Monkey wrench Emojis & Symbols

⠀⠀⠀⢀⡴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⣤⡀
⠀⠀⣰⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢦⡀
⠀⣼⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢷⡀
⢰⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⡄
⡾⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠹⣆
⡇⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⠹⣦
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢸⡟⠓⠘⢷⡀
⠈⢷⣄⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⡿⣷⠀⣿⠋⠿⠛⠿⠁
⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⢸⡇⣌⠁⣽⢠⡌⣿⠀⣇
⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⢸⣷⣿⣾⣷⣾⣧⣿⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⢸⣿⢛⣟⢻⣟⢻⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⣀⣀⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢠⡿⠁⠀⠉⠆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⣇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⡆⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡄⠀⠀⠀⠀⠆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢳⡀⠀⠀⠀⠀⠂⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢧⠀⠀⠀⠀⠈⠄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢇⠀⠀⠀⠀⠈⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡈⣆⠀⠀⠀⠀⠘⠄⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠘⡄⠀⠀⠀⠀⠘⡄⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⡀⠀⠀⠀⠀⠘⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢱⡀⠀⠀⠀⠀⠸⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢳⠀⠀⠀⠀⠀⠹⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⢧⠀⠀⠀⠀⠀⠹⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⣆⠀⠀⠀⠀⠀⠻⡌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⡆⠀⠀⠀⠀⠀⢻⡌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⡄⠀⠀⠀⠀⠀⢳⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡄⠀⠀⠀⠀⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠻⣆⡀⢀⣀⡼⠃⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⠙⢉⣥⠶⠶⠶⢤⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⡟⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⡇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⣀⡼⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⣽⣛⣋⣭⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⢀⣀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣄
⠀⠀⠻⣿⣿⣿⡟⠀⢀⣾⠿⣿⣿⣷⣀
⠀⠀⠀⠈⠻⢿⡇⠀⢸⡇⠀⣿⠁⠉⠻⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣾⣯⣀⡀⠀⠀⠈⠉⠛⠷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⢶⣤⣀⠀⠀⠀⠈⠙⠻⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣤⡀⠀⠀⠀⠀⠉⠛⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣤⣀⠀⠀⠀⠀⠈⠙⠻⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⣩⣿⡷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⢰⣿⠁⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣴⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣶⣄⡀
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⢰⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⡿⠛⢷⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣴⠟⠻⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⡿⢋⡴⠛⠀⠈⠻⣿⣷⣄⣻⣿⠟⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢿⣿⣁⣴⣿⡿⠋⠀⠘⠳⣌⠻⣿⣿⣿⣿
⣿⣿⠟⢋⡴⠋⠀⠀⠀⠀⠀⢙⣿⣿⠟⠁⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠙⢿⣿⣏⠀⠀⠀⠀⠀⠈⠳⣌⠻⢿⣿
⠈⠳⠞⠋⠀⠀⠀⠀⣠⠴⣶⡿⠋⠀⠀⠀⠀⠀⠀⣠⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡀⠀⠀⠀⠀⠀⠉⠻⣷⡶⣄⡀⠀⠀⠀⠈⠓⠶⠋
⠀⠀⠀⠀⠀⠀⣠⠞⣡⡾⠋⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠈⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢼⡋⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢦⡀
⠀⠀⠀⠀⠀⢾⣥⡞⠁⠀⠀⠀⠀⠀⠀⠠⣾⡏⠀⢀⣴⡿⢿⣆⡀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣶⣄⠀⠈⣿⡦⠀⠀⠀⠀⠀⠀⠈⠙⣦⣽⠆
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠲⠶⣦⡈⠻⣶⡿⣿⣿⣦⡙⢷⣆⠀⠀⠀⢀⣶⠟⣡⣾⣿⡿⣷⡾⠋⣠⡶⠶⠂⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⡀⠀⠀⠈⠻⣦⡉⠐⢿⣿⣿⣿⣎⣿⡆⠀⠀⣿⣏⣾⣿⣿⣿⠇⠈⣠⡾⠋⠀⠀⢀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⠀⠀⠈⢷⣄⠀⠻⣿⡛⠟⠋⠀⠀⠀⠈⠛⠟⣿⡿⠁⢀⣾⠋⠀⠀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠙⢷⣄⠈⢻⣦⡀⠀⠀⠀⠀⣠⡾⠋⢀⣴⠟⠁⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠙⢷⣄⠙⢿⣆⡀⣠⣾⠟⢀⣴⠟⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠙⢧⡀⢙⣷⠟⠁⣴⠟⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣹⡟⠁⣠⠞⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠋⣠⡾⠋⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣠⡾⠋⠀⠀⣠⣾⠋⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⡴⠋⠀⠀⣠⡾⠋⠈⠳⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⡴⠋⠀⠀⣠⡾⠛⠻⣦⡀⠀⠈⢷⣄⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢃⣴⠟⠁⠀⣠⣾⠟⠀⠀⠀⠙⢿⣦⡀⠀⠙⢷⣌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣰⠟⠁⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠙⢦⡈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣠⠞⠁⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠙⢦⡘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣡⡞⠁⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠹⣦⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠿⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠸⠇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡻⠋⠻⣷⣜⣷⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢶⣞⣵⡿⠋⠹⣛⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡩⠊⠀⠀⣀⣼⠟⠁⠀⠀⣱⡀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠉⠻⣄⣀⠀⠈⠳⢝⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡪⠊⠁⠀⠀⣠⠞⢁⣹⡂⠀⠀⠀⠀⢸⣅⠙⢦⡄⠀⠀⠈⠳⣕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠐⢦⡙⠶⣿⣧⡻⣆⠀⢀⡴⣫⣾⡷⠾⣡⠶⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠙⢦⡘⢿⡿⠞⠀⠈⠳⣿⠟⣡⠞⠁⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢦⡙⢦⠀⢀⡴⢃⡔⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢆⡷⢋⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢎⡴⠋⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡱⠋⢀⡴⠻⣜⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠊⢀⡴⠋⠳⣄⠈⠳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⢁⡴⠋⠀⠀⠀⠈⠳⣄⠈⠢⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⠞⠁⣠⠟⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄⠘⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣈⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⣁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⢴⣿⠏⠀⢀⡜⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠉⠳⣦⣼⡀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠙⡳⢦⣤⡀⠉⠻⠿⡶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠴⢴⣠⠁⣪⠛⠻⣶⣤⣼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⡿⠃⠀⠀⠀⠉⠙⢋⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠄⠀⠀⠀⠀⠀⠚⣁⡀⠤⠤⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡁⠐⠁⠀⢀⠄⣊⠴⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⡡⠊⠀⠀⡠⢖⡥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣩⠔⠁⠀⠀⡴⣊⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⡵⠊⠁⠀⢀⠔⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣫⠖⠉⠀⢀⡠⢚⡥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⡵⠊⠁⠀⣠⡴⢋⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣫⠖⠋⠀⢀⣴⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⣠⢖⡵⠞⠁⠀⣠⡶⢋⡥⠊
⠀⠀⠀⢀⡤⢊⠴⠋⠀⠀⣠⠾⢋⠔⠋
⠀⢀⡴⣫⠜⠁⠀⢀⡴⠚⣡⠞⠁
⣼⣡⠞⠳⣄⠀⠴⢋⡤⠊
⢿⡻⣄⣀⠾⢂⠔⠋
⣈⣻⣶⣵⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡴⠶⠛⠛⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡉⠻⣧⡀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠻⣦⡈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠻⣦⡀⠀⠀⠈⠻⣦⡈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢷⣌⠻⣦⡀⠀⠀⠈⠻⠾⢿⡄⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠙⢷⣌⠻⣦⡀⠀⠀⠀⠀⣹⡷⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⡴⢦⣄⠀⠀⠙⢷⣼⠋⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⡾⠋⠀⠀⠙⢷⡀⠀⠈⠻⣄⣠⡾⠋⠀⠀⣄⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⡾⠋⠀⠀⠀⠀⣀⣨⡿⠀⠀⠀⠈⢿⣄⠀⢤⡀⣸⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⡾⠋⠀⣀⣠⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⢳⣤⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠸⣯⡴⠞⠛⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⠉⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⣠⡾⠋⠈⠳⣄⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⢀⡾⠋⠀⠀⢰⣄⣸⡿⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⢦⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣠⡾⠃
⠀⣠⡾⠋⠀⠀⠀⠀⠀⢀⣴⠋
⣼⠋⣴⠞⠳⣆⠀⠀⣰⠟⠁
⢿⡀⠻⣤⣴⠏⢠⡾⠋
⠈⠳⣦⣤⣤⡴⠋
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠔⠂⠀⠉⠉⠁⠀⠒⢤⡀
⠠⣊⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢄
⠀⠀⠀⠀⠀⠉⠁⢢⠀⠀⠀⠰⠀⠀⠣
⠆⠀⠤⠤⠀⣀⡀⡌⠀⠀⠀⠘⠀⠀⠀⠐⢄
⠈⢆⠀⠀⠀⠀⠀⠅⠀⠠⢒⢗⡀⡤⢀⣀⠀⠢⡀
⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⣾⠀⣸⠁⠂⠑⠠⣈⠑⢄⡀
⠀⠀⠀⠀⠈⠒⡆⠀⠀⡐⠄⣇⠀⡟⢰⠀⠀⠀⠀⠉⠒⠬⣑⠂⠄⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⠋⠁⠐⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠌⡐⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠁⠁⠈⠉⠀⠀⠤⠍⡑⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⡒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠌⡑⠠⢄⡀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠌⣐⠢⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠌⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢉⡒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢌⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢌⡑⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢌⡒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠬⣁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢌⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣉⠠⠤⡉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠠⢄⡀⠀⠀⠁⡜⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢌⡑⠢⠆⢣⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⣁⠒⠒⡁⠔
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⡹⠦⠄⠀⣤⡀
⠀⠀⠀⠀⠀⢀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⣇⡠⠴⠎⣁⡀⢀⣠⢊⡴⠃
⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠁⠁⠒⠀⠀⠰⡉⠀⠀⠀⠀⠀⠈⡥⡀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠠⠤⡤⠤⠀⠀⡴⠷⠀⠀⠀⠤⠁⢰⠆⠀⠠⠀⠢⠠⠄⠤⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠀⢆⣀⣀⣀⠄⠘⠀⠀⠀⠀⠠⡐⣠⡤⠴⡓⠠⢀⡠⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣿⠀⢰⠀⠈⠘⠃⠀⢳⡄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢋⠁⠈⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡠⠋⣀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠠⡀⢀⠀⠀⠀⢀⠠⠄⠐⠒⠉⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠁⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡚⠉⣡⣤⣤⣍⡢⡀
⠀⠀⠀⠀⡠⣔⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠤⣄⣠⣵⣶⣿⡇⣴⣿⡿⠟⣻⢿⣾⡄
⠀⠀⠀⢸⡟⠁⠀⢹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢽⣻⣿⣿⡿⠀⠀⠈⠉⠛⠟
⠀⠀⠀⠘⢿⣤⣤⣾⣇⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡭⠾⢿⣿⣿⠃⢰⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠶⢿⣿⣿⣧⣾⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣮⣿⠯⠙⠛⠛⠛⠉


⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣷⠦
⠀⠀⠀⢰⠿⠟⠻⣿⣿⣿⣆⢠⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣶⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣇
⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠘⣿⣿⣿⣷⣶⣶⢋⠟
⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣵⣯⣷⣶⣾⣽⣯⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠛⢉⣴⣾⣿⣿⣿⣿⣿⣗⡄
⠀⠀⠀⠀⠀⢀⣞⣾⣿⣿⣿⣿⣿⣿⣿⣿⣮⣦⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⢿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⢀⣞⣾⣿⣯⣶⣯⠭⢝⡻⠟⠛⠋⣡⢽⣫⣟⡛⢿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⢸⣿⡋⠁⢀⣀⣤⣶⣿⡇
⠀⠀⠀⢀⣞⣾⣿⣿⡿⠋⠀⠀⢀⡙⣦⠴⢻⣾⢿⣿⣿⣷⡀⢫⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢿⣿⣿⣿⣷⣺⣽⣿⣿⣿⣿⠃
⠀⠀⠀⢾⣴⣿⡟⠋⠀⠀⠀⢠⣾⣿⢃⣾⣿⣿⡿⣽⣿⣿⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⣟⣡⠌⢿⣿⣿⣿⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⢫⣾⣿⣿⣿⡵⣵⡷⡲⣄⠀⠀⠀⠀⠀⠀⢀⣾⣾⣷⣿⣷⢂⠘⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣿⣿⡿⢋⣴⣿⣿⣿⣿⢯⡾⣋⣾⠋⣨⡳⡄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣯⣄⣸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠈⠉⣰⣿⠟⠉⠻⣿⣯⢯⣾⠟⣠⣼⢋⣼⢗⣦⣠⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠿⠿⠿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢝⢿⣇⠀⠀⠀⢈⡻⣷⣝⢾⣟⣵⢟⣵⣧⢼⢿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡳⣄⠀⣾⣿⣾⣿⢷⣿⣾⣿⡿⣳⢿⣮⡻⣿⣿⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣳⣬⡻⣿⣿⣷⣝⢿⣿⡏⣿⣳⣝⢿⣮⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢝⢮⡻⣿⣿⣷⣝⢿⡹⣿⣶⡳⣭⢟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣿⡳⣝⢿⣿⣷⣝⢮⡳⣍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣮⣳⣽⡻⣿⣷⣝⢮⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠳⢝⢮⡻⣿⣷⣝⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠙⢿⡺⣝⢿⣷⣝⢷⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠫⣷⢝⡿⣷⣝⢿⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢝⢮⡻⣿⣝⠳⣕⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡮⣝⢿⣎⡳⣝⢦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣷⣝⢿⣮⡳⣝⢦⡀
⠀⠀⠀⠀⣴⣿⣿⡼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣕⢮⡻⠎⢿⣯⣢⣄
⠀⠀⠀⢸⣿⠋⠀⠈⢻⣿⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢟⢶⣿⠟⠻⢿⣵⡄
⠀⠀⠀⠸⣿⣧⣀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⢧⣀⢀⢽⡞
⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⠿⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠛⠛⠛⠛⠛⠛⠷⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣧⣤⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠙⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣰⠋⠉⠛⠓⠶⠶⠦⣤⣤⣬⣙⣛⣛⣛⣛⣂⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡏⠀⠀⢀⠀⢤⣄⣈⠙⠯⡷⠀⠿⠀⣖⣲⠶⠆⠀⠀⠀⠀⠀⠀⠀⢠⡏⢹⡰⠃⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣴⠟⠉⢻⠁⠀⠀⠈⣭⣭⣭⣿⡿⠶⠀⠀⠀⠀⠠⢴⣶⣶⣿⡿⢿⣯⣥⠀⠀⣸⠀⠀⣧⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⣿⣀⣘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠄⠀⠀⣏⠀⠀⣿⠀⡰⢚⠋⠻
⠀⠀⠀⠀⠀⠀⠈⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⠶⠛⠚⠙⠏⠀⣠
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⡿⠟⢿⣄⣠⣄⢀⣼⠗⠒⣲⠟⠻⢯⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠓⠒⠛⠋⠁⠀⠀⠀⠀⠉⠛⠛⠿⠶⠶⠚⠉⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⣀⡀⠀⠀⢀⣸⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡦
⢀⡤⣤⣯⠹⠶⠞⠋⠁⢘⣇⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡼⠓⢦⡀⠀⢀⣀
⣹⡶⠄⠀⠀⠀⣀⣤⠶⠛⢻⣆⣀⣤⡴⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡼⣧⡀⠀⠉⠲⠋⢽⠦
⠛⢲⡾⢃⡶⠛⠉⠀⠀⠀⠀⠉⠉⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠳⣤⠾⠁⠀⠙⠳⢤⡀⠀⠀⠴
⠀⠈⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⡶⠤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⢀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣠⣀⣀⣈⣉⣉⣉⣉⡉⠉⠙⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢰⡏⠈⠉⠉⠉⠉⠉⠙⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠉⠉⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠚⠋⠉⢹⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠤⠤⠤⠤⡀
⠀⠀⠀⢀⡔⣚⠯⠭⠉⠉⠉⠑⠂⠒⠒⠒⠒⠒⠒⠀⠀⠀⠉⠁⠀⠈⠑⡀⢹
⠀⠀⠀⡎⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣀⠀⠀⠀⢁⠈⡆
⠀⠀⠀⢳⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡔⠉⠀⠀⠀⠈⠀⢧
⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠑⠶⠚⢳⠀⠀⠆⢸
⠀⠀⠀⠈⡇⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢠⠖⢀⣬⣀⣀⡤⠊⠀⠀⢰⠈⡇
⠀⠀⠀⠀⢧⠇⠃⠀⠀⠀⢀⡖⠋⠉⠓⠋⡄⠕⣁⠔⠉⠀⠈⠀⠀⠀⠀⠀⠀⡆⢧
⠀⠀⠀⠀⢸⡀⠠⠀⠀⠀⢸⡤⠒⠢⡀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣧⠘⠀⠀⠀⠀⠁⣠⠼⠁⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡆
⠀⠀⠀⠀⠀⡇⡄⠃⠀⠀⠀⠀⠈⠓⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠜⢸⠃
⠀⠀⠀⠀⠀⢹⠁⠰⡀⠀⠀⢀⣀⣀⠤⠤⠤⠶⢲⡒⣶⡖⠶⠬⠽⠽⠶⠒⠒⠋⠳⡄
⠀⠀⠀⠀⠀⠈⢳⡤⠭⠭⠥⠐⠒⠒⠂⠈⠉⠁⢀⣀⣀⡀⢠⡤⠴⡖⠚⣫⡉⠠⣄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢸⠱⡀⠀⡔⠒⠲⠒⠂⢺⣉⠉⠡⡄⠜⢆⠒⠛⣄⡉⢤⠠⠑⢖⠊⣳⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠘⣆⠑⡄⠘⠖⠂⠛⡈⣁⠹⡄⠀⠲⠒⠘⢇⡁⠨⢦⠀⢳⣀⡈⠧⠠⠈⠳⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠘⡄⠈⢂⣉⢧⠤⠀⠘⡆⣈⣹⡤⠤⠗⢒⣋⡩⠭⠾⠒⠒⠊⠉⠉⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⡄⠀⠳⣖⣃⠬⠭⠛⠒⠒⠒⠊⠉⣁⣀⣀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠀⠀⠀⣀⣀⠷⠀⠀⠀⠀⠀⢀⣀⣀⡤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⣈⠉⠀⠀⣄⣐⣒⣬⠭⠭⠵⠖⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣜⠦⠤⠄⠄⠐⣒⣀⣁⠭⠭⠔⠒⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠰⠆⢠⠄⠠⠄⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡒⠀⠀⢇⠀⠄⡌⡀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢁⠉⠀⢀⠏⢀⡀⠊⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢨⡤⠀⠸⠀⠀⢱⠰⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠤⠄⠤⠈⠀⠀⠘⢄⠀⢻⠃⠒⠠⠤⠤⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⡖⠀⠨⠀⠀⠉⠛⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠠⠀⢀⠂⠀⠀⠀⠀⠀⡨⠒⠤⢀⠀⠀⠀⢀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡦⠀⠀⢀⠤⠖⠀⠰⠀⠠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⠀⠀⠀⡀⠈⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠘⡄⣤⣴⠂⡞⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠜⡴⠁⢀⢿⡟⠈⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡜⠈⢈⠱⠀⠐⡖⠀⣀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠰⠂⠀⣌⠅⠀⠀⠀⠀⠁⠓⡈⠀⠸⠁⠉⣈⠠⡃
⠀⠀⠀⠐⢚⠁⠈⠃⠀⠠⠆⠀⢠⠀⠸⠀⠀⠀⠂⠉⠁⠀⠁
⠀⠀⠀⠀⠠⠀⠀⠀⢠⠀⢀⣐⠆⠀⡆⠀⠀⠘⢠
⠀⠀⠀⠀⠀⠀⢰⠀⡎⠀⠈⠉⢢⠤⠦⡜⠀⡧⠆
⠀⠀⠀⠀⠀⠁⠏⠆⠀⠀⠁⠀⠘⠀⠀⠉⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣤⠤⠴⠶⠶⠒⠒⠒⢒⣛⣛⣉⣉⣉⡉⠉⠩⠭⠭⠭⠍⠀⠀⠠⠤⢤⡀⠙⡆
⠀⠀⠀⠀⡼⠁⡖⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢷
⠀⠀⠀⢸⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠛⠛⢲⡄⠀⠀⠀⠘⡀⢸⡀
⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠀⣰⠋⠀⢀⡀⠀⠀⢃⠀⡇
⠀⠀⠀⠀⢣⢰⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠘⠲⠴⠚⢳⠀⠀⢸⠀⢻
⠀⠀⠀⠀⢸⠈⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡆⠀⢀⠁⠀⠀⣠⠞⠀⠀⠀⡆⢸⡀
⠀⠀⠀⠀⠘⡇⡇⢃⠀⠀⠀⠀⠀⢀⡤⠶⠦⣤⠔⢁⢔⡌⠁⣠⠔⠉⠙⠒⠋⠁⠀⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⢣⠓⢸⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠓⢁⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻
⠀⠀⠀⠀⠀⢸⢸⠈⠄⠀⠀⠀⠸⡶⠋⠉⢳⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠘⡏⡀⡇⠀⠀⠀⠀⠀⢠⡤⠋⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⡇
⠀⠀⠀⠀⠀⠀⣇⡇⢰⠀⠀⠀⠀⠀⠀⠉⠒⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠆⢸⠇
⠀⠀⠀⠀⠀⠀⢻⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠤⠤⠤⢶⣒⣒⣚⣉⠭⠤⠤⠖⠒⢯⡀
⠀⠀⠀⠀⠀⠀⠘⢦⣀⠙⠒⠒⠒⣒⣒⣫⡭⠥⠤⠔⠒⠒⠈⠉⠉⠉⠉⠀⠀⣀⣠⠤⢤⠒⠢⣀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⡉⠉⠉⠉⠀⢀⣀⣀⣀⣀⠤⠤⢤⠶⠒⢲⡒⠉⢻⣍⠩⢦⡀⠂⠳⡀⢉⣳⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠱⡀⠀⢏⠉⠉⢣⡀⣀⠘⢦⠄⠒⠳⡂⠈⢹⡉⣀⠸⢦⠄⠳⡆⠉⠉⣦⡀⠬⠷⣄⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠱⡀⠈⢻⠉⠉⢱⣀⠀⠄⢦⠀⠒⢻⡉⣉⠹⣄⠀⠼⢶⡂⢙⣇⣀⠤⠷⠐⠂⠉⢳⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠹⡄⠀⠹⣉⣉⣧⠤⠰⠚⢷⠂⢉⣹⣤⠤⠼⠖⠚⣋⣩⠥⠴⠞⠒⠒⠛⠉⠉⠉⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⡄⠀⠙⣤⠼⢒⣒⣀⡽⠭⠭⠤⠖⠒⠒⠋⠉⠁⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡄⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠊⠉⠉⠁⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⡠⠤⠖⠒⠚⠁⠀⢀⣀⡀⠤⠤⠴⠒⣒⣋⡭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠤⣒⣒⣠⣭⡭⠤⠶⠖⠒⠚⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⣒⠒⠒⠒⣊⣉⣉⡥⠤⠤⠶⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⡿⠟⠿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡸⣿⣿⣿⣿⢱⣿⣿⡞⣡⢨⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠌⡛⠿⣿⣿⣿⣿⣿⡿⠟⣫⣴⣿⡇⣿⣿⣿⣿⢸⣿⣿⢻⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠟⣉⢸⣿⣷⣶⣶⣤⣤⡰⢶⣿⣿⣿⡿⢰⣿⣿⣿⣿⢸⣿⣿⢸⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⣶⢰⣿⡆⣿⣿⣿⣿⣿⣿⣿⣶⡆⢠⣬⣄⢻⣿⣿⣿⣿⢸⣿⣿⠸⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⣿⣮⣭⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⢸⣿⣿⠟⣵⣾⡿⢿⣷⣌⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠈⠉⢡⣿⣿⣿⢸⣿⠣⣶⣶⢹⣿⡆⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡘⢿⣷⣭⣥⣾⡿⢡⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢸⣿⣿⣿⣿⣦⣝⣛⣛⣩⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣦⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⢶⣿⣿⣿⣿⣿⣿⣷⡆⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⠟⣫⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⣠⣶⣿⣿⣿⣿⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠶⢤⡀
⠀⠀⠉⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣻
⠀⠀⠀⢀⣀⣠⣼⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢦⣤⠴⠋
⠀⠀⠀⠉⠛⠛⠛⠋⠁

⠀⠀⠀⣀⣴⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠚⠛⠉⠉⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡟⠁⠀⠈⢢
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣧⡀⠀⢀⠞
⠀⠀⠀⠰⠶⠾⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⢀⣤⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣧⣄
⠀⠈⠙⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⡏⠁
⠀⠀⠀⣟⢿⡇⠀⠀⠀⣠⣶⣶⠶⠤⠀⠀⠀⠀⠀⢀⡄⠀⣀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠰⣿⣿⣿⡃⠀⠀⠿⠛⢿⡃⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣈⣧⡀⠀⢸⡻⣧⣸⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠈⢹⣿⣿⣷⡀⠀⠀⠀⠘⣇⠀⣀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠏⠳⡄⣸⣽⣿⣿⢻⡄⠀⠀⢀⠀⣴⣿⣟⡃
⠀⠀⠈⣿⣿⣿⣷⡄⠀⠀⠀⢻⡇⠈⢱⣄⠀⠀⠀⢠⣿⣿⠃⠀⠀⣹⣿⠋⠉⠀⢸⡇⠀⣴⠋⠀⠘⣿⣿⣿
⠀⠀⠀⡷⣿⣿⣿⣿⣄⠀⠀⢰⣿⣶⣾⢿⣤⡀⠀⢸⣿⣿⠀⠀⣰⣿⢋⣣⣦⡀⢸⣧⠞⠁⢀⣠⣾⣟⣻⣿⣷
⠀⠀⠀⣿⣾⣿⣿⡿⡿⠲⡄⠸⣿⡯⣿⠞⠋⠻⢦⣼⣿⡏⠀⣼⣿⠃⠀⠸⣿⣿⣮⠏⠀⠴⠿⠋⢹⣿⣟⠿⠋
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠦⠶⠶⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡔⡇⠀⠀⠀⠀⠀⡞⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐
⠦⠀⠀⠐⢠⢎⢠⣇⠈⠉⠉⠉⠉⠁⠃⠠⠃⡃⠀⠀⠀⠀⠀⠐⠒⡀⢱⠂⠀⠉⠈⠉⠉⠁⠈⢀⣶⡔⠀⠀⠠
⠀⠀⠀⠀⢈⡄⠻⢿⠀⠀⠂⠀⠀⢸⠀⡇⠀⡇⠀⠀⠀⠀⠀⡄⠀⡀⠸⠀⠀⠀⠀⠀⠀⠀⣠⠎⠐⠇
⠀⠀⠀⠀⠀⠁⠚⠀⢧⠀⢀⣀⠬⠒⠲⣹⣶⢀⠉⠀⠀⠀⡔⠁⠀⣓⠖⠀⠀⠀⠘⠋⠙⣾⠃⠐⠂⠁
⠀⠀⠀⠀⠀⠐⢤⡀⠈⡓⠈⠀⠀⠀⠐⠛⡏⠈⠀⠐⡐⠉⠀⠐⠀⠀⠒⠂⠀⠐⠓⠀⠐⠁⢀⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠅⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠂⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠘⡀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂
⠀⠀⠀⠀⢄⠀⢂⣀⠀⡀⠀⠀⠀⠀⣠⠶⣿⡄⠀⢫⠃⠀⢸⢤⢤⣴⠷⣄⠀⠀⠀⠀⠀⠀⣄⠔⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠒⠒⠂⠀⠈⠁⠀⠱⡀⠘⡆⠀⠨⠀⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡠⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣎⣐⣓⣈⣡⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⠒⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣠⠖⠉⣠⠔⠁⣼⡡⠖⠚⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠘⠋⠀⢀⡞⠁⢠⠞⠉⣀⡤⠤⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⣠⣯⠴⠖⠛⠚⠊⠁⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⡠⠠⢄⡀⠀⠀⢀⣀⣼⠁⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠣⠤⠤⠼⠆⠀⣴⠟⠙⣦⠟⠁⠀⢀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠾⠤⠤⠴⠃
⠀⠀⠀⠀⠀⢠⠤⠤⣤⠀⠀⢰⡇⠀⡜⠁⠀⠀⠀⠸⠟⣋⡉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⢻⡀⠀⠀⢠⡤⠤⢤
⠀⠀⠀⠀⠀⠘⠶⠚⠁⠀⠀⢺⠏⣸⡇⠀⠀⠀⢠⠖⠋⠉⠙⠳⣄⠀⠀⠀⠀⠀⠀⢀⣀⣉⠀⠈⣇⠀⠀⠀⠙⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⣇⠀⠀⢠⡏⠀⠀⠀⠀⣀⣹⡆⠀⠀⠆⠀⠀⣿⣁⣈⣳⠀⢿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠖⢻⡀⠀⠈⣷⠖⠚⠋⠛⠋⣼⠀⠀⠀⣦⠀⠀⠣⡌⡩⠋⠀⢸⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣆⠀⢧⠀⠀⠘⠦⣤⣀⡤⠞⠁⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣁⣠⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⡴⠋⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⡼⣿⣿⣷⣼⠆⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠰⠦⣌⣻⣿⡟⠀⠀⠀⢀⡴⠋⠀⠀⢀⣸⡁⢸⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣤⣄⡀⠀⢸⡆⠀⠀⠀⠀⠀⠈⠙⠁⠀⢀⡴⠋⠀⠀⠀⠀⡏⠀⢈⡇⢀⠎⢙⡆
⠀⠀⠀⢀⣠⠤⠤⠴⠊⣉⠴⠚⠉⠉⠉⣿⣉⠉⢻⡄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢳⠀⠙⠉⠙⡶⠊
⠀⠀⡾⠋⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠈⠉⣻⠇⢛⡗⠶⠦⠤⠴⠚⣏⠙⣦⠀⠀⠀⠀⢀⡰⠋⣠⠖⠀⠁
⠀⠀⠳⡞⠀⠀⠀⡀⢀⠇⠀⠀⠀⠀⠀⠀⢠⡟⠁⠘⠀⠀⠀⠀⠀⢨⢻⡷⠛⢶⣦⣤⣶⡯⠴⠋⠁
⠀⠀⠀⠙⢾⣁⣸⠋⠉⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡀⠀⣸⠇⢰⣦⢀⠘⡇
⠀⠀⠀⠀⠀⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣀⡈⠃⠐⠾⠃⠐⠀⡇
⠀⠀⠀⠀⠀⠀⣰⠟⡹⣧⠀⠀⠀⠀⢰⡿⠶⣤⣤⣉⣉⠛⠛⠚⠚⠃⢠⡇
⠀⠀⠀⠀⢠⡾⢣⠞⢀⡼⡆⠀⠀⠀⣸⣯⠀⠀⠀⠈⠉⠙⠛⠳⠶⠦⣼⠃
⠀⠀⠀⣴⠟⡴⢁⣶⣿⡔⢻⡄⠀⣾⠉⠙⢦⣀⡀⠀⠀⠀⠰⢤⣤⣴⡋
⠀⣠⡾⣡⢮⠀⢠⡿⠁⠹⡄⠙⣛⡿⠳⡤⠞⠉⠛⠷⢦⣤⣀⣤⣶⣿⣧⣤⣀
⣴⢏⡔⢁⣠⡷⠋⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⢳⡦⠉⡷
⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠯⣴⣿⣶⡶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠢⠤⢤⡬⣉⣀⣀⣰⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠺⠶⠶⠦⣤⣭⣭⣉⣗⣻⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀
⠀⠀⠀⠀⠀⢠⠖⠲⣤⡞⠁⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣬⣭⣀
⠀⠀⠀⠀⠀⡾⠀⢀⡟⠀⡶⡀⠘⣇⠀⠀⠀⡰⠂⠀⢠⠞⠋⠀⠀⠀⠀⠙⢷⡀
⣜⣙⣦⠟⠺⠃⠀⠚⠣⢄⡓⠛⢀⣿⡀⠠⠞⠁⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠋⢸⣤⠞⣦⠀⠀⠀⠀⠀⢹⣄⢯⡿⢧⣠⠞⠋⠙⢻⡤⣤⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠁⠀⠸⣆⣠⠖⠛⠧⢶⠛⣦⠀⠈⡟⠀⠀⠀⢼⠟⠛⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢂⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣻⣭⡇⠀⠙⠦⣤⠴⣿⠦⣤⣀⣀⣀⣀⣴⣏⣀⣀⣤⣤⠤⠤⠤⠶⠤⢤⣤⣀⠀⠀⠀⠀⠀⠁⡖⠒⢦⡀⢀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⣭⣭⣝⠳⠀⠀⠀⠀⠔⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⢧⠀⢀⡿⣼⢦⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠉⠉⠉⢈⡀⠀⡟⣄⠀⢠⡀⠀⠀⠀⠻⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⢀⡞⢻⣿⡆⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠙⣇⠀⠉⠁⠀⢧⡷⠀⡰⢄⠀⠈⠛⠦⣄⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣠⠟⠛⣿⢃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣦⠀⠀⠀⠀⠑⠚⠀⠀⠀⠀⠈⠙⠲⠦⣤⣘⣂⣳⣄⣲⣠⣴⠞⠃⠀⠀⢢⣇⠻⢤⣤⡼⣿⠱⢃⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠦⠗⢰⢻⡄⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠉⠉⠉⠧⠟⠀⠀⠀⠀⠀⠀⠀⡤⠾⣶⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠒⣷⠀⠀⠀⠈⠉⠀⠳⠞⠀⠀⠀⢀⣀⣀⠤⠄⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢢⣼⠖⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠳⠤⣿⡀⠀⠀⠀⠀⠀⠀⣠⡴⢶⠋⠙⡇⠙⢦⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠀⢹⠶⠶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣐⠠⠉⠙⠒⠦⠤⣴⣋⣁⡀⠈⠳⣤⣤⠴⠛⠀⠀⠀⠈⣯⣽⣟⠛⠉⠙⣯⠁⠀⠀⣠⠞⠋⠉⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠙⠲⠦⣤⣀⡀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⡛⠙⠲⣤⣿⠂⢀⠀⠀⠀⠸⣆⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠛⠲⠦⣤⣀⣀⠀⠀⠀⢠⡇⠀⠀⠙⢻⠁⠀⠳⣤⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠉⠙⠓⠲⠾⠃⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡄⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⡤⠤⠤⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢲⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⣆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢤⠶⢶⠛⠉⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠺⡆⠀⢸⡗⠒⠒⠛⠛⠛⠉⠉⠉⠀⠀⢀⡞⢠⠏⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠽⡄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢠⠟⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⡟⠂⠀⠀⢸⠆⠀⠀⠀⠐⠒⣲⠶⢶⠿⠶⠿⠒⠒⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠥⣶⣯⣤⣄⣀⣀⣀⣀⣀⣤⡾⣶⣶⡖⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⣋⣉⠉⠉⠃⠸⢧⠀⠀⠀⠀⠀⣠⠴⠚⠁⠙⠒⠢⣤⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢷⠞⠛⠙⠲⣬⡙⣦⣀⣀⡼⣧⠀⠀⠀⢠⣧⣀⣀⣀⡤⠖⣋⡭⠿⠯⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣓⣖⣚⣓⣋⣉⣩⡭⠥⠚⠉⠀⠀⠀⠈⠓⠦⣤⣉⡁⠛⠳⠶⠦⣤⣤⡗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⢻⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿⣿⣧⠀⣾⣷⣶⡆⠀⠀⠀⠀⠀⣠⣿⣿⣷⣄⠁⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆⣼⣿⣿⣿⣾⣿⣿⡿⠃⣠⣶⡀⣀⡴⠋⠀⠈⣿⣿⣿⣶⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⠛⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⠃⠀⠀⢀⣴⡿⠿⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣾⠟⠁⣀⡀⠤⠖⠀⠀⠀⠀⠈⠹⣿⣿⣅⣀⣤⠀⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃⠀⣠⡤⠀⠀⠀⠀⠀⠀⣠⠆⠀⠈⢿⣿⣿⣿⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠇⠀⣼⡿⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⣿⣿⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⢰⣿⡇⠀⠀⠀⣠⠞⠁⠀⠀⣠⣾⠇⠀⢸⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣟⠀⢸⣿⠀⢀⣠⠞⠁⠀⠀⣠⣾⠟⠁⠀⠀⢸⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠀⠸⣿⡷⠋⠁⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⡴⠋⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡽⠋⠀⠀⢀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣴⡿⠋⠙⠳⠤⠀⠀⠀⠀⠀⣠⣾⣿⣷⣶⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡾⠋⠀⠀⢀⣴⡿⠻⣿⣶⣤⣀⣀⣀⣀⣠⣴⣾⣿⠟⢿⣿⡿⠛
⠀⠀⠀⠀⣴⡿⠿⠿⢿⣿⣿⣇⠀⣀⣴⡟⠉⠀⢸⣿⣿⣿⠿⣿⣿⣿⡿⣿⣿⣿⣷⠀⠉
⠀⠀⠀⢸⠏⣤⠖⠂⠀⠈⢻⣿⣿⣿⠏⠀⠀⠀⠈⠻⠿⠋⠀⣿⣿⣿⡇⠘⠿⠛⠛
⠀⠀⠀⠠⡾⠁⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠛⠋


⣶⢶⢾⢷⢿⠿⢾⠿⠿⣿⣷⣶⣿⣿⣿⣶⣶⣾⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣷⣾⡿⢶⣿⣶⠶⢶⡿

⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠣⣄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠤
⠀⠀⠀⢠⡀⠀⠀⠀⣀⣄⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠠⡄⠀⠀⠀⢺⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠈⠀⠀⠀⠰⣷⠀⠀⠀⣆⣀⠀⠀⠀⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠸⢠
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠉⠉⠀⠀⠀⡬⠅⠀⠀⠀⣾⠆⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⣠⣸⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠆⠀⠀⠀⠰⡇⠀⠀⠀⠘⠀⠀⠀⠀⢸⡄⢿
⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡏⢣⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣽
⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠁⠠⡇⠀⠀⠀⡇⠀⠀⠀⠀⣄⡄⠀⠀⢰⠁⠛⢀
⠀⠀⠀⠀⠘⠀⠀⠀⠈⣄⠀⠀⠀⠀⣀⣠⠀⠀⠀⣀⡇⠀⠀⢀⣠⠀⠀⠀⣀⣄⠇⠀⠀⠀⢿⠀⠀⠀⠀⠉⠁⠀⠀⢸⣀⠴⠃
⠀⠀⠀⠁⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈

⠀⣠⣶⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⡀
⠞⠛⠛⠉⠻⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠉⢻
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣸
⠀⢠⣤⣶⣾⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠁
⠀⠀⠀⠉⠉

⠀⣀⣴⣶⣾⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠘⠛⠋⠉⠻⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡟⠁⠀⠈⢳
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢠
⠀⢀⣤⣤⣶⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠚⠁
⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⣠⣴⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠂⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⡖⣆
⠀⠀⠀⠀⣸⠏⠀⠀⠀⠹⡆⠀⠀⠀⣦⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⢸⠁⢹
⠀⠀⠀⠀⠸⣇⠀⠀⠀⣸⠇⠀⠀⠀⣿⣇⠀⠀⢀⣴⣿⣿⠀⠀⠀⠐⠀⠀⠀⣃⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⠀⠀⠀⠀⠀⢸⡀⣸
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢠⡀⠀⠀⠀⠈⠉⠛⠻⠿⠄⣀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡿⣏
⠀⠀⠀⠀⠀⢸⡿⠛⢿⡏⠀⠀⠀⠀⠀⠙⣷⣶⣿⣿⡟⠀⠀⠀⠀⠈⢿⣷⣶⣦⣴⣤⣦⣀⠀⠀⢀⠃⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⢹⡇⠀⠀⠀⠀⠀⠈⢻⣿⣿⣏⠀⠀⣙⣿⡷⠁⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⡉⠛⠛⢻⣿⠁⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⣶⣷⣿⣶⠀⠀⠀⠀⠀⣀⣇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⢰⣿⠋⢻⣷
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⣿⡄⢰⣿⠀⠀⠀⠀⠘⣿⠀⢸⡿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣦⣾⡇
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⣿⠋⠹⣿⡆⠀⠀⠀⢀⣿⠟⢿⣧
⠀⠀⠀⠀⠀⢸⣧⣀⣼⡇⠀⠀⠀⠀⠀⠀⣼⣷⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡏⠀⠀⢹⣷⠀⠀⠀⣼⡟⠀⠈⣿⡄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣿⠃⠀⠀⠸⣿⡄⠀⢀⣿⠃⠀⠀⢹⣧
⠀⠀⠀⢠⣾⣽⣿⣿⣿⣾⣗⠀⠀⠀⣾⣿⣿⣿⠿⠿⠿⣿⣷⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣄⠀⠀⣰⣿⡇⠀⢸⣿⡀⠀⠀⣸⣿
⠀⠀⠀⢸⣿⡟⠋⠉⠉⠹⣿⡇⠀⠀⣿⣿⡟⠁⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣿⡾⠟⠻⢷⣿⠀⠀⠀⠀⢸⣿⣿⡿⢿⣿⣿⠃⠀⠘⣿⣿⡶⣾⣿⡿
⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢻⠇⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⣿⡁⠀⠀⢈⣿⠀⠀⠀⠀⠀⣿⣯⠀⠀⣽⣿⠀⠀⠀⣿⣏⠀⢈⣿⡇
⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣴⡿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃⠀⠀⠀⠘⠿⣿⣿⠟

⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣥⣿⣶⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣾⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠇⢀⣬⣙⣿⣇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⢋⣶⠘⢿⣿⣿⣿⢰⡏⠀⠈⠙⠏⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣿⡏⣠⡀⣼⣿⡇⠀⠙⢿⡇⠸⣇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⢰⣿⡇⣈⡉⠀⠀⠀⠀⠛⠦⣌⣳⣦⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠉⠁⠀⠀⣤⠀⠀⠠⡦⠉⠁⣿⣆⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣸⢣⡿⡇⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⣰⠿⠛⢀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣷⣄⠀⠲⠿⠿⣷⣦⠀⠀⢀⣴⡿⣶⣿⡟⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣿⣷⣤⣄⣀⣀⣀⣤⣶⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⣯⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠷⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡀⠀⠀⠀⣶⣶⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣿⠋⠁⠀⠀⠀⢀⠀⠀⠀⣠⣾⠟⠻⣿⣄⡀⣸⣿⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢰⣶⣦⠀⠀⣶⣦⠀⣿⣇⠀⢾⡿⠿⠿⢿⣿⡟⡻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⢰⣿⡀⠀⢸⣿⠀⠀⢹⡟⠀⠙⢿⣿⣶⣦⠁⠀⣸⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⡍⠁⢸⣿⠇⠀⣼⡇⠀⠀⢸⡇⣄⣾⣷⠀⡙⣿⠆⠀⣿⡿⢠⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣴⣿⠟⠀⠀⢿⣷⣤⣴⣿⣿⡀⡻⢿⡿⠿⠋⠀⠀⢻⣧⡈⠛⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠦⠙⠛⠛⠛⠁

⠀⠀⠀⠀⢣⣴⣶⠿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢨⣴⣦
⠀⠀⠀⠀⣼⣿⠀⣴⡄⣹⡇⢃⣴⣾⡿⣷⣤⡀⠀⣴⣶⣦⠐⣂⣀⠠⠀⠀⠀⠀⠂⣿⡏⠀⠀⣽⣿⠁⠐⢣⣶⣿⡿⣷⣄⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢹⣿⠀⠙⠛⠋⢠⣾⢋⠁⠀⠈⠹⣿⣆⠸⣿⣿⡿⠛⢿⣷⠀⠀⠀⠀⠀⣿⡷⠀⠀⣿⣿⠀⢠⡟⡉⠀⠀⠈⢹⣿⡄⣶⣶⡆⠀⢰⣶⡄
⠀⠀⠀⢠⣼⣿⣤⡤⣤⣤⢸⣿⡄⠀⠀⠀⠄⣹⡿⠀⣿⡏⢸⣿⣴⡿⠀⠀⠀⠀⠀⠻⣷⣵⡾⠋⣿⠀⢿⣧⡀⠀⠀⠠⠀⣿⡇⠀⣿⠇⠀⠈⣿⠁
⠀⠀⠀⠈⢩⣿⠉⠀⠘⠉⠀⠻⣿⣦⣬⣥⣾⡿⠃⠀⣿⡇⠂⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⢶⡌⠁⣿⡆⠈⢿⣿⣦⣭⣤⣾⡿⠁⢠⣿⠀⠀⠀⣿⢠
⠀⠀⠀⠀⣼⣿⡖⠂⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣇⠘⠃⣰⣿⠇⠀⠀⠉⠛⠛⠛⠉⠈⠀⠸⣿⣧⣴⣾⣿⣇
⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣯⣿⣷⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣾⣶⣶⣾⣿⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠈⠃⠭⣍⡉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⣉⡭⠅⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣰⠛⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡀⠙⡄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢻⡓⢦⣀⠀⢀⡔⠁⠀⢠⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢃⠀⠈⠢⣀⠀⣀⡤⠞⡻
⠀⠀⠀⠙⢄⠈⠙⠉⠀⣀⠔⠁⢠⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠈⠳⢄⡀⠉⠋⠉⣠⠞
⠀⠀⠀⠀⠀⠉⠐⠂⠀⠀⠚⠲⣄⠈⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⢠⡶⠓⠂⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⣏⠉⠻⣦⡀⠀⠀⠀⠀⠀⣠⠞⠉⢉⡅⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠻⣦⡀⠀⣠⠞⠁⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠈⠻⣾⡁⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠈⠻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢙⣿⣦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⡿⠋⠙⢿⣦⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⡿⠋⠀⠀⠀⠀⠙⢿⣦⡀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠙⣷⣄
⠀⠀⠀⠀⢀⣀⣤⣤⣠⡄⢶⡿⠃⠑⠚⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⠒⠈⢻⡷⢤⣄⣤⣤⣀⡀
⠀⠀⢀⡴⣫⣶⠾⠷⣶⣭⡳⣄⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢀⡔⣫⣶⠾⠷⢶⣭⡳⡄
⠀⠀⡞⣼⠏⠀⠀⠀⠀⠙⣷⡸⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣼⠏⠀⠀⠀⠀⠘⣷⣹⡄
⠀⠀⡇⣿⡀⠀⠀⠀⠀⠀⣿⢇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⠀⠀⠀⠀⠀⢀⣿⢿⡇
⠀⠀⠹⣜⢿⣤⣀⣀⣠⣼⢏⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡜⢷⣤⣀⣀⣠⡾⢏⡞
⠀⠀⠀⠈⠳⠬⢭⣭⠭⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⣭⣭⠭⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢹⠿⣯⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡴⣲⣦⠀⠀⠀⠀⠀⠀⢀⣽⣇⠈⢳⣦⡞⠉⢀⣾⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⢤⣤⣀
⠀⣀⣠⣾⣏⣀⣀⣨⣇⣉⣙⣶⣴⣋⢠⣤⡿⣿⣄⠀⠀⠀⠀⠀⠉⠻⣦⡞⠋⠳⣤⠟⠉⠁⠀⠀⠀⠀⢀⣴⣏⣁⣀⣤⣿⣤⣤⣾⣿⣶⣶⣦⣤
⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⢸⣿⡆⠀⠀⠀⠀⣴⡾⠉⠻⣶⣟⠁⢳⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⠿⠿
⠀⠛⠻⣏⢎⡿⠛⠛⠛⠛⠛⠛⢿⣱⣺⠟⠋⣿⢹⡇⠀⠀⠀⠀⠈⢠⡖⡿⠈⠻⡷⡄⠉⠀⠀⠀⠀⠘⠿⢿⣗⣎⡟⠉⠉⠉⣽⣿⣿⣁⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠴⣚⣭⣟⡶⢤⣤⣀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⣰⣶⣿⡟⣻⣿⣿⡻⣿⣷⡶⣼⠀⠀⠀⠸⣷⣾⠅⠰⣤⡦⠀⡀
⠀⠀⢀⣶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠟⠀⠀⠀⠈⣧⢿⡆⣈⠉⠉⠉⢁⡄⣿⢣⠇⠀⠀⠀⠀⢹⡏⠀⠀⣿⠀⠀⢻⡟⠀⣤⣠
⠀⠀⣼⡏⠀⢠⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⠖⠀⠀⠀⠀⠘⣞⣷⣀⣶⣶⣶⣄⣰⢏⡞⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⢷
⠀⣴⣿⡀⠀⣿⠁⠀⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠀⠀⠀⠀⠀⠘⣮⢿⣿⣿⣿⣿⣫⠞⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⢸
⠀⠿⠿⠀⣾⣿⠀⣰⡇⢠⣤⣤⣤⣤⣀⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠿⣛⡵⠋⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢠⡿⣄⠀⡸⣇⠀⣸⡇⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⠛⠉⠉⠙⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠈⠀⠁⠀⠑⠋⠀⠙⠊⠀⠑

