Germans Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⢾⣿
⠀⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⣿⣿⣄⠀⠀⠀⣠⣶⢫⣿⣿
⠀⠀⣿⠟⠋⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢣⣿⣿⣿⠏⠻⠖⠺⠟⢿⡾⣿⣿⡄
⠀⠀⣿⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠹⡜⢿⣿⢷⡶⣠⣠⠀⢀⠙⠆⠻⡞
⠀⢀⡗⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠟⣿⣧⡙⠶⠼⢋⣚⢉⣵⣴⣿⡆⢰⣼⣿
⠀⢠⡿⠀⣿⣿⣧⡀⠀⠀⢀⣠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣶⣿⡏⠀⠀⢹⣯⠀⠛⠁⠈⠛⠺⣿⣿⣿⣷⣼⢷⣿
⠀⢸⣧⠀⢿⣿⣿⣿⣷⣿⣿⣧⡾⠟⠻⢿⣿⣿⠟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢉⣽⡿⠏⠀⠀⠀⠈⠿⠀⠘⢰⢷⡤⢠⡿⠞⣿⣿⣷⣦⢸⣦⡀
⠀⠀⣿⣦⡈⠛⠿⠿⣿⣿⣿⠋⠀⠀⡦⠈⢻⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣼⡟⠀⠀⠀⢀⣄⠀⢠⣷⡄⢀⠸⡏⣷⣦⣙⣿⣿⣿⣿⣿⣽⣿⣦
⠀⠀⠀⠙⠻⠷⠶⠶⠿⠻⣿⠀⠀⠀⢰⡇⠸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⢰⡇⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠹⠿⣽⣧⣷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⢨⣇⢠⡆⠘⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣿⡅⠀⠹⣧⠐⣄⠀⠘⠻⣧⣤⡀⠀⠘⣿⣿⣷⣮⣭⣝⣿⣯⣛⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠛⠀⣿⡄⠻⢿⣿⣿⣿⣿⣿⣿⠂⢸⣿⡏⠀⠀⠈⠀⣿⣧⡀⠀⢿⣿⣿⡀⠀⠈⢻⣟⠉⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⣘⠻⠆⠈⠻⠻⣿⣿⣿⣿⡅⢰⣿⣄⠀⠀⠀⠀⠈⢻⣧⠀⠈⣿⣿⡇⠀⢠⣿⡏⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⡿⡗⠀⠀⠀⠀⡿⠀⣀⡀⠀⠀⠀⠉⠹⣿⡦⢸⣿⣿⣇⠀⠀⠀⠀⢸⣿⠇⠀⠈⢻⠇⢀⠾⠫⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠙⠀⠀⠀⠀⠀⠁⢀⣿⣿⣷⣤⣄⡀⠀⠘⠓⠀⠈⣿⠀⠀⢠⣶⣀⡿⠋⠀⠀⠀⠈⣀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣤⣀⣀⣀⣴⣿⣿⣿⣿⠿⠋⠹⣦⡀⠀⠀⠘⡿⠀⣠⡼⠿⠿⢤⣶⣤⣀⣴⣾⡟⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠙⢻⠀⠉⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣼⡟⠀⠠⣤⣤⠦⠄⠀⠀⢀⣀⠀⠈⠙⠻⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠃⠀⣾⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣀⣀⢀⣀⣀⣀⣀⣀⣉⡙⣦⣄⠀⠈⠙⠳⠶⠦⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠁⢰⢾⢠⡏⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⡿⢯⣍⣉⠉⠉⠉⠉⢿⡿⣦⣀⠀⠰⠦⣔⠲⣽⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠷⢶⠆⢸⢦⣦⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠳⡈⠙⠿⣷⣦⡞⢿⣙⠧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⡟⠳⣄⠀⠀⠙⣄⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⣌⡁⢀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣷⡀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣀⢈⠑⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣠⢽⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣾⣇⢻⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣦⣄⡀⢠⠃⠀⠀⠀⠀⠀⢲⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⣿⣤⠀⠀⠀⠀⠀⠈⣧⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠛⠛⠷⠶⢦⣶⣴⣶⣿⠶⣷⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⣁⣶⣤⣤⣀⣤⣤⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢻⣿⠁⢀⣤⣤⡀⢠⣤⣤⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠉⠀⠉⠈⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣠⣦⣄⣀⣴⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠙⠛⠛⠋⠉⠛⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢶⣄⠀⠀⠀⣴⣻⢰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠙⠓⠒⠚⢋⡟⠈⠀⠀⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠁⠀⠀⠀⠀⠀⢴⣤⠤⠀⠀⢸⣦⠀⠀⠀⢰⣈⠉
⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⢲⣴⠀⠀⠀⢸⠃⠀⠀⠀⠀⣿⠟⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⡄
⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⡀⣼⠀⠀⠀⢰
⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢻⡇⠀⠀⠀⢸
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠘⣿⡇⠀⠀⣀⣀⣤⣤⠤⣤⣤⣤⣄⣀⡀⠿⠇⢨⡃⠀⠀⠀⢸⡀
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⢣⠇⠀⠀⣭⡝⣾⣿⡿⣿⣟⠉⢠⡙⢹⡿⢻⣿⡇⣿⡆⢸⡇⠀⠀⠀⠚⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠀⠀⠘⣿⡿⣿⣿⣷⣼⣿⣁⣠⣀⣸⣿⣷⡿⠧⣙⣥⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⣯⠀⠀⠀⠀⠀⢶⣶⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿⠀⡇⠀⠀⠀⠀⠘⡂
⠀⠀⠀⠀⢀⡀⢀⡴⠖⠛⠀⢿⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠲⢶⣗⡒⠀⢻
⠀⠀⠀⠀⡞⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢨⡶⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠃⡘
⠀⠀⠀⢸⠀⣿⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠛⠘⠒⢀⠀⢶⡶⢶⣶⣶⣶⡶⠂⠀⢀⡀⠐⠁
⠀⠀⠀⠨⠀⠸⣇⠛⢰⡗⣀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢹⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠒⠤⣤⣁⣈⠀⠉⠉⠒⠀⠀⠀⠀⠘⢳⣖⠁⠀⠀⠈⠀⠀⠀⣿⠸⠄⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠙⠒⠶⠶⢶⣶⠤⠤⢄⣈⣀⣀⣠⡔⢦⠤⠀⢻⡆⠀⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠀⠀⠀⠘⠛⠀⠀⠀⠀⠉⠉⠀⠀⠘⠒⠃⠘⢧⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠿⡿⠿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠉⢉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢿⡿⠷⣄⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⡴⠿⡿⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠋⢰⠏⠀⠀⠒⠲⣿⡶⠒⠀⠀⠈⢧⠈⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢠⡿⠀⠀⠠⣄⢰⠃⢣⢀⡴⠀⠀⠸⣧⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⣿⠃⠀⠀⠠⠇⠹⣇⡿⠁⠧⠀⠀⠀⢻⣇⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢠⡶⠾⡦⡀⠀⠹⠁⠀⡴⡾⠶⣦⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢀⡀⠙⠂⠔⡿⠁⠀⠀⠹⡕⠄⠚⠁⣀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢳⠀⡏⠙⡆⠀⣰⣇⠔⠉⠑⢄⣷⠀⠀⡞⠉⡇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⢧⡇⠀⢱⣾⡿⠏⠀⠀⠀⠈⠿⣿⢶⠁⠀⣧⣯⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡸⠀⢧⠀⠀⣿⠁⣔⣁⠀⢀⣰⡄⢹⡇⠀⢠⠏⠸⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⠁⠀⠀⠀⠀⡏⠀⠩⠴⠃⠳⠬⠀⠘⡇⠀⠀⠀⠀⠉⠉⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡙⠧⣀⣀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⢀⣀⡠⠟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠉⠂⠀⠀⢠⡀⠛⠶⣯⣥⣀⣠⣭⡷⠞⠃⣠⠀⠀⠀⠊⢁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣩⣤⠤⠤⣤⣼⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣤⡤⠤⠤⣬⡹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣡⠤⠀⠀⠀⠀⣸⠇⠰⠊⣲⠀⢰⠊⠰⠀⢿⡀⠀⠀⠀⠠⢤⡁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⠈⠹⢠⠀⠐⠛⠶⣄⠈⢇⠀⢀⠋⢀⡴⠞⠓⠀⢠⠸⠉⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣵⠏⠰⠤⣀⠀⠈⠳⡀⠀⠀⡰⠋⠀⢀⡠⠴⠈⢷⣀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢶⣤⡉⠢⡀⠑⠀⠐⠁⡠⠊⣡⣴⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡲⠚⠛⠉⠉⠉⠉⠒⠒⠤⢤⣀⠀⠀⣀⡠⠴⠒⠊⠉⠉⡩⡍⠉⠉⠑⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠂⠉⠰⣢⠀⠀⠀⠀⠵⠀⠀⠀⠈⠐⠪⠙⢿⣁⠀⠀⠐⢖⠄⠀⠈⠀⠀⠀⣴⡀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⢀⠔⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢶⠄⠀⠈⠳⡔⠄⠀⠀⠀⠐⡂⠀⠀⠈⠀⠀⢠⢄⠀⠀⣀⡙⢦⡀
⠀⠀⠀⡰⠃⠀⠀⠉⠁⠀⠾⠆⠀⠄⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠀⠀⢰⠆⠀⠀⠀⠐⠗⠀⠈⠋⠀⠀⠛⠁⠀⠱⡄
⠀⠀⡜⠁⠀⠀⠀⠰⡤⠀⠀⠀⠈⠋⣀⣤⠤⠭⣅⡂⠀⠀⠀⠀⠐⠱⠀⠀⡀⠈⠌⢷⣀⡠⠤⠴⠤⢭⣐⠀⠀⠀⡤⠀⠀⠀⡀⠀⠀⠹⣆
⠀⡼⢡⠁⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠉⠢⡀⠰⢦⠀⠀⠀⠘⠚⠀⠀⠈⢧⠀⠀⠀⠀⠀⠈⠑⣄⠀⠉⠀⠀⠘⠂⠀⣠⡄⢹
⢠⠇⠆⠀⡴⠀⠀⢲⠄⢀⢢⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢆⠀⠀⠀⠀⣤⡀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⣆⢢⠀⠀⠀⢠⡄⠈
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠘⣆⠐⠀⠀⠋⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⡈⡆⠀⠀⠘⠁⠀⠀⡇
⡏⠀⠀⠀⠈⠁⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢦⡀⡀⠀⠀⠠⣖⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠈⠇⠀⠀⠀⡄
⡇⠀⢀⣀⠀⠀⠀⠘⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢰⣆⠀⠀⠃
⡇⠀⠈⠊⠀⠀⠶⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡼
⢿⠀⠀⠀⠀⢀⣀⠀⠀⢀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡀⠀⠛⠆⠀⠀⢀⡀⠀⠈⠑⠢⣄⡀⠀⠀⠀⠀⣰⢣⠀⠟⠀⠀⠀⠀⠀⠀⢠
⠸⡆⠀⠀⢀⠀⠊⠀⠀⠈⢆⢳⡀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠑⢄⠢⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠉⠲⢄⠀⣰⠏⠠⠀⠀⠀⠸⠃⠀⢀⠃⣼
⠀⢳⠰⡀⠈⢆⠀⠈⠛⠀⢀⠀⠳⡄⠀⣀⠴⠊⠁⠀⠀⠲⠄⠀⠑⢬⡀⠀⠀⣴⠀⠀⠀⠪⠇⠀⠀⠐⠙⠣⣀⠀⠀⠀⠀⠀⠀⢀⠎⢰⠇
⠀⠈⢧⠱⡀⠀⠁⠀⠀⠀⠀⠀⣠⠿⠊⠁⠀⠀⠘⠇⠀⠀⠀⠀⠀⣀⡽⠒⠤⣉⠀⠀⢀⣄⠀⢶⠄⣀⡄⠀⡈⠳⣄⠀⠀⠠⠁⠈⢠⠏
⠀⠀⠈⢣⡀⠈⠂⠀⠀⠀⣠⠊⠁⢀⣄⠀⢀⡀⠀⠀⠐⠀⣈⠔⠊⠁⠀⠀⠀⠈⠑⠦⣌⠙⠀⠀⠀⠀⢀⣀⠈⠀⠪⡳⡀⠀⠀⣠⠏
⠀⠀⠀⠀⠹⣄⠀⠀⠀⡼⡡⠀⠀⠈⠁⠀⠈⠁⠀⣠⠴⠯⢤⣄⣀⣀⣀⣀⣤⠤⠤⠒⠋⠑⢦⡀⢄⠀⠈⠛⠁⠀⠀⠐⠙⣄⡴⠋
⠀⠀⠀⠀⠀⠈⠳⣄⣸⢡⠃⠀⠰⠀⠀⡠⢀⡴⠋⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠝⠢⣅⠀⠀⠀⢀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠨⡃⠀⠀⠀⢊⡔⠉⠈⠀⠀⢨⡆⠀⠀⠀⢠⡆⠀⠀⠀⠀⠛⠂⠀⠀⣀⠀⠈⠓⢄⡈⠂⠁⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⢀⣴⡋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠓⠃⠀⠀⢀⡽⠶⠤⣤⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠉⠓⠦⢅⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⣀⣠⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⠀⠹⣿⣌⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⣠⡿⠋⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠰⠑⠀⠙⢾⣿⡈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢁⣴⣟⡁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣷⣿⣑⠀⠀⠀⢿⣷⢼⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢁⣼⣯⠍⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣦⡑⡄⠀⠘⣿⣈⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣼⡿⠁⠀⠀⢀⣠⣾⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢆⣿⣿⡝⣷⡈⠢⠀⣿⣿⣿⣿⡟⠛⠶⣤⣤⣴⣶⠶⣶⣤⣤⣤⣤⡴⠿⣿⣿⣷⣿⡇⠀⢀⠔⣩⣾⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠹⠏⢻⡎⢿⣦⡰⣿⠟⠀⠈⠁⠸⣷⡄⠀⠀⢸⣴⠀⠀⠀⢠⣶⠀⠀⠛⠛⣿⣿⡃⠐⣡⡾⢋⣾⣿⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣿⡇⢸⣏⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠹⣷⣴⣾⣿⣴⣤⡾⠛⠁⠀⠀⠀⠀⠘⠛⢡⣾⠏⣴⠏⠸⠛⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣈⠙⢷⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣇⣿⠀⣼⣷⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣾⣿⣿⠛⠀⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠈⡇⢰⠃⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢻⣿⣧⣴⠞⠋⢉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡀⢀⠀⠁⠈⠠⠀⠀⣿⣿⣿⣧⣤⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⢰⣿⣿⢿⣿⣿⣿⣿⣷⣾⠀⠀⠀⠀⣧⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⠟⠀⠀⣀⣀⣴⣿⣿⢿⣿⣽⣿⣿⣿⣿⡏⠀⠀⠀⠀⢻⣿⣿⣿⣽⣯⢟⣿⣿⣧⠀⠀⠀⠀⢹⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⣠⣾⣿⣿⣿⣿⡄⠀⠙⠿⡿⢿⣿⠟⠀⠀⠀⠀⠀⠈⢻⣿⡿⡿⡿⠟⠉⢻⣿⣷⣤⣦⡀⠀⠻⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⢀⠞⠉⠀⢙⠁⠉⠹⣿⣦⣀⡔⢁⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠈⠀⠀⣠⣿⠟⠛⢻⠿⢿⣄⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡂⠀⠀⠀⠙⣿⣿⣧⣾⡿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢿⣇⢸⣾⣿⡟⠀⠀⠀⠀⠀⠉⢧
⠀⠀⠀⠀⠀⠀⢀⠥⢐⠀⠀⠀⠀⠀⣀⡿⠂⣀⠀⠀⠀⢈⣿⣿⣿⡇⠀⢀⠔⠀⠈⠃⠀⠀⠄⠀⠘⣿⣿⣿⠃⠀⠀⠀⠈⣶⡄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⡠⠀⡴⠊⢁⣴⣾⡏⠀⣀⣠⣾⣿⣿⣿⣇⡔⢁⣴⣶⣶⣶⣶⣶⣦⡐⢄⣿⣿⣷⣇⠀⠀⠀⣀⠻⣅⠀⠀⠀⠀⠀⠀⡑
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠈⣠⣶⡿⠿⣿⣿⡯⣴⣿⣿⣿⣿⣿⠁⢛⡿⣫⣭⣭⣭⣭⣭⣝⢿⡏⠹⣿⣿⣿⣷⢀⣀⣿⣷⡈⠱⣄⠀⠀⠀⠀⢈⠂
⠀⠀⠀⠀⡠⠖⡈⢀⣤⠀⠤⠛⢉⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⣷⣿⡿⣿⣿⣿⡿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⡻⣿⣦⡈⠢⠙⠢⠀⣄
⠀⠀⢀⠠⠂⠁⠔⣩⠤⡠⠚⢉⡤⠐⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⡗⠂⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠠⣿⣿⣿⡿⠛⠿⣿⡗⠢⠉⡻⢦⡁⠀⠑⠈⠁
⠀⠀⠀⠀⠀⡰⠋⡠⠂⡠⠊⠀⡀⠀⠀⠠⠀⠀⠀⢿⣿⣿⣿⣿⡥⠀⠀⠀⠙⠻⣿⣿⡿⠿⠋⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠈⠁⠠⠈⠐⠀⠈⠘⢆
⠀⠀⠀⣀⠎⠀⠌⢀⠜⢠⠊⠔⣠⢊⠄⡀⡤⠠⠀⠈⢿⣿⣿⣿⣞⡀⠀⠀⣀⣠⣶⣤⣶⣄⠀⠀⠀⣺⣿⣿⠟⠀⠀⠀⢄⠀⠀⠀⠀⠐⢄⠀⠢⣈⠃
⠀⠀⠄⡉⢀⠆⠀⠎⢠⠃⡌⡰⢁⠎⠐⠁⠀⠁⠀⠀⠀⢻⣿⣿⣿⡙⣏⠙⠛⠋⠉⠈⠉⠉⢙⣿⣿⣿⣿⠋⠀⠠⡐⠊⠢⠉⠢⠀⠀⡀⠀⠱⣄⠀⠧⡀
⠀⠀⡰⠀⠌⠀⡌⠀⠆⡸⢀⠃⡜⢀⠀⡀⠐⢀⠀⡀⠀⠀⠹⣿⣿⣿⣿⡢⢄⣀⠀⠀⣀⠤⣺⣿⣿⣿⠃⠀⠀⠁⢌⠀⠀⠀⠂⠈⠀⠀⠀⠀⢌⠠⠈⢄
⠀⢠⠁⡘⠀⠰⠁⡘⠀⠃⠘⠀⠃⠘⠀⠁⠃⠘⢀⠀⠀⠀⠀⠹⣿⣿⣌⠻⢷⣶⣿⣷⣶⡾⢋⣿⣿⠃⠀⠀⢀⠀⢀⠀⠄⠀⠠⡈⠢⡈⢀⠐⢄⢣⡘⠄⠢
⠀⠁⡀⠀⠀⡔⢀⠃⠸⠀⡇⢰⠀⠇⠈⠸⠀⡆⠀⠀⠃⠆⠀⠀⠙⠿⣿⣗⣒⣒⣛⣒⣒⣾⡿⠟⠁⠀⠀⠑⢄⠀⠀⠀⢄⠀⡄⠐⡄⠐⠄⠑⠄⠱⡍⠢
⠀⠌⠀⠀⢀⠁⡸⠀⠀⠀⠃⡸⠀⠀⠀⡀⢠⠀⡆⠘⢸⠀⠀⠈⠰⠀⡀⠉⠉⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡀⠣⡈⢦⠘⢄⠐⡀⠄⢂⠘⡄
⠀⠀⠀⢀⠊⠠⢁⠀⡄⠀⠀⡇⠀⠸⠀⡇⠀⠘⠀⡃⠘⠀⠀⠀⠀⠀⢀⠘⠀⠀⡀⠀⠰⠀⠀⠀⠀⠈⠢⡀⠙⢢⠐⠄⠀⠀⠱⡀⢣⠈⢆⠐⡀⠂⠀⡀⠄
⠀⠀⠀⠈⠀⠃⠸⠀⡇⢠⠀⡇⠀⠀⠀⠃⠀⡇⡀⢁⠀⠀⠀⠃⢠⠀⠌⠀⠀⠀⠃⡀⠀⠙⡄⠐⠰⠁⠀⠀⢌⠀⠁⠈⠄⠀⡀⠱⡀⢣⠈⠀⠰⡀⠂⠙⠻⡄
⠀⠀⠊⠀⠀⠀⠀⠀⠀⡆⠀⠃⠀⠂⠈⠀⠀⠃⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠈⠂⠀⠀⠀⠈⠆⠠⠀⠀⠀⠁⠀⢡⠀⠀⠠⠀⠃⠘⡀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠲⣤⣀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣓⣢⣄⣀⣀⣀⡀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢛⣽⣿⠛⠋⣉⠉⢉⡻⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢵⣾⡿⢿⡇⠀⠘⠋⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣊⣿⣤⣄⣤⣾⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠱⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠄⠙⢿⣿⣿⣿⣿⡿⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⣿⡟⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣇⠀⠀⠀⠈⠉⣛⠇⠤⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⢸⣿⡧⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠁⠉⠙⠳⣶⣦⣄⢸⠈⠁⠀⠀⠀⠙⣧⡀⠀⠀⠄⠀⠀⠀⠀⠀⠔⠁⠈⣯⣿⣷⡟
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣻⣏⣠
