Talisman Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠍⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡄⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⢶⠊⠙⢦⣀⣀⠐⠂⠐⠒⣊⣡⠞⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢰⡇⠀⠀⠀⠈⠉⠈⠁⠀⠉⠉⠀⠀⢀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⣠⠞⠁⠙⢦⣀⡀⠠⠤⠀⠀⠀⠠⠤⠔⢚⣡⠞⠁⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢠⠁⠀⠀⠀⠀⠈⠁⠒⠂⠀⠀⠀⠐⠒⠊⠉⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢡⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠹⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠏⡄⠀⠘⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠸⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠸⡰⣆⠀⠱⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢃⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠱⡌⢦⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢦⡙⠢⣄⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⠀⠀⠀⠀⠙⠦⡀⠉⠒⠤⣀⡈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠸⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢇⠀⠀⠀⠀⠀⠙⣷⣤⣀⠀⠉⠑⠂⠤⢄⣀⡉⠉⠛⠛⠻⠿⣁⡀⠀⠀⠀⠀⠇⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⣆⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⣤⣄⣀⠀⠀⠉⠉⠐⠒⠠⠤⣈⡑⠢⡄⠘⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢦⡀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠟⠓⠂⠤⢄⡀⠀⠀⠉⠐⠻⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡦⢌⣉⡐⠒⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⡖⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⠉⠒⠒⠒⠒⠒⠒⠒⠈⢉⠄⡰⠄⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠄⢀⣀⣀⠠⠤⢒⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⠂⠤⠤⠤⠄⠒⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣤⠤⢤⡄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡜



⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⣿
⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠻⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠰⠀⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡂⢰⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢻⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⢇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠘⢀⣀⠘⢧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢰⠀⠀⠠⠈⢣⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠒⠒⠀⠀⠀⠙⢦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⢀⠀⡀⠀⠙⠦⣌⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠒⠤⣬⣉⣛⠛⠛⠻⠿⠟⢋⣀⢀⢤⣴⠶⢛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠐⠂⠤⠀⠀⠀⠈⠉⠉⠉⠁⠀⠉⡍⡉⠁⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠓⠤⢤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡠⢴⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⣿⠈⠉⠉⠉⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠰⠀⠀⠀⠀⠀⠀⢀⠁⠄⠀⠀⠀⠀⠀⠀⣠⠂⠈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡐⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢐⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡟⢠⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡋⠰⣇⠘⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⣧⡘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⠀⠰⠏⢠⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣇⠐⢳⠀⡉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠋⠀⣼⠃⠀⢻⠀⠀⠀⠀⠀⢀⣁⠀⠀⠀⠀⠀⠀⢸⠇⠀⣾⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠐⠀⣽⠂⠀⣸⠀⠀⠀⠀⡴⠋⠙⢷⣄⠀⠀⠀⠀⢘⣀⠁⡼⡇⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢰⡟⠀⠀⡏⠀⠀⢀⡞⢡⠀⠠⠌⡬⣷⠀⠀⠀⠸⡇⠀⠸⣏⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠘⠀⢃⡼⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢱⠀⡄⠹⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠠⠀⣠⠞⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠻⣆⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠚⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠳⢄⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠚⠀⠀⢀⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠈⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣟⣝⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣛⣛⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢖⣚⣛⡺⣿⣹⢋⣼⣿⣿⣷⡝⣏⣿⣗⣛⣓⡲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡗⣩⢴⢿⡟⣽⡏⣿⣯⣼⣷⣽⣿⢹⡯⢿⡿⠧⣍⢪⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⢷⢧⡟⣾⣿⣾⣇⣿⣼⣿⣿⢯⣿⣸⣿⣿⣿⢸⣾⣿⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢸⣿⣿⣿⣷⣿⣿⠹⣿⡟⢣⣞⢻⣿⡇⣿⣿⣿⣿⣷⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢿⣷⣿⠿⢋⣿⣰⣶⠄⣿⣿⠢⣴⡆⣿⡙⠿⡻⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣽⣿⣿⣞⣻⡽⣶⡿⣿⣻⢿⣶⡻⣟⣳⣶⣿⣟⣿⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣼⣿⢊⣿⣛⣯⣿⡿⣶⣿⣼⣶⢿⣿⣭⣛⣿⡛⣿⣧⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⣾⣷⣾⣇⣽⣦⣾⣿⣿⡟⢻⣿⣿⣿⢡⣯⣚⣷⣾⣷⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣼⣿⡛⣽⣯⣿⣿⣮⣷⣟⣻⣾⣧⣿⣿⣭⣯⢙⣻⣧⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣚⣻⣿⣿⢯⣽⣿⣉⣿⣿⣿⣿⣿⣾⣿⣿⣙⣿⣯⡽⢿⣿⣟⣓⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣕⢷⠿⣽⣿⣾⣿⣿⣯⣟⣾⣿⣿⣿⣿⣧⣻⣽⣿⣿⣷⣿⣯⠿⣿⣩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⣿⣿⡊⣿⣽⣟⣿⡿⣾⣿⠿⣿⣿⠿⣿⣷⢿⣿⣿⣯⣿⢁⣿⣿⡑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡞⣿⡿⢿⣞⣿⢻⣿⣿⣶⣿⣿⣶⣿⣿⣿⣿⢷⣿⢿⣿⢳⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣷⢻⢸⣆⢽⡛⣶⣹⣦⢟⢿⡟⡟⡝⣍⣦⣿⠿⢷⡷⠿⣿⣾⠀⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠈⣼⡾⣼⣷⠷⠯⣽⣿⡿⠾⢿⡷⠷⢿⣿⣯⣿⠶⢶⢶⣶⠿⠻⠀⠗⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢳⢻⣿⣶⣂⣿⠿⡋⠻⣧⣷⠟⠑⢿⣿⣿⣿⣿⡟⡿⡿⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠯⣷⡻⣿⣿⣴⣿⣿⣷⣿⣿⣾⣷⣿⣬⣿⣿⢟⣼⠝⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠱⡕⢿⣿⣿⣿⡿⣯⣿⣿⡿⣿⣽⣿⣿⡯⢻⠎⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢝⠻⢿⣿⣏⣿⣟⣱⣿⡿⢟⡫⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠠⠁⠀⠀⠈⠓⡶⠭⠽⠯⠭⢶⠚⠁⠀⠀⠈⠄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⠀⡾⡀⠀⠀⢠⢷⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⣌⢲⠇⠙⠶⠾⠋⠘⡖⣡⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡤⠟⢸⠀⠀⢀⡀⠀⠀⡇⠹⢤⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣰⢿⠀⠀⠠⠄⢼⠀⡀⠂⠀⢀⠀⡧⠠⠄⠀⠀⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⢶⠏⢸⡇⠠⠄⠠⡀⠐⠇⢀⡀⠸⠊⠀⠄⠠⠆⢰⡇⠹⡶⣶⣿⡀⠤
⠀⠀⠀⠀⠀⠀⠀⢔⠒⠶⠾⠃⢿⣻⣠⣾⡇⠦⠰⢄⠂⠈⠀⢸⡇⠀⠁⠐⠠⠆⠰⢸⣧⣄⢿⡻⠼⠷⠶⠒⡂
⠀⠀⠀⠀⠀⢠⠤⠈⡴⢆⣠⣀⣺⣷⣿⣿⠧⣀⠓⠌⡂⠕⣡⣸⣇⡬⠈⢐⡡⠚⣀⠼⣿⣿⣾⣗⣂⣄⡰⢦⠁⠤⡄
⠀⠀⠀⠀⣄⠘⣆⠉⢛⣾⣿⣿⣿⣿⣿⣿⠛⢩⣁⠶⠒⢛⣿⣿⣿⣿⡟⠒⠶⣈⡍⠛⣿⣿⣿⣿⣾⣿⣷⡟⠈⣰⠃
⠀⠀⠀⠀⣠⡴⣿⣅⢠⣺⣿⣿⣿⠿⠾⢿⣷⡋⠀⠀⢠⣿⡟⠁⠈⠻⣿⡄⠀⠀⢙⣾⡿⠷⠿⣿⣿⣿⣟⡄⣠⡿⢦⣄
⠀⠀⠀⠀⠙⣿⣻⠵⣻⣿⣿⣿⣿⡄⠀⠈⠙⠿⣶⣤⣈⣿⣕⣤⣤⣮⣿⣁⣤⣾⠿⠋⠁⠀⢠⣿⣿⣿⣿⣿⠯⢟⣿⠋
⠀⠀⠀⢀⡴⠛⠁⣀⣿⣿⣿⣟⣿⣿⡀⠀⠀⠀⠙⢌⠻⠿⣿⢿⠿⣿⠿⠟⡹⠋⠀⠀⠀⠀⣿⡿⣻⣿⣿⣿⣂⠈⠛⠦⡀
⠀⠀⠀⠑⠂⣱⠖⣿⣿⣿⣿⣿⣿⡿⡇⢀⣂⡄⡄⡀⢹⠒⢬⣀⣀⡥⠒⡋⢀⢡⢠⠐⡄⢸⢿⣿⣿⣿⣿⣿⣿⡲⣎⠰⠊
⠀⠀⠀⢠⣒⠘⠋⠚⢳⣿⣿⣿⡾⣃⡙⠒⢷⡟⠷⣧⡎⡦⣹⡋⢙⣯⢴⣱⣼⠾⢻⡾⠂⢋⣘⢷⣿⣿⣿⡟⠛⠉⠃⢒⡄
⠀⠀⠀⠀⠉⠙⣾⣤⣬⣟⢿⣿⣿⣩⣿⣴⣶⣿⢛⠶⠿⣿⢅⣿⣿⡈⣿⠿⠷⡟⣿⣶⣤⣿⣯⣿⣿⣿⣻⣥⣤⣶⠛⠉
⠀⠀⠀⠀⠀⠠⣿⣿⡏⠀⠿⠟⣋⣥⣴⡶⣶⣏⣁⣀⡜⡙⠻⢹⡏⠟⣛⢃⣀⣈⣻⣷⢶⣤⣬⣙⠻⠿⠁⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⡀⠉⣵⢀⡴⠟⠉⠁⢉⠉⠉⠉⡉⢀⠁⣧⣤⠎⠹⣤⣽⠀⠀⠉⠉⠉⠁⡉⠈⠉⠻⣦⡰⣮⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣱⠿⠣⣄⣀⣤⣶⡶⠶⠶⠓⠛⠠⠋⣯⣇⣸⣽⠙⠄⠻⠚⠶⠶⢶⣾⣦⣀⣠⠜⠿⣎⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡏⡐⠛⠻⠓⠋⠁⠀⠀⠀⢠⡄⠀⠀⣻⣿⣿⣟⠀⠀⢠⠄⠀⠀⠀⠈⠉⠚⠛⠛⢀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠿⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⡆⣿⠞⠛⡁⣮⣵⢨⠛⠳⢿⢰⠀⠀⠀⠀⠀⠀⠀⣁⠴⠚⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠦⣤⣤⣄⠂⣰⠁⣺⢰⡱⢹⡏⢎⡆⣷⠈⣄⠐⢤⣤⣤⠴⠂⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠘⠲⠿⠄⢿⠀⢣⡿⢷⡜⠀⣿⠠⠾⠖⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠈⡁

⠀⠀⠀⠀⠀⠀⡠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀
⠀⠀⠀⠀⠀⢰⣷⠀⠀⠈⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⢰⠁⠀⠉⡇⠀⠀⣤⣀⢀
⠀⠀⠀⠀⠀⣸⠿⢷⣦⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣾⠿⢿⡄⠀⠀⢀⡔⠀⠚⢦⡀⠀⡇⣠⡼⠋⠀⠀⢳
⠀⠀⠀⠀⢰⠉⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠈⢱⠀⠀⠘⣇⡀⣀⢀⣹⣾⣷⣧⣤⡤⠤⠤⢾
⠀⠀⠀⠀⣏⣴⡦⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⣶⣌⡇⠀⠀⠋⠉⠉⢈⡵⢛⢿⡟⠑⢦⣀
⠀⠀⠀⢸⢱⣟⠃⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡗⠛⣧⢻⠀⠀⠀⠀⠴⠋⣴⠋⠈⡇⠀⠀⢏⠁
⠀⠀⠀⢸⢸⣷⠲⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡷⢲⣿⢸⠀⠀⠀⠀⠀⠘⠁⠀⢀⣇⣀⣠⠜
⠀⠀⠀⢸⠸⡏⠃⢐⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠰⠁⠩⡿⢸
⠀⠀⠀⢸⡀⢣⢺⠙⣄⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⣼⣹⢢⠃⡸⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⠌⠉⢓⣲⣶
⠀⠀⠀⠀⢧⠈⢣⡁⠀⣣⡀⠙⠢⠤⣀⣀⣀⠠⠴⠚⠁⣠⣃⠀⣱⠃⢰⠇⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⢠⡼⠁
⠀⠀⠀⠀⠈⠻⣄⠱⣜⠦⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠡⣞⡾⢁⡼⠃⠀⢀⡴⡯⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⠋
⠀⠀⠀⠀⠀⠀⠙⠦⣌⢉⣁⡀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣉⢁⡴⠟⠀⠀⠀⣼⠋⠀⣠⠀⠀⠀⢀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣀⣀⠄⣀⡀⢄⣀⣴⣻⠟⠉⠀⠀⠀⠀⠀⠣⠤⢮⣥⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡷⠿⠿⢦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⣼⠃⠘⣧⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣲⡿⢶⢮⣽⡏⣀⠸⢋⡘⠆⣀⢻⣯⡵⠶⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠲⣤⠀⠹⡇⠉⢆⡄⢠⡸⢉⢸⡗⠀⣥⠖⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⢴⠞⢋⡇⠳⢬⢰⡆⡵⠚⠀⡝⠳⡦⢈⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠻⣇⣠⣼⡇⢀⣾⣿⣿⣷⡐⠁⢧⣄⣹⠏⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣜⠋⠁⢰⡅⣿⣿⣿⣿⣿⣿⡆⡆⠈⠙⣣⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⠹⢀⣴⠾⡑⢿⣿⣿⣿⣿⡿⡇⠷⣤⣀⠋⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠷⣤⡋⠀⣰⠸⠛⣿⠟⠻⣿⠛⡇⣆⠀⢙⣤⢼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡈⠱⡼⠋⣼⣞⢹⡶⣶⡏⣳⡇⠙⢦⠏⢀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠛⢦⡆⠘⣽⣼⡟⢠⡅⢹⣧⣏⠃⢰⡴⠛⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⢿⣯⢿⣿⣦⣀⣶⢿⣟⠞⠀⠼⠇⠀⠳⢣⠳⣶⣀⣴⣿⡿⣽⣿⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢻⡎⣿⣽⣝⠂⣾⡎⢠⣄⡞⢳⣠⡄⢋⣧⠘⣿⣿⣿⣱⡟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣿⡽⣿⣿⡶⢿⣗⠾⣯⠆⢸⣽⠷⣻⡿⢾⣿⣿⢧⡿⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⣷⣿⡟⣿⡄⠉⣺⡏⡰⢆⢹⣗⠉⢠⣿⣻⣿⣾⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⢸⣇⠙⣿⡻⠞⣷⡄⢿⢸⠀⠈⡀⣿⢠⣿⠳⢟⣿⢋⣸⡅⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠾⠛⢻⡿⠛⢦⣤⡤⠠⡸⢯⠉⢾⣿⠷⠌⠻⣾⣿⠀⠀⣿⣷⠟⡡⠼⣿⡗⢉⣭⢃⠄⢆⣠⡴⠟⢿⡛⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡍⠉⠀⠦⠅⠀⠙⠷⣦⣵⣬⣙⢇⡸⣋⣤⣮⢴⠾⠋⠀⠨⠴⠀⠉⢩⣀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣷⣶⣶⣶⣿⡿⢿⣿⣶⣶⣶⣾⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⣿⣿⢿⣿⣿⡟⣽⣿⠿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢏⡿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢛⣉⣭⣤⢤⣶⣶⣤⣤⣭⣉⡛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣡⡴⣾⣟⣿⣧⡟⠀⢳⣿⣯⡿⣿⣹⢿⡷⣦⣌⠛⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣟⣿⡻⡏⡶⣻⣽⠇⠀⠈⠷⢿⣝⢆⣷⣿⣿⣯⣿⣿⣦⡙⢦⡀
⠀⠀⠀⠀⠀⢠⠟⣠⡿⣗⣿⣿⡿⣿⠞⣋⣥⠀⢠⡄⠀⢶⣦⣬⣁⠙⣿⢟⣋⢻⠿⢿⣄⠻⡄
⠀⠀⠀⠀⢰⠏⣰⣿⡿⣋⡟⣿⢊⣴⣾⣿⡇⠀⣼⣷⡀⠈⣿⣿⣿⣷⣦⡑⣦⡿⣿⣟⣿⣆⠹⡄
⠀⠀⠀⢀⡟⢰⣿⣷⡿⣷⡾⢡⣾⣿⣿⣿⠃⠀⣿⣿⣷⡀⠘⠿⠛⠛⠛⠛⠈⠉⠉⠉⣩⠾⡆⢻⡀
⠀⠀⠀⢸⠃⣿⡒⠛⠛⠛⠁⠉⠉⠉⠉⣀⣀⣀⣀⣩⣽⣷⡀⢠⣴⣶⣶⠖⠀⠀⣠⣞⣿⠿⣿⠘⡇
⠀⠀⠀⣿⢠⣿⣽⣷⢤⡀⠀⠛⠿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣧⢸⣿⠟⠁⢀⣴⡆⢹⢏⡍⣿⣿⡄⣿
⠀⠀⠀⣿⠘⣿⣭⢿⠻⣇⢸⣦⣄⡀⠉⠁⢰⣿⣿⣿⣿⣿⡿⠟⠁⠀⣴⣿⣿⡇⣸⣾⣾⣿⣾⠃⣿
⠀⠀⠀⢸⡄⣿⣻⣥⢭⢿⡘⣿⣿⣿⡗⠀⣸⣮⡙⠻⣿⣋⣠⣤⣄⠀⠹⣿⣿⢁⣇⣿⣿⡿⣿⢠⡇
⠀⠀⠀⠈⣧⠸⡟⣾⣿⣷⢷⡘⢿⣿⠇⠀⣿⣿⡷⠀⠀⠉⠛⠿⣿⣆⠀⠹⢃⣾⢩⠳⣿⣾⠇⣼⠁
⠀⠀⠀⠀⠸⣆⠹⣽⣛⡿⢫⢻⣌⠛⠀⢸⡿⠋⠀⣠⣾⣷⣦⣄⡈⠙⠀⠀⢺⣻⣾⡾⣷⠏⣰⠇
⠀⠀⠀⠀⠀⠘⣦⠙⣧⣿⣼⣾⣻⡇⠀⠈⠀⠠⠾⠿⠿⠿⠟⢛⣡⣴⣤⣀⠀⢻⣻⣷⠋⣴⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣷⡿⣿⠃⠀⣠⣚⡿⢗⢲⣖⣿⣟⢏⠭⡹⡶⣿⡗⣦⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡙⠛⢴⣏⣿⢟⣯⣍⣾⢿⣿⣿⢿⡟⣾⡷⠟⢋⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣬⣉⣛⠛⠳⠿⠷⠛⠓⣛⣉⣥⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⣀⠀⠈⢉⠁⡐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⣤⡴⣫⠾⢋⠀⣤⢂⡦⣄⠀⣦⡙⢆⡀
⡀⠀⠉⠈⠃⠀⠁⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⢀⠔⣡⡀⠚⢉⠠⠤⠘⠇⢀⡈⢽⢆⡉⢉⣭⣭⠭⠄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⡇
⡇⠀⠒⠒⠒⠒⠂⢈⡂⠄⣀⣀⣀⡀⠀⠀⣀⣀⡈⢸⣏⡀⠀⠀⢀⡀⣀⣬⣴⣼⣶⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠓⠀⢐⣚⣁⡩⠀⠴
⠁⠙⠒⠖⠒⠛⠛⠻⣷⢆⡀⢀⠀⠈⢉⠉⡉⠉⡁⣠⣿⣿⣧⣴⣶⣯⣿⣿⣿⣿⣿⣤⣶⠲⠶⠖⠒⠒⠒⠁⠰⡌⠣⡱⠀⣩⣥⣶⣤⡀⣼
⠰⠠⠤⠤⠾⠛⠒⠀⢿⠈⡇⢸⠀⣾⢸⠀⡇⢸⡇⣷⢻⡿⠿⠿⢿⣿⣿⡿⢿⣿⣧⢻⣿⢘⣴⡄⠈⢰⢰⣆⢡⠹⣜⠋⣀⠀⠀⠀⣒⢦⡶
⠀⢠⣄⣀⣠⠴⠒⠃⡀⠴⠇⣿⣀⣯⢸⣠⡇⢸⡇⡟⣾⡇⠀⢀⣤⣶⡖⠁⠉⣹⣿⡏⣿⣾⣧⣷⠘⣘⡎⣿⣼⠟⡁⠀⠈⠙⠒⢤⣐⡽
⠀⠀⢀⠀⢀⣠⠴⠋⢀⡴⠈⠁⠈⠻⠾⠿⢧⠽⢷⡏⣿⣿⣶⣯⣿⣿⣥⣰⣼⣿⣿⡇⣿⣿⣿⣿⠿⠿⠛⠋⠘⠀⠈⠳⡄⠀⠀⣐⡼⠁
⠀⠀⠀⠹⢍⡀⣀⠴⠋⡠⠊⢀⠜⠀⠆⠀⠀⠀⢠⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠉⢡⠀⠀⠂⠀⠀⠁⡀⠁⠀⠌⣲⡴⠚
⠀⠀⠀⠀⠀⠉⠓⠢⢾⣅⣠⣋⢀⡌⠀⠀⣌⡞⠻⢿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣇⢈⡇⠀⡢⢠⣤⣤⣵⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠓⠲⢜⡳⡹⢿⣿⣿⣿⣿⣿⣿⣿⣃⣥⠾⠶⠿⠓⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠯⣽⣛⣛⣭⠵⠞⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢖⠟⠛⢛⣿⣽⡽⣛⣛⠻⠓⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠻⣫⡷⠾⢚⠻⢱⣼⢟⡻⢝⡻⣯⡦⠛⡛⠿⢮⣍⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣛⣭⣶⢰⣝⢰⠀⠀⢑⡾⡑⡥⠀⠀⢼⢎⢷⡊⠀⠀⡆⢩⡄⢶⣮⡻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢫⣵⢻⠀⠒⢢⣿⢟⣻⣛⡻⠱⢸⣿⠀⠀⣻⡆⠟⢟⣛⣟⡳⣿⡅⠐⠈⡙⣧⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡟⣠⡀⣥⡘⡀⣠⣼⣧⡟⡀⢈⠹⡇⣏⠙⢞⡳⠻⣹⢸⠏⡉⢀⢻⣼⣧⣀⣀⠇⣨⡅⣴⢿⣆
⠀⠀⠀⠀⠀⠀⣰⢋⡾⠉⠅⢬⣻⢟⣿⣽⢸⡇⣷⡾⠟⡇⣿⠎⠢⠐⠹⡿⢸⠹⣷⣾⢸⡇⣯⣿⡿⣿⡭⠤⠉⢷⣜⣧
⠀⠀⠀⠀⠀⣼⢏⣾⠁⡀⠀⣰⣿⠻⠾⣿⢸⣇⡿⣷⣄⣿⡏⢂⢹⡥⣰⣸⣾⣠⣾⢿⣨⡇⣿⠷⢟⣿⣂⠀⢠⢈⣷⡹⣧
⠀⠀⠀⠀⢰⠏⣠⡈⠷⣾⣷⣿⣽⡙⠛⢿⢸⡀⠁⠀⠀⣿⡗⣿⠟⠻⣦⢸⣿⠀⠁⠈⢀⡇⣿⢰⢛⣭⣿⢞⣥⡶⢖⣴⠻⡄
⠀⠀⠀⠀⣾⢥⡏⠐⠊⠒⣷⠯⠿⠿⠤⣿⢸⣇⠂⠀⠀⣿⣿⠇⠀⠀⠘⣾⣿⠀⠀⢐⣹⡇⣿⠥⠿⠿⡿⣾⠟⠒⠂⢹⡖⣯
⠀⠀⠀⠀⣟⢸⡇⢤⣀⣰⠿⡗⠖⠉⠙⣿⢸⡇⠄⣴⠆⣿⣿⠀⠀⠀⠀⣿⣿⠢⣶⠠⢸⡇⣿⣅⠀⠶⢛⣿⣆⣀⡠⢨⡇⣿
⠀⠀⠀⢸⡏⢋⡑⠶⠾⣿⣽⣿⣷⠆⡇⣿⢸⡟⠰⢈⢠⣿⣍⣴⢛⡛⣦⣙⣿⡄⠁⠆⢻⡇⣿⣹⠠⣾⣯⣿⣯⡷⢶⣍⠁⣺⡂
⠀⠀⠀⠀⣿⢹⠋⠔⠉⢹⣯⣭⣴⣷⠷⣿⣼⡃⢧⠈⠃⣿⡿⣡⣿⣿⣌⢿⣿⠘⠁⡼⢚⣧⣿⠦⢤⣤⣬⣭⡏⠉⠢⠘⡷⣻
⠀⠀⠀⠀⣿⣸⣇⠀⢀⣠⣿⣶⣾⣿⣔⣦⡌⠳⣄⣦⣠⡟⠐⢿⠿⠿⡟⠂⢻⣤⣄⣠⠞⢩⡴⣢⣟⣷⢶⣿⣄⡀⠄⣸⠇⡿
⠀⠀⠀⠀⠸⣎⡛⠱⡶⢟⣳⣿⣛⣡⢿⡜⣿⣆⠹⣇⣾⢀⡿⣶⣶⣶⢶⢿⡈⣷⣹⠏⣰⣿⢣⣿⣌⣛⣾⣾⡶⢶⣜⠟⣺⠇
⠀⠀⠀⠀⠀⢻⣎⢿⡀⠋⠀⠹⣿⣡⣾⣧⠹⣿⡆⢹⡇⣘⣘⡑⠓⠋⢊⣃⣃⢸⡏⣰⣿⡇⣼⣷⣮⣿⠍⠀⠑⠀⣾⢰⡏
⠀⠀⠀⠀⠀⠈⢿⡪⢷⡀⠂⢒⣿⢿⣶⣻⡆⢙⡻⠄⠀⠈⠻⣍⠉⠉⣩⠟⠁⠀⠠⢟⡋⢰⣿⣿⣯⣟⡒⠐⢀⡾⣣⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣬⠟⢑⣛⢡⠋⠙⢿⣷⠘⢿⢳⢶⢦⣤⣤⣙⣋⣤⢤⡤⡶⡞⡽⠃⣾⡟⠋⠙⡍⠻⠜⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣌⠻⣌⠀⠀⠔⣿⣧⠙⣾⣾⣼⣽⣇⣟⣹⣼⣫⡧⣧⣾⢋⣼⣏⠢⠄⠀⣧⣿⢫⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠾⣝⠳⠶⠘⣋⠸⠳⣌⡙⢿⣿⣿⣿⣿⣿⣿⡷⢋⣠⠞⡇⡛⠇⠶⠾⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣎⣛⢧⣤⣬⣽⢦⣌⣙⣛⠛⣋⣡⢶⣯⣡⣴⠾⢛⣠⡵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢧⣭⣦⣝⣛⣯⣛⣿⣣⣨⣽⠴⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⡶⣶⢶⢷⢿⠿⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⡶⢶⡾⣶⣶⣶⠶⢶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠠⠷⠤⠰⡆⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡃⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢂⠄
⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⠠⠀⠀⠀⠀⠀⡀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢶⣾⡆⠀⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁
⠀⠀⠀⠀⠀⠀⠉⠑⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣾⣦⠁⠀⠁⠂⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⡿⢂⣼⣿⡇⠀⠀⠀⠀⡀⠂⡀⠀⢀⠀⠀⠀⠀⢀⡀⢴⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⢹⣿⣿⡧⠀⡀⢠⡀⠀⠀⠀⢀⡀⢀⠀⠶⡿⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠉⠉⠉⠙⠋⠁⠀⠀⢻⠉⠈⠉⠉⡛⠃⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡏⠉⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠛⢾⡎⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠰⣾⡧⣷⣄⣀⡀
⠀⠀⠀⠀⣾⢲⣦⢀⣬⣤⣾⣄⢽⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢩⣷⡶⠿⠷⠶⣄
⠀⠀⠀⠀⠻⣿⠿⣉⣁⣆⣸⣿⠂⢥⣄⠛⠷⢶⣀⣄⣤⣄⠀⠀⠀⡀⠀⠀⠀⠀⢈⠀⠀⠸⢻⣴⠛⡻⠛⠛⠳
⠀⠀⠀⠀⠀⠰⠶⠿⠟⠉⠉⠁⢀⣄⣬⣄⣠⣘⣾⢉⠁⢠⠶⠤⠀⡄⠀⠀⠀⠀⠰⠀⠀⠀⠠⠻⠂⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⠁⠈⠙⢹⣿⣮⡤⠰⣤⣄⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠹⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢀⡰⢛⠀⠀⠀⢈⠷⣰⣶⣤⢀⠄⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⠲⢤⣀⡿⣁⠔⢁⣄⠈⣶⠃⠀⣹⠃⡀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠻⣶⣽⡾⢋⡴⠋⢹⣷⣿⣤⣤⡇⢤⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣤⣤⢻⣇⡜⢀⠒⣦⡹⣿⣿⣾⣿⡈⠷⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠀⣶⣿⣦⠟⠀⠀⢷⣿⣿⣿⣿⡿⣤⠐⠀⠉⠁⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢸⣿⠃⠀⣤⠀⡘⠛⢿⣿⢟⣵⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⣠⡾⣿⣆⠇⠸⣏⣴⣽⣿⡯⠀⠀⠀⣀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣿⡿⠀⠀⠏⣵⣮⢻⡇⠀⠀⠀⣿⠸⣇⣠⣄⣸⣧⢤⠀⣤⢠⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠘⠃⠀⠀⠀⠸⣿⠀⢈⠁⠀⠀⠀⠟⠠⠟⠹⠄⠈⠟⠸⠟⠳⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡼⢡⠀⠀⠀⠀⡌⢧⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠈⡝⢦⡇⠘⠷⣦⣴⠾⠁⢸⡴⢫⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡿⠘⡇⠀⠀⠀⠀⠀⠀⢸⠇⢿⡀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠛⠋⠁⠀⡇⠀⠀⡰⢂⠄⠀⢸⠀⠈⠙⠛⠉⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠛⠒⠰⡇⢐⡐⠀⠀⢂⡀⠸⠖⠒⠛⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠀⠲⡀⠈⠳⠀⢀⡀⠀⠞⠁⢀⡔⠀⠶⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢤⡀⢤⡀⠠⠀⢃⠀⢸⡇⠀⡘⠀⠜⢀⡤⠀⡤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠰⣄⠑⢄⠡⡸⠀⣿⣯⠀⢃⠌⡠⠊⣠⠎⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠴⠦⣄⠙⣂⣭⣬⣷⠿⢿⣾⣥⣭⣐⠋⣠⠴⠮⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢰⡶⠖⠋⠉⢡⣾⢟⡿⣿⡻⣷⡌⠉⠙⠲⣦⡄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢾⣿⡋⠀⠀⠀⠀⣿⢱⠃⠀⠀⠘⡎⣿⠀⠀⠀⠀⢙⣿⡷⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠐⡀⢸⠙⠻⡷⣦⣄⠀⢿⣯⡣⣄⣠⢜⣽⡿⢀⣠⣴⢿⠿⠋⡇⢀⠂⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⡅⠰⠀⠀⠈⢹⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⡏⠁⠀⠀⠆⢨⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠀⡀⠀⠀⠰⠀⠰⠅⠁⠝⠉⠉⠫⠉⠈⠄⠀⠆⠀⠀⢐⠀⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠒⣅⠤⠘⠁⠀⠐⡇⠀⠂⠀⠀⠐⠀⢘⠀⠀⠈⠃⣤⣨⠒⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠨⡄⠀⠀⠀⠀⠇⢠⠀⠀⠀⠈⠁⠸⠁⠀⠀⠀⢠⡏⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⠀⠘⣄⠀⠀⠀⢒⠆⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠋⠀⠐⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⢃⠀⠀⠉⠐⠀⠉⡄⠀⠀⠀⠀⢰⠋⠑⠛⠉⠀⢀⡘⠋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠞⡠⣀⠀⠀⠀⠈⠢⢀⡀⠔⠁⠀⠀⠀⡀⠄⠓⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢀⠀⠐⠉⠖⣀⠀⠀⠀⢈⡁⠀⠀⠀⣀⠲⠉⠀⠀⡀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⠴⡀⡤⠀⡄⠊⡠⠀⡎⠱⡀⢤⠑⣠⠀⢤⢈⠦⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⠉⡚⠢⢿⣲⣍⠀⠀⣩⣖⡽⠔⢛⡉⠁⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠉⣀⣈⣿⣞⣷⣿⣑⣀⠉⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⠭⠿⠿⠭⠴⠶⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠲⢄⡀⠀⣀⣀⣀⠤⠤⠤⠤⠤⠄⣀⡀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠐⠂⠀⠠⠂⠀⠀⠈⠉⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⢠⠤⠶⠤⣄⠀⠀⠀⠀⠀⠀⢀⣠⠤⠦⢤⡄⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⢸⡀⠰⡷⢈⡄⠀⠀⠀⠀⠀⢌⠀⣿⠀⢠⠃⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠠⠤⠀⠀⠀⠀⠉⠒⠒⠉⠀⣤⣤⣤⣤⠀⠈⠙⠒⠚⠁⠀⠀⠀⠤⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠑⠤⣄⣀⠤⠚⠁⠀⠙⠦⣀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠈⠑⠦⣄⡀⠀⠀⠈⠒⢤⣀⣀⡀⠀⣀⣠⠤⠊⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢣
⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠰⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢃
⢀⠜⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠃⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⢆
⡌⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠊⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘
⡇⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠄⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⡀⠠⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆⠀⠀⡀⠀⠀⠠⠂⠀⠀⠀⠈⠙⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠊⠉⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠄⠀⠀⠀⠀⡸⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⢄⠀⣀⠠⠚⠁⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠘⢦⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⠄⠀⠤⠄⡞⠁⠀⠐⠒⠂⠀⢄⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠠⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⣀⠀⠀⡄⠀⠀⢀⣸⡆⠀⠀⠀⢾⣤⢀⡀⠀⡀⠈⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠐⢼⡧⠠⢿⠂⠘⠀⠀⠀⠀⠈⠀⢹⠃⢸⠋⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠠⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡠⠀⠀⠀⠀⢀⠀⡺⠂⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡠⠚⠉⠀⠀⠀⠠⢣⡔⢠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠌⠐⡏⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⢠⣿⣿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⢀⠀⡄⠀⠀⡀⠀⠠⢠⠀⠀⠀⠀⣿⣿⡄⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣾⣿⡏⠀⢀⣄⠀⠀⠀⠀⠀⠢⠀⠀⢹⠢⢀⡆⠸⢀⠁⠀⠀⠙⠀⠀⠀⠀⣠⡀⠀⢹⣿⣷⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⣿⣿⠁⠀⣾⣿⠂⠀⠀⠀⠒⠀⠀⠀⠀⣤⣯⣄⠋⠉⠀⠀⠉⠀⠀⠀⠀⠐⣿⣷⠀⠘⣿⣿⡄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⣿⣿⠀⠀⠈⠛⢠⡄⠀⣀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠛⠁⠀⠀⣿⣽⣇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣹⣿⠰⣆⣀⣼⣿⠏⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⣿⣧⣀⣰⠆⣿⣯⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢻⣧⡿⣤⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠃⠀⠁⠀⠀⠀⢠⣿⣿⣿⣿⣤⢟⣾⡟⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣜⣯⡻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⣿⣿⣿⢟⣽⣣⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣭⣿⡙⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢋⣿⣭⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠡⠶⢈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡁⠶⠈⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⡞⠍⠀⠉⠁⠀⠈⠙⠉⠛⠉⠋⠛⠋⠀⠉⠉⠁⠁⠈⠉⠉⠉⠘⠛⠋⠉⠙⠁⠉⠉⠛⠁⠈⠃⠘⠋⠛⠛⠚⠛⠉⠁⠚⠉⠉⠁⠚⠉⠙⠒⠊⠉⠀⠁⠆
⠀⠀⢁⠀⠀⣦⠄⠠⠼⠷⠤⠤⠶⠷⠘⠲⠶⠶⠤⠶⠞⠃⠲⠶⠖⠶⠶⠛⠤⠀⠠⠾⠲⠶⠾⠓⠒⠂⠐⠀⠒⠛⠚⠓⠒⠶⠿⠶⠖⠚⠓⠒⠒⠚⢷⡎⠀⡦
⠀⠀⠈⠓⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠘⣇
⠀⠀⣱⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠃⢀⡁
⠀⠀⡇⠁⠀⣀⠃⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⡀⠀⠀⠀⢡⠆⠀⠀⠀⠛⠀⢠⣤⠀⠘⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢷⠀⠈⠃
⠀⠀⡰⠀⠀⢹⡀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⠀⠀⢠⡌⣁⡠⠤⢤⣉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠚⠀⢠⡇
⠀⠀⠃⠀⠠⣴⠁⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢠⠁⠉⠀⠐⢐⢉⡀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣱⠀⠀⡗
⠀⠀⢰⡄⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠉⠀⠆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⡇
⠀⠀⢸⠃⠐⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠞⡀⠀⠠⠀⠀⠀⠘⠒⠆⢀⣀⣀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡆
⠀⠀⣸⠃⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡁⠀⠀⠐⠄⠁⠀⠀⡀⠜⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣿⠇⠀⡀
⠀⠀⢸⠀⠀⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠐⠃
⠀⠀⣼⠓⠤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠘⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⢀⡷
⠀⠀⢸⣀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⣇⠀⠀⠂⠀⠀⠀⠀⢴⣦⠀⠀⠀⠀⠈⠀⣙⠢⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠀⠸
⠀⠀⢻⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠈⠀⠀⠄⢀⠄⠀⠸⢻⡄⠀⢠⠀⠀⠎⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠢
⠀⠀⡼⠄⣀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⠃⠀⠀⠀⠃⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⡀
⠀⠀⢻⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠃
⠀⠀⢸⠅⠠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⡀
⠀⠀⢘⣄⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠑⢈⡀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⠀⠰⠇
⠀⠀⢸⠏⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⡇
⠀⠀⢰⠷⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⡁
⠀⠀⢸⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠻⡇⠀⠀⠀⠀⠄⠀⠀⢀⣞⣧⠠⠘⠛⠛⠛⠋⠁⠀⠀⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠠⡇
⠀⠀⢸⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⣠⠞⠀⣼⠃⠀⠀⠀⠀⠀⠀⢐⣯⠅⢠⡖⠂⠀⠒⠈⠀⠀⠀⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⡏
⠀⠀⢸⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠟⣀⡀⠀⢈⠡⠤⠾⠋⠀⠀⠀⠀⡄⠀⠐⠀⢀⣴⢄⡤⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⡅
⠀⠀⢸⠀⠉⢻⡄⠀⠀⠀⠀⢴⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⢀⠂⠰⠦⠚⠀⢰⣾⣁⣨⠵⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠀⢀⡄
⠀⠀⢰⠣⢀⣸⡇⠀⠀⠀⠀⠰⠆⠀⠊⡠⠀⡀⠀⠀⠀⠀⠄⠈⠀⣴⡀⠰⣦⣄⠰⡟⠁⢀⣱⠀⠀⠑⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠈⡷
⠀⠀⢸⡀⠀⢉⡆⠀⠀⠀⠀⠀⠀⠀⠉⠠⣀⠀⠀⠠⡀⢾⠀⠀⠘⠉⢠⠐⠀⠉⠻⠿⢿⠁⠀⠗⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠇
⠀⠀⢸⠓⠠⣬⠇⠀⠀⠀⠀⠀⢧⡀⠀⠀⠈⠉⠉⠊⠁⠀⠀⠴⡆⠀⠀⠀⣀⣄⠀⠀⢘⡆⠀⠀⠀⣄⠀⠀⠰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⡄
⠀⠀⢸⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠂⠌⠀⠀⢁⡀⠀⣁⣢⠤⠊⢿⡧⠤⣦⣄⠉⠣⠄⠀⢸⣇⠀⠀⠁⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠐⡇
⠀⠀⢸⠀⠀⢾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡼⠉⠀⠀⠀⠀⢀⠀⠈⡿⢷⣦⡀⢀⠀⠘⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⡟
⠀⠀⢸⡧⢀⣈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢹⡅⠀⠈⠀⠀⠀⠀⠀⠀⠠⣤⠹⢿⣿⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠄⠈⡅
⠀⠀⢸⡇⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⢹⡆⠈⠀⠀⠄⠀⠀⠀⠀⠀⠛⡻⠏⠍⠀⠀⠀⠀⠀⠀⠀⠢⠀⠄⠀⠀⠀⠀⠀⠀⢠⡏⠀⠰⡇
⠀⠀⢸⠀⠐⢟⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⢘⡁⣤⠀⠀⠀⠈⢡⡀⠀⠀⠀⠀⠀⠀⢀⣆⣘⠃⣠⠁⠀⠀⡀⢠⠀⡀⠰⠒⠈⠀⠀⠀⠀⠀⠀⠀⢈⣿⠁⢀⡇
⠀⠀⣾⠦⠀⢻⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠷⡄⡠⠬⠛⢀⠀⠀⠀⠀⢠⠚⣹⢯⠜⠡⠄⠀⡼⠉⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠈⠁
⠀⠀⣹⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠗⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠⡏⠀⠀⡆
⠀⠀⡟⠑⢠⣼⣷⣤⣤⣤⣤⡤⢀⣠⣤⣶⡤⣤⠲⣤⣤⣤⣤⣄⢀⣄⠀⠀⣠⣤⣤⡄⣠⣤⣄⣀⡠⠄⠀⢀⣄⣀⠀⣠⢠⠀⢄⡤⠒⠤⣀⠀⢤⠼⣧⡀⠚⢀
⠀⠀⠃⠀⣼⠟⢯⣁⠀⠈⠀⠀⠀⠉⢀⡁⠀⠀⢀⠀⠀⠉⡁⠀⠈⠁⠀⠀⠁⢀⡀⠀⠱⣄⠀⠈⠁⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠈⢂⡀⠀⠹⣧⣀⠰
⠀⠀⠀⠀⠉⠀⠀⠉⠓⠂⠐⠋⠉⠉⠉⠀⠀⠉⠋⠉⠐⠀⠁⠀⠀⠈⠉⠀⠀⠈⠉⠁⠁⠙⠛⠀⠈⠁⠀⠀⠀⠉⠛⠀⠈⠉⠒⠚⠋⠙⠒⠛⠋⠀⠀⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⢛⣿⣛⣛⣛⣻⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣬⣭⣿⣿⣯⣽⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠠⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠐⠀⠠⡦⢀⢀⡼⠧⠄⠚⢲⣶⠚⠂⠰⢧⠄⠐⡆⠀⠀⠀⠀⠀⢀⡴⠀⠀⠤⡀⠀⣀⣀⠠⠒⠉⣤
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⡉⠙⠀⠘⠃⠀⠀⢀⣤⣤⡘⠁⠀⠀⠀⠔⠊⢹⡼⠗⠲⠤⣄⡀⠓⢂⡑⠚⠉⠀⢄⠀⠀⢀⠀⠉⠲⠛⠉⠀⠀⠛
⠀⠀⠀⠀⠀⠲⢀⠰⠆⠀⣀⣀⠒⢜⠋⠀⡻⢃⣀⠤⠓⠀⠀⠀⠐⡖⠀⠀⠰⠇⢀⡀⢀⡑⠀⡘⠳⠂⢀⡀⠘⠓⠀⡘⠆⠀⠀⠀⠁⣤
⠀⠀⠀⢠⠰⣶⢻⠆⠀⢀⣽⠏⠀⢀⣘⡟⣁⠈⡽⠤⠦⣄⣀⡆⠛⠀⠀⠀⠀⠀⠀⠁⠈⠣⠤⠛⠲⠚⠉⠱⠞⠋⠙⠗⠐⠂⢶⡞⠀⠈⣠
⠀⠀⠀⠀⠀⠈⠉⠀⣄⢁⠙⠓⠦⠤⠉⠙⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠙⣧⣀⣀⠀⠋⠉⠄⠈⠑⠀⢈
⠀⠀⠀⠀⢀⣤⣄⣰⠏⠙⢉⡴⠀⢀⡰⠟⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣤⠐⠉⠀⠈⠁⣠⣿⠁⣀⠉⢡⠀⢦⡀⢤⡀⠈
⠀⠀⠀⠀⠙⠀⠘⢧⠖⢈⣭⠀⠺⠇⠘⠁⡈⠀⠐⠆⠀⡴⢊⡙⢛⢀⡀⠀⠀⠀⠀⢠⡞⠁⠈⠁⠀⠐⠛⠀⣀⠀⠡⠚⠉⠀⡀⠀⢧⡟⠸⠄
⠀⠀⠀⠀⠀⠀⢠⠀⠠⡎⠁⠠⣄⠀⢀⠀⢡⡀⠀⠀⣤⠄⠈⠀⠈⢻⡁⠀⠀⠀⠐⠀⡟⠀⠀⠀⢰⡆⢠⡼⠃⢰⡇⣠⠆⣾⢱⠆⣸⠃⣦⣤⠄⠘⠂
⠀⠀⠀⠀⠀⠀⢀⡀⢸⡇⠠⡄⢹⡄⠘⣇⢸⡇⢠⣾⣿⣰⡆⠘⢷⣼⡇⠀⠀⠀⠀⠀⣷⡆⠘⠀⠘⣿⠈⠀⠀⣾⡇⠩⠀⣿⡄⢸⣿⠀⢹⣅
⠀⠀⠀⠀⢀⣤⡞⠋⠀⣸⡀⠁⢸⡇⠀⠈⢸⡇⠀⠀⣿⠋⣠⠇⢀⣿⢻⡄⠀⢀⠀⠀⣿⣇⠀⠀⠀⢻⡆⠀⠀⣿⠿⠆⢆⡸⠷⢾⣯⡘⠀⠰⠇
⠀⠀⠀⠀⠀⠘⠛⣀⡔⢉⡅⠀⣸⡇⠰⠟⢿⡇⠀⣤⢻⡆⠉⠀⡻⠿⢼⡇⢠⣾⡦⠄⠏⢉⣆⠀⠻⢸⣧⠸⠃⢹⡇⢀⣾⡷⠀⠘⠏⢹⠂⠀⢀⡀⠀⡄
⠀⠀⠀⠀⠀⠰⣄⠁⠸⣯⠉⡀⢘⣷⠄⠀⢸⠀⡁⢸⣿⠀⠀⢸⡟⠀⣬⠃⠀⠙⠁⢰⢠⡜⢻⡆⠀⡛⢻⡆⠿⣻⠀⢠⣿⠷⣶⠛⠀⠹⠆⠀⠈⠁⠀⢀
⠀⠀⠀⠀⡀⠀⢉⣡⠄⢠⡌⢻⠉⢵⣤⠈⠹⣷⡀⠻⣷⠀⠂⣾⣷⡾⠁⠀⠀⠀⠀⠈⢎⠀⢸⣯⠈⢀⣾⢤⣤⠟⢉⣶⣽⡶⠃⠀⠈⣦⣡⡖⠀⠘⠀⡌
⠀⠀⠀⠀⠑⠶⠈⠁⠀⠄⡄⠈⠱⡄⢿⣆⢠⣄⣇⠰⣿⠀⠀⢻⡍⠃⠀⠀⠀⠐⠀⠀⠀⠑⢠⣿⠀⠘⢿⠈⣧⠐⣞⠁⠉⠀⠀⠀⢠⠀⣾⣅⡐⠛⠆⠈
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠘⠠⠀⠘⢄⣙⡀⠛⢹⡄⢹⡄⠘⠘⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠛⢺⡇⠘⢀⣩⠜⠋⠀⠀⠀⠀⠀⡀⢐⣅⢈⠋⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠂⠀⠈⠻⢆⣰⣧⠾⡇⣶⣴⣇⠀⠀⠀⡂⠙⠉⠁⠀⠀⠀⢀⣽⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠰⢡⡛⠉⠉⣰⣶⠀⢹
⠀⠀⠀⠀⠀⠠⡟⠛⠀⠉⠙⠓⠀⠄⡀⠀⠀⠀⣝⣿⣧⣿⣿⣿⡟⠀⠀⠀⠀⠆⠊⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⣇⠀⠀⠀⠀⣜⠃⠈⡑⠀⠀⠀⡸⠗⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⣠⡟⠛⣿⡟⠻⣿⣇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠿⣤⣏⠀⠐⣿⣧⠀⠀⠰⡏⠁⠸⠷⠂⠀⠰⡁⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠱⠂⠀⠀⣀⡀⠀⠀⠠⣴⡾⠃⢀⠈⣷⣦⠹⠇⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢐⡾⠃⠀⠀⠈⣿⣶⠊⠀⠉⣠⠀⠀⠄⠀⠀⠛⠉⠂⠁
⠀⠀⠀⠀⠄⠀⠠⣄⠀⠀⠈⠀⢀⠉⣎⡰⢿⠗⢤⣾⣷⠿⠾⠂⠀⠀⠀⠀⠰⠀⠀⠆⠀⠀⠀⢀⣿⢧⣶⣁⣀⠸⠉⢻⣥⡀⠀⣈⠁⠰⣦⠖⠊⠈⠂
⠀⠀⠀⠐⠀⠉⠀⠈⠗⠀⣴⠋⠻⠑⣈⣻⠟⠀⠀⠁⠈⠃⢴⡿⣶⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣾⡏⠀⠈⠉⠁⡀⠿⠛⢿⣷⠀⠙⠂⣤⣈⠁⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠐⠓⠀⡰⣂⠁⠀⢠⣯⡽⠀⠀⠛⠀⠀⠀⠀⠛⢻⣐⠂⠄⡤⠄⠃⠀⠄⠐⣾⡿⢉⡆⢠⣀⡰⠿⠀⠂⠰⣟⣧⠀⠀⠀⢉⠀⠀⡤⠀⠐
⠀⠀⠀⠀⠀⡀⢀⡴⣎⠁⠉⠀⠀⣾⣏⠀⣄⣠⣠⡔⠦⢌⣠⠾⢿⣽⡄⠀⠀⠀⠀⠀⠀⣸⡽⠙⠛⠀⢀⠛⠁⠆⠳⠀⢲⢉⣽⣄⠀⠀⠈⠳⡄
⠀⠀⠀⠀⢠⠉⠈⠀⠀⠀⠀⠂⣻⡿⠃⠀⢉⡙⠋⠀⠀⣤⣥⠀⠄⢉⣷⣆⠀⠀⡄⠀⢸⣿⠇⠠⡄⠀⢸⣆⠀⠀⠰⠐⠈⠃⠹⢿⣄⠰⡆⠀⠏
⠀⠀⠀⠀⠘⡆⠈⠀⠁⠰⠀⣾⣻⣇⣘⡃⠈⠉⢀⡇⠀⢹⣿⠇⢠⣾⣷⣿⡀⠀⠃⢀⣼⣯⣦⢴⡇⡄⢸⡏⠀⠙⢰⣇⣼⣷⣴⣿⣽⠂⢧⣀⡼
⠀⠀⠀⠀⠈⠳⠀⠄⠀⡀⠸⡿⠿⠿⠛⠟⢂⠀⢾⢧⡞⠛⠉⠋⠹⢿⡙⣿⡟⠀⠀⢹⣧⠈⠙⢿⠋⠓⣨⡟⠐⠖⠈⣇⠉⠙⠛⣿⣿⡟⠀⣰⣷⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡟⠀⡀⢀⡀⠨⠁⠀⠤⠀⣶⠀⠖⠱⠆⣤⣨⣿⠀⠀⠈⣿⣶⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⠇⠀⠛⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠒⠀⠀⢀⡄⠀⡇⠈⣡⣴⣶⣶⣶⣶⡄⠀⣽⣿⣿⣷⣾⣆⢀⡇⠀⠀⠀⢫⣅⣴⣶⣿⣿⣿⣆⠀⠀⣶⣾⣶⣶⣶⡄⣾⡄⡀
⠀⠀⠀⠀⠀⠑⠒⠦⠀⠀⢲⡄⠰⣿⣿⣿⣿⣿⡿⢠⣦⢿⣿⣿⣿⣿⣿⣩⡄⠀⠐⠀⣎⠀⣿⣿⣿⣿⡿⢋⣤⣄⠻⣿⣿⣿⣿⠏⠘⣼⢇⣀⡀⢨⠅
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣼⡇⠤⣍⠛⠛⣫⣵⠀⣿⣿⣆⠈⣩⠉⢉⣁⠙⢑⡆⠀⢤⡛⠀⢈⣉⠁⢠⣶⢼⣿⣿⠀⣶⠤⠈⠁⠲⠄⣸⡆⠩⠝⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠰⠄⠀⠀⠈⠉⠧⣄⢀⣄⣠⣄⠀⢸⡿⢿⣿⣄⣡⣄⣼⣧⣾⠟⢠⠀⣀⠷⣄⠉⠛⠃⢀⠀⣾⡿⣿⢗⡀⠐⣿⡃⠀⡰⠋⠀⢀⣸⠆
⠀⠀⠀⠀⠀⠀⠳⢶⡞⠀⠘⠃⠉⠀⠀⢀⣿⣤⣀⡄⠀⣠⣌⣁⣿⡈⠀⠘⣆⣠⠟⠣⣠⡟⠙⠃⢻⣀⣀⠀⣀⣀⣀⣁⣠⡏⠉⠛⣁⣀⣶⠋⠀⠁
⠀⠀⠀⠀⠀⠀⣤⡤⠆⣤⠀⠐⠈⠉⣤⡀⠙⠿⠙⠙⣋⣹⣟⡿⠋⠀⣀⡴⢛⡀⠀⢀⡟⠣⣄⡼⠛⢿⠉⣟⠉⣋⣛⣩⡝⠋⢀⡔⠋⠁⠙⠀⢰⡟⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣴⠉⠀⠷⠀⠈⠁⢠⣧⢰⡎⠀⠁⠀⠻⠁⠘⠙⣏⠁⠈⢻⢉⡿⠉⠉⡙⠁⠀⠀⠚⠉⠛⠉⠉⠁⠀⢀⣙⡀⠀⠀⢚⠀⣀⠟⠀⢡⣤
⠀⠀⠀⠀⠀⢴⣄⠀⠈⢤⡠⢄⡀⠠⠀⠘⠉⢫⡁⣤⢀⠄⢀⣀⣄⠰⣮⡶⠆⢡⢖⣠⠖⠀⠀⠀⠀⡖⠳⣦⠛⣀⡤⠠⠴⡿⠉⠉⠁⠀⠹⠟⠋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⢁⡀⢠⠀⠈⢿⡀⠀⠀⠀⠘⠀⠀⠈⠷⠈⠉⡠⠀⠀⠀⠀⠚⢨⠙⣆⠀⣄⠸⠖⠛⠋⠁⠐⠻⠀⠀⢘⡗⠒⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠳⠤⠀⠀⠠⠀⠀⠞⠁⢀⠃⠀⠳⣄⠀⢀⡀⢢⣀⠀⡔⠒⢂⡤⠀⠀⠀⠈⠀⠀⠒⠰⠇⠀⠀⠀⠐⠲⡂⠀⠥⠄⠀⠠⠤⠚⠉⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⠀⠀⠙⢷⡀⠶⣀⣈⣀⠠⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠤⠤⠤⠴⠀⠙⠒⠚

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠛⠻⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠈⠀⠠⡀⠀⠀⡴⠀⠀⠀⢠⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⢀⠀⢂⠀⢅⠀⠀⠀⠈⠉⠑⠀⠀⠀⡰⠀⠀⠀⢀⡞⢀⣀
⠀⠀⠀⠀⠀⠠⠀⠀⠎⠀⠀⠀⠈⠀⠀⢸⡇⠀⠀⢠⣰⣤⠀⡼⢦⡌⠁⠘⠀⠠⠀⠀⠀⢀⠀⠀⣤⠁⠐⠀⠀⢈⡀⠀⣀⠡⠀⠐⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠃⠀⠂⠤⠒⡀⠐⠀⠀⠈⢀⡀⠀⢤⠠⠀⢠⡄⠀⠑⠀⠀⠀⠀⢈⠁⠀⠀⢀⠁⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠻⠦⡰⠏⠙⠀⠀⠔⠀⡀⠀⠀⠐⠀⠰⠚⠁⠁⠀⡐⠀⢈⠀⠛⢲⡀⠐⠀⠀⠀⠂⢁⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⡀⠁⠀⠀⢂⠀⠀⠀⠞⠁⠀⠀⠀⢠⠃⡐⠀⢠⠇⠀⠀⣀⠀⠘⠁⠀⠀⢠⠀⠀⠸⣒⠷⠄⠐⠒
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠢⠐⡎⠀⠀⡄⢠⠀⠻⠀⠀⢠⠘⠊⠀⠀⠌⠆⠁⠀⡟⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⣀⠰⠈⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⠀⠀⡃⠀⠀⠡⡜⠀⠀⡀⠀⢨⡀⡇⠀⠀⠀⠁⠀⢰⡁⠖⡁⠀⠀⠀⠀⠀⠘⠀⡰⣉⠑⠒⠒⠢⠤⡜⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⢸⠇⠀⠀⠀⠀⠀⢈⣠⠀⢀⠀⢾⠀⣌⠀⠐⠀⠂⠀⠀⠀⢀⡀⠈⡴⠁⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠌⠉⠅⣄⠐⠀⠀⠂⠀⠀⢸⠀⢸⠀⣻⡀⠛⠄⡁⠠⠤⠤⠀⠀⠊⣀⠐⠀⠁⠀⠀⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⢀⡄⠀⢣⠀⠀⠀⠀⢈⢸⡀⢈⠔⠙⡇⠀⢈⡇⠀⠀⠀⠠⠄⠠⠁⠀⣀⣄⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡠⠊⠁⠄⢤⡁⠀⢀⠉⢸⢀⡴⠀⠀⢸⠀⢀⡁⠖⠃⠀⢀⠀⡾⠋⢻⠅⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠀⠀⠀⠀⠀⢌⠙⠊⠀⠀⠛⠀⠀⠀⠸⠊⠁⠠⠖⠂⠁⠀⣸⠃⠀⣸⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡁⠄⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⡿⠀⢠⡋⠀⠀⠀⡀⠀⡅⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠙⠒⢆⢠⠀⠀⠀⠀⣀⣠⣠⣀⣀⡀⠀⠀⠀⡔⠁⠀⠀⠀⢀⡼⠃⠀⠻⣄⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠑⠐⡃⠀⠳⠄⣠⠴⠐⢚⠛⠉⠉⠙⠛⠻⠷⣦⣤⣤⣀⣤⠴⠛⠁⠀⠀⠀⣼⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠘⠀⠀⠋⠀⠀⠀⠀⠀⢀⠀⠑⡆⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠠⠎⠉⠦⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⠀⠧⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠠⠀⠀⠀⠀⠌⠀⢠⠀⢀⣀⣤⣴⠷⠾⡖⠀⠀⠀⠀⠀⠀⠀⠠⡄⠱⠀⠐⠂⠀⠀⠂⠶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠉⠀⠀⡠⢯⠀⠀⡆⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠠⠯⠃⡰⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠶⠐⠀⠀⠀⠀⠀⡸⠁⡆⠀⠁⠀⡀⠀⣁⠀⠁⠀⠀⠀⠀⡐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠄⢥⠄⠁⠀⢀⠄⠛⠁⡠⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⢉⠀⠀⠀⠀⠀⠔⠀⠐⠀⠀⠂⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣴⠋⠀⠊⠁⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠤⣤⠶⠾⠷⡄⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢢⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡛⠋⠉⠀⠀⠀⠀⠊⠁⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠠⠐⠀⠀⠒⡀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢒⣀⠀⠀⠀⠐⠣
⠀⠰⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠿⡀⠹⣆⠰⠀⠡⠀⠀⠒⡠⠄⠀⠀⢀⠀⠀⠠⠀⢤⣀⣀⠦⠤⠀⠁⠀⠁⠀⠠⡌
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠚⠉⢦⡌⠳⠷⣀⠀⠀⠚⠰⠆⠀⠀⠠⠀⠁⡘⠀⠀⠀⡀⠀⠀⠀⠋⠈⠈⠉⠀⠤⠀⠀⠴⠋⠉⠒⠠
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠲⠀⡵⠄⢀⣀⡀⢡⠀⡀⠀⠠⡀⠀⠀⠁⠈⠉⠀⠀⠀⠆⠀⡀⠀⠀⠀⡀⢀⠆⣠⠀⠸⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢀⡴⠈⠀⡀⠀⠀⣀⠀⣁⠀⠜⠀⠠⠈⠁⠀⢅⠀⠀⠠⠐⠚⠓⠠⠀⠁⠈⡀⠸⠀⠀⠉⠀⠀⠀⠀⠄⣀⠀⠐
⠀⢸⠀⠀⠀⠀⠀⠘⠂⠠⢄⠀⢈⠀⠒⠉⠓⠀⠈⠉⠤⠉⠀⠠⢈⡀⠃⢀⡄⢠⡀⠁⠐⠀⠠⠀⡠⠅⠀⢀⡀⠀⠀⠀⠀⠈⠀⠀⠠⠩⡄
⠀⢙⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⡄⠈⠈⠉⢉⠆⣠⠒⠋⠁⠀⠉⠉⠀⠀⠀⠄⠁⠀⠂⠐⠀⠀⠀⠀⠀⠐
⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⠀⠁⠀⠀⠰⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠻

⠀⠀⠀⢀⠀⠀⠀⠿⠀⠀⠀⡀⢠⣯⡀⠀⠘⠃⠀⡀⠀⠀⣀⠀⠘⠃⠀⠀⠀⠀⢀⠸⠀⠀⠀⢹⢀⡇⢀⠀⠀⠀⠀⠀⠀⢈⣧⣀⡠⠶⠦⠀⠂
⠀⠀⠀⠠⠀⠘⠳⡆⠠⠠⣦⣽⠻⠉⠙⠀⠀⠈⠃⢨⣠⠉⠹⡖⠂⢀⡰⠲⣦⠈⠰⠀⠀⢠⠀⡀⠋⠀⠛⢸⡆⢘⢰⠖⠊⠀⠀⢁⡠⣀⠀⢀⡄
⠀⢸⡄⠀⠀⡆⠀⢀⣄⢘⣿⠟⠀⠀⠀⠀⠀⠀⡀⠀⠉⢀⣠⢴⠀⠀⢇⣠⠛⠁⠀⡀⢠⡄⠀⠃⠀⢀⡧⠈⡵⠉⠀⠀⠀⠀⠃⠈⠁⠀⣦⠘⠃
⠀⠠⠀⢰⡄⠸⣀⣾⠀⠀⠼⢦⡴⠊⡃⠀⢈⠘⣅⢶⡞⢿⡄⠀⡶⣾⣇⡄⠀⡷⠀⡉⠃⠀⠀⢾⡔⠶⣠⠎⠁⠀⠀⠀⠀⡀⣠⡤⠒
⠀⠀⠀⣤⠀⠀⠋⠀⠀⠀⠀⣿⠳⢋⣷⢶⡿⠋⠉⠀⣜⡉⣑⣶⠈⡁⠁⢶⠀⠱⣶⠛⢉⡇⠀⠘⢁⣼⠁⢀⠀⠐⠂⠉⠀⠀⠾⠄⢀⡀⠠⣤⡆
⠀⠰⢶⠈⠀⠀⢠⡖⣢⢴⠆⠈⠀⠀⠙⡈⠁⣀⣠⡀⠛⢻⡉⠩⣎⠁⠀⠸⣫⣄⡹⠋⣾⠡⣄⣰⣾⣿⣚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠀⠈⡃
⠀⠀⠈⠀⡠⢦⣦⠀⠻⢀⣠⣄⡶⠚⠆⣻⢺⣷⠿⢦⡤⢍⠁⢀⣀⡤⠉⠉⢩⣏⢳⠟⠉⠃⠿⠙⣿⢿⡍⠉⠉⠁⠀⠀⠀⠉⠙⢷⡶⠒⠓⠀⡀
⠀⢀⠀⣀⠁⢈⣫⠄⠐⠛⠊⠰⠦⠄⠨⠤⠈⠳⠄⠀⡉⠭⢠⠄⠿⡷⣦⣠⠶⣿⡃⠀⠀⠀⠤⠀⠌⠀⠀⢀⠀⠠⠄⠐⠂⣐⠤⢄⠀⣀
⠀⠈⢿⡉⠀⢵⠆⠤⣀⣂⣴⠆⠀⣤⣤⣤⣭⣤⡾⢿⡛⠘⠳⢦⣾⣿⣦⢧⣾⣿⠖⠒⠓⠘⠳⠛⢰⡖⠲⠒⠒⠢⣀⠲⢆⡀⢠⣾⡏⠈⣡⡀⡄
⠀⢰⡤⢡⣼⠋⠀⢦⠄⠉⠰⢆⡿⡖⣷⡇⠈⡿⠳⡄⣤⠀⠀⠀⠀⢸⡟⠘⢿⣤⡀⠀⠸⠃⢢⡜⠛⡇⠀⠀⠀⠀⠈⠲⢄⣀⠀⠈⢛⣿⠉⠸⠇
⠀⠀⠀⣲⡏⡀⡤⠀⠼⢰⢆⣌⠷⢋⣼⡁⠈⡇⠀⠃⣉⠘⢣⣸⠙⣾⠃⢀⡷⠑⠾⣧⣄⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⣀⠼⠛⢤⠀⠚⠷⠆
⠀⢀⠀⣩⡇⠷⠀⠠⠀⠈⣈⣡⠜⣛⡟⡇⠀⢱⡀⠀⠛⠳⣌⠻⠀⠐⠂⠈⠁⣸⠃⠙⠛⣧⡶⣤⡼⠀⠀⠀⢀⡴⡚⠧⡖⠱⠆⡘⠃⠰⠆⠈
⠀⠘⠁⠀⠗⣠⡤⢢⠀⢀⡉⢄⠞⣉⡇⣧⠀⠈⢇⠀⠀⠀⠈⢳⣄⣤⠤⠀⣀⠿⣋⢨⣶⠿⠕⡿⠄⠀⠀⠤⡁⣐⠃⠀⠀⢠⡀⠁⠀⣘⡃⠀⠁
⠀⠀⠀⠀⠀⠁⠈⠁⠀⢸⠚⣈⡌⡸⡇⢹⡄⠀⠀⢧⡀⠀⠀⠀⢺⡙⠧⠿⢸⠒⠬⢤⡉⣠⡼⠁⠀⠀⠈⠷⣋⠟⠡⠦⣤⠄⠀⠀⠠⣾⠁⡀
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠸⠁⠬⣤⢡⡧⠀⢃⠀⠀⠈⢣⣀⠀⣠⣾⡀⣽⡀⢸⣧⣠⡞⠡⠋⠷⡁⠀⠀⠆⠀⠀⠀⠀⠀⡙⢢⣿⣶⠟⠁
⠀⠠⠀⠀⠀⠀⠈⠁⠀⣻⠁⠶⢠⡂⣷⠀⠀⠀⠀⣀⠼⠿⡟⠁⠀⢸⡌⢻⣾⣧⠴⠋⠀⠀⠀⠃⠀⠠⠀⢀⣠⣞⡶⠟⠘⠉⠛⣶⠂⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⡀⠰⠂⢠⠖⢊⡡⠔⣼⡇⣀⣀⢦⡍⠐⠀⠉⠀⠀⢀⣏⠄⠙⢠⡀⠀⠖⠀⠀⢠⠀⣠⣰⠍⠃⡀⠀⠀⠀⠀⣴⣩⡆⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠳⠴⢃⡴⢇⠀⡀⠉⢷⡀⠀⠀⠀⠀⢨⡍⢀⡀⠠⠙⣄⠀⠀⡠⠀⠰⡤⠁⠀⠀⡀⠀⣀⠺⢷⣿⣧⡤⠶
⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠀⠀⣀⠀⠘⣿⡆⠁⠠⠤⠁⠀⠀⠘⠂⢸⡇⡀⠀⢿⠂⡘⢀⠊⠀⢨⣙⠀⠀⠀⡐⡛⠆⡹⠀⢻⡿⠉⠀⠀⠀⠀⠂
⠀⠀⠀⠀⡐⠰⠛⣄⡸⠉⠀⠀⠐⠈⠰⡼⠁⠀⠀⠨⠀⠀⠀⢄⠀⣤⠁⢀⡴⠞⠀⠁⠂⠀⠀⢴⠇⢀⠔⢪⡉⠙⢷⡄⠠⢛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⣠⣄⢀⢾⣶⡦⣴⠀⠀⠀⠀⠀⠀⡀⡐⢁⣤⠞⢤⣴⠆⠀⠀⠀⢀⣩⣞⣡⠀⢀⠀⠀⢸⡿⠀⠀⠑⢯⡤⡀⠀⠰⠆
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⣴⢻⡟⠛⠃⠛⣰⣆⣰⡦⠀⠋⠀⣃⠸⡷⣄⠀⠉⠀⠈⠉⢉⢻⣿⡏⠹⠀⠀⣠⣤⡾⠁⠀⢠⡄⠀⠐⢫⡀
⠀⠀⠀⠀⠀⠀⠐⠂⠈⠰⢴⠙⠞⠣⠀⠀⡞⠁⠈⣥⣤⠀⠀⠀⠙⢷⠇⠹⣗⣰⣧⠀⡚⠈⣹⣻⠇⢁⣴⠟⠋⡿⠀⠀⠀⠀⢀⡀⣷⣬⡴⠐⠃
⠀⠀⠀⠀⠀⣰⢦⣀⠀⠳⠂⠀⠀⠀⠀⠀⠁⠀⠀⣈⢵⣿⣿⣶⣠⠤⣠⡀⠁⠉⢉⣉⢀⡴⣾⡏⠀⢘⣷⣠⡜⠀⢀⡔⠀⠉⠀⣬⣼⣛⠖
⠀⠀⠀⠁⠀⢱⠟⣵⠐⢋⡥⠀⠀⢡⠀⡄⠀⢠⣤⡿⠿⠋⠉⠷⠏⠚⡹⣿⣾⠶⡿⠿⠛⠛⠋⠀⠀⠀⠹⠿⢄⣀⣈⣀⡠⣾⣥⡿⠹⡇
⠀⠀⠀⠀⣰⠟⠀⣽⠟⡀⢱⠂⣠⢘⣧⣷⠆⣴⣿⠇⠐⣤⠀⠀⣿⠈⢁⣆⣁⢌⣀⡄⠀⠀⠀⠀⢠⣂⣤⣷⠞⢏⡟⠉⠃⡉⠌⠁⠀⠙⣆
⠀⠀⠀⠀⣿⠀⠀⠘⢾⣳⣦⣉⠉⠻⣷⣯⣄⣿⡗⠀⠀⠹⡛⡿⠗⠚⠟⣿⠉⠊⡹⣟⢢⡀⣇⢰⣾⣿⠏⠁⢠⡿⠃⠈⠓⣀⡆⠀⠀⠀⢹
⠀⠀⠀⠀⠇⠀⠀⠀⠀⣩⡿⢿⣁⡀⣸⣿⣿⣿⣇⢠⡀⡄⠐⠃⢤⣦⠀⣹⠀⢺⠅⠻⠀⠃⣞⣿⠃⠟⠀⢀⠟⢡⠖⠀⢹⠝⠁⠀⠰⠄⣸⠀⡇
⠀⠀⠀⠀⠈⢧⣆⠐⢠⡷⢤⣼⣿⠿⡟⠛⠿⡯⣿⣋⢡⣇⡀⠀⠀⢩⡴⠿⣦⠀⠀⠀⢀⣰⣾⣽⢰⡆⠀⠠⠶⡌⡀⢀⠜⠁⠲⡎⠀⣠⠆⠀⠃
⠀⠀⠀⡤⠀⠀⠻⣄⣰⠧⠀⠹⠿⡢⢇⣠⣠⡀⣿⠟⠋⢻⣗⠐⠂⢤⡁⠈⣁⡠⠜⠿⠈⠉⣉⠁⢺⣇⢰⣍⠡⢚⡿⠿⠄⠠⢾⣆⡴⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠩⡿⠆⢼⣄⡀⠈⠀⢄⡠⠐⠋⠀⠠⣀⠀⠀⠀⠸⣷⣤⣎⠀⠀⠀⣀⡀⠀⠀⠀⠙⠣⠀⠀⠉⠀⣴⠀⠀⠈⠉⠁⡀⠉⠀⡀
⠀⠀⠀⠁⠘⠁⠐⠿⢃⠀⠈⠁⠀⠀⠐⢾⠖⠂⠐⣮⠟⠋⣥⠀⠀⢶⡿⠾⣷⠀⠀⠀⠙⠻⣖⠀⠀⠶⢶⡴⠀⠀⠘⠁⢀⠀⡀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠠⠘⢫⣄⠀⠀⡐⠀⠀⠀⠄⢀⡘⠇⠀⠀⠀⠀⠀⠘⢃⣀⠌⠀⠉⠂⠀⠀⢫⡷⣠⢄⡆⢠⣀⣠⣖⠷⠛⠐⠛⡴⠃⠀⢱
⠀⠀⠀⠀⠂⠀⠀⠀⠛⠴⠧⣠⠄⣤⣀⣤⢰⠎⠉⠀⠂⢀⡂⠀⠀⡠⠉⠛⠀⠀⠀⣄⠒⡀⠀⣰⠘⠘⠃⠻⠞⠏⠁⠲⡄⠐⡂⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠁⠀⠀⠈⠀⠁⠀⠀⠉⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⠀⣀⡀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀⡀⢀⣤⡼⠀⠀⢈⠉⠋⣤⠛⠋⠁⠀⠞⣤⡄⢀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠿⢻⣮⠙⠁⠀⠀⣿⣧⢠⣄⡄⠀⠀⠀⠀⠉⣵⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣆⣀⣴⡄⠀⠀⠙⠙⢋⠉⣧⣀⠀⣰⣶⣠⠤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠋⠉⣿⡆⣶⣆⡀⠘⡀⣿⣿⡃⣡⠋⠛⡟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠐⠿⢃⠿⣹⡇⠆⢀⣼⡟⢁⡜⢠⠈⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠟⡁⠀⠶⢳⣎⠳⠈⠰⠶⢬⣝⣋⠘⣷⠶⣄⣼⣿⠀⠀⠀⣀⠸⠿⣶⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣶⠇⣼⣾⡧⡤⠶⣴⠛⣦⠇⣼⡀⣿⣏⠱⠒⠇⠠⡽⠁⢱⣷⢀⣼⡿⣼⠋⠈
⠀⠀⠀⠀⠀⠈⢀⣴⣄⡶⣶⢛⣁⠘⢁⡟⣱⣷⡆⣶⠶⠖⣀⠘⠃⣿⡿⢣⡞⠇⣾⡛⢂⣶⣭⣬⣛⠿⢷⡶
⠀⠀⠀⠀⠀⠀⠈⠁⠘⡁⣰⣿⣿⣷⡝⠿⣋⢁⡇⢸⡿⣤⡿⠋⠸⠻⡿⠸⠿⣂⡿⣿⡌⣭⣴⠋⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡷⢿⠟⢻⣽⠋⠿⣘⡏⠛⠑⠊⠁⢰⠂⠀⠀⢈⠀⠀⠀⣍⠸⢿⣿⡿⠓⠃⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢵⣿⡇⠿⣿⠀⠀⠀⠸⣶⣤⣀⠀⢀⣤⡶⣿⠃⠘⠃⠈⠁⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣤⡤⠿⣧⢸⢿⡀⡀⠀⡈⢻⣌⣽⡿⣿⣅⣼⠇⠀⠀⠁⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠺⣿⣼⣿⠁⠁⣠⣴⣟⣿⡓⣠⣠⣭⡟⣶⣤⡀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢹⡟⠛⠀⠉⠉⠉⠉⠛⣿⠙⢩⡟⠉⠛⠉⠉⠀⢀⢀⡄⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢀⠀⠀⠀⠀⣤⠄⠘⠹⣧⡞⠁⠁⠀⠀⠀⣀⣀⠸⡇⢸⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠑⢸⡄⣸⠀⠀⠀⣀⣀⣀⠀⠀⠙⠀⢀⣠⣤⣶⣶⣦⣍⠀⣿⡟⠀⠀⠀⠀⠄⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠠⢴⣈⠁⠠⠀⡈⢃⣻⢀⣴⣿⣿⣿⣿⣿⡄⢀⠀⣾⣿⣿⣿⣿⣿⣿⡀⢿⠃⡠⠒⣀⣠⣴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣌⣉⣻⠋⠘⣿⣿⣿⣿⣿⡿⢡⣿⣿⡜⢿⣿⣿⣿⣿⢻⠇⠈⣯⣴⢸⣏⠁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⡻⠟⢻⠀⠀⠈⠋⡛⠛⣫⡿⣼⣿⣿⣧⢲⣬⡉⠉⡉⣀⣠⣶⠟⢁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⠃⢠⣾⡷⡖⠲⣶⣧⠀⠀⢠⣿⣿⠿⣿⡄⠉⠁⣶⣿⡇⢼⠃⢘⡻⠟⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣉⢉⣔⢃⠀⢋⡝⠀⠀⠀⠘⠁⠐⠓⠂⠀⠀⢹⣬⠠⣸⠓⠀⠁⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣌⣩⠠⣤⣼⡇⠘⣷⡜⣃⠐⠄⣃⢀⡄⢀⢠⣴⢸⡏⢠⡟⠿⢡⡘⣦⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⠝⠛⠉⣠⠿⣦⢻⡧⣄⢠⣼⡉⠉⠑⠙⣧⣴⣿⡇⣼⠃⢀⡈⠑⠛⠋⢼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⣀⠤⠄⠀⠉⠀⠁⢸⡎⠓⠤⠼⣸⣿⡃⣸⢉⡍⠘⠃⣸⠏⠈⠀⠉⠁⠒⠀⠀⣈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠉⠉⠉⠁⠐⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠄⠀⠠⠀⠀⠠⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢛⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠆⡀⠄⠱⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⡄⠈⣻⡛⣮⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⠁⣤⠀⠐⠂⡀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠐⠂⠀⠀⠀⢂⡘⠀⠺⠫⣅⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠘⢯⣤⠀⠐⣀⠀⠙⠠⢄⡀⠃⠀⠀⠀⠀⠀⡄⠀⢀⡀⠀⠠⢌⢡⡆⣴⡶⠊⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡆⠀⠀⠊⡀⠉⠐⠀⠀⢠⡄⠃⠀⣀⡀⠀⠀⠀⠰⠀⠀⢋⠰⡦⠰⡜⠁⠀⠀⠶⠞⠁⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣀⠀⠀⠀⠁⠀⡤⠁⠀⠀⠀⠟⠀⠉⢀⠈⠁⠀⠀⠀⢠⠀⠀⠀⠀⠈⠁⠀⠀⠐⢤⣀⠀⠀⠀⠀⠰⣦⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠀⠀⢀⡁⠰⠂⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⢶⡄⠀⠀⢀⠀⠀⠄⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣠⡄⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡄⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⡀⠀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⢠⠧⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠠⠎⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠋⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠃⡤⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠴⢶⡆⢰⠾⠿⠐⣿⠉⠀⢠⣀⠔⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⢲⣤⠘⠃⠤⠠⣴⠆⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢱⡇⠀⠀⠁⠄⠀⠀⠀⠐⠒⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠰⠇⠀⠀⠀⠉⠑⠀⠀⠀⠘⠀⠀⠚⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⠀⠀⠀⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣤
⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠊⠀⠀⠀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠇⠐⠛⠀⢀⠀⠀⠁⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠠⠀⠀⣀⠂⢀⠔⢀⠞⠀⢈⣀⣀⣀⡠⠛⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠈⠁⠀⠀⠀⠀⢰⣦⠀⠀⠀⠐⣄⠁⢠⡾⢋⣯⠙⢯⡠⣾⣀⡴⠋⢀⣴⡈⡡⣌⢡⠀⠀⢀⣾⠂⠀⠀⡰⢁⠠⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠰⠗⠂⠀⠀⠈⢿⣷⢤⣄⣤⣤⣀⡹⣴⠎⠫⢾⠌⣷⣸⡟⠀⣴⠋⠉⠙⠱⠋⡠⠷⢴⣯⠟⠀⠀⠀⠀⠈⠀⠀⠈⣸⠃
⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠳⠟⠻⠀⠀⠀⠀⠀⣿⢹⣿⠁⢸⡉⠀⠀⠀⠀⠠⠃⠛⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠰⢖⡻⠘⢏⠓⠘⠃⠴⠦⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠒⠁⢠⡤⠀⠈⠠⢄⣈⠀⠒⠁⠀⣠⣤⣀⡀⠙⢦⣤⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠘⡧⠤⠴⠶⠖⢁⠀⠁⠀⠀⠈⠛⠀⠀⠀⠀⠁⠀⠉⠉⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⢴⠶⣶⣦⡀⠀⠉⠈⠉⠁⠀⠀⠀⡶⠿⠀⠀⢀⠾⠤⠀⠄⠀⠀⠀⠀⠀⣀⠰⠴⠛⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⠀⠉⠠⠔⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠔⡠⠍⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡄⠀⠀⠀⠈⢇⣀⣀⠐⢤⡀⠀⠈⢆⡀⠀⠀⣄⠀⣱⠀⠀⠀⠄⠀⢠⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠘⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⡄⠀⣐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡋⠑⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠐⠀⠀⠀⠄⠘⠁⠀⠀⢄⠐⠀⠠⠄⠀⠀⠀⠰⠄⠀⠀⠀⠄⠀⠀⠀⢀⡀⠀⠀⢀⡉⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⡙⠛⢤⣦⠀⡔⠈⡀⠰⠀⠀⠀⠀⣠⣆⠀⠤⠛⠋⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣈⣷⣬⣟⣧⠿⣉⣻⡾⢿⣿⡿⢿⡉⠘⠆⠀⠉⠆⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡁⠟⠉⠀⠀⠛⠟⠁⠛⠈⠛⠈⠱⠋⠈⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⢀⡫⠤⢀⡀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠲⠈⠝⣣⡌⢡⣬⠉⠀⣄⡈⢻⠀⠀⡀⢀⡤⠘⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠘

⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣭⣭⣭⣭⣬⣭⣥⣬⣭⣽⣭⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣿⣿⣶⣶⣿⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⢀⡢⠀⢀⠄⠀⠰⢤⠐⠘⡁⠠⠀⡀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⣄⠀⠐⠤⠎⢫⡴⠊⡩⠰⠄⠖⠓⠙⢥⣤⠀⡀⠋⢀⠀⠄⠀⠀⠠⣶⡀
⠀⠀⠀⠀⠀⠀⢰⡏⠠⠀⠀⠀⠀⠂⢀⠊⠀⠀⠀⠀⠀⠮⠖⠀⠀⠉⠈⠁⠮⠀⡀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⡠⠞⠓⠃⠈⢰⠉⢻⡔⢂⠀⠀⠉⠀⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠈⠐⠝⡶⠀⠀⠀⠀⠀⠸⠟⠆
⠀⠀⠀⠘⠁⠰⠒⢦⠀⢠⡴⠃⠴⢂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡁⢡⣤⢀⠀⣠⡤⢆
⠀⠀⠀⠀⠀⠀⣠⠀⠀⣼⠃⢀⣬⣅⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⡽⡷⢸⡃⠀⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡤⠋⠀⠛⡶⠮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⢸⠀⠀⣲⠌
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⠐⢠⣲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢾⢣⣾⡸⣖⣈⠂
⠀⠀⠀⠠⠒⠀⢀⡾⣩⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⢨⣸⠏⠃⢹⠂
⠀⠀⠀⠀⡆⢠⣿⡸⣧⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢢⠽⣆⣀⡼
⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⢂⠀⠀⠀⣴⠓
⠀⠀⠀⠠⢿⣿⡿⢻⢍⣏⡀⠀⡁⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡀⠈⠶⢀⡯
⠀⠀⠸⣷⠟⡡⡤⣛⣶⠁⠀⠀⣱⢀⠐⠀⠀⠂⠈⠀⠢⠀⠀⠠⠀⠀⡆⠈⠢⣴⡌⠒⢄⣁⣤⠄⠂
⠀⠀⠐⠁⠔⠶⣨⡏⣀⠀⠀⠀⠈⡯⠅⡶⢅⡀⠒⠃⠺⠂⠘⠒⠠⠄⠚⡀⡀⠉⠲⣄⠈⢣⢈
⠀⠀⠀⠙⢈⣸⠿⠓⠛⠀⠀⠀⠀⠀⠀⣤⠞⣶⠳⠁⠤⠀⠀⠀⠀⠀⠀⡁⢀⠀⠀⠉⠀⠀⢀⠀⠐
⠀⠘⠙⢀⢺⠻⠀⣻⣽⡀⠀⠀⠀⣤⢾⣿⣇⢄⠀⠀⠀⠀⠀⣀⠀⡀⢀⠄⠁⠀⠀⠀⠀⠀⣨⠆
⠀⠀⢀⠋⠐⣶⣧⡴⣿⣇⢰⢁⢠⣿⡞⠈⢁⠄⠀⠀⠀⠄⠀⠉⠀⠠⠈⠊⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⡀
⠀⠀⢸⠄⠠⠭⠭⢥⣟⣻⠋⠸⠷⠾⠁⠀⠂⠈⠂⠀⠀⠈⢁⢰⡶⠁⠒⠀⠀⢀⡂⢠⣄⠀⡠⢀⠔⠚⡂
⠀⠀⠀⠆⠀⠀⢐⣏⢼⣟⣧⢀⢐⣮⠌⠁⡔⢀⣧⠀⣌⣄⡁⢸⣅⠀⠡⠘⣧⢙⡘⢯⣷⣚⡦⣉⣶⡈
⠀⠀⡃⠘⢣⣀⣈⠙⠃⢷⠻⠎⠈⡻⣤⣽⣋⠢⣴⣶⣄⡭⠃⠾⢉⣷⠶⣧⣌⡶⠹⠙⣗⣆⢾⣿⣕⠳
⠀⠀⢃⣰⢬⠉⠀⠀⠀⢺⢰⠄⠾⠗⠏⠿⠋⠛⠋⢉⠉⠁⠋⠁⠛⠻⢦⣝⣼⠇⠀⠀⠀⠸⢡⡟⢛⡀
⠀⠄⠀⠗⠻⣇⠀⠀⠀⣈⣧⡀⠀⠀⢀⡀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⡀⠠⡋⠀⠀⠀⣠⡀⢀⠖⠸⡃
⠀⡂⠀⠤⠘⡜⡄⢠⣴⣿⠛⠙⠀⠀⠀⠈⠀⠠⠀⠠⢠⠀⠀⠀⠀⠀⠸⠹⠿⣾⣤⠸⠛⣻⣈⠏⠙⠁
⠀⠀⠀⠴⡀⠍⠸⡞⠁⠬⠀⢠⠀⢀⡀⢰⣳⢈⣟⣃⣚⣻⠦⠈⠀⢠⠀⢑⣀⠻⣿⢢⠟⢁⣌⠁
⠀⡆⠀⠈⠻⠚⢓⡙⢤⡞⡦⠠⢄⠃⢠⣖⡓⠷⣛⠛⠛⠋⡞⡙⡸⢿⣳⣋⣯⣴⡧⠒⠴⡽⠁
⠀⡆⠀⠀⠀⠰⠌⠉⠌⡅⠄⢀⠉⠃⠀⠋⣨⠛⠀⠀⠀⠐⠛⠺⠁⠛⠻⠾⠏⠘⠷
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠛⢻⣿⣿⡟⠛⠿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠁⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠈⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⢀⣴⣿⣿⡿⠋⢸⣿⣿⡇⠙⢿⣿⣿⣦⡀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏
⣴⣦⣄⢇⣰⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣴⣿⣿⡿⠋⠀⠀⢸⣿⣿⡇⠀⠀⠙⢿⣿⣿⣦⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣆⡸⣠⣴
⠙⠻⠿⢿⡿⠛⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡋⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢙⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠿⠟
⢠⣤⣤⣀⢣⢠⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣀⣀⢸⣿⣿⡇⣀⣀⣤⣶⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡄⡜⣀⣤⣤
⠘⠻⠿⠿⠿⢿⠛⠉⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠉⠛⡿⠿⠿⠿⠟
⠀⢠⣤⣤⣄⣈⢧⢰⣿⣿⡿⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢿⣿⣿⡆⡼⣁⣤⣤⣤⡄
⠀⠹⠿⣿⣿⣿⡷⠿⡟⠉⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠉⢻⠿⢾⣿⣿⣿⠿⠏
⠀⠀⠀⠀⣠⣤⣄⣀⡘⢦⡸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢇⡴⢃⣀⣠⣤⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡿⢿⡋⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⢿⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠙⠺⣿⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⢺⣷⣀⠔⠋⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠟⠛⢻⣷⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⣠⣤⣄⠀⣿⣭⡿⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠲⣶⣾⣋⣠⣿⣧⣿⠇⢰⣿⠛⣿⠇⠰⣿⡀⠋⣀⣿⠿⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠉⠉⠉⠻⠷⣦⡶⠿⣷⣾⠿⣧⣤⡿⠛⠛⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣯⣿⠾⢾⣯⣿⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⢀⣿⠋⢻⣧⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢬⣩⣿⢿⢿⡷⣍⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣶⣿⡶⣾⣷⣆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⢤⡀⢸⣿⡆⣿⢌⢼⡇⣾⣿⠀⣠⢤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⢟⣿⡞⣾⢟⣡⢟⣙⣉⢣⣘⢿⣎⣾⢯⠗⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣻⡹⣿⡵⢉⣤⣌⠱⡝⡿⣼⣻⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣗⡿⢌⣧⠞⣩⡻⢶⣇⠾⣟⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢶⣞⠉⣥⠖⡶⡶⢤⡉⢛⣶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣸⣲⣎⣙⣿⣛⣉⣶⢾⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠚⠉

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⣞⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠈⠁⠀⠛⣀⣠⡤
⠀⠀⠀⠀⠀⠉⠍⠀⠄⡀⠀⢀⡀⠀⠀⠀⠠⠌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣉⠆⠀⠀⢉⣤⠖⠉⠛⠛
⠀⠀⠀⠀⠀⠒⣂⠛⣦⣮⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⢀⠀⠢⠐⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠻⢛⣾⣿⣯⣠⣫⡕⠂⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤⠃⠀⢣⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠠⠞⢡⡄⠀⢄⣀⡀⣀⠑
⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⡾⠛⢻⣏⠠⡀⠀⠔⠀⠀⠀⠀⠀⠐⢀⢀⡀⠀⠀⠀⠀⠁⡴⣦⠀⠀⠀⠀⠀⠀⠀⣀⡶⠉⡟⠧⠎⠙⠉
⠀⠀⠀⠀⠀⠀⠈⠁⣿⣠⣘⣟⡳⠏⠰⠄⠀⠔⣣⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠷⡄⣀⠀⢤
⠀⠀⠀⠀⠀⠠⠀⠀⢈⣹⣦⣤⣿⡤⠀⠀⠄⠸⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠘⠀⠀⠀⠀⠀⠀⢂⣠⣴⡆⠀⠴⠷⠖⠒⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠹⢟⣳⣘⠻⠎⠀⠲⣄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠉⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠈⠛⣛⣙⣃⠀⠰⢿⡎⠦⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢣⠀⠀⠀⠀⠀⠐⣉⠛⡉⠹⠂⠀⣸⣈⣀⡄⢠⠀⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⣯⣽⠀⠀⡿⣷⡀⢂⠀⠀⠀⡀⠀⠀⠀⠿⠐⠧⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⡼⠃⠐
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⢹⡆⠀⠐⠋⠁⠈⠀⠀⠘⡅⠀⠀⠀⡀⠀⡀⠀⡄⠀⢀⡀⢀⣀⠠⠤⠄⠒⣀⠀⡌⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠙⢦⣤⣤⠤⠤⠤⠒⠀⠀⠀⠠⠀⠓⣈⡁⠴⡿⠀⠈⠉⠫⡉⠁⠂⠠⣤⡴⠶⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠀⠘⠲⠆⠀⠀⢀⣀⠀⠊⢀⡜⠋⢠⠙⠋⠀⠦⡄⢀⡀⠉⠀⢀⠀⡀⡀⠤⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⢈⠀⠢⢉⡙⠏⡯⢉⣐⢀⠘⠄⠙⠒⢤⡀⠈⠁⢀⡀⠄⠂⠀⠀⢀⠀⠀⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠄⠤⠊⠀⠀⠀⠀⠀⢀⠀⠀⢰⡏⣳⣆⣸⣿⠄⠀⠐⠄⠀⠀⠉⠀⠛⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣉⣩⣠⣿⢀⠀⢀⠄⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠈⢀⡤⠀⠀⠀⠀⠐⢻⡿⠥⢄⣈⠛⠭⠉⠥⠀⢰⠎⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠾⣷⢶⠶⠤⠛⡦⠰⠖⠀⠘⠂⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠦⠀⢀⡀⠐⢁⣠⠶⡄⠀⠐⠒⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠠⠞⠉⠉⠉⠁⠀⡔⣴⡄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⣿⠉⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢠⡄⠀⠀⠀⠀⠀⠛⢠⣼⠋⢲⣄⡰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡼⠀⠀⠀⢀⡀⠀⠀⢘⡃⠀⠀⢿⣧⢈⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢈⠁⠀⠀⢸⣏⠀⠀⢀⠀⠈⢣⠀⠀⠀⠂⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⠀⠀⠀⢨⡻⠄⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠈⠀⠀⡿⠀⠀⠀⠘⣵⠆⠀⠀⣿⠀⠘⠠⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠤⠆⠀⠦⠆⠴⠒⠠⠿⠀⡜⠀⠀⠀⠀⣇⠀⠀⠀⠀⢹⠆⠀⠀⢹⠄⠀⠴⠆⠛⠷⠀⠤⠀⠐⠠⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠇⠀⠂⠀⢀⣬⠀⠀⠀⡀⣿⠇⠀⠀⠘⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠋⠀⠀⠉⢁⠙⠋⠹⠀⠀⠀⢁⠀⢠⡀⠉⠁⠀⠀⠀⠀⢠⠀⠂⠀⠀⢆
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠘⠀⠾⠀⡀⢀⡆⠘⠆⢨⠧⠀⢀⠀⠀⠰⢜⡧
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡠⠀⠀⠠⠂⠁⠠⢀⠁⠜⠁⠀⢀⡁⠀⠈⠃
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣄⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠇⠀⠄⠀⠀⠆⠀⠋⠀⠈⠡⠀⠘⠉⠀⠀⠀⠀⠠⠀⠄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠠⠀⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠒⠛⠊⢠⡀⠲⠶⢄⠀⠁⠲⠊⠑⢀⡀⠀⣀⠴⠾⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠆⠀⠀⠸⡅
⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡺⠃⠀⣼⡟⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠔⠂⠘⠈⡄⠉⠡⢅⠀⠄⠈⢁⣀⡀⠀⠀⢸⣧⡀⠀⠛⠄
⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣠⣤⠘⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠃⢤⣄⡀⠘
⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⢸⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣽⣽⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢖⣲⢄⡚⢿⣿⣿⠟⣃⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⢸⡇⠀⠀⠀⠀⢿⠈⠉⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⢸⢸⢫⣿⣯⢻⢿⠒⡶⠞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⢤⣤⡞⣼⣿⡞⢿⣿⠜⢦⣤⣴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⣩⣿⣿⡏⠀⠈⣿⣾⣯⡉⢉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⣶⣤⣄⢸⢵⢡⢦⣿⣿⣿⠀⢰⣻⣿⣧⠤⢸⣼⣇⣤⣴⣲⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣽⣟⣿⡿⣟⢿⣿⣾⣜⣻⡿⡿⠍⣟⡛⠹⡿⣿⡝⣸⢼⣿⠞⣿⣿⣿⣷⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⡿⣻⡯⠻⣳⣾⢏⣿⠃⠘⠞⠃⠹⠞⠁⢹⣣⡿⡷⡺⠻⣽⡽⣟⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⣴⢤⢱⣟⢀⠉⢸⡇⠀⢀⡀⠀⢾⠀⢁⢀⢷⢝⢤⣎⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⠛⠉⠫⠨⠀⡝⠀⡔⡟⡏⡤⠈⡅⠨⠈⠋⠹⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠉⠀⡅⣸⢸⠇⠀⠈⣹⡉⠁⠀⢻⢨⡀⡁⠀⠡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢦⢻⠸⡀⢸⣶⣻⣿⣿⢲⣶⡀⡸⢱⠇⡆⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠙⡈⢯⡱⣵⣿⡿⠿⠿⣿⣷⣷⣯⠎⣸⠁⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⢠⡙⢦⣽⢟⠅⢺⣿⣿⠒⢝⢿⣭⡿⢩⡀⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣌⣈⣻⣿⡳⢀⢸⣧⣿⢃⠠⠛⠉⢛⣿⡰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠏⣽⣷⡠⢻⢷⣽⣛⣻⣵⠿⢓⣴⣿⡫⢵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⢛⠿⣾⣤⣮⣍⣩⣦⣴⡿⠟⠇⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢌⡎⠊⡩⢝⠭⢉⠚⣌⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⠙⣿⣿⣿⡏⠹⠿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⢷⡀⠉⠀⢨⡄⢀⠄⣰⡀⠀⠈⠛⠿⣿⢏⣿⣿⣿⣿
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⡇⠠⠖⣻⠃⡜⠠⣉⠙⠳⢶⡤⣶⠈⣸⠗⣿⣿⣿⡄
⠀⠀⢻⣿⣿⣿⣿⣿⡄⠋⠀⠀⣠⢾⠇⢰⠞⠁⠈⢀⢔⣴⣾⡇⣦⠁⡟⢠⡇⢸⢿⣿⣿⣿⣆⡀
⠀⢀⣿⣿⣿⣿⣿⡿⢇⠰⣤⠞⠡⠊⠀⠀⠠⠂⢐⣡⣾⣿⣿⡇⡟⣸⠀⣾⠀⣿⠀⠹⣿⣿⣿⣷⡄
⠀⠇⣾⣿⣿⣿⡿⠁⠸⡆⠙⠈⠁⠀⠀⠀⠀⢠⣿⣿⣿⣻⣿⡇⠃⡿⠀⡇⢸⣿⠠⡄⠘⣿⣿⣿⡇
⠀⢠⣿⣿⣿⣿⠁⠀⣰⢷⣄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣽⣿⡇⠀⠃⢸⠁⢸⠻⠀⣴⡀⠸⣿⣿⣿⡄
⠀⣿⣿⣿⣿⠃⠀⣰⠃⠄⣼⠟⠀⠀⠀⣠⣿⣿⣿⣿⡟⣿⣿⣧⠀⠀⠀⠀⠈⠀⠀⡇⢱⠀⢹⣿⣿⣿
⢰⣿⣿⣿⡏⠀⠠⠟⠐⠛⠁⠀⢀⣤⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⡆⠀⡆⠈⣿⣿⡿
⢸⣛⡛⠛⠁⠀⢰⠗⠲⠆⠀⠰⣿⣿⡟⢶⡄⠀⠉⠻⡻⣿⣿⣿⣷⠀⠀⠀⠀⣰⡟⢰⠀⢹⠀⣿⣿⣇
⣿⣿⣿⣷⠀⠀⣀⠀⠉⠀⠀⠀⠀⣻⡇⠀⠁⠀⠀⠀⣈⠈⠛⢿⣿⡆⠀⠀⠀⢻⣄⠀⠀⢸⠀⢸⣿⣿⡆
⣿⣶⠄⠉⠀⠀⠈⢀⣈⣉⢀⣵⣾⣿⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠻⣦⡀⠀⠘⡿⠀⡇⣼⠀⢸⣿⣿⡇
⢸⣿⡿⠶⢤⠀⠀⠠⠤⠄⠀⣻⣿⡿⠀⠀⠀⠀⠀⠐⡌⠀⠀⠀⢀⣴⣿⣷⠀⠀⢃⢠⣇⣿⠀⣼⣿⣿⠏
⣾⣿⣿⣿⡒⠒⠰⡖⢲⣾⣿⣿⣿⣇⣀⡄⠀⠀⠀⢐⠆⠀⢀⣠⣼⣿⣿⠇⢠⡀⠀⢸⣿⡏⠀⣿⣿⣷⠆
⢻⣿⣿⣿⡇⠀⠈⣇⢸⣿⣿⣿⣿⣿⣿⣷⣶⣦⣴⣯⢮⣼⣿⣿⣿⣿⠃⠀⠙⢿⠀⣾⣿⠇⢠⣿⣿⣷
⢸⣿⣿⣿⣷⠀⠀⠋⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇⠀⠀⢳⡈⢰⣿⡟⠀⣼⣿⣿⡟
⠘⢿⣿⣿⣿⣦⢀⠀⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣾⣆⠀⠳⢸⣿⠃⢠⣿⣿⣿⡇
⠀⢸⣿⣿⣿⡇⣸⠀⡇⢰⢠⣾⣀⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣙⠲⠤⢤⣀⢺⣿⣿⣿⣷
⠀⠀⣿⣿⣿⡇⢿⠀⢃⢀⢢⠙⢿⡃⠀⠀⠈⠻⠿⠛⢛⡋⠻⣿⣿⣿⣛⠶⣍⠙⠒⠒⠤⣤⣿⣿⣿⡅
⠀⠀⢸⣿⣿⣇⢸⣆⣿⡸⠎⠳⣄⠙⢿⢃⣀⡐⡄⢀⠈⠁⠀⡛⠟⢋⣼⣷⠎⠉⣩⣿⣷⣿⣿⣿⠟⠁
⠀⠀⠘⠿⢿⣿⣼⣷⣿⣾⣦⡀⠈⠳⢦⣌⣙⡇⠰⠀⢂⠘⡂⢻⣿⠿⠋⢀⣠⣾⣿⣿⣟⣿⠉⠉
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⣄⡀⠈⠉⠙⢄⢣⡘⡆⠋⠀⢀⣠⣴⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠽⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣧⣴⣾⣿⣿⣿⣿⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⣿⣿⣿⣿⡿⠛⠻⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡠⡀⠤⢀⣐⠀⢀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣋⠑⡟⢉⣛⠧⠤⠍⠰⣼⣿⠆⡔⣻⠁⡫⢄⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠈⡰⢷⣹⢆⡊⠀⠀⠀⠘⠁⠀⠚⡴⢷⡎⢟⠲⣷⢀⡄⠲⠫⠄⢄⡀
⠀⠀⠀⠀⠀⣠⣖⢽⡆⢭⣤⡁⢉⡐⠀⠊⠈⠀⠀⡀⠀⠀⠈⠀⠀⠒⠚⠂⡦⡄⠗⠀⢈⡡⣀
⠀⠀⠀⢀⠀⣽⣟⣏⡱⣇⠏⠁⠀⠀⠀⠀⠀⠸⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⠍⣀
⠀⠀⠀⠎⠀⣴⡭⡩⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⢂⡰⠀⠀⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢠
⢀⡯⠦⠱⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⢀⣀⣍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠋⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⡀⢣⠄⢠
⠀⠀⠂⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠈
⠀⡀⠀⠥⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⡄⠠
⠀⢋⠁⠄⡜⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠰⠁⠀⠀⠳⠂⠀⠠⠘⢃⠀⠄⠀⠀⡄⠀⠀⠀⠠
⠀⠈⢧⣀⠀⠉⠶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⣾⠃⠀⠀⠀⠟⠠
⠀⠀⠀⠂⠀⢡⣡⢀⣠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠋⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠟⡇⠆⣸
⠀⠀⠀⠀⠄⠀⢻⠟⡚⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⢜⢰⡀⠀⠀⠀⠀⠀⠠⠂⠂⠄⠳⢁⣠⡧⠄
⠀⠀⠀⠀⠈⠀⠨⣀⡀⠀⠀⠀⠀⠉⠀⠠⠄⠀⠀⠤⠷⠇⡠⠀⠀⠈⠀⢀⠈⠘⢃⡑⠀⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠐⠍⡀⡀⠘⡂⠂⠁⠀⡴⠆⠀⠀⠂⠀⠐⠀⠈⡀⠹⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢈⣀⠸⠋⢣⠠⠀⡕⠀⠀⠓⠀⣩⠇⠈⠁⠌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠲⣀⣉⢀⣤⢀⠂⠃⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠉⠁
⠀⣤⢸⢰⡇⣄⠀⠀⠀⣿⡇⠀⠀⠀⡄⢰⢸⠆⡄⠀⠀⠀⠀⠀⣄⣶⣰⣄⡀⠀⠀⢸⣿⠀⠀⠀⣀⣶⣴⢠⡀
⠀⢸⢸⡄⡇⡧⠀⠀⠀⣿⡇⠀⠀⠀⡇⣾⢸⢰⠇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣇⠀⠀⢸⣿⠀⠀⢠⣿⣿⣿⣿⠁
⠀⢎⠈⢃⣁⣉⣀⣀⣀⠀⢀⣀⣀⣀⣁⣁⠋⢈⠗⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⠦⢤⡀⠀⡠⠤⢾⣿⣿⣿⣿⠆
⠀⠀⠑⠀⠀⠉⠁⠀⠀⠁⠊⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡌⠊⣾⣿⣿⣿⣿⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣉⣀⣀⣉⣛⣉⣉⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣄⡀⢀⡾⠋⣐⢁⣁⠈⠻⣆⠀⣀⣤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢱⣯⣵⣎⢻⡟⡙⢾⠛⠉⡙⢳⠎⡙⣿⣋⣶⣭⣷⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣼⢿⣯⣨⣿⠋⠃⠚⣠⠘⢀⡘⠃⢉⢻⣯⣨⣿⢿⣴⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣿⡿⣼⣿⣿⣿⠏⢀⣘⠁⣤⡄⢉⣈⠈⢿⣻⣿⣾⡼⣻⣧⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡶⠛⢐⣉⣤⣿⠰⢈⠛⠉⠩⠍⠙⢛⡠⢸⣧⡈⣑⠚⠳⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣁⡟⠈⠋⣰⣾⠁⣨⠈⠀⠈⠀⠈⢨⡀⢹⣶⡌⠋⠘⣇⡹⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡴⠀⢴⠇⢌⢸⣿⠊⠔⠀⡤⠚⠢⡄⠐⠍⢺⣷⠈⠄⣷⠆⠢⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢫⣶⣤⡳⣠⢻⡇⠀⠠⡀⠇⢰⠀⠇⠠⠀⠀⣿⢳⡢⣧⣴⣊⠋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣉⡉⢉⣻⢻⡟⠏⢀⢋⡀⠊⠂⣈⢃⠈⠗⣿⢹⣋⠉⢉⣹⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠄⢾⡿⡅⣫⠹⣗⡶⠋⠘⠁⠀⠀⠉⠍⠳⣖⡟⢹⡇⠽⣿⠆⠬⣿
⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⣤⣀⠀⠀⢸⠩⠄⣈⠄⠘⠠⣴⡏⠀⢑⠆⠀⢀⠀⠀⢆⠁⠀⣷⡤⠘⠀⢄⡀⠬⢹⠀⠀⢀⣠⣤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⣰⠏⠀⣠⣀⠦⠈⢳⡄⣸⡀⣰⠧⢀⡃⠷⣼⠓⠂⠱⠃⢀⠘⠁⠀⠳⠍⠒⢿⡔⠇⣁⠀⢷⠀⣸⣂⣴⠋⠠⢀⡠⠀⠈⢳⡀
⠀⠀⠀⠀⢀⡏⢢⣤⡀⣣⡚⠃⣇⣼⡇⠀⠙⣦⢨⠏⡃⠻⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⠃⡈⢭⢠⡞⠀⠀⣿⣴⡃⠛⣢⡃⣠⡤⠊⣇
⠀⠀⠀⠀⠈⠉⠻⣄⢀⡈⢃⢀⠉⠷⡵⢄⡘⢫⠓⠋⠉⠀⠈⠈⠀⠀⡀⢰⠀⡀⠀⠀⠈⠀⠈⠉⠓⢋⠛⣐⠲⠽⠏⢁⢀⠋⣀⢄⡼⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢻⡼⡇⣼⡈⠮⡕⢰⣆⡈⠀⢠⣄⣀⠤⠬⠶⠠⡄⠔⠹⠑⠄⡤⠰⠮⠤⢤⣀⣤⠀⠈⣡⣶⠰⡭⡎⣸⡄⡵⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣖⠘⢃⠈⠀⢡⣃⠈⠰⡀⠀⠉⠁⣀⡀⡀⢱⡈⣠⠒⢢⡈⣰⠁⡀⢀⡲⠉⠁⠀⡰⠈⢀⡃⠁⠈⣈⠛⢐⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢾⣝⠀⠡⣀⠃⠄⠷⠀⠨⠉⢅⣘⠠⠆⠀⠁⢀⠀⠁⠀⠠⢑⡠⠍⠩⠀⠰⠇⠆⢃⡠⠁⠐⣽⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠂⠁⠰⠤⠟⠃⠐⠓⢄⣀⢀⡀⠃⠀⠂⠛⡀⠀⠀⡘⠃⠂⠀⠁⣀⢀⣀⠔⠛⠀⠛⠶⠴⠀⠁⠂⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠁⠀⠭⠰⡆⠀⡸⢃⠈⠁⢈⡻⠠⢇⣈⣀⠠⢀⠀⢈⣌⣀⠇⠸⡋⠀⠁⢀⠁⡀⢀⡄⠨⡥⠀⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠀⠆⠀⡋⠀⠉⡆⠹⠆⠘⣣⠈⢺⡟⠠⠀⠽⠄⠠⠘⣷⠊⢠⠙⠀⡾⠁⡎⠁⠈⡃⠀⠇⢈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⢿⠏⠀⡬⢁⡄⠁⢀⠁⠊⠈⢉⠚⠀⢠⠀⠐⢪⢉⠈⠢⢁⡄⠁⣄⠡⡄⠈⢿⠿⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠐⠂⣈⠐⢄⠉⠒⠿⡷⠀⠀⠀⢳⠈⢸⠀⢰⠂⠀⠀⠰⡿⠇⠊⢁⠔⢈⡀⠓⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⡆⠣⠘⢀⠀⠐⠂⠀⠆⠠⠄⠁⣸⡀⠁⠤⠀⠂⠀⠒⠀⢀⠘⠀⠃⣮⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠿⣀⢂⡀⢰⠗⠂⣀⢙⡛⢛⢁⡀⠚⢷⠀⣀⢎⢹⠃⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣂⠀⠸⠓⢦⠈⠋⠑⠈⠋⢠⠖⡻⠀⢀⣢⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠬⢤⣐⣤⣷⣧⣔⣣⠬⠴⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⣀⡀⠀⠀⣀⠀⠀⢀⠀⠀⡀⣀⠀⠀⡀⠀⣀⡀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢠⠛⠀⠀⠀⠈⠁⡉⠉⠉⢉⠉⠉⢂⠀⠉⠀⡀⠈⠉⠀⠈⠉⣉⠈⠉⠀⡈⠉⠁⡈⠉⠉⠉⠉⠉⡉⣉⠉⠉⡁⠈⠉⢉⠉⠉⢉⡀⣀⠈⡆
⠀⠀⣄⠀⢸⡖⠐⠚⠛⠒⠒⠛⠑⠛⠛⠛⠲⠛⠋⠛⠛⠛⠛⠛⠋⠒⠪⠖⠛⠛⠛⠋⠛⠉⠈⠈⠉⠉⠙⠋⠙⠛⠒⠛⠉⠓⠑⠊⣻⠇⢠⡇
⠀⠀⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⡄⠈⠇
⠀⠀⠧⡀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠠⠄⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢾⡀⠴⡆
⠀⠀⡇⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⡀⠀⠀⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⣬⠀⢀⡆
⠀⠀⡏⠀⢀⡇⠀⠀⠄⠀⠐⠂⠀⠀⠀⠀⠰⠀⠀⢀⠐⠈⠀⢀⢀⣀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⢀⠀⠀⠀⠀⠀⠀⠀⢣⠄⠘⡇
⠀⠀⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⣐⠉⠈⡅⠐⣷⠛⡳⠀⠀⠀⠀⠐⠀⣀⠚⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠐⡆
⠀⠀⡟⠁⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢁⠀⠀⠠⠀⠉⣼⢁⣙⠳⢛⠻⠟⠓⡄⠚⢒⣀⡐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡆
⠀⠀⡟⠂⢈⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠈⠀⠖⠺⠁⡌⠻⡘⠓⣹⢶⠔⠹⡆⠐⡄⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⡄
⠀⠀⣳⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠁⣤⠙⠓⠃⠀⠛⢾⣀⡴⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⠈⡇
⠀⠀⡿⠀⠲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⢀⡀⠐⢀⣖⢛⣜⡫⠤⠟⠀⢀⡀⢀⡠⠒⠁⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣾⠀⠠⠇
⠀⠀⣿⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠀⢮⣍⡈⠀⢠⠼⠋⣷⡆⠀⣨⠶⠛⠟⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠀⡀
⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠀⠈⢀⠀⠀⠀⠈⢣⠘⠀⠛⠰⠇⡎⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣻⡁⠀⡁
⠀⠀⣇⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠂⠀⠀⠀⡥⢸⠀⠀⠀⠀⠐⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡇
⠀⠀⣿⠄⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⠀⠀⠀⠀⠀⠘⡀⠈⢒⠂⣴⡔⣺⠃⠀⠀⠀⠀⠀⠀⠲⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣽⠃⠀⡀
⠀⠀⢣⡀⠈⣣⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⣠⡆⠀⠁⢠⢻⡷⡍⣩⢝⣀⠀⢠⡄⢷⣀⡀⢀⡏⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⡇
⠀⠀⢿⠁⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠒⣞⠓⠀⠀⢠⠏⠀⠀⢀⠙⢏⢿⡄⠢⠐⠈⡿⠃⠘⠷⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡁
⠀⠀⣼⠢⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠂⠀⠐⠀⠀⠹⡇⠀⠀⠀⠀⠀⣠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⡇
⠀⠀⢻⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣴⠷⢴⢶⣿⣦⢻⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣽⡀⠐⡇
⠀⠀⢿⠁⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⢠⣏⠀⠀⢺⣆⡍⢸⠀⠀⠀⠀⠀⠐⠶⠄⠀⠜⠁⠀⠀⠀⠀⠀⠀⣯⠀⢀⡇
⠀⠀⡜⠢⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠸⣬⠛⢦⠀⠘⢻⣶⢘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠀⠇
⠀⠀⢡⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡾⣄⢈⡇⠀⢸⢻⣴⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡇
⠀⠀⢇⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣵⠋⣹⠁⢠⡏⢸⣼⠀⢀⡴⣶⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢀⡇
⠀⠀⡾⠆⣀⣿⠀⠀⠀⠀⠀⠀⢀⣠⢄⠀⠀⡊⠀⠠⠒⠖⣩⠎⢰⠁⢠⠉⢡⡟⣏⡀⠌⠢⠀⢴⠄⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⡆
⠀⠀⣵⠀⠨⡇⠀⠀⠀⠀⠀⣠⡘⠃⠀⠂⡘⠋⢳⣐⡊⢠⣁⡴⢿⠀⣮⡆⠘⢇⣻⡗⣀⠀⢰⡼⢶⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠰⠉⠀⠐⡇
⠀⠀⣿⠀⠸⡇⠀⠀⠀⢀⡀⠉⠶⠀⡼⠖⣉⣤⠀⠌⢇⠘⠉⠀⠀⡆⣻⣿⡄⢀⣈⣁⣠⡴⠏⠀⢀⡁⣤⠦⢀⡀⡠⠖⠀⠀⠀⠀⡖⠀⢐⡇
⠀⠀⣿⠄⢠⡟⠀⠀⠀⠈⠀⠀⠄⠀⠀⠘⠃⣸⠀⡠⠈⢡⣈⠃⠀⡁⢬⢨⡗⠘⡙⠉⣈⠙⢦⠀⠀⢡⡀⠘⣀⠃⡀⠄⢀⠀⠀⠀⣿⠂⢨⡇
⠀⠀⡿⠀⢰⡇⠀⠀⠀⠀⠡⠄⠁⠰⢠⢀⡚⠁⠘⣰⡶⠈⠉⠉⠘⡡⠉⠀⠳⣞⠅⢀⡀⠀⢀⠆⠔⠆⣵⣆⠈⠙⠲⠖⠈⠀⠀⠀⣿⠀⢘⡆
⠀⠀⣧⠀⢹⡄⠀⠀⠀⠐⠠⠧⣴⣀⠂⠆⢈⣵⠛⠋⠜⠁⠀⠘⠃⣉⠉⢁⡄⢀⣦⠘⠣⣨⡄⢀⠛⢷⠎⠑⠂⠀⠠⠂⠦⠀⠀⠠⣧⠀⠸⡇
⠀⠀⡟⠃⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠐⠂⠀⠐⠓⠀⠐⠂⠊⠈⠈⠢⠼⠫⠼⠋⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⡇
⠀⢠⡷⠄⠈⣷⢀⣀⠀⣀⠀⠤⢄⣀⡀⢀⣀⠀⠀⢀⣀⠄⢀⡠⠤⠠⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⡀⡀⠀⣀⠀⡀⢀⣀⠀⠀⠀⠀⡗⠀⢠⡁
⠀⠘⠀⢀⡿⢿⡛⠉⠙⠉⠉⠙⠋⠋⠁⠉⠈⠉⠙⠛⠙⠉⠋⠀⠙⠛⠋⠉⠙⢏⠙⠙⠋⠉⠀⠋⠈⠁⠙⠉⠉⠈⠁⠀⠉⠃⠀⠉⢹⣧⠀⠠
⠀⠀⠀⠘⠃⠀⠙⠂⠄⠐⠒⠒⠒⠋⠐⠲⠒⠂⠤⠘⠂⠠⠄⠓⠀⠀⠙⠒⠂⠘⠓⠂⠐⠃⠀⠀⠐⠒⠂⠐⠒⠦⠘⠓⠒⠶⠛⠂⠀⠙⠛⠁

⠀⠀⡤⠒⠂⠤⠤⢤⡀⠀⠀⠀⠂⠐⠀⠀⠀⢀⣴⣦⡄⣠⣴⣤⠀⢠⣶⣤⣤⣤⣴⡆
⠀⠀⠈⠒⠦⠤⣤⠊⠀⠀⠀⠊⠂⡀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⡀⡀⢠⠆⢄⡀⠶
⠀⠀⠀⠀⠀⠀⠉⠻⠇⠀⠀⠃⠘⠃⠀⠀⠀⠸⠿⠿⠇⠻⠿⠿⠇⠸⡿⠛⠛⠛⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢰⡏⠀⠶⠀⠀⠀⠶⡶⢶⡶⢶⠶⠂⠀⠀⣰⣿⣷⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣸⣉⡇⠀⠀⠀⠀⠀⠐⠒⣶⣄
⠀⠀⠀⠘⠿⡶⠄⠀⠀⠀⠈⡏⠉⠉⢹⠁⠀⠀⠀⢸⣿⣿⠇⠀⠀⠠⢊⠉⣉⠉⡓⠆⠀⠀⠀⣿⣿⣇⠀⠀⠀⣸⣿⠀⠀⠸⣿⠃
⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠧⠀⠀⠼⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠈⠻⠧⣤⢀⠁
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⡀⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠁⡧⠃⢃⠁
⠀⠀⠀⠀⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠐⠁
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠋⠖⠀⠀⠀⢠⣾⣧⡀⠀⠀⠀⠀⠀⠐⠂⠄⡀⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⣤⣧⡄
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢰⠜⠢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠗⠥⠚⠳⠊⠰
⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⠘⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣖⢲⡒⢲⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣠⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠩⣁⡀⣄⠭⠂⠀⠀⢀⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇

⠀⠀⠀⠀⠠⠦
⠀⠀⠀⠀⠀⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣨⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⣀⠁⡀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡄⣤⠔⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⢀⠠⠀⢠⠀⢄⢁⠀⠄⡀⠀⠀⠈⠐⠒⠀⠀⠀⠚⠁⢀⢐
⠀⠀⠀⠀⠀⠀⠀⢳⠆⠀⠐⠠⠀⠀⠀⠀⠀⠁⠀⠀⠈⠆⠀⢀⠀⠀⠀⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠒⠆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⠒⡇⠚⢃⠹⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡏⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠈⡀⠠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣤⡀⠀⠈⠑⣀⠖⣙⢲⡠⣄⡔⢈⣽⡂⡀⠀⠀⣰⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⢀⠄⠁⠀⠉⠀⠀⠘⢿⣶⢶⣦⠄⠃⠈⢣⢷⡟⢠⠋⠀⠙⠡⢚⡾⠋⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠰⡼⠚⠣⠓⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⣈⡀⠒⠀⠁⠰⠆⠀⠀⠖⠢⠤⠡⠖
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡤⡤⠀⠁⠈⠀⠀⠀⠴⠀⠀⠤⠄⠀⠀⠀⠀⣀⢤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠁⠀⠀⢄⠀⠀⡠⢠⠀⠀⠀⠀⠐⠀⢀⠊⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠁⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣸⣲⢄⡤⣄⣥⣤⣤⠴⢄⠈⠁⠀⠠
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠉⠙⠛⠘⠙⠃⠙⠉⠳
⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠅⡀⠤⡠⣆⡤⠀⠀⠄⠀⠀⠀⠰
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠁⠀⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣠⠤⣼⢱⢹⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣼⠁⠈⣦⢤⡀
⠀⠀⠀⠀⣇⠖⣽⢲⢺⡇⢆⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⣆⣰⡏⠈⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⡇⠀⣿
⠀⠀⠀⠀⣧⣛⣼⢠⢺⣇⢣⡇⠀⠀⠀⠀⠀⠀⠀⢸⢳⢻⠈⠁⡟⡞⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⡇⠀⣿
⠀⢀⣤⠤⣯⡂⢻⠀⢸⡏⣊⡧⠤⣄⠀⠀⠀⣠⠤⠼⣼⢸⠀⠀⡇⣧⠧⠤⣀⠀⠀⢀⡤⠤⢿⡀⢸⠀⠀⡇⣀⡿⠤⢄⡀
⠀⠛⢦⡰⠄⢙⣿⠤⠼⣟⠁⠰⣠⠞⠓⠀⠘⠻⣏⠲⡌⣻⢴⡦⣟⢡⠖⣹⠟⠃⠀⠛⠦⡀⠀⣈⠿⠶⢶⠿⣁⠀⢠⠔⠛
⠀⠀⠀⢷⠰⣏⠰⣿⣤⠈⡳⢠⡇⠀⠀⠀⠀⠀⢸⡖⠻⣄⢿⡷⣠⠟⢺⡇⠀⠀⠀⠀⠀⢱⠊⠁⠒⡠⠚⣂⠈⠳⡏
⠀⠀⠀⠸⣴⠎⡓⠚⠒⡋⠴⡼⠀⠀⠀⠀⠀⠀⠀⢷⡒⠺⢍⡩⠗⢒⡾⠀⠀⠀⠀⠀⠀⠘⣆⠠⣈⣱⠋⣤⠇⣸⠁
⠀⠀⠀⠀⠈⠓⠧⣜⡤⠗⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢾⡷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡠⠴⠚⠁


⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣄⣼⠩⢹⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠋⠙⣆⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢠⠖⢲⡄⡀⡀
⠀⠀⠀⠀⡏⠖⢹⠀⢸⡇⠆⡇⠀⠀⠀⠀⠀⠀⠀⣾⣀⢸⠞⠛⡇⣀⣷⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⠀⠀⡏⠙⡆
⠀⠀⠀⠀⡇⠀⢸⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⢉⣸⠋⡤⣏⡉⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇
⠀⢀⣠⣤⣇⠀⢸⠀⢸⡇⢀⣧⣤⣀⠀⠀⠀⣀⣤⣿⡈⢸⠶⠶⡇⢁⣿⣤⣀⠀⠀⠀⢀⣠⣼⡀⢸⠀⠀⡇⢀⣧⣄⡀
⠀⠿⢤⠃⠈⠑⣾⣤⣬⡗⠉⠀⢒⠼⠗⠀⠺⢥⡘⠠⡝⣾⣤⣤⣷⢫⠄⢓⡬⠗⠀⠰⠧⣄⠀⠉⣺⡤⢤⣗⠉⠀⣠⠿⠆
⠀⠀⠀⣧⠠⣞⠱⣿⣤⢉⡦⢠⡏⠀⠀⠀⠀⠀⢻⠁⣞⠁⣾⣧⠈⣳⠈⡟⠀⠀⠀⠀⠀⢸⡆⢾⡁⢿⡧⢈⡗⢰⠇
⠀⠀⠀⠸⡄⠈⠓⢺⠒⠋⠀⣸⠁⠀⠀⠀⠀⠀⠘⣦⠎⣙⠺⠓⣋⠱⣴⠃⠀⠀⠀⠀⠀⠈⢧⡀⠈⠙⠋⠁⢀⡾
⠀⠀⠀⠀⠙⠢⢤⣈⣠⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⢭⣀⣀⡭⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⡴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣦⠀⠀⠀⠀⠀⠸⣶⣤⢁⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢀⣀⠘⡈⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⣀⣙⣛⠳⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣀⣴⠟⠉⣉⠙⠻⠷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠉⠀⢀⣴⠾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠶⣶⣶⣶⣶⣶⡶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠐⠷⠶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠛⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⣼⡏

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄⡀⠀⣷⣾⠿⣷⣾⡆⠀⢠⣶⢾⣶⡀⠀⠀⢀⣶⣷⣄⠀⠀⣾⢛⣷⣟⢻⡇⠀⣰⣾⡓⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡍⣬⡿⠇⠀⣿⡇⠀⢼⣿⡇⠀⣻⡟⡘⣿⡇⠀⢰⣿⡿⠸⣿⣷⠀⣿⡘⠁⠉⢸⡇⠀⢻⠃⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣛⣯⣶⣽⣶⡆⠀⣿⠀⣄⢀⣿⡆⠀⢸⡧⠁⢿⡇⠀⢸⡏⠁⠈⠻⣿⠀⣿⠀⠀⠀⠛⡇⠀⢸⠃⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣯⡁⠀⠀⣿⠀⠀⠈⣻⡆⠀⠾⠂⣤⠾⢇⠀⢸⣧⣄⣀⣠⣿⠀⣿⡀⠀⠀⢰⡇⠀⣼⠀⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⠛⢻⡇⠀⣟⣿⣿⡏⢽⡇⢰⣿⣿⣿⣿⣿⠀⢠⣤⣬⡿⢿⡿⠀⠀⢀⣾⣦⠀⠀⠀⠀⢰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⣧⡄⢠⡇⠀⣿⠿⠟⣀⢀⡁⢸⣿⠁⠀⠀⠹⠀⠸⢿⣧⡉⠀⣶⠀⣶⠟⠉⠉⢻⡆⠀⣶⣿⡟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⣽⣈⡇⠀⣇⡄⠀⠀⣈⠁⢸⡟⠀⠀⠀⢸⠀⢰⠞⣻⠇⠀⣿⠀⣿⣇⠀⢀⣸⡇⠀⣿⢻⡀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣯⣴⣼⡇⠀⣿⣆⣄⡎⢠⠀⠸⣟⣀⣠⣴⣿⠀⠀⠀⠉⠀⠀⠉⠀⠻⢿⣿⢟⡭⠄⠀⡿⠿⠇⠻⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⣀⣀⡀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠆⢉⣽⡇⠀⣿⢛⣭⣷⣄⠀⠀⣠⡿⠛⣷⣤⠀⢰⡾⣿⣿⢿⣮⠀⠀⣴⠛⢷⣤⡀⠀⠀⠹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠘⣿⡇⠀⣷⠁⠠⠀⣛⠁⢸⣿⡄⠀⣾⣿⠀⢸⠁⠙⠘⠀⢻⠀⠻⢿⡇⢹⡿⠇⠀⠀⢰⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⣿⡇⠀⡤⠀⠒⠐⢻⡇⢸⣿⡆⠀⢸⣿⠀⢸⡴⠋⠈⠃⣼⠀⣴⡾⠃⢸⣷⡆⠀⢀⢾⣿⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠐⠈⠿⠇⠀⠇⠼⠧⠀⠸⠇⠀⠠⠤⠤⠴⠦⠀⠘⠟⠶⠶⠾⠋⠀⠿⠥⠀⠀⠌⠇⠀⠀⠘⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⡄⠀⠀⢠⣄⣤⡤⠀⢀⣤⣤⣤⣤⣤⠀⠀⠤⣤⣤⡄⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠿⠿⣿⣇⠀⣶⣿⣯⡁⢤⡄⠘⠛⠁⠄⠛⠛⠀⠀⡴⣿⣶⣄⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣸⡏⠀⣿⣿⣿⡇⢸⡇⢀⢤⣴⡤⠤⣤⠀⣶⣿⡆⢼⣷⡆⠀⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣶⣾⣿⣯⠀⠛⠯⣭⣤⡾⠇⠀⠀⡀⠀⠤⠀⠀⢼⣿⣿⣼⠟⠁⠀⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠐⠶⠛⠛⠁⠀⠀⠀⠀⠀⠒⠐⠀⠒⠚⠻⠋⠀⠂⠀⠰⠿⣿⠿⠆

⣿⣿⡿⢿⣾⠿⣿⣿⣿⢾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣷⣶⣾⣶⣾⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⢀⠴⢤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⠀⠘⠤⡼⠁⠀⢴⣹⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠉⠉⠁⢀⣀⣀⣀⣀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁⠀⠀⣼⠟⠉⠉⠉⠙⢿⣷⣦⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣄⠀⠀⠀⠀⠀⠠⣾⣆⣄⣐⠲⣶⠀⠩⢿⡷⣄⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣋⡄⠀⡀⠀⠀⠀⣿⣿⣿⣱⣧⡉⠀⠀⠀⢻⣿⡀⠀⠤⠾⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠁⠈⠀⠀⡁⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠘⣿⡇⠀⠀⠘⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⢻⡟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡄⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⢾⣿⠟⠀⠀⠀⠀⠀⠀⠉⠓⠆⣀⣤⣤⣀⣀⠀⠀⠀⠘⢻⡟⠁⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠋⠀⠀⠀⠀⠘⠃⠀⠀⠀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣎⣿⣿⣦⡀⠀⠀⠀⣤⡄⠀⣀⣤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣴⣾⣿⣫⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡻⣿⣿⣦⡀⠀⠈⠀⢸⡇⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣷⡂⠀⠀⠀⠀⠐⠀⠂⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡧⠀⠸⣿⣿⣿⣿⣿⡿





⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⡿⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢀⡇⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢸⠁⠀⠀⣾⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⢀⡏⠀⠀⢰⠇⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡜⠀⠀⢀⡞⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡰⠁⠀⠀⡼⠁⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠐⠀⠀⢀⡜⠁⠀⢀⡞⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠠⡘⠆⢠⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⠎⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣤⡙⠆⠰⢫⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡐⠁⢀⣾⣿⡏
⠀⠀⠀⠀⠀⢀⡈⠳⢄⠠⠏⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⣸⢃⣴⠾⠿⢷⣶⣄⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠟⠀⠒⠂⠀⠈⠻⣷⣄⠠⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠈⢿⣦⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠹⣷⡈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢀⡶⢶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠈⠻⣌⣿⠀⠀⠀⠀⠀⠀⠀⢐⣿⠏⢀⡾⠁⠸⠁⠠⠞⠉⣩⡿⠛⠳⢶⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⡞⠀⠀⠀⠀⠀⠐⠋⢁⣠⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠏⠤⠔⠊⠀⠀⠀⠀⣀⡠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⠀⠀⠀⣀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡈⠰⠀⢰⣶⡂⠰⠀⡄⠀⠀⠀⠺⠃⣷⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃
⠀⠀⠈⠀⠀⠀⠿⡿⠏⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣾⢀⣆⣶⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡇
⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠆⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⡛⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⢠⠀⠛⠀⠀⠀⠀⠀⠀⠐⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⢀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠠⠄⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠿⠀⡀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠈⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠠⠀⡄
⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⡄⠠
⠀⠰⢠⣘⠻⣿⠏⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣍⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣠⣞⢳⣠⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠁⣶⣶⠏⠡⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣀⢫⣀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢙⡋⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠔⣉⠠⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠰⣄⡀⠀⠈⣛⠔⠀⢀⣠⠄
⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠉⠙⠓⠒⣶⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣦⠙⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠋⢀
⠁⣴⢠⠒⠈⠁⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠀⠀⠀⠀⠉⠘⡤⢢⠈
⠃⠉⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣶⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⣠⠎⠀⠐⠘⠀⡀⠀⠀⠀⠀⢠⡄⠀⢠⡀⠀⠀⠀⣀⠀⠘⣿⣿⡗⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⣧
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣦⡭⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⢀⠾⠒⠑⠀⠙⠳⣦⡀
⣿⣿⣿⣿⢟⣏⣴⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢐⣴⡟⠃⠀⠀⠀⠀⠀⠙⣏⡠
⣿⣿⣿⡿⣻⣽⢿⡿⣿⣿⣿⣿⣿⣃⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣬⠼⣠⠀⡀⠀⠀⠀⠀⢠⣿⢰
⣿⣿⣿⣿⡿⣿⣟⣿⣿⣿⣿⠟⣻⣿⣿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠁⡂⢘⣔⢀⡀⠀⢀⣤⠊⢜⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⠌⠘⠈⠀⡯⠥
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠥⢌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠀⠸⠀⠂⠐⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠠⢀⠀⠐⠀⠀⢠⢀⠀⠀⠀⠀⠁
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⡄⢸⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠁⠂⠑⠈
⠀⠀⠀⠀⠱⢵⣰⣄⣀⡀⢐⣁⣹⠀⢀⣀⣴⣰⠵⠅⠀⠀⠀⠀⠀⠀⠁⠘⠀⡔⠁⠀⠈⡆⠀⠀⢀⠄⢸⠀⠀⡄⡠⡀
⠀⠀⠐⠀⠀⡐⠥⡝⢹⠃⠆⢀⠀⠦⢻⠝⡥⠕⡀⠀⠐⠀⠀⠀⡀⠁⠀⠐⠎⡢⠀⠀⡰⠀⠀⠀⠸⠺⠂⠸⠇⠀⡀⠄
⠀⠀⠀⠀⡧⠀⠀⠈⠒⢻⠀⠼⡀⢸⠓⠊⠀⠀⠂⡅⠀⠀⠀⠘⢀⠀⠀⠀⣯⣆⢡⠎⠀⠀⣀⠀⣰⠺⠀⠀⠁⠀⡃⢄⣀⣀
⠀⠀⠀⠀⠈⢶⠮⢒⡤⠊⢂⠛⢃⠈⠣⡔⠢⢶⠎⠀⠀⠀⠀⠀⠀⠀⠰⠀⣉⡁⠰⠀⠀⠀⠡⡀⠉⠀⠀⢀⢀⠀⠑⠉⢀⠌
⠀⠀⠀⠀⠀⠀⠒⠊⠁⠀⠀⠐⠂⠀⠀⠉⠒⠊⠀⠀⠀⠀⠀⠈⢄⠘⠀⢠⠍⡡⠃⠀⠀⠀⠀⠱⡀⢠⠚⢳⡖⣛⠆⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠊⠀⠀⠀⠀⠀⠀⠀⠑⠀⠊⠉⠀⠁⢠⠊
⠀⠀⠀⠀⣠⠥⠺⢤⡐⣀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣴⠀⣀⠀⠀⠀⠀⠑⠂⠐⠊
⠀⠀⠀⣻⠁⣒⡥⠒⠛⢠⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⢀⠺⣤⠚⢵⣼⡆
⠀⠀⠀⠋⠍⠁⠈⠀⠐⢸⠀⠀⠀⠰⠄⠀⠀⣀⣀⡀⠀⠀⠶⠀⠀⠀⡄⠘⣜⠀⡜⣀⠅⡂⠀⠀⡐⠄⢡⠤⢤⣊⡠
⠀⠀⠀⠀⠚⠀⠀⠀⠃⡘⠀⠀⠀⠦⠀⡔⢉⠄⠂⢌⠑⡄⠠⠆⠀⠀⢃⠀⠈⠉⠉⠁⡐⠀⠀⠀⠀⠀⢠⣘⣣⠀⠃
⠀⠀⠀⠈⠢⠄⢉⠠⠊⠀⠀⠀⠀⠀⠚⠂⠒⠀⠀⠐⠂⠚⠒⠀⠀⠀⠀⠑⠤⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣒⣒⣒⣒⣒⣒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⣽⣿⣿⣿⣅⣿⣿⠟⣩⡝⢿⣿⣧⣽⣿⣿⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠰⠆⢹⣿⣿⣿⡀⠉⠁⣸⣿⣿⣿⠁⠶⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡦⢴⣟⣿⣿⣶⣻⠇⢿⣳⣾⣿⣟⣷⠤⣾⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣭⣬⣞⣿⣿⣿⣾⣷⣿⣿⣿⣿⣷⣯⣬⣵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢿⡟⠻⣿⣿⣿⣙⢻⣛⣻⣿⣿⣿⠛⣿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣏⣱⣎⣹⣿⣿⣿⣍⡵⣋⣽⣿⣿⣿⣙⣷⣉⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣍⣵⣏⣹⣿⣿⣿⣭⣧⣯⣽⣿⣿⣿⣩⣗⣍⣽⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢹⡅⣹⡋⣿⣿⡏⣻⡇⣽⡋⣿⣿⣏⣻⡁⣽⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⡜⢿⠟⣿⣿⡿⢿⡟⣿⠿⣿⣿⡟⢿⠟⣽⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡶⠾⢿⣿⣿⣷⣿⣦⣾⣷⣿⣿⣷⡿⠶⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡷⢯⡷⣶⣿⣿⣿⡦⢇⠧⣾⣿⣿⣿⠶⣯⠷⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡚⣫⣟⢺⣿⣿⣿⠵⢾⠷⢼⣿⣿⣿⣚⣯⡛⣺⣿⡇
⠠⣴⣶⠒⠒⠲⠶⢤⣀⣿⣿⣻⠿⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠿⢿⣛⣿⣇⣤⠴⠖⠒⠒⢲⣶⡄
⠀⠘⣮⢷⡐⠿⠂⣠⢿⣿⣿⠿⢿⣿⣿⡍⢹⣻⣿⣿⣿⣾⣫⠉⣿⣿⣿⠟⢿⣿⣿⢧⡀⠺⠗⣰⢯⡞
⠀⠀⠘⣧⢿⣶⣿⢵⣫⢏⣿⣦⣾⣿⣿⡷⢾⣿⣿⢣⢻⣿⣿⠶⣿⣿⣿⣦⣾⣏⢯⣳⢽⣷⣾⢧⡟
⠀⠀⠀⠸⡞⣿⣛⣭⣶⡻⡟⣿⡇⠎⠻⣦⣼⣿⠇⢀⠀⢿⣿⣤⡾⠋⠄⣿⡟⡿⣳⣮⣝⣻⡟⡾
⠀⠀⠀⠀⡇⢿⣿⠿⣷⣿⣿⣿⡇⠀⠀⠌⢻⡟⠘⢠⠘⠘⣿⠋⠄⠀⠀⣿⣿⣿⣵⡿⢿⣿⠇⡇
⠀⠀⠀⠀⡇⢸⢿⡌⠂⠉⠛⠿⣧⠀⡘⡔⢦⡇⠘⣸⡘⠀⣧⠒⡜⡀⢠⡿⠟⠋⠁⠊⣼⢿⠠⡇
⠀⠀⠀⠀⡇⢸⠸⣿⡀⠔⢭⢠⢸⡄⢀⡺⡜⡇⠚⢸⠘⠂⡞⡼⣀⠀⡼⢠⢨⡕⠄⣸⡿⢸⠀⡇
⠀⠀⠀⠀⡇⢸⠀⡟⢷⡀⠂⡱⢆⠿⣄⣡⣵⢿⣶⣿⣦⣾⣿⣥⣉⡼⢇⡶⠁⠂⣰⠟⡇⢸⠀⡇
⠀⠀⠀⠀⡇⢸⡀⢱⡈⢿⣄⡘⠈⣷⣿⠟⣻⢋⢔⡤⣔⢍⢻⡛⢿⣶⡏⠘⢂⣼⠏⣰⠁⣸⠀⡇
⠀⠀⠀⠀⣿⢸⡇⠀⢧⡀⠙⢿⣿⣽⢇⠀⣿⢨⢿⣤⣼⠯⢸⡇⢀⣽⣽⣿⠟⠁⣠⠏⠀⣿⢸⡇
⠀⠀⠀⠀⢹⡄⣿⣶⣾⣷⣄⠀⠉⠻⣿⣷⣾⢦⣑⣙⣃⣡⢾⣶⣿⡿⠋⠁⢀⣴⣷⣶⣾⡇⣼⠁
⠀⠀⠀⠀⠘⣷⠘⣿⣇⣼⡿⢿⣶⣿⣵⣯⣿⡿⣾⡶⣿⣿⣿⣯⣷⣽⣷⣾⡿⣿⣄⣿⡟⢰⡟
⠀⠀⠀⠀⠀⠹⣷⡌⠻⣿⣿⣦⣄⡀⢸⠁⣹⠛⢻⠟⢿⠛⢻⠁⢹⠀⣀⣤⣾⣿⡿⢋⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢿⡶⣌⠙⠻⠿⣿⣿⣿⣿⣶⣾⣤⣼⣶⣾⣿⣿⣿⡿⠿⠛⢉⡴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣝⡲⢦⣤⣍⣉⣉⢙⠉⡋⡋⢉⢉⣉⣉⣥⣤⠶⣚⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠮⢭⣭⣍⣛⣛⣛⣋⣭⣭⠭⠶⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡏⢹⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢻⡇⢸⣿⠀⣿⡶⣶⡄⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⣿⠀⣿⠀⢸⡇⠀⠀⢀⣤⣿⡏⢹⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⣿⠀⣿⠀⢸⡇⠀⠀⢺⣯⣭⡀⢀⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⣿⠀⠘⠁⠀⠁⠀⠉⠀⢸⡇⠀⠀⠀⠉⣿⡇⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠉⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣤⣿⠷⠾⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠉⣁⣠⣤⣤⣄⡉⠙⠛⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢰⠶⠟⠛⠉⣁⣈⠉⠛⠻⠶⣦⣤⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⡶⠶⠶⠾⠟⢁⡄⠀⠀⠀⠀⢾⣏⣻⡇⠀⠀⠀⠀⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⢿⡇⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⢸⡷⢶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠈⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣶⠶⠟⠛⢻⡇⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⢼⡟⠛⠻⠶⣦⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣶⠶⢿⡇⠀⠀⢠⣿⠋⠁⠈⠙⣷⠀⠀⠀⢸⡿⠶⣦⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⣸⡇⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣸⣇⣀⣀⣸⣧⣀⣀⣀⣀⣿⣅⣀⣀⣼⣇⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡷⢿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣿⣻⣿⣿⣿⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣦⠀⠀⢀⡴⣾⣿⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⡗⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⢠⣿⣿⡀⠀⠀⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠀⣹⣿⣿⠿⠿⠿⢧⡈⢻⡟⢻⣷⠀⠀⢰⣯⢽⡇⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⢨⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠀⣼⣿⣡⣶⣾⣟⠛⠲⠌⢧⠈⣿⡄⠀⣸⣿⣾⣷⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⣿⣀⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣷⠟⠛⣙⣿⣟⠛⠼⡇⢧⣠⣿⣿⣿⠟⠀⠀⠀⣾
⠀⠀⠀⠀⡇⠀⠀⠀⠹⣿⣿⣿⣿⣿⡟⠉⢰⡋⣹⡆⠙⣿⣿⡟⠀⢻⣿⣿⠁⢀⣾⡟⠉⢻⣿⠲⠀⣾⣿⣿⣿⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⣬⣽⣯⣤⣴⣿⣿⡇⠀⢸⡿⣿⡀⠈⢿⣷⣤⣾⣿⣧⣸⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡇⠀⠸⣇⠈⢻⣦⣄⣉⣻⣿⣿⣿⣿⣻⣿⠟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⠀⠀⠀⣿⢦⣀⢻⣆⠙⣿⡟⢹⣯⢿⣾⠏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢿⣿⣛⠛⠿⣧⣿⠀⢘⣷⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣷⣶⡆⠀⠀⠀⠈⣿⣿⠻⠷⣶⣾⢇⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣽⣷⣶⣤⣬⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢻⣯⡇⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠈⡇⢠⠀⠀⠀⠀⠀⠀⠀⠄⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⣸⣠⠤⠒⡊⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼⠀⡤⠤⠙⠁⠀⢘⠀⠀⠀⠀⠀⢀⣰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⢶⢛⠉⡇⠀⠐⠀⠀⠀⡸⣐⣠⣤⣴⣶⣿⣿⣿⣿⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠯⣬⡅⣿⠈⢀⣅⣀⠀⠀⠀⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣠⣾⣾⣿⣿⣿⣿⣿⠀⠀⠀⣸⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠛⠋⠉⠁⠀⣿⠀⢀⣀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣽⣋⣠⣄⣉⠘⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣍⡿⣿⣿⡖⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣅⣨⡑⡌⣿⣿⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣏⣉⣝⡃⣿⣿⣇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⣿⡟⢱⡇⢻⣿⣶⡄⠀⠘⠢⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡀⠘⡃⢸⣿⢿⡏⠀⠀⠀⣿⡎⡓⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⢸⣿⣄⡙⢦⣀⣠⣿⣿⠀⠀⠀⠀⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢸⠉⠛⢻⣦⣘⣿⢸⣿⠸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⢸⣿⣿⡏⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⢿⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⠏⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢼⣿⣿⣿⣿⣇⡀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣧⠀⠀⠀⠀⠘⠿⣿⣿⠿⣿⠟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣻⢸⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡂⠀⠀⠀⠀⠀⠐⠒⠛⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⡏⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⢳⠃
⠀⠀⠀⠀⠀⠀⢂⠢⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡱⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠱⡑⢜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⢞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⢖⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢭⡢⢝⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣫⢕⡪⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⢍⣒⡩⠭⢟⣛⣛⣛⣻⣿⠯⠭⣓⣪⠭⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠚⠉⠉

