Computing Emojis & Symbols

⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣖⣒⣒⣒⣛⣒⣒⣒⣒⣒⣒⣒⣒⣒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⢶⣄
⠀⠀⠀⠀⠀⣼⠋⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⣿⠀⡶⠒⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⠀⣿⣿
⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡏⣿
⠀⠀⠀⠀⠀⣿⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇⡿
⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⡇
⠀⠀⠀⠀⠀⢻⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⡇
⠀⠀⠀⠀⠀⢸⡇⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣓⣊⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⢀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⢻⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣋⣙⣛⣛⣛⣃⠟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⠤⠤⠤⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⠶⣖⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⣲⣒⣶⣒⣶⣒⣶⣒⣶⣒⣲⣒⣲⣒⣲⡀⢰⠛⣿⠛⣿⠿⡍⢷⠀⠀⠀⢸⣣⠖⠋⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠟⡼⠲⡿⠲⠗⠒⡷⠒⢿⠒⢾⠖⢺⠖⠺⡧⠸⡇⠈⣯⠽⣭⠽⣭⠽⡌⢷⡀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢠⣏⣀⣉⣉⣉⣉⣉⣉⣙⣛⣉⣛⣛⣛⣛⣛⣛⣙⣛⣋⣀⣸⣶⣿⣶⣟⣶⣟⣈⣷⠀⠀⠀⠹⣦⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠸⠧⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠁⠀⠀⠀⠈⠛⠶⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⢶⡦⠤⠤⠤⠴⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡿⠻⠿⠿⠿⠿⠿⠿⠻⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⡷⠶⠶⠶⠶⠶⠶⠶⠶⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⠶⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢚⠃⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣙⡁⣿⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣬⣥⣿⠀⠀⠀⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡖⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⣛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣧⣀⣀
⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠁⢙⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠴⠶⠶⢦
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡿⠥⣽⠥⢼⠯⢬⡯⠬⡿⠭⣿⠥⢽⠥⢽⡯⠬⣯⠬⢿⠤⢽⡭⠽⣷⠀⠀⠀⢸⣀⣠⣄⣀⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⠿⢿⡿⢿⣿⣿⠀⠀⠀⣾⠓⠒⠛⠒⣿⠒⣺⡓⢺⡟⠒⣿⠒⢺⡗⠚⡗⠒⠛⠒⢻⡗⠚⣷⠒⢻⣇⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣤⣾⣦⣼⣧⣼⣿⣿⠀⠀⢰⣏⣉⣽⣉⣹⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⣿⣿⣄⣈⣿⣉⣹⣏⣉⣿⡄⠀⠘⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠬⠭⠭⠭⠭⠭⢭⣭⣽⣿⣟⣛⣓⣒⡶⠶⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣉⠉⠉⠉⢻⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣼⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⠂⠀⣿⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋⠉⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡿⣿
⠀⠀⠀⠀⠀⢠⣶⣿⣭⣤⣤⣤⣄⣠⡴⠟⣿⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡤⢤⡀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇⡟
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⢸⠃⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡇⢸⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣾⢸⠇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⢸⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⣸⠀⡿⠀⠀⢿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⡇⣿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢉⠁⠀⡇⠀⠀⢸⡄⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⡇⠀⢸⡇⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⣄⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⣹⡇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠙⠋⠀⡧⠀⣀⡼⠷⢲⡶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠦⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣬⣥⣤⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⣤⣤⣴⡷⠛⠉⠀⠀⣏⡀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠓⠒⠒⠒⠒⠒⡺⢻⡇⠀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣉⡓⢲⡀⠀⠘⣯⡭⠭⠭⠭⢭⣭⣭⣭⣭⡷⠋⠀⠀⠀⠈⠉⠓⠒⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠿⣿⠯⢭⡭⢭⡭⠭⣍⣙⣛⣛⣛⣛⣒⠒⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣄⣀⠀⠀⠀⠀⣈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⠼⢭⡽⢭⣭⣏⣩⣟⣛⣲⠓⢒⡞⠓⡾⠧⢬⡽⢭⣭⢇⣼⣹⣗⣲⣒⡶⠀⡼⢻⡀⢀⡴⠛⣿⣏⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣀⣉⣉⣚⣓⡒⠓⠴⠯⠤⠽⣭⣩⣏⣉⣟⣓⣲⠟⠒⣲⢣⠾⣴⢧⣴⣁⡜⢁⠞⣴⠏⠀⣼⠗⠉⠱⠒⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠶⠶⠦⠭⣭⣭⣍⣉⣙⣒⡒⠒⠶⠦⠤⠭⠭⣭⣍⣁⣛⣚⣓⣾⠶⢎⡴⢋⡼⠃⠀⠀⢿⣇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠒⠶⠶⠶⠶⠤⠤⣤⣤⣬⣭⣹⣴⠛⠀⠀⠀⠀⠘⣯⣢⣤⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢻⠉⠙⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⡆⠀⢸⡄⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠍⠉⠤⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⣀⣀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⡛⠛⠛⠓⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡾⠛⠛⠛⠛⠋⠉⠉⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⡴⠯⣴⠿⠶⡾⠷⠶⡴⠧⢤⣤⢏⣍⣽⣏⣉⣩⣏⣉⣉⣟⡓⣳⣶⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⡜⠓⡲⠛⢒⠞⠓⢒⠞⠓⢲⠞⠓⢲⡞⠳⢶⡾⠷⢤⡴⠯⢤⡝⡿⠁⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⡟⣠⠞⠓⡾⠛⢲⠟⠒⢲⠟⠒⣲⠟⠒⣶⠟⠶⣶⠿⠶⣤⠾⢤⡴⣫⠟⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⣠⠟⠀⠿⠶⠾⠷⠶⠿⠶⠾⣿⣶⣺⣗⣒⣺⣗⣒⣺⣗⣒⣾⣷⠶⣯⠟⠁⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⢠⡿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠉⠁⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⡇⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠞⠛⠉
⠀⠀⠀⠀⠸⣧⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠒⠶⠶⠶⠶⠶⠶⠶⠼⠷⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢉⣉⣁⣀⣄⣒⣒⣒⣒⣂⣐⣂⣀⣦
⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⢀⡀⠀⡠⠐⠀⢀⣉⠁⠐⢄⠀⠀⣠⢘⠃⠀⠀⠀⠈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⠀⠀⡄⢘⢈⣅⠊⢀⠔⠉⡀⠀⣉⡆⠠⠆⠀⣭⢰⡆⠀⠀⠈⠀⡅⣺
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢼⠀⠀⡆⠼⠀⡀⠀⠼⣀⣀⣀⣀⣤⣰⣀⣼⡆⢠⠠⠆⠀⢀⠀⠀⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⡃⣒⢘⠀⠀⠐⠀⠀⢠⠠⡄⠀⠀⠀⠁⠃⠀⠀⠄⠀⠶⠀⠄⠖⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠁⣭⢘⠀⠀⠀⠀⠀⢸⣅⡎⠀⠀⠀⠀⠗⠀⠐⠇⣄⠺⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠄⡇⠀⠰⠄⡄⠀⠀⠀⣼⡥⣷⠀⠀⠀⠀⡄⠀⠀⠂⣉⠀⠈⠁⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠰⠀⠀⢇⡄⠀⠀⠃⠛⠙⠡⠖⠛⢅⠀⠀⢀⡇⠉⠀⠀⠁⠁⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣁⣉⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⠀⠀⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡨⠶⠤⠶⠶⠂⢀⡈⠛⣿⣛⣟⣯⡿⣿⣿⣿⡺⢻⣺⣖⣶⡷⡿⡿⢿⣭⡿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⡐⡠⠀⠀⠀⠀⠠⠴⠤⠴⠤⠴⠤⠾⠦⡮⢠⠰⢤⠠⢄⠠⢄⡀⣄⡰⡄⠀⣀⢸⣕⡀
⠀⠀⠀⠀⠀⠀⠐⡐⠀⢀⠀⣐⣀⣱⢀⡑⣈⣱⠀⣱⠈⡒⠉⣎⢓⣜⢲⡚⠒⠆⠲⢶⠒⢦⠐⠃⢘⣇⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠒⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣬⣶⣤⣤⣤⢤⣤⣤⢤⣄⣷⡇
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⢀⡄⠈⠒⢤⡑⢌⡓⢤⡀⠈⢗⠤⢺⠠
⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠈⠀⠈⢡⠀⠀⠀⠉⠲⢌⠒⢌⡓⠄⣌⡳⠅⠆
⠀⠀⠀⠀⠙⠛⠿⠭⠽⠯⠟⠝⠝⠴⠵⠴⠶⠔⠶⠲⡶⡶⡖⡖⡔⡖⡖⠶⡶⡶⢶⢾⢿⡾⡶⡳⡶⠿⠗⠃
⠀⠀⠀⠀⠀⠄⠀⠀⠠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡍⠉⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠂⠀⠴⠶⠤⣤⣤⠠⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡞⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠀⠀⡌⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡼⠀⣀⡤⠒⠋⠑⠒⠢⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠀⠠⠩⠐⠀⠀⠈⠉⠀⠀⠀⠀⠀⠇⠀⠀⠂⠀⠀⠘⠀⠀⡇⢸⠁⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣀⠀⠀⠑⠁⠀⠀⠀⢀⠶⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⢀⢀⡇⡈⠀⠀⣸⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⠈⠈⠁⠀⠀⠀⠀⠀⠀⣠⡤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⢸⠃⡇⠀⠀⡟⠀⠀⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠴⠒⠄⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡞⢰⠁⠀⢸⠀⠀⠀⠀⠀⡀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⠀⠀⠀⠁⠈⠀⢀⠰⠖⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣿⠀⠤⠍⠁⡠⠠⢠⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠔⠏⢤⣾⣀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠀⠀⡏⠁⠀⠀⠀⠀⢰⠁⡟⠀⠀⡇⠀⠒⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣤⡇⠀⢠⣇⠀⠈⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀⣀⣀⣠⡴⣞⡁⠀⠀⠈⣠⣖⣀⣀⣀⣀⣀⢠⡤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⢥⡎⢨⡀⠀⢸⠇⠈⠀⠀⠰⠀⢰⠃
⠀⠀⢀⡤⠔⠒⠋⠉⠉⠉⠉⠉⠓⠢⠄⣉⠛⠀⡁⠀⣀⣀⠀⠀⠀⠀⠀⠆⠤⠐⠀⠀⠀⠙⠳⠤⠄⠲⠒⠲⢤⣧⠀⢸⠀⠀⠀⠀⠀⠀⣸
⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡙⢦⠉⠑⠉⠏⠉⠉⢉⠹⠃⠀⠀⠀⠀⠀⠠⠅⠀⠀⠐⡄⠀⠀⠀⡛⢆⢸⠀⠀⠀⠀⣩⠁⣯
⡎⠀⠀⠀⠀⠀⠠⠊⠀⠃⠀⠀⠀⠀⠀⠀⠈⠈⡇⠀⠠⠁⢘⠃⠀⠀⠠⠀⠀⠠⡄⠁⡰⠂⠂⠀⠀⣀⠤⠖⠊⠀⠘⣾⣀⣀⣀⣀⣀⣠⡏⢧
⢇⠠⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠠⢠⠇⢀⠂⠀⠖⠀⠈⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣤⣠⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣧
⠘⢦⡁⠀⠀⠀⣸⠁⢀⢈⢻⡀⠀⠀⠀⢐⡵⠋⠛⠛⣛⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣸⡊⠅⠀⠀⠠⠶⠶⠄⠀⠀⠀⠀⠀⣿
⠀⠀⠙⠲⠤⣄⣼⠀⠀⠀⢸⡧⠄⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⢽⠉⠉⠁⠀⠉⠉⠻⠤⠬⠛⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠋
⠀⠀⡀⠂⠀⠀⠸⡇⠀⠀⠘⡧⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠙⡖⠒⠒⢒⣇⠀⠀⠀⣷⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣸
⠀⢰⠞⠉⠉⠉⠉⢹⢀⣀⠀⣿⠀⠀⠀⠠⠈⠉⠙⣦⠒⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠈⠣⣄⣀⣀⣀⠊⠁⢀⣤⣟⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⢀⠈⡇⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠉⢱⡆⠉⠉⡏⠉⠉⢨⡟⠛⠁⠀⠀⠀⣠⣁⣠⡇⠀⠀⠀⠀⣧⠀⣀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⡃⣉⠀⣇⠆⡀⢾⡁⠀⠀⠀⠀⠀⡹⠁⠀⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠐⡇⠠⠀⡟⠢⠰⣏⡀⠃⠀⣄⠠⠔⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⢈⡇⠠⠀⡇⠰⠀⠩⡈⡁⠀⠺⢿⠛⠻⠶⠶⠶⠶⠶⠿⠿⠿⠿⠏⠛⠃
⠀⡀⣀⣀⠀⡀⡀⢈⡇⠒⠀⣏⣀⣡⣄⣤⣡⠶⠴⠶⡀
⢾⣁⣀⣭⣭⠭⠭⠥⣤⡄⢤⣤⠔⠐⠒⠢⣤⡤⢠⣶⣾