⠀⠀⠀⠀⠀⠀⣼⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⣀⣠⠍⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⢻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠰⠀⠀⠀⠀⠀⠀⠜⠉⠁⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⣦⣄⣀⠀⠀⠀⠹⡀⠀⠹⡄⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠙⢻⡯⠉⠋⠀⠀⠘⠈⢓⣙⡷⣛⣉⡀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠛⣦⣨⢦⣄⣠⣴⠋⠁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⣀⣀⣠⣾⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠘⣆⠀⠀⠀⠀⠀⢠⡟⢼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⣿⢀⠀⠀⠀⠀⢰⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⡷⠋⠀⠀⠀⠙⢿⣹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⠛⠀⠀⠀⠲⠆⠀⠀⢹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⣱⠄⣀⣶⣦⠀⢠⣾⣆⣘⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠻⣿⣿⠟⣀⠈⣿⣿⡿⠋⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣏⣈⣿⣿⣡⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⢟⣛⣛⣿⣿⠂⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠀⠀⠀⠀⠘⢿⣷⣿⣿⣿⣿⡟⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠰⣦⡸⣄⠀⠻⡇⠻⠏⠉⣅⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⡤⠀⠠⠞⢷⣼⣧⣄⣿⣤⣤⠼⠃⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡉⠛⠀⠀⠀⠀⠀⠈⠛⠀⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣇⣐⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⠻⢷⣾⣶⣧⣄⣠⡀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠉⠿⢿⣷⠀⠈⠀⠙⠛⠛⠛⠛⠲⠶⠄⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣄⠀⠈⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡠⠀⠀⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡟⠀⢀⡿⠄⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⢰⣶⠏⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠿⣄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⣿⣿⠏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⣷⣆⣠⣦⣿⣦⠀⠀⠀⠀⢿⠇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣬⡟⠙⠋⠉⠀⢻⣇⠀⠀⠀⠸⡆⠀⠀⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⠆⠀⠠⡄⠀⠀⠀⠀⣠⣆⣴⠟⠋⣡⡀⣠⠔⠈⠙⢿⡀⠀⠀⠀⢳⠀⠀⠱⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢐⠊⠋⢀⣄⡀⣀⣶⣆⣀⠀⢻⡗⢸⣿⣿⣿⡴⠞⠛⠉⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⡇⠀⠀⠡⡀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⣀⣤⣤⡶⠻⢾⣿⠿⠛⠛⠿⠛⠉⠂⠀⢦⡀⠈⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠸⡄⠀⠀⢻⣀⠀⠀⠑
⢀⣠⣴⣾⣟⣿⣿⣃⠀⢀⡘⠋⠀⡀⠀⠀⠀⠀⡀⠀⠀⡵⢤⡷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠱⡀⠀⠀⠻⣧⠀⠀⠀⠢⢀
⠛⠿⢿⣟⣿⣹⣏⢀⣴⡯⢂⢀⣼⣀⢄⣄⣀⣤⡿⠦⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠢⡀⠀⠘⠷⣶⣆⠀⠀⠀⠀⠒
⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⡈⢳⣴⡀⠀⠀⠀⠉⠙⠛⠳⠾⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣄⣀⣤⣤⠬⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠉⠁⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⣮⣏⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⡄⠀⠶⣶⡀⠀⠻⠟⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠸⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣠⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡛⠉⢀⡠⠀⠀⢠⢇⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡴⠂⡠⣿⣾⣷⡄⠀⠔⣡⣾⠟⠉⠀⠀⠌⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣰⣿⠁⠉⠀⣼⡏⠀⠙⠁⣰⣿⠁⠀⢀⠔⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⣿⠇⠀⢀⣾⣿⡄⠀⠀⢸⣿⠃⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣼⣿⠀⠀⣾⣿⣿⠀⠀⢀⣿⠇⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⣿⡿⠀⠀⣿⣿⣿⠀⠀⣸⣿⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠀⡀⣿⡇⠀⢰⣿⣿⣿⠀⠀⣿⡟⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿⣼⡟⠀⣾⣿⣿⡟⠀⠀⣿⡇⠀⠀⠀⠀⠈⢑⣶⣤⠀⠀⡠⢀⣀⡀⠤⠤⠬⠷⣄
⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠈⠀⠀⠀⢹⣿⡇⠀⣿⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠈⢼⡟⡟⠐⠾⡵⣱⣿⡵⠶⠛⠛⠓⢦⢱
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠛⣿⠋⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠉⡜⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣯⠄⠀⠁⢸⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⠐⣿⣿⣿⣿⠀⣂⠀⢸⣿⣧⣄⣀⣀⡀⣀⣠⣿⣿⠀⡀⢀⣴⠍⡀⠈⣦⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠒⢾⣿⣿⡆⢰⠓⠉⠉⣿⣦⣿⣿⠿⠛⣻⣿⣿⣿⣟⠛⠻⢟⣭⡤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⡃⡟⠀⠀⠀⢻⣿⣿⣿⠀⠀⡉⢿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠸⣯⠥⠤⠀⠀⠀⢸⠛⣛⣒⣶⣶⣶⣶⡖⠒⣉⠄⠈⡄⠀⠀⣠⣶⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣏⣠⣤⣆⣀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣯⣿⠟⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠾⠿⠿⠿⠿⠟⠛⠋⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣦⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⣤⡀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⢀⡀⢀⣈⣙⢿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣶⣿⣿⣿⣶⣿⣿⣿⡿⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣿⣿⣿⡿⠿⣿⣿⡏⠀⠀⠐⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣇⣙⠸⣿⣷⣄⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣎⣡⣿⣿⣿⣷⣾⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⠠⠀⠐⠻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣷⣅⡱⣄⠀⠀⠉⠻⢿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⡰⠊⡙⢻⣿⣮⡛⢦⡄⣀⡼⡼⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠬⠄⢱⡀⠉⠨⢻⣿⣦⣠⠯⠳⡀⠈⠠⠙⢷⣤
⠀⠀⠀⠀⠀⣻⠐⡡⠴⠀⠀⠀⠀⡅⢰⠀⣉⢊⢻⣿⣄⢂⢑⡄⠄⠰⡜⢿⣆⢀
⠀⠀⠀⠀⠀⢈⣁⡄⠺⠊⠀⡛⠠⠄⢸⠃⠘⢆⢣⠙⣿⣦⢡⡬⠦⠢⠺⠞⢿⣆⡀
⠀⠀⠀⠀⠐⢩⣠⠄⣚⠉⠀⠉⠐⠀⢨⢁⣬⢉⣤⠠⠘⣿⣧⠠⠰⣦⣤⣤⣼⣿⣏⡀
⠀⠀⠀⠀⠀⣰⠞⠇⠉⠀⢈⡁⢓⢐⡒⢸⣿⣶⠔⠖⢣⠹⣿⣶⣿⣿⣿⣿⣿⣿⣿⠥⠀⠠⠤⠄⠤⠤⠄
⠀⠀⠀⠀⠀⠴⡆⣪⠤⠄⠰⠦⠤⣄⣅⣽⡧⣷⠔⢛⠘⡋⣿⣿⡿⠿⠿⢿⠟⠉⣿⠇⠀⠀⡉⠛
⠀⠀⠀⠀⠀⠰⡇⢀⠄⢓⠀⢓⡀⡇⣒⣿⣿⠤⡶⢾⠊⢉⢹⣿⡇⡶⢺⣺⠍⠘⠋⠸⠇⣦⣭⡀⢀⠀⣤⣰
⠀⠀⠀⠀⠀⠀⠸⣈⠆⠊⣁⠸⠂⢚⣨⢿⣃⣴⡓⢺⣋⣿⢽⣿⣷⣯⠽⠃⠀⠀⠀⠀⠰⠻⣿⣿⢿⠂⢹⡏
⠀⠀⠀⠀⠀⠀⠒⢡⡸⠎⠁⣤⠾⣁⣴⠞⠉⠀⠀⠀⠀⠘⠈⠉⠁⠀⠀⠀⠀⠀⠐⠂⠐⢰⣿⣿⠈⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⢇⠈⡆⠡⠶⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢰⡾⠜⣿⡄⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣆⣇⠈⠳⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣽⣧⢁⣀⣡⣾⣿⡇⠃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣬⣧⣜⠙⣆⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⡏⣽⣯⣽⢴⣿⡏⠀⠀⠙⢘⣻⣻⡻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣬⣡⣙⣷⣶⣶⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠟⠹⠆⠀⠀⠴⣟⣿⣿⣃⣉⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⡿⠿⠛⠿⠟⠩⣿⣿⡿⠿⠟⠋⠉⠉⠉⠁⠀⠐⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⡦⣾⡷⣶⠀⠀⢀⡼⠅⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠽⠄⠈⠁⠀⠋⠀⠀⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠐⡆⢀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⢀⡁⠈⠀⢈⡄⠀⢠⡄⠀⠀⠀⠀⠃⡀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⣄⠀⣠⡀⠀⠄⠀⠀⠈⣧⡀⠂⠈⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⠂⠀⠘⠃⠀⠀⠀⠀⠀⠀⢀⠒⠀⢀⠀⠀⠀⢀⢀⣀⣀⡀
⠀⠀⠀⠀⠉⢻⣉⢤⡀⢀⠀⠀⠀⠘⣿⡄⠀⠀⠏⠀⠀⠀⢄⡀⠀⠀⠀⠀⠿⠊⠀⠆⠠⠀⡀⠀⠀⢀⣀⠀⠀⢰⡎⢠⡴⢿⠿⠿⠿⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠸⠋⢸⢦⣤⠀⣻⣢⣀⣀⢈⡗⠀⠀⡄⢠⠀⠀⠀⢹⣀⡀⠀⠘⠀⠀⠀⠀⢦⡿⠁⠀⠀⠀⠀⢀⡠⢿⣿⡉⢴⢺⠲⠖⢲⠃⠘⠛⠋
⠀⠀⠀⠀⠀⡀⢼⢸⣿⢰⠟⠉⠉⣛⣿⠁⠀⠀⢌⠂⠠⠄⠀⠈⡈⢀⣄⠀⣰⠀⠈⠁⠀⠇⠀⠀⠀⠀⠀⠹⢷⣤⢸⡇⢺⡟⠃⠀⠸⠃⠀⠈⠁⠘⠆
⠀⠀⠀⠀⠾⠶⣮⠘⠛⢸⠋⢠⢆⣄⣿⠀⠀⢀⣅⡀⠀⠿⡇⠀⣿⣷⡟⠀⠏⢠⣤⣤⣼⠀⠀⠀⠀⠀⠀⣀⣗⢺⡼⠇⠀⠅⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⣰⠟⢸⠶⣾⡆⠯⡾⠀⠀⠀⠀⠁⠚⠂⠀⠲⣄⣙⡃⢀⢢⡤⢤⣭⣽⠀⠀⠀⠃⠙⣛⣥⣮⠀⢀⣤⡀⠂⠀⠀⠀⠀⠀⠀⢠⡼⠄
⠀⠀⠀⠀⠀⠀⠀⡏⠀⢨⡄⢰⡀⠀⠀⠀⠀⢀⠄⣴⠊⠿⠿⠋⠘⠉⠃⠀⠐⡀⠈⢐⢋⠀⠀⠘⠀⠀⣉⣉⣠⣶⡀⠠⠄⠀⠀⠀⠀⠀⠀⣾⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡛⡏⣱⣷⣶⠿⠂⠀⢀⣀⠀⠁⠋⠐⠀⣿⣧⠂⠀⠀⠒⠀⠈⠀⠀⠀⠀⡅⠀⠙⡟⢫⠀⠀⠀⣸⡇⠀⠀⠸⣇⣐⢻⡏⠤⣵⡄
⠀⠀⠀⠀⠀⣄⠀⣿⣿⣿⡧⠹⠭⠛⡟⠃⠀⠀⠀⠈⠀⠀⢙⡂⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠮⠷⠎⠀⡦⠆⠞⣥⠀⠼⠿⢻⠁⡈⣀⣀⠀⠁
⠀⠀⠀⠀⠀⣿⠀⣇⠀⢁⡗⠂⡐⠀⠛⠀⠀⠀⠀⠀⢸⣈⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⢴⠀⡦⣄⠉⠁⠀⠀⢶⠀⠀⠀⠌⠄⣰⠄
⠀⠀⠀⠀⠀⠙⠀⣿⢀⠇⣭⣴⠶⠀⠛⠃⠀⠀⠀⢤⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⠀⣆⠀⠀⢿⠈⢀⠀⠈⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠛⣎⢀⡇⣠⣀⠠⠶⠂⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⣧⠀⠀⠀⠁⠈⠈⠟⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠩⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤

⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠆
⠀⠀⠀⠰⠁⠈⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠈⠋⠘⠑⡀⠀⠀⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠈⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠔⣠⡎⢁⡁⠀⣁⢀⣀⣈⠦⠐⢻⣟
⠀⠀⠀⠀⢰⠆⠀⢠⠄⠁⠀⠀⠀⠀⡠⢚⣌⠀⠐⠼⢲⠼⡁⠤⠠⠀⠀⢤⠔⠉⠀⣴⢶⢺⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢢⡤⣢⠟⢰⣾⡎⢇⠲⠚⠩⠀⠀⣠⡦⠤⢠⡟⠀⠊⣀⡀⠀⠘⡿⣧⠀⠂
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣯⠜⠣⠀⠸⠛⠇⠠⢳⡀⠀⢠⣜⡁⣤⣴⣯⡴⠞⢋⣽⠇⢰⢤⣷⠏⠀⠀⠐
⠀⠀⠀⠀⠀⠃⠀⠀⠃⡀⠹⠁⠻⠀⡟⡀⠷⠘⢺⣟⣖⠡⠔⠊⢹⠟⠡⠰⠒⡋⠀⠀⠀⠈⢃⡀⠈⠀⣀⣴⡇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠀⢰⠿⠇⠟⢓⠒⠛⢁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⢥⣰⣾⡿⣿⣿⣦
⠀⠀⠀⠀⠀⡖⠀⠀⠦⠀⠰⡶⢾⠶⣲⡖⣞⢒⣶⡤⢤⠔⠒⠛⠙⠀⠀⠀⠀⠀⢀⠀⠁⠐⠆⠀⠙⢿⣿⣝⣿⣿⡅
⠀⠀⠀⠀⠀⠤⠀⠈⠀⡠⠈⡄⢀⡿⢧⠿⣇⠀⡁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠋⠈⠁⠈
⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⣇⣤⠦⠤⢤⣧⠤⡴⠀⠒⠀⠀⠀⠠⠒⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠰⡆⠀⠸⠄⠈⠀⠐⠖⢶⠀⠀⠀⡃⠀⠠⠀⢀⠀⣄⠀⠀⠀⠀⡆⠰⠀⠀⢸⠀⠀⠠
⠀⠀⠀⠀⠈⠈⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠐⠀⢤⣤⡄⠀⠀⠀⠀⠇⠸
⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠂⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡟⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⢀⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠆⣼⠃⡀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡐⠀⣸⣿⡿⣿⡀⠀⠀⠀⠀⠀⡸⢰⠃⢀⣷⠀⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡇⠀⣿⣿⡇⣿⡇⠀⠀⠀⠀⣸⠇⡎⠀⢸⣿⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣿⠀⠀⣿⣿⣇⢼⣿⠀⠀⠀⣰⣿⠀⡇⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡏⠀⠈⠛⢿⣿⠸⣿⡀⠀⣰⣿⣿⠀⣧⠀⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⡇⠀⠀⠀⠀⠛⠀⣿⣧⠸⢿⣿⣿⣈⡟⠿⣧⡈⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣿⣾⡇⠀⠀⠀⣠⣿⠟⠓⠀⠀⠈⠛⠋⠑⠒⢥⡈⠫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢀⣾⣿⣿⡄⠀⠀⣼⡟⠁⠀⠀⠀⡀⠀⡘⢶⣤⠀⠀⠙⢆⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢳⡀⠀⣼⣿⠟⠁⠀⠈⠛⠀⠀⠀⠐⠚⠉⠉⠉⠛⠻⣿⣶⣶⣄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣸⠇⠼⠟⠋⠀⢀⡤⠀⣠⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⠈⢿⡆⠉⠃⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣷⣤⡴⠀⢠⠟⢀⣾⠃⠀⠀⠀⠀⠀⣠⡿⠟⠋⢁⣄⠈⢿⡄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠉⠀⢰⡿⠿⠟⠀⢠⡟⢠⣿⣧⡀⠀⠀⠀⠀⠼⠋⢀⣀⣴⣿⣿⣇⠈⠳⡀⢀⡜⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⢿⠁⠀⠀⠀⣸⠇⣼⣿⣿⣿⣿⣿⡆⠀⢀⡔⣩⣤⢤⠹⣿⣿⡆⠀⠀⠀⢻⢱⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢃⡀⠀⠀⢀⡿⠀⠀⠀⢀⣿⠀⠀⣿⢿⣿⣿⡿⠶⠾⠯⣾⣙⣷⡼⠆⠈⢿⣇⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠋⠀⠀⠀⠘⣧⠀⠀⢠⣿⠋⠀⠀⠀⠈⠟⠁⠀⠀⠀⢀⠀⠉⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣃⡄⠀⠀⢀⡟⠁⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢦⣤⠄⠀⠀⠀⠀⣶⣤⡀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠶⠶⣶⣶⡿⠃⠀⠀⠀⢀⣼⣿⣿⣿⣦⣄⣀⣀⠻⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⡄⠒⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⢤⣾⣥⣤⣤⣄⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⣦⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⢀⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⣶⣦⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣝⠻⢿⣿⠛⣩⡉⢻⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⡀⢉⣴⠋
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⠹⣿⣿⣆⠀⠉⠙⢷⡉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠫⠛⠁
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⢻⣷⣽⣦⡀⠀⠀⠙⢧⣤⡰⢦⣬⣭⣿⣿⣟⣛⠉⠉
⠀⠀⣠⣾⣿⠿⠿⢻⣿⣿⠟⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣙⣀⠙⢷⣍⣻⣶⣦⣤⣤⡈⠿⣦⣬⠙⠿⣿⣿⢻⣧
⠀⣰⡟⠉⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠃⠀⠉⠉⠛⢿⣿⣧⠀⠀⠀⢹⣇⠀⠈⢿⡈⠿
⠀⡿⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠈⠛⠿⠷⣦⣤⠘⢿⠀⠀⠘⢁⠖
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢀⣤⣶⠶⠀⠀⠀⠀⣤⠀⠀⠀⠑⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢻⠻⣿⣿⣷⣤⣀⠀⠀⠈⠙⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡿⠋⠁⠀⠀⠀⢀⡆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣦⣄⡀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⣄⠀⣾⠃⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢿⡷⣿⠀⢀⣘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣷⣄⠀⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢤⡀⢸⡇⠀⠰⣾⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠘⣷⣼⠃⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀⡿⠀⠀⠀⢰⡟⠁⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡛⠂⠀⠀⠀⣸⠁⠀⠀⢠⡟⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠁⠀⠀⡰⠋⠀⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⠀⠀⠀⠀⠠⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠂⠁⠀⠀⠂⠁⢢⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣟⣾⣯⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⣤⣄⣤⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⢂⣏⣽⣷⣶
⠀⠀⠀⠀⢀⡄⣇⢑⣄⠀⠀⠀⠀⠀⠀⠀⡼⠿⠟⠻⣿⣿⣾⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣰⣄⣒⣿⣭⡿⠷⠟⠛⠛⠛⠛⠛
⠀⠀⣀⠀⠷⠯⠙⡀⠩⠐⢦⡉⠐⠄⡀⠈⠁⠀⠀⠀⠈⣛⢠⣠⡋⠘⢳⡀⠀⠀⠀⠀⠀⠀⠼⠉⠉⠉⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⣾⣯⣴⣮⡶⢀⡙⣌⢣⠐⣌⢦⡀⢀⠀⢀⠀⠀⠀⠀⠉⣀⠀⢉⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠈⠉⠉⠁⠉⠉⢻⡏⠈⠷⠛⠦⣬⡐⠁⠀⣸⡿⢏⡶⠀⠀⣿⢋⠀⠀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠐⠃
⢀⣀⣀⣀⣀⣀⣸⣇⣀⣩⣦⣴⣬⠿⢷⡦⡵⣖⠸⠒⠀⠀⣿⡘⠀⢰⣟⠀⠀⠀⢀⣼⡄⠀⠀⢀⣀⠀⢠⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠉⠉⠉⠉⠉⢹⡏⠉⡋⢨⣉⡛⠎⣿⣿⡷⢻⣷⣦⣄⡀⠀⠀⠀⠘⠉⠀⠀⢠⣿⣿⣿⡄⠀⠸⠈⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢸⡃⢸⣸⡄⠘⡇⠀⠀⣿⣤⣽⣿⣷⣯⠃⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠘⠀⢈⣻⡀⢰⡇⠀⡆⣹⣅⣿⣿⣿⡏⣿⠱⢀⣰⠀⠀⢠⣤⠀⢸⣦⠈⡇⠀⢀⣀⠀⠀⠀⢠⣤⠀⢰⡆⠀⠀⢸⣟⣿
⠈⠉⠉⠉⠉⠉⢻⡏⠉⣿⠛⠻⠿⣷⣵⣇⣿⠀⠀⠛⠀⠀⠸⠿⠀⢸⣧⠀⠃⠀⠉⠉⠀⠀⠀⠸⠬⠀⢘⡇⠀⠀⠘⠛⠋⠀⠀⢀⣠
⢀⣀⣀⣀⣀⣀⣸⣇⣀⡹⠀⠀⢀⠀⡿⢹⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣴⣿⣿