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

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

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⡁⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒⠂⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡠⠠⡄⠘⠀⢀⡀⠀⠀⠀⠀⠀⠐⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠜⠀⡀⣂⡄⢛⣃⡸⠆⠀⢀⠀⠈⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢈⠐⡀⠀⠚⣠⠤⡙⢩⡈⠏⠈⡀⠀⢉⠄⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⢄⡙⡄⠁⣸⠓⠇⠈⡃⠈⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠇⣁⢀⣉⠛⢀⡴⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢠⡄⠀⡀⡬⢉⣌⡡⠒⠊⢀⡠⢄⢀⡠⠂⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠡⠀⢺⡥⣉⠀⢠⠊⠉⠱⡆⠀⣠⠖⠎⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡀⠀⠀⢀⡈⢃⠈⠁⠉⠀⠃⠘⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠂⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣤⡄⢼⠃⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡠⠀⠀⠀⢠⠐⠇⡌⠅⠀⣀⠀⠀⠀⠰⡄⠀⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠰⣌⠇⠀⠀⠀⡃⠀⠀⠁⠈⠀⢻⡄⠀⠀⠀⠙⠇⠀⠻⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠂⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣰⠶⣤⢦⡾⣤⠸⡄⠀⠀⠀⠀⠘⠁⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⢀⣍⠀⠀⠸⡇⢨⠠⡅⠀⡀⠀⠀⠀⢦⣄⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣈⠟⣤⠀⠀⢿⣴⠀⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⡀⠘⠧⠀⢸⢿⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⢀⣣⡟⢸⠇⠀⡼⢈⣹⡇⠀⣠⣴⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠞⠀⣀⡤⡤⢚⡉⢠⠏⢀⠞⠃⣼⡿⡃⡀⠃⡀⢠⡅⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠰⠀⠐⠗⠄⠀⡀⠀⠋⣀⣼⠀⠘⡄⠀⣧⢻⡣⠀⠀⠀⠀⢦⠐⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢋⡠⠐⢢⡔⠓⡀⠈⢑⡆⠘⠚⠁⠈⠄⢾⡳⠀⠈⠚⠓⠒⢀⡰⠓⠈⠣⠀⣤⠀⠄⠀⢠⠄
⠀⠀⠀⠀⠀⠖⠀⢀⠀⠈⠁⠰⡉⢸⠀⢀⠐⠀⠲⠤⠀⠃⢘⠹⣷⠐⠶⠖⠛⠻⣄⠀⠈⠑⠈⠁⠄⡈⠉⠁
⠀⠀⠀⠀⠀⠀⣂⠀⠂⢀⠀⠀⡤⠋⢠⢉⣤⠽⠗⡶⠲⢎⠸⠀⢇⣰⠖⠀⠀⠀⠄⡆⢠⣀⠓⡀⠀⠰⠦⢀⡄⠂
⠀⠀⠀⠀⠀⢄⢩⡄⠀⠀⠋⡀⠰⠦⢤⠞⣫⠄⠀⠀⠄⠢⠤⠄⠀⠙⡁⠠⣆⠀⢂⠙⠁⣠⡠⠛⣁⠀⠀⢈⢀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠃⠶⠀⠠⠴⠋⢀⠀⢁⡁⠀⢀⡀⡑⠀⠐⠃⠀⡎⠃⡴⠑⠋⢀⠢⠈⡁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡛⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄⣀⣠⣴⣶⣿⣿⠿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⣀⣤⣶⣿⡿⢿⣿⣿⠁
⢠⣴⣿⡿⠟⠋⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⢿⡟⠁⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢀⣾⡿⠉⣷⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣾⣿⠁⠀⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣠⣶⣿⡷⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠰⠶⠶⣿⣿⣤⣤⠀⠀⣸⣿⡇⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠇⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⣿⣿⠇⣴⠟
⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠉⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⣿⣿⡿⠃
⠀⠀⣴⣾⣿⠿⠋⢹⣿⡇⠀⠀⢀⣴⣿⣿⠿⠿⢿⡆⠀⠀⣼⣿⠃⠀⠀⠀⣿⣿⠇⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠉⠋⠁⠀⠀⣿⣿⡇⠀⠀⣾⡿⠋⠀⠀⢠⡆⠀⠀⢸⣿⡟⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠸⣿⣿⠋⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢰⣿⠁⠀⠀⣰⣿⠃⠀⢀⣾⣿⡇⠀⠀⠀⣸⣿⡇⠀⠀⠀⢀⣼⣿⣿⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢿⣦⣴⣾⢿⣧⣠⣴⠏⠹⣿⣧⣤⣴⠞⠉⢻⣷⡀⣀⣠⠞⠁⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠙⠉⠀⠈⠛⠋⠁⠀⠀⠉⠋⠉⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠈⠛⠃⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡿
⣿⣿⠛⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⢛⡛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⠶⢶⣤⣾⠁⣔⠂⠇⠀⠵⠆⢳⣄⣤⣶⣶⣶⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣡⠶⢻⡿⣶⢹⡇⠈⣼⠀⠀⠂⠀⠄⢸⡟⣿⡃⠀⠀⢈⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡞⠀⠀⠀⠀⢸⣟⡷⠋⠁⠢⡀⠀⢀⠄⢹⣷⡿⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠁⠀⠀⠀⣤⣞⡇⣿⠀⡀⠀⠀⠄⠀⢀⢸⣿⡵⠀⠲⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡇⠀⠀⢠⠋⠀⢸⡇⣿⢄⣄⠀⠀⠀⢀⡦⣼⣿⠆⠀⣸⠈⠀⢄⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠄⠬⠙⡆⠀⠀⡿⣿⠀⠀⠐⠀⠀⠀⢀⣿⣿⠀⢠⡅⠞⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠈⠚⡦⠐⣶⣿⢀⡄⠀⠐⠂⠀⡈⢿⣿⠟⢟⠀⠀⢀⠀⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠐⡤⣄⢀⡂⠀⣽⣿⠋⠐⢀⠀⠀⠀⠀⢻⣿⡀⠸⠇⠲⢤⡜⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠈⠁⠀⠈⣡⡀⣾⣿⡀⠀⠸⠒⠰⢶⣤⣽⣿⡿⢿⣤⡤⠄⢘⣹⣿
⠀⠀⢀⣠⡶⠟⠛⣛⣛⣛⣛⠶⠶⣦⣄⣸⣿⣿⠀⠀⣤⠚⠁⠀⣿⣿⠛⠒⠪⠀⠀⠀⠈⣿⣿⡇⠀⢨⣉⡉⢅⣸⣿⠀⢀⣤⣴⣶⣶⠶⠶⠶⠶⢦⣤⣀
⠀⣴⢟⡥⠒⠀⠀⠀⠀⠀⠈⣍⠁⠒⢬⣹⣇⡼⠀⣈⢤⠀⠀⠀⢿⣿⡖⠉⠀⠀⢀⣀⣠⣿⣿⣇⠀⠀⠀⠈⢦⣽⣿⡿⣫⠽⠟⣯⢭⠟⠛⡉⠉⡒⢦⣙⢷
⣼⠁⣏⡀⠀⠘⠀⣄⠀⠀⢀⠀⡁⠀⠀⢿⣿⣠⠞⠀⡀⢀⠀⠋⣺⣿⠀⠀⠀⠐⠶⠶⠦⣼⣿⣯⣴⣤⣄⠀⢸⣿⣿⡸⠁⣸⣶⣁⠉⢠⡏⣀⠷⠻⣠⡼⢂
⠉⠛⠾⣧⣅⡠⡀⠘⠀⡼⡶⠀⠉⣴⠄⣻⣿⣿⠀⡺⣭⠉⣷⠀⢸⣭⣷⡤⠄⠀⠀⠀⠀⢿⡼⢿⠉⡅⢣⣣⠞⣿⣿⡇⡾⢃⠟⡸⡆⢰⡀⢹⣦⣾⡷⠚⠋
⠀⠀⠀⠈⠙⢿⡜⣄⢀⣅⠀⢠⡔⠰⣦⣽⣿⡟⢦⣀⣀⣴⠟⢠⡀⣉⢀⡓⣠⣖⣠⣆⡠⠼⣁⠈⠓⠞⠛⠁⠀⡏⣿⣷⠄⣾⢶⠇⣇⠀⢸⣯⡿⠁
⠀⠀⠀⠀⠀⠈⢿⡜⣿⠙⠀⣯⠐⣆⠃⣿⢻⠋⣀⠸⢉⡠⠖⡞⢿⡉⠠⠀⡛⡇⠡⠈⣿⠊⣽⣅⠀⢀⠀⠀⠀⡟⢻⢏⣼⠃⢸⡂⢹⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠸⣧⢻⣶⠀⣷⠀⣿⠸⣿⠯⢿⣭⠽⣢⠄⠀⠃⠀⠹⣆⡐⠁⠘⡄⠀⠀⠀⡟⢀⣉⣬⢦⣌⠅⢿⣿⡆⢿⠀⠀⡇⢼⣏⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣸⣇⣔⠙⠛⡋⣤⡧⡴⠀⠻⡄⠀⠑⢼⡒⠄⠀⡌⠳⣤⡟⠀⡆⠀⢠⣷⠔⠉⠀⣸⡟⣆⠀⡡⣏⠘⠦⠔⠀⡸⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣏⣀⠁⣀⣨⠽⠁⠁⠀⢻⡄⠀⠀⢉⢢⠁⡀⢀⠛⠀⠀⣶⡔⣛⠁⠀⠀⣰⠏⠐⠞⠷⣹⡿⠳⠶⠞⢙⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣾⠘⢧⣀⣝⠟⠀⠂⠀⢠⠂⠉⠦⡀⠀⢎⣱⢡⢸⡀⢀⣠⣿⡼⠃⢀⣠⡞⠃⠀⠀⠈⠁⢾⣷⠷⠖⢿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⡠⢩⠇⠐⠂⠀⡀⠀⢀⠀⠀⢩⡔⠚⠛⣋⠙⠙⠛⠛⠛⠴⣞⠋⠈⠀⣠⠄⠐⠋⠁⠀⠸⠓⡆⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠃⣮⢠⡖⢤⢀⡐⠀⠀⡹⠛⡡⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⢶⡋⠁⠀⠔⢂⣀⠠⠐⡃⠙⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣦⡄⠃⠀⣁⡤⠀⠁⣀⠆⠠⢊⢠⡤⢲⡶⢸⣿⣿⣿⣶⣶⠤⢄⡈⠂⠹⣤⠀⠈⠉⠁⠀⣈⡧⢈⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣇⠘⡗⠛⠨⣤⣄⣀⣩⡰⣟⠁⠀⣆⠘⣷⣼⣿⣿⣿⣿⡿⠀⠈⠀⢹⡇⢸⠃⠀⢈⠴⢄⡁⣶⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡤⡤⢁⠀⡀⢀⡀⢀⠉⣮⡙⠷⣌⠢⣈⠻⠿⣿⡿⠟⡁⠀⡴⠞⣩⣄⢹⡄⠈⠳⢄⠀⢀⡟⡙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣸⡨⠛⠁⢡⠖⠁⢐⣡⠴⠓⠂⢁⣈⠙⠶⠶⠶⠊⢀⡀⡀⠚⠷⣌⣋⠢⢁⡖⠺⠆⢘⢠⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠃⢿⡬⠄⠀⠷⢶⠟⠁⠀⠀⠰⣠⠏⠀⠈⠀⢉⠀⠀⡇⠈⠆⠀⠈⠙⢷⡼⠃⡀⢀⠃⡶⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡯⠞⠂⠹⣤⣠⡾⠃⠀⠀⠠⠄⠀⢹⡆⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⢠⠀⢳⣄⣠⡈⣾⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡽⡏⢱⣸⣿⣷⣤⠂⠤⠄⠀⠤⡬⢷⡄⠀⠀⠀⠀⣰⠷⢶⠦⠤⠴⠤⠶⣶⣿⡧⢸⢯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣍⢆⠙⠉⠉⢂⣄⠠⠀⠀⠀⠀⠢⡱⣄⠀⢢⣔⠟⠀⠄⠀⠀⠀⠰⢎⠁⠀⣠⣣⡟⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡱⣄⠀⠘⢈⠐⠂⠄⣁⠰⠆⠘⠻⡿⠿⠊⢠⠘⣠⠄⠊⠁⠄⠈⣠⢞⣵⠟
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣕⠢⣄⠀⠑⠤⠈⠁⠛⢶⡀⠀⣀⡰⠋⡁⠀⠀⠁⣀⡤⣞⣷⠟⠁
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣭⣓⡢⠴⠅⣀⣀⠈⣁⣁⣀⣀⣤⠤⢒⣫⣵⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢷⣶⣶⣦⣬⣶⡶⠶⠾⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢒⠲⣄
⠀⠀⠀⠀⠀⢠⢞⣛⡃⣿⢹⡆⣛⡓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⠏⣶⣶⠹⠿⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢚⣛⢱⣿⣷⢘⣛⠳⡄
⠀⠀⠀⠀⠀⣿⠸⣭⠇⣬⢻⡄⣯⡽⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⡆⠻⠟⢰⣿⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⢸⣿⣿⢸⣿⡇⣿
⠀⠀⠀⠀⠀⣿⠸⣨⠇⢹⣻⠀⢇⡷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⢛⡄⣷⣾⢠⡛⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⢸⣿⣿⢸⣿⡇⣿
⠀⠀⠀⣠⠤⠿⢨⣟⡅⣿⣾⡇⣝⣧⠸⠦⢤⡀⠀⠀⠀⢀⣤⡶⠿⢸⣿⡇⣿⣿⢸⣿⡇⠿⢶⣦⡀⠀⠀⠀⢀⣤⠤⠇⣿⣿⢸⣿⣿⢸⣿⡇⠿⠤⣄
⠀⢠⡞⠱⣷⡾⣶⣍⠃⣿⣿⡇⢋⢴⡟⣾⠗⠙⣦⠀⣰⣟⠁⢺⣛⢷⣌⠃⣿⣿⠘⣡⡾⣛⡗⠈⣻⣆⠀⣴⠋⠶⣿⣷⣮⡙⠸⠿⠿⠘⣫⣶⣿⣷⠎⢳⣄
⠀⠈⠉⠙⣆⢳⣿⡿⠛⣩⡭⠙⠿⣿⣶⢣⡞⠉⠉⠀⠉⠉⠛⣧⠹⢧⡿⠋⣵⡎⠙⢿⡼⠏⣴⠟⠉⠉⠀⠉⠉⢳⡜⠿⣋⣵⠶⣶⣒⢶⣭⡻⠟⣱⠋⠉⠁
⠀⠀⠀⠀⢸⡘⣿⣧⣀⠻⠿⢃⣤⣿⡏⣼⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿⣿⣦⣝⣫⣴⣿⣿⢠⡟⠀⠀⠀⠀⠀⠀⠀⣧⢸⣿⣷⡟⢸⠗⣋⢻⣿⢀⡇
⠀⠀⠀⠀⠈⢧⡩⣼⠿⣿⣿⡟⢿⡼⣡⠇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡘⠷⣮⡙⢫⣵⠞⢂⡾⠃⠀⠀⠀⠀⠀⠀⠀⠸⣌⢿⣝⠻⣃⣮⣛⣴⠟⡼⠁
⠀⠀⠀⠀⠀⠀⠙⠦⢍⡛⢞⣋⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣥⣈⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⣙⠻⢛⣩⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⣤⡞⣵⢲⡙⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⡏⣴⣦⠹⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡟⣫⡛⣿⣤⣄
⠀⠀⠀⠀⠀⣼⢱⠶⡆⣿⣿⡇⡖⣦⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⣷⡄⣯⣙⢠⣾⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢫⣭⠀⣿⣿⢨⣭⠻⡷
⠀⠀⠀⠀⠀⣿⢸⣿⡇⣿⣿⡇⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢨⣽⡁⣋⣿⢈⣯⡅⣽⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⣿⣿⢸⣿⠀⡿
⠀⠀⠀⠀⠀⣿⢸⣿⡇⣿⣿⡇⣿⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⣩⡁⣿⣴⢈⣭⡅⢻⡀⠀⠀⠀⠀⠀⠀⢀⣸⣿⢸⣿⠀⣿⣿⢸⣿⠀⣷⣀⡀
⠀⠀⡴⢚⣭⣭⣘⠻⡇⣿⣿⡇⡿⢛⣬⡭⣝⢶⡀⠀⢀⡴⢋⡭⢥⣘⢾⡇⣥⣽⢸⡿⣃⡬⢭⣙⢦⡀⠀⢠⣾⢛⣩⣭⣘⢿⠄⣿⣿⢸⢟⣠⣭⣝⠻⣦⡀
⠀⠚⠣⢬⡱⣿⣿⣷⣤⠿⠿⢧⣴⣿⣿⡖⡩⠄⠛⠀⠛⠒⢬⡳⠿⣏⣷⡴⢟⡻⢤⣾⣹⠿⢞⡥⠒⠓⠀⠿⠷⣬⡙⣿⣿⣷⠴⣛⠻⢴⣿⣿⡟⣩⡶⠿⠷
⠀⠀⠀⠀⢻⢸⣿⣏⠀⣿⣯⡄⢈⣿⡿⢸⠁⠀⠀⠀⠀⠀⠀⢷⢰⣿⡁⠰⣿⣧⠄⢈⣿⡆⡾⠀⠀⠀⠀⠀⠀⠘⣷⢸⣿⣇⡘⢿⠦⢀⣽⣿⢀⡏
⠀⠀⠀⠀⠸⡇⣿⣿⣷⣮⣭⣶⣿⣿⠇⡟⠀⠀⠀⠀⠀⠀⠀⠸⡌⣏⣿⢷⣮⣵⡾⣿⣹⢣⠇⠀⠀⠀⠀⠀⠀⠀⢻⡆⢿⣿⣿⣷⣿⣿⣿⠏⣼⠁
⠀⠀⠀⠀⠀⠙⢮⡛⠿⣿⣾⡿⠟⣫⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⡓⠶⣿⣿⠶⢚⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣉⠛⠿⠟⢋⣡⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣿⣆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⡿⡿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠚⡿⢰⡤⣦⠸⡓⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠈⠀⠀⠀⣀⣀⡐⢾⠦⢴⣅⠀⣄⠙⢿⠰⣸⠞⢁⡀⢠⣷⠴⣿⠖⣀⣀⠀⠤⠀⠁⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠣⢠⡤⠐⠚⠐⠊⣕⣤⣔⡀⠈⠁⠻⡄⢻⡷⣹⠃⡼⠋⡁⠀⢲⣤⣔⡍⠂⠐⠒⢠⣤⠼⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⠂⠶⠄⠀⠈⠑⠂⠛⣮⠿⢷⣾⣿⠈⣹⣿⣶⠿⣭⠟⠂⠒⠉⠀⠠⠶⠂⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠨⠖⣺⡿⣿⡄⣿⢿⣯⡒⢄⠀⠀⠀⠀⡀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⡘⢀⣴⠂⠀⠀⠀⠀⡽⢠⢹⣷⡿⠥⠹⡄⠀⠀⠀⠀⣾⣀⠘⣤⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢈⡻⠁⠁⠄⠀⠀⠀⠁⠀⡾⢻⢻⠆⠀⠃⠀⠀⢀⠌⠁⠹⡇⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣧⣠⠤⢀⠀⡀⠘⡇⠀⡀⠀⠀⠙⣠⠘⠀⠁⠀⡀⠨⠃⠀⡀⢠⠠⣤⣠⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡻⡧⠘⣾⡣⠒⠄⣀⠐⠀⠉⢲⣀⢤⡭⡥⢆⣴⠉⠁⠰⠀⠀⠕⢦⡻⠊⠰⣿⣙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⣗⠡⠈⠈⣡⠀⣀⠂⠐⠀⠀⠀⢠⡎⢻⠛⡄⠀⠀⢀⠐⠐⢃⠀⢀⡉⠈⡰⢱⡷⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⡈⢹⣶⠳⠄⠨⢦⠙⣇⣰⢠⠌⡑⢳⢤⠟⢦⡔⢀⡀⣤⢰⣀⡎⣴⢮⠠⠞⣳⡟⢁⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠿⠶⣌⡓⢼⣦⡀⠀⢿⡻⣾⠓⠙⠋⠀⠀⠀⠈⠻⠗⢻⡾⣻⠆⢀⣀⣦⠘⣣⡴⠿⠧⠞
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠈⠩⡈⢁⠙⠳⣌⣛⠻⠆⢃⡤⣾⣋⣲⠦⣀⠋⠟⢙⣡⠴⠛⠁⠉⡄⠁⠐⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠂⠀⠌⢀⡀⠠⢙⣧⠴⣿⣷⣾⣷⣿⣾⣿⡶⣴⣟⠁⠀⠀⠀⠀⠐⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠐⠀⠚⠓⠒⠀⣀⣀⣦⣶⡿⠞⠛⢻⣿⣿⡀⣿⣿⣿⠛⠓⢿⣷⣴⣀⣀⣀⡒⠘⠓⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⠾⠛⠛⠛⠛⠛⠛⡻⠛⢷⣄⡀⠀⢍⢻⣿⣽⢻⠀⢀⣠⣴⠟⠛⠛⠛⠛⠛⠛⠛⠻⣶⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠉⠁⠀⠐⠂⠀⠂⡸⠁⠈⠅⣸⡿⢯⣴⡿⣿⡻⣦⣽⢿⣯⡠⠍⠀⢳⡐⠀⠀⠂⠀⠉⠉⠙⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡀⡤⠀⠠⠘⠛⠆⠛⢁⠜⣡⣾⠋⣀⡞⠳⠾⣿⡷⠶⢹⣎⠈⢻⣦⡁⢀⠙⠢⠚⠃⠠⠀⠠⡄⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠄⣠⠂⢀⠀⢰⡟⠮⠀⠸⠁⠀⠞⠛⢛⠂⠊⠙⠀⢹⠞⣷⠘⣤⠀⢦⡀⠤⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⣿⡄⢃⣠⠀⠀⠀⠀⣲⡁⠀⠀⠀⢠⣜⠃⣿⡀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢷⣤⣭⠭⢤⠀⠆⣿⠃⠀⣤⠼⢭⣥⣾⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠾⠟⠛⠛⠀⠀⠀⠁⠀⠘⠛⠛⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⣿⣿⢟⠛⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣞⣩⣷⣶⣦⣴⣶⣶⣦⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠦⢤⣀⣀
⠀⠀⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣠⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀⠰⠚⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠖⠒⠛⠉⠁⠀⠀⢰⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣡⡄⢿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣌⢿⡆⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣎⢷⠀⢸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢸⣿⣎⢷⡶⠿⠟⣛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣫⣶⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣫⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣢⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠄⠂⠀⠀⣐⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣗⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣠⢠⡀⠀⠀⣐⠃⠈⠀⢀⠐⠀⠖⢀⣀⣀
⠀⠀⠀⠀⠀⢀⡞⢱⣜⡉⠴⠈⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⠀⠀⠀⢀⠀⠉⡿⢲⣤⡞⣠⣶⢲⣶⠦⠥⠒⠛⠉
⠀⠀⠀⢀⠠⠾⠶⠯⠤⠤⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠸⠋⠀⠈⠉⡀⢄⠀⠀⢀⣴⡌⡟⠉⠉⡉⠛⠟⢛
⠀⠀⠀⠀⠀⠀⡀⠀⣴⡶⡖⡄⠀⢿⠀⠀⠀⠠⡔⠂⣤⠀⠀⠀⠀⢳⡆⠙⢃⣠⣿⣆⠀⠀⠙⢛⣷⣶⣤⠖⠒⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠧⠴⣾⠇⢱⡱⡶⣾⠀⠀⠶⡶⠁⠀⣿⠤⠤⠀⠀⠀⠙⠛⠉⠉⠁⠙⠷⣦⣄⡾⠣⠄⠀⢀⣀⡀⠀⠀⠒⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠑⠈⠛⠀⠴⠊⢀⡁⠀⡟⠀⠀⠀⠀⠀⠀⢠⡞⠆⠀⠀⠀⠈⠉⠓⠦⣤⣤⣀⣀⣉⣀⡁⠂
⠀⠀⠀⠀⠀⠀⠀⢸⢿⢿⡄⠱⡀⠀⠀⠐⠒⣉⣞⡌⡅⠘⡇⢠⠄⠀⢀⣿⡞⠈⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢰⢶⡆⠀⢸⠄⠀⠹⣦⡀⡙⠒⠂⠚⢫⠎⠀⣷⠀⠹⡀⠀⠀⠉⠉⢧⡀⢨
⠀⠀⠀⠃⣰⠁⠀⠘⠂⠀⠰⠃⠁⣾⠿⢿⠀⠘⡤⠰⠇⠀⠀⠑⢼⣄⠀⣑⣦⡙⠞⠀⠀⢀⡴⠚
⠀⠀⠀⠀⣿⢆⠀⡁⠀⠀⠀⠀⠀⡿⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠟⠋⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⣎⢴⣦⣤⣴⡶⢿
⠀⠀⠀⠀⣡⠤⢌⡱⣄⠂⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⢖⣴⣦⡄⢠⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⣷⣆⠀⡀⠀⡀⢼⠃⠀⠁
⠀⠀⠀⠈⢁⣴⣦⣸⠏⠀⡄⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡀⠥⠀⣢⠏
⠀⠀⠀⠀⠀⠀⠀⢣⠀⣰⠃⠀⠀⡗⠀⠸⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠈⣿⣟⣻⠁
⠀⠀⠀⠀⠈⠆⠀⠈⠛⣿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣤⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⣂⠀⠀⠀⠀⠀⠘⣿⠅
⠀⠀⠀⠀⡐⠆⢀⠀⢠⣿⣀⠀⠀⡆⠀⠀⠀⠀⢀⣲⢮⣽⣛⣿⣷⣦⡀⠀⠀⢀⡠⠒⠉⠤⠶⠛⠿⡆⠀⠀⠀⢀⣠⡆⣠⣄⡀
⠀⠀⠀⠀⠛⡶⠦⠍⣛⠟⠈⠀⠀⡇⠀⢀⠀⠀⣸⡯⠤⣶⣶⣝⣻⣛⣛⠂⡐⠁⣀⣀⣴⣶⣿⣚⣻⣿⡄⠀⠀⠈⣑⣀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⠀⠚⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠰⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠈⠀⠀⠀⡀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠌⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡐⠈⠈⡁⠀⢰⡼⠚⠄⠀⠀⠀⠀⠐⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠐⠲⡬⠖⢸⠎⠠⠀⠤⠤⠀⠀⠌⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⢀⡀⠐⠄⠁⢼⠂⢦⠀⣧⣀⡀⢠⠀⢀⠀⠁⠎⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⢣⣈⡓⠀⢰⣎⠍⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠄⠀⢀⣤⠀⣤⠁⠃⣤⠂⠀⠄⠀⠐⠀⣀⠐⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠠⡀⠀⠀⢠⠀⢀⠠⠶⢅⣶⣤⠒⠒⠀⣠⠀⢄⣀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠐⢦⠤⣁⠀⢠⠚⠉⠈⢓⠀⢀⡴⠲⠈⠰⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⢆⠀⠀⠀⠀⠈⢳⠘⠁⠘⠁⠻⢐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⠀⠀⠀⠀⠀⠣⢸⠀⠀⠀⠀⠀⠸⠀⡀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣤⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈⠦⠀⢠⣄⠰⡾⠁⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠠⡔⢙⠛⠀⠅⠀⠀⠀⠀⣤⡀⠀⠀⢁⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣷⠀⠀⠠⡀⢞⠅⠀⠀⠁⠞⠀⠈⠂⠀⠇⠘⣆⠀⠀⠀⠈⢳⠄⠀⢸⡁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠘⠂⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠠⢹⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⢀⣀⠀⠀⣀⣀⠈⣇⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⠀⢼⠉⠚⠇⣿⠹⡇⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣸⠠⣶⠀⠀⠀⢻⡄⠃⢸⠀⠀⡄⠀⠀⠀⠙⠓⠂⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢩⣤⠋⢳⡀⠀⠘⣷⡟⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡤⣄⠀⡻⠀⠀⡟⢧⢰⠄⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⢀⣧⡏⢸⠇⠀⣰⠇⢣⣼⠀⠀⣠⣼⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡁⠀⠀⠀⠁⢀⣠⠄⡰⢊⡍⢀⡏⠀⡐⠛⣠⣿⠟⠀⡀⢇⡀⢀⣤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠐⠀⠢⠶⢦⠁⠠⠈⠠⠋⢀⣼⠀⠐⣇⠀⢸⡀⣟⣧⠀⠀⠀⠈⢧⡀⠄⠀⠀⣤
⠀⡀⠀⠀⠀⠀⠀⠐⢋⣀⠀⢠⣈⠲⠀⠈⡒⢲⠀⠰⠲⠉⠈⠃⢰⡿⠄⠀⠑⠾⠄⠑⠂⢰⠭⠂⠻⠄⠀⡀⠀⠀⠀⢀⡀
⠀⠁⠀⠀⠀⢠⠄⠀⠈⠀⠀⠋⢡⡖⢻⠀⠀⠈⠣⠈⢤⡀⠀⠞⠸⠏⣶⡀⢢⣤⠴⠶⢿⡁⠀⠀⠲⠄⠛⢃⡁⡆⠒⠊
⠀⠂⠀⠀⠀⠀⢠⠄⠀⠀⠄⠀⠀⢉⡼⠀⡒⠂⠐⣲⣤⣁⢀⡤⢐⡁⢺⠁⣠⡄⠀⠐⠀⡉⠄⢈⠀⠢⠀⢈⠐⣁⡀⠐⠂
⠀⠠⠀⠀⠀⠀⠐⠂⠀⠀⠓⠤⢀⠏⡀⢀⢠⣾⠏⠀⠀⠀⠈⣁⠀⠀⠈⠲⡹⠀⢀⡄⠀⡀⡶⠰⠘⠢⢼⡦⠀⠀⠀⠛⠁
⠀⠖⠀⠀⠀⠈⠢⠽⠴⢂⣀⠀⠂⠀⣵⠊⠉⠠⠊⡀⠄⠀⠑⠀⡁⠈⣠⠀⠀⢣⠘⣣⢀⠴⠀⢈⠘⠷⠂⠐⠂⠀⠀⠄⠰⠆
⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠂⠀⠀⠒⠂⠀⠐⠀⠀⠀⠀⠁⠀⠜⠥⠨⠛⠀⠀⠀⠈⠁
⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡡⠽⠫⢍⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡚⠜⢀⣙⣋⡀⠧⢳⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⠖⣉⣳⣽⣥⡯⣛⠹⠍⡛⢽⣬⣯⢞⣉⠲⠟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⣥⠾⢛⡉⢸⢊⢀⠀⢎⣱⠀⡀⠱⡇⢉⡓⠷⣬⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠋⠪⣦⢾⢸⡳⣄⣁⣿⣿⣀⣠⢞⡇⡷⣴⠵⠉⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢱⢆⡌⢰⢸⢑⠺⠽⠇⠸⠯⠷⡊⡇⡄⠡⡰⡮⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠕⣼⠈⢇⠈⢾⠈⢰⡛⢠⡄⢛⡆⠁⡷⠁⡸⠁⣧⠪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡪⣜⣄⠈⠣⣸⠈⢎⡃⠃⠨⢘⡱⠁⣇⠔⠁⣠⣣⢵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠭⢙⠛⢷⣽⠀⡸⣥⡰⢆⣮⢧⠁⣧⡾⠟⣃⠭⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡕⠠⢈⠐⢽⠎⠀⢫⡷⢾⡝⠀⠱⡏⠂⡡⠄⢢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡼⠾⢴⡌⣬⡠⠚⡱⠋⠙⢎⠳⢄⣥⢣⡦⠷⢧⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢷⠃⡒⡆⠉⢿⣡⢚⠣⡰⠆⠜⡒⢈⡿⠉⢐⢒⠘⡾⢂⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣦⡝⡐⠀⣉⠻⡄⠬⡁⡘⠒⠦⠅⡰⢃⡘⢆⡩⠴⠒⢃⢈⠤⢠⠟⣉⠄⢂⢩⣵⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠑⣨⡘⠋⢘⣊⢳⣿⠁⣯⢀⢰⣃⠾⠶⡘⡖⡀⢝⠈⣿⡞⢑⠃⠛⢃⣅⠊⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠣⠜⡒⠠⣇⠙⠩⠭⢓⢧⡞⠰⢮⡝⠂⢳⡼⡚⠭⠍⡇⣸⠔⢒⠣⠜⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠢⠜⡠⠼⣎⠖⢒⣢⣵⡿⠾⠟⠿⠷⢿⣮⣔⠒⠲⣱⠧⢄⠣⠔⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢁⢌⢢⢣⣈⣾⣾⠛⢹⠁⠰⣿⣽⡆⠀⠏⠛⢷⣿⣁⡜⡔⡡⡈⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠉⡰⠫⠎⠴⠛⢶⣄⣣⣀⠉⠉⣀⣜⣠⡶⠛⠶⠱⠝⢆⠈⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⣨⡍⠀⣰⢚⣛⣩⠿⠻⠿⠿⠟⡿⣍⣛⡒⡆⠀⢩⣅⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠁⠐⠃⢒⠾⡁⢸⢇⢸⡷⡰⡇⢈⠷⡒⠚⠂⠈⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣘⠘⠸⢌⡥⠊⡼⠢⠔⢡⠑⢬⡡⠃⠃⣃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⢃⣌⡀⠺⠗⢀⣡⡘⠤⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⠴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⢀⣠⠤⠖⠚⠋⠉⠉⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣀⣾⢸⡇⠀⠀⢀⡀⠀⠀⢀⣾⡿⠁⢀⡇
⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⣀⣠⡤⠤⢬⣿⣼⣹⣻⢸⠃⠀⢠⣿⡇⠀⢀⣾⡿⠁⢀⠞
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣁⡀⠀⠀⢸⡟⠁⠀⢀⠀⠀⠈⠻⣿⠟⠓⢲⣿⣿⣄⣀⣼⣿⠥⠚⠁
⣧⠀⠀⠀⠀⢀⣀⠤⠴⠒⣻⣿⠏⠉⢻⡆⢀⣴⡶⢢⣰⣿⠃⠀⢀⣴⡏⠀⠀⣼⣿⠁⠀⢠⣿⢇⣴⣶⡆
⠙⢦⡀⠀⠀⠉⢉⠉⠓⢦⣿⡟⠀⠀⠀⠀⣾⡟⠀⢸⣿⡟⠀⠀⣸⣿⠇⠀⠀⣿⡏⠀⠀⣼⣿⡞⣸⣿⠃
⠀⠀⠙⠳⠶⡶⠟⠀⠀⢸⣿⢧⠀⠀⠀⠀⣿⡇⣠⢻⣿⠃⠀⢠⣿⡿⠀⠀⣼⣿⠀⠀⢠⣿⡟⠀⣿⡟⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⡆⠀⠀⠀⠈⠉⠁⢸⣿⠀⡰⠃⠹⣧⣠⠞⠘⣿⠀⡰⠃⠙⠁⠀⣿⡇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢠⠇⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠈⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠄⠀⠀⠀⠘⠛⡇⣀⣰⠀⣠⣆⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣎
⠀⠀⠀⠀⢤⡘⡣⣷⢶⡍⢠⣴⣶⣯⣟⣛⠠⠼⢟⢤⣷⣥⣔⣒⠺⡟⠀⠀⠀⠀⠀⠀⣠⣴⠛⠋⠛⠘⠀⠁⠀⠀⠈⣨⡟⠈⠉⠉
⠀⠀⠀⠀⣶⢿⢷⠉⣱⡿⢧⣬⠙⡞⠓⠎⠓⠒⠾⠳⠿⠿⠓⣟⢳⠟⡋⠁⠀⠀⢀⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⣀⣀
⠒⠓⠒⣰⠻⡈⣧⡶⣫⠚⣍⣁⡈⠀⠴⠆⣤⢤⠔⠦⠤⡤⣤⣼⢿⢏⡉⠂⠀⠀⡞⠠⠆⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⣴⣿⠟⢻⡇
⢠⣀⡀⠀⠙⣤⢡⡾⠃⢀⠾⠷⣴⡟⣋⣭⢭⣌⣙⣓⠘⠖⣦⠹⠏⠉⠋⠂⠀⠀⡇⠀⠀⠀⣀⣠⣴⣾⠟⠋⠁⠀⠀⠀⢳⣿⡀⠩⡇⠀⣠⣲⣶
⠀⠈⣹⠟⠴⠿⣄⠴⠭⢒⡿⠗⠶⠶⠶⠒⡶⠁⠈⠋⠙⠋⢹⣄⠞⠙⠋⠁⠀⠀⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⢷⡾⠟⠉⠈
⠀⠀⠀⠀⠀⡰⡻⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣯⠿⠀⠈⠀⣀⠴⠚
⠀⠐⠒⠒⠯⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠖⠖⠻⠻⠈⠈⠹⣧⠀⠀⠀⠀⠀⠀⠀⣤⣶⡾⠿⠋⠁⠀⢠⡀⢸⡀
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡖⢶⣄⠀⠀⠀⢧⢻⣷⠀⠀⢀⠴⣿⡇⠈⡇
⠀⠀⠀⠀⣠⡶⠻⠿⢶⣤⡟⠋⠀⠀⠀⠙⢷⡄⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⡴⠖⠖⠖⢦⣄⠀⠈⢻⣧⠀⠀⠘⡆⣿⣆⡴⠃⠀⡿⡇⠀⢹
⠀⠀⠀⣼⠁⠀⡴⡄⠈⣿⠃⢰⠟⢶⡀⠀⢘⣷⠀⠀⠀⠈⡆⠀⢀⡾⠋⣠⣶⣟⡿⣷⣦⢘⡷⢖⣿⠟⠛⢦⡀⠳⠾⠋⠀⠀⠀⣿⣷⠀⠈⡇
⠀⠀⠀⡇⠀⠘⣧⣄⣴⠟⢧⣈⢃⣸⠃⠀⢠⣿⠀⠀⣠⢾⣧⠀⡼⢠⣾⣿⣿⣯⣴⣿⣿⢋⣴⠟⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢹⡀
⠀⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠉⠉⠁⠀⠠⣽⠇⠀⣼⠁⠀⠉⠀⠃⣿⣿⣿⣿⣿⣿⣿⡏⣼⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢈⣧
⠀⠀⠀⠘⣧⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⡏⠀⠀⠀⠀⡀⢻⣿⣿⣿⣿⣿⣿⡇⢿⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣿⣿⣄⣠⡿⠋
⠀⠀⠀⠀⠈⠛⠿⠶⣧⣶⣴⣠⡄⡀⢸⠃⠀⠀⠀⣷⠀⠀⡀⠀⣧⠈⠛⠻⠿⠿⠛⢋⢳⣬⣳⣄⣀⣀⣴⡏⠀⠀⠀⠀⠀⠀⠀⣷⣿⠿⠋
⠀⠀⣔⡒⣤⣄⡀⠀⠀⠀⠀⠉⠙⠻⣿⢠⡖⢦⡀⢹⣦⡦⠷⠶⠿⠿⣮⣦⣴⡶⠾⠾⠛⠋⠉⠉⠛⠛⠋
⠀⠀⣯⠹⣎⢫⡹⣄⠀⠀⠀⠀⠀⠀⠈⠧⣭⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⣤⣄⣀⡀⠀⣀⣀⣀⠀⠀⠀⣀⣤
⠀⠘⣾⠳⡎⢷⡱⢌⠳⣔⡟⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠉⣉⡩⣤⠀⢲⠂⢠⡏⣹⡷⠚⢩⡉⠉⣍⣉⠩⡃⢸⣶⣄
⠀⢠⢏⣧⠻⡄⠻⡌⢧⡘⢧⡸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢁⡆⠀⠉⠉⠋⠁⣈⡀⢸⡀⠨⣇⠀⠘⠋⠀⠛⠙⢂⠁⢸⡏⡏⢻⡆
⠀⣾⢸⡟⣇⠘⠦⠙⠆⠉⠀⠙⠎⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠑⢘⠃⠀⠈⢛⠀⠀⠒⠒⢸⡇⢹⡇⠀⠈⠐⠆⠀⠈⠐⠀⠈⣇⡇⢸⡇
⠀⢹⡄⠻⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠀⠀⠈⠐⠀⠀⠐⠒⠀⣏⢸⡧⠀⠀⠤⠅⠀⠀⠈⠀⠀⣿⡇⠈⡇
⠀⠀⠻⡄⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡌⠀⠀⡀⠀⠀⠀⠀⠀⢀⡇⣸⡯⠀⠀⠐⠐⠀⠀⠀⠂⠀⣿⡇⠀⡇
⠀⠀⠀⢧⠀⠘⣷⣂⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⠇⠀⠀⠠⠀⠀⠀⠒⠒⢸⠇⢸⣟⠁⢀⣁⣀⣀⣀⠈⢁⣠⠿⣇⠀⡇
⠀⠀⠀⠘⣧⠀⠈⢷⣖⡀⠀⠀⠀⠀⠀⠈⢙⡦⠀⠀⠀⠀⠀⡿⣾⣿⣷⣦⣤⣤⣶⣶⣾⣿⣿⣿⣶⣬⣷⣾⣿⣿⣷⣿⣿⣿⣯⣤⣤⣾⣿⡇
⠀⠀⠀⠀⠈⢧⠄⠀⠈⢿⣧⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣿⣫⠞⣿⣿⣿⣯⣿⣿⣿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢼⡁⢀⣠⡾⠋⠈⢷⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠘⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⣿⢿⠿⠿⡿⡟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣿⣟⣿⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠆⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠄⠀⠠⠦⡈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠠⢀⠎⡔⣡⡤⡜⢣⠦⣌⢂⠱⡀⠄⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣠⠤⡤⠈⢸⠀⡔⢢⡈⢁⡜⢢⠀⡎⠁⣤⠤⣄⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡾⡄⠁⠡⠠⣇⠀⠀⠐⠃⠘⠂⠂⢀⣸⠄⠎⠈⢤⢷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠂⠀⠄⠀⠀⡏⠀⠠⠀⠈⠀⠀⠄⠀⢻⠄⠀⠔⠀⠄⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣟⠀⠂⠘⠀⠀⡿⠀⠀⠰⠀⠐⠄⠀⠀⢿⠀⠀⠃⠐⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡧⠀⠀⠆⠀⠀⡧⠀⡐⠁⠀⠠⠈⢂⠀⢼⠀⠀⠶⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣟⠀⠀⠰⠀⠀⡇⢰⠀⡎⢰⡄⢡⠀⡆⣈⠀⠀⠆⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣯⠀⠀⢠⠀⠀⣝⠸⡄⠁⠈⠀⠊⢠⠃⣏⠀⠀⡄⠀⠀⣽⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠐⠂⠆⣧⠀⠀⠠⠀⠀⡵⠀⠙⡤⠞⠳⣤⠃⠀⢆⠀⠀⠄⠀⠀⡌⠰⠄⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣠⠔⠚⠉⠉⠉⠒⠢⠦⣀⠀⡝⢀⠊⠄⠀⠀⠈⠑⡀⣇⠀⣀⠦⠔⠒⠉⠉⠉⠒⠦⣀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢈⠀⢆⠀⠀⠀⢀⡀⠀⠀⠈⠱⣼⠃⢠⠀⠀⠀⢠⡀⠸⣧⠎⢁⠀⠀⢀⡀⠀⠀⠀⡰⠀⡁⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢳⡆⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢰⡦⢴⡎⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢰⡞⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣⠀⠀⠁⠀⢸⢀⡠⠄⠘⣧⠟⠀⠀⠹⣾⠃⠠⢄⡀⡇⢀⠀⠀⠀⡜⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⡇⠀⠀⠈⠲⣈⠀⠄⢈⡟⠀⠁⠀⠀⢻⡁⠰⠀⡑⠖⠁⠀⠀⢸⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡷⠀⠀⠠⠀⠁⠂⠀⠀⠙⢄⠀⠀⡠⠋⠀⠀⠐⠉⠀⠤⠀⠀⡾⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠈⠀⠠⠄⠀⠀⡈⠖⠲⣅⠀⠀⠠⠄⠀⠀⠀⠀⠀⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡄⠀⠀⠀⠀⠀⢀⡠⠋⢀⡠⢄⡀⠙⢄⡀⠀⠀⠀⠀⠀⢠⡃⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣧⠀⠄⠐⠀⢆⡌⠀⠀⠄⠀⢀⠠⠀⠀⢣⡰⠀⠂⠠⠀⣼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠹⡀⠀⠀⠀⠀⠡⡈⠀⡀⠔⠢⢀⠀⢁⠌⠀⠀⢀⠀⢠⠏⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⢦⡙⠂⡭⠀⠔⠉⡰⠄⡠⢌⠍⠢⠀⣥⠘⢋⡤⠋⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠉⠛⠶⠲⠦⠤⠶⠖⠴⠾⠤⠴⠶⠖⠛⠉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠶⠆⠒⠂⠉⠉⠐⠒⠲⠶⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡉⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢰⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠇⠈⠿⠿⠟⠀⢰⡿⠳⡄⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠷⠀⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠤⢤⡸⠞⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇⠋⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢠⣄⣠⣿⠇⠓⠛⠁⠀⠀⣄⣠⣤⣶⣶⡆⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⠾⣄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⣿⠀⠀⢿⣻⠟⠛⠛⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⣿⣿⣿⠀⠀⠀⢸⣿⠀⠀⣼⣳⣄⣀⣀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡟⠀⢹⣿⣿⠀⠀⠀⢸⣿⣧⠀⠉⠉⠹⣭⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠟⠃⠀⠀⠁⠀⠘⣿⣿⠀⠀⠀⢸⣿⣿⣷⠆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⢰⣶⠀⠀⠀⢿⣿⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⠀⠀⠀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠠⣄⡤⠂⢉⠖⢢⡊⠀⣠⡶⠚⢉⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠐⠒⢳⠐⡏⠀⠀⠀⠠⠞⢳⡞⠁⠀⡰⠁⠀⠀⠈⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⢙⣰⣆⠀⠀⠀⠔⠘⠇⠀⣀⠀⢀⣴⣤⣀⣴⠻⡄⢀⡀⠄
⠀⠀⠀⠀⠀⠄⠀⠈⣧⡀⠐⣦⡀⠐⡋⢉⢷⣄⠀⠠⠐⣿⠾⠍⢉⣉⢿⠻⠋⠛⢣⡐⣼⡀⡁⠀⡀
⠀⠀⠀⠀⠘⠀⠀⡰⠋⠹⣿⢁⣶⣿⣷⣿⣇⣉⠷⡐⣰⠟⣧⣼⣿⣏⣉⣑⢃⡀⠀⠻⣩⠅⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⣰⣧⠤⠘⠋⠙⠋⠉⠁⣀⠘⢻⡏⠻⠋⠙⢻⣽⠍⠀⠁⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⢤⡦⠀⠁⠸⠀⠀⠀⠀⢅⣀⣀⣈⠈⢹⡗⠈⠇⡀⠀⢙⡄⠀⠈⠀⠀⠀⠴⠀⠈
⠀⠀⠀⠀⠠⠀⠠⣁⠀⠉⢀⣤⣶⡄⠀⢺⣿⣿⣿⣿⠄⢹⣄⠲⠠⢰⣬⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠗⠔⠀⠀⢰⡘⢛⡿⢣⣤⠀⠻⠿⠟⡡⠀⠀⣉⠉⠂⣸⠏⢀⠃⠀⠀⢐⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⢃⣪⢀⢌⣹⣿⡞⠃⠎⣁⢐⣰⡇⢠⣏⠙⠀⣾⡶⠀⢠⣤⠄⠀⣠⠊
⠀⠀⠀⢠⠀⠀⢠⣿⠁⠀⠈⠛⠢⡀⠄⠠⣀⢀⣰⠥⢚⡿⠀⠀⡟⠀⣸⠟⠁⣠⡿⠋⢀⣴⣿⠀⠀⡤
⠀⠀⠀⠸⠀⠈⠛⡝⠃⣀⣀⡀⣴⡍⠋⠋⢈⡌⠾⢊⠜⠀⢀⠘⠀⠀⠛⠀⠰⠁⠀⣠⡾⠋⠀⠀⠈
⠀⠀⠀⠀⠱⠚⢃⡑⡞⠀⠀⠅⠼⡻⠏⠉⠩⠇⠀⠁⢀⡾⠁⠀⠠⠀⠀⠀⠀⠀⠐⠉⠀⡤⠂
⠀⠀⠀⠀⢰⠋⠉⠉⠁⠀⠀⠰⢂⠈⡆⣀⣀⡀⠀⢀⠿⡀⠊⢉⣉⣉⣰⡆⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠘⠁⠆⢀⣈⠃⠀⠰⠟⠁⠸⠋⠛⠇⢳⡼⠆⣠⡾⠛⠉⡙⠛⢡⠀⢀⠐⠛⣦
⠀⠀⠀⠀⠀⠊⠈⢁⣴⠇⠀⠀⠀⠀⠘⠙⣾⡕⣶⡖⠚⠉⠀⠀⠀⠄⠀⠀⠀⠾⠧⠤⠽⢦⣀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⣸⡇⠈⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣦⣤⣀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠄⣡⡀⠀⠐⠀⠀⠀⣿⣷⢰⡞⣭⣿⣦⣶⣿⢶⡶⠒⢺⣿⡿⠁⢨⠏⠳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠲⣶⣦⡴⠿⡿⣛⣿⣱⣶⡿⣶⠈⢿⠠⡧⠀⠀⡏⠀⣤⠋⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠠⠄⢿⣧⡜⣟⢿⣻⣿⣿⣤⣬⠁⠂⡰⠒⠋⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠘⣿⣴⠛⣿⡱⠟⠛⠉⡁⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠛⢾⠉⠢⣤⠤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢

⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⢠⡄⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⣤⠀⠐⠂⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⣀⡮⠣⢆⡀
⠀⠀⠀⠀⢀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⢀⡄⠠⠑⢀
⠀⠠⠄⡀⠀⠀⣾⡷⢈⠀⡂⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⣀⡈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠃⠀⠀⠉
⠀⠀⠀⠘⠹⠆⣤⡄⠮⠘⠀⠀⠀⢀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠁⢐⣾⣦⠈⠀⢢⠀⠀⢀⡀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⡀⠀⠀⠀⠺⠡⡇⠻⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠆⢀⠀⡀⠲⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡇⠀⠀⠀⠀⠠⣶⡦⠀⠀⠀⠀⠉⠲⢤⣻⣻⡤⠚⠁
⠀⠀⠀⠘⡋⠀⠀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⠀⠀⣾⣿⣿⡓⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣋⣈⠂⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠔⡒⡈⠂⠀⠀⠀⠀⣤⢀⢀⡄⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠘⢜⡋⠂⠀⣠⠤⠀⠀⠋⢸⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⣠⠿⣄
⠀⠀⠀⢰⣆⡆⠀⠀⠀⠉⢙⠛⢲⣖⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⣶⠆⠀⠀⠀⠀⠀⠠⣀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠊⢹⣾⡝⠣
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠑⠄⢦⠀⠀⠀⠀⣠⣄⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⡈⠛⢁⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢛⠓⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⢀⣤⠈⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠂⠀⠸⠿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣚⣓⠢⠀⠀⠀⠀⠀⠀⣶⠄⠈⠀⠀⠀⠛
⠀⠀⠀⠀⠀⢄⠘⠋⠛⠀⠀⠀⠀⠀⣠⣄⠀⠀⢸⣶⡟⠀⠀⠀⠀⠀⢀⣀⠀⠈⠀⠀⠻⠋⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⣀⠀⠃⠀⠀⠀⠀⠙⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣹⣷⠀⠀⠸⡦⠀⢿⣿⡷⠆⠀⠀⠀⢻⣧⠀⠀⠀⢀⠐⢻⣷⠄⢠⡀⠛⢀⣤⡆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⢯⠀⠀⢸⠛⠍⠀⠀⠀⠀⠀⢀⡈⠀⠁⠀⠀⠞⣷⡈⠁⣠⣤⡄⢉⡀⠛⠋⣠⣿⡄⠀⠸⠻⠄
⠀⠀⠀⠀⣦⠀⠀⢹⣷⠆⢠⡀⢀⠀⠀⢀⡀⠀⠐⠠⢼⡏⠄⠰⠟⣿⠂⠈⠁⣤⡘⢷⡡⠄⠀⠈⣄⠚⠉⠀⣠⡇⢹⡇⠀⠀⠀⢀⣴⡇⠀⠰⠃
⠀⠀⠀⠀⠁⠀⠀⠛⠙⢣⠀⠁⠈⠃⠀⣿⡻⠀⠀⢰⣶⡿⠀⠀⠀⠐⢷⡇⠀⣿⣷⣌⢣⠀⣀⣤⣾⣦⣤⣼⡿⢷⣼⣷⠀⢠⡖⣀⣿⠇⢀⣀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⣶⠀⠐⠀⠀⠉⠁⠀⠢⡜⠋⠀⠀⠀⠐⣀⠀⡀⠀⢱⣏⢽⣧⣶⠋⠉⠋⢻⣿⣟⣷⠀⣿⣿⡴⢋⣶⣿⡿⢶⣾⡿
⠀⠀⠀⠀⠀⣿⡏⠁⠘⠇⠈⠒⠂⠒⠻⢄⠰⡂⠘⠛⠆⠈⡧⠤⣄⡀⠁⠙⠲⡤⣿⠀⣻⠋⠘⢦⡶⢸⣿⣿⡇⠀⣿⣿⠁⣼⣿⣿⡇⠀⠁
⠀⠀⠀⡄⠀⠙⠓⠀⣀⡀⠀⢠⡄⠀⣰⡾⢎⠁⠀⠀⠀⢷⠄⣴⣿⣧⣀⢰⣾⡛⣹⣶⡏⠀⠀⠂⠀⢸⣿⣿⣇⣠⣿⣿⣰⣿⡿⠟⣻⣿
⠀⠀⠀⠀⠀⠀⣀⠙⠋⠃⠹⣦⠉⠀⠙⠋⢀⠀⠸⣦⣤⣿⠀⢻⣿⣿⣿⡟⠋⠉⣿⣿⠁⢀⡄⠀⢠⣾⠛⠉⢉⣿⣿⡿⢻⣿⡦⢀⣿⣿⢀⣴⡇
⠀⠀⠀⠀⠀⣼⡟⠃⠀⠀⡈⢻⡆⠀⠀⠀⢨⡄⢰⡌⠛⣿⣧⠀⠿⢻⣿⡇⠀⢠⣿⣿⣷⣾⠅⢠⠋⣽⣆⢠⣾⣿⣿⠀⢀⡿⢴⠞⣿⠏⠙⣭⡄
⠀⠀⢀⡀⠀⠘⠃⡀⠸⠒⠃⠀⢃⢰⣾⣿⣦⠑⢼⣷⣴⡾⢿⣧⣶⣾⣿⡀⠀⠀⡿⠋⢹⣿⣷⡏⢠⣿⣿⣿⣿⣾⣯⠀⠊⠀⣠⣿⣿⣶⡄⠉
⠀⠀⠀⠉⢀⠀⣿⡟⠀⠀⠀⠀⠈⠺⡗⡨⠀⠁⢀⠈⢻⡆⠈⢿⣿⡟⣿⣧⠀⣶⠀⠀⠈⣿⣿⡆⠈⣿⣿⣿⣿⣿⡷⢤⣤⣸⣿⣿⣴⣿⣿⡆
⠀⠀⢠⡶⠟⠁⠀⠀⠀⢢⡀⠀⠀⠐⠋⢀⡀⠀⠈⠀⠘⣿⣦⡟⠻⣿⣿⣿⣷⠿⣏⠀⠈⣿⣧⠄⢰⣿⡿⠟⠀⠀⠀⠰⣿⣿⣏⠈⣿⣿⣿⡇⣀
⠀⠀⠉⠀⠀⠀⣠⣤⠀⠀⢱⡀⠀⠀⠐⠞⠓⠂⢶⣀⠾⠋⠙⠳⣤⣾⣿⣿⣿⢀⣼⠛⠹⣿⣿⢣⠜⠋⠀⠀⠙⣦⣶⣿⡿⠿⣧⣶⣿⣿⣿⠛⠋
⠀⠀⠀⣤⣶⡄⠙⠁⠀⠀⡌⠛⠶⣤⠤⠀⠀⠀⠀⡏⣰⡏⠙⠷⠿⣡⢙⡿⣿⠛⠉⠛⡷⢿⣿⣁⠔⠀⠀⣠⣾⣿⣿⣋⣤⣾⡿⢿⣝⠋⠉⢳⡶
⠀⠀⠘⡙⠀⣁⠀⢀⣀⡙⠆⠒⢀⠃⠀⠀⠀⡻⠎⠸⣿⣤⣶⣶⡄⠀⢨⣄⠀⢠⡀⠀⠀⠶⠈⢷⠤⠴⠋⣿⣿⡿⠛⠛⠀⠉⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠁⠀⠀⣀⠀⣭⡙⠓⠂⡀⢀⣤⡄⠀⣽⠇⢰⣿⣿⣿⣿⠇⠀⠈⠛⢠⣟⡿⢀⣾⣿⠀⠐⢠⡆⣰⡞⢿⠉⢷⠊⠹⡝⠛⠛⠛⠏⠉⢻⡄
⠀⠀⠀⠀⣿⡟⠁⠀⠀⠑⠀⠀⠛⢾⢿⣶⠞⣡⣶⣴⣏⣉⠉⠁⣰⣶⠀⠀⠨⣯⣿⣿⣿⣿⣶⡶⢈⣀⠉⠈⣶⡄⠈⠻⣄⠀⠘⣶⣦⢠⡀⠸⣷
⠀⠀⠰⢦⠈⠃⣶⡷⢀⡀⠱⣄⡖⠠⠘⣣⣿⣿⣿⣿⣿⣿⣧⣨⡿⢿⡶⠆⣀⣈⣈⣋⣉⣍⠛⣴⡟⠻⠧⣀⠀⠉⠂⠀⠻⡷⠀⣿⣿⣿⣭⡀⢸⡆
⠀⠀⠀⠠⣦⣤⡴⠁⣨⠃⡷⢌⡑⠂⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⡙⠋⣡⣼⣿⣿⣿⣿⣿⣿⣄⠘⠓⢀⣤⠄⠀⣂⠀⠈⠁⠀⠊⠙⠛⡟⠛⢀⠈⡇
⠀⠀⠀⢀⣈⠙⠀⠀⠀⠈⠁⠀⠉⢩⣿⡁⠙⠻⠿⣿⣿⣉⠀⠐⠈⠎⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⣨⣴⠖⠸⣿⣄⣀⡀⢠⡀⠀⠀⠁⠀⠉⠠⠁
⠀⠀⠀⢸⣯⡤⠖⠀⠀⣀⡀⠀⢲⣾⠀⡀⠘⡷⢰⣿⣿⣿⣦⠈⣤⠓⠻⣿⣿⣷⣿⣿⣿⣿⣿⠟⠀⢿⡷⠖⣻⣷⣿⣿⣤⣘⣛⡂⠀⠀⠋⠀⣾
⠀⠀⠀⣾⡟⠃⢀⣤⠐⠈⠙⠉⣹⣧⣶⠀⠀⣴⣿⣿⣿⣿⣿⠸⣿⣤⡄⢉⢉⣉⡛⠻⠿⠛⢁⡰⠀⣸⣿⣿⠿⠛⠛⠛⠛⠛⣿⣿⣿⠂⠀⢀⡎
⠀⠀⠀⠙⠧⠀⠘⠇⠀⠈⡁⢠⣿⣿⠟⣀⢈⠛⠙⢿⣿⣿⡿⣞⢻⣽⣦⣜⠉⣹⢃⣄⡀⡄⢀⣤⠶⢿⣿⣷⡶⢶⣿⣿⡿⠟⡛⠉⠉⠰⠆⢸⡇
⠀⠀⠀⠳⠆⠀⢠⠈⠐⣤⡈⠀⣿⠿⢿⣯⣮⣤⣴⣂⡟⠲⠞⠈⠀⣿⠹⢿⣿⣿⣿⣿⡗⡘⠒⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢠⣴⣤
⠀⠀⠀⠀⣶⣴⣗⣀⠠⠏⢙⠹⣇⣆⡘⢁⠟⡸⠟⣿⣴⡗⠂⡌⢰⠏⡄⢺⣯⡽⠛⣿⡏⠛⠃⠀⠛⢻⣿⣧⣉⣿⣿⣿⣃⠀⣚⡤⠒⠀⠉⣟
⠀⠀⠀⣾⣈⠿⠋⠁⣤⣦⠀⠰⠞⣿⣿⣿⣶⣿⣴⠃⡼⢻⡟⣠⣧⣸⢁⠛⡏⣀⣾⣿⣶⣋⠁⠞⣃⣉⣉⠻⣿⣿⡟⣿⡯⠙⡿⢷⡗⠀⡰⠟⠛
⠀⠀⠀⠉⠉⣤⡶⠖⠛⢭⢤⣴⣞⡏⢹⣿⢿⣿⡏⢾⣧⣞⣠⠏⢹⡟⢿⣤⣇⣯⣿⣿⣿⣿⢿⣿⡿⢋⡍⣷⣿⣿⣧⣿⣿⣷⣤⡏⢁⣾⠇⣠⣷
⠀⠀⠀⢀⠀⠘⢛⡄⠀⢀⠀⠀⢹⠉⣂⠈⢾⣟⠛⠛⣿⢿⣿⣶⢛⣴⣦⣸⣯⣇⣿⣿⣿⡇⠀⣤⠀⠈⣰⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⡟⠀⠛⠁
⠀⠀⠀⠀⢴⣶⣾⠃⠐⢾⡖⠀⢸⠀⢻⠀⠀⣬⠀⠸⠿⣶⣿⣝⠋⢹⣿⣟⣿⣿⣿⣿⣿⢁⡾⢁⣶⠿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠘⡉⠀⢙⡿
⠀⠀⠀⠀⠈⠉⠁⣤⡈⠀⢸⠖⠘⢒⡈⠑⡒⠧⠀⢃⡀⢀⢀⣼⠀⣾⡋⠛⠋⣿⣿⠛⢇⡾⣠⡎⠁⠀⠀⠈⠙⠋⠉⡉⠉⠀⠀⠰⠎⠉⢷⢀⣦
⠀⠀⠀⠻⠆⠀⠀⢿⣧⠀⠸⢦⠰⠾⣧⡶⠿⣤⢀⠀⠑⣌⠉⢯⡤⢿⡿⣤⡼⢿⣿⡟⠸⣱⠿⠇⠀⠐⠀⠀⡀⠀⠀⠘⠢⠀⠀⠀⠀⣾⠇⠘⠋
⠀⠀⠀⠀⠺⢿⣆⠀⠀⢠⢴⡾⢁⠀⢀⣤⠄⣿⠘⣿⠀⠛⣶⢈⡔⠻⠖⠾⠶⢞⡾⠁⢀⡏⢠⣤⢱⠆⠁⠈⡏⠀⢰⣆⣠⣧⠄⠀⠀⠁⢀⣴⡆
⠀⠀⠀⠀⠀⠸⢻⠀⣠⠃⣸⣇⠙⢃⠈⢁⡄⢤⡴⡇⠹⠶⢩⡏⠁⠀⠘⡓⠀⠈⠐⠆⠈⠀⢿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⢡⣤⡄⠀⠀⠚⠛⠁
⠀⠀⠀⢾⣤⣦⢀⣨⠷⠂⠀⠙⣶⡿⢶⣦⣄⠈⢻⢃⢄⡀⠙⢁⡉⠀⠈⢀⣴⠾⠀⢀⣧⡀⠈⠉⢐⠀⠀⠠⠀⣷⡄⢀⣀⣰⡿⠛⠃⠘⣷⡆
⠀⠀⠀⠀⠉⢀⣤⠤⣤⠄⢶⠞⢩⢅⡀⢁⡄⠛⠋⠙⠃⠀⡄⠙⠻⠶⠿⡮⢥⢴⠖⠛⠏⠙⠀⢀⡆⠀⠰⠂⠿⡄⠀⣌⠁⠀⠿⢁⣄⠠⡾⠃
⠀⠀⠀⠀⢠⣦⡔⠈⠛⢢⡄⠠⠅⠊⠀⣤⣶⢀⡀⣼⣇⠘⠀⠀⣤⣀⣤⡀⠀⠾⠟⢢⡶⠋⠀⢸⣿⣿⣧⡄⠀⢀⣤⣾⠀⠀⠠⣤⡿
⠀⠀⠀⠀⠀⠻⡶⠂⠀⢿⡄⠀⠀⠀⠀⠈⠛⠛⠁⠘⠛⠃⠀⠀⠛⠁⠚⠁⠀⠀⠀⠘⠛⠁⠀⠀⠉⠛⠛⠁⠀⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠂⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠡⡀⠀⠀⢀⠀⠀⠈⢀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠂⠄⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠁⠠⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠄⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣈⡀⠀⠀⣀⣀⣀⣀⣸⣿⣿⣄⣀⣀⣀⡀⠐⠀⠀⢀⣀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢀⠀⠠⠀⠐⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⠾⠿⠿⠿⠿⢶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢄⡿⣄⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠄⠒
⢉⠁⡃⠀⠀⠀⣼⡟⠋⠙⢿⣆⠀⠀⡰⠋⣠⡈⢲⠀⠀⠀⠉⠉⠀⠀⠀⢰⣶⣶⣀⡁⠀⡀⢠⠀⡄
⠀⠀⠀⠀⠀⠘⣿⠀⠿⠂⢸⡿⠀⠀⠰⡀⠙⢁⡜⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠈⠀⠀⡽⣾⣿⣟⡧
⠀⠀⠁⠀⠀⠀⠙⠷⠶⠶⠟⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⣤⡄⠈⠁⠀⢀⣀⠀⠀⠀⠈⠑⠛⠛⠋
⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠘⠳⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀
⠘⠿⠞⠁⠀⠀⠀⠀⣖⣲⢦⡀⠀⠀⠀⠰⡁⠿⠄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⣉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⠿⠏⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠊
⡄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠻⠿⢏⡠
⠴⣿⣿⢃⠆⠀⠀⠠⡶⠊⢰⣿⡿⣧⠛⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠈⢉⣀⣉⠀⠀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣝⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⢎⣾⣷⡎⠇
⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠈⠻⠏⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣶⠦⣄⣀⠀⠀⠀⠀⠈⠙⠟⠊
⡀⠀⠀⠀⠀⢰⠟⣋⡛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡏⠀⢾⣛⡗⢀⡟⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠
⠁⠀⠀⠀⠀⠘⠀⠻⢇⡼⠃⠀⠀⢠⣾⣿⣟⡦⠀⠀⠁⠘⠒⠶⠖⠚⠉⠀⠀⠀⢀⢥⡈⠥⠚⢠⠀⠀⠉
⣿⡟⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⢩⠩⠄⠀⢀⡴
⣿⣷⢚⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠓⠚⠒⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢾⡇⠀⠠⠀⠀⠀⢰⣶⣦⡀⠀⠀⢀⣠⡤⣤⡀⠀⠘
⠀⠀⠀⠀⠰⣟⡷⠀⠀⢠⢏⣤⣤⡱⡀⠀⠨⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠀⠀⢠⡎⢠⣶⣦⢹⡏
⠈⠀⠃⠀⠀⠈⠀⠀⠀⠈⢮⣻⣿⡵⠁⠀⠀⠁⡐⢀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠾⢧⡘⠯⢥⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠠⣤⣤⣤⠄⢠⡄⢠⣤⠀⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⣤⣤⣴⡄
⠀⠀⢐⣀⣄⣿⠶⠎⠹⠾⠉⠀⠀⠐⠃⡜⠻⠀⢤⠀⠀⠀⠀⠀⠃⢳⣄⠀⠀⠤⠉⠠⠄⠉⢢⡀⠉⠠⠴⠀⠈⣠⠿⢿⠃
⠀⠀⠈⠒⢶⣶⢀⣀⣀⣀⠀⠀⢀⠋⣼⣛⠁⣈⠀⢀⠀⠀⠀⠘⢷⡀⠆⠀⠀⠀⠀⠀⢘⡛⣛⡛⠀⠀⠀⠀⠀⠙⠖⢿⣷⡀⠤⢠⠤⡄
⠀⠀⢠⣖⠛⢁⡾⠄⠷⠬⠄⠀⡈⢠⣛⣃⣀⡈⢠⠃⠄⠀⠀⢠⡃⠉⡙⠀⠀⠀⠀⠀⢀⣇⣇⣛⠀⠀⠀⠀⠀⣾⣻⣿⡿⠱⠞⠹⣤⣘⣆
⠀⠀⠰⣎⣏⠿⠀⠀⠀⠀⠀⠤⢠⣎⣹⣿⣿⣿⣃⠛⡴⡄⠀⠀⡝⠶⠶⠀⠀⠀⠀⠀⣿⠉⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⡏⢈⣿⡆
⠀⠀⠘⣽⣬⡵⠀⠀⠀⠀⠐⢂⣾⣉⠋⠉⠉⠉⠁⡿⢣⢇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢈⣄⣰⣿⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⣷⣿⡉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⣠⠖⠉⠉⠑⠦⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡐⣄⠀⡠⢄⡀⣠⢂⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣠⣶⡦
⠀⠀⠀⠀⠙⠢⢄⣀⡤⠞⠉⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠤⣽⠿⢫⡴⢦⡝⠿⣯⠤⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⠀⢀⣀⠀⠀⠀⠀⠶⠥⠤⢤⣭⣭⣤⠤⠬⠶⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣛⣿⣷⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠘⠘⡇⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡀
⠀⠀⠀⠀⠀⢸⣯⣶⣽⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⡿⠃⠈⢿⡆⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⣸⡿⠿⢿⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⣿⣿⣷⡆⠀⠀⢀⣸⣷⠇⠸⣾⣇⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⢰⣿⣁⠀⢀⣽⣧⠀⠀⠀⠀⠐⣿⠂⠀⠀⠀⠀⢸⣟⣻⣿⠃⠻⣿⣛⣿⡇⠀⠀⡿⠟⠋⠀⠀⠙⠻⠷⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡏⣾⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠛⠁⠀⠀⠙⠛⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄
⠀⠀⠀⠀⠸⣿⣶⣿⣷⣾⡇⠀⠀⠀⠀⠀⢀⣄⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠉⣿⠻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣀⣤⣶⣷⣶⣤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⣿⢻⣿⣿⡇
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⣤⣶⣿⣿⣟⣧⣷⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢸⣿⣿⣿⠀⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠙⠻⢿⣿⣿⣿⣿⠟⢋⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣷⣶⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡈⠛⢉⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠛⠿⠛⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢰⢷⡀⣸⠧⠞
⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠶⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠉⠉⠙⠳⣦⡀⠘⠾⣳⢏⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠶⣶⣦⣄⡀⠀⠀⠀⠀⣼⠋⠀⠀⠀⢠⣦⠀⠈⠻⣦⣤⡏⠻⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⠀⢰⡟⠀⠀⠀⠀⠈⠉⠀⠀⣰⠏⢸⢣⡴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣴⡶⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠉⠛⠛⠻⠶⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣠⣴⠾⠋⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢠⡞⡉⠁⠈⠙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢰⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⡶⢶⡄⢠⡟⠉⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⠶⠶⠛⣡⡶⠶⣦⠀⢠⡾⠋⠉⣼⡆⠀⢰⡏⠀⠀⠀⢿⣤⠾⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣟⣀⣴⠏⠀⣿⡇⠀⣠⣿⣦⣠⡼⢷⣤⣤⠶⠛⠛⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠈⠻⣭⣤⡴⠞⠋⠙⠛⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


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

⠀⠀⠀⠀⠀⢀⣶⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠸⣿⠟⣿⣿⣟⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⢠⠀⠀⠀⣆
⠀⠀⠀⠀⣠⣾⠟⡛⠻⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠠⠈⠀⠀⠠⠀⠀⠈⠣
⠀⠀⠀⡴⣻⠇⠐⡄⢰⣄⠠⠹⣿⡄⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⡀
⠀⠀⢸⣿⣿⠠⣴⠳⣬⢷⡔⢢⣿⣿⠀⠀⠀⠀⠀⠀⠈⠳⣌⠰⣀⠀⠁⠀⢀⡄⢊⣠⠛⠁
⠀⠀⠘⣿⣿⣷⣄⣀⠀⣈⣠⣾⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠮⣤⣬⠴⠖⠉
⠀⠀⠀⠈⠙⠿⠿⠿⣿⡿⠿⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠠⠀⠀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠠⣷⠀⠀⣀⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠁⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣽⢿⠉⠉⠙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⡏⣿⣶⢦⠀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠸⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⢀⣀⣀⣀⠐⠀⠐⠠
⠀⠀⠀⠀⠀⢰⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠰⠲⡀⢰⠋⠉⣻⠀⠔⢆⠀⣇
⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡎⠀⠃⢈⠀⠀⠀⡀⡀⠘⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢘⠡⠐⢤⡀⠄⠠⢄⠠⠔⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠲⢉⠩⠉⠁⠈⠛⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢈⠗⠀⠀⠘⠁⠀⣀⠀⢟⠀⢀⡤⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⢶⡶⠖
⠀⠀⠀⠀⠈⢻⡟⠤⢶⣄⡠⠀⡀⠀⠂⠀⠀⠀⠌⠀⠀⠈⠁⡀⠀⢸⠀⠀⠟⢠⠈⠀⠸⠀⢠⡄⠀⠀⡤⠀⠀⠆⠀⠴⠶⠛⠀⢁⢀
⠀⠀⠀⠀⠀⠀⢁⢀⢀⠬⣿⡶⠖⡈⠢⢄⣴⡠⠢⠀⠀⡏⠸⣇⡼⣄⠀⣦⣤⠬⢂⠰⠇⠀⡄⠀⠀⠄⢁⠀⠻⢥⡖⠂⠒⠚⢄⠈⢻
⠀⠀⠀⠀⠀⠀⣼⠂⠁⠀⠀⠀⢫⡴⠶⣄⠻⣧⡂⢨⢢⣤⠀⡤⣀⠀⢰⡇⢀⡶⠲⠾⠢⠦⣄⣰⠆⣠⠈⣿⡖⠀⠣⡄⠀⠀⠈⠱⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⡞⠀⢠⠀⣬⣤⣘⢣⡴⠋⠉⠀⡀⠠⠤⠺⡅⢈⠀⠀⠀⡀⠀⠀⠘⠛⣅⠚⠓⠉⠤⠀⠀⢀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠏⡠⣶⠇⠺⠟⣿⡿⠋⠠⠀⠀⠀⠘⠳⠖⠢⣉⢻⣓⡛⠙⢛⢿⠂⠀⠀⠹⣦⠀⠈⡄⢳⡄⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡠⠠⡽⠆⠀⣿⠛⣅⠀⠀⠀⠀⠀⣀⠤⠀⠀⠍⢩⡍⢀⠀⠈⣿⡃⠰⠶⢀⠼⣶⢆⡈⣛⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⢀⠀⣤⠀⠋⡗⠸⠿⠿⠉⢙⣉⣥⣬⣍⣉⣩⣬⣤⡍⠀⠀⠸⠙⠳⠶⢶⡶⢀⣸⣿⣤⡠⠜⠉⠀⠄
⠀⠀⠀⠀⠀⠀⣨⣦⡾⢁⣄⠉⢠⠼⢢⠖⣷⠄⡄⠀⠀⠘⠉⠀⣀⢉⣙⡁⠀⠃⠤⡄⠀⢠⠀⠀⠐⠋⡀⠀⠁⢀⡟⡤⠤⣤⣀⢰
⠀⠀⠀⠀⢀⠌⠠⠡⠠⡏⠡⠀⠀⠀⠚⠁⠄⣐⠀⠀⠀⠚⠦⡤⢤⣴⣴⣴⠖⡶⡦⠁⠀⠶⣈⡶⠃⠉⠙⡄⠀⣼⠿⠧⠶⢀⠈⠻⢦
⠀⠀⠀⠀⡌⠀⠀⠀⠋⠀⠀⠐⢻⡗⠙⠉⠂⢨⠟⠓⠚⠛⢳⡀⠀⢤⠬⠶⠂⠿⠢⠀⣄⢀⠋⠀⠀⠀⠀⣷⡟⠉⠀⠀⠀⢻⠃⡀⢨⡇
⠀⠀⠀⠀⡇⠀⢐⡠⣅⠀⠀⠀⢠⡆⠀⢀⡔⠁⠀⢀⣄⠀⠄⠙⠚⠖⠳⠟⠛⢿⣳⠇⢀⢈⡁⢠⣄⠀⠛⠓⠛⠖⡀⠀⢀⡼⠃⠉⡾
⠀⠀⠀⠀⢡⠀⠈⠀⡁⢤⡀⠉⠀⠀⣄⣨⣇⡤⠀⢀⢤⡄⠤⢀⡁⠁⠈⠀⡀⠀⠧⣄⣉⡸⠦⢀⠈⠀⢠⠄⠀⢰⡟⠛⠛⢡⡤
⠀⠀⠀⠀⠀⠁⢘⡦⠄⢠⠔⢶⢰⣧⠈⢠⡟⡇⠀⣿⣶⣶⣤⡤⣜⣢⠀⠀⠁⠀⡞⣻⣯⡤⡛⠿⢇⠘⠂⠀⣤⣿⠁⢀⠒⠀⠀⠶⠋
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢠⣠⡆⡿⠀⢼⠃⡇⠀⢉⣀⢠⣼⣷⣯⡿⠀⠀⠀⠀⢿⣷⣬⡁⢹⡆⣀⠀⣠⣠⡿⠃⠀⢿⠆⡀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠳⣄⣴⡂⣽⡅⣿⠀⣹⠷⡷⠀⢼⠙⠚⠛⠌⠁⠀⢀⣾⣗⠈⠀⠈⠉⠀⠙⠋⠛⠙⡻⠃⠂⠚⠀⢆⠠⢆⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠓⠈⠁⣾⠀⢸⠄⠁⢻⣾⡁⢀⡀⠉⠓⠀⣾⡟⣿⡇⠀⠀⠛⠀⣀⡀⢴⠎⢀⡀⢰⡇⠀⠂⢀⣀⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠶⠉⢄⣽⠀⠾⠒⢻⣶⡙⣇⡸⣷⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠼⠟⣰⢚⠃⠈⠔⣢⠟⠼⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣀⣠⣸⣏⠐⣏⣯⠞⣏⠀⢁⠇⠋⠁⠐⢸⠰⠠⠄⡆⠄⠀⠤⢉⢀⠂⠁⠶⡈⠁⠛⠀⠘⢀⣀⠘⣆
⠀⠀⠀⠀⠀⠀⢠⣰⠞⠉⠉⢉⣉⢻⡆⢻⠁⢐⠞⢠⡄⢸⠥⣐⡓⢸⠀⠀⠀⠀⡆⠰⢃⣥⡈⠁⠖⢺⠟⠃⠠⠴⠘⠉⢻⣆⠈⡄
⠀⠀⠀⠀⠀⠀⢺⡇⢠⡠⠀⠿⣹⠾⡀⡸⢃⢆⡴⢋⠐⢄⠀⠀⠉⠉⠒⠀⠀⠂⠁⠉⠁⢀⠈⠑⡒⣯⡒⠈⠳⣬⡀⠀⣿⡏⣠⠃
⠀⠀⠀⠀⠀⠀⣿⡄⠀⢠⡌⠞⣻⠦⣈⠀⣠⢞⠴⠀⠀⠠⠰⠄⠀⠀⠋⠀⠀⠀⠀⢀⠤⠆⡤⠜⢉⣀⠛⣀⣄⡀⠻⣴⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠙⣇⣤⠏⠀⡢⠙⠓⠉⠀⠣⣀⠀⡤⠀⠤⠀⣃⠀⡛⡉⠙⣛⣯⣿⡯⠴⢈⣶⠞⠂⠉⢀⣼⢿⡿⠆⢹⣆⠀⠓⢄⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⣉⠛⡿⢏⢱⣤⠞⣻⢦⡄⢨⡫⡶⢠⠶⣄⠩⡈⠉⠐⢾⣩⠖⢁⣤⡼⠿⢣⣄⣤⣤⡞⠁⢠⠙⠋⠂⣻⠀⠀⢀⢺⣿⡇
⠀⠀⠀⠀⠀⠀⡌⢀⠔⠁⡴⠚⡁⢑⠀⢾⣽⢦⡐⠈⠢⡁⢸⡃⠔⠢⠒⢈⠠⠒⠁⡉⠀⣀⠴⠈⠁⠀⣆⠔⠋⣀⣸⣷⡟⣠⠤⣿⣤⠟
⠀⠀⠀⠀⠀⠀⢇⡈⠠⣄⠄⠘⠂⢀⠀⣄⢨⣮⠝⢷⣤⣌⠊⣡⠐⣤⠞⠁⣀⠀⣀⠤⠊⠁⠔⢲⠆⢙⣅⣮⣝⠋⣭⡃⠀⣠⡴⠟⡿
⠀⠀⠀⠀⠀⠀⠈⠃⠂⠀⠠⡀⠀⢈⠔⠉⠀⠀⠀⠀⠀⠀⠚⢤⠉⠀⠀⡴⠍⠊⡡⠄⢀⣤⡾⠿⣗⠦⡰⣷⠶⣤⣎⣷⠚⣇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣤⠀⠀⣠⠀⢶⠂⠁⡤⢖⣡⠔⣾⢷⡎⣻⢦⠛⣷⢷⣻⠶⣻⣴⣿⣿⣏⢶⣮⣷⡠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠸⠀⠀⠀⠀⢀⣠⠥⠒⠋⠀⠁⠐⢒⡬⠓⣯⠷⠋⠁⣠⡿⠁⣸⠇⣿⣬⠯⠦⣽⣈⣛⣋⣿⣆⠉⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠼⠀⠀⠀⠉⢠⠅⠀⠀⠀⠀⠄⢸⣡⠶⠋⠁⣠⡴⠊⠀⢠⣶⣻⣄⣻⣿⣿⣯⡿⣟⣏⣺⣷⡾⠽⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠛⠁⣠⠴⠋⠁⠀⠱⠄⢀⠉⠉⣸⣿⣿⣏⣉⡙⠋⠉⠉⠛⣠⡖⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡠⠴⠋⠸⠄⢀⢠⡤⠲⡶⠮⣉⣀⢀⣼⣯⠾⠙⠏⣿⠏⡉⢙⣻⠃⢲⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡖⠋⠀⠀⠀⠀⠂⠀⠀⠀⠸⣇⠀⠀⠀⢀⣼⡄⠄⠀⠐⡏⢀⡿⠄⠉⠀⠘⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠁⠈⠱⢤⡴⠀⠀⠀⠛⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣶⣶
⣿⣛⣛⣛⣛⣛⣛⣙⣛⣻⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣉⣩⣿⣭⣿⣿⣿⣙⣿⣯⣿⣏⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠶⢦⣤⡏⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⠀⣰⠋⠉⠛⢶⡇⣠⠴⢤⣄⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⣠⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠛⠁⠀⠀⠈⣷⠀⠀⠀⠀⣰⣿⣿⠟⠉⠀⢹⣦⣾⣿⡟⠋⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣈⣳⣄⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⣸⣿⣿⠏⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠳⠤⠤⠖⠋⠁⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⣸⠇
⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⢸⣿⣿⠇⠀⠀⣠⡾⣿⣿⡏⠀⠀⣀⡼⠋
⠀⠀⣼⣿⣿⡇⠀⢀⣤⣄⠀⠀⠀⢸⡇⠀⠀⣀⣤⣤⣤⣄⣼⣿⣿⣿⠀⠀⣿⣿⣿⠶⠖⠛⠁⠀⣿⣿⡷⠶⠛⣩⣶⣶⣦
⠀⠀⣿⣿⣿⠀⢠⣿⣿⣿⠀⠀⠀⣸⠁⠀⣼⣿⣿⠟⠉⢹⡟⣿⣿⡏⠀⠀⣿⣿⣿⠀⠀⠀⠀⠰⣿⣿⡇⢀⡾⠁⣽⣿⣿
⠀⠀⢻⣿⡏⠀⣼⣿⣿⡿⠀⠀⠀⡿⠀⠀⣿⣿⡏⠀⢀⡿⠀⣿⣿⠇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣧⣟⣀⣼⣿⡿⠃⠀⣤
⠀⠀⠈⠻⢿⠶⠋⣿⣿⡇⠀⠀⢰⡇⠀⠀⢿⣿⡇⢀⡾⠁⠀⣿⣿⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⢸⣿⣿⡿⣟⠛⠛⠉⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⢀⡾⠀⠀⠀⠈⠙⠓⠛⠁⠀⠀⢻⣿⠀⠀⢠⡇⠈⠻⣷⣄⣀⡴⠏⠀⠉⠀⠹⣄⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣠⠟⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠄⠒⠃
⠀⡎⠈⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠠
⠀⠡⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⢠⠋⠁⠀⠙⡄⠀⠀⠀⠀⢸⣿⡟⠛⠻⣷⡆⠸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⠋⠉⠉⢳⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠓⠺⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡿⠀⢿⣷⣄⣼⣿⠏⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠋⠁⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢸⣿⡇⣀⣼⡿⠃⠀⠀⠉⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠖⠒⠒⠋⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠿⣷⡄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣾⠿⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⢠⣶⣄⠀⠀⠀⠀⠀⣀⣀⡀
⠀⢀⣠⣤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⢀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣠⣿⣿⠛⠛⢷
⣼⣿⣿⠁⠀⠀⠀⠀⣴⣤⡀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠦⠤⣿⣿⣿⣀⣀⣠⣿⣿⠟⠀⠀⢸⡇
⢿⣿⣿⡀⠀⡿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣀⡀⠀⠀⣿⣿⡇⠀⠀⣽⣿⡇⠀⠀⠀⣼⠁
⠈⠛⠿⠿⠞⠁⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⣿⣿⡇⠀⠀⣿⣿⠃⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⣴⣶⡄⠀⠀⣿⣿⡇⠀⠀⣿⣿⠗⠛⢁⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣤⠶⢾⣿⣷⠤⠤⠤⠶⠃⢀⣴⣿⡿⠷⣾⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡀⢠⡞⢰⣿⣿⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⣾⣿⡟⠀⢰⡇⢿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⣧⡞⠀⢸⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⢻⣿⡇⠀⡾⠀⢸⣿⡇⠀⠀⣿⣿⣇⠀⣰⢿⣿⡇⠀⠀⡟⠻⠟⠀⠀⠈⣿⣿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠛⠷⠞⠁⠀⠈⣿⡇⠀⢰⠇⠈⠛⠛⠋⠈⢿⣧⠀⣸⠃⠀⠀⠀⠀⠀⢿⣿⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠘⢿⣄⣠⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣸⣿⣠⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣻⣿⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⢻⡿⣯⣿⢿⣯⠿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣽⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⣿⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢿⠻⠯⠋


