Particular Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⢲⣄⣀⡴⠚⠉⠀⠀⢀⡤⠆
⠀⠀⠀⠀⠀⠀⡀⠤⠀⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⡄⣴⡛⠬⠨⠝⣳⡄⢀⣶⠋⠀⠀⠀⠀⠀⠀⡀⡀⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠘⡄⠀⢠⠁⣀⠐⠍⠓⠢⢄⡀⠀⠀⠀⠸⣇⣹⣿⣦⣷⣿⣿⣁⣼⠏⠀⠀⢀⣀⠤⠖⠉⠀⠐⠀⣀⠀⡀⣵
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠁⠀⠈⠑⠂⢤⣀⣸⣿⣵⡟⠿⠽⢙⣿⢁⣠⠴⠚⠉⢀⠄⠰⡀⠈⠁⠉⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣁⢀⠄⠄⠐⠀⠀⢀⡀⠠⢬⣿⣿⣷⣂⣠⣨⡅⠨⠦⢀⣀⠀⠀⠠⠀⠠⠐⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣌⡁⡀⢀⣀⣁⣨⠴⢞⣿⣿⣿⣭⣯⣿⢗⠢⣴⣄⡀⠀⠈⠉⠍⢀⡩⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣤⠿⠁⣤⣴⣿⣿⣿⣷⢿⣿⣷⣤⡙⢧⣈⠉⠘⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠃⠀⣼⡏⣾⣿⣏⣿⣿⣿⣿⣿⡎⡿⡄⠙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⢸⣽⠀⢿⣿⣿⣗⣶⣆⣾⣿⠇⠰⣶⡀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠸⣿⣷⣟⣿⣖⣿⡿⠀⠀⠹⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠀⠀⠀⠹⣿⣿⣷⣿⡿⠃⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠀⠒⢫⣴⢶⠾⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠴⣿⠋⠁⠀⠀⠀⠀⢩⡗⢀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢓⡾⠏⠁⣠⣔⢂⣒⡦⠀⣤⣜⢻⣄⠶⢤⠿⡈⠙⠂⠓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢊⣶⣿⣿⢹⣿⡟⢛⣷⣆⣿⣛⡏⠀⢁⣾⣿⣷⣦⡞⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣫⣿⣿⣏⣠⣿⣿⣿⣏⣽⣿⣯⣾⣿⢿⣋⣿⣿⣻⡟⣶⣷⢦⣤⣌⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣙⣿⣿⣿⣿⠿⢿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⢿⣿⣷⣿⣿⣧⣼⣟⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡿⣿⣿⣿⣿⣏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠦⣽⣮⣿⣭⢶⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢯⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣢⣽⣿⠗⠊⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠋⠝⣽⣿⣿⣦⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣧⣤⣶⣤⣄⣶⣆⡀⠀⢾⣿⠿⠛⠉⠁⠉⠀⠀⠀⠀⠘⣷⣽⣿⣶⡿⠶⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢼⣣⣭⣿⣉⣛⣿⡅⠀⠈⣧⣶⣿⣉⣈⣿⣦⡆⠀⠀⠀⢸⣿⣿⣿⢿⠂⠀⡀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡿⠻⠛⠛⠛⣻⠀⠀⠀⠨⣍⠉⠉⠉⠀⠁⠀⠀⠀⠀⠐⠿⢛⣿⡀⢰⡿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡏⠙⠶⠖⠀⢰⡏⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⣽⠛⡋⠱⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⣿⠗⠀⠀⠀⢠⠟⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠌⠁⠘⣿⣿⠋⠀⠸⠳⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠴⠂⠰⣾⣿⡆⠀⠀⠀⢶⣧⣄⣤⣠⣤⣤⠚⠀⠀⠀⠀⠀⠀⠀⣠⢰⡞⢓⣤⠶⢤⣿⢿⣿⡀⠀⡀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋⠁⣿⣿⣇⠀⠀⠀⣼⣿⣿⢿⣿⡆⠀⠀⠀⢠⡀⠀⠀⠀⠀⢫⣾⣼⠾⣿⣷⣾⣿⠆⣍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⢀⠰⣿⣿⣿⣾⣶⣿⣿⣿⣧⣤⣼⣿⣿⣶⣦⣼⡇⠀⠀⠀⣠⣶⡼⢿⣿⣿⣿⡏⣿⡀⠃
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⣁⣨⢿⣿⣿⣷⢀⣤⣧⣼⣿⣿⣧⣿⣿⣯⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣿⣿⣿⣿⣧⢘⣿⠟⠛⠛⠉⠁⠀⢠⣿⣿⠟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡟⣉⣽⣿⣿⣿⣿⣯⣀⡀⠀⠀⠀⠀⢠⣽⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣹⠃⠘⢩⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣼⣿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣸⡟⡤⣀⣀⣐⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⡛⠈⠻⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢃⣾⢋⠀⠉⠾⣿⣿⣿⠟⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⢻⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣟⣿⣿⣿⡏⠀⣾⠷⠁⠀⢈⣼⡿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠟⠁⠀⠀⠉⠉⠛⠿⢿⣿⣿⣿⣿⡿⠛⢁⣴⣿⡿⣳⣿⣿⠀⢺⡟⠀⠀⣠⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⣀⠲⠟⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⢫⣿⠿⢾⣴⣟⣽⣿⣼⠟⠀⣋⡰⣻⡄⠀⣠⣾⡏⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣄⣌⡠⢠⡜⢓⣩⠔⣫⣿⣿⢩⣿⠋⠀⠀⣿⣿⣿⣃⠀⡴⠿⠃⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⢀⡴⢙⡷⣿⠿⣿⣾⣿⣥⣾⠟⣫⣴⣿⠇⠀⠀⣠⣿⡿⣭⣅⠀⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣠⠏⠔⣢⣠⠦⣀⣶⣼⣿⠟⣁⣸⣿⣿⡃⢀⡴⠟⠁⠀⠚⠉⢀⡠⠄⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣴⠟⢂⣽⡼⠃⣾⣿⣿⣿⡏⢠⣿⣿⣿⡏⡠⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣠⠉⠈⠕⠺⣹⣿⡫⣻⠄⠈⢻⣿⣫⠖⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡾⠁⠐⠀⠀⣵⣰⡿⠋⣙⣴⠀⠁⢸⣿⠟⠁⠀⣠⣴⡄⣒⠠⣤⢤⣄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠀⠀⠁⠠⢶⣼⣋⠀⠠⠟⠀⠒⠀⣸⡿⣀⠀⡴⣜⡵⠶⠉⠀⠀⠀⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡴⠿⠿⠯⠚⠠⠀⣄⢘⣠⡟⠁⠀⠀⠀⠁⠀⣀⠔⣈⠡⢁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⢠⠀⠐⣽⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⠀⠁⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠀⣹⡿⠆⠿⣛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡞⠀⠈⠛⠀⢺⡆⢿⣟⠀⠀⠒⠂⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠛⣿⣿⡆⠀⠖⣱⣀⣷⠘⣿⠐⣿⠃⣰⣿⠧⣼⡿⠿⠿⣿⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢁⣀⠘⠹⣿⠀⠘⣤⢿⣿⣿⣶⠋⠀⣼⣅⠙⢩⣿⡝⠃⠐⣶⠛⠀⢼⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠈⠁⢰⡴⣆⣴⠀⢀⣤⣽⣼⣧⣄⠼⢿⣿⣇⣸⣯⡏⠀⣠⣿⣦⣶⡾⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠖⠻⠷⣤⣄⡀⠶⠿⢃⡞⠉⠀⠀⠀⠈⠻⣧⣄⡈⠙⣿⠟⠁⢺⣿⣿⣿⣿⡄⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠚⣻⣿⣶⣶⣿⠁⢀⡀⢠⡎⢩⠿⢻⡟⢻⣿⣧⠀⡀⢾⡽⣿⡿⢿⣥⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣾⠀⠀⣤⣀⡤⡌⢹⠟⢹⣿⣿⠃⠀⠼⠃⠘⠃⣬⣿⣿⣷⣸⣿⣷⠊⠁⠀⣀⠈⠻⢿⡀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣿⣿⣀⡀⠙⠃⠀⠰⣆⡀⠈⢉⣄⡀⣀⠀⠀⠈⠙⣿⡏⣡⣮⣿⠿⡇⠀⣰⡆⢠⡿⣦⣼⡃⠐⢠⡄
⠀⠀⠀⠀⠀⢀⣾⡿⠿⠇⢠⣬⣿⣷⠀⠀⢀⣙⣣⣼⡿⠀⡀⢈⡻⠿⣿⣾⣿⣿⣿⣿⣿⠿⣿⠿⠁⠀⣉⣡⠈⣷⠿⠿⣿⠛⠈⢻⣇
⠀⠀⠀⠀⠀⠸⣿⠀⢤⣄⢈⣍⠉⠀⠀⠀⠟⠛⡿⢿⣁⡀⣧⣼⣇⠀⠈⡏⠉⠀⠈⣀⣽⢠⡿⠀⠠⣿⠋⡙⠚⢁⠀⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣼⡆⠘⠁⠈⣁⣠⡆⠀⠀⣀⣠⣄⣸⣿⣿⣿⣿⣿⣶⣇⠀⠀⠀⣈⡋⠁⠘⠡⠄⡀⠀⠶⠿⢶⡌⠀⠐⠀⠀⢠⣤⣁
⠀⠀⠀⠀⠀⠀⠈⠛⢄⡀⠀⠉⠁⠀⠀⠀⢀⣻⣿⠟⠁⠈⠉⠉⠻⢿⣿⣷⣤⡼⠋⠁⠀⠀⠀⠀⠀⠐⣀⣠⣤⣧⣄⣤⣤⣄⡈⠛⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠰⠒⣺⣷⠶⠦⣼⣿⠏⠀⠀⠀⠀⢠⡄⠚⢿⡿⢿⠁⠀⠀⣾⣦⡄⢀⣤⠄⢹⣿⠟⠉⠉⠁⠘⢿⣷⡄⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⢰⡟⠀⠀⠀⢸⣏⠀⢠⣄⣨⠀⢈⠰⠟⠈⣷⡄⠀⠀⠀⠨⣿⠀⢻⣥⠀⣾⣏⠈⠁⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣾⠀⠀⠀⠒⣀⣿⡄⠀⢈⡉⢤⣾⣷⠶⢦⡋⠙⠀⠀⢀⡤⢀⣠⣄⠁⠀⠿⠉⠀⠀⠀⢀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢻⣿⡄⢸⣦⠀⠙⠛⢷⣄⠈⠃⣤⣤⣤⣆⠀⠹⠃⠀⠀⠁⣀⣌⣉⣉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠀⢸⣤⣼⠂⣌⠙⠄⠀⠈⠻⣿⡿⠀⣸⡀⠀⠀⢼⠟⠿⢿⣿⠀⢰⠀⠀⠀⠰⣶⠐⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⣩⠤⣼⡿⠀⠀⣶⣄⢀⣀⣠⣴⣿⣇⠀⠀⠀⠀⣀⡼⠃⢀⣿⣿⣆⡀⠀⠹⢦⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣉⣻⡿⠾⢇⣼⣿⣿⣿⣿⣿⣿⣿⠛⢀⠀⠀⣀⡄⢀⣴⣿⣿⣿⣿⣿⣶⣶⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⣏⡙⠛⠁⣉⣅⣀⠀⡀⠘⠚⠉⢹⣿⣿⡿⠟⠋⠈⠻⣿⣿⣇⢻⣿⣿⣿⣿⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠈⠙⠛⢛⣿⣿⣿⠋⣠⣶⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⢻⣿⣏⢻⣿⣿⣿⣿⣿⡏⠀⠐⠟⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡛⠛⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢤⠒⠀⠉⡀⠉⠡⣤⣤⠀⠺⠯⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⡄⠀⠒⠛⢻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⢀⡇⠀⡄⠀⣿⣿⠋⣠⣿⣿⣬⡁⠀⢀⣄⠀⣤⠀⢠⡄⠠⡄⢀⡀⡀⠀⢀⠀⡄⠀⢠⡄⠀⣤⠀⠀⠀⣤⣄⣿⣿⡿
⠀⠀⠀⠀⣾⠀⢤⣤⣴⣿⣷⣦⡈⠙⣿⣿⠀⠀⠸⠃⠀⠿⠄⠘⠧⠀⠀⠀⠂⠦⠀⠸⠀⠲⠀⠘⠓⠀⠛⠀⠀⠀⣿⡿⣧⠙⠁
⠀⠀⠀⠀⢛⡠⠶⢶⣮⣙⣿⣿⠿⢷⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣤⣄⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡄
⠀⠀⠀⠀⠋⠀⢀⣀⣹⣿⣿⣿⣦⡘⠛⠀⠀⠀⠀⠀⠀⢸⣿⡇⠿⠼⠃⠸⠇⠸⠇⣿⡈⣰⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡧
⠀⠀⠀⠀⠘⣿⣿⣿⣿⢿⣿⣿⣿⡿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠘⣻⣿⣿⠁
⠀⠀⢀⠄⠺⢟⣿⣿⣿⣿⣆⠀⠙⣴⣦⡈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿
⠀⠀⠋⠀⣰⣿⣿⣿⡿⠿⠿⣷⡄⠸⣿⡻⣷⣄⡀⠈⠉⠛⠻⠶⠶⠴⠶⠿⠛⠻⢿⣤⣄⣤⣴⡶⠟⣿⣿⣯⣨⣿⣿⣿⣿
⠀⢸⠀⠰⢿⣿⣿⡿⠃⠀⠀⠘⠇⠀⣻⠀⠹⡟⠻⢿⣶⣶⣤⣄⣠⣄⣤⣀⡀⣀⣠⣍⣡⣥⣶⣶⣿⣿⣿⣿⣿⡿⠿⢿⣿⠂
⠀⢸⠇⠀⣀⣼⡿⠆⠀⠀⠀⠀⠁⢠⡟⠀⣶⡟⣦⡀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠇⠀⠀⠾⠟
⠀⠀⢦⡀⠘⣿⣄⣤⠀⠀⠀⠀⢀⣿⡟⠚⠛⠧⠌⢿⣄⠀⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣭⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠈⠹⣦⣏⡁⠀⠀⣀⣤⣴⣿⡿⠃⠀⠀⢀⠀⠀⣿⠃⠀⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣀⠀⢠⣶⣴⣾⣿⣿⣿⣯⠉⠀⡀
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠈⠀⢿⣄⡀⠀⢀⡀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣨⣴⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠓⠀⠈⢿⣿⠿⢿⣿⣿⣿⣿⣄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⡀⣀⠀⠸⣿⠀⠀⠈⠿⠿⠿⢿⣦⣄⣼⣿⣿⣿⣿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣼⣦⣤⣶⣿⣶⣶⢶⣿⣻⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣍⠁⢹⣿⣿⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⣡⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⡾⠟⠋⠁
⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡟⠁⠈⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣾⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢠⡁⠀⠀⠠⣁⠀⠀⢀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠚⢿⣀⠚⠓⢻⡦⠘⠚⢻⣇⡐⠛⢛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠁⠁⣰⡟⠛⢀⣾⡛⠉⠉⠉⣩⣷⣿⣿⣉⣻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⡏⠉⠉⢹⡏⠉⠉⠉⠉⢹⣿⣿⠉⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⢠⠄⠛⡀⠀⠠⠟⢁⣠⣤⣤⣤⣤⣿⠿⠆⠀⠙⠗⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠷⣿⡿⠿⠷⠾⠃⠘⠋⢩⣉⠋⢉⣉⣻⠀⢳⢾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣛⠆⠻⢷⣟⡶⣄⣠⡆⢠⣭⠿⡅⣿⣭⣿⣷⣼⣿⡿⢿⣿⡄
⠀⠀⠀⠀⢀⠠⠐⠒⠒⠢⡄⠀⠀⠀⠀⢠⡄⠁⠀⣴⣦⡀⣠⠳⠿⠁⠀⠀⣀⠀⣏⡀⣨⡿⡿⣿⣇⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠆⠀⢀⢠⣦⣤⠛⠀⠀⠀⠀⠈⠻⣶⣺⣿⣿⣿⠟⠀⠘⢁⢠⣿⡿⣿⣿⣿⡟⢠⠃⡌⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣳⣀⣀
