Rune Emojis & Symbols

⠀⢠⡆⢀⣠⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⢸⡷⣋⡴⠀⠀⠀⠀⠀⡇⠈⢹⡆⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⢸⣍⡳⡄⠀⠀⠀⠀⢸⠉⣳⡄⠀⠀⠀⠀⠀⣴⠋
⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⢸⣧⠟⠀⠀⠀⠀⠀⢸⠀⠙⠦⠀⠀⠀⠀⢸⡟⠉⠀⠀⠀⠀⠀⠘⢧⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠛⢦⡀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢠⡀⠀⢀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⡤⠀⢠⡄⠀⠀⠀⠀⣀⢰⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⡤
⠀⠀⢳⣠⡞⠀⠀⠀⠀⠀⣿⠙⣳⡄⠀⠀⠀⠀⣷⡀⢸⡇⠀⠀⠀⠀⠙⢻⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠰⣏⠀⣀
⠀⠀⢠⢿⡄⠀⠀⠀⠀⠀⣽⠞⠁⠀⠀⠀⠀⠀⡯⠙⢾⡇⠀⠀⠀⠀⠀⢸⠉⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠳⠈⠳⣄
⠀⢠⠏⠀⠳⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⢀⡀⢀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣿⠳⠀⠀⠀⠀⠀⡟⠶⠋⠀⠀⠀⠀⠈⢷⣼⣴⠃⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠚⢹⡟⠂⠀⠀⠀⠀⢸⡟⢶⣄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠼⠷⢶⠄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣷⣋⠁
⠀⠠⣤⣿⠀⠀⠀⠀⠀⠀⣧⣤⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣇⡼⠗
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠉

⠀⢸⡷⣤⢾⠀⠀⠀⠀⠀⡷⣤⢾⡇⠀⠀⠀⠀⡷⢦⡀⠀⠀⠀⠀⠀⠳⣄⣰⠃⠀⠀⠀⠀⢀⡞⢧⡀⠀⠀⠀⠀⢰⡀⢀⣶
⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⡷⠛⢾⡇⠀⠀⠀⠀⡇⠀⠙⠂⠀⠀⠀⠀⣠⠟⢧⡀⠀⠀⠀⠀⠘⣆⡼⠃⠀⠀⠀⠀⢸⢳⡞⣿
⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷⣏⠀⠀⠀⠀⠀⢀⡞⢧⡀⠀⠀⠀⠀⢸⡟⠹⣿
⠀⠘⠇⠀⠸⠀⠀⠀⠀⠀⠇⠀⠸⠃⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠃⠀⠀⠀⠀⠋⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⠚⠋⣿⠉⠉⢁⡞⠳⣄⠈⠉⠉⡟⣳⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⡏⠁⠀⠸⡓⢤⡋⣇⠀⠈⢳⣴⠋⠀⠀⣸⣯⠵⠋⠀⠈⢹⡶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠒⡾⠲⡄⠀⠀⢧⡞⠙⢿⣀⣠⣎⣙⣦⣀⢠⡏⠀⠀⠀⠀⢠⠏⢹⠄⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣥⣀⡀⠀⠓⢲⠷⢤⣤⠼⠗⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠦⣄⣰⠋⢠⠏⠀⠀⢀⡼⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠳⣄⠀⠀⢀⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⢀⣴⣋⣿⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣶⢤⣀⠀⠈⢳⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⢀⣹⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠯⣔⢻⠋⠉⠑⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣀⡤⢺⠏⡇⠹⣆
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠙⠲⢄⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠘⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡾⠹⣏⠓⠲⠦⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⣀⣤⢤⣷
⠀⠀⠀⠀⠀⢸⡧⢶⣍⣀⡀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⡽⠉⠲⠏⢸⡆
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢸⣠⠷⣄⡼⢳⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡴⠶⣄⠀⡇
⠀⠀⠀⠀⠀⢸⡟⠛⠋⠉⠋⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⠘⢲⡇
⠀⠀⠀⠀⠀⠸⡇⠀⡤⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⡀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢷⣾⡶⠚⠋⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣠⠽⢶⠖⠚⡿
⠀⠀⠀⠀⠀⠀⠘⣇⠉⠃⢀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠈⠳⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⠘⣯⡉⠉⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠈⢳⡦⣄⡀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠉⠉⠁⠀⠀⣨⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠙⢦⡀⠀⠧⠞⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡴⠖⣶⡞⠁⠀⠈⣷⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠙⢦⣀⠤⠟⢦⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣱⠇⠀⠀⠀⠙⢒⡿⠙⠲⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⡤⠖⠛⠉⢦⠀⠀⠀⠙⢦⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠴⢦⢠⠏⠀⠀⠀⢰⢷⡀⠉⠉⢉⠝⠉⠉⠀⠈⣇⠀⠀⠰⠶⢷⡶⠆⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢾⣏⡀⠀⠰⣄⡟⠀⠁⠀⠀⠹⣌⠢⡄⠀⠀⢸⡄⠀⠀⢀⣠⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢼⣧⣀⣀⡀⠀⠀⢈⡖⣁⣀⣀⣸⡧⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⡄⢀⠄⠀⠀⠈⠄⠀⠀⠀⢸⠳⢤⡴⢳⠀⠀⠀⢀⠀⡧⠀⠀⠂⠀⠀⢸⡇⢠⡏⠀⠀⠸⠋⠃⠀⠀⢸⠀⠉⠈⠁⠀⡏⠙⠦⠄⠀⠀⣿
⡀⠜⠓⠀⠀⣀⠀⠈⢀⡀⠀⢸⠀⠀⠀⢸⠀⢀⡴⠋⠀⠀⠀⠀⠈⠀⠀⢸⡇⢿⡀⠀⠀⠀⠀⣀⠀⠀⢸⡄⢀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠁
⠁⡀⠀⠀⠀⣏⣹⠶⢏⣧⠀⢸⡀⠀⠀⢸⠀⠙⠦⣄⠲⢤⡀⠀⠀⠀⠀⢸⡇⠈⢧⠀⠀⠀⢘⠤⠁⠀⠸⠇⢺⠀⠀⠀⣿⠀⠀⢠⠀⠁
⢰⠃⠀⠀⠀⡏⠀⠀⠀⣿⠀⠀⠀⠂⠀⠈⠀⠀⠀⠈⢁⡴⠋⠀⢤⣀⡴⠀⠁⢀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠋⠀⠀⢀⡖⠉⠳⡄⠠⡁⠀⠀⢀⡀⠀⠀⢸⡏⠳⠀⠈⠀⠀⠀⢌⢀⠄⠀⠀⢀⡠
⠀⡴⠛⡟⠳⡍⠀⠀⠀⠈⢠⠖⠁⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⢉⡶⣏⡀⠀⠈⠂⠀⠀⠀⠀⠀⢸⡟⠳⠀⢀⠀⡀⢠⡞⠙⢦⠀⠀⢸⠗
⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⢙⡦⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⡇⠀⠀⠀⠂⠀⠀⣩⠾⣅⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠉⡄⡃⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡢⠴⡀⠀⠀⡇⠀⠀⠀⠀⢠⢤⣀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⣷⠀⠀⠚⠀⠃⠀⠀⠀⠀⠀⣄⡀⢀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠁⠀⢸⠆⠀⢸⠀⢩⠇⠀⣀⠴⠀⢸
⠀⠀⠀⠛⠀⠀⠀⠀⢰⢦⡀⢀⢶⠀⠉⠛⢾⠀⠀⠀⡟⠳⢤⡀⠀⣆⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢸⠀⣟⠀⠀⠓⢤⡀⠘
⡀⠀⠀⡟⢢⠀⠂⠀⢸⠀⠉⠁⢸⡆⠁⠀⠈⠀⠀⠀⣧⡴⠚⠀⠀⡯⣕⠦⣀⢸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⡆⠘⣆⠀⠰⠊
⢄⠀⠀⡇⣎⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠙⠪⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⡁⠀⠀⠇⠘⠄⠀⠀⠸⠆⢘⠲⠘⠇⠀⠮⡁⠀⠸⢂⠉⠀⠀⠀⠀⡧⠀⠀⠀⠸⠇⠀⠀⠀⣄⡀⠀⠀⡇⠀⠀⠢⣄⡴⠂⠀⠀⠀⠁
⠁⠀⠀⡀⠀⣐⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠁⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠲⣤⡇⠀⠀⡟⠪⣗⢤⡇⠀⠀⣔⠋⠙⡦⠀⠀⠀⠀⠀⢺
⠀⠀⠀⡇⠀⠠⠀⣆⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠀⠀⠸⠄⠘⠀⠀⠃⠀⠀⡇⠀⠈⠙⡇⠀⠀⢈⡷⢯⡀⠀⠀⡷⢄⠀⢸
⠆⠀⠛⣷⠤⠀⠀⣿⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⠀⠀⠀⠀⠀⠨⠀⢀⡀⠀⠀⢀⣀⠁⠀⣀⡀⠁⠀⠀⠉⠀⡀⠁⠀⠀⡷⠊⠀⢸
⠀⠀⠀⣿⠀⠀⠀⢹⡴⠋⠀⠀⠀⠀⢀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⡷⠀⢸⠀⠀⠀⡏⠙⣶⠄⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⢀
⠀⠀⠀⣿⠀⠀⠀⢸⠀⣤⠀⣠⠆⠀⢸⣗⣤⡀⡇⠀⠀⠀⠀⠀⠀⢨⠂⠀⢸⡏⠀⠀⣸⡀⠀⠀⣿⠞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢸
⠂⠀⢀⠀⠀⠀⠀⠀⠀⢸⠞⣡⠆⠀⢸⡀⠉⠻⣯⠀⠀⠀⠀⡄⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠙⢦⠀⠀⣀⠀⠀⠻⣄⡐⢤⡀⠀⠘
⠆⠐⢥⡀⠀⠀⠠⠀⠀⢸⡞⠁⠀⠀⠘⠃⣀⠀⠘⠀⠀⠀⠀⣯⣲⣄⡀⡇⠘⠃⠀⠀⠀⠀⠀⠀⢿⠖⠋⠀⠀⠀⠀⠀⠀⠈⠙⠀⣩⠇
⠀⠀⠊⠁⠀⢰⠀⠀⠀⢸⡀⣶⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⡇⠈⠑⢿⡇⠀⠀⠀⠀⢼⠀⠀⢰⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⢀⠞
⠀⠀⠀⠀⠀⢸⡓⢦⣀⠈⠀⢿⠀⠀⠀⠀⠁⠀⠀⠀⢀⢀⠀⠃⠀⠀⠀⠇⠈⠁⠀⠀⠈⠀⠀⢸⠀⡸⠁⠀⠀⠀⠀⡟⢲⣤⡞⡇⠀⠈⠲
⠃⣳⣠⠞⠁⢸⠉⠳⢮⡓⢦⣼⠀⠀⡟⢲⡄⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠴⠀⣧⡀⠀⠀⠀⢸⠀⢧⠀⠀⠘⠀⠀⡟⠉⠀⠈⡇
⠀⣿⣣⠞⠁⢸⡀⠀⠀⠉⠳⢿⠀⠀⣷⠊⠀⠸⠀⠀⠀⡶⢤⠀⠀⠀⢀⠀⠀⠀⣇⣩⠇⢀⣀⡼⠄⠈⠆⠀⠀⠀⠀⡇⠀⠀⠀⣧
⠀⣾⠁⠀⠀⢸⡇⠀⢀⠀⠀⢸⡄⠀⣿⠀⠀⠀⠀⠀⠀⡟⠦⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣸⠉⠀⠀⠀⢠⣤⡀⠀⠀⠃⠠⡆⠀⠛
⠀⠛⠂⠀⠀⠰⠄⠀⠈⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⠲⣤⡁⠀⠀⢸⠀⠀⠀⠀⢸⣄⡁⠀⠀⠀⣀⡁⠀⠀⠀⢢

⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠄⠀⠠⠤⠀⠠⠤⠤⢤⣤⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡆
⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿
⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠀⠀⣰⠎⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣀⠀⣴⠃⣿⣶⣶⢶⡟⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡾⢁⣼⡷⣾⣿⣿⣿⣾⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⡷⢸⡏⣿⣹⣿⣿⢡⣿⢗⡤⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢘⣿⢻⣿⢿⣷⠿⢃⣿⣿⠿⣇⣼⣧⣿⣿⣭⣠⡖⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣫⣿⢻⣿⡀⠈⣿⡿⣰⡿⣿⣿⣿⣿⣿⣿⣿⣯⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣏⣿⠋⣹⣧⣾⣿⣿⣿⡷⢛⣿⢻⣽⣿⣿⣿⣿⣇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣼⡿⠛⢿⣶⡟⠿⠟⠋⢉⣵⣿⠿⣿⣿⣿⣿⣿⡿⣿⠋⣡⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⣧⣴⣿⢛⠈⠁⠀⠰⣾⣿⣧⣼⣿⣿⣿⠿⢟⣿⠟⢻⣿⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣯⡍⠁⠲⠀⠀⠀⣹⣿⣯⣵⠟⠋⢻⡴⠋⣼⣥⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣏⣰⣿⠀⠀⢠⠎⢁⣶⣾⣧⣤⣶⠾⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣟⡇⠀⣿⠂⣿⡟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⣿⣿⢀⣶⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏⠟⡘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⣿⣾⣿⣿⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⣀⣤⣤⣶⣿⣶⣤⣀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢷⣿⣿⢿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣶⣶⣿⠿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡷⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣈⡓⠭⠖⠆⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣺⣿⢿⣏⢻⣥⢝⡻⣿⣿⠿⠳⠛⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⢚⣯⣶⡈⠀⠉⠛⢿⣅⡋⠀⠒⠀⠀⠀⠀⠀⣿
⡏⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡏⠛⢻⢧⣄⠐⠲⠠⠁⠀⠀⠀⠀⠀⠀⠀⣿
⡗⢈⠀⠀⠁⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣽⢿⣿⣿⡕⣧⣽⣿⣦⣬⠝⠓⠂⠀⠀⠛⠃⠀⠀⠀⠀⠀⣿
⣇⠀⠀⠀⠀⠈⠉⠛⢋⣹⣿⠟⣿⡟⣻⣿⠃⣿⡆⣿⣿⣷⡘⣷⣾⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡄⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠛⠽⣿⣿⣟⣾⣷⢿⡿⠟⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠉⠉⠙⠓⠒⠒⠂⠐⠶⠶⠶⠤⠤⠤⠴⠴⠶⠴⠷⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⣤⠀⠀⠀⠀⢀⢠⡀⠀⣄⡀⠀⢠⠀⠀⣀⠀⠀⢠⠀⢀⣤⡄⠀⣶⠀⠀⢂⡀⠀⠀⠀⠠⠆⠀⠀⠠⠀⠀⠤⠄⠠⣄⠀⠀⢤⠀⠠⠆⠄⠀⢠
⠀⠈⠁⠀⢀⠀⠈⠀⠀⠀⠈⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠿⠆⠀⠻⠀⠀⠠⠀⠀⠯⠂⠀⠟⠅⠀⠉⠀⠀⠺⠀⠀⠺⠀⠀⠓⠂⠀⠡⠁⠀⠻⠀⠀⠬⠆⠀⠃⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠈⠒
⠀⣤⠀⢠⢤⡄⢀⡔⠀⠀⢤⠀⠠⣦⡄⠀⢠⠀⠀⠄⠀⠀⠐⠀⠀⡤⠀⠀⣶⠀⠠⠀⡄⠀⢂⠀⠀⢀⠀⢀⣤⡀⠀⣶⠀⠀⣀⠀⠀⣤⠀⠀⡲
⠀⢀⠀⠀⣀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠈⠁
⠀⠽⠀⠀⠚⠁⠀⠚⠂⠀⠚⠃⠀⠛⠀⠀⠀⠂⠀⠐⠀⠀⠛⠂⠘⠓⠃⠀⠛⠀⠀⠛⠂⠈⠁⠆⠀⠙⠀⠀⠛⠁⠨⠋⠇⠀⠂⠃⠀⠒⠀⠈
⠀⣶⡀⠀⣤⠀⠀⠰⠀⠀⣤⠀⠀⠰⠀⠀⣤⠀⠐⠂⠀⢠⣤⠀⢀⡀⠆⢀⣴⡀⠀⣤⠀⠀⡰⠀⠀⠶⠀⠀⡤⠀⠀⣄⠀⠀⠆⠀⠠⢄⡀⠀⣤
⠀⠈⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠠⠀⠃⠀⠤⠀⠀⠉⠁⠀⠘⠀⠐⠆⠃⠀⠐⠀⠀⠲⠀⠐⠟⠁⠀⠀⠀⠀⠁⠁⠀⠻⠂⠀⠦⠀⠀⠙⠀⠈⠩⠁⠀⠒⠄⠀⠓⠂⠀⠒⠀⠀⠒
⠀⢠⠀⠀⠰⠀⢀⣤⡄⠀⣴⠀⠀⡰⠄⠀⣠⠀⠀⣖⠄⠀⣤⡀⠀⡄⡄⠀⢀⠀⢀⣰⡀⠀⢶⠀⠀⣤⠀⠀⣤⠀⠠⣖⡄⠀⡀⠆⠀⣀⠀⠀⣤
⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⡀⠀⠀⢀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠉⠀⠀⠈⠀⠀⢈⠀⠀⠉⠀⠀⠀⠁⠀⣉
⠀⠻⠅⠀⠓⠃⠀⠟⠀⠀⠔⠁⠀⠴⠀⠀⠓⠀⠀⠣⠄⠀⠰⠀⠀⠛⠀⠀⠐⠀⠀⠛⠁⠐⠐⠀⠀⠶⠀⠀⠙⠀⠀⠘⠀⠀⠙⠀⠀⠠⠀⠀⠉
⠀⣴⡄⠀⣰⠀⠀⣀⠀⠀⣠⠀⠀⣤⠄⠀⢒⡄⠀⢰⠀⢀⣤⡀⠀⢠⠀⠀⣀⠀⠀⣠⠀⠀⣀⡀⠀⠠⠄⠀⡀⠀⠀⣀⠀⠀⠀⠄⠀⢤⠀⠀⢄
⠀⠉⡀⠀⠉⠀⠀⡉⠀⠀⠈⠀⠈⠀⠀⠀⠈⠀⠀⢈⠀⠀⢀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠠⠙⠄⠀⠑⠀⠀⠁⠀⠈⠉⠁⠀⠰⠀⠀⠹⠁⠀⠺⠀⠀⠠⠀⠀⠸⠀⠀⠖⠀⠀⠛⠀⠐⠛⠂⠀⠀⠀⠀⠺⠀⠀⠁⠀⠀⠉⠀⠀⠉⠀⠀⠙
⠀⣐⠀⠀⠄⠀⠀⠤⠀⠀⢴⠀⠀⣤⠄⠀⡄⠀⠀⣄⠀⠀⣠⠀⠀⣤⡀⠀⣀⡀⠀⣄⠀⠀⡤⠀⢀⣴⡄⢀⡔⠄⠀⢠⠄⠀⣤⡀⢀⠰⠀⠀⣠
⠀⠀⠀⠀⠀⠃⠀⢈⠀⠀⣀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠈⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠂⠂⠀⠹⠀⠀⠾⠆⠀⠍⠀⠀⠃⠀⠀⠒⠀⠀⠺⠀⠀⠰⠀⠀⠣⠃⠀⠽⠀⠀⠿⠀⠀⠙⠀⠀⠣⠀⠀⠚⠂⠀⠶⠄⠀⠥⠇⠀⠤⠀⠀⠴
⠀⣠⡀⠀⠀⡀⢀⣤⡀⠀⠀⠀⠀⠄⠀⠀⣠⠀⠀⠀⠀⠀⡀⡀⢀⣰⡀⠀⠠⠀⠀⠀⠀⠀⠠⡀⠀⣠⠀⠀⣀⠀⠀⢀⠀⠀⢢⠀⢀⣐⠀⠀⢦
⠀⠉⠁⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠐⠻⠂⠀⠲⠀⠀⠙⠀⠀⠐⠂⠀⠒⠂⠀⠼⠁⠀⠒⠀⠀⠹⠀⠀⠽⠀⠀⠰⠀⠀⠒⠂⠀⠂⠀⠀⠀⠀⠀⠙⠁⠀⠈⠄⠀⠀⠀⠀⠖⠁⠀⠃
⠀⢤⠀⠀⠤⠆⠀⢠⠀⠀⣀⠄⠀⠤⠆⠀⠠⠀⠀⠐⠆⠀⠀⠀⠀⡐⡂⠀⢲⠆⠀⣴⡄⠀⡀⡀⠀⣀⠀⠀⠠⡄⠀⣀⡄⠀⡠⣀⠀⠀⠀⠀⣠
⠀⢈⠀⠀⠁⠀⠀⠈⠀⠀⢈⠀⠀⠀⡀⠀⢀⠀⠀⢈⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡉⠀⠀⢉⠀⠀⠁⠀⠀⢀⠀⠀⢀⠀⠀⠁
⠀⠳⠂⠀⠐⠂⠀⠀⠃⠀⠀⠄⠀⠈⠃⠀⠍⠄⠀⠸⠀⠀⠥⠃⠀⠒⠃⠀⠸⠂⠀⠸⠀⠘⠉⠁⠀⠟⠇⠀⠯⠃⠀⠠⠀⠀⠺⠂⠀⠻⠂
⠀⡰⡀⠀⢤⡄⠀⡄⠀⠀⠶⡆⠀⣴⡀⠀⢠⠀⠀⢀⠂⠀⡄⡄⠀⢢⡆⠀⠠⡄⠀⡐⠀⠀⢀⠀⠀⣀⡄⠀⣴⡄⠀⢠⠀⠀⠐⡄⠀⠤⠄⠀⢠
⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠐⠂⠀⠋⠂⠀⠇⠇⠀⠛⠃⠀⠙⠁⠀⠆⠁⠀⠛⠃⠀⠙⠁⠀⠃⠃⠀⠛⠃⠀⠖⠀⠐⠂⠀⠀⠛⠁⠀⠁⠂⠀⠛⠂⠀⠓⠂⠀⠸

⠀⢰⢔⡀⠀⠀⠀⠀⡶⢄⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⢰⢦⡀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠐⣄⠆⠀⠀⠀⠀⢰⣢⡄⠀⠀⠐⠦
⠀⠸⠉⠀⠀⢦⡀⠀⡇⠀⠀⠀⠀⠀⡟⠁⠀⠠⣴⠀⠀⡏⠂⠀⠀⠀⠸⠛⠄⠀⠀⠀⠘⠢⡀⠀⠀⠀⢠⠛⠄⠀⣠⣤⡀⢸⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠺⡌⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⡀⠀⠸⠓⠀⠀⠀⢴⡀⠀⠀⠀⠀⠀⠲⡦⠂⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣤⠀⡆
⠐⡷⠂⠀⢠⣀⠀⠀⠀⠀⣀⡍⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠘⠁⢠⣄⡀⠀⠀⠀⠇⠀⠀⠀⢸⠀⡇⠀⢰⢤⡀⠀⠁⢀⠀⠀⣿⠙⡇⠀⢀
⠀⠃⠀⠀⢸⠀⣧⠀⠀⠀⠈⠁⠀⠀⠈⠋⠁⠀⠘⢳⠆⠀⠀⠀⠀⢸⠛⠁⠀⠀⠀⠀⢀⡤⡀⠈⠀⠀⠀⢸⣻⠄⠀⠀⢸⣢⠀⠀⠀⠀⣼⠥
⠀⠀⠀⠀⠈⠀⠉⠀⡤⡀⠀⠀⡱⣆⠀⠀⡀⢀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⣄⣠⠀⠀⢈⡷⣅⠀⠀⣄⡀⠈⠀⡀⠀⡀⠘⠀⡀⢀⠀⠀⠛
⠀⢠⠖⠁⠀⠀⠀⠀⡷⢅⠀⠀⠜⠣⠀⠀⢘⣎⠀⠀⠀⠀⠀⡟⣤⢻⠀⠀⡇⢸⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡽⢮⡅⠀⠀⡏⠉⠀⠀⢰⣢⣔
⠀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⢰⢤⡀⠀⠋⠀⠙⠀⠀⠁⠀⠀⠀⠰⡻⠆⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠓⠓⠀⠀⠸⠁
⢀⣤⡀⠀⣆⢰⠀⠀⠀⠀⠀⠢⡇⠀⠀⠀⢰⠀⠀⢸⠻⡄⠀⢀⢦⠀⠀⠀⠀⠐⠦⠀⠀⠇⠀⠀⡟⠚⠀⠀⠀⠐⣴⠔⠀⠀⠀⠀⣰⠂
⠈⢸⠁⠀⠇⠹⠀⠠⣴⠄⠀⠀⠏⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠳⠍⠀⠀⠀⠀⠲⠄⠀⢀⡀⠀⠀⠧⠤⠀⠀⠀⠀⠸⠀⠀⡀⠀⠀⠉⠟⠀⢀⡔
⠀⠈⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣖⠢⠀⠀⠀⡷⠶⡇⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⢸⠬⠁⠀⠀⠀⢰⡀⡆⠀⠀⠀⢠⠬⠆⠀⣀⢀⡀⠀⠑
⠀⢠⡤⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠇⠉⠀⠀⠀⠃⠀⠃⠀⠀⠀⠐⣭⠂⠀⢀⣀⠀⠘⠀⠀⠀⠀⠀⢸⠉⡇⠀⠀⠀⠀⠑⠀⠀⡏⠉⡇
⠀⢸⠊⠁⢠⠀⠰⡋⠀⠀⠀⡏⢱⠀⠀⠀⣀⠀⠀⠀⣠⠤⡀⠀⠀⠈⠀⠁⠀⢸⠞⠁⠀⠀⠀⣓⠤⠀⠀⠀⠀⠀⠘⡴⠃⠀⢠⡀⠁⠀⠁⠀⢀
⠀⠈⠀⠀⢸⠀⠀⠈⠃⠀⠀⠃⠘⠀⠀⢀⡏⠁⠀⠀⣨⠾⡁⠀⠀⡶⠴⠀⠀⠘⠀⠠⣄⠀⠀⠭⠃⠀⠀⢸⠀⠀⠔⠙⠆⠀⢸⠈⠀⠀⠀⣿⠥
⠀⠀⡀⠀⠀⠀⠐⢾⡀⠀⠀⢄⢠⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⠀⠀⠀⠀⢮⡊⠀⠀⢀⡀⠀⠀⠸⠀⠀⣀⡀⠀⠀⠈⠀⢠⠀⠀⠃
⠀⠀⡽⠂⠀⣀⠀⠘⠈⠀⠀⡨⢧⠀⠀⡦⣀⠀⠀⣄⣠⡀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⢸⢞⠁⠀⠀⠀⠀⡷⠬⠀⠀⠀⠀⢸⠀⢀⠀⢀
⠀⠀⠀⠀⠈⡏⠁⠀⢠⣇⡀⠁⠀⠀⠀⣟⠧⠀⠀⡇⠀⡇⠀⠀⣷⠶⡇⠀⠀⢹⠦⠀⠀⠒⠄⠈⠀⠁⠀⡲⡀⠀⠁⠀⠀⠸⣹⠆⠀⠀⢸⡳⣞
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⠈⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠞⠉⠂⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀⣀⠀⣀⠀⠀⢀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠁⠀⠘⡆⠹⡄⢀⡏⢸⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠸⣄⣀⣰⠃⠀⢳⡼⠀⢘⣇⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠃⠀⠀⠠⠀⠀⠁⠀⠀⠀⠈⠀⠑