⠀⠀⠀⠀⠀⠀⣀⢀⠀⡀⣀⠰⠀⡀⣀⢀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠨⠀⠅⠈⢸⠀⠅⠁⠨⠀⠅⠀⠀⠀⠀⠀⣈⠇⢴⠀⠀⣼⠀⣤⢰⠟
⠀⠀⠀⠀⠀⠀⠸⠈⠆⠣⡈⢺⠚⡠⠃⡜⢠⠁⠀⠀⠀⠀⠀⠉⡓⠂⣀⢰⡿⢀⠻⡳⠞
⠀⠀⠀⠀⠀⠀⢀⣁⣘⣠⠬⣹⣍⠤⠊⡠⠃⠀⠀⠀⠀⠀⠀⠀⠙⠙⠁⠈⠏⢿⣨⡖
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠒⢺⠖⠒⠉⣤⣌⠀⠀⠀⠀⠀⠀⠀⢦⣾⣿⣿⣷⣤⡁
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣠⣼⣤⣄⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠉⠉⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢰⡦⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠊⠒⢤⡇⠀⠀⢠⢇⠀⠀⢈⣦⠒⠈⠒⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⠚⠑⢲⠀⡇⠀⢠⠏⠈⢆⠀⢸⡇⡴⠚⠙⢦⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡏⠀⣀⠈⠀⡇⢠⠃⠀⢀⠈⢧⢸⡇⠀⢀⢀⠘⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠘⣞⠡⠊⠀⠀⣷⠃⠀⣧⣰⡆⠊⣾⡇⠀⠘⠠⢱⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢏⡰⢉⠙⠀⣿⠀⡄⢆⣡⢆⠆⢸⡇⠺⡉⠳⣸⠃⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣈⢧⡁⢤⡀⡿⠀⠨⣶⣷⡮⠀⢸⡇⡤⠌⣶⢋⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠱⢷⡝⢦⠀⡇⢠⡾⠋⠉⠳⣄⢸⡇⢠⢟⣽⠞⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣲⠄⠩⡀⢀⡿⣿⡱⠐⡇⠱⢹⡿⡇⠀⡉⠀⣷⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡼⣠⠀⡘⢠⡈⣧⣇⣇⣠⣄⣀⣿⣧⡇⣨⠘⡐⢠⡹⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣱⡷⣋⠁⢙⣿⡏⠛⣿⣼⣿⣜⡞⠉⣿⡏⠀⢍⠺⣧⣻⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⢊⣩⣍⣉⡩⠍⠲⣄⣸⡿⡱⣁⠀⠀⢀⢣⣼⡏⣨⣯⠈⣿⡄⢁⢠⠈⢀⢣⣹⣿⣠⠔⠊⠍⣉⣉⡍⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠊⢉⡉⠉⣛⠻⡷⢦⡌⢹⡟⣡⠋⢢⡇⠀⢸⡜⣠⡟⠹⣧⡙⣁⠀⠘⣤⠋⢇⠳⣿⢁⣬⢶⠿⢛⡉⠉⣉⠑⠚⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠩⢿⣾⣇⣄⡰⠈⠢⡙⢾⣵⠃⠈⠘⢠⣇⣿⣸⠋⡴⢦⡘⢧⣸⣯⣆⠋⠈⠀⢷⣿⠟⡡⠁⠂⣠⣠⣷⣞⠯⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣀⢙⡎⢰⠻⣌⡇⠀⢈⢠⠛⣡⠟⠡⠊⠁⠀⠝⠤⡙⢏⡙⢂⠀⢈⠘⣇⡾⢶⢉⣞⢁⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⢻⣾⠀⠙⡇⢀⣾⢬⡴⡡⠊⠰⢀⠀⡀⢠⠂⠉⢮⡳⣬⣾⣄⢠⣏⠀⢸⡾⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⢀⠐⣅⡵⢋⢎⡲⠁⢰⣀⣾⣤⣧⣼⣀⣠⠀⠱⡘⣌⠓⣀⡆⢀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠄⠀⣿⢧⡎⠰⠁⣤⣞⣿⡿⠛⠛⠻⣿⣙⢦⡄⢱⠘⣬⢿⡇⠀⢼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠠⢀⠏⢺⠁⠇⢸⠋⠈⡴⠀⢾⣹⠀⣀⡏⠑⣷⠘⠀⠱⢨⢇⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢎⢧⣸⡄⠀⢘⣷⣤⣳⢥⣤⣤⣤⣾⣤⡾⡟⠸⠀⡞⣰⣣⠞⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⠈⠈⠦⣳⡀⠀⠀⡏⡹⡟⠙⡟⢙⣯⡉⡇⠀⠀⣰⣡⠆⠁⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡖⠂⠀⢉⢷⣤⠀⠉⠀⠹⠀⠃⠘⠀⠈⠀⢀⣜⠏⠀⠀⠲⣦⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⠊⢉⠉⣼⠦⣓⣤⣀⠀⠀⠀⠀⣀⡤⣜⠃⢿⠈⠉⠑⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡄⠀⢰⣯⢡⠃⢠⣌⡇⠀⠀⠰⣇⡤⠈⣏⠋⣧⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⢸⣿⡜⣷⢡⡄⠄⠀⡆⠃⢄⢤⣰⡏⣶⡿⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣷⣿⠢⣽⡄⣥⣷⠁⣺⡦⣻⣼⣿⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⠿⠿⠿⠿⠿⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠟⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⢿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠚⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠰⠶⢶⡴⣜⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠛⢹⠙⢻⣦⡀
⠀⠀⠀⢠⡄⠀⢷⣞⣁⣬⠭⡅⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⣴⣶⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣈⣳⣼⣴⣋⠈⣷
⠀⠀⠀⠈⢷⣄⣠⠟⣅⠀⣠⠇⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣤⠟⠛⠿⣯⣿⣟⣿⣷⣆⠀⠀⠀⠀⠀⠀⣯⢉⡽⢿⢫⡉⢉⡟
⠀⠀⠀⠀⠀⠉⠁⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣶⣶⡿⣿⣿⡿⢿⣧⠀⠀⠀⠀⠀⠈⠻⣤⣿⣠⡽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣾⣟⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠉⠉⠉⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⣠
⠀⠀⠀⠀⣄⣀⡿⡴⣿⣀⣀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣟⣦⣀⣀⣀
⠀⠀⠀⢀⣨⡿⠁⠀⠈⢿⣁⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠹⣿⠏⠉⠉⢿⣽⠃
⠀⠀⠀⠈⠉⣿⣤⣠⣼⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⣴⣟⣦⣀⣀⣾⣳⡀
⠀⠀⠀⠀⠘⠁⠈⠟⠀⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠉⠙⣯⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡴⢉⣿⠳⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠐⡀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢣⣾⣇⡴⢸⡇⠀⠀⠀⠀⠀⠀⢡⠡⠁⣀⣀⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢛⣼⣟⣥⣴⠟⠀⠀⠀⠀⠀⠀⠀⠈⡄⣾⣟⣙⣿⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡏⠷⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢜⡛⢛⡫⠊

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

⠀⠀⠀⠀⠀⢀⡀⡠⠐⣲⣷⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣯⡭⠁
⠀⠀⠀⠀⡴⠛⣻⣟⣿⣡⢾⡟⠚⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⢏⣯⡙⠐⠀⣠⣶⡒⣀⣤⣤⡀
⠀⠀⠀⠀⠀⢸⠟⠙⠛⡹⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⡙⢄⠀⠀⠋⠘⢷⠈⠋⠟
⠀⠀⠀⠀⢀⠎⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⡀⠀⠑⠤⡀
⠀⠀⠀⡠⠃⠀⡠⠊⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⡈⠑⠄⡀
⠀⠠⠞⢁⣤⠞⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠗
⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⢾⣻⠛⣉⠉⢽⣭⢹⣍⡍⣭⢯⠛⣿⠶⣦⣤⣀
⠀⣤⣤⣤⣤⣤⣶⢾⢛⡟⣉⠧⣞⣒⣣⣥⣽⣽⠶⠾⠶⠶⠷⠶⢿⣬⣥⣞⣁⡵⠨⢝⣻⣶⣤⣄⡀
⠀⠈⠻⠷⣶⣶⣿⠶⠶⠿⠟⠛⠛⣉⣉⣤⣤⣶⢶⠶⠶⢶⣶⡶⢶⣦⣤⣭⣉⡛⠛⠷⠶⢶⣷⣽⡿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠟⡛⡭⠕⠀⠀⠀⢝⣾⣿⣿⣿⣷⡖⡒⢾⣯⢟⢛⠷⢶⣤⣄⣀⣀⣀⣀
⠀⣿⢶⢶⣦⣶⠶⠟⠛⢩⠂⠤⡸⣶⣽⢙⣛⣆⠀⢸⣿⣿⣿⣿⣿⣷⣌⣌⠁⠨⢒⡺⠔⣠⣈⡭⢩⡛⣿
⠀⣿⢴⡀⣀⣂⡐⠰⢫⣀⠂⠻⠋⠙⣋⠄⠕⢳⠌⠚⢿⣿⣿⣿⣿⣃⡴⢙⣏⢪⠭⣛⠳⣉⣭⣼⡶⠶⠿
⠀⣿⣰⡟⠉⠉⠛⠷⢮⣤⣁⠨⢌⡥⠭⣠⠎⡳⡛⢓⡤⠎⢫⡽⣦⡬⣉⢤⡯⠀⡈⣣⡾⠋⠁
⠀⣿⡏⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣬⣥⣀⣖⣈⣠⣉⣈⣒⣒⣩⡤⡀⠣⣡⡇⡆⣿⠀⠀⠀⠀⠀⢀
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠋⠉⣽⢟⢐⢜⣼⢿⡧⢄⡻⠶⣶⡶⠶⣿⠏
⠀⠀⠀⠀⣠⣴⠶⠿⠷⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⠘⣹⡟⠁⢸⡇⠀⣟⠐⠪⢽⣾⠃
⠀⠀⠀⣼⠟⢀⣯⣴⠭⠓⡨⢻⣆⠀⠀⠀⠀⠀⠀⢀⣴⢟⡉⣫⡾⠋⠀⠀⢸⡟⠰⠌⣒⢂⣾⠃
⠀⠀⢸⡏⠈⣾⣿⠁⠠⢐⣈⣓⣿⠀⠀⠀⠀⢀⣴⡟⡁⣠⡾⠋⠀⠀⠀⠀⢸⣇⡇⡰⢻⡿⠁
⠀⠀⠘⣷⢱⢻⣿⣦⣂⣐⣡⡾⠃⠀⣀⣤⡾⠻⢙⣽⠾⠋⠀⠀⠀⠀⠀⠀⢸⡇⡢⢩⡿⠁
⠀⠀⠀⠹⣯⡧⢙⣿⠿⣿⡷⠶⠶⢿⠻⣅⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣰⡟⠁
⠀⠀⠀⠀⠈⠛⠿⢾⣤⣭⣤⣵⡷⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡄⣶⣯⣽⣶⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡍⣭⡆⣿⣿⣿⣿⢰⣭⢩⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⣿⡿⢿⣿⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⣿⡇⣩⣾⣷⣍⢸⣿⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡟⣼⣿⢟⠻⣿⣧⢻⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣿⣿⣷⣿⢣⣿⣿⡜⣿⣾⣿⣿⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢱⣶⣾⣝⠻⡏⣿⠿⠿⡿⢹⠟⣫⣷⣶⡎⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣎⢿⣮⢛⣥⡶⠾⠿⠿⠷⢶⣬⡛⣱⡿⣱⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣡⣵⣶⢰⣿⢵⡮⣿⡆⣶⣮⣌⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣌⠻⣿⣎⡻⣶⣶⢟⣵⣿⢟⣱⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣊⣿⡶⣭⣟⣓⣚⣻⣭⢶⣿⣑⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠫⢲⣯⡭⢭⣽⡖⠝⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠮⠵⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠶⣤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢷⠃⡏⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⢢⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢦⠀⠁⣤⠀⠀⣤⣠⠄⠀⠰⠬⠀⠀⠄⠀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠉⢻⠄⡟⠀⠀⠀⡄⠀⠀⠀⠀⠠⡄⢛⠂⠘⠴⢆⠙⠆⡀⠀⠁⠀⠀⠀⠇⠀⠀⠘⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡀⠀⠀⠀⠀⠘⠀⡒⠙⠂⠀⠀⠀⠄⠠⡙⠛⠀⠀⠀⠀⠀⠀⠐⡀⠀⠰⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠀⠆⠡⠀⠀⡄⠘⠀⠄⡀⠀⠻⣨⠆⠀⠴⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠁⢀⡄⠀⠀⢂⠀⠀⠀⠁⠂⡀⠀⢈⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣘⡣⠀⠠⣅⠀⠒⠀⠂⢰⡫⠀⠀⢈⡅⠌⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⢂⠈⠁⢀⠀⠑⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠈⠀⠈⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⠀⠄⠀⠂⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣩⠀⠀⡄⠀⡀⠀⠞⠀⠀⠰⠀⠀⠁⠀⠂⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠒⠀⠀⠰⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠠⠠⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⢀⠀⠀⠀⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠀⠶⠚⠚⠀⡂⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠀⡘⠁⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⣤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⠁⠀⠀⠀⠀⠂⠀⠀⠐⠰⠄⠀⠀⠀⡀⠄⠀⠀⠈⠀⠀⢀⡠⠁⠂⠀⠀⠀⢰⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠀⡠⠀⠀⠀⠀⠂⠠⠈⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠙⠃⡄
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢄⡀⠀⡐⠀⠀⠄⣀⠴⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⡠⠀⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡘⠀⢀⠈⠘⠧⣄⠀⠘⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⢀⡽⣀⠀⠀⠀⠀⣠⡴⠂⣀⠄⠁⠀⢩⠄⢀⡀⠁⠄⠀⠈⠰⣄⡈⢷⡄⠀⠀⠀⣰⠠⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢠⣦⠀⠀⠀⣿⠠⡈⢿⣄⡀⠤⠀⠀⠀⠁⠀⠀⠀⠒⣤⠋⠁⠨⣇⠀⠀⠠⠔⠋⠀⠈⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢃⠀⠀⢹⡀⠈⠀⣈⡗⢂⠀⠀⠀⠀⠀⠀⠠⡀⠒⠖⠀⢸⠇⢀⣴⠎⠁⠀⠀⠀⣀
⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡶⠀⠀⠃⢀⣀⡬⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠑⣀⡔⠋⣁⠐⠀⠀⠀⠀⠀⠀⠁
⠀⡘⠀⠀⠀⠰⠁⠀⠀⠀⠀⡘⠀⠀⠁⢐⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢚⠣⠀⠴⠫⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠒⠂⠀⠀⠀⠂⠀⢀⠀⣀⠀⠀⠂⠢⠀⠀⠀⠠⠀⢀⠀⠀⠀⢀⠀⠀⠈⠀⣀
⠀⢨⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠀⠂⡃⠀⠀⠀⠀⠀⢀⡅⠂⡱⠀⠄⠀⠀⠐⠑⠀⠀⢀⡀⠈⠑
⠀⢉⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠲⠁⠈⠀⠠⠉⢀⠀⠀⠸⠀⠀⠀⠁
⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⣿
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢻⡀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⡇
⣿⣷⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⢷⡀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠘⢿⣿⢸⣿⡄⢠⣄⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢣⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⣠⣤⣬⣾⣿⡟⢸⣿⣷⡀⠀⠀⣀⠼⠤⣄⡀⠀⠀⢠⠞⣡⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠈⠛⠿⣿⠟⢧⡘⣿⣿⡇⢀⠌⠁⠀⠀⠀⠹⣦⣰⠋⡼⠁⠀⠀⠀⠀⠠⠴⢶⡍⠁
⠀⠀⢀⣤⣶⣶⣾⣿⡿⣤⣸⢤⠸⠗⠀⠀⠀⠘⠃⠸⠀⠀⠀⠀⣀⣠⡤⠴⢋
⠀⠀⠙⠿⣿⣿⡿⠋⠀⠀⠗⠚⠗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⡈⢳⡀⠀⠀⢀⣀⠴⠊⠁⠘⠀⡄⣧⣹⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡏⠉⠉⠀⠀⠀⠀⠀⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⢠⣾⣄⠀⢠⣶⡄⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⣿⣷⣦⣼⣿⣿⣷⠿⠿⠿⠿⠿⣿⣿⣶⣿⣿⣆⣠⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣸⣿⣿⣿⡿⢉⣁⣀⣠⣿⣯⣤⣤⣤⣤⣴⣿⣏⣄⣀⠉⠙⢛⣿⣿⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⠟⠋⢁⣸⣿⡷⠛⠛⢉⣿⣅⣠⣄⣤⠿⣤⡠⣀⣉⣿⠛⠛⠶⣿⣿⣉⠉⠻⢿⣧⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⣿⣇⣤⡾⠛⣉⣹⠶⠄⢋⣈⣁⣤⣤⣤⣤⣤⣤⣥⣬⣉⣈⡓⠶⠦⣤⡟⠛⠷⣤⣀⣽⣿⣿⣇⣀⣠
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠟⠁⣀⡿⠟⢻⣤⠜⢍⣡⣴⣾⣿⣿⣿⠿⠿⠿⣿⣿⣿⡿⠿⣿⣿⣿⣷⣦⣄⡙⠖⣂⣈⣿⢿⣍⠉⠻⣿⡿
⠀⠀⠀⠀⠀⣤⣤⣼⣿⡇⢠⣾⠋⢠⡤⠋⣠⣾⣿⣿⢿⣿⠿⣅⠀⠶⠀⠻⠭⠿⠯⠾⠂⢤⠈⣉⣛⣿⣿⣿⣷⣌⡉⢷⡀⠙⢷⣄⠈⢻⣶⣶⡆
⠀⠀⠀⠀⠀⢻⣿⠏⢹⣿⠟⢹⡟⢋⣴⣿⣿⡿⠋⣴⣏⡽⢖⣉⣤⢶⣶⣶⣾⣻⣷⣶⣶⣤⣄⡙⢧⣽⠻⡟⢿⣿⣷⣦⠙⠦⣤⠿⣿⠿⠿⣿⣀⣀⡀
⠀⠀⠀⣶⣾⣿⠏⢀⣼⠋⢀⡙⢠⣾⣿⡿⢋⠰⠶⢀⣤⠶⣟⣿⠾⠟⠛⢋⡿⢿⣟⠛⠻⠿⢿⣟⣷⢮⡛⠉⣤⡙⢿⣿⣷⣄⢻⡀⠘⢿⡀⠘⣿⣿⠇
⠀⠀⠀⢈⣿⣿⣄⣾⠛⣶⠛⣰⣿⣿⡿⣾⣹⢇⣴⢿⣽⣿⠉⣠⡶⢶⡾⣿⣾⣷⣾⣷⣶⠶⣦⡉⢻⣿⡻⣦⡀⢰⠦⣹⣿⣿⣎⠱⣄⣸⣷⣀⣘⣿⣤⣤⡀
⠀⠐⢿⣿⡏⠙⣿⠋⢰⠆⣴⣿⣿⠋⣭⡼⢣⡿⣷⠟⢁⣼⡿⢋⡴⣯⣴⡿⠋⠈⠻⣦⣼⣷⣌⠻⣯⡀⠹⣿⡿⣌⢶⡿⢿⣿⣿⣆⠰⠁⠘⣿⠿⢿⣿⠏
⠀⢀⣨⣿⠁⢠⣿⣤⡾⢠⣿⣿⡏⣤⡀⢠⣯⣿⠋⢀⡿⢁⡴⣿⣴⡿⢋⣴⠟⢷⡄⠘⠿⣷⣿⢷⣌⢻⡄⠘⣿⣽⡎⠱⠂⢹⣿⣿⡞⢧⡀⣹⡇⠀⣿⣷⣶
⠐⢿⣿⣿⡀⣾⠇⣸⠇⣸⣿⡿⢀⡀⢀⣞⣿⡇⠀⣸⡿⣿⣶⡿⠋⣀⠈⠻⣷⡀⠀⣤⡄⠈⢿⣶⣿⢿⣿⠀⠘⣿⣿⡄⢴⠄⢿⣿⣇⢠⡟⠙⣿⡀⢹⣟⠁
⠀⣠⣼⡿⢿⣿⣀⣼⠂⣿⣿⣷⣬⣱⢼⣯⣿⢀⡶⣿⣶⡿⠋⣀⡀⠛⢁⣤⠞⠻⣦⡈⢀⣴⣦⡈⠻⣶⣿⢷⣄⣿⣿⡇⢠⣦⣸⣿⣿⠢⡗⠀⣿⡿⣾⣿⣷
⠘⢿⣿⡇⠀⣿⠋⢳⠀⣿⣿⡿⣁⡽⠻⣿⣿⢏⡸⣿⣏⡀⢼⣏⣀⣴⢿⡁⠺⡷⢈⣿⠟⠁⣩⠟⠀⣸⣿⠟⣹⣿⡿⣿⣰⣼⣿⣿⣿⠀⣟⠴⣿⠖⢹⣿⡁
⠀⣠⣿⡇⠀⣿⠀⢹⡄⣿⣿⡇⢈⡁⢘⣿⣿⠈⠻⣿⠛⣷⣄⠙⠟⠁⣀⠻⣦⡶⠛⠁⣤⡀⠉⣠⣾⠛⣿⠟⠁⣿⣻⡏⠱⡬⢻⣿⣿⡠⡅⠀⣿⠀⢸⣿⣿
⠘⠻⢿⣿⣷⣿⣷⠾⡄⢸⣿⣿⡈⠋⠀⢷⣿⣇⠀⢘⣿⣿⡛⣿⣆⡈⠛⠀⡈⢻⣦⡀⢉⣠⣾⠟⣿⢿⡏⠀⣰⡿⡿⠁⣴⢀⣿⣿⡏⠰⣆⢠⣿⣧⣼⣏⠁
⠀⢀⣾⣿⡄⠙⣧⠀⢟⠞⣿⣿⣧⢞⢻⡌⣿⡿⣆⠀⢻⣌⠻⣿⠛⣿⣦⠘⠳⡾⠋⣠⣿⠟⣿⠟⣡⡾⠁⣠⣿⣿⢃⣤⡄⣼⣿⣿⢁⡞⠉⣿⡇⠙⣿⣿⠷
⠀⠙⠛⠿⣧⠀⢹⣆⣠⢇⠘⣿⣿⣿⣯⣹⠈⢷⣝⣷⣄⣼⣷⣌⡻⣿⠛⣿⣦⣠⣾⡟⣻⠟⣡⣾⢿⣁⣴⣿⡾⢡⢾⣼⣿⣿⣿⠃⣸⠁⢠⡟⠀⣸⣿⡀
⠀⠀⢀⣾⣿⣶⣦⣿⣇⠈⠓⡘⢿⣿⣧⡈⢠⡄⠙⢾⣝⢿⣤⡉⠛⠿⠻⣿⡻⢟⣿⠟⠻⠛⢋⣠⣾⣟⡷⠋⠀⠻⢾⣿⣿⣿⠃⡰⠻⣶⣿⣇⣰⡿⠿⠟
⠀⠀⠈⠉⠙⣿⣏⠉⢻⣆⠈⣷⡈⢻⣿⣿⣦⡀⣴⠢⣍⠻⢿⣿⣿⣶⣦⣬⣿⣾⣥⣴⣶⣾⣿⣯⠗⢋⣀⠘⢃⣴⣿⣿⡟⢁⡖⠃⣠⡿⠉⣿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣦⡀⠹⣿⡉⠉⢶⡙⢿⣿⣿⣷⣾⡏⢱⡆⢈⡉⠛⠛⣛⣷⣛⠛⠛⠛⠉⣁⡖⣿⣥⣿⣴⣿⣿⡿⢋⡴⠛⢧⣴⠏⠀⣴⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠈⣿⣷⣶⣾⣿⣦⣀⣿⢦⡈⠻⢿⣿⣿⣷⣤⣌⣁⡐⢿⣩⣿⡩⠿⠀⠛⣀⣩⣵⣿⣿⣿⡿⠟⣁⡴⠛⢁⣴⠿⣿⣄⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠐⠿⠛⢻⣧⣄⠈⠻⣷⣄⠈⠛⣦⣈⡙⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⠴⠆⠉⣹⣶⠟⠋⢀⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢷⣄⣬⣽⣷⣶⣶⡉⠉⠇⠲⣤⡍⣭⢉⣉⢉⣉⠻⡉⣉⢡⡴⠶⠚⠋⣯⣀⣴⠾⣿⠇⣠⣶⡟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⡀⠉⠻⢳⣶⣶⣾⣀⣀⡀⢉⣟⠁⠀⢀⣈⣿⣤⣤⣶⡾⠛⠉⢉⣸⣿⡿⠻⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠸⣿⣿⡿⣾⣿⣟⣁⠈⠉⠙⠛⢛⣿⡿⠛⠛⠋⠉⠉⣼⣿⣤⣶⣾⡟⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⢾⡿⠟⠻⣿⣿⡿⢶⣿⣿⡷⣶⣶⣾⡿⣿⣿⡟⠉⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⡴⠛⠛⢤⡈⠻⠏⠀⠈⢻⡟⠙⢾⣽⣃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡃⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⡀⢈⣷⠀⠀⠀⠉⢳⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠂⠀⠀⠀⢀⠿⠧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣟⣯⣽⣏⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⣰⡇⣶⣄⣈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⢿⣿⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣠⠴⣿⣿⡀⠀⠀⠀⠀⣿⠜⢹⡏⣈⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⣾⠧⢤⡈⢿⡟⠁⠀⠀⠀⢀⡿⠌⠁⢰⡿⠛⢿⣦⡀⠀⠀⣿⡤⠊⣿⠀⠀⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣿⣉⣋⠙⣾⠇⠀⠀⠀⢀⣎⠀⠀⣠⣿⢿⣦⣄⣙⡅⠀⠀⢸⣧⣞⡹⣿⡟⠻⠶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⢀⣿⠉⠑⠚⢼⡇⠀⠀⠀⣀⢁⣴⣿⣿⡿⢠⡌⡉⣿⠃⠀⠀⠸⣿⡴⠋⠈⢷⣤⣄⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣅⡈⢾⡃⠀⠀⠀⣾⣿⣿⣿⣿⠷⣤⣄⢰⣟⠀⠀⠀⠀⣻⠇⠀⢀⡴⢿⣦⣤⣌⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠉⢿⣀⡀⠙⣾⠇⠀⠀⠀⣿⣿⡿⠋⣿⣤⠀⠈⠙⢿⣆⠀⠀⠀⠈⣿⡀⠖⣠⡾⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣀⣸⣿⣿⣦⣸⡆⠀⠀⠀⣿⡟⠁⡀⣻⣇⠈⠁⠐⠂⣿⡆⠀⠀⠀⠘⢷⣰⡟⢁⡾⢿⡀⠙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣡⡞⠋⠙⣍⠹⣿⡿⠀⠀⠀⠀⠈⣧⡾⢠⢹⣿⣿⡶⢢⣤⣨⣿⠀⠀⠀⠀⠈⢿⣁⠎⢠⣻⣿⣶⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⢀⣠⣴⡉⠀⣼⠁⠀⠀⠀⠀⠀⢸⣕⡂⣼⣿⣿⡿⢷⣮⡻⣿⡄⠀⠀⠀⠀⠸⣿⡐⢆⣹⣿⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢡⣞⡱⣾⣿⣷⣘⠀⠀⠀⠀⠀⠀⠀⠉⡁⣿⣿⢙⣿⣄⠈⠛⢾⡇⠀⠀⠀⠀⠀⢿⢷⡄⢳⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠢⡔⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⠈⣿⣿⣿⣦⣸⡇⠀⠀⠀⠀⠀⠈⢿⣽⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