Bike Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣶⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⡀⠀⠀⠿⣿⣻⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣤⣾⣿⣶⣌⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣷⣶⣶⡶⠂⠉⢩⡟⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣼⣿⣮⣻⠋⠀⠀⣠⣿⣷⠀⠳⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣏⠃⠀⠀⠀⠘⣷⡀⠀⠈⠙⠢⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠘⣿⣿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠙⢶⣤⣾⡙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠻⢿⣿⣷⡀⠀⠀⠀⠀⠀⢠⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣴⣽⡿⠃⠀⠀⠀⠀⠀⠀⠘⣷⢶
⠀⠀⠀⠀⢀⣤⣴⠶⠶⠶⠶⣦⣄⡀⠀⢨⣿⡿⠟⠁⠀⠀⠀⠀⢀⠌⠀⠀⢘⣼⣆⣶⠶⣦⣤⣀
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣽⣻⣶⡛⠹⡄⠀⣿⠀⠀⠀⡐⠁⠀⢀⣴⠟⢱⣿⡏⠀⠀⠀⠙⠳⣄
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡀⢷⠀⢿⠇⡠⠊⠀⠀⠀⣾⠃⠀⠀⢻⣻⡄⠀⠀⠀⠀⠹⣧
⠀⣼⡇⠀⠀⠀⠀⢠⣾⣲⡄⠀⠀⢻⣿⣿⡇⢰⣶⠆⠀⣀⡀⠀⠀⠠⣿⠀⠀⠀⠘⣷⣷⡄⠀⠀⠀⠀⢹⡇
⠀⢿⡇⠀⠀⠀⠀⠘⠋⢭⣥⠀⠀⠀⠀⢰⡞⢿⣿⣷⣾⣿⣿⣿⡶⠆⣿⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠘⣷⡀⠀⠀⠀⠀⠀⢀⣇⠀⠀⢀⣠⡿⠁⠈⠛⠛⠉⠀⠉⠁⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠉⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣀⠀⠀⠀⠀⢀⣤⡾⠁
⠀⠀⠀⠀⠈⠛⠷⠶⣶⡶⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢁⠄⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⢄⠀⠀⠀⠀⠀⠀⠉⠛⢻⡟⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣤⣤⣏⠳⣄⠙⠢⣀⠀⠀⠀⠀⠀⡞⠑⢄⣠⣤⣴⣶⣶⣤⣤⡀
⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⡜⠙⢿⣎⠳⣄⠈⠑⢤⡀⠀⣸⠁⣠⡿⠻⡁⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⡼⠁⠀⠀⢻⡆⠈⠳⣄⠀⠈⢲⠃⢰⡟⠁⠀⠘⢄⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⣿⠀⠀⠀⠀⠰⠾⠁⠀⠀⠀⢸⡷⠀⠀⠈⡷⣞⠟⣄⣻⣇⣀⠀⠤⠴⢷⠆⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣧⣞⣣⡏⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠐⠚⠀⠀⠀⠘⠒⠈⢷⣄⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠰⠶⣶⣶⣶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠶⠶⠶⠶⠶⠶⠶⠶⢶⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣰⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠻⣆⣠⣤⡶⠶⠶⣦⣤⡀
⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⢩⣿⢷⡌⠻⣦⡀⠀⠀⠀⠀⣸⡏⠀⣰⡿⢿⡄⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣠⡿⠃⠈⢿⡆⠘⢿⣄⠀⠀⢠⡿⠀⣸⡟⠀⠀⠹⣦⡀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⢸⡟⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⢸⣷⠀⠀⠙⢷⣽⣿⠃⠀⣿⣇⣀⣠⣤⣼⣿⠆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢿⣿⠟⠛⢻⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠹⣷⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⢀⣠⡿⠃
⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣤⣴⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⣶⡶⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠭⣯⣭⠭⠽⣛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠤⠤⣄⠀⠀⠀⠲⠭⠷⢒⣊⠭⢿⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡖⢒⠉⠉⣀⡠⠤⠖⣚⡩⠥⠒⣪⠟⣣⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⣊⡩⠤⠒⠊⠉⠀⠀⣠⠞⡡⠊⠘⡖⢇
⠀⠀⢀⡤⠒⠊⠉⠉⠑⠒⠤⣀⠔⣡⡇⢸⠀⠀⠀⠀⠀⠀⡠⠞⣡⠞⠀⠀⣠⠼⡘⡏⠉⠑⠒⠤⣀
⠀⡴⠁⡠⠖⡏⠉⡏⢉⢖⠔⣡⠾⡁⡇⢸⠀⠀⠀⠀⣠⠎⣡⠞⠁⠀⣠⠊⢀⢴⣣⢻⡏⠉⡶⢤⡈⠳⡄
⡼⠀⡼⢅⡀⠸⡄⣧⠟⣡⢞⣱⡀⣱⣇⣸⠤⡤⢒⠾⣡⠞⠁⠀⠀⢰⠃⣰⢅⡀⠹⡎⣇⡜⠀⣀⡹⡄⠸
⡇⠠⡧⠤⠬⢷⡟⢻⣞⠛⠋⠉⠉⠀⠀⠀⠀⢧⣻⢳⡇⠀⠀⠀⠀⡎⠀⣇⣠⠬⢽⣾⢙⣮⡭⠤⠤⡇
⢧⠀⢇⣠⠔⢉⠿⡻⡙⠫⢍⣹⠋⢹⠋⠉⠉⠉⠹⠿⠛⠀⠀⠀⠀⢸⠀⢣⡠⠒⢩⠎⡟⡄⠉⠒⢤⠇⢠
⠈⢦⠈⠣⣄⡎⠀⡇⠙⣄⠔⠁⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠳⢤⡃⢠⠇⠸⣀⡠⠊⢀⠎
⠀⠀⠑⠦⣀⡉⠉⠉⢉⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡉⠉⠉⠉⢁⡠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢖⠀⠀⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠒⠒⠂⠈⠉⠹⡖⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⣠⠇⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⠀⠀⠀⠀⠀⢠⠧⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡏⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣼⠤⠤⠒⠒⠊⠁⠀⢷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠸⡀⠀⢀⣤⢠⡀⠀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⢀⡀⠀⠀⢀⣀⡤⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠙⠒⠛⢻⡇⠘⡏⢁⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣻⢿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⣾⢿⡧⣼⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣀⠀⡔⣱⠃⠸⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠚⠁⡗⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡠⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⡼⢱⢧⡀⠀⢇⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⢉⡠⠔⡾⠃⣠⡟⢿⡡⢽⠦⠤⠤⠤⠤⠄⢀⡀
⠀⠀⠀⢀⠔⠋⠀⢀⡤⠖⠊⠉⠉⠉⠉⠑⡾⢡⠃⠀⠙⢦⠈⠀⢃⠀⠀⠀⠠⠆⠐⠈⢉⣀⠤⠒⠋⠁⣠⠎⢀⡞⠉⣇⠸⡇⣀⣤⣤⣤⣀⣀⠀⠀⠈⠑⢄
⢀⠔⣤⠊⠀⢀⠖⠁⠀⠀⢀⣤⠤⠤⢤⡜⢱⠋⠳⡀⠀⠈⠃⠀⠙⠂⠁⠀⢠⠤⠐⠊⠁⠀⠀⠀⢀⡔⠁⣰⠋⢠⠂⠍⠀⣟⠉⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠱⡄
⡞⣸⠇⠀⢠⠋⠀⠀⢀⠔⠁⠀⠀⠀⡼⢠⠛⡄⠀⠙⣄⠀⠀⢳⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⠜⠁⣰⠃⣠⢗⠀⢻⢀⡤⠖⠒⠒⠤⡀⠀⠈⠳⣄⣀⡽
⠳⣿⠀⢠⠇⠀⠀⣰⠃⠀⠀⠀⠀⡸⢡⠃⠀⠘⣆⠀⠘⣆⠀⠀⣇⠸⡀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣠⠋⠀⢠⠃⢰⠃⢸⠀⢸⠏⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠈⣇
⠀⣇⠀⡟⠀⠀⢰⠃⠀⠀⠀⠀⡰⢡⠏⠀⠀⠀⠘⡄⠀⠘⡄⠀⢹⡀⡇⠀⠀⠀⠀⠀⢀⠊⠀⡀⠁⠀⠀⡎⢠⡇⠀⠈⡇⠘⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸
⠀⠸⢴⡇⠀⠀⡞⠀⠀⠀⠀⣰⢣⠏⠀⠀⢀⣀⡤⠷⠐⠒⠛⠲⢌⣧⣸⣄⣀⡀⠀⡰⠋⢀⠞⠁⠀⠀⢀⠇⣼⠀⠀⢠⠿⡀⢇⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⡇⠀⠀⡇⠀⠀⣠⠴⣧⣯⣤⣴⣾⡁⢨⠆⠀⠀⠀⠀⠀⣼⢻⠁⠀⠈⠻⣾⠁⣠⠏⠀⠀⠀⠀⢸⠀⡏⠀⠀⣾⠀⢳⠘⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⡇⠀⢰⡷⣉⢌⡇⠀⡏⠧⣼⠞⠀⠀⠀⣀⣤⣾⣿⠈⢇⠀⠀⠀⢸⡶⠁⠀⠀⠀⠀⠀⠈⣤⠃⠀⠀⡿⠀⠈⢧⠹⣄⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠇⠀⠀⣧⠀⠈⠣⠥⢼⣇⡀⠓⠾⠿⠶⠒⡿⠋⠉⠀⣿⠉⡇⢸⡦⡀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣿⠀⠀⠀⠻⣽⣸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⢀⡇⠀⠀⢠⢻⠀⠘⢾⣡⠇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⢆⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⠳⢤⣄⡾⠀⣇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠘⢆⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢸⠋⠓⠿⢦⣀⣀⣀⣴⠏⢱⣀⡀⠠⠤⠤⢤⡀⠀⠘⡄⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠈⢣⡀⠀⠈⠲⣄⣀⣀⣀⡤⠞⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠉⠉⡴⢆⠈⡇⢳⠀⣀⣀⡤⠇⠀⠀⠹⡄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠈⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠹⣤⣀⣀⣀⣠⠞⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠉⠉⠉⠁⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢤⣀⣀⠀⣀⣠⠔⠋

⠀⠀⠀⠀⠀⣤⣤⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠛⠶⠶⣽⣻⡟⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⢸⡇⠀⠀⠀⠀⠀⠀⢠⡞⣫⡟⣿⣓⠶⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⢧⡤⠤⢤⣄⡀⣠⡟⣸⠋⠛⠋⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⣀⣀⣸⣯⣥⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠬⣿⠉⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⠤⠤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⣟⡓⠶⢤⣄⣀⡀⠀⠘⠿⣷⣖⣲⠤⠤⠤⢬⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣼⢯⡹⣟⠓⠶⠮⣭⣟⣲⠶⢤⣾⢩⡟⠛⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⡏⢉⡏⠈⢷⡘⢧⡀⠀⠀⠈⠉⠛⣿⢃⣾⠳⣄⠀⠀⢀⣠⡴⠶⠒⠒⠒⠒⠶⢦⣄⡀
⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⣁⣿⠀⢸⡗⢦⣄⢻⡌⢷⡀⠀⠀⠀⢰⠏⣼⠉⠻⣮⡳⢶⡛⣡⡴⠾⡟⠋⢹⠛⠳⣦⣈⠙⢦⡀
⠀⠀⠀⣠⠞⠉⣠⢶⡛⠻⣽⡇⠀⣾⢶⣄⠈⠻⣿⡌⢳⡄⠀⢠⡟⣸⠃⠀⠀⣸⠟⣦⡙⢯⣧⡀⢣⠀⣸⠀⡼⠀⠈⢳⣄⠙⣦
⠀⠀⣰⠏⢀⡾⢅⠀⠱⡀⣿⠁⢀⣿⢋⠜⢷⡀⠹⣿⣆⠹⣆⣾⢡⡏⠀⠀⢰⡇⣴⠛⠿⣦⣝⠷⣼⡄⡏⢰⠃⢀⡔⠁⠹⡆⠘⣧
⠀⠀⡟⠀⣾⠥⢄⣙⡲⣽⡟⠀⢸⣷⣥⣒⣉⢿⡄⢹⡿⣶⢟⣉⡛⢿⡉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠓⠳⠷⣴⢋⣠⠴⠚⢻⠀⢸⡄
⠀⢰⡇⠀⣿⡤⠤⠤⠞⣻⣧⣣⣿⣯⡭⠤⠤⠼⡇⢸⡷⡇⣇⣶⢎⠆⣿⣉⣉⣉⣉⣛⣛⣛⣛⠛⠛⠒⣲⡄⠈⣯⣄⣀⣀⣸⠃⢸⡇
⠀⠈⣧⠀⢻⣀⡠⠖⢋⡔⢹⠹⡌⠳⡍⠓⠦⣤⡇⢸⠃⠻⣾⣯⣭⣾⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣠⣴⠓⢤⡀⢈⡿⠀⣼
⠀⠀⠘⢧⡈⢻⣄⢠⠎⠀⡼⠀⣇⠀⠈⠢⣠⠞⢠⡿⢦⣾⡁⠀⠀⠈⠉⠉⠉⢻⡉⠻⣽⠋⢉⠝⠉⡽⠉⢱⠀⢳⡀⣨⠟⠁⣼⠃
⠀⠀⠀⠈⠳⣄⠈⠻⢦⣀⣇⣀⣸⣀⡤⠞⢁⣰⠟⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⣄⡈⠓⠯⣤⣀⣇⣀⣸⣤⠴⠛⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠛⠦⣤⣈⣉⣉⣉⣡⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣉⣉⣉⣁⣠⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⣋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⣶⠒⠒⠲⠶⣶⣦⣴⡒⢋⣉⣡⣤⡖⣾⡏⣿⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣩⣭⡿⠛⢋⣉⡥⢴⣚⣟⣏⢻⣇⢹⣳⣸⡽⠼⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠙⠓⠒⠶⠤⢤⡀⠀⠀⠀⠀⠯⠤⠶⠓⣋⣩⡤⠶⠚⣇⢹⡾⠞⣛⣋⣭⣴⢶⢾⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢶⡴⢶⠒⠚⠉⠁⠀⠀⣀⣠⠤⠖⠚⢉⣩⡤⠶⢚⡽⠻⣄⣿⣿⠹⣽⡘⣾⡌⣿⣎⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣀⣤⠤⠖⠚⢉⣁⡤⠴⠚⠋⠉⠀⢀⡴⠋⣠⠞⢧⣀⣿⣇⣻⣧⠽⢗⣚⡭⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡇⢸⣁⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⠀⠈⣏⢹⡥⠒⠒⠋⠉
⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠛⠲⢤⣀⢀⡴⠋⣰⡇⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠞⠁⠀⠀⠀⣠⠼⡄⢿⠉⠉⠉⠛⠒⠦⣄
⠀⠀⣠⠞⠁⢀⣠⡴⠒⢲⡖⠒⢦⣄⣀⡼⠋⣠⠞⠁⡇⢸⡇⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠔⠁⠀⠀⢀⡴⠋⠀⢀⣷⠘⡖⣶⠲⠦⣤⡀⠀⠙⢦⡀
⠀⡼⠁⢀⡴⠋⠀⢹⡀⢸⡇⢀⣞⡴⠋⣠⡞⠉⢧⠀⡇⢸⡇⠀⠀⠀⣴⠦⣴⡋⢀⠔⠁⠀⠀⠀⢠⠎⠀⢠⠞⠙⣎⣇⢻⣿⠀⣰⠃⠈⠳⡄⠀⠱⡄
⣸⠁⢀⡞⠦⣄⡀⠀⢧⢸⣧⡾⠋⣠⢞⣁⣷⡀⢈⣇⣧⣼⡧⠤⡴⠒⣿⢹⣶⠟⠁⠀⠀⠀⠀⢀⡏⠀⢰⠧⣄⡀⠈⢻⡄⣿⡰⠃⠀⣀⣠⠼⡆⠀⢹
⡏⠀⢸⣁⣀⣀⣙⣲⣼⠛⢻⣤⠾⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⣇⢰⣧⠏⡟⠀⠀⠀⠀⠀⠀⢸⠁⠀⡟⠀⠀⣉⣛⣲⡿⠛⢷⣒⣋⣁⣀⣀⣻⠀⠈
⣇⠀⢸⡅⠀⠀⢀⡤⢟⣦⣾⠷⣶⠶⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⠬⠷⠖⠊⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⠉⠉⣉⡥⢞⡿⡶⡟⠛⢭⣉⠉⠁⣹⠀⢀
⢹⡀⠀⢣⡠⠞⠉⢀⡞⢸⡈⢧⠈⠙⠲⣄⡞⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠸⡶⠚⠁⢀⡞⠀⡇⢻⡀⠀⠈⠙⢲⠇⠀⣸
⠀⠳⡄⠀⠳⢄⡀⡼⠀⠘⡇⠈⢳⣀⡠⠎⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠘⢦⣠⠞⠀⢸⡇⠀⢷⠀⣀⡴⠃⠀⡰⠃
⠀⠀⠙⠦⣀⠀⠉⠓⠲⠶⠷⠖⠚⠉⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠓⠲⠼⠦⠤⠞⠛⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠈⠙⠲⠤⣤⣄⣠⣤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⣀⣤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠒⠉⠉⢒⡄⢉⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠒⠒⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡴⠋⠑⢦⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢠⡇⡜⠀⠀⠀⠀⢧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢲⠦⡴⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⡀⠀⠀⠀⢇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡞⣠⡐⢖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢢⠘⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢉⣉⣡⣤⣍⣉⡙⠚⠴⣃⠱⡌⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠛⣉⣉⣥⣬⣍⣉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⣡⠔⠋⠁⠀⠀⡇⠀⠀⡝⢱⠦⡈⠳⣜⢆⠱⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠖⠹⡈⢇⠀⡇⠀⠀⠉⠒⢤⡉⠢⡀
⠀⠀⠀⠀⠀⠀⡜⢁⠞⠱⣄⠀⠀⠀⠀⡇⠀⡼⢠⠇⡠⠊⢢⡈⢦⠳⡘⢦⠀⠀⠀⠀⢠⠎⣠⠊⠳⣄⠀⢳⠘⡄⡇⠀⠀⠀⢀⡴⠛⢆⠙⡄
⠀⠀⠀⠀⠀⡼⢀⠎⠀⠀⠈⠳⣄⠀⠀⡇⣰⢁⡧⠊⠀⠀⠀⢳⠈⢇⠙⣄⠳⡀⠀⢀⠏⢰⠁⠀⠀⠈⠳⣌⡆⢹⡇⠀⢀⡴⠋⠀⠀⠈⢆⠘⡄
⠀⠀⠀⠀⢀⠇⣸⠀⠀⠀⠀⠀⠈⠳⣄⣿⣣⣞⣀⣀⣀⣀⣀⣀⣇⣸⣀⣈⣦⢙⠴⠚⢢⡇⠀⠀⠀⠀⠀⠈⠻⣄⣇⡴⠋⠀⠀⠀⠀⠀⢸⡀⢇
⠀⠀⠀⠀⠸⡀⢿⠒⠒⠒⠒⠒⠒⠒⡺⡿⣷⠶⠶⠶⠶⠶⠶⠶⡧⢼⠦⠤⣴⠃⠀⣠⠞⡗⠒⠒⠒⠒⠒⠒⢒⡿⡿⣖⠒⠒⠒⠒⠒⠒⢺⠇⡸
⠀⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⡠⠊⠀⡇⠈⠳⣄⠀⠀⠀⠀⢠⠃⡸⠀⠀⠙⠖⠚⠹⡀⢣⠀⠀⠀⠀⢀⡴⠋⠀⡇⠈⠳⣄⠀⠀⠀⠀⡸⢀⡇
⠀⠀⠀⠀⠀⠘⡄⠱⡀⠀⡠⠊⠀⠀⠀⡇⠀⠀⠈⠳⣄⠀⢠⠏⣰⠃⠀⠀⠀⠀⠀⠀⢳⡈⢧⠀⢀⡴⠋⠀⠀⠀⡇⠀⠀⠈⠣⡀⠀⡰⠁⡜
⠀⠀⠀⠀⠀⠀⠘⢦⡙⢮⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣈⠗⢁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠑⢏⡀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣨⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣈⠑⠲⠤⠤⠯⠤⠤⠖⢋⣡⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣉⠒⠢⠤⠤⠧⠤⠴⠒⢉⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠶⠖⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠶⠶⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⣹⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠚⠚⠛⠓⠾⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⣠⡟⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠄⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⢹⡇⠐⠂⠁⠀⠀⠀⠉⠑⠂⠄⣠⡀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠐⠀⡀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠐⠁⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡣⣆
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣧⠀⠘⣧⠀⠦⡰⠀⠀⢀⡴⠋⠀⠀⠀⠁⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠐⢄⣤⣿⠀⠀⠀⠀⠀⠀⠀⣘⡇⠀⠸⣧⢸⠃⢀⡼⠋⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⢻⣄⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠲⠂
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠐⠀⠻⣿⡿⠿⠿⠿⣷⣦⣤⣤⣉⣯⣄⣠⣛⣯⡿⣏⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠹⢷⣆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠠⠤⠄⢈⣩⡏⠉⢿⠟⣿⣧⣹⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⠀⠀⠀⠀⠀⠀⡀⣸⠃
⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠈⢻⡗⠚⠁⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⡀⠀⠀⠀⠀⢀⡀⣤⣶⣋⡁⢀⢀⠀⠀⠀⠉⢁⠀⠀⠀⢀⠀⡀⠄⢀⠀⠈⢻⣯⣒⠠⣄⣀⣀⣀⣀⠀⠀⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⣿⣿⣿⣷⣶⣿⣿⣿⡿⠿⠟⠿⠋⠩⢤⠄⠤⠀⠄⠀⠀⣿⣉⠆⡬⡙⠛⠓⢶⠘⠷⣽⣿⣶⣶⣶⣶⡶⣶⣿⣿⢷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠀⠀⠀⠷⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠗⠂⠀⡄⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠂⠒⠀⣀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⠩⠄⠐⠒⠒⠚⡔⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⠊⠁⠀⠀⠀⠀⡀⠐⢈⡆⢀
