Hands on face Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣔⢿⣵⣺⣟⣛⣻⣿⣿⣿⣿⣿⣽⣿⣿⣃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠜⣁⣷⣾⡿⠝⣿⢻⡟⠿⢿⢿⣿⣿⣛⣿⣿⣷⣕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣁⠀⢜⣛⣽⣭⣽⣶⢿⢻⡾⠃⡼⣤⡿⢹⣿⣿⣦⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣯⢸⡿⣁⣤⣤⣭⠙⠆⠑⢲⠿⠓⠃⣹⣼⣜⡻⢿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⣾⣚⡉⣠⣿⣷⢶⢳⣄⣒⣾⣿⣿⣿⣷⣿⣷⣾⣝⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠰⣻⠟⢁⣦⠶⡻⢏⣁⡙⠛⠻⢮⣝⠳⣾⣏⢺⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣃⡼⠃⣴⡻⣺⢋⣵⠋⠉⢴⣶⠶⠦⠌⢳⣌⢻⣿⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠱⣾⢱⢰⡇⠘⠛⠒⠶⢤⣙⢳⡄⢠⣤⡙⠿⣏⣰⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡟⣿⣹⡈⠿⠾⠛⣩⠭⠭⡛⢷⡝⡆⣿⠿⣬⠃⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣰⡗⣱⣵⣝⡲⠶⢟⣡⣶⢶⡌⢰⢣⠃⡏⠀⣏⡌⡥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⠈⢢⣝⠻⠭⣭⣥⣤⡀⣿⢸⠸⣤⢁⣿⢨⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢤⣄⣽⣷⡄⠀⠙⣿⠇⡜⡸⣄⡿⡸⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠈⠛⠛⣁⣤⠾⡫⣚⣵⡟⠨⡳⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣛⣛⠛⣠⡵⢞⣫⠯⣠⣾⣗⣡⡾⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠛⢯⣭⣭⣯⣭⡶⢾⣿⣿⡿⠋⠀⣰⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢧⡀⠀⠻⣿⡿⠷⣿⠿⠟⠁⠀⣠⠾⠋⠘⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠇⠀⠀⠈⠻⣶⣄⠀⢀⣠⡾⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣀⢽⣷⣿⣏⡁⠀⡀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⣿⣿⡓⢮⣿⣿⣿⣦⣄⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠀⣀⣤⡾⡿⢿⣿⣿⣿⡟⠉⠉⠙⠛⠿⣦⣻⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠿⢛⠉⠈⠀⠀⠈⢸⣿⣹⣧⢀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⠀⠀⠀⢠⡆⣻⢿⡧⢿⡆⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠔⣡⢞⡤⣻⣷⠀⠀⢈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⠀⠠⣏⣞⡵⢋⡼⠁⣿⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⠀⢸⣿⠊⠀⠏⠀⠂⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠃⠀⠘⠃⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠙⢋⣿⠿⣿⣿⡏⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠐⠀⠀⠀⣀⣯⡉⠉⠑⠲⠄⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣠⠴⠖⠋⠉⡿⠛⠷⣶⣤⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠒⠉⠀⢀⣤⣴⣿⣿⣤⣀⠀⠉⠙⠲⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠐⠚⠉⢀⣨⣿⣿⣿⡋⠙⠲⠤⣀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣄⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⣿⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⡦⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣄⣈⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⡾⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀⣈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⠙⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⡟⠛⠟⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡦⣲⣽⢵⣷⣷⣶⣾⣶⠥⣯⣲⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠟⠉⠁⢀⣞⣿⣿⣿⣭⣿⣷⠀⠉⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣗⠤⣀⠀⠀⠘⣿⣿⣿⣿⣿⡿⣽⠀⠀⠀⠀⣀⠽⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠎⡉⠒⠦⠽⣿⣿⣿⣿⣿⣡⡤⠴⢒⡫⠕⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡱⡲⠢⠠⢤⠤⠄⠄⠆⠋⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠉⠀⠀⠀⠀⠈⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⢀⢀⡀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⣭⣴⣶⣦⣭⠵⢤⡀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⣠⠶⢭⣴⣶⡶⣶⠄⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠼⣿⣿⣿⠏⠀⠀⠈⠀⠀⠈⡄⠀⠀⠰⠀⠀⠈⠁⠂⠬⣿⣿⣿⡟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⡀⠊⠁⣀⠀⠉⠁⡀⠀⠀⠁⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠉⠁⠀⢀⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠉⠢⡄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣾⠿⠿⠿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠁⣀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠻⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⣠⡾⣿⡟⠀⢰⣧⢿⣇⠀⠀⠘⣷⠀⠀⠈⠙⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣠⡾⠋⠀⣼⠃⠀⢸⡿⠀⠻⣦⡀⠀⠙⣧⡀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⣠⡾⠋⠀⠀⣼⠃⠀⢀⣿⠇⠀⠀⠈⢿⣄⠀⠈⢻⣄⠀⠀⣄⠀⠈⠙⢿⣆
⠀⠀⠀⠀⠀⣼⡟⠁⣠⡾⠛⠀⠀⣠⡾⠃⠀⠀⣸⡏⠀⠀⠀⠀⠀⠻⣧⡀⠀⠙⢷⡄⠈⠳⣄⠀⠈⢻⣧
⠀⠀⠀⢀⣾⠟⢀⣼⠟⠀⠀⢀⣴⠟⠁⣠⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠙⣦⡀⠈⠷⠀⠀⢻⣇
⠀⠀⠀⣸⣿⠀⢸⡟⠀⣠⡶⠟⠁⣠⡾⢣⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠈⠳⣄⠀⠀⠀⠘⣿⡀
⠀⠀⢠⣿⠃⠀⢨⡅⠀⠋⠀⢀⣾⠏⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠉⢠⡄⠀⠀⢻⡇
⠀⠀⢸⣿⠀⠀⣿⠁⠀⢀⣴⠟⣷⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣶⣦⠈⣿⡀⠀⠈⣿⡄
⠀⠀⠈⣿⡄⠀⣿⠀⢠⣾⠋⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠈⢿⡄⠀⠘⢳⣄
⠀⠀⢀⣼⣷⠀⢻⡄⠀⠀⣾⡟⠁⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠈⢿⣦⠀⢻⣄⠀⠀⢻⣇
⠀⠀⣿⠏⣿⡆⠸⡇⢀⡄⠉⠀⢠⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣷⡆⠀⢻⣆⠀⠹⣦⣤⠘⣿
⠀⢸⣿⠀⠸⣷⠀⠁⢸⣿⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⢸⡎⣿⡄⠀⠈⣿⠀⣿
⠀⢸⡇⠀⣄⠙⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢸⡇⠀⠀⣿⠀⣿⠆
⠀⢸⣧⠀⣿⡀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣇⠀⢸⡟⠀⣿
⠀⠘⣿⠀⠘⣷⡀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠘⣿⠀⣹⠀⠀⣿
⠀⠀⢹⡇⠀⠈⢷⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢻⡆⠋⠀⢀⠁
⠀⠀⠈⢿⡄⠀⠈⠳⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠈⠻⠄⠀⠘⣷
⠀⠀⠀⠈⢻⡄⠀⠀⠀⢀⡈⢿⣆⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⣠⣤⡀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⣴⡆⠀⠀⢻⡇
⠀⠀⠀⢠⣶⠁⠀⠀⢰⣄⢻⡌⢻⣧⣄⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⢰⡿⠀⠀⢠⢸⣧
⠀⠀⠀⣾⡇⠀⠀⠀⢸⣿⢸⣷⠀⠉⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⣿⠇⣀⠀⣿⣾⡿
⠀⠀⠀⣿⡇⢸⡀⠀⣸⡇⢸⡟⠀⠀⠀⠙⢻⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡾⠟⠁⠀⠀⠀⠀⠀⣿⢠⡟⢠⡿⢸⡇
⠀⠀⠀⠸⣷⠀⠃⢀⡿⠀⣿⠃⠀⠀⠀⠀⢸⣿⠋⠛⠿⠿⣷⣶⡾⠿⠿⠟⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠚⠁⣾⠁
⠀⠀⠀⠀⢻⣇⠀⠈⠁⢰⡟⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⣰⠋
⠀⠀⠀⠀⠀⠻⣦⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⡿⢡⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⣀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⣿⡆⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣿⡆
⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠸⣿⠀⠀⣴⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣦⣤⣀⣀⣀⠀⢠⣿⠆⢠⡏⢻⡇
⠀⠀⠀⠀⢸⣿⠀⣾⠃⠀⠀⣿⠿⠿⠟⠀⠉⠙⠻⠿⣿⣶⣦⣤⣶⣶⣶⣾⣿⠿⠟⠁⠀⠈⠉⠛⠛⠛⠃⢸⡿⠀⢸⠇⣸⡇
⠀⠀⠀⠀⢸⡇⠀⣿⡆⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠏⠀⣾⠁
⠀⠀⠀⠀⢸⣧⠀⢹⡇⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢰⡇
⠀⠀⠀⠀⠈⣿⡀⠈⠉⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⡾⢿⡄
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠉⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠄⢸⣧
⠀⠀⠀⠀⠀⣴⡌⠙⠃⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢀⡀⣿
⠀⠀⠀⠀⢀⣿⠁⢀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⣿⣿
⠀⠀⠀⠀⢸⡏⠀⣸⡷⠆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣿⣿
⠀⠀⠀⠀⢸⡇⠀⡿⠁⣰⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⢿⡏
⠀⣀⣀⣀⣸⡇⠀⠁⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠻⡾⠁⢀⣤⣾⣿⣧
⣸⣿⣿⣿⣿⣧⡀⠀⠈⣩⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⢷⡈⠉⠛⠛
⠀⠙⠻⠿⣿⣯⠘⣿⡉⠿⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⢠⣿⠀⢘⣧
⠀⠀⠀⠀⣿⣿⡇⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⣿
⠀⠀⠀⠀⣿⡟⣿⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢹⣿
⠀⠀⠀⠀⢸⡇⢻⣆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⣿
⠀⠀⠀⠀⠈⣷⡀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⡿⠿⣿⣿⡦⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⢀⣭⠋⠉⣀⣴⣀⣽⠁⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣝⠻⣿⣾⠐⢦⠿⢟⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢥⡀⢠⣴⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢿⣶⣿⣽⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⣩⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⣩⣿⣷⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠠⡄⢀⣟⢲⣆⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣤⡠⠀⢀⣠⣟⣁⠀⣸⣍⠀⣆⠀⠀⠀⠂⠀⣀⣺⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢞⣿⣿⠟⠁⣠⣿⡿⠟⠟⣀⣏⣼⣿⣿⣄⠀⠀⠈⠉⣹⡿⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢉⢑⣤⣾⣿⡟⠀⠀⢰⣿⣿⣿⢿⣿⠿⡇⠀⣂⣠⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣈⣴⣿⣯⣿⣿⣦⣼⣿⣾⣿⣿⣿⣼⣿⣯⣰⣬⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣵⣿⣿⣿⣿⣿⣿⣿⣏⠉⠉⣹⣿⣿⢿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣉⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣵⢯⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡾⠟⣻⢿⣷⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠷⠆⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⣾⣿⣠⣾⣿⣿⣟⣽⠟⠿⠛⣉⡡⠀⠞⣡⣾⣧⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣸⣼⣿⣿⣿⣿⣿⣿⣧⣤⣞⢛⣽⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠐⠊⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣅⣙⡛⠉⣴⣼⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠲⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢿⡿⣿⢻⣿⣯⣿⣿⣿⣸⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠄⢀⣴⣿⣹⣾⢳⢊⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⠏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠈⣠⣿⣿⣿⣿⣿⣿⣿⡛⠛⠻⢿⣿⠷
⠀⠀⠀⢰⣿⣿⠿⠛⠉⠀⠙⣿⣿⣾⣯⣿⡏⠀⢠⣐⣿⢻⣿⣧⣿⣿⣿⡿⠟⢃⣀⣠⠿⠿⣿⣿⡿⢛⣁⡈⠁
⠀⠀⣠⣿⡟⢡⠆⠀⡀⠄⠀⠀⠉⠻⠿⣿⠇⠀⠀⠙⢿⡎⢿⣿⣿⠟⣩⣴⣾⣿⡿⠁⠀⠀⠀⢹⣷⣷⣾⣿⣿⣿⣶⡀
⠀⣸⠿⠛⠀⠇⠀⠈⠿⠞⠋⠀⠀⠀⠀⢸⣷⠀⠀⣰⢄⡁⠀⠉⣵⣾⡿⣿⣯⡎⠃⠀⠀⠈⠳⣦⠉⠉⠉⠉⠛⠛⠉
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣴⠋⠀⠀⠀⢸⡿⣿⠃⣿⣿⡧⠀⠀⠀⠀⠲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⢱⡟⢸⣿⣿⡇⢰⠦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠸⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠢⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡘⠀⠀⠀⠀⠀⠀⢂⡀⢰⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣄⠀⠀⠀⠀⢀⣴⣞⣻⣾⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⡿⠋⠀⠀⠀⠀⣠⣶⠁⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠋⠁⠀⠀⠀⠀⠈⠋⡀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢰⠀⠀⠀⢣⡀⠀⠠⠀⠀⠀⠀⠿⠿⠛⠁⠀⠀⢀⣠⣴⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⣀⠀⢤⡐⠀⠀⠀⠀⠀⢀⠃⠀⠀⣴⣿⣿⡿⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠀⠫⠀⠀⠀⠀⠀⠀⣡⠀⡀⠠⠀⠠⢸⠀⠀⣸⠛⠉⡉
⠀⠀⠀⠀⠀⠀⣠⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣈⡄⣡⡁⢂⠔⣰⣼⣾⣿⣒⣁⣪⡴⠋⠀⠀⢧⠀⢀⡀
⠀⠀⠀⠀⠀⠘⠀⠑⠂⠃⠐⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠙⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡀⠸⠒⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⣶⣆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⢀⣶⡤⠠⢿⠛⠉⠉⠁⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⡀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⢰⡶⠂⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣾⠀⠀⠁⠀⠈⠉⠊⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣤⣄⣦⣦⣴⣿⣿⣤⣿⡿⣷⣰⣶⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⣀⣿⣿⣽⣶⣾⣳⣼⣟⡷⠋⠉⠉⢻⣻⣧⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣟⣿⣿⠻⠛⠛⠫⠙⠉⠃⠄⠀⢀⣠⠿⠏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠙⠉⠀⠀⠀⠀⠀⣀⠀⠔⠐⠊⣥⣖⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣤⠶⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣀⣴⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⡟⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠀⠀⠀⠘⣿⣿⣿⣯⣽⣿⣷⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⢿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⣿⣿⣒⣿⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣁⢸⣿⣿⣿⣿⢿⣿⠟⣴⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⢿⣿⣿⠏⣾⣴⣶⣷⣼⣯⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⡻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⣤⢤⡄⣤⣤⣤⣤⡀⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡛⠛
⠀⠀⠉⠈⠉⠁⠉⠉⠉⠁⠉⠈⠉⠉⠉⠉⠉⠁⠀⠈⣧⡀⠀⠀⣿⣶⡄⠀⠀⢀⣠⣤⣶⠀⠀⢉⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠹⠃⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠉⠈⠇⢈⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⢿⣿⣿⣆⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢿⣷⣿⠆⠀⠙⠀⠀⠀⠛⠒⠒⠀⠀⠀⠀⢰⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣎⢻⣿⣧⡀⠀⢀⣤⢄⣉⣈⣁⣠⣄⠀⠀⣼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⡎⣿⣿⣿⣄⣼⠀⠀⢉⣉⣉⠁⠀⠀⢰⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣼⣿⣽⣿⣿⣿⣿⡄⠀⠀⠙⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠋⣿⣿⡿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠾⠋⣼⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡏⢴⣿⣿⣷⣬⣍⡻⢿⣿⣿⣿⣿⠇⠀⢰⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣦⣝⠛⢿⣿⣿⣿⣷⣽⣿⣿⣿⠇⠀⣾⣿⣿⣿⣿⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠛⠻⣦⣍⡛⢿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣷⣮⣽⡛⢷⣄⠠
⠀⠀⠀⢀⣤⡄⠀⢸⣿⣿⣿⠟⠉⢻⣿⣿⣿⣿⣿⣶⣤⣸⣿⣿⣶⣌⡻⣿⢿⣶⣿⣿⣿⣿⣿⣯⢻⣿⣿⣿⡆⣿⢇⣤⡀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠘⠿⠇⠀⠀⠙⠛⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠛⠛⢿⣿⣿⣿⣿⢿⣿⣿⣿⡆⣿⣿⡟⣸⣿⣿⣿⣿⣷⣶⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⢹⣿⣿⣿⣆⢿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠻⣿⠏⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⠛⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⣶⠀⠈⢻⣿⠋⠀⠀⠀⢻⣿⣿⣿⣿⡟⠉⢹⣿⣿⡏⠀⣿⣿⡁⠀⠘⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⢸⣿⣏⠀⠀⠿⠿⠃⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠈⢿⣿⣿⣧⠀⠀⠘⠻⠋⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⢿⣿⡿⠛⢿⠿⣿⡿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠿⠋⡁⠀⠀⠀⠀⠀⠘⣁⣀⠸⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢿⣼⣧⣴⣷⣤⣤⣤⣶⣾⣿⣿⣿⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠙⠿⡟⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠉⠛⠻⢿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⢿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠉⠉⠁⢻⡞⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠗⠛⣛⣛⣿⣶⣦⡄⠀⠀⠸⣿⡻⢷⡾⠛⠛⠀⠀⢀⡘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠛⠉⠙⠋⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠀⣼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠀⠀⠀⠀⠀⣰⣿⡇⠀⠙⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⣠⡿⠿⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⣴⡟⠄⠀⠀⠀⠀⣾⠟⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠙⠎⠂⠀⠠⢤⣤⡤⠶⠛⠉⠟⠁⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⡶⠿⠟⠳⡆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⢉⡠⢹⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣰⠏
⠀⠀⢀⣠⣾⠟⠻⠁⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⡀⢀⣦⠀⠀⢀⠀⢠⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⢠⣿⠛⠿⡄⢤⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢱⣿⡟⠃⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⢀⣠⣾
⠀⢸⡟⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠓⣶⠀⠀⠀⠀⢸⡇⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠋⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠛⠰⠂⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⡟⠁
⠀⠀⠇⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡟
⣀⣀⣀⣀⣀⡀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⣸⣄⣀⡀⣸⣁⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⡿⠿⠿⠿⠟⠻⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⡿⠿⢿⣿⣿⣿⡿⢿⡿⠿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣰⣦⡿⣹⡟⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⣦⠔⠛⠿⠿⠾⠋⠀⠀⠀⠀⠺⢧⣀⡀⢀⡾⠁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⣠⣾⡷⢷⣄⠀⠀⠀⠀⢀⣤⣠⡤⠞⠉⢀⠁⠱⠀⠒⠂⣴⡄⠀⠀⠀⠀⠀⢈⠿⠋⠀⠰⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⢀⡼⠋⠁⠀⠀⠙⢻⣦⠀⠦⣉⢱⡏⣀⠄⠀⠋⠐⣠⡄⠀⢀⣿⣷⠀⠀⠀⢀⣴⣯⠖⠀⠀⠀⢀⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⣀⡀⠂⢀⡀⠀⣻⣿⠀⣼⣿⣿⡀⡤⠚⠥⠋⠀⠘⢀⣼⠠⣻⣿⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠋⣠⠆⠠⠀⠀⠀⠀⠀⠀⢦⣼⣿⠞⠁⠀⠔⠉⠀⠠⣾⣿⣷⣿⣿⣿⣇⠀⠀⠀⠀⡠⠌⢉⣠⡾⠿⠋⠁