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⡙⠓⠒⠲⠯⢼⣼⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠸⠿⠀⠀⠀⢰⡆⠀⢸⣟⠀⠀⢸⣿⡏⠙⠉⠉⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡭⣿⢻⠀⠀⠀⠀⠀⢀⡴⠀⠀⠐⢢⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠸⠛⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⡿⢸⠀⠀⠀⠀⠀⠘⣇⠀⠀⠺⢻⡆⠘⣷⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣷⣯⢼⠂⠀⠀⠀⢶⠰⠿⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠀⠉⠀⠀⠀⢰⡆⠀⢹⡇⠀⠀⢀⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣾⠋⠉⠛⠉⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠈⠥⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⢹⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠂⠒⠂⠐⠀⠀⠀⠀⡃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣸⣿⣷⣄⣄⠀⠀⠀⠀⣠⣠⣾⣿⣇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⣿⠟⣿⣿⡿⠣⣤⣤⠜⢿⣿⣿⠻⣿⠆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⢏⣾⢟⡵⠃⠈⠃⠘⠁⠘⢮⡻⣷⡹⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣱⣿⠋⢈⠷⠀⠀⠀⠀⠀⠀⠾⡁⠙⣿⣌⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡿⡇⡴⠁⠀⠀⢰⡆⢰⡆⠀⠀⠈⢦⢸⢿⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡇⢠⡴⣿⠓⠚⡇⢸⠓⠚⣿⢦⡄⢸⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠉⠈⢀⠜⠁⠈⠣⡀⠁⠉⠀⢸⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⡿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢿⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢯⣍⠛⣱⡇⠀⠀⠀⠀⢸⣎⠛⣩⡽⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡕⣠⠖⠛⠃⣿⠙⢯⠉⠉⡽⠋⣿⠘⠛⠲⢔⢪⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡺⢦⡝⠳⣄⠀⠻⣄⣈⠙⠋⣁⣠⠟⠀⣠⠜⢩⣤⢿⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠏⠠⢤⡘⣦⠀⠘⠲⠦⠴⠖⠃⠀⣴⢃⡤⠄⠹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣤⣄⡀⢳⡘⣇⣸⡆⠀⠀⢰⣇⣸⢃⡞⢀⣠⣤⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⣥⣼⡟⠸⣇⠻⠃⡇⠀⠀⢸⠘⠟⣸⠇⢻⣧⣬⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣿⣦⣀⢣⣀⣀⡼⣀⣴⣿⡴⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⢿⣀⣙⣉⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠋⢻⡿⠿⠿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⣷⣤⡴⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⠛⣿⣿⣿⠛⠛⠛⠉⠛⠻⢿⣿⣿⣿⣿⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠘⠁⠉⣴⣾⣶⡤⠈⠓⠀⠀⠀⣭⣷⡋⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⢀⢀⡄⠀⠁⠀⢸⠟⠁⠀⠀⠀⠀⠀⠀⠉⠙⡆⢸⡏
⠀⠀⠀⠀⠀⠀⢀⠀⣠⠶⠛⠉⠉⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠒⠚⠻⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠠⠀⠀⠀⠀⠀⠂⠷⠶⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠘⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⢿⡿⣿⣶⣤⣠⣤⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟⢁⠃⠀⢤⠘⠀⡀⡀⠉⣿⣏⠉⢠⠀⠀⢈⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡋⠀⡀⠁⣀⣤⣄⣄⡀⠉⠀⢀⠈⣻⣆⣄⣘⡀⠀⡄⠁⠘⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠂⠁⠈⣠⡾⠟⣻⣿⡏⠻⣿⣆⠈⠈⢸⣿⠛⠛⠿⣿⡅⢀⠀⢨⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⡀⠀⣶⡟⠀⠀⢹⣿⣿⡄⠙⣿⣆⠀⣿⡇⠀⠀⠀⠈⣷⡈⠀⢀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡂⠙⢀⣿⠇⠀⠀⠈⢻⣿⣾⠀⢋⠛⣿⣿⠃⠀⠀⠀⠀⢸⣇⠰⠁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢀⠘⡇⠀⠀⢀⣼⣿⣷⣤⠀⠀⠂⠈⢳⣄⠀⠀⠀⣬⡿⠀⠰⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠈⠈⢿⣄⣠⣿⣿⠷⡍⢻⣷⣤⡃⠀⠰⠙⢷⣄⣴⡿⠁⠐⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡂⠆⠘⢿⣿⣿⠓⠀⣴⣿⠋⠻⣿⣶⣤⣤⣼⣿⡟⠁⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⡄⠘⣿⣿⣶⣾⣿⣧⣤⣤⣤⣿⣿⣿⣿⣿⠇⢀⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣈⠙⠛⠁⠜⠈⠋⠁⠍⠰⠠⠄⠀⣀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣶⣷⣶⣶⣶⣶⣶⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⢷⣶⣤⣤⣶⠿⠟⠛⠛⠛⠉⠙⠛⠛⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠘⠿⠂⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣠⡄⠈⠻⣷⣄⣀⣀⣤⣄⣀⣀⠀⠀⠀⠘⠷⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⢀⣤⣶⠿⠟⠛⠛⠛⠛⣻⣿⣷⣦⡀⠈⠁⠀⠀⠙⣿⡟⠋⠉⠉⠛⠻⢿⣦⡀⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⢀⡀⠀⣠⣿⠋⠁⠀⠀⠀⠀⠀⢰⣿⠃⠀⠙⢿⣆⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠘⠃⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣀⠈⣿⡆⠀⢤⡄⠀⣿⡇⠀⠀⠀⠀⠀⠀⢹⣷⠀⠛⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠛⠀⣿⡇⠀⠈⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠸⣿⣄⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⡦⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⠛⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⣀⡀⠀⠘⢿⣦⣀⣾⡟⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣦⡉⠁⠀⠀⠀⠙⠻⣿⣤⡀⠀⠀⠀⠀⠀⢰⣿⠃⠀⣀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⣀⠹⣿⡄⠀⠀⠀⠀⣀⣴⣾⠟⠉⠻⣷⣄⠀⠀⠐⠛⠀⠈⠙⠿⣷⣦⣄⡀⣠⣿⠏⠀⠈⠋⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠉⠀⣙⣿⣦⣴⣶⠿⠟⠋⠀⣤⠀⠀⠈⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣶⣤⡀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⣴⡿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠙⠻⣷⣤⣀⠈⠳⠀⠀⠀⠀⣰⡄⠀⠹⣿⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠐⠓⠀⠀⠀⠶⠂⢀⣠⣴⡾⠟⠉⠀⠀⠀⠀⠀⠙⣻⣿⣶⣤⣄⣀⠀⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⣀⣀⣤⣴⣶⠿⠟⠻⠿⣶⣶⣶⣤⣴⣶⣶⡾⠿⠛⠋⠁⠉⠛⠻⠿⢷⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣏⡉⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢶⠄⠀⢀⣤⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⡀⠀⠙⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢷⣶⣶⣶⣶⣶⣶⣶⣶⡾⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠶⠶⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣩⠔⠚⠉⠉⠉⠉⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡱⠋⠀⠀⠒⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠙⢆⢀⣠⢴⣒⠯⠍⠉⠀⠈⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⡄⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡩⠖⠉⠀⠀⠀⠀⢠⠄⠀⠘⠁⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠃⠀⠀⠀⠀⢀⠴⠚⠛⠓⠒⢦⡀⠀⢠⢞⡵⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠙⣴⠃⠉⠐⠃⠀⣀⢤⠚⠉⠉⠉⠙⠢⣄⠀⠀⠈⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⠀⣀⡴⠊⠁⠈⡇⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣀⠀⠘⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠘⣇⡧⠚⠁⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠉⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠰⠇⠀⠀⡞⢳⡀⠀⠀⠀⠀⢰⠃⡀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠰⠂⠀⠀⢳⡀⠀⣠⠔⠋⠀⠀⠀⠀⣠⠀⠀⣠⠞⠀⠀⠹⣄⠀⠀⢀⡎⠀⠀⠀⠘⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣹⠚⠁⠀⠀⠐⠋⠀⠀⠀⣠⡾⡁⠀⢤⠀⠀⠘⣆⢀⡞⠀⠀⠆⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢆⠀⠀⠀⡼⡹⠁⠀⠀⠀⠀⠐⣚⠴⠋⠀⠱⡝⣄⠀⠀⠀⠀⠘⡎⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢣⡀⠀⢇⠳⢄⣠⠴⣚⠵⠊⠁⠀⠀⠀⠀⠈⢮⡃⢀⠀⠀⠀⣿⠀⣀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡑⣄⠀⠙⠒⠒⠋⠙⠦⣄⡀⠀⠀⠀⠀⠀⣀⡽⠮⣙⣋⡴⠃⠀⠈⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡓⢄⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠠⠆⠀⠀⠛⠀⠀⠀⠀⠀⢀⡠⠖⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⢤⣀⣀⣀⣘⣀⣖⣒⣋⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠉⠁⣠⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠋⠉⠀⠀⢀⣡⣶⡶⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠄⢰⡴⠂⠒⠠⡀
⠀⠀⠀⠀⠀⡠⠄⠀⠠⠤⣀⣡⡀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⣠⡴⠿⠛⠒⢠⣔⠂⠐⠲⡖⠉⠀⠀⢸⠃⠀⠀⢠⠞⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠎⠀⠀⠀⠴⠚⠉⠀⠉⣠⣶⡶⠦⠠⢄⣀⡀⠀⠀⠁⠀⠀⡴⠾⠛⠋⠁⣲⣟⡀⠀⠀⠀⠏⠀⠀⠰⠋⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠠⠞⠋⠁⠀⢀⡴⠟⠋⢉⡡⠶⣤⠔⠒⠒⠒⠒⢘⡾⠛⠛⠛⠒⣦⠔⠒⠒⠂⣤⠄⠀⠀⠀⣰⡖⠡⢄⡀
⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣰⡋⠀⠀⠀⠀⠀⠼⠁⠀⠀⢀⡼⠃⠀⠀⠀⣰⠏⠀⠀⢀⡴⠋⠀⠀⠀⠘⡄
⠀⢀⣠⡴⠞⠋⠉⠉⠀⠒⠤⢀⣀⢀⡠⠒⢰⡖⠉⠉⠉⠉⠁⣼⠋⠁⠀⠀⣰⠔⠉⠉⠉⢩⡗⠉⠉⠉⣩⡟⠀⠐⠲⡋⠀⠀⠀⠀⠀⢀⠟⠶⣤⣀
⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢠⠏⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⡴⠋⠀⠀⠀⢠⠏⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⣀⡀⠀⠤⠄⠊⠀⠀⠈⠹⣷
⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠐⢍⠻⢷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠀⠶⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠄⠐⠒⠒⠉⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⢛⠝
⠀⠀⠉⠒⠬⢉⡛⠻⠶⠶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡴⠶⠾⠛⡋⠭⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠂⠤⠍⢉⡉⢙⣛⡛⠛⠛⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠚⠛⠛⢛⠉⠉⠉⠉⠥⠀⠒⠀⠈⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣿⣷⣶⣶⣶⣶⣾⣷⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣶⣿⣿⡿⠿⠻⢿⣿⢿⣭⣷⣤⠀⠀⠀⠀⠀⢀⣀⣤⣖⣲⣶⣶⣿⣿⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡴⢞⣿⡟⠋⠉⠹⣿⠀⠀⠀⠸⠇⠀⠀⠸⣿⠁⠀⠀⣠⣺⡽⣿⡏⠁⠀⣿⠃⠀⠀⠀⣿⠋⠙⠶⣽⣦
⠀⠀⠀⠀⢀⣰⣏⡴⠋⢿⡇⠀⠀⣀⣭⠴⠶⠶⠶⠶⠶⠶⠿⠛⢀⢔⣿⣿⠙⠀⢿⡃⠀⢀⣿⣀⣠⣀⣀⣟⠀⠀⠀⣰⣟⠂
⠀⠀⠀⢠⣿⣿⠏⠀⠀⠘⣣⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠘⣿⠀⠀⣸⡶⠞⠛⠉⠉⠉⠉⠉⠉⠛⠳⠿⠟⠁
⠀⠀⢀⡿⣼⣿⡇⠀⢀⡼⠋⠀⠀⣀⣀⣤⣀⠀⠀⠀⠀⠀⣰⣽⣿⠀⠀⢿⣴⠞⠋⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⣼⣿⠃⢹⡇⢠⡞⠀⠀⣴⣾⡽⠟⠋⠻⣷⠀⠀⠀⢸⣿⡇⢻⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢰⡿⠚⠻⢿⣆
⠀⠀⣿⡏⠀⠀⢠⡟⠀⢠⣾⢿⣅⠀⠀⠀⠀⡿⠀⠀⠀⣿⠇⠀⠸⢃⡾⠁⠀⢀⠀⠀⠀⠀⣠⣾⡷⠶⠄⠸⣿⠀⢀⣤⣤⣤⣀
⠀⠀⠹⣧⣀⣠⠞⠁⢠⡿⠇⠀⠙⠂⠀⢠⡞⠁⠀⠀⠀⢿⡀⠀⣀⡼⠁⠀⠀⠈⢤⡀⢡⣸⣿⣏⣁⠀⢀⣼⠏⢀⣾⡓⠋⠁⠹⡆
⠀⠀⠀⠀⠉⠁⠀⢠⣿⡿⣦⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⠛⠚⠋⠁⠀⠀⠀⠀⢈⣿⣿⣧⣾⣯⣽⣦⣾⠋⠀⣼⣧⣀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠻⠀⠈⠁⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡿⢻⣭⣿⡿⠛⠛⠓⣺⣿⠿⢿⣿⣿⣿⣯⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣧⡄⠀⠀⠀⢸⠁⠀⠀⠀⢀⡴⠶⣾⣿⣿⢿⡿⣿⣿⣿⣧⡈⠿⠋⠀⣀⡐⢟⠋⠀⠀⢤⠿⠿⠛⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠀⠀⠀⢸⠀⠀⠀⢰⣿⠊⠉⢁⣹⣿⣦⣤⣬⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⢠⣬⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣼⠀⠀⠀⠈⠷⣤⣤⣿⣿⡿⣍⣻⣿⣿⡉⣛⣻⣿⣿⣧⣶⣾⣿⣿⠏⢻⣿⡿⠿⣦⡀⠀⠈⢯⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠹⣏⠙⠒⠚⠛⠻⠿⠿⠿⠿⠿⠿⠏⠉⢁⣠⠴⠚⠉⠀⠀⠈⢻⡄⠛⠛⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣤⣤⣤⣤⡤⠤⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⣀⣀⣤⣤⠤⣤⣤⣀
⠀⠀⠀⠀⢸⠃⠀⠀⠀⣠⠞⠛⠛⠛⠳⢦⣄⠀⠀⠀⠀⢹⣦⠶⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⢶⣄
⠀⠀⠀⠀⣿⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠉⢳⡄⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⢹⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠁⠀⠀⠀⠀⣀⣤⠶⠖⠲⢦⣄⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠘⣇⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣀⡴⠟⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⢸⡇⢀⣤⠞⠋⠀⢠⡏⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠈⠛⠦⣤⣀⣀⣼⠿⠋⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣤⣀⣤⣤⠶⠛⠁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠠⢀
⠀⠀⠀⠀⠔⠁⠀⠀⠀⠁⠁⠀⠀⠐⠡⠀⠁⠢⡀⠀⠀⠀⠀⠀⠊⢈⠂⠀⠀⢰⠀⠀⠁⠁⠀⠀⠀⠢⡀
⠀⠀⡐⠁⠀⠋⠀⠀⠘⠔⠀⢀⡀⠀⠀⠀⠀⠀⡈⠢⡀⢀⠊⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⢀⡢⠀⠈⠀⠀⠄
⠀⠐⠀⠀⠀⠀⠄⠀⡀⠄⠀⠈⢌⠀⠀⠃⠂⠀⠀⠀⠈⢆⠀⠀⠀⠈⠀⢀⠔⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠈⡄
⠀⠄⠈⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠢⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠈⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠁⠀⠀⠀⠀⠑⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠈
⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠢⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠑⠄⢀⠠⠀⠉⠀⠀⠀⠨⠀⠀⠀⠄⠰⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠀⠊⠁⢀⡂⠀⠁⡀⠀⠀⡀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⢄⠀⡀⠐⠡⡀⠀⠀⠀⠀⠀⢢⠀⡠⢌⠈⠀⠀⠒⠀⠀⠀⠀⠊⠀⠀⠁⠀⠂⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠒⠀⡀⠈⠀⠀⢈⠀⠀⡀⠐⠂⡀⠀⠈⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠐⡀⠂⠁⠀⠀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠁⠄⣀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⣤⣾⣤⠀⠀⢀⠎⢠⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡼⠆⠈⠀⠘⠁⢀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠃⠄⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡆⠀⠒⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠳⠴⢓⡀⠀⠀⠀⠘⣧⡀⠀⢰⠂⠀⢀⠀⠀⠀⠀⢤⡔⠀⡀⡀⠀⠀⠀⠀⠀⠀⣤⠆⣠⣶⣤⣴⡶⣴⣶⠶
⠀⠀⠀⠰⠋⢥⣅⡀⣶⣀⡀⠘⡇⠀⢀⢀⠀⠀⢑⣀⣀⠀⠂⠀⠀⠀⣰⠇⠀⠀⠉⠀⣠⣿⡞⢡⣼⠤⠶⠆⠐⠒⠂
⠀⠀⠀⠀⠠⡆⣿⢰⡋⠉⠹⣿⠁⠀⢠⠑⠰⣀⠀⢈⢠⡄⢠⡆⠈⠀⠈⠀⠀⠀⠀⠸⠷⡄⣇⠸⠟⠂⠀⠃⠀⠈⠈⠆
⠀⠀⠀⠘⠛⠷⣩⢸⣅⣐⢶⢼⠀⠀⠢⠄⣈⠛⠀⠙⢿⠁⠈⠰⠶⠶⠃⠀⢀⠀⣠⡴⡏⠹⠛⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⢰⠃⠘⠉⡛⠈⠉⠀⠀⠀⢠⠠⣴⡤⠳⠶⠀⠀⠉⠛⠿⠃⠀⠠⠀⠰⠛⢓⡀⠘⡂⠀⠀⠀⠀⠀⣄⣿⠁
⠀⠀⠀⠀⠀⢸⣶⡾⡗⣽⣤⢶⠀⠀⡁⠉⠀⠂⠀⣶⡄⠀⠀⠀⠀⠉⠁⠀⠀⡀⠘⠻⢏⠉⠀⢰⡆⠀⠀⣆⡰⣿⢉⣡⡀
⠀⠀⠀⠀⡄⢸⠿⣿⡏⠯⠙⣿⠁⠀⠀⠈⠀⡀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠎⢰⠒⢘⡅⠠⠟⢻⠁⠁⣤⠀⠁
⠀⠀⠀⠀⣇⢸⡄⡀⡏⣠⠁⣩⡄⠀⠀⠀⣠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠸⢲⠀⡀⠀⢘⠅⠀⠐⠂⠘
⠀⠀⠀⠀⠀⠸⢷⠂⡏⠋⠁⣀⡅⢠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡃⠀⠉⠀⠐⢠⠄⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠖⢻⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉⡳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠚⢳⠶⣄⡀
⠀⠀⠀⠀⠀⢀⣠⠶⠒⠒⠺⢤⡺⣻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠥⠧⢼⣏⡟⠃
⠀⠀⠀⠀⣠⠟⢁⣀⣀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⣰⠏⣰⠏⠀⡿⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡞⠙⡆⠀⠀⠀⠀⠘⣆
⠀⠀⢰⠏⢀⡏⠀⡜⢀⣀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⠁⢰⠁⠀⠀⠀⠀⠀⢻
⠀⠀⡾⠀⣸⣠⣼⠿⠛⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣾⡶⠟⠋⠁⠀⠀⢸⡀
⠀⢰⠇⠀⡏⢹⠃⠀⠀⠀⢀⠔⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡾⡏⠀⠀⠀⢀⠞⠀⢸⡇
⠀⣿⠀⢸⠃⡎⠀⠀⠀⠀⠁⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⠀⡇⠀⠀⠀⠂⠀⠀⢸⡇
⢸⡇⠀⡼⢰⠁⠀⠀⢀⡠⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⡇⠀⠀⠀⣠⠀⠀⢸⡇
⣸⠃⠀⡇⢀⣠⣤⡶⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀⠇⠀⣠⡼⠁⠀⠀⠈⡇
⣿⠀⢸⠛⠛⠉⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣇⣴⠾⠋⠁⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠸⡆
⢿⠀⠀⠀⠀⣀⣠⣴⡿⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⢳⡀
