Vinegar Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⡇⣿⣷⣶⣶⣶⣶⣶⣶⣿⡇⣿⣿⣷⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠛⠛⠛⠛⠛⢣⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢧⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣼⣡⣆⣀⣧⡀⣴⣤⡘⣷⠄⢰⣦⢠⣦⣀⣴⠀⣠⡄⣰⡄⣰⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣻⣻⣿⠟⣻⣟⠻⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣅⠻⣿⣹⣿⣗⣻⢯⣀⣼⢿⡟⢙⣭⠭⠿⢛⣽⠏⢍⣭⠿⠻⣿⣿⢹⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣌⠻⡟⣛⣻⠵⠿⣷⠟⢱⣬⣝⡿⣛⢚⣾⣫⣖⢙⡿⢧⣥⣽⣾⣿⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢯⢣⣙⣿⣷⣿⣿⣝⣾⣷⣿⣿⢿⣰⣉⣿⡿⣿⣟⣶⣿⣺⣭⣿⡿⢧⡧
⠀⠀⠀⠀⠀⠀⠀⠀⣗⢐⠀⠀⠈⠹⠿⠯⢭⡙⣻⣿⣿⣳⣾⣿⡾⣿⡿⣫⡿⣿⣿⣿⢟⣭⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣅⡇⠀⠀⠀⠀⢠⠀⠈⠏⢦⣙⡻⠿⠿⠿⠿⢿⠿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⡿⠿⡷⣶⣤⣤⣤⣤⣬⣼⣤⣿⣶⣾⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢶⣇⠀⢀⡀⠀⠸⠉⠉⢉⣹⠹⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣙⢀⠀⠘⠃⢀⣀⠺⠇⠠⢹⢸⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣶⣴⣤⣌⣛⣀⡀⡀⢾⣚⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢀⠀⠈⠉⠉⠛⠛⠛⠿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⠂⠀⠀⠀⠀⡄⠀⠀⢻⣏⣧⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣷⣦⣤⣀⣀⣀⣀⠀⢰⡾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⢉⣽⣿⣿⣿⣿⣿⣽⣾⠦⡶⠴⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣍⣉⣉⣀⣋⡘⠟⣻⣿⣻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠉⠉⠙⠛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⡿
⠀⣀⠀⠀⠀⠀⠀⠀⠰⠤⠤⠤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⡀⣼⠿⠿⣷⣤⣀⣀⣀⠀⠟⠿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⢀⡀
⠈⠠⢴⣶⣦⡰⠆⣀⠀⠀⠀⠄⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⡋⣄⣀⣤⣉⣀⣠⠴⠶⠇⠀⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠙⠋⠙⠁⠦⠦⠴⠶⠶⠿⠣⠶⠅⠓⠬⠭⠋⠥⠽⠟⠻⠛⠚⠛⠚⠋⠉⠀⠀⠉
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⢤⡤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⡀⠚⠛⢀⣀⠐⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠯⠥⠤⠤⠬⠭⠤⣤⠿⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣏⣉⣉⣉⣉⣉⣉⣉⣉⣹⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡖⠒⠒⠒⠒⠒⠒⠒⠒⠲⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠶⠶⠶⠶⡆⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⡇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠗⠚⠛⠉⠉⠉⠙⠛⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⡀⢸⡧⡤⠤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠲⠂⠀⠀⠀⠲⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠈⢷⡟⠀⢤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⣤⠀⠀⠀⠀⢠⣀⣠⡀⠀⠀⠀⠀⠀⢀⣤⣄⣤⣀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠚⠛⠛⠂⠀⠀⠀⠀⠀⠀⠴⠤⠀⠀⠀⠀⠀⠀⠶⠶⠂⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠴⠦⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⣀⡀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁⠲⠶⠆⠠⠭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣛⣛⣛⣛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⡉⠉⠉⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢀⡞⠁⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⠟⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢸⠋⠉⠉⠉⠉⢉⣩⣍⡉⠉⠉⢹⢀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⣾⡵⠛⠁⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣠⠞⠙⠻⠋⠙⢦⡀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢠⡇⡶⠀⠀⠀⠀⠈⣧⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠘⣇⢻⡀⠀⠀⠀⢀⡟⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠀⠀⠘⠦⣤⠶⢦⡤⠞⠀⠀⢸⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⠤⣤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠀⠀⠀⠀⠀⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⠛⠛⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⢉⣉⣉⠉⠉⠉⢻⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠉⠁⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⣈⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠖⠒⠛⠛⠒⠶⠦⠤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠶⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠐⠛⠛⢀⣀⡀⠀⠒⠂⣀⣀⡀⠀⠐⠒⠀⠠⣤⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠀⢀⡈⠉⠀⠴⠦⠀⠉⠉⠀⠀⠰⠶⠖⢀⣀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣬⣭⣭⣤⣤⣤⣤⣬⣭⣤⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⢤⠄⢀⡴⢶⣦⠄⢠⡤⣤⣤⣤⢤⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣴⣋⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣴⠛⠉⠻⠟⠉⠛⣦⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⡖⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡄⢷⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣀⡀⠻⣌⣁⣠⣄⣀⣠⠞⢀⣀⣀⣀⣀⣠⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣤⣤⣤⣤⣤⣤⣬⣭⣥⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠁⠀⠐⠒⠂⠀⠀⠀⠙⠛⠁⠀⠀⠴⠶⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠲⠖⠀⠀⠐⠶⠂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣀⡀⣀⣀⡀⣀⣀⣀⣀⣀⣀⢀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢺⣿⡇⠀⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠋⠀⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣤⡇⠀⠀⠀⠀⢹⣩⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠿⢸⣿⣿⣿⣿⡆⠷⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⢾⣿⣷⣄⡀⢀⣠⢾⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⡡⠀⠘⠻⣿⡿⠟⡁⠀⠘⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⡇⢸⡇⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣾⠁⠀⠀⠀⢸⡇⣾⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡇⢸⡇⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡇⢸⡇⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⢛⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡇⢸⡇⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣽⠄⠀⠀⠀⢽⡇⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⡇⢸⣇⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢿⠀⠀⠀⠀⢸⡇⢿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠛⠉⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠛⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⣠⣶⣾⣿⣿⡿⠿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠻⠿⠿⣿⣿⣷⣶⣄
⠀⢀⣾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣷⡀
⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⠀⠀⠀⠀⠀⢻⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⢸⣿
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿
⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿
⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⣿⣿
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⠀⣼⣿⡿
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⣿⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⢰⣿⣿⠃
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿
⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣸⣿⡿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⣿⣿⡇
⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢀⣿⣿⠇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠏⠀⠀⢸⣿⣟
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⡀
⠀⠀⠀⣠⣿⣿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣄
⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠐⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠘⢿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣴⣾⣿⡿⠋
⣀⣀⣀⣀⣀⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠛⠀⠘⠛⠛⠉⠉⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⣀⣀⣈⣀⣀⣀⣀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⣶⣆
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠘⣇⠀⠀⠀⠀⠀⠀⣸⠇⠹⣦
⠀⠀⠀⠀⠀⠀⣸⠋⢻⣄⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠹⣄⠀⠀⠀⠀⣰⠏⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⢷⣄⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠹⣆⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠳⣄⢸⠃⠀⠀⣴⡀⠀⠀⠀⢹⡾⠃⠀⠀⢠⣦⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⡇⠘⢦⣀⠀⠈⠀⣀⣠⠔⠃⠈⣆⠀⠀⠀⠀⠈⢷⡀
⣤⣄⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠙⠓⠛⠋⠉⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣧
⠈⢿⡛⠶⢦⣄⡈⣇⠀⠀⠀⢀⡀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣴⠒⠒⡶⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⢳⡀⠀⠈⠙⠻⠆⠀⠀⠀⡏⠳⢄⣸⠁⠀⠀⠀⠀⢰⠃⠀⡼⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠁⢀⡀⠀⠀⠀⢸⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⠲⣄⡀⣸⠀⡟⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡄⠀⠉⣙⣻⣧⣀⡀⠀⠀⠀⠰⠿⠞⣷⠀⠀⠀⠀⢰⠇
⠀⠀⢰⡇⠀⠀⠀⠸⠓⠦⣤⣸⠀⠀⠀⠀⣠⠖⠛⠶⢤⣄⣠⠟⠉⠙⠳⣄⠀⠀⠀⠀⡏⠀⠀⠀⢠⡏
⠀⠀⢸⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡜⠁⢀⣤⣄⠀⣠⠶⠖⠶⣄⠀⠈⢧⠀⠀⠀⡇⠀⠀⠰⢿⣤⠴⠶⠛
⠀⠀⠈⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰⠁⠀⡏⠠⠎⢻⠁⠀⡶⠀⠈⡇⠀⠘⡆⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠸⡆⠀⠙⠦⠤⠾⣧⡀⠀⢀⣰⠇⠀⠀⣷⠀⠈⣩⠗⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠁⠢⣀⠀⠹⣄⠀⠀⠀⣀⡈⠉⠛⠉⣀⡄⠀⠀⣽⠖⠋⠁⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠙⠒⠾⠷⢦⡀⠈⠉⣟⠛⡟⠉⠀⠀⠀⡇⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠹⣄⠀⢸⡖⣇⠀⠀⠀⣼⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠲⠶⠶⠖⢻⡄⠈⣧⢹⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⠿⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠠⠤⠆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠛⠲⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣶⠓⠟⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢦⣤⣤⣤⣤⡤⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⣄⡄⠀⠀⣤⢠⣤⠀⠀⠀⢀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠔⠎⡛⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠃⠇⠘⠂⠉⠈⠀⠀⠀⠀⠀⠁⠈⠁⠀⢀⡀⢸⣳⡠⠔⠁⠀⠀⢀⡜⠓⢤⡀⢰⠋⠀⠀⠠⠀⠠⠀⡇⠀⣰⣦⡁
⠀⠀⠸⢇⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⣼⣠⠏⠀⠀⠀⠐⠀⠈⠀⠀⠈⠀⠸⡆⢄⠀⢅⠀⣐⠼⠁⠀⡇⡜⠃⠀⠀⠀⣀
⠀⠀⠀⠘⡆⡜⠀⠀⠀⠀⣤⠀⡰⠂⠀⠀⠼⢾⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⠒⠒⠟⣶⠛⠛⠉⠁⠀⠀⠘⡇⣇⠀⢀⡰⢶⣝
⠀⠀⠛⢄⢻⡇⠀⠀⣠⡞⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⢉⠦⠤⠤⠴⢠⡤⣧⡄⢼⠆⠀⠀⢰⠉⠂⠌⣐⠀⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢳⣈⠉⢉⣀⠜⠋
⠀⠀⠀⠈⢢⡇⠀⠀⢈⣧⡀⠀⣸⡄⠀⠀⡠⡄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠘⣆⠌⠐⠀⢁⠂⠈⠂⡼⠀⣠⣤⣄⠀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢳⠀⠀⠁⠘⠛⠙⡍⠀⠀⠀⠉⠁⠀⠀⠀⣄⠀⠀⠀⠀⠀⣴⡽⠀⣀⡠⢤⠤⡬⠷⣤⠤⠤⠤⠖⠋⢀⢮⠞⠁⠀⡴⢹⢀⣤
⠀⠀⠀⠀⠀⢹⢠⠶⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⡿⠋⠀⢸⠁⠠⠀⠀⠀⠄⠈⢦⠀⠀⠀⠀⢸⠘⠒⣦⡸⠁⠓⠊⡼⠀⣾⣿⡆
⠀⠀⠀⠰⢆⢸⡏⠀⠀⡠⠤⠤⣀⣀⠴⠊⠳⡀⠀⠀⠀⠀⣀⡠⣤⠄⣀⣀⣀⠘⣦⠈⡀⠀⠀⠄⠀⣘⠇⠀⠀⠀⠈⠓⠊⠡⡷⠒⡆⣸⠁⠀⠙⠛⠋
⠀⠀⠀⠀⠈⢾⡇⠀⠘⣅⠀⠀⡼⠉⠓⢦⣸⠁⠰⠀⠖⠉⠃⠀⠀⢀⠠⠐⠉⠉⠉⢷⠚⠛⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣵⠖⢲
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⢦⡞⠁⠺⠇⢠⡏⢢⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⣤⣤⠀⢠⣤⡄⠀⣠⣤⡀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠠⠂⢼⠙⠲⢄⣠⠋⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣠⡇⠀⢸⢹⠀⠀⠀⡿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡴⠞⠁⠉⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣾⠀⢸⣿⡇⠀⢸⢸⠀⠀⠀⣙⠇⢸
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡀⠠⠍⡇⠀⠀⢀⢏⣌⢧⠀⠀⢸⣿⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⣧⠄⢾
⠀⠀⠀⠀⠠⠀⡷⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠛⠀⡇⠀⠀⢸⢸⣸⢸⠀⠀⠘⠋⠀⠈⠛⠁⠀⠘⠛⠀⠀⠀⠿⠤⠼⠰⠆
⠀⠀⠀⠀⠀⠀⡧⡇⠀⠁⠀⠀⠀⠀⠀⠀⠐⡿⠷⢶⠂⠀⠀⢸⢰⡏⠙⢆⡇⠀⡆⠘⣆⣡⠞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⢹⡇⠀⢠⠀⠀⠀⠀⠀⠀⢀⠇⠂⠘⡆⠀⠀⠀⠶⠁⠀⠘⠃⠘⠃⠀⠀⠀⠀⠀⠀⢠⣾⡁⢀⣀⣠⠔⠛⠓⢤⣠⠎⠀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⡴⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⢹⡀⠀⠀⠀⢀⠀⢀⠀⠀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠈⠛⠿⡀⠀⠀⠀⠈⠋⠀⠤⠆⠀⠈⠁
⠀⠀⠀⠀⢀⡞⠤⣈⣆⠀⠀⠀⠀⠀⠀⣸⠃⠠⣕⠀⠀⢧⠀⠀⠀⠉⠀⣿⠀⠀⠀⢰⡆⠀⢇⣤⣤⣀⣀⣀⣀⡀⠘⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡞⠈⠘⠋⢹⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⠀⣤⡄⠀⢶⠀⠀⠀⠣⠤⠄⣀⣀⣀⣀⣤⠀⠀⢠⣤
⠀⠀⠀⠀⢣⣀⠀⠀⣸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⠐⣿⠀⠀⠀⢸⢿⠀⣿⡇⠀⣿⠀⠀⡤⡄⠀⠀⠀⣶⠀⠀⡷⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣠⡀⠙⠿⢿⡿⢿⡿⠿⠿⠿⠿⠛⠀⠀⠀⢿⠀⠀⠀⢸⢸⠀⡇⡇⠀⣿⠀⠀⡇⡇⠀⠀⠀⢸⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢖⣢⣀⣀⣹⣷⣤⡤⠼⢷⣦⠤⠤⠄⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⢸⢸⠀⡇⡇⠀⡿⡆⠀⡇⠇⠀⠀⢸⢸⠀⠀⣷⡇⠀⠐⢶
⠀⠀⠀⢀⣤⣤⠔⠊⠉⡿⠃⠀⠐⠖⠁⠀⠺⠊⠀⠀⠀⠀⠀⢤⡄⠤⠤⢾⠤⠶⠶⠾⠾⠶⠷⠷⠶⠻⠗⠦⠗⡖⠒⠒⣼⠛⠒⠲⠟⠓⠿⠆⠙
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠘⠈⠀⠈⠉⠁⠀⠁⠀⠀⠈

⠀⢠⣤⣶⣴⣤⣦⣤⡀⠀⢠⣤⣦⣴⣤⣦⣤⡀⠀⣠⣤⣦⣴⣤⣦⣤⠀⠀⣠⣤⣦⣴⣤⣦⣤⠀⠀⣠⣤⣦⣴⣴⣤⣤
⠀⠿⠁⠀⠉⠉⠀⠘⡇⠀⠿⠁⠀⠉⠁⠀⠙⡇⠀⡟⠁⠀⠉⠁⠀⠙⠇⠀⡟⠁⠀⠉⠁⠀⠹⠆⠀⡟⠁⠈⠉⠁⠀⠹



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦
⠀⣤⠖⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡞
⠘⢧⠾⠻⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢷
⠀⠀⠀⠀⠀⠹⣆⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⢧
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠙⣦⣄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠉⠓⠒⠒⠒⠋⠉⢳⡀
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠤⠀⢀⡴⠃⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⡷⣄⣀⣀⣠⣤⠶⠭⣀⡠⠒⠀⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢋⣉⢳⠀⠀⡸⠓⠒⣲⡞⢑⡒⠐⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡘⠀⣤⠀⠈⡇⠰⠄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⠋⠈⠉⠀⠀⡇⠀⠀⠉⣇⡈⠀⢀⠞⠀⡀⠸⡇⠠
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠱⡀⠀⠀⣠⠗⡤⡀⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡏⠀⠀⠀⠀⠀⠘⢲⣴⣚⣉⣉⣉⠥⠔⠊⡇⢠⡇⠄
⠀⠀⠀⠄⠀⠀⠀⠻⡄⠐⢷⠬⢥⣉⣁⣀⣀⡀⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢸⡇⠀⠀⠀⢀⠄⠀⠸⣀⢈⠀⠀⢀⣀⠠⢾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⣄⣀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢹⠉⠉⢉⠀⠀⠐⠉⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡼⠳⢤⣀⡀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⡟⠉⠉⠉⢻⠈⠉⢻⠀⠀⢀⡤⠒⠒⠺⣅⠀⣀⣈⣉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠐⠀⠀⠸⠋⠀⢀⠈⠱⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆⠀⠀⠀⠀⠀⢸⢀⠀⠘⢄⡀⠀⣀⣧⠀⠀⠀⠀⢠⠃⢠⠏⠀⠑⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠣⣀⠀⠀⠀⡠⠎⠈⠀⠀⠀⠈⣹⡁⠀⠑⠦⠤⠒⣁⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⠀⠀⠀⠀⠀⢉⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣈⠁⠉⠉⢉⣀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢿⡿⠿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⢠⠋⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠚⠁

⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢠⣀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⢉⣤⣌⠃⡀⠀⠀⠀⠈⢋⣬⡟⠀⠀⠀⠀⠀⢠⡶⣦⠀⠀⠀⠀⠀⢀⣼⣶⠂⠀⠀⠀⠀⢀⡬⠛⡀
⠀⠀⠁⠝⠃⠔⠀⠀⠀⠀⠀⠺⠏⠸⠇⠀⠀⠀⠀⠸⠦⠾⠀⠀⠀⠀⠀⠘⠾⠞⠀⠀⠀⠀⠀⠰⠆⠸⠽


⠀⠀⠀⣰⣦⣄⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⣶⣦⠄⠀⠀⠀⠀⢿⣗⣻⣿⠀⠀⠀⠀⠀⠀⣷⠐
⠀⠀⠸⣻⣿⠞⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣰⣧⣯⠀⠀⠀⠀⠀⠘⢸⡇⠀⠀⠀⠀⠀⠀⣷⣶⡆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⢠⣤⣠⣤⠀⠀⠀⠀⢀⣀⠠⡄⠀⠀⠀⠀⠀⣤⣄⣤⡄⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠄⣤⣤⣄
⠀⠀⣠⠉⡌⢡⠀⠀⠀⠀⢀⡁⠶⠸⠆⠀⠀⠀⠀⠈⢡⠉⠀⠀⠀⠀⠀⣀⢙⣫⡴⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠙⠀⠋⠘⠁⠀⠀⠀⠈⠁⠘⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠋⠙⠉⠀⠀⠀⠀⠀⠉⠉⠻⠋

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠁⠻⢋⠀⠀⠀⠀⠀⠈⠋⡝⠃⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠀⠀⢰⣉⡟⢳⠄
⠀⠀⠰⡶⠷⠚⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠺⠷⠗⠁⠀⠀⠀⠀⠺⠿⠷⠋⠀⠀⠀⠀⠈⠻⡹⠏