⠀⠀⠀⠀⠠⣿⣧⣤⣀⡐⢡⡔⡁⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠃⠀⠀⠀⡀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠀⡀⢀⠴⠶⠟⢁⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣮⣼⣿⣦⣤⣀⠀⠀⠀⠀⣤⣿⡷⠀⠀⣀⠀⠂⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡷⢾⣷⢀⣴⣦⣀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢢⣌⠙⠛⠁⠀⠀⠹⣷⣄⣤⣿⡿⠋⠀⠀⢰⣿⣆⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⣿⣿⠿⠛⠋⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠢⣤⡀⠀⠀⠈⠉⠻⠿⠦⢶⣤⣶⣾⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠐⠠⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⣀⣠⣄⠖⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠈⠻⠟⠑⣷⣄⣀⠀⠀⠈⠓⢶⣤⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠐⠋⠟⠋⠀⠀⠀⠀⠀⣀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠛⢦⣦⢀⣄⠙⠷⣦⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠻⣿⡷⢿⣿⣿⣿⣿⣿⣿⠛⠋⠋⠁⠙⠁⠀⠀⠀⢀⣴⣧⣠⡴⠂⠀⠀⣠⠞⠋⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢁⣾⣿⣿⠿⠛⠛⠛⠀⠀⠀⠀⠀⠔⠀⢀⠾⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠳⣿⣇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣶⣦⣤⠤⠒⣶⣶⣾⣿⣿⣟⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠉⠙⠛⠃⠀⣟⠉⠁⠛⠛⢻⠈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢀⣾⣀⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢶⣾⣿⠃⠀⢠⣾⣿⡟⢻⡿⣟⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣾⣿⣿⣷⣤⣸⣿⣿⣿⣼⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠉⢭⣭⣽⣿⢀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣄⡘⢋⡉⣉⣈⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⢟⣫⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣟⣋⣿⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⠛⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⡃⠀⣀⣤⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣇⣀⣶⣾⣿⣿⣿⣿⣿⣿⡿⣿⣤⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠿⢿⣿⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⢿⣭⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣶⣶⣶⣦⣝⢿⣿⣿⣿⣿⣿⡿⠋⣼⣿⣿⡇⢾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠻⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠸⣿⣿⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠃⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⣿⣿⣿⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡟⡱⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⣿⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣀⣀⡀⠀⠀⠀⢀⣤⡶⠿⠛⠛⠂⠀⠀⠀⣿⣧⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡟⢛⠿⣿⣿⣆⠀⠈⠽⠖⠉⢛⣋⣛⣀⡀⣸⡟⠛⠀⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣀⢠⣤⣠⣤⣤⣭⣤⣮⣤⣤⣶⡿⠟⢛⠉⠉⣙⣿⠿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣈⡙⠛⠌⢻⣏⣉⣠⣤⣤⣽⡏⠀⠙⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢰⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠒⡈⠙⠓⣦⠀⢸⡄⠘⠀⢻⣏⣿⣿⣿⠿⣅⣀⣤⠼⠿⠿⠿⠿⠛⠁⠀⠀⠀⣠⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠃⠙⣿⡄⢸⣧⡀⠻⠶⣀⡞⠙⢻⡍⢃⡀⢈⡉⢁⣀⣀⠄⠀⠀⠀⠀⠀⠀⣠⡟⠀⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠟⠉⠉⡄⢾⠋⠀⠀⣸⣿⣄⠋⠉⣉⣉⣩⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⣿⣿⣿⣷⣄⠉⠉⠁⠀⠀⠀⠀⢀⣴⣿⡿⠛⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⣀⠀⠀⢀⣀⣴⣿⣿⡏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠁⠀⠀⠀⠀⣿⣿⣿⣿⢟⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⢾⠁⢀⣴⣿⣎⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣏⢿⣿⣿⡿⣿⣿⣿⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠇⠀⠀⠀⠀⠀⣀⣴⣿⣿⣯⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢘⣿⣿⣿⣿⣷⣽⣿⣷⣄⣛⡁⠛⢩⣿⣿⣿⣷⣦⣤⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡀⠀⠀⠀⠀⢠⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⡙⠦⢀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⠉⠿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⠀⠀⠀⣼⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠛⡓⣹⣿⣿⣿⣿⣆⣠
⠀⠀⠀⠀⣠⣾⣿⠟⠀⢤⣶⣄⠀⢀⡛⣿⣿⣌⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⢔⣼⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠠⠙⠿⢁⡰⣌⣿⣿⣷⡀⣼⣿⡽⣿⣿⣮⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣮⣻⣷⣹⣿⣿⣷⡙⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡌⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣸⣿⣿⣿⣿⢟⡿⣾⠇
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⣿⣿⢿⣿⣿⣿⡎⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢇⣿⣿⣿⣿⡟⣼⣻⡟
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣎⢿⡿⣿⣿⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⣿⣿⣿⡿⠀⠃⠈
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡌⠃⠙⠿⣧⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣌⢢⣇⣿⡟⣰⡟⣡⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠰⡲⡏⢏⣿⣀⣿⣿⣀⠛⢾⣟⣵⡦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡈⡇⢳⣵⡞⣩⠝⠋⠉⠉⠙⢿⣿⣃⣤⠄⠀⠀⠀⠀⠀⠀⠀⣦⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⢠⡾⠀⠀⠀⢠⣄⡈⠙⣿⣾⣵⠟⠳⠀⠀⠀⠀⣠⠶⠈⢿⣴⣶⣬⠀⠀⠀⠀⠀⠀⢿⢀⡟⣠⣤⠖
⠀⠀⠀⠀⢠⣄⡀⠈⢻⣆⣾⡃⠀⠀⠀⡯⠬⢽⡞⣼⡟⢁⣄⡐⠀⠀⠀⠀⣠⠖⠲⡌⢿⣿⠚⠁⠀⠀⠀⠲⣤⣾⡿⣿⣋⡀
⠀⠀⠀⠀⠀⠈⠙⠳⣤⣿⠉⣿⡀⠀⠠⠲⠖⡸⣼⡿⠀⢾⣿⡷⠀⠀⠀⢸⠃⣶⡆⠀⢸⣿⠀⠀⠀⠀⠀⠀⠸⣿⣴⠟⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠶⠾⠟⠋⣿⣶⣿⣷⣄⠀⠀⠀⣿⣿⠃⠀⣄⠉⣡⠀⠠⠆⠀⠳⠭⠔⠁⠸⢻⣦⠀⠀⠀⠀⢠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠹⣌⢳⡄⠀⢿⡏⡆⠀⠀⠉⠀⣀⣤⣤⣤⣀⠀⠀⠀⢰⡿⡿⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⢦⡙⣦⡌⢳⡘⣄⠀⠀⢰⢿⠟⣙⣻⣿⠇⠀⣰⣿⠟⠁⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠙⢦⡳⣌⡓⢦⣝⠛⠿⠗⠛⢉⣠⣾⠟⢁⣤⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⠀⠹⡟⠻⣾⣯⣍⣓⣦⣤⣿⠛⢁⣴⠋⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠙⢷⡀⢀⡟⠀⠘⠿⡯⠃⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣴⠀⠀⠈⠻⣿⠁⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠉⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢠⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠖⠶⠶⢶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⡤⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠈⣹⣿⣷⠒⠲⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣯⣀⣀⣼⠏⢸⣿⡄⠀⢳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⠻⠿⠋⠀⠈⣿⢷⣶⡾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠦⣄⣠⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣴⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⠖⠛⠻⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⡤⠶⠞⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠻⢦⣨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣦⠾⠟⠟⠋⠉⠉⢉⠋⠉⠻⢛⣲⣶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣿⣿⠃⠈⠁⣀⠀⠀⠐⢯⣀⡀⠀⠀⠈⠂⡑⠼⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⢻⣿⣧⡴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⢿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣟⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⠏⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡄⣌⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⣿⣿⣿⡟⢡⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡄⣽⣇⣸⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⢃⡞⠁⢀⣀⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⢠⣧⣼⣷⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠊⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣾⡁⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⣻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡓⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠿⠛⠛⢉⣝⣩⣰⣾⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣸⢇⢴⡿⢻⠂⠀⢈⣾⠐⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⡆⠀⠀⣾⣧⣤⣾⠃⡞⠀⠀⠘⠏⢺⣿⣿⢠⣿⣿⣿⣿⣿⣿⣧⢼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⡏⣿⣿⡿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⣽⣿⡿⢻⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠏⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣟⣿⣿⣾⣾⣿⣧⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢉⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⡘⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣽⣯⢁⣀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠽⢿⣿⣿⣿⣿⣊⠿⣿⣿⣿⣿⣍⣹⣿⣶⣬⣻⣿⠿⠏⠃⠀⠀⠀⠀⢀⣙⠛⣶⣋⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡻⠻⣿⣿⣿⣿⣷⣟⢿⣿⣿⣿⣿⣿⣿⣭⢉⣁⡐⠆⣶⡖⣤⠄⢀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠝⢿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣁⡉⠉⠁⠴⠶⠀⢳⣀⠙⠿⠿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡟⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡐⣹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡟⣁⠀⠛⠓⣀⠈⠉⠤⠤⠀⠴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣟⣿⣤⣤⡀⠐⠂⠲⣹⣿⣿⣿⣿⣿⣹⣿⣿⣿⡚⢿⣿⣿⡹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠙⢿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣏⣹⣙⠛⠛⠛⠛⠏⠐⠒⠊⠉⣽⡿⣿⣿⣿⣖⣻⡟⢯⣹⣿⡙⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⢠⣶⣤⣬⣭⣿⣿⣿⣾⣿⣿⣏⠉⠛⠙⠋⠛⠿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⠿⢿⣿⡿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⢀⣴⠿⢋⣥⣾⣿⣿⣟⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠿⢿⣿⣿⣏⢹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢨⠟⠋⠈⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠶⠶⠶⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣤⣶⣾⡟⠛⠛⡉⠀⠠⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠈⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡿⠁⠀⠀⠀⠀⣠⣴⣾⡿⠟⠛⠻⠿⣷⣤⠀⠀⠀⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⣤⣀⠀⠀⠀⣼⣿⣿⡃⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣟⠋⢻⣷⡀⠀⢿⣿⣿⢟⣛⣣⣤⣤⣶⣾⣿⠁⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⢻⠟⠀⠀⠾⠻⡿⠂⠀⠟⣻⡿⠋⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠶⠶⠶⣦⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡿⠛⠛⠛⠛⠓⢦⠀⢹⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⠿⠿⠿⠋⠀⠀⣀⣤⣤⣤⣤⠁⢈⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣶⠶⠟⠛⠋⢩⣴⡿⠀⢸⡟⠀⠀⠀⠀⢿⣇⣠⣶⠀⠀⢠⣿⣿⣿⣿⣻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠊⠹⠋⡡⠁⠀⠈⠀⠀⠀⠀⠀⠘⣿⣿⠉⠂⠀⣿⣿⡿⠋⠗⣨⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠿⠀⠒⠉⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⠀⢰⣿⠋⠀⢀⣼⣿⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡶⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡿⠁⢀⠟⠀⠀⣠⣿⠿⢛⡃⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡟⠁⡐⠁⠀⢠⣴⣯⣥⣶⠿⠁⠀⣀⣤⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⣄⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣴⣿⣿⡿⢛⣀⣤⣶⣿⣿⡿⠿⠟⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠉⢛⢿⣿⣶⡶⢿⣿⡿⠿⠟⠉⠙⢣⠿⠃⠀⠀⣀⣴⣾⣿⣿⣿⣷⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⡃⠀⠀⢿⣿⠻⣿⣿⡄⠙⣿⣦⡀⣴⣖⠂⠀⠀⢀⣾⣿⣿⣿⣿⡟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢉⠈⠁⠀⠀⠈⣿⣷⣼⣿⣇⢠⣬⢿⣿⡿⠃⠀⣠⣶⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⠟⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣾⠟⠁⣠⣾⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⢀⣴⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏⣠⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠈⢻⣿⣿⣼⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⡿⣶⣿⣿⣿⣿⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠋⠀⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠵⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⢀⠄⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠿
⣠⣤⠤⠀⢀⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⢀⡼⢂⡤⣀⡀⠀⢀⡀
⠉⠁⠀⢀⣨⠁⠀⠀⠟⠀⠁⢀⣀⣀⣤⣤⣤⣦⣴⠄⠘⣶⡙⠀⠆⣰⣿⣾⡣⡘⠯⠗⠚⠻⣿⡄
⣿⣿⣿⣿⡟⢣⣤⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠰⠻⠋⠀⠠⠿⠯⠓⠀⠀⣠⡆⠀⠁⠙⢿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣀⣠⣶⠀⠀⠋⠀⠀⠀⠀⢀⣠⣿⢷⡀⠀⠀⠀⣉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠡⠄⠻⢿⠿⢻⠀⢰⣶⠀⠀⠀⠀⠈⠏⠀⣼⣿⣦⣵⢾⣿⠀⣀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢉⡀⠀⠐⣤⠀⠀⠀⣿⣧⣌⣉⡀⣦⢠⡼⣦⣤⢹⣿⣿⠿⣿⣿⣿⢻⣿⠆⠀⠀⠀⠤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣾⡿⣁⠀⢀⡟⠀⠀⠐⣿⢹⡿⣈⠹⡿⢻⠙⣯⣷⣏⣿⣿⡀⢹⣿⡿⠟⡁⠀⡀⠰⡂⠀⠀⠀⣀⣀⡠
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣿⣿⣷⣯⣴⣾⣆⣀⣀⣸⡿⠾⠦⣾⣶⣷⣬⡶⠾⠋⠁⢻⣿⡇⠘⣿⠃⠀⠐⠔⠁⠀⠉⣼⣿⡶⡩⢿⣧⣶⣿
⠙⠻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠈⣿⣷⠀⠙⡀⠀⠀⡶⠀⠀⠀⠈⠻⣷⣇⣾⣿⣿⣿
⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢿⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠉⣛⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠸⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⠀⠀⠉⠁⠀⠀⠀⠀⢰⣿⡿⠿⠛⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣆⠀⢿⣧⠀⠀⢳⡀⣀⠀⠀⠀⠀⠰⣦⠀⢀⡈⠻⣿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⠀⠀⠀⠀⠀⠉⠁⠀⠈⣿⡆⠀⠀⢧⣸⡆⠀⠀⠀⠀⢻⣷⠀⠘⡀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡜⠁⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠘⣷⣷⡄⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣾⡇⠀⢠⡄⠘⠀⠀⠀⠀⠀⢸⣿⣄⡁⠀⠙⣿⢿⡄⠀⠀⠀⢿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⠀⢸⣷⣾⡆⠀⠀⠀⠠⣾⡿⠿⣧⠀⠀⠙⠉⠛⢧⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠸⣿⣿⠃⠀⠀⠀⠀⠈⠋⠀⠙⢶⣤⣶⣦⠀⠀⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⡇⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢀⠤⠖⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣰⠇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠉⠉⠀⠉⠙⢦⡀⠂⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⢄⣠⠀⠀⠈⡇⠀⠀⠀⠙⢳⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠁⡕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠈⠲⣿⠀⢀⠀⠀⠀⠀⠀⣿⠉⠛⠒⠀⠀⠀⠀⠀⡀⠀⡐⠁⠤⣀⠎⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡇⠀⠀⠀⠀⠀⠀⠉⠀⡼⣧⠀⠀⠀⠐⠩⠲⡀⠀⠀⠀⠀⠀⢠⣙⠐⢧⢤⡀⠈⠑⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢷⡀⠀⠀⣀⠀⠀⢀⡜⠀⣿⡄⠀⠀⢠⣐⡰⠃⠀⠀⠀⠀⣰⠃⠘⢧⡀⠀⠈⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠙⠛⠛⠁⠀⣠⠎⠀⣸⡇⡇⠀⠀⢀⡿⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠙⢶⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢀⠞⢇⣀⣴⡿⠀⡇⠀⠀⣸⠃⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠋⠀⢀⡇⠀⢰⡏⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⢋⡀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠞⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠒⠻⢄⡀⠀⠀⠀⠀⠀⠀⣀⡤⣞⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⡅
⠀⠀⠀⠀⠀⠀⡆⢡⠘⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⢲⡖⠛⠉⠁⠀⠘⡄⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠹⡄
⡀⠤⠀⠀⢉⠁⠙⡦⠁⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆⠀⣧⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⡱⠦⠤⠀⢼⠀⢸⠁⠀⢸⠀⠀⠀⠀⠀⢀⣀⡤⢶⡄⠀⢄⠀⠀⠸⡄⠀⠀⠀⠀⠀⢣⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠈⠐⡒⠂⠈⠀⢀⣀⡤⠺⣦⣤⠶⠖⠛⠉⠁⠀⠘⡇⠀⠀⠢⡀⠀⢳⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠐⠄⣷⡀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠈⢳⡶⠶⠶⠶⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⢳⡀⠀⠀⠀⣀⣠⡴⠞⠛⠲⠤⢤⠤⠤⣄⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣤⣤⣤⣤⣤⣽⡆⠀⠀⠀⠀⠳⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣀⣀⣀⣀⣀⣤⠴⠖⠋⠁⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⠛⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⣠⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⡥⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣉⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠉⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⣀⡴⠖⠛⠛⠛⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⠔⢲⣦⠀⠀⠀⠀⢸⡇⣠⠒⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢸⡗⣾⣿⠃⠀⠀⠀⢸⡇⠻⣙⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⣠⣴⣶⣶⠶⠶⣤⡀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠀⠀⠀⠈⠛⠷⠶⢤⡤⠶⠶⠛⠁⠀⠀⠀⠀⠀⢀⡤
⠀⠘⠉⠀⢻⣦⠀⠀⠉⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠉⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢠⡏⠉⠛⠳⠶⠦⣤⣤⣤⣤⣤⣤⣤⣴⣶⠶⠶⠒⠛⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣈⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡛⠛⣿⠛⠛⠋⠙⠛⢻⠋⠉⠉⠉⠉⠙⣿⠉⠉⠉⠉⠉⠉⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣏⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⣀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠾⠷⢤⣤⣤⣤⠤⠾⠷⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣖⣚⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⠿⢟⣛⡫⣍⣲⠾⣝⣛⠿⢗⡲⠶⣖⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⠿⣟⣛⢫⣍⣶⠦⣝⣞⠯⣝⣲⠯⢝⣒⣦⠽⠖⠚⠋⠉⢸
⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⠿⡿⡛⢯⣙⡽⢦⣓⣚⠭⣕⣲⠯⢽⣒⣫⠭⠷⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸
⣤⡖⢺⣿⠿⠛⢋⣉⠡⢌⣒⡲⠭⣒⣲⠬⢵⣒⣫⡭⠷⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⠙⠳⠮⣽⣛⡶⢶⠬⢕⣒⣫⡭⠶⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⡇⠀⠀⠀⠈⠉⠻⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡟⠲⢤⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣷⠀⠀⠈⠙⠲⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡛⠦⣄⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠘⡇⠀⠈⠙⠲⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢻⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⡏⠓⢦⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⢦⡀⠈⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⣿⠀⠙⠲⣄⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠖⠚⠛⠉
