Skates Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢳⣀⡤⠴⠶⠒⠚⠛⠉⡯⠁⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⡇⢠⣦⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢠⠀⣤⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠸⡀⠙⠁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣠⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠤⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⠿⠁⠈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⣿⡏⠙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠓⢤⡉⠁⠀⣸⣯⠑⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠌⠀⠁⠌⠑⠦⣌⡋⠀⠀⣾⣿⠓⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠂⠠⠀⠌⢀⠂⡐⢀⠍⠁⠒⠠⠤⠤⠤⠤⠚⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠂⡐⠁⠄⠠⠁⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⠂⠠⠤⠤⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠂⠐⠀⠌⠀⠁⡐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⡀⢀⣀⣀⣀⣀⣀⠀⠈⠉⠉⠁⠒⠒⠒⠤⠆⠤⢡⡆⣠⢁⡌⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣄⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣁⣉⣉⣉⣉⡉⠉⠁⠐⠒⠀⠀⠒⠒⠐⠒⠒⠚⠛⣛⣋⣉⣭⣴⣶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣻⣻⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠉⠛⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠛⠋⠉⠉⠛⠿⣟⠋⠁⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡔⢊⡉⣉⠒⢄⠈⢿⡉⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣛⣹⠏⢀⠔⢊⡉⣉⠲⡄⠈⢦⠀⠀⠀⢀⣱
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠘⠐⢡⡖⣦⢱⠈⡆⠈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡏⠀⡎⢰⢡⡖⣦⠑⠈⡄⢸⡀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠰⡈⠢⠛⠉⠌⡠⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠡⡈⠌⠙⠩⠂⡰⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠒⠂⠒⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠒⠂⠒⠊⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣂⣤⣶⡢⣄⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⣿⣿⡿⠿⠞⠳⠦⢤⣤⠞⠁⠀⠀⠀⠈⠀⠺⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠻⣥⣶⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡈⢛⠛⠻⠿⣿⣿⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠉⠁⠀⠀⠀⠀⠀⠌⡉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⣀⣀⠤⠤⡶⢟⣵⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣗⣻⣭⣿⣶⠤⠤⠾⠻⢤⣭⣤⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢚⣥⠈⢷⣒⠞⠿⢸⡦⢠⣤⣤⢜⣻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣋⠀⢈⣇⡴⣫⡶⢤⣤⡕⠛⣿⣯⠉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡥⢎⡐⠛⢀⢂⣀⣀⣀⠜⣡⠛⢿⠿⠲⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⢁⣴⡡⠌⡛⡩⢁⠘⣁⣘⣿⡶⣣⠶⠟⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⢀⢻⣵⣿⡿⠟⡉⢋⡼⠛⣿⠟⣴⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢇⡴⣉⠳⡿⣣⡾⢭⠍⣩⣷⣮⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣏⢶⣄⡶⢢⡞⣋⣉⢻⣿⠌⣷⡄
⠀⠀⠀⠀⢀⣠⡄⠒⠈⠉⠉⠓⢮⡱⣌⢻⠋⠙⠻⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠲⠦⣭⣉⣡⣤⣿⣿⡄
⠀⠀⠀⢠⠏⣡⣤⣅⠢⡄⠀⠀⢸⣷⣾⢤⡔⠲⠷⠾⢿⣽⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⡇
⠀⠀⠀⣾⢠⣿⡁⢀⠑⠎⣆⢀⣼⣿⣧⡘⠿⣶⣤⣤⣾⣿⣿⣿⣿⡿⠃⠈⠉⠛⠿⢿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢰⣾⣿⣿⠁
⠀⠀⠀⢹⠈⣿⣅⠘⠂⣆⢹⣾⣿⣿⡇⠈⠁⠒⠉⠽⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⠯⢽⣿⠿⠿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⡿⠋
⠀⠀⠀⠀⢣⡘⢿⣣⡴⢏⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣋⣿⠀⢠⢇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠈⠒⠠⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠉⠉⠳⢾⡄⢨⠉⢹⣿⣿⣿⠟⠛⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠠⠄⠉⠒⢄⠀⠀⠀⠹⣄⠀⢹⣿⣿⣿⡆⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⣾⡇⠀⠀⠀⠡⡀⠀⠀⣻⣿⣿⣯⢻⣿⣿⠀⢀⡤⢞⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⣟⠀⠡⠀⣇⣠⣾⣿⠅⠀⢹⣾⣿⣛⣛⣭⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⢻⢿⣧⡄⢠⠀⣯⣿⣿⡿⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⠶⠶⠋⣰⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣄⣀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡦⠄⠀⠀⢠⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠾⡿⠉⠀⠀⠈⠉⣛⠛⢻⣿⣿⢻⣿⡀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠄⠀⠈⠁⠀⢀⣈⣩⣿⡿⢁⣾⡉⠙⠫⣿⣿⣷⣶⣦⣶⣾⣿⣿⡿⠋⠁⠀⠠⢿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠤⢤⣍⣀⡐⠲⠐⠒⠚⠙⣿⣦⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠐⡾⢽⣿⣿⣷⣦⣄⠀⠈⢿⣿⠀⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⠉⠛⠿⢿⣿⣿⣿⡆⠀⠀⠁⢠⢿⣿⠏⠀⠀⠀⢠⣶⣤⣤⣄⠀⠀⢀⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⣷⣲⣤⣀⡀⠀⠈⢻⡿⠀⠀⠀⠀⢸⣎⣿⣦⣀⣀⣤⣾⣿⣿⣿⣿⡇⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣩⣐⠶⣤⣌⡉⠛⢿⣷⣦⣴⠃⢰⡿⡦⠀⢿⣿⣾⣿⣿⣿⣿⡿⠋⠀⠈⣿⣷⣤⣄⣤⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⣡⣜⡲⣄⡉⠳⣮⣝⣿⣷⣦⡈⣩⠿⠦⣤⠤⠶⠊⠀⠛⠿⣿⣿⣿⣏⠀⡀⠀⠀⠛⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠠⠶⠶⠿⢿⣿⣿⣮⣿⣦⡈⢻⣿⣿⣯⠞⢁⣠⣴⣶⡷⠀⣰⣿⣶⣤⣄⠀⢻⣿⣿⣿⠆⢤⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣤⣴⣤⣤⣀⣀⣀⣀⣀⠀⠉⠛⢿⣷⣤⠿⠋⠀⠾⠿⠿⠛⠛⠁⠀⠙⠛⠉⠉⠁⠀⣼⣿⡿⠋⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣋⡽⣿⣛⣛⣛⣛⣛⣛⣓⣀⣈⣁⣠⣤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⠤⠤⣤⣄⣚⡉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⢿⣡⠝⣿⠟⢁⣼⣿⣿⣿⡿⠏⠉⠛⠛⠓⠒⠂⠈⡀⣀⣀⣤⣴⣶⣶⣿⣿⣿⣶⡄⠙⠯⣥⣱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡇⡎⢠⠐⠛⠋⣉⣉⡀⠀⠀⠀⡀⠀⠀⠀⣀⣈⡉⠉⠉⢉⡉⠉⠉⠉⡉⣉⡉⠁⠀⣀⠑⠰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⠗⣀⠐⣨⣽⣿⠅⢻⣿⡶⢂⡤⢐⣾⣿⡿⠈⢿⣿⠖⣀⠄⣰⢞⣿⠇⠹⣿⣫⣔⠀⣔⠳⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⡿⠿⠋⠀⠀⠉⠻⠿⣿⣾⠿⠟⠁⠀⠀⠙⠻⢿⣷⡿⠿⠋⠀⠀⠈⠛⠿⣾⣿⡾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⡒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡊⠁⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠫⠊⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠔⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⡤⠶⠞⠓⠲⣄⠠⠀⠀⠀⠀⠈⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠐⠒⠋⠉⠀⠀⠈⠁⠀⠀⣸⠇⠀⠀⠀⠀⠀⠐⠀⣈⠓⠢⠄⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⠁⠁⠀⠈⠉⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠉⠑⢢⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣶⡶⣿⡛⠋⢹⠀⠀⠀⢀⠤⠊⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠙⣾⣿⣿⡏⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠙⡆⠈⠑⠂⠉⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⠄⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡌⠫⣀⠀⠀⠀⠀⡀⠄⠒⠉⣀⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣌⣁⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠛⢛⡿⠿⠏⠀⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣀⡤⠤⢤⣀⣀⠄⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⠿⣶⣶⣶⣦⣤⣤⣤⣤⣤⡌
⠀⠀⠀⠀⠛⠳⠶⣤⣄⣀⠀⠀⠀⠉⠒⠲⠦⠤⢄⣀⣀⠀⠀⠀⠀⣸⠀⢰⠋⠈⠉⠉⠙⢿⠛⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⢤⣤⣄⣀⣀⠀⠀⠈⠉⠉⠉⠁⠀⠈⠓⠄⠤⠄⣀⡼⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠒⠶⠶⠶⠶⠦⠤⠤⣤⡤⠤⠤⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⠈⣼⣿⠛⠉⠉⠉⠉⠩⠍⠛⣛⣛⣛⣛⣛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠓⠃⠀⠀⠰⠒⣶⣿⠇⣴⡏⠻⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⣀⣀⡐⠀⢈⣉⣉⣥⡴⠿⣷⣄⠀⢉⣭⣭⣭⠿⠿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠀⢙⣛⣷⣶⣶⣤⣄⠀⠀⠙⣿⣷⠀⣶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠰⠾⣭⣿⣿⣿⣿⣷⡀⠀⠈⠛⠀⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣄⣀⠀⠀⠈⠙⠻⣿⣿⠃⠀⠀⠀⢸⡍⣥⠶⠖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣉⡘⠓⠻⢿⣶⣤⣄⡀⢸⡏⢀⣤⡀⠀⢸⣿⡵⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣍⡐⠳⣮⣝⢿⣶⣤⣈⠙⠿⢟⣿⡀⠘⠛⢗⡠⠹⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠐⠒⣉⣡⣴⣿⣮⣿⣦⣄⠙⢿⣾⣿⣿⣷⣤⠟⠉⢛⣛⡛⠉⢀⣀⡀⠙⠛⠳⠦⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⡼⣀⣀⠀⠉⠉⠙⠛⠛⠛⠻⠿⣿⣿⣦⡈⢻⣿⡿⠛⣁⣴⣾⣿⣿⠏⢠⣾⣿⣿⡿⠶⠀⣸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⣷⣿⣷⣶⣦⣴⣦⣤⣀⡀⠙⣿⠷⠚⠉⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⣀⠤⠔⠓⠷⡾⠯⣭⣭⣭⣭⣭⠭⣭⠤⢴⠶⠶⢒⣒⠲⠶⠶⠛⠛⠛⠛⠛⠛⠓⠲⢶⠚⠣⠤⣀
⠀⠀⠀⠀⠀⡰⠁⢠⠄⡲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣤⠈⠱⡄
⠀⠀⠀⠀⠀⡇⠀⠆⡎⠠⠄⠀⣀⣠⣤⣀⣀⠀⠀⠠⠀⠀⢀⣀⣠⣀⣀⠀⠀⠤⠀⠀⢀⣀⣀⣀⣀⡀⠀⠄⠑⠸⠀⢱
⠀⠀⠀⠀⠀⠘⣄⠘⠏⣖⠚⡭⠀⢀⠇⠹⡄⠙⠃⣖⠒⡿⠁⢠⠃⠣⡀⠻⢃⣖⢲⠞⠁⣰⠀⢣⡀⠨⠕⢒⣎⠟⠀⡸
⠀⠀⠀⠀⠀⠀⠈⠑⠤⢄⣀⠠⠔⠋⠀⠀⠈⠓⠤⢄⣀⠤⠔⠁⠀⠀⠈⠒⠤⢀⣀⠤⠖⠁⠀⠀⠙⠢⠤⢀⡠⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠿⠿⠿⠿⣿⠛⡛⣿⠋⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠈⠋⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠘⠃⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⢿⠘⡃⣻⣀⣠⣴⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢸⡌⢻⠿⣏⣉⣛⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⣇⠛⠋⣻⣭⣭⣽⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠟⠉⣷⡿⣶⠾⢯⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠛⢁⣿⣿⡾⢛⣓⡿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠈⢢⡉⢠⡽⢿⣩⣟⣭⣿⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⢧⣀⠀⠙⢲⣄⡀⠀⠀⠀⠀⠀⠉⠳⣄⡶⠏⣳⣼⣋⣉⣩⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠙⠲⠼⠗⠦⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠙⠲⠯⣄⣈⣉⣁⣀⡽⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠈⢧⡀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠉⠳⢤⡙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠈⠓⢮⣙⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠈⠉⠁⠀⠙⡟⠛⠛⠛⡿⠛⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠙⠲⢤⣤⠇⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⣶⠟⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⣄⡠⠔⠤⣀⣀⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢈⠀⠀⠀⠈⠀⠀⠈⠉⠙⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⢣⣼⠆⠀⣶⣶⢲⡇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣵⢟⣿⠻⢿⡀⡞⣿⠙⣿⡿⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⢜⠊⠙⡀⠷⠁⠀⠀⣑⡿⢢⡷⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠘⡀⣥⡄⠚⠃⠀⠀⡠⡊⠀⢢⡆⠀⠀⠀⠀⠀⢀⡠⠀⡶⠸⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢂⠠⠶⠀⠀⠀⠀⠀⡀⠠⣴⡎⢀⣀⣤⡇
⠀⠀⠀⠀⠀⠀⣯⣴⣶⣒⣤⣠⣄⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣀⣡⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⠿⣿⣿⣿⣿⠿⣿⣧⠀⢀⣀⠠⣤⣰⣦⣤⣴⣿⡿⠿⠛⢻⢿⣿⠆⠁⠀⠉⢦
⠀⠀⠀⠀⠀⢼⡁⠂⢹⠃⡄⢁⠀⢄⠙⣿⣾⣿⣿⣿⣿⡟⠛⠿⢿⡀⠀⠀⢹⡀⠫⡀⢁⢃⠅⢸
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⣄⠀⠈⠔⢀⣧⠽⡙⢿⠋⠜⠀⢀⡐⠀⣣⠀⠀⠀⠉⠒⠓⠤⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠋⠘⠤⠠⠼⠳⢤⣀⡀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⠉⠉⠁⠀⠀⠀⠀⠙⠻⣾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⣀⣠⣤⣬⡁⢀⠀⠀⣀⡀⠐⠓⠲⠶⢿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⡿⠟⠋⠉⠁⠀⠀⠀⠈⠛⢶⡟⣷⠈⠛⠛⠛⠛⠛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⣤⠶⠶⠶⠶⠶⠶⠦⣼⣧⣿⠈⠛⢛⣻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠁⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⢻⡆⣰⠟⠁⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢛⣿⣿⣿⠃⠉⠀⠛⣩⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⢸⡇⠀⠀⣠⡾⠋⣠⡾⠛⢷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠐⣯⡀⠀⠀⠀⣠⡴⣄⠀⣼⠃⠰⠛⣻⣷⡀⠀⠉⢀⡴⠋⠀⠀⠘⠃⠈⢹⡟⠳⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠦⢀⡀⠹⠶⠟⢠⡟⠀⣠⠞⠉⣸⠟⢶⣤⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣼⠛⢷⣄⠈⠙⠳⠶⠶⠛⠁⣾⠋⢀⡾⠋⠀⠀⠿⠉⠛⣷⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠉⠻⢦⣤⣄⣀⡀⠈⠻⡾⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠛⠉⠙⠻⢶⣄⠀⠾⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣙⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⡶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠿⠿⢷⣶⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⠶⠶⠛⠀⠀⠀⠉⢻⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠶⣤⣶⠶⠶⣬⣿⠶⠶⠄⠀⠀⠀⣀⣼⠷⠿⢯⣍⣉⣁⣠⣿⡶⠶⢦⣄⠀⠀⠀⣤⠾⢁⡿