⠀⠀⠀⠀⠀⠀⢀⣀⣦⣤⢶⢶⣶⣦⣤⣀⡀⠀⠀⠀⢱⣀⡀⠀⠀⠠⢀⠤⠊⠀⠀⠀⠀⠀⠀⡀⠊⣠⠖⠋⢹⡟⢤⣸⣦⣶⣤⣤⣤⣤⣤⡄
⠀⠀⠀⡠⣢⡾⠟⢋⠅⠀⠀⢠⠦⢄⠩⡙⠻⣷⣦⡀⠘⢩⣇⠐⡡⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⣠⠞⠁⠀⠀⢀⣽⣆⠙⣽⣧⣠⠤⠤⠈⢍⡉⠲⣤⡀
⠀⢀⣬⣾⠋⠱⣦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠕⠀⠙⡷⢔⣤⢏⠋⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⡼⠁⠀⠀⠀⣰⠟⡁⠚⣧⡘⠟⠁⠀⠀⠀⠀⠂⡄⢠⠙⢦
⠀⠀⡸⠁⢀⠐⠀⠙⢦⣀⡀⠀⠀⠀⠀⣀⠤⡛⣽⡞⢻⡄⠈⣦⣄⠀⣀⣀⣀⣀⢀⠊⣠⠎⠀⠀⠀⠀⣸⠧⣐⡀⠀⠘⣷⣌⢄⠀⠀⠀⠀⠀⠀⠘⢧⠀⠳⡀
⠀⠠⠅⠀⡎⠀⠀⠀⠀⠜⠛⠲⣴⢻⣬⠖⠋⡀⠉⠉⠉⠉⢠⣿⣃⣀⠀⢤⠔⠲⣌⠰⠁⠀⠀⠀⠀⢰⡇⠀⡞⠉⠐⠤⣈⢻⣦⢢⡀⠀⠀⠀⠀⠀⠈⡳⠀⣧
⠀⢸⠀⠀⡄⠀⠀⠀⢀⠀⠾⢦⣶⡍⠛⣟⢋⣉⣩⣿⣿⣿⣞⢫⡌⠛⢏⠳⣼⡧⠈⡇⠀⠀⠀⠀⠰⢺⠅⠐⡇⠀⠀⠈⠲⠿⣿⣿⣿⠔⠃⠀⠀⠀⠀⣤⠀⢸
⠀⠘⡄⠀⠆⠀⠀⠀⠀⠒⠀⢄⠀⠀⢀⠀⠈⠃⣀⣀⣀⡀⠀⠀⠀⠀⠈⠠⣤⣀⠜⣡⡧⠀⣀⠀⠀⠘⡇⠀⢃⠀⠀⠀⠀⠀⠀⠉⡁⠐⠂⠀⠀⠀⠈⠏⠀⣼
⠀⠀⢻⡀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠙⠛⢻⠋⠉⠉⠉⠛⠒⠒⠒⠒⠊⠁⠘⠛⠛⠁⠀⠀⠹⡀⠈⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡐⠀⢀⠇
⠀⠀⠀⠳⣄⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⣀⡴⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢆⣀⠀⠀⠀⠀⠀⠀⢀⣤⠚⠀⣠⠏
⠀⠀⠀⠀⠈⠓⢤⣀⠉⠽⠓⠶⠖⠒⠚⠁⣀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠈⠔⢒⣒⠤⠖⠞⠋⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠶⠶⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⣿⣾⣿⣿⣧⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣟⡋⢐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡻⣮⣿⡆⠙⣇⡀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢉⣱⣿⣿⣿⣼⣷⡄⡀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡔⢮⣿⣶⣿⣿⣿⣿⣿⣵⠶⠀⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣠⣄⣶⣾⣿⣿⣶⡿⠟⣁⣼⣿⣿⣿⣿⠽⢖⣹⠟⣡⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⣟⣛⡟⡫⡬⠛⠛⣟⡛⢻⣿⣿⣷⣤⣭⣤⠀⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠙⢧⣴⡿⠟⢼⣿⣓⠐⠈⡀⠀⠈⢿⣿⣿⣯⠀⠀⣀⣀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⠟⣷⣶⣦⣄⡈⠉⠙⠂⢉⠓⢤⣘⣿⣿⠟⢲⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢶⠖⠿⢿⣿⣿⣿⣶⣤⣀⣀⣑⡤⠉⢛⣙⠍⢉⡁⠀⠠⢄⡈⠈⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣀⣤⣶⣤⡈⠐⠣⡟⡒⡾⣿⣿⣿⡿⣿⣿⣿⠟⠉⠀⠠⠼⠛⣿⣆⣁⣨⠶⢒⣘⣶⣾⣿⣦⣤⣄⡀⠀⠀⠈⢑⣦⣀⡀
⠀⠀⣠⢞⣯⣽⠿⣯⣠⣥⣽⣾⠻⡦⣿⣾⣏⣴⠟⡋⣀⠐⠀⠀⣠⣤⣬⡝⠋⠉⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣶⠾⠟⠻⣿⣿
⠀⢰⣿⣿⠋⠀⣴⣿⡿⢹⣿⣾⣶⢡⣸⣷⠏⠁⢤⣗⣾⣤⣶⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⣫⣽⣿⣿⠿⠿⣿⠋⠀⠀⠀⠘⠇
⠀⣿⣿⣿⠡⣰⣿⣿⣅⣶⣿⢿⣿⣿⣿⡿⠀⠻⠦⣞⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⡿⠛⠛⠛⠄⢀⣾⣷⡀
⠀⣿⣿⡇⣵⣿⣿⡟⣿⣿⣇⢸⣿⣿⣿⠃⠀⠀⠀⠸⣡⢟⢹⣟⡻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣥⣾⡄⠀⠀⠀⢻⣿⣯⣇⡀⠀⠑⠀⠘⠆
⠀⢸⡿⢷⠹⡿⡻⡗⠿⠧⡈⢻⡛⣿⠿⠟⠀⠀⢠⣿⣿⣿⣜⠃⢹⢿⣻⣿⣿⣿⣿⣿⣿⡟⣿⡿⣿⣄⣠⣤⢤⣿⣛⣾⣿⡀
⠀⠀⢻⣨⠣⡈⢩⡄⢹⡤⠴⣾⣿⡜⠀⠀⠀⠀⠀⢹⣿⣿⣿⡖⠲⢟⣛⣿⣿⣿⣿⣿⣿⣇⣿⣦⣿⣿⣧⣤⢸⣿⣿⣿⣿⣷⡀⠀⠀⢸⡄
⠀⠀⠀⠻⣶⡈⠺⣤⣤⢷⣾⣿⡿⠃⠀⠀⠀⠀⠁⠘⢿⣿⡿⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⢿⡟⢿⣿⣿⡿⢃⡾⣿⣿⣿⣿⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠋⣀⠳⢤⣄⣀⠛⠻⠒⠲⠿⠤⠀⠰⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡶⣭⣭⣴⣯⡀⢻⣌⣿⣿⣿⣿⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⣾⣿⣿⣶⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣻⡯⠽⢿⣿⣽⣿⣿⣿⣿⢃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⠿⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣒⣒⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢺⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣔⠒⠒⠒⠒⠒⠒⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠯⠤⠤⠤⠤⠤⠤⠤⠀⠠⠬⠍⠲⠆⢀⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠲⠤⣤⡏⢀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⠛⢄⠙⢄⣀⡤⠴⠒⠒⠒⠒⠤⢄⡀
⠀⠀⠀⢀⠔⠋⣀⠤⠖⠒⠒⠒⠒⠤⡄⠀⠺⡕⢄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡎⠀⢈⠗⠈⢁⡤⠒⠒⠒⠒⠒⠦⢄⡈⠓⢄
⠀⠀⡰⠃⣠⠊⠁⠀⠀⠀⠀⠀⠀⢰⠃⡴⢄⠈⢮⠳⡄⠑⢄⠀⠀⠀⠀⠀⢰⠁⡸⠀⣠⠋⣠⠚⢆⠑⣄⠀⠀⠀⠀⠀⠀⠉⢢⡀⠳⡀
⠀⢰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⡴⢃⡼⠁⠈⢣⠀⢧⠈⢢⡈⠳⣄⠀⠀⢀⠇⢠⠃⢠⠃⢰⠁⠀⠀⠳⡌⢢⡀⠀⠀⠀⠀⠀⠀⠱⡀⢱
⠀⡇⢀⡇⠀⠀⠀⠀⠀⢀⠴⢋⡤⠊⠀⠀⠀⠈⡆⢸⠀⠀⠙⢦⠈⠻⡏⡟⠀⡞⠀⢸⠀⡎⠀⠀⠀⠀⣘⣦⠑⢤⠀⠀⠀⠀⠀⠀⣇⠀⡇
⠀⡇⠘⡇⠀⠀⠀⠀⠀⠘⠦⠎⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠑⣄⠀⠀⠸⠥⠤⠾⠀⠓⠚⢉⣉⣉⡤⠤⠤⠜⠀⠀⠀⠀⠀⠀⡟⠀⡇
⠀⢱⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡼⠀⠀⠀⠀⠀⠈⣇⠀⣠⠤⠖⠲⡀⠸⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⠁
⠀⠀⢣⡈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡰⠁⠀⠀⠀⠀⠀⠀⠈⢧⢧⠀⠀⠀⠱⡀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣠⠃
⠀⠀⠀⠑⢄⠈⠲⠤⣀⣀⣀⣀⣀⡤⠔⠉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⠼⠆⠀⠀⠀⠘⢦⡈⠑⠦⣄⣀⡀⢀⣀⡠⠔⠊⣀⠔⠁
⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⣉⣁⣀⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⣀⣉⣁⣀⡠⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⠒⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⢤⡈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠹⡄
⠀⠀⠀⣠⠴⠖⠒⠓⠒⠲⣴⣇⠀⠀⠉⠲⣄⠈⠙⠲⠤⣄⣀⣀⣤⠔⡿⠀⠀⠙⢦⣠⠤⠴⠶⠦⢤⣀
⠀⣠⠊⠁⠀⠀⠀⠀⠀⣰⠃⠈⠳⣄⠀⠀⠀⠙⠦⣀⠀⢀⡄⠀⠀⣸⠁⠀⢀⡴⠋⠑⣄⠀⠀⠀⠀⠈⠙⢦⡀
⣸⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⠓⢬⣳⣤⣤⡇⠀⢠⠏⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢳
⡇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢹⠋⣻⠓⠒⠦⢤⣀⡀⢳⠀⠀⠀⠀⠀⠈
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⢯⡀⢸⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⢠
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⢀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢚⣉⣩⠭⣥⢞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠉⠀⠀⠀⠸⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡣⣉⠑⠒⠲⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣙⡒⢒⣒⣒⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣖⡪⢽⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡹⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣚⠭⠒⠋⣁⢔⡽⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣒⣊⣙⣒⡒⠾⢵⡁⠀⠀⠀⢀⣀⠤⣒⣫⢵⠊⠉⠀⠀⣠⢞⡵⠋⣀⡵⠷⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⢀⡴⣫⠔⠊⠏⠀⠀⡇⠀⣸⣿⠲⢍⠲⣔⣺⠭⠒⠉⠁⢸⢸⠀⠀⣠⢞⠕⣩⠔⣫⠵⠖⡟⡏⡏⠉⠑⡲⢬⡓⢤⡀
⠀⠀⠀⣠⢋⠞⠁⢄⠀⠈⡀⠀⡇⣰⣿⠃⢀⠔⠑⣌⢦⠀⠀⢀⣀⣼⣼⡠⢞⠕⢁⡜⡡⠮⡀⠀⠈⣧⢣⡇⠀⠠⠁⠀⡈⠳⡙⢆
⠀⠀⢰⢣⠯⢄⡀⠀⠑⠄⠰⡀⣷⣿⢃⠔⠁⠀⢀⠨⣆⢧⡔⠉⠀⠀⠀⠙⢇⢀⡎⣜⡁⠀⠈⠢⡀⠸⡜⡇⢠⠃⡀⠊⠀⠀⠘⡌⢇
⠀⠀⡎⡸⠀⠀⠀⠁⠂⠠⢀⣵⣷⣟⣡⣤⡦⠭⠤⠤⢾⢸⡥⣀⣰⣹⠆⠀⢸⣸⢸⠁⠈⠁⠂⠤⣈⠢⣣⣧⡂⠂⣀⠤⠐⠈⠁⢸⠸⡄
⠀⠀⡇⢿⠈⠉⠉⠁⢀⠡⢜⢧⣴⠿⠟⡛⠋⠉⠉⠉⢹⢹⢤⡈⠳⣅⠀⣠⠎⡇⢸⠀⠀⠀⠀⠀⠀⡠⣏⣉⡷⡁⠀⠀⠀⠀⠀⢸⠆⡇
⠀⠀⢹⠸⡤⠔⠂⠉⢀⠔⢁⠃⡎⠆⠀⠀⠉⠒⠠⢀⡜⡸⠀⠉⠢⠞⠉⠀⠀⢸⢸⡀⡀⠤⠐⠈⡁⠀⠍⡏⠌⠢⡁⠒⠤⢀⠀⢸⢰⠁
⠀⠀⠀⢣⠱⡀⢀⠔⠁⢀⠂⠀⡇⠘⡄⠀⠠⠀⢀⡜⡱⠃⠀⠀⠀⠀⠀⠀⠀⠈⢇⢣⡀⠀⢀⠄⠀⡐⠀⡇⠐⡀⠈⠢⡀⠀⢨⢃⠎
⠀⠀⠀⠀⠱⣌⠣⣀⠀⠌⠀⠀⡇⠀⠰⠀⢀⡴⢋⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡑⢆⡀⠀⡰⠀⠀⡇⠀⠠⠀⠀⢈⠖⣡⠊
⠀⠀⠀⠀⠀⠀⠑⠦⢍⣒⠢⠤⠧⠔⢒⣋⠥⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢍⡒⠥⣄⣀⣇⣀⡤⠷⣚⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠛⠳⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣀⣀⣤⣤⡤⠤⠴⠀⠀⠀⠀⠀⢿⡀⣀⣀⣀⣀⡀
⠀⠀⠀⢀⣠⣴⣾⣿⣟⣛⣛⣓⠒⠀⣸⣿⣿⠉⠁⠀⠀⠀⠀⢀⣠⠖⢁⣤⠾⣟⡭⠭⠤⠤⠍⡁⠂
⠀⠀⣴⣿⣿⠿⢋⠁⠀⠀⠀⠀⢨⣾⠟⠁⠈⢣⡀⠀⠀⣠⡶⠋⠀⣴⣿⡵⠊⠹⡄
⠀⠺⠿⣿⢋⡞⠁⠀⠀⠀⠀⣴⢟⣁⣀⠤⠄⢠⡿⣟⣿⡟⠀⠀⣼⣿⡞⡀⠀⠀⠑⠀⣀⣀
⠀⠀⢠⣇⣿⠀⠀⠀⣠⢀⣾⣿⣏⣉⣀⠀⠀⠸⣇⠻⠟⡠⠀⠰⠿⢿⢱⡇⠀⠀⠀⣾⣿⠋
⠀⠀⢸⣿⣿⡀⠀⠀⢷⣬⣿⣿⣋⣀⣀⣤⣶⡆⠈⠉⠉⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠛⠷⠞⠀⠀⠀⠀⢰⣿
⠀⠀⠀⢿⣿⣷⣄⠀⠀⠉⠉⠉⠉⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠈⠻⣿⡟⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣀⡀⠀⠀⠀⢀⣠⣴⠿⠃
⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠛⠛⠋⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡀⠀⣸⠃⣰⠞⠀⠀⠀⡠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤⠀⠀⠀⠀⠀⢀⣴⡯⣰⣿⠁⡴⢡⡞⠁⠀⠀⠀⢠⢆⡴⠃
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⠇⣤⠀⢀⡀⢠⣾⡟⣿⡿⠋⠸⠘⠋⠀⠀⠀⡠⢂⡾⣫⣶⠏
⠀⠀⠀⠀⣾⣽⡿⢀⣤⢀⡀⠀⠀⢀⠀⠀⠀⡞⢰⠏⡼⣣⢰⡿⣿⣿⣯⣾⣿⣣⡼⠁⠀⢀⠀⣠⢞⣴⣿⡾⠟⠀⣤⡤
⠀⠀⠀⠀⢻⣷⣶⠆⢹⣿⣧⠀⠀⠀⠈⢓⠖⠱⢇⣾⢳⣷⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣠⢆⣵⣿⣿⣛⣦⣮⡾⠏
⠀⠀⠀⠀⠀⠈⣷⣦⣾⣿⣿⡇⠀⡄⠀⠀⢀⠀⠀⢻⣷⣿⣿⣿⣿⣿⣿⣿⣛⢁⣠⡴⠊⣰⣿⣿⣋⢉⣠⣿⣡⠞⠁
⠀⠀⠀⠀⠀⢈⣿⡿⠛⠛⠻⡇⠀⣵⣶⣖⣿⡄⠀⢀⣿⣿⣿⣷⣿⣿⣿⣿⣶⡟⢠⣶⣾⡿⢿⡟⣽⢟⡵⠛⣁⡀
⠀⠀⠀⠀⠀⢸⠋⠁⠀⠀⠰⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣟⣿⣿⣉⣽⣿⣿⣥⣾⡿⣫⣷⣭⠞⠛⠀⢀⣀
⠀⠀⠀⠀⠀⡇⠀⠀⢀⢠⡁⠀⠀⣿⣿⡿⡿⠀⠀⣿⣿⣿⣿⣿⣟⣿⣵⣿⣿⣿⣡⣾⠿⢿⡾⠏⠉⠀⣠⣴⡿⠛⣉⠠
⠀⠀⠀⠀⠀⣇⠀⢀⡼⠎⠁⠀⠠⣿⣿⣿⣯⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⡿⠿⠔⠋⢀⢀⣴⣾⣧⣴⣾⠭
⠀⠀⠀⠀⠀⣃⡴⢿⡆⠀⠀⢠⣾⣿⣿⣿⣿⣦⡀⠘⣿⠟⣽⣿⡿⡿⢿⣿⣿⡃⠋⠀⣠⣶⣿⣿⣿⣿⡿⠟⠋⣶⠖
⠀⠀⠀⠀⠀⠉⠁⣿⡒⢦⣤⣆⣽⣿⠿⠝⡿⣿⣇⠀⢸⣼⣿⣿⢟⣿⡿⢑⣋⣤⣾⣿⣿⣿⣿⠿⢋⣡⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡏⢹⣿⡙⣿⡿⢦⣄⣿⣳⣿⡄⠀⠄⣨⡙⢿⣿⣷⣿⢿⡿⣿⣿⣿⣿⣯⣶⡉⠁⠀⠀⣤⣆
⠀⠀⢀⣠⣤⠶⠾⣿⣧⣈⠤⠛⣯⡇⠀⣿⣿⣿⣳⣧⡀⠸⣿⢿⣌⢿⣿⡷⣛⣿⣿⣿⣿⣟⣉⣽⣿⡿⠿⠟⠋⢀⡀
⢀⣴⠟⠉⠀⠀⢸⣿⠈⠙⢷⣼⣿⠟⢦⡟⣻⣿⠿⣿⠇⢰⣿⡏⢥⠘⣷⢿⣿⣿⣿⠿⣿⡿⡿⢿⣃⣀⣤⣖⠚⠉
⣼⡏⠀⠀⠀⠀⢸⣿⠀⠀⠀⠹⣷⣶⣶⣿⣾⣿⣿⣉⡴⠻⠙⣷⣿⢀⡿⢿⣿⣿⣗⠋⠛⠙⠛⠉⠉⠁
⣿⡀⠀⠀⠀⠀⣾⠻⠀⠀⠀⠀⣿⣿⣿⣿⣿⣙⣿⣮⡲⣶⣷⣾⢏⣼⣷⠿⠟⠋
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣻⣿⣿⠟⠷⠶⠶⠛⠉
⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠿⣿⡿⠿⠛⠁
⠀⠀⠈⠻⠷⣶⣤⣤⣴⠾⠟⠁⠀⠈⠁
⣿⣟⣻⠿⠿⠿⠻⢻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡄⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⠿⠿⠛⠛⣹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⣠⣾⡿⠋⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⠀⠀⣠⠟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠀⠀⠀⠀⠹⣧⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⣠⣴⣿⣿⠿⠛⠛⠙⠛⠛⠿⣿⣿⣿⡃⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢀⣴⣾⣿⣿⠿⠛⠛⠛⠛⠿⢿⣿⣦⣄
⠀⠀⢀⣼⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⢿⣿⣦⠀⢻⣧⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⠀⠀⣴⣿⡿⠋⠘⣿⡆⠀⠀⠀⠀⠀⠀⠉⢻⣿⣧⡀
⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠹⣿⣧⣬⣿⡄⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⢸⣿⡏⠀⠀⠀⠀⠀⢀⣴⣶⣴⡟⠀⢀⠀⠀⠀⣻⣿⣝⣿⣷⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⢹⣷⣀⡀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⢸⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⡷⠤⢤⣤⣤⣤⣼⣿⣇⣼⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⡷⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣧⠀⠀⠀⠀⠀⠈⠛⣿⡏⠀⠀⠀⠀⠀⠀⣾⣿⠋⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠤⣼⣿⠟⠀⠀⠉⠉⠉⢳⣄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠻⣿⣷⣄⡀⠀⠀⠉⠁⠀⠀⠀⢀⣤⣾⣿⠋⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣤⣤⣤⣶⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣶⣦⣤⣤⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣷⣶⣤⡀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡉⠉⠉⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠟⠛⠛⢁⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⡟⢻⣿⣿⣿⣦⠀⠘⠻⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠋⣿⣿⡿⠋⠀⢸⣿⣷⠀⠙⣿⣿⣿⣧⠀⠀⠀⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⣿⡿⠀⠀⠀⠘⣿⣿⠀⠀⠈⢽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⢀⣿⣥⣄⣀⠀⠀⣿⣿⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣤⣤⣠⣾⣿⣿⣿⣿⡿⠿⣤⣿⣿⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⠈⢿⣆⠙⠻⣿⣷⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⣿⡏⠀⠀⠘⣿⡆⠀⠈⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠻⠀⠀⠀⠀⢹⣿⡀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⡛⠉⠙⢧⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢁⠀⠀⢘⢃⣿⢷⣦⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠿⢸⣿⡃⢿⣿⣿⣿⡻⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣤⣬⣿⣟⠧⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣷⠆⠹⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣹⣿⡀⢠⠘⣿⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡌⠛⢿⣿⣿⡟⠅⠀⠀⠙⠉⡛⢶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡧⠀⠀⠈⠟⢀⣠⣴⡾⢿⣿⠂⢿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠁⠀⠀⣠⠖⣋⣥⡶⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⣴⣿⡇⠴⠶⢊⣵⣿⣿⣿⠁⠀⣰⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣽⠶⣶⣿⡏⠀⠀⠃⠀⣠⣾⣿⣿⡿⠃⠀⠸⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⠀⠀⢀⣴⣞⢿⣿⠟⠁⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣿⠻⣦⣀⣸⣧⣯⡷⠝⢂⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⠀⠀⢠⡶⣿⡿⠉⢷⡘⡿⢿⣿⣿⣿⡶⠹⡿⠋
⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣾⢿⣽⠏⠀⠀⠀⠻⡜⣆⠻⣿⡇⠀⠀⢠⡀⠀⠰⣴⣄⡀
⠀⢀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠈⣿⣿⡜⣿⣄⠀⠀⠀⠀⠹⣼⣆⢹⣟⢆⠀⠈⣿⣆⠀⠀⢻⣿⣦⣀⡀
⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠘⣿⣿⣦⠀⠀⠀⠀⠹⣿⣎⣿⡏⠃⠀⣙⢿⣧⠀⢸⠟⠻⠿⠛
⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠹⣿⣿⣿⠒⠀⢻⣷⣮⣧⡆⢀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄
⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⣺⣿⠋⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠘⣿⡿⢿⣯⡌⠿⢉⣿⣷⣝⢿⠿⠛⠋⠉⠉⠉⠙⠛⢿⣿⣿⣦⡀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⠎⠙⠋⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣀⣨⠵⠚⠛⠀⠳⢸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠈⢉⣒⡒⠒⠒⠒⠒⠒⠲⣿⣿⣿⢿⠛⣿⣅⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆
⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠈⠛⠛⠛⣿⠟⠛⠸⣿⣿⣧⣦⣬⣬⣽⣿⣷⡄⢦⡀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⢻⣿⣿⠙⠶⠶⠶⠾⡿⢿⣷⣾⣗⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠙⢿⣿⣿⣷⣶⣤⣤⣤⣴⣶⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠘⠠⠤⠴⠁⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣴⣾⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠄⠐⠀⠀⠟⡀