⠀⠀⠀⠀⠀⠘⠶⣄⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣸⡇⠀⢀⣀⣠⡤⠶⠖⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣅⡤⢾⠟⣁⠀⠀⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣄⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠄⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⣴⠂⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠚⠁⠀⠈⠁⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣙⠉⠃⠀⠀⠀⠀⣤⣾⠆⣤⠀⠀⠈⡎⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⢀⢺⡿⣧⠀⠙⠀⣠⣼⡅⢛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡔⠉⢿⡖⢎⡚⢹⣿⡿⢋⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠠⣤⣄⠀⣴⠀⣿⡄⠿⡷⢶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⢾⣿⠀⠀⠀⠀⠀⠈⠙⠷⡘⢧⣈⣉⣀⣡⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣰⣾⣇⠀⠀⠀⠀⠀⠀⠈⣩⣭⣽⣿⡿⠿⠿⣿⣿⣿⣷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⠀⠰⣿⠟⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠉⢿⠋⠀⠀⠀⠀⠀⠿⡄⠀⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠐⣦⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⢀⣀⣤⣤⢀⣁⣠⣤⣤⣤⠀⠀⠀⢸⣯⡙⢿⣿⣿⣿⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣽⠉⠛⣿⣿⣿⡅⠈⠙⣿⣷⡀⠀⠀⠀⠀⠘⣿⣷⡌⢿⣿⣿⣷⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⢠⣿⡿⠋⢹⣆⠀⠸⣿⣷⠀⠠⠀⠀⠀⣻⣿⣿⣀⣿⣿⣿⣧⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡄⠸⣿⡇⠀⠘⣿⡄⠀⢿⣿⠇⠀⠆⠀⠀⢸⣷⡙⣿⣿⣿⣿⣿⣷⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⢸⣇⠀⢹⣇⠠⠀⠸⣿⡀⠈⣿⡄⠀⠀⠀⠀⠈⣿⣷⢸⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠸⡟⢰⠀⠀⢻⣧⠀⢼⡟⠢⠀⡀⠀⠀⢿⣿⣼⣿⣿⣿⣿⣿⣿⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣿⡠⣤⡇⢸⠀⠀⠈⣿⣷⡀⢀⣀⠀⠂⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⠠⣿⣧⢛⠇⠈⠀⠀⠀⢹⣿⣷⠘⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⣿⡇⣿⣠⠀⠀⠀⠀⠈⣿⣿⣧⠸⣇⠀⠘⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡆⠀⣿⡇⢹⠸⡇⠀⠀⠀⠀⣿⣿⣿⡇⢹⣦⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⢻⣧⠈⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠈⣿⣧⡀⠀⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠏⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡈⢿⠻⡇⢀⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠻⠿⠷⢸⡇⠁⠈⠉⠛⠛⢿⣿⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣿⣶⣤⣖⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡰⠞⢻⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡄⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⢿⡇⠀⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⣤⠀⠀⠉⠁⠀⢶⠖⠀⠀⠀⠀⠈⣿⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢛⡿⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⡀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡾⠿⠿⠟⠛⠛⠋⠁⠈⢋⠀⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⢀⣀⣤⣤⠀⠀⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠙⠛⠃⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢸⡀⠀⠀⣠⣾⡍⠳⣾⡉⢀⡠⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠑⠂⠉⠁⢈⣿⡠⡞⠉⠀⡀⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⢋⣩⠁⢱⠱⣀⠤⣄⡜⡀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⢁⡆⢹⡀⠀⣽⡁⣀⣜⠠⠄⠤⠀⠒⠒⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠂⠻⠁⠤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣩⡽⠃⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠤⡤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢱⣴⢌⡇⠀⠀⠀⠀⠀⠀⢀⠀⢠⢤⢤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣀⣀⣉⣉⣐⡂⠨⠱⢤⡈⠐⠠⢀⡀⠀⠀⠀⠀⠚⠲⠼
⠀⠀⢠⠖⠒⠦⢄⣀⠀⠀⠀⠀⡇⡎⠀⠈⠻⣿⣿⣿⣿⣿⡖⠂⠓⠦⢄⡀⠁⠀⢰⠒⢤⡤⡖⠆
⠀⠀⢸⡀⠀⠀⠀⠀⠉⠒⠠⢤⡇⢸⡀⠀⠀⠈⠟⠛⠻⢿⠃⠀⠀⠀⠂⠌⠑⢦⡾⠤⠞⣟⠷⠴⠤⡴⢖⠶⢢⠀⣠⠤⡤⣤
⠀⠀⠀⠣⣠⠂⠉⠀⠈⠹⠐⠂⠀⠐⠃⡠⠊⠀⠀⠀⠠⡋⠀⠀⠚⠓⠠⡀⢀⣮⠵⠂⠀⠀⠘⣀⣞⣓⢈⠦⢾⣐⣳⠦⣧⢫⡱
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢣⡀⢀⡴⠊⠉⠒⢄⠀⠀⢰⠋⠀⠀⠀⠀⠀⠄⠉⠁⠀⢀⡄⠠⡤⢨⠛⠋⠙⡶⡸⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣀⡧⠔⢊⠷⢄⠀⠀⢀⡤⠀⠠⡀⢀⣈⠁⠠⣣⠎⠀⠀⡰⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⠀⢀⡴⠋⠀⠀⠈⠉⠘⠓⠦⢄⠑⢾⠛⠒⠏⠃⢄⠀⡰⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠈⠢⡀⠈⣦⡸⢡⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣴⣮⣰⠡⢱⡟⣋⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⢀⠤⠀⢄⡀⠀⠀⠀⣈⣼⠭⠭⠐⢛⣁⠨⡤⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠠⠃⠀⠀⠀⠙⢦⠀⣬⡾⢄⠶⠊⣭⠿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠛⠒⠁⠐⡿⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣝⣿⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠟⠻⣄⠀⠀⣈⠿⢿⣷⣄⡀
⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⠁⠀⠀⠀⠘⢧⡾⠃⠀⠀⠈⠉⠛⠷⢦⣤⣀⡀
⠀⠠⠔⢺⡟⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣘⣇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢹⡉⠁
⠀⠀⠀⠘⡷⠒⠒⠊⠙⢏⣹⣯⠿⠁⠀⠀⠀⣿⠀⠙⠭⠯⠟⠉⠉⠉⠀⠒⢺⠇
⠀⠀⠀⠀⢿⡤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠠⠀⠀⠀⠀⠀⠀⣀⠀⢀⡟
⠀⠀⠀⠀⠈⢳⣄⠀⢀⡮⠀⠀⠀⠀⠘⣠⠞⠙⠦⣀⠀⡇⠀⠀⠀⢈⡽⠏
⠀⠀⠀⠀⠀⠀⠉⠳⠶⣇⣀⣀⡤⠴⠚⠁⠀⠀⠀⠈⠉⠓⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠴⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⣀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⣯⣛⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠈⠙⠻⢯⣿⣻⣶⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣿⣻⡽⣲⢤⣀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢽⣞⠯⣡⣴⠦⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠹⣿⣷⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠴⠚⠁

⣿⣿⣦⣀⣈⡳⠿⣜⡓⠺⡴⠈⢱⡟⢿⣼⢿⣿⢿⣯⣽⣯⣯⠙⠀⠘⠟⠠⠆⠹⡇⠙⠃⠙⠹⣷⣦⠼⠃⢑⠟⠀⣹
⣿⣿⣿⣿⣿⣶⡤⢭⣁⠀⠉⠳⠀⡟⠾⢋⣾⣻⣿⠟⡟⣷⣿⢶⠍⠁⣿⣠⣀⡰⠀⠀⠀⠀⠀⢹⠃⠀⠓⢠⡡⠴⢛
⣿⣿⠿⠿⢷⡼⣿⠎⠿⠲⠖⠀⠐⠀⣾⠾⡍⠙⠛⠙⠙⠛⠇⠈⠀⣗⠛⢷⢟⢇⠀⠰⢀⠀⠰⠄⡀⠈⢁⣠⣧⠴⠟
⣿⣿⣷⣶⣿⣰⣿⠾⠟⣣⣤⣆⠀⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡴⠂⠛⠚⠀⣤⠁⡀⠈⠤⣤⠀⠀⠈⡿⠀⠀⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣕⠖⠋⠉⠁⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣓⢒⡆⡽
⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⡇⣤⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢾⣟⣼⣿⣷⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣰⡆⠀⠀⠀⠐⠚⠙⠀⠀⠀⡀⠀⠃⠀⠨⠷⠊⣮⠹⢓⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠗⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢀⣄⠈⠃⠀⢀⡩⠔⢸⠁⠀⠁⠀⠀⠘⢿⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⣫⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠱⣆⡀⢀⠄⠀⠀⠀⠈⠉⣵
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢞⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠉⠀⡄⠐⠲
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡏⠀⠀⠀⢸⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠆⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠿⠻⡶⠚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠺
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠂⠀⠀⠀⠰⠖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠀⠡⡶⣂⣂⠀⠐⡟⢦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣡⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢋⣀⣏⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣿⠃⠼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠦⣎⣴⣷⣗⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣿⣻⣼⡿⠗⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠀⠀⠀⠀⠀⠀⢠⡕⠛⣯⣰⡿⠳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢡⡦⠦⠙⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣣⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠘⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⡄⠠⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⢋⣰⣿⢿⣿⡿⠿⢠⡄⠀⠨⠽⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠉⠛⠛⠓⠀⠀⠀⣀⣀⣀⣤⠴⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠈⠀⠀⠀⠀⠀⠀⠀⣤⣦⣠⣤⡀⠀⠀⠀⠀⠀⠐⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⢿⠃⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠰⠶⢲⠄⠀⠀⠀⠀⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠃⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⢀⡴⠁⠀⠀⠀⢀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣁⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠙⠛⠓⠚⠛⠤⠀⠤⣀⡈⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠈⠰⢺⣿⠀⠀⠀⠀⠀⠀⠀⢀⡺⠁⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣤⣉⡀⠀⠀⠀⠀⠙⣤⣯⡄⠀⠀⠀⠀⢀⡴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡇⠁⠀⠤⠀⠀⢻⣝⣿⠇⠀⠠⠄⠀⠛⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⣄⠀⠀⠀⠀⠘⣷⣤⢀⣀⠄⠀⢛⣿⣿⠀⠀⠀⠀⠀⠢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠛⠓⠀⠀⡿⠛⢯⡀⠀⣤⠻⣿⣿⣷⡀⠀⠀⠈⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣤⣁⣴⣀⣰⣿⣿⣿⡇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⣿⣭⣁⣍⣙⣻⣿⣿⣧⣀⣴⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠶⠰⠦⠀⣄⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣠⣤⠴⣆⣴⣤⠦⣤⣼⣃⣠⣷
⣀⣀⣀⣠⣤⣠⣴⣶⣤⣴⣶⣦⣤⣴⣀⣠⣤⣴⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣼⣭⣥⣴⣧⣺⣿⣿⣶⣿
⠁⣉⣹⣍⣉⣉⣉⣉⣉⣙⣋⣉⣩⣌⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣟⣿⡿⣯⣶⣾⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠑⢄⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⡐⡡⠚⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠲⠤⣀⢹⣷⡷⠓⠲⢦⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠑⢨⣻⣿⣀⣙⠢⢀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⣿⣿⣷⡀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠖⠛⠓⢶⡀⠄⠀⣠⠿⠇⠷⢹⢻⣆⡀⠠⢤⡤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠚⠁⢶⣻⣶⣤⣉⢒⣿⣃⡀⠀⠀⣀⣀⣫⡉⢉⣡⣶⣶⣦⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠙⣿⣿⡟⣿⠿⣿⣿⣿⡷⠺⣿⣿⠿⠹⣿⣿⣿⣿⣯⠔⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⣿⡟⠁⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠙⡿⢿⢡⡥⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⡀⠀⠈⢿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠈⠛⠟⠛⠛⠛⠛⠋⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠧⠴⠶⠶⣶⡤⠤⠤⣤⣶⠒⠒⣶⠒⣶⡖⠒⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢦⡀⠀⠀⣿⠇⠀⠀⣿⡿⠀⠀⣿⠀⢹⣇⣴⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠹⢦⣤⣿⠀⠀⠀⢸⡇⠀⠀⢻⠀⢈⡿⠁⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠈⠻⣿⣄⣠⣤⣼⣿⠿⣿⡿⠷⠞⠁⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠞⣿⣿⣷⣖⢦⣤⣔⣒⣻⣿⣿⢿⠿⣿⣿⢷⣤⣀
⠀⢀⣀⣠⣤⠶⠟⠛⠋⠉⠵⣾⢻⡟⢻⡿⠟⠋⠉⠉⠉⠉⠉⢠⡾⠏⢹⡟⠀⠙⠻⠿⠷⠶⣤⣤⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣀⠀⠀⠀⢀⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣤⣶⠂⣤⡠⠄⠀⠀⡈⠀⢰⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠈⠀⠀⠀⠠⠀⠀⠏⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡾⠿⠏⢠⣆⠀⠀⠀⠑⠀⠀⠀⠀⢃⣌⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠆⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣤⡨⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣄⡀⠀⠀⠀⠀⠘⠿⣧⣄⠀⢀⣤⠀⠀⢀⡄⢀⠹⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⡙⠂⣠⣿⣿⣣⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠁⢆⠠⣀⠃⠀⠀⠈⢻⣿⣿⡙⠛⠿⠿⠇⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄⠃⠐⠒⣾⡀⠀⠀⠉⠢⡄⡀⠀⠀⠀⠉⠉⠙⢓⠢⠄⠀⠀⠀⠈⢷⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠸⣿⡢⠐⢄⠀⠀⠅⠀⠀⠀⠐⠠⠄⣀⣠⠔⢋⣀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢙⣛⡀⠀⢱⢄⠠⢀⡄⡀⠀⠀⡰⠋⣹⠿⢏⣡⡄⠈⣡⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⣿⡅⠀⠀⠁⠀⠁⣌⡉⡀⠀⠀⠀⢀⠀⣨⡿⠟⠚⠳⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⢩⣉⠁⠀⠁⠀⡲⣾⣿⣷⡾⠛⣠⣤⣿⡋⠀⠀⠀⡂⠀⠀⠀⠰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⠒⠤⠤⠾⠿⠿⠏⠉⠉⣟⠁⠘⢥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠘⠒⣄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠁⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣠⣤⣶⣾⣟⡲⠦⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡛⠋⠉⠉⠉⠐⠶⣦⣤⠌⠁⠑⠌⣧⣄⠙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣌⡙⠷⠤⠤⠴⠶⠶⣤⡙⢷⣄⠀⡀⠸⣿⡆⢱⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⡛⠷⣶⣶⣶⠶⣦⣤⣳⠀⠟⣰⠟⢐⡿⢀⡈⢘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣠⣄⣙⡛⣻⣦⣭⣤⣛⣀⣬⣹⢇⡼⣋⣤⠟⣡⠞⣡⡾⣽⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠟⠛⠉⢝⣿⣿⡟⠻⠿⢿⣿⣿⣶⣿⣵⣾⣿⣼⣟⣁⢶⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠗⠀⣀⣴⣿⡟⠉⠛⢶⣄⠈⠻⣏⠉⠉⣿⠻⢙⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠆⢈⣴⣿⠋⠙⠷⣦⡀⠙⣧⡀⠸⣧⠀⠸⣇⠀⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡧⠀⣽⣟⠉⠛⠷⣦⣈⠻⢦⡈⠻⣄⠘⣷⠀⢻⡄⠈⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣋⡛⠛⠶⣤⣙⠳⣤⡙⣦⣘⣧⣸⣧⢈⣿⠀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠙⢻⠁⣻⣏⣉⣉⣛⠓⠶⢭⣹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠊⣥⣾⠀⢻⣯⣥⠬⣌⡉⣹⣿⣿⠛⡛⠙⠜⡍⢋⢹⠛⣿⣾⣿⣧⣜⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠂⢹⣯⠀⢻⣷⢞⣛⠶⢤⣿⠿⣄⣤⣴⣤⣄⡁⠊⠸⠀⠃⣼⢻⣯⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⣿⣠⠀⢻⣯⢞⣽⣸⣿⣿⣿⣿⣷⣮⠙⢻⣦⡀⠘⢠⠇⣼⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⣷⡄⠀⠹⢯⣿⣿⣿⣿⣿⣿⣿⠙⣧⢀⠻⣷⠀⠀⢸⡟⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⣴⣏⡶⢠⢉⡛⡛⢛⢋⣿⠏⠘⣿⠀⢊⣿⣇⠀⠈⣷⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠛⠿⠿⠾⠿⠿⠟⠛⠁⢐⣛⠏⠀⠤⣾⣯⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⣄⠀⣀⠠⣄⠲⣌⡳⠎⠀⠒⣚⣿⣇⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⣬⣷⡈⢷⡌⠷⣈⣁⣀⣈⣉⣿⣿⣿⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣼⡇⠈⠛⢷⣈⠁⠒⠠⣤⣤⣠⣿⠟⠟⠉⣦⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠟⠀⠹⡇⠀⠀⠀⠙⢿⣶⣦⣤⣾⠟⠁⠀⠀⢸⡿⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠙⠻⠟⢁⠀⠀⠀⢠⠟⠁⠀⠀⠹⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⠀⠈⠢⠀⠀⠳⣄⠀⠀⠀⠀⠀⠌⡀⠀⠐⠁⠀⡄⠃⠀⠀⠀⢉⣷⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠶⠾⡟⢫⢻⡄⠀⠀⠀⠀⠈⠑⠠⠀⠈⠲⣄⣀⠀⠀⣴⣤⠴⠂⠀⠀⠀⠀⠀⠀⠀⣸⣇⠈⠳⠶⢤⣤⣤⣀⣀⡀
⠀⠀⢀⣠⠶⠛⠋⠉⣁⣀⠤⠄⠘⠉⠦⠁⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠝⣇⣼⡫⢄⠀⠀⠀⠀⠀⠀⠀⡀⢠⡿⡏⠀⠀⠒⠒⠶⠤⠤⠭⣉⠓⢦⡀
⠀⠀⠘⠃⠠⢤⡔⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣭⣿⣿⣦⡀⠀⠀⠀⠀⡠⣱⡟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠶⡘⠛⠱⠶⣤⣀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣴⣿⠷⡜⣿⣿⣿⡿⣻⠿⣧⠀⠀⠀⠀⢱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⡾⡁
⠀⠀⠀⠀⠀⠀⠈⠳⡆⠀⠀⠉⠳⠄⠀⠀⠀⠀⠀⠸⡇⠀⢀⣼⠏⠁⠀⢹⢿⡝⣿⣳⡟⠀⠙⢷⣄⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⣫⢣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠾⠋⠀⠀⠀⠀⠈⢻⢿⡿⠋⠀⠀⠀⠀⠉⠳⠴⠃⠀⠀⠀⠀⠀⠀⡔⢋⡡⢂⡼⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡑⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣿⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠍⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢰⣽⡿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠐⠟⢹⠆⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀⠀⠀⠀⣞⣿⣷⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⣿⣿⣿⣿⣿⣿⣿⡿⣿⣰⣴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢿⣿⣿⣿⣿⣿⣷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⢿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠙⠛⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⣠⣤⡌⠀⠀⠀⠈⠛⠛⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠈⠉⠙⠂⣤⡄⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⠚⢲⣤⣴⣿⣿⣿⣷⣤⡄⠂⢀⡀⠁⠚⠓⠒⠲⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠏⠉⠉⠀⠀⠀⠈⠉⠙⠿⣯⣤⡔⠒⠂⠀⠘⠟⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⢶⣤⣄⡀⠀⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣉⡉⠙⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠙⢳⣄⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣸⡏⠙⣦⡀⠀⠈⠛⠛⠛⠛⠓⠀⠀⣠⣿⣿⣿⣶⣤⠛⠃⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠻⣿⣆⠈⠻⣄⠀⠀⢤⡤⢤⡄⢠⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠈⠻⣷⣤⡈⠷⡀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣿⣄⠀⣠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠙⢿⡉⠉⠉⠉⠙⣿⣿⣭⢋⣽⡿⠋⣿⡉⠉⠉⠉⠹⣿⣿⣯⠉⣻⡏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠈⠻⣦⣤⣀⣀⣀⣉⣹⣿⠋⠀⠀⠈⠿⢦⣤⣄⣀⢈⣩⣭⡿⠟⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⣹⡿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣠⡶⠶⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⡀
⠀⠀⠀⠀⠀⢴⡋⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⢹⣶
⠀⠀⠀⠀⠀⠘⣥⣀⣀⣽⡇⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⣀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⣦⣤⣴⣶⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣦⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣴⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⠶⣶⣶⣦⣤⣤⣤⣀⣠⣤⣤⣤⣤⣤⣴⢶⣶⢶⣶⡾⠿⠿⠿⠿⠛⠛⠉⠉⠉⢹⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡟⢫⡄⠀⠀⠀⢨⡏⠁⠀⠀⠀⠈⡿⠉⠀⢘⣿⠁⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⣷
⠀⠀⠀⠀⠀⠀⣠⡟⠀⢸⠇⠀⠀⢀⡸⠃⠀⠀⠀⠀⠀⡿⠀⠀⣸⡟⠀⠀⠀⠀⠀⠛⣧⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠘⣿
⠀⠀⠀⠀⠀⢀⠿⠁⢠⡎⠀⠀⠀⢸⡗⠀⢀⡀⠀⠀⠀⢿⠄⢠⣿⠀⠀⠀⠀⢀⣀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢹⡆
⠀⠀⠀⠀⠀⢿⠆⠀⢼⡇⠀⠀⠀⢸⣷⣶⡟⠁⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⢙⡿⠷⠿⠃⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢸⣿
⠀⠀⠀⠀⣠⠀⠀⣀⡈⠁⠀⠀⠀⠘⠁⠸⣿⡀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⢸⣿
⠀⠀⠀⠤⣿⣶⡶⠛⢛⣠⣤⡀⠀⠀⠀⠀⠸⢿⡀⠀⠀⢸⣿⡏⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⣙⣿⡇⠀⠀⢸⣿
⠀⣾⣃⣤⠟⠁⠀⣰⡿⠛⠋⣷⡆⠀⠀⠀⠀⠘⢿⡄⠀⢸⣿⠀⠀⠀⣀⣿⡁⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠻⣿⡇⠀⠀⠀⣿⣦⠀⠤⠄
⠀⣸⠟⠃⠀⢀⡾⠋⠀⢠⡾⢿⣧⡀⠀⠀⠀⢀⣴⣿⣦⣾⡿⠦⠶⠚⠿⠛⠛⠛⠛⠛⠛⢻⣿⣷⣤⣤⣤⣤⣿⡇⠀⠀⠀⣾⣷⣤⣤⣴⣆
⣤⠛⠀⠀⣤⡟⠁⠀⢰⠏⠁⢀⣽⡿⠘⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⢸⣿⢀⣀⠀⠘⠿
⠻⣤⣀⣾⡏⠀⠀⡺⠃⠀⢀⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣿⠿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠛⢻⣷⡀
⠀⠈⣿⠋⠐⣤⡾⠃⠀⣠⣾⣧⣤⣤⣤⣤⣤⣤⡶⠿⠿⠿⣿⠋⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⣶⣤⣦⣾⣿⠀⠃⣤⡀⠈⡷
⠀⠐⣿⡄⠀⢿⣧⣤⣼⡟⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⣄⣈⣿⠿⠃
⠀⠀⠉⠿⢷⣾⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡀⠐⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠈⠋
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⠰⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢿⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣤⣤⣴⣶⣶⣾⣿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⢿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣿⣿⠿⣿⡿⠿⠛⠛⣿⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠿⣄⠼⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠶⠂⠉⠹⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠉⠙⠿⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠶⠋⠙⠁⠀⢀⣰⣶⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣿⣦⣀⡀⠀⠀⠀⠉⠉⠛⠷⠶⣦⣀⣀⡀