⠀⠀⠀⠀⠀⠀⣼⠋⣤⠶⢶⣤⣽⣧⣾⣏⣤⡶⢶⣄⣸⣧⣴⣏⣠⡶⠶⣦⣹⣿⣽⣏⣸⣷⠶⣤⠙⠁⠀⠀⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠐⣯⠸⣇⠐⠋⣿⠀⣿⣯⠸⣏⠛⢛⣿⠉⣿⣿⠉⣿⡛⠛⣹⠏⣹⣿⠙⣿⠛⠂⣼⠃⣾⠂
⠀⠀⠀⠀⠀⠀⠹⣦⣈⠛⠛⣁⠘⠋⠹⣦⣈⠛⠛⣁⣼⠋⠘⢧⣈⠛⠛⡉⠰⠏⠹⣦⣈⠛⠛⣁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⣿⣿⣿⣧⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢰⢇⡿⣿⡹⡆⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠴⠋⢸⡇⡏⡇⠹⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⠃⢿⠀⠙⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⢺⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠫⠄⡟⠩⢒⠂⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢂⡇⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠠⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡛⢧⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠸⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠈⠀⣇⠒⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣟⠿⠈⠀⠀⣾⡞⠂⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣝⡻⠗⠁⠀⠀⠀⢀⡟⣌⠃⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠈⢄⠃⠀⠀⠀⢀⣤⡊⢳⠊⠀⠀⠀⠀⠠⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠈⠀⣠⣤⠟⠣⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠶⢶⣤⣤⣴⠶⠃⠈⠱⡍⢄⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠤⠀⠀⠰⠁⠀⠀⠀⠀⠈⠀⢀⠀⣀⣀⣤⡾⠿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠀⠀⢶⣶⣶⠶⠶⠿⠟⠻⠛⠛⠉⠁⠀⠀⢀⡀⠜⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⠀⠀⠀⢀⡠⠀⠀⠒⠂⣐⣀⡩⠥⠔⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⣤⠤⠤⠖⠒⠚⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⡔⣢⣽⣷⡢⡀⠀⠀⠀⠀⣀⡤⠴⠶⠶⠦⣤⣄
⠀⠀⠀⠀⠀⠀⣾⢸⣿⡿⠟⠛⣛⣒⣲⣤⡞⠁⠀⠀⠀⠈⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡀⠙⢶⣿⣿⣿⣿⣿⣿⠰⠀⠀⠀⠀⠀⠀⠀⢻⣿⡝⣦
⠀⠀⠀⠀⠀⠀⢻⣻⠦⠄⠬⠉⢉⠛⠻⠿⣸⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣸
⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠁⠀⢌⠑⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢀⣀⣸⢟⣿⣷⡄
⠀⠀⠀⠀⠀⡰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⢿⣒⣛⣉⣉⣩⠥⣟⣛⣛⣽⡀
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢽⣉⠹⣯⠥⠶⡤⢦⠉⣩⣭⡽⣷⡀
⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣋⠈⢙⣀⢔⣥⣤⣭⡉⠚⣻⣏⢳⣄
⠀⠀⠀⠀⠀⣏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⡤⣃⠘⢂⠄⣀⣀⣀⠞⣵⠻⠿⠗⠶⢤
⠀⠀⠀⠀⠀⢸⣞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⣴⣆⠆⢉⡩⢁⣊⣀⣼⣿⢞⡴⠾⠓⢦
⠀⠀⠀⠀⠀⠈⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡙⣷⠯⣟⠫⢌⣵⢏⣹⠟⣵⣿⣷⡶⢄
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠳⠚⢤⡹⠣⠞⢒⢒⣴⣿⣷⣌⢦⡀
⠀⠀⠀⠀⣀⡤⠤⣌⡻⡛⢿⣿⢿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⠳⠘⢡⣛⠠⠤⠼⠿⢀⣿
⢀⠴⠊⠡⢀⠀⠀⠀⢹⣎⣇⡏⣄⣠⣝⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠺⣿⣿
⡞⢰⡟⠛⠢⡕⡄⠀⣸⣿⡏⢿⣤⣈⣉⣩⣿⣿⣿⣿⣿⠟⠛⠿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⣼⣿
⡇⢹⣟⠀⠆⢠⢸⣴⣿⣿⡏⠒⠨⠙⣛⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠈⢙⣟⣻⣿⣶⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣾⣿⣿
⠸⣌⢿⣣⣤⡼⣼⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠖⣿⠀⢈⠏⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠈⠓⠠⠤⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠉⠉⠙⠲⣿⡈⡋⡙⣿⣟⣿⡿⠛⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡁⠤⠀⠐⢄⠀⠀⠀⢳⡀⢈⣿⣿⣿⣇⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣶⡿⠀⠀⠀⠑⡀⠀⢀⣿⣿⢿⡜⣿⣿⡄⣀⠴⣊⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡘⢿⣏⣸⠁⠀⠀⢠⣴⣿⣿⠀⠀⢿⣿⣯⡭⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⢝⡷⠆⡀⠀⣶⣿⣿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣒⣍⣉⣡⣾⣟⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣴⡶⠶⠾⠛⠛⠛⠿⠿⣶⣶⠶⠿⠶⣦⡀
⠀⠀⠀⢀⣾⠟⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣷⣶⣶⣿⣧⣤⣄⣠⣤⣤⣄⡀
⠀⠀⠀⢸⡏⠀⠀⠀⣠⣴⠾⢿⣷⣶⣤⣴⣾⡿⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠛⢿⡉⠉⣹⡷
⠀⠀⣀⣻⡇⣠⣴⠿⠋⠀⠀⣾⡟⠁⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⢈⣷⢀⣿⠃
⢠⣾⠟⠙⠛⠋⠁⠀⠀⠀⢰⣿⠀⠀⣠⡴⠟⢫⡽⠶⠶⠶⠮⣭⣽⣿⣶⣤⣀⣸⡏⡼⠃
⣾⡟⠀⠀⠀⠀⠀⠀⢀⣾⠿⠿⠴⠞⠋⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠶⣷⣆
⢿⣇⡀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠈⠛⢻⣿⣶⡀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣿⣶⣶⣦⣤⣀⣼⣿
⠀⣴⣿⣿⣿⠇⠀⠘⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠿⠿⠿⢿⣿⡟⠋
⢸⣿⠀⠉⠀⠀⠀⠀⢈⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠀⠀⠀⠀⣠⡴⠟⠿⣦⣄
⠘⢿⣶⣤⣀⠀⠀⠀⣸⡏⠙⠛⣿⡇⢠⡶⠟⢶⣄⠀⣿⠃⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⢙⣷
⠀⢸⣿⠈⢻⡆⠀⠀⣿⣧⣀⠀⢸⣇⠹⣧⣤⣼⡟⢠⣿⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣠⣶⣿⠿⣶⣤⣀
⠀⣾⠿⠶⠟⠁⠀⠀⠀⢹⡿⣿⣆⠹⣷⣤⣉⣁⣴⣿⣿⣀⣼⠏⣄⣀⣠⣴⠆⣠⣿⠿⠿⠷⠶⣦⣍⣛⠻⠶⣶⣤⣤⣤⣀⣀⣀
⠀⣿⡆⠀⠀⠀⢠⠄⢀⣾⣇⣼⠟⠀⢸⡟⠛⣛⣛⡉⠘⣿⣿⣤⣉⣉⣭⣥⣾⠿⠷⢶⣦⣤⣀⠀⠈⠙⠷⣶⣤⣤⣤⣿⠟⠋⢹⣷⣦
⠀⠘⢿⣦⣤⣴⠟⠀⢸⣇⠉⠁⠀⠀⣼⠃⣼⣿⠛⢿⣦⠈⠻⣯⡉⠉⠉⠉⠀⢀⣠⣤⣤⣬⣉⡻⠶⣦⣤⣤⣀⣀⣀⣤⣤⡶⠟⠁⠈⣷⡀
⠀⠐⣿⣿⠋⠁⠀⠀⠈⣿⣆⡀⠀⣰⠏⠀⢻⣇⣀⠀⠹⣷⡄⠈⠛⢶⣤⣀⣸⡿⠛⠋⠉⠉⠙⠋⢱⣦⣈⠉⠉⠉⠉⠉⠁⣀⣀⣤⣴⣿⡇
⠀⠀⠘⢿⣷⣦⣄⡀⠀⣼⣿⡿⠟⠁⢀⣀⣀⠉⠛⠛⠛⠛⠋⠀⠀⠀⠈⠉⠛⠛⠿⠷⢶⣤⣤⣶⣤⣬⣭⣤⣤⣤⠾⠿⠿⠛⠛⠉⠁⢈⣷
⠀⠀⢀⣴⡟⠉⠉⠛⢛⣻⣿⣷⣄⠀⠸⣯⡿⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣦⣤⣄⠀⢠⡶⢶⡄⠀⠀⠀⢀⣀⣀⣤⣤⣾⠿
⠀⣰⣿⣿⠃⠀⣀⢠⣿⣿⣷⣼⠛⠻⠷⢶⣦⣤⣤⣤⣤⣶⣾⣿⡿⠿⠛⠋⠉⠉⠉⠉⠙⠛⠿⢿⣷⣮⡛⣋⣠⣶⠟⠛⠛⠻⣿⣭⣥⣀
⠀⣿⣿⣿⠀⣿⣛⣷⠙⣿⣿⡟⠀⠀⠀⣠⣾⣿⡟⠛⣿⣷⣦⠀⠀⠀⣴⣾⣿⡛⠛⣿⣿⣶⡄⠀⠈⠛⣿⣿⡍⠙⣿⣦⡀⠀⠀⠻⣾⣿⣷
⠀⢻⣿⣿⣧⣉⣋⣥⣼⣿⣿⡇⣰⡶⣦⡻⠿⠿⠟⠛⠛⠛⠉⣠⣶⣦⠙⠛⠾⠷⠼⠿⠟⠋⣰⡶⣶⡀⠉⠛⠛⠛⠛⠛⠃⣠⡶⣦⡜⣿⣿
⠀⠀⠙⢿⣽⣿⣿⣿⣿⣿⣿⣧⠙⠷⠟⢁⣀⣤⣴⢶⣶⣤⣄⡻⣶⠿⢁⣠⣤⣶⢶⣶⣤⣀⠙⠷⠟⢁⣠⣴⡶⢶⣶⣦⣄⡹⠷⠿⢃⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠻⣯⣿⣿⣿⣿⣿⣫⡿⠃⠀⢻⣿⣿⣿⣿⣿⡿⣿⡿⠃⠀⠻⣯⣿⢿⣿⣿⣿⣿⡿⠁⠀⠻⣿⡿⣿⣷⣶⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠒⠒⠒⠒⠋⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⠤⠤⠤⠤⠤⠤⠤⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⣀⣀⣀⣀⣀⣀⡠⠞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠐⠀⠀⠂⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠉⠁⠑⢄⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⠤⠔⠋⠀⠀⠀⠀⠀⢀⠀⠂⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠶⠤⠤⠴⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⢲
⠀⠀⠀⠀⣀⣀⡠⠖⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣯⠉⠉⠁⠙⢆⡼⠒⣖⣒⣒⣒⣖⣒⣒⣖⣲⠒⠒⠒⠒⠒⠒⠒⠒⠒⣖⣛⣛⣛⣋⢙⣉⣋⣙⣹⠉⠁
⠀⠀⠈⢦⠀⠔⣡⠋⠀⠀⢀⡵⠚⠉⠉⠑⢦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡥⠛⠋⠙⠛⢭⠁
⠀⠀⠀⠀⠑⠜⠁⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠠⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠐⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠍⠉⠑⢆⢀⡤⠴⠒⠒⢲⠖⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⢀⡈⠁⠀⠀⠠⠀⠀⠄⠀⠉⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠤⠄⡀⠀⣠⠶⠃⠀⢀⣤⡀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⡘⠰⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣁⠀⠀⠀⠀⠀⠀⢰⠈⡐⢀⠠⠄⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢤⠄⠀⠈⠒⠋⠃⣰⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣇⠀⠀⠈⠀⠀⠀⠈⠒⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⡠⠜⠈⡹⢆⡀⠀⠡⢀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄⠀⠈⡍⠉⠉⠙⠂⠀⢀⠀⠀⠀⢠⣀⠀⠉⢁⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠤⠊⠁⠀⠀⠂⠀⢰⠀⠀⠀⠀⠀⠀⠑⠤⣀⢀⡟⢉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⣠⠤⠖⠚⢿⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠁⣀⠀⣀⡹⠀⠀⠀⡀⠈⠁⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠃⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⣀⠀⠀⣠⠃⠀⢦⡴⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⣀⠀⠀⡗⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠐⠒⠊⠉⠈⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⡄⢀⣄⢡⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠉⠓⠢⠤⣀⣀⡀⠀⠀⠀⠀⠀⡼
⠀⢀⠀⣀⠀⠀⠀⠡⠄⣁⡼⠚⠉⠙⡆⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠛⠛⠛⡅
⢀⠠⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⡧⢄⣀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠘⡀
⠁⠐⠀⡀⠀⢠⠞⠁⠀⠀⠀⠀⢠⠊⠀⠀⠈⠉⠓⠶⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢣
⠀⠢⡈⢁⡼⠁⠀⠀⠀⢠⣀⡴⡏⠀⠀⠀⠀⠀⠀⠀⠨⠑⠂⠀⠀⠀⠀⠀⠀⢀⡴⢇⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⢀⠜⠀⠀⣀⡦⠼⠊⠁⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠘⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣀⣀⡀⢀⣀⣠⠴⠯⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠁⠈⠀⠀⠀⠀⠒⠁⠋⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡬⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢉⡉⢫⠉⠉⠉⠉⠁⠜⢉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢈⡁⠌⠀⠀⠀⠀⠀⠢⣈⣁⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡓⠦⠶⢦⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⠐⠒⠶⠤⠬⠉⣅⢚⡓⠋⠵⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣅⠦⠀⡠⠆⣠⠀⠀⠇⠀⠀⠀⠉⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⣀⣈⠻⣝⣃⣄⠀⠀⢂⡀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣻⢮⡛⢭⣍⣧⡀⠀⠀⠀⠈⠢⠄⡀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢯⡻⢛⣝⣱⠮⠕⡤⠁⠢⢄⠀⠀⠀⠈⠑⢄⡸⣇⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡻⠒⠽⢟⣯⣏⠹⠝⠂⠀⠀⠀⠀⠉⠢⡀⠀⢀⠜⠕⠙⠈⠣⠭⢉⣋⡙⠓⠲⠤⢶⣒⣀⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⢯⡅⠉⣗⡴⢆⣈⠉⠃⠀⠢⡀⠀⠀⠀⠀⠀⢈⡶⡫⠊⠀⢺⠠⡀⠀⠀⣘⣀⠉⠉⢒⠒⠂⠤⠌⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠚⠁⠈⢙⢧⡶⢧⠿⢦⣄⡀⠀⠀⠀⠈⢢⠀⠀⠀⡰⡫⠊⠀⠀⠀⣮⠭⠭⢭⢯⡗⠊⠀⠀⠈⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⡉⠈⠉⠂⠀⠈⡷⠃⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠑⣄⠜⠜⠀⠀⠀⠀⢠⣧⡲⣤⠼⠛⠾⠓⠀⠀⠈⠢⣄⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠉⠙⠛⠒⠒⠒⠚⢷⡄⠀⠐⣭⠶⣄⠀⠉⠳⣄⡀⢀⠎⠎⠀⠀⡀⣠⣶⡻⣾⣿⣦⢝⡻⠟⡃⠤⣀⠀⠀⠀⠉⠢⢄⡀⠀⣧
⠀⠀⠀⡘⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢷⡄⠀⠸⣧⠈⠳⣦⡀⠈⣫⡮⠼⠒⠈⠁⢠⣼⢷⣯⢮⡲⣟⣿⡭⡁⠚⠀⠀⠑⠢⣀⠀⠀⠀⠈⡖⡹⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢷⡀⠀⢹⡂⣀⠴⠛⠉⣀⠠⠤⠐⣲⢾⠩⠿⠄⣈⡟⠛⠷⡌⠣⡄⠀⠀⠀⠀⠀⠈⠑⢄⡠⢎⠜⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⣉⠤⠒⠉⠙⡀⢀⡴⠚⠠⣿⣄⣨⡻⣖⣉⠓⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⡠⣊⠕⠁⠀⠀⡄
⠀⠀⠀⠘⣌⣀⠒⠀⠤⠤⠤⠤⠤⠤⠤⠀⠒⢂⡉⠥⠒⠉⠀⠀⠀⢀⡠⠚⠃⠄⣀⡀⢶⣝⠇⠑⠁⠙⠳⣦⣀⠀⠀⠀⠈⢆⠀⢀⢜⠔⠁⠀⠀⠀⠀⡃
⠀⠀⡠⠊⠉⠐⠨⢍⣀⣈⠠⠒⢋⠵⠒⠭⣁⡌⠀⠀⠀⠀⢀⠤⠚⠙⠲⠤⣤⣤⣤⣤⣼⣯⠀⠀⢀⢠⣀⠀⠙⢷⣄⡀⠀⠀⠳⢣⠂⠀⠀⠀⠀⢀⠴⡇
⠀⠸⡀⠀⠀⠀⠀⠀⡘⠀⢀⡜⠁⠀⡠⠔⠚⠛⠢⡀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢻⡉⠳⣄⡀⠈⠻⣦⣀⢃⣃⣀⡤⠤⠐⠊⠁⠀⡅
⠀⠀⠈⠓⠤⢀⣀⡠⠧⠤⡞⠠⢤⠊⠀⡠⠒⢲⠀⢸⠀⠈⠉⠐⠒⠒⠀⠀⠀⠒⠒⠀⠈⠉⠻⡆⠀⠘⣧⠀⠈⢻⣦⠤⠐⠋⠉⠁⢀⣀⣀⠤⠄⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠆⠀⣌⠀⡠⠊⠠⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢹⠆⠂⣉⠄⢤⠒⠐⠂⠈⠩⡁⠀⡀⠤⠧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣈⣁⠠⠔⠁⠀⢦⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠼⠓⣈⠤⠒⠉⠀⠀⠈⠢⡀⠀⠀⠰⣀⠊⠁⢐⠤⠐⠛⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠯⣒⠠⠤⠉⣉⣉⣉⣉⢉⡡⠤⠄⠲⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣀⡅⠀⠠⠋⢠⠖⠉⢣⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠉⠐⢢⠔⠊⢁⠔⠋⠉⠑⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡇⠀⢇⣀⡠⠊⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠄⠀⢰⠃⠀⣠⠚⠉⢀⡀⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⡠⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠔⠃⠒⡇⠈⠲⠁⢀⠞⠁⢘⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡃⠀⠠⡀⠔⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⡄⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⠶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣟⣀⣀⣀⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠿⢿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣶⣶⣿⣿⡏⠀⠀⠀⠀⠀⠀⢹⣿⣷⣴⣾⣧⣌⠹⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣿⣻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⣿⢻⣿⡇⠻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢂⣭⣭⣖⢤⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⣿⣿⣿⣿⠧⠽⠦⣤⣄⣀⣠⠊⠁⠀⠀⠀⠘⠒⠺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⠟⣁⣤⣴⣶⣶⣶⣶⣿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠶⠮⠐⠠⠄⢉⡉⠛⠻⠧⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡋⠘⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣆⣀⣀⣤⠤⠴⠶⠶⢫⣶⣿⣿⡿⡀
