Shot Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠁⠙⣆⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣤⣈⢳⣄⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⣠⠤⣍⠉⢳⠒⠒⣇⠀⠀⣾⠛⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⢧⡀⠀⠉⠒⢠⡀⣈⣀⣠⠷⣄⢸⡟⠉⢛⡶⢤⡀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢤⣵⣤⣤⣤⡁⠀⠀⠀⢮⡷⠬⠉⠀⠰⡇⢠⣬⣙⣷⣶⣤⣀⣀⠀⢹⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠏⠈⠑⠂⠖⠋⠀⠀⠀⠀⠀⠐⣄⣀⡸⣄⠀⠀⠈⡿⢋⢹⣻⡙⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡌⣿⠉⠙⠓⠲⢇⣸⣈⣱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠿⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢧⡟⠀⠀⠀⠀⢀⡿⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⡇⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡤⣤⣤⣤⣀⣀⠈⣳⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠾⠶⠶⠶⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠙⠻⠷





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

⠀⠀⠀⠀⠀⠰⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⠶⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣭⣤⡾⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃⠉⠀⠈⠉⡠⣤⣤⣤⣒⣒⣢⣤⣤⡤⠤⣤⢤⣤⡤⠤⠤⠤⠤⠤⠦
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠑⠀⠀⡀⠀⠀⠀⢨⠯⠉⠋⠉⠉⠁⠀⠀⠀⢀⠀⠂⠁
⠀⠀⢀⠀⠀⠀⠑⡶⠖⢚⣩⠉⠀⠀⣾⣥⣀⠀⠀⠀⠀⠀⠀⠀⣀⠁
⠀⠀⠈⠄⠀⠀⠈⠳⣞⡟⠁⠀⠀⣰⡟⠿⠿⢷⣶⣤⣴⡶⠚⠛⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⣴⡟⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠈⠀⠈⠳⣄⡀⠀⣠⣾⣿⠁
⠀⠀⠀⠠⠀⠀⠀⠈⠻⠿⣿⣿⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⣻⠿⠟
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠐⠒⠦⠴⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣼⣷⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣶⣾⣿⣿⣿⣿⣿⡿⠿⠒⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠆⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣭⣴⣶⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣥⡈⣡⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠴⢾⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡯⠀⠐⠀⠀⣀⡠⠤⠄⠒⢚⣶⣠⣤⣤⣶⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⢀⣴⣿⠏⣰⣿⣿⣷⡄⢹⣿⣿⣿⣛⣳⠾⢠⡖⠋⢡⣠⡄⣠⣶⣼⣿⡹⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠈⠛⠟⠀⠙⣿⣿⣿⣧⡸⣿⣿⣿⡿⠁⠀⢿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⣶⣦⣶⣿⣿⣮⣿⠿⠶⢤⣥⠾⠁⠀⠀⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⣠⠆⠀⠀⠀⣴⣿⣿⣿⣿⣿⠿⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⢻
⠀⣠⠞⠁⠀⠀⢀⣄⡻⣿⣿⣿⡟⠁⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⣰⣿⣿⣻⡆
⣸⠇⠀⠀⠀⢠⡾⠉⢳⣿⣿⣿⠀⢀⣴⣿⣷⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⢻⡗
⣿⠀⠀⠀⠐⠋⠀⠀⢾⠟⠋⣿⣤⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⠷⢻⡆
⠙⠳⠶⠦⢤⡴⠚⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣤⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⡄⠀⠐⠻⣿⣿⡇⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣄⣀⣾⣿⣿⡇⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣻⣿⣿⡿⠿⠿⠿⠿⠿⠿⢟⣛⣋⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⠷⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇⠛⠂⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠉⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⠈⠀⠀⠀⠀⠀⠀⣆⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣭⡉⢿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⢀⣀⡤⠔⠚⠛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠛⠻⢿⣯⣛⢿⣿⣿⣿⣋⡴⠟⠉⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠈⣿⣿⡎⠿⠛⠛⣫⣾⣷⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠘⠇⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣶⡀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⡀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⣠⠞⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⣭⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣭⠶⠞⣩⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡾⠛⠋⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣩⣽⣿⣿⣶⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⣶⣤⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣉⠛⠋⠙⣛⣡⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠛⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣛⠛⠛⠻⠿⠟⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⣴⡿⠿⠿⠁⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢷⣦⣤⣤⣤⣴⡿⠝
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣶⣬⣿⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⣿⡿⢿⠿⣿⠿⡙⠿⠿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⡟⠛⠛⠛⠻⠟⢿⣿⡿⠿⠿⠿⠿⠿⣿
⣿⣶⣷⣶⣾⣶⣷⣶⣿⣶⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢱⠋⠋⠀⠴⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⣿⢆⠂⠀⠀⠼⠊⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣤⣈⣈⣁⣁⠂⠠
⠀⠀⠀⠀⠀⣠⣴⠶⠟⠛⠉⠉⢀⣀⣀⡀⠉⠉⠉⠛⠳⠶⢦⣭⣀⡀
⠀⠀⠀⣠⡾⠛⠁⢂⣠⣴⣾⣯⣽⣿⣿⣿⣟⣛⣓⠀⠀⠀⠀⠀⠈⠙⠳⠦⣄⣀
⠀⠀⠘⠻⢷⣖⡔⡫⢚⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣥⣄⣀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀
⠀⠀⠀⠀⠈⠘⠛⠑⢿⣿⣿⣿⣿⢏⣽⣶⣿⣾⡇⠈⣉⡙⢛⠳⣦⣄⣀⡂⢀⣀⠀⠀⠀⠀⠉⠙⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢛⣮⣿⣿⣿⠏⢱⣿⣿⣿⡶⠲⣿⣿⣿⣿⣿⣿⣷⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⢉⡉⠀⠀⠻⠿⠛⠀⣠⡾⠟⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣦⣄⣴⣠⢥⣀⣿⣃⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢠⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⠸⠷⠉⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⢀⣀⣠⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠙⣶⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡅⠉⠩⠉⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠗⠢⠬⣯⣤⠀⠈⣄⠐⠐⠊⠀⠀⠀⠀⠘⠛⠛⠂⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣄⠛⢿⣿⣦⡀⠐⠀⠀⠀⢠⣿⢃⣦⣄⢀⠈⢲⢢⣀⣘⣿⣩⣤⠂
⠀⠀⠀⠀⠀⢀⣾⣆⠙⠿⣽⣷⢦⣉⡙⠻⣦⣤⠀⠀⢹⡿⢸⣿⡿⢹⡦⠀⠒⠀⠉⠹⠗⠔
⠀⠀⠀⠀⠀⠈⠟⢿⣷⣦⣈⡁⠙⠟⢽⢷⣦⣝⠳⡄⠸⡏⣼⡿⠁⢸⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠻⠶⣤⣄⡀⠉⠷⠜⠂⠇⠋⠀⣸⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡿⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢻⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠸⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣀⣉⣉⣉⣁⣀⣤⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠜⡐⠀⠀⠀⠉⠉⢢⡖⡠⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⡀⠈⠱⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠁⠪⡈
⠀⠀⠀⠠⡀⠀⠀⢀⡄⠀⠀⠀⠀⢠⣲⣀⣢⣀⣀⣀⣀⣀⣀⡠⠈⠀⠀⠀⣠⣈⣁⣂⣀⡀⣀⣰⣀⡖⠂⠀⠀⠀⠀⠐⠢⢄⡀
⠀⠀⠀⠀⠁⠀⠤⠤⠆⢀⠠⡀⠰⡥⢼⠦⠴⠧⠤⠤⠤⠤⠠⢇⠀⠀⠀⠚⠤⠤⠤⢬⠦⠤⠤⠍⡂⣇⡀⠀⠀⠀⠀⢀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠊⠃⠅⠀⠀⠀⠀⠰⠊⠁⠘⡄⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣄⠀⠀⠀⠀⠀⣠⠃⠊⠀⠀⠀⠀⠀⠣⣀⠀⢀⡀⠀⠀⡀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⢄⣀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠩⡁⠈⢉⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠰⠕⢀⡀⡔⠁⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠘⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⡰⠋⠉⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠔⡒⠁⠀⠀⢠⣀⠀⠀⢀⠇
⠀⠀⢠⠆⠁⠈⠉⠚⢊⡖⠀⠀⠀⢠⠊⠀⠀⠀⣄⠀⢀⡟⠟⠀⠀⠀⡠⠂⢀⢀⠀⠀⡏⠁⢠⡥⠀⠠⠇⠀⠀⢠⣀
⠀⡈⠃⠀⡠⠊⠐⡀⠀⢱⠀⠀⠀⢸⡀⠀⠈⢠⠃⣽⢈⣇⠀⠀⠀⡞⠀⠀⠀⠈⠀⢠⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢓⠄
⠀⢑⠀⠀⠱⠖⠐⠋⠀⢠⠃⠀⠀⢰⠇⠀⠀⠀⠧⡔⢠⠃⠀⠀⠀⠳⡀⠀⠀⠀⠀⠠⣁⢀⠊⠒⠀⠰⠀⠀⠀⠐⠊⠁
⠀⠘⢄⣀⣄⠀⣀⠀⡀⠎⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠎⠈⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⡷
⠀⠀⠀⠃⠀⠀⠛⠀⠀⠀⠀⠀⠀⢠⣬⡦⠀⠀⣀⡠⢻⠟⠃⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⣱⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢⡀⠤⠤⡄⠠⣄⡀
⠀⠀⠀⣠⠰⠊⠉⠓⠦⡀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⠱⡀⠀⠀⠀⠀⠀⢀⢀⡭⢲⠂⠀⠀⢀⡀
⠀⣀⢰⠁⢀⠴⡀⠀⠀⡇⠀⠀⢐⠁⠀⢀⠄⠄⢒⡀⠀⠀⠀⣃⠀⠀⠀⡠⠂⠉⡀⡀⠈⠉⠦⡀⠃⠀⠄
⠀⢹⠊⢠⠁⡰⠁⠀⠀⢹⠓⠒⡊⠀⠀⠘⣒⡐⢸⡃⠀⠀⠚⡞⠙⠀⢠⡇⠀⢠⠁⢈⡲⠀⠀⡇⠈⠉⠀⠀⠀⠀⠀⠉⠑⠦
⠀⡁⠀⠑⠒⠁⠀⢀⠠⠊⠀⠀⢠⠀⠤⠂⠀⠄⠈⠁⠀⠀⡜⠁⠀⠀⠸⡀⠀⠈⠐⠉⠀⠀⢀⠇⠠⠤⠤⠀⠀⠀⠀⠤⠄⠊
⠀⠈⠒⠐⠉⢠⠍⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠈⠦⡀⢀⠤⠐⠻⠆⠀⢒⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠀⠐⠒⠂⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠞⠛⢉⣱⠆⠀⢻⠋⠀⠀⠀⠀⠀⠀⠈⢹⠃
⠀⠀⢠⣀⣀⣀⣠⣤⠀⠀⢀⣠⡀⠀⠀⢤⡀⠀⠀⠀⠸⠀⠀⠀⣿⠀⠀⠀⠳⣬⡭⠥⠬⢗⣸⡿⢃⣀⠀⠀⣀⣀
⠀⠀⣸⠩⡉⢉⣧⣿⠀⢰⣿⡟⠓⠲⠖⠒⢷⠀⠀⠀⢠⡀⠀⢠⡏⠀⠀⠀⠀⠙⢧⣄⡀⢀⡽⢁⠋⠉⠉⠉⠉⠻⡆
⠀⠀⢻⡅⠀⠀⠁⣿⠀⣸⣯⣀⣀⣀⡀⠀⢸⡀⠀⠀⢸⡧⣧⣼⡇⠀⠀⠀⠀⠀⠀⠙⡤⠏⠀⡌⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠸⣧⠀⠀⣰⠇⠀⣿⠋⠉⠉⠙⡇⠀⢸⡇⠀⠀⠘⠇⠠⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠙⢻⣿⠏⠀⠀⢿⠀⠀⠀⠀⠃⠀⣸⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⣀⣀⣿⣿⡿
⠀⠀⠀⠀⠁⠈⠀⠀⠀⢸⡀⠀⠀⠀⢦⣤⡿⠀⣰⣫⣭⣅⣀⣁⣀⣨⣆⠀⠀⠀⣀⡀⠴⢤⠄⡀⠈⠉⣙⣛⠛⠉⢰⡆⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣿⠇⠀⣿⠋⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠁⠒⠚⠓⠐⠁⠀⠀⠀⡊⠀⠀⢸⣷⡀⠀⠀⠀⠤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢨⣿⠀⠀⣿⣷⣦⣤⣴⣶⡄⢰⣿⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣧⡀⢀⣀⣀⢠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡏⠀⠀⡏⠀⠀⠈⠉⣻⠀⢸⣿⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⡟⢿⣿⡷⣾⣿⣿
⠀⠀⠀⣀⡤⣤⣀⡀⠀⠀⢸⠀⠀⠀⣸⡇⠀⠀⡇⠀⠀⠀⠀⢹⠀⢸⣿⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⠚⠷⠾⠟⠧⠀⡇⢸⡏⠀⢻⡏⡿
⠀⠀⠂⠘⠓⠚⠒⠊⠁⠀⢸⣷⠶⠾⢿⣧⠀⢠⡇⠀⠀⠀⠀⣸⠀⢸⣿⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠒⠋⠀⢸⣇⡀⠀⠀⣠⣿⢀⣸⣿⡶⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⣇⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⠿⠿⣿⣿⣿⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⠿⠶⢿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠿⠥⣉⣹⣿⡤⠴⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⣠⣤⣀⣤⣤⣤⣤⣄⡀⠀⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⢤⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⡞⠁⢸⠁⠀⣀⣀⣉⣹⡇⠀⣿⣛⣻⣟⣻⣫⣋⣹⣭⣻⣿⣿⡏⠛⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣶⣾⣿⣿⣿⣶⣶⣾⣿⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠙⠻⣿⡯⠿⠿⠿⣿⣿⡛⠃
⠀⠀⠀⠀⠀⢀⡤⢞⠛⢻⣿⡄⢀⣤⠴⠒⠒⡏⠉
⠀⠀⠀⠀⢠⠟⢠⣿⣷⣿⠟⠁⢯⠻⠀⠀⢀⠃
⠀⠀⠀⢠⡏⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣾⠃⢸⣿⣿⡏
⠀⠀⣰⣿⠀⢸⣿⣿⣧
⠀⢰⡿⠿⠶⠿⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣀⣀⡤⠀⢀⣀⠀⠀⠀⡀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣀⣒⣣⣤⣴⣶⣶⣿⣿⣿⣷⣿⣷⣏⠀⣸⣿⣗
⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢀⣀⣤⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣭⣭⣥⠀⠀⠉
⠀⠀⢀⣞⣯⣭⣵⣶⣾⣿⣯⠿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠋⠉⠉
⠀⠀⠈⠉⠉⠻⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⢻⣿⡟⠉⠀⠀⣿⠁⠀⠀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⠀⠀⢀⡿⠄⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠿⠟⠛⠛⠉⡀⢀⣀⣀⣠⣤⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⣴⣶⣶⢸⡗⢸⡏⣿⠋⣿⠁⣿⠇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣤⣉⣾⡷⣿⣷⣿⢰⣿⢀⣿⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠇⢠⣄⣹⡏⣿⠃⣿⠹⣿⠾⠏⠘⠟⠀⠛⠁
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠈⠛⢋⡁⢩⠀⠰⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠙⡿⠛⠛⠛⠛⠛⡋⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢳⡆⢰⣿⡆⠀⣿⠳⣦⠚⢷⡟⠒⣷⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡓⠒⠒⠒⠒⢲⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣸⡇⣸⠙⣇⠀⣿⣤⡟⠀⢸⡇⠀⢹⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠈⠉⠙⠃⠅⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⣿⠶⣿⠀⣿⠈⣿⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠀⣀⡠⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠋⠀⠙⠁⠋⠀⠛⠀⠘⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡟⠛⢸⡇⢸⣿⣧⠀⣸⣿⡇⢸⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⢸⣿⣿⠀⣿⣿⡇⢸⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣠⣤⣤⣴⣰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⢸⣿⢹⣦⡟⣿⡇⢸⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠉⠀⠀⠀⡀⠼⡧⠤⣁⡒⠠⢄⡀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⢸⣿⢸⣿⡇⣿⡇⢸⣧⣀⣀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠁⠀⠈⠉⠉⠁⠁⠀⠀⠀⠁⠠⡈⠱⡄⠀⠀⠀⠉⠁⠀⠈⠁⠈⠉⠀⠉⠀⠉⠁⠈⠉⠉⠉
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠸⡆
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⡀⠃
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠁⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠠⠀⠀⠁⠀⣼⠀⡀⠀⠀⠐⠇⡁
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡏⠀⠀⠀⠀⠀⣎⡈⠀⠀⠈⡁⠀⢀⠇
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⠇⠀⠀⠀⠀⠀⠜⠁⠀⣤⠊⠁⠈⠉
⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡾⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⢀
⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⠇⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⢎⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂⠠⠰⢣⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠈⠢⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠖⠒⠒⠲⠦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⣀⣤⠤⠀⠀⠀⢀⡀⠀⢀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢠⠖⠁⠀⠀⢸⠀⠀⠀⠀⠀⢳⠴⠃⡔⠉⢦⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣠⠤⣄⠸⡖⠁⡀⠀⠸⠀⠀⠀⠀⠀⠘⠀⠘⣄⣠⢎⡞⠀⢀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⢀⡞⠁⢰⠁⢀⡤⡄⠳⠊⠀⣀⡤⠴⠒⠒⠒⠒⠦⢤⣀⠀⠀⠸⣤⠴⠋⡴⢦⠈⢳⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠸⣄⣀⡴⠃⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠰⢛⠗⠉⠀⠀⢳
⠀⠀⠀⠀⠀⡼⠀⠀⢀⠀⠀⠁⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠸⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢀⡇⠀⠀⡼⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡶⠋⢶⡄⠀⢹⡀⠀⠀⠘⡆⠀⠀⢸⡀
⠀⠀⠀⠀⢸⠁⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠴⢏⣀⠜⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⢳⠀⠀⠈⡇
⠀⠀⠀⠀⢸⡀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⣴⠚⢁⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢀⡇
⠀⠀⠀⠀⠈⡇⠀⠀⠘⡆⠀⠀⠈⢇⠀⠐⣶⣾⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⡞⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠘⠆⠀⠀⠘⢦⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢀⡀⠐⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⡀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈⠁⢀⡰⠛⢦⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠉⢨⠷⠄⠀⠉⠓⠲⠤⠤⠤⠤⠖⠊⠉⠀⢀⢰⡃⠘⡆⠀⠓⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⡰⠓⠂⣠⠋⠀⢠⠀⠀⡀⠀⢀⡤⠆⠀⣆⡸⣆⠳⠤⠃⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠰⣅⡄⠀⡇⠀⣸⠁⠀⠘⠒⠒⣦⠘⠄⠈⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠓⠚⠁⠀⠀⠐⠒⠋⠁⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠦⠤⠤⠴⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠂⠉⠉⠉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢤⡴⢷⢱⡄⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⣀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠉⠁⠀⠀⠀⡇⠘⣎⡟⠢⡀⠀⠀⠀⠉⠀⠐⠀⠉⠁⣬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠘⡆⠀⠈⣑⡒⠤⠤⠤⠤⠤⢔⣺⠁⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢱⡀⠀⠈⠛⠛⠓⠀⡀⠛⢫⠃⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡗⢄⠀⠀⠀⡀⠀⣧⢴⡁⠀⠀⠀⡄⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢳⠀⠓⢤⣍⣿⠟⠁⢸⡇⠀⠀⣰⠁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⢀⣀⡀⠸⠄⠀⠀⠀⡜⠀⣀⣨⠇⠀⢠⠃⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢶⡂⠉⠉⠀⡇⠀⠀⢠⠁⠈⠙⡲⠄⢠⠏⠀⢠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⢸⠀⢠⠇⠀⣠⠞⠁⢀⡎⠀⢀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠘⡆⡎⢠⠞⠁⠀⠀⡾⠀⠀⠜⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣿⡴⠁⠀⠀⠀⡸⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣾⠉⡶⠲⠦⠤⠶⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⣁⣀⢼⡇⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⣸⢿⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠛⠉⠁⢀⡠⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠃⠘⡇⡇⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠔⠋⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣰⠃⠀⠀⡇⢰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⣾⡁⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⢸⠀⠰⠁⠀⠀⠀⡇⢸⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠋⠀⡟⠓⠲⠤⣀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⣀⡄⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠓⠢⢄⣀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠶⣄⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⡀⠙⢖⣠⣁⡾⢰⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠠⢀⡰⠾⠓⠤⣹⣌⠻⣼⡀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡘⡇⡞⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡺⣛⠭⠑⠒⣒⠩⢝⡻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣾⢏⠞⠱⡙⡆⢸⢠⢃⣤⠼⠚⠛⠛⠛⠒⠒⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⡏⣎⠤⠤⣈⢺⣴⣧⣿⣥⣤⣤⣤⡤⠤⠴⠶⠖⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⢉⣉⡭⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⣷⡱⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣔⣒⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣒⣢⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⡸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠄⠠⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⡋⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠨⠋⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄⠀⠀⠀⠀⢀⠒⠤⠀⠀⠀⠀⠀⠀⠀⠴⠊⣁⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠘⠿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⠀⠂⠀⠀⠀⠀⠀⠂⠁⠀⢉⡿⠃⠁⢹⠀⠀⠀⠀⡒⠀⡀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠀⠀⠀⠀⠸⣦⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆⣀⠠⠔⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠃⠈⠐⠚⠀⠀⠘⠐⠂⠀⠀⠀⠉⠀⠀⠤⢀
⠀⠀⠀⠀⠠⡓⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠐⠀⠀⠀⠀⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠢⠄⠠⣤⠶⣄⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠒⠀⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠧⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⣟⠁⠈⠁⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⠁⠀⠀⠀⣠⠀⢀⡀⠈⠀⠀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⣉⣣⠞⠃⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠇⠀⠘⠀⡄⡆⠸⠀⠋⠀⢸⠁⠀⠀⡄⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⢰⡃⡼⠮⠁⣀⣰⣟⠋⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠀⡀⢐⠀⠀⠀⢇⠀⢰⡇⢀⠀⢠⠇⠀⡴⢁⠄⠀⡀⠁⢀⡀
⠀⠀⠀⠀⠀⣾⡜⠛⡀⣤⡼⠧⠁⣄⣴⡛⠋⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠆⠀⡀⠓⠈⢣⠐⡄⠸⡄⣼⠁⠀⡰⢠⢆⡜⣠⠏⣠⠊⠀⠀⠉
⠀⠀⠀⠀⠀⢋⣁⠿⢏⢉⣄⡴⠾⠿⠁⣀⡼⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢴⡀⠘⢇⢹⡀⢿⡟⠀⡼⢀⡦⣾⠞⢁⡼⢁⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⣏⠛⢦⣞⠛⢥⡀⠀⠶⡟⢥⣄⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣄⠀⢃⠀⠈⢺⠁⣸⡇⣰⠃⣸⠟⠁⣰⠋⣴⠧⠀⠀⠈
⠀⠀⠀⠀⠀⢭⡕⢣⣙⠿⣮⢑⣶⡶⠻⣌⢻⣜⡫⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠐⣌⢣⡘⣦⣄⢸⡆⢹⣧⡏⢀⡿⠀⡾⢁⣴⠯⠀⢀⡠⠔⠒
⠀⠀⠀⠀⠀⠰⣌⡗⢬⣷⢾⡛⢧⠘⣧⡙⢦⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠳⣼⣻⣿⡆⠈⣿⠀⣼⣧⣾⣴⣻⠥⠐⠈
⠀⠀⠀⠀⠀⠀⠘⠾⢧⣉⢷⣙⠣⠿⠬⠷⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠒⠒⠛⠷⠤⢄⣉⠛⠻⣦⡏⣰⡿⠟⠋⠋⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⣠⠔⠒⠂⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠀⠀⠤⢤⣄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠤⠖⠲⠶⠛⠂
⠀⠀⠀⠀⠀⠨⡃⠢⠤⠀⠀⠀⠀⠤⣤⡎⠀⠀⠀⠀⠘⡿⢤⢀⢀⣀⡀⢀⣠⣿⡇⠀⠀⠀⠀⠐⢀⢄⠀⠀⠀⠀⠂⠀⠀⠁⡠⠃
⠀⠀⠀⠀⠀⠀⣷⣤⣀⡀⠀⠀⠀⢀⣹⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⣷⠦⢤⣤⣤⣤⠴⢶⡎
⠀⠀⠀⠀⠀⠀⢹⣿⡉⠙⠻⢿⡏⢻⡟⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⠲⠆⠰⡦⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠈⠈⢁⣾⠃⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⣀⣀⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⡛⢹⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⢸⠿⣿⣧⣨⣽⣾⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⣧⠻⣧⣀⡤⢊⣿
⠀⠀⠀⠀⠀⠀⠘⠀⠀⡏⠉⡍⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠿⣇⠀⠀⣀⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠣⣄⡅⢀⣇⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠤⠤⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠛⠿⣄⣀⠀⠀⠀⠀⣀⣸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢿⣿⡁⠀⠀⠀⠂⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢓⠂⠠⠄⠀⠄⠒⢿
⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⡀⠀⢀⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣶⠄⠠⣤⡶⢾⡇⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠠⣼⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢄⡽⡇⣿⠁⠀⠀⠀⠀⠀⠀⠈⠸⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠿⡟⣵⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠀⢀⣶⡿⠀⠀⠀⠀⠉⠓⠚⠳⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡛⠛⠛⢻⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠦⣤⣠⠾⠛

⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⣤⣤⣤⣤⣤⣤⣤⡄⢠⣤⣀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠿⠿⠿⠿⠿⠇⢸⣿⣮⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⡦⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⢠⣶⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⡅⢸⣿⣿⣿⠇⣼⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠇⢘⠟⢋⣡⣾⣿⣿⣯⣗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡆⠀⠀⣤⡟⢨⡙⠿⣿⣿⣿⣿⣿⣷⣍⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⡶⠟⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡄⣰⣶⡀⣶⣶⣦⢰⡆⣴⠆⠀⢰⡆⢰⡆⣶⢰⡆⣶⡄⣶⢴⣶⠆⣶⡶⢰⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⢹⡇⣿⣆⣿⢸⣿⣿⠀⠀⢸⣧⣼⡇⣿⢸⡇⣿⣿⣿⠀⣿⠀⣿⣄⢸⡇⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⢾⡗⣿⠉⣿⢸⣟⣿⣀⡀⢸⡏⢹⡇⣿⢸⡇⣿⢿⣿⠀⣿⠀⣿⠁⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠃⠟⠘⣧⡿⢖⣻⠽⡷⠾⠻⠭⠿⢷⡺⠇⠻⠿⠃⠻⠈⠟⠐⠿⠀⠿⠿⠸⠇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠈⠀⠉⣻⣿⠟⢿⣿⣿⡿⠚⠉⢀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⣀⣤⡶⢶⣾⡋⠁⠀⠀⠀⢀⠀⠀⠬⣝⣿⠻⠶⠭⠁⠀⢠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣾⣶⣿⣿⣤⣼⡟⠀⠀⠀⠀⠀⠀⠁⠀⠱⣮⣻⡂⠀⠀⠀⡠⠚⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣭⣿⣿⣿⣥⣦⡔⠀⠀⠀⠀⠀⡄⠀⠙⣝⣿⣦⠀⣾⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠁⠀⢠⠀⠀⠀⠃⣠⢻⣎⢻⡎⢅⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣫⣿⣿⣾⡿⣿⣿⢿⢹⠀⠀⠘⠀⠀⣴⢀⣿⣿⣿⣦⣧⠈⣮⠻⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⢏⣼⣿⣿⣾⠈⠎⠀⣰⢃⣼⣿⡎⣿⣿⡿⣿⣿⣦⡉⣦⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠟⡩⢋⣿⣾⣿⣿⠟⠁⢠⡄⠀⣱⣿⣿⣟⣼⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡤⠤⢂⣨⣿⣿⣿⣿⣼⣰⢟⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣙⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢤⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣢⣟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣏⣾⢻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡉⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣩⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠾⠛⠛⠛⠛⠛⠛⠷⠶⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠉⠀⠀⠀⠀⢰⣼⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⣼⣦⡀⠀⠀⢾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣄⣘⣿⣿⣿⡆⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣦⣀⣄⠿⠛⠉⠀⢀⣤⣶⣤⣤⣤⣀⠘⢷⣄
⠀⠀⠀⣴⠟⠀⠀⠠⣆⠀⣶⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠻⣦
⠀⠀⣼⠏⠀⠀⠀⠀⠹⣿⣿⣧⡠⠖⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠛⠉⠁⠀⠀⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠹⣧
⠀⣸⡏⢀⠀⠀⠀⠀⠐⠛⠛⠉⠁⠀⠀⠀⠀⠈⠙⠛⢉⡏⡰⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣇
⢠⡿⠀⢿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⢿
⣸⡇⠀⠀⠀⠉⠙⠻⠿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣿⣿⣿⣿⣿⣷⣦⡀⢸
⣿⠀⢠⡀⣤⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣶⣦⣤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⠀⠈⢻⣿⣦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⠀⠀⠉⠁⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⡄⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⡇⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⡿⣿⣿⣷⣀⣀⢰⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⣷⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣽⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⢻⣆⡄⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠻⣧⢸⡄⠀⠄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣴⣶⣾⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠙⢿⣇⢰⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⣾⣿⡆⡆⡄⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣧⣱⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣝⣻⣿⣿⣿⣿⣿⠛⠋⠉⠀⠘⢿⣿⣿⣿⣿⣿⣿⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠾⢿⣿⣤⣴⣴⣦⣶⡶⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⣾⠇⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠀⢰⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⠋⠉⠉⠙⠛⠛⠻⠶⠶⢶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢄⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠶⠶⠶⣤⣦⣤⣄⣀⣀⢀⣀⠄⠀⠀⠀⠀⢀⣠⣷⣶⡿⠿⠋⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⡝⠛⠛⢏⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠃⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣦⡀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠛⠛⠛⠛⠛⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⠂⢉⣀⣀⣄⣀⣤⣄⣠⣄⡀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠾⠛⢋⣡⣤⡴⠶⠶⠶⢶⣶⠶⢬⣭⣱⠶⣤⡀⠀⠈⢳⡄⢠⡀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⠀⠀⢈⡛⠉⠀⠀⠀⠀⠀⠘⠛⢷⡄⠀⠈⠙⢳⣎⢳⣦⡼⠇⠘⠃⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⡇⠀⢀⣴⠖⠀⠀⣠⣤⣄⢻⡆⠀⠀⠀⢿⠋⠀⠀⠀⢸⠇⠾⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣴⠛⢉⣩⣶⣟⡁⣠⣄⣙⣿⡀⠀⠀⠈⣧⣠⣄⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠀⢸⠁⢻⣿⣿⡇⠀⢼⣿⣧⣿⠀⠀⠀⠀⠟⠀⣹⠀⡇⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣠⣄⠷⠦⠾⠏⠙⢦⣄⣩⣿⠟⠀⠀⠀⠀⣀⣤⠟⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢶⣄⠀⠀⠀⡋⠀⠀⢠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⣠⣴⠟⠑⢦⡌⠈⠀⠀⢠⡇⠀⠀⢸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⠶⠶⠶⠛⠛⠉⠀⠀⠀⠀⠈⢢⡀⠀⢸⠀⠀⠀⠘⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡤⠶⠚⠁⠈⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠞⣶⡖⢠⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣄⣀⠀⠀⠀⣠⣴⠟⠉⣶⣤⡀⠀⣀⡀⢠⣤⣤⠾⠋⠀⢸⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡙⢋⠁⠀⠀⠉⠁⠀⠀⠈⠉⠉⠉⣠⡟⠀⠀⠀⡼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⠲⣤⣀⣀⠀⠀⠀⢀⣀⣤⡾⠋⠛⢧⡀⠀⡷⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢰⡇⠈⠉⠛⠛⠛⣿⠉⠀⠀⠀⢠⡖⢱⣄⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠶⠚⠃⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⠁⠃⠸⡁⢸
⠀⠀⣠⡴⠶⠒⠛⠉⠉⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢠⡇⠘⠀⠀⢻⣾
⠀⣾⠁⠀⠀⣠⡀⠀⠀⠀⢀⣀⣤⣴⠛⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⠶⠶⠀⠈⢿⡄
⠀⠻⣆⠀⠀⠉⠻⣯⡉⠉⠉⠉⣰⠏⠀⠸⡀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠸⡇⠀⡆⠀⠀⠘⠹⣦
⠀⠀⠘⢷⣄⠀⠀⠈⠙⣦⢀⡼⠋⠀⠀⠀⢹⡄⠀⠀⡇⠀⠀⠀⠀⠀⢀⣠⣾⣥⣰⣂⣡⠀⡇⠀⠀⠀⡇⠈⢷⣄
⠀⠀⠀⠀⠙⣧⠀⠀⠀⢘⡿⠁⠀⠀⠀⠀⠀⠻⣀⣠⠋⠀⠀⠀⠀⠀⠈⢱⣯⠀⠉⠉⠁⠀⠑⢦⣀⠀⠀⠀⠀⠉⢷⣄
⠀⠀⠀⠀⠀⣸⠆⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠠⣎⠁⠀⠀⠀⠀⠀⠀⠈⣿⠉⠻⢦⣤⣀⣀⠉⢷⡄
⠀⠀⠀⠀⢸⡷⠒⢺⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠛⢦⠀⠙⢦
⠀⠀⠀⠀⡞⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⠄⡀⢸⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡀⣀⣤⡤⠒⠾⣅⡀⢀⡀⠀⠀⢀⣀⣠⣤⡴
⢇⠘⠓⠈⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡇⠀⢩⠀⠀⠀⠀⠀⠉⢹⡏⠉⠁
⠈⠓⠶⠶⢺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠑⠒⠒⠒⠒⠒⠒⠉⠉⢹⡇⠀⠀⡄⠘⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠓⠶⠤⣤⣤⣤⣤⠶⠖⢺⡇⠀⠀⠐⣄⠡⣄⠀⠀⠘⣷
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢦⡈⠓⢤⣤⠿⣀
⠀⠀⠀⠘⠒⠒⠒⠒⠒⠲⢦⣤⣤⣤⣉⣉⣉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠉⠒⠒⠖⠒⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠉⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠛⢶⡶⠦⣤⣤⣄⣀⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠙⡞⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠛⠓⠒⠒⠒⠒⠀⠤⢶⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣧⠀⠀⢀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣟⠁⠙⠳⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠒⡶⠒⠒⠒⠒⠲⢤⣤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⣀⣤⡤⠔⠒⠛⠓⠛⠒⠀⠀⠀⠀⠈⠓⠚⠻⠛⠛⠋⠉⠉⠙⠛⠛⠻⠿⢶⣶⠶⠾⠃
⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣷⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠙⠢⠤⠤⠤⠤⠤⢤⣤⡴⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣞⣀⣀⣀⣀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⣠⣤⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣮⣛⣛⣯⣽⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣧⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣿⣿⣿⣿⣿⣦⠹⣿⣷⣌⡙⠻⠿⣿⣿⣷⣍⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣽⡳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣿⢿⣻⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻
⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⣿⣿⣿⠶⣶⣦⣤⣭⣭⣍⣉⣙⣛⣛⣛⣛⠛⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣭⣽⣿⣿⣿⣿⣿⢏⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢫⣭⠭⠭⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣻⣿⣿⣿⣿⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣤⣀⣀⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠰⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠆
⠀⠀⢻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡟
⠀⠀⠀⢻⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣶⡄⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⢠⣴⣾⣿⣿⣿⣿⣿⣻⠟
⠀⠀⠀⠀⠀⠘⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣫⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣭⣝⣛⣛⣛⠿⠿⢛⣛⣛⣫⣭⣶⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣦⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⡿⠀⢸⣿⡿⠉⢿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⠟⠁⠀⠘⠋⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠸⠛⠁⠀⠀⠀⠀⠠⠄⠀⢀⣸⡟
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠋⠛⣿⣿⣋⣀⣀⣀⣈⣉⣁⣀⣀⣤⣤⣤⣶⣶⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁⠀⠀⠘⠛⠻⠿⠿⠛⠛⠛⠛⠛⠛⠉⡉⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠈⠠⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠛⠉⠀⠀⢀⣤⣶⣾⣧⣀⠀⠀⠀⠀⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢨⣤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠄⡳⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⡀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢺⣿⣿⣿⣿⣿⡿⠀⠹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠓⠆⠀⠀⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣿⡀⠀⠀⠀⣼⣶⣶⣶⣾⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⣿⣿⣿⠿⠷⠀⠀⠀⢿⣿⣿⡿⣿⣿⣿⣿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣖⣺⣭⣭⣭⣟⣒⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⡿⠿⠛⠛⠉⠉⢁⣀⣀⣀⣀⣀⣠⣤⣀⣀⣈⣷⣽⣒⣶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⣿⣀⣤⣤⣶⣶⣿⣿⣿⢿⣿⣿⣿⡿⠿⣿⢽⣿⣿⣿⣿⣻⣿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣵⡿⣿⣿⣿⠿⣿⣾⣿⡿⠋⠀⠙⠛⠯⠛⠳⢾⣿⣿⣿⣿⣿⣻⡿⢯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⢻⣿⣿⡟⠀⠒⠂⠿⠉⠀⡲⣦⣄⠀⠀⠀⠀⠀⠈⠉⠟⢿⣟⣋⣚⣓⣒⣲⢤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢞⣻⠿⠿⠿⣧⠀⠀⠘⠄⠀⠀⠀⠙⠁⠀⠀⠀⠀⠉⠁⠈⠁⠉⠉⠉⠉⠉⠉⠉⣷⢳⠀⠀⢠⢞⡭⣭⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠮⣛⠶⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠚⢒⢀⢀⡷⠶⣶⣶⣶⣾⠶⢫⠏⠀⢠⢯⡞⠀⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣖⣲⣶⣶⣶⣦⣗⣻⡀⢀⡀⠄⠀⣙⣦⢀⠀⠀⢀⡴⠃⡎⠀⡎⠉⠉⠉⠉⠉⠧⠭⣭⣭⠟⠀⢤⣿⣙⡲⣄