⠀⠀⣀⣀⠀⠶⠖⠉⠉⠀⢀⣀⣠⣴⣶⠟⠋⢹⣷⣀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⠷⠀⠈⠛⠳⢶⣶⣦⣄⣀⣀⣀⣀⣀⠹⢿⡿
⣀⣈⣻⣿⣷⣶⣶⣶⣶⣾⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠉⠁⠹⠋⠙⠋⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⢀⣀⣀⣀⣀⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠛⠛⠛⠛⠿⢿⣿⣿⡿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣴⣶⣦⣄⡈⣹⣯⣴⣿⣿⣿⣿⣿⣛⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⣫⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⣿⡿⠟⠋⠀⠀⢸⡟⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠌⠻⣿⡟⠁⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⣾⣯⣸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠦⠄⠀⣻⣇⠀⠀⢛⣩⣿⠿⣿⠾⠃⠘⡟⠃⢀⣀⣬⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠈⠙⠀⠀⠈⠈⠉⠀⠈⣀⠀⢀⣰⣾⠿⢿⣻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⠋⠙⢷⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣽⡿⣛⣥⣽⣿⡟⠂⢐⡀⠝⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡟⢓⠓⠀⡡⠀⠙⢮⣻⣿⠿⠟⣛⣛⣉⠀⢀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡟⡀⡀⢼⡄⠀⠘⣦⣄⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠳⣄⠀⠀⠀⠠⢀⣀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣶⣿⡏⠀⠀⡼
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⣿⣿⣿⣿⢻⡄⣰⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢠⠃⠈⣿⣿⣇⠀⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢲⣴⣿⣿⣿⣿⣶⣆⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣷⣀⣤⣴⣶⣿⣧⣉⠘⢿⣿⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠘⠛⠁⢿⠿⠛⠿⠛⠛⠛⠀⠉⢿⣷⡾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠆⢰⣖⣂⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡀⠈⠛⢉⣁⣀⣀⠀⠀⠀⠀⣠⣤⣤⣴⡄⣿⣿⣜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⣦⠠⠟⡩⢽⠋⠣⣄⠀⣠⣿⠋⣄⠈⠛⢽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣷⡇⠀⣰⠃⠁⡔⠀⣈⠀⣿⡀⠩⠉⢇⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣽⡇⢀⡱⠃⡀⠀⢠⠉⡃⣻⠗⣄⡀⠉⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣟⠁⣾⠁⣸⡁⠀⠀⢀⡟⠉⢄⠘⠌⣤⣄⣿⣏⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⣾⠀⠃⠀⠾⠦⢰⠖⡄⠈⢃⠐⡜⠏⢸⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡿⠀⠀⠀⠁⠀⢸⠀⠑⠀⠈⢆⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⡇⢀⢢⠈⠘⠀⢿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠘⡄⠀⠗⠈⠀⠀⠀⠃⢸⡟⣿⣯⠛⠷⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣹⡟⠀⡆⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⢸⡇⣿⠈⢧⠀⠀⠀⠀⠀⠉⠛⠦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠉⢀⠞⠁⠀⠀⢻⠀⠀⠀⠀⢰⠁⠀⡰⠀⠀⠀⣼⠇⠸⡀⡸⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠉⢳⡄
⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠈⠣⡀⠀⡴⢈⠀⠀⠀⢀⡟⠀⠀⠃⠀⠀⢀⡟⠀⢠⡈⠓⠦⢄⠀⠀⠀⠞⠁⠀⠀⠀⣴⠃⠀⣇
⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⡈⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⡟⠀⣬⣷⠀⠀⢰⠂⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⣿
⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⣰⠋⠉⠉⠛⠒⢾⡀⠀⠀⠀⠀⠀⢿⡇⢸⡏⠊⠀⠀⣿⣼⡟⠀⠀⠀⣰⠏⡄⠀⠀⠀⣿
⣼⠁⠀⠀⡇⠀⠀⠆⠀⠀⠀⠀⣠⠷⠤⠠⢀⣀⡀⠀⣸⣷⡀⠀⠀⠀⠀⢸⠃⣸⠃⠀⠀⢸⣿⣿⠁⠀⢀⣾⢇⠼⠀⠀⠀⠀⣿
⣿⠀⠀⢰⣷⡀⠘⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠙⠳⣿⣼⡇⢀⡠⠤⠶⠾⠶⣿⡇⠀⢠⣿⣿⠇⠀⢀⣿⡏⠀⠀⠀⠀⢀⣶⣿
⣿⠀⠀⠀⢻⣷⡀⠀⠀⡆⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⡯⠀⠀⠀⠀⠀⢙⣇⣠⣿⣿⠃⠀⠀⣸⣿⠇⠀⠀⠀⠀⣾⡟⢸
⣟⠀⠀⠀⠀⠙⣿⣆⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢡⢌⣷⣠⠴⠚⠉⠀⠀⠀⠹⣿⠃⠀⠀⢰⣿⣿⠆⠀⠀⠀⠸⠏⠀⢸
⣹⣀⣀⣀⣀⣀⣈⣿⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣾⣚⣋⣁⣀⣀⣀⣀⣀⣀⣀⣹⣀⣀⣀⣾⣿⣏⣀⣀⣀⣀⣀⣀⣀⣼
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠴⠶⠶⠆⠀⠀⠀⠀⠰⠶⠶⠶⠀⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠒⠛⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣰⣿⣧⡄⠀⣰⣯⣿⡇⠀⠀⠀⣀⠤⣐⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣍⣿⣿⣿⣿⣿⣧⣾⣿⣿⣟⣠⣴⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣩⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⢀⣠⠴⠒⠒⠒⠲⢦⣍⠓⢌⠙⠛⠃
⠀⠀⠀⠀⠀⢠⡾⣫⣾⣿⠏⡡⠒⠉⠉⠉⠉⠉⠓⢏⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠑⢄⠀⠀⢀⡴⠶⠶⠶⣦⣄
⠀⠀⠀⠀⢠⣯⣾⣿⣿⠃⡜⠁⠀⠀⠀⠀⢀⣤⣄⠈⢧⢀⣀⠀⠀⠀⠀⠀⢸⣧⠀⠈⢣⣴⠋⠀⣤⣤⣤⡀⠙⣦
⠀⠀⠀⠀⣿⠟⢹⣿⡏⢸⠁⠀⠀⠀⠀⠀⢸⣿⡿⠀⢸⠿⣿⡇⠀⠀⠀⠀⣼⡟⠀⠀⠀⠁⠀⠘⠋⠀⠈⢻⡄⢸
⠀⠀⠀⠐⠋⠀⣼⠟⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠤⣄⣀⣠⣤⣴⣾⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡾
⠀⠀⠀⠀⠀⠀⠉⠀⠃⢿⡀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣡⣤⠶⠚⠛⠉⢿⠉⠉⠉⠙⡗⢦⡀⠀⠀⢀⣀⣠⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⣧⡀⠀⠀⣀⣠⣶⡿⠋⠉⠀⢺⠀⠀⠀⠀⠘⡆⠀⠀⢀⣅⣀⣷⡄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⣀⣤⣼⡆⠈⠙⠛⠋⢩⠟⠁⢇⠀⠀⠀⠸⣆⡀⠀⣀⣠⣿⣶⣾⣿⣿⣿⣿⣿⡄⠀⢳
⠀⠀⠀⠀⢠⠞⠁⣀⣀⠀⠀⠀⠀⣰⢻⠀⠀⠸⡄⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠸⡇
⠀⠀⠀⠀⣾⠀⢸⡟⠁⠀⠀⠀⠀⡇⠈⢆⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣷
⠀⠀⠀⠀⢻⡄⠸⣧⡀⠀⠀⠀⠀⢻⣤⣼⣿⣿⣿⣿⣿⣿⠋⢹⠋⠀⠀⠀⠀⢙⡿⠿⣿⡋⠹⣿⠇⠀⣿
⠀⠀⠀⠀⠈⢷⣄⠈⠙⠃⢀⡴⡄⠀⢿⣿⣿⣿⣿⣿⣿⠃⢀⣈⣀⡀⡴⠛⠳⣿⠀⠀⠹⣧⠀⡟⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠙⣄⠈⢿⣿⣿⣿⡿⠛⠳⡏⠉⠈⢳⣷⠀⠀⠙⣇⠀⠀⢻⠞⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢻⣿⠋⠻⡄⠀⢳⡀⠀⠀⠹⣇⠀⠀⢘⡦⠖⣡⠄⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠷⣄⠘⢦⡀⠑⣄⣀⡠⠽⠒⣊⣡⠴⢚⣡⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠉⠉⠉⠁⠀⠀⣀⣀⣭⣴⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠾⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠘⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⣚⣡⣵⣶⣿⣿⣿⣿⣷⣄⠀⣀⣠⣤⣴⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣇⠀⠀⠀⣀⣤⠶⣛⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠤⣀⠀⠐⠀⠄⠀⠀⠐⠀⠀⠀⠀⣰⣾⣷⣦⡄⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠘⣟⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣄⣼⠟⠋⢻⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⣿⡟⠋⠙⠑⠶⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠘⣿⣿⡄⠰⢀⣶⡆⢹⣿⣿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⡀⠤⣒⣨⣥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⣾⣿⣦⢛⣿⣶⠾⠉⠛⠀⠈⢙⠿⣿⡟⠀⠀⠟⠿⡿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⡿⠋⢁⣠⣐⣪⣵⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣿⣯⣿⡙⢿⣿⣧⣄⣨⣁⠀⠸⣧⠘⠁⠀⡴⢋⡼⠋⢀⢠⠶⠖⠀⠀⠈⡁⠂⠀⠘⠀⠉⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢻⣦⣸⣿⣿⣧⡙⢿⣿⣯⣽⡛⠁⡀⢹⠀⣇⠠⢒⣩⣴⣶⣦⣤⡖⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣄⠹⣿⣿⣿⣿⣷⣼⣿⣿⣯⡁⠘⣷⣆⣤⣿⣯⣽⠶⠷⠶⠶⠀⠀⠀⢀⣦⣀⢶⣴⠖⢧⡛⣫⡴⣶⣦⡄⢰⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣌⣻⣿⣿⣏⠻⠿⣿⣿⣤⡄⣿⣿⣿⣟⣉⣖⣀⣶⡶⠀⠀⠀⢀⣾⢱⣾⡷⠉⠀⠾⠇⣸⣿⣿⣿⣧⠞⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⡀⢠⡏⠀⠀⠈⠻⠿⠿⠿⠻⠿⠛⠀⠀⠀⢀⣾⣿⣆⣙⡁⠀⠄⠀⣴⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡿⠟⠉⠚⠿⠿⣿⣿⣶⣶⣄⣸⣀⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⢁⠀⠀⠞⢡⣿⣿⣿⡿⢣⡞⣠⣿⣿⣿⣿
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢚⣡⡴⠶⠛⠿⠋⠀⠉⠙⢿⣍⣉⣽⣿⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⡟⠁⣠⡏⠀⢀⣴⣿⣿⣿⣿⣧⣿⣿⡿⠟⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢘⣋⣩⣴⡶⠞⢁⡴⠟⠃⠀⠀⠉⠉⠉⠁⠈⠉⠀⠙⠻⠿⠿⠿⠟⠃⣄⣠⡿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣴⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣛⣩⣤⣧⣯⠄⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣤⣿⣿⡜⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⢟⣿⣭⡭⠀⠉⠀⢈⠀⢀⣾⣷⣶⣄⡀⠀⠀⢀⠄⣀⣴⣶⣿⣿⡋⢉⠁⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠶⠞⠋⠉⠁⠀⠀⠀⠀⣀⡴⠊⠋⠙⠿⢿⣿⣶⣴⣆⠔⣻⣿⣿⠸⣿⣧⣾⡀⣰⡶⢻⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⡶⠶⠟⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣍⡼⠒⠀⢀⣈⣦⣤⢶⣶⠾⣿⠿⢿⣦⣀⠀⠀⠀⠈⠻⠭⠶⢛⠟⢈⣽⢶⣜⣿⣿⣿⣿⠀⣿⣥⣿⣷⡟⢃⠀⠀⣿⣿⣿⣧⣠⣠⣤⣤
⣿⣿⣿⣿⣿⣿⠟⢉⣤⣴⡾⢟⣩⣤⣴⠆⢑⣾⣵⣾⣿⣿⣿⣷⣦⣄⡀⠀⢀⣀⣡⣾⡿⢁⡴⠟⣵⣿⠿⢻⣤⣟⣼⠟⠋⠀⠈⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣇⣼⠛⣿⡿⣶⣿⣿⣫⣷⣿⣭⣾⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣻⣴⣋⣀⠀⠰⣷⣻⠶⠞⠉⠁⠀⠀⢀⣈⠀⠀⢻⣿⣿⣿⣿⠿⠿⠟
⣿⣿⣿⣿⣿⣿⠋⢂⣡⡅⢫⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⢻⣿⣿⣿⡿⠋⠛⠿⠿⢟⣩⣿⣯⣭⣤⠐⠉⠁⠀⠀⠀⠀⠀⠚⠃⣠⠶⠒⡀⣿⣿⣿⣿⡜⣤⣤
⣿⣿⣿⣿⣿⡏⠀⠀⢛⣶⣾⡟⠻⣿⡿⢿⣿⣿⣿⡆⠀⠀⠻⣿⠏⠀⠀⠀⠀⠀⠤⣄⡀⣀⣠⣤⣤⣶⡄⠀⠀⠀⠀⠀⠀⠘⢡⠖⢀⠀⠹⣿⣿⣿⣷⠀⠻
⣿⣿⣿⣿⠏⠀⠀⠲⠊⣿⠻⣷⣆⠸⠇⣼⣿⣿⡀⢻⡀⠀⠀⡐⢆⡠⣿⡷⠖⠀⠀⠀⠉⠉⣉⣙⣛⣛⣿⣦⠀⠀⠀⠀⠀⠀⠀⠘⢥⡆⠀⠘⢿⣿⣿⣧⡤
⣿⣿⠟⠀⠀⠀⠀⠀⣸⣿⣿⣎⣻⢶⠀⣿⣿⣿⠇⢠⠁⠀⠀⢷⣤⡅⢦⡉⠰⢎⠉⠀⠐⠶⠶⢏⡛⠋⣽⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠻⣿⣿⣷
⣿⣿⡆⠀⠀⠀⠀⠀⢻⣿⣿⣛⢛⡿⠀⠛⠉⣠⠄⠀⠀⠀⠀⠀⠻⣿⣦⠙⢷⣄⡀⠺⣇⣠⣤⣈⡁⠾⠿⠞⠛⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⣿⣿⣿⣤⣀⡀⣠⡶⣿⠟⢻⡿⠋⠀⠀⢀⣾⠏⣀⡀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠈⠻⢶⣌⠻⢿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣻⡇⠘⣷⠌⠀⠀⣶⢀⣿⣡⣾⡿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡧⣀⠀⠀⠹⣿⡂⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠁⠀⠀⢠⠃⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡹⣧⠀⠀⠈⢷⣄⠀⠀⠙⢿⣿⣿⣿⣿⣯⣤⣈⡻⢿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⢲⣿⣿⣷⣾⣷⣤⣶⣤⣄⣀⡀⠀⠀⢸⣿⡇⢻⡄⠀⠀⠀⠹⣷⣤⡀⠀⠻⣿⣿⣿⣿⣿⣿⠓⠀⠙⠿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⡟⣠⣤⣴⣤⣿⣿⣯⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⢿⣦⣸⣿⣧⠈⣿⣦⡀⠀⣀⣹⣿⣿⣶⣤⣌⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠉⠢
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡿⠛⠁⠀⢠⣶⣿⣿⣿⠿⠀⣾⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣷⣶⣦⢤⣤⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣤⣿⣿⣿⡃⢠⣶⣠⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣰⡟⠁⠀⠉⠛⠿⠿⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣽⣿⡇⠀⢀⣉⣿⣶⡆⢠⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⢽⣧⣀⡀⠀⠀⠁⠟⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠻⣿⠀⠀⠀⠻⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⣦⣀⠛⠀⠈⢩⣥⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢻⣿⣇⠙⣿⣷⣦⣄⣀⠀⣬⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⣾⡿⣿⡄⠈⠻⣿⣿⠏⢠⣿⣿⡀⠑⣶⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣍⡴⢻⡿⣿⣷⣦⣵⣦⡀⠙⠃⠀⢾⣿⢟⣼⣿⣾⣿⡆⠁⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢿⣿⣿⣷⣼⣧⣿⣿⡿⣿⠿⠿⣄⠀⠀⣼⣿⡿⠉⠉⢻⣿⣿⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢐⡤⡀⠹⣿⣿⣿⡿⣿⣍⠀⠀⠀⠐⠈⠀⢠⡟⣾⠃⠀⠀⠈⣿⣿⡇⢰⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠇⢹⣿⣿⣧⣀⠉⠛⠦⡀⠀⠀⠀⣼⢸⠇⠀⠀⠀⠀⠸⣿⣧⣽⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⣿⣿⣿⡏⠙⠓⠦⠀⠀⠀⢠⡇⠀⠀⣠⠤⣤⣄⣀⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⢠⣾⣿⣿⣷⣀⣠⠤⠶⠺⣆⠀⡇⠀⠀⣯⣤⣄⣤⣤⣿⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣏⣿⣿⣿⣿⣧⣤⣴⣶⣿⢀⠃⠀⠀⣿⠻⣿⣿⣿⣿⣿⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣾⡿⣽⣿⣾⣾⠀⢀⣿⣷⣝⢿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⠿⠿⠿⠿⠿⠂⠸⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠚⠋⠉⠀⠀⠀⠀⠉⢉⠒⠈⠷⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠟⠲⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣆⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⠢⠉⠂⠀⠀⠑⠀⠀⠀⠀⠐⢤⠀⢸⡾⠊⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡑⣄⣣⣼⠰⠘⡉⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⡂⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣲⠭⡃⢈⡉⠉⠁⠀⠐⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⣰⠲⠂⠀⠓⠈⠓⠒⠀⣀⠌⠩⠃⠀⠁⠉⠀⠒⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡗⠦⡀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠤⠴⣶⠃⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⠖⠊⣯⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⠔⢚⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⡂⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠿⢼⡛⡃⠀⠀⣠⡤⠤⠦⠤⣤⣤⡀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠖⠲⠶⠀⣦⢠⣀⢀⡠⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠈⠀⡠⠊⠀⠀⠒⠒⢢⣀⣠⡤⣤⢤⠌⠁⠀⠀⠀⠀⠀⢀⣤⢶⣶⠶⣦⡀⠀⠶⠋⠀⠀⠀⠢⡀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⡀⠈⠀⡀⢀⣀⡴⢤⣴⡾⠶⣤⣄⣄⣍⣵⡶⠾⣶⡖⠲⣶⡶⠛⠛⠿⠟⠲⣾⡖⣾⣥⠤⣤⣀⡀⠀⠀⠐⣄⠀⠐⢄⡈⠲⠄
⠀⠀⠀⠀⢠⡴⢰⠞⣯⡏⢹⡞⠁⢸⡇⠀⢠⡏⣿⡌⣿⠘⣿⡀⢸⡇⠀⣾⡇⠀⠀⡰⠄⢨⣿⠀⣿⠀⢀⣽⠃⣿⣳⢲⣶⣼⠦⣤⣀⠁⠀⠀⢀
⠀⠀⠀⠀⠘⢧⠘⡀⢻⡇⢸⡇⠀⠸⡇⢀⢼⣁⣿⣿⡜⣆⠘⣷⠈⣿⠀⢻⠃⠀⠀⣷⠀⢸⡇⢰⡏⠀⣸⡇⢸⣯⡟⢼⣿⠋⣴⡟⢈⡽⠳⣤⡏
⠀⠀⠀⠀⠀⠈⢳⣷⡸⠿⠶⣻⠛⢛⡛⠉⠉⠉⢷⡝⢷⡘⠇⠹⠀⠙⠀⠸⠀⠀⠀⠙⠀⠾⠃⣾⠇⢠⣿⠖⢿⣍⣀⡾⠁⣠⠞⣠⠊⣡⢞⡴⠋
⠀⠀⠀⠀⠀⠀⢸⠁⠘⢦⡀⠘⣆⠘⣧⠀⣷⠀⠘⡇⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⡾⢃⡸⠃⢩⣭⠛⠰⠷⣤⣯⣞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠳⢦⣀⠙⠆⠘⠂⠘⡆⠹⠀⠀⠉⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠁⣠⡞⢁⡴⠋⠀⣠⡯⢙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠒⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⠈⠁⠀⠶⠋⠰⠛⠀⠀⠀⠉⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢀⣠⣔⣒⠒⠒⠒⢒⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡌⡀⢌⠉⠉⣉⠉⡏⠀⢸⡆⠀⠀⠀⠀⠀⠀⣴⣢⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡇⠤⠜⠁⢨⣀⡕⣇⠀⠀⡇⠀⠀⠀⠀⠀⢰⠁⢿⠀⡜⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⢄⡀⠀⢀⣀⡀⢻⠀⠀⣷⠀⠀⠀⠀⠀⢈⡶⣄⢉⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠲⢼⣭⣅⠧⠇⢸⠲⣀⣹⠀⠀⠀⠀⢀⢮⣉⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⢲⡶⢶⠶⠤⠾⠞⠋⠀⠀⠀⠀⣠⢏⢒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠾⣧⣼⠶⠤⠤⢤⣄⣀⡀⢀⣴⠓⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠓⠲⠶⠦⠤⠤⠤⠤⡤⠴⣖⣯⢿⡙⣦⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⢸⠀⠀⠤⠤⠤⠤⠤⣄⠀⡇⠀⡿⢬⡦⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢙⡇⠀⣴⣉⠆⠀⠀⢸⣦⣷⠀⠓⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣹⡇⠀⣇⡠⣄⠀⠀⢸⣅⢺⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠓⡇⢹⠀⢻⣓⣃⣀⣀⣸⠟⣼⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⢺⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣻⢀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢺⡀⠸⣇⣒⣒⣒⣒⣒⣒⣒⣿⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡗⢚⠘⡄⠈⠭⢽⠁⠀⠀⠈⠉⡟⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠁⠈⣦⡇⢰⠒⢾⠀⠀⠀⠀⠀⣏⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⣉⡿⠀⠀⠀⠀⠀⢷⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⡇⠀⠀⠀⠀⠀⢸⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⠒⣧⡀⠀⠀⠀⢠⢺⣉⡗⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⠛⣀⡇⠀⠀⠀⢹⢤⣀⣀⣀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣑⣒⣊⡩⠕⠁⠀⠀⠀⠀⠑⠲⠤⠭⠴⠂

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠰⣸⣿⠀⠀⢠⣴⣦⣄
⠀⠀⠀⠀⢹⣿⡀⠀⢾⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⠀⠀⠀⢀⣘⡟⣧⡀⠀⠛⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⡵⣤⠄
⠀⠀⠀⠈⢻⣧⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣤⣤⣷⣤⣶⣾⣿⣦⣤⡤⠀⠀⠀⠀⣾⠏⠀⣶
⡀⠀⠀⠀⡰⣿⠈⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⡿⠷⠦⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠈⠓⡇⠀⠀⢨⡄
⡇⠀⢰⠁⢀⣯⠄⠴⢿⣿⣿⣿⣿⠿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠛⢿⣿⣿⣿⠛⠿⠿⠗⠂⠀⠸⠁
⣷⣾⣷⣦⣿⠃⠠⠴⠛⠉⠉⠀⠈⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠴⠆⠀⠈⠃
⣿⣿⣿⣿⡏⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⣿⣿⣿⣿⡇⣼⣿⣯⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡤⠴⠿⣿⣿⣷⣄
⣿⢿⣿⣿⣣⣿⣿⣿⣿⣦⡀⠀⠀⠀⢺⣿⣭⣍⣷⣄⣀⣤⣴⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⡟⠂⣀
⢀⢘⣿⣿⡏⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⠀⠉⠁
⣼⢸⣿⢿⡇⠀⠻⠿⣿⡿⠛⠿⢷⣄⠀⠀⠘⢻⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⢸⡀
⡇⠀⢫⠀⢻⠀⠀⠴⠯⠀⠀⠀⠀⠈⠳⠄⢀⣼⣾⣿⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣴⣿⣧⢸⣷
⠃⠀⠀⠀⠀⠆⠀⣠⡄⠀⠀⣤⣄⣠⣤⣤⣼⣿⣿⣿⣿⣠⣤⣴⣦⣴⣶⣶⣦⡄⠀⢀⡃⣿⣿⣿⣿⣿⡏
⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠛⢻⡟⠿⢿⠿⠿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠏⢲⠛⣱⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⠋⠀⠀⣀⣀⡀⠀⠈⠉⢃⣀⣻⣿⣯⣰⡿⣧⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⣀⡀⠀⠀⠛⠙⢿⣿⢿⣿⠟⠋⢹⡍⠹⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠉⠁⠀⠀⠀⢀⠘⡀⣠⠁⠀⠀⠀⠈⠀⢀⣰⣿⣿⣿⣿⣿⣯⣿⡆⠉⣿⣿⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⣿⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣔⣢⣼⣿⡿⠟⣿⠗⢠⡿⡟⠁⠀⠸⣿⡿⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⢿⣽⡟⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⢟⠁⠀⠁⢘⠏⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⠇⢸⣿⣿⡿⠃⠋⠀⠀⠀⠉⠐

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠰⣄⡀⢘⡯⣤⣠⢼⢷⡖⡀⣰⡁⢼⣠⢔⡵⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣙⡒⣜⣿⣾⣯⡫⠿⠿⡶⢷⣶⣷⣵⣼⣿⣿⣾⣗⡉
⠀⠀⠀⠀⢤⣀⠀⢀⣔⠪⡍⡁⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⢿⣿⣭⠑
⠀⠀⠀⠀⡉⠙⣳⢿⠿⣿⠟⠛⠛⠓⠺⢄⣀⡀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⢛⣿⠁
⠀⠀⠀⠀⠀⡔⠁⡾⠋⠀⣔⠀⠀⠀⡂⠐⠉⠋⠉⡛⠒⠀⠉⡻⢛⣛⢿⣦⣈⡏
⠀⠀⠀⠀⢰⡇⢸⡇⢈⢼⣵⣿⣟⡆⠁⠘⠀⠀⡆⠃⠀⠀⠨⢠⣽⡃⡘⠀⠀⠅
⠀⠀⠀⠀⠸⣇⢸⢧⠀⡌⠛⠿⢗⠁⡘⡘⢰⣼⠁⢰⠀⠀⠀⠒⠒⠊
⠀⠀⠀⠀⢰⡸⢢⣙⠶⢤⣀⣀⡤⠾⣊⣴⡟⡙⠀⢸⣀⣀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠈⢱⢃⢨⠓⠂⡤⠒⠒⢉⠏⠷⠃⠆⠀⠈⢃⣀⣀⣀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⡏⢸⠀⡎⡄⢣⠰⡀⡇⢸⠀⡇⠀⠀⠀⠋⠀⣈⡁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢈⡆⠘⡇⠃⠀⡇⠃⠀⢀⡧⠄⠊⠀⠀⠀⡁⠂⠀⠀⠈⠣⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⡠⠄⠒⠂⠉⠀⠀⠀⠀⠀⠀⢰⠁⡄⡀⠀⠀⠀⠀⠀⠈⠉⠊⠛⣲⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡸⠀⣅⢙⣀⣀⣀⡀⠀⠀⠀⠀⠤⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠂⠀⠀⠉⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠋⠁
⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⣿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠉⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢀⠀⠀⢀⠀⡯⠁
⠀⠀⠀⠀⠀⠀⢀⡟⢠⡄⠀⣤⠄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⣘⡣⠤⢙⣱⢹
⠀⠀⠀⠀⠀⠀⢸⢷⡊⣉⣉⣉⣲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡁⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠘⡈⠳⡒⠒⢂⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⣄⣀⡠⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⠉⠡⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣣⣼⡄⠀⢀⣷⣽⣷⣦⣄
⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⢀⣼⣿⣿⡆⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⣠⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⡟⢻⣿⣿⣿⣶⣿⣿⣿⡛⢃⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⡇⠸⣿⣿⣿⣿⣿⣿⡿⡇⢸⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣁⡇⢿⣿⣿⣿⣿⣿⡇⢡⣸⠀⠀⣾⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⠿⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠚⠛⠛⠉⣿⠿⠿⢿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⠀⢠⣤⡄⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⠿⠟⠛⠉⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⣿⡿⢿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣾⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠙⠦⣴⠿⠋⠀⠉⠉⠛⣻⠏⣿⠋⠉⠻⣅⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⡟
⠀⠀⢀⣠⣶⡟⠛⠛⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠃⠀⠀⠀⠀⠙⢶⣄⣀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⢰⠇
⠀⢠⠟⠁⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠙⠛⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⠒⠒⠒⠚⠛⠛⠛⢉⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⣠⠖⠛⠉⠉⠛⠓⠶⡋⠁⠀⠈⠙⣷⣴⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣼⡁⠀⠀⠀⠀⠀⠀⠀⠈⣀⣠⣤⠾⠋⠁⢰⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠶⠖⠛⠛⠉⠉
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⢘⣿⠰⠂⠀⢠⠤⠀⡀⢀⠤⣀⡀⣀⣀⣁⡀⠀⢀⠀⡀⠀⠀⠀⠈⡀⠀⠀⠠⠎⡇
⠀⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⠸⠔⢸⠶⢸⠂⡽⠧⠽⠎⠸⠀⠿⠞⢽⡇⠀⠷⡌⠈⠏⠀⠶⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⡀⢠⣤⢤⡀⢀⡠⢀⡠⢉⡭⢡⠤⡀⣠⡄⣀⢀⣀⡤⢠⠤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⡎⠀⠀⠬⠗⠾⠡⠜⠱⠸⠂⠸⠂⠸⠀⠸⠛⠄⠧⠀⠟⠶⠣⠄⠾⠅⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⣣⣘⣤⣀⡀⣀⣀⠀⠀⠠⠤⠤⠀⣀⣀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣓⣸
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠉⠀⠀⠀⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⣠⡔⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠉⠀⠁⠀⠈⠓⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠦⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⢏⡀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⠢⡈⠒⢄⣸⣢⡄⡀⠀⠀⡰⠒⠂⠤⠤⠤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠁⠀⠀⠀⠀⠑⠶⢿⣧⡀⠈⠉⠰⠃⠀⠻⠿⠁⢠⣜⣀⣀⣠⣄⡀⣤⣤⣤⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣤⠤⢴⠾⠟⠛⠿⠭⠽⠛⠛⠛⠒⠐⠛⢳⣿⣦⣤⡇⣤⣴⣶⣤⣠⣏⣭⣭⣭⡿⠛⠒⠦⠶⠶⠶⠶⢶⡤⣭⣽⣯⣛⣶⡀
⠀⠀⠀⣀⣠⣀⣥⣶⣿⣇⠀⠀⠀⠀⠀⠾⠽⣤⣾⣉⡇⠀⠀⠀⠀⢘⢻⣡⠾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣤⢧⡔⢲⣴⢆⣶⡄⢲⠖⠀⠉⢩⠃
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀⡀⠤⠿⠛⠁⠐⠒⠒⠒⠒⠒⠂⠚⠓⠚⠒⠚⠛⠛⠛⠛⠛⠛⠛⠻⠿⠧⠤⠤⠍⢈⣁⣉⠈⠉⠉⠃⠘⠁⠀⠀⠇
⠀⠀⣿⣿⣿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠂⠚⠲⠤⢄
⠀⠘⠉⠙⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡇⠀⠀⠀⢸⠁⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⢀⣤⡴⠶⠛⠛⠛⠛⠻⠶⣦⣄⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⡀⠀⠘⢿⣄⠀⢀⡤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣈⣛⣶⣤⣤⣤⣴⠾⠟⠉⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠛⠛⠉⠉⠙⠛⠛⠛⠿⢷⠶⠶⣶⣶⣶⡾⡟⠋⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠘⣦⣀⠀⢿⣿⡿⠃⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠈⠙⠀⠀⠘⢯⣙⡉⠉⠀⠀⠀⠀⠀⢀⣴⠿⣶⣄
⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣿⡆
⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⣴⠟
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠘⢷⣤⣀⣀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣶⡶⠾⣿⠛⠉⢁⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⢠⡟⠁⠀⠀⠀⠀⠉⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠚⠀⠠⠶⠿⠿⠶⠦⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣆⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢠⡾⠁⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⢹⡇⠘⣧⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⠏⠀⠀⢸⡟⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠾⠟⠉⠀⠀⢸⡇⠀⠈⢷⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡾⠃⠀⠀⠀⢸⡇⠀⠈⠉⠛⠛⠛⠒⠒⠚⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢻⡄⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣠⠟⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠹⣆⠙⣦⣀⣀⣀⣀
⠀⠀⠀⢀⡴⠋⠉⠳⠞⢁⡴⠋⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⡦⠘⠉⠁⠀⠉⠛⢦⣄
⠀⠀⢠⠟⠀⠀⠀⠀⠀⠘⠳⣄⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⢀⣹
⠀⠀⢿⣤⡄⠀⠀⠀⠀⣰⢂⡾⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⢳⡀⠲⣄⠀⠀⠀⢠⣬⡟
⠀⠀⠀⠀⠳⢻⡂⣠⠀⣇⠈⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣷⠀⣹⡄⣠⡶⠟
⠀⠀⠀⠀⠀⠀⠙⠙⠶⠟⠶⠟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠹⠶⠋⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠰⡏⢹⠏⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠻⡏⠉⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣀⡤⠴⠖⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠳⠶⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠿⠶⠶⠛⠛⠛⠛⠉⠉⠉⠉⠉⠛⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠿⣥⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡼⠟

⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⣀⣀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⡤⠤⡤⢤⣤⣀⡀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⣼⣿⣿⣿⣶⣶⣶⡟⠉⠉⠀⢻⡘⢷⣶⣶⣶⣶⣶⣶⣶⣶⡾⢳⣶⣶⣶⡞⠷⣷⣶⣶⣶⣶⣶⣿⣷⡟⡄⠀⠀⠈⣿⣶⣦⣴⣿⣿⣿⡄
⠀⠀⠀⠀⢹⣿⣿⡟⠛⠛⠛⠻⣶⠲⠂⠚⠃⢸⠟⠛⠛⠛⠛⣋⢻⢋⡇⢸⠋⡿⠉⡇⣸⠘⣻⣟⠛⠛⠛⠻⣟⠀⢷⡾⠿⢶⡟⠛⠛⠻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢠⣿⣄⣀⠀⢠⠟⠀⠀⠀⠀⣰⡟⡇⠀⠀⠘⡀⡇⢰⠃⠋⠀⣹⠙⣆⠀⠀⠀⠙⢦⡀⠀⠀⠀⣷⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⠃⡇⠇⠀⠟⠂⠁⠁⠈⠸⢰⠀⠉⠀⢸⠀⠀⠀⠀⣾⣷⣶⣶⣶⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠀⠀⡰⢺⠀⠁⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠰⠈⡆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⢐⠃⠈⠀⠠⠾⢿⣇⡀⠀⠀⡰⠟⠻⡄⠈⠀⠀⠸⣄⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣁⠀⣿⠸⡄⠀⠀⣠⢤⡈⢳⡀⢀⠁⣀⢀⠘⠄⠧⠀⣄⢹⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡁⣻⣿⣿⠀⠋⡀⢸⣤⡄⢹⠀⡇⢠⣾⢠⡄⢱⠀⠀⡀⢿⠈⣿⣿⡛⡟⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣩⣵⣷⣿⣿⣿⠀⢸⡇⠈⠶⠴⠃⠀⡇⠘⠈⠒⠶⠋⠀⠀⢡⢸⡄⡧⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡎⣧⠀⠀⠀⣀⡀⠇⠀⠀⠀⠀⠀⠀⠀⡸⠘⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡟⠁⢠⠇⢸⡄⠀⣾⠁⠀⠀⠀⠀⢙⡆⠀⠀⢠⠁⠀⢻⡀⠸⣿⣿⣿⣿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⠀⠀⣼⠀⣸⠁⠀⠈⠲⡆⠀⢀⡖⠚⠁⠀⠀⢸⡄⠀⠘⡇⠀⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⢀⡇⢀⡟⠀⠀⡰⠀⠘⠒⠚⠀⠀⢣⠀⠀⢾⠇⠀⢀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣴⣧⡞⢸⠁⢸⠃⠀⢸⠇⠀⠀⠀⠈⠩⠕⡞⢧⠀⠸⡆⠀⢸⠃⠀⣧⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠘⠀⢸⠀⠀⠀⠀⠐⢤⣀⣀⠀⠔⠀⠀⠀⢀⠀⠀⠊⠀⢰⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⡀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠄⡐⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣴⠀⣀⠀⢢⡀⠀⠀⠀⠀⠀⠀⢀⡄⢈⡁⠀⠀⣁⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⣾⡟⢠⠂⠙⠶⠤⣤⣤⣤⠶⠋⡀⢸⢻⢸⣤⠿⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡟⠛⠉⠀⠀⠻⣌⠸⡀⡇⣤⠀⠰⡀⡆⢰⠁⣸⣿⠋⠀⠀⠀⠉⢻⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠈⢷⣷⣇⠸⠀⠄⡇⡇⠸⢰⠟⠁⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠙⣾⠳⣆⣰⠵⠋⡰⠋⠀⢀⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠈⠻⣯⣀⣴⠏⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢸⣿⡇⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣀⣼⣿⣧⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⣿⣿⣿⡟⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣿⣿⣿⣿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠛⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣦⣴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⣿⣿⡿⠿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⡀⠀⢀⣀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠶⠀⠀⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣦⣄⣀⣤⣴⣶⣶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⡀⠀⣀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣰⣿⣷⠀⣿⣿⣯⠀⣿⣿⡇⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢭⣤⣬⣤⣬⣭⣥⣤⣭⣭⡤⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⠉⠉⠉⠛⠀⠛⠀⠙⠁⠀⠒⠘⠋⠉⠘⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢤⠀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⡄⠀⠦⠀⠀⠀⠀⠀⣰⣶⢀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠴⠈⠹⣷⡀⠀⠀⠀⠀⢀⠆⢹⢿⣿⣿⣶⣶⡶⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠛⠉⠀⠒⠚⠓⠶⣦⠀⣨⣷⠆⠀⠀⠀⠁⠀⠀⠹⠿⠙⢿⠙⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠿⠋⠁⠀⠀⠀⠀⡀⣀⣤⡴⣿⠋⢻⣿⠀⠀⣄⠀⢀⣀⣀⠀⢀⠀⠀⢀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠰⠼⠃⠀⠹⢶⡈⠳⣾⡧⠀⠀⢹⣦⡌⠛⠁⠀⠸⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣦⠀⣠⣀⠀⠐⠀⠀⠘⠣⢀⠀⠀⠉⠰⣿⠿⠓⠀⢨⣿⠟⠀⢀⢶⡇⢠⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢿⣇⠘⣿⠆⠀⠀⠀⠀⢰⣿⡿⢶⡄⠀⠀⠀⠀⠀⣸⡇⠀⠐⢃⣼⣇⣦⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⡄⢹⣿⠏⠀⠀⣀⠀⡀⢸⡿⠀⢠⡛⢷⣄⠀⠀⣴⣏⠀⠀⠀⠸⠟⠋⡉⠀⠀⣠⠂⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣽⣾⠍⠀⠀⠐⣿⠀⢱⡾⠁⠀⣼⠁⠀⢿⡆⠋⠹⣯⠀⠴⢄⡀⠀⠘⠃⡀⠀⠟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠋⠀⠘⠀⠀⠀⠋⠀⠀⠀⣶⡄⠀⢉⠀⠀⠉⠀⠀⠀⠀⡁⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠴⢦⣨⣿⣤⠞⠁⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡆⢀⠀⠀⠀⠰⠶⠀⠀⠀⢹⣿⠋⠁⠀⠀⢀⡀⠀⠈⢱⡏⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⡇⠀⠰⣶⣾⣿⣷⡾⠆⠶⠶⠄⠀⠀⠀⣠⣿⣿⣧⣄⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⡀⠀⠀⠀⣿⣦⣄⣈⣿⣿⣿⠃⠀⢠⣤⠀⣀⡄⠐⣿⠟⣻⣿⣿⣷⡄⠀⠀⣼⠇⠀⠀⠀⠀⠉⠠⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⡷⠀⠀⠀⣿⣿⣿⣿⣿⡿⢃⣄⠂⠀⠃⠀⢻⣇⡀⠀⢾⣟⠛⠛⠁⣀⣠⣾⣁⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣦⠀⠀⠀⣿⣿⣿⣿⣿⡇⠘⢿⣧⠀⢰⡀⠈⢿⣷⡀⠀⠙⠷⠖⠀⠉⠛⠛⠿⠛⠛⠢⠄⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⠀⢿⡆⠀⢷⠀⠈⠛⣷⣄⠀⠀⠀⢀⣠⣴⣀⡀⠀⠀⠀⠀⠀⠓⢴⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⡙⠀⠀⠁⠀⠀⠈⠉⠀⣠⠔⠋⠙⠿⣿⣿⣷⠄⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠘⢣⣿⣿⠟⠟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⠛⠋⠀⠀⠒⠀⠀⠈⠛⠦⣀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⢈⣿⣿⣿⠉⠉⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠚⢋⣈⡭⠙⠋⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢋⣩⣴⣾⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⢟⣿⡟⢻⣯⡅⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠑⢋⠁⠙⠛⢿⡟⠫⠈⣛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣦⣀⣶⣤⣶⣴⣴⣆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡌⣷⣾⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣷⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣪⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠘⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⠃⢀⣠⣴⣶⠶⠤⢄⣀⣠⠴⠋⠉⠀⠀⠈⠻⣶⣄⠀⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣻⣿⣿⠃⣰⣿⡿⠋⠀⢀⣀⣀⡀⠀⠀⣠⠤⠶⠦⣄⠀⠹⣿⡆⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠸⣿⠟⠀⣠⠾⠋⠉⠉⠙⢷⡞⠁⠀⣀⠀⠈⣷⠀⠀⠀⠀⠈⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⣼⠃⠀⠀⢾⡆⠀⠸⣧⠀⠀⠻⠃⠀⢸⡇⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡏⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣰⠿⣦⣄⡀⠀⣀⣾⠃⠀⠀⠀⠀⢸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠻⣦⣤⣤⣶⣿⠁⠀⠈⢿⣿⣿⣏⣁⠀⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢔⡿⠏⣼⠀⠀⠀⠀⢿⣏⡑⠒⠀⠀⠀⠀⠀⡂⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢀⡴⢿⠀⠀⠀⠀⢸⣿⠙⣦⠀⠀⠀⠠⠄⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣸⣇⠀⠀⢀⣼⣿⣤⡿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡉⢙⢻⡿⡟⠋⠀⠁⠀⠀⠀⠀⠀⠈⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⠄⠀⠄⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⢀⠀⠀⠀⠀⠀⠀⢀⠐⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠳⣄⠀⠀⠒⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⢤⣤⣤⣤⣄⣀⣀⣝⣠⣿⣴⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉


⠀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠠⠄⠄⠀⠀⠠⠠⠀⠁⠀⠀⠚⠳⠿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠛⠉⠉⠀⠀⠀⠀⠈⠈⢲⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠁⠀⠋⠂⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠜⠀⠀⠀⠢⢀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⡠⠂⠀⣀⠀⢀⡀⠄⠀⠑⠠⡀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠠⣴⣾⣿⣟⣿⣦⡄⠀⠀⡘⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡈⠀⠀⠈⠛⠻⠿⠛⠁⠀⠀⠀⢠⣾⣿⣿⣦⣤⣤⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢀⡴⣲⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⠠⠒⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⢿⡻⣿⣿⣿⣿⣿⣷⣄⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠙⣻⢟⣿⣿⣿⣿⣿⡿⣡⠏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢮⡻⣿⣿⣿⣿⣿⣷⣬⣑⡒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠒⣒⣩⣴⣿⣿⣿⣿⡿⣋⡾⠣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠲⢝⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠟⠩⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠁

⠀⠀⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠁
⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄
⠀⠀⣿⠟⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡈⣿⣿⣿⡟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠟⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠳⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠛⠻⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⡀⠻⣦⡐⢷⣄⠘⢷⣄⠺⣦⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠛⣀⣾⠟⣠⡿⠃⣠⡿⢃⣤⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠋⠁⠈⠉⠀⠘⠋⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⢿⣿⣿⡇⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⣿⣿⣿⣧⠀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠙⠛⠛⠛⠛⠛⠛⠛⠻⠛⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣴⣶⣴⣒⡀⠀⠈⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣴⣿⣿⣿⡿⠛⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⡿⠋⠀⣀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠿⠋⠀⠀⠀⠉⠀⠀⠀⠙⠿⠀⠀⠐⠿⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣶⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣌⢿⣿⣿⣯⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣯⣷⣿⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⢧⠉⠻⣿⣿⣛⣻⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠋⠈⠀⢺⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠒⠀⠀⠈⠛⠀⠁⠀⠀⠚⠫⠝⠀⢀⡀⠀⠀⠀⠀⣿⠿⠻⠿⠿⠃⠀⠀⣀⠀⠈⠉⠉⠉⠉⠛

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠀⠀⠀⠀⠀⠀⠀⠀⣘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣀⣀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⡀⣀⣀⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⢀⣀⣤⣤⣤⣀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡶⣤⣄
⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⣭⣿⢻⡍⣧⣬⣿⣷⡀
⠀⠀⣾⣿⣿⣿⣿⢿⣿⣿⡿⠁⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠈⠂⠈⢻⣯⣿⣿⣟⢷⣹⣧
⠀⠀⢿⣿⣿⠏⢠⡾⡕⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⡇⢲⡿⣯
⠀⠀⠈⠛⠋⠀⠨⢐⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣧⣾⣤⠇
⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⡄⡽⠛⢁⣠
⣿⠿⠿⣷⣄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠓⢀⣴⠿⠛
⠁⠀⠀⠈⣿⣧⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣿⠃
⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⣷⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡾⠿⠿⠿⠿⠿⠿⠿⠿⢷⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀⡀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⣬⣉⣙⠛⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⣀⣀⣀⣀⣀⣀⣀⡀⠉⠉⠙⠛⠿⢷⣦⣌⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠈⢻⣯⣍⡛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⣶⣦⣤⡀⠈⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⣠⣿⠛⠛⠿⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣈⡙⠻⣷⣦⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣷⣤⡙⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠛⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣇⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣇⡀
⠀⠀⠀⠀⠀⢀⣤⣾⠿⠟⢻⣿⢠⣿⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠘⣿⡀⣿⡟⠻⠿⣷⣤⡀
⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣼⣿⠀⠀⠀⠀⢿⣧⡀⠈⠉⠀⣠⣾⠏⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⠉⠀⣠⣿⠏⠀⠀⠀⠀⣿⣧⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⢸⣿⠋⠉⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠾⠿⠛⠁⠀⠀⠀⠀⠀⠉⠙⣿⡇⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠘⣿⣦⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠾⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣶⣶⣶⣶⣶⣶⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠎⢉⡆⣀⣄⡀⢫⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢧⢀⣀⠀⠀⡠⢄⠀⡤⠒⢣⣀⡙⢾⡄⣸⠏⠁⣠⡷⣾⣏⠉⠀⠀⢀⡞⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠈⢾⡅⠑⡞⠀⠀⣹⣀⠀⢫⣀⣉⣵⣋⣹⣞⣉⠙⢲⣀⠞⠒⣆⣠⠞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠱⢄⣳⡤⠒⢧⡟⠒⢻⠋⣀⢜⠏⢹⡀⢹⢝⣩⢻⠛⠻⡍⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢠⠊⠛⣄⠀⢀⣇⠀⢈⠏⠁⠸⣀⡰⠉⠉⠀⢇⢈⢷⠐⠁⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⣜⢊⡟⠀⠉⠁⢈⡭⠥⢤⡀⠀⣠⠔⠒⠒⢄⠀⠁⠈⣇⣀⠴⠾⢥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢉⡭⠔⠚⣻⠀⠀⠀⢀⡏⠀⢀⣀⠱⣴⢁⣀⠀⠀⢸⠀⠀⠀⠸⣍⠙⠒⠦⣈⠑⢦