⠀⠀⠀⠀⠀⠀⠸⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢮⣛⣫⣭⣿⣶⣤⠬⠽⠞⠳⣤⣭⣥⣴⣧
⠀⠀⠀⠀⠀⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢿⣅⠈⢧⡴⠶⠶⡬⢴⡉⢀⣤⣤⡴⢿⣷⡀
⠀⠀⠀⠀⠀⠀⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢹⠞⠀⠙⠷⠀⡤⢊⣄⣈⣉⡋⠴⠿⠿⡝⠓⣄
⠀⠀⠀⠀⠀⠀⢻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠖⣩⠏⠰⡏⠉⡉⠉⠉⠀⢈⡿⢋⣴⣿⣧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣑⡦⠀⣀⠏⠰⣶⠶⢶⠋⡴⠋⣀⣽⣉⡠⠉⠹⣆⣀
⠀⠀⠀⠀⠀⠀⠀⣿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⡞⠑⣤⡕⢡⣮⣤⣤⣶⣿⠟⣵⣟⣋⡉⢉⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⡙⣷⠿⢭⣙⠳⠆⣡⡾⣁⣻⠟⣡⣿⣿⣿⡶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⡿⢥⠪⣅⡙⡿⢋⡾⠷⠶⠖⣠⣿⣷⣍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣎⣏⡴⣤⢰⡖⣡⣞⣉⡉⠙⣿⣷⠘⣷⡀
⠀⠀⢀⣠⠤⠖⠛⠉⠙⠳⣮⡳⣬⢹⡿⠙⠛⠻⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠘⠓⠦⣭⣍⣉⣉⣡⣼⣿⣿
⢠⠞⠉⣁⡐⠠⣀⠀⠀⠀⠈⣿⣼⣿⣇⠦⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠙⣿⣿⣿
⡾⢰⣿⠟⠙⠓⣌⢣⠀⠀⢠⣿⣿⠉⢿⣤⣀⡀⠀⣠⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿
⡇⠘⣿⠒⠨⢆⠈⡄⣗⣴⣿⣿⣿⠓⠢⢍⡛⠻⢿⣿⣽⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠉⠙⢻⡿⠿⣿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⣿⣿⣿⡟
⢰⡀⢿⢷⡤⠊⢠⡇⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣍⣋⣿⡏⠉⠙⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠱⢄⡳⠿⠷⠛⣰⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣭⣼⣇⠀⣼⢣⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠑⠊⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⣉⠁⠀⠀⠀⠉⠛⢧⡄⢠⠠⣌⣿⣫⣷⣿⠋⠉⠙⢿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠤⠤⢈⠑⢦⡀⠀⠀⠀⠹⣆⠀⢈⣿⣿⣿⣿⡄⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠰⣴⣾⣃⠀⠑⠀⠙⣄⠀⠀⠀⣻⣶⣿⣿⡘⣿⣿⣷⠀⠀⣀⡴⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⠒⢶⠀⠠⠀⠘⠀⢀⣾⣿⡏⠉⠹⣧⣽⡿⠿⠗⣋⣡⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢻⣿⣶⡋⠀⡄⡀⠀⣛⣿⣿⣿⠃⠀⠀⠛⠛⠓⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠢⣝⣛⢃⣤⠁⣸⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣌⣉⣀⣴⡿⠟⠋
⣿⣿⡿⠛⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡠⠞⠁⠘⠿⡿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢈⢣⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡞⠛⠛⠋⠛⠛⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠻⡏⠉⠛⠒⠶⣤⣠⡀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢠⣾⡇⠰⣄⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⢦⠀⡏⡾⣿⡄⢡⡿⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣧⡠⠤⠶⠒⠲⠖⠂⠀⠛⠃⠃⢧⠀⠁⠀⠀⠉⠓⢒⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢦⠴⠖⠊⠉⢸⣿⣿⣿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢙⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢛⢂⡅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠐⠀⢉⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠃⠰⠖⠔⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⠀⡜⠏⠣⢡⠅⡩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⢀⡔⠸⢯⠗⢬⣣⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡻⣿⣷⡚⠣⡉⠆⡶⠊
⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣿⡦⣦⢣⠋⠀⣉⣌⢶⠀⠀⠀⠀⠀⠀⠀⠠⣿⢿⡿⠏⢷⣪⡅⢋⠘⠇⠁
⠀⠀⠀⠀⠸⢿⡷⣮⡿⢿⠫⢎⣀⠥⠰⣉⣈⣲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⣸⠃⡱⢃⣴⠜⠁
⠀⠀⠀⠀⠠⣓⠼⡮⢜⣀⢲⡭⠆⠛⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠁⠉
⠀⠀⠀⠀⠀⠘⠋⠘⠓⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣘⣛⣿⣿⣷⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⢤⣄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⡆⢿⡿⠿⠃⠚⢿⠇⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣌⠀⠀⠘⡇⣦⡠⠐⢢⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⣆⣿⣶⣾⢋⣷⣤⣀⠀⠉⠓⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣢⡰⠋⢸⡿⢡⣿⣿⣿⣿⣷⣶⣄⣀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠘⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢀⡏⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠀⣀⣾⣧⣠⣐⣖⣲⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠁⠀⠀⠀⠀⠀⠈⣿⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⢌⣿⠹⣿⠇⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠎⠉⠉⠉⠉⠛⠛⠛⠛⢺⡟⠀⠏⣿⠀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠸⡟⠠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠤⢤⠤⠼⣿⣶⣶⣶⣦⣤⣤⡤⠤⣤⣤⣾⡁⠀⠀⢠⠇⢸⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⢿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⡁⠀⣰⣿⣿⡇⠀⠀⠸⢠⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠀⣴⣿⣿⣿⡇⠀⠀⡇⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠈⣿⣿⡇⣾⡇⣿⠯⢍⣿⣿⣿⣿⠟⠋⢀⣾⣿⣿⣿⣿⠃⠀⠀⠃⠘⣿⣿⣿⡉⠻⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⣿⣯⣤⣿⣿⣧⣿⡇⢿⣷⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⠐⣿⣿⣿⡇⠀⠀⠙⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣿⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠀⠀⠈⡇⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿
⠀⣠⣤⣤⣤⡶⠟⢛⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢛⣩⣽⣿⣿⣿⣿⣿⣿⣿⠙⣷⠀⠀⠀⠠⡟⢻⣿⣿⡷⠀⠀⠀⢴⡿⠿⠏
⣰⣿⡏⣴⠋⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣋⣥⣶⣿⣿⣿⣿⣿⣿⡛⢻⣿⣿⡇⠀⢸⡇⠀⠀⣀⣼⣿⣿⣿⣷
⣿⣿⣇⣻⣀⣀⡀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣷⣀⡿⠀⠀⠺⠿⠿⠟⠛⠛⠁
⢻⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⡟⠀⠈⣿⣿⣿⣿⣿⣿⠿⠛⠉⠈⠉⠉
⠈⠻⠿⠿⠿⠛⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⢻⣇⣸⣿⣿⡇⠀⠀⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣧⠀⢠⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠙⠛⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⣤⣤⣴⡶⠶⠿⠿⠿⠿⠿⠿⣷⣶⡶⠾⠿⢶⣦⡀
⠀⠀⠀⠀⢰⣿⠻⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣿⣶⣶⣶⣿⣧⣄⣀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⣿⠇⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⣀⣠⣴⣾⡿⠟⠛⠉⠉⠉⠉⠉⠉⠉⠛⠻⣿⡛⠛⠛⣿⡇
⠀⠀⠀⢸⣿⠀⢀⣠⣶⠟⠋⠁⢰⣿⠟⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢠⣿⠇
⠀⣠⣶⠾⣿⣴⠿⠛⠁⠀⠀⠀⣿⡏⠀⠀⠀⣀⣤⠶⢞⣛⣛⣛⣛⠻⠶⢦⣤⣀⡀⠀⠀⣿⠇⣾⠇
⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⢀⣴⠾⠋⠁⣰⡏⠉⠉⠉⠉⠙⠛⠛⠻⠿⢿⣷⣾⣟⣸⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠛⠛⠋⠁⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣆
⠻⣷⣄⣀⡀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠉⢹⣿⣿⣦⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⢿⣿⣿⣶⣶⣄⣠⣿⡇
⠀⣴⣿⣿⣿⡿⠀⠀⠸⣿⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠿⠿⠿⣿⣿⠛⠃
⢰⣿⡇⠉⠁⠀⠀⠀⠀⢈⣽⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠉⠀⠀⠀⠀⢀⣤⡾⠿⢷⣦⡀
⠘⣿⣧⣤⡀⠀⠀⠀⠀⣸⡟⠙⠻⢿⣷⠀⣠⡶⠶⣶⣄⠀⣸⡟⠀⠀⠀⠀⢀⣤⡾⠟⠁⠀⠀⠀⠉⢻⣷
⠀⠀⣿⡏⢻⣷⠀⠀⠀⣿⣇⠀⠀⠸⣿⡀⣿⣀⠀⣨⣿⠀⣿⡇⠀⠀⢀⣶⠟⠉⠀⠀⠀⠀⠀⣀⣴⣾⣿⣦⣀
⠀⣼⣿⣇⣴⡿⠀⠀⠀⠙⢻⣿⣶⣄⠹⣷⣌⠛⠛⠛⢁⣴⣿⡅⢀⣴⠟⠁⠀⠀⢀⣄⠀⣠⣾⣿⣿⣯⣤⣙⠛⠿⣶⣤⣄⣀⣀
⠀⣿⡇⠉⠉⠀⠀⢀⠀⠀⣸⡏⠙⣿⠆⠈⣿⣿⣷⣾⣿⠿⠛⣷⣾⡏⠘⠷⠶⠾⠟⣫⣾⣿⣋⣁⠉⠉⠉⠙⠻⢷⣦⣈⡉⠛⠛⠛⢿⣷⡾⠷⣦⣀
⠀⢻⣷⡀⠀⠀⢀⣾⠀⢰⡿⠿⠾⠋⠀⠀⣿⠀⢠⣶⣶⣦⡀⠘⢿⣿⣶⣶⣶⣾⠿⠟⠛⠛⠛⠛⠛⠷⢶⣤⣀⠀⠈⠙⠻⠿⠿⠟⠛⠉⢀⣠⡿⠟⢷⡀
⠀⠀⣹⣿⣷⣶⠟⠁⠀⢸⣷⠀⠀⠀⠀⢠⡟⠀⣿⡟⠁⠙⢿⣆⠀⠻⢷⣄⡀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣌⡙⠛⠷⠶⣶⣶⣶⣶⡶⠾⠛⠋⠀⠀⠈⣷
⠀⠀⢻⣿⣍⠀⠀⠀⠀⠀⢿⣷⣄⣀⣠⡟⠁⠀⠻⣧⣄⣀⣈⣻⣷⡄⠀⠙⠻⢶⣤⣼⣿⡋⠀⠀⠀⠀⠈⠙⠻⢷⣦⡀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⠾⢿⡇
⠀⠀⠀⠻⣿⣿⣦⣄⣀⣀⣼⣿⡿⠛⠋⠀⢀⣤⣀⠀⠉⠉⠛⠛⠛⠁⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠷⠶⠶⠶⠿⠿⠶⠿⠿⠟⠛⠛⠋⠉⠁⠀⠀⢈⣷
⠀⠀⢀⣤⣾⠉⠉⠉⠙⠛⣻⣿⣿⣤⡀⠀⢻⣭⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣿⣶⣶⣦⣤⣄⠀⢀⣶⠶⣦⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣿⠿
⠀⢠⡿⣻⡏⠀⠀⠀⢠⣿⣯⣿⣤⡟⠛⠿⣶⣦⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⠿⠟⠛⠋⠉⠉⠉⠙⠛⠻⠿⣿⣿⣦⡛⠛⢋⣠⣴⠿⠛⠛⠻⣿⣯⣭⣁
⠀⣾⣿⣿⡇⢰⡟⢻⣎⠛⣿⡿⣿⠁⠀⠀⠀⣠⣾⣿⡟⠛⢿⣿⣭⡉⠁⠀⠀⣠⣴⣶⡶⠛⢻⣶⢶⣤⡀⠀⠙⠻⢿⣿⣿⠛⠛⣷⣤⠀⠀⠈⠙⢿⣽⣿⣦
⠀⢻⣟⣿⣧⡈⠻⠟⢃⣰⡿⣾⣿⠀⣠⣤⡐⣿⣿⣾⣷⡶⠾⠿⠿⠃⢀⣤⣀⠻⢿⣾⣿⣄⣾⣿⡿⠟⢡⣤⣤⡀⠙⠷⠿⠷⠾⢿⣿⠇⠀⣠⣤⡈⢿⣿⣽
⠀⠈⠻⣮⡿⣿⣿⣿⣿⣿⡿⣿⣯⠸⣿⣬⡿⠀⠉⠀⣀⣀⣀⣀⠀⠀⣿⣉⣿⠂⠀⢀⣀⣩⣁⣀⠀⠀⢿⣧⣽⠇⠀⢀⣀⣠⣤⣀⡀⠀⢸⣯⣹⣿⠈⣿⣧
⠀⠀⠀⠈⠛⠿⠶⠷⠿⠛⣿⡽⣿⣦⣬⣭⣤⣶⡾⣿⡟⠙⢿⡿⣿⣶⣬⣭⣥⣶⣾⢿⡿⠋⠻⣿⣿⣶⣤⣬⣥⣤⣾⣿⣿⠃⠹⣿⣿⣷⣦⣭⣉⣡⣼⡿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⣿⣻⣯⡾⠋⠀⠀⠈⠻⣮⣟⣿⣿⣿⣻⣵⠟⠁⠀⠀⠙⠷⣯⣿⣿⣿⣯⣷⠟⠁⠀⠀⠙⠿⣯⣟⣿⣿⣿⣯⡾⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣯⣥⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡛⠻⣿⣿⣶⣶⢦⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣟⢂⠀⠀⢩⣭⣿⣧⣤⣤⣤⣤⡀
⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⣠⣾⢀⣯⣿⠈⠀⠀⠛⣚⣦⡉⠉⠛⠛⠓⡻
⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⢀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⢃⠆⣼⢻⡰⠀⣰⣤⢰⣷⠰⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⡥⢠⡾⠙⠷⠃⣸⣟⡉⣸⢿⢀⠃⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⣀⡤⠖⢾⠟⡱⣂⡴⠏⠁⠀⠀⣴⠏⠋⣵⢻⣜⡌⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠉⣵⠲⠟⠁⠀⣀⣴⢞⡁⣉⡽⠏⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⣻⢀⢋⣵⠏⠁⠀⠀⠀⢱⠄⠀⠀⠀⢸
⠀⠀⠀⠂⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣄⣀⠀⢠⠏⠀⠀⠀⠀⠾⠋⠁⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⢚
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣦⣤⣀⣀⣀⣠⡀⢀⠄⠫⢙⣛⡿⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣋⣤⣶⣶⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠉⠉⠁⠀⠀⠈⠉⣷⠈⠄⠀⠀⠈⢹⠋⣨⠟⠻⢷⣦⣤⣤⣤⣴⣖⣯⢥⠖⠫⣝⡟⢁⡴⠒⠓
⠀⠀⠀⠀⣮⣿⣶⣦⣧⣤⣤⣤⣴⠒⠻⣷⣺⡃⠀⠀⠀⠀⠁⠸⢰⡇⠈⠂⠢⣼⣫⠽⠛⣫⣅⡀⠀⠀⠀⠈⢆⣼⠰⠿
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠉⡌⠖⠈⠁⠀⣸⡄⠀⠀⠀⠀⠀⠙⠻⣤⣀⠀⡀⡏⢠⠞⣁⠈⢱⠀⠀⠀⠀⠀⠙⠦⣀⠤
⠀⠀⠀⢸⠠⠀⢠⠴⠒⠘⣿⣿⣏⣡⡴⠾⣩⡻⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢼⡈⠛⠁⣸
⠀⠀⠀⢘⠀⠀⠸⣀⠔⠊⠅⢀⣀⣧⠀⠀⢎⡔⠣⢉⢗⡲⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠉⠀⠀⢀⠻⣀⡴⠞⢏⡭⢅⡘⠒⠒⠘⠬⠥⢜⣺⠑⠂⠈⠑⢦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⢱⠤⢄⣉⠔⠁⢀⠀⠘⢱⠛⠎⠽⡑⣲⠤⣀⡀⠀⠀⠀⠐⠤⠀⣹⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠈⠀⠀⠇⠀⠀⠀⠑⠒⠊⠢⠤⠰⣑⠏⠛⠀⠉⠳⣬⣤⣤⠤⢖⡽
⠀⠀⠀⢰⡀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢈⣷⢶⠒⢏⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠁⠠
⠀⢀⡠⠤⠷⣤⣅⣀⣀⣉⣉⣙⡓⠒⠶⠤⠭⠭⠥⡤⠤⠖⣹⢏⣁⠀⡈⣧⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⡰⠁⠠⢓⣤⠊⠉⠀⠐⠒⠠⠍⠭⢍⣛⣛⡒⠲⠶⢖⠖⢫⣤⣎⠹⡉⠒⠃⡠⠃⠀⠀⠀⠀⠂
⠯⣉⣱⡎⢧⠘⠮⠖⣩⣏⠢⣦⡄⢒⣦⢁⠠⣈⠭⣥⡀⢀⢹⡌⢧⠈⠉⠉
⠀⠀⠀⠀⠀⠑⠒⠚⠁⠈⠢⠤⠴⠋⠘⢦⣉⣁⠴⠉⢆⠑⠛⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣄⣶⠠⡆⣀⢠⡄⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣹⠆⠀⠙⡇⠀⣧⢿⣾⢀⡾⣴⠇⣰⠶⠶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠰⣏⡤⠀⢻⡓⠶⠼⡇⠀⠙⠈⠃⠘⠡⡏⢰⡋⠛⢀⣾⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡘⣆⠘⢧⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠏⢻⡀⣠⣟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⡈⠙⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⠔⠀⠀⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⡔⠁⡘⠀⣀⠀⠀⠀⠐⠄⠀⢀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠳⡬⠇⠀⠀⠀⢨⠃⠀⢀⠘⠌⠀⠀⠀⠀⠀⠈⠑⠰⠖⠒⠶⠀⠏⠂⠀⠀⠀⠀⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢿⠶⡄⠀⠀⠀⠗⠀⠖⢀⣰⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⢂⡒⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠂⠈⠐⠙⠃⠀⠚⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣐⢦⣀⣀⣄⣀⠀⠀⠀⢠⠖⠰⠀⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣀⠈⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣠⠶⠈⠇⠀⠀⠀⠉⠀⠀⠀⣈⢁⠂⠄⡰⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⡀⠋⢉⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢃
⡄⠀⢀⡴⠖⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⢀⡈⠑⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠼⠀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢹⠀⠈⣷⠒⠀⠀⢻⡀⠀⠀⠀⠀⢀⡠⠏⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⢀⠤⠀⠄⠠⠗⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠸⠞⠁⠘⠒⠃⠀⠀⠀⠀⢀⣀⠒⠄⠡⠈⡀⠉⠀⠀⠀⠀⠀⠀⠴⡈⠉⠑⠐⠒⠀⠈⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣀⡀⠠⢀⠅⠘⢄⡐⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⡜⠑⡊⠀⠀⡄⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⡠⠀⠒⠀⠉⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⢁⡀⠄⠂⠀⡄⢠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠰⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢀⠔⠂⠁⠉⠀⠠⠂⠈⠁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠑⠄⠀⣀⡀⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠂⠐⠂⡖
⠀⠀⠀⠀⠀⠀⠀⠊⠁⡠⢄⡈⠰⠠⠀⣀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠴⢀
⠀⠀⠀⡘⠀⠀⠈⢧⠾⠁⠀⠁⠂⢤⡀⠀⡈⠛⠠⢦⠄⠰⢶⣌⠀⠀⠀⠂⠀⠀⢀⣀⡀⠀⠠⠠⣠⡄⣀⠀⣀⡀⢀⠡⠐⠾⠀⠀⣤⡀
⠀⠀⠀⠇⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⢆⣈⣠⠃⠘⠋⠐⢦⡀⠐⡇⠩⠀⠀⠀⠀⠀⣿⠀⠀⠀⡠⠐⢆⡀⠈⢀⠖⠀⠀⠈⠠
⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⢠⠊⡣⠀⠀⠀⠀⡆⠀⠀⠉⠀⡀⠀⠀⠀⠀⠙⢂⠡⠐⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢨⠀⠐⠃⠀⢄⠶⡀
⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡌⠀⠀⢀⣤⡀⠀⠀⡆⠈⠙⠒⠀⠈⠀⠀⠀⠐⠀⠀⠀⠘⠀⢀⠀⠀⠛⠲⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠈⠳⠤⠃⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠣⣄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡀⠠⠢⠀⠀⠀⠀⠀⢀⢠⡠⢀⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠈⢠⠀⠀⠀⠀⠀⡀⠁⢈⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠀⠠⠈⠉⠀⠀⠀⠀⠀⠈⠄⠐⠀⠒⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⢌⠚⠞⠁⠀⠀⠀⠀⠀⠆⠈⠠⣀⠀⠂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠍⢐⠀⡰⠀⠀⠀⠀⠀⠀⠰⠂⠠⠀⠋⢉⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⡂⢄⢠⠀⠀⠀⠀⠀⠀⡰⡒⠐⡀⠊⠑⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠢⣠⠀⠀⠀⠀⠀⠀⣀⣴⣂⠐⠀⢊⠀⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡉⠃⠀⠀⠀⠀⠀⠀⠀⡀⡀⠘⣀⣀⠄⠉⠔⢤⡄
⠀⠀⠀⠀⠀⠀⠠⡴⠎⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠢⠆
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠒⢰⠄⠀⠈⠒⠁
⠀⠀⠀⠂⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⢠⠉⢀⠠⡀⠡⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠸⠴⠃⠀⠉⠀⢤⡀⠀⠀⠂⢨⣅⠐⢂⣄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⣄⡴⠃⠈⠃⠑⢦⡀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⡠⠚⢧
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠎⠕⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢐⡞⣦
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢀⣱⢳⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣱⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠉⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣦⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣀⠉⠛⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠉⠉⠓⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠓⠲⢤⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠛⠶⢤⣈⠹⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣛⠉⠳⣄⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⢿⡛⠙⢯⡀⠀⠈⠓⢦⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠉⠁⠀⠀⠙⢦⡀⢻⡄⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣶⣿⣿⠃
⠀⠀⠀⢀⣴⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⣠⡿⠃⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⢀⣼⠏⠀⠀⠀⠀⠀⠀⢨⣿⣿⠁⢀⣴⡿⠿⠛⠻⠿⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⣿⠿⠟⠛⠻⠿⣿⣭⡉⠉
⣾⣇⡀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣴⡟⠁⠀⣀⣤⣤⣀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⣀⣤⣄⡀⠀⠙⣿⡄
⠘⢿⣿⣶⣤⣀⣀⣀⣼⣿⡿⢰⡟⠀⠀⣾⠋⠀⠈⠙⣧⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⡾⠉⠀⠈⠻⣆⠀⠸⣿
⠀⠀⠉⠛⠿⢿⣿⣿⣿⡿⠁⢸⡇⠀⠀⢷⠀⠀⠀⢀⡿⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣇⠀⠀⠀⢠⡏⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⣄⠀⠈⠙⠒⠒⠋⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠓⠒⠚⠋⠀⢀⣼⠏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⣀⣀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠄⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠑⠚⢃⣸⣧
⠀⠀⠀⠀⢀⣴⣿⣶⣈⣥⣖⣀⣀⣐⣦⣄⡉⠻⣷
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣀⣹⠆
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣧⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⢆
⠀⠀⠀⠀⠀⢸⣿⣿⠟⠋⠁⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⢘
⠀⠀⠀⠀⠐⠛⠋⠁⠀⠀⠀⣀⣤⣴⡿⠟⠋⠉⠈⣿⠟⠋
⠀⠀⢠⠁⠀⣦⣤⣤⣴⣾⣿⠿⠉⠁⠀⡀⠀⠀⠀⠙⡦
⠀⠀⠘⠀⠐⠠⠚⠹⠘⠛⠛⢃⡀⠀⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠈⠐⠂⠂⠀⠀⠀⠐⢀⠣⡈⡆⠀⠀⠀⣠⣜⡄⠀⠀⠀⣠⠀⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⢱⡸⡀⣰⢋⠏⢛⣚⣦⡄⠰⠏⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡇⣤⢼⣯⣾⣿⣿⣶⡝⢯⣴⣯⣤⣄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣿⣿⣿⣿⣿⢻⣿⣿⡛⢷⣄⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⢟⣉⠏⠀⠀⢀⠀⠀⠀⠀⢠⠄⠐⠀⠉⠉⢁⣒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⣾⣿⣶⣷⣮⣾⣿⣿⣧⠤⢐⣒⣭⣽⢶⣾⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣭⣶⣿⢿⠿⠛⣛⣭⣿⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢉⣾⢿⠿⠋⢈⣤⠾⢟⣋⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⡿⠛⠉⣀⣤⠶⠋⣉⣥⣶⣿⣿⣿⠷⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠿⠿⣛⣉⣥⠶⠞⣛⣩⣤⣶⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢦⣭⣭⠥⠤⠶⠖⠛⣉⠥⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣒⠢⣔⡲⠎⢛⣁⣀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠂⠐⠧⠉⠀⠀⠁⠠



⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠉⠉⠉⠉⠁⠈⠫⣉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⢲⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠐⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠖⠛⠒⠒⠒⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠧⣤⠤⡀⠀⠀⠀⠀⠈⢆⠉⠉⠉⠁⠀⠀⠀⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡼⢻⡿⠒⠷⠤⠃⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⣇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⢂⢴⢸⣏⣳⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⣿⠁⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢢⠞⣠⠏⣼⣾⣇⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢲⠤⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠏⢠⣏⡰⢣⡿⠤⣧⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡳⠠⠐⣸⡗⡦⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⠁⣠⠟⢘⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢡⡴⡋⠀⢾⣩⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠋⠡⣨⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠛⢄⡴⡄⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⣾⠿⠋⠑⡤⢦⠈⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⢶⠿⠟⠛⠉⡁⠀⣤⠶⡀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡇
⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠀⠈⡴⢦⠀⢟⡹⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠁
⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠾⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⣒⣒⠲⢤⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⠩⠅⠒⠈⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⢶⠤⠍⣑⡒⠂⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⢒⡩⠥⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠸⡉⠒⠒⠀⠭⠭⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠵⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠲⠤⢼⠇
⠀⠀⠀⠀⠀⢟⠓⠤⡀⠀⠀⠀⠀⢀⡠⢶⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣤⡀⠀⢸
⠀⠀⡰⠉⠑⠚⠑⠢⡬⠽⠋⢉⡠⠟⠓⠚⠓⠛⢶⡒⠦⠤⣤⠠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⣀⠀⠤⣤⠤⠚⠚⠓⠛⠶⢎
⠀⢼⠀⠀⠀⠀⠀⠀⠑⢦⡴⠁⢀⡤⠖⣒⠲⢤⡀⠈⢦⠤⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⢤⠞⠁⣀⠴⢒⡒⠢⢄⠀⠑⡄
⠀⠈⠓⢄⡀⠀⠀⠀⠀⡿⠀⢠⢃⠞⣁⣤⣈⠲⡘⡄⠀⣦⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡎⠀⡴⠡⠊⣡⣄⡙⢦⢱⠀⠘⡀
⠀⠀⠀⠀⠈⠑⠒⠰⠊⡇⠀⢠⢸⡀⢿⣥⡿⢀⡇⡇⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢇⠰⣯⣼⠇⢸⢀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⢦⡑⠤⠤⠤⢊⠔⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⢌⠲⠤⠤⠔⣃⠎⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠈⠑⠒⠊⠁⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠉⠒⠒⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⡀⠀⡀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣶⢾⣿⠓⠛⢰⠈⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣟⢘⣋⡁⠀⠈⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⢘⣭⣅⠀⠀⠗⠊⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡨⡶⠾⡄⠀⠘⡁⠜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣿⣿⣿⣶⠜⢛⣝⢆⠈⠰⢎⡹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣾⣥⣔⠶⢟⣢⠌⠡⢎⡽⢶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣜⡛⠻⠿⠿⣿⣿⣿⣿⣾⣯⣕⣾⣶⣌⠠⠟⠁⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠐⡬⠷⠒⠲⢬⣛⠿⣿⣿⣿⢸⣿⣿⣇⢀⣀⣠⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡘⠀⢀⠇⠀⠀⠀⠀⠰⠛⠀⠛⠭⢉⡍⠁⠺⠀⢠⣎⣁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢣⠀⢠⠀⢠⠧⠤⠴⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠍⣙⣙⠓⠒⠶⠤⢤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡢⠄⡠⠄⡄⠀⠉⠈⠁⠐⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣄⣉⢯⡧⠄⠀⠠⣀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⣽⡾⡕⡶⡗⠄⠀⠀⠀⠑⠢⢀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⢟⣾⣺⠺⡠⠄⠁⠀⠁⠢⡀⠀⠀⡱⡹⠒⢖⡒⠶⠶⠤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢼⠜⢓⡠⠎⡉⠓⠀⠀⠀⠀⠀⠀⠀⠠⡪⠊⠀⡆⡀⠀⢈⠃⠐⠀⡖⠈⠭⠉⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡔⠉⠀⣙⢶⠛⠟⠢⣄⡀⠀⠀⠑⢄⠀⢀⠴⠊⠀⠀⢰⠍⠉⢽⡷⡃⠁⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠐⠫⢄⣀⣀⣀⣨⡏⠀⠀⢄⡀⠙⠦⣄⠀⠀⠠⠁⠁⠀⠀⣀⣾⣮⣓⣭⣩⣍⠀⠀⠈⠐⠤⡀⠀⠈⠆
⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢻⡙⠢⣄⠈⠳⣠⣠⠀⠀⠒⣸⣿⢮⣻⣿⣕⡊⠢⠈⠒⠄⡀⠀⠀⠑⢤⢺
⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠱⡀⠀⣧⢀⠨⠗⢈⡁⠀⠤⡴⠎⢿⠌⣑⠟⠯⡙⢌⠀⠀⠀⠀⠈⠢⢄⡠⡢⠀⠆
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⡈⠅⠂⠀⠡⣀⠔⠛⠸⢅⡨⣻⡮⡑⠀⠀⠀⠑⢄⠀⠀⠀⡠⡮⠊⠀⠀⡆
⠀⠀⢀⠷⢈⠡⠀⢀⢀⣐⢀⣀⠠⠍⠄⠂⠁⠀⠀⢀⣤⡊⠉⠀⠒⠈⡧⠁⠀⠀⠉⠓⢤⡀⠀⠀⠠⣀⠎⠊⠀⠀⠀⢀⠁
⠀⠐⠁⠀⠀⠈⠐⠒⠀⡴⠊⠀⡁⠾⢄⠀⢀⠐⠀⠀⠀⠈⠉⠛⠛⠛⠛⡄⠀⠠⡗⢤⡀⠙⠦⡄⠀⠁⠁⠀⣀⠠⠐⠈⠂
⠀⠈⠐⠤⢀⢀⠸⠀⠜⠀⡔⠁⡠⠠⡄⢰⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠒⠻⡄⠀⢻⠀⠙⣢⡀⠼⠓⠊⠉⣀⡀⠠⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠀⢈⡀⠔⢀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠨⠖⣈⠄⠀⡂⠈⠁⠈⡁⢀⡄⠚⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⠄⠂⠁⠀⢣⣒⡠⠄⠀⠠⠤⠀⠀⠠⠄⢐⡊⠅⠂⠉⠀⠀⠀⠑⠠⡀⠀⠲⠁⢀⠝⢀⠈⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠂⠍⣀⡀⠒⡃⠔⠊⢅⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠣⠀⠁⢰⠁⢀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠌⠀⡔⠀⡠⠒⠉⡉⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⣉⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠐⠁⠘⠀⠐⠀⡐⠉⡈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠃⠊⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⠿⠟⠟⠟⠟⠟⠟⠟⠛⠻⠻⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⢿⣿⣿⡶⠶⠶⠶⠶⠷⠶⠷⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠇
⣶⣶⢶⣶⢶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⡇⣿⣿⠿⠛⠛⠿⠿⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢃⣿⠁⠀⠀⢀⣀⣀⣤⣤⠀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⠀⠀⠀⠀⣉⣥⣿⡿⠃⣼⣿⠀⠙⠣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣀⣀⣤⣈⠀⠠⠤⠤⠴⠶⠾⠛⣿⣷⡄⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⣙⣛⡻⢷⣶⣦⣄⡀⠀⠀⠘⣿⣿⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠠⣮⣭⣛⡿⣿⣿⣿⣿⡆⠀⠀⠘⠿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⡇⠀⠀⠀⠀⣰⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⣽⣛⡶⣤⣀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⣿⡎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢒⡁⢶⣤⣄⡈⠙⠳⢿⣿⣶⣤⣠⡟⠀⣴⣶⡆⠀⣸⣿⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⡚⢥⣀⠈⠻⢶⣬⡻⣿⣷⣦⣀⠉⠻⢟⣿⣅⠀⣙⣘⣡⠔⠈⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠤⠤⠐⠒⠉⣁⣠⣾⣿⣶⣝⢷⣄⠀⠻⣿⣾⣿⣿⣿⢇⡴⠋⠉⠉⣛⣋⠉⠀⢀⣀⠀⠉⠛⠛⢄
⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠈⠀⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣷⡀⠘⣿⣿⣿⠟⠋⣠⣴⣾⣿⣿⡟⠀⣴⣿⣿⣿⣷⣦⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣇⣾⣭⣵⣶⣖⣠⣤⡄⣀⣀⠀⠀⠀⠙⠻⣿⣷⣠⠿⠋⠁⠲⠿⠿⠟⠛⠛⠉⠀⠀⠉⠉⠉⠉⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢟⣛⣛⠿⠿⠿⠿⠿⠿⠿⠿⠶⠄⠀⠋⠉⢀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⣠⡤⠊
⠀⠀⠀⠀⢀⡤⠚⠉⠁⠈⣩⠟⠛⠛⠛⠛⠛⠛⠛⠛⢻⡛⠛⣻⣉⠭⠭⠑⠐⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⠈⠉⠒⢄
⠀⠀⠀⢀⠎⠀⠰⠆⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠆⠀⢳⡀
⠀⠀⠀⢸⠀⠰⠇⠊⠠⠄⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠄⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠤⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠠⠄⢘⠸⠆⠀⡇
⠀⠀⠀⠘⡄⠀⠶⠑⡀⠒⣋⠉⠀⣰⠀⢧⠈⠳⠖⣂⠄⢒⡜⠋⢀⠇⠸⡀⠙⠶⢒⠀⢐⣆⠋⠁⡼⠈⡆⠀⠉⣑⠂⢐⠊⠖⠀⣰⠃
⠀⠀⠀⠀⠘⠦⣀⠈⠉⠀⠁⣀⠔⠁⠀⠈⠳⣄⡀⠉⠁⠈⢀⡤⠊⠀⠀⠑⢄⡀⠈⠁⠈⢀⣠⠞⠀⠀⠈⠦⣀⠈⠀⠉⠁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⢀⣤⣤⣄⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢸⣿⣿⣿⠸⣿⣏⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢁⣾⣿⣿⣿⡄⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣠⣿⣿⣿⣿⣿⣧⠘⣿⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⢿⣯⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣌⠛⢿⣿⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⣾⡿⢃⣴⣬⠉⣙⡛⠛⣃⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⠟⣠⣿⡿⢡⣾⡿⢃⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⠋⣴⣿⠟⣠⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣛⣛⣛⣛⣛⣛⣁⣛⣛⣁⣘⣛⣃⣘⣛⣋⣐⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⠄⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣡⣤⣤⣤⡙⠻⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠿⢋⣡⣤⣤⣌⡉⢡⣾⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠛⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠛⢿⣿⡆⠹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⣸⣿⡿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⡨⠭⠭⡲⡄
⠀⠀⠀⠀⣠⣲⣤⣶⣾⣿⣷⣶⣶⣶⣎⠀⠀⠀⠀⠀⠀⠀⢸⠲⡄
⠀⠀⠀⠀⡟⠿⣿⣿⣿⣿⠿⠿⠿⠿⢿⠒⠒⠒⠢⣄⠀⠀⠈⡆⣇
⠀⠀⠀⠀⡷⠦⣤⣀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣤⣄⣨⣇⣠⣤⣷⢿⠁
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⡾⢟⠉⢹⠉⠀⠸⡘⣆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⣀⠈⣇⣠⣴⡿⢟
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣫⣿⠟⠹⠁⠀⢸⡸⡀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡈⠁⠀⠀⢇⠀⢈⣶⣿⠂
⠀⠀⠀⡜⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢣⢐⣿⣿⠾⡟⠋⠹⡈⢆
⠀⠀⢰⠁⠀⠙⣄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⡌⠚⠂⠀⠱⡀⢀⣳⡾⣇
⠀⠀⣾⠀⠀⠀⠘⡆⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢱⡀⣾⣿⡶⣷⠛⠉⢣⠘⣆⡀
⠀⢠⣿⠀⠀⠀⠀⠘⡈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⢃⠋⠉⠀⠈⣆⠀⣀⣷⡟⢣⡀
⠀⢸⣿⡆⠀⠀⠀⠀⢣⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⡄⠀⣶⣶⡾⢿⠋⠁⠘⢦⣹⣶⡀
⠀⢸⣿⣿⣦⣀⠀⠀⠈⡜⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡰⡄⠫⠍⠀⣀⠱⣄⣠⡾⠛⠣⣹⣶⣄⢀
⠀⠘⣿⣿⣿⣿⣿⣶⣤⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢆⠀⠊⢻⠟⠋⠙⢦⣀⣴⠟⠓⢤⣽⠗⠦⡀
⠀⠀⢹⠿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠳⣌⠉⠀⢠⣶⣾⠟⠉⠒⣤⣴⠋⠉⢲⡼⠤⣀⡀
⠀⠀⠈⣧⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠦⣴⡛⠏⠀⣼⢿⡏⠀⣧⢖⠉⠀⠀⠀⠈⠉⠒⠶⢤⣀
⠀⠀⠀⣿⣷⣦⣄⡈⠑⠒⢭⣝⡻⠿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠽⠯⢅⣜⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠈⠛⣿⣿⣿⣿⣶⣾⠙⣿⣷⣦⣍⡻⢿⣿⣿⣿⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡆
⠀⠀⠀⠀⠀⢹⡿⠻⠿⢿⣿⢶⣾⣿⣿⣿⣿⣶⣍⠻⢿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⠀⠀⢰⣿⢃⡀⠀⠀⢀⣀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣇⣀⣀⢸⠇⠀⠉⠻⣿⣿⣿⣃⡈⠻⣶⣬⣙⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣭⡚
⠀⢀⡤⠴⠶⢿⠟⠛⠉⠙⠦⣀⣠⣤⠼⠛⣋⣩⣽⢢⠉⢻⣉⠍⠙⡶⠲⠯⠭⣍⣛⣛⡛⠛⠛⠛⠛⠛⠛⣛⣛⣋⣭⣭⣵⣶⠞⠿⡛⠋⠉⠑⢄
⢠⠋⠀⢀⡠⢄⡙⢦⠀⠀⠀⠹⣭⣴⣶⡟⣏⠉⠀⣸⠀⣸⣤⣆⣠⠏⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⠛⢻⡟⠙⢿⣿⣿⣿⣿⠏⠠⢄⠈⢆⠀⠀⠈
⣇⠀⢀⡏⠀⣴⡝⡌⡖⠃⢀⣠⡇⠀⠻⣿⣮⣙⣚⣥⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⢀⣀⣠⣬⣷⣶⣶⣾⡀⠀⠀⢉⣻⣿⠿⠸⣗⡀⠇⠨⡆
⢏⠁⣀⢷⠛⣿⢃⡇⢸⠀⠀⣀⡇⠀⠀⠈⠛⠻⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠈⠙⢿⡿⠿⠟⠛⣶⢛⣋⣥⣴⣶⠛⠛⠁⡇⣼⣏⣤⣤
⠘⣏⢀⣴⣿⣿⢪⡀⣾⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⠒⠒⢢⡀⢳⠖⠀⠀⠸⡛⠉⠁⠙⢿⣷⣶⣾⣿⣿⣿⡿⠋
⠀⠈⠻⢷⣿⣿⡦⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢷⣦⢿⣷⢓⠀⡗⠋⠀⠤⠇⠀⠀⠀⠀⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠘⢦⣞⣠⠏⠀⣇⣠⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⣷⢀⣼⡿⠟⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠉⠁⠀⠉⠀⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠁⠀⠀⠖⠠⢀⡀⢸
⠀⠀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠐⠠⡀⢠⠛⠁⠀⠀⠀⠀⠀⠀⠠⠚⠛⠚
⠀⠀⠁⠀⠀⠈⠀⠈⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠲⠄⠉⣀⡀⠀⠀⠀⠀⠶⠤⢠⡤
⠀⠀⣷⠶⠶⣄⠀⠀⠀⠀⠈⠀⡀⠀⠀⣀⣤⡄⠀⠀⠁⠀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠑⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠘⠻⠏⠀⠈⠲⢄⡀⠀⢱⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⡃⠀⠆⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢀⠀⠘⠿⠛⠲⠶⠖⠉
⠀⠀⠀⠀⠀⠀⠑⠀⠊⠀⠀⣾⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠘⠂⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠈⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠐⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠿⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠔⠀⠠⠐⢶⢶⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠉⠒⠂⠤⠄⠀⣀⣀⣀⣀⡀⠀⠤⠐⠶⣈⠀⠀⠀⠀⠀⠡⠤⠔⠁⠀⠀⠠⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⠀⠈⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⢇⠀⠀⠀⠀⢠⣿⣿⠔⠁⠀⠀⠀⠀⠁⠢⢀⣾⣦⠀⠀⠀⠀⠀⡹⠛⠁⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠈⠧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⡐⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠒⠒⠲⠆⠓⠄⠀⠀⠐⠁⠈⠒⢤⡀⠀⠐⠄⡀⡀⢀⡴⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠦⣀⠀⠒⠒⠀⠀⠄
⡠⠦⠡⠤⠄⠀⠀⠀⣀⣰⡁⠀⠀⠀⠀⢀⠀⠀⠄⠐⠀⠁⠀⠀⠀⠀⠈⠑⠐⠢⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠈⡂
⠁⠀⠐⠒⠒⠒⠀⠀⠀⠀⣩⣭⣬⣄⠈⢉⣁⢀⣀⣀⡀⠐⠒⠒⠒⠂⠀⠠⠠⠤⠤⠤⠀⠀⠈⣁⣀⣁⡀⠒⠀⠁⠀⠀⢀⣀⠠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠑⡄⢡⠀⠀⠀⢡⠀⠀⠀⠉⠉⠉⠉⠁⠀⢀⠴⠒⠒⠶⣖⠲⢄⠀⠀⠠⡤⠤⠄⠀⠒⠈
⠀⠀⠀⠀⠀⠄⠀⠀⡇⠀⠀⡎⠉⡆⠀⠀⢠⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⣀⠀⠈⠃⠈⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠢⢀⠀⠡⡀⠀⠈⠉⠀⢀⣠⠚⠁⠈⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠇⠀⠀⢣⡠⠃⠀⠸⢠⡅⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⢀⣠⡶⠊
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⠖⣨⣭⣷⡦⡀⠀⠀⠀⠀⠀⣠⡤⠴⠶⠶⢤⣤⣄
⠀⠀⠀⠀⠀⠀⢠⡇⢸⣿⣿⠿⠗⠛⠒⠦⢤⣤⠋⠀⠀⠀⠀⠉⠑⢿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠘⢿⣵⣶⣿⣿⣿⣿⣿⡇⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⡖⢦
⠀⠀⠀⠀⠀⠀⠸⣿⣦⣀⣍⡛⠛⠻⠿⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡜⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠟⠀⠁⠀⠁⠀⠀⠂⠌⠙⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⡆⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢽⣄⣀⣠⡤⠤⠶⠶⣫⣾⣿⡿⡄
⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⡯⣿⡷⠶⠦⠴⠛⣳⠶⠧⡶⣷⡀
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣫⣄⡘⢶⡛⠛⠟⣻⠄⢴⣶⡶⣻⣿⣄⡀
⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣅⣀⣹⣥⢞⡵⠶⠦⠬⠌⠩⣿⡮⣹⣦
⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⡖⣡⠈⠃⡔⣀⣀⣀⡤⢣⡞⠻⠿⠗⡲⢦⡀
⠀⠀⠀⠀⠀⠀⣿⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠠⠈⠐⠈⡉⠉⣀⢉⣈⣽⣿⢟⡴⠶⠛⢲⡄
⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠯⠠⢝⠛⢭⢍⣵⠏⢹⡿⣣⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢤⠪⣍⡻⢏⡼⠷⠶⢂⣾⣷⣍⠲⣄
⠀⠀⠀⠀⠀⠀⣀⣈⠙⣿⢿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⡵⢦⢴⢞⡴⢏⣉⣉⣿⡷⢸⣷
⠀⣠⡤⠒⠊⠉⠀⠉⠙⣮⡳⡌⡿⠉⠉⠻⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠈⠙⠒⠶⣤⣤⣴⣾⣿
⣸⢁⣤⣤⣍⡲⡄⠀⠀⢸⣷⠷⣷⠙⠛⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣿
⡇⢼⣯⡀⣄⠹⡘⡄⣠⣿⣿⡦⣙⠻⢶⣶⣶⣿⣿⣿⣿⡿⠟⠀⠀⠉⠛⠻⠿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠀⣾⣿⣿
⢧⠸⣿⣤⠹⠀⡇⣿⣿⣿⣿⠇⠀⠉⠐⠒⠛⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣟⠶⣿⡟⠛⢿⣿⣷⣶⣶⣶⣦⣴⣶⣶⣶⣿⣿⡿⠛
⠈⠧⡙⢯⠶⠞⣱⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣭⡇⠀⡼⡣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠈⠑⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠚⢉⡀⠀⠀⠉⠛⢧⡄⡥⢌⣿⣿⣷⣟⠋⠉⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡠⢀⡠⢈⠑⢄⠀⠀⠀⢳⡄⢀⣿⣿⣿⣿⡀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣾⣿⣤⠀⠀⠀⢣⠀⠀⣠⣿⠿⢿⡏⣿⣿⣧⣀⡤⢞⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡘⣿⣏⣈⡇⠀⠀⠀⣤⣼⣿⣿⠀⠀⢿⣿⣻⣭⠵⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡜⢻⡻⡶⢐⠀⢠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢝⣛⢋⣡⣾⠿⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣾⣯⡈⠢⡀⠀⠀⣀⣠⡤⢤⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣷⡀⠈⠲⣾⣯⣿⣿⡤⠙⢦⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡷⡄⠀⠀⠙⢾⣿⣿⣄⠀⠙⢄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣜⡄⠀⠀⠀⠙⢿⣿⡄⠀⠈⠳⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠙⢿⡄⠀⠀⠈⢦⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠑⢌⠢⡀⠀⠙⢦
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣟⣫⣧⢄⠀⠀⠀⠀⠈⠳⣌⢦⡀⠀⠱⣄
⠀⠀⠀⠀⠀⠈⢟⣿⣿⣿⣿⣿⣿⣾⣿⣇⠱⣄⠀⠀⠀⠀⠈⠢⡙⠄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⣿⣿⣿⣿⣿⣿⣦⠈⢧⡀⠀⠀⠀⠀⠈⠲⣄⢀⠀⣈⣑⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⢻⣿⣿⣿⣿⣿⣷⣄⠳⡄⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣶⣜⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣶⣾⣯⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣧⣴⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠀⠈⠀⠤⠀⢄⠀⠲⠄⠀⠢⠘⠒⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠀⠀⢄⡀⠐⠀⠂⠀⠀⠐⠁⠂⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠟⠓⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠨⠂⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⢀⠀⢀⠄⠰⠀⠠⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢁⠆⠂⠁⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠁⢀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⢤⢈⡀⠐⠐⠁⠨⠐⠠⠈⠀⠁⡁⠀⠌⠒⠂⠀⠀⠀⠀⢠⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⡄⠈⠀⢐⡄⠀⠈⠳⣨⠳⠄⠀⠀⠑⢀⠀⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠁⠐⢌⠌⠢⠀⠊⠈⠠⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠎⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⡂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁
⠀⠐⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤
⠀⠂⠁⠀⠀⠁⠀⠔⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡪⣽⠃⢠
⠀⠔⠀⠀⠄⠈⠀⡠⠀⠁⠀⠠⠀⡀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢕⢝⠀⢘
⠀⠀⠀⠀⠀⠔⠀⢀⠠⠂⡀⠐⢀⠄⠂⠠⠀⠁⠐⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣵⡁⢨⣧⣠
⠀⠀⠀⠂⠀⠠⠀⡁⠀⠈⠄⢂⠀⠀⡁⠀⠈⠄⠊⢀⠄⢂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢩⣧⡴⠿⢸
⠀⠀⠀⠀⠁⠀⢈⠄⠊⠠⠂⡁⠐⢈⠄⠊⠀⠄⠊⠠⠀⠁⢀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⡄⠀⠰⠮⠆