⠀⠀⠀⠀⠀⠀⢀⠀⢀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣴⣄⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢈⣷⣿⣧⡾⠋⠀⠀⠀⠀⠀⠀⢀⡴⢻⡟⢦⡀⠀⠀⠀⠀⠈⠻⣟⣹⣉⡿⠋⠀⠀⠀⠀⠀⠀⠻⢦⣸⣠⡾⠃
⠀⠀⠀⠀⢀⡴⠟⣡⣾⡿⣄⠀⠀⠀⠀⠀⢀⡴⠋⣠⢾⡷⣄⠀⠀⠀⠀⠀⠀⠀⢈⣿⣏⡀⠀⠀⠀⠀⠰⣄⠀⠀⣨⣿⣯⡀⠀⢀⡴
⠀⠀⠀⠰⣿⡁⠺⣏⠘⡇⢈⡷⠀⠀⠀⠰⣿⡁⢾⣏⢸⡇⢈⡷⠀⠀⠀⠀⠀⠐⢯⣸⣭⡷⠀⠀⠀⠀⠀⠈⣳⣾⠋⢸⠀⢻⣶⡋
⠀⠀⠀⠀⠈⠹⣦⡈⢻⣷⠋⠀⠀⠀⠀⠀⠈⠻⣦⡙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⡀⠀⠀⠀⠀⢠⡾⠋⠈⠳⣾⡶⠋⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢀⡿⣿⡟⠳⣄⠀⠀⠀⠀⠀⠀⠈⠻⣾⡇⠀⠀⠀⠀⠀⠀⣠⡾⢯⣼⣦⠿⢦⡀⠀⠀⠀⠀⠀⢀⡴⢻⠷⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠁⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠘⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⢰⣄⡀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⣤⡀⠀⠀⠀⠀⣀⡀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢶⣄⠀⢀⣴⠆
⠀⠀⠀⢸⡏⠙⠶⣄⣤⠞⠋⢸⡇⠀⠀⠀⣿⠙⠷⣄⣤⠞⠋⠀⠀⠀⠀⠀⠙⣦⡞⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⢙⣷⣟⠁
⠀⠀⠀⢸⣇⣤⠞⠋⠙⠶⣄⣸⡇⠀⠀⠀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⡞⠁⠙⣦⡞⠁⠙⣦⡀⠀⠀⠀⠀⣴⠟⠁⠙⢷⡄
⠀⠀⠀⢸⡏⠁⠀⠀⠀⠀⠈⢻⡅⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠞⠉⠙⣦⡾⠋⠀⠀⠀⠀⠀⠙⢷⣄⣴⠟⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣠⠶⠋⠙⢶⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⠋⠙⢦⡀⠀⠀⠀⠀⢀⣴⠟⢷⣄
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠋⠁⠀⠀⠀⠀⠈⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠛⠁⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠶⠛⣿⡛⣻⠟⠛⠶⣶⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⣴⠋⠀⠀⠀⠈⣷⡏⠀⠀⢠⣏⣷⡄⠈⠉⢻⡶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⣭⣤⠄⠀⠀⠺⢧⣄⡀⠀⠀⣰⠟⢷⡀⠀⣼⠉⠀⠀⠀⢠⣿⡄⣸⠟⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠈⢷⣯⣄⣀⠀⣀⣤⣬⠿⠷⠛⠛⠛⠛⠛⠛⠿⠶⣤⣄⣠⡟⠈⣿⠃⠀⠀⢠⢙⣷⣄
⠀⠀⠀⠀⠀⣠⡾⣿⣛⣩⡤⠀⠀⢻⣬⡿⠟⠉⠁⠀⠀⠀⢠⠀⢠⡄⠀⡄⠀⠀⠀⠈⠙⠻⢾⣇⡀⠀⢀⣼⠟⠁⠙⢷⡄
⠀⠀⠀⠀⣴⠏⠀⠈⠻⣇⡀⢀⣴⠟⠁⠀⢤⡀⠀⠀⠀⠀⢻⠤⢼⡧⠤⠇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡟⢻⠀⠀⠀⣀⡿⣦
⠀⠀⢀⣾⠳⢶⣞⣿⠀⢈⣿⠟⠁⢀⣠⣄⠀⠙⢦⡀⠀⠀⠀⢐⣺⣗⠂⠀⠀⠀⠀⠀⢀⡴⠋⠀⡀⠈⠻⣦⣀⣴⠞⠉⠀⠙⣧⡀
⠀⢀⣾⠃⠀⠀⠙⠳⣦⡾⠁⠀⠀⢿⣳⣼⣇⡀⣠⠟⠀⠀⠀⠨⢽⡯⠄⠀⠀⠀⠀⠀⡆⠙⣶⡞⠁⣠⠀⠈⢿⡄⠀⣤⣀⡀⣼⣧
⠀⣼⣧⡞⠛⠳⠶⣤⡟⠀⠀⠳⣄⠀⠉⠹⣻⣾⣁⣤⣄⡀⠀⠀⢸⡇⠀⠀⢀⣤⣤⣄⣹⢾⣁⡙⠞⠁⠀⠀⠈⢿⣼⠃⢮⣵⠇⠸⣇
⢠⡟⠉⠛⠳⠦⣤⡿⠀⠀⠀⠀⠈⠳⣤⠞⠁⣼⠻⣄⠀⠁⠀⠀⢸⡇⠀⠀⠀⠘⣠⠞⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⢿
⣼⣧⢠⡄⠀⠀⣸⠇⠀⠛⠛⡇⠀⠀⠀⠀⠀⠻⠄⠈⠳⣄⠀⠀⣸⣇⠀⠀⣠⠞⠁⠉⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠀⠀⢀⣸
⣿⣼⣆⣿⣀⣀⣿⠀⠀⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠛⠉⠉⠛⣾⠁⠀⠐⢦⡄⠀⠀⢠⡴⠂⢰⠒⠒⠀⠀⣿⠛⠛⠛⣹⠏
⣿⠀⠈⠉⠉⠉⣿⠀⠀⠶⠶⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡇⠀⠀⠀⠀⢸⡷⠶⠶⠶⣿⢾⡷⣿⠶⠶⢾⠶⠶⠀⠀⣿
⣿⢤⣶⠖⣺⠟⣿⠀⠀⣤⣤⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣷⣄⣀⣀⣠⣞⠀⠀⢠⠴⠋⠈⠁⠙⠦⡄⢸⣤⣤⠀⠀⣿⣿⠓⠶⠶⣴
⢻⡄⢈⡿⢯⣄⢻⡆⠀⣀⣠⡇⠀⠀⠀⠀⠀⠀⣟⠀⣠⠞⠁⠈⢹⡏⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠞⠁⠀⢾⣽
⠸⣷⠿⠗⠚⠛⠙⣷⠀⠀⠀⠀⠀⠀⢀⣤⡀⢤⣹⢾⣁⣤⠄⠀⢸⡇⠀⠀⠀⠈⠳⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣿
⠀⢻⡄⠀⠀⠀⢠⡿⣧⠀⠀⠀⢀⡴⠋⣄⣹⢾⡙⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢋⡿⣯⡤⣠⣤⡀⠀⠀⠀⣼⢧⣄⣀⣀⣤⣴⡏
⠀⠈⢿⣠⡟⠛⡿⠶⢽⣧⡀⠀⠋⠐⣷⣞⠓⣠⠟⠀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⠀⠀⠘⣫⠿⣇⡀⠙⠀⢀⣾⠃⠀⠈⢻⡿⢶⡿
⠀⠀⠈⢿⡟⠾⠃⠀⠀⠈⢻⣄⠀⠈⠁⣩⠞⠁⠀⠀⡏⠛⣩⣭⣽⣯⣭⡍⠛⣻⠀⠀⠀⠳⣄⠈⠁⠀⣰⡟⠁⠀⠀⠀⠀⢻⡿⠁
⠀⠀⠀⠈⢻⣆⠀⠠⡤⠤⣤⠙⠷⣄⡀⠀⠀⠀⠀⢠⣧⠀⢿⠀⢸⡇⠀⡏⠀⣽⡄⠀⠀⠀⠀⢀⣤⠞⠉⣷⣠⣾⡄⠀⣰⠟
⠀⠀⠀⠀⠀⠙⢷⣸⣇⣀⣿⠀⠀⠈⠛⣶⣤⣀⠀⠐⠓⠀⠀⠀⠘⠃⠀⠀⠀⠛⠂⠀⣀⣤⣾⠛⠁⠀⠀⠘⠋⠈⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣴⠏⠈⣿⠛⠷⣶⣤⣤⣤⣤⣤⣤⣤⣴⢶⣾⠛⠉⠁⠙⣆⠀⣀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣻⡾⠃⠀⢰⣏⠀⢀⡟⠀⢸⡇⠀⢹⠀⠀⠰⢿⣽⡆⠀⠀⢤⣼⣷⣇⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣤⣿⠟⢿⣿⠁⠀⢸⣇⣄⣼⠀⠀⠀⣾⣯⣷⢀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⣤⣼⣯⣬⣿⣤⣤⡤⠶⠾⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠾⠟⠛⠛⠛⠛⠻⠷⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠉⢡⣄⠀⠀⠀⠀⠚⡟⠀⠀⠀⠀⡄⠈⠉⠛⢷⣤⣀
⠀⠀⠀⠀⠀⠀⣠⡾⠛⡁⠀⠀⠀⠈⢷⠆⠀⠀⠀⠀⠃⠀⠀⠀⠘⢛⠇⠀⠀⠀⢈⠛⢷⣄
⠀⠀⠀⠀⣠⡾⠋⠀⠲⣷⠀⠀⠀⣀⣤⣴⠶⠾⠛⠛⠛⠛⠷⠶⣦⣤⣀⠀⠀⠀⡸⠓⠀⠙⢷⣄
⠀⠀⢀⣼⠟⠀⠀⠀⠀⠈⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣦⡈⠀⠀⠀⠀⠀⠹⣧⡀
⠀⢀⣾⠋⢤⡤⠤⠄⢀⣴⠟⠁⠀⠀⣴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⣠⠴⠦⠙⣷⡀
⠀⣼⠇⠀⠀⠈⠁⢀⡾⠃⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠃⠀⠀⠀⠸⣧
⢰⡟⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⢻
⣾⠇⠰⠾⠦⠀⣸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠁⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠰⠦⠖⠶⠸
⣿⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢿⡆⠹⠷⠤⠂⢹⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⣶⡄⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠒⠿⠂⢰
⠸⣧⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⣼
⠀⢻⣆⠀⠀⠀⣄⠈⢿⣄⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⣠⡿⠁⣄⠀⠀⠀⣰⡟
⠀⠀⢻⣆⠸⠛⠁⠀⠀⠻⣧⣀⠀⠀⠻⠿⠀⠸⠿⠃⠀⠀⠀⠈⠻⠟⠀⠀⣀⣼⠟⠀⠀⠀⠉⠂⣰⡟
⠀⠀⠀⠻⣦⡀⠀⠀⠀⢠⣈⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠡⡀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠘⢷⣄⠀⢀⠜⠁⠀⠀⠈⡙⠛⠷⠶⢶⣶⣶⡶⠶⠾⠛⠋⡁⠀⠀⠀⠛⠧⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⢀⡷⠃⠀⠀⠀⠀⡆⠀⠀⠀⢳⣦⣧⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣁⠁⠀⠀⠀⠀⢴⡇⠀⠀⠀⠀⠃⣈⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣶⣶⣶⣶⣶⣶⡶⠾⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⢠⠀⡄⣠⣄⢠⠀⡄⢀⡠⡄⡤⣀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⢸⠀⡇⠠⡆⢸⣀⡇⢨⠐⡇⣧⠊⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠐⠃⠘⠀⠃⠀⠀⠃⠋⠓⠐⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠋⡁⠀⠀⡏⠑⡆⠀⢸⢦⡀⠀⠀⣏⠒⠀⠀⣏⡱⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⡇⠀⡇⠀⢸⡔⠁⠀⠀⡍⠑⠀⠀⣽⢄⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢆⠖⠀⠀⡟⢦⠀⠀⢸⠀⣶⠀⠀⠀⡀⠀⠀⠀⡆⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠀⡷⠁⠀⠀⢸⠓⣯⠀⠀⠙⡧⠀⠀⠀⡇⠀⠀⢰⡢⠊
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠑⠀⠀⠃⠀⠀⠀⠘⠀⠋⠀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⡀⠀⡶⠴⠀⠀⢠⣀⡤⠀⠀⠀⡄⠀⠀⡠⠶⡄⠀⢰⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⡇⠀⠀⠀⠀⢻⠀⠀⠀⠺⡤⠀⠀⠀⣆⠀⠀⢸⠮⡁
⠀⠀⠀⠀⠀⠀⠀⠑⠾⠀⠀⠀⠷⠢⠀⠀⠀⠘⠀⠀⠀⠎⠀⠀⠀⠀⠿⠀⠀⠸⠖⠁

⠀⠀⠀⠀⠀⠀⠀⢠⡀⣠⠀⢠⣄⢀⡄⠀⢠⢄⡀⠀⠀⢄⡤⠀⠀⢀⣄⠀⠀⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⠈⠹⡀⠀⡼⠣⡄⠀⢸⠀⠁⠀⠀⡞⡦⠀⠀⠺⣼⠃⠀⢸⢱⣻
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠇⠀⠀⠀⠇⠀⢸⠀⠀⠀⠀⡜⢧⠀⠀⠰⠻⠄⠀⠸⠃⠹


