Butterfly symbol Emojis & Symbols

⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠛⢻⣿⣷⣶⣄⡀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⢀⣠⣴⣾⣿⡟⠛⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢃⣶⠶⣽⣻⣯⡻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢟⣿⣟⣯⠴⣦⡘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣌⣛⣿⣿⣿⣟⢋⣌⣩⣛⢷⣄⠀⠀⠈⡀⠀⠀⢀⠀⠀⠀⣠⡾⣛⣍⣡⣙⣿⣿⣿⣿⣛⣡⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⢻⣿⣿⣿⢿⣿⣮⣳⣌⠳⣄⠀⠀⠀⠀⠂⠀⣠⠞⣡⣾⣵⣿⡿⣿⣿⣿⡟⠻⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠿⣩⡾⣿⣶⣝⡻⣿⣾⣷⣜⢦⠀⠀⠀⢀⡴⣡⣾⣿⡿⢟⣩⣾⣿⢶⣝⠿⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡶⢙⣻⡟⠿⣿⣿⣶⣽⣻⢿⣆⠁⣼⣦⠊⣴⡿⣟⣭⣶⣿⣿⠿⢻⣟⡋⢶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢿⣿⣿⢼⣿⣿⠿⠶⠶⠦⠀⠀⣾⣧⠀⠠⠴⠶⠶⠿⣿⣿⡇⣿⣿⡿⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⢂⣩⣥⣶⣶⡶⠖⣫⣵⣶⡦⠸⣿⣿⠀⢴⣶⣮⣛⠳⢶⣶⣶⣭⣍⢐⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣸⣿⣟⡿⣡⣢⢞⣵⠟⣫⢖⣄⣿⣏⣨⡳⣝⢿⣮⡳⣴⡍⣻⣻⡿⣃⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢇⣼⣿⣽⠵⢋⣵⣾⣿⢸⡇⣿⣷⣮⡙⢯⣯⣿⣧⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⣿⡟⣨⣛⣛⣵⡞⣾⣿⣯⡿⢸⡇⢿⣿⣿⣧⢳⣬⣛⣛⣅⢻⡿⣻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣼⢿⣿⣿⣿⣷⣾⣹⣿⣿⣿⡇⠘⠃⢸⣿⣿⣿⣏⣿⣾⣿⣿⣿⡿⣧⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡷⣝⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢟⣫⢾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣝⣿⡿⣯⣻⣿⡟⠁⠀⠀⠀⠀⠈⢷⣿⣟⣿⢿⣿⣫⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣽⣯⣽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉

⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⢠⣿⠟⠛⠛⢛⠛⠷⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠛⠛⠛⢿
⠘⣏⠀⠀⠀⠀⠈⠑⠠⢀⠉⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠋⢁⠤⠐⠉⠀⠀⠀⠀⢈
⠀⢻⡀⠀⠁⠀⠐⠒⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠒⠀⠈⠁⠀⣸
⠀⠀⢳⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⠃
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠳⢮⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣫⡶⠛⠻⣷⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠐⠂⠐⣿⠀⠀⠠⡀⠨⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡩⠂⡠⠀⠀⢸⡗⠀⠀⠂⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⣐⣀⣀⣀⠘⢦⠀⠢⣈⠢⡈⢢⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣡⠊⡠⢊⠠⠊⢠⠞⠀⣀⣀⠐⠂⣀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣀⣀⠀⠀⠀⠈⠑⢄⠠⡑⢌⠢⡱⣌⠂⠀⠀⠀⠀⠀⠀⠀⠀⢊⡔⡡⢊⠔⡡⢀⠔⠉⠀⠀⠀⢀⣀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠐⠌⠢⢕⢌⢊⢆⠀⠀⠀⠀⠀⠀⠀⢀⣎⢎⢔⠡⠊⠔⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠐⠒⠀⠈⠉⠀⠐⠒⠒⠂⠀⠄⠢⠑⠑⢹⠇⠀⠀⢀⡀⠀⢠⡟⡅⠐⠡⠂⠀⠀⠒⠒⠒⠀⠉⠉⠀⠐⠒⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠐⠀⠉⠉⠉⠉⠉⠉⣉⡁⣐⠒⠂⠀⠊⠀⠂⡀⢿⠇⢠⣿⠀⠀⠀⠒⠒⠀⣉⣉⠉⠉⠉⠉⠉⠁⠐⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣐⣋⣉⣉⣉⣩⣭⣭⡭⠭⠀⠔⠦⣤⣵⣾⣶⣿⡷⠶⠀⠬⠭⢭⣭⣭⣉⣉⣉⣉⣒⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⣩⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⠿⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣬⣉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣫⡍⠀⢀⣠⣶⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⡟⣦⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⠀⠈⣭⣓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣩⡭⠖⠚⠉⢉⣽⡄⠀⠀⠀⠀⠠⠀⡼⠀⣿⡇⠸⡀⠡⠀⠀⡀⠀⠀⣼⣍⠉⠙⠒⠮⢽⣁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣵⡄⢀⣠⠶⠛⣹⣷⡀⠈⠀⠀⠁⣰⠃⠀⣿⡇⠀⢷⡀⠁⠀⠀⠀⣰⣿⡙⠳⢦⣀⠀⣤⣭⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠾⣟⠋⠀⠁⠀⣠⠞⢩⠟⣿⣦⣤⣀⣠⡟⠀⠀⠛⠁⠀⠘⣧⣀⣠⣤⣾⡟⢯⠙⢦⡀⠀⠁⠈⢛⠿⠊⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣥⣶⠀⣠⠞⠁⡰⠃⢀⡟⠀⡟⢸⣿⠁⠀⠀⠀⠀⠀⠀⢹⣿⠘⡇⠘⠇⠀⠳⡀⠙⢦⡀⢰⣶⣝⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠺⣟⣡⠞⠁⠀⠜⠁⠀⠀⠀⢰⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢱⠀⠈⠀⠀⠙⠄⠀⠙⢦⣝⡿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣡⣶⢠⠎⠀⠀⠀⠀⠀⣾⣤⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⢠⣜⡆⠀⠀⠀⠀⠈⢆⠰⣮⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠻⢟⣷⣟⣴⣶⢠⢀⣄⢠⣇⠿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⢏⣷⢀⣤⢠⢰⣿⣮⣷⣝⠗⢹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢹⡿⣳⡏⣾⣿⣿⡿⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣿⣯⣿⡏⣷⡻⣿⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠠⣾⣿⡧⢡⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⣿⣿⡦⠘⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠐⠂⠀⠀⠀⠂⠀⠂⠂⠀⠂⠃⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠈⠂⠃

⠀⠀⠀⢀⣴⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤
⠀⠀⠀⣿⣽⡏⠁⠉⠙⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠉⠁⠉⣻⣼⡇
⠀⠀⠀⠘⢿⣟⣿⠛⢻⣿⠟⠛⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⢿⣿⠟⢻⣟⣿⡟
⠀⠀⠀⠀⠈⢿⣩⣿⣉⣿⢦⣤⣤⣄⣹⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣋⣤⣤⣤⢼⣏⣹⣿⣹⡟
⠀⠀⠀⠀⠀⠘⣿⣉⣟⠀⠀⠀⠙⢿⠏⠉⠙⠻⣦⡀⠠⢄⠀⠀⠀⠀⠀⠀⠠⠀⢠⡶⠟⠉⠉⢻⠟⠁⠀⠀⠈⣏⣹⡟
⠀⠀⠀⠀⠀⠀⢸⡟⣿⠟⠛⠓⠲⠾⣧⡀⠀⠀⠈⠻⣆⠀⠁⠀⠀⠀⠀⠂⢀⡴⠋⠀⠀⠀⣠⡾⠶⠒⠛⠛⢻⡟⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡟⢻⣦⣤⣀⣀⣀⡀⠉⠓⢦⣀⠀⠈⢷⡀⠀⠀⠀⠀⣠⠟⠀⢀⣠⠔⠋⠁⣀⣀⣀⣀⣤⣾⠛⣿
⠀⠀⠀⠀⠀⠀⢸⣏⡟⠁⠀⠀⠀⠉⠉⠙⠓⠶⣬⣷⣤⡀⢻⡀⣶⡦⣰⠏⣀⣴⣯⡵⠖⠚⠉⠉⠁⠀⠀⠀⠉⣏⣻
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣀⣀⣀⣠⣤⣤⡤⠤⠤⣭⣹⣿⣮⣷⣿⣷⣿⣾⣿⣯⡥⠤⠤⣤⣤⣄⣀⣀⣀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⡿⠿⠧⠿⠦⣤⡤⠶⠛⠛⣻⣿⣿⣿⣿⡛⠛⠓⠦⣤⡤⠾⠯⠽⠿⢯⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣿⡤⠤⢖⡶⠛⠁⠀⢀⣤⠞⣹⣿⣿⣿⣿⡝⠷⣄⠀⠀⠉⠳⣖⠢⠤⣴⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋⢀⡤⢚⡏⠀⠀⢀⣰⠋⠁⡼⣱⢿⢿⣿⡿⡝⣆⠈⠳⣄⠀⠀⠈⢛⠢⢄⡈⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣻⠏⠁⢀⡼⢷⡦⢴⡟⠁⠀⡜⢠⡇⡟⣼⡏⣇⢳⠘⣆⠀⠘⢷⠤⣶⠿⢆⠀⠉⢻⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡶⠁⢠⠋⢀⡎⠀⠀⡼⠀⣸⡿⠁⣿⡇⠹⣿⡆⠘⣆⠀⠈⢇⠈⢣⡀⠙⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣲⣇⣀⣼⠀⠀⣼⠁⢀⡿⠁⠀⠈⠀⠀⠹⣧⠀⢸⣄⠀⢸⣆⣀⣿⣲⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣽⠿⣟⣻⣿⡷⠟⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣛⣿⠿⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡷⠶⠿⣿⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡖⣿⠿⠿⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⣻⣿⣾⠿⣟⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢿⣿⣾⣟⣻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡉⠉⠙⢿⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠋⠙⠋⠉⠉⣹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠉⠀⠀⠀⠑⢄⡀⠈⠻⣦⠀⠀⠀⠀⠀⠀⣰⠟⠁⢀⡠⠊⠀⠀⠈⠉⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠖⠀⠀⠀⠀⠀⠈⠂⢄⠙⢷⡄⠀⡀⢀⡼⠁⡠⠒⠉⠐⠒⠒⠒⠀⠐⢿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⣀⣤⣤⣤⣄⣀⣀⣀⣁⣨⣿⣿⣿⣞⣁⣊⣀⣀⣀⣀⣤⣤⣤⣤⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣿⡿⠛⠛⠛⢻⡿⠿⠿⠿⣿⣿⣿⣿⠿⠛⠻⠿⡛⠛⠛⠛⠻⣿⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠛⠁⠉⣉⠥⠃⠀⠀⢀⠀⢈⣿⣿⢿⡀⢀⠀⠀⠈⠣⣉⠐⠂⠛⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠓⠂⣉⠤⣁⡀⣤⠔⠊⢁⢄⠅⣿⣿⠸⡇⡀⠁⠢⡤⢀⣈⣢⡉⠐⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⡆⠀⢀⡔⠀⢠⠂⠀⡠⢃⠊⠀⣿⣿⠀⢻⡘⢄⠀⠐⢄⠀⠡⣀⠑⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⡎⠀⡐⠁⠀⡰⢁⣾⠀⠀⣾⣿⠀⠀⢿⡄⢳⡀⠀⢀⠀⢈⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣤⣌⢀⡾⠃⠀⠀⣿⣿⠀⠀⠈⢿⣄⣡⣤⣴⣾⣿⣷⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣾⠟⠁⠀⠀⠀⠘⠏⠀⠀⠀⠀⠳⢿⣾⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄

⣾⣿⣶⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⣾⣿
⢹⣿⢿⣷⣶⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠢⣄⣈⣀⠀⠙⢦⡀⣠⠞⠉⢀⣀⣀⡤⠂⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣿⢿⣿
⠀⠙⣿⣾⣶⣾⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠉⠛⠛⠷⣄⠹⢁⡴⠟⠛⠋⠁⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⢿⣶⣿⣿⡟⠁
⠀⠀⠈⠻⣿⣽⣿⣝⣿⣿⣿⣿⣿⢾⣽⣻⢿⣦⣄⠀⠀⠈⠁⠈⠀⠀⢀⣴⣾⢿⣻⣽⢾⣿⣿⣿⣿⣟⣽⣿⣽⡿⠋
⠀⠀⠀⠀⠲⣷⣶⣺⢾⠟⡛⠿⣯⣟⡺⣽⡳⣮⡻⣦⠀⣴⣭⣵⡄⢠⡾⣫⡷⣻⡽⣚⣯⡷⠟⡛⢿⢿⣒⣶⡷⠂
⠀⠀⠀⠀⠀⠈⢿⣿⡿⠸⣱⣿⣟⡻⢿⣷⣽⣷⡽⣜⣇⢹⣧⣿⢁⣟⡼⣵⣿⣵⣿⠿⣛⣿⣷⡼⠸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠋⣴⢃⡬⢿⣯⣍⣒⠍⠻⣝⣿⣾⣿⣟⡏⣟⣿⣿⣾⣟⡽⠋⢕⣊⣭⣿⠯⣄⢳⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠐⡾⣈⣷⣤⣬⣿⣿⣿⣿⣯⡌⠹⠟⢈⣟⣏⠘⠿⠉⣬⣿⣿⣿⣿⣯⣤⣴⣯⡸⡖
⠀⠀⠀⠀⠀⠀⠀⣸⠇⣳⣾⣽⣽⣿⣿⣿⣿⡿⠛⣡⣶⣾⡿⣿⣶⣦⡙⠻⣿⣿⣿⣿⣿⣽⣽⣶⡃⢿⡀
⠀⠀⠀⠀⠀⠀⠘⠿⠸⠭⠯⠯⢿⣟⣛⣉⡱⡶⣿⣏⡏⢸⡇⣿⠈⣏⣿⡷⡶⣉⣙⣛⣿⠯⠯⠯⠽⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⢟⡻⢾⣿⣷⣿⡟⠀⢸⣇⣿⠀⠘⣿⣷⣿⣿⠾⣛⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⢏⡾⣡⣿⢛⣿⣿⠏⠀⠀⢸⣿⣿⠀⠀⠈⢿⣿⣟⢻⣧⡹⣎⢿⣿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣚⣿⡏⣾⣿⣿⣻⣻⡿⠁⠀⠀⠀⠘⠿⠟⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⡎⣿⣟⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠾⢛⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣗⠻⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆

⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⣼⣿⡿⠿⠿⠿⠿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⠿⠿⢿⣿⣧
⠀⠀⠀⢹⣿⣦⣀⣀⣀⣄⣀⠀⠉⠙⠛⠶⣤⡀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⣠⡴⠞⠛⠉⠁⢀⣀⣀⣀⣀⣠⣿⡟
⠀⠀⠀⠀⠹⣿⣷⣈⣉⠀⠉⢉⣓⣶⣤⣄⣈⠙⠳⣄⡀⠀⠀⢳⡀⠀⠀⠀⡼⠁⠀⠀⣠⡴⠛⢁⣀⣤⣴⣒⡋⠉⠉⢉⣉⣽⣿⡟
⠀⠀⠀⠀⠀⠹⣿⣿⠿⠿⠿⠿⠶⣤⣄⣀⠉⠛⠶⣬⡙⢦⡀⠀⢣⠀⠀⡼⠁⠀⣠⢞⣡⡴⠚⠉⢁⣀⣤⣴⠿⠿⠿⠿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢹⡅⠀⢀⣀⣀⠀⠀⠀⠉⣙⣲⣦⣀⠙⠷⣝⣦⡀⢧⣰⠁⣠⣾⡵⠛⣁⣤⡖⠚⠉⠁⠀⠀⢀⣀⠀⠀⢈⡟
⠀⠀⠀⠀⠀⠀⠈⣷⠋⠁⠀⠈⠙⠛⠋⠉⠉⢉⣉⣙⣛⣲⣬⡛⢿⣿⣷⣾⢟⣡⣶⢛⣛⠉⠉⠉⠙⠛⠒⠚⠉⠀⠈⠙⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⣴⣶⡶⠖⠒⠉⠉⠉⠉⣉⣉⣉⣉⣽⣿⣿⣿⣷⣟⣉⣉⡉⠉⠉⠉⠉⠑⠒⠶⣶⣶⣦⡀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡉⠀⠀⣀⣠⡤⠖⣛⡭⠟⠛⢉⣩⠝⣻⣿⣿⣟⠯⣍⠛⠛⠯⢽⣓⠶⢤⣀⡀⠀⠈⠉⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡽⢉⡠⢲⠟⠁⠀⢀⠴⢋⡴⠋⢠⣿⡟⡆⠑⢮⡙⢦⡀⠀⠈⠻⡶⢌⠉⢿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣩⠖⠁⠀⣠⠞⣡⠴⡟⠀⢠⠇⣿⡇⠸⡄⠀⢻⠦⣌⠳⣄⠀⠈⠢⣍⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠋⠀⠀⣠⠞⠁⠀⠁⡼⠁⣠⠏⠀⣿⡇⠀⠹⣄⠈⢧⠈⠃⠈⠳⣄⠀⠀⠑⢦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣠⡴⠁⠀⠀⣠⠞⠁⣰⠏⠀⠀⣿⡇⠀⠀⠹⣆⠀⠲⢄⠀⠀⠘⠷⡀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣯⡀⠀⣠⠞⠁⠀⡰⠃⠀⠀⠀⣿⠃⠀⠀⠀⠘⢇⠀⠈⠳⣄⠀⢀⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣥⣌⣀⣠⠴⠊⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣁⣈⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⢀⡤⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠆⢁⠀⠐⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠚⠀⠀⢃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣦⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠤⡎⠀⠃⠀⠣⠀⠀⠂⠀⠀⠀⠈⣰⠋⣴⠿⢿⣿⠂⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⢁⠀⡜⠈⠂⢀⣐⠁⢀⠀⠀⠀⠀⢁⠳⠀⠀⠀⢰⢃⡞⠁⠀⣰⡏⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⢀⠀⢈⠔⠃⠀⡀⠀⠉⠂⠀⠀⠀⣵⠀⢰⣤⣏⡞⢀⣠⡾⠋⣳⡆⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠘⠢⢼⠀⡀⣸⠃⠀⠀⣴⣾⣤⠀⠀⢱⣟⢻⣿⠿⠛⢉⣠⡴⠛⠀⠀⢀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⡠⠐⠋⠀⠙⡿⠟⠲⣖⢈⣿⣿⠣⡀⠙⠛⢦⣀⠀⠀⠀⠘⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣶⡄⠀⠘⡝⠀⠀⠀⠟⢿⣿⣿⣧⣹⣦⣄⡀⠙⣧⠈⠀⠀⠚⢶⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣋⠀⡆⠐⣤⠀⠆⡀⠀⠋⢷⣰⠋⠉⠉⠛⠿⠿⠿⡇⠀⠀⠀⡿⢧⡀
⠀⠀⠀⠀⠀⠀⠐⣠⠞⠉⠐⠫⡄⠌⢉⡀⠠⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠙⢎⠀⠀⣷⢿⠁
⠀⠀⠀⠀⠀⠀⠘⠁⢀⠀⠀⠀⠳⠀⢀⠀⠉⢔⠀⠀⠀⠀⠀⠀⠀⣶⣏⣙⣿⣠⠬⣆⠀⣯⣼⠃
⠀⠀⠀⠀⠀⠁⠀⠒⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠋⠙⢿⣤⡶⣾⡥⣭⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⢯⣂⣷
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣠⣶⣶⣶⣶⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⣶⣶⣶
⠿⣿⠟⠻⣿⣿⡿⠛⠿⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠟⠻⢿⣿⣿⠟⠻⣿
⠀⠀⠀⠀⠈⢿⣧⠸⣿⣶⣦⡉⠻⢯⡙⠷⣤⡀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⢀⣤⠶⢋⡽⠟⢉⣴⣶⣿⠇⣼⡿⠁
⠀⠀⠀⠀⠀⠈⣿⣧⣌⠻⠿⠿⠂⠀⠙⠦⣈⠙⢦⡀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡴⠋⣡⠴⠋⠀⠐⠿⠿⠛⣡⣼⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⡟⢛⡒⠀⢤⣀⣀⣀⣹⣷⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⣾⣏⣀⣀⣀⡤⠀⢒⡛⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠘⣿⣆⠻⣿⡷⠀⠙⣿⠋⠉⠙⠛⢷⣄⠻⣆⠀⠀⠡⠀⠀⠈⠀⠀⣰⠟⣠⡾⠛⠋⠉⠙⣿⠋⠀⢾⣿⠟⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣤⣤⣴⣿⡀⠀⠀⣠⣄⡈⠳⣜⢧⡀⠀⠀⠀⠀⢀⡼⣣⠞⢁⣠⣄⠀⠀⢀⣿⣦⣤⣤⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡅⢶⣶⣦⠉⠻⣦⣄⡈⠛⠿⢷⣌⣻⣷⠀⢀⡀⠀⣾⣟⣡⡾⠿⠛⢁⣠⣴⠟⠉⣴⣶⡶⢨⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣤⣴⣞⡛⠉⠉⠙⠓⠒⠬⣽⣿⣧⣹⣇⣼⣿⣯⠵⠒⠚⠋⠉⠉⢛⣷⣦⣤⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣭⠬⠽⠟⣛⣛⣻⣶⣿⣿⣿⣿⣿⣿⣾⣟⣛⣛⠛⠭⠭⣭⣉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢁⣠⣴⠾⠛⣩⣵⡾⢟⣽⠟⣽⣏⠻⡯⡻⢷⣮⣍⠛⠷⣦⣄⡈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠞⠁⣾⠁⠀⠾⠟⢋⡴⢣⠏⠀⣿⣿⠀⠹⡌⢦⡙⠿⠷⠀⠈⣷⠈⠳⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢡⡿⣸⡿⢤⣀⣀⣴⠏⢀⡟⠀⠀⣿⣿⠀⠀⢻⡀⠹⣦⣀⣀⡤⢿⣇⢿⡌⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡈⣡⠋⣠⡄⢹⡿⠁⠀⣼⠁⠀⠀⢿⡟⠀⠀⠈⣧⠀⠈⢿⡏⢠⣄⠙⣌⢁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢰⣿⠇⣸⡁⢀⡼⠁⠀⠀⠀⠸⠇⠀⠀⠀⠈⢧⡀⢈⣇⠸⣿⡆⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠘⠋⣰⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠙⠃⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⣀⣼⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣧⡀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠂⠀⠀⠀⠀⠐⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢰⣆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢸⡇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⣠⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⡀
⠀⠀⢿⢹⣹⣻⣿⣏⣿⠗⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⢾⣯⣻⣷⢲⡎⢹⠂
⠀⠀⠸⣟⢻⣹⣍⣽⣇⢠⣿⠟⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⠉⢻⣷⢀⣿⣭⣹⣿⢻⠏
⠀⠀⠀⠹⣟⣿⣧⠼⠛⢻⣿⡀⣿⠃⢙⣷⣄⠀⠀⠀⢀⠀⠀⠀⢀⣴⣤⡀⢻⣷⣴⡛⠛⠓⣴⣯⣿⠋
⠀⠀⠀⠀⠙⢏⣿⣷⣖⠋⣉⠽⠻⣦⣿⡿⠋⠑⣦⣰⣻⣷⣠⠞⠉⠙⢻⣷⠞⠛⠤⣌⣁⣾⢧⣯⡏
⠀⠀⠀⠀⠀⠈⢧⣿⣟⣯⣡⠴⠚⠉⢹⣿⠒⠢⣼⣿⣿⣿⣿⣀⠠⠐⢿⡍⠉⠛⠒⠦⣾⣏⣟⡿
⠀⠀⠀⠀⠀⠀⠈⢻⣽⣭⣷⡀⠀⠒⣻⣿⣇⣨⣿⣿⣿⣿⣿⡯⣀⣰⣿⣛⠒⠒⠒⣼⢧⣎⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣥⣾⣿⡇⣀⠄⣸⣯⣿⣿⣿⣯⣿⠒⢤⣄⠉⣴⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣋⡿⣠⡎⠀⢻⡿⢿⣿⣿⡟⣿⡀⠀⢳⡀⢼⡛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢾⣿⣽⡗⢺⣧⣴⣿⡇⢾⣿⣿⡇⢻⣧⣤⣾⠃⣾⡛⣾⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠛⣷⢾⣶⠉⡙⠁⠿⣆⠘⢿⡟⠀⣾⡏⢀⣉⣰⢿⣹⣿⣉⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣻⣯⣞⣹⢟⣧⡤⣼⠀⠈⠀⠀⣧⣶⡟⣿⣹⣾⡿⣥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣟⣉⣟⢿⣿⣟⡟⠀⠀⠀⠀⢿⣻⣷⢻⣏⣙⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠉⠉⠿⠏⠉⠀⠀⠀⠀⠀⠀⠉⠿⠏⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⢀⣤⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣤
⢸⣿⣿⠛⠛⠿⠿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⡿⠿⠛⠛⣿⣿
⠀⢿⣿⡆⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠛⠋⠁⠀⠀⠀⠀⢰⣿⡿
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⣿⣿⡀⢀⣿⣿⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⢸⣿⣧⣼⣿⡇⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡄⠀⣰⣿⡿⠟⠻⢿⣿⣆⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⣿⣿⣀⣀⣀⣀⣿⣿⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠈⠻⣿⣷⣦⣤⣀⣀⣀⣠⣤⣶⣾⣿⡿⠿⠛⣛⣿⣿⣿⣿⣿⠇⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠸⣿⣿⣿⣿⣿⣛⠛⠿⠿⣿⣿⣶⣤⣄⣀⣀⣀⣤⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠛⠋⠉⣀⣤⣶⣿⡿⠟⠛⠉⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠉⠛⠻⢿⣿⣷⣦⣀⠉⠙⠛⠿⠿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⣀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣄⣀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⣀⣠⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣽⣿⡁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠤⠐⢀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢿⠉⠁⠨⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣀⣴⣾⣿⡂⢸⣿⣿
⠀⠀⠸⣷⣦⣀⠀⠀⠀⠀⣰⠋⡠⠂⠀⠀⠒⣿⡇⠀⠀⠀⣿⣿⣿⡈⠙⠻⣿⣿⣿⣿⡿⣱⣿⣫⣿⣿⣿⡅⢘⣿⣿
⠀⠀⠀⠙⢿⡟⢅⡢⣠⣦⡧⢈⣠⣴⣾⣿⣭⣅⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⣼⢿⣻⠛⠃⣿⠿⠛⢋⣁⠒⠺⣿⣿⣿
⠀⠀⠀⠀⠀⠉⢣⡀⠚⣿⣽⠛⣐⠒⠣⡄⠈⢿⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⠿⣯⡇⠐⢠⡿⣿⣽⣛⣿⣦⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠱⢾⡿⠋⠑⢌⠡⡙⠉⢈⢘⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⢪⣶⣾⣝⢿⣿⣷⣿⣿⠄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠂⢐⠀⠑⠴⣤⡶⠟⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⢯⣿⢿⣷⣟⡛⠋⣁⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣤⡀
⠀⠀⠈⠻⣿⣷⡄⠀⠀⠀⠀⣀⣀⣤⡄⠀⠀⠶⣶⣶⣶⣶⣤⣤⣶⣿⣿⡿⠛⠁⠀⣿⣿⣧⡀⢀⣠⣴⣶⡆⠀⢻⣷⣶⣤⡀⠀⠀⠀⢀⡀⠀⠀⣰⣾⣿⡆
⠀⠀⠀⠀⠸⢿⣿⣆⢀⣴⣿⣿⠿⠛⠁⢀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣤⣿⣿⣿⣷⣿⣿⣿⡟⠀⠀⠀⢻⣿⣿⣿⣆⠀⢰⣿⣿⣷⣶⣿⣿⣟⠁
⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⡉⠁⠀⠀⢰⣿⣷⡄⠀⠀⠙⢿⣿⠀⢿⡿⠃⠀⠀⠘⣿⣿⠟⠁⠀⠹⣿⣿⣷⠀⠀⣴⣿⣿⣿⠟⠛⠀⠀⠙⢛⣿⣿⡿⢿⣿⠇
⠀⠀⠀⠀⠻⡿⠟⠉⣿⣿⣿⠀⠀⢀⣸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠙⠻⠏⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠀⣀⠀⠀⠀⠀⠈⠉⠁⠀⣴⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⣴⣿⡆⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⣠⣶⡇⠀⠀⠀⣀
⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⢈⠻⠿⠛⠉⠈⠀⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⢰⣿⣿⣷⣤⣤⡀⠀⢻⣿⣷⣦⣀⣠⣾⣿⠿⠇⠀⠸⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⢸⣿⣿⡆⠀⢀⣴⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⠀⠀⠀⠺⡿⣿⣿⣿⡿⠃⠀⠈⣋⣽⣿⣿⣿⣿⣿⣷⣆⠀⠀⠹⣿⡿⠉⣿⣿⡿⠋
⠀⠀⠀⢀⣛⣿⣿⣴⣿⣿⡿⠁⠀⠀⢦⣤⡀⠀⠀⠀⢀⣀⣤⠌⠉⠉⠁⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠈⠻⣿⡿⠟⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠐⣿⣿⣿⡿⣿⣿⣦⡀⠀⠀⠀⣿⣿⣿⣧⣀⣴⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⡆⠀⣤⣶⣿⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⣠⣴⣶⡆
⠀⠀⠀⠙⠛⠋⠀⢿⣿⣿⠃⠀⠀⠀⠉⣽⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠻⢿⣿⣿⣾⣿⣿⣿⠆⠀⢿⣷⣦⣄⠀⠀⢀⣴⣿⡿⠀⠀⠀⠀⣴⣿⣿⣿⡟
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠻⢿⣿⠹⣿⡿⠟⠀⠀⠀⠀⠀⢰⣿⣿⢻⣿⣿⡉⠁⠀⠀⠀⠻⣿⣿⣷⣤⣿⣿⠟⠁⠀⠀⠀⠚⠟⠻⠿⠿⠃
⠀⠀⢿⣿⣶⣄⡀⠀⠀⢀⣠⣶⣿⣿⠀⠀⠀⠋⠀⠋⠀⠀⠀⠀⣀⣀⠀⠈⠉⠉⠈⠻⠿⠁⠀⠀⠀⠀⠀⠈⣿⣿⡏⣿⣿⠂⠀⠀⠀⢀⡀
⠀⠀⠈⠻⣿⣿⣿⣦⣴⣿⣿⣿⠟⠁⠀⠰⣶⣦⣤⡀⠀⠀⣠⣾⡿⠃⠀⢀⣀⣀⣀⠀⠀⠀⢀⣤⣾⣿⠃⠀⠈⢻⠃⠘⠇⠀⠀⠀⠀⠘⣿⣷⣄⠀⣠⣴⣶⠆
⠀⠀⠀⠀⣴⣾⣿⡿⢿⣿⣿⣦⠀⠀⠀⠀⠈⠛⣻⣿⣷⣾⣿⣯⣄⠀⠀⠀⠙⢿⣿⣿⣶⣤⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠀⠀⠻⣿⣿⣾⣿⣿⠏
⠀⠀⠀⠀⠘⢻⣿⡇⠸⣿⡿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⣿⣿⣿⡇⠀⠀⠀⠈⢛⣿⣿⡿⢿⣿⣷⣦⠀⠀⠀⠀⢠⣾⣿⣿⣿⡏⠀⠀⣾⣿⠏⢻⣿⣧
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠸⡿⠿⠃⠀⠈⠛⠛⠃⠀⠀⠀⠀⢻⣿⣿⠃⠈⠻⠿⠃⠀⠀⠀⠀⠻⠛⠛⠛⠉⠀⠀⠀⠛⠉⠀⠈⠻⠏
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⣠⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢠⠎⠉⠱⣄⣄⡀⠀⠀⠀⢇
⠀⠀⠀⢀⡞⠁⡀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⣸⠁⠀⠀⡠⠼⠀⠀⠀⠀⠀⢘⡆⠀⠀⢸
⠀⠀⠀⠘⢧⣠⠇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⡇⠀⠀⠰⡅⠀⠀⠀⠀⠀⠠⡎⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⠁⠀⠀⠀⠈⢲⠀⠀⢀⣀⣀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⡴⠒⠛⠛⠓⠦⣄⠈⣇⠀⢸⠀⠀⠀⠀⠀⠈⠲⠤⠚⠀⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⣜⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣀⣀
⠀⠀⠀⠀⡏⠀⣀⣄⠀⠀⠀⠀⠀⢀⣼⣛⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠁⠀⠉⠳⡄
⠀⠀⠀⠀⢧⠀⠁⠈⡧⠀⠀⠀⠀⡞⢿⣧⠇⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⠀⡷⠃⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠒⣄⡀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⢈⡆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠴⠚⠋⠁⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣀⠀⢸⠄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⠴⠚⠉⣀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠉⠙⠉⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡴⢶⠋⠁⢀⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢈⡿⣴⠚⠉⠀⢈⡧⣀⣀⣀⣀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠙⠒⠾⢍⣀⣼⣳⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃

⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣴⣦⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣙⣛⣛⣻⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣻⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣿⡿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢸⡇⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⡿⠟⠋

⠀⢀⣴⡾⠿⠿⠿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠿⠿⣶⣄
⣰⣿⠋⠀⠀⠀⠀⢠⣿⡿⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⣿⣥⠀⠀⠀⠀⠈⢻⣷
⣿⡇⠀⠀⠀⠀⢠⣿⠋⠀⠀⠈⠻⢿⣦⡀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⣠⣾⠿⠋⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⣿
⢻⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠉⠻⣷⡄⠀⠀⠀⠀⣴⡿⠋⠁⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿
⠈⢻⣷⡀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠹⣿⡄⠀⠀⣼⡿⠁⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⣰⣿⠋
⠀⠀⠙⢿⣦⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⢹⣷⠀⢰⣿⠁⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⢠⣾⠟⠁
⠀⠀⠀⠀⣿⡇⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣌⣿⣿⣿⣯⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠘⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣷⣶⣶⣶⣤⣤⣤⣶⣶⣶⣶⣾⣿⠀⠀⠀⢸⣿⣶⣶⣶⣶⣦⣤⣤⣤⣶⣶⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢻⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⡟⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⣿⡄
⠀⠀⠀⠀⠀⠀⢿⣇⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⠛⢿⣦⣄⡀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⣀⣤⣾⠟⠋⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠈⠙⠻⠿⣶⣶⣶⣿⡟⠁⠀⠀⠀⠀⠙⣿⣷⣶⣶⡾⠿⠛⠉⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣄⣀⣀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⣀⣀⣤⣴⣾⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠈⠛⢿⣷⣄⠙⠲⣄⠀⠀⡰⠚⠉⣠⣿⡿⠋⠁⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠙⢿⣦⠀⠘⣆⣸⠁⢀⣼⡿⠋⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠻⣧⠀⢸⡇⠀⣾⠏⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢹⡆⣾⣷⢸⡏⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠀⠀⠀⣿⣿⣿⣿⠀⠀⢀⣠⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣦⣿⣿⣿⣿⣴⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡽⠛⣿⣿⣿⣿⠛⠯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⢀⣿⣿⣿⣿⡀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⢀⣼⣿⣿⣿⣿⣧⡀⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⢀⣤⣿⡿⠁⢿⡿⠈⢿⣿⣦⡀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⢠⣶⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣷⣦
⣿⣿⡟⠉⣉⡙⠛⠿⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠟⠛⣉⡉⠙⣿⣿
⠹⢿⣿⣦⡙⣿⣿⣿⣶⣮⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⣶⣾⣿⣿⡟⢡⣾⣿⠟
⠀⠀⠉⢿⣷⠘⣿⣿⣟⣉⠉⠛⠻⢿⣿⣷⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣴⣿⣿⠿⠛⠋⢉⣙⣻⣿⡟⢰⣿⠏⠁
⠀⠀⠀⣸⣿⠀⣿⣿⣿⣿⣿⣷⣶⣤⣉⠻⣿⣷⣄⠈⠙⢦⡀⠀⠀⠀⠀⣠⠞⠋⢀⣴⣿⡿⢛⣡⣴⣶⣿⣿⣿⣿⣿⡇⢸⣿⡄
⠀⠀⠀⢹⣿⣆⠸⣿⣿⣧⣄⣀⣀⠉⠙⠻⢮⣿⣿⣦⠀⠀⠱⠀⠀⠀⠰⠁⠀⢠⣾⣿⣯⡾⠛⠉⢁⣀⣀⣤⣿⣿⡿⠀⣾⣿⠁
⠀⠀⠀⠀⠛⢿⣦⢹⣿⣿⠋⠉⠉⠉⠀⠀⠀⠙⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠟⠁⠀⠀⠈⠉⠉⠉⢻⣿⣿⢁⣾⠟⠃
⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⡏⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣦⡴⠖⠂⠀⠀⠀⠀⠀⠀⠀⠻⣷⢠⣶⣦⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠒⠶⣦⣾⣿⡇⢸⣿⡀
⠀⠀⠀⠀⠀⠸⣿⣷⣌⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣿⣧⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⢋⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣶⣶⣶⣶⠶⠖⠛⠛⠛⠒⠒⠀⣿⣿⣿⡇⠐⠒⠚⠛⠛⠓⠶⢶⣶⣶⣶⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠉⠀⠀⠀⣀⣤⠶⠒⠀⠀⢹⣿⣿⠃⠀⠐⠲⢦⣄⡀⠀⠀⠈⠙⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣾⡿⣱⡿⠁⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠙⣷⡽⣿⣆
⠀⠀⠀⠀⠀⠀⢸⣿⢣⣿⡿⠋⠀⠀⠀⠀⠻⣧⣤⠖⠀⠀⢀⡇⣿⡇⣧⠀⠀⠘⢦⣤⡿⠃⠀⠀⠀⠈⠻⣿⣷⢹⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡸⣟⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⣷⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣿⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣻⡷⣹⡟⠁⠀⡀⠀⠀⡀⠀⠀⠀⠀⣼⡇⠀⣿⡟⠀⢿⡄⠀⠀⠀⠀⡀⠀⠀⡀⠀⠙⣿⡁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡸⢷⣶⡾⠁⠀⢰⣇⢀⣠⠄⣼⡿⠀⠀⢿⡇⠀⠘⣿⡄⢠⣀⢀⣿⠀⠀⠙⣶⣶⡿⣡⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣌⠿⢶⣶⡈⠛⠋⢁⣾⡿⠁⠀⠀⠈⠀⠀⠀⠹⣿⣦⡉⠛⠋⢰⣶⡾⢟⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⡻⢿⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣾⠿⣣⡶⠿⠋

⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⣏⠑⢤⡄⠊⣽⠀⠐⣆⠔⠀⠀⠀⠐⢄⡖⠀⢸⣷⣄⠀⢠⣾⡏⠀⠀⢻⣿⣦⡀⢠⡶⣿⡗⠀⠘⣿⣦⡄⠀⢀⣴⣿⠇
⠀⠀⠀⢰⡀⠈⠁⢠⠄⠀⠀⠁⠀⠀⢺⠂⠀⠀⠁⠀⠈⢷⣿⣶⡿⡿⠃⠀⠀⠘⣷⢺⣿⣿⣗⣾⠃⠀⠀⠙⢿⣿⣶⣿⣿⠟
⠀⠀⠀⠀⠟⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⢻⡗⠀⠀⠀⠀⠻⠿⠉⠹⠿⠟⠀⠀⠀⠀⠻⠋⠀⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢠⣤⠀⢤⠤⣀⠀⠀⠀⢀⠤⣤⠀⢶⣄⡀⠀⠀⢀⣤⡦⠀⢤⡄⠀⠀⠀⠀⠀⢠⡤
⠀⠸⣿⢟⣴⣴⣦⣦⠻⣿⠀⠘⣵⣾⣶⣾⣶⡝⠀⠀⠀⠈⢳⣄⡶⠁⢀⠀⠀⠈⣿⣷⣄⣠⣾⣿⠁⠀⠀⣿⢷⣀⣀⣀⢼⢿⠁
⠀⠀⠘⣾⣿⣿⣿⣿⡞⠁⠀⠀⠀⣿⠛⢻⡇⠀⠀⠀⠠⠀⠀⣿⡀⠈⡆⠀⠀⠀⣼⣿⠿⠿⣿⣧⠀⠀⠀⢀⢠⣼⠿⢧⡆
⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠑⠘⠀⠀⠀⠀⠻⠇⠀⠀⠻⠟⠀⠀⠀⠀⠪⠅⠀⠈⠅⠂
⠀⢰⠐⠀⡀⠀⠀⢀⠄⢂⠆⣤⣄⠀⠀⠀⢀⣤⡄⠘⡟⣦⠀⠀⠀⢰⢺⡇⠀⣤⠀⠀⠀⠀⠀⠠⣤⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠳⣤⣾⣆⣴⣷⣤⠎⠀⢿⠛⣷⡤⣴⡟⢻⠇⠀⠃⠁⠓⣴⠜⡌⠈⠁⠀⠀⠀⠲⢠⡔⠗⠀⠁⠀⠘⠀⠀⠀⣶⠀⠀⡀⠋
⠀⠀⠀⣴⢿⡟⢿⣿⣦⠀⠀⠀⠆⠹⠻⠻⠁⠆⠀⠀⠰⡀⠀⠙⠁⢀⡆⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠘⠁⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢶⡶⣀⠀⢀⡀⢶⡶⠀⢻⡷⢄⠀⢀⠔⣿⠃⠀⡆⡀⠀⠀⠀⢀⣄⠂⢿⣷⣄⠀⠀⠀⢀⣴⣾⡟⠸⡿⠲⣄⠀⣠⠖⠯⠇
⠀⠀⠸⡷⠿⢷⡾⠿⣾⠃⠀⠈⣡⣿⣷⣿⣧⡋⠀⠀⠃⢻⣤⣴⣴⡾⠘⠀⠀⢳⡙⣷⣤⣤⣾⢫⡟⠀⠀⠑⣂⣿⣶⣿⣄⠊
⠀⠀⠀⠻⢤⠂⠠⡴⠟⠀⠀⠀⠏⡻⠀⠙⡋⡇⠀⠀⠰⣘⠈⠈⠁⢃⠄⠀⠀⠀⢴⣉⠓⢋⣉⡦⠀⠀⠀⠈⠩⠝⠀⠙⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⠙⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣵⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠁⠀⢰⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⢠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠤⠞⠁⢀⡼⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠄⠀⠀⠈⠀⠀⢀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⠀⠀⢀⠠⠀⠂⠀⠀⠀⠀⠀⠈⠉⣻⡄
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠶⠒⠛⠛⠛⠛⠿⠿⠿⠷⣶⣶⣤⣴⣿⣿⣿⣿⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⡇
⠀⠀⢀⣴⢟⠋⠑⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠻⢿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡇
⠀⠀⢾⣷⣃⡔⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⠃
⠀⠀⠈⠛⢿⣷⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠨⠋⠉⡿⣿⣿⣿⡙⠦⣄⠀⠀⠀⠀⠀⠀⣤⣉⡿
⠀⠀⠀⠀⠀⠈⠙⠳⣶⣶⠀⢀⣠⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢿⠹⣷⡄⠀⠉⠀⠀⠀⠠⠄⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢟⣁⢀⣤⣠⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣇⣠⣇⠀⣰⡄⠀⠀⢠⣄⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠓⠚⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡍⠀⠾⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡙⢢⡀⠀⠀⠀⠀⠀⣴⠋⡣⠔⢒⣐⣶⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠸⠟⢀⢳⠀⠀⠀⢠⠞⡠⠋⢀⠔⠉⣀⣼⠶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠒⡴⠲⡌⡈⣇⠀⢠⢋⡜⢁⡔⢁⠔⠉⢀⣼⠤⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢴⠁⢰⢳⢉⡸⣄⢏⠎⣠⢋⠔⠁⢀⠔⠉⠘⣆⣤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠀⡆⠘⡄⢤⠹⣌⡴⣡⠋⣠⠞⠁⠀⠀⣠⠽⣍⣠⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠀⠇⠀⢸⠖⢤⡊⡳⣅⡼⠁⢀⣠⠖⠋⠀⢀⡬⢧⣠⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢸⠀⠀⠀⢸⠀⠀⢧⡐⠺⣄⠴⠋⠀⠀⢀⠔⠁⠀⡞⣈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⡇⠀⠀⡴⠉⣧⡇⠀⣀⡤⠚⠁⠀⢠⢞⡁⢉⠜
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡄⡆⠀⠀⡇⠀⡜⠀⢠⢿⠷⠊⠁⠀⠀⡠⠖⠥⡨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢧⢱⢸⠀⡇⢰⠇⢠⢃⡏⠀⣀⣀⣀⡞⢑⡤⢾⣒⡹⠍⠭⣝⠉⡙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠸⣾⢸⠀⠀⣸⢀⡇⡼⢚⠭⠚⠢⣜⡋⠣⡝⢉⣠⠤⠖⠒⠈⠱⣄⠒⠀⡃
⠀⠀⠀⠰⠖⠶⠤⣀⠀⠈⠢⢄⡀⠀⢻⣿⢸⢀⡇⡜⣰⡗⠁⢀⡠⠚⠁⢙⣥⣬⠻⣄⣠⠴⠖⠒⠒⠉⢳⣤⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⣈⡑⣾⣿⣾⡸⣸⣱⢃⢇⡴⢋⡀⠔⠊⠁⣀⣈⣢⢛⣇⠴⠖⠒⠒⢲⠎⣄⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⢹⣿⣧⡿⢃⠞⠁⠉⠁⠉⠑⣖⣉⠁⠀⠈⡏⣗⠒⠦⣄⠀⡎⢠⡅⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⠺⣟⠿⣤⠃⣀⡀⠠⠤⢴⣈⣀⠈⠉⠓⠲⡟⢹⡀⠀⠀⢑⢏⣄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⣹⠟⠺⣏⡿⡤⣄⡀⠀⠀⠀⠈⠉⠑⠢⣠⢋⢸⠛⢦⣠⠋⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡸⠁⠀⠀⠙⠎⢳⣄⣈⠙⠓⠒⠤⠀⣀⠀⢸⠀⢸⠀⡴⣃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠑⣭⢛⢶⢤⣀⠀⠈⢳⠞⣤⡧⠞⢄⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢾⣧⠈⠉⠓⠛⠚⠓⠲⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣿⣿⣿⠲⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⣺⣿⣿⣿⠷
⢻⡼⣿⣿⣿⠳⠀⣿⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠻⣿⣷⠀⢸⣿⣿⣿⣼
⠀⢳⡼⣿⣿⣷⡜⢿⣿⠀⠉⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⢿⡟⣚⣿⣿⡿⣾⠃
⠀⠀⠳⣼⣿⣷⣦⣜⠛⠂⢸⣿⣿⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⠀⢿⣿⣦⠔⠈⢉⣹⣿⡿⢶⡏
⠀⠀⠀⢻⡽⣿⣧⣈⣒⠀⠀⢻⡏⠀⣼⣿⣿⣷⣄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣠⡾⣿⣿⡇⠀⢘⣿⠅⠁⠶⣿⣿⣿⢗⡿
⠀⠀⠀⠀⢧⡿⣿⣿⡉⠀⠀⠨⠿⠄⣿⣿⠅⠈⢻⣷⣄⠀⡀⠙⠄⠀⠀⠀⠀⢀⠜⢀⠀⣠⣾⡋⠀⠘⢿⣷⠀⠛⠀⠀⠐⣲⣾⣿⡟⣾⠃
⠀⠀⠀⠀⠘⣇⣿⣿⣿⡎⠛⢂⢀⡀⣹⣿⢀⠀⣼⣿⡿⢷⣌⠁⡌⠒⣤⣤⠔⢁⠔⣪⣾⣿⣿⣗⠀⠀⠸⡿⠀⠄⠀⠀⢉⣿⣿⣿⢷⡇
⠀⠀⠀⠀⠀⠸⣇⢹⣿⣿⡾⠶⠒⠀⡹⠁⠀⠀⣿⣿⡃⣈⢿⣷⣜⢜⢟⣿⢿⣮⣾⡟⠁⠘⣿⣿⠀⠁⠴⠀⠀⠀⡀⢾⣿⣿⡿⣳⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣙⢿⣷⣶⣦⡤⠀⠀⢀⢀⣿⡏⠀⠐⢿⣿⣿⣾⣿⣿⣟⣿⣿⢃⢀⣀⣸⣿⣇⡀⠀⠀⠐⢶⣶⣾⣿⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠧⠶⠦⢭⠄⠛⠋⠐⣶⡀⡈⠨⣿⣿⣿⣿⣿⣿⡆⠈⢻⣿⠀⠈⠉⣿⣅⡲⠶⠾⠿⠿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⡶⢀⢠⠂⠘⣇⠀⠀⠉⡘⣿⣿⡿⡟⠆⠀⢸⡏⠀⢘⢦⣽⣟⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⣿⡟⠁⠀⠘⠀⠀⠹⡄⠈⢠⢱⢿⣿⠇⣧⠐⠀⢸⠃⠀⠀⣦⣨⣿⣿⡏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣾⣿⣿⡗⡐⠻⣆⠀⠀⡗⠀⠘⠸⢸⣿⡖⠟⡆⠀⡞⠀⠀⣸⠇⢼⣿⢡⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠙⣿⣿⣧⣤⠀⡠⠀⠀⣿⠀⡆⣿⠸⣿⠂⡗⡄⢠⡇⠀⠾⠉⣤⣠⣿⣾⡟⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣿⣿⣿⡏⢠⣤⠀⠀⣿⠀⠇⠀⠀⠀⢠⡗⡇⣸⠁⠀⣆⠀⠈⣿⢣⣾⣷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢹⣿⣿⣿⡟⢀⠀⣹⡀⢚⡒⠀⠀⠘⣃⠁⣿⣇⡀⠉⣿⣿⣿⡞⣡⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⣟⢻⣿⡿⣿⠟⢿⣛⡃⠀⠀⢈⣟⣴⣿⣿⡿⣾⣯⠶⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣌⣻⣏⣬⣿⣤⣿⡏⠁⠀⠀⠀⢨⣋⣘⣿⣀⣸⣧⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⡟⢉⣻⣫⣼⠃⠀⠀⠀⠀⠘⣿⣝⣟⠛⠻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⢷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠿⢯⡻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢟⣽⠟⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣙⠯⣽⠦⣤⠼⣍⣻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣩⡧⣤⠤⣯⠥⣏⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣗⣚⡧⣭⣯⠀⠁⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠈⠀⣹⣯⣼⣛⣪⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣻⡛⣻⡄⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢁⣽⢛⣟⣩⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢨⠋⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⡥⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⢷⡤⣄⣀⣀⠀⠀⠀⠀⠠⠄⠀⠀⢄⠀⠀⠀⠀⠠⠀⠀⡠⠄⠀⠀⠀⠀⣀⣀⣠⠤⡾⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡲⣃⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⣆⢸⡷⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣜⢶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣗⣢⣤⣤⡤⠀⠐⠀⠀⠀⠀⠀⢿⣾⣷⡿⠁⠀⠀⠀⠀⠒⠀⠤⣤⣤⣔⣺⡵⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠋⣡⠊⠀⠀⡀⠄⠀⠀⠀⢠⡜⣿⣿⠳⡄⠀⠀⠀⠀⢄⡀⠀⠐⣌⠙⢥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡿⠂⠀⠀⠈⠁⣐⠀⠀⢀⡠⠖⠁⠀⢻⡟⠀⠀⠳⠤⡀⠀⠀⢀⠈⠁⠀⠀⠀⢿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣁⠀⠀⠀⠀⠀⠰⠆⠚⠁⠀⠀⠀⠄⠈⠇⠐⡀⠀⠐⠈⠓⠐⠆⠀⠀⠀⠀⠀⣀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣥⡏⢀⠀⠀⢀⠀⠀⠔⠀⠐⠀⠁⠎⠀⠀⡆⠀⠐⡀⠄⠁⠀⠡⡀⠀⡀⠀⠀⡀⢸⣬⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⢛⠏⢀⡀⠀⢀⠄⠀⡀⠀⣰⡎⠀⠀⠸⠇⠀⠀⠱⣤⡀⢀⠀⠐⡀⠀⢢⡀⠘⡏⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣟⣽⣄⡴⣇⡠⢾⣽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⡷⢤⣸⢦⣠⣯⣹⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿⡷⢛⣿⠋⠙⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠋⠙⢿⡟⢿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆

⠀⠀⠀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄
⠀⠀⢠⣿⣯⣿⣿⣿⣿⣿⣦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣴⣿⣿⣿⣿⣿⣽⣿⡄
⠀⠀⠈⢿⣷⣿⣿⠛⢿⣧⣼⣿⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡾⠟⣿⣧⣼⡿⠛⣿⣿⣾⡿⠁
⠀⠀⠀⠈⠻⣿⡿⣷⡟⠛⠻⠶⣿⠛⠓⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠚⠛⣿⠶⠟⠛⢻⣾⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠖⠂⠠⠼⢄⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⡠⠧⠄⠐⠲⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣀⣀⣀⣀⠀⠽⠦⢄⡀⠈⠻⣦⡀⢠⡄⢀⣴⠟⠁⢀⡠⠴⠯⠀⣀⣀⣀⣀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣌⠳⣾⣷⠞⣡⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣽⣿⣿⣯⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⡶⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢉⣉⣉⣉⡝⠒⠚⠉⣉⢽⣿⣿⡯⣉⠉⠓⠒⢫⣉⣉⣉⡉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⣁⠤⡴⠋⠀⢀⡠⠊⢡⢾⣿⣿⡷⡌⠑⢄⡀⠀⠙⢦⠤⣈⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⠊⢉⡠⠼⡤⠤⢲⠋⠀⡰⢃⠎⣿⣿⠱⡈⢆⠀⠙⡖⠤⢤⠧⢄⡉⠑⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣖⠉⢀⡜⠀⢠⠃⢀⠞⠀⡎⠀⣿⣿⠀⢱⡀⠳⡀⠘⡄⠀⠣⡀⠉⣲⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣎⢀⣠⠃⢀⠎⠀⣼⠀⠀⢸⡇⠀⠀⢧⠀⠱⡄⠘⣄⡀⣱⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⣿⣷⣿⣀⡼⠁⠀⠀⠈⠁⠀⠀⠈⢧⣀⣿⣾⣿⣿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁

⣴⠉⡟⠳⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠛⡏⢱
⣿⣿⡿⣾⣿⠿⣎⡙⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⣉⡾⢿⣿⡾⣿⣿
⠹⣿⣍⣿⣿⣷⣿⠙⠋⠉⠊⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠈⠉⠛⢹⣷⣿⣿⣏⣽⡿
⠀⠹⣿⣿⣿⡗⠈⠭⣶⣶⠞⠳⣾⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣹⡶⠛⢶⣶⡮⠭⠐⣿⣿⣿⡿⠁
⠀⠀⢹⣿⣄⣽⣷⣶⣿⣿⠿⠾⡿⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠻⡿⠾⢿⣿⣷⣶⣿⣅⣼⣿⠁
⠀⠀⠀⢻⣷⢰⣿⠟⠋⠁⠀⠀⠙⢄⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠜⠁⠀⠀⠉⠛⢿⣷⢰⣿⠃
⠀⠀⠀⠸⣿⢿⣛⡤⠄⠀⠀⠀⠀⠀⠐⠂⠀⠈⢿⣷⣄⣤⣄⣴⣿⠏⠀⠀⠒⠀⠀⠀⠀⠀⠀⠤⣜⣻⢿⡿
⠀⠀⠀⠀⢹⡶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣾⠁
⠀⠀⠀⠀⠘⢿⣿⣶⣦⣤⣶⣶⣶⣾⣷⣶⣴⡶⠶⠛⠻⣿⡟⠛⠳⠶⣶⣤⣶⣷⣶⣶⣶⣦⣤⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣩⣟⣉⠛⠄⠈⠉⣡⠛⠁⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠙⢣⡉⠉⠀⠜⢋⣙⣯⡁
⠀⠀⠀⠀⠀⠀⢰⡽⡯⠀⠀⠀⠀⢈⣍⣀⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⣈⣍⠀⠀⠀⠀⠨⡿⣵
⠀⠀⠀⠀⠀⠀⢸⣆⡟⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⣇⣾
⠀⠀⠀⠀⠀⠀⠀⢹⡧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢿⠇⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣵⣄⣀⠀⠀⢀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠑⡀⠀⠀⢀⠀⠀⢀⣀⣴⣥⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣿⡦⣴⡬⢦⣴⠞⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⠮⣴⡤⣾⡷⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠶⢶⣾⡷⢶⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⣿⣭⣿⣧⣴⢿⣦⡾⠿⣶⣾⡿⣾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠛⠉⠀⠿⠿⣟⣿⣭⣿⣄⣠⣿⣧⡀⢈⣿⣟⣣⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣿⣿⣧⣤⣆⠀⠀⠀⠀⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⣤⣾⡋⠀⢘⣿⡿⠛⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢿⠿⠿⣿⣿⣿⣭⣟⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⣠⡞⣋⣿⣿⣿⣾⣿⣿⣷⣶⣶⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣽⣿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠟⢛⡿⠿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⡄⠀⠀⠀⢠⣾⣿⣯⣿⣯⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠛⠋⠁⠀⠉⠀⠀⠀⢀⣠⣿⡿⠋⠉⠉⠙⠻⣿⣿⣿⠀⠀⠀⠀⣸⣧⣀⣽⣋⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠛⠿⠿⠿⠶⠶⣶⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⠉⡻⠁
⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⣀⣤⣶⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⠛⡿⢿⡿⠁
⣤⡀⠀⠀⣰⣿⡿⠋⠀⢀⣀⣤⣴⣾⠿⠟⠛⠛⠛⠛⠛⠻⠿⠶⣶⣶⣤⣤⣤⣀⣀⠀⠀⠈⣈⣿⣿⡾⣷⡾⠁
⣿⣿⠇⣴⡿⠋⣠⣴⡾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⢿⣿⣿⣿⣿⣤⡾⠁
⣿⣿⣾⡿⠗⠛⠉⠁⢀⣤⠀⣐⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣤⡿⠁
⣿⣿⣿⡿⣩⠉⢉⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣶⣶⣶⣦⣤⣄⣀⣀⣀⣠⡟⢡⡿⠁
⣿⣿⣿⣍⡿⠳⢶⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⣿⣿⡏⠙⢿⣦⣀⠈⠉⠙⠛⠿⠿⢿⣿⣿⣿⣯⣀⠀⠀⠀⣈⣿⣿⣿⣇⠹⣿⡟⢿⣍
⣿⣿⠃⠠⡀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠈⠹⣆⠀⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣻⣿⡒⢿⣆
⡉⡅⠀⠀⢱⡄⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠹⡶⠶⣤⣴⣾⣿⡻⣿⠋⠽⠿⣿⣿⣧⣼⣿⡆
⣦⡆⠀⠀⡀⠹⣄⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠘⢦⡀⠈⠉⠛⠻⠿⣶⣤⣀⣸⣿⣧⣾⡏⢻
⡏⠀⠀⠀⠰⡀⠘⣦⠀⠀⠀⠻⣿⣿⣷⣶⣦⣤⣾⣿⣧⣄⣀⠀⠀⠀⠉⠛⢿⣿⣿⡿⠷⣿
⣅⠀⠀⠀⠀⢳⡀⠘⣧⠀⠀⠀⠘⢿⣷⡄⠈⠻⣿⣄⠈⠙⠻⣿⣦⣀⠀⠀⠈⣿⠿⣿⢾⣿
⡇⠀⠀⠀⠀⠈⢷⡀⠈⢷⡀⠀⠀⠈⢻⣿⣄⠀⠘⢿⣷⣄⠀⠀⠙⠻⣿⣦⣀⡿⣿⣏⣸⠏
⡗⠀⠀⠀⠀⠀⠀⢷⡀⠈⢷⡀⠀⠀⠀⢻⣿⣆⠀⠀⠹⣿⣦⡀⠀⠀⠈⣿⢿⣿⣏⢹⠏
⡅⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢻⡄⠀⠀⠀⢻⣿⡆⠀⠀⠈⢿⣿⣄⣀⣴⣿⣼⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⢻⡄⠀⠀⠀⢻⣿⣆⠀⠀⣰⣿⣿⣿⣼⣿⢿⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⢻⣄⣀⣀⣼⡿⣿⣿⠛⢛⣿⣶⣿⣿⣥⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠛⢻⣄⣤⣿⣿⠿⢿⣥⣤⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣬⣿⣟⣛⣿⣀⣼⡿⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣽⣿⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣁⣾⣡⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣽⠋⣿⠋⣹⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣾⣷⣾⠷⢾⡿⣿⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠻⢁⠏⠀⢀⡼⠻⣯⣿⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣠⢧⡎⣠⠖⠁⢀⡠⠟⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⢸⠏⠀⣠⠔⠋⠀⢀⣼⣿⠏
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⣼⣿⠀⠀⣾⠴⠊⠁⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠸⣿⣿⣿⣟⠻⣷⡶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢰⠃⠀⠀⣠⣾⣿⣟⣥⣤⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠘⢿⣿⣌⣻⣏⣉⣲⣿⡿⣿⣿⣶⣄⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡏⠀⣠⣾⣿⣿⡿⠋⢁⣀⠬⠛⠻⢿⣻⣧
⠀⠀⠀⠀⠀⠀⠙⠿⣿⢧⣭⡿⠯⢭⣐⡽⠛⠿⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠃⢀⡏⠀⣼⣠⣾⡿⢟⡟⣁⠤⠒⢉⡠⠤⠒⠊⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⡷⣄⣀⣀⣽⠦⢤⣀⠈⠉⠛⠿⢷⣦⣄⠀⢤⣦⣼⣧⣴⡿⠟⣫⠔⠋⠉⠓⠒⠛⠧⣤⠤⠖⠒⠈⢹⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⣀⣀⠤⠼⠛⠓⠒⠲⠦⠬⠽⢿⣾⣿⣿⣿⣥⣴⣊⣁⣤⣤⣤⣤⣴⠒⠉⠉⠙⠒⠦⢤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣄⣀⣀⣀⣀⣀⣠⣤⣶⣾⣿⠟⣹⣿⣯⣿⣷⠦⣤⣀⠀⠀⠀⠀⠉⠑⠲⢤⣀⢀⣮⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣟⣛⣩⠴⠊⢁⠏⡿⣿⣿⣝⡢⢌⡉⠓⠲⠤⣄⣀⠀⠀⣸⢿⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⢀⡼⢻⡀⠀⠀⡞⠀⢧⠹⡜⢿⣍⠒⠛⠶⣄⡀⠀⠈⣹⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠀⡼⢃⠀⣼⠁⠀⢸⡀⢱⡸⣿⣦⠀⠀⠀⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣅⠸⡛⣄⠣⡀⠀⢇⠀⢳⡏⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢧⣌⣆⡙⢄⠘⡀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⡿⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣿⣙⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⡷⢬⣷⣄⠀⠀⠀⠀⠴⠀⠀⠀⠀⠦⠀⠢⠀⠀⣠⣾⡥⢾⣛⣯⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣟⣺⣿⣬⣷⣄⢣⠀⠀⠀⠀⠀⠀⠀⠀⡘⣠⣾⣥⣿⣗⣻⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣟⣧⣽⣭⣿⡇⢀⣼⣷⡅⠂⢄⠀⠀⡠⠐⢨⣾⣥⡀⢸⣿⣭⣯⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣼⡦⠤⢿⣿⠁⠀⣹⣆⠀⣣⣴⠀⣰⣟⠀⠀⣿⡿⠤⢴⣧⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⢿⠤⠤⣿⣿⠿⣯⣘⣷⠏⠻⣾⣃⣽⠷⣾⣿⠤⠤⡿⡼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣾⣶⣖⣒⣻⣦⣤⢬⣿⠀⠀⣿⣥⣤⣴⣟⣒⣲⣶⣷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⡧⢼⡧⠤⢤⡶⠚⣩⣾⡧⢼⣷⡍⠓⢶⡤⠤⢼⡧⢼⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠗⣿⣒⣺⡋⠀⣰⣿⠟⡇⢸⠻⣿⣆⠀⢙⣗⣒⣿⠺⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣏⣹⡅⣺⢏⡽⣿⠋⠀⠻⠟⠀⠙⣿⢯⡹⣗⢨⣏⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢯⣬⢛⠟⠉⣷⠃⠀⠀⠀⠀⠀⠀⠘⣾⠉⠻⡛⣥⡽⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠚⠛⠿⡿⠖⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⢿⠿⠛⠓⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠨⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⣺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢠⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠒⢤⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⣆⠀⠀⠀⠀⠘⢆⡀⣀⡀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⣷⣼⠃⠀⠀⠀⠀⠀⠀⢀⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⣰⡟⠃⠀⠀⢰⣶⠀⢢⠄⣾⠏⢐⣡
⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡘⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠠⢞⣦⣬⠖⡂
⠈⠣⣀⠐⠃⠀⠀⣤⣄⠀⠀⠀⠐⠁⠐⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠟⣡⣶⠿⠃
⠀⠀⠉⠲⠤⠄⠀⠀⠙⣷⣄⠘⠁⠀⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡄⠀⢈⠀⣿⣷⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠃⠀⠀⣱⠟⣡⣾⠆
⠀⠀⠀⠀⠀⠈⠣⣄⡀⠈⢛⠗⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠰⣀⠹⣿⢀⡼⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⢘⣽⢀⣀⣘⠁⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠶⣄⠀⠀⡴⠿⠋⣷⠀⠀⠀⠀⠀⠉⠑⠋⢻⣴⣶⣶⣶⣿⡿⠿⠃⠀⠒⠂⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⢀⣀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢠⣠⡄⠄⠀⠀⠀⠀⠀⠀⢠⠦
⠀⠀⠀⠀⠀⠀⣀⠶⠛⣿⡆⠀⠀⠀⠀⠀⢀⣠⡶⠀⠀⣿⠛⠻⠛⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⣼⠿⣻⡄⣛⡃⠀⠀⢠⣠⣴⡿⢯⣀⣠⣼⣿⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡁⠂
⠀⠀⠀⠀⠸⠃⠀⢐⣣⠽⣄⡀⠀⢸⣿⠋⠀⠀⠉⠉⠉⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⡖⣿⣿⠀⠀⠀⠀⢀⡀⣈⠤⠀⠅⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠇⠀⠘⣉⠓⠢⠈⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠛⠀⣠⣿⠃⠀⠀⠀⢰⡊⠀⡴⢁⠀⣠⡾⠂⡀⠀⠁⠖⠠⠌
⠀⠀⠀⠻⢿⣼⠇⡀⢀⡀⢷⠤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢦⠸⣦⠐⠠⠰⢾⣮⠭⠴⠠⣾⢿⡅⠀⠀⠀⠀⠘⠆⠂⠳⢄⣴⠋⠀⡄⠁⠈⠂⠈
⠀⠀⠀⠀⠈⠉⠣⢷⡄⠀⢎⡼⠃⠀⠀⠀⢀⣴⣄⠄⠀⠀⠀⠈⠻⢿⣦⣤⣤⣀⣔⠋⠘⣿⡬⠀⠀⢁⣼⠁⠀⠀⠀⠀⠀⠉⠰⠀⠺⠁⢑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⡛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣍⡂⢀⠀⢙⠟⢤⡴⠟⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠃⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠢⡰⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⢀⣀⠀⠀⠉⢹⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠃⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⣢⢠⡀⠀⠙⢷⡄⠀⢀⢉⡿⠀⠀⠀⢀⣦⣄⠀⠀⡠⠊⢳⡄⠀⠀⠀⠀⠀⠀⡀⢾⣫⠀⠀⠰⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡤⠄⠀⠀⠙⠂⠼⠿⠁⠀⠀⠀⢸⠀⠈⠳⣄⠀⠀⠀⣷⠀⠀⠀⠀⢠⢎⡞⠀⠀⡠⡄⢰⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⡿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⢆⡀⢀⠟⣄⠀⠀⠀⠀⢋⡰⢀⠈⠘⠡⣝⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⢺⠀⠀⠀⠀⠀⠐⠛⠄⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠈⣇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠲⠶⣤⠐⠋

⠀⠀⠀⠀⠀⣀⡤⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⢤⣀
⠀⠀⠀⠀⢸⣯⣱⣞⡢⣽⢿⣷⣤⣀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⣀⣤⣾⡿⣯⢔⣳⣎⣽⡇
⠀⠀⠀⠀⠸⣷⢿⡖⢺⣿⣞⣿⣿⣙⣷⣤⡀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⣤⣾⣋⣿⣿⣳⣿⡗⢲⡿⣾⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣟⠁⠀⠉⠙⠻⣅⠈⠙⢷⣄⠀⠀⠸⡄⠀⠀⠀⠀⢠⠇⠀⠀⣠⡶⠋⠁⣨⠟⠋⠉⠀⠈⣻⣟⣿⠋
⠀⠀⠀⠀⠀⠀⠘⣿⢿⡿⠶⠶⠦⠤⠼⣦⡀⠀⠻⣷⡀⠀⢧⠀⠀⠀⠀⡜⠀⢀⣼⠟⠀⢀⣴⠧⠤⠴⠶⠶⢿⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⣀⣀⣀⣀⣀⣙⣦⡀⠈⢿⣆⠘⣆⠀⠀⣰⠃⣰⡿⠁⢀⣴⣋⣀⣀⣀⣀⣀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠛⠉⠉⠉⠉⠉⠉⠉⠙⢦⡀⠙⣧⡈⣻⣟⢁⣼⠋⢀⡴⠋⠉⠉⠉⠉⠉⠉⠉⠛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠈⠂⠈⢷⣿⣿⡾⠁⠐⠁⠀⠀⠀⢀⣀⣀⣀⣠⣤⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣛⣛⠛⠻⣿⠿⠿⠿⠿⠿⠟⣻⣿⣻⣟⠻⠻⠿⠿⠿⠿⣿⠟⠛⣛⣛⣻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣉⣉⡉⣤⠴⠃⠀⠀⢀⡠⠔⣫⡿⣿⣿⢿⢝⠲⢄⡀⠀⠀⠘⠦⣤⠈⣉⣉⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠋⠉⣀⣰⣁⣠⢴⠖⠋⠁⢀⠞⣽⠃⣿⣿⠘⣧⠳⡀⠈⠙⠲⡦⢄⣈⣆⣀⠉⠙⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡴⠋⠁⡰⠋⢠⡏⠀⠀⡴⢃⡼⠃⠀⣿⣿⠀⠘⢧⡘⢦⠀⠀⢹⡄⠘⢆⠈⠙⢦⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⠞⠀⢠⠟⠀⣠⠞⢠⠞⠁⠀⠀⠘⠃⠀⠀⠈⠳⡄⠱⣄⠀⠻⡄⠀⠱⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣴⡇⢀⡴⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢢⡀⢸⣦⣤⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⣿⠿⠟⠋

⠀⠀⠀⢀⡔⠒⣶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢶⡶⠒⣤
⠀⠀⠀⢸⣿⣹⣿⡓⠒⠿⠛⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠘⠛⠗⠒⣻⣿⣹⣿
⠀⠀⠀⠀⢿⡤⣇⣿⡀⠁⢠⣿⡇⠑⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⠁⢿⣷⠀⠁⢸⡇⡧⣼⡇
⠀⠀⠀⠀⢸⡇⢧⠸⡆⠀⠪⠅⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⡄⠩⠂⠀⣼⢣⠇⢿
⠀⠀⠀⠀⢸⡇⢺⠈⣷⠀⠀⠀⠈⠉⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠈⠉⠀⠀⠀⢰⡏⢸⠃⣿
⠀⠀⠀⠀⠀⢿⠘⡆⢻⡆⠀⠀⠀⠀⠀⠈⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⣾⡇⡾⢸⡏
⠀⠀⠀⠀⠀⠘⣎⢣⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣄⢲⣷⣆⣼⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⢹⡟⢡⢋⡾
⠀⠀⠀⠀⠀⠀⠈⠂⠑⠙⠂⠀⢀⣤⣤⣀⣀⣀⣹⣿⣿⣿⣯⠈⣿⣿⣿⣿⣁⣀⣀⣠⣤⣤⠀⠀⠸⠁⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢸⡗⠀⠉⠉⠉⠉⠉⠉⣿⣿⣷⣶⣿⣿⡏⠉⠉⠉⠉⠉⠁⠀⣿⠑⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠹⡶⢸⡇⠀⠀⠀⠀⠀⠀⠀⠻⢛⠳⠵⢛⡻⠃⠀⠀⠀⠀⠀⠀⠀⣿⠐⣾⠡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢠⣟⢹⡅⠀⠀⠀⠀⣀⠀⠀⠀⢨⡙⠚⣋⠇⠀⠀⢀⡀⠀⠀⠀⠀⢸⠙⣧⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣧⢸⣇⠀⠀⠀⠀⠉⠀⠀⠀⠸⡆⠔⡺⠀⠀⠀⠀⠁⠀⠀⠀⢀⣼⢀⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠻⣆⠹⣦⣤⠀⠘⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠙⠀⢀⣤⡾⠃⡾⠃⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠐⠛⣶⣄⠸⠷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⢁⣴⡞⠓⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠗⠛⠻⣄⠙⠛⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠞⠛⢁⡼⠛⠓⢿⣆
⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠙⢧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⣩⢍⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⢝⣀⠇⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣋⣉⢙⠲⣄⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠘⠷⠈⡇⢡⠈⢣⠀⡀⣄⢻⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣧⡈⠒⠚⠡⠂⠀⠐⡀⠥⠞⠸⡇⠀⠈⠁⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢟⣁⣀⢉⠲⠀⠀⠀⠀⠀⢀⠐⣈⡆⡇⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⡏⡻⠆⢈⡇⠱⠀⠀⠀⡀⠀⠈⢂⠀⡀⠇⠘⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢺⠀⠈⠒⠋⠔⠁⠀⠀⠀⠀⠀⡀⠀⡈⠐⣀⣀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡿⠛⢦⡀⠀⠸⣗⣻⠀⣮⠑⡄⠀⠀⠀⠀⠤⠀⣠⣼⣶⣿⡿⠻⠋⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢠⣬⣶⣿⠻⡄⠀⢻⣉⡆⠐⠚⠑⠀⣀⡠⠔⣺⣿⡿⠋⠉⠁⠀⠀⠀⢀⣴⡏⠛⢛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢯⣿⣘⢯⡹⡝⣦⣿⡄⠈⠻⠽⣦⠤⠖⠊⠁⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⢠⡿⢮⣱⠔⠋⣉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢋⣡⣬⣿⡿⢾⡏⢿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷⡞⣹⣷⠚⢉⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣟⣳⠘⢿⣟⠻⣧⠈⣧⣨⡇⢧⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡽⣞⣿⢃⡼⣗⢁⡸⢏⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡏⠐⢶⣿⣃⠚⢧⣸⣿⣿⣸⡄⠘⡇⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⣛⡻⣿⣫⠴⢻⣅⠔⠋⡿⣳
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⣾⣿⡿⠯⢁⠀⠻⣿⠟⣧⣇⠀⠀⠀⠀⠘⠀⠀⠀⠀⣶⠀⠀⠀⡸⢫⠳⠤⣹⢸⣡⠖⠈⣹⡦⠎⠠⢿⡆
⠀⠀⠀⢀⣀⣤⣤⣤⣤⣍⣻⣿⡅⠀⠂⠀⠀⠛⠆⢻⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⣰⠊⢳⢝⣿⣥⠖⠋⠉⣳⠖⠀⠼⠒⢦
⠀⠀⡴⠿⣽⢛⣿⣭⢘⣷⣍⣻⡻⣶⣾⣲⣤⡀⢰⣌⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⢀⢈⡼⣿⣷⡶⠚⠋⣃⠤⣂⡀⠀⠉⠻⣦
⠀⢰⣟⣻⢃⣤⣤⣁⣶⠛⠉⠍⣙⠻⢼⣿⡲⣝⣮⣹⣿⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠎⡈⠀⠉⣹⢶⡖⠊⠁⠚⣹⣦⣤⠤⠾⠳⣄
⠀⠘⣧⠾⣇⠠⠶⢀⡚⢛⠶⢶⣶⣚⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣄⡐⠀⠀⠀⠻⣧⣓⢶⡉⠁⠀⠸⣧⠄⠀⠐⠿⠀⣀
⠀⠀⠘⠮⣝⣷⣐⣿⣫⡿⠖⠚⠉⠉⠁⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣧⣄⣀⣈⡻⣞⣻⣉⠁⠉⣹⡓⠂⣰⠶⣶⠋⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣦⡀⠉⠉⠙⠓⠾⣭⣉⣹⠛⣲⠮⠷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠉⠛⠒⠫⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣟⣻⣏⣟⣟⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣭⣭⣿⣿⣽⣯⣭⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣸⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢮⣿⡾⠄
⠀⠀⠀⣀⣀⣀⣀⠤⠤⠤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠈⠁
⣰⡾⠯⠒⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢀⡠⠔⠊⠉⠁⠀⠀⠀⠀⠀⠈⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢻⣿⢥⠀⠀⠀⠀⠀⠀⢰⡿⣽⠃⢀⡤⠋⠀⣀⠴⠒⡊⠉⠉⠉⠉⠁⡐⠠⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠈⠓⠦⢄⣀⡀⠀⠀⢸⣇⣿⠀⡻⠂⢀⡞⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢴⣦⠒
⠀⠀⠀⠀⠀⠈⠉⠙⠒⠺⣿⠃⠀⡇⠀⢸⠀⠀⠀⢰⠀⠃⠀⠀⠀⠀⠘⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⠿⢗⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢹⡀⠸⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢧⠀⢳⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣞⠙⠋⠀⠀⠀⠀⣵⣦⣤⡄⠊⠀⠀⡜⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠙⢆⠀⠀⠈⠙⡆⠀⠀⠀⠀⠋⣙⣛⠇⠀⠀⣰⠁⠀⠀⠀⣰⠉⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠤⠤⣙⠢⢄⡀⠘⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⣀⠴⢊⡥⠂⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠑⠢⢌⣙⠋⠙⢏⠛⠛⡋⠟⠍⢭⡌⢹⣉⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠒⠁⠀⠀⠀⠀⠀⠀⢳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡟⠓⠲⠤⠤⠤⠴⠖⢛⣯⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢦⠀⠀⣾⡁⠀⢀⣀⣀⣀⣠⣴⣎⣀⣞⠓⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⢆⠱⠞⢁⠝⠞⠁⠀⠈⠁⠀⠀⠈⢉⡬⢋⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⡁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢗⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠟⠛⠛⠆⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣬⣀⣮⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⢻⣧⠀⠀⠀⠀⠀⠀⣀⣤⣬⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠎⠁
⠀⠀⣠⣶⣖⣷⠋⠉⠉⠉⠟⠧⡀⠀⢰⣿⠏⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⡀
⠀⠀⠈⠛⠻⠿⡾⢴⡆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠀⡄⢱⣶⣦⣤⣀⠀⠀⠉⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⢸⣿⣿⣿⣿⣷⡄
⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠃⠈⠛⠽⠿⠿⣿⣿⣿⣿⣿⣿⣌⣿⣿⣿⣿⣿⠟⠁⢠⣶⡶⣤⣀
⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⣀⣈⡉⣯⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠈⠻⣷⣏⡹⠿⣦⣀
⢹⡿⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣀⣉⣹⣿⣷
⠀⣵⡀⠀⠀⠀⠃⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣉⡡⠈⣿
⠀⢿⣧⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⢀⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⣠⠻
⠀⠈⠙⠛⠀⠀⠀⢠⣴⠿⠃⠀⠀⠀⠀⠀⢻⣿⣿⢿⣻⣦⡀⠀⠀⠀⠀⠀⠀⣠⣤⣖⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⣿⣽⣿⣄⠀⠀⢀⣴⡿⣿⣿⣻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢛⣿⣿⣄⣈⣽⣿⣶⣶⣿⡿⠛⢿⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⡄⢰⣿⣿⣿⡉⢉⠉⢻⣿⠿⠿⢶⣶⣾⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⢀
⣦⡀⠀⠀⠀⢀⣶⣿⡿⢿⣿⣯⣿⣿⠟⠁⠈⣿⣿⣿⣧⣦⡞⣹⠟⣷⣑⡠⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋
⣿⣿⣦⣴⣴⣿⣿⣿⣀⢀⣿⣿⣿⠇⠀⠀⠀⠈⠻⢿⣿⡿⠃⠸⠀⢾⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⣀⣀
⠉⢉⣿⣿⣿⡁⠀⠉⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠃⠀⠀⠀⠀⠠⢶⣶⣶⡄
⣶⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢸⣷⠇⠀⠀⠀⢹⣿⠃
⣿⣿⡿⠉⠻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠁
⠿⠿⠁⠀⠀⠙⠿⠛⠉⠉