⠀⠀⠀⣠⢤⠆⠀⠀⠀⠉⠉⠀⠉⠉⠉⠀⠀⠁⠀⠉⠉⣉⠉⠙⡆
⠀⠀⢀⡃⢸⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⡇⠀⡇
⠀⡴⢹⡇⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⡟⢹⡇⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⡇⢸⡅⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣻
⠈⡇⢸⡯⢸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠐⢇⡐⡇⢼⡇⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠑⢯⣈⡅⠀⠁⠑⠠⠤⠀⠀⠀⠤⠀⠤⠄⠀⠀⠀⠀⠃⠀⢹
⠀⠀⠀⠀⠙⠣⣤⣤⣤⡤⡠⣤⡤⣤⣠⠄⡤⡤⡤⣡⣬⣬⠀⠅⠊
⣴⢦⡀⠀⠀⠀⢀⣦⣍⡁⠙⠛⠪⠁⠈⠈⠙⠉⠉⣉⡯⠑⠒⠤⣀
⣿⠀⠙⠢⣄⠀⠀⠈⠉⠙⠛⠛⠒⠂⠐⠒⠒⠛⠋⠀⠀⠀⠀⣀⡀⡉⡒⠦⣄⡀
⡗⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⢲⢠⡇
⠳⣄⠑⠕⢸⠁⠉⠀⠘⠛⠛⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⢺⢸⡇
⠀⠈⠢⡀⢸⠉⠉⠉⠉⠀⠀⠉⠉⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢨⢨⡅
⠀⠀⠀⠈⠺⢄⣄⡀⣀⣀⣀⣀⣀⢀⣀⣈⣉⣁⡁⢁⢀⣀⣀⢀⣀⣀⠈⣼⡀⠇
⠀⢀⣤⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⡀⢀⣀⣤⣤⣤⣤⣄⣠⣠⣄⣀⣀⣀
⠀⠀⠓⢝⡪⣒⢝⡐⠾⠓⣀⣷⠀⣀⣟⠀⠺⡓⢀⢹⡒⢒⢟⡒⠒⢟⣒⡺⠷⠤⠭⡮⠽⠷⣤⡀
⠀⠀⠀⠀⠑⢮⣓⣭⡲⠀⠀⢿⡐⣈⢻⡂⢁⠲⣌⣃⣀⣈⣃⣀⣈⣃⡀⠻⠅⠲⠙⠉⢺⣛⣻⣛⣷⢤⣀
⡀⢀⣀⣀⡀⣀⣈⢳⣿⣥⣄⣤⣤⣸⣫⣠⣭⣠⢎⡽⠣⠷⠃⠘⠽⢏⠬⢭⣍⣨⣁⠃⠎⠞⠻⠡⠥⠪⠝
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⠛⠓⠶⠶⣤⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡿⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⣾⠋⢰⡾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⠾⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⣿⠀⢸⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠉⣙⣛⣛⡓⠒⠒⠒⠒⠒⠃⢀⡏⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⣿⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⠃⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡟⠀⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⠒⠒⠚⠛⠛⠀⠀⢀⡟⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢀⡟⠀⣾
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢿⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⢸⣇⡴⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⣸⠃⠀⢀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠩⠉⠉⢩⠉⠉⠉⠛⠛⠛⠛⠛⠀⠀⡿⠀⣠⡿⠉
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣀⡴⠋⠈⠉⠉⠉⠛⡟⢛⡿⠛⠓⣷⡒⠶⠶⠶⠶⢶⣦⢤⣤⣤⣤⣤⣴⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⣠⠖⠚⠁⢾⣇⡀⠀⠈⠙⠛⠓⠒⠒⠛⠁⠀⣠⡿⠀⠈⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣧⣀⣀⣀⠈⠻⠯⣿⣶⣶⣶⣶⣶⣒⣒⣾⡽⠟⠁⠀⠀⠀⠀⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡤⠤⡤⢤⡤⠤⣤⢤⣤⣄⣈⣉⣉⣉⡉⠙⠛⠛⠛⠷⠾⠯⢭⣭⣥⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠋⡿⠓⡾⠓⣲⠓⡶⠧⢤⠧⢤⡯⣤⣽⣉⡟⢀⡖⢲⠖⢦⠤⢤⡀⠀⢩⠟⣇⠀⠀⠀⠀⣠⡴⢶⢞⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⣉⡽⠉⣹⠛⢲⠟⢲⠓⢲⠿⢤⡾⠤⣼⢥⣼⠁⡜⢉⡟⢒⠟⠒⡾⠁⡰⠋⣴⠏⠀⠀⣠⣞⣁⣠⣧⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⠤⠬⣉⣉⣉⣙⣛⣒⠛⠒⠛⠲⠾⠤⠴⠧⠴⠃⢸⣉⣽⣉⡟⠓⡾⢀⡜⣡⠞⠁⠀⠀⣰⠋⠀⠀⠀⠈⠉⠲⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠓⠒⠶⠶⠶⠯⠭⣭⣭⣉⣓⣒⣒⣒⠒⠒⠦⠤⠤⢭⣭⣉⣠⢋⡼⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠲⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣴⣲⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⢰⠒⠒⠒⠒⠒⠒⡆⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⢹
⠀⢸⠀⠀⠀⠀⠀⠀⡇⢘⣀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠸⠤⠤⣤⣤⠤⠤⠏⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠛⠀⣤⡀⠀⠀⢀⣀⣠⠖⠒⠢⣤⣄⡀⠀⠀⠀⠀⣠⠄⠀⢸⣦⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠏⢀⠔⠒⠉⠑⠒⢤⡙⠲⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡜⠀⠀⠀⠀⠀⠀⢳⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠤⠬⠭⠭⠭⠭⠭⠭⠭⠤⠴⠃⠀⠀⠀⠀⠀⣀⣠⡤⣤
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⠀⠴⠃⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⡷⠒⠚⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣮⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣧⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⣠⣤⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⠟⠃⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡇⠀⠀⢀⢀⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠖⠞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣀⠀⠀⠀⢀⡀⠀⣀⡀⠀⣀⡀⢀⣀⡀⠀⣀⡀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⢀⣬⠉⢠⣭⠁⠤⠤⠀⠤⠄⠀⠤⠤⠀⠰⠦⠀⠒⠂⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠒⠂⠀⠐⠒⠀⠒⢀⣄⣚⣃⣀⣈⠉⠀⠈⠉⠀⠈⠉⠁⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⣼⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣶⣶⣶⣶⡾⠧⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⠤⣤⣤⡷
⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢠⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⡒⢦⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡧⠄⠠⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣧⠄⠀⠀⠠⠤⠤⢨
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠁⠀⠀⠀⠀⣷⡀⠀⢀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠁⠀⠇⠉⠐⠀⠀⠀⠀⠀⠊⠁⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠃⠀⠐⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⡀⣀⣀⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠤⠤⠤⠤⠤⠈
⠀⠀⡀⠀⠀⢀⠀⢀⠀⡀⠀⡀⠀⠀⢀⠀⣀⠀⣀⡀⡀⠀⠀⢀⠀⣀⠀⠀⠀⡀⠀⠀⢠⠈⡥⣄⠀⢆
⠀⢰⠁⠒⠒⠒⠒⠂⠐⠀⢒⡐⡒⢀⡒⠒⢂⠐⠒⠐⠀⠒⠒⣒⣁⠂⠂⠀⠂⠘⠀⡀⠈⠉⠑⠚⠒⠊
⠀⠈⠉⠉⠉⠀⠉⠉⠈⠉⠉⠉⠉⠉⠉⠁⠉⠉⠁⠉⠉⠉⠀⠁⠉⠁⠉⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣹⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⣏⡀
⠀⠀⠀⠀⠀⢀⡼⢋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⢦⡀
⠀⠀⠀⢀⡴⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢦⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠳⣆
⠀⠀⠀⠀⢠⡞⠉⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠚⢻⡄⠀⣿
⠀⠀⠀⠀⡟⣷⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹
⠀⠀⠀⠀⣿⢹⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⢻⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⢸⡜⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⢸⡇⣷⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⢸⡇⢿⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡆
⠀⠀⠀⠀⠀⣇⢸⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠸⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡄⣧⠀⢸⡆⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠴⠶⠶⠶⠶⠶⠖⠒⠒⠛⠛⠛⠛⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠸⡇⢻⠀⠈⠛⠋⠉⠉⠉⢉⣉⣉⣁⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠤⠤⠶⠶⠶⠶⠖⠒⠒⡚⠛⠛⣧
⠀⠀⠀⠀⠀⠀⣧⣼⡶⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⠤⠤⡤⠬⡤⠤⡤⠤⡀⠘⣧
⠀⠀⠀⠀⠀⠀⢻⡀⢷⡀⠀⢠⡤⠤⠴⡖⠒⢲⠒⠺⡍⠉⣏⠉⢻⣀⣸⣄⣀⣱⣤⡬⣦⠤⢷⠤⠷⡤⢷⠶⠳⡀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⢷⠈⢧⠀⠀⢻⡤⠤⠷⡖⠚⢷⠒⠻⡉⠹⡍⠉⢣⣀⣹⣄⣀⣧⠤⠼⡦⠐⢷⠒⢳⠚⢷⠚⢳⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⢷⠀⠀⢳⡤⠤⢷⠤⠞⢷⠚⢻⠉⠹⡉⠉⢇⠀⢱⡀⠈⣆⣀⣘⣄⣘⣆⠬⢧⠬⡦⠴⢧⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⢷⡀⠈⢷⣀⣀⣧⣀⡸⣦⠤⢧⠄⢷⠒⠚⣏⠋⢿⠉⠉⡏⠀⠹⡀⣹⣀⣈⣆⣼⣤⠬⠇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⢷⡀⠈⢯⣁⣈⣇⣀⣨⣦⣬⡦⠼⠧⠤⠾⠶⠾⠗⠒⠛⠛⠋⣉⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠈⣧⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠖⠒⠚⠛⠋⠉⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢀⣀⣀⣀⡀⠤⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠭⠭⠭⠥⠻⠒⠒⠒⠒⠚⣁⣀⣀⣠⣤⣤⣴⣶⣒⣛⣯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠘⢧⡀⠀⢀⣀⣀⣀⣠⣤⣤⠤⠤⠶⠶⠶⢒⣒⣛⣛⣿⣭⡭⠭⠿⠶⠚⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢯⣭⣭⠭⠭⠽⠶⠶⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⠤⢤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠤⠤⠤⠭⠭⠍⢉⣉⣉⣛⣛⡛⠛⠛⠛⠛⠿⠷⠶⠶⠶⠶⢶⣶⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠒⠒⠒⠂⠠⠤⠤⠬⠍⢹⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⢀⠀⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡠⠚⠀⠀⠀⠀⢸⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠨⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⡉⠐⡒⠢⠤⠭⢉⣁⣒⠒⠠⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠡⢄⠘⢠⠆⠊⠠⠔⣀⡀⠄⢀⠬⠍⣑⠒⠒⠲⠠⠭⢉⣐⣒⡒⠂⠤⠤⢀⣀⣀⡀⠀⠀⠀⢸⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⡐⠢⠉⠔⢂⡠⠀⠁⡂⠀⠃⠠⠆⢒⡠⠉⣁⠄⠮⣈⠡⠒⠀⢀⡄⠡⢉⠉⠐⠒⠂⠬⠭⠍⣉⣒⠚⢸⣀⡇
⠀⠀⠀⠀⠀⣀⡤⠚⠁⡠⠤⠉⡀⠐⠉⠠⠂⠐⡠⠁⡈⡀⠘⠃⠨⠔⠀⡤⠒⠉⡄⠀⠁⠀⠰⠄⢀⠰⠀⠁⡀⠀⠰⠤⠀⠰⠀⢀⡠⣼⣏⣡⠇
⠀⠀⠀⠀⠀⡏⠉⠑⠒⠒⠤⠭⠄⣚⣀⠈⠒⠀⢔⣈⠀⠀⠀⠀⠉⠀⠒⠀⡤⠀⡀⠀⠘⡀⠀⠘⠀⣤⠀⠈⡅⠒⠊⢀⡠⠔⢊⣵⠶⠚⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠢⠤⠤⣄⣀⣉⠉⠙⠒⠒⠤⠤⢤⣀⣐⡂⠂⠡⠔⠀⢤⠄⢈⢀⠒⠠⠀⠖⠀⢁⡠⠔⠊⣁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠤⠤⣀⣀⡈⠉⠑⠒⠒⠦⠤⠥⣀⣀⣁⡠⠔⠊⣁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠢⠤⠤⣄⣀⣏⡤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡤⠤⠤⠤⠄⠀⠀⠀⠠⠤⠶⠶⠶⠤⡤⠀⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⢒⣀⡈⠉⠁⠤⠄⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠚⠛⠓⠚⠚⠚⡘⡙⠋⠙⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⢁⠂⢡⢂⡕⠐⠌⠠⠰⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠂⠂⠀⢢⠁⡐⡀⠠⢡⠞⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠂⠌⠀⠜⠠⠡⠐⠠⢁⡞⡐⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡐⡀⢈⢈⠀⠁⠀⢁⡞⣠⠊⣷⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡐⠐⡜⢠⠂⠀⠀⢀⠁⡾⢠⠇⢌⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⠐⡁⢠⠇⠈⠄⢀⠂⡼⢡⠏⠀⠂⢴⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡐⢀⠎⠀⠌⠀⠀⡼⠁⡟⡄⠀⢤⢾⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠡⠠⠠⡊⠐⠀⢄⠄⣸⢁⠞⡐⡐⢔⢣⢺⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢠⠱⢡⡑⠀⡐⠀⠂⡰⢃⡎⠌⡐⢈⠀⠢⢻⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠂⠠⠠⡐⠀⠄⠌⢘⠃⡜⠙⠐⠖⠀⠈⠦⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠂⠃⠰⠡⠠⠔⠌⠀⠦⠮⠘⠀⠌⠌⠄⠂⠞⢼⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡂⣐⣆⣀⣀⣉⣉⣉⣁⣀⣀⣀⣀⣒⣒⣒⣒⣒⣓⣒⣓⣒⣒⣒⣒⣒⣓⣓⣒⣒⣒⣶⣾⣿⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠄⠁⠠⠤⠤⠤⢀⠀⢀⡀⢀⣉⡙⣿⠍⢭⡮⠍⢭⠭⠨⠭⡉⠭⠭⠭⠭⠭⠭⠬⠭⠵⢤⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠊⠀⠀⠀⠠⠤⠤⣤⠤⢥⠤⠮⡤⠤⠤⡤⠼⠀⡁⠥⢴⠦⠧⠈⢀⠵⠤⠭⡬⠦⠄⠤⠤⠀⢥⠀⠻⣄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡌⡠⠁⠀⠀⠀⢀⣀⣀⣰⢀⡀⠬⣀⠠⠆⢤⠤⠤⠤⠤⠧⢠⠀⠦⠨⡩⠈⢦⠳⢂⡑⡄⣀⣀⠀⢀⡀⢦⠓⠘⢦⠱⡀
⠀⠀⠀⠀⠀⠀⡜⡐⠁⠀⠠⠤⠄⠤⠬⢠⠄⠜⠀⠀⠦⠀⠠⠼⠤⠤⠠⠄⠠⠠⠤⠤⠤⠬⠬⠦⢁⠤⠵⠶⠬⣦⣄⠋⠢⠤⢤⣥⢌⢷⣳⡀
⠀⠀⠀⠀⠀⡜⠀⠤⠀⠀⠀⠀⠀⠠⠤⠶⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠤⠠⠀⠷⠂⢈⠨⣟⢮⣠⢥⣬⠢⢥⠤⣴⣏⢣
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⠉⠉⠈⢣⠀⠀⠀⠀⠀⠁⠢⢅⡂⢔⡨⢑⠢⣌⢀⠈⡀⢣
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡒⢅⠢⢐⠡⢀⠌⠆
⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣅⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣀⠬⡂⠅⠘
⠀⠀⠳⡤⠠⢤⢄⣀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣊⣱⡾
⠀⠀⠀⠀⠉⠉⠛⠓⠛⠛⠚⠚⠛⠐⠂⠲⠂⠊⠴⠳⠳⠓⠣⠞⠞⠞⠰⠳⠷⠓⠒⠲⠾⠶⠾⠂⠎⠖⠰⠛⠴⠣⠳⠶⠛⠂⠋⠒⠚⠚⠛⠛⠉
⠀⠀⠀⢀⠀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠠⢀⠠⣀⠠⣀⠀⡀⠀⢀⠠⣠⠤⢀⠠⠀⠀⠀⡂⠄⢐⠀⠀⡀⠀⣀⠠⢐⠠⢀⠠⢀⠠⠀⡀⠀⡀⢀
⠀⠀⠀⠀⠁⠉⠂⠉⠀⠀⠁⠀⠩⠀⠈⠀⠠⠀⠀⠐⠈⠀⠀⠐⠉⠐⠨⠀⠈⠀⠂⠁⠐⠨⠀⠊⠅⠂⠡⠐⠠⠐⠀⠄⠀⠅⠂⠡⠂⠁⠂⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢨⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠉⠉⡇⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⡀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⣿⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⢿⡟⠀⠀⠀⠀⣿⣷⡅⠀⠀⠀⠀⣿⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⠀⢀⠀⠀⠀⠉⢀⡀⠀⠡⠀⣿⠀⠀⠀⠘⠋⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠰⠀⠀⠀⠀⣀⣀⠠⡠⠚⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⠀⠠⠤⠀⠀⠀⠀⠄⠀⠐⠂⢀⡇⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠁⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠀⠠⠶⠀⠀⠀⠀⢠⠄⠀⠀⠀⣤⠀⠀⠀⠀⠀⣰⠤⢅⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⢥⠘⠂⠤⠤⠤⢄⣀⣀⣀⣀⣀⣀⣈⣀⣀⡁⣀⣀⠀⠠⠞⠁⠀⠀⠉⠚⡤⢾⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠰⠥⠮⠼⠀⠀⠐⠃⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠴⠀⠀⠀⠄⣶⣰⠆⠂⢀⠔⡇⡇⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⢀⣀⣈⣈⡀⣀⣀⣁⣉⡉⣀⣀⣀⣀⣂⡀⢀⣒⣀⣀⠀⠀⠀⠐⠒⢂⣀⡇⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢹⠇⠀⡼⣧⠽⣬⢷⡭⢦⡵⣮⠷⣭⢦⡽⢮⠿⣎⢷⡘⠀⢰⡿⠹⡄⠀⠀⢸⣁⠕⠀⠀⠀⠨⡀
⠀⠀⠀⠀⠀⠀⠀⡜⣡⠋⢸⣧⢻⡜⣯⡝⣮⢻⣭⢳⡽⣮⢽⣭⢷⡽⢯⡽⡮⠷⠀⣼⠇⠀⢸⠀⢐⠸⢁⠎⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡜⠰⠃⢤⢤⠾⠤⠤⠴⠷⠴⠷⠠⠾⢠⡄⠀⠾⠇⠼⠇⠮⠇⠼⠅⠿⢀⠀⢠⠂⢀⡳⣷⣤⡀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⡌⠀⠁⠀⠘⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡠⠋⠉⠉⠁⠀⠉⠉⠁⠀⠈⠈
⠀⠀⠀⠀⠀⠂⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⡳⠂⠀⢠⠎⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡼⠁⠀⢀⠴⠚⠛⠛⠃⠀⠀⠈⠙⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢠⠏⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡞⠀⠀⠀⠀⣀⣤⡤⠄⠀⠠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢠⠃⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡞⠀⠀⢠⠔⠊⠙⠿⡇⠀⠀⢰⠁⢰⣷⣄⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⠃⠀⠀⠀⢠⠏⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡜⠀⠀⠀⠉⠉⠉⠙⠒⠅⠀⠀⢸⣀⣀⣉⣈⡗⠀⠀⠀⠀⠀⡼⠀⠀⢠⠇⠀⠀⠀⢀⡎⢠⢠⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣸⠁⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⣸⠁⠀⢀⠏⠀⠀⠀⠀⡼⠀⡞⡸⢸⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠇⢰⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡞⠀⠀⠀⠀⢠⠃⠰⢡⠃⡞⡸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠀⠈⠳⢄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢀⡠⠂⠀⡞⠀⠀⡸⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡏⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⣸⠁⠀⢠⠃⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠳⢤⣀⠀⠀⠀⠙⠲⠤⢤⣤⡤⠤⠖⠊⠁⠀⠀⠀⠀⢀⡏⠀⠀⡜⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠈⠙⠒⠦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢠⠃⠀⠀⠀⠀⢰⠁⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⡀⠀⠀⠉⠉⠓⠒⠲⠦⠤⠤⢤⣄⣀⣀⡟⠀⠀⡼⠀⠀⠀⠀⠀⣮⡤⠖⠊⠉
⠀⠀⢀⣠⠤⠴⠶⠆⠒⠤⢄⣀⣀⡤⠶⠤⢤⣴⣒⣒⣉⠉⡒⢦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⣠⣤⠖⠳⡄
⢀⡔⠉⠀⠀⠀⠀⠀⠈⠂⣠⠝⡡⠞⢉⣕⠒⡢⠬⢭⣁⣒⠓⠮⠤⠧⢀⣉⡙⠿⠶⢒⣶⣦⣤⠤⠤⡼⠗⢻⡁⣠⠇⠀⢀⣟⠢⡄
⢸⡀⠀⠀⠀⠀⠀⢀⡴⢋⠕⠛⠠⠤⢀⡀⠈⠁⠐⠓⠢⠌⠉⣉⡛⠐⡲⠀⡨⠭⣍⣙⣒⠒⠢⢬⢍⠉⠉⣉⣉⣠⠤⠒⢁⡏⢱⢸
⠀⠙⠒⠢⠤⠖⢾⠁⠀⠓⠠⠄⣀⡀⠀⠀⡡⠋⠒⠴⠁⣀⡐⠁⡠⠋⡠⠚⠀⠴⠁⢀⡴⢃⠔⠁⠀⢯⣉⣀⣀⣀⡤⠔⢋⡠⢋⡜
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢄⣀⠀⠈⠑⠚⠢⠤⣈⣀⠀⠀⣠⠟⣠⠾⢁⡠⠀⢀⡴⢋⠔⣁⠀⠤⠤⠮⠤⠤⠤⠔⠒⣚⡩⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⣀⡀⠀⠈⠍⣚⠁⠴⣁⣠⠊⠀⡰⢋⠔⠁⠀⠓⣲⠖⠂⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⣈⡁⠒⠨⠭⣉⠐⠁⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢬⣄⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠀⠀⠀⠠⠤⢄
⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠁⠀⠈⠉⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⣀⢀⡀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⡊⠀⠀⠠⠤⠬⠭⠤⠤⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⢀⣀⡀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⣀⢎⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⠤⠤⡤⣤⣤⣤⣤⡄⠑⠄⡀⠀⠀⠀⠀⠀⠀⡀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠀⣄⠈⡀⠸⡀⠀⠀⢀⠀⠀⠠⡀⠀⠀⠁⠉⠉⠈⡀⠉⢉⡉⠉⠉⠙⣷⠀⢀⠂⡠⠊⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠁⠈⠀⠇⠀⠀⢎⡖⠈⢶⢰⠀⠀⠀⠀⠀⠀⡖⠛⠛⡀⠀⠀⠀⣏⠐⠃⠁⠀⠀⢀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⡔⠊⠁⠀⣀⣀⠀⠀⠉⢆⠀⢰⠀⠀⠀⠙⢣⣠⡎⠐⠀⠀⠀⠀⠀⠀⢆⠀⢀⠀⠀⠀⠀⡇⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢈⠀⣀⣀⣈⢣⢬⠝⠑⠂⡀⢀⣀⣀⡔⣏⣻⣏⣒⣠⡀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠉⠉⠉⠁⠀⢀⡀⠈⠀⠀⠀⠠⠂⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⠀⠀⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠑⠀⠠⠀⠀⠀⢀⡀⢇⠀⢸⠀⠀⠀⠍⠓⠚⠐⠀⠀⠀⠀⠀⠀⢰⡉⠀⠈⢓⡆⠀⠀⣿⠀⠈⠐⠄⠀⢀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠓⢸⠀⠀⠀⢈⡦⣤⣎⠀⠀⠀⢰⠀⠀⠈⣛⢦⡴⢏⡃⠀⠀⣿⠀⣤⣊⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⠀⠁⠐⢂⣤⣼⣦⣤⣼⣤⣤⣬⣬⣦⣴⣥⣬⣤⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡨⠥⠬⠤⠨⠌⠬⠹⠏⠼⣽⣻⢏⣿⡻⢟⡚⠉⠛⠻⠏⣍⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡐⠀⠀⠀⣀⣉⣎⣹⣉⣋⢹⡉⢏⠿⣿⢝⣿⢍⠉⢁⡌⠏⠙⢯⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⡀⠐⣀⣀⣀⡀⣈⣐⠚⡚⠀⠘⣒⠓⠚⠒⠚⣒⣳⣎⣂⣈⣓⣘⣙⣈⣻⣳⡄⠀⠂⠂
⠀⠀⠀⠀⠀⠈⠐⠀⠂⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠉⠉⠉⠙⠛⢢⠀⠀⠀⠁⠉⠈⠉⠉⠉⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⢙⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠓⠓⠛⠛⠛⠷⠿⠷⠿⠷⠾⠾⠿⠮⠿⠶⠾⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⡀⢀⡀⠀⡠⠀⠀⠀⠠⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⠸⠭⠭⠭⡇⠀⠀⣿⣿⡇⣶⣶⣶⣶⣶⣶⣶⣶⣶⢸⣷⣒⣚⣒⣾⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠨⠭⠭⠭⠇⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣒⣒⣒⣿⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣶⣶⣶⡆⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡏⠛⠛⠛⢹⣿
⠀⠀⠘⠓⠒⠒⢒⣦⡒⠒⠒⠒⠋⢸⣿⣭⣿⡇⠀⠀⣿⣿⣧⣭⣭⣭⡭⠽⢭⣭⣭⣭⣼⡇⠀⠒⠀⢸⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⣀⡀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⣀⣀⡀⣠⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠃⠀⠐⠒⡄⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⣇⠉⠁⠁⠉⠉⠉⠁⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⡇⡤⠤⠤⠄⠀⠀⠤⠄⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠁⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡼⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠠⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠂⠀⠀⠀⠀⠀⡤⠤⠠⠠⠄⠤⠤⠤⠤⠤⠤⠴⠭⠼⠁⠀⠀⠀⠀⠀⠀⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⠀⠀⠉⠀⠀⠀⠀⠀⡀⣀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠦⠠⠠⠄⠀⠀⠠⠄⠤⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠤⠤⠤⠤⠤⠴⠤⠤⠔⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠒⢂⠀⠐⢠⡀
⠀⠀⡰⠁⡄⡀⡀⠂⢀⡐⠀⡐⠂⢀⠒⢀⡐⠄⡄⠲⢀⠒⠐⡒⠒⣐⠒⣀⣂⣀⣒⠆⣐⠒⠀⠒⢀⠀⡆⠠⠀⠘⠯⠤⣏⣙⣆⢀⡰
⠀⠰⣧⣠⣠⣤⣍⣤⣬⣠⣤⣥⣤⣭⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣠⢅⣤⣭⣤⣤⣍⣀⣀⣁⣠⣬⣤⠄⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⡿⠛⠛⠛⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠠⠀⢾⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⣿⡟⠙⠋⠀⠀⠀⠈⠛⡙⣿⡇⠀⠀⠀⢸⣿⡀⢸⣿⡇⢨⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣾⣿⣿⣇⣀⡅⠀⠀⠀⠀⣄⣀⣿⣧⣄⣀⣠⣾⣿⣿⣦⣷⣼⣿⡿⣛⣛⣛⢷⣄
⠀⠀⠀⠀⠀⣴⡟⣭⠉⠉⠉⠩⡙⢿⣿⣿⣿⡿⢛⠛⢿⣿⣿⣿⣿⠟⣛⣛⣛⠻⣿⣿⣿⣿⡇⣿⣿⣿⢼⣿⣦
⠀⠀⠀⢀⣾⣿⠁⠈⠖⣰⣆⠾⠀⢸⣿⣽⣿⡇⣬⡄⢸⣿⣇⠀⠀⠀⣿⣿⣿⠀⠀⠀⣸⣿⡇⠀⠀⠀⣾⣿⢿⣷⡀
⠀⠀⠀⣾⣿⣿⣤⣼⣿⣿⣿⣿⣦⣼⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣇⠀⢀⡀⣿⣿⣾⣿⣷
⠀⠀⢸⣿⢩⣭⣭⣭⣭⣭⣭⣭⡍⣿⣿⣿⡇⣶⣶⣶⣶⣶⣶⣶⣶⢸⣿⣿⠿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇
⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣧⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣬⣼⣿⡇⣿⣿⣿⣿⣏⣽⣿⡇⣿⣿⢸⣿
⠀⢰⣿⣿⢼⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠇⠛⠛⠛⠛⠛⠛⠛⠛⠸⣿⡿⣿⣿⣿⣧⡿⠟⠻⠿⠿⠿⠿⠇⠿⣿⣼⣿⡆
⠀⢨⣿⣿⣶⣶⣶⡖⠒⢲⣶⣶⣶⣿⣿⣿⣿⣿⡿⠿⠿⠶⠶⢶⣶⣿⣿⣷⣿⠟⠛⠻⢿⣿⣿⣿⣿⢿⣿⢿⡿⢿⣿⣿⡅
⠀⠘⣿⣿⠛⣿⣿⡶⠶⡶⠿⣿⣿⣼⣿⡟⠃⣤⡃⠀⠀⠀⠀⢸⣤⠛⢿⣿⠁⠀⣤⠀⠈⣿⣿⣿⣿⢸⣿⣿⣿⢸⣿⣿⠇
⠀⠀⣿⣿⣤⣿⢏⣾⣿⣿⣿⣮⢿⣿⣿⣧⣠⣿⣷⣄⣀⣀⣠⣾⣿⣀⣼⣿⣦⡀⠀⢀⣴⣿⣟⣿⣿⢸⣿⣿⣿⢸⣿⣿
⠀⠀⠸⣿⣿⣿⢼⣿⣿⣿⣿⣿⠾⣿⣿⢿⣿⡿⠿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣛⣀⣛⣿⣿⣿⣿⣿⢸⣿⣿⣿⢸⣿⡇
⠀⠀⠀⢻⣿⣿⠀⢹⣿⣿⣿⠃⢈⣿⣷⣾⣿⡆⠀⠀⣠⠀⠀⢸⣿⣿⣬⣿⡿⠿⠿⠿⠿⠿⢯⣽⣿⣘⣛⣛⣛⣼⡟
⠀⠀⠀⠀⢻⣿⣶⣼⣿⣿⣿⣴⣾⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⡇⣾⣿⣿⣶⣶⣾⡟⢉⡉⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠻⣿⣿⠛⣿⣿⡏⠈⢩⡍⢹⣿⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡇⢴⡆⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠉⢉⡉⠉⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣶⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⡶⠐⣶⣿⣻⣿⡟⡛⣿⠀⠙⠛⠀⣿⢛⢻⣇⣉⣉⣉⣉⣉⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⡇⠁⣿⠀⠀⠀⠀⣿⠈⢸⣿⣟⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣶⣿⣿⣿⣿⣿⣿⣶⡾⠿⠟⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣺⣿⣿⣷⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢠⢮⡾⠛⠉⢟⡻⠉⠛⢷⡷⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣦⣤⡶⠶⠚⠻⠿⣧⣠⢔⡾⢷⡢⣄⣼⠿⠟⠒⠶⢦⣤⣴⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⣶⣄⠁⠀⠀⡸⡳⠁⣨⣅⠈⢞⣇⠀⠀⠈⣠⡲⡟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠄⠙⡻⢎⣤⡶⢽⡷⢋⣉⣉⡙⢾⡯⠶⢤⣱⢟⠋⠠⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣿⡆⠀⠀⣠⡯⡮⣿⡟⡔⣵⢚⡓⣮⢧⢻⣿⣽⢿⡄⠀⠀⢰⣿⣲⢄
⠀⠀⠀⠀⠀⠀⢀⣞⣾⠋⠉⣻⢶⡪⢷⡷⢛⡭⡁⠀⣑⡃⢘⣊⠁⢈⠭⡛⢾⡾⢗⡶⣟⠉⠙⣷⣳⡀
⠀⠀⠀⣀⣀⣀⣾⣾⣧⢄⣐⡵⡁⣠⡏⢰⡱⢛⠵⠼⢿⣷⣾⡿⠇⠮⡛⣎⡆⢹⣄⢈⢮⣂⡠⣼⣷⣷⣀⣀⣀
⠀⠀⠀⠉⠉⠉⢿⢿⡟⠊⠹⡳⡁⠙⣇⠸⡹⣬⡲⢰⣾⡿⢿⣷⡆⢖⣥⢏⠇⣸⠋⢈⢾⠍⠑⢻⣿⡿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⢯⢿⣄⣀⣽⠾⢵⡾⡷⣭⣲⡁⢀⡩⡅⢨⣍⡀⢈⣖⣭⣾⠷⡮⠷⣯⣀⣠⡿⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠯⣿⠇⠀⠀⠘⣇⣗⣿⣧⠧⡻⢬⡥⢟⡜⣼⣾⣻⣺⠃⠀⠀⠸⣿⠽⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠂⣠⣵⢎⠛⠷⣺⡷⣌⣉⣉⣡⢾⡟⠶⠚⡳⣮⣄⠐⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣼⠾⠋⡀⠀⠀⢱⡵⡀⢙⡋⢀⢮⡎⠀⠀⢀⠙⠷⣧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠟⠛⠷⠶⢤⣴⣶⡟⠙⠪⢳⡾⠗⠋⢻⣶⣦⠤⠶⠾⠛⠻⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠘⢞⢷⣤⣀⣮⣵⣀⣤⡾⡳⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢽⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⠿⠿⠿⠿⠿⢿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠿⠛⠛⠻⠿⠿⢶⣶⣶⣦⣤⣄⣀⠀⠀⢈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⢀⣠⣤⣶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣷⣆⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣴⠿⠛⠉⠀⠀⠀⠀⣀⣤⣤⣶⡶⠶⠶⣶⣶⣦⣤⣄⡀⠀⠀⠙⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣤⡶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠠⡀⠀⠀⠀⠉⢻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⢀⣤⣶⣶⠿⠿⠿⠿⠿⠿⠿⠿⢿⣶⣤⣀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣠⣤⣶⡶⠶⠶⠶⠶⣦⣤⣄⡀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢀⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠋⠉⠉⠉⠉⠙⠛⠷⣶⣄⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠛⠋⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⣶⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠠⠀⠀⢤⡀⠀⠀⠀⢀⡠⠄⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⢰⡱⢹⠀⠀⠀⠀⠀⡇
⠀⠂⠐⠒⠒⠒⠐⠐⠀⠀⠀⠀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⢀⠢⡄⠀⠀⠀⢫⡉⣀⠄⠀⠀⠀⡇⠀⠀⢀
⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠉⠁⠀⣀⣈⠉⠀⠀⡇⠀⠸⠀⠀⡠⠁⠀⠀⠀⠠⡀⠶⠚⠃⠀⠀⢇
⠀⢠⣤⠀⠀⠀⡠⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⡎⠀⠀⢣⠀⠀⠈⠊⠠⠀⠈⠁⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠸⠷⠀⠀⠀⢰⠀⠀⠀⠀⡠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠈⠀⠀⠚⠉⠀⠀⡀⠂
⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠸⠾⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄
⠀⢀⡴⠓⠈⠉⠉⠀⠳⣄⠀⠀⠠⠀⠀⠀⠍⠁⠠⠀⠉
⠀⠎⣀⣠⣄⠀⠀⣤⣄⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠉⠁⠈⠑
⢸⡔⡀⢈⣿⠀⢘⣿⠀⠘⣺⠀⠈⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⢠⢴⣷⢧⢤⣤⢴⢶⢧⣄⠀⠀⠆⠠⠈⠀⠀⠀⠀⠠⠀⠂
⠀⠃⢿⣿⡿⠀⠈⣿⣿⠧⠋⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣁⣁⣁⣀⣀⣀⣀⣈⣁⣉⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠏⡆⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢦⡀⠀⠀⠀⠀⠀⠀⡜⡎⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⢆⠳⡙⢆⠀⠀⢀⡤⡞⡜⠀⠚⠂⣿⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⠀⢀⡱⠬⠖⠛⠉⠈⣻⣰⡥⠤⠖⠊⢫⢳⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠴⠗⠒⠉⠁⣀⡠⠤⠖⠒⠉⠉⠈⢷⡀⠀⠀⠀⢇⢇⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠉⢁⣀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⢀⣀⡬⣧⢀⣀⠤⢾⣿⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡳⣤⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⡺⠯⠗⠓⡏⠀⠀⠀⢸⣼⣿⡿⠟⢡⡾⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⠼⢄⡀⠳⠗⠀⠀⠀⠀⠀⣀⡤⡴⣖⡹⠷⠛⠋⠁⠀⠀⠀⠀⣇⣀⣠⣴⣾⠟⠋⢀⡴⠋⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⣾⣷⡜⠌⠓⢖⣉⡹⠀⢀⣤⢴⣲⣹⠓⠛⠉⠀⠀⠀⠀⣀⡠⠤⠔⠒⠻⡍⠳⡜⢯⠁⣠⠴⠋⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠚⠉⠁⠀⢀⣀⡤⠴⠒⢎⠉⠀⠀⠀⠀⠀⠀⠈⢦⡈⢢⡙⣅⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣽⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⠀⢀⠀⢀⣀⣀⡠⠤⠶⢶⣯⣭⣭⣭⣭⣭⡭⠭⠶⣒⡤⠄⣀⠀⠀⠀⠑⣄⠙⣌⣳⡀⠀⣀⣴⣿⣿⠟⣡⣾⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠁⠒⠫⠵⠶⠶⠮⠍⠚⢉⣤⣾⣿⡿⠋⣁⣾⠟⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠉⣠⡾⠋⠁⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠉⣀⡴⠚⠁⠀⠀⠀⠀⢀⣴⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠋⠁⣠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣤⠿⠋⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠋⠉⢀⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠉⠉⢁⣀⣠⡤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⣠⣶⡿⠋⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⣀⣴⡾⠛⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⢀⣠⣴⠾⠛⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣶⡶⠶⠞⠛⠋⠁⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣀⣀⣀⣤⣴⣶⡾⠟⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠥⠄⣀⣀⣀⣀⠠⠤⠴⠒⠋⠁