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠼⠥⠖⢋⣴⡇⠀⠀⠀⠀⣇⠀⠸⣿⡇⡇⣿⡿⠀⠀⡼⠀⠀⠀⠀⣏⠳⣍⠓⠺⠃⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠾⠤⠤⠤⠶⠚⠉⢸⠀⠀⢀⣀⡤⠼⠶⠖⠚⠒⠓⠚⠒⠒⠾⠧⢤⣀⡀⠀⢹⠀⠈⠙⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠹⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⣤⣄⣀⣀⣠⡤⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣤⣶⣾⣿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⡿⣶⣤⣀
⠀⠠⠖⠛⠛⠛⢻⡿⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⣤⣤⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⠿⠿⠛⠋⠙⣧⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣿⣿⣴⣶⣿⣿⣿⣿⣿⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠻⣄⣨⣿⠟⠉⠉⠛⠻⢿⡿⢻⡟⠛⠛⣿⠉⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠈⢷⡄⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣠⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣤⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠖⠚⠛⠛⠓⣶⠶⠶⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠋⠉⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠙⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡏⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⣀⣿⣀⣀⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠓⠒⠚⠛⠛⠛⠛⠻⣿⠛⠛⠛⠋⠉⠉⠉⠉⢹⡏⠉⠁⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⢀⠀⠀⣿⠋⠻⣄⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣀⣀⣸⡏
⠀⠀⠀⠀⢸⠀⠀⢻⣆⠀⠙⣧⣠⣤⠶⠶⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠒⠦
⠀⠀⠀⠀⠸⣇⠀⠈⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠳⠶⢤⡤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣀⢸⣷⣦⣤⣄⣠⣤⣤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡸⣿⠿⠿⠟⠛⢻⣉⣭⣥⣴⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣿⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⢸⣿⣩⣭⣧⣙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣼⣿⣿⣿⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠲⠈⢀⣀⠀⢀⠀⠀⠉⠀⠙⡄⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣀⡀⣄⣀⣡⣴⣾⡟⠛⠂⠀⠀⠘⣿⣿⠟⣩⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⠋⠈⠛⢿⡟⠉⠉⠓⠂⠀⠀⠀⢸⠏⢿⣯⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣄⣽⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡞⢹⡀⠘⠉⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣤⣤⠶⠔⠀⠀⠀⠀⠐⠻⠏⠀⢸⣿⡶⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⢸⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⢿⣿⣿⣥⣤⡝⠙⠛⠛⡿⠷⠶⠃⠀⢠⡾⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣧⣀⠀⠀⢀⣗⣀⣤⣴⣶⠟⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠃⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⡿⠋⣼⠃⠀⠀⠀⢀⡟⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣺⣿⣿⣿⣿⣿⣿⠋⠀⠰⠃⠀⣀⠀⠀⣼⠁⠀⠀⠀⢹⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣧⣤⡄⠀⢀⣾⠃⠀⢰⡇⠀⠀⠀⣰⣿⢏⡾⡍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⡇⠸⣿⡟⣀⣠⠿⠧⠴⠶⠿⠷⣾⣴⣾⣟⣵⣿⣿⡧⠄⠈⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠈⢹⣶⣿⡯⠵⠂⠀⠀⠀⠀⠀⣠⣾⢟⣵⣿⣿⣷⡟⢠⣄⣴⣉⠄⣈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⢀⣾⠁⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠉⠉⠉⠀⠉⠉⠀⠉⠀⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡑⡾⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠂⢹⠟⠉⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠺⡟⠙⢮⠤⢔⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⣆⠤⠀⠀⠠⠖⣤⠤⢹⡤⢸⠋⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠄⠀⠰⣇⣐⢺⣀⣮⠧⠀⢉⡿⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠂⠀⠀⢐⠀⠀⠀⠲⠀⠀⠀⠵⠂⣈⣩⣼⣾⣚⣇⢤⣤⣌⠁⠒⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠩⠀⠀⢀⡀⡆⡐⠷⣤⣷⣴⣿⣿⣿⣿⣿⣀⢙⡿⣤⢖⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢴⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⢈⠇⢀⣛⣾⡿⢿⣿⣽⢻⣿⣵⣟⣿⣥⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢀⢀⡀⢈⡷⣍⢹⡏⠀⠘⢿⣿⣍⣻⣟⣿⣿⣟⡻⡶⠒⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡙⠣⠌⣁⠈⢷⢞⣾⣧⡄⠀⠙⢿⣿⣟⣿⣟⣻⣿⣽⣭⠉⢀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠴⠋⢤⠀⡄⠀⠀⢰⣿⡶⢂⣉⣸⣇⣽⣸⡿⠋⠄⠀⠀⠻⣭⣿⣿⣿⣿⣷⣦⡤⠐⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠑⢞⠀⠀⡀⠀⠇⢠⣸⣤⢬⣗⣸⣾⣿⣿⣿⡿⠀⠀⠀⡠⠐⠀⠙⣿⣿⣻⣿⣿⣟⡝⣠⡌⠀⠞⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢄⠰⠀⠠⢘⠀⢀⣵⣼⣌⠁⠐⠋⠁⠙⠻⠿⠿⠛⠁⠀⠀⠀⡐⠁⡀⠁⡃⢸⣿⣿⣿⣿⣿⢷⠟⠋⣔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⠁⢹⢤⡟⢶⣵⡿⠂⡀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣤⡀⠑⢠⡿⣿⣿⣿⣷⡾⣵⠄⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣿⣿⣴⣆⢯⣯⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡔⢪⠀⠀⠈⠻⣰⠆⢸⣿⣿⣿⣿⣿⣧⡷⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠲⡖⢢⣸⣟⣟⣫⢾⣷⣮⡟⠁⠀⠀⠀⠨⠄⠈⢑⠀⠀⠀⠀⠁⡄⠀⠀⠀⣹⣯⠖⣾⣿⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡻⣷⣿⣿⣯⠿⣿⣛⢿⡿⠃⠀⠈⠀⢀⡀⠄⠀⠀⠄⢃⢀⠀⠁⠀⠐⡤⠀⢨⣿⢻⣟⣛⣿⡋⠙⢷⣾⡇
⠀⠀⠀⠀⠀⠀⢡⠀⣹⣿⣿⣿⣿⣶⣿⣽⠟⠀⠀⣠⣾⣷⡌⠴⣎⠰⢀⠘⡄⠈⠆⣠⠐⢀⢺⢠⡌⣿⢿⣿⠿⠿⣻⣦⣴⢻⡷
⠀⠀⠀⠀⠀⠀⠈⣠⣼⣿⣿⣿⣿⣿⠿⠀⠀⠀⣾⣿⣿⡷⣷⠀⠁⠰⢠⡀⠀⠀⠡⠄⡁⠰⡊⣼⣿⠡⣾⣿⣿⣿⣿⣿⣼⣾⣏⠢
⠀⠀⠀⠀⠀⠀⢲⡟⢻⡉⣿⣯⣬⣧⣴⣶⣶⣶⣿⣿⣿⡷⢿⡷⣶⡆⠠⠘⠈⠰⠠⠰⠄⣳⣿⣿⣧⣿⣿⣿⣿⡿⢿⡿⢩⣯⣿⣍⠦⣄
⠀⠀⠀⠀⠀⠈⠀⠀⠘⠉⠉⠉⠉⠿⠿⠿⢉⡻⡏⢹⣿⣿⢿⡟⢻⣿⣷⣶⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⢿⡿⢿⣗⠈⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠚⠻⢇⡉⢈⣿⢿⣿⣋⠻⣿⢿⡿⠋⠙⠛⠻⠋⠁⠀⠀⠁⠀⠈⢹⣽⠏⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⡉⠑⠸⣿⠗⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣶⣾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣠⣴⣶⣿⣿⣿⣿⣷⠀⠀⣼⣷⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠿⠟⠛⢛⣿⣦⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠁⠀⠀⣀⠀⠸⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣳⡆⠛⠉⠀⣰⣿⣿⣷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡷⡆⠄⣼⢟⣽⣿⢿⡿⢟⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣷⣿⣤⣼⣫⣿⠾⣻⠏⠸⣿⣿⣿⣿⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠈⠍⢉⣹⡿⢳⠀⠁⠀⠚⠁⡴⠲⣿⣿⣿⡟⠙⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣥⣶⠖⠿⢿⡇⠌⣦⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢻⡄⠀⣄⠀⠀⢹⡇⠤⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⣀⣹⡼⣷⠀⠉⠀⠀⠀⠀⠀⠀⡀⣤⣄⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠟⣡⢾⡇⠀⠀⠀⠀⠀⠀⠈⢈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⠤⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠧⣖⣾⣿⣛⣀⣀⠀⡀⠀⣀⡄⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡘⠛⢿⣿⣯⠈⠉⠉⠁⠀⠉⠀⠀⢨⠹⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠍⢁⣶⡿⠷⣄⣀⣀⠀⠀⠀⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠄⢀⣾⣿⠇⠀⠙⠉⠁⣠⣴⠞⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⠃⠀⣾⣿⡟⠀⢀⣴⣾⡿⠋⠀⠀⢀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⢭⣿⣿⣿⣿⠀⠀⠘⢿⣷⣶⣿⣿⡟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⣰⣿⢟⣿⣦⣀⣶⡿⣿⣿⡿⠋⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊⡥⢴⡶⠿⢛⣻⡿⢋⣡⣿⣿⣷⠟⠁⠀⠀⠀⠈⣿⣿⣯⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠖⠊⠁⠀⠀⢀⣤⣶⣿⠟⣻⠟⠁⠀⠀⢀⣼⡇⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣻⡿⠟⢡⡾⠃⠀⠀⠀⣴⣿⣿⣷⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠟⣡⡾⠋⠀⢠⠏⠀⠀⠀⢀⣼⡿⠉⢿⡿⣧⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠈⠓⠉

⠀⠀⠀⠀⠀⠀⠀⢠⣴⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⣤⣤⡤⠄
⠀⠀⠀⠀⠀⢰⡆⣿⡏⣿⣇⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡴⠋⠁⣀⣀⣀
⠀⠀⠀⠀⠀⢹⣿⠻⢟⣙⣫⣴⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣿⣵⣾⠿⠛⠛⠻⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢻⣸⣿⡿⢁⠔⠛⠛⠛⠦⣌⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⣤⣀⠘⣿⣿⣧⡇⡴⣊⣉⣑⢦⠈⠳⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣾⣿⣿⣷⣾⣿⣿⣿⣶⣅⣀⡉⢢⠳⡄⠙⢦⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⡿⠿⠛⠛⠉⠉⠉⠀⠀⠀⢠⣾⠋⠉⠀⠉⠉⠻⣿⡉⠛⠛⠿⣿⣷⣬⣢⡀⠳⣄⠙⢿⣷⡀
⠀⠀⠀⠀⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣟⡳⢸⣧⠀⠀⠀⠀⠉⠻⣿⣯⢦⡈⢳⡈⣿⣷
⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⠀⠉⣁⣼⠇⠀⠀⠀⠀⠀⠀⠈⢻⣷⡝⡀⡇⣿⣿
⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡇⣿⡿⣇⣀⡄
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⢻⣧⠈⠉
⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣤⣀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣽⣷⣤⢀
⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣏⠙⠂⠀⠀⢀⣴⡾⠿⣿⣿⣿⣿⡵⠃
⠀⠀⠀⠈⠻⢿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⢀⣾⠋⠀⢿⡄⠀⠀⠀⠈⠉⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠉⠉⠁⠀⠀⠀⢀⣴⠟⠁⠀⠀⢸⡇⠀⠀⠀⢀⣄⡀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⢤⣤⣤⣴⠶⠞⠋⠁⠀⠀⢀⣴⣿⡇⠀⠀⠀⠈⠿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿⡄⠀⠀⢀⣀⣴⣿⠿⣿⠃⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⣠⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠋⠀⣴⡟⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⢀⣤⣶⣤⣠⣼⣿⣤⡾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠈⠛⢿⣤⣤⣤⡾⠋⠀⣰⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣤⡾⠟⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⣠⣴⡿⠿⠿⠟⠁⠙⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢧⣤⣤⣬⣉⣉⣉⣛⣻⣦⣀⣴⠟⠛⠓⠛⠛⠛⠛⠋⢹⣿⣇⣠⣴⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠒⠶⠶⠶⢾⣿⠛⠋⠉
⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⢀⣾⠋⠈⣿⠄⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣸⡏⠐⠀⣿⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⢿⡇⠀⢰⡏⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣴⠾⠗⢠⣟⣀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢿⡴⠶⠛⠛⠻⠿⢿⣶⣦⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠷⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠶⢦⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⡆⠈⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⡾⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠿⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷
⠀⠀⠀⢠⠖⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⢆
⠀⠀⠀⠘⠦⣤⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣤⡤⠏
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⡀
⠀⠸⢤⠤⠔⠂⠀⣀⣀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠂⠉⠙⠂⠀⠀⠀⠀⠀⢸⠀⠀⠒⠤⢄⠼
⠀⠀⠀⢀⣴⢾⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⢰⠟⠁⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠉⠛⠲⠶⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢤⣄⣀⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⠉⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣠⣽⣧⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⢠⠔⠋⠉⠉⠉⠉⠛⢏⠀⠀⠀⠈⣻⡾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣏⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠟⠋⠀⠸⠿⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣤⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡾⠟⠿⠿⠿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢡⣿⣿⣾⣶⡂⢀⣈⣉⣭⣵⡾⠿⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣸⣿⣿⣿⣿⣿⣴⣿⣿⣟⣡⣄⣴⣾⣿⣿⣿⣿⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡿⠋⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠉⠙⠛⠋⠉⠙⠁⢻⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⢻⡇⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢻⠇⠼⢩⠽⣿⣿⣿⣶⠦⢶⣴⣶⣶⣿⣿⡻⣧⠀⣾⣡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⡄⠀⠀⠀⠀⠀⠉⠋⠀⣼⡿⠿⠆⠉⠈⠉⠈⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣷⡄⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⢀⣴⣿⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠻⣦⠀⠀⠀⠠⢶⣄⣴⣶⠄⠀⠀⢠⣾⠿⢹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠀⣤⣶⣾⣿⠛⣻⣿⣶⣄⠀⢀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⠀⠀⠛⠉⠉⣉⣉⣉⡉⠉⠛⢣⡎⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠈⠳⣄⠀⠀⠈⠉⠉⠉⠁⠀⠀⣼⣣⣾⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠁⡇⠀⠀⠿⢷⣄⣀⣤⣶⣤⣤⣴⣾⣿⣿⡿⠿⣿⣿⣿⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣸⣿⠉⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⡿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⡟⠉⠀⠀⠀⠘⠻⠛⠀⡠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃

⠀⠀⠀⢀⠴⠚⠉⠙⠒⢄⠀⠀⠀⠀⡠⠒⠋⠉⠓⠦⡀⠀⠀⠀⣠⠔⠊⠉⠑⠲⣄⠀⠀⠀⢀⠴⠚⠉⠉⠒⢄⠀⠀⠀⠀⡠⠒⠋⠉⠓⠦⡀
⠀⠀⢀⠏⠀⠤⠀⠀⠆⠀⢣⠀⠀⡜⠀⠰⠄⠀⠤⠀⢹⡀⠀⢰⠁⠀⠆⠀⠰⠀⠈⡆⠀⢀⡏⠀⠤⠀⠠⠆⠀⢣⠀⠀⡜⠀⠰⠀⠀⠤⠀⠹⡀
⠀⠀⠘⡆⠀⣠⠖⠦⡀⠀⡼⠀⠀⢧⠀⠠⠔⠲⠄⠀⢸⠃⠀⢸⡀⠀⠤⠤⠤⠀⢀⡇⠀⠘⡇⠀⠠⠤⠤⠂⠀⡸⠀⠀⢧⠀⠠⣄⣀⡄⠀⢠⠃
⠀⠀⠀⠘⠦⣀⣀⣀⣠⠜⠁⠀⠀⠈⠣⣄⣀⣀⣀⠴⠃⠀⠀⠀⠑⢄⣀⣀⣀⡠⠊⠀⠀⠀⠘⠦⣀⣀⣀⣠⠔⠁⠀⠀⠈⠣⣄⣀⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠚⠛⠉⠉⠉⠉⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠒⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⢀⡴⠞⠛⠉⠉⠙⠛⠁⠀⢣⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡼⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡀⠘⠦⣀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣴⡏⣠⠾⠭⠟⠁⠀⠀⠀⠀⠀⠀⠙⠒⢬⣆⠀⢨⡉⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠈⠁⢸⠟⠁⠀⣀⣂⠀⠀⠀⠀⠀⠀⢀⣐⣂⡀⠉⠀⠀⡷⣄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⣸⠀⠰⠟⠁⠤⠑⠀⠀⠀⠀⠀⠏⠡⠌⢻⡆⠀⠀⣇⣠⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣽⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢠⠞⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣻⠆⠀⠀⠀⠀⢨⢠⠇⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠛⠁⠀⠀⢀⡤⣴⠋⠀⠀⡇⠀⠀⠀⠀⠀⠶⢤⣀⠔⠁⣠⠋⠀⢀⣠⠶⣦⡤⠎⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠐⠛⠛⢦⠈⠳⡀⠀⠸⡀⠀⠀⠀⠀⢀⡔⠁⢀⡴⠁⢀⠔⠋⣠⢮⡽⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣲⠆⠀⠀⠀⢸⡆⠀⡇⠀⠀⢳⠀⠀⢀⡴⠋⠀⢠⣾⡤⠖⠁⢠⠞⢁⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⣠⠂⠀⠀⣼⠁⠀⢧⠀⠀⠀⣣⠴⠋⠀⠀⣰⠿⠋⠀⢀⠔⠀⣠⠊⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⣰⣏⡴⠖⠋⡏⠀⠀⢀⡇⠀⠀⠈⠓⠒⠋⠁⠀⠀⠀⠀⠉⠀⢀⠔⠁⢀⠜⡡⠞⡹⢳⡀⠀⠀⢸⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⣸⠃⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⣰⠟⠋⡠⡞⠀⢠⡇⠀⠀⢸⠀⠀⠀⢢⠸⣄
⠀⠀⠀⢠⡞⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣠⠞⠁⢿⠀⡾⠀⠀⠀⡌⠀⠀⠀⠈⢶⠛⠉⠉⠉⠛⠶⣄
⠀⠀⢀⡿⠀⠀⣤⠴⠋⠀⠀⠀⢀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⡼⠁⠀⠀⠀⢳⠀⢠⠀⠀⠈⠳⡄⠀⠀⠀⠀⠈⢳⡄
⠀⠀⢸⣃⣀⠀⠈⠓⠦⠤⠴⢚⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⠈⠃⠀⠁⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢸⡇⠈⠙⢦⡀⠀⠀⣠⠞⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠤⠄⠀⠀⠀⢿⠀⠀⠀⠀⠀⣱⠦⣄⡀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢻⡀
⠀⠀⣼⠀⠀⠀⠀⠙⢦⡈⠑⠶⠶⠖⣻⠇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢅⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣼⠃⠀⠀⠉⠲⣄⡀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣼⠋⢆⠀⠀⠈⠑⠒⠂⠐⠒⠒⠋⠁⠹⣄⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹⣠⠟⠀⠀⠀⠀⣀⡤⣶⠟
⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡟⠀⠀⠀⠀⠀⠀⠀⣸⠿⣆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣇⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀⡴⠞⠁⠀⢿
⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⡟⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡟⠘⢷⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣽⣿⣭⣿⣿⣿⣿⣰⡟⠙⢿⣿⣮⠙⢀⡀⣌⢿⠑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿
⣿⠿⣌⡹⣿⣿⣿⣿⣿⣿⣷⣾⣿⣥⣿⣿⣿⣿⠟⠉⢿⣦⠈⠿⢿⣿⠿⣿⣿⣧⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣅⣾⣿⣿⣿⣿