⠀⠀⠀⠀⠀⠀⢀⣰⢳⡏⠉⠉⠉⠉⠉⠉⠉⢉⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⢾⠀⢇⠱⠀⠱⡀⠀⠀⡠⠀⣤⣤⣤⣴⣀⣀⣆⣀⣿⡿⣸
⠀⠀⠀⡴⣺⡽⠿⠷⠾⠧⠀⠀⠐⠒⠒⢠⠋⠀⠀⠀⠀⠠⠤⠤⠄⣀⠏⣿⣦⡃⠀⠀⠀⠀⠀⠉⠉⣀⡼⣳⠛⠒⠒⠿⠿⠽⠿⠞⠚⠁
⠀⠀⣞⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠤⣀⠀⠉⢻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣸⣿⣜⣇
⠀⠘⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣶⣤⣀⠀⠠⠤⢤⣼⣿⣿⣿⣿⣿⣿⣿⣇⠒⢹⡯⠹⣿⣿⣮⢣⡀
⠀⠀⠙⢜⡶⣤⣄⣀⣠⠄⠒⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠈⠙⢿⣷⡝⣆
⠀⠀⠀⠀⠉⠓⢲⡹⡄⠉⠉⢩⣿⣿⣿⡿⡟⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣎⣇
⠀⠀⠀⠀⠀⠀⠀⠱⢝⡶⠴⢾⣻⡯⠗⠚⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢻⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⢠⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡻⣧⡈⢉⣼⢫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠙⠺⠭⠭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⡭⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠯⠭⣟⣛⣛⣛⣻⣿⣛⣛⣛⣻⠭⠽⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣖⣮⣭⣽⣭⣭⣵⣖⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣾⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⣯⠧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡟⢉⣁⣠⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣽⣾⣿⣿⣿⣿⣿⣿⡿⠃⠙⠿⠟⠳⢯⣷⣿⣿⣿⣿⣿⣿⢟⠞
⠀⠀⠀⠀⠀⠀⠀⣸⣱⣿⣷⣿⣿⠋⠖⠈⠿⠋⢐⣦⣄⡀⠀⠀⠀⠀⠙⠻⢿⣿⣋⣚⣷⣶⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢾⣻⣿⣿⢿⡀⠀⠐⠀⠀⠀⠈⠋⠀⠀⠀⠀⠉⠀⠉⠘⠉⠉⠉⠉⠉⢹⣽⠀⠀⣠⣫⢿⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠯⣓⣆⠀⠀⠀⠀⢀⡀⠁⠀⠀⠋⢍⣍⡽⠲⣶⣶⣶⣞⡋⣎⣀⣰⣳⠃⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡴⣯⠭⠿⣭⣯⣷⣯⣽⠄⠤⠐⠀⠠⠵⠠⠤⠀⡴⢋⢪⠀⢆⠀⠀⡀⠀⠙⠒⠓⢲⠃⠐⡞⢷⣭⣢
⠀⠀⠀⢀⡤⣤⣴⣹⡇⠀⠀⢀⣀⡀⡠⠀⠀⠀⠀⠀⠐⠒⠀⡤⣴⢣⠈⠀⠀⠈⠂⠤⠄⠃⡴⣲⣒⣚⣦⣴⣷⣾⣟⡽
⠀⠀⡴⡵⠋⠉⠉⠉⠉⠁⠀⠀⠀⠰⠁⠀⠀⠀⠀⠉⠉⠠⢾⣥⣿⣷⣄⠀⠀⠀⠀⠀⣰⣞⣵⠁⠀⠀⠉⠉⠉⠉⠁
⠀⢸⣹⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄⡀⠀⠀⣈⢀⣀⣾⣿⣿⣿⣿⣷⣤⠤⢤⡔⢿⣿⣮⢦⡀
⠀⠈⢫⡳⣤⣀⣀⣀⡠⠄⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠁⠀⠙⢿⣷⡵⣄
⠀⠀⠀⠈⠓⠺⡽⣍⠉⠉⣹⣿⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⢿⣞⣆
⠀⠀⠀⠀⠀⠀⠙⢮⣲⣶⣛⡯⠗⠊⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢹⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⠠⠄⢀⣴⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡷⢝⡦⠤⣞⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⡯⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠭⠽⢿⣿⣟⣿⣿⣿⠯⠭⠟⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡼⠛⢿⢿⡇⢀
⠀⠀⠀⠀⠀⠀⢠⣤⠀⢠⡘⠿⠷⠀⠀⣾⣿⣿
⠀⠀⠀⢀⣤⣌⠲⠟⠒⠚⠧⣶⠷⠾⣿⣿⣿⣅
⠀⠀⠀⢼⡿⢋⡉⠧⠀⠀⠀⠈⠳⣾⣿⡿⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣶⣦⡤⣄
⠀⠀⠀⠘⠃⠈⢁⠀⢰⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⣀⠀⣤⠔⠶⣻⠀⠀⢀⠔⠉⣹⣀⠐⠀⠠⠊⠁⠰⠤⠤⠤⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠠⠩⢘⠏⠁⢀⠏⠃⠀⢹⣦⣴⣿⠃⠀⠐⢠⣴⠃⠀⠀⠀⠀⠀⠉⢫⠙⡀⠀⠈⠻⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⠀⠀⢸⡀⠀⠀⢰⡿⠿⠃⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⡇⢃⢀⡀⠢⠹⣿⣿⣿⣿⣝⡂
⠀⠀⠀⠀⠂⠄⡀⠋⠀⠀⠀⢸⣧⡀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⢡⡀⡇⠙⠌⠻⢿⣿⣿⡧
⠀⠀⠀⠔⢢⠀⠉⠂⠀⠀⠀⠀⠙⣿⠿⠿⠛⠋⠁⠀⢢⠀⠀⠀⠀⠀⠀⠀⡠⠒⢢⠀⠐⡀⠀⠰⣿⣦
⠀⠀⠀⡀⣰⣧⡀⠀⠀⠀⠀⡆⠀⣿⡆⠀⠀⠀⠀⠀⠀⣁⣀⠀⠀⢀⡠⠞⠁⠀⠀⢆⠀⠬⠛⣿⣿⠛⠁
⠀⠐⣀⣿⠿⠁⠙⢦⣀⣀⢴⣁⡀⠘⣿⣄⢀⣀⡠⠔⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠐⠒⠀⠈⠁
⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣶⣶⣶⣶⠶⠶⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣀⡹⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣻⣾⣿⣿⣿⣿⣿⠃⢠⡄⢤⣀⣦⡀⠀⠙⣿⣶⢄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣂⣿⣷⣀⣙⣿⣿⣦⣤⣿⣿⡆⢻⣄⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⣸⣿⣿⣿⡿⠿⠿⠿⠚⠛⠉⣉⣩⣿⣿⣿⣿⡟⠋⠉⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠘⣿⣌⠛⠛⠛⠛⠓⢀⣠⣤⣶⣿⣿⣿⣿⡿⠋⣿⣧⠀⢀⣴⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⢸⣿⣿⣷⣶⣿⣿⣿⠿⣻⣿⣿⣿⣿⠟⠁⠀⣿⢸⡀⡛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣾⣿⣿⣿⣿⣿⠟⢡⣾⣿⣿⣿⣿⠋⠀⠀⠀⣿⠀⣯⣻⡿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣉⣛⣛⣽⠟⠁⣴⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢻⠀⢸⣿⡇⢸⡟⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠰⣦⣤⣬⣿⠟⠋⢀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⠀⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣦⣴⣶⣿⣿⣿⣿⡟⠉⠀⠸⣿⡇⠀⠀⠀⠀⢸⣿⣦⣸⣇⣀⣠⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿⣿⣄⠀⠀⠀⠈⣿⣿⡟⢻⡁⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠻⣦⡀⠀⠸⣿⣿⣦⡀⠀⠀⣿⣿⡇⠀⠃⠀⡀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠈⠻⣦⣀⠻⣿⣿⣿⣷⣦⣸⣿⣷⡇⣀⣸⣷⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⢸⡀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⠋⠉⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢻⣿⡇⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠈⣿⣷⣄⠀⠀⠀⣿⣆⠀⠀⠘⣿⣿⣿⡃⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠸⣿⣿⣷⣤⣤⣿⣿⣷⡀⠀⢿⣿⣿⣿⡀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⠻⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠋⢉⠉⠉⠙⠲⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣀⣠⣤⡴⠖⠉⢀⣤⣶⣶⣶⣶⣶⣶⣤⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠸⠿⠟⠉⠀⠀⣴⣿⣿⡿⠛⠛⠉⠙⠛⠉⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⡴⠚⣉⣠⣤⣤⣬⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⣰⠋⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀
⢠⠃⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠙⠒⠦⢤⣀⡀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠀⠀⢀⣀⣠⣤⣤⣴⣉⣹
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣴⣶⢿⣿⣯⣴⠾⠿⢛⣫⠭⠗⠋⠉
⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣩⣭⠶⠟⠛⠉⣁⡤⠖⠊⠉
⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠋⣉⣠⠤⠖⠊⠉
⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣚⣉⠤⠴⠒⠋⠉⠁
⠀⠀⠀⠙⠢⣅⣂⠤⠤⠀⠀⣀⣀⡶⠼⠒⠋⠉⠁
⣶⣶⠶⠶⠶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢰⣿⢛⣛⠛⠻⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠟⠛⠛⣿⣿⡛⠋⠙⠛⠛⠛⠛⠛⠛⠟⠛⠻⠿⠛⠻⠃
⠀⠀⣿⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠷⠖⠂
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⣄⠀⠐⡀⠀⣠⡀
⠀⠀⡆⢀⣄⣤⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢼⠇⠈⠉⠁
⠀⠀⡷⠀⠉⠙⣦⢀⣠⠈⢛⣿⡟⢻⣦⡶⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃
⠀⠀⡇⠀⡀⠀⠀⢈⠙⢧⡌⢻⡷⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇
⠀⠀⠇⠀⠁⢀⡴⠋⠀⠀⠁⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗⣠⡀
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣠⣤
⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿
⠀⠀⡁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠰⣷⣾⣿⣿⡿⠋⠉⠉⠉⠉⠛⠷⢄⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠸⢛⣿⣿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠹⣶⣾⣿⣿⣿⠃
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⢀⠈⣹⣿⣿⠃
⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠙⣿⣦⣴⣿⣿⣿⣿⣿⠀⠀⠀⢠⡄⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠀⠀⠾⢿⣾⠟⠉⢠⣿⣿⡁⢀⣄⠰⠆⠙⣿⣿⣆⠀⠀⠀⠀⢠⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣿⣷⣦⣤⣴⣦⠄⠉⠀⣀⣻⣿⣿⣿⣿⣟⣷⣾⣷⣾⡿⠃⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡃⠀⢀⣾⡿⠧⢾⣿⣿⣴⣶⣾⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⣿⠇⠀⢀⣸⣿⣿⣉⣿⠷⣬⣿⡿⠿⠉⠛⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⡇
⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⡀⠘⣷⣶⡦⣤⠄⠐⣿⣿⡤⠛⠁⠈⠻⢿⣿⣿⠃⠀⠀⠀⠉⢹⣿⣧⠀⠀⠀⠀⠀⠀⠁
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣬⣀⣴⣿⣿⠇⠀⠀⣼⡿⠋⢀⡄⠀⠄⠀⣸⣿⡅⣘⠀⠀⠀⠠⣤⣿⣿⣆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⢿⣟⠀⠀⠰⠏⣀⣼⣿⠀⠀⠈⠹⠛⢻⣿⣉⠀⠀⠻⣆⠸⢿⣿⣿⣤
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣴⣤⣏⠀⠀⠀⡿⢀⡀⢀⣾⣏⣴⢿⡆⠀⠀⢀⡀⣿⣿⠿⠏⢠⣾⣿⣿⣿⣿⡟⠉
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⠃⠀⢸⣿⠀⣀⣿⠿⣄⡀⠸⣷⣄⠀⠀⢀⣿⣿⣷⣶⡟⠻⣿⣀⣹⣿⣇
⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢸⣿⣘⡿⣿⣷⠉⠛⠦⠙⠿⣤⡞⢿⣿⣿⡿⠋⠈⣷⠞⠛⠛⠛⣿⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⣿⠻⣷⣾⡟⠉⠁⠀⠀⠀⢹⣧⣴⣿⣿⠀⠀⠀⠁⠀⠀⠈⠙⢿⣧
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣿⣴⣾⣿⠃⠒⠂⣀⣄⢰⣏⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣷⣶⣶⣦⣹⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢀⡄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⢻⣿⢿⣿⣿⢻⣼⢿⠿⢿⠟⣿⣿⣿⡀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠰⢄⣤⠀⠀⠀⠀⢀⠈⢹⡿⠃⠀⠁⢠⣿⡆⠈⣿⣿⣷⠀⠀⠀⢈⣿⠋
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠘⢣⣾⣿⣶⣶⣦⣼⣿⠆⢠⣶⠀⠀⠆⠀⣿⣿⡶⢻⣿⠟⠀⠀⠀⡿⠃⣠
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡀⠘⠛⢻⠛⠿⠋⠉⢉⠛⢿⠛⠆⠀⠸⣿⣄⠙⣷⣾⡏⠀⠀⢠⣾⠀⠚⠉
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣾⠿⠃⢰⣶⣦⣿⣷⡼⣿⡆⠰⣦⣼⠟⠛⡤⠛⠁⠀⢰⣿⣿⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⢀⣷⣾⣿⣇⡠⠉⢠⣿⣷⣤⣼⣁⣸⠿⠁⠀⠀⠀⠀⠻⣿⣿⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⠿⣿⣿⣷⣦⣿⣿⣛⣀⡙⣿⣿⢷⣶⠀⣠⣰⣶⡀⠘⢿⣿⡀
⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⢹⣿⣿⣿⢿⣿⠿⠏⠉⢨⡿⣷⣿⣿⡿⣧⡀⠘⢿⠃
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠠⣼⠃⠀⠀⠀⠀⠘⢿⣿⣦⣿⡇⠀⠈⣠⠂
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣟⡿⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⡟⢿⣷⡀⠀⠙⣧
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣶⠀⢠⠾⣷⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠈⣿⣷⠀⠀⠈⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⡄⠀⠀⠁⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠸⣿⣇⠀⠀⠀⢷⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢁⣀⣄⣰⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢻⣿⣄⠀⠀⠈⢿⣧
⠀⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⢠⣄⣼⠋⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢿⣿⡀⠀⠀⠘⡇⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢀⣽⣇⠀⠀⠘⣿⣷⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢿⠟⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⣿⣿⣄⠀⠀⢹⣿⣇⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠛⢻⣦⠀⠀⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠈⣿⡇⠀⠀⢻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣶⠻⣿⠀⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣿⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢸⢋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⣀⠀⢈⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣭⣿⣷
⠀⠀⠀⠀⣠⣤⡿⠀⢾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣽⣿⣄
⠀⠀⠀⣀⣈⣉⣠⣤⣴⣿⣿⣿⣄⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠉⣿⣿⣦⡤⠀⠀⠘
⠀⠘⠛⠛⠋⠙⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠓⠀⢸⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠈⠛⠒⠒⠒⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠉⠁

⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠹⣿⣿⣿⣷⣸⣷⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠛⠉⠉⠉⠛⢿⣿⣿⣆
⠀⠀⠀⢀⣶⣶⣤⣤⣬⣭⣉⣉⣛⡛⠃⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⢀⣠⣤⡀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠈⠙⠛⠋⠛⠱⠿⠿⠿⣿⠟⠀⠀⠀⠀⠀⢰⣦⣤⠀⠀⢀⣙⣛⣿⣿⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠛⠿⠟⠁⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⡀⠀⠀⠀⣀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⣉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⣿⣿⣿⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣶⣤⡄⢠⣀⣴⣿⠟⢉⣴⣿⣿⣿⣻⣷⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣁⣉⠉⣉⣠⣾⡿⠿⣄⣠⣿⣿⣿⠋⠙⢻⣿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡶⠻⣿⣿⣿⣿⣶⣶⣶⢿⣿⣏⣀⠈⠻⣿⠏⢿⣿⠋⣠⡄⢀⣿⣿⡿⠿⠞⠛⠉⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⣼⣿⡇⢶⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣷⣦⡑⢼⣿⣿⣿⣀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢠⣿⣿⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⠘⠉⠉⠁⢳⣈⣿⣆⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⢘⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣸⣿⣿⣿⡿⠀⡿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⣸⣿⣿⣿⠿⣿⣾⣿⣿⠏⠉⠛⠿⢿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣅⠻⠁
⠀⠀⠀⠀⠀⣠⣾⣿⠟⢁⣼⣿⣿⠟⠁⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢻⣯⣿⠏⠙⣿⣧⡄
⠀⠀⠀⠀⠀⠉⠉⠀⣸⣿⡟⠋⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⣠⣤⣤⣤⣠⣬⣻⣲⣶⣤⣤⣤⣤⣄⡀⣀⢠⣤⣤⣤⣹⣷⣤⣤⣤⣶⣌⣿⡄⡀⣀⣀⡀⣄⡀⣴⣿⣧⣠⣶⣶⣶⣦⣦⣦⣦⣠⣤⡀⡀
⠀⠀⠈⠉⠀⠉⠉⠁⠀⠀⠈⠈⠁⠀⠁⠁⠈⠉⠉⠉⠛⠙⠋⠙⠛⠛⠋⠉⠉⠋⠀⠙⠛⠛⠛⠛⠋⠀⠈⠉⠁⠉⠉⠉⠁⠈⠉⠁⠁

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

⠀⠀⠀⢀⣤⣖⣒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢤⣤⣤⣤⣤⣤⣤⢖⣶⣶⣶⣶⣶⣶⣾⣿⣷⣦⡄⠀⣀
⠀⠀⠀⡟⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢰⢰⠀⠀⠉⠉⢙⣩⠿⠃
⠀⠀⠀⢧⢸⠛⠛⠛⠓⠒⠲⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⣨⣼⣠⣠⣀⣀⣨⡇
⠀⠀⠀⠸⣼⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣐⣀⣀⣀⣀⣀⣀⠀⠀⣀⣤⣤⣤⣶⡄⠀⠀⠀⠑⢦⢄⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⡍⢻⡿⠛⠛⢿⡉⠻⣿⡄⢹⣿⣿⣨⣿⣿⡄⠀⢀⣴⡯⠖⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡀⠀⠀⢸⡇⢀⡿⡁⠀⢿⣿⣿⣿⣿⣷⡀⡏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣙⣒⣒⣚⣐⣙⣷⣥⣀⠘⣿⣿⣿⣿⣿⣧⢧⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠑⣽⣧⢹⣿⣿⣿⣿⣿⣏⣯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡌⣿⣿⣿⣿⣿⣿⡜⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠸⣿⣿⣿⣿⣿⣿⡹⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢻⣿⣿⣿⣿⣿⣷⢱⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡘⣿⣿⣧⣼⣿⣿⣧⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣙⣿⠭⠭⠭⠗⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⡶⠿⠿⠿⠿⠿⠿⠿⢷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠁⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠻⠿⣿⡿⠿⠟⠋⠁⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⠓⠒⢢⣠⣾⣟⣛⣿⣆⣀⣀⠛⠐⣿⢿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⣴⠟⠛⠻⣿⣿⣷⡶⢷⣷⢾⡟⢻⡇⠈⣿⠀⣿⠉⠉⠙⠛⠳⡶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⢀⠔⠛⠚⢿⣿⡟⣞⠛⣿⢹⣱⠈⠇⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠠⠒⠙⠛⣿⡇⠡⡻⢟⣫⢜⡾⠀⢸⣿⠁⠀⡇⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠈⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⢀⣶⣿⣿⣿⡟⢄⡈⠁⠈⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢹⣿
⣼⡏⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠘⣿
⣿⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡄⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⣿⠛⠷⣦⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⡟⣿⣿⣦⣄⣠⣾⣿⠀⠀⠀⠉⠛⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⢻⣏⠛⢦⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣿
⢹⣧⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣿
⠸⣿⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⢻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⡿
⠀⢿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢻⣿⣿⣿⣿⣿⣿⡇⢠⢸⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣿⠇
⠀⠘⢿⡎⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣾⡟
⠀⠀⠘⣿⡄⠙⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⠤⠀⢀⣴⠂⣾⠇⠀⠀⣼⡟
⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣧⣾⡟⠀⢀⣾⡟
⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⣀⠀⠀⠀⠀⢻⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣦⣤⣈⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠔⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⢟⣭⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡦⠊⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢙⣷⣿⣿⣿⣎⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠛⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⣠⣾⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡖⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢈⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠢⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⢿⣿⡿⣣⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⢿⢿⠇⠀⢰⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡿⠿⢿⠀⠀⠾⠇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣇⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡐⠀⠲⡾⠐⠆⢁⣈⣩⡩⣛⣺⣧⣵⣲⣱⠗⠂⣉⡵⠿⢾⣿⣮⠹⡟⠛⣟⣿⢙⣉⢬⣥⠇⠁⢸⠃⡃⠾
⠀⠀⠀⠊⠀⠀⠂⠀⠈⢁⠀⢍⠈⢣⣨⡟⠅⡙⣾⣷⠾⠋⠁⠀⣉⡩⠕⠛⠩⢭⠀⠀⣘⠀⠠⠉⢀⡔⠛
⠀⠈⠀⠀⠀⠀⠀⡁⣀⣁⠷⠞⠂⢸⠿⢀⣂⣼⣿⣁⣔⢡⣶⡍⠁⠀⠈⣿⠀⠀⠃⠐⠁⣠
⠀⠀⠀⢂⡠⠈⠀⠈⠉⢁⣀⠁⣰⣺⢠⠊⣉⡿⡟⠻⢿⡯⠁⠀⠀⢠⣠⡿⣾⣤⢀⠀⣄⣽
⠀⠀⠐⠀⠀⠀⣈⡤⠶⠟⣫⢅⠈⠀⣝⠖⠋⠈⣯⡐⢋⡡⠄⠀⠀⢺⣿⠏⠠⡁⠀⠀⡍⠹
⠀⢀⠀⠀⠂⠙⠄⠀⠘⠀⡏⠁⠓⣴⡇⠀⠀⠸⡿⠻⢾⢦⠀⡀⠀⢸⠇⠾⠆⠻⣄⠀⢸⠀⡀⠀⠀⠀⠌
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢆⡟⠉⠁⣀⣈⠵⡇⠐⠀⣽⠉⠀⠀⢠⡴⠧⠀⠏⣿⠀⢘⡄⡀⠁⠀⢀⡀
⠀⠀⠀⢀⠀⠀⡐⠀⢻⣤⡏⡸⣧⣾⣿⣿⡧⣀⣷⣈⠣⣄⠀⠀⢈⠋⡑⠈⢃⡄⢸⠀⠰⡿⣐⠨⣠⡜⣴
⠀⠀⠀⠘⠀⡐⠀⠀⢂⢻⠀⣿⡏⠁⠀⠈⢰⢿⡗⢌⠳⠂⠂⠲⠎⠲⠴⡀⣀⠤⣩⣧⢈⠓⠤⠒⣿⣿⡿
⠀⠀⠀⠀⠀⠃⣂⡀⢸⣧⣴⣿⣷⠈⣀⣀⣼⢀⣳⢈⡇⢈⠀⠀⠀⠀⠀⠀⢹⡄⣿⡇⢸⠀⡀⠈⣽⣿⠃
⠀⢀⠀⠀⠉⠀⠀⠀⠈⠻⢍⣟⡖⡈⢀⣁⢀⣴⢿⡈⠀⠈⠀⠀⡁⠀⠀⠀⠈⣶⣻⠀⣿⠑⠓⠶⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠂⠀⠀⢹⡆⠁⠹⡏⢠⠄⠀⠠⠀⠀⠀⠄⠀⣿⡇⠀⠼
⠀⠀⡀⠀⠀⠰⠀⠀⠀⠀⠀⠙⡀⠀⢀⣾⡇⠀⠠⢷⠈⢀⡠⠀⠀⠀⠀⠈⢀⠃⣷⠀⡇⠀⠀⠄⢤⣶⡄
⠀⠀⠈⢢⣄⣧⡬⠖⠉⠀⠀⠀⠁⠀⡼⠀⠇⠀⠀⡗⠀⠀⠀⠀⠄⠀⠀⠀⠈⢀⣿⠀⡆⠀⠨⠉⠈⢉⠄
⠀⠀⠀⠀⠋⠉⡒⠒⠀⠀⠀⠀⠀⠀⡗⠐⢹⠀⠀⣷⠀⠀⠀⠀⡧⠀⠀⠀⠠⠈⣡⠀⢧⡄⠔⠀⣠⠏
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⢐⠀⠀⣿⠀⢸⠃⠀⠛⣀⣓⢰⣈⣀⠃⠀⡂⢰⣻⣷⠀⠘⠦⠆⢂⡁
⠀⠀⡀⠀⠀⠀⠀⠀⠈⢃⢰⡇⠀⢀⣏⡀⢩⣴⢴⣧⠿⡏⡁⠉⠡⣴⠧⡙⢀⣡⠋⢶⠀⠀⠀⠈⠸⠒⢣
⣆⡠⣅⠒⢦⣰⣄⢷⠐⢻⡯⠇⠀⠿⢛⠃⠨⠟⠛⣯⠈⠈⠰⠀⠀⠣⠀⠘⠀⠉⠀⠈⠀⠀⠁
⠃⠐⠀⠈⠀⠁⠃⠀⠚⠋⡘⠀⣴⢀⠨⣃⣼⣇⠀⢸⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠀⠀⠀⠐⠁⠈⠀⠀⠀⡠⠅⢠⡯⠩⠴⠦⠤⠏⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠤⣀⣀⠀⠴⠷⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢰⡤⠠⠀⣀⢀⣠⠖⠫⠉⠉⠘⢧⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠂⠌⣎⡫⡕⠠⣀⠐⠂⡜⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠩⡽⠀⠈⣢⣄⣼⡓⢼⠦⡀⠁⠺⢄⠘⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠉⢲⣴⡋⠁⠀⠀⡽⠀⢀⡤⠓⠀⢀⠉⠪⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⠧⠤⠴⠻⡙⠁⢀⣠⠞⠁⢀⡽⠀⠀⠀⢀⠰⠄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠉⠀⠀⠀⠀⢀⠜⠀⠙⠋⠉⠀⠀⠀⢜⣣⣤⡑⣄⠸⠓⠢⢌⠮⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⡀⠀⡠⠔⠉⠢⡀⠀⠀⠀⠀⠀⣠⢈⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠑⠤⠤⠴⡒⠉⠀⠀⠀⠀⠈⠳⣄⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⡤⡎⢳⡀⠀⠀⠀⠀⠀⠀⠈⢢⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠥⠲⠦⠒⠉⠀⢻⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⠑⠢⣄⡦⣄⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⣠⠜⢉⠇⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣛⡆⠀⣠⠔⠊⠁⢠⠊⠑⠢⢤⡀
⡴⣦⣶⡄⠤⠄⠀⣀⠀⠀⢀⠤⣤⡀⢶⡒⠈⣠⣤⠹⠞⡀⠒⢄⣰⠃⠀⠀⠀⣸⡁
⠣⠻⣤⣇⣀⣀⣀⣀⢠⠚⡜⠶⣏⠉⠉⠉⠓⠃⠛⠀⠐⠛⠉⢰⡦⢾⣁⡦⠴⣟⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠤⢆⣷⠤⣤⠠⢬⠿⠯⠥⠤⠉⠉⣉⣙⣟⠛⡟⠺⡄⠈⠉⢃⣠⠔⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠁⠐⠀⠠⡤⠤⢼⣏⢢⡈⠙⣟⢯⢮⠣⠄⠁⠈⠽⠵⡧⠛⠉⣶⠁⣔⣠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣔⡇⣰⢶⡟⡆⢅⠊⣿⡓⠋⠉⠒⠒⠒⠒⠚⢧⠀⠀⠈⠉⠉⠀⠈⠑⠾⠓⠢⢤⣀⡀
⠀⠀⠀⠀⢀⠤⣄⣤⢞⣦⢸⣧⢧⣇⣀⠽⢾⠭⠑⠒⠂⠠⠤⠤⣤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢠⠧⣠⡞⠪⠚⠉⠉⢈⡅⠁⠀⠀⠯⣄⣀⣀⣀⣠⡤⡾⡏⠙⡑⣄
⠀⠀⠀⠀⠑⠂⠭⠄⠀⢀⢉⣩⠇⠀⠀⠀⠀⠀⠀⠈⡏⠉⣹⣠⠇⣄⢰⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⣆⣊⠟⠀⣎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣈⣁⢶⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⠟⡘
⠀⠀⠀⡀⠀⠀⠢⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠘
⠀⠀⠀⠹⢶⣤⣤⣀⣈⣉⣙⡛⠛⠿⠿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⣿⣿⣿⣿⣶⣤⣈⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⣀⣀⣀⣈⣉⣙⡛⠷⡄⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⢦⠀⣿⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠿⠿⠟⠛⢛⣒⠀⠸⡆⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠶⠾⠛⠋⣉⡭⠄⠈⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠁⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠠⢤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⡆⢠⠠⡀⢤⡀⢤⣌⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⢰⠇⢸⡇⢻⡆⢻⣦⠹⣿⣦⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠸⠀⢸⡇⢸⣿⠘⣿⣇⠹⣿⣷⡘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠀⢻⣿⡀⠹⣿⣷⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⡄⠘⣿⡇⠀⢻⣿⣧⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⣿⡀⠀⠻⣿⣆⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠄⠀⠙⢿⣦⣀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⣰⣿⣿⣿⡶⣷⣿⠁
⠀⢠⣾⣿⣿⣿⣿⣾⣫⣾⣇
⢴⣿⣿⣿⣿⣵⣟⣽⣿⣿⣿⠆
⠀⠀⠉⣩⡾⢫⣿⣿⣿⡿⠋
⠀⠀⠀⠙⠁⠘⣿⣿⠟⠁
⣤⣤⣤⣤⣤⣤⣬⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣥⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡞⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠲⣭⣓⢦⣝⣻
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠈⠙⠻⠿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠾⡦⣤⣀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢯⣝⡒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⠙⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣬⣙⠲⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⠷⢴⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠧⣥⣒⠤⣀
⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣄⡀⠀⡿⠀⠀⠀⠉⠙⠻⠧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢮⣝⡉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡧⢤⣀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣬⣙⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢮⣙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡾⠉⠚⣯⣝⡒⠦⣄⣀⠀⠀⠀⠀⡆⢩⠛⢳⠦⣍⡛⠶⣄⣀⡠⠞⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡉⠛⣶⠀⠀⠈⠳⢤⣴⣿⢁⠏⠙⢲⣼⣃⢀⠀⢠⡇⣾⠀⢸⠀⡏⡝⠛⠶⢿⣿⣒⠻⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠻⢦⣄⣀⠀⠀⠈⠉⠙⠓⠶⠞⠀⡸⠋⠙⠾⣅⡃⠀⣹⣘⡇⣿⢸⡇⣼⠀⡍⠙⠓⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⡟⠉⠛⠳⠶⢤⣤⣀⢾⠀⠀⠳⢤⣀⠀⢿⡫⢿⡉⠉⣟⣿⣸⠃⣿⠀⡇⠀⠀⣿⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣼⠃⠀⠀⠀⠀⠀⢀⡝⠙⠲⣦⡀⠀⢈⡷⠚⠁⠀⠳⣤⣈⣉⣩⢷⣿⢤⣇⠀⢠⡟⠚⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿⠀⠀⠀⠀⢀⣠⠞⢁⣀⣤⣾⡆⠀⣸⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠛⢷⡍⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢿⣄⠀⠀⠀⠛⠙⠋⠉⠉⠀⢸⡇⠀⢸⣧⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⣠⣼⣯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣹⣷⣤⣄⡀⠀⠀⠀⠀⠀⢸⡟⢠⡟⠁⠀⣯⣟⣩⡗⠀⠀⠀⡴⢋⣥⣌⣁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣷⠶⣤⣤⣀⣼⠃⣏⣹⠇⠀⠈⠉⠉⠀⠀⠀⣼⠁⡟⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⢭⡁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠘⣳⠄⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡏⠀⠀⣠⠖⣳⣦⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢧⣼⡇⠀⠀⢻⣞⣁⡼⠃⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⣬⣙⡓⠶⢤⣌⣉⣀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⠯⣽⣧⣴⣟⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣀⣀⣀⢀⡀⣀⣀⣀⢀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠶⠆⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⢀⡠⠤⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⢒⡾⠘⠧⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⠫⠑⠀⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣠⣤⠶⠞⠒⠛⠛⠛⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠍⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠦⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣴⠟⠁⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣇⠈⠉⠒⠒⠒⠒⠒⠒⠒⠋⠉⠁⣠⠞⠁⠀⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⣀⣀⣀⣀⡀⠤⣤⡾⠋⠀⠀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣫⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣤⡄⢀⣀⣀⣀⣀⣤⣾⠿⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠁








⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣅⠀⠀⠀⠀⠀⠀⠐⠫⠟⠉⠙⣟⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠈⠁⢠⡃⠀⠀⠳⣍⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠘⢸⡆⠀⢶⣿⠀⠉⠓⠺⢷⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢶⣤⣶⣺⣷⣤⠘⣆⣄⠀⠄⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⣤⣾⡉⠙⢋⡙⠻⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣿⣦⣀⠉⠻⢿⣿⡆⣀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⣿⣶⣾⣿⣿⡏⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⣿⣿⣧⠠⠒⠂⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⣤⡶⠂⠀⠙⣀⠀⢘⣁⠂⠈⠛⠀⠀⠀⠀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣆⠀⠸⠁⠀⢀⠀⠂⢹⡇
⠀⠀⠀⢠⣾⣿⠟⠀⠀⠀⠈⠛⠃⠈⢋⣤⡀⢸⣶⣶⣀⣤⡼⣿⣿⡿⣿⠿⠟⠙⠿⠋⠀⠀⠐⠚⠛⠛⠳⢾⡇
⠀⠀⠀⢘⣿⡗⠀⠀⠀⠀⠀⠐⠂⢠⠄⠈⠀⠘⣿⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠞⢷⡀
⠀⠀⠀⠀⢻⣿⠆⠀⠀⠀⠀⠀⠛⣤⡀⠀⢀⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡐⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣍⡙⠳⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠙⠻⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣦⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠉⠛⠻⠿⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡆⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠉⠉⠁⠀⠀⠉⠉⠉⠙⠛⠛⠿⠿⡶⢤⣀⠀⠀⠀⠀⠀⠀⠴⠶⠒⠛⠋⠉⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣿⠛⠛⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡀⠈⣿⡏⠛⠲⢤⣀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⣺⠋⣁⠀⢸⡇⠀⠈⢷⢍⠛⣦⣄⠀⠀⠘⢿⣿⣷⣾⣿⡿⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢗⣟⡉⠀⣼⠃⠐⣦⡘⡌⢣⡝⢌⢿⡦⣄⠈⠳⣮⣿⣿⡄⠀⠀⠈⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⣿⠥⠀⣿⠠⡤⠼⡏⢿⠈⠃⠘⣾⢦⡽⡷⣦⣈⠻⣿⣷⣄⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢞⠁⠀⢀⡿⠀⠘⢆⡀⠀⠀⠀⠀⡿⢄⠛⠸⠇⣿⠓⠮⣿⣿⣷⣄⠀⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣻⣇⠀⣸⡇⠀⣠⡤⠛⠒⠀⠀⢀⣇⡬⠷⣄⠀⡟⠀⠀⠀⠉⠙⠛⠿⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣽⣏⢀⡿⠀⢠⣞⣉⠀⠀⠀⣇⣼⢷⡞⠃⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡍⣼⠁⠀⢠⠞⠁⠀⣰⡸⠙⣹⠋⢀⣤⡘⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⢻⣄⡿⠀⢰⠃⢀⠜⣣⠻⢃⣴⣿⠖⠋⣨⡾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢹⡇⠀⣧⠖⠁⠘⢣⡶⠛⠁⣧⠀⣿⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⢸⣇⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣾⠟⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠻⣿⣄⣀⣀⣀⣼⡿⠁⠀⠀⠀⢻⡀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡏⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⠉⢧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠦⢤⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣳⠤⣄⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠓⠺⢭⣛⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣤⣤⣤⣤⣀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⡀⠈⢹⣶⣍⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠒⣚⣋⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣾⣿⣿⠷⣌⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⢻⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠓⠦⣄⣉⠛⠛⠓⠒⠶⠖⠒⠒⠛⠛⠛⢉⣉⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠒⠚⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⠆⠀⠀⠀⠀⢴⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣄⢀⠐⡀⠀⠀⠉⠉⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣴⣴⣦⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣾⣿⣿⣿⣿⣿⡿⣿⣛⣛⣛⡛⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣏⣹⡏⠉⡟⢻⣦⠀⠙⡿⠿⢭⣟⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣿⢿⡟⢿⡇⢀⡟⠁⠸⠀⠀⠁⠐⠸⠎⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣾⣿⢻⣎⢻⣏⣧⣾⠁⠰⠀⠀⣀⣠⣴⠶⡟⠛⢹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣜⣧⣷⣿⣿⣧⡛⠛⣛⣫⣭⡶⡾⠁⣠⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡷⠀⠘⢥⣤⣼⠀⣈⠉⣀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣻⣿⣿⣿⣿⣿⣧⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠄⠺⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣴⣶⠎⠐⣿⡀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠻⠿⠻⣿⣿⣿⣿⣿⡇⠀⢰⣿⡛⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠆⠀⣿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡷⠦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢇⢀⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣮⣿⣽⣿⡝⡿⠛⠟⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣻⣿⣷⠀⠀⠀⠀⣴⣿⣿⡏⡯⠍⠧⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣮⣭⣿⣿⣿⡇⠀⠀⠠⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠋⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣷⣤⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⣤⣤⡸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣴⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢛⣛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠟⢿⣿⣷⣦⣤⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣀⠀⠈⠓⠀⣉⠻⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⣤⢀⣤⣶⣶⣶⣿⣣⣾⣿⡧⠀⠀⢸⠃⠀⢹⣆
⠀⠀⠀⠀⠀⠀⣴⣘⠷⢎⠸⠋⠀⠀⠺⢯⣿⡿⣿⣿⣿⣿⠀⡄⠘⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠉⠉⣼⣿⡇⠀⠀⠀⠀⠀⠈⠙⢮⡝⢿⣿⣾⡇⠀⠀⡐⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠢⠙⠿⢁⣤⡴⠃⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣀⣀⠜⢸⡇⠀⠀⠀⠀⠀⠀⢰⡷⠛⠉⠀⠀⢀⣴⡇⢘⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠁⣩⠟⠁⣀⣨⣧⠤⠔⠒⠉⢀⡔⠁⠀⠀⣠⠤⠞⠀⠠⣴⠺⠿⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠈⠉⠁⠀⣿⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⢰⠟⠂⠀⠻⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠘⣧⣹⣿⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠉⢻⠀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠶⠿⠶⠆⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢀⣾⣿⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡠⠚⠛⠛⠒⠒⠒⠂⠀⢻⡀⠀⠀⠀⢀⣠⣴⠞⠋⠀⢠⠂⠀⢸⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣴⠾⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢰⠋⠉⡄⠀⠀⢰⠏⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⢰⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢸⡀⠀⠹⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⢸⣇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣧⠀⠀⠹⣆⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠿⠟⠀⠀⠀⢹⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣷⣦⡀⣀⠀⠀⠀⢀⣴⣾⡿⠋⠁⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣷⡀⢀⣾⣟⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠟⠋⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⣠⠤⠒⠂⠠⢤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡛⠀⠀⠉⢳⠐⠞⢥⡈⠀⠀⣀⠀⡿⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠢⢄⠀⠑⢳⣞⠳⡈⠑⠠⡀⣰⠃⣀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⠚⠁⢀⣠⠵⠖⠺⣄⠑⠛⠂⢄⠀⠉⠢⣄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢈⣶⣿⠀⠀⠀⠀⡜⠀⢀⠄⠂⠁⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡬⠷⠤⠴⠛⢁⠁⠀⢀⡠⠊⠀⠀⡠⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠀⠀⠀⠀⠀⣃⣀⣀⠀⡀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠈⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠢⣀⣀⣀⡬⠔⠊⠁⠀⠀⠀⠀⠠⣀⠀⠀⠀⡴⠋⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣠⠶⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣀⡀⣀⠴⠊⢹⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⡆⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠑⠠⣀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣴⢶⣤⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠤⠃⠨⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠑⠀⠀⣀⡤⠂⠁⠀⡀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⢀⢤⣤⢤⠄⡀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠠⠀⠀⠀⢠⡈⢷⡞⠁⠢⢄⣀⡔⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⢨⠼⣇⢽⠂⠀⠀⠀⠀⠀⡀⢀⣠⣀⡀⠂⠘⠻⠦⠼⠱⠏⠈⣠⢟⣒⣂⣭⣡⡖⢠⣀⣠⡿⢣