⠀⠀⠀⠊⠠⠊⠀⠂⠁⠔⢈⠠⠂⠀⠠⠊⡀⠔⠁⢀⠀⠊⠠⠐⢁⠔⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠇⡣⠂⠀⢅⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠀⢁⠄⠈⠀⡠⠂⡀⠀⢁⠀⠊⠠⠂⢁⠀⠂⡀⠀⠀⠀⠀⠀⠀⠁⢲⡂⠀⠨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠔⠈⡠⠂⡁⠀⠁⠠⠐⠁⠄⠈⡠⠀⠀⠀⠀⠀⠀⠀⠐⣓⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠊⠀⠄⠈⡠⠂⢀⠄⠀⠠⠊⢀⠄⠀⠠⠀⠀⠀⢉⠀⠀⢪⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⡠⠀⢁⠄⠀⡠⠀⡠⠀⠁⠀⠀⠄⠀⢀⣶⠀⠘⢃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠐⠀⠄⠀⠀⠂⠠⠂⠔⠀⠘⠀⠟⡀⠀⣿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⡠⠀⡠⢂⠀⠀⠂⢸⡇⠀⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠁⢀⠀⠐⠈⣫⡄⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣑⠀⣀⣠⣶⣶⠒⠚⠻⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠿⠿⣿⣿⣭⣽⠛⠻⣷⣬⡂⠈⢿⡯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠗⢠⣾⣿⣿⣿⣿⣟⡠⣿⣿⣿⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣿⡇⢨⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠛⢷⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠁⠚⠓⠲⠾⠚⠛⠉⠛⠋⠁⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⢡⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⣈⣛⣿⣿⣿⣿⣿⣿⣿⣻⣼⡿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣟⣷⣼⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⣩⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠛⣿⡟⠿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣄⡀⠻⣿⣾⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⣤⠀⠉⠙⠻⢷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠛⠻⢿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣤⣤⣤⣴⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⠤⠤⠤⠤⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠤⠤⠤⠤⠤⣤⣤⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠁⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣠⣶⡶⠒⠛⠓⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⢿⣄⠀⠀⠀⠀⠀⠀⠈⠷⣄
⠀⠀⠀⣴⠛⠁⡿⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⠳⠖⢲⠟⠷⣄⠀⠀⠙⢦⡀
⠀⠀⠀⣿⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠶⠛⠻⣦⠘⢧⡀⠀⠈⢳⡄
⠀⠀⠀⣿⣀⠜⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⢷⡄⢠⣄⠻⣆
⠀⠀⠀⢻⣇⠜⠻⣆⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢻⣆⠻⠄⠹⣦
⠀⠀⠀⠈⢿⣄⣀⣈⠛⡶⢶⡄⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⣿⡄⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠈⠉⠉⢿⠀⢸⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢿⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢠⣦⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠸⣧⠀⢀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢺⡇⠈⠿⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⢹⡄⠸⣷⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡠⠋⣧⠀⠈⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⡿⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡠⢿⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠊⣿⠀⠀⣿⡀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢸⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⠀⠀⠈⠁⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠃⠀⣶⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⡆⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⣼⠀⠀⠛⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢨⣇⠀⠀⢠⣄⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⢀⡏⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢹⡀⠀⠀⢿⡄⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡞⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢨⣧⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⠞⠀⠀⣠⡄⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠁⠘⣧⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⠁⠀⠀⢰⡿⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡴⠙⣧⡀⠀⠀⠻⣦⡀⠀⠀⠙⠶⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠎⠻⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢳⣄⡀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠁⠀⢀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣼⣀⡙⠦⣄⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣉⣛⠒⠶⠤⣤⣤⣤⣤⣤⣶⣶⣶⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠏⠉⠙⢦⠀⠤⠔⠒⠂⠰⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠄⠀⠨⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠁⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠄⠂⠄⠀⠠⠖⠁⠀⠠⠤⠄⠀⠀⠀⠀⡓⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠏⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡄⠀⠀⠀⠀⠀⠰⠀⣬⠀⠐⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⢉⡀⠀⠘⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣱⡀⠀⠀⠄⠀⠀⠀⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠂⠤⠤⡤⠴⣊⠀⠐⠛⠂⠀⠈⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠂⠀⠐⡀⠀⠀⠀⠀⠈⢀⠀⠀⠐⢦⠤⠤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠈⠀⠀⢀⣠⣤⠀⠁⠀⠀⠀⠀⠀⠀⠈⠒⠂⠃⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢤⠈⠉⠀⠀⠿⠐⠒⠲⠖⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠖⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠎⢀⣈⣰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠐⠂⠀⠊⠀⠀⠙⣍⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠤⠤⠒⠫⢀⠠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠐⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⢄⠁⢀⡥⠖⠛⡆⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⠤⠴⠶⡏
⠀⠈⠀⠁⠀⠀⠀⣠⠜⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⡀
⠀⠀⢀⠀⠀⢠⠞⠁⠀⠀⠀⠀⡰⠁⠀⠉⠑⠲⠤⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢳
⠀⢀⠁⢁⡾⠁⠀⠀⠀⠄⣀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⢀⠞⠀⠀⣀⡤⠖⠋⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡀⠘⠒⠒⠉⠁⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⣤⠔⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⣀⣀⣀⣀⡠⠴⠫⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠈⠀⠩⠛⠢⣄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠠⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠃
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠁⠀⠀⠀⠀⠀⠈⢠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢳⣀⡤⠴⠶⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡄⢠⣦⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢠⠀⣤⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠘⡀⠙⠁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣠⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠤⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⠟⠁⠈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⣿⡇⠑⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠓⢤⡉⠁⠀⣸⣯⠑⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠌⠑⠦⣌⡋⠀⠀⣾⣿⠓⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠂⠠⠀⠌⢀⠀⡐⠀⠉⠁⠒⠀⠤⠤⠤⠤⠚⠉⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠀⡐⠁⠄⠠⠁⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⠂⠠⠤⠄⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠂⠀⠀⠌⠀⠁⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⢀⠀⣀⣀⣀⡀⠀⠈⠉⠉⠁⠒⠒⠒⠠⠆⠤⢠⡄⣀⢁⡈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣄⣄⣄⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣉⣉⣉⣉⡉⠉⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⣉⣀⣠⣤⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠉⠛⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠛⠊⠉⠉⠛⠿⣟⠋⠁⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡔⠊⡉⣉⠒⢄⠈⢿⡉⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢉⣹⠏⢀⠔⢊⡉⣉⠒⡄⠈⢦⠀⠀⠀⢀⣱
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠘⠀⢡⡖⣦⢱⠈⡆⠈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡏⠀⡎⢰⢡⡖⣦⠑⠈⡄⢸⡀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠰⡈⠢⠛⠋⠌⡠⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠡⡈⠨⠙⠩⠂⡰⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠒⠂⠐⠊⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠒⠂⠐⠊⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠚⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠈⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⣀⣤⡤⠤⣀⡀⠑⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⣀⡤⠴⠚⠋⠩⠀⠄⠀⠀⢳⡄⠀⠀⠀⠀⠀⢀⠁⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠉⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⢿⣍⡈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠾⡟⠋⠁⠀⠀⠀⡹⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢄⣀⣤⣤⣤⣶⣶⣶⣶⣶⠿⠿⠿⠛⠛⠿⣄⠀⢧⣀⡀⠄⠂⠈⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢻⣿⠟⠂⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⢀⣠⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⢤⣀⠀⠀⡀⠄⠒⠉⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢤⣤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠻⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⡻⠟⠻⠃⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⣠⡴⠶⠦⢤⢀⠜⠀⠘⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⠿⠿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠉⠛⠳⠶⣤⣄⣀⠀⠀⠀⠉⠑⠒⠢⠤⠄⣀⣀⣀⠀⠀⢀⡼⠀⢰⠁⠀⠀⠉⠉⢻⡉⠙⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⢦⣤⣄⣀⣀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠑⠒⠠⠤⠤⠴⠃⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠲⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠴⠚⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⠙⠻⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⣻⣷⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡛⠛⣳⣶⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠿⣧⣤⣶⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠛⢿⣿⣿⣿⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠷⢶⣾⣿⣟⠛⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡟⠀⠀⠀⢹⣿⣿⠀⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣤⣤⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⣀⣀⣼⣿⣿⠀⠹⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣼⣭⣴⣬⣬⣿⣼⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣴⣶⣿⡆
⠀⠀⠀⠀⢠⣿⣿⣿⠉⠂⢀⠀⠀⠀⠀⣠⡴⢦⡀
⠀⠀⠀⢀⠎⢻⣿⡇⠀⠀⠀⠉⠒⢴⠚⠁⠀⠀⠈⢳⣄⠀⣀⡀
⠀⠀⠀⣼⣀⠈⣿⡇⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⡀⢀⡽⠋⠀⠈⣠⡄
⠀⠀⢰⡿⠙⢧⠘⣷⡀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠟⠁⠀⠀⣠⠾⣟⢁⢄⡀
⠀⢀⣾⡇⠀⠈⠳⣾⣿⣿⣿⡿⠿⠿⢿⡿⠋⠁⠀⢀⣠⣾⣁⡠⠈⠁⠀⠀⣄
⠀⣼⠻⣷⡀⠀⠀⠈⠉⠉⠀⠀⠀⣠⠊⠀⠀⢀⣴⣿⣽⡿⠋⠀⠀⢀⡴⠾⣿⠤⠶⠄
⠀⢻⠀⠀⠑⠢⢄⡀⠀⠀⠀⣀⣴⣇⡀⣠⠾⠛⣽⠟⠁⠀⠀⣠⣶⣋⡀⠈⠁⠀⢀⣠⡶
⠀⠈⢧⡀⠀⠀⠀⠈⠓⠶⣿⣿⡿⠛⠉⢁⣠⡞⠁⠀⢀⣤⣾⡿⠛⠉⠀⢀⣤⠶⠋⣿⣿⣦⣄
⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠈⠙⢤⣤⣶⣿⣿⠿⠖⠀⢈⣿⡁⠀⠀⣠⣶⣿⣷⣤⣾⣿⡿⠟⠈⠑⠢