⠀⣰⣶⣦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡶⣆⠀⠀⠀⣴⣦⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣶⡄⠀⢠⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶
⠀⣿⣻⡮⠀⠙⠀⠀⠀⠀⠀⠐⠋⠀⢵⣞⣿⠀⠀⠀⢹⣿⠰⡇⣿⠂⡀⠀⠀⠀⠀⡀⢺⡇⣿⢸⣿⠁⠀⠘⢟⡩⣴⠺⡗⢀⡀⠀⠀⠀⢀⡀⢚⠗⣦⢍⡻
⠀⠹⣿⣅⡁⠀⠀⠀⢀⡀⠀⠀⠀⢀⢈⣿⠇⠀⠀⠀⠀⢿⣧⠉⢙⣲⣷⣄⣰⣄⣤⣷⣚⠉⢁⣿⠇⠀⠀⠀⠸⡅⠀⠁⠀⠀⠙⢆⠀⡠⠋⠀⠈⠈⠀⢨⡏
⠀⠀⠙⢺⠶⠒⠒⠒⣾⣷⠓⠒⠒⠶⡟⠋⠀⠀⠀⠀⠀⠀⢙⣿⡏⠁⠀⢨⣿⣯⠀⠀⠉⣿⣟⠁⠀⠀⠀⠀⠀⠓⣦⠶⠶⡶⠤⠴⣿⡧⠤⢶⠶⠶⣴⠚⠁
⠀⠀⣰⡁⠀⠀⠀⢀⢺⡟⠄⠀⠁⠀⠘⣆⠀⠀⠀⠀⠀⠀⢽⣿⣇⠘⠃⠿⢸⠸⡇⠟⢀⣿⣿⠅⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢈⣿⡁⠀⠄⠀⠀⢨⣇
⠀⠀⠸⣦⣀⣀⣠⡆⠈⠃⢰⣄⣀⣀⣴⠇⠀⠀⠀⠀⠀⠀⠰⢻⣿⣷⡶⠆⠀⠀⠶⣶⣾⣿⠳⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⠃⠿⠘⠂⠀⠀⢀⣰⠏
⠀⠀⠀⠈⢋⠏⠉⠀⠀⠀⠀⠉⠹⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠋⠁⠀⠀⠀⠈⠉⠋⠻⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⣶⣶⡦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⣶⣶
⠰⣿⣖⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣲⣿⠆⠀⠀⢦⢀⣀⡱⡶⠤⣀⠀⠀⠀⢀⡠⠴⡖⢁⣀⢠⠖⠀⠀⢻⣿⡟⢯⣼⠗⢄⠀⠀⠀⡠⠾⣧⣽⢫⣿⡟
⠀⠙⣿⡏⠉⠘⠢⡀⠀⠀⢀⡴⠃⠉⢹⣿⠋⠀⠀⠀⠈⢳⣴⣌⠁⠀⠈⣳⣶⣶⡋⠀⠀⢉⣴⣴⠋⠀⠀⠀⠈⢿⣿⣆⠹⢳⣿⣷⣶⣾⣿⡖⠏⣰⣿⡿⠁
⠀⠀⢿⠦⠀⠐⠀⠙⢤⡤⠋⠀⠂⠀⠼⡿⠀⠀⠀⠀⠀⠀⠹⣟⣧⠴⠖⠚⣿⡟⠒⠶⢤⣟⡿⠃⠀⠀⠀⠀⠀⠀⠻⢿⣇⠘⠛⢹⣿⡏⠋⠃⣸⡿⠟⠁
⠀⠀⢘⣿⠋⠙⠛⠛⢻⡟⠛⠛⠋⠙⣿⡃⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠐⠰⢻⢻⠐⠀⠀⡸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣰⣾⣿⣷⣆⣠⣿⣿
⠀⠀⠨⣷⣀⠐⠘⠀⠻⠿⠀⠃⠂⢀⢾⠇⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⡄⠀⠀⣤⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡜⠀⢢⣿⣿⡿⠏
⠀⠀⠀⠙⠯⠾⠦⠎⠀⠀⠱⠴⢷⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠀⠀⠀⠈⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠁⠀⠀⠀⠈⠉⢳

⠀⠀⠰⣶⠂⠀⠀⠀⠀⠀⠀⠀⢲⡶⠀⠀⢠⡄⠀⠀⠀⠀⠀⢤⠀⠀⠀⢶⢠⡦⠀⠀⠀⠀⠀⠀⢰⣄⡶⠀⢶⡖⠂⢄⠀⠀⠀⢀⠄⠒⣶⠆
⠀⠀⠀⠈⢧⡀⠀⠀⡄⠀⠀⣠⠏⠀⠀⠀⣾⢠⠞⡄⡀⣜⢢⠘⡇⠀⠀⠀⠘⡇⠀⠀⢀⡀⠀⠀⢰⡏⠀⠀⠈⡦⠀⠀⠱⣤⡄⠀⠀⢠⡏
⠀⠀⠀⠀⢠⡍⠁⠀⡇⠀⠉⣥⠀⠀⠀⠀⢛⣸⠐⣇⣿⡾⢸⣅⡃⠀⠀⠀⠀⢀⠀⠀⠸⡇⠀⠀⡀⠁⠀⠀⠀⠹⣤⢄⣠⢿⢦⡀⢤⠞⠁
⠀⠀⠀⠀⠘⣧⣄⣠⣡⣀⣤⠟⠀⠀⠀⠀⠘⣿⣆⠈⡏⠀⣼⡿⠁⠀⠀⠀⠀⠻⣧⣀⡀⢁⣀⣴⡿⠀⠀⠀⠀⠀⣇⠸⢣⠘⢘⠿⣸⠇
⠀⠀⠀⠀⢀⡼⠉⠀⠀⠈⠹⣄⠀⠀⠀⠀⠀⠈⠛⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⢨⠛⠁⠈⠛⢧⠀⠀⠀⠀⠀⠀⠈⠳⠋⠀⠘⢶⠉
⠀⠀⣀⣀⠈⠁⠀⠀⠀⠀⠀⠉⢀⣀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠸⡇⠀⠀⠀⠀⡀⠀⠀⠀⢨⡟⠀⠘⣦⢨⣸⢦⣄⣠⡴⣇⡄⣴⠃⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⣧⣤⠀⢀⠀⣠⣴⡏
⠀⠀⠀⠹⣄⡀⠀⠸⡇⠀⠀⣠⡾⠀⠀⠀⢸⣆⡀⢠⣸⣇⡄⢀⣰⡇⠀⠀⠸⣇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡙⠉⠀⢹⠀⠉⠙⡀
⠀⠀⠀⠀⢻⣤⡀⠀⠃⠀⣠⣿⠀⠀⠀⠀⠀⠧⡳⢤⠈⠁⡤⢞⡼⠀⠀⠀⠀⠉⣇⠀⠀⠙⠀⠀⢰⠏⠀⠀⠀⠈⣦⠀⠀⠈⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠙⠛⠃⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠐⠚⠃⠀⠀⠀⠀⠀⠈⠛⠒⠀⠘⠻⠋
⠀⠀⠰⡆⢀⠄⡀⠀⠀⠀⡀⡄⢀⡆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠹⣄⠈⠁⢀⡄⠀⠉⢠⡞⠀⠀⠀⣀⠀⠀⠀⢿⣿⠀⠀⠀⢀⠀⠀⠀⠈⢰⡇⠋⢠⡄⠑⢸⡆⠁⠀⠀⠀⠀⢀⡐⣿⡇⡀
⠀⠀⠀⠀⢨⡃⠀⠸⡇⠀⠀⣋⠀⠀⠀⠈⠓⠂⠀⠀⠸⠇⠀⠀⠀⠚⠁⠀⠀⠀⢠⠁⠀⢸⡇⢀⢘⡄⠀⠀⠀⠀⢠⣌⣛⣛⣚⣫⣤
⠀⠀⠀⠀⠈⠳⢴⠄⠁⠦⠶⠃⠀⣤⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⢳⡄⠀⠀⢀⡄⠀⠀⠀⠀⣼⡷⢯⣽⣿⣯⣭⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠘⠛⠛⠛⠛⠈⠋⠁
⠀⠀⣴⡶⢂⣄⠀⠀⠀⠀⣀⡀⠶⣦⠀⠀⣴⣤⣀⡀⠀⠀⠀⢀⡀⣤⣄⠀⠀⣶⣦⣄⠀⠀⠀⢀⣴⣶⡄⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⢹⠁⢸⡏⠀⢀⡀⠀⠈⣿⠀⡿⠀⠀⢹⡁⠋⡸⠷⣤⠺⡏⠛⢸⡏⠀⠀⣿⠀⠀⠠⡀⡐⠁⠀⣿⠁⠀⠀⢻⡎⠀⠀⣠⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠘⢷⡀⢸⡇⠀⣰⠇⠠⠀⠀⠀⠈⠧⠠⡿⠁⠗⠘⣿⠠⠞⠀⠀⠀⠹⣄⡀⠀⣿⠀⢀⣀⡝⠀⠀⠀⠈⢳⡄⠀⢻⠀⢠⡼⠃
⠀⠀⠀⠀⢰⣌⠛⠋⠙⠛⣁⣴⠀⠀⠀⠀⠀⠸⣆⣀⠀⠀⠀⣀⣴⠃⠀⠀⠀⠰⡇⢀⡄⠙⠠⡀⢀⡇⠀⠀⠀⠀⠸⣇⢠⡞⣄⢀⡿
⠀⠀⠀⠀⠈⠙⠻⠁⠀⠻⠛⠁⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠈⠉⠑⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⠉⠛⠀⠙⠋

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⣤⣤⣤⣀⡀
⠀⠀⣰⣾⠏⢹⣿⣟⠁⠚⠋⠉⠙⢛⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⣚⠋⠉⠑⠂⠀⠈⣻⣿⡅⠘⣿⣧
⠀⠀⢿⣿⠢⢼⣏⡉⠉⠉⠀⠀⢀⣨⣿⠟⠒⠫⠕⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠯⠛⠐⠹⣿⣤⠀⠒⠒⠒⠀⠤⠸⠟⣁⣿⡟
⠀⠀⠈⢟⡯⣤⡒⠀⢤⡄⠒⠀⠈⠉⠢⣄⡀⣠⣤⣴⣿⠝⠦⣀⠀⠀⠀⠀⠀⠀⣠⢞⠽⣿⡷⣶⣦⣀⡠⠖⡉⠈⠀⠀⣛⡖⠪⣽⣶⣿⠝⠁
⠀⠀⠀⠀⠘⢯⡿⢶⣿⡉⠴⠖⠀⠀⠠⣾⣿⣉⡿⠤⣍⣀⠝⠮⣢⡀⡤⣤⢀⣾⡷⢥⣚⠥⠤⠾⢩⣿⡿⠆⠀⠉⣙⠓⠨⣭⣖⢻⠼⠋
⠀⠀⠀⠀⠀⠀⠙⢏⡭⢔⣊⡉⠤⢖⠂⣹⣿⣷⠦⠤⠤⢭⣛⣶⢮⣿⣆⣺⣿⠷⢛⣋⣁⠒⠒⠒⢿⣿⣧⡉⣒⠢⠍⣩⡶⢮⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢻⣤⣔⣛⡡⠬⠛⣛⣉⡭⠤⠐⠀⠖⢊⢞⣻⣟⠳⡈⠀⠐⢒⠠⠌⡉⠁⣓⠂⢤⣙⣷⣶⣏⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣿⣿⡣⠎⢁⡀⠄⠀⠀⠄⠀⠘⢁⢼⢱⠈⡀⠈⠓⢄⠂⠀⠉⠔⠀⡀⠐⣩⣯⠿⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡔⣋⠉⣀⡌⠀⠀⢀⡠⠊⠀⠀⠎⠀⢨⠀⢁⠠⠐⡄⠑⢄⡀⠀⠄⢹⠄⡠⠍⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⡿⠖⠉⣠⠟⢐⠊⠁⠀⢠⠃⡜⠀⠀⠠⠀⠀⠆⠁⠘⡄⠰⠀⠙⡆⠙⢄⠈⡙⢿⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣧⣊⡴⢁⠀⡞⠀⢸⠀⡏⡸⠠⠀⠀⠀⠀⠀⠈⠄⠃⠹⡀⠁⢀⠸⡀⣌⠳⣝⣾⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣩⣿⣠⠏⡼⢠⠇⡞⣸⢀⠃⠆⠀⠀⠀⠀⠀⠀⠀⠈⠄⣧⢸⠈⡇⢳⣘⣦⡿⣧⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⣏⣿⠳⡾⢴⢷⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⡾⡷⢿⡞⢿⡘⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣴⢛⣩⣍⣉⣉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢋⣉⣉⣩⣙⢳⣄
⠀⠀⠀⠀⢸⣁⣠⣤⣤⣤⣤⣉⡑⠲⢍⡳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣋⠵⢒⣉⣡⣤⣤⣤⣤⣀⣻
⠀⠀⠀⠀⢸⣇⢀⣠⡤⠤⠤⣤⣍⣙⣶⠛⠉⠛⢦⡀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⣠⠞⠋⠙⢳⣞⣉⣥⡤⠤⠤⣤⣄⣀⣿
⠀⠀⠀⠀⠀⢿⣄⠠⠤⠤⠤⠤⠤⣬⣿⣄⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⢀⣼⣧⡤⠤⠤⠤⠤⠤⠀⣼⠇
⠀⠀⠀⠀⠀⠈⢿⡀⠠⠤⠶⠒⠒⠒⠶⠾⠷⣄⡀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⣀⡴⠿⠶⠖⠒⠒⠶⠦⠤⢀⣸⠏
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠤⠒⠒⠒⠒⠒⠒⠒⠾⠿⠶⢤⣄⣀⠙⣄⢠⢤⢀⡞⢁⣀⣤⠶⠿⠷⠖⠒⠒⠒⠒⠒⠒⠶⢄⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣖⠒⠒⠒⠒⠶⠦⠤⠤⠬⠿⠷⣾⡾⠰⣮⠾⠿⠧⠤⠤⠤⠶⠖⠒⠒⠒⢒⣦⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡩⠛⠛⠛⠛⠓⠶⠟⠛⣛⣛⢿⣧⢠⣿⢟⣛⡛⠛⠷⠖⠛⠛⠛⠛⠯⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣠⠤⠒⠒⠚⣛⡷⠒⠊⠉⣠⢻⡇⠋⡏⢣⡈⠉⠒⠺⢟⡛⠃⠒⠢⢤⣈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠋⡠⠔⠚⠋⡽⠁⠀⠀⢀⡴⡃⢸⢇⠀⢿⠀⡷⣄⠀⠀⠀⠹⡉⠓⠒⠤⡈⠳⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠀⡠⠔⢊⡷⣄⡤⢶⠋⠀⠁⡾⢨⠃⠸⡄⠁⠈⢳⠦⣄⡴⣏⠒⠤⡀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡰⠋⢀⡴⠋⠀⡜⠀⡏⠀⠀⣰⠇⠸⡔⠀⢧⡀⠀⠈⡇⠘⡆⠈⠓⣄⠈⠳⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⡞⠀⠀⡜⠀⢀⠀⠀⢀⡟⠀⠀⠀⠀⠘⣇⠀⠀⣦⠀⠘⡄⠀⠘⡆⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣅⡀⢸⠀⠀⠀⡇⣠⡞⠁⠀⠀⠀⠀⠀⠙⣦⡀⡏⠀⠀⢸⠀⣀⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠾⠛⠉





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

⠀⠀⠀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄
⠀⠀⢠⣿⣯⣿⣿⣿⣿⣿⣦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣴⣾⣿⣿⣿⣿⣽⣿⡄
⠀⠀⠈⢿⣷⣿⣿⠛⢿⣧⣼⣿⠻⢷⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡾⠟⣿⣧⣼⡿⠛⣿⣿⣾⡿⠁
⠀⠀⠀⠈⠻⣿⡿⣷⡟⠛⠻⠶⣿⠛⠓⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠞⠛⣿⠶⠟⠛⢻⣾⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠖⠂⠠⠼⢄⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⡠⠧⠄⠐⠲⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣀⣀⣀⣀⠠⠽⠦⢄⡀⠈⠻⣦⡀⢠⡄⢀⣴⠟⠁⢀⡠⠴⠯⠄⣀⣀⣀⣀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣌⠳⣾⣷⠞⣁⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣽⣿⣿⣯⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⡶⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢉⣉⣉⣉⡝⠒⠘⠉⣉⢝⣿⣿⡯⣉⠉⠓⠒⢫⣉⣉⣉⡉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⣁⠤⡴⠋⠀⢀⡠⠊⢡⢾⣿⣿⡷⡄⠑⢤⡀⠀⠙⢦⠤⣈⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⠊⢉⡠⠼⡤⠄⢲⠋⠀⡰⢃⠎⣿⣿⠱⡈⢆⠀⠙⡖⠤⢤⠧⢄⡉⠑⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣖⠉⢀⡜⠀⢠⠃⢀⠞⠀⡎⠀⣿⣿⠀⢱⠀⠳⡀⠘⡄⠀⢣⡀⠉⣲⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣎⢀⣠⠃⢀⠎⠀⣼⠀⠀⢹⡇⠀⠀⢧⠀⠱⡄⠘⣄⡀⣱⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⣿⣷⣿⣄⡼⠁⠀⠀⠈⠁⠀⠀⠈⢧⣀⣿⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁

⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⢀
⠁⠀⠀⠀⠀⠀⠀⠀⢄⢿⡇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣦⡁
⠆⠀⠀⠀⠀⢀⣆⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⡓
⠀⠀⠀⠀⠐⠺⡷⠣
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣡⣿⣔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⣴⠖⢓⠈⠈⠉⠉⣼⣿⡀⠀⠀⢀⡠⠄⠒⠀⠀⠀⠀⠀⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⢿⡗⢀⠀⠀⢀⠀⠀⣿⣽⠁⢀⠜⠁⢀⠄⠀⠀⠀⠀⠀⢀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣎
⡁⠀⠀⠀⠉⠚⠠⢟⣀⠀⠀⣿⡿⠀⠗⠀⡔⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣻⠃⠀⡄⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⢱⠀⢧⠀⠀⠀⠀⠀⠀⠄⠀⠀⠨⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢀⠧⠼⠃⠀⠀⠠⣄⣀⡀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠓⢔⠀⠈⠓⣆⠀⠀⠀⠸⠛⠿⠇⠀⠀⡌⠀⠀⠀⡞⠀⢠
⠀⠀⠀⡄⠀⠐⠀⠀⠀⠐⢧⡤⠤⡑⠤⣀⡙⣦⡀⠀⠀⠀⠀⠀⣀⠞⠀⣀⠴⢊⠕⠐⠊⠁⠀⠀⠘⢀
⠀⠀⠀⠁⠀⠀⠀⠱⠀⠀⠈⢴⠀⠈⠓⠤⢨⠋⠉⠉⠉⠉⠙⠃⢩⡉⡭⠄⠊⠁⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⢠⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢠⠀⠀⢠⠀⠀⠳⠀⠀⢀
⠀⠀⠀⢀⠀⠀⠔⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠸⠀⠀⠘⠄⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⢤⣄⣀⣀⣀⣤⢤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠄⠀⠀⠀⠀⠀⢢⠀⠀⠀⢀⣰⡷⣄⠀⣴⡃⠀⢀⣀⣀⣀⣤⡞⣩⠓⠤⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀
⡄⠤⠀⠆⠲⠂⠐⠒⠒⠀⠀⠚⠋⠳⡌⠞⢁⠽⠾⠭⠭⠭⠥⠬⠭⣭⢙⡤⠯⠤⠤⠤⠤⠤⠤⠤⠬⠤⠤⠴⠄⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠴⣷⣯⡀



⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣭⣷⣶⣶⣶⣶⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣆⣤⡿⢿⣿
⣾⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⢿⣿⣏⣉⢈⣙⣷⣾⣿
⢿⣿⣿⣿⣧⣀⠠⢴⣿⣿⠛⢿⣗⠦⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⢀⣠⣾⣯⣾⣯⠄⠚⣿⣅⣀⣈⣹⣿⣿⣿
⠸⣿⣿⣿⣯⣤⣤⣴⣾⣍⣁⡾⢿⣿⣟⡛⠦⣄⠀⠀⠀⠈⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⣠⡴⣋⣥⣾⣿⣋⣉⣽⣿⡏⠉⠉⣹⣿⣿⣿⡟
⠀⠹⣿⣿⣿⣿⣀⢀⣀⣿⣿⠶⠦⢾⠿⠿⠶⠾⢿⣦⣀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⣠⡾⠟⠋⠉⠉⠉⠙⠉⠉⠉⢻⣿⠷⠾⣿⣿⣿⠏
⠀⠀⠘⢿⣿⣿⡟⠋⢻⡟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⡼⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⣿⣤⣤⣿⣿⠋
⠀⠀⠀⠀⢹⣿⣷⡶⠾⣿⣄⡀⠀⠀⣤⠷⣄⡀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠘⣆⠀⠀⠀⡼⠁⠀⢠⡾⠋⠀⠀⠀⠀⢀⠔⠋⠉⠉⠀⠐⠲⣿⣿⣭⣿⣿⡿
⠀⠀⠀⠀⠈⢿⣿⣤⡤⣿⡟⠁⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠙⢷⣄⠀⠘⡆⠀⡸⠁⢀⣴⠟⠀⠀⠀⠀⠀⣰⡁⠀⠀⠀⠀⠀⠀⠀⣸⣉⣈⣿⣿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣦⣼⣧⡀⠀⠀⣀⣀⣤⠤⠶⠓⠠⠀⠀⠀⠀⠀⠈⠳⡄⣸⣶⣇⢠⠞⠁⠀⠀⣀⠀⠀⠉⠉⠉⠛⠒⠒⠒⠀⠤⣶⣏⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠙⢿⣿⣿⠃⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⢁⣠⣾⣿⣿⣆⣈⣁⣀⣀⣀⣤⣤⣴⣶⣶⣶⣦⣤⣄⣀⣠⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣶⣶⣿⠿⠿⠿⣿⣿⡿⠿⠿⢛⣛⠛⠛⣿⣿⣿⣿⣿⡛⠛⠋⣉⣉⠙⠹⢿⡟⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⢀⡴⠋⡟⣿⡟⢏⠳⢄⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣀⠀⢀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠶⠆⠀⠂⠀⠸⠁⠀⠀⠀⠀⣠⠔⠋⠀⠘⢀⣿⣷⠈⠄⠀⠙⠲⢄⡀⠀⠀⠀⠈⢄⠀⠀⠀⠈⠉⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁⠀⠀⠀⠀⣠⠃⠀⣀⡤⠶⠋⠁⠀⠀⡴⠀⡜⣿⣿⢀⠈⢦⠀⠀⠀⠉⠛⠲⢤⣤⠈⠷⢄⠀⠀⠀⠀⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣠⠀⠀⠉⢉⠬⠋⠁⠀⠀⠀⠀⢀⠌⠁⡼⠁⣿⣿⠈⢦⠀⠱⡀⠀⠀⠐⡀⠀⠈⠑⠀⠀⠀⠈⠛⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⠊⢀⣶⠃⠀⢹⡏⠀⠘⢷⡄⠈⢀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣤⣤⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣷⣾⣏⠀⠀⠀⠀⡀⠀⠀⣠⠃⠀⣸⠃⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢢⡀⠀⠀⣦⣀⣀⣀⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⣶⣿⣅⣀⣴⠃⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⠿⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣴⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠏⠈⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣤⡀⠀⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢻⠃⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⡠⠁⠀⠀⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠹⠟⢿⣧⣴⢟⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣤⣄⠁⠀⠀⠀⠀⠀⠘⢿⣯⣰⡆⢸⡿⠋⢐⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡏⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣇⡜⠀⢼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⢿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⢭⠒⠋⠁⠀⠀⠀⠀⠀⢠⡾⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢹⠆⠁⠀⠀⠀⠀⠀⠀⣀⣄⢁⢸⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⣤⣤⣴⣴⡦⠒⠛⣶⣞⡁⠁⢀⣈⣀⣠⠀⢰⠀⠀⠀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⠟⠉⠁⠀⡀⠰⣾⣶⣯⠛⢫⠟⠁⠀⠀⠀⠀⢀⣼⣴⣿⣠⣦⡔⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠘⣿⣶⣼⣋⠡⠴⠹⠁⣶⢆⠄⣤⣴⠂⠹⠿⡿⠛⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡴⠂⠀⠀⠀⠘⣿⣇⣠⣆⡴⠀⠀⠀⠀⡀⠴⠧⠀⢠⣤⠇⠠⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠘⣿⠄⠀⣴⡿⠀⣹⡏⢩⣶⡶⠀⣴⡆⠀⡠⣶⢤⠀⠸⣇⠖⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⣦⣤⣤⠀⠐⡦⠤⢄⢸⣷⣬⣽⣾⣦⣭⣥⣤⣤⠴⢀⡤⣀⠀⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢶⣦⠔⣶⣽⣆⣞⠻⢿⣿⣿⣿⣿⢿⡿⢟⣡⡟⣥⣿⣗⣢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠖⠹⠾⠙⠛⠋⠛⠛⠛⠛⠦⠖⠻⠿⠿⠿⠛⠛⠋

⠀⠰⣄⠀⣴⣠⢀⠀⠀⠀⠀⢀⢠⣰⡄⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡰⢤⠀⠀⠀⠀⠀⠀⠀⡀⡤⠆⣹⠇
⠀⠈⢏⣷⡀⠉⠚⢹⣶⣷⡿⠙⠈⠁⣰⢯⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡥
⠀⠀⠀⠑⠻⣦⠀⠾⠉⡇⠻⠂⢠⡞⠓⠁⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⣿⡇⠀⠀⠀⡠⠂⠀⠀⠀⠀⠐⠆⠀⣀⠀⠀⢘⡃⠀⠀⣀⠀⠰⠓
⠀⠀⠀⠀⠀⢿⣄⡀⠀⠁⠀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠉⠘⠃⠉⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⣨⠙⠛⠀⠀⠙⠛⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡡⠀⠀⢀⠈⠁⡀⠀⠀⢄⡞
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠖⠀⠀⠀⠀⠲⠶⠚⠁
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢤⣤⡀
⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠉⠀⠀⠙⣿⣬⡿⠛⠛⠓⠤⠀⠀⠀⠀⠀⠔⠛⠛⠻⣿⣸⡿⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠈⢻⣏⠀⠀⠀⠀⢀⣠⣧⣀⠀⠀⠀⠀⢀⣿⠏⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⢠⡄⠐⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠶⡶⠾⠟⠙⡟⠛⠿⠶⡶⠦⠼⠃⠀⠀⠈⠑⢄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⣇⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⢸⡇⠀⠀⢠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠏⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣨⣅⣀⣠⠃
⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠉⠛⠶⠞⠃⠀⠛⠷⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁
⠀⠈⢿⣧⠀⠙⠂⠀⠀⠀⠀⠀⠐⠋⠁⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤
⠀⠀⠈⢿⣧⠀⠀⠀⠀⣀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠙⢥⡗⣠⣄⠀⠀⠀⢀⣤⣲⣧⠟⣱⣿⠏
⠀⠀⠀⠈⠛⠢⠄⠀⠀⣻⠀⠀⠀⠄⠛⠋⠀⠀⠀⢺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠘⣿⣧⡀⠉⢻⣥⣷⣼⣴⣧⣿⠋⠁⣰⣿⡏
⠀⠀⠀⠀⢸⡅⠀⠀⠀⠈⠀⠀⠀⠀⣗⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠙⢳⣦⠀⠀⣾⢹⢹⡄⠀⢰⣶⠛⠉
⠀⠀⠀⠀⠀⢻⣦⣴⠆⠀⠀⢶⣤⡾⠃⠀⠀⠀⠀⠀⠙⣳⠖⠀⠀⠀⠞⠀⠀⠐⠲⣞⠃⠀⠀⠀⠀⠀⢸⣿⡀⠀⣇⠸⢸⠇⠀⣼⣿
⠀⠀⠀⠀⠀⡜⠉⠁⠀⠀⠀⠀⠉⠹⡀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠈⢻⣿⣷⠀⠀⠠⣰⣿⣿⠃
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁
⠀⠀⡀⢀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤
⠀⠀⠀⠠⠀⠀⠀⢁⠀⠀⠀⠀⣀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠢⠀⠈⠀⠀⡀⠀⠉⠀⠰⠀⠀⠀⢀⣴⢦⣶⢇⣶⣶⣦⣼⣤⣶⣶⣦⠶⣶⢴⣤⡀⠀⠀⠈⣿⠴⠂⠐⠂⠢⠀⠀⢀⡀
⠀⠀⠀⠀⠁⢄⠀⠀⠀⠒⠐⠀⠀⠀⠠⠀⠀⠀⠙⠿⣿⣿⣌⣥⡘⠻⣿⡿⠛⣁⣍⣰⣟⣿⠟⠃⠀⠀⠸⣇⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡇⠀⠀⣄⠀⠁⠀⠀⠀⠀⠀⠙⢿⢿⣿⣦⣾⣧⣴⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠻⣶⣀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⡤⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣎⣙⡏⢿⣉⣵⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁

⠀⠀⢰⣧⠶⠀⠀⠀⠀⠴⢦⣷⠀⠀⠀⠀⢿⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣧⠀⠀⠀⠀⠀⠘⣿⡟⢴⠦⣄⠀⡠⢶⡌⣻⡟
⠀⠀⠸⣭⣀⠀⢀⣄⠀⢀⣩⡝⠀⠀⠀⠀⢸⡆⠀⠁⠀⠀⠐⣻⠂⠀⠀⠀⠀⠀⠀⡼⠖⢺⣷⠄⠀⠀⠀⠀⢸⣧⣀⣠⢘⣟⢠⣀⢠⣿⠁
⠀⠀⠀⠀⠨⡁⠸⠇⡀⡭⠱⠀⠀⠀⠀⠀⣸⡇⠀⢠⡿⢄⡴⠃⠀⠀⠀⠀⠀⠀⠀⠁⢠⠞⠻⣄⠀⠀⠀⠀⠀⠉⣿⠩⠘⡏⠨⣹⠏⠁
⠀⠀⠀⠀⠢⠄⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠘⠿⠔⠈⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠀⠈⠃⣸⠂⠀⠀⠀⠀⠀⠘⠛⠋⠈⠛⠋
⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⡤⣀⠀⠀⠀⠀⠀⠀⣀⢤⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⡀⡀⠉⠁⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⣤⣮⠠⢷⡄⠀⠀⠀⢹⡝⢟⡦⠀⢀⢔⡻⢫⡇⠀⠀⢡⡁⠀⠄⠀⠀⠔⠁⣁⠉⠀⠀⠀⠐⠀⠈⠇⠀⢠⡄⠀⠀⠀⡀⠁⠀⢹
⠀⠀⠀⠀⣾⠀⠰⢨⠇⠀⠀⠀⠘⢦⠶⠉⢻⡟⠡⢦⡔⠃⠀⠀⠀⢿⣀⠀⣠⡌⠀⣨⡇⠀⠀⠀⠀⠐⠃⢌⣷⠀⠀⠡⠀⣀⣾⣠⠤⣶⣌
⠀⠀⠀⠀⢸⡶⠚⠥⡑⡄⠀⠀⠀⢿⢄⢀⡎⢡⣀⢨⡇⠀⠀⠀⠀⢠⡈⠁⠸⠁⠉⣡⠀⠀⠀⠀⠲⡄⠀⢨⡽⠀⠀⠀⠀⠟⣘⡙⣄⢭⣬⠄
⠀⠀⠀⠀⠀⠙⠒⠶⠛⠁⠀⠀⠀⠈⣙⠉⠀⠀⠉⡏⠁⠀⠀⠀⠀⠈⠻⠊⠀⠈⠲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⢁⠀⠀⠀⠀⠀⣖⣣⡀⠀⠀⠀⠀⢸⣧⠐⠀⠀⠀⢀⠂⣽⡇⠀⠀⠀⣼⣻⣾⡳⡀⢀⣤⡤⠄⠀⠀⡈⠝⢿⡇
⠀⠀⠈⠹⣁⣀⠀⡀⠙⠂⢹⠀⠀⠀⠀⠀⣿⡄⠁⠀⠀⠀⠀⠘⠧⣬⠀⢰⡆⢀⡥⠾⠃⠀⠀⠀⣨⠶⠶⢿⣿⣟⣻⠃⠀⠀⠀⠀⢺⣹⣧
⠀⠀⠀⠀⠀⠰⠴⢻⡒⠖⣩⠀⠀⠀⠀⢐⠈⡳⡄⠸⡀⠀⠀⠀⢀⢪⡱⠜⠣⣟⡃⡀⠀⠀⠀⠀⣿⡢⣠⠎⡉⡹⠁⠀⠀⠀⢾⡀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠙⠈⠀⠁⠈⠀⠀⠀⠀⠈⠂⠀⠠⠋⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠈⠉⠁⠀⠷⠃⠀⠀⠀⠀⠀⠉⠀⠀⠈
⠀⠀⠐⠀⠀⢀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⣾⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡴⠖⠀⢀⣀⠀⠐⠶⡄⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡆
⠀⠀⠀⠈⣏⠀⠁⣀⠈⠀⢹⠃⠀⠀⠀⠀⡇⠀⠈⠂⠀⠀⠀⠀⡔⠀⠀⠀⠸⣷⣀⡤⣚⣟⣠⣀⣼⡿⠀⠀⠀⠀⢖⣠⣄⠀⠀⠘⣫⣯⣷
⠀⠀⠀⠀⢨⠛⠉⢿⠉⠛⠅⠀⠀⠀⠀⠠⠓⠘⠀⢴⠧⠀⡠⠊⠀⠀⠀⠀⠀⢰⠁⡅⠸⡃⢀⠍⠇⠀⠀⠀⠀⠀⠀⠙⢙⣀⣰⡆⠁⠀⣦⡀
⠀⠀⠀⠀⠘⠶⠆⠀⠐⠶⠇⠀⠀⠀⠀⠘⠤⠀⠀⢻⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡟⠠⠀⠿

⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⠯⠀⠙⠆⠀⡀⠠⠊⠁⠹⠋⠀⠀⢹⡇⣿⡶⢀⠀⠀⢀⢰⣾⢇⡿⠀⠀⠈⣿⣿⣶⠀⠀⡀⠀⢰⣾⣿⡏
⠀⠀⠰⢤⣀⣀⣼⣯⣀⣀⣠⠶⠀⠀⠀⠀⠻⣦⡽⠿⢶⣷⠾⠯⣥⡾⠃⠀⠀⠀⠩⣷⡷⠐⢶⣷⠆⠰⣷⡯⠁
⠀⠀⠀⡊⢀⠄⢸⣯⠀⣄⢈⠂⠀⠀⠀⠀⠰⣿⡇⠶⡟⠿⠶⠆⣿⡷⠀⠀⠀⠀⢠⣾⣿⡥⠉⡏⠣⣿⣿⣦
⠀⠀⠀⢸⣀⡐⠀⠀⢀⣀⡞⠀⠀⠀⠀⠀⠀⢻⣿⠾⠃⠀⠳⣿⣿⠃⠀⠀⠀⠀⢸⣝⣥⠇⠀⠀⠀⢧⣝⣽
⠀⣀⣀⠈⠀⠀⠀⠀⠀⠁⠘⣀⣀⡀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⢻⣿⠛⠢⡄⠀⠀⢀⠴⠚⢻⡿⠁⠀⠹⣖⡹⠋⠒⢄⣄⠔⠈⢿⣓⡞⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⣿⡄⢐⣈⣢⣴⣃⣐⢀⣿⠇⠀⠀⠀⠘⣿⡶⠖⢻⣿⠳⠶⣾⡏⠀⠀⠀⢦⡄⠀⠀⠀⢼⣿⠄⠀⠀⠀⣤
⠀⠀⢸⡋⢉⡉⢹⣿⢉⣉⠉⣿⠀⠀⠀⠀⠀⠹⡆⠈⡉⠋⡉⢀⡾⠀⠀⠀⠀⠀⠙⠠⠔⠀⢸⣿⠀⠐⠤⠘⠁
⠀⠀⠘⢷⣤⣄⠉⠉⢤⣤⣶⠟⠀⠀⠀⠀⠀⢠⠟⠋⠀⠀⠙⠛⡆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠓⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⡀⢠⣄⡀⠀⢀⣄⠀⠀⣀⡤⠠⡀⠀⡼⠂⠀⠑⠠⣀⣄⡠⠒⠁⠀⠲⠀⠀⢻⡽⠟⠇⢤⠀⠀⢠⠄⠟⠿⣽
⠀⢳⣆⠀⣀⡤⢘⡗⠠⣄⠀⢀⡾⠃⠀⠈⠒⡶⠤⠤⣾⣿⠦⠤⠴⠖⠊⠀⠀⠀⠇⣀⣀⣀⣱⣶⣁⣀⣀⡀⠇
⠀⠀⠉⡀⠀⠀⡰⢧⠀⠀⠁⡉⠁⠀⠀⠀⠘⠀⠔⢀⠞⠟⢆⠐⠄⠈⠀⠀⠀⠀⣼⠁⠀⠀⣸⣿⡀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠐⠃⠀⠁⠀⠀⠈⠁⠀⠓⠀⠀⠀⠀⠘⠶⣤⠄⠀⠁⠁⢤⡴⠞
⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⢠⣶⣦⢄⠀⠀⠀⠀⢀⢴⣶⡆⠀⠀⣤⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤
⠘⣗⠀⠙⢦⠀⠀⠀⠀⡠⠚⠁⢰⠇⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠸⣿⡿⣽⣢⣄⢀⣤⣻⡽⣿⡿
⠀⠈⡁⠐⠀⠁⢠⣆⠜⠀⠐⠀⠉⠀⠀⠀⠘⡻⠖⠈⢺⣷⠂⠐⠾⡛⠀⠀⠀⠀⠹⢿⣆⠿⢿⣿⠿⢧⣾⠟⠁
⠀⠀⠀⡰⠒⠐⢺⣿⠂⠐⠢⡀⠀⠀⠀⠀⢸⣇⠀⢀⡈⠏⣄⠀⢀⣷⠀⠀⠀⠀⠀⢸⣿⣤⡾⠿⣦⣼⣿
⠀⠀⠀⠇⣠⠀⠈⠋⠀⣠⠨⠃⠀⠀⠀⠀⠀⠙⡛⠉⠀⠀⠈⠹⡛⠁⠀⠀⠀⠀⠀⠀⡻⠛⠃⠀⠛⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣿⣿⣿⡟⡏⠏⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣶⣀⠀⢸⡋⠙⡏⠉⡟⢻⣤⢧⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣏⠉⢢⠘⡍⢻⢷⣿⠇⠀⠇⠀⠃⣘⡿⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣈⠑⢄⡦⢥⠘⢠⣿⠀⠀⢀⡴⠋⢉⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠧⣌⡙⠷⣥⠀⠙⣾⡿⠀⢀⠏⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⣉⠳⢄⠱⣄⢸⡇⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣾⣿⣷⣶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⠀⢈⠩⢛⢿⡟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣉⡤⠞⣁⠔⠁⣸⡇⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣠⠶⢋⡍⢀⠴⢿⡇⠀⠈⢆⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⡠⠔⠉⠋⡅⡀⢸⣿⠀⠀⠈⠳⣄⣘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣇⣴⣁⣼⣠⣷⠟⣿⠀⢰⠀⠰⠀⢫⡷⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠀⠀⢿⣤⣼⣄⣰⣧⣞⢻⢻⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣼⣠⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢻⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣉⣩⡷⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣽⣷⠋⢁⣠⡿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣩⣿⣤⣿⡟⠉⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢚⣏⣹⢋⡿⠟⠛⢻⣯⣼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢠⣿⠟⠉⢹⡟⠀⢀⣠⠾⣿⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠋⠀⠀⣼⠶⠚⠁⠀⠀⢈⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⡿⠃⠀⠀⣰⠃⢀⣀⡤⠶⠚⠉⠻⣿⣼⡆
⢠⣶⣞⣻⠟⠛⢻⣯⣭⣿⣛⣶⣶⡆⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡾⠁⠀⢀⡼⠛⠋⠉⠀⠀⠀⠀⠀⣠⣿⡽⠁
⠘⣿⣽⢷⠖⠋⢉⣹⣧⣼⣿⠛⢢⠞⠛⠛⠻⠾⠷⣦⣤⣀⠀⠀⠁⡀⠀⠀⡼⠁⠀⣠⠊⠀⠀⠀⠀⣀⣠⣤⣶⣿⢿⣏⡀
⠀⠈⠳⣾⣗⣟⠉⣁⣽⠟⠋⠙⠛⠷⣄⡀⠀⠀⠀⠀⠈⠉⠛⠶⣄⣸⣿⣷⡃⣠⠞⠁⠀⣀⣤⣶⡿⠟⠋⠉⠈⣹⣶⣿⣱⣦⡀
⠀⠀⠀⠀⠻⠿⣟⣻⣷⣄⣠⠴⠒⠛⠉⠉⠛⣲⠦⠤⢤⣤⣀⣀⣀⡸⣿⣿⣯⡵⠾⠿⠿⠛⠛⠥⣄⠐⠛⠉⠉⠁⠀⢉⣿⣷⣇
⠀⠀⠀⠀⠀⠈⢫⣿⣹⣏⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⣀⣴⡶⠛⣿⣿⡦⢀⡀⠀⠀⠀⠀⠈⠉⠛⢯⡉⠉⠉⠉⠉⣿⣽
⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⣷⠞⠉⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠟⠉⠀⠰⢹⣿⣿⡒⢌⡉⠓⠒⠶⢤⡤⠖⠚⠛⠦⣀⠀⢀⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣹⣿⣀⠀⣀⣠⣴⣾⠿⠛⠋⡩⡞⠁⠀⠀⣰⠃⡆⢹⣿⣷⣄⠉⠲⣄⡀⠀⠹⣶⣄⠀⠀⠈⢹⣿⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠿⠿⣿⢿⡏⠀⡠⠖⠉⣰⠃⠀⢀⡼⠁⢸⠁⢸⠻⣿⡌⠳⣄⠀⠙⢦⡀⠈⠙⢦⣀⣠⣾⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣾⡟⠉⢀⡴⢻⠇⢀⡠⣾⠀⠀⡾⠀⢸⠀⢹⣧⡀⠈⠳⣄⡀⠈⢲⣄⣼⣟⣿⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣄⡴⠋⠀⢈⠏⠁⢀⣿⡄⠀⡇⠀⣸⠀⠀⣿⣷⡀⠀⠈⠻⣶⣺⣯⣼⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⡀⠀⠀⡼⠀⠀⠈⡟⠀⢰⠀⠀⣿⠀⠀⠘⢿⠇⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⢿⣶⣴⣧⣀⠀⢀⣧⣀⣼⣤⡴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢽⣿⣿⣿⣿⣯⣽⣧⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀⡀
⠀⠀⠀⣠⣧⣼⣥⣽⣯⣨⣿⣉⣿⣖⣶⣤⣄⡀
⠀⠀⢠⣇⣘⣻⣟⠀⣠⣿⣅⣹⣯⣿⣻⠾⠯⠉⠛⠲⣄⡀
⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⠿⠟⠛⠛⠀⠀⠀⣰⣦⣼⣿⣷⠦⣄
⠀⠀⠈⣧⣸⣾⣿⡿⠛⠛⠓⠀⠀⠀⠀⠀⢀⣾⣿⡃⠉⣿⣿⣀⣨⣿⣦⣄⠀⠀⠀⠀⠰
⠀⠀⠀⢻⢾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣷⣄⡀⠀⠀⠁
⠀⠀⠀⠈⣿⢻⡏⢙⣟⠀⠀⠀⢹⣿⣿⡿⠛⠛⠻⢿⣯⡉⠉⠀⠉⠛⠉⠛⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠘⣟⢻⣿⣿⣦⣤⣴⣿⣿⣿⣶⠶⠶⠶⠾⠿⠿⣶⣄⡀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⢿⠛⡟⢿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣈⣙⣿⣷⣦⣄⡀⠀⠀⠈⠹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠈⢿⠛⣿⣿⣤⣥⣤⣤⣴⣶⠶⠾⠟⠛⠛⠉⠉⠉⠉⠉⠙⠻⠿⢶⣤⣄⡈⠛⢿⣦⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠛⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠛⠷⢬⣿⣧⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⣧⡀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣿⠖⠒⢒⣺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⡷⠶⢞⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣻⣟⣡⣿⣿⣿⣄⣀⣀⣠⡿⢿⣿⠿⠟⠛⠋⠉⠀⣠⣶⠟⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢩⣿⣿⣿⣿⣿⣿⣿⡋⠁⢀⣀⡞⠁⠀⠀⠀⠀⣠⣾⠟⠁⡰⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣻⣿⠏⠉⣀⣿⣿⡿⠿⠛⢉⠟⠀⠀⠀⠀⣠⣾⠟⠁⢀⡜⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣿⣿⣿⡿⠟⠋⠉⠀⢀⣠⣴⣯⣀⣠⣤⣴⣾⠟⠁⠀⢠⡞⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢾⡿⣟⠀⠀⣀⣤⣾⠟⠋⢠⣿⠟⠉⣱⣿⠋⠀⠀⢠⠏⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⣿⣿⣤⣾⠟⠋⠀⢀⣴⡿⠃⠀⣼⡿⠁⠀⠀⣰⠋⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣼⣿⠻⡇⠀⠀⣰⣿⠏⠀⠀⣼⡿⠁⠀⠀⣰⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣉⣿⣿⣳⣾⣿⠁⠀⠀⣼⡿⠁⠀⠀⣼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⣻⣿⣯⣿⠷⢶⣾⣿⣇⠀⢀⡼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢧⣨⡿⢷⣾⣯⣤⡟⠛⣿⣷⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠿⢤⣿⣍⣿⡧⠴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣶⠤⠀⠀⠀⢀⠠⣴⠆⠀⠀⠀⢀⣴⣀
⠙⠦⡀⠘⠷⢦⣤⠶⠟⠀⡠⠞⠀⠁⠀⠀⢰⡆⠀⠀⠈⠀⠀⠘⣦⠘⣦⣴⠏⣰⠏⠀⠀⠀⠀⢈⣿⡿
⠀⠀⠈⠱⣾⣿⣿⣿⡶⠉⠀⠀⠀⠀⠢⡀⠘⠀⣀⠄⠀⠀⠀⠀⠸⣷⡜⠇⣴⡗⠀⠀⠀⠀⠲⡴⢇⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⣠⡄⠀⠀⢠⣄⡀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠶⣆⠀⠀⡄⠀⢀⡶⠆⠀⠀⠸⣿⣶⡀⣠⣾⣿⠁⠀⠀⠀⠻⣿⣦⣴⣿⡟⠁
⠀⠀⠀⠸⣏⡯⠫⣏⡟⠀⠀⠀⠀⠀⠉⠉⢽⢏⠉⠁⠀⠀⠀⠀⠀⣿⠏⠷⢿⠃⠀⠀⠀⠀⠁⠀⢰⣦⡄
⠀⠀⠀⠀⠊⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠁⠈⠀⠀⠀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠶⠛⠉
⠀⠀⠘⢶⡀⠀⠀⠀⣰⠏⠀⠀⠙⢿⣳⣦⣀⣀⣴⢟⡿⠁⢲⣟⢶⣄⠀⠀⣠⣶⣾⡷
⠀⠀⠀⢨⠃⠠⡧⠀⢫⠀⠀⠀⠀⠀⢽⡌⢹⠏⣡⡏⠀⠀⠀⠹⣄⡹⣧⣄⠏⣁⠝⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠓⠂⠀⠒⠃⠀⠀⠀⠀⠀⠘⠛⠀⠈⠛⠁⠀⠀⠀⠀⢻⡿⠈⠁⢿⡿⠀⠀⠀⠀⠀⠻⠶
⠀⠀⣤⣦⣄⠀⠀⢀⣤⣦⡄
⠀⠀⠈⢿⣿⣷⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠙⠻⠻⡟⠟⠛⠁⠀⠀⠀⣮⡀⣀⣤
⠀⠀⠀⠀⢣⠋⠉⢣⠇⠀⠀⠀⠀⠀⠢⠄⠀⠀⠠⠞⠀⠀⠀⠀⠈⠶⠂⠀⠶⠃⠀⠀⠀⠀⠐⠿⠿⠇⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⢠⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣤⣤⣤⣀⣽⡜⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⠛⠿⠇⢸⡀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠙⠷⡶⣬⣉⡋⠻⣿⢄⡀⠀⣀⣤⣴⡄⠀⠀⠀⣿⣿⣿⣶⣄⣉⡙⠲⢶⣧⡀⣸⢿⣿⡿⠟⡛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠛⠿⡷⠀⠀⠀⣠⢞⠉⠙⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣄⣀⣿⣿⣿⡿⢫⣶⣶⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣿⣿⣗⡚⠛⠛⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿⠇⠁⠈⢯⣿⣴⡜⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠂⣇⠩⠛⢦⣤⠿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣿⡏⣷⣦⣌⠟⣁⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠶⠾⠶⠶⠾⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠚⣙⣁⣘⣛⣉⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣝⣿⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⢿⣿⣿⣷⣦⡀⣸⣿⣿⣿⡟⠀⠀⠘⣧⣻⣿⣽⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⣀⡀⠈⠛⠉⠙⢿⣷⣿⢿⣿⠋⣥⣄⠀⢀⣸⣿⣥⣩⡿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣙⡇⠀⠀⠀⠀⠀⢻⣿⣼⣿⣾⣿⣿⡿⠉⣏⣹⡁⢀⣾⣿⣿⣿⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣿⣿⣷⠀⠀⠀⠀⢸⣿⠏⣿⣿⠿⠿⢧⣾⣿⣿⣿⣾⡿⢟⡿⠿⣿⡿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠉⠛⠁⠀⠀⠀⠀⣿⡽⣸⠛⠋⠀⠀⠀⢹⣿⡿⢐⣀⣀⣠⡷⠀⠹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⢿⣿⣶⣷⡇⠄⣿⡧⠛⠀⣠⣶⠀⠂⠈⠹⣧⣾⣷⣏⡉⣴⣤⠄⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢹⣿⣿⡀⢨⣿⡇⠀⣰⣿⢟⠂⠀⡀⠀⢻⣿⣿⢋⣤⡀⠀⢀⣴⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢾⠛⣿⣿⠺⣿⣇⢠⣿⣿⠃⣠⡈⠁⠀⣼⣿⣷⣿⣿⣿⠀⢚⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠹⣿⣿⣦⣽⠩⢰⣿⣿⣀⡽⠁⡠⢞⣽⣿⣿⣿⣿⣫⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣽⣿⡆⠈⣿⣿⡟⠁⠊⢾⠿⣛⣯⣿⠿⠿⢶⣾⠿⠛⠛⢻⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢸⣿⣿⠁⣀⡠⠶⠛⠉⠀⠀⠈⢉⡥⠤⠬⠍⠩⢭⣽⣿⣿⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣶⣿⣷⠞⠉⠀⣀⣤⣴⣶⣶⡖⢉⣭⡉⠉⠳⡦⠀⠼⠿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣷⡿⠿⠿⢿⣿⣿⣿⣉⠀⢿⣿⡀⡀⠀⣿⠀⠲⢼⣿⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠓⠶⠦⢭⣍⡛⠛⠧⠈⠉⠠⠀⠋⣀⣲⣾⣯⣁⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢌⠉⠓⢶⣤⣐⠶⣦⣴⣾⠟⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠙⠿⣿⣿⡼⠿⠛⠋⠉⠀⠀⠀⠉⠛⠻⢶⣶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣷⠀⠈⠉⢇⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠄⢀⡀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠹⢿⡇⢘⠟⠀⠀⠁⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⣀⠀⣀⣄⠀⠀⠻⠉⠛⢾⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠌⢳⡀⠸⠧⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣋⡙⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⢠⣠⣴⣶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣴⣿⣿⠙⠿⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣵⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⣿⣿⣿⣶⣿⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⠀⣆⠀⢸⡟⢠⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠊⢻⣿⣿⣿⣿⣿⣷⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢈⠃⠂⢐⢀⡀⢀⣺⣏⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⣽⣗⣿⣿⣿⣿⣿⣗⠄⠤⠦⠄⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⢘⠀⣂⣸⠋⠉⣿⣿⣧⣷⡄⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠹⣺⡟⠁⠀⠟⢍⠋⠻⣧⡀⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣾⣿⣿⡹⠆⠀⠀⠀⢤⠀⢀⣾⠶⠖⠒⠒⠂⠀⠐⠚⠛⠛⠷⠠⠄⠀⣼⡆⠀⢸⣿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⢸⠛⠶⠶⠶⡶⣶⣿⠶⠶⣿⣿⣷⣶⣶⠖⠛⠁⠀⢈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣛⠛⠻⢿⣿⣿⣿⣿⣿⠿⠛⠲⣤⡜⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⢹⣿⢁⡀⠀⠀⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⣧⣞⣋⣉⡀⠀⠤⠶⠦⣄⣠⣀⣀⣀⣤⣤⣄⣀⣀⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⡋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣇⠀⣂⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠙⣧⣈⢰⣶⣿⣌⡽⣿⡿⠟⠉⢸⣿⣿⡇⣿⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣇⠀⣘⣿⣿⣿⣿⣿⡿⣿⣿⣷⣷⣿⣿⣿⣷⣿⠀⠀⡌⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢉⣿⣿⣿⣿⣇⠈⣻⣿⠁⠀⣠⡟⣹⣿⣿⠿⠿⠛⠉⠉⣯⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣧⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢯⣏⢼⣿⣿⣿⣿⣿⣼⣿⣯⡃⠘⣿⡷⣿⣿⣇⠀⠀⠀⠀⠀⡇⢀⣿⡏⢙⠛⣿⡗⠶⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣽⣷⣾⣿⣿⣿⣿⠋⢈⣿⣿⣿⣤⡛⣻⣿⢿⣏⠀⠀⠀⡀⢠⡇⢸⣿⣷⡂⠀⢀⡿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⢻⠘⣇⢸⡿⠿⠿⠃⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣸⠀⣿⡸⣇⣠⣦⡴⣿⡿⣯⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⢿⣿⣿⣿⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣯⡠⠿⣯⡛⡿⣿⣿⠟⠋⠉⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠻⢿⣿⣿⣿⡄⠈⠉⠛⢻⡛⠛⠿⢿⡷⠴⠒⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠓⠈⠉⠉⠛⠛⠂⠀⠀⠀⠉⠁⠁⠀⠒⠒⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠤⠀⠀⠀⠀⠠⠠⠀⠠⠠⠐⠀⠀⠠⠄⠄⠀⠀⠀⠀⠐⠀⠀⠐⠐⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⡄⢀⢠⠀⠀⢠⠀⢠⠄⣤⠀⢠⢠⠀⡄⠀⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⣿⠀⠀⠀⠀⠀⠀⠏⠱⢸⠀⠀⢸⠀⢸⠀⠿⠀⠸⠹⠘⡇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⢴⣦⣄⣀⣴⣿⡯⠀⢾⡄⢀⣶⠀⠀⠀⢀⡀⠀⠀⢲⣿⣾⣿⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⠻⣿⣿⣿⣿⠁⠀⠀⣠⣿⣿⡗⠀⠀⠀⠁⠀⠀⠀⠉⠻⣿⣿⣶⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠙⣛⡉⠁⠶⠾⣿⣯⠉⠃⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⢉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⣠⣾⣿⣿⣶⣿⡦⠉⣁⠀⠀⢚⠻⣦⣀⡉⠒⢴⡿⠗⠂⠉⣴⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠐⠛⠛⠉⠉⠻⣿⣷⠀⠻⣧⠰⣾⣤⣦⣭⡿⠷⠀⠳⢀⠀⣄⡉⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⣯⣵⠖⢸⣷⡌⠹⢤⣤⡸⣇⣻⠃⠀⢀⡀⣺⡿⠶⢢⠀⠀⢽⠷⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⠈⣠⠀⠀⢽⣿⣆⠀⠀⠁⢻⠏⢀⣀⡘⣿⠋⠀⢰⣿⠇⢀⡈⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⢀⣤⣿⣇⣀⣀⠿⣿⣿⣿⣶⣶⡿⢟⣋⣀⣀⣠⣤⡿⡷⢠⣿⣟⣦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠿⠋⠉⠉⠼⠿⠋⠉⠉⠉⠀⠈⠙⠋⠻⠿⠛⠷⡄⠛⠛⠛⠛⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⢿⣦⣤⣾⡗⠀⠀⠀⠈⠙⠐⠒⡚⣳⠒⠒⠋⠁⠐⠀⠀⢘⡶⣤⠶⡗⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠘⡿⠛⠟⣴⣄⣀⣀⠀⠐⣶⢄⠁⢹⣶⣿⠇⠀⣀⣀⣶⣎⠄⠛⠠⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⣾⢳⢴⡞⢷⣧⠹⣿⢿⣦⡀⠠⠀⠈⡿⣀⣴⣿⣿⠃⣫⣾⣦⡐⢷⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠙⣈⠀⠀⣸⣿⡀⠈⣧⣿⡇⠀⢀⡀⠸⢿⣿⡟⠁⠀⢸⣿⡿⠃⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⣿⣷⠀⠀⠹⡿⢢⠹⡿⠀⠈⠩⠇⢀⠸⣿⡇⠀⡴⣿⣿⡿⢠⣶⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⢰⡋⣿⡧⢨⡗⠛⠛⠃⠀⢀⠀⣶⡄⠀⠀⠉⠀⢼⡿⠋⠉⡴⠀⢆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠈⠗⠋⣀⣿⡗⢿⣿⣷⣦⡈⠀⠙⠋⠀⢀⣴⣶⣿⡷⣾⣷⣄⠀⠸⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⢀⡛⠻⣿⣇⡀⢻⣿⣿⣿⣦⣀⣀⣴⣿⣿⣿⠟⠀⢹⣏⣽⣷⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⢴⣟⠁⠀⣸⣿⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣤⡶⠞⣿⡿⠍⢰⣦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⢿⣤⡄⢙⠁⠀⠀⠀⠻⣿⡿⠛⡟⠿⣿⠟⠙⠛⠛⠺⣿⠟⣴⡋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠄⠠⠤⠤⠄
⠀⠀⣞⣝⢤⡀⠀⠀⡰⠒⠂⠶⡶⠨⣄⣀⠀⠀⠀⠀⣀⣀⠀⠀⡀⠄⢀⣥⠁⢶⠶⠐⠒⢆⠀⠀⢠⡤⣿⣳
⠀⠀⠘⠚⣷⣍⠛⣆⠈⠲⠤⣶⣇⣸⣧⣏⠀⠂⣵⣤⣿⢹⣤⣮⠀⠀⣹⣼⣆⣸⣶⠤⠖⠁⣰⠛⣭⣾⠛⠃
⠀⠀⠀⠀⠀⠛⢷⣤⡻⢄⡀⠀⠀⠙⠴⠷⠀⠠⠼⢿⡿⢿⣿⠧⠄⠀⠾⠤⠛⠀⠀⢀⡠⢞⣤⡾⠛
⠀⠀⠀⠀⠀⠀⠈⠉⢿⣦⣍⡢⣄⡠⠀⠀⠀⠀⠀⡼⡧⢽⢧⠀⠀⠀⠀⠀⢄⣠⢔⣫⣴⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣟⠉⠀⠀⠀⠀⠀⢠⠃⣏⣸⠘⡄⠀⠀⠀⠀⠀⠉⣻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⣺⣆⡀⠀⠀⢀⣀⡘⠀⢻⡟⠀⢃⣀⡀⠀⠀⢀⣰⣗⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠛⢿⣺⡇⡿⠿⣃⡄⠀⠀⢠⣘⠿⢿⢸⣗⡿⠛⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣬⠠⢨⡏⠙⠶⠁⠀⠀⠀⠀⠈⠶⠋⢸⡧⠄⣥⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢀⣤⡀
⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⢿⣷⣾⠿⠀⠀⠘⣿⣦⣶⡶⠒⠀⠀⠸⣿⣿⠀⠀⠘⠿⠃
⠀⠀⠺⣿⣿⡟⠛⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠘⢻⣿⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⣤⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠟⣻⣿⣆
⠀⠀⠀⠀⢠⣶⠀⠀⠀⠸⢿⣿⣾⣿⡟⠀⠀⠀⠀⠀⠠⣾⣿⡆⠀⠹⠿⠟
⠀⠐⢿⣿⡿⠿⠁⠀⣀⠀⠈⠋⠙⠁⠀⢀⣿⡆⠀⠀⢸⣿⣿⣦⣤⣄
⠀⠀⠀⠀⠁⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠘⠿⠻⠟⠛⠁
⠀⠀⠀⣀⣀⣀⣈⣿⣿⣯⡀⠀⠀⢀⣀⣘⢿⣿
⠀⠀⠚⢿⣿⣿⣿⣿⠛⠛⠁⠀⠀⢸⡿⠟⢻⣿⣶⣶⣶⣦⣤⡀
⠀⠀⢠⣦⣍⠉⠛⠋⠀⠀⣀⣤⣷⢀⡄⠀⣼⣿⠈⠻⠿⠟⠛⠋
⠀⢀⣿⣿⣿⣷⣦⠀⠀⣤⣿⣿⣿⠟⠀⠀⠁⠁
⠀⢸⣿⣿⣿⣿⣿⣷⠀⠉⠉⠉⠁
⠀⠀⠙⣿⣿⣿⣿⣿⣇⠀⢀⣤⣶⣾⣿⣿⣿⣿⡿
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢠⣾⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⡿⠻⠟⠁
⠀⠀⠀⢻⣿⣿⡟⠀⣸⣿⣿⣿⡟⠛⠁
⠀⠀⠀⠈⠉⠋⠀⠀⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣴⣾⡷
⠀⠀⠀⠀⠰⣿⣿⣷⣦⣀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷

⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠈⣿⣿⣦⠀⢀⣼⣿⣿⠀⠀⠀⠀⣶⡶⠤⣀⠀⠀⣀⠤⢶⣶⠀⠀⠀⡏⡵⢆⡀⠀⠀⠀⡰⢮⢹⠇⠀⠀⣿⣿⣤⡀⠀⠀⠀⣤⣼⣿⠃
⠀⠀⠀⠀⠀⢙⡇⠹⣿⣿⠯⢜⠋⠀⠀⠀⠀⠻⢥⣠⣈⣶⣿⣁⣠⠸⠏⠀⠀⠀⠻⠿⣶⣒⣦⣤⣖⣷⠾⠟⠀⠀⠀⠸⣿⣿⣿⣦⣤⣾⣿⣿⠿
⠀⠀⠀⠀⠀⣿⠁⡖⠹⠉⠦⢸⣧⠀⠀⠀⠀⢀⣿⠷⣻⠟⠿⣛⠾⣿⡀⠀⠀⠀⠀⣶⣿⣶⢽⡏⣱⢿⣶⠄⠀⠀⠀⠀⣾⣿⣌⠿⠻⣃⣾⣿⡆
⠀⠀⠀⠀⢀⠿⡿⠀⠀⠀⠐⠿⢯⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠙⠿⠍⠀⠀⠈⠿⠟⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠙⠿⠟
⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣾⢶⣤⡀⠀⠀⢠⣤⢶⡧⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣿⠟⠤⠀⠀⠀⢀⠤⠚⣿
⠀⠀⠀⠀⠸⣿⣷⣿⣶⣴⣷⣿⡿⠃⠀⠀⠀⠈⠀⠐⠀⢰⡆⠀⠄⠀⠁⠀⠀⠀⠻⠄⣀⢈⣶⣶⣁⣀⠠⠿⠀⠀⠀⡔⠁⠙⠛⢳⣶⠘⠛⠁⠢⡀
⠀⠀⠀⠀⠀⣾⣾⡷⠻⠟⣾⣷⣷⠀⠀⠀⠀⠀⢠⠂⠀⢹⡏⠀⠠⡄⠀⠀⠀⠀⢠⣞⠁⣼⡿⣿⣧⠈⢳⡆⠀⠀⠀⠙⢳⠾⠾⢿⡿⠷⡶⡶⠚
⠀⠀⠀⠀⠀⠻⠿⠋⠀⠀⠛⠿⠋⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠽⡷⠟⠁⠂⠻⢶⠿⠀⠀⠀⠀⠀⠀⠑⠒⠃⠈⠒⠒⠁


⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⢻⡉⠘⢷⣴⣴⡟⠅⣫⡇⠀⠀⠘⠧⡉⠀⠛⢾⡾⠋⠀⣉⠼⠃⠀⠀⠀⠀⠀⠐⣤⣤⠂⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣶⣶⢾⣿⣶⠸
⠀⠀⠀⠀⠀⣳⢤⣔⢿⠿⣠⢴⡏⠀⠀⠀⠀⠀⠈⢷⣤⡞⢳⣤⡿⠁⠀⠀⠀⠀⠀⠀⢀⢤⠀⠃⡤⡀⠀⠀⠀⠀⠀⠀⢸⣿⡻⢿⣿⠇⣽⡯
⠀⠀⠀⠀⠀⠈⠓⠃⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠙⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⣤⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠁⢻⠀⠀⠘⠙⠁⠉⢻⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⢰⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣠⠴⠊⠲⡄⠀⣀⠀⠀⢀⠕⠚⠳⣄⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠃⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⡟⠲⠀⢀⡴⠃⠀⠘⣧⣀⠀⠀⠀⠀⣻⠀⠀⢰⣦⣀⣤⡈⠙⠋⠰⣄⠀⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣃⠀⣸⠁⠀⢰⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠶⠋⠀⠀⣿⢁⣿⠉⢳⣄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠓⠖⠋⠀⠀⠀⠀⢀⡀⢀⠴⠦⣄⠀⠀⠀⠀⢇⡌⠀⢀⡴⠫⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠏⠁⠀⠀⠀⠈⠓⢤⣀⠀⠈⣧⡔⠋⠀⠀⢀⡇⠀⣰⠖⠲⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⠀⢀⠀⠀⢹⡇⠀⣸⠀⠀⠀⢷⠀⠀⠀⠀⠙⠃⠀⠀⠈⠀⠀⠐⠊⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠱⡇⠀⠠⢾⡁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⣄⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠃⠀⠀⢀⡇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠈⣴⣧⠀⠀⠀⠘⣿⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡴⠀⠘⠶⠚⡀⠀⠀⣸⠃⢠⠏⠁⠀⠀⠀⣾⡁⣀⢹⣿⣷⣤⡀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣴⠲⠃⠀⠀⠆⢠⠀⠀⠀⠘⠃⠀⢉⠅⠙⠋⠴⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⡀⠀⠷⣶⠶⠛⠛⠚⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⣤⣤⠀⠈⠷⣀⠀⠀⢀⣠⠦⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡾⣵⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⡳⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠉⢷⡀⠀⠀⠀⠀⣀⡼⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣤⡁⠈⢿⣧⡀⠀⠀⠀⠀⠶⣏⣀⠀⠀⠈⠓⠀⠀⠚⠃⢂⠀⣠⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡃⢀⡀⣻⡇⠀⠀⠀⠀⠀⠀⠈⠉⢦⣀⣠⠀⠀⠀⠀⠈⠉⣿⡏⠀⠀⠀⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠁⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠀⠀⠀⠀⠈⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁

⣶⣯⣾⣷⣤⣽⣶⣿⣷⣿⣮⣅⣤⣀⣀⣤⡀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣄⣄⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣐⣷⣶⣶⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣦⣤⢀⣄⡤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣫⣐⣀
⣟⢛⡛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣧⣿⡅
⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⠃⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣟⡤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⡀⠀⡘⣿⣿⣿⣿⣿⣿⣿⣿⣥
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣠⣀⠐⣠⣤⠄⠀⠀⠀⢸⣿⣿⣿⡿⠛⢁⣴⣤⣿⠿⣿⣿⣿⣿⣿⣽⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⢷⡎⠉⢹⣷⣦⣀⣄⡉⠉⠉⠀⠀⠀⢀⣿⣿⣿⢿⣿⣽⣟⣛⣥⣼⣧⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢳⣏⠁⠈⠉⠙⠻⠿⣿⠀⣀⣀⣤⣾⣿⣿⣿⣿⣟⣫⣉⡿⢿⣿⣷⡺⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣳⡄⠀⠀⠀⠀⣰⢯⢸⣿⣿⠋⠀⣹⣿⣿⣿⣿⣯⣿⣿⣿⡿⣿⣽⣟
⣾⣭⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠉⠁⠘⢋⣥⣠⣾⣿⣿⠿⣿⣿⣿⣿⠛⢻⣿⣟⣛⣿
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣤⣤⣀⠀⣀⣼⣿⣿⣿⣿⠟⢻⠓⣿⡿⠛⠋⢳⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠉⠉⠉⠉⠈⠃⠟⠍⠻⡷⠺⠟⠙⣰⡿⠟⣿⣿⣿⡏⠁
⣿⣿⣿⣿⣿⠋⠻⢿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠉⠀⠀⠀⢿⣿⠁
⣯⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃
⡾⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⢡⣾⣿⣹⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠽⠻⣷⠆
⠻⢋⣼⣥⣄⡬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢀⣴⣳⣶⣆
⢰⣿⡟⠻⢋⣤⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢨⣿⣿⣿⣿
⣤⣾⣷⣾⣿⣿⡿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠻⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣻⣿⢟⣿⣿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁⣠⣴⠆⠀⠸⠟⠉⠀⠀⠀⠀⠀⠀⣄⠀⠈⠛⢿⣿
⣿⡿⠣⠸⣿⣿⣀⢠⣔⠀⠀⠀⠃⣠⠀⠀⠀⣸⣿⣿⢥⣴⣿⠟⣀⣤⣴⣿⢀⣴⣿⣷⣦⣤⣀⠈⠻⣷⣦⣀⣈
⣿⠁⡤⣰⣟⣯⣿⣿⣿⣦⣄⠀⠐⠁⠀⠀⢨⣿⣿⣿⣿⣿⣷⣾⣿⣿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿
⣶⣗⣾⣷⣾⣿⣿⣿⣿⣿⡟⢀⣀⡔⠀⢀⣼⣿⣿⣿⣿⡿⢯⠀⠝⠁⠈⠀⠀⠀⠈⠉⠙⠉⠋⠉⠑⠛⢿⠟⠋
⣾⣝⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⣦⣾⣿⣿⣿⣿⣷⣴⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣵⠀⠀⠂⠀⠐⢠⣄⣴⣄⠀⢀⣤⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣷⣶⣶⣶⣶⣶⣿⣧⣄⣤⣼⣦⣤