⠋⠀⠬⠘⠀⠀⠀⠀⠀⠁⠀⠀⢰⠂⡀⠀⠀⠀⠊⠀⠀⠀⠉⠀⠀⠁⠀⠁⠈⠈⠀⢈⠉⠀⠀⢮⣭⡷⠂⠀⠀⠈
⠏⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠨⡀⠆⠀⠀⠀⠀⠤⢀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁
⠀⠀⣀⡔⢒⠙⠛⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⣀⠂⠀⠀⠀⠀⠀⠀⠀⢲⠄⠀⠀⠀⣠⠂⡸⠂⠀⠰
⠁⠀⠈⢁⡉⠀⢬⠀⡀⠀⠀⠀⡐⠀⢀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠘⠁⠀⠀⠁⠀⠀⠸⠁⠐⠂⠀⠀⠀⠀⠐⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⣑⢶⣄⡀⠀⠀⠀⠀⠘⠁⠀⣀⡀⠀⠀⠀⠀⠘⡅
⠃⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠂⠀⠀⠀⠀⠀⢀⠠⠝⠛⠖⠀⣀⠀⠀⠃⢀⠀⢰⠒⢰⡄⠀⠲
⠁⠠⠂⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⠁⠀⠀⠀⠉⣈⠁⠠⡌⠁⠀⠀⠀⠂⠙⠀⠀⠁⠩⠤⠀⠀⠁
⠀⢀⡀⠀⠀⠀⡆⠀⠉⠠⠄⠈⠇⠀⣠⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠠⠓⠀⠀⠀⠀⠠⣤⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰
⠰⠈⠀⠀⠀⠀⣁⠀⠀⠀⠀⢱⠀⠀⠀⣔⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠋⠙⢠⠁⢐⣀⡄⠀⠀⡀
⠀⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⢠⠴⣶⣧⠀⠀⠨⠁⠀⠀⠀⠂⠀⠀⠑⠘⠾⡌⠁⠙⠀⠀⠀⠋⠠⡐⠆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠖⠐⠀⠀⠠⠀⠀⠈⠙⠉⠁⠀⠀⠀⢠⠄⠀⠐⠀⠤⠄⠲⠄⠀⠀⠀⠀⢠⠆⠀⠀⠀⠡
⠲⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣛⡂⡀⠀⠀⠀⣀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠈⡀⠀⠈
⠄⠀⠠⠤⠒⠂⠀⡀⠘⠠⠐⠂⡀⠈⠁⠀⠻⢡⡗⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡅⠀⠀⠈⠀⠀⠀⢃⠀⡅⠀⠀⠀⠀⠶
⣄⠀⠳⠀⠀⠀⠙⠉⠁⠀⠂⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⡁⠀⠀⠢⠀⠀⠀⠀⠖⠆⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⠳⠀⠀⠀⢠⠁⡸⡃⠀⠀⠀⠀⠀⢠⡄⠘⠁⠀⠀⠐⠀⠀⠀⢀⣀
⠷⠃⠀⠀⣀⣤⠊⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢲⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢲⡀⠀⠀⠀⠀⡠⡐⢰
⠀⠀⠀⠯⠽⠞⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠑⢀⠀⠀⢀⠀⠀⠀⠀⠆⠀⣀⠀⠀⠀⠀⠈⢄⠸⠀⠀⢱⠈⠁
⠀⣶⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠈⠩⠡⢤⠄⡀⠀⠀⠲⠊⠁⠀⠀⠀⠉⠁⠀⠀⠠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠁
⢠⠋⠠⠂⠀⠀⠸⠸⢰⠋⠁⠀⠄⠀⠀⠰⠂⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⢠⡄⡀⠀⡄
⡮⢄⠢⠀⣤⡴⢒⢉⣢⠖⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠎⠸⠇⠀⠈⠠⠀⡀⠐⠚⢀⣠⠀⠀⠀⠠⡌⠀⠀⠀⠁⠈⠐⠆
⠁⠁⠀⠀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠂⠛⠠⠁⠀⠈⢃⣘⠆⠀⠦⠄⠀⠀⠀⠁⠀⠀⠄
⢀⠀⠀⠀⠀⢰⠂⣢⡀⠀⠆⠀⠀⠀⠀⠀⡀⠀⡒⡒⡄⢂⠀⠀⠀⠀⠀⠀⠴⡿⠞⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⣀⠁⠀⢀⠀⠀⢈⢉⡅⠁⠀⠁⠀⡇⠀⠀⠟⢋⠄⢁⠪⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣀⣠⠐⠂⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢒⣁⡤⣠⣦⣤⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠛⠛⢻⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⣤⠀⡏
⠀⠀⠀⠀⠀⠀⢠⡲⣿⣿⡆⠀⠀⠀⣠⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⣀⢰⣿⡄⠀⠀⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠟⠁⢀⣀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠀⠀⠀⡀⠒⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣇⠂⠀⠀⠀⠁⡴
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⡻⣿⣿⣷⣄⡀⠶⣾⡁
⠀⠀⠀⠀⠀⢀⠖⠛⢢⡀⠙⢮⡛⠿⠿⢿⣿⣿⡻⡢⡀
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠹⡄⠀⠱⡄⠀⢰⣿⣿⣿⣇⢡⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢹⡀⠀⠙⣄⠀⢹⣿⡿⣿⣿⣇⠀⠀⠀⠀⢾
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⢧⠀⠀⠈⠉⠈⢿⣷⠘⠿⠸⡄⠀⠀⡰⠛
⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢱⠀⡸⠁⠀⠘⢿⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣰⠁⠀⣤⠀⢺⡏⠁
⠀⠀⠀⡇⠀⠀⠀⢀⠀⢾⣣⡟⠀⠀⠀⢀⣀⣠⠀⠁⠀⠈⢁⡠⠂⠀⠙⠛⠛⠀⢀
⠀⠀⠀⢡⠀⠀⠀⡡⢖⢛⡷⠒⠘⠉⠉⠁⠀⢰⠀⠀⠀⠀⢸⣽⣇⠤
⠀⠀⠀⠘⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠖⠒⠉⢸⠐⠄⣾⣛⠛⠛⠛⣛⠛⠃⡄
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠏⠀⢀⠀⠀⢸⡄⡆⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠈⠲⠆⠶⢶⠒⢚⠋⠍⠅⠤⢨⡧⠷⠶⠶⠾⠶⠶⠞⠣⠶⠀⠀⠀⠀⠀⠠⠀⠼⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡆
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⣿⣿⠟⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢈⢀⡃⠀⠀⠀⠀⠸⢤⡀⠀⠐⡄⢿⠂⠀⠀⠞⠭⠇⠀⠀⢐⢠
⠀⠀⠀⠀⠀⠐⠞⣮⢫⢧⡀⠀⠀⠀⢂⠀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠢⢀⡀⠀⠠⣄⠠⠤
⠀⠀⠀⠀⠀⠀⢠⡹⣇⣩⣆⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⡄⠠⣤⠀⠀⠀⠁⠲⠦⠤⠶⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠶⠇⢠⠀⠀⡀⠀⠀⠀⠀⠉⠰⠬⠀⠀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠸⠒⠒⠚⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠀⠀⠀⠀⠆⠐⢢⡀⠀⡀⠈⢲⠑⢲⣀⡐⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠄⡀⣠⠔⠀⠀⢠⠞⠁⠀⠀⠀⣀⠔⠚⠓⢲⣷⡀⡉⠫⢿⣤⡧⠉⣹
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⠇⠀⠀⢠⠟⠘⠁⠀⠀⠆⠀⠀⣰⣦⣆⣿⣿⣿⣠⣴⣿⣤⣤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠴⣴⣻⠖⠁⠀⠀⢟⣃⠈⠀⠀⠀⠀⢶⣶⣿⠿⠛⠋⠉⠙⠛⠛⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠠⢀⣈⣥⡴⠶⠚⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠚⠻⣦⣀⡀⠀⠀⢀⣄⣹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠐⠂⠐⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣀⠀⠀⡥⠀⠀⠀⠀⠤⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢠⡎⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢃⠀⠀⠊
⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠁⠠⠀⠄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠐⠂⠄
⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡜⠀⡃⠀⠀⠀⡤⣖⡧⠀⣂⣀⠔⢒⠄⢀⣠⠀⠀⠀⢠⠀⢀⢢⣄⣐⠄⠀⠀⡅⠀⠀⠀⠀⠀⠀⡐⠀⠀⠁
⠀⠀⡀⠀⠠⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠨⢰⡌⢇⠀⢀⣼⠀⢠⠔⠀⢠⡜⢸⢀⠇⢀⡠⠮⠂⠀⠀⡇⠀⠀⠙⠦⠀⠙⠁
⠀⠀⠃⠃⠉⠁⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠋⠁⠚⠁⠐⠚⠀⠙⠀⠋⠀⠀⠊⠁⠛⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⠆⠀⠀⠁⠶⠦⠠⠶⠦⠼⠿⠧⠷⠄⠴⠶⠴⠄⠠⠶⠶⠦⠿⠇⠀⠘⠀⠀⣖⠀⠤⠀⢐⡀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢀⣀⠀⠀⠈⢠⡄⠀⠀⠈
⠀⠀⠒⠀⠀⠀⠀⠄⠀⠀⠀⢠⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠇⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠣⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⠉⢀⣀⣀⡀⠊⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠢

⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡶⠒⠒⠒⠒⠒⢲⡄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⡇⠠⠴⠆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠒⠒⢒⣒⡒⠒⠒⠒⠒⠒⠚⠃⣠⠗⠒⠒⠒⠒⠒⠚⢧⡀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣠⡽⣛⡛⢦⡀⠀⠀⣞⢓⣒⣒⣒⠒⣒⣒⡒⣒⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠿⠞⠹⡟⠁⠉⢻⠇⠀⠀⠉⠉⠉⠉⠉⣩⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⡶⢶⣏⣀⠀⠀⠀⠀⠀⢯⠀⢸⠆⢸⡄⠀⠀⠒⠒⠲⡖⠒⢲⡖⠒⢶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠉⠉⠀⠈⣷⠀⠀⠀⠀⢘⣀⣤⣄⣘⠁⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⡴⠀⠀⡶⢳⢄⣿⡀⠀⣀⣰⠯⠥⠤⠤⠭⢷⣀⡀⠀⠀⠀⡇⠀⣸⣇⠀⢸
⠀⠀⠀⠀⠀⠀⢸⣅⠀⠘⠳⢦⣠⣟⡏⢠⠈⠓⠶⠯⣿⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⡞⢹⣆⡏⢹⢰⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⡶⠴⣇⣹⠇⢺⠶⠶⠶⠶⢾⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠙⠚⠙⠓⠚⠈⠓⠋
⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠳⠶⠶⠶⠶⠞⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⢸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⣶⠶⠶⣶⣶⡶⠶⢶⡶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⣿⠸⠭⠇⠸⠭⠭⠿⠀⠯⠟⢸⠀⠘⠚⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢛⣛⣛⣛⣛⣛⣛⣛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣈⣉⣉⣉⣉⣉⣉⣉⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠁⠀⠀⠐⢀⠠⣄⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠈⠁⠈⠃⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⢠⡄⢠⡄⠀⠀⠀⠀⢸⠀⡯⠀⠀⠀⢀⣤⣤⡤⢤⡄⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣯⣤⣤⣤⣤⣬⡇⠀⠀⠀⢸⣇⡇⠀⠀⠀⢿⣿⣯⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⢫⡄⠀⢼⡇⠀⠀⠀⠀⣿⠇⠀⠀⠀⣾⣿⣿⡟⢳⡆⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⢸⣇⡀⣸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣇⣸⣇⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠿⠃⠘⠇⠀⠻⢧⡀⠀⠀⠀⣿⠀⠀⠀⢠⣿⡿⠛⠇⠘⠃⠀⠿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⢸⣿⠀⠀⣰⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⢀⠀⣄⣀⠀⠀⡔⢹⠀⢸⣿⠀⢰⣹⣷⡄⠀⠀⠀⣴⡄⠀⡀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠱⣼⣷⣿⣿⣴⣧⡄⠈⠄⠸⣿⠀⢨⣼⠛⠁⠀⠀⠀⡀⢽⠈⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠊⢛⣛⣿⣿⣭⢃⣿⣷⡀⢠⡁⠈⣿⠀⠘⠏⣶⢶⠀⠀⢀⡴⠈⠢⣤⠐⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⡽⢿⣿⡿⣿⣿⢯⠇⠀⠀⡅⠀⣿⡂⠀⣴⣿⣤⣤⡖⡉⠀⠀⠀⠡⠉⢠⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣆⠀⠈⠘⠃⢊⢹⠰⠀⠀⠀⠇⠀⣿⡇⠀⡼⣿⣾⣾⡿⣮⣴⣦⡤⣀⣵⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣶⣤⡄⢀⡀⠈⡈⠶⠁⠀⢀⣄⣤⣿⣦⣤⣄⣿⣿⣻⣶⢟⢥⣰⡌⢸⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⡟⠻⣿⣿⣿⣿⠋⢻⣿⢉⣿⡿⠉⠀⣿⠉⠙⡟⣿⣿⣿⣻⡛⠛⡿⠙⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⣷⢾⣟⢻⣿⣿⣟⣽⣴⣿⣿⠁⠄⠀⣿⠀⠀⡆⣿⣿⡾⠿⣶⠡⡀⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⣿⣿⣤⢿⢿⣿⣿⣿⣸⣿⣿⡿⠀⡆⠀⣿⠀⠀⣹⡿⠃⠀⠀⠘⠇⢠⡙⣿⣿⠟⢿⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣹⡟⡇⠈⠑⢿⣿⣿⣿⣿⣿⠋⠀⡇⠀⣿⠀⢨⢿⡇⠀⠀⠀⠀⠀⠜⠀⠀⠉⠀⢸⡅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣷⡅⠀⠐⠠⡙⢿⣿⠋⠈⠁⠀⣦⣤⣿⣦⣴⣸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣷⣦⣤⣤⣼⣾⣷⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠁⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠚⡿⠉⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠺⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢃⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣵⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠋⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⢿⠿⢿⣶⣶⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢸⡇⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢤⣬⣭⠤⠶⢴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠀⠀⢠⣤⣀⣤⡄⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣄⣀⣉⣠⣄⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⡇⠀⠀⠀⠀⣈⣉⣩⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠟⠀⠀⠀⠀⠈⠉⣉⡀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠘⠉⠉⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡀⠀⠀⠀⠀⠀⠒⠛⠉⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣼⠃⠀⠀⠀⠀⣘⣋⣉⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠃⠀⠀⠀⢀⣀⣩⣭⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⡏⠀⠀⠀⠀⢈⣉⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡟⠀⠀⠀⠀⠀⢬⠍⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣠⠏⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⠁⠰⠋⠀⠀⠀⠛⠓⠚⢒⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⣀⣘⣛⣛⣓⣺⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠤⣤⣉⡉⣩⣭⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠶⢦⣤⣤⣠⣤⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⢀⣀⣀⠙⠛⠶⠦⣴⣾⡿⠋
⠀⠀⠀⠀⣀⣠⣶⠿⠋⢀⡀⠴⢤⣀⣉⣉⣙⣛⣿⣿⠟⠁
⣀⣤⣾⡿⠟⠋⠁⠠⣤⣀⠙⢻⣶⣤⣽⣿⠿⠛⠋⠁
⠛⠻⠿⣶⣶⣴⣿⣿⣿⣿⡿⠿⠛⠛⠉
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠓⣤⠤⠤⠤⠤⠶⢍⡀⠀⠀⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠃⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠓⢄⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⡤⠃⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢚⠦⣀⢀⠓⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢸⠀⠀⠁⠀⢱⠀⠑⠂⠔⠂⠤⠤⠤⠴⢂⠀⡖⠊⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣂⣀⡀⢀⡤⠐⠒⢷⡦
⠀⠀⠀⠀⠀⠀⠘⠴⡀⢠⢧⡄⢆⠰⠤⣀⠀⡷⠶⡏⠁⢠⠀⣹⠹⣄⠀⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠳⡳⠈⠉⠀⠀⠘⠦⠤⠔⠣⢀⠌⠁⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⡀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠂⠽⣝⠉⣝⢉⡎⡃⠀⢤⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠛⢆⠈⣷⡱⣤⠜⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⢄⠀⠀⠀⠀⠉⠀⠀⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⡦⠤⣀⠀⠀⣀⣰⠖⠉⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠤⠇⠀⠀⠀⠀⠘⡇⠀⠀⠀⣸⡑
⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠜⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⢸⠀⠀⣀⣀⠤⠄⠃⠀⠰⡤⠤⡴
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⡩⢷⠈⠉⠉⣹⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⠀⠀⢀⠔⠊⢆⡀⠑⠒⢶⡁⠉⣉⠽⣄⠀⠀⠀⢀⠄⠀⢇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠡⠀⠈⠀⠀⠀⢋⠉⠉⣷⠀⠙⠤⠤⠋⠳⡀⠐⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⠑⠊⠁⢇⠀⠀⠀⠀⡰⠁⠃⠀⠀⠀⠀⢀⠇⠐
⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠤⠊⠀⠑⢄⡀⠀⠀⠀⡠⠊⠒⠒⠒⠉⠀⠀⠈⠢⠄⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠑⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢙⠓⣂⠠⠐⠂⠁⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡱⠀⠀⠀⢠⠀⡈⠀⠀⠐⠒⠀⠤⠀⡀⠀⠀⠀⠀⢀⣠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⡀⢠⠏⣿⣿⣶⡀⠀⠀⠀⠀⠀⠈⠑⠠⠀⢀⠈⢷⠱⠄⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠍⠀⠈⠼⠟⠀⠛⠃⠀⠀⢠⡶⠿⠿⢦⡀⠀⢀⡁⠀⢁⣵⣄⠀⣛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢀⠈⠀⠠⠀⠀⠀⠠⠄⠀⠀⠀⢠⣿⣿⣶⣾⠄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠇⠀⠀⡜⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢀⡀⠜⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠟⡯⢩⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣰⣶⠘⠚⠛⠷⢤⣤⣬⡀⢀⠀⠀⡀⠀⠀⠈⠀⠄⢀⠆⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠠⣦⢶⡇⠀⠊⠀⡰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⠀⢀⠀⠀⣀⠠⠄⠀⢈⠄⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠄⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢄⠰⡐⠄⠀⠀⠈⠀⠠⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠸⠆⠀⠀⠀⢀⡶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠆⠈⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠴⠴⠅⠤⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣐⠤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢰⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣁⢀⣤⡀⢀⣀⡀⣠⣤⣀⣤⡀⠀⠀⣠⣤⡀⣰⠿⠂⠈⢹⡏⠁⠶⣆⣠⣤⣤⣿⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣽⢿⡗⣛⣾⠋⣷⣙⣶⣿⡛⡃⠀⠰⣏⣼⠟⣿⠃⠀⠀⢸⡇⢸⣿⣿⣙⣷⠸⣇⣤⣿⢚⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⣩⣴⣧⣭⣁⠈⠉⠁⠀⠀⠈⠁⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣠⣴⠖⠲⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣾
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡴⠞⢛⡿⠿⢿⣷⣄⣈⠳⢶⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠉⠀⢀⣀⣀⠀⢀⣀⣉⠻⣿⣿⣇⢠⡏⠀⠀⢸⡄⠀⠀⠈⠳⣭⣉⣻⣏⢹⠇⢸⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣾
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡄⠀⠐⢿⣿⡿⠀⠹⣿⣿⠗⠀⣸⣻⢿⡀⢀⣴⠏⠙⠲⠤⣤⣤⣤⣴⣿⠵⢛⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣆⠀⢀⡀⠀⠚⡂⠀⠀⢀⡴⠋⠁⢈⣿⠋⠀⠀⣀⣤⣀⠀⢀⣤⣌⠓⠦⣼⣇⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣶⣶⠋⠀⠀⠀⢻⣽⡄⠀⠘⠿⠿⠟⠀⠹⠿⠿⠃⠀⣼⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⣿⣿⣿⢷⡄⠀⠀⠀⠈⠹⣆⡀⢀⣀⠀⢉⡀⠀⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣧⣈⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⠙⣿⢿⣿⣿⣷⣶⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠼⣱⣿⣿⣿⣿⣿⡇⣧⠀⠀⠀⠀⠀⠀⡼⢹⣿⣿⣟⣾⢿⡏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣴⣿⣿⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠘⡷⢾⣦⣬⡽⠀⠀⢷⠤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠈⣧⠀⠀⠀⠀⢠⡇⠚⣸⠃⠀⠀⠀⢸⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠚⠃⠀⠀⠀⠀⠈⢧⣼⣇⣀⣀⣀⣀⣤⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⠏⠀⠙⠀⠘⢦⣧⠈⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⣿⣿⣿⣿⣿⣿⣿⠲⠾⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢻⣿⣻⣿⣿⣿⣿⣻⣿⣷⡀⠀⠀⠀⠀⡀⢸⣀⢀⣀⣿⠀⢀⣄⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠿⡿⢿⠿⢼⡿⠿⣿⠿⢿⡇⠀⠀⠀⠈⠳⣾⣿⣿⣟⣿⢻⣿⣿⣿⠓⠚⢶⣠⢀⢀⣄⣀⣸⣃⣀⣀⣸⡄⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠚⠓⠒⠒⠒⠚⠃⠀⠀⠀⢠⡞⢹⣯⣽⡟⢿⢻⣿⣽⣿⢳⡀⠾⠛⠋⠘⠙⠟⠙⠋⠋⠙⠙⠑⠃⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠾⠭⠭⠷⢾⠾⠯⠭⠿⢾⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠉⣭⠙⠒⠒⠒⠒⢉⠙⠒⠒⠒⢻⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⠉⠉⠙⢢⣄⣀⣤⠬⣁⣀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢃⡼⠁⡏⠀⠀⡀⠈⢻⢶⣧⠀⠀⢀⣠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠋⠩⡀⠀⣸⣤⣀⣉⡰⠃⠸⣄⡤⠖⣉⡽⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⠀⠀⠀⠀⠑⠦⣝⡋⠀⠀⠀⠀⠀⠀⢀⣀⣤⢚⡡⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣞⠣⣄⠀⠀⠀⠀⢀⡎⠉⣿⣒⢒⣺⠋⡹⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡆⠗⠀⠏⠑⠠⡔⠉⡇⢀⠂⠀⢀⠃⣞⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠳⠞⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⡘⢠⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠒⠤⢀⡀⢴⢰⠃⠀⠀⠀⡇⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡠⠴⠋⠀⠀⠀⠀⡆⡃⠀⠀⠀⠀⢠⠔⠋⠂⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡕⠀⠀⠀⣀⠴⠥⡀⠀⠀⠀⡠⠃⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⡶⢄⣈⡷⠀⠀⠈⣢⠤⠒⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠀⠀⠈⢳⠧⠶⢊⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠃⠀⠀⠀⣴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⡟⠀⢦⣀⣀⠀⠀⠀⣀⣀⠠⠜⢄⠀⠐⡠⠀⠀⠀⢢⠉⠉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⢧⣀⠀⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠈⠉⠒⠒⢲⠁⠀⠀⠀⠠⠀⢀⠀⠀⡇⠀⠀⢸⠀⠈⡇⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡞⠒⠦⢄⣀⠀⠀⠀⠀⠀⢱⠀⡠⠊⠀⠀⠑⢄⠀⠀⠌⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⠀⢀⠂⠀⠐⠒⠩⡉⠉⠉⠉⢣⡀⠀⠀⠀⠈⠉⠉⠙⣳⠃⠰⡀⠀⠀⠀⠀⠀⡆⢠⠂⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠸⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠊⠉⠁⠀⠈⢆⠀⠀⠀⠀⡰⢁⠆⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⢨⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣼⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣿⢿⡿⠿⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠂⠀⢀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣶⠳⡖⢋⠕⢊⣾⠟⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⢀⣿⠃⠀⠉⠉⢠⣿⠇⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢧⠀⠀⠀⣾⣿⣀⣀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⣧⣀⣼⡇⢀⣿⣿⣿⣿⣿⣿⠀⠀⠸⣄⠀⣀⣠⣤⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣟⢶⡀⠀⠀⠀⡼⠁⠀⣰⠋⠉⠁⢸⡇⢐⣿⣿⣿⣿⣿⣿⠀⠀⠀⣽⢫⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢠⠿⠀⢹⡆⠀⡰⠁⢀⡜⠁⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣇⢀⣠⠗⠉⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣦⡿⣄⢀⣀⡼⢀⠔⠁⠀⡎⠀⠀⠀⠀⠀⠈⡇⢸⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⢷⡿⡆⣤⠀⢁⠀⡀⢰⠃⠀⠀⠀⠀⠀⣰⠃⣬⣿⡿⠟⠋⠉⠀⠀⣀⣤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢹⡿⣔⣿⡻⠛⢡⡀⠀⠀⠀⠀⡴⢁⣾⣿⣿⡄⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⠀⠈⠁⠙⠉⠳⠤⠏⢹⣄⣀⢀⣜⡡⢟⣫⣥⣾⣤⣤⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠉⠀⢀⣺⣷⣤⣦⡀⠀⢀⣿⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠸⣿⣏⠀⠉⣷⢀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠓⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣙⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡛⠘⢏⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠡⠀⠈⠙⠢⢤⣀⠰⠜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢱⣄⠀⠀⠀⠙⣷⣾⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⣠⡆⢀⠤⠀⣐⣈⡀⠐⢿⡇⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠰⠖⠉⠒⠓⠳⢰⠋⠉⣩⡶⠛⠁⠙⢦⠀⠀⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠃⠀⠀⠀⠀⢨⣷⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠞⠃⠀⣼⠀⠀⠀⠀⠀⡸⠃⢸⡥⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠞⢃⣀⣀⣤⠶⠛⠁⠀⣠⣵⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣴⠖⠲⠤⠔⠒⠒⠒⠐⠂⣀⡤⠖⠛⠉⠉⠁⠀⢀⣀⣠⣤⣿⣷⣟⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠉⠢⣄⡀⠀⠀⠀⣠⣴⠟⠁⠀⠀⢀⣀⣤⣴⣾⡿⢟⡭⠏⠹⡋⣋⢱⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠻⠟⠗⠖⠻⠟⠻⠋⠉⠁⢀⠈⢰⣞⣴⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⡦⣤⣄⣠⡄⠀⣀⠀⣀⢀⣀⣀⣁⣘⢠⣂⣠⣤⣼⣾⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢣⢹⡄⢹⡟⠈⢿⠛⣿⠛⢻⡟⢻⠟⢻⡟⢻⡏⢩⣇⣸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡎⣧⠀⣿⠀⢸⡀⢿⠀⣸⡁⢸⠃⢼⠀⢺⠇⢺⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⣸⡆⢿⠂⠸⡇⢸⢢⣼⡇⢸⠂⣿⡀⣿⢓⣿⣿⣿⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣿⣇⢺⣧⠀⣷⢾⡧⣿⣧⣿⢲⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⡸⣿⣿⡧⣿⡎⣿⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣿⣿⣿⣧⣿⣿⣷⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣶⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣾⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢟⡛⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠿⠟⠻⠛⠛⠙⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠺⡶⠿⠿⡷⠀⢠⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢠⠄⠤⣀⡤⠔⠢⡀⠀⠀⠘⡷⠤⠒⠉⠀⡀
⠀⠀⠀⠈⠁⡀⠀⡿⠀⠀⡇⢀⡀⠀⠀⠀⢀⡎⠉⠑⡍⠀⠀⠀⢀⣧⢠⡼⠵⢤⣀⣱⠀⠄⠊⠀⠀⢘⠉⢵⠈
⠀⠀⠀⠀⠀⠁⡰⠥⠀⠀⢣⠀⠁⡠⢤⣴⠟⠻⠀⢀⣼⠀⠀⠐⣅⠈⢲⡀⠀⣤⠑⣾⠉⠛⢆⠀⢠⢬⠉⢁⣤⡀
⠀⠀⠀⠀⠀⢠⠃⣠⣀⣠⣌⣆⠀⠣⡴⡅⠀⠀⠙⡿⠿⠀⠀⠀⢈⡉⢶⠬⠦⣄⣠⡏⣦⠤⠚⠀⡈⢊⡴⣋⠜⠁
⠀⠀⠀⠘⢁⣿⣿⣿⣿⣿⣿⣿⣆⠀⣷⠹⡄⠀⣐⣒⣒⡄⠀⠀⢀⣸⠈⢆⣀⡤⠛⠒⠊⠀⠀⠀⡤⣋⠜⢁⣀⡀
⠀⠀⠀⠀⢎⢻⣿⣿⣿⣿⣿⡿⢈⠀⢸⠖⠚⠚⠒⠒⠒⠚⣆⠀⠀⠙⠀⠀⠀⠀⣠⣴⠶⢒⡚⠿⣿⠁⠀⠀⢸⣏
⠀⠀⠀⠀⣈⠑⠒⠂⡒⠒⠒⠈⠁⠉⠘⠒⠦⠤⠤⠦⠆⠒⠃⠀⠀⣤⡆⠀⠀⡾⢋⠖⡒⠒⡌⢢⠙⡄⠀⠀⢸⣍⠁
⠀⠀⠀⠈⠚⠁⠀⠀⠻⠯⠀⠀⠀⠀⠀⠄⠀⠀⠀⠔⠒⠄⠴⠀⠀⠈⠁⠀⠀⡇⠸⣀⡛⠘⡜⡸⠀⡇⠀⡆⢸⠮
⠀⠀⠀⠀⢴⢀⡶⠚⢲⡀⠀⠓⠺⠰⠀⠀⠀⠀⠀⠊⠒⠤⠴⠂⠀⢠⣤⠀⣄⠹⣤⠈⢉⠉⠐⣡⠞⠀⠀⠧⠿⠁
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠁⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠀⠀⢠⠴⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⣠⠞⠛⠳⡄⠀⢰⡆⠉⠂⣴⠶⠶⠄⠈⢀⡀⠀⢀⡄⠀⣠⠤⢤⡒⠀⢸⣂⣀⣀
⠀⠀⠀⠀⠐⣆⠀⠀⢀⡇⢰⡇⠀⠀⠀⠉⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⣾⢧⠀⢸⡇⢰⡇⠀⠈⠁⠀⢸⠉
⠀⠀⠀⠀⠀⡌⠛⠒⠋⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⣟⠛⠛⠂⠀⡟⠈⣇⢹⠃⢺⢁⠀⠀⢀⡀⠸⠶⠖⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢧⣀⣀⣠⠄⠀⣸⡇⠈⠁⣿⣤⣤⡄⠸⠃⠀⠘⠟⠀⠈⠦⠤⠖⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⢒⣤⣤⡄⢀⡴⣊⠥⢄⠉⢆⠲
⠀⠀⠀⠀⢠⣄⠈⠉⠀⠀⠠⡆⠀⠀⠘⠀⠈⢫⡂⢠⠛⠒⠠⢼⡀⠛⠑⠙⠀⠀⣀⠁⡀⡄⢠⣾⡔⠁⠀⢀⠃⣼⢠⣤
⠀⠀⠀⠀⠀⠁⢰⠭⡆⢰⠯⡆⠀⡟⡇⠀⠀⠀⡸⠡⠀⠀⠠⠀⠈⠟⢠⣄⣤⠀⠳⣄⢀⡗⠋⠀⠀⠀⢴⣧⡴⠃⢸⠏
⠀⠀⠀⠀⢶⣾⣯⡿⡽⠽⠯⡿⠿⠿⣿⠟⠀⠘⢇⠀⠀⠀⠀⠀⠀⡆⠈⠻⠃⠀⢂⡬⠛⠬⡀⠀⠀⠀⠔⠈⠀⢻⣮⡄
⠀⠀⠀⠀⢸⣿⠀⣶⡇⠀⣿⡇⢠⣿⣿⠀⠀⢀⡠⠣⡀⠀⠐⣂⣀⡇⠀⠀⠀⡰⠉⠀⠀⠀⠉⢦⠀⠀⠀⠀⣦⣠⠟⠁
⠀⠀⠀⠀⢸⣿⠀⣿⡇⠘⠿⠇⠀⠿⣿⡆⠀⡉⣀⢀⠙⠉⠀⠀⠀⠀⠐⠀⠙⠁⠀⠀⠀⠀⠀⠸⠟⠄⠀⠀⠸⠛⠋
⠀⠀⠀⠰⠿⠿⠿⠯⠿⠿⠯⠿⠽⠽⠽⠿⠃⠈⠁⠈