⠀⠀⣠⡴⠞⠛⠛⠶⣦⡀⠀⠀⠀⠀⠀⠀⣾⣶⢤⣤⡤⣶⣷⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣄
⠀⣼⠏⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⢀⣼⣁⣙⣋⣀⣀⣀⣉⣋⣈⣷⡀⠀⠀⢠⣿⣟⡽⢿⡿⣽⣿⣷⡀⠀⠀⠀⠀⠀⢀⣛⣻⣟⣛⡀
⠀⣿⠿⢿⣿⠿⢿⣿⠿⢿⣷⠀⠀⢸⣿⠋⣩⡉⢿⣿⡟⣉⣉⠻⣿⡇⠀⠀⣿⣿⣿⢳⣿⣷⠞⣿⣿⡇⠀⢀⣄⣠⣴⣿⣿⢯⡽⣿⣿⣦⣤⣀⣤
⠀⢿⣦⣴⡷⠶⠶⣿⣄⣼⠇⠀⠀⢸⣿⣌⠻⢋⣼⣿⣇⠉⠉⣡⣿⡇⠀⠀⠹⣿⣿⣀⣉⣉⣘⣉⣿⠇⠀⢸⣿⣿⣿⣿⠋⢸⡇⠹⣿⣿⣿⣿⣿
⠀⠀⠙⠿⣦⣤⣤⡾⠟⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠙⠿⣿⣯⣧⣽⠏⠁⠀⠀⠀⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⢸⠿⣷⣶⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣺⡀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠳⣄⠀⠀⣴⣦⠀⠀⠀⠀⢰⡟⠛⠻⠛⠛⣷⢰⣿⣿⣿
⠀⠀⠀⢘⠛⠋⠉⠛⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡹⡇⠀⠀⠀⠀⠀⠀⢸⡇⠰⡿⠀⢸⠆⠀⠘⡏⠀⠀⠀⠀⢸⡇⢿⠶⢾⠇⣿⠸⡿⣿⢿
⠀⠀⠀⠘⢆⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢹⣦⠀⠀⠀⠀⠀⠀⠈⢳⣤⣤⣴⠟⠀⠀⠀⣧⠀⠀⠀⠀⠸⣿⣷⣶⣾⣿⡿⠀⠀⣿
⠀⢀⣀⣤⣾⣝⢛⣵⣦⣄⡀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠘⠛⠛⠳⣿⠀⠀⠀⠀⠀⢹⣟⡻⣛⣿⠁⠀⠀⢸⡄
⣴⣿⣿⣿⣿⣿⢸⣿⠿⢿⣿⣷⡀⠀⠀⠀⢰⣿⠁⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠈⠙⣿⣍⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⢸⣯⣵⣭⣿⠀⠀⠀⠘⡇
⣿⣿⣿⣿⣿⡿⢸⣷⣶⣶⣿⣿⠇⠀⠀⠀⢰⣄⣀⣤⠴⠶⠶⢦⣄⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣠⡴⠋⠀⠀⠀⠀⠀⠈⠉⢻⡏⠉⠀⠀⠀⣰⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢿⣿⣦⡀⣠⣿⠶⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡶⣾⠋⠀⠙⢦⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⣀⣠⣤⡀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⣉⣉⣿⡇⠀⢳⣶⠃⢀⣿⣼⡿⠀⠀⠀⠀⠀⠀⠀⠠⣁⣈⣁⣒⣲⡄⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠻⠛⠿⠿⠿⠿⠇
⠀⠀⠉⣿⣿⡇⠀⠀⣿⠀⠈⢻⣿⠁⠀⠀⣠⣄⠀⠀⠀⠀⢀⢂⠀⠀⠚⠧⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⣾⣶⣿⣿⣿⡇
⠀⠀⢸⣿⣿⡇⠀⠀⣿⠀⠀⢸⣷⠀⠀⢰⣿⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣭⣭⣍⡄
⠀⠀⢸⣿⣿⡇⠀⠀⣿⠀⠀⢸⣿⠀⢰⣿⣿⣿⣿⠀⠀⢀⡀⠀⠀⠀⢀⣤⡀
⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⣼⣿⠀⠘⣿⣷⣿⣿⠀⠀⣾⣾⠀⠀⠀⢸⣯⡇⠰⠿⣿⠷⠂⢰⣶⣶⣷⣶⡆⢰⡶⣶⣿⣶⣄
⠀⠀⢸⣿⠀⠀⠀⢀⠿⠀⠀⣿⣿⠀⠀⣿⣿⣿⡇⠀⠀⣾⢿⡄⠀⠀⢸⣿⡇⠀⠀⣿⠀⠀⠘⠛⢿⡟⠛⠃⠀⠀⢸⣿⠀⠈
⠀⠀⢸⣿⠀⠀⠀⠈⠂⠀⠀⠫⠜⠀⠀⣿⠉⣿⡇⠀⣸⠇⠘⣧⠀⠀⣸⠛⣧⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢘⣿
⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠀⣿⡇⠀⣿⠀⠀⣿⠀⠀⡏⠀⢹⡀⠀⣿⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠉⠀⣾⣿⣿⣿⣿⣿⣿⣿⡆⠘⠈⠀⢹⠃⠀⢸⠀⠀⣿⠀⠀⣇⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠁⠀⠙⠀⠀⠙⠀⠀⠘⠀⠋⠀⢀⣿⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣿⣿⣿⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢀⢀⠀⠀⠀⢀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣿⣿⣿⣿⠀⢠⣦⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⢸⣿⠀⠀⠀⠘⣿⡇⠀⢸⡇⠀⢸⠆⠀⣸⠀⠀⠀⠀⠀⠀⠸⠿⠃
⠀⠙⢿⣿⣿⠟⠋⠀⢸⣿⠀⠀⠀⠀⠀⢀⠃⢡⠀⠀⠀⠈⠀⠀⠀⠀⣹⠁⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⢸⡆⠀⠀⡆
⠀⠀⠀⢹⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⡀⠀⢠⣿⡇⠀⢨⡇⠀⠈⣆⠀⣿⠀⠀⢸⡇⠀⠀⡇⢹⡏⣿
⠀⠀⠀⣿⣇⠀⠀⠀⠸⠿⠀⠀⠀⠒⠶⢦⣤⣄⣀⢀⡤⠶⠖⠃⠀⠀⣿⡇⠀⢸⡇⠀⢠⣧⠀⣿⠀⢀⣼⣧⡀⠀⡇⢸⡇⣿
⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠖⠛⠛⠛⠛⠓⠉⠒⠲⠀⠀⣿⡇⠀⢸⡇⠀⢸⣿⠀⣿⠀⢿⣿⣿⡗⠀⠇⠸⠃⠟

⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⣿⡿⠋⢀⡀⠙⢿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⡇
⣿⣧⠀⠛⠛⠀⣼⣿⣿⣧⣤⣤⣤⣼⣿⣿⣿⣷⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣾⣿⣿⣿⣿⡇
⢿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⣿⣿⠛⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣇⠀⠻⠟⠀⣸⣿⣿⣇⣀⣀⣀⣸⣿⣿⣿⣯⣀⣀⣀⣀⣿⣿⣿⣿⣀⣀⣀⣀⣽⣿⣿⣿⣿⡇
⣿⣿⣷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠈⠉⠙⠻⣿⣿⣿⣿⣿⣿⠇
⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠙⢿⣿⣦⡀⠘⠛⠛⠛⠛⠉⢠⣶⣿⡿⠋
⢠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⢠⣷⣄⠀⠀⣹⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⣠
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢻⣿⣷⣾⣿⣿⣯⡀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣾⣿
⣿⣏⠀⢾⡷⠀⣹⣿⣿⡏⠀⠀⠀⢹⣿⣿⣿⣟⠁⠀⠀⢀⡀⠉⠛⠛⠋⠻⣿⣿⣦⡀⠀⢴⣿⡿⠋⠙⠛⠋⠁
⣿⣿⣦⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣄⠈⠻⣿⣿⣦⡀⠉
⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⢀⣄⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣴⣿⡿⠃⠀⠈⠻⣿⣿⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢿⣿⣿⣏⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⢀⣽⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣅⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠘⠻⣿⣷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢩⡇⠀⠀⠀⣼⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⣇⠀⠀⠀⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠉⠉⠛⠋⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢠⠶⠋⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⠏⠀⣠⣤⣤⣀⡀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⡀⢸⠇⠀⠀⠈⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢷⣼⠀⠀⠀⢠⣇⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠉


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣶⡟⠉⢀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣿⣿⣿⣄⠐⠠⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⡟⢃⡀⢂⢃⡠⠠⣀⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠢⣿⣿⠀⠦⠆⢸⢝⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠻⠀⠑⠒⠁⠀⠈⠁⠀⠀⠀⠀⠒⠂⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠁⡄⠀⠂⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠐⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠀⠘⠂⢄⡀⠀⠀⠘⠀⠀⠀⠆⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠃⢠⠀⠠⠀⠀⠀⠈⠑⠒⠠⠤⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡔⠀⠐⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠑⢤⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠒⠄⣀⣀⠠⢖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡍⠑⠂⠂⠀⠀⠀⠀⠀⠀⢰⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠢⠃⢸⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈⡡⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⣰⢯⠀⠀⠀⠀⠀⠀⠀⣀⠔⣁⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⢫⠬⠛⠔⢡⠘⠂⠀⠚⠓⠀⠀⠔⠑⠢⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⡠⡀⢸⠀⠐⡀⠄⠀⠀⠀⠈⢊⣔⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡞⣀⠇⠀⠙⢯⡾⢀⣈⡧⢴⣂⣚⣧⠶⡄⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡑⠚⠀⠐⠶⠿⢿⣿⡠⣯⣼⣿⣿⣾⣶⣧⣤⣼⣿⣿⣿⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢡⠃⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⡠⢔⣠⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⢀⣀⣀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢰⣀⣴⠛⣹⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣂⣛⣃⣸⡆⢾⣧⠽⠏⠑⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠈⠙⠛⠉⠳⡀⠀⠀⠀⠀⠀⠀⢀⣶⠄⠀⠀⠀⠀⣼⣹⡿⠀⠠⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣓
⠀⠀⠀⠀⠀⠙⣦⣴⠢⠤⠔⠛⢻⣿⠀⠀⠀⠀⢐⣿⣿⠍⠀⠠⡇⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⣴⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠔⢦
⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⣲⠀⠀⠒⠲⢦⣤⣭⠅⠀⠀⣧⠤⠀⠀⠀⠀⠀⠀⢸⠃⠀⡿⠀⢠⡇⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡄⠁⠀⡗⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠈⠓⠀⠙⠀⠀⠃⣠⠞⠀⠀⠀⠀⠀⠐⠂⠀⢠⠚⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⡰⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⢒⡂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠃⠚⠦⠤⠧⠤⠭⠷⠒⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠡⡓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠣⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊
⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⢳⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠠
⠀⠀⠀⠀⠀⠠⠽⠇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢢
⠀⠀⠀⠀⡼⠆⠀⠾⠀⢀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢐⠀⠀⡇
⠀⠀⠀⠠⡇⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⢸⡇⠀⠀⣆⠀⠀⠀⠀⠀⠠⠀⢈⡤⠄⠁
⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠐⠁⠀⠛⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢨⠇⠀⠀⠀⠀⠀⠖⢼⡇⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀
⢠⢀⣾⠀⠀⠀⠀⠀⠀⠘⠤⡟⠀⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡄⢈⡇⠀⠀⠀⠀⠀⠀⠀⢵⠹⡇⠀⠀⠀⠀⠨⠀⠇⠀⠀⠀⠀⠀⠀⢀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠠⠀⡄
⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⠀⠘⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠻⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠦⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠈⠉⠀⠈⠉⠉⠉⠉⣠⠁⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡏⠀⠠⣤⣜⡓⠘⠰⣶⣂⣀⣀⠀⢠⠋⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⠃⠀⠈⠉⠉⠀⠀⠀⠉⠉⠛⠋⠁⢸⠀⣇⣠⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⣀⠀⠀⢀⠀⢀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠒⠋⠉⠒⠶⠮⠜⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠢⠤⠚⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡦⡠⠖⠋⠙⠶⠊⡇⠀⠀⠀⠀⠀⠀⠀⡶⢄⠜⠲⢄⡉⠙⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠒⠂⠤⠤⠤⠤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡼⠉⠉⠉⢹⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢤⡇⠀⠀⠀⠀⣇⡸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠓⠒⠚

⢾⣿⣿⣿⣿⡆⠀⣿⠀⡆⠰⡟⢿⡶⠀⢠⠀⢸⡇⠀⠉⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠛⠛⠛⠓⠀⣤⠀⠠
⣿⡿⠿⠛⠛⠃⠀⡟⠀⡇⠀⠈⠁⠀⠀⣿⠀⢸⣧⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠
⣟⠀⠀⠀⠀⠀⠀⠃⠀⠇⠀⠀⠀⠀⠀⡿⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠡⡤⠀⠀⠐⠀⠀⠀⠀⣿⠀⢸
⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈
⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⠃⠀⠀⠸⡦⠀⢀⠀⠀⠀⠀⣿
⠀⠀⡇⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠖⠀⠀⠸⣿⠀⠀⠀⠾⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠡⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⠀⠀⠀⠀⠀⠃⠀⠃⠀⠀⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⡀
⠀⢿⣄⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣭⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⢿⡏⣠⣴⣞⢦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⣹⣇⣿⣿⣿⣎⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠉⠁⠈⠛⠛⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠛⣿⡿⣦
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠄⣿⠋⠀⠘⠂⢿⣏⡿
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⢹⠀⠁⠰⠦⢤⣄⡀⠀⢂⠀⡀⠘⢠⠀⡁⡙
⠀⠀⠀⠀⠀⠀⢇⠀⠐⠁⠿⠀⠀⡀⠀⠀⠀⠈⠓⠋⠀⠐⠀⠈⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠈⢆⠀⢃⠀⠀⠀⠘⠀⠀⠀⢀⢀⣀⡀⢀⠠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢦⠀⠀⠀⠀⠀⠀⣠⣴⠜⠣⠂⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠳⡈⠢⡀⢀⣶⣶⣿⡉⠙⠀⠀⠀⠀⠀⠀⢂⠀⢀⠀⡠⠐⢒⣀
⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⠈⠒⠶⢿⠿⢿⣿⣮⣶⡦⣤⡴⠦⠤⠤⠤⣀⣀⣀⣈⣉⡡⠤⠔⠚⣷
⠀⠀⠀⠀⠀⠀⠀⣰⣷⣄⡀⠀⠀⠨⠀⠀⠀⠀⠀⢠⡞⣧⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⡰⠀⠀⠀⠀⠀⠘⠒⢿⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⣿
⠀⠀⢰⡋⡏⠇⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠇⠘⢄⠸⣿⣿⣿⡟⠿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡐⠁⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠄
⠀⠀⠀⠷⠄⢗⣾⣧⣻⣿⣿⣷⢲⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠸⠆
⠀⠀⠀⠘⣄⣘⢿⣿⣿⣿⣿⣿⠀⠙⢧⣀⣀⣀⣀⣀⣀⢘⡇⠀⠀⠀⠀⠐⠂⠀⠏⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⡀⠀⡹⣿⣿⣿⣿⡄⡄⢀⠈⠛⢿⣿⣿⡇⠈⣍⠟⠒⠒⣲⠒⠤⠤⢾⠐⠂⡉⢓⠑⠖⠄
⠀⢆⠠⠈⠀⠐⠂⠀⠣⠛⠿⠟⠀⡉⢁⡊⠹⠎⠏⠉⠔⠄⠠⠈⠀⠈⢋⠉⠉⠒⠦⡇⠀⠔⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢱⡀⠀⠀⠀⠀⠀⠀⢀⣔⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⣀⣀⠀⢀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠔⡿⠋⠐⠁⠖⠂⠒⠀⠀⠀⠴⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰⠋⠀⡠⠖⡛⣤⠊⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠀⠀⠀⣀⡽⠻⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢠⠀⠀⠀⠀⠀⢶⡤⠤⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⠤⠶⣄⠀⠀⠀⠀⠀⠀⢀⣀⡸⠋⠀⣤⣖⣺⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⠛⠋⠉⠁⠀⣠⣀⠈⠻⠶⢤⣤⡔⠛⣯⠙⣧⣤⡟⢯⡩⣄⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣻⠖⠂⠳⣤⣄⡴⠋⠙⠙⢦⡀⠀⠀⢹⣧⠀⣹⢾⣟⡟⢳⣼⢧⠀⠉⢳⡀⠀⠀⡷⡀⠀⠀⠀⢀⡴⣤⠴⣄
⠀⠀⠀⠀⠀⠀⠘⢷⣟⣤⡞⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⣾⢻⠀⠛⢠⡷⡇⠈⣿⡼⣶⣄⠀⠳⡀⠀⠑⠃⢀⣠⣶⠏⢰⣱⣋⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣼⠃⢠⢀⣦⡆⢸⠇⢰⡏⣧⢸⡿⣦⡚⠛⠓⠒⠒⠋⢹⣙⣷⢀⣝⣓⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣤⠾⢯⠻⡅⠀⣾⠟⣿⣧⠎⠀⣼⠃⣿⠛⡇⠘⣧⡀⠀⢀⣀⣤⠼⠋
⠀⠀⢀⡀⠀⢀⣀⣠⡤⠞⠉⣿⠀⠀⠻⣏⠀⠀⠀⠑⠈⣰⢿⣀⡼⠋⢀⣼⣡⡾⠃⠀⡇⠀⠈⠉⠋⠉
⠀⠀⠸⣍⠛⢻⡍⢻⣿⣀⡰⠛⣧⠀⠀⠙⣦⡀⠀⣠⡾⣧⢶⣶⣶⣛⣿⣿⡿⢶⣤⣶⣛⣀
⠀⠀⠀⠈⠙⠻⠟⠋⢉⡟⠀⠀⠘⣆⠀⠀⠈⣛⢾⡯⠚⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⡿⠘⠞⠉⠁⠀⠀⢀⠴⠿⠿⠚⠛⠛⠛⠛⠛⢲⣾⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⣀⣸⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠋⠉⢨⢟⢟⠟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣀⣠⡭⠭⠴⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠄⠈⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⠔⠚⠀⠐⠢⣄
⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢠⠄⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠑⠦
⠀⠀⣤⣀⣄⠀⠀⠀⡀⢀⣤⡤⣤⣀⡀⠀⠀⢸⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⢲⣄
⠀⠀⢈⣿⡇⠀⠀⠀⣱⡞⠛⠳⠛⠈⠁⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣏⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢑⢴⠁⠈
⠀⠀⠚⠛⠛⠀⠀⠀⠿⠁⠀⠈⠁⠀⠀⠀⠀⢸⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠙
⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠞⠁⠉⠀⠈⠀⠓⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⢀⣀⣀⣀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠠⣼⡆⠀⠀⢿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣻⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢴⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠉⡠
⠀⠀⠀⠀⣀⡀⣤⣤⠤⡤⡤⢥⢤⡤⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠐⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡨⠯⡀⣀
⠀⠀⠀⠀⣼⡇⠿⠿⠶⠮⠷⠷⠶⠶⠾⠿⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠆⠀⢶⣽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠂⠀⠀⠃⠀⠀⠀⠑⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣠⠀⠄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠄⢄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⠦⠑⠈⠁⠀⠈⠁⠚⢴⡢⡀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠊⠁⠀⠀⠀⠀⠉⠄⣀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⣀⣀⣀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠄⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢰⣦⣄⡀⠀⡇⠀⢀⣠⣴⡜⠇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣟⠉⣿⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠛⠀⡇⠀⠳⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠿⠿⠿
⠀⠀⠀⠀⠀⣹⣿⣿⣅⣇⣯⣭⣹⣥⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⢶⣿⡇⠀⠀⡇⠀⠀⣽⣿⠆⠀⠀⠀⠀⠀⠀⢘⣷⣿⣿⢙⠋⡟⣙⠻⢻⢹⣿⣿⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⠉
⠀⠀⠀⢠⣤⣀⡀⢀⣼⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠙⠿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢣⣶⡜⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣝⣛⣫⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⢟⣛⠿⣿⣿⣿⣿⡿⢛⡛⠿⣿⠇⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⠈⣛⣫⣴⣿⣿⣷⣬⣛⣛⣫⣾⣿⢏⣴⣶⣶⡌⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢈⣿⣿⠟⣫⣭⡙⢿⣿⣿⣿⡿⢣⣾⡟⣛⡻⣿⣎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢨⣭⣵⣾⣿⣿⣿⣶⣭⣽⡟⣱⣿⣿⡄⣿⡇⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⢏⣼⣿⣿⣿⡇⣿⢡⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢨⣿⣿⣿⣿⣿⣿⡿⢣⣾⣿⣿⣿⣿⣧⠻⢸⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣭⣭⣭⣭⣽⣿⢱⣿⣿⣿⣿⣿⣿⡿⣫⢻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣎⠻⠿⠿⠿⠿⠿⠜⠿⠿⠿⠿⠿⠿⠿⠶⠾⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠢⠀⠐⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡤⢄⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠛⡈⠀⠀⠀⠃⡄⢈⠙⠋⡄⠀⠀⠀⠀⠡⣀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠁⠀⠀⠂⣈⠉⠦⠠⢤⡀⠀⠀⠀⠀⠀⠀⡉⠉⠉⡁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⡱⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠈⠀⠀⠠⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⡆⠈⠂
⠀⡀⠀⢀⠐⢄⡀⠲⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢣⢆
⠀⠀⠈⡹⠁⠀⠀⠐⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡀⠁⠆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠙⠋⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠅⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡤⠆
⠀⠀⠠⠀⠀⠀⠀⠀⡠⢆⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠶⡵⠠⠀⠈⠂⠀⠀⡀⠀⢀⠀⢀
⠀⠀⠀⢠⠀⡀⠂⠀⠁⡨⢄⠀⠀⠀⠰⠐⠂⠀⠀⠀⠀⠀⠀⠚⠈⠀⠀⠒⠎⠂⠀⠂⠀⠀⠀⠈⠁⠘⠁
⠀⠀⠀⠀⠐⠀⢠⡜⠂⠡⣌⡀⠀⡀⠢⡀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒
⠀⠀⠀⠠⠀⠀⠛⠁⠰⠄⠃⠀⠀⡽⠀⠀⠀⠀⠘⢀⠄⠀⠀⡉⠁⠀⡀⠀⠀⠂⠁
⠀⠀⠀⠐⠂⠀⡀⠀⠀⠁⠀⡰⠊⢀⠀⠁⠀⠀⠘⠡⠤⠀⠀⠠⠀⠀⠘⠂⡀⠀⠀⠛⠡⠀⠡⣀⠒⠄
⠀⠀⠀⠀⠀⢉⠇⠲⣆⡶⠚⠁⠖⠁⠀⠀⠀⠀⠀⠛⠂⠀⠀⠰⠡⡄⠀⡀⠐⡀⠀⠀⠀⢈⠀⠀⠡⠄⠨
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⣠⡦⠀⠀⠒⠂⠀⠀⠐⣀⡬⠀⠆⠀⠀⠀⠁
⠀⠀⠀⠀⠂⠤⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠅⠤⠷⡶⡂⡂⡐⡋⠉
⠀⠀⠀⠀⠀⠂⠂⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉

⠀⠀⠞⣛⠻⢿⣷⣶⣄
⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠔⢶⣤⣤⣴⣿⣿⣿⣿⢳⣦⡀
⠀⠀⠙⠟⣛⣻⣿⣛⠵⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠹⣻⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⢟⡿⣿⣿⣿⡷⢲⠶⠶⣒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢿⣿⡁⠀⠈⠀⠉⠱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡱⣄⠀⠀⢀⢔⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢏⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⡄⢀⣀⣸⠘⠜⡆⠀⣀⠤⢤⡀⠀⠀⢀⣤⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠛⠛⠉⠉⠀⢱⠃⢸⠁⣂⣆⣧⠤⠴⠗⡒⣮⢣⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣐⣢⣄⠀⠀⠀⠀⠑⠢⣤⣍⠑⡄⠹⡖⢚⠉⠉⠁⡄⡆⣆⣽⣿⡟⢸⢀⢹
⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⢠⠏⠀⣸⣧⡽⣀⣤⠤⠒⠚⡉⠉⢹⢣⡠⡖⠿⠿⢟⣴⡶⢷⢩⡷⢲⣿⡖⢸⠀⣺⡄
⠀⠀⠀⠀⠀⠘⢦⢦⣘⣦⠤⠤⣀⠤⠜⠒⡟⠀⠉⠙⢶⣨⠷⣗⡓⠓⠦⣌⡬⠵⡇⠀⠨⢹⣯⢸⢸⣾⠁⠘⠿⠷⣶⣧⣾⢻⣷
⢀⣤⡀⠀⠀⠀⠈⣾⢉⠬⣝⣦⠼⣦⢀⢸⣍⣖⡀⠀⠀⠀⡀⢀⠉⠲⢄⠀⠑⢤⡇⠀⠀⣸⢧⡎⣾⠃⢾⣯⣷⣒⡒⢤⢀⣾⣏
⢣⡤⠙⢆⡀⠀⠀⠘⢸⡒⡾⢸⢀⢽⠶⡞⢚⣛⣻⠀⠀⢐⠯⡒⢽⣦⣄⠑⣄⢦⡱⡀⡐⢺⣇⠻⡿⠀⣀⡉⢙⣛⢛⢋⣼⡏⠘⢆
⠀⠙⣿⡂⠙⠢⣄⡁⠀⠈⠀⠸⠀⢿⠀⡞⠰⠶⡏⢳⠤⣄⡉⣺⡿⢋⣈⢧⡈⣎⣿⣧⢘⣺⣷⡂⢱⠤⢤⠞⠋⠉⠉⣽⡿⡙⣄⢜⣆
⠀⠀⠈⠙⢾⣔⣤⡇⠀⠀⠀⡆⠨⢽⡀⠙⡖⣖⣁⡆⠀⠧⡟⣛⡽⠋⣽⡿⣿⢸⣿⠟⣐⣺⣯⡊⣇⣨⡴⢚⠂⠀⢬⣷⣽⣄⢸⡄⣺⡄
⠀⠀⠀⠀⠀⢈⡿⠁⠀⠀⠀⢡⢐⢽⣿⣉⣛⠻⣶⠁⠨⢱⣟⢽⡇⢾⣿⣿⡾⠋⣹⣶⣾⠿⣿⡿⠀⣿⣽⠧⠴⣶⣾⠽⠗⡞⣹⠀⢚⡅
⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠘⡄⠙⢻⡿⡿⣷⡇⠀⠥⣾⣷⢷⡷⡀⣈⣛⣁⣴⣿⣇⡽⠶⢛⣋⣽⠿⡶⢊⠟⡔⢡⢊⠎⡔⣹⠨⢽
⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠙⣇⣿⠀⠨⢽⣯⢮⣿⣌⣠⡿⢿⣿⡿⢯⠗⣲⢋⡝⣡⢃⠜⡠⢫⠎⡰⣣⢃⠞⡐⢹⢒⣽
⠀⡇⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⠳⣀⡭⠿⢷⣿⡯⡽⣲⢋⢝⡝⣡⢊⡔⣡⢊⡔⡡⢋⠔⡱⢁⠞⡴⣡⢋⠜⡰⣫⢐⣺⢄⣀⠀⠀⠀⣀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠂⣿⠲⢫⢫⢊⢞⡔⡡⣫⢊⡜⣡⢋⠔⣡⢊⠜⡡⢊⠞⡰⢁⠞⡰⢡⢎⠜⡔⡿⠀⣾⣒⣒⡀⠭⢅⣐⡒⠦⠤
⠀⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡇⡉⢽⠔⡵⡡⢣⢊⠔⡕⣡⠊⡔⡡⢋⡔⡡⢋⠔⡡⢃⠞⡠⢃⠜⡰⢣⢊⠜⡐⡯⢕⣿⣂⠀⠬⠅⢒⣒⣀⠉⠉⠥
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⠍⡏⡔⡱⣡⢪⢊⡔⢡⢊⠜⣡⢊⡔⢡⢊⠜⡠⠃⡴⣡⢃⠜⡰⢡⣊⣾⣿⠇⣽⣗⠒⠬⠭⠤⠀⣐⣒⠒⠪⠍
⠀⠀⣀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢒⣨⢋⠜⡔⡱⣡⠊⡰⣡⠋⡔⣡⠋⡴⣡⢊⡜⢁⠎⣴⣡⣮⠞⣗⣯⢗⣛⣛⣛⣂⣘⣂⠈⠩⠭⠤⠄⠐
⠀⢠⠉⠀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇⠕⣺⢃⢎⠞⡴⢁⢞⡔⢡⢎⡔⢁⠎⡴⢡⣊⡴⣗⣞⣭⡜⣚⣛⣛⣁⣈⡉⠭⠤⠤⠶⠶⠶⠒⠀⠁
⠀⠘⠢⢌⣓⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠥⢺⣠⢋⠎⡰⢣⠎⡴⣡⢊⣰⢧⠞⣟⣯⠷⠟⠋⠉⠂⠀⠀⠐⣒⣐⡒⠒⠚⠛⠓
⠀⠀⠀⠀⠈⠉⠒⠤⣀⠀⠀⠀⠀⠀⢀⠀⣇⡨⠝⡱⢃⠞⣰⡧⢾⢲⣻⣽⠼⡾⠏⠩⠭⠭⠥⠬⠭⠭⠭⠭⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠈⠀⡇⢒⡷⢶⣻⣭⢷⣶⣟⣛⣂⣐⣀⣶⣦⡀⠠⠍⠭⠭⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⠯⣷⣟⣛⣒⣒⡋⠭⠭⠭⠭⠭⠤⠶⡖⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⢦⡀⠀⠀⠀⠹⡄⢲⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣙⢶⣄⠀⠀⠹⡌⢧⠀⠘⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣭⡳⡄⠀⢁⣤⣴⣶⣾⡿⠻⣦⣦⣀
⠀⠀⠒⠶⠤⣤⣄⣀⡀⠀⠀⠶⠤⣤⣀⣀⠀⠀⠁⠀⣴⣿⣿⣿⢏⡿⠿⠶⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠶⠤⣬⣍⣙⡛⠂⢸⣿⡿⠛⠛⢿⣇⡀⠀⠀⠀⠙⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⣿⡧⡔⠉⠩⠂⠒⠒⠒⠂⠀⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⣷⣶⢶⡜⠀⢺⣟⣛⠂⠈⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢫⠾⢤⡀⠀⢠⢟⣻⠿⢸⠘⠢⠉⠙⠁⠀⣵⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣠⣞⡒⢌⣂⠸⣄⠀⠀⣼⡀⢀⣄⠀⠀⡴⡆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣉⠦⢤⢹⡀⠀⠀⢹⡄⠚⠁⣁⣀⣈⠁⣇⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢸⡄⠈⠓⠅⡖⠀⠀⣿⡀⠉⣓⣂⣉⠀⡟⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣀⡴⡏⢷⠀⠀⣼⢻⡆⠀⢻⣅⠀⠈⠋⠀⢀⡾⠁⡏⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⣷⠖⡇⠘⠆⠘⠁⠘⣷⠀⠸⡿⡷⣶⠶⠶⠋⢀⡼⣥⡈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢷⡶⠦⢤⡀⠀⠈⢧⠀⢳⣷⠈⢷⡄⣰⠋⢀⣾⠟⢮⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠛⠲⠶⠶⠶⠿⢷⣄⣿⠁⢠⡾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⡀⠁⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠶⣕⠀⠀⠀⠀⠀⣠⢾⠟⠛⠚⠚⣦⡆⠁⠺⠶⠶⣦⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠄⠀⠀⠀⠈⠳⢮⡢⣀⣺⡑⣷⣆⠀⠄⢀⡇⠀⠀⣸⠀⢸⣇⣿⡈
⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢀⡤⠐⠛⠂⠠⠀⠀⠈⠉⠉⠈⠉⠉⠉⠁⠀⣉⡅⠀⡞⠛⠑⠋⠀⠉⠁⠈⠁⠠⠄⠶⠶⣶⣤⣤⣤⣄
⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⢐⣉⡀⢀⣔⣿⣿⣟⠢⡀⠤⠀⡆⠠⠀⠀⠀⠀⠀⣶⠆⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢸⡏
⠀⠀⠀⠀⢸⡀⠀⣠⠖⠲⠘⢝⠁⣿⢋⡤⡴⢌⢳⣗⠒⠒⡷⠒⢒⠒⠒⠒⠲⣗⠂⡇⠀⠀⠃⣤⡀⠀⠀⠀⢀⠀⢰⠀⠀⠀⠃⣿⡇
⠀⠀⠀⠀⠀⠙⢲⡗⠖⠚⠀⠀⠀⠀⢌⠫⠏⢃⡜⠈⠀⠀⠀⠉⠹⠶⠶⠶⠶⢶⡘⢧⠀⠀⣀⡈⠁⠀⠀⠓⠋⠻⠁⡏⣹⠁⢀⣿⠃
⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⢀⣀⣀⣀⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⠠⠀⠀⡀⠀⠀⠄⣂⠐⠂⠤⠄⣀⡀⠀⢸⣿⣀
⠀⠀⠀⠀⠀⠀⢨⡁⠀⠀⠠⣟⣿⣿⣗⣗⣷⣾⣾⣾⣶⣓⣓⣛⣿⣿⣿⣷⣾⡿⢫⣽⣴⣠⢠⢸⣷⣶⣦⣤⣉⡀⠀⠀⠀⡄⣸⣿⢯⣧
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠝⠘⠈⠉⠙⠻⣖⣀⢤⠕⠁⠀⠀⣉⣑⡳⠳⠾⡮⣭⣿⣟⢻⣿⣷⣶⣦⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢅⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⡠⠋⠂⠀⠀⢺⠭⢗⠊⠉⠙⣚⠒⠑⠉⣋⣟⠛⠾⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣘⠄⠀⠀⠀⠀⠀⢀⠴⣨⡭⣁⡉⠭⡝⠺⠶⡬⠤⢲⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠶⣤⣀⡀⠠⣃⠀⠀⢈⡖⠈⢻⠓⠦⢤⣁⣰⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⡳⠶⣥⣀⡊⠀⠀⠘⡄⠀⠢⠹⣽⡷⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠛⠛⠶⣶⣷⣤⣧⣼⡿⠥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠒⠈⠉⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⢀⣀⡀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣇⣴⠿⢿⣿⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠛⠋⠀⠀⠀⠈⠀⠀⠀⠐⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⡿⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠁⠀⠀⢙⣦⠀⠀⠀⠀⠠⠀⣤⡤⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠄⣸⡇⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⢀⠔⠀⠘⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠙⠿⠿⠀⠀⢀⠔⣡⣤⣶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡀⠉⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠙⣿⣁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡇⢸⣿⣿⣦⣄⣀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡟⠁⣸⠟⢿⣿⣿⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⣀⣼⣿⣿⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠸⢿⣿⣿⣿⣿⣿⡿⠁⢠⣾⣿⠿⠿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠁⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠠⠤⠀⠴⠀⠴⠧⠄⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⢀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠾⠿⠿⠿⠿⠿⠿⠶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⢩⣿⡄⠀⣀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠿⠛⠉⠀⠀⠀⠸⠿⠃⣸⣿⡆⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠈⣿⡆
⢰⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠃⠀⠀⠀⠀⠈⠙⠻⢶⣦⣤⣤⣸⣷
⣼⣿⣶⣶⣶⠶⠾⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿
⣿⡀⠀⠀⢠⣦⠀⠀⠀⠀⠛⠛⠉⣁⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣈⣉⡉⠉⠀⠀⠀⠀⠀⣼⣿⡄⠀⣿
⣿⡇⠀⢀⣿⢿⡇⠀⠀⠀⠀⣾⠟⠛⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠛⠛⢻⡇⠀⠀⠀⠀⠀⣿⣽⡇⠀⣿
⢹⡇⠀⠀⠿⠾⠇⠀⠀⠀⠀⢻⣦⣀⣀⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⣀⣠⣾⠃⠀⠀⠀⠀⠀⠉⠉⠀⢀⣿
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⣶⡆⠀⠀⣶⡆⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣸⡟
⠀⢻⣇⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⢀⣿⠃
⠀⠈⢿⡆⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠀⢀⣾⠏
⠀⠀⠈⢿⣆⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣠⣤⣤⡀⣠⣤⡀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⠛⠛⠁⠙⠛⠛⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⡆⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡛⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣣⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢄⡀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡇⠀⠀⠘⠄⣀⠀⠀⠀⠀⠀⣀⠤⠃⠀⠑⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⠄⠄⠀⠠⠀⠠⠀⠄⠀⡄⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⢒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡿⠻⣿⣷⣦⠆⠀⠀⠀⠀⠀⠐⠀⢠
⠈⠂⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡁⠀⠀⡇⠀⢹⣿⣿⠀⢀⠄⠆⠀⡤⠌⢄⠀⡀⠀⠀⠀⠀⡠⣤⣤⣶⣀⠄
⠀⠘⡄⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⡆⠀⠀⡇⠀⢸⣿⣿⢠⠃⠘⠀⠀⠀⠀⠂⠁⠁⠀⠀⠀⠐⣼⠋⠀⠀⠹⣯
⠀⠀⠘⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⣷⣶⣶⣶⣶⣶⣶⣶⣶⡇⠀⠀⠀⠀⢸⣿⣿⠸⡀⠈⠒⠐⠃⠀⡌⠀⠀⠀⠀⠀⠷⡻⣆⣀⢀⣴⢏
⠀⠀⠀⠱⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⢠⠀⠀⠸⣿⣿⡆⢳⠠⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⡭⠟⠑⠁
⠀⠀⠀⠀⠡⠀⠀⣍⠐⠤⣀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢸⠀⠀⠀⣿⣿⡇⠀⠂⣤⠀⠃⠀⠀⠀⠀⠀⠀⡜⣰⡟⡰⠁
⠀⠀⠀⠀⠀⢃⡘⠀⠁⠂⠤⢈⡁⠒⠠⣀⠈⢃⠀⢠⠃⠀⠀⠀⠀⠘⠀⠀⠀⣿⣿⡇⠀⠀⣿⢠⠀⠀⠀⠀⠀⠀⠌⣼⡟⡐⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⢾⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⢸⣿⠈⠀⠀⠀⠀⢀⠌⣼⠟⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⢰⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⣿⣧⢸⢸⡏⠀⠀⠀⠀⢀⠎⣼⠏⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠚⠀⠀⠀⠀⠀⠃⠀⠀⠀⠚⠛⠛⠒⠚⠃⠒⠀⠀⠀⠒⠚⠛⠘⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣄⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣀⡠⠔⠒⠿⠟⠣⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠀⠠⣄⠀⣀⢀⡒⠢⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠃⠀⢨⡍⠉⢉⠀⡼⠀⠀⠉⠉⡁⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢀⣀⠀⣣⠀⠒⢰⢡⠆⠀⠀⠀⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡇⠁⢠⡄⠸⠀⢄⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠐⠀⠀⠀⠀⠑⠀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⢀⣙⣓⠢⢟⠃⠀⠀⠀⠀⠀⠤⡤⢤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡉⠑⠚⠙⠀⠀⠀⠄⠀⠠⢤⠴⡃⠀⢀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠐⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⢀⠟⠒⠤⣀⡐⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢄⢀⠞⠁⠀⠀⢀⡀⢀⠀⣀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⢸⡼⠃⠀⠀⠠⠂⠈⠀⠈⠈⠁⠈⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠘⠀⠀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣀⡐⠀⠘⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡀⠀⠀⣃⣤⠃⡀⠀⠈⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠚⠡⠮⡀⠀⠈⣉⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠈⠁⠀⠀⡏⢰⠰⡶⠒⠶⡤⢀⡤⢄⠀⢀
⠀⠀⠀⠀⡎⡇⠀⣰⡑⢩⠁⠀⠀⠀⢠⠁⡌⢰⠀⠀⠰⢁⢄⢀⣴⠀⢎
⠀⠀⣖⡂⠘⠀⠀⠡⠀⡘⢀⠔⣄⠀⢸⠀⠁⠃⠀⠀⠀⡈⠈⠀⠢⠁⠉
⠀⠀⠀⠠⣂⠄⠀⠀⠀⢶⠁⠀⡟⣤⡇⠀⠒⠀⢀⠀⢠⠃⡆
⠀⠀⠀⠀⠀⠀⠀⡄⢀⡞⠀⠰⠶⢶⡀⠀⢰⠀⡜⠀⡌⢰
⠀⠀⠀⠀⠀⡄⢰⠇⠘⠇⠀⢠⠂⠀⢸⠓⠋⠐⣇⣀⡏⠁
⠀⠀⠀⠀⠀⠀⢸⠺⡤⢤⡔⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠂⠇⠀⠙⠓⠢⠼⢃⣐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣷⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡟⠉⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⠈⠉⠀⠙⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣽⣿⡄⢳⠜⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⠏⠀⠀⢰⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠄⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣿⣿⣿⣿⣤⡀⢀⢾⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣶⣿⣿⣿⣿⣿⣿⡅⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣄⡀
⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⡿⢏⣿⢀⢿⣿⣿⣿⣿⣿⡿⠛⠶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢉⠁⢸⡏⡸⠀⠙⣽⣿⣿⡿⣿⣿⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⣇⠔⣿⡿⣁⠀⢀⣿⣿⡿⠀⠀⠉
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⡏⢉⣿⡁⡀⢹⠛⣿⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢌⠋⡀⠜⠉⠁⣸⣧⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠁
⣿⣿⡿⢿⣿⠿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣷⣷⣾⣶⣾⣶⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠤⠴⠶⠶⠶⠶⢶⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⣾⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣘⣫⣴⡞⠻⠿⣫⣿⣧⣠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⡆⢠⠞⠟⠈⣿⣡⣴⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⣿⣿⢿⠿⠿⣿⡿⡇⢀⣴⣶⣌⡉⠻⣿⣿⣿⣯⡁⠈⣠⣷⣤⣀⣿⠃⢼⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡼⠋⢰⣾⣿⣶⣶⣴⣦⣻⢿⠟⣿⢿⡄⠐⢿⣿⣿⣤⡀⠁⠀⠈⠉⣿⠠⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⣽⣿⣿⣿⣿⣿⣿⣯⡍⢻⠀⣀⡼⠃⠀⣰⣿⣿⣿⣿⣿⣤⣤⣴⢻⡀⢻⣿⣿⠘⣻⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢀⣬⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡈⢿⡋⠀⠀⠒⠿⠿⢿⣿⣿⣿⣿⣿⣿⣈⡇⢺⣿⣿⠀⣽⣿⣿⣿⣧⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⢿⣯⣿⣿⣧⠈⡆⠀⠀⠐⠿⠿⣟⡿⠫⢽⣇⣠⠔⠉⢀⣾⠿⡟⠀⢻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⡟⢀⣴⡚⠛⠿⡞⢿⣿⡂⢸⣿⣶⠶⠶⣮⣤⣤⣠⠇⠀⠐⢄⢺⣿⡫⠊⠱⣄⡼⣫⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣿⣃⠸⣿⣿⣆⢿⣿⡆⢳⡀⠀⣀⣤⣿⣿⣿⣷⣦⣶⣿⣿⣟⢿⣆⠀⣸⠂⣹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣿⣿⡟⠀⡏⠉⢙⣀⠉⠉⣛⢿⣿⣷⠀⢻⡏⣩⣏⣁⣀⣈⣉⣉⣩⣿⣧⠹⣆⢻⡄⣿⠀⣹⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣜⣿⣿⣿⣿⣷⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠿⣿⠛⠛⠛⠛⠋⣭⣹⠀⣷⡏⢈⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⠿⡇⢠⣷⣙⢿⣿⣿⣿⣿⣿⣶⣞⣿⣿⣿⣿⡿⢋⣼⡇⠘⠃⣼⣠⡾⢏⣿⣰⣿⡇⢈⣽⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⣿⣿⡷⠀⢸⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⡶⠿⠿⠗⠛⠛⢡⣟⣽⡇⠀⢾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⣿⣿⣅⠀⣼⣿⠛⠛⠛⠛⠛⢛⣻⣟⣛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⢉⣭⣿⣗⣿⡇⠀⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣯⠀⣿⣿⣿⣶⢶⣶⠒⠒⠚⠛⠳⢶⣶⣶⣶⠶⣶⣶⣶⠶⠶⠶⠿⠟⣿⣿⠀⣸⡇⠈⣩⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⠶⠀⣿⣿⣿⣿⣷⣭⣤⣀⣀⣒⣚⣛⣛⣫⣥⣾⣿⣿⣿⣷⣶⠶⠶⢿⣿⣿⣻⣿⡇⢠⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣼⣟⠛⠀⣿⡇⢰⣶⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⡭⠴⠶⠶⠶⠶⢶⣶⡈⣿⣽⣿⣿⠇⢠⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⠟⠀⡿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠸⣿⣧⡻⠿⠾⢿⠀⢤⣶⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢋⣀⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣾⡿⢟⡍⢐⣻⣶⣶⣽⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⢿⣿⣿⣦⣬⣞⡹⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡽⠛⠛⠋⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⠀⠘⠁⠀⠀⠀⠀⢠⠾⠃⠀⠀⠀⠀⠀⠀⠖⠓⠲⣜⢦⠀⠀⢀⡀⠀⠀⡠⠒⠈⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠘⢧⡀⠀⢹⠋⠀⢠⠀⡁⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠒⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⡂⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠄⠀⠂⠔⠀⠀⢠⠀⠀⠁⠊⠁⠀⠚⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢀⠁⡘⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⡀⠠⠤⠄⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢘⡔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠐⠀⠀⠉⠀⠈⠈⠂⠀⠀⠄⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢿⠃⠀⠰⠁⠀⠀⠀⠀⠀⡀⠐⠊⠀⠀⢀⣤⣆⠀⠀⠀⠀⠀⠀⠀⣀⠈⠑⠒⠋⠁
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠆⠀⠀⠀⠀⠀⠈⢂⠀⠓⠀⠀⢻⣿⠀⠀⠀⠀⠀⡤⠊⠠⠆⠀⠀⠅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢸⠀⠀⠀⠀⠈⠀⠀⠀⠣⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠓⠀⠀⠀⢸⠋⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠑⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠈⠒⠒⠁⠀⢀⡠⠤⠄⠰⢀⡤⣀⠀⠀⠀⠀⠀⠂⡼⠁⡀⠀⠀⠇⠈⡄⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⡀⠀⠈⠁⢀⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡟⠃⠀⠀⠀⠀⠐
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠀⠀⠀⠉⠳⠀⢠⠂⠀⣸⡟⠀⠀⣀⠀⠀⠀⡀⠀⠀⠁⠀⠠⡒
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢀⠄⠀⠇⠀⠀⡞⠀⠠⠏⠀⢀⠀⠈⠣⡀⠀⠀⠀⢀⠀⠰
⠀⠘⠲⠶⠖⠶⠀⠸⠉⠀⠀⠀⢠⠁⠀⠀⠀⠠⠴⠑⠋⠀⠀⢠⡇⠀⠀⢀⡀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠔⠁⠀⠀⠠⠁⠀⠀⠀⠂⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠖⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠁⠀⢀
⠀⠀⡀⢀⠀⠀⣰⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣿⠆⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢠⣄⡀
⠀⠀⠀⡀⠀⠀⠀⠈⠹⠃⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⡄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠈
⠀⠀⠀⠁⠀⠈⠀⠠⠂⠀⠀⠀⠀⢀⣾⠏⠀⣀⣤⣉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⡐⠈⠑⠢⣅⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠙⠂⢤⡰⠃⠀⠀⠘⠀⠙⠳⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣟⠀⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⢹⣿⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⣿⣷⣾⣿⠃
⠀⠀⠀⢀⣤⣾⣿⣿⡿⣿⣿⣿⢁⣿⠇⢸⣿⣿⣿⣿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠁
⠀⢴⣷⣾⣿⣿⣿⣿⡁⠈⠉⠛⠿⢿⣶⣿⣿⣿⣿⣿⣭⣝⡻⠿⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠻⠿⠀⠈⠹⣿⣷⡄⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠿⠷⣦⣄⣀⣼⣿⡟⠁⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠃
⠀⠀⠀⢀⣀⣤⣤⣽⣿⣿⣄⣀⢀⣀⣠⣤⣤⣤⣴⣾⣿⣿⠿⢿⣿⣿⣿⣿⣁⣤⣶⠾⢿⣿⣿⣿⣦⣤⣴⣶⣿⣿⡿⠋⠛⢿⣿⣿⣧⡀
⠀⠀⠀⣠⣀⣭⣅⢀⣩⣿⣿⣿⣿⣋⣙⣛⡛⠋⠉⠉⠀⠀⠀⠀⠉⠙⠻⠿⣿⣍⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠙⣿⣿⣿⡄
⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⢿⣿⣯⡙⠻⠿⠿⢿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⣀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠻⣿⣿⣿⣿⡿⠟⠉
⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠟⠋⠉⠁⠀⠀⠀⣠⡴⠚⠙⠟⠉⠁
⠀⠀⢿⣏⣽⣿⣿⡿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⢀⣤⠶⠋⠁
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⣛⣫⡿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⠞⠀⣠⠞⠉⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣅⣏⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣍⠀⠀⠀⠈⠉⠙⠛⢷⣦⣤⣀⣀⣀⣠⣤⣾
⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠟⢻⣛⠋⣻⢙⣭⠍⢉⡏⠀⠀⠀⢳⡖⢄
⠀⠀⠸⠀⠀⢻⡎⠛⢦⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡏⠀⠈⠛⣛⡥⠀⠙⠛⠛⣩⡀⠀⠀⠈⣷⠋
⠀⠀⡞⢧⡤⠞⢙⡆⠈⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⢸⡁⠀⠀⣀⠀⢀⠎⠀⠀⠀⢠⡿
⠀⠀⢳⡈⠳⠖⠋⠀⢠⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠛⠶⠟⠃⠀⣉⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠙⠦⣤⡤⣄⠘⣆⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠈⠉⠀⢀⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣆⢘⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡇⠈⡟⠓⠶⠶⠴⠶⠶⢾⡏⠩⣏⠳⣄
⠀⠀⠀⠀⣠⠤⣀⠀⠘⢯⢦⣀⣤⡜⢧⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⢦⡈⢳⡀
⠀⠀⠀⢸⠁⠀⠈⡇⠀⡜⠉⣀⠤⢛⡆⢷⠲⢤⣄⣀⣀⡴⠋⠀⣀⣼⠁⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⠀⠙⣆⠹⣆
⠀⠀⠀⠘⠦⣤⠶⠃⠀⠉⠻⢤⣒⣉⣽⣼⣀⣀⣀⣉⣉⣀⡤⠞⠋⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⠀⠀⠘⣆⠹⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠼⢆⠹⡄⠉⠉⠉⠉⠉⠀⠀⠀⣿⠀⢸⠷⠶⠶⠶⠶⠿⠛⠛⣃⡀⢸⠀⠀⠀⢹⠀⢹⡄⠀⠀⠀⠀⠀⣴⢉⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣧⠘⠋⢙⠳⣆⠀⠀⠀⠀⢸⣞⣉⠆⠀⠀⠀⠀⠀⠀⠸⣅⡞⢸⡀⠀⠀⠘⣧⠀⣹⣄⡀⠀⢀⡴⢛⡘⠷⠖
⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⡇⠀⢰⡇⢠⡖⠋⢧⣸⠀⠀⠀⠀⣾⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣧⠚⠁⠀⠙⠻⣥⠖⠋⠉⠛⠋
⠀⠀⠀⢸⠁⠀⠈⡇⠀⠀⠀⠀⡇⠀⠀⠳⣄⣙⡄⠈⠋⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣸⣁⡴⣆⠀⠀⠀⠈⢧
⠀⠀⠀⠘⠲⣤⠞⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⢤⡤⠞⣁⣤⠿⢀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣸⣡⠴⡄⠠⡞⠉⠈⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠈⠀⡠⠛⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢀⡏⣇⠀⠀⠀⠀⠀⠘⠒⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⢦⣤⣀⣀⣀⣀⣿⡄⠸⣤⣀⣀⠀⢀⣒⣊⡽⢧⣄⣀
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⡴⠶⠶⠏⠁⠀⠀⠀⠉⣉⣹⠁⠀⢰⡇⠉⠉⡉⠉⠉⠁⠒⠉⠁⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡀⠀⠀⠀⠀⢀⣠⣴⣯⣭⡽⠀⠀⠸⣛⣓⣚⣻⠦⣤⣤⣤⣤⣤⠤⠶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠭⠝⠛⠛⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⣠⣤⠶⠶⠶⠶⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⠛⠉⠀⠀⣠⡤⠀⠀⠀⠀⢤⡉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠐⣻⡟⠀⠀⠀⠀⠘⢋⡄⣠⠄⠐⣤⡀⠻⠂⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣎⣼⠙⡄⠀⡏⣷⣴⡀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⠀⣠⠿⠿⢠⡇⠀⡇⠻⠿⣧⠀⠀⠈⣷⢊⣦
⠀⠀⠀⠀⠀⠀⣀⠿⣇⠀⣼⡏⠀⢈⢧⣀⡼⠀⠸⣿⣄⡠⢃⠀⢸⡄⣿⣯⡿
⠀⠀⠀⠀⠀⠰⡇⡰⣌⠂⠈⢀⡄⠈⣓⠂⠠⠄⠀⠙⢦⡘⠥⠀⢸⡇⢸⣦⠇⠀⠀⠚⠩⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠹⣏⠻⡆⠀⢸⡇⠀⠀⣠⠀⠛⠲⠦⢄⡉⢳⠀⠀⠁⠀⣿⠀⠀⢀⡤⠒⠉⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡄⣚⡇⠀⢠⡇⠀⠀⠀⠀⠀⠈⠉⣄⠀⠀⢠⡿⢀⡴⠋⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠈⠀⠀⢸⠀⣀⣀⣀⠀⣀⡀⠀⢸⡄⢀⣾⢃⡞⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣦⠀⠀⡼⠛⠉⠀⠒⠒⠂⠀⠙⠺⣣⡾⠃⢸⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⢦⣄⣢⣄⡀⠀⠀⠀⣀⡴⠛⠋⠀⠀⠘⠦⡤⡀⠀⠀⢀⡴⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⡟⣳⡤⠜⣙⣿⠛⢻⡟⣯⣀⠀⠀⢀⡔⠢⣼⠟⡟⠒⠊⣁⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠈⠁⠐⣯⢽⢿⡯⡛⠷⠃⣼⢧⠀⣾⡙⡖⠈⠻⣇⠀⠨⠖
⠀⠀⠀⠀⠀⠀⠀⠠⣾⠷⠒⠋⠀⠀⠀⣠⠟⣸⡄⠀⠀⠀⣿⢧⠗⢻⡘⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⡤⠚⠛⠉⣏⣠⣿⣧⣀⣀⠀⠸⡇⢰⠉⠙⢂⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠉⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣦⣴⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣭⣤⡤⠤⣤⡌⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢽⡟⣻⡟⢸⡇⠀⠀⢸⡇⣸⡟⣯⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢭⣁⠤⠶⠶⠛⠀⠁⠀⠀⠈⠁⠷⢬⢤⣙⠖⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠈⠓⢄⣀⠠⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡂⠀⢀⣁⡀⡈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠴⠖⠀⠂⠀⡠⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠞⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⡤⠤⠀⠀⠸⠆⠀⠀⠀⢀⡾⠳⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠄
⢤⣤⣀⣈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠂⠀⠄⠈⠉⠉
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠄⠀⠀⡉⡁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠒⡿⠄⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠂⠤⠤
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⠄
⠒⢦⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠐⠂⠈
⡀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠓⠚⢣⣴⣖⣒
⠛⠋⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢋⡉⠉
⠀⡀⠒⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢻⣤⠄⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠙⠋⠉⠁
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠐⠐⠀⠀⠘⠛⠀⠀⠃⠀⢀⠀⠀⠎⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⢄⠀⠀⠀⢀⠧⠤⠀⢹⡄⠶
⠀⠀⠺⢤⣄⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠂⠀⠀⢰⡷⢄⠀⢐⣿⡂⠸⠀⠀⠀⢰⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠢⠀⣰⣖⣛⣤⡀