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠐⠧⣌⢠⠁⣟⣒⣐⣒⣒⣒⣶⣂⡐⠒⠢⠰⠤⠿⣴⡶⣾⠤⣴⠛⠈⠗⢲⠀⢀⣀⣀⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠉⣵⠿⡀⠻⡦⣙⣯⢅⢠⠀⠋⠬⠉⠵⢶⣲⡦⣴⢇⣀⠁⠀⠀⠀⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠉⢒⠓⣧⠁⢄⣀⢻⣅⡚⠌⣀⣀⠀⠒⠒⠁⢱⠀⠀⠀⠁⣠⠼⠴⠂⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡝⠀⢿⡿⠉⡆⠰⣅⠀⢎⡀⠀⠀⠀⠀⠈⠉⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠙⠂⠠⢄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢦⠠⡺⣍⡄⣿⢁⢠⣁⠠⠀⠘⠗⠀⠈⠙⠂⠀⠐⠒⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠧⣠⡎⠓⠒⠃⠈⠁⢀⠀⠀⠀⠀⠸⢤⣀⣀⣀⣀⣠⡴⢾⣭⠉⠫⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠍⢀⡐⣒⡒⠈⣩⠆⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⢹⣀⣜⢠⡀⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠊⠙⣇⠤⣺⠇⢸⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠈⠓⠤⣤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠪⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠆⠂⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⣲⡶⠶⠚⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠤⠤⠤⠤⠄⣀⠀⣀⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠦⣀⡀⠀⠀⢠⣿⣿⡿⠿⠿⠓⠂⠀⠐⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢭⣁⡀⠈⠉⠒⠀⠀⠀⠀⠀⣀⡈⠥⠖⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡿⡿⣿⣿⣛⠋⢫⣥⣾⣟⠂⠘⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠘⠙⠋⠻⠀⠈⠛⠉⠁⠀⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⣶⠆⠀⠀⢀⠀⠀⢤⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⠐⡟⠇⠸⠛⠶⠞⠃⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣶⠿⠄⠷⣦⠘⠀⡀⠈⠁⠀⠀⣀⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⠈⠙⠲⡤⠀⢻⣄⠀⠀⠀⠀⠀⡼⢿⡵⠃⢀⡶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣿⠁⠈⠙⣷⠶⠤⠐⠋⣿⣷⣶⣶⣶⠊⠁⣼⣷⡐⠯⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠐⣿⠄⠐⠲⢾⡦⠠⠔⢺⣿⣿⣿⣄⡀⠀⢀⡿⠉⣿⣷⢈⠆
⠀⠀⠀⠀⠀⠀⠐⠃⠀⢸⠂⢻⣆⠀⠠⣼⣦⡠⠤⣾⣿⣿⣿⠛⠉⠑⠚⢃⣼⡿⣿⠏⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⣄⠀⢘⣤⠠⠚⠙⣿⣷⣹⣷⣦⠀⣔⣼⣿⠁⠟⠀⠀⢠⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⠀⠀⠘⠋⠉⠉⠀⠀⠀⠀⠈⠻⣿⣟⢿⢹⣿⣿⣿⠀⠀⠀⠀⠀⠃⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⢈⣱⣿⡟⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⡄⠀⢟⢿⣿⡄⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣹⠟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⣤⣄⠀⠸⣿⣿⢷⢀⣴⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡏⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⠻⣧⡄⠀⢀⢴⣫⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣽⣤⠎⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣵⣿⠀⠀⠀⢀⣰⣞⠉⠉⣀⣿⡑⠀⣀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠴⠞⢩⣤⠽⠛⠋⠁⠈⣳⡞⠁⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠻⡟⠉⣿⣿⡧⠞⣠⠶⠊⠁⠀⠀⠀⠀⢸⠛⣷⡐⠶⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⣉⠢⣜⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⢳⡱⡄⢀⣀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⠼⣿⣿⣿⣿⣿⣿⡋⠙⠻⠾⠿⠦⠤⠤⠤⠤⠤⢤⣤⣞⣉⣧⣀⣝⡿⢭⣁⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠒⠒⣶⡟⠻⠿⢶⣿⣿⣿⢳⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠛⢼⣏⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⣿⣿⣿⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⡽⠉⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣰⠁⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣾⣯⣋⣞⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⢀⡀⠀⠀⢀⣠⡾⠁⠀⣿⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⠿⠛⠷⠟⠀⠀⠀⠹⣇⠀⠀⣤⠀⠀⠀⠀⠠⣤⣀⢿⡃
⣘⣏⠛⠋⠈⠁⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣿⡀⠀⠀⠀⠀⠀⠙⢷⣽⣦
⠘⠧⣄⣠⠀⣠⣶⠊⠈⠀⢹⡄⠀⠀⠀⠀⣼⠛⠻⣿⡉⠛⢿⣿⠒⠶⠤⠤⠤⠶⠒⠛⠋⠉⠛⠓⠒⠓⢦⣄⣀⡀⠈⠉⠓⢦
⠀⠀⠙⠛⠛⠁⠁⠀⠀⠀⠀⣷⠀⠀⢀⡾⠁⠀⠀⠀⠻⡄⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡈⠉⣹⢶⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⡟⠀⠀⠀⠀⠀⠀⣷⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣩⡏⠀⠙⢁⡼⠻⣄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠎⠀⠀⢹⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡙⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣉⣙⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣉⢹⣆
⡶⠒⠲⡶⠶⡶⡶⡶⢶⡶⣶⡶⡶⣿⢿⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠝⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡮⡳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣽⠿⣝⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠚⠻⢕⣌⡓⢤⡀⠀⠀⠀⢶⠤⣀⠔⠉⠉⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⣀⠿⣮⠶⠭⣧⣀⠀⠀⢣⡄⠢⡀⠀⡠⠖⠋⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠀⠀⠹⠀⠀⠀⠈⣄⡄⠻⠀⢷⣄⠀⠉⠢⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⠤⠳⢶⣤⠔⠛⣯⣁⠀⣈⡿⢳⣦⣀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣛⣲⣤⣄⣧⣷⡀⣈⣻⣷⣭⣴⢼⡛⠉⠉⠁⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⠠⠽⣿⣿⠿⣛⡻⣄⠐⢄⠀⠉⠳⠘⢮⡉⠀⠀⢇
⠀⠀⠤⠤⢤⣀⣀⣀⣀⣤⠤⠤⠤⠤⠶⠤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠈⠀⠀⠀⠀⠉⠢⢳⠈⠀⠀⠀⠀⠀⠳⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⡒⠒⠚⠓⢤⡈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠓⠦⣉⠙⠲⠦⣭⣉⣙⠛⠛⠒⠒⠒⠂⠀⠀⣂⣉⣀⣀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠈⠑⠲⢤⣀⠈⠹⣿⣿⣷⣶⣶⡀⠀⠀⠀⠘⡀⠀⠄⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⡀⠀⠀⠀⠀⢹⢦⠀⠀⠀⡀⠀⠀⠀⣿⠣⣀⡘⣆⠰⡄⠈⠙⠀⣿⠉⢿⡏⢿⣿⡶⢄⠀⠁⠐⠀⠘⡀⠀⠀⣧
⠰⠀⠂⢄⠀⡿⣀⠀⢹⠱⡙⢂⢠⣠⠀⢹⠀⢣⢠⡀⣿⣆⠈⠂⢸⠀⠙⠹⣿⢆⠃⠀⠀⣠⠟⠀⠀⢿⠈⢳⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠉⠀⠉⠁⠂⠀⠉⠀⠃⠁⢐⡼⠀⠀⠷⠡⡿⠘⠇⠀⢺⠀⠀⠀⠁⠈⠃⠀⠀⣿⡀⠀⠀⠈⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⡠⠔⠚⡆⣀⡀⠀⠀⠀⠀⠀⣈⠦⢤⡀
⠀⡻⠋⢀⠔⠉⠁⢱⠀⠀⠀⠀⡼⠁⠀⠀⢹
⡼⡶⠀⠈⠑⣄⠀⠸⡀⠰⡤⠤⢷⡂⠀⣀⡼
⢰⢀⠔⢳⡤⡹⡄⠀⠑⢼⡁⠀⢠⣽⠉⠁
⠈⠋⠀⠀⠀⠙⣿⡆⠀⠀⠙⠲⣶⠎⣀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣑⠆
⠀⠀⠀⠀⢀⡴⣚⡻⠄⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠒⢲⠒⠋⠉⠁
⠀⠀⠀⠀⢈⠘⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠑⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⣀⡤⠤⠤⠴⡇⠀⠀⠀⠀⡠⠆⢡⡵⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⣺⠭⠭⠤⠄⠀⣀⣸⠝⠂⡒⠈⠉⢉⣉⡉⠉⡇
⠀⠀⠙⠲⣿⡶⠤⣇⣀⠀⠀⠀⠑⠜⠁⠀⠀⠀⠀⠀⣇⠀⢀⣀⣤⠀⠀⠀⠀⣎⣠⣌⡷⠒⠲⡒⠚⠁⠀⠀⠀⢃⠀⠙⢆⡉⠁⠀⡇
⠀⠀⠀⠀⠓⢤⣄⡀⠈⠉⠉⠁⠒⠒⠠⠤⠤⢤⣀⡀⡿⠿⠁⠀⣏⡉⠭⠤⠬⢤⣿⣿⣋⣲⠎⠀⠀⠀⠀⠀⠀⢸⠀⠀⣫⡍⠀⠀⠇
⠀⠀⠀⠀⠀⢸⠀⠈⠁⠒⠠⢄⡀⠀⠀⠀⠀⡰⠀⣙⣿⢳⡄⣀⡶⠀⠀⠀⠀⠀⠛⠛⠧⠜⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⢭⡀⠀⠀⢸
⠀⠀⠀⠀⣸⠍⠀⠂⠀⠸⡦⠀⠈⠉⠒⠤⠴⠁⠀⠈⠁⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠂⠉
⠀⠀⠀⠸⣁⣀⡘⠦⡤⡤⠗⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠤⠤⠤⠤⠠⠤
⠀⠀⠀⠀⢳⠀⣉⣉⡇⣏⠉⠉⠉⠉⠉⠉⠉⠉⢉⠇⡞⠀⢠⠁
⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⡘⢀⠇⠀⡌
⠀⠀⠀⢀⣎⠈⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠶⢷⣞⣛⣡⠷⠤⠄⡀
⠀⠀⠸⠥⠷⠛⠓⠒⠒⠶⠦⠤⠤⠤⠤⠶⠖⠒⠈⠁⠀⠈⠉⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣀⡤⠤⠔⠒⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⣆⣀⣀⣠⠤⠖⠊⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢀⡤⣶⠯⠉⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⣄⣀
⠀⠀⠀⢀⠔⡹⠊⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣈⣙⣶⣤⣀
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⣶⡿⠿⠛⠉⠀⠀⠀⠀⠊⠭⢓⠦⣄
⠀⣰⢃⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢀⡇⡈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡭⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⢸⠃⠁⠀⠀⠀⠀⠀⠀⠀⠚⠉⠓⠦⠬⣍⣉⣉⣉⣉⣉⡽⢋⠔⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠒⠒⠛⠛⠛⠛⠛⠋⠁
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠁⠀⠀⠀⠀⣀⣠⠴⠚⠉⠁
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡫⠚⠁⠀⠀⣀⡤⠶⠛⠉
⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⠕⠋⢀⣠⠤⠖⠋⠁
⠀⠀⠈⠳⣄⡀⠀⠀⠀⣀⣀⠠⢖⣋⡬⠴⠒⠋⠉
⠀⠀⠀⠀⠈⠙⠓⠲⠖⠛⠛⠉⠉
⠀⠄⠠
⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣤⣤⣶⣶⣶⣶⣶⣦⣤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣛⣯⠭⠭⠤⣄⡉⠙⢿⣿⣿⣧⡉⠛⢿⣿⣿⣿⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡯⠗⣋⣩⣤⣤⣤⡀⠀⠈⣷⠀⢸⣿⣿⣿⣷⠀⠈⣿⣿⣿⡇⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⣫⣵⡾⠟⠛⠛⢻⣿⣿⡇⠀⣰⠇⠀⣼⣿⣿⣿⡿⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣺⠷⣫⣶⠟⠋⠁⠀⠀⠀⣠⣿⣿⠟⢀⡴⠃⢀⣾⣿⣿⣿⣿⠃⠀⣼⣿⣿⠏⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢞⣵⡿⠋⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⣠⠞⠁⣰⣿⣿⣿⣿⡿⠃⢀⣼⣿⣿⠏⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⢟⣴⡿⠋⠀⠀⠀⠀⣠⣴⣿⣿⠟⢉⡴⠞⠁⣠⣾⣿⣿⣿⣿⠟⠀⣠⣾⣿⡿⠁⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⣴⢋⡞⢁⣾⣿⣅⣀⣠⣤⣶⣿⣿⠿⢋⣡⠖⠋⢀⣴⣾⣿⣿⣿⣿⠟⠁⣠⣾⣿⡿⠋⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢠⣿⢣⡟⠀⢸⣿⣿⣿⣿⣿⠿⠛⣉⣤⠖⠋⢁⣠⣾⣿⣿⣿⣿⣿⠟⠁⣠⣶⣿⡿⠋⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⢠⣿⡇⠸⣇⠀⠀⠈⠉⣉⣠⡤⠶⠛⠉⣀⣤⣾⣿⣿⣿⣿⣿⡿⠋⢁⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⢀⣿⣿⣧⠀⠈⠙⠛⠛⠉⠉⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⠛⠁⣠⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⣠⠾⠃
⠀⠀⢀⣾⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⣠⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⣾⣿⡁⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⣀⣤⣶⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⢸⣿⣿⣧⡀⠈⠉⠛⠛⠛⠛⠋⠉⠁⣀⣠⣤⣶⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋
⠀⠘⣿⣿⣿⣿⣶⣦⣤⣤⣤⣶⣶⣾⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋
⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠁
⠀⠀⠀⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠉⠁
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠙⠛⠛⠿⠛⠛⠛⠻⣿⣿⣉⠛⢛⢛⣿⢿⣿⣿⡟⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠻⠉⣿⣿⣿⡏⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣶⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣄⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠳⢤⣌⢻⣿⠻⡶⠋⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠖⣀⡄⠉⠛⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠡⣾⣤⡾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠨⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⣿⣿⠛⠛⠛⠛⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠉⠓⣶⣦⣭⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⣀⠀⢈⣻⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢈⣿⣷⣿⣿⣿⡿⠀⠀⣲⣦⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆⣤⣬⡯⠟⢁⣾⣿⣷⣾⣿⣿⣿⣿⠇⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣬⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢠⣤⣿⣿⢷⣽⣿⣿⠟⣿⣿⣿⣿⠿⠛⢿⣿⣆⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⣠⣿⣿⠯⢶⡿⢋⣤⣾⣿⣿⣿⣿⡦⣰⣾⣿⣿⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⡿⠛⢿⡏⢠⣟⣴⣿⣿⣟⣿⣿⣿⣿⣷⣽⣿⣿⣿⡿⠆⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⠀⣸⡿⣻⣿⣿⣿⣿⣭⠙⣻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣿⣿⣿⣟⣿⣟⣿⣿⣿⣿⡿⠉⠿⠛⠿⣿⣿⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⣿⣿⡿⢿⣽⣭⣿⣿⣿⣿⠇⠀⠘⠃⠈⢹⣿⠇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣇⠀⠀⠁⠀⣾⣿⠂⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣏⣿⣿⣦⠀⠀⢶⡾⠃⢀⠖⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠰⣿⣿⣷⣶⣾⣿⣿⣯⠸⣿⣿⣿⣾⣿⡏⢰⣶⣿⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣉⣿⡿⣯⣉⣿⣿⣿⣦⣼⣷⣼⣿⡿⢀⣼⣿⠟⠁⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣋⣦⣿⣿⡛⢿⣿⣿⣿⣻⣿⠃⠀⠀⠻⣿⣧⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⠿⢿⣷⣾⣿⣿⣽⣿⣿⣿⣴⣿⣧⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢀⣿⣿⡛⣿⡿⡷⠟⠻⣿⣿⣿⡇⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⣈⣷⣾⣿⠋⠀⠈⢳⠀⠀⠀⠸⣿⣿⣷⡄⠘⣿⣆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⣿⡇⢻⣷⠀⠙⢿⣦⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣺⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣿⡇⠈⢿⣧⠀⠘⣿⡆⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠁⠀⠀⠀⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣧⠀⠘⣿⡆⠀⠘⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⣿⣿⣆⠀⢹⣿⡀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠏⣬⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⢻⣧⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⣐⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡀⠀⢿⣇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠈⢿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣚⣋⣤⣿⣿⣿⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣿⣷⣤⡤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣿⣿⠋⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣨⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣋⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣃⣽⣯⣹⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡶⠶⢦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠋⠀⠀⠀⠈⠉⠉⠻⠿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⠿⠿⢿⣿⣿⣷⣶⣿⣿⣿⣇⠀⠀⠀⣶⣀⣀⣴⣶⣶⣿⠟⠛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠈⠉⠓⠮⣙⢿⣦⡀⠀⠈⠉⠛⠋⠁⠀⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢝⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠑⣽⣷⡀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣹⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢁⠈⣿⡿⢷⣤⡀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⠄⢸⡇⠀⠙⢿⣷⣤⡀⠀⠈⠿⣿⣿⣶⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡷⠂⣸⡇⠐⢶⢶⡟⢿⣿⣷⣄⡀⠙⣿⣿⣿⡟⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠁⣿⠃⠶⢬⡇⠻⠀⢻⣿⣿⢿⣶⣌⡳⣿⣱⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣢⢠⣿⠀⠈⢢⣁⣀⠀⢸⣿⣟⠺⣿⣿⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢷⡍⣼⡇⠠⢞⡻⠆⠀⢠⣸⡟⠛⠂⣿⡿⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢠⡿⠀⠀⣛⡽⡃⢰⣾⣿⢧⢰⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⣼⡇⠀⡴⢁⣼⡟⠀⣼⣿⠸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣿⠀⠰⠗⠋⠀⣴⣾⠿⢻⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣼⣿⠀⠀⠀⠀⠀⣼⡗⠀⢸⡿⡿⠁⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⢿⣷⣤⣀⣤⣾⡿⠁⠀⠀⢿⡀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠋⠉⠉⠉⠉⠀⠀⠀⠀⠈⠳⣄⠀⠀⢻⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣤⡀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣦⣀⠉⠿⠉⢸⣿⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣫⢷⣄⡀⠀⠀⠘⠋⠻⣦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣝⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣦⣤⣄⣀⣀⣈⠙⠛⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣷⣌⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢹⣷⣤⣶⣿⣛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠶⣍⡓⠦⢤⣀⣀⡀⠀⢀⣀⣀⣠⣤⣤⠶⣿⣿⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⠤⠭⢭⣭⡭⠤⠶⠶⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣷⣄⡀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣯⣤⣽⣿⣿⢂⡄⣻⣿⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣯⣭⣍⣉⢉⣩⣭⣶⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣿⣿⣿⣿⢸⣿⡟⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⣿⣿⣿⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⡿⠋⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣘⣆⣀⣤⣤⣴⠶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠛⠛⠛⠻⣿⣷⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠋⠀⣿⠋⠛⠀⠀⣰⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣶⣴⣿⣦⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⠆⠀⠐⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽⣿⣿⣿⣿⠟⠙⢤⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⣄⠀⠰⠦⠤⠤⠤⡔⠋⠁⠈⠉⠉⠉⢻⡿⠁⠀⠀⠀⠉⠻⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣤⠬⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠗⠦⣬⣁⠀⢴⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠛⠓⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⣞⡄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣡⣄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⡜⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢋⡿⢻⣿⡿⠀⠀⠨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠃⠀⢻⣿⢿⠒⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠟⠳⣦⣤⣄⣀⠀⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢻⣿⡇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢼⡛⡆⠀⠀⠀⠀⠉⠀⠀⠴⢢⣄⣆⣸⡇⡀⠀⠀⠀⣀⡀⢦⣴⣾⡇⠀⠀⠘⣿⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣉⠛⠿⣇⣿⣴⣾⣦⣿⠹⠎⣿⣾⠁⠀⠀⠀⢿⣗⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣾⡟⠛⣁⣠⣠⣤⣤⣬⣿⣷⣄⠀⠀⠀⠘⠋⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠁⠉⠁⠀⠛⠛⠛⠛⠛⠁⢿⣿⣿⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⣿⠻⠀⠀⠀⠀⢁⣬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣸⡇⠀⠀⠀⠐⣼⣾⡇⠀⠀⠀⠀⢸⡏⢸⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡷⠀⠀⠀⠀⠈⠹⣷⡀⠀⠀⠀⠈⠃⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠀⠀⠀⢸⡏⡗⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢸⡇⢸⡄⠀⠀⠀⠀⠀⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣾⠃⠀⡕⠀⡀⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⡀⠀⠀⣹⠆⠀⢷⣀⠁⠀⠀⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⢴⣶⣾⡇⠀⠀⠘⢧⠖⠀⢀⠀⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⣿⠀⠀⠀⠀⠈⢷⣭⡥⢶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⣿⣿⠀⠀⠀⠀⠀⠀⢻⣷⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣇⣿⣿⠀⣠⡀⠀⠀⠀⠈⣿⡈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣾⠿⣿⣿⣿⣿⣿⣷⠀⡀⠀⠀⢸⣧⢹⣿⣇⢀⣶⣤⡆⠀⣀⢀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢰⣿⣇⠰⣟⣵⣿⠟⠋⠛⠁⠀⠀⠀⠀⠀⢸⠟⣿⣿⣿⣾⣿⣿⣷⣿⠟⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣴⠈⠀⢺⣾⣿⠿⣾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⣀⣾⣧⣼⢿⣽⣿⣿⠿⠛⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⡿⠋⣠⣿⡿⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡟⢹⡿⢟⣿⣿⠟⠁



⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⡂
⠀⡴⠴⢑⢠⠀⡊⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣩⣭⣭⣍⡛⢿⣦
⠀⢀⣁⣐⠈⢀⡇⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢡⣿⠋⣥⡌⠹⣷⠘⣿⡇
⠀⠀⠘⢃⣀⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⢿⣤⣉⣡⡼⠏⣰⣿⢁
⠀⠀⠲⠬⠗⠚⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣬⣭⣭⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣩⡅
⠀⠀⢰⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢿⡇⢹
⠀⠀⡜⠀⠀⠀⠀⢈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠳⣄⠀⠀⠀⠀⠀⠀⣸⠁⢸⡇⢸⡇
⠀⢸⠀⠀⣀⣀⣠⠆⠀⠀⠉⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⠀⠀⠀⠀⠀⠈⢠⡏⠀⢸⡇⠀⣧
⠀⠀⠓⠦⠬⠥⠥⠤⢄⣀⣀⣀⡖⠁⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠃⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣤⣤⣌⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⢠⡿⣫⣭⣻⣧⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⢀⣿⣻⢋⢹⡇⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠄⠀⠰⡴⠋
⢸⡿⣏⠿⢻⣧⡟⠀⡋⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡘⡠
⢸⣧⠻⠶⢟⡿⠁⠸⡇⠀⠀⠀⠀⠈⠂⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶
⠀⢻⣶⣶⣿⡁⢰⡇⢻⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠘⠷⢬⣥⡞⢻⠙⡇⠸⡄⠀⠀⠀⠀⠀⢰⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⢰⠁⠸⢰⡇⠀⣇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀
⠀⠀⠀⡞⠀⠀⢸⡇⠀⢻⠀⠀⠀⠀⠀⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⠃⠀⠀⢸⡇⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡟⠀⠀⠀⣼⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⡈⢆⠀⠀⠀⡀⠀⠀⠀⠀⡄
⠀⠘⠁⠀⠠⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠘⠁⠀⠀⠀⠀⠁⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⠓⠶⠤⣤⣤⣤⣤⣤⠤⠶⡚⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡰⠂⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⣆⠀⠀⠙⠳⠤⠤⠤⠖⠋⠀⠀⡜⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⣷
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⡞⢯⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣀⡴⠞⢉⡟
⠹⡈⢧⠀⠈⡳⣤⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣤⣴⣾⠩⠀⠀⡞⡸
⠀⢳⡘⢧⠀⠀⠈⠉⠉⠉⠉⢹⡷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢻⠉⠉⠉⠁⠀⠀⠀⠀⡼⣱⠃
⠀⠀⠳⡌⢧⠀⠀⠀⠀⠀⠀⠘⠳⣄⡉⠓⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠚⠉⣀⣤⠞⠃⠀⠀⢀⠀⠀⠀⡜⡵⠃
⠀⠀⠀⠹⣌⢧⠀⠀⠀⠀⢀⠀⠀⠀⠉⠛⠲⠦⣤⣈⣉⡉⠉⠛⠛⠛⠛⠛⠉⠉⣉⣁⣠⡤⠴⠖⠛⠉⠀⠀⠀⠀⡴⠃⠀⢀⣞⡼⠁
⠀⠀⠀⠀⠘⢦⡳⡄⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⣫⠞
⠀⠀⠀⠀⠀⠀⠳⣝⢦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⣞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⣀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⣠⢴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡑⠦⣄⡀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡠⠖⣫⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣍⣓⠲⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠴⢒⣋⣥⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡆⣀⣀⣀⣀⣀⣀⣀⣠⣴⣒⣒⣻⣆⡐⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠉⠉⠉⠉⠋⠙⠛⠛⠛⠛⠿⢿⣿⣿⣿⡟⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡃⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠁⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡆⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣀⣀⣠⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣿⣿⣇⠀⠻⢿⣿⣿⠿⠿⠛⠛⠋⠉⠉⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠟⠛⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠟

⣶⣶⠶⣶⠶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⢀⡴⢒⣶⠶⣤⡀
⠀⠀⠀⠀⠀⣠⠏⣰⠟⢡⡭⣬⢷⡄
⠀⠀⠀⠀⣰⠏⣼⠇⡴⠁⠀⠈⢷⣿
⠀⠀⠀⢠⡟⢰⡯⡞⢠⠖⠒⢦⠘⣿⡇
⠀⠀⠀⣼⢀⡿⢸⢡⢯⡞⠙⢶⣧⢻⣷
⠀⠀⠀⡏⢸⡧⡞⣾⡟⣴⣿⡆⣿⢸⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⡤⢤⠤⣶⠂
⠀⠀⢸⡇⢸⠒⡇⣷⣿⡿⣿⣷⣿⣼⣿⡦⠤⠤⠤⠴⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠻⣿⢻⡟⠛⡻⣇
⠀⠀⢸⡇⢸⡽⡇⣿⡟⣟⡟⡏⣿⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⡇⢸⡗⡇⣗⣇⠙⠚⣸⡏⣸⣿
⠀⠀⠀⢿⠘⣧⢻⠸⣎⠢⠴⡿⢡⢿⡇
⠀⠀⠀⢸⡇⢿⢪⣆⠈⠓⠋⢁⡎⣾⠁
⠀⠀⠀⠀⢳⡘⣇⠈⠢⣤⣴⣛⡿⠃
⠀⠀⠀⠀⠀⠳⣜⣷⣔⣊⣥⠟
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡒⠉⠉⠙⠒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠯⢀⡈⠙⠦⣀⡄⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⣤⣄⡀⠀⠀⠉⠂⢄⠁⠂⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⠉⠀⠁⠢⢄⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣦⢂⣀⠀⠀⠀⠉⠒⢄⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠉⠁⠈⠋⠁⠀⠀⠀⠀⠀⠹⣦⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⡀⠀⠀⠀⠠⠀⠀⡀⠀⢈⠉⢣⠼⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠉⠛⠻⠿⢷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠤⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⡠⠀⠀⢇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⣀⡤⠖⡻⠁⠱⠀⢰⢶⠀⠁⠀⠀⡠⠊⠀⠐⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠃
⠀⣀⠤⠒⠉⠁⠀⠈⠑⠀⠀⢀⠋⠈⡇⠀⠀⠈⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠄⠆
⠀⢆⠀⠀⠀⠀⠄⢲⠀⠀⠈⠀⠀⠀⠀⠐⠒⠒⠀⢠⢄⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠍⡘⠨
⠀⠀⠣⡀⠀⢀⡴⠀⠀⠀⠀⠀⠀⢈⠀⠐⠀⠐⠀⢰⠀⠉⠢⢄⡀⠀⠀⠀⠀⠑⣤⡀⠀⠀⡸⢰⢃⠃
⠀⠀⠀⠈⣀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠂⠀⠀⢣⡀⠀⠀⠉⠢⣄⠀⠀⢀⠆⢹⢀⠊⠑⢆⡌
⡦⠓⠲⠞⢹⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠑⢦⡊⢀⡃⠀⠀⢤⡜⠁
⠑⠄⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⢯⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⠃⡼⠄⢍⠁
⠀⠀⠀⠀⡀⠈⠳⠶⢴⡦⠄⠠⠤⠶⠿⠛⠋⠁⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠰⠸
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠔⠀⠹⠀⢣⠃
⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠄⢰⠃⠀⠦⡀
⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⡜⠀⠉
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠂⠄⢀⣀⣀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠐⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣀⠀⠀⢀⠊
⠀⠀⢀⣀⠤⠄⠒⠘⠂⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠑⣤⠆
⠀⢮⣥⣤⣤⣤⣤⡤⠤⠴⠶⠂⠴⠤⠤⠤⠤⠤⠴⠞⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠞⠛⠛⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣤⣄⣀⣀⠀⠀⠀⠈⠙⠻⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣼⣿⣿⠷⠶⣽⣿⠛⠻⠿⣷⣶⡞⠁⠀⠀⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠈⣿⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⠀⣀⡀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⢀⣴⡾⠋⠉⠙⣻⡷⠚⠛⠛⠛⠉⠙⠙⠛⠒⠲⠤⣄⣀⠹⣿⡟⠁⠀⠀⠀⠀⠘⢿⣆
⠀⠀⣠⡟⠁⠀⢀⣠⣾⣯⣤⠄⠀⠀⠀⢀⣀⣠⣴⣆⣠⣤⣤⣬⣷⣿⣿⣦⣤⣤⣤⣄⣀⡈⢻⣧⡀
⠀⣼⣯⣴⡶⠿⠛⠛⢉⣿⣷⣶⠶⢿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣿⣿⣷⣄
⠀⠉⠉⠀⠀⠀⠀⠀⠈⠀⢀⣴⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⣠⣤⣄⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⢠⣾⠋⠉⠻⡿⠋⠉⠉⠉⠉⠩⠭⣍⠉⠛⠛⠻⢷⣶⣤⣄⣀⡀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠒⢾⣿⠛⠀⢀⣸⣿⠛⠛⠻⠿⠿⣷⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣼⣿⡿⣶⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣴⡶⠿⠛⣿⣿⠁⠀⣽⣇⠀⠀⠀⠀⠱⢶⣶⣾⣿⣾⣿⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣉⣭⣭⣶⣶⣶⠶⣾⣿⣧⣤⣤⣿⣿⣷⣶⣤⣀⣀⠀⠀⠐⠛⣉⣉⣉⣿⡆
⠀⠀⠀⠀⠀⢠⣿⠟⠋⠉⠀⠀⢀⣀⣤⣤⣶⡾⠿⠿⠿⣿⣿⣦⣈⠉⠛⠳⢶⣬⣉⣩⣽⣟⠁
⠀⠀⠀⠀⠀⣸⡿⠀⣠⣤⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣀⠀⠈⠙⠋⠘⢻⡄
⠀⠀⠀⠀⢀⣿⣷⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⠀⠀⢸⡇
⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⣼
⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⢠⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⠈⠻⣷⣄
⠾⠿⠿⠿⠿⢷⣶⣦⣤⣄⣀⣀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠟⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠿⠿⢿⣾⣿⣦
⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣻⣿⣿⣿⣿⣟⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡭⠄⣀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠀⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠳⠂⢠⡀
⠀⠀⠀⠀⠀⠀⠗⢆⠘⠆⢣⣀⠤⣄⣀⠀⠀⠀⡆⡌⠑⠦⠀⠀⡠⠀⠀⠃⠠⡄
⠀⠀⠀⠀⠀⠀⣰⠋⠉⠑⣰⡌⠆⠀⠈⠉⠓⠦⢅⠁⠀⠠⠒⠪⠴⠒⠁⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⡇⢀⡔⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣉⠀⣤⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠈⢃⡞⠈⠁⢸⠇⢀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⢄⠘⣆⠀⠘⠆⠀⡀⡀⠠⣄⠀⠀⠀⠀⠀⠀⠀⡠⢞⡭⢭⣉⡓⠲⢖⡂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⣸⣧⠔⠋⡁⠀⠀⠀⠀⠀⠀⠘⠦⠘⢦⠀⠀⠐⠁⠸⡔⢬⣀⠑⠒⠒⠶⢴⣋⠐⠁⠐⠒⠦⢭⣉⣩⠴⠂
⠀⠀⠀⠀⠀⡞⠀⠉⢉⠀⠀⠚⠉⠈⢧⠀⠙⠄⠀⢀⣠⠴⠂⠀⠀⠀⠈⠳⣄⠈⢦⡀⢻⡀⠈⣳⠀⠠⡀⠀⠈⠳⣄
⠀⠀⠀⠀⢸⡇⠺⢀⡝⠊⠀⠀⠀⠀⢸⡆⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠱⠀⢏⠉⠁⠀⠀⢹⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠘⠃⠀⠀⠀⠀⠀⠈⠋⠉⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡄⠀⣀⠘⢦⡀⠀⠀⠀⠇⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠁⠤⠤⠤⠄⠀⠀⠀⠀⠀⣸⢠⠇⠀⢹⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⠀⠐⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡾⠀⠀⠀⠁⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢠⣀⠀⢳⠀⡇⠀⠈⠉⠁⠀⠀⢀⠞⠀⡇⠀⢀⣀⡤⠴⠚⠁⠀⠀⠳⡀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⡁⠸⠀⠇⠑⠲⠤⠤⢀⡴⠟⠒⠀⠀⠉⠉⢠⣀⣀⣀⣀⣀⡤⠤⠌⠓⠒⠀⠀⢠⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡄⢰⠆⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠈⠀⢈⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡄⠀⠀⣧⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣈⠀⠠⢿⣀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⡄⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠈⠋⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⢸⡇⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡟⠹⣆⠀⠀⠀⠀⠀⢀⣴⣋⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠳⣦⣀⣠⣤⣤⠶⠶⢶⣶⠀⠀⠀⠀⠀⠀⢠⡾⢁⡞⠀⠀⢹⡄⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⡞⣿⠀⢠⣶⡶⠶⠶⢯⢤⠜⠀⠀⠀⠀⢻⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢻⢠⡟⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟
⠀⠀⠀⠀⠀⠀⣾⣯⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⢸⣿⠁⠀⠀⠀⠑⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣱⠋
⠀⠀⠀⠀⠀⠀⠈⠳⣌⠉⠓⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣸⡟⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⢹⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣙⣧⡄⠀⠀⢠⠟⠀⣀⡤⢖⠟⠳⣄⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⢸⠀⠀⠀⠀⢠⠞⢹⠀⠀⠀⠀⠀⠉⠉⠉⣋⠀⢀⡾⠖⠋⠁⢀⠎⠀⠀⠀⠙⠦⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣸⠀⠀⠀⣴⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡸⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢀⡽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢀⡞⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢠⠏⠘⣇⠀⠀⢀⠎⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠋⠀⠀⠀⠀⠀⢹⡀⠀⢀⣀⣀⣠⠏⠀⠀⠘⢤⣤⣮⣤⣤⣤⠀⣠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠈⡟⢯⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⢾⣏⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⡿⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢇⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢠⠞⢁⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢸⡀⠀⣰⠃⠀⠀⣀⡀⠀⠀⠈⢧⡼⠃⠀⠀⠸⣟⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⡇⣰⢃⣠⠔⠋⢹⠉⠓⠦⣄⡈⢷⡀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣧⢟⠁⠀⠀⠀⡏⠀⠀⠀⠀⢉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢸⣼⠀⠀⠀⢸⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⣿⠀⠀⠀⡟⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣿⠀⠀⢰⠃⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢹⠀⠀⡏⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⠀⣸⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⢀⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⣁
⠀⠀⠀⣀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠃
⠀⠀⠀⠇⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣭⠉⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⢀⡀⠒
⠀⠀⠉⠁⠀⠁⠀⠐⠁⠀⠀⢠⣀⠀⠤⠀⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠠
⣀⣀⣀⣄⣘⣁⣀⣀⣀⣐⣒⣀⣀⣄⣀⣐⣛⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣐⣂⣒⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⣠⣔⣂⣀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡴⠋⢀⠀⠀⠈⠑⢦⡀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡵⢉⡴⠚⠋⠉⠙⠳⡀⠀⢣⠀⢹⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡰⢋⣤⠖⠒⠲⢄⠀⠸⣆⠀⢇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⢁⠞⠁⣀⠀⠀⠀⢆⠀⢹⠀⠘⡀⠘
⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣇⡏⣰⠋⠀⠉⠆⠀⢸⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠠⠤⠤⣤⣄⣀⣀⣀⠀⠀⠀⣼⣰⣿⢀⠇⠘⠉⡀⢸⠀⢸⠃⠀⡇⠀⡇
⠀⠤⠼⣷⣶⣶⣾⠶⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⣲⡿⠿⠿⠿⢿⣿⣿⢿⠷⡷⠂⡇⡈⠀⢸⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠸⣿⣽⠘⡀⠑⠊⢠⠇⠀⡎⠀⣸⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠆⠑⠤⠔⠁⢀⡼⠁⢀⠇⠀⡼⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⡝⠣⣤⣤⠴⠋⠀⣠⠊⠀⣰⠁⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣍⠢⣤⣀⣀⠤⠊⠁⢀⡾⠁⢠⣾⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⡓⠤⣤⣤⣤⡤⠚⠁⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡲⣦⡀
⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⣉⣉⣉⣉⣉⣉⣉⣉⣉⠁⢸⣿⣿⣦⢻⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠇⣸⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢸⣿⣿⣿⡟⢀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠸⠿⠟⢋⣠⣿⣿⣿⣽⡳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⣰⡷⠐⠾⣿⣿⣿⣿⣿⣿⣿⣷⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠘⢛⣡⡾⠃⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣼⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣍⡓⠂⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣴⣶⣾⠿⢿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠨⠙⠀⠀⠙⠁⣀⢄⣀⣀⣀⠤⠤⠦⣤⣀⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠁⠀⠀⠀⠀⠁⠉⢻⠟⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠉⠀⠄⠚