⢫⣵⢤⣤⣤⣤⣤⠀⣠⣬⡙⠿⢀⠙⠿⣷⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠⠀⠀⣀⠐⠟⠁⠀⢠⠊⠁⠀⠀⠡⣀⣀⣸⡿⠀⢠⣤⠈⠉⠀⠀⠀⠈⠨⢀⢁
⠀⡝⢟⣿⢵⣿⣿⣿⣿⣷⡶⠀⠀⢰⣄⠘⠃⠍⠑⡀⠈⠇⠀⠀⠀⠀⢤⠀⠁⢀⠇⠀⡩⠤⠄⠄⠀⢀⠠⠀⠸⢦⠀⠈⣉⠉⠀⢠⣶⠿
⢦⣀⣀⠈⠙⠇⢻⣷⠈⣿⣿⡦⠀⠀⢑⡙⠄⠸⡀⠀⠀⠀⠀⢀⠀⠀⡈⠃⢠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⡐⢒⠉⠀⣀
⠻⠿⠟⠁⠀⣀⠀⠈⠀⠙⢋⠀⠶⠀⠈⢴⣤⠄⣀⣈⠀⠐⣤⡀⠀⡼⢇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢠⠘⢇⣴⡀
⢀⡆⠰⡄⠺⠿⢣⠀⠀⠀⠘⠁⠀⠀⢴⡄⠙⣀⣴⠋⠀⠀⠹⣤⣶⣤⣮⠤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠑⠊⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣡⠆⠚⠛⣦⢾⣿⣿⡛⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⢯⠿⣦⣀⡠⠤⠄⠀⠀⠀⠀⠀⠀⣠⣾
⣶⣲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠞⠫⠰⠀⠚⠋⠈⣿⡿⠁⠲⠷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣽⡿⢷⣿⢋⣠⡤⡠⠀⢤⡶⠄⣀⣴⣿⢋
⡧⣷⣞⡷⠀⠀⠀⠀⠀⠠⠀⣸⣿⡆⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠀⠂⠀⠁⠀⠀⡀⡈⠙⠋⠁⠈⠛⠀⠀⠘⠉⠀⠈⠉⠉⠀⢸
⠛⠚⠋⢀⣀⣠⣤⠶⠆⠁⠀⢈⢉⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣶⣾⡇⠀⣤⡀
⠚⠛⠛⠛⠉⠁⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠑⠂⠀⠀⠀⠈⠉⠀⢠⣽⠗
⠀⠰⠂⣀⡴⠟⠉⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠰⠃⠀⠀⠀⠉⠢⠀⠀⠀⠈⠁
⠀⠒⠛⠉⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠈⢻⠓⢄⡛⠁⠐⠒
⠰⠵⠀⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⢀⡐⣷⠄⠒⠲⠈⡆⠀⠉⢀
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣦⠉⠀⠀⠀⠀⠀⠀⠔⠂⢀⣈⣿⣿⣤⣿⣿⢷⣟⢷⢛⠁⠀⢠⠃⡀
⠀⠀⠒⠰⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠚⠀⡀⠀⣄⠀⣠⣽⣿⢋⣽⡿⢾⠏⠋⠁⠘⣠⢀⠠⠂⠀⠀⠁⠀⠰⠆⠐
⠀⠂⠀⠀⠘⠒⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⠀⠀⠀⡈⠆⠙⠻⠟⡘⠋⠁⠀⠀⠐⢾⣶⡷
⠒⠀⠀⠀⠀⠀⠰⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠠⠖⠂⠀⠀⠷⠀⡰⣧⠀⠀⢀⡀⠀⠀⠉⠀⠀⠠
⠀⠀⣉⣀⡀⢀⣤⡄⠈⠁⠀⠀⠀⠲⠶⠤⠰⢂⠀⠀⠠⡤⠀⠀⠠⣤⠆⠀⠀⠀⣤⣀⠀⠀⠀⠐⢶⠶⢤⣇⣤⣔⣨⠟⠀⢆⡀⢧
⣿⣿⢿⣿⣷⣿⣽⣶⡖⠀⠧⡀⠇⢀⡈⠑⠚⠁⠀⠚⠀⠀⠐⠈⠢⡀⠀⠀⠀⠢⠛⠛⠀⡯⢸⡆⢀⠔⠚⣿⣿⣿⣁⣤⣄⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⡠
⠚⠙⠽⣟⢿⣷⡌⣿⣿⡗⠀⠀⠀⠈⠘⠐⣶⣄⣠⠞⠀⠀⡄⠈⢢⢑⡄⠀⢠⡄⠐⠂⠀⢤⡍⠐⠛⠓⠈⣿⣟⠏⢿⡌⠁⠐⡦⠀⠀⠀⠐⠦⢤⣤⡴⠶⠚⣁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠿⢟⡿⠿⠗⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣟⣛⣋⣭⣴⣾⣿⠿⠛⠓⠶⠦⢴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣻⣵⣾⣿⠿⠟⠛⠋⠀⠀⠀⢀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠠⣶⣾⣿⡟⠛⠛⠛⣿⡟⠛⢻⣿⢶⣀⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⠉⠀⠀⣠⣴⣾⣿⣿⣿⣿⡏⠤⣛⢿⣿⣦⡀⢠⣿⣃⣴⣿⡿⠛⢉⣛⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⠿⣗⣾⢿⣿⣿⣿⣿⣿⣿⣿⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣽⣿⣿⣿⣿⣿⣿⣧⣸⡏⠈⠙⠛⠶⠥⢭⠥⠴⠒⠋⣉⡴⠾⣿⣿⣿⣿⣿⣿⣿⡷⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣀⠀⢀⣀⣀⣤⣴⣾⣍⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣥⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣮⣼⣤⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣾⣿⣇⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠋⠉⠻⣿⣿⠋⠙⣿⣿⣿⠷⠀⡀⠉⠛⣿⣿⣿⠟⠋⠉⢿⣿⣿⠟⠉⠻⠋⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡂⠀⣷⣾⣿⣿⠂⢸⣿⣿⣿⠀⢸⣿⡆⠀⣿⣿⣿⠀⣸⠇⣀⣽⣿⣿⠀⢸⣿⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡅⠀⣿⣿⣿⣿⡁⢸⣿⣿⣿⠀⠸⣿⡇⠀⣿⣿⡿⠀⢰⣾⣿⣿⣿⣿⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⡀⢉⣩⣿⣿⣇⢈⣽⣿⣿⣤⣄⠈⣡⣤⣿⣿⣯⣄⡈⣉⣥⣿⣿⣯⣄⠀⣩⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡟⠻⣛⢻⢛⡟⣟⡟⢿⠛⠿⢛⣿⣻⢻⣿⣟⡟⣿⣿⣿⣿⣿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣾⣷⣿⣶⣿⣶⣾⣶⣿⣷⣾⣿⣾⣷⣿⣿⣿⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣽⣯⣿⣿⣿⣿⣿⢇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠻⣿⣿⣿⡇⠀⠀⠠⠀⠀⠀⣤⠀⠆⢤⡀⠀⠀⡄⠀⢸⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣿⣿⣇⣀⠀⣀⣄⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣷⣝⠪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣭⣽⣿⣿⣿⡿⠿⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠰⠁⠸⠗⠇⠿⠰⠘⠅⠿⠾⠃⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠤



⣾⣶⠟⠷⣶⣾⣷⣶⣷⣶⣿⣶⣷⣿⣶⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣷⣷⣿⣶⣿⣾⣷⣶⣶⣿⣶⣶⣶⣾⣷⣶⣶
⣿⣷⣏⣹⣧⣤⣤⣴⣤⣦⣤⣮⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣿⡯⢚⣭⠶⠞⣛⣉⣉⣩⣭⣄⢠⣿⣿⣿⣿⡿⠇⠀⠸⠀⠀⢠⠃⠀⠀⢀⣼⡿
⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⢟⣩⢞⣩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟⠉⠀⠀⠀⠃⠀⠀⠁⠀⠀⣠⡾⠟⠁
⣠⡶⠋⠁⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⡿⢟⡵⣋⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠁⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⣫⡶⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⠞⣡⣾⣿⡿⠟⠉⢁⣾⡟⣹⣿⣿⣿⠟⠁⠀⣠⠤⠶⠚⠛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣥⠞⣡⣾⣿⡿⠋⠀⠀⠀⣼⣿⣿⣿⡿⠋⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⣫⡶⢟⣵⣾⣿⠟⠉⠀⠀⠀⢀⣼⣿⣿⡿⠋⡠⠒⠋⣁⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⣿⣿⣿⡿⣛⣵⠿⣫⣶⣿⡿⠛⠁⠀⠀⠀⠀⣠⣾⣿⣿⢋⠔⠈⠐⠉⠙⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣶⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿
⣿⢟⣭⡾⢟⣵⣾⣿⠿⠋⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⣑⣥⣴⣤⡻⠄⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠿⣫⣶⣿⠿⠛⠁⠀⠀⠀⠀⢀⣴⣿⣿⢿⣯⡾⠛⠛⠛⠛⠛⠛⠳⠷⢦⣤⣈⠛⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣵⣿⠿⠋⠁⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡷⣋⠥⠒⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠛⠳⠦⣬⡓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠟⡃⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣭⣓⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣻⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢭⣓⠦⢄⣀
⠀⢁⣴⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠭⣕⣒⡠⠤⢀⣀⣀⣀
⣴⣿⣿⣿⡿⠋⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶
⣿⣿⠿⣋⠤⠂⠁⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⡀
⢟⡷⠊⣁⡤⠴⠚⠋⠉⠁⠉⠙⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠉⠁⠀⠠⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⣸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣉⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⡿⢿⠙⠶⠖⠀⠀⠈⢛⡄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣌⣀⠠⠀⠀⠀⠀⠈⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⣄⠤⠀⠠⠀⠀⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣠⠆⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⢷⣶⠾⠛⣉⡤⠞⣡⣿⡿⠛⠷⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢡⣾⡟⠁⠀⠿⢤⡼⠥⢤⣴⣿⡿⠁⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠀⠀⠀⠀⣾⣿⠁⠄⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣯⡀⠀⠀⠀⠀⢀⣼⣿⡏⠀⠀⠀⠀⠀⠈⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠈⢷⡀⠀⠀⠀⣸⡿⠛⢿⣶⣤⣤⣤⣤⣤⣼⣿⣿⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⣷⡀⠀⣠⣿⡇⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⣷⡀⠀⠀⢀⣀⣀⣀⣀⣿
⢀⠀⠀⠀⠀⠀⠀⣾⣆⢠⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⠞⠿⠷⠛⠉⣿⠃⠀⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣷⣴⠞⠋⠉⠉⠉⠙⣿
⣼⡄⠀⠀⠀⠀⠀⢻⡜⠛⣧⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣻⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢰⡿⠹⡆⠀⠀⠀⠀⠀⢿
⣿⣧⠀⠀⠀⠀⢠⡜⡇⠀⠸⣷⡇⠀⠀⠀⡼⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⡧⠚⠁⠀⠀⠀⠀⠀⣸
⣿⣿⣆⠀⢰⡄⠸⠀⠀⠀⠀⣸⠇⠀⢀⡼⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡿
⢹⡟⢿⣆⣺⢷⡜⢦⣀⢀⣠⠟⢀⡴⠋⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠈⣧⣈⣿⣯⣠⣿⡆⠈⢉⠁⠀⢸⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⠛⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠁
⠀⠘⣿⣇⠙⢿⠙⢿⣀⣿⡀⢀⠘⡴⢀⣴⢆⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢀⣶⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡏
⠀⠀⠈⣿⡄⠀⠀⠘⣿⡿⣇⢸⣷⡙⡟⠼⠋⢁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⢠⣾⣿⣿⣿⣡⡄⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡀
⠀⣧⠀⢹⣷⡀⠀⠀⢻⡇⠹⣾⡏⢿⡆⠀⠀⣼⠿⣆⠀⠀⠀⠀⠀⢀⡾⠁⣤⣿⡿⠿⠛⢉⡝⠀⡔⣠⢀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⢳⣾⣿⡇⠀⠀⠀⠀⡀⠈⠀⠀⠙⠳⠛⠋⠀⢹⣧⣀⣀⡀⢠⣞⣡⠶⠋⣁⣤⣶⣿⣿⣤⣾⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢿⡀⠈⠋⠋⠀⠀⠀⢦⣿⣦⡀⢠⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠘⣧⠀⠀⠀⠀⠀⣷⣾⣿⡯⠻⢿⣿⣦⠀⠀⠀⢸⣿⣿⡿⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠘⣧⡀⠀⠀⠀⢻⣿⣿⡁⠀⠀⠈⣿⣿⣀⣤⢿⣛⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⢷⣤⣤⣀⠹⠿⠿⠶⣖⣒⣭⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠈⢻⡉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠙⠷⣤⡀⠛⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣭⣷⣶⣾⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⠃⢀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠁⠀⢀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠋⠀⠀⡀⠈⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠘⠂⠀⣤⡀⠈⠁⠀⢀⣀⠀⡗⢄
⠀⠀⠀⠀⠀⢠⠎⠀⠑⠄⣀⣀⣀⣀⡀⠀⠄⣀⡀⣉⠰⣡⢀⠦⡀
⠀⠀⠀⠀⠰⠁⢄⠐⠃⠀⠀⠀⠒⠀⠀⠠⠤⠀⠀⠀⢀⠤⢊⣀⠈⢆
⠀⠀⠀⠀⡆⠀⢀⠉⠒⠀⠤⠀⠀⠀⠀⠤⠄⠀⠒⠈⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠈⠁⠀⠀⠋⠀⠀⠀⠐⠒⠀⠀⠘⠁⠀⠉⢁⡀⡴⠁
⠀⠀⠀⠀⠀⠸⠈⠉⠀⠀⠉⠉⠀⠉⠉⠀⠉⠁⠀⠉⠁⠉⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠩⠶⠦⠀⠀⠀⠀⠤⠶⠭⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠢⠴⠃⠀⠀⣀⢄⢶⠋⢉⠔⠖⢄⣀⡀⠤⠐⠈⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡤⠠⠤⠄⠀⢀⠔⢠⢂⡆⢀⠊⠘⠀⠱⠱⡤⠄⠒⠁⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣀⠧⠀⠀⠀⠤⡜⠀⠂⢻⠃⣬⡀⠿⠆⠀⠀⡇⠠⠤⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⢄⢰⠀⠸⠈⣻⠏⢀⠀⡜⡰⠁⠒⠠⢄⣀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠈⢀⡠⠘⢢⡀⠀⠄⠰⡀⠈⣎⠔⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠒⠁⠀⠀⠀⠈⠑⠚⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠆⠙⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⠀⢀⡠⠚⣩⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⢋⣡⡴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠓⢀⢐⡉⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠵⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣴⠁⣀⠔⠈
⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠠⠴⠞⠋⠀⠀⠀⠀⠀⠀⢀⣒⣼⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠳⠇⠿⠇⠟⠁⠀⠀⠀⠀⠀⠀⢠⡠⠀⣐⡤⠤⡂⠀⠀⠀⠀⠀⢀⠀⣀⡀⡀⣀
⠀⠀⠘⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢌⣐⠀⠀⠀⠁⣈⠀⠀⠀⠀⠰⠈⠀⠁⠀⠁⠀⠁⠐⠦⢀
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠱⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉⠚⣻⠧⡀⠀⠀⣈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢿⠋⠀⠀⠀⢣⡌⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣯⠏⠁⡀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠁⠨⣯⣆⠹⣆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⡆⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠹⣧⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠙⢿⣾⣆⢄⢀⠀⠠⠖⠂⠈⠀⠀⠀⠀⠰⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⡠⠈⠛⠣⣂⣀
⠀⠀⠀⠀⠀⠄⠐⠁⠁⠐⠆⡀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠛⢿⣷⡀⠀⠀⠀⠀⢠⠀⠠⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠤⣀⠀⠈⠉⡁⡀⣀⣹⠀⠀⠀⠰⠇⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠩⠈⡅⠱⠀⠀⠀⠀⠀⠀⠰⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⠀⠀⠀⠠⠠⠧⠚
⠀⠀⠀⠀⠀⣰⠤⠀⠀⠀⠀⠀⠀⢰⣰⡶⠄⠁
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢰⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡈⠒⠄⢶⣦
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢄⣀⠀⠀⠀⠀⠀⠀⠀⢠⣬⡁⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠬⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠒⠾⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢛⡛⠛⢛⡛⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⠋⠙⠛⠛⠙⠓⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠳⣄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣧⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡴⠺⡆
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⡟⠀⢀⡟⠀⠀⠀⠀⠀⢠⠞⠁⡀⠀⣿
⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠛⠀⢀⡾⠁⠀⠀⠀⠀⢠⡏⢀⡾⠁⢀⡟
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣴⠞⠋⠉⠉⠉⠉⠉⠉⠛⢦⡀⠀⠀⠀⢀⡴⠋⠀⠀⠙⠲⣄⠀⢸⣁⡾⠁⣀⡾⠁
⠀⠀⠀⠀⠀⣇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢸⡇⠀⢀⠀⠀⠀⠈⢷⠘⡿⠷⠞⠉⢀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡼⠟⠋⠉⠉⠙⠓⠶⣼⣧⡴⠖⠛⠉⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠞⠋⠀⠀⠀⣿⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠻⣦⣀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁⠀⠀⠀⠀⠉⠙⠛⠋⠉⠉⠉⠛⠛⠉⠁
⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠇⠰⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡠⠤⡀⡀⠠⡤⠀⣤⠀⢠⢤⡀⡄⠀⡠⠄⠀⢠⠄⠀⣤⠄⣠⢀⠀⡠⢤⡀⡄⣠⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⡇⠀⡕⡇⠀⡇⢰⠂⠆⠘⢚⡄⡇⠀⡗⠂⠀⠈⢲⡄⢸⠀⠗⢺⠸⡀⢀⠇⡟⢇⠸⠒
⣶⣶⡶⠶⠶⢶⣶⣾⣿⣷⣿⣿⣶⣿⣶⣷⣿⣶⣾⣾⣿⣶⣿⣿⣿⣿⣶⣾⣿⣶⣾⣶⣿⣾⣷⣿⣿⣶⣷⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣬⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠚⠻⡇
⠀⠀⠀⠀⠀⠀⢀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢀⣀⣸⣻⠷⠋⠀⠀⢀⠠⠶⡒⢀⣴⣮⠤⠾⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⢀⠔⠁⠀⠀⠁
⠀⠀⠀⠀⢪⡿⠛⠁⢀⡀⣁⢿⣼⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠧⣝⣋⣀⣀⡴⡂⢀⠀⠀⠀⠄
⠀⠀⠀⠂⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⣀⣨⣶⣾⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢶⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢼⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⣦⡤⠀⣀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⠀⠀⠀⠀⠀⠁⠀⠀⠤⡀⢀⣀⠤⠞⠉⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢀⡀
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⣀⣀⣀⣠⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐⠀⠤⠁⠂⠠
⠀⠀⠀⠀⠘⠒⠒⠀⠀⠀⠀⠻⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠉⢀⠘⠄⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⡴⢊⡴⡶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢤⣴⠇
⠀⠀⠀⢀⠰⠛⠟⠙⠃⢀⣠⠴⢤⣶⡆⠀⣴⣂⣄⠀⠀⠀⠀⠀⠲⢤⣤⡦⠞⠁⢀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣳
⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠰⣦⣌⠈⠩⠝⠡⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠿⠂⠀⠀⠀⠀⢣⠀⠀⢶⣶⣧⣀⠀⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣠⣉⡵⠶⠒⠉⠉⠉⠀⠀⣠⡶⠖⣒⡒⠉⣉⡩⠴⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠂⠀⠈⠉⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡵⠿⠏⠁⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣂⠤⣄⣀⠜⠁⠀⠀⠈⠂⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠰⡀⢂⠐⠀⠀⢦⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠒⠀⠀⠐⠀⠈⡇⠀⠀⠀⠸⠀⠁⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠀⠀⠈⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢢
⠀⣬⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⡠⠿⡟⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢱⡐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠀⠀⠀⢈⠀⠀⠄⠀⠀⠀⠄⠀⠙⠀⢡⡀
⠀⠀⠀⢠⡈⠰⠄⠀⠀⠀⠀⠀⠀⠀⣀⣈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⡠⠛⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢈⡀⠀⠈⠃⠀⠀⠀⠀⠃⡀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠹⢿
⠀⠀⢰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡔⢉⡈⠉⠀⠐⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢷⣧⡀
⠀⡘⠈⠀⠀⠀⠈⡄⠁⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⡀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢳⠀⡀
⠀⣕⠢⠀⠀⠀⠘⠁⠀⣀⣀⠀⢀⠀⠄⠀⠀⠀⠀⡀⠀⢀⣀⢑⡂⡙⠃⡄⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⡀⠀⠀⠃⠀⠈⣦⠁
⠀⠐⠦⠀⠀⠀⡠⠀⠀⠀⠁⠀⠀⡀⡆⠀⠀⢀⠲⠤⡄⠀⠀⢀⡄⣀⠀⢡⠀⠀⢀⢀⣀⠈⠃⠀⠀⠀⠀⠀⠀⠟⣶⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠀⣀⡈⢴⡄⠀⠈⠛⠁⣴⠏⠀⣀⡭⠃⣈⣿⠆⠀⠀⠀⠀⠀⠀⠠⠻⡧
⠀⠀⠀⠀⠂⣤⠀⠀⠀⠀⢰⣷⠰⡂⢠⠀⠀⠀⠿⣿⣆⣀⠀⠀⠉⠉⠁⡠⠀⡈⣑⢙⠝⡡⡃⠀⡀⠀⠄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣔⣤⣉⠄⠀⠀⠀⠈⢎⠻⣶⠎⢐⡠⣺⠁⡔⠃⠚⢹⠀⡇⠁⣴⡁
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠹⠁⠁⠀⠀⢀⠠⡀⢚⠓⡄⠑⠦⢀⡡⣾⣖⢀⡄⠈⠓⠾⣷⣄⣤⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡗⢤⣀⣀⠀⠀⠈⠂⢐⠀⠀⠘⢦⡀⠀⠀⠀⠬⠕⢦⣀⡀⠀⣀⡙⠣
⠀⠀⠀⠀⠀⠀⠀⠐⠂⣶⠀⠀⢴⠈⠿⢴⣦⠃⠀⠀⠐⢄⣀⣀⡙⠢⣀⠀⠀⠀⠀⠈⠣⠀⠈⠙⠳⡀⠘⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠠⠆⠀⠀⠀⠀⠀⠘⠃⠀⠑⠤⠀⠀⠉⠀⢠⠀⠄⠋⡀⢨⡷⢀⡗⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠲⠶⠷⠐⠁⠘⣧⣤⡥⠜⢁⠂⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁⠐⠈⠁⠀⠀⠆⠙⠛⠉⠁⠀⠁⠀⠰⠾⢷⡆

⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀
⠀⠀⠀⢙⣽⣿⡗⠀⠀⠀⣠⠀⠠⢤⡼⡇⠀⠀⠀⠀⠀⠀⠀⡗⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣿⣿⡷⣦⠄
⠀⠀⠀⠈⢻⡿⠀⠀⠀⠘⠁⠀⠀⢸⠃⡇⠀⠀⠀⠀⠀⣰⣧⣿⢀⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢟⣃⡌⣵⣽⣶⣿⠃
⠀⠀⠀⢸⣿⡇⠀⠀⢰⣷⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡾⢯⡏⠸⡄⢻⡀⣿⠇⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⠿⠟⢁⣀⠔
⠀⠀⠀⠘⣿⢧⠀⠀⢸⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠃⠀⠿⢀⣧⡇⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣤⣶⣶⣯⡇
⠀⠀⠀⠀⣿⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤⣤⡄⠀⠈⠤⢦⣅⣠⡀⡇⠀⠀⠐⣿⠀⠀⠀⢠⣿⣿⣿⡽⣿⠃
⠀⠀⠀⠀⢻⡇⠀⠀⠐⣿⠇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠁⠀⠀⢰⣿⣿⣯⣽⣿⡿
⠀⠀⠀⠀⢹⣿⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡿⠏
⠀⠀⠀⠀⠺⠹⠀⠀⢰⡇⠀⠀⠀⠀⠀⡆⠀⠀⠑⠾⠂⠀⠀⠀⠀⢸⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠁
⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⣷⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⠀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⡿⡏⠀⠀⠀⣶⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣶⣇⡈⠋⠉⠉⠁⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⢸⡇⡇⠀⠀⠀⣿⠀⠀⠀⣿⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⣼⣧⣼⡁⢻⡇⠀⠀⠀⠀⠀⢴⣾⠇⠀⠙⠉
⠀⠀⠀⠘⢰⡇⠀⢠⡤⣿⡄⠀⠀⢿⣿⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⢸⡇⣾⣅⠛⠛⠋⠟⠁⠀⠀⠐⣷⣿⣁⢀⣦⡀
⠀⠀⠀⠀⢸⠁⠀⠘⠃⠛⠂⠀⠀⠉⢹⡇⠀⠀⠀⠀⠛⠄⠀⠀⠀⠀⠀⠘⠃⠈⠙⠳⠀⠀⠀⠀⠀⠀⠘⠿⠋⠁⣾⣿⣿
⠀⠀⠀⠀⠛⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠰⠀⠒⠀⡀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⢀⡀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠰⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⢻⠆⠀⠀⣀⠃⢀⠆⠀⠀⢰⡟⠀⠀⠈⠃⣿⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀⠠⣴⠇
⠀⠀⠀⠀⠈⠳⣄⡄⢻⣧⢚⣶⢀⣠⠞⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠠⠀⢄⣀⠀⢀⣠⣀⣤⠼⠷⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣤⣤⠀⠀⠈⣽⣿⣿⣿⣛⠛⠃⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⡀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡛
⠀⠀⠉⠉⠉⠉⠉⢢⢆⢤⡕⠌⠓⠒⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠒⠀⠀⠀⠀⠀⠀⠀⠐⢰⣄⣀⣰⠆⠴⠁
⠀⠀⠀⠀⠀⣀⣠⠋⠘⠸⡏⠂⢠⡔⠀⠀⠀⣴⡆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠉
⠀⠀⠀⠀⠼⠟⠁⠀⠰⠔⠃⠀⠈⠀⠀⠀⣀⣍⠁⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡦⠛⣿⣰⣿⡏⠀⠀⣤⡀⠀⠀⠀⠚⠋⠁⠀⣤⣤⣀⠀⠙⡾⠂⠀⠘⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠘⡤⠔⠀⠉⠉⠉⠉⠉⠁⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⣀⢀⣀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠏⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣠⡟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢸⠀⠀⠀⠀⠠⣄⡀⠀⠀⠤⠄⠒⣩⠏⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢻⠀⠀⠀⠀⠀⢀⠶⢆⡀⣀⠰⠚⠁⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⣷⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠖⠓⠀⠤⠤⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡛⢌⡦⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠻⣦⠈⠡⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⢶⠶⠶⠶⠆⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠘⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠔⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢋⡆⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡺⠏⠦⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠰⠀⠳⣄⠀⠀⠀⣀⢠⠔⠒⡝⠁⠈⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⠈⠷⠐⠊⠁⠀⢀⡠⠟⠲⠦⣤⣀⡀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠈⠱⣤⡜⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⡠⠛⠻⠉⠉⠉⠁⠀⠀⠈⠒⢄⡀⠀⢀⠞⠀⠈⠙⠃⠤⠤⠌⠿⠿⣧⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠙⠓⠤⣀⣀⣀⣀⣀⣀⣀⡀⠀⡨⠛⢷⣄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢈⣳⣤⡚⢠⡄⠀⠀⠀⠀⠀⣤⣤⣤⡀⣀
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠛⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠒⣒⣖⡚⠛⣻⣿⡛⢻⣿⣖⠒⠒⠖⠲⠚⠷⠲⣶⣶⡶⠶⣶⣶⡿⢵⣶⣤⢶⣤
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠘⠋⠉⠃⠈⢉⠉⠃⢹⣏⣹⠃⠀⠀⠀⠀⠀⠰⣿⣽⣿⢺⣿⣹⡿⢺⣯⣽⣿⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡦⠤⠴⣦⠤⠤⠤⣤⡤⣥⣤⣤⣤⠤⠤⣤⣤⣤⣬⢭⣥⣤⣭⣭⣥⣬⣭⣉⣠⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⢠⣀⣀⣠⣘⣀⣀⣀⣀⣀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣻⣛⣛⣛⣻⣻⣯⢸⡇
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠐⠒⠚⢦⠔⡄⠤⠀⠀⠀⠤⠄⠤⠤⠤⠤⠤⠤⠤⢤⣤⣄⠀⠀⠈⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠰⣆⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠍⢹⠀⠀⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⢘⣄⠀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠭⠥⠤⠬⠤⠤⠤⠤⠤⠤⠤⢴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠓⢤⣤⣇⠀⠱⢦⣄⠀⠠⣄⡀⢀⣀⣀⡀⢀⣀⣀⣀⣀⣀⢀⣀⠀⠀⢀⣀⣀⣀⣀⡀⣀⣀⣀⢀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠛⠓⠒⠚⠛⠛⠛⠒⠛⠓⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣠⣤⠀⠀⠀⠄⠧⠽⠄⠠⠤⠀⠀⢤⣤⣄⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠙⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠒⠒⠒⠒⠒⠛⣿⢻⡟⠓⠒⠒⠒⠒⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡞⠒⠒⠒⠒⠓⠺⣷⣿⡶⠒⠒⠒⠒⠓⢳⡎⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢰⡶⣶⣾⣿⣿⣿⣼⣿⣿⣿⡷⠶⠶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⡇⠼⠇⡄⣐⣀⡂⢀⠀⢐⢀⡀⣀⢀⡼⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢠⡤⡄⢀⢀⣀⣀⣀⣀⣀⣀⡀⠤⡤⢺⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⢸⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⡇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨⡇⢸⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡗⣍⣉⣉⠉⠛⠓⠘⠻⠋⢉⣉⣙⣉⣡⢽⡇⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⢡⣿⣼⡇⣵⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠬⠭⢭⣍⣟⣛⣛⣉⣋⣉⣉⡭⠭⢭⣭⣭⡤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⡹⠋⠀⠀⢹⢻⣿⠃⠀⠀⠙⣿⡖⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠁⠀⠀⠀⢸⡾⣿⠀⠀⠀⠀⢸⣥⠀⠀⢠⠈⠻⠏⡀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⣿⠀⠀⡁⢀⣀⣰⣾⣶⣦⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⠀⠀⠀⠈⢿⡿⠀⠀⠀⠀⠀⣿⣴⠀⢹⡏⠁⠙⢿⣧⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠳⠄⠤⠞⠉⠃⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⣠⠄⠈⡁⠀⠘⠀⠾⠂⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣦⡠
⠀⠀⠀⠀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣷⡀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣝⠷⠾⢓⣆⠀⠂
⠀⠀⠀⢠⡤⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⣦⠖⣾⣯⡉⠀⠀⡀⠀⣴⣷⣿⣼⣿⠗⠀⠀⡀⠀⠀⠀⠀⠀⠟⣷⢿⡏⠉⠂
⠀⠀⠀⢀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋⠐⣴⣷⣏⣩⡅⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⣄⠀⠀⠀⠀⠀⢤⠿⠙⠍⠋⠈⠀⠀⠀⠀⣠⣤⢰⣦⣄⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⡀⢰⡤⠀⡀⠀⠀⠀⠀⢺⡃⢰⡆⠀⠈⡀⠀⠀⠀⠻⠀⠀⡅⢻⠋
⠀⠀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠁⠰⠆⠀⠀⠈⠁⠀⠙⣦⡀⣿⠀⠀⠀⠘⢶⡰⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠿⠿⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠁⡀⠀⢠
⠀⠀⠀⠰⠂⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢶⣴⣇⣀⠀⠀⠀⡀
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⣾⣭⣻⢻⡷⡔⠚⠁
⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠈⠀⠀⠐⣟⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⣼⣩⡄⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢿⡿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠁⣼
⠀⣀⠀⠀⠀⠰⣄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠐⢀⣠⣞⠁⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠘⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠿⠀⠀⠀⠀⠄⠉⠙⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢈⠀⠀⠀⠀⠰⠦⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⠈⢘⠀⠀⠐⠀⠀⠀⠘
⠀⠀⠀⠀⠁⡄⠃⡀⠀⠀⠀⠀⠀⠀⠀⠂⠐⣲⣾⣿⣞⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡚⢲⡄⠀⠀⠘⠟⢻⣿⠛⠃⠀⠀⠂⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠎⣰⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠉⠳⡖⠀⠀⢠⠆⠀⠈⠀⠂⠀⠀⠀⡀⠠⢥⣮⣿⣿⣷⡀⠀⠠⠄⠀⣇⠀⠉⢸⠀⠈
⠀⠀⠀⠀⠂⠀⠀⠀⢠⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠛⠻⠻⠟⠑⠦⠀⠀⠀⠉⠀⠀⡀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣴⠆⠁⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠒⠀⠀⠠⣤⠉⠀⠀⠄⠀⢨⠀⠀⡀⠀⢼⡴⠁⠀⣀⣀
⠀⠀⠀⠀⠁⠀⠀⡼⠀⠀⠀⠀⠀⣀⡴⠒⣫⡟⠁⠀⠀⠀⠀⢰⡀⣠⡔⠀⠀⠀⠛⠀⠀⠀⠀⠀⠗⠀⠞⠀⠀⠈⠁⠐⣴⠛⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠞⣁⡴⠚⠁⠀⠀⠀⠀⠀⠐⠘⢿⣿⣆⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡾⠁⠀⠀⢄⣸⣿⣯⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣸⣧⠖⠚⠙⠋⠉⠁⠀⠀⠀⠀⠀⠙⠲⠦⠤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢴⣶⡄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠙⠛⠛⠁⠀⢀⠀⠀⠀⠀⠘⠑⠛⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠘⠙⠛
⠀⡀⣀⢂⠀⠀⠀⠁⠀⠀⠀⠀⢠⣠⠄⢙⡇⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢂⠀⠀⠈⠁⠀⠀⠀⠀⢠⣀⠄⢙⡇⠀⠀⠀⠐⣶
⠀⠱⠂⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡄⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠈⠓⠇⠀⠀⠀⠻⠲⡰⠦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠓⠇⠀⠀⠀⠻⠲⡰⠶⠀⠀⠀⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃⠢⠤⠴⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠦⠤⠴
⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠂⠈⠁⠀⢀⡀⠠⠀⠀⠀⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠂⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠂⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢤⠤⠤⣄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠁⡁⠀⠀⠀⠀⠀⠀⢐⠀⠀⡀⠀⢀⢤⠤⠤⣄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠋⠁⠀⡋⢠⠂⠂⠀⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠁⠀⡋⢠⠒⠂⠀⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡠⠃
⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢠⢚⠶⢀⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢰⠾⠶
⠀⠀⠀⠸⣦⣀⢀⣡⠟⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠠⠀⠐⠀⠈⠁⠀⢸⠁⠀⠀⠸⣦⣀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠠⠀⠂⠀⠈⠁
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠘⣿⣧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠀⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠉⠁⠀⠐⠀⠀⠸⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⠤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠘⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⣤⢤⣼⠁
⠀⠀⠀⡀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠡⠆⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⣀
⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣡⠰⠇⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣡⡰⠆⠀⠀⠀⠀⠟
⠀⠀⠀⠑⠀⠀⠀⠠⣄⡄⠀⠀⠀⡯⣑⠀⣘⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⢀⣄⡄⠀⠀⠀⣯⣁⠀⣘⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠐⢦⣀⡀⠁⠀⠀⠀⠈⠈⠘⠁⠀⠀⠤⡄⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡀⠁⠀⠀⠀⠈⠈⠊⠉⠀⠀⠤⡄⠀⠃⢀
⠶⠀⠄⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⢀⠄⠀⠀⠃⡀⠠⠍⠉⠁⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠠⠉⠉⠁
⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈
⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⠤⠆⠀⡴⠉⣉⡉⠃⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣀⠤⠇⠀⡴⠉⢉⡉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠄⠀⡅
⡆⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⢶⡽⠒⠀⠀⠀⠀⠘⠀⠤⠰⠊⠀⣠⣀⠀⠂⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⠶⠽⠖⠀⠀⠀⠀⠈⠀⠤⠰⠊⠀⣠⣀
⠃⠀⠀⢸⡀⠃⠀⠃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⣀⠀⠸⡄⠉⢰⠇⠀⠀⢸⡀⠃⠀⢃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠸⡔⠉
⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⢰⣿⡇⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⠀⣿⡇⠀⠀⠀⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢿⠟
⠀⠀⠀⠤⠄⠀⢀⣀⠀⢸⣶⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠂⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⢰⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢰⡖⠤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡶⠋⠀⢀⢸⠀⡠⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⡴⠿⠛⠛⢛⠛⠛⠋⠈⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠖⠉⢰⡴⠚⠁⠀⠀⠀⠘⠾⠊⠀⡠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠐⡀⣰⠁⠀⠀⡠⡀
⠀⠀⠀⠀⡀⠄⢠⣤⠔⠉⢀⡤⠞⠁⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠳⠿⠥⠀⠀⠶⠃⠸⠀⠀⠈⠣⣀⠀⠀⢀⣨⠔⠁⠀⠀⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⠋⠀⠀⠀⠀⠶⠂⠀⢠⠏⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⡤⡸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⠀⠸⢦⣾⣥⢤⣄⠀⠀⢆⠠⢴⢀⢀⠎⠀⠉⠒⠢⢄⣀⠀⢀⣀⠀⠀⠀⠀⠙⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠋⠀⢰⣖⡒⢤⡀⠀⣄⢠⠀⠀⠀⣾⢁⣀⣠⣟⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⠟⠢⠀⣠⠀⡮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡮⠉⢹⠗⠋⡀⠐⠂⠂⠀⠀⠉⠛⢡⠏⢀⣽⠂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⢀⠤⠐⠋⠁⢀⠏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⡿⠋⠉⢳⡄⠀⠀⠈⠢⡐⠀⠀⠠⢀⡸⠀⠀⠀⠀⢀⠀⣹⠧⢒⠆
⠀⠀⠀⢰⣿⣶⣤⣄⡀⠘⢄⠀⠀⡆⢀⠀⠀⢠⡄⠀⠀⠀⠸⣅⠀⠀⠚⠀⠀⠀⠀⠀⠘⡀⠒⠊⠉⠀⠀⠀⠀⠀⢺⡷⠛⠊⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣷⣦⡼⠀⠀⡏⠙⠀⠓⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡞⠙⡆⠀⠼⠁
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢄⣀⠀⢠⣤⡌⢀⡄⠀⠀⠀⠸⡿⠀⠀⠀⢀⣼⠙⡣
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⠖⠛⠃⡀⣀⠀⠀⠀⣞⣽⣿⣿⣿⢶⠚⠁⠘⠀⠉⠀⠁⠀⠀⠀⠀⢿⠀⢀⣴⠟⠊⢹⡅
⠀⠀⠀⠀⠄⠸⣔⡽⠀⠀⠠⢀⣘⣧⣤⠤⠴⠞⠁⠀⠀⠀⠀⢹⢹⠁⣀⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠭⡟⠁⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢳⣏⠉⠐⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⡞⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠉⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠹⣧⠀⠀⠂⢉⣻⣦⣤⡼⠀⠀⣀⡀⠀⠀⠀⠀⢃⢿⣿⠀⣿⠀⠀⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠸⣍⣀⠈⡧⠀⠀⠘⠉⠀⠃⠀⠀⠄⠸⡸⣿⡇⢻⡇⠘⠋⠉⠻⡁⢘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠧⡡⠃
⠀⠀⠀⠀⣀⡀⠀⠀⢈⡴⠒⠛⠻⣏⠃⠀⠀⠀⣀⡠⢠⡀⠀⠠⠀⡇⢿⣿⣘⣿⠀⠀⠀⠀⢸⢎⠀⠀⠀⠀⠑⠁⠈
⠀⠀⠀⠀⠛⠁⠀⠀⠾⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠘⢼⣟⡩⠋⠀⠀⠀⠰⠃⠀⠳⣄⠀⠀⣠⣐⣒⣒⢒⡄
⠀⠀⠀⠀⢞⣹⠀⢴⡦⠀⠀⠐⠋⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢠⠆⠀⢀⡠⠖⠉⠀⠀⠈⡟⠛⠛⡟
⠀⠀⠀⠀⠀⣀⣶⠶⠦⢤⣤⡤⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠙⠙⠈⠂⠾⠃⠀⠀⢰⣹⠆⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⣀⣤⣤⣤⣤⣖⠚⠉⠉⠉⠉⠉⣷⠴⠶⠶⠶⠶⠚⠃⢀⠀⠀⢠⢊⣁⣀⣨⣌⣲
⠀⠀⠀⠀⡀⠸⣄⣀⣀⣠⠇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠸⢿⣿⣿⣿⣿⠟⠰⠄
⠀⠀⠀⠰⠀⠀⠀⠀⠘⠒⠒⠒⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠊⠁⠯⠽⠭⠇⠤⠀⠿⠿⠼⠵⠸⠥⠁⠸⠿⠫⠼⠯⠼⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠁⠀⠀⠀⠀⠈⠶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⠂⠀⠀⠀⠢⣤⣀⣠⣴⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠙⠢⢄⣀⡀⠀⠀⢀⣀⣠⠟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠅⠠⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢁⡄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡊⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠒⠂⠀⣀⠄⠀⠀⣀⡄⣀⡤⠞⣹⠁⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠣⠀⠈⠉⠁⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⣉⠙⠛⠛⠉⠁⠉⠁⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢅⣀⠤⣤⢤⣤⢤⡤⠼⢑⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⡄⠀⠜⣀⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠼⡇⠂⠀⠒⠒⠒⠒⠂⠀⠐⡁⠀⠠⠀⠀⠀⠀⠀⠀⡆⠀⠃⠀⠀⠀⠀⠀⠰⠀⠀⠀⠣⠜⠉⡤⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡖⠂⠉⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠢⣀⡠⠆⠀⠀⠀⠀⠀⠀⡠⢶⠁
⠀⠀⡀⠀⠠⠀⠀⠀⠀⠇⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢⠀⠀⢀⣀⠀⠀⠀⠀⠑⠢⢄⣀⣀⣀⣀⣀⣤⣤⡲⠯⠖⠋
⡀⢀⡍⠀⠐⠒⠒⠂⠀⢸⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⢀⠀⣤⢯⡑⢢⣀⠀⢀⡔⠙⣏⡙⠉⢹⠏⠀⠀⠙⣄⠀⠀⡀
⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡦⢀⢀⡀⠀⠀⡀⠀⣀⠠⠤⠌⠀⠠⠀⠘⢯⣻⠄⠈⠒⠚⠳⠆⠀⠀⠉⢹⠀⠀⠀⠀⡇
⡇⠠⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠓⠠⠀⢀⠀⢀⡀⠀⢀⠤⢄⠀⠀⢀⡤⠴⠖⠊⠷⠀⠚⠀⠘⠒⠲⠤⢄⣸⠒⠤⣀⡴⠁
⡇⠀⠈⠦⠤⢤⠤⠤⠴⠏⣄⠀⠀⢩⠀⠀⠀⠁⠀⡯⠉⠐⠅⢤⣀⠀⠀⠘⡄⠀⠈⠁⠠⠠⠠⠄⠀⠀⣀⡤⠞⠋⠀⢠⣧⡴⢲
⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣟⠀⠀⠀⠐⡬⠤⣈⡤⠼⡆⠁⡀⠀⠀⠀⣺⠌⠉⠀⠀⢀⣠⠔⠉⢹⣤⠶⠥⠤⠤⠤⠤⠤
⡇⠁⠈⠁⠀⠉⠁⠉⠉⠉⠉⠁⠈⠈⠉⠁⠁⠁⠁⠉⠐⡀⠀⡌⠁⠉⠉⠁⠉⠁⠐⠠⠄⢀⣀⠳⣊⣤⠤⠚⠉⠀⠀⠀⠸⡎⡅⠉⠁⠉⠉⠈⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡖
⡇⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠉⢹⠃⠀⠒⠂⠒⠁⢧
⡇⠀⠀⠀⠀⠀⠈⠉⠒⠂⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡄
⠧⠤⠤⠤⠤⡌⠀⠀⠀⠀⢨⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡔⠉⠑⠲⠤⢤⡼⠀⠀⠀⠀⠀⠀⠀⢣⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢣⡀⠀⠀⠀⢀⠷⠤⣄⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠐⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⣄⠀⠀⠀⠉⠉⠐⣒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠂⠉⠁
⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⡀⢀⣤⣄⠀⡀⢀⣴⢦
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣈⠀⠀⠀⠀⢀⡴⠖⠻⠶⣦⣄⡀⠀⢀⡾⣿⠀⣶⠋⣹⣇⣇⠎⢠⣿⢀⡇⢸⣇⠉
⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠛⢲⡾⠷⣿⠀⣏⣰⡿⢹⣟⣠⡿⠃⢸⡇⣴⢋⡄
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠈⢄⠀⠀⠠⠊⠀⠀⠀⢀⡁⠀⠀⡾⠁⠀⢿⠀⡏⠁⠀⢸⡏⠁⠀⣶⡿⢦⣬⣯⣤⡶
⠀⠀⠀⠀⠀⠠⠃⠀⡀⠀⠀⠣⠀⠈⠉⠐⠀⠉⠁⠀⠀⠀⣴⡾⢷⣦⠀⡁⠀⣠⣌⡁⢠⠶⡆⣤⠴⢬⡍⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠀⠀⠁⠀⢂⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠁⢠⡿⠈⡏⠙⣷⠸⣦⡀⢸⣤⡴⠋
⠀⠀⠀⠀⠀⡇⠐⠡⠀⠁⠀⠠⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠻⢦⡤⠖⢸⡇⠀⢧⡼⠃⢾⣋⡀⢸⠘⣷⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡈⠀⠀⠀⠰⠤⠦⠲⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠉⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⡄⠀⠈⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠀⠀⠀⠀⠀⠀⠴⠴⠞⠗⠒⠛⠒⠓
⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠔⠀⠀⠁⠂⡀⠀⠂⠀⠀⠀⠀⠀⢠⣎⠷⡤⠶⡼
⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠘⠂⠀⠠⠀⠀⠐⡀⠀⠂⠀⠀⠀⢑⡀⡄
⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⡐⠀⡀⠁⢂⡀⢐⠀⢅⡀⠀⠁⠀⠢⡀⢀⠃⠢⡀⠀⢠⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠈⠀⠀⠈⠀⠈⠀⠀⠡⠀⠀⠀⠀⢐⠀⠀⢀⣠⠀⢺⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⢧⡀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠉⢈⢀⣀⢄⠈⠊⠿⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠺⠷⣶⣤⡄⠀⠀⠈⠉⠁⠐⠒⠒⠒⠒⠒⠀⠒⠂⠂⠁⡍⠀⠖⠐⠀⠀⠐⣛⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⠄⠀⠀⠀⢠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠉⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠘⠟⠉⣀⡀⠀⠸⠠⠤⢀⡀⡁⢀⠀⢀⠀⠀⠀⡀⡀⣠⠠⠰⠐⠂⠀⣄⡀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⢼⠟⠀⠤⠈⠒⠠⠄⢀⣀⠀⠀⢀⣈⠀⢀⡀⠀⡤⠄⠄⠂⠀⠀⠸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠈⠀⠀⠈⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⠰⠖⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢠⡂⠴⠋⠈⠀⠀⠀⢁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢀⣀⣀⡈⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⢁⡀⠀⠔⠁⠀⠀⠤⠞⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⡛⠿⠁⢀⠀⠀⠀⠀⠤⠀⠀⠙⢄⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⡀⠀⠀⠀⠀⠀⠢⠆⢨⠻⠷⠀⠀⠀⠀⠀⠀⣠⣤⡠⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠢⡄⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⡗⠀⠀⠀⡠⠊⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⡠⡰⠗⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣮⠆⠀⢠⡞⠀⠀⣄⣑⠀⠙⠻⠶⠤⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠚⠁⠀⠀⠀⠀⠉⠒⠄⢀⠔⠈⣒⠤⠚⢁⠴⠃⠀⠀⠀⠀⠈⠢⡀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⢈⠢⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⢰⠀⠈⠢⡀⠀⠀⠀⠀⠀⢣⠀⠀⢈⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠑⢄⠀⠀⠁⠀⠀⠀⣸⠁⢀⠂⠐⢄⠀⠁⠀⠀⠀⠀⠀⠀⢃⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠄⠀⠀⠀⡡⠖⠉⠀⢤⠀⡇⠀⢸⠀⠀⠀⠉⠚⠀⠀⠰⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠘⠀⠀⠀⠀⠀⠆⠐⠤⠁⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⢤⣀⣀⣘⣀⣀⣀⣀⣀⡤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⡿⢿⡿⠿⢿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⣰⡦⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠁⠀⠏⢀⣴⠞⠋⠉⠁⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠲⠶⠦⣄⡉⠀⠀⠀⢛⠃⠠⠀⠀⠈⠀⠀⠀⠀⠀⣤⣶⣾⢿⣦⡀
⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢿⡄⠀⢵⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢿⣇
⠀⠀⣾⡏⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡆⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⢀⣄⣠⣀⣤⣶⣶⣿⣿⣿⣿⣿⡿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⡟
⠀⠸⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣾⠟⠁
⠀⠀⢻⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠙⠿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⠏