⠀⠀⠀⠀⢀⢀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⡓
⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠇⠀⠀⠀⠀⢆⡈⠀⢀⠠⠐⠒⠛⠲⠆⠀⠀⣀
⠀⠀⠀⠈⠄⡀⠀⠀⠀⠁⠀⠀⢛⠂⠀⢀⡀⠀⢀⠾⡟
⠀⠀⠀⠀⠀⠛⢦⡔⠦⡇⢤⠀⠘⠂⠠⠊⣸⣦⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠁⠀⠀⣰⡿⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⣠⣾⣿⣧⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠋⠉⢃⠀⡥⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⠛⢄⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢰⡿⠀⠑⠤⣀
⠀⠀⢀⠀⠤⠄⠒⠈⠀⠠⠀⠂⠀⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⠢⣀⠀⠉⠒⠠⠤⢀
⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠈
⣐⠡⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⢆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠷⠶⣤⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠨⠒⠤⡀⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠁⠀⠈⠐⠱⠲⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢩
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡔⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⡠⠶⠂⠐⠋⠑⠢⠀⠀⠀⠀⠀⠈⠢⠶⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡤⠂⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡷⠲⠶⢤⣀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⣶⣠⠞⠋⠉⠁⠂⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠪⡀⠓⠇⠀⠀⠀⠨⠓⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠄⠼⢄⠂⠀⠘⣂⣀⣀⣀⣤⣥⢤⡟⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠒⠂⠲⠶⠖⠂⠉⠀⠀⢂⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠓⠀⠀⠤⠠⠀⣬⡥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⡰⠒⠒⠶⡀⠀⠐⠂⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡂⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠄⠀⡐
⠀⠀⠀⠀⠀⡀⠀⢀⠀⡐⢠⣀⣤⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠐⠒⢊⠀⢀⠀⡀⠀⠀⠀⠀⠀⡀
⣿⠿⠿⠿⠿⠿⠻⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣛⣿⣿⣟⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⢦⣤⣤⣸⣦⣶⣦⣶⣶⣴⣶⣷⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣾⣷⣛⣛⣿⣿⣶⣶⣾⣿⣧⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣯⣭⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⠟⢃⣭⢿⣿⣿⣿⣿⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣭⠡⡟⠿⢻⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣎⠉⠉⠉⠉⠉⠉⢉⣙⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣻⣿⣿⡿⠿⠿⣯⣭⣉⠉⠛⣿⠿⠿⠿⠿⠿⢟⣀⣀⣤⣶⠄⣀⣘⠿⢾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢟⣽⡿⠿⢿⠷⠖⠀⢠⠀⠀⣄⣀⠈⠩⢤⣤⣿⠆⣈⠉⣿⣿⣿⣿⣻⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣧⣤⣉⣋⡀⠀⠀⠙⠛⠶⣤⣶⠂⠀⢊⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⠾⢛⣁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⢞⡙⠛⠛⣛⣛⣋⣩⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡛⠁⠀⠘⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣦⣄⠀⠈⠛⠿⣿⣻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠉⢻⡿⠧
⠀⠀⠀⠀⣴⣾⠟⠋⠉⠉⠉⠙⠻⣷⣄⠀⠀⢿⣯⣿⣿⣿⣧⠤⠶⠒⢒⣿⣿⣧
⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⡀⠀⠀⠈⢿⣧⣴⣿⣿⣿⣿⠿⠁⠀⣠⡾⠟⠁⠀⢹⡀
⠀⠀⢸⣿⠀⠀⠀⢀⣾⣾⣽⡷⠶⢶⣾⣿⠋⢻⠋⣼⡟⢀⣠⠾⠋⠀⠀⠀⠀⢸⣷⣄⣄⣀⣀
⠀⠀⠸⣿⡀⠀⠀⠈⠻⣿⡟⠒⠶⢤⣽⣿⣧⣿⣾⣿⣷⣟⠁⠀⠀⠀⠀⣠⣾⠿⣿⠛⠉⠛⠻⢿⣦⡀
⠀⠀⠀⠻⣷⡀⠀⠀⠀⠻⠀⠀⠀⣠⣿⠏⢛⡟⠿⠿⠟⠛⠛⠀⠀⠀⣼⡟⠁⠀⢹⡆⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣤⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⣷⣶⣢⠀⠀⠀⢸⣿
⣀⣀⣀⣀⣠⣤⣤⣉⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⢿⡛⠯⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣄⣀⣀⣀⣤⣶⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣀⡀⠀⠉⠙⠛⠛⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⡭⠁⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣉⡙⢎⠐⠁⠀⠀⠀⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⢊⠕⢿⡼⠀⠀⠀⠔⠀⠀⠀⠀⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⠛⢦⡀⠙⢶⡋⣱⠄⠀⠠⠔⠪⠂⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⠀⠀⠀⢠⣶⣿⡦⣄⡙⣇⣠⣤⣀⣤⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠻⣹⢠⣀⠀⠸⣙⢖⡵⠋⣹⠻⣤⣦⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠐⣾⡗⠀⠀⢣⠔⣎⠉⠀⠀⡿⣤⣽⡯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠰⢴⠇⣙⢵⣶⢠⠞⠀⠋⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠈⠒⠴⢲⠚⠋⠀⠀⣠⡾⠋
⠀⠀⠀⠀⣠⠖⡒⠲⡜⠁⠀⠀⠀⠀⠀⠀⠀⣔⠠⠇⠀⣠⡾⠋
⠀⠀⠀⠀⡇⠸⣿⢿⡇⢠⣤⣤⠞⢦⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⡹⠶⠞⣆⠙⠺⣍⣈⣧⠞⠀⠀⢀⣠⠾⠋
⠀⠀⠀⡸⠀⠀⣴⠋⠢⣀⠈⢻⡁⠀⢀⡔⠋⠁
⠀⠀⢠⡇⠀⠀⠈⠲⠖⠋⢳⡾⣭⡝⡏
⠀⠀⠘⡇⠀⢀⠀⠀⠀⠀⠘⢦⣹⠕⠁
⠀⠀⠀⠱⣄⠀⠀⠠⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠈⠓⠦⢤⣤⣴⡵⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠤⠈⠀⠀⠀⠀⣀⣀⣀⣀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢁⡾⠤⠒⠒⠚⠛⠛⠓⠒⠒⠤⢄⡀⠉
⠀⠀⠠⢤⡀⠀⠀⠀⠐⠢⠄⣸⢀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⡐
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠉⠒⠤⠤⠒
⠀⠀⠀⠀⠀⢱⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠠⠐⠈⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣠⠒⠁
⠀⠐⠓⠲⠄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠙⢦⡀
⠀⠀⠀⠀⠈⠢⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⢤⣀⣀⡄
⠀⠀⠀⠀⡠⠔⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠜
⠀⠀⢠⠊⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣀⠀⠄⠂⠁
⠀⢠⠇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⡄
⠀⠸⠈⠁⠉⠐⠠⢀⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠔⠂⢙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡈⠀⠈⠙⠲⠦⢤⣀⣀⣀⣀⣀⣀⣤⡤⠖⠋⠁⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⣀⠤⠚⠩⡀⠀⠀⢀⠞⠉⠢⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡴⠋⠁⠀⠀⠀⠀⠑⢄⡀⠏⠀⠀⠀⠈⠑⠒⠠⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⠱⣆⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠻⠄⢠⠀⡀⠈⡼⣋⠄⠀⠂⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⣉⣉⣉⡛⠷⢦⡈⣵⠲⣮⢢⣞⡼⠋⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡒⢼⣋⣬⢿⣾⣲⠅⠒⣝⢾⣷⢳⣿⣋⣴⣆⣽⣥⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⡴⠄⣶⣤⣠⠆⡀⠀⠀⠉⠒⠾⣾⣷⢿⣭⣭⣍⡤⢀⣀⣈⠁⢀⠙⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠛⠓⠒⡯⠔⢊⢹⡇⠡⠀⡀⠈⠉⠒⠶⣤⡛⠘⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⣠⡾⣫⣿⠁⠀⠏⠙⠑⠴⠄⣀⠀⠉⠢⡀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢁⡼⢋⡴⣇⡟⡧⠀⢸⠈⡓⠀⠀⠁⠘⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠰⢋⣴⡟⢰⢸⠇⣶⠀⠘⢀⡋⠀⠀⠀⠀⠀⠘⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣠⡟⠉⠀⠆⣿⠀⢹⡀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣤⠋⠀⠀⠀⠘⡿⠀⠋⠷⢀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢰⢰⣷⠀⠀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠻⣷⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣄⠻⣿⡆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡇⢰⡿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠉⠉⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⣀⣠⣤⣬⣥⣤⣄⣀⠀⢴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡦⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠾⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠈⠉⠛⠛⠻⠟⠛⠛⠉⠁⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣆⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠀⠀⢠⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⢤⡧
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠠⢾⣿⣯⡀⠀⢸⣟⡁⠀⡀⠀⠀⠀⠀⠀⢀⣿⠙⠣⡀
⠀⠀⣄⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⠐⠀⠀⠙⠙⠛⠀⢠⡌⠓⠀⠀⠀⡀⠀⠀⢠⣼⣿⠄⠐⡋
⠞⠛⠋⠙⢷⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢸⡀⠒⠂⠀⠀⠀⠀⢀⣴⣿⣿⣤⣾⠇⠀⠀⠀⠁⠀⠀⠿⠿⣷⣄⠀⢼⡧⠀⠀⣤⣿⣿⣦⡄⠡
⠀⠀⠀⠀⠀⠻⣾⡁⠀⠈⠀⣰⠧⠀⠐⠀⠀⠀⢀⡜⠀⠈⢦⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣷⣦⠄⠀⠀⠐⠀⠀⠀⠈⠉⢀⣿⠇⠢⠐⣾⣿⣃⡘⠟⠟
⣅⣰⢄⡀⠀⠀⢻⣶⣤⣠⣚⣀⠿⢿⣦⠀⠄⠀⣾⣄⢀⣄⣈⡆⠀⠀⣀⠀⠀⠘⠻⠛⣿⣿⡤⠀⠀⠀⠀⠡⡀⠀⠀⠀⣬⡟⡓⢡⣴⣟⣿⢿⣧⢺⣦
⢿⢿⡿⠧⠘⠂⠀⠉⠙⠗⠉⠈⣢⠀⢰⡄⢀⠈⡃⠉⢻⠟⢻⠟⠒⠀⠉⠡⠀⠀⠀⢀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⡿⠤⠤⢹⡏⣴⠆⢁⣆⢡
⣼⣮⡀⠀⠀⠀⠘⠲⠂⡀⠀⢺⣿⠀⠈⠰⠀⠀⠰⡀⠀⢀⡞⠀⠁⠀⠀⠀⠀⠤⠄⠈⠽⠿⣶⣀⣀⠀⠀⠒⠀⣄⣤⣿⢟⡉⠘⠻⣀⣾⣷⣴⣮⣽⡷
⢿⣿⡇⠆⠆⢘⣁⠠⠀⢆⠀⠀⠀⢀⡞⠁⠀⢀⡤⠴⠒⠁⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⢤⣀⡀⠀⣀⣄⢀⠀⠀⠀⠴⣿⣷⣶⣄⠀⠘⠛⠙⠛⠛⠛⠉
⠀⠈⠉⠁⠀⠘⠁⠀⠀⣈⣀⣀⡠⠊⠀⠀⠀⠈⠐⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⡀⠀⠀⠉⠀⠠⢿⣿⣿⡇⠀⠀⠀⠈⠉⢹⣤⠀⠀⠀⠀⣀⡴⠆⡀
⠸⠗⠛⠉⠀⠀⠙⠠⠀⠠⠜⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠈⢉⡀⠀⠀⠈⠉⠀⠐⣤⠤⠄⠀⠈⠉⠙⠀⠠⢤⣦⣄⠙⠧⠔⠀⠀⣠⣴⣷⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣓⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠁
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⢿⣷⣄⡉⠛⢿⣿⡄
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⠀⠈⠿⣿⣷⣌⡛⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣭⣿⣿⣿⣿⣿⣍⢻⡋⠉⠁⠀⠀⠀⠉⢉⣹⣿⣿⡄
⠀⠀⠀⠀⣴⡾⠛⠉⠉⠉⠻⣿⣿⣿⣿⠊⣇⠀⠀⣀⡤⠶⢻⣿⠿⣿⡆
⠀⠀⢀⣾⠏⠀⠀⢀⢀⡀⢀⣈⣿⣿⣟⠀⣻⡿⠋⠁⠀⣴⡟⠁⠀⢸⣇
⠀⠀⢸⡟⠀⢀⣼⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣶⣄⣠⡾⠋⠀⠀⣀⣀⣿⣤⣀⣀⣀
⠀⠀⢸⣿⠀⠀⢽⡗⠙⠛⢻⣿⣿⣿⣿⣿⣿⣋⣽⠟⠁⠀⠀⢀⣠⣾⣿⣿⠿⠿⠿⢿⣷⣦⣄
⠀⠀⠀⠻⣧⣀⠀⠀⠀⣀⣼⡿⣿⣿⡿⠛⠛⠉⠁⠀⠀⠀⣠⣿⠟⠉⢸⣿⠀⠀⠀⠀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠸⣿⣷⣦⡄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠁⠉⠁⠈⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⣀⢀⣀⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠂⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠶⠊⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⣴⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠠⠤⠀⢀⣼⠃⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢷⣦⣄⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣄⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠁⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⡠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠒⠎⠀⠀⡀⠐⠀⣁⠠⠤⠤⢀⡉⠑⢄⡀⠀⢀⠌⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢄⡤⠄⢀⡠⠒⠁⠀⠀⠀⠀⡊⠀⠐⢄⠑⡄⡌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⠆⠁⠀⠀⠀⠀⡄⠂⠁⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠈⠆⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⣠⣄⢤⢠⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠉⠀⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠓⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡛⠀⢀⡀⠀⢈⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⢀⠠⠀⡏⠀⠈⠁⢀⠒⠀⡏⠀⢀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢻⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠅⠀⠀⠀⠠⠂⠀⣁⠀⠀⢢⡤⠖⠚⠒⠒⠒⠒⠒⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠄⡄⠀⠀⠄⢀⠀⠀⢠⢹⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢠⠉⡇⠀⠀⠀⠀⠀⠀⡇⢠⡇⠰⠋⠀⠀⠀⢀⣮⠚⡀⠀⠀⣠⠾⡟⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢡⠀⠀⡀⠤⢄⠀⡇⠈⠁⠀⠐⡀⠀⠀⡜⠁⠀⡇⠀⠟⠁⢰⠇⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠰⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⣷⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⡀⠀⣧⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⡾⠀⠀⠀⡞⠙⡅⠰⢠⡆⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠙⢄⣸⠀⠀⠀⠀⠀⢀⣇⣀⡴⠀⢰⡇⠀⠀⢰⡁⠀⢻⠐⠃⢇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⢹⡀⠤⠔⠂⠈⠁⠀⠐⡄⠁⢸⠀⠀⠀⣸⠅⢀⠸⡆⠐⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣿⠀⠀⠀⣿⠀⠀⠀⠩⡄⢤⡄⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡀⠠⣤⠒⠁⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢏⣀⣀⠜⠀⠀⠀⢸⡀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠳⣄⡀⠀⠀⠀⣀⡴⠞⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢳⡀⠀⠀⡇⠀⢸⠃⠀⢀⠞⠁⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠐⢄⠈⢢⠀⡀⢣⠰⠀⡇⠆⢏⠄⣠⠋⠀⣠⠖⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀⠀⡑⢄⠑⠌⠀⠂⠀⡇⠘⠌⡔⡡⡢⠞⢁⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⡀⠁⡂⠀⠀⠀⠀⣀⠀⠁⠀⠪⡈⢀⠠⢚⡡⠀⣀⡤⠴⠂
⠀⠀⠀⠀⠀⠰⠶⠦⠤⢤⣀⣀⡉⠑⠀⠀⡐⢡⠊⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢉⣀⣠⣄⣠⣤⡴⠒⠦⠤⠴⠒⠶⠄
⠀⣀⣀⣀⣀⡤⠴⠤⠤⠤⢄⡀⠀⠄⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣁⠀⣀⠴⠉⠉
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⢄⣀⣀⠨⠤⠤⠤⠤⠔⠊⠉⠀⠀⣀⠜⢁⡤⠂⠀⠐⠀⢀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠁⠐⠄⠄⠀⣀⠐⠄⠀⠀⠵⢦⣒⠂⠀⠀⠐⠒⡴⠊⢉⠁⠂⠉⠀⠶⠀⣒⠍⣁⣀
⠀⠀⠒⠐⠒⠐⠀⠐⠂⠐⠒⠀⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⣰⠚⣡⣀⣄⣖⡰⠀⠀⠂⠁⠚⠀⠘⠲⠗⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⢤⣭⠴⠞⠟⠋⣁⣉⠁⣀⣈⠛⠛⠛⠋⠉⠉⠓⠒⠛⠛⠚⠓
⠀⠀⠀⠠⠶⠒⠒⠛⠒⠛⠓⢾⣍⠉⠁⠀⠩⠤⠬⢒⣐⠈⠁⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠰⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣽⣦⠀⣿⣡⣴⣿⡿⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣟⣥⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⢿⣿⣿⣿⣿⣟⠛⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣴⠏⠀⣀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠁⣼⣿⠿⡿⠙⣿⡆⠀⠈⠋