⠀⠀⠀⠸⢆⣐⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⣯⣶⣤⣁⣠⣼⡟⠛⠛⠛⣋⣳⣌⣈⣠⡿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠉⢛⠿⢇⡥⠀⠀⠀⠀⠀⠀⢀⣤⢀⣿⣿⡇⠸⠬⠏⠈⣿⡟⡲⠴⣛⣿⠿⠉⣿⢿⡇⠻⣷⢸⣿⣷⣄⠀⠀⠀⠀⠀⠀⠙⣿⣛⠛⣙⣻⠇
⠀⠀⠀⠀⠀⠀⠘⠀⢸⣷⠀⠀⠀⠀⠀⢶⣟⡇⢸⣿⣿⣿⠃⠀⠐⢀⡿⢇⠀⠋⠌⢃⡰⠀⣿⣾⣇⣴⣿⠸⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣿⣯⣬⠍⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣧⡀⠀⢀⠀⠈⠹⠇⢸⣿⣿⣿⠀⢡⢸⣿⣷⣾⣷⠖⢾⣟⠀⢸⡷⠿⣧⣿⣿⢰⣿⣿⣿⣿⣇⢀⡀⠀⠀⣠⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡝⠻⢿⣶⠀⠮⠅⠀⠀⠘⣿⣿⣿⡒⣀⢹⡿⣿⣏⣿⠀⡀⢹⠀⣼⣡⡀⣿⡿⡇⢨⣿⣿⣿⣿⣿⡘⠁⠀⢄⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠉⠁⠀⢀⣠⣶⠷⠼⣿⡿⢿⣧⠈⢸⡇⢀⡬⠋⠀⠑⠊⠀⣿⣽⣧⣿⢿⣇⣤⣿⣿⠛⣛⠛⠃⠀⡀⢀⣿⡗⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣆⣤⣶⣿⡏⠁⠀⠀⠀⠁⣠⣿⣧⣈⣿⣿⢶⣄⡤⢆⣠⠀⢉⠡⣽⣶⣾⠿⢿⠷⣤⣤⡉⢻⣦⡀⢰⡾⠏⠀⢀⣤⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢫⣿⣿⡿⢿⣿⣿⣷⣾⣿⡿⣿⣿⣿⣿⣿⡃⢸⡿⣶⠞⣡⡾⢿⣆⣰⣯⣤⣀⣈⣤⣽⣿⣿⣦⣿⣿⣿⣥⣤⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠟⣸⣿⣧⣼⣿⢃⣿⣿⣿⣿⣿⣿⣿⣇⣿⠃⢸⠏⠈⣷⣿⣡⣾⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡠⠎⣠⠎⣰⢿⣿⣿⡿⢣⣾⣿⣿⣿⣿⣿⡟⢻⣿⣧⡄⢀⣴⣾⣿⡿⣿⣾⣿⣿⡟⠁⡻⡉⢿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⡄⠛⢀⡼⠃⣸⠇⣼⡿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣧⠑⢊⣩⡵⠏⢈⣉⣡⣾⣿⣿⠿⠋⢀⠀⢹⢧⣸⣿⣯⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⡇⠰⠟⢀⣼⠏⠸⠋⠀⣠⣾⣿⣏⣽⣿⣿⣿⢋⣉⠶⠃⠀⠈⠀⠀⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠿⣧⡹⣿⣏⣀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⠀⠠⡾⢃⡜⠀⢀⣴⣿⣿⢻⣾⣿⣿⠋⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣧⠙⢿⠙⠓⠀⠀⠹⣿⣿⣿⣿⡏⠀⢈
⠀⠀⠀⠀⠀⠘⠆⣠⣾⡿⢀⣼⣿⣿⢁⣾⣿⡟⠁⠀⠀⠀⠙⣿⣿⡷⢶⣴⣴⠉⢀⣤⠄⢐⡉⠀⠀⠀⠀⠀⠀⢻⣇⠀⣧⡀⠀⠀⢀⣿⣻⠟⢻⠇⠀⠛
⠀⠀⠀⠀⠀⠁⣴⣿⠃⢠⣾⣿⣿⣷⡿⠛⠉⠶⡄⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡀⣸⣷⣾⣿⣅⣠⣀⣄⣀⡀⣀⣈⣙⣆⠻⣿⣤⣾⣿⣿⣧⣈⠋⠀⢸⢇
⣤⣤⣤⣤⣤⣾⣿⠇⣠⣿⣿⣿⡿⠋⠀⠀⠀⠠⠙⣦⡀⠀⠁⠀⠀⣼⣿⣿⣿⣿⠿⠿⠿⠛⠻⠿⠿⠿⣿⣿⡿⣿⣿⣟⣧⠽⣿⣿⣿⣿⣿⠏⠀⢀⠆⣨
⣿⣟⠛⣿⣿⡟⢉⣴⣿⣿⠋⣿⣧⡀⠀⢀⠀⢀⣴⣿⡿⢷⣤⣀⣰⡿⠿⠛⠋⠀⠐⠂⠂⠀⢀⣠⣴⠾⠟⠁⠀⠀⠀⠙⢿⣏⢿⣷⡀⣭⣿⣷⡾⣷⡷⣿
⣻⣿⣥⣽⠛⣡⣾⣿⣿⡃⠀⠘⢿⣇⠀⠀⣠⠟⢫⣿⣇⠸⠉⠋⠈⠑⠒⠖⠢⠴⠶⢒⣶⡾⠟⠛⠋⠀⠀⠀⠀⠀⠀⢳⣿⣿⡄⠺⣷⣾⣿⣿⣷⣿⣧⣿
⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠶⠶⡎⠻⣇⣰⣧⣀⣘⠛⠃⠀⠀⢀⣀⣀⣤⣤⣶⡶⠿⠛⠉⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠘⣯⣿⣿⣎⣹⣿⣿⣿⣿⣿⣿⣿
⠆⠀⡀⣠⣼⣷⣤⣤⣅⣽⣄⣰⡋⠀⠈⠛⠛⠛⠋⠀⣰⣦⠀⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⣸⡏⢸⣿⣿⣶⣿⣿⣿⣼⣛⣿⣿
⡆⢸⣧⣭⣹⣿⠋⠉⠉⠉⠉⠉⣁⣀⠲⠀⣰⣶⣶⣾⣿⠋⠀⠀⠀⣴⠀⠀⠀⠀⠀⢀⣿⣆⡀⠀⠀⢰⣿⢻⣿⠀⠈⣛⡁⣹⠿⠉⠉⠉⠉⢹⣿⡿⡿⢿
⠃⠘⠯⠙⢿⣿⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠟⠡⢿⡟⢳⣀⣀⣤⡼⣟⢣⡀⠀⢀⣠⡾⢋⣿⡟⠶⣶⣿⠏⢸⡟⠀⠀⣿⡿⢿⠂⠀⠀⠀⠀⢸⣿⣯⣿⣿
⠀⠄⠰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⡾⣿⣦⣈⣭⣥⠾⠁⢈⡙⠿⠿⠋⣠⣾⠁⢠⣶⣶⣤⣴⣿⣤⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿
⠀⠀⠖⠉⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉⣿⣿⠙⠋⠉⠷⡾⠟⠛⢷⣴⡿⠛⠛⢻⡿⠋⠈⠙⠛⠋⠉⠉⠛⠛⠃⠀⠀⠀⢀⠀⠀⢸⡷⠤⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⣠⡀⠀⠀⠉⠂⠀⠂⣀⡀⣀
⠉⠉⠉⢁⣶⠖⣶⠦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⢀⠁⢀⠀⠀⢹⡆⠀⠀⠀⠀⣄⡤⠤⠤⠤⣴⣶⡒⠶⠒⠶⠛⠙⠉⠉
⡲⡀⠀⢈⡠⣻⣿⢇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠒⠒⠒⠒⠲⡄⠀⠀⠀⠁⠀⠀⠀⠀⣀⣀⣀⣈⠤⢀⠀⠀⠀⡟⠑⠘⠉⠉⠉⠄⠉⠀⠉⡇⠀⠠⠈
⣠⡟⠀⠺⣧⣿⣾⡞⣾⠀⠀⠀⢐⡆⠁⠀⠀⢰⠀⠀⡇⠀⢀⣀⡀⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⣸⠀⠀⠀⡟⠀⠀⠀⡀⠂⠀⠀⠄⠀⣧⠈
⠞⢴⣤⠈⣦⣌⣇⡼⢻⠀⠀⠀⢸⡄⠐⠠⠄⢠⠀⠀⡗⠀⢸⣿⡇⠀⡇⠀⣀⠐⣀⠀⠀⠀⢀⣿⣂⡈⠐⡠⢸⡀⠀⠀⡜⠀⡀⠂⠀⠀⠄⠁⠀⠀⡧⠈
⠉⠛⠉⠛⠉⠙⠿⠋⠉⠀⠀⠀⠸⡅⠀⠀⠒⢀⠀⠀⡗⠀⠘⠛⠃⠀⡇⠐⢠⠀⡀⠀⠀⡙⢁⠚⠋⢠⠄⡤⠾⣄⠀⢀⡻⠀⠀⢀⠔⠁⠀⠀⠀⠈⣇⠀⠀⠀⢂
⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⢠⠀⠀⣱⣀⣀⣀⣀⣀⡇⠐⢸⠀⣇⠀⠀⡴⢂⣀⣤⡼⢧⠀⠂⠀⠰⠀⢸⣾⣿⣿⡷⠀⡀⠂⡀⠒⢉⠀⠀⠐
⠀⠠⠈⠁⠈⠅⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠀⡀⠀⡙⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣣⠀⠀⠄⠀⠰⢑⠠⠈
⠀⠀⠀⠀⠈⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠋⠺⠂⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣮⣾⣿⣿⠂⠋⡆⠀⠠⢈
⠀⠀⠀⠀⢀⣠⠄⠠⠖⠀⠈⠁⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠴⠀⣠⡀⠀⠀⢸⣇⣽⣿⣿⢰⠚⢛⠙⣼⠟⢒⢌⣤⠄
⠀⠀⠀⠀⠀⠀⠀⢣⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠃⡠⢀⡎⠁⠜⠛⣗⠃⠠⠃⢸⠿⢿⣿⣿⢸⡀⡀⠁⣵⠊⢀⡛⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢂⡵⢩⡀⠁⢀⠒⢀⣿⣿⠐⢸⣏⠀⠀⠉⠉⠛⠉⠚⣥⠤⢤⠃
⠀⠀⠀⠀⠐⠀⠀⠀⠈⢆⠀⠆⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⢀⠂⠀⠀⠀⠀⢀⠘⠋⠘⠋⠑⠺⢀⠐⢀⣿⣿⠰⢸⣿⣿⣿⣿⣿⣶⣶⣶⡇⠠
⠀⠀⠠⣄⣈⣀⣠⠔⡠⠈⢣⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠀⢀⠐⠀⠀⣀⣰⣦⠀⠀⠉⠦⠀⠀⣈⣿⣿⠠⢸⣿⣿⡏⠉⠉⠂⢈⢺⡇⠀⠀⠐
⠀⠀⠾⣋⣽⠏⠵⣊⠡⠇⠲⠤⠈⠄⠁⣠⣀⠤⠔⠲⠎⠀⠔⡒⠊⠉⠀⠈⠀⠀⠠⠀⠐⣟⠀⢈⣀⢤⣀⣶⠄⠤⢄⡀⠈⠁⠈⠙⠛⠃⠤⡀⠠⣉⣸⡇
⠀⠰⣄⡀⠈⠉⠀⢀⠠⡔⠁⠀⠀⡍⠋⠉⠀⠀⠈⠀⠀⠀⠀⠀⠄⠐⠀⠀⡀⠀⠀⠀⠠⢜⣉⠀⠉⠘⠋⠠⠀⠂⠉⠀⠀⠀⠀⠀⣀⣡⡆⠉⠙⠛⠿⡇
⠀⠀⢸⠀⡇⡏⠁⠀⠀⡜⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡇⣤⣀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡀⣤⡼⠿⣟⢹⡇⠒⠤⠀⣤⣇
⡀⠀⢸⠆⣿⡇⢠⣀⠀⡼⠀⠀⠀⡀⠂⡀⠀⠀⠀⢀⣀⣀⣤⠤⡴⠶⢺⡟⠛⠿⣭⡒⢤⡀⠀⠀⠀⡀⠀⠀⢀⣤⡶⠗⠛⢁⠀⠐⠈⢸⡇⢂⡀⠀⡀⠟⠀⠉⠃
⠀⠀⠀⠇⠙⡇⢨⣜⢋⡟⣦⣄⡆⢁⣶⠶⠶⢛⠛⠋⠙⠁⡤⠀⠀⠀⢸⡧⣠⠜⠉⢙⠻⣶⣭⣲⣦⣔⡾⠞⠋⠁⣀⡀⠀⢀⠀⠀⠀⣈⠃⢆⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⢀⣀⣄⣤⣤⣠⣀⡀⠀⠀⠀⣿⣦⣄⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣿⣿⣿⠀⠀⠀⣷⣦⣤⣀⠀⠀⠉⠋⠁⠀⢀⣠⣤⣶⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⠘⠿⣿⣿⣿⣶⣦⣤⣶⣾⣿⣿⡿⠟
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⣿⣶⣤⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣿⣿⣿⣧⣤⣴⣿⣿⣿⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⠘⠃⠀⠀⢰⣦⡀⠀⠀⠘⠫⠃⠂⠀⠀⠀⠀⠀⠀⠨⡿⠃⣦⣶⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠉⢀⡀⠀⠀⠀⠁⡋⠉⢆⠠⣄⠈⠀⠀⠀⢀⡄⠀⠀⠀⡀⠙⠂⢸⢷⠇⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⢨⠃⠡⠀⠈⠚⠹⣷⠆⠛⠁⠀⢀⠀⡇⠐⠀⠀⢠⠈⠀⠀⠺⠇⠈⡃⠀⣘⠘⠦⠀⠀⠜
⠀⠠⠚⡇⠀⠀⠀⠘⢁⠁⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠃⠀⢤⡤⠗⡄⠄⢤⡄
⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢂
⠀⠀⠀⠠⣼⡗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣶⣴⣟⠟⠓⢤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠻⡽⠇⠲⣰⠖⣯⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠉⠀⠀⠻⠾⠋⡒⢶⠀⠚⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⡀⠂⢛⣬⢖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡺⣿⢋⡀⢦⣬⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⠾⠿⣾⠷⣆⠀⢚⡏
⠀⠀⠀⠈⢀⡀⠆⠠⢀⡀⠀⠀⢄⠀⠀⠀⠀⢀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠮⣠⣄⠡⠀⣹⣾⣁⣤⠋
⠀⠔⣀⣤⣬⣴⡷⣷⡟⠀⠂⠀⠠⡄⠀⠀⢈⡙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠁⣴⣏⣛⢉⠓⢏⠟⠉⠁
⠀⠀⡏⠉⠀⠹⠋⠀⠀⠀⢂⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡗⠓⠀⠀⠀⠘⣦⡄
⠀⠀⠁⠀⠀⠀⠀⠀⠒⡀⣀⡗⠀⠐⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠉⠀⠀⠀⠀⡸⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢬⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣆⣀⣶⣀⣴⣆⣰⣆⣀⣶⣀⣶⣄⣰⣆⣠⣶⣀⣶⣄⣰⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⢻⡟⠛⣿⠛⢿⠟⢻⡟⠛⣿⠛⢿⠟⢻⡟⠛⣿⠛⣿⠛⢻⡟⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⣆⢠⣦⠀⣴⡀⣰⣄⢠⣦⢀⣴⡀⣰⣆⢠⣦⢀⣴⡀⣰⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⡟⠛⢻⡟⠻⣿⠛⣿⠟⢻⡟⠻⣿⠛⣿⠛⢻⡟⢻⣿⠛⣿⠛⢻⡟⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠛⠛⠓⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠠⣴⣶⠶⠾⠿⠟⠀⠀⠀⠻⠿⠷⠶⣶⣦⠄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠛⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⢀⣾⠏⠁⠀⠉⢻⣦⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢸⣿⠰⣷⡾⠟⢀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠾⠃⠀⠻⣷⣤⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠴⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠶⠶⠦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⠾⠃





⣶⠒⠲⣶⠶⡶⣶⠶⠶⣶⠶⣶⠶⡶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⠶⠶⣶⠶⣶⠶⠶⢶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⣤⣤⡤
⣄⠀⠀⠀⡄⠀⢠⡄⡻⣿⣶⡾⠈⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁
⠻⣶⣦⣾⠻⣶⣾⣿⡿⠏⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖
⠀⠈⣸⡿⣿⣽⣿⣿⣇⠀⠀⠘⣆⠀⡀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢐⣸⣷⣿⣿⣿⣿⣿⣿⣗⡉⠀⠹⣯⣷⡦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⣀⣩⣿⣿⡿⣏⢻⢿⡟⠻⣦⡠⠄⣿⣏⠀⠀⠀⢰⡂⣀
⣿⣿⣿⣿⣯⠹⠈⣻⣧⣠⣾⠿⣦⣸⣿⠀⠀⠀⠈⢻⣿⣿⣥⡄
⠀⠉⠙⢦⣀⠤⠈⠙⠉⢻⣙⣦⡿⣿⣿⣧⣄⣠⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠈⠙⢹⣿⣿⣿⣿⣿⣯⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠛⢟⣿⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠀⢿⣿⣿⣤⣄⣠⣤⣶⣶⣶⣿⡇⠈⠁⣠⠖⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣦⠦⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⣠⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⣸⣿⡿⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠉⡿⢋⡉⠙⠛⠛⠿⣿⢴⣿⣿⣿⡿⠟⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣦⣄⠀⠐⠟⠁⠸⣿⣿⣿⣿⣿⣿⣇⠀⡐⣿⡿⠆⣶⣶⡄⡟⢸⣿⡿⢿⠰⠆⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣄⣸⠟⠋⠉⢉⣉⣻⣆⣠⣤⣤⣤⡈⠻⠛⠛⠿⠿⢿⡀⢳⣶⡶⢦⣈⣋⣡⣦⣬⣿⣁⣼⣷⣶⡗
⠀⠀⠀⠀⠀⠀⠀⢺⣿⡿⠟⣛⠿⣿⣯⣶⣷⠄⠘⠛⠛⢿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⣿⣿⡿⢿⣿⣿⣿⠟⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣏⣠⣤⣶⣿⣿⡗⢀⣠⡾⢷⣤⡔⠋⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠿⠛⣅⠀⢈⠙⣿⣾⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⢻⣾⣍⠄⣇⣈⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠒⣠⣶⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⣽⣿⡟⠀⠈⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣻⣡⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⡿⠿⠿⠟⠻⣿⣿⡟⠿⠀⠀⠀⢸⣿⠛⠑⠆⠀⠠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣷⣿⠟⠋⠀⠀⠹⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⠀⢸⣿⣥⣶⣶⣷⣦⣬⣿⡇
