Pet name Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣦⣤⣤⣀
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠈⢀⡀⣿⣦⣶⡄⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢠⣶⣴⣿⢀⡀⠠
⠀⠀⠀⡄⠀⢘⣿⢯⣷⣾⣷⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣾⣷⣾⡽⣿⡃⠀⢠
⠀⠀⠀⠃⠀⠘⠻⠻⣿⠟⠁⠀⠀⠀⢠⣴⣶⣦⣤⡀⠀⣠⣴⣶⣦⣄⠀⠀⣠⣴⣶⣴⣶⠄⠀⠀⠀⠈⠻⣿⠟⠟⠃⠀⠘
⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠉⣿⣿⢰⣿⡟⠉⠹⣿⡇⢸⣿⠏⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣀⣾⣿⠸⣿⣧⣀⣰⣿⡇⢸⣿⣆⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⡄⠀⢠⣴⣴⣿⣦⡀⠀⠀⠀⠸⠿⠿⠿⠟⠃⠀⠙⠻⠿⠿⠋⠀⠀⠛⠿⠿⠟⠻⠇⠀⠀⠀⢀⣴⣿⣦⣦⡄⠀⢰
⠀⠀⠀⠃⠀⢨⣿⣞⡿⢿⡿⠀⠀⡠⠠⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⢠⠀⠀⢿⡿⢿⣳⣿⡅⠀⠘
⠀⠀⠀⠀⢀⠈⠁⣿⠟⠿⠃⢀⠌⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠣⡀⠘⠿⠻⣿⠈⠁
⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠀⢀⡄⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⠈⠋⢀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⡄
⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢿⣦⣤⣤⣤⣀⡀⠀⠀⠉⠀⠀⠀⠀⢀⡀⠀⠀⣀⣠⣄⡀⠀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠶⠿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢈⡟⡌⠻⡀⠀⠀⠀⠀⠀⠈⣹⡿⠙⢻⣟⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⣆⠀⠀⠀⠀⠀⢀⠀⠀⣠⣤⡀⠉⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⢠
⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⣿⡇⢸⣿⣿⠃⠀⠘⠁⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣦⣤⡀⠉⠉
⠀⠀⠀⠈⢹⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣿⣿⣾⣿⣿⣆⠀⠀⠀⠀⠀⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡟⠛⠛⢻⠟⡇
⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠋⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣄
⠀⠀⠀⢀⣤⣄⡀⢀⣀⡀⠀⠀⠈⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠈
⠀⠀⠀⠀⠉⡻⣿⣿⣿⣿⣧⡀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣦⣤⡶⠶⠶⣤⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⣷⠀⠀⠀⠀⠀⠀⡽⠙⢿⣿⡿⠇⠀⠀⠀⠈⠁⠀⠀⠀⠐⠛⠛⣿⡿⠿⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠁⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⡀⠀⠀⠙⣿⡟⠁⠀⠀⢀⣠⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣿⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠂⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠤⣄⡈⠉⠉⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠉⠉⣁⡠⢤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠙⡄
⠀⠀⠀⠀⢀⡤⠒⠒⠒⠒⡇⠀⠀⠀⠀⠀⢸⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠀⠀⠀⠀⠀⠀⡸⠒⠒⠒⠢⣄
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠈⢧
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠙⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠚⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠉⠉⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⢤⣤⠴⠶⠞⠋
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠸⣏⠻⣦⡀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢀⣴⠟⣹⠇
⠀⠀⠀⠀⠀⠀⠹⣆⠈⠙⢦⣾⠁⠳⡄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⠟⠈⣷⡴⠋⠁⣰⠏
⠀⠀⠀⠀⠀⠀⣾⠾⣷⣄⠀⢹⡄⠀⠙⢦⣀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣀⡴⠋⠀⢠⡏⠀⣠⣾⠷⣷
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠉⠓⢦⣽⣄⠀⠀⠈⠛⠲⢤⣈⠻⠃⢻⣿⣿⣿⣿⣿⣿⡟⠘⠟⣁⡤⠖⠛⠁⠀⠀⣠⣯⡴⠚⠉⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣶⠻⠷⢤⣘⣦⡉⠙⠂⠀⠀⠀⠀⠈⠇⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠰⠁⠀⠀⠀⠀⠐⠋⢉⣴⣃⣤⠾⠟⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⢹⡛⠓⠂⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠐⠚⢛⡏⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣲⠿⠶⠶⠲⡀⣀⡄⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠢⣀⢐⠖⠲⠶⠿⣖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣟⡭⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⢭⣻⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⣉⣉⠉⣉⣉⠉⠉⠉⠉⢉⣉⣉⡉⠉⢉⡉⠉⠉⢉⣉⡉⣉⣉⠉⠉⣉⣉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠉⢹⡟⠛⠘⣧⣰⡏⠈⢹⡏⠁⠀⢸⡇⠀⡟⣿⠛⠋⣿⠉⢳⢸⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡷⠶⠀⣽⢿⡀⠀⢸⡇⠀⠀⢸⡟⠻⣏⣿⠶⠆⣿⢶⡏⢸⡷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠸⠷⠶⠸⠏⠈⠷⠀⠘⠃⠀⠀⠸⠇⠐⠃⠿⠶⠆⠛⠀⠷⠸⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⠶⠶⠶⠶⠶⠶⢶⣶⣾⣶⣶⣶⣶⣶⣶⣾⣷⣷⣶⣶⣶⡶⠷⠶⠶⠶⠶⠶⠶⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣀⣀⣠⣤⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣆⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⡀⠄⢰⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⡨⠋⠉⠙⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⣶⣶⣶⣶⣤⣄⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠠⠉⣿⣿⣿⣿⣿⣿⣿⠉⠉⣿⣿⣿⣿⣿⣦⣤⠀⠀⢤⣿⣿⣿⣷⣄⣀⡀⠔⢻⣿⣿⣤⣤⣤⣤⠄⠊⢹⣿⣿⣿⣿⣿⣿⡏⠉⠹⣿⣿⣿⣿⣆
⢀⣦⣴⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣷⣴⣾⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⡆
⢸⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⡿⠛⠻⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⡇
⠸⣿⣿⣿⣿⣦⣀⣼⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⣠⣿⣿⣿⣿⣿⣿⣿⣄⣠⣾⡇
⠀⢻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⠀⣀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢢⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⠀⣈⠇⡔⠒⡄⢺⠂⠈⢠⣒⠂⠀⠀⢸⠀⠀⡧⢒⣒⣄⠖⢢⠔⢢⣠⠒⢢⠀⠀⠀⠈⢃⠀⣨⣿⣿⣿⡿⠁
⠀⠀⠀⠙⠻⣿⣿⣿⣿⠀⠀⠀⠀⠸⠀⠀⠈⢏⣉⠄⠸⣀⠀⠠⣀⡱⠀⠀⢸⠀⠀⡿⣅⡀⡿⠆⠈⠀⠸⠫⣉⡡⠀⠀⠀⠀⢸⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⡠⠀⢹⣿⠀⠀⠀⠀⢀⢄⠀⣀⢀⠀⣀⣀⠀⠀⠀⡀⢀⣀⠀⡀⡀⠀⠀⠀⣀⠀⣀⡄⠀⣀⣀⢠⢀⡀⠀⠀⠀⢸⣿⣿⣧⣀
⠀⠀⢀⣴⣿⣷⣶⣿⣿⣧⡀⠀⠀⠣⡠⡇⠣⡠⡆⠅⠰⣆⣀⠈⠙⠁⡠⠁⠀⢎⣀⣀⡔⢹⠀⠖⠂⣴⠗⢢⠀⡐⠁⠀⠀⣠⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠂⠐⠁⠂⠐⠀⠂⠚⠀⠀⠀⠀⠪⠄⠈⠂⠚⠀⠀⠉⠙⠁⠂⠒⠁⠢⠊⠀⠁⠀⣠⣴⣿⡟⠉⠻⣿⣿⣿⣿⣧
⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⡿⠟⢿⣷⣶⣦⣤⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣤⣴⣶⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿
⠀⣿⣿⣿⣿⣆⣀⣸⣿⣿⣿⣿⣿⣿⣇⣀⣠⣿⣿⣿⣿⣿⣿⣧⣀⣠⣿⣿⣿⣿⣿⣿⣧⣀⣠⣿⣿⣿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⣿⣿⣄⣀⣼⡟
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⣀⠘⠳⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠂⠉⠉⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣾⣿⣿⡆⠀⠈⠀⠀⠀⠠⣾⣿⡆⠀⠀⠀⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⢠⣿⣀⣠⣄⡀⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡾⠏⠀⠀⠀⠀⠹⣿⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠋⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠤⣔⠀⠀⠀⠀⠀⠘⣦⡆⠀⠀⠀⠀⠀⠀⣴⣶⣆⠀⠀⠀⠀⠀⢀⣔⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣠⣾⠠⠀⠌⠳⣀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠘⠟⠃⠀⢀⣠⣴⠄⠁⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠻⣄⠀⠀⡸⠀⠀⢀⣀⣿⡟⢱⣶⣤⣤⣤⣤⣤⣦⣶⡾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣦⣼⣶⣾⣿⣾⡟⢩⣿⣿⣷⣾⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣽⣿⣿⣿⡿⠳⠺⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⠙⢛⣉⠀⠀⠀⠈⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣿
⠀⠀⢸⣷⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⢘⣿⣧
⠀⠀⠸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢠⣠⣆⣀⣤⣄⡈⠙⡆⠀⠀⠀⠀⠀⠚⢿⣿⣶⣄
⠀⠀⠀⢘⡛⠻⠿⠅⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⣇⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⡴⠃⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣷
⠀⠀⠀⠉⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠉⠁⠀⠀⠀⠈⠛⠙⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠽⠛⠋⠉


⢀⣀⣲⣶⣄⠀⠀⠀⢲⡀⠀⠀⠀⠀⣀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣖⣀⣀⣐⣰⣄⠀⠀⠀⠀⠀⠀⢀⢤⡄⠶⣐⠢⡀
⠸⠿⢿⣿⣿⣶⣶⣶⡿⠁⠀⠀⠀⠀⠁⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⡸⠯⢥⠿⠤⠤⠬⠝⠇⠀⠀⠀⠀⠀⡃⠸⣤⣶⠌⡆⠅
⠀⠀⠘⣿⢿⡿⢿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⢡⠀⠈⣴⣷⣦⡄⠀⡄⠀⠀⠀⠀⠀⢣⠖⠻⢄⡀⠷⠁
⠀⠀⠀⡧⢨⡇⠰⠃⠼⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠑⢄⣈⠉⣀⡠⠊⠀⠀⠀⠀⠀⠐⠤⠈⠈⡉⡑⠤
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣴⣾⣾⣿⣿⣿⣆
⣠⣿⣛⣉⣉⣉⣉⣛⣻⣄⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⣿⣿⠃⠀⠀⣰⣿⠛⠛⠋⠉⠙⠛⠛⣿⣦⠀⠀⠀⣿⣟⣻⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣏⣻⣿⣿⣿⠟⠀⠀⠀⠸⣿⠟⠉⠉⠉⠉⠿⡿⠃⠀⠀⠘⠿⣏⣿⣿⣛⣿⣿⣹⠿⠇⠀⠀⠀⣿⡇⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠿⠿⠛⠋⠉


⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣄⣀⡀
⠀⢀⣼⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⠿⡆⠀⠀⠀⠀⡶⠿⣿⠿⣿⡿⠟⠻⢳⠀⠀⠀⢠⣿⣛⣉⣉⣉⣉⣙⣻
⢠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠆⡇⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠈⠛⠻⠿⣿⠿⠿⠿⠟
⠻⢿⣿⣏⣩⣍⣹⣿⡿⠟⠀⠀⠀⢂⡼⣿⡐⢈⣳⡶⠾⠃⠀⠀⠀⠀⠧⣤⣀⣿⣀⣤⠦⠖⠋⠀⠀⠀⠀⠀⠀⣼⣿⣧

⠀⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠹⣝⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣝⠦⣄⣠⠴⣒⡒⢦⣀⣤⢖⡯⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⢿⣭⣭⣭⣷⢹⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠀⠈⠉⠙⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡴⠚⢲⡀⣠⠞⠳⡄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⣀⣀⣇⠀⠀⡗⣇⠀⢀⡟⣀⣀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⡇⠀⡞⠀⠈⣏⢓⡾⠷⠾⣖⠋⡟⠀⠈⡇⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠻⣄⣠⣿⠋⠀⠀⠀⠈⠳⣧⣀⡴⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢧⡀⠀⢀⣀⣀⡀⠀⣠⠏⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⣀⣀⣤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠁⠀⠈⠀⡎⠀⠀⠱⠀⡏⢢⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢄⣀⠠⠀⠣⣀⣀⠜⠀⠇⠀⠑⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠒⠄⢰⠒⠒⠂⠀⠒⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠐⠐⡄⢸⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠰⠃⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠅⠀⠄⠁