⠀⠀⢀⣤⣶⣾⣿⣿⣶⣿⣾⣿⣾⣟⠿⢿⣆⠀⠀⠀⠀⣿⠁⠀⣰⣿⠇⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢀⣴⠶⠿⣿⣿⣿⡉⠛⢷⡀⠈⠀⠀⠀⠀⠃⢀⡼⢹⠏⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⢰⡟⠁⠀⣼⠛⢿⣿⣷⠀⠀⡇⠀⠀⠀⠀⠀⢀⡾⠁⠈
⠀⠀⠀⠀⠈⠁⠀⠀⢿⡆⠈⢷⡙⡇⠀⠁⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣾⣷⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣼⣿⣿⣿⣼⣿⣷⣯⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⠿⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢴⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠠⢶⣾⠿⢿⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠖⠶⠦⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠑⠾⢟⠛⠛⠃
⠀⠀⠀⢀⣴⣞⡭⢴⣟⣠⠤⢤⡶⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⡜⠰⠿⡛⠛⠃
⠀⠀⠐⠋⠿⠋⠀⠛⠁⠀⡴⠟⠒⠒⣲⠶⠂⠀⠑⣄⠀⠀⢰⠳⢾⣟⠋⠀⢀⣠⠤⠐⠒⠒⠒⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⣉⣷⠶⠌⢣⡀⡆⢠⠞⢁⠴⠊⠁⠠⣤⣄⣀⠘⣶⠦⣄⣰⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⣉⡓⠦⡳⣵⢃⠞⠁⣐⣶⣖⠒⠺⢷⡀⠉⠺⠇⠀⠙⣧⠙⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢉⠀⢀⣤⠎⢉⣷⣼⡏⣹⢾⡿⠷⠮⠭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⣶⣰⢻⡴⢩⠗⠊⢹⠋⠻⣩⠩⡷⣅⢰⣧⢠⡆⢉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣤⡜⠸⠇⠘⠁⠀⠀⠀⠈⣻⠁⡟⠒⠁⠈⠻⢹⡼⢻⢸⡇⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡋⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠈⠀⢸⡎⣧⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣇⠀⠀⠀⠀⠀⠀⠈⠀⢸⢿⣾⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠈⠁⢀⣸⣆⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⢻⠀⠈⠉⠀⠘⣆⠀⡾⡔⣇⠔⠡⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢆⣣⠈⡧⠴⠖⠚⠁⠈⢣⣧⠃⠻⢰⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠙⠀⣸⠄⠀⠀⠀⠀⣠⠭⠷⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣯⣀⣽⣭⣭⣭⣭⣿⣽⣭⣭⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣯⣽⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡾⠓⠒⢄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⡷⣤⣀⡠⣾⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⣠⡤⠒⠉⠉⠉⠉⠙⣿⣾⡟⢛⠀⠁⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢂⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡳⣈⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣴⣿⣄⡀⣀⣠⣄⣠⣴⣤⣶⣤⣤⣤⣤⣷⡉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡟⠋⠛⠛⢿⣿⣿⣟⠋⠀⠀⠀⠉⠿⣄⠀⠙⢶⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢷⣶⡞⠀⠀⠀⠈⠉⢻⣶⣤⡀⠀⠀⠈⠑⠶⢤⣈⠙⠷⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣛⢦⣄⣀⣀⣀⠀⠉⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣷⣮⣭⡙⣻⣿⠿⢟⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⢉⣥⣶⠿⠛⠛⠛⠭⢶⣬⣩⣾⠟⠁⣠⣼⣿⣿⣀⣠⣤⠶⠖⢚⠷⠛⢿⣄
⣿⣿⣿⣿⣿⣿⣿⣿⠟⣵⡟⠉⠀⠀⠀⠀⠀⠀⢀⡽⠋⢀⠰⢚⡿⣿⣿⣿⢉⣀⣠⣤⠖⠁⠀⠀⠀⢹⡀
⣿⣿⣿⣿⣿⣿⣿⡟⣸⡟⠀⠀⠀⠀⣠⣤⣄⣰⣿⣤⣤⣻⣧⣤⣷⣼⣿⣿⠛⣩⡿⠋⠀⠀⠀⠀⠀⠘⣧⣄⣤⣤⣀⡀
⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇⠀⠀⠀⣼⣿⣿⣿⡿⢿⣿⣿⣽⣿⠀⣿⣸⣿⣿⣿⠉⠀⠀⠀⠀⠀⣠⣶⠟⢻⠏⠉⠉⠙⠫⢗⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣇⢻⡇⠀⠀⠘⠌⢿⡿⠟⠛⠒⠛⠻⣿⣿⣾⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⣼⠟⠁⠀⠈⡇⠀⠀⠀⠀⠀⠙⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣋⠈⢿⣄⠀⠀⠀⣼⠀⠀⠀⠀⠀⣠⡿⢁⣙⣛⣩⣭⡍⠉⠉⠉⠀⠀⢸⡟⠀⠀⠀⠀⣿⣶⣦⡄⠀⠀⡄⠘⣿⠗⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⡛⢷⣤⣄⣀⣀⣀⣠⣴⡾⢋⣴⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢿⡇⠀⠀⠀⠸⣿⣿⣿⡷⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣍⣙⣛⣛⣭⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⣀⠸⣷⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠐⣰⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⠛⠿⢧⣖⣤⣤⣤⣴⡶⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣤⠛⢤⢻⠉⠹⣉⠘⢹⠸⢹⢹⢹⣿⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠉⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣤⣛⣼⣀⣀⣟⣧⣸⣸⣸⣸⣘⣻⣘⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⠏⣿⠋⢿⣭⡍⡏⡛⢹⠉⣽⡉⢿⡉⢿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣾⣾⣶⣾⣷⣿⣾⣶⣶⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠿⡻⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⠉⣨⣴⣿⣿⣿⣿⣿⡟⢿⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣷⣶⣼⣾⣷⡟⢩⣿⣿⣿⣛⢛⣭⣿⠎⣉⡠⢧⡃⠉⢻⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⣟⣟⣭⣭⣽⢻⣬⣷⣾⣿⣿⣿⣿⢿⣿⣿⣿⣭⠤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠞⠁⢁⣬⣶⣿⣿⣿⣿⣿⢿⣯⢿⣿⢿⣿⣿⣦⡥⢿⣿⢿⣿⣿⣿⡿⠒⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⡟⣼⣿⣿⢿⣹⣿⣟⣻⣿⢹⢿⣿⣿⣿⣿⢸⢻⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠷⣼⣿⣿⣿⣆⣻⣿⣿⣖⢿⡜⠘⠏⢹⣿⣿⣌⡀⠀⠹⠛
⠀⠀⠀⠀⠀⣼⠿⠟⠏⠈⢠⣿⣿⡝⠯⡧⢹⣿⣯⡓⠸⣷⠀⠀⠏⢿⢿⡓⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡿⡝⠆⠀⢸⣿⣦⡉⠀⣿⢀⣤⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠑⠈⠀⠀⠀⠻⠇⣩⣶⣿⡝⢿⣿⡿⣶⣤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⡿⣫⠵⢿⣹⡇⠀⠀⠹⡄⢹⠷⠄⠀⣀⣙⠿⣔⣲⣤⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠛⠛⠛⣿⣷⣾⣖⣲⣤⣄⣿⡗⠛⠛⠓⠂⢲⣶⡼⠶⢦⣸⣷⣶⡿⠿⠿⠿⠛⠛⠛⠻
⠀⠀⠀⠀⠛⠉⠛⠒⠒⠒⠚⣁⣤⣤⠤⠤⠌⠈⠉⠒⠒⣿⡇⠉⢁⣷⡆⢠⠈⣡⠈⣋⣉⣣⠤⠴⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡾⢸⢻⢃⣮⢞⢥⠀⠠⠤⠭⣭⣿⣿⣯⣈⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣬⡏⢀⡟⣯⣈⣬⡃⢻⣾⣯⣀⠀⠐⣾⠿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠒⠊⠉⢛⡽⠋⠋⠀⠸⠘⠹⠉⠳⠈⠙⠛⠿⠿⢽⣿⠿⠿⠓⠒⠓⠂

⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠈⠟⢳⡄⠀⠀⢀⣠⡤⠀⢴⠖⠀⢄
⠀⠀⠀⠀⠀⠀⡰⠁⢀⣠⡀⠀⡀⢹⣄⢴⠗⠀⠀⠁⢛⣢⢀⣜
⠀⠀⠀⠀⠀⡼⠀⢰⣿⢋⠏⠆⠇⡘⠡⠄⠄⠀⣤⡬⠚⢿⣏⠙⡖⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡌⠀⢰⡷⠤⢷⡴⠄⠁⠐⠀⠈⠣⣠⡴⠋⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠟⢶⡴⡏⠁⠀⣠⢂⠀⠀⢀⡿⠋⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡴⣏⡼⢇⢳⠚⠙⢢⡾⠅⡾⠓⢢⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠶⣿⠋⠒⠌⢁⢄⢀⡸⠇⠀⣧⡄⢀⠃⠠⢰⠁
⠀⠀⠀⠀⠀⠀⢠⣀⣼⡟⢀⡡⠜⡆⢐⡵⠻⠐⠦⣬⠓⠄⣙⡑⢿⡅⡠⠊⢠⡤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠁⠀⢸⡅⠀⡸⠇⢱⣌⠃⠰⢦⡉⣱⣯⠳⣄⣀⣵⢻⢭⡓⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡚⣛⡛⠓⢷⡄⢳⣟⠆⠘⠻⣿⣿⣷⠿⡉⠘⢷⡂⢹⣦⣾⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢿⠿⠚⡛⠿⣶⣼⡘⣦⡑⠛⢶⣌⠻⣭⣧⡴⢷⣌⡛⣦⡶⠏⣤⣽⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢰⣏⠀⢌⢻⣹⡄⢯⠷⠸⣽⣆⡀⠈⠳⣌⡑⠲⣅⣼⣻⢿⢶⣿⣾⠳⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⠐⠠⡽⣿⣟⠘⢳⣶⣿⣿⣷⡄⠀⣼⠛⡴⠯⠳⡝⠒⠈⠫⢫⠑⠴⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⡄⠀⠀⢻⡿⣅⠛⡃⣿⠋⣷⡈⢄⠙⠀⠻⠖⠀⠙⢦⡁⠈⡄⠀⠁⡆⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⣆⠀⢨⣡⡜⠶⢔⡹⢆⠘⢷⡌⠆⠀⠀⠀⠀⠀⠀⠈⠳⢮⣓⣲⣅⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠘⠣⡀⠀⠃⠀⢘⠦⡀⠆⠸⡟⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⡘⠢⡄⠀⠀⠁⠀⠄⢲⡇⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⣀⡑⠲⠤⠤⢔⣋⠄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠷⡅⠤⠴⠒⠁

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

⠀⠀⠀⠀⠀⢀⡀⡠⠐⣲⣷⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⠭⠁
⠀⠀⠀⠀⡴⠛⣻⣟⣿⣡⢾⠏⠚⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣷⠏⣯⡉⠀⠀⣠⣶⡒⣂⣤⣤⡀
⠀⠀⠀⠀⠀⢸⠟⠹⠓⡹⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁⡙⢀⠀⠀⠋⠘⢷⠈⠋⠗
⠀⠀⠀⠀⢀⠎⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⡀⠀⠑⠠⡀
⠀⠀⠀⡠⠃⠀⡠⠋⠀⠤⠀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡈⠑⠀⡀
⠀⠰⠞⢁⣴⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠗
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⢿⠿⡿⡿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣾⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣶⣾⣿⣆⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣇⠀⠀⠀⣰⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠙⢿⣿⣿⣿⣦⠀⣰⣿⣿⣿⠿⢁⣠⣤⣶⣶⣶⣶⣶⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⠋⠛⠛⠛⠻⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠀⠀⣡⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣿⣿⣿⣿⠋⢸⣿⣿⣿⣿⡇⠀⠙⣿⣿⣿⣟⠻⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠃⢀⣿⣿⣿⠉⠀⢸⣿⣿⣏⣿⣷⠀⠀⠙⢿⣿⣿⡄⠀⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⢸⣿⣿⠃⣿⣿⠀⠀⠀⠀⠙⠟⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠋⠁⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣜⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⠉⠙⠛⠧⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⠋⠁⣹⣿⣿⣿⣶⣤⣶⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⣴⣾⠆⠀⠀⠀⠙⠛⠉⣈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡁⠉⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠠⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣛⣿⡿⠃⢀⣾⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣤⣼⣿⣥⣤⣿⣧⣤⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠙⠁⠀⠛⠹⠀⠺⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣙⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⡟⣧⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⣀⡴⠞⡻⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡇⣿⠀⠀⠀⠀⠀⠀⣿⢸⣿⣇⠀⠀⠀⢠⡾⢋⡴⢋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢻⣄⠀⠀⠀⠀⠀⣿⢸⡇⣿⡀⠀⠀⣾⠀⡾⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⡀⠀⠀⠀⠀⢹⣄⠻⣄⠉⠛⢶⣤⡀⢸⡇⢸⡇⠸⣇⠀⠀⢻⡄⣷⠈⣷⠀⠀⠀⠀⠀⢀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠲⣤⡀⠀⠈⠻⢦⣈⠙⠲⣄⠈⢿⣾⠁⢸⡇⠀⢿⡄⢀⣼⠃⣿⠀⢹⡆⠀⠀⣀⡴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡑⢤⡙⠳⢤⣀⠀⠙⣦⠀⢻⣆⣸⣿⣤⣼⣧⣤⣼⣿⡟⠉⣠⠟⢀⣾⣃⣤⠞⢩⡖⣿⠋
⢀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⢦⡀⠉⠓⢶⣿⡶⠟⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠻⢿⣤⣾⠟⠋⣠⡴⢋⡼⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⣶
⠾⠛⠳⢦⡈⣷⣄⠀⠀⠀⠀⢀⣀⣹⣆⠀⠻⣦⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⢠⡞⠁⠀⢀⣴⠞⢛⣉⣉⣉⠙⠛⣩⡿
⠀⠀⠀⠈⢷⡹⣍⡛⠿⠿⠛⠋⠉⠙⠻⢷⣴⠟⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣴⣿⣀⣀⣴⠟⣡⠞⢩⡴⠶⠶⠶⠟⠋
⠀⠀⠀⠀⠈⠻⣦⣉⡛⠚⠛⠛⠛⠲⣦⣼⠏⠀⣠⠶⠚⠋⠻⠿⢷⣴⣿⠀⠀⣾⣆⣤⣶⣟⣿⠓⠶⣦⣄⠹⣏⠉⠉⣁⡴⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⣦⣼⠇⠀⣀⠁⢠⣶⣿⣿⣿⣿⣿⣿⠂⠀⢻⣿⣿⣿⣿⢿⣿⣦⠈⠙⣆⢻⡟⠛⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠛⣿⠀⠀⠻⠷⠿⢿⣷⣯⣻⡼⠟⢿⠀⠀⢸⡏⠿⣿⣧⣽⣽⠿⠳⠞⠃⠈⣿⠻⠿⢧⣤⣀⣀
⢀⣀⣠⣤⣶⣒⣚⣛⣉⣀⣀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⢸⣷⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣈⣉⣙⣛⣿⣶⣦⣤⣤
⠈⠙⠛⠛⠻⠿⠧⣤⣤⣀⣀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣏⠀⠀⠀⣿⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⣀⣀⣤⡤⠴⠶⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⠿⠛⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⠟⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡶⠚⠛⠛⣷⡀⠀⠀⠀⠀⠀⣠⣀⣀⣠⣴⣤⣤⢶⣤⣄⣤⣄⠀⠀⠀⠀⠀⢠⣿⣄⠀⠻⢤⣤⣤⣤⣄⣀
⠀⠀⠀⣀⣀⣀⣠⣼⠇⣰⠏⣠⡤⠴⣾⠟⠳⣄⠀⠀⠀⠀⠟⠛⠛⢻⣿⠿⠷⠛⢻⡟⠛⠛⠀⠀⠀⠀⣠⣿⡀⠉⠳⠶⠤⠶⠶⠶⢬⣙⠳⣄
⠀⣴⢟⡭⠥⠤⡤⠴⠞⢁⡾⠃⠀⣰⠏⠀⢀⣿⢦⣄⠀⠀⠀⠀⠀⠀⠙⠷⣶⠶⠟⠁⠀⠀⠀⠀⣠⣾⡃⠈⢿⣦⣤⣠⣤⠴⠶⠾⠶⣌⢣⣹⡆
⢸⣷⡶⠟⠛⠓⠶⠶⠟⠋⠀⠀⣰⠏⠀⣴⠟⠁⣀⡽⠷⣤⣀⡀⣂⣤⡀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣅⠈⠙⢦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠙⢦⣽⣟⣳⣶
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⠞⢁⣴⢾⡟⠀⢠⡟⠉⣙⣿⡿⠿⣶⡶⠶⢾⠟⢻⡇⢀⡟⠉⠻⣦⣀⠙⢦⡘⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣼⣥⠞⠋⠀⠘⣷⠀⣾⠀⣸⠏⠈⣿⠀⢹⠀⠀⣿⡄⠈⢧⠈⢷⣄⠀⠀⠙⠳⢦⣉⢮⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⢸⡇⢸⡆⣿⠀⠀⢹⡆⢸⠀⢰⡟⢳⣄⠈⠳⢤⡉⢷⡄⠀⠀⠀⠈⠛⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⡼⢀⡿⠀⠀⠘⣧⢸⠀⣾⠁⠀⠉⠛⠷⣦⠹⡄⢿⠀⠀⠀⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⡜⢁⡾⠃⠀⠀⠀⢻⣼⢰⡇⠀⠀⠀⠀⠀⢸⡆⠇⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⣯⡾⠋⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⢸⣰⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠈⢿⣗⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣘⣷⣤⣧⣾⡻⠋⠉⠙⠛⠲⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣣⠅⠼⠘⠞⠒⠝⠸⠞⢀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣬⣭⣻⣿⣿⠵⠾⠿⠿⠒⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⢾⡿⢛⡿⣯⣛⠟⠻⣶⣤⡈⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⠞⣵⣿⡆⣇⠙⣦⡀⠈⠙⠆⠙⠅⠀⠀⠀⠀⣠⣼⣷⡛⣚⠷⠦⠀⢀⣳⣠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⡄⣤⢻⣿⡄⠹⢶⣽⣿⣄⠀⠀⠀⠐⡀⠀⠐⣣⢞⢿⣷⣬⠈⢳⢤⣴⠞⣻⣿⠛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢡⠻⠘⣿⡇⠘⠀⢹⡷⠙⠐⠆⠀⠀⠀⠀⣸⠧⠈⠈⣿⣟⢷⡀⢸⡏⢺⡏⢌⠓⣆⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⠀⠀⢠⡀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠸⣿⡁⠀⠀⠀⠁⠂⠰⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠦⣄⡀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠠⡀⠀⠈⠙⢦⣀⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠤⠬⠯⠿⠿⠤⠴⠖⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣆⣤⡀⠀⠀⠹⣿⡄⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢈⣉⡀⣠⣤⠴⠚⠋⠀⠀⢠⢴⠖⠈⠀⠀⠤⠐⠂⠙⢿⣯⢦⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣤⡤⠶⠶⠶⣶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠈⠐⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣯⣷⡀⠀⠈⠳