⠀⠀⠀⠀⠀⠀⠰⣿⣿⠿⢿⣿⣿⣿⡆⠀⠀⠀⠰⣿⣿⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠱⠆⠀⠉⠁⠸⢟⡟⠙⠿⠿⠿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡈⠛⣿⡟⠀⢀⣠⣤⣾⣿⣿⣦⠀⠀⣿⡿⠛⠛⠻⣤⣤⡀⢀⣀⣄⣀⡀⢸⠣⠆⠀⢠⠄⡿⠙⢷⣤⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠁⠀⠙⠇⠀⢾⣿⣿⣿⡏⠉⠀⡳⠶⡟⠀⠀⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣧⣴⢶⣄⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⠲⣿⣿⣿⣟⠃⠂⠈⣙⣛⡃⠋⣀⡀⠃⢀⣿⡇⠀⠀⣹⣿⣿⣿⣿⠿⠿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⣿⣿⣿⠀⠀⣀⣛⣛⣛⣿⣯⣿⣿⠿⣿⣧⡀⠀⠈⢾⣿⠟⠋⠀⠀⠈⠃⠀⠀⠀⠀⠘⠲⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣉⣉⣉⣉⣉⠛⠋⠉⣾⣿⣿⣿⣶⠄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠁⠀⠀⠘⣾⠀⠀⠀⣨⣽⣿⣶⣤⣶⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⢰⣿⡄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠠⠀⠀⠈⠁⠒⠒⠤⣄⣀⣴⣦⣀⡤⠔⠒⠂⠉⠀⠀⠠⠀⠀⠀⣀⡀
⠀⠀⠀⢀⣼⣿⣷⠀⠀⠀⠀⠀⠀⡠⠄⠰⠒⢽⣿⣿⣿⣿⠕⠒⠀⠤⡀⠀⠀⠀⠀⠀⢰⣿⣿⣄
⠀⠀⠀⠈⠻⣿⣿⣷⠀⠀⠀⣷⡆⠀⠀⠐⠀⡼⣿⣿⣿⡿⡆⠐⠀⠀⠀⢶⡇⠀⠀⢰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⢿⣧⡀⠀⠀⠀⢠⣶⢠⠎⠀⢿⣿⣿⡇⠘⢦⢰⣦⠀⠀⠀⠀⣠⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠈⠀⠈⣠⠋⠀⠀⢸⣿⣿⠐⠀⠈⢣⡈⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠶⡤⠊⣌⠔⠃⠈⢸⣿⣿⠘⠀⠓⢌⡌⠢⡴⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠁⠀⠀⠀⠈⢿⠏⠀⠀⠀⠀⠉⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⢀⠀⠀⠀⢀⠆⢆⠀⠀⠀⢀⠀⠈⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠁⠀⠀⠁⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣯⣽⠄⠖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⡏⢻⠿⢷⣰⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣦⣿⣿⣿⣼⣷⣤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⢿⣿⣷⣿⣑⡙⢛⣿⡿⢿⣿⣿⣿⣦⡀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⢶⡄⠠⠀⠀⣻⡛⣿⠛⣿⡧⠿⢿⡿⠿⠦⣸⡿⢿⡟⠻⢿⠀⣀⣾⠱⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡸⠇⠀⠀⣘⠛⢛⠛⠛⣻⣿⣴⡄⣼⣶⢸⣾⡗⠂⠁⡘⠋⣀⠀⠛⣄⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠁⠀⠀⠀⢹⡄⠘⠇⢠⡾⢷⡀⢰⡷⠀⣼⣿⣆⠀⢸⠁⠸⠃⠀⠀⠈⠻⢄⣀⢀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠍⠀⠀⠀⠀⠀⣠⣿⣲⣛⣹⢧⡴⣟⡐⢃⣺⣅⣠⣯⡔⣊⠻⢧⡀⠀⠀⠀⠀⠻⣿⡝⠁
⠀⠀⠀⠀⠄⡒⢀⠀⡇⠀⠀⠀⣠⣾⡕⢩⠃⣿⡟⢀⣶⡌⢉⡉⢩⣿⠇⣴⣿⠘⠁⢂⠹⡄⠀⠀⢀⡀⠉⠷⠤
⠀⠀⠀⠀⠀⡸⢋⠀⣻⠄⠀⠀⢧⡿⢦⣜⣻⣿⢢⣿⣿⡟⠦⠶⢛⡟⠀⣸⣿⠤⡄⠀⣏⠘⡆⠀⠿⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠈⢻⠁⠉⣰⣴⠾⡿⢶⣧⢼⣿⡟⠈⢹⡇⠀⠀⢸⡇⢦⣿⣿⣙⣁⣤⢼⣿⣤⣤⣀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣰⠾⠷⠛⠮⢧⡴⠐⢊⡝⠉⠁⢸⠇⡇⢰⣿⠠⠷⣲⡿⠋⡉⠁⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢏⠀⠀⠀⠀⣀⢸⢽⠆⠹⡗⢐⡔⠉⠙⣿⠛⠉⠉⠁⠉⠀⠉⠁⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⡦⢣⢨⣷⡆⣿⠀⣶⢠⣴⡟⢻⣶⡖⢤⣤⣤⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⡟⣿⢹⣾⣿⠁⣯⠀⣿⣴⡏⠃⣸⣿⡇⢦⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢙⡏⣿⠀⡇⣹⣿⣿⠙⢿⣦⡼⠛⢣⡖⠁⣿⣗⢺⠝⣿⣿⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢌⠑⡿⢀⠧⣾⣿⡏⠢⡀⢛⢶⣖⠊⠀⢀⣿⣿⡚⠆⠹⣿⣇⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠄⠉⢀⢀⠀⡱⠆⡘⢲⣿⡏⢳⣅⠀⠀⢀⡈⠿⠿⣴⠛⣿⣏⢙⠀⢻⡜⠆⠀⠈⢿⢦⣄⣀
⠀⠀⠀⠀⡄⢰⡚⡍⠂⠀⠀⠀⡠⠍⠀⠠⣿⣿⠁⢠⠙⠻⢦⡀⠑⠀⠈⠔⠁⣿⣿⣅⠧⠀⠹⣄⠀⠀⣿⣸⡇⢸⠁
⠀⠀⠀⠀⠑⡀⢳⡘⢆⠀⢀⡤⠀⠠⠦⣼⣿⣁⡦⠀⣁⣀⢾⣿⣷⢤⣤⢶⡂⡟⢿⣿⠴⢃⠀⠈⢦⣬⠃⣸⠇⢸
⠀⠀⠀⠀⠀⠐⣄⠙⢦⣱⡞⠀⠀⠣⣼⣿⠉⣯⡶⡿⠏⣼⣿⡿⣿⢂⣫⠿⣴⡇⠘⣿⣷⠊⡄⠀⠀⠙⣔⡟⠀⡞
⠀⠀⠀⠀⠀⠀⠈⠃⢄⡙⠀⡀⠁⣜⣿⠏⠀⡇⠘⠀⢐⡾⢻⡇⠿⣃⠀⠟⢽⠀⠀⢹⣿⣏⢰⡄⠀⠀⢸⡡⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣉⣾⣿⠃⠀⠀⣧⠄⡰⠏⠀⠘⠁⠀⠻⢂⡀⡾⠀⠀⠀⢻⣿⣦⠘⡄⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⠯⡽⠁⠀⠀⢀⡷⠮⢁⡐⢦⣤⠶⣶⢾⣯⡧⡗⠀⠀⠀⠀⢻⣿⣆⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠠⣈⣞⣿⠿⢄⠖⣠⠛⠡⡝⠲⢤⣈⣉⣉⣉⣛⣹⡝⠒⠞⢣⡜⣁⠖⠀⢿⣿⣗⡊⣢
⠀⠀⠀⠀⠀⢀⠐⠀⣽⡟⢃⣳⡌⠶⢠⡜⠁⢶⡤⠤⠆⣽⠉⠹⣯⠙⠛⢶⡳⠆⣤⣉⠶⠂⣼⠻⣯⣥⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠻⣾⣀⢉⡇⢸⠃⠎⣁⠶⣀⣀⠤⢿⠀⢰⠏⠑⠢⣼⣱⠛⣤⢉⡴⠐⠇⢀⠈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⢤⠇⡸⣉⠖⢈⡰⡥⠄⠰⠟⡦⠛⠶⠤⠤⡿⢡⡛⠰⢏⣐⠀⡗⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠠⠥⠉⠰⠦⠀⠆⠠⢰⠌⠱⠏⠠⠲⠴⠗⠠⢌⡁⠮⠉⠦⠼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⢦⣤⡠⣤⣤⡤⣤⣤⣴⣦⣀⡤⣦⣤⣤⣤⣤⣤⣽⣦⣤⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠤⣤⣷⣀⣡⣼⣿⣤⣼⣿⣤⣤⣿⣯⣤⣼⣥⣬⣿⣤⣿⣧⣼⣤⣿⣤⣧⣤⣥⣽⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⢿⣿⠟⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡀⠀⠀⠀⠁⠀⢀⣠⣾⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⢶⣶⣶⣦⣤⣀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠛⠛⠉⠉⠉⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠟⠛⠛⠛⠻⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣀⣀⣀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃⠀⣠⣶⣿⣿⣿⣶⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⠿⠛⠉⠀⠈⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠋⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⢻⣿⣿⣶⣤⣤⣴⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⢠⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠒⢐⣶⣶⡿⣷⣒⡆⠐⠂⠀⠀⠐⠀⠀⠐⠒⠂
⠀⠀⠑⢠⡄⠀⠼⡇⢸⡷⠀⠀⡄⠈⠀⠀⢀⠀⡼⠀⠀⠀⠀⠀⠐⠀⠘⠉⢿⠈⠉⠀⠂⠀⠀⠀⠀⠀⡸⠦⣤⠴⢇
⠀⠀⠀⠀⠐⠒⠓⠉⠋⠚⠃⠂⠀⠀⠀⠀⠘⣄⢣⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠈⠂⠀⠈⠀⠐⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠀⢀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠐⣀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠁⢂⣰⠉⢧⣀⠈⠀⠀⠀⠀⠐⢄⡁⠶⢈⡠⠄⠀⠀⠠⠀⠀⠀⠀⢸⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⢠⠘⠧⡙⠂⢛⡠⠃⡠⠀⠀⠀⠀⡀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠰⣞⣆⡈⢿⠁⣰⣳⠆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠙⠒⠀⠈⠀⠐⠊⠁⠀⠀⠀⠀⠀⠑⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠂⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⣀⡀⢀⢠⢠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⢠⠠⢤
⠀⠀⠀⠀⠈⠀⡿⠩⡇⠈⠀⠀⠀⠀⠀⠀⠐⢀⡠⠧⠀⠆⠀⠀⠀⠀⠈⡇⠀⠀⣀⣀⠀⠀⢸⠃⠀⠀⠀⠀⢸⡆⢸
⠀⠀⠀⠀⠀⢠⡚⠚⢢⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠬⠁⠀⠀⠀⠀⠀⢸⡄⣐⣿⣿⢂⢀⡟⠀⠀⠀⠀⠀⢸⡧⡞
⠀⠀⠀⠀⠀⠘⠄⠀⠜⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⢂⠀⠀⠀⠈⠸⣇⠈⠛⠟⠁⣰⠗⠃⠀⠀⠀⠀⢸⡓⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⡇⠀⠀⠀⠀⠈⠀⠘⠂⠀⠀⠐⠃⠀⠁⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂
⠀⠀⠀⠀⠢⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⠄⠄⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⢀⠯⠼⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠒⣋⠉⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠊⠇⠀⠀⠀⢤⡀⠐⠒⠀⣠⠄
⠀⠀⠀⠀⠎⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠠⠀⢀⠘⠶⠃⡀⠀⡄⠀⠀⠀⠀⠀⠚⠑⠂⠱⠂⠀⠀⠐⠐⠀⠋⢈⠁⠀⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⣈⠁⢁⠈⢠⠇⠀⠀⠀⠀⠀⢠⠀⠀⠰⠈⠀⠀⠀⡀⢂⠀⠊⠀⠂
⠀⠀⠀⠀⠀⠦⢤⠤⢚⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠰⠆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡁
⠀⠀⠀⠀⠀⠈⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢄⠀⠰⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣁⠁⠀⠀⠀⠈⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣓⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢡⠀⠠⢠⣬⣤⡤⢤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⠀⠇⠀⠀⠀⢷⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠰⠹⠃⠿⣿⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⠀⣬⣷⠀⠀⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠂⠀⠀⠀⢀⣠⣤⡘⠰⠇⠀⠀⠀⠀⠎⡷⣄⡀⠀⠀⡼⠀⢷⡀⢄⣰⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⡀⠦⠀⡈⠉⣷⠀⠀⠈⠀⡇⠀⠂⠤⠜⡀⢮⠓⢨⠁⠐⣆⣸⠃⠀⠉⠗⢠⣎⢀⡠
⠀⠀⠀⠀⠀⠀⠁⢀⣬⠄⡻⠤⠁⢰⢠⠔⠉⢀⣄⠀⠀⢻⠀⠀⠂⠳⣾⠄⠈⠁⠀⠀⣽⣇⣀⢀⡀⠠⢃⠿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣴⢀⣠⡴⠆⠢⡘⠛⠡⠀⢸⠏⠀⡐⠉⠀⠀⡄⡠⠀⢨⣩⠃⠻⠆⣹⢦⣍⠀⠈⡃
⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠋⠙⢉⡀⠤⣀⠈⠁⠀⠒⢉⣲⣿⡣⡀⡠⢀⡇⠀⠐⢥⣊⠇⠀⡔⠁⠘⠟⠟⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⢤⡄⣰⡊⠁⠲⠬⠭⠅⠤⡄⠉⢀⠀⠙⠺⣵⡿⠁⠀⢠⠏⠉⠀⠜⠀⠀⡈⠲⣷⡀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡯⡤⠀⠀⠀⠀⠀⢰⢡⠠⠘⠃⡠⠀⢸⡇⠀⣰⠆⠀⢠⠂⠚⠀⠠⠂⠀⠀⠹⣤⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⡜⡶⣡⠃⠀⠀⠀⣠⠚⠈⠀⠻⠃⠀⣀⣾⣪⠷⠯⠄⠀⠻⠀⣿⠃⢤⣧⠀⢀⠀⢈⣹⣿⠧⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠼⣱⡣⠆⠀⡐⢉⡁⠀⠀⠀⠀⣁⠀⡟⡣⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠤⡌⢼⣿⡟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢀⡾⠻⢼⣢⣋⡁⠀⢀⣞⡢⠏⠁⠀⠀⠀⠀⠀⠀⠑⠒⠶⠢⣴⣶⣷⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡤⡤⡄⢀⠀⠠⢠⣼⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠿⠿⠛⠀⠀⠛⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠂⠀⠀⠀⠀⠀⣠⠖⠲⣄⠠⣖⠒⠒⠒⢒⡆⠀⠀⠀⡦⠴⠤⣄⢀⠤⣄⣀⢶⡄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⡇⠀⠺⠥⠤⡄⠹⠀⠏⢩⡄⠀⡤⠄⡇⢠⢤⠈⠇⠑⡌⢡⠊⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⠀⠙⠛⠒⢲⡀⢈⡀⡇⡆⢣⠱⠴⠃⡄⡆⢸⡤⢀⡆⠀⡇⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣰⡒⢒⣊⡠⠞⠀⠧⠇⠈⠓⠶⠚⠁⠷⠤⠔⠋⠀⠀⠙⠚⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣇⢸⠩⠀⢰⠋⣉⠆⢰⠋⡉⠳⡠⡖⠒⠲⢤⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢅⠀⡆⢸⠀⣉⠘⡌⠀⠋⣴⠃⠀⢘⢢⠀⢣⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⢸⠀⡇⢸⠀⡟⡄⢻⠀⡷⡈⢢⡇⠘⠋⢀⡎⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠈⠳⠃⠘⠚⠁⠘⠋⠓⠃⠙⠊⠱⠶⠖⠋⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠴⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣄⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣑⠲⠦⠤⠤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠉⠚⠒⠒⠒⠒⠲⠶⠄⠀⠤⠤⠄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⣤⣤⣤⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⣉⢉⡀⠀⠀⠁⠀⠐⠀⢀⣨⣤⣤⠀⢸⠀⠄⠁⠀⠸⠿⠿⠿⠷⠠
⠀⠀⠀⠀⡀⠀⠁⠀⠐⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠦⠶⠰⡴⠀⠀⠀⠀⠂⠀⠨⠍⠩⠍⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⢦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠴⠒⢀⡄⠀⠠⡀⠀⠀⠀⠀⠐⠋⠈⠋⠀⠀⠘⠀⢸