⠀⠀⠀⢀⠀⠀⠀⠱⣄⣀⣀⣀⣠⠤⠴⠖⠀⢀⣤⠄⠘⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠂
⠀⠀⠀⢰⠀⠀⠀⠀⢸⠋⠉⣁⣤⠖⠀⠀⠀⢈⣿⣿⣤⡄⠀⡀⡀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠓⣿⡿⠃⠀⠀⠀⢰⣿⠛⠻⠿⣿⣿⣿⣅⣴⡿⠿⠋⠛⠉
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠈⠙⠛⠉
⠀⠀⠀⠀⠸⠀⠈⠲⣄⠀⠀⠀⠀⢀⣴⣿⡏
⠀⠀⠀⠀⡂⠀⠀⠀⠘⢿⣶⣤⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣟⡇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣛⡽⠿⡛⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋
⣶⣶⡶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠛⠁⢿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⣠⣴⣾⣿⣷⣶⣦⣤⣼⣿⠄⠀⠀⠘⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣶
⠀⠀⠀⢸⠛⠉⠉⠉⠉⠉⠙⠛⠛⠁⠀⠀⠀⣰⣿⣿⣿⡟⠋⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⠛⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠙⠛⢿⣿⣿⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣈⢁⣀⣠⣴⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠉⠉⠉⠉⠉⠛⠻⠿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⢠⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣜⡛⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣷⣌⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠸⠀⠀⠀⠀⠁⡦⠨⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⣶⢶⣾⣖⢦⣄⣤⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠰⣿⣽⣟⣿⣿⣻⣿⠖⣿⣽⣶⣦⡬⠿⠝⠒⠒⠺⠯⠭⢵⡲⢤⡀
⠀⠀⠀⠀⠀⠈⠛⠛⠛⢻⡟⠛⠻⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣴⣦⣴⣤⣤⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⣠⣿⣿⠿⠛⠋⠉⠉⢁⡀⢠⢬⣭⠉⢙⣛⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠋⠀⠀⣄⣴⣤⣬⣽⣤⠴⣛⣿⣷⣾⢶⣷⣆⣭⡻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣈⣿⡟⠿⠿⠛⢃⣉⣤⣤⣤⣬⣤⣬⣭⣭⣉⣻⣽⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣆⣤⣶⣿⣿⣟⡭⠭⠽⠭⠭⢭⣭⣛⣻⣿⣿⣿⣿⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠄⠂⠀⠀⠐⠀⠀⠀⢀⠒⠄



⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣦⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⣿⣛⣋⣉⠉⠙⠛⠛⠿⠿⣿⣛⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣤⣤⣄⣀⣈⣉⣛⣠⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢿⡆⠫⠀⠀⠀⢸⠀⠀⢺⣼⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠿⡇⠘⠖⢿⣷⠀⠁⠀⠀⠸⡄⠀⣬⢿⡇⠀⠀⢀⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢺⡇⠀⠀⠹⣿⠀⠀⠃⠰⣶⡇⢀⣿⣎⠃⠀⠀⡬⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⡇⠀⢠⣤⣿⡆⠀⠀⠀⢸⡇⠈⣿⣿⠁⠀⠰⣀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⢸⠁⣿⡇⠀⠀⠀⠈⡇⢠⣋⣅⠀⠀⣄⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣿⠀⠈⠀⠸⡇⠀⠀⠀⠀⠇⢼⣿⡉⠀⢰⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠀⠀⠻⠀⠐⡶⢶⣾⠀⠀⠀⢲⣷⢸⣿⡇⠀⠈⢻⡟⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠰⠀⢦⡀⢀⣠⡀⠀⠁⢸⣿⠀⠀⠀⠈⣿⢸⣿⡇⢀⡇⢸⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣟⠃⠀⡇⣀⣻⠀⠀⠀⠀⢻⢸⣿⡅⢸⢹⣿⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⡇⠈⡄⠉⢿⣿⠆⠀⠀⠀⠸⣾⣿⡆⣗⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠀⠘⢡⡆⠸⡏⢢⡆⠀⠐⣾⣿⣿⡟⡃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠠⣾⣧⣤⠁⢉⣴⠀⠀⢽⣷⣿⠆⣿⣾⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⢀⠔⢨⣿⡿⠀⣀⠞⢀⠀⠸⣿⡏⠳⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣆⠀⠀⠀⠙⢻⡿⣷⣶⣿⣶⡞⠀⠀⢸⠓⠜⣻⠿⢿⣿⠿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠻⣿⣿⣦⣠⠞⠋⣼⣿⣿⣿⣿⡆⠀⠀⣼⠆⢠⣿⢶⡶⠒⡖⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠐⠾⢿⣿⠃⣿⠟⠠⠠⢾⢿⣀⣛⣿⣿⣿⢻⣧⣉⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠋⢠⣿⣿⣄⣚⣎⠁⠈⣁⣄⠈⠻⡿⠋⠁⣼⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠓⢤⣴⡿⠋⠈⣿⣿⣷⣼⣿⣿⣿⣿⣷⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⣽⣥⣂⣀⡀⠙⡁⣐⣶⣿⡿⠟⠉⠀⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣉⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⡀⠀⠘⣿⣀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⣛⣫⣥⣤⣶⡾⣡⣤⠤⠀⠁⠈⠀⢀⠀⠀⠀⠀⢀⡿⢹⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣤⠀⠘⠡⠭⠭⠵⠶⠶⠦⢤⣴⣟⣿⣶⣤⡀⢀⡴⠋⠀⠀⢀⣀⠉⡰⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠶⠄⣸⠷⠂⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠙⠻⠷⢏⠀⠀⠀⠀⠀⢠⠞⠀⠀⡴⠁⣲⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣿⠟⠛⠉⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⢠⡿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⢶⠻⠿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⣿⣿⠁⠳⡀
⠀⠀⠀⢀⣿⡇⠀⠀⠀⣠⡆⠐⢲⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠣⣴⣶⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⣿⡟
⠀⠀⠀⡞⢿⣦⣤⣴⡾⠛⣠⡴⠋⠀⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⢦⣈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⣻⠀⠀⠈⠙⣧
⠀⠀⣼⣇⣀⣼⢛⡥⠖⠋⠁⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠬⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠘
⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣠⠄⠀⠀⢀⣤⣾⡇⠀⠀⠀⠀⠀⠀⠘⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣂⣠⠴⠖⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠛⠋⠁⣀⣤⠶⠛⠉⠀⠉⠀⠀⠀⠀⠀⢀⣀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣰⣾⣿⣿⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⢀⣀⠤⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢹⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣾⡇⠀⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠒⠛⠛⠛⠓⠒⢲⠖⠉⠉⠓⣄⡠⠴⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⣀⣀⡤⠤⠴⠖⠒⠒⢾⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⢀⡠⠴⠚⠉⠁⠀⠀⢀⣀⡤⠤⠤⠼⣆⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⢰⠉⠢⡀⢀⠔⠉⠀⠀⠀⢀⠴⠚⠁⠀⠀⢀⣠⠴⠒⠉⠁⠀⠀⢀⣀⡠⠬⠦⡀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠙⢇⠀⠀⢀⣠⣾⠥⣄⠀⢀⡠⠖⠉⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠈⠓⠚⠉
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⠄⠐⠈⠀⠀⠀⠀⠠⠐⠉
⠀⢀⣀⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⢀⡠⠈
⣖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⡰⠋
⠈⠉⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢀⠞
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠱⡃
⠀⠀⠀⠀⠀⢸⠀⠀⢀⡴⠋⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠈⢆⣠⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⣀⠦⢤⣤⣤⣄⣴⣶⣶⣦⠖⠋⠉⠉⠒⠤⠀⠐⡧⢌⣛⣹⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⠈⠱⠛⣵⠐⠂⠘⠶⠿⠏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⢈⣆⣀⣀⣀⣤⣤⣤⣤⣶⡆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢸⢿⣿⡿⠿⠿⠿⢿⠟⠻⣤⣤⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣶⣶⣶⣶⣶⣷⡏⣶⣤⣈⣁⣀⣤⣤⣶⣶⣤⡴⢶⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⡇⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣤⣽⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⢻⣿⡇⣿⣿⡟⠉⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣠⣾⣿⡇⣿⣿⠀⣰⣿⣿⣿⢟⣿⣿⣿⣿⠉⠻⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣤⣿⣿⣿⣽⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠿⠿⠭⠍⠛⠓⠀⠉⠉⠉⠙⣿⣿⡟⣿⣿⡿⠿⣿⡿⠀⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣶⠶⢂⡴⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣦⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⢤⠀⠀⠈⠛⠀⡙⣿⣿⣿⣿⡿⠟⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡆⠀⠈⢇⣀⠐⣠⡀⠘⣿⣿⣿⣿⣟⠂
⠀⠀⠀⠀⠀⢀⠀⣿⣷⣷⠀⠀⠀⠀⠀⢀⣤⣶⣾⡌⣟⠱⣾⣿⣶⣭⡍⢷⣽⡸⣿⣤⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠄⠒⢠⣄⣿⣿⡟⣿⣧⢸⡏⣿⠙⢻⣭⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⡿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⡀
⠀⠀⠀⣰⣆⣸⣿⡿⠃⠀⢀⠀⠀⠀⠛⠻⣿⣦⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣿⣿⡆
⠀⠀⢰⠉⢻⣿⣿⠃⠀⣼⣿⣤⣴⣿⣶⣶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⣿⣻⣿⣿⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟
⠀⠀⢸⣄⣻⣿⡇⠀⠀⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣶⣿⢱⡿⣿⣿⣿⡁⢂⠀⠀⠀⠀⠁⠀⠀⠀⢠⣶⣾⣿⣿⡿⠿⠇
⠀⠀⠾⢿⣿⣿⠁⠀⠀⠀⠀⠀⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⢿⣿⣧⣿⢳⣿⣿⣛⠲⢄⡐⠲⣔⠠⠀⠀⠀⠀⠈⢻⣏⡿⠋
⠀⠀⠰⣿⣿⣿⠆⠀⠀⠀⠀⢀⣌⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⡁⡰⢀⣿⣿⢇⣾⣿⣿⣿⣿⣿⡜⠓⠉⢻⣡⣄⢠⣤⣝⣻⣿⣿⣧⡀
⠀⠀⠀⠹⢟⠽⠀⠀⠀⠀⠀⠺⣧⣘⣛⣿⣿⣯⣿⣿⣿⣿⡯⢸⠓⠒⠀⠿⢏⣾⣿⣿⣿⡟⠀⣽⣷⣶⣶⣴⣿⣿⣿⣽⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣠⡴⠄⠀⠀⠀⢺⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⡄⠀⠀⠀⠈⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⡀⢀⢠
⠀⠀⠀⠀⠀⠉⣠⣜⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⡇⠀⠀⠀⢰⡼⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣟⠛⠀⠀⠀⠟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣩⡁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣿⠀⠀⠀⢸⣷⡛⠉⣿⣿⣿⣿⣏⡀⣿⠟⠛⢸⣿⣿⡇⢸⣿⣿⣿⠃
⠀⠀⠀⠀⣀⠀⠀⠝⢿⡆⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣄⣠⣤⡌⣿⢁⣼⣿⣿⣿⠇⢹⣇⣼⣇⣠⠎⢹⣿⠇⠈⠻⣿⣿⣄⡰⣶⣤⣶⠄
⠀⠀⠠⢼⣿⣄⢀⣄⠀⢒⣿⢸⣿⣿⠀⣿⣿⠋⡹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣹⣿⠇⠉⠀⠀⠈⠁⠿⣿⣿⣤⣼⣿⣆⣦⢰⣛⡿⢛⣷⣿⣿⣿⡀
⠀⠀⠈⠄⣻⣿⡟⠻⠻⠟⢹⣾⣿⠏⢰⣿⣿⣦⢶⣿⣿⣿⡿⡁⠀⠛⠉⢹⣿⣿⡇⠀⠀⠀⠀⢀⡀⢚⣿⣿⣿⡿⠟⢛⣻⣿⣿⣿⣶⣾⣿⣿⣿⣿⣤⡶
⠀⠀⠠⡶⠛⠛⢿⢦⣄⠀⠸⣿⣿⠀⢎⣿⡏⠈⢸⣿⣿⣿⣿⣇⡘⢀⣀⠸⣿⣿⡇⠀⡀⠀⢠⣼⣿⠟⠋⢀⣾⣷⣾⣷⣿⣿⣿⠿⠿⠛⠛⠟⠛⡻⠿⠂
⠀⠀⠀⢻⣿⣾⣿⣶⣿⣧⣀⠝⠻⠀⣿⣿⣿⢆⣿⣿⣿⣿⣿⣿⣿⡌⣿⣀⣹⣿⢏⠀⠀⠀⠙⠉⠀⢀⣀⣈⣿⣿⡟⠉⠿⠛⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⣄⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡿⣠
⠀⠀⠀⠀⠀⠀⠀⢩⣷⣿⣿⣿⡟⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠿⡟⠣⠀⠀⠀⠀⡀⠀⡄⢸⣇⣦⣸⣿⣿⡾⢛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣾⢿⣿⣿⣿⣿⡿⡿⠶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡦⢀⢸⣿⣿⢶⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣔⣶⣶⣶⣶⣿⣿⣷⣾⣿⣿⣿⣿⠿⠿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣥⣤⣬⣔⣙⣛⣛⣉⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣄⣀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠦⢌⡀⢼⣿⡄⠀⠀⠀⢀⣤⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢫⣤⡀⠾⠗⢨⣟⣃⣠⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣅⣴⣶⣾⣿⡿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣷⣿⣿⡟⠀⡽⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣛⣡⢾⣿⠿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣉⠑⡄
⠀⠀⠀⠀⢴⠖⠂⠀⠀⠀⠀⣆⡂⡀⠆⠐⠛⣿⣿⡿⠁⠻⣿⡄⠈⠋⠀⠀⠀⠀⠀⢀⡜⠁⠀⠙⢄⡀
⠀⠀⠀⠀⠈⢷⣄⣠⣤⣤⣤⣼⠿⠛⠘⡛⠖⠉⢠⠃⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠿⠛⠛⣏⠁⠀⠈⣰⡄⠀⠀⠀⠄⣸⠀⠀⠀⡸⠀⠠⠤⢀⠀⠀⠄⢰⣏⣀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠐⠛⠻⢃⡄⣠⣄⣸⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣴⡄⡀⠀⠀⢀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠙⠿⣿⣿⡇⠉⠀⢸⡀⠐⠐⠆⠤⠄⠀⠀⠀⠉⠉⡼⠉⡳⠂⠉⠁⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⢀⣼⠻⣷⠂⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠃⠀⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡇⣠⠆⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡶⠈⠀⠀⠀⢀⠀⠐⠶⠤⠄⠀⠀⢧⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠙⢻⣛⡭⢅⣠⣤⣶⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣶⣤⣀⠀⠀⠘⠄⠒⠻⢋⢁⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠀⠈⠉⠉⠀⠈⠉⠉⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠠⠀⠀⠀⠀⠀⡄⠀⠀⢠⠠⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠉⠈⠉⠛⠛⠟⠛⠙⣿⣿⡄⠀⢀⠀⠀⠀⠀⠀⡘⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠁⠢⣶⣾⣿⣿⣿⣦⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡖⠶⠀⠀⡀⠈⠋⠀⠀⠀⡤⠀⠀⠀⣠⠀⠀⢋⠁⢂⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⡩⠁⠈⠀⡄⠀⠀⠀⡀⠀⢀⡀⠈⠀⠀⠋⣰⣿⣷⣄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⢄⠀⠀⠘⢄⠀⠀⢻⣤⣤⡔⠀⢀⠌⠀⢻⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠁⠂⢠⠀⠑⣄⠀⠉⢉⠠⠐⠁⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠬⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢤⠀⠀⠀⠀⠀⢀⠈⡄⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠿⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠘⣠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣤⣤⣤⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠂⣀⣻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠟⠛⠛⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣨⠀⠀⠀⠀⠀⠀⠀⠄⠻⣿⣯⣟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠁⠀⠀⠠⠀⠀⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠔⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢶⣾⣿⠉⠻⢿⣿⣶⣤⣤⣀⣠⣤⣴⣿⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉⠁⣄⡈⠂⠀⠀⠤⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⣤⣤⣶⣿⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠁⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠄⠄⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠒⠀⠉⠁⠒⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠹⣿⣧⠀⠀⣀⠠⢆⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠁⠐⠛⠛⠃⠉⠀⠀⠀⠱⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⡰⠀⠀⠀⠀⠠⠂⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⡄
⠘⢿⣦⣤⣄⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⡏⠀⠊⠀⠀⠀⠀⠐⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠑⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⣀⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣮⡏
⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠒⠒⠚
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠉
⣿⣿⡿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡻⠟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣬⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠻⣷⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠯⣍⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠛⣯⣿⣿⣯⣱⣎⡛⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⠱⣾⣤⣀⠀⠀⠀⠀⠀⢠⡾⡿⠻⡉⢠⣿⣿⣿⣿⣿⣿⣽⣶⣏⣬⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣌⠉⢉⢩⡁⠈⢰⣤⣿⣿⠁⠀⠁⠾⠁⣿⣿⠉⠦⣈⠉⠛⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⠀⢀⣼⣿⣅⠁⢈⠀⠀⠀⠀⣠⣾⣿⣿⣷⣤⡀⠘⠀⠀⢠⡇⠀⠀⠀⠉⣙⠿⠟⠻⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠁⠀⠀⠀⠈⠙⢿⣾⣿⣟⣿⡗⠐⢾⣷⡦⠀⠘⠑⠽⠿⠿⠏⡈⠉⠛⠛⠒⢾⣗⣒⣒⠒⠌
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⠀⣠⡶⠂⠀⠀⠀⠀⢻⣿⣿⣾⣟⠀⠀⢶⣾⣿⠆⠀⠀⠀⠀⠀⢻⣦⠀⠀⢀⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠁⠁⡄⠀⠀⠀⣀⠀⠙⠻⠿⣿⣆⠀⠈⠁⡟⠀⠀⠀⠀⠀⠀⢸⡏⠀⠒⢿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣧⡀⠀⠀⠀⠉⠉⠐⠒⠺⣿⣷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡋⠁⠀⢀⣬⣶⣦⣾⣧⣴⡄⣤⡀⢰⢿⡏⠀⣰⠀⣰⠀⠀⠀⠀⠀⢀⣀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡀⠀⡸⣿⣿⣿⠟⠻⣿⣿⣿⣷⣸⣾⣷⣏⣹⣾⣿⠀⠀⠀⣠⣶⣿⢏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⡰⢡⡟⠉⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣏⢸⣿⣿⠀⠀⠠⢿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢀⡜⣱⣿⡏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛⠛⢻⡄⠀⠇⠸⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡇⠸⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⠀⠀⠈⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣄⠀⡀⠀⢻⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠘⣻⣦⡀⠈⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢤⣀⣀⣀⣻⣎⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣧⣡⠀⠈⠙⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣯⠇

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠰⡀⠀⠀⠄⠀⠀⠀⣤⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤
⠀⠀⢀⡄⠀⠀⠀⠉⣀⠈⢣⡀⠀⠀⢺⡀⢧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠
⠀⠀⣸⠅⠀⠀⠈⠀⠐⠄⠀⠁⠀⠀⠀⢷⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⢹⣇⠀⢢⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠉⢦⠀⠱⠀⠸⡆⠀⠀⠀⠈⠀⠀⠀⠪⡇⠠⣴⣾⢁⠰⣾⡿⣶⣶⣄
⠀⠀⠀⠀⠀⢳⡀⠀⡀⠻⡰⠄⠀⠀⡄⠀⣈⠳⣽⡀⢸⣿⠠⣤⣼⡷⡿⠟⠿⠂
⠈⣉⠀⠀⠀⠀⣿⣄⠈⠚⣤⠀⠸⠀⠀⠀⣼⣿⣿⣇⣾⣯⣦⡽⠿⢃⠷⠃
⠠⢋⡕⢂⣠⡄⠈⢻⣆⡀⠀⣦⢀⠀⠀⠁⠈⠿⣿⡿⠿⣿⡿⣿⡇⣷⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀
⠐⡻⢆⡨⢃⣀⠀⠈⠛⢷⡄⠠⠋⡀⠀⠀⠀⣿⣿⠟⠷⢮⣿⣯⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⡀⠀⠀⠀⠀⢀⣔⡑⠂⠠⠄⠁⠀⢙⡃