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠤⢤⠤⣤⡤⠂⢀⡤⠖⠲⠿⠟⠛⠋⢀⡾⠀⠀⠀⠀⠀⢀⣤⠖⠒⠈⠁⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⢞⣋⣩⣥⣤⡤⣶⢶⡶⠶⢶⣦⣤⣄⣀⣀⣀⣠⣤⡤⡶⠚⠋⠀⠀⣀⣀⠤⠚⢉⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠠⠤⠤⠤⠤⠴⠶⠖⠒⠒⠂⠉⠉⠉⠉⠀⣀⣀⡤⠨⠀⠀⠴⠒⠊⠁⢀⡀⠄⠚⠁⠈⠀⠀⢀⣜⠛⣋⣲⣆⣤⡤⠟⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢉⣉⠥⠀⠀⠀⠀⣀⠠⠄⠀⠲⠉⠀⠀⠀⠀⠀⣠⡖⣥⣦⡾⢸⡞⠈⢙⡓⠢⠄⠀⠀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠒⠒⠒⠒⠚⠉⠀⣠⠴⠒⠋⠉⠀⠐⠂⠀⠠⠂⠀⢀⠤⠐⠛⠱⠋⠽⠇⠀⠙⠋⠙⠛⠑⠃⠀⠀⠀⠀⠈⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠉⠉⠉⠙⠛⠛⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⣀⡀⣾⣿⣤⡦⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⣀⣾⣶⢶⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⡀⠄⠀⣾⣿⣿⣇⣰⣿⢉⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⣆
⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠤⠄⢿⣿⣿⣯⠹⡷⠋⠈⠻⣧⠀⠀⠀⠰⠄⠀⠀⡩⠀⢹⣆
⠀⠀⠀⠀⠀⢠⣿⣾⡆⠀⠀⠀⠀⢰⣿⡿⢿⣷⣅⣀⣀⣀⣈⣣⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣯⡀⠀⠀⠀⠈⢿⣷⡌⢻⡏⠉⣸⡋⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣄⠀⠀⠀⠈⢻⣿⣿⣧⣴⣿⠓⣆⠀⢀⣀⠀⠀⠀⢠⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣀⣠⣿⣀⣿⡿⣾⢃⣷⣿⣿⣿⣆⠠⢂⣠⣾⣿⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⠙⠛⢻⣿⣿⢿⣿⣿⣷⣿⣇⣸⡆⢈⣿⣦⣽⣿⣿⣯⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡝⣶⠀⢸⣿⣿⣾⣿⡟⢹⣿⠻⣿⣿⡿⣻⣿⣵⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣻⣾⣿⣿⣉⣿⣿⢷⣶⣿⡿⠿⠿⣸⣿⣿⣿⡿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣉⠛⢿⠿⣶⣟⣑⣾⣷⣾⣾⣿⣿⣃⣰⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣧⣤⣈⣙⣻⣿⣿⣿⣿⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉

⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⣿⣿⣿⣿⣿⣇⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⡿⠿⠛⠛⠛⠛⢿⣿⣿⣧
⣿⣿⣿⢿⣿⣿⣆⠀⣼⣿⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣤⣤⣴⣿⡿⠏
⣿⣿⣿⠈⢿⣿⣿⣾⣿⣿⡿⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡟⠛⠛⠛⠛⠛⠻⣿⣿⣦
⣿⣿⣿⠀⠀⢻⣿⣿⣿⡟⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣤⣴⣶⣿⣿⡿
⠙⠛⠛⠀⠀⠀⠛⠛⠋⠁⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⣠⣤⣤⣀⣰⢟⣿⣟⠷⣆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⢲⣤⣤⣬⣯⠁
⠀⠈⠉⠉⢻⣿⠿⣾⣦⣷⢿⠆⠀⠀⠀⠀⠀⣴⠿⠿⣶⣟⠿⣿⣿⠇⠀⠀⠀⢠⠖⢛⡿⣽⠄⣰⡿⣟⠳⣄
⠀⠀⠀⠀⠘⣯⡳⠯⠽⣟⡿⢷⣶⣶⡄⠀⠘⠁⠀⠀⣩⣷⣿⡻⢿⠿⣦⡀⠀⢻⡀⠛⢓⡿⢿⠻⡄⠙⠀⡼
⠀⠀⠀⠀⠀⠈⠙⠓⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠁⠁⠀⠀⠀⠉⠀⠀⠙⠛⠋⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⣀⢰⣤⡀⣠⣴⠀⡀⠀⠀⠀⠀⠀⠀⢐⣳⣶⡄⢀⣤⡤⠀⠀⠀⠐⠲⣶⣠⣶⣖⠀⠀⢀⣀⣀⣻⣆⠀⣀⣄⣀
⠀⠀⠀⠀⠘⢻⣿⣷⣿⣿⣿⣥⡀⠀⢀⠀⠀⣰⣯⡿⠛⣿⠿⣿⡻⣆⠀⠠⠾⢿⣿⣿⣿⡿⠷⠄⢴⣿⣿⠟⣯⢼⡿⠿⣿⠂
⠀⠀⠀⠀⠒⠛⡿⠁⠙⣿⣍⠁⠀⢠⣾⡿⢶⠿⣟⠣⣼⠃⠀⣸⠇⠋⠀⠀⠐⠛⠁⣿⠙⠀⠀⠀⠘⠹⡏⠀⡇⠈⢿⡆⠈
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠉⠀⠀⠈⠸⡇⠀⢧⠈⢰⡟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠃
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⠀⠀⠀⠀⠀⢠⣤⣤⡀⣠⣶⣖⣀⠀⠀⢀⣤⣶⣦⣀⣠⣴⡦⠀⠀⠀⠀⠀⡀⢠⣶⣦⠄
⠀⠀⠀⠀⣀⠀⣴⢠⣬⣿⣿⣶⣞⣀⡀⠐⣻⣿⣿⣿⡿⠿⣯⡅⠀⠀⣴⢿⣿⣿⣿⣿⣿⣷⡄⠀⢠⢶⣿⣿⣿⣿⣿⠯
⠀⠀⠀⠤⣿⣷⣿⠛⠛⡠⢻⣏⠛⠿⡄⠀⣟⠰⡏⠀⢆⠀⠀⠙⠀⠀⠀⢸⡟⣸⠃⢻⡿⠛⠀⠀⠰⠛⣿⠀⠱⡀⠈⠃
⠀⠀⠀⠸⠋⣿⠀⢳⡾⠁⢰⠟⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠁⡟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠈⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⠿⠸⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠖⠉⣭⠿⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣎⣀⣼⡠⠃⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣛⠯⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢆⠀⠀⠄⠐⢄⠹⡂⢠⠈⢖⣠⠂⢠⠀⡀⣀⡀⠤⢀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠑⠦⣸⡄⢄⠻⣇⡜⢣⣸⢁⢂⣾⠟⠋⠁⣀⠔⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⡀⠰⠈⢻⣬⣶⣿⠞⠳⣿⣿⡏⠀⠰⢀⣼⡉⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⠛⠻⣦⣴⣯⠝⠉⠀⠀⠀⠀⠉⢋⣷⡶⣋⠥⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢈⣑⠶⠔⣿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⠶⠿⠒⠉⠉⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠠⠐⠉⠀⠔⠚⢅⣻⣿⠀⠙⠒⠂⠀⠀⠑⠚⠋⢈⣿⢌⡵⠒⢀⣤⠄⠂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣀⣠⠔⢲⡿⢋⡹⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣒⠻⢍⠑⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠖⠁⢂⣡⡞⠟⢭⣷⣤⣀⣀⣀⣠⣴⣯⠟⠉⠻⣦⡅⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⢋⠘⠂⣡⣿⣿⡟⠷⢻⡟⢯⠻⣅⡘⠀⠘⣷⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠄⠉⣀⣠⣴⢾⠏⠌⣸⡏⢶⠋⣿⣄⠣⠹⡏⢦⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⠁⠀⠀⠐⠀⡔⠁⢧⠀⠀⠸⣄⠑⠄⢩⠀⠌⠢⠑⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⡶⢯⠀⠀⠀⠁⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢐⠙⠃⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢾⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠙⠶⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⣈⣉⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠇⠸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⡠⠐⠒⠤⣌⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣧⡀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡠⠞⠉⠙⢦⡈⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠐⢤⡊⠀⠀⠀⠀⠀⠉⠓⠦⠈⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣠⠝⢦⡀⠀⠀⠀⠀⠀⠀⠀⢘⣆⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠇⠀⠈⡇⠀⠀⠀⠀⠀⣠⡴⠋⣽⣞⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⡄⠀⣼⣇⠀⣀⣠⣴⠞⠁⣠⠞⠁⠘⣆⢀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⠖⣫⠍⠁⠈⠉⠙⠳⣦⡼⠁⡼⢻⠉⠙⢿⠀⣿⣠⡾⠃⠀⢀⡴⣻⣯⠉⠀⠉⠉⠛⢶⣄
⠀⠀⠀⡴⣡⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⠴⠋⠀⠀⢸⣧⡿⠋⠀⠀⢠⢋⠞⠁⠘⠄⠀⠀⠀⠀⠀⠹⣦
⠀⠀⢰⢣⡇⠀⠀⢰⠋⠀⠀⠔⠚⠻⡉⣿⢀⡤⠀⣠⡼⠋⠀⠀⠀⠀⡏⡞⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠘⠸⡇⠀⠀⠘⢦⣀⡴⠃⠀⠒⠚⣿⠺⣖⣛⡁⠀⠀⠀⠀⠀⠀⠇⣧⠀⠀⠀⠳⣄⣤⠆⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⡀⣰⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⡠⢠⡟
⠀⠀⠀⠀⠀⠉⠳⠦⢤⣤⠤⢖⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⣤⠤⠴⣊⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠋⠉⠁


⡟⠉⣿⣛⣻⣛⣻⣛⣟⣛⡟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⣟⣻⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠳⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⢯⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⣆⠚⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠟⣿⣿⣽⣎⣿⠿⣿⣿⣿⣽⣿⣽⣿⣬⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣈⣽⡿⣻⡿⡿⢞⢿⠿⣿⣏⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠽⣡⣾⢿⣿⣿⣝⣳⡽⣪⣾⣷⣿⣿⣷⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣶⣏⣼⣧⣬⡟⣹⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣯⣭⣿⣿⠿⣿⣿⣿⣿⣿⣿⣧⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣾⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⢃⣨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣢⣾⠿⢋⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⣭⣟⣿⣿⡋⢉⡉⡤⠒⣙⠡⢄⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣛⡿⢿⡻⢿⠚⠃⣀⡀⠈⢷⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣈⣧⣭⣶⣶⣿⢅⡷⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢶⣻⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣻⣯⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣁⣿⣿⣾⣾⣿⢿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣖⣱⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠷⣿⣼⣿⣿⣿⣷⣶⣾⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⢶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⠒⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⢤⡈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠹⡄
⠀⠀⠀⣠⠴⠖⠒⠓⠒⠲⣴⣇⠀⠀⠉⠲⣄⠈⠙⠲⠤⣄⣀⣀⣤⠔⡿⠀⠀⠙⢦⣠⠤⠴⠶⠦⢤⣀
⠀⣠⠊⠁⠀⠀⠀⠀⠀⣰⠃⠈⠳⣄⠀⠀⠀⠙⠦⣀⠀⢀⡄⠀⠀⣸⠁⠀⢀⡴⠋⠑⣄⠀⠀⠀⠀⠈⠙⢦⡀
⣸⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⠓⢬⣳⣤⣤⡇⠀⢠⠏⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢳
⡇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢹⠋⣻⠓⠒⠦⢤⣀⡀⢳⠀⠀⠀⠀⠀⠈
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⢯⡀⢸⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⢠
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⢀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣆⣀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣆⣴⣾⣿⣯⡈⢄⠻⣿⡯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡍⢩⣽⣿⣿⣿⡄⠸⣿⣶⣄⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⣿⣿⣿⣿⣿⣷⣮⡿⣿⣿⢐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢮⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇⡘⠷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠐⠒⠶⠶⠶⠶⠛⠛⠟⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣦⡈⣉⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣈⣻⣿⣿⣿⣿⣿⣿⣻⣼⡟⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣟⠉⢙⣟⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣌⠙⠻⠿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⡗⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣿⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⠿⠃




⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⢤⣀⣀⣀⣀

⠀⠀⠀⢠⣄⣰⣆⣠⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣆⡠⠀⠀⠀⠀⠀⡄⣐⣰⣂⡀
⠀⠀⠀⣤⣿⣿⣿⣿⣤⠀⠀⠀⠠⣿⣿⣿⡧⠀⠀⠀⠀⠬⣿⣿⣿⡎⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⣽⣿⣿⣿⡧⠀⠀⠀⠐⣾⣿⣿⣿⡒
⠀⠀⠀⢠⠿⢿⡿⢯⠀⠀⠀⠀⠀⠹⠿⠟⠃⠀⠀⠀⠀⠈⠽⠿⠿⠓⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠐⢚⠿⡿⢟⠓⠀⠀⠀⠈⠝⢿⠿⠛⠌
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⣄⣂⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⡄⣂⡐⢀⠄
⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢾⣿⣿⣿⣿⡇⠀⠀⢀⢹⣿⣿⣿⡧⠄⠀⠀⠀⠠⣿⣿⡯⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⠀⢒⢾⣿⣿⢀⡒
⠀⠀⠀⠻⣿⣿⣿⣿⠏⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠚⡻⢿⠿⡓⠂⠀⠀⠀⠀⠋⠛⠑⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠁⠆⡉⢁⠲⡌⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⢤⣰⣦⡆⡀⠀⠀⠀⡀⢦⣴⣤⠆⡀⠀⠀⠀⡀⢆⣶⣠⢄⠀⠀⠀⢀⡄⣴⣇⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⣅⡂⠄
⠀⠀⠀⣽⣿⣿⣿⣿⡥⠀⠀⠠⣽⣿⣿⣿⣿⠥⠀⠀⢠⢽⣿⣿⣿⡯⡄⠀⠀⣀⣻⠁⠀⠙⡟⠆⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠅⣾⣿⣿⣿⡎⠄
⠀⠀⠀⠚⣻⢿⣿⡟⠂⠀⠀⠀⠊⣻⢿⢿⡛⠂⠀⠀⠀⠊⡻⠿⢿⡓⠂⠀⠀⠈⣙⣷⢤⡞⠻⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠂⠝⠿⠿⠟⡔
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁
⠀⠀⠀⠠⡰⠘⠰⢢⠀⠀⠀⠀⢀⣠⣤⣆⡀⠀⠀⠀⠀⣠⣢⣖⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣷⣶⣄⠀⠀⠀⢀⠰⣀⣆⡐⢀
⠀⠀⠠⠅⢰⣿⣷⡄⠩⠀⠀⠀⢺⡏⠀⠈⣿⡂⠀⠀⢸⣹⠁⠀⠈⣳⡅⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠌⣿⣿⣿⣿⡅
⠀⠀⠀⢫⢌⠙⢋⠠⠑⠀⠀⠀⠙⠷⡦⣶⠟⠀⠀⠀⠘⢭⡷⢤⡴⡿⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⠿⠀⠀⠀⠐⢙⠿⠿⢟⠐
⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣵⣭⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣷⣿⣿⣾⣽⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⡛⠹⣦⡀⠈⠀⠀⣀⣠⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⢿⣿⠿⠲⠇⡆⠀⠀⠙⠓⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠠⣞⡿⠟⠛⢩⠟⠉⠀⠀⢼⡇⢸⡌⠻⣿⡶⣠⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠙⠀⡑⢼⣿⡄⠈⠰⢽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢸⡇⠈⣿⣇⠀⠀⠈⠙⠉
⣀⣀⣀⣀⣀⣀⣀⣀⡼⠀⢀⣀⡀⣀⠘⣀⣈⢇⣀⣿⣷⠀⢀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀
⠉⠉⠉⠉⠉⠉⠙⠉⠇⠀⠺⠛⢋⠦⠉⠙⠷⠶⠇⠸⡿⠀⠈⠉⠉⠉⠛⠛⠛⠛⠟⠋⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⣀⣀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠐
⠒⠠⠤⠤⠤⠤⠀⠀⠀⠀⡔⠈⠈⠉⠀⠀⠘⠓⠒⠀⠀⠀⠀⠀⣀⣀⣉⣘⡋⠢⠤⠤⠄⠀⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠓⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣦⠈
⠀⠀⠀⠠⡀
⠀⠀⠉⠀⠴⠦⠤⠲⠒⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡄⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣴⣿⣄⡀⣼⣿⣇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢻⣿⣿⡿⣻⣿⣿⣯⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣋⣠⣿⣿⣿⡟⠀⠀⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡿⠿⠟⠛⠛⣦⣴⣿⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⣤⣭⣥⣤⠴⠶⣶⣾⣿⣿⣿⣿⣧⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡿⠛⠉⠉⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣷⣄⣤⠆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣤⣴⡾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⣿⣿⣿⣯⣤⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠟⠉⠉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣽⠿⠛⠛⠋⠉⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⣤⣴⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⢉⡩⠿⢍⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢗⡟⣿⢻⣄⠢⡀⠀⠀⠀⠀⠪⣄⢠⣀⠀⢉⣤⣤⣾⣧⣀⣠⣇⣽⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⡞⠀⠘⣇⢹⡑⢌⠢⡀⠀⢠⣤⣿⣿⡿⣷⢾⣿⣿⡿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⢶⣤⣤⣶⣿⠀⡞⠀⠀⠀⠘⢆⢳⡀⠙⢮⣢⡀⢿⠐⣿⣷⡗⠀⢴⣿⠁
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠈⢹⣿⡏⣴⡁⠀⠀⠀⠀⠈⢧⠙⡄⠀⠙⢮⡻⢦⣼⡝⠟⠉⢹⡿
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⡽⣿⣿⡄⠀⠀⠀⠀⠈⢳⡈⢦⡀⢀⠙⣶⣴⡁⠀⠀⡿⠁⢀⣠⣤⣶⣶⣶⣶⠂