⢸⡄⠀⠾⠛⠛⠉⠁⠀⠈⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⣀⣴⡾⠛⠁⠀⠀⢀⠀⠳⣄
⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠘⣆⠀⢀⣤⡴⠾⠟⠋⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢀⣤⠾⠛⠁⠀⠀⠁⠂⢀⠀⠈⠙⠶⣤⡀
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⢠⣶⠆⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠉⠀⡇⠀⡔⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⡠⠀⢀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠀⠀⠁⠀⣴⣠⡆⢀⡀⠹⣄
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⠉⠁⠀⠀⠀⠁⢠⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⢸⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⡴⠂⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠀⠀⠀⢠⠾⠋⠀⠀⠈⠀⠀⠀⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⣠⡿⢿⠁⣰⠃⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠘⠛⠉⠀⠀⠀⠁⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢤⣤⡤⠶⠖⠈⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠂⠒⠛⠻⣌⣨⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠖⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠓⠢⠤⣄⣀⡀
⠀⠀⠀⠀⠀⣠⠒⠋⠭⠭⠅⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠈⠁⠂⣀⣀⠀⠉⠙⠒⠢⣤⣀⡀
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡆
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⡐⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠒⠒⠠⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠡⡀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠆⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⡀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚
⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠂⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄
⠀⣀⠤⠔⠒⠊⢳⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠒⠒⢤
⡎⠁⠀⠀⠀⠄⠀⢀⣉⠉⠉⠉⠉⠉⠓⠋⠀⠀⠘⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠂⠀⠀⣤⣤⡤⢄⣀⠗⠂⠀⢀⡠
⠈⠓⠢⠤⢄⣀⠀⠀⠀⠈⠉⠉⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠐⠒⠂⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣒⡤⠦⠤⠭⠖⠒⠚⠉⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⣿⣦⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⢦⡀⣿⡿⢻⡆⠒⠲⣶⣶⠿⢄⡀
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣹⣿⠁⠈⣿⣀⣀⣹⡿⢿⣛⠋⠑⡀
⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⣿⣿⣀⡀⠸⢀⡜⠉⠛⢾⠋⠀⠈⠇
⠀⣠⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣿⣿⣷⣷⣏⣀⠤⠤⠦⠶⠒⣶⣾⡀
⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠟⠋⠉⠀⠀⢀⡠⠄⠀⢹⣿⣷⡀
⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⣢⣭⠍⠁⠀⠀⠫⠚⠛⠂⢰⣿⣹⣶
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣽⣿⣧⠀⠀⠀⠀⠤⠤⠖⠂⠀⠀⢠⣿⣿⣿⣧
⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⢠⡶⠖⠲⠖⠀⢠⣿⣿⣿⣿⣿⣧⡀
⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⣀⠀⠀⠉⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⣿⣿⣿⣿⣏⠳⣦⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠒⠂
⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⣿⣿⣿⣿⣿⣿⣦⡈⠙⠻⢿⡿⢿⣿⣿⣿⣿⣿⣿⠿⠂
⠀⠀⢸⣿⣿⣿⠀⠀⢀⡠⠒⠀⠒⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣠⣶⣿⣧⣿⣿⣿⣿⡿
⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣎⠻⣟⣿⣿⣿⢿⡏⠹⡿⠋⠉⠛⠛⠋⠛⠁
⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠂⠘⠛⠉⢸⠁⠀⢹⡄
⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠇
⣤⣠⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰
⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⣿⣿⣿⣿⣿⣿⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠉⠉⠉⠉⠛⠲⠤⣄⡀⠀⠀⠀⣠⡤⠖⠋⠉⠉⠁⠉⡉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠙⢦⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⣰⣤⣄⡀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠿⠿⠿⣿⣿⣦⡀⠀⠀⠀⠈⢿⡿⠿⠛⠛⠛⠿⣿⣿⣷⣄⠈⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠄⠀⠀⣾⣿⣿⡟⠁⠀⠀⠀⠀⣼⣿⣟⢿⣆⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⢰⣿⠿⠿⢾⣿⡄⠘⠂⠀⢷⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⠀⢦⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⠁⠈⠂⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⠙⣷⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⣿⣿⡁⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⢻⣶⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⢿⣿⡇⠀⠀⠀⠀⠀⠀⢳⡄⠈⠁⠀⠀⠀⠉⠻⢤⣀⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⡀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⢀⡤⠾⣿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⣰⣿⣿⡇⠀⠀⠠⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⠂⠀⠀⣿⣿⣿⣧⣀⡤⠖⠋⠀⠀⠈⣷⣽⣷⣾⣷⣦⣄⣀⠀⠀⠀⠀⠀⢀⠈⠙⠿⣿⣇⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⢀⣿⣿⠿⠛⠁⠀⠐⠀⠀⢀⣴⡿⠟⠁⠉⠛⠿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠛⢉⠄⠀⠀⠀⣀⣠⣴⣾⡿⠿⠤⠤⠤⠤⣤⠤⠤⠼⠻⣿⣿⡿⣿⣶⣄⡈⠂⠀⠀⠀⠈⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⠀⣠⣶⣿⣿⣿⡟⠁⠀⠀⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠙⣮⣟⢿⣿⣦⣀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⣠⣾⣿⣿⣿⣵⣿⣿⣤⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣷⣦⢀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣿⡿⠟⠋⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁⠈⠛⢿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⣤⣄⣀
⠀⣀⠚⠋⠉⠉⠁⠀⢀⠀⠄⠘⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⣿⣿⣿⣿⣿⣯⣿⣷⣤⡀
⠀⣠⡄⠀⣀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⣾⣧⠀⣿⣿⣻⣿⣿⠀⠰⣆⠀⢰⣦⡴⣶⣶⡜⠉⠙⣿⠀⢠⣿⣿⣿⠀⠆⣹⡇⢰⣿⠀⢦⣽⣦⠀⣾⣿⣿⣧
⠀⢸⣿⣿⣿⣿⣟⣿⡏⠀⢰⣿⣤⣿⣿⣿⣿⣽⡇⠀⢀⣿⠀⣾⣾⣿⣿⠀⡆⢹⡇⢰⣿⠓⡆⢹⣿⠀⣿⣿⣿⣿⡆
⠀⠘⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣽⣿⣿⣿⣿⣧⣤⣾⠏⠀⣿⣿⣿⣿⠶⠶⣿⡷⠶⢾⠷⠶⣿⠿⠶⣿⣿⣿⣿⡇
⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠸⣇⢿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⢿⣿⣿⣿⠀⠇⣸⡇⠰⣿⠀⠾⡿⠐⠇⣼⣿⣿⣿⡇
⠀⠀⢸⣟⣿⣿⣿⡇⠀⠀⠀⠙⠺⣿⣷⣾⠟⠋⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠇⢸⡇⠸⢿⠀⠻⣿⠀⡇⢸⣿⣿⡟
⠀⠀⢸⣿⣿⣾⣿⡇⠀⠀⠀⠀⢀⣛⣛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⢸⣿⣼⣿⣿⡇⠀⠀⠀⠀⢀⢟⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠲⠾⠓⠞⠛⠛⠿⠷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢤⣴⣶⣶⣶⣶⡦⠄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣯⣽⣿⣿⡿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⣠⣶⠾⠻⠟⠛⠛⢛⣋⡩⠩⠉⠉⡈⣉⡰⠆⠀⠀⠀⢠⣾⣏⣻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢻⣦
⠀⠀⣀⡤⠾⠗⠀⠊⣀⣉⣤⣤⣴⣶⣶⣾⣾⣤⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⡼⠿⠛⠿⢿⣧⣽⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣏⣿⣿⠏⠀⠀⠀⠀⠀⠙⣿⣿⡟⣻⡿⢿⡆
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣷⣾⡇
⠀⠀⢀⣤⣶⢻⣿⡿⠿⠛⠚⠛⠛⠻⠿⣯⣿⣗⣦⣄⠀⠀⠀⣿⣉⣿⣿⣶⣿⣦⡀⠀⠀⠀⠀⣠⣿⣴⣿⣿⡿⣿⠃
⠀⢰⣿⣽⡿⣿⣟⣷⣶⣤⣤⣤⣤⣶⣶⣿⣽⣿⠿⣾⣷⠀⠀⠸⣿⣿⣿⣩⣿⣿⡿⢶⣶⣶⣾⣿⣿⡿⣿⣿⣶⡟
⠀⢸⣿⣿⣿⣿⣿⣯⣿⣷⣿⣿⣿⣾⣿⣯⣿⣿⣿⣿⡿⠀⠀⠀⠙⢿⣟⣻⣿⣿⣿⣿⣿⣧⣾⣿⣿⠿⣿⣿⠏
⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠙⠻⠿⣿⣥⣿⣿⣯⣽⣿⣿⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⣤⣤⡤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⣭⣷⡾⣟⠉⠩⠙⠻⢿⡿⢿⣾⣽⣲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣵⡾⠟⢩⣄⣠⣿⣔⡶⠶⠤⣏⣀⡀⠈⠙⠻⣷⣭⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣵⣿⡯⠀⠀⠀⠙⢿⣯⠉⠶⠂⠙⠋⠛⠁⠀⠀⠀⠀⠙⢿⣜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡾⠟⣿⡇⠀⠀⠀⠀⠸⢿⣄⣠⡀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡞⣾⣇⠠⢿⣷⡆⠀⡀⠀⡀⣿⣴⣨⣷⡾⠋⠓⠤⣄⡀⠀⠀⠀⠀⠀⢻⣧⢧
⠀⠀⠀⠀⠀⠀⠀⢰⢹⡟⢻⣈⣹⠛⠃⢒⠟⣭⢁⡘⢯⢿⢝⡁⣰⣶⠆⠀⠉⠙⠢⡄⠀⠀⠈⣿⡼⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡜⠀⠀⡏⠙⣾⠿⠷⢾⡏⠈⠀⠀⠃⠀⠉⣻⡏⠀⠀⠀⠀⠀⠘⣦⠀⠀⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢰⡟⠉⢳⡀⠀⣿⡇⣀⣤⣤⣤⣀⠀⣿⠀⢸⡀⠀⡠⠖⠉⢉⡙⣦⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣿⠀⠈⡇⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⣟⣠⠀⠄⠒⠒⠻⣿⢻⠁
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢠⣏⣴⣷⠷⠞⣿⡏⠙⠛⠻⠛⠋⠹⣿⠀⠀⠀⣿⠿⠒⠁⠀⠀⠀⢻⡎
⠀⠀⠀⠀⠀⠀⠀⣼⡉⢻⢻⣿⡉⠁⠀⠀⢹⡇⠀⡤⠒⠦⠤⠤⠿⠤⠖⠊⠀⠀⠀⠀⢀⡀⠀⡸⢠⡆
⠀⠀⠀⠀⠀⠀⠀⣻⣇⠀⡱⣝⢿⣦⡀⠀⠸⣧⠎⡄⠀⠀⠀⠀⠀⢀⡀⠀⣀⣒⣈⣩⡤⣶⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣧⡌⠲⣝⢿⣶⣄⣿⡚⠷⣥⣆⣰⣾⣿⣿⣿⣿⣯⣿⠿⡫⠞⢥⣸⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣗⣂⣀⠙⠪⢽⣻⠿⢿⣿⣿⣿⣿⣿⡿⠿⣛⡯⠗⠋⣀⣐⣺⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠉⠉⠒⠒⠒⠒⠚⠉⠉⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠸⡸⢣⢇⢔⡀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢱⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⣄⠀⠈⠀⢀⠴⠙⢻⣶⠖⠀⠀⠀⠀⠐⠀⢀⠀⠀⢉⢶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠓⠀⠀⠀⠀⢠⢡⣛⠻⠿⣧⠀⠀⠀⠀⠸⠋⠘⢀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠘⠁⠀⠀⠀⠀⠀⠈⢀⠐⠿⣳⡆⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⣢⠇⢰⡄⠀⠀⢩⠶⣝⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠌⠀⠀⠀⣠⠀⡌⠣⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⡴⣰⢰⡇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠉⡀⠀⠀⠀⠈⠀⠠⢦⣄⠉⠀⠂⠀⠁⣊⢞⢔⣭⣠⠄
⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠒⠲⢿⡢⠀⠀⠀⠜⠁⠄⡽⠷⠭⠤⠀⠄
⠀⠀⠰⠄⠀⠈⡹⢦⠀⠀⠀⢀⠀⠄⠀⠋⠸⣄⡄⠂⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠨⠙⢒⣨⣍⡙⠃
⠀⠀⠀⣤⢀⠀⢨⠷⠠⠀⠀⠐⠻⠀⠀⠀⠁⡤⢿⣥⠀⠀⠀⠨⠀⠀⠠⠄⢀⠀⢀⠀⠀⠲⡤⡙⢾⣛⠅
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠶⡶⠀⠄⠀⠀⠠⠂⢴⡧⡎⠀⠸⠀⠁⠀⢘⡛⠓⠭⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠓⠁⠃⠀⠀⡟⠋⠀⠀⠀⠀⠀⠀⠊⠐⠀⠀⠀⠀⣶⠀⣤⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠚⠘⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣀⡀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠗⠒⠛⠋⠉⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⠭⠿⠷⠶⠄⠀⠀⠀⠀⣰⣃⡤⠴⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠈⠻⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢥⡀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠉⠲⣤⣄⣀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣶⠶⠦⠄⠀⠀⠀⠀⠀⠀⣠⠎⠀⠉⠑⠻⢷⡀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⠙⣓⣶⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⢀⣨⡷⠤⢤⣤⢤⣤⣤⣤⣤⣤⣤⠶⠒⠛⠛⠛⠛⠛⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠈⠉⠛⠂⠀⠀⠉⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠒⠋⣁⣀⣀⣀⣀⣤⠤⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠈⠳⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠖⠒⠋⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⡄⣴⠋⠀⠈⠂⢰⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢰⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡾⣸⠃⠀⠀⠀⠀⣼⠀⠀⣰⠀⣿⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡤⠀⠀⠀⠀⠀⣸⡥⠄
⠀⢰⡇⣿⠀⠀⠀⠀⠀⡿⠀⠐⠁⠒⣿⠁⠈⠀⣠⣰⠉⠈⠀⢸⡇⠄⠀⠀⠀⠀⠀⠀⡤⠄⣠⠄⢸⠇⢠⣯⠀⠀⠀⢀⡈⠁⢸⠀⠀⠀⡀
⠀⢸⡇⣿⠀⠀⠀⠀⢰⡇⠂⠀⠀⠀⣿⠀⠀⠀⣷⡇⠀⠀⠀⢸⡇⠀⠀⠀⣠⠖⠀⠀⠣⢠⡇⠀⢹⠀⢸⡈⠁⠀⢰⡃⠀⠀⠸⡇⢀⠜⠁
⠀⠸⣧⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠹⡀⢀⠴⢿⠁⠀⠀⠀⢸⢿⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⣼⠀⠀⠉⠂⠀⠀⠙⢶⣄
⠀⠀⠙⠶⠤⠀⠀⠀⠟⠀⠀⢆⠀⢀⡄⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠁⠀⢸⡇⠒⠊⠀⡀⠘⠄⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠂⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠨⠀⠀⠰⢀⠀⠠
⠀⠀⠀⠀⠀⡤⡺⣡⠵⠋⠀⠀⠀⠀⠀⠙⠯⣌⢆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠤
⠀⠀⠀⡄⠇⡢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠕⠝⢠⠀⠀⠀⠀⠀⠀⠀⠠⠈⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⢄
⠀⠀⢆⠌⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠣⠲⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠂
⠀⡈⠎⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡐⢁⠀⠀⠀⢠⠠⠀⠀⠀⠀⢠⠀⠂⠀⠀⠠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⢈⠄⠀⠀⠄⠀⠀⠀⠀⣀⠘⠀⠀⢀⡀⢀⠀⢀⠀⡀⠀⢀⢀⡀⠀⠀⠀⠠
⠰⢈⢊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠰⡡⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠐⠠⡁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣸⠐⠂⠀⠀⢠⢀⠀⠀⠀⠀⠸⠁⡀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠃⡀⠀⠀⠀⡀
⠀⠂⠣⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠣⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠃
⠀⠘⢀⢑⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠊⡈⠁⠀⠀⠀⠀⠀⢈⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⡁
⠀⠀⠈⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠐⠒⠀⠀⣀⠀⠀⠀⠀⠒⠲