⠀⠡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⣶⠦⣤⣶⠦⠀⠛⠋⠀⠸
⠀⠀⠀⢰⡀⠀⢠⣤⣀⣀⣀⣀⣀⣀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠦⠤⠀⠀⠴⠶⣶⠞⠋⠁⢠
⠀⠀⠀⠀⠁⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣤⣤⡤⠶⠶⢤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠎⠉⠉⠁⠀⠀⠀⠀⠀⠉⠳⡀⠠
⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠞⠋⣹⣿⣿⠟⠀⣤⠀⠰⣄⣄⡀
⠀⠀⣸⣿⣿⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠒⠁⠀⠀⡉⠉⢠⣄⠸⡿⠟⣁⠐⢂⣽⠀⠀⢸⡌⠃
⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⢶⣄⣈⣛⣀⣠⡤⠈⠀⣼⢋⡆⠀⢸⣀⡀⠂
⠀⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠦⣀⣀⣠⣄⣤⣤⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣸⣅⠙⡇⠀⠀⢻⠇⢠
⠀⣿⣿⡿⠃⢀⠀⠀⠂⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠟⠉⠁⠀⠰⢾⡄
⠀⠻⠟⢁⣴⣿⠀⠀⠀⠀⠀⠈⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⠀⣰⣾⣿⡏⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆
⠀⢰⣿⣿⣿⠃⠀⠀⠀⢸⣧⠀⠀⠹⣿⣿⣿⣦⠀⠀⠀⠀⠀⢶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⠟
⠀⣼⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣄⠀⠘⢿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠉⠉
⠀⢸⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣦⠀⠈⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⣀⡀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡆
⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣄⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣬⣦⠀⠀⣿⡇⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⢙⣿⣇⣼⣿⣇⣀⠀⠤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠈⢻⣷⡀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⢿⣿⣿⣿⣿⡷⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠿⣦⡀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⢲⣶⣤⣴⣶⡿⣿⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠻⣦⣀⠹⢷⣄⠀⠀⠀⠀⠀⠀⠉⠛⢋⣡⣴⣿⣿⣿⣿⣿⡛⠛⠛⠋⠙⠶⠶⠾⠿⠛⠛⠛⠛⠛⠉⠛⠀⠈⠲⡄⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⣀⣤⣶⣿⣿
⠀⠀⠀⠀⠉⠙⠾⢿⣷⣄⣀⣀⣠⣤⣴⠶⠟⠉⣴⣿⡿⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣍⠉⠁⠀⠀⠀⠀⣿⡏⠀⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢠⣶⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢠⡿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠠⠤⠶⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠈⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠆⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣀
⠀⠀⠀⢾⣿⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣿⣿⣿⣿⣶⣶⡿⠛⠁⠀⠀⠀⠀⠀⢴⣦⣤⣄⣠⣴⣿⣶⣶⡀⠀⠀⠀⠤⠤⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢋⣿⣿⣿⢿⣿⣿⠿⣿⣧⡀⠉⠁⠒⢀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠻⠀⠉⣿⣷⡀⠛⣷⡏⠀⠀⠀⠐⠆⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⢩⣿⠛⡋⢻⠀⠀⠀⣿⣿⡇⠀⠀⠀⢇⠀⠀⠀⠀⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡀⠁⠸⡄⠀⠀⢻⣿⠇⠀⠀⠀⢸⡇⠀⠀⠘⢿⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣇
⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣷⡀⠀⢸⡿⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⣧⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⡀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡞⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣆
⣿⣿⡿⠛⠻⠟⠛⠻⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠⣾⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣆
⠛⠀⣀⣤⣶⣶⣶⣶⣤⣀⡀⠙⠻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣦⡤⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⣿⣿⣿⣿⣿⡟⠀⠈⣿⣿⣿
⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠿⠿⡧⠀⠀⢿⣿⣿⣶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠈⠻⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣦⡀⠀⠠⠀⠈⠿⢀⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠙⠻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣦⠀⠘⢂⠀⠸⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠙⠛⠿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡀⠙⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⡄

⠀⣴⣶⡶⣆⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣰⢶⣶⣦
⠀⣿⣿⠶⣾⡉⣉⣉⣟⡿⠗⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⢿⣻⣉⣉⣉⣷⠶⣯⣿
⠀⠸⣟⣲⣾⣍⣠⣤⣼⣛⢀⣽⣷⣿⠷⣤⡀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⢀⣤⠾⣷⣾⣯⡀⣛⣧⣤⣄⣩⣷⣖⣻⠇
⠀⠀⠹⣯⣤⡿⣅⠀⣨⣿⣿⣤⣭⡿⠿⠦⣽⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣯⠴⠿⢿⣭⣤⣿⣿⣅⠀⣨⢿⣥⣽⠏
⠀⠀⠀⠹⣯⣿⣿⠉⢻⡏⠀⠀⠀⠁⠄⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠠⠈⠀⠀⠀⢹⡟⠙⣿⣿⣽⠏
⠀⠀⠀⠀⠈⣿⣿⡶⢿⣷⣄⣀⠀⠤⠴⢄⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡤⠦⠤⠀⣀⣠⣾⡿⢶⣿⣿⠁
⠀⠀⠀⠀⠀⢻⣿⣿⠿⣿⠁⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠘⣿⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣶⠿⣦⣀⠠⠤⠤⠤⠖⠚⠓⠠⠄⡀⠀⠀⠙⢦⣀⣤⣤⣀⡴⠋⠀⠀⢀⠠⠔⠚⠓⠲⠤⠤⠤⠀⣀⣴⠿⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠹⣿⣿⠏⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⣿⣄⣀⣀⣤⣤⣴⣶⣶⣤⣤⣄⣀⣀⣀⣩⣴⣿⣿⣧⣍⣀⣀⣀⣠⣤⣤⣴⣶⣦⣤⣤⣀⣀⣠⣿⣷⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠿⠛⠛⠋⠉⠛⡻⠏⠉⠩⠭⠉⠉⢩⢿⣿⣿⡿⡍⠉⠉⠩⠍⠉⠹⢟⠛⠉⠙⠛⠛⠿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣽⣧⣀⣀⣀⠤⠄⡔⠀⠀⠀⠀⠀⣀⠔⠁⡜⣹⣏⢣⠈⠢⣀⠀⠀⠀⠀⠀⢢⠠⠤⣀⣀⣀⣼⣯⣈⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⡿⠋⠀⠀⠀⠀⡰⠀⠀⢀⣠⠴⠊⠁⠀⠀⠀⣿⣿⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⢆⠀⠀⠀⠀⠙⣿⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠨⣿⣷⣧⣀⣀⠄⠐⠚⢀⠔⠛⠉⠉⠀⠀⢀⠞⢀⡄⣿⣿⢠⡀⠳⠀⠀⠀⠉⠉⠛⠂⡀⠓⠂⠀⣀⣀⣼⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣟⠁⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢸⡏⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠈⣿⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣿⣿⣷⣿⡁⠀⠀⢀⡄⠀⠀⡀⠀⢠⡟⠀⠀⠈⠀⠀⠀⢻⡄⠈⢀⠀⠀⢠⡀⠀⠀⢈⣿⣾⣿⣿⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣭⡿⢯⣿⠶⣶⣿⣄⣠⡾⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⢷⣄⣠⣿⣶⠶⣿⡽⢿⣭⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠯⣿⣿⣲⡿⣽⣻⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣯⢿⣖⣿⣿⠽⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⢠⡤⣤⣤⣀⡀
⠀⣧⣿⡾⣷⣶⣦⡉⠳⢶⢄⢀
⠀⠻⣷⣿⣾⣿⣽⣿⣷⣶⣯⣿⣶⣦⣀⡀
⠀⠀⠸⣯⣄⣿⣿⣏⣙⣛⣮⣿⠿⠿⣿⣿⣶⡀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⣹⣿⣻⣛⣿⡤⠬⣿⡁⠀⢀⣼⣿⣾⣻⣦⡢⡀⠀⠀⠀⠀⠙
⠀⠀⠀⠈⢻⣿⢋⣹⣷⠒⠚⠧⡀⠿⣿⡟⠃⢈⣻⡙⣌⢀
⠀⠀⠀⠀⠘⢻⣟⠻⣿⣶⠒⠉⠋⠓⢬⡀⠰⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠘⢟⣛⣛⣿⡟⠉⠉⠙⠛⢿⣷⣮⣝⡛⠉⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⢳⣻⣛⣋⡿⠃⠀⠀⠚⠻⣿⣿⣿⠉⠲⢤⡹⣿⡇⠀⢀⣼⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣀⣼⣏⣉⣷⣏⡀⠀⣉⣉⣉⣙⣛⠒⠦⠴⣿⣿⣿⣶⣿⣽⣿⡏⠀⣀⣀⣠⣤⣴⣶⣿⣾⣶⣾⡶⠿⠯⣭⣟⣓⣲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⣶⣟⡙⣯⣠⠤⡾⠋⠉⠀⠀⠀⠀⠀⠀⠉⣛⣶⣭⣿⣿⣿⣿⣿⣿⣟⣿⣉⣉⠀⠀⠀⢨⣿⣿⠋⢙⣿⣷⠖⠲⣭⣝⡛⡶⣬⣿⢷⣦⣀
⠀⠀⠀⠀⠀⢿⡿⠛⣿⣤⠐⣇⠀⠀⠀⠀⠀⣠⡴⠞⠉⠁⢀⣾⡿⢮⣿⣯⣝⡛⠿⣭⠉⠉⠙⢀⣶⣾⣟⣋⠀⢴⣻⠟⠀⠀⢸⣏⡉⠛⣿⢿⣿⣬⣻⠗⣄
⠀⠀⠀⠀⠠⣾⡷⡟⣿⣇⡔⠉⢻⠶⠶⢾⡿⠋⠀⠀⣠⠞⣱⡿⣽⣿⠟⠛⢁⠀⠐⢬⡑⠲⣤⣾⣿⣿⡿⠻⠟⠻⡷⠶⢶⡖⠫⣄⣹⣾⡿⠿⢿⡏⢹⣶⣿
⠀⠀⠀⠀⣀⣿⣧⡾⣿⣿⣤⣴⡇⠀⢰⠟⠀⠀⢀⣼⠏⣴⣿⣽⣿⠋⡄⠀⢸⡆⠀⠀⠉⢦⣌⠙⢿⣏⡠⣄⠀⢄⠈⠢⣀⣹⣷⢟⠿⣧⢸⣷⠬⣿⠏⠉
⠀⠀⠀⠀⠈⢷⣶⣷⡿⣿⣿⣿⣷⣞⣿⣷⣄⡰⣻⠁⣸⣿⣿⡟⢁⣼⠁⠀⢸⣷⡀⠀⠀⠈⣿⡙⢾⣻⣿⣿⣦⣴⠿⣿⢻⣿⣼⣿⣾⠟⠓⠉
⠀⠀⠀⠀⢠⡿⢛⠋⠳⣼⣿⣏⣽⡟⣷⣾⡿⢽⠃⢠⣿⠟⠋⢠⣟⣀⠀⠀⣸⠁⢷⠤⡄⢴⠉⣷⢾⣻⣿⠙⠻⠿⠻⠿⠛⠻⠏
⠀⠀⠀⢀⣞⠔⠁⠀⠈⠊⠁⠈⣿⠀⠀⠸⠼⠃⠀⠀⠀⠀⢀⢼⣯⣻⣷⠀⡿⠀⣸⠀⢀⣠⣷⣿⠙⣿⡟
⠀⠀⢠⢿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⢋⣼⣿⣶⣇⢀⣿⣶⣿⠿⣦⢙⣿⣿⡧
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠟⣿⣿⣿⡟⣿⠙⣿⣂⣽⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣴⣿⣙⣿⣾⣿⡶⣿⣿⣿⠚⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⢝⣿⣿⣿⠟⠛⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⡣⠋⠀⠀⡰⠁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠄⠀⠁⠀⠀⢰⣶⣿⡃⠀⠱
⠀⠀⠀⠀⣀⠁⠰⠀⠔⣐⡋⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠦⠒⠉⠄⠐⠚⠁⠀⢸⡟⠁
⠀⠀⠀⢰⣿⡶⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠚⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡾⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡇
⣀⠀⣠⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋
⠿⣿⣿
⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠆
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄⣙⠌⡄
⢀⣤⣀⣠⠄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⣿⣿⠉⠀⢠⠀⠀⡠⠈⠠⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣾⠏
⠞⠋⠀⠀⠀⣀⣀⣴⠀⠈⡀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠛
⠀⠀⠀⠤⠄⠛⢿⠷⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡄⠀⠀⠐⠋

⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⣼⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⣤⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡅⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠂⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⠄⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⣶⣿⣿⣿⣿⣷⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⡏⠉⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿

⠀⢀⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⡀
⠀⠘⢿⣧⡈⢫⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⡝⢁⣼⡿⠃
⠀⠀⠈⠻⣿⠈⡇⣠⣤⣕⢄⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⡠⣪⣤⣄⢸⠁⣿⠟⠁
⠀⠀⠀⠀⣿⡤⢧⢻⣿⣟⡒⢝⢦⡀⠐⣅⡰⠈⢢⠀⠀⠀⠀⡔⠁⢆⣨⠂⢀⡴⡫⢒⣻⣿⡟⡼⢤⣿
⠀⠀⠀⠀⣿⡇⠈⠾⡀⠓⠃⢸⡀⠙⢄⠀⠀⠀⠀⢣⠀⠀⡜⠀⠀⠀⠀⡠⠋⢀⡇⠙⠚⢀⠳⠁⢸⣿
⠀⠀⠀⠀⣿⠗⢺⠀⠈⠁⢉⣁⡉⠢⡀⠳⡀⠀⠀⠘⡆⢰⠃⠀⠀⢀⠞⢀⠔⢉⣈⡉⠉⠁⠀⡗⢺⣿
⠀⠀⠀⠘⣿⣀⣈⣳⠀⣴⣿⠟⢫⠑⢮⡢⡙⡄⠀⠀⡇⢸⠀⠀⢠⢋⢔⡵⠊⡝⠻⣿⣦⠀⣞⣁⣀⣿⠃
⠀⠀⠀⠀⢸⣿⠀⠸⠀⠹⣿⡓⠊⠀⠀⠙⢿⣌⢆⠀⣸⣇⠀⡰⣡⡿⠋⠀⠀⠙⢊⡿⠏⠀⠇⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠒⠛⢄⠀⠀⠈⣽⣿⣶⣶⣤⣻⣿⣆⣿⣿⣰⣷⣯⣤⣶⣾⣿⣯⠁⠀⠀⡠⠛⠒⣿⡇
⠀⠀⠀⠀⠀⠙⠷⠤⠾⢭⣒⣦⣽⣿⡿⠭⠭⢥⣵⣿⣿⣿⢿⣦⡬⠭⠭⢿⣿⣯⣴⣒⡭⠷⠤⠾⠋
⠀⠀⠀⠀⠀⠀⢀⣠⠖⢩⠋⠉⡬⣫⠕⠒⢉⣟⡝⠀⣿⣿⠀⢫⣻⡉⠒⠪⣝⢭⠉⠙⡍⠲⣄⡀
⠀⠀⠀⠀⠀⢠⣿⣟⠀⠈⠓⢋⣾⣇⠀⢀⣿⢻⢇⠀⣿⣿⠀⣸⡟⣿⡀⠀⣸⣷⡉⠚⠁⠀⣻⣿⡄
⠀⠀⠀⠀⠀⠸⣿⣿⣟⠀⠀⣀⡻⠿⠷⠋⡇⠀⣿⡆⠸⠇⢰⣿⠀⢸⠙⠾⠿⢿⠤⡀⠀⣻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⠋⠘⢤⠜⠀⠀⠀⠈⠉⢠⣇⠀⠀⣸⡀⠉⠁⠀⠀⠀⠑⠤⠃⠙⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⣠⣄⠀⣠⣄⣀⣶⣦⣾⡟⠀⠀⢻⣷⣴⣶⣀⣠⣄⠀⣠⣄⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠻⣿⣿⠿⣿⣿⠟⠿⠿⠃⠀⠀⠘⠿⠿⠻⣿⣿⠿⣿⣿⠟⠿⣿⡄
⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⢹⣿⣄
⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣯⣭⣭⣭⣽⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣯⣭⣭⣭⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣶⣦⣭⣿⣯⣛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⢟⣛⣛⣛⠁⣙⣛⣻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣻⣦⣤⣌⣽⠻⢿⣿⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣡⢀⡘⠉⢨⡉⠉⠛⢿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡀⠀⠀⣨⣤⡀⠉⠂⢨⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⡁⡴⠋⠀⠀⠀⠈⠹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⣡⠀⠀⠀⠀⠙⢶⣄⡈⠛⢿⣿⣆⠀⠀⠀⠀⠀⣰⣿⣿⠋⡈⠐⠁⠀⠀⠀⠀⠲⣄⡈⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠘⠻⢦⣼⣿⣿⣧⠀⣀⠀⣼⣿⣿⡷⠊⠁⠀⠀⠀⠀⠀⠀⠘⠻⢿⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⢠⣤⣤⠤⢤⣾⣧⣬⣿⣿⣿⣿⣿⣿⣿⣥⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠛⣷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣻⠿⠿⠛⠋⠉⠀⣉⣹⡿⣿⢿⣿⣿⣿⣿⣿⣿⣍⡉⠀⠘⠓⠶⢦⣤⣤⣤⡶⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠛⠁⠀⣴⣿⣿⡿⣿⣿⣿⢧⡀⠈⠉⠀⠀⠀⠀⠀⠈⠛⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡠⢰⡿⣿⣿⠀⠃⢹⣿⣣⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⢠⠏⢂⣿⠃⠀⠀⠀⣿⣿⡆⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⣡⠄⠀⠀⠀⣠⠃⢀⣼⡏⠀⠀⠀⠀⠹⣿⣿⡄⠀⠒⣤⣄⠀⡀⠀⠀⠈⠶⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⣯⣡⣴⡄⠘⣧⣶⣿⠟⠃⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠈⢿⣿⠉⢳⣦⣄⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣧⣼⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣄⣈⣿⣷⣤⣬⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢾⣾⠀⢀⡏⡹⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢾⡿⢖⠂⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠛⠀⠃⠲⡛⠿⣦⣤⠄⠀⠀⠀⠀⠀⠀⢀⠴⠻⣶⠾⠳⢽⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⣲⠀⠀⠀⠙⣆⠀⠙⢷⡀⠀⠀⠀⢀⠔⠁⢀⠼⠓⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣃⠈⢀⣀⣉⣕⣦⣄⡙⢦⣶⡷⠃⣀⣒⣉⣀⣀⣀⢀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢛⢻⠿⠟⢉⡙⠫⡭⣿⣿⣷⣯⣋⡉⠉⠛⠿⣿⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣉⠀⠀⡐⠊⡠⠂⠐⠸⣿⠘⠂⠠⡈⠲⠀⠀⠘⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠁⠒⠀⡐⠀⢀⣿⠀⠠⠠⠈⠒⢤⠀⠀⢐⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢢⡄⡀⠀⠀⠠⠀⠄⠈⣿⠀⢀⠂⠡⠀⢂⠁⣈⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢇⣞⣦⣷⡾⠀⠀⠁⠀⠀⠺⢦⣷⣾⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⡎⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣤⢻⣧⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣋⡟⠢⠃⠀⠀⠀⡀
⠀⣀⡤⠤⠖⡂⠛⠋⠉⠉⠉⣉⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⡠⠤⠶⠒⠒⠒⠒⠒⠶⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⣾⡿⠒⡈⠁⠀⠀⠀⠀⠀⠀⢰⣿⣻⡇⠀⠀⢀⡤⠒⠉⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠘⣦
⠹⣿⡂⢅⠀⠀⡀⠀⠀⠀⠐⣸⡇⣾⠇⢀⡴⠚⠁⠀⡠⠖⠋⣉⠀⠀⠀⠀⠀⠀⠨⡉⠲⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡄
⠀⠀⠙⠲⠤⣠⡀⠀⠠⠀⠀⣾⣧⣿⠀⢸⠗⠀⢠⠎⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢆⣷⠔⠐⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⢿⣿⠇⠀⢾⠀⠀⣾⠀⠀⠀⠀⠊⠐⠂⠀⠀⠀⡀⠀⢃⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣚⢿⢧⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠸⡄⠀⢻⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠐⠀⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢳⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢈⣧⣄⡹⠂⠀⠀⠀⠢⡀⠀⠀⠀⠀⡠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠸⣄⣀⠀⠀⠀⠀⠀⢀⣿⣿⣶⡾⠁⠀⠀⢀⡏⠀⠀⠀⠀⢀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠈⢣⡀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠈⠏⠉⠉⠇⠀⠀⠀⡼⠀⠀⠀⠀⢀⡎⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣤⠤⠤⣉⠳⢄⣀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⡠⠖⣉⡴⠂⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠑⠦⣈⡉⠲⠚⠛⢗⠒⠒⠒⠒⡲⢞⣋⣕⠒⢒⢋⣉⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠜⠁⠁⠁⠁⠀⠀⠀⠸⡦⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣏⠙⠓⠲⠶⠶⠶⠶⠒⠛⣻⢥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢳⡀⠀⠠⣞⠁⠀⠀⣀⠀⠀⠀⣀⣠⡴⠃⢠⠞⠓⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠙⢆⠹⣤⠞⠉⣳⣤⠞⠉⠉⠉⠉⠉⠀⠉⠙⠋⣩⠗⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⠿⠛⠛⠦⣄⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣠⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⣀⠀⢀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⣷⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠛⢿⣿⣉⢉⣹⠏⣻⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣄⣻⣿⢺⣿⣤⣧⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⣷⣿⣿⣿⣿⣿⠟
⠀⠀⣸⣿⣿⣿⣿⣄⣤⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⡈⢿⣿⣿⡟⢯⡀⠀⠈⢿⣿⣿⣿⡀
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣷⠀⠉⠻⡗⢧⡉⠳⢶⣿⣿⣿⣿⣿⡄
⠀⢸⡟⠈⢿⡿⠻⢻⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠋⠙⠇⠀⠀⠈⠳⢬⡳⣾⣿⣿⣿⣿⣿⣷
⠀⢸⡇⠀⢸⠁⠀⢸⡇⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣷⠀⢀⣄⣰⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠟⠃
⠀⢸⣤⣀⣼⣄⣀⣼⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣿⣿⣿⡇
⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉⠉⠁⠈⠛⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡰⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⣤⠠⠄⡁⠀⠀⠀⠒⠀⠰
⠀⠀⠀⠀⠀⢡⠀⢀⠀⡀⠳⡄⠁⠀⠀⠀⠀⠀⠀⠈⢈⡇⠠⣀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠰⣄⠀⠀⠀⠀⠀⡀⠀⢀⣠⣬⡁⠟⠻⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠈⠓⠉⣿⣶⡆⠨⠀⠀⠈⠺⡁⠂⠀⠁⠌⡆
⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠨⡆⠰⠿⠟⠁⠀⠈⠲⠀⠒⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣨⠤⠄⣀⠄⠉⠀⠀⢄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠂⣁⣀⠖⢙⠁⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢸⠏⢠⠈⡆⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠰⠀⠈⠀⠀⠀⠀⠀⠀⡆⢑⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠧⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠤⣤⣷⣠⠀⠀⠀⠐⢀⠀⠊
⠀⠀⠀⠀⠀⢈⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠚⠀⠀⢀⠾⢾⡿⠓⠂⣄⠀⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢤⠀⠀⠀⠀⡀⠀⠌⡔⠁⠀⢠⡾⠃⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠃⠙⠈⡄⠙⢄⣀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⢢⣀⣄⠐⠁⢀⡀⠁⠀⠚⠻⣞⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⠈⠀⠀⠀⠑⣾⡋⢹⡧⠂⠉⠂⠠⡆⣀⠀⠉⠀⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⢁⠬⠩⢿⠒⠂⠈⠛⡿⠟⠙⣄⠀⡀⠐⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠘⢈⣢⡶⢀⠐⡀⠠⡄⡈⠂⠄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⢈⡞⠈⠁⠀⠃⠐⠀⠀⠀⠈⠑⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⡀⠀⡀
⠀⠀⠀⠀⠠⠤⠠⢍⠄⠒⠀⠀⠀⢀⠤⠀⠀⠀⠂⠀⠀⠀⠠⢤⠈⠀⠀⠀⠀⠤⠀⢀⠀⠀⠀⠀⠠⠀⠱⠐⠣⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠴⠤⠋⠁⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠢⢀⡀⠀⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡄⠠⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠄⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⣈⡂⠀⢐⠂⠀⠀⠀⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠠⠏⠃⠀⠀⠀⠀⠀⠁⠀⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠁⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠄⠐⠀⠀⠀⠩⡊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⡠⠐⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠘⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠄⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⣀⠀⠀⠈⠠⠀⠁⠂⠀⠀⠀⠀⠀⠈⠁⢈⡄⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⢑⠀⠀⠀⢨⠀⠈⠀⠀⠀⠀⠀⠀⠄⠤⡀⠀⡠⠀⠀⠀⠀⠁⠀⠀⠑⠤⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⡀
⠀⠀⠀⠀⢀⠄⢓⡀⠢⠀⠀⠀⠰⢀⡀⠒⠀⠡⡀⠀⠈⠉⠀⠀⡰⠈⠀⠀⠀⠀⠐⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⢀⠴⢁⠀⠌
⠀⠀⠀⠀⠀⠀⠤⠀⠃⠐⡒⠘⠐⠂⠀⠁⠁⠀⠈⠢⢀⠀⠀⡘⠀⠬⠑⠘⠀⠠⠠⠄⠁⠀⠃⠘⢨⠁⠀⠄⠀⠠⢈⢠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⢼⣶⢆⡀⠀⠀⠀⠀⣀⢶⣿⠆⠀⣿⠷⣂⡀⠀⠀⠀⠀⢀⡰⢶⣶⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢻⣿⣶⢄⠀⠀⠀⢤⣾⣿
⢸⠟⣛⣫⣤⣀⢠⣬⣩⣛⢿⠀⠀⢹⡟⠛⠻⣦⠀⠀⡴⠟⠟⢻⡏⠀⠘⣵⠛⣴⣿⣶⣄⣦⣤⣾⣷⡙⣺⠇⠀⠈⣿⣇⠀⢣⣤⠀⢀⣿⡟
⠈⣻⣿⣿⣿⢿⢿⣿⣿⣿⡍⠀⠀⠈⢂⠤⠀⠈⢳⡞⠁⠀⠀⠐⠁⠀⠀⠀⢼⠛⠟⢋⣿⣿⣏⠛⠟⡳⠀⠀⠀⠀⢠⡿⠋⠙⠟⠉⠙⢷
⠀⠈⢿⣿⠃⠸⠀⢹⣿⠏⠀⠀⠀⠀⡎⠀⠄⠀⠸⠁⠀⠠⠀⢱⠀⠀⠀⠀⠀⠀⠀⢀⠻⠾⠋⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡄⠀⢦⣤⡿
⠀⠀⠀⠉⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠳⢤⣄⠀⠀⠀⢀⣠⡤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠀⠀⠈⠉⠅
⣠⣤⣄⠀⠀⠀⠀⠀⢀⣤⣴⡄⠀⠠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤
⡟⠉⢀⣀⡄⠀⠀⢄⣀⡀⠈⠻⠀⠀⢻⢟⢻⣦⡀⠀⠀⠀⣰⡿⡉⡏⠀⠀⠀⠀⠀⠢⡀⠀⠄⠀⠀⠀⠀⢰⡟⠋⠁⢀⡀⠀⢀⡀⠈⠉⠛
⠀⣰⣿⣧⠀⢰⠀⢠⣿⣿⠂⠀⠀⠀⠈⣿⣯⠿⣷⣄⣠⡾⣟⢀⢻⠁⠀⠀⠀⠀⣠⣾⣷⣾⣶⣀⠀⠀⠀⠀⠀⣴⣿⡉⠀⣄⠀⢈⣿⣷
⢸⠻⣿⣧⣴⢾⢷⣤⣿⡿⠪⠀⠀⠀⠀⠜⢉⣠⠙⠛⠟⠏⣠⡈⢲⠀⠀⠀⠀⢸⡋⠻⠉⠙⠟⢹⡆⠀⠀⠀⠼⢿⣿⣅⣠⣿⣄⣈⣿⡿⢡
⠀⠀⠈⠙⠋⠈⠈⠛⠉⠀⠀⠀⠀⠀⠀⠘⢠⡴⠟⠀⠀⠺⢧⡐⠁⠀⠀⠀⢀⡜⠛⠃⠀⠀⠙⠛⣄⠀⠀⠀⠀⠀⠙⠻⠋⠉⠙⠟⠋
⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣯⡶⠛⠋⠀⠀⠀⠀⠀⠉⠛⠳⣜⠀⠀⠀⢻⣄⠀⠀⠀⠀⢠⣎⠀⠀⠀⢻⣮⢷⣄⠀⠀⠀⠠⣴⢿⣾⠃⠸⣅⢥⣦⡀⠀⠀⠀⢀⣤⡀⣩
⢻⣀⣴⣶⣷⢶⣶⡶⣾⣷⣶⣤⡼⠀⠠⣶⡾⢷⣷⡀⢀⣴⡿⢷⣶⠆⠀⠈⢿⡌⢻⣧⣠⣴⣟⠉⡾⠋⠀⠀⢻⣭⠻⣷⣄⠀⣠⣾⠟⣵⡏
⠀⣰⢸⣽⣞⣷⣿⢾⣳⣿⡇⣶⠀⠀⠀⠙⣳⣶⣿⣿⣿⣿⣶⣞⠋⠀⠀⠀⢠⣾⠀⡼⠿⢧⡀⢹⡄⠀⠀⠀⠀⢙⣤⣿⣿⣿⠿⣧⣀⠋
⠀⠹⣿⣿⡻⠃⠰⠈⢿⣻⣵⠏⠀⠀⠀⢸⣏⣻⠟⠹⠏⠻⣟⣹⡗⠀⠀⠀⠐⢿⡇⠁⠀⠈⢳⡿⠀⠀⠀⠀⠀⠘⢿⡵⠀⠏⠠⣷⠿⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠃⠈⠈⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠈⠑⢢⣄⠀⠀⠀⠀⣤⡰⢉⠀⠀⠀⢶⣶⣤⡀⠀⠀⠀⣠⣶⣿⠂⠀⢰⠖⠀⠀⠀⠀⠀⠀⠀⠠⠶⡄⠀⢰⣶⡶⢆⣀⠀⠀⠀⢀⠀⣤⣴
⠀⠀⠀⠀⠑⣴⣴⠉⠀⠀⠀⠀⠀⠀⢸⣉⠁⠉⠀⣀⢈⠀⠀⢻⠀⠀⠘⡀⠀⠀⠀⢀⡄⠀⠀⠀⢘⠁⠀⠀⢿⡅⡞⠉⠀⣴⠀⠈⢻⣎⣿
⠀⠀⡨⠓⠐⢻⠛⠐⢏⡠⠀⠀⠀⠀⠈⣙⡶⠾⢿⡿⡿⠶⣆⡉⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣵⡿⠟⠉⢻⠉⠳⣾⣆⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣰⠖⠓⢲⣀⡼⠃⠀⠀⠀⠘⢀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠻⣀⣀⠀⠙⠀⣀⣈⠟