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⠀⠀⢀⠀⠀⡀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠓⠀⠀⠘⠀⠑⠀⠀⠀⠋⠀⠈⠓⠃⠀⠂⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⢿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠈⠙⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠚⠋⠉⠉⣹⡟⠁⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢁⣀⣄⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⠔⣊⣉⣀⣠⠤⠤⠴⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠒⠸⠲⢄⠀⠀⠀⠼⣫⡶⠮⠡⠄⠀⢀⣀⣀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣬⡋⢶⣦⣄⢠⡍⠩⠐⠒⠂⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣹⣿⣆⠁⠠⣦⠄⠀⠉⠉⠀⠀⠠⠊⠀⠀⢠⣤⣶⣾⣿⣦⡀⠈⡄⠀⠀⠹⣿⡄
⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠴⡁⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣠⣴⣤⣤⣄⣀⠀⠀⠻⣿⣿⣿⣿⣷⣼⡇⠀⠀⠀⢻⣷
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣠⠖⢋⠕⡡⢾⢱⠘⠪⠻⢯⡓⠂⣤⠀⠀⢠⣾⡿⠛⠛⠛⠿⣿⣿⣿⣤⡀⠀⠀⠉⣿⣿⣿⡧⠄⠀⠀⢸⣿⡄
⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⣠⠎⠁⣰⠃⣸⠁⡎⠀⠀⢰⡆⠀⠀⠀⠉⠀⠀⠩⣥⣤⣄⡐⠢⣤⡈⠻⣿⡛⣿⣶⣴⣶⣿⣿⡟⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⣼⠃⠀⢠⠏⢠⠇⠀⡁⢀⣨⠀⠀⠀⣠⣴⡾⠃⠸⣧⠈⠻⣿⣿⡗⢸⣿⡄⠘⠷⠈⠻⢿⣿⣿⣉⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢻⣿⠀⠀⣰⣏⣀⡴⠋⢀⡏⠀⣸⠁⠸⠃⠀⠀⣼⣿⠟⠁⠀⠀⣿⡇⠀⠀⢻⣿⡀⢸⣇⠀⠀⠀⠀⠀⠈⠉⠭⣁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⡀⠰⠟⠛⠉⠀⢠⡾⠷⠚⠁⠀⠀⠀⢀⣼⣿⡏⠐⠀⠀⣸⣿⣷⠀⠀⣼⣿⡇⣼⡿⠀⣀⠀⢀⠀⠀⠀⠀⠀⠳⢆⡀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣼⣿⣿⠇⠀⠀⠘⣫⣿⣿⠀⣼⣿⣿⠃⣿⣧⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⣥⠀⣾⡟
⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣿⠟⠁⠀⠀⠀⢠⣾⣿⣿⢃⣾⣿⣿⣿⠈⠙⢿⣿⣶⣶⣿⣶⣄⠀⠀⠀⠀⡠⠃⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠉⠛⢿⣿⣿⣿⢿⡏⠀⠀⠀⠀⣰⣿⣿⡟⢹⡿⢫⣿⣿⠛⠄⠀⠈⠛⠿⣿⣿⣿⣿⡆⠀⠁⠈⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠈⠉⡠⠎⠀⠀⠀⣀⣬⣿⣿⣿⠁⠛⠀⣾⣿⣿⠀⠀⠀⠀⠀⢄⡈⠛⠛⠛⠓⠀⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⣼⣿⣿⣿⣿⢀⠀⢰⣿⣿⣯⠀⠐⠀⠁⠀⠀⠙⢦⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡌⠂⢸⣿⣿⣿⣿⣴⣾⣿⣶⠆⠀⠀⠆⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣀⠀⠀⠀⠿⠋⠁⠀⠀⠀⠰⠏⠹⣿⣿⡿⠋⢀⣠⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣦⣤⣤⣤⣤⣠⣤⣤⣤⣤⣴⣶⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡥⢤⣟⡛⠛⣛⣛⠛⠛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠞⣉⣉⡽⠖⠛⣋⣉⣙⠓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡤⢤⡿⢡⡞⣱⠋⣠⠚⣯⡥⠤⠬⢽⣲⣄⣹⠖⣋⣉⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⡇⣼⠶⢋⣠⢿⣙⠳⠾⠾⢳⣌⠲⣏⣛⣲⣌⢿⣵⣞⠻⠦⠽⠾⠷⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⡤⣶⣶⣧⣿⣛⣭⠴⠚⣩⠟⠛⠛⠉⠉⠓⠮⣭⡭⠜⢦⡙⢮⣙⠒⡶⢺⣛⣛⣦⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡞⣡⢞⣡⠶⡶⢶⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣌⣉⣿⠘⣧⢤⣬⣍⡙⠲⣄
⠀⠀⠀⠀⠀⠀⢀⡴⢺⡇⣾⣤⣧⣾⡤⠞⣡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣻⢁⣿⣉⠳⡌⢳⡀
⠀⠀⠀⠀⠀⠀⣾⢠⢾⡆⣿⢶⠶⡶⠒⠉⠁⠀⠀⠀⠀⠀⢰⠋⢹⡄⠀⠀⠀⠀⡾⠉⢳⠄⠀⠀⠀⠀⠀⠀⠀⢿⢸⡷⣌⢳⣹⡀⣷
⠀⠀⠀⠀⠀⠀⣷⠸⣆⣷⠼⢋⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡇⠀⠀⢀⡼⠁⢠⠟⣠⠶⢦⡀⠀⠀⠀⠀⠸⡌⢧⣸⠈⣿⡇⢸
⠀⠀⠀⠀⣰⠋⡹⡆⠹⠶⠖⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢀⡴⠋⢀⡴⠋⣴⠃⠀⡼⠃⠀⠀⠀⠀⠀⠙⠲⡞⢸⣏⣛⠛⠶⣄
⠀⠀⠀⠀⢿⠘⣇⡇⢠⢂⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠗⠋⣀⡴⠋⢀⡼⠃⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢰⡇⣶⢸⠀⡏⢲⠘⡇
⠀⠀⠀⠀⠈⣳⣌⠳⣯⠞⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠚⠉⢀⡴⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢻⡘⣦⡙⢯⣴⠃
⠀⠀⠀⠀⣰⠃⣼⡿⢫⡞⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠿⠚⠋⠉⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠟⣡⡟⢦⠙⣆
⠀⠀⠀⠀⡟⢰⢻⡇⡏⢹⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⡤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⢧⠙⣮⠇⣻
⠀⠀⠀⠀⣷⠸⡄⢧⢹⡞⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠟⣋⣷⠘⡶⠋
⠀⠀⠀⠀⠘⣆⠳⣼⠋⣴⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⠴⡏⣯⢸⡇⣷
⠀⠀⠀⠀⠀⡞⢳⣌⠛⠾⡇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣏⡸⢃⡏⡼⢀⡇
⠀⠀⠀⠀⠀⢳⡈⠯⣝⣳⢃⡾⣍⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⢓⣫⣴⣫⠞⢡⠞
⠀⠀⠀⠀⠀⠀⠙⠲⣬⣉⡙⠳⣌⢧⠈⣆⠀⠀⠀⠀⠀⠀⠘⢦⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⢚⡭⢧⠹⣥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣉⣧⣘⣮⡇⢻⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣼⣡⡞⣠⠞⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣥⡤⢶⣿⡘⣏⠓⣎⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣴⢋⣥⡤⣤⢽⠀⣿⢚⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠳⡑⢬⣙⣚⡆⢻⣉⡛⠲⡴⢚⣋⣛⠓⠦⣴⣋⡴⠒⢲⡤⣍⠛⡾⢀⣏⡾⢠⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⠲⠤⣭⣧⠼⠶⢛⣧⡘⣯⡭⠭⢭⣷⢦⣍⣛⣛⣩⠞⣉⡿⢁⣞⣉⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⣶⠶⢾⣭⠴⢋⣡⡶⠒⣶⢃⣞⠛⠛⠓⠒⢚⣡⠴⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠒⠛⢧⣉⣉⡥⠞⠉⠉⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠀⣾⠟⠉⠑⠂⠀⠠⣴⣿⣷⣦⣤⣤⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⣽⣿⡇⠀⠀⠀⠀⠀⠐⢻⣶⣿⣷⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠟⠈⠁⠀⠀⠀⠀⠀⢰⣾⣟⣛⡿⢿⣯⣽⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣴⣯⣽⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⠟⠋⣽⣆⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⡀
⠀⠀⠀⠀⠸⠺⣇⣈⣭⣩⡇⠀⠀⠀⠀⠀⠈⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠹⡟⠃
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⢣⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣿⣷⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠾⠿⠿⠿⠿⠿⠟⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⡢⠄⠉⡁⠀⠀⠀⢀⠉⢀⠰⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢦⠁⡠⠐⠅⠀⡀⠔⠁⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⡄⠊⠴⠔⠐⣡⢠⠀⠀⢀⠔⢈⠉⠈⡦⠊⠐⠃⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⡔⢁⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠫⣀⡟⢃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠓⠲⠒⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠀⠈⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⢦⡇⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠈⢨⠱⡀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⣀⠔⠊⠉⠀⠉⢆⢀⠾⡖⠋⠀⠀⠈⠢⣸⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠸⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠐⠊⠉⠀⢀⠀⠐⢉⡀⠑⠊⢈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢐⢰
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠉⠈⠁⢊⣆⠀⠀⠈⢯⡡⢬⠉⠑⢆⠀⡠⠈
⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⠀⠀⠀⠀⢀⠆⠀⠀⠇⠰⡀
⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡑⣄⠠⠖⠁⠀⠀⠀⠈⠱⢰
⠀⠀⠇⠀⠡⠀⠀⠀⠀⠀⠡⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠆⠇⠀⠀⠀⠀⠄
⠀⠀⠀⡄⠀⢰⠀⠀⠀⠀⠀⠅⠒⠄⠠⠄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⠀⠀⠀⠈⢠⠃
⠀⠀⠀⠱⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠐⣌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⣨⢁⠒⡁⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠐⠄⡀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⡠⠊⠁⡣⠀⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣈⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀⣀⠤⠒⠁⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠤⠀⣀⣀⡀⠀⠀⣀⣀⠠⠤⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⣹⣿⣿⣿⡿⠿⠟⢿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⣀⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢻⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⡟⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⣨⣤⣶⡄⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣦⣄⡀⠀⠀⠀⠹⢿⡦⢻⣿⣿⣿⣟⢳⡄
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠤⣿⣿⣿⣿⣷⡀⠠⣤⡀⠀⠘⠿⠙⢿⣿⣿⣿⣿⡆⣠⣤⠀⠀⢦⣄⣻⣿⣿⣟⠣⣿⣆
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣽⣧⣴⣺⣷⣦⡀⠉⢻⣿⣿⠀⠙⠁⠀⠀⠘⠯⠛⣿⣿⡇⢠⣽⣿⣆
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⣼⣿⠆⠈⠿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣼⣿⣿⣿⡄
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣷
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡿⠉⠉⠋⠉⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣇⠀⢀⡀⠀⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣇⣀⡀⠀⠉⠁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇⠀⢀⣠⣤⣤⣤⣤⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⡿⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣰⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣤⠄⢨⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣶⣶⣶⡄⡿⢿⣙⣇⢻⣿⣆⣿⣿⣿⣿⠇⠀⢀
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣠⠄⠀⠀⠀⠈⣿⣿⣿⢿⡟⣟⠁⡧⢻⠘⣿⠈⣿⣯⣹⡟⢻⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠐⠉⠛⠀⢀⠠⠀⠀⠠⠟⠃⢸⡁⣾⠁⠁⢰⢈⡿⢀⠸⣿⣿⢄⡠⢿⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠉⠀⠉⠙⠈⢠⡆⠄⠀⠀⠀⠈⠀⠀⠀⢸⡗⠃⠀⡀⢼⣸⢃⣿⡆⢹⣿⣾⡅⣾⠅⡀⠈⢷⠤⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠄⠀⠀⡀⠀⠀⠀⠉⡇⠀⠆⡃⣸⠇⢸⣿⡇⢸⣿⣏⣠⣿⣢⡇⠀⢰⣀⠀⡄⢉⡹⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠃⢄⠂⣼⡿⡇⠀⣤⣤⢉⢙⠛⠲⠧⠈⡡⠃⠀⠠⠘⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠁⠈⣰⣿⣷⢧⠸⣯⣿⣡⣾⣿⡟⢀⣟⡁⠀⠀⠐⠘⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⣸⢠⣿⠏⣿⡺⢞⡇⣌⠛⢣⠀⠀⠂⢀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠰⠀⣿⡿⣿⠠⡃⠸⢁⡴⠟⠀⣿⠇⠀⠘⠀⣧⡴⠂
⠀⠀⢳⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣿⣏⡿⠃⠁⠘⡡⢖⠀⠄⢩⠆⠀⠀⠀⠇
⠀⠀⠘⠀⡖⠉⠀⠀⣠⢦⣤⣀⣀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡠⡠⢀⡰⠛⡯⢅⠀⠀⠀⠑⠀⠀⠀⠈
⠀⠀⠀⠀⡇⠀⠀⠈⠀⢸⣿⣿⣿⣿⡯⣷⣿⡿⢓⠲⣖⠢⠀⠀⠀⡩⠈⣤⠟⠀⠀⠡⠀⣠⠂⣄⣂⢀⠀⣦⠀⠐⠀⠀⠀⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⣏⢿⣿⢩⢸⣿⣻⢠⠘⠀⣠⠀⣀⡤⠊⡠⠊⠀⠀⠀⠀⠒⠸⠀⣴⣌⠙⢾⡧⢹⣤⡄⠤⠀⠔⠁
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠈⠵⢾⠿⣁⢸⣿⢿⢀⠔⠋⠹⠃⡀⠀⢘⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣹⣿⣆⠙⣯⡇⣡⣷⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠜⣿⠴⠫⢸⠀⠀⠁⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⡹⢟⠹⣧⠈⡟⢿⡇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⢸⠀⠀⠰⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠐⠀⠐⠁⠀⠀⠀⣿⣧⠘⣿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢪⠹⡄⢻⡏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢱⢨⡀⠀⠀⠀⠉⠂⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⡀⠁⠀⠀⣠⠔⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠰⡄⢀⡴⠞⠙⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡞⢫⡌⣿⣿⡿⠁⠈⠀⠀⠀⠈⠉⠓⢲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠈⢸⣿⣷⣾⡟⠆⢻⡿⠃⠀⠀⠐⠆⠀⠈⠙⢄⠀⠛⡙⠿⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⣿⣿⣿⣿⠿⠂⠀⠃⠀⠀⠀⠀⠀⢀⣀⡆⠀⣠⣤⣿⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⣇⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠁⠐⠂⠆⠀⠀⠀⠀⠀⠀⠤⠀⣾⣿⡿⠋⠉⡟⡬⢿⡿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣾⣿⣷⣽⡀⣈⣀⡄⣠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣿⣿⡟⠉⠉⠀⠛⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠿⣧⣤⣶⣈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠙⣿⡁⢀⣄⣀⡀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣸⣟⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣿⣿⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢐⠀⠀⠀⠀⠀⠃⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⡿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠉⠀⠐⢸⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⢁⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⢻⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⠂⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠁⠀⠀⠻⠳⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⠒⠴⢦⣾⣄⠀⠠⠄⠀⢀⣤⣶⣶⣾⣶⣶⣶⣞⡁⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠻⠿⢿⣿⣷⣶⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠒⠒⠒⠒⠒⠀⠀⠀⠐⠒⠒⠒⠂⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠄⠤⠤⠀⠀⠀⠀⠄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣈⡉
⠀⠀⣀⠀⢀⡀⢀⠀⡀⠀⠀⠀⠀⣈⢡⢎⣼⣿⣾⣷⣤⡒⢤⡀⠀⠀⠰⠶⠒⠀⢀⣐⠂⣠⡀⠀⣀⠀⠄⢀⡀
⣀⣨⣤⣤⣀⠒⣀⣈⣉⣀⡈⢉⣿⡏⠐⣩⢟⣵⣿⢟⣿⣯⠀⠤⣄⡀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡷⢶⠛⡿⡲⠮
⠈⠙⠉⢉⣁⣈⡉⠛⠙⠛⠋⢉⡭⢒⡠⣸⡽⣛⣵⣿⣿⣿⣦⠀⠈⢝⣆⢠⣰⣠⣀⣀⣈⠉⠛⠛⠛⠻⠿⠾⠞⠛⠒
⠤⢆⣚⠫⠭⣛⣫⣭⠤⣒⡨⠔⣪⠵⣚⣽⣾⣿⡿⣿⣿⣿⡟⣄⡀⠐⠛⠍⡿⢤⡈⣍⣬⠍⣉⠈⠙⢃⡴⣧⡦⢴⡃
⡉⠄⣐⣚⣭⠵⠶⢚⣩⠵⣒⡭⣒⣯⣟⠽⣛⣥⢞⣿⡿⣿⣿⣯⣋⣐⢀⠀⡉⠐⠉⠁⠉⠩⣘⠉⠈⠉⠀⠀⠩⠟⠉
⣫⣭⣽⡥⢴⣒⣫⢽⣾⡿⣗⡿⢟⣫⠥⣐⢭⡾⢟⣽⡾⢟⣽⢿⣿⣿⣶⣤⣬⣥⣤⣬⣴⣮⣵⣍⣱⣀⣒⣵⡶⣾⢣
⠩⠽⠖⣛⠭⢑⣚⢭⣶⠏⣁⡡⢈⣠⡴⠞⣉⡤⠞⣩⠾⣫⣵⠟⣫⣴⠟⣋⢜⣹⠟⡯⢚⣿⣿⣻⣿⣿⠿⣫⠾⢃⣟
⣶⣶⠶⠶⠿⣷⣶⣿⣶⣿⣷⣿⣿⣷⣶⣿⣷⣶⣾⣶⣾⣿⣶⣾⣿⣷⣾⣷⣿⣶⣿⣶⣿⣿⣿⣿⣿⣷⣶⣷⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢻⡟⠻⢵⣶⣮⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⢉⣥⣴⡾⢟⣴⣿⣏⣀⣴⣶⣿⣯⣿⣿⣿⠟⠁⣴⡀⠘⠋⠉⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠈⠹⠋⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣟⣏⡻⠋⠚⠋⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣠⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓⢀⣤⡤⡄⢀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠚⠛⠉⢿⡿⢿⢶⡦⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢰⠀⢰⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠳⠶⣤⣤⣠
⠀⠀⠀⠀⠀⠀⢸⣦⣬⣁⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣯⠁⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣧⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣃⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣰⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠛⠛⠋⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡗⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠠⠲⣤⣤⣀⢀⠀⠀⠀⣀⣠⣦⢠⣶⣀⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡗⡏⣯⣼⡆⢿⣿⡼⣿⣿⣿⠃⠀⠄
⠀⠀⠀⠠⠀⢸⠆⠀⠀⠀⠀⠀⣠⢔⡤⣶⠀⠀⠀⠠⠚⢻⡟⢹⠏⢿⠅⡇⠏⢸⡇⠘⣿⣿⢟⠙⣻⣇
⠀⠀⠀⠈⠀⠀⠀⠀⣀⠠⠄⠀⢸⢠⢈⢁⠀⠀⠁⠀⠀⠉⠁⢨⣶⡿⠀⠆⣧⣾⢣⣆⠹⣧⣯⠔⣹⡇⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢼⠀⠀⠀⠀⠀⠀⠈⠸⡇⠀⡀⡆⣵⡇⣾⣿⠀⣿⡿⠃⣼⡅⡤⡀⢙⠉⢀⠠⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠁⠂⢄⠀⠀⠃⠀⠀⠇⣋⠆⣿⣿⠂⢉⡙⠒⠿⠿⢥⡆⠸⢺⢼⠀⠋⣿⢿⣶
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠐⣰⣿⣽⠀⣸⣯⣟⢰⣶⡦⠈⣶⡃⠠⢀⢀⢨⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⣴⣯⡿⢹⠃⣹⡿⣷⣿⣿⣇⡸⠯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠸⣿⢿⡟⢨⠁⠋⣰⠿⠃⣿⡇⠈⠃⢸⣆⡤
⠀⠈⢦⣀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⡻⢋⡟⠈⠀⣀⠥⠀⠠⢙⠄⠀⠀⠸⠋
⠀⠀⠘⢸⡹⠁⠈⢀⢰⣤⣀⡀⣀⣀⣄⡀⠀⠀⠀⠀⠀⢀⠀⠄⣀⠤⢀⡴⢹⡉⠂⠀⠀⠏⠀⠀⠀⠈
⠀⠀⠀⢸⡇⠀⠀⠁⠈⣿⣿⣿⣿⡾⣿⣿⠟⡂⢶⡶⠂⠀⠀⣨⠈⡤⠋⠀⠈⠀⢀⡖⠠⣆⠁⡀⣔⠀⠐
⠀⠀⠀⢨⡀⠀⠀⠂⠀⣿⣿⣻⡩⢺⣿⡏⡆⠑⣈⡄⢀⠰⠊⠠⠊⠀⠀⠀⠈⠂⠍⣠⣦⡉⠻⣧⣸⡤⣶⠇⠠⠊
⠀⠀⠀⠘⡇⠀⠀⠀⠈⠰⠻⠟⣥⢸⡿⣧⠰⠚⡅⡏⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣧⣿⣷⡈⢯⣇⣽⡿⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠊⡿⡚⠁⢉⠀⠀⠄⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⢟⡇⠻⠉⣷⡄⢽⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⣰⠀⠀⠃⠀⠀⣀⠈⡄⠀⠀⠀⠀⠀⠁⠀⠁⠃⢀⠀⡝⣿⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡊⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠅⠘⡇⠸⠃⠁⠁⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡂⢸⠐⠂⠀⠀⠀⣠⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠰⣊⡠⠬⠓⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⡤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⢀⣩⣽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠛⠧⠋⠘⠁⣭⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣝⣛⣷⡎⠙⠁⠀⠀⠘⢉⣹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⡄⣹⣿⣭⣛⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠠⢿⣯⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢸⣷⢿⣻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⢆⡴⣤⣼⣇⡤⢀⠄⢣⢺⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⡞⣏⣻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠿⢿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠟⠹⣷⡿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣴⢋⢻⡿⢿⣿⣿⡿⠛⣿⣿⡿⣿⠃⠀⢰⡿⠙⡟
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠰⠋⣎⣿⣥⣿⣿⣿⣧⡦⣟⡿⠾⡋⠀⠀⠈⡃⢀⣷⡄
⠀⣤⣶⡖⣶⣾⣷⣶⣶⣶⣶⣿⣷⣿⣿⣷⣿⣻⣄⠙⣿⣦⣼⣦⣀⣴⣷⣾⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡤⠶⠒⠚⠛⠛⠒⠒⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠈⢉⣀⡤⠤⠴⠤⣤⣄⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡾⠉⠀⣀⡤⠤⠤⢄⡠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠙⢶⡄⠀⠈⠉⠉⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠉⢠⠔⠋⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣄⣤⣄⣀⡀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⣀⡤⠖⠛⠋⠛⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠈⠉⠳⢦⡀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⠇⠀⡠⠊⠁⠀⠀⠀⠀⠀⠢⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡈⠉⠛⢦⡄
⠀⠀⠀⣠⠖⠋⢀⣼⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠈⠉⠳⣄⠀⠹⣆
⠀⢀⡞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⢰⡄⠀⠀⠀⠀⠈⣧⠀⢹⡄
⠀⡾⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⡆⠘⡇
⠰⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠾⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⡇
⠀⠻⣄⠀⠳⣄⡀⠀⠀⣀⡴⠞⠉⣭⠗⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢀⡞
⠀⠀⠈⠻⠦⣬⠉⠉⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠛⠉⣹⠶⠤⠤⠴⠚⠉⣠⠟
⠀⠀⠀⠀⠀⣿⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡋⠀⠀⣀⣠⠴⠋⠀⠀⠐⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠈⠛⠓⠶⠶⠤⣤⣤⠤⠶⠞⠛⠦⣤⣀⣀⣀⣀⣤⣴⠖⠋⠁⢀⡬⣭⣭⣅⣤⣤⣤⠴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢤⣀⡀⢀⡟⠀⠀⠀⠀⡾⠒⣒⣚⣭⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠚⠋⠁⠀⠀⠈⠉⡿⠁⠀⠀⠀⠘⠛⠋⠉⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣤⠴⠖⠛⠋⡇⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣯⢡⡴⣶⣤⠞⠛⠋⠉⠛⢳⡄⠀⠀⢀⣤⡴⠟⠛⠛⠷⣤⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠸⠧⠴⢬⣩⣽⡛⣦⡀⠀⠀⠈⢷⠀⠀⠘⠷⠶⠶⣦⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⡟⠀⢀⡤⠟⠀⢀⣤⣤⣤⠶⠟⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠳⠤⠟⠀⠀⠀⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⢰⡇⠀⠀⢀⣠⣤⣤⣀⣀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣉⣛⡛⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⣼⠛⢶⡀⢸⠁⣠⣤⣈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣤⣾⠁⠀⠉⠛⣯⠉⢿⣄⠀⠀⣴⠲⣄⠀⠀⠀⣧
⠀⣿⠀⠸⣧⣿⣀⣉⡙⠛⠃⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠸⠿⠶⣦⣻⣤⣤⡿⠀⠈⣿⠀⢀⣿
⠀⣿⠀⣰⠟⠀⠀⠈⠙⠦⣤⡀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⠇⠀⠻⡴⠋⡿
⠀⢿⠀⣿⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⠈⠛⠛⠉⠛⢷⡄⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢠⡇
⠀⢸⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣶⠛⠀⠀⣠⠖⠒⢺⠇
⠀⠀⢻⡙⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⢿⣤⡄⢼⡋⠀⢠⡿
⠀⠀⠈⢷⡀⠀⠙⢳⡄⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣼⣿⡇⣸⡃⣠⡿⠁
⠀⠀⠀⠀⠳⣄⠀⠀⣿⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡟⢹⠟⠁⢉⡽⠋
⠀⠀⠀⠀⠀⠈⠳⣄⠹⣆⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡴⠟⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣳⣾⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠒⠶⢤⣀⣀⣀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢸⠁⠀⢀⡴⠋⠉⠀⠉⠙⢾⡁⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢇⣀⠤⠐⡲⠊⠁⠀⢠⠖⠛⠛⠒⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣸⠁⠀⢠⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡄
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢇⠀⠀⠀⠀⡠⠤⡀⠀⠀⢀⠀⠈⢢⢀⠖⠀⠀⠀⠀⠀⡝
⠀⠀⠀⠀⠀⠀⠘⢦⣀⡔⠁⠀⠀⠀⢸⠁⢄⠀⠀⠀⠀⠹⣄⠀⠀⠀⣴⠁⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠂⡐⠀⠀⠀⣐⠠⣉⡇⢠⢋⡁⠀⠀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠳⢤⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⢹⡍⠁⠀⠫⡈⠀⠐⢤⠜⠁
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠘⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠚⠁⠀⠀⠀⡄
⠀⠀⠀⠀⢀⣀⠇⠀⠀⠀⡀⠈⠓⢧⡀⠀⠀⠀⠀⠀⣠⠞⠀⡴⣻⢟⡳⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠓⢤⣀⠈⠑⡏⠁⠀⠀⢇⠉⠆⠙⠲⣉
⠀⠀⠀⠀⠀⢀⢆⠀⢀⣠⠇⠀⠀⠀⠀⠀⠈⢉⠉⠙⢦⡀⠀⠀⢹⠈⡀⠀⠠⡉⠢⢄⡀⠀⠀⣀⠤⣄⡀
⠀⠀⠀⠀⠀⠁⣿⠀⠀⠁⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠁⡇⠠⠀⠀⠈⠲⣄⢈⣑⠊⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢄⣀⠇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠐⠚⠀⠀⡀⠀⠀⢈⠀⢳⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠃⠀⠀⢀⠀⠀⠀⠀⠀⠠⠜⢀⡼⠆⠀⣀⡤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠈⢆
⠀⠚⠁⠀⠘⠀⠀⠠⠁⡸⠀⠀⠀⠀⠀⢠⢪⠀⠀⠤⠆⠀⠀⢀⡏⠀⠀⠀⠁⠀⠀⠀⠀⠈⠳⠀⠀⠒⢄⠀⡜
⠀⠀⠀⠈⢠⠂⠀⢤⣠⠇⠀⠀⠀⠀⢀⠁⢸⠀⠀⠀⠀⠜⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁
⠀⠀⠀⠢⠞⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠲⠊⠀⠀⠀⠀⢸⠀⠀⠻⢦⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠲⠊⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⢀⡴⠓⠒⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣒⣛⣉⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖⠲⢾⡉⠉⣩⠝⠛⠋⠙⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠈⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄
⠀⣀⡀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠳⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡷⣄⣠⠞⠁⠀⠀⠀⠀⠘⣧⠀⠀⢀⣀
⠀⠈⠉⠛⠲⠇⣼⠳⢤⡀⠀⠀⡰⠋⡇⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢰⠇⠈⢧⠀⠀⠀⣀⡤⠞⢧⠸⠒⠋⠉
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠉⠳⡶⠁⠀⢳⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡞⠀⠀⠈⢧⡴⠚⠉⠀⠀⠘⣇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢰⠃⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⡠⠧⣀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢹⡀
⣀⣀⣀⡀⠐⡇⠀⣴⠤⣤⡎⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠉⠀⠀⠈⠉⠓⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⣇⣠⡤⠴⢦⠀⢸⡇⣀⣀⣀
⠀⠀⠉⠉⠉⠁⢰⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸⡆⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠀⠀⠀⠀⠀⠀⠀⢈⡟⠉⠀⠀⠀⠀⠀⠀⡼⢀⣀⣀⡀⠀⣧
⠀⠀⠀⠀⢀⣀⣸⡀⣾⠉⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡼⠉⠉⠉⠁⣿⠀⣏⣀⣀
⠀⠀⠈⠉⠉⠉⠀⠀⣿⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⣸⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⢀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠒⢒⠎⠉⠋⠋⠽⠛⠷⣖⠒⡦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢉⠽⠂⠒⠁⠩⠝⠒⢶⣔⠚⠋⠉⠐⠭⠓⢤⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠐⠎⠁⠀⠈⠉⠓⠢⠤⠴⠒⠒⠉⠤⠲⢤⡉⠙⠓⠲⠿⠄⠾⠿⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢠⢆⣄⠔⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠌⣑⠢⣄⠤⠤⠤⠬⠦⣄⠉⠓⠤⡉⠳⡄⠀⠤⠵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⢾⣾⣿⣺⣯⡄⢀⢀⠀⠀⢀⣡⣤⣀⣀⡀⢀⡀⠀⡀⠠⠤⠤⠬⠷⠀⠈⠲⢤⡀⠙⢦⠠⠬⠤⣙⣆⠲⠤⡀⢝⢦⡀
⠀⠀⠀⠀⠀⢀⡠⡎⠁⠀⠀⠀⠪⢻⣿⣯⣿⣷⣀⣤⣴⢞⡻⢿⣿⡛⠻⣿⣯⣛⢿⠟⠛⢙⣃⣀⣀⡀⢤⣤⠠⠭⢦⡬⠷⠘⠢⣀⠈⢙⡒⢼⡒⠤⢹
⠀⠀⠀⣠⠎⠁⠀⢸⠀⣤⡀⠀⢀⣀⣬⣿⣿⣿⣿⣿⣿⡟⠉⠿⠿⢃⣠⣿⠏⠒⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠭⡀⠀⠀⠀⣐⠦⣈⠃⠀⠈⠢⡱⡄
⠀⢠⠞⠀⠀⠀⠀⢠⠀⠸⠃⠀⠘⠿⣿⣿⣿⣿⣿⣿⡟⡛⠓⠒⠿⠛⠋⠀⠀⠐⠒⠀⠀⠀⢄⣀⠀⠀⠀⢐⠂⠠⠄⠁⠤⠀⠑⠈⠫⡓⢶⡤⢄⣸⡄
⢠⡏⠀⠀⡀⠀⠀⠀⠓⠄⠀⠀⠀⠈⠉⠿⠿⠫⠼⣿⠻⢽⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⠑⢦⠈⠁⠀⠀⢀⡀⠐⢈⠀⢱⣄⡉⠢⡍⠁
⣿⠀⢀⠄⠀⠀⢀⣀⣀⠠⠤⠤⠤⠄⢚⣓⣒⣒⣬⣥⣤⣥⡍⡐⡄⠐⠒⠶⣤⣀⡈⠓⠢⠀⠄⠤⢄⡀⠀⠤⣀⢀⡀⠀⠀⠠⡐⢄⠑⠪⠇⠉⠢⢽
⣿⠀⢲⣾⣭⣭⡉⠀⠀⠀⣀⣀⣀⠀⠀⠠⠀⠀⠀⠸⣆⠃⠀⠈⠀⠀⠀⢠⡀⡈⠙⡒⠄⡂⢀⡀⠤⣌⣉⡀⠀⢉⠉⠑⢈⠳⡂⠌⣦
⣿⠀⡏⠀⠀⠉⠉⠉⠉⠉⠀⠀⠁⠀⠠⡄⠘⢦⠀⠘⢮⠢⠑⠀⢣⠘⡀⠀⢡⡢⠄⡁⢠⡀⡈⠉⠉⠀⠈⠉⢫⠁⠐⠄⠂⠁⢹⠄⠉⠁
⠸⣄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⠀⠀⠳⠀⠀⠃⠠⡀⠠⡁⠸⡨⡦⣵⡀⠈⠢⡑⠈⠒⠠⣀⠣⠐⠠⠥⠀⠀⠀⠂
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡐⡐⠆⠐⡙⡆⠏⢳⡘⣗⠷⣙⡌⢏⠱⠀⠌⠓⠤⠆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠐⠄⠈⠀⠀⠑⢸⡂⠩⡙⠺⠘⡆⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠇⠀⠢⠄⠀⠀⠘⠃⠀⠈⠓⢤⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣐⡏⠙⠿⣭⠼⠋⠉⠉⢿⠋⠙⢻⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⡤⣭⣥⣤⣬⡶⠶⢤⣤⣤⡆⠀⠀⣠⣭⣿⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⣿⣶⣿⣿⡶⢮⡭⣴⣶⣿⣿⣟⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡤⠉⠙⣠⣽⣶⣿⣿⣿⡟⠿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡸⠉⠻⣦⣽⣿⣿⣶⠾⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢿⡄⣶⣌⣦⡙⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⡀⠀⠸⢿⣷⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⢸⣿⣿⡆⣦⣿⣿⣿⣿⣿⢂⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⣠⡾⣿⣿⣧⣽⣿⡛⢿⠻⢇⠈⢦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⣷⠟⣧⣿⣿⣿⣿⣿⣧⣾⣷⣼⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣶⣿⡍⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡿⢹⡅⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠘⠉⠀⢸⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣾⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠁⠀⠀⢀⠴⠖⠒⠉⠙⠳⠤⠤⠀⠀⠀⠀⢽⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⣤⡀⠀⠄⣀⣙⣳⡄
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡀⢰⡎⠀⠀⠀⠀⠀⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢛⣀⣉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⢠⠃⢀⡶⠷⣿⣧⢠⠀⠀⠀⠀⠀⢠⣤⣾⣶⣠⣤⢄⣀⣀⣀⣀⠀⡉⠻⣶⣽⣧⡀
⠀⠀⠀⠀⠀⠀⣸⠔⠋⠁⠀⠈⢻⣇⠀⣠⠀⢀⣴⡟⣿⣟⢃⣿⠟⠁⠈⠁⠀⠀⠀⠈⠙⢾⣿⣿⣷⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⠦⠀⠑⡽⠿⢿⣿⣿⣿⣿⣶⣤⣾⠿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄
⠀⠀⢀⡜⠁⠀⠀⠀⠀⠈⠣⡀⠀⢅⠀⠀⠙⢿⡿⠿⢶⣒⡀⣀⡀⠀⠀⠀⠈⠙⠒⠦⣤⣀⡀⠈⢿⣿⣿⣦⡀
⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢁⣀⣠⠴⢉⠆⠀⠀⠐⠉⠁⠶⢦⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣿⣿⣮⡙⢄
⠀⠀⣿⠀⠀⠀⢀⡀⠤⠤⠾⠥⢖⣫⣥⡤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠤⣄⡈⠲⣤⣌⡻⣯⣹⡻⣿⣮⣢
⠀⠀⢹⡀⢠⡾⠓⠊⠉⣿⣿⣿⣿⣿⣷⢖⣥⣆⡄⣤⡀⣀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠈⠻⣷⣜⣿⣿⣿⣿⣿⡄⠉⠣⣄
⠀⠀⠀⠣⡏⠀⠀⠀⢠⠿⢿⢿⣿⣿⣷⣿⣿⣼⡇⣿⣿⡜⣆⠀⣀⠀⠰⣄⡀⠙⣶⣤⣄⠙⢿⣿⣿⣼⠿⣿⣝
⠀⠀⠀⠀⠀⠀⠀⢠⠯⠒⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⣿⣿⣿⡄⢸⣧⡀⠘⣿⣦⡈⣿⣿⣟⣾⣿⡿⡟⠀⠈⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣷⢠⣿⣿⣷⣼⢿⠇⠈⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⡟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢾⢿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠘⢿⢿⣿⠺⣿⣿⣿⡿⣿⠟⣸⡿⡫⠋⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠝⡀⠈⠳⣉⠔⠁⢰⡭⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⣀⣠⣤⣤⣤⣤⡤⣤⣬⣍⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠐⡿⠛⡟⠉⠏⢀⡀⠬⠷⣌⡉⣫⣹⢿⣿⣷⡒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⡄⠀⠀⠀⠀⠀⠀⠁⠛⢿⣧⠀⠀⠀⠈⣿⣿⠛⣿⣿⣿⡝⣷⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⣀⣤⣴⠃⣨⡿⠀⠀⠀⣸⡿⣻⣿⣿⠟⠙⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡿⠋⠀⠀⠀⠀⠀⢿⡿⠁⢹⣤⣿⣇⣠⣴⣾⣯⣾⠁⣉⡸⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣷⠏⠀⠀⠀⠀⣴⡀⠀⠈⠁⠀⠈⢸⣿⡿⠉⠙⠛⠋⠚⢓⢿⣧⣤⢀⣾⠆⠀⠀⠀⠀⠀⣠⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠋⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⢰⠖⢙⣿⡗⠺⠇⠠⣤⣤⡎⣠⠹⠟⠈⠁⠀⡠⠃⢠⢆⡀⠈⢧⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⢠⡞⠀⢀⠀⠀⠀⠀⠀⠀⣡⣶⣷⡶⠞⠛⠋⠃⢀⠀⠀⠸⣿⣿⣿⡅⠀⠂⠀⠀⠄⠀⠘⠟⠋⠀⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⣤⠀⢀⣾⣇⠀⠀⣀⡀⠀⠀⣠⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠶⠀⠀⢘⣿⡟⠛⢀⣴⣿⣾⣕⣆⣰⣤⣴⡆⠀⠀⢦⡾⣧
⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⠁⢰⣿⣟⣹⡆⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⣿⣷⡤⠐⠛⠋⠉⣿⣿⡟⠛⠛⠡⡀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣄⠺⠋⣿⣭⣛⠿⢿⣿⣗⣀⠀⠀⠀⠀⠀⠀⠀⠋⠁⣾⣿⠟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⢢⡀⠀⣾
⠀⠀⠀⠀⠀⠘⠀⠛⠻⣿⣿⣷⡄⢹⡟⢿⣤⣄⣻⣶⣾⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠞⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠘⣿⣧⣘⠳⠛⠻⠿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⢀⣟⠀⢆
⠀⠀⠀⠀⠀⠀⢙⠀⠀⠐⠦⢈⢻⣷⠇⠀⢀⣀⢈⡙⠿⣿⡁⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣼⠃⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠈⣦⣢⡀⠀⠘⣿⠃⡤⠒⠃⠈⠀⠁⠀⠈⢹⡄⠀⠀⠀⠈⠁⠈⣿⡧⠀⠀⠀⣀⣠⣤⣀⡄⣶⣷⡟⠉⠣⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢈⠡⡈⠢⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠠⠄⠘⠲⢤⣔⣦⣦⠀⠀⠀⠻⠷⢿⣿⣿⣿⣿⠇⠋⣬⠀⠀⠀⠀⣛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠐⠿⣦⡀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠛⣧⣤⣄⠀⠀⠀⠀⠈⣽⢿⣿⣿⢸⣥⠀⠀⢸⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠢⠀⠀⠈⢳⡄⢹⡿⢶⡄⠀⠀⠀⠀⠀⣴⣾⣿⠋⠁⠀⠴⠂⠀⠈⠠⡜⣿⠏⠀⠙⠂⣠⢿⠁⡘⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡂⠀⠠⣌⣇⢾⠀⠈⢱⠿⠂⣀⣄⣤⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡇⠀⢀⡼⠋⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⢈⣿⣼⠀⠀⠀⢁⣠⣿⣿⡟⠀⠀⠀⠀⢀⡠⠴⠒⠚⠋⠉⣻⣿⣡⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠳⢌⣿⣦⠀⢰⣾⣿⣿⣷⣦⣴⣊⣅⣀⣠⣄⣘⡁⢀⣦⣴⡿⠛⠙⠋⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠻⢾⣿⣿⣿⠿⠟⠛⠋⠉⠋⠉⠉⡉⠛⠙⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢀⡟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⡀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢡⠀⠀⠀⠀⠀⠀⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣆⠀⠀⠀⠀⠀⠀⡼⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢇⠀⠀⠀⠀⢸⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡹⡄⠀⠀⠀⠀⢸⣇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⡄⠀⠀⠀⡾⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢹⡀⠀⠀⠀⢻⡟⢿⡀⠀⠀⠀⢀⣀⣤⣤⣄⠀⠀⠀⠀⣼⠁⡜⠁⠀⢀⡼⣷⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣶⢄⡀⠈⢳⣄⠻⣄⠀⢠⡿⠛⠉⠉⠈⠻⣆⠀⣰⢣⠞⠀⢀⣴⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣕⠠⡙⣦⠹⣤⡿⠁⠀⢀⣀⣀⣀⣘⣶⡣⠋⣀⣴⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣝⣷⣼⣷⠾⠛⠛⠉⠉⠉⠉⠉⠻⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣽⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣶⣤⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⡤⠊⢰⠏⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠐⡄⠙⠷⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡽⠀⠀⡟⠰⠁⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠘⣆⠘⣄⠀⠘⣧⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠃⠀⢸⣷⡇⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠘⣿⢛⡇⠀⢸⣆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡀⠀⢸⡆⢧⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣧⠀⡇⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⢿⣈⣇⠀⠀⠀⠘⢿⣷⡀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣿⣠⣧⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈⢻⣍⢧⡀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⢠⡏⣼⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠂⠈⠉⠀⠠⢤⠔⡉⠉⠐⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠼⠃⠀⠐⠁⠀⠀⡆⣄⣶⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠴⠀⠀⢀⠔⡌⣡⠦⠂⠉⠈⠋⠀⠀⠀⠀⠀⠀⢩⡄⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⢀⡄⢠⣯⣔⣐⠽⠚⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢸⡌⢂
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⢚⣞⠄⠾⠁⠊⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⠉⠀⠀⠀⠀⠀⠀⠀⠠⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⢀⡞⢁⡀⠀⠀⠀⠀⢀⣤⣠⠀⢀⢄⠦⠠⠆⠀⠀⠁⠀⠀⠀⠀⠡⠀⠀⠀⠀⠐⠱⡀
⠀⠀⠀⠀⠀⠀⠌⠀⠀⢠⢊⡯⠚⠉⢷⣗⢆⣀⣰⣈⣉⣑⠆⠈⢂⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⠀⢣
⠀⠀⠀⠀⠀⠐⠀⠁⠀⡸⠭⠄⠐⡄⠀⠋⢸⠏⢯⣴⣤⠄⠀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠈⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠼⠀⠀⠀⠀⠑⢄⣠⠤⡀⠀⡀⡸⠀⠀⠀⠀⣀⢀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠐⠉⠣⣄⢈⠀⡅
⠀⠀⠀⠀⠀⠀⢠⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣶⣄⠀⠀⠀⠂⣉⣋⠗⢉⠉⠈⠻⠴⡄⠀⢀⠀⠀⠀⠘⡊⢁
⠀⠀⠀⠀⠀⠠⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⢷⣀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠙⢏⠈⠀⠀⠀⠀⠘⠚⡆
⠀⠀⠀⠀⠀⠰⠈⢧⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠹⣧⡀⢀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⠀⠀⠉⢳⠁
⠀⠀⠀⠀⠀⠀⢃⠈⢿⣢⡀⣀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⢋⡉⡄⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠉⠷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠁⠀⠀⠈⡠⠄⠂⠀⠒⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡸⣀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⣈⠆⠀⠀⠔⠁⠀⠀⠁⢀⠠⠒⠁⠠⠒⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠢⡀⠈⢿⣶⣂⠀⠀⠀⣄⠛⠀⢠⡁⠀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠀⡄⠈⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⡀⠀⣿⣹⠀⠀⠀⠈⠀⢦⠈⢢⠀⠀⠤⡀⠀⠀⠀⠀⠀⠠⢊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣌⡳⣎⣤⠀⠀⠀⠀⡴⠊⠀⣧⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠶⣶⣼⣃⣈⣀⣈⣀⠤⠤⠖⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠟⢻⣿⠉⠙⣧⡟⠀⣸⠏⢹⡏⠛⣿⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⠁⠹⣦⣾⠛⣧⣴⠟⠻⣶⣏⠀⢸⣧⡾⠃⢠⡟⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣁⠙⢿⣄⠀⠘⣿⠀⠘⣿⠀⠀⠈⠻⣷⣿⡋⠀⠀⣿⢀⡴⠋⣨⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣻⠉⠙⢷⣿⠉⠛⠿⣿⡷⣶⣿⡆⠀⠀⣰⡿⠛⠿⣶⣴⣿⠟⠀⣼⣃⣴⠟⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠈⢻⠛⠶⠶⢿⣦⣄⣠⣿⠀⠀⠹⣿⣄⣰⣿⣥⣤⣤⣼⣿⣿⠶⢿⡟⠉⠀⢠⡟⢹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡯⣷⣤⣼⣧⠀⠀⠀⢿⡍⢻⣿⠀⠀⠀⢻⣿⣿⠋⠉⠉⠀⠀⣾⡇⣀⣼⣧⣴⠶⣾⠷⠛⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣀⢸⣦⡈⠙⢷⣄⠀⠘⣧⠈⢿⣦⡀⠀⠈⣿⣿⠀⠀⢀⣠⣴⣿⠿⠟⣿⠁⠀⢠⣟⣠⡶⠋⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠉⠉⠉⢻⣦⣄⣩⣿⣶⣿⣄⠀⠙⢿⣷⣤⣿⣿⣷⣶⠿⠛⠻⣿⣤⣼⣏⣀⣤⣾⣿⣥⣤⣶⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣉⡛⠛⠛⠋⣉⣩⣿⣧⣼⣿⠙⢷⣦⣄⣹⣿⣿⣿⡟⠁⠀⠀⠀⢀⣿⣿⣿⡉⠀⠀⣠⠾⠋⠉⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣉⣩⣷⢶⣿⣉⣁⣀⠙⢿⣿⣦⡀⠈⠙⢿⣿⣿⣿⣆⠀⢀⣀⣰⣿⡏⠈⠻⢿⣶⡟⠁⣀⣤⠶⠶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣥⣤⣶⠛⠛⠛⣻⣿⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣿⡿⠛⠛⠛⠿⣶⣤⣤⣤⣭⣿⠿⣯⣄⣀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢁⡿⠉⠙⣶⡾⠋⠁⠀⠀⠀⠉⠉⠉⠛⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡋⠉⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⡀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉




⠠⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠊⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠈⠓⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠄⠠⢀⠀⠀⠀⢀⠀⢑⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄
⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠁⠀⠀⠁⠀⠠⠀⠀⠀⡐⠀⠐⠦⣀⣀⡂⠀⠀⠈⠣⢄⠤⣁⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⠁⠀⠀⠠⠀⠀⣀⠀⠐⠀⣀⡠⠒⠊⡑⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⡤⡤⠤⣤⣤⠊⠙⠀⢘⡆⠀⠀⠌⢳⢠⣍⠲⢠⠀⠬⠄⠐⠚⠒
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠈⠁⠀⠀⠀⠈⠁⠀⠀⠋⠩⡀⠐⠓⠀⢀⣀⠀⠀⠤
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠤⠀⠀⠀⠒⠄⠀⠀⠈⠭⠀⠦⠤⠄⠤⠅⠃⠈⠈⢄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⢀⠤⠂⠀⠀⢀
⠀⠀⠀⠀⠠⠌⠀⠐⠀⠀⠠⠀⠀⡀⣀⣀⡠⠀⠀⠈⠒⠂⢦⠀⠀⡠⠐⠈⠢⠐⠒⠀⡁⠀⡠⠀⠁⠈⠀⠁⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⠀⠀⠀⢀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠔⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢠⣴⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣄⣴⣿⡿⣠⣶⣶⣿⣿⣿⠁⠀⢀⣴⣿⣿⣿⣿⣦⡀⠠⣤⣶⠷⠶⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⣷⣿⣿⡿⢿⣿⣷⣿⡇⣸⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⡄⢀⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢰⣾⣿⣿⣿⣿⢛⣷⣿⢿⣷⣟⣿⣿⠿⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⣨⣿⡿⠃⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢀⣾⣿⣿⣿⡿⡀⠘⢿⣥⣾⡿⣏⣽⡏⠀⠀⠀⠈⠛⠻⠟⠛⠁⠀⠀⣰⣿⠏⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠸⢻⣿⣿⠋⢠⣿⣶⣿⡿⢏⠗⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣶⣿⣿⣿⠀⣿⡟⡏⢹⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⡶⠉⢳⣇⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣶⣶⡄⠙⠿⣿⠿⠟⣛⣻⣿⣿⣿⣿
⠐⢿⡿⢿⣿⡟⠀⣴⣿⣟⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⡀⠓⢶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⢫⣵⣿⢏⡉⠀⠉⠻⣦⣀⣀⣀⢠⣾⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⡆⡹⢟⣉⣿⣶⣿⣿⣿
⠀⠀⠀⠀⠾⣿⣿⡿⠁⠀⠀⠀⠈⢉⣿⣿⢸⣿⡇⢀⡄⠴⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⣿⡇⣧⣼⣿⡎⢾⣿⣿⣿
⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢸⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣯⣶⢻⣷⢹⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣭⡅⢨⣉⣿⣿⡿⠛⠉⠉⠛⢿⣿⣿⣿⣿⡷⣶⣿⣿⢿⣾⣿⣧⣟⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⢹⣿⣿⣽⣷⣿⣿⣿⣼⣿⣷⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⠀⠀⠀⣠⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣿⠋⠀⠀⠀⢸⠘⡆⠀⠀⢀⡵⠀⠀⠀⠰⣏⠀⠀⠀⠀⣏⠇⠀⠀⠀⢠⠎⠀⠀⠀⠀⢡⡆⠀⠀⠸⡦⡀⠀⠀⣀⠀⠀⠀⠀⢸⣀⡇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠈⠣⠀⠀⠀⠀⠌⠣⠀⠀⠀⠃⠀⠀⠀⠀⠈⠇⠀⠀⢸⠈⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠰⣭⠀⠀⠀⠀⢸⠀⠀⠀⠠⢨⠆⠀⠀⢸⠓⠋⠀⠀⠀⠈⡟⠀⠀⠀⠀⡐⠀⠀⠀⠀⠙⠂⠀⠀⢘⡢⠀⠀⠀⡟⠢⢺
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠸⠀⠀⠀⠑⠄⠀⠀⠀⠸⡤⢄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡑⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⣀⠀⢀⠀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣯⠓⢹⠀⠀⠹⡇⠀⠀⠀⠸⠁⠀⠀⠀⢸⢎⠀⠀⠀⡆⣀⠀⠄⠀⠠⠆⢑⠄⠀⠀⢠⣔⠇⠀⠀⢸⢱⡄⠀⠀⠘⢣⠀⠀⠀⢸⠐⡄
⠀⠀⠀⠀⠇⠀⠸⠀⠀⠀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠘⠣⠀⠀⠀⠁⠀⠐⠃⠀⠀⡜⠣⠀⠀⠀⠸⠁⠀⠀⠀⠨⠸⠄⠀⠀⢸⢾⠀⠀⠀⢸⠒⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⣏⢱⠀⠀⠀⡇⣧⠀⠀⠀⠀⡇⡇⠀⠀⡿⠩⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢿⠀⠀⠀⠐⠀⠀⠀⠀⢰⠖⠀⠀⠀⢲⣶⠀⠀⠀⢠⣤⠂
⠀⠀⠀⠀⠈⠦⠀⠀⠀⠇⠹⠂⠀⠀⠰⠇⠇⠀⠀⠇⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠦⠀⠀⠀⠸⠀⠀⠀⠀⡹⢿⠀⠀⠀⡸⠛⡀

