Greens Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⠀⠀⠀⠺⡀⢀⣷⠄⢠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠉⠙⠻⢦⡄⠀⢿⢋⢿⠂⠀⡇⠀⠰⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠉⠻⣤⡀⠀⢠⢹⣤⠇⣼⡋⠀⠠⠀⢈⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠈⠻⡀⠀⠀⢻⡄⠹⡧⠂⠀⣶⠉⣠⠿⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⡀⠀⠀⢹⡆⠀⢸⡇⠘⡇⠀⠀⡇⣼⢃⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⣿⡀⠸⣇⢠⡏⠀⠘⢰⡗⠀⠀⠀⠀⠀⠀⠀⠠⠒⡄⠀⣀⠼⠀⠀⢰⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠸⣇⠀⠻⠸⡏⢰⠃⡼⠇⠀⠀⠀⠀⠀⠀⠊⠠⣼⠏⣾⠁⠀⠀⠀⡴⠀⠀⠷⠈⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠘⣇⠀⣦⢡⢏⡼⠃⠀⠀⠀⠀⠀⠀⢘⠀⠐⣿⡽⠃⠀⠀⠀⡠⠀⠀⢀⡞⠀⢘
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢹⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣿⡟⠀⠀⠀⣰⠕⠀⠀⢠⡇⠀⣈⡀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣾⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣯⠀⠀⢀⡟⡀⠀⠀⣸⠀⢠⡟⠁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠹⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣳⣦⡌⣥⡄⣠⠾⢁⣠⠟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢙⠋⠉⠡⠔⢳⡖⠛⡁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠐⠁⠐⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣙⡼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣷⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢻⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⣀⡀⠈⠁⠀⢨⣳⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠳⡄⠀⠀⡌⠀⢀⠀⣾⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠶⠖⣒⣸⠶⢄⠀⠀⠈⢢⣰⡟⠀⢠⣾⣯⠴⠖⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡀⠀⠀⠙⠻⣦⡰⠙⢶⣄⡘⠘⠛⠓⠞⠋⠁⠀⢀⣠⠄⣒⡭⠟⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠀⠀⠙⢻⣄⡂⠈⠻⣯⡑⠯⠠⠤⠛⠷⠧⠚⠈⠁⡠⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⣤⣉⣀⣙⣓⣤⣜⡨⣿⢲⠦⣀⣠⣤⣤⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢀⡀⠤⠟⠋⠀⠉⠻⢊⣁⡀⡀⠀⠀⢀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⣠⠖⠓⠒⢒⡦⠒⠒⠒⠒⠚⠉⠉⠳⠤⣀
⠀⠀⠀⠀⠀⣀⣠⠊⠁⢠⠞⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣆
⠀⠀⠀⠀⡼⠉⠁⠀⠀⢧⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⡔⠒⠚⠉⠉⠛⢤⡀
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⢱⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠓⢢⠀⠀⢰⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠈⡆⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣳⡀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣄⣀⠀⠀⠘⠦⣄⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠲⡄⠀⠉⣶⠚⡄⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⠤⠟⠀⡇⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡽⠛⠒⢄⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⢷⡤⠀⣤⠀⠉⠺⠋⠒⢐
⠀⠀⠀⠀⠀⠀⠢⠤⠥⢈⠁⠀⠛⠞⠒⣶⣆⠀⠀⢻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠏⠀⠓⠲⣿
⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠙⠀⠀⢒⠀⠀⠀⠀⠀⢠⢠⠀⣀⡀
⠀⠀⠀⠀⣤⠔⢄⠀⠀⠄⠀⠀⠀⠀⠀⠐⠢⠔⠁⠀⠉⠙⢞⢀⠎⢀⠀⠂⣎⡀⢶⢉⡒⠀⠀⣀⡄
⠀⠀⠀⠀⡇⠀⠘⠳⢤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡳⠛⠀⣐⢋⡉⠀⠛⠠⢁⡀⠈⠉⣀⠤
⠀⠀⠀⠡⡇⠐⠀⠀⠀⠑⠂⠉⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠘⢀⣈⣩⣥⡉⠠⠀⠀⡀⠀⠀⠀⠠⠄⠶⣴⣤⡄
⠀⠀⠀⠀⠈⠀⢀⡀⡀⠀⠀⠀⠀⠀⢶⣿⠀⠂⠀⠀⠀⠀⢀⡤⢹⠓⠘⠃⠀⣠⣬⣍⠉⠄⠀⠁⠀⠬⢿⣿⣧⣀
⠀⠐⠀⠀⠀⠠⣼⢣⡷⠚⠀⠀⠀⡠⠆⠀⠀⠂⠀⠀⠀⠀⠁⠀⣠⣄⠀⢀⣬⡀⠀⠀⢹⠍⢉⠀⠲⠶⠼⠉⠁
⠀⠀⠀⠀⠀⠀⠙⢉⣁⠹⠉⠀⠞⠁⠀⠀⠀⠀⠀⠄⠀⣣⠀⠀⠉⠁⠀⠈⠀⠀⡀⠀⠀⣴⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠉⠁⠀⠂⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠘⠒⠚⠁⠀⠀⠈⠉⠉

⠀⠀⡀
⠀⠀⣸⡍⠢⠴⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡳⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡈
⠀⠀⢧⡁⢕⣶⣟⣋⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠁⠈⠉⢢⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣽⣷⣿⣻⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠉⠉⠉⢳⡼⠁
⠀⠀⠀⠉⠙⠛⠛⠻⠶⠀⠀⠀⠀⠀⡈⠃⠀⠀⠀⠀⣠⠏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠷⠀⠀⠀⠀⠀⢸⣻⠁⠴⠶⠖⣢⣞⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠋⠓⣂⡡⠋⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣿⣛⣛⣻⠟⠁⡾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠍⣱⠏⠉⠀⢀⣾⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣻⡟⠉⡕⣀⣼⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣮⣷⣶⣿⡛⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣿⣶⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⢀⢦⠠⣀⣀⣈⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡡⣤⣄⣉⣛⣷⣤⡀
⠀⠀⠀⠀⠰⠁⠀⠳⢀⠀⠈⠉⠁⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢻⣦⡈⠉⠉⠉⣉⢿⣄
⠀⠀⠀⠀⠸⡄⠀⢰⠀⠁⢤⡀⠀⠈⢈⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⣸⡇⠉⢦⣔⡚⠋⢣⣻⣷⡀
⠀⠀⠀⠀⠀⠙⠷⣾⢄⣀⡀⠈⢱⣤⣤⢤⣭⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣠⣶⣉⢻⣶⣿⣿⣯⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠫⠏⠤⢿⣽⣿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⠿⣿⣿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠈⠻⣧⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣴⠀⣰⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣇
⠀⠀⠀⢐⠀⠀⢰⠉⠛⡙⠦⡀⠈⣙⣀⣮⠀⡄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡎⣠⡿⢀⢻⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣦⡆⠀⠈⣼⡀⠀⠈⢢⠱⠆
⠀⠀⠄⠈⢳⣄⡼⣄⠀⠐⠄⠸⣄⣸⡟⢉⢫⣧⠀⠠⢀⠀⠀⠀⠀⠀⠈⠃⣻⡇⡾⣧⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢐⣾⠄⠀⢸⣧⣄⠀⣿⣶
⠀⠀⠠⠌⠉⠙⠋⡈⠷⣿⡆⣀⠀⠁⠈⠑⣷⣌⢷⡀⠈⠧⠀⠀⠀⠀⣠⠈⣿⣶⣵⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⣣⣌⠰⡿⢡⠂
⠀⠀⠀⠀⠲⡀⡈⠀⠀⠈⢹⢿⣣⣘⡀⠀⠈⣋⣄⡳⢄⡀⠄⠀⠀⠀⣿⢸⢠⡻⠉⠄⣴⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠢⠘⡦⠀⠉⡉⠰⠃⠁
⠀⠀⠀⠀⠀⠘⢹⣤⣀⠀⠀⠀⢴⠿⠟⣁⠰⢏⡁⠐⠄⡈⢁⠄⠀⠀⡟⣶⡈⠃⣰⢧⡿⠀⠀⠀⠉⠸⢷⡀⠈⠃⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁
⠀⠀⠀⠀⠐⠒⠤⠬⠟⠋⠈⠁⠋⠁⠀⠀⠀⠀⠈⠁⠀⡈⢦⣦⠂⠀⠘⠌⠻⠜⡷⠋⠀⠀⠀⠀⠣⠀⠛⢯⡀⠀⢰⠀⠀⢠⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⢿⣧⠂⠀⢸⣇⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠘⡄⠀⠀⠙⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣷⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠰⣏⠀⢸⡃⢁⠀⣾⠃⠀⡹⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢉⢀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣄⠀⡆⠀⠙⢷⣼⠃⢸⡏⠁⠀⠀⡇⣧⠂⠀⠀⠀⠀⠀⠀⠑⠂⠀⢾⠣⠠⠇⠂⠀⠀⠈⣱⡇⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢄⠖⡔⠈⠂
⠀⠀⠀⢰⣇⠀⠾⠀⠿⠀⣾⣝⠀⠂⡠⢡⣵⠀⠀⠀⠀⠀⠈⠀⠀⡠⠀⢠⡀⠀⠀⠀⠠⠤⠀⢈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦⢀⡼⠃⢆⡤⠄
⠀⠀⢸⠀⣿⠀⠂⢠⡆⢻⣫⠋⣴⠟⢁⡿⡙⠀⠀⠀⠀⠀⠀⢘⡀⠁⠀⡘⠷⠈⣀⣀⣀⠀⣠⣠⣴⡷⢒⠎⠀⠀⠀⠀⠀⢀⢠⣇⣴⠟⠩⢅⡈
⠀⠀⠀⠳⣟⡆⠀⣨⠄⢼⣇⠀⠁⣄⣾⡷⠀⠀⠀⠀⠀⠀⢰⠈⠟⠀⣠⡴⠂⠈⢀⢩⣿⠻⠋⠙⠋⢀⠘⡀⠀⠀⠀⠀⠀⣬⣵⡟⠉⠄⠊⠁
⠀⠀⠀⠀⠸⣧⡤⠀⡠⠀⢈⠀⣴⢿⡟⠁⠀⠀⠀⠀⠀⠀⠘⢠⡙⢴⢫⠁⠀⢰⡏⠀⠀⠀⢀⠀⠀⣸⠦⠀⠀⠀⠀⣰⡾⠋⠈⠁⠀⠰⣄⢀⡀⡄
⠀⠀⠀⠀⠀⣿⠁⣼⣁⠤⡌⠀⡷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢠⡉⣠⣴⣯⡀⠐⣆⠀⠈⠲⣴⡷⠁⠀⠀⢀⡶⠉⠀⠀⠀⠀⠀⠄⠁⠀⢹⠗
⠀⠀⠀⠀⠀⢷⠺⣘⠋⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠂⠒⠲⠽⡟⣛⠡⠾⠛⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠐⣦⣤⠏⠈
⠀⠀⠀⠀⠀⢸⠀⡍⢡⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠤⢸⡼⢖⠸⠀⠄
⠀⠀⠀⠀⠀⠀⠛⠡⠿⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢔⣂⡱⠄⡀⠄⠀⠀⠀⢀⣉⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢏⡅⢨⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠰⠀⢀⣙⡛⠀⣿⢿⣋⡁⣴⠁⡂⠀⠀⠀⠀⠙⣤⢀⣤⠀⠖⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢦⠁⣦⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡾⣷⡀⠀⡉⢀⡌⣁⡤⢹⠟⡬⢤⡄⠀⠀⠀⠀⣀⠘⠐⢿⣛⣾⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⡸⡇⠠⠀⢹⣠⠂⠐
⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⢀⠌⣼⣧⣌⡿⠌⠛⠭⠂⠉⡤⠋⠂⠤⠴⠆⠀⠀⠀⠀⠀⢤⠀⠄⠠⠝⠳⢿⡷⣌⠃⠀⠀⠀⠀⠀⠏⡇⠀⠀⠻⡟⢁⡇
⠀⠀⠀⠀⠀⣨⠁⠀⢠⠾⣋⣀⣡⣬⠒⢀⣾⣌⠙⠃⠉⠁⠲⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡴⣿⣿⠈⠓⢄⠀⠀⠀⠀⠘⣿⡆⣰⢘⡀⡼
⠀⠀⠀⠀⠒⠒⠒⡀⠀⠹⠄⠀⠄⠰⢥⠀⠀⠰⠐⠶⠦⡔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣡⠃⠀⠀⠀⠱⡀⠀⠀⠀⠈⠓⠇⠞⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠲⠀⠘⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠁
⠀⠠⠀⠂⠠⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡼⠀⣤⠈
⢠⠀⠀⡀⡀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢧
⠀⢀⠐⡇⠃⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠁⠀⠂⣀⠐⣀⠐⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⡀
⠀⠈⡄⣐⢁⡼⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠄⡄⢀⡀⠀⠠⠁⠀⠀⠀⠀⠄⠀⢀⢈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢀⢀
⠀⠀⢣⢸⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠
⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠣
⠀⠀⣸⣿⢣⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠀⠀⠐
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠂⠈⠃⠀⡀⠀⠀⠄⠢⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂⠀⡉⠠⠀⠐⠂⠁⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⣜⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡇⢹⡴⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⣇⠀⠀⠀⠀⠀⢠⠌⠳⣊⡐⠻⢄⣮⡁⠀⠀⠀⠀⠀⠀⢀⠀⡈⠤⠀⢠
⠀⠁⠀⠀⠀⢤⠀⠀⢠⠀⠀⠀⠀⠀⠸⡿⠘⠀⠀⠀⠀⠀⠀⠹⠁⡁⠈⡎⠀⠈⠁⠃⠀⠀⠀⡀⠀⠀⠀⠁⠀⢠⠈⢠
⠀⠀⠀⠄⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠋⢀⠀⠶⠊⠀⠀⠀⠀⠄⣀⠁⠀⠠⠖⠀⡀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠈⠠⠀⠀⠀⠀⠀⠀⡀⠀⡀⠈⠂⠂⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢐⠀⡇⡇⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⣻⠀⡁⠀⠀⠀⠀⠀⠄⠁
⡌⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠀⢠⠙⡄⡅⠀⠀⠀⠀⠀⠂
⠀⡀⠐⠁⠀⠀⠀⠀⠠⠀⠀⡎⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠁⠀⠀⠀⠀⠀⠄
⡈⢀⠄⠀⠀⠀⠀⠀⠀⠀⡴⢀⠀⣁⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣩⠔⢉⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠠⠒⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠝⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠈⠀⠀⠀⣘⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠢⢀⠄⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀⣠⣶⣿⡆
⠀⠀⠀⠀⡀⠀⣠⠁⠈⡀⡀⠣⣰⡥⠔⢢⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣽⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⡀⠢⣨⠂⢅⠗⡀⠤⢋⢀⠂⠈⡤⠊⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿
⠀⠀⠀⠀⠡⡀⠘⢀⣠⠋⠐⠀⠃⡾⠂⠀⣄⡃⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⢀⠐⠁⠡⡇⠀⣤⣠⠤⣠⠋⢀⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢃⠒⠺⠠⡀⠁⠀⠛⡁⡄⠀⠠⠂⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠑⢄⠳⡈⠰⠁⡀⡰⢂⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢨⣦⠈⢢⣠⠟⠶⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⡷⢷⣖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢷⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡏⢨⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠲⠼⠦⠴⠤⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡬⠐⠋⢠⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⢻⡬⣧⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⣷⠘⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠏⠉⢠⠀⠀⠀⠢⠀⠀⠀⠀⣇⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⣠⠞⠒⠀⠁⠀⠈⠀⠀⣌⠈⣠⣄⠀⢏⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣑⢄⣼⠀⠀⣘⠆⠠⠘⢶⠒⠈⢲⣽⡉⠓⢾⡇⠀⢈⡴⠋⠀⠀⢄
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⠝⠢⠗⠀⠤⠯⣶⣾⠿⠶⢄⢈⢫⠁⠀⠀⠈⠀⠏⠁⠀⠀⠀⠈
⠀⠀⠐⣧⣤⣂⠈⢤⠀⠋⣄⡀⠀⠀⠀⠀⠈⢧⠀⣀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠠⣾⠀⠘⠏⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣴⡆⠈⠁⠁⢀⡀⠀⠀⢀⣠⣤⡤
⠀⠀⠈⢃⡤⠚⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⡇⢠⣤⡤⣾⡿⠀⠀⠻⠷⣇⠀⠀⡘
⠀⠀⠀⠊⠑⠆⠉⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⣧⠘⠿⠃⢿⠗⠀⠀⠐⠈⠉⠃⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣀⣠⠫⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⡖⠁⠀⠀⠀⠐⠁⠀⠀⠀⠄⠄⢀⢀⡀⠐⠀⠀⠀⣠⠀⢘⢠⡍⣀⠖⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢂
⠀⠀⠀⡗⠄⠀⠀⠀⠊⠜⠀⠀⠜⠀⢀⠌⠀⢈⠂⠀⠀⠀⠛⠛⡌⠒⢆⣈⠙⡀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠂
⠀⠀⡨⠰⠀⠀⠆⠀⣘⠮⡀⠈⠠⠀⢌⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠤⠀⠱⠈⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⠁
⠀⠀⢇⡀⣄⠁⠁⠀⠰⠊⠀⠈⠂⠁⠘⠂⠰⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠀⠕
⠀⠀⠀⠀⠈⠀⠀⠐⠒⠁⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠀⠀⠀⣀⠀⡀⢀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡰⠦⠤⣀⠠⢄⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠙⠃⠀⠀⠚⢢⠈⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡆⠃⢠⠊⠉⢂⣀⣠⠇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠚⠙⠆
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠐⣄⠄⠍⠀⢈⢨⠑⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⢰⠃⠀⡥⡝⢢⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⠀⠀⠔⠀⠂⠀⠀⠀⠠⢒⠀⠀⠀⠀⠈⠀⠀⢢⡀⠀⠀⠎
⠀⠀⠀⠀⣠⠭⢶⠀⣺⣉⡦⢀⠀⠀⠀⠀⠀⣀⠛⠁⠀⠀⡜⠀⠐⠀⡐⠁⠠⠊⠀⠀⠀⠀⡃⠀⢠⠀⢀⠀⠠
⠀⠀⠀⠀⠘⢆⠙⠈⠃⠀⠹⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢇⠌⠄⠔⠂⠂⠂⠂⠤⠀⠅⢀⠀⠙⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⣏⠉⡠⠎⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⣡⠊⠔⠈⠤⠒⠊⠁⠀⠀⠰⢄⠁⠧⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠈⠘⠥⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⢻⠷⠛⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⡷⠐⠀⠀⠀⠀⠀⢠⠪⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠔⠈⠉⠈⠑⠢⠤⣀⣀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠢⢀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⡀⠒⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣠⢀⡀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⡀⡀⠈⠈⠀⠉⠀⠉⠀⠀⠈⠉⠀⠡⠀⠀⠀⠀⠀⠀⠐⠤⣀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠒⢠⡀
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⢀⡰⠒⠦⠤⠤⣀⠀⡀⠀⠀⠘
⠀⠀⠀⡀⠀⠀⠀⠀⢀⡤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠚⢄
⠀⠐⠁⠀⠀⠀⠀⣀⠞⠀⠑⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖⠁⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠤⣀⢀⠀⠀⠀⠀⠦⠁⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸
⡅⠀⠀⠀⠀⠀⠀⢀⠀⠃⠂⠀⠀⠀⠀⠀⢯⡙⢔⠆⠂⢂⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠀⠑⢌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣆⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡒⠉⠀⠘⢆⠀⠀⢹⠀⠀⠐⢀⠀⠀⠀⠀⡠⠂⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠈⠒⠢⣀⠀⠀⠀⡄⠃⠀⠈⢦⠀⠀⠈⢣⡀⠘⡆⠀⠀⠀⠙⠉⢚⣉⠤⢊⡅⠈⢆⣀⠀⠀⠀⣀⡀⣄⡠⢄⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠚⠉⠀⠀⠀⠀⠀⢳⡀⠀⠀⠳⡀⡇⠀⡀⠀⠀⡔⠉⢀⡶⡎⠀⠐⡡⠋⣱⠛⠋⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠱⡇⠀⢁⠀⠹⠀⠀⠜⠐⠀⠀⡞⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠐⡄⢇⠀⠈⠀⢠⠀⠈⠀⠄⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢹⣸⠀⠀⠀⠀⣖⣀⠄⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢿⠀⠀⢀⠃⠉⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⡇⢰⠁⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠱⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣤⣀⣀⣀⣀⡀⠀⣈⣧
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⠦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⢀⡤⠖⠒⠚⠁⠀⠀⠀⠀⠈⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠘⣷⠋⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⣸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠓⠚⠋⠉⠉⠛⠒⠢⡄⢀⡤⠴⠚⠁⠀⠀⠉⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢢⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠄⠀⠀⠀⠸⡄⠀⢀⡠⠄⠂⠁⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⢰⠧⠂⠁⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⣼⠀⠀⢀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠛⠉⠓⠦⣄⡉⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠐⡀⠀⡇⠀⠀⠸⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠙⠢⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⢹⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠠⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⣀⣀⡜⠀⠀⠁⠀⠀⠒⠂⠒⠂⠀⠀⠀⠀⢀⣠⠊⢀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⡴⠋⣩⠋⢀⣴⠖⠒⠲⠤⠤⠤⠴⠦⠤⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣞⡠⠞⣡⠔⢁⡞⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠢⡀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣵⠟⠋⣠⢞⣡⠔⢋⡠⠔⣒⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⣠⠴⠶⠟⠛⢉⣠⣴⡯⠖⢋⣠⠖⢉⡴⠞⠉⠀⠀⠉⠛⠶⢤⣄⡐⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃
⠀⠀⠀⠀⠀⢀⡷⠤⠶⠶⠚⢛⣋⣥⢴⡾⠋⣡⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠶⠤⠤⠴⠶⠖⠲⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠖⠛⣉⡥⠖⢋⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠴⠞⠋