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⡁⡸⠁⠘⣿⣿⠀⠀⠀⠀⠀⠈⢷⣀⣹⡻⢿⣿⣿⡟⠀⢠⣧⣴⡻⣿⠟⠛⠋⠉⠉⠉⠓⠶⣄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿⣼⣴⡅⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠈⣿⣶⣭⣀⢻⠻⡿⣤⣿⣿⣯⡙⠃⠀⢀⠀⠀⠀⠀⠀⠀⠈⠱⣆
⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⠀⠀⢈⡗⠀⠀⠀⠸⡇⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⣧⣬⣿⣇⡜⢠⣑⢤⡀⠑⠠⢤⣄⠀⠀⠀⠀⠈⢧
⠀⠰⡀⢫⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣟⠓⠿⣟⣿⣶⣾⠆⢙⡀⠀⠀⠀⠀⠈⠇
⠀⠀⠱⡄⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠉⢻⡏⢿⠛⠓⠿⢧⢭⣽⣖⠚⠃
⠀⠀⠀⠙⢦⡑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⠂⠀⠀⢀⡞⠛⠉⠛
⠀⠀⠀⠀⠀⠙⠶⣤⣀⡀⠀⠀⠀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣉⠂⠀⠀⠈⠁⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁

⠀⠀⠐⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡔⣚⣮⢮⣠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢢⣀⢀⣜⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠦⠁⢸⣇⠠⠄⣤⠀⠀⠀⠀⡘⢫⢥⡈⡴⠆⡤⣸⠊⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢗⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣄⢄⡈⠒⢦⣸⡿⡛⠍⠂⠀⠀⠀⠀⠀⠓⠦⣴⣧⡄⡇⣏⡡
⠀⠀⠀⠀⠐⠀⠀⠠⠆⠘⠀⠈⠑⢤⡀⠀⠀⠀⠂⠀⠀⠀⠌⠓⢺⣵⡀⣍⣔⣢⣄⣀⢄⣀⠀⠀⠀⠙⠒⠋⠁⠟⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⣦⠤⠀⡀⣙⣉⠓⠒⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⣀⣲⣂⢈⠟⠙⢙⠳⢕⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⢉⠻⡦⡈⠑⠀⠂⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠜⡇⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠑⠟⠷⠆⠀⠀⠀⠀⠀⣀⠀⠄⠠⠂⠄⣠⢳⠞⢀⡄⠉
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⡀⠀⠀⢉⡔⠊⣠⠃⡀⡑⡖
⠀⠀⠀⠀⢀⠐⠀⢀⠈⠣⡀⠄⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠓⠪⣽⣳⣶⠋⠀⠀⠀⠀⠈⠀⠀⢈⡻⠮⣱⣣⢰⣡⡊⠙⠂⠅
⠀⠀⠀⠀⠌⠀⡀⣀⡑⠤⠄⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⢧⣈⣉⣩⣽⣇⠿⡃⠀⠀⠀⠀⠀⣀⠐⠒⣀⣀⣽⣿⣾⡁⠀⠁
⠀⠀⠀⠀⣩⠞⠉⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠺⠍⣀⣹⣆⠿⠋⡧⠚⣿⡏⠑⠀⠀⠀⠀⠈⠉⡵⢨⢟⠞⣹⢏⠃⠈
⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢒⡠⠔⢋⡏⡈⣢⣃⠄⠓⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⢛⠃⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠑⠜⠠⠂⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣝⣫⣽⣩⣟⣹⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠀⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠲⢤⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⡴⠃⠀⡴⠀⢠⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡼⠁⠀⡼⠁⠀⡾⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠸⠁⠀⢰⠇⠀⢠⡇⠀⠀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⠀⠘⠁⠀⢸⡇⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣰⡇⠀⢀⡇⠀⠀⢀⡀⠀⠀⢸⣧⡀⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⣀⣼⠀⠀⠀⣼⠀⠀⠀⢸⡏⠙⠛⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠃⠀⠀⢰⢿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⠀⢠⡏⠀⠀⢀⡟⢸⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢻⣿⡟⠀⠀⠀⣼⠁⣿⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠻⣇⣀⣠⠞⠛⢿⡇⠀⠀⣰⠧⢤⣤⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⠀⠉⠀⠀⠀⠀⠙⠲⠖⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⡽⠟⠳⠶⣦⣤⣀⣀⡀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣟⢟⣿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄⠀⡄⠀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠸⡆⢹⡄⢻⠀⡇⢰⡇⢸⠁⡼⠀⡴⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠹⠆⠻⠄⠧⠸⠆⠧⢸⠃⡟⢰⠃⡼⠁⡔⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣒⣒⣒⡒⠀⣿⣿⡎⢰⠃⢏⣐⣥⣬⠀⠎⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⣒⡒⣒⣤⣶⣾⣿⠿⠷⠶⠾⠛⠡⢻⡃⠀⣔⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣒⣒⡒⢂⣤⣿⣿⡿⠛⠁⠀⠀⠀⠀⢀⣴⣿⣻⣞⠋⠉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠰⢿⣿⣿⣿⣶⣶⣷⣶⡄⢀⣴⢏⣼⠃⡟⠈⠳⣴⣦⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠠⠭⠥⠝⠛⠛⠛⢹⡟⠀⢘⢿⣞⢁⡾⠁⠀⢿⡀⠀⠉⠉⠀⢀⣾⠁
⠀⠀⠀⠀⢀⣤⣬⣭⣭⣭⣭⡍⢀⣴⠞⠛⠛⢻⣷⡴⠛⢧⣉⢯⣷⣾⠀⠈⠛⠶⣤⣤⠶⠟⠁
⠀⠀⠀⠀⠀⣤⡤⠤⠶⠖⢒⡃⣾⠁⠀⢀⣤⣼⣿⣿⡿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⢠⠴⠖⠚⣉⡭⠄⣿⠀⠀⠙⠟⠋⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠴⠚⢉⡡⠖⢋⡌⢷⣤⣀⣀⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠚⢉⡠⠞⢁⠔⢡⠎⡩⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⠞⢁⡔⢁⠜⢠⠃⠌⠐⠀⠀⠀⠀⠀⠀⠀⠠⡀⣠⠊⠀⠈⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⢠⠊⢠⠃⠐⠀⠀⠀⠀⢠⡀⠀⣠⣶⣶⠏⣡⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣻⣷⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠋⠻⠟⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠢
⠀⠀⠀⠀⢘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠪⣷⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⣄⠀⠀⠀⢀⡀
⠀⠘⠷⣄⠂⠈⢽⣇⡕⠏⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠳⠲⠄
⠀⢠⣄⠙⢻⡀⠱⢃⣴⣤⣧⣦⣶⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⣀⡀
⠀⠀⠨⢧⠈⢥⣠⢀⣌⡀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠑⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠈⣣⠜⣉⡁⠙⠛⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡜⢲⠟⠛⠷⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡀⡆⠃⢀⢠⠇⡄⢠⠃⡄⢀⠄⠀⠀⠀⠀⠂⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡰⣌⢆⠣⢱⣠⣘⣘⡰⠡⢃⠎⡰⢋⠀⠀⣀⠀⠀⠀⢀⠀⡀⠀⢀
⠀⠀⠀⠠⠔⠀⢀⠀⠈⣇⠀⠀⠀⠐⠦⣄⡐⠦⣀⠦⡑⢌⠾⠋⠉⠀⠀⠀⠉⠓⢮⢊⠴⣋⠴⠊⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⠀⢈⠳⠀⡿⠒⠒⠒⠠⠤⣀⣉⠓⠬⡑⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠵⢚⣁
⠀⠀⠀⠀⠁⠀⠈⠂⢧⢈⡉⠓⠶⠆⠀⠠⠬⢭⡑⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠭⠤⠒⠒⠉⠁
⠀⠀⠀⠛⠀⠀⠀⠨⠶⢦⡉⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈
⠀⠀⠐⠂⠀⠈

⠀⠀⡀⡀
⣤⣤⣉⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠚⢿⣿⡏⣠⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡍⣷⣶⡝⢻⡆⠀⠀⠀⢀⣀⢠⠼⣿⢣⠈⠂⠿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣧⣾⣧⣭⡛⣾⣿⡇⠀⠀⠀⢠⡥⠁⢘⣿⣄⣀⣨⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠿⠟⢉⡉⠻⠿⠏⠁⠀⠀⠀⠈⠙⠶⠿⠉⡉⠹⠿⠋⠁⠀⠀⠀⠈⠹⠿⠏⢉⠉⠿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⢰⣴⢷⣦⠸⠟⢰⣾⣧⡄⠀⠀⠀⢰⣴⣷⣶⠸⠟⢰⣾⣧⡄⠀⠀⠀⢠⣼⣷⡆⠻⠃⣶⣿⣦⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣷⢶⣿⣿⣻⡇⠀⠀⠀⠈⠁⠀⠙⢷⣶⡿⠟⠛⠃⠀⠀⠀⢸⣿⣿⣿⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⣤⣿⡯⣿⡇⠀⠀⠀⠰⠶⡠⠄⡾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢼⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣠⣽⣿⡟⢡⠘⡇⠀⠀⠀⢈⠂⠐⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣃⣳⡷⣷⣃⡿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⣉⣉⣤⣿⣧⡁⠀⠀⠀⠸⠿⢿⣿⣧⢀⣴⣶⡄⠀⠀⠀⠀⢸⣿⠏⠁⠀⠀⠈⢳⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣌⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⠈⠛⠉⣴⡇⠀⠀⠀⢸⣿⡄⣠⡄⢠⣄⢸⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣼⡏⢸⠿⠛⠻⡇⠀⠀⠀⢠⠀⣶⡆⣿⣗⢲⡄⠀⠀⠀⠀⠀⢸⣿⣧⣉⠘⠋⣉⣼⣿⠅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣥⣴⣾⣿⣷⡆⠀⠀⠀⢀⠙⠀⠖⣿⣿⡀⠁⠀⠆⠀⠀⠀⢸⣿⣿⣌⠁⠈⣰⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣿⡁⣹⡇⠀⠀⠀⢈⡁⢀⣠⣿⣿⣿⣶⣤⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⢉⣍⣹⣿⡿⠇⠀⠀⠀⠸⠿⣿⣿⢉⣍⢹⣿⡿⠇⠀⠀⠀⠸⢿⣿⡏⣉⡉⣿⣿⠿
⠀⠀⠀⠀⠀⠀⢀⣀⣭⣄⠘⠋⢠⣭⣅⣀⠀⠀⠀⢀⣀⣭⣤⠘⠋⢠⣭⣀⡀⠀⠀⠀⢀⣈⣭⡀⠙⠁⣬⣭⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣛⡻⡛⣿⣿⣻⡇⠀⠀⠀⢸⡏⠁⠙⣿⠟⠉⠙⢻⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣋⡼⣿⠧⢻⣿⣿⠃⠀⠀⠀⠀⣯⠁⢹⣏⢒⢀⡀⣸⠃⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⢿⣿⣀⠀⣤⡼⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⢸⣿⡀⢸⣦⡀⣿⡏⠀⣾⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠹⣶⣦⣿⣷⣤⣿⣿⣿⣷⣼⣿⣧⣾⡟⠀⣤⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣼⣿⡿⠟⠛⠋⠉⠉⠙⠛⠿⣿⣿⣧⣾⣷⣀⣴⣾⠟⠁
⠀⠀⠀⠒⠶⢶⣤⣤⣀⠀⠙⠾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⡿⠀⢀⣀⣀⣀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣉⣩⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣤⠤
⠀⠀⠒⠛⠛⠛⠛⠛⠛⢛⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠿⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡛⠛⠛⠻⠿⠷⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠓⠂
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⢸⣿⠟⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⠻⢿⣦⣄
⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠋⢁⣴⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⢦⠀⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⢻⡟⠀⣿⡿⣿⣿⠻⣿⡿⣿⣧⠈⢻⣷⡀⠀⠀⠀⠀⠀⠈⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠈⠀⣸⣿⠁⠉⡏⠀⣿⣇⠀⠘⠂⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⢾⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣷⣦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⣿⣷⣿⣷⣿⣿⣿⣿⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⣿⣿⣿⣿⣿⠖⢿⡟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠈⠻⠿⡛⠋⣿⣿⣿⢀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠓⠀⠛⣿⡿⠙⢹⣿⠏⢹⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠆⠀⠀⠀⣿⡇⠀⠻⠋⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣼⣿⣿⣿⣿⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢹⣿⡿⠻⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠘⣿⠀⠀⢸⡏⠁
⠀⠀⠀⠀⠀⢀⣤⣴⣷⣶⣶⣤⡀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢾⣯⡿⣿⣯⡿⠿⠃⠀⠀⠀⠰⡿⠛⢰⡟⠻⣿⠖⠀⢸⡇
⠀⠀⠀⠀⠀⠘⠿⠃⢸⡟⠀⠈⠛⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠿⠄⠀⠸⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡀⠀⠐⠀⠀⠀⠀⠀⣀⡀⠘⢃⣀⣉⠀⠰⠇⠀⠀⠀⠀⠀⠀⣄⠹⣇⠹⠀⠀⡜⢀⡀
⠀⠀⠀⠀⠀⡀⣄⣴⠟⢯⡓⣄⠀⠀⠀⠀⠀⠀⠙⠁⡄⠀⡀⠀⡀⠀⠀⠶⠂⠀⠀⠀⠲⠀⠁⢈⣤⣤⣄⠐⣡⡴⡆
⠀⠀⠀⠀⠀⢀⠸⡧⣸⣅⣼⣇⠀⠀⠀⠀⠀⠛⠃⠰⠀⠢⣀⣀⡡⠄⠁⠠⡤⠀⠀⠨⠭⠄⠀⣿⣿⣿⣿⡇⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠛⠏⠀⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠶⠒⣡⠌⠛⠛⠋⠀⢌⡙⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⣤⠀⠲⠄⠀⠀⠀⠀⠀⠀⠺⠃⠀⡆⢰⠀⠘⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠲⠄⠸⡄⠀⢠⡟⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢃⣀⠃⢠⠆
⠀⠀⠀⠐⠶⣄⠢⠄⠐⠈⡔⢀⣤⡄⠀⠀⠀⠀⠀⢠⠿⣟⣛⡛⠿⣷⡀⠀⠀⠀⠀⠀⠐⠄⢠⣾⣿⣿⣿⣦⡀⠤
⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠠⠄⠀⢤⠄⠀⠀⠀⠀⢿⣜⢾⣜⣟⡶⣿⠁⠀⠀⠀⠀⠀⠐⠂⣿⣿⣿⣿⣿⣿⡇⠤⠄
⠀⠀⠀⠙⠋⠁⠔⡀⠄⣀⠢⠘⠲⠄⠀⠀⠀⠀⠀⠈⠮⣯⣿⣿⡵⠃⠀⠀⠀⠁⠀⠀⠐⠂⡘⠿⣿⣿⣯⠟⠡⢄
⠀⠀⠀⠀⠚⠁⡸⠁⠀⠘⡄⠐⠦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⢀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄⠀⣠⠆
⠀⠀⠀⠀⢠⣀⢻⠢⡸⠣⠞⣏⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣤⣀⣹⠟⠀⠀⠀⠀⠑⠧⢤⣤⣤⠀⠀⠀⠀⠀⢀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠍⠫⣝⢦
⠀⠀⠀⠀⠈⣲⠄⠀⠀⠀⠀⠀⠺⣍⣀⠀⠀⠀⠀⠀⢰⠡⠄⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⢆⢞⣻⣾⣿
⠀⠀⠀⠒⠛⣡⣟⣀⠀⠀⠀⠀⣏⠈⠁⠀⠀⠀⠀⠀⠈⠢⣈⣋⠠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠧⣂⠥⣕⣵⠏
⠀⠀⠀⠀⠘⠉⠀⡶⠉⣦⠲⡏⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

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

⠀⠀⢀⡀
⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣀⣠⠤
⠀⠀⠀⠀⢾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡟⠉
⠀⠀⠀⠀⠸⠻⣏⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣿⣿⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠈⠻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀
⠀⠀⠀⠀⠰⠀⠈⢻⣇⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⣸⠀⠀⢸⠃⠀⢀⡿⠀⣀⣠⡤⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣶⡖⠉⣤⢤⡤⠤⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣧⠀⠀⢿⠀⠠⠘⠿⢧⣍⣁
⠀⠀⠀⠀⠀⠀⡀⠀⠘⢿⣄⡀⠀⠠⡄⠀⠀⠀⢀⠀⠀⠀⣿⣷⣴⣷⣬⡹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢻⠀⡠⠚⠧⣤⡀⠀⠂⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣅⠀⠀⠀⠀⠀⠀⠘⠢⠐⠊⠀⠟⠁⠀⠙⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠸⣧⣀⣀⣀⣀⣉⣉⠓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠒⠉⠳⢤⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⣰⢙⡆⠀⡴⠀⡄⠐⡾⠒⢰⠀⣠⠀⡦⡠⡆⡠⣆⠀⡄⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠉⠹⡄⠧⠞⠧⠐⠧⠄⠺⠔⠻⠀⠟⠁⠻⠁⠸⠄⠟⠀⠳⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠆⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠰⠀⠀⠀⠀⠀⣼⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⠀⠀⠀⢠⠇⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠁⠘⣇⠀⠀⠠⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⣴⠃⠀⠀⠀⠀⠀⠈⠓⡄⠀⣸⠀⠀⣠⠔⠉⠀⠀⠠⣄⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢠⡟⠀⠀⠤⠀⠀⠀⠋⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠉⠁⠀⠀⠀⠀⠀⠠⣄⡘⣦⣯⡴⠋⠁⠀⠀⢀⠀⠀⠈⠑⠢⣄⡀⠀⠀⢻⡄⠀⠀⢰⡟⠀⠀⠀⠀⠀⣀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠛⣿⣏⡉⠉⠉⠉⠉⠁⠀⢀⡐⠀⠀⠙⢷⣄⠘⣿⠀⣠⡟⠁⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⢸⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢿⣿⣦⡟⣀⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⠿⠿⠿⢯⡤⣤⣀⣀⣀⠚
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⣀⠈⠉⠉⠒⠤⣀