⠀⠀⠀⠀⣷⢄⡇⠀⠀⠰⣴⠂⠀⠀⠠⡇⠀⠀⠀⢀⡄⠀⠀⠀⢀⣶⡀⠀⠀⠀⠲⡶⠀⠀⠀⠠⡂⠀⠀⠀⢀⣠⠀⠀⠀⢀⣆⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠇⠀⠇⠀⠀⠈⠉⠁⠀⠀⠀⠋⠀⠀⠀⠈⠃⠀⠀⠀⠈⠺⠃⠀⠀⠀⠀⡇⠀⠀⠀⠠⠞⠀⠀⠀⠈⠉⠀⠀⠀⠈⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣠⠄⠀⠀⠀⡀⢢⠀⠀⠀⠰⡆⡀⠀⠐⢴⡆⠆⠀⠀⠀⠶⠀⠀⠀⠀⢰⡀⠀⠀⠀⠠⣀⠀⠀⠀⢀⡀⠀⠀⠠⢀⡀⡆⠀⠠⠄⡀⠄
⠀⠀⠀⠀⠁⠜⠀⠀⠀⣠⠫⣀⠀⠘⠉⣏⠉⠀⠀⠀⡇⠀⠀⠀⡀⢨⠀⠀⠀⠀⢾⢇⠀⠀⠀⢸⠿⠂⠀⠀⠒⡟⠂⠀⠘⠂⡅⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠈⠉⠈⠂
⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⡄⠀⠀⠀⣄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⠀⠀⠀⢠⠚⠀⠀⠀⠰⠀⠀⠀⠀⠀⣇⠀⠀⠀⠄⠀⠐⠂⠀⠀⠀⣼⠁⠀⠀⠀⡇⠀⠀⠀⠈⡇⠀⠀⠀⢹⠵⠆
⠀⠁⠀⠀⠈⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠹⠏⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⣸⡄⠀⠀⠀⠀⡟⠂⠀⠀⣴⢠⣰⠂⠀⠀⡆⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠰⠇⠀⠀⠀⠸⠚⠀⠀⠀⠀⠟⠀⠃
⠀⠂
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⠷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠿⠿⠿⢷⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠛⠻⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠙⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣤⣄⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⣠⣾⠿⠛⠉⠉⠉⠙⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠉⠛⠿⣶⡄⢀⣤⣶⣶⠶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠀⠀⠀⠀⠀⠙⠻⣷⡀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⠶⠶⣶⣤⣄⠸⣿
⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠻⣿⣿
⠀⣠⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷
⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠙⠻⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠟⠉
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⢀⣠⣾⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠟
⠀⢠⣾⡿⠋⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁
⠀⣿⡏⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷
⠀⢿⣧⣀⣀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣷⡀⠀⠀⠀⠀⣀⣴⣾⠟⠉⣿⡆
⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⣿⡇⠀⠀⢀⣾⡿⠋⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡆⠀⠈⢿⣦⣄⣠⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢹⣿⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿⡆⠀⢸⣿⡁⠀⠀⠀⣸⣿⠁⠀⠀⠈⠻⣷⣤⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁⣿⡇⠀⠀⠻⣷⣦⣤⣾⠿⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣴⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⡾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣀⣀⣄⣤⣤⣤⣶⣿⣷⣤⣀⣀⢤⣤⣤⡄⢀⡀⣀⣞⠋⠉⠛⠃⣕⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⢀⡀⢀⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⠿⣿⣿⣿⣿⡿⣶⣀⣤⣤⣴⠀⢽⠋⠉⢛⡳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠂⠀⢽⠿⠛⠛⠿⣿⣭⣴⣿⡿⢟⠾⠿⠻⣿⣿⡿⢟⣿⡷⢶⠪⠆⠈⠳⡶⣿⣟⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡆⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⢉⣉⣁⡀⠀⠀⠐⠈⠉⢡⠴⠛⠒⠚⠃⢠⠄⠠⢶⣟⠊⢿⣷⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣿⣹⣷⣴⣾⣿⡛⠉⠠⠤⣤⣤⠂⠀⠠⠤⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡀⠠⠰⢶⣆⣈⣿⣷⣿⣟⣯⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠿⠿⠟⠛⠋⠉⠛⠿⣿⣿⣭⡉⢑⡲⢤⠤⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠐⠈⠁⠀⠈⠛⡙⢿⣯⣙⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡯⠉⠀⠀⠀⠀⠀⠀⠈⠙⠇⠉⠐⣦⡤⣿⣶⣬⣿⢿⣷⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠀⠀⣈⡏⠘⣯⡉⠉⠻⣍⣻⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⢀⠀⢹⡆⠀⠀⢳⣄⠀⠹⣄⣸⣇⡀⢰⡿⢯⢿⣿⡛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⢻⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠈⠁⠀⢶⣌⣿⢙⡗⢻⣧⣾⡟⢈⣿⣷⣶⣦⣭⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⣴⣿⣠⡄⠰⠛⢆⣀⠁⠀⡀⠀⢸⣿⣁⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⣰⠋⢀⣾⣿⣿⣿⣿⣿⣇⠀⣾⣿⡄⠀⢳⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣯⣴⣿⣿⣿⣿⣿⢿⣿⣿⣶⣿⣿⣿⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢻⡝⠹⣦⠀⣸⠛⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠘⠀⠰⣦⣤⡟⢀⢀⣿⠇⣿⠀⡴⣪⠙⡋⠻⣷
⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠄⠡⣼⣦⠀⠀⠀⠀⣧⠀⠀⠈⠉⠁⠈⠻⠿⠀⣿⢠⣫⣿⣿⣿⣦⣼
⠀⠀⠀⠀⠀⠀⠀⣹⣷⣤⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠛⢿⠀⠀⠀⠀⠁⣠⣤⠀⠀⠀⠀⠀⠀⠀⡿⠟⠉⣸⡟⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⢀⡄⢠⣄⡄⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡇⠋⠙⢹⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠀⠀⠲⠀⠀⠀⠀⠀⠐⠈⠄⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⢸⠃⠒⠀⠀⠹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⠃⠀⠀⠀⠃⢀⠊⣗⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠆⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠁⡀⠀⠀⠀⠀⢹⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⣀⣤⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣇⠀⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠒⠚⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⡇⠿⠟⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣄⢤⣀⡐⠁⠀⠀⠀⠙⣿⡿
⠀⠀⠀⠀⠀⠀⢀⡴⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠂⠀⣠⣤⣿⣿⣿⣿⢸⡇⠄⠀⢀⣀⠀⣶⠉
⠀⠀⠀⠀⠀⢠⠈⣄⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡇⢸⣷⣄⠴⣾⠿⢿⡇
⠀⠀⢀⠀⠐⠁⠈⠛⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢰⣿⣿⡿⣻⣿⠿⢛⡇⢸⣿⣿⣿⣿⣶⠈⠳⣆
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣼⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⠋⠀⠀⠈⠩
⠀⠀⠈⠉⣵⣀⢠⡦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⡟⠀⠀⠂⢨⣿
⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠀⠁⠀⠀⠀⡀⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⢼⣿⢀⣿⣿⣿⣿⠁⠀⠀⠀⣿⡿
⠀⡠⢧⣀⣤⠤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⠄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⡿⢸⣿⣿⣿⠿⠀⠀⠄⢰⣿⣿
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣾⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⡠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢋⠀⠀⠀⠀⠀⠠⠤⠀⠀⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⣽⠃⢴⠛⣃⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠖⢉⡀
⠀⠀⠀⠌⠉⠄⠠⠄⠄⠀⢲⣇⠀⠀⠠⢀⠀⠐⠉⠡⠘⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⠄⡍
⠀⠐⡋⠀⠀⠠⢀⡀⡄⠀⠘⡉⢀⠀⠀⠈⠃⠀⡀⠀⠀⠀⠉⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⢠⡈⠂⠀⠀⠀⣠⠚
⠀⠀⠁⠀⠀⠁⠤⠀⢀⡀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⡩⠀⠀⡄⠀⠀⠀⣐⡖⠡⡄
⠀⠀⣿⡀⠘⠂⠁⠀⣼⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⡇⠁⠀⢀⠃⠀⠀⢀⠞⢉⠀⠀⠁
⠀⠀⢸⠀⠀⠀⠂⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⠀⠀⠀⡞
⠀⠀⢠⡄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠘⡅⠀⣸⠀⠀⠀⣀⡀
⠀⠀⠀⣧⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠧⠶⠛⠛⠛⠛⠛⠋
⠉⣩⡍⣹⣾⠻⠓⣾⡗⠙⠒⠉⠙⠛⠛⠿⠿⠻⠋⠉⠉⠛⠛⡉⣛⠛⠉⠉⠛⠛⠋⠙⢋⡁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂
⣴⢱⣿⣿⣷⣇⣶⣾⡗⢠⡀⡀⠐⠒⠀⡀⢀⠀⢀⣀⣀⡀⠈⠉⣤⣾⣯⣠⡀⢀⣠⣄⣠⣴⡤⠤⠐
⣷⣿⣿⣿⣿⣿⣿⣿⡇⣸⣟⣴⣦⡀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠺⠟⠻⠻⠿⠿⢦⠝⠋⠉⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆
⣿⣿⣿⣿⡿⠿⢿⣿⠂⠿⠿⠿⠋
⠉⠀⠉⠉⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⢅⣀
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠊
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣦⣶⣴⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⡤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠛⠛⣛⠟⠁⣿⣿⡿⠿⠃⠀⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⡇⠀⠻⠟⠁⠺⠇⠀⠀⠀⠀⠀⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⡴⣿⣿⠋⠁⠀⠈⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠀⡜⠸⣿⣧⣀⡀⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠈⠟⠿⣿⣿⡄
⠀⡸⠁⠀⠈⢹⣿⠁⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣤⡈⠉⢷⡀
⢀⠇⠀⠀⠀⠀⢹⡀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢣⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿
⠸⡀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿
⠀⢇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇
⠀⠘⡆⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟
⠀⠀⠘⣆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟
⠀⠀⠀⠈⢦⡀⠀⠀⢸⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠑⢄⠀⠘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⣀⠀⠀⠀⠀⠀⠀⣀⠤⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢇⠀⠈⠉⠐⠂⠈⠉⠀⢰⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠐⡶⡒⠂⠉⠁⠀⠀⠀⢀⡴⢹⠉⠑⠒⠒⠊⠉⡝⠱⣄⡔⣶⣶⣾⢇⠒⠒⣠⡤
⠀⠀⠀⠀⠀⠀⢱⠈⠑⠤⣀⣀⡠⠔⠉⠀⢠⢷⠂⠤⠤⠄⢺⢇⠀⠈⠓⠿⣉⡀⣈⠴⠊⢱⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠓⢄⣀⠠⠔⠁⢈⡦⠀⢀⢤⡇⠀⠑⠤⢀⣀⠤⢻⠁⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡜⠀⠀⡉⠲⣤⠔⠃⠼⡂⠀⡾⢈⠒⢤⡤⠊⠅⠀⠸⡀⠀⠀⠈⢆
⠀⠀⠀⠀⡠⠊⠀⠀⠀⡰⠁⠀⡸⠀⠀⠇⠙⢣⣔⢫⢸⢓⡤⠛⠁⢆⠀⠰⡀⠀⠣⡀⠀⠀⠈⠢⡀
⠀⠀⠶⣞⠒⠒⠒⠒⢾⡒⠒⠾⡒⠒⢾⠲⡗⠶⢒⣿⣿⡗⢲⠖⡷⢺⠖⠒⡳⠖⠒⡷⠶⠲⠶⠶⢺⠷
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠑⡄⠀⠱⠀⠀⡃⡠⢜⠃⡼⢸⡌⠛⣦⣀⠆⠀⡰⠁⠀⡜⠁⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢡⠀⢀⣡⠔⠉⠒⢄⡹⠒⠀⢷⢁⠔⠊⠑⢤⣆⠀⢰⠁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⡔⠋⠈⠉⠢⡀⢠⠋⠉⠉⠙⢇⢀⡤⠊⠉⠉⢑⣼⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡸⢀⠴⠊⠀⠈⠉⠒⢄⠀⠈⡾⠒⠂⠐⠒⠺⡎⠀⣠⠔⠋⠁⠀⠈⠑⠤⡘⡄
⠀⠀⠀⠀⠀⠐⠛⠃⠢⠤⣀⠀⠀⠀⠀⠱⣼⡠⠤⠤⠠⠤⠤⣱⡔⠁⠀⠀⠀⣀⠤⠒⠒⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠇⣀⠤⠤⠠⠀⠤⠤⣀⠘⡄⡜

⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠐⠀⠀⠐⢄⠀⠸⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠁⢘⣿⢮⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢰⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣸⣿⣿⣷⡙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⣿⣿⢷⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡼⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣯⢷⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡡⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡖⠁⠀⠀⠀⠀⠀⠀⢀⣀⣬⣾⣿⣿⠏⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣧⢠⠀⠀⢠⣴⣾⣿⣿⣿⡿⠋⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠋⣁⣶⣿⣿⣿⣿⠟⠁⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠋⠉⣤⣴⣾⣿⣿⣿⡿⣿⠃⠁⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⣤⣭⣿⣿⣿⣿⣷⣿⡋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢾⠿⠿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢀⣼⠟⢻⣿⣿⣿⣿⣿⣿⢿⣷⣖⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣜⣚⣵⣆⡾⣋⠼⠿⠏⠉⠀⠀⡉⢿⠳⢤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠻⠑⠀⠃⠈⠉⠁⠀⠀⠀⠁⠉⠉⠉⠀⠀⠀⠐⠁



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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠱⡀⠀⠀⠀⠀⢀⠴⡶⢄⠀⠀⠀⠀⠀⡰⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣇⠀⠀⠀⢠⠋⢰⣷⠈⢣⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⣶⡄⠘⡿⠀⣴⠆⠀⠀⣸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢙⡙⢦⠀⠸⣷⢸⡟⢰⡿⠀⣠⠞⣉⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡳⣄⢹⣆⢃⣾⢁⡴⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⠿⠿⢿⣿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡶⠲⣦⢶⢺⣿⣤⠆⠐⢦⣽⣿⣶⢶⡶⠲⣶⣶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠿⠈⠃⠈⠁⠀⠀⠀⠀⣠⢺⣿⣵⠏⠈⢷⣽⣿⢢⡀⠀⠀⠀⠀⠉⠀⠊⠸⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⣽⡛⢿⣡⣇⢀⣧⣹⡟⣻⣝⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⡋⠀⣰⠋⡔⢠⣿⣿⣾⣿⢧⡐⡌⢷⡀⠈⡛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠷⠊⠀⢰⡋⡸⠀⢿⠀⠙⠏⠁⢹⠇⠸⡈⣳⠀⠈⠲⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⠀⠀⠀⠀⣼⡿⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⠹⣿⡆⠀⠀⠀⠨⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠍⠁⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⢀⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣁⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇




⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⢠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⡈⠿⡛⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠠⠄⠀⠠⢠⣀⣾⡿⠟⠙⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡀⠀⠀⠀⠐⠿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⠗⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠱⠆⠀⠀⢀⠀⡀⠀⠀⣐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢾⣿⣿⣿⣿⣴⣥⣦⣤⣯⡸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡎⢻⣹⣷⣿⡿⣿⣿⣿⣿⢻⣿⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡗⣩⣿⠿⣾⡽⣯⡘⣿⣽⣿⣿⣿⣿⣿⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⡟⣿⣿⣶⠒⢫⡷⣮⣙⠻⣿⣿⣿⣿⡿⡿⢟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⠿⢹⣿⣶⣏⣌⢧⡓⣼⣯⣾⣿⣿⣿⣿⣷⢟⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣏⣿⣮⣿⣿⣿⢫⠟⣶⣹⣿⣿⣿⣼⣿⣿⣿⣶⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⢏⣽⣻⣿⡿⣷⣦⠻⣬⣿⣿⣿⣿⣿⣿⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣧⣿⣿⣏⢲⣎⣿⠧⣍⣽⣿⣿⣿⣿⣿⠮⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠷⣺⣿⠾⢫⣜⢿⡷⣾⣿⣿⣿⣿⣿⣿⢴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣾⠟⣩⢿⡷⢯⣼⡷⣻⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣳⣾⣿⠿⣶⣿⢻⡿⣿⠈⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⢿⣿⢺⢡⡟⣼⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢿⡎⢁⠈⣾⣾⣿⣿⡟⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⣿⣿⡿⣷⣿⣥⣷⣷⣿⣿⣿⡿⢿⡇⠀⠀⠀⡤⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⡽⣧⣿⢿⣿⠟⠉⠉⠙⠿⣇⣤⡙⣤⢦⣀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣣⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠿⣷⡘⣿⣦⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢠⣾⣷⣀⠸⢙⣷⢊⣄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⢿⣿⣷⠀⠀⠀⠀⠀⣸⠋⠛⠷⠒⠺⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠏⣾⣿⣿⠇⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠸⣿⠏⢹⡄⠀⠀⠀⠤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⣼⠏⠀⠈⢿⢠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣶⠏⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢲⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠟⠻⣿⠟⢻⠿⢿⢿⠻⣿⡿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣿⣶⣿⣶⣿⣷⣾⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⡭⣭⣅⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢒⣶⢏⡼⣋⠄⠀⢄⠈⢻⣆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣱⡾⢿⠏⣮⣾⡿⠋⣩⢼⡷⢀⣿⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠋⢀⠊⡾⢻⠋⠀⣾⢡⡾⣥⡾⡿⡻⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⣁⡴⠷⠊⠀⠎⠀⡀⠸⠘⢍⣒⡩⠔⠙⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡜⣴⠂⣶⣶⣭⣥⣭⢥⡆⠡⢙⠂⠐⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡒⣛⣀⣴⡿⢷⣾⢝⢧⣕⣚⣛⣛⣉⣉⣛⢶⣄⢬⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡛⢢⣼⡎⢣⢸⡿⣛⡭⠭⡙⢿⣷⣄⣹⠄⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢱⡟⡆⡙⡠⢣⢺⣿⣿⡳⠀⠀⠀⢻⡿⣿⢰⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⣭⣽⠶⣿⣿⢟⡡⠣⢃⣵⠇⣿⣿⡄⠁⠀⠀⠀⠀⡇⣿⣿⠀⠣⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣠⡞⢻⡇⠘⡧⠋⠑⡩⠀⣰⡟⠙⣷⣌⣙⣻⠦⣄⠀⠀⠀⠃⣿⣿⡈⢷⡚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣼⡏⢷⡀⢻⡞⣵⡴⠋⠀⡼⣿⠁⠀⢹⣿⠷⣾⣷⡄⠀⠀⠀⢄⠙⠙⢿⣦⠙⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⠁⢻⣈⣷⡿⢸⡏⢀⡼⠀⣇⠃⠀⠀⣾⠏⠀⠈⣧⢿⣄⠀⠀⠀⢱⡀⠀⢻⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⡟⣧⢮⠁⠻⡇⠈⠇⣾⢠⡇⢹⡀⠀⠚⠉⠠⢲⣞⣹⣷⣝⣷⣤⣤⡼⠃⡀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⢸⣷⡘⣿⠧⡞⣙⣄⠀⠘⢺⣍⢢⠷⡀⠀⠠⠐⠋⣩⣿⣿⣿⠋⢻⣷⢂⡞⠀⠀⠈⢧⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠜⣯⢷⣱⠀⠙⡋⢟⣿⣦⡀⢻⣾⠀⠀⠀⠐⠒⠛⣿⣿⣿⠋⢡⣿⡇⣾⠀⠀⠀⠀⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠉⠀⠀⠀⢻⡌⢿⠦⡀⠀⣈⠚⣿⠸⣾⡿⡀⠉⠒⠒⠒⠒⢿⣿⡿⢉⣾⣿⣷⠸⡆⠀⠀⠀⠘⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢛⡌⣿⣮⣧⠘⢿⣷⣀⣿⡇⢻⣷⣄⡀⠐⠀⠀⠤⢾⣿⣿⣹⣿⣿⣿⡆⢻⣀⡀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⣶⠿⠛⢜⢿⣿⣷⣼⣿⣷⣾⣷⡈⢻⣦⣙⣦⣉⠀⠀⠘⣿⣿⣖⡹⢻⣿⣿⣿⣿⡀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠋⠉⠀⠀⠀⠈⢢⣝⠿⣿⣿⣿⣿⣿⣷⡀⢻⣿⡄⠀⠀⠀⢠⡜⢻⣿⣷⣼⡿⠛⠁⠀⠀⠈⠻⣄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠵⠾⠓⠂⠉⠛⢛⣛⣛⣿⡶⠽⠿⠒⠲⣤⡼⢿⣦⡑⡙⣿⣶⠀⢰⠀⢠⠀⡀⢈⢷⣚⣳⡄⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢈⣉⣉⣩⣥⣤⣤⣶⣶⣿⣿⣷⣶⣶⣿⣟⢿⣎⢊⠻⡇⢻⣄⢺⣠⢠⣸⢎⠿⠿⣇⡿
⢀⣀⡀⠤⠀⠂⠀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⡛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣿⣷⡑⣙⠀⢠⠄⣡⢂⡅⠈
⠈⠋⠠⢄⣈⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⠋⠁⣀⣴⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡻⠓⠛⠴⠛⠉
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠤⡄⣀⣀⣀⣀⣀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⠀⠄⠀⠀⠀⢢⠀⠀⠀⠀⣿⠲⣄⡴⢻⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⢸⠀⢀⡟⠀⠀⠸⠏⠙⠀⠀⠀⣿⠈⠁⠊⠁⠀⡇⠙⠶⡄⠀⠀⢹⡏
⠀⠠⠓⠄⠀⠀⠀⠀⠁⠀⠀⠀⣿⠀⠀⠀⢸⠀⢀⣠⠞⠁⠀⠀⠀⢈⠀⠀⠀⢸⠀⢾⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢠⠏⠹
⠁⠀⠀⠀⠀⢸⠓⣦⣴⣛⡆⠀⢹⠀⠀⠀⢸⠀⠘⢧⣀⠲⣄⡀⠀⠀⠀⠀⠀⢸⡆⠈⢧⠀⠀⠀⠐⢩⠐⠀⠀⠿⠀⡇⠀⠀⠀⣿⠀⠀⠀⡀⠈
⠀⠇⠀⠀⠀⢸⠋⠀⠀⠈⡇⠀⠈⠀⠄⠀⠈⠀⠀⠀⠉⠃⢀⡽⠂⢀⡀⠀⡠⠘⠃⠀⡈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰⠋⠀⠀⢀⡼⠻⣄⡀⢀⠞⠁⠀⠀⠀⠀⠀⠀⣾⠳⣄⠀⠃⠀⠀⠀⠙⠀⢀
⠀⢀⡴⡶⢦⡘⠀⠀⠀⠀⠇⢀⡤⠂⠀⠀⠉⡍⠂⢀⠀⠀⠀⠀⠀⠙⠦⣠⠞⠁⠈⠲⡄⠀⠀⠁⠀⠀⠀⣻⢦⣄⠀⠀⠀⠀⠀⣹⠶⣅⠀⠀⠀⣇⠄
⠀⠋⠀⣷⠀⠉⠀⠀⠀⠀⠀⠛⢤⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢂⡄⠀⠘⠣⣤⠞⠁⠀⠀⠇
⠀⠀⠀⣿⠀⠀⠠⠄⠰⠀⠀⢀⠔⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠚⠉⠑
⠀⠀⠀⣿⠀⠀⢀⡜⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠃⠀⠀⠸⠄⠀⣀⠀⠀⢾⠙⢦⡀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣄⣀⠀⠀⠀⢿⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⠀⠀⠀⠀⠀⡞⠁⠀⢸⠀⣰⠃⠀⣴⠊⠀⠀⡁
⠀⠀⠀⣈⡀⠀⠀⠀⠀⡟⠲⣄⡴⢻⠀⡄⠈⠻⡄⠀⠀⢸⠉⢛⡦⠀⢸⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠇⠀⠀⢸⠀⢷⠀⠀⠈⢓⡄⠀⠁
⠄⠀⠀⡇⢙⡆⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⢸⡷⢭⡳⢤⣸⡇⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⢸⡄⠀⢣⠀⠐⠁
⢢⠀⠀⡇⢸⠀⠀⠀⠀⣷⠀⠀⠀⢸⡄⠀⢀⡀⠀⠀⠀⢸⡀⠀⠀⠀⢸⡇⠀⠉⠳⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈
⡁⠀⠀⠿⠀⠃⠀⠀⠀⠛⠀⡏⠒⠘⠃⠀⠫⣄⠀⠀⠃⠈⠀⠀⠀⠀⠘⠇⠀⠀⢀⠸⠇⠀⠀⠀⣧⢄⡀⠀⢸⠀⠀⠀⠲⣤⡔⠁
⠁⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⢸⠱⢦⡄⠀⠀⢽⠒⢽⡢⣼⠀⠀⠠⣞⠉⢙⡦⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⣿⠀⠨⠀⢰⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠆⠀⠀⠬⠀⠘⠀⠀⠑⠀⠀⣺⠀⠀⠉⢻⠀⠀⠀⢈⡷⢯⡀⠀⠀⢰⢄⡀⢸⡇
⠄⠀⠒⢿⢤⡄⠀⢸⡗⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠈⠀⢀⡀⠈⠀⠀⠀⠉⠀⠀⠉⠀⠀⢸⠔⠁⠸⡧
⠀⠀⠀⢿⠀⠀⠀⢸⣇⡴⠋⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣦⠀⠀⡅⠁⠀⢸⠙⠲⣤⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⣀
⠀⠀⠀⢸⠀⠀⠀⠐⡏⠀⡄⠀⣠⠄⠀⢸⡦⣄⠀⢸⠀⠁⠀⠀⠀⠀⠀⢐⠄⠀⢸⡞⠁⠀⠀⡅⠀⠀⢸⣤⠞⠁⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⣧⡞⢋⡄⠀⢸⠉⠛⢽⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠃⠀⠀⢨⡓⢤⣀⠀⠀⠀⠀⠀⢾⡁⠀⣀⠀⠀⠀⠄
⡄⠠⡞⠁⠀⠀⠀⠀⠀⠀⣯⡴⠋⠀⠀⠸⠄⠀⠀⠸⠀⠀⠀⠀⢸⠦⣀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⢸⣇⡴⠋⠀⠀⠚⠀⠀⠀⠙⠶⠌⠛⣦
⠀⠀⡨⠆⠀⢀⡀⠁⠀⠀⣿⠀⡄⠀⠀⠀⠀⣖⠄⠀⠀⠀⠀⠀⢸⠙⠺⣕⣾⡃⠀⠀⠀⠀⢠⠀⠀⠀⣈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴
⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠿⠀⣧⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⡇⠠⠄⠀⠀⠨⠀⠀⠀⡏⠙⣦⠀⠀⠀⠀⢀⣄⡀⠀⣠⡄⠀⠠⣏
⡄⢀⠀⢀⡄⠸⡷⣬⡓⢦⣀⠀⣿⠀⠀⣤⡀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⣀⠀⠘⣇⠼⠛⠷⣾⠀⠀⠈⠑
⠀⢸⡴⢋⡤⠀⡇⠀⠙⠲⣭⡛⣿⠀⠀⣇⣹⠆⠠⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠀⡷⢦⣀⠀⠀⠀⡇⠘⢆⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠠⠄
⠀⢸⡴⠋⠀⠀⡇⠀⠀⠀⠀⠙⣿⠀⠀⣿⠁⠀⠈⠀⠀⠀⢸⡙⠂⠀⠀⠀⠂⠀⠀⠀⣧⠔⠋⢰⠤⠖⠃⠀⠈⠀⠀⠀⠀⠀⡇⠀⡀⠀⢸
⠀⢸⡇⠀⠀⠀⠗⠀⠀⠄⠀⠀⠻⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠙⠂⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⠀⠀⠀⡶⢦⡀⠀⠀⠁⠐⡃⠀⠈
⣀⣈⣁⣀⣀⣘⣃⣀⣀⣁⣀⣀⣀⣀⣀⣛⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣸⣛⣦⣄⣀⣀⣸⣀⣀⣀⣀⣀⣷⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣄⣰
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣼⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⣼⣧⣴⡄⠀⠀⣠⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⣿⣿⣿⣿⣷⠀⠘⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠉⠀⠘⢿⣿⣷⣦⣤⣌⠻⠿⠿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⡇⠀⠀⠀⠉⢻⣧⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡟⠋⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠠⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⡀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠊⠁⢀⡀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⡖⠀⠒⠒⠒⠉⠁⠀⠀⠀⡠⠚⡓⢄⡀⠀⠀⠈⠙⠓⠒⠲⠶⢶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢦⣀⣀⣀⣀⡠⠔⠒⠉⠀⢀⡇⠀⠉⠒⠢⠄⣀⣀⢀⣀⡴⠋⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢸⠙⢄⠀⠀⠀⠀⠀⣀⠤⠊⠙⠦⣀⠀⠀⠀⠀⠀⣠⠊⡇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡎⠀⠈⡷⣖⠒⠈⠉⠀⠀⢀⡀⠀⠀⠉⠑⠒⢲⢾⠁⠀⢹⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠞⠀⠀⢠⠃⠈⢢⣄⡀⠤⠖⠉⠉⠒⠤⢀⣀⡔⠁⠘⡄⠀⠀⠳⡀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡠⠋⠀⠀⣠⠃⠀⠀⡰⠉⢦⡀⠀⠀⠀⠀⠀⡰⠋⢆⠀⠀⠘⣄⠀⠀⠙⢄⡀⠀⠈⠢⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠛⢇⠒⠒⢶⡯⠤⢤⣤⣞⣁⠀⣀⠔⠁⠀⠀⠑⢄⠀⠀⡠⠊⠀⠀⠈⢢⡀⠀⣈⡳⣤⡠⠤⠽⡶⠒⠒⡸⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠱⡀⠀⠈⢃⠀⠈⠙⡍⠉⠒⠒⠒⠤⣷⣾⡄⠒⠒⠂⠈⢉⠏⠁⠀⠰⠁⠀⢀⠎⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢣⠀⠀⠈⡀⠀⠀⡆⠀⢀⡤⠒⠉⡎⢱⠉⠒⢤⡀⠀⢸⠀⠀⠀⠃⠀⠀⡜⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⠀⠀⠀⡇⠀⣀⡴⣚⠁⠀⠀⡸⠀⠀⢇⠀⠀⠈⣑⣾⣀⠀⢸⠀⠀⠀⠇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠁⠀⣠⣷⣊⠁⠀⠀⠑⢄⢠⠃⠀⠀⠘⡄⡠⠊⠀⠀⠈⢑⣾⣆⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⣼⢔⠉⠀⠀⠈⠑⢦⡀⠀⠀⠟⠉⠉⠉⠉⠻⡀⠀⢀⡤⠊⠁⠀⠀⠉⡢⣧⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡚⠁⠀⠀⠈⠐⠄⡀⠀⠀⠙⢆⡐⢀⣀⣀⣀⣀⡀⢃⣠⠋⠀⠀⢀⠤⠂⠁⠀⠀⠈⢓⣼⣄
⠀⠀⠀⠀⠀⠀⢠⠴⠊⠁⠀⠈⠉⠒⢤⡀⠀⠀⠈⠢⡀⠀⢸⠋⠀⠀⠀⠀⠀⠈⠙⣇⠀⢀⠔⠁⠀⠀⢀⡤⠖⠋⠁⠀⠈⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠐⣠⣇⡠⠤⠤⠤⠤⠤⠤⢄⣸⣠⠋⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣔⡠⠄⠐⠒⠒⠂⠐⠒⠒⠂⠤⢄⣣⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⢀⣤⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠉⠛⠿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢀⣴⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢠⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⡟⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠈⢂⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠁⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣥⣀⡙⠛⠷⣦⣤⣄⣤⣶⣦⣴⣶⣶⣶⣿⣿⡿⠿⢟⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣦⣤⣄⣉⣉⣛⣛⣛⣛⣋⣩⣭⣴⣶⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠂

⠀⠀⠀⠀⣀⠀⢸⡟⠉⣍⠉⢡⠀⠀⠀⡄⠀⠀⡉⠉⠉⠍⠉⢠⠀⠀⢨⠅⠉⢡⠈⠈⠍⠉⢩⡍⠉⠉⢩⠉⠉⠉⢬⡌⠉⠙⢿⠇⣀⣀
⠀⠀⠀⢺⣤⢿⢸⡍⠀⡀⠀⢈⡀⠀⠀⡀⠀⢀⠁⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⡀⡀⠀⠀⡀⠐⠀⡈⡁⣀⠈⢽⢸⣷⡾
⠀⠀⠀⠀⠏⣿⢸⣏⠠⡇⠀⢸⢮⠀⢰⣇⠀⢸⣹⠄⢚⠃⠀⡿⠊⠀⢸⠀⢈⠾⢉⠀⣇⠀⠀⡗⠁⠀⢠⡈⠀⠀⠑⡗⠁⢸⣿⢸⡿⠓
⠀⠀⠀⠰⠀⣿⢸⡍⡀⠀⠀⠉⠁⠀⢈⠁⡐⠐⢀⠢⠈⠀⠀⠁⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠈⢸⣷⠘
⠀⠀⢀⣴⣾⣽⢸⣿⣋⣋⣀⠀⠀⠀⢸⠊⠀⢘⣽⠂⢼⠒⠀⠀⠀⠀⠪⡇⠂⠀⠀⠀⢻⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⢸⣿⣿⣦
⠀⠀⡸⣧⣽⠿⢸⡍⠠⡄⠀⠀⠀⠐⠀⡖⠒⠒⠀⡆⠈⢡⠀⠀⢠⠂⠀⢡⠂⠒⢂⠀⠀⣀⠀⠀⢠⠀⠀⢀⡆⠀⠀⠀⡄⠀⢰⢸⣟⣯⡝⠃
⣴⢄⡕⣛⣻⣿⢸⡆⠐⡖⠒⠒⢊⠒⠀⡇⡀⠀⢀⠐⠐⠀⠐⢒⢃⠀⠀⣀⠀⠛⢚⡒⠀⣠⠒⠒⢠⠒⠂⠛⡟⠛⠛⣲⡒⠒⠀⢸⣿⣛⣃⠐⢰
⠈⠀⢢⠻⣿⣻⢸⡃⠀⡟⠂⠀⢽⠀⠀⠒⡁⠰⢹⠀⠀⡓⢌⠀⢩⡃⠈⢸⠀⠀⡇⡇⠀⣮⠀⢀⠇⢣⠀⡄⡗⠄⠈⣰⠎⠁⢰⢬⣹⡿⠣⠃
⠀⠀⠻⢷⣯⣼⢸⡗⠒⡁⠒⠒⠂⠐⠒⡓⡒⠒⠂⠐⠒⡁⠈⠀⡀⢀⠒⠒⡒⠒⠒⠓⠂⢀⡐⠂⡀⡈⢀⠒⠲⠌⠠⠬⠤⠄⣠⢠⣏⣷⠞⠁
⠀⠀⠀⢈⠙⣿⢸⡆⠀⡷⠀⠀⠀⠀⠀⡶⡁⠀⠀⠀⢀⣫⠀⠀⡏⠁⠀⠋⡇⢀⣤⡭⠀⢾⡈⠀⡽⡧⡄⠀⠀⠀⢀⣀⣀⠸⢽⠀⡟⢣
⠀⠀⠀⠈⣁⣿⢸⣟⠂⠓⠤⠤⠄⠀⠄⠁⠈⠀⠠⠤⠤⠉⠄⠤⠥⠤⠤⠄⠁⠤⠄⠀⠤⠌⠡⠄⠅⠁⠉⠀⠤⠤⠤⠄⢀⣀⣠⠭⣗⡄
⠀⠀⠀⠀⣷⣻⢸⡟⠀⠀⠠⠀⠀⠀⠀⣀⠀⠠⠦⠀⠖⠂⠂⠀⠰⠘⠀⠂⠄⠀⠐⠯⠲⠀⠇⠀⠏⠂⠀⠊⠀⠀⢀⡀⠀⣀⣘⡏⡯⢷⡀
⠀⠀⠀⠀⠀⠙⢰⣇⣀⡀⠀⠤⠀⢀⣀⣀⣀⠀⠀⠀⠒⠲⣶⡶⠒⠂⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢐⣀⣀⣉⣙⣇⠑⠚⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⣀⡠⠀⠠⠢⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠤⡀⠀⠀⠀⠀⠑⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠈⣢⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢁⠔⠂⠐⠖⠉⠢⠤⢄⣀⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠠⢀⡀⠄⡀⢀⣀⡀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠄⠀⠀⠀⠈⠀⠀⠈⠁⠀⠈⢦⠀⠉⠉⠉⠢⠄⠤⠒⠉⠁⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠉⠀⠐⠢⣄⠀⢀⡀⠀⠀⢀⣀⠤⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠪⡀⠀⢀⡀⠀⠀⠀⢹⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⡀⠀⠀⡀⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣨⣴⡀
⢀⣀⠤⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠁⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⣀⣀⣀
⠤⡀⠀⠀⢀⠄⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⠤⠂⠉⢉⡋⠀⠀⠀⠁⠀⢹⡀⠀⠘⠷⠦⠀⢀
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠻⡀⠀⡀⠉⠳⣀⡴⠉⠑⢤⡠⠃⠀⠀⠀⠀⠰⡉
⠀⠀⠀⠀⠀⠀⡔⠉⠙⠁⠀⠀⣑⡢⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠃⠀⠙⠢⠄⠤⠚⠀⠛⠋
⠀⠀⠀⠀⠀⠰⢅⠀⠀⡀⠀⠙⠹⠗⡅⠀⠀⡔⠲⣥⣄⠉⠉⢢⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠁⠈⠁⠉⠀⠀⠀⢎⠈⠉⠁⠀⠀⢈⠆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠛⠻⡄⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⠀⠀⠀⠁⠒⠈⠀⠀⠀⠀⠀⠀⢀⡠⠔⠢⡀⠀⠀⠀⠀⠀⠀⢘⠷⠂
⠀⠀⠀⠀⠀⠸⠛⠷⠀⠀⠀⠀⠀⠀⠋⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀⣀⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠶
⠀⠀⠀⠀⢠⣤⠄⠀⠀⠋⠙⠀⠀⠀⠀⠠⣤⡀
⠀⣄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⣀⠘⡄⠀⡆⢀
⠀⠀⠀⣤⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠘⣶⡊⠀⠀⠀⠈⠓⣷⣾⠕⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⣤⠔⠁⠀⠀⠀⢦⣤⠖
⠀⠀⠀⠻⠇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠊⣿⠓⠄⠀⠀⠀⠱⣿⣿⢧⠁⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠺⣿⠦⠀⠀⠀⠀⠺⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⣀⣈⢑⣄⠀⣰⣽⡩⠭⣍⡐⣄⠀⠀⠀⠀⠀⠀⠐⢤⣀⣇⣼⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⣶⠏⠓⠐
⠀⠀⠀⠐⠉⠁⣼⡿⠟⠻⠿⠋⠑⡄⠀⠀⠈⠂⠀⠀⠀⢀⠀⠚⠛⠛⠟⠻⡒⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⣿⣿⡿⢶⡤⢄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠠⠔⠒⠛⠩⠏⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠛⠉⠉⣹⣿⣿⣏⠉⠉⠛⠛⠿⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⢠⣿⡟⢻⣿⡄⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣏⣀⣀⣀⣀⣀⣀⣹⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠈⠛⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠛⠁⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠉⠻⣿⣷⣤⡀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⢀⣤⣾⡿⠟⠉⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄⣀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣀⣠⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠉⠻⢿⣷⣦⡀⢀⣤⣾⡿⠟⠉⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠸⣿⡧⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⣠⣴⣿⠿⠋⠙⠿⣿⣶⣄⠀⠀⠀⠀⠲⣷⡆⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⣾⣿⠁⠀⣀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠈⠛⢿⣿⣦⣀⠀⠘⣿⣿⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⣸⣿⣇⣤⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣤⣸⣿⣇⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠁⠀⠀⠀⠉⠳⣶⠟⠉⠛⢶⡶⠛⠉⠉⠉⠉⠙⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡶⠾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠛⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡟⠀⠀⠀⠀⢀⣱⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠠⣴⠿⠛⠀⠀⠀⠀⣀⢀⡝⠀⠀⠀⠀⠀⠀⠛⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⣀⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠬⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢦⢠⣀⣔⣬⡺⠛⠁⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠈⢻⡄
⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠢⡄⠮⠓⠋⠁⠀⠀⠀⠀⣠⡌⠁⠀⠀⠀⠀⠀⠀⢈⣀⢀⣠⡴⠎⠀⠀⠀⠀⣀⠠⣴⡍⠀⠀⠀⠀⠀⢀⣿⣀
⠀⢸⡇⠀⠀⠀⠠⡀⢀⡠⢀⠎⠀⠀⠀⠀⠀⠀⡠⠀⠈⠒⠙⠞⠓⢉⠀⠀⢠⣠⡀⣄⣦⣴⣿⣿⡿⠋⠀⠀⠀⠘⠎⠳⠿⠿⠊⠀⠀⠀⠀⠀⢸⠀⠉⠉⠛⢷⡄
⠀⠀⠻⢦⣤⣤⡄⠉⠀⠉⠀⢄⠀⠀⡀⢄⠠⡔⢊⡄⡀⢀⡦⣄⡴⠄⢠⡐⢤⣙⢻⣿⠿⣫⣾⠛⡱⣄⢦⡠⣰⣵⣦⢂⡄⠀⠀⠀⠀⠀⠠⡰⡅⠀⠀⠀⣠⠼⣿
⠀⠀⠀⠀⠀⢸⣇⠀⡀⠀⠀⠀⠀⠁⢈⣤⣤⢾⢯⡳⣝⡲⣝⡲⠭⢃⣼⢯⣷⡿⢋⣡⣾⡿⣷⣷⠙⠮⠻⠟⠛⢉⣥⣿⢭⡇⡄⢐⢦⡪⣕⠿⠃⢀⡠⠖⢉⣼⠏
⠀⠀⠀⠀⠀⠀⠻⣦⣍⡂⠓⠌⣓⣩⡀⣀⡀⣀⠓⠝⠳⠝⢒⣉⣴⣿⣿⠟⠉⠀⠋⠉⣠⡾⣟⢮⣿⡿⢲⣖⠿⡛⣝⣾⡟⣰⣏⣪⣕⠎⢚⢁⣬⣤⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠙⣷⣀⣑⣈⡙⡿⢿⢿⢿⡻⣽⣿⣥⣶⠖⠀⣠⡾⣏⣽⣮⡿⢏⣴⣟⢮⣓⣽⡾⢟⣴⡿⣤⣥⣶⡾⢋⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⢷⣌⠈⢯⣷⣻⣾⡿⠋⠀⣠⣾⢻⣶⣤⣤⣴⣾⣿⣏⣈⣓⣉⣥⡶⣿⢯⡷⠾⠛⣩⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⡽⠋⠀⣠⣾⣿⣿⣿⡿⠿⠟⢛⣿⣿⠿⠟⣽⠿⠷⠶⠶⠶⠶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣤⢦⣄⢀⡰⠋⠀⠀⠔⠛⠛⠉⠁⠀⠀⢀⣴⣟⣋⣤⠴⠋⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠛⠋⠀⠀⣹⠟⠀⢀⣀⣤⣴⣶⣾⠏⠀⢀⣴⠿⠋⠥⠅⠂⠀⢀⣤⣾⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠿⠃⠀⠀⠀⠀⣠⣼⣧⣶⡿⢟⣿⠁⣲⠞⠁⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣠⡐⢶⡄⢄⡐⠻⡿⠯⢠⡿⠋⢁⡼⠋⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢶⣤⣥⣴⠞⠛⠛⠋⠀⣠⠟⢁⣴⡿⡫⠂⠀⠀⠀⠀⠀⠀⠀⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡾⢋⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣼⠟⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠉⢤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡾⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⡟⠁
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠟
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁

⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢀⣀⣀⣈⣉⡛⠻⠷⣦⣤⣀
⠀⠀⢉⣉⣉⣩⣿⣿⣿⡿⠿⣿⡶⣿⣿⣷⣿⣿⣶⣤⣄⡀
⠀⠬⠽⢿⣿⣿⣿⣿⣿⣽⣿⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⡀
⠀⢀⣠⣤⣶⡶⠶⢿⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣤⣤⣼⣿⣶⣾⡷⣦⡀
⠀⠀⠀⢠⡤⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⣨⣩⣿⣿⣿⣿⣞⣿⡆
⠀⠀⠀⠀⠀⠀⠚⠛⣿⣿⣿⡿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡿⠟⠚⠛⢻⣿⣿⣩⣾⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢻⣿⢿⣿⣿⣿⣿⣿⡟⢁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⠋⣀⣤⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠋⡹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⡿⠉⠟⠿⠿⣛⣿⣿⣿⣿⠁⠹⢿⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⢴⣶⠋⠉⠿⠿⠟⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠊⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠯⡉⠛⠒⢤⡮⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠴⠒⠛⠉⠁⠀⣠⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⣅⡏⠂⠀⡾⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣠⠴⠖⡻⠻⡀⠀⠀⠈⠑⠒⠠⠤⠤⠤⠶⠒⣲⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣼⢧⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⡀⡀⠈⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⡀⠀⠀⢀⣠⠶⠋⠉⠀⠀⢀⣠⠴⡾⠀⠀⠀⠀⠀⠐⠢⣄⣀⣀⣀⠴⠋⠀⠀⡏
⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⠀⢹⣿⡞⠋⠀⠀⠀⠐⠒⠋⠉⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢟⡏⠀⠀⠀⠀⠁
⠀⠀⠈⠉⠛⠶⠦⢄⣀⣀⡄⠀⠀⢻⣆⣿⣶⡀⣠⡟⠁⠈⠳⣖⠒⠒⠚⠋⠉⡀⠀⠀⠀⠀⠰⣄⣀⣀⡠⠖⠉⠀⠸
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠉⠛⣿⣻⣿⣿⣿⣿⣿⢟⣁⣀⣠⡴⠼⠶⡖⠒⠖⠚⢁⠀⠐⢆⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣷⡴⣶⣿⣿⣿⣿⣿⣿⣟⠛⠻⠭⣵⣊⣀⣴⠂⢙⣓⡒⣄⡸⠶⠒⢊⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⣠⡴⠋⠉⢰⣿⣿⣿⣿⣿⣿⣿⣧⠠⡀⠀⠰⡄⠘⠯⡿⢿⠿⠛⠛⠑⠛⠒⡺⠗⠦⠤⠤⣧⣄⣀⣀⣀⡀⠀⠳⣄⠀⠀⠀⠈⠳⣄
⠀⠀⣠⠞⠁⠀⠀⠀⠘⢿⣿⣿⣿⡿⣻⠇⠈⢳⡇⡀⠜⠋⠀⢤⡸⠇⠀⢺⡠⠄⠈⠚⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⠒⠒⠒⠀⢈⣳⠤
⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⡟⠀⡠⠾⠿⢆⠀⠀⠀⢸⣧⠔⠈⠀⠁⠀⠀⠀⠀⠀⠴⠾⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣸⣗⠈⠀⠀⠀⠀⠁⢀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⡤⠒⠉⢸⡏⠉⠢⠀⠀⠀⠀⠀⢳⣄⣤⠄⠀⠀⢠⣻⠃⠀⠀⠀⠀⠀⠀⠀⢠⠤⢵⣿⣄
⠀⠀⢀⣠⡤⠾⠷⣾⣭⣭⣀⠀⠉⢸⠇⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠈⠑⠲⢤⡀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠳⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠹⠀⣤⠔⠈⠀⠀⠀⠀⠉⠳⣞⡿⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠮⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣴⡯⠖⠂⠀⠀⠀⠀⠀⠀⠀⠒⠤⢌⣧⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠠⠒⣐⣢⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⡀⠤⠀⠀⠠⢄⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠉⠢⠀⣄⠀⡤⠊⠀⢄⠀⠀⠀⠀⠑⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⠺⠀⠀⠀⠀⠀⠀⠶⠄⢁⠀⠄⠀⠀⠀⠀⠀⠁⠀⠱⢤⡀
⠀⠀⠀⠀⡤⠠⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢀⠀⠀⠐⠀⠀⠀⢻⡀⠦⢄⡀⠀⠀⠀⠀⠆⣀
⢠⠐⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠦⡀⠀⠈⠀⠀⠀⠈⠒⠒⣄⣀
⠈⠂⠖⠢⠤⠄⢤⠔⠢⠀⠠⠄⠒⠤⠀⢀⡠⠄⣀⠀⠀⡀⠠⠀⣠⠠⡔⣤⠤⠔⠁⡀⢄⣀⢀⡀⣤⣀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠈⠒⠓⠠⠤⠀⠒⢲⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠁⠀⠀⠀⠀⠀⡈⢉⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠑⠒⠒⢒⠿⠖⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠲⣌
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠘⠅⣀⡀⢀⣀⣀⠠⠄⣀⣀⢀⡀⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠒⠒⠉⠁⠒⠒⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠄⠄⠐⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠒⠒⠒⠂⠴⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⡀⢤
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠀⠀⠀⠀⠀⠀⡠⢚⡹⢯⠓⢄⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠌⢀⣿⡇⠘⣧⡀⠱⡀⠀⠀⠀⠀⢺⠃
⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠈⠟⢄⠀⠀⢸⠀⢈⣿⠇⠨⢻⣇⠀⣧⠀⠀⠀⡐⡿⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠙⢀⢀⠀⠀⠀⠀⠈⠻⡁⠠⣼⠀⢈⣺⣿⠐⣞⡁⠀⣏⠠⢶⡦⠖⠁⠀⠀⠀⣠⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠙⢇⠠⠀⠀⠀⠀⠀⠐⠼⣧⡈⣩⣥⢀⣬⣥⣼⡧⠖⠉⠀⠀⠀⣀⢀⡦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣅⠠⣤⡄⠀⣶⣮⣿⡟⢻⡿⠛⢿⣡⠦⣯⣈⣿⣵⣫⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢁⡴⣺⢿⣧⣘⢧⣤⣿⡽⣖⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠼⢫⡞⣯⣸⣺⣸⡟⣯⠪⡓⢬⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩⠛⢁⣣⠎⠐⡏⠙⠙⠁⠇⠈⠣⣴⢆⠈⠣⣕⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠞⠁⢠⢎⠏⠀⠀⠘⠀⠀⠸⠀⠀⠀⠈⢣⡠⡀⠈⠳⢯⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠟⠁⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⢂⠀⠀⠉⠢⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡼⠁⠀⠀⠀⢺⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣏⢁⣤⣤⣤⣤⣠⡶⠛⠛⠛⠉⠙⠛⢷⣄⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⢀⣾⠏⠀⠙⠦⡿⠱⣯⣤⣄⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢰⣿⣋⣁⣀⣀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠈⠛⠛⠛⠛⢷⡶⠿⢶⣆⠹⣦⣀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠈⠉⠛⠛⠛⠻⣷⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢠⡟⠀⠀⢀⣠⡾⠋⠀⠀⣀⣀⣀⣀⣤⣶⡿⠏⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠸⣧⡀⢀⡿⠁⠀⠀⠀⢸⣿⡛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠶⠶⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⣼⣧⣀⠀⠀⠀⣴⡿⠛⠛⠛⠷⢶⣤⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⣦⣄⣻⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⢀⣿⠉⢻⣷⠀⢸⣿⠀⣠⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⡟⠀⠀⠀⠀⠉⠻⣷⡀⠉⣿⣆⠀⠀⠀⢰⡟⠛⣷⡀⠀⠀⠀⠀⣿⡀
⠀⢸⡿⠀⠀⣿⣄⢸⡿⠀⠙⠛⠻⢯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠉⠁⠀⠀⠀⠀⠀⣾⣿⣤⣤⣈⢻⣇⡀⠀⢸⡷⠀⠈⠙⣷⠀⠀⠀⢹⡇
⠀⢸⡇⠀⠀⠘⣿⡘⠳⠞⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⢿⣿⣛⣿⣯⣥⡀⠀⠐⣿⠀⣠⣴⣾⠇
⠀⢸⡇⠀⢀⣾⠟⠀⠀⠀⠀⠀⠘⠻⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣽⠃⠀⠀⠹⠷⠟⠀⣿
⠀⠸⣧⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠛⠻⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠀⠀⠀⠀⣀⣀⣀⣸⡏
⠀⠀⢹⣇⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⢠⡾⠋⠉⠉⣽⡇
⠀⠀⠀⢿⡌⠻⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣄⠀⣴⡟⠁⠀⠀⣰⡿
⠀⠀⠀⠈⢿⡀⠀⠉⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⢙⣷⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠉⢿⡆⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠀⣼⣇⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⢸⣿⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡿⠋⣿⡷⠋⠀⠉⣩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⣿⡀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⡀⠀⣠⣾⠃⠀⠈⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡘⢿⣦⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠾⠋⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣍⠻⣮⡛⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⡶⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⣶⠶⢶⡶⠶⠶⠶⢶⡶⠶⢶⣶