⠀⠀⡆⠀⠈⠈⣀⣤⠀⠀⠀⠀⠈⢿⡏⠋⠉⠀⠁⠘⣿⡎⠘⣯⣥⣄⣠⠀⠈⢹⣭⣤⣴⣦⣭⣟⣤⣼⣿⣄⣉⣉⡉⠀⠁⠀⠀⣾⣼⡇⡄⠉⠈⢹
⠀⠀⠁⢀⡴⠟⠉⠀⠶⠚⠀⠀⠀⢸⡇⠀⠀⠀⢀⣼⠿⠇⢀⣿⣠⠟⠁⢀⣴⢻⣍⢀⣤⡶⠋⠋⠁⠠⣈⠀⠉⠛⠻⣦⣄⠷⣴⣯⠛⣿⡰⢺⣄⣼
⠀⠀⠚⠀⠀⠤⠀⠈⠀⠀⠀⢠⡶⢾⡟⣷⢀⣴⣿⡏⢡⣾⠋⠀⣿⣟⠿⠛⠿⠾⣿⠾⠋⡀⠘⠂⡀⠰⠤⣤⠄⠘⠂⠈⠻⣦⠌⣠⣍⣸⡇⣤⣿⣹
⠀⠀⠀⠐⠈⠀⠀⠀⢠⠄⠠⠂⠛⠋⢠⡌⠛⣿⠃⠀⠹⠿⢀⠀⣤⡿⠀⠀⠀⢰⡿⠀⠙⠃⢠⠀⠁⢸⣷⡎⠀⠶⠀⠸⠄⢻⣟⣡⣼⣥⡶⢳⣮⣭
⠀⠀⡆⠀⠀⠀⡄⠀⡀⣠⠇⣤⡾⠛⡄⣤⠈⣿⡇⣀⠀⠀⣿⡆⠀⢀⣠⠴⠀⠺⢣⣤⡀⠀⢤⠀⠀⢸⣿⠇⠀⠤⠠⠀⢶⠘⣿⢀⣿⣷⣤⣺⡷⣿
⠀⠀⡄⠀⠀⠀⠀⡀⠁⣿⠟⠉⢓⣀⣾⡿⠐⠛⣷⢀⣴⠿⠋⠀⢰⡿⠁⠀⣤⣤⡄⢈⣿⡆⢀⠀⠐⠂⠀⠀⠀⠀⢀⠀⠀⢠⣟⠀⣽⡆⠀⠉⢠⣾
⠀⠀⡀⢀⡀⠀⢀⣴⢿⡟⠆⠀⠀⢉⡟⢻⠁⠀⡟⠈⠋⣤⡀⠀⣿⣄⠸⣤⣷⣿⣧⣈⣽⣇⠀⠉⠀⠀⠀⠀⢀⠀⠀⠙⢂⣾⣿⣿⣿⣾⡓⣴⡏⢻
⠀⠀⡇⣩⣤⢾⡟⠓⢈⣁⣀⠀⣼⡿⠿⣧⠀⢰⡇⢠⡴⠟⣡⢾⠓⠈⠀⠙⠛⠋⢁⡀⢹⡟⢻⣦⠀⢀⡓⠀⠀⠘⢂⣴⡟⠳⣽⢟⣉⣹⡟⠻⠟⣾
⠀⠀⣿⣿⡏⢘⣵⠿⠛⠿⣯⡈⣳⣷⣾⠃⠀⢸⣷⠟⠑⣾⣁⠜⣄⠳⠤⠄⠀⠀⠡⢤⣾⣡⡞⢉⣿⣈⣁⣀⠰⠚⠋⠀⠁⠀⠀⢚⡹⣯⡇⠛⠀⣸
⠀⠀⠈⠛⠁⣾⠁⣘⠂⣶⢻⡏⠀⠀⠀⠀⠀⠸⡷⠚⡀⠋⡡⢀⡾⠃⠀⠀⠀⠀⡀⠘⣏⣿⠿⣿⢻⡇⠀⠀⣀⠤⡖⠓⡒⠦⣄⠊⠠⣼⣧⡾⠷⢿
⠀⠀⠇⠀⠀⠉⠉⠙⡿⣧⣾⣃⠀⡀⠀⠀⠀⢘⣷⣿⠀⢘⡶⠾⢿⣾⠃⠀⠉⠉⢡⣜⠛⢛⣼⣿⡙⢃⣤⠸⢿⣄⣠⡆⣴⠃⣘⣧⠀⠉⢻⡇⣠⢸
⠀⠀⡇⣠⣤⡄⠶⠿⢰⣯⠙⢻⡛⠛⠛⠛⠻⠛⣹⠟⠂⢾⣿⣶⢡⣤⠀⠀⠀⠀⠀⠸⠶⠀⣅⡸⣏⢸⣇⠀⣰⣤⠶⠷⢰⡞⠉⣿⠀⠐⡿⣿⢒⠺
⠀⠀⡇⢻⣧⠀⠀⠀⢀⠀⠀⣸⡇⠀⠀⠀⠀⢸⣧⠀⢠⣾⠉⢳⡾⣿⠄⣀⣀⣆⢀⡠⢻⣿⣾⣿⣿⣤⠟⠀⣧⠉⢀⡐⣿⣁⣴⡟⠀⠐⠂⢹⡏⢰
⠀⠀⠾⡏⢀⡴⢚⠛⠻⢦⣄⣿⡀⢀⡤⢚⣀⣸⠿⣶⡿⢙⠗⢺⣄⠻⣦⣀⠉⢏⣀⡴⠿⠹⡆⠛⠿⣿⠷⡄⣿⠀⡈⠷⠛⢛⠯⠀⠀⠴⢦⣿⢐⣼
⠀⠀⡆⢧⣿⡴⠏⣉⠙⡶⠋⠉⠁⠛⡷⠋⠈⢹⡶⠈⡀⠻⡆⠀⠉⠀⠀⠉⠛⡋⠉⠀⣤⡶⠀⠀⠸⠋⢴⣄⣧⣄⡁⡞⠃⠀⢠⣤⠚⠀⢸⣏⡾⣽
⠀⠀⠁⢨⠻⣧⠰⣿⣄⣻⣦⡀⢁⣾⡄⠀⢀⣸⠃⠀⠀⣖⠓⢶⡶⢤⠐⣳⣶⣡⣤⣖⠀⠤⠋⢹⠂⠀⠀⢹⣿⣭⣤⠟⢠⠀⢰⡏⣸⠋⠶⢻⣟⢿
⠀⠀⠇⢰⣤⢀⡷⢄⣌⣽⣻⠃⠈⣿⠀⠀⢸⡏⠀⡄⢀⣿⡀⠀⠀⠀⠀⢉⣸⣯⡟⠁⠀⠀⠀⣸⣧⠀⠀⣼⣿⡿⣿⠀⣠⡰⠟⣡⣽⡲⡖⣚⡙⢻
⠀⠀⡀⢈⣽⠻⣇⠈⢁⡴⠿⡇⠰⠇⠀⠀⡸⢧⣀⣠⣾⢁⣿⣶⣶⠿⢿⣯⣽⣿⣧⣴⡤⢶⡾⣿⠛⣧⢠⣿⣿⠃⣼⣿⣯⣤⢾⣯⡍⠁⠈⣿⢻⣴
⠀⠀⣷⡍⢈⠁⣍⢘⣯⣽⣻⡟⢶⠀⣠⣴⠀⠀⠈⣿⠳⠛⢧⣿⣇⠀⠘⢣⣤⠛⢢⣿⠁⢾⡇⢻⣤⡈⢿⣿⡿⢾⡿⢀⣭⡁⠾⠈⡷⢼⣿⡟⠃⣽
⠀⠀⢯⣠⠚⠀⠉⠈⣿⡅⠀⠀⢀⠀⠸⣯⣀⠀⠐⣿⣀⢤⣼⡽⣿⠀⠀⣿⣽⢀⣻⣯⠀⣾⡇⠈⣉⠟⢀⣙⡿⡿⠋⣿⠁⡐⠀⢀⡇⠰⠘⠟⠸⣿
⠀⠀⠄⠀⠀⠇⠀⢀⠙⢃⠀⠰⣆⠀⠀⠃⠹⣧⠚⢁⣹⣶⡁⣼⣿⠀⠀⢽⣿⠿⠛⢿⣦⣼⣧⡐⠻⠿⠿⠋⣻⡆⡐⠃⠸⠃⢀⡈⡇⠀⢀⠀⠶⢹
⠀⠀⠘⠃⣠⠀⠀⠘⠀⠉⢠⠀⣿⣶⡄⢠⣶⠟⠛⠿⣶⣿⡙⢻⣿⣤⣼⠟⡉⠘⠀⣻⣔⣺⣯⡙⠛⠒⢆⠀⢿⣁⣈⣀⠀⠀⣎⠁⣧⡀⢠⢁⣀⢸
⠀⠀⣿⠀⣿⠀⢰⡇⢰⡦⣾⠆⠈⢻⣿⣿⣯⠀⠀⠀⠈⠉⠻⢶⣿⣿⣅⣀⣉⣤⣾⠋⢉⣤⡀⠀⠀⠐⠄⠉⠢⣝⠛⣿⠀⠀⢿⠀⣯⡀⠀⢛⡛⢨
⠀⠀⣿⠀⣿⢀⢸⡇⠈⠃⢸⠀⢰⣄⣬⣿⣯⣤⠶⠖⠲⠖⠁⠀⢀⣤⡝⠋⢉⡏⠈⠉⠉⠁⠀⠙⢿⠓⠚⠷⠷⠺⠦⣿⠀⠀⠾⠀⠛⣧⣤⡄⣬⢻
⠀⠀⣿⡤⣦⠉⣈⣥⢴⣿⣿⠄⢸⠉⢸⣿⣿⣶⣤⢠⣼⡤⠤⠤⢾⣯⢄⢦⣸⡇⢀⠀⡂⠀⠄⢠⡄⣧⡘⠻⣿⡇⠀⣿⣤⣛⡿⠽⢻⡟⠉⠑⣾⠘
⠀⠀⡇⣀⣿⠀⣿⡿⢷⡝⣷⣤⣤⣜⠃⠈⠉⠙⠛⢰⡟⠀⠀⠀⠀⣀⣌⣁⡸⣿⣿⣧⣭⣤⣤⣿⣤⣩⣷⠀⣿⢄⡴⠏⠙⠋⢁⣀⢿⣤⣄⣤⠟⢺
⠀⠀⠛⠀⣹⡇⠙⠿⣦⡄⠀⠀⠉⣴⢶⣤⣀⠀⠀⠈⠛⠿⠛⠛⠛⠋⠛⠙⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⢀⠀⠘⠿⠂⢰⣾⣻⠹⢷⡉⠉⠁⠀⢸
⠀⠀⡇⢀⡿⢁⠀⠀⠈⣿⠀⠀⣼⢻⠀⢠⣿⠃⠀⠀⠀⡼⠶⠾⣷⠀⠀⠀⠀⣼⡟⠟⢿⡆⠀⠀⠀⣤⡾⣿⢷⡀⠀⠀⠈⣿⡀⠆⠘⣧⠀⠚⠀⢸
⠀⠀⠁⢘⣅⣼⣷⣶⠾⠋⠀⢰⣏⣌⣳⡟⠀⠀⠀⠀⣼⡇⡘⢠⣿⠂⠀⠄⢰⣏⠘⢃⣿⡇⠀⠀⠰⠿⣧⠐⠈⣧⠀⠀⠀⠻⣷⡴⠶⠟⠃⠀⠀⢸
⠀⠀⡿⣿⡟⣻⠁⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠀⠀⢰⣿⣦⣤⣾⠟⠐⠂⠀⢾⣇⢠⣸⣿⢇⠀⠀⠀⠀⠹⣷⡖⠛⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡆⠙⠿⠋⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠌⠛⠁⠈⠁⠀⠀⠄⠈⠙⠋⠙⠛⠴⠃⠀⠀⠀⠀⢀⣠⣆⠀⠀⠀⠀⠀⣴⠶⠶⣶⠄⠀⢸
⠀⠀⡿⠷⠶⢷⣆⠀⣱⡟⢩⢷⣶⠀⠐⠆⠀⢠⡄⣴⠷⢢⣤⡐⠀⠀⠰⠄⢠⣤⣴⣶⣆⠀⠀⠣⠀⢀⣴⠟⠙⣿⠀⠀⠈⠤⢸⡟⢢⣄⣿⠆⠀⢸
⠀⠀⠂⠀⠀⠀⠋⣰⠟⢀⠐⣿⠏⠀⠀⠀⠀⠀⣸⡿⢸⢧⣿⡇⠀⠋⢠⡆⣾⡇⢴⠘⣿⠀⠳⢶⣤⠘⣿⡈⠁⠸⣧⠀⠀⠀⣼⡇⣛⢠⡟⠠⠤⣸
⠀⠀⠃⠀⠀⠀⠀⢻⣦⣤⣼⡿⠀⠀⠀⠀⠀⢀⣽⣁⡀⢸⠿⠓⠀⠀⠘⠃⢾⡇⠀⠈⣿⡄⠐⢶⠋⠀⠸⣧⣀⣤⣿⠇⠀⠤⠟⠛⠛⠛⠃⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⣀⡈⠉⠉⠀⠀⠀⠀⠀⠀⠘⠻⢟⣛⠋⠀⢀⡀⠁⠠⠆⠸⢷⣾⠿⠟⠀⢾⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠁⠀⣠⡶⠛⣧⣨
⠀⠀⡄⠀⠀⢀⡾⢹⡿⣶⡄⠀⠀⠀⠀⠀⠀⣴⠿⡟⢻⣄⡀⠈⠀⠀⠈⢄⣠⡴⠿⢷⡤⢄⠘⠀⠀⠀⠀⢀⣴⠲⠒⢶⡀⠀⠀⠀⢈⢿⡦⠖⠀⢻
⠀⠀⡃⠀⠀⡼⠁⠹⢇⣿⠀⠀⠀⠀⠀⢀⡠⢿⣼⣙⢨⣿⢷⠶⢀⠀⠀⠘⣿⢀⡴⠈⢿⡯⠃⠀⠀⢀⣀⣸⡯⠰⠂⣿⠆⠀⠀⠀⠀⠛⢿⡄⣀⣴
⠀⠀⠧⠀⢿⣧⣤⡀⣼⠇⠀⠀⠀⠀⠀⠀⠁⢸⣧⣨⣄⣽⡆⠀⠻⠷⡷⡶⢿⣧⠀⢈⣹⡇⠀⠀⠈⠀⠀⢼⣇⠀⠀⣟⠀⠀⠀⠀⠀⠄⠈⠻⠋⢸
⠀⠀⡁⠀⠀⠈⠉⠉⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠟⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠁⠀⠀⠀⠀⡀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠉⢉⣉⠉⠙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠶⠄⢀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣤⣿⣤⠀⠀⠀⣴⡿⠃⣀⣼⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⣼⡿⠁⠀⠉⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣾⠇⠀⣠⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠉⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⣴⣶⣿⣿⣿⣿⣷⣶⡦⠀⠀⣠⣾⠟⠁⠀⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣶⣤⣉⣙⣛⣛⣛⣉⣩⣤⣶⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣛⣉⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢗⢀⠃⠁⢐⠸⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣴⡰⣲⡄⠀⣤⡟⣥⡏⠐⢀⠀⠄⠂⣺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢹⠐⣿⠁⣼⡟⠫⣴⠀⢃⠄⠐⠈⡌⠎⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⡜⢳⡆⠀⡖⣿⠀⣞⣀⠢⠉⢐⠒⠠⠁⢀⡂⠠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⢳⡄⢰⡀⡇⢸⠀⣿⠂⡎⠃⠐⠂⠠⠀⠠⢂⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠻⡄⢳⣼⢸⡄⣸⣆⢣⠁⠀⠀⠄⠀⠄⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠹⡄⠹⣶⠇⠘⣷⡾⡊⠲⠆⢀⠰⢠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡄⢿⣷⠀⠘⡿⣵⣬⢁⣂⡲⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡿⣎⠌⢦⠀⠈⢮⠉⢉⠽⠊
⠀⠀⠀⠀⠀⠀⡇⠘⣇⠈⢳⡀⢸⠀⡌
⠀⠀⠀⠀⠀⠀⡇⠀⠻⣆⠀⠙⣯⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠒⠒⠖⠒⠤⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠷⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡰⠋⠑⠀⠀⠀⡀⠂⠀⠀⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⡁⡄⣷⡀⠀⠀⠈⢦⠀⠀⠀⢀⡞⠀⠀⠀⠈⡶⠊⠀⠀⠀⠀⢀⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⣿⣧⠀⠀⠀⠀⢣⠀⠀⡸⠀⠀⠠⠐⠈⠀⠀⠀⠀⡠⠂⠁⠀⠀⣀⡤⠞⠓⠒⠒⠒⡆⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⣧⣿⣿⢧⡀⠀⠀⠀⢳⡖⠉⠚⣅⠀⠀⠀⠀⠀⢀⠚⢄⠀⠀⣠⠞⠁⢣⡀⠀⠀⠀⠈⠙⠢⡈⢢⡁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣏⠁⠘⢷⡄⠀⠀⠀⠱⡀⠀⠘⣦⠄⠀⠀⠀⠂⠀⠀⠑⠴⠁⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⢸⡄⢀⡔⠛⠃⠷⣄⠀⠻⣄⠀⠀⠀⠰⡄⠀⠘⡄⠀⠀⡆⠀⠀⠀⢀⣧⡠⠔⠈⠻⣿⠿⠗⢀⠀⠀⡀⠄⠛⠿
⠀⠀⣠⠒⠉⠁⠙⣇⠬⠷⢀⠀⠀⠈⠳⠤⠈⢂⠀⠀⠀⠘⢄⡰⠃⡏⠈⠁⠂⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⡇⠀⠀⠀⠈⢧⡀⠀⠀⠉⠒⠤⠄⠀⠀⠘⢧⡀⠀⠀⠈⢣⣀⠘⢆⠀⠀⠐⠀⠀⠷⢄⠀⢀⣀⣄⠀⠀⠀⠀⡀
⠀⠀⢣⡀⠀⠀⠀⠎⠙⡇⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠹⡑⡄⠉⠈⠹⠤⠤⠒⡀⠙⣿⣿⣿⣷⡄⠀⠈⠁⠀⢄⣀⡰
⠀⠀⠀⠉⠀⠀⠒⠛⢄⣘⣦⡽⠂⠀⠀⠀⠀⠀⠀⠀⣈⣦⠀⢀⠀⣧⣸⠀⠔⠃⠀⠀⠀⠑⢤⠘⠻⠿⢏⠀⠀⠀⠀⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⠊⠑⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠈⠧⠄⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤
⠀⠀⠀⠀⠀⠀⡔⠶⢖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠐⠈⠱⡤⢈⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡆⢠⣻⣓⢲⣹⡇⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠘⠙⢻⠶⠊⡁⠀⠙⠃⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⠄⣿⠡⣤⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠴⠆
⠀⠀⠀⠀⠀⡿⠿⣶⣷⣾⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠿⠜⣿⣧
⠀⠀⠀⠀⠀⠌⠂⠐⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠚⠻⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⢀⠀⠺
⠀⠀⠀⠀⠀⠞⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⣭⣧⣀⠀⣀
⠀⠀⠀⢰⣤⣀⡈⣯⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣖⣐⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⡽⢉
⠀⠀⠀⠀⠉⠙⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣾⣇⣹⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⢤⣶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠙⠋⠉⢉⣿⣷⣴⣿⣦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢴⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⠁⠉⠉⠢⠀⢀⡐⢀⠀⠀⠀⠀⠀⠀⢠⡟⠘⠛⣿
⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠁⠀⠀⠀⠀⠘⠂⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⠆
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⢤⣦⡀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⣴⣥⠞⠀⠀⡠⢴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠚⠶⡀⠀⠀⠀⠀⠠⢿⣀⠸⣄⣁
⠀⠀⠀⣀⠀⡷⠟⠁⠀⠀⠀⠹⠯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠁⠠⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠈⠳⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡬⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠀⠈⠹⠾⠇
⠀⠀⠀⠀⠀⢻⣿⣶⣤⣀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⣇⡀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣮⠁⠀⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡟⠛⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣽⠟⢋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⠉⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠈⡹⣿⣴⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⣤⣧⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⢰⣥⣤⠆⠈⠙⠂⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⠶⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠘⠀⠜⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠒⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠴⠖⢒⠒⠲⠤⣄⠈⠀⠀⠀⠀⠀⠀⠀⢳⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣪⣴⣿⡟⢻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢸⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛⠁⠘⢿⣷⣾⣿⣻⣿⠏⠈⠙⢿⠃⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣈⡉⠉⣉⣀⣠⠤⠞⠋⠀⠀⠀⠀⠀⠀⣼⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢠⡟⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠈⠳⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⡿⠛⣡⠖⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠉⠛⠿⠿⣿⣾⣿⣿⣿⣿⠿⠟⢛⣩⠴⢛⣡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⡳⢶⠦⠶⠤⠤⠤⠤⡤⠤⠴⠴⢟⢒⣛⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣿⣾⣧⣿⣼⣿⣾⣿⣿⣼⣧⣿⣼⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣮⣽⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣥⣶⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⠀⠀⠀⠀⠀⢐⡀⣠⣾⣿⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⣰
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠈⢻⣤⠂⠀⠀⠀⢸⣿⠟⣼⣿⠏⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡞⡆⠛⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠁⢀⠀⠙⣾⣿⣄⣰⡄⢰⣾⡟⣼⠿⣾⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣿⣈⣤⡠⢃⡔
⠀⠀⠀⠀⠀⠀⠀⢀⠓⣤⣾⣎⣿⣋⡿⣾⡏⣰⡿⠷⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠠⠻⡿⣸⣽⡏⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠸⣿⣿⣿⣯⡶⠛⢱⣿⣩⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠒⢤⣤⠸⢏⡿⠐⠁⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣈⠃⠻⣂⡴⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣍⡀⢻⣷⣻⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠸⡇⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⡌⡧⢂
⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⢳⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣇⠘
⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣾⣿⣯⠗⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⡴⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣾⣟⣿⣧⠈⢀⣿⣿⡗⡆⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣠⠦⠰⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣯⣹⠇⢼⣿⣿⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⠀⠀⠿⠁⠀⠀⢻⣥⠙⠀⢀⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠐⣬⣹⣿⣾⣾⣿⣿⢋⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢸⡇⠠⣷⠀⡖⠀⣾⡇⣀⣠⣼⡟
⠀⠀⠀⠀⠀⠀⠢⠀⣈⣙⣿⣹⣿⡋⠻⣿⠆⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣞⣀⢛⠀⡼⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣿⣿⠿⠶⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠷⠘⢿⡿⢾⣤⣿⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⡘⡰⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⢀⡀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣀⡘⣁⡀⢀⣾⣿⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠾⠟⠀⠀⣠⣾⣿⣿⣿⠟⣿⠿⢿⣿⣿⣧⣌⣉⣩⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡎⠉⣩⣤⣔⠀⠈⠀⢸⣿⣿⡟⠉⠀⠀⠀⠀⠈⠿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⡇⣿⣿⣩⣽⣦⠑⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠦⠀⠀⣨⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣮⣿⣿⠿⠿⣲⣄⠸⣿⣿⣿⡖⣀⡀⠀⢠⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣩⡽⠛⣋⣵⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⢻⣿⣿⠿⣿⣿⣿⣿⠋⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣿⣿⡿⠋⣿⣿⠋⠵⠚⠉⠁⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠻⠿⠧⠈⠩⠀⠀⢀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣅⠀⡰⠀⠀⢠⣠⣾⣿⣄⣴⠾⢋⠁⣁⣀⠀⠤⠛⣁⢀⢰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⣸⣤⣾⣿⣿⣿⢏⣁⠴⠊⠀⢹⡄⠡⠠⢾⣟⢢⣸⣿⠀⠀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢘⣭⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡿⠉⠂⢀⣐⣢⡶⣾⡇⢸⣿⣨⣿⣿⠿⠛⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⣿⠏⠁⢸⡏⠙⠛⠟⠛⠋⢀⣉⣠⣽⣿⣷⣦⣍⡤⠿⢿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠁⠀⠀⢤⣷⣤⡀⠀⠀⠀⠈⡻⢿⣿⣿⣿⣿⣿⡇⢠⣿⣬⠀⠄⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣭⣶⣾⣾⡇⣤⠀⠈⠋⠁⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⠀⢾⣿⠇⣀⣴⢀⡀⠀⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡛⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⢿⣦⣉⣀⣿⣿⣿⣷⣤⣶⣶⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠒⠠⠘⠟⠀⠀⣸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠹⠿⠿⠿⠿⠒⠾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡛⠻⠿⢿⡿⠿⠷⠄⠜⠛⠛⣿⣿⣶⡞⠃⠀⠘⣿⠿⣿⣋⣉⠚⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣿⣿⣦⡜⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⠀⡀⢭⠔⠀⠀⠉⠁⠀⢹⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣿⣿⡀⠀⢀⠴⢤⠄⢀⣴⣋⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠿⢿⣿⣶⣄⣤⠀⢀⣀⣼⣿⣿⡿⠛⠁⣠⢄⠱⣦⣄⣴⡿⢟⣛⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣶⣆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣾⣿⣾⡿⢟⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠙⠿⣿⣧⠉⠙⠛⠋⠺⠷⠿⠛⠻⣽⣴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⡀⠤






