Land Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡧
⡀⠀⠀⠀⣤⣎⡀⠀⠀⠀⢺⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠙⠙⠀⠀⠀⠀⠀⠀⢀⡤⠾⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⠀⠀⠠⠠⠀⠒⠀⠀⠨⠟⠓⠒⠊⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠠⠄⢀
⠐⢤⡐⠨⢔⡂⠄⣀⡀⠠⣀⠠⠄⢄⡀
⠣⣄⠉⠲⣄⡉⠓⠤⣍⠙⠲⠤⣁⠓⠠⢭⡐⠲⠭⣐⣒⠠⠄⣀⣀
⠀⠈⠳⣄⠈⠙⢦⣀⠈⠙⠶⣄⡀⠉⠳⢦⣄⡉⠓⠠⢬⣉⠛⠒⠮⣍⣙⠲⠤⣤⣄⣀
⢷⠀⠀⠙⢶⣄⠀⠙⠣⣄⠀⠈⠙⠷⣤⡀⠈⠙⠳⢦⣀⣈⠉⠒⠦⣤⣈⡉⠛⠶⢤⣍⣛⠒⠦⢤⣀⣀⡀
⠘⢷⣄⠀⠀⠻⣦⡀⠀⠈⠳⣄⠀⠀⠈⠛⢷⣄⠀⠀⠈⠙⠿⢤⣤⡀⠈⠛⠷⢦⣀⡀⠙⠓⠦⢤⣄⡈⠙⠛⠶⣤⣄
⠀⠈⢿⡃⠀⠀⠈⢷⡄⠀⠀⠈⢻⣄⠀⠀⠀⠙⠿⣦⡄⠀⠀⠀⠙⠻⣶⣄⡀⠀⠈⠛⠶⣦⣀⠀⠈⠙⠳⣦⣀⡀⠉⠙⠳⢦⣤⣀
⡄⠀⠘⢷⣄⠀⠀⠀⢻⣇⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠉⠻⣦⣄⡀⠀⠈⠉⠳⢶⣀⡀⠀⠉⠛⠷⣤⡀⠀⠈⠙⠳⢶⣄⣀
⣇⠀⠀⠘⣿⡄⠀⠀⠀⠹⣧⡀⠀⠀⠀⠙⣡⠄⠀⠀⠀⠈⠙⢿⣇⠀⠀⠀⠀⠀⠙⠻⣶⣄⠀⠀⠀⠉⠛⢶⣤⣀⡀⠈⠙⠳⢷⣦⣄⠀⠈⠛⠳

⠀⡀⢀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢀⣴⣶⡀⠀⠀⠀⠀⠀⢀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⣴⣶⡄
⣿⣷⣾⣿⣿⣷⣶⣿⣷⣿⣿⣿⣼⣿⣿⣷⣾⣷⣾⣿⣿⣷⣤⣥⣤⣾⣿⣿⣾⣿⣿⣷⣶⣾⣿⣾⣿⣿⣶⣿⣿⣧⣾⣿⣾⣿⣿⣷⣤⣼⣬⣼⣗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⠀⢠⣾⣿⣿⣧⠀⠀⠀⡀⠀⠀⢀⣀⠀⢀⡀⠀⣴⣷⣤⡄⠀⠀⢀⡀⠀⣤⣿⣿⣿⡄⠀⠀⠀⡀⠀⠀⠀⡀⠀⢀⣠⣴⣶⡄
⠰⠄⢴⣷⣿⣧⣼⣿⣷⣿⣿⣶⣴⣦⣿⣿⣧⣿⠿⡿⢿⣷⣰⣿⣟⣿⣿⣃⣶⣿⣷⣤⣯⣿⣽⣿⣿⣤⣤⣨⣁⣈⣣⣿⣿⢄⣿⣿⣟⣿⣷⡀⣀
⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣶⡀
⠀⣤⣤⢀⣠⡀⢰⡆⠀⠀⠀⣼⣿⠿⢿⣿⢿⡄⢀⣴⣄⣀⠀⣤⣼⣿⣿⣿⣆⠀⣰⣦⣀⣤⣶⣄⢠⣤⣀⣶⠀⡀⠀⢰⣿⠿⠿⣿⠿⡇⠀⣴⣄
⠈⠻⢿⣇⣤⣤⣤⣧⣤⣶⣾⣿⣿⣿⣶⣿⣶⣶⣶⣿⣿⣿⣇⣛⣻⣟⣭⣽⣿⣤⣻⣿⣛⣛⣛⣿⣶⣿⣿⣷⣶⣿⣾⣿⣿⣿⣿⣿⣷⣿⣶⣯⣭
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠉⠛⠋⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠋⠋⠉⠉⠉⠛⠛⠋⠋⠉⠉⠛⠛⠛⠛⠛⠛⠛

⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⣾⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣦⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣶⣾⣿⣿⣿⡀
⣿⣿⠿⣿⣿⣷⡀⠀⠀⠀⠀⣶⠀⣀⣄⠀⢠⣤⣦⣀⠀⣠⣦⠀⠀⢠⣿⣿⣿⣿⣿⣠⡄⠀⠀⢀⣠⣦⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷
⣾⣷⣶⣷⣿⣿⣶⣤⣾⣦⣄⣿⣽⣿⣿⣡⣿⣿⡿⣿⣿⣿⣿⣷⠀⣿⣿⠿⠟⢿⣿⠿⡷⢰⣿⣿⣿⣿⣧⣤⣭⣭⣿⣥⣴⣦⣿⣿⣤⣄⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣶⣼⣷⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠿⡿⠻⠿⢿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣾⣿⣷⣮⡹⣧⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⢿⣿⣿⠿⢣⡟⠀⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠈⠙⠛⠚⠋⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⢰⡆⠀⠀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣹⡏⢻⣏⢿⣦⡀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣰⣿⣇⣼⣿⣎⣿⡿⣀⣴⣶⣄⢠⣾⢏⣿⢹⣏⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣼⢋⣿⢹⡝⢷⣝⢿⣿⣼⣿⣦⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⣧⣾⣿⣼⣿⣼⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣿⡿⢛⣭⣍⠻⢿⡿⢛⣩⣍⡸⢿⡿⢟⣩⣍⡻⢿⡿⢟⣩⣍⡻⢷⡝⢛⣩⣍⡛⢿⡿⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣾⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣬⣛⣋⣴⣿⣷⣬⣙⣋⣵⣾⣷⣮⣙⣋⣵⣾⣷⣮⣙⣋⣵⣾⣷⣮⣙⣋⣵⣾⣷⣦⣙⣋⣥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠻⣿⡿⢛⣭⣍⠻⢿⡿⢟⣩⣍⡻⢿⡿⢟⣩⣍⡻⢿⡿⢟⣩⣍⡻⢿⡿⢟⣩⣭⡛⢿⡿⠟⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣾⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣷⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠈⠙⠋⠁⠀⠀⠈⠙⠋⠁⠀⠀⠈⠙⠋⠁⠀⠀⠈⠙⠋⠁⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⣠⠀⠀⣀⠀⢸⣇
⠀⠀⢠⡆⢠⣿⡇⢀⣿⡄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣴⡀⣼⡀⣀
⠀⠀⣼⣿⢸⣿⣇⢸⣿⡇⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡇⣿⣧⣿
⠀⠀⢿⣿⠘⣿⠟⠈⣿⠃⢹⡏⢀⡀⠀⠠⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣸⣇⣽⠀⣻⣁⡟
⠀⠀⢈⡇⠠⠘⠒⣒⣉⣩⣤⣤⣶⣶⣶⠿⠿⠿⠛⢛⣋⣉⣭⣭⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤
⢾⣿⣶⣶⣶⣾⠽⣿⣿⣿⣛⡛⠀⣀⣠⠶⢶⡿⠟⠛⠋⢉⣉⣭⣴⣶⡾⠿⠟⣛⣫⣭⣷⣾⠿⢛⣛⣭⣿⡿⠿
⠈⣿⣿⣿⣿⣛⠻⢶⣾⣭⣛⠻⢷⣶⣬⡙⠛⠶⣦⣤⡛⠻⠛⣉⣡⣠⢔⣾⣯⡭⣭⣭⠽⣠⣤⠍⣩⡽⠓⣾⠇
⠀⠘⢿⣿⣯⣙⠻⢷⣦⣉⠛⠿⣦⣌⡉⠛⢷⣦⣄⠉⠻⢿⣦⣈⠉⠰⡿⠋⣡⡾⠛⢁⣾⠟⢁⣼⠟⠁⣼⠏
⠀⠀⠈⢷⣭⡛⢿⣦⣌⠙⠿⣦⣄⠙⠻⢶⣄⡈⠛⢿⣶⣄⠉⠻⢷⣦⡀⠘⠏⠀⣴⡿⠁⢠⣾⠏⠀⡼⠃
⠀⠀⠀⠀⠙⢿⣷⣌⠛⢿⣦⡈⠛⢿⣦⡀⠙⠿⣶⣄⠈⠻⢷⣦⡀⠙⠻⣷⣄⠙⢿⠀⢠⣿⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢻⣷⣄⠙⢿⣷⣄⠈⠻⣷⣄⠀⠙⢿⣦⡀⠈⠻⣷⣆⠑⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⡀⠉⢻⣦⡀⠈⢿⣧⡄⠈⠹⣷⣄⠀⠙⢿⣦⡀⠀⠙⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢆⠀⠙⢿⣦⠀⠈⠻⣷⠄⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠑⠂⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣄⣀⣠⣤⣷⡀⣀⣬⣴⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣴⣶⣺⢶⣿⣯⣍⡽⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⠄⠀⠀⠀⠀⠀⡆⠀⣄
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠀⠀⠀⠀⠀⠀⠀⠀⢇⠰⡆⢄⠀⠋⣈⠑⠀⣙⠶⠀⠀⠁⣀⠠⠄⠘⠂⠲⠄
⠀⠰⠀⠀⠀⠀⠘⠇⠀⠀⠀⠆⠇⠀⠀⠈⠂⠀⠀⠀⠈⠀⠑⡠⠤⠀⠈⠂⠀⠀⠀⠀⠛⣰⠀⠀⣴⣶⣾⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⡀⠀⠀⠀⠀⠸⣧⢤⡤⠤⠄
⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠛⠛⠛⠃
⠀⠘⠫⣀⠀⣠⣠⡝⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠤⠄
⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢦⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⢶⡄⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⢀⠀⣤⣼⡄⠀⣤⢀⡄⢠⡄⠀⢠⢸⡇⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⣿⣠⣇⣿⢷⣧⣿⣿⣯⣿⣿⣇⣿⣿⣿⣷⣿⣶⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣆⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠣⣼⡥
⢘⣿⣿⣿⣿⣿⣿⡿⠿⡿⠿⠟⠉⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠰⠦⣄⣼⡴⠞
⠀⠀⠀⠈⠉⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠈⠉⣐⣢⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠈⠓⠖⠭⣽⣗⣊
⠀⠀⣼⣦⢀⣰⡋⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣽⡶⠈⠷⢏⡉⣻⣟⣠
⠐⣼⣿⡿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢍⠰⢾⡗⣋⡀⢈⡉⢻⣟⣡
⠀⣺⣿⣿⣿⣿⣿⣿⣤⣤⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠠⠀⢨⣉⣿⡯⡥⠄⠈⣙⣿⣿⣁
⠀⢿⣿⣿⣿⣿⠯⠞⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⣦⣿⣶⣂⠐⠖⢿⣛⠐⠂⠴⣻⡿⣿⣏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⡶⢶⣦⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⣠⣶⠿⠉⠈⠋⠀⠀⣿⣆⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢁⣴⣾⠋⠛⠋⠈⣿⣆⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⢀⣿⠆⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢸⣧⣀⣀⣀⣀⣀⣨⣿⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠁⠀⠀⣀⣀⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠉⠉⠉⠉⠛⠷⣦⡀⠀⠀⠀⠀⢀⣴⠿⣷⡄⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⢀⣾⠋⠀⠈⢿⣦⣤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠉⠉⢙⡛⠿⢶⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣶⣿⠃⠀⠀⠀⠀⢿⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣴⡟⢿⣆⠀⠈⠙⠻⣶⣄⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⢿⣆⠀⠀⠀⢀⣾⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢀⣾⠏⠀⠀⠻⣧⠀⠀⠀⠀⠙⠿⣦⣄⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠙⠿⣶⡾⠛⠁⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡛⠛⣿⡏⠀⠀⠀⠀⢹⡧⠀⠀⠀⠀⠀⠈⠙⠿⣧⣄⣀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣷⠶⠾⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠹⣧⡀⠀⠀⢀⣾⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣶⣶⠿⠛⠉⠁⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠉⠛⢿⡿⠛⠁⠀⠀⠉⠛⢷⣦⡀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⢢⠀⡦⢢⠀⡔⢺⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠄⠫⠸⠀⠇⠘⠀⠣⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⢀⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⡄⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⣻⡇⢸⠠⡇⠀⢀⣧⠀⠀⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⡇⢠⠀⢠⣄⣿⣽⣿⠀⠀⠀⠀⠀⠀⢰⡆⢸⡇⢠⠀⠀⠀⢤⣿⣼⣷⣾⣿⣷⣶⢫⣿⣰⣿⣾⣽⡏⢷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣼⣿⠨⡀⢼⣷⣿⣿⣿⡇⠀⠀⠈⣵⣶⣾⣧⣾⣷⣾⠆⠐⠒⠞⢿⡿⠋⠉⢻⣿⣿⢨⡍⣷⡿⣷⡾⢣⡎⢻⣇
⠀⠀⠀⠀⠀⠀⠀⡀⢧⣴⣿⣿⣿⣿⣦⡆⠸⣿⣿⣿⠻⠁⠀⠀⠀⢹⠛⣿⣿⣿⣿⠷⠀⣀⣀⡀⣰⠃⠀⠀⠀⢹⡶⢞⣿⣧⣬⣿⠇⣵⣦⠘⣻⠄
⠀⠀⠀⠀⡄⣠⣤⣿⣿⣿⣿⣿⣿⠯⣾⠁⠀⢹⣿⡏⠈⢢⠇⡀⣀⡞⠀⠸⣿⣿⠃⠀⠀⢩⡉⠙⢿⢀⠠⠤⡄⠉⢹⣾⠋⣷⣈⣷⣾⣛⡿⡃⠛
⠀⠀⠀⢰⣿⣿⣿⣽⣿⣿⣿⢟⣿⣰⡟⠀⠀⠀⣿⠂⠁⢸⡟⢿⣟⠀⠀⠁⢿⣿⣦⣀⣀⣄⢀⡖⠉⠀⠀⠀⠈⠀⠻⣿⣄⡿⣿⡯⠘⣿⡄⣍⠃
⠀⠀⢠⣿⣿⣿⠿⠟⠛⠉⠉⠁⠈⠉⠀⠀⠀⠠⠏⠐⠤⠄⢿⢾⠁⠐⠂⢈⡀⠙⠉⠉⠙⠟⠿⠄⠀⠀⠶⢾⡷⠶⣾⣉⣿⢿⣞⠡⠘⢻⡉⠙⡄
⠀⠀⠛⠋⠁⠠⡶⢀⠀⠀⠔⢺⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠠⠄⠐⠞⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠓⠚⠓⠂⠚⠻⠷⢴⠄⠐⢓⣒⠾⣒⣶⡀⠵⢤⡀⠀⠀⠤⣀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠠⢤⡀⠐⠢⢤⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠚⠃⠒⠒⠛⠛⠷⠖⠒⠾⠧⠀⢀⣠⣀⣀⢉⡛⣋⣉⣳⡔⠈⢙⣆⣀⣈⣷⠤
⠀⠠⠀⠰⠆⠀⣀⣤⣤⠤⠀⠀⢠⣀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠈⠉⠛⠻⢻⢾⣿⣷⣼⣿⣠⣀⠸⣿⡄⣠⢠⣞⣆⣦⢀⢹⣇⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣦⣼⣿⣿⣯⣿⣾⣿⣼⣽⣹⣆⣿⡆⠀⠀⢀⡀⠀⠀⠀⣿⣾⡄⢡⢸⣆⠀⠀⠀⢘⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠛⢿⣿⣥⣤⣼⣿⣿⢿⡇⠀⢰⠸⣿⢷⣴⡄⣿⣿⣇⠈⡏⣿⠀⠀⠀⡇⣟⡄⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠿⠇⠉⢭⣍⠙⢭⡀⠔⠀⣋⣿⣿⣿⣿⣿⣿⡤⠷⠛⠀⣀⠀⣿⣻⣿⣄⣽⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠒⠒⠂⠀⠀⠀⠀⢳⣶⠶⡒⢬⡿⠿⠶⠶⣶⣿⠁⠀⠛⢛⣿⣤⣤⠤⠿⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠐⠀⠀⠁⠀⠀⠈⠐⠛⠛⠛⠛⠛⠛⠞⠙⠚⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣈⠧⢵⣶⣤⣠⢄⡀⠀⣀⡀⢀⣤⣴⣾⣶⣄⠀⠀⠤⠤⠶⠤⢤⡤⠤⠤⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠈⣈⣛⠋⠉⢱⣼⣿⣿⣿⣟⣻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⢄⡹⠿⣿⠿⢿⠿⠟⡛⡉⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠑⠲⠦⠶⣤⡝⠃⠲⢀⣛⣸⣣⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠁⠀⠀⠀⠉⠹⡇⡚⠤⠤⣿⠿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠀⠠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠈⢦⠁⢈⣀⠀⠄⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠁⠈⠀⠀⠀⠂⢨⠀⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣬⠐⠀⠐⠀⠠⣢⢠⢁⠀⠀⠢⢠⡾⠀⠉⠠⠿⠇⠄⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠾⢮⡁⠁⠈⠉⠹⣘⠉⣀⡄⠀⠀⠆⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠐⠁⠐⠋⠓⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⡇⢀⠶⣾⣤⣤⡿⡦⠀⡀⣾⣷⣴
⠀⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠉⠉⠀⠀⠁⠘⠚⠀⠀⠀⠀⠙⠀⠈⠁⠈⠋
⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠂
⠀⠀⠀⠂
⣀⣀⣀⣐⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣐⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡆⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠉⠀⢡⡀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⢫⠁⠀⣘⠶⣷⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠏⢀⡾⠾⠁⣠⣿⠿⣮⡓⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⣴⠻⢀⣠⠞⡽⠋⠀⡾⣿⢸⠟⣧⢻⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⢉⠻⠁⣼⡇⢸⠁⠀⠀⢹⡏⣠⡆⢸⣃⡁⣠⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢠⣌⠈⢳⡶⢼⣧⠀⠀⢰⡀⠘⠟⠉⢿⣏⠉⣧⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢠⠚⢠⢏⠄⢻⣇⠄⠹⠄⢀⣾⠛⣶⡄⠀⢀⡟⠼⢻⡇⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⣴⣧⡟⢷⣼⣮⠙⡆⣠⡟⠁⠀⣿⠇⠀⣿⢵⡆⢸⡇⣿⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⢸⡏⣼⣷⡀⠈⠻⠀⣤⣿⡄⠀⢨⣿⠻⣤⡀⠈⣿⣾⣥⠈⢹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢲⠧⣿⡈⠁⢰⡀⢸⠁⢈⠿⠀⠀⠹⡄⠙⡗⠠⠋⢰⡏⣼⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡈⢳⣌⠻⣄⢶⡇⡿⠆⠸⡄⠀⠀⠀⢁⡼⣇⠀⠀⡾⣿⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢮⡻⣄⠈⠛⢦⡙⠄⠰⡛⠀⠀⠀⠈⠀⠁⠠⠞⢛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠈⢻⣶⣥⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡻⠟⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⢢⡤⠔⠋⠈⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣤⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣊⣥⣤⡦⠞⠑⠂⠀⠀⠀⠰⠿⣶⣿⢷⠿⠿⣿⠋⠛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⢀⡤⠒⠚⠉⠁⠀⢠⣤⣶⣿⠷⢶⣿⣶⣶⡴⠶⠚⠋⠛⠒⠶⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣦⣶⣿⣟⡋⢙⣻⡁⣾⢛⣿⣿⢻⣿⣿⣴⣶⣦⣶⡷⡯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠙⠛⠻⠟⠛⠉⠳⠷⠞⠛⠈⠛⠃⠀⠀⠈⠉⠙⠛⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠚⡦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⣄⡀⠷⠀⠀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠉⠀⢺⡆⣴⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⡄⢂⢠⣦⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠠⠂⢀⡿⠀⠀⠀⡘⠃⠈⠀⢻⡆⢀⡡⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠁⣇⢺⡀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠈⠁⣀⠈⠀⠴⠁⠀⠀⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠄⠀⠠⠻⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠁⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠈⠀⢠⡟⠀⠀⠀⠀⠀⠀⣤⡴⣣⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠓⠄⠀⠀⢂⡴⠉⣴⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡷⠇⢀⣠⠏⢡⡎⢶⣄⠀⠀⢙⢦⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣴⣿⣿⣣⣷⣿⠕⣠⣻⣿⣄⢹⣄⠀⡼⢤⣙⠶⣤⣰⡟⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣿⠉⣿⣿⣿⣯⣾⣻⠥⣎⠉⠹⠥⢈⡽⠿⣦⡀⣀⣽⡷⠶⢄⣨⣯⣡⡀⠑
⠀⠀⠀⠀⠀⠀⠘⠛⢻⣋⣡⣌⣉⣙⠛⠿⠤⠿⠿⣠⠤⠾⠷⠦⠼⠿⠻⣉⡐⣢⡏⠥⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠉⠓⠒⠀⠶⠶⠶⠖⠒⠒⠌⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔⠀⢀⠀⠀⢰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠂⣸⠱⡄⠈⠈⢳⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠂⠀⢦⢄⡀⠂⠄⠀⣤⣀⠀⣠⠗⠊⢺⠀⠀⠌⡢⡈⠀⠀⡰⡅⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⣇⠉⢦⠀⠀⢸⠈⣶⠇⠐⠃⢸⠀⠊⣠⠥⢹⠀⠀⠁⢃⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡧⣒⠀⠀⣼⠀⠈⢇⠀⢸⠁⠆⠌⠀⠆⢸⠀⡰⠁⠀⢀⠁⣰⢀⠌⠀⢱
⠀⠀⠀⠀⠀⠀⠈⠀⠱⣌⢉⣳⣿⠂⢀⡈⠢⣜⠘⢄⠀⠀⠠⠈⢶⢃⠀⠀⣜⡴⡽⠀⢀⢦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⢧⡀⠤⢀⡈⠂⢈⠀⠀⠀⠀⠀⠡⠀⠀⠀⠐⠟⡞⢁⠔⠉⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⢱⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠈⡤⢰⡔⠁⠔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢵⣸⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠰⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢠⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠁
⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⢀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢉⣝⠁⠀⢀⡤⡤⢄
⠀⠀⠀⠀⠀⢻⣡⣀⣠⣠⡤⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠤⠶⠲⠶⠶⣿⣿
⠀⠀⠀⠀⢀⡏⠉⠉⠀⢀⠀⢀⠖⡎⠙⢦⡀⠠⣴⠅⠂⠺⡦⣷⡄⠀⠈⠉⠩⠉⠒⢦⡄⠀⠀⣸⢉⣸⡆⠀⠀⠀⡀
⠀⠀⠐⠒⠤⠆⢾⡆⠀⣉⣨⣿⡀⠁⠀⡂⡅⢀⡷⠀⠀⢵⠚⠉⢷⡐⠷⣤⣀⡂⠀⣾⡷⠛⠛⡿⢎⡼⠁⠁⠀⠁⠈
⠀⠀⠀⠀⢀⡘⠶⠓⠛⠉⠉⠉⢻⣀⣢⣴⡧⠛⢛⣉⠠⠛⣢⡒⠛⢦⣰⠀⠹⡁⠿⡿⠷⠤⣤⣶⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠹⠷⠶⠿⠁⠀⠀⠈⠙⠲⠆⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂⠀⠒⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⢄⣀⣀⠉⠁⠀⠀⠀⠀⠀⣈⣉⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⠦⣈⠛⢦⠀⠀⢀⠴⣶⠶⢤⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠁⡎⣸⣿⣿⣆⠙⣿⣞⡾⣥⣿⣍⣉⣙⣒⠲⢤⣤⣄⢂⡤⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⡆⢫⠙⣿⣟⠛⢋⢿⣝⢿⣿⣶⣌⠙⢷⣶⣖⠚⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⣹⢿⣿⣷⠘⢧⡀⠛⠃⣸⡼⣯⡼⡟⢻⣿⣿⣦⣮⡻⣿⣃⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣱⣾⣃⣏⣻⣿⣇⢀⣹⣧⠀⢡⠄⡏⠃⣿⠇⠟⢉⣿⣿⣿⣮⡻⣤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠴⠶⠶⠦⢬⡻⠿⢿⣶⡶⠚⡈⠀⣼⣿⡿⠄⠀⠉⢹⣿⣷⣽⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣤⣒⠢⡀⢀⡀⢄⠒⠤⠄⡴⢺⣿⠟⠋⠀⢀⣴⢻⡿⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣌⡑⠛⠦⠕⠒⠊⠀⠀⠈⠓⠂⠒⢫⡍⣨⢁⣾⢻⣿⣯⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣷⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢈⡏⣼⢿⣿⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡞⢆⠰⣤⣄⣀⠀⠀⠀⠀⠀⣉⣾⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢧⡀⠳⣜⠿⣮⠙⢶⣄⠀⠀⢈⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠋⣻⣷⣦⡈⣆⣼⣿⣷⣬⡛⠿⣄⢼⣿⣷⡄⠈⣺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠿⢿⣿⣿⢿⣿⣿⣿⣾⡿⡟⠿⣹⣷⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⡀⢀⡀⠀⠀⠛⣦⣹⣿⣷⠻⣿⣿⡈⠁⠉⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢶⣷⠀⠀⣿⣿⣿⣿⣯⠈⢉⣟⣿⡶⠒⢹⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⢀⣮⣿⠀⣷⣿⣿⣿⢿⡟⢰⠿⣫⣟⣡⣤⣼⣶⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⠏⣼⣿⢇⣿⡟⣼⡠⠴⣺⢿⣻⣿⣿⣿⠻⣿⣟⠀⠙⠆
⠀⠀⠀⠐⢶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⡶⠶⠀⠀⠀⣠⣴⣶⣿⣿⡿⠋⣰⡿⢋⣾⠟⡼⠉⠀⢘⣵⢿⣿⣿⠿⠏⠙⠈⠛⣿⡄
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⢂⣼⡿⣛⣽⣾⣿⠁⡈⠉⣴⡿⠃⠈⠀⠀⣰⠟⣵⠿⠟⠉⠀⠀⠀⠀⠘⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠫⣾⣿⠿⠋⠛⠛⠁⠸⠋⠀⠀⠀⠀⠈⠁⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠈⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⠊⡀⠐⠒⠀⠘⠓⠂⣀⡀⠀⢬⡇⠀⠀⠀⠀⢠⠀⢰⡏⠁⢀⢰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠉⠀⠆⠴⠀⠀⠀⣤⣦⢼⡇⠀⠀⠀⠀⢼⠀⢸⣟⠓⠈⢾⡄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⢀⠀⠀⢸⡇⣼⣼⡆⠀⢸⣇
⠀⠀⠀⠃⡇⢸⡀⡄⢀⣆⣸⠀⠀⠀⠀⢺⣧⡀⠀⠀⠀⠀⠂⠀⠀⠘⢿⢿⣿⡐⠂⠈⠀⠀⢸⢢⣿⣿⣷⠀⢸⡟⠁
⠀⠀⠀⠀⢰⠸⠛⠃⣾⡿⢿⠆⠀⠀⡷⢸⠇⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⢠⠈⠻⠀⠈⠀⠀⢸⣈⣿⣿⣿⣶⢸⡷⠀⢠
⠀⢀⠀⠀⣸⣄⠀⠀⠻⣷⣿⠆⠀⠒⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣸⣷⢀⣿⣷⢶
⠀⣿⢃⢆⠀⠙⠀⠀⣀⠛⠛⠂⠀⣀⠄⠀⢀⣀⡀⢰⠀⠀⠠⡆⠀⠀⠀⣿⡗⠛⠃⠙⠒⠒⠒⠶⢾⣷⣶⣿⣿⣿⠆⢁⠀⣀
⠀⠉⠹⠟⣭⣴⡀⢈⣻⣦⣀⣀⡐⠋⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⢃⣠⣤⣄⠀⠀⡀⠐⠚⠿⠿⠿⠋⠁⠈⣉⣥⣶
⠀⠀⠀⠀⠈⠉⠁⠈⠉⠛⢿⢆⢈⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠁⠈⠁⠀⠀⠠⠶⡄⠀⣈⣄⢀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣤⣄⠈⠁⢠⣴⣶⣄⠀⠀⠀⠀⠀⣰⣲⣀⡀⠀⠀⠀⠙⠈⣛⣟⣓⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⢛⣛⠁⠀⠈⠉⠁⢧⡀⠀⠀⠀⠀⠻⠟⠋⢩⣍⠀⠀⠀⠀⠈⠛⠹⣿⣷⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣡⠀⠀⠀⢠⡆⠀⣻⣿⡆⠀⠀⠀⠉⠙⣿⣶⣄⡀⠀⠀⠀⠀⠙⢯⣭⣼⣦⠀⠀⠀⠀⠀⠉⣰⡶
⠀⠀⠀⠀⠀⢀⠀⣪⣶⠏⠀⠀⠀⣠⣿⠇⠐⣿⣿⣇⠀⠀⠀⠀⠀⡤⠻⣿⣿⣄⠀⠀⠀⠀⠀⠋⠉⢷⣼⣶⣦⡀⠀⠀⠀⠺
⠀⠀⢀⠤⣒⣵⣾⠟⠁⠀⠀⢀⣞⣿⣟⠀⠀⢸⣿⡞⠇⠀⠀⠀⠀⠀⠀⣤⣿⣿⣷⡄⠀⠀⠀⠀⠀⠸⠿⡟⢋⡑⠄
⠀⠀⢠⡾⠟⠋⠀⠀⠀⠠⠀⠜⣿⣿⡏⠀⢐⣾⣿⡧⢃⠀⠀⠀⠀⠀⠀⡻⡿⢿⡿⠿⠀⠀⠀⠀⠀⠀⠈⠟⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⢴⣟⣿⠁⠀⢸⣾⣿⡟⣈⠀⠀⠀⠀⠀⠀⢧⣿⣟⢰⠀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠇⠹⠺⠃⠀⢦⠙⠇⢻⡴⠿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⠘

⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢠⣔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢷⣀⠀⡀⠀⠀⠀⠀⠤⢖
⢈⡇⡀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⢠⣔⠀⠆⠋⠡⡀⠈⠘⠠⠀⠀⠀⠀⠀⠈⠁
⡎⣧⢻⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠙⠀⠀⠈⠀⠰⠀⠀⠄
⢠⠏⣄⠠⡀⡄⠠⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠤⠤⠀⠀⠀⠀⢠⣤⣮⡔⠀⡀⠀⠀⡀⠆⠀⠀⠀⠂⠃⠀⠰⠂
⠤⠄⠛⠓⠛⠁⠈⠀⠀⠀⠀⠈⠉⠛⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢧⣶⣄⣀⣠⣤⣞⠀⠀⠀⠀⠐⠡
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⢁⣰⡦⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡿⠿⠿⠟⠛⠂⠈⠀⠠
⠐⠁⠀⠀⠀⠀⣁⣀⣠⣤⣤⣶⠤⠭⠶⠶⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠤
⢀⣠⣴⣶⣾⡿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣛⠉⠀⠠⠀⠀⠀⡀
⣿⣿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣶⣦⣤⣀⠀⠀⡀
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣶⣤⣄⣀⣀⣤⡴⣤⣶⣷⣤⣦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣄⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣻⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣻⣾⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣻⣾⣿⣿⡿⠿⠿⢃⣀⣀⣀⣀⣤⠴⠶⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⢟⣩⣴⣶⣶⣾⣿⡿⣻⣿⡟⠡⡜⢿⣼⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣤⣀⠸⣿⣿⠟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⢷⢹⢸⡿⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣮⡉⣸⣿⡿⣿⣿⣿⣿⣿⣿⡟⠛⠀⡟⠀⡟⡾⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣇⢹⣿⡇⣿⣿⣧⣈⣉⡙⢿⣷⣶⣤⣀⡀⠀⢙⣦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣛⣛⣻⣽⡔⢮⣅⣈⣛⣛⣫⣥⣾⣷⣮⣝⠻⢿⣿⡿⠿⠛⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠿⡿⣦⣝⡛⠿⠟⠛⢋⣛⣿⣿⡿⠿⠿⠟⠁⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠈⠉⠉⢉⣴⠟⠉⠉⠀⠙⠃⠀⠀⠀⠐⠳⣶⣶⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣼⡿⠿⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣿⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⣤⣤⢀⣤⣤⣤⣤⢠⣤⡄⣤⣤⡤⣤⣤⢠⣤⣄⢠⣤⡄⢀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡌⠁⢸⣿⠘⢿⣷⣬⣉⠀⣿⣦⣼⣿⠀⣿⡇⢸⣿⣿⣦⣿⠀⣿⡏⢈⣛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⣾⣿⠼⡿⢶⣿⠿⢰⣿⡆⢼⣿⠤⣿⣧⢼⣿⠌⠿⣿⠀⠻⣿⡶⢿⡏

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⢤
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣿⠟⠟⠋
⠀⠀⠀⠀⠀⠀⢸⣇⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠰⣿⠀⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⣠⣤⢤⠤⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠦⠤⠤⠤⠤⢤⣤⣤⣤⢄⡀⠀⢂⣡⡼⠟⠋⠀⠀⠀⢀⠄⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡴⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠖⠛⠛⠿⠿⠿⢿⣧⣤⣼⣷⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣀⠀⠑⠲⠂⠤⠰⠇⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠋⠉
⠀⡀⠀⠀⠀⠙⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣶⡶⠶⠹⢯⣿⣿⣟⠉⠁
⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠄⠀⠀⠈⠿⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣁⣠⣀⠀⠀⠀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⡁⠀⠀⠈⠛⠛⠻⣿⡿⡿⠞⣭⣤⣾⣄⣸⣶⡶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠁⠈⢻⠀⠀⠀⣤⡀⠀⠀⢸⠋⠀⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢎⣦⡴⠋⠀⠀⣰⣿⣷⡄⠀⠈⠳⣤⣎⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠁⠀⠀⢀⣼⣿⣿⣿⣿⡄⠀⠀⠀⠁⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⢻⣿⣷⣿⡄⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠈⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠍⠙⠚⠻⢦⣄⠀⢀⣠⠾⠟⢊⠭⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⣅⡀⠀⠄⠐⠩⢝⠲⠥⣀⣤⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠉⡙⠓⣶⣤⣄⡀⠀⢀⠀⠈⣙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣷⣾⡿⠟⠛⠉⠙⠻⠶⣭⣵⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢒⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠉⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡿⢤⡀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢼⣆⣀⠀⣀⣤⣶⣿⣟⣛⣿⣿⣿⣿⣿⡟⢿⣷⣶⡅⠀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⣼⣷⡘⢷⣽⡿⠿⠿⣛⡋⡉⠛⠛⠻⣾⣿⠏⣴⣿⣶⣄⠀⠀⠓⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣞⠉⠀⣴⣿⣿⣿⡿⠈⠁⠀⠀⠀⠀⠀⠈⣉⣉⣉⣡⣄⣜⣷⣿⣿⣿⣷⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡁⢈⣧⣾⣷⣾⣿⡯⣄⡀⠀⠀⣠⠴⡒⡋⠉⠉⠀⠈⠠⠀⠀⣉⣻⣿⠿⢿⡿⢿⡿⣿⡿⢟⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢍⣉⣿⣿⡿⠋⠀⠀⠉⣽⠟⠉⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠀⠀⠞⠐⣛⠛⠈⢀⡼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⢠⠀⣱⠟⠇⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣅⠀⠠⣄⡈⣿⣦⣾⣥⣇⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢫⠀⣒⠿⡈⠠⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⣛⠶⣄⠈⠁⠉⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢦⣄⡠⠀⠀⠀⣤⠀⠀⣰⡦⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡷⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⢖⣛⣿⡦⠖⢻⣿⣿⣿⠦⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡿⠉⣹⠃⠀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣄⠀⠀⠀⠀⠀⡀⢰⢖⡓⢲⠶⠚⠁⠀⢰⠇⣰⠏⢇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⣿⣿⣿⣷⣀⠀⡀⡠⠚⠙⡟⡇⡿⠀⠀⠀⠀⢸⢞⣅⣴⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢻⣿⣿⣿⣷⣾⣄⡜⡇⢀⣿⠇⠠⣀⠀⣀⡄⠀⢻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⡱⢄⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣆⠀⢹⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡐⠀⡄⡌⠙⢻⣶⣯⣿⠿⢿⣿⡿⠿⠿⣿⣿⣿⠿⠛⡿⠲⠁⠀⡠⡀⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢳⠮⠥⡘⠃⠀⠀⠀⠙⢿⣷⣆⠈⠚⢿⣿⣿⣿⡿⠁⠀⡰⠃⠀⠀⠀⠈⠛⣀⠭⢶⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⣤⠀⠀⠀⠀⠙⢻⡄⠀⠈⠹⣿⣿⠀⠐⡏⠀⠀⠀⠀⢀⡔⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣥⠀⠀⠀⠀⠀⠀⠿⠒⠦⣤⣿⢟⠁⠀⣧⠀⠀⠀⠀⢤⡁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠻⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡾⣇⠒⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢀⡄⡄⢠⢀⡀⡀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡙⢳⣶⡿⢾⠷⠦⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣼⣾⣧⣿⣾⣾⣧⣝⣯⢼⣤⣸
⠀⣴⣦⢺⣦⣀⢠⣤⣿⣦⣤⣬⣿⣿⢿⡿⣿⠱⣤⡠⠀⣤⠀⣄⡀⢠⣄⣰⣾⣿⣿⣿⡷⣿⡿⢸⣿⣿⠷⣿⡟⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣼⣿⣿⣿⣾⣻⣾⣺⣷⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣅⣿⣟⢻⣿⣿⡓⣿⡷⣿⣿
⢻⣿⣿⠟⡀⣀⣙⣿⣭⣭⣭⣽⢻⣿⡿⢿⣿⣿⠿⢿⣿⣿⢿⡿⡿⣿⣿⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢨⣴⣶⠿⠟⢛⣙⠿⠶⠀⠲⠾⠟⠋⠁⣚⢟⣠⢞⡵⠁⡇⢹⡙⢦⡑⠬⣓⠮⢽⣛⠯⠬⣝⣛⡛⠿⠿⢶⣿⣶⣾⣿
⠨⠶⣒⣫⠭⠀⣀⠠⠄⠀⠀⢀⣠⠶⣋⡵⠛⢠⡾⠀⢰⡗⠈⢳⡌⠻⣪⡈⠙⢦⣌⡙⠳⠶⣭⣙⡛⠛⢶⣾⣭⣭⣉
⢘⣩⡵⠒⠚⠉⠀⠀⣀⡤⠒⣩⡴⠞⠁⠀⣠⠟⠀⠀⡾⣶⠀⠘⣇⠀⢈⢿⡶⣄⠈⠙⣷⣦⣀⠙⠿⠴⣦⣄⣌⡙⠒
⠈⠁⠀⠀⢀⣠⡴⠚⢁⡴⢚⠍⠀⠀⣀⣾⠃⠀⠀⢸⡷⢻⠀⠀⢹⡅⢀⠀⠙⢻⣵⢤⣀⠈⠛⢿⣆⠀⡤⠽⠳⢦⣄
⠀⣀⣴⡾⠛⢉⣤⣾⠏⠈⠀⠀⢠⡴⣻⢧⡆⠀⠀⢼⠂⣻⢐⡀⠀⢻⣜⢧⡀⠀⠹⣦⡿⢤⡀⠀⠉⢛⣶⣶⡤⣔⣚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢠⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⢠⠀⠀⠸⠀⠀⠀⠀⠀⠀⢿⣿⣤⠀⠀⣠⣾⣤⠀⠀⡀⠀⠀⠀⠀⠀⠆⠈
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⣀⠀⠀⣸⡿⠛⠁⠀⠀⠁⠀⢶⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠂⠁
⠺⠷⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠤⠄⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠒⢺⣷⣶⣦⡀⠀⣐⠀⠉⠁⠒⠂⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⣄⡀⠀⠀⠈⠻⢻⣿⣿⣿⣿⣿⢶⠀⠀⠀⠀⠀⠀⠀⠘⠖⠖⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠑⣷⣿⣛⢟⣿⣮⠁⠀⠀⠙⠜⢿⣿⣿⣾⠀⠀⢠⣤⣄⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠁⠙⠛⣿⣿⣾⠟⠒⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠾⢿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⢀⣿⣿⣯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠻⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐




⠀⠀⠀⠀⠀⣀⡠⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢐⠒⠂⠉⠂⠀⠀⠉⠀⠀⠀⣀⣀⣤⣤⡤⠤⠄⠤⠤⠄⠄
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⣤⣀⣠⣶⣤⣀⣀⡀⠀⠀⠀⠀⢒⣀⠀⣀⣠⣶⣶⣶⣄⣤⣽⣤⣖⣂⣐⣚⣍⣉⣉⣉⣴⣶⣶⣦⣶⣦⣤⣤⣀
⠀⠀⠀⠞⠿⠿⠛⠛⠛⠛⠛⠛⠛⣛⡛⠉⠛⠙⡉⠉⠽⠛⠛⠛⠍⠛⠋⠉⠛⢉⣉⠉⠹⠛⠋⠉⠉⠉⡉⠙⠉⠉⠩⠉⠉⢉⠩⠂⢋⠨⠸⠏⠉
⠀⠀⠀⠀⠋⠈⠀⠈⠀⠀⡀⠀⢀⡤⠀⠄⠄⠀⠀⠀⠀⠀⠀⢀⡁⢠⠀⠀⠀⠠⠐⠀⠀⡉⡈⢀⠀⠤⠦⣂⠀⢤⡀⠠⠁⠀⣀⡰⢯⠁
⠀⠀⠀⠀⠂⠀⠈⠉⠀⠀⠁⠀⠀⣀⠀⠀⢤⠄⠀⣄⠀⠀⠀⠉⢀⡀⠠⡁⠠⠔⠀⠄⠀⠈⢁⡀⣠⠄⠰⠄⠂⠀⠀⢠⡀⠲⡟
⠀⠀⠀⠀⠀⣰⡄⠀⡀⠮⠀⠀⠄⠉⠁⠀⢀⡀⠀⣤⡄⠀⠲⠀⠈⠁⠀⠁⢀⡀⠀⢴⠆⠀⠛⠠⠘⠀⠀⠀⣀⣄⠈⢷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢾⠀⠀⠸⠇⠀⠈⠀⠀⢀⡀⠀⣴⠀⠈⠻⠁⠀⠰⠂⠀⠀⠀⢀⣀⠀⢐⣯⡍⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠄⠀⠀⠀⠀⠀⢀⡀⠄⠐⢾⠂⠀⠸⠀⠒⠁⠀⠀⠀⠀⠀⠀⠐⠂⡦⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠁⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠔⠁⠀⠀⠀⠘⡄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠣⡀⠀⠀⠀⠀⠀⠘⢆⠀⠣⡀⠀⠰⠀⠇⠀⠂⠀⡁⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢀⠜⢠⡀⠠⠀⠘⠀⠈⢣⠀⠈⠢⣄⠀⠀⠀⠀⠀⠣⡈⠢⡀⠀⠃⢠⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣆⠈⠢⠀⠀⠀⠀⠀⢹⠀⢡⠀⠀⠀⠈⠁⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣸⠻⣆⠀⠀⠀⠀⠀⢀⠎⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠑⠄⠀⠀⠠⣤⣀⠀⠀⠀⢀⣴⠏⡪⢚⣇⠀⠀⠀⠀⢈⠀⠃⠀⠀⠀⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠳⡄⢠⢛⠅⠋⠈⠡⢺⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡰⢀⠎⠀⠊⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢠⠀⠀⠀⠻⡳⣄⠀⠀⢸⡗⠹⣗⠅⡠⣾⠀⡪⠼⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⠀⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢇⠘⣄⠀⠀⠀⢻⠝⠆⠀⢰⡫⠂⠁⡜⠁⠺⠈⠕⡃⠀⠀⣸⡄⠀⠀⢀⣾⡇⠀⢠⠀⠳⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⢢⠀⠀⡖⠐⣺⡄⢸⢗⠁⡼⠀⠀⠸⡀⠪⠅⠀⢠⠖⡹⡄⠀⣸⣻⠁⢀⢎⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⡰⠁⢰⡫⢊⡀⢷⣼⡻⢀⠃⠀⡀⠀⢣⠘⣧⠀⡾⡊⣣⢾⠀⡿⣺⣴⢟⠅⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡛⠔⡬⠱⡄⢙⡞⢸⠀⠀⢀⠀⠀⢣⠘⣶⡋⢰⡀⠹⢠⡋⠞⢑⡡⢊⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⠁⡆⠀⠈⠳⣌⢸⠀⠀⠀⡠⠀⠈⣇⠹⠃⣼⡀⣽⡿⢋⡄⡩⣪⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⡵⠀⡃⠀⠀⠀⠈⢾⡄⠀⢀⠔⠁⠀⢩⢈⠞⡸⢸⡿⣠⢺⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢸⡵⣿⡄⣇⠀⠀⠂⠀⠈⠃⠀⠀⡫⠂⠀⢸⡏⠀⣆⢀⡔⠁⢸⠨⣧⢔⡽⠛⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢦⡼⠆⠀⠐⡡⠀⠀⠀⠠⢖⠄⠀⠈⠁⡆⠈⠁⠠⠂⢸⠐⢱⠎⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⢀⣀⣀⣀⣉⣲⣌⠦⠀⠈⠔⡡⠀⠀⠀⠁⡁⠀⠀⡬⠂⠀⠀⠚⠀⡼⢸⡇⣀⣀⡀⠀⠀⢀⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣔⣍⢈⡶⢳⡲⡶⠈⢣⠀⢀⢀⡠⢄⣶⠤⡀⠀⢀⢀⠈⠀⠐⣲⢖⡺⣅⣞⣙⠀⠀⢳⠀⠠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⠒⣀⢹⢟⠕⠜⡰⣁⠄⡏⡐⠕⠇⣉⣘⣆⡤⠣⢪⢀⢀⡧⡜⣈⣸⡣⠡⠣⢢⢸⡤⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⢋⡹⠷⣤⡷⢾⡿⠾⢛⠋⠝⢛⠳⢤⣲⠈⠡⢂⡿⠐⡛⠿⣚⠾⣕⣒⣹⡟⠿⠻⠟⠛⠿⠊⠕⠂⠝⠂⠭⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠉⠀⠒⠁⠠⠛⠉⠵⠊⠁⠀⠉⠔⠊⠽⠋⠡⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠔⢡⠔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠎⡐⢁⠞⡠⣊⢔⡁⢒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣰⣘⣃⣃⣊⣐⣡⣊⣊⠕⡰⠲⠕⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⣴⣲⠟⢹⣿⡿⡪⠋⠛⣽⣽⡝⡟⠹⣷⣷⣯⣀⣠⡾⠪⣂⢐⢌⢂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⢀⣤⠔⠛⠿⠱⠑⡀⡢⠙⣿⡇⡄⠜⣾⣿⣟⣄⣜⣿⣿⣻⡟⠊⠻⣿⣷⡶⢵⣯⡇⠀⠀⠀⠀⠀⠀⣠⣶⣫⡂
⢀⣠⠖⣁⠣⠖⠇⠀⠀⠀⠀⠍⠉⠎⢅⢄⠌⢜⣿⣞⢀⠈⠜⣿⣽⡟⠔⠴⣽⣽⣟⣔⢜⣿⣿⢁⣦⣿⣷⣘⡮⣿⣥⠂⢀
⠫⣁⠑⢬⡁⣧⢠⠆⠀⠂⡙⠂⠃⣀⢂⡎⠊⠊⠻⠏⠔⠐⡠⣻⣿⠏⡊⡀⣾⡿⣫⠎⠨⠛⠋⠉⣉⢉⢩⣉⡡⣿⣆⠡
⢀⠈⣢⢤⠳⡈⠜⣠⠤⠈⠀⡠⢠⡾⠛⠇⠀⠀⡐⠄⡸⡈⢀⡿⠋⠈⢁⠁⠙⠈⢄⢠⣢⣥⢾⣉⠀⠀⠀⠀⠙⠯⣍⠒⡠
⠀⠀⠈⠁⠉⠐⠲⠤⢤⡄⠀⠐⠰⡋⠅⠲⡀⠄⡠⢑⠀⠀⠎⡈⣖⣱⣕⡤⠴⠚⠻⠛⠈⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢺⣷⠖⣚⣒⠛⠒⠓⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣲⡽⡟⢌⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⠉⠹⣷⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⣿⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢸⡇⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠁⢸⣿⠀⠉⣀⣀⠀⢀⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣄⠀⢀⣀⡈⠁⢸⣿⠀⠉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢷⣆⢀⣶⠟⢻⡿⠀⣾⣯⣤⣤⣄⣤⣤⣤⣤⣤⣤⣄⣤⣬⣿⠂⠸⣿⠛⢷⣆⢀⣶⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⣶⠿⠻⢷⣾⣛⣁⠀⠈⠉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⣉⠉⠉⠀⢀⣙⣿⣶⠿⠸⢷⣾⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⢷⣆⢀⣾⠟⣻⣿⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠹⣿⡛⢷⣆⢀⣶⠟⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⡶⠿⠹⠷⣾⣟⣁⠀⠀⠀⠀⠻⣧⣄⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⢀⣙⣿⡶⠿⠹⠷⣾⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡛⢿⣆⢠⣾⠟⡻⠋⣠⣴⠶⠿⢦⡈⠙⠻⠿⠿⠛⠋⣠⠾⠷⣶⣤⡈⠻⡛⢿⣦⢀⣾⠟⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠟⠹⠷⠾⢁⣾⠟⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢿⡄⠹⠶⠟⠙⠷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⢸⡏⠀⠀⣀⣀⣼⣧⣀⣀⣀⣀⣀⣠⣿⣄⣀⡀⠀⠘⣿⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣾⣿⣶⡄⢸⡇⠀⠀⣿⡏⢹⡏⠉⠉⠉⠉⠉⠉⣿⠉⣿⡇⠀⠀⣿⠀⣴⣾⣿⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣴⣾⡇⢸⡇⠀⠀⣿⡇⢈⡁⠀⠀⠀⠀⠀⠀⣉⠀⣿⡇⠀⠀⣿⠀⣿⣶⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣟⣉⣈⡀⢸⣧⣀⠀⣿⡇⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⣿⡇⢀⣠⣿⠀⣈⣈⣹⣿⣧⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉