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣴⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⢤⠄⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣞⠛⣳⠄⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⢁⣠⣎⣉⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠍⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢯⣁⠡⠝⠃⠀⠀⠀⠀⣐⠲⣤⠤⠀⠒⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣩⣤⡤⣀⡀⠀⠀⣀⣀⡀⠤⠤⠐⠒⠂⠉⠉⠈⢁⣀⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣡⡼⣇⠈⢊⡾⠋⠉⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⢠⠞⠻⠮⡺⣝⢷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⣯⠀⠀⢘⡩⣲⡟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢠⣞⣳⠄⠀⡀⠀⠀⠀⠀⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣴⡟⢫⡄⠐⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠉⠁⠀⠀⠀⠀⠀⢀⡞⠛⠥⣀⡝⢵⡄⠀⠀⠘⠀⠀⠀⠈⠈⠑⠈⠁⠀⠀⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢆⠦⠀⠘⣧⡀⠀⠀⢊⡼⠁⠀⠀⠀⢰⠒⣦⠀⠀⠀⠀⠀⠀⠀⠓⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠠⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠋⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠴⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠴⠒⠒⠂⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠒⠒⠃⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠈⠑⠂⠂⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠖⠢⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⣤⣄⢀⣀⡀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠻⠿⠿⣿⠿⠃⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠶⣶⣶⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣄⠀⠀⠉⣩⠟⠛⠻⢅⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⢡⣶⡿⠀⠹⣾⣿⣂⣀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⢀⣀⣄⡀⠀⢸⣿⡿⣶⡄⠙⣿⣿⡋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⡶⠷⠒⠀⠀⠀⠀⢽⣿⣿⣿⣿⡇⠀⠙⠛⠛⠁⠀⠈⠻⠿⠷⠟⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⡇⠀⠀⠀⠈⠿⠿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢴⠂⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣮⠀⠀⠀⣸⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠠⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⡀⠀⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣀⠀⠀⠙⢟⠛⢋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⡳⠀⠀⠀⢤⣄⠀⠈⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⢀⣀⣿⣿⣧⠀⢿⣿⠏⠙⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡰⠁⢻⣿⣿⣿⠏⠀⠈⠙⠀⠀⠖⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣀⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣠⠴⠋⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠶⡤⠀⠀⣀⣠⠤⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⡀⠤⠐⠂⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁



⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠦⠠⠠⠰⠖⠰⠶⠠⠶⠤⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡠⢀⠠⠀⠄⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠤⠀⠠⠤⢸⣿⣕⣂
⠀⠀⠀⠶⠇⠃⡈⠀⢀⣴⣬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣛⣋
⠀⠀⠀⠀⠂⠀⡁⡀⠿⣿⣿⣿⠇⠾⠿⠋⠙⠻⠂⠀⠀⠖⢀⠍⠃⡀⠀⢰⣿⣿⡅⠀⠀⠈⠀⠀⠀⠀⢀⣤⣤⣤⢀⠀⠀⠀⠀⠾⣿⣿⣦
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢡⡀⠀⠮⠄⠀⠀⢀⡇⠀⠘⠛⠛⠛⠁⠀⠀⠐⠛⠀⠀⠊⢀⠀⠈⠂⠀⠀⠈⠂⢘⢿⡿⠟⠄
⠀⠀⠀⠀⠀⢀⠀⣆⣣⢀⣆⠀⢀⠀⠀⠈⠋⢀⠁⢖⡆⠐⠰⠷⢞⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠲⠀⠀⠀⠀⡀⠀⠀⢀⣼⣷⠂
⠀⠀⠀⠀⠀⠸⠇⣉⡙⠉⠠⡤⠈⢰⡰⠤⣤⠥⠀⠀⠀⠸⠿⢿⠃⠀⠀⢠⣿⣶⣶⣶⣷⠀⠀⠈⠀⢢⣤⣤⣦⡴⠾⠇⠀⠰⠋⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠚⠋⠉⠠⡤⠚⠀⠘⠷⣶⣧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⣛⠁⠀⠀⠀⠀⢀⣿⣿⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠤⠤⠤⠤⠠⠤⠄⠀⠀⠀⠀⠠⠾⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠙⠚⠂⠐⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⡤⠶⢤⡀⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠍⠀⠀⠀⠈⠳⠤⠔⠁⠀⠀⠀⠀⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠈⢁⡴⠒⢲⣤⠤⠤⠴⠖⠂⠀⠃⠀⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⠷⠖⠒⠒⠋⠉⠉⠳⠤⠼⠁⠀⠀⠀⢀⠤⢤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡼⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⢠⠖⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⡼⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠐⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡋⠁⠉⢻⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠿⠿⠟⢛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠿⠿⠟⠁⠀⠀⢩⣭⣭⣴⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⠋⠙⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠘
⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠙⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣟⠛
⣤⣤⡬⠙⠓⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣦
⠭⣷⣤⣄⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠉⠉⠀⠀⠀⢠⠖⠒⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⡀⠙⠢⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠐⠛⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣋⠀⢀⡀⠀⠀⠈⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠠⠾⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢿⣤⣀⣤⣀⣀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⣨⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⣄
⣤⣤⣶⣶⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢻⡏⠀⠀⠿⠉⠉⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⣧⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣾⠟
⡬⠶⠒⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡋⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠕⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⠴⡞⠉⠀⣿⡠⢊⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⠀⠀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⢠⠀⠀⢀⣷⣤⣴⡿⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⢿⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠷⠾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡶⠿⣦⡀⠀⠀⠛⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⠉⡈⠄⠀⣀⡿⠃
⠀⠀⠀⠀⢀⣠⢤⠶⢶⡶⠠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠰⠄⠀⣠⠴⠁
⠀⠀⠀⠀⣾⠀⠀⠀⠈⠀⠀⠼⠋⠙⢱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠄⠀⠀⡴⠁
⠀⠀⠀⠀⠘⠲⠤⠤⠤⠤⢄⣀⠀⠐⠉⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠛⣿⠿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡽⣿⠁⠀⠸⣃⣀⣀⣉⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠉⠀⠻⣠⠖⠋⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠶⠜⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠋⠉⠙⠛⠓⠶⢤⣀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠛⠓⠶⢤⣄
⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠙⢦⡀⠀⢀⣀⣤⡤⠤⠤⣤⣄⡀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⡾⠁⠀⠀⠀⢀⣴⠞⠉⠁⠀⠀⠀⠈⠉⢳⣦⡀⠀⠀⠀⠻⡞⠋⠁⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⢹⡄
⠀⢸⠇⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠙⢧⡀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣷
⠀⢸⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢹
⠀⢸⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠈⢷⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣼
⠀⠸⡇⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡿
⠀⠀⢿⡀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣆⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣸⠃
⠀⠀⠘⣧⠀⠀⠀⠈⢷⡀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠈⢷⣄⠀⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⣠⠟⠀⠀⠀⢰⠏
⠀⠀⠀⠘⣧⠀⠀⢀⣬⠷⠶⠛⠉⠁⠀⠀⠀⠀⢀⣤⠞⠙⠳⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⣦⠀⠀⣰⠏
⠀⠀⠀⠀⠈⢷⡀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠹⣦⡛⠶⠶⠶⠶⠾⢿⣍⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡭⠿⠗⠶⠶⠶⠶⠞⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠒⠒⠒⠒⠚⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢱⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠀⠀⠂⢸⠆⠀⠀⠀⠀⢸⡇⠃⠀⠀⡫⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⠃⠈⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢧⠀⠀⠐⠁⠘⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣐⣮⠋⠱⡀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢠⠋⠙⣴⣂⠄
⠀⠀⠀⠀⠀⣀⣠⡤⢼⠏⠉⢻⡀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⢀⠟⠉⠻⠶⢤⣄⣀
⠀⠀⢀⠔⡛⠁⢺⠀⠈⠆⠀⠀⠀⠀⣀⠴⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢦⡀⠀⠀⠀⠀⠐⠀⠈⡇⠈⢛⠢⡀
⠀⢰⡁⢀⠁⠀⠀⠀⠀⠀⢀⣠⡶⠞⠁⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠈⠳⢦⣄⠀⠀⠀⠀⠀⠀⠈⡀⢈⡆
⠀⠈⠣⢄⣀⣀⡠⠽⠛⠋⠉⠀⢂⠀⠀⠀⡀⠃⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠂⠀⠉⠙⠓⠧⢔⣀⣈⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠣⠀⠀⠀⠀⠖⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠤⣀
⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⢀⡿⠀⠑⠦⣠⠖⠤⢀⡀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠘⢃⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⡀⠤⠒⣄⡤⠊⠁⢿⡀
⠀⠀⢞⠁⠀⠀⠀⢸⠃⠔⠂⠁⠉⠉⠐⠒⠤⠤⠤⣀⣀⣆⣀⣀⣡⣀⡀⠤⠦⠄⠐⠂⠉⡉⠀⠑⠲⠘⣇⠀⠀⠀⠈⣹⠂
⠀⠀⠀⠑⢄⡀⠀⣯⣄⡀⠀⠀⣎⡇⢰⠖⠐⡦⠀⡤⡀⣀⡀⣠⠀⡤⡀⡤⠀⣖⠀⢇⠈⢧⠀⠀⢀⣡⣼⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠉⠒⠿⣦⣉⡶⠤⣌⡀⠉⠃⠀⠃⠘⠤⠃⢟⠇⣯⠀⠟⠄⠋⠀⠓⠂⠛⣁⣀⠤⢶⣋⣰⠿⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠙⠒⡶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⢖⠛⠋⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠁⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠓⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢄⠀⠀⠀⠀⢀⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡀⠉⠲⢄⠑⠀⠐⠚⠛⠛⠛⡃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⣏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣀⣤⣼⣧⠤⠤
⠀⠀⠀⠀⠁⠒⠢⢤⣀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠔⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⢀⢰⠀⠀⠇⡀⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠡⡈⢆⢃⣬⠬⡴⠚⠙⠲⡥⠤⣘⠰⢁⠌⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡑⢄⡱⡜⣢⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢢⡡⢎⡠⢊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠌⣂⠥⠺⣿⣧⣤⣠⣶⣦⣀⠀⣠⣤⡀⢼⠓⠬⣐⠡⠔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡡⠊⠁⠀⠀⣈⡇⣿⣿⣿⡟⠻⠟⠋⠙⣿⣾⡶⠒⡎⠑⢌⠠
⠀⠀⠀⠀⠀⠀⢠⡤⣤⣾⣁⠀⢀⠴⠛⠿⣇⣿⣈⣿⡇⠀⠀⠀⡠⠼⠿⢿⣭⣍⠲⣌⣧⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⢳⣹⣷⣿⣷⣏⣾⣿⣷⣿⢸⣿⣿⡇⠀⠀⠎⡔⣿⡿⣦⣹⣿⣿⣿⡀⠀⡎
⠀⠀⠀⠀⠀⢀⣴⣪⣾⣿⢿⣿⠿⢿⡏⣼⣿⢸⣿⣿⣇⠀⠀⢧⣣⣿⣧⣙⣛⠛⣿⣿⣿⠀⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⢾⣿⣿⣾⣷⣜⣿⣟⣿⣿⣿⣦⠄⠈⣻⣋⣥⡿⠟⠋⢹⡿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⣇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢻⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠛⠿⠿⠦⣤⣤⣀⣀⣤⣤⠤⠤⠴⠒⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠂⢰⠀⠀⡆⠐⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⠤⡀⢀⣀⠥⢤⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠞⠀⠀⠀⠙⢶⡗⠋⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡀⢈⣀⠉⠄⢹⠀⠀⠀⠀⠀⠀⠀⡠⠾⣄⠀⠑⠆⠀⢀⠏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀⠧⣀⣀⠀⠀⡸⠀⠀⠀⠀⠀⢀⠼⣅⠀⠈⠑⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠉⠑⠒⡄⠑⡀⡇⠀⠀⠀⠀⣀⣇⡀⠀⠙⠂⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠂⠀⠈⠉⠄⠔⢹⠁⠀⠀⠀⣜⠀⠈⠛⠦⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⠐⡀⠀⠀⠀⠀⡼⠀⠀⠀⡔⠉⠳⢤⡀⠀⠀⠀⠀⢀⠞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠐⢠⠀⠁⢆⡇⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠦⡄⠀⠁⠤⠚⣸⠁⠀⢨⠀⠈⠑⠀⠀⠀⠀⠀⢠⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣰⠇⠀⢐⠙⠲⢤⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡶⢙⡆⢀⡴⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡼⠉⠁⠀⠀⠀⢀⡀⠳⢤⡀⠀⠠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡸⢠⠇⠀⠀⠀⠀⠀⠈⠙⠤⠜⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠢⣀⠀⢙⣋⡘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡀⠀⣇⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢠⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡂⠀⠀⠀⠀⣀⡤⠠⠤⠤⠤⠤⠤⠤⠖⠒⠊⠁⢠⠀⠀⠈⠢⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⠞⠁⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠂⠀⡠⣡⠏⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⡄⠀⣿⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⢧⠀⠀⠀⠑⠄⠀⠀⢀⡨⠞⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠰⠀⠈⠦⡀⠀⠀⠆⠀⠀⠘⡄⠀⠀⠈⠃⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢶⠖⠁⠀⠀⠀⠈⠑⠢⠤⢄⣀⣀⣀⣀⣠⡤⡤⠖⢛⠩⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢁⠀⠈⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⠄⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠄⠀⣀⡀⠤⣀⣀⣀⣀⠀⢀⣀⠀⠉⠢⢄⡀⠀⠀⠃⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⠴⠛⠺⠁⠀⣀⣀⠀⠀⠀⢤⡀⠀⠱⡄⠀⠀⠑⡀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠐⠀⡇⠀⠀⠀⢡⠀⢹⡀⠀⠲⠿⠲⠦⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⣀⢀⣀⠀⣀⡠⠼⠥⠖⠊⠀⠀⠀⠀⠀⢇⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⡀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⣧⢸⣸⡸⡰⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠌⣀⣈⣤⠀⢀⠀⠀⠀⢬⣓⣬⣾⣿⣿⣟⣥⣐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡆
⠀⠐⠀⠀⠁⢡⣙⠃⠒⠤⠀⠀⠀⡦⠶⣿⣿⣿⣿⣿⣿⡭⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠠⠀⠀⠀⠄⠁⠀⡀⠀⠀⠀⠐⢛⡩⢻⢿⢿⢟⢮⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⢀⠀⠀⠀⠈⠐⠻⠸⠸⠘⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⡄⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠐⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⡈⣐⡚⣼⣄⣤⠁⠀⠠⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠐⠂⠄⠀⠀⠀⠀⠄⠪
⠀⠀⠀⠀⠀⠀⠀⠂⡩⠙⣿⣿⡁⠖⠀⠠⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠄⡡⠢⢓⠉⠀⠂⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠂⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠐⢀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣄⣀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⡿⢯⣉⣦⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡚⢀⣸⣿⣿⣿⣿⣿⣏⣩⣿⣿⣴⣾⣿⣿⣿⣿⣟⢟⣿⣷⣿⣿⣛⡿⣿⣷⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣤⣴⣿⣷⣼⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠟⠿⠟⠛⠻⢿⣿⣾⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡼⠃⠀⢿⣽⣿⠟⠋⠉⠉⠻⢿⣿⣿⣯⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢻⣰⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⠊⣻⡿⠏⠀⠀⠀⠀⠀⠀⠉⢻⣯⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⢁⡀⣼⣿⠅⠀⠀⠀⠀⠀⠀⠀⢰⣦⣉⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡅
⠀⠀⠀⠀⠀⠀⠀⠀⡯⢶⣿⣦⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⠉⣿⣵⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣮⣿⣿⣿⢻⣿⡆⠀⠀⠀⠀⠀⣠⣴⣷⡀⢼⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⠀⢀⣠⣶⣿⣿⣿⣿⣧⡈⢿⣿⣿⣽⣿⣷⣄⡀⠀⠀⠀⠀⢀⣿⣿⣿⢿⣾⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣟⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠆⠙⢿⣿⣿⣯⣿⣿⣶⣤⡀⢀⣾⣿⣿⣿⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣽⣿⣏⣿⣿⢿⣿⣯⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⠋⣙⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣩⣿⣿⣿⣿⣿⣿⢿⣿⣿⣶⣤⣤⣄⣀⣀⣀⣠⣤⣾⣿⣿⣷⣌⠓⡊⠉⠉⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⣌⣉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠉⠻⣿⣿⣍⡙⣿⣛⣿⣿⢽⣿⣿⣾⣿⢿⣿⣿⣿⡾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠶⠒⠒⠒⠒⠒⠒⠒⢶⣶⣒⣒⡲⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠉⠀⠀⠀⠀⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠂⢤⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡏⠀⠀⠀⠀⠈⠉⠒⡤⣄⡀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠁⠂⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡂⠀⠀⠀⠁⠒⢤⡜⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⢀⣀⠀⠀⠀⠘⠓⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⡇⣿⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠰⣇⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⡇⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⡇⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣶⣷⠀⠀⠀⠀⡀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⢷⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡁⠰⠀⠀⠀⡀⠀⠀⠉⠀⡤⠀⠀⠀⠀⠀⠀⢧⣾⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⢈⠂⢄⠀⠀⠾⠃⠀⠀⠀⠀⠐⠀⡄⠀⠠⡿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣌⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠘⠇⡇⠀⠀⢠⠃
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣤⣤⣀⣀⠀⠈⠙⠓⠤⣐⡠⠐⢶⡄⠀⠀⠀⣣⠀⡠⠃
⠀⠠⣾⣿⣿⡟⣿⢻⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠉⠑⠒⠤⢤⣠⣼⣿⠔⠁
⠀⠀⣿⣿⣿⣏⣿⣣⣃⣕⣩⣼⣿⣿⣿⠇
⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣷⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⢇⣸⣿⣿⣿⢻⣿⣿⣿⣶⣦⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣟⠿⣿⣟⣷⣧⣧⣿⣿⣿⡿⣻⠿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣦⣴⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⡉⠛⢯⣿⣿⣿⣿⣧⣿⣾⣿⣿⣿⡿⡷⣯⣿⣿⢿⣽⣿⣿⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⣬⣉⠛⠿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣾⣧⣾⣿⣿⣷⣾⣿⣿⣿⣿
⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣛⣻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣿⣷⣶⣶⣶⣶⣬⣭⣿⣿⣿⠿⠋
⠀⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠹⣿⣷⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠙⢿⣿⣿⣧⣿⣿⣿⡛⡿⣿⣾⣿⡿⣿⣻⣿⣿⣿⣶⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠙⢿⣿⣿⣿⢷⡟⢰⣿⣿⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣾⣧⣿⣿⠏⣤⢼⣿⣿⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⡿⠛⠟⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣾⣟⣾⣽⣟⠁⣾⣿⣿⡟⣾⣿⣿⣿⠟⢻⣿⣿⣿⢋⢿⡯⢿⢿⣛⣿⣿⣿⣿⣄⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣶⣝⣿⣿⡧⠙⣙⣿⣿⠅⣯⢿⣿⣿⡘⠎⡿⣚⣦⣌⠊⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣦⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⠋⠉⠉⠀⠀⠠⠠⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠋⠁⢀⠠⠔⠐⠒⠒⠒⠰⠤⣀⠈⠙⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠒⢄⡈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢠⠜⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠂⠀⠀⠐⢄⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠏⢐⠋⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⢆⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡏⢀⡎⠀⠀⠀⢀⡔⠀⠀⢉⡡⠤⠤⢬⡁⠀⠀⠰⡄⠀⠀⠀⠈⡆⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠁⢸⠀⠀⠀⠀⣪⠁⠀⢰⡋⢀⣀⡀⢈⣳⠀⠀⠀⢻⠄⠀⠁⠀⢔⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢰⠀⠀⠀⠐⡏⠀⠀⠀⣿⣼⣿⣿⣿⣇⣀⠀⠀⢘⡇⠀⠀⠀⢚⠀⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠐⡆⠀⠀⠀⣧⠀⠀⠀⡿⢸⡇⢹⠈⣿⠉⣷⠀⢐⠇⠀⠀⠀⡆⢠⠇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠘⡄⠀⠀⠻⠀⠀⢸⡇⢸⡇⢸⠀⣿⠀⡿⠀⠨⠀⠀⢀⡜⢀⡾⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⠖⠈⠿⠷⡄⠈⢆⠀⠀⠀⠀⢸⣧⣼⣇⣼⣄⣿⡶⠃⠀⠀⠀⣠⣎⠠⠾⠇⠐⠲⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠗⠁⠀⠠⠀⠀⢀⢀⠉⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠈⠁⡀⠠⠤⠀⠀⠀⠺⣽⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣛⠅⠀⠀⠀⠠⠊⠜⡄⠈⡉⠁⠂⠐⡄⡖⠀⣒⡂⢆⡆⠐⡬⠀⡇⠀⠁⠂⠀⠂⠀⠀⠨⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣤⣤⣀⣀⣀⠀⠁⠀⠁⠈⠈⠃⠁⠃⠂⠀⠀⠀⠁⠀⠁⢀⣀⣠⣤⣤⣶⣾⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢶⡷⠶⠶⠶⠶⠶⣾⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠜⠷⠀⢠⡅
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣴⣿⣿⣷⠀⠀⣆⣴⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⣿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣘⡁⠀⠀⠹⢿⡿⢿⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠶⣾⠶⠶⠶⠶⠶⢶⡷⠶⠶
⠀⠀⠀⠀⠀⠀⠒⠛⠓⠀⠀⠐⠂⠀⠚⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠋⠀⡯
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣧⣀⣀⣸⣿⣿⡿⠛⠉⠀⠀⠀⠀⠸⠿⡿⢇⣈⣉⣉⡁⠠⣴⠃⠀⢸⣷
⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⡇⠀⠈⠀⠀⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⢀⣤⠖⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠀⢀⠀⠀⠀⠴⠒⠒
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⡄⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⡄⢀⣿⣿⣿⣿⣿⠶⠶⢶⡄⠀⢸⣿⣼⣿⣿⣿⠀⠀⢸⣆⣸⣿⣿⣿⠀⢘⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣷⣾⣿⣿⣿⣿⣿⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣤⡾⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⡾⠃⠀⠀⢸⣿⣿⣿⠁⠀⠀⠘⢞⣿⣟⣿⡿
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢷⠾⠋⠁⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠛⣻⡧
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⠳⠿⠛⠀⠀⠀⣀⣀⣹⣿⣦⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⣤⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⣿⣷⣄⣰⣿⡿⠿⣿⣷⡀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣌⠻⣿⣷⡙⣦
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣶⡆⡀⢁⡄⠀⠀⣾⣿⡇⠀⠈⢻⣿⣿⠋⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⢹⣿⣷⢸⡆
⠀⠀⠀⠀⠀⠟⠛⠻⠿⠿⣿⣿⢧⣶⠟⠀⠀⠀⠸⣿⣧⡀⠀⣾⣿⣿⡇⠀⠀⣼⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠸⠁
⠀⠀⠠⣤⠚⠛⠛⠛⠒⠲⠶⠶⢶⣿⣤⠀⠀⠀⠀⠹⣿⣿⣿⡿⠻⣿⣿⣶⣴⣿⠟⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣇⣸⡿⢣⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠙⠛⢿⣿⣷⣶⣿⣿⠟⠛⠀⠀⠀⠀⠀⠰⡯⠻⠛⢿⣿⣿⣿⡷⠞⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂

⠀⠀⠀⠀⠰⣦⣄⣀⠀⢀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠐⠦⣀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣯⣿⣿⣻⣷⣦⣄
⠀⠀⠀⠀⢨⣿⣿⣿⣧⣼⣿⣿⣿⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⡇⠀⠀⠀⠘⢧⡀⠀⠀⠀⢀⣾⣧⣿⣟⣿⣿⣿⣿⣿⣿⣽⣧
⠀⠀⠀⢀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣴⣶⠆⠀⠀⣼⡇⠀⠐⡄⠀⣧⠀⢀⠐⠀⠘⣧⠀⠀⠀⡾⣿⣏⣿⠟⠁⠀⠀⠙⣿⣿⣻⡿⡇
⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⡆⠀⠀⠹⣤⣿⣠⠆⠀⠀⢀⣿⠀⠀⠀⣿⢿⣿⣻⡄⠀⠀⠀⠀⣼⣻⣿⣿⡇
⠀⠀⠀⠚⠿⢿⣿⣿⣿⣿⣽⠿⠿⠧⠀⠀⠀⠀⠹⣧⡀⠀⠀⢻⣿⡟⠀⠀⢀⣼⡏⠀⠀⠀⢻⣿⣿⣻⣿⡦⣤⣤⣾⣿⢿⣿⣽⠃
⠀⠀⠀⠀⢠⣾⣿⣿⡿⣿⠃⠐⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠈⠙⠁⠀⠀⠻⠋⠀⠀⠀⠀⠀⠙⢯⣽⣿⢿⣿⡷⣿⣟⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠾⠛⠛⠛⠛⠛⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠉⠛⠳⣦⣤⣴⠟⠋⠁⠀⠀⠀⠰⠆⠀⠶⠀⠀⠀⠀⠈⠙⠻⣦⣄
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢀⣾⠋⠀⠀⠀⠠⡄⠀⠀⠀⠘⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠀⠀⠀⠉⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠈⢷⣤⣴⠶⠟⠛⠳⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⣰⠟⠀⠀⢀⡄⠀⠀⠀⠀⠁⠀⣠⡴⠾⠛⠉⠉⠉⣹⠟⠻⢶⣄⡀⠀⠀⠀⠀⠀⡀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⣠⠀⠰⠀⠀⠀⠸
⢠⡿⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠉⠻⣄⠀⠀⠈⠭⠵⠶⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠹⣧
⣼⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⠁⠀⠀⠀⠀⠀⢠⡿⠀⢀⡄⠀⠀⠀⣽⣧⠀⠀⠘⠋⠁⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⡀
⣿⠀⠀⠀⠛⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡟⢹⡆⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠃⠀⢰
⣿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⠐⠂⠀⠀⠸⣇⢸⡇⠀⠀⠀⢤⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣾
⢹⡇⠀⣠⠞⡄⠀⠀⢠⡄⠀⠘⣷⡀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠘⠀⠀⣼⠏
⠈⣿⡀⢁⡞⢡⠆⠀⠀⠀⣀⠀⠈⠻⣦⣀⠀⠀⠘⣧⠀⠀⠉⠀⠀⠀⠀⢳⣄⠀⠈⠃⠀⠀⣀⢀⣿⠀⠀⠀⠀⣠⣴⢟⡁⠀⠀⠀⠀⠀⣼⠏
⠀⠘⣧⠈⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣿⣆⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⣤⠀⠁⣼⠃⢀⣠⣴⠞⠋⠀⠈⠁⢠⡄⠀⢀⣼⠋
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⠘⠂⠀⠈⠻⣦⣄⠀⣼⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠈⢻⣆⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣀⡀⠀⠀⠀⠀⠛⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠛⠀⠀⠀⠈⠀⠀⠀⠀⣠⡾⢷⣄⠀⠀⠐⠃⠀⠀⠀⠀⠠⠦⠈⠉⠛⠛⠶⠶⠶⢦⣴⣾⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠉⠳⣦⡀⠀⠀⠖⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣄⣀⣀⣠⡶⠛⠁⠀⠶⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡈⢧⠀⠀⠀⠀⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠛⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢙⣳⣦⣄⣀⠀⠀⠀⠀⠹⡄⠳⠘⡆⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠈⠀⠀⠀⠀⠛⠀⠀⠀⢀⣾⠛⠛⠛⠉⠁⠈⠙⠛⠷⢶⣤⣤⣀⣀⣀⣀⣀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠘⠃⠀⠀⢠⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣷⣶⣾⣿⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⢰⠄⢇⣞⡸⡣⠏⠴⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⡆⣣⡼⠞⠛⠉⠉⠉⠉⠛⠲⢦⣞⢡⠖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢦⠥⠞⠛⠒⠒⠤⢀⠀⠀⢀⠠⠐⠒⠛⠳⠬⢥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠁⠀⠀⣀⠀⠀⡤⠚⠆⠀⠟⡦⡄⠀⢀⡀⠀⠀⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⣉⠀⠲⡀⠙⠊⠉⠋⠁⠀⠀⠀⠑⠋⠉⠚⠁⡰⠀⢐⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢠⠀⣼⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠄⠀⢽⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⠀⢀⡗⡄⣤⣄⢠⣶⡄⢿⣿⢇⣴⣦⢀⣤⡀⢂⣇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⠀⢘⣿⣬⣿⣿⣶⡿⣿⣿⣿⣿⣷⡷⣾⣿⣯⣸⡿⠀⢠⠇
⠀⠀⠀⠀⠀⢀⣠⠄⣠⣴⣿⣿⠻⣿⢹⣏⣹⣧⣧⣧⣴⣯⣾⣇⣿⣹⣿⠻⡿⣷⣮⣀⢀⣀
⠀⠀⠀⠰⣿⣿⣯⣾⣿⣟⣩⣿⣶⡿⠿⠛⢛⡛⢫⡝⠻⣍⠛⣛⠛⠻⠿⣷⣧⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠈⠛⠉⠈⢻⡿⠛⠉⠀⠘⢡⠗⡬⢙⠛⡛⣛⢛⢉⡍⢶⠹⠈⠀⠈⠉⠻⠟⠁⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠘⠃⠛⠘⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠶⠒⠚⠋⠉⠉⣡⠞⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠴⠶⠶⠶⠒⠚⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢸⡿⠥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠉⠛⠲⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⢦⣤⣄⣀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡴⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇⠀⠀⣾⡷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣞⣱⠆⠀⠀⠀⠀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣅⣠⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠺⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠶⢤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠦⣤⣤⣀⣀⣀⣿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠀⠀⢣⠀⢠⠀⢠⠁⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠂⢸⠀⠂⢀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⠀⠀⢉⣀⡀⠀⠀⠐⠈
⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣼⡿⠛⣷⡀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⢀⡀⡼⣿⣦⣤⣝⠂⠐⠀⠀⠐⠠⠄⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠔⠈⡀⠐⠁⡉⠀⠑⢀⠁⠈⠀⢀⠐⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⠄⠀⠀⠁⠠⠀⠡⠀⠀⠀⠑⠄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠂⢠⠀⠀⠀⠈⠀⠀⠁⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⡀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠊⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠠⠀⢠⠃⠀⠀⠂⠀⠀⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠄⠂⠠⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠠⠁⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠩⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠜⡇⠀⠀⠀⠀⠀⠀⢀⡴⡉⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠘⠦⡀⣀⣀⠤⠶⠃⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠈⠒⠁⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠆⠀⠀⠉⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠑⢤⡀⠠⠤⠀⢀⡀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠉⠀⠃⠈⠀⠀⠀⠠⠄⠀⠓⠐⠈⠁⠀⠈⢣⠀⠀⠀⠀⠀⠈⠁⠃⠂⠆⠤⠤⠔⠒⠉
⠀⢀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠂
⠀⠁⠀⠀⠀⠐⠄⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢀⡜⠁⠰⠛⠀⠀⠀⠶⠄⠈⠣⡀⠀⠀⠀⡠
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⡜⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢱⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⡀
⠇⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠸⠇⢠⠸⠇⠀⠀⠀⠘⠁