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣴⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⢀⣽⠷⠖⠒⠶⠦⣤⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⢰⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⢢⡂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣠⠀⡠⠀⠀⠀⠀⢿⠀⠀⠀⣀⣠⡤⠴⠖⠚⠛⠛⠙⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⣿⡐⠀⠀⠀⠀⠀⣸⡧⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠉⠀⠀⢺⡄⠀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢀⠔⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣙⣆⠀⠀⠀⠀⣧⠀⠀⠀⠈⠳⡀⠀⠀⡄⠀⠀⠀⠀⢹⡇⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣴⠖⠋
⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠉⠛⢧⠀⠀⠀⠘⣇⠀⠀⣤⠤⢙⡄⠀⠀⠀⠀⠀⠀⠀⣷⠋⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣡⠶⠋⠁
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠂⠀⠀⠈⢧⠀⠀⠀⠘⢧⡀⠀⠀⠀⠘⡌⠀⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠄⢀⡴⠛⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘⣧⠀⠀⠀⠀⠙⢦⣀⠀⠈⠁⠀⡜⠁⠀⣀⡴⠋⠀⠀⠀⢡⡀⠀⣠⠶⠋
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠠⠀⣣⡀⠀⠈⢷⡀⠀⢠⣀⠀⠈⠙⠓⣶⠀⣤⠶⠛⠉⠀⠀⠀⠀⠀⢸⢁⡼⠋
⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠙⠂⠤⡀⠙⢦⣀⠀⠉⠳⠐⠀⣟⠀⣿⠠⠀⠀⠠⠀⠀⠐⡀⣱⠏
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⠖⠒⠒⠈⠲⠄⠉⠳⢦⣄⡇⠀⣯⠀⣿⣤⣤⡴⠖⠛⢲⠀⢡⡏
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⢰⣄⢠⡞⡇⠀⣿⠀⣿⠀⡇⠑⠤⠔⢻⠃⡾
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠉⠀⠀⣰⠟⣆⢷⡇⠀⣿⠀⢸⠀⡇⠀⠀⢀⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀⡀⠀⣀⣠⠞⠁⠀⠹⡞⣷⠀⢹⡇⠸⣇⡇⠀⢀⡞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢻⣿⡀⠘⣷⠀⢻⡇⢠⠟⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠹⡆⠀⢻⡋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠻⣄⠀⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⠼⠛⠋

⠀⠀⢀⠤⡂⠀⠉⠙⡌⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠉⠙⡆
⢀⠔⠁⠀⠈⠓⣄⠀⠸⠀⠀⡜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠙⢗⠢⢌⣒⠒⠁⠀⣀⣀
⡄⠀⠀⠀⠀⠀⠈⣇⠀⡇⡼⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢠⣚⡉⣉⣥⣴⢾
⡧⢄⣀⠀⠀⠀⠀⢸⠀⣧⣃⡠⠞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡴⣩⠖⠊⠉
⢱⡀⠀⠈⠉⠐⠒⡞⢺⠛⡅⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⡴⠁
⠀⠑⢤⣀⠀⣠⠞⠀⡞⢀⡹⠖⠁⠀⠀⠀⠀⣠⠔⠐⠒⢔⠒⠦⢤⡀⠀⠘⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠟
⠀⠀⠀⠀⠉⠉⠒⠋⠉⠉⠀⠀⠀⠀⠀⢠⠎⢀⡉⢦⡀⠈⢂⠀⢡⠈⢣⡀⠘⠢⢝⡒⠶⠤⠤⠴⣒⡪⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡔⠁⢀⡠⢷⠀⢸⠀⠘⠀⢸⢳⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢀⣀⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⡠⠃⠀⣀⣄⢸⠀⢠⣀⣸⠼⡇⠀⠀⠀⠀⠀⠀⠀⡠⢚⣅⡢⠾⣿⠷⢦⡑⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠸⡇⠀⠁⢀⠾⢄⣀⠈⠉⠉⢀⣀⢴⠇⠀⠀⠀⠀⠀⢠⢊⣴⢷⡏⢨⠞⢄⡜⠃⢼⠸⡆
⠀⠀⠀⣠⠔⠛⢧⡔⠋⠀⠈⢟⠦⡀⠀⠱⣄⠀⢸⠤⣀⠈⠉⠉⢉⣁⣠⠞⠀⠀⠀⠀⠀⠠⢡⣳⠗⢅⡜⠑⠦⣎⠈⡰⢺⠀⡇
⠀⠀⡌⠀⠀⠀⢠⡀⠀⠀⠀⡘⠲⣝⣆⠀⠈⠑⠮⢷⣤⣬⣭⡭⠽⠛⠁⠀⠀⠀⠀⠀⢠⢣⡿⠋⡦⡂⢈⠖⢄⡠⠋⡲⡞⢰⡇
⠀⡜⣸⠦⠤⡖⠁⠙⠦⡠⠴⣄⠀⣸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣼⢉⠶⣀⡼⠓⣤⡊⢩⠾⡕⡡⢣⠁
⠀⣏⠀⠀⠀⠱⣄⣀⡜⠀⠀⢠⠋⠈⢦⡇⠀⠀⠀⠀⠀⢀⠤⠀⠲⠤⡀⠀⠀⠀⠀⠀⡇⡧⠉⣦⡎⠑⠴⣁⡜⡳⢊⠔⡱⠃
⠀⠘⣄⢀⣀⠊⠀⠀⠑⢢⠖⠣⣀⠀⡼⠁⠀⠀⠀⠀⢠⠊⠐⠢⡀⠀⢈⣆⠀⠀⠀⢐⢠⡘⠶⣁⣨⡳⠤⢛⡩⢔⡡⠊
⠀⠀⠈⠳⢄⣇⠀⢀⠠⠯⣄⣰⠧⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠆⠊⢸⠀⠀⠀⠌⣬⠶⢋⡠⠴⡒⠉⠁⠂⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣠⠔⠋⢉⡷⠃⠀⣀⡌⢈⡵⢊⠕⠊⠁⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠒⠒⠲⢤⠤⣀⡀⠀⠀⠈⠉⠉⠁⠀⢀⠾⠉⢍⢫⠈⠀⠀⠀⠀⢸⡀⣀⣸
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⣀⣠⢤⠴⠤⢦⠈⡍⡗⠀⠀⠀⠀⠀⡰⡉⢦⢱⢸⠎⠀⠀⠀⠀⢀⣼⣖⣗⣮⣄
⠀⠀⠀⠀⢀⠜⢄⠀⡠⠴⠎⠙⠉⠀⠀⠀⠀⠀⠇⢸⠃⠀⠀⠀⠀⡾⠤⡙⡼⡤⠃⠀⠀⠀⠀⠀⣾⢏⡏⢿⢙⢿⣄
⠀⠀⠀⠀⠊⠡⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠑⢄⣈⡷⠁⠀⠀⠀⠀⢀⣎⡟⣹⠓⢼⠉⣍⣷⢆
⠀⠀⠀⠀⠀⠀⢱⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠾⢁⣼⠑⣾⠁⣤⢹⠛⠧⡀
⠀⠀⠀⠈⠢⢄⣼⣠⠁⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠘⣅⠘⡹⠃⠫⣀⣨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡖⣾⣿⣧⣿⣿⣿⣿⣿⣧⣽⣿⣗⣲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣭⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⣿⣿⣿⣿⠿⠛⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣟⣳⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣯⣄⠀⠀⠀⠻⣿⣿⣿⣿⣿⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⣸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠙⣿⣿⣿⣿⠁⣠⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇
⠀⠀⠀⠀⠀⠀⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡈⢿⣿⡟⣰⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣻
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠉⠉⠀⢈⣿⣿⣧⣄⡈⠉⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⢿⢻⣟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣽⣶⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣟⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣼⣿⣤⣀⡤⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⢀⣾⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⣿⣿⣏⣀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣉⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣶⣿⣿⣭⣿⣿⣭⣿⣿⣷⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢴⡶⢫⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣭⡍⢩⡍⡉⠉⡿⠉⡍⢩⣷⣿⣿⣿⠿⡟⢸⣿⠟⣥⣯⣭⣭⣿⣿⣿⣿⣿⣿⣶⠸⣧⠻⠴⠛⠚⢻⢀⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⠾⠛⠓⠈⠁⢀⠈⠉⠁⠈⠉⣉⠝⣡⡖⢁⡇⡟⣵⠋⠀⢋⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠓⠀⢸⢸⢿⡵⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⢀⣤⠠⢄⠀⠀⠀⠀⡜⣠⣾⣿⢃⡼⢱⡿⣿⠀⠀⢸⠀⠀⠀⠀⣄⣠⣤⣠⢤⠀⠈⠀⠀⠀⠀⡜⡜⣼⣏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠁⠠⠄⠀⠀⠸⣌⣉⠴⠀⠀⠀⢸⣵⣿⣟⣡⢃⡆⣼⣼⣿⡄⠀⡼⠀⠀⠀⣸⢻⣿⠿⢯⠏⠀⠀⠀⠀⠀⡰⢱⠡⣽⣾⣏⡝
⠀⠀⠀⠀⠀⠀⡠⢣⠎⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡻⣿⡿⢁⢾⡇⣿⣾⡿⠁⡴⠁⠀⠀⣆⠀⢀⠄⣀⡀⣤⣤⣤⣴⡖⣰⢣⢣⡜⣽⣿⡿⠁
⠀⠀⠀⠀⠀⡔⣱⣿⣶⣶⣶⡖⠰⡆⣴⣠⣞⣪⠉⡥⣤⣤⣤⣿⡟⢠⢋⣾⡇⣿⣧⣶⡧⢤⣤⢤⣤⢀⣤⣤⠄⠤⢤⡥⣭⠵⠒⠺⢃⣯⢻⣰⣿⡿⠁
⠀⠀⠀⠀⢾⢤⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⣤⣤⣭⣉⣉⣉⣹⣡⣏⣤⣼⣄⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣭⣭⣭⡴⠶⠒⣒⣛⡿⣱⣷⣿⡿⠁
⠀⠀⠀⠀⢘⡿⣿⡿⢗⣒⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠤⢤⣬⣿⣿⣋⠈⠀⡐⠒⠲⠶⠶⠶⢴⣶⣿⣽⣿⠃
⠀⠀⠀⠀⠸⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣽⢿⣯⠿⠿⠿⠿⠿⠿⠦⠶⠶⠿⢿⣾⣿⣿⣿⣿⣿⢿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠁⠀⠈⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠘⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠐⠀⡸⠢⠀⠀⠀⠀⠡⡀⢀⠠⠤⠄⠃⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠘⠆⠀⡄⠀⠈⠀⠀⣓⡤⠀⠄⠀⢣⠈⠀⠀⠐⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠳⠗⠀⠀⠈⣀⠦⡄⠀⠀⠐⠀⣌⢆⠀⠀⠠⠈⠈⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠐⠢⢀⠀⠔⠠⠀⢀⠀⣹⠁⠃⡠⢀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠄⠀⠢⣄⡀⠐⠀⠀⠀⠀⠠⡈⠢⠀⢸⠈⠀⠀⠀⠀⠀⡆
⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠘⠫⠰⠀⠀⠱⡀⢧⡀⠀⠀⠀⠀⠱
⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠁⢦⡀⠈⠀⠀⠀⠁⠀⠁⠀⢠⠸⡀⣆⠀⠀⢀
⠀⠀⠀⡞⣆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠛⢶⡤⡀⠈⠙⠦⡀⠀⠀⠀⠀⠀⢸⠄⢠⠰⠀⡐⠛⠠
⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⠄⡀⠈⢦⡀⠀⠀⠀⠈⢢⠘⡂⢉⠁
⠀⠀⠀⠀⠐⡜⡄⠀⠀⠀⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⢦⡀⢀⠀⠀⠣⡛⡹
⠀⠀⠀⠀⠀⡬⢎⠢⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠢⡀⠀⠀⠳
⠀⠀⠀⠀⡀⠀⠁⠙⠢⢉⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢔⡂⠑⢜⢄⣀⠈⠃⡀
⠀⠀⠀⠀⢡⡀⠀⠀⠠⠠⠌⠙⠒⢀⡀⢀⠀⠀⣀⣀⣤⣀⠶⠖⠊⠑⠛⠱⢽⡚⣈⣓⣹⡓⢛⣻⣷⣄
⠀⠀⠀⠀⠀⠈⠰⠦⣦⠶⠒⠛⡋⠁⠀⠠⠤⠭⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⣭⣭⡭⣤⣭⣭⢭⣽⢿⣗⠦⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠉⠄⠀⢀⢀⡀⠀⠄⠀⠀⣀⠀⠀⠤⠄⠒⠀⠠⠀⡏⢰⢱⡗⢌⠳⣌⠲⢍⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠐⠒⠢⠤⠟⣁⣂⡁⠁⣀⢀⣀⣀⣀⣀⣀⣄⣀⣀⣀⡠⠄⠐⠒⠙⠧⢦⡉⢶⠣⡉⢷⣄⠑⢄⡈⠲⢄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠹⢌⠦⠙⢷⣄⠙⢮⣐⣙⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⢦⣀⣌⣻⠷⠾⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠗⣟⠛⠛⠁