⠀⠀⠠⠀⠄⣄⣠⣀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠄⠉⢀⡈⠁⠀⢀⡀⠉⠀⠀⠠⠄⠀⠂⠀⠀⠐⠐⠒⠒⠂⠀⠐⠀⠈
⠈⠈⠁⠉⠈⠈⠁⠁⠁⠀⠠⠀⠀⠘⠂⠀⡉⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠒⠅⠀⣤⠁⠂⡠⠂⠀⠀⠀⠂
⠀⢄⣀⡑⢦⡴⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠠⠀⠀⠀⠀⠐⠀⢠⢀⠁⠀⠀⠠⠀⠀⠀⠠⠀⠸⢸⠶⡇⡿⡄⠁⠠⡄
⠀⢀⡤⣤⣤⠶⠤⠼⠁⠈⠀⠀⡄⠀⠉⠀⠘⡇⠈⠀⠀⠀⢀⣡⣀⠈⣈⣄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠂⠁⠀⠋
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢠⠀⠀⠁⠴⠋⢳⣄⣸⠀⠉⠻⠉⠙⢁⠔⠀⠐⠲⢀⠐⠀⠀⠀⠄
⠀⢀⠀⠀⠐⠘⠂⠀⠀⢀⠀⠀⠧⠀⠔⠀⠶⢲⡤⠤⢄⠈⠁⠐⣂⡀⠢⠀⠁⠈⠀⠈⠁⠉⠀⠀⢦⠀⢠⠒
⠀⠉⠉⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠃⠐⡂⠠⠅⠀⠀⢆⠀⠐⠂⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⢐
⠀⠈⠂⠘⠀⠀⠐⠅⠀⢃⠀⠀⠀⣀⠀⠀⠀⠀⠆⠈⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠖⠀⠰⠄⠀⠀⠤⠀⢀⠄⠀⠁⠘⠀⠄⠈⢀⠀⠀⠂⠀⣀⣀⡀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⣤
⠀⣀⠀⠠⣤⠄⡀⢀⠀⠈⠀⠁⠠⠀⢳⠃⠀⠀⠀⠀⠀⠀⠀⣼⠄⠨⢹⠀⡴⠚⠁⠠⢀⠀⠀⠀⢀⣈⡁⠀⠀⢀
⠀⠁⠀⢱⣶⡶⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠁⠀⠀⠡⠀⠘⠓⠒⠋⠀⠧⠀⢥⡀⣤⠁⠀⡠⣶⡒⣠⣔⢤⣤⠀⠠
⠀⢐⠀⠸⣛⡇⠀⠏⠀⣁⢈⠀⠀⠁⠀⠀⠈⠐⠈⠀⠀⠈⠀⠈⠉⠉⠀⢸⢂⡲⢒⡄⠀⠀⠀⠓⢻⡳⢚⠷⣸⢁⡀⢼⣇
⠀⠨⠄⣀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠐⠃⠠⢱⣶⡆⠄⠀⠁⢰⠒⠀⠒⢂⠈⢉⡥⠟⠉⡅⠉⠙⠀⠁⠚⠛⠉⠀⠋⠁⢀⡻
⠀⠀⠀⠀⠡⠀⠀⣀⣂⠋⢀⣠⡀⠀⠀⠀⢸⣿⣷⢀⣤⠀⠀⠠⠀⠀⢀⠀⠾⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠘⢦
⠀⠠⠤⠤⠤⠥⠀⠉⠁⠁⠈⠀⠁⠀⠀⠀⣺⡟⣿⠀⠉⠀⣉⠀⠀⢀⢘⠀⠀⠀⠀⠂⡃⠀⠀⠀⠀⠀⠀⢀⠀⡅⠀⣠⠟
⠀⢠⣤⣤⣤⣤⣄⠀⠹⣦⠎⠀⠀⠀⠀⠠⠉⠁⠁⠠⠀⠀⠈⠃⠉⠁⠁⠀⠀⠀⠀⠀⡅⠀⠀⠚⠛⠀⠴⠿⠂⠉⠀⠉
⠀⠀⠘⢛⠋⣿⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡎⠀⠂⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠂⡀⠐⠀⠰⠨⡀⠐⠳⢆⡀⠀⣀⣀⣠⣠⡀⠀⠀⠂⠁⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠛⠁⠉⠁⠀⠙⠂⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠀⠀⢰⡄⢀⢉⠛⠋⣉⡀⢰⠆⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⡄⠀⠀⠀⠀⠀⠑⢤⣆⠀⠀⣰⡤⠊⠀⠀⠀⠀⠀⢠⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⡸⡯⣴⣦⡽⢇⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⣠⢶⢟⣶⣾⡿⣿⣶⣾⡳⡦⣄⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⡁⢰⡿⣷⣿⡇⢸⣿⡾⢿⡅⢸⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡇⢸⠁⣿⣿⡇⢸⣿⡿⢈⡅⢸⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢉⣩⣿⣽⣇⠸⣿⡇⣾⣿⠇⣸⣷⣿⣉⡙⢷⣄
⠀⠀⠀⠀⠀⣀⣀⡠⠤⠴⠾⠟⠁⢠⣯⣤⡄⠉⠙⢿⡦⡤⢤⢴⠿⠋⠁⢠⣬⣽⠀⠙⠿⠷⠦⠤⠄⣄⡀
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠘⡏⢿⣷⢴⣾⡦⡿⣱⣮⢞⣰⠷⡖⢿⡿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢘⡯⣯⡽⣷⣌⢡⡆⢰⣾⢞⣳⣿⠇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢚⠛⣛⣿⣽⣯⢸⡆⣿⣗⣿⢹⠧⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣀⣃⣻⢤⣻⣹⡷⣏⡖⣟⣭⣤⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⣙⣶⡢⡿⢸⡇⢺⢳⡇⡅⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡉⡍⠫⠃⢸⡇⠙⠫⣵⢛⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠙⠛⠓⠒⠛⠓⠒⠚⠛⠋⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣄⠀⠀⠀⠀⠀⣀⠴⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠖⢉⣀⠀⠀⠀⠀⠀⠀⠑⢤⣄⡀⠀⢴⣂⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡐⠆⠀⢠⠃⡴⠉⠈⢦⠀⢶⡀⠀⠀⠀⠈⣿⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢰⠃⠀⠀⠀⠑⢬⣯⠲⣄⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⠀⣀⠀⠀⠀⠈⡁⠀⢙⠒⣼⠀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠉⠀⠠⠀⠀⠁⠀⢸⡞⠓⢤⡈⢃
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠢⣄⠀⠀⠀⢀⣠⠔⠣⣀⠀⢀⡧⠃
⠀⠀⣖⣩⡄⢨⣑⣩⣭⣍⡒⡄⠀⡸⠁⠀⠀⠈⠛⠝⠁⠀⠀⠀⠈⢆⠀⠀⢸⣉⣉⣉⣉⣉⣹
⠀⠀⠈⢎⣀⣀⣀⣀⣀⣱⣈⠁⠀⡇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡧⠤⠤⠤⠤⠤⢼
⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⡇⠀⣟⢀⣪⣆⣀⡀⠀⠀⣀⣀⠤⠤⢄⠇⢀⠋⠉⠉⠉⠉⠉⠹
⠀⢀⣠⠚⠒⠒⠒⠒⠒⠒⠒⣇⣀⣻⣲⠫⠝⠀⠈⡉⡉⠤⠹⢯⣂⣸⣀⣀⠗⠀⠀⠀⠀⠀⠰⣄⣀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠊⠉⠀⠉⠒⠊⠉⠉⠉⠋⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⢰⣿⡆⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢶⣢⣶⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣖⣶⠄⡀
⠀⠠⢀⡐⣒⠚⠍⠁⠀⠁⢳⣿⡞⠀⠀⠉⠹⠓⣀⢀⡀⠄⠀⠈⢧⣕⣿⣯⣃⣹⢁⣸⠗⠠⢶⠂⠚⣏⡀⣏⣘⣽⣿⣚⣵
⠀⠀⣀⣀⠈⠁⠐⠄⠀⠂⢾⢛⡇⠐⠀⠠⠂⠉⠁⣀⡀⠀⠀⠀⠀⠀⣿⣿⣖⡠⣶⣪⣿⣷⣈⣼⣿⣯⣲⢄⣰⣿⣿⠃
⠘⠓⠒⠂⠀⠀⠀⣀⣁⡤⠜⣍⠢⢤⣈⣀⠀⠀⠀⠐⠲⠚⠀⠀⠀⠀⠉⠻⢿⢷⠏⢡⡟⣯⣟⣯⡟⡄⠙⣶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡄⠑⠊⢀⡟⡄⠙⠒⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢶⢄⠃⠀⠀⠈⠻⡶⣶⠞⠁⠻⣶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠤⠄⢀⣑⣾⣭⣿⣊⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢹⢿⣤⡿⠏⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠇⢨⣿⡇⠘⣾⢃⠀⠀⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⢠⠀⡏⠀⠀⠀⢸⠀⣆⠀⠀⠀⠀⠀⠀⠁⠃⠃⢸⣿⣇⠈⠈⠈⠀⢸⠀⠀⢠⣄⠀⡇⠀⣰⠏⠙⠒⠂⢄
⠀⠀⠀⢀⣠⡤⠑⢮⡀⣶⣀⠥⠊⠠⣄⡀⠀⠀⠀⠀⠀⢀⡜⢻⣿⡟⢢⡀⠀⠀⠸⡗⠀⣼⣿⣿⣷⣶⣇⣠⠴⠓⠒⢤⡀
⠀⠀⠀⠀⠤⡤⠀⣀⣻⠭⣿⣁⠀⠤⠤⠀⠀⠀⠀⠈⠓⠋⠀⠀⠉⠀⠀⠙⠚⠁⠀⣈⡛⢻⣿⣟⢟⣿⣽⡷⠚⢧⡀⠀⠈
⠀⠀⠀⠀⢠⠘⠋⣩⣤⠛⣡⣍⠙⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠈⢉⣻⡿⣿⠿⠇⠀⠀⢱
⠀⠀⠀⠀⠘⠰⡄⠙⢿⣶⠿⠋⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⣰⡏⠘⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⣿⣿⣿⣛⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣻⣿⣛

⣿⣿⣿⣿⡸⣷⡙⢷⣜⣿⣌⢳⣌⠛⠿⣶⣦⣬⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⡿⣛⣵⣿⣵⣾⣿⢟⣠⣾⣣⠟⠁⣠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠹⣷⣄⠈⢿⣿⣷⣿⣷⣦⣄⡈⠙⢻⣿⠉⣿⣿⣿⣼⣿⣿⣵⠟⣋⣭⣿⣿⣿⣿⣿⣿⡟⢁⣤⣾⡿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠹⣿⣆⠀⣿⠇⣈⡉⠛⠿⣿⣦⡈⣿⣇⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⠟⠋⢉⣁⢹⣿⣴⣿⡿⠋⣴⡿⢹⣿⣹⣿
⣿⣿⣿⣿⢿⣿⡄⠹⣿⣼⡿⢸⣫⡩⠳⢦⡀⠙⢿⣿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠋⢀⡴⢚⡩⣹⡎⣿⣿⠋⣠⣾⠟⢁⣿⢣⣿⣿
⣿⣿⢿⡿⣧⡻⣿⣆⢻⣿⡇⣿⢸⢹⣞⡎⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡪⡾⡿⠆⡇⣿⣧⣾⠟⠁⢀⣾⣧⣿⣿⣿
⣿⣿⡎⣿⡜⢿⣼⣿⣷⣿⣇⣿⢸⡶⣫⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣭⣻⣿⡆⣇⣿⡿⢋⣠⣾⣿⣿⢟⣽⣿⣿
⣿⣟⢿⣜⣿⣄⠙⢿⣿⣿⣿⣾⣆⣶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⣼⣿⣿⣿⣿⣿⢟⣫⣴⣿⢟⣿⣿
⣿⡛⠿⣿⣿⣿⣷⣦⣽⣯⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣿⣿⣿⣿⣿⣟⣵⣿⣿⣿
⣿⣉⣥⣤⣭⣽⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⣄⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⣦⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣭⣉⣹
⣿⣥⣶⣶⣾⣿⣭⣭⣽⡳⠆⠀⠀⠀⠀⢸⠀⠀⣿⠊⢳⡀⠀⠀⠀⢠⠏⢸⣇⠀⢈⠇⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣯⣿
⣿⡿⢛⣽⣿⣯⣭⠴⠂⠀⠀⠀⠀⠀⠀⠈⢧⡀⣿⠇⢠⠇⠀⠀⠀⢸⡀⢸⡿⢀⠞⠀⠀⠀⠀⠀⠀⠀⠲⢦⣭⣿⣿⣯⣍⡻⢿
⣿⣴⠟⣫⣿⡿⠁⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠛⠒⠉⠀⡠⡤⠀⠀⠙⠒⠛⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠙⢿⣿⣿⣷⣍⣻
⣿⣷⣿⣿⡟⣠⠞⢋⣽⠿⠛⠉⣿⣷⢦⣄⡀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⢀⣠⡶⣾⣟⠉⠛⠿⣭⡛⠶⣌⢿⣿⠿⣿⣿
⣿⣾⣿⣿⣿⠃⣰⡟⠁⠀⢀⡼⠛⡏⠀⠈⠙⠳⠦⣤⣀⣀⢴⣿⠤⣀⣀⣤⠴⠞⠋⠁⠀⢻⠻⢦⡀⠀⠈⠻⡄⢹⣿⣿⣾⣿⢻
⣿⡿⡏⢸⣿⠀⡟⠀⠀⠀⠉⠀⠼⡏⠀⠀⠀⠀⠀⠀⠀⢉⡟⡏⢿⠁⠀⠀⠀⠀⠀⠀⢀⣿⠆⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⣧⣾
⣿⣿⣷⠀⢻⡆⠀⡀⠀⠀⠀⣀⣼⣿⡇⠀⠀⠀⠀⠀⢀⡞⢀⣷⡘⣧⠀⠀⠀⠀⠀⠀⢼⣿⣦⡀⠀⠀⠀⢀⠀⣸⢏⡟⢸⣿⣿
⣿⣿⣿⣧⡈⢻⡀⣷⣄⡐⠋⢉⣴⣿⡇⡄⠀⠀⠀⢀⡾⢠⠏⡏⢳⡈⢧⠀⠀⠀⠀⢠⣸⣿⣮⡉⠁⢀⣠⣿⢰⠟⠈⢠⣿⣿⣿
⣿⣿⣿⣿⣷⣦⣁⣙⣮⣉⣽⠟⠋⠈⣿⡇⡄⠀⠀⡼⣱⠏⠀⡇⠀⠳⡌⣇⠀⠀⢤⢸⡟⠀⠙⢿⣭⣩⣽⣃⣀⣠⣴⣿⣿⣿⣿
⣿⡻⣿⣯⡙⠛⠛⣿⣿⣿⣿⡛⠻⣦⠘⣿⠁⠀⢸⣷⡁⠀⠀⣇⠐⠀⢹⣾⡄⠀⢸⡿⢁⡴⠟⣛⣿⣿⣿⣿⠛⠛⢉⣿⡿⢛⣿
⣿⣿⣎⢻⣧⢠⣾⡿⢿⣟⢿⣿⣷⡘⣇⢸⣧⡀⠘⣿⡇⠀⠄⠉⠠⠀⣸⣿⠀⢀⢾⡇⣾⢡⣾⣿⡟⣿⡿⣿⣷⣀⣾⠏⣴⣿⣿
⣿⣿⣿⠀⢻⣿⣿⣷⣺⣿⣼⣿⢹⣿⣿⣾⠁⠙⣆⢻⣷⠈⠀⠳⠀⠀⣿⡇⡴⠃⢸⣷⣿⣿⡟⣿⣷⣿⣁⣿⣿⣿⠏⢰⣿⣿⣿
⣿⣿⣿⡇⠀⣿⣇⢿⣿⡟⢻⣷⣴⣿⣿⠏⠀⠀⠘⣾⣿⡀⠗⠀⠪⢰⣿⣿⠁⠀⠀⢿⣿⣿⣤⣿⡟⢿⣿⠓⣾⣿⠀⣸⣿⣿⢹
⣿⠹⣿⣷⡀⠹⣿⣿⣟⣁⣐⣬⣽⣿⣿⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣿⣿⣯⣥⣑⣞⣻⣿⣿⠇⢠⣿⣿⠃⣼
⣿⣷⣾⣿⣿⣦⣬⣝⣛⣻⠿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣇⠀⠀⠀⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⣛⣋⣭⣤⣶⣿⣿⣷⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠔⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣿⣇⣀⣏⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢨⠇⠀⠀⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣏⣌⠥⠀⠈⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠈⠀⢸⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⣀⣠⡤⠀⠀⠀⢀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣀⣤⣤⣶⣾⣿⣿⡿⢃⣗⠀⠄⢼⡄⢃⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠀⢀⣄⣾⠀⠈⠁⣤⡀⣀⡭⠴⣙⡻⣿⣿⡿⡟⢿⠿⣿⡟⠋⠁⢸⣿⠀⠂⣿⡇⢠⡄⠈⠑
⠀⠀⠀⠀⠀⠀⢀⣿⡀⣼⠿⠓⠁⡠⠟⠪⠴⠀⠩⠈⢁⣄⢂⡡⠈⠉⠉⢠⠀⢀⡄⠁⠊⠲⠄⢄⣠⠈⠙⠛⠻⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠸⡟⠁⠡⠀⠀⢊⠑⢠⡆⡀⠠⣀⠀⠈⠁⠀⠀⡠⠐⠈⠀⡄⢸⣇⠀⠀⠀⠀⠀⠘⠩⢂⠀⠀⡉
⠀⠀⡀⠀⡀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠸⡧⠓⠀⡀⠀⠀⠀⣸⡆⣀⠄⠀⣰⣷⣾⠿⠄⠲⢂⠀⠂⠀⠄⢸⠀⠀⠇
⠀⠀⠃⠠⠀⠀⠤⠈⠐⠒⠀⠾⢿⣿⠿⠆⠀⠈⠉⠀⠚⠃⠾⢧⠈⣉⠈⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠁⠲⠄⠀⢰⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠒⢂⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠋⠀⠙⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠿⠀⢑⠂⠀⠤⢤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⠀⠀⠈⠀⠀⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⢳⠈⢦⡀⠦⣤⣡⡎⠙⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠈⢆⢀⡝⠓⣯⣤⠃⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠏⠐⠢⣄⠀⠀⠀⠀⠉⢹⣾⡇⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⣾⠀⢻⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⣾⣿⣿⣿⣿⣿⠟⡙⠻⣿⡟⢻⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣼⢿⡿⢛⢏⡤⣪⡿⢤⠙⠃⠸⠿⢛⣒⠦⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⣇⠀⠀⡇⣼⢀⡿⢩⠞⢉⠔⣿⠀⠀⣠⠀⡴⠈⣧⠀⢀⠀⢉⣹⣆
⠀⠀⠀⠀⠀⠀⣾⠴⠖⠒⠲⢤⡇⠀⠀⠀⢸⡆⠀⣿⠀⠀⡇⡏⠀⠀⠀⠀⠕⣻⠋⠀⣠⠃⢀⡇⠀⣟⠲⣼⣦⣷⣻⣿⠳⣄
⠀⠀⠀⠀⠀⠀⣷⢀⠤⠶⠤⣄⢡⠀⠀⠀⢸⡇⠀⢹⠀⣀⢇⡋⠀⢀⡤⠶⠋⠉⣖⠋⠳⣄⡞⢄⡀⢸⠀⠈⠁⠙⠷⠃⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⢸⠻⢭⣓⣒⣺⠇⠀⢸⡿⢿⢸⠃⠀⣿⠀⠀⠀⠀⠘⣴⠿⣽⡃⣰⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⡘⠀⠀⠀⠀⠈⢣⣀⡾⠀⢸⡿⠀⠀⣿⠀⠀⠀⠀⠀⠈⠀⠸⣍⣽⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡇⠀⠀⣿⠀⢀⡀⠀⠀⠀⠀⡼⠁⢹⡀⠐⠒⠶⠒⠂⠀⢱⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢸⠘⡇⠀⢰⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠃⠀⠀⢹⠀⠈⣧⠀⠀⠀⢰⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⠀⠀⠀⢸⡄⢠⡟⢳⡀⠀⣼⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣤⣤⣤⣀⣇
