣠⠖⠋⠙⠲⣄⠀⠀⡤⠒⠋⠓⠢⡀⢸
⠀⠀⠀⠀⣧⣮⣀⣀⣀⣀⣀⣧⣠⣇⣀⣀⣀⣀⣈⣇⣰⣇⣀⣀⣀⣀⣸⣄⣼⣀⣀⣀⣀⣀⣹⣼

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

⠰⢿⣿⣿⣿⣿⣿⣏⡉⠈⠑⠌⠉⠙⢿⣿⡷⠶⣄⣀⡀⠉⠉
⠀⠸⣿⣿⣿⡉⣹⢿⢿⣶⣶⣦⠀⠀⠀⠈⠙⠛⠀⠉⠛⠿⣶⣀⣀⡀
⣠⣴⣿⣿⣿⣿⣏⡀⠉⠙⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠦⠄⠀⢀
⡉⣹⣿⡿⣿⣿⠿⠿⣷⣦⡠⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⣄
⣿⣿⣿⡇⠁⠈⠷⢶⡄⠙⠷⣿⣿⣶⣾⣂⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡟⣽⣷⣶⡀⡀⠀⠀⢀
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣧⡉⣡⠀⠀⠀⢀⣤⣠⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣶⣤⣼⡀⠀⠀⡆⢀⣀⣤⣬⣧⣭⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠙⠿⣿⣿⢿⣿⣟⠻⠏⣉⣉⠛⠂⢀⣠⣤⣤⣦⣀⡀⠀⠀⠀⢠⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⡿⡿⣿⡿⣿⣿⡿⠟⠛⠋⠉⠉⠉⠙⠛⠛⠷⡆⠀⠀⠉⠉⠛⠉⣴⣷⣾⣛⠛⠿⠿⠿⠻⠿⠿⠿⢿⣿⠿⢳⡄⢸⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠉⠀⠀⠀⠈⠈⠛⠻⠿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠉⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⣾⣧⣿⣿⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠀⣾⣰⠀⠀⣾⠀⠀⠀⠈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿⣿⣧⣿⣿⣿⣿⣿⣇⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡄⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣾⢰⣤⣷⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣧⣼⣇⣿⣾⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠴⠾⠿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⠻⠟⠉⠉⠛⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠛⣿⣿⣯⣽⣭⣽⣿⣿⣯⣷⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠿⠿⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣴⣤⣄⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿
⣿⣿⣿⣿⣿⣿⣿⡟⠿⢻⣿⠿⣿⠟⠉⠹⠛⠋
⠏⠈⠻⣿⠛⠿⠿⠇⠀⠸⠉⠀⠉

⠀⠀⠀⠀⠀⠀⡞⡒⣄⠀⡀
⠀⠀⠀⠀⠀⠰⡗⠳⡌⠳⣍⠳⣄
⠀⠀⠀⠀⠀⠐⢣⣉⠙⢄⡀⠑⢄⠑⢦⡀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⢀⣼⠠⢧⠀⠙⢆⠘⠳⡄⠛⢶⡏⠨⣇
⠀⠀⠀⠀⠀⠸⠈⣆⠀⠣⠀⠂⡷⡀⠘⣆⠈⢧⡀⠹⡀
⠀⠀⠀⠀⠀⠇⠘⠉⣦⠀⠵⢄⠀⠹⡄⠈⠢⠀⠙⢦⡹⡄
⠀⠀⠀⠀⢸⠀⢀⡅⠘⡆⠀⠙⢦⠀⠙⠀⠀⠀⠀⠀⠈⠊⢦⡀
⠀⠀⠀⠀⠘⣄⠀⠈⢆⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢆⠀⠸⠁⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢧⢹
⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠁⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⡄⢇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢⠵⠊⠉⠉⠙⡶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⠀⠀⠃⢹⣦⣄⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢠⠊⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢧⡀⠀⢘⣡⢿⠟⠢⣀⠀⣠⠋⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⠀⢀⠵⠁⠀⠄⢨⠳⠁⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣿⠦⠴⠒⠁⠀⠀⠘⡠⠃⡇⠠⡀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠀⠀⠀⢀⠤⠊⠀⢀⡸⠦⠙⠃⠈⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠈⠀⠀⠀⠀⢺⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⢁⠐⣿⣿⣿⣤⡄⠀⠀⠀⠁⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠑⡀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⠀⠀⠄⠀⠠⠂⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠄⠀⠀⠀⠐⡄⠀⠀⢻⣿⣿⣿⣿⠇⠘⠀⠠⠀⠠⠂⠀⠐⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠀⠀⠁⢀⠀⠢⠀⠈⠀⠀⠀⠀⠈⢿⣿⣿⠏⠀⠀⠠⠁⡐⢀⠔⠁⠀⠔⠁⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠈⠂⢀⠑⢄⠀⠄⠈⠂⠁⠀⠀⣸⣿⣿⣄⠀⠀⠡⠂⠀⢁⠀⠊⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⡀⠀⡑⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠊⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠩⠀⠀⠈⠀⠀⢀⠀⠠⠄