⠋⠀⠈⠛⠙⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣆⠀⠀⠺⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠹⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣷⣶⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣨⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⡈⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⣀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠈⠉⠉⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⣷⠴⡛⣿⣿⣿⣧⡀⢸⣴⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⣶⠶⠀⠀⠠⠀⣀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠲⢶⣦⣀⠀⠀⠈⠉⠛⠛⢿⣧⠹⠻⣿⣿⣿⣿⣾⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⢿⡄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⠀⠀⢠⣀⣘⢻⣿⣿⡇⠀⠀⠀⠀⠀⢸⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠛⠷⠋⠀⢀⠀⠀⠀⠀⡈⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢠⡅⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⣉⣉⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⡏⠙⠛⢷⣶⣤⡄⣀⣠⣤⣦⣸⡟⠀⠀⠀⠠⠜⠏⡹⢿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⢳⣿⠿⠋⠀⠀⠀⠉⠛⠿⢿⠃⠀⠀⠀⠀⠀⠀⣡⠀⠉⠁⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠻⠷⣶⠿⠛⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠰⣦⣅⠀⠄⠀⢀⣀⣀⣠⣴⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⠛⠛⠿⠛⠛⠿⠛⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠛⠻⢿⡟⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⡿⠉⠀⠈⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⠀⠀⠀⠀⠈⢯⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣾⣿⣿⠁⠀⠀⠀⠀⠉⣿⠀⠀⠀⢠⣄⣀⣀⣀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⣡⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣯⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠠⡀⠀⢻⣧⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⢠⠾⠋⠉⠉⠉⠛⢻⡟⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⣦⢛⡟⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠋⠁⠀⠀⠘⣏⠀⣠⣿⡄⠀⠀⠀⠇⢬⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣀⣤⣆⣠⣿⣿⣿⡄⠀⠀⠀⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢹⣿⠟⠁⠀⠀⠀⠘⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠘⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⣴⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠿⠀⠀⠀⠀⣰⠎⣹⣿⣿⣿⣿⣿⠿⣿⣿⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣦⣀⠀⣀⠀⠀⢀⣸⢯⣤⠀⡟⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠋⠀⠀⠀⢀⣴⢣⣼⣿⣿⣿⣿⠿⣧⣼⣿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣷⣶⣷⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⡆⢰⣿⣿⣿⣿⡟⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣭⣻⡟⣿⡄⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⠇⠀⢀⡴⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣿⢇⣾⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⡿⠿⠿⠋⢾⣇⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⠀⠀⠙⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⣤⣤⠀⠠⠟⠓⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⠀⠀⠀⡔⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⢹⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠙⠸⠃⠰⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣶⣤⣤⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣠⠟⠋⢳⠛⠀⠀⠈⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⣼⣿⠁⢀⠴⠚⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢀⢠⣿⣿⣿⣿⣿⡆⠀⢀⣀⠀⠀⣰⣿⣿⣯⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⣴⣿⢸⣿⣿⣿⣿⣿⣿⡇⠀⠈⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢿⡏⠈⢿⣿⣳⣿⣿⣿⣧⠀⠀⢀⣰⣿⣿⣿⣿⣿⡿⠟⢋⣿⢷⡙⢸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠛⣿⣿⣿⣿⣿⠀⠀⣰⣿⣿⣿⣿⣿⣵⣶⣿⣿⡿⠧⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠋⢻⡇⣰⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠟⠛⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠿⠿⠿⠿⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⣀⣴⠆⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠀⠀⠀⠀⠀⠀⠀⢠⣶⣏⡉⠛⣿⡦⣠⣴⠿⣿⡁⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⠳⣼⠀⣟⣠⣤⠖⠛⣆⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⣠⣿⣧⣬⡥⠀⠀⠀⠀⠀⠀⠀⠹⣍⡛⢁⣀⡞⠀⣿⣿⡿⠀⢀⡟⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣸⡏⠀⢘⣿⠀⠀⠀⠰⣄⡀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠶⣤⣶⣋⠀⠀⠀⣿⡷⢦⡄
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡿⠀⠀⠀⠀⣨⣿⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣦⣿⡇⠈⣧
⠀⠀⠀⠀⠀⣿⠀⠀⣸⡇⠀⠀⠀⣰⠟⣿⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣿⡀
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⢰⣿⠀⣿⡀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⣿⢡⡖⠉⢿⣄⠀⠀⠙⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⣿⠈⢧⢀⠌⠙⢷⣄⠀⠀⠈⠉⠛⠿⠿⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⡶⠿⠛⠋
⠀⠀⠀⠀⢸⡟⠀⠀⣿⠇⣀⠀⢿⣄⠜⠳⣄⡀⠀⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡏⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⢰⣿⡄⠹⡄⠈⠻⢦⣤⣬⣭⣷⠶⠚⠋⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠸⣿⠻⣦⡙⢦⣄⣀⡀⠀⠀⠀⠀⠀⣠⠶⠛⠶⣄⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⠇
⠀⠀⠀⠀⢸⡇⠀⢠⣿⣄⣈⡛⢶⣬⣉⠁⠀⠀⠀⣠⠞⠁⠀⠀⢀⠸⣆⣀⣤⣤⣤⣶⠿⠗⢀⡿
⠀⠀⠀⠀⢸⣧⠀⠀⠈⠉⠉⠛⠻⠿⣿⣛⠳⢶⡾⠁⠀⢀⠀⢣⠈⡄⣿⠉⠉⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠛⠷⢶⣶⡶⠶⠶⢦⣤⣀⠉⢿⠏⠀⠀⢀⣼⡆⠸⠀⣇⡟⠛⠛⠛⠛⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠉⠻⢦⣀⣤⡴⠋⡘⢣⠀⣷⠟⠁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⢉⡹⠀⢰⠁⢈⠷⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⣇⣠⡇⢀⣾⣠⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⠙⠛⣿⠿⠿⠿⠿⢿⡿⢿⡿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢹⡆⠀⠀⢀⡞⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⢻⡄⠀⣼⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠾⠟⠛⠛⠛⠉⠀⢻⣴⡇⠀⠈⠉⠙⠛⠛⠛⠛⠛⠲⠶⠶⢦⣤⣤⠶⠚⠋⢩
⣏⠙⠲⢦⣤⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠉⠀⠀⢀⣴⠟
⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⠿⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⢀⣀⣴⠿⠃

⣟⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠈⠁⠈⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⢀⡠⠄⠚⠤⢀⡊⠀⠀⣀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠐⠌⢀⡁⠀⢀⠄
⠀⠀⠀⢀⠀⠘⠁⠀⢐⠂⢀⠐⠃⠀⠀⠀⠙⢤⣀⣀⣀⣠⣤⡽⢛⣋⡓⠘⣩⡃⠀⠤⠀⠀⠊⠀⠀⠀⠠⠻⠃⠀⢲⡟⠂⠀⡄⡀
⠀⠀⣊⠀⠀⢁⣤⡀⠿⠞⠀⠰⠀⠀⠀⠀⠀⠀⠸⡏⠀⠩⠿⠃⠜⠿⠂⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⡠⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠔⠊⠉⠈⠁⠄⠠⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠐⡄⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⡠⠀⠂⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠄⠀⠈⣆⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙⠂⠠⠠⠆
⠀⠀⠀⠀⠀⠀⠹⠿⠀⡿⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠐⠒⠀⢄⠟⠓⠒⠁⠀⠀⠀⠀⠀⡀⠀⠀⢴⠄⠀⠰⠆⠀⠀⠇
⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡎⠐⢠⣤⡀⠛⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠉⠀⠀⠀⠐⠄⠀⠀⠠⠄⠀⠀⠰⠀⠀⠀⣀⠀⡀⣀⡰⠋⠠⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠒⢆⡀⠀⠈⢐⡕⠀⠀⠀⠀⠓⠀⠀⠀⣀⠀⠀⠀⠀⠐⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡀⠀⠐⠆⠠⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠁⠀⠀⡌⠈⠩⢠⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠤⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠄⠀⠀⠀⠀⠀⠀⢀⢡
⠀⠀⠀⠀⢰⡄⣠⢀⢠⡄⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⣦⠀⢿⣇⠀⠀⠀⠀⠂⠋⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠠⠀⠀⠀⠀⠐⠘⡄
⠀⠀⠀⠊⠀⠀⠻⢄⠊⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⣀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⣾⣿⣷⣶⡄⢰⣿⣿⠀⠀⠀⣠⣦⡄⠀⠀⣤⣄⠀⠀⠀⠀⣠⣤⠀⠀⢠⣴⣄⠀⠀⠀⣿⣿⡆⣠⣶⣾⣿⣷
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣤⣀⣴⣿⣿⠃⢀⣼⣿⣯⣶⠄⠠⣾⣽⣿⣧⡀⠘⣿⣿⣦⢀⣤⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣇⣠⣾⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣷⣄⣸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣷⣤⣄⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡟⠁⠀⠈⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠈⢿⣿⡇⠀⣤
⠀⠀⠀⠀⠀⠄⠀⠀⠗⠓⠶⡶⠀⠀⠀⠐⠛⠛⠇⠀⠀⠀⠈⢻⡇⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⣩⣄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢁⡀
⠀⠀⠀⠀⠛⠋⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⣦
⠀⠀⠀⠀⠀⠛⠀⠀⠐⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠰⠚⠀⠉⠉⠛⠆⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⡀⠀⠸⠂⠀⠀⠀⠀⠀⠀⡀⠀⢀⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⡀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢤⠢⣀⠀⠀⠀⣠⡔⠁⠀⡀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠑⢄⠉⠉⠙⠟⠁⠀⠀⠀⠀⠀⢠⠁⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⡀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢖⡀⠀⠀⠄⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠛⠀⠀⠎⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⡉⠀⠀⠀⠀⡶⠲⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⡟⣷⠀⠀⠀⠀⣀⣬⣿⣦⣤⣄⣀⣷⠈⠚⠁⠀⢠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⢦⣤⣶⠿⠛⠉⠉⠉⠙⠛⠿⣿⣦⡀⠀⠀⢈⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣀⡀⠀⠀⣠⣾⠟⠁⠠⣖⡛⠀⠀⣀⣀⢦⡈⠻⣿⣦⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡼⠉⢧⣀⣴⡿⠁⠀⢠⠎⣩⣍⢳⡎⢡⣌⢹⡇⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⢀⣘⣆⣉⣉⣼⣧⣈⣉⣼⣁⡀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢰⠋⣴⡈⢻⠋⣴⣌⢹⡏⢠⣦⠹⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⢦⣉⣡⣾⣦⣉⣡⣼⢷⣄⣡⠼⠁⠀⣿⡇⠀⠀⠀⠠⡖⢦⢸⠀⣇⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢠⠏⢴⣂⢹⡏⢰⣦⠘⡆⠀⠀⠀⣿⡇⠀⠀⠀⠀⠹⡄⠻⠃⠟⢀
⠀⢰⡶⢄⠀⡴⠒⡆⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⢧⣌⡡⠞⠳⢤⣤⡼⠅⠀⠀⠀⣿⡇⠀⠀⢠⠖⠒⠾⠋⠀⠀⣯
⢏⡑⢷⡌⣣⠇⡞⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠚⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣿⠁⠀⠀⠈⠋⠉⢧⡀⠀⣠⠏
⢠⠟⠊⠰⡇⠀⠹⡆⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠆⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠉⣿⠁
⠈⠉⠉⢦⣄⣀⣴⡃⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣿⡇⠈⠲⢤⣀⣀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠁⠀⠻⣦⡀⠀⠀⠀⢸⣿⡤⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⣿⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⢻⡏⠀⠀⠀⠀⠀⠈⠳⢤⣤⠤⠞⠋⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⢦⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠒⠦⣄⡀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⠾⡷⠶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢸⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⡀⠠⣞⠁⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⠤⠗⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⡴⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣀⣄⡀⠀⠀⣾⠁⠀⣿⠀⠀⠀⣴⠞⢶⡄⠀⢠⡟⠉⢻⡄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣼⠃⠈⣷⠀⢸⡇⠀⢰⡇⠀⠀⢸⡇⠀⠈⣇⠀⢸⡇⠀⠈⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡿⠀⠀⡏⠀⣸⠇⠀⢸⡃⠀⠀⢸⣧⠀⠀⣿⡀⠈⣧⠀⠀⠹⢷⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠟⠁⠀⠀⠻⠾⠟⠀⠀⠸⣧⣄⡀⣠⣿⠀⠀⠘⢷⣾⠏

⠀⠀⠄⠐⠀⠈⠦⠀⠤⠤⣄⡀⠀⣀⡤⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠠⡀⠀⡄
⠀⠈⠀⠠⠀⠈⠃⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡂⢠⠤⢤⣤⣄⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠀⠂⠸⠿⠯⠿⠇⠁
⠀⠀⠄⠀⠄⠀⠐⠄⠀⠀⠀⠀⠀⠀⢀⠀⠂⠰⣄⠤⠀⠀⠰⠖⠀⠠⢄⣹⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⠉⠀⠙⠛⠋⠓⠀⠈⠁
⠀⠀⠀⢀⡀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠂⠀⠀⠐⠒⠀⠀⡀
⠀⠀⠠⠁⠀⠀⠀⠀⢂⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠁⢤⣤⠄
⠀⣠⣁⠀⠀⠀⢀⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢇⣀⣀⣀⡀⠀⡰⠁⠀⠰⠤⠆⠤⠤⠶⠀⠀⠠⠤⠇⠀⣂⡀⢀⠀⠀⢀⡀⠀⢂⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠠⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠉⠉⠀⠀⠋⠀⠁⠈⠉⠘⠀⠈⠉⠃
⠀⢰⣧⡀⢀⠀⠀⠀⠨⡄⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⢀⠀⠀⠀⠀⣄⠀⢀⠄⠀⠀⢀
⠀⢸⢤⡈⠛⡊⢁⠀⠂⢓⠀⠀⠀⠀⣾⡏⠀⡇⡌⠃⣼⠈⣽⠰⣿⣿⡄⠀⠀⡧⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐
⢠⠏⠀⠈⠓⠉⠀⠠⠊⠉⡁⠀⠀⢸⣿⣧⣠⣦⣱⣦⣽⣤⣼⣁⣿⣿⡇⠀⠀⡗⠀⠀⠀⠀⢈
⠘⠄⠀⠀⠀⢈⠁⢀⠀⠈⠀⠀⠀⠀⠻⣿⣿⡀⡀⡀⢀⠸⢸⣿⣿⡟⠀⠀⠀⢪⠀⠀⠀⠀⢠⠀⠀⣤⠤⣀⣀⠠⢼
⠀⠀⠀⠀⠂⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡀⠀⠀⠀⠄⢀⠀⠀⡀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠐⠀⡇⠔⠇⠀⠘⠈⡅⢠⠀⠔⠐⠃⡄⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠈⡂
⠀⠀⡶⠀⠀⠆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄
⠀⠀⡀⠀⠀⠆⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠠⠀⡄⠀⡂⠀⠠⠀⠂⠀⠀⡐⠐⠀⡀⠀⠀⠀⠠⣄⣂⣤⢤⠤⣤⠀⠀⢩⠀⠀⠀⡀⣀⡀⠀⠀⣀⠀⠀⠀⢀⠂
⠀⠈⠀⠁⠉⠁⠀⠉⠀⠁⠈⠀⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠈⠉⠁⠀⠀⠈⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠿⠿⢶⣶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣡⠖⠂⠉⠐⠒⠢⠤⠤⢀⣩⣉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⡐⡡⠄⠈⣥⣄⣤⣤⣤⣤⣌⡙⢎⢿⣆
⠀⠀⢀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣧⣤⣧⣶⠿⠛⠛⠉⠉⠙⠛⠿⣿⣿⣶⣝⡻⢶⣶⡇⠀⠀⠀⢀⣴⢶⣤⣤⡄
⢠⣿⢿⣿⡛⣿⠀⠀⠀⢀⣀⣀⣠⣤⣤⣿⠿⠿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣯⣯⣾⣅⠀⠀⢸⣿⢹⡿⢿⠿⠿
⣶⣻⣷⢩⣽⣿⠀⠀⠀⠛⠛⣻⣿⣿⠋⠀⢰⣟⡄⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠙⢿⣷⠀⠘⠷⣾⣶⣿⣿⠟
⠈⠛⠰⣿⠿⣷⡀⠀⢀⣴⡿⠋⠉⣷⠀⠀⠀⠉⢀⣼⣇⡀⠀⠀⢀⣠⣤⠄⠀⠀⠉⠀⠀⠀⣸⣿⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠉⠻⣶⡿⠋⠀⠀⠀⠙⠳⠶⣶⡶⠟⠋⠁⠀⠀⡴⠛⣿⠷⣤⡀⠀⠀⠀⠻⠿⠛⣿⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠻⣦⡀⠀⠀⠀⠀⠈⢿⣆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢦
⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⢷⡀⠀⠀⠀⠀⠈⢿⣆⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠘
⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢀⡿⠀⠀⠀⡼⠈⣷⠀⠀⠀⠀⠀⠈⢿⣦⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣠⣟⠀⠀⠀⡴⠁⢀⣿⠀⠀⠀⠀⠀⢀⡼⢿⣏⠉⠉⠛⠛⠷⢶⣦⣤⣤⣴⡶⠟⠋
⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣧⣀⠀⠛⠁⠉⠛⠶⢿⣤⡶⠞⠁⠀⠀⠀⣀⠔⠋⣠⠾⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠁⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠚⠳⠴⣞⠁⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠈⠻⣷⣤⣤⣤⣴⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⢀⣴⡏⢀⣿⠦⠴⠚⠋⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⢀⡠⠴⡏⠀⣹⠛⠁⠀⣠⠞⠛⠛⣦⠀⠀⠀⠀⠀⠀⠻⣷⣴⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⣟⣉⡴⠾⣧⠞⣡⠞⠛⠛⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⢊⡼⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⣿⡍⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⠀⢊⠴⠋⠀⣀⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢹⣧⠀⠀⠀⠀⠙⠦⣄⣀⣀⢀⣠⠞⣠⣤⠔⡡⠞⠁⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠁⣠⠤⠞⠁⣠⠋⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⠃⠀⠀⢀⡼⠁⠀⡼⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⣷⠤⠤⠒⣁⣴⡿⠛⢻⣧⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠈⠳⢤⠴⠞⠁⠀⣸⠇⠀⢻⣧⠀⠀⠠⢂⡡⠐⠈⠀⣠⣴⡾⠛⠁⠀⠀⠀⣻⠓⣆⡴⠚⠉⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢿⣦⠔⠊⠁⣀⣤⣶⢿⣏⠁⠀⠀⠀⠀⠀⠀⠈⣦⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠚⠛⠁⠀⠀⠀⠀⠈⢿⣦⣶⠿⠛⠉⠀⠈⢻⣦⣄⠀⠀⠀⠀⠀⠀⢷⣬⠟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣹⡄⠸⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠉⠁⢀⡴⣤⠞⠁⠀⠀⠀⢠⡄⠀⠀⠀⠈⢧⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⣀⣠⠶⠋⠀⠀⠀⠀⠀⢷⡀⠀⠹⣆⠀⠀⠀⠈⢳⡘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠘⢦⠀⠀⠳⡄⠀⠘⣦⠀⠀⠀⠀⠳⡌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠹⣆⠀⠈⢧⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⣆⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⢉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣏⠀⠀⠀⠐⢻⣕⣆⠀⢀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⡇⠀⣤⠈⡌⠀⢠⣄⣾⣿⣿⣶⡀⢺⣿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣤⣹⣿⣿⡇⠀⣿⣠⣄⣄⠸⠿⢿⣟⣿⣿⡇⠘⢛⢣⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠿⠞⡛⠛⠿⠓⠈⠲⣾⣿⣿⣿⣾⣿⣿⣿⣿⣧⠄⠛⠛⠛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣇⣶⡄⣶⣿⡏⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⡆⣄⢿⢡⠇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣷⡿⢉⣧⡙⠇⠀⢉⣉⣤⣤⣤⣤⣈⡉⠛⠿⣷⣭⣿⣙⡾⠃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⠛⠋⣿⣿⡶⢞⣿⠷⠿⠾⠿⠿⠿⣿⣷⢤⣀⣈⡉⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣻⣿⣿⣿⣯⣿⢁⣤⣶⣿⣿⡏⠐⠈⢿⣷⣾⣯⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡿⠿⡿⢟⣇⠈⠁⠀⠀⠻⠇⣚⣷⣎⣛⣛⣛⣿⣿⣿⡿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠛⠛⠋⠉⠉⣯⣶⣦⡀⢠⣀⣏⣛⣙⡉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡨⠟⢻⡙⠿⢿⣿⣯⣭⣿⣖⣄⡀⢀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⠞⢫⡆⠀⣄⣹⣶⡄⠙⣻⠟⠿⠿⢿⠞⣋⣻⣯⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣻⣏⡿⠲⠯⣴⠬⠅⠀⢿⣿⣿⣿⣾⣏⣙⣣⡜⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠛⠀⠁⠀⠀⢀⡀⣶⣼⣿⣿⣿⣿⣯⢯⠶⣞⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠈⠙⠛⠛⠁⠀⠀⠀⠉⠈⠁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣻⣻⠷⢶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⠶⢟⣛⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣶⣮⣍⣭⣩⣭⣭⣭⣫⣭⣭⣭⣵⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣶⣿⣿⣿⡻⠿⢷⣶⣶⣤⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⠿⠿⣿⣿⣷⣦⣌⠉⠻⢿⣿⣿⣿⣿⣿⢿⣿⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠟⠋⢉⣀⣤⣤⣤⣄⡉⠻⣿⣿⣷⡄⠀⠙⢿⣿⢿⣿⣧⠹⡇⠀⠀⠑⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⡆⠀⠀⢻⡆⠹⣿⡄⠁⠀⠀⡀⠈⣧⠀⡀⠈⠙⢿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠋⢀⡴⠾⠛⠉⠉⠀⠀⠀⠀⠉⠛⢿⣿⣿⣷⣻⣿⣿⡄⠀⠀⢿⠀⠹⣷⠀⠀⢀⡇⠀⢹⣇⠘⣧⠀⠘⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣿⢋⣀⣀⡉⠀⠀⣠⣶⣶⣿⣶⣶⣦⣄⡀⠀⠙⣿⣿⣿⣿⣿⣧⠀⢀⢸⠆⠀⢻⣇⠀⢸⡇⢀⣿⣿⠀⢿⡆⠀⢹⣿⣿⡟⢿⣷⡀
⣦⠀⠀⠀⠀⣿⣿⠟⠋⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠘⣿⣿⣿⣿⣿⠀⢸⡼⠀⠀⢸⣿⠀⣾⠃⣼⣿⡏⠀⣼⡇⠀⢸⣿⣿⣿⠈⣿⢱
⣧⠙⢦⡀⢀⣫⣴⠆⢀⣴⣿⣿⣿⣿⣿⣿⣏⠛⠿⣿⣿⣿⣿⡄⢸⣿⣿⢹⣿⠀⣸⣗⠀⠀⣸⣿⣾⢏⣾⣿⡟⠁⢠⣿⠃⢀⣿⣿⣿⣿⣤⣿⣶⣦
⢻⣷⣤⣈⣉⣩⣤⣶⣿⣿⣿⣿⣿⣏⡉⠛⠻⣷⣄⠈⠙⢿⣿⣷⠀⣿⣿⠀⡿⠀⣿⣿⠀⢠⣿⣿⣿⣿⡿⠋⠀⣠⡿⠋⢀⣼⡿⠟⠉⠁⠒⠲⢤⣭
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡙⠛⠻⠶⣦⣍⢳⣄⠈⠻⣿⡆⣿⠇⠀⠁⣸⣿⡇⣰⣿⣿⣿⠟⠋⠀⣠⣾⣟⡁⣤⢾⣿⠀⠀⠀⠀⠶⣤⣀⠙
⠀⠸⠷⠶⠶⢦⣤⣽⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠁⠈⠉⠉⠉⠀⠀⠀⣿⡈⢿⣷⡀⠀⠀⠀⠈⠻⣷
⠀⣾⠶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡌⠻⣿⣶⣤⣀⣀⡀⠈
⠀⣿⣿⣿⣿⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣀⠉⠛⢿⣿⣿⣿
⠀⢯⣡⣤⣴⣶⡏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⣀⠀⠙⠻
⠀⠈⣥⣶⣾⣿⠇⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡹⣿⣿⣿⡍⠙⠷⣦⡀
⠀⠀⠟⢋⣩⣿⠀⠀⠀⠃⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣤⣴⣶⣤⡀⢀⡀⠠⠀⠈⢷⡌⠻⣿⣿⣦⡀⠈⠹
⠀⠀⣿⣿⠿⠿⠀⠀⡴⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡌⠳⡄⠀⠀⢸⣿⣆⠘⢯⡙⢿⣄⡘
⠀⣀⣘⡀⢀⠀⠀⠀⠀⠀⠋⠀⠀⠉⠉⠉⠙⠻⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⢿⣿⣧⡈⠳⣄⢹⠇
⡜⠉⠈⠙⣴⠀⠀⠀⠀⠀⠀⠀⠠⢄⣐⣲⣤⣀⣄⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⡠⠀⠀⣠⢴⡶⠶⣶⣢⠤⠀⠀⠀⠀⠀⠀⠀⣆⠙⢿⣷⠖⠙⢯
⣧⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠊⠁⠉⠶⠈⠈⠛⠀⢹⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⡐⠖⡈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡏⠀⣀⠘⠆
⢿⣿⣆⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣼⣿⠙⡇
⢸⣿⠈⢳⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣟⠁⣿⢀⡇
⠈⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⣤⢻⠘⡇
⠀⢹⡇⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣿⢸⠀⡃
⠀⠈⣿⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢣⡿⠛⢁⡼⢰⠁
⠀⠀⠻⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⣸⠧⠒⠋⢀⠎
⠀⠀⠀⢣⡀⠀⣿⢿⡄⠀⠀⠀⠀⠀⢀⣠⣴⠞⠁⠀⠂⠰⣄⠀⢀⠴⠀⢀⠾⠃⠀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⠀⠴⠞⠁