⢸⣟⣇⢠⠀⠈⡂⠀⠀⣿⣿⣣⡀⠀⠀⠀⣸⡏⣧⠀⠀⣀⠀⠉⠿⢏⠹⣇⠁⠀⠀⠀⠀⢀⣠⡆⢾⡀⠘⠢⢖⠙⢦⡀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠘⠃
⠘⠷⡀⢣⠐⣠⡄⠀⠀⠘⢙⢻⣷⡀⠺⣶⣏⣀⠀⠑⠂⣼⠀⠀⠠⡈⡄⢰⠏⠀⢀⡀⠀⠈⠈⠁⠀⠈⡀⠀⠀⠀⠀⠃⠀⠀⠀⠆⢀
⠀⠆⠲⢀⡂⠶⠀⠀⠀⠀⠀⠸⣻⣷⣼⣿⡿⠿⣿⣶⣤⣿⣷⣦⡀⠋⠁⠀⠀⠀⢀⣁⣌⣁⠓⠂
⢀⡗⠰⣀⠁⠐⠀⠈⠀⠀⠀⠀⠀⠸⣿⣿⠶⠂⣫⡍⣹⣿⣙⠟⠛⠶⣶⣄⣀⠄⠈⢓⣀⠉⠂⠄⠰⠂⠀⢰⠒⠦⠀⠀⠀⡠⢄
⢸⠁⢀⠁⠔⠂⠀⣷⡰⠆⠀⠀⠀⠀⣾⠟⠀⢠⣭⣝⣿⣿⣛⠀⠀⠘⢊⠙⠿⠐⠆⠀⢉⠀⠀⠀⠠⠛⠀⠘⢛⠀⠠⠤⣤⠷⠞⠃⠠⡄⡀⢠
⢈⡶⢝⡁⠹⡆⠀⠐⢆⢀⠀⠀⠀⠀⢿⣷⣤⠀⠈⡾⠘⠿⣯⣥⣄⣤⣆⣠⠀⢀⠀⠀⠠⠀⠀⠀⠰⢀⠶⢬⣠⣦⠀⠉⠙⠶⠦⠒⠃⠈⠁
⢘⡶⠈⢛⡀⢹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣶⣴⣋⣤⡠⣬⡟⢹⡟⢿⡷⠉⠀⠀⠁⠀⠀⠀⢶⠁⢠⣄⠁⠘⣆⠀⠀⠐⠟⢛⡃⠀⠀⠀⠀⠀⣀
⢸⠇⣤⡀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣹⣿⣶⠋⠀⠀⠙⢦⣭⠀⠀⠀⠀⠀⠀⠀⡼⠀⠃⣹⡶⢾⡍⠀⠰⠀⢤⠈⠁⠀⠀⠀⠀⠀⠀⠐
⢈⡞⠛⢡⠄⠐⢿⣆⠀⠀⠀⠀⠀⢀⠀⠀⡄⠈⠛⣿⣿⣿⣿⡀⠀⠀⠀⠢⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠴⠀⠔⠂
⢀⢸⣷⣆⣤⠀⢈⣿⡀⠄⠀⠀⠀⠀⠠⠀⢳⡀⠀⣿⣿⣿⢻⣷⣆⠀⠀⠀⠉⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠔⡂⢀⠀⠀⠀⠀⠀⠦
⢸⡾⣬⣏⢈⡀⠸⣦⠳⠰⠂⠀⠀⠀⠀⢠⡀⠀⠀⢽⣟⣿⠸⣿⣿⣷⣶⠀⠀⠀⠄⠀⠀⢐⣀⡀⠀⠄⠈⣼⠀⠀⠀⡟
⢘⣷⠸⣿⡁⠀⠀⠙⡶⢁⡀⠰⠄⠀⠀⠀⢉⠀⠀⢼⡝⢻⠀⣿⡛⢿⣿⣷⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠠⠍⠃⠘⠃⠀⠚⠒⠢⠤⠰⠂
⠨⣿⣼⠃⣍⣃⠀⠘⠦⣽⢆⠉⠀⠠⣄⠀⣍⠀⡀⠺⣇⠈⢃⣠⣧⣘⣿⣵⠶⠖⠉
⠀⡟⢿⡄⣛⠃⠀⠀⠀⢰⡎⠿⠀⠀⢻⠀⠘⢷⡯⠂⠈⣽⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀
⠰⡗⣿⣷⢻⣾⠀⠁⠀⠨⣿⡆⡁⠀⠀⢧⡀⠀⠉⠀⠀⣻⣿⡿⣿⣿⣿⢿⣶⣄⠄⠀⡄
⠈⣿⣿⡛⣧⣿⢷⣶⡀⠀⢿⡇⢀⠀⠀⠐⠿⣆⡀⠀⠀⢸⣿⡇⣿⣿⣿⢿⣿⣿⠓⡠⠃⠐⢒
⠀⡜⠙⠧⣯⡙⠾⠻⠏⠀⢸⣧⠘⠀⠀⠀⠰⢌⠳⡒⠀⢸⣿⣷⡏⠉⠁⢸⣿⣿⣿⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡇⠀⠀⢱⠊⠀⠀⢿⡀⠈⢿⣧⡀⠀⠀⠀⠀⠀⢿⠀⢬⣿⣿⡏⠀⠀⠀⣿⣿⡟⠀⠀⠀⠈⠲⢄⠀⣀⣀⠁
⠀⣠⡾⠃⡞⠀⠀⠀⠹⠅⠀⠈⣱⣟⣦⢠⣤⠀⠀⢸⡇⠘⢛⠿⣧⠀⠀⠀⢹⣿⠃⣀⠀⠀⠀⠈⠀⠀⣄⠀⠉⠉⠉⠁
⠀⠴⡷⠃⠀⠂⠀⠀⢀⠀⠀⠀⢱⣌⣷⣻⣿⣇⠀⠀⢁⢦⡀⣿⣾⣧⡀⠀⢸⡿⠀⠈⠛⠒⣀⡀⠀⠀⣨⡄⠄
⠀⠀⠑⢀⠀⠀⠀⡠⠠⡆⣧⠄⢤⣿⣿⣿⣿⣿⠶⠀⠈⢄⡁⠘⣿⣿⢧⠀⠀⡅⠀⠀⠄⣜⡈⢿⣆⠀⡿⠁⣲⠶⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠿⡄⢈⢻⣿⡿⠰⠃⠀⠀⠀⠈⠀⠰⡟⣿⡈⣇⠀⢁⠀⢠⣤⡿⠇⠘⠃⠀⠀⠀⠙⠀⢠⢀⠀⠤
⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⢀⡶⡈⢠⣿⡁⠀⠀⠀⢶⣧⡄⢶⠄⠸⣿⣷⣼⡆⠘⣦⣀⠉⠃⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⡄⠀⠄⢀⡀⠺⠷⠦⡩⡕⠂⠐⠦⢀⠈⠉⠁⠀⠐⠛⠿⣿⣷⣘⣌⡿⢿⠶⠦⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⠁⠄⢑⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡖⠀⠀⠀⠀⠀⡀⢄⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠾⡟⠀⠀⠀⣀⡤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣠⣤⡖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣮⣴⣾⣿⣿⣾⡟⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⣿⠟⠁⣾⡇⣮⣿⡟⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣵⠟⠁⠀⢸⠟⠰⡹⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢰⣚⠛⠿⢿⢶⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠠⠚⣛⠻⠶⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠸⡏⣏⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠁⣌⣉⣠⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠠⢶⣿⡏⣿⡟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠠⣤⣿⣯⢿⣿⠁⠀⠟⡄
⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⢈⠟⣷⣟⣧⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢘⣽⠇⣼⡏⢩⡄⠀⠘⢀
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⣸⢶⢻⣿⠀⠀⡖⡀⠀⠟⠆
⠀⠀⠀⠀⠀⣴⡎⠀⠀⠀⠀⠀⠶⠿⣯⣾⣧⣤⣤⣬⣽⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠋⠉⠁
⠀⠀⠸⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠶⠖⠒⠒⠶⠶⠶⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠉⠑⠒⠒⠒⠒⠒⠂⠀⠀⠁⣀⠠⠐⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢉⣠⠴⠚⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⢤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠟⣡⠖⠉⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣇⡀⠤⠭⠽⠶⠤⢄⣀⣀
⠀⠀⠀⠀⢀⡼⢁⡞⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⣀⣀⣠⠤⠤⠤⠤⠤⣤⣄⣈⡙⠲⢄⡀
⠀⠀⠀⠀⡾⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠞⢉⣠⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠹⡄
⠀⠀⠀⣸⠃⡾⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠛⠛⠿⠿⣿⣿⣖⣞⣻⣿⣟⣛⣓⣛⡿⠟⣁⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⡿⢸⡇⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢋⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⠴⠞⠃
⠀⠀⢰⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢛⣡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⠛⠛⠉⠉⠉⠉⠉⠁
⠀⠀⢸⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣩⠶⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁
⠀⠀⠘⣇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣫⠴⠋⠁⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢞⣫⠖⠋⠁⠀⠀⠀⣀⣠⠴⠞⠋⠁
⠀⠀⠀⠘⢷⣄⠀⠒⠤⣤⣤⠤⠴⠚⠋⠁⠚⠉⠀⠀⣀⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠛⠷⣤⣄⣀⣀⣀⣀⣀⣀⣤⡤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣄⣠⡤⠀⢀⣀⠀⠀⠀⡀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞⠁⣀⣚⣣⣤⣴⣶⣶⣿⣿⣿⣷⣿⣷⣏⠀⣹⣿⣗
⠀⠀⠀⠀⣀⣀⣀⣀⢀⡀⣀⣠⣴⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣭⣭⣥⠀⠀⠉
⠀⠀⢀⣞⣿⣭⣿⣶⣿⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠝⠋⠉⠉
⠀⠀⠈⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠉⠁⠀⣿⠁⠀⠀⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣀⡿⠄⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠿⠟⠛⠛⠉⡁⢀⣀⣄⣠⣤⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣶⣶⣸⡗⢸⣿⣿⠋⣿⠁⣿⡇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣤⣍⣿⡷⣿⣿⣿⢰⣿⢀⣿⠀⣚⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⣹⡟⣿⠃⣿⠻⣿⡾⠏⠘⠟⠀⠛⠁
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠘⡛⢋⡁⣩⠀⣴⠂⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠛⠛⠛⢛⡻⠀⠘⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⢲⡤⣄⠂⠀⠀⡀⡤⠤⠤⠀⠠⠤⠠⠤⡤⢤⡤⡀⠀⠀⠀⠀⡄⠀⠀⡀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⢀⠄⠂⠉⡟⢆⠀⠀⠀⠀⣿⣧⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠇⠀⠀⠀⠐⢀⠀⠈⠂⠜⠉⠁⠻⠉⠉⠉⠙⠉⠀⠀⠀⠘
⠀⠀⢠⠃⠀⡀⢠⡇⣠⠉⢍⡡⠟⠛⠉⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠖⠂
⠀⠀⡋⠀⡁⠀⡇⠐⢽⣷⡶⠁
⠀⠀⡇⠐⠄⢀⠃
⠀⠀⠑⠠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⣀⡀
⠀⠀⠀⣀⣠⣤⣀⣀⣀⣀⡈⠿⠿⢿⣿⣟⠃⢹⡄⠀⠀⢀⠀⠀⠀⠤⠤⠄⠀⠀⢀⣀⣀⣀⣴⣷⣿⣿⡟⠋⠉
⢀⣴⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⠇⠀⠀⠀⠀⠀⣠⠴⣖⣻⣯⣭⣭⣽⣿⠿⠿⣿⣿⠿
⠈⠻⠿⢿⣿⣿⣿⣿⣟⢋⣭⣉⢉⣹⠛⢋⠁⠤⠶⣺⣿⣿⣯⣍⢁⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠹⣿⣿⣿⣿⣿⢋⡼⠋⠀⣠⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡷⠋⠀⣀⣤⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠁⠀⢈⣡⣾⣿⡿⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠏⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠿⢿⡿⠉⡽⠁⠀⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⣿⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠲⣤⢄⡀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣆⡆⠄⢀⠀⠈⠈⢃⣶⣀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠋⠀⠀⠀⠠⢠⠀⠈⢧⣛⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⢀⡰⢂⠀⠈⠉⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡟⠋⠱⠏⢀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣰⢿⠄⠀⣠⣇⣼⣤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠀⠀⠉⢹⣧⣄⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠊⠣⠀⠀⠀⠀⣸⡏⠈⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠟⢃⡄⠀⠀⠒⢀⣀⣤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢐⡾⡶⣾⣝⣷⣤⡴⢺⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣿⠙⠛⠋⠿⣴⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡘⡶⢶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣆⣘⢸⣆⡀⢳⣹⣿⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⢸⣿⣿⡇⠉⠀⠀⠀⢱⣶⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡲⢲⣮⠀⠀⢀⠏⠀⡈⣿⣿⣿⡀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡠⣄⠠⢶⣄⣀⠀⠀⠹⣿⣿⣧⠀⠀⠀⠀⠈⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠰⠿⡆⢧⣿⣦⠀⢦⣟⣽⣧⠀⠸⣿⣿⣿⠰⠀⠀⠀⠀⠘⡟⠄
⠀⠀⠀⠀⠀⠀⠀⣉⢛⡑⠀⣖⣿⡄⠇⠹⡛⠁⠀⠀⢹⣿⣿⡀⠠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣄⢹⣿⣧⠠⣮⣿⣶⡄⠀⣶⠙⣿⡇⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⣿⣿⡆⠘⣿⣿⣿⣦⣼⠁⢻⣇⡀⠈⢶⣄⠤⡄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠤⢀⠀⠙⠿⢛⣛⢿⠀⠸⣿⣿⣦⠈⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⠀⢸⠄⠠⠉⠉⠙⠓⢧⠀⢦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠟⠉⡰⣾⡷⠀⠀⠀⠀⡀⠀⠉⡉⠀⢀⠀⠙⠻⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠼⠟⣿⣶⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠒⠂⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠄⠐⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠲⣤⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢮⣦⣀⠀⠀⠀⠀⣻⠐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣤⣄⣬⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣿⣿⣿⡆⢀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⣿⢍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠹⣾⡼⣿⣯⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡾⠁
⠀⠀⠀⠀⠀⠘⠿⣮⡯⠉⠛⠛⠿⠿⠿⠟⠋⠉⠉⠛⠛⠓⠒⠾⠿⢿⣿⣿⠿⠋⠁⠁⣿⣿⣯⣿⣿⣿⣿⣿⣟⣿⡿⠿⢿⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⡟⠋⠁⠀⠀⠀⠠⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢉⣠⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣾⣯⣷⣶⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡾⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣶⣶⣿⣼⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡛⠋⠁⠀⠀⠀⠀⠀⠀⣀⣶⡿⣫⣴⣿⡿⣿⣿⣤⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣅⣀⠀⠀⠀⠀⠀⠀⣰⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣄⠠⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡄⠀⠀⠀⠀⢲⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣤⡾⡇⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⣈⣿⣿⣼⣿⣿⣿⣿⣿⡿⣯⠉⢙⠿⣿⣿⣹⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠆⠀⠀⢺⣿⡇⣿⣿⣿⣿⣿⣿⣿⡜⡗⣩⠞⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⣄⠀⣽⣿⣷⣿⣿⠻⣿⣿⣿⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣾⠿⣿⣿⣿⣿⠞⢛⢿⣯⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠋⢿⡛⣻⣤⣀⣼⣿⣦⣀⠉⠈⠙⠻⠷⠌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⡿⠁⠀⠀⠈⠻⣿⣿⡝⢿⣿⣿⣿⣷⣶⣤⣤⣀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠟⠛⢻⣿⣿⣧⣄⡀⠀⠀⠀⠈⠻⣿⡀⠹⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⣤⣴⣶⣿⣿⣭⡉⠙⢿⣶⡆⠀⠀⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡟⢿⣷⡀⢸⣿⣿⣤⣼⣿⡇⠀⣼⣿⡿⣿⣿⣿⢻⣿⣶⣶⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⠘⣿⣿⣧⣿⣿⣷⣼⣿⣿⣿⣿⣿⣥⣼⣥⣽⡿⠛⠁⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣿⣿⣿⣿⣷⣼⣿⣇⣹⣤⣿⣿⣿⣿⣿⣿⣿⣿⡟⢏⠁⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⢻⡟⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠈⠛⠿⣿⣾⣷⠀⠀⠀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠈⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣈⣿⣿⣿⣿⣿⣿⡟⢿⡿⠉⠀⠀⠀⠀⠀⠀⠀⣹⣿⡅⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠟⠛⠛⠛⠿⠿⠛⠓⠈⠀⣴⣾⣿⣤⣤⣴⣶⡿⠿⠛⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢻⣿⡿⢿⣿⣿⣿⣿⡆⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠟⠋⠁
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⢿⣿⠀⠀⠀⣠⣾⣿⣿⣌⣻⣿⣿⣿⣿⠾⠃
⠀⠀⠀⠠⠶⠟⣻⣿⣿⣿⣿⣿⢸⠏⠀⠀⠀⣹⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠃⠘⠀⠀⣶⣿⣿⣿⣿⠿⢿⣿⣧⡀
⠀⠀⣤⣭⣿⣿⣿⡿⢻⡿⠃⠀⠀⠀⠀⢹⣿⣿⠟⠀⠀⠀⠈⠛⣿⣷⡄
⠀⠀⠀⠈⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢁⡀⢄⣀⠄⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡿⠿⠿⠛⢦⣵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣀⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠻⠟⡂⠛⠋⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣼⣦⣴⣴⣿⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠟⠉⢉⣿⣿⣿⡷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣿⣽⣿⣿⣿⣿⣿⠛⠀⣤⣾⣿⣿⣿⣋⣤⣤⡤⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠂⠈⠛⠛⠛⠛⠛⠻⢿⣼⠟⠉⠀⠀⠀⠈⠉⠉⢱⣧⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⣿⡿⣾⣤⣤⣠⣀⣤⠀⠀⠀⣤⣻⣶⣶⣶⠶⢾⣶⣠⡧⣼⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣻⣿⣿⣿⣿⡾⠛⠋⠉⠁⠉⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣶⣶⠀⣀⣠⣴⣶⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣵⣿⣿⣿⣿⣿⣿⣷⣤⣼⣶⣄⡀⣤⡤⣼⡆⢀⣴⣦⣶⣦⣤⣄⢾⣿⣿⣿⣿⣿⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣬⡍⠛⣿⣿⠟⠋⠀⢹⠟⡉⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⠟⣉⣸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢿⡏⠀⠀⠀⢻⣆⣡⠆⡄⣄⣤⣤⢀⣀⣤⣴⣖⣿⠟⢛⡫⠈⠙⢍⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⢸⡇⠀⠀⠀⠸⠛⠋⠉⠉⠉⠛⢿⡿⠟⠋⣉⣉⣤⣶⣿⠧⣀⠈⣀⣁⣼⣿⡥
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠾⣇⡀⠀⠀⠀⣶⡏⣤⣶⣯⣶⣞⣷⣾⣿⣿⠟⢛⡏⠀⢀⡀⣹⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⠟⠉⠉⠀⣤⣿⣿⣶⣿⣿⣿⣿⠿⠛⠉⢩⣼⣶⣿⣿⣿⡿⠰⠏⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣶⣏⠻⣟⢁⣽⠏⠁⠀⠀⠀⠀⣀⣀⠟⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢾⣧⣴⢿⡷⢀⠀⢿⣯⡴⣦⣶⡄⣰⣆⣿⣿⠃⣌⣶⣐⣶⣿⡾
⠀⠀⠀⠀⠀⣀⠶⠛⢉⣠⣤⣾⣿⡟⠓⠈⠁⠀⠈⠛⠛⠉⠉⠹⠁⢉⣰⠃⡘⠉⠉⠉⠡⠁⠐
⠀⠀⠀⠀⠀⢣⣤⣶⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢠⣤⣶⣾⣿⣿⣿⠃⣴⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠿⢿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⢃⣼⡿⢱⡿⢻⣿⢿⣿⡿⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⣰⡿⠣⠊⠉⠀⠋⠀⠸⠋⠈⠉⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣄⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡿⠟⢻⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣊⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⡴⠂⢋⠏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀⢀⣴⠟⢀⡴⠇⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡆⠀⠀⠀⢀⣀⣀⠹⠿⠋⢈⣁⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣠⡤⠶⣿⣫⣿⣥⠴⠓⠁⢨⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⡿⠟⠉⠀⣠⠲⠛⠛⠟⠃⠀⠀⢀⣿⠠⠄⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠉⠀⠀⢠⠞⠁⢠⣰⠆⠄⢠⣶⠞⠋⠁⠀⢀⣠⠖⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢿⠿⠋
⠀⠀⠀⠀⠀⠀⢀⣄⣴⣿⣿⠟⠁⠀⠀⠀⢠⡇⠔⠀⣼⠏⠀⢠⡿⠛⠿⠳⠟⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃⠀⠀⢰⣠⣾⡏⠀⣴⣾⡟⠐⢦⡿⠁⠀⠀⠀⠀⠀⢀⢀⣀⣄⣿⠁⠀⠀⢠⡆⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⢀⣀⣾⣿⣿⣿⡗⣾⣏⠛⠃⠀⡿⣧⣶⣶⣶⣶⣿⣏⣿⣿⣿⣟⡍⠀⠀⠀⢀⣿⠛⠿⠛⠋
⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⣶⣿⡿⠋⠉⠈⣷⣼⡏⠑⠲⡄⠘⠛⠻⠿⢿⣿⡿⠿⢿⣿⣯⣿⡇⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠸⣿⠃⠐⠋⡙⢏⠀⠀⠀⢠⣿⣿⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢠⣾⣾⠋
⠀⠀⠀⠀⠹⠿⣿⠖⠋⠉⣿⣺⣆⠀⣠⣾⡿⡩⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅⣇⠚⢿⣿
⠀⠀⠀⠀⠀⠀⠙⣷⣶⣞⣽⣿⡶⠖⢻⣿⠃⢷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⡀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⣧⠀⠀⣿⡄⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠀⣼⠇⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣷⡍⠀⠀⢸⣷⡀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠷⠙⢿⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠈⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⢼⣿⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⠂⠀⠸⢋⠅⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢠⠄⠀⠁⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣶⣶⠶⠦⣤⣤⠆⣤⣤⣤⣤⣶⡷⣿⠿⢗⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣡⣾⣿⢟⣭⠞⣩⣭⣿⣯⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣿⡿⣡⡞⢁⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⣿⡷⠋⢠⣿⣿⠟⠉⠈⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠉⠀⠀⠈⠋⠁⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⣀⣀⣀⣀⡀⣀⣠⢴⣶⣶⡆
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣯⠉⠉⠉⢉⠝⣫⣾⣿⣿⡿⠁
⠀⠀⠀⢰⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢿⣿⣦⣤⠖⣱⣿⣿⡟⣽⠏
⠀⠀⠀⢸⣿⣿⣯⠀⠀⠀⠀⠀⠀⢸⣝⡦⣀⡠⠀⢀⣀⣀⣀⣠⣴⡿⣿⣻⣽⣻⣿⣿⡿⣣⣾⣿⠛⣣⠞⠁
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣻⢿⣗⢰⡛⣂⢸⣿⣿⣿⣿⣿⣷⣿⣿⡿⡿⢋⣾⡿⡟⣫⡾⠋
⠀⠀⠀⠈⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠓⠾⢬⣙⣋⣽⣿⣿⡿⠷⠞⠛⠋⡡⢚⣴⣿⢏⣥⠟⠋
⠀⠀⠀⠀⠈⢯⡻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣋⣴⣿⣿⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠙⢮⣳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣊⣵⣾⣿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⢭⣗⣒⣲⠶⠶⠖⣒⣚⣩⣥⡶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