⠀⠀⠀⠀⡀⢤⣤⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡼⠛⢄⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⡠⠆⠀⠐⠻⣄⠀⠀⠀⠉⠀⠀⠀⠀⠀⡈⠁⠀⠀⢹⣄⠀⠀⠀⢀⡴⢺⣹⡄⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠗⢀
⠀⠀⡄⠀⠀⠀⠀⠈⠛⢆⠀⠀⠀⠀⠰⡄⠀⣄⠀⠀⠀⢸⣸⣦⠔⠊⠁⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠒⠸⡆
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠑⢤⡤⠴⠒⠊⠀⠁⢀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠠⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠴⠾⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠙⠳⣤⣀⣀⣀⠀⢀⠀⣵⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⢠⠤⢲⡴⠊⠠⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⢀⡀⠈⠉⠉⠛⠋⠉⠉⠀⠀⠀⠀⠀⡀⠈⠉⠛⠓⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⣦⡤⠴⠒
⠀⠈⠀⠀⠀⠈⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡎⠂⡱⢸⢰⢰⡼⠄⣖⡁⢠⠁⡎⢀⠇⣄⡖⠈⡇⡇⢈⠀⠓⠀⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠄⠁⠈⠸⡘⠀⢰⠄⡀⠸⡀⢅⠘⢆⠏⠱⠀⡄⣇⢀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠄⠀⠀⠀⠀⡞⠉⠈⡳⣄⠀⠐⢠⣀⠄⠀⠀⠀⠀⠀⠀⣠⢴⡖⠲⠆⠀⠀⠒⢤⡞⠣⠘⠳⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠤⠀⢸⠁⠀⠀⠀⠈⢦⣶⠟⠁⠀⠀⠀⠀⠀⡠⠊⠱⠈⠁⠀⠀⠀⠀⢠⡝⠆⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠉⢦⠀⠀⠀⠀⣀⠊⠀⠀⠈⠁⠀⠀⢅⣶⡔⠁⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣴⡞⠁⠀⠀⠀⠀⠀⠀⠀⢠⣁⡇⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⡀⠀⢠
⠀⠀⠤⠐⠀⠀⡀⠀⢀⡤⠀⠀⠀⠀⠀⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠾⠶⣤⡀⡀⠀⢤⣀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠳⠖⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠉⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠛⢿⢛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠴⠖⠃⠈⠉⠁⠀⣠⡾⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠉⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢸⢻⡆⢀⡤⢤⣄⠀⣿⢻⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠛⢶⠀⢸⢸⡇⢸⠁⠀⣿⠀⣿⢸⠀⣴⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠘⣮⠻⣮⡳⠶⣫⠞⣡⠞⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⠳⣌⠙⠛⢉⡼⠋⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢀⡟⠀⠀⠸⣇⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⣾⢁⡞⠻⣆⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⣠⠞⠁⠀⠀⠀⠀⠀⣸⢃⡾⠀⠀⠹⡆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠀⠀⠀⠀⠀⠀⢠⡏⣾⠁⠀⠀⠀⢹⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠃⠀⠀⠀⠀⠀⢷⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢠⠤⢄⢀⠤⢤⡀⡤⠄⢠⠤⢄⢠⠤⠀⡤⠤⡀⡄⠠⡤⠠⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠋⢸⠓⡅⠸⣀⣠⠃⣉⣳⢸⠓⠉⢸⣈⡀⡟⢪⠀⡇⠀⡇⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠚⠛⠛⣛⣷⠴⠶⠶⠶⠶⠶⠖⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣿⠀⢀⣀⣤⠞⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⣧⢠⡞⠁⠀⠀⠀⢀⡌⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢻⡋⠉⠉⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⢀⣼⠀⠀⣠⠞⠁⠀⢀⠄⠀⠀⠀⠀⢸⡇⠀⠀⢀⠆⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⢰⡀⣠⡾⠁⠀⠀⠀⠀⠀⠘⡇⠀⣴⠃⠀⠀⢀⡞⢀⡠⠀⠀⠀⣼⠃⠀⢀⡞⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠉⠁⠀⠀⠀⣠⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⢠⣇⡼⠁⣀⣤⣶⡛⠋⠁⠀⠀⠀⢀⡿⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢸⣿⡵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⡟⢀⣠⠄
⠀⠀⠀⢠⣤⣤⣤⣠⡿⠀⠀⠀⢀⣤⠴⠟⠀⠀⢀⣇⣠⠔⠁⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡼⠛⠛⠉
⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⣠⡟⠀⠀⠀⡀⠀⠀⠀⣠⠄⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠙⣧⠀⠀⠀⠠⣀⠀⠈⢻⡆⠀⠀⠀⢀⣷⠀⠀⠀⣰⠏⠀⠀⢀⡞⠀⠀⣠⠞⠁⠀⠀⠀⢸⡇⠀⠀⣤⠂⠀⠀⠀⠀⣴⠋
⠀⡶⠛⠛⠛⠉⠀⠀⠀⠀⣨⠿⠛⠛⠁⠀⠀⠀⠀⠙⣧⢀⣼⠃⠀⢀⣀⣾⣤⠴⠞⠓⠂⠀⠀⠀⠀⣸⡧⠶⠾⠷⠒⠂⠀⢀⡾⠁
⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡿⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠙⠳⣦⣄⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⡀⠀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢀⡤⠖⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⢀⣴⠏⠀⢹⣆⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⡿⠶⠞⠛⠛⠂⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⢀⣴⡿⠃⠀⠀⢠⣿⡿⠴⠶⠦⣤⣤⣀⠀⠀⠀⠀⣾⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⢋⡾⠷⠶⠶⣶⢿⣟⠀⠀⠀⠀⠀⠀⠉⣻⣦⡀⠀⣿⣤⣤⣤⠶⠂⠀⠀⠀⣀⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣠⠞⣰⠏⠀⠀⢀⡼⢣⡟⠈⠳⣄⠀⠀⢀⣤⡞⠁⠈⠛⠞⠁⠀⠀⠈⠓⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⢀⡼⢁⡼⠃⠀⠀⢠⡞⢡⡟⠀⠀⠀⢈⣧⢞⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢠⠟⢀⡞⠁⠀⠀⣴⠋⢀⡞⠀⠀⣠⠖⢋⡴⠋⠙⠛⠓⠲⠦⣄⠀⠀⠀⠀⠀⣰⠶⢦⣄⣀⣸⠇
⠀⠀⠀⣰⠏⢀⡞⠀⠀⢠⡾⠁⢠⡟⢀⡴⠛⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⢸⡄⠀⠀⠉
⠀⠀⣰⠏⢀⡾⠀⠀⣴⠏⠀⣰⣟⡴⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⠁
⠀⣰⠏⠀⣼⠃⣠⡾⠁⠀⣴⡿⠋⠀⣠⠟
⢰⡏⠀⢰⣏⣴⠋⠀⢠⡾⠋⠀⢀⡼⠃
⢿⣀⣀⣾⣟⣁⣀⣴⡏⢀⣀⣠⠟⠁
⣶⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣦⡙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⠟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠒⠛⠛⠛⠛⠛⠛⠃⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠰⣌⡻⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠘⠿⠿⠿⠿⠟⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠀⠀⣰⣾⠤⠀⠀⠀⠀⡠⠄
⠀⠈⢄⠀⢀⣀⢀⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⣀⠔⠉
⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⢀⠀⠀⠡⣀⠀⠀⠀⠔⠀⠀⠠⢂⠴⠊⠀⠀⠀⣠⠆
⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⡠⠈⠀⢀⠀⠀⠉⠈⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠰⠛⠂⠀⢴⣢⣄⠀⠀⠉⠁⠀⠈⠀⠀⠀⠙⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠁⠀⠀⠀⠈⢪⠘⣆⠀⠀⠀⠀⣦⠀⠀⠀⠠⠀⠀⢀⠔⠉⠀⠀⠀⠞⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣺⠟⠓⠲⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⡀
⠀⠀⠀⠐⠤⣤⠠⠔⠀⠀⠑⠤⠀⠀⢠⠏⠓⣾⠇⠀⢇⠀⠀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⡞⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢜⠁⠀⣠⠞⠀⠀⠀⠀⠀⡰⠀⠐⠂⠀⠀⠀⠀⠀⠃⠀⢠⡞⠀⠀⠀⠀⢐
⠀⠀⠀⡯⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠊⠀⠀⢠⣼⡦⠀⠀⢃⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⢤⡦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠈⠲⢄⡀⠈⠀⠀⠀⣀
⠀⢸⡀⠀⠀⠀⠀⣠⣾⡀⣀⣀⠀⠀⠀⠀⠈⠁⠘⠉⠉⠀⠀⠀⠀⠴⠠⢰⠄⠀⠀⠀⡠⠀⢀⡴⠶⠀⠀⠒⠁⠀⢠⠀⡄
⠀⠀⠳⢤⠀⠐⠛⠛⠻⠻⣛⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⢀⡀⠀⠀⡐⠂⠀⢀⣠⠤⢤⠀⢀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠣⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡈⠁⠀⢀⣿⠇⠰⣿⣿⡆⠃⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣶⣤⠄⡘⠁⠀⠀⠈⡉⣄⣀⠉⠩⣠⡤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠹⢿⣿⠂⡴⠿⠁⠀⠀⠀⡇
⠀⠀⠁⠀⣀⠀⢀⠀⠀⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⡀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠚⠁⠀⠀⠀⠀⠠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠊⠁⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠉⠈⠉⢱⠀⡖⠒⠒⠒⠋⠉⠉⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢢
⠀⠀⠀⠀⠉⡏⠀⠀⠀⠀⠘⠛⠀⠀⠈⠛⠂⠀⠀⠀⠀⠛⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠂⠀⡀⠐⢀⠀⢀⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⠂⠘⠃⠠⠛⠀⠀⠀⠀⠀⢀⣁⠤⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⠔⡇⠀⠀⠀⠢⠀⠠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⢄⣨⠐⢁⠃⠐⠒⠀⠐⠒⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠑⠒⠀⠠⠤⡄⠀⠀⡅⠀⠈⣠⢤⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⢼⡒⠉⠀⠀⢰⣏⠀⣲⠄⠀⠀⠀⢰⠤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⡜⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⡘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠳⣄⠀⣠⠴⠖⠒⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠙⡟⠁⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⢦⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⡿⠴⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⡏⠀⠀⠀⢀⠀⢠⠄⠀⠀⠸⡄⠀⣠⠶⢤⡴⠉⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢇⡴⠁⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⣧⠀⡀⠈⠀⠀⠀⢈⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⠀⠻⣄⠀⠀⠀⠀⠇⠀⠄⠀⠀⡟⡠⠁⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠃⢞⠳⢤⣀⠀⢠⠀⠀⢀⡰⠇⠁⠀⢸⠀⠈⡄⠁⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣴⣦⠸⡇⢄⠈⠙⣄⢾⠉⠉⠀⢸⠀⠀⠀⡆⢠⠃⠀⠄⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡤⢾⡏⠀⢧⢳⠈⢆⢀⠸⡘⡄⠀⢀⠀⠀⠀⢠⢷⡏⠀⠁⠀⠀⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠘⡎⡇⢸⣆⡴⣇⢹⠒⡏⡷⠦⣶⢋⡾⡇⠰⣴⠛⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢳⢹⠈⡿⡇⣿⢸⣷⡇⡇⢰⠏⡾⢠⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠘⡆⡇⡇⡇⣿⢸⣿⢀⣧⡟⣼⠁⣸⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⢳⢹⣇⢻⡏⢸⡿⢸⡟⣰⠃⢠⠏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⡎⣿⢸⡇⢸⠇⡾⢁⡏⣠⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⣬⠃⢸⡼⠁⣼⠟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡘⢦⠞⠀⣼⡵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡎⢱⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⢴⣄⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⢿⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡿⠀⠀⠀⠀⣰⡿⠁
⠀⠀⢶⣦⡀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠛⠁⠈⠛⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠰⣿⣿⣆⢀⣿⣿⣿⣷⡀
⠀⢀⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⡀⠉⠛⠻⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡀
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⢿⣷⣶⡄⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢸⣿⣿⡄⠀⢀⣤⣀⠀⠀⠀⠀⠻⣿⣿⣷⡀⠈⠻⠟⠃⢀⣿⣿⣿⠟⠀⠀⠀⠀⣀⣤⡀⠀⢠⣿⣿⡇
⠀⠈⣿⣿⣷⡀⠸⣿⣿⣷⣄⠀⠀⠀⠙⠿⣿⣿⣷⣆⣰⣾⣿⣿⠿⠋⠀⠀⠀⣠⣾⣿⣿⠇⢀⣾⣿⣿
⠀⠀⠹⣿⣿⣿⣄⠉⠻⣿⣿⣷⣤⣀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⠀⠀⣀⣤⣾⣿⣿⠟⠉⣠⣿⣿⣿⠏
⠀⠀⠀⠘⢿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣯⣤⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠈⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⢄⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⣦⣠⣴⣿⣿
⠀⠀⠀⠀⠄⠀⡀⠀⠀⠠⠈⢠⡷⠐⠱⠀⠀⣀⠀⠀⠀⠀⣠⣤⣤⣯⣾⣿⣿⣿⣿⣿⣿⡆⠀⢀⡀
⠀⠀⠀⠀⠈⠢⣀⠑⢘⡅⠐⠎⡀⠂⠀⠒⠀⠐⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠁
⠀⠀⠀⠀⠡⡀⠈⢢⢎⠀⠀⠀⠘⠉⠀⣰⠆⠂⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣉⢂⡀⠀⠺⠀⢳⡧⠒⠲⠃⠠⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠠⡈⢡⡐⢄⣀⠈⠐⡘⢠⡔⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⢆⠑⢄⠁⢰⣌⡁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣷⣥⣄⣶⣯⣼⣝⣁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣃⣠⠄
⠀⠀⠀⠀⠀⠀⠀⡳⢻⠛⠟⡻⠋⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⡀⣘⠀⣁⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⡠⠖⠋⠉⠈⠉⠒⢄⠀⠀⠀⣠⠔⠋⠉⠉⠉⠒⢤⠀⠀⠀⢀⠔⠚⠉⠁⠉⠓⢤⡀⠀⠀⢠⠖⠉⠒⠢⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⡰⠁⠘⢡⡄⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠑⢦
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡇⠴⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢣⡀⢿⠧⠀⠀⠀⠀⠀⡼⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠑⢄⠀⠀⠂⠀⠀⣀⠇
⠀⠀⠀⠑⠤⣀⣀⣀⣀⡤⠊⠀⠀⠀⠙⠦⣀⣀⣀⣀⡠⠚⠁⠀⠀⠈⠢⣀⣀⣀⣀⡠⠞⠁⠀⠀⠀⠀⠀⠙⠢⠤⠠⠔⠁
⠀⠀⢀⡀⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡤⠄⣀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣀
⠀⠀⡇⠀⠀⣠⠄⠉⠲⢄⠀⠀⠀⢀⣴⠁⢀⣴⣶⣄⠀⢱⣄⠀⢀⡔⠋⠀⠀⠀⠀⠀⠙⢆⠀⠀⡴⠋⠀⠀⠀⠀⡈⠙⢆
⠀⠀⢧⠀⠀⠀⠑⢕⠄⠀⠱⡀⠀⣾⠃⠀⠈⠿⠿⠏⠀⠀⠹⡄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠁⠀⠄⠀⢠⠀⡀⠂⠈⡆
⠀⠀⠈⢆⠀⠀⠀⠀⠈⡦⠀⢱⠀⠇⢠⣤⣄⠀⠀⠀⣠⣤⢀⠇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠐⠑⠁⠀⠇⠈⠌⠀⢂⡇
⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⢘⠀⠈⢾⣿⣿⠆⠀⠐⢿⣿⠏⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠳⡀⠀⠀⠀⡐⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠀⠀⠀⠉⠒⠢⠤⠔⠊⠁⠀⠀⠀⠀⠉⠒⠶⠤⠔⠚⠁⠀⠀⠀⠀⠉⠓⠲⠦⠖⠚⠁
⠀⠀⠀⢀⠴⠂⠉⠉⠑⠢⣄⠀⠀⠀⢀⡤⢆⣉⠉⠓⠢⣄⠀⠀⠀⢀⡴⠒⠻⠉⠓⠲⢄⠀⠀⠀⠀⠀⠀⡐⠳⢤
⠀⠀⡰⣡⡶⡻⣶⡄⠀⠀⠈⢣⠀⠰⠃⠀⠀⠀⠀⠀⠀⠈⢣⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢰⠄⠀⠀⠙⢄⡀
⠀⠀⡃⣿⠰⢡⣾⡇⠀⠀⠀⠈⠃⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⣿⣦⡀⠀⠀⠀⠑⣄
⠀⠀⢣⠈⠛⠟⠋⠀⠀⠀⠀⣰⠁⢣⠀⠀⢀⠀⠁⠀⠀⠀⣰⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⢴⣿⣿⣿⣿⣦⣄⠤⠔⠊
⠀⠀⠀⠳⢄⡀⠀⠀⠀⣀⠔⠁⠀⠀⠑⢄⡁⠀⠀⣀⣀⠔⠁⠀⠀⠑⢤⣀⠀⠀⠀⣠⠔⠁⠀⢀⠀⠙⡿⠛⠉
⠀⠀⠀⠀⠀⣉⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠁⠈⢀⣀⣀⣀⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠢⡀⠀⠀⡴⠋⠀⠘⣄⠀⠈⠲⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠓⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠑⣄
⠀⠀⡞⢰⡷⠀⠀⠀⠐⣷⠆⢹⠀⡜⠀⠀⠀⠀⠈⠱⢄⡀⢘⡄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡗⠒⠠⣀⠀⠀⠀⠀⠈⠉⠇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⣄⠀⠀⣠⣀⠀⠀⢀⠎⠀⠘⣄⠀⠈⠳⡄⠀⠀⢀⠜⠀⠘⢄⠀⠀⠀⠀⠀⠀⢀⡜⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠈⠑⠶⠬⠥⠤⠚⠁⠀⠀⠀⠈⠑⠢⠄⠼⠤⠒⠁⠀⠀⠀⠀⠑⠢⠤⠤⠤⠒⠉⠀⠀⠀⠀⠑⠢⠤⠀⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠚⠀⠀⣠⠞⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡰⠞⠛⠋⠛⠛⢶⣄⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣰⠎⠳⣆⠀⠀⠀⣠⠞⠙⣧⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡟⠀⠀⠈⠳⣤⠞⠁⠀⠀⢸⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢻⣤⣤⣤⣤⣤⣾⠁⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣗⠲⠶⠤⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡁⠸⣏⣉⣛⣛⣺⡇⠀⢛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⡜⠁⠀⠻⣍⣉⣉⡽⠁⠀⢯⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⠀⠀⢠⡄⠈⠉⠉⠀⠀⠀⠈⣇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⢤⣜⡧⠤⣤⣤⣀⡀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠉⢧⡀⠈⠓⢺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠻⢤⡀⠀⠀⠀⢠⠇⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡇⠀⢀⡴⠋⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣀⡀⠀⠙⠚⢉⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠠⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⢀⡠⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠀⢠⠟⠻⣽⣿⣿⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢚⣶⣶⣿⣿⡇⠀⠀⣄⡀⢠⡶⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣀⠠⣀⣄⣾⣾⣿⣿⠟⠀⠀⠀⢹⢷⠌⣷⠸⣶⡖⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠘⠰⠟⠀⠈⠰⠏⠀⠀⠀⠀⠀⠀⣠⠀⣸⠡⡄⢠⠀⡠⢠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⢠⠈⡇⣎⠈⠂⠷⠰⠄⠻⠐⠫⠀⠺⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡏⠉⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢀⣀⢀⡆⠃⠀⠀⢇⠀⠛⠀⠀⢈⠴⠲⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠉⢀⡔⡦⠀⢢⠀⡜⠀⡐⢠⡆⣼⠀⠭⠘⠂⠀⠀⠀⠀⠀⠀⢀⡔⠁⡠⡐⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⠖⠉⠀⠘⠀⠓⠀⠙⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣸⣇⠁⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢠⣾⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠔⠒⠈⠁⠀⠀⠀⣽⣄⡂⠀⠀⠀⠠⠦⠤⢄⣀⣀⢀⣈⣺⠿⢛⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠰⣿⣷⣤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣾⣿⣗⣿⢸⢀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣦⣶⣴⣱
⠀⠀⠀⠀⠀⠈⠛⢿⣟⣦⣇⣤⠀⠀⢀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡿⢻⣿⣾⣸⣷⣾⣿⣻⣅⣿⣲⣻⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣧⡿⣿⣧⡠⢰⣷⣤⣴⣶⣞⡟⠛⠒⠶⠶⠤⠤⠴⠖⠚⠙⠻⠜⠿⠿⠽⠟⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣿⣾⢹⡿⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣸⡆⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠟⣻⠛⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠟⣋⣥⣾⣿⣷⣤⣙⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣩⣶⡿⣩⣿⣿⣿⣷⣤⣈⠻⣷⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⡔⣸⣟⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣼⣿⣿⠟⣱⣿⠟⠙⠻⢿⢿⣿⣾⣿⡇⣿⣿⣧⡀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣙⠒⣫⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣵⣶⡟⠁⠀⠀⠀⠀⢂⣉⣻⣿⣿⡸⣿⣿⣿⡆⠀⠀⠀⠀⠘⣿⣿⣿⣿⣻⢿⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣱⣿⡟⠶⣾⢿⢶⡆⠀⠀⡶⠛⢻⡶⢿⣿⣧⢻⣿⣿⣇⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣼⣡⣿⣧⡀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣌⢧⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⣿⣿⣿⡇⠀⠀⢁⠀⠀⠘⠂⠠⠤⢺⣿⣿⣮⡻⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠘⣿⣿⣧⣿⣄⠀⢸⣴⣶⣶⡄⠀⢀⣾⣿⣿⣻⣿⣷⣭⡻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣿⣿⢩⣿⣿⣷⢎⠈⢹⠁⡠⢾⣏⣬⣿⣧⣿⣿⢿⣿⣿⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢀⣿⣿⣿⣿⣿⣿⣷⣶⠉⣵⣶⡿⣿⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⣿⣿⣿⣿⣿⣝⠷⣄⣈⣉⠉⠉⣥⣾⣯⣾⣇⠙⣿⣭⣿⣙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠿⠿⠿⠿⡿⠿⢿⣷⡬⠋⣡⣄⣠⣴⣾⣿⣿⣿⣆⠘⠏⢸⣿⣿⢦⡙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠛⠛⠛⠛⣿⣿⡿⢿⣿⣿⣀⠀⢸⣿⣿⣰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⡀⠀⠀⠀⡀⠀⢾⣉⡟⠿⠃⢀⡀⠀⠉⠉⠛⠶⣌⡁⢹⣦⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣍⠉⠙⢤⡀⠀⢿⠦⣬⣿⡀⠀⠀⠘⠛⠓⠀⠀⠀⠀⠈⢻⣆⠙⠀⡏⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣷⣶⣾⣧⢱⣾⣄⡛⠿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡿⢿⢰⠀⢸⡆⠀⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡹⣶⣿⡷⢋⣼⣿⣿⣿⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢘⠋⠁⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣾⣿⣤⠇⠄⠤⠈⣸⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⢿⣦⡀⠀⠀⠀⠀⠀⠐⢿⡿⣧⣿⣿⣿⡷⣤⣤⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠙⢦⣝⠻⠷⣶⡶⠶⠖⠚⣩⠴⣿⣿⣿⡿⣿⣶⣶⣾⣿⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠉⠙⠳⠶⠶⠶⠒⠋⠁⠀⠈⣿⣯⢿⣝⠻⠿⠟⣡⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣽⣻⣶⣟⣫⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠛⠛⠉
⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⢶⣶⣶⢶⣶⣶⢶⣶⣶⣶⣶⣶⡶⣶⣶⡶⣶⣶⣶⢶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣧⣙⠻⣧⠻⣿⠱⣿⡿⣿⣿⣿⣿⣿⢸⣧⡗⢸⡷⢙⢸⣿⣿⣿⣿⣿⡇⣿⡇⣻⡌⢻⡇⢸⣿⣿⣿⣿⣿⡇⣿⣴⣶⣝⠿⡇⣾⣿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣷⣿⣷⣿⣿⣿⣿⣿⣾⣿⣷⣾⣷⣿⣾⣿⣿⣿⣿⣿⣷⣿⣶⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣾⣿⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⠛⠉⠉⠉⠉⠉⠙⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣴⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⣡⠿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⢽⣒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣾⣣⡼⠖⠋⠉⢑⡜⠂⠀⠀⠀⠉⠉⡿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢙⣲⣤⠤⠤⠤⣿⠄⠀⣀⣀⠞⠁⠀⠀⠀⢀⡀⠀⢧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠘⠄⠀⠁⠀⠀⠘⠉⠉⠁⠉⠁⠀⠀⠀⣠⣏⠏⠁⠀⣁⡤⣤⣄⠹⡄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⢛⡶⠋⠀⠀⣀⣀⣉⣁⣠⠼⠁⠀⣠⠀⠀⣦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠉⠒⠦⠤⠤⠤⠤⠴⠒⠋⠉⠀⠘⢉⡴⠚⠑⠺⣄⠀⠀⠈⠑⠢⠤⠴⢻⣀⣠⠿⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⣀⣀⣀⣀⣠⠤⠤⠒⠚⠙⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡖⠒⢜⣯⣍⣠⣟⠔⠊⣉⠆⠀⠀⠀⠀⠀⠀⠀⣠⣞
⠀⠀⠀⠀⢀⣴⣾⠏⠁⠉⠹⣿⣦⣄⠀⠀⠀⠀⠀⠀⠸⣉⣱⢇⡠⠤⣽⠏⠇⠠⢐⠄⠁⠀⠀⠀⠈⣦⡠⢾⡿⢿
⢠⠞⡑⠠⡟⠋⠑⣄⣀⣀⡠⠋⠙⠻⣆⠀⠀⣀⠤⡄⠀⢉⣒⣊⣀⣴⠃⠀⠢⢈⡁⠔⠁⠀⠀⠀⠀⣧⢠⠼⠀⠘⠓⡄
⡄⠀⠇⣶⡅⠀⢠⣿⣿⣿⣇⠀⠀⠀⣻⡀⢈⡟⢠⢞⣆⣸⡄⠀⠀⠈⠙⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢻⠈⢆⣠⠤⠚⠁
⠃⠀⠀⢻⡟⠀⠙⢿⣿⠿⠛⠒⢤⣴⣿⠁⠈⠻⡈⢧⠚⢇⣻⣄⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠚⠛⠦⡄⢡⣊⠙⢶⡄⠀⠀⢄
⠀⠀⠀⠈⢅⠀⠀⠀⡇⠀⠀⠀⢰⣿⠃⠀⠀⠸⢵⡌⠁⠛⠁⠈⠳⡀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⢺⠉⣇⠀⢣⠀⠈⡆⠱⠀⠀⠑
⠀⠀⠀⠀⠀⠑⠲⠾⢿⣶⡶⠖⠋⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠒⠊⠁⠀⠱⣄⠀⠀⠀⠀⢀⣴⣿⣁⡼⢾⡽⣇⢈⡿⠀⠁⠀⠄⠘⠃⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠤⠖⠉⠀⠀⠀⠀⠈⠫⣈⡼⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢠⠀⡄⢀⠀⡄⢀⠀⡄⠀⡀⠀⡀⠀⡀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣌⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣦⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠐⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡀
⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣙⠿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠟⡋
⠀⠘⢷⣦⣭⣭⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣴⣿⣄⣀⣀⣀⣀⣀⣈⣉⣉⣉⣭⣤⣶⡾⠁
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠹⣿⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀⠀⠀⠀⠀⠠⠓⠀⠀⠀⢀⡤⠀⠀⠀⢠⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠠⠬⠂⠀⠒⠒⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣾⠀⠀⡀⠂⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⡿⠂⠁⠀⠀⠀⠀⠀⢠⠞⠋⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⢀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠐⠤⢴⡀⠀⠀⣸⠀⠀⠀⠀⢠⠔⠀⠀⠋⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠹⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⢀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⡀⠂⠀⠀⠻⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⢀⡴⠋⠀⠀⠀⠀⠌
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡏⠀⠀⠀⡄⠀⣴⠀⠀⠀⠠⠆⠀⢀⣀⠈⣀⡴⠏⠀⡀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠾⠚⠛⠛⠻⠉⠀⠀⠰⡇
⠀⠀⠀⠀⢠⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠃⠀⢿⡀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⡴⠋⠁⠉⠀⠀⠀⠀⠀⠀⢳⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢟⡀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢃⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡄⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⢀⠀⠀⠙⡙⠶⠶⢦⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢠⠟⠃⠀⠀⠈⢁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⣾⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠
⠀⠡⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡀⢤⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠓⣤⠘⠁⠉⠁⠂⢀⠀⢨⠁⠀⠘⢂⡀⠀⠀⢀⡅⢤⡄⠒⠐⠢⣀⣠⡶⢢⡔⠀⠀⠀⡄⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⢹⡍⠁⠈⠁⠀⠉⠁⠀⠀⠅⠀⠀⠀⠀⠀⠈⣠⠀⠓⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⣀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⣤⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡰⠂⠁⢀⡀⡀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⢀⠀⠈⢦⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⡠⠶⠂⠀⠀⠈⠀⣀⣠⠴⠖⠛⠛⠋⠃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠈⠃⠰⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⠾⠁⠀⠀⠀⠀⣠⠖⠛⢉⣀⡀⠀⠀⠠⠤⠠⠧⡀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠈⠇⠀⠀⠀⡀⠀⠀⢀⠀⠀⠙⢤⣄⠀⠀⠄⠋⠁⠄⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠈
⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣥⢀⠴⢀⣠⠚⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⡰⠞⣁⣬⠿⠷⠒⠳⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢤⡈⠂⠀⠀⡀⢂⠀⠀⠀⠀⠀⣀⡸⠁⣰⣧⠚⢁⠠⠄⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠉⠀⠀⢀⡀⢦⠀⠀⠀⣿⡗⠚⠁⠀⠀⠀⢀⠀⢀⠲⠀⠀⠁⠠⠤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⠠⠀⠈⠉⢆⡤⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠂⠀⢿⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢠⠈⡀⠀⠀⠐⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠿⠀⠁⢀⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⡄⢀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡄⢀⡟⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣴⣧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡅⢺⡄⢠⠀⣶⠂⡀⢾⠀⡼⢃⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠛⠓⠉⠈⠓⠋⠁⠉⠉⠀⠀⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠋⠈⠛⠛⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣴⣶⣴⣶⣶⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠉⠙⠒⠉⠀⠀⠈⠙⠛⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡤⠒⠚⠓⠒⢤⡀⠀⠈⠧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢠⠏⠀⢠⠖⠒⠀⠀⠹⡄⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡄⠀⢸⠀⠀⠈⠓⠲⡄⠀⠀⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⠀⠈⢧⡀⠐⠒⠚⠁⢀⡼⠁⠀⠀⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠉⠒⠲⠖⠒⠉⠀⠀⢀⣀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠹⣄⡴⠋⡟⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠤⠤⢤⠤⠤⢴⠧⢤⠤⡼⠦⠤⡤⠤⠤⡤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠲⡖⠓⢺⠖⠒⢾⠖⢶⠚⠒⠚⠒⡷⠲⡶⠓⠲⡖⠓⠲⡗⢲
⠀⠀⠀⠀⠀⠀⠀⠓⢶⠓⣶⠚⠒⡶⠚⠒⡞⠲⡄⠀⠀⢳⠖⢳⠒⠒⢳⠒⠚⣷⠚⡆
⠀⠀⠀⠀⠀⡖⠒⡖⠚⠲⡟⠒⢲⠓⢲⠒⠓⢲⠓⠒⡶⠚⡖⠚⠒⡖⠚⡖⠒⢻⠖⠓⢲⠒⢲
⠀⠀⠀⠀⠀⠓⠒⠓⠒⠚⠓⠒⠚⠓⠚⠒⠒⠚⠒⠒⠛⠒⠃⠀⠀⠓⠚⠓⠒⠚⠓⠒⠚⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠫⠉⢍⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⠍⠊⠀⠀⠀⠀⠈⠒⢍⡻⢿⣿⣿⣿⡿⠟⢋⠡⠐⠀⠀⠀⠀⠉⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄
⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦
⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠟⡁⠤⠈⢐⡀⠀⢀⡀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀⠄⠀⠒⡈⠈⠀⠙⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠟⡡⠊⠀⠀⠀⠀⠀⠁⠀⠒⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢩⠄⠒⠀⠈⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⢸⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⠁
⠀⠀⠀⠸⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠉⠉⠉⠉⠉⠉⠩⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠇
⠀⠀⠀⠀⢳⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡞
⠀⠀⠀⠀⠀⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠞
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⣤⣤⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⣤⣤⣶⣶⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⠬⠥⠤⠤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⠀⣀⣤⠀⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⣤⣄⡀
⠀⠀⠀⢀⡘⣿⡙⠷⠟⢸⣧⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡾⠋⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⣿⣙⠛⢡⡶⠶⣦⠉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⣨⡾⠛⠛⠛⢷⣄⠀⠀⢹⣆⡀
⠀⠀⠀⣩⡽⠇⠸⣧⣤⠿⢀⣙⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠻⣆
⠀⠀⠀⠹⡷⠞⣷⢀⣴⣄⢻⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠁⠀⠀⠸⣧⠀⢤⣴⡄⠀⣼⠃⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠘⠻⠾⠃⠹⠛⠁⣶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⣿⢰⡞⠁⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⣦⣀⣿⠀⢀⣿⣠⡾⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⢰⡾⠷⠿⠾⣷⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠉⠁⠀⠀⠉⠋⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣭⡛⢿⡄⠀⠀⠀⠘⠿⣾⠿⡷⠟⠀⣼⠛⠛⠉
⠀⠀⠀⠀⠀⢠⣤⣤⣴⠟⠀⢠⡾⠛⠛⢶⡄⠈⢿⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⣨⡗⠈⠻⢶⣤⡴⣦⣀⡀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⢸⣇⣀⣀⡀⠀⢿⡄⠀⠀⢸⡇⠀⢀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠉⣩⡷⠀⠈⠛⠶⠾⠋⠀⠀⣿⡉⠉⠉⠀⠀⣀⣀⣀⣀⡀⠀⠻⠶⠟
⠀⠀⠀⠀⠀⠀⠀⢾⣏⠀⣠⡶⣤⠀⠀⣴⢦⣄⠈⣻⡦⣠⡶⠛⠉⠉⠉⠉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⣿⠀⢀⣿⠀⠙⠿⠋⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠰⠟⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣄⣠⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⢠⡞⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⣀⣤⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠃⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣄
⠀⠀⠀⠀⠀⢠⣄⢀⡤⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⢸⡇⠀⣀⣀⡀⣠⣤⡀⠀⠀⠀⣴⠶⠾⠋
⠀⠀⠀⠀⠀⢀⡭⢨⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⢸⡏⠉⠿⠋⢀⣿⠆⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⣨⡿⠃⠀⠀⠘⣧⣀⡼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠻⣦⡀⠀⠀⠀⠀⠉⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣷⠄⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠈⠻⣦⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣧⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣄⣼⡇⠀⠀⠀⠈⠻⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⡂⠀⠀⠀⠀⠉⠠⣤⠀⠈⠁⢀⠄⢒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠡⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠐⠼⠁⠂⢄⠑⢄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⠀⢈⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠊⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠑⠂⠁⠀⠀⠀⠀⠀⣇⠂⠒⠐⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠆⣁⡀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢹⠅⠡⠄⠤⠠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠂⢀⠇⠀⠀⠔⠀⠀⢄⠀⠀⠀⠀⠀⢠⠨⢄⠀⠀⢌⠀⡀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⢁⡀⠀⡀⠀⠔⠀⠀⠨⠀⠀⠃⠀⢈⠀⠀⠀⢐⠂⢁⠀⠀⠀⡇⠀⠈⠀⠂⠀⠀⠀⠉⠀⠀⠧⠀⠀⠠⠌⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡡⠀⡐⠀⠀⠀⠀⠀⢰⠈⡀⠀⠀⠠⠄⠀⠀⢈⡀⠸⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠈⠂⠀⠒⠁⠀⠀⠢⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⡂⠈⢀⢐⠀⣀⣀⢀⡀⢀⡀⠔⠁⠀⢀⣉⠔⠀⠀⠃⢊⠂⢀⠐⠀⡄⠀⠰⠄⠀⠠⠀⠀⡀⡀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠂⠀⠂⠀⠀⠈⠀⢀⠐⠄⠀⠀⠀⠀⠀⠀⡐⠁⢀⠀⠀⢀⠀⠀⡉⠀⠀⡈⠉⡉⠀⠀⠀⠀⠀⢠⠁⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⡆⢰⠂⠈⡄⠀⢁⠤⠄⠀⡄⠀⠀⠁⠀⠀⡆⠐⠀⠀⠀⠎⢀⠀⠄⠀⠐⠀⠤⠤⠴⠤⠄⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠦⠤⠶⠀⠆⠀⠀⠀⢴⠂⡀⠀⠅⠘⠂⡐⠀⠀⠈⠀⡀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠚⠒⠀⢄⠀⠀⠀⣸⡀⠂⠀⡅⠸⠁⠰⠀⠀⠀⠡⢀⠀⠡⠈⢁⠔⠀⠈⠐⠔⢀⠁⠀⡀⠘⠀⢠⠀⠀⠀⠀⠀⠄⡀⠀⡠⠠⠀⢀⠀⠄
⠀⠀⠀⠀⠈⠁⠁⠉⠢⠀⠀⠀⠀⡨⢄⠀⠀⠐⠂⠈⠈⠠⡈⠀⠈⠁⠀⠀⠀⢀⡀⠀⠀⠒⠂⠁⠰⠀⠈⠐⠀⠁⠀⠀⠀⠀⠀⠡⢈⣎⠠⠁⠀⢸
⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠂⠀⡐⡀⢀⠬⡀⠀⡀⠀⡀⢠⠀⠀⡀⠠⢀⠀⠀⡆⠀⠀⡠⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢢⡄⠄⠁⠀⠘⠀⠀⠀⣀⠀⠀⡆⠂⠄⠁⢢⡠⢃⢴⠀⠑⣠⠆⢀⠈⠀⠰⠀⠁⣸⠀⣀⠋⡹⡀⠁⡤⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡌⠂⠀⠀⠈⠛⠀⠀⠦⠁⢀⠨⠃⠍⠀⠉⠊⠁⠘⠀⠱⢗⠋⠓⠀⠊⠀⠒⠀⠃⠈⠀⠉⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠁⠀⠀⢠⢀⠀⠚⠂⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⢠⠊⠙⠇⠀⡄⠀⠀⠀⠀⠀⠀⡐⠀⠠⢂⠄⢀⠤⠀⠀⠰⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡑⢄⠔⡁⠔⠀⠀⣀⠀⠀⠀⠀⠀⠘⠐⠊⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠔⠁⠀⠊⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠹⠀⠀⠀⠀⠀⠈⠆⠀⠀⠠⣀⠢⠀⠀⠀⠄⠃⠀⠀⠀⠀⠀⠠⣂⡁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠐⢄⠀⠐⠀⢈⠄⠀⠀⠀⢈⠄⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠰⣘⡀⢀⡀⠀⡄⢱⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠀⠀⠀⠀⠀⣀⠐⡰⢅⠈⠀⠀⠔⢡⢳⣤⠀⡀⢄⠀⠀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠁⠁⠀⡄⡄⢰⡴⠁⠦⠄⠐⠂⠀⠀⠂⠸⣆⡈⠄⡀⠀⠉⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠎⠠⠄⠀⠀⠀⠀⡀⠀⢐⠬⠥⢈⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠂⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣧⣤⣤⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣼⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠠⡄⠸⣧
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠲⠀⢻⡆
⠀⠀⠀⠀⠀⠀⢰⡟⠀⢘⠀⠀⠀⠀⠹⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡤⠤⢤⣤⣤⡤⣤⠇⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⣴⣿⣤⣤⣤⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⢾⡇
⠀⠀⠀⠀⠀⢻⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⠉⠉⠙⣿⠉⠉⣿⠏⠉⠉⠉⠉⠉⠉⠉⢹⡏⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠇⢀⡀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣼⠏⠀⠀⠀⠀⠀⠀⠀⠹⠷⠶⠟⠀⠸⠃⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢻⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⣀⣀⣀⣀⣻⡆⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠛⠷⠶⠶⠶⠶⠶⠆⠀⠀⠀⠘⢿⡏⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣤⡄
⠀⠈⠀⠀⠀⠀⠀⠀⢸⡏⢠⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⣿
⠀⠐⠀⠀⠀⠀⠀⠀⢸⡇⢐⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣼⡷⡂⢸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣀⣿⣯⣇⣸⡟
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠉
⠀⢠