⠀⠀⠀⠀⠉⠉⠹⡆⠻⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⡀⠀⢿⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⡟⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠰⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢻⡄⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢷⡀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠄⠀⠀⠀⠠⠤⠤⠛⠁⠀⠀⠀⠀⢀⠌⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢷⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣷⡀⠀⠀⠀⠀⠠⣬⣿⣿⣿⣿⣿⠿⣿⣭⠄⠀⠀⠀⠀⢀⡾⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠁⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢤⣤⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⣿⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣿⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠈⠇⠀⣀⣀⣉⣀⡀⠀⠏⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠷⠀⣀⣀⣉⣀⡀⠀⠞⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠙⠀⣠⠴⢺⠟⠋⠉⠉⠉⠛⢿⠲⢤⡀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣠⡴⢚⡿⠋⠉⠉⠉⠛⢿⡓⢦⣀⠰⠃
⠀⠀⠀⠀⠀⠀⡠⠚⠁⢰⠋⠀⣴⣶⡟⠲⡄⠀⢳⠀⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡏⠀⣠⣶⡿⠒⣄⠀⢹⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠸⣅⠀⠀⢸⠀⠀⣿⣿⣷⣶⡇⠀⢸⠀⠀⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⡀⠀⢿⣿⣿⣶⡿⠀⢸⡇⠀⠀⡼⠂
⠀⠀⠀⠀⠀⠀⠈⠳⣄⡘⢧⡀⠈⠛⠛⠋⠀⣠⠟⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⢷⡀⠈⠛⠛⠛⠁⢀⡞⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠿⠶⣤⣤⡴⠾⠗⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠿⠶⣤⣤⣴⠾⠿⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⡇⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⢀⣾⠁⠐⣆⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⢀⣾⠃⠀⠀⠘⣷⣄⠈⠙⠻⣿⣶⣦⣄⡀⠀⠀⠀⢹⣿⡀
⠀⠀⣰⠟⠁⠀⠀⠀⠀⠈⠻⢷⣦⣀⠀⠈⠙⠻⢿⣷⣄⠀⠀⣿⡇
⠀⣾⠏⠀⠀⠀⠀⠀⠀⠄⠀⠀⠍⡛⢿⣦⣄⠀⠀⠙⣿⣧⠀⢹⣿
⢸⣿⠀⠀⠀⠠⣤⣤⣶⣿⢿⣿⣶⣤⡀⠈⠻⣧⡀⠀⠘⣿⣇⠸⣿⡄
⠀⢿⡆⠀⠀⠀⠈⠻⢦⣄⣎⣥⠆⠀⠉⠀⠀⠈⢷⡀⠀⢸⣿⠀⢿⣇
⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⢸⣿⡆⠘⣿⡄
⠀⠀⠀⠈⠢⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠙⢿⣦⣀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠚⠛⠻⠿⠿⠛⠛⠳⠂⠀⠀⠙⠿⣷⣤⣀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠑⠶⢤⡤⠶⠊⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⣀⣀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⢸⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⡤⠶⠞⣛⣷⠶⢶⣤⣄⣀⣿⣿⣿⣿⢸⠈⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣠⠤⣌⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡾⠋⡡⠊⠀⠋⠁⠀⠉⠉⠉⠉⠹⢿⣿⣿⣿⢸⠀⠀⠀⠀⠀⢠⣾⣷⣦⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⠀⠀⠙⠻⣼⠀⠀⠚⠹⠷⠾⠇⠈⠿⣃⣤⣤⡤⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠀⠀⠁⠀⠀⠀⢀⣤⡤⠊⠀⠄⠋⠘⢒⢂⡀⠀⠀⠈⠳⡄⠀⠠⣤⣤⣰⣶⣄⣻⠿⣿⣇⡀⣤
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣸⠁⠀⠀⢀⣤⢖⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⢻⠿⣷⣄⠀⠀⢽⣶⠛⣛⣛⣛⣛⣛⣛⣬⣭⠭⠭⡿
⠀⠀⠀⠀⠀⠀⣼⣄⣀⡇⣿⠀⠀⢾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢝⡦⡉⠙⠀⠀⠻⣟⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢀⡏⠀⠉⠉⠘⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⡉⠓⠦⠀⠀⠀⢺
⠀⠀⠀⠀⠀⢸⢁⣴⠀⠀⠀⠀⢀⠔⠑⢛⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢌⠳⢬⣐⠀⢀⠄⠀⢬⣷
⠀⠀⠀⠀⠀⣼⢱⡏⠀⠀⢀⠜⠃⠆⢀⣾⠀⢀⡀⠊⠁⠐⠃⠀⠀⠀⠀⠀⣍⡒⠂⢀⠏⣰⡿⣾⢿⡇
⠀⠀⠀⠀⠀⢹⣾⡇⠀⢀⠎⠀⠀⢀⣿⠃⡠⠁⠈⠁⢀⣴⠃⠀⢀⠠⠤⡀⠈⠿⣷⣜⢰⣿⠙⢮⠈⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠞⠀⠀⢀⣾⣿⠜⠁⠀⠀⢠⡾⠁⡠⢊⡁⠀⢒⠇⠀⠀⠘⣿⡆⢻⣰⡀⠀⣻
⠀⠀⠀⠀⠀⠀⣿⣿⣸⠀⠀⠀⠀⣿⠏⠀⠀⠀⣰⡿⡣⠊⠀⠀⠁⣰⡋⠀⠀⠀⠀⢩⡾⠘⠟⣧⢠⣿
⠀⠀⠀⠀⠀⢀⡟⢿⡿⠀⠀⠀⢸⡟⠀⠀⠀⢸⣿⠏⠒⠦⠀⣠⠊⠙⠛⠁⣀⡀⠀⠀⠀⠀⠀⠁⣾⣿⣄⡀
⠀⠀⠀⠀⠀⢸⠇⢸⠇⠀⠀⠀⡼⠁⠀⠀⠀⣿⠋⠀⠀⢀⣼⠃⣀⣤⠔⢁⠃⣸⠀⠀⠀⠀⠑⠶⢻⣿⣿⣿⣿⣶⣶⡶⠶⢤⣤
⠀⠀⠀⠀⠀⣼⢀⡾⠀⠀⠀⢀⡤⠀⠀⠠⠞⠁⠀⠀⠀⣾⡿⠟⠋⠀⠀⢈⠞⠁⠀⣠⡏⠀⠀⠀⠀⠹⡏⣿⣿⣿⣿⣿⣾⣦⣄
⠀⠀⠀⠀⠀⣿⡿⠁⠇⠀⠀⢸⠀⠀⠀⡞⠁⠀⠀⠀⠀⠁⠀⠀⠀⣨⢖⡁⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣯⠻
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⣀⣴⠋⠁⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣀⣿⠀⠀⢠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠰⠛⠀⡴⠁⠈⢀⡘⠉⠂⠀⠀⠀⠀⠀⠀⠀⢠⢀⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣠⣴⣿⣿⣿⡆⠀⢸⠀⢀⠂⠀⠀⠂⠀⠈⠀⠀⠀⣸⠁⠀⡰⠋⡈⠁⠉⢙⡓⠆⠀⠀⠀⢀⡞⡜⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣇⣠⡧⠈⠀⠈⠁⠀⠀⠀⠀⡼⡹⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠟⠉⠉⣀⣀⡴⠃⠀⠀⠀⢀⠀⣴⠁⠀⠀⣼⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣈⠋⠀⠀⠀⠋⠉⠁⠀⠀⠀⠀⠀⣼⡾⠃⠀⠀⢰⡏⡿⣿⣿⣿⣿⣿⣿⣿

⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠀⠀⠐⠂⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣇⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣷⣶⡂⠀⠰⣿⠇
⠀⠀⠀⠀⠀⣾⣉⠐⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⢆⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠿⠻⠿⠿⣿⣿⣶⣄⣹⡄
⠀⠀⠀⠀⢸⣿⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⡀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⡀⠈⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣄⢻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣦⡙⠳⠀⠀⢠⣿⣿⠁⠀⣠⣶⠾⠛⠛⠛⠛⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⣻⠸⣿⣿⣿⣿⠟⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣾⣿⣿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⢀⣿⡉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣼⡁⠈⠛⢿⣷⣶⣶⣶⣶⣶⣦⣤⣤⣼⣿⣿⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣦⡈⣿⣶⣿⣿⢸⣿⣿⡻⠿⢿⣿⣿⣿⣿⣿⠃⣠⠼⠋⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢿⢃⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢌⠙⠿⣿⣿⣏⢸⣿⣿⣷⡀⠀⣹⣿⣿⣿⡷⠚⠁⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢠⣿⢩⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣄⣿⣿⣿⡇⣰⣿⣿⣿⣿⣀⣀⣠⠴⠖⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠏⢀⠄⢸⡟⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⣀⠠⣴⠟⠁⠀⠀⠀⢹⣿⣿⡳⢂⡼⠁⣴⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⠁⠀⣿⣦⣀⣀⣀⣴⣾⣿⡿⠛⠉⠀⢠⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠉⠉⠀⣹⡿⠁⠉⠛⠉⠉⠉⠉⠛⠛⣿⠋⠀⠀⠀⠀⡼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣭⡇⠀⠀⢀⣴⣯⣄⣀⡠⠤⣤⣴⣷⠂⠀⢠⡟⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⠁⠀⠀⣀⣤⣤⣶⣿⣿⡿⢀⣴⣿⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣶⣿⣿⣿⣧⣽⣫⣶⠋
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣴⣿⣿⣟⢹⣿⣿⣿⣿⣿⣆⠀⡀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡓⠂
⠀⠀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀
⣴⣿⣿⠿⠋⠉⠉⠉⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⣿⡟⠁⠀⣀⣦⣴⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⡿⠛⠉⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠟⠁⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⣝⣛⣒⣒⠶⠦⢤⣤⣤⡤⢤⡄
⠀⠀⣤⣦⣀⡀⣤⠀⠀⠀⢀⣠⣤⣤⡀⢰⠄⠀⣰⣆⣴⣧⣠⢆⣦⣤⣤⣴⡄⠀⠀⠀⠀⠀⠀⢰⣯⣽⣒⣲⠮⢭⣭⣭⣽⣹⣿⣼⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣷⣷⣾⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣻⣇⢀⣀⠀⢸⡷⣶⣿⣽⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠛⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⠒⢻⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠶⠬⠄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋⣠⣿⣶⡿⣦⣶⣆⣉⠉⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢉⣹⣿⣶⣶
⠸⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⡓⢾⣿⣿⠏⢀⠉⣫⣿⠟⢼⣤⣌⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢛⣩⣴⣶⢉⣀⡼⠟⡉
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣻⣿⡿⡟⠻⢿⣤⣤⣿⡿⠟⠻⢶⣄⡙⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣩⣴⠚⢹⣿⣿⡿⢿⣿⣷⣾⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣷⣤⣼⣿⣿⣿⣷⣤⣙⠻⣿⠟⢋⡡⠴⠺⢿⣿⣿⣶⣶⢻⣷⣶⣿⣿⣿⣿
⠀⠈⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣈⠙⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠶⣤⡙⠿⢿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣷⣤⣴⣿⣿⣷⢤⣌⠛⠧
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣏⣿⣿⣇⢀⢩⣿⣦
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠙⠛⠃
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠛⠛⠉⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣦⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡏⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⠋⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠋⠛⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⣻⡀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣿⡀⢠⣶⣶⣦⠐⣿⣿⡷⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣝⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣀⣀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢦⣄⠀⠀⠀⢀⣶⣶⡌⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠟⠀⠀⠈⠙⠒⠒⠒⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⣿⣶⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡀⠀⠀⠀⢀⣴⣿⣿⣿⡿⣷⣾⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⡇⠀⠀⠀⠸⣿⣿⣿⣿⡋⢿⣿⣿⣶⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣠⡶⠗⠀⠀⠠⣤⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠰⠋⣀⡀⠀⠀⣀⡀⠙⠆⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⡞⢱⣿⣦⡎⢹⣿⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢋⠓⠀⠀⠀⣧⣈⣛⠟⠣⣄⡡⠇⠀⢀⣀⣼⣷⠶⠾⠟⠛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠲⠄⠀⠀⠀⠀⠉⠀⠀⠀⠠⠶⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠋⠀⠀⢲⡖⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⡀⠈⠛⠷⢶⣤⣤⣤⣤⣤⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢧⡀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠳⣄⡀⠀⠈⠛⠶⠴⠞⠁⠀⠀⠀⠀⢀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⢁⣿⠀⠈⠙⠳⠶⣷⣶⣶⣃⣠⣤⣤⡶⠾⠛⠉⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣠⡶⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠳⣄
⠀⠀⢠⠦⣄⠀⠀⣀⣀⣠⠞⠋⣠⠞⠉⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⣦⡈⠳⣄
⢀⡤⠤⠵⠦⠙⠋⠉⠀⠁⢰⡟⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣦⡈⠳⣄⠀⠀⠀⠀⢀⢄
⣜⠂⠒⠂⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⣦⡈⠻⠶⠶⠔⠡⠟⣊⠇
⠈⣟⣉⣉⡥⠔⠒⠉⠉⡆⠸⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⡴⠃⣀⠀⠀⠀⠐⢛⣑⣹
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠹⣤⣨⠗⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⣤⡤⠞⠉⠓⢤⡈⠙⠻⢭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠙⠷⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠹⣄⠀⠀⠀⠀⠀⠀⢸⡍⢿⠙⠛⠛⠛⠛⢻⡟⣿⠋⠉⠁⠀⠀⠀⢀⣤⠞⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠻⡷⢶⣤⣤⣤⣤⣼⡇⢸⡀⠀⠀⠀⠀⢸⡇⣿⣀⣀⣀⣀⣀⣴⠟⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢷⡀⠀⠀⢰⠟⠀⠉⠉⠉⠉⠉⠉⠁⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠶⠶⣦⣤⣤⣤⡾⠃⠀⠀⠘⠷⣤⣤⠤⠶⠶⠶⠶⠾⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣬⣭⣭⣭⣭⣭⣙⣓⡂⠐⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠭⠍⠉⢀⣀⣤⡭⠭⠙⠳⢆⢻⣿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠛⠛⠉⣁⣀⣤⡴⠶⢚⣩⣵⣲⠄⠉⠀⠀⢦⡀⢿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣾⣿⣿⡿⠟⠂⣠⣶⣿⡿⠋⠁⠀⣠⣶⣿⣾⣷⣼⣿⡟⢠⡟⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣟⣛⣛⣋⣭⠵⢂⡤⢞⣿⡿⠋⣠⣶⣤⣴⣿⣿⡿⠿⢿⣿⡟⣠⣿⠛⢶⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠀⠊⢉⣠⠾⢋⣤⣾⣿⣿⡿⠛⠛⠋⠁⠀⠀⠈⠻⣿⣿⣧⡈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣶⠶⢛⣉⡚⠩⢥⣶⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣎⢳⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣥⠶⠚⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⢷⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⣶⣯⠭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣋⣤⣴⣀⣤⣾⠶⢶⣶⣦⣤⣤⣴⠀⠀⠀⢀⣴⣶⣶⡶⠿⠶⣦⣼⣇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡟⠀⠁⠀⠤⠼⠿⠟⠋⣿⡿⠿⣿⣿⡏⣭⠤⣤⠄⠀⠀⠘⣿⠟⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⣦⣿⡇⢿⠀⠀⠀⠀⠐⠂⠀⠀⡿⠀⠀⠘⣿⠄⠀⠊⠂⠀⠀⠀⢸⣿⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢃⢻⠀⠸⣄⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢀⡼⣿⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡸⢼⠀⠀⠈⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠃⠈⠉⠛⠛⠛⠉⠉⢠⡿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⣐⡇⠀⠀⠀⠀⠀⠀⣀⠄⠀⢄⠀⡀⠀⡀⠠⣄⠀⠀⠀⠀⠀⣾⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠈⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢦⣤⣤⣤⣤⣥⣤⣤⣴⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠈⠿⠛⠛⠛⠛⠛⠿⠃⠀⠀⢠⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠠⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⢠⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣁⡀⠀⠀⠀⠀⠀⢀⣀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣶⠀⠀⠀⠀⠀⢀⡴⠛⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿⠀⠀⠀⠀⠀⠈⠱⠴⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠒⢻⠆⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⢸⠃⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠳⡄⣠⠹⠄⠀⠀⠀⠀⣿⡿⣿⣷⣄⠀⠀⠀⢸⠀⢸⠁⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣿⡇⠈⠻⣿⣿⣶⣤⣘⣀⣀⣠⣴⣿⣧⡀⠀⠀⣀⣠⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠙⠿⣿⣿⡿⠿⢻⣿⡟⠁⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⣀⣀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿⡉⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⢻⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠶⠚⠛⠛⠛⠳⠦⣤⣼⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⣦⡿⠁⠀⢀⣠⣤⣀⠀⢀⣼⠏⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠁⠀⢀⣼⣿⣤⣤⣀⡀⢰⣿⢟⢧⡀⠀⠀⠀⡿⠀⠀⠹⣟⡄⣾⠟⢋⣩⡿⠿⠞⠛⢻⣿⡆⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣶⣾⡿⠋⠁⠀⠀⠀⠈⠙⠻⣿⣮⠂⠹⣆⠀⢰⡇⠀⠀⠀⢻⣞⣏⠀⢸⡇⠀⠀⠀⠀⢸⣿⣧⣀⣀⣿⠙⠃
⠀⠀⠰⡦⠴⢳⠀⠀⠀⠀⠀⠀⠈⣻⣿⡀⠀⢀⣤⣤⣤⣄⠀⠀⠈⢷⡵⠀⠈⠳⣼⠁⠀⠀⠀⠀⢳⡟⢦⣬⣷⡾⠟⠀⠀⣸⠋⣿⠁⠀⢸⣦⣤⣀⣀⡀
⠀⠀⠀⡾⢄⣠⠂⠀⠀⠀⠀⠀⠀⣿⣿⡕⣼⡟⠉⠉⠉⠻⣧⠀⠀⠘⣷⢤⡀⠀⠉⠀⠀⣼⡄⠀⠈⣿⡀⠀⠀⠀⠀⣀⣴⢏⣤⣿⣤⣶⣾⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣠⣾⣿⣿⣯⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣸⣿⣆⠀⠀⢀⡟⢿⡀⠀⢘⣷⣷⣶⣾⠿⠛⢡⡟⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣻⣧⣀⠀⢀⣠⡞⠀⠀⠰⡿⢿⣿⠻⣿⣿⣽⡇⠸⣿⠿⠟⠛⡈⢦⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢀⡿⠶⣶
⠀⠀⠀⠀⠀⠒⠛⠋⠉⠉⠉⠉⣉⠟⢿⣿⡿⣆⠉⠙⠋⠁⠀⠀⣠⣆⣲⣠⡰⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢙⣦⡿⣶⡛⠛⠺⠿⠦⠀⠀⠀⠈⠳⣴⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⠞⠛⢿⣶⣧⣄⣀⣠⣤⣾⣿⡿⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠻⡈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⢀⡴⠋⠀⠀⢀⠼⠋⠛⠛⠉⠉⠀⠪⡝⣿⣦⠀⠀⠀⢀⣴⢶⡳⣞⡤⡲⢦⣶⠒⠀⠳⣶⡐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣰⠟⠁⢀⣤⠶⠶⢄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠜⡀⠀⠀⠀⠀⠀⠀⠸⡇⣰⠏⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⣨⠓⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠗⠚⢦⡇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣶⣿⣤⣴⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⠋⢿⣿⣻⣾⣿⡿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⡸⣿⣧⣿⣤⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣟⣱⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⢿⠈⠻⣿⣿⣿⣿⣿⣿⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠋⠀⣿⣿⣿⣿⣿⣿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⢰⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠐⠀⠀⠆⠀⠀⠀⠀⠀⠘⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠱⠆

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠰⣉⠄⢀⢀⠶⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠰⠷⡏⠀⢵⠋⡙⣡⡁⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⠀⠀⠀⠀⠀⠀⠈⠋⣁⠐⠀⢃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣫⣭⣶⣮⣭⡛⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠄⠘⠀⢀⠀⠀⠀⠀⠀⠐⡶⢻⠆⠦⠀⣄⠂⢠⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣼⠟⠉⠉⠉⠙⢿⣝⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠋⠠⠚⡀⠰⠶⠀⠀⠀⢶⠄⠃⡤⢗⡔⠘⢋⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠋⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠋⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⢣⠀⠘
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠃⠀⠄⠀⠀⠀⠐⠂⠈⠁
⠀⠀⠀⠈⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡸⣇⠰⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⢻⢀⠔⠋⣯⠞⠀⠈⠳⠀⢱⡴⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠞⠹⠃⠀⠘⢇⠀⠀⠑⠀⠀⠀⠀⠀⠘⠀⠈⠟⠑⣄⢀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢆⠈⠁⠀⠀⠀⢀⡤⠖⠒⠒⠒⠢⣄⠀⢸⡀⠀⠀⢀⡤⠖⠒⢤⠀⠀⠀⠀⠀⠀⡷⢦
⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⢸⠀⢀⠔⠁⠀⠀⢀⡀⠀⠀⠀⠙⠢⠿⠖⠊⠁⠀⠀⠀⠀⠑⢄⡀⠀⢢⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢤⣬⡄⢈⠢⣓⡞⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣶⣤⣀⠀⠀⠀⢦⠀⠀⠈⢳⣾⠀⠀⣸⣠⠄
⠀⠀⠀⠀⠀⠀⢀⣧⠀⠠⣕⢒⡇⠀⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣶⣤⣄⡑⠀⠀⢸⣷⡇⠐⠉⡼
⠀⠀⠀⠀⠀⠀⢸⠛⠀⠨⠔⣛⣻⠀⠀⠀⠀⠀⣀⣤⣴⡄⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⡆⠀⠀⣿⠂⣠⠞⢳
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠁⡞⠀⣤⣤⡴⠶⠿⠿⡿⠿⠃⠀⠀⠀⠀⣁⣀⣀⣀⣀⣀⣈⣁⠀⢸⡇⠀⡀⡰⠃
⠀⠀⠀⠀⠀⠙⢧⣀⡄⠀⢸⠀⠠⡦⠤⣤⠤⣤⠶⠾⡶⠖⠀⠀⠀⢸⡏⠉⠙⠿⠛⠀⣰⠇⠀⣸⠀⡼⠰⢅
⠀⠀⠀⠀⠀⢠⠞⣉⢳⠀⢻⠀⠀⠱⣄⡈⠙⢉⣀⣼⠃⠀⠀⠀⠀⠈⣿⡲⠦⣤⡴⠾⡿⠀⢀⡏⡸⢀⢤⠘⡆
⠀⠀⠀⠀⠀⢸⢰⡁⠈⡆⠘⠀⠀⠀⠘⠛⠛⠛⡻⠃⢀⡤⠶⠶⠦⣄⠘⠌⠀⠀⠀⠀⠀⠀⢸⠃⡇⠘⠹⢰⠇
⠀⠀⠀⠀⠀⠘⣿⣝⠛⣇⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⣠⠦⠀⢸⢀⣇⣄⡴⠋
⠀⠀⠀⠀⠀⠀⠙⢮⡓⠙⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⢀⣠⠞⠁⠀⠀⠙⢋⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠖⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⢚⣩⡿⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⣭⣭⣥⣶⠶⡿⢋⡿⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠈⠛⠿⢤⠤⠖⣋⢀⡀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⣄⠀⠀⠈⠙⣋⡵⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣚⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