⠀⠀⠀⠴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠂⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢾⡦⠀⡀⠄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⠂⠙⠆⢀⡸⠁⣴⣆⢹⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠴⠒⠂⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⢤⠁⠈⠀⠀⠀⠀⠑⣄
⠀⠀⢠⣢⡏⠥⢃⡚⠀⠛⠋⠁⠀⠀⠲⠤⠄⠊⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠰⣷⡀⠀⠀⠀⠁⡄
⠀⠀⠈⠋⠲⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠠⠄⢤⠀⠀⠀⠀⠀⠀⣷⡀⠀⣿⠇⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⢷⠀⠉⠀⠀⠀⠀⠀⠀⠐⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠄⠈⠲⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣰⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠰⣄⠀⠀⠀⠈⠢⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⡀⠉⡇⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢻⣿⣶⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠁⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡐⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠔⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣷⣄⣀⠀⠀⠀⡀⢀⠀⠀⢴⣿⣿⡃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⡿⢿⣿⣿⠟⢡⣴⠦⣵⣤⡰⣿⣰⣿⣿⣿⠀⠀⠱⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠈⠉⠉⠰⢾⣿⣿⣷⣺⣿⣿⡿⣿⡿⠃⠀⠀⠀⠎⠀⢈⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣤⣀⢀⡙⣿⣿⣿⣿⣿⣫⠿⢡⣾⣤⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠟⠻⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣃⣾⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣰⣶⣾⣿⣿⣿⣿⣿⣿⣵⣶⣠⣤⣶⣾⡉⠀⠒⠖⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⣿⣟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⠻⠛⢁⡜⠙⣠⣿⣿⣿⣿⣿⣿⡟⠻⣿⡿⢿⣿⣿⢿⣿⠄⠀⢄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⢀⣴⡯⢀⣾⢿⣿⠿⠛⣿⡿⣿⣿⣮⣽⣻⠈⠹⣿⣿⣿⣿⡢⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⡿⠇⠀⠀⢠⣾⠟⣠⣯⢿⣟⠏⠀⢰⣿⡇⠙⣿⣿⣿⢿⣿⡆⠸⠛⣿⡾⣿⣆
⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⡾⠃⠀⢿⡸⠈⠀⠀⠀⢸⣿⠀⠀⠈⠇⢻⣿⣳⢯⠀⠀⠁⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠸⠁⠀⠀⠀⠀⢸⣻⠀⠀⠀⠀⠈⠛⣾⣗⠄⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣡⣿⣿⠿⢿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢉⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠁⠀⠀⠀⢀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢰⣾⣿⡛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿⡒⠤⣄⣀⡀⣸⣿⣿⣷⡀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠈⠙⣦⡴⣮⣙⠻⣿⠟⣿⣷⡾⠋⠉⠉⠉⠛⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⢶⣶⣿⡀⠀⠀⠀⠘⢧⡀⠉⢻⣿⣦⣿⣿⣇⡀⣠⣄⡀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⠀⠀⠀⣸⠋⢻⣦⠀⠀⠀⠈⢻⣷⣾⣿⣿⣿⣿⣬⢽⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢠⢲⣶⣧⠀⠀⢻⡆⠀⠀⠀⠘⠛⠛⠿⠿⢻⣇⠀⠀⠰⠋⠉⠀⠀⣸⡯
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠐⠿⠿⠋⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⣀⣠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⢶⡶⠾⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⣰⠖⠀⠀⠐⠂⠄⣄⡀⡇
⠀⠀⠀⠀⠄⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠢⠔⠉⠀⠀⠀⡀⠀⠐⠉⠉⠁
⠀⢰⣤⣄⡀⠀⠀⠈⠢⡀⠀⠀⣀⡀⠀⠐⠀⠀⠉⠀⠚⡀⠀⠀⠀⠁⠀⢉
⠀⠈⠂⠀⠙⠀⠀⠀⠀⠀⠁⠚⠿⠀⠐⠢⣀⠈⡅⠀⢤⣀
⠀⠀⠀⠀⠀⡀⢨⠈⢢⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢄⠈⠋⠙⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠰⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡧⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⠀⠀⠀⠀⠀⠈⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁⠁⠄⢀⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠑⠒⠀⠠⠤⠠⠠⠤⠀⠀⠒⠬⡀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⢠⣷⡄⠀⠀⠀⠀⠂⠀⣠⣆⠀⠀⠀⠘⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠙⡻⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡐⣀⠀⠀⢀⠤⠀⡀⠀⠀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⠄⠀⠉⠈⠀⠀⠀⠀⠀⣈⠁⠀⠁⠀⠀⠘⠑⠂⠰⢤⡀⠠⠄⠄⠀⢀
⡤⠆⠠⠄⠀⣀⡀⣴⠁⠀⢀⣀⡀⠀⠀⠐⠀⠀⠀⠈⠉⠒⠂⠤⠀⡀⡀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠈⢢
⠉⠁⠀⠀⠐⠒⠂⡀⠴⠶⠦⡀⢤⠀⠍⢥⠭⠀⠀⠐⣂⣂⠀⡀⠂⣀⡀⠀⠁⠀⠀⠈⠀⠀⠄⠀⠴⠃
⠀⠀⠀⠀⠀⠀⡐⠀⢠⠤⡀⠈⠄⡆⠀⢀⠀⠀⠀⠀⠀⡀⠀⠔⠉⠉⠉⠁⡈⠃⠀⠈⠃
⠀⠀⠀⠀⠄⡀⠡⡀⠈⠂⠁⢀⡖⠒⠂⠁⠀⠀⠀⠠⠀⡀⠈⠀⠀⢍⠁⠀⠀⣀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⣀⡴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⢊⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⡠⠔⠒⠒⢻⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣕⠢⠤⠤⠤⢽⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⡱⠋⠉⠉⠀⠀⠀⠀⢀⣀⠤⣒⠭⠓⢋⡵⡻⣸
⠀⠀⠀⠀⠀⠀⣀⠤⢐⣒⣒⣒⣒⠰⢵⡁⠀⠀⠀⣀⡠⣔⣪⢽⠊⠉⠀⢀⢔⡯⠊⣀⠧⠧⠤⠤⠤⢀⡀
⠀⠀⠀⠀⡴⣪⠔⠉⠅⠀⢠⠀⢠⣿⠓⢬⡲⣖⠭⠒⠉⠁⢸⢸⠀⢀⢔⡵⣫⢔⡩⠒⢺⢹⠉⠉⠉⠒⠪⣑⢤⡀
⠀⠀⢀⢎⠜⠁⠠⡀⠈⠀⠠⢠⣿⠃⡠⠊⠙⢌⢆⠀⣀⠤⠼⠾⣔⠕⠋⡔⡱⠛⢄⠀⠘⡏⣆⠀⠠⠁⢀⠌⠳⡑⡄
⠀⠀⡎⡜⠐⠠⠀⠈⠢⠐⢰⣿⡧⠊⠀⠀⠄⠚⡎⡾⠁⢀⣀⠀⠈⡆⡜⡼⠠⢀⠀⠑⢄⢹⢹⠀⢀⠔⠁⠀⠀⠸⡘⡄
⠀⠀⡇⡇⠄⠀⠀⠀⠁⢰⡛⣻⣶⣿⡶⠭⠭⠭⡧⡟⠲⢞⠖⠁⠀⣿⡇⡇⠀⠀⠀⠀⠂⢱⠿⢧⠁⠐⠀⠀⠀⠀⡇⡇
⠀⠀⡇⡇⠀⠀⠄⠂⡨⢊⠙⢉⠀⡐⠀⠄⡀⠀⡇⡏⠓⢄⡱⠔⠊⠀⡇⡇⠀⠀⠀⠄⢀⠘⠓⠛⢄⠀⠀⡀⠀⠀⡇⡇
⠀⠀⠸⡘⡄⠀⡠⠊⠀⠂⢀⠀⠄⠀⠠⡀⠀⡜⡸⠀⠀⠀⠀⠀⠀⠀⢱⠱⡈⠀⢀⠔⠁⠌⠀⠈⠀⠑⢄⠀⠀⡰⢱⠁
⠀⠀⠀⠘⢌⠪⣀⠀⠄⠀⢈⠀⠘⡀⢀⡠⢊⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⢄⠁⠀⡐⠀⠀⠀⠐⠀⠀⢁⠔⡱⠁
⠀⠀⠀⠀⠀⠙⠪⢝⣒⠢⠴⠤⢒⣚⠭⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢝⡒⠤⢄⣠⣀⠤⠗⣚⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠈⠉

⠀⠀⠀⠀⠀⠰⣆⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⢻⣦⢰⣿⣿⠏⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⣼⣿⡿⠀⣀
⠀⠰⠟⣻⣿⣷⣌⢻⣿⣿⣵⣾⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢛⣿⣿⣦⡘⣷⣿⣯⣴⣿⡃
⠀⠀⠀⣀⣤⣭⣿⣿⣿⣿⣿⣯⣭⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⣽⣿⣟⣿⣿⣯⣥⣤⣤⣤⣄⡀
⠀⢠⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡻⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⢿⣟⡿⣿⣯⡟⠛⠉⠙
⠀⣼⣿⡿⠋⣿⣿⡿⠀⢿⣿⣷⡝⢿⣷⠀⠀⠀⢰⣄⠀⣠⠄⢀⣀⠀⠀⠀⢠⣿⡿⠟⢹⣿⣿⠇⠘⡿⣿⣦⠛⢿⡄⠀⠀⠀⣤⡀⢀⡄⠀⣀
⠸⠋⠁⠀⠀⣽⣿⠇⠀⠘⡆⢿⣿⣀⣨⣶⣶⣤⣄⣿⣾⣿⣿⡟⠁⠀⠀⠀⠟⠉⠀⠀⢨⣿⡻⠀⠀⢳⠘⣿⣆⣀⣵⣶⣤⣤⣸⣧⡟⣴⡟⠉
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣷⢀⣻⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢸⡄⢈⣿⣿⣿⣿⣿⣿⣿⠋⣼⣳⣿⠿⡿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠉⠉⠉⠉⣉⣿⣿⣿⣿⠻⣿⣿⣦⡝⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈⠉⠉⣩⣿⣿⣿⡟⠻⣿⣶⣌⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⢴⣿⣿⠏⣿⣿⡇⠈⠉⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⣾⣿⡟⢸⣿⣿⠀⠁⠙⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠃⣾⣿⠋⠀⠸⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠞⢠⣿⡏⠁⠀⢿⣿⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡞⠁⠀⠹⠃⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠃⠀⠀⠛⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣯⣞⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⣿⣵⣃⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠠⢀⡄⣠⢔⣒⠰⢱⣦⣄⢄⠀⠀⠀⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡄⣴⣾⣿⣿⣦⣼⣿⣷⣿⣿⣿⣦⡀⠀⠀⠈⡈⠉⠂⠄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣦⣾⠇⠀⠂⠀⠈⠐⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⢨
⠀⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⠙⢿⣿⣏⡀⡀⠀⠁⠀⠀⣇⠀⠀⠀⡰⠀⠀⣀⠀⣠⢀⢀⡀⠀⠀⠀⠀⢀
⠀⠈⠢⣠⠸⣿⣿⠋⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⠿⠿⣿⠀⠀⢄⠀⠸⡄⡀⢠⠃⠀⣠⡔⢊⣤⠲⣪⣍⢊⣿⣿⣿⣿⡑
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⠀⠀⠀⣤⣾⣿⡂⢀⣀⣀⠀⠘⣆⢇⡤⠞⠁⣃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠃⠀⠀⠀⠀⠹⠿⠛⠁⠀⠀⠀⡠⢁⠿⠇⠀⣀⠔⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢠⣄⣧⡼⢸⡤⠿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠀⠈⠢⢄⠀⢀⠔⠁⠀⠀⠀⠀⠀⠘⠂⢙⠛⢦⣒⣒⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠄⢹⠇⠀⠂⠀⠠⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⠀⠀⠉⠛⠛⠁
⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠂⠀⢰⠀⠀⠀⠈⢳⡄⠀⠀⠀⢀⣠⣿⡿⣿⣿⡿⠻⠿⠿⠷
⠀⠀⠀⠀⠈⠁⠀⠁⠀⠊⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⢹⡿⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣄⣆⣴⡾⠷⠢⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣬⣝⣿⣿⡷⡶⠶⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⢏⡽⡛⣝⠙⠷⣦⠠⡀⠀⠀⠀⠀⢠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠀⣿⡇⢱⣘⣷⡀⠀⠁⠀⠀⠀⠠⣸⣿⣏⠀⢍⠁⣠⢾⣶⠧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠇⢹⡗⠀⠈⣿⠙⠠⠀⠀⠀⠀⡾⠉⠘⣿⣧⡈⢸⠳⡟⢹⡆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠉⢿⡅⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢄⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠓⡂⣂⣀⣲⢄⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⡀⠀⠀⠈⠑⢦⡀⠀⠀⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢻⠛⠚⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠈⠳⣄⡀⠀⠘⢦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⡀⡀⠀⠙⣶⠄⢴⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠲⣀⠹⣷⣄⢀⡈⠈⣲⣄⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠆⠀⠀⠀⠀⢀⣠⡴⠞⠊⠉⢉⡽⠥⣑⣌⠿⣹⡽⣟⣠⣶⠉⣿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⣴⠾⠉⠀⡄⠀⣴⠖⡀⡄⢠⣆⠹⣿⣇⣤⡀⠘⢙⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢋⣭⢏⣴⣾⢐⡋⢡⢰⣷⡷⡿⠿⢿⠿⠿⠿⠿⢷⣄⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠟⠛⠿⠿⠷⠿⠷⠿⠋⠃⠀⠀⠀⠀⠀⠑⠀⠐⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠰⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠛⠛⠓⠒⠲⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡾⠉⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣠⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃⠀⠀⠀⣿
⠀⠀⠀⠀⣶⣶⣿⣿⠷⠄⠀⠀⠀⠀⣧⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢿⠓⠦⣤⣀⡀⠀⠀⣀⣠⣴⡾⠁⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠙⠛⠿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⢤⣀⠈⠉⠉⢉⣩⡵⠋⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣤⣤⣤⣤⡤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢤⠄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⡀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠄
⠀⠀⠀⠀⢀⠀⣴⠦⠀⢀⠂⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠳
⠀⠀⠀⠉⠊⠐⢋⠃⠂⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡐⠀⠈⡀⠀⠀⠤⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⠠⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠐⠀⠠⠐⠀⠀⠂⢰⠀⠀⠀⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠀⠀⣡⠘
⡳⠉⠔⠆⠀⠢⠐⠁⠀⠁⠉⠙⠀⠀⠄⠀⠈⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⠁⠀⠊⠁⠀⠠⡀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠠⠒⠉⠂⠀⠐⢤⠀⠀⠄⠀⠀⠈
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠘⠀⠈⠀⠂⠠⠀⢀⣠⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⢁⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠂⠀⠈⣀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠰
⠄⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠄⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡀⢈⠁
⠐⠀⠀⠈⠁⠀⠀⠀⠈⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⡀⠄⠀⢁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⢁⠀⠠⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢸⠂⠀⠒⠐⠄
⠀⠀⠰⠀⠀⠀⠀⠀⢠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠠⣠
⠀⠃⣄⠘⠂⡀⠀⠄⢨⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠠⠀⠉⠀⠀⣰
⠀⠀⠈⠀⠀⠑⠐⠄⢌⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⢆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠛⠒⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣤⡀⢠⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣆⡏⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⢀⠇⠘⢀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠱⠄⠀⠁⠀⠀⠀⡀