⠀⠀⠀⠠⠤⣤⣾⣤⠇⠀⣾⣀⣀⡀⠀⢀⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇⣼⠀⠀⢻⣀⠇⠀⠀⡼⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿⣁⣤⣤⣄⡈⢻⡀
⠀⠀⠀⠀⣀⣤⡞⠁⢠⡀⢸⡉⠉⠉⠉⠛⠛⠛⠛⠒⠒⢿⠀⠀⠀⢸⠍⣧⣄⣀⣀⡟⢀⣴⠶⠧⣄⣀⣀⣠⣤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠹⡄⡇
⠀⠀⠀⢰⣿⡯⠕⠊⢹⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⡾⣰⠃⢀⡼⠛⠁⣾⣵⣶⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠁⠠⣯⣀⣀⣠⠽⠻⣏⠐⡻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠠⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠁⠀⠀⠀⠀⠈⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⢀⡀⠘⡄⠀⠀⠀⠀⠀⢀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠄⠂⠀⠁⠀⠃⠨⠐⠀⠀⠒⠚⡑⠈⢤⠊⠡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣈⠀⠀⠀⠐⣄⡄⠀⠀⠢⡀⠀⢀⠀⠀⠀⠔⠀⡀⠁⠐⣰⠃⣀⣀⣡⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠈⠉⢁⠃⠠⣌⠉⡀⠠⠀⠈⠂⠤⢠⣤⠤⠔⠊⠀⠐⠊⠉⠉⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠤⡂⢀⠲⣌⠓⣷⡦⠴⠦⠤⠀⠠⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠙⠊⠡⠄⠀⠀⠀⠈⠁⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⣒⢒⣒⣒⣀⣀⣈⡁⠀⠐⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠠⢀⣄⠸⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⢰⡿⠁⢰⣶⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣿⣿⠚⠣⠾⢇⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⠇⠀⠠⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢀⡘⣿⣦⣴⠶⣤⡄
⠀⠊⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣮⡇⠀⠈⠁⠀⠲⠈⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣄⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢧⡀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠠⣿⢀⠁⣀⠙⣿⣿⣿⡞⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠛⢿⣿⣅⠉⠀⠀⢀⣰⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠓⠀⢙⣿⣿⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡇⣀⣀⣨⣇⡁⠀⠀⠀⠀⠀⠀⠰⣶⡄⢀⠀⠀⠀⠀⠈⣾⣿⣦⣤⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠙⣿⣿⣃⠈⠀⠀⠀⠀⠀⢠⣀⠹⠟⣾⠗⠀⠀⠀⠀⡟⠻⠿⠿⣧
⠀⠀⠀⢙⠿⣧⡀⠀⠀⠀⠀⠀⠙⠇⢀⡑⢆⡈⠀⠀⠀⢠⠀⠠⣿⡇⢀⣠⣦⣄⢠⡤⠚⠁⠀⠀⢀⣈⠇
⠀⠀⠠⡎⠀⠈⠁⠀⠀⠀⠀⠀⠋⢠⠞⢿⣄⠁⠀⠀⢠⠄⢀⡄⠉⠀⠻⠙⢻⠁⠀⠀⠀⠀⠀⠀⠀⡝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢷⢄⠈⠀⠀⠀⡆⠠⢅⣹⣤⡄⠘⣶⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡄⠈⠀⠀⠀⠺⠀⠀⠀⠸⠿⠃⠀⡾⠁⠀⠰⠦⠀⡀⠀⠀⢢⡀⠄⠄
⠀⠀⠀⠀⡜⠀⠉⠀⠀⠀⠖⣠⠗⠈⠀⠀⠀⣀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠃⠀⠸⠿⠦⠀⡀⠋
⠀⠀⠀⠀⠱⠀⠀⠈⠋⣀⡰⠀⠀⠉⠀⢠⠞⠉⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣀⣄⢰⣶⣦⢔⠰⡒⣤⠄
⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⡈⢉⣻⢿⠟⠛⢻⢦⠋⡉⠈⠓⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢠⣿⡶⣷⣶⣾⠢⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣀⣀⣀⠄⠀⠀⠀⠀⠿⠿⠋⠉⢿⣰⡿⠿⢿⣄⠁⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⡈⢡⡆⠀⠀⠹⠀⠐⠠⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣀⢦⣴⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠀⠀⠈⠃⠀⠻⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟⠀⠠⠶⠀⠀⠀⠠⣎⠉
⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣧⡀⠈⠁⠀⠀⠀⠀⠀⢹
⠀⠀⠐⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠐⠶⠀⠀⡈
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠶⢀⠀⢀⠀⠈⠋⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢠⣤⠂⠀⠀⢠⡄⢠⡄⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢀⠀⠀⠊⠻⠇⠄⠸⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⡀⠀⠀⠈⠉⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡁⠐⠀⠉⠀⠐⠀⠂⠐⠷⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠂⣠⡆⠀⠀⠀⠀⠀⠀⠀⠰⠞⠀⠀⠀⠈⠁⠀⡄⠂⣈⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⠁⠒⠀⡉⠀⠀⢀⡄⠀⠀⠦⠀⠀⠰⠟⠀⣀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⣀⣐⠨⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⡀⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠹⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣴⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠻⠙⠻⠟⠙⠟⠻⣟⠟⢻⡿⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣶⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⠔⠒⠒⠢⢤⡀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠉⠉⠙⠲⣄
⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠁⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠑⢤⣀⠀⠀⠀⠀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⣠⠖⠒⢄⠀⠀⠀⠀⠀⠀⠀⠶⠁⠈⠒⠊⠉⠀⠀⠀⢀⡔⠛⠳⣄⠛⠲⣄⠀⠀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠇⠀⠀⠀⠀⠘⠁⣠⠒⠑⢢⠀⠀⠀⠀⠀⠸⡄⠀⠀⡸⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣓⣶⣉⠀⠀⠀⠀⠀⠀⣠⢧⡀⠀⣸⣇⠀⠀⠀⣀⣤⠬⣭⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠠⢄⣹⡓⢤⣀⣀⣠⣤⡏⠀⠉⠉⠀⢹⡦⠖⠊⢱⠇⠀⢠⠎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠈⢷⠦⣄⣤⠤⢴⡟⠋⠀⠀⠀⡞⣧⠀⠀⡞⠀⡰⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸⣇⠀⠀⠀⢰⠧⢼⠤⢶⠗⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⠘⢦⡀⢠⡏⠀⢸⡇⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢈⡇⠀⠀⢸⡄⠀⠀⠀⠀⢀⣠⡇⠘⣶⠶⠶⠶⢤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠒⠛⢻⠋⠁⠀⠀⠀⠉⠙⡏⠉⠉⡏⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠂⣾⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⠗⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠓⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠃⠀⠛⠋


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

⣀⡀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣀⡀⠴
⢩⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣫⣁
⠈⠉⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⣿⠞⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡙⠈⠉⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⢀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⣿⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡿⠇
⠀⢀⠠⠒⣀⡰⡖⠀⠀⠘⡀⠀⠰⢦⣀⠐⠢⠀⠀⠀⠀⠀⠊⣌⠋⣬⠁⠀⠀⠀⢀⠠⠒⣀⡐⡖⠀⠀⠈⠀⠀⠲⢦
⠀⠁⠀⠀⠀⠀⠀⠐⡇⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⠠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⣀⣤⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣄⡀⠀⢀⠀⡀⠀⠀⣀⣤⣤⣦⡄
⣩⡽⣻⣿⠟⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡽⣦⣍⡓⢄⢤⡤⣒⣩⡽⣻⣿⠟⠀⠀⠀⠀⡀⠄⢀
⣦⣼⡿⠃⠀⠀⠀⠀⠀⣤⡀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣶⠛⣩⢤⡙⠳⣦⣼⡿⠃⠀⠀⠀⠀⠀⣤⡀⢠⣄
⣤⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡞⠹⡜⡌⣦⣮⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃⠸⣟
⠾⠋⠀⠀⠀⠀⠀⠀⠀⠠⠃⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠩⠃⠘⠏

⣀⡀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣀⡀⠘
⢵⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⢧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣰⢵⣥
⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠿⠋⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡌⠀⠈⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠢⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂

⠀⠀⢀⡴⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢸⣯⣿⣁⠀⣀⣨⣿⠉⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢤⣴⣿⣿⣯⡤
⠀⠀⢸⣿⣿⣿⣿⡟⠛⠛⠿⢶⣶⣾⠛⠛⠛⠶⢤⣄⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠛⣈⠃⢀⣤⡀⠶⠀⠀⠉⢀⡍⣩⣡⣾⡀
⠀⠀⢸⣿⣿⣿⡿⠿⠶⠤⣄⣀⠈⠉⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠟⢡⡎⢀⡴⠿⠿⠶⣤⣠⢄⡿⣋
⠀⠀⢸⣏⣿⣿⣄⡀⠀⠀⠀⠈⠉⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⣿⣇⣀⣤⠀⢀⠀⠀⣀⣀⠀⠀⠀⠄⠀⠙⣤⢈⢠⣤⡀⠀⠀⠘⢘⠛⠁
⠀⠀⣿⣿⣿⠋⠉⠉⠉⠉⠙⠒⠦⢄⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣀⣿⣿⣿⣿⣾⠀⠆⢀⣀⠛⠟⠻⠧⡁⠀⢀⡠⠋⢀⠀⠙⣿⣇⠀⠀⠀⠘
⠀⠀⢻⣿⣿⠷⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢐⣒⣛⣻⣿⣿⣿⣟⣘⢷⣦⠿⠿⠷⣦⣤⣄⠀⠀⠀⠁⠀⠀⠙⠀⠀⣉⠀⠀⢀⣤⡍
⠀⠀⠈⢿⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣿⣿⣿⣿⣿⣿⢿⡿⠋⢀⣠⡀⠀⢘⣿⡂⠀⠐⡿⠷⠀⠀⠀⠀⠠⣇⠰⠟⠁
⠀⠀⠀⠀⠉⠛⠛⣿⣿⣿⣿⣿⣿⡷⠶⠶⠓⠚⠛⠛⠛⢉⣉⣭⣿⣿⣿⣿⣿⣿⣿⠂⠈⠛⠙⢌⣩⣿⣧⣤⡄⣀⢀⠀⠀⠀⠈⠛⠉⠋
⠀⠀⠀⠀⢀⣴⣿⡿⠟⠛⠉⠁⠸⣄⣀⣀⣀⣠⠀⠐⠊⠉⠁⢘⣿⣿⣿⣿⡏⠁⠙⢦⣤⣴⣷⡾⠟⠹⡇⠀⠀⠘⠉⠳⣦
⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⢈⡭⠋⠉⠀⠀⠀⠀⢀⡴⠿⠿⣿⣿⣿⣷⣀⣰⠞⠛⠋⡿⠛⣷⡀⠈⠠⣴⣶⡄⠀⢿⣄⢲⣿⣿⡄
⠀⠀⠀⣼⣿⣿⠖⠒⠀⠁⠀⠀⣠⠎⠀⠀⠀⠀⠀⣠⡴⠋⠀⢀⠔⣹⣿⣿⣿⣯⠀⠀⢠⣤⡄⠀⡿⣧⣤⠀⠉⠛⡀⣀⣾⣧⠀⢁⠀⠀⡀⠛⢦⡀
⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⣸⡯⠀⠀⢀⣠⠴⠛⠁⠀⠀⠠⠂⠀⠀⣿⠟⣿⣧⠀⠄⠹⠟⢏⡁⢀⣿⣿⣻⣶⠞⠛⠋⢨⡖⠂⠉⣷⣿⣿⣧⢄⣵
⠀⠀⣿⣿⣿⣷⠂⠀⠀⠀⠠⠈⠚⠓⡿⠋⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢸⡇⠘⢿⣆⣀⣸⣄⣠⣷⣾⠿⠟⠟⢠⠀⠀⠀⠀⠂⠀⠈⠉⣿⠙⣿⣷⣾⡆
⠀⠰⣿⣿⣿⣧⡀⣀⡴⠊⠀⠀⢀⠊⠀⠀⠀⠀⣀⠄⠛⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⠻⣿⡿⠯⠛⠁⠀⠘⠀⠀⠁⠀⠀⠀⠀⢀⠀⣸⡟⢠⡾⠗⠚⠃
⠀⠀⢿⣿⣿⣿⣿⣿⣆⣀⣀⣤⣾⠀⠀⣀⡠⠂⠀⢠⣤⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠷⣶⡦⢠⠄⠀⠀⠈⠁⠀⠀⠀⠀⣸⣿⣟⢲⣦⠄
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢀⣤⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⡾⠻⣥⣟⣳⠘
⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣉⣽⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⡶⢾⣟⡻⠿⢵⣦⠜⠿⠇⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⡀⠶⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠒⠲⡖⠀⡀