⠀⠀⠰⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠰
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠈⢉⡽⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠒⠒⠒⠒⠒⠒⠲⠦⠤⠤⢤⡞⠉⠀⢀⡤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆⢸⡇⣠⠔⠉⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠦⠤⠤⠤⠤⠤⠤⠤⠴⠃⢸⠁⠀⡠⠞⢁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆⢸⠀⠋⢀⠔⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⣓⣒⠲⠶⠶⠶⠖⣒⣒⡃⢸⠀⠖⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠉⠉⠉⠉⠁⠀⠈⡇⢸⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⢸⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡞⠉⠑⡄⡴⢦⠀⠀⢸⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⠦⠖⠁⠉⠉⠀⠀⢸⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣀⣀⠀⣀⣀⠀⢀⣀⡀⠀⢸⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠤⠤⠸

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

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠀⠀⠀⠀⡀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠒⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠄⠀⠂⠀⠀⢂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡖⣂⡀⠈⣴⡤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⡟⠛⠃⠀⡉⠁⠀⠀⠀⠀⠀⠀⠐⠐⠓⠀⠀⣶⡀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣐⡲⠀⠙⡐⣂⣄⠀⠃⠀⢂⡀⠀⠶⠀⠀⠀⠀⠊⡠⠈⠄⠀⠀⠀⠀⠸⢤⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠢⠀⠀⠘⢄⡀⠀⢈⣒⢌⠫⣀⡄⠁⠰⣋⢼⣰⣼⡱⡁⡄⠀⠀⠀⠀⠀⠈⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠧⡥⠀⣤⣄⢈⡀⡏⠀⠙⠢⠀⠂⠀⣀⠀⠁⠀⠈⢠⡁⠣⡀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣨⠀⠢⠀⢀⠀⠀⣄⡇⡠⢀⢀⡑⠶⠚⢁⠙⢢⣴⡡⠈⠀⠀⢸⢴⠍⠆⠀⣠⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠈⠀⢆⠙⠃⠁⢠⡍⠈⠙⣻⠁⠀⠀⠀⠀⠄⡐⢋⠹⡄⠀⠀⠀⠀⠔⠉⠙⠁⠀⢹

⠀⠀⢸⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠂⢋⠀⠀⢀⠔⠇⠀⠀⠀⢰⠠⠀⠘⡀⠙⢧⣀⢀⡀⠀⠠⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠐⢁⠸⠀⠀⠀⠀⠀⠄⠀⠀⢇⠀⡀⠀⢻⣀⡘⠓⡄⢰⡀⠄⠂⠁⠉⠉⠒⢜⣤⠀⠁⣼⠄⢇⠀⠀⠀⠁⠀⢸
⠀⠀⢸⠀⠀⠀⡀⠁⠀⢀⠀⠀⠠⠦⠀⠠⠀⠀⠀⠈⠑⠺⡀⠀⠀⡠⠊⡀⠴⠂⡠⠃⡠⠀⢀⠋⠓⠌⢀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⡠⠀⠠⢄⠀⠀⠀⠤⠤⣀⣒⣀⣀⠠⠤⠦⣀⣱⡀⠰⠇⠞⠀⠼⡀⡆⢨⠀⢴⠃⣤⡤⠠⢬⡀⠘⡄⠀⢹⠀⠀⢸
⠀⠀⢸⠀⡀⠀⠠⣉⣉⣉⡉⠉⠉⠛⠓⠒⠒⠒⠒⠲⠤⠤⠤⠬⢪⠭⠤⡤⣤⣈⣙⠈⠑⢺⡊⠡⠀⠀⠀⠘⠉⠋⠀⠈⠑⠒⢸
⠀⠀⢸⠐⠓⠒⠒⠢⠤⢬⣭⠉⠉⢠⡀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠄⠄⠀⠀⠀⠀⠀⠃⠀⠘⠣⣄⠀⠀⠀⢰⠀⡀⠀⠀⠀⠀⢸
⠀⠀⢸⠁⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠉⠑⠒⠂⠀⠤⢼
⠀⠀⢸⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⢰⠶⠶⠶⢶⡄⠀⠀⢀⠔⠓⠘⠦⡀⠀⠀⡞⠛⠛⠛⠛⠛⢻⠀⠀⠀⣀⣀⣄⣀⣀⠀⠀⠀⡴⠒⠒⠒⠒⠒⡆
⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⠄⠀⠀⢠⡅⠀⠀⡏⠀⠀⠀⠀⠀⢸⠀⠀⢸⣯⣿⣿⣿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⢸⣤⣤⣤⣼⠇⠀⠀⠘⠃⠀⡀⠸⠃⠀⠀⠛⠒⠒⣶⡖⠒⠚⠀⠀⠸⠿⠇⠀⠻⠿⠃⠀⠠⠿⠶⠶⠶⠶⠶⠧
⠀⠀⠀⠁⠁⠁

⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⢀⣀⣀⣠⡀
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣤⣤⡷⢶⣴⣦⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣿⣿⢾⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠸⢿⣿⡟⠃
⠀⠀⠘⠻⠿⠿⠟⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠘⠛⠟⠀⠀⠀⠀⠉⠉⠩⠿⠍⠉⠉⠀⠀⠀⠰⠾⠿⠣⠂


⠀⡖⠒⠒⠒⠒⠒⠒⣶⠀⠀⠀⣸⣿⡆⠀⠀⠀⠀⢠⠟⠛⠛⡆⠀⠀⠀⢰⠖⠒⠒⠲⣄⡀⠀⠀⢰⠶⠶⠶⠶⢶⡀
⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢐⠁⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢰⠖⣶⣼⣿⡟⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠳⠶⠶⣶⡶⠶⠶⠟⠀⠀⠀⢲⣶⡞⠀⠀⠀⠀⠘⣄⣀⣠⡇⠀⠀⠀⢠⣀⡀⣸⣿⡟⠀⠀⠀⢸⣦⣤⣤⣤⣼⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⢀⣀⣸⣉⠉⠉⠉⠉⠉⢹⠀⠀⠀⡴⠶⠶⠲⠶⠿⡆⠀⠀⠀⠀⢀⣀⣇⣀⠀⠀⠀⠀⣀⣀⣠⡖⠒⠒⠒⠒⢲
⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠚⠂⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠁⠀⠘⠒⠀⠀⠖⠻⠀⠀⢹⠀⠀⠀⠀⢸
⠀⢸⣤⣤⣼⠛⣛⣿⡛⠻⣤⡄⠀⢀⣳⣶⣶⣶⣶⣶⣏⠉⢹⣿⡉⢹⣤⣤⣤⣠⠀⠀⣤⣤⣤⣤⣼⣶⣶⣶⣶⣾⡀

⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠁⠀⡠⠤⠤⠤⠤⠄⠀⢨⠞⠃⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠉⠩⠭⠭⠭⠽⠶⣤⡀
⠀⠀⠀⡇⠀⠣⠤⠤⠤⠤⠜⠀⢸⠀⠖⠒⠒⠀⠐⠒⠀⠀⠒⠒⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⣀⠀⢣
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡇
⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠁
⠀⠀⠀⠀⡇⠀⠀⠀⡠⡀⠀⠀⠀⢸⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡸
⠀⠀⠀⠀⢰⠀⠀⠀⠉⠀⠀⠀⠀⠀⢇⠀⠂⠄⠀⠀⠀⠠⠤⠤⠀⣀⣀⣀⣀⣀⣠⠤⠁⢀⠇
⠀⠀⠀⠀⢸⠀⠀⢰⠁⢱⠀⠀⠀⢰⠀⠉⢒⡤⠤⠤⠤⠤⡤⠤⠤⡤⠤⣤⣤⡤⠤⠤⠤⠊
⠀⠀⠀⠀⠀⡆⠀⠀⠉⠁⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⢀⡇⠀⠀⢇⠀⠈⠯⠷⠾⠯⣷⠖⣒⠶⠦⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠓⠒⠚⠁⠀⠀⢀⡠⠋⠉⠉⠀⠀⠀⠈⠉⠙⢶⢶⠦⡀⠈⠛⢄⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠈⢉⣉⣉⣉⣉⣽⢛⣻⣿⣿⣿⣾⣵⣃⣀⣀⡀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠘⠃⠐⠂⠒⠂⠐⠖⠠⠶⠀⠤⠄⠤⠄⠠⠤⠠⠄⣉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠥⠀⠉⠁⠈⠉⠀⠁⠀⠀⠀⠀⠀⠠⠤⠘⠃⠺⢓⡐⠒⠂⠧⠇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠛⠛⠛⠓⠒⠀⠀⠀⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⢸⠿⠿⠿⠿⠿⣷⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⣆⣀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡆⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⠶⣤⡀
⠀⠀⢸⠀⣴⣶⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⣿⣷⡄⠀⠀⠀⠀⠀⣾⡟⠛⠃⢀⠀⢳⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠱⡀
⠀⠀⢸⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠶⡶⢶⠾⡟⠛⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⡇⠀⡇⠀⣧⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢸⠀⠸⣦⣤⡀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣩⣏⣿⣩⡗⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢠⠇⢀⡇⠀⡞⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡀⠀⠈⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣒⣗⣺⣒⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⡆⠀⠀⠘⠀⠰⠃⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⢀⠀⠀⣠⠟
⠀⠀⠸⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉


⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡄⠀⠀⠀⠀⢀⣤⢠⣤⣤⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠉⠉⠉⢻
⠀⣯⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡏⣾⡏⠉⢉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠂⠒⠂⢸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢈⣼⣿⡇⠀⠈⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣶⣶⠶⡶⣿
⠀⢻⣤⣤⣤⣤⣤⣤⣤⣼⠀⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢍⣿⢽⠝⡯⢿
⠚⠻⠳⠗⠥⠇⠥⠹⠟⠿⠗⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⣤⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢥⡯⢽⠭⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⢀⣀
⠀⠀⢺⣿⡉⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠈⢿⣷⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠒⠲⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⠷⣛⡺⣧⣼⣿
⠀⠀⠀⠘⣿⣧⠀⠘⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣤⣤⣤⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠒⠋⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠎⠀⠈⡆⣿⣿
⠀⠀⠀⠀⣸⣿⠷⠛⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⣿⢻⡿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣧⣓⠤⢔⣵⣿⣿
⠀⠀⢀⣼⣿⣋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠯⠿⠿⠯⠟⠛⠋⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣵⣿⣿⣶⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⡾⠿⡆⣼⠿⢷⢰⠿⠿⡆⠀⠀⠀⠀⠀⠀⠐⠄⢒⣒⠂⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠦⢦⣤⣤
⠠⡇⠀⡇⣿⠀⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇
⠀⣿⣿⡇⣿⣿⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣟⣿⣷⣶⡶⢶⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠶⢶⣤⣤
⠀⣏⣹⡇⣿⣉⣿⢸⣏⣹⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⢴⣾⣿⣟⣛⣛⣿⠿⣟⣿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⠀⠀⠐⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⣀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⣀⠀⠀⠀⠅⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢠⠀⢌⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⢀⡀⠒⠠⠀⡀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⢉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠠⠂⠀⠀⠤⠀⠀⡄
⠀⣀⡠⣀⡀⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠃
⡔⠀⠀⠀⠑⡤⠤⠤⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠊⠁⠉⠣
⢁⠀⠀⠀⠀⠁⠀⠀⠘⢢⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⡀⠀⠀⠴⠁⠁⠀⠃
⡜⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⡀⠀⠀⠀⣀⠀⢓⠉⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠘⠠⠤⠆⠀⠀⠀⠀⠀⢠
⠀⠈⠑⠀⠉⡋⢀⡀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠊⠀⠘⠗⠆⠠⠐⠂
⠀⠀⠀⠀⠀⠁⠈⠉⠂⢄⡑⢄⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡠⠖⣁⠔⠒⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣅⠢⣇⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⢊⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⢀⠚⠀⡐⠀⠀⡈⠀⠀⡂⠀⠀⠀⠀⡀⠀⢃⠀⠘⣂⠀⠳⡀⠈⢢⡀⠱⠄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠴⠁⠀⠬⠀⠀⢠⠀⠀⠀⠂⠀⠀⠀⠀⡂⠀⠐⠂⠀⠐⡀⠀⠐⡀⠀⢐⣄⠈⢠⠀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⣀⣀⣀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣛⣛⣛⣻⣟⣛⣛⣛⣛⣛⣛⣛⣿⣿⠀⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠸⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⢿⣿⣆⣛⣛⣛⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣿⣿⣾⣶⣶⣾⣷⣾⣷⣿⣿⣾⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⡿⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠸⣿⣿⣯⣭⣭⣭⣭⣭⣭⣽⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣛⣛⣛⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣘⣛⣛⣛⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠋⣭⣭⣭⣭⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢨⣭⣭⣭⣭⣭⣭⠙⣿
⣿⣿⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⠀⣿
⣿⣿⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⡻⠿⢇⣿⣿⡆⠿⢟⣙⢿⣿⣿⣿⠀⣿
⣿⣿⠀⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠰⣿⣿⣶⣿⣿⣿⣿⣷⣿⣿⠇⣹⣿⣿⠀⣿
⣿⣿⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢃⣿⣿⠟⣩⣶⣶⣬⡹⣿⣿⡜⠿⢿⣿⠀⣿
⣿⣿⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡟⢸⣿⣿⣿⣿⣷⢸⣿⣿⣿⠀⣿⠀⣿
⣿⣿⠀⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⡙⣿⣿⡜⢿⣿⣿⡿⢏⣾⣿⠏⣭⣴⣿⠀⣿
⣿⣿⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣷⣦⣴⣶⣿⣿⣿⡦⢹⣿⣿⠀⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣝⣫⣭⡙⣿⣿⡟⣭⣬⣉⣴⣿⣿⣿⠀⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣭⣴⣿⣿⣿⣿⣿⣿⣿⠀⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⣿⣿⡀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢀⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣭⣭⣭⣭⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⡇⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣴⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣫⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⠟⣚⣓⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣯⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⢸⣏⣹⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣾⣿⣿⣦⣍⣩⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⣉⣩⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣨⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣶⣾⣷⣶⣶⣾⣶⣶⡄
⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⢹⡏⠉⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⣿⠀⠠⣄⣠⠀⠸⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠐⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣐⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠺⠤⢤⣤⣤⡤⣤⣼⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠿⠟⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⣿⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⣰⠐⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠒⣤⠄⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣳⣟⡿⡇
⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣿⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢞⢀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠷⠊⠃⠫⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡏⣭⡍⢻⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⢛⢛⡛⣻⣿⣟⢙⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢴⣷⠀⢘⡇
⠀⠀⠀⠀⠀⠀⣿⣧⣉⣁⣼⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣶⣶⣾⣷⣿⣶⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣢⣀⣀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣍⠀⠁⠀⢂⠲⢀⡀⠀⠀⠐⡡⢂⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢊⠈⠁⠀⠀⢠⡀⣋⠀⠈⠉⠚⠠⠅⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠐⠐⠀⠄⢀
⠀⠀⠀⠀⠀⡀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⣼⢤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⡀⠉⠀⠂⠀⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠀⠈⠈⠐⠘⠀⢰⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡐⠈
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢇⠀⠀⠀⣠⡿⢛⡿⠈⠐⡠⢄⣠⠀⠀⠀⠀⠀⠈⠉⠚⠃⢇⡖⡤⣀⠀⠀⠀⠀⠀⠑⠀⣀⠔⢢⠜⢠⡀
⠀⠀⠀⠀⡠⠸⣽⣿⠿⠗⢛⣴⣵⢄⢘⡶⣶⡇⢀⠄⡀⠀⠀⠉⠀⠀⠒⠠⠀⡀⠀⠀⠀⠀⠁⠙⠽⢃⢂⢤⠀⠀⠄⠠⠃⠀⠂⣴⣿⣦⡀
⠀⠀⠀⢠⠽⡺⢁⡰⣢⣿⡶⢣⡇⠘⡻⢀⣍⡗⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⢄⣀⡀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡆
⠀⠀⠀⡾⠄⢡⣳⠉⠹⡿⠂⢴⡟⢲⠁⠀⢖⠅⢀⡐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠠⠀⠀⠀⠀⠠⠿⠿⠿⠛⠁
⠀⠀⢰⣄⠀⣘⠉⢠⡔⠂⢹⣿⠰⠈⠓⠊⠀⠁⠀⠀⠐⠀⠀⠀⠉
⠀⠀⢸⠀⠩⡇⢐⡺⠋⠸⠃⢐⣤⡟⡄⠀⠀⠈
⠀⠀⠀⠀⠴⠇⢈⡟⢶⣤⠶⢦⣟⣴⠇
⠀⠀⠀⠈⠠⢺⡞⢛⠻⠮⡶⡾⢞⣁⠈
⠀⠀⠀⠀⠘⠿⠗⠀⠀⠒⢃⣍⣄⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠘⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠄⠀⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣡⡀⢠⠴⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠞⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⡤⠄⠤⠤⠤⠤⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⠶⠦⠤⠤⠤⠤⢤⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣧
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣺
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠦⢤⠞⠁⠀⠀⣠⠄⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢰⠋⠉⠛⠛⠛⠛⠛⠋⠙⠙⢳⠀⢸⠀⣀⡴⠊⠁⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠀⢸⠀⠉⢀⡴⠞⠁⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⡖⠒⠒⠒⠒⠒⠶⠖⠒⠒⢦⠀⢸⠀⠴⠋⢀⡴⠂⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⣀⣀⣀⣀⣀⠀⠀⣀⣀⣸⠀⢸⠀⣀⠴⠋⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡴⠤⠬⠭⠭⠭⠤⠤⠤⠤⢤⠀⢸⠀⠁⠀⠀⠀⠀⡿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⢉⣉⡉⠉⠉⠉⠉⠉⠀⣿⠀⠀⠀⠀⣠⠏
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠋⠀⠙⡆⣴⠲⡄⠀⠀⣿⠀⠀⠀⡴⠃
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠳⠤⠞⠁⠈⠉⠁⠀⠀⣿⠀⢀⡞⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣠⠏
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣈⣉⣁⣀⣉⣉⣀⣈⣉⣉⣠⡤⠟⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡇

⠀⠀⠐⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

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

⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣴⣶⣿⣿⣿⣷⣶⡄⠀⠀⡇⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠛⠫⠿⠿⠿⠿⠛⠁⠀⠀⠁⠒⠒⠒⠒⠒⠳⠂⠀⠀⠀⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢀⣴⣶⣶⣶⣶⡶⡶⣄⠀⠀⠀⠀⡏⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣸⣿⣿⣿⠟⢻⣿⣿⣿⡀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⣠⡦⣾⣿⣿⣿⣆⠴⣄⠀⠀⠀⣾⠛⠛⡆
⠀⠀⠀⣿⡿⠋⠉⠀⠈⠉⠻⣿⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠿⠀⠻⠿⠿⠿⠟⠀⠿⠀⠀⠀⢻⣀⣠⠇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣇⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠁⢹⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣟⣿⣿⠀⠀⠀⣶⣷⣶⣶⣶⣟⡆⠀⠀⠀⠀⣿⡟⢹⣿
⠀⠀⠀⠘⠧⠤⠤⠤⣤⠤⠤⠤⠾⠀⠀⠀⠀⠛⢿⠟⠃⠀⠀⠀⣾⣿⣿⣤⣿⣿⡇⠀⠀⣿⡇⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠦⠀⠀⠀⠀⠀⠀⠀⠒⠛⠒⠂⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠙⠁⠈⠉⠉⠉⠈⠋
⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢰⣶⣶⠶⣶⠀⠀⠀⠀⡏⠉⠉⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⣶⣿⠀⠀⠀⠀⡇⣀⡀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠘⠿⢿⠿⠟⠀⠀⠀⠀⣿⣷⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠻⢿⡄⠀⠀⠀⣿⠿⠀⠀⠀⠀⠀⠀⠛⠃⠀⠈⠁⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣤⣤⣀⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠁

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣤⣤⣄⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠴⣟⣿⠸⠿⣿⣻⠉⢡⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣼⣿⡿⢾⣿⣶⣤⡉⠉⣠⣾⣿⣿⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡀⠛⢿⣿⣿⣿⣿⣿⣿⣏⡉⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣆⠀⠙⠻⠿⡿⢿⡿⣷⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡏⠀⡀⠿⢿⣿⣷⣿⣦⣶⣿⣇⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⠿⣿⣟⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁⢸⠟⢛⣿⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠾⣿⣄⣐⣶⡆⠋⠁⠀⠀⠀⠰⣦⣤
⠀⠀⠀⠀⠀⢀⠆⢀⠀⠀⠀⠀⠀⠀⠄⢰⣿⣿⣝⣛⣷⣬⣶⣦⣄⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⠇⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠸⠿⣿⣿⠿⠿⠓⠁⠀⠋⠂⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣶⠜⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢀⣿⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣿⣶⠄⠀⣐⠂⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢸⣿⣿⡒⠛⠁⠙⢳⡤⠂⠈⣻⣾⣿⣧⠀⡀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⢰⣷⣄⠀⠀⠀⠀⠻⠿⠿⠿⠿⠶⠤⠈⠁⢽⢤⣿⣿⣿⣿⣷⡰⣄⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠉⠛⠷⠦⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⣤⣀⢀⣾⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣸⣿⣷⡀⠀⠀⢀⠔⠋⠉⠙⢢⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⡰⠚⠉⠉⠲⡄⠀⠀⠀⢀⣿⣁⠀⠀⠀⠀⠀⢰⣿⣿⣦
⠀⠀⠀⣘⣿⣟⡀⠀⠀⠘⣌⠁⠉⢈⡰⠃⠀⠀⠘⢿⣿⠛⠀⠀⠀⢧⡉⠈⠀⣁⠆⠀⠀⢰⡎⠉⢹⡦⠀⠀⠰⣶⣿⣿⣿⠇
⠀⠀⠿⠿⠇⠿⠿⠇⠀⠀⠘⠋⠉⠉⠀⠀⠀⠸⠷⠜⠏⠾⠷⠀⠀⠔⠋⠉⠉⠁⠀⠀⠠⠾⠷⠤⠾⠷⠆⠀⠀⠉⠛⠏⠁
⠀⠀⢀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠠⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⡇⠀⣄⠀⠀⡇⠀⠀⡏⠩⠭⠍⢹⠀⠀⠀⡏⣩⣬⡉⡆⠀⠀⡔⢒⠛⠓⠲⡆⠀⠀⣴⡄⠁⠀⠀⡁⠀⣠⣇⣀⣀⣀⣧⡀
⠀⠀⣧⣤⣬⣤⣤⡇⠀⠀⣇⣀⣀⣀⣸⠀⠀⠀⡟⠿⠿⠋⡆⠀⠀⡇⠘⠈⠛⠀⡇⠀⢸⣿⣯⣿⣿⣽⣿⠀⣿⡿⠿⠿⠿⣿⡇
⠀⠀⠀⠠⠿⠧⠀⠀⠀⠰⠿⠿⠿⠿⠿⠆⠀⠀⠳⠰⠦⠶⠁⠀⠀⠙⠻⠛⠛⠟⠁⠀⠈⠿⠏⠛⠛⠛⠋⠀⠀⠣⠤⠤⠤⠃
⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⢠⠤⣄
⠀⠀⣿⡷⢀⡲⡝⡄⠀⠸⠿⠉⠉⡹⢿⠆⠀⢠⡏⠀⠀⠈⣆⠀⠀⠀⡇⠀⣀⡇⠀⠀⠀⢰⢸⣤⣧⡦⠀⠀⢀⣀⣤⠴⣒⣺
⠀⠀⠙⢷⣄⣠⣤⡀⠀⠀⠀⢘⣁⡇⡆⠀⠀⠘⠋⠀⠀⢀⠟⠀⠀⠀⣇⣚⣛⡇⠀⠀⠀⠈⠾⣛⠯⠃⠀⠀⠘⢻⣻⠿⢿⣿
⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠘⠒⠒⠒⠃⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠓⠓⠓⠃⠀⠀⠀⠀⠐⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⠤⠤⡀⠀⠀⠀⡴⡀⠀⢀⢦⠀⠀⠀⣀⡤⠤⣀⡀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⡠⠤⢄⠀⠀⠀⠀⢀⡤⠤⣀
⠀⠀⡾⠟⠁⢠⣿⡆⠀⠀⢧⠇⣛⠸⣸⠀⠀⣴⠉⡜⠛⡌⣳⠀⠀⠀⣿⣦⡝⠎⠇⠀⢰⠊⠀⣀⠀⠈⡇⠀⢔⠉⠀⠀⠀⢙⠆
⠀⠀⢧⡰⣷⠆⣸⠇⠀⠀⠀⣰⣟⣆⠀⠀⠀⠸⣹⡖⠖⣟⡝⠀⠀⠀⢈⠻⠿⠆⠀⠀⠀⠉⢡⣿⡇⠉⠀⠀⠈⠉⢺⣿⠊⠁
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠈⠛⠘⠋⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⢠⣒⣒⣒⣀⠀⠀⠀⡔⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠦⢤⠀⠀⠀⢠⣴⡶⣦⣄⠀⠀⠀⣀⣀⣠⢄
⠀⠀⢸⣿⠋⢻⣿⠀⠀⠀⡇⠍⠅⠄⢹⠀⠀⢨⣛⢿⣿⢟⣥⠀⠀⣜⠨⠭⠭⢈⡇⠀⢠⣿⡵⣯⢲⣻⡆⠀⠀⠒⠒⠲⠔⠁
⠀⠀⢸⣏⣀⣉⣿⠀⠀⠀⡇⠭⠭⠅⣸⠀⠀⢸⣫⣷⣶⣯⣻⠀⠀⡿⣻⣮⣶⣛⡇⠀⠀⠻⣬⣾⣷⡿⠁⠀⠀⠾⣿⠶⠿⠆
⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠁
⠀⠀⢠⡤⠤⠤⠤⡄⠀⢀⠔⣒⠒⠒⢦⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⣀⣤⣤⣤⣤⡀⠀⠀⣠⠖⠒⠲⣄⠀⠀⢠⣴⣤⣤⣤⣦⡀
⠀⠀⠹⣐⣂⢒⡀⡅⠀⢸⠀⢬⡙⢦⠀⡇⠀⢰⣷⣖⠉⠉⠉⠀⠀⣿⣙⣉⣩⣹⡿⠀⢰⡧⠘⠶⠊⠬⡇⠀⢈⣉⡏⡏⡏⣿⡇
⠀⠀⣈⠻⠿⠈⠇⡇⠀⠸⣄⠃⠙⢘⣢⠃⠀⠀⠉⠀⠙⣿⣷⠀⠀⠈⠹⠟⠋⠋⠁⠀⠀⠳⣄⣀⣠⠜⠁⠀⢸⣇⣇⣇⣇⣉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠠⠖⠀⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡆⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⠿⣽⢿⣿⣿⣿⣿⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣻⡾⢟⡵⠆⡴⢆⠠⠔⠢⠐⠙⢿⣿⣿⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⣿⣿⢻⡿⣻⡎⣠⠖⢋⡁⠰⠆⠀⡍⢷⣾⣿⣿⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣮⠓⠋⢰⣯⣾⣿⣿⠜⡂⣾⠡⢘⣣⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⣿⣿⡟⣃⣾⣇⠀⣝⣛⣯⣭⡆⣡⢐⣼⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡅⢷⡜⠿⠄⠙⠛⣻⣿⣱⣶⣿⣟⢌⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⣿⣶⣿⣾⣟⢷⣬⡙⠋⢩⣬⠿⣫⣾⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣷⣬⣴⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢍⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⡀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠁⠂⠤⠤⠤⠤⠦⠤⠤⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢄⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠒⠂⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⠒⠒⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠤⠄⠀⠀⠀⠀⠒⠂⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⠤⠄⠀⠀⠀⠀⠒⠂⠀⠉⠉