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣶⣦⣤⣈⠙⠯⣀⣠⣴⣿⣿⣿⠿⠛⠼⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⡈⠙⠿⣟⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀⠀⠈⠙⠒⠤⣄⣀⡀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⠴⠒⠉⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠉⠁⠀⠀⢀⣀⡴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⢄⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⠿⣟⣯⠥⢆⣀⣤⣶⣾⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡎⠿⣛⣯⠵⢚⣫⣭⣶⣿⣿⠿⠛⢋⣩⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣟⣫⠵⢒⣨⣥⣶⣿⡆⠿⠛⣋⣡⣴⣶⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⣫⣴⣿⣿⣿⠟⣛⣩⣤⢺⣿⡿⠿⢛⣋⣥⣴⣶⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣛⣭⠥⢶⣾⣿⣿⣿⣿⣿⣿⣿⠿⢛⡈⣥⣶⣾⣿⡿⠟⣛⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣥⣏⡴⠶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠟⢘⣋⣭⣴⣾⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⢵⣏⣹⢓⡆⢿⣿⣿⣿⣿⣿⣿⣯⣭⣴⣶⣎⣿⠿⠛⣋⣭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣹⠟⣼⣿⡘⣿⣿⣿⣿⣿⣿⠟⣛⣋⡉⣤⣶⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣯⣼⠛⣐⢿⣿⣷⣌⠻⣿⣿⣿⣿⣿⡿⠿⠃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⣷⡟⢉⢸⣿⣆⢻⣿⣿⡷⣠⣬⣭⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡞⠛⣧⡾⠛⡁⢴⣿⣇⢻⣿⡎⢿⡿⣱⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⣯⣨⠿⠏⣡⣴⣿⣿⡌⣿⣿⡜⣿⡿⠘⠃⠿⠛⣋⣭⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⡴⠛⢁⣴⣿⣿⣿⣿⣿⣧⠸⠿⣃⣥⡶⣞⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣧⣾⠁⠰⠿⠿⠿⠿⠟⣛⣩⣥⡶⠟⢫⣷⣿⡟⢰⡶⣯⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡷⠶⠶⣶⠿⠛⢿⣯⡛⠀⠀⠈⠻⣿⣧⣈⣾⣿⣿⡟⡩⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠺⣷⣿⣿⣿⠷⢯⡀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⡐⣇⠞⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿⢡⠖⢦⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣈⠓⢋⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⣃⡀⠀⠀⠀⠀⠀⠀⢠⣬⢉⣿⣆⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠶⠿⠿⠟⠛⠛⣻⣿⣿⣿⣧⡿⠃⠀⠀⠀⠀⠀⠀⠈⣻⠾⢻⣻⣿⣿⣉⠉⠛⠛⠛⠛⠿⠷⣦⣄
⠀⠀⠀⠀⠀⣴⡟⠉⠀⠀⢠⣴⣶⣤⣞⣿⠻⠟⠿⢅⠈⠙⠲⢤⣀⠀⣀⡤⠖⠋⠁⢀⠀⢉⠉⠈⢻⣭⡴⢿⡷⡄⠀⠀⠈⢹⣷
⠀⠀⠀⠀⠀⢿⣧⣀⡀⠀⢸⣾⢠⣾⡿⢯⣄⠐⠉⣀⣿⢂⣠⣴⠞⠋⠁⠀⠀⠀⠀⠻⣤⡈⣁⡴⠚⠹⣿⣆⣿⠇⢀⣀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣟⣻⡛⠀⢀⣌⣽⡾⠿⠛⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠀⠀⢈⣭⣿⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢠⡞⠉⠛⢿⣿⣿⠿⠛⠋⠉⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣛⣡⣴⠿⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠺⣿⣶⣴⡟⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠋⠀⠀⠹⣏⢶⣄⢤⣾⡇
⠀⠀⠀⠀⠀⣠⠶⠿⠿⠿⠷⢶⡑⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣬⣾⠟⠋⠀⠀⠀⣀⣀⠔⣥⣾⣯⡿⢿⣦⡀
⠀⠀⠀⣠⡞⠁⠀⠴⠆⣠⠔⠊⠻⣧⣝⣦⡐⡀⠀⠀⣠⠂⠀⠀⠀⣀⣤⣮⣿⠟⠻⡀⠀⠀⠀⠄⣠⢞⣱⠾⠣⢄⡀⢠⡄⠈⠙⢦⡀
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠈⠳⣤⡀⢈⣻⣷⣹⠴⢶⣶⣌⠁⢀⣠⣮⣷⣿⣫⡀⠀⠐⢉⣴⣦⣬⡞⣽⡿⠁⠀⣠⡴⠛⠀⠀⠀⠀⠈⢻⡄
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠑⣿⡁⠀⠀⣘⠿⣧⣿⡿⠋⠀⠉⠻⣷⣵⣴⣿⠏⠁⠀⠙⣏⠛⠷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⣿⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⡾⣻⡽⠛⠁⠀⠀⠀⠀⠀⠈⠙⠿⣮⡿⣦⣦⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⣿
⠀⣿⡞⠁⠀⠀⠀⠀⠀⠘⠿⡟⠃⠀⠀⢀⣼⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣯⠀⠀⠀⠐⢶⣶⠆⠀⠀⠀⠀⠀⠙⢦⣿
⠀⢹⣗⠁⡀⠀⠀⠠⠂⠀⠐⠶⠖⣠⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣄⡀⠤⣴⡅⠀⠠⡄⠀⠀⠀⠐⣼⡟
⠀⠀⠹⣾⡄⠀⠀⡠⠀⠀⢠⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⡀⠀⠀⡀⠀⠀⢈⣶⡟⠁
⠀⠀⠀⠈⠛⠷⣾⣶⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣶⣶⣼⣦⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠉⢻⣷⠶⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⣼⢋⡿⠷⠶⠴⠿⠿⠷⠞⣃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣎⣙⡟⠀⠱⠿⠤⣄⠀⠀⠀⠀⠘⣋⣉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⣋⣩⣿⡟⢙⡛⣶⣸⣇⠀⠀⠀⠀⠉⣥⠾⣇⣴⠿⣷⣴⣦⣤⡤⠶⠶⠖⠚⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡿⠣⣤⣿⣿⣿⠾⠟⣹⠛⣷⡶⠶⢦⣄⢀⡿⠁⠀⣈⣿⡿⠿⢿⣿⣿⣶⠞⢂⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠛⢿⡶⠿⠛⣿⡿⠀⠀⢻⡄⢿⡷⢶⣤⡬⠙⠷⠶⢛⣻⣇⠀⠀⠀⠀⠀⢈⣉⣉⣩⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣍⢻⡏⠛⠿⠃⢀⣼⡏⠀⠀⢀⡟⠠⠤⣽⣿⡿⠃⠼⢿⡿⠿⣿⠇⠀⣀⠀⠀⠀⠀⣩⣩⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⣏⣈⡿⠟⠛⠃⢀⣠⡶⠿⠁⠀⠀⠀⠸⣇⠠⢴⣏⣵⣾⠃⣾⡌⢷⡞⠃⠀⢠⠝⠛⠷⣤⣤⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⣤⣶⠟⠛⠉⠻⠶⣬⣭⠴⣤⡴⣾⢿⣅⣀⠤⣤⣀⠀⠀⢀⣿⣶⢿⣍⠁⣾⠀⣿⡇⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠈⣉⣿
⠀⠀⠀⠀⠀⣼⠋⣙⠳⡶⠶⣦⡄⠠⣤⡴⣿⡳⣮⣻⣮⠉⠳⣦⣽⠶⣾⠿⣥⣹⠆⠛⠛⠘⠷⠾⠷⠟⠀⠀⢰⠏⠀⠀⠀⣠⣴⣞⣩⢾⡇
⠀⠀⠀⠀⠀⣿⢠⣏⢰⡇⢸⠆⣴⡀⣷⣿⣼⣿⣾⣿⣿⡷⣾⠛⠛⠶⡾⠆⠈⠉⠀⢀⣠⠞⠓⠀⠀⠀⠀⠀⢸⡀⠀⢀⣾⣋⣤⣤⡤⢾⡗
⠀⠀⠀⠀⠀⢿⣄⣉⠛⠛⠛⠓⠿⠿⠿⣿⡿⠋⣥⡀⠀⠀⠈⠛⠒⠲⠶⠶⣦⡴⠞⠋⣁⣤⡶⠦⣄⠀⠀⠀⠈⢳⣄⣼⠁⢠⣤⡤⠶⢻⡇
⠀⠀⠀⠀⢠⡞⠉⠉⣻⣶⣶⡶⢶⣶⣶⡿⢀⡖⣠⢉⠛⢛⠶⢶⠶⢤⣴⡾⣋⣶⡜⣷⡝⣿⣿⡀⠈⢳⡄⠀⡀⠀⢙⣿⡀⢀⣠⣴⡾⢫⡅
⠀⠀⠀⠀⠈⠷⣤⣀⣈⡻⠟⠀⢺⣿⣿⣧⣼⣁⡏⢸⡇⢸⠁⣾⠀⣶⢸⣇⣿⣸⣿⣿⣿⣾⡿⠗⠐⠛⠛⠛⠿⠿⠟⠛⠛⢛⣉⣡⡀⠿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠘⠿⣿⣧⣬⣙⠛⠻⠷⠾⠆⠹⠶⠿⠿⠿⠿⠿⠛⢛⣉⣡⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⢩⣻⣿⣶⡶⣶⣶⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠐⠦⣬⣭⡿⠀⣿⣿⡿⣻⡿⠟⠛⠛⠿⣿⣿⡟⠋⠉⠀⠸⣿⣿⣿⣿⡟⠉⠀⣠⡴⠿⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠟⠻⠣⠶⢿⣄⣼⢃⣤⣠⡤⠤⠤⣸⣿⡥⠶⠆⠀⠀⠈⠛⠻⡟⢠⡆⣾⠃⣠⡴⠄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣷⣟⡾⠁⣠⣤⣄⠈⢿⡶⠶⠖⠛⠋⠀⠀⠀⢿⣾⡄⣧⠀⢿⣤⠆⣠⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⡇⠀⣿⠀⣸⠇⢹⡷⢚⣃⠀⠀⠀⠀⠀⠀⠉⠛⠿⠦⠀⢤⡶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣀⣀⣡⣶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁⠀⠀⢀⡞⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡶⢄⠀⣠⡴⠶⢤⡀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁⠀⠀⠀⠀⣠⢏⡜⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠢⡹⡍⠙⠳⣄⠹⣤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠋⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣖⣚⣉⣙⡟⣦⣤⡾⠷⠿⣤⣀⣀⡀⠀⠀⠀⣀⣠⠴⢋⡵⠊⠀⠀⠀⣠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⣿⣿⠬⡑⠾⣧⡙⠮⡙⢍⢲⢦⡭⢍⣉⣉⡉⠭⠴⠚⠁⠀⠀⠀⠀⣰⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡷⢬⡓⢾⣳⡾⡌⡷⠬⠞⠘⣾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠳⢬⡒⢏⣲⠿⠃⠸⡿⡀⠀⠀⠈⠳⢝⡶⣄⡀⠀⠀⠀⠀⣠⠶⢋⡤⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠢⣍⠺⣹⡾⠟⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠉⠲⡝⣆⠠⠔⣋⣥⠞⢋⡤⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⡟⠁⠀⠀⢿⢻⡿⠁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢹⣽⡶⠞⣉⡤⠞⠉⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⡽⠋⠀⠀⠀⠀⠈⢧⡙⢄⠀⣀⠴⣚⡭⠶⠛⠻⣭⢲⣠⣴⡾⠟⣡⣴⣊⢁⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⢀⡴⣋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡭⠖⠋⠁⠀⠀⣠⢞⣽⣿⠟⣩⠴⠋⠁⠀⠉⠉
⠀⠀⠀⠀⢀⣴⣯⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣵⡿⠊⣡⠈⠁
⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣋⢴⡿⢋⡴⠊
⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣩⢞⡡⠊⣠⠋
⠀⠀⣾⢷⡱⢤⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠶⣛⣫⡵⢒⣩⠵⠋⣠⠞⠁
⠀⠀⢻⣶⣝⡓⡮⣭⣭⣭⡭⡽⢷⡖⣶⣛⣫⠽⠖⠚⣉⣠⠴⠋
⠀⠀⠀⠙⠿⣯⣉⠓⠚⠚⠒⢛⣉⣉⣉⡥⠤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⢀⣠⡤⠶⣤⣄⠀⠀⠀⠀⠀⢠⣤⣄⡀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢠⣯⣨⣶⣾⣶⣌⣳⡄⠀⠀⠀⠸⣿⠋⠀⠀⠈⢻⡿⠁⠀⢤⡀⠀⠐⠘⠂⢃⠀⠘⡐⠀⠀⣠⠄
⠀⠀⠀⢸⢿⣿⡿⠿⣹⢾⣟⡇⠀⠀⠀⠀⣀⡈⠛⡜⠋⣈⡀⠀⠀⠀⢙⡶⣤⡀⠀⠀⠀⠀⣠⢴⣟⠁
⠀⠀⠀⠘⢿⡻⣿⣿⣥⣾⡿⠁⠀⠀⠀⣸⣿⣿⡂⠀⣸⣿⣿⡀⢀⢠⣈⣀⡀⠀⠛⢂⠚⠃⠀⢀⢉⠀⡀
⠀⠀⠀⠀⠀⠙⠛⠓⠛⠁⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠉⠛⠛⠁⠀⠈⠈⢀⣥⠖⠦⠈⠠⠓⢦⣀⠈
⠀⠀⠀⠀⢀⣤⠶⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠉⠇⠀⠀⠀⠀⠀⠰⠏⠙⠲⠄
⠀⠀⢠⣴⣟⣷⣶⣶⡶⣿⣷⣄⠀⠀⢀⢀⠀⣴⣾⣿⣽⣷⣄
⠀⠀⠘⠻⣿⡶⠷⠾⠿⣶⡾⠋⠀⡠⡏⢇⣹⠉⣣⣋⡋⢛⣜⣪⠊⡵⠀⠀⠐⢶⠴⠤⠾⠦⠤⠓
⠀⠀⠀⠀⠈⠻⠶⠶⠾⠋⠀⠀⠀⠈⠉⠑⠋⠻⣿⣿⣿⡿⠋⠀⠋⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⣡⣔⡂⢀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⣿⣿⣻⣿⡆⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⣹⣾⡀
⠀⠀⠀⠀⠈⠑⠨⠾⠋⢻⣿⣿⣿⣿⠏⠚⠼⠕⠈⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⢸⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠋⠅⠂⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠊⠈⡁⠀⠀⣻⡈⣿⠁⢂⣠⢴
⠀⠀⠠⢄⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢹⡄⠉⠈⠂
⠀⠀⠠⢤⡤⠤⠤⣤⡤⠆⠀⠀⠠⣤⡤⡤⣄⠀⣿⣿⡇⢀⡤⣤⣤⠄⠀⠀⠀⠀⠀⠈⠧⡷
⠀⠀⢀⡤⣤⣤⣤⡤⣄⡀⠀⠀⠀⠈⠙⠀⠈⠛⣿⣿⡟⠉⠀⠚⠉⠀⠀⡄⣤⠠⠄⢠⠀⢠⠄⣤
⠀⠀⠘⣓⣛⣛⣛⣳⡛⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡧⠀⠀⠀⠀⠀⣀⡓⠛⡐⠀⠞⠡⠙⢃⣋⣀
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠈⠙⠛⡟⠓⠚⠛⠛⠛⢻⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⣠⠴⠒⠋⠉⠉⠉⠉⠓⠦⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠶⠤⠴⠖⠒⠒⠊⠉⠉⢀⠴⣫
⢾⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣥⡾⠋
⠀⠉⠛⠓⠒⠶⢦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣚⣥⡾⠟⠁
⠀⠀⢠⠴⣄⠰⡜⠁⠈⣙⣻⣿⡿⠟⣶⣶⠦⣤⣤⣀⣀⡀⠀⠀⠀⠀⣀⡀⠀⣀⡀⠀⠀⠤⢐⣒⣬⣵⣶⡟⠛⢅
⠀⠀⠀⣿⢺⣆⡇⠀⠈⠁⠀⠈⠳⠾⠟⠉⠀⠀⠀⠉⠉⠙⠛⠛⠲⠶⠶⢶⣶⣶⡶⠶⠿⠟⠛⠛⠛⢽⠎⢿⡄⢘⡇
⠀⠀⠐⠬⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠐⢦⣤⣬⠜⠋⠀⠀⣀⣠⠤⠶⣞⠁⢀⣼⣧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠒⣎⢻⡄⠈⣧⣥⣴⠶⠛⠋⠉⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠐⠂⢸⡇⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠾⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠀⡀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠠⠂⡴⢡⠆⡤⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠞⣡⠏⡼⠃⠀⢠⡟⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⣠⠟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢠⣦⠄⠀⠀⠀⠀⠀⠀⢀⡴⢋⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⠀⠀⠺⠃⠀⠀⣠⡶⢋⣴⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⢋⡴⢫⡔⠶⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⣡⠶⢯⣂⣈⣛⣳⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⢉⡤⠞⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢋⡡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣡⠔⠋
⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠁⠀⢾⠆⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣋⡴⠋⠁
⠀⠀⢀⡶⠟⠋⠉⠀⣠⠄⠀⠀⠀⠀⠀⠴⠂⠀⠀⣠⣾⠟⣡⠞⠉
⠀⠀⣼⠁⠀⠠⢔⡾⢗⣢⠆⡀⠀⠀⠀⠀⢀⣤⡾⠋⣡⠞⠁
⠀⠀⢹⣧⠀⠀⠁⢊⡭⠴⣋⠴⠀⠀⣀⣴⠟⣉⡴⠾⣅
⠀⠀⠀⢉⠻⢦⣤⣀⣀⣀⣀⣠⣴⠟⣋⣴⢾⠉⣠⢤⡜⣦
⠀⠀⠀⠀⠙⠒⠤⣭⣉⣉⣩⣥⡴⠞⠻⣏⠈⠣⠈⢉⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡉⠙⣿⣿⣿⣿⠛⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⠟⢉⣽⠿⠿⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⡈⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧⠘⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋

⠀⠀⠀⢀⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄
⠀⠀⠀⠸⣎⠉⠻⣆⠀⠀⠀⢀⡴⠊⠁⠀⠀⢳
⠀⠀⠀⠀⠈⠳⣦⣹⠖⠒⢹⡏⢉⡙⣆⢀⣀⣨⡟⢆
⠀⠀⠀⠀⠀⠀⠈⣧⠤⠤⠼⣇⠻⠟⢻⡫⢟⢽⣿⡸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⡀⢞⣩⣾⣡⣿⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢷⡀⢰⣿⠿⣍⡿⢉⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠻⣦⣁⢴⡿⠛⣿⣋⣤⣻⢦⣄
⠀⠀⠀⠀⠀⠀⢸⠉⠲⢄⠀⠀⠀⠀⠈⠙⢷⣄⡸⠟⢫⢷⣧⡎⡾⣷⣦⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢳⠀⠀⠀⠀⠀⢀⡞⠛⢶⣌⡋⢹⡸⢿⣿⣏⣈⣑⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡇⠀⠀⠀⠀⡎⠀⢠⠏⢨⡟⢳⠶⣬⣟⣀⣈⡏⡟⠪⣕⠪⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡟⠦⢄⣀⣸⠀⠀⠀⣸⠀⠀⡞⠀⣼⠀⡎⠀⠀⠀⠉⠉⠉⠀⠀⠈⢳⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠈⢲⠦⠤⢌⣉⡑⠒⠧⣀⣰⡇⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠀⠀⢘⠆
⠀⠀⠀⠀⠀⠀⣠⣔⠚⠶⣖⢢⣄⣈⣉⣶⠦⣭⣑⡒⠷⢤⣇⡀⢀⠔⠊⠉⠉⠉⠉⠉⠀⢀⣀⠤⢺
⠀⠀⠀⠀⢰⢫⠔⠲⢝⢦⠈⢿⣑⣒⣫⢿⣠⠇⠀⠉⠉⠒⠒⢬⠭⢭⠽⢭⣯⠭⠭⠭⠭⠥⠔⢺⠁
⠀⠀⠀⠀⢸⡸⡀⠀⢈⠎⡆⡼⠘⠟⠒⠋⠁⠀⠀⠀⠀⠀⠀⣈⣽⠛⠻⢭⡛⠦⣀⣀⣤⡤⠖⠋⠓⢄
⠀⠀⠀⠀⠀⠑⠭⣒⣫⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡥⠬⣍⢢⡀⠹⣄⠇⢏⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠈⡇⢣⠀⡷⠒⢻⡀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡑⠤⠴⢃⡾⠞⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⣿⡏⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⣾⣟⠲⣄
⠀⠀⠀⣴⠟⢠⠙⣦⠈⢣⡀
⠀⢀⡼⠁⢀⡎⡇⠘⢷⡀⠹⣄
⢠⣿⠃⠀⡜⢸⠀⣴⠈⢻⣄⠈⠳⡀
⣾⡏⠀⠀⡇⡇⠀⣿⠀⠀⠻⣆⠀⠹⡄
⣿⡇⠀⠀⠀⢧⠀⡏⠈⡆⠀⠙⢦⠀⠘⢆
⢿⣷⡀⠀⠀⠀⠀⣇⠀⡇⢄⠀⠘⣧⠀⠀⢣⡀
⠸⣟⢿⠢⣄⠀⠀⠀⠀⠇⢸⠀⠀⠘⣆⠀⠀⢷
⠀⢿⡀⠑⢬⡓⢄⠀⠀⠀⠈⠀⠀⠀⠹⣇⠀⠈⢳⡀
⠀⢸⡝⢶⡴⠿⢦⣷⣤⡄⠀⠀⠀⠀⠀⠹⡄⠀⠀⢣
⠀⠈⡇⢸⡇⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠹⡄⠀⠈⡇
⠀⠀⡇⢸⡇⠀⠀⠀⠀⠈⠛⣷⡀⠀⠀⠀⠀⢳⠀⠀⠘⡄
⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⣧⠀⠀⢻⡀
⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢻⡆⠀⠀⠱⣤⡀
⠀⢀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⢿⣆⠀⠀⠀⠈⠓⠶⠦⠤⠴⠶⠤⠤⣄⡀
⠀⢰⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠈⠙⠲⣤⡀
⠀⠸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠙⣿⣦⡀⡀⣀⠀⣀⣠⡤⠖⠛⠑⠶⣤⣤⣄⣉⣓⠦⢄⣀
⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠈⠉⠙⢿⣿⡛⢯⣙⠷⣦⣄⠀⠀⠀⠉⠙⠛⠛⠛⠃⠈⠙⢲
⠀⠀⣇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠈⠛⢷⡮⠓⠾⠙⠿⠶⠄⣤⣤⠤⠴⠂⠀⣀⣀⡼
⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠶⠿⠛⠋⠉⠁
⢀⣀⣀⡀⡀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⡛⠓⢒⣶⠶⠒⢚⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⢀⢀⣀
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣾⡿⢛⣓⣂⣤⡔⠁⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣈⠻⠿⢿⣿⣿⡁⠀⠀⠀⠀⠈⢿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠁⠀⠀⠀⠉⡙⠄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠈⣄⢀⣀⣠⣤⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⢿⣲⣒⣋⠓⠷⢺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣊⣹⡤⣮⢬⡿⠵⣿⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢖⡄⡁⠢⠤⡴⠱⢻⣟⡛⣲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⢾⡼⡶⠟⣫⢛⡷⣴⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠊⣬⣋⢚⣛⡴⠿⣮⣢⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠐⠂⠐⢺⡫⢻⠛⣻⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠚⠦⢥⣭⣼⣿⡄
⠀⠀⠀⠀⠀⢠⢣⡖⠒⣔⡀⢰⣿⠻⣬⣉⣉⣿⣿⣿⡿⠻⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠃
⠀⠀⠀⠀⠀⠘⡘⣯⡘⢀⣷⣿⣿⠁⠀⠉⠛⠛⠋⠉⠀⠀⠀⠀⢸⣯⣿⠻⢶⣶⣦⣤⣤⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠑⠘⠛⠵⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠛⠻⢤⡋⡛⢿⣿⣿⠿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⣀⠀⠐⠄⠀⠹⣄⣹⣿⣿⡀⠀⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢢⠀⠀⣀⣴⡟⠛⣯⣿⣷⣲⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡻⣷⣄⠀⣿⣿⠇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠹⣶⢶⣖⠒⠒⠒⠒⠲⠶⠶⠦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠶⠤⣤⣄⣀⠀⠀⠀⢠⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⡆⠉⠛⠲⠦⣤⣤⣀⣀⣀⣤⣤⠬⠟⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⣙⠳⣄⠀⢹⡄⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣙⡻⢿⣦⣽⣦⡀⠀⠀⠀⠀⠈⠳⣶⠶⠶⠶⢶⡶⠶⠶⠖⣛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠉⠉⠉⠉⠉⠙⠛⠛⠻⣿⣿⣧⣀⡀⠀⢀⣸⣧⣶⠆⣾⠁⠀⠳⠆⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⣻⠏⢹⣍⡉⠉⠉⣩⡽⡏⠀⠀⣧⠀⠙⠂⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠒⠛⣻⠟⠋⠁⢀⡾⠁⠀⣾⣬⡿⠶⢿⣅⣠⣇⣤⡄⠘⢦⣄⣉⣋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⠟⠀⠀⣸⠛⢦⡀⠀⢀⣠⢼⡏⠉⠀⠀⠀⠈⠉⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡼⠃⠀⠀⢰⣯⣤⡤⠟⢾⣍⠁⢸⢁⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⠟⠀⠀⠀⣠⠟⣇⠀⠀⠀⠀⠈⢙⣿⠟⠛⠀⢀⣠⣤⣤⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⣯⣤⣬⣧⣤⡤⠶⠚⣿⠁⠀⢀⡴⠋⠁⠠⡄⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣞⣁⠘⢧⡀⠀⠈⠓⢲⣾⡷⠿⠂⢸⡇⠀⠻⠀⠉⠀⣏⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⡖⠚⡟⠉⠙⣿⠛⢻⡁⠀⠀⠀⠙⢯⡉⠛⠛⢿⣛⡿⢋⠀⠀⠀⠀⠘⢧⡈⠳⣤⡴⠚⠋⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠙⠳⣤⡁⣰⠆⣽⠀⠈⠛⢦⣄⣀⣀⣀⣹⣦⡴⠚⠉⠙⠛⣃⣀⠀⠀⠀⠈⣻⠞⠁⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠻⣇⣠⠟⠀⠀⠀⣀⣀⣀⠀⠀⠀⠈⠿⠃⣴⠞⠉⠉⠉⠙⠲⣄⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣰⠟⠉⡉⠉⠙⠲⣄⠀⠀⣼⠁⠐⣆⠘⠇⢠⡄⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢰⡇⢠⠀⠇⢠⠇⠀⠹⡆⠀⢻⡀⠰⣄⡀⣀⡼⢃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠸⣆⡀⢀⣠⠆⢰⠇⠀⠀⠹⢦⣌⣉⣉⣤⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⣸⡄⠙⠓⠶⢤⣀⣀⣀⣀⣸⣇⣀⣙⣶⣄⣭⣭⠥⠴⠿⠒⠚⠒⠲⠶⠦⢭⣤⣀⣸⣇⣀⠀⠀⢀⣀⣠⡤⠖⠋⣸
⠀⠀⠀⢀⡴⢛⣩⡿⢦⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢶⡄⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⢠⣏
⠀⠀⢠⠏⣰⠋⠀⠻⡄⣽⣹⣷⠶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣍⢳⡀⣿⡀⠀⠀⡀⢀⣀⣀⣀⣀⣤⠶⢻⡈⢧
⠀⠀⣼⠀⡏⠙⢛⡷⢟⣩⣴⠖⢲⡶⠦⣍⡳⢦⡀⠈⡏⢸⣛⣿⠿⣛⣋⣭⣙⣛⠻⢿⣈⡇⠈⢹⡋⣿⡽⢛⣋⣭⣭⣭⣛⠲⢤⡇⢸⡆
⠀⠀⠸⣆⢻⣶⡞⢡⣟⡁⠘⣧⣸⡇⣀⣤⠿⣄⢳⡄⢷⣼⠋⡴⠛⠻⡅⢠⡏⠉⠛⣦⡙⣧⠀⠘⡿⢋⡴⠋⠹⣄⢸⡇⠈⣛⣦⠙⣾
⠀⠀⠀⠘⠷⣽⡇⢾⠉⠉⢙⣯⣉⣹⣧⣤⣤⣿⢸⡇⢨⡇⢸⠷⠶⢤⡿⠛⢷⡞⠛⠉⣷⢸⡇⢸⡇⣾⠓⠲⢶⡟⠛⢷⣛⣉⠉⡇⢸⡄
⠀⠀⠀⠀⠀⠈⢷⡘⢷⡚⠋⠈⡯⠉⠳⣄⣼⢃⡼⠁⠘⣇⠸⣄⣠⡴⠛⡶⠾⣍⠛⢻⡟⢸⠇⠸⣇⢻⣤⡤⠞⠛⡶⠻⣍⠉⣹⠇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⡓⠶⠷⠶⠞⣋⡴⠟⠁⠀⠀⠘⢦⡙⠦⣄⣀⣇⣀⣨⡷⢋⡴⠋⠀⠀⠙⢦⡙⠶⣤⣠⣇⣠⣬⠟⣋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠬⣭⡭⠴⠖⠋⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⠴⠶⠚⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣃⣉⣋⣠⣋⣹⣏⣣⣿⣻⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣴⣷⠛⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠳⡴⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⢳⢧⠞⣻⣟⣶⡦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣥⣶⣒⣛⢦⣹⣿⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⣏⣩⣬⣙⡻⠿⠿⣓⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠺⢿⡿⠟⣋⣀⣁⡻⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣙⠻⡟⢾⣿⣿⣿⣿⣿⣦⡺⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⢆⣴⣿⣿⣿⣿⣿⡆⢷⠿⣋⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠠⣄⡙⠻⢿⣿⣿⠟⣃⠉⣥⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣦⣍⠋⡘⢿⣿⣿⡿⠟⠋⣠⠄⢾⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣮⢸⣿⣾⣝⡦⣄⣀⡴⠞⣉⣄⠻⠿⠿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⠏⣀⡙⠲⣌⡉⣠⢴⣫⣷⣿⢊⣵⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠘⣿⣿⣿⣿⣌⠻⡄⢾⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⢠⣿⣿⡆⢠⠟⣱⣿⣿⣿⣿⠘⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡆⢻⠘⣿⣿⡇⢶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⢸⣿⣿⢃⡏⣰⣿⣿⣿⣿⣿⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢘⣛⣻⣭⡄⢻⣿⣿⣿⣿⡿⢸⣇⣈⣤⣤⠈⢉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⣈⣉⣁⣸⡇⣿⣿⣿⣿⣿⡇⣨⣝⣛⡻⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠸⣿⣿⣿⡿⢁⡞⢉⣤⣤⠐⢿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠷⠀⣤⣍⠉⢧⠙⣿⣿⣿⣿⠁⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢙⣯⣶⣾⣇⢿⣿⠟⣡⠞⣠⣾⣿⠇⠸⢿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⠇⢹⣿⣷⡈⢷⡈⢿⣿⡟⣼⣶⣬⣝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢟⣫⣌⠁⡴⢋⣼⣿⣿⠃⠸⣿⣶⣮⡍
⠀⠀⠀⠀⠀⠀⠀⠈⢫⣥⣶⣾⠆⠹⣿⣿⣆⠹⣦⠉⣥⣝⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⡇⠠⢿⠿⢟⣵⣿⣷⣮⣝⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣫⣵⣾⣷⣜⠻⢿⡷⠀⣼⣿⣿⡷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⣭⢀⡔⢋⡳⡀⠙⠻⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠿⠟⠋⢠⢖⠐⢄⠀⡍⠹⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢚⣩⢿⢌⢰⣿⣿⠘⣤⣇⢤⡙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢂⣤⣳⣠⠃⣿⣷⠌⠃⡏⣙⠲⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡩⠔⢺⠘⡄⢿⡏⢰⡿⠺⣕⢬⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡽⣪⠟⣿⡆⢿⡟⢠⢱⡗⠬⣙⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣶⣶⣶⣄⠘⣌⡰⢁⠕⠒⢌⠣⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠞⡡⠤⢬⡘⣌⣡⠃⣠⣶⣶⣦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠿⠟⠀⢀⡀⠇⣤⣤⣄⠱⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠎⣠⣤⡄⢱⢈⠁⠘⠿⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⡇⠸⠟⢺⠀⣿⣿⣿⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢰⣿⣿⣇⠀⡞⠿⠄⢻⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⣛⣉⣉⠀⣼⣿⡄⠀⣿⣿⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⠀⢃⣿⡇⠈⣙⣛⣛⣛⡻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟⢀⣿⣿⠇⢠⣿⣿⣿⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠘⣿⣿⣿⠀⠹⣿⣿⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣿⣶⣶⣶⡆⢸⣿⣿⣀⠈⢉⡉⢄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠬⣉⡋⠀⠀⣿⣿⡆⢰⣶⣾⣯⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠀⠿⠿⠃⡬⠿⣛⣻⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣭⣓⡺⠯⣅⠸⠿⠇⠈⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠴⢖⣒⠓⠦⠤⢜⣁⣤⠥⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⢭⣀⣘⠦⣀⡴⠒⣒⠤⠬⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠭⠴⢒⣚⣉⣭⣵⣶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣦⣭⣍⣛⠒⠶⠬⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢶⣤⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡠⡄
⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣶⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡲⠃
⠀⠀⠈⠀⠀⠀⠟⢛⣦⣄⣀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠩⢗⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣅⠐
⠀⠀⠈⠀⠀⠀⡀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠀⡀⢈⢀⡠⢄⡀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠠⠀⠀⠀⠘⠋⢀⡒⠂⠘⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠐⡀⢙⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠠⠰
⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠉⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂



⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠁⣰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡤⠅
⠀⠀⠀⠀⠀⢺⡠⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⠾⣯⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣯⡟
⠀⠀⠀⢠⠀⠀⠣⣃⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠬⠀⠈⠙⠲⠀⠀⠀⠀⠀⠀⠀⠴⠞⠉⠁
⠀⠀⠀⠀⢀⠄⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠂⠈
⠀⠀⠰⢂⡑⠠⠐⣂⠤⠀⢠⠀⠀⠀⠀⠀⣀⠁⠦⠴⠨⠠⠐⠀⠄⠈⢡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠌⡀
⠀⠀⠀⠀⠀⠀⠈⠆⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠆⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⣠⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⡁⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣂⠀⢄⠚⡿⣓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢀⣾⣿⣜⡥⢻⡿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣻⣿⣿⣿⣴⢞⣿⣯⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣧⢎⡕⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡯⣿⣿⣿⣿⣿⣿⣷⡌⠐⢫⠖⡵⣶⠤⢄⣶⠆⠂⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣇⠀⠐⣾⣷⣿⣿⣿⣿⣷⣀⣄⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⠿⠟⠛⠻⠿⢿⣷⣓⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⡟⠛⠛⠛⠉⠉⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⢿⣬⣙⣷⣶⣶⣶⣶⣶⣖⣲⣶⣿⣓⣶⣶⣶⣶⣶⣶⣶⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣻⣿⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠉⠉⠏⠉⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⣷⣿⣦⣄⠀⠀⡠⠐⠓⠲⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠺⣿⣭⣶⣶⣾⣿⠀⠀⠀⠀⠀⢹⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠤⠉⠛⠛⠿⠧⡄⠀⠀⠀⠀⠀⢻⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣲⣶⣦⡄⠀⠀⠀⡠⠤⠤⣤⣤⣸⣦⣠⠀⠀⣼⠷⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⣿⣟⣫⣭⣴⣶⠋⠀⠀⠀⠀⢻⣿⣉⣯⡽⠯⢥⢉⣙⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣈⡛⠿⠿⢿⣿⡃⠀⠀⠀⠀⠀⢻⣷⣣⠽⢔⡵⠢⡿⠛⡿⢭⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠈⣿⣿⠿⢆⣅⡔⢖⠞⠴⢫⣯⢍⣽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣠⣿⣿⣦⠈⠋⠰⣿⠾⣟⠏⣡⣛⠟⣰⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠐⠿⠟⢿⢲⣶⣚⠓⠲⢾⣧⠀⠀⠀⠀⠻⠊⣲⣏⣚⡉⡵⠿⢷⡜⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡎⢱⣄⣴⣅⣝⡻⣿⣗⣦⠀⠀⠀⠀⠈⠀⠁⠘⠺⠭⣭⣵⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡅⢙⢀⣬⣤⢊⠜⣿⣓⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⣶⣡⡶⢖⢛⡵⢿⢏⣿⣶⣦⣤⣤⣤⣤⣶⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡶⢫⡿⠣⠟⢋⣽⣾⡙⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⣀⡠⠤⠖⠦⡛⣟⣿⠻⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠱⠿⣒⣘⣛⣼⣧⠀⠙⣦⡀
⠀⠀⠀⠀⠼⣩⡥⢆⡄⠀⢸⣾⢿⡒⠒⢻⣿⣿⣿⣿⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⢤⣺⠝⠁
⠀⠀⠀⠀⠀⢿⡂⡆⠉⣦⣿⣿⠚⠟⠛⡿⢿⣿⠿⠏⠀⠈⠉⢙⣟⣷⣶⣤⣤⣀⣀⣀⣀⣀⣠⣼⣿⠏⠉
⠀⠀⠀⠀⠀⠘⠿⠴⣳⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣻⡇⡹⢹⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⡀⠈⠳⡔⠀⣿⣿⡏⠉⠙⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣶⣀⠀⠑⡄⠀⣹⣾⣿⢿⣷⣀⡠⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣼⠀⠀⢰⣾⣿⠀⠘⠿⠛⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠑⣂⣿⠟⠁