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠊⠁⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠖⠓⠒⠒⠒⠿⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠐⠆⠀⠖⠈⣦⠠⡄⠶⠴⠲⠊⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢄⣀⠜⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⢠⠀⠀⡤⠀⢰⡇⠐⠟⢠⠦⡀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⡆⠀⡄⡛⠉⠉⠙⠛⠛⡟⠒⠒⠒⠒⠒⠀⠐⠚⡇
⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠂⠀⠘⠀⣈⣤⣤⣶⡄⠘⢧⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠐⠾⣄⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠿⠍⡉⠛⠛⢳⠲⠶⠤⠤⠤⠄⠀⠀⡏
⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⡔⠉⠙⣿⣿⣿⣿⣿⣿⣇⠐⠂⠈⠁⣀⡠⠂⠑⡀⡀⠀⠀⠀⡀⠃⠀⠘⠀⢘⠁⠀⠀⠀⠻⣦⣄⡘⣄⢂⠀⠀⠀⠀⠀⠀⢃
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⡄⠐⠡⠀⠀⠀⠀⠀⢁⠀⢰⡆⠠⢘⡇⠀⠀⠀⢠⡏⠀⠈⠹⠃⠠⠠⢤⡜⣉⠀⠉
⠀⠀⠀⠀⠀⢶⠀⠀⠀⣀⣘⣧⠀⣴⣿⣿⣿⣿⣿⡟⢂⣠⡀⠀⠀⠀⠀⠄⠂⠀⢠⣴⠀⠈⡀⢸⣇⡔⠋⠀⠒⠒⠂⠈⠀⠀⡀⡀⠀⠀⠀⠤⠀⠀⣀
⠀⠀⠐⡀⠀⠈⠇⠀⠀⠀⠉⠙⡆⣿⣿⣿⣿⣿⣿⢀⡀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢀⢧⡽⠂⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠉⠒⠂⠒⠀⠀⠉⣤
⠀⣴⣿⣿⣦⠀⠈⠀⠛⢲⠀⠀⠡⢈⣿⣿⣿⣿⡇⠈⡀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠆⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠈⠁⠒⠀⣤⣀⡀⠀⢹
⠰⣿⣿⣿⣿⣷⠀⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡿⠽
⠀⣿⣿⣿⣿⣿⣿⠀⡸⢃⣠⠀⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢣⠃
⠀⠟⠛⠛⠛⢻⣉⢱⣧⣬⠛⠋⠉⠉⠉⠉⠛⠟⣿⣷⣶⣾⣿⣾⣿⣿⣻⣿⣶⣶⣸⣿⠀⡄⠚⢼⠀⠀⠰⠃⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣼
⠀⠠⡄⠡⠀⢀⡛⠉⠘⠙⠀⣀⠀⠀⠀⠀⢀⣀⠉⠋⠛⢻⣿⡶⠒⠛⠻⡿⠿⠶⢿⣿⠀⠘⠀⢸⠷⠠⠃⠘⡎⡐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡏⣞⠁
⠀⠀⠚⠓⢶⣤⣄⣤⣦⡤⠞⠛⢪⡵⡟⠻⢿⣿⣿⣶⡤⠞⠙⡁⠤⠀⠀⠀⠀⠀⠈⠉⠐⡂⠐⠿⠃⠀⠀⠄⠘⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣼⠿⠁
⠀⠀⠀⠰⠹⢿⣟⣿⡿⠓⠈⠑⠀⣠⣄⠀⠤⠦⠀⠈⠚⠓⢀⠀⠀⠀⢠⣢⠄⠀⠶⠖⠂⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠄⠀⠀⠀⠀⢠⣿⡦⠤⠶
⠀⠀⠀⢢⡀⠏⢡⡏⠵⠀⠠⡄⢀⣉⠀⠀⠀⠀⠀⠀⠀⠉⠏⠀⡀⠀⠀⠐⠂⠀⣼⡟⠀⣤⣄⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠶⠆⣁
⠀⠺⠊⠉⣭⡄⠁⠐⠀⠤⡀⠁⠀⠀⠉⠀⠀⠀⠀⢠⣀⢠⣄⠀⠀⢀⣀⠀⠀⠀⠈⠉⠀⠻⢿⣯⡛⠦⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⢋⣀⣀⡤⢤
⠀⠀⠀⠀⠛⠘⠛⠲⠦⠄⠀⢹⣶⣶⣤⣀⣀⢀⣠⣤⡞⠁⠉⠍⠉⠉⠉⠀⠤⠠⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⣶⢤⣄⣠⣶⠟⠿⠶⠶⠖⠋⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠛⣿⠛⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣿⢿⣥⢀⠀⠀⠀⠀⣀⣨⠥⠆⠒⠛
⠀⠀⠀⠀⠀⢤⠤⢄⣀⠩⠁⠒⠀⠀⠀⢀⠏⠐⠀⠀⠀⠀⠓⠒⠒⠒⠒⠀⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠀⠈⠙⠖⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠤⠄⠠⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣂⠠⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠵⢻⠆⠁⣸⠇⠀⠀⠀⠀⠀⠀⡀⡀⢀⣀⣬⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡫⠉⢁⡭⡝⡆⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢈⣿⠟⠉⠋⠙⠻⣾⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⡀⠀⠀⠀⣨⠟⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠒⠲⢤⣾⠃⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠰⡀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣩⡄⣀⠲⢠⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠭⠖⠁⠀⠀⠀⠀⠰⡞⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣦⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⣠⣾⣿⣴⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⡁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠾⠛⣻⣿⣿⡿⠿⠋⠋⠀⠀⠀⠀⢻⡉⠀⠀⠀⠀⠀⢿⣹⣇⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠱⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⠀⠀⡀⠀⠢⡈⠁⠀⢢⠂⠀⠀⠀⠀⠀⢿⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡰⣱⡀⠀⠀⠀⠀⠀⢸⠟⢻⠷⡀⠀⠑⠀⠀⠀⠈⠄⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡅⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠤⠂⠀⠠⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠈⠒⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠜⠃⢶⠦⢄⠀⠁
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠉⠉⠉⠁⢹⣿⣿⠄⠀⠀⠀⠀⠸⣴⡢⢀⣼⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢀⡀⠀⠀⠀⢸⣿⣿⠀⠀⠘⣅⠁⠃⠘⡆⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣛⣿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠀⣿⠎⠈⡀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠘⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⠀⠀⠀⠀⠀⠒⠐⠀⠀⠐⠒⠒⠒⠒⠒⠒⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⣾⣷⣜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣾⣿⣿⣿⣿⣦⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠛⠿⠟⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣦⣤⣀⣠⣿⣿⣿⣦⡀
⣀⠀⠀⢀⠀⠀⠠⡦⡀⠀⠀⠀⠀⣴⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣦⠀⠀⠀⠀⢀⠴⠄⠀⠀⣀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢴⣦⡀⠈⣿⠃⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠹⡟⠁⢀⣰⡦⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠢⠄⠀⠀⠸⣿⣾⣿⡟⢻⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡄⢻⠟⢫⣿⣷⣿⠇⠀⠀⠀⠄
⠀⠀⠀⠄⢄⠀⠀⣀⡻⡄⠀⠉⠉⢁⣼⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣼⣤⡈⠉⠉⠀⢠⡟⣀⠀⠀⡠⠠⠂
⠀⠀⠀⠡⠀⠀⠀⠻⠿⠿⠂⠀⠀⠘⠿⢷⡄⢿⣦⣤⣴⠀⠰⣤⣤⣾⠇⢱⡟⠿⠃⠀⠀⠐⠻⠿⠟⠀⠀⠀⠄
⠀⠀⠀⠀⠈⣢⣤⣤⣤⡤⣤⣤⡀⠀⠀⣼⠁⠀⠛⠋⠁⣾⣇⠉⠛⠋⠀⢈⡇⠀⠀⢠⣤⡤⢤⣤⣤⣤⣔⠁
⠀⠀⠀⠀⠀⠻⣯⣭⣀⡀⠰⠶⠋⠀⠀⠘⢦⣤⣤⠀⠀⠛⠛⠀⢠⣤⣴⠞⠁⠀⠀⠙⠲⠆⢀⣀⣭⣽⠟
⠀⠀⠀⠀⠀⠀⢢⣍⣉⣁⣀⣤⣶⣿⣿⣿⣾⣧⠛⣖⡂⠀⠀⣐⣺⠟⢸⣾⣿⣿⣿⣶⣤⣀⣈⣉⣩⠔
⠀⠀⠀⠀⠀⠀⠑⠮⣛⡻⠿⠿⠿⠿⠿⠿⠟⢯⣀⠁⠀⠉⠉⠘⠀⢠⡾⠿⠿⠿⠿⠿⠿⠿⢟⣋⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠒⠚⠛⣻⠏⣹⣷⣤⣀⣀⣠⣴⣏⠻⣟⠛⠓⠲⠖⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠋⠐⠚⠛⠛⠻⠛⠛⠿⠿⠛⠟⠛⠻⠛⠓⠂⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠒⠋⣉⡡⠤⠤⠤⠶⠶⠶⠶⠶⠶⢤⣄⢠⡤⠶⠶⠶⠶⠶⠶⠤⠤⠤⢭⣉⠙⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣹⣦⣄
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣶⣾⣿⣿⡿⣆
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣄⡀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣈⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣟⢻⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⣿⣿⡿⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⢀⢨⢿⠠⠶⣾⣁⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣼⣿⡎⠀⠀⠀⠐⣿⣿⡄⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢆⠀⣿⣿⣷⢀⢀⠀⣨⣿⣿⡇⠀⠀⢀⡴⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⡆⠈⢛⣾⣾⣾⣷⣿⣭⣤⣁⡀⢀⢾⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠟⠙⠛⠉⠁⠉⠉⠙⠹⣿⣻⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢊⣶⠔⢀⡀⠀⠀⠀⠀⠀⣀⢠⣿⣤⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⣤⡴⠊⡐⡂⣺⢿⣿⣾⣤⣴⣆⡄⣄⣮⣷⣿⣭⠭⢳⠀⠈⠳⣤⣄⣄⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠚⠓⢚⠍⠁⠀⡰⠃⠉⢉⣹⣿⠻⠿⡏⡿⣟⠿⢻⣙⣓⠚⠃⢣⠀⠀⠈⠛⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⢁⣞⣯⣽⡃⠀⠀⠁⠀⠈⠁⣀⣠⡏⠿⣭⣛⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠈⢠⡙⣆⠋⠹⠓⠶⠠⠖⠒⠚⠛⠉⠁⠘⠄⢀⡀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢠⠁⠀⠈⣇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠇⠀⠰⠀⠘⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠞⣽⠟⠒⠀⠄⠄⠀⠀⠤⠠⠐⠊⠈⠙⠀⢲⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⢇⡾⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣧⠀⠀⠻⣝⡂⠃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠿⠾⠧⡀⠀⠀⠀⠀⣀⣀⡥⠘⠏⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⢀⡰⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⠶⠤⢄⡀⣄⠤⡤⠾⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠄⠒⠦⠤⢤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠖⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⢉⡉⠉⠉⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⡓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⡞⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣼⡃⠀⠀⠀⠀⠀⢠⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⡖⠉⠀⠑⠢⠤⠤⠤⣀⣸⠇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣄⡀⠀⠀⡠⢶⡑⠒⠖⠒⣉⡉⠉⡆⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣀⡀⠀⠀⠀⠀⠀⠀⠈⢹⡴⠣⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⠫⠭⠤⠊⠳⢄⡤⠊⠀⣹⡀⣇⣨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡾⠭⠭⡷⢉⡿⢭⣉⠙⠢⣿⠁⠀⢹⣗⡢⠤⣄⣀⣀⣀⡤⢾⠑⡄⠀⠀⠀⠀⠀⠀⠀⢠⠦⠔⠉⠳⠿⣿⡖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⠖⢉⣧⡜⠓⢦⡈⠀⠀⠸⡄⠀⠀⢳⠉⠓⢦⣄⡀⠀⠀⠀⣇⢱⡀⠀⠀⠀⠀⣀⡠⠃⠀⠀⠀⠀⠀⢨⠃⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠩⡇⠀⠀⠀⡇⠸⣼⣷⡀⠀⠀⠀⠀⡇⠀⠀⠀⢇⠀⠀⠙⢍⢳⠀⠀⡌⠓⠺⠿⠿⡉⠉⠁⠀⠀⠀⠀⠀⠀⡴⠃⠀⢀⣀⣤⣤⢽