⠀⠀⠀⠀⠀⠀⢀⣤⠔⠊⠹⠖⠒⠶⠐⠒⢦⢤⡄
⠀⠀⠀⠀⠀⡴⠏⠁⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠱⡆
⠀⠀⠀⠀⠀⡇⢀⠔⠊⢀⣀⣀⡀⠀⠀⠐⣐⣈⣠⡙
⠀⠀⠀⠀⠀⣧⣤⠖⢻⡉⠑⠋⣹⠋⠙⣏⠩⠁⣹⡇
⠀⠀⠀⠀⢸⠉⠀⠀⠀⠙⠛⠛⠉⢤⣤⠌⠙⠚⠈⡇
⠀⠀⠀⠀⠈⠒⣆⠀⠀⠀⠰⣂⠀⠀⠀⢀⣀⡀⢰
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠹⢟⠋⠉⡙⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⡀⠀⢀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣇⠀⠀⠀⢘⠉⢦⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠾⠇⡀⢀⣰⣆⣀⣈⡴⠶⠆⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠴⠶⠺
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠈⠁⠈⢻⡤⡏⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢠⠾⠛⠉⠉
⠀⠀⠀⠀⠀⣸⠁⠀⠀⢀⠀⠀⠀⠸⠀⠰⠀⠀⣀⠦⡆⢈⡇⠀⠀⠀⠀⠀⠀⢸⠘
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⡇⠈⣦⠀⠀⠀⠀⢀⣀⣼⡄⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⡷⠐⠒⢐⡆⠀⠀⠀⠀⠀⠀⣿⠀⠈⠁⡇⢀⣿⠐⢒⣊⣛⠋⠨⡇⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⣸⢃⠀⠀⠀⠸⠀⠀⠏⠀⠄⠈⠁⠺⣅⣀⠈⠉⠡⠄⢠⣷⠀⠀⢠⣤⠤⡤⣠⣄⣻
⠀⠀⠀⠀⢀⠃⠀⢠⡇⢸⠀⠀⠀⠀⡄⠀⠠⠀⠀⠀⢠⣀⣀⣈⣉⣓⣲⣤⣴⡄⠠⠶⣉⣌⣠⡤⠴⠊⠁
⠀⠀⠀⠀⢸⠀⠀⢸⠃⠀⡇⠀⠀⠀⢡⠀⠀⡀⠀⠀⠈⠁⠉⠀⠈
⠀⠀⠀⠀⡀⠀⠀⣾⠀⠀⣇⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠇⠀⠀⢿⡀⠀⠋⠙⠒⠒⠒⠀⠀⠀⠀⠀⢹
⠀⠀⠀⣼⠀⠀⠦⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⢠⡖⠚⢧⡀⠄⠐⠀⢱⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⡼⠄⡐⣤⡉⠳⠚⠀⠈⠙⢦⡀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢃
⣇⠀⠙⢢⠀⠂⢀⠠⠄⠀⠀⢹⡆⠀⠀⠀⠸⠁⠀⠀⠀⢸⡆
⠈⠛⢮⡀⠀⠒⢠⡄⠠⠀⠐⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠙⠲⣄⠈⠀⢀⣀⣀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠈⠛⠋⠉⢹⠉⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣀⣤⣀⡀⡀⣸⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠈⢋⠈⠉⠉⠉⠉⠉⠉⠁⠂⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⣶⣀⣀⣀⣀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⠀⠀⠀⣂⠜⠉⠉⠉⠉⠀⠉⠈⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣁⣀⣀⣀⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⣽⠏⢍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⢀⠐⠈⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣞⡀⠈⠁⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⣤⡀⠀⢠⡄⠠⣤⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢆⠹⠠⠀⠈⢀⠄⠚⠠⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠠⠾⠓⠚⠀⠂⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠈⠀⠈⠀⠁⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢨⠠⠄⠒⠠⠀⢄⠀⠉⠁⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢚⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀⡊⣈⡓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠑⢿⠀⠀⢂⠀⠀⠀⠀⠀⠀⡀⠠⠔⡄⠀⠀⢠⣋⠁⠈⡀⠈
⠀⢀⠤⠤⠤⠤⢄⠀⢠⡄⡤⠄⣆⡁⠳⡄⠀⠀⠀⠀⠀⡄⢆⡲⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠁⠀⠂
⠀⠇⠀⠀⠐⠰⣾⣞⢸⡏⠱⡀⠀⠀⠀⠨⢂⠀⠀⠀⠀⢆⣨⠋⢅⣀⣀⣄⣀⣀
⠀⢈⣀⣤⣬⣬⣽⣿⣾⣷⣄⡀⠀⠀⠀⢀⣧⣴⣄⡀⡤⣬⡷⡖⠚⠫⡉⠙⠖⠟⠁⠀⠀⠀⡀
⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣋⣉⣉⣍⣄⢸⣄⣼⣿⣿⣶⣶⣶⣶⣶⣾⠿⠗
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁

⠀⢀⣠⣤⣄⠀⠀⢀⡀⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀
⠀⠨⣭⣽⣧⠀⠹⣾⠁⢼⣷⡿⠀⠀⠀⠀⠀⠀⠀⠧⠤⠈⠁⣶⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⢛⠛⣛⣛⡛
⠀⠀⢸⣿⣿⠀⠀⡟⠀⠈⣿⠁⠀⠀⡴⣦⠀⠀⠀⠀⠠⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠿⠻⠿⠿⠿
⠀⠀⢸⣿⡇⠀⠀⡇⠀⠀⣿⠀⠀⣼⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒
⠀⠀⢸⡟⠃⠀⠀⡇⠀⠀⣿⡄⠨⣿⣟⣿⡇⠀⢠⣶⠀⠀⠀⣼⣳⠀⣤⣤⣤⠀⣤⣤⣤⣤⡄⣀⣀⣤⣤⡀
⠀⠀⢸⡇⠀⠀⠀⣇⠀⠀⣿⡇⠀⢻⣿⣿⡇⠀⣸⣿⡄⠀⠀⢻⣿⠀⠀⣿⠀⠀⠿⢿⣿⠿⠇⠈⠉⢻⡏⠉
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠘⠉⠃⠀⣿⠉⣿⠁⢀⡿⠘⣧⠀⠀⡿⢻⡀⠀⣿⠀⠀⠀⢨⣿⠀⠀⠀⠀⢸⡇
⠀⠀⢏⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⠀⢸⡇⠀⣿⠀⢸⠇⠈⡇⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠁⠀⣿⠀⢠⡇⠀⣿⠀⠘⡇⢠⠇⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠁⠈⠀⠀⠿⠀⠀⠀⠘⠛⠀⠀⠀⠀⣿⡇
⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⢠⣴⠀⠀⠀⢰⣶⠀⠀⡆⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⣿⡇
⠀⢿⣿⣿⣿⡿⠀⢸⡇⠀⠀⠀⠀⠀⠟⡄⠀⠘⠻⠀⠀⠀⠘⡿⠀⠀⡇⠀⢸⠀⢰⡇⠀⠀⡀⠀⢀⠉⠁
⠀⠀⠙⣿⠁⠀⠀⢸⡇⠀⠀⠀⠀⠘⠀⠐⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⡇⠀⢘⡀⢸⡇⠀⠈⡇⠀⢸⠀⣶⣶
⠀⠀⢠⣿⠀⠀⠀⠸⡇⠀⠀⠠⠤⣤⣄⣀⠀⢀⠴⠤⠃⠀⢸⣿⠀⠀⡇⠀⢸⡇⢸⡇⠀⣰⣇⠀⢸⠀⡇⢻
⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠟⠛⠛⠓⠖⠒⠲⠀⢸⣿⠀⢰⣿⠀⢸⡇⢸⡇⢸⣿⣿⡇⠸⠀⠇⠸
⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣥⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⣲⣶⣦⣄⡀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢃⣵⣄⠀⠀⢠⣺⠋⠀⣰⠟⠁⠀⠀⠙⣿⣦⠀⠀⣠⣮⡘⢷⣄
⠀⠀⠀⠀⠀⠀⡀⠴⢃⣴⣿⡿⠋⠀⢠⢣⠃⠀⢰⠃⠀⠀⠀⠀⢈⡙⣿⣷⠀⠙⢿⣿⣦⡘⠦⣄
⠀⠀⠀⠀⢀⡜⠁⣠⣾⣟⠉⠀⠀⠀⣸⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠄⢿⣿⡄⠀⠀⠉⣻⣿⣄⠈⢣⡀
⠀⠀⠀⢠⢏⣤⣾⣿⣿⣿⣷⣄⡰⣉⠈⠒⠤⢀⣀⣀⣀⣀⣀⣀⣤⣤⣾⣿⠗⢄⣠⣾⣿⣿⣿⣷⣤⡹⡄
⠀⠀⢠⢣⡞⠁⠙⢿⣿⣟⡷⢋⣠⠞⠉⠑⠒⠀⠂⠈⠛⠋⠉⠉⠉⠙⠛⠛⠳⣄⠙⢶⣻⣿⡿⠋⠈⢻⡜⡄
⠀⠀⣞⡟⠀⠀⠀⢀⡽⢋⣤⠞⠁⠀⠀⠀⠀⠀⢰⠀⣀⢰⠀⡀⠀⠀⠀⠀⠀⠈⠳⣄⡙⢯⡀⠀⠀⠀⢻⣳
⠀⠸⠟⠀⠀⢀⡴⢋⣴⠟⢁⣄⢠⡞⣷⠀⢤⠀⡄⣤⡄⣀⠀⠠⠀⡄⠀⣾⢳⡄⣠⡈⠻⣦⠙⣆⣀⣀⠀⠻⠇
⠀⠀⣀⢤⣤⠮⣰⣟⣁⣀⣈⣉⣀⣉⣉⣀⣀⣁⣁⣉⣀⣉⣈⣈⣀⣁⣀⣉⣉⣀⣙⣁⣠⠔⠛⠛⢛⣛⣛⣦
⠀⢰⣷⢋⣭⠟⣎⠁⠁⠀⠄⠀⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠈⠀⠀⢸⠻⣯⡙⢦⡀
⠀⠸⣿⣌⣛⡦⢃⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣠⣠⣤⣤⣄⣤⣤⣴⣆⣒⣲⣾⣀⣀⣀⣀⣘⣶⣿⣥⣼⡅
⠀⠀⠈⠛⠒⣞⠹⣯⡉⠉⠉⠉⠉⣤⣭⣍⣭⣭⢩⣭⣭⢁⣭⡍⢩⡍⣭⣭⡌⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣌⠻⣦⡘⢿⡆⣿⣿⡿⣿⣧⢸⣿⣿⢿⣿⣷⢸⡇⣿⣽⡟⢰⡿⢃⣴⠟⣠⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⢦⡀⠿⠿⠧⠿⠷⠼⠿⠁⠾⠟⠿⠼⠇⠿⠿⠧⢀⣴⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⣌⠻⢦⡀⠰⠂⠺⠀⠶⠀⠆⠰⠀⠆⠰⠆⢀⣴⠛⣡⠾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⡖⢁⠜⠈⣳⣄⠻⢦⡀⠀⠉⠉⠉⠉⠉⠁⠀⢀⣴⠏⣠⣞⠁⠣⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣤⣴⡿⠋⠳⣄⠙⢦⡀⠀⠀⠀⠀⢀⡴⠋⣠⠎⠙⢿⣦⣤⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠈⠳⢄⠙⢦⡀⢀⡴⠋⡠⠞⠁⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⠋⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠀⠠⠔⠁⢀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠈⠉⠐⠛⠁⠀⠀⠀⠀⠀⠀⠈⠁⠂⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣯⠆⠀⠀⠀⡨⠍⠀⠀⠹⠐⠢⣀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⡀⢱⠀⠀⠀⠀⠀⡀⠂⠈⠂⢀⠀⢀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⠀⠀⢀⠄⠬⢀⠀⠀⠄⠀⠑⢄⡱⠤⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⠀⠀⠀⢀⡨⠄⣀⠂⠀⠀⠀⠈⠧⠈⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠂⠈⠀⢀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⢤⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡔⢣⠀⠀⠀⡘⠀⠠⠀⠀⡌⠣⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⡤⠘⣿⡆⠀⠀⠀⠀⡠⢊⠀⠀⠱⢄⠀⠀⠀⠀⡠⠀⠀⡐
⠀⠀⠀⠀⠀⠀⡀⢠⠤⠀⠐⠉⠀⠀⠠⠀⡖⠉⡗⠒⢀⢀⠠⠲⠉⡠⣝⠀⢸⠀⠁⠂⠐⠊⠁⠀⠀⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠋⢋⠵⠘⢆⠀⠀⠀⠀⠠⣄⠰⡁⠈⠁⠀⠀⠀⠀⠘⠀⠈⠀⠑⠋⠀⠓⠒⠂⠀⠀⠘⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡘⣀⡈⠀⠀⠀⠱⢄⠀⠀⠀⢀⠷⡜⠦⡀⡀⠀⠀⠀⢀⠀⠄⠄⠢⢸⠀⠒⠒⠒⠒⠒⠒⠀⠀⠠
⠀⠀⠁⠠⠁⠂⠀⠀⠀⠀⠀⠈⠰⠤⠀⠀⡈⢒⢀⠬⡄⠠⠂⣠⠂⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠁⠢⠁⠀⠀⢑⢀⢀⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠆⠀⠀⠀⠀⠡⢄⠀⡤⠁⡾⠻⠚⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸
⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠈⠍⠁⠀⠀⠀⠀⠀⠀⣄⡄⠣⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⢀⣰⠞
⠀⠄⠀⠀⠙⠦⠲⠤⠀⠀⠰⠀⢤⣀⡼⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⢊⣇⣀⣁⣀⣈⣀⣀⣀⡷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠚⠯⢿⣿⢸⣀⠿⠿⠘⠘⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⣿⣿⢸⡏⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠴⠂⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠃⣐⣂⣀⣀⡀⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠋

⠀⢀⡀⣀⠀⢀⢀⠀⣠⣤⡠⢀⠀⡀⣾⣶⣤⣀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⣾⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣸⣿⣷⣶⣤⣀⣀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣴⡋⢸⡇⠀⠀⠀⢹⣿⣮⣷⡄⠀⠀⠀⣴⡃⢸⡇⠀⠀⠀⣿⣿⣮⣷⡄⠀⠀⠀⣠⢟⣦⠀⠀⠀⠀⢻⡌⠻⣿⣽⣦⡀
⠀⠀⠀⠀⠀⠀⢰⣿⡇⢸⣧⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⡇⢸⣧⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢠⡟⢸⣿⣧⠀⠀⠀⣀⣷⠀⠙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠸⣿⡇⠈⠻⠶⠶⠞⢻⣿⣿⣿⣿⠀⠀⢸⣿⡇⠈⠻⠶⠶⠖⢫⣿⣿⣿⡿⠀⠀⢸⡇⠘⣿⣿⣧⠶⡛⠉⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠈⣿⣿⣿⠃⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⠘⣷⠞⠋⢉⣚⠭⣤⡀⠀⠀⢠⣿⢧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠄⠀⠀⠀⠐⣿⡟⠁⠀⠀⠀⠀⠀⠙⣷⠄⠀⠀⠀⠐⣿⡟⠁⠀⠀⠀⠀⠻⣆⢸⣝⣧⡶⣿⣧⠀⢠⣿⣣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⣀⣀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⣀⡀⢰⣿⠀⠀⠀⠀⠀⠀⠈⢿⣆⠻⠷⠛⠉⠀⠀⡾⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠀⢸⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣾⠀⢸⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢰⡿⣦⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠸⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠸⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣽⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⢸⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣹⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⢸⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢸⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢸⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠨⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠁⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⣸⠃⣀⣤⣄⡙⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠤⣒⡢⢄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⣴⠶⠶⣾⣿⣿⣿⠀⠀⠀⠀⠀⣸⢣⡞⠉⠀⠈⢻⡎⢷⠀⠀⠀⠀⠀⠀⠀⢸⡇⢱⠏⠉⠙⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢸⡇⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⢻⡸⣧⠀⠀⠀⢨⡇⡾⠀⠀⠀⠀⠀⠀⠀⠀⢷⡘⢦⣀⣠⠟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣿⠀⠀⠘⣯⠟⠀⠲⠲⠒⠂⠀⠚⠷⣾⣿⡶⣾⣿⠾⠃⠀⠀⠀⠂⠀⠀⠒⠂⠈⠓⠦⢤⠴⠞⠁

⠀⠀⠀⢀⢔⡂⢄⡂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠐⠒⠠⡀⠀⠀⠀⠀⠀⠀⢰⢀⡆⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⢃⠁⠀⠀⠈⢆⠂⠀⠀⠀⠘⠄⠀⠀⠀⠐⡀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠀⠀⢂⠀⠈⠢⠀⠀⠀⠀⠀⡌⠀⠅⠀⠀⠀⠀⠀⠀⢸⠸⡆
⠀⠀⠀⡌⢄⠀⠀⢀⢆⡄⠀⠀⠀⡃⠘⢄⠀⡀⠔⠁⢠⡇⠀⠀⠀⠀⠀⠀⠌⠀⠀⢁⠀⢈⠀⠀⠀⢃⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⢸⢰⡄
⠀⠀⠀⢸⠂⠍⠉⠅⢺⠀⠀⠀⠀⢳⠄⡀⠀⠀⢀⠠⢢⠁⠀⠀⠀⠀⣀⠠⠦⠀⠒⠈⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢸⣶⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠔⠒⠄⡄⠀⠀⠀⠀⠀⠑⡄⠉⣉⡀⢠⠁⠀⠀⠀⠀⠘⢄⠀⠠⡂⢤⣔⣦⠐⠀⠈⡠⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢋⠙⣞⣯⣯⣿⠀⠀⡐⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠛⠊⠉⠁⡘⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡔⣈⠉⡑⠂⠀⠀⠀⠀⡐⣈⠉⡁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡇⠀⠀⠠⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢡⢀⠇⡆⠀⠀⠀⠀⠀⢠⢀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠠⠀⠁⠀⠃
⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡜⠈⠆⢰⠀⠀⠀⠀⠆⡘⠀⢂⠠
⠀⠀⠀⢀⠁⠢⠤⠂⠇⠀⠀⠀⠀⠀⢀⠆⠈⠂⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⡀⡆⠀⠀⠘⠰⠀⠀⠈⠄⠆
⠀⠀⠀⠌⠀⡀⠀⣀⠘⡄⠀⠀⠀⢠⢂⠄⠒⠈⠉⠐⠊⢤⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⠘⠀⠀⠀⠃⢠⠀⠀⠁⠀⠀⠀⠀⠀⠠
⠀⠀⣊⠖⠁⠀⠀⠀⠁⢺⡄⠀⠀⡥⠁⠀⢀⡀⠤⠤⡀⠀⢣⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠠⣀⠀⡠⠃⢸⠀⠀⠀⠰⣀⠀⡠⠂
⠀⢰⠇⠀⠔⠉⠀⠈⠆⠀⡃⠀⠀⡇⠀⠐⠁⠀⠀⠀⠐⡀⠈⠀⠀⠀⠀⠀⠀⡣⠐⠒⢌⠁⠇⠀⠀⠀⠀⠀⠁⠀⡠⠠⢄⠀⠀⠀⠀⠄⠀⡠⠀⢄⠀⠈
⠀⠀⢢⠀⠂⠀⠀⠀⠘⣀⠃⠀⠀⠘⢄⠈⠄⠀⠀⠀⠀⠐⠌⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⢀⠀⢄⣀⡸⠀⠇⠀⠀⢠⠀⢄⣀⡸⠀⠇
⠀⠀⠀⠑⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠑⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠅⠀⠕⠁⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠤⠊⠀⠀⠀⠀⠁⠄⠀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⣏⢸⡇⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡇⣿⠀⡇⣿⠀⠀⠀⠀⠀⠀⢀⣿⣧⡄⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⣇⢻⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣄⠀⠀⠀⠀⠀⠀⠀⣽⢸⡇⢿⡇⣷⢸⡆⠀⠀⠀⠀⠀⠛⢯⣤⡌⠀⠀⠀⠀⠀⠀⢸⡇⣿⣸⣿⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣿⣸⣇⣽⣧⣽⣼⣧⠀⠀⠀⠀⠀⢀⣿⠈⢷⡀⠀⠀⠀⠀⠀⠈⠉⢹⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣴⣿⡟⠉⠉⠹⣏⠉⠀⠀⠀⠀⠀⢀⣼⠃⠀⠈⠻⣄⡀⠀⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⢁⣴⣴⢿⣿⡟⠁⠀⠀⠀⠘⠷⣤⣠⣄⣀⣤⡟⠁⠀⠀⠀⠀⠈⠛⠷⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣥⣴⣾⣿⣿⣷⡿⠀⠀⠀⠀⠀⢀⣠⣾⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣿⣿⣿⣟⣉⣠⣶⣆⣴⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣝⠻⠿⠿⠿⠿⠿⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣌⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣤⣍⣙⡛⠛⠛⣿⣿⣿⡌⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⠹⣿⣿⣿⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⢻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣀⣻⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣽⣿⣷⣤⣀
⠀⠀⠀⠀⠀⢰⡿⠛⠛⠋⠙⣿⡋⠛⠛⠛⠙⣿⡟⠛⠉⠛⢻⣧
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢹⣷⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣁⣀⣀⣀⣀⣀⣘⣿⡄
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⠟⠛⠛⠛⠛⠛⠛⠛⠻⣷
⠀⠀⢰⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⢀⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡆
⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣤⣄⠀⠀⠀⣀⣀⠀⠀⢀⣀⡀⠀⠀⢀⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠀⠙⠿⣿⣿⠛⠉⠀⠀⠀⠐⠒⠒⠒⠒⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⢸⣿⣿⠀⠀⠨⣿⡿⠀⠀⢽⣿⠇⠀⠀⠀⠰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⢸⣿⡿⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢻⠛⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠻⣿⣿⣿⣿⡀
⠀⠸⢿⠗⠀⠀⠀⠸⠀⠀⠀⠀⠇⠀⠀⢀⠀⠀⠘⠁⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠉⠛⠛⠛⠛⠛⠛⠒⠚⠀⠀⠀⠀⠸⠿⣿⣿⡇
⠀⠀⢸⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⢸⡟⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⢸⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠏
⠀⠀⢸⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⢸⡇⣿⡄⠀⠀⢠⠒⢢⠖⢢⡔⢦⠒⣤⢢⡔⣤⢠⡤⡄⠀⠀⠀⣄⡄⡄⡄⣄⣄⡄⣄⣄⡀
⠀⠀⢸⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⢸⡇⢻⡇⠀⠀⠀⠑⠉⠑⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⣸⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣼⣷⣿⡇⠀⠀⠀⣠⡀⣀⣀⣀⡀⣀⢀⣀⣀⡀⣀⢀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠈⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠀⠀⠈⠛⠛⠁
⠀⢠⣤⣴⣶⣤⣄⠀⠀⠀⣤⣤⡴⠚⠲⣄⠀⠀⢀⣶⣦⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠀⣦⡀⠀⢀⡴⠀⢀
⠀⠀⠁⠈⣿⠀⠈⠀⠀⠀⢸⣿⣧⣀⣠⠟⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⣿⣶⣿⠇⠀⠀⢻⣶⠟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣿⡿⠉⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢹⣿⡁⠀⠀⠀⠀⢰⣯⡇⠀⠀⠀⠙⣷⡏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢴⣿⣷⠄⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢠⡟⠙⣧⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⣿⠁⠀⢹⡆⠀⠀⢀⡾⠋⢷⡀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢰⣿⡇⣿⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⣼⠃⠀⠈⣷⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⠁⢸⣿⡄⠀⢸⡇⠀⠀⠀⣿⠀⠀⢿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⣼⣿⡄
⠀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⠀⠀⠸⠇⠀⠀⠿⠀⠀⠘⠇⠀⠀⠀⠏⠀⠀⠘⠇⠀⠰⠇⠀⠀⠀⠯⠇⠀⠀⠈⠃⠨⠏
⠀⠀⠀⠀⠀⠀⠀⡶⠲⠒⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⣼⡟⠛⢻⣿⣿⣿⣿⣿⣿⡿⠛⠻⣦⠀⢠⣴⣤⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢻⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⡿⠀⠹⣿⣿⣾⣸⣿⣿⣥⠀⢀⠞⠉⠉⠉⠉⠉⠉⠉⣿⠉⢻⣷⡀⠀⠀⢿⣿⣿⣿⣿⣿⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⣿⣿⢸⣿⠟⠁⠀⠮⠤⠤⠤⠤⠄⠀⢀⣠⡟⠀⠈⣿⡇⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣤⣤⣤⣤⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠶⠀⢸⠇⠀⠀⠀⠀⢸⡇⣐⠖⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣩⣿⡇⣼⣿⣿⣿⣿⣇⢸⣿⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠃⠸⡄⠀⠀⠀⠀⣠⠇⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣾⣇⢻⣿⣿⣿⣿⠟⣸⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡻⠿⠿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⡤⣶⡞⠁⠀⠉⠁⠀⠘⣳⡶⢤⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣛⡉⢡⣿⣿⣿⣿⣿⣷⡌⢙⡛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡞⠁⠀⢸⣿⡀⠀⠀⠀⠀⢀⣿⡇⠀⠈⢳⠀⠀⠀⠀⢸⣿⣿⣿⣿⢱⣛⣻⠃⠘⣿⣿⣿⣿⣿⣿⠀⢸⣿⣷⡜⣿⣿⣿⣿
⠀⠀⠀⢸⡇⠀⠀⣽⣿⣿⣶⣤⣴⣶⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⡅⠀⠈⠉⠉⠙⠋⠁⠀⢸⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⢸⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢾⡗⢈⡑⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⠈⠯⠶⣬⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢻⡄⠀⣠⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠠⠀⣹⣿⡀⠀⠀⢀⠄⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢠⠤⢦⡀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠚⠀⠀⠀⠄⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠶⠔⠐⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠈⠉⠀⢀⣠⣽⣦⣵⣯⡀⣄⠀⠀⠀⡈⠉⠉⠁⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣧⣩⣈⣷⣽⣧⠈⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡯⢉⡛⢻⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⠔⢊⠉⣁⡀⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡛⠫⡒⣼⡀⡔⠁⡴⠗⠁⠀⠀⢀⠠⠄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠛⠦⠄⠉⠙⠀⠉⠀⠈⠁⠀⠀⠠⠚⢉⣀⣂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠒⢒⡶⠤⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠒⠒⡺⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠊⠁⣀⣠⠄⠀⣠⠔⠊⢁⡠⠔⠒⠒⠒⡲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠔⡽⣩⠄⢤⠊⢀⡠⠎⠁⠀⡠⠂⠒⣀⠇⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠸⠏⠀⠀⢈⠷⢏⣀⠆⠀⢸⣁⠠⠖⠉⠱⠢⠄⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠢⠒⠈⠀⠀⠈⢇⣀⢀⡞⠶⠖⠒⠋⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠢⠭⢤⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠢⠔⠁⠀⢸⠀⠀⠀⡰⣤⣖⠄⠀⡖⠢⠤⠀⣀⣀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⢤⣼⠀⠀⣼⡏⡀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣲⠤⣾⡿⢆⠀⠓⠁⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠉⠁⡎⠀⣴⠋⠀⠈⠳⠤⠤⠊⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠉⠙⠁⠁⡠⠀⡀⠀⠀⠤⢴⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⡀⠀⢀⣠⠄⠊⢀⠜⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⢷⡴⠃⠀⠀⡴⠁⠀⣀⡀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡌⠛⢿⣲⡞⠀⠀⠀⠈⠉⠙⡟
⠀⠀⠀⠀⠀⠀⠀⠔⡶⢢⣄⡹⠏⠐⢣⡉⠁⠒⠂⠄⣠⠖⣱
⠀⠀⠀⠀⠀⠀⠐⡘⠀⡼⠱⣸⢧⡀⠀⢫⠙⠹⡍⢙⣡⡟
⠀⠀⠀⠀⠀⠀⠤⡇⣠⡄⢀⡴⠓⠳⡤⠀⢣⠀⠈⠓⠊
⠀⠀⠀⠀⠀⢪⠿⠘⢫⡟⢫⣄⣀⣀⣀⢰⡁
⠀⠀⠀⠀⠘⠳⢤⣔⡿⣉⣠⣯⢋⡹⣵⠯⡚⠒⢤⣤⣄⣀⣀⣀
⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣥⣶⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠈⠃⠢⡀⠀⠀⠀⠴⣦⠀⠀⠀⡰⡟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠻⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⡤⠋⠀⠀⢀⠇⢷⡖⢃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠠⢂⠀⠐⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣁⠀⠀⣼⠀⣸⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⣆⢀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⢰⣿⡇⠁⢀⡇⠀⣹⡧⢤⣦⠀⠀⠀⠀⠐⠒⠀⠀⠀⠁⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠘⠛⠃⠀⠘⠇⠻⠄⠘⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠘⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣶⠀⠀⣀⣀⠀⡀
⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⢈⣤⣤⣤⡤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⡄⠄⣀⣧⣤⣤⣇⡀
⠀⠀⠀⠀⠀⠀⢀⠁⢀⣴⣿⡘⡟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠟⠻⡤⠄⠐⣦⣄⠀⠀⠂⠀⠂
⠀⠀⢀⣀⡀⢀⠀⠻⠯⢟⢥⡀⢾⠂⠀⠊⠁⠀⢐⡚⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⡀⢾⣿⣿⠄⠀⠠⠤⠤
⠀⠀⠀⠠⠀⠐⠆⠈⢻⠀⠚⣿⣾⡇⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠛⢴
⠀⠀⠀⠐⠀⠀⠈⡄⠀⡀⠈⢿⣿⣿⠀⡀⣀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠐⠀⠁⠀⠁⠀⠉⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠲⣿⠁⠀⣠⠃⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣀⣀⠈⠛⠶⣧⣀⣀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⠉⠗⢶⣄⣈⠛⢛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠈⠙⢾⣴⠎⠈⠙⠻⢾⣿⠷⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⡠⠏⠀⠀⢀⣀⡤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠡⠂⠀⠀⠠⢀⡠⠐⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠔⠁⠀⢀⢔⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⡡⠊⠀⠀⡠⢊⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⠔⠉⠀⢀⠴⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡥⠚⠁⠀⣠⢖⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣪⠔⠋⠀⢀⡴⢊⠴⠋
⠀⠀⠀⠀⠀⠀⠀⣀⢔⡭⠞⠁⢀⣠⠞⣫⠖⠁
⠀⠀⠀⠀⠀⡠⢚⠵⠋⠀⢀⣴⢟⡡⠊
⠀⠀⢀⡴⣫⠖⠁⠀⣀⡴⢋⠔⠋
⢀⠔⣫⡊⠀⠀⣠⠞⣡⠖⠁
⣷⣞⠀⠈⡆⢈⡡⠊⠁
⠙⢮⣗⣫⠴⠋