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣎⡛⣻⢿⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣽⣿⡟⠛⠛⠋⣹⣿⡜⣿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣒⣫⣿⣿⣿⣿⣦⣰⣿⣯⣭⠀⠀⠙⠛⠁⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡟⠛⠛⠀⠀⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⡿⠋⠀⠈⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣷⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣷⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠁⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄⡀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢶⣤⣤⣿⣧⠤⠴⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢶⣩⣿⣿⠻⡗⣶⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⠤⡤⠶⡶⡉⠁
⠀⠀⠀⠙⠚⠻⡶⣶⠶⣤⠶⠶⠶⣒⠒⠛⠛⠭⣳⣨⡟⢤⣯⣇⣧⠚⢫⣠⠋⠀⡰⠋⠁⢸⠀⣧⠇⢰⠃⠀⡇⣧
⠀⠀⠀⠀⠀⠀⢱⠈⢣⡉⠙⠲⠤⣈⢳⡀⠀⣠⠞⠉⠛⡟⢦⣇⡨⣶⠊⠉⠢⣀⡇⠀⢀⠇⡜⠈⣠⠃⠀⢰⢃⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣⠀⠀⠀⠈⡷⣧⡾⣥⣄⠀⡸⠀⠀⣯⠀⠘⡄⣠⡶⠟⠣⡀⢸⡜⣠⠞⠁⠀⢠⠏⡜
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡆⠀⢀⣠⣏⣁⡀⠈⠛⣷⣃⠤⠤⣿⠴⠒⠺⡏⠀⢀⠤⠬⠾⣷⠁⠀⠀⡴⠁⡜
⠀⠀⠀⠀⠀⠀⢠⠃⢠⣿⢁⠴⠋⠉⠉⠓⠫⣷⣄⠏⢀⣀⠀⢸⠀⣠⠤⢵⣰⠃⠀⠀⠀⠈⠳⣄⣸⠀⢸⡟
⠀⠀⠀⠀⠀⢀⡜⢀⣾⠿⠓⠒⠢⢤⡀⠀⠀⠀⡝⠋⠉⠈⠻⠿⡞⠀⠀⠀⢻⡀⣠⠖⣩⠭⠽⠛⠛⢤⣸⠇
⠀⠀⠀⠀⢀⣸⣷⣟⣉⠉⠉⠉⠉⠓⠿⣦⠀⢰⠁⣀⣀⣀⡀⠀⡇⠀⠀⢀⣀⢿⣡⠊⠀⠀⠀⡠⠾⠟⠙⠧⡀
⠀⠀⠀⠈⠉⠀⠈⠓⢌⠙⠢⣄⠀⠀⠀⠘⢷⡟⠋⠁⠀⠀⠉⠓⣷⠺⠏⠉⠀⠀⠻⡀⠀⠀⡞⠀⠀⠀⢀⠔⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠣⡀⠀⠀⣜⣠⠤⠔⠒⠢⢤⡀⢸⠀⠀⠀⠀⠀⠀⠳⡀⢸⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢱⡀⢰⠋⠀⣀⣠⣤⣀⡀⠈⢻⡀⢀⣴⡿⠒⠒⠒⢷⣽⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⣧⣧⠖⠉⠀⠀⠀⠀⠈⠳⣄⡇⡻⠋⣠⠔⠚⠉⠉⠹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣿⠕⠶⠛⠛⠛⠋⠒⠦⣄⠈⢿⢁⠞⠁⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣾⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡰⠛⢉⠟
⠀⠀⠀⠀⠀⠂⠀⠈⠉⠉⠉⠉⠀⠀⠀⠘⣷⣟⡀⠀⠀⢼⡧⠂⠀⠠⣶⣴⠃⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠒⠒⠀⠀⠘⠛⠛⠛⠀⠀⠘⠛⠁⢛⣶⣆⠀⢼⡯⡀⢶⣶⡋⠻⠁⠀⠀⠀⠀⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠃⠀⠀⠀⡤⣤⠀⣈⠙⢦⣸⣇⡴⠋⢋⠁⠀⠀⢀⠀⠀⠘⠀⢀⠟⠙⠛
⠀⠀⠀⠐⠒⠒⠚⠛⠛⠛⠛⠓⠀⠚⠿⣗⣻⡞⡿⠗⢒⣿⣿⡒⠿⢓⢷⡿⡿⢿⠒⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⣴⣶⣶⣦⣤⣤⣤⣶⠶⠀⠀⠀⠙⣧⡘⣶⣶⠋⣸⣇⢝⣦⣴⢋⡼⠁⠀⠀⠀⠀⠀⠙⡄⠉⠉
⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣀⠀⠀⠀⠀⣴⢾⡿⣿⣍⣉⣻⣟⣋⣭⡿⣿⣒⡄⠀⠀⠀⠐⠂⠐⠛⠒⠒
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠸⠻⠟⠃⠈⣭⣿⣿⣩⠀⠈⠛⠛⠆⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠐⠒⠛⠛⠛⠛⠳⠖⠂⠀⠀⠀⠀⢀⣰⠂⠰⢿⣹⣇⡸⠂⢸⣀⠀⠀⠀⠀⠀⠠⠤⠱⡤⠤⠤⠀⠀⠠⠠
⠀⠀⠀⠒⠛⠛⠛⠛⠓⠚⠒⠀⠀⠀⠀⠀⠘⠻⠉⡏⠩⢽⡯⠍⢹⠹⠛
⠀⠀⠐⠒⠒⠒⠒⠚⠚⠒⠒⠂⠀⠀⠀⠀⠀⠀⠈⢋⠀⣼⣧⠀⡙⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠿⠿⠷⠶⠶⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⢻⢾⢋⣽⣯⡙⣷⡟⠀⠀⠀⠀⠀⠀⠈⠃⢌⠉⠉⠉
⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠰⢯⣺⡗⡵⠆⠁⠀⠀⠀⠀⠀⠀⢠⡔⢀⣀⣀⣀
⠀⠀⠀⠤⣤⣤⣤⣤⢦⡤⠴⠀⠀⠀⠀⠀⠀⠀⠠⣠⠾⢹⡏⢦⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⡠⣾⣷⢄⠉⠀⠀⠀⠀⠀⠀⠀⠈⠶⡀⠀⠱⢮⡀
⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠊⠀⢾⣗⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⢀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣊⠈⠠⢴⡉
⠀⠀⠀⠉⠉⠉⠉⢉⣉⣉⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⡀⠙
⠀⠀⠀⠒⣚⣛⣛⣛⣛⣛⡛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠈⠦
⠀⠀⠀⠈⠉⠉⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠜⠁

⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣿⣿⣛⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢴⣶⣤⣤⣀⠀⠁⠀⠀⠀⠀⠀⢀⡀⢀⣠⣤⣶⠶⠿⠻⡿⠓⠲⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⠛⠀⠀⠘⠷⠀⠀⢀⣼⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠉⠙⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣀⣤⣀⣀⣦⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣧⡀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠁⠈⠀⠉⠀⠈⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣅⢀⣤⡐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠉⠻⣿⣿⣶⣦⣤⣀⣀⣀⣀⣀⣠⣤⣴⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⢇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⢰⠀⠘⣦⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠃⠀⢂⠀⠀⠀⠀⠀⢰⢃⠀⠀⠀⠸⡆⠈⠀⠀⠸⣆⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⢄⠀⢠⠀⠀⢢⡀⠀⠀⠀⢸⠈⢆⠀⠇⠀⠹⢆⠁⠀⠀⠙⠆⠀⠀⠈⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠻⢦⣆⠀⠀⠙⢦⡀⠀⠸⠀⠈⢦⠘⡀⠀⠑⢿⣄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢢⠀⠀⠀⠉⠢⣄⣇⠀⠀⠑⢧⠀⠀⠀⠈⠙⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠱⡀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠈⠙⠓⠢⠀⠁⠀⠀⠀⠀⠀⠀⡞⠃⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠂⠤⣈⡀⣄⡀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠈⠴⠁⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠉⠙⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠄⣀⠈⠑⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠈⠉⠁⠉⠉⠓⠀⠀⠀⠀⠀⠘⠆⠀⠂⠀⠙⢄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠤⠤⠄⣀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠛⠦⠀⠉⠉⠀⠀⠀⠀⣈⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡤⠛⠀⠀⠀⡀⠀⠀⠀⠀⠈⢠⡄⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠄⠊⠀⠋⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠘⠀⠀⠀⢻⣄⣙⣻⠲⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠖⠁⠀⠀⠐⠀⠐⠀⢀⣉⠥⠣⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢱⢄⠀⠐⠈⠁⠒⠒⠈⠉⠀⠀⠀⠈⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢁⠖⠁
⠀⠀⠀⠀⠀⠀⠘⢆⠠⢀⣍⣒⣦⢤⡤⠒⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠖⠉⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠉⠲⠄⢉⣉⠁⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⠦⠤⢠⣤⣬⡁⠀⢀⠀⠤⠄⢒⠁⠀⠀⡀⠀⠠⠤⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠛⠳⠶⠆⣠⠴⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢈⡦⢌⣉⡀⢀⣀⠀⡇⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⠴⢊⡡⢤⠀⠹⠈⢉⠈⡴⠖⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠡⠂⠁⠀⢠⠴⠳⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠨⠄
⣀⣀⣀⢀⣀⢀⣀⣀⠀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢸⡿⠀⠀⣿⡇⠹⣿⠿⢶⣶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⣀⣤⣴⡾⠛⣿⡟⠿⣶⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣤⡶⠟⢻⣿⣿⡄⠀⠀⠀⠸⣷⡀
⠀⣸⡇⠀⠀⣿⠁⠀⣹⣷⡀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⢀⣿⣦⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢠⣿⠃⠘⣷⡄⠀⠀⠀⠙⣷⣄
⢠⣿⠃⠀⢠⣿⠀⠀⢿⡿⣷⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⣿⡏⠻⠷⣶⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣷⠀⠀⠈⢿⣆⠀⠀⠀⠈⠙⠿
⡿⠁⠀⠀⣾⡇⠀⠀⢸⡇⠘⣿⡉⠙⠛⠛⠻⠷⠶⠶⠾⠿⠟⠋⠉⠀⠀⣿⡇⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠛⠛⢉⣿⠋⠻⣧⡀⠀⠀⠙⢷⣄
⠀⠀⠀⢀⣿⠀⠀⠀⢸⡏⠀⠘⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⠘⢿⣄⠀⠀⠀⠙⢷⣦⣀
⠀⠀⠀⢸⡟⠀⠀⠀⢸⡇⠀⠀⣿⣿⡿⢶⣦⣤⣤⣤⣤⣶⠶⠾⠟⠋⠀⣿⡇⠈⠙⠛⠿⠶⣶⣶⣶⠶⠾⢟⣿⣿⡄⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠈⠛⠿
⠀⠀⢀⣿⠁⠀⠀⠀⣿⠃⠀⠀⣿⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏⠈⠻⣶⣄⠀⠀⠀⠈⠻⣦⣀
⠀⢀⣾⠃⠀⠀⠀⣰⡟⠀⠀⠀⣿⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⣿⡿⣦⣄⣀⣀⣀⣀⣤⣴⠾⣿⣿⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠈⠛⢷⣤⣄⣀
⣠⡿⠃⠀⠀⠀⢠⡿⠁⠀⠀⢠⣿⢠⣿⠘⣿⡟⠛⠛⠻⠛⠛⠛⠁⠀⣠⣿⣧⠀⠉⠉⠉⠉⠉⠁⠀⣼⠏⢻⣧⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠉⠛
⠟⠁⠀⠀⠀⢀⣿⠃⠀⠀⠀⣾⠇⣸⡏⠀⣸⣿⣶⣤⣤⣤⣤⣤⣶⠿⠋⣿⡟⢷⣄⡀⠀⢀⣠⣴⣾⣿⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣶⣤⣄
⠀⠀⠀⠀⢀⣿⠇⠀⠀⢀⣾⠏⢠⣿⠃⠀⣿⠋⢿⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠛⠛⠛⠋⣽⡟⢻⣇⠀⠀⠈⠙⠻⣦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣿
⠀⠀⠀⢀⣾⠇⠀⠀⢀⣾⠋⢀⣾⠇⠀⢰⣿⠀⢸⣿⣤⣄⣀⣀⣀⣠⣤⣿⣷⣤⣄⣀⣀⣤⣶⡟⠀⠀⠙⢷⣤⣀⠀⠀⠀⠉⠙⠛⠷⢶⣶⣶⡾⢻⡿⠁
⠀⠀⣠⡾⠃⠀⢀⣴⠟⠁⢀⣾⠏⠀⠀⣾⠇⠀⣿⠋⢿⡏⠉⠉⠉⠉⣠⣿⡇⠈⠉⠉⣉⣹⣿⣆⠀⠀⠀⠀⠉⠛⠻⠿⢶⡶⣶⣶⣶⠿⣿⡏⠀⣿⠁
⣶⣾⣟⡁⢀⣴⡿⠋⠀⢠⣾⠋⠀⠀⣼⡏⠀⣸⡟⠀⠈⣿⣦⣤⣤⣴⠟⣿⡿⢶⣶⡾⣿⣿⠈⠻⢷⣤⣀⣀⣀⠀⣀⢀⣠⡶⣿⡏⠀⢰⡟⠀⢸⣿⠀⢀
⢿⡍⠙⠻⣿⣿⣤⣤⣴⡟⠁⠀⢀⣴⠟⢀⣼⠟⠀⠀⠀⣿⢿⣍⠉⠁⢠⣿⡇⠀⢀⣰⡿⢻⣦⡀⠀⠈⠙⠛⢛⣿⣿⠟⠉⠀⣿⠀⠀⣾⡇⠀⢸⡟⠀⢸
⢸⡿⠀⠀⢻⡇⠈⠹⣿⠻⢶⣶⣿⣋⣰⡿⠋⠀⠀⢀⣼⠟⠈⣿⣷⠾⠟⣿⡿⠶⣿⣿⡁⠀⠙⠿⢶⣶⣶⡾⠛⢹⡟⠀⠀⠀⣿⡀⠀⣿⠀⠀⢸⣧⠀⠘
⣼⡇⠀⠀⣾⡇⠀⠀⢻⡇⠀⢻⡏⠻⣿⣷⣶⣿⣿⠛⠁⠀⢀⣿⢿⣶⣶⣿⡷⣾⡿⠉⠻⢷⣶⣶⠾⢿⡇⠀⠀⢸⡇⠀⠀⠀⢹⣇⠀⢿⡄⠀⠈⣿⡀
⣿⠀⠀⢀⣿⠁⠀⠀⢸⡧⠀⢸⣿⠀⠘⢷⡄⠉⠻⣿⡿⣾⣿⣁⣈⣿⣇⣹⣧⡿⢿⣶⣶⠿⣿⠁⠀⢸⣧⠀⠀⠘⣿⡀⠀⠀⠈⢿⡄⠈⢿⣄⠀⠸⣷⡀
⠃⠀⢀⣾⠇⠀⠀⠀⣾⡇⠀⢸⣿⠀⠀⠘⣷⠀⠀⢸⣇⠀⢹⡿⠛⢿⣿⣿⣿⣿⡟⢻⣯⣀⡙⣷⣦⣀⣙⣷⣦⣄⣸⣷⣄⣀⣀⣘⣿⣄⣈⣻⣦⣀⣹⣷
⠶⢶⣾⠿⠶⠶⣶⣿⠿⠷⠿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⡿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣟⣿⡟⠛⠛⣿⡿⠛⠛⠛⣿⠟⠛⣿⡿⢛⣿⠟⠉⣿⠟⠉⢉⣿⠟
⠀⠀⠻⣷⡀⠀⠈⠙⢷⣄⡀⠈⠙⠻⣶⣄⡀⠉⠻⣷⡙⢷⣄⣽⣿⣟⣹⣿⡟⣿⣿⠿⠿⢿⣦⣼⣿⠀⠀⢀⣾⠏⠀⣼⠏⢠⡾⠃⠀⣼⠏⠀⣰⡟⠁
⠀⠀⠀⠈⢿⣆⠀⠀⠀⠙⠻⣦⡀⠀⠀⠙⢿⣆⠀⠈⣿⣼⣿⢿⣯⣽⣿⣿⡿⠻⢿⣄⣠⡾⠟⠛⠛⢷⣶⣾⣏⠀⢸⣿⠀⣾⠃⠀⢀⣿⠀⠀⣿⠃
⢷⣄⠀⠀⠀⠻⣧⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢻⣧⢀⣿⡿⣷⣤⣈⣿⠿⣿⣷⡶⠾⣿⣿⠀⣠⣶⠿⠛⠛⠛⠛⣿⣾⣿⣄⣿⡄⠀⢸⣿⠀⠀⣿⡄
⠀⠙⢷⣄⠀⠀⠙⣷⡀⠀⠀⠀⠀⠹⣧⠀⠀⢰⣿⣿⣥⣄⠀⠈⣿⣿⢶⣾⡇⣠⣤⣬⣿⣾⡟⠀⠀⢀⣤⣾⠟⠋⠉⣈⣽⣿⣷⣦⣬⣿⡀⠀⢻⣇
⠀⠀⠈⢿⣆⠀⠀⠘⢿⡄⠀⠀⠀⠀⢹⡇⣠⡿⠋⠉⠉⠙⣷⣼⣿⣀⣀⣹⡿⠛⠉⠉⠉⠻⣧⠀⣠⡿⠋⠀⢀⣤⡾⠛⠉⠀⢀⣨⣽⣿⣿⣦⣌⣿⣆
⠀⠀⠀⠈⢿⡄⠀⠀⠈⢿⣆⠀⠀⢠⣿⣿⣿⣶⣶⣦⣤⡀⢸⡿⠉⠉⠙⣿⣧⣴⣶⣶⣶⣶⣿⣷⣿⠁⢠⣾⠟⠁⠀⢀⣴⡾⠛⠉⠁⠀⠀⠀⢉⣿⣿⣷
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠈⣿⡀⣠⣾⠋⠀⠀⠀⠀⠀⠙⣷⣿⠁⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠹⣿⡀⣾⠃⠀⠀⣰⡿⠉⠀⠀⠀⠀⠀⣠⡾⠛⠉⠁
⣄⠀⠀⠀⠀⠸⣷⠀⠀⠀⣿⣿⠿⠛⠛⠻⢶⣤⡀⠀⠀⣿⡿⠛⠛⠛⠿⣿⣷⠾⠿⠛⠛⠿⠿⢶⣼⣿⡏⠀⢀⣾⠟⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⣠
⠻⣦⡀⠀⠀⠀⢹⡇⢀⣴⠟⠁⠀⠀⠀⠀⠀⠉⠻⣦⣼⣿⣤⣤⣤⣤⣀⢹⡇⠀⣀⣀⣤⣤⣀⣄⣉⣛⣿⡄⣼⡏⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⢀⣾⠋
⠀⢹⣧⠀⠀⠀⣼⣿⣿⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠉⠀⠀⠀⠈⠻⣿⣿⠿⠋⠉⠉⠉⠉⠉⠉⠙⠻⣿⣿⡇⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⣠⡿⠁
⠀⠀⣿⡇⢀⣼⠟⠉⠉⠉⠛⠻⠷⣦⣄⡀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢠⣿⠃⠀⠀⠀⣰⡿
⠀⠀⣿⣷⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⣾⠃⣀⣀⣀⣀⣀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⣼⡇⠀⠀⠀⢰⡿⠁
⠀⣴⡿⠋⠉⠛⠛⠿⢶⣤⣄⡀⠀⠀⠀⠀⠈⣿⡿⠟⠋⠉⠉⠉⠉⠻⣷⣼⣇⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⢻⣧⠀⣿⠀⠀⠀⠀⣾⡇
⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⢿⣷⣿⠀⠀⠀⢠⣿⠀⠀⠀⠀⢀
⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⢸⡏⠀⠀⠀⠀⣼
⠉⠉⠉⠙⠛⠻⢷⣤⡀⠀⠀⠀⠀⠀⢹⣿⣷⠿⠛⠛⠻⠿⠷⣦⣄⡀⠀⢸⡇⠀⠀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⠘⣷⡀⠀⣾⡇⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣼⣧⣴⠾⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣾⣿⡄⣿⡇⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⣼⡏⠀⢀⣀⣤⣠⣀⣀⣀⠀⠀⠀⠀⢻⡟⠁⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠙⢿⣿⡇⠀⠀⢸⡟
⠶⢶⣤⣄⡀⠀⠀⠀⠀⠀⠘⣿⣰⣿⡶⠟⠛⠉⠉⠉⠉⠉⠛⠛⠷⣦⣄⣸⡧⠀⠀⢀⣠⣴⡶⠟⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⠿⣾⣿⣆⠀⠀⣼⡇
⠀⠀⠈⠙⠻⢷⣤⡀⠀⠀⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣯⣴⡾⠛⠋⠁⠀⣀⣤⣤⣶⠶⡶⣶⣤⣤⣄⣀⡀⠀⠈⠙⢿⣦⠀⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠀⠀⣠⣶⠖⢦⣄⠀⠀⠉⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⠀⣞⡿⢳⡾⠟⢃⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⣧⡀⠐⠛⢳⡶⠛⡣⠈⠆⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠈⢿⣿⣿⣯⣶⣀⣿⣺⣷⣿⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣄⠙⢿⣿⣿⣿⣿⣿⠿⠋⠁⣀⣀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣿⣿⢿⣿⣿⡶⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠾⠿⣿⣶⢿⣿⣿⡿⣿⣿⣷⣦⣶⡿⠿⠿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣰⣿⣿⣿⣴⣿⣯⡿⢷⣶⣶⣶⣤⡀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⢀⣿⠋⢿⡍⠉⠿⡥⠗⠀⠀⠀⠈⠙⣷⡀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠟⠋⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⣴⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⣰⠏⣠⡟⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⣰⠏⣴⠏⣠⢖⣫⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⡟⣴⣫⣾⣵⣯⣷⣻⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣾⣿⣿⡿⠿⢛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣾⠫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⢶⣶⣾⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠤⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠒⠄⠀⣀⠀⠀⠀⣀⣠⣤⡿⠿⠶⠖⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⢿⡿⢿⣟⡻⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠛⠉⢤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⠯⢄⣀⠀⠀⠀⢿⣯⣿⠏⡙⢛⡘⣻⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⡒⠒⠀⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠧⠉⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⢯⠲⠽⠿⣿⡿⢍⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⡍⠈⠂⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣶⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡍⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠈⠲⠀⠀⠁⠀⠄⠀⠀⠀⠀⢾⣷⢽⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢄⣼⣿⣿⣶⠀⠀⠀⠀⠑⢦⢘⠃⡴⠊
⠀⠀⠠⠀⠀⠀⢢⡔⠀⠄⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠈⠘⠀⠀⠀⠒⠒⠒⣷⣾⠒⠒⠒
⠀⠀⠀⠀⢀⠄⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣴⣖⢚⠕⢬⡓⣲⣦⠀⠀⠀⠀⡴⠋⢰⠄⠙⢆
⠀⠀⠀⠐⠁⠀⠀⠄⠀⠈⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠈⢹⠷⠾⠋⠁⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⣸⣀⠈⠀⠀⠀⠀⠀⠀⢸⣧⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⠀⢀⣤⠉⢡⢤⡀⠀⠀⠀⠀⠐⠄⠀⢺⠂⠀⠔⠀⠀⠀⠀⠀⢸⡇⢺⡄⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠈⠑⢄
⠀⠀⢀⠴⣿⣼⣀⣸⣼⡷⢄⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠀⠀⠀⠀⠀⣼⢸⡇⡇⠀⠀⠀⠀⠀⡇⠀⢠⣟⣟⣆⠀⠘⡆
⠀⡄⠘⠛⠛⢯⣾⣮⠛⠛⠛⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⢟⣋⣜⢿⣤⣤⣤⠀⠀⢧⠀⠈⠿⠿⠏⠀⣰⠁
⠀⢠⠸⣯⣿⣷⠭⢵⣿⣯⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⣩⣕⢚⣞⣼⢃⣫⣍⣹⠀⠀⠈⠓⢤⣀⣀⡠⠖⠁
⠀⠀⠡⡈⠻⣼⠀⢸⡼⠋⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢾⡷⠛⠉⠉⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠐⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠈⠀⠘
⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣀⠀⡀⠀⡀⠀⠀⠂⠘⠀⠐⠂⠐
⠀⢀⣠⣉⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⣤⠁⣤⠀⠀⠀⠀⠂⠰⠀⠐⠀⠀⠀⠐⠀⠰
⠀⠸⠤⠈⠧⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠐⠤⠂⠄⠀⠄⠀⠐⠀⠰⠆⠐⠀⠀⠄⠠⠀⠤⡄