⠀⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⠁⠀⠀⠀⠤⠈⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣐⠀⠈⠁⠄⠤⠄
⠀⠀⠀⠀⠠⠤⠄⠠⠤⠤⠤⠤⠀⠄⠠⠤⠀⠀⠀⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠒⠂⢀⣼⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⠁⢸⣿⣧⡀⠀⠂⠠⠀⡉⡁⠀⠀⠀⠀⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠁⠀⢀⣴⣿⣿⠟⠁⠀⢠⣿⣿⣿⣿⣿⣯⠀⠈⢿⣿⣷⡄⠁⠂⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣤⣿⡿⠋⠁⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠙⠻⣿⣦⣤⣴⡀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠀⠀⠂⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡻⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠙⠂⠀⠀⠀⠀⣁⣠⠤⠖⠚⠉⠁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⣠⠞⠁⠀⣀⡤⠴⠚⠋⠁⠀⠀⠀⣀⣠⠤⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣦⠀⢰⠀⠈⠀⢀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⠁⠀⠀⠀⢸⣿⢧
⠀⠀⠀⠀⠀⠀⠀⡈⠳⣷⠀⠸⡆⠀⠀⠀⠀⠀⠀⠴⠚⠉⠁⠀⠀⠀⢀⠀⠀⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢡⡀⢸⠦⡀⡇⢀⣀⡀⠒⠀⠀⠀⢀⠀⡄⢀⠀⠀⠘⠳⡌⠀⠀⠀⣿⡆⠀⠀⣄
⠀⠀⠀⠀⠀⠀⣐⣹⡆⠈⠃⢱⣷⣭⠾⣭⣽⣷⡂⠀⠀⣄⠷⣬⠀⠀⠀⠀⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⢔⡡⠛⣿⣯⠵⠚⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡇⠀⠶⣄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠳⠶⠖⠋⠀⠓⠀⠀⠘⠂⡀⠀⠀⠀⠀⣾⡆⠀⠀⠀⣿⣆⠀⠀⠀⢸⣿⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡀⠀⢀⣾⡇⠀⠀⠀⠀⢹⣿⠀⠀⠀⢻⣏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⢀⡹⠓⠇⠀⡾⢹⠃⠀⠀⠀⠀⠈⣿⡄⠀⠀⠘⣿⡄⠀⠀⠀⣿⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡏⠀⠀⠀⢠⡇⣾⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢹⣷⠀⠀⠀⣿⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⠁⣿⡀⠀⠀⠀⠀⢠⢿⣷⡀⠀⠀⠀⢿⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⣠⡟⠀⠹⣷⣤⣀⣠⣴⡟⠀⠻⣿⣶⣶⣶⠟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠙⠻⠿⠟⠋⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠆⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣇⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣄⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢠⣼⣿⣦⠀⠀⠀⠀⠀⠀⠠⣶
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣹⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⣼⣿⠄⠀⠀⠀⠀⠀⢠⣿⣿⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣽⣿⣧⠀⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⣠⣄⢸⣿⣤⡀⠀⣠⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡀⠀⣠⣄⢹⣿⣁⣤⡀
⠀⢰⣿⣿⣿⣿⣿⣿⢰⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣀⣿⣿⣿⣿⣿⣿⣇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠻⠿⠿⠛⠛⠛⠛⠿⠿⠿⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⢻⡌⠉⠛⠲⠦⣤⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⠒⠒⠒⠲⠶⠦⣤⣀⡀⠀⠀⠀⠀⠀⢀⡤⠶⠚⠋⠉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡀
⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠉⠛⠶⣤⣀
⠀⠈⠛⠶⢤⣄⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠲⠶⠶⢤⣤⣀⣀
⠀⠀⣀⣤⠴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⢤⣄⡀
⠴⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠖⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠲⠶⠦⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠳
⠀⠀⠀⠀⠀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠦⣤⡀
⣤⣤⣤⣤⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣦⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢄⡑⢤⡁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠱⣄⠈⠓⢦⡀⠐⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠢⠀⠑⠤⡀⠀⠣⣄⠈⠑⢬⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢦⡀⠐⢄⠀⠹⣦⠀⠙⠦⡀⠀⠳⡄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢼⠳⡀⠀⠣⠀⠈⠳⣄⠀⠁⢠⠀⠙⠢⡀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠿⣆⠀⠘⢄⠀⠈⠃⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢢⣿⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠑⠸⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠉⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣔⡄⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡸⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣼⣦⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣍⢛⡵⠊⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠉⢻⡟⢲⢤⡀⠀⠀⠀⠀⠀⠖⢡⡞⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣤⣤⡀⠁⠀⠋⠳⣦⣠⠌⢀⣴⠏⠀⡿⢋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠌⠛⠦⡀⣠⣾⠟⣡⣴⠿⠛⠀⣰⢁⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣆⠀⠀⠀⠀⡿⢣⢞⣿⣃⣤⡶⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⠌⠑⠀⠉⣸⣡⣟⣼⣿⣿⣿⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⣿⣿⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠐⠀⢀⡴⢠⡀⠀⠰⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢠⠀⠀⠀⠏⠀⠀⠉⢠⠀⡀⢰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⠀⠀⠀⡃⢿⡀⠀⠀⠀⡄⠀⠀⠀⡄⢡⠀⠀⠀⠀⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠐⠤⢄⣀⠁⠌⠁⠀⠀⠀⠓⠂⠀⠁⠙⠚⠟⠒⠀⣀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⡀⢸⡀⠂⠀⠀⠀⢀⣀⠀⠀⠀⠔⠀⠀⢀⡠⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠙⠛⠀⢀⡜⠊⠀⠀⠀⠀⠀⠠⠴⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⠠⠐⠈⢱⠄⠀⠀⠀⠈⠀⠀⢀⣀⣈⣲⡒⠀⠶⠓⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⢀⠟⠢⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣤⠄⠀⠀⠀⠀⠐⠉⠁⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠐⠂⠛⠁⠀⠀⠀⠀⠀⠓⠒⠛⠓⠲⠶⡒⡟⠀⠈⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠛⠒⠒⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠻⡀⠎⠀⠀⠀⠀⠀⠉⠉⠓⠐⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⡀⠀⠐⠂⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢠⣾⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣀⣀⣼⠁⢠⡀⠢⠶⣠⡁⠀⡀⣤⣀⣶⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠙⢿⠀⢸⡇⠀⠀⠁⡈⡀⠑⢛⣿⣿⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣧⡒⠂⠛⠢⣀⠀⠀⠀⠣⠀⡍⢩⠟⡀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠠⡺⡄⠸⡄⠀⠹⣿⣷⡀⠈⠉⠀⠀⠀⠀⠠⣀⣀⠀⠁⠀⠀⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣽⣦⡀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠿⠟⠃⠀⢀⣼⠿⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠄⠈⠙⠻⣽⡟⢿⣾⣷⣦⠀⠀⠈⠀⠀⣀⡤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠢⣀⠀⠉⠻⣿⣯⣄⡀⠀⠀⠤⠾⣿⡟⠀⣠⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠢⣄⠈⠙⣿⢿⣶⣦⣤⣾⠋⠴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡀⠀⠘⢿⠿⠛⠁⠚⠛⠉⠉⠁⠀⣔⡴⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⡴⣿⢿⢿⡿⣿⣿⡿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠸⣇⣿⢿⣸⡿⣿⣧⡿⣿⣸⢿⡇
⠀⠀⠀⢀⣀⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⡀⠀⠀⢀⣀⣀⣙⣿⣿⣿⣿⣿⣾⣿⣷⣿⣟⣀⣀⡀
⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⠟⠉⠻⣿⠁⠀⠈⢻⡿⠋⠙⢿⡇⠀⠀⠀⠀⢸⣿⠟⠛⠻⣿⠉⠀⠈⣻⡿⠋⠻⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⡆⠀⠀⣿⣧⠀⠀⣾⡇⠀⠀⠀⠀⢸⣿⡄⠀⢰⣿⡆⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠇⠀⠀⣿⡿⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⡇⠀⢸⣿⠇⠀⠀⣿⡇⠀⠀⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⣶⠶⣦⡀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣆⠀⠀⠴⠟⣉⠉⠉⠃⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠈⠳⡶⢶⣤⣤⠅⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡀⠀⠀⠀⠀⠀⠀⠈⢀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠈⢳⣤⣤⣤⣤⠄⠀⠈⢀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⣙⣷⣿⠁⣠⡴⠶⠋⠉⣉⣯⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠉⠻⣅⣾⣥⡴⠂⠀⢉⣉⣤⣤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⡘⣿⡏⠀⠐⡛⠉⠁⠀⢠⣶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣶⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⢄⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠈⠀⠀⢠⡟⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡖⣠⠞⠀⣠⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣈⠻⣦⣤⣤⣦⣴⣤⡶⠋⠀⣸⣟⣁⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣯⣭⣤⣴⣦⣀⣠⣾⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣲⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⣿⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⢣⣿⢵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⠀⠈⠱⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡔⠁⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⠏⠀⠀⣠⢰⣿⣏⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡻⠀⢀⣼⢣⣿⡿⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣧⡀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣗⣅⣠⣿⣟⣿⣿⣵⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣄⠀⢈⠻⣶⡀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣿⣿⣟⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠟⠛⠳⢄⠀⠀⠀⠈⡇⠀⠀⠀⠘⢻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