⠐⠀⠂

⠀⠀⠀⠀⠀⠀⣀⡶⠋⠛⠶⣦⠴⠒⠶⡄
⢄⡀⢀⡴⠒⣤⡏⠁⠀⠀⠀⠁⠀⠀⣠⠇
⠀⠈⠻⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⠖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠃⠀⠀⠀⠁⠘⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢀⣀⣤⣤⣄⣀⣠⣴⠞⠁
⢀⣀⣀⣀⣴⡶⢦⠶⠶⣄⣀⣀⡀⢀⠀⠀⡀⢠⡤⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡙⠛⠛⢿⡻⠙⣿⣿⣋⡔⠚⢲⣄⠀⠀⠀⢀⣠⢤⣀
⣻⣿⡛⠋⠀⠹⠈⣶⣤⣤⠟⣛⣛⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠀⠀⠹⠆⠠⣤⣧⠀⠏⠉⠀⠀⠀⠈⠸⠓⠶⠶⠞⠀⠀⠈
⠙⢧⣬⣉⣛⠁⠀⠟⠋⣠⠞⢉⣉⣉⣉⣉⡙⠋⠉⠀⠀⣀⣛⣻⣶⡦⠀⢴⣶⣛⡒⠒⠒⠒⠚⠒⠒⠲⡟⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⢹⡄⠀⠀⢰⠁⢀⣀⡠⠤⣶⣶⣿⣶⡶⠿⠟⠛⠋⠉⠁⢀⡀⠤⠄⠙⠛⠀⠀⠀⠤⠶⠛⠛⠓⠲⠞⠏⠉
⠀⠀⠀⠀⠈⡇⠀⠀⠘⠋⢉⣴⣾⠿⠛⠋⠉⠀⠀⠀⠀⣤⠂
⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀
⠀⠀⠀⠀⠀⡏⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢥⣒⡠⠄⣀⡀
⣴⣶⣶⡤⠴⠇⠀⠀⠀⠿⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠯⢭⣒⡦⢤⣀⡂⠠⠄⣀
⠉⠀⣠⣟⣡⡤⠤⠤⠒⠒⠒⠉⠉⠑⠨⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠯⢽⣒⡦⢍⣑⠒⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠯⣗⣢⢌⡙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢥⣂⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠉⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠤⠶⠶⠖⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠒⠛⠉⠁⠀⠀⠄⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠙⠷⣄⡀⠀⣿⠉⠍⠩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠈⠛⢦⣟⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣄
⠀⠀⠀⠀⠀⠹⣧⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠘⠿⢻⡧⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣠⣀⣀⣀⡀⠀⠀⣿⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡃⠐⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣀⣀⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠠⡀⠀⠑⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡟⠉⣶⣄⣻⣯⣿⠿⢟⣻⡦⠀⠀⠰⡆⠀⠄⠀⢨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡇⠀⢿⣿⣿⣏⠀⢸⣽⣽⠇⠀⠀⠀⠀⠀⠸⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡰⣿⣿⣦⣀⣬⣙⣛⣷⠿⠋⠁⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠙⣻⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠻⢿⣿⣿⠟⠛⠋⣀⣤⠞⠋⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣀⣤⠤⠴⢶⣯⣵⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⣿⣿⣿⣷⡕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⣿⣿⣿⣿⣿⡟⣻⠷⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡹⣿⣿⣿⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢁⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠖⠙⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⡌⠀⠹⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠐⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣆⣶⣶⣶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣧⠺⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠀⠉⠁

⠀⠀⠀⠀⣔⠀⠀⠀⠐⠀⠀⣀⠀⠠⠈⠲⢄⣀⡀⠀⢏⠆⠀⠐⢂⠀⠐⠀⠀⢀⣀⣴⡎⠀⢸⡀⠀⣇⡴⠋⠀⠂⡀⡇⠀⣀⡁⠴⠂
⠀⠀⠀⠀⡋⢤⡀⠀⠀⠀⡀⠄⡀⠀⣀⢀⠀⢢⠈⠲⠬⡎⠆⠀⠈⣅⣩⢴⡿⠫⣗⢰⢅⠤⠺⣇⡴⠋⠐⠤⠄⡇⠿⢒⠸⠋⢰⣦⡆
⠀⠀⠄⢠⢀⠀⠃⠙⠀⠂⠁⡴⠁⠀⢠⣀⡠⠬⠷⠤⣤⡙⡆⢰⠞⡋⢀⣼⡂⠫⢡⠎⠈⠀⢠⢻⠇⡀⠀⠀⠀⠀⠀⢣⠗⠤⢏⠈⠳
⢀⠄⠀⠀⣀⠡⡀⠀⠀⠀⢰⠁⠀⢀⠄⢹⡀⢰⠁⠉⠀⠈⠙⢆⠑⣤⠋⠁⠀⠠⠎⠀⠀⠀⠰⢘⠀⠓⠉⠘⠄⢢⠀⢸⠂⠐⠀⠁
⠈⠑⠈⢃⠀⢀⡀⠂⠀⢸⣀⡤⠎⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠑⢤⡀⠶⠇⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠈⡄⠃⢸
⡀⢤⣦⡈⠁⠀⠈⢜⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⣁⡀⠀⠤⣿⠸⡄⠀⠀⠀⠀⡃⠀⠘
⠉⢰⡟⢁⠤⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠙⠀⣷⠀⠀⠀⠀⡇⠀⢰
⠀⢿⡷⣄⣀⣀⢐⠀⠀⠒⠁⠠⣠⡄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠉⣂⠀⠀⠀⢳⠀⠈⡃⠀⠀⠂⠠
⢔⣙⠁⠀⠈⡁⠀⠀⠀⠂⠀⠂⠁⠀⠂⡄⡢⢄⣀⣀⡀⠀⠀⠀⠈⠁⠀⣀⠀⢀⡀⠀⠀⢀⡠⢫⡆⠙⢣⠄⠀⠀⣠⣠⠇⠰
⢀⠈⠁⠒⠀⠀⠀⠀⠁⠀⠀⠉⠚⠓⢀⠀⠁⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⢠⣄⠀⣊⡘⠂⠀⢢⣰⣶⣿⣯⣴⠶⠔⠂⡢
⠛⠏⠐⠻⠁⠀⠀⠀⠀⠀⢀⣀⡐⠰⡤⠤⠄⠉⠭⠶⠖⠒⠒⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⢀⠨⣥⢰⠞⣀⢂⣻⡟⢿⠶⠂⠞⠒⡩⠤
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⡨⠔⠁⠂⠠⠠⠜⣣⢋⠈⠖⠋⠉⠁⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢊⠴⠊⢠⣤⠠⠰⠆⣠⡄⠉⠉⠁⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣊⡡⠔⢉⡄⠀⠀⢠⠴⠁⠉⠀⠀⠀⠀⠀⠀⢀⡠⠁⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢁⡴⠁⠀⠀⠉⠀⠀⡺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⠃⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⣿⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠸⡇⠀⠀⠹⣧⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡿⠀⣿⠀⠀⠀⢹⣇⠀⣾⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⡇⠀⢹⡇⠀⠀⠀⢻⡄⢻⡀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠀⠀⠀⠀⣿⠀⠘⣷⠀⢸⡇⠀⠀⠀⠘⣧⠘⣧⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢻⣆⠀⠀⠀⢹⡀⠀⠙⠷⠾⠃⠀⠀⠀⠀⣿⠀⢸⡇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⡇⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡿⠁⠀⢸⡇⠀⣠⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣿⠀⢸⣷⣄⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣼⠋⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠈⣿⠙⣷⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢸⡏⠀⢠⣿⠀⢸⡇⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣸⡇⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⢸⡇⠀⠀⠈⣷⣤⣾⠃⠀⠸⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠟⠁⠀⠀⣿⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠿⣯⣅⡀⠈⠻⣦⠀⢸⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⣴⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⢹⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣾⠋⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⢸⡇⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⢺⣿⡇⠀⣸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢷⡀⢀⣷⡄⠀⢸⡇⠿⠾⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⣧⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠻⠾⠋⣿⠀⣼⠃⠀⠀⠀⣿⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡌⣿⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⣧⣼⠏⠀⠀⠀⠘⣷⣼⢸⡇⠀⠀⠀⠀⡀⠀⠀⢻⣿⡟⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢸⣷⣦⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣸⡇⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⠹⣧⡀⠀⣀⣤⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣀⣤⡾⠋⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡇⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⣤⣤⣤⣤⣤⣀⣀⣘⣿⣄⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣣⣴⠂⠀⠀⠀⠀⠀⣠⣿⣃⣤⣤⣤⣤⣤⣤⣤
⣼⡯⠒⢿⣆⣀⣈⣉⠉⠉⠻⣷⣄⠀⠀⠀⠈⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣽⠏⠀⠀⠀⢀⣤⡾⠛⠉⠉⠉⠀⠀⠀⣿⠀⠘
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠙⠻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠟⠁⣀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠰⡇
⢻⣆⠀⣾⠁⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣷⣦⣤⣽⣿⣷⣦⣤⣤⣀⣠⣠⣤⣤⣶⣾⣿⣿⣥⡶⣾⡿⠿⠿⠶⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⠐⣷⠢⣺
⠀⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⣀⣤⡶⠟⠋⠀⠀⠀⠈⢉⡽⢿⡿⠛⠉⠛⠙⣿⠉⠙⢧⡈⠀⡀⠀⠈⠛⠷⣤⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⣠⠔⠀⣠⡾⠋⠁⣼⡇⠀⠀⠀⠀⣿⠀⠀⠈⠻⣦⡈⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣟⣁⠄⠀⠀⠀⠔⠉⠀⣠⡾⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⡀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⣾⢏⡥⠖⠠⠈⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢸⡏⠀⠀⣠⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢷⣄⢀⠀⠀⠈⢳⠞⣉⣉⡻⢷⣄
⠀⠀⠀⠀⢸⣿⠸⠀⠀⠀⠈⢘⡇⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⣿⠃⡀⡀⠃⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣰⡏⠓⠀⠀⠙⣎⣿⡄
⠀⠀⠀⠀⠈⢿⣄⠁⢰⠀⠀⣸⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣟⣠⣧⣥⣤⣄⣀⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⡀⠀⠀⠀⠀⢉⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠷⢾⡷⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣉⠢⠀⠚⠀⠙⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣾⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⣓⠀⠀⢈⣩⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠘⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢿⣇⠀⠀⢻⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⢿⣷⣴⣿⡟⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢿⣧⠀⠀⠈⠉⠉⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⢀⣶⠀⢸⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⣿⡇⢸⣧⣄⣠⡾⣻⠀⢸⣿⣿⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢸⣿⠀⠀⠀⢀⣿⠇⢸⡇⠈⠀⢠⡏⢠⣿⢿⣿⠃⡀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⢻⣧⣀⣠⣾⡟⢀⡾⠀⠀⢠⡟⢠⣿⡏⣼⡟⢸⣿⡄⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⠙⠛⠛⠉⣠⡾⠁⠀⠀⢸⡅⠈⠻⠿⠟⢀⡾⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣟⠳⠶⠶⠞⠉⠀⠀⠀⠀⠘⣧⡀⠀⠀⣠⡾⠁⠀⣽⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠹⣆⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠉⠛⠋⠉⠀⠀⢀⡿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠉⢀⣠⣤⠶⠲⣾⠋⠉⣿⣆⠀⠀⠀⠀⠀⢀⣼⠃⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⢛⣉⣁⣤⡶⢻⢿⡀⢀⠇⣿⠻⡓⡶⢦⣤⣈⡁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣟⣛⣉⣉⠉⠑⠄⣰⢟⡱⢣⢾⣇⣎⠎⢹⡷⣷⣮⣲⣤⣈⣉⠛⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣏⡉⠉⠉⢱⢨⢂⣼⣾⣭⣕⡊⠁⣰⣿⡟⡠⠚⣇⠈⢙⣦⣍⣉⣛⣛⣛⠛⣿⠟⠛⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⡆⢀⣾⣷⣾⡿⠒⠉⠀⢹⣼⠃⢻⣷⠁⠀⢻⡀⡏⢀⡽⠿⣶⣤⣄⣸⡇⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠲⠟⠛⢡⡾⠗⠂⠉⠉⢢⡾⠁⠀⢸⣿⣤⠤⣼⡿⣧⢸⠀⣀⣬⣷⣄⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠓⢦⣤⡞⠁⠀⠀⠘⣿⠁⠀⠈⣷⠈⢿⣤⡏⣀⡼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⠴⠞⠛⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠀⠀⠙⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠎⡉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠉⠒⠠⣍⡒⠤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣠⣤⣄⡀⠀⠈⠑⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣄⠀⣤⢄⣀⡜⠁⢀⡴⠋⠁⠀⠈⠙⢦⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢥⡇⠀⠀⠀⠀⡔⠉⢆⠀⠀⠀⠀⠀⡠⡫⠊⠁⠀⢀⡠⢘⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⣠⠔⣉⠉⠭⣉⣿⡲⡀⠀⠀⠹⠛⡟⠀⠀⠀⠀⠀⣿⠁⠀⢀⠔⠁⢀⠎⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⡾⠁⠀⡌
⠀⠀⠀⠀⢰⠁⡜⠁⠀⠀⠈⡇⢸⠄⠀⡠⠃⡰⠁⠀⠀⠀⠀⠀⣾⠀⢠⠋⠀⠀⡌⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠋⠀⠀⡰
⠀⠀⠀⠀⣾⠟⢆⠀⠀⠀⡠⠃⡜⢆⠉⣀⠔⠁⠀⠀⠀⠀⠀⠸⡆⢠⠃⠀⠀⡸⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⠀⢠⢻
⠀⠀⠀⠀⠑⠵⢤⣉⣉⣉⣤⠮⠒⡌⢷⠻⠼⢖⢄⠀⠀⠀⣠⡳⢁⠏⠀⠀⢀⣇⡀⠀⠀⠀⡰⠁⠰⠁⠀⠀⠀⠀⢀⠇⠈⡄
⠀⠀⠀⠀⠀⢀⠊⢣⠰⡊⣴⣺⢷⢘⡌⢆⠀⠀⠑⠵⣖⠮⠛⠁⡘⠀⠀⣠⠋⠑⡦⠕⡒⡴⠁⣰⠁⠀⠀⠀⠀⢠⠋⠀⢠⠁
⠀⠀⠀⠀⠀⢸⠀⡇⠱⡑⣌⠥⠒⢉⣠⡼⣆⠀⠀⠀⠀⠀⠀⠀⢧⠀⢠⠁⠀⠀⠉⠒⠼⠁⢠⠏⣒⠤⣀⠀⡰⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠘⡄⠸⡄⢱⣠⣴⠾⠛⠉⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢰⠁⢠⠋⠒⠢⠟⠒⠿⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠘⣶⠛⢹⠻⡀⠀⠀⠷⣄⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⠀⠀⢠⠃⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠱⡀⠀⠸⡈⠢⡀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢃⠀⠘⣦⠜⠓⢒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠜⢆⠀⠈⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡡⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠋⠐⠐

⠀⠀⠀⢤⣷⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⡀
⠀⡀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠠⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠰⠆⠀⠀⠈⠘⠇⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢾⡼⠃⠇⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠈⡁⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠄⠁⠂⠁
⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠂⠀⠀⢀⣤⣄⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⣀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡁⠼⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⣤⡀⡀⢀⢣⡯⢆⡆⠄⠐
⠀⠀⠀⠐⠣⡟⠿⠷⠤⠤⠤⠤⠄⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠃⠀⠘⠛⠈
⠂⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠠⠠⠀⠀⢀
⠀⠀⡀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠇⠀⠈⠀⠀⠄⠀⠀⠚⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠀⠀⠰⡆⠀⠠⠀⠀⠃⠀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⢀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⠁⡀⠉⠰⠇⠀⠤
⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣆⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣟⠿⣿⡿⠿⠿⢯⣭⣿⣿⣾⣿⣮⣜⡿⣆⠀⢉⣦⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣒⠚⠓⠈⢀⣀⣀⡼⢥⣿⣿⠼⠋⠛⠻⢿⣼⡄⠀⢀⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢠⣾⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠒⠒⠢⣤⣤⠰⢿⣿⡦⠌⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⣤⣤⣤⡾⠙⠏⢀⣼⠅⠀⠀⠒⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⣀⡔⢿⣿⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢰⣇⠀⣴⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀⠀⢠⣶⣾⣿⣿⡍⡅⠀⠀⠀⢠⡀⣼⣟⣩⣁⡀
⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⢿⣿⣿⣿⣧⣗⠀⠺⣿⣄⠀⢹⣿⣟⣋⣠⡐⢀
⠀⠀⣴⡈⢤⠄⠀⠀⠀⠀⠈⠁⠀⠉⠀⢻⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣟⣁⣀⣿⡟⢩⢹⣿⣟⣟⠩⢉⣯⡤⠆
⠀⠶⠿⣭⡤⠶⢖⠉⠉⠉⠁⠈⠃⠋⠉⠙⡛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠋⠀⠈⠙⠛⠛⠐⠊⠐⠀⠉⠀⣀
⠀⠀⠀⡔⣤⠡⢄⠀⣤⣀⡠⣤⠀⡠⠤⢠⢄⢄⠀⠀⠀⢀⣠⡄⢠⢀⠀⠀⠀⠠⣀⢀⡀⢀⡀⡀⡀⢀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⢹⡶⡌⢱⢛⠀⣿⡌⠉⣢⡽⢇⣦⡏⢀⢜⣮⣆⠀⢇⢁⠡⣉⣬⢠⣴⡧⣌⡅⣚⡤⣡⡶⠗⠠⠄⠀⠵⢹⣅⡠⡄
⠀⠀⣹⣽⣇⣷⢡⠲⣦⡆⢼⡣⡇⣻⡀⣧⠨⠞⢸⡟⠻⣤⡘⢹⣫⡍⣊⡡⢿⢳⣼⠽⢿⠾⠃⢜⡋⠦⠟⠃⢨⣏⣀
⠀⠀⢸⡇⡏⢾⢸⡟⢫⡇⢻⡟⣇⢺⣆⡟⢻⡯⢱⡏⠿⡳⢻⢾⣷⡄⣿⢠⣬⡀⢸⡷⣺⣧⡴⣦⢶⡖⡐⢺⣫⠌⠐
⠀⠀⣹⣏⡉⣺⠘⠒⢸⡇⢤⠮⣯⣼⠇⣷⢸⣋⣸⠎⠑⡟⡴⠛⢻⠂⠿⠠⠾⠉⣸⣏⠟⣻⣐⡗⣻⡿⠆⣶⠡⡆
⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠚⠷⠤⠤⣭⣤⣄⣹⡅⠉⠈⣿⠀⠉⠃⣛⠄⠣⠀⢸⡇⢀⣹⣎⡟⣹⠛⠁⡿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠒⠒⠦⠿⣀⣀⡀⢸⡇⠀⠀⢸⡏⠈⠐⠋⡟⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠓⠒⠓⠠⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡇⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠀⢸⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠘⢼⠄⡸⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠁⠋⢸⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⢠⡼⠀⠀⠀⠀⢀⠀⡄⠂⠀⠀⠈⠀⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠁⠀⠀⠀⢰⡋⠁⠀⢀⡦⠀⢸⣿⡟⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⠀⠀⣴⣒⡶⣛⣷⡀⣰⣟⠷⣄⢈⠽⣯⣑⡦⣀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⣀⣀⡥⢛⡡⣠⠖⢛⣻⣿⣿⡀⠈⠻⡾⠉⠉⠙⣦⣙⠢⣄⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⢿⣧⣶⣿⣿⠃⠀⠸⠇⢠⣼⣷⣀⠀⣌⠻⣿⣿⣿⣿⣦⣶⣟⠓⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠦⠀⠀⠀⠉⠹⠿⠿⠷⢤⡴⣿⣾⣿⣿⣿⣧⣹⣴⣿⡏⠉⠁⠀⠀⢁⡀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⡂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⠀⠀⠀⠀⢀⡴⠛⢳⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⣸⡷⠄⠀⠀⡄⠀⠀⢠⡶⠛⠿⡗⠰⠿⣧⡀⠀⠀⠀⠀⠀⠀⠑⢌⢢⡀⠀⢀⠀⣠⢔⡵⠊
⠀⠀⠀⠈⠉⠈⠉⣿⣧⠄⠀⠀⡀⢀⣠⠋⠀⠀⠀⠳⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠑⢽⣖⣸⣿⡕⠁
⠀⠀⢱⣶⣶⣶⣿⣿⡧⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣏⣱⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣃⡀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⠿⣦⣄⡀⠀⠀⣠⢞⠽⠉⠙⣿⣿⢏
⠀⠀⢸⣿⣿⣿⣿⣿⡁⠀⠀⠀⣿⢿⠇⠸⠇⠸⠁⡿⠈⠀⢾⠆⢠⡄⣤⡌⣭⣷⢒⠔⠁⠀⠀⠀⢸⣿⡗⠀⡀
⠀⠀⢸⣿⣿⣿⣿⣿⠋⠁⠀⠀⣿⣸⡀⣀⣀⣸⣀⣿⣀⣀⣀⣀⣈⣃⣿⡇⠛⣃⢈⠀⣠⡦⠤⠤⣴⣿⣇⠈⠁
⠀⠀⠘⢿⣿⣿⡿⠿⠛⠛⠛⠫⠋⠉⠁⢀⡠⠐⠋⠁⠀⠀⡀⠊⣉⠛⠁⠀⠀⠀⠈⠠⠈⠁⠘⠤⠚⠿⠷⠶⠂⠐⠐⠂⠀⠀⠤⢤⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠀⣀⠀⠀⠀⠀⠐⠀⡀⠀⠈⠀⠒⠀⡔⠂⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠈⡉⠉⠐⡒
⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⡀⠀⠀⠀⠊⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠩⡄
⠀⠀⠀⠁⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠐⠂


⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣴⠆⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁⠀⠀⢰⣿⣿⠀⣿⣿⣿⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣇⠀⠀⠀⢾⣿⣿⠀⣿⣿⣿⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡃⣿⠀⢸⣦⢸⣿⣿⡇⢻⣿⣿⠿⣿⡟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡟⢀⣿⣿⠀⣿⢸⠃⢸⣿⡏⢠⡟⣴⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⣾⣿⠃⣸⢣⡏⠀⢸⣿⢠⠏⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⣰⠏⣼⠀⠀⣼⠏⣾⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡿⠋⠉⠛⠄⠀⣿⣀⡼⠋⠀⠙⠦⠞⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢸⣿⠃⠀⠀⠀⠀⢠⠟⠋⠀⠀⢳⡀⠀⠀⢀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⣶⡶⠶⢾⠌⠳⠼⣇⠀⠀⣴⡀⠀⠋⠀⠀⠀⣸⡎⡇⠀⢀⡾⠋⠉⠁⣀⣀⣀⡀⠀⠀⠹⡆
⠀⠀⠀⢀⡟⠀⣿⠀⠀⠀⠀⠀⠙⢦⡀⢻⡙⣦⡀⠀⠀⢠⣿⠇⣠⣴⡟⠓⠒⠂⣠⡤⠤⠤⠄⠀⠈⠙⡇
⠀⠀⠀⢸⡇⡀⣿⠀⠁⢙⣶⣶⣶⣶⣿⣶⠿⠿⣧⠀⡴⠛⡿⠻⠉⠙⠻⠟⠻⠿⣿⣿⣿⣿⣶⣦⣤⡼⠁
⠀⠀⠀⠀⣷⣴⣿⠶⠚⠉⠁⠐⠲⠿⠛⠋⠀⠀⠈⣿⡿⣦⣀⣔⡢⢄⡀⠀⠀⠀⠀⢉⣻⠶⢤⡀
⠀⠀⠀⣼⠁⠀⢈⢳⡀⠠⠤⠖⠐⠀⠀⢤⣖⣂⡤⣿⣿⡂⡙⠻⢿⡳⢤⡀⠈⢱⣶⠏⠁⡀⢲⣽⡄
⠀⠀⠀⡇⣆⠄⠇⡇⣧⠀⠀⠀⣀⡶⢿⣿⡿⠿⠚⠁⠀⠉⢻⠰⣶⣌⡑⣀⠀⢸⣧⠃⢘⡇⠀⣼
⠀⠀⠀⢻⣳⡑⠀⠃⡟⠀⣒⣫⠵⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⠈⠙⠳⢤⣼⡻⡶⠖⢄⡼⠃
⠀⠀⠀⠀⠻⢯⣤⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠚⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⡠⡔⣒⣁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡄
⠀⠀⠀⠀⢀⡀⠤⢠⠀⣀⠀⠀⠀⠀⠰⠒⣟⣏⢭⠅⠃⠀⠀⣀⣀⣤⡤⠚⡁⠀⠊⠁⢲⠶⢤⠤⣄⡀⠀⠀⢍⠀⠈⠁
⠀⠀⠀⢸⢼⢶⢺⠻⠉⠉⣀⠀⠀⣀⡄⠀⠀⠀⠀⠀⢠⡖⠋⠹⣉⡀⠀⠀⠁⢀⢀⣀⣉⠤⠖⠁⣢⣿⠂
⠀⠀⠀⠘⠚⠉⠈⠀⠀⠀⠘⢻⡿⢛⠁⠀⠀⠀⠀⠀⠈⠻⠶⣤⣄⣈⣉⣉⣉⣉⣡⣤⣤⣤⡶⠾⠋⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⣤⠆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠉⠉⡤⠑⠀⠀⢀⡀⡔⢢⠀⠻⠇⠀⠀⠀⠀⠀⠠⠀⠀⡤⡤⣦⢤⢤⡀⠺⠿⣀⡖⡗⢟⢻⢹
⠀⠀⢀⡠⡤⠴⠀⠀⠀⡈⠀⠀⠀⠁⠀⠀⠀⠀⠛⠀⠈⠐⠃⣠⠀⣇⢀⢹⣸⡼⠋⠀⠀⠏⠳⠶⠼⠼⠼⠄
⠀⠀⠀⠀⠀⠀⣀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠼⠀⠠⡻⠀⠈⠉⠉⠀⠀⠀⢐⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠄⠖⠈⠀⠀⠀⢀⢀⢀⠀⠀⣈⠋⠀⠀⠀⠸⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠈⠑⢦⣀⣷⣦⡀⠀⠀⢠⣶⣶⣄⣴⠀⢰⣷⠿⣿⡟⠃⣿⣯⣩⢹⣿⣹⠗⠀⠀⠀⠈⡗⠒⠂⢰⠂⠐
⠀⠀⠀⠠⢤⣀⣀⠀⠀⠀⠉⠃⠀⠀⣿⣁⣿⡏⢿⣧⣼⠏⠀⣿⡇⠸⣯⣤⣤⠸⡿⠙⠷⠀⠀⠠⠀⢹⠋⠓⣾
⠀⠀⠀⠀⠀⠈⠃⠀⠀⡤⠶⣄⠀⠀⠘⠛⢉⣤⣄⢩⡤⢤⡀⣴⢶⡀⢀⣴⢶⢠⡶⠖⡂⠈⠀⠀⠀⠾⠲⠶⠻⣤⠐⠐⠂
⠀⠀⠰⠖⠶⢶⡀⠀⠀⢧⣀⠞⠀⠀⠀⠀⠘⠳⢦⣼⣷⠾⠛⣿⠾⣧⢼⣇⣠⠜⣟⢋⣁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠻⠷⠖⠀⠀⠀⠀⠀⢈⠠⠂⠈⠛⠛⠁⠛⠀⠀⠉⠀⠀⠀⠉⠁⠀⠛⠉⠉⠀⠀⠀⠀⠉⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀⡰⠀⣔⢈⠀⢀⡀
⠀⠀⠀⠀⠉⠰⠶⠀⠀⠀⠀⢀⠖⠉⢠⠄⡀⠀⢁⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢦⣀⡀⠉⠀⠄⠈⢁⣸⠔⠋⠀⠀⠇⠀⠀⠄⠀⠈⡷⠀⠀⢀⠀⠄
⠀⠀⠀⠃⣄⡀⢀⡀⠤⢾⡁⠀⠀⠀⠉⠋⠍⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠁⠀⠀⣌⠀⠄⠀⢀⡀⠀⢠⠐⢀
⠀⠀⠀⣴⠇⠀⠈⠇⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠋⠀⠐⢦⣤⠄⠒⠾⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢲⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠰⠀⠇⠀⠤⠄⠀⠀⠀⠀⠀⠤⠄⠇⠰⠇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠆⠤⠀⣼⠁⠀⠀⢹⣄⠀⠤⠶⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠉⠉⠁⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠀⡠⢴⣶⢶⣤⡀⢠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠴⠀⠘⡄⣠⠟⠋⠛⢆⡁⡜⠀⠰⠤⠤⠤⠤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⣠⠀⠀⠀⠘⠈⠂⠀⢀⡀⡤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠈⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠐⠒⠂⠲⠀⣶⡇⠀⠀⠀⠀⠀⠀⣷⡆⠰⠂⠂⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣾⣿⡷⠒⠒⠒⠒⠒⠲⣿⣿⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⠿⠁⡇⠀⠀⠀⠀⠀⠀⡏⠹⢯⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠲⡞⢣⣤⣧⠤⠤⠤⠤⠤⢤⡧⣤⢛⡶⠒⠒⠒⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠾⠉⡈⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⡘⡹⢦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⢞⡥⠊⠀⠇⢠⣇⣀⣀⣀⣀⣀⣀⣇⠀⢇⠈⠢⣙⢦⡀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠴⠋⢀⠔⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠈⠑⢄⡈⠓⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⠔⠉⣠⠜⠁⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⢤⡀⠓⢌⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠔⠁⠀⢸⠀⠀⠀⠀⠀⠀⢈⡯⠉⠉⣩⡉⠉⠨⣏⠀⠀⠀⠀⠀⠀⢨⠀⠀⠑⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⡎⠀⠀⠀⣿⡇⠀⠀⠈⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠛⠛⠛⠛⢻⠛⠛⠛⠛⠛⠛⡇⠀⠀⠀⣿⡇⠀⠀⠀⡟⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠲⡞⠀⠤⠄⠀⣀⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⡇⢀⣀⠀⠤⠄⠐⠶⠂⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠤⠤⠤⠬⡇⢹⡶⠶⢾⣿⡗⡶⠶⢺⠃⡻⠤⠤⠤⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢁⠀⣿⣿⣿⡿⢿⣿⣿⣿⠀⠃⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣰⣄⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣴⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠉⠉⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠉⠉⠉⠉⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⠀⠀⠄⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠄⠤⠤

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡎⢠⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⠀⠀⠀⠀⢀⠀⠀⠐⠀⣀⠤⠀⠀⢀⠀⠀⠀⠘⠄⠀⡀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠾⠀⠉⠠⠀⠘⠓⠁⠀⠀⡐⠂⠾⡐⢄⡀⠀⢠⡘⠀⠄⢼⣷⡤⠅⢷
⠀⠀⠀⠀⠀⠠⠄⢰⡀⠀⠂⠠⠀⠀⠀⠀⣰⡿⣣⣾⣧⣁⣈⠀⣇⠉⠀⠀⠀⡀⠱⡄⠈
⠀⠀⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢀⠀⠈⠆⠠⠄
⠀⠀⢳⡀⠀⠈⠛⠿⠿⠷⠾⠲⠶⠶⠶⢶⣶⣒⣛⣛⣛⣛⣛⣉⠉⣿⠀⡸⠃⠀⠀⠀⠀⠂
⠙⠛⣉⣉⣐⣛⣛⣛⣋⣁⠀⠀⠈⠛⠿⠯⠭⢥⣤⠀⠀⠀⠀⠀⠀⢹⡗⠰⠀⠀⠀⢱⠀⠀⠁⠀⠀⣠⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣦⣬⣭⠉⠁⠀⡀⢠⠀⠀⠀⠬⣤⣴⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠘⡉⠀⡀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣧⠀⠀⠀⣿⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠃⢰⡀⠀⠀⠛⠁⠀⠀⢰⡀⠐⠠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣄⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡼⠁⠠⠄⠀⠀⠀⠀⠀⠀⠁⠐⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡓⠀⠀⠀⠄⠈⠛⠉⠓⠄⠀⠠⠀⠀⠐⠊⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠽⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⣀⣀⣤⣤⣶⣦⣀⣀⣀⣓⣠⣤⣤⣤⡀⠀⠀⠶⠿⠿⠛⠛⠉⣉⣀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠉⠉⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠘⠀⠀⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣠⣾⣟⣷⡄⠀⠀⠌⠀⡔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⣠⣤⠶⠶⢶⠾⠛⠉⠉⣻⣿⣍⣁⣀⣀⣀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣍⠉⠙⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠁⠀⠀⣿⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⢱⣿⣿⣿⣷⡄⠀⠘⠳⢦⡀⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⣿⠀⠀⣀⣀⣀⣀⠀⢘⣿⣿⣿⣧⣿⣿⣿⡇⠀⠀⠀⠀⠈⢿⡄⠀⢀⡠⠄
⠀⠀⠀⠉⠒⠀⠀⢀⣤⣤⣤⡶⠿⠶⣤⡈⠉⠉⠁⣠⣾⣿⡿⠿⠿⢿⣿⣛⣛⣒⣶⣶⣦⣤⣾⣧⡀⠀⠀⠒⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠉⠀⠀⠀⠈⠷⣦⡀⣴⣿⣿⠏⢀⣠⣤⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣉
⠀⠀⠀⡀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡿⡿⠃⠀⢸⣟⣻⣷⠀⠀⠻⣿⣿⢿⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠤⠄⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⣿⣟⡟⠁⠀⠀⠘⠛⠛⠛⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⠏⢀⣴⣶⣏⠉⠉⠉⠉⣡⣶⣦⡄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣙⣻⣿⠀⢸⣿⣹⣾⠀⡀⢀⠀⣿⣏⣿⡇⠀⠀⣿⢩⣭⣭⣭⣽⣿⡛⡟⠛⠁
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⡁⠀⠀⣿⢸⣿⣿⣿⣿⣿⡧⡇⠀⠀⣼⣧
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⠠⠤⠤⠤⣤⡤⠖⣒⠦⣤⣄⠠⠤⠄⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⣰⣿⣿⣆
⠀⠀⠀⠀⣿⠿⠿⠿⠶⠶⠒⠒⠂⠀⠀⠈⣿⠀⢠⡾⠿⠷⠾⠿⣿⠿⠦⠬⠿⢶⡄⠀⣿⣿⣿⣿⠿⠿⢿⡏⡇⢈⣿⣿⣿⣿⡁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⣿⠀⢸⣷⣮⠛⢫⣶⣿⣦⡝⠛⣵⣾⡇⠀⣿⣿⣿⡗⣼⢟⣻⣷⣏⣹⣿⣿⣿⣿⣍⡀
⢠⡄⠀⠀⣿⠀⢠⡄⠀⠀⠀⣸⡀⠀⢸⣀⣿⠀⢸⣏⡱⣿⡷⣊⣿⡱⢾⡿⢎⣹⡇⠀⣿⣿⣿⡿⠐⢿⡧⣿⣿⣭⣽⣿⣿⣯⣭⣵⡆
⢸⣯⣭⣽⣿⣭⣽⣯⣭⣭⣭⣿⣯⣭⣭⣽⣿⠀⢸⡿⢟⣤⣼⠻⣿⠟⣧⣤⡻⢻⡇⠀⣿⣿⣿⡇⢿⣦⣯⣿⡿⠿⠿⢻⡿⠾⠿⠟⠁
⠚⠛⠒⠚⠟⠖⠺⠷⠶⠶⠒⢛⣳⣶⣶⣾⡿⠶⠾⠷⠶⠶⠶⠶⠿⠶⠒⣲⣶⣾⣷⠶⠿⠿⠿⠿⠶⠶⠾⠿⠿⠶⠶⠿⠗⠲⠒⠒⠒
⠀⠀⣀⣤⡤⠶⠖⠚⠛⠉⠉⠉⠉⠀⢀⡀⢀⣀⣤⡤⠶⠶⠒⠛⠉⠉⠉⠁⠀⠀⠀⣀⣠⣤⡤⠶⠶⠶⠶⠖⠒⠶⠶⠶⠶⣤⣤⣀⡀
⠚⠋⠉⠀⠀⠀⠀⠀⠀⣀⡤⢔⣺⡽⠖⠛⠉⠁⠀⠀⠀⠀⠀⢀⡤⠔⠊⠠⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⣿⡇⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣧⢀⣻⠄
⠀⢰⣽⣿⣦⣿⡀⠠⣷⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⢀⡇⠀⡀⠀⢑⢾⣏⠲⣾⣭⠄
⠀⣺⣿⣿⣿⣿⣃⠾⣿⣓⣾⣴⣴⣼⣤⢰⢠⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⠀⠀⠠⣺⡧⠀⡄⢸⣇⠀⢽⣿⢰⣿⡀⢬⣿⣯⣿⣿⣭⠵
⠐⠿⣿⣿⣯⣿⡏⢻⣿⣷⣎⠁⠉⠙⠛⠛⠻⠷⠆⠤⠤⠤⠤⢠⣄⣷⢰⣿⣧⠀⢼⣾⣿⣟⣇⢽⣿⠋⠺⣿⣿⣿⣧⠬⣿⣿⣓⣾⣿⣯⠄
⢀⣹⣿⡏⠀⡿⠅⠘⣿⣿⠏⠀⢀⣀⠀⠀⠀⠐⠀⠀⠠⢄⡘⣛⠛⠿⠻⠟⠛⠤⢸⣿⣷⡾⢧⣽⣿⠬⠽⢿⣻⣿⠸⣿⣿⡻⣿⣾⣿⣗⡂
⠈⢉⣿⡇⢤⡾⢄⡤⠖⠋⠁⠀⠄⠛⠿⠂⠀⠀⠀⠀⠀⠀⠉⠉⠚⠒⠂⣀⠄⣀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧⠤⡿⣟⣻⣶⣿⣷⣦⠄
⠀⠈⠙⠉⠀⠀⠠⣀⠤⢦⠀⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢞⠥⣾⠦⠤⣄⠀⠀⢀⡄⣀⠀⢀⡀⠀⠈⠁
⠀⣤⠰⠚⠒⠉⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠛⠘⠋⠀⠘⠝⢦⣹⡆
⠀⠉⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠃
⠀⠀⠀⠀⣠⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠁⠀⣀⣦⡀
⠀⠀⠀⠀⠘⠓⠂⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠋⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠳⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣄⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠘⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⢱⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⢻⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⡀
⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠟⠁
⠀⠀⠀⠀⣿⠃⠀⢠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⢀⣄⠀⢁⣤⣄⡀⠀⣤⡀⢠⣄⠀⣤⡀⢠⣄⠈⢩⡍⠁⢀⣤⣀⡀
⠀⠀⠀⢰⡟⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠋⠀⣿⠀⣿⠀⠘⠇⠀⣿⠀⢸⡇⠀⣿⠁⠸⣿⡄⢸⡅⢠⡟⠀⠘⠇
⠀⠀⠀⠸⡇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣴⠀⠀⣿⠀⢿⣦⣀⠀⠀⣿⣀⣸⡇⠀⣿⠀⢀⡻⣇⢸⠇⣾⡷⣆⣀⣀
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠛⠀⠀⣿⠀⠈⠙⢻⣷⠘⣿⠉⢹⡇⠀⣿⠀⠰⡆⢻⣼⡅⢿⡇⠀⢸⡏
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣆⠀⠀⣿⠀⣿⠀⢸⡇⠀⣿⠀⢰⡇⠘⣿⠀⠸⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣃⡀⠀⢐⣟⣀⠛⣳⣖⠁⢀⣿⣆⢸⣷⡀⣿⣇⣸⣧⡀⢹⡀⠀⠙⣳⣞⢿
⣿⣿⡿⠿⣿⡿⠿⠿⡿⣿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣾⣶⣾⣶⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⣀⣼⣷⠶⣄⡀
⠀⠀⠀⠀⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠛⣡⠞⠁⠈⠢⡀
⠀⠀⠘⠤⡐⢂⣉⡵⠦⡀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⣀⣠⣴⣾⡿⠛⠉⡠⠞⠁⠀⠀⠀⣠⣾⣆
⠀⠀⠀⠀⠈⠢⢔⢛⣮⣍⠶⣄⠀⠀⣜⣀⣠⣤⣶⣾⣿⠿⠛⠋⠁⠀⠔⠉⠀⠀⠀⠀⣠⣾⡿⢋⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠑⢶⣊⣻⣭⠵⠴⡿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢋⣴⠟⠁⠘⡀
⠀⠀⠀⠀⠀⠀⠄⠒⠒⠻⣦⣀⡛⣮⠡⠤⣐⢖⠐⡤⢄⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⣁⠴⠋⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⠟⡆⠘⢸⡟⣿⠃⡇⢸⠚⣀⣤⣴⣾⣿⠿⠛⣁⠤⠊⠁⠀⠀⠀⣠⡾⢋⡅
⠀⠀⠀⠀⢀⡇⠤⣐⣒⠆⠀⢰⣿⢀⣧⣥⣬⠷⢷⡛⣿⡿⠛⠋⣁⠤⠐⠈⠀⠀⠀⠀⢀⡠⠞⢁⣴⣿⠃
⠀⠐⠽⠋⠉⠹⡀⠀⠀⠀⠀⡼⠉⠀⢛⣉⣁⣲⣉⠽⠦⣚⣺⡉⠀⠀⠀⠀⠀⢀⡠⠒⠉⢀⣴⠟⠋⠎
⠀⠀⠀⠀⠀⠀⠈⠢⠄⠤⠚⠁⠀⠀⠀⠻⡉⠁⠈⠒⢍⡩⠷⣜⣓⣄⡠⠔⠊⠁⣀⣤⠾⠋⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣤⠴⠷⠮⡩⠕⠐⠛⢢⠶⠛⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣤⣶⠾⠒⠋⠉⠀⠀⢀⡀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉

⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⢸⣿⣿⣿⣀⣴⡾⠿⣦⣄⡀⠀⠀⠀⠰⢚⣹⣿⣿⣿⣿⣿⣯⣈⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠀⣾⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡄⠀⠀⠒⠉⠉⠁⠘⢒⡇⠺⠏⠀⠙⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢰⣶⣿⣷⣦⡄
⠀⠀⢸⣿⣿⣿⣿⡏⠉⠉⠉⣿⣿⡏⠀⠀⠀⠀⠸⢧⠀⢘⡏⠀⠙⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢦⡄⠂⠉⠋⠀⠀⡆⠀⠀⠀⠀⠀⣴⣿⣦⠀⠉⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠘⠛⠛⠛⠛⠓⠒⠒⠒⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⣀⡀⠀⠟⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡷⠻⠞⡴⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⢀⡴⣶
⠀⠀⢱⠃⢠⢴⣶⢶⣄⢀⡄⡀⠀⠀⠀⠀⠀⠀⠐⢲⡃⠰⠆⣾⢿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠸⠁⠉⠉⣠⠘
⠀⠀⠘⠀⠿⣶⣿⣾⠿⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠩⠟⡷⣾⠯⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⣰⣻⠂⣀
⠀⠀⠀⠆⢈⣿⣿⣿⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⠬⠟⢁⠈⠙
⠀⠀⠀⡆⠸⣿⣿⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠻⠶⠦⣰⢶⣿
⠀⠀⢀⣇⡀⣿⠁⣿⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠔⠳⣴⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠻⠛⢁⡀
⠀⠀⢸⣿⣧⣿⠀⣿⣤⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣖⡲⠻⢏⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠈⠳⠿⠛⠉⠇
⠀⠀⠀⠉⠀⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠞⢧⡀⡨⢪⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⡀⡀⡀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠙⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⡹⣿⡯⣯⣿⣯⡄
⠀⠀⠀⠀⢠⣶⣦⣴⣶⡄⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣄⣤⣤⠀⠀⠀⠀⠀⠀⠀⢉⡷⢉⣙⡙⢙⢙⡹⠙
⠀⡈⣵⣶⣾⣿⣯⣝⣿⣦⣄⢀⠀⠀⠈⠑⠦⣀⠀⠀⠀⣠⣿⣯⣿⣿⢟⣁⡀⠀⠀⠀⢠⣄⠁⡷⠀⠀⣘⡀⠀⣾
⠮⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠵⠀⠀⠀⠀⠈⠑⢦⣶⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⠤⠾⡿⠧⠤⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⠹⠿⣿⣿⠿⣛⠉⠀⠀⠀⠀⠀⠀⠰⣦⣟⣶⣲⣞⣷⣯⠆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠠⠊⢹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠐⢿⣾⣽⣭⣷⣯⣿⠇
⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠁⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⠘⠻⠷⣾⢵⡾⡶⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠠⢠⠥⢠⡀⠀⠀⠀⢀⡀⢐⠀⠀⠀⠀⠀⢇⣠⠆⠀⠀⠀⠠⣀⣸⣇⠀⠀⠂⡀⠀⢂⠎⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⡼⠻⠹⠀⠄⠀⠀⠐⠚⠛⠃⠀⠀⠀⠈⠉⠉⠐⠀⠀⠀⠀⠀⠀⠌⠒
⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⢤⣄
⠀⠀⠀⢠⢲⡀⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢉⡟⢂⣮⠖⣶⡖
⠀⠈⠸⠛⢲⠚⠑⠈⠄⠰⠀⠐⠁⠐⢄⠂⠀⠀⠐⣂⠁⠓⠥⠔⠀⣾⢉⠺⠁⠀⠀⢠⣴⣀⣀⣀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⢿⣍⣹⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡿⠟⠛⠉⣉⡠⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠏⠛⠁⠀⣄⣠⢦⣿⣿⣿⣿⣿⣿⡿⠿⢛⡛⢿⣿⣿⣿⣿⠋⠄⠠⠀⠀⢽⣦
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠊⢴⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡆⡛⢿⢣⣻⣿⡏⠃⠀⠀⣤⠀⠁⢸⣿⡇
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠿⢿⣿⣷⡤⡺⣿⣿⣿⣿⠟⠋⠉⠈⠀⢀⠀⠀⠃⢧⠈⡆⢿⣿⣇⣄⠀⠃⢬⢀⣸⣾⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣥⣠⣼⣿⣿⣿⡿⠿⠛⠉⡏⢰⠻⣦⠀⠀⢸⡀⠀⠰⡘⣤⣇⢸⣾⣿⣮⣻⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣸⣿⣿⣏⣽⣿⡟⢻⠟⠋⣹⣀⢀⣀⣬⣧⣺⡟⢿⣴⣴⣾⣷⣤⣤⣵⣌⣛⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⣿⢻⣿⣿⣷⣾⣷⣿⣿⣿⣤⣿⣿⣿⠟⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢭⡏⠈⠢⠤⠌⠻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⢿⡿⠫⢉⡉⠉⠉⠛⣿⣿⢿⣿⣿⣿⣿⡏⣾⣛⡄⢉⣿⣿⣦⢹⣟⢇⣠⣿⣯⠂⠀⠀⠀⠀⠀⠄⢻⠿⣿⣿
⠀⠀⠀⠘⠟⣽⢛⠀⠉⠉⠀⠂⠀⠀⠀⠻⣷⡽⣿⣯⣿⣿⣿⣿⣇⡸⣿⣿⣿⣷⣬⡠⣉⡛⠿⢦⣥⣀⣀⠀⠀⠀⠘⠓⠈⠙
⠀⠀⠀⠀⣼⣛⣊⠈⠁⠀⠀⠀⠀⢠⡾⠟⣉⢹⣽⣿⡟⢯⠍⠛⠾⠦⠉⣽⠛⣿⣿⣿⣿⣿⡗⠲⣤⣍⣙⠛⠻⠿⠁⠀⠘⠀⣱⡄
⠀⠀⠀⠀⣿⣅⣈⣀⣤⣤⣤⢶⣦⠟⠁⠀⠁⣼⣿⣿⣿⢋⣶⣶⣶⣶⣶⣶⠐⣿⣿⣤⣿⣿⣿⣤⣀⣠⣍⣝⣿⣿⣶⣤⣶⠛⠓⠚
⠀⠀⠀⠀⣿⡆⠛⠮⠙⠷⠟⠋⠁⠀⠀⠈⣠⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⡅⢿⣿⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣉⣿⣿⠏
⠀⠀⠀⠀⠘⢧⣤⡀⣀⠀⠀⠀⠐⢀⣤⢾⣵⣿⣿⣿⢃⣷⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⡿⡃⠀⠁⣉⣾⣿⢻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠳⠶⢆⣀⣂⣈⡿⢞⣿⣿⣿⣿⡏⡸⣻⣿⣻⣿⣿⣿⣿⣿⡆⣿⣿⣟⣫⣅⣤⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⣶⣿⡿⣾⣿⣿⡿⢃⣷⣿⣿⣿⡟⣿⣿⣿⣿⣧⡸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡷⣻⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣷⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢹⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠉⠉⠉⠉⢧⡙⠉⠉⠉⠉⠈⢿

⣿⣿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣶⢦⣀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣄⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣿⣟⣷⣼⣿⣿⣿⡿⢟⣯⣽⡟⠟⡴⢋⣴⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣛⢷⣶⣿⣿⣿⣿⡿⠋⠑⢋⡽⢛⡅⣠⢿⠟⡉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠻⠿⣿⡘⣿⡇⡄⠀⠀⠀⢉⡞⠟⢸⣴⣷⣦⣿⡤⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⠀⣘⣶⣶⣝⠻⡧⢳⣄⣀⢀⣼⣤⡄⣨⡭⠭⣂⣀⡨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠽⣿⣿⠀⢷⣆⠛⠛⠛⢛⡭⠀⢰⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣾⣷⣤⠙⠿⠶⣟⣫⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠩⢭⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⣠⠴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣣⣸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢹⣿⣿⡿⠃⠀⠀⠀⠀⢀⣾⣿⢿⣿⣿⡏⠉⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⢿⣿⣧⠀⠀⠀⠀⠀⠈⣿⡇⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠈⠻⣿⣦⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠈⢻⣷⣄⠀⠀⠰⠿⣧⣀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡇⠀⠀⠀⢿⣿⣷⣦⡀⠀⠀⠉⢿⣿⡟⠋⣉⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⡇⠀⣦⡄⠈⠿⠟⠙⠻⣷⣤⣤⣾⣟⠳⣌⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣧⠀⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⢻⡝⢳⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⢦⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢲⡀⣀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠳⢦⣄⡈⠀⠈⠁⠀⢀⡀⠀⣾⠀⠀⡇⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢠⠄⠀⠀⠀⢰⡆⠠⠉⢶⣶⢲⡶⡾⣟⡀⣿⣲⢴⡇⡞⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠤⠤⠤⠤⠤⠤⠷⠲⣿⡿⠓⠒⠒⢶⢸⡇⠀⡀⣿⠿⠾⠷⠷⠷⠶⠻⠷⣾⡇⣻⠀⠐⣏⠉⢯⡌
⠀⠀⠀⠀⠀⡈⢲⡀⠑⣤⠒⣦⠤⣄⣀⣤⢤⣰⣯⣤⣤⡤⣤⣟⣸⣧⡘⠇⠉⣙⠶⣄⣀⣠⣤⣤⣤⣬⣽⣿⣷⣦⢿⣷⣾⣿⠃
⠀⠀⠀⠀⠀⠼⠷⣿⣿⣾⣿⣯⣦⣙⠻⣷⣶⣾⣷⣶⡦⣰⣯⣯⣰⣍⣻⣆⣅⠿⣆⣿⣿⣯⣽⠿⠯⠽⠿⠛⠚⠉⠉⠁⠤⢠⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠨⠥⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⢶⠖⠣⣬⠾⢄⣒⡬⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣄⣀⢀⢀⠈⠁⠀⠀⠈⠍⠙⠛⠛⠒⠶⠶⠶⣶⣿⣭⢉⡫⣄⠰⢦⣿⡤⢓⣢⠴⢚⡩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠠⠴⠒⢃⣀⣀⣀⡀⢨⡤⠤⠤⠲⠒⠒⠒⣒⣂⣘⣛⣛⢷⣾⣭⣴⣞⠋⡤⠞⠁⣤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠋⣥⣤⣴⠶⠶⠖⠚⠛⣉⣩⣭⡥⠤⠤⠔⠒⠒⠒⠛⠿⠛⠑⠊⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠂⠘⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢄⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡈⠀⠀⠀⢠⠇⠀⡠⣠⠀⡠⢖⡨⢒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢰⢰⢠⣾⠀⢸⣄⠀⠀⠁⣎⣶⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⢢⠸⡜⡜⣿⡄⢆⠉⠳⣄⢂⠘⠯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⢀⠠⠌⡢⣧⡱⣼⡘⣿⣎⣇⡀⢹⡎⣇⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⢎⡦⢾⣜⢿⣿⣷⣘⣿⢸⣿⣿⢃⡟⠔⠀⣸⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠙⡎⢻⣿⣿⣿⡇⣾⣟⡵⢋⣤⣢⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠇⠀⠀⠀⡀⢠⡇⢀⠻⣿⣿⣿⣿⣿⣴⣯⣍⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡛⢳⠶⡗⡤⢿⣦⣷⣿⣿⠿⠛⠛⠛⠉⠩⠑⠂⠫⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠺⠇⣸⠀⠈⠁⢂⡼⠛⠁⠁⠀⠄⠀⠀⠠⠆⠀⡀⢀⡞⡐⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠿⠧⠀⣶⠯⣥⣄⠀⠀⠀⢠⣞⣅⢠⠖⠶⠖⣾⣮⣧⣑⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⡖⠄⠀⣷⠀⠘⣮⡇⣀⠼⠛⠙⣯⣼⠀⠂⢠⡿⣿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⣈⣃⣠⣴⣿⠶⢦⣾⣟⣿⣆⠘⣀⣼⠟⠙⠏⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠚⠉⠶⠀⠀⠘⠋⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠶⣦⡼⣴⡖⣰⣂⣿⡺⣧⠀⠖⠈
⣾⡾⢷⣿⣷⢾⣕⣾⣞⠇⠾⢛
⡁⠴⣹⠿⡿⣾⣿⢻⣿⠃⠰⠞⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⢱⡄⠒⡼⠗⡆⢹⠿⡟⢳⡁⠐⠿⠀⠀⠀⠀⣠⠾⣛⣛⣻⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡺⡛⠿⠍⠉⠀⠐
⠀⠈⠙⠖⠀⢸⠈⡗⠀⠈⠀⠀⠀⢀⡤⠶⠟⠯⡁⠀⠀⠐⢒⠒⢒⣘⡳⠶⢦⣤⣤⠶⠟⠉⠉⠡⠀⠀⠒⠀⡀
⡠⢤⣤⠤⠤⠾⠐⣧⣤⣤⣤⣀⣼⣉⣀⣐⡘⠒⠒⠆⠤⠤⣭⣤⣄⣤⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠡⣤⠀⠸⢿⣤⣟
⠉⠙⣛⠚⠉⠛⠒⠛⠞⠉⠽⠯⠝⠛⠻⠿⡶⢤⣄⣒⣲⡔⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⡤⢂⠔⠁⠈⢲⠊⠻⣿⠓⠛
⠉⢉⠩⠖⠒⠒⠤⣀⣈⠁⠒⠒⠆⠈⠀⠀⠀⢈⠉⣻⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡃⠸⠀⣀⠈⠉⠀⢆⠀⠈⠃
⠀⠀⠀⠀⠐⢢⡖⠀⡰⠅⠐⢤⡀⠀⠀⠀⢸⠵⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⡸⠴⠋⠀⠀⠀⠀⠀⠈
⠒⠀⠀⠀⠲⢭⣵⡶⠶⡖⣶⣾⠛⣣⣴⡟⠋⠈⠀⢀⣀⠀⠀⡠⠀⠀⢀⡠⠊⠁⠀⣠⠔⠉⠀⠀⠀⠀⠀⡀⠀⣰
⠀⠀⠀⠀⣀⠘⠋⠁⢠⣿⠗⠛⠉⠑⠁⠑⠄⠀⠀⠃⡁⠉⡶⠁⢠⠊⠁⠀⡠⠔⠊⠀⠀⠀⠀⡀⠀⠀⠀⠈⣧⠃
⠄⠄⡰⠚⡄⠐⣄⠂⢿⠀⠑⠄⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠣⠀⡃⣀⡜⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡲⠐
⠀⣰⠁⠨⠁⣰⠃⠀⡐⠀⠜⠀⠀⣔⣈⡀⠤⠈⠀⠀⠞⠉⠃⠘⣧⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠧⠀⡀⠨⠀⠀⠀⡀
⠀⢉⠀⠰⣤⡃⠀⣀⡁⠀⠀⠀⠀⠀⠀⢀⡤⠖⠀⢠⢄⠂⠀⠀⠇⡏⠟⠀⡀⠀⠀⡀⠀⠀⠀⠁⠀⠀⡠
⠀⣸⣧⣀⣸⠉⢘⡟⠇⠀⠀⠀⠀⠀⡴⢋⠇⢠⣤⠊⠀⠀⠁⠀⡀⡄⡀⠀⠀⢀⠠⠀⢀⠀⠀⠀⡠⠀⢀⡀
⢸⣿⡗⢻⣀⣄⣾⣇⢘⠦⠀⠀⠀⠀⡟⡙⣴⣭⢨⢆⢣⠊⢀⠀⢂⠈⣅⢠⠀⡇⠀⠨⡆⠂⡠⠀⠁⠀⠫⣄⠀⠀⠈
⢹⣿⠀⠀⡤⢿⣾⢝⠳⠃⠈⠢⡀⠀⠻⣗⣿⡥⠈⣡⣡⠓⡠⠀⢨⡄⣠⡨⡄⠁⢔⡂⠉⠀⠀⠀⠀⠀⠀⡈
⡼⠏⠀⡜⠀⠀⣿⡤⢟⣤⡴⠶⠌⢦⠀⠭⣼⣟⠈⠀⢈⠋⠁⢀⡜⡠⠈⠁⠈⠁⠐⠀⠀⠀⠀⠀⠀⢰⠀⢠⡄⠊⢁⠀⢠
⠼⣷⡿⠬⠥⠀⠘⣅⠀⠸⡏⠲⠶⣒⠒⣀⢉⠩⣷⣦⣼⡀⢄⡥⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⡇⠀⠠⠀⢀
⢰⠋⠀⠈⠅⠀⢐⠈⢦⡒⢷⣦⣤⣍⠁⠀⠀⠀⠀⠉⠀⣀⡐⡀⠠⠤⠀⠀⢐⡀⡈⠡⠤⠀⠀⡀⢀⡴⠉⣁⢓⡀⠀⣧⢠
⣿⣤⣀⠈⠃⠀⠀⢀⣚⣿⠤⠤⠭⠼⠒⣒⣀⣀⣈⠭⠤⠄⠒⣒⣒⣂⡩⠭⠄⠒⠒⠂⠈⣁⣀⣀⣀⣩⢍⣁⣰⣶⣤⣤⢤
⢤⣬⣽⣃⣛⡉⠭⠤⣐⣒⣒⡢⠬⠬⠥⣐⣒⠒⣒⠂⢬⡭⣁⣂⠠⠴⠖⠒⡖⠚⢲⣍⣉⣁⣠⣾⣁⣹⡧⠤⠤⠶
⠽⣟⣒⣒⣲⣭⣭⣑⠖⠐⠶⠶⠺⣿⣿⣿⣞⡋⣉⣉⣽⣯⡿⣷⠶⠖⢞⠛⡟⠛⡋⠁⠙⠃⠀⠂

⠀⠀⠀⢀⣤⡀⠀⠀⢀⣤⣶⡟
⠀⠀⠀⠨⣿⣷⢀⣼⣿⡿⠿
⠀⣀⣤⣶⣿⣿⣿⣿⣿⣶⣤⣤⣄⡀
⠐⢋⣥⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢠⣿⣿⣿⠛⢿⣿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠠⣤⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⡾⣿⠛⠀⠀⢸⣿⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣿⣥⡄
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠺⣷⣶⣤⣀⠀⠀⠛⡿⣿⢿⠁
⠀⠀⠀⠀⠀⣻⠇⠀⢀⣀⣠⠤⠶⠚⠛⠉⢻⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣠⠶⠋⠁⢀⣀⣀⠈⠻⣿⣿⣿⣦⣤⣀⢻
⠀⠀⠀⠀⠰⡿⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣿⣿⣶⣶⣤⣤⣤⠤⠶⠚⠋⠙⢿⣿⠟⠋⠁⠀⠀⠀⠨⢿⡿⠁⠀⠈⠻⣝⠻⢿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⣼⡏⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⠁⠀⠈⠁⠘⣶
⠀⠀⠀⣴⣿⣾⣷⠾⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⠶⣷⣶⣶⢿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⠻⡶⠶⢷⣶⡶⣿⣷
⠀⠀⠀⣿⣿⡟⡡⠒⠒⠛⠛⠻⣿⣿⣿⡟⠻⣽⣿⡿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⣖⠿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⣛⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣟⣅⣀⣀⡰⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠿⠿⠯⠭⠀⠀⠀⠀⠘⣿⣿⣿⡏
⠀⠀⠀⣿⢣⣯⣤⣼⠯⠤⠤⠈⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣀⡀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠓⠒⠀⠀⠒⣒⣒⣒⣒⡒⠀⠀⠈⠙⠲⣄⣀⢠⣈⣿⣿⠇
⠀⠀⠀⣿⣿⣿⣿⢁⡀⠀⠀⢀⣴⠀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣄⣀⣀⣀⣀⣀⣀⣈⡉⠉⡉⢉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠻⣿⣿⣽⣿⡇
⠀⠀⠀⢸⣿⣿⡷⣋⣴⠶⠓⣸⠇⣰⣯⣶⣿⠏⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⡄⢠⣡⣦⠀⡀⠀⠀⣼⣵⠋⡀⠀⠀⠀⠀⠀⣉⡉⠉⠉⠙⢷⣼⣿⣇
⠀⠀⠀⣿⣿⡿⠟⠻⠶⢾⣯⣿⣿⣿⠟⢋⡡⢸⡏⠁⢀⡠⠀⠐⠀⠀⢄⡙⣾⣿⣿⣿⣷⠞⡓⠀⢰⣿⣷⣿⡿⠉⠉⠋⣀⣀⣤⣽⣿⣶⣤⡈⢿⣿⡏
⠀⠀⠀⠹⠛⠛⠛⠛⠛⠛⠚⠛⠛⠓⠀⠛⠛⢿⠇⠐⠋⠓⠒⠒⠒⠚⠓⠋⠛⠻⠿⠟⠉⠋⠉⠀⠻⠛⠛⠛⠓⠂⠀⠀⠉⠉⠛⠛⠛⠉⠉⠙⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⣶⣶⣾⣷⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⢋⡄⠀⠀⠀⠀⢲⡴⠋⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⣠⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠘⢃⠀⠿⢿⣿⣿⣦⡾⠁⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠻⣿⣿⡥⢾⠀⢠⣶⡇
⠀⠀⠀⠀⠠⠀⠀⠸⠀⠀⠀⠀⠀⠈⠳⠀⠘⠟⠁⣿⠇⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠙⠻⢿⣧⠀⠀⠀⠀⠀⢠⠀⠀⠉⠉⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣷
⠀⠀⠀⠀⣰⠿⠁⠀⠈⠉⠀⠀⠀⠀⠈⠈⠓⢀⡀⠤⠤⠄⠻⠀⠀⠀⠸⠹⡤⠀⠀⡐⠀⠔⠈⡇⠀⠀⠀⠀⠀⠲⠈⠀⢀⣀⡀⢻⡆
⠀⠀⠀⢠⠏⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢰⡆⢛⠶⠄⠀⠀⠀⠀⢀⠑⠀⠀⣶⣄⠀⠁⡠⠂⠀⠀⠀⠀⠀⠀⠠⣍⠡⣦⡁⣀
⠀⠀⠂⠈⢢⡈⠛⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠰⠋⣠⡎⠀⠀⠀⠘⢿⡀⡠⠖⣁⡠⣴⠊⠀⠀⠀⠀⠀⠀⠙⢶⣾⣿⣶⣶⠟
⠀⠀⠐⠗⠤⣅⣀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠀⠀⠀⠀⠀⠀⠀⠻⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠉⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣽⣯⣯⣭⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠀⠈⠉⠉⠉⠉⠁⠂⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⣀⡀⠠⠤⢀⣀⣀⡀⠤⠄⢀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠉⠙⢿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡟⣿⣿⣧⣄⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣄⠙⢿⣿⣯⣝⣛⠛⠻⠿⠿⠟⠛⣛⣫⣽⣿⡿⠋⢠⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣉⠀⣿⣷⣄⠉⠛⠿⠿⣿⢿⣿⣿⡿⢿⠿⠻⠛⢉⣠⣶⣷⠀⣈⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⣿⣿⣿⣿⣿⣸⠑⠒⠂⠤⠄⠄⠀⠤⠤⠐⠒⠋⣿⣿⣿⣿⣿⣿⢿⡋
⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⢷⠸⠘⡀⠈⠙⠻⢧⣖⣀⣀⣀⣀⣀⣀⣀⣀⣲⡬⠟⠋⠁⢀⠃⠆⡾⠉⠒⠢⢄⡀
⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⢸⠀⡆⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⢠⠀⡇⠀⠀⠀⠀⠙⢢⡀
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢠⡇⠀⠀⠀⠀⠀⠀⠹⡄
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡔⠉⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠉⢢⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⡀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠄⠀⠀⡤
⠀⣼⡁⠀⣠⣟⡅⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠘⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣴
⠀⠀⠁⠀⠀⠈⠙⢧⡖⠀⠀⠀⢀⠀⠉⠙⠦⠀⠀⠀⠀⢠⢶⣀⠀⠀⠈⢽⠦⠀⠱⣄⠀⠈⠓⠂⣤⣶⡀⠀⢀⡀⠸⢳⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⣰⡟⠂⠀⠀⡴⠛⠀⠀⠀⠉⠈⠉⠂⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⢐⠝⠋⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠀⠀⢲⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠐
⠀⢀⣄⠀⠀⣀⢀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠛⠛⠉⠁⠙⠛⠛⠿⠶⢷⣶⣷⣤⡀⢀⡄⣄⡄⠀⠀⠀⠀⠈⠛⠲⠶⣤⣤⣀⣀
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠘⠘⠻⠟⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠳⠶⢦⣤⣤⣀⣀⣀⡤
⡀⠀⠀⠒⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⣟⠉⣹⣛⣟⣟⣛⣻⣛⣛⣛⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣛⣛⣿⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⣿⣿⠿⠿⠿⠿⠀⠺⠿⠿⠿⢿⣿⡆⠀⢀⣿⣿⡆⠀⠀⢸⣿⡿⠿⠿⠿⠇⠀⠿⠿⠿⠿⠿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⣧⠀⠀⢸⣿⡇
⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡟⢸⣿⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⢸⣿⡿⠿⠿⠿⠃⢰⣿⡇⠘⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⣼⣿⢠⡆⣿⣿⠀⢸⣿⡇
⠀⠀⠀⣶⣶⣶⣶⣿⡿⠀⢸⣿⠀⠀⠀⠀⠀⣿⡿⢸⡇⢸⣿⡄⢸⣿⣷⣶⣶⣶⡆⠀⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡛
⠀⠀⠀⠀⠀⢀⣤⣄⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⣤⠀⢰⣶⣾⣿⣿⣷⣤⣄
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣧⣤⣀⣠⣄⠀⢀⣤⣀⣸⣷⠀⢀⣀⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⣶⠶⠆⢲⣄⡶⢰⡶⢶⡆⣶⠀⠀⡶⠶⣦⢰⡶⢶⡄⢰⣶⠀⠶⣶⠖⢰⠀⣶⠶⣶⢰⣆⢰⡆
⠀⠀⠀⣿⣦⠄⠈⣿⡃⢸⣇⣸⡇⣿⠀⠀⡇⠀⣿⢸⣧⡼⠇⣼⢻⡄⠀⣿⠀⣼⠀⣿⠀⣿⢸⣿⣼⡇
⠀⠀⠀⣿⣀⡀⣸⠟⣧⢸⡏⠀⠀⣿⣀⡘⣧⣀⣿⢸⡏⢷⢀⡿⠺⣇⠀⣿⠀⣿⠀⣿⣀⣿⢸⡇⢿⡇
⠀⠀⠀⠈⠉⠁⠉⠀⠈⠈⠁⠀⠀⠉⠉⠀⠉⠉⠁⠈⠀⠈⠀⠁⠀⠁⠀⠁⠀⠈⠀⠉⠉⠁⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⠀⠀⢀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠄⢤⡄⠠⠄⠄⣀⣀⣀⣠⣤⣠⣤⣤⣤⣴⣦⣿⣇⢶⣶⣶⣍⣙⠲⣾⣶⣿⣿⣷⡷⠙⢋⡄⠈⠀⠈⠀⠀⠀⠀⡀
⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣷⣶⣤⠘⠳⣷⠃⣔⡤⠠⠀⠉⠀⠂⠀⠀⠀⠀⠀⠆⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣻⣿⣷⣤⡘⢻⠿⢰⣧⣳⡅⣴⠀⣒⣜⢡⣶⢐⢀⡖⣠⡱⣶⠀⢔⠀⣀
⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣻⣦⣆⣉⠽⢻⢿⣾⣹⡁⣼⡷⣸⢎⣏⡿⠿⣇⠠⣽⠆⠈⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣫⣴⣾⠊⡹⠓⠘⢻⣷⠷⢠⣧⣤⣼⣏⣼⣧⢖⠐⡊⢰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣄⣵⢾⡋⣬⡖⠳⡼⣻⡿⣼⡿⣈⠛⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣯⣿⣿⣻⣧⣿⣧⣮⡐⠖⠏⣿⣾⣿⢴⡖⢡⢶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢼⣹⣿⣿⣿⣿⣿⣿⣷⣿⣼⣿⣿⣿⣿⢿⣿⣷⣿⣯⣿⣿⣿⣿⢂⠀⠉⠕⠐⠨⠏⣾⠆
⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⣏⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⡾⣿⣿⣿⡿⣿⢋⡽⠫⢽⡤⣶⡀⠰⠷⠀⠀⠛⠠
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠑⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠑⠛⠁⠚⠛⠘⠓⠛⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⣿⣿⡿⠁
⠄⠀⠀⠀⠰⠾⣧⠀⠀⠀⠀⢸⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⢀⣀⠀⠀⠀⡠⠤⠄⠀⠀⠀⠀⡩⠟⠃⠒⠒⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠉⠁


⢀⠀⢁⡂⠤⣀⠀⢄
⡀⠙⢦⣈⠙⠢⢭⡑⠲⠤⣀⡒⠠⠍⣱⠲⠇⣐⢂⠠⠄⣀⡀
⠙⢦⡀⠈⠓⢤⡀⠈⠓⠦⣀⡉⠛⠢⢤⣉⠓⠲⠬⣉⡓⠲⠮⣍⣓⡒⠶⢤⣀⣀
⠀⠀⠙⢦⡀⠀⠙⠦⣄⠀⠈⠙⢶⣄⡀⠉⠛⠶⣤⣀⠉⠑⠲⢤⣌⠉⠛⠲⠤⣍⣙⠓⠶⠤⣤⣀
⢳⡀⠀⠀⠙⣦⡄⠀⠈⠳⣤⡀⠀⠈⠛⢶⣦⣀⠀⠉⠛⢶⣤⣀⡀⠉⠓⠶⣦⣀⡉⠙⠻⠶⣦⣍⡙⠛⠶⢦⣤⣀⡀⡀
⠈⢿⣆⠀⠀⠈⢿⣄⠀⠀⠈⠳⣦⠀⠀⠀⠉⠻⢷⣀⠀⠀⠀⠙⠻⠧⣠⣄⠀⠉⠛⠷⣦⣄⣀⠈⠙⠳⠦⣤⣀⠈⠉⠛⠷⢦⣤⣀
⠀⠈⢿⡇⠀⠀⠀⠹⣧⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠉⠛⢿⣦⣄⠀⠀⠈⠛⠷⣦⣄⡀⠀⠉⠛⠶⣴⣀⠀⠉⠙⠛⠶⣤⣀⡀
⡀⠀⠈⢿⣄⠀⠀⠀⠈⢶⡆⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠉⢻⣶⣄⠀⠀⠀⠀⠈⠛⢷⣆⣀⠀⠀⠀⠙⠻⢧⣤⡀⠀⠈⠙⠻⢷⣄⡀⠀⠀⠉⠛⠷⣦⣄⡀
⡇⠀⠀⠈⣿⡇⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠈⠻⣷⣦⡄⠀⠀⠀⠀⠉⠻⢿⣦⣄⠀⠀⠀⠉⠻⢿⣥⣄⠀⠀⠈⠛⠷⣦⣄⣄⠀⠀⠙⠻⢶⣄
⣿⣦⣤⣤⣼⣿⣦⣤⣤⣤⣬⣿⣷⣤⣤⣤⣤⣴⣿⣶⣤⣤⣤⣤⣤⣬⣿⣷⣶⣤⣤⣤⣤⣤⣤⣽⣿⣷⣤⣤⣤⣤⣤⣬⣿⣷⣶⣦⣤⣤⣤⣽⣿⣿⣷⣶⣤⣤⣬
⣿⣿⣇⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⡏⠀⠀⠀⠀⠀⠓⢻⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⡜⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡔⢹⡀⡇⠀⠀⠀⠀⡜⡇⠀⠀⠀⠀⠀⢇⣠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠁⠀⠀⠀⢸⠁⢱⠀⡀⠀⠀⠀⠘⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⣀⣊⠀⠀⠀⠀⠀⡜⠑⠃⠀⠈⠚⠹⡀⠀⠀⠀⠀⢱⢀⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠈⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠖⠇⠀⠀⠀⠀⠀⠀⠀⠸⠊⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠈⠒⠤⣀⣀⡠⠴⠊⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⣟⠒⠤⢄⣀⣀⠤⠤⢄⣀⣀⡠⠤⠒⣾⡵⢮⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡏⠏⡿⡀⠀⠀⣰⠁⡆⡞⢝⡟⣄⣀⡀⠀⡿⣝⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠯⠼⠒⠊⠉⠘⢄⠣⣉⡭⢃⠇⠀⠈⠉⠉⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⣿⣼⣿⣻⣟⣩⣿⣾⣀⢴⡿⣿⠯⢀⣩⡐⢺⠦⣿⠙⣽⣾⢻⣟⣹⡯⣿⢻⠤⠦⢀⣶⣂⠤⣤⣤⣄⠀⠃⣶⣴⡄
⠀⠀⠀⠀⠉⠉⠁⠉⠉⠁⠉⠉⠀⠀⣩⣿⣿⡿⠻⠙⢷⣄⡙⠀⣇⣼⡞⣯⣿⣷⣿⣨⣧⡏⠈⠁⠝⡘⣯⡿⢹⠋⢰⢞⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡟⢋⣠⣤⣦⣿⣿⣿⣀⠈⠭⠵⠯⠌⠸⠟⠟⠇⢤⡤⠙⣿⣿⠇⠀⣴⣗⠼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣧⠀⢸⡧⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣯⣿⣻⣿⣿⢟⣻⣿⣿⣷⣯⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣼⡇
⠀⠀⠀⠀⢀⡀⣴⢿⣿⣿⣿⣿⣿⣿⡛⠩⡽⠎⠈⠓⠊⠁⠉⠙⢼⡽⠃⢘⣿⡯⣵⠉⢻⣿⣷⣄⡀⠀⠀⠀⠀⠉⠛⣻⡷⠇
⠀⠀⢰⡾⣏⡿⢋⣿⣿⡟⣿⣿⣿⠁⠂⠀⠀⢀⣀⠀⠀⠀⠀⢠⡶⠁⠀⠈⠉⠁⠀⠀⠈⠴⠬⠻⣧⡀⠀⠀⠘⠳⠾⠙⠇
⠀⢠⢤⣷⡽⣤⣿⣿⣿⣴⣿⡿⠉⠀⠀⠀⣀⠜⢿⣦⠙⢷⣤⣸⣿⠳⠤⠀⠀⠀⣀⡔⠀⣀⣤⠴⠒⣬⣶
⠀⢹⡄⢿⣷⣿⣿⣿⣿⡿⣿⡇⠀⠀⠘⠋⣁⡈⠀⠙⠃⠘⠻⠿⠌⠳⣄⠈⠶⠬⠄⡁⠈⠙⢁⣴⢋⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠈⣿⣷⠙⣿⣻⣿⣿⣿⣿⣷⣤⠶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣤⣴⣽⣿⣦⣤⣤⣬⣷⣤⣥⣈⣙⣳⢦⡆⠀⠀⠀⣿⣿⠋
⠀⢀⣿⣿⠀⠛⢿⣿⠻⣿⣿⣾⣧⢀⣿⣿⣿⣿⣿⡿⣻⣾⣿⣿⣿⣿⠿⣻⣯⣥⣼⡿⣿⣦⢈⠉⢭⡿⠎⠀⠀⠀⠀⠈⠉
⠀⠈⣽⣿⣧⢀⣠⠙⣷⡌⠛⠚⢿⣿⣿⡇⣾⣿⣿⠷⠾⣫⣾⣿⣿⣿⣿⠿⠋⢉⣿⣿⠇⠀⠀⠀⠄
⠀⠀⠸⣻⣿⣦⡈⢁⣙⠻⣦⣀⠈⠉⢻⡇⢸⣿⣯⡿⠟⣵⣿⡿⢃⣴⣶⣶⠏⢉⠛⠉
⠀⠀⠀⠹⣿⠿⠿⢦⣝⣄⠀⠉⠳⣀⣠⣇⢸⣿⠏⢀⣠⣷⣻⠿⠋⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠸⠟⠙⠋⢂⠰⡶⠾⣿⡧⢿⡇⠀⣰⣶⣦⣤⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠵⣮⠓⠉⠁⢾⡇⠀⠀⠀⠤⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡤⠀⠀⠀⣻⠟⠃⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡠⠊⣀⡿⠒
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⣀⡀⠀⠀⠀⠀⠐⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢐⠋⠉⠻⠆⠰⠀⠽⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⡟⠑⣶⠃
⣀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆
⠀⠀⠈⠉⠐⠂⠀⠨⠉⠉⠉⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⣠⣤⣤⣄
⠀⠀⢀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠾⠟⢻⣎⠉⠁⠀⠉⠀⠀⢀⣦⣤
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠞⠉
⠀⡠⣠⡀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⢤⣶⠂⠀⠀⠉⠉⠀⠤⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠚⠛⠂⢀⣄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⢷⣦⣀⠖⠀⠀⠀⠈⠉⠀⢀⠐⠓⠓⠂⠀⠀⢀⡀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣷⣦⣤⣄⣀⠉⠛⢲⣿⣧⣀⣤⣟⣉⣁⣄⡀⠀⢀⡟⣿⡀⠀⠀⠺⠛⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⣷⣦⣤⣤⣀⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣛
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡰⠄⢞⠃⠀⠘⡟⢠⣖⣐⣾⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⢟⡿⠖⠀⠀⠀⡀⢠⠄⢘⡃⢘⡆⠀⣰⣷⣷⣿⢷⡿⢯⡍⢿⣶⣿⣿⠆⠀⠀⣀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⣨⡍⠛⠚⢃⣨⣿⡛⣤⡼⠏⠈⠉⠉⠉⠈⠳⠏⠈⠉⠙⠛⠛⠿⣿⠾⣁⡄⣾⡿⢴⣿⣶⣷⣶⣷⣤⠀⡠
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣿⣷⣿⣮⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠀⠉⠻⣿⣿⣀⣌⠉⠉⣽⡿⢿⠿⠟⣧⠆
⠀⠀⠀⠸⣥⣰⣷⣷⣿⣿⣦⣻⣿⣿⣿⣿⢋⢶⣶⣤⣄⠀⠀⠀⣀⣠⣤⣾⡿⠈⣿⣿⠅⢠⡄⠀⢀⠀⠈⢿⣿⣿⢆⠼⢿⣿⡌⠘⠀⠉
⠀⠀⠀⠀⢍⠛⠛⠉⣉⣉⣉⣻⣿⡟⠋⠁⠓⠶⠤⠷⠶⠶⠉⠛⠋⠉⠋⠉⠿⠷⠶⠦⣤⣭⣄⡀⡠⠠⣀⠀⠉⣽⣶
⠀⠀⠀⠺⡀⢰⡿⠛⠿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠣⢶⡦⠀⠀⠀⠀⠩⠀⠰⠿⣿⡿⣶⠶⠶⢦⣄⣠⡄
⠀⠀⠀⢦⣼⡃⣀⡀⠀⠘⢫⣟⣿⣿⣿⣷⠀⠀⠀⠀⢀⠘⣾⡴⡆⣳⠿⢀⣀⠤⡇⠀⠀⢀⣀⠀⠀⠓⠠⠂⠀⠘⠻⣏⠀⠀⠀⠀⠉⠻⣶
⠀⠀⠀⠙⠁⠉⣉⠉⠙⠛⣛⣯⣿⣿⣿⣿⣷⡄⠀⠍⠙⠓⠿⢦⣯⣿⠾⠿⡦⠄⠀⠀⠈⢡⣏⠀⠀⠀⣠⣶⣶⣶⣶⣿⠀⠀⣾⣿⣿⠀⠈
⠀⠀⠀⠀⠀⠉⠛⢿⢄⣙⣿⣿⣿⣿⣭⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠨⠉⠀⠀⠈⢁⣤⣬⣻⣿⣿⣿⣿⣿⣧⣀⣀⡈⠉⠀⠈
⠀⠀⠀⠀⠀⢲⣦⣸⣿⠿⣭⣴⣿⠿⠤⣄⣉⠈⠉⠙⠿⣿⢷⣶⣶⣦⡤⠤⣤⡤⢤⣶⣶⢶⡁⠼⣿⠟⠧⣤⣤⡭⢛⡿⣟⠉⠹⠛⠗
⠀⠀⠀⠀⠀⠀⠙⠿⠒⢄⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠋⣄⢺⣿⢿⣦⣤⣿⣷⣤⠀⠤⣼⡧⠀⠙⠂⠀⠈⠛⠀⠀⠙⠚⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠀⢶⡦⠀⢫⣽⣽⠲⠆⠴⠿⠆


⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉


⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠂⠀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠤⠄⠀⠀⢀⣤⣤⣶⣶⣶⣶⣶
⣾⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠁⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣿⠿⡿⢿⣽⣿⡿⢷⣶⣶⢶⣦⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⣿⣿⠿
⣛⣃⣤⣜⢢⣍⡑⠠⣴⣰⣖⣲⡾⠿⣋⠽⠿⠻⣿⡻⣟⠛⡫⢽⢶⡄⣶⣶⣦⣤⣤⣀⣀⣀⣀⣀
⠋⠹⣿⣿⣿⣶⣾⣿⣾⣿⣿⠿⣿⣷⣾⣾⣟⣸⣿⣷⣿⣿⣿⣿⣷⡿⣽⣳⣟⣻⢹⢭⢙⢙⡻⠻⠿⢿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⡀
⣿⣿⣿⣿⣟⣛⣿⣿⣯⣭⣿⣴⣾⣿⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣷⣿⣖⣻⣿⣯⣤⣖⣾⣿⣯⣭⡿⢶⣦⣄⣀⣠⣭⣛⣾⠖⠛⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⢿⣿⠿⡻⣿⣿⣟⣻⣿⡿⣿⢿⣟⠻⣿⣿⣬⣭⣍⣽⣿⣶⣿⣿⣿⣿⣿⣿⣭⣿⣿⣽⣶⣿⠿⣯⣴⣮⣽⣿⣿⣽⣿⣭⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢾⣟⣿⣿⣿⣿⣾⣿⣿⣿⣯⣿⠿⢿⣷⣶⣶⢬⣭⣻⣿⣶⣏⢭⣭⣤⡼⠟⠿⡻⡿⣿⣿⣿⣿⣿⣿⣟⠛⣛⡿⢿⣿⣿⣷⣾⣿⢿⣿⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣿⣿⣿⣿⣶⣯⣿⠻⣿⣿⢿⡻⢻⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣿⣿⣽⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣾⣿⣱⣾⣮⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⢿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠃⠛⠛⠛⠛⠛⠂⠙⠛⠓⠓⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠠⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⣀⡀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⣀⠀⠀⠀⠠⡴⢦⡚⠟⠛⠛⠋⠁⠉⠙⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⢉⢄⠠⠄⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠦⣼⢇⣨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣯⡇⠁⢘⡆⢀⠀⠀⡬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠈⠉⠉⠋⠙⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠆⠦⠼⢀⠀⢠⠃⠐⠃⠀⠀⠰⠃⠀⡠⠈⢠⠰⢿⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡜⠂⡌⡿⠃⠀⠀⠀⠀⡀⠀⠐⠈⠃⠀⠘⢸⡄⢣⢹⡀⠀⡤⠂⡄
⠀⠀⠀⠙⠀⠀⠀⠀⢠⡄⠉⠙⠛⠋⠁⢀⣀⡠⠄⡎⡠⣶⡄⠀⠀⡇⠀⠀⡎⢧⣸⠀⡇⠀⠉⠈⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⠉⠀⠀⣰⣆⢷⠶⣧⣣⣪⣼⣄⢀⣘⣌⠻⡦⠇⠀⢀⠀⠀⡀⠆⡀⣀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⡶⠿⠷⣿⠿⠿⠿⠾⣇⢽⠟⠛⣿⠋⠛⠟⠟⠛⠈⠀⢠⢿⠉⠁⠀⠈⠁⠘⡁⠈⠀⣸⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠉⠉⠀⠀⠀⠀⠀⣄⠀⠀⠈⠓⠓⡟⠃⠀⠀⠀⠀⢀⣠⣴⣿⣭⡀⠀⢀⣠⡴⢟⣣⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢹⢿⠁⠀⢀⣄⣰⡗⠆⠀⠀⠀⣴⣿⣿⡁⠹⠿⢗⣛⡩⡵⠞⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣤⡀⠀⠀⠀⠀⠀⠈⠛⢹⡇⣁⠀⠀⠀⠈⠀⠉⠉⠁⠈⠛⠳⠦⠒⠊
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠠⠆⠀⠀⠀⠁⠛⣷⡄⠀⠀⢀⣀⠶⢿⣛⡈⠓⣄⡀⠀⠀⢀⣤⠐⠀⠄
⠀⠀⠀⠀⠀⠀⢰⣏⠤⠀⠀⠀⠀⠀⣾⠋⠤⢻⠀⠀⠘⢭⣕⢠⣬⡍⢘⡏⠀⠀⢀⣿⠃⠀⡀
⠀⠀⠀⠀⠀⠀⣿⡏⠒⢀⣀⣤⣶⡶⠃⠀⠀⣺⡇⠀⠀⣸⡏⢸⣿⠇⠈⣿⠀⠀⠀⢿⣧⣠⡁⢀
⠀⠀⠀⠀⠀⠀⢸⣏⡾⠯⠛⠟⠋⠀⡀⢐⣴⡿⠀⠀⠄⡿⠁⢸⣿⠀⠀⢹⡄⠀⠀⢠⡙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⡀⢴⣮⠟⠀⠀⠀⣸⠇⠀⢸⣿⣀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠘⠚⠉⠀⠀⠀⠀⢀⢃⣠⠞⠁⣿⠈⠳⣄⢸⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣮⠿⠁⠀⠀⠀⠀⠀⠈⠹⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠀⠀⠀⠀⣼⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠇⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⠀⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣀⡀⠈⢯⡝⣳⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣛⣳⣦⣌⠳⠃⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠹⣟⡯⢷⣚⣷⡀⠀⠀⠀⣰⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⡀⠀⣰⢿⣇⠀⠀⣿⠀⣮⡐⢊⣷⡀⠀⠰⡏⢞⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⠇⣰⡯⠥⢜⣦⢀⣿⠁⡇⢳⠐⣚⡇⠐⡶⣝⢦⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⣟⣠⡈⢗⣻⠾⠒⣸⠁⢸⡇⣲⡇⠀⣿⠽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠧⠹⡾⣟⠦⣘⠽⣰⢏⡯⢸⢠⣿⢁⡼⠗⢀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠰⣦⣄⡀⠀⢻⡍⢆⠈⢳⡇⢰⡃⡟⠬⣧⣞⡫⠁⠰⣲⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣳⠙⣞⣹⡄⢸⠿⢸⠰⡀⠁⡼⠂⡇⢨⠿⠵⠂⣐⡽⠃⣼⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⣿⣢⠷⣛⡉⡾⢘⠅⠀⠊⠀⢳⠈⢡⣶⣨⡟⢠⡀⠳⡚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣦⢻⣓⡜⢁⣵⠂⠀⠀⣰⡀⠳⡏⢻⢰⡇⢸⡽⣦⠹⠃⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⠾⠍⡜⠘⣇⠞⠀⣸⠅⠀⠀⠀⠀⢻⣀⠀⠸⡍⣻⣞⡫⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢉⡞⠀⠸⡏⠀⠀⣺⠀⠀⠀⣀⠀⠀⠡⡷⠀⢻⠉⠭⣲⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡒⡾⢀⡧⠀⠀⠀⠀⢀⠀⠀⠀⠐⡀⠀⠀⠀⠅⣸⣸⡟⠁⠘⠋
⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⡀⣯⡀⢷⠨⠯⠄⠀⠀⢀⡽⠀⠀⠀⠪⢆⠀⠀⠀⢠⣏⣾
⠀⠀⠀⠀⠀⡞⡅⡀⡀⠈⠉⠙⢷⠘⡆⠀⠁⠀⠀⣲⠍⠀⢀⡀⠘⡵⡃⠀⢀⣘⣧⢿⠑⠒⠒⠋⠉⠛⠒⡆
⠀⣀⣀⣀⡠⣗⡁⣇⣧⣷⣒⡒⠛⠉⠉⢱⣀⡀⢈⣴⣖⣋⡉⠙⠲⡌⠕⡾⠁⠈⠉⠉⠓⣶⢦⡆⣦⡆⣄⣿
⠉⠛⢛⡿⠿⣿⣻⣿⢯⠧⣖⢝⣆⢠⣄⣄⠀⡽⢁⣖⣒⡀⢍⢻⣧⣽⣎⣴⣰⡖⣤⡄⣾⣯⣧⣿⣿⣇⣏⡿⠮⠤⠤⠄
⠀⠀⠀⠀⠉⠁⢺⣏⣇⢧⣏⡼⣿⡾⣿⣿⣇⣇⢯⠯⣿⣳⢮⠇⣿⡿⠯⠽⢿⣇⣿⢸⣿⣧⣼⣿⡛⢻⠩⠭⠭⠭⠭⠭⠭
⠀⠀⠀⠀⠀⠭⢿⣛⣯⡿⠥⢾⠟⠻⣿⠟⢙⣟⠦⢍⣓⣒⣫⡾⠯⠍⠀⠀⠉⠭⢛⣾⣯⣭⣿⣷⣾⣦⣤⣀
⠀⠀⠀⠀⠀⠉⠩⠒⠲⠾⠟⠋⠉⠀⠀⠀⠀⠀⠬⠭⠭⠿⠭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠙⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢳⣄⠀⠀⠀⠀⠀⠀⢀⡤⢚⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⣾⣾⣿⣿⣿⣶⣤⣤⣾⣍⣴⠋⣀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡱
⠀⠀⢀⣠⣤⣶⡾⢿⣿⢿⣿⠿⠿⣿⡿⡿⢿⢿⡟⣻⣿⡿⣿⢿⢿⡿⣿⣿⣷⣤⣄⣀⣀⠀⠀⠀⣰⠋⣰⠁
⠰⢾⠋⠩⣽⣿⠃⠀⠉⢷⣶⣶⠶⠾⠿⠷⠚⠳⢶⣷⡌⠀⠁⠉⠚⠁⠉⠁⠋⠙⠛⠛⢛⣿⣾⣾⣿⣿⠃
⠲⢦⣧⡶⠈⠁⠀⢀⡄⣾⣸⣄⣀⠄⠒⠒⠒⣈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠟⠋⠋⠉⠉⠙⣿⠿⢇
⠀⠀⠈⠉⠛⠲⢦⣼⣿⣟⣻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣶⣶⣶⣾⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣟⠿⡿⠛⠛⠛⠛⠋⠉⠉⠉⠙⠯⣉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢬⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤
⣠⣄⣀⠀⠀⠀⠤⢤⣄⣀⡀⡀⠉⠉⠉⠑⠓⠲⠶⢶⣄⡲⠶⢴⣶⣶⣤⣂⣤⣴⣄⣀⣀⠀⠀⠐⠂
⠀⠈⠉⠉⠑⠀⠀⠤⢤⣀⠈⣩⡉⣢⣄⣠⣶⡲⣾⣷⢾⣽⣿⣷⡄⠈⠉⠛⠛⠿⣿⣿⣿⣿⢶⡄⠠⠦⠾⠿⠛⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⣶⣿⣿⣿⣿⣿⣽⣿⣿⣿⢿⣻⠟⠋⠀⠀⠀⠀⠀⠀⠀⣘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⡛⢷⣿⣿⣛⣛⠴⠟⠉⠀⠀⢠⣀⣐⣤⣤⣶⣶⣾⣿⣷⢦⣾⣷⡛⢶⣶⡿⠿⣷⣀⣀⡀
⠀⠀⠀⠀⠀⣤⣼⣿⣿⡿⢈⣱⣿⣾⡿⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣷⣿⡿⣿⠏⣥⢚⠻⠗
⠀⠀⠀⠀⠈⠙⠛⢷⠔⠶⠞⣙⣿⣟⢷⣤⣶⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠻⣿⡿⡛⠿⢿⡿⢿⣟⢾⢯⣷⠄⣀
⠀⠀⠀⠀⢀⣶⣿⣿⣿⣷⣶⣟⣿⣿⣧⣿⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠛⢳⣦⢺⣯⣲⡿⠉⠙
⣿⣦⣾⣿⣿⣿⢆⣝⣿⣿⣿⣿⣿⡿⣽⣾⣵⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡤⠆⠻⣦⣠
⠟⠉⠈⠿⣿⣿⣮⣿⡿⠿⣋⣉⣼⣿⣿⣿⡿⠏⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡇⢀⣄⡀⠘⢿
⠢⢠⣛⣛⠡⠼⢿⣟⠋⢅⡾⢿⡿⡛⠟⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠤⣾⣿⣷⣶⠿
⣥⣭⣷⣶⣶⣾⣯⡿⠇⠈⠂⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠋⠉⠹⠿⠆⣶⡿⠛
⣭⣭⣿⣏⣁⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣧⣤⣤⣤⣤⣴⣯⣀⣀⣀⣀
⣿⣿⣫⣛⣩⣛⣿⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⡀⡀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠬⣠⠃⠀⠀⠾⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣻⣯⣿⣦⣿⣿⣏⣿⢿⣿⣿⣿⣷⣾⣿⣽⣻⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡝⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢿⣿⣿⣿⣟⣶⣹⣟⣾⣿⣿⣿⣿⣿⣻⣛⣿⣱⣟⣿⣿⣿⡿⣟⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣻⣿⣿⣿⣿⣿⡟⣉⣉⡛⠽⣿⣿⣿⣿⣿⡿⣻⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣾⣽⣽⣿⣳⣶⣯⣿⣿⣶⣿⣿⣿⣿⣽⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢀⣾⣿⣿⣿⣯⣉⣽⣿⣾⣿⡷⣠⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣮⣿⣧⢸⡇⢙⣿⣿⡿⣥⢻⣿⣿⣿⡄⣿⢰⣿⣷⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣯⣿⣧⢸⡇⢼⣯⣟⣽⣏⣶⣿⣽⣿⠁⣿⠈⣿⣯⣿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣫⣿⣯⣿⣟⠘⠃⠙⢟⠋⣿⡏⠻⡟⢹⣿⠀⠛⢠⣿⣯⣻⣟⣽⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡹⣿⢟⣿⣏⢰⣶⣶⢲⣶⣶⣶⣶⣶⡖⣶⣶⣶⠘⣿⡿⣿⣟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⣿⡿⣿⡿⢸⣿⣿⣿⣾⣟⣿⣫⣷⣟⣿⣿⣿⠀⣿⣟⢿⡿⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⢿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠰⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠲⠀⠀⠀⠘⢀⠀⠀⠀⠆⠠
⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠐⠀⢉⠀⠈⡀⠀⠀⠀⠁⠀⠀⠀⠢⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡰⢄⠀⠌⠀⠀⡇⢀⠀⢰⡄⠀⠀⢀⡂⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠐⡀⠅⠈⢳⡀⠀⠀⡇⠈⢠⠇⢸⠀⢂⡜⠁⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⠦⣀⡅⠀⠀⠁⠀⠀⢰⠀⡌⠀⢸⡀⠀⠀⢀⠃⠀⠀⢈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠣⠀⠀⠀⠀⠀⠀⠑⡄⠀⠘⡄⠀⢀⠎⣄⣀⣬⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠠⠖⢀⠞⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠐⠀⡐⠀⢀⡀⠀⠀⠃⠂⢀⠈⠁⠀⢈⠸⠃⠔
⠀⠀⠀⠀⠀⢀⣰⡤⠤⠶⠲⢶⣀⠀⠀⠀⠀⡀⠀⠈⢁⠀⠀⡀⠈⠀⠀⠀⢀⣭⣤⣠⡀
⠀⠀⠀⠀⠀⢸⣇⣻⣿⡤⢎⡏⣁⣦⡴⢶⡊⣯⢉⡿⠻⢤⠉⢱⡄⠨⠅⡆⠀⢀⣬⢯⣳⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠛⠳⠚⠁⠘⠦⠴⠞⠙⠲⠼⠏⠉⠛⠚⠚⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡈⠙⠻⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠉⠻⢿⣶⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⢿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡿⠿⢿⣶⣦⣄
⠀⠀⠀⠀⠈⢿⣧⠉⠛⠻⢿⣶⣄⡀⠀⠀⠀⢀⣤⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠈⠙⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠉⠻⢿⣦⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⢀⣠⣴⡿⠟⠛⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⢀⣿⣟⣠⣴⣾⡿⠟⠋⠀⠀⠀⠀⠈⠙⠿⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⢾⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⡿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⣀⣤⣾⡿⠛⠛⠛⠿⠿⠿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢰⣹⠉⠉⠉⢿⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⠋⠉⠉⢽⠉⢹
⠀⠀⠀⣀⣀⣀⣀⣯⠀⠀⢀⣺⣤⣼⡀⣄⠠⠀⢄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣼⠀⠀⢀⣺⣤⣼⡀⣤⡠⠀⢤⡀
⠀⢀⣨⣽⣿⣿⣿⡏⠀⣰⣿⣽⠿⢿⣿⣿⣷⢺⣿⣧⡀⠀⠀⣬⣽⣿⣿⣿⡏⠀⣰⣿⣿⡿⢿⣷⣾⣷⣻⣿⣷⣄
⠀⣼⣿⠋⠉⢻⣿⣷⣶⣿⣿⡇⠀⠀⢹⡿⠛⢿⠿⣿⣿⠆⣴⣿⡏⠉⠹⣿⣿⣶⣿⣿⣿⣴⣶⣿⣟⠛⠿⠿⣿⣿⠦
⠀⠁⠹⢷⣤⡾⠃⠉⠈⠁⠙⢿⣶⣶⠿⠃⠀⠀⠉⠉⠃⠀⠀⠙⢷⣤⡾⠏⠀⠈⠸⣍⣿⣿⣿⣿⣩⠆⠀⠉⠁⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⡀⠀⠀⠀⠀⠐⣦⠄⠀⠀⢀⡀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣭⣯⣭⣭⣽⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠐⠀⢉⣯⣽⣿⡏⠀⢀⣀⣸⠃⠀⢀⣹⣧⡀⠀⠀⠀⠀⠀⠽⠏⠉⢈⣭⡉⡏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⣄⣽⣿⣿⣿⡏⠀⣼⣿⣿⣿⣿⣯⠔⠛⣿⣆⣿⣿⠀⠀⠀⠀⠀⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⢿⣾⡿⡿⣿⠛⠉⣹⢷⡄⠈⢿⣿⣿⠃⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣾⣿⣿⣿⡆⠈⠁⠀⣿⠉⠻⡇⢠⡇⠀⠚⠛⠓⠒⠀⠀⣸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠘⠿⠿⠛⠀⠀⠀⠀⠈⠓⠶⠿⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠢⣄
⠀⠀⠀⠀⣠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠴⣿⠖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠙⡗⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤
⠀⠀⠀⠀⠰⠝⣒⣶⢠⡿⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠐⠄⠀⠀⠈
⠀⠀⠀⠀⠈⢁⢻⠿⣿⣟⣿⣗⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠚⠀⠀⠈⡷⠀⠀⠀⠈⠰⡐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⢓⡀⠂⠀⠀⠐⠛⠛⠛⠒⢚⠂⠒⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠉⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⠈⠙⠒⠲⠶⠶⢶⣤⡤⠤⠴⠂⠀⠀⠀⢉⠒⠤⠄
⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣖⣉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣁⡀⠀⠀⠠⠤⢤⣄⣀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠂
⠀⠀⠘⠛⠟⠛⠏⠋⠉⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⣀⡀⠀⠈⠉⠙⠛⠛⠛⠛⠓⠲⠿⠟⠉
⠀⠀⠀⠀⢤⠀⡤⠄⠀⠘⠂⠐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠙⠟⠃⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⢀⡀⠀⠀⣀⠘⠻⢛⠛⠋⠉⠉⠉⠉⠁⠐⣒⡄⠀⢄⣀⡀
⠀⠀⠀⠀⠸⣷⡴⠾⠳⠦⠾⠛⠉⠉⠀⠀⠀⠀⠈⠀⠀⠄⡀⠀⠀⠑⠿⢿⡯⠝⣚⣩⢠⠄
⠀⠀⠀⢀⡀⢽⡇⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠀⠃⣀⣉⡴⢀⣤
⠀⠀⠀⠸⠟⠛⠋⠀⠀⣠⣌⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠀⠠⢐⠁⠀⠈
⠀⠀⠀⠀⠀⠐⢠⠤⢿⠀⢦⠀⠀⠀⠠⣲⡦⠤⠄⠀⠀⠀⠀⣰⡠⠟⠓⠁⣰⠄⡿⠤⡄⠂
⠀⠀⠀⠀⡠⠒⠁⠀⠸⠖⠉⠁⠀⠀⠲⠀⢀⡀⠀⠀⠀⢀⡀⠀⠔⠂⠀⠈⠉⠲⠇⠀⠈⠐⠄⡀
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠒⠒⠒⠒⠒⠒⠂⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣝⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢔⡣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠑⠤⣉⡁⠒⠚⠟⠭⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠛⠒⢉⣁⡴⠋⠒⢤⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡠⢤⣀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠲⠤⠐⠒⠒⠒⠒⠒⠊⠉⠁⠀⠀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠀⠸⢍⡤⠤⠒⠒⢉⣢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⠃
⠀⠀⠀⠀⠀⠀⠑⢶⡊⢉⣳⣤⡬⠗⢭⡭⢑⣒⠂⠴⠤⠤⠄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠠⠤⠤⠴⣒⣒⡊⠭⠭⠒⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⠤⠛⢶⣈⣙⣦⡀⠀⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠉⠉⠁⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢖⡉⣉⣷⣤⠜⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠲⢽⣦⡠⠬⠷⣶⣎⣉⣫⡲⢶⡒⠒⣤⣤⡤⠤⠤⠤⠤⠄⠖⠒⠚⠉⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠈⢟⢾⡉⠙⡍⣤⣖⣶⣽⠝⡄⠘⡀⣱⣀⣀⣀⣠⣀⠠⠤⠴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⡎⣷⠀⢸⢹⡏⠻⣆⠀⡇⢀⠋
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⢒⣊⣩⠇⠀⠀⢰⣿⠀⣸⠤⠄⢒⣚⠿⢿⡓⢒⣄
⠀⠀⠀⢸⣹⠞⠓⠊⢫⠉⠀⠀⠀⠀⠀⠀⣾⠋⠉⠀⠀⠀⠈⠲⣖⠊⣙⣧⣤⠝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡤⠼⠳⣎⡉⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡂⢉⣙⢦⡤⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠤⠛⠒⠉⠉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⡿⢾⣿⠆⢸⡄⣿⡟⣿⣷⠸⣿⡇⣿⡷⣟⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⣾⣇⣿⣷⣿⣿⢸⣿⡇⣿⡇⣽⣼⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁⠉⠉⠁⠈⠉⠁⠉⠀⠉⠉⠉⠁⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⢰⣿⣷⣿⣸⣿⣾⣿⣿⡆⣿⡇⠀⣼⡄⠸⣿⡼⣿⠇⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⠀⣿⠀⣽⣇⡏⠀⣿⠀⡇⢸⠀⠀⡟⡇⠀⢸⣧⣿⠀⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠀⣿⠀⣿⢹⡇⠀⣿⠀⡇⢸⠀⢸⠷⣷⠀⢸⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣿⡌⡇⢰⣿⣶⠇⣾⣶⣿⣶⣿⣶⣿⡆⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠈⠉⠉⡭⣤⢠⡄⢉⡁⡤⡤⡄⡤⣤⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠉⠈⠁⠈⠁⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠤⠀⠀⠀⠒⠒⠐⠒⠁⠐⠐⠐⠀⠀⠐⠃⠚⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠒⠒⠐⠂⠂⠀⠒⠀⠃⠈⠀⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⠀⠐⠀⠂⠒⠘⠀⠀⠘⠐⠂⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⢰⢲⣶⢺⡆⣷⠀⡀⢺⣷⣷⣿⣿⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠺⠿⠿⠹⠞⠟⠺⠷⠗⠀⠀⠺⠟⠛⠓⠛⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠉⠀⠹⠿⠏⠤⠄⠎⠦⠆⠠⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠁⠈⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠋⠉⠁⠉⠀⠉⠀⠉⠉⠉⠁⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢰⣿⣷⣷⡄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⣀⣤⣤⣤⡀⠀⣀⣤⣤⣤
⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⡟⣽⣿⡏⣧⠀⠀⢠⣤⣤⣿⡟⣿⣿⣿⡿⠃⠀⠀⠻⠿⠾⣿⣿⣤⣟⡿⠶⠿
⠠⣿⣿⣿⣿⣷⠀⠀⠀⠀⢠⡿⢸⣧⠀⠀⠀⠘⠛⠋⠹⠇⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃
⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠸⠇⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⣲⣶⣶⣄⠀⠀⠀⠀⠰⣿⣷⢀⣀⣄⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣷⣦⡴⣶⡦
⠀⠀⠈⣿⣏⣹⣿⠁⠀⠀⠀⠀⠈⢱⣿⡿⢃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡶⢿⣿⣷⡩⢿⠃
⢀⠄⠀⠿⢿⠿⠿⠂⠠⡀⠀⠀⠀⠪⠉⠰⣿⣷⠆⠀⠀⠘⢿⣿⣿⣿⡿⠏⠀⠀⠀⠸⡟⣧⣜⣻⣿
⠈⠁⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠄
⠀⠀⠀⡀
⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⢠⣾⡂⢀⡀⠉⠉⠀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣿⡆⠀⠠⣄⡀⢀⣠⣤⣀
⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⣇⢻⣿⢾⣃⡤⣶⠀⠀⠀⠀⣸⣿⣿⡿⠿⠻⠛⠁⠀⠀⠀⠑⣾⣿⣿⣿⡆
⢀⡾⢿⡿⢷⡆⠀⠀⠀⣿⣿⡿⠉⠀⠉⣾⣿⡷⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠻⠷
⠘⠁⠘⠇⠈⠇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⡀
⢀⣴⣿⡾⣶⣄⠀⠀⠀⠀⢠⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣷⣦⣀
⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⢿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣠⣼⣿⣟⣟⣧⣄⡀⠀⠀⠀⠀⠀⠈⠿⠈⠉⠑
⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⡟⠿⠿⠗⠀⠀⠀⠙⢻⠿⠿⡿⠿⠿⣟⠋⠀⠀⠀⠀⢀⠂⠂⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢸⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡀⠀⠑⠀⠀⠀⢸⠃⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠐⠀⠀⠀⠀⢀⣴⢋⠀⡠⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠃⠀⠀⠀⢣⠸⠃⢂⠞⠐⠃⡸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠜⢆⠀⠘⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⢀⠤⠖⠒⠒⠒⢯⠀⠀⣠⣴⣕⣦⡀⠀⢠⢀⢳⠀⠀⠀⡇⡴⠋⣰⠋⠉⠑⣄⠤⣄⣀
⠀⠀⠀⢀⠤⠾⢄⠀⣠⣤⣤⣜⣁⣀⣿⣶⣶⣧⠝⠊⡀⠠⡂⠀⢀⣀⠸⠃⣤⣬⣷⡀⡀⠀⠀⢀⣌⣧⡀
⠀⠀⡴⠁⠀⠀⠸⣿⣷⣷⣿⣿⣿⡟⣫⡿⠋⣁⠀⠈⣁⡶⢀⠀⠠⡱⡅⢿⣿⣿⣶⣠⣼⣖⢤⠞⠉⠁⠉⠑⢆
⠀⠀⠹⣤⣀⣀⣠⣟⠈⠉⠙⢛⡿⢟⣧⡶⠾⢥⣾⡿⣸⣇⣾⣷⣶⣿⣝⢶⣿⣿⣿⣿⣿⣇⡰⣀⡀⣠⣂⣠⡼
⠀⠀⠀⢠⡏⠁⠀⠀⠉⢢⣶⣯⣶⠟⡥⠶⣿⣿⣿⣿⣿⡟⠉⢙⡿⠙⣿⣿⣷⣦⡿⣿⣿⡉⠣⠖⠉⠀⠈⢻⡄
⠀⡀⣤⢿⣷⡀⠀⢀⡠⠜⠛⠯⣥⢸⣿⡂⢙⣿⣿⣯⡿⠻⠉⠉⠁⠀⠙⣿⣎⡽⡠⠚⠛⠒⠒⢤⠀⠀⠀⣸⣇⡀
⠀⠀⠀⠀⠀⠉⢩⡏⠀⠀⠀⠀⠈⡷⢈⣛⡫⣜⣷⣼⣇⠀⠀⠀⢀⠤⠶⠾⠯⠮⡀⠀⠀⠀⠀⠘⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠷⠧⣤⣶⣤⡖⢉⠁⠀⠀⠀⠉⢏⡉⠡⠀⠐⠁⠀⠀⠀⠀⠀⠸⣤⣔⣤⣾⡷⠿⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡰⣼⣾⣄⠀⠀⢠⢏⢀⠀⠀⢠⡀⠀⠀⡀⣀⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷⠒⠙⠲⢶⠿⠖⠛⠛⠓⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠂⠀⠰⠆⢠⣩⣅⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠄⣀⣀⠀⠁⠀⠀⠀⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⣿⠋⢀⣴⡄⠠⣀⡀⣀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣿⠏⠒⠛⠉⠀⠀⠀⠀⣉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣠⣾⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢿⡄⠀⠀⠐⣦⣿⣧⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠈⠻⣦⡀⠀⠀⠈⠛⠋⠓⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠹⣿⣿⠀⠀⠀⢀⣠⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣦⡀⠀⠉⠁⠀⠀⠀⠈⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠁⠙⠷⣦⣄⣀⡀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣀⠈⠁⠤⠤⠤⠄⠐⠢⠶⠖⠒⠠⠤⡀⠠
⢴⣮⣷⣆⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠽⠯⠉⠁⠀⠐⠀⢀⠀⠀⠀⠐⠤⢶⡖⠐⠒⠷⢶⣶⣢⡶⠴⠆⠒⠒⠂⠀⠠⠤⠶⢤⣤⣀⣀⡀⠀⣄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠦⠀⠀⠀⠀⠁
⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣠⣴⣷⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⠴⠖⠶⠶⠿⠉⠉⠉
⠀⠀⠀⠐⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠴⠖⠛⠋⠉⠉⠀⠀⠀⠤⠀⠈
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⡴⢶⠿⡍⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠉⠀⠀⠀⠀⡀⣀⣀⣀⣀⡀⢀⡀⡀⣀⢀⣀⡀⣀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠉⢉⣀⣉⣉⣁⣀⡀
⣿⣿⢛⠿⢛⠛⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣄⡀⢀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠈⠛⠓⠚⠻⠿⠿⠿⠷⠷⣤⣶⣄⠀⢾⠀⢸⡇⠀⠀⠀⠀⠐⡓⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣄⣀⣤⣤⣶⣤⣤⣶⣶⣀
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠈⠁⢺⡄⢨⡇⢀⠶⣄⢀⠀⠙⣄⠁⢸⣖⣶⠂⠀⠀⠀⠀⠀⡤⠤⠔⠒⠛⠋⠀⠈⠉⠁⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠐⠌⠂⠁⢄⠨⠀⠀⠈⠒⢈⡂⠀⣹⡇⢸⡧⠉⠒⠊⠲⠒⢶⢺⡀⢸⠀⠠⣶⣤⣆⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠚⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣁⣀⣤⠤⠶⠟⠻⠶⠹⠤⠼⠧⠗⠒⠂⠀⢀⣛⠛⠃⠈⢀⡒⠛⠓⠿⠿⣾⣭⣵⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠀⠰⠶
⠀⠀⠀⠀⢀⡤⠐⠂⣠⠈⡤⢀⠄⠀⡀⠀⢄⠑⠢⠄⠀⠐⠲⠤⠈⠑⠲⠄⠈⡉⠁⠂⠠⢍⣉⠒⠀⠈⢕⡒⠦⠀⠩⠭⠒⠢⠤⢀⠀⠀⠠⠴⠒⢒⣒⣾⠂
⠀⠀⠀⠀⣡⠞⠁⠊⢠⠞⠁⣾⠀⠀⠁⠀⠈⠳⡄⠀⠉⠲⣤⡀⠀⠀⠒⢤⣀⠈⠛⠳⠤⣀⡈⠉⠒⠂⠀⠈⠀⠀⢠⠈⠉⣩⣷⣶⣶⡄⡀⠀⠀⠀⠒⠒⠂
⠀⠀⠠⠚⢁⡴⠁⠀⠛⠀⠀⡿⠀⠀⠸⡆⠀⠀⠙⣦⠐⠀⠀⠙⠣⠄⠀⠀⠈⠁⠦⣀⠀⠀⠉⠀⠀⢊⣾⣷⠐⠀⢸⣄⣀⣿⣿⣿⣿⠁⣌⡑⠢⠁⠢⢄⡀
⠀⠀⠀⢠⠟⠀⢠⡆⠀⠀⢠⡀⠀⠀⠀⢿⠀⠀⠀⠘⢷⡑⠀⠀⠀⠈⠲⣄⡀⠀⠀⠈⠛⢦⡀⠀⠈⢘⣿⣿⡆⢠⣾⣧⠙⢸⠎⠉⠁⢄⡈⢉⣦⡤⠬⣄
⠀⠀⠐⡉⠀⢠⡟⠀⠀⠀⣸⠇⠀⠈⠀⠸⡇⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠙⠦⣄⣼⣿⣿⠟⣴⣿⠟⠀⣿⠲⢀⠜⣅⣿⠻⠻⠟⢛⣿⡀
⠀⠀⠀⠐⣡⢿⠂⠀⠀⠀⣿⠈⠀⠄⢠⠀⢧⠀⠀⠀⠀⠀⠻⣄⠣⡀⢄⠀⠀⠈⢳⡀⠀⠀⠀⠐⣈⣁⣯⡥⠤⠶⣇⣚⣒⣿⣨⣷⢾⡚⠿⠶⢶⣿⣿⣿⣇
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠟⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠠⠀⠘⠆⠱⠀⠀⠀⠀⠀⠙⠠⠴⠒⢋⡭⠴⠚⠟⠛⢉⡛⣉⠉⡩⣝⠻⣛⢧⡐⢦⣔⢦⣖⢼⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠰⠜⠎⠳⡝⠮⠳⠽⠮⠻⣮⠷⣿⠷⣬⣿⡿⠿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⣠⢤⢤⡼⠶⢄⣀⣀⣀⣀
⠀⠀⠀⢠⣾⡅⠷⢨⣌⣁⣐⠛⡶⠧⠎⠈⠑⠄
⠀⠀⢠⡿⣙⣡⠀⣠⡆⠄⣀⣈⠁⠐⠀⠸⠃⠀⠄
⠀⠀⠈⠉⠉⠳⠓⠚⠻⣦⡉⣤⣀⡅⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠄⣠⠖⡠⠒⣠⠒⢀⠔⠀⡴⠂⢀⡐
⠀⠠⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⢦⠄⠀⠉⠁⠉⠈⠁⠚⠁⠜⠁⡰⠃⢠⠊⠀⠠⠈⠀⣠
⠒⠤⣤⡶⠊⠸⣦⣤⢄⣤⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠃⠀⣰⠃
⠀⠀⡼⣇⣀⣰⡄⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⡶⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠉⠉⠑⠒⠻⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡄⠰⠆