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠖⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⠲⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⣤⣀
⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄
⡀⠀⠀⠀⠈⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⡄⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠈⠻⣿⣿⡄⠀⠀⠀⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠙⢿⡷⠀⢀⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠻⠿⠿⠿⣿⡷⠀⣠⣤⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠀⠙⠟⠃⢠⣶⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⠟⠉⣰⣦⠀⣶⣦⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣴⣿⡇⠀⢸⣿⣷⣄⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⢀⣾⣿⣿⡇⠀⠈⣿⣿⣿⣦⡀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃⠀⠀⢹⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈⠛⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤

⣴⣦⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣴⣴⣶
⣿⣿⣿⡟⠉⠛⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠛⠉⠉⣀⣀⠀⢀⣀⡉⠉⠙⠛⠿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋⢁⣠⣴⣶⣿⣿⣿⣿⠀⢸⣿⣿⣿⣷⣶⣤⣀⠉⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⣠⣶⣿⠿⠛⠉⠀⠀⣿⣿⠀⢸⣿⡇⠀⡈⠙⠻⢿⣷⣦⡀⠙⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣤⣾⡿⠋⢀⣠⣶⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⣿⣷⣦⣀⠈⠻⣿⣦⣴⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⠏⢀⣽⣿⠋⢀⣴⣿⡿⢻⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠻⣿⣷⣄⠈⢻⣿⣅⠈⢿⣿⡄
⠀⠀⠀⠀⠀⣸⣿⠏⢀⣾⣿⢃⢀⣈⣉⢁⣀⢸⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⢀⢀⣉⡉⠀⣀⢻⣿⣆⠈⢿⣿⡀
⠀⠀⠀⠀⢠⣿⡟⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⣿⣿⠀⢸⣿⡇⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠄⠘⣿⣧
⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⠀⢠⣤⣤⣤⣤⣤⣤⣤⣾⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⢸⣿⡏⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⠀⢸⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠘⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢠⣿⡟
⠀⠀⠀⠀⠀⠹⣿⣆⠀⢿⣿⡄⠀⢀⣀⣀⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⢀⣀⣀⠀⠀⣼⣿⠇⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣆⢈⣻⣿⣦⠀⠙⢿⣷⣼⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⣴⣿⠟⠁⢠⣾⣿⣇⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠛⢿⣷⣤⡀⠙⠻⢿⡇⠀⣿⣿⠀⢸⣿⡇⠀⣿⠿⠛⠁⣠⣶⣿⠟⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠙⠻⣿⣷⣤⣀⡀⠀⣿⣿⠀⢸⣿⡇⠀⣀⣠⣴⣿⡿⠛⠁⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠉⠛⠿⢿⣿⣿⣿⠀⢸⣿⣿⣿⠿⠟⠋⠁⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⣠⣴⣶⡿⠿⠿⢿⣶⣦⣄⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠟⠉⠀⣀⡀⢀⠀⠀⠉⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢰⡶⣛⣛⢿⣥⡀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⢰⣿⠁⠀⠀⢿⡗⣿⣿⢇⣿⡅⠀⠀⠀⠈⣿⡆⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⠀⠀⢀⣜⣟⣶⡶⢿⢽⠞⣓⢷⡄⠀⣿⡇⠀⠀⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠠⡏⣶⢝⡇⣠⣤⣹⣮⣋⡼⠃⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠙⠖⠛⠙⢿⣻⠇⠈⠁⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡷⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⢾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠙⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣿⣶⣶⣶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⣿⣿⣿⣿⡯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡍⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡾⠁⠀⠀⠀⠉⠐⠠⡀
⠀⠀⠀⠀⡠⠂⠁⠌⢦⣀⣀⡤⠤⠤⠤⣄⣀⡀⢺⢠⢥⡟⢠⣴⣐⠒⠂⢰⠀⠈⢆
⠀⠀⠠⠂⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠚⠧⠖⠟⢻⠀⠀⠈⡇⢀⡀⠣⡀
⠀⠀⠀⠀⢂⠐⠀⢠⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢱⡄⠀⣀⢈⠳⡄
⠀⠀⠀⠐⠀⠀⠀⠤⠀⠀⢾⣶⣦⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡯⠝⠛⠋⣘⣒⣒⣒⡤⠟⢀⠄
⠀⠁⠀⠀⠀⡠⠀⢂⡒⠚⡂⢷⢸⡿⢠⡄⠩⠴⣦⠤⣤⣤⡤⠖⠛⠚⣛⣚⡈⠁⠈⠀⠈
⠀⠀⠀⣪⠵⠚⠛⠉⠉⠉⠉⠉⢩⠀⠈⠒⠚⠀⠉⠀⢀⠉⠁⠀⠪⣹⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠉⠛⠛⠿⣿⣿⣿⣷⣶⣦⣼⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⢀⣴⣿⣿⠿⠋⠑⠂⠐⠂⠢⡠⠀⢤⠀⠀⠀⠀⡤⠝⠻⢿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡟⢉⠢⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠁⠲⠀⠻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⡘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿
⣿⣿⣿⣿⡛⠉⠁⠀⠀⠀⠀⠀⠈⣿⣿⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠠⠌⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠘⢄⠀⠀⢠⠀⠂⠐⢊⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⢄⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣤⣤⣀⣻⣇⠀⠀⠀⠀⠈⠑⠒⠚⢼⡅⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠉⠂⠤⠀⠀⠀⠀⠀⡀⢀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⠀⢀⠎⠀⠀⠀⠀⢰⣡⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⡠⠼
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠈⡄⢀⠔⠁⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠤⠒⠉
⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣶⣤⣄⣀⣀⣣⣃⣀⣠⣤⣴⣾⣿⣿⣿⠟⠁⠸⣿⣿⡿⠟⠛⠉
⣿⣷⣦⣄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠁
⣿⣿⣿⣿⣿⣶⣄⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣶⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⢋⡅⠚⠏⠉⡙⠆⠀⢛⠿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠫⠓⠁⠀⠀⠀⠚⠀⠀⡠⠀⠀⠀⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⣵⠞⢋⢄⡊⠀⢠⣧⣤⠐⠀⢄⠀⣠⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠧⠒⠁⠀⠀⠀⡈⠁⠀⠀⠀⠀⠑⢄⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣆⣀⣀⣤⣬⡁⠀⠒⣒⣀⡘⢦⡈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠫⠓⠪⣍⠀⠈⠀⢩⡥⠚⠶⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⣴⣿⣭⡀⢀⣿⣄⠀⣨⣙⣄⠀⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⣦⠙⠃⢀⠶⠷⣿⣽⠷⠄⠰⠿⢁⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣟⠁⠀⠀⠀⠲⣿⡶⠀⠀⠀⠀⢸⣿⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⠟⢿⢹⢿⡼⠆⠀⠀⡋⡀⢉⠀⠀⠀⢀⣿⢿⠛⣿⢿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⢹⣿⠸⣿⠟⠁⠀⠈⠈⡌⠻⣆⠀⠈⠙⠻⠟⠁⠀⢀⣾⠏⢸⠀⠃⠀⠈⠹⢿⡟⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠘⡆⢿⡀⠙⢃⣠⠴⣛⠀⠀⢸⠈⠁⠠⡈⠉⠁⠀⢰⢾⡧⠀⡎⠠⠤⣄⣀⡀⠈⠁⡠⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠐⠘⣧⠾⠫⠐⣫⣭⠀⠀⠘⡄⠀⠀⠀⠀⠀⠊⠁⡸⠄⠀⠁⠉⣉⣒⡲⠭⣗⢦⡀⠀⠀⠀⣽⣿⣷⡀
⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⢀⣼⡫⠖⢈⡁⠄⠂⠀⠀⠀⡃⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠀⠀⠀⢈⣉⠉⠚⢿⣿⣦⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⣿⣏⣶⡆⠀⠀⣴⣿⠥⠒⠉⡥⢀⡠⠀⠀⠀⢀⡨⠔⡒⢋⠉⠉⠙⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⢉⡓⠻⢿⣿⣆⠀⠀⠙⢧
⠀⠀⠀⠀⢿⣿⠟⠀⠀⣰⠏⠀⠀⠉⠅⠵⢉⠀⠀⢀⡴⠋⠐⠉⠀⠀⠀⠀⠀⠐⠀⠀⡙⢦⡀⠀⠀⠤⣤⡀⠭⣁⠀⠈⢻⣷⡶⣶⣿⠃
⠀⠀⠀⠀⣸⣯⠞⠀⢠⡟⠀⠀⡰⠀⠀⡀⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢳⡈⠐⠒⠂⠤⡀⠀⠑⠀⠀⠙⣷⣾⠿⢶⡄
⠀⠀⠀⣰⡿⢋⣠⣶⣿⠃⠀⣠⠇⢀⢸⣧⠀⠀⢸⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⢰⡇⠀⠀⠀⠃⠀⠈⢻⣿⣿⣷
⠀⠀⢠⣿⣿⣿⣿⣿⠛⣚⣯⣷⠀⠁⠀⣿⠀⣶⢸⢰⣶⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡇⠀⠀⢠⠀⣿⡆⣰⠂⣿⣷⣶⣿⣽⣿⣿⣦
⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⡇⠀⠀⣿⡇⢸⠘⡄⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⢰⠃⠀⠀⢸⠀⣿⣿⢱⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷
⢰⣿⣿⣿⣿⣿⣯⢿⣿⡿⠁⣼⡅⡄⠀⣿⡇⢸⠀⠘⢆⠻⢿⣷⣶⣶⣦⣤⣀⣤⣾⣿⠏⡰⠃⠀⠀⠀⢸⠀⣿⣧⢸⠀⠀⠙⢳⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⠈⠁⠀⠀⣿⣷⠀⠀⣿⡇⢸⠀⠀⠀⠙⢮⣍⣉⠙⠻⠿⠟⣛⣩⠴⠊⠀⠀⠀⠀⠀⢹⠀⣿⣿⠚⢧⠀⠀⠘⣿⣿⣿⣿⣿⣿
⢿⣿⣿⡿⠿⢿⠇⠀⠀⠀⠀⢿⣿⣧⢠⣿⣇⣾⣀⢀⣀⠀⡈⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣈⣀⣹⣿⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠒⠒⢺⣿⣉⠉⣉⣻⣿⡗⠒⠲⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠮⠤⢒⣋⣉⣉⣉⣉⣂⣀⣤⠥⠴⠚

⠀⠀⠀⠀⣀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⡠⠊⢱⠤⠒⠒⠈⠙⢆⠀⠀⠀⣰⣿⡘⣯⣵⣶⣶⣶⡄⠀⠀⢀⢾⡁⠀⢀⡤⠤⣌⢢⡀⠀⢠⡸⣿⣿⠿⣛⣿⡛⢦
⠀⢠⣇⠀⠀⠣⣀⡠⠤⣄⣈⡆⠀⠐⢿⣿⣷⣝⠿⣟⣛⢛⡿⠀⠀⡎⠀⠉⠉⠁⠀⠀⠀⢣⠇⠀⣿⣿⣶⣶⣿⣿⣿⣿⡎⡇
⠀⠘⡏⠣⣄⢀⡎⠀⠀⠇⢀⠇⠀⠸⣮⡻⢿⡿⣸⣿⣿⣸⣿⠀⠀⢇⠀⣀⣀⡀⠀⠀⠀⡜⡆⠀⣿⣿⠿⠿⣿⣿⣿⣿⢇⡇
⠀⠀⠙⢄⣨⠏⠀⠀⣼⡠⠊⠀⠀⠀⠙⢿⢗⣵⣿⣿⢇⡿⠃⠀⠀⠈⠻⣀⠀⠈⠙⠒⣫⠜⠀⠀⠈⢵⣿⣿⣶⣭⣯⣥⠞
⠀⠀⠀⠀⠉⠑⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉
⠀⠀⠀⣠⡶⠓⠀⠺⣶⣤⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡙⠻⣦
⠀⠀⠈⠁⢀⣤⣤⡆⠀⠉⠀⠀⠀⢀⣾⣿⡟⠛⠛⣿⣿⣾⣧⠀⠀⠀⠀⠀⠀⠀⣠⡦⠙⢷⡄⠀⠀⢀⣾⣿⣿⠏⢙⣷⣌
⠀⢰⣦⠀⠸⢿⡿⠇⠀⣠⣶⡆⠀⠈⠻⣿⣅⡀⢀⣽⣿⠟⠙⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣾⣿⡿⠁⣰⣿⣿⠏
⠀⠀⠿⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⢦⣀⠙⠁⠀⠀⠀⠀⠀⠀⢈⠻⣿⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠐⢾⣿⡶⠁⠀⠀⠀⠀⠀⠀⠙⠏⠁⠈⠹⠟⠉⠀⠀⠀⠀⠀⠉⠳⠦⠀⠀⠀⠀⠀⠀⠘⢷⣦⡉⠻⠛⠁

⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣴⣶⣷⣶⣦⡀⠀⠀⠀⠀⢀⡤⠂⠈⠀⠒⢀⡀⠀⠀⠀⢀⢰⣶⣿⣷⣶⠄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠠⢁⠇⠀⠀⡴⠚⠉⠑⡀⠀⣰⡇⣾⣿⡿⢫⣴⣾⣷⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣿⣿⣿⣿⣿⣿⠀⠀⣧⡞⠀⠀⢸⠁⠀⠀⠀⡁⠀⠛⣼⣿⣿⠅⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣿⣿⣿⣿⠟⠀⠀⠸⡀⠀⠀⠸⡄⠀⠀⢠⠃⠀⠸⣿⣿⣿⡅⢿⣿⣿⡿⠁
⠀⠀⠀⠁⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠈⠂⠄⠀⠉⠶⠒⠁⠀⠀⠀⠈⠻⠿⢿⡦⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⣀⡠⠄⢀⡀⠀⠀⠀⢀⣿⣧⢀⣀⣀⣀
⠀⠀⠈⢿⢆⠀⠀⠀⠀⡀⠀⠀⠀⠙⣤⡀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠈⢀⠞⠁⠦⡀⠀⠈⢢⠀⠀⣾⣟⣵⣿⣹⢿⣿⣷⣄
⠀⠀⠀⠀⢂⠁⠒⠈⠁⣔⠀⠉⢐⡄⠘⣿⣶⣶⣿⡟⠽⣷⣶⣶⠦⠀⠀⢸⠀⠘⠂⠉⠀⠀⠀⡇⠀⢿⢹⣿⣧⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠢⠀⢀⣀⡁⠤⠀⠀⠀⠀⠘⠿⠿⠿⠷⠽⠟⠛⠛⠉⠀⠘⢌⣄⠀⠀⠀⠀⢀⡔⠀⠀⠘⠯⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁
⣟⣿⣻⣻⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣿⣿⣛⣛⣟⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠋⢵⡉⢥⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠂⠀⠀⠘⠁⠂⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠆⠀⠀⠀⠀⠀⢀⡍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠂⠀⠀⠠⢂⡀⠀⢀⠈⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣂⠀⠀⠀⠀⠀⠩⠁⠰⣦⡄⠀⡀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠚⠿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠂⠃⣤⠁⠓⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠙⣧⢀⢀⡇⡐⠂⠀⢀⣍⣸⠆⠀⡤⠴⡊⠑⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⡈⠛⠀⠘⡭⢇⢈⠀⡇⢰⠆⢉⣴⡟⣀⣠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠆⠀⠀⡄⢀⠤⠈⣅⠄⠠⢶⣊⡍⠜⠡⡏⢠⣖⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠀⠈⣐⠀⠁⠀⠀⠀⠀⠉⠙⠒⠤⢌⣻⣛⠀⠀⠀⠤⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠀⢀⣿⡂⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢶⡈⠄⡀⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄⠒⠋⠙⡆⠉⠉⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠄⠀⠃⠁⣀⠀⠠⠅⠀⠄
⠀⠀⠀⠀⠀⠀⣶⣛⠍⠉⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠐⢠⠌⠑⠌⠡⠆
⠀⠀⠀⠀⠀⡤⠥⠍⡳⠢⠄⠱⠶⣀⡀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⡀⡀⢀⡀⢈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣀⡃⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣀⣑⡐⠿⣄⡁⢁⠚⠃
⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢀⠊⠹⡀⢩⡟⠃⠀⣀⣀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡴⠋⣵⢀⡉⠛⢀⠉⠉⢂⠀⠈⠁⠀⠀⢀
⠀⢠⣁⡀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠙⡓⠻⠺⠄⠈⢀⠀⠀⠀⠀⠀⠀⢀⢤⣚⢽⡓⡇⡌⠀⠀⣀⡄⠀⠀⠀⢀⡀⠳⠀⠶⠈⡐
⠀⠁⠀⠔⠀⠀⠁⠐⠀⠀⠀⠀⠀⣸⢭⠔⠂⠘⠀⠉⠀⠁⠀⠀⠀⣴⠉⠀⠁⠀⢈⣧⠀⢀⠀⠤⢤⡺⡅⠀⡈⣠⠤⠁⠀⡈
⢠⠀⠠⡀⠀⠠⢄⠐⢃⠀⠠⠤⣿⡜⢂⢀⠀⠈⠁⠰⢀⡀⢀⡀⠀⠈⠲⣄⣀⠀⣴⠊⠂⢀⠀⠀⠀⡀⠠⠀⡀⣭⠄
⠀⠀⠀⠁⠰⠐⠦⢀⡒⠓⠀⠲⠀⠀⢉⠈⠀⠊⠀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠉⠀⡆⠁⠥⠀⠉⠀⠀⢣⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠆⠀⠉⠂⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡄⠀⠀⠀⠀⠀⠐⠁⠈⠋⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠋⠉⠉⠑⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠍⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢨⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠻⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⣻⡿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⡌⢁⡙⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⡠⠁⠙⠸⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡄⠀⠀⠀⠀⠈⠀⠀⠀⠨⠙⠠⠖⠉⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣂⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡂⠈⠕⢹⠇
⠀⠀⠀⠀⠀⠀⠀⣧⠈⣧⢻⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠳⣯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢄⠻⣦⠀⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⠀⣤⣤⣤⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠰⢤⠙⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠤⠙⢷⣦⡘⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡩⠻⢶⣤⣤⣀⣀⡛⠈⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣯⡀⠈⠉⠛⠿⣿⣭⣙⣧⣄⠀⠈⠒⠀⢠⡄⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡈⠠⣲⢌⠻⣿⣿⣿⣿⣶⣧⣄⣂⠉⠂⠀⠀⣀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⢡⠑⡙⢿⣿⣿⣿⣿⣷⣿⣿⣷⣲⣦⣀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⢃⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣴⣃⡀⠀⡍⠉⢉⠉⠁⠴⢆⣺⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠓⠒⠒⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⡄⠀⢀⡀⢠⠀⠀⠀⠀⠀⠠⠖⡲⢦⡀⠀⠀⠀⠠⡊⢳⣿⣿⣿⡏⢑
⠀⠀⠀⠀⣤⣷⣿⠇⠀⠀⠀⠀⠀⠈⢃⠀⣾⡿⠟⠀⠀⠀⠀⣈⠂⠜⢠⠃⢽⠀⠀⠀⠀⠙⠺⣿⣿⣿⠗⠁
⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀⠠⣶⣾⡟⢀⣤⡀⠀⠀⠀⠀⠘⢕⣒⣣⡈⠎⠀⠀⠀⠀⠀⠀⢉⣿⠁
⠀⠀⠀⠀⠘⠿⠛⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛

⠀⠀⠀⠀⢀⠀⢀⣤⡄⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣧⡂⠀⠀⠀⠀⠀⢠⣤⡄⣀⡀
⠀⠀⠀⠀⢨⣿⣿⡿⠋⠀⠀⠘⢟⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣰⣿⡣⣹⣦⡄⠀⠀⠀⢻⣿⣧⣿⡿⣦
⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠻⡟⠉⠉⠉⠀⠀⠀⠀⠀⠰⢟⣱⠿⢛⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⡥
⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠞⠋⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⢀⡈⣶⣷⠆⢳⡀⠀⠀⠀⢀⡤⡴⠗⠳⠤⢤⠄⠀⠀⠀⢠⣿⢃⡼⣭⣤⠀⠀⠀⠀⠀⣼⡙⣎⣣⠶⣀
⠀⠀⠘⢇⣈⠉⢰⣶⠀⠀⠀⠀⠀⠙⣷⠀⠀⣴⠃⠀⠀⠀⠀⠰⣟⣶⣽⣷⡝⠀⠀⠀⠀⡼⠡⠗⢉⣤⠶⣉⡄
⠀⠀⠀⠈⠛⠃⠈⠁⠀⠀⠀⠀⠀⠰⠿⠛⠛⠾⠀⠀⠀⠀⠀⠀⠈⠛⠓⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⡿⣶⠄⠀⠀⠀⠀⠀⠘⣿⠇⣴⡿⠃⠀⠀⠀⢐⣴⣿⠟⢻⢹⣿
⠀⠠⠞⠋⡳⠤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⣶⣿⣿⡾⠋⠀⠀⠀⠀⠀⣾⢏⠘⡐⣼⡼⡋
⠀⠀⠀⠀⠁⠢⢈⠗⢦⡄⠀⠀⠀⠀⠛⠛⠟⠃⠀⠀⠀⠀⠀⢻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠿⠦⠾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣿⣿

⠀⠀⠀⣀⣀⣄⢄⡀⠀⠀⠀⠀⠀⠀⠤⠄⠤⠄⠄⠀⠀⠀⠀⠀⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡐⢆⣀
⠀⠀⠰⣿⠸⡏⢸⡇⠀⠀⠀⠀⠀⠿⢂⣠⡖⠁⠀⠀⠀⠀⠀⣤⣎⣽⡻⠦⠀⠀⠀⠀⠀⣘⣿⣴⠻⣮⣛⡀⠀⠀⠀⠀⠘⠿⣃⣵⠟
⠀⠀⢰⣷⣼⣿⣴⡧⠀⠀⠀⠀⢸⢿⡏⠁⠀⠀⠁⠀⠀⠀⠀⠀⠓⢋⣃⠤⠆⠀⠀⠀⠀⠿⠿⠇⠀⠸⠿⠿⠀⠀⠀⠀⢿⢿⣿⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢰⣶⣄⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠤⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⣠⣄⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡄
⠀⠀⠀⠙⢿⣻⠿⠃⠀⠀⠀⠀⢠⣧⡃⡖⢤⠉⠆⠀⠀⠀⠀⣄⢡⣤⢀⡝⠿⠃⠀⠀⠀⣤⡿⣯⡃⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⣫⣷⡀
⠀⠀⠀⠀⠻⢭⠀⠀⠀⠀⠀⠀⠈⣿⠨⠍⡏⢀⡇⠀⠀⠀⠀⠙⢿⣵⠟⠁⠀⠀⠀⠀⠀⠀⢘⣡⣵⣾⢧⡀⠀⠀⠀⠀⠘⡯⣘⣿⣿⡁
⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⢠⣀⡄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢀⢀⣪⡅⠀⠀⠀⠀⠠⣦⣊⣿⢿⡁⠀⠀⠀⠀⠀⢾⡗⣀⣂⠀⠒
⠀⠀⠀⣰⣶⣶⣦⡀⠀⠀⠀⠀⠀⠈⠉⣐⠊⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠆⠀⠀⠀⠀⠐⠾⠞⠛⡛⠓⠀⠀⠀⠀⢰⣿⣿⠋⠑⠚
⠀⠀⠘⠃⠛⠛⠋⠃⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢠⡞⠷⠄⠀⠀⠀⠀⠀⠀⢰⡛⣧⠀⠀⠀⠀⠀⠀⠀⢘⠿⡿⡋⠀⠀⠀⠀⠀⠀⢦⣶⣂⡝⠟⠀⠀⠀⠀⠀⣀⣶⣠⣠⣤⠶
⠀⠀⠠⣶⢌⡛⢳⡆⠀⠀⠀⠀⠀⢠⡾⣿⡿⣆⠀⠀⠀⠀⠀⠀⢁⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣻⡇⢺⠀⠀⠀⠀⠐⢾⡿⣅⢀⣰⣟⢣⣤
⠀⠀⠀⠈⠋⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠦⠼⠀⠀⠀⠀⠀⠀⠈⠱⠋⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⣤⡠⣷⣆⠀⠀⠀⠀⠀⠠⣤⣺⣶⡃⠀⠀⠀⠀⠀⠀⢠⣴⣯⡁⠀⠀⠀⠀⠀⠀⣲⣂⣬⢈⡉⠀⠀⠀⠀⠀⠀⡟⢤⡄
⠀⠀⠀⣘⣛⣥⡷⠋⠀⠀⠀⠀⠀⠀⠤⠟⠓⣆⠀⠀⠀⠀⠀⠘⠇⡝⢷⢀⡀⠀⠀⠀⠀⠰⣿⡇⠠⠀⠄⠀⠀⠀⠀⠀⢸⠑⢲⠁⡀
⠀⠀⠀⠿⠻⠿⠀⠀⠀⠀⠀⠀⠸⠿⠦⠀⠀⠰⠖⠀⠀⠀⠀⠀⠀⢁⠄⠈⠀⠀⠀⠀⠀⠀⠿⠇⠂⠀⠁⠀⠀⠀⠀⠀⠁⠉⠟⢸⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢸⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠘⡇⠀⠀⠘⠀⠀⠀⡟⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⣀⡴⠖⠛⠛⠛⠓⠶⣄⡀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⡀⠀⣠⠞⠁⣀⡀⣰⡖⣶⡀⣀⡀⠙⢦⡀⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⠏⠀⠸⣏⠙⣩⠤⢬⡙⢉⡿⠀⠈⢷⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⡀⣿⠀⢰⠞⠃⡞⠁⠀⠀⠙⡆⠓⢦⠀⢸⡇⣠⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⠳⣆⠳⡀⠀⠀⡰⢃⡶⠋⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠘⣧⠀⠸⣧⡴⢬⠉⢩⡴⣤⡿⠀⢠⡟⠰⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣈⢳⡀⠀⠀⠘⠓⠛⠀⠀⠀⣴⢋⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢹⡄⠀⠀⠀⠀⠀⠀⣼⠁⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣛⣛⣛⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠤⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣛⣙⣛⣋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠐⢀⣠⡄⠀⢀⠤⣤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣼⣿⣦⣄⣀⣼⢠⠁⣠⠛⠒⠪⣕⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠸⣿⣿⣇⣿⣿⣿⣿⣿⣿⠀⠀⡆⠀⠀⢸⠹⡌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⠐⣾⣿⣿⡯⠿⣿⣿⣿⣿⡟⠀⢤⠃⠀⠀⠚⢴⠃⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢰⣿⣿⣿⠀⠀⣿⣿⣿⣿⣧⠤⠼⠧⠀⠀⠀⢸⠀⠈⠹⡾⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠳⠧⣀⣸⣿⣿⣿⠀⠀⣯⣿⡿⠛⡇⠀⠀⠀⠔⠚⠋⠉⠓⢄⠀⢷⢸⣇
⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⠄⠠⠬⣍⣛⡟⠀⠀⣿⣿⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠘⢹⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠁⠙⠊⡌⠣⡀⢀⣿⣿⡇⢸⡷⠖⠒⠒⠮⠀⠀⡀⢀⡦⠴⠃⣰⠞⢳
⠀⠀⠀⠀⠀⠀⢸⣯⢁⠎⠀⠐⠄⠀⠀⠀⢘⡞⠛⢻⡇⢸⡇⠀⠀⠀⠀⠀⢀⡍⠉⠁⠀⠀⠻⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠌⠙⢀⠔⠈⠀⠀⠀⠘⣷⠀⠈⡇⢸⡇⠀⠀⠀⢰⡴⠊⠀⠀⠴⠒⠢⡄⠀⢼⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢄⠤⠐⠀⠀⠀⠹⡄⠀⠃⢸⣷⠶⠶⠶⣏⠀⠀⠀⠀⢀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⣽⠀⠀⠀⠀⣤⣤⡇⢀⠄⠀⠁⠀⠀⢸⡇⠀⠀⢀⠈⠙⠒⠒⠒⠋⠀⣶⠋⣸⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⣷⣷⡇⠈⠒⠀⢈⠀⠀⢸⡇⠀⠀⣸⠷⠤⠤⠾⠦⠤⠤⠻⣦⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠧⣍⡇⠀⠄⠀⢸⣴⣷⣼⡇⠀⣰⠏⠀⢰⣄⣠⠖⠒⢦⡀⢘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢝⣦⢬⡑⣴⠁⣠⠂⢀⡏⢻⣿⣿⣧⠾⠯⡤⠔⠋⠈⣷⠀⠀⢀⣷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡙⣧⡰⢋⣴⣾⣷⣿⣿⣿⡇⠀⠀⢤⣄⣴⠾⠋⢀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡿⣿⣿⣿⣿⠏⡷⠶⠛⡋⠉⢀⣆⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠋⠀⠘⠦⠶⠿⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣌⢻⣿⣿⣿⣿⣿⣄⣠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠋⠀⠻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣰⣿⣧⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠁⠀⠀⠰⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⢀⣼⣿⣿⣿⣁⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢤⣤⣤⣤⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⡿⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⡛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⣀⣀⣈⣛⣩⡔⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣤⡴⢏⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠀⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣶⣿⣆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⢻⡟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣆⢠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣄⣀⠀⠸⣀⣷⠀⡆⠀⡈
⠀⡀⠐⢦⣀⠈⠻⣄⢋⡏⠀⡇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠸⣆⠀⠉⢆⠀⢹⠌⡗⠀⡶⣡⠚⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣧⠶⠰⢧⡷⣿⠞⢼⣱
⠀⠀⠈⠐⠄⢸⣇⢸⣆⡇⠀⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠈⠋
⠀⠀⠀⠀⠈⢦⠹⡄⢛⠇⡊⡼⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠣⢳⣼⣼⠞⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⢤⡀⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠶⣤⠦⡀⠉⢦⣼⠀⣠⣿⣁⡤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢆⡀⠈⢷⣍⡳⣬⢙⣛⠋⣁⣀⣤⡴⡶⢶
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⣄⣹⣧⣜⢷⢤⣉⣉⣉⠉⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠒⠀⠀⠄⠀⠈

⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠔⠁⠀⢀⠀⠉⢢⡀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⠔⣅⠀⠀⠀⡨⠢⡀⠀⠀⢀⣤⣻⣿⣿⣿⢟⣤⡀
⠀⢸⠀⠀⠀⠀⠀⠠⠄⡃⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡌⠀⢸⡄⠀⢠⡇⠀⠁⠀⠀⢸⣿⣧⣿⣿⣿⣼⣿⡇
⠀⠸⡀⢀⠀⠂⠀⠀⣠⠃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠰⡀⡸⠁⠀⠀⢇⢀⠆⠀⠀⠸⣿⢏⣿⣿⣿⡹⣿⠇
⠀⠀⠈⠲⠤⠠⠤⠞⠁⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⠣⠄⠀⠠⠜⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠗⠁

⠀⠀⢀⠄⠒⠀⠒⠠⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⡀⠀⠀⠀⠀⠠⢔⣂⣀⠐⠢⢄⠀⠀⠀⠀⣠⡴⠶⠿⣶⣦⣄
⠀⢰⠁⠀⠀⠀⠀⠀⠽⠆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣻⡄⠀⠀⣐⡝⠁⠀⠀⠙⡄⠈⠂⠀⠀⣰⢫⣾⣿⣿⣮⢻⣿⡆
⠀⢀⠀⠀⠀⠀⠀⠀⡅⡂⠀⠀⢸⣿⣿⣿⣿⣿⣿⢾⡇⠀⠀⠌⡇⠀⠀⠀⠀⠸⣄⠘⠀⠀⢿⢸⣿⣿⣿⣿⣧⢻⡿
⠀⠈⠢⡀⠀⠀⠀⣀⠜⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⢜⣦⡀⠀⠀⠀⡨⠁⠀⠀⠈⢷⡻⣿⣿⣿⣿⡷
⠀⠀⠀⠈⠁⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠂⠂⠁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠋⠁
⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⢀⢔⣿⣷⡆⠀⠉⠦⡀⠀⠀⢀⣴⠋⠉⢽⣿⣷⣤⡀⠀⠀⢀⠎⠁⠀⠐⠆⣌⠱⡀⠀⠀⢀⣾⣿⣿⣯⣹⠻⣷⡀
⠀⠈⠀⠙⠉⠀⠠⣴⣆⠁⠀⠀⢸⣿⣶⣾⣿⣿⠋⠹⡇⠀⠀⡂⠀⠀⢀⣄⠀⠀⠀⢠⠀⠀⣾⣿⣿⣿⠻⣿⣿⣿⣧
⠀⠸⣤⣶⣄⠀⠘⠻⢋⠇⠀⠀⠸⡟⠙⠿⣿⣯⣄⣴⠇⠀⠀⠡⡀⠀⠀⠉⠀⠀⢀⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠈⠛⠏⠤⠤⠖⠁⠀⠀⠀⠀⠈⠲⠾⠿⠛⠋⠁⠀⠀⠀⠀⠑⠢⠀⣀⠠⠔⠂⠀⠀⠀⠀⠘⠻⢿⣿⡿⠟⠋

⠀⠀⠀⠀⣀⠄⢤⡄⠤⠀⠀⠀⠀⠀⠀⣀⣤⡀⢤⣤⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⢠⠊⠀⢀⣤⠙⢶⠀⠀⠀⠀⢠⣾⣿⡿⢛⣷⡙⠀⠀⠀⠀⠀⡠⠊⠀⢠⠂⢘⠀⠀⠀⠀⠀⣠⣾⣿⣟⣿⣿
⠀⠀⣇⠀⣴⠟⠁⠀⡔⠀⠀⠀⠀⠻⣿⠋⣡⣾⣿⡟⠀⠀⠀⠀⠎⠀⣀⠔⠁⡠⠊⠀⠀⠀⠀⣼⣿⢿⣿⣿⡿⠃
⠀⠀⡉⢳⣄⣀⠤⠊⠀⠀⠀⠀⠈⣷⣝⠻⣿⠿⠋⠀⠀⠀⠀⠀⠳⣈⢀⡠⠊⠁⠀⠀⠀⠀⠀⠹⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡾⢶⣶⣶⣶⣶⣶⣶⣾⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠋⠉⠉⠉⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣿⣹⣿⣿⣿⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣾⣶⣶⣶⣶⣶⣶⣾⡶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⣿⡖⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣿⣮⣿⣯⡟⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢟⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡿⢿⣿⣿⡿⣟⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣾⣿⣿⣿⣿⣿⡿⣿⣿⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠙⠋⠛⠉⠙⠛⠛⠋⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠷⠿⠧⠿⠿⠷




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠉⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣿⣷⣄⠀⢠⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣾⣿⣿⣿⣶⣿⡷⣿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡷⣶⣽⣿⣿⣵⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣟⣿⢿⡿⣿⣿⣽⢿⡿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣽⠏⠻⣿⢿⣿⣟⣿⣿⡾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠾⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⠏⠀⢀⣴⠟⠁⠙⠿⠏⠁⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠋⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡁⠀⠀⠀⠀⠀⠀⢀⣴⣴⣶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⢦⣤⠀⢰⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⣿⣷⣜⣧⣾⣵⠀⢠⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣤⣶⣾⣿⣿⢿⠟⠛⠷⣤⣿⣶⣦⣦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢉⣾⢻⢏⡇⠞⠈⡠⡀⠀⠆⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠸⣿⠛⠘⠀⣴⠀⠀⡀⣤⣾⣤⣤⣤⣤⡔
⠀⠀⠀⠀⠀⢀⣤⣲⣖⣂⡖⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⠈⠀⠀⢿⡷⠖⠋⠉⠅
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⢠⣶⡀⢀⡀⢰⣿⣀⣤⣤⣤⡾⠁
⠀⠀⢸⣿⡿⢿⣿⣿⣿⡇⡇⠈⣄⣀⡀⠀⠀⠀⠀⣠⣼⠏⠀⠀⢀⣴⣿⣿⢿⠿⠟⡩⣥⠀⠀⠀⠀⠘⡄
⠀⠀⢘⠛⠀⠈⠈⢠⢸⡇⡇⠀⢻⣀⣀⡀⢠⣴⣿⣿⠋⠀⠀⠀⠸⣿⣅⣿⢰⣄⡀⣠⣿⣤⣤⣤⣴⣶⡇
⠀⠀⣼⣀⣧⣧⣼⣾⣿⣇⢳⠀⠸⣿⣛⣿⣭⣽⣿⠇⠀⠀⠀⠀⠀⠙⢿⣿⣼⣟⠙⠉⠉⠉⠉⠉⠉⠉⢱
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠸⡀⠀⢻⣿⠿⠿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡇⢧⠀⠘⡇⠀⠀⢸⡃⠀⠀⠀⠀⠀⢠⣶⡄⣿⠀⣿⢘⣿⡄⠀⠀⠀⠀⢤
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡸⣄⠀⠙⠀⠀⢺⣵⣄⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣾⠿⠁⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣷⣻⠀⠀⠀⠀⣄⣿⣿⣿⣶⡄⠀⠀⠀⠀⠻⠉⠿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠆
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣦⣤⣤⣤⣄⡀⣀⣀⠀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁


⣾⣷⠿⠿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣤⢴⡤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣊⡥⢚⡟⠣⣜⣾⣧⣆⡌⠙⢿⣟⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢡⡜⠁⣴⠏⠬⢝⣾⠟⡻⢿⣿⠊⠀⢹⣧⠙⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢰⡟⠀⢼⡟⠀⠙⠛⠳⣜⣧⣾⠟⠁⠀⠀⣿⣧⠈⢧⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡇⠀⣿⡇⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀⢸⣿⠅⢸⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢸⣧⠀⢹⣧⣀⣀⡤⠲⠇⣸⡖⢤⣄⡀⠀⢸⣿⡅⢸⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⣆⣠⣻⠟⠉⠉⠀⠀⠀⠀⠘⠻⢿⣷⣾⣿⠃⣸⢗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⣇⢻⡶⣶⣤⡀⠀⠀⡴⠶⠶⠆⠀⢹⡟⢻⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡀⣀⠀⠒⠒⣒⢂⡀⠂⢐⠒⠀⠀⠀⢸⣶⠀⡟⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⡒⠒⠲⢯⣼⡧⢿⣄⠀⠀⠀⢼⠁⢀⠀⡁⠀⠀⠀⣸⠏⣼⣧⡼⠒⠛⢛⣳⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠛⠛⠻⢿⣶⣦⣄⠙⠐⢿⡏⠛⠞⣡⣦⣀⣼⣾⠋⠲⠴⠃⣿⠟⡸⣣⣴⠾⠛⠉⠉⠉⠉⣛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣤⣴⣶⣶⣤⣄⡈⠙⠻⣿⣦⣼⣿⣆⠓⠛⣋⣭⣙⠛⠷⠤⠀⣰⡟⣿⣷⠟⢁⣀⣠⣶⣿⣷⣿⡿⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠛⠛⠛⠿⢿⣷⣤⣈⠙⢿⣙⣿⣷⣄⣀⣀⣀⣀⣠⣤⣾⠟⢰⣿⢃⣴⣿⣿⠟⠋⠁⠀⠀⠀⢀⣈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣡⠎⠀⠀⠀⠀⠀⠀⠀⠉⠛⢛⡆⢌⣻⡿⣿⣿⣿⣿⣿⠿⠛⠁⢰⣾⣧⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣸⢱⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣴⡿⠟⠚⣭⣭⣝⣓⣲⡶⠶⠿⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣰⢣⣿⠇⠀⠀⠀⠀⠀⢠⡔⠋⠁⢀⠀⡀⠈⢈⡹⠆⠀⠈⠉⠻⢿⣿⣌⣀⣀⣀⠀⠈⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⢠⠏⣿⡏⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠉⠉⠉⠛⠩⡀⠀⠀⠀⠀⠀⠀⠦⣐⠎⠉⠉⠀⠀⠀⠀⠹⡄⣦⠀⠀⠀⠀⠀⠀⠙⠺⣿⣇
⠀⠀⠀⠀⣠⠏⣼⠟⠂⠀⢀⣀⣐⣶⡿⠀⢀⣀⠀⠉⢙⠲⢆⠁⠀⣀⣤⣤⣤⣤⡀⠀⠘⠂⠶⠓⠚⠉⠉⠀⠀⢧⠸⣆⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⣰⠏⠘⣉⣀⣀⣀⣀⣤⣭⠿⠃⠀⠀⡀⠉⠑⢶⠊⠉⣠⠊⠁⢾⣟⣿⠸⠙⢳⡌⠑⠲⢤⡴⠖⠚⠂⠀⢺⡀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⣰⠃⠰⢚⣿⢿⢿⡿⢧⠏⠀⢀⣀⠀⠀⣹⣿⢂⡒⠃⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠢⢭⣀⣠⣤⡆⠀⠀⠓⠾⢿⣦⣀⣀⡀⠠⣄⣀⡀⠘⣧
⠀⣼⠃⠀⡴⠋⢠⠏⠀⠀⣿⣤⡾⣫⣵⠞⠛⠉⣿⡜⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡀⠀⠀⠀⠀⠈⣷⢺⡿⣟⣿⠿⠿⠿⡟⠻⢷⡀
⢰⡏⢀⡞⠁⢠⡏⠀⠀⢸⣿⣵⣾⣿⡿⠅⡀⠀⠹⣷⡘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠙⢿⣤⣞⡷⢶⣆⣿⡆⠀⠀⠸⣇⠀⠀⠀⠀⠀⢹
⣿⠁⠈⠀⠀⣿⠀⠀⠀⠘⣿⡿⣿⣿⣟⣯⠴⠂⠀⠙⣷⣌⡻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⣷⣾⣿⣿⣷⣮⣽⣧⠀⠀⠀⢿
⣿⡄⠀⠀⠀⣿⠀⠀⠀⠀⠉⣸⣿⣿⣿⣶⣞⡥⢂⡀⢀⣿⠉⠒⠮⠥⢤⣤⣤⡤⠤⠶⠚⠋⣻⣄⠠⣐⢬⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀
⠘⢿⣄⡀⠀⠹⡄⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⡟⠙⠢⢴⣤⣀⣰⣀⣠⣤⡴⠶⠚⠙⠻⠷⣮⣿⣿⣿⣿⣿⣿⠹⠿⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼
⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⡏⢸⡇⠀⠠⣀⣈⣁⣉⣀⠀⣀⠤⠂⠀⢈⣦⡘⣟⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠛⠒⠂⠐⠶⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠖⠋⠑⠦⢤⣀⣰⠉⠉⠹⣄⣀⣠⡤⠞⠋⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣦⣤⣞⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣷⣾⣶⣶⣷⡞⠛⠃⠀⠀⠀⠀⠀⠁⡂⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣞⠀⣀⣶⣄⠃
⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠘⠀⠀⠃⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡸⢀⠸⡹⠃⠈⠻⣿⠁⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣍⡉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠄⢹⠁⠏⠀⡼⠀⠀⠀⠀⢻⣄⣬⠷⠤⢤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡄⠀⠼⣷⠀⣾⠁⠀⠀⠀⠈⠈⣿⡁⠀⢀⠀⣿
⠀⠀⠀⣀⠀⠀⠀⣠⠄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⣀⣀⠘⢀⠀⠀⠀⢲⠘⣇⠀⠀⢿⢳⡇⠀⠀⠀⠀⠀⢠⠚⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠈⠢⠀⠀⠀⠀⠀⡄⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⣆⣹⠀⠀⢸⡀⡇⠀⠀⠀⠀⠠⠀⢸⠃⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠠⠀⠅⢠⠀⠀⠀⣄⠳⣄⠈⠣⡙⢆⠀⠀⠈⢁⡰⠋⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠈⠓⠦⣌⡆⠀⠀⢀⡀⠀⠀⠀⠰⣄⠀⣠⠖⠁⠀⠀⢀⠀⠀⣄⠀⠂⢰⠀⠆⠀⠈⠂⠭⢿⣦⣿⣦⣶⣰⣷⣿⣤⣤⣐⣚⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠛⠷⠶⠾⠖⠊⠁⠀⠀⣎⠀⠀⠉⡀⠒⠀⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢇⠈⠁⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠘⡀⡼⡄⠀⠀⠀⠀⠀⡇⣶⣼⡟⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢀⠀⠠⠀⠀⠀⠀⠁⢄⣀⠤⢈⢠⠃⠀⠀⠀⠀⢰⡿⣿⣿⡇⠀⠘
⠀⠀⠀⠀⠀⠀⣀⡄⠛⠛⢡⡳⣄⠀⠀⠀⠐⠀⠀⠠⢈⢀⡴⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢄⠀⠀⠀⠀⠀⠈⠉⠙⠁
⠀⠀⠀⠀⠀⡾⠃⠀⣀⡴⢖⡿⠋⠛⠦⢥⣤⣤⣠⣴⠗⠊⠀⠈⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠘⠛⠐⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠚⠐⠊⠉⠑⡀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠀⣤⡀
⠀⠀⠀⠀⠀⣀⠀⢀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣀⣀⣀⣠⣠⣿⠏⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣆⠀⠀⢀⡶⠀⢀⡀⣸⠃
⠀⠀⠀⠀⠀⠀⠐⠆⠼⣷⡀⠀⠀⠀⣠⠰⡤⠛⠋⢠⣿⣿⡊⣏⠁⠀⠀⠀⠀⠀⠀⠀⠳⣦⡀⠀⢀⠀⠀⣠⡴⠚⠁⠀⠀⠘⠦⣉⠉⠉⠷⣾⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠓⠚⠀⠀⠤⠒⠛⠛⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡛⣋⡙⢛⡟⣫⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⡔⠉⡍⠁⠀⠑⡄⠀⠀⠀⡠⠚⠡⠊⠀⠉⢢⡀⠀⠀⠀⡴⠋⠀⢀⠀⡙⢆⠀⠀⠀⡠⠊⠁⣀⣀⠈⠳⡀
⠀⢸⠲⠀⠀⠐⠐⠐⢽⠀⠀⠀⡇⠈⠦⠀⠀⠐⠀⡇⠀⠀⢸⠀⠀⠀⠈⠠⠄⢸⠀⠀⠀⡃⢰⠋⣶⡆⢱⠀⠁
⠀⠘⣄⠅⠀⠣⡀⣀⡜⠀⠀⠀⠳⣁⡆⠀⠀⠀⣰⠃⠀⠀⠘⣄⠀⠀⠀⠀⢀⡜⠀⠀⠀⢣⡈⠢⠽⠥⠊⣠⠃
⠀⠀⠈⠑⠒⠒⠛⠁⠀⠀⠀⠀⠀⠈⠉⠓⠒⠉⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁

⠀⠀⡠⢚⡽⠋⠉⠲⡄⠀⠀⠀⢀⣴⠉⠉⠉⠑⣄⠀⠀⠀⠀⣠⠖⣩⡙⢉⡲⡀⠀⠀⠀⣀⡠⠤⠤⠤⢤⣀
⠀⢰⢱⠃⠀⠀⠀⠀⣸⠀⠀⠀⡏⢸⠀⠀⠀⠀⡇⡇⠀⠀⢸⠡⠈⢄⠈⠀⠐⢻⠀⣠⢚⠁⠀⠀⠀⠀⠀⢸⡓
⠀⠸⡈⢆⡀⠀⣠⠞⡸⠀⠀⠀⢧⠈⠲⠤⠤⠞⢠⠃⠀⠀⠸⡔⢢⡀⠡⠀⠤⡻⠀⠘⢿⣍⠉⠀⠀⠉⣉⡽⠟
⠀⠀⠙⠢⠬⠭⠕⠊⠀⠀⠀⠀⠀⠑⠦⠤⠤⠒⠁⠀⠀⠀⠀⠈⠳⠤⠦⠥⠊⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⢀⠤⡶⢲⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢤⢒⢢⠛⡿⠾⡆⠀⠀⠀⠀⣠⠴⡶⠖⡤⣀
⠀⢠⠃⠐⠀⠀⠀⠀⢳⠀⠀⢠⡖⠉⠁⠀⠀⠉⠑⢦⠀⠀⠀⢸⣿⢸⢰⢻⣸⠁⠀⠀⠀⡜⢡⠛⢊⡅⠂⡏⡆
⠀⠸⡀⠀⠀⠀⠀⠀⡿⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢻⣾⣾⡾⠃⠀⠀⠀⠀⣗⠰⠐⡆⠐⣩⠒⡇
⠀⠀⠑⠤⣄⣀⡤⠔⠁⠀⠀⠀⠈⠉⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⡿⠿⡇⠀⠀⠀⠀⠀⠘⠲⣥⣔⣷⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⡠⠤⠤⠤⣀⠀⠀⠀⠀⠀⢀⠤⠄⠒⡢⢄⡀⠀⠀⠀⠀⠀⢰⠉⡆⠀⠀⠀⠀⠀⠀⠈⢣⣈⣯⡰⠋
⠀⢠⠊⠀⠀⠀⠀⠈⢳⠀⠀⠐⢭⣁⣠⠀⣟⣭⠆⢱⠀⠀⠀⠀⠀⢸⠉⡇⠀⠀⠀⠀⠀⠀⡠⠚⠛⠙⠣⣄
⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⣲⣷⣖⣧⣜⣋⡇⢸⠀⠀⠀⠀⢀⣎⣀⣧⠀⠀⠀⠀⠀⡜⡐⢡⡀⠀⠈⡘⡄
⠀⠈⠣⣄⡀⢀⣀⠴⠃⠀⠀⢸⣞⣻⢽⣿⣁⠤⣤⠎⠀⠀⠀⠀⢸⠦⠢⢾⠀⠀⠀⠀⠀⢡⡀⠀⠀⢀⠠⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠃⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⣭⣭⠿⠀⠀⠀⢠⠈⠡⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠠⢜⠀⠀⠶⠖⣿⠍⠡⠤⠤⢵⠀⢚⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣉⣿⣉⠘⣛⣛⣽⡇⠀⢒⣚⡿⠀⠥⠵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢤⣷⠀⢈⣁⣀⢽⡇⠈⣉⣭⣿⡀⢒⣲⡧⠀⠀⡀
⠀⠀⠀⠀⢀⠀⡀⡤⢄⡐⢒⡲⠒⠚⠓⠒⠛⠂⠀⠂⠤⢾⡃⠀⠶⣺⣯⠄⣉⡥⣿⢉⣽⡇
⠀⠀⠀⠀⠘⠙⠓⠻⢶⠀⠈⠀⠀⠀⠀⣀⣠⠴⠖⠒⠈⢉⠉⠈⢻⣽⡻⡤⠤⣶⣷⢴⣿⣧⡆⢀⣤⣤
⠀⠀⠀⠀⢀⡀⠀⢈⢹⠀⠀⠀⠀⠀⠀⡿⠿⠷⢤⣬⣦⡶⣰⣭⣿⣶⣒⣖⣒⣲⣾⣽⠿⠿⣯⣽⣿⣿⠷⠂
⠀⠀⠀⠀⠈⠙⠊⠛⠻⢦⣀⠀⠀⠀⠀⣏⠂⠀⠀⠀⠀⡏⠁⠉⠈⠉⠩⣏⠉⠉⠾⣁⣐⣐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠤⠹⠷⢶⣦⣄⣀⣇⣠⣤⣶⣾⠽⠏⠂⠐⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣿⣟⠛⢻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠟⡡⣰⠷⢰⡘⣿⣟⡻⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⣿⡳⣣⡔⠀⠛⠁⠀⠈⠙⢷⣡⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣸⣿⠒⣢⠀⠀⠀⠀⣴⠒⠻⡷⣉⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈⣫⣿⡧⡟⡇⠁⠈⠁⠀⠀⠀⠁⠈⢀⠹⣶⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠄⢠⣿⣿⣿⡇⠷⣃⠀⠀⠀⠀⠀⠀⠀⢀⢮⠇⣽⣗⠃⠀⠽⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⢂⣰⠻⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣫⣁⡀⠀⠀⠀⠀⠈
⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣥⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠙⠿⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⠟⢀⣀⣀
⠀⠀⠀⠀⠀⢄⣠⣿⣿⣿⣷⣶⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣠⣾⣿⣿⣇⠀⠤
⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀
⣷⣄⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠠⠰⠀⠈⠄⢡⠀⠀⠀⠀⢉⡀⠀⠀⢀⠈⢠⠂⡀⠃
⠀⠀⠀⠀⢀⠀⠀⠀⣴⠒⣦⠀⡠⠄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠢⠀⢰⠀⢆⠀⠀⠀⠠⡄⢀⠄⠀⠀⠘⠎
⠀⠀⠀⣀⠙⠛⢷⣄⠸⣞⣿⠂⣗⢠⡎⢀⠀⠀⠀⠀⠀⠀⡤⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠃⠈⡄⠘⠀⠀⠀⠀⢣⠈⠀⠀⠠⠀⠃
⠀⢰⣅⠈⠳⣆⠀⢻⣶⢸⣇⠀⣃⠟⢠⣇⠀⠀⢀⠀⢀⡞⠀⢘⡄⠹⡏⢣⠀⡀⠀⠀⠀⡽⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⣠
⠀⠀⠻⣄⠀⠈⣦⠀⣽⡃⡇⠀⣟⣴⠋⠠⠃⢠⢡⣿⣾⢁⠀⣸⠃⠀⣧⠀⣦⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠙⣄⢸⡆⢹⣿⡏⢰⢱⡟⠀⠀⠀⢰⡂⢻⣿⠋⠑⣿⠃⠀⢻⡄⣿⢡⠇⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⢦⠻⡄⣽⢃⣅⡾⠁⠀⠀⠀⠀⠳⠈⢿⣿⡄⢻⣆⣡⡿⣴⢇⠜⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠠⠄⠀⣤⡀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠣⢷⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠙⠴⢿⠿⡿⠋⠎⡿⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⢽⡆⠀⢳⠂⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠘⠈⠀⠀⠀⠀⠀⠀⡴⠶⠶⣦⡷⢤⡀⠸⢦⣾⡆⢸⣾⡷⠶⠛⠀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠆⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⢳⣾⠙⠷⣬⡭⣯⣋⣁⣠⣤⣶⣾⠛⠋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣄⡰⠚⢿⣤⣼⣻⣮⣍⡃⣈⣋⡀⠁⠚⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠄⠠⠈⠽⠋⠉⠛⠟⡀⢁⢈⠉⠉⠉⠁

⠀⠐⠒⠛⠛⠙⠀⠀⠀⠐⠦⠤⣠⡤⠤⠶⠶⠒⠒⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠛⠉⠑⠀⠒⠂⠤⠀⣀⠀⠒⠀⢄⠀⠤⠀⠀⠈
⠀⢀⣠⣦⣤⢦⡤⠠⣤⣀⣀⣤⣤⣴⠶⠶⠶⢦⣤⣤⠄⠄⠠⠤
⠀⢸⠥⣤⢤⡥⣤⣤⣀⣀⠈⠉⠉⠦⠀⠀⠤⠀⠤⠴⠆⠒⠂⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⢈⡑⠀⠠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠒⠒⠤⠄⣀⣠⣀⣤⣰⣆⡠⠄⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡠⠤⣠⡠⡄⣤⠀⡀⢤⡠⢄
⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠⣀⣠⣠⣄⣀⢀⡀⠀⠈⢉⠙⠉⠉⠉⠉⠒⠂⣼⣄⠘⠁⢉⣤⣰⣰⠡⣖⢶⡧⣯⠿⠰⣦⢽⣷⣿⣘⣃⣿⣧⣛⠳⢾⣾⠇
⠀⣠⡤⠀⣒⣈⠭⠟⠟⠋⠉⠉⠉⠈⠉⠉⡙⠛⠷⢶⣾⣴⣴⣒⠠⠤⠤⡒⠀⠀⠀⠀⠀⠉⠉⠙⠷⣿⣷⣾⣄⣟⣿⣿⠿⠞⢋⡋⠭⠤⠐⡓⠥⠠
⠀⣶⠾⠟⠉⠀⠀⠀⠀⠤⠀⠀⠠⢠⡆⠀⠁⠀⠀⠀⠀⡈⠉⠙⠓⢖⠤⡹⠿⠾⣁⢠⣏⠀⠠⢄⣴⣾⠿⠛⢉⢉⡤⠔⠒⣉⡥⠴⢚⣉⡥⠄⠚⣫⠄
⠀⣸⣴⣔⣦⣰⣦⣦⢦⣖⣶⣭⣷⠿⣗⣶⣤⣀⠨⠀⡴⣛⣦⣀⣀⣀⡀⠀⠴⢤⡟⣾⣯⡀⠚⠋⢁⠠⠄⠊⠉⣀⠤⠖⠊⣠⠤⢚⣉⠤⠒⣈⣠⠴⠆
⠀⣙⣷⣽⣯⢹⣽⣽⡆⠤⣿⣮⣭⠽⠹⠶⠶⠷⠶⠶⠶⠿⠎⠈⠡⠭⢴⢆⣲⣺⣷⣿⡿⠿⢠⣊⡴⢒⣡⠔⠈⣁⡴⠚⢃⡤⠚⣋⡤⠖⢋⣡⡤⢞⡀
⠀⣿⣻⣿⣿⡿⣿⠾⠟⠋⠉⡀⠀⠀⠀⠀⠂⠀⠀⡀⠠⣐⠢⣐⠀⢀⠢⠤⡈⠭⣉⡉⠙⢃⠠⢉⢀⠉⠓⠐⠛⠥⣴⣞⣉⣤⢛⣥⠤⢚⣩⡴⠖⢋⡁
⠀⠽⠛⠉⠁⠀⠁⠀⠀⠄⠀⠐⡀⠀⠀⢀⠐⢀⠳⡌⠲⢌⡑⠮⡙⠦⣁⠢⣌⡙⠦⣍⡓⠤⣙⡒⢤⣀⠢⠌⠒⠆⣄⡢⠉⠙⠛⠻⠟⣯⣀⢴⣺⡭
⠀⠀⠀⠄⠀⢠⡔⠒⠀⠄⠀⠁⡋⠀⢣⠈⢦⡈⠠⣌⠳⣄⠙⠦⣌⠳⠌⠓⢦⡝⠲⣤⡉⠲⢌⣙⠳⣬⣙⠲⢌⡐⠢⢍⡓⠶⢬⡲⠥⣂⠀⠍⣙⣂⠄
⠀⢰⣏⠌⣰⣿⣷⡄⠀⠀⢂⢠⠉⠄⠈⡆⠀⠷⠀⠻⣕⠬⠳⣄⠈⠳⡀⠛⢤⣙⠲⣄⠙⠶⣄⠈⢓⠦⠋⠙⠢⣌⡓⠦⣄⠐⠲⢍⡳⢦⣕⡲⢭⣛⠅
⠀⢪⣋⣀⣿⣿⣟⣿⣷⣈⣿⣇⡙⢀⠀⠙⢀⡖⢠⢶⣨⣤⣀⡘⢧⣀⠙⢦⣀⢹⣦⣈⡙⢦⡊⠑⠮⠁⢀⣤⡞⡆⢁⡀⠈⠙⠦⡄⢉⠓⢬⣙⡶⣮⡁
⠀⣰⣧⡈⠹⣿⣿⣿⡿⢿⣿⢿⣿⣿⣷⣤⣾⣿⣾⣦⣿⣷⣿⣿⠶⠦⠒⠀⠀⠀⠀⠀⠉⠑⣉⣀⠀⣰⣿⣿⣿⣿⣌⢿⣯⣷⣺⣷⣄⢀⣤⡈⠓⡀⠁
⠀⣙⣻⡟⣃⣤⣤⡾⠶⠚⠛⠛⣉⣅⣀⡀⠀⢀⠀⠀⠈⠉⠉⠒⠦⠤⣉⡒⠠⠀⠀⠀⠤⣀⠠⠩⣛⢿⣿⡿⠟⠛⠛⠛⠛⠿⠿⠿⠿⢯⣿⣿⣹⣧⡀
⠀⣽⣿⣿⣿⣭⣦⣶⣿⣷⣾⣿⣿⣿⣾⣯⣮⣷⣥⣵⣿⣭⣷⣾⣶⣶⠀⠉⠓⠲⣤⡀⠑⠂⠍⡒⢯⡻⣿⠿⡷⢕⠀⠙⠹⣿⣭⣾⣶⣶⣴⣶⣭⣭⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⢀⡤⠴⠶⠦⠤⢤⣖⠢⣤⣀⡀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠉⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣾⣶⣾⣾⣷⡀⠀⠀⠀⠀⠀⣠⠴⠶⡿⠷⠦⣄⡀
⠀⠀⠐⠦⠀⠀⠀⠈⢳⣀⣈⣉⢀⠞⠀⠀⠀⣀⠴⠚⠁⠀⠈⠉⠉⠉⠉⠀⠙⠢⣄⠀⠀⠀⠈⢦⣀⣀⣀⣀⡜⠁⠀⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⣀⠼⢿⠿⢿⠿⣄⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠿⢿⡿⠿⠿⢷⡀
⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⢀⡤⠶⠧⢤⠀⠀⠀⠀⠀⠀⠈⢷⠁⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⡴⠁⠀⠀⠀⣠⠤⣄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣠⠦⡄⠀⠀⠀⠈⢢⡀
⠀⠀⢸⠁⠀⠀⠀⠀⢧⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢆⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⣅⣀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⣇⢀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⡏
⠀⠀⠘⡄⠀⠀⠀⠀⠈⠛⠉⠀⠀⠘⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠉⠛⠁⠀⠀⠀⢀⡜⠁
⠀⠀⠩⢞⡦⢄⣀⣀⣀⣀⣀⣀⣀⣠⣼⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣴⡯⣀⡀⠀⠀⠀⠀⢀⣠⡔⣋⣀⣀
⠀⠀⠀⠀⠀⠞⠁⠼⠋⡽⢋⠜⢋⠔⢁⠔⢁⠞⣩⢟⡿⢛⡿⠋⣹⠟⣹⠋⡹⢋⡽⠁⠎⠰⠋⠞⠙⢫⠟⡽⠁⠞⠙⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠀⠈⠀⠀⠁⠀⠁⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠉⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢆⠺⣷⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣴⠒⠒⠀⢸⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠈⠳⣄⣸⠀⠀⠀⣀⣤⠀⠀⠀⠀⢀⣠⣼⡀⢸⠀⠀⠀⢸⢁⣧⡤
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠻⡟⠣⣀⣠⠞⠋⣁⡈⣇⠸⢒⣒⣒⢾⢸⢁⣇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣠⡶⠛⠒⠦⣄⡀⠀⢠⠃⠀⠀⣻⡴⠚⠁⢷⢹⠀⢒⣿⡤⣿⢸⠘⣸⠆
⠀⠀⠀⠀⣀⡤⠖⠋⠑⢦⣤⣴⣋⣠⠁⠀⠀⠀⠀⠙⠆⣇⠀⠀⠛⢅⠳⡀⠀⣸⢼⡆⢀⡀⠀⣿⠘⡰⠁
⠀⠀⠀⠞⠁⠀⠀⠀⢀⠞⠀⠀⠉⠛⢶⣄⣠⡤⣤⣀⣓⡙⣆⠀⠀⣸⢣⣱⡞⠁⢸⡇⠙⠓⠀⣿⠰⠥⠤⢤⣀
⠀⠀⠀⠈⠢⢀⣄⢀⠎⠀⠀⠀⠀⢀⡜⢩⣴⢦⠇⠙⣄⠀⠘⡀⠀⡈⢣⠳⣷⠀⢸⡇⠀⠀⠀⢻⣷⠄⠀⢀⠴
⠀⠀⠀⠀⢀⡔⠁⡌⠀⠀⠀⠀⠀⠸⠤⠴⠒⠉⠀⠀⠈⣆⡠⠷⣾⡿⠀⠡⠸⡇⢸⠃⠀⢀⡠⠞⢁⠤⠊⠁
⠇⠀⠀⠤⠤⣍⣉⣓⡒⠂⠤⠤⠤⠴⣧⣀⣀⣀⣀⡀⠀⠀⠀⠀⣮⠀⠀⠀⢁⣿⣼⠴⢚⡡⠔⠊⠀⠀⡠⠔⡎
⠀⢤⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠀⠐⠒⠒⠂⠀⠤⠌⠭⠭⠭⠭⠭⠭⠭⠍⠉⢡⡔⠊⠁⠀⣀⠤⠒⠉⠀⠀⡇
⠀⢸⡀⠀⠀⠀⠀⠀⠈⠁⠁⠈⣷⠒⠒⠒⠒⠶⢦⠤⠤⠤⠤⢤⣤⣄⣀⣀⣀⣀⡁⠤⠒⢉⡠⡀⠀⠀⠀⠀⡇
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣄⢊⠔⠀⠤⢄⠀⢠⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡀⠨⠂⠀⠤⠺⡀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠐⡜⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⣀⡸⠐⢢⠀⠁⢔⠇
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠤⠄⠊⠓⠊⣀⠴⠋
⠀⠀⢠⠎⠁⠀⠀⠀⠀⠉⠉⠉⡓⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢀⠠⠐⠊⠁
⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⣶⣹⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠄⠒⠈
⠀⠀⠸⣄⣀⡀⠀⠀⠀⠀⠀⢿⠈⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⣈⢁⠈⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠎⠘⠓⠆⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶
⠀⠀⠀⢰⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉
⠀⠀⠀⠘⣷⣾⢻⢘⡢⡾⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡄⡄⣄⢤⠔⠂⡤⠰⡄
⠀⠀⠀⠀⠘⠃⠈⠈⠁⠁⡜⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣻⣮⢻⣽⡟⣟⣸⣿⣟⣽⣧⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⠛⠛⠛⠻⠛⠋⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⢡
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠊⢐⡞⠀⡀
⠀⢸⡒⡆⢰⣄⠰⡔⡇⢙⢵⢴⡆⢠⡆⠀⠀⠀⠀⠀⠀⠸⣷⡿⠑⠂⠀⠀⠀⠀⠀⠀⠀⣄⡄⡄⡀⣠⢤⡀⡄⠲⢀⣤⣰⡀
⠀⠈⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠝⢓⠀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠃⠉⠉⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣰⢶⠐⡠⢲⠂⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠉⠁⠁⠈⠀⠃