⢠⡇⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢸⡇⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣐⠀⣀⠀⠀⠀⠀⠀⠀⠘⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⠛⢿⣿⣦⣿⣤⣤⣀⠀⠀⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣆⠈⠉⠙⠻⢿⣿⣿⣷⡄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣭⣄⡀⠀⠀⠀⠀⠉⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡿⣥⣄⣤⣀⠀⠀⠀⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⠚⠛⠂⠼⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣶⠀⠀⣸⣷⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠘⠇⠀⠀⠀⠘⣃⣤⠶⠞⠋⠉⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣠⠞⠁⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⣀⣤⠀⠸⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣤⠶⠛⠉⠀⠀⣀⣤⠶⠛⠉⠀⢀⡴⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠂⣈⠀⠑⢀⡀⠈⠀⠀⠀⣤⠶⠛⠉⠀⠀⢀⣤⠶⠋⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⢀⡆⢡⠂⢀⠀⢇⠻⠈⠠⢀⣦⡀⠀⠀⠀⡀⣀⢠⠀⠀⠀⠀⣰⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠈⢄⢦⡀⠘⠈⡳⢍⣃⣐⣛⡚⢻⣷⡄⠀⠀⣧⣹⣼⡇⠀⠀⠀⣿⣯⢙⣿⣿
⠀⠀⠀⠀⠀⠸⠥⠦⢽⡿⠿⠯⠥⠴⠿⠷⢶⣆⠹⣿⡀⠀⠈⠁⠈⠀⠀⠀⣀⣼⣿⠿⠛⠁⠀⠘⢳⡜⠃
⠀⠀⠀⠀⠀⢸⡅⢁⣸⣿⠒⠲⠶⠶⠖⢚⣭⠊⠀⢹⣧⠀⠀⠀⠀⢀⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠲⠖⠇⠘⠟⠠⢌⣃⣁⢤⣤⣨⠅⠀⠀⣿⡀⠀⢀⣴⣟⣉⣀⣠⣤⣤⣤⣴⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠲⠥⠚⠷⠀⠂⠐⠂⠒⠁⠐⠂⠀⠀⢸⣇⠠⣿⠿⠿⠿⠿⠛⢛⣛⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡜⣿⡄⠀⠀⢀⣤⣶⣾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠶⠂⠀⠀⢀⣠⡞⢀⠘⣿⣄⠀⣾⣏⠉⠀⠀⠀⠰⢶⡜⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⡀⠀⠀⣀⣤⣶⡿⠋⠀⠿⠇⠘⢿⣿⣿⣿⣿⣷⣶⣤⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠟⠋⠁⠀⠀⠘⣆⠀⠀⠀⠈⠙⠻⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⢠⠋⠑⡄
⠀⠀⠀⠦⡜⠉⢾⢁⠦⣀
⠀⠀⣤⠀⡹⢆⠀⠳⣄⠈⠳⣄
⠀⠘⠙⠇⡀⠈⢷⡀⠈⢦⡀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣀
⠀⢀⣀⡧⠼⡄⠀⠳⡀⠀⠈⢦⠀⢤⡳⣄⠀⣀⣴⠁⢎
⠀⡞⠉⢇⠀⢻⡀⠀⠙⣆⠈⠂⢳⡀⠀⠈⢿⡁⡄⠀⠀⡄
⢠⢧⡀⠾⡄⠀⢳⡀⠲⡼⢇⠀⠀⢻⣄⠀⠈⢷⡇⠀⢿⡃
⢈⠀⣿⠂⠹⡄⠠⡽⣆⠀⠀⣧⠀⠀⠻⣦⠀⠘⣧⠀⠘⣇
⡄⠀⣿⠰⡄⢹⡀⠁⠸⣄⠀⠈⠳⠄⠀⠀⠀⠀⠹⣷⡄⠹⡄
⢧⠐⠟⢳⢰⡀⢇⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢻
⠈⢧⠀⠈⢧⢳⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠈⣇⠀⠈⠋⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⣿⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⢻⠀⠀⠀⠀⢼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⢸⡆⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣳⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠈⣸⣿⣦⣄⣀⣀⣀⣀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⢴⣿⣿⣿⣧⣠⣤⠶⣄⡀
⠀⠀⠀⢀⡼⢿⣄⠀⠀⠀⠁⢻⣿⣿⣿⣿⡟⠁⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⠟⠁⠀⠙⢿⣦⡀
⠀⠀⣠⠞⠀⢀⡿⠻⠃⠀⠀⠀⠉⠁⠀⣨⣿⣆⠀⣠⣿⣿⠃⠀⠀⠀⠀⠀⣱⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿
⠀⣰⠋⠀⠀⣼⣇⠀⠀⠀⠀⢀⣠⡴⢫⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⢀⣠⠔⠒⠈
⢠⡟⠀⠀⠀⠹⣿⣷⣶⣶⣾⡿⠏⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⣠⣴⠟⠁
⢸⡇⠀⠀⠀⠀⠈⠛⠛⢛⣩⣴⠞⠸⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⣠⣾⡟⠁
⣼⠇⣀⡀⣶⣶⣶⣿⣿⣿⠟⠁⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⢀⣼⡿⠋