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⢀⠀⠀⢠⡇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⡀
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢒⣦⣄⣀⡀⠠
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠐⠦⠄⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡠⠔⠒⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠘⠢⡀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠙⠦⣤⠶⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⣀⠀⡠
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⡰⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⡞
⢀⡴⠀⠀⠀⠈⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠉⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⣀⣤⠤⠦⢤⣌⡉⠒⢄⡀
⠀⠀⠀⡠⠊⢀⣀⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠞⢁⡔⠋⡉⠀⠈⠻⣆⠀⠙⢆
⠀⠀⡼⠁⠴⠛⠉⠉⠛⠛⠛⠿⠿⢿⣶⣶⣄⡀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⢰⠃⢠⣎⣠⣀⡘⠒⠀⢢⢹⡆⠀⠈⢣
⠀⢰⠁⠀⠀⡀⠀⣀⣠⣀⣰⣄⣀⡀⠀⠉⠛⠿⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠀⠘⠉⠀⠘⢧⡀⠀⢸⢸⠇⠀⠀⠈⡆
⠀⢸⠀⠠⣤⣽⣿⠿⠛⣿⣟⣹⣿⣿⡷⣄⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠑⠦⠀⠀⣈⠃⠀⠀⢸⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠈⠹⣄⠀⠙⢯⣟⣯⠞⠀⠈⠳⣄⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠃⡼⠡⠆⠰⠀⣼⠀⠀⠀⠀⡇
⠀⠀⢧⠀⠀⠀⠀⠈⠓⠶⠤⠤⠤⠶⠒⠂⠀⠉⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⢀⡾⠁⠀⠀⠀⡼
⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣀⠤⠒⠊⠉⠉⠉⠉⠑⠒⠤⣀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠉⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣡⠞⠁⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠈⠳⣌⠓⢤⣀⠀⠀⠠⠤⠴⠋⠀⢀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠖⠒⠒⠉⠁⠀⡜⠁⠀⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⢧⠀⠈⠉⠒⠒⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡠⠀⠀⠀⠀⠈⢳⠀⠀⣄⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠒⠢⠤⣀⠀⠱⡀⠀⠸⣇⢀⣀⣀⠀⠐⣋⣀⡀⣸⠂⠀⢀⠎⠀⣀⠤⠔⠒⠖⠒⠲⠤⢄⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡘⠦⣀⠀⠈⠉⠉⠀⠈⠉⠉⠁⠀⣀⠴⣃⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠑⠢⠤⣤⣀⣀⣤⠤⠔⠊⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⢀⠎⠀⠀⠀⠀⢀⣠⣤⣀⣀⣀⣤⣀⡀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⢸⢀⣀⣀⡴⠞⢯⠁⡄⠉⠉⣡⡀⠨⡟⠶⣤⣀⣀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠈⠉⠣⡒⠒⠒⠚⠛⠶⠶⠿⢛⠓⠒⠛⡢⠛⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠸⡄⠀⠀⠙⢤⣈⡀⠀⠃⣉⠘⠀⢀⣀⠔⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⠇
⠀⠀⠱⡀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠞
⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠙⠲⢄⣀⡀⠀⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠀⠀⢀⣀⡤⠖⠋
⠀⠀⡀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⣉⡉⠉
⣿⣏⢹⣟⡛⣛⣛⡛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠂⠠⢄⣀⣀⠠⠤⠶⢀⠀⠤⡄⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠟⠒⠻⢤⣀⠀⣀⣐⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠰⢠⠏⢿⠁⠀⠉⠙⣶⣦⡀⠀⠀⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠊⢅⠀⢘⢦⣀⣁⡼⠁⢇⣠⠔⣉⢽⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡰⠀⠀⠀⠈⠑⠬⣙⡁⠀⠀⠀⢀⣀⢠⠤⠎⠡⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡾⣝⠢⣀⠀⠀⢀⡼⠀⡝⠭⠭⡟⣁⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⢈⢀⠃⠈⠐⠁⢰⠀⠃⠀⠰⢀⠇⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢱⡀⠀⠀⠀⡘⡠⠀⠀⢠⡆⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡜⠀⠈⠁⢉⠁⠁⠀⠀⠀⠃⡠⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢈⠄⠀⠀⢀⣠⡊⠀⠀⠀⢀⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⢄⡀⡯⠀⠀⠑⣄⠠⠊⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠏⠀⠈⢳⠤⠖⡊⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡄⠀⠣⠄⠀⠀⠀⢠⠄⠒⠁⠤⠀⠤⠀⠀⠀⠳⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠤⢀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⢸⢄⡀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⡀⠀⠀⡈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠠⠀⠀⠉⠒⠒⠠⠐⢦⠀⠈⠉⠒⠒⠂⠤⢤⡄⠐⠁⠀⠀⠀⠈⠀⢀⠄⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠒⠉⠀⠨⡀⠀⠀⠀⢀⠂⠂⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠁⢠⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣟⣟⣿⣻⣛⣟⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣟⣻⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢈⣙⠻⢦⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣶⣟⣉⣈⣭⣿⣿⣿⣿⣯⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠟⠉⢭⣽⣿⣿⣿⣿⣿⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃⠀⠀⠐⣺⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣯⠁⠀⠀⢤⣼⣿⣿⣿⣿⡭⠪⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⢒⠺⢿⣿⣿⡋⠐⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⡀⣸⣿⣿⠿⡉⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠈⣙⣿⡿⠓⠚⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⣀⣂⣈⡉⠉⠁⠀⠀⢴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣴⣿⡿⠇⣠⠄⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠉⢀⣤⡾⠃⠀⣠⣴⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣀⣤⣶⡿⠟⣡⣶⣿⣿⠏⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⢀⣾⡿⠛⢉⣐⣿⣿⣿⠿⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠚⠉⠀⠀⢀⣤⠞⠋⠀⣀⣼⣿⣿⣿⣣⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠖⠒⠋⠉⠀⠀⠀⢀⣠⣤⡤⠈⠉⠀⠠⢀⣨⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⣤⣾⡿⠋⠉⣀⣠⣤⣶⣶⣾⣿⣿⣿⣟⢛⣋⣀⣤⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠰⣿⣏⠠⠚⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⣀⠀⠀⡔⠋⠱⠤⢶⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢠⠒⣄⠀⠀⢀⠚⠠⠂⢀⠤⣄⠓⠛⢢⡑⢄⣆⠀⡀⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠑⣄⠀⡴⢶⠲⡒⠒⡖⡚⢲⠲⣆⣠⠙⠚⡗
⠀⠱⡤⡑⠼⠟⢍⡑⣄⡀⠘⠂⠊⠀⢤⡠⣵⠋⠁⢌⣀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢠⣍⣘⢃⠀⢣⢸⠀⢰⢀⠀⡇⣿⣁⠀⠘⢆
⢀⠞⠀⠀⠀⠀⣠⠞⠳⠥⠀⠀⠀⢇⣠⠇⢸⡧⣵⠀⢠⠒⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⢀⡀⢹⣤⠂⢸⢀⠀⢸⡄⢠⢇⡿⢹⣱⠒⠋
⢇⠀⢀⡴⢾⡉⠙⠢⣄⣔⡇⠈⡂⢀⠤⣤⠋⠧⣈⣁⡜⠀⠀⠀⠀⢀⡀⠤⡤⠤⠤⢤⣤⣤⣀⡀⠀⠈⠓⢻⠃⣧⠜⠋⠈⠒⠞⠚⠒⠛⠛⠉⠉⠀⠀⠉
⠀⠉⠉⠀⠀⠑⠤⠴⠛⠾⣌⣉⣠⡧⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⡎⠑⠠⠀⣌⡶⡷⠷⢶⡀⠤⣍⣣⠀⠀⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⡒⠲⠤⠤⡶⠤⠤⠼⠖⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⠀⠀⠀⡇⠀⠀⢀⡄⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠒⡩⠤⢕⢤
⠀⠀⠀⠀⠀⡰⠲⡋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠠⠤⠤⠧⠤⠤⠚⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢡⠀⡠⢀⠑⠤⠚⠀⣣
⠀⠀⠀⠀⠀⣧⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠒⠁⠀⠢⠴⠃⢀⣀⠸⣄⡆
⠀⠀⠀⠀⠀⢏⠳⣄⠈⠓⠤⣀⡀⠀⠀⢀⣰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⡄⠀⠐⢄⠸⣄⡨⠃⡠⠁
⠀⠀⠀⠀⠀⠘⢦⠀⠑⠢⠤⣀⣉⡉⢉⢉⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⣀⡴⠂⠈⠉⠑⠢⠤⣀⠀⠀⠈⠣⣆⠀⠁⡁⠄⣀⣰⠻⠥⣀
⠀⠀⠀⠀⠀⠀⠀⢹⠦⣀⡀⠀⠈⣉⡡⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠂⠁⠀⠒⠀⠤⣠⠀⠈⠁⠀⠀⠀⠀⣀⠄⠊⠉⡄⠀⠀⠀⠀⠈⠈⡑⢟⡉⢠⠻⡖⠊
⠀⠀⠀⠀⠀⠀⠀⡼⠀⡰⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣎⣁⡘⠊⢇⡀⠀⠀⠀⠂⠀⠀⠀⠉⠓⢂⠔⠊⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠉⠁⠈⠙⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣰⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠈⠉⢀⠿⣓⣦⣀⠀⠀⢀⣤⣀⠂⠀⢸⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠱⡀
⠀⠀⠀⠀⠀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⣞⡋⠤⡀⠀⠀⠀⠀⠀⠔⠅⠈⠀⠀⠀⢉⡹⠓⡄⠉⡆⠀⣸⠗⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠱⣄⢑
⠀⠀⠀⠀⢰⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⢬⠑⠒⠒⢮⠒⠉⠓⢦⠒⠒⠲⣤⠐⠀⢠⠀⡾⠊⠀⠀⠀⠀⠀⣠⠔⠛⠉⡡⢈⠩⠛⢦⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣆⠠⣀⣔⠠⠤⢄⣍⣀⠀⣀⣸⠀⡠⠞⠉⠁⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⣀⣀⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⢏⠠⡀⢀⡴⢋⢀⢔⢹⡏⠩⢑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠚⡟⠀⠰⡉⢁⣼⢅⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠤⣼⠗⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠈⢷⡤⢀⣷⣀⠴⡻⠉⠈⠎⣆⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠚⠉⠙⠢⡈⢆⠀⠈⣷⡖⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣄⣙⣺⣤⡤⠴⡟⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠐⠚⠉⠁⢄⠙⢆⠀⠀⠀⢹⠈⣄⡴⠙⠛⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⢆
⠀⠀⠀⠀⡀⠠⣀⡀⠀⠀⠔⠉⢍⡉⠲⣌⠢⠀⠀⠀⠑⠘⣀⣠⠤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠳⡄
⠀⠀⠀⠐⡀⠀⠸⡈⠑⠚⠒⢀⡀⠱⠀⠈⢆⣷⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠘⢆
⠀⠀⠀⠀⡝⠉⠀⡨⠛⢳⠔⠣⠤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠘
⠀⠀⠀⠀⢇⠀⣸⠽⠛⠉

⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⣖⣫⢿⠿⠃⠀⣀⣴⣦⡀⠀⠀⠀⡴⠋⠉⠀⠀⠀⢀⣴⣛⣶⣄⠀⠀⠉⠙⠒⠂⠀⠀⠀⣀⣀⡆⠠⢄⢀⣴⡿⠁
⠀⠀⠀⠰⠋⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⢔⠿⣓⢿⡯⣟⣶⣄⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣷⠀⢖⣻
⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠻⠡⣗⢫⠬⡝⢧⣚⠽⠀⢁⠀⠀⢠⡿⡿⠻⠟⢸⡟⢰⣯⣿⣦
⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢴⠸⣹⣷⠀⢹⣿⣗
⠀⠀⢠⢷⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣖⡰⡠⢀⢀⣀⡆⠐⣒⣦⠀⠀⡀⢀⢀⠁⠀⢀⡀⠀⠸⠈⢹⡏⠘⡿⣿⠏
⠀⠀⠠⡿⠠⠃⠰⡟⠟⠛⠓⡄⠀⠀⠰⠯⠒⠓⠛⠻⠚⠁⠀⠃⠻⡾⠻⠚⠋⠙⠀⠀⠨⠽⠃⢀⣤⡈⠇⠀⠈⠿⠄
⠀⠀⠀⠀⠀⠀⣴⢰⡄⠘⠻⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠁⠀⠀⢀⡀⠀⠀⢠⡖⢄⠈⠿⠇⠀⣶⣦⠀⠀⢀
⠀⠀⠀⠺⢿⣿⣿⣿⣧⡄⠀⠘⠀⢠⠀⠀⠀⠈⣀⣀⣀⠀⠀⣼⠀⠀⠀⠀⢿⠶⣿⠇⠈⠙⠉⢀⣠⣤⣤⣄⣀⡀⠀⠻⠆
⠀⠀⠀⠐⠿⣿⣿⣿⣿⣧⠀⠀⠀⠈⢰⠀⠀⣿⠉⠉⠁⠀⠀⠁⢨⣽⣶⠀⣤⠀⣤⣄⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⡀⠀⠀⣿⠀⣽⠉⡏⠀⠀⠀⢰⠎⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⣭⡇⠀⠉⠉⠉⠁⠸⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠰⡟⠀⣿⠀⡇⠀⠀⠀⠈⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⣴⣿⠇⠀⣁⡀⢀⣀⠀⠀⠙⠉⠉⠀⠀⠃⡏⠙⡿⠁
⠀⠀⠀⠀⠀⠇⠀⣿⠀⠇⠀⢰⡀⣰⣆⡀⠀⠀⠑⠀⠀⠀⣀⠰⠾⠉⠀⠀⣿⢹⣿⣯⠀⠀⠰⢄⣀⠀⠀⠀⢀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⠖⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⣰⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀⣞⡅⠀⠀⠀⠀⠀⠀⠀⡴⠂⠒⢦
⠀⠀⠀⠀⠀⠀⢮⣀⣄⣄⡼⠀⠀⠀⠀⠀⠀⡰⠣⠭⢽⡀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠦⡀⠀⠀⠀⠀⢀⠷⠔⠒⢺⡄
⠀⠀⠀⠀⠀⠀⠈⢟⠢⡱⠁⠀⠀⠀⠀⠀⠀⣗⢒⠒⢒⡇⠀⠀⠀⠀⠪⣀⢤⣀⡠⣀⡼⠀⠀⠀⠀⠸⡀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢸⠁⠀⠀⠀⠀⠀⠸⣴⣤⣦⠇⠀⠀⠀⠀⠀⠀⠈⢹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣱⠦⡀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠢⡀⠀⠀⠀⠀⢀⡠⠚⣧⣼⣦⠀⠀⠀⠀⠀⡰⠁⠉⠉⡖⠁⠀⠀⠀⠀⠀⡴⠷⠞⢄
⠀⠀⠀⠀⢰⣫⢤⣀⠤⣄⡠⢌⣲⠀⠀⠀⣏⣩⣭⣤⣤⣤⡇⠀⠀⠀⠀⠳⡒⠒⢲⠞⠀⠀⠀⠀⢀⣠⣏⣙⣉⣹⣄⡀
⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠯⠭⠭⠭⠤⠴⠃⠀⠀⠀⠀⠀⢹⢻⠏⠀⠀⠀⠀⠀⠈⠑⠒⣢⣖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠽⠦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠤⠶⠭⡉⠑⣄⠀⠀⠀⠀⢰⠣⢢⢔⢮⢣⠀⠀⠀⠀⠀⡔⠉⠀⠈⠱⡄⠀⠀⠀⠀⢸⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⡏⡞⡉⠳⡈⡆⢸⠀⠀⠀⠀⠸⡰⣐⣾⣮⡾⠀⠀⠀⠀⢰⠳⡖⢶⢲⡖⣷⠀⠀⠀⠀⢸⣿⣿⡛⢻⡇
⠀⠀⠀⠀⠀⠱⣱⢌⣒⣁⠷⠊⠀⠀⠀⠀⠀⠉⢲⣿⠃⠀⠀⠀⠀⠀⠈⣇⣧⢸⢸⢧⡏⠀⠀⠀⠀⢸⠋⠛⠻⠿⣯
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡔⠋⠀⢙⠲⡀⠀⠀⠀⠀⠀⠀⢀⣠⡓⢤⠀⠀⠀⠀⠀⠀⡞⡁⡙⡆⠀⠀⠀⠀⠀⠀⣀⣄⣤⣀
⠀⠀⠀⠀⠀⡜⠡⠆⠀⠈⠀⢳⠀⠀⠀⠀⠀⡴⡏⢣⠈⡞⠁⠀⠀⠀⠀⡖⠚⠈⠘⠒⡆⠀⠀⠀⣰⡎⢱⡀⢀⠎⢑⡆
⠀⠀⠀⠀⠀⢸⣀⠀⠀⠙⢀⠎⠀⠀⠀⠐⣮⠣⣸⡤⠋⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢏⠀⠀⠀⠀⠙⢌⡏⠙⠋⢹⣡⠃
⠀⠀⠀⠀⠀⠀⠈⠙⣾⠉⠁⠀⠀⠀⠀⠀⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠋⠳⠬⠇

⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢰⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠠⠁⠠⠬⠐⢂⠀⠀⠀⠀⡦⠀⡐⠉⢂⢀⡀⠀⠀⠀⣀⣧⡵⠖
⠀⠀⠀⠀⠀⣥⠜⠁⢢⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠸⡀⠁⠐⠀⠂⢉⡜⠀⠀⠀⠈⢢⡠⣯⡟⣍⣈⠏⠀⠀⢼⠁⢈⡇
⠀⠀⢀⠀⡀⠀⠀⠰⢀⠀⠀⠘⡁⣤⠁⠠⡀⢀⠀⠀⠑⠤⡄⠀⡠⠎⠀⠀⠀⠀⠐⣁⣸⠨⢵⡋⠑⡀⢀⡴⠚⢯⠉
⠀⠀⠈⡉⠅⠀⠠⢂⠌⠀⠀⠀⠐⢄⠀⠀⠐⣄⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⠈⠉⠀⠀⠈⠉⠚
⠀⠀⠀⢰⣶⣿⣷⣷⣀⠀⠀⠀⠀⠈⠰⢤⡾⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠤⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠀⠑⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⡈⢀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠃⠀⠐⠃⠀⠀⠀⠠⠘⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⡠⢊⠆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠐⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠉⡇⠠⠀⠀⠀⠀⠀⢻⣷⠄⠀⠠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡦⢄⠤⠖⠁⠀⠀⠀⠀⠀⠠⣠⣶⣶⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠢⠄⠄⠊
⠀⠀⠀⠀⠀⠠⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡡⠀⠀⠀⠀⠈⢤⡐⢀⠆
⠀⠀⠀⠀⢠⠁⠏⠀⠀⠀⠄⢀⠴⠀⠀⠀⢠⠿⢿⠿⣿⣿⡿⠷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠂⠐⠔⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠒⢀⠸⠈⠂⠀⠀⠀⠀⠀⠀⢂⡀⠀⡰⢀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣈⠁⢈⣤⣤⡗⠀⠀⠀⠀⠀⢠⠊⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡍⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂
⠀⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠀⡆⠀⠄⣀⠀⠀⠀⠀⠀⠀⡠⢐⣤⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠠⠐⠁⠀⠑⢩⠁⠀⠀⠀⠀⠀⠀⠈⢿⡇⣀⣀⠀⡄⠀⠀⠀⢰⡸⠏⠟⢀⢰⠀⠀⠀⠈⡀⣬⢢⠀⠀⡠⡂⠥⠀⠠⠁
⠀⠀⠀⡁⡄⠀⠐⠒⠄⠅⠀⠀⠀⠀⠀⣠⣴⡾⠛⠻⢿⠃⠀⠀⠀⠀⠀⠃⠈⠄⢉⡀⠂⠀⠀⠀⠐⠠⢙⢠⣌⠜⢁⠠
⠀⠀⠀⠑⠈⠀⠒⠨⠈⠀⠀⠀⠀⠀⠠⡈⠛⠃⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⣀⣰⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠓⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣶⣶⡂⠀⠐⠐⣓⣓⠐⠒⠒⠂⣾⣶⣦⣤⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠶⠤⠤⠶⠶⠾⡿⣿⠶⠤⠴⠶⠴⠶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣯⢱⠖⠒⠒⠒⠒⠶⣵⣷⠖⠒⠒⠒⠒⢲⡉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣶⠾⠿⠿⠿⢧⠿⠿⠿⠿⠷⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠐⠛⠠⠴⠤⠄⠄⠀⠄⠄⠴⠄⠒⢼
⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⠀⣶⡖⠒⠒⠂⠆⠐⠒⠒⠒⠒⠚⢸⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⢔⣙⠒⠒⠒⠒⠒⠒⠒⠒⠒⣈⣉⢿⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀⣟⠉⠉⠉⠋⠉⠉⡋⣉⣉⣩⣿⣾⠀⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⠭⣭⣭⣭⠭⠭⣭⣭⡭⠭⠭⣴⣶⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡱⠁⠀⠀⢿⢻⡇⠀⠀⠈⣿⡖⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⢸⡇⠀⢠⠃⠈⢁⣢⢰⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠘⣿⡀⠸⣶⠖⠺⣿⡛⠛⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠿⡧⠀⠻⣄⣀⡼⠛⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢞⣁⣠⠶⠿⢿⣷⣶⣶⣾⣷⠶⠶⢦⣤⣬⡿⠂⠀⠀⠀⠀⢀⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⢰⣿⡵⠁⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠴⣾⡴⠋⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣤⣀⡤⠤⢤⡀⠀⠀⠀⠙⠆⣰⠿⠃⠀⠀⠀⢀⣶⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢱⠀⠀⢀⡀⠀⠀⢠⡞⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠞⠀⠀⢸⣧⠀⢰⡿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⣦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⢧⡉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⡴⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣇⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⠃⠀⠀⠀⠚⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠈⠲⠤⠞⠋⠛⠿⠿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠈⢧⣄⣀⣠⡞⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⢀⣤⠀⠀⠀⢸⡏⠉⠉⠉⠉⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⡂⠀⠀⢸⣧⣤⣤⣤⣄⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣴⣿⣿⣿⣄⡀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⠇⠀⠀⢸⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⡿⣿⣿⣿⣷⡀⠀⠀⠛⠻⠛⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣇⣽⣿⣿⣿⡇⠀⣀⣀⣀⢀⣀⣀⠀⠨⣿⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡿⠁⠀⢨⡟⣱⣿⡇⠀⢿⣿⣿⣿⣿⣿⠁⠈⣿⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣷⠀⠀⠀⠁⠙⢿⡇⠀⠀⠛⠻⠿⠛⠁⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣯⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢠⣾⡇⠀⢀⠀⠀⠤⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣷⣾⣿⣿⡇⠀⢈⠀⢁⠀⠆⠀⣠⣴⣿⣤⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠄⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⢠⣄⣀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠙⠻⠟⠛⠁⠀⠀⠀⠙⠛⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣀⣀⢀⠀⣠⢤⡀⣀⠀⣴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠅⠠⡅⠀⠀⠀⠀⡴⠞⠛⠻⢶⣤⣀⠀⢠⡿⣿⢰⡗⢁⣿⢸⡼⠁⣾⠇⣿⠐⣧⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠉⢻⡟⠒⣿⢸⣧⡾⠋⣾⣧⡾⠋⢀⡟⢰⣏⡅
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠐⢄⠀⠀⣈⠀⠀⢀⡠⠜⠀⠀⠟⢀⡀⠿⢸⡇⠀⠀⣿⠀⠀⠸⠿⠳⢶⣭⡶⠞
⠀⠀⠀⠀⠀⡘⠁⠠⠀⠀⠈⠂⠀⠀⠀⠉⠀⠀⠀⠀⢀⣾⠟⢻⡇⢠⡄⢠⣴⣤⡀⡞⠻⠰⡞⠓⣦
⠀⠀⠀⠀⠀⠀⢀⠀⠠⠅⢀⢰⠀⠘⡀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣸⡇⢸⡆⣸⠇⣹⠆⢰⡷⣞⠋
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡂⠀⠀⢡⠀⠀⢀⠀⠀⠀⠈⠛⠚⠁⠸⡇⠈⠓⠋⠘⠷⠄⠘⠃⠹⣦⣤⡶
⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⢀⠀⠀⠀⠉⠉⠉⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⡀⣤⣀⣀⣀⣀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠤⡀⠒⠀⡀⠀⠀⠀⠀⠀⢉⣉⡉⠉⠁⡀
⠀⠀⠀⠀⠀⠑⠢⠤⠤⠄⠒⠋⠀⠀⡄⠀⠀⠀⠀⢀⠂⠀⡀⠀⠀⠀⠄⠀⠘⠃⠙⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠠⢀⠀⠠⠈⠁⠀⠀⠂⠀⣀⠀⠈⡤⠐⠈⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⢎⠀⠐⠀⠀⡆⠸⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠒⠀⠀⠁⢀⠆⠀⡄⠀⣀⣴⠆
⠀⠀⠀⠀⠀⣶⣤⣌⣀⠀⠁⠒⠒⠀⠠⠤⠤⠤⠤⠠⠄⠠⠤⠄⠀⠒⠈⠉⠀⠀⠀⠤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠈⣉⣉⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⠿⠷⣶⠄
⠀⠀⠀⠀⠀⠴⠿⠋⠁⠀⠀⠆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠴⠀⢀⡀⠘⠷⠆
⠀⠀⠀⠀⠀⠀⠀⣾⠏⢀⡀⠡⠄⢀⠀⡈⠈⠀⠈⠀⠀⠀⢁⢁⡀⠀⠤⠌⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠁⠀⠉⠉⠁⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠐⠂⠐⠚⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⡀
⠀⠀⠀⠄⠠⣾⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⡀⣩⠴⣺⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⡴⠟⠏⡹⢿⣿⣦⣤⡀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠞⢁⠞⡽⢠⣿⣿⣿⠀⠀⠀⣠⣶⠃⠀⠀⠁⣴⣣⣤⠿⢿⣿⣷
⠀⠀⠀⠀⠄⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⠁⡰⠃⡜⡰⠋⣷⣿⣿⠀⠀⠀⠻⢯⣄⣀⡀⠀⢀⣿⣥⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠹⢛⡿⣿⠿⠋⠀⠀⠀⠀⠀⣧⠞⣀⠴⠀⠀⠀⣻⣿⠇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠘⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤
⠀⠀⠀⠀⠀⢰⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⣿⣟⡛⢚⣭⠀⠀⠀⠀⠀⢀⣤⡤⣿⣿⣿⡿⢀⣠⣄⡀⠀⠀⠀⠀⢀⠰⢶⣶⣶⣾⡀⡰⠶
⠀⠀⠀⠀⠀⠀⢸⣟⠒⠈⠋⠀⠀⠀⠀⠀⢿⣿⡿⠋⠉⠁⠀⢾⣿⣿⡶⠀⠀⠀⠀⢸⣦⣤⣤⣤⣤⣤⠄⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣴⣶⡄⠀⠉⠉⠀⠀⠀⠀⠀⠈⢿⡷⠆⠛⠛⠁⠀⠀⡎
⠀⠀⠀⠀⠀⠀⣿⣧⣀⣀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠃⢀⣾⣿⣦⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣟⡷⠾⠿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠀⢠⡛⠋⠉⠁⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠳⠿⠿⠵⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠶⢦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡴⠃⠘⢦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠞⠛⠉⠀⠀⠂⠀⠀⠀⠀⠀⠉⠛⣷⡶⠶⠶⠶⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠙⠳⢤⡀⢀⡤⠞⠋⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⢀⠀⠀⠀⢳⡞⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⣀⡴⠛⢧⣄⡀⠀⠀⠀⠀⢀⣼⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⣦
⠀⠀⠈⠙⣦⠞⠉⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⢴⣦⣀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣾⣿⡆⠈⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠉⢉⣀⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠁⢀⣴⡄⠀⠀⠀⠀⢻⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡈⠷⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⣈⣉⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣄⡀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠉⠁⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢻⣆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠷⣤⡀
⠀⠀⠀⠀⠀⢠⡟⠉⠉⠻⡦⠴⢿⣆⠀⠀⠀⠙⠋⠀⠀⠀⠀⠉⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⠀⣰⠟⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣈⡿⠂
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⣤⣿⣧⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠈⠿⣤⣤⣤⡾⠉⠀⠀⠈⢻⣦⣄⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣡⡶⠿⣥⣀⠀⠀⠀⠀⠀⠀⢀⣈⣠⣄⠀⣻⠿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣩⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣤⣤⣴⣶⠟⠉⠀⠀⠈⠙⠛⠻⠶⠶⠞⠛⠛⠉⠁⠛⠶⠋⠀⠈⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠾⠿⠛⠛⠉⠈⢻⣌⠻⣿⣿⡿⠋⠉⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⣼⠟⠁⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⡞⠁⠀⠈⢳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣏⢈⣷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⣀⣴⣞⣡⡄
⠀⠀⠀⠀⠀⠀⡔⣿⣿⣮⢞⡵⣫⠝⠁
⠀⠀⠀⠀⢠⢖⡣⣞⡿⠁⢀⣜⣡
⠀⠀⠀⠀⣎⣿⣿⣿⣷⣠⠜⠻⡀⠀⠀⢀⡀⠀⠀⠀⠈⠁⢀⣀⡀
⠀⠀⠀⠀⠙⠿⣿⣿⠿⣇⠀⠀⠱⣄⠀⢉⡤⢤⣀⠀⢀⠞⠁⠀⢸⠂
⠀⠀⠀⠀⠙⠓⣼⣅⠀⠀⠑⢢⡄⠀⠙⠺⣆⠀⠈⠳⣼⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⡎⢻⣷⣄⠀⢀⣭⡓⢤⡀⠈⢣⡀⠀⠈⢳⡭⠥⠤⠤⠤⠤⠔⠒⠚⠉⠁⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠳⢿⡈⢮⣑⢮⡀⠈⠙⢿⣦⡀⠙⠲⢤⠤⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⢀⡤⢔⡛⠿⠆⠀⠀⠀⠃⠀⠀⢹⡉⠁⠐⠒⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠘⠋⣉⠴⠚⢇⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠻⠻⡖⠚⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢣⠀⠀⠙⠛⠓⠿⣶⡄⢀⣤⣶⠿⠛⣁⣸⠾⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠹⡭⠞⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠧⣄⠀⠀⣾⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠷⢖⠤⣈⣋⢿⠢⣄⠀⠀⠀⠀⠀⠀⠀⠸⡁⢠⡼⠂⠀⠙⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠔⣊⡥⠄⠒⠂⠀⠀⢲⡀⢱⠀⠁⠀⠱⢬⡑⢤⡀⠀⠀⠀⠀⠀⣈⡦⡇⠀⠀⠈⠳⡞⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣈⠳⢄⡠⢊⡵⠊⠁⠀⠀⠀⠀⠀⢰⡣⠷⠞⠀⠀⠀⠀⠈⢯⠓⢬⡒⠒⣒⡩⠥⠒⠊⠹⣤⣀⡀⠠⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠑⣒⣻⡀⠀⣀⠉⣉⣉⡹⠂⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡩⠕⠊⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣤⣀⣀⣀⣀⣀⣀⢤⠤⣴⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⠧⡔⠈⢋⠫⠏⠉⢖⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣍⠋⢀⢿⣾⣿⡛⠺⣧⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷⣶⠷⠞⠿⢳⣾⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⡴⢩⠃⠀⠀⣠⠎⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⡀⠈⣛⣦⣀⡀⣀⣨⠇⣴⡉⠀⠀⢉⣑⣾⣖⠒⣒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣏⡫⢟⣽⡩⠊⠀⠀⠉⠒⠭⣓⠿⣏⡩⣟⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠽⠷⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠈⠛⠟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡔⠀⣼⣃⠀⢀⠈⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⠛⢅⣺⣤⣴⡿⢿⣿⠭⠭⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⣄⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡏⠀⢠⣤⣤⣄⠀⠑⢆⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠋⠉⠙⠳⣤⡴⠄⣇⠀⠈⣿⣿⣿⣷⣆⠀⠑⠆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢰⡈⠛⢿⣷⠀⠀⠙⠙⠛⢿⣿⣷⣄⠀⠉⠠⣀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⢘⣇⠸⠁⠀⠀⠀⠀⠀⠀⢻⣿⠿⣳⣤⡀⠈⠙⠾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡘⠎⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠈⠈⠻⣿⣷⣤⡀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⣀⣀⠀⠀⠀⠀⢀⣾⣿⣷⣦⣄⡀⠉⠑⠻⣷⠀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⣴⣾⣿⣿⣿⣶⠾⣿⣽⣿⣿⣿⣿⣟⣿⣿⣷⣤⣿⠂⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠛⠿⠿⢿⣿⣦⡀⠀⠀⠸⠙⣿⣿⣿⣾⣽⡿⠏⠀⣴⣿⣇⠀⠀⢀⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠄⠀⢀⡀⠉⠙⠷⣦⣴⣄⣫⣿⣿⣿⣿⠀⠀⠞⠁⠉⠉⠉⠀⠁⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢀⠉⠙⠛⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠉⠉⠉⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠈⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⢀⣀⣛⣛⡛⠻⠿⢯⣛⠿⢦⠀⢦⣼
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⢁⣀⠀⠀⠚⠋⣉⣩⣭⣟⠛⠶⠀⠈⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⣿⣡⠴⠀⠀⠸⠟⣦⣀⣀⣠⣴⣶⡄⠀⠀⠀⠀⣠⡶⠶⣤⠀⣿
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢠⡟⣡⠾⠋⠀⠀⠀⠀⢉⣩⢴⣿⡽⠂⠀⠀⠠⡾⠋⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢸⠃⣠⠆⠀⠀⣀⣾⣯⣶⣯⡵⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢻⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣴⠶⣠⣾⠃⠀⠀⠀⣉⡵⢞⣽⠏⠀⠀⠀⠀⠀⠀⠀⠿⠶⡶⠿⠦⠀⠙⢷⣄⠀⠀⣠⣄
⠀⠀⠀⣀⡴⠿⠀⠀⣠⢴⡟⠀⢸⠇⡿⠀⠀⠀⠀⠉⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⡟⠉⠻⢶⣤⡶⢶⣄⣀
⠀⠀⣾⠋⣰⠟⢡⣾⡟⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠀⠀⠀⠀⠙⠷⢾⣧⣽⡿⣦⠀⠀⣠⠶⡆
⠀⠸⣏⠀⠿⠀⡿⠘⠇⠀⠀⠰⣄⡀⠀⠀⠀⠲⣦⣀⣀⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢿⣶⣞⣡⠞
⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣦⣤⣴⢿⣯⣩⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣙⠶⢤⣤⣤⠶⠋⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢧⣤⣤⣤⡤⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡖⢀⡟
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠙⠻⠶⠖⠂⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠿⣤⡌⠻⠓⢶⣦⣤⣀⡀⠀⠀⠀⢠⡏⢀⡾
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣴⠏⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡅⠀⠀⠈⢻⡄⠀⠀⠓⠀⠈⣹⠟⠲⠦⣼⣁⡾⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⣶⣶⠶⠛⠁⠀⣀⣼⣄⣠⣄⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⣿⠆⠀⠀⢀⡾⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠉⠛⠉⠈⠙⢷⣶⣶⣤⣴⣟⠉⠙⠛⠓⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠄⠶⢦⣤⣄⣀⠀⠀⠙⠿⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠉⠙⠻⠶⣤⡀⠀⠸⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣴⠶⠛⠂⠀⠀⣀⠀⠀⠈⠙⢦⠀⣤⣿⣧⡀⠀⣀⡠⣄⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠴⠋⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⢶⣶⡉⣿⠟⢋⣡⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠗⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠀⠀⠈⢻⣽⡏⠀⡈⢻⡿⠳⠶⢦⣤⣤⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣿⠛⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠛⣅⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⣼⠂⠀⠀⠀⠀⢿⣿⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣗⠀⠈⠙⠷⣦⡀⠀⠀⢀⣴⣿⠋⠀⠚⠁⠀⠀⠀⠀⠀⠈⢿⡉⠉⠛⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⠛⣷⠀⠈⠙⠆⣴⡟⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠙⠳⢶⡶⣦⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⣰⠿⡅⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⠀⠀⠀⠀⣼⠃⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢿⣀⡞⠁⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠏⠉⠙⠲⣄⠀⠀⣿⡧⠒⣫⠉⢛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣤⣓⣮⣷⣾⡟⣠⠊⠁⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠉⢷⣻⣟⣵⠞⢉⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⠞⠉⠀⠀⣠⠉⠁⠀⠀⠈⠑⢆⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⡾⠿⣛⠿⠷⢶⣤⣤⣀⠀⠀⠀⢰⠃⠀⣈⠁⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⢇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⣴⡿⠋⠙⣿⣿⣿⣍⣭⣍⣉⣁⠒⠌⠛⢷⣄⠀⡏⠀⠀⠇⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣷
⠀⠀⠀⠀⠀⣾⡟⣠⠴⣚⠹⡽⢏⢳⠦⠀⠀⠨⠍⣓⢤⡈⢻⣧⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡿
⠀⠀⠀⠀⢐⣿⡶⠁⠀⢃⡳⠝⠺⠉⠀⠀⠀⠀⠀⢬⠑⡝⢆⣤⣳⡀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡸⣸⣻⠁⢧⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⣠⠏
⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣀⣼⣵⣿⠀⠈⠳⡈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⡴⠁
⠀⠀⠀⠀⠀⣿⠱⣌⠻⢷⣦⣤⣀⣀⣀⣀⣀⣤⣴⡾⠛⢡⠋⣼⠀⠀⠀⠈⠻⢄⣀⠀⠀⠀⠀⠀⠀⣀⡴⢟⠋⡄
⠀⠀⠀⠀⠀⢻⠀⠈⠓⠆⢀⣉⠉⠉⠉⠉⠉⣉⣠⠴⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⡿⠿⠩⠕⠚⣩⠞⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠉⠁
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⡟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣸⠁⢀
⠀⠀⠀⠀⠀⠀⠘⢮⡑⠦⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⢀⡴⢃⠆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣌⣉⣙⣛⣓⣀⣉⡥⠖⣋⠴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠐⠺⢿⣦⣿⣿⣿⢿⣱⣿⡃⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣠⣾⣿⣿⡇⣄⣾
⠀⠀⠀⠀⢸⠇⢙⣯⣽⣿⣿⣻⢿⣿⣧⡈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⣿⣿⣿⣿⡿⣥⣼⡏⣿⣻
⠀⠀⠀⠈⢳⠶⣿⣾⣿⡟⠉⠉⢘⣵⠽⠀⠀⠙⠳⣦⣤⣤⣄⣠⣴⣿⠛⠋⠛⣿⣿⣿⠏⡴⠋⠙⢿⣿⣿
⠀⠀⠀⠀⢸⣞⣿⣿⣿⡿⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠛⢛⣉⡁⢀⣿⣒⣂⢺⣿⠟⢿⣸⣇⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠓⠹⣿⢏⡏⠁⠀⠀⠀⠀⠐⠀⠀⠀⢀⣠⡦⠾⣻⣯⣟⠛⠁⠈⠉⣡⣴⠻⣿⣿⣖⣰⡿⢿⣿
⠀⠀⠀⠀⠀⡠⠔⠛⠁⠀⣠⡺⢀⣴⣦⠤⠶⠛⠉⢩⣴⢞⣽⠋⠉⠑⠒⠃⠉⠀⢸⡷⢻⣿⣽⣭⠒
⠠⠀⠀⠂⣈⣤⣀⣤⠴⠾⢛⣡⣾⠟⣡⡶⠟⣃⣤⣿⡾⠋⠸⣀⠀⠀⠀⠀⠀⠔⠋⣷⢇⣴⠶⣾⣦⡄
⣦⡤⢠⣵⣶⣮⡻⣤⣴⡾⢟⣩⡴⠞⣫⣴⣫⣿⣿⡾⢯⠀⢠⠛⠛⠒⠂⠀⠀⠀⠀⣻⣾⣥⠀⠀⠙⣿⠘
⣙⣉⣉⣠⣤⣌⣹⣿⠷⠞⠋⠁⢴⢿⠿⠿⠯⠿⡯⠉⠀⠀⡀⣳⣤⣠⡀⠀⠀⠀⡀⣿⣿⡀⢳⠀⠀⢹⡇
⣛⠛⠛⠉⠀⠀⠀⠜⠲⠤⠶⡶⠲⠶⠻⠿⣿⢆⣃⠀⠀⠀⠽⡍⡛⠩⠟⠳⡖⢿⣿⡿⣅⡀⣾⣴⠿⠀⠃
⠙⣳⠆⠀⢠⢔⣠⠤⠤⠖⣩⡴⢞⡛⠛⠓⢦⣝⠻⣿⣉⣒⠃⠘⠏⠁⠀⠀⠈⠀⠉⠁⡿⢰⢟⣯⡀⠀⠒
⣧⣙⣤⣠⠴⣿⠛⢶⣶⢟⣥⣀⣤⡤⣶⡶⠶⣶⣜⡳⢟⡚⠀⠀⠒⡀⠀⠀⠀⠀⡀⠘⡛⣴⢾⣏⣀⡠⠄
⠉⠉⠉⠁⠀⠀⠀⠈⠛⠳⠤⠴⣬⣿⣽⣤⠀⢈⡎⣰⡟⠷⠀⠀⠐⠛⠒⠤⠖⠛⠻⢿⣺⠋⣾⣯⣹⣦⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠁⠈⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⣽⣟⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢠⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢟⣿⣿⣿⣾⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⡄⢿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣴⣿⣿⢟⣫⣽⣶⣾⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠈⠉⠉⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⡟⠉⠉⠉⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣯⣿⣽⡟⣿⣿⣿⠟
⠀⠀⠀⠈⠂⡀⠀⠀⠀⠘⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⠀⢀⡄⠀⠀⠀⠲⠾⣿⣿⣿⣿⣯⣿⣧⣿⣿⣿⡿⠟⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠘⣷⣿⣷⣦⣀⠀⠀⠀⠉⠉⠹⠋⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠙⠭⠽⠯⠿⠿⠛⠋⠁⠀⢀⣀⠻⠿⣿⡷⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡄⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠰⣶⣬⠙⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠘⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⡆⠐⠀⠀⠀⣀⠰⠀⠆⠀⠀⠀⠀⠈⠳⣌⠳⢄⡀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠐⠦⠈⠀⢀⡁⠀⢀⠄⠀⠀⠀⠀⠀⠱⣤⣿⣦⣹⣿⣃⣠⣴⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠒⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣟⠻⣅⠀⠀⠀⠀⠈⠛⠛⠛⣻⡿⠿⡟⠻⠏⠙⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠉⢰⠈⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠻⣝⠻⣷⣜⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢹⣿⠿⣿⣿⡿⡿⠟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠌⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢧⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿
⠀⠀⠀⠀⡀⠀⠈⡌⠀⣸⣯⠟⠁⠀⠀⠀⠀⠀⠀⣴⣾⢿⣿⣧⣡⣄⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣴⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠏
⠀⠀⠀⠀⠈⢼⠀⡇⢰⣟⢕⣡⡄⠀⠀⠀⠀⠀⠀⠉⢡⣾⣏⢻⣿⠻⠗⠀⠀⠀⠀⠀⠀⣿⣻⣟⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣠⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠸⡆⣷⣿⣟⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⢻⣿⣞⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢠⣤⣶⡾⠿⠿⢛⠻⢿⠋
⠀⠀⠀⠀⠀⠀⣿⡿⠝⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⣽⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣿⣷⡆
⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠗⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠛⠃


⠀⠀⠁⠁
⠀⠐⠒⠃

⠀⠀⠀⠀⠀⣠⡔⠉⢢⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⢅⡀⠪⠂
⠀⠀⠀⠀⠀⡇⠇⠀⠰⢂⠀⠀⢐⡡⠂⢚⣝⠒⠨⡀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⡤⠬⢀⠁⢁⠨⢦⠄⠀⠀⠀⠈⡢⠀⠆
⠀⠀⠀⠀⠀⢃⠀⠀⢀⠄⠀⠀⠈⠐⠠⠉⠑⠡⠐⠁⠀⠀⠰⡀⠀⡀⢀⠀⠀⠀⠈⠢⣀⠤⢀⠀⠀⠀⠀⠀⠀⠀⡀⠬⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⠿⠿⢟⣛⣛⣭⣭⣭⣭⣝⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢗⡲⣶⣿⣿⣿⣿⠫⣵⡤⣶⠝⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⣵⣿⣿⡿⠿⢿⣿⣮⡃⠫⠾⠇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣛⣫⢍⢠⣚⠿⢸⣿⣿⣿⣮⡻⢏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣝⡋⠾⠇⢟⣣⣿⣿⣿⠿⠿⣿⣿⣿⣀⠀⣤⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⢟⣼⣿⣿⣿⡤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⢰⣭⡝⠻⠿⣿⣿⣿⣿⠛⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣮⣣⣿⣿⣌⢵⡞⣴⡹⣶⢎⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣤⣭⣥⣶⣿⣿⡿

⠀⠀⠀⠀⠀⠀⢀⠤⠔⠢⢤⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡤⣤⢤⠀⡠⢠⣄⢀⣄⢀⣀⠀⢠⣤⡀⣠⠄⡤
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠁⠉⠉⠉⠉⠈⠁⠁⠈⠉⠀⠉⠉⠁
⠀⠀⢀⡴⠊⠁⠀⠀⢀⡤⣄⠀⠀⠀⠀⡼
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠙⠒⠒⠊⢳
⠀⠀⠈⠓⠒⢷⠒⠂⠸⠤⠖⠉⠛⣄⢀⡮⣀⡀
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⡨⠋⠀⠀⠉⠓⢤⡀
⠀⠀⠀⢀⠎⠈⠙⠒⠒⠒⠒⠋⠁⢀⠀⠀⠀⠀⠀⠀⠘⢉⡆
⠀⠀⠀⡜⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠑⠄⠀⠐⠠⠀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⢳⠤⣀⣈⡠⢄⣀⠈⣀⣤⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠈⡄⢀⠀⠀⠀⠀⠀⢧⠀⠤⠀⠀⠈⠉⠁⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢐⠀⠀⠀⠀⡀⠠⠤⣌⠢⡀
⠀⠀⠀⡇⠠⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠉⠉⠉⢨⠀⡠⡖⠁⠀⠀⠀⠀⠀⠈⡖⢄
⠀⠀⠀⡇⢠⠀⠀⡆⠀⠀⡇⠀⠀⢀⡀⢀⡠⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠳⡌⠉⠭⠭⠭⠭⠍⠋⠥⠎
⠀⠀⢴⠃⠠⠀⠀⠀⠀⢀⣇⣀⣀⠀⠈⣉⣠⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡀⠀⠀⠉⠳⣄⣠⣄⡀⢸⠉⠙⢏⠙⠛⢡⡟⢦⡀⠀⣀⠄⢢⢧⠄⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⣤⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢘⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠘⡏⠉⠀⠰⠚⠂⠀⠀⢴⠋⠂⠄⡀⠀⠀⠐⠂⠤⠔⠒⣁⠀⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠶⣼⣄⣀⠤⢞⡥⠒⡂⠉⠁⠙⠁⠀⠊⠓⠲⠦⣅⠀⠀⠁⠀⠀⠀⠈⠀⠈⠗⠀⣤⠀⠀⣸⡀⠀⠀⠀⠀⠀⢸⠗
⠇⠀⠀⠀⠀⠀⠈⠂⡤⠏⠡⡀⠈⠑⠨⠑⣒⠀⠀⠀⠀⠒⠒⠂⢁⠅⠀⠀⢠⣀⣀⣀⣠⠀⠰⡀⠀⠇⠀⡀⠩⣷⣦⣤⣤⣤⣶⡧
⡆⠀⠀⠀⠀⠀⠀⡔⢸⠁⠀⠘⠢⠄⡀⠀⠀⠰⢁⠀⠀⠀⢀⡠⠂
⠁⠀⠀⠀⠀⠀⠀⣗⠃⣀⠀⠀⠀⠀⠈⠐⠒⠚⡳⣶⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡬⠤⠭⢭⡭⠄⡾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠙⢄⠀⢀⡾⠄⡀⢥
⠁⠀⠀⠀⠀⢀⠞⠁⠉⢦⠤⠉⣽⠂⠠⠐⠚⡦⡀
⠁⠀⠀⠀⠀⠘⡀⠀⠀⠘⢄⣀⠼⡄⠀⠉⠈⠀⠈⠆
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡳⠄⣀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠚⠁
⣶⡶⢶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⢶⣶⣶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣰⠂⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢠⡏⠀⠀⠀⠀⠀⠻⣦⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣻⡇⢸⡇⠀⡀⠀⠀⠀⠀⠈⠙⠛⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠛⠛⠛⠻⠇⠈⠀⣸⠃⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠴⠚⠉⢀⣠⠴⠖⠚⢻⡆⠀⢈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠠⠴⠚⠋⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⢀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡾⢷⠄⠀⠀⠀⠀⠀⠈⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⢀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣀⣀⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠠⠞⠋⠁⠀⠀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣦⣤⣤⣤⣤⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⡟⠀⡰⠀⠀⢻⡇⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⣦⡀⠀⠀⠁⢠⡇⠀⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠀⢰⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡀⠀⠻⣤⠏⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡾⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠊⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠸⠆⠀⠀⠀⠀⠀⢠⠑⠶⠃⠀⠀⣳⠖⠀⣠⡄⡐⠀⠴⠉⠒⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠈⠁⠀⠀⡤⣻⠀⠀⠊⠳⣼⠀⢀⡴⢡⠞⠁⡜⠒⠁⠙⢡⣀⠀⠒⢀⡈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⣀⡄⠀⠐⠀⠀⡔⠀⠀⣰⡃⠀⢸⠀⠠⠀⠀⡘⢰⠃⣠⠞⠀⣀⡆⠂⠀⠀⢀⣀⠈⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠀⠁⠘⣆⡰⠀⣼⠀⠀⠀⢱⠃⠀⠀⡇⠀⢂⡔⠀⠈⢉⣥⠔⠈⠁⠀⠀⠀⠀⠀⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠄⠀⢦⠈⠀⠴⠁⠀⠀⣠⠏⠀⠀⠀⡟⢠⢎⣠⠴⠊⠁⢠⡖⠒⠒⠒⠀⠤⣤⣀⣙⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⡀⢀⠀⠂⢀⠗⢀⡴⠊⠀⠀⠀⠀⣀⡴⠟⠋⠩⠍⣀⠐⠂⠩⠉⠉⠩⠿⠥⢤⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠱⡈⡞⡰⠋⠀⠀⠀⡠⠔⠋⠁⠀⠀⠀⠈⠢⣄⠈⠐⢀⠀⣀⣀⠀⠀⣠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠃⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⡎⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡠⠊⠀⠀⡠⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⠀⢀⡴⠊⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢤⣀⠀⠀⠐⢂⣠⣥⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢈⠑⠢⢄⣈⠁⠀⠈⢉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠸⡳⡀⠀⠙⠂⠠⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠣⣄⠁⠙⠓⠤⢄⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢸⣿⠙⣷⢀⣀⡀⢀⡀⣀⢀⣀⠀⣿⡇⢀⣀⡀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⠃⣸⢼⣿⢸⣿⠁⠹⣧⡁⣿⡇⣿⠜⠃⠹⣇⢸
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠛⠚⠛⠘⠛⠀⠐⠚⠃⠛⠃⠙⠓⠂⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⣿⣿⣿⠀⠀⢸⣆⣠⢊⣿⣫⣶⣯⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⣿⣿⣿⠀⠀⢸⣿⣿⣿⡹⠀⠈⠯⠁⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠈
⣿⣿⣿⠀⠀⢸⣿⣿⡟⠀⠀⡰⠀⡇⠀⠠⠀⠀⠀⠰⠀⢄⡀
⣿⣿⣿⠀⠀⢸⣿⣿⣇⠀⢠⠃⠀⡅⠂⠀⠀⢀⣀⣠⡄
⣿⣿⢿⡄⠀⠀⣿⣿⣿⡀⣼⡄⠀⣱⡤⠖⠊⠉⠠⠊⠈⡆
⣿⣿⡄⠀⠀⠀⢹⣿⣿⣿⣿⣧⣼⡏⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡄
⣿⣿⣷⡀⠄⠀⠀⢿⣿⣿⣿⣿⣇⢻⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⡇
⣿⣿⣿⣷⡄⠈⠀⠈⣿⣿⣿⣿⣿⣸⡇⠀⣀⡀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣛⣃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡀⠀⣀⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣸⣄⣦⣀⣠⡀⠀⣠⣴⣶⣶⣶⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣻⣿⣿⣿⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠉⠙⠻⠿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠒⠒⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⢀⡀⠀⠀⠀⠀⠀⠀⠘⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⣰⠋⠉⠉⠉⣗⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣧⣄⠀⢀⣀⡈⠲⣀⣤⡀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢘⡷⡆⠀⢠⣄⠁⠀⠋⠀⣹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⢸⠀⣄⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⢰⡬⢤⡤⠀⠀⠀⣿⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠼⣆⠓⠊⠀⠀⣠⠔⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⠒⠉⠀⠀⠹⣷⣄⣀⣠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⢧⡀⠀⠀⢀⡏⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠓⠢⠞⠀⠀⠀⠀⠀⠘⢆
⣠⠶⠿⣽⡲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠁⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣧⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠦⣤⣀⣤⠖⣫⡥⢭⣑⣶⠚⠛⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⣤⠖⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠔⠋⠀⠀⠀⢯⠙⢷⡄⠀⢧⡀⠀⠀⠀⠀⠀⠀⣸⠀⣞⢇⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠻⣦⡀⠱⣄⠀⠀⠀⠀⠀⡇⠀⠘⢾⣷⢄⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠻⡓⠾⢦⣄⣀⣀⠀⣷⣄⠀⠀⠻⣮⠳⡄⠀⢀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⣄⠈⠓⠒⢶⡙⠲⣝⢷⠈⠓⠦⣄⣹⡷⡈⠢⣼⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠑⢄⡀⠀⠹⢤⣈⣻⡆⠀⢀⣀⣩⢃⣷⣄⣈⣳⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠛⠛⠛⢻⠿⠆⠀⠀⠉⡿⠯⢭⣉⣽⣟⣋⣉⣉⣉⣉⣉⣉⣉⣩⠭⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠘⠒⢾⠀⠀⠈⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠤⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡏⠙⠲⢤⣄⡀⠀⠀⠉⠳⣶⣦⣤⣤⣤⡤⢤⣤⣤⡶⢶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⠒⠤⢄⣉⠉⠉⠁⠀⠀⢀⣤⣤⣭⣭⣥⣤⡴⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣤⡤⠶⠴⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀⣼⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇⠉⠓⠲⢤⣉⡙⠲⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⢸⠅⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡇⣰⡄⣄⢠⡀⣼⡇⢸⡇⢢⣰⢠⢠⣠⠋⡿⡶⣤⣬⣙⡶⡦⣶⣀⣠⠂⣼⣦⣾⣾⢦⠀⠀⡀⠀⠀⢉⠧⡎⡦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠁⠈⠀⠁⢹⠉⢹⡏⠉⠈⠈⠈⠈⠀⠁⠁⠋⠉⠉⠛⠧⢬⣉⠉⠛⠻⠇⢿⢹⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣦⣄⣸⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣋⠁⠀⠀⠀⢢⡀⠀⠀⠀⠀⡀⠂⠐⢀⠀⠀⠀⠠⣠⠀⠀⠀⠀⠀⠉⢠⡈⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠐
⠀⠀⠀⠀⢰⢄⠀⢀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⠀⠑⢤⠀⠈⠃⠐⠄⠀⠀⠀⠀⠀⠑⠠⠐⠐
⠀⠀⠀⠀⣀⠀⠀⠘⣃⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠰⢢⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣷⡾⠛⠀⠀⠀⠈
⠀⠀⢰⢔⣽⠄⢀⣼⡿⣶⡄⠀⠹⠛⠿⣷⣦⠀⠀⠀⠀⠁⠀⠀⠀⠐⠈⠋⠁⠀⠀⠀⠀⠱⠒⠀⠀⠀⠈⢿⡽⠷⠛⠉⠀⠀⠀⠀⠀⠀⢶⣢⠀⠀⠉
⠀⠀⠀⠉⠁⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠁⠀⠘⠀⠢⠄⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉
⠀⡠⠖⠀⠀⠀⠀⣿⣿⣻⣿⡆⠀⠀⢀⣰⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⣤⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠈⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⡄⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠈⠃⠀⠀⠼⠂
⠀⠀⠀⡀⠻⣄⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠬⣤⢁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠁
⠤⢄⠀⠀⠀⠐⠀⠀⠀⣠⣶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠰⠀⠀⠀⠀⡀⡤⢦⡄
⣰⠀⠌⠁⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⣆⠰⠂⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀
⠀⠖⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠉⠀⠀⠀⠀⠤⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡎⠀⠀⠊⠂⠀⠠⠀⢀⡀⠑⠕⠢⢀
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⣲⣿⣾⢁⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⡴⢂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠀⠀⠋⠚⠁⣿⣀⣠⠟⢚⡷⠶⢀⡤⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠐⠋⠤⣀⠐⠀⠀⠀⠀⣀⠠⠖⠨
⠀⠀⡄⠀⠀⠀⢀⡀⢀⠀⠀⠀⢰⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠁⠠⠎⠀⣰⠟⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠠⠀⠀⡁⠀⠂⠀⠀⠀⠀⠐⠆
⠔⠈⠀⠀⠀⠀⢸⣥⡛⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⢂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢀⠈⠀⠀⢀
⢀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣤⣤⣤⣤⡄⠞⠃⠘⠀⠀⠀⠀⠱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣄⡤⠀⠀⢀⣾⣄⣀⠀⠀⠀⠀⠸⠷⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾
⣿⣿⣯⣿⣿⡇⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⣯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⡀⡖⠀⠀⢀⣾⣿⣿
⣿⣿⣿⣿⡿⠁⠀⣀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠁⠄⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⣉⣡⠔⠂⠀⠀⠹⣿⣿
⠛⢉⡉⠉⠀⠀⠀⠜⢀⠂⠀⡄⠀⠤⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⡠⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠉⠀⠐⠇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠲⡄⠐⠀⠀⠄⠀⠀⠀⣠⠁⠀⠀⠠⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⢠⠀⢒⠀⠀⠀⠀⠐⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠈
⠀⠡⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠐⠂⢀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠠⠀⠀⠇⠀⠌⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤
⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠞⢀⠈⡁⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠂⠸⠀⠀⠀⠄⠀⠀⠀⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⢀⡶⣆⣀⠐⠃⠀⢀⢰⣯⡴⢇⡴⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁⠀⠀⠀⠀⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠈⠁⠓⠆⢹⠄⠀⣴⣟⣸⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠠⣤⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⡼⣿⡿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⡛⡛⣿⡇⡀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⡁⢈⡀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⡇⢈⠰⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠹⣦⠟⠛⠀⠀⢂⣹⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠹⣆⢰⢤⡹⣷⡀⠀⠀⠘⠋⠙⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢙⣿⠀⠓⣄⠹⣧⠀⠀⠀⢀⢴⡿⠂
⠀⠀⠀⠀⡀⠀⠀⠀⣄⣈⣈⠀⠹⡅⠀⠈⣦⠛⠀⠀⢲⠞⠋
⠀⠀⠀⠀⠸⠀⠒⠐⢄⠀⠀⢀⢏⡿⣠⢡⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⣄⠐⠀⠠⠈⡕⢽⡏⣴⢃⣾⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡇⡄⠀⠀⠀⠀⢚⡀
⠀⠀⠀⠀⠀⠁⠀⠘⠁⠀⠀⠐⠞⣡⢋⡼⣫⣾⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢦⡜⢇⢱⠀⠀⠀⣒⣁⡓
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⢢⣤⡥⢯⣴⣿⣿⣿⡧⣖⣀⡀⠀⠤⢤⣤⠀⡡⡆⢲⣯⣾⠀⠀⣀⢨⡐⢥⡀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠮⠀⠀⣿⠁⡟⢿⣿⣿⣿⣷⣾⢒⡏⢸⠃⡇⢚⡙⣿⣿⠀⠀⠉⢸⣯⣿
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠄⡀⠀⡿⠀⣧⣻⡏⠙⠛⠟⠒⠿⢿⠧⠒⠛⣂⡝⠿⠛⠲⢾⣿⣯⢺⣿⠳⠶⠄⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⡇⠀⢹⣿⣇⠀⠄⡤⣤⡃⠰⠷⡶⣴⣼⡧⠔⠺⠷⠾⠾⣽⢿⡟⠻⠷⣄⣡
⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⠈⠿⠃⠠⡴⠖⠀⠲⠠⣾⣾⣿⣿⡟⠻⠷⠖⡿⠇⠿⠾⠻⢷⣦⠀⠸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⡀⢁⡀⠼⣴⠆⠒⣄⡆⠀⣿⣽⢿⡯⣷⣤⡄⡀⢰⣶⣆⡠⠶⠤
⠀⠀⠀⢸⠘⠀⠀⠴⠀⠀⠀⢀⡛⣀⠄⠀⠀⠘⠇⠀⠣⢄⡀⠀⣀⣀⣿⣿⢸⣧⣄⠀⠉⢃⡘⢿⣿⣁⣀⠀⠁⠉⠐⠆
⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⢀⡽⠋⠀⠀⠉⢽⠋⠀⠀⠀⠹⠗⢦⡬⣿⣿⠈⠙⠛⠋⠂⢀⡄⢸⣿⡏⠛⠳⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠰⠆⣀⣀⠀⠀⠀⠠⠾⢿⣯⣤⣀⡀⠀⠀⠘⠃⠿⠍⠿⣦⣤⣀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⣶⣾⣿⣷⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢀⠀⣀⣀⣀⣀⡀⣀⣀⣠⢀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠠⠄⡀⢄⠀⢀
⠀⠀⠀⠀⠠⠬⠥⠤⠖⠒⠩⠭⢍⠀⠈⠲⠖⠋⠠⠀⠆⠀⠉⠓⡜⠒⣾⣔⡾⠄⠀⠀⠀⠀⠀⠀⡀⠰⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⡄⠂⠈⠄⠁
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡀⠆⠀⠀⢀⡀⠀⠀⠀⠀⢠⠤⠔⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠈⠟⠛⠁⠀⠀⠀⠠⠼⠀⠉⠀⡀⠣⠀⠐⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠋⠀⠀⠀⠀⠒⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⡀⠀⠠⠀⠀⠀⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠘⠰⢄⣀⠀⠀⠀⠀⠀⠀⠀⠠⢐⠠⢀⠀⠀⠀⠀⠈⠀⡀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠈⠉⠱⠣⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠤⠀⠀⠀⠀⠀⠀⠠⠽⠛⠀⠄⠀⠁⠈⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢄⡆⠀⠀⢘⠀⢀
⠀⠀⠠⠀⠠⢂⣀⣀⣀⠀⠈⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⠀⣀⠀⠀⠀⠈⣑⡀⠇⠀⠀⠠⠄⠠⡰
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⢀⠀⠀⠀⠀⠉⠄⣊⣁⡄⠀⠉⠀⠀⠀⠁⠀⠈⠀⠀⠀⢀⡂⠁
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⠙⠒⠈⠀⠀⠀⠈⠁
⠀⠀⣿⣿⢿⡿⣿⣿⡿⣿⢿⣿⡇⠀⠀⣠⣄⣤⣤⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠆⡄
⠀⠀⠘⣿⣶⣾⣷⣿⣷⣾⣾⡿⠃⠀⢿⣿⣿⣿⡿⠟⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠈⠻⠿⣿⣿⣿⠿⠛⠁⠀⠀⠈⠂⠀⠀⠂⠑⠀⠁⠀⠀⠀⠀⠀⠠⠄⠤⠁⠀⠀⠠⠤⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠈⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢜⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂
⡒⠶⢶⠤⢄⠀⠀⠑⠊⠀⠀⠀⠀⠀⢀⠔⠂⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠠
⠠⢀⠀⡇⠀⠀⠀⡠⢤⣶⣆⢀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡾⠀⠀⠀⠀⠱⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡸
⠀⢀⠆⠀⠀⠀⠀⠀⠉⢄⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠙⠋⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠸⠀⠀⠠⠂⠀⠀⠀⠀⠑⠇⡀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠐⠄⠉⠉⠉⢠⠁⠀⠀⠀⠈⠳⣤⣀⣀⢴⢣⠀⠀⠀⠀⠀⠠⠎⠉⠑⡄⠀⠀⠀⠀⠸
⢀⠇⠀⠀⠐⠄⣀⡀⠀⠀⢀⡀⠈⡀⠤⣄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡼⠶⣄⠀⠰⠋⠙⠳⡄⠀⠀⠈⠧⣀⡴⠁⠀⠀⠀⠀⠠⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢺⡀⢀⡸⠀⠘⢤⣀⠴⠃⠀⠀⠀⠀⠀⡶⠖⠢⡀
⠀⡠⠶⢤⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡞
⠀⣇⠀⣠⡦⣀⣀⢄⡀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠒⠚⠁
⢀⣈⠉⠁⡸⠋⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠁⠈⢳⠄⣇⠀⠀⠀⢠⠆⠀⠀⠀⢀⠄⢤⡀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡠⠀⠈⠀⠀⠀⠘⠀⠙⢄⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⠀⠀⠀⣀⠠
⢤⠴⠋⠀⠈⠲⠤⠖⠃⠀⠀⠀⠀⠹⢄⣀⠇⠀⠀⠀⠀⢀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢣⡀⠀⠀⠆⠻⠟⠀⠚⠂⣤⠄⠀⠀⠀⠀⠠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠁
⢀⣤⠆⢱⠀⠀⠀⠀⡀⠤⡀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⠠⠉⠐
⠉⠀⠀⠀⢢⠀⠀⠘⢅⣀⡼⠓⠢⢼⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠄⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⠾⢁⡤⠦⠠⣄⠀⠀⠀⡠⠄⢄⠀⠀⠀⡠⠲⠶⠟⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⢄⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠈⡧⠀⠘⢅⣀⡼⠀⠀⠀⠈⡂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠙⠢⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠢⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞⠁⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢁⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠁
⠀⠀⠘⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢠⠋⠀⠀⠀⠉⢣⠀⠀⠀⠀⢠⠀⠲⡎⠉
⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⡄⠀⠀⠘⢄⡀⠀⠀⣠⡎⠀⠀⠀⠀⠸⡀⡀⠆
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠃⠀⠀⠀⣸⣙⠉⠉⢱⠼⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠲⢴⠘⠣⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⢱⠀⢣⡀⠀⣨⠃
⠀⠀⠀⠀⠀⢠⠒⠒⠄⠀⠀⠀⠀⠀⣇⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠡⠀⢀⠌⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠀⠀⡄⠀⠀⠠⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠉⠉⠄⠉⠑⢄
⠀⠀⠀⠀⢠⠖⠉⠀⣀⠠⠂⠁⠀⠄⠐⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂
⠀⠀⠀⠀⠇⠀⠈⠀⠈⠀⠀⠂⠀⡀⠀⠰⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣬⡀
⠀⠀⠀⠀⢇⠈⢁⠀⠉⠀⠋⡀⠀⠀⠀⠀⠀⡠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⡎⠐⠾⠓⢹
⠀⠀⠀⠀⠈⡆⠀⠀⠀⣀⠀⠁⢠⠋⠉⢆⠀⠰⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠠⠀⠀⡧⣀⣀⣀⠜⢦⡀
⠀⠀⠀⠀⠀⠸⡌⠀⣾⠉⠙⠒⠋⠀⠀⠈⠂⠐⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠐⠀⡁⠐⠀⠬⠀⠈⠀⠠⠙⡄
⠀⠀⠀⠀⠀⠀⢣⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠈⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡠⡶⡻⠃⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⣩⠊⡴⠁⠀⣇⠑⠤⣆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡰⠁⢰⠁⠀⠀⠈⢦⡀⠀⠉⠳⣽⢦⡔⢢⢀⣤⡀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣀⣠⡎⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⣦⣹⠃⢸⠤⠻⢖⠚⠉⡦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠔⠒⡏⠀⢸⣇⠀⠀⡇⠀⠈⢆⠀⠀⠀⠀⠀⡇⠀⠀⣠⠣⣈⠒⢛⣦⣀⠠⢵⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣱⢄⣾⠀⠑⠒⠚⠢⠄⣈⠷⠤⠠⡤⠚⠉⠉⠉⠀⠏⠉⠣⡀⠀⠑⢦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠓⠂⠉⢸⡁⠒⠪⠕⣤⠊⠉⠢⠔⠉⠑⠤⠊⠉⠲⠊⠙⢦⠖⠻⢶⣀⣉⠯⠽⠥⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠏⠀⡏⠒⢶⠐⠸⡆⠀⠐⡄⠀⠀⡆⠀⠀⡛⠀⠀⣇⡤⠚⡉⠀⠠⠀⠀⠂⠀⠠⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠰⠀⠘⡄⠀⢣⠀⠀⠇⠀⠀⡇⠀⠀⡇⠀⡼⠃⠀⠂⠀⠀⣠⢄⠴⠛⢤⣀⠀⢁⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢣⠀⢱⣀⡈⡆⠀⢸⠀⠀⠁⠀⢰⠁⡜⠀⠈⡁⠉⢔⡲⠗⠂⠉⠉⠐⢦⠇⠄⠐⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠲⣧⠄⠠⢔⠂⠘⠿⠁⠀⠈⢷⡀⠀⡆⠀⠀⠀⡸⢠⠅⠈⠀⠀⠂⡈⠣⢄⣀⣀⡀⠤⠊⡀⠀⢠⠲⢄⡨
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠘⡂⠀⠀⠀⠀⠀⢸⠥⠄⠁⠀⠀⠤⠅⠤⠵⡞⠦⠚⢢⠀⠀⠀⠁⠄⠐⢀⠁⢀⠤⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡸⣁⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠓⠒⠉⠑⠤⠔⠉⠁⠉⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⠶⠖⠊⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣀⣀⣀⣀⡀⠤⠔⠊⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠑⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⠓⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠃⠀⠀⠀⠁⠒⠤⣀⠀⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⡈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉⠀⢹⠁⢰⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠛⠉⠀⠀⠀⠀⠀⢀⣬⣀⡟⠀⠉⠲⢤⣀
⠀⠀⠀⠀⠀⢀⣠⣤⡶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⢙⣦⡄
⣀⣠⣴⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠉⡇
⣿⠿⠿⣶⣤⡀⠀⢀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠛⠋⠁⠀⠀⠀⠀⣷
⣿⣧⠀⠀⠙⢿⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⡟⣧⠀⠀⣾⠇⠉⠛⢷⣄⣀⣤⣴⣶⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⢹⡆⠀⣿⠀⠀⠀⣠⣿⡟⠋⢉⠤⠒⠀⠀⠀⠠⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⢿⡀⣿⠀⢀⡼⠋⢸⡇⠈⢿⡛⠛⠒⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠈⣷⣿⠶⠋⠀⠀⢸⣷⠀⠀⠙⠳⠶⠤⠤⠤⢤⣹⣷⠐⠂⠉⠉⠒⢦⡀⠀⠀⠀⣿
⢸⡇⠀⠀⠈⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢹⣤⠀⠀⠀⠀⠀⠱⡀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⡿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⡇
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⠀⢀⣀⣴⠇
⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠈⠳⣄⣀⣀⣠⡤⠖⠋⠀⠀⢀⣠⣴⠾⠛⠉
⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣀⣤⡴⠾⠛⠉
⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣠⣴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣼⣧⣤⣤⠶⠾⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠸⠧⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⣀⣀⢀⣁⡦⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⡁⠝⠂⠠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠰⣬⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠒⢠⡂⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⣡⡂⠀⠀⠀⠀⠀⡣⢠⡀⣄⣤⢀⡄⠘⣂⢉⣿⠁⣤⠲⠆⡾⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢺
⠀⠀⠀⠀⠀⠉⣽⡍⠓⢀⠀⡀⠀⠀⠀⠤⠀⠠⠆⠀⠽⠇⠀⠀⠀⢀⠎⡀⠸⠶⠿⣯⣊⣁⣈⡁⠐⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠁⠴⠤⠼⡄⣠⠄⠀⠀⡀⠀⠤⣡⠉⠁⠀⠀⠀⠀⠀⠈⠓⠛⠁⠒⠯⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠀⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⡀⠤⢦⢀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀⣀⣀⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠸⠂⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠠⠂⣀⡀⠔⠀⠍⠀⠂⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠁⠀⠉⠑⠚⣶⣷⡞⠤⠀⠀⠀⠀⠀⢀⢾⣁⠉⡀⠜⠁⠁⣠⠀⠀⠴⡌⡀⣢⢴⡇⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠘⡷⣤⣤⡀⡀⢠⣴⣶⠿⣾⣷⡀⠄⠀⠀⠀⠀⠈⠀⠈⠑⠐⠇⠤⠤⠤⠤⠼⠡⢚⡫⠕⠈⠓⠀⠀⠀⠀⠀⠁⠀⠐⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠁⠘⣟⠁⠀⠀⠟⢱⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠀⠉⠁⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢀⣄⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⠓⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠀⠀⠀⠐⠒⠈⠁⠀⡀⠄⣀⠀⠀⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡴⠿⠗⠀⠀⢠⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢠⡰⠃⢀⣀⡀⢠⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠛⠛⠉⠉⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠊⠁⠀⠀⠀⠀⠀⠀⣬⠂⣠⠅⠀⠀⠀⠀⠀⠀⠀⢺⠀⢀⣄⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣀⠠⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⢤⢥⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠖⠂⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣈⠁⠈⠀⠀⠀⢐⠄
⠀⠀⠀⠀⠀⠁⠀⠀⢀⣦⡀⠀⠀⠀⣼⢂⡀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠆⠀⠀⠀⠈⠳⠾⢄⣗⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠤⢤⠤⣀⠄
⠀⠀⠀⠀⢠⡀⠰⣎⡀⠂⠻⠶⡶⠚⡋⡿⠥⣠⡆⠀⠀⠀⢴⠋⠁⠀⠀⠀⢠⠌⢁⣆⠄⠀⠀⠈⣭⡩⣷⡆⠀⠀⢀⠀⠀⠀⠀⠀⠘⠁⠀⠘⢦⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠃⠍⠄⠐⠊⡝⠀⠀⠀⠀⠱⠒⠄⢐⣀⢁⠀⠉⠉⠉⢀⣀⣰⣦⠴⠟⡟⠀⠀⠀⠀⠛⠀⠰⠆⠀⠀⠀⠀⠀⠘⠙⢿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⡠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣄⠀⠀⠀⠀⢀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⣀⣤⠤⠐⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣵⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢄⡀
⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡯⢄⠑⣽
⠀⠀⠀⠀⢊⠔⠊⢏⣩⣻⠒⢌⡂⠀⠀⠀⠀⢠⣔⣊⡈⠑⠉⣈⣒⣤⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⢀⣿
⠀⠀⠀⠀⠈⠢⢄⣛⣉⣊⠤⠒⠁⠀⠀⠀⠀⠀⠱⣄⠈⣉⣉⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢌⡡⠊