⠀⠁⢹⠀⠀⠈⠃⠂⠀⠀⣰⢆⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⢀
⠀⠀⢸⠤⢀⡠⠀⠀⠀⠨⣏⠑⡃⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠆⠀⠂⠀⠰⠀⣿⣀⡉⠸⡘⢣⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡔⠊⠲⣤⢄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠄⠋⠢⡘⠢⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⣆⠰⠤⠮⠴⡆
⠀⢠⠃⠀⠀⠀⠂⠀⠐⠀⠸⡗⠁⠀⠀⠀⠈⠒⢌⡐⠠⢀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⠀⢀⡔⢁⡜⠀⠠⡖⠁
⠀⠈⠑⠒⠤⠤⠤⠤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⠀⠄⠴⢋⠔⢁⠉⢐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠈⡀⠀⠀⠒⠀⠀⠀⡁⡀⢸⠁⠉⠀⠂⠁⠀⠨⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠁⠀⠉⠀⠉⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⢀⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⠤⠆⠀⠀⠀⠀⠀⠠⠆⠃⢸⠀⠀⠀⠀⠀⠘⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠤⢠⣤⠤⠤⢶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣓⢲⠒⠳⠒⣒⣊⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⣁⣀⣀⣀⣈⣉⣠⢸⣈⣉⣀⣀⣀⣀⣈⣲⡄
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢠⢰⣖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢠⣤⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⡀⠀⠀⠀⢽⣦⠄⠀⠘⢉⣿⣖⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢢⣌⠈⠀⢁⠝⠳⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⠂⣸⢰⣾⢏⡞⣼⣷⠀⡠⠀⠖⠀⠀⠀⠀⢉⢿⣭⡅⠀⡀⠀⡀⢠⢠⢠⡆⡀⠀⢀⣉⠋⠁⠀⠀⠐⠿⣝⠿⠄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡙⢦⠡⣔⠈⡏⡾⠐⡿⣫⠞⣠⠾⡃⣷⡟⠰⠀⠈⢷⠿⠷⢤⡈⠃⠳⠈⠋⣘⢿⢔⣅⣤⡄⢆⣠⠀⡀⠀⢦⣛⢿⡃
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣖⠌⢀⠁⠘⠀⠃⠀⣠⠖⡡⠎⣲⡿⢷⣛⡯⠀⢰⠁⠎⠀⠒⠾⢿⠎⠀⠀⠐⢁⠤⠿⠿⢾⢵⠟⠇⠌⠃⣀⠘⢏⠛
⠀⠀⠀⠀⠀⠐⠛⠛⢺⣿⡢⠶⠁⠀⠀⠀⠔⠁⠀⢴⣩⣵⣾⣯⣶⣤⣤⠀⠀⠀⢀⠰⠆⠀⠀⠀⠀⠀⠐⠉⢹⣥⣭⣉⠀⠀⠀⠀⠇⠘
⠀⠀⠀⠀⠀⠀⣂⣀⠀⠈⠉⢀⠀⠉⠀⠀⠀⢐⣚⣢⣬⣤⠤⠶⠶⠶⠶⠤⠀⠀⠀⠀⡀⢚⣤⡆⢠⡀⠀⠙⢦⣍⢾⣕
⠀⠀⠀⠀⢀⣩⠴⠒⠂⢀⡠⠄⠀⠀⠈⠀⠀⠀⢈⠄⠀⠤⣽⡿⢶⣦⣤⡀⠀⠀⠀⠈⠐⡿⡻⣠⠈⠇⢀⠀⠸⣌⠛⠂
⠀⠀⠀⠀⠀⠔⠪⠀⠀⠜⡴⡀⣢⡄⠄⣀⠀⠀⠀⠑⢧⣤⡉⢙⣯⣓⠶⠄⠀⠀⠀⢠⠀⡇⣥⡟⣦⣶⣿⢿⡷⠈⡀
⠀⠀⠀⠀⠀⠠⢒⡥⢆⣴⡾⣣⣾⠃⠀⢿⡄⠘⠀⠀⢀⢰⡾⣷⢭⡛⠗⠀⠙⢦⣄⠲⡙⡆⢸⠛⣽⢃⣿⡿⣠⠞⢁⣤
⠀⠀⠀⠃⠀⠀⢉⠔⠹⣿⡟⣹⠏⢠⠀⠉⡇⠀⡐⠀⠀⢳⣧⡀⠑⢽⣲⢦⣐⠢⠙⠧⠙⡆⢸⠀⠁⠐⢋⡼⣫⢖⣫⣤⣀⣠⠤
⠀⠀⠀⠀⠠⠀⠀⠀⠔⠙⠰⡝⠰⢀⠇⠀⠅⠠⣧⠂⢈⣆⠉⢳⣀⣤⣄⣙⣶⡕⠠⠦⡀⠀⠈⠀⢀⠼⠋⠜⣉⠵⢚⣫⡭⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠆⠄⠀⡆⡆⣴⣿⣄⠘⡜⠃⠀⠉⠰⠾⠻⠿⣍⠺⠃⢀⡀⠀⠐⠋⠀⢀⠚⣶⠾⠿⠿⠿⠿⠶⠶
⠀⠀⠠⡀⡀⠶⢲⣧⡀⠀⡀⠈⠳⣅⢘⠏⡏⠏⣽⣣⢊⡤⠀⢀⣀⡩⠤⠀⠀⠀⠀⠤⠀⠀⠐⠈⠀⠨⣍⡽⠶⠶⣖⣚⠛⠛⠛⠻⠆⠀⢠
⠠⠭⣧⡄⠀⠀⠚⣉⣿⣞⣛⢳⡄⢈⠀⠃⠁⢠⠖⠕⡹⢾⡶⢯⡤⠖⡛⠋⢉⡠⠄⠀⠀⠀⠁⠀⠀⠀⢀⠠⡀⠰⢶⣽⠻⠷⣶⣤⡄
⠀⠴⠀⠀⠀⠀⠈⠛⢪⣭⣛⠛⠿⠃⠠⠀⠀⠁⢠⣓⡚⣛⣛⣳⣖⠩⢒⠀⡴⡠⠂⢘⣔⠠⠂⡆⠀⠀⠀⠙⢮⣦⣄⠩⣻⢭⣓⢶⣄⢀
⠀⡀⣺⣴⠂⡆⠀⠙⣾⣿⡿⠂⢀⠀⠀⠀⠀⠀⠙⠈⠩⣵⢶⣤⣩⢖⣥⠖⣠⣶⢏⣼⡿⠀⠀⣷⠀⠘⠆⠀⢀⠨⣪⣷⣮⣝⠿⡿⠋⡛⢦⣆⡀
⠀⠀⠋⠃⠀⠀⣰⠠⡙⠊⠃⠰⢶⡀⣦⠀⣄⠀⡀⠙⢶⣌⡹⣿⡓⠙⡡⢊⣿⣵⢏⡿⠁⠀⠀⢻⡇⠀⠀⡀⠀⠱⣽⡈⠛⢿⡳⠆⠈⢠⣞⡿⣷
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠊⢴⣿⢺⠃⠀⢻⠀⠀⠀⢸⣏⠻⡪⠁⠈⠀⡠⠫⢃⢮⢃⠆⢣⠀⠈⡇⠀⣇⠇⠀⣦⠙⠻⣏⡀⢀⠀⠀⠀⣄⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠎⠂⡇⠀⠀⣿⠀⣦⠙⢆⠀⠀⡀⠀⠁⠀⠈⢎⢊⢀⠊⢠⠀⠁⣰⣿⠀⠈⡞⡧⠐⠻⣿⡟⠐⡇⠀⠨⡟⠪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠈⡘⠀⢸⢸⠀⠹⠁⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠇⠸⠌
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣠⣶⣻⣿⣿⣿⠟⠻⡿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⡟⠛⠛⠛⢻⣿⣿⣦⠀⢀⣼⢿⣿⣿⡿⠁⠀⠀⠀⠁⠀⠙⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠩⣛⣿⣿⠆⠀⠈⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⠀⢋⣿⣿⣷
⠀⠀⠀⠀⠀⠀⡇⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠂⠀⣾⠙⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⢠⣽⣿
⠀⠀⠀⠀⠀⠀⡇⠘⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣄⠈⠿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⡏⠀⣄⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢈⣳⣄⠀⡿⣿⣷⠀⠀⠀⠀⠀⢀⠃⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⠈⣿⣿⣷⣤⠘⠏⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣟⠻⢿⣷⢄⡀⠠⣾⣆⠈⠹⣿⣿⣷⣄⠀⠀⠀⢀⣼⣿⣻⣿⣝⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠶⠻⠈⢁⠰⣏⣽⢻⠷⠀⠙⢻⣿⣿⣶⣤⣤⣾⢟⢹⣿⡟⢚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠁⢀⡨⡿⢇⡷⣿⣧⣄⣀⣀⣈⣁⡍⠻⢿⠙⣿⣻⣿⣧⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣠⡔⣠⡚⠁⢰⣽⢿⣿⣿⣿⣿⣟⠋⠀⠄⠀⣹⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠙⠃⢉⣁⠀⠀⠀⠈⠛⠣⠀⠁⠐⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠐⠒⠚⠉⠁

⠀⠀⠀⢀⠀⢄⠀⠠⠀⠂⢀
⠀⠀⠀⠀⠂⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠈⠀⠈⠀⠊⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢣⠀⠀⠀⣀⠀⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⡒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠐
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠢⠀⡀⠁⠈⢈⠀⠀⠀⢀
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠄⢀⠁⢀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠰⠄⡃⠰⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠺⠄⢳⡀⡿⠣⢀⣗⠅⡀
⠀⠀⠀⠀⠂⠀⠤⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⢒⡙⠦⠤⠅⠀⠐⡡⠴⠃⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢋⠛⠇⠀⢀⢩⣐⢒⡖⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⡟⣤⠊⡠⢸⣀⣈⣳⡈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⢁⡜⢣⢘⠁⣯⠛⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣿⣥⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡙⢿⣿⣿⣿⣻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡆⢻⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡙⢻⣿⣿⣯⣿⣹⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⣯⡟⣻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣫⣾⣿⣿⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣏⣵⣽⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣽⠻⣿⣿⣿⣿⣧⣿⣿⣿⠛⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣝⣿⡏⣩⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡏⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣾⣝⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣭⣾⣿⡟⣥⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣟⣻⣿⣿⡟⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣅⣡⣿⣿⡏⢳⢿⣿⣿⣿⠉⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣮⣿⣿⣿⣿⣶⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠉⠀⠀⠀⠉⠉⠛⠷⣶⡞⠋⠉⠀⠀⠀⠀⠀⠈⠉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⢿⣤⣤⡶⠶⠶⠶⣤⣄⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⣠⣴⠞⠋⠉⠉⠙⣿⣦⡄⠀⠀⠹⣇⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⢸⡏⠈⢻⣦⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣿⢻⡆⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⣧⡾⠁⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢻⣇⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣧⡀⠙⢿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣸⠇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢿⡄⠀⠀⠀⣀⣴⠾⠋⠀⠀⢈⣿⣦⡀⠉⠛⠶⣦⣤⣀⣀⠀⠀⣠⣾⢁⣀⣰⣿⣤⡶⠾⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⣈⣿⣶⠶⠛⠋⠀⠀⠀⢀⣴⠟⠁⠈⠙⠷⣦⣀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⠋⠉⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠉⠛⠶⣶⣦⣤⣤⣀⣀⣠⣤⣤⣤⣴⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⣀⣤⡶⢿⣯⣄⣀⡀⠀⠀⣀⣀⣀⣠⣤⡶⠞⠋⠁⠀⢈⣩⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣠⣤⣴⠶⢿⣏⡁⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠛⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⡶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⣤⣀⣀⣀⣀⣀⡀⠤⠀⡠⣶⣿⢶⣧⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⣧⣬⣰⠿⡶⠀⡆⠀⣼⠓⠛⢦⠑⣟⢿⣿⡟⢩⣿⣗⣠⠤⢄⣢⣼⣯⣿⣿⠶⠖⢃⣃⠀⢲⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣇⠛⠋⢨⢻⡀⠀⠈⢉⣠⠋⢊⠀⠀⢳⣌⣘⠉⣹⡾⣿⣷⣶⣿⣿⡿⠿⣛⣭⣶⠉⠁⠀⣸⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣂⠁⠞⠏⢿⡷⢾⣽⢧⠀⣸⣧⣰⣶⡨⣿⣿⣿⣿⣮⠻⠟⠋⣡⣾⣿⡟⢹⣗⣬⡖⠡⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣾⠈⠹⢧⡼⠿⠾⣿⣉⣭⣦⣽⣿⣥⣤⢾⣿⡹⣯⢹⣿⣾⠃⣿⣃⣼⠷⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡏⠀⠀⠀⠀⣽⢃⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠊⢁⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠸⣇⡀⠀⠀⢘⣵⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣥⠀⠀⠀⠉⢷⠀⠋⠉⠿⣿⣯⢝⣟⡛⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠄
⠀⠀⠀⠀⠀⢠⠤⠄⠖⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠀⠀⠀⠀⠀⠀⠙⡀⢀⣀⠔⠁⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠒⡄⠀⠀⡘⠘⠄⠠⠤⠒⠂⢠⡄⠖⠛⢄⡀⣀⠀⠀⠀⠀⢀⢀⡴⠠⢀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⡴⠈⠸⠎⠁⣤⠀⠀⠀⠀⠀⠄⢸⡇⠀⠀⠀⡇⠈⠓⠒⠔⠉⢹⠀⠐⠀⣠⣕⡒⡒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣛⠉⠉⠁⢺⠍⠀⠀⠀⠀⠀⠀⣾⠀⢰⠀⠀⠃⠀⠀⠀⠀⠀⢸⡆⠠⣼⠁⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣌⣉⣉⣉⡗⠀⠠⢀⠀⠤⣀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢸⢂⡆⠀⠀⠀⠀⠀⠁⢰⢲⠁⠀⠀⡄⡄⠠⠂⠒⠀⠀⠁⠀⠘⡤
⠀⠀⠀⠀⠀⠈⠉⠓⠒⠊⠉⠂⠀⣸⢠⡛⠁⠀⠀⠀⣤⠋⣸⠀⠀⠀⡇⢧⠀⠀⠀⠀⢀⡀⢸⠄⠳⣤⣀⡀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣹⡛⠳⠶⡶⢶⠛⡹⠀⠀⠀⡷⢤⣝⣀⣀⣀⣤⠴⢸⠆⠊⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠑⠌⠶⠯⠭⠥⠰⠆⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠈⠉⠓⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣊⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠃⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠘⠤⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠤⠚⡆
⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⣀⣀⣀⣀⠤⠤⠤⠖⠒⠒⠒⠉⠉⡩⠓⠦⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠚⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⡴⠊⠲⡄⠀⠀⠸⡄⠀⢀⡗⠀⠀⠀⠀⢠⠊⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠳⢄⡤⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠈⠉⠢⡀⢰⠁⠀⠀⢳⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡇⠈⠢⠤⠤⠃⠀⠀⡔⠉⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡴⠒⠀⠒⢦⡀⠀⣉⣒⣋⣀⠤⠤⠤⠃
⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⢀⣀⣀⣀⠠⠤⠤⠤⠒⠊⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠘⠒⠒⠂⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠶⠶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣟⠒⢶⣶⣤⣄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣄⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⢀⣴⣿⣇⣸⣿⣿⣿⣏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣰⣿⣿⣤⣾⣿⣿⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣧⣬⣷⡄
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢧⣼⣿⣿⣯⣉⣿⣿⣁⣹⣿⣦⢰⣿⣿⣿⡋⣹⣿⣿⣧⠞⠉⠉⠉⠉⠻⣷⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢀⣿⠿⣿⣿⣏⣠⣿⠟⠁⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡻⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⣿⣿⣿⣿⣷
⠀⢸⣿⣤⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠻⣿⣿⣿⣿⣿⣝⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⣿⣿⢿⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣼⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢉⣿⣿⣮⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢿⣿⣷⣯⣻⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣠⣴⣾⣟⣿⣿⣿⣷⣼⣿⣿⣿⡟⢳⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⠟⢻⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣶⣤⣄⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣟⣵⣿⣿⣿⣿⣿⣾⣿⣿⠻⣷⣽⠿⣥⣬⣿⣿⣿⡛⢻⣿⣿⣿⣟⣩⣿⣷⣶⣿⣟⡿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⢯⣿⣿⣏⣼⣿⣿⡿⢿⣿⣿⣿⡷⠟⠉⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⡻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠻⢿⣿⣿⣿⣧⣾⡿⣿⣽⣾⣷⣤⣀⠀⠀⠀⠈⠙⠛⠿⣿⣏⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⢿⣻⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣟⣿⠿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⣠⣤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⣄⣀⠀⠉⠛⠶⢤⣀
⠀⠀⡇⠈⠉⠓⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠙⠲⠦⢤⣽⠗⠀⠀⠀⠀⠉⠛⠶⣤⣀
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠉⠙⣳⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠉⠛⠶⣄⡀
⠀⠀⡿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠴⠛⠒⠶⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⡀
⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⠖⠒⠛⠛⠒⠦⡀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣽⠒⠒⠒⠲⠶⠦⠤⠤⢽⡇
⠀⠀⡇⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⠁
⠀⠀⡇⠀⠀⠀⢀⡴⠛⠉⠳⡄⠀⠀⠀⠀⠸⣄⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡏
⠀⠀⡇⠀⠀⢰⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⣿⠀⠀⠈⠳⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡏
⠀⠀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠓⠶⠤⣤⣄⣀⡀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠐⠃⠀⠀⠀⠀⣰⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠿⣅⣀⡀⠈⠉⠳⡄⠀⠀⠀⡀⠀⠀⠙⢦⣄⣀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠦⣿⣀⡀⠈⠛⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀⠀⠀⠀⠉⠙⠛⠲⠦⣤⣄⣀⢸⠀⠀⠀⠯⠭⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⣶⢾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠸⣇⠘⠃⡉⠉⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⡾⠟⠒⣾⣥⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⠋⠀⠀⣉⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠰⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⢿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠂⢀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣷⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠀⣠⠤⠀⠀⠀⠀⠚⠛⠁⠈⠙⠻⣿⢿⣗⣲⢟⣧⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣀⡠⢾⣶⣿⡟⠀⠉⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠲⠉⠁⢸⣿⣿⣷⣶⣆⣴⣿⡟⠃⢸⡋⠉⠀⠀⠀⠀⣿⡅⠘⠓⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⡿⠟⠻⣿⣿⡟⣩⡟⠀⠀⠘⠋⠁⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⣤⢆⣰⣿⣏⣴⠋⠀⠀⠀⠀⠀⠈⠉⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣅⣤⡀⣴⠀⠀⠉⠹⠛⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣤⡀⢀⣗⠀⠠⠂⠀⠀⢂⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⡄⠀⢰⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣤⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠙⢿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⢾⣷⣄⡀⠀⠀⠀⢀⣠⣠⣾⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠛⠛⠛⢿⣿⣿⡿⣿⣿⣿⡿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠸⡄⢰⣄⣼⣿⣿⣧⣼⣿⠟⣁⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠛⣿⣿⣿⡏⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢸⣿⢁⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣗⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢟⠛⣦⠀⠀⠀⠠⣤⡤⠀⠀⠀⢠⡟⣻⠟⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠊⣰⠏⠀⠀⠈⠙⡿⡟⠉⠀⠀⠈⣧⠈⠄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⢠⡿⠀⠀⠐⡦⣸⡁⣷⡴⡖⠀⠀⠸⣧⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣢⣿⠁⠀⠀⠐⠃⠘⢧⡏⠀⠓⡀⠀⠀⢻⣦⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠐⢟⡛⢋⣆⠄⠸⠀⣤⡞⢛⣻⠗⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠃⢠⢤⡀⠀⠀⡎⢀⡠⡀⠘⡎⠀⠀⣠⢤⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡈⡇⡏⠀⢣⣠⣼⡷⠁⠀⠀⢱⣷⣤⣰⠃⢸⢀⣏⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⡞⣧⠀⠘⣿⡏⢁⠀⠀⠀⣀⠙⣿⠋⠀⢸⠏⡏⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠤⠞⠁⠘⠀⠀⣿⠀⠳⣒⡄⣔⡺⠀⢸⡇⠀⠈⠀⠹⠦⠼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠠⡄⠂⠀⠀⠀⡏⠀⠀⠉⠀⠈⠁⠀⢸⡇⠀⠀⠀⢂⡤⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠣⠌⠑⠂⢀⠀⢿⣠⡤⠔⠀⠒⠤⣄⣸⠃⢀⠀⠒⠉⠴⠋⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢂⢀⣀⡀⢈⣷⡄⠈⠙⠛⠛⠛⠛⠁⠀⣴⣏⠀⣀⡀⣀⠺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠘⣋⣉⠉⠉⠛⢠⡇⢠⠄⢄⠀⢠⠤⢄⠀⣇⠙⠋⠉⢉⣉⡓⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡞⠅⠒⠄⡀⠀⠾⣄⡀⠐⡎⠀⠈⠖⠀⣠⡽⠆⠀⡀⡖⠊⠙⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢈⡼⠀⣀⠀⠀⠙⢦⠀⠀⠈⣠⠞⠁⠀⢀⡄⠹⣄⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠷⠶⣤⣉⠒⢄⠀⠣⠀⠰⠁⢀⠔⢈⣤⡴⠶⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠁⠀⠀⠀⠐⣡⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠙⠦⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣀⡤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⢶⡛⣶⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⠛⠹⡽⠈⠳⠟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢤⣈⠙⠒⠚⢙⠄⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡊⠳⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠙⠢⢤⣈⣉⡹⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠑⢄⡉⠓⠶⢤⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢯⠀⠀⠀⠀⠀⠉⠓⠒⠊⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠉⠈⢣⡈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⢻⠀⢧⠀⠀⠀⠀⠙⠲⠤⠜⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠳⡀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁
⡔⠶⢤⣀⡀⠀⢸⢷⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠈⠲⣄⠀⠉⠓⢿⣻⠽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠉
⠀⠀⢸⠀⠀⣠⠞⣟⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠸⡄⣰⠃⠀⠈⠳⢄⣀⡀⣀⣀⡤⠴⠒⠉⠁
⣶⣶⠶⢾⡷⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠂⢻⠀⠀⠀⠀⡔⢻⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢚⡷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠘⡆⠀⠀⠀⠹⠷⡶⠿⠒⠒⠀⠀⢵⣄⣀⣸⠀⠀⠀⢀⡞⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢣⣄⡀⢀⠤⢾⣤⣄⣀⣤⣤⣀⣸⣙⣉⠉⠀⠀⠀⡞⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠹⣄⡤⠺⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⢴⣶⢐⣦⠴⠛⠒⠶⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⢉⣡⠴⠚⠉⠙⡾⠁⠀⠀⠀⠀⠀⠸⡍⠉⠉⣹⠋⠀⠀⠀⠀⠙⢷⣄⢳⡶⠤⣄⣀⡀
⠀⠀⠀⢀⣀⣠⣴⣶⣿⣷⠖⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠙⢳⡞⠋⠀⠀⠀⠀⠀⠀⠈⣿⠋⠉⠓⢄⡉⠙⠤⣄
⠀⣠⣋⠝⠛⠋⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠙⢦⣤⠞⣧⣀
⠎⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⡄⠀⠫⢟⣶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡞⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣋⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣸⠁⠀⠘⣄⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢀⡾⠉⢷⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡇⠀⠀⠀⠈⠁⣀⣀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣀⡴⠋⠀⠀⠈⡄⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡶⣄⣸⠀⠀⠀⠀⠀⠀⠀⠻⢿⣏⣑⣿⢐⡛⣻⡶⠟⠋⠁⠀⠀⠀⠀⣧⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡛⠹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⢤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠳⠎⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠘⣻
⠀⠀⠀⠀⠀⠀⠀⡴⣫⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⡧⢷⡀
⠀⠀⠀⠀⠀⢀⣾⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⢧
⠀⠀⠀⠀⠀⡼⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣧
⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⠤⠴⠒⠋⠁⠀⠀⠀⡎⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣾⢰⡇⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⠀⡆⠀⠀⠘⡇⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢸⡀⢇⠀⠀⢸⡇⡎⠀⠀⠀⠀⡎⢠⠀⠀⠀⠀⢰⠃⡆⠀⠀⠀⢻⠘⡇⠀⠀⠀⢳⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠱⢄⠀⠀⠘⣇⠹⠀⠀⠀⠘⡇⢸⠀⠀⠀⠀⢸⡄⢧⠀⠀⠀⠘⣆⠑⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⠀⠘⢆⠀⠀⠀⠀⢻⡈⠂⠀⠀⠀⠀⢳⡈⠀⠀⠀⠀⠈⠃⠀⠀⢀⡀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣤⣄⣀⣀⡙⠂⠀⠀⠀⠀⠀⠈⣁⣀⣀⡤⠤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣿⣶⣶⣦⣤⣀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣾⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠉⢁⣾⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣇
⠀⠀⢰⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣛⣵⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣴⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⢣⣾⣷⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡏
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣄⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣙⠻⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⡙⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⢋⣥⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠋⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠈⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠷⠤⣄⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢴⣎⣀⣤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣀⣀⣠⠤⠴⠚⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡎⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠚⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠚⠉⠉⠀⠀⠙⢤⣀⡤⠋⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⣀⣀⡤⠤⠶⠒⠊⡟⠉⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡶⠓⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠳⠤⠖⠃⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⡴⠋⠙⢦⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠁⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠹⢤⣠⣞⣠⠇
⠀⠀⠀⠀⠀⠉⢲⡀⠀⠀⠣⣀⣀⡰⠃⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠒⠋⠉⠁
⠀⠀⠀⠀⠀⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢹⠀⣀⣀⡤⠤⠒⠚⠉⠉
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠿⠔⠒⠋⠉⠁
⠀⠀⠀⠀⠀⡇⠀⣀⣀⠤⠴⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣸⢷⣷⡀
⠀⠀⠀⠀⠀⢠⣘⠋⠈⠀⣡⣇
⠀⢠⣲⣿⣿⣏⡈⠒⠂⣙⢉⢈⣦⡀
⠀⠀⠙⠛⡻⢿⣿⣿⣆⠀⠄⣤⣿⣷⡄
⠀⠀⠀⠀⠾⠛⢛⠟⠃⠀⢨⢯⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⢣⣶⣿⠟⠛⢻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢾⢡⣶⡾⠉⠉⠀⢀⣩⣯⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⣠⡶⠟⠛⠁⠉⣻⣿⠟⠉⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡰⣦⡾⠛⠉⠀⣠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢣⡀⠼⠁⠏⠀⢀⣠⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢻⣅⣰⣆⠀⠈⠘⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢻⡿⠯⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⡇⠀⠀⣿⠟⠛⠉⠉⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠂⠀⠃⠀⠀⣤⣤⠄⠙⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣼⠀⠀⠐⠀⠀⢰⠫⣁⡀⠉⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠴⠋⠀⠐⠁⠀⠀⠀⠀⠄⢸⠀⠉⠀⠀⠀⠈⣫⣽⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠐⠒⠀⠀⠀⠀⠐⠒⠒⠒⠓⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠚⠛⠋⠉⠉⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⢀⣤⣶⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢄⢀
⠀⠀⠀⠏⠙⣻⣿⣿⠛⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠘⠄⠀⠘⠘⠀⢀
⠀⠀⢸⡀⠘⠟⢿⣁⠀⠀⢙⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠡⡀⢀⠀⠀⠀⠀⡀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠂⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠐⠄⠀⠀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⢨⣄⡠⣄⠉⠒⠋⠀⠀⢰⠢⠤⣤⣴⠆⠠⠄⠀⠀⠀⠀⢀⡄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⡀⢀⠀⢀⠁⠀⠠
⠀⠀⠀⠀⠀⣿⣿⣿⣯⡷⣷⣶⣾⣿⣿⡇⡏⠈⠋⠓⠐⠃⠀⠀⠀⠀⠀⠐⠓⠙⠉⠉⠉⠉⠩⣹⠀⠀⠀⠰⠀⠐⠀⠘⠦⠤
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣟⠛⠛⠋⠉⡇⡇⢀⣔⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⣀⠀⡎⣇⡔⠉⢀⠀⠀⠀⣀⡆
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠿⠁⠀⠍⣉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠻⠶⡦⠀⠡⠹⢣⡀⠂⢀⠖⠉⠉
⢀⣴⣿⣿⣿⣿⣿⣿⣿⠉⠙⠿⢿⣿⡶⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⡠
⠈⠻⢿⣿⣿⣿⣿⡿⠀⠀⡀⠀⠀⠈⠀⠀⡆⠂⣀⣀⣐⣒⣒⣒⣴⣶⣷⣦⣤⡀⢀⡀⠀⠀⢀⣀⠤⢒⠁⠀⠠⡀
⠀⠀⠈⢮⣙⠛⠋⠀⠀⠀⠉⠢⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠚⢁⡠⠟⠁⠀⠀⣼⠇
⠀⠀⠀⠀⠉⠙⠦⠤⢀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣥⡀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⡞⠘⠒⠋⡼⢄⡀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠈⢳⣤⠔⢀⡆⢉⠒
⠀⠀⠀⠀⠀⠇⣀⣀⣈⡀⠀⠙⠆⢀⣜⣠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡴⠂⡠⠞⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠐⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠰⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⡾⠶⠲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡎⣸⠛⠛⢶⣷⡘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣯⣜⠻⣧⡇⠀⠀⢀⣻⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⡟⠿⢿⣄⣿⡗⠦⠀⠈⠚⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣤⠿⠿⣷⠀⠀⢸⣿⣧⡀⠀⣀⠀⢹⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠿⢿⡇⠀⠀⢹⣿⣶⣿⣿⣷⣌⠛⢁⣴⢿⣿⣿⣿⣿⣿⣎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⡄⠀⠀⢿⣷⣶⡛⠛⠻⡿⠿⢿⣿⣩⢤⠶⠛⣿⣿⣿⣿⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠘⣿⣶⠾⠻⢟⢧⡠⢜⣠⡶⠟⠉⢠⠃⠀⠀⠜⠛⠿⣿⣯⠝⣾⡇
⠀⠀⠀⠀⠀⠀⠀⡆⠠⢀⠊⠁⠉⠈⢢⡄⢀⣹⣲⠟⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢻
⠀⠀⠀⠀⠀⠀⠀⡇⢰⣾⠀⢈⣉⣡⣤⣿⣿⣿⠁⠀⠀⢀⣀⠀⣸⣦⡀⢀⣀⠀⠀⠿⡟⠛⠃⣸
⠀⠀⠀⠀⠀⠀⠀⡇⣸⣿⡁⢸⣿⣿⡟⣿⠿⡹⠀⠀⢰⠀⣀⣀⠀⠀⠀⠀⠈⣳⡤⠄⠘⣆⣀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡖⠚⣇⠸⠿⠿⢣⣷⣯⣀⣀⠀⣰⠀⣤⣏⠀⢈⣉⣉⣁⣄⣷⣾⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀
⠀⠹⡸⣥⣤⣶⣖⣸⡇⢿⣿⡏⠀⠉⠙⣿⣿⣿⣿⣿⡏⠀⣿⣿⠇⢸⣿⣿⣿⡇⣿⣿⡇⣧⠀⣼⣶⣶⣤⣬⣷⡢⠁
⠀⠀⠈⢝⢿⣿⣿⣯⣭⣞⣙⣳⡀⠄⠠⣿⣿⣿⣿⣿⣷⣐⠿⢿⡠⠸⣿⣿⣿⢇⣟⣋⣵⣃⣢⣭⢹⣿⣿⡿⡑⠁
⠀⠀⠀⠀⠪⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣧⣍⣛⣋⣩⣼⣿⣿⣿⣿⣿⣿⢸⣿⣟⢎
⠀⠀⢀⢔⣼⣿⣿⢹⣿⣿⣟⣯⣝⢿⢻⣽⡿⣿⣿⣿⢿⣽⣿⣿⣿⣿⠿⣽⣿⣿⢿⡟⣿⣿⣿⣿⢸⣿⣿⣯⡢⡀
⠀⠠⢑⣿⠿⢿⢿⢸⣿⣿⣯⡻⢾⣼⢸⢺⡇⣿⢳⣝⢨⣎⡻⣼⡏⣷⡖⡏⢶⡧⢏⣇⣻⣿⣿⣿⢸⡿⠿⢿⡟⠮⠆
⠀⠀⠀⠀⠀⠉⢨⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠐⠋⠩⠝⠛⠻⠛⣛⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡛⠛⠛⠻⠯⠍⠛⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠚⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠢⠞⠣⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠵⡄⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠠⣄⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠉⠛⠲⠤⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⢲⡄⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠒⠒⠒⠒⠐⠊⠉⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣠⠞
⠀⠀⠀⢀⠤⣀⠀⠎⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⡣⠀⠙⢄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠘⢦⠀⢀⣜⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁
⠀⠀⠀⠀⠐⠄⠊⠀⠓⢄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠤⠤⠤⠤⠤⠄⠀⠒⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠒⠀⠀⠴⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠠⠊⠐⠀⠀⢈⠀⠀⠴⠀⠐⠄⠀⠈⠑⡀⠀⠒⠀⠀⠀⠈⠀⠀⠠⠂⠀⠆⠀⢈⡁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠒⠀⠀⠂⠂⠀⠀⡀⠀⠀⠁⢀⠀⠄⠀⠒⠒⠀⠀⢀⠀⠁⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡀⠀⠀⠐⠳⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠢⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠂⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠐⠂⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠤⠀⠀⠐⠀⠀⠀⠀⠄⠁⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠌⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠈⣀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠦⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠈⠀⠀⠀⠰⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠆⠀⠀⠋⠀⠀⣀⠀⠁⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂



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