⠀⠀⣠⣤⣶⣶⣶⣦⣤⠴⠖⠛⠛⠛⠻⠿⣶⣶⣤⣀
⣠⣾⣿⠿⠿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣤
⣿⣿⠁⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠘⠿⣷⣄⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠈⠉⣧⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣶⣶⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢀⣀⡀⠘⢧⠀⠀⢻⣄⡀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⢀⣀⣤⣤⣤⣴⣟⠁⠈⡙⠛⠛⢳⣦⡀
⠀⠀⢺⣿⣍⡉⠚⠃⠀⠀⢻⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⡴⠛⠁⠤⠂⠀⠀⠙⠆⠑⢾⠁⣼⣵⣿⣷⡤⢀
⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠟⢛⣻⡿⣫⠖⠀⡠⢒⣒⢦⣠⣾⡄⠐⠚⢻⣟⠁⡈⠹⡄⢾⣁
⠀⠀⠀⠀⠈⢻⣿⣿⣷⣦⣀⢀⣿⢿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠈⡟⠁⠀⠀⣧⣿⡿⠋⠁⠀⠀⠀⠠⠼⢿⣤⣟⠀⣷⣸⠟⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠉⠁⠀⠙⢿⣿⣿⡇⠀⠀⠀⠀⢀⡼⠋⢀⣠⣶⠿⣷⡄⠀⠀⠀⠴⣣⠖⠀⠀⠀⠀⠀⠀⠉⢙⣋⠴⡻⠳⣿⣿⣞⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠙⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⢀⣔⣋⣤⣴⡟⠛⠁⠀⠈⠁⢰⡀⠀⠀⠁⠀⠀⠀⠀⡀⠀⢀⣀⣠⠄⠤⠄⢺⡟⣏⣧⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⣴⠀⠀⠉⠀⠙⢛⣷⢦⣤⣤⡄⠀⠃⢀⠀⠀⠀⠀⠀⠈⠀⣴⢋⡘⣍⣧⠿⢷⡈⣿⣮⣀⣄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⡏⢸⡏⢸⣷⣆⢀⡘⡀⠀⠈⠋⢀⡄⢰⣿⠟⠛⠛⠛⠛⠓⠿⠿⠀⢨⣻⣷⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠀⠀⠃⠘⠁⢸⣿⣿⣿⣷⣄⠀⠀⠀⠸⣿⢸⣿⣀⡐⠇⠯⠷⢎⣆⣀⣀⣼⣹⠧⢖⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢿⡏⠻⣿⣿⣟⠓⠀⠀⠉⠟⢿⡶⠶⠛⠉⠩⢝⣛⣛⠳⠟⢃⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠃⠀⠈⠻⢿⣷⣶⠄⠀⡀⠀⢀⣤⣤⣤⠶⠶⠶⠶⠿⠿⠟⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠀⠀⠀⠀⠀⢀⢄⠀⢀⠀⠦⣀⣀⡤⣀⣈⣉⡿⠛⢲⣾⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡄⠀⠀⠀⣀⣼⠆⠙⠊⠳⢤⠈⠙⠁⠀⣉⣁⣠⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠃⠻⣶⣤⣭⣭⣤⣤⣴⣶⣦⣤⣀⣠⣤⠶⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢠⡀⣌⣻⡇⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠙⠟⠻⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⣶⣿⣿⣶⣾⣿⣿⣿⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⢀⣠⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣄⡀⠻⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠀⣾⡿⠛⠈⠻⣿⣿⣿⠟⢿⣿⣿⣿⡿⠀⠛⢿⣷⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⢀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠰⣯⣿⠆⠀⠀⠀⠀⠀⠀⠰⣿⣽⠆⠀⠀⠀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⣤⣤⣬⡅⠀⣀⠀⣲⣖⠀⣀⠀⢨⣥⣤⣤⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡧⠤⠬⣍⠉⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠉⣉⠥⠤⢼⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⢳⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⠁⠀⠀⠀⠙⣏⠁
⠀⠀⠀⢠⠖⠉⠁⠈⠙⡄⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⠀⠀⠀⢀⠏⠈⠈⠉⠲⡄
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠲⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⠖⠋⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⡛⢄
⠓⠐⠴⠶⢤⠂⠂⠀⠀⠀⠀⠀⠚⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⣫⠸⠑⠄⠀⠀⠀⠀⠀⠢⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢠
⠀⠂⠀⠉⠀⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠔⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣢⡒⠀⠀⠀⠀⠈⡄⠀⢠⠄
⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⣤⣈⡈⣵⣆⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⢠⣤⡄⠙⠁⠀⠀⣤⣀⣠⣄
⠀⠀⠀⠉⠉⡁⠉⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉
⠀⠀⢠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠒⠃⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀
⠀⠀⣨⠰⠦⠙⢂⠀⠀⠀⠀⠀⣀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠀⠀⠀⠀⠄⣂⡒⣀⡐⢤⠀⠀⠀⠀⠀⠀⠀⠀⡴⢾⢰⠆⠂
⠀⠀⠡⣄⢀⠄⠀⠀⠀⠀⢀⠠⠃⠚⠂⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⣥⣴⡄⠁⡀⡅⠀⠀⠀⠀⣰⡒⠀⠃⢸⠈⢀⣠⡴⠃
⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠉⣦⠀⠀⠀⠀⠀⣇⠈⠛⠁⡀⠁⢃⠀⠀⠀⠀⠁⠀⠀⣀⠀⡄⠸⠿⣄
⠀⠀⠀⠀⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⡆⠲⢆⡁⣀⠀⠀⠀⠈⠑⠒⠂⢀⠀⠀⢡⡀⠀⠀⠀⠀⢤⣿⡀⡇⢠⠀⠀⠁
⠀⠀⠀⢰⣶⣦⣧⣄⠀⠀⠀⠀⠀⠐⠆⠀⠀⠢⠀⠀⠀⠀⠀⠀⠁⠡⠜⠥⠉⠁⠀⠀⠀⠀⠸⠇⠸⠸⠇⠈⠉⠀⠀⠀⠀⠀⠙⠀⠁⠉⠁
⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠴⢒⠶⣀⠀⠀⠀⠀⠀⢠⠀⠚⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⡛⠆⠀⠀⠀⠀⠀⢀⡇⠦⠡⢜⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⡀⠐⠓⠀⠀⠀⠀⠈⠀⠀⠀⠀⢘⠻⠟⠐⠇⠀⠀⠀⠀⠁⠀⠠⡄⢺⣆⣀⠀⠀⠀⠠⠓⠁⠀⠄⠠⠀⣤
⠀⠀⠀⠉⠠⣄⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡘⡰⠆⢀⠀⠀⠀⠀⠀⠩⠉⠃⠀⢸⡀⠀⠁⠀⠀⢰⠤⠴⠂⠘⡀
⠀⠀⠀⠀⢀⡁⢀⡀⠀⠀⠀⠀⢀⠀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡌⢴⠁⡞⠼⠒⠀⠀⠀⠀⢠⣄⡉⣥⡌⠁⠀⠀⠀⠀⠭⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠡⠠⢤⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡖⢄⠄⠀⠀⠀⠀⠀⠀⠀⢒⣦⠳⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠨⠰⡿⠂⠀⠀⠀⠀⠀⠀⠄⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠀⡀⠀⠀⢄⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠔⢠⠄⠳⡀⠀⠀⠀⠀⠀⠜⡥⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⢄⡀⠃⠀⠀⠀⠀⠀⠄⡔⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠓⡄⠀⠀⢠⠚⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⢀⠆⠀⠀⢣⡀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⣀⠀⠀⠉⠉⠉⢆⠀⠀⡰⠉⠉⠉⠀⠀⣀⠤⠤⠤⠤⠤⠄⣀
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⣀⣀⣀⡀⠀⠉⠢⡀⠀⠀⠘⡄⢠⠃⠀⠀⢀⠔⠉⠀⣀⣀⣀⡀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⡰⠁⠀⠀⡠⠒⠉⠀⠀⠀⠀⠉⠒⢄⠈⢢⠀⠀⢡⡂⠀⠀⡔⠁⡠⠒⠉⠀⠀⠀⠀⠉⠒⢄⠀⠀⠈⢆
⠀⠀⢰⠁⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠗⠊⠉⠉⠑⢾⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⡆
⠀⠀⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢀⡀⢀⡀⢀⡀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆
⠀⠀⠈⢆⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠄⣀⣀⣀⣀⠠⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⡰⠁
⠀⠀⠀⠈⠢⡀⠀⠙⠢⢄⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⡠⠔⠋⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠈⠑⡲⠂⢀⠜⠁⠀⠀⠀⠀⠀⠸⠉⠐⠒⠒⠒⠒⠂⢉⠃⠀⠀⠀⠀⠀⠈⠳⡀⠐⢖⠊⠁
⠀⠀⠀⠀⠀⠀⡔⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣀⣀⣀⣀⠠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⢣
⠀⠀⠀⠀⠀⢰⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⠀⠀⠀⠀⠀⠀⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡆
⠀⠀⠀⠀⠀⠘⡄⠀⠈⢆⠀⠀⠀⠀⠀⣀⠔⠁⡰⢗⠒⠒⠒⠒⡺⢆⠈⠢⣀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠑⠒⠒⠒⠉⠀⡠⠎⠀⠘⣄⣀⣀⣠⠃⠀⠳⢄⠀⠉⠒⠒⠒⠊⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣀⣀⣀⠤⠒⠉⠀⠀⠀⠀⠘⢄⡠⠃⠀⠀⠀⠀⠉⠒⠤⢀⣀⣀⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠀⢠⠀⣆⡄⢘⡆⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣦⡇⠠⡇⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠤⠀⠀⠀⠀⠰⡀⠹⠃⠘⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⡹⠀⠆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢀⣠⣌⠃⠀⠀⠀⡡⣤⢤⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠂⠉⠀⠁⣖⠀⠀⢸⠃⠉⠀⠀⠀⠙⠲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⠀⠀⠀⠀⠉⠁⠀⣠⠂⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠤⠤⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢒⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠈⣀⠀⠀⠤⠒⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠏⢀⣀⣀⡀⢀⠀⢠⡀⠀⠀⠀⠀⠀⡀⠈⠙⠿⠂
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠅⠀⠀⠁⢬⠿⠿⠟⠉⠁⠀⠈⠹⡟⠗⡄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠚⢁⠀⡀⠀⣠⣀⣀⡀⠤⢦⣀⡆⠉⣸⣯⣾⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢶⣷⣦⠉⡏⠀⠀⠀⠀⠉⢹⣾⠟⠛⠊⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⡈⠁⠘⠆⠐⠀⠀⠀⠠⠏⠀⣤⢤⡸⢿⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠴⠆⠀⠜⢀⠤⢤⣀⠀⠀⣀⡆⠀⡀⠀⠀⠶⠦⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢳⠋⠉⠛⠀⠓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈


⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢷⣫⠄⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⣿⡟⠛⠦⣀⡠⠄⠀⠐⠒⠒⠚⠛⠋⠁⠀⠸⣷⣃⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣸⠷⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⣀⣯⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠁⠀⠀⠀⠀⢀⡈⠃⠀⠀⠀⠀⢁⣴⣶⡶⡀⠀⢀⡀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢰⣿⡝⣦⠀⠀⡀⠀⠸⣯⢻⡷⠃⠀⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠠⠀⠀⠳⠭⠔⠋⠀⡢⠤⠀⣄⠀⠀⠀⢀⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡞⠀⠀⠀⠀⠀⠁⠀⢘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠘⣤⡀⠀⠀⢀⠴⠻⠦⡤⠀⠀⠤⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣆⣠⡀⠀⠉⠉⠀⠀⠠⠤⠄⠀⠀⠀⡤⢴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⣾⠁⠀⠁⠸⠷⠤⠠⢤⣤⣴⠦⠾⠉⠀⠀⣏⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⢹⣤⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⠀⠘⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡀⠀⠀⠀⠛⢫⣧⣤⡀⢀⠀⢠⡆⠰⣄⠀⢤⣰⠞⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠈⠿⠻⠛⢷⠂⠑⠒⠚⠟⠋⠉⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢻⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠘⡀⣭⡀⠀⠀⠀⣀⠄⡴⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠃⠈⠷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠁⠤⣤⣄⡀⠀⠀⠀⠀⢈⡇⠀⠀⢀⢀⣤⡜⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠄⠀⠀⢘⣁⡀⠀⣀⡀⠀⠀⠈⣷⣴⠀⠀⠀⢸⣶⠻⠚⠋⠋⠛⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣀⠀⠈⢿⠛⠚⠋⠀⠀⠀⠀⡏⠋⣿⡄⢰⡇⢸⠄⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠆⠀⢸⠀⢀⡀⣤⡰⣤⣼⡇⠀⠘⣇⣸⠇⢸⣇⣀⡤⠴⠞⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠘⢧⠈⠁⠀⠀⠀⢻⡆⠀⠀⠙⠿⠀⠸⡏⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⢤⡤⠖⠛⢳⡀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⣴⠊⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⣷⡀⠀⠀⠀⣀⠀⢸⡄⠠⡄⠀⠀⠈⢣⠀⠀⠀⠀⠀⢨⡇⠀⣰⡶⣄⣼⡇
⠀⠀⠀⠀⠈⠛⠂⠐⠶⠒⠒⠄⠘⠓⠶⠶⠴⣏⠈⣡⠖⠃⠉⠐⢶⡇⠀⠀⠀⠀⠀⢿⡀⢀⠴⠿⣶⢾⡇⠈⠁⠀⠈⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⢠⠀⠀⣄⣧⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⣤⢨⡇⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠼⠆⠔⢿⠃⠀⠀⠒⠓⠒⠀⠻⡌⠉⠉⠉⠉⢡⡀⠀⠀⠤⠀⠀⠙⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢦⠀⠘⣇⠀⣀⠇⠀⠀⠀⠀⠀⢧⡀⠀⠀⡄⠀⢘⣀⣀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠓⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠛⠒⠛⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣯⣿⣹⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣯⣿⠇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣜⠿⠿⡿⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠋⠁⠠⣤⠀⠸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⢹⣦⣜⣛⣛⣿⣿⣿⣿⣿⣿⡿⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⢀⣀⣻⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠙⢋⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⣿⣿⣿⣦⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠺⠿⠃⠀⠄⠀⠀⠀⢠⣿⣄⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣇⢾⣿⡟⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣾⣿⣿⣿⡄⠀⠀⡀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⢀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣿⣿⡏⠉⠉⢻⣿⠀⠀⠃⢀⣿⡿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⣿⣿⣶⣤⣦⣿⣆⠀⠀⢸⣿⣧⣄⣠⣴⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠻⠟⠛⠛⠛⠛⠉⠁