⠀⠀⢴⣒⢆⢱⢝⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡈⠀⠀⠈⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⠙⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣶⣤⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠉⠿⠿⢿⡿⠟⠛⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣀⠀⠀⠀⠀⠀⠈⣿⡿⢟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⢶⣶⣶⣶⠶⠋⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠚⠛⠂⠘⢿⡀⠀⠙⠛⠋⠀⢠⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠉⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⣠⣶⣿⣿⣿⡟⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣾⣿⣿⣿⣿⡿⠋⠴⠁⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣤⠀⢀⣀⣤⠞⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⢉⣿⠟⠁⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠿⠃⠀⣠⠞⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⡄⣴⡞⠁⠀⠀⠀⢀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⢻⡏⠁⠀⠀⠀⣤⡌⠹⠟⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠛⠻⣧⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⣠⠏⠀⠀⠀⠀⣸⠛⠁⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⠏⠀⠀⠛⣄⢀⣀⣠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣠⡾⠁⠀⠀⠀⠀⡼⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠾⠛⣉⣴⣿⠏⠀⠀⠀⠀⠈⢿⣿⠟⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⡾⠋⠀⠀⠀⠀⠀⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠶⢛⣉⣿⠏⠀⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⡄⠒⠋⠁⠀⣾⠀⠀⠀⠀⠀⠀⢀⣿⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣪⣿⡇⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠹⠀⠘⢀⣀⣠⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣈⠳⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣀⣾⣿⣿⣿⣶⣦⣄⠀⠀⣼⣿⣿⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⠀⠀⠀⠀⠀⠠⠟⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠳⣾⣿⣿⣿⣿⡿⢷⣾⣿⣿⣿⣿⣿⣿⠁⠀⠉⠛⠳⠶⠶⢦⣤⣄⣉⡉⠛⢻⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠻⠿⠿⢷⣬⣿⣿⣿⡟⠛⠻⢿⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠈⠙⠃⠾⣿⢄⡤⢀⣴⣀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣶⡿⠿⠿⠛⠛⢻⣿⣿⣿⣶⣶⠂⠀⠁⠈⠉⠙⢃⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣼⣿⣿⣿⣿⣿⣿⠻⣿⣿⡇⠉⠀⠠⠴⠾⠟⠛⢿⣿⣇⣀⠙⢧⣀⡀⣠⡀⢶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠏⣠⣾⣿⣷⡖⠀⠀⠀⠀⠀⠀⠈⠛⠉⢹⣧⠈⠉⠉⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠏⢰⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⢠⣿⣿⣿⣿⡿⠁⣿⠓⡄⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣛⣛⣋⣴⣦⣸⣿⣷⠈⣿⠃⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⠟⣿⣿⡿⢿⣿⣿⡟⣷⠘⣇⠀⢹⣯⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠇⠀⣿⣿⣿⣿⣿⠛⣷⣿⣧⠸⠇⢈⡟⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣷⣶⣤⣿⣿⣿⣿⠃⠾⢿⣯⢿⣧⠘⢋⣷⣤⣤⣀⡀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢠⣿⣃⣶⡀⣾⣿⣾⣿⣇⠘⣭⡄⠀⠈⠉⠛⠷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡯⠛⠿⢿⣿⣿⠸⣧⠹⣿⡇⢿⣿⣿⣿⣿⣧⢻⣷⡄⠀⠀⠀⠀⣿⣋⡁
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⢁⣾⣿⣿⠁⢀⣄⣠⣀⣤⡄⢀⡀⢹⣇⢸⣿⣿⣿⣿⣿⣇⢹⣷⡀⢀⣀⠀⠘⢿⣿⣶⣀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋⢀⣾⡟⣾⠋⠀⠈⠛⠛⠙⠛⠑⠏⠀⠘⠏⠘⣿⠟⠻⣿⡟⢻⡆⠛⠇⠸⠧⠀⠀⠈⠻⠿⢿⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠁⠀⣼⡟⣿⠋⠀⠀⠤⣶⣿⡿⢠⣷⣄⡀⢰⣷⢰⣿⣿⣿⣿⡷⠻⣷⣶⣿⣆⠀⠀⠀⠀⢼⣿⣿⣷
⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⠀⣼⣿⣿⠃⠀⠀⠀⢠⣿⡿⠁⣸⣿⣿⣄⢸⣿⠀⣿⠛⢿⡿⠀⣼⣿⡏⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⢸⣿⠃⣰⣿⡿⢻⣿⠀⣿⠀⣿⠀⠸⠇⢰⣿⣿⣧⠻⣿⡆⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⣸⣿⣿⣿⣿⠁⠀⢰⣿⣿⠿⠀⢠⣴⣆⣀⣿⠃⡀⢿⣿⡇⢸⣿⠀⣿⠀⣿⣤⠇⢰⣿⣿⣿⣿⡄⣿⡇⠀⠀⢰⣿⣿⣿⣿⡇
⠀⢠⣿⣿⣿⣿⣧⣤⣴⣿⣷⡏⠀⣴⣟⡁⢈⣿⠃⢰⡇⣼⣿⠃⢸⣿⣤⣽⠀⣿⡏⢰⣿⣿⣿⠟⢿⡿⢸⣿⠀⠀⡈⢿⣿⣿⣿⡇
⠀⣼⣿⣿⣿⣿⠿⠿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡃⣠⣿⠃⣿⣿⠀⢸⣿⣿⣿⡀⢹⣷⣿⣿⣿⡿⠀⣤⣀⣸⡟⠀⠀⡇⣼⣿⢿⣿⠁
⢠⣿⣿⣿⣿⠇⠀⠀⠈⠻⠟⢹⣿⣿⣿⠹⣿⣿⣿⣿⣾⣿⡇⠀⣼⣿⣿⣿⡇⠘⢿⣿⣿⡿⠀⣰⣿⣿⣿⣇⣀⣤⣿⣿⣿⢸⡿
⢸⣿⣿⣿⣿⠀⠀⠀⠀⠈⠀⢸⣿⠻⢟⣀⠉⢹⣿⢿⣿⣿⡷⠀⣿⣿⣿⣿⣷⣶⣾⣿⣿⢱⠿⠿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣼⡇
⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⢿⣴⡞⠁⢀⣿⠏⣼⣿⠇⠀⠀⢿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣶⣶⡦⠙⠛⢫⣽⣿⠿⢿⣿⣧
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣰⣿⡏⠀⠀⠀⠀⠀⠉⠀⣾⣿⣿⣿⣿⣿⣿⡿⢻⠀⠀⠀⣼⣿⣿⠀⣾⡏
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⣞⡀⠀⣼⣿⣿⠉⣶⡿
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢁⣾⣿⣿⣿⣿⣿⠟⢰⡿⠁
⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⡿⢋⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⠃
⠘⣿⣿⣿⣿⣇⠀⠀⠀⢀⣠⡾⢋⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠾⣿⣿⣿⣿⣩⡤⣴⣿⠃
⠀⢹⣿⣿⣿⣿⣷⣶⣿⣿⣭⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⣈⣽⣿⣿⣿⣿⡿⠁
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠉⠛⠿⠾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣤⣄⡉⠉⠙⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⣶⣄⠀⣿⣧⠀⠀⠀⢀⣤⣶⡶⣶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠂⠀⠀⣀⣀⣠⣤⣖⡢⢿⣿⣦⠈⣿⠀⢀⣴⡿⠿⠍⠉⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⢁⣴⣾⣿⣯⠛⠟⡻⡫⢿⣷⡝⢿⣇⣽⢠⣿⣟⠃⠀⣠⢄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣰⣿⡿⠛⠀⠀⣀⣀⣀⡀⠉⢴⣿⣆⢿⣿⣿⡟⢡⣴⣾⣿⣿⣿⣟⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢸⣏⢠⣿⡿⢋⣴⣾⣿⣿⣿⣿⣿⢿⣶⣍⡻⣼⣻⠟⣡⣾⠿⠿⠿⡟⠛⠻⠃⠁⠀⠉⠳⡄
⠀⠀⠀⢠⡶⠛⠉⠛⠻⠿⣿⣦⣄⢿⣼⡟⣰⣿⣼⢻⣿⣿⣿⣶⣬⣦⣅⣭⣿⣶⡦⣬⣕⠿⣿⠿⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣰⣦⣬⣝⠻⣿⣿⣇⡿⢋⡯⢾⣟⣛⣻⡿⠏⢩⣿⣿⣿⡟⢿⡎⠻⣿⣮⡳⡀⠉⠙⣿⣿⣦⡀
⠀⠀⠀⢀⣴⣿⣿⡿⣿⣿⠿⣿⣷⡮⣛⣭⣴⣿⣏⡛⣻⠿⣿⣿⣋⣿⡟⢹⡿⠀⠘⣿⡀⠉⢿⣻⣿⣦⡀⠘⠁⡿⣿⡄
⠀⠀⢠⡟⠁⠈⠈⢀⣥⣾⠿⡻⣩⡾⢻⣾⠿⣿⣿⣿⣦⡀⠁⠊⢹⣿⢹⠈⠇⠀⠀⢻⡇⠀⠰⣿⣿⣿⣇⠀⠀⠀⢹⣷
⠀⠀⠈⠀⠀⠀⣰⣿⠏⠁⣠⣴⡿⠁⢸⡇⠀⠉⣷⠀⢻⢿⣆⠀⢸⣿⠆⠁⠀⠀⠀⢸⣷⠀⠀⠸⢹⣿⣯⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢰⣿⠙⠀⢰⣿⣿⣇⠀⠘⣿⠀⠀⠉⠀⠈⢸⣿⡆⠈⣧⣃⠀⠀⠀⠀⣸⣿⠀⠀⠀⢱⣿⣿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢿⡏⠀⠀⢸⣿⣿⡃⠀⠀⢿⣇⠀⠀⠀⠀⠈⠋⣿⠀⠹⡾⠀⠀⠀⠀⡿⣿⠀⠀⠀⢉⣿⡟
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⣷⡁⠀⠀⠘⣿⡆⠀⠀⠀⠀⠸⢿⠀⠀⠱⠀⠀⠀⠀⡏⣿⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠹⣼⣄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⣜⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⢧⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠳⣄⠀⠀⠀⠀⠀⠀⡾⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣄⡄⠀⡀⠙⣶⡈⢶⣧⢴⢇⣎⣼⢃⣾⣿⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠦⣝⢾⣿⣄⠈⣻⣿⣷⣀⡙⢿⣿⣿⠃⢸⣿⣿⣿⣿⠷⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢙⠛⣿⣿⣿⣶⣾⣬⣻⣿⣧⣄⣽⠃⣀⣿⡿⣿⣿⣟⣉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣧⣽⣿⣿⣝⣯⣅⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠈⠉⠓⠦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⢿⣿⡿⡿⠋⠈⠉⠀⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢿⢿⣿⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡇⣶⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⢰⣧⣿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣴⡗⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣸⡵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⣀⣤⡆⣼⠃⣼⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣟⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣖⣮⣷⣿⣿⣿⣿⢠⣿⢀⣿⡟⠀⣾⣿⣿⣿⣿⡿⠟⠋⠛⠹⢿⡿⣿⡇
⠀⠀⠀⠀⠀⢀⣀⠤⣔⣮⣽⣾⣿⣿⣿⣿⣿⣿⣿⡏⣼⡇⣸⣿⢃⣸⣽⡿⠟⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⣖⡹⢱⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣠⡽⠞⠛⠉
⠀⠀⠀⠈⠛⠛⠳⠾⣿⣿⣿⣿⣿⣿⠿⣿⣟⣿⡿⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠛⠉⠛⠛⠶⠿⣿⣾⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠷⠃

⢋⣤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠧⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁⡈⠉⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠘⠀⢀⠀⠀⡀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⣐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⠂
⠀⠠⠀⠀⠀⠴⠂⠈⠉⠻⢃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⢀⠀⠀⠀⠀⢈⠀⠘⠀⠀⠀⡁⠀⠀⠲⣶
⠀⠀⠀⠀⠀⠐⠊⠀⠰⠀⠘⠀⠀⠀⠀⠀⠀⡀⠉⠑⠒⢄⡀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡄⡀⠸
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠀⠀⠀⠀⠐⠈⢉⡀⢌⠠⠱⠃⠤⠂⡀
⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⢠⠠⢀⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠶⠬⢾⣷⣿⣴⠤⠴⠄
⠀⠀⠀⠦⢬⡄⠀⠀⠀⠀⠐⣠⡴⡋⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣍⠤⠖⢂⠖⠟⢟⢣⡀⠀⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠞⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂⠀⠀⠠⠂⠷⠐
⠒⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⢀⠤⠠⠀⣀⡀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠄⠀⠀⣀⠄
⠀⠀⠀⠀⠈⠀⠈⡈⠰⡏⠀⠠⠂⠀⠀⣀⡡⠎⠀⠀⠀⠀⠀⠼⠛⠆
⠄⡄⠀⠄⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠛⠶⠤⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠁⠀⡀⠀⠀⠀⠀⠁⢒⠂⠐⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠈⠀⣒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⣠⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠐⠀⠒⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⢀⠀⠐⠀⠀⠀⠒⢤⡀⠀⠶⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⡙⠳⣤⣄⡀⠢⠄
⣒⣖⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣈⣛⣂⣀⣁⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⠸⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣶⠄⣉⣤⣭⣥⣌⡁⢴⡞⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡌⢛⡷⠞⠛⠛⠻⢾⣟⢉⣾⣿⣿⣿
⠀⠀⠀⠀⠦⢀⣾⣿⣿⣿⣿⣿⣿⣆⠠⠆⠀⠀⠀⠀⣿⣿⣿⣿⣟⣻⠏⠀⠀⠀⠀⠀⠀⠙⣿⣻⣿⣿⣿
⠀⠀⠠⣴⣆⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⡤⠀⠀⠀⣿⣿⣿⣛⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠍⢛⣻⣿
⠀⠀⠀⠀⣤⠘⣿⣿⣿⣿⣿⣿⣿⡟⢠⡄⠀⠀⠀⠀⣿⣿⣿⣿⣟⣻⡄⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⣿⣿
⠀⠀⠀⠀⢀⣴⡌⠛⠿⠿⠿⠟⠋⣴⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⠇⢹⡦⠄⠀⣀⣠⣴⣟⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠁⠀⠚⠀⢶⠆⠘⠃⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣯⣾⡏⢩⣿⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠷⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠈⢿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⡇⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⠄⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⠦⣤⣄⡀⠀⢠⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⡤⠒⢀⣸⣷⡶⠾⠛⠛⠛⠓⠲⠦⢄
⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠷⠒⠒⠒⠒⠤⢄⣀⣼⣿⠟⢷⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⣀⣴⡿⣋⣽⡖⠛⠉⠉⠉⠉⠒⠢⢄
⠀⠀⢠⣾⣿⣿⡿⠋⣁⠄⠀⠀⠀⠀⠀⢀⣴⡿⠛⠀⠀⠈⠳⡄⠀⣀⣀⣠⣾⠟⠁⠀⣴⣿⢫⡞⠉⠀⢳⡀
⠀⠰⢿⣿⣿⠟⣠⠞⠁⠀⠀⠀⠀⠀⢠⣾⠏⠀⣀⣀⣤⠄⢀⣼⡿⡿⣿⡿⠁⠀⠀⣼⣿⣳⠏⡀⠀⠀⠀⢣
⠀⠀⠀⢠⡟⣼⡏⠀⠀⠀⠀⠀⠀⣰⠟⣷⠟⠛⠉⠁⠀⠀⢸⣟⠸⣿⡿⠁⠀⠀⢰⣿⣿⡟⣰⠇⠀⠀⠀⠀⣵⣿⣿⠃
⠀⠀⠀⢸⣧⣿⡇⠀⠀⠀⣰⠁⣸⣿⣾⣿⠤⠶⠒⠒⠀⠀⠀⠻⢦⣬⣤⠞⠀⠀⠈⠉⢹⡇⣿⡄⠀⠀⠀⠸⣿⣿⣃⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠸⣷⣶⣿⣿⣯⣤⣤⠤⠴⢾⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⢿⣿⣿⣧⣀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏
⠀⠀⠀⠀⠈⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⣄⣀⣀⣀⣀⣤⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣤⣤⣤⣤⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠛⠙⠛⢿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠋⠀⣠⣴⣾⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⢀⣼⣿⣿⣿⣿⣯⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⠁⣾⣿⣿⣿⣿⣿⣿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣬⣭⣷⡶⣥⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⢄⣿⣿⣿⣶⣼⣻⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣸⣿⢻⣿⡿⣿⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣿⢃⣼⡟⠀⣿⣿⣿⡧⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⣼⡏⣾⡿⠀⢠⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣹⡟⠁⢠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢁⣾⣿⣿⠏⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠇⣼⣿⣿⣿⠇⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣠⣿⣿⡿⠁⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⢾⣿⣿⣿⡇⠀⠀⠀⠀⣰⣿⢟⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⡿⠟⢿⠀⠀⠀⢠⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠃⠀⠀⣸⣧⣶⠀⢼⣿⣿⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠀⠀⢠⣿⣡⣀⣀⣄⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⢿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⣛⣿⡿⣿⣿⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣐⣌⣛⡛⣓⣶⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⣿⣿⣿⣿⣿⡏⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⢋⣾⡇⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡇⣻⣿⣿⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠻⠿⠿⠿⠿⠃⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣤⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣾⡿⣿⠿⠷⢬⡀⠤⣽⡿⠿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⢸⢃⠀⠈⠃⢰⠋⢠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠁⠀⠀⠀⢇⠀⠀⠈⠀⢸⠀⠴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⢿⡋⠒⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠀⢡⡀⠀⠀⠀⠀⠀⢳⠀⠀⣀⣿⣄⣦⣤⣀⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣤⣙⣶⣆⣶⣶⣶⣼⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠿⠟⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣛⡛⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄⠀⠀⠀⢀⣀⡄⢀⡉⠉⠑⠒⢒⣒⣂⠠⠄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⢒⣒⠒⠀⠠⣤⣄⠀⣀⣀⣀⢉⣉⣉⣉⣀⣀⣤⣤⡶⠞⠋⣉⠁⠀⠀⢐⢒⡟⠉
⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣐⣠⣬⡥⠶⠶⠶⣾⣻⠻⠿⠟⠛⠛⠋⠉⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠈⠁⢲⣶⡤⣄⠀⠁⢠
⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⣶⣶⣶⠿⠟⠛⠉⠀⡾⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠛⢛⣉⣉⣉⣥⣤⣤⣤⣤⣤⣤⡀⡀⠀⠈⠀⠀⠀⢀⣀⣠⣤⠶⢻⡅⠀⠀⠀⢀⡠⠔⢋
⠀⠀⣶⠀⠐⠒⢀⣉⣉⣙⣋⣉⣭⠭⠤⠶⠞⠛⠛⠋⠉⠁⠛⠙⣋⣁⣀⣉⣉⡽⠟⢛⠛⠛⢛⡭⠍⠑⠊⠉⠒⠺⠿⠖⣒⠉⠀⠀⣀⣈⡦
⠀⠀⠛⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠖⠒⠋⠩⠍⠐⠊⠉⠀⠂⣉⡠⠖⠋⠀⠀⠀⠀⠀⠀⡀⢀⡀⠐⢺⣷⡞⣶⣯⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠀⠀⣀⣠⡤⠖⠒⠉⠔⠈⢂⠀⡀⠀⠀⠀⢀⡤⢺⣼⣮⡿⠯⣵⣿⠁⣤⣴⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠋⠉⠉⠁⠀⠠⠴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠈⠉⢀⡛⠴⠞⠃⠀⠀⠉⠀⠉⡁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