⠀⣀⣤⣖⣥⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⣀⣐⣂⣔
⣤⣤⣤⣤⠀⠠⠤⠤⠖⠒⠒⠒⠒⠀⠑⠒⠛⠛⠛⠓⠒⠒⠒⠒⠚⠛⠛⢒⣒⣒⠒⠒⠒⢒⣒⣒⢒⣒⠒⠀⠒⠂⠂⠉⠉⠁
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠉⠑⠒
⠀⠀⢀⡾⡀⡀⠀⣀⣠⣆⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠈⠰⣤⡤⣤⡀⠀⠀⠀⠀⠠⡤⠄
⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠈⠛⠶⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠁⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠠⠤⠤⠄⠀⠐⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣬⣤⡀⢠⣾⣦⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠙⠿⠋⠀⢹⣿⣿⣧⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⡀⠀⠀⠀⠀⠈⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣿⡄⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡇⢸⠟⢱⠜⣇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⠛⠁⡠⠁⠀⠈⠀⢹⡀⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠁⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣿⡷⢧⢀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣴⢪⠣⣿⡜⣿⠀⠀⠀⠀⠀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡲⠟⠁⠀⠁⡘⢧⢿⡀⢸⠀⠀⠀⡟⢟⢧⣼⡗⠀⡖⣢⡀
⠀⠀⠀⠀⠀⠀⠀⡔⣊⡀⠀⠀⢹⣷⠞⣱⣾⡷⠁⣰⡇⢸⠀⢻⣾⣄⠀⣴⡿⡀⠀⢙⢻⣼⠏⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣸⠟⠻⢮⣦⣤⡾⣣⡾⡿⣹⠁⢠⣿⠀⡇⠀⢿⣦⡌⡟⢻⣷⠸⣦⡀⣱⣭⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⣻⡿⠾⠷⢧⠤⠿⠾⢴⣧⠀⣸⣿⣷⣳⠀⠼⣷⠟⠋⠁⢉⠑⣤⠀⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⢐⣿⠀⠈⠀⠀⠀⠀⠀⢺⢻⡏⢀⣀⠀⠀⠀⠀⠘⡇⠐⠋⠁⠀⢸⣷⠊
⠀⠀⠀⠀⠀⠀⠑⢄⡀⢀⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⡴⢸⡇⠋⠀⠀⠀⠀⠀⡠⠓⠠⡤⣀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠉⠂⠦⠄⠠⠄⠀⠀⢰⡧⠹⢅⣀⣀⣀⣀⣀⡔⠀⠀⡜⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣦⠐⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠠⠠⣔⣀⠀⠀⢀⠀⠀⡀⠀⠀⠀⢀⠤⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡊⣴⣦⡄⢣⢰⣴⡐⣄⣴⡤⠰⡾⣟⢥⣧⣔⠦⢤⣀⡀⠁⡀⠀⠀⢠⣂⣨⣲
⠀⠀⠀⠀⠀⠀⠀⣠⣦⣔⣶⣯⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣾⣾⣾⣿⣶⣷⣶⣾⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠈⠈⠉⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠋⠉⠈
⠀⠀⠀⠀⠀⠁⠈⠁⠦⢶⣶⣤⣄⣰⡄⠀⢰⡄⢀⢀⡄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣷⣬⣿⣿⣾⣯⣿⣴⣿⣥⣧⠀⠀⢀⠀⠀⣰⣴⡀⠰⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠿⣷⡶⠿⢿⣿⡀⠰⢿⣷⣶⣿⣿⡇⢛⣷⠀⠀⣜⣧⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠁⠈⠂⠉⠀⠀⢰⣈⣛⣟⣿⣿⣍⣩⡔⠒⠿⢿⣓⢓⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠐⠙⠛⠒⠚⠿⠯⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢗⠿⣶⡦⠄⣠⣤⣠⣶⣿⣷⣄⠀⠂⠚⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠎⢿⣿⣿⣿⠿⠟⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⢜⣘⢂⠟⢷⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠉⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣫⣛⣉⣛⣟⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⣀⣁⢨⡭⣄⣁⣂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⢈⣤⣊⠽⠓⠛⠉⠉⠙⠛⢚⠿⣷⣪⡐⠄
⠀⠀⠀⠀⠊⣴⡵⠋⣡⡤⣴⣒⢲⠢⢤⢄⡀⠀⠈⠙⢿⡌⠄
⠀⠀⠀⠀⣼⣟⣴⣯⠶⡋⣁⡼⣿⡿⡿⢳⠿⣶⣄⠀⠀⠙⢀
⠀⠀⠀⢰⣿⡍⠇⠀⠀⠣⡙⠾⠿⠛⣡⠎⠀⠀⢙⡧⠀⠀⡘
⠀⠀⠀⢸⣷⠉⠓⠶⣶⣦⡬⣝⣲⣯⣥⡔⠾⠛⠉⠀⠀⠀⡄⢀
⠀⠀⠀⢌⠻⣆⠀⠤⢄⣀⡀⠈⠀⠀⠀⠀⢀⡀⠔⠃⠀⡜⠀⢸
⠀⠀⠀⠈⠳⡈⠑⠦⢄⡀⠉⠉⠉⠉⠉⠉⠀⠀⢀⠄⠃⠰⠃⢨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⠭⢍⣛⢒⠩⠍⠖⠋⣠⠂⠠⠋⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⠀⡐⠐⠁⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠁⠀⠈⢀⠜⠁⠀⡔⠡⠀⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠀⣀⣴⠓⢃⠀⠀⠀⠀⠠⠐⠀⠂⠒⢂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠠⢄⡀⠀⠀⠀⠒⠀⠁⠢⠰⠤⠦⠾⠖⠛⠉⠀⠀⣼⠀⠀⠀⠈⠀⠀⠉⠈⠉⠀⠀⠀⠀⢀⢠⠤⠐⡀
⠀⠀⠀⠰⠀⠐⠀⠈⠀⡀⠂⡁⠐⠢⡀⠐⢄⠀⠀⠀⠀⠁⠀⠈⣼⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠓⠒⠚⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⣀⣀⣐⣤⣾⢻⡄⢄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠋⠀⠁⢰⠇⠶⠌⣁⠠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠁⠀⠀⠀⠀⠀⣤⢠⡤⡈⠢⡶⠶⠛⣦⣴⣀⣶⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡥⢼⠔⣵⣄⣴⡿⠋⠁⠱⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⠛⠾⠁⠀⠘⠀⠠⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⠤⠤⠤⠤⠴⠶⠟⠁⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠳⣄⣀⠀⠀⢀⣀⡰⠋⠉⠓⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠈⠉⢡⠏⠀⠀⠀⠀⠤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠠⣾⣷⣶⣶⣶⣶⡄⠀⠀⣴⣯⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠘⣧⠀⠀⠀⠀⠀⢸⣇⣸⡇⠀⣏⣀⠀⠀⢹⡏⢹⡟⠙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⣼⠇⠀⠀⠀⣠⣟⣿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⡄⠀⠙⣦⠀⠀⣴⣿⠏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⢧⣄⠀⠀⣧⠀⣠⡼⢃⣾⡿⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣀⡇⠀⠙⣶⣿⡿⠁⢼⣷⣾⡆⠈⢿⣿⣿⣦⡀⣿⣷⣿⣿⣿⣷⣄⢰⣿⣾⡏⣿⡀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⡤⠀⠀⠠⣏⠀⠀⠀⠈⣇⣀⣴⣿⠟⠀⠀⣾⣿⣿⡇⠀⠀⠹⣿⣿⣷⣿⣿⣯⣿⣿⣿⣿⣶⣿⣿⣧⣿⣿⣦⣼⠁⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢾⣿⣯⣾⣿⠋⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢷⡀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⠁⣀⣀⣀⠀⣀⣀⣀⣀⠀⣀⣀⣀⠈⣿⠉⠉⠉⠉⠉⡉⠉⠉⠛⣛⣛⡛⠛⠛⡿⠁⢠⠀⠀⠈⠳⡄
⠀⠀⠀⢀⡤⠖⠚⠛⠓⠒⠄⢸⡇⣹⠀⣿⣿⣿⢸⡏⠉⠉⢹⡇⣿⣿⣿⠠⣿⠀⠀⠀⣾⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⣧⣤⣼⣤⣤⣤⣤⠏
⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⣼⣧⣿⠀⣿⣿⣿⢸⡇⠠⠄⢸⡇⣿⣿⣿⠈⣿⠀⠀⠀⣿⣿⣿⣿⡇⠀⣿⣿⡿⠀⢀⣿⣠⣿⣏⡠⠀⠀⠀⠐⢤
⠀⠀⢀⣀⣠⠖⠒⠲⠞⠁⠀⢸⣿⣿⠤⠤⠤⠀⢸⣇⠀⡀⣸⡇⠠⠤⠀⢰⣿⠄⠀⠀⣿⣿⣿⣿⡇⠰⢦⣤⣴⣴⣿⣿⣿⣯⣿⣿⣷⣶⣏
⠒⠒⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠽⠯⠭⠭⢭⣭⣭⡭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⣭⣭⣭⠭⠭⠭⠿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠿⢭⣉⣉⣩⠽⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⢸⢻⢰⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣼⠀⠀⠀⠀⠀⠀⠀⡾⢸⠀⢇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢻⠀⠀⠀⠀⠀⠀⣼⠃⣾⠀⠸⠄⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠁⢸⡆⠀⠀⠀⠀⣰⠋⢠⠉⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠀⠈⢷⣄⣀⣠⡾⠃⣠⠏⢸⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠀⡆⠀⢨⠉⠁⢀⡜⠁⢠⠃⠀⢀⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢤⣀⠀⠀⠀⠀⢹⣿⡀⡇⠀⠘⡆⠀⡜⠀⣰⠃⠀⠀⢸⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠳⣄⠀⠀⢸⡇⣇⠰⡄⠀⡇⠀⠃⣰⢃⣴⣿⠁⢸⠀⣼⠃⠀⠀⢰⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠆⢠⡀⢹⡆⠀⢸⡇⣿⠀⠸⠀⠃⣀⢠⣇⡞⢹⡇⠀⢸⠀⣿⠀⠀⠀⢸⡇⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⠆⠸⡃⠀⢿⣤⡾⠁⡿⠀⣠⠞⣻⠟⢸⣾⠁⠈⣧⠀⢸⠀⢿⡀⠀⢀⡾⢡⠃⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣼⠀⠀⢇⠀⠀⠀⠀⢠⢃⡾⠁⣼⠋⠀⠈⢿⠀⠀⠹⣆⠀⢃⠈⠛⠞⠛⠁⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⡇⢠⠀⠘⢆⡀⠀⠀⠸⢸⡇⢰⡏⠀⠀⠀⣼⠀⠀⠀⠹⣇⠘⡄⢀⡄⣰⡆⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⢸⠀⠀⠀⠙⢄⠀⠀⠈⣧⠈⢷⣄⠀⣠⡟⠀⠀⠀⠀⣿⠠⢃⡜⣰⢿⡇⠀⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠈⡆⠀⣠⡄⠈⢣⠀⠀⠹⡆⠀⠉⠛⠉⠀⠀⠀⠀⣰⠇⡠⢊⡴⠋⢸⡇⢀⡏⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⣇⠀⢳⡀⣿⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠰⣯⣤⡶⠋⠀⠀⣿⠀⡸⠁⢿⡀⠀⣀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠹⡄⠈⣧⣻⠘⢧⣀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣰⠃⠀⠈⠙⡛⢉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠙⣄⠀⢿⣇⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠋⣀⣠⡀⠀⣠⠞⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⠳⣄⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⣛⣽⢟⡡⢚⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣌⡁⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋⢁⣠⡴⠛⠁⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⣴⠞⠛⠲⠶⣤⣀⡀⠀⠀⠙⠓⠦⣭⣛⠷⠶⠶⠶⠶⠶⠶⠶⠚⠛⣉⣁⣠⡤⠶⠛⣋⣁⣤⡤⠶⠚⠛⠉⠉⠹⣆
⠀⠀⠀⠀⢰⡏⠀⠒⠤⢀⡀⠈⠙⠻⠶⣤⣄⡀⠀⠉⠉⠛⠛⠛⠚⠚⠛⠛⠛⢋⣉⣥⡤⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢸⡇⠀⢀⡀⠀⠈⠙⠀⠀⠰⠤⣈⠉⠛⠶⢤⣤⣀⡀⣀⣠⡴⠞⠛⠉⠉⠀⢀⡀⠀⠀⠀⠉⣀⠠⠔⠚⠉⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠷⣤⣀⠀⠈⠁⠲⠦⢤⡀⠀⠀⠈⢃⣀⣤⠴⠟⠋⣉⠠⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣄⣀⣀⣀⣤⠶⠞⠋⠉⠀⠐⠚⠁⠀⠀⠀⢀⣀⣤⠄⠀⠀⠀⣀⣀⣤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠛⠉⡀⠠⠄⠀⠀⣀⠀⠀⣀⠤⠐⠊⠉⢁⣀⣤⠴⠶⠛⠉⠉⠛⠛⠶⢦⣤⣀⡀
⠀⠀⠀⢀⣠⣤⣴⠶⠛⠋⠁⠀⠠⠒⠋⣁⡤⠖⠊⠀⠀⠀⠈⣀⣠⡤⠶⠚⠋⠉⠀⢀⡀⠀⠈⠉⠓⠲⢄⡀⡀⠉⠙⠳⠶⢦⣄⣀⡀
⠀⠀⢠⡟⡁⢤⣉⠳⣤⠀⠀⠠⠐⠊⠉⣀⣀⣤⣀⣠⡴⠖⠛⠉⠙⠓⠶⢦⣤⣄⣀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⣠⠞⣉⣉⠙⠳⣦
⠀⠀⣿⢡⠛⣲⠈⡇⣽⠀⠀⡤⠔⢒⣩⡽⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢤⣄⣀⠀⠀⠤⢀⡀⠀⢰⡏⡘⣡⣬⣄⠀⢸⡇
⠀⢀⠿⣌⣳⡤⢚⣰⣯⡤⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣄⡁⣿⡀⢣⣙⢂⣨⣃⡼⠃
⠀⠠⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠦⠤⠴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡊⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣚⡀⠀⠀⠀⠀⠀⠀⠬⡭⣷⡄
⠀⠀⠀⠀⠀⠔⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢐⡀⠀⠚⠁⢠⣾⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣯⡄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣀⣀⢠⡀⠀⠐⠈⠅⡴⠂⠀⠀⡸⠒⠤⣶⠦⠤⠶⠶⠤⠴⠶⠎⠉⠲⣿⠀⠀⠀⠤⡀
⠀⠀⠀⣧⠀⠀⠀⠀⠄⠀⠸⠛⢶⠐⠀⠀⠀⠀⠀⠠⠄⠀⠀⠷⠀⠀⠀⠣⣀⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⣀⠠⣿⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠈⠃⠠⠤⠴⠖⠀⠀⠀⠈⢴⣶⠄⠶⢦⠴⠶⠦⠮⠯⠬⠥⠈⠀⠄⠀⠂⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠀⠊⠀⠒⠁
⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠘⠃⠀⠀⠋⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣄⠀⢀⣀⠀⢀⢀⠀⢀⡈⢻⡄
⠀⠀⠀⢀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡄⠀⠔⠀⠸⠁⠀⠛⠀⢠⣄⠀⢸⢭⡀⠘⠋⠈⢻⣦⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠄⠂⠘⠑⠒⠢⢄⡀⠀⠁⠠⣀⠀⠙⡷⠐⠃⣴⢺⢲⢚⣷⠀⠀⠐⡶⢸⡇⠀⡀⠀⠈⠁
⠀⠰⣜⠒⡄⠀⢀⣀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⠰⣶⣄⢀⣀⡀⣠⣹⠆⡈⡏⠀⠀⣼⣶⣺⣈⣿⠄⠀⠀⠛⢺⡇⢀⣆⣀⣤⢀⡀
⠀⠀⠀⠐⠛⠛⠂⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠉⠁⠈⠉⠉⠉⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⡶⣶⢶⢷⠷⠷⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⢿⣶⣶⢶⣶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣘⡻⢝⠻⡙⣌⠦⠰⠒⠋⠋⡇⠉⠉⢉⣀⣂⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠒⠦⢈⠓⢓⡙⠁⢠⣇⠀⠀⠀⠁⠀⠀⡟⢟⢉⡇⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠼⠷⠭⢍⣢⣄⣀⠴⠋⣢⠔⠃⢹⡄⠀⢀⣄⠀⠀⠙⠒⠛⠁⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠠⣞⣭⣍⣑⡶⠋⢀⣤⣶⠛⠁⠀⠀⢰⡇⢠⠞⠉⠳⣄⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡯⠖⠒⣲⠏⠀⠀⠐⠋⠛⢷⡀⠀⢀⡾⡿⡧⡀⠀⠀⠈⣇⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠁⢠⠟⠒⠒⠶⣤⣄⡀⠀⠿⡤⠋⠈⠀⠁⢳⣠⠖⠀⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠙⣏⠢⡄⢻⠀⠀⣀⡤⢺⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡧⢴⠶⣾⠲⠒⠢⢜⡄⠘⣼⣠⠚⠁⠀⢸⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠱⡀⠀⠀⢡⢀⡟⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⠦⣧⣄⡀⣼⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢀⣾⣿⣦⣑⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⢻⢿⢿⡺⣷⣤⣤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠟⡇⠃⠈⢫⡳⡑⠢⡉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡇⠘⠀⣀⣳⡈⠢⣈⠂⠌⣢⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡜⠀⠀⡸⠀⠈⠑⠤⠤⠌⠭⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻⠜⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⣏⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢐⡋⡠⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⢃⡄⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⡆⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠟⠀⠀⡾⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⢠⡇
⠀⠀⠀⠀⠀⠸⠆⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠄⠀⠐⡟⠁⠀⠀⠁⠀⠥⡄⠀⠀⠸⠣⠀⠀⠋⠂⠀⠀⠀⠈⠑⠂⠀⣀⢀⡇⠈
⠀⠀⠀⠀⠠⠀⠐⠁⠀⠀⠀⡆⢀⠃⢀⡀⣴⠀⠙⠃⠀⠀⠉⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⢀⢀⡇⢹⠢⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⠂⠀⢸⡃⠁⠀⠀⢾⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢺⣜⢏⢢⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠸⠂⠀⠀⠀⢉⢆⡧⢌⣁⣤⣶⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢨⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠈⠛⡁⠙⢿⡉⣞⡄⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⠀⢀⣾⠉⢌⢈⡂⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠒⠛⠛⠛⣃⡤⠆⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣥⡀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⣀⣀⣀⣤⣤⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⣤⣤⣀⣄⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣶⣾⡟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠻⣿⣿⣿⣷
⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠈⠁⠀⠀⢀⣠⣴⣾⣿⣿⠿⠯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞
⠛⠈⠁⠁⠘⠐⢦⣿⢛⠻⢿⣿⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠚⠀⠙⠀⢶⡿⠗⠂⠀⠀⠀⠀⠀⢀⣤⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠋
⠀⠀⠲⢷⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠛⠛⠉
⠀⠀⠀⠈⠙⠀⠀⠂⠀⠀⠀⠀⠀⠀⠚⠻⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⠿⠟⠛⠉
⠀⠀⠀⠁⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠓⠖⠄⠀⢀⠆⠀⢲⣿⣿⣿⣿⡿⠛⠉⠁