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢰⡇⠀⠀⢰⠁⡰⠃⢹⣷⡀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠘⡟⠦⣀⡇⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⡠⠊⠀⣠⠾⠛⠉⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠣⣀⣸⡇⠀⠀⡸⠋⠀⠀⠀⠁⢱⡀⠀⠀⢸⠀⠀⢠⢸⠀⠀⠀⢠⠇⠀⠀⢣⠀⠀⠀⠀⠀⠑⠦⠤⠴⠒⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢇⠀⣿⡸⠀⠀⠀⢸⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡄⡎⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡆⣿⠇⠀⠀⠀⡼⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠹⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣇⠀⠀⢰⠃⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⡀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢻⣆⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⡎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡜⣀⣀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠶⠟⠋⠁⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⣀⡦⢄⠠⠔⢗⡤⠤⢚⠏⠀⠀⠉⠙⠒⠒⠢⠤⠒⣷⡤⠤⠤⠾⠤⣀⣀⣀⣀⠤⠔⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠾⠉⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠲⠒⠒⠒⠒⠒⠒⠚⢧⣀⣀⣀⣀⣀⣀⣀⣀⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⠛⠛⠛⠉⠉⠑⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏⢀⣀⣠⣤⣤⣤⣄⠀⠀⠀⠀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⣷⠒⠁⠀⠀⠈⠁⠀⢄⠙⠋⠢⣄⠐⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣻⣷⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⡈⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣾⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⣀⠀⠙⡄⠀⠀⠀⠈⠙⠿⢿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣾⣿⣿⣿⣿⣿⢿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠻⡿⢿⣿⣿⣿⡇⠀⠀⠀⣀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣀⣤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡐⠁⠀⠉⠛⠛⠃⣤⣶⣿⣿⣿⣧⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣧⣄⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣀⣠⠴⡚⠛⠉⠉⠁⠀⠀⢠⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⡿⠻⢿⣿⡿⠿⣿⣿⣿⣷⣾⣿⡿⠿⢳⡀⠀⠉⠅⠀⢠⣦⠄⡀⠀⠀⠀⡴⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢀⠀⠉⢰⡄⠇⠙⢛⠉⠁⠐⠀⠀⢇⠁⠀⠀⠀⠀⠀⠿⠿⠃⠀⢀⡜⣃⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡤⣷⡘⢗⣤⣆⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠂⠀⠏
⠀⠀⠀⠀⢤⠤⠶⠂⠀⠀⠈⠁⠹⠇⠸⣾⡿⡌⣿⠸⠀⠀⠀⠀⠀⠰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⢁⠀⢀⡘⣀⠅⢀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⢀⡄⠘⢣⡭⠀⡈⢡⣯⠄⠀⠀⠀⠀⠀⡀⢄⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⢀⡀⢀⠀⣰⣇⣀⠀⣴⠀⠀⠙⠁⠄⠀⠐⣆⠘⠿⠐⠟⣧⢶⠄⢠⣦⠶⣻⠀⡘⢷⣾⠁
⠀⠀⠀⠀⢀⣾⠟⠁⣼⣿⡟⢁⣶⡟⠀⠀⠀⠀⠀⠀⠄⠈⠀⣘⠛⡒⠃⠀⢴⡜⡙⠎⠏⠀⠉⠂
⠀⠀⠀⢠⣾⠏⠀⢀⣿⡟⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⣿⣟⡄⠀⣼⣿⡏⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦
⠀⠀⢰⣿⣿⣿⣐⣿⣿⡇⠈⣿⣧⠀⠀⠀⠀⠀⠀⢰⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡇
⣦⡀⣼⣿⣿⣿⠙⣿⣿⣿⠀⠙⣿⡄⠀⠀⠀⠀⠀⠈⣷⡀⠉⠑⣲⣄
⠻⣿⡘⣿⣿⣿⣦⣿⣿⣯⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠈⢻⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠠⠂⡄⡰⠋⣁⡠⢒⡄⢠⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⡆⠐⡷⠀⠘⡃⠀⢸⡅⠂⠙⠅⢨⡗⢀⣸⡤⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠢⣤⠀⠘⡧⠀⢐⣁⣈⠁⠀⠏⠀⠀⢀⠬⠿⠂⠀⣙⠃⠤⣾⠐⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠢⣇⠀⠘⣂⣤⣶⣾⠟⠁⠀⠀⠓⠶⠴⠶⠦⠴⠖⠛⡡⠄⠈⠱⣦⣍⡐⠏⠁⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢦⠀⣊⣴⣾⣿⠿⠿⠛⠀⢃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠳⢀⠀⣿⣿⣿⣷⣬⡛⢀⢠⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢦⠄⣡⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⡀⠀⣀⠠⠎⠀⠁⠈⠙⠻⣿⣿⣦⡑⣀⣠⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠦⡠⢀⣾⣿⡿⣝⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡏⢽⣿⡏⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⢹⣿⣷⣌⠣⡀⢠⡇
⠀⠀⠀⠀⠀⠀⢀⠀⢡⡀⡀⢡⣿⣿⣟⠓⠄⠧⢌⡄⠀⠀⠀⠀⠀⠀⠄⢸⣏⠁⢨⢽⡇⣰⠀⠀⠀⠀⠀⠐⢀⠰⡶⠃⠀⣿⣿⣿⣧⠙⠁⠀⣀⠁⠠
⠀⠀⠀⠀⠂⠀⠀⠆⠀⠸⢠⣿⣿⣿⡏⠁⠀⠀⠘⠂⠃⡆⣀⢠⡀⡄⠐⠈⠳⣧⣼⠟⣁⣄⡁⠀⠀⠀⡀⠰⡍⠃⠁⠀⠀⣿⣿⣿⣿⣧⠘⠟⠛
⠀⠀⠀⠀⠀⢄⠀⠀⣻⢁⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠠⡀⠀⠀⠀⠀⠙⠂⠙⠚⠓⠉⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⢰⠤⠶⠅
⠀⠀⠀⠀⠀⠀⠷⣰⠇⠼⠿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⡆⠀⢦⡀⢀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡄⣄⣠⡔⠂
⠀⠀⠀⢀⡠⠃⢁⣠⡴⠦⢶⡤⣌⡙⢿⣷⣤⡀⣀⠀⠀⠀⠀⠉⠈⠁⠀⠈⠀⠀⡀⠉⠀⠀⠀⠀⠀⠁⠲⢶⣤⣤⣴⠿⠋⢉⣉⣉⣉⠙⠻⣷⠈⠉
⠀⠀⣰⠎⣠⠖⠋⠠⠋⣠⠞⠀⠀⠉⢢⡘⣯⠁⠀⢀⡀⠄⠒⠀⠀⠀⠀⠀⡈⠈⡣⠔⠃⠸⠒⠐⠤⣤⣄⣀⣨⢟⣡⠶⠋⠉⠁⠀⠀⠉⠲⣈⠻⡛⢃
⠀⡴⣧⠞⠁⠀⠀⠀⠨⡗⡀⠀⠀⠀⠀⠑⣌⣷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣷⠞⢳⡄⠀⠀⠀⠀⠀⠀⠂⠈⣆⢱⡖⠃
⣾⣾⣇⢠⣆⡀⠀⠀⠀⠈⠀⠂⠂⠤⣰⡆⠈⡓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⣿⡶⢺⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⡀⠹⣆
⠈⠙⠳⢷⠚⠁⠗⠂⠆⠶⠐⠂⠂⠓⠉⠙⠉⢡⠀⠁⠒⠠⢄⡀⠀⠆⠀⠀⠈⠁⠀⠨⠘⠀⣀⠔⣒⣉⣤⣴⠁⣯⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⣈⠆
⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⡍⢩⣍⠐⡟⠄⠀⠀⠀⢤⣀⠀⠁⠀⡼⢁⠜⢉⣴⣀⢸⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠢⠤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠐⠊⠘⡇⠘⠀⠀⠀⠀⠀⠀⢀⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⢠⠏⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡄⠀⢀⣴⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠀⠀⠀⠀⢀⠐⢛⣿⣿⣿⣦⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⠌⠛⠻⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⡴⢖⣉⣽⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⣟⠛⣤⣀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣋⣥⠐⢻⣿⣿⠖⠠⣀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⣸⣉⡻⠦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠶⢈⠜⠁⠀⣠⣿⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⣧⢿⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣤⠀⡸⠟⣿⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡿⠝⠊⠀⢐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⣉⡗⢸⡇⣷⠀⠀⠀⠀⢗⠀⠀⠀⠀⠀⠘⡃⠀⣰⡿⠿⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠉⣇⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⠑⠚⠓⠴⡴⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠜⠃⠀⣀⠀⠀⠀⣠⠤⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣌⣛⣦⡤⢤⣀⣀⣀⣀⠀⣀⣀⡀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣳⣿⠻⢷⣽⠭⣿⠼⣯⣉⠉⠳⣦⠀⠀⠀⠉⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣿⠉⠻⢦⣉⠲⣄⡀⠀⠉⠓⠒⠛⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣹⠆⠀⠀⠀⠉⠳⣬⡙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢻⡄⠀⠀⠀⠀⠀⠀⠙⠶⣬⡑⢦⡀⠀⠀⠀⣤⡄⠒⢲⣬⣭⡵⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢾⠛⡛⢻⣿⢦⡀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣌⣩⣭⣵⠶⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⣽⠀⢃⠸⡇⠙⢧⣙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡏
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠙⢦⣈⣹⡄⠀⠀⠙⠳⢦⣝⣛⣡⠤⠤⣤⣤⣛⡿⠛⢳⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠙⠉⠉⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠰⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢢⠀⣿
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣅⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣸⣿⠀⢸⡇
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠻⣆⠀⠀⠀⠀⠀⠈⢿⣿⠏⠀⢸⡇
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠉⠉⢻⡿⠛⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢧⡉⠛⠛⣷⠀⠀⠀⠀⠀⢻⡋⠉⣠⡶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⡎⠙⠷⣾⣧⣦⣀⣤⣀⣀⣌⣷⡞⠋⡆⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠁⠀⠀⠈⠉⠈⠛⠈⠉⠉⠉⠉⠀⠀⠁⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠻⣷⡶⢤⣀⣀⠀⠀⠀⠀⣀⣠⣷⣶⡿⣋⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠲⠶⠶⣶⠛⠉⠁⠀⠀⠀⠈⠉⠑⢲⡖⠋⠉⠁⠀⠀⠀⠉⠉⠓⣶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠘⢷⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⣷
⠀⠀⢀⡼⠃⠀⠀⠀⠀⠐⢷⡄⠈⢉⣉⣉⣉⡀⠶⢀⡀⠘⠃⢀⡀⠤⣄⣉⣉⣉⡉⠁⢠⣾⠀⠀⠀⠀⠀⠘⢇
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠛⠛⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠟⠛⠿⠿⠿⢾⡇⠀⠀⠀⠀⠀⠀⠘⣦
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠙⠶⢤⣤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡤⠶
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢉⡏⠁⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁⠈⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠿⠓⠉⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠓⠚⠿⣦
⣤⣤⣤⣤⣤⣤⣤⣽⣷⣶⣶⣮⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣵⣶⣶⣾⣭⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠟⠻⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣀⣤⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⢶⣄⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠈⠄⠈⠁⠈⠀⠀⠄⠀⣣⣶⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣶⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢸
⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠉⠀⡽⣿⣿⣿⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠋⠻⣿⣿⣿⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⢿⣻⡧⠉⠀⠀⠻⣼⣿⡿⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡧⠈⠀⠀⠘⢤⣿⣿⠃⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡤⠠⡤⢴⣿⡛⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣦⢄⣠⢴⣾⡟⠉⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣞⣿⡅⢀⣼⣯⣤⠲⠤⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡳⢿⡔⢒⢽⡟⠩⠑⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢽⠛⠀⠘⠇⠀⡇⢸⠇⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⠀⣨⠿⣤⡜⠣⡀⡇⠠⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⠌⠀⠀⠇⠸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠺⠋⠹⣀⠿⠷⣴⠟⢳⠄⠂⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠀⠉⠁⠉⠈⠉⠈⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠉⠀⠀⠉⠁⠀⠈⠉⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢆⣤⣾⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠄⡀⠀⠀⠀⠙⠿⣦⣿⣆⢀⣴⠶⣦⣠⣴⣾⣿⣿⣿⣷⣦⡀⣴⣿⣿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠛⠋⠉⠸⠇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡿⠟⠁⠘⠿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠀⠀⠀⠀⢣⣼⢟⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣯⣟⠁⠐⠚⡃⢀⣼⣿⠟⠀⠀⢰⣾⣿⣋⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶⣶⡆⠈⠙⣻⣷⣶⡭⢔⣫⠿⠋⣱⠀⡀⠀⠙⠇⠉⠉⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠙⡿⠀⢠⣶⠗⠚⣷⠤⠯⢹⠃⠀⠈⣷⠈⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋⠀⠀⠀⠀⢀⠏⠸⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣎⣴⣤⣼⣁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣿⣿⣴⣴⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢁⣂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⡟⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣛⣶⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢉⢹⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣿⣿⠿⣿⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠩⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⠿⠏⠙⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⣶⢰⡄⠀⡶⣤⠀⢠⡆⢠⡆⡄⢶⣰⡆⣶⠀⣴⢤⢰⡖⡆⠐⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⣿⠀⢻⠀⣇⡿⠀⠸⡇⠸⣇⠇⠸⢻⡇⣿⠀⢿⠸⢸⡏⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⠀⠀⠀⣠⠀⣠⢠⠀⡄⠀⠀⣀⠀⣄⢠⠀⢀⠀⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣆⠀⠀⢸⠀⣿⢸⠀⡗⠀⠀⣿⠀⣿⢸⠀⢻⡄⣿⢸⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠈⠀⠉⠈⠀⠁⠀⠀⠉⠀⠉⠈⠀⠈⠁⠈⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠙⢿⡿⠁⠀⣀⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⡀⠹⣿⣷⣶⣶⣶⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⠛⠿⠿⠿⠿⢛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⠉⠉⠀⠉⠻⣿⣿⠀⠀⠀⢠⣴⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⣰⣿⣿⣿⡿⠋⠉⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣿⡿⠋⠀⠀⣠⣿⣿⣿⡟⠁⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣿⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢰⣧⡀⠀⠀⠀⣀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⡿⠀⠀⠘⠿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣏⣀⣽⣭⣭⣭⣭⣯⣯⣭⣽⣍⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣽⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⣯⣭⠽⠷⠶⠯⢭⣽⣲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣫⠷⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣫⣾⣥⣤⣴⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⣷⢻
⠀⠀⠀⠀⠀⠀⣰⢫⣾⣿⣿⠉⠉⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠁⠸⣿⣿⣷⣾⢸⠇
⠀⠀⠀⠀⠀⠀⣇⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢷⠄⠀⠀⢻⣧⣿⡏⢿⠤⣄
⠀⠀⠀⠀⠀⠀⠈⠲⣽⣛⢻⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣿⡆⠀⠀⠀⠀⠘⣿⣬⡿⠋⠙⢮⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⣿⣿⡟⠇⡇⢸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡞⣿⠈⠻⠛⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠱⠀⠃⣸⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⣇⢸⠀⠀⠀⠋⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⣰⢏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡽⣾⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⣫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢿⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡄⠀⠀⠀⠀⢠⣇⣀⣀⣀⣤⢀⡀⠀⠀⠀⠀⠀⠀⢸⡟⠧⠤⢄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢸⡼⡇⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠠⡙⢦⡀⠀⠀⡠⠤⠘⠛⠛⠛⠳⣮⢳
⠀⠀⠀⠀⡴⣫⢵⡶⠶⠶⣽⡣⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠱⣖⠉⣷⠀⠒⠒⠒⠂⢰⡿⣾
⠀⠀⠀⡼⣹⠁⠎⠀⠉⠰⠈⣷⢰⡝⢦⡀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠹⣄⠈⠁⢻⡀⠀⠀⠀⠀⢸⡇⣧
⠀⠀⠀⡇⣇⠀⢆⠀⠀⢀⠃⣸⢷⠟⢮⡻⣦⡀⠀⠀⠀⠀⠳⡄⠀⠠⠉⠳⣄⡀⠑⠀⠀⢀⡰⠋⣹⢏⡇
⠀⠀⠀⠹⣜⢦⣈⣒⠶⢛⣴⢏⡞⠀⣞⢵⣿⣿⣷⣦⣤⣀⣀⣹⡆⠀⣷⡀⠀⠀⠉⠉⠉⠉⠀⠎⠙⢮⢷⡀
⠀⠀⠀⠀⣨⢷⣨⣿⠼⡟⣷⢹⠀⠀⠈⠓⢲⡭⢛⣿⣿⣿⣿⣿⣀⠀⢻⡈⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⢧⢳⡀
⠀⠀⠀⠀⡇⣿⣁⢤⠂⡇⠘⢮⡳⡴⢶⣚⣩⣾⣿⣿⣿⣿⣿⣿⣿⠀⠈⣷⠄⠀⠀⠀⠀⠀⡌⢳⠀⠀⠈⣧⢳
⠀⠀⠀⠀⣇⢿⡤⠶⡁⠇⠀⢸⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠈⡄⠀⠀⢸⣏⣇
⠀⠀⠀⠀⢸⡸⣧⠚⣷⡄⠀⠀⠉⠙⠛⠟⠀⠉⠛⠻⠿⠟⠋⠀⠌⢧⣀⣀⠄⢀⠀⠉⠑⠒⠲⠶⡇⠀⠀⣸⣿⡸⡄
⠀⠀⠀⠀⠀⠹⣭⣛⣻⡏⠈⠓⢾⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⡼⠀⠀⡰⢻⣿⣧⢧
⠀⠀⠀⠀⠀⠀⠀⠈⡟⣇⠀⢀⠆⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠏⠤⠶⠚⠁⣸⣿⣿⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡹⣶⣮⣤⣤⣤⡤⠔⠃⡜⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⣿⣿⣿⣇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⢺⢆⣿⣿⣷⠀⠀⠁⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢠⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣖⣭⣾⣿⣿⡿⢘⠃⠀⠀⠶⠁⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⢀⣠⠀⠀⠈⣾⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢺⣿⣿⣿⣿⠿⢳⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⣿⣿⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣻⢿⣛⡵⠺⣜⢷⣄⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣿⣿⣿⣿⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠑⠮⣝⣓⠶⢤⣤⣀⣀⡀⠀⠙⠒⠀⠠⠤⠄⠒⢁⣀⣀⣤⣿⣿⣿⡟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠺⠭⠭⠭⠽⢿⣿⣿⣿⣿⠯⠭⠭⠟⢶⡹⡿⣋⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁

⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢸⡅⠀⠀⢀⡗⠀⠀⠀⣠⡤⢠⢀⢄⣠⣀⢄⢠⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⢢⠀⢻⢛⡛⢿⠃⡀⠀⠀⠙⢦⡀⢢⣾⡷⣿⣦⠂⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠣⢼⣯⣽⡯⠔⠁⠀⠀⠀⠀⠹⣦⣀⣹⣋⣠⡼⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢎⠉⠁⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⢀⣺⣿⣷⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠸⣶⣧⡶⠄⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡠⣼⠿⢿⢤⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⣯⡧⠀⠀⠀⠀⠀⠈⡜⢹⣄⣸⢱⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠈⠁⠈⠻⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣽⣷⣇⠀⠀⠀⠀⠀⢐⣛⡂
⠀⠀⠸⣿⠇⠀⠀⠀⢀⣿⣷⣾⣧⠀⠀⢀⡜⠻⡿⠛⡀⠀⠀⠀⣿⣽⣿⣯⡟⠀⠀⠀⠀⠹⣿⣿⡐⠄
⠀⠀⠀⣉⠀⠀⠀⠀⢸⢀⢠⢇⣭⡇⠀⠀⣵⣦⣶⣶⡉⠀⠀⠪⡇⠈⠛⢀⣶⡅⠀⠀⠈⠁⢸⡏
⠀⠀⠀⠉⠀⠀⠀⠀⠰⠻⠾⠾⠣⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠰⡹⣴⣶⣿⢿⠃⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣟⣛⣛⣛⣛⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠄⠀⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⢻⡋⠛⣛⣛⣒⡢⢤⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⡀
⠀⠀⠀⠀⣜⣀⠊⢗⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⡱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢈⣾⣹⣿⣶⣤⣁⢋⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠈⠑⠢⡤⠤⠀⢀
⠀⠀⣤⠒⠨⡙⠻⢾⣿⢏⠂⡀⢀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠈⠑⢂⠀⠀⣀⢀⠀⢸⡾⠀⠀⠀⠠⠀⠀⠀⣸⡆
⠀⠜⡈⠀⠴⣠⢮⠕⢀⠀⠉⣸⠩⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⣑⠈⢸⡇⠀⠀⠀⡄⠀⠀⢠⣿⠁
⠀⠑⠚⠵⣢⠄⡉⢀⠂⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠛⠉⠙⠛⠓⠳⠶⣾⡇
⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠀⠀⠀⠴⠶⡈⢦⠀⠀⠠⠀⢀⡠⠊⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣒⣒⡠⡞⠊⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠸⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡴⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣆⣾⣀⣸⣆⣆⣀⣀⣀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⢤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⡀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣷⣄⣤⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⡀⠀⣀⣠⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣄⣀⣀⣀⣀⣠⣤⣴⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠄⠀⠀⠀⣤⡤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠤⣶⠄⠀⠀⢀⠊
⠀⠀⠘⡂⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⣀⠀⠀⠀⠀⣠⡴⣦⡄⠀⠀⠀⣀⡤⠔⠂⢀⣯⡄⡀⠀⠀⠠⠦⠀⠀⠀⡠⠃
⠀⠀⠀⠈⢆⡀⠀⠀⠐⠀⠀⠻⠱⡂⣄⡀⠂⠨⢉⣓⣼⣏⣥⣼⣧⣉⣾⣴⡫⢛⠀⠀⠰⠬⠛⠊⠀⠀⠠⠀⠁⠀⠠⠔
⠀⠀⠀⠀⠀⠱⠄⢆⠀⠀⠀⠀⣠⣀⣀⣀⣀⣈⡠⠾⣤⣘⣿⣷⣿⣟⣃⣽⠷⣤⢤⢄⣠⣤⣄⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠙⢢⣿⣿⣿⡛⢻⣿⣿⣿⠂⠁⠉⠁⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⡿⣿⣿⣟⢿⢿⡋⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢣⣿⠿⣿⡿⢿⡗⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⠛⠂⣽⡀⠙⡑⠏⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⡿⠿⢿⣋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠑⣒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠋⠠⠄⠛⠋⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠺⠃⠀⠀⠀⠉⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠠⡄⠀⢠⡄⠈⠇⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⡼⢣⡀⠀⠀⠃⠀⠀⠀⣠⢶⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠰⡎⠀⠀⠙⠢⡤⠒⠂⠉⠉⡤⣣⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢴⣆⠀⠀⠀⠘⠢⠤⣀⢸⠀⠀⢠⠔⠺⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⣣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠤⢄⣀⣀⡇⠹⣷⡄⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡇⠀⠘⢾⣆⠀⠀⠀⢸⠀⠀⠀⠀⢱⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠚⠧⢤⠴⠊⣉⡳⢦⡀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⡀⠀⢻⠀⠀⠀⠀⠀⢠⠀⡒⠤⢽⣾⡃⠘⣄⠀⠀⠀⢀⠇
⠀⠀⠀⢸⠉⡩⠭⠛⠉⠉⠉⠁⠀⠉⠉⠉⠚⢿⡒⠠⠤⠰⠚⠢⠭⠿⠞⠙⠿⠉⠉⠙⠒⠶