⠀⠀⡴⠉⠉⠑⠒⠦⣄⡀
⠀⢸⡇⡰⠚⠛⠓⠲⢤⣙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣀
⠀⢸⡇⡇⠀⠀⠀⠀⠀⠈⠑⣮⣣⣤⠤⢤⠴⠖⠒⠶⠤⠤⠤⣄⣀⡀⠀⠀⣠⠴⢊⣉⣠⣠⣄⡀⢸⡆
⠀⠈⡇⡇⠀⠀⠀⣀⡤⠖⠛⢯⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠓⠾⣵⡚⠉⠁⠀⠀⠀⢱⢸⡇
⠀⠀⡇⡇⢀⣤⡞⠁⠀⠀⠀⠀⠈⠓⢤⣸⠀⠀⠀⠀⠀⠀⠀⡇⣀⡠⠔⠊⠁⠈⠳⢄⠀⠀⠀⢸⢸
⠀⠀⢷⣼⠋⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⡱⣄⠀⡇⡟
⠀⢀⡾⠥⣄⣀⣀⣈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⢳⣁⡇
⢀⡟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠤⣤⣄⣀⣹⡀
⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⣧⠤⠤⠖⠒⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢳⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⣸
⠘⣆⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢨⠇
⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣀⣀⡴⠧⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢠⠏
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊
⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣲⠶⠤⢤⣤⣀⣀⣀⣠⣤⡤⠴⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡰⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⠃⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⠃
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⢸⠃⢀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⢧⣏⠀⢀⡞⢰⢁⡄⠀⠀⠀⠀⠀⢠⣇⣀⣀⠴⠒⠊⠉⠑⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢿⣀⠀⡞⠀⠀⣠⠎⠀⠈⠹⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠾⠒⠋

⠒⠈⢁⣠⡷⠊
⠂⠉⢁⠈⠀⠀⠀⠀⠀⠈
⡀⡠⠀⠀⠀⠀⠀⠀⡀⠀⣠⡀
⢿⠀⠀⠀⢀⠀⠀⠐⠁⠀⣿⣙⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦
⠌⣆⠀⠀⠀⠀⡀⠀⠀⢀⢿⢼⣿⣿⣷⣄⣀⣄⣀⣀⣀⣀⠀⠀⢠⣾⣿⠎⡏
⠀⠀⠰⡄⠀⡀⠃⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⣿⣿⣼⠇
⠀⠀⠂⠀⡀⠀⡀⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡟
⠀⠀⢸⡄⣗⠈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠠⠈⠡⠀⠀⠀⣠⢀⣿⣿⣿⣿⣿⣿⣿⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣇⣠⣤⣤⣦⣴⣤⣦⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⢠⠆⠐⠒⠐⢀⣿⣿⣿⡿⠛⠛⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⡏⢰⡤⢃⠀⠀⣸⣿⣿⡟⠓⠒⠀⢲⣦⠔⠀⠀⣈⠙⣿⣿⠟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⡀
⣀⠀⠀⣵⡟⠐⠉⠀⢀⠰⠐⣿⣿⣦⢤⡄⠚⠳⢀⡈⡁⣰⡞⠉⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣷⡎⠀⠀⠀⢠⠀⢠⡟⠠⢸⣿⣿⠃⠈⠙⠒⠒⠋⠹⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦
⣢⣞⣷⣦⣶⣶⣾⣋⠁⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣛⣿⠛⠉⠀⠀⡻⣁⣐⣾⡵⠿⠋⠀⢀⡀⠀⢰⣀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⡘⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⠁⠀⠴⢂⣠⠾⠛⠉⠀⠀⠀⠀⠀⠘⢻⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠉⠃⠙⠛⠟⣿⠉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡄⢂⡴⠞⠋⠁⠀⢄⣀⣀⣤⣤⣤⣤⣶⢾⡿⣿⠃⠀⠀⠀⠀⠀⣀⣴⣶⣾⣶⣦⢀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⡿⠙⢿⣿⣿⣿⣿⣿
⡀⣿⣗⣆⣀⣶⣼⣿⠿⠋⣁⣴⣀⡄⠠⠐⠐⢺⠇⠀⠀⠀⠀⡼⠟⠛⠞⠛⣿⡛⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡌⠋⠉⠻⣿
⠙⠟⠿⣿⣿⠿⠻⣁⠀⠀⠀⠯⠁⠤⡆⠀⠀⣾⡄⠀⠀⠀⡼⠀⠀⠀⠀⠂⠀⠉⠁⢈⣻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣓⡍⠀⠀⠰⠽
⣵⠖⣶⠚⠋⠔⠛⡁⠛⠀⢀⠀⠀⠀⠁⡠⠘⣿⡇⠀⠀⠀⢹⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⣻⡗⠃⢄⠀⠀⠀⠀⠀⠈⢹⣉⣀⣠⡔⠋
⣶⣶⣿⣭⣭⣉⣼⣷⣤⣤⡅⠀⡾⠋⠁⠀⠀⢻⣿⣄⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡎⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⠉
⡛⠿⠛⠛⣻⠛⠛⣯⡏⠛⠛⠉⠀⠀⠀⠠⠄⠘⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⢁⡀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣨⡅
⣷⣤⣤⢴⠟⠐⣸⡏⠀⠀⠀⣀⣴⡤⠴⠶⠟⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⣴⣶⣶⡾⠟⠟⠉⠛⠉⠋⠉⠁
⡝⣩⣤⣍⣿⣯⣽⣡⣤⣺⣿⣿⣿⣭⣁⣈⢓⣀⣤⡀⢠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣿⣿⣉⣭⡟⢻⡏⢉⣽⠋⢉⡹⠋⠡⠉⠉⠉⢩⠍⠁⠤⠇⠀⠀⠀⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠛⡹⢉⠽⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠋⣉⠤⠔⠊⠉⢣⠚⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⢡⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣫⣽⠿⡆⢹⡟⣻⣶⣭⣓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣺⣵⠾⢻⣿⣿⣿⣿⣉⣻⣿⣿⣿⣯⡻⣮⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢫⣾⣿⣵⣶⣾⣿⠿⢛⠂⢹⡙⠯⣙⠻⢿⣿⣾⣿⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⣿⣿⡿⠟⠋⠉⠀⠀⠁⠀⠀⠂⠀⠈⠣⡀⠉⠻⢿⣿⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⢟⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢢⡙⢿⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⡄⠁⠋⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠈⣿⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣌⡙⠒⠤⠴⣁⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡇⢀⣀⣸⠧⠟⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠐⠬⢭⣒⠲⠦⠤⢤⣬⣍⣉⣉⣉⣉⣉⣉⣉⣉⣭⣭⣥⠤⠤⠶⠒⠋⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⢍⣀⠀⠡⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢊⡩⢄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⣿⣿⣷⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⡾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⣿⣿⡿⠁⣶⢻⣿⣿⣿⣿⢛⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣹⡇⠀⣿⣿⠟⣻⣿⠇⢾⣿⠃⣾⣿⢿⣿⣿⣿⣟⡟⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⣧⠀⢿⡿⠀⣿⣟⠀⣿⡇⠰⡟⢀⣿⠋⣾⣿⣿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⣿⣷⣾⣿⣶⣾⣿⠐⣿⡇⢿⣿⣶⣿⣶⣿⣿⡏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⡏⢹⣻⡟⢿⣿⡷⢼⣿⡶⠿⡿⢿⢫⣿⣿⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⣿⣇⢸⣿⣧⠸⣷⡇⢘⣿⡇⢠⣿⡮⢸⣿⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣶⣾⣿⣦⣿⣧⣼⣿⣧⣷⣾⣶⣿⡿⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⣿⣿⡄⢹⢿⣿⠍⢸⣇⠰⣿⢰⣿⣿⠇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡘⣿⣿⡇⢸⡏⣿⠀⢼⣿⡀⠋⣾⣏⡿⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢻⣿⣷⠘⢧⣿⢰⣾⣿⣷⢤⣿⣹⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⣯⣿⣴⣿⣧⣸⣇⣿⣬⣾⢧⡟⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣍⣛⡛⠿⠿⣿⣿⡿⠿⠟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⣿⡛⢧⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢷⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠾⠓⠀⡀⠈⢛⡌⢳⣤⣄⣄⢀⣤⣦⣤⣄⣤⣶⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⣞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠆⠀⡀⢈⠀⠙⠛⣿⢿⣿⣿⣿⣿⣟⣯⣿⣿⣿⣿⣦⣤⣄⣤⣠⣤⣤⠶⢚⣭⢾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠂⠐⠒⢒⠉⣮⣉⣤⣱⣿⣾⣿⣿⣿⣿⣿⣯⣮⣷⡿⠻⠛⡽⣿⣿⠟⠋⡥⠚⠻⣣⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠬⠱⠖⠀⢭⣶⢾⣿⢿⣿⢿⣿⣿⡿⢿⣿⣿⣿⣿⢭⣿⣷⣯⡾⠾⠱⠂⢀⡀⠀⠀⠀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠎⠀⣐⡟⠛⠀⢠⣄⠳⠼⠻⢧⢉⠈⣿⣿⢿⣿⣿⣾⣿⢟⣠⣦⠶⠚⠋⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣦⢴⡏⠉⠀⣀⡀⢠⡜⠛⢺⣷⡀⠘⣷⠛⣿⣿⣿⡏⠛⣉⣸⣯⡌⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⡶⢾⡿⣼⡏⣉⣀⠒⠒⠦⣤⡉⠒⢠⠸⠀⠁⡟⡏⠀⣰⠟⠀⣵⠖⢀⣶⡄⠀⣠⡎⣠
⠀⠀⠀⠀⠀⠀⢀⣙⣿⣿⢖⣶⠮⢭⣿⣟⣘⣶⣦⡄⠙⡆⠈⠀⡪⠨⠅⠃⣀⣤⣤⠄⠀⠀⣈⣒⠁⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿⣶⣶⣾⣟⣻⠳⠾⡗⣸⡿⠠⠀⠁⠀⠀⣼⠃⢠⠤⡄⠤⣤⣈⣹⣟⢄⣘⣋
⠀⠀⠀⠀⠠⠄⠤⠤⠀⠀⢈⢙⣓⣻⣧⣮⣶⡞⠂⢍⠀⢹⣿⠤⠀⠠⣤⣿⣰⣿⢯⣽⣯⢭⠻⢬⣻⢓⠏⠁⠤
⠀⠀⠀⠀⠀⠐⠒⠒⢉⣿⠿⣿⣿⣿⠟⣷⣶⣦⡈⠀⠀⠈⢿⣿⣿⡿⠃⠉⠡⠾⠋⠥⠭⢦⣤⣄⣘⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⡋⢉⢗⡒⣈⣿⠉⠉⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⢀⠭⣥⣴⣾⡿⣿⣿⡽⠿⠟⠿⢿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⣤⠿⢿⠟⡡⣪⡞⠛⡉⠁⠂⣀⠀⠈⠉⠛⠛⠳⠄⡀⢠⣄⠹⠟⠋⠀⠀⠈⠋⠭⢓⠂⢄⣀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠫⠞⠀⠀⠀⡀⠐⠀⠄⢿⣶⣤⣄⣀⣀⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠋⠛⠑⠀⠰⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⡆⠀⠀⢠⡄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠉⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣥⠤⠞⠓⠒⠒⠚⠓⠢⢄⣙⢢⠀⠀⠀⠀⡠⠐⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣫⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢃⢀⡴⠋⠀⢀⡠⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡈⠋⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⡄⣤⠠⠤⠒⠊⠁⠀⡼⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠋⠀⠀⡀⠠⠄⠀⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⠻⢷⣄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡀⠐⠄⣴⠋⣰⠊⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠃⠀⠀⢹⣇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⠇⠀⢇⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⢦⡀⢀⠄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⠬⠷⣖⠊⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⠠⠄⠐⠂⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠑⠄
⢀⡤⠤⠀⠶⣀⠠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⣀⣀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠑⠠⡰⠆⠠⠤⣄
⣏⠀⠀⠀⠀⢸⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢸⠀⠀⠀⠀⢈
⠈⠁⠀⠀⠐⠊⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠈⠒⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⢍⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡰⢲⠶⢿⢷⡠⢌⠁⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠴⣒⡁⠒⠈⠙⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢰⠃⠀⠇⡎⠎⠞⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢞⣠⡿⢻⠛⢷⠚⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠒⠚⠀⠗⠁⣀⡠⠤⠐⢒⡈⠀⠀⠀⠘⠢⢀⡀⠀⠀⠈⢸⣰⠸⡀⠀⡇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠤⠒⢊⠩⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠍⣐⠢⠄⣉⠀⠉⠊⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠨⠙⠒⠤⠴⠋
⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣯⣭⣬⣭⣬⣭⣭⣥⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣾⣿⣶⣾⣿⣶⣶⣶⣾⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠉⠓⢄⠀⠀⠀⠀⡰⠚⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⡃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⣤⡎⠀⠀⠀⠀⢳⣄⣀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠒⠒⠒⠒⠢⠤⣄⡀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠒⠒⠒⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⢳⠀⢀⡏⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣀⠴⠒⠉⠉⠉⠉⠉⠑⠲⢄⡀⠈⠳⡀⠀⠀⠈⡆⢸⠀⠀⠀⢀⠜⠁⠀⣠⠖⠊⠉⠉⠉⠉⠙⠒⠦⣀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⢀⡎⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠹⡄⢀⣠⠧⠤⣄⡀⢠⠏⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢱⡀
⠀⠀⠀⡼⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣠⠟⠁⠀⠀⠀⠀⠉⠻⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢧
⠀⠀⠀⡇⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⠀⠀⠀⠀⣀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸
⠀⠀⠀⣇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢯⡽⠀⠉⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⣸
⠀⠀⠀⢹⡀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⡇
⠀⠀⠀⠀⢣⡀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠒⠲⠤⠤⠤⠤⠤⠤⠖⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠑⠦⢄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠊⠁⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣤⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠲⠦⠤⠤⠤⠤⠴⠖⠒⡛⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠒⠤⠤⠤⠤⠤⠤⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⢨⣷⠲⠤⠤⠤⠤⠖⣾⡅⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠘⠦⣀⣀⣀⣀⡠⠴⠚⠁⢀⡴⠃⠸⡄⠀⠀⠀⠀⢀⠇⠈⢦⡀⠈⠓⠦⣄⣀⡀⣀⣀⠴⠋⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠱⡦⠤⠤⢴⠎⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣤⣀⣀⣤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠙⣄⡤⠋⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⣤⠤⠖⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣤⣄⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⠻⡟⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣀⡻⠿⣘⡿⢫⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣄⣠⠧⣤⡜⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠿⠗⣼⣿⣷⣜⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⣸⠃⠀⠈⢧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢾⣿⣿⣿⣿⠇⠀⢠⣶⡄⣀⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡁⠀⠀⠀⢀⣽⣿⡟⠛⢿⠿⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡜⠿⣃⢿⠿⣁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⢧⣠⢼⣀⣠⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣣⣾⣿⣇⢾⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠞⠁⠀⢹⡁⢀⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠿⠅⠀⠀⠀⠀⠠⣿⣾⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠀⠀⠀⡿⠋⠻⣷⣄⣠⠤⢼⣿⡦⠤⣄⣠⣾⠟⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢿⣌⣓⠂⢉⡻⠀⠘⣿⠀⠀⢟⡉⠐⣒⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠞⠛⠁⠀⠀⠉⠀⠀⠉⠛⠳⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠛⠀⠀⡞⠿⠄⠀⠀⠀⢠⠿⢱⠀⠀⠛⠛⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠷⣦⡤⠤⠄⠀⠑⢐⠀⠀⠀⠀⠀⠒⠂⠀⠠⠤⢤⣶⡾⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀⢀⡤⠂⠀⠁⠐⠢⣎⡀⠀⠀⠀⢀⣣⠔⠂⠈⠁⠒⢄⠀⢀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠈⠀⠀⠠⡄⠈⠀⠘⢿⣿⣿⣿⡿⠃⠀⠀⡠⠄⠀⠀⡇⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠘⡄⠀⠀⠘⡄⠀⠀⠀⠀⣿⡀⠀⠀⠀⡰⠁⠀⠀⣠⠃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⢦⡀⠀⠈⢲⠀⠐⠊⠀⠉⠀⠀⡍⠀⠀⢀⡴⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢼⡀⠀⠀⠀⠀⠀⢠⠧⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣒⠒⠒⠒⣂⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠁⠀⠉⠻⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡋⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠸⠟⢁⣴⡦⠀⠀⠀⠀⠀⠀⠘⠻⠟⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠒⢦⠘⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⣸⢶⣿⠗⠀⠀⠀⠀⣤⢠⣶⠀⠀⠀⠀⠀⠠⢾⣷⣬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⠀⢀⡀⣠⡄⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⢀⣿⣤⡅⠻⠃⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣙⡑⢟⡀⠀⠀⢸⣃⣉⠰⠶⠀⠀⠀⠀⠀⣤⢴⢀⡀⠀⠀⠈⠻⣿⡿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡟⠋⠀⠀⠻⣿⣿⢶⡶⠀⠀⠀⠀⠠⣶⣶⣨⡄⠀⠀⠀⠀⠈⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠋⠀⠀⣀⢠⡄⠀⠀⠈⠋⠀⠀⠀⠀⠀⢀⡤⣼⡛⠱⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⣌⠱⠿⠂⠀⠀⡦⠴⢂⡀⠀⠈⣿⣿⡦⠂⢶⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠰⣿⠇⠙⠃⣾⡿⠀⠀⢀⡤⢴⠆⠀⠙⢿⣿⡷⠾⠂⠀⠰⣿⣷⣬⡄⠀⠀⠻⠭⠺⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⢩⠀⢠⣾⣿⣿⣷⣀⣴⡄⠀⢴⣷⣦⣚⡃⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠻⣿⣿⠋⠋⠀⠀⠀⠻⠋⠉⠁⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⠀⠀⠈⠁⢀⣤⢀⣤⠀⠀⠀⠀⠀⠰⣷⠬
⠀⠀⠀⠀⠀⠀⣶⠀⣤⠀⣨⣧⣍⠞⠀⠀⠀⣠⡼⣥⡘⢩⣤
⠀⠀⠀⠀⠈⣛⢠⣶⣿⡆⠈⠻⠛⠋⠀⠀⠀⠘⣇⣠⣯⣬⡁
⠀⠀⠀⠀⠀⠙⠙⠿⠋⠀⣠⣀⣤⠀⠀⠀⠀⠀⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⡉⠾⠆
⠀⠀⠀⢶⣃⣃⠀⠀⠀⠈⢻⣿⠷⠶
⠀⠀⠀⠘⠛⠋⠀⣀⠀⠀⠀⠁
⠀⠀⡀⠄⡀⠀⢸⣤⡥
⠀⠀⢶⣶⠄⠀⠀⠈⠁