⠀⠀⠀⠀⠀⠀⣾⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⢴⣤⡓⠃⡀⠀⠃⠀⠀⠀⠀⢀⠐⠒⠶⣃⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠉⠀⠀⢘⣀⣴⣿⠃⠰⠀⠀⠀⠀⠀⠀⡁⠀⢠⣿⠟⠋⢀
⠀⠀⣦⡄⠀⠀⠀⠀⠀⠀⠀⣴⠶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣉⣩⣀⠀⠀⠀⠀⠀⠀⠀⠻⡶⣿⡟⠀⠀⣬⣷
⠀⠀⢹⣷⡀⠀⠤⠔⠋⠓⠿⠮⢕⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠂⠀⠀⣷⡀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠺⡼⠐⠔⠋⣷⠀⡀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⢠⣆⠀⢸⣿⠁⠀⠲⠂⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⣀⣶⠒⠉⠉⠉⠁⠘⠀⠘⠋⡁⢀
⠀⠀⠀⠀⣴⠖⠲⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢺⣁⣀⣀⠀⠀⠀⠀⠀⢢⠈⠀⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⣀⠀⠀⢀⠗⠶⠒
⠐⠶⢶⡌⠁⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠓⠓⠀⢿⣯⠭⡀⠐⠢⣦⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣄⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡅⠀⠀⠀⠀⠀⠀⢳⠾⠿⢭⣯⣶⣷⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠃
⠀⠀⠀⠀⢠⠶⣿⡆⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍
⠀⠀⠀⠀⠀⠀⠈⣓⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣚⡺⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠘⣿⣤⡀
⠀⠀⠀⠀⠀⠈⡱⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣿⣵⢾⡀
⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⠈⠛⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⢀⣾⣿⠟⠋⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⣿⠆⠘⠛
⣄⠠⠂⠀⠀⠀⠀⠀⠸⡇⠀⠀⣼⣿⠃⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⣬⣥
⠫⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠻⡟
⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣼⣯⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣍⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠿⠟⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢈⡙⠿⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⢟⣡⠆
⠀⠀⠀⠀⠀⠈⢻⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⠀⠀⠀⢀⣼⣿⣀⣀⣀⣀⣀⣀⣀⣛⣭⣥⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⠀⠀⠀⠙⣿⣿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢻⣷⡄
⠀⠀⠀⠀⠀⠀⡀⠀⣴⣾⣿⣷⣌⡻⣆⠀⣀⠠⠄⣀
⠀⠀⢄⠀⠀⡠⠄⢸⣿⣷⣏⢷⣽⣿⣮⡇⠀⠀⠀⠈⣷⠞⠉⠑⠢⣄⠀⢀⡀
⠀⠀⠀⠑⠄⣈⠀⣸⣿⡟⣟⣸⢿⠇⢸⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣼⠋⠁⠉⠑⣄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⣶⡿⠄⠛⠠⠤⠄⣀⠼⠄⣄⡀⠀⠀⡏⠀⠀⠀⠀⣸⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠛⠙⠛⡟⠀⠀⠀⠀⠀⠸⠁⠀⠀⠠⠁⠀⢸⠁⠀⠀⠀⢠⡏⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠂⠀⠲⡀⠀⠀⠀⠀⠰⠧⣀⣴⢇⡜⠀⡟⠀⠀⠀⢀⡞⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠀⠲⣤⣀⣀⣀⣤⣴⠋⠀⢸⡇⠀⠀⢠⡞⠀⠀⠀⣴⣷⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⣿⡿⡟⠹⡿⢶⢶⣿⣷⣴⢶⣯⣀⣀⣠⣾⣽⣏⢻⣾⣿⡿⠋⣡⠄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠞⠁⠃⠀⠀⠈⡇⠛⢿⡀⠀⠉⢍⣍⠙⢿⣯⣻⣫⣾⡿⠀⣮⡈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢻⢢⠈⠻⣿⣿⣿⡇⢸⣿⢃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠓⡄⠀⠀⠀⢡⡀⠙⠹⣿⠀⠘⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣀⣀⡀⠀⢀⣀⣼⣣⠀⠈⠀⠀⣀⠀⣟⠦⣀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠙⠛⠛⠛⠋⠥⢤⣤⣤⣤⡼⠋⠦⠖⠚⠙⠓⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠆⠀⠀⣠⠴⠀⠀⠀⠀⠈⣿⠘⠀⠀⠀⢹⣿⠀⢨⡧⣤⡠⠀⠀⣬⣴⠶⠶⣶⣦⡤⠾⣿⢤⣤⣀⡀⠀⡀⠀⣿⣿⠀⠀⡀⢘⡀
⠀⠀⡀⠰⠊⢁⠀⠜⠉⠀⢀⡀⣿⣀⠀⠀⣠⣾⠙⣀⡬⠷⣋⣀⣠⣾⢻⡆⣰⠎⢃⠉⠀⠒⠤⠀⠀⠈⠻⣦⡙⢾⠃⠉⣿⠉⢧⣴
⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠸⣯⠿⠸⣧⣾⡟⠃⣼⡏⠀⠘⢿⡍⠁⠙⢛⡿⠃⡔⠀⠀⠆⢠⣌⡟⠀⠉⠀⡌⢿⣀⠾⢦⢛⢰⡛⠽⠃
⠀⠀⡀⠀⠀⠀⠀⠀⠁⡜⠁⣤⠀⠐⠀⢿⡄⠀⠈⢡⣖⠘⠻⠀⢀⢀⣿⡁⠀⠀⠀⠂⠀⢼⣿⠄⠈⠁⠀⢑⠈⣧⠰⣾⡛⢡⣿⣶
⠀⠈⠁⠀⠀⠀⠀⠃⣾⣤⠉⡥⢈⡮⡄⢼⣿⠉⣠⡤⠋⠀⣴⠊⠁⠀⣀⡈⢳⣄⠈⠁⢀⠘⠉⠀⠀⠀⠁⠘⢈⡷⠀⢿⠙⠱⠋⣹
⠀⠀⠀⢀⠀⠀⣡⣶⡇⡄⠀⠙⣻⠟⠆⠀⡷⠘⢧⡄⠀⢸⡇⢐⣸⣽⡿⣄⢛⣿⠈⠐⠁⠀⠀⠀⠀⠀⠐⠆⣼⢳⣶⣿⣶⠀⣼⠛⡀
⠀⠀⠄⣩⡴⡿⠋⢈⡁⠀⢀⣶⡿⢾⠀⠀⡏⢀⠰⢏⡴⠾⠉⢉⡙⠛⠋⢉⠙⠟⠓⣦⠀⢈⠂⠀⠐⠢⢠⡾⠫⣺⣿⡉⣹⠒⠿⠶
⠀⠀⣿⣿⠁⣣⠾⠋⡟⣫⣊⢵⣷⠋⠀⠠⣷⢛⠠⣅⠡⠂⡐⠃⠀⠀⠀⠀⣼⣆⠾⢁⣿⣨⡅⡠⠆⠘⠁⠀⠀⠀⢫⡿⣵⠐⠀⢠
⠀⠀⠈⠁⢸⣇⣘⣁⡾⢻⡃⠀⠀⠀⠀⠀⣯⢘⠖⢉⠆⣸⠃⡀⠀⣀⣀⠀⢃⣸⠟⢻⣝⡧⠀⢀⡤⠚⠛⠛⡓⢤⠁⠰⣽⣾⠻⠾
⠀⢀⠁⠀⠀⠀⢈⡿⣷⣯⣁⣀⣠⣀⣀⣀⣿⣷⡀⢢⡶⡚⠛⡁⠀⠀⠀⠰⣌⠙⠢⠟⣤⢹⡆⠛⠳⢤⣛⡰⢀⡜⣷⠈⠀⣯⡰⢈⠃
⠀⠐⠤⣾⠟⠙⠛⠹⠿⠈⣯⠉⠉⠉⠉⢩⡌⠳⠘⣻⣞⣡⣾⠀⠀⠀⠀⠀⠛⣄⣳⣼⣦⠻⡇⢀⠿⠋⠉⣰⡏⢀⣿⠀⢘⢸⣍⡄
⠀⢘⣠⠟⠃⣀⠤⣠⡀⠀⣟⠀⠀⣀⠠⠬⣿⣀⡞⢧⢀⡟⢿⡍⠂⠸⠇⠐⣡⡿⡏⢿⣻⣾⣅⢰⡃⠀⢆⢿⠷⣾⠃⠀⣈⢠⠛
⠀⠈⠹⣀⣎⡤⠒⠢⣌⠗⠳⠒⣾⡥⠊⢻⣭⠁⠛⠾⡀⠙⠂⠙⠳⠶⠶⠛⠉⣀⠌⠀⢻⠟⣈⢸⠃⡾⢠⠕⠐⢁⡀⢼⠁⣿⣢⢿
⠀⠐⠀⠳⣿⠀⣾⡀⣸⣄⠀⢠⣦⠀⠀⣈⠇⠀⢀⡔⣤⣤⠄⣠⢢⡔⣀⣤⠸⢉⡄⠒⠁⠂⠘⢿⣿⢠⡼⢦⡀⢸⡇⡴⢠⠴⡵⢺
⠀⠀⠂⣤⠀⡐⢩⡌⢹⣿⠃⢘⡏⢀⠀⡷⢀⠀⢘⣇⠀⠀⠀⠀⠛⣸⣿⠛⠁⠀⠀⢠⣦⠀⠀⣿⡳⢿⡇⢀⡐⠞⣡⣿⣤⡂⢛⠻
⠀⠀⠀⣨⣟⢧⡀⣉⡼⢻⡆⠘⠁⠰⠸⢧⣄⣠⡟⣉⣷⡖⠛⠷⣍⢹⡿⡆⣦⠲⣶⡿⠟⣧⢸⢾⠛⢨⡷⣯⣥⣾⣽⡅⠀⢰⠟⣤
⠀⢸⡯⢄⠉⢠⡨⣭⣜⠽⠙⠦⢠⣀⠇⠀⠀⣽⠙⠙⠎⣿⠀⠘⣇⡄⠛⣴⠉⠸⡏⠱⣦⣌⢿⣟⡛⣸⢇⡴⠦⠚⠘⡗⢾⣭⡁⣼⠁
⠀⠘⠗⠂⢁⠀⠄⢿⡄⠀⠂⠀⠀⠹⢦⡀⢀⠳⣄⠲⣇⣽⡁⠀⢻⡦⣬⣿⣀⠰⡏⠀⣤⣋⣠⣍⡓⠋⣿⠁⡌⠈⢸⡇⠁⠚⠁⢿
⠀⠀⣤⠀⠈⠀⢠⡀⠠⠀⢘⣀⠀⠀⠀⢿⡁⣀⠽⠷⣤⣾⡁⢀⣯⠶⠋⡉⠻⠛⣻⠆⣝⡛⠋⢠⡿⠀⠀⠈⠁⡰⢸⡇⠀⢆⠘
⠀⠀⣤⢐⡇⠀⣤⢀⡀⣸⡈⢿⣳⣤⣿⠀⠀⠉⠻⢯⣄⣛⡷⠞⠓⡀⠀⣤⠿⠐⠛⠃⠀⢀⠣⠌⡳⢶⣶⡆⠀⣷⠘⡗⠀⣢⢶⠄
⠀⠀⣧⠘⡇⠀⡏⠈⠇⣿⠀⠄⠙⢿⣟⡀⣀⣤⠀⡔⠈⠙⢙⡿⠶⠶⣿⠋⠒⠚⠒⠲⣤⢤⢤⣤⡜⣦⣸⠄⠀⣽⠰⣿⡀⣀⠘⡀
⠀⠰⣏⣈⡑⠣⢛⣠⣴⣾⠀⡿⠋⣿⣽⣍⣀⠀⣔⣄⣀⢠⣟⠁⠐⢰⡇⠀⠀⠀⠀⠀⡈⣆⠰⢶⣾⡀⢽⡄⠶⣶⡶⣧⠜⠛⢈⠁
⠀⠀⢁⣸⣇⢲⡿⣾⡝⣫⣤⣭⠶⠈⠛⠛⠛⢈⡃⠀⠠⠀⠉⢒⡉⡹⣷⢦⢩⣉⣈⣩⣍⣙⣆⠀⡟⣠⠌⠙⠛⢁⣐⢯⣀⣀⡸⠲⠁
⠀⠐⠛⠀⣿⠘⠳⢶⡄⠈⠉⢡⡶⣤⣄⠀⠀⠘⠛⠛⠛⠻⠛⠛⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁⡀⠁⠻⠆⠀⣾⢋⠹⣆⠉⠉⠀⠰
⠀⠈⡁⣸⠇⡀⠀⠈⣿⠀⢀⠞⡣⢠⡾⠃⠀⠀⢀⡞⠛⣻⡆⠀⠀⠀⣾⣻⢹⣇⠀⠀⠀⣴⢻⠟⣦⠀⠀⠀⣻⡄⠂⠼⣆⠐
⠀⠀⣀⣭⣤⡷⠶⠛⠁⠀⢸⣥⣦⡟⠀⠀⠀⢀⡾⠁⠄⣿⠃⠀⠀⢸⡁⠈⣠⣟⠀⠀⠔⠹⣧⠈⢘⣦⠀⠈⠙⠿⠖⠛⠉
⠀⠀⠛⣿⣭⠇⠀⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⢿⣷⠶⠟⠈⠁⠀⢹⣦⣶⣾⡏⡀⠀⠀⠀⠘⠟⠉⠉⠀⠁⠀⠀⢀⣀
⠀⢈⣤⣈⣁⣀⠀⠀⣴⢶⣄⡀⠀⢀⠀⠀⠀⢀⣀⣁⠠⠀⠀⠀⠀⠀⠀⣁⣈⠉⠉⠀⠀⠀⣀⣴⢷⡄⠀⠀⠀⢠⣿⠉⢙⡇
⠀⠀⠉⠉⠉⠿⢂⣾⠃⣈⣾⠇⠀⠈⠀⠀⢳⣿⡥⢜⣷⠀⠤⠀⡂⣸⡟⡋⣿⡆⠄⢀⣀⢸⣏⠰⠜⣷⠀⠀⠀⢹⠇⠮⢸⡧⠀⢀
⠀⠀⠄⠀⠀⠀⠾⣇⣀⣰⡟⠀⠀⠀⠀⠄⣺⠃⠈⣼⡏⠐⠂⠘⡏⢹⠌⠃⢸⣇⠠⣏⠉⠀⢻⡄⠀⣸⣇⠀⢀⡻⠷⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⠐⠻⠟⠛⠇⠀⢀⠀⠀⠀⠸⣧⡶⠾⡇⢐⠈⠀⠀⠀⠛⠋⠉⠁⠀⠀⠀⠀⠀⣠⡔⢦⡀
⠀⠀⠀⠀⠀⣰⢛⡟⣶⡄⠀⠀⠀⠀⠀⣴⠾⡟⢷⣀⠀⠁⠀⠀⡄⣠⡴⠿⣆⣀⠈⠈⠀⠀⠀⢀⡦⠤⢶⡄⠀⠀⠀⠸⣯⢤⠌⠹
⠀⠀⠀⠀⣰⠃⠨⢁⣿⠀⠀⠀⠀⠀⡠⢿⡗⣁⢺⣞⠆⢁⣀⡀⠁⣧⡰⠆⢹⣿⠙⠀⠀⣀⠀⣻⡇⠒⢸⠇⠀⠀⠀⠀⠙⣧⣠⣴
⠀⠀⠁⠘⠿⣶⣤⣼⠃⠀⠀⠀⠀⠀⠀⠘⣷⣬⣴⠿⠀⠈⠛⠻⠉⠺⣷⣀⣤⡿⠀⠀⠀⠀⠀⢾⣄⣀⣿⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⠋⠁⠀⠀⠀⠀⠄⠀⠠⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⡿⠿⠿⠿⠿⡹⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣟⣻⡛⢛⣛⣿⣿⣿⣟⣿⣛⡛⠿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠸⠀⠁⠈⠀⠀⠀⠀⠡⠀⠀⠀⠀⢠⠃⠀⠄
⠀⠀⠀⢠⣄⣀⠀⢪⣩⣆⢠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⡇⠰⠀⠀⠀⠐⠀⡔⠀⠁⠀⡁⠀⠆
⠀⣀⠐⢦⡀⠙⣷⣘⢳⠏⢰⢀⠏⠰⠀⠀⠀⠀⠀⡔⠡⡐⢤⠤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⠀⠀⠈
⠀⠸⣆⠀⠙⠄⠈⣿⢼⠇⠰⣎⡴⠓⠄⢀⢔⣤⡞⠀⢀⠇⠈⡆⢠⠔⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠈⠑⢄⢸⣆⢻⣶⡆⠐⣽⠁⠀⠀⢈⠘⣿⡇⠀⢨⡆⠀⢿⢀⡟⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⢦⠻⡈⣻⢃⢣⡏⠀⠀⠀⠀⠄⠻⣷⡄⠸⣆⣀⡞⣼⢁⠇⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⡄⠀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠣⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠣⠼⡿⠿⠋⠜⡿⠑⠁⠀⠀⠀⠀⠀⠀⠈⠯⣤⠀⢳⠆⠀⢰⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠘⠀⠀⠀⠀⠀⠀⡴⠒⠂⣴⠶⢄⡀⢳⣼⣇⣼⣿⠷⠚⠃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠑⣄⡀⠈⠷⣇⡙⢶⣍⢯⠉⣤⣤⡶⠿⢛⡋⡳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⠀⠹⢦⣦⡭⣷⣶⠽⠷⠦⣤⠤⡤⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣉⣀⣀⣑⣀⣐⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⣟⡛⢻⡟⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⣿⣿⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠧⠼⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣌⣢⣀⠉⠇⠀⠀⠸⠉⢀⣔⣡⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢰⣍⣩⣭⣶⡷⣤⣤⣤⣤⢾⣶⣭⣝⣩⡆⣀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡔⢫⣄⢻⣦⠹⣿⠇⠈⣳⠼⠿⠿⠧⣞⠁⠘⣿⠏⣴⡟⣠⡝⢠⡄
⠀⠀⠀⠀⠀⠀⢀⡿⣼⠁⠙⣱⣿⣷⠟⡸⠋⠀⠀⠀⠀⠀⠀⠙⢇⠻⣾⣿⣖⠋⠈⣧⢿⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣼⣿⡟⢹⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡏⢻⣿⣧⣀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠋⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠙⠻⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢊⡠⠖⠋⠑⢄⠀⠀⠀⠀⠀⢀⡠⠊⠉⠲⢄⡑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡉⠀⢶⠾⠁⢠⠋⠐⡀⢀⠂⠙⡄⠘⠷⡶⠀⢉⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⠤⣀⣠⠇⠀⠀⠙⠋⠀⠀⠸⣄⣀⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠬⣻⣷⣶⣶⣶⣶⣾⣟⠥⠔⠊

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⢸
⠀⠀⠄⣡⠆⠐⢄⠠⠀⠀⠀⠀⠄⡥⠂⣄⣀⡀⠀⠀⠠⢀⡴⠶⢦⡅⠄⠀⠀⠀⠠⢀⢔⠀⡢⡀⠄⠀⠀⠀⠰⢉⠞⠒⣷⣈⠂⠀⢸
⠀⠀⠐⡃⠀⠀⢀⠀⠂⠀⠀⠐⠀⡀⠀⢈⣿⠁⠀⠀⠂⢸⡁⠀⢀⡂⠐⠀⠀⠀⠂⢀⢆⡋⡠⡀⠐⠀⠀⠰⣈⢀⠀⠀⠀⡇⣁⠆⢸
⠀⠀⠀⠑⣀⣠⠃⠀⠀⠀⠀⠀⠀⠱⣀⠟⠉⠁⠀⠀⠀⠀⢩⣙⡙⠁⠀⠀⠀⠀⠀⠀⢃⣿⡘⠀⠀⠀⠀⠀⢠⡀⢂⣀⡜⣀⡆⠀⢸
⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⢸
⣿⣿⠟⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