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠛⠛⠛⠛⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢸⠛⠛⠛⠛⢛⣛⠛⠛⠈⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣷⠿⢓⠻⢿⣿⣦⡀
⠀⠀⠀⢸⠁⠀⠀⡔⠁⣤⠙⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⡆⠀⠀⠀⠀⢰⣶⣾⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⡞⢩⢹⡎⣿⣿⡇
⠀⠀⠀⢸⠇⠀⠀⠘⠤⠭⠜⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡻⠶⠞⣡⣿⣿⡇
⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠈⠉⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠
⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣉⣉⣉⣩⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣤⣴
⠀⠀⠀⠀⠛⣿⣿⣿⣤⣤⣤⡄⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⡈⠀⠈⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢸⢉⣭⣭⣭⣭⣭⡍⢿⣻⣿
⠀⠀⠀⠀⠀⣀⣀⣂⣐⣂⣿⡿⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣫⣿⡝⡯⠿⠿⣽⢫⣿⣝⡇⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⣝⣟⣣⡇⠖⠶⢸⣜⣛⣫⡇⠀⠀⠀⠀⠀⠀⠀⢸⠸⠿⣿⣿⣿⠿⠏⢸⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠷⠿⠿⠿⠷⠿⠿⠷⠿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡖⣠⠤⡀⢢⠀⡆⡠⢤⡐⡆⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡶⢶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⡗⢅⣒⡼⢿⠀⡿⣅⣃⡟⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢏⣴⣾⣷⣮⡹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣔⣁⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⡇⡴⠒⢄⣻⠀⣇⡴⠲⣄⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⠗⢼⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⢻⠛⠛⠛⠛⠛⢿
⠀⠀⠀⠀⡏⢢⣭⠞⡻⠀⡟⢦⣡⢛⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣝⡒⢞⡹⢮⡁⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⠶⠶⠶⠚⡶⢲⣶⣢⡲⣿
⠀⠀⠀⠐⠛⠛⠛⠛⠛⠐⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⣤⣧⣼⣷⣝⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⣰⣿⢻⣿⢻⣿⢻⣟⣿⣟⣿⣟⣿⣻⣿⣻⣿⣻⣟⣿⣟⣿⣟⣿⢛⣿⢻⣿⢻⣟⣿⣟⢿⡟⣿⣆
⠀⠀⠀⠀⣰⣿⢣⣿⣿⡿⣿⡏⢿⡟⢿⣿⢿⣿⢿⣿⠿⣾⠻⣾⡹⣾⡹⢾⡿⣾⣿⢾⣿⢾⣟⢿⣿⣿⢿⡜⣿⣆
⠀⠀⠀⣰⣿⢧⣿⣿⢻⣯⠯⣿⡏⣯⡿⣿⣼⢻⣼⢻⣿⡯⣷⡏⣧⣿⢿⣿⢹⣼⡿⣾⡏⣯⣿⣿⣿⣿⡍⣿⡼⣿⣆
⠀⠀⣰⣿⣯⣿⣿⣾⣿⣼⣿⣧⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣧⣿⣧⣿⣿⣼⣿⣧⣿⣷⣿⣿⣽⣿⣆
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠦⠔⠒⠒⠢⠴⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⣠⢤⣔⣒⣒⣈⣉⣭⣭⣭⣀⣒⣒⡢⠤⣄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠉⠙⢻⣿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣻⣿⣿⡀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⠘⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣀⣥⣭⣭⣅⢉⣾⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⣷⣄⠙⠻⢿⣿⣷⣶⣤⣤⣄⣀⣀⣠⣤⣤⣶⣿⣿⣷⡧⠄⣠⣴⠬⠶⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣄⣀⣿⣿⣷⡄⣀⡉⠙⠿⢛⣻⣿⣿⣿⣿⣛⡛⠿⠛⢉⣀⣴⣾⣿⣿⣀⣠⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠿⡿⣿⣿⣿⣿⣗⡏⠉⠑⠒⠲⠶⠤⠤⠤⠤⠶⠒⠒⠊⠉⢹⣿⣿⣿⣿⣿⢿⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢻⡆⠇⢱⠀⠀⠙⠿⢿⣴⣒⡠⠀⠀⠀⠤⠄⠀⠀⠀⢄⣒⣾⡿⠿⠋⠀⠀⡎⠸⢰⡟⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢸⡇⢰⠈⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠀⡇⢸⡇⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢸⠃⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠸⠀⠀⠀⡰⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⢦⠀⠀⠀⠇⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⡖⣂⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡆
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⠸⢀⠰⠆⠪⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠚⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠐⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⡠⠒⠀⠂⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠐⠆⠀⠀⠀⠀⠀⠀⢴⠀⠉⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⣇⣑⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣂⣁⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠣⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣓⣒⣒⣒⣒⣒⣒⣒⣒⣓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠂⠀⠐⠒⠒⠒⠲⠖⠖⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡤⠴⠖⠒⢚⣛⠢⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⢀⡠⠀⠠⡀
⠀⠀⠀⠀⠀⢠⢺⠛⣈⠭⠄⠐⠒⠈⠉⣀⡘⠂⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠋⢆⠀⠀⠀⠈⡊⢡⠙⡎⡀
⠀⠀⠀⠀⠀⠘⣼⠸⠀⣠⠔⠒⠒⠀⡮⠀⠃⡇⢸⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢸⠀⠀⠀⠘⠬⣒⠪⠴⢍⡓⠤⡀
⠀⠀⠀⠀⠀⠀⣟⠀⡆⠃⠀⠀⠀⠀⢠⣃⡸⠄⢀⠀⠀⠀⠀⠀⠀⠞⠁⢏⠉⠀⠐⠒⠀⠈⠉⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠟
⠀⠀⠀⠀⠀⠀⢸⡄⠣⠸⠂⠆⠒⣒⣉⣥⡴⢮⡥⣕⠦⡀⠀⠀⠀⠀⠀⠈⠑⠠⠠⠤⠤⠤⠄⠂⠦⡎
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⡩⠭⣙⣺⡍⢷⡹⣧⠻⢯⣣⠌⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⡩⠩⠁⠐⠒⠒⠀⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢪⠣⡑⢮⠙⣶⡽⢒⠩⢔⡂⠭⣄⠀⠀⠀⠑⡄⠀⠀⢀⠤⢒⣨⣥⣤⡀⠭⠂⠀⠀⡌⠀⠀⠉⠉⠀⠒⠒⠐⠆⠠⢬⠁
⠀⠀⠀⠀⠀⠔⣒⢆⠀⠑⢌⢆⠙⠚⠉⠀⠈⢣⣀⠤⠜⢁⡠⠐⣪⠇⠀⠀⠳⢪⣶⢛⠢⡐⠚⠀⠀⠀⠀⠃⠀⠛⠚⠐⠴⠴⠀⣤⣤⠀⠘
⠀⠀⠀⠀⠀⢃⠛⢹⠀⠀⠈⠣⡑⢄⣀⣀⠀⠤⢰⣶⠿⠛⠃⠉⠀⠀⠀⠀⠀⠀⠁⠘⠒⠁⠀⠀⠀⠀⠀⠆⠤⠤⠄⢀⣀⡀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⢦⡎⠀⠀⠀⠀⠈⠲⠀⠰⠒⠊⠁⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠦⣀⠀⠀⠀⠀⠀⠀⡘⠁⢀⣀⠘⠦⢄⡀⠀⠀⠀⠘⡀⠘⣉⣍⠹⢝⡈⡅⠀⠀⠀⢀⠤⢀⡒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⢠⣞⠐⠀⠐⠒⠂⣳⠀⠀⠀⠀⡖⠁⠀⢣⠀⡃⠀⢀⡱⠀⠀⠀⠀⢃⡇⢋⢀⣉⡀⠇⢡⠀⠀⠀⢿⡄⠂⢐⣒⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⠒⠤⠤⠺⡀⡄⠀⠀⠀⠉⠒⠒⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⢀⣀⣈⠄⠀⠀⢡⢋⣍⣐⠆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣃⡀⠠⠤⠤⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⢠⡄⢄⣀⠀⠩⠭⠍⠐⠋⠉⠀⠀⠀⠀⠘⣜⠀⠰⠉⠛⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠵⠇⠠⠟⠀⠀⠀⠀⠀⢀⠔⠒⠒⢄⠀⠀⢡⣷⠤⢭⡥⠐⠇
⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⡀⠒⠠⠬⠉⠉⠄⠀⠐⠒⠉⠛⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⢰⠢⢲⠀⢡⠀⠈⢾⣦⠬⠿⠒⠃
⠀⠀⠀⠀⠐⣷⠉⠁⠀⠀⠀⠀⠀⢀⣀⣈⠀⠤⠤⠰⠐⠒⠚⠂⠀⠀⢠⠖⢋⠉⢫⢡⠀⠀⠀⠀⠀⢦⠀⠑⠁⣀⠎
⠀⠀⠀⠀⠀⡘⠒⠒⠒⠀⠈⠉⠉⠀⡀⢀⢀⣄⣤⣠⡄⠀⠀⡇⠀⠀⠀⡘⠴⠖⢊⡀⠆⠀⠀⠀⠀⠀⠞⠉⠉⠀⠀⠀⠀⠀⡴⢤
⠀⠀⠀⠀⠀⡇⡇⠀⡖⠈⠉⠹⡀⠀⠖⠉⣉⡈⠠⠤⠤⡄⠀⢁⠀⠀⠀⢡⢸⣉⠃⠅⡸⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⡼⠄⠴⣄
⠀⠀⠀⠀⠀⢠⢰⠀⢠⠀⠀⠀⢇⢚⠉⠁⠀⠀⠀⠀⠀⢰⣀⣸⠀⠀⠀⠀⠓⠒⠂⠉⠀⠀⠀⠀⠀⡰⣇⢘⣄⠁⠊⡀⠈⡀⠸⠇⢸
⠀⠀⠀⠀⠀⠘⠈⠄⠸⠄⠀⠀⣘⡸⠤⠤⠐⠀⠒⠂⠉⠁⠀⢸⠀⠀⣠⠔⠒⠒⢄⠀⠀⠀⠀⡠⠊⠀⠻⣅⠼⡃⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢇⠖⠒⠉⠉⠁⠀⠀⠀⠘⠟⣀⣀⠤⠄⠒⠂⠁⠀⡐⠁⡠⢦⠀⢠⠀⠀⠀⡞⢄⠀⣀⠠⠽⠦⠇⠀⠀⠀⢀⠒⢄
⠀⠀⠀⠀⠀⠀⠈⠛⠒⠆⠀⠒⠒⢲⠸⠉⠁⠀⠀⣄⠀⠀⠀⠀⠀⠡⡀⠓⠉⠒⠁⠀⠀⠀⠈⠉⠫⢽⡦⠀⠀⠀⠀⠀⠀⣌⠈⠈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⠺⠆⠀⣀⡀⠭⠃⠀⠀⠀⠀⠀⠉⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠂⠒⠤⠤⠤⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠌⢀⠤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠐⠀⠀⠂⢅⠢⢁⠀⠄⡠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⡀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠐⠠⢐⠤⢐⠀⡁⠀⠤⣒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢈⠀⠉⠐⠀⠒⠀⢀⠀⠚⠠⠀⡀⠀⠀⠀⠀⠠⢀⡆
⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⠀⠐⠀⠀⠂⠈⠁⠀⠀⠀⠂⠉⠀⠀⠀⡀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⡀⠐⠀⠁⠒⠀⠄⠂⠁⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡠⠔⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢉⣵⠟⢹⡏⠻⣮⡉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⡞⠁⠀⢸⡇⠀⠈⠻⣄⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⣰⠏⠀⠀⠀⢸⡇⠀⠀⠀⠹⣆⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡆⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠗⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠤⠤⣿⠤⠤⠤⠤⠤⢼⡧⠤⢀⣴⠞⠛⠛⠛⠶⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇⣰⠟⠀⠀⠀⠛⠳⣄⠈⢷⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠸⣇⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⢻⡄⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠻⣆⠀⠀⢸⡇⠀⠻⢦⣄⣀⣀⣀⣤⠾⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠙⢷⣄⢸⡇⣠⡾⠃⣈⣭⠉⠉⠁⠀⢼⡋⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠿⠿⠿⠿⠚⠛⠉⠀⠀⠀⠀⠀⠈⠙⢦⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢀⣀⡀⣀⡀⢀⣀⠀⣀⢀⠀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡿⣾⠀⣿⠀⣿⡂⢸⢾⡃⣿⣾⠃⣿⡂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠈⠀⠉⠁⠈⠀⠁⠁⠈⠀⠉⠁⠀⠁

⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⠀⢀⡀
⣼⣿⣿⣿⣿⣾⣶⣿⣿⣦⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠐⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⢢
⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣷⣦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⡿⠿⠂⠘⠿⢿⣿⣿⣿⣿⣿⣿⠟⠿⣷⣾⡿⠻⣷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣷⣶⠄⢠⣶⣾⣿⠛⠛⠛⠛⣿⣦⣴⡿⢿⣦⣴⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⣿⣷⣾⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦⣴⣿⠟⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠛⠛⠛⠛⢳
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣤⣤⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⢿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⣑⣒⣒⣒⣒⣂⣿⣿⡟⢭⣭⣭⣭⢹⣿⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠈⠻⠿⠿⠿⠽⠉⠿⠿⠿⠟⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣇⠽⠿⠿⠿⢼⣿⡇⠀⢸⣧⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⡿⣵⣿⣿⣮⣿⣿⠀⠀⠀⠀⣤⠀⠀⢰
⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣷⡿⣿⣿⣿⣾⣿⠀⠀⠀⢀⣿⡀⢀⣼
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠘⠿⠿⠿⠾⠷⠿⠿⠿⠀⠀⠀⠸⠿⠟⠸⠿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⡀
⣾⣿⣿⣿⣿⡏⠁⠀⠨⢻⣿⣿⣿⣿⡇⠀⠀⣴⠟⣛⣛⣛⣛⡛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⡂⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⣿⢨⣿⣿⣿⣿⡇⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⢷⣤⣤⣤⣼⣿⣿⣿⣿⠇
⠹⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⠁⠀⠀⣿⡦⢭⣭⣭⣭⣵⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠛⠿⠿⣿⠿⠿⠛⠁
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣶⣾⣾⣦⣄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣐⣶⣶⣂⣿⣿⣇⣀⣀⠀⠀⠀⠉⠂⠀⠀⠀⠰⠾⠆
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⢉⣉⣉⡉⠛⢦⡄⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⢠⣾⠟⣛⣛⠻⣷⡄⠹⣆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⠴⢘
⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⢫⣶⣾⣶⣯⢻⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⡇⣾⣿⣿⣿⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣏⣿⡇⠀⠀⠀⠀⠀⢸⡟⣭⣭⣹⡀⢻⣧⡙⠿⠿⢋⣼⡟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣜⡻⠿⠿⣫⣾⣿⡇⠀⠀⠀⠀⠀⢸⣗⣿⣿⣿⣿⣄⠙⠻⠿⠿⠟⠋⣠⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠻⣾⣿⣾⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⡿⠃
⣾⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⢂⣴⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣴⣾⡟⣼⡇⣾⣿⣿⣿⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠚⢿⠟⠂⠀⠀⠀⠀⠀⣀⡤⣷⣿⣿⣿⣇⣿⣇⢿⣿⣿⣿⣿⣦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢖⣵⣿⣷⣾⣿⣿⣿⣿⣿⡸⣿⡌⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⣠⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣦⡙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⠿⠶⣤⠄
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢿⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣌⠻⣿⣿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣾⣿⠀⠀⢀⣰⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⠀⣶⣿⣿⣿⣿⣿⣿⡿⠟⢩⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⡀⠙⠟⠉⠁⠈⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⣼⣿⡟⠀⠈⠙⠉⠸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣰⣦⣀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣧⠹⣿⣆⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠹⣿⡆⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⢻⣦⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣶⣶⣾⣿⣶⡶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠀⠘⠀⠘⠃⠀⠛⠀⠀⠀⠸⠀⡂⠀⠀⢀⠀⠁⠘⢀⠀⠠⠀⠀⠐⠂⢐⠀⠀⠀⠐⠟⠀⠘⠃⠀⠃⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣿⣽⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠙⣿⣿⣿⡏⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠁⠈⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⣶⠀⠀⠀⠀⠀⢰⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢶⣶⠿⠛⠛⣿⠛⠛⠿⣶⡶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⢀⡴⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢦⡀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣼⠃⠀⠀⠀⣀⣀⣀⣀⣿⣀⣀⣀⡀⠀⠀⠀⠘⠆⠀⠀⠀⠈⠻⣄
⠀⠀⢀⡞⠁⠀⠀⠀⣀⣼⠧⠶⠚⠛⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠙⢛⣤⡶⠿⠛⠛⠛⠛⠿⢶⣤⡀
⠀⢠⡟⠀⢀⣤⠶⠋⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⡾⢠⡶⠋⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⢸⣷⠋⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⣿⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠤⢤⣤⣤⣤⣤⣼⡧⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣿⢤⠌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⢹⣧⡀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏
⠘⣏⠳⣄⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣄⣠⣤⣤⡶⣿⣿⡿⢋⣤
⠀⠹⡆⠈⠛⢦⣄⡀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠍⢉⣩⠴⠂⠈⢡⣾⣿⣿⣷⡄
⠀⠀⠙⣄⠀⠀⠈⠙⠳⢷⣤⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣤⡿⠚⠋⠁⠀⠀⣰⠊⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠈⢧⠀⠉⠉⠛⠛⠓⠒⠒⣿⠒⠒⠚⠛⠋⠉⠉⢀⡾⠁⠀⠀⠀⢀⡾⠁⠀⠀⠘⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠻⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠏⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣬⣿⣦⣄⣀⠀⣿⠀⣀⣠⣴⣫⡥⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣴⣶⡇⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⢰⣯⣿⣽⡦⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⣶⡄
⠀⢸⣿⠉⠁⠀⠀⠁⠀⠀⢹⣿⣿⣿⠀⠀⠀⣿⣀⣀⣀⣿⠀⠀⠀⢸⣿⡿⣡⣾⣿⢦⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿⡇
⠀⢸⣷⠀⠀⠀⠀⡀⣀⠀⣸⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⢘⣿⣇⠻⣿⣿⡵⢠⡆⠀⠀⠀⠘⠛⣿⣿⣿⠛⠁
⠀⠘⣿⣶⡆⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠻⠿⠷⠮⠭⠶⠿⠃⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠈⠙⠃⠀⠀⠀⠀⣴⣦⣀⣀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⠀⠀⠀⠛⠛⠛
⢀⣤⣶⣶⣄⠀⠀⠀⢸⣿⣿⣿⣿⡿⠉⠀⠀⢩⣿⡇⠀⠀⢰⣶⡶⠿⣶⣶⠶⠶⠶⣶⣶⠾⢷⣶⣦
⣿⣿⠋⢻⣿⡇⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⣸⣿⡇⠀⠀⢸⠏⠀⠀⠀⢻⣀⣀⣀⣸⠁⠀⠀⠈⣿⠀⠀⠀⠀⠀⠄
⠹⣿⣿⣿⡿⠃⠀⠀⠸⣿⣿⠿⠿⠿⢷⣦⣶⡿⠿⠃⠀⠀⢸⣧⣀⠀⣀⣾⣿⣿⣿⣿⣄⡀⢀⣠⣿
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠰⠛⠛⠛⢳⠀⠀⠀⠀⠀⡶⠚⠓⢲⡄⠀⠀⠀⢸⠉⠉⠉⠉⢹⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇⣀⣀⣠⡆⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⣁⣀⣀⣀⣀⡷⠀⢐⡆⠈⠁⠉⡆
⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣯⣽⣽⣽⡇⠀⠀⠀⢸⣶⣶⣶⣶⣾⠀⠀⢸⣉⣹⣉⣏⣉⡧⠀⠈⡇⠀⠀⠀⡇
⠀⠸⣶⡶⣶⡾⠀⠀⠀⠀⠀⣗⣻⣺⣺⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣥⣼⣤⣥⣤⣗⠀⠀⣿⣿⣿⣿⡇⠀⢠⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠸⠤⠼⠤⠧⠤⠏⠀⠀⣿⣿⣿⣿⠇⠀⠸⠇⠾
⢀⣠⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⣿⣷⣿⣷⣾⣿⣿⣿⣿⠀⠀⡀⢠⡿⠿⢿⣿⣿⣷⡀⠀⠀⣴⡾⢷⣾⠿⠿⢿⣷⣦⠀⠀⠀⣾⣭⡽⢦⠄⡀⢀⣀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠻⠿⣿⡿⠿⠋⠀⠀⠀⣿⣷⣾⡏⠀⠀⠀⢹⣿⠀⠀⠀⣿⣿⣼⣿⣿⣼⣿⣿⣿⣿⣿
⠙⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⣟⣛⣷⣤⣀⣤⣾⣿⠀⠀⠀⢿⣿⣮⣟⣫⣾⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⢰⣾⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣦⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣤⣤⣤⣤⣤⡀
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⣼⣾⣿⣿⠀⠀⠀⠀⣠⡴⣶⣶⣶⣶⣦⡶⢦⡄⠀⢸⣿⣿⡿⢿⣽⣿⣿⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⣿⣻⣟⣿⠀⠀⠀⠐⣯⡅⣩⣿⠿⠿⣯⡽⣯⣽⠀⢸⣿⣿⣷⣾⣿⣿⣿⣿⡇
⣘⣿⣦⣶⣶⣤⣤⣤⣤⣤⣼⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣋⣁⣀⣀⣈⣓⣛⣁⣀⣈⣛⣛⣛⣛⣛⣛⣛⣛⣁
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⢖⣒⣒⣒⡒⣒⣖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⠒⠒⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣗⣻⣷⣲⣿⣒⣾⣗⣲⣾⣓⣶⣛⣶⣞⣳⣶⣛⣷⣞⣛⣒⣻⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣰⣛⣓⣓⣛⣿⣛⣻⣟⣒⣿⣓⣺⣟⣚⣿⣛⣺⣷⣛⣿⣛⣻⣚⣚⣛⣛⡄⠀⠀⢂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠛⠙⠛⠋⠙⠛⠛⢻⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠛⠛⠛⠛⠋⠛⠛⠛⠛
⠀⠀⠀⠀⠀⢰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠈⠉⠉⠉⠐⠒⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⢋⣷⢄⡀
⠀⠀⠀⠀⢀⣀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠛⢁⣴⠟⠁⠀⠉⠢⡀
⠀⠀⠀⠻⣥⠄⠒⠒⣉⣱⢤⡀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠉⢀⡴⠋⠀⠀⠀⠀⠀⣠⣿⣄
⠀⠀⠀⠀⠀⠑⢯⣁⡩⠶⣔⠛⢢⣀⠀⠀⠀⠀⡜⠀⠀⠀⣀⣠⣤⣶⣿⣿⡿⠟⠋⠁⠀⡠⠞⠁⠀⠀⠀⠀⠀⣠⣾⣿⡿⣩⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣖⠊⣛⣥⡨⠷⢄⠀⣼⣶⣶⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⢉⣼⡿⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡤⠝⠶⣊⠙⣻⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⢁⣴⠟⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠉⠙⠿⣖⣒⡭⢿⠤⠤⢐⣒⡺⡍⠑⡔⢢⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠋⣠⠶⠋⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠙⡹⣆⠈⠆⢿⠿⣿⠛⢸⠀⢱⠴⠁⢀⣠⣤⣶⣿⣿⠿⠛⠉⣀⠴⠊⠁⠀⠀⠀⠀⢀⣼⠟⢉
⠀⠀⠀⠀⠀⠀⢈⠀⣀⣀⠤⢄⠀⠀⢡⢻⠀⢸⠘⢃⣉⣶⣼⣤⣾⣶⣿⣿⡿⠟⠛⠉⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⣠⡶⠋⢁⣴⡿
⠀⠀⡠⢤⣤⡲⢯⠭⠔⠒⠊⠉⠀⠀⢸⡿⠄⢺⣿⣿⠿⣉⡩⠷⢤⡚⢫⣄⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⣴⡿⢛⠇
⠀⠀⠑⠉⠀⠀⠀⠣⡀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢣⣤⣤⣼⠶⣖⠊⢙⣧⣤⠵⢤⡀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⣠⣶⠟⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠖⠁⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠙⢤⡤⠜⠳⣎⣉⡦⣤⠔⠚⠉⠀⢀⣠⣴⠟⠋⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣤⣤⣤⡼⠶⣉⣉⠹⠤⠘⠓⢤⣴⠾⠛⠉⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢍⣀⣀⣀⣬⣷⣤⠴⠖⠚⠉⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠫⠥⠀⣀⣀⣀⠠⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⡹⠛⡟⢯⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢅⡀⡼⠁⠀⡇⠀⢳⣀⣠⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢹⠛⠒⠲⡗⠒⠛⣏⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⢺⠒⠒⠲⡗⠒⠒⣿⠒⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣀⠼⡖⠒⠚⡟⠂⢶⠦⠤⣄⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡁⠀⢳⡀⠀⡇⠀⠸⡆⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣙⣤⣧⣞⡥⠳⠋⠲⣌⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠤⠤⠤⠤⠤⠶⠶⠖⣒⡒⠒⢦⡀
⠀⠀⢀⠔⢚⣛⠉⣉⡭⢭⠭⠭⠭⠜⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉⠉⠉⠉⠉⠉⠁⢽⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡀⢉⢑⡒⠢⠤⢄⣀⡀
⠀⠀⢸⠰⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⣼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠇⠀⠈⠀⠉⠀⠒⠢⢤⡉⣆
⠀⠀⠸⡆⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⠀⠀⠀⠀⠀⠀⠀⠀⣾⢱⡇
⠀⠀⠀⢷⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢐⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇⠀⠀⠀⠀⠀⠀⠀⢐⡏⣾
⠀⠀⠀⠸⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⠁⠀⠀⠀⠀⠀⠀⠀⠸⢰⠇
⠀⠀⠀⠀⢳⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇⣇⡴⠀⠀⠀⠀⠀⠀⢀⡇⡾
⠀⠀⠀⠀⠘⡇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡇⠀⠀⠀⠀⠀⠀⠀⣾⢸⡟⠁⠀⠀⠀⠀⠀⠀⠸⢸⠇
⠀⠀⠀⠀⠀⢹⡈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢸⡇⠀⠀⠀⠀⠀⠀⢠⡇⡼⠁⠀⠀⠀⠀⠀⠀⠀⢇⡿
⠀⠀⠀⠀⠀⠀⣇⠹⣠⠆⠀⢀⠀⣀⢀⠀⠀⢀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠤⠤⠒⣋⣸⠃⠀⠀⠀⠀⠀⠀⢸⠁⠃⠀⠀⠀⠀⠀⠀⢠⡼⣸⠃
⠀⠀⠀⠀⠀⠀⠹⣄⣙⡛⣋⣉⡉⠉⠉⠭⠭⠤⠤⠴⠶⢶⡒⢒⠚⠋⠉⠉⠉⠭⡉⠁⠳⣄⠀⠀⠀⠀⠀⠈⠑⠊⠭⢀⣘⡃⠤⠔⡚⢃⡟
⠀⠀⠀⠀⠀⢀⠜⠁⢠⠆⠀⠀⠀⠙⠀⠀⠀⡁⢀⡉⡁⢀⠀⢀⠀⠀⠀⠰⠄⠉⠁⡄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠚⠁
⠀⠀⠀⠀⡰⠃⠀⠠⠋⠀⠠⠄⠀⠀⠀⠃⠒⠒⠂⠐⠒⠋⢀⣉⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣂⣀⣀⡀⣀⠀⠀⢀⡓⣀⠀⢀⣀⣀⣀⣠⣤⣠⣤⣤⡼⠆
⠀⠀⠸⣄⡀⣀⣀⠠⠄⠠⠤⠤⠄⠒⠒⠒⠒⠙⠒⠒⠒⠚⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠴⢁⠿⠶⢮⣉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣤⢤⡤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡴⠏⠀⢐⠀⠀⠀⠉⠙⠓⠦⢍⡛⠲⡄
⠀⠀⠀⠀⠀⠀⡟⠠⠀⠉⠉⠉⠉⠉⠉⠒⠒⡶⢒⣂⣀⣀⠤⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠋⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⣠⠞⢡⡞⠁
⠀⠀⠀⠀⠀⢰⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⢾⡇⠇⢠⣶⣀⠨⣩⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠞⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⡴⠃⡴⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⢘⢿⣿⣤⣿⠇⡿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣠⠎
⠀⠀⠀⠀⠀⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⣘⣂⣀⡙⠛⣉⡹⢰⡇⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⠞⢡⡾⠁
⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⠟⡿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⣆⡀⠲⣶⣭⡒⠤⣄⣀⣀⣰⠋⣠⠟
⠀⠀⠀⠀⠀⠙⠛⠛⠛⠲⠶⠤⠤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣭⣥⡒⠀⠉⠉⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⠰⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⡀⠸⢀⣄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⣀⣴⣾⡯⠤⠾⠛⠁⠀⠀⠀⢠⣿⣥⣿⣿⡷⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠛⠋⠂⠈⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣇⠄⠈⣾⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠏⠈⡄⠀⡠⠔⢏⠉⢹⣇⣇⣦⠖⠁⠀⣴⣿⣿⣿⡆⠀⠀⠀⠤⣿⡀⠃⣼⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⣢⣇⣴⠿⣦⣤⣼⣿⠾⣷⣆⠀⠹⣿⣿⡿⠃⠀⠀⠀⡖⣿⣏⣹⣿⢸
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠄⠄⠀⠀⠈⠁⢻⣄⣽⣿⡿⠧⠤⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⢸⣿⣹
⠀⠀⡤⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⢰⠀⠀⢻⣥⣽⣷⠀⣠⣴⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⡷⡿⡇⢸⢿⢾
⠀⠀⣤⣀⣴⡇⠀⠀⠈⠉⠛⠶⢶⣾⣄⣤⡀⠀⠈⢿⡏⠀⣿⡓⣶⣵⡀⠀⠀⠀⠀⠀⠀⣿⠃⡇⢸⡎⢿⡀
⠀⢾⣿⠋⠙⠳⣄⠀⠙⠓⠶⠟⠋⠀⠈⠉⠉⠀⠀⠈⢿⣤⣴⡝⢿⣟⠁⠀⠀⠀⠀⢀⣾⣹⡿⢹⣾⢹⣿⣷⡄
⠀⢠⣿⣦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠁⠘⣉⠀⠀⠀⠀⠀⠀⠁⣟⣷⢿⣿⢾⣘⠆⠁
⠀⠉⠀⠀⠉⠳⢴⣄⣀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠻⢥⡑⣄⠀⠀⠀⠀⠘⠁⠈⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢂⠀⠀⠉⠁⠀⠀⠀⠀⣠⡖⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠈⠙⠲⢕⢄⠀⠀⠀⠀⠀⠀⠀⢹⣤⡆⠀⠀⣰⠃⠀⠘⠋⠉⠠⠂⡀⠐⠀⣆
⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⢀⢈⠀⠀⠀⣄⠁⠰⠆⠀⡀⣐⣰⣿
⠀⠀⠀⠀⠘⢠⣤⡀⠀⣠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣤⠶⠷⠛⣨⠀⠀⠈⣶⣴⣰⣾⣥⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠛⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡤⠚⠋⠀⣀⣴⣾⣯⣄⣀⠀⠈⠙⠛⠛⠛⠛⠁⠀⡀
⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠚⠋⠀⢀⣤⡼⢟⣋⠥⠭⠯⠥⠭⠵⠖⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠈⠁⠀⠀⢀⠐⠁⠀⣠⢾⠀⣀⣴⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠔⠂⠄⠀⠄⠒⠐⠁⠀⠰⠿⠷⠛⠋⠉⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢠⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠒
⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠤⡀
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠏⠀⠀⣼
⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠍⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠒⠒⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡒⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣠⠆⠀⠀⠂⢄⠀⠀⠀⠠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠓
⠀⢀⡠⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⢤⣀⠀⠀⠀⢀⠤⠖⠒
⠀⠂⠀⠀⠀⠀⣨⣉⣉⣏⣙⣛⣟⣛⣻⣛⣛⣻⡒⠀⠀⠀⠀⠀⠀⣩⣭⡭⢷⠀⠈⢣⠀⠀⠻⢤⡀
⠀⠀⠀⠀⠀⠠⡯⠤⢼⡧⠬⢽⠭⠥⢼⠤⠤⢼⠄⠀⠀⡀⠀⠀⠀⠠⢾⡶⠘⡇⠀⠈⡆⠀⠀⠀⡸
⠀⠀⠀⢓⣒⣛⣉⣉⣉⣉⣉⣛⣛⣓⣒⣒⣒⣒⣒⣒⣒⣁⣀⣀⡀⠀⠀⠁⠐⠃⠀⠀⢸
⣀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣖⣁⣀⣤⣤⣄⣀
⣿⣯⣬⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⢛⣿⡿⠋⠉⣿⠉⠙⠻⣿⡛⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⢠⣿⠋⠀⠀⠀⣿⠀⠀⠀⠙⢿⣄⠀⠈⠙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣋⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣘⣿⣀⣀⣀⣀⣀⣘⣿⣆
⠀⠀⠀⠀⠀⠀⣰⡿⠛⠛⠛⠛⠛⠛⢻⡿⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⠛⠛⢻⣟⠛⠛⠛⠛⠛⠛⢿⣆
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⠇
⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⣷⣶⣤⣈⡁⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⡉⠉⠉⠉⠉⢻⣏⠉⠉⠉⠉⠉⣿⠉⠉⠉⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⢿⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠘⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⠈⢻⣦⡀⠀⠀⣿⠀⠀⢀⣴⡟⠀⢹⣿⣿⠟⠙⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢶⣿⣿⣶⣤⣿⣤⣴⣿⣿⣶⠷⠀⢿⠏⠀⠀⠀⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠟⠃⠀⢰⣤⠀⠀⠀⠚⢉⣈⣷⣀⠀⠀⠀⠀⣰⠆⣠⠀⢀
⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠤⠤⠄⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⡀⣀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣤⠐⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⣶⣴⡄⠀⠀⠀⠀⠀⠢
⠀⠀⢿⠀⠖⠉⠁⠀⠀⢠⣀⣠⣄⣤⣤⣤⣴⣶⣶⣶⡇⠀⠀⠀⡀⠀⠀⠀⣿⡿⠛⠛⠋⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠽⠿⠍⠀⠀⠀⠀⠀⠀⣉⢙⡆⠀⠀⠉⠁
⠀⠀⠀⠀⠑⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣍⣩⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢻⡶⠁
⠀⢠⠀⠀⠀⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣐⣊⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀
⠀⠀⠀⣸⡀⠀⠀⠀⠦⠤⠤⣴⣶⡶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣄⢀⣤⠀⠀⢠⢶⣍⢺⣿
⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠈⠉⠉⠀⠉⠀⠀⠀⠤⠊⠉⠁
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠠⠴⣶⠖⠢⠄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⣠⠶⠛⠒⠆⡀⠐⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠒⠒⠁⡄⢠⡄⢠⠈⠒⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠌⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⠀⢀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠁⡌⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢰⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠈⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⠔⠁⠀⠀⡠⠂⠀⠀⠋⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠘⢤⡀⠀⠈⠢⡈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀⠔⠁⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠢⡄⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠂⠁⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠈⠐⢄⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠊⠁⡠⠊⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠧⠤⠄⠠⠤⠤⠠⠤⠤⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄⠈⠑⢦⡀
⠀⠀⠀⣰⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠀⠠⠠⡔⢢⠤⠄⠀⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠢⡀⠙⣆
⠀⠀⠀⢹⠀⠃⠒⠀⠒⠒⠒⠀⠒⠂⠀⠒⠒⠒⠒⠒⠒⠂⠖⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠲⠐⠀⠒⠒⠒⠒⠀⠐⠂⠒⠂⠒⠒⠒⠂⠒⠘⠀⡏
⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠚⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠃⢤⣀⣀⡀⣴⢿⡿⢠⣀⣀⣀⠠⡞⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣀⣭⣹⣯⣨⣅⣀⣈⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡿⠋⣍⣀⣀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠯⠿⠯⠽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣶⣶⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠀⠀⠀⠀⠀⠀⠀⣾⡟⢛⠛⠛⠛⣿⡀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢘⣿⠁⠀⠀⠐⠀⡿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠈⠻⣄⣠⠄⠀⢠⠗⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣈⡟⠀⣄⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣆⠀⠟⢦⣳⠖⠏⢹⠆⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣷⣀⠀⠀⠀⠀⡿⠄⠐⢘⣈⠐⠀⡃⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠿⠿⠿⠷⠟⠛⠛⠛⠻⠿⠿⠽⠯⠿⠿⠿⠭⠿⣽⠿⠽⠿⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠩⠒⠻⠟⠿⠛⠿⠳⠦⠅⢘⣊⣐⣦⣭⣼⠿⠿⠿⠿⠿⠿⠿⠞⠳⠌⠢⣄
⠀⠀⠀⠀⠀⢠⣴⠩⠤⢤⣤⡤⡤⠤⣤⡤⣤⣤⣤⢤⣯⡽⡽⣿⠿⡾⢤⢤⡤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣵⣦
⠀⠀⠀⠀⠀⠈⠛⠚⠛⠛⠛⠓⠚⠚⠛⠛⠓⠛⠛⠛⠛⠚⠒⠛⠛⠓⠚⠛⠚⠓⠛⠚⠛⠛⠛⠓⠚⠛⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⡀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⠀⠀⡘⠁⠀⠀⣀⠈⠹⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡁⠀⠸⠠⠄⠀⠀⠀⠀⠀⡠⡻⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⡀⠀⠀⠀⠂⠀⠐⠀⠀⠀⡀⠄⢀⡄⠀⠀⠀⠀⣀⠀⠀⠃⠸⡆
⠀⠈⠀⡀⠀⠊⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠛⠉⠉⠉⠀⠀⠋⠀⠀⢠⠐⣷⠀⠀⢰⠷
⠀⠀⠀⠼⣿⠧⡤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡎⠀⠀⠀⠀⠀⠀⠀⢬⢾⡆⠀⠈⡇
⠀⠀⠀⠀⠠⢑⣬⡇⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠈⢾⠃⠀⠀⠆⠀⠀⠠⠐
⠀⠀⠀⠀⠀⡸⡛⡇⠀⠀⠀⠀⠀⠂⠁⡀⠀⠀⠀⠀⠀⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠁⢱⠀⠀⠀⡤⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⢞⢃⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⢀⠷⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠔⢀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠄⠀⠀⠀⠀⠀⢐⣒⣾⡄⠁⢀⡄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠧
⠀⠀⠀⠀⠀⠀⠀⢉⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⢱⠄⢡⠐⠂⠀⡌⠐⢄⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠄⠀⠀⠘⢇⠀⠀⠀⠀⠀⢠⡈⡀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⠀⠀⠀⠀⠀⠀⠈⡄⠰⠘⠀⠀⢤⠀⢠⡈⡆⠀⠀⠀⠀⣈⠃⠐⣤⠐⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢇⠈⠈⠀⠀⠈⠓⢄⣳⡙⡄⠀⠀⠀⠈⠋⠀⠉⢦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡅⠀⠀⠀⠀⠀⠀⢈⠀⠀⡀⠀⣿⣤⠀⠉⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠔⠈⠑⠀⠙⠋⠂⠀⠈⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠢

⠀⠀⢀⡔⡾⠻⡗⢦⠀⠀⠀⠀⣠⠞⠳⣶⠀⠀⠀⠀⠀⢠⢞⠟⠓⣄⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⣠⣾⣷⡀⠀⢀⣠⠚⠉⠓⠤⣄
⠀⠀⡞⠉⡟⠉⡟⠉⡇⠀⠀⠞⣡⣾⣷⣌⠳⠀⠀⠀⠀⢸⡈⠀⢀⡝⠀⠀⢾⣏⣿⣹⣏⣿⡇⠀⠠⣤⡻⣿⣿⣿⡿⠀⠀⢻⣀⣀⣀⣀⣀⡼
⠀⠀⠹⣏⣯⢉⣏⡽⠁⠀⠀⢸⣿⠉⠉⣿⡇⠀⠀⢠⣾⡟⠉⠉⠉⠀⠀⠀⢨⠿⠿⠿⠿⠋⠀⠀⠀⠙⠻⠷⡿⠛⠁⠀⠀⠀⠀⠸⣿⡷
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⢠⠀⢀⣀⠀⠠⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⢰⣝⢿⣿⢟⣵⡄⠀⠀⠰⠋⠭⠭⠹⠆⠀⠀⢀⣾⠫⣗⢪⢻⡆⠀⠀⠘⠀⠈⣟⠀⠰⠁⠀⠀⣠⣉⠙⢷⣄⠀⠀⠀⠀⣠⣄⠴⠚⠛⠁
⠀⠀⠸⣫⣶⣶⣮⡻⠅⠀⠀⣿⡦⡉⢉⢴⣿⠀⠀⠈⢿⣑⡭⣬⣾⠇⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⠀⢀⡉⢻⡄⢹⡆⠀⠀⠀⠙⠋⠳⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠞⠛⠛⠳⠉⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠐⠋⠉⠙⠀⠀⠀⠀⠙⠁⠀⠋⠀⠋⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠌⡀⣐⠀⠀⠀⠀⠀⠀⠀⠁⣷⠉⠀⠀⠀⠀⠀⠀⣴⢂⢶⡄⠀⠀⠀⣀⡀⢀⣼⣇⣀⡀⠀⠀⢸⣿⣿⣷⡆⣿⡆⠀⠀⣀⣠⢿⣆⣀⡀
⠀⠀⠀⠁⠸⣿⣷⠂⠀⠀⠀⠀⣦⣿⣿⣶⠀⠀⠀⠀⢠⣦⣙⣰⡆⠀⠀⠀⣿⡇⣿⣿⣿⣿⠃⠀⠀⠿⢿⣿⡿⠇⠿⠇⠀⠈⠻⣷⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠉⠙⠷⠄⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠈⠿⣿⡿⠃⠀⠀⠀⠛⠃⠛⠛⠛⠛⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠰⠟⠛⠻⠇

⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⢀⣴⣶⣶⡆⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⢰⣶⣤⣤⣄⡀⠀⢠⣶⣦⣠⣶⣆⠀⠀⠀⠀⢀⣴⣶⣶⣄
⠀⠀⣶⣰⣿⣾⡇⠀⢢⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⣿⠟⣩⡛⣿⣷⠀⠀⡖⠚⠚⣚⠒⣾⡏⠀⠀⣽⣯⣬⣿⡅⠀⡀⠀⠀⣿⣿⠈⠙⣿⡇
⠀⠀⠉⠈⠻⣿⡇⠀⠂⠀⠀⢠⣾⡇⠸⡿⠇⠀⠀⢸⣿⣎⣛⣣⣿⣿⠀⠀⣧⣾⣷⣤⣦⡟⠃⠀⠀⣿⣿⣿⣿⡇⢿⡇⠀⠀⠻⣿⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⣤⠤⠤⠤⠤⠤⡄⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⢠⠤⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠀⠀⠀⠀⡤⠤⠤⢤⣄
⠀⠀⣿⠀⠰⠦⠀⠀⡇⠀⠀⣿⠀⠰⠆⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⣶⢸⣿⣷⣶⣶⣶⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⡇⣉⣉⣨⠙⡆
⠀⠀⠻⠶⢶⣶⠶⠶⠇⠀⠀⣙⣶⣶⣶⣶⣏⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⢹⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⡇⣒⣒⣒⠀⡇
⠀⠀⠀⠰⠾⠿⠶⠀⠀⠀⠐⠿⠿⠷⠾⠿⠿⠂⠀⠀⠘⠿⠟⠿⠟⠀⠀⠀⠘⠾⠿⠿⠿⠿⠟⠀⠀⠘⠿⠿⠿⠿⠃⠀⠀⠀⠳⠶⠶⠶⠶⠃
⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⣠⣮⣀⣀⣀⣧⡄⠀⠀⠈⣧⣤⣤⣤⣤⠄⠀⠀⢀⣀⣀⢻⣽⠇⠀⠀⠀⠀⣠⣾⠅⣸⠂⠀⠀⢠⠞⠋⠋⠲⡄⠀⠀⠀⢰⠁⠀⠈⡃
⠀⠀⣿⡿⢿⣿⡿⣿⡇⠀⠀⠀⢹⣿⡿⠿⠟⠀⠀⠐⣾⠟⢻⣷⠀⠀⠀⠀⢠⠞⣟⣡⣾⠇⠀⠀⠀⢿⡀⠀⠀⢀⡟⠀⠀⠀⠀⢣⡀⡰⠃
⠀⠀⠉⢣⣘⣓⣠⠟⠁⠀⠀⠀⠠⡶⠠⣶⠀⠀⠀⠈⠻⡷⢿⠛⠀⠀⠀⠀⠸⢤⣬⠟⠁⠀⠀⠀⠀⠈⠓⠴⠶⠚⠁⠀⠀⠀⠀⠀⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡭⠍⠩⠭⠉⠉⠙⣛⣛⣓⡒⠒⠶⠶⠶⠦⠤⠤⢤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠒⠂⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⡀⠀⠀⠀⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡃⢒⡉⠭⠥⢂⣒⠀⠤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⡉⠐⡈⠘⠠⠐⢫⠐⡂⠔⡀⠌⢉⠁⢐⠂⠈⠭⠅⠐⣒⠂⠤⠄⠀⣀⢸
⠀⠀⠀⠀⡀⠔⢊⠁⢒⠀⠒⠠⠀⡁⠀⠃⠰⠈⠡⢈⠅⢊⠀⠃⠀⠄⢈⠅⠊⠀⠀⠄⠀⠍⢀⣰⡞⡺
⠀⠀⠀⠸⣉⣁⠒⠒⠠⠴⠤⣐⣀⠆⠁⠀⠀⠀⠀⢀⠐⠀⠄⢈⠄⠘⠀⠄⠨⠄⢈⡠⠔⣊⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠠⠤⠌⣉⣁⠒⠒⠠⠤⠦⣀⣂⡤⠀⠥⢀⡡⠔⣊⠥⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠀⠤⠄⣉⣹⡡⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⣠⠤⠔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⢠⡤⣼⢾⢚⢚⡹⠀⠀⠀⠀⠀⢀⣀⠠⠆⠀⢠⠛⠠⣤⣄⣀⣀⠀⠀⠀⠀⠙⠃⠀⠚⠁
⠀⠀⠀⠀⣆⣏⣥⡦⠶⠒⠒⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠀⢀⣠⠴⣺⠟⠋⠉⠀⠒⠀⠀⠀⠠⠞⢈⣹⠎⠍⣳⡄⠀⠀⠁
⠀⠀⠀⠀⠏⡧⠖⠗⠁⠀⠀⠳⣶⣤⣶⡏⠀⠀⠀⠀⠀⠀⢺⣧⣀⠈⠓⠒⠖⠒⠶⠒⠶⠒⠚⠋⢉⣀⣤⡾⠟⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠔⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⣀⢤⠶⠋⠁⠀⠀⠀⠀⢀⡀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⢀⠠⠀⠫⡽⢀⠘⠋⠀⣠⣀⠀⠀⠠⡀⠀⠀⣖⢻⢻⠚⡗⣦⡄⠓⠋⣼⡏⠇⢹⠸⠸⡇
⠀⠀⠀⠴⠶⠚⠃⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⣵⠀⢸⣸⣸⡤⠷⠋⠀⢀⠘⠀⠙⠛⠚⠒⠗⠃
⠀⠀⠀⠀⠀⠀⣀⣬⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⢀⠝⠀⠈⠈⠀⠀⠀⠀⠀⣸⣣⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠔⠐⠒⠀⠀⣀⡀⢀⣀⣀⡀⠀⣀⣘⠁⠀⠀⠀⠸⡀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠉⠓⢤⣸⣷⣦⠀⠀⠀⢀⣴⣶⣦⣰⡆⠀⣾⣷⠿⣿⡟⠃⣾⣯⣭⡍⣿⣏⣽⠇⠀⠀⠀⠀⡗⠒⠒⠀⡖⠀⠂
⠀⠀⠀⠀⣤⣀⠀⠀⠈⠀⠀⠙⠣⠀⠀⢼⡇⣸⣿⠹⣿⣤⣿⠇⠀⣿⡇⠀⣿⣥⣤⡀⣿⡟⠻⠦⠀⠄⣀⠀⢹⠚⠓⢺⡃
⠀⠀⠀⠀⠀⠈⠙⠡⠀⠀⣠⢤⡀⠀⠀⠘⠛⠛⢁⣀⡈⣩⣥⣄⠀⣬⣤⡀⠀⣠⣶⡄⣠⡴⠶⠆⠀⠀⠀⠀⣼⠦⠤⠼⣇⡈⠄⠤
⠀⠀⠀⡤⢤⣤⡄⡀⠀⢸⡁⢀⡽⠀⠀⠀⡀⠀⠿⣤⣅⢹⣧⣾⠗⣿⡾⣇⢸⣏⠀⡀⢿⡷⠚⠃⠀⠀⠀⠀⠀⠁⠒⠀⠉⠁
⠀⠀⠀⠀⠠⢶⣷⣤⡄⠀⠉⠉⠀⠀⠀⠁⡀⠀⠶⠶⠿⠸⠿⠀⠀⠛⠀⠉⠈⠛⠛⠁⠸⠷⠚⠛⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠘⠈⢀⡄⠂⡂⠠
⠀⠀⠀⠀⠰⠄⣀⡉⠀⠀⠀⠀⠀⣀⠤⠆⠚⠀⠈⠒⠂⠀⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⡁⠀⡀⠧⠖⠀⠀⠤⠀⣀⢀⡼⠀⠀⠆⠂⠀⢀⠀⡐⢶⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠑⠦⠦⣄⣀⡑⠠⠤⠠⠚⠉⢀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⠀⠰⠀⠑⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠈⣶⠆⠀⢤⠒⠈⠳⣤⡀⠀⠀⠀⠀⠐⠒⠘⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠑⠀⠀⠸⣦⣄⡒⠀⠠⠤⠀⠀⠂⣀⡜
⠀⠀⠀⠸⠇⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠳⠶⠖⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢨⢐⠛⠀⡅⠀⠀⠹⠀⠀⠀⠀⠆⣤⠀⠀⠆⠀⢀⡴⠋⠀⠀⠀⠙⢦⠀⠀⠀⠀⢌⠭⠤⢔⠀⠀⠀⠀⢀⡗⠛⠷⢶⣤⣀⢀⣀⡀
⠀⢹⠸⠀⠀⡇⠀⠀⢒⣆⡐⠀⠀⠇⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣻⣤⣤⣼⠀⠀⠀⠀⢸⡀⢻⠄⠀⠀⠀⢉⢮⡏
⠀⢘⠀⠄⠀⡇⠀⠨⢨⢼⡄⠀⠀⠀⢀⡀⠀⠀⠀⢧⣤⠀⠀⠀⠀⠀⣤⡇⠀⠀⠀⣿⢿⣿⣿⠀⠀⠀⢀⣬⡧⢄⣀⠀⠀⠀⡸⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⡆⠀⠀⠀⢰⠟⠁⠀⠀⠀⠙⠛⠓⠋⠀⢀⣠⣾⣶⢬⣍⣒⠭⢕⡲⢳⠇
⠀⡔⡀⠀⠀⢀⣒⠠⠀⠸⠐⣒⠒⠒⠒⠒⠒⠒⡆⠀⠀⠁⠂⠀⠀⠀⠀⢠⠄⠀⠀⡀⠀⠀⢰⡤⢜⡛⠿⣽⢺⣾⣽⣻⠗⣼⠟
⠀⡟⡧⠭⣤⣼⠙⠀⡆⠀⡆⠙⠂⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡰⠁⠀⠀⠀⠈⠑⠲⠥⣐⡨⢙⢚⢥⠔⠁
⠀⡆⠀⢤⡄⠀⡆⠀⡇⢀⠂⠀⠠⠤⠤⠤⠤⡴⠁⠃⠀⠀⣰⢠⠎⠁⣉⠉⠀⣀⣧⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠤⠀⠤⠃
⠀⡇⠀⠴⠀⠈⡇⠀⠃⠈⠁⠀⠀⡆⠀⠒⠀⠀⠉⠀⠀⠀⣇⡸⠤⠀⣀⠉⡙⠙⠁⠀⠶⠀⠀⠀⣊⠤⢄⠀⢠⠀⢠⡝⢒⡤⠖⠠
⠀⡗⠀⠀⠀⢜⣧⠎⠀⠀⠀⠀⠔⣃⠀⠼⡄⠀⠀⠀⠀⠐⠋⠀⡀⠀⢀⣜⠥⠶⠔⠒⠁⠀⠘⣐⠡⠀⢀⠥⠊⠀⠣⠄⠈⢠⠀⡠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⠇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠄⡸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠁⠀⢈⠁⠀⠀⠀⠀⣀⠠⢸⣿⠄⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠉⠉⠒⠒
⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠢⠥⠀⢊⠅⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢀⢤⠤⡤⢀⠀⠄⠤⢵⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢸⡋⢹⡟⣽⠖⠠⣌⣋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⣄⣀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠈⠐⠀⠀⠀⢠⣶⣆⣀⣀⣰⢶⡾⠰⠷⠿
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠉⢡⠌⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⡀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠄⢠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠂⠉⡉⣉⠀⠀⠀⠀⠀⠠⠤⠤⠤⠴⠧⠤⠤⠤⠤⠤⠤⠤⠘⠖⠒⠒⠓⠓⠒⠓⠓⠚⢳⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠆⡈⠌⡀⢌⠂⠀⠁⠌⣼⢜⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠌⡐⠐⡰⡀⢃⠆⠄⠜⡰⢾⣺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠠⠰⠠⠁⠑⠀⢈⠄⡈⣰⢏⢼⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢁⠡⢁⠒⡐⠀⠈⡐⢰⠏⡰⢘⡕⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⢡⢂⢂⡮⠀⠐⡐⡐⢠⠏⣸⠂⢊⡯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠂⢂⠂⡾⢁⠁⠐⠠⢀⡟⣸⠃⢄⢉⣧⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢄⠂⡜⢁⠂⠒⠠⠁⡞⣠⠃⠌⢄⢪⣟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠌⢈⠄⢜⢂⢀⠂⠠⢁⡞⢠⠍⠌⠌⠬⢃⣧⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠌⢀⠂⢈⠂⠀⠀⢀⠃⡾⢡⠏⠎⢌⢎⢂⠣⣯⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠄⠄⠜⠀⠌⠈⢀⠁⡼⢡⡣⠊⢌⠀⢂⢂⡡⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⡐⡈⠈⠌⠌⢌⠂⠂⢊⠠⡕⠌⢌⠎⢂⢂⢂⡕⣺⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⣀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⣀⣰⣀⣀⣀⣌⣈⣤⣉⣈⣀⣈⣉⣈⣀⣊⣈⣀⣀⣈⣬⣜⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠒⠐⠿⠲⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⡲⢶⢶⡶⡲⢖⠶⡾⡶⢶⠶⠶⠶⢖⠶⡷⢶⢞⠿⡿⠚⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⠄⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⡲⠀⠲⠂⠈⠻⠁⢘⠟⢃⡙⡎⣪⣛⢢⠙⣋⡓⡍⠈⠋⠛⠙⡊⠋⠳⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠞⡠⠋⠀⠀⠀⠀⠒⠒⢒⠒⠛⠒⠀⠁⠒⠘⠒⠒⠛⠐⡂⠛⠘⡛⣙⣒⠲⡚⣓⡒⢲⡚⡓⡒⠒⠂⠐⢂⢩⡙⢦⠑⡄
⠀⠀⠀⠀⠀⠀⢀⠎⡐⠁⠀⠀⠀⠀⠤⠤⠐⠃⠶⠐⠓⠖⠚⠓⢲⠒⠓⠒⠒⠓⠲⠒⠛⠐⠲⡷⠚⠾⠳⠶⢳⡶⣦⢦⣠⣶⣾⣧⣡⡈⢷⣽⡄
⠀⠀⠀⠀⠀⠀⠎⡐⠀⠀⠀⠒⠒⠒⠊⠃⠖⠈⠉⠀⠈⠁⠀⠐⠚⠒⠒⠚⠒⠒⠒⠓⠒⠒⠒⠓⠒⠒⠖⡒⠚⠛⠺⠛⠿⠉⠛⠛⠿⣿⡾⢿⣿⡄
⠀⠀⠀⠀⢀⠞⠀⠓⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⡉⠉⠉⠉⠉⠉⠀⠀⣀⣂⣐⣒⣒⡒⡒⠒⠒⠚⠛⠓⢎⡛⢟⠻⢟⠿⠿⡛⠻⠻⠿⢯⠳⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠉⠒⢬⡐⠡⣂⠌⡓⢄⠄⡂⠌⣣
⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠆⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣑⠨⡐⠨⡊⠕⡂⢃
⠀⠀⡎⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠭⠄⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠠⠶⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢅⡢⢑⠨⢘
⠀⠀⠙⢖⣀⢠⢤⣤⢤⠤⡤⣠⣤⣠⢄⡤⡠⣠⣠⢀⣄⣄⡄⣤⢠⣄⣄⡄⣠⣀⣀⣀⡀⢀⣀⣀⣀⣀⡀⣀⣀⣀⢀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣨⣡⣵⠞
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠘⠓⠓⠓⠚⠒⠚⠒⠓⠒⠚⠒⠚⠒⠓⠒⠒⠒⠒⠒⠓⠓⠒⠚⠒⠒⠒⠚⠒⠐⠒⠓⠒⠚⠚⠛⠚⠒⠒⠃⠓⠉⠉⠉
⠀⠀⠀⠀⡀⡀⢄⡠⢄⡀⠄⠂⠄⠐⠠⠀⡀⠤⡀⠄⡀⠄⠀⠐⡂⠤⡐⠤⢐⠠⢐⠠⠐⡂⠤⢐⠠⠐⡂⠄⢒⠠⠐⡀⠔⡠⠔⡊⠄⢒⠠⢂⠄⢂⠄⡀
⠀⠀⠀⠀⠁⠈⠀⠈⠐⠀⠀⠂⠈⠂⠀⠀⠀⠐⠀⠐⠀⠑⠈⠁⠂⠁⠂⠡⠀⠈⠐⠀⠁⠂⠨⠐⠀⠀⠀⠨⠐⠈⠔⠂⠁⠂⠡⠐⠈⠐⠈⠐⠊⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡀⠃⠐⠒⠀⠀⠀⠀⠀⠒⠘⢀⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠒⠒⢰⣏⣀⣀⣹⡤⠀⠒⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢀⠀⢘⠠⢨⡿⠭⡓⠆⠃⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠈⠁⠁⠰⠆⠌⠈⠁⢀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⢹⡖⠂⠐⠂⠀⢲⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠰⢀⣾⡇⠀⠀⠀⠀⢸⣷⡀⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⢿⡃⠐⠀⠀⠀⢘⡻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣰⡏⢸⡇⠀⠀⠀⠀⢸⡅⢹⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⠘⢙⡏⠉⠉⠉⠉⢹⡋⠃⢿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⢖⠁⠁⡆⢘⡃⠀⠀⠀⠀⢘⡁⢸⠈⠨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⠁⡠⠊⠀⢨⡇⠉⠉⠈⠉⢸⡇⠀⠑⢄⠈⠪⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠔⢁⠄⠊⠀⠀⠀⢸⣇⢀⡀⢀⡀⢸⡇⠀⠀⠀⠁⠠⡈⠢⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⡡⠂⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⢴⡦⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢳
⠀⠀⠀⠀⠀⠀⠀⣧⣉⣉⣉⣉⣭⣉⣉⣍⣉⣩⡇⠀⠀⢸⡇⠀⠀⢸⣍⣉⣉⣉⣉⣭⣉⣉⣩⣉⣼
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⠀⢀⡤⠀⣀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣀⠀⢠⡀⠀⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⣤⠤⠬⡆⢲⠤⢤⣾⡷⣤⠤⠾⢰⠏⠤⣬⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢁⠈⣿⣿⡿⠿⣿⣿⡇⠘⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣶⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣶⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠿⠉⠉⠉⠉⠉⠉⠉⠉⠉⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡶⢶⢶⢶⠶⡶⡶⢶⢶⡶⠶⡶⡶⡶⢶⠶⡶⡶⢶⢶⠶⣇
⠀⠀⠀⠀⠀⠀⣠⢞⢑⠛⠗⡛⡻⣻⢚⠟⡱⠫⠞⠸⠇⠳⢻⡙⡊⢳⠓⡙⡚⡎⢛⢊⢧
⠀⠀⠀⠀⠀⢾⠓⠒⣶⡖⠒⠲⠶⠖⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⠖⠒⠚⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢋⡽⠟⢻⠛⢿⣍⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⣰⠟⠁⠀⢸⠀⠀⠙⢷⡀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⣼⠃⠀⠀⠀⢸⠀⠀⠀⠀⢻⡄⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢻⡄⠀⠀⠀⢸⠀⠀⠀⠀⣼⠃⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠹⣆⠀⠀⢸⠀⠀⢠⡾⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣈⣳⣤⣸⣠⣴⣋⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠋⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣧⡇⢹⡏⣾⠍⢸⣽⠂⣷⣾⢸⡯⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠁⠈⠁⠉⠉⠈⠈⠁⠁⠉⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠭⣽⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠟⠋⡁⢄⣤⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⡛⢉⡁⢀⣢⣤⢶⣿⣿⣿⣿⣿⠿⠟⠟⣩⣝⢿⣿⣿⣿⠟⠡⠀⠂⣀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣏⠰⠟⣡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠀⠀⠀⡝⢏⢷⣻⣿⠰⠀⢀⠰⡆⠀⣂⣿⡇
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠛⣿⣿⣷⣨⣿⣿⣿⣿⠛⣉⡀⠀⠀⢀⠀⠀⡄⣇⢸⡄⣏⣿⣷⣄⣈⣐⣆⣬⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣶⣿⣿⣿⡟⠋⠉⠁⣌⢰⣉⣽⢀⠀⢸⡄⠀⠸⡜⢾⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⣿⣿⣴⣿⣿⠉⣟⣡⣴⣧⣤⣤⣶⣿⣼⣇⢹⣿⣿⣿⣿⣶⣶⣾⣢⣭⣿⣿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⢏⠉⠛⢿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣽⣿⡿⢻⣿⠟⠿⣿⣿⡻⣿⣿⠛⢻⣿⢍⠚⠀⠈⠉⠑⠈⢻⣿⣿⣿⡄
⠀⠀⠀⢸⡿⣫⡟⠡⠤⠖⢀⠀⠀⠘⢻⣯⣻⣿⡿⣿⣧⣽⣶⡇⢰⣿⣿⣷⣜⠪⠐⢾⣿⣇⠠⠀⠀⠀⠀⠉⢈⣭⡛⢿⠁
⠀⠀⠀⠈⣱⣏⠀⠠⠀⠀⠀⠀⠀⣀⣤⠹⠷⢻⣷⣿⣿⠿⢿⣳⡊⠛⢿⣿⣿⣷⣧⣶⣦⣍⡛⠳⠶⢦⣤⣤⡄⠀⠐⡀⢄
⠀⠀⠀⢠⣿⠒⠖⠀⠀⢀⢀⣀⣠⡟⠈⢘⢸⣿⣿⣷⡝⣅⣀⣭⣭⣁⣟⢡⣿⣟⢻⣿⣿⣄⠙⠛⠳⢶⣦⣬⣤⣀⢀⣠⣾⣫
⠀⠀⠀⠸⣿⣳⣖⡻⢿⣿⡿⠟⠋⠀⠁⠀⣺⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠘⣿⣿⢿⣿⣿⣿⣿⣶⣿⣿⣿⡟⢿⣿⡿
⠀⠀⠀⠀⢻⣅⡀⠉⠀⠉⠀⠀⡀⢀⣢⣞⣿⣿⣿⡇⡌⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣷⠛⠛⠛⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠿⣤⣀⠀⠠⠀⢶⣯⢷⣿⣿⣿⣿⢡⢿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⠿⣛⡀⠀⣸⣿⣿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢬⣭⣭⣭⣶⣿⣿⣿⣿⡏⣌⣾⣿⣿⡿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣿⣿⣿⢶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢦⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡟⣼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣧⢻⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠙⠛⠛⠛⢇⠛⠛⠛⠛⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⠖⠚⠛⠋⠉⠉⠙⠓⠒⠲⠶⢤⣀⡀
⢀⣤⣤⠤⠶⠒⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠴⠒⠊⠉⠉⠀⢹
⡇⠀⠀⠉⠐⠒⠤⢄⣀⣀⠠⠤⠖⠒⠒⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⡖⠒⠂⠤⠤⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀
⣷⠙⠢⠤⠤⠀⠸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⣴⠛⠛⠛⠓⠒⠛⠉⠩⠭⠭⠍⠉⠉⠉⠉⢉⡉⠉⠉⠉⠉⠩⠭⢭⢍⡙⢦
⢸⠀⠐⠒⠂⠤⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠠⠒⠒⠀⠠⠤⠤⠄⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⠀⠈⢁⣀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠢
⢸⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⠇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡆⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⡇⠀⠀⠀⠀⡅⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⡇⠀⠀⠀⠀⠉⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⢣⠀⠀⠠⠒⡄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⠀⠀⠘⠒⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠾⣿⡙⡇⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠸⣄⠀⠐⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠊⠉⣀⠠⣴⡿⢁⢧⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠒⠒⠠⠦⠤⠴⠃
⠀⠀⠈⠓⠶⢤⣀⠀⠀⠀⡇⠀⣀⣀⣠⡤⠤⠶⠒⠋⠉⣀⣀⣀⡤⢂⣭⡴⠞⣻⠞⠓⠋⣉⣉⡻⢿⣶⣶⠖⠒⠶⡶⠤⠤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠾
⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⡰⣋⠥⠶⣏⡀⠙⠓⣚⣛⣛⣭⠭⠽⠖⠒⠋⠉⢁⡤⠤⠤⠤⣟⡀⠀⣫⣀⣀⠈⠦⢬⣷⣒⡺⠭⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣠⠤⠴⣶⡶⠿⠛⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣔⣁⣀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉⠉⡷⠀⠀⠉⠓⡌⠱⡄
⠀⠀⠀⠀⠀⠀⣀⡤⠶⢯⡉⠉⡎⢳⠋⠉⠀⠀⠀⣠⢖⡒⡒⠂⠤⠤⠤⠤⠤⢤⣤⣄⣀⣀⣀⣀⣀⣉⡉⠉⠉⠉⠉⠛⠛⠒⠒⠒⠒⠊⠀⠀⠀⢀⣠⠇⠀⡇
⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠉⣉⡤⠞⠀⠀⠀⢀⠜⠁⣜⡖⢁⢙⠘⢀⣂⠒⠐⠒⠆⠀⠦⠄⠠⢤⠄⣭⣉⢉⣉⠉⠙⣶⣒⠒⠲⠶⠶⠶⠒⠒⠋⠉⠀⣀⡴⠁
⠀⠀⠀⠀⠻⣄⣀⣀⣀⠴⠊⠁⠀⠀⠀⢀⠔⢁⣀⡛⠃⢀⣦⠆⠠⠯⠈⢠⣬⠙⢁⣐⠒⠂⠠⠶⠄⢀⠤⣄⠀⡠⢊⡼⠉⠉⠉⠉⠛⠛⠒⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠰⡧⢀⣈⡁⠁⠈⠋⠩⠭⠭⠍⠁⠲⠖⠒⠂⠀⠰⠐⣓⣐⡁⠉⠴⣡⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠤⠤⠬⠭⠭⣭⣉⣀⣒⣒⣒⠒⠒⠒⠒⠂⠠⠭⠭⠥⣀⣔⡵⠋