⠀⠀⠀⠀⠀⢀⣠⢺⢱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢸⢸⢸⢸⢸⠤⡄⠀⠀⠀⠀⠀⠰⡿⢶⡶⢶⠶⣶⡾⡷⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡜⠀⠀⠀⠀⠀⠀⠀⠈⠢⡇⠸⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠘⡦⠤⠤⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡐⠢⠠⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣄⡜⢧⡄⠀⡄⢠⠈⡛⢠⢖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣏⠙⠁⡏⠁⢰⣄⢰⠃⠀⠀⠛⠃⠀⠰⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠁⠀⠚⠃⠀⠀⡄⣸⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⣌⠀⠀⠀⠀⠀⢰⡥⠃⠂⠀⠀⣾⠽⠂⢁⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡀⠀⠀⠀⡇⠀⠀⡴⠀⠀⠀⠀⠀⠀⠨⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠆⣤⡀⠀⠀⠀⠀⠀⣵⠀⠀⢠⠀⡠⠊⠁⠀⠀⢾⡭⠄⠂⠒⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣴⣆⣘⡅⠠⠆⠙⣤⡀⠀⠀⠀⠈⠀⢀⣦⠎⠀⠀⠀⠒⠀⠀⠀⠀⠰⠒⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢏⠁⠘⢍⢡⡀⠀⠘⠃⠀⠀⠀⠀⠀⣰⢏⡀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⠲⠼⠂⠀⠀⢈⢁⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠘⠁⠢⣀⢀⠀⢰⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡛⠀⠀⠀⠀⠀⠀⠀⠈⡉⠘⠋⠓⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⣉⡍⢩⣠⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡘⠂⠀⠀⠀⠀⠀⠉⣠⣄⢰⡀⠀⠀⠀⢠⡤⣄⡀⠀⠀⠰⡔⠉⣤⠇⠀⡠⢫⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡦⢚⡷⠀⠀⠁⢸⠁⠈⠀⠃⠀⠀⠈⢋⡄⠀⢈⠙⠓⠦⣤⣄⠀⠀⠀⠉⠀⠀⠁⠘⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡯⠀⠀⢁⡄⡆⢰⠇⠀⠀⠀⠈⠙⢤⠂⠀⢸⠃⠀⠀⠀⠁⠈⠑⠒⠀⠀⠀⠋⠁⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣶⠴⡶⠋⠿⠀⠘⣻⠀⠀⣴⣄⢰⠄⠀⠀⠰⠀⢀⣬⠂⠀⠦⠉⠂⠀⢠⣤⣅⢠⣤⣤⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠶⡆⠈⠃⠀⠀⠀⠀⣄⠸⣼⠀⠀⠀⠐⠆⠀⠈⠛⠿⠧⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢹⣰⡄⠀⢞⠰⢶⠶⡶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⣀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠁⣼⡄⣿⡇⣤⢰⣄⠠⣦⢰⣦⢰⣿⠐⣴⡤⢆⣴⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣿⡇⣿⡃⣶⣸⣿⢀⣿⢀⣿⢸⣿⠀⣿⠀⠸⣿⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠂⠈⠀⠉⠀⠉⠁⠉⠀⠉⠘⠁⠀⠉⠀⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠈⣻⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠀⠊⠑⠀⢙⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⣀⠄⡀⠀⠀⠀⠀⠀⠘⣰⣤⠔⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠋⠐⠲⠟⡩⢆⠈⠀⠀⠀⠀⠀⠀⢇⣀⠀⢠⡴⠂⠸⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⣢⣬⣤⡀⠀⡄⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⡇⠀⠙⣄⠀⠀⢨⡛⢷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠈⢣⡀⠀⠁⠀⠀⠀⠀⠀⡴⠀⠀⠀⠈⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠱⣤⠁⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠳⣄⣀⣴⠃⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⣀⣤⣤⣄⣀⠀⣀⣀⡄⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠙⡆⠀⠀⠀⠀⠀⠀⣴⣻⠃
⠀⠀⠀⠀⢀⣠⣴⣶⣆⠈⣿⡟⢳⡄⠈⢹⣿⠿⠟⠋⢉⣭⣅⣀⡀⠀⠀⢰⢻⣆⡀⠀⠀⠀⡠⠀⠁⠀⠀⠀⢀⣴⡾⢡⡏
⠀⠀⠀⠀⠈⢛⠻⣿⣿⣶⣿⣭⣬⠍⢠⡀⢲⣿⠻⠇⠈⠳⣮⣛⣄⡈⠀⣿⠘⠿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠁⣼
⠀⠀⠀⠀⠀⠈⢦⡹⣿⣿⣿⣿⣿⠦⣈⠁⣼⠿⠀⠀⠀⠀⠈⣿⣾⢁⣀⡏⠀⠀⣿⣿⣦⣄⠀⢀⣤⣶⣿⡿⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠿⠛⠛⠛⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⣾⣿⠇⠀⠀⠈⠙⢿⣿⣿⣿⡿⠛⠉⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣜⡻⢿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠈⡹⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠸⣀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉
⡌⠛⠛⣱⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠄
⠀⠀⢀⠃⠀⠀⠀⢀⣀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⡸⠀⠀⠀⡎⠘⠛⠉⠐⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡇⠀⠀⠀⠡⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢠
⠀⢸⠀⠀⠀⡀⠀⠑⢄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠃⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜
⠀⡇⠀⠀⠊⠀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠣⠈⠉⠁⡄⠀⠀⠀⠀⠱⢤⠤⠴⡟⡀⠀⠀⠀⠀⡜⠁⠈⡆⠀⠀⠀⢠⠃
⡜⠀⠀⠀⠒⠠⠤⠀⠀⡤⠠⠌⠒⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠚⠓⣄⠀⡏⠈⠙⡆⠀⠀⠘⠢⠾⠁⠀⠀⠀⠀⠢
⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠦⠤⠃⠀⠙⠒⠊⠀⠀⠀⠀⠀⡜⠉⠑⡄
⢰⠋⠈⣷⠀⠀⠀⠀⠈⠙⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠴⠃
⠀⠑⠒⢉⡞⠉⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠉⠙⣦⢸⠀⠀⠀⢠⠂⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠂⠈⢄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠤⠔⠉⠈⠳⠤⠖⠁⠀⠀⠀⠘⢄⣠⠃⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠘⠛⠃⠓⠀⡤⠀⠀⠀⠀⠠⣶
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⣤
⡤⠞⠁⢆⠀⠀⢠⡔⠒⣄⢀⣀⡴⠛⠲⡄⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠐⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠀⠀⠈⢂⠀⠈⠓⠒⠉⠀⠘⣄⠀⠀⣽⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⡀
⠀⠢⠀⠀⠀⠱⣀⡀⡀⠀⠀⠀⠈⠉⠉⠀⡞⠁⠀⠱⣄⡀⣞⠁⢘⠄⠀⠀⢢⠈⠀⢸
⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠙⠦⣄⠴⠁⠀⠀⠁⠁⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠁⠈⡄⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⡀
⠀⠀⡇⠀⡀⠀⠐⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡞⠉⣢⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢰⠀⠀⠀⢀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠰⣄⡠⠇
⠀⠀⠰⡀⠁⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡎⠀⠀⠀⠈⡆⠀⠀⠀⠐⠒⠚⡄
⠀⠀⠀⠈⠂⠤⢀⣀⡀⠄⠊⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀⠀⠀⠑⢤⣤⠤⠞⣀⠀⠀⠀⠈⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⣧⡀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠀⠀⢀⠔⠛⢄⠀⡎⠉⠉⡆
⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⡏⠀⠈⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠠⠞⠀⠘⠒⠚⠁
⠀⠀⠀⠀⠘⡀⢀⡅⠀⠀⠀⠀⠉⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⢉⡇⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡀⠐⠀⠇⢀⠀⠀⢀⡀⢠⠉⠒⢁⡀⢈⡼⡃⢀⠴⣀⠂⠐⠃⠈⠁⣁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠔⣸⠀⠀⠀⠙⠾⠀⢠⠊⡴⠋⢠⠋⠁⠀⠁⠄⠄⠀⠀⣤⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢢⠴⡀⠀⠃⠀⡞⠀⠀⠴⡇⠀⠀⡆⠀⡀⢀⠌⢸⣠⠴⢁⣠⠼⠊⠀⠀⠀⠤⣄⡠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⡀⠙⠤⠀⠰⠁⠀⠀⢰⠃⠀⠀⣧⠀⢡⠊⠀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⢀⣀⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠚⠀⠈⠂⠀⢘⠃⠀⢀⡴⠃⠀⠀⠀⠟⣰⣥⠶⠋⠁⣀⠸⢭⣤⣤⢌⣭⠒⠶⠤⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠎⢀⠖⠁⠀⠀⠀⣠⠴⠚⠉⠀⢐⡀⠀⠀⠈⠁⠀⠀⠘⠑⠈⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠱⡜⠐⠁⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⢖⠦⠤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⠔⠉⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠔⠁⠀⣠⠖⠁⠘⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠐⠊⠀⢀⡠⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠓⠠⢀⠀⠀⠚⠉⠁⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣦⡀⠉⠲⣄⢀⠀⠰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠡⡀⠀⠱⢤⣀⡀⠈⠁⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠢⠖⠢⠒⠚⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣶⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⠀⠟⢘⡩⣿⠈⣿⠉⠸⣯⠀⣿⢠⣾⠙⠋⢿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠻⠗⠿⠀⠟⠀⠀⠾⠃⠿⠈⠻⠆⠀⠘⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈

⡿⢿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⢿