⠀⠀⠀⠀⠀⠀⢀⢴⠞⠛⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⢙⡆⢰⡋⡉⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⠛⠷⡦⡀
⠀⠀⠀⠀⠀⢠⢡⠃⠀⢎⣉⡜⢀⣠⣤⢄⡀⠀⠀⠀⠀⢰⠁⠀⠈⠉⠀⠀⠉⠁⠀⠈⡆⠀⠀⠀⠀⢀⡠⣤⣄⡀⢇⣉⡗⠀⠘⡈⡄
⠀⠀⠀⠀⠀⡄⠈⣦⣤⣤⣵⡾⣽⣿⠻⣦⡾⠷⣦⡀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢀⣴⠾⣷⣼⠛⣿⣯⢶⣦⣤⣤⣼⠁⢡
⠀⠀⠀⠀⠀⣶⣟⠻⣷⣌⣻⣧⣬⣿⣦⣹⣧⡀⣷⡾⢦⡀⠱⡄⠀⠀⠀⠀⠀⠀⣠⠊⢀⡴⢷⡾⢀⣾⣏⣴⣿⣥⣾⣟⣡⣾⠟⣻⣶
⠀⠀⠀⠀⠀⣿⠛⢟⣩⣍⠂⢐⢒⠊⠀⠩⠍⠛⠻⢷⣸⡟⣆⠈⢢⠀⠀⠀⢀⡔⠁⣴⢻⣇⡾⠟⢛⠩⠉⠀⠐⠒⡂⠘⡩⣍⡿⠛⣿
⠀⠀⠀⠀⠀⠹⣻⡟⠛⠲⣅⢦⣤⣤⣀⡠⡀⠀⠀⠂⠌⡓⢼⣆⠀⢱⠀⢀⠎⠀⣸⡧⢋⠡⠀⠀⠀⠀⢀⣀⣤⣤⡔⣡⠖⠛⢻⣟⠋
⠀⠀⠀⠀⠀⠀⠹⣛⣿⠟⠛⢦⢠⢸⡉⣹⠳⢶⣄⠀⠀⠀⠀⠙⣆⠀⡇⢸⠀⣰⠋⠀⠀⠀⢀⣠⡾⠞⣏⢉⣇⡄⡼⠛⠻⣿⣛⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡶⠚⢛⢠⠀⠉⠛⢤⡔⠛⢻⡆⡀⠀⠀⠸⣤⡿⢿⣴⠇⠀⠀⠀⣴⡟⠛⢦⠴⠛⠉⠀⡄⡛⠓⢶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⠶⠏⢘⠨⠁⠢⠑⢝⢶⠋⠙⣾⠆⣀⣴⣿⡿⢿⣿⣦⣀⠴⣵⠋⢙⡶⡫⠂⠰⠈⠀⡃⡿⠶⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⠾⣆⠑⠤⠀⣌⠢⠡⢓⡫⢶⣿⣿⣿⣿⣯⢿⣿⣿⡿⣿⡾⢗⡋⠌⢔⣠⠀⠤⠊⡰⠷⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⣤⣿⢉⣶⣒⣲⣾⣩⣿⢷⣿⣿⣿⣿⢿⣷⣾⡿⣿⣿⣾⣿⡾⣿⣍⣷⣖⣲⣶⡉⣿⣄⣼⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢻⠟⡶⠛⠋⣩⣵⣾⣿⣿⣿⢯⠇⡟⣿⣿⢳⠹⡹⣿⣿⣿⣷⣮⣍⠙⠛⢾⠻⡟⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡜⠀⣰⣿⣿⣿⣿⣿⡿⢡⠏⣸⠀⢸⠇⠀⡇⠹⡍⢿⣿⣿⣿⣿⣿⣆⠈⢧⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠁⠀⠙⠿⠿⠿⢛⣡⠴⠋⡠⠃⠀⠀⠀⠀⠘⣄⠙⠦⣍⡻⠿⠿⠿⠋⠀⠨⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣀⡤⠴⠒⠛⠋⣉⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⡉⠙⠓⠒⠤⢤⣀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠘⠋⣠⠤⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠤⣀⠙⠃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠔⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢂⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠸⣤⠌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠡⡠⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠄⠠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄⠠⠀⢚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⡄⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠐⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢄⠐⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡉⠀⡀⠎

⠀⠀⠀⠀⢠⣶⣟⣿⡒⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⣻⣟⣿⣦
⠀⠀⠀⠀⢺⣿⣟⣿⡛⠲⠤⠀⠀⠈⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠉⠀⠀⠠⠴⠛⣻⣟⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠓⠀⠀⠈⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠀⠀⠒⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣯⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡋⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣧⣤⡤⠤⠀⠰⠒⠈⠀⢸⣿⣿⠀⠀⠐⠲⠀⠠⠤⣤⣤⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⠀⠀⠀⠀⠀⠀⠀⡀⣼⠉⢹⡄⡀⠀⠀⠀⠀⠀⠀⠐⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣶⡞⠀⢀⡄⠀⣼⠗⠋⠀⠈⠓⢷⡄⠀⣄⠀⠘⣶⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⣿⣷⠿⠛⠀⠀⠀⠀⠀⠘⠻⢶⣿⣿⣶⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁




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

⠀⠀⣴⠻⣦⣄
⢴⣿⣿⣿⣿⣽⢇⣀⣀⡀
⠀⠻⣿⣛⣶⣿⣷⣽⣿⣿⠄
⠀⠀⠀⠙⠛⠻⠶⠟⠛⠋
⠀⠀⠀⠀⠀⢠⡀⣠⡀
⠀⠀⠀⢶⡄⢸⣇⣿⣡⡶
⠀⠀⠀⠘⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣤⣶⣶⣦⡀
⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⢀⣀⣤⣶⣿⣀⣀⣙⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣷⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠘⢿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠟⠛⠋⠀⠀⣀⣤⣿⡿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠠⡸⠋⡡⠍⡆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠤⠃⠘⠴⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⢌⠙⢇⠄
⠀⠀⠀⠀⢀⠁⢅⠀⠑⣒⣥⣲⣿⣏⣲⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣤⡖⣹⣿⣖⣌⣒⠂⠀⡸⠈
⠀⠀⠀⠀⣠⣤⢾⣯⡛⣿⣍⣻⣇⠹⣿⠈⣯⣦⡀⠘⡄⠀⠀⠀⠀⠀⠀⠠⠃⢀⣴⣽⠀⣿⠇⣸⣟⣩⡿⢛⣽⡷⣤⡅
⠀⠀⠀⠀⢸⡿⢶⣚⡛⠍⠉⠉⠉⠙⠛⠳⢿⣇⢻⢦⡈⢢⠀⠀⠀⢀⠔⢁⡴⡏⣸⠷⠞⠛⠋⠉⠉⠉⠩⢛⣓⡶⢿⡇
⠀⠀⠀⠀⠘⣻⡞⠛⠢⡀⣤⣤⣀⢀⠀⠀⠀⠊⠙⠢⣷⡀⠱⠀⢀⠎⢀⣾⠜⠋⠐⠀⠀⠀⡀⣀⣤⡤⢂⠔⠛⢳⡿⠃
⠀⠀⠀⠀⠀⠘⢛⡿⠛⠛⡆⠤⣋⣹⠓⢶⣄⠀⠀⠀⠈⢣⠀⢇⡘⠀⡜⠁⠀⠀⠀⣠⠶⠚⣍⣙⡠⣰⠛⠛⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠈⢻⡟⠋⡁⠄⠀⠉⠲⢜⠉⣳⣄⠀⠀⠈⣧⣟⣻⡼⠁⠀⠀⣠⣞⠉⡢⠞⠉⠀⢀⢉⠙⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡝⠛⣃⠡⠀⠀⠨⡪⠻⣀⡙⢆⣠⣴⣿⣟⣻⣿⣦⣄⡰⣋⣀⠟⢄⠀⠀⠀⠄⣘⠛⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣻⡟⢉⡦⢀⣂⢒⣀⠤⣴⣊⣿⣻⣿⣿⡟⢻⣿⣿⣟⣿⣑⣦⠥⣑⡒⣐⡠⣴⡉⢻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠛⣷⣞⣷⠾⠿⣚⣯⣿⣿⣿⡿⡹⣿⣿⠏⠿⣿⣿⣿⣽⡓⠿⠷⣾⣳⣾⠛⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⠞⢀⣴⣾⣿⣿⣿⡟⣹⠃⠇⢹⡏⢸⠘⣟⢿⣿⣿⣿⣷⣦⡀⢳⡙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡞⠀⠘⢿⣿⡿⠟⣉⠔⢁⡜⠀⠈⠀⠀⢣⠈⠦⣙⠻⢿⣿⡿⠃⠀⢳⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⣠⠤⠆⠒⠂⢉⡠⠔⠋⠀⠀⠀⠀⠀⠀⠑⠢⢄⡉⠒⠒⠰⠤⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡈⠛⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠁⠀⢀⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠀⠀⠈⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠈⠒⢊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡐⠒⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢨⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠇⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠬⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠐⠥⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠈

⠀⠀⢀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⡀
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢸⣿⣿⣿⣿⣿⣟⣉⠉⠂⢠⣽⣷⣄⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣾⣯⡄⠐⠉⣉⣻⣿⣿⣿⣿⣿⡇
⠀⠈⣿⣿⣿⣧⣤⣀⣀⣈⣹⣾⣿⣿⣿⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⣿⣿⣷⣏⣁⣀⣀⣤⣼⣿⣿⣿⠁
⠀⠀⠘⣿⣧⣄⣀⣿⣿⣿⠛⠛⢿⡟⠿⠿⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠟⢻⡿⠛⠛⣿⣿⣿⣀⣠⣼⣿⠃
⠀⠀⠀⢸⣿⣿⣿⡟⠛⠛⠃⠂⠚⠓⢄⡀⠀⠈⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⢀⡠⠚⠓⠂⠘⠛⠛⢻⣿⣿⣿⡇
⠀⠀⠀⠀⢹⣿⣷⣤⣀⣄⣀⣀⣠⣤⠤⢽⣦⣄⠀⠀⠈⠻⣄⠀⠀⠀⠀⣠⠟⠁⠀⠀⣠⣴⡯⠤⣤⣤⣀⣀⣀⣀⣤⣾⣿⡏
⠀⠀⠀⠀⠈⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⠘⣆⣤⣴⣰⠃⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣉⣢⣬⣿⣿⣥⣔⣉⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⡟⠉⠉⠙⠫⢭⡍⠉⠉⠉⠽⠯⣻⡏⢸⣟⠽⠯⠭⠉⠉⢩⡭⠝⠋⠉⠉⢻⣯⣍⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠃⠀⠀⠀⡀⠁⠀⠀⠀⣀⠴⢂⠟⣿⣿⢻⡐⠦⣀⠀⠀⠀⠉⢀⠀⠀⠀⠘⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣧⠤⠒⠀⠀⡀⠀⠀⢀⡠⠚⢁⡴⠱⠀⢻⡿⠀⠎⢦⡈⠓⢄⡀⠀⠀⢀⠀⠀⠒⠤⣼⣿⣯
⠀⠀⠀⠀⠀⠀⠠⣿⣿⠀⣀⡤⠒⠉⡹⢋⡴⣿⠀⢠⠞⡰⠁⠀⢸⡇⠀⠈⢄⠳⡄⠀⣿⢦⡙⢏⠉⠒⢤⣀⠀⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠾⠁⠀⣠⠞⠀⣶⠔⠁⣠⠋⠠⠁⠀⠀⢸⡇⠀⠀⠈⣄⠙⣄⠈⠦⣶⠀⠳⣄⠀⠈⠷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣶⣶⡁⠀⣸⠃⠀⡰⠃⢰⠇⠀⠀⠀⢸⡇⠀⠀⠀⠸⡆⠘⢆⠀⠘⣇⠀⢈⣶⣶⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⣷⣿⣯⣄⣼⣁⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣈⣧⣠⣽⣿⣾⡿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠛⠃




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

⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢯⡅⠓⣦⡀⠀⠀⠀⠀⢀⣴⠚⢨⡿⠀⠠⣿⣿⡶⠄⡀⠀⠀⠀⠀⠀⠀⠠⢶⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣦⣤⡆⢀⡀⠀⠀⠄⠀⠀⠀⢸⡓⠀⠀⠙⢆⢀⡀⡰⠋⠀⠀⢚⡃⠀⠀⠹⣿⡾⠿⠟⢦⡀⠀⢀⡴⠻⠿⢷⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢺⣿⠂⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⠤⠤⡠⣻⣟⠔⠬⠤⣤⡼⠁⠀⠀⠀⢻⣂⣀⣀⣀⣹⣶⣟⣀⣀⣀⣀⡟
⠀⠀⠀⠀⠀⠀⡀⠀⠓⠃⡇⠓⠋⠀⡐⠀⠀⠀⠀⠀⠀⡞⣱⣊⠐⢹⠻⠟⡏⠠⣑⣌⢳⡀⠀⠀⠀⠀⢽⠉⠩⠉⠀⣿⠀⠉⠍⠉⢯
⠀⠀⠀⠀⠀⠀⣱⠤⠄⠀⠀⠀⠄⢴⡁⠀⠀⠀⠀⠀⠈⢇⣍⣀⢀⠂⠀⠀⠐⡀⣀⣡⣸⠁⠀⠀⠀⠀⠱⣆⠀⠀⠀⠛⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠈⠋⠁


⠀⠀⠀⠀⠀⢾⡿⠶⣤⡀⠀⠀⠀⠀⠀⠀⣠⡴⠾⣿⠂⠀⠀⠀⠀⢠⡾⠟⠲⡀⠀⠀⠀⠀⠀⠀⢿⡏⠳⣦⣄⡀⠀⠀⢀⣠⣴⠞⢹⡟
⠀⠀⠀⠀⠀⠈⢳⠔⢣⡈⢦⡀⠀⢀⡠⢊⡼⠣⡽⠁⠀⠀⠀⠀⠀⣉⣤⡆⢀⣥⣀⠀⠀⠀⠀⠀⠈⢿⠀⠘⢸⠃⣶⣶⠘⡇⠃⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠡⣄⠩⢾⣯⣶⣮⡾⠍⣐⠼⠁⠀⠀⠀⠀⠀⠀⢉⠁⣰⠋⠀⠈⠇⠀⠀⠀⠀⠀⠈⣡⠀⢘⢀⢿⠿⡀⡇⠐⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠯⠌⡟⢻⢻⠉⡜⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⡼⠀⠀⢧⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⠂⠀⠀⢶⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⠁⠀⠀⠈⣿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠙⠀⠀⠀⠘⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⢤⣤⣀⠀⠀⠀⠀⠀⠀⠹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠀⠀⠀⢀⣾⡿⠃⠰⠿⠁⢀⣀⢠⠤⢤⣿⡤⠤⠄⢤⣄⡈⠛⠃⠀⢹⣿⠂⠀⠀⠀⠀⠀⠀⢯⠁⠀⠀⠀⠀⡦⠀⠀⠀⠀⢩⡇
⠀⠀⠀⠀⠀⠹⢃⣀⢀⣤⠚⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠳⣤⡴⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢀⣏⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠹⡧⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠋⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⠤⠀⠀⠀⠈⠢⠴⠘⠁

⢠⣤⣶⣶⠶⠶⠶⠦⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠶⠶⢶⣶⣦⣤
⣼⡿⠛⠀⠀⠀⠀⠀⠀⣀⡀⠈⠙⠛⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠀⢀⡀⠀⠀⠀⠀⠀⠘⠻⣿
⣿⠗⠒⠂⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠁⠀⠈⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠀⠀⠠⠀⠀⠀⠄⣈⡀⠀⠀⠈⠀⠐⠒⢿
⠸⣦⠄⠒⠈⠉⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠈⠉⠉⠐⠂⢤⡾
⠀⠈⠻⢶⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠰⣾⡶⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⣶⠾⠋
⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⢀⣀⠠⠐⠈⠀⠉⠀⠂⠤⢄⣀⣀⣻⣷⣾⣿⣿⣶⣿⣋⣀⣀⠠⠄⠀⠈⠁⠈⠐⠠⢄⣀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠉⣶⡶⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠛⠽⢟⢿⣿⣿⠟⡻⠝⠛⠳⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠲⣦⡎
⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠀⣀⠀⠀⠀⢀⡿⣿⡿⣇⠀⠀⠀⢀⡀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣦⣤⣶⠟⠉⠀⠄⠂⠁⠀⠈⠀⠀⠀⣾⢸⣿⣿⢸⣆⠀⠀⠈⠀⠀⠁⠂⠄⠈⠙⢿⣶⣤⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⡤⠂⠀⠀⠀⠀⠀⠀⠀⠠⣾⡟⠘⣿⡟⠘⣿⡦⠀⠀⠀⠁⠀⢀⠀⠀⠢⣄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⢠⠊⠀⡀⠁⠀⠈⠀⢻⣧⠀⢿⠇⢀⣿⠃⠀⠀⠀⠁⠀⠀⢣⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡠⠁⠀⡔⠀⠀⠀⠀⠀⠈⣿⠀⠈⠀⢸⡏⠀⠀⠂⠀⠀⠐⡄⠀⠱⡀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⡰⠁⠀⡜⠀⠀⠀⠠⠀⠀⠀⠸⡀⠀⠀⣸⠀⢸⠀⠐⠀⠀⠀⠘⡄⠀⠱⡄⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⡸⠀⠀⠁⠀⡄⠀⠀⠀⠀⡇⠀⠀⡇⠀⠈⠀⠀⡆⠀⠀⠀⠸⡀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣠⠁⠀⡜⠀⠀⡇⠀⠀⡇⢠⡇⠀⠀⣷⠀⡆⠀⠀⡇⠀⠈⡄⠀⢱⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡀⢠⠁⠀⠀⠁⠀⠀⣷⠟⠁⠀⠀⠙⢷⣇⠀⠀⢠⠀⠀⢱⠀⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣀⣀⣸⣄⣠⣾⡇⠀⠀⠀⠀⠀⠀⣿⣦⣄⣼⣀⣀⣸⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⢻⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

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

⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⣤⡀⣀⠀⢀⡀⢀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⡀
⠠⡟⠉⠋⠀⠀⠀⠀⢸⡇⠀⣼⠋⢹⣇⢸⣍⣵⠀⢹⠀⣿⣧⣼⣧⣿⢋⣡⡄⠀⢸⣌⣿⠀⣞⠉⢱⣀⣾⠉⢻⣄⣿⣾⠏
⠀⢿⣶⡷⠀⠶⡶⠆⢸⣷⣦⠻⣷⣾⠇⣼⡟⠿⠂⣾⠀⣿⠙⢿⡏⢿⣬⣽⠇⠀⢸⣯⣽⡗⠹⣶⡾⠈⠿⣷⡿⠃⣿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡠⢀⠠⣁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢙⠄⠪⠐⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⢠⠚⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠈⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⠀⠐⠀⢀⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⠐⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⠀⠢
⠀⠀⠀⠀⠔⠐⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠄⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⢐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠞⠂⠀⡶⠆⠀⠀⠀⠀⠀⠈⠀⠀⣒⠠⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⠠⠀⠁⠁
⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠈⠦
⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢌⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠐⠁⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⢂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢁⠄⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠢⠀⣀⢀⠁⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠐
⠀⠁⠀⢠⡀⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠉⠑⠠⡠⠚⠀⠀⡀⠀⠀⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠒⠀⠰⢈⠁⠀⡀⡀⢉⠐⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠊⠠⡀⢀⠀⠀⠈⠁⢀⠀⡀⢀⠀⠄⠀⠀⠀⠐⠀⡎⠁⠄
⠀⠀⠀⠈⠀⠈⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡒⠈⢀⠀⠈⠁⠃⠀⠀⠀⠀⠀⠒⠀⠐⠂⠈⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣟⣛⣛⣏⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣽⣿⣯⣽⣿⣽⣽⣿