⠀⠀⠀⠀⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠿⠿⠿⠿⠷




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⠢⠄⠀⠀⠀⠀⡄⠩⡒⣂⠥⠀⠣⠄⠚⢃⠀⠀⣄⠀⠙⠀⢸⠀⠀⠘⡇⠀⠀⠀⠀⡇⡇
⣈⠔⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠂⠉⣀⠠⠄⠒⠁⠀⠀⠙⠐⠒⠒⠁⠀⠀⠀⡝⠀⠀⠒⣊⡡⡣⠀⠀⠠⡀⠴
⡉⣦⠄⢻⠂⠀⢀⠔⣊⠩⠤⠤⠍⣑⠢⡀⠀⠀⠑⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠘⠓⠛⠋⠉⠁⠀⠀⠀⣵⠀⡦
⢡⠠⠄⡜⠀⠰⡡⡪⢒⠩⠥⠬⢅⠢⡑⠬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠋⠉⠉⠀⠑⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢣⡠⠰
⣢⡴⠔⠀⠀⠀⠘⠔⡡⢊⠉⠉⡒⢝⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢨⡇⢀⣋⣀⣘⠀⠀⠀⠀⠀⢠⡐⠊⣩⡞⠉⠀⠀⠀⣠
⠉⠀⠀⠀⠀⠀⠀⠀⠉⢡⠂⢲⠈⠁⠀⢀⠤⠠⠔⠒⠚⠉⠉⡆⠀⠈⠋⠀⠈⠀⠰⠄⢠⠄⠀⠀⠀⠀⠀⠀⠈⣏⢔⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⢰⠀⠀⢀⢀⠂⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠤⠐⠢⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠂⢖⠏⠫⠆⠀⠀⠀⢀⡠⠄
⡐⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠌⠂⠄⡌⠉⡄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡠⠄⢒⡢⠉⢰⠊⠙⣞⠀⠀⠰⡄
⠈⠀⠀⢸⠄⠀⢠⡋⠉⠒⠘⠀⠀⠀⠀⠀⠀⠀⠚⠓⠊⠉⠉⢀⢀⠤⡖⡀⡰⠠⣀⠀⠀⠈⡄⠩⢐⡂⠁⠀⠦⠤⠘⠇⠀⠀⠇
⠀⢤⠉⠀⠁⠀⠀⠉⢀⡐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣁⣤⠰⡊⢢⠠⠘⠥⠀⠀⢡⠈⠒⠂⠈⣀⡠⠄⠒⠁⠀⠀⢃⠒⢀
⠂⠉⠛⠀⠀⠀⡔⠉⡍⠑⣤⠀⠀⠀⠀⣀⣀⠀⣀⠀⠀⠀⠀⠱⡀⠴⠖⢙⡀⡥⢤⠀⠀⠀⠀⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⢰⠀⠀⠃⠀⠏⠀⠀⣪⠓⠀⠁⠀⢾⣇⠀⠀⠀⠀⠑⢄⡜⠣⣱⠄⠀⠀⠀⠀⢀⠠⢐⡂⠀⢐⡂⠄⡀
⠤⠀⢠⡀⠀⠈⠑⠂⠤⠚⠀⠀⠀⣹⠀⠀⣠⠀⠸⡹⡀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⡔⡡⡪⠕⢐⣒⠒⠬⡑⢌⡢⠀⠀⠀⣀⡠
⠀⡤⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠁⠀⠀⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠔⡩⠔⣒⠨⢕⠬⠂⠀⠀⠀⡐⠁
⠐⠃⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⣀⣀⠤⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⡡⠤⡉⠒⠀⠀⠀⠀⠀⡤⠥⠈
⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⡠⠴⠶⠛⠃⠀⠀⣀⡀⣀⡀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠃⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣒⣒⡒⠤⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⣤⠐⢒⢤⢄
⠤⠤⠄⠔⠀⠀⢠⢊⢔⡨⠐⠀⠐⠠⢅⠢⣑⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⢀⢇⢀⡕⢣⠆⡦⠴⠄⠀⠀⢠⠀⠄
⠂⠀⠀⠀⠀⠀⠈⠡⡡⢊⠥⠀⠠⢍⡢⠕⠀⠀⠀⠀⠀⠀⡀⠙⠻⡛⠹⢉⡙⠙⠀⠂⠀⠀⠀⢰⣀⣨⣁⡛⢰⠇⢀⡡⠀⠀⢸
⡀⣀⣀⠂⡄⠀⠀⠀⠀⠣⢊⠥⢍⠓⠃⠀⠀⠀⠀⠀⠀⠘⠢⠤⠴⠥⠤⠤⠁⠀⠀⠘⠁⠀⠀⠈⠧⣘⣸⠕⢣⡠⠊⠃⠀⠀⢸
⠇⢉⠉⠁⠐⡄⠀⠀⠀⠀⠘⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⢯
⠀⠘⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠉⠉⠀⠀⠀⣀⠠⡄⢒⣂⡲⡄⠀⠀⠓⠒⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⣰⢶⠀⠀⠀⠀⣀⡠⠤⠐⠒⢏⠏⠯⢇⠀⠀⠀⠀⠉⣡⠘⠀⠀⠀⣇⠁⠀⠀⠀⠸⣽
⠀⠀⠀⠀⠀⠀⡀⠈⠢⡤⢪⠃⠈⠀⠀⠀⢸⡀⠤⣐⠈⠉⠰⠈⠉⢶⠀⠀⠀⠰⠁⣼⠈⢻⠄⠀⢹⠀⠀⡞⠄⠀⣿⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠛⠀⠀⠀⢘⠃⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⡦⠀⠀⠀⢸⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠀⠀⠀⠈⠈⠓⢄
⠀⠀⠀⢄⠀⠠⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠃⠀⣶⠀⠹⡀⠀⠀⠀⡁⠀⠀⡑⡀
⠀⠀⠀⠈⠐⢌⠲⣅⢄⡀⠀⠀⠁⠀⠉⠑⠒⠒⠒⠒⠒⡊⠉⠁⠈⠒⠀⡚⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠐⢬⡓⣎⣢⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⣤⣤⣿⣤⢤⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠹⡶⡞⠀⠄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠛⠛⠛⠿⠿
⠀⠀⢀⡤⠴⠀⠀⠃⠀⠿⠐⠀⢞⠻⣕⠢⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠀⠉⠂⡙⢭⡣⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠈⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠚⠛⢻⡧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠁⠀⠀⠰⣛⡧⢨⡇⠀⠉⠓⠶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⣀⣤⣤⡤⠴⠶⢦⡄⠀⡇⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⠦⠤⠤⠴⠶⠶⠞⠃⢀⣷⣀⣀⣤⣤⠴⠶⠖⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉⠉⠉⠙⣯⠉⠳⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣀⣤⣤⠤⠶⠒⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⠴⠶⠶⢶⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⣀⣠⣤⠶⠶⠚⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠉⠉⢈⢵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⢸⠀⠐⡃⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠖⠉⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠛⠶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⡀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠶⠶⣤⣤⣀⣀⣀⠀⠀⠀⠀⢀⣰⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠓⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠉⠉⠁⠀⠀⠀⠀⡷⠀⠀⠀⠀⢀⣠⠶⠞⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣀⣀⠀⠀⠀⠀⠉⠙⡿⠒⠶⠶⠶⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣀⣤⠞⢻⡥⢤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠉⠀⠀⠀⠀⠉⠙⠓⠲⠶⢤⣤⣳⣤⣤⣤⣤⣤⣄⣀⣈⡉⠙⠛⠳⠶⠶⣤⣤⠶⠚⠓⠋⠉⠀⣠⠞⠁⠀⠀⠀⠉⠳⣤
⠀⠀⣀⠴⠊⠉⠀⠀⢀⡤⢶⣙⣶⠶⣀⡀⠀⠀⠀⣠⠔⠀⠀⠉⠉⠙⠛⠒⠶⠤⢭⣭⣙⣛⠒⠒⠒⠲⠶⠶⠶⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⣼⠁⠀⠀⠀⠀⠚⠛⠶⠾⣉⠽⠛⠉⠀⢠⡴⣯⣷⢛⣧⠶⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠉⠛⠓⠲⠦⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠈⠓⠲⠤⣤⣀⡀⠀⠀⠈⠁⠀⠀⠀⠙⠙⠺⣥⠼⠋⠁⠀⠀⠀⠀⣠⠶⣛⡷⣶⣤⣂⡀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠇⠀⠀⠀⠀⠀⣀⣀⣀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⢤⣄⣀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠜⠛⠧⢮⣥⣛⣩⠞⠋⠀⠀⠀⠀⠀⠀⣠⠞⠁⡾⠀⠀⠀⠀⠰⠿⠿⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⠤⣤⣄⣀⣀⡀⠀⠀⠐⠉⠀⠀⠀⠀⠀⢀⣠⠴⢋⣀⡤⠾⠃⠀⠀⢀⣀⣤⡤⢤⣄⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⣤⣄⡀⠀⡴⣚⣭⠶⠚⠉⠁⠀⠀⢀⡤⠞⠛⠳⢤⡀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⣆⠠⠖⠀⠀⢹⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣤⣤⣤⣤⣤⣤⣤⣄⣤⣀⣼⣿⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣇⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣿⣿⣿⣿⣿⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⢟⡛⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠿⠿⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣤⣤⣼⣿⣿⣿⣿⣿