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

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠁⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠉⠉⠛⢦
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⠀⠀⠙⢿⣶⣤⣄⣠⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠛⠻⢿⣶⣄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠘⠿⣷⣶⣤⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⠀⠀⢀⣀⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠈⠁⠀⠁⠀⠀⠈⠉⠀⣀⣿⣿⣿⣿⡿⠋⠁⠈⢮⠩⠁⠀⠃⠉⠙⠗⠒⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠀⠀⣀⠁⢀⣁⣀⣤⣤⠤⠤
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣯⣽⣠⣖⣀⠀⠀⠀⠀⠀⡄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠳⢄⢠⣄⡀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠉⠙⢷⡄⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢰⢦⣄⢒⠟⠁⠀⡐⠋⢤⣭⡓⣦⡄⠀⠐⠀⠂⢀⣀⠀⢀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⣸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⢯⠀⣤⣊⠀⢀⡴⠛⠛⢿⡆⡀⠀⠀⠀⠚⣿⠾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢃⣿⣿
⠀⠀⠀⠀⠀⠀⠠⠲⠻⠁⠀⠙⢵⣿⣧⣎⡀⠀⠠⠊⠀⠀⠀⠀⡀⠌⢱⡄⠀⠉⢓⠤⣀⠀⠀⠀⢀⣼⣿⣷⣿⡿⠁⣠⡆
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢻⡼⣽⠛⠂⠀⠀⠀⠀⠀⠈⠂⣀⢹⣶⣦⣤⣽⣮⣽⣶⣦⡟⠛⣿⣿⠟⡑⣨⣻⠟
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠐⣀⣠⣴⡿⣿⣿⣛⡏⡁⠘⠁⠁⠉⠁⠁⠈⠐⠒⣠⣴⣿⡯⠊⣬
⠀⠠⡀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣿⣶⣿⣿⣿⣿⠿⡿⣶⠤⠂⠒⠒⠺⠟⠛⠛⢯⠵⠛⠠⡚⠁
⠀⣴⣄⣄⠢⠀⠀⠀⠈⠁⠀⠀⠉⠉⠸⠿⣿⣿⡿⠿⠊⢉⠀⠉⠹⡦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡦⠖⠁⡀⢀⡔⢠
⠀⢈⠻⣿⣶⠤⠀⠤⣀⠀⠀⠀⠀⠀⠐⠈⠉⠈⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠁⣠⠆⣪⡀⡊⠄
⠀⠀⠁⠉⣭⣷⣶⣄⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠛⠉⠀⢀⣠⣶⣦⠀⢿⡛⠀⠙⠒
⠀⠀⠀⠀⠨⠙⠛⠿⢿⣇⣄⣴⣏⣿⡇⠀⢈⣦⠀⣀⠀⠀⠀⠀⢒⡂⢠⣴⣾⣷⠶⣿⣷⡤⣿⣿⣿⣿⠗⠀⣝⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠥⠀⠀⠛⠁⢲⣿⡝⠻⢿⡖⠿⡿⠿⢿⣿⠿⠻⢿⣿⠿⠉⠉⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠀⠐⠒⠒⠒⠂⠐⠒⠐⠂⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠋⡡⠔⠋⠁⠈⣹⠀⠀⠀⠀⠒⠒⠀⢠⡞⠁⠈⠁⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢟⠆⢀⠔⠋⠑⠀⣀⠤⠊⠀⠀⠀⠀⣀⡀⠀⠀⠀⠧⢄⠢⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⡤⠚⠰⡧⠒⠒⢫⠄⢢⠀⠀⠀⢠⠖⠉⠈⠉⠹⣦⠞⠉⠚⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢠⠤⠜⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠰⣃⠀⣞⠧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠅⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣠⣘⡛⣋⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠙⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠱⢼⠇⢀⡀⢀⣤⡀⠀⠀⠀⢀⣄⡀⠀⠀⠀⢰⠈⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⢀⠴⠊⠉⠁⠀⠑⢆⠀⢠⢮⠒⠚⠀⠀⠀⠘⠢⡀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⣜⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠍⠥⠴⠒⠠⡄⠀⠀⠨⠗⠊⠈⠣⠤⠊⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⡀
⠀⠀⢠⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⠀⠀⠀⠀⠀⠘⠒⠄⡀⡴⠶⠤⢀⣀⡀⢇⠀⠀⠀⠀⢰⠦⡄⠀⣄⠤⣄⠀⠀⠀⠰⠘⡇
⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡆⢇⠀⠀⠀⠀⠳⠴⠊⢓⠀⠈⠙⣄⠀⡼⠀⡇
⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢳⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⠆⠰⡄
⠀⠀⢠⠀⠷⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡑⢆⠀⢀⣠⠞⠁⠀⠀⠀⠀⠑⢂⡇⡆
⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⢄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡛⠥⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠇⠀⠈⢀⠀⠀⠀⠀⠀⠰⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⠃⠊⠀⠀⠀⠀⠀⢠⠃⡘
⠀⠀⠀⠸⡄⠀⠀⡧⠀⠀⠀⠀⠀⠐⠦⡙⠒⠂⠀⠒⠂⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠹⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎
⠀⠀⠀⠀⠀⠱⡌⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⠤⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢇⡘⠁⡌⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⣎⡤⠊⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢀⡠⠚⠀⠀⠀⠀⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠋⠉⠀⠀⠀⣀⠠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠘⠓⣶⣿⣿⣷⣼⣠⣤⡀⠀⠀⠀⠀⠀⠐⠒⠐⡆⡇⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⡴⠶⣿⣿⣿⣿⣷⣟⡴⢧⡀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣖⣠⠖⠚⠉⠀⠀⠀⠀⢀⣀⣨⠀⣰⠃⠀⠀⠀⢀⣀⡠⠤⠀⠒⠒⠒
⠀⠐⢁⣤⢾⣿⣿⣿⣿⣯⣀⠀⠙⣆⠀⣠⠋⠉⣿⣿⣿⠿⣟⠛⠑⢦⣀⠀⠀⡔⠊⠹⢋⣧⣦⣧⣀⣠⣔⣊⣁⡀
⠀⢀⡞⠁⢸⡟⣿⠟⠿⡇⠹⣇⠀⠀⠠⠋⠀⠀⠉⢩⠏⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⢰⣿⣿⣿⣿⠿⡍
⠀⡞⠀⠀⢸⠇⣿⠀⠀⡇⠀⠸⡆⠀⠀⠀⠰⠖⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⢻⠀⠘⢄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠋⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⢰⡄
⠀⠀⠀⠐⢀⠀⣀⣤⣄⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣠⣤⡄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⡇⠀⠀⣼⠑⡀
⠀⠀⠋⠲⡘⣴⣿⣿⣿⡇⡌⡰⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡥⠀⡇⠀⠀⡇⠀⣼
⠀⠀⠀⠀⣘⣷⣽⣿⣿⣥⣗⣁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢮⣿⣿⣿⡥⠞⠉⠉⠉⠀⠁⠀⠀⠀⠀⣞⠉⠦⣷⣾⣧⣧⠜⠉⣦
⠀⢀⡴⠋⢸⠙⢻⠟⠻⡟⢹⠉⠳⡀⠀⠀⠀⠀⠀⢀⡴⢒⣿⣿⣿⣿⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡞⣿⣿⣿⡟⣶⠞⠁
⠀⡌⠀⠀⢸⠀⠁⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢀⠖⠁⠀⡿⠈⠋⠉⠁⡇⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠻⣿⡿⠁⢸
⠀⠁⠀⠀⠈⡆⠀⠀⠀⠀⢨⠀⠀⠈⠆⠀⠈⠁⠀⠀⡞⠀⠀⠀⠀⠀⢡⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠱⣌⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠈⢧⠀⠀⠠⠙⣄⠀⠀⠈⠑⠲⠤⣄⣀⣀⣀⣀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠙⠒⠤⣄⣀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠘⣆⡠⢇⠀⠈⠳⢄⡀⠀⠀⠀⠀⣀⡠⠝⠋⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠈⠉⠓⠒⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠸⣆⠀⠣⣄⠀⠀⠉⢓⣴⡶⠋⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠉⠙⠒⠲⠤⣶⢭⣀⣀⣀⠀⠀⠀⠄⠘⡆⠀⣀⡭⡶⠊⠉⣸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⢆⠀⠀⠉⠉⠉⠙⣛⡷⠾⡿⢧⣤⣅⣀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⣀⡠⠖⠋⠑⢠⡀⡇⡠⠙⢦⡈⠉⡻⠲⠤⣄⣀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⣀⡴⠶⠯⣅⣀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠙⣶⠀⠀⠀⠀⠈⠉⠛⠲⠤⢤⣑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣘⣀⡤⠒⠋⠁⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⡇⢀⡴⠂⠉⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⡠⠴⠚⠉⠉⠉⠉⠙⠒⠒⠦⢄⣀⠀⠀⠀⠀⠑⣌⡧⠋⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠓⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠙⣾⡇⠀⣀⡤⠶⠒⠒⠒⠒⠒⠢⢜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⢸⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡏⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠖⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠦⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠿⠿⠛⢛⣿⣿⡛⠛⠿⠿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⣼⡿⢿⣧⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣃⣀⣀⣀⣀⣘⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠃⠙⠻⣿⣦⣀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⣀⣴⣿⠟⠋⠘⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠿⣷⣦⡀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⢀⣴⣾⠿⠋⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⣶⣄⠀⠀⠀⠀⠀⠀⣠⣶⡿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠉⠻⣿⣦⣀⣀⣴⣿⠟⠋⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⢀⣽⣿⣿⣯⡀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⣸⣿⠁⠀⢀⣠⣾⡿⠛⠁⠈⠛⢿⣷⣤⡀⠀⠈⣿⣇⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⢠⣿⠇⣠⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⠸⣿⡄⠀⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣾⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣀⠀⠀⣠⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⣷⠙⡏⢟⠡⢴⣾⠿⣁⣠⡜⠁⢀
⠀⠀⠀⠠⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣠⣿⠃⢬⡇⣍⠀⠈⠉⠋⣛⢏⣠⡖⠁
⠀⠀⠀⠀⠺⣿⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⡋⠐⠈⠀⢢⠀⠀⠀⢀⣤⣾⣳⡆⠀⠄
⠀⠀⠀⠀⢀⠙⢿⣮⠀⠀⠠⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣾⠋⠀⡽⠀⢠⠈⠀⠀⣀⣠⣼⣿⣿⡦
⠀⠀⠀⠀⠀⠐⠀⠹⣦⡴⣀⠘⠦⣄⠀⠀⠀⠀⡀⠠⣤⢤⠄⢀⣀⣤⠁⣼⣿⡤⠾⢁⡼⠁⠀⠀⠠⠴⠾⣿⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠈⣷⣦⡤⠀⠀⠀⢹⠋⠀⢾⣿⣷⣞⣯⣽⠗⢂⡞⠁⠀⠀⣤⡤⠲⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠼⠁⠀⠀⠀⢰⣜⠠⠀⠀⣹⣿⣷⣯⣴⣶⡟⠁⣰⡷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⢽⣶⣿⣿⣇⡀⠀⠠⠉⢏⢻⡍⠙⠳⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⣷⡤⡀⢈⢰⠖⢋⣀⣧⠀⠀⠀⢠⣤⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠿⠃⠀⢀⢰⣿⣿⣯⡀⠂⢀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⡄⢾⣿⣿⣭⣤⠂⠨⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⢀⠀⠉⠕⠢⡤⠀⠀⠈⠻⠟⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢁⠀⡂⠀⠂

⠀⠀⣀⠀⢠⡀
⠀⠀⢈⣉⡒⠬⣒⣄⡣⣄
⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠈⠹⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠈⠙⠿⣷⣿⢻⣭⣮⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⢀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣺⣙⣿⡿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣦⠔⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣬⣿⣿⣿⢻⡿⠟⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⢠⡸⣇⢸⢃⡄⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⢆⢄⠀⠀⠀⠀⠀⡷⢓⡛⢾⠀⠀⠀⠀⠀⡠⡰⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⣰⣚⣟⠔⠒⢄⠀⠐⡖⠊⠑⢲⠆⢀⡠⢒⠢⣻⡓⣎⢘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠩⡏⢀⣥⡀⠈⠒⠾⣲⣖⠷⠒⠁⢀⣮⡀⢩⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⢹⡧⢹⠷⣦⢤⣯⣽⡤⠔⣊⡉⢸⣯⢄⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠙⡂⢽⣄⠆⢺⠋⠀⠀⠁⠈⠁⢨⡾⢐⠋⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠠⠳⢳⠞⠈⠉⠁⠀⠀⠀⠀⠀⠀⠑⠿⠞⣞⠄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣑⡇⡄⠀⠀⡀⣄⣀⣀⣠⢀⡀⠀⢠⢸⢎⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⢹⡇⣿⣶⣿⡿⠛⠿⠿⠛⢿⣿⣶⣿⢸⡏⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⡀⢸⡇⣿⣿⣿⡁⣀⣀⣀⣀⢀⣿⣿⣿⢸⡎⢀⡎⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⡳⠿⣦⣇⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣸⣴⠿⢟⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠞⠡⢀⣄⣹⣃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣘⣧⣠⡀⠌⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⡀⣼⣿⣣⣿⣿⣙⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣼⣿⣇⢀⠨⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠎⠖⢠⣿⠋⣼⡽⢻⣿⣿⣞⣿⣕⢿⣿⣿⣽⣿⡟⣯⣷⡙⣿⡆⠲⠡⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠀⠀⠈⢿⣻⡜⡟⠈⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠁⢻⢳⣾⠿⠁⠀⠈⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⠀⠘⣼⡇⣜⣴⣆⠺⣿⣿⣿⣛⣿⣿⣧⣰⣦⢣⢸⢯⠃⠀⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡢⡀⠄⡀⢱⢻⡗⠿⣿⣇⢹⣿⣿⣿⣭⣾⣿⣿⠿⢻⡏⡎⢀⠰⢀⢄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢪⡐⣝⡄⢿⣷⠰⠟⠉⠀⢙⣉⠭⠄⡘⠉⠻⠆⣿⡿⢠⣦⢟⡕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⡿⠀⢸⣿⡄⠀⠀⠀⢀⠭⠓⠲⠃⠀⠀⢀⣿⠇⠈⠿⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠄⠀⠀⠀⡇⠃⠀⠀⠀⣨⣷⣿⣋⣡⣄⠀⠘⢸⠀⠀⠀⠢⠠⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠺⠡⠃⠀⠀⣸⣿⣿⣽⡿⣽⣿⠀⠘⠨⠓⠃⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣵⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠛⠂

⠀⠀⠀⠀⢸⣿⣿⣿⣦⣴⣿⣶⣤⡄⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣰⣿⠟⣠⣿⣿⣿⣿⢟⣫⣥⣬⣿⣿⣿⣿
⠀⠀⣴⣿⣷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠉⠉⢿⣿⠟⠛⠉⠛⢻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠛⢻⠋⠀⠈⠻⠿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣠⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⣰⣾⣿⣶⣶⣾⣿⡌⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⣷⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢰⢰⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣴⣶⡵⠂⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠻⣿⣿⣿⡧⣄⡠⣤⣷⣶⣃⡀⠀⠀⠀⠀⣠⣴⣿⣧⣾⡆⣿⣿⣆⢈⡁⠀⠀⠀⠀⣼⣷⣿⣶⣶⣿⣿⠇
⠀⠀⠀⠀⠈⠉⠁⠀⢳⣽⡛⠛⠛⢡⡳⣶⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⠟⠟⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠾⣿⡿⢿⣮⣛⣿⣿⣿⣿⡿⠋⠀⢹⡛⠿⠛⠛⠁⠉⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠧⠀⠀⠻⡄⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠉⠉⠁⢠⣷⣴⣄⠀⠹⡄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⣟⠛⠓⠀⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠉⠁⠀⠀⠀⠀⠼⣏⣿⠿⠖⣲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⢠⣤⡘⡛⢁⠀⠰⠜⠛⠁⢶⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣤⣶⣤⣀⣤⣤⡐⠿⠿⠃⣡⠠⠾⢷⣄⠀⠀⠘⣿⣿⣿⣿⣿⡇⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⠖⠀⣀⣿⣇⠀⠀⣻⣿⣿⣿⣿⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⣵⣾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⢁⣤⣈⠛⠁⠀⠀⢿⣿⠋⠉⠁⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⡼⠐⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⣨⣿⣷⣶⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣟⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠈⢙⡇
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣿⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⣁⡉⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠋⠉⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⣀⡀⠀⠀⠸⣿⠁⠀⢀⣀⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠁⠢⢴⡤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⣀⠀⠀⠀⠈⡶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠒⠀⠀⠛⡤⡆⠀⠀⢘⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⣈⢀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠳⢬⢀⣀⠰⠂⠀⠀⠀⠉⠡⠄⠀⠀⠀⠈⠁⠳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⡽⢛⣞⣐⢦⠌⢻⠶⣤⣴⡴⠀⢀⣀⢈⣤⠎⢳⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⣿⣎⡁⠀⣀⠀⣈⣼⡻⢟⣥⣤⣾⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⡤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣵⣿⡿⠚⣿⣯⣤⣼⣿⣿⣿⣿⣿⡁⠀⢄⣀⠀⠠⠀⠀⠀⠀⠀⠀⠋⠋⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢚⡟⠿⢿⡯⢶⣽⣿⡟⢻⠟⠠⠀⠰⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⣘⠁⣀⠂⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣆⡀⣄⡀⠲⠶⠓⠀⠀⠀⠀⠀⠀⠠⠀⠀⠜⠛⠟⡛⣦⠶⣿⡙⠁⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠟⠙⢲⠶⣆⡀⠀⡀⠀⠀⠺⢅⡀⠀⠈⠃⢼⣶⣶⣾⣿⣻⡂⠻⢶⠟⠉⠀⠠⠖⠀⠁⠀⠀⠀⠀⢀⡀
⣶⣾⣉⠁⠹⢦⣄⣼⣶⣤⣤⢠⣄⣵⠲⠖⠚⠋⠉⠛⠉⠉⣁⣠⡀⠀⠀⠀⠀⢀⠠⣦⣠⡄⠀⡿⣾⣿⣶⡖⡀⢀⡀⠀⡀⡀
⣫⣿⡿⡿⠳⠦⢘⣧⣟⠿⠿⠿⠖⣶⡶⠇⠀⠀⠀⠀⣀⢚⣩⣤⡀⠀⠀⠀⠀⠀⣤⣠⣿⣥⣴⣦⣶⡏⠙⢿⣇⡀⣐⡁⠚⡛⢦⠟⢿
⣟⠋⠁⢀⣤⣴⡀⣄⡀⠀⠴⠾⣲⠂⢀⠀⠀⠀⠀⠂⣘⣉⣁⠀⢀⠀⠀⢀⡀⢀⣈⣤⣄⣭⣠⣵⣶⣶⣦⣶⣭⡛⠺⠷⢿⣿⡿⢿⣷
⣬⣄⢠⣀⡄⢤⣤⣉⠽⠃⠀⠀⠉⠁⠁⠂⠀⠀⠀⠀⠲⠄⠘⠉⠉⡟⡿⣴⣿⣟⣩⣬⣵⣾⠿⠙⠙⠷⠶⠷⠈⠽⣴⣂⠸⣋⣥⣾⣛
⣿⣶⣿⣽⣏⣸⠛⢶⣤⡤⣀⠀⠀⠠⡀⠤⠀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠉⠛⠛⠻⠛⣛⠉⢀⠀⠀⣦⣀⡀⠠⣤⡀⠀⠀⣄⣄⣌⠙⠛
⠶⠙⠛⠻⣏⣛⣯⣉⣸⣤⣠⣠⣤⣤⣀⡀⢠⡂⠀⢂⣀⠚⠄⣐⠃⢨⡄⠀⠈⠀⠸⣦⣥⠴⠿⠂⣀⣈⣞⣳⣶⣍⣀⠄⣽⢿⠿⡂
⠀⠀⠀⠀⠉⠉⠀⠉⠙⠁⠈⠛⠋⠉⠋⠉⠛⠛⠒⠛⠛⠚⠚⠓⠚⠛⠉⠀⠀⠀⠁⠀⠭⢔⠄⠋⠉⠛⠉⠩⠙⠋⠹⠯⠛⠻⠿⠤⠚
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣵⣬⣴⣮⣼⣯⣾⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠠⠄⠀⠒⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠈⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⡀⠀⠈
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠒⠀⢠⡄⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠂⣀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠘⠁⠀⠀⢀⡀⠀⠀⠤⠀⠀⠀⠀⡀⠀⢠⣤⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠁
⠀⠀⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠔⠁⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣈⣤⣬⣷⣤⣶⢿⠞⠃⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠀⠀⢀⠔⣡⣴⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣡⠄⠀⠀⠀⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⢃⣼⣿⣿⣿⣿⣿⣿⡄⠀⠀⣠⢖⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⠃⠞⣻⣽⣿⣿⣿⣿⣿⡏⠀⢠⣵⣿⡿⣿⣿⣿⣿⣿⡿⢿⡿⠉⠙⠛⠻⡿⠇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢛⡙⢿⠐⠀⠐⠂⣽⠋⣀⣤⣠⣀⠑⢄⠀⠀⠀⠀⠉⠀⠀⠰⠀⠈
⠀⠀⠀⠀⠒⠀⠀⠀⠐⠘⠐⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠑⠌⣻⣶⣆⣨⣁⣿⣿⣿⣿⣿⣷⡈⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⢠⡀
⠀⠠⠄⠀⠀⠀⠀⠀⣠⣆⣾⡿⠿⠿⢟⣛⣿⣿⣿⣿⣿⡟⠉⠒⢲⣶⣥⣿⣿⡿⠛⣿⣿⣿⣿⣿⣷⠺⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠒⠉⠉⠙⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠆⠐⢄⠀⠀⠀⠉⠀⠀⠈⠃
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⣿⣿⣿⣿⣷⣄⣁⠂⡤⠔⠀⠀⠀⢠
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢻⣷⣌⠻⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠾⠆
⠀⠀⠒⠀⠀⠀⠀⠀⠠⠄⠀⠀⠙⢿⣿⣶⣭⣽⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢿⠟⢍⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⡀⠉⠀⠀⠉⠺⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠡⣿⠈⠣⡞⢸⠀⠀⠀⠰
⠀⠀⡀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡄⠀⠀⢠⡍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢛⡻⢆⠀⠀⢘⡇⠀⠀⠀⠈⠁
⠀⠀⠁⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠤⠊⠁⠈⢂⠀⠀⠛
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠎⠀⠀⠀⠀⠀⠡⡀⠀⠀⠢⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠈⢢⣦⢀⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⣿⣷⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡾⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣾⣿⣟⣿⣟⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀



⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⣠⢾⣧
⠉⠉⠳⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠉⠊⠁⠄⠸⣇⡀⢀⡀⠁
⠀⠀⠀⠈⠓⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣳⣿⠀⠀⠀⡐⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠭⠅⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡣⠂⢁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠓⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢠⢀⡀⢄⡀⠠⠐⠀⠀⢀⡺⠃⡀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠶⡀⠀⡨⢐⣺⡿⠫⠰⡄⢈⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⠲⢶⣻⢷⣸⠀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⢠⣰⡤⠂⠀⠠⠤⠀⠘⠄⢀⣀⡐⠐⠅⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠠⠀⡆⠀⠀⠈⢀⣀⠀⢀⣀⣀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠀⠐⠲⠠⢑⠦⠀⠉⢁⠀⠂⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢳⣆⣎⠓⠀⣠⠶⠀⠀⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠛⠆⠈⠉⠙⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢚⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠐⠒⠀⠐⠒⠀⠉⠀⠀⠘⠀⠈⠑⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢃⠀⠀⠀⠀⠀⢀⠠⠐⡗⢄⠀⠀⠀⠀⠀⠉⠉⢀⠜⢃
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⡠⠊⢣⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠒⠤⠄⠀⢀⡠⠂⠀⠈⢄
⠀⠀⠀⠶⢄⡉⠀⠀⠀⢀⡠⠂⠁⠀⠀⢣⣀⣀⡀⠠⠔⢺⠃⢄⠀⠀⠀⠀⢀⠊⠘⡀⠀⠀⠈⠢⡀
⠀⠀⠀⢘⠀⠈⢲⠆⡈⠀⠀⠀⠀⢀⠔⠁⢣⠀⠀⠀⠀⣸⡀⠀⠈⠁⢀⠜⠇⠀⠀⠐⢄⠀⠀⠀⠈⠑⠤⣀
⠀⠀⠀⡐⠀⠀⢨⠀⠀⠑⠤⢔⠂⠁⠀⠀⡰⢣⠀⠈⠁⠀⠈⠒⠂⣤⠂⠀⠈⢄⠀⠀⠀⠑⠤⣀⠀⣀⠠⠄⠸⠃
⠀⠀⢠⠁⠀⠀⡈⠀⠀⠀⠀⠀⠁⠲⢖⠉⠀⢀⠷⠀⠒⡇⠤⣠⠊⠀⠣⡀⠀⠀⠑⣢⡤⠄⠒⢈⠋⠀⠀⢠⠁
⠀⡠⠁⠀⠀⠔⠀⠀⠀⢀⠃⠀⠀⡸⠀⠈⡟⢥⡤⠳⢨⢁⠶⣁⠑⣄⠠⠘⠗⠈⠀⠰⠀⠀⠀⠈⠀⠀⠀⠈
⠺⡐⠂⠐⢪⠀⠀⠂⠰⡃⠀⠀⠰⡅⠀⠰⡄⠴⠤⢱⣷⡧⢂⣙⡁⢀⠀⠀⡄⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠈⠄⠀⠀⠱⡀⠀⠀⠘⡄⠀⠀⢰⠀⠀⣅⠔⠢⣡⠙⠉⣠⡎⢩⠚⠓⠂⡴⠄⢀⡀⢣⠀⠀⠈⢆⠀⠀⠀⢃
⠀⠀⠈⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⣔⠈⠀⠑⢦⠥⠌⣇⠤⠓⢆⠀⢠⠊⠀⠀⠀⡠⠘⠉⠐⠂⠀⣧⣄⣀⠀⠣⡀

⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡦⠐⠁⠀⠀⠑⢆⠠⠁⠀⠀⠀⠀⢀⣀⡱⣄⠀⠀⢀⠊⠀⠀⠀⢀⠔⠉⠀⠀⠀⢈⠡⠛
⠀⠀⠀⡄⠀⢀⣃⠄⠊⠁⠀⠉⠐⢄⠀⠀⢨⠋⠈⠀⠐⢰⠊⠁⠀⠀⠈⠢⡀⠀⠀⠀⠀⠔⠁⠀⠀⠀⡠⠊
⠀⠀⢀⡁⠔⠊⠀⠁⠒⢄⡀⠀⠀⠀⠱⣀⠃⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠠⠐⢦⠀⠀⡌⠀⠀⠀⢠⠊
⠀⠀⠈⠐⠒⠠⢀⠀⠀⠀⠈⢢⠀⠀⢀⠋⠁⠀⠀⠈⠁⠢⡤⠊⠀⠀⠀⠀⠀⠀⠑⢄⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠡⢀⠂⠀⣀⣀⠀⠀⠀⠀⢀⠀⠀⢀⠠⠄⠐⠒⠀⠠⠷⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⢀⠋⠁⠀⠀⠀⠀⠉⠐⠤⣘⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠂⣀⠀⠀⠀⠀⣀⠀⠀⠀⠈⡀⠀⠀⡀⠄⠒⠈⠉⠀⠀⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢄⣃⠔⠈