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠈⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⣀⡀⠈⢧⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⡼⠁⢀⣀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⢀⣤⠂⠀⠀⢸⠉⠉⠀⡾⠀⢠⠊⠀⠀⠀⡱⢎⠁⠀⠀⠑⡄⠀⢧⠀⠉⠉⡇⠀⠀⠰⣤⡀⣿⢀⡀
⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠸⣾⡇⠀⠀⠀⠀⠑⠀⣈⡤⣄⠰⣀⠚⠇⠀⡇⢸⠀⠸⠓⣠⠇⣠⢤⡁⠀⠊⠀⠀⠀⠀⢸⣷⠇⠈⠁⠀⠀⠀⠀⠀⠀⢄
⢠⠋⠀⡀⠤⣄⡀⠀⠀⠉⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠛⠂⠈⡇⠀⠀⠀⠀⢣⡜⠀⠀⠀⠀⢸⠁⠐⠛⠀⠀⠀⠀⠀⠀⣼⡿⠟⠋⠉⠀⠀⢀⡠⠄⡀⠀⠑
⡏⠀⢀⠀⣤⡄⠹⡄⠀⠀⠀⠀⠀⠉⠓⠤⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠡⠌⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠚⠉⠀⠀⠀⠀⠀⢠⠏⢠⣤⠀⡂
⢻⡄⠈⠓⠚⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠓⠚⠁⢠
⠀⠉⠒⠂⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠐⠒⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢠⠖⠉⠀⠈⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠉⠲
⡟⠀⠰⠋⣽⠆⢈⡆⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⢀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠰⣯⠙⡆
⢳⠀⠈⠠⠀⠤⠞⠀⠀⠀⠀⢀⣤⣶⠋⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈⢢⡔⠁⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠙⣶⣤⡀⠀⠀⠀⠀⠳⠤⡀⠄⠁
⠀⠑⢄⠀⠀⠀⠀⠀⠀⠒⢛⢛⣿⡇⠀⠀⠀⠀⠀⠀⠻⣥⡴⢃⠤⠀⡀⢀⡇⢸⠀⢀⠀⠤⡘⢦⣬⠟⠀⠀⠀⠀⠀⠀⢸⣿⡛⡛⠒⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣾⠹⣧⡀⠀⠀⣰⠊⠀⠐⣄⠀⢨⠀⠙⠃⠀⢇⡸⠀⠘⠋⠀⡇⠀⣠⠂⠀⠑⡆⠀⠀⢀⣼⠏⣧⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠁⠀⠀⠘⠶⠖⠀⣿⠀⠀⠑⠀⠀⢀⣮⡑⠀⠀⠀⠈⠀⠀⣏⠀⠲⠾⠃⠀⠀⠈⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠈⠋⠋⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⡠⠛⠁

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⠐⡀⢀⠀⠀⢠⠀⡈⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠈⠀⢰⣧⣷⠀⠀⢰⣮⣷⠀⠘⠀⠀⠀⠀⢼⣥⡧⠀⠀⣠⢠⡐⠄⠀⠀⠀⠀⠈⠀⢰⣧⣶⠀⠀⢠⣴⣦⠁⠰
⠀⠀⠀⠠⠶⠉⠉⠘⠛⠈⠉⠱⠦⠀⠀⠀⠀⠶⠂⠉⠁⠶⠆⠻⠿⡃⠀⠀⠀⠀⠀⠀⠠⠦⠙⠉⠰⠖⠘⠛⢫⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡄⡆⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢒⣫⡀⠀⣜⡒⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢁⡀
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠀⠀⡀⠰⠋⠗⠀⠀⠀⡀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣰⡓⣰⠀⡿⠃⠀⠈⠁⠁⠀⠚⠋⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⠆⠀⠙⠉⠀⣴⡶⠦⠄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⠀⠀⠀⠜⠋⠁⠂⠀⠀⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⢶⡶⠄⠀⠀⠐⠿⣶⣤⣀⣀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣷⣆⠀⠀⠀⢈⠶⣿⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠛⠛⠃⠙⠋⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠿⠛⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣙⣥⣦⠒⠿⠛⠛⢷⡆⠀⠀⠀⠀⠀⠀⣦⣽⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⣀⡀⠀⠀⢦⠀⠀⠀⠀⠀⢠⣻⢿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢜⢿⣿⣿⣿⣧⣴⣦⠊⠀⠀⠀⠀⢀⣸⠿⠶⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⣿⠏⠋⠋⠀⠀⢀⠀⠀⠀⠘⠁⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡀⠀⠀⡀⣾⣿⣿⡎⠿⠀⠀⠀⠒⠛⠟⠀⠀⡀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣷⡦⠈⠓⡿⢛⣿⣿⡆⠀⠀⡀⢀⡀⠰⠘⣿⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠐⠻⠤⠾⠿⢿⡿⠁⠀⠀⢀⣠⣿⡟⠁⠀⠀⠀⠀⢀⠀⠀⠈⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠉⠃⠀⠄⠀⠤⠦⠴⠤⠠⠄⠸⠷⠾⠿⠄⠀⠀⠀⠘

⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢉⣥⠸⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠸⠇⣠⣶⣦⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⠿⠿⢃⣍⠁⠀⠀⠀⠀⠀⠀⠀⢰⣦
⡜⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⣴⡦⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠰⣿⡌⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣴⣶⣜⡋⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣸⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⢛⣣⣶⣦
⠀⠀⠀⠀⠀⣼⣿⣿⠇⣬⣭⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠛⠛⢋⣬⡋⠀⠀⠀⠀⠀⠀⠀⠀⢉⣥⡘⠿⠟⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⣅⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠙⠋⣵⣦⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⡇⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢱⡿⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣴⡌⠛⠃

⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄
⣁⠛⣫⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⡘⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡍⣡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠺⠿⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⡇⣙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⣌⠛⣋⠸⡗⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⠿⢣⣬⣍⠀⠀⠀⠀⠀⠀⠀⠀⠺⢇⣾⣿⣷⡝⠛⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠟⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠌⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣠⡄⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣤
⠀⠀⠀⠀⠀⣀⣉⢡⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣶⣎⢛⣁⣙⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⣿⠇⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣥⣌⠱⣾⡇
⠀⠀⠀⠀⠐⠛⣋⣈⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢨⣅⢾⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠘⠿⢂⣭⡁⣶⡌⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⢇⣤⡄
⠀⠀⠀⠀⠀⠀⠛⠋⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠈⠉

⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣈⡛⢡⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⡌⣫⣥⡀⠀⠀⠀⠀⠀⠀⠀⠸⠛⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡧⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡌
⣿⣿⡌⠉⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⣃⡙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢇⣭⢉⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣫⣍⢱⣶⡍⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱
⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣴⣶⣦⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠿⢇⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢶⡟⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣾⣷⣤⣄
⠀⠀⠀⠀⠠⣶⠜⢿⣿⠟⣥⡀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣩⣥⣶⣄⠛⣋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠻⣿⡿⢟⠁
⠀⠀⠀⠀⠀⠁⣾⡦⣰⣦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣈⡻⢿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣤⡄⠿⠃
⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⢸⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠆⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⡇⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠒⠲⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢀⡤⠴⢦⣀⡤⠴⠚⠉⠁⠀⠈⠁⠀⠀⠀⠀⣀⡤⠤⢦⠀⢸
⠀⠀⠀⠀⢀⠞⠉⠉⠉⠉⠉⠉⠙⢲⠞⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡧⡏⠀⡞
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡼⠁
⠀⠀⠀⠀⠀⣇⠀⠀⠒⡖⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠸⡄⠀⠀⢱⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢶
⠀⠀⠀⠀⠀⠀⠱⡀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⣦⣀⡄⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⠀⠀⠘⠛⠀⠀⠘⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠓⠆⠀⠀⠀⠀⣀⣠⡟⠁⠙⣦⠀⠀⠀⣀⢘⣾⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⢀⡤⠤⢤⡀⠀⠀⠀⠈⠑⠲⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠊⠁
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⢢⡀⠀⠀⠀⡤⠒⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠹⠒⠲⣘⣆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠊⠑⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠉⠑⡄⠀⠀⢦⠀⢸⠸⢄
⠀⠀⠀⠀⠀⠘⢤⣀⡀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠒⠋⠉⠛⠉⠓⢤⣄⡀⠀⠀⠠⢮⡱⡀⠈⡇⢸⠀⠘⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠶⠤⠶⠛⠙⠒⠓⠋⠙⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣤⣤⣄⣠⠶⡚⠻⠻⢷⣶⢄⣀⠔⢚⣟⣍⣋⡖⠤
⠀⠀⠀⠀⠀⠀⢀⣴⡿⡫⢏⣗⢄⣰⡿⠁⠉⠉⠉⠉⠓⢾⠁⠀⠀⠀⠈⠹⡥⠂⠀⠀⠀⠀⠉⠓⣾⣆
⠀⠀⠀⠀⠀⢀⡿⣫⠞⠁⠀⠈⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠖⣶⣒⠦⣄⡀
⠀⢀⣤⡤⠤⣼⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣴⡢⡀
⣰⣿⣿⠃⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣧
⣟⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⢻⢼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠃⡀
⠀⠉⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢥
⢠⠞⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⣾⣫⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢹⠁⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡟
⠘⡠⠋⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⢀⢀⣠⠔⣫⠟
⠀⠘⢾⠟⣉⠛⢦⢄⡀⢀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣄⠀⠀⠀⠀⠀⠀⠀⢠⢿⡏⠑⠶⠶⠿⠒⠋⠁
⠀⠀⠀⠉⠒⠶⠧⠯⠤⠯⠥⠖⠋⢿⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢈⠙⡶⠤⠤⡤⠤⢶⣫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢌⣙⠶⢂⠤⠦⠄⣠⣂⣼⠟⠀⠀⠉⠓⠛⠚⠒⠛⠋⠁

⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⠟⠃⠙⠻⣿⡇⠀⣴⣾⣿⣿⡆⢢⣤⠤⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡖⠀⠹⠀⠀⠹⣿⣿⣿⣿⢻⡏⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⣿⣇⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢸⣷⣀⣤⣤⣴⣤⣾⣿⣾⠆⠐
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣀⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣯⣿⣿⣥⣄
⠀⠀⠀⠀⠀⠺⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠺⠿⠿⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠹⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠈⣤⣤⣤⣌⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡝⠉⠻⠟⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡿⣄⠘⣆⠀⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣷⡈⡿⠿⣿⠀⢀⡴⣄⠉⠙⠋⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠿⣷⢿⣿⣧⡷⡄⠙⠀⣞⠶⡼⠀⠀⠰⠿⢛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣶⠥⠴⠦⠤⣙⡷⠿⢿⡇⠀⠀⠀⠘⠋⢀⡠⠔⠚⠳⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⡴⠁⠀⠀⠀⠒⠾⢏⣉⡀⠀⠀⠤⠀⡀⠀⠐⠒⢦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣖⠿⣶⠀⠀⠀⢠⡺⣏⡷⣄⠀⠀⣀⣾⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⣠⠀⠀⠀⠀⣨⠟⠛⠀⠀⠀⣌⠓⠋⠀⠋⠀⡀⠛⠉⢡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⣯⠀⠀⣴⠀⣻⠃⠀⢠⡀⠀⠟⠀⢰⡟⢤⠀⠠⠴
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣤⠀⠙⠀⠀⢻⡆⢉⡁⠀⣿⡁⠀⣤⡁⠈⣀
⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⡀⠐⡞⠇⠀⣇⠀⣽⡿⣄⠈⠛⠐⠿⠶⣦⣤
⠀⠀⠀⠀⠀⠀⢇⠀⠈⠄⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣀⠀⠀⠀⢳⡄⣽⠘⠀⠸⡀⠘⠁⠀⠀⠐⢢⣤⠀⢀⢉⡀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠁⠀⠀⠀⠀⣌⠀⠀⠀⠀⠨⡀⣀⠈⡄⢧⠸⣆⠀⢻⣇⠀⠘⢆⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢣⠘⢦⢲⡌⣧⠹⡆⠀⢿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠳⡈⢿⢳⠈⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠀⠀⠁⠀⠀⠣⠁⠀⠀⣦⣤⡤⠦⢀⣀⠀⠀⣠⣶⠿⠟⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠶⠚⢛⢋⠀⠀⠇⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀⡀⠀⠀⢘⣷⣾⣿⡿⠛⠃⠐⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⣀⣀⣀⠀⠀⠀⠀⠀⠉⠛⠒⠲⣶⣧⣄⣤⣴⠀⠀⣴⠄⠀⣴⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⢤⣀⠀⠀⠀⠀⠘⡍⠉⠁⠈⢁⡀⣿⣤⣰⠏⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠣⠤⢼⣦⣀⣤⣼⠷⢬⠛⢥⡀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠃⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠊⠉

⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃
⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠋
⠀⠀⠀⢀⡴⠖⠒⠒⠒⠒⠒⠒⠒⠦⣄⠀⠀⢀⠴⠖⠒⠒⠒⠒⠒⠒⠲⠦⡀⠀⠀⢀⠴⠒⠒⠒⠒⠒⠒⠒⠲⢄
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⠀⠀⣴⠛⠛⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠔⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⢾⠆⠀⣿⠤⠄⠀⠀⢸⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠸⠁⠀⢷⣤⣤⡄⠀⠸⣤⣤⡤⠀⠀⢧⣤⣤⠀⠀⠘⢦⣤⠜⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣤⢤⣄⡈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠊⣉⢤⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣞⠀⠀⠀⠙⠳⣄⠈⠳⣄⠀⢀⣀⣀⣀⣀⡀⢀⡤⠚⢉⡤⠖⠋⠁⠀⠈⡣⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠧⠀⠀⠀⠀⠀⣠⠗⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠰⣉⠀⠀⠀⠀⠀⢀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠰⡆⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⡼⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠲⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⣸⠃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⠝⠦⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠊⠉⠀⠒⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣽⣯⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀
⠀⠀⠀⣠⣶⣶⣤⡀⠀⣸⣿⣿⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣿⠀⣀⣤⣀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⠘⢿⣿⣿⠏⢰⣿⣿⣿⡆
⠀⠀⣀⣀⠈⠛⠿⠿⠃⠀⠀⠉⠀⠀⢻⣿⣿⣿⡏
⠀⣾⣿⣿⣿⣶⡄⠀⢀⣴⣿⣿⣿⣶⣌⠿⣿⠟⠁
⠀⠘⢿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠉⠛⠛⠉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣠⣤⣄⠀⠐⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠛⠛⠛⠛⠉⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠹⣿⣿⠟⢡⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠿⠿⠿⠛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢛⣤⣶⣿⣿⣿⣦⡀⠀⣰⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣼⣿⢸⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠈⠉⠻⣦⡄⠘⣿⣧⡀⠀⠀⠀⠀⢠⣿⡇⠘⣿⣧⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⢻⣆⠘⢿⣿⣦⣤⣤⣶⣿⠟⠀⠀⠹⣿⣷⣄⣀⣠⣴⣿⡟⠁⢀⣠⡴⠖⠛⠓⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠋⠀⣰⡟⠁⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⢀⣠⠴⠶⠒⠲⠶⣤⣀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⣸⣿⠇⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣶⣤⣴⣾⣿⠏⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⣀⣀⣠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⠿⢿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⣀⣀⣀⣤⣤⣶⣶⣿⣿⣿⠿⢿⣿⣷⣦⣤⣀⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠋⠁

⠀⠀⠀⠀⢀⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿
⠀⠀⠀⣿⡟⠁⠸⣿⣿⣿⣄⢰⣦⣶⣶⣶⣤⣤⡴⣾⣿⣿⣿⣿⠃
⠀⠀⢀⣿⣿⣶⡦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣽⣿
⠀⠀⢠⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡇
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠉⣽⠻⣿⣿⣿⣿⣿⠛⣿⢹⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣄
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣦⣴⣦⣤⣄⣀⣀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢿⣿⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠋⠙⠟⠙⠞⠻⢛⣿⣿⡿⣟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠏⢶⠛⡐⠒⠐⠄⡀⠉⠈⠙⠂⠱⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣄⢀⠘⣄⠄⠀⠈⠲⡤⠀⠐⠄⠁⣘⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣷⣦⡁⠻⠆⣀⠄⠀⠀⠀⢤⣈⠲⠄⠳⣦⡉⣹⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣾⣧⣄⣤⣼⣶⣯⣷⢦⣤⣮⣛⣢⣥⣝⡿⠿⡿⣿⡻⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠿⠛⣋⡙⠇⠄⠀⠢⣢⢸⣝⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠐⠉⠀⠀⠀⠀⠀⠈⠛⠂⠹⣿⢻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡻⢎⠀⠀⠀⡀⢸⠂⠀⢤⠐⠆⢢⢹⡾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠻⠏⠴⠀⠀⠀⣀⡀⠄⠠⠙⠀⢐⠀⡻⣿⣧⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣀⣀⠏⠁⠀⠀⠈⠀⠀⠀⡄⠌⠣⠉⣿⣿⣝⣿⣟⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠄⠀⠀⠄⠀⠀⠀⠀⠦⠀⠁⠀⠀⡱⠀⠍⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⠁⡇⠀⠀⠀⡀⠀⠀⠂⠀⠐⡽⠦⠀⣼⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣦⣁⢠⠀⠀⠙⢢⠀⡀⠠⢄⡄⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⡆⢢⢠⠈⠀⠀⠁⠈⣦⣶⡟⣮⣿⣳⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⡊⢘⣻⣿⣟⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣤⡚⢀⣀⠀⣛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣶⣿⣳⣦⣤⣽⡻⣽⣿⣿⣟⠛⠻⢿⣿⡿⡾⠿⢷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣽⣿⣿⣿⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠛⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⣦⡄⠘⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣄⣀⣀⠀⠀⠀⣠⠖⠉⠀⠉⠉⠉⠻⢶⣤⣀⣀⠀⢀⣦⣤⠤⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠻⣿⣿⡇⢀⣈⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣶⠀⣸⠇⠀⣀⡀⣀⣰⣖⠀⠀⣻⣏⢻⡌⠛⠉⠀⠀⠀⠞⢛⠛⢷⣄⠺⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡿⣟⣭⣭⡙⠳⢦⡀⠀⠀⠀⠀⠀⠀⢾⣿⡏⣴⣶⡌⣻⣿⡄⣿⣧⣾⣿⣿⣿⣷⣶⡄⠀⠛⣿⠀⠙⢦⡀⠀⠀⣴⢟⣿⣷⡊⣿⢀⡼⠁
⠀⠀⠀⠀⡠⠊⣿⠓⣿⣽⣏⣿⠀⠀⠹⣆⠀⠙⣧⠀⠀⢴⣾⣷⣮⣭⣶⣿⣿⣥⢰⣾⡯⢡⣶⡦⢹⣯⣄⠀⠀⠈⠓⠀⠀⠙⢦⡀⠻⣿⣷⢿⣽⡟⠊
⠀⠀⠀⢰⣷⣀⣿⣎⣙⣟⣛⣁⣀⣀⣀⣈⣳⣄⣿⣟⣁⣠⣤⣽⣿⣻⣟⣉⣨⣿⣠⣿⣷⣶⣭⣥⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣙⣷⣶⣶⣟⣻⣤⣦⣄⡀
⠀⠀⡴⠋⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⢦
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠳⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⣀⣠⠞
⠀⠀⠀⠈⠙⠻⠛⣯⣽⠿⠿⢿⣍⠉⠉⠉⠉⠉⠉⠉⠉⣩⣿⣿⢛⣛⠿⢿⣿⠋⣿⡏⣉⣽⣯⣿⣟⠛⠋⢉⣽⣿⠙⢯⡉⠉⠉⠉⢉⣭⣽⣍⠹⣿⠉⢿⠇
⠀⠀⠀⠀⠀⣠⣼⣟⣷⢾⣿⣦⠈⠳⣄⠀⠀⢤⡀⠀⠀⠙⣻⣇⠺⠿⢃⣺⡿⠇⢛⣿⣿⠿⣛⡻⢿⡿⠇⠀⠀⢻⡄⠀⠹⣦⠀⠀⣿⣹⣟⣿⢤⣿⡠⠊
⠀⠀⠀⢀⡞⠁⣿⡨⢿⣿⣵⠟⠀⠀⠈⠳⣄⠀⣿⣤⠀⠘⠿⢿⣿⣿⣿⡿⢻⣿⠘⣿⣯⡘⠿⠟⣸⣿⡷⠀⠀⠀⠀⠀⠀⠈⠛⢦⣌⣛⣛⣽⣾⠃
⠀⠀⠀⢸⣿⡦⠙⢷⣤⣡⡴⠀⠀⠀⣀⣤⡝⣧⣹⣷⠀⠐⠽⠏⠉⠈⠁⠀⢸⡟⠀⠻⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡉⠁⢸⣿⣿⣦
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠒⠛⠻⠁⠀⠉⠉⠛⠷⣦⣀⣀⣀⠀⣀⠴⠋⠀⠀⠀⠉⠉⠙⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡄⠘⠻⠉
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣤⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠻⠀⣿⠁⠈⣷⠰⠟⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣇⠀⢰⠾⢿⣼⣧⡿⠷⡆⠀⣸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠹⡙⢦⡀⠀⡿⣿⠀⢀⡔⢉⠏⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠈⣛⣠⣿⠁⠈⢛⡷⠛⣡⣌⠛⢾⡛⠁⠈⣿⣆⣉⡁⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⢿⠶⠞⠡⠚⠉⠉⠙⠳⠶⠞⠁⠀⠙⠋⠀⠈⠳⠶⠞⠋⠉⠉⠛⠌⠓⠶⠿⠟
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣶⠶⢦⡐⢦⣀⣀⣤⠴⠶⢦⡀⠀⣰⣄⠀⢀⡴⠶⢦⣄⣀⣀⡤⢂⡤⠶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢀⣭⠙⣿⡀⢀⣼⡷⢤⡙⢋⣥⢾⣥⡀⢀⣿⠋⣭⡀⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣰⣡⠞⠁⠀⣷⣿⠀⠈⠳⣈⣆⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠀⠸⠶⣾⢻⡟⣷⠶⠇⠀⢹⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣴⠀⣿⡀⢀⡿⠐⣦⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊

⠀⠀⠀⠀⠰⠆⠚⠚⠐⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠈⠂⠤⢀⡀⠀⠻⣯⣍⡁⠄⡀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⣓⡲⢾⡿⠛⠒⠒⠚⠛⢦⠀⠀⠀⠀⠀⠀⠀⠈⠓⡀⠀⢠⣶⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣽⡻⣦⣿⢋⠀⠤⠄⠀⡄
⠀⠀⠀⠀⠀⠀⣀⣄⣂⢀⡄⡤⣤⣬⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠛⢿⣼⠟⠋⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠤⡴⠃⣨⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⣠⣼⡇⠀⠀⠀⠩⡂⡀⠉⣡⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⣴⣦⣴⣿⣿⣿⠤⠀⠀⠀⠀⠀⠐⢂⠃⢿⡇⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣟⣠⠄⠈⠐⠀⠀⠀⠁⢨⠀⣸⠟⢀⣤⡞⠱⠀⠐⡀⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⢂⠤⠤⠤⣄⠉⢛⣿⣿⣟⣿⠛⠛⠛⢿⣿⣿⡀⠀⠀⠳⠄⡁⣀⣞⣠⣿⡿⠛⠙⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⢻⠋⠁⠀⠀⠀⠀⣿⠙⠿⣿⣿⣿⣀⠘⣃⣼⣿⡿⠃⠀⠀⣄⢀⣴⣿⠯⣄⣈⠋⠉⠓⢄⠁⡀
⠀⠀⠀⠀⠀⠀⠐⢄⠁⠤⠤⠤⡊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⣿⣿⣿⡟⠛⡡⠶⠶⢾⣷⣟⣣⡄⠈⠲⢸⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣈⣄⡀⠀⠀⠀⠀⢀⡤⠴⠚⠉⠁⡰⠋⠰⠿⣿⠁⠸⢿⠿⢅⡈⣄⣉⣲⡀⠀⠹⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣷⣤⡠⠀⠈⠉⠀⠀⠀⠀⢀⡠⠐⠁⠀⣉⣤⣮⣤⡿⢦⣥⣄⠠⠄⠱⢀
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠜⠉⢸⠻⠿⡏⠀⠀⠀⠀⡠⠄⠰⠎⠁⠀⠀⠀⢠⡿⠋⠀⠘⡁⢠⢿⠿⣷⣦⣄⡀⠈⠠
⠀⠀⠀⠀⢠⠁⠀⠀⠔⠀⠀⢀⠄⠠⠃⠐⣶⡤⠾⣿⡀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⡄⠀⠀⠀⢇⠀⠙⡯⢉⠑⠢⡘
⠀⠀⠀⠀⡀⠀⡀⠁⠀⢀⠐⠁⠀⠅⠀⠀⠌⠁⠀⢈⠀⠀⠀⠀⠀⠀⣿⠁⢠⠀⠀⠖⠀⠀⡀⠀⢻⡄⠀⠀⠀⠙⢦⠘⡄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠅⠀⠌⠀⠀⢰⠆⠀⠀⠀⠀⠀⢠⢴⡶⢋⡀⠂⠀⠀⠀⡀⠀⠀⣗⠀⠀⠂⠀⠀⠁⠻⠆
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⢉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠔⠢⡑⢦⣀⣠⡤⠶⠴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⠴⢊⠉⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⢒⠈⠈⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣜⣵⣿⣿⣧⡀⠀⠀⠀⢀⣀⢄⠀⠀⠚⢤⠀⠀⠸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⣿⣧⠀⡄⠀⠀⣾⣿⡿⠷⠄⠀⠀⠈⡃⠠⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠐⠪⠑⢨⢄⠀⠠⠻⢿⡧⠀⡀⠀⠀⠀⠉⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠪⠅⡥⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢑⠆⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡜⠀⠀⠀⠀⠀⠀⠈⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢷⠀⠀⠀⠀⠀⢀⠐⠀⠢⢸⢠⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠑⡄⠀⠀⢠⣀⠀⠀⠀⡜⢀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⣎⣁⣁⠤⢾⣁⣁⡠⠞⠯⠥⠤⠊

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


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣠⣤⠴⠶⠶⠶⠶⠶⣤⣄⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⡜⠡⢞⣴⠞⡉⠁⠠⣐⠙⠀⠀⠃⣂⠄⠈⢉⠳⣦⡳⠌⢣⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣬⣐⡈⠑⠄⠐⠀⡴⠋⡇⡀⢡⠂⠀⠀⡤⠆⠰⢤⠀⠀⠀⡌⢀⢸⠙⢦⠀⠂⠠⠊⢁⣂⣥⣤⣄⣀
⠀⠀⠀⠀⢀⣴⠿⢩⠀⠀⢀⠈⠙⠳⡄⠀⡞⠁⠀⠙⢦⣀⣂⠀⣀⣳⣤⣤⣞⣀⠀⣀⣁⡴⠋⠀⠈⢳⠀⢠⠞⠋⠁⡀⠀⠀⡍⠿⣦⡀
⠀⠀⠀⣰⢻⠉⠙⢻⣤⣀⣀⣈⣶⣃⣛⣘⣀⣀⣀⣀⠀⠀⣉⠉⠉⡉⠁⠀⢉⠉⠉⣉⠀⠀⣀⣀⣀⣀⣃⣛⣘⣴⣁⣀⣀⣤⡟⠋⠉⡟⣆
⠀⠀⢠⠇⣼⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣧⠸⡄
⠀⠀⠸⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡇
⠀⠀⠀⠃⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘
⠀⠀⠀⡄⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢠
⠀⠀⢰⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡆
⠀⠀⠘⡆⢻⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⡀⢀⡀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡟⢰⠇
⠀⠀⠀⠹⣼⣀⣠⣼⠛⠉⠉⢉⠹⡍⣭⢩⠉⠉⠉⠉⠀⠀⣉⣀⣀⣁⠀⠀⣈⣀⣀⣉⠀⠀⠉⠉⠉⠉⡍⣭⢩⠿⡉⠉⠉⠛⣧⣄⣀⣧⠏
⠀⠀⠀⠀⠈⠻⣶⣘⠀⠀⠈⢁⣠⡴⠃⠀⢣⡀⠀⣠⠞⠉⠍⠀⠉⡽⠛⠛⢯⠉⠀⠉⠉⠳⣄⠀⢀⡜⠀⠘⢦⣄⡀⠀⠀⠀⣃⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢛⠩⢁⡠⠂⠀⠀⠳⣄⡇⠁⡘⠀⠀⠀⠓⠆⠰⠚⠀⠀⠀⢃⠈⢸⣠⠞⠀⠄⠐⢄⡈⠍⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⢣⡐⢮⠻⢦⣁⡀⠐⠫⣠⠀⠀⣄⠍⠂⢀⣈⡴⠟⡵⢂⡜⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠙⠛⠶⠶⠶⠶⠶⠶⠛⠋⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⣾⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣏⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢠⣴⣿⡅⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡄
⢤⣾⣷⠆⠀⠀⠀⠀⢠⣿⠏⠀⠙⠿⣿⣷⣄⣠⡤⠶⠖⠺⣿⣿⡿⠒⠶⠤⣤⣀⣴⣾⣿⠟⠁⠈⢻⣿
⠀⠟⠛⠃⠀⠀⠀⠀⠨⣿⡄⠰⢦⣤⣤⠉⠛⢿⣿⠀⠀⠀⣿⣿⠇⠀⠀⢸⣿⠟⠋⢩⣤⣤⠴⠂⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⣦⣤⠄⢀⣠⣤⡙⠀⠀⠀⢹⣿⠀⠀⠀⠙⣡⣤⣀⠀⢤⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠀⡼⠟⠛⠛⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠛⠛⠷⡄⠈⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣤⣄⡀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⣀⣤⣤⣷⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠉⠁⠀⠀⢸⡇⠿⠏⣇⠀⠀⠀⠀⠀⠀⢀⡏⠿⠃⣿⠀⠀⠀⠉⠉⠉⠉⠙⢆
⠀⠀⠀⠀⢀⣴⣯⣄⣀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡤⠏⠀⠀⠠⠤⠄⠀⠈⠧⣄⡴⠃⠀⠀⠀⠀⠀⠀⣀⣀⣬⣷⡄
⠀⠀⠀⢠⠞⠛⠛⠿⠿⠿⠟⠛⠛⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠘⠛⠻⠿⠿⠿⠟⠛⠛⢦
⠀⠀⢀⡏⠀⠀⠀⢀⡤⠖⠛⠉⠉⠉⠓⠲⢄⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠴⠚⠋⠉⠉⠙⠓⠦⣀⠀⠀⠀⠈
⠀⠀⢸⣷⣶⠆⣰⠋⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⣿⣶⣦⣤⣤⣤⣤⣴⣶⣾⡏⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢴⣶
⠀⠀⢸⡀⠀⠀⡇⠀⠀⠀⠀⠒⢦⡀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠈⠁⠀⠦⣠⠖⠂⠀⠀⠀⢸⡇
⠀⠀⠀⢧⠀⠀⢹⡀⠀⠀⠀⠀⠀⢻⡀⠆⠀⠀⠉⠀⠀⠉⠙⡏⠉⠁⢠⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡼⠁⠀⣠
⠀⠀⠀⠈⠳⣄⠀⠹⣄⠀⠀⠀⠀⠈⢧⡀⠀⠀⠠⠀⠀⠀⣸⠛⣄⠀⠀⠀⠀⢀⣡⠞⠁⠀⠀⠀⠀⢀⡼⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠈⠛⠦⣌⣢⣀⠀⠀⠀⠀⠙⢲⠶⠤⠴⠶⠚⠁⠀⠀⠉⠛⠛⠋⣏⠀⠀⠀⠀⠀⢀⣴⣋⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⣀⡤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠮⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⣀⣀⣤⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣤⡀⠀⠀⢀⣠⣴⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠋⠉⠻⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣟⣛⣁⣀⣀⣀⣀⣀⣙⣻⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠔⢉⣉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣀⣈⠉⠀⣿⣿⣿⣦⠈⠦⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⣿⣿⣧⠀⢿⣿⣿⣿⡇⠀⣴⣦⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⣿⣿⣿⡇⠈⠻⣿⣿⠇⠘⣿⣿⣿⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠜⢁⣀⡈⠻⣿⣿⠇⢠⣤⣄⠁⠀⠀⠹⣿⣿⡇⢀⠇
⠀⠀⠀⠀⠀⠀⢸⠀⢼⣿⣿⡄⠀⠀⢀⣿⣿⣿⣿⣦⣄⡀⠈⠉⣀⠜
⠀⠀⠀⠀⠀⠀⠈⡀⠸⣿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠙⠻⠛⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠀⣿⣿⣿⣿⣿⠟⠛⠉⠉⢉⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠎⢀⣴⣿⣿⡆⠈⠋⣁⡈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠛⠛⠋⢀⠔⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⣄⠀⣼⣿⣿⣿⠇⢠⣾⣿⣿⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣰⣿⣿⡿⠀⣿⣿⣿⠟⠀⣾⣿⣿⣿⠇⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⣿⣿⣿⠃⠀⠈⢉⣡⣄⠀⢿⣿⣿⠏⢀⣀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠈⠋⠁⢀⣠⣴⣿⣿⣿⣧⠀⠉⠁⣴⣿⣿⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣤⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⢰⣿⣿⣿⠁⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⠿⠟⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡈⠙⠛⠛⠻⢿⣿⣿⣿⣿⡇⠀⠤⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠢⣄⠉⠛⠿⠛⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠠⢀⠀⠀⠳⡀⠢⡀⠀⠀⡌⢀⣎⠄⠀⠀⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⢤⣤⣀⠀⠈⠒⠄⡀⠠⠀⣌⠂⢌⢮⡀⣧⠚⠁⠀⣀⠀⠀⠀⠠⠔⠁⠀⣀⣄⠤⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣠⠄⠀⠀⠤⠀⠀⢀⠁⠓⢈⢉⢢⡙⠷⡟⢠⡄⠉⡁⠚⠀⡁⠀⠀⠠⠀⠀⠀⣈⡉⢁⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠔⠐⠋⠽⠿⣋⡠⢀⠠⠛⢋⡅⠀⠀⠀⠀⢀⣠⠬⠋⢤⡇⠙⢥⣄⣀⠀⠀⠀⠀⢨⡙⠻⠀⠠⠈⠙⢿⢭⡙⠂⢠⣀
⠀⠀⠀⠀⠀⠀⠈⠁⣒⣦⠒⠀⠀⣁⣊⢁⣰⠂⠊⠀⠖⠀⣀⠒⡡⢋⣠⠒⠁⠈⠲⣀⠙⢍⠲⣐⠀⠶⠀⠁⠰⣄⠁⡛⡈⠀⠀⠢⣤⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡈⢶⠈⢱⡀⣾⢹⠁⢸⡄⡳⢰⠃⠀⡿⢲⠀⡞⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⡏⠀⠘⠆⠈⠀⠂⠸⠀⠙⠉⠀⠀⠇⠘⠂⠓⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠸⠛⠢⠀⢀⡈⣌⣀⠙⠄⣀⠐⠶⡀⠱⠤⣡⡀⠙⠤⡀⢠⠔⠉⣠⢊⠤⢉⠀⠴⠀⡀⠰⠋⣀⡉⢁⠀⠀⠤⠻⠅⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠠⣬⣲⣶⣄⡀⠂⠐⣦⣌⠃⠀⠀⠀⠀⠈⠙⣢⡤⠚⡧⣠⡚⠋⠁⠀⠀⠀⠀⢚⣡⣤⠊⠀⢂⣡⣶⣒⣡⠄⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠁⢈⡉⠀⠀⠀⠒⠀⠀⢈⠀⡤⣡⡠⠉⣡⡾⣇⠙⠃⣠⡁⣤⢀⠀⠀⠀⠒⠀⠀⠐⠋⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠋⠁⠀⡀⠔⠈⠀⠈⠐⢉⠔⢩⠋⡀⡟⣤⠁⠈⠙⠈⠢⡈⠐⠤⡀⠀⠙⠛⠓⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠂⠀⠀⢀⡔⠡⠒⠁⠀⠀⡀⠘⠕⠀⠀⠀⠀⠀⠀⠂⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠸⠂⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠜⠀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢡⠀⡌⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠘⣮⡖⠁⠀⣀⡤⠤⠤⠤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡃⡠⠊⠁⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠉⠉⠉⠉⠉⠉⠀⠚⠀⠀⠀⠀⠙⠀⠀⠀⠠⠎⠀⢠⣤⠤⠖⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⢖⡁⠀⠀⠀⠀⠀⠀⠀⡤⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣄⣀⣀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠐⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⡀⠀⠀⠀⠀⠀⠁⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⡠⠒⠈⠐⠀⠠⠤⠐⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠑⠢⠄⠂⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢒⠢⠋⣀⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁⡂⠅⠐⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠊⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠂⠳⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢁⡨⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣂⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣋⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⢈⣁⣀⣤⣭⣤⣌⣑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⢉⠑⠠⡀⠀⠀⡀⠀⢉⢑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠑⠤⢈⣀⠡⠔⠐⢤⠃⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠈⠁⠀⠁⠢⣀⠀⠀⠀⣀⠤⠀⢄⡏⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠅⠒⠒⠂⠀⠤⢀⡀⠉⠈⡁⠤⠐⠒⢸⠁⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢰⣋⠉⡋⠍⣋⠉⣉⠉⠙⣒⠢⠤⣼⠁⣀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣼⠀⠀⢀⠤⣄⠀⠀⠘⣆⠔⣲
⠀⠀⠈⠲⢖⡢⠁⠒⠒⣀⠐⠢⠀⠈⠀⣿⡊⠄⠒⠂⠠⠄⣁⠒⠠⠀⠠⠐⡩⠐⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠓⡄
⠀⠀⠀⠀⠀⠈⠉⢓⡲⠖⠈⠀⠀⢀⠍⢿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⢀⡠⠤⢼⠀⠀⠀⠀⠀⠀⠀⢰⠉⠒⠒⠃
⠀⠀⠀⠀⠀⢀⡴⡫⢂⠀⠀⠡⠐⡡⢂⡩⢦⡠⠂⠈⠁⠐⠠⢄⡀⢀⡠⠔⢁⠔⠀⠘⣆⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⣞⢁⠄⡡⣒⠒⣂⠼⠚⠉⠀⠀⠳⣎⠀⠀⠀⠑⠀⠠⠤⠄⠂⠁⠀⠀⠀⢘⣆⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠘⠳⠮⠾⠖⠋⠁⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⡀⠤⠤⠄⢀⣀⠠⠔⡡⠊⠳⢄⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠮⣤⣂⠈⠀⠒⠒⠒⠈⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⡃⠘⣉⠉⠩⠋⠉⡩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠤⠤⠖⠋

⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣏⣛⡙⠿⠗⠀⠀⠀⠀⠀⠀⠀⣷⠺⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡻⠷⢠⣤⡀⠀⠀⠈⠛⠁⠻
⠀⠀⠀⠀⠀⡀⠀⠐⣤⣀⠀⠀⠀⣿⣷⢸⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠰⣾⣿⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡋⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠘⢿⣷⠹⠿⠀⣀⠀⠈⠁⠻⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⡾⠿⣴⠆⠀⠈⠛⣥⣦⡝⠛⠃
⠀⠀⠀⠀⠙⢿⣷⠀⢰⣶⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡆⣀⣀⡀⠀⠀⠀⠀⠷⣰⣿⣶⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⣶⡾⣆
⠀⠀⠀⠀⠀⠠⣶⣶⠌⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣮⣍⡉⢿⣿⡿⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠉
⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⢿⣿⡟⣱⣶⣮⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣸⣿⢧⣿⡆
⣟⣙⠻⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣝⢛⣰⣶⡄⠀⠀⠀⠉⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡿⢧⣭⣌⣛⡋
⠿⣿⠈⠉⠀⠀⠀⠀⢀⣄⠀⠀⠀⢬⣽⡅⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡈⣶⣤⠀⠀⠀⣿⡿⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣸⠟⣛⠾⠆⠀⠀⠉⠶⣷⠎⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣈⡻⠿⠙⠛⣀⣤⡀⠀⠀⠙⠻⠛⠉⠀⠀⠀⠀⠀⣀⡀
⣤⣤⡄⠀⠀⠀⠀⠛⢾⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣧⡀⠀⠀⠀⠀⠈⠻⠿⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⠿⣟⡁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⠿⠀⠀⠀⠀⠀⠀⠐⠾⠿⠂⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣦⣄
⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿
⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⡼⣿⢳⣿⣇⠀⠀⠀⠀⠀⠀⠀⣀⣠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠈⠻⣿⠏⣼
⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣼⣟⢳⣶⣦⣭⣅⠀⠀⠀⠀⠀⠀⢀⣯⣬⢹⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡮⣩⣾⣿⡆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢦⣤⣄⢿⣶⠀⠀⠀⠻⠟⣼⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠻⠿⠀⠀⠀⠀⠀⢠⣠⣄⢀⠀⠀⠲⣾⠆⣙⢿⣿⡇
⠀⠀⠀⠀⢄⣤⣽⠛⢈⣉⣤⣶⣄⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⢫⣭⣻⡃⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠙⣛⣀⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡻⠿⠿⣧⣶⣶⡆⠀⠀⠀⠈⠉⠩⠟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠛⠛⠁⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⠙⣻⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢏⣈⡻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣷⡄⣤⡀
⣦⢴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⢀⣀⠀⠀⠀⠙⠿⠃⢾⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡞⠿⠹⣿⡧
⣷⣶⠨⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠗⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⣾⣯⢹⣿⣷⣶⣦
⠉⠛⠀⠀⠀⠀⠀⢠⣴⣷⣠⡀⠀⠙⠿⣃⣬⡻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⣦⢻⣿⠀⠀⠀⠛⠋⢾⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⣮⣡⣶⣭⡀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠲⣶⣮⠉⢠⣤⣶⣿⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣤⣶⣦
⣿⣿⠇⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⠃⠀⠀⠀⠀⠀⢈⣭⣤⡘⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣝⡛⠛
⣽⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⣿⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢸⣿⣆⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⢫

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⡸
⠀⠀⠀⠀⠀⢀⣀⠉⠉⠓⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⡴⠁⠒⠒⢂⣠⠶⠀⠀⠀⠀⢀⣠⠴⠖⠚⠋⠉⢁⣀
⠀⢸⡀⠀⢀⠤⠮⠷⣄⠀⠀⠀⠀⠉⠛⢦⣀⠀⢀⡴⠊⠀⠀⠀⠈⠀⠀⠐⠚⠫⠤⣀⠀⢀⣤⠞⠉⠀⠀⠀⠀⢀⡾⠧⠦⢄⠀⠀⡇
⠀⠈⣇⠀⠁⠀⠀⢀⣈⣆⠀⠀⠀⠀⠀⠀⠿⠓⠁⠀⠀⠀⠀⠀⠀⠀⠶⡶⠦⢤⣄⣈⠳⡿⠁⠀⠀⠀⠀⠀⢠⢋⡀⠀⠀⠀⠀⢰⠇
⠀⠀⠹⡄⠀⠀⡴⠉⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠈⢆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢠⠏⠉⠉⠳⠀⠀⢀⡟
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⣈⣧⡀⠀⠀⠀⠐⠛⠓⠆⠀⠀⠀⠀⠀⠀⠈⠑⠬⠂⠀⠞⠛⠒⠀⠀⠀⠀⣰⣃⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠵⠢⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠥⠾⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠙⣶⣄⠀⠀⠀⠀⠀⢠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢼⣥⡄⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⣸⣷⡀⠀⠀⠀⠀⠀⣼⣯⠀⠀⠀⢀⣿⣿⡆⠀⢠⠀⠠⡤⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠣⢾⠀⢸⣿⣿⣿⣷⣶⣾⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣿⣿⣿⣿⠀⣸⠼⠢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⣏⣿⡟⠛⠻⢿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⠟⠛⠛⣿⢯⡏⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠹⣎⡣⣄⣀⣨⣿⡟⠀⠀⠀⠀⠀⠀⠹⣿⣇⣀⣀⠴⣣⡞⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠈⠙⠒⠛⠛⠉⠀⢀⢀⡠⢤⣀⣀⠀⠈⠛⠛⠒⠋⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠀⣠⣤⣖⡚⠋⠉⠉⠙⠲⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠉⠑⠔⠉⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠔⠛⠉⠉⠉⠓⣲⣦⣄⡀
⠀⠀⠔⠋⠉⠉⠉⠉⠙⠒⢦⡀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠃⠒⠒⠒⠒⠂⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠊⠉⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠉⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡁⡀⠀⠀⡠⠖⠚⠉⠉⠙⠒⢤⡀⠀⠀⡈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠓⠚⠁

⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠾⠻⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⣼⣶⣟⣷⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠉⣿⠂
⠀⠀⠀⠀⠘⢀⣀⠀⠈⠙⢦⠁⣀⢶⠓⠶⣧⡑⢠⣤⣤⠄⠀⠀⠚⠙⠗⠀⠹
⠀⠀⠀⠀⠀⠛⣽⠗⠀⠀⠘⠠⣾⣾⠳⣿⡿⢿⣿⡟⢡⠔⠀⠀⠀⢠⡆⢠
⠀⠀⠀⠀⠈⠂⠈⠀⠴⣄⢠⢴⣿⣿⣰⣿⣿⣾⣿⣿⣧⠀⠀⠀⢠⣿⠉⠃
⠀⠀⠀⠀⠀⠄⠀⠀⠘⠉⣶⣜⣻⣿⣿⣿⣿⣿⡟⣟⠛⠲⣇⡀⠈⠙⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠻⢹⡟⢿⣿⠙⠛⠉⠙⠂⠀⢁⡄⠀⠀⡜
⠀⠀⠀⠀⢠⣤⣄⡤⠀⣶⣿⣱⡆⢸⠉⢨⡇⢀⡤⢤⣄⠀⢤⠆⠀⠀⣾⠳
⠀⠀⠀⠀⢿⣍⠷⢾⣦⡐⠾⠿⢳⣔⠁⢸⣀⣼⣿⣿⡋⠀⢀⣵⣐⠀⣿⡇
⠀⠀⠀⠀⠈⣿⡐⢰⣾⣿⣷⣶⣿⡟⠀⢠⣿⣷⣦⣤⡤⢴⡯⢈⣛⣉⡟
⠀⠀⠀⠀⠀⠘⣿⣤⣶⣁⠈⠀⠀⠙⣧⠟⠁⢉⣽⣿⣷⡿⢷⡀⣿⣟⣧
⠀⠀⠀⠀⠀⢀⠬⡿⠿⢟⣣⣤⡖⠒⠻⠶⠒⣈⣽⡿⢋⣢⣌⢿⣹⣿⣿⣽⡤
⠀⠀⠀⠀⠀⠈⡀⢀⡀⠀⣧⠉⠿⢾⡶⣶⡿⠛⡻⠗⠏⠛⠙⢟⣿⣿⣿⡬⣡⡔
⠀⠀⠀⠀⠀⣐⡀⣾⠆⠠⢿⠎⠘⠚⠁⡀⠀⠀⠐⣠⡎⢢⡙⠌⣻⢿⢿⡯⠘⠛⢂⣀
⠀⠀⠀⠀⠈⢠⣇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠧⠘⠀⣣⠇⠘⠛⠂⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠁⠀⠀⢰⡦⠀⣤⠀⠀⠀⠀⠀⠀⠈⠀⣀⣄⣾⠇⠀⠀⠀⠀⠁⢀⠀⣠⣄
⠀⠀⠀⠀⠀⣤⠀⠀⠈⠃⠘⠛⠀⠀⠀⠐⠦⠆⣰⠻⠿⠁⠉⠂⠀⠀⠀⠘⠤⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠙⢣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠔⠇⠠⡄⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⢐⣶⣆
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏
⠀⠀⠀⠀⠀⠀⣰⠠⡀⠀⢀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠠⠤⠔⠂⠀⠀⠀⠋⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠈⠈⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⡀⠘⢺⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣴⣿⣆⠀⠀⠀⠀⠀⠀⠈⡀⠀⡀⠀⠀⠀⠀⠀⢀⡿⠀⢪⠁⢨⣺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡄⠆⠙⢺⣿⡇⠠⡀⡇⠀⡀⡆⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⡈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣧⡇⠀⠑⢤⡇⣆⣿⣇⠀⠀⠀⠀⠀⠀⠀⠆⠀⠃⣶⠀⠹⠀⢸⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣹⠈⠛⡀⢆⠸⡷⡼⢯⣇⠀⠀⠀⢸⡆⠀⡿⣾⣷⠐⣣⣦⡀⠠⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⡄⠐⠊⣃⢘⣇⣾⢿⡇⠀⠀⠀⠀⠀⢆⢹⣿⡇⠺⣿⠃⡸⣧⣼⡏⢎⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⠈⡆⣄⣫⢀⡭⣫⣬⠃⠀⠀⠀⠀⠀⠀⣸⣿⣿⣤⠁⠀⠊⣌⠁⣰⠏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣾⠋⢶⣿⣿⣿⣯⣶⡷⠮⠀⠀⠀⠀⠀⢀⡀⢿⣿⠉⠉⠀⠠⣘⡹⠇⠋⠈⠁⠀⣄
⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⢀⢉⣴⠾⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⢁⣹⣿⠷⢲⠿⠀⠀⠀⠁⠀⣀⠱⠂
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⢠⡀⢤⣸⣡⣼⣶⠿⠟⠛⠛⠀⠀⢀⣤⡶⠶⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠛⠛⠛⠻⣿⢶⣤⣀⣤⠶⠶⠟⠛⠛⠷⠶⢦⣤⣀⣴⢾⣿⠟⠋⠛⠛⢛⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠈⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠈⠁⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠃⠀⣠⣶⣿⣿⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⣶⣄⠀⠘⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⣴⣾⣿⣿⡇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⣟⣿⣿⣦⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⡄⠀⢻⣿⣷⡇⠀⠀⢠⣤⣄⠀⠀⣠⣤⣄⠀⠀⢀⣤⣄⠀⠀⠀⢻⣾⣿⡟⠀⢠⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣄⠙⢿⡇⠀⠀⢶⣿⡿⠃⡞⢹⣿⣿⣧⠀⢦⣿⣿⠇⠀⠀⢸⡿⠋⣠⣦⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⡇⠀⠀⠀⠀⠀⠀⣇⢸⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⣿⡷⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠙⠿⠿⠿⠋⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⠶⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠉⠉⠉⠀⠠⢤⣄⣀⡀⢡⡀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢷⣄⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠙⣿⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠑⣌⠛⠶⣤⣄⣀⣠⣤⡿⢉⡇⢀⣴⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠈⢣⠀⠀⠈⠉⠉⠁⠀⣸⡿⠋⠀⠈⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⣰⣿⠁⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣶⣿⣿⣦⣀⣀⣏⠀⠊⠀⣿⣶⣶⣿⣿⣿⣷⣤⣾⣷⣶⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣶⠒⡲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⠟⠉⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠉⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡕⠁⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠉⠻⣶⣶⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⢃⠊⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡇⢆⠀⠀⠀⠀⢀⣦⣌⡈⢻⠉⠳⠈⢻⣿⡟⢡⡾⠋⠏⣾⣿⣆⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⡳⠄⠀⠀⠸⠟⠛⠛⠀⡀⣤⣀⠀⢿⠁⣠⡖⣀⠀⠈⢭⣤⣄⠰⣶⣦⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢰⣶⠄⣰⡖⠂⠀⡀⢀⣿⠈⠻⣧⢸⠰⡿⢰⣿⡀⢷⡤⠈⠉⠓⠀⣿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⣿⠀⠉⠀⡀⠛⠛⠀⣿⢀⡀⠙⢸⡆⢃⡀⠛⢃⣠⣤⣈⠙⠳⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣵⠋⠠⠾⠋⣠⠖⢺⡶⢦⡀⢻⡄⠀⠁⣾⠁⡴⠋⣿⠀⡝⠀⠨⡛⢦⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠃⣀⡤⢀⠀⠈⢧⡘⢃⣠⣿⡄⠁⠀⠀⠻⢰⡿⠖⠚⠊⠀⢀⣠⡷⠌⠧⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠋⢰⣿⣦⣀⡀⠈⠉⠁⢹⣿⡀⠀⠀⠀⢸⣧⡀⠀⢶⣬⣉⡁⢀⣠⠤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢿⣄⣀⣄⣀⣀⣀⠀⠀⠀⠈⠉⢧⣀⠈⢀⡼⠉⠁⠀⠀⡈⠉⠉⠉⣠⠆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠉⢽⡛⠋⠁⠀⠀⢐⠀⢀⠀⠀⠈⠒⠁⠀⠀⠀⠀⢀⡇⢀⣤⡾⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⡀⠙⢿⣷⣄⣙⠦⣆⣀⣀⡤⠴⠿⠶⢤⣄⣤⠴⠊⢀⡼⠟⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠾⣿⣿⣏⠙⠻⣿⣦⣄⡈⠉⠙⠛⠙⣦⣄⡀⠀⠀⠀⢀⣠⡴⣀⣀⣠⣤⠶⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣀⠙⠻⣿⣝⠛⠿⢿⣿⣿⣿⡿⠿⠿⠛⠉⠀⠸⠟⠉⣠⣾⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣷⣦⣀⠀⠉⠉⠓⠒⠀⠀⠀⢀⣠⣴⣾⣯⣁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠈⠙⠻⠷⠆⠀⠀⢀⡀⠈⠍⠋⠁⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢠⢁⠉⠉⠙⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠁⢀⠀⠀⣀⣠⡴⢦⡤⠤⠠⠳⢤⣀⠀⢄⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣌⣂⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⠆⠀⠀⠀⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡀⡠⠈⠰⠁⢀⠀⠨⡔
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠡⠀⠀⠈⠁⠀⠐⠂⠉⠠⠠⠀⠂⠠⡄
⠀⠀⠀⠄⠲⠐⣀⠀⠀⢀⣠⠴⠒⠚⠛⠛⠒⠢⢤⡐
⠀⠀⠀⠀⠀⠀⠁⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢
⠀⠀⠀⠴⠀⠀⠀⡞⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⢀⠀⠀⢀⠀⡀
⠀⠀⢤⠀⢠⠀⣸⠁⠀⠀⠀⢸⠐⡅⢿⠀⠀⠈⠓⢦⢸⡇⠀⠑⢄⠀⡇⠀⡀⠓⢦⢈⠒⡄⠀⠀⠙⢦⡀⠀⠹⡄⠈⢣⡀⢸⠀⡇
⠀⣠⡀⠐⠄⠐⢺
⠀⠈⠄⠀⠒⠀⠘⣆
⠀⠀⠀⠐⠀⠒⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢤⠀⠰⠆⠀⠀⢙⠦⣄⣀⠀⠀⠀⣀⣠⠴⠊⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠈⠉⠙⠉⠁⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⠀⠁⠈⡀⠀⡄⡀⠰⠆⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠄⠀⠈⠀⠀⠀⠐
⣶⣶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣄⠀⠀⢠⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⢾⣿⣿⣿⠇⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣄⡈⢿⣿⣿⠇⠀⠈⠻⠿⠋⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⣴⣿⣿⣦⡀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⢀⣼⣿⣿⣿⣿⣿⣦⣈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⣰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠹⣿⣿⡿⠁⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠈⠉⣡⣤⣄⠘⠛⠛⣡⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⢀⣼⣿⣿⣿⣷⡀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧
⠀⠀⠀⠀⠀⢶⡄⠖⢳⠄⣀⡀⠀⠀⠀⠀⢠⣴⣬⢿⣏⣴⣤⠀⠀⠀⠀⠀⣀⡀⠴⠓⠆⣴⠆
⠀⠀⠀⠀⠀⠰⡞⢓⣦⣼⠰⠼⡖⢀⡠⠔⣌⠛⠈⣽⣽⢘⠋⡔⠠⣀⠐⡾⠴⢸⣤⣖⠋⡷
⠀⠀⠀⠀⠀⣀⣀⣇⣆⣛⣓⣒⣂⣁⣀⣈⣁⣈⣉⣾⣎⣉⣀⣉⣀⣀⣉⣒⣒⣛⣃⣆⣕⣀⡀
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡀⢇⣃⣶⠖⠒⡂⢄⡀⠠⣄⠠⠦⣻⣣⢦⠀⡤⠀⣀⠤⡒⠒⢶⣆⣃⠕⡀
⠀⠀⠀⠀⠀⢘⡳⠖⢋⠸⣘⡹⠓⠀⠈⠑⢡⣶⣠⢽⣽⣰⣦⠑⠉⠀⠐⠻⣙⡰⢉⠓⠶⣟
⠀⠀⠀⠀⠀⠛⠁⠓⠞⠁⠀⠀⠀⠀⠀⠀⠈⠙⢡⣿⣧⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠖⠃⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡀⠀⠀⠰⠀⠀⡇⢀⣀⣀⣤⢴⡇⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠈⡀⢠⠐⠆⠀⠀⠁⠉⠉⠀⠀⠈⠁⠐⣤⠀⠖⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠻⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠄⠀⠀⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⢀⣤⡞⠩⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠦⠀⠘
⠀⠀⢠⡼⠋⠀⠀⠀⠞⠓⠒⢒⡶⠂⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⣾⢴⠇⣄⢀⠀⠀⠀⠀⣧⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠳
⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⠉⣦⡀⠀⠀⠀⢸⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡼⢶⡐⠢⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢸⡇⠀⠀⠀⡿⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣧⡙⡆⣸⠀⠀⠀⠀⠀⠀⣠⣤⠤⠀⢀⡄⠀⡘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢠⠇⠠⡇⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢺⣿⣟⣿⣇⢸⡟⠀⠀⠀⠀⢀⣾⠿⣦⣦⡀⢸⠟⢠⠃
⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⡞⠀⠀⢇⡸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡼⠁⠀⠀⠀⠀⢸⣿⣤⣿⣿⠇⠏⢀⠎
⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡴⠋⠀⠀⠀⡼⠁⠀⠀⠈⢷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⣠⣤⣤⡈⠻⣿⣿⣿⠋
⠀⠀⠙⠲⠦⣤⣤⢤⡤⠖⠋⠈⠉⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠈⠑⠻⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⢀⠠⠝⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⣀⡀⠀⠀⠀⠀⠀⠱⣤⣀⣤⢴⡋⠀⢀⡤⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠈⠉⠉⠑⠂⠤⣀⠀⠀⠈⢆⣀⠍⠉⠉⢀⠀⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠑⠦⢤⣤⣤⡤⠴⠚⣷⠒⠚⠉
⠀⠀⠀⠀⠀⢀⡟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢠⢏⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⢀⠄⣰⠃⠀⠀⢀⣠⡤⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠙⣆⠀⣠⣯⠔⠋⠳⡄⠂⠁⡴⠂⠀⢀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢸⠃⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢻⢠⡀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠘⠞⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⣧⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢧⠳⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣸⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⣀⠀⠀⠀⢀⣠⠖⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠉⠉⠲⣌⠓⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠤⠀⣀⠰⡀⠀⠀⠀⠀⠀⠈⣆⠀⠳⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⠀⣌⢣⣹⡄⠀⠀⠀⢀⠀⠘⢦⣤⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⡀⠀⠀⠀⠟⠀⠀⢸⡇⢸⡆⠉⢳⠀⠀⠋⠉⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠶⠶⠂⠚⠃⠞⠁⠀⠸⣄⠀⠀⠀⠀⠀⠀⢰⠈⢆⣿
⢀⣀⣀⡀⣀⡀⢀⡀⣀⣀⢀⡀⣀⠀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⢛⠒⣦⣤⣤⣄⣸⣤⡜⠁⢀⣀⠀⠀⠀⡀⠀⢀⠀⣀⣀
⣿⣿⡿⡿⢿⡿⠻⠿⢿⢿⠿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣶⣷⣾⣶⣾⣷⣾⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