⣿⣎⠻⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⠀⣠⣿⣿⠇⢀⣴⣿⠟⠁
⠙⢿⣷⣤⡉⠉⠀⠀⠀⣀⡀⣀⡀⢀⣾⡿⢿⣿⠇⠀⢀⣀⣤⣾⣿⣿⣿⣾⡿⠟⠁
⠀⠀⠙⢿⣟⣃⣀⣠⣿⣿⣿⡿⠿⠛⠁⠀⠈⠛⢶⣬⣿⣿⣿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣬⣅⡀⠀⢸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠉⢁⣠⣤⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣿⠿⠛⠛⠻⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠙⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⢠⡀⢸⡆⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢾⣤⡼⢷⣶⠃⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠹⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⠙⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⡄⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⣾⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠈⠁⠀⠀⠀⠀⢻⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠁⠐⠀⠁⠀⠀⠠⠀⠀⠀⠈⡟⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠤⠐⠉⠀⠳⠜⠀⡒⠒⢂⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠈⠀⠀⠐⠁⠀⠂⠀⣀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣅⠀⠀⣷⠀⠐⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠇⠀⠻⠿⠄⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢀⠄⡺⠁⠀⠀⠀⠀⠀⠀⢀⢶⡀⠀⠀⠀⣰⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⣀⣤⠾⠓⠁⡠⠔⢢⠀⠀⠀⠀⠀⠸⠀⢇⠀⠀⠀⠈⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⢠⣞⢶⠀⠘⠀⠀⠀⠀⠀⠀⢃⠘⡀⠀⠀⠀⠹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⡞⡏⠘⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⢁⠀⠀⠀⠀⠹⣿⣷⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⣼⣿⠀⠀⠀⣸⣿⠁⢢⠰⠀⠀⠀⠀⠀⠀⢀⠆⠀⠘⣄⠀⠀⠀⠀⠘⣿⡇⠘⠷⠖⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠀⠀⠀⢹⣿⣧⣀⣴⣿⡏⠀⠸⡄⠳⠄⠀⠀⠠⠐⠁⡰⠉⠢⣽⣷⣶⣶⣶⣾⣿⠇⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⢻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠋⠀⠀⠀⠙⠲⠤⣄⣀⠤⠔⠊⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠈⢻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠓⠀⠀⠀⢠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣍⠁⠀⠀⠀⠀⠹⣄⠉⠒⠤⣀⠀⠀⠀⣦⠀⠰⠘⣧⠀⠀⠀⡠⢞⡽⠋⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⡀⠀⠀⠀⠀⠑⠠⡈⠣⠁⠀⠀⠻⡶⠤⠷⠟⢀⠀⢊⠔⠋⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠶⣄⣀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⠲⢄⠀⠀⠀⠀⠈⠀⠀⠁⠢⠀⠀⠀⢀⠔⢁⠔⠁⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⡀
⠀⠀⢠⣏⣤⠤⠄⠀⠀⢀⣶⠀⠀⠈⠓⢄⠁⠢⡀⠀⠀⠀⠀⠁⠠⠀⠀⠀⢁⠐⠁⠀⠀⠀⠀⢀⡴⣻⠟⠁⠀⣀⡘⠛⠛⠉⠉⠉⠟⢭⡝⠛⠙
⣤⠆⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢀⡀⠀⠉⠢⠀⠑⠄⡀⠀⠀⠀⠈⡀⠄⠀⠑⠄⠀⠀⢀⡠⢊⠞⠁⠀⠀⡦⠀⠉⠓⠢⢤⡀⠁⠀⠈⣀
⠙⠷⠤⠄⠀⠀⠀⠀⠀⡔⠀⢀⠔⠋⠀⠀⠀⠀⠀⠑⠄⠈⠀⠊⡠⠊⠀⠀⠀⠀⢄⠈⠒⢉⠔⠁⠀⠀⠀⠐⠃⠘⠲⣄⠀⠀⠈⠑⡄⠀⢹⡀
⠀⠀⠀⠀⠀⣠⣄⠀⠸⠣⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠤⢄⠀⠢⡀⠀⠀⡀⠔⡡⠲⣌⢢⡀⠀⠀⠀⠀⠀⢠⣀⡈⠀⠀⠀⢸⡼⠶⠛
⠀⠀⠀⠀⠀⠘⠿⣍⣁⣀⣿⠇⠀⠀⠀⠀⠀⢀⡴⢊⠔⠁⠀⠀⠀⠢⣀⠑⠊⡠⠊⠀⠀⠈⠳⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣰⣋⡔⠁⠀⠀⡤⢲⠀⢀⢎⡴⢄⡱⡄⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⢀⣼⡠⠃⢠⣧⠋⠀⠀⢱⣿⠀⠀⣤⠦⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠛⠋⠀⠀⢸⣿⣤⣤⣴⣿⡿⠀⠀⠙⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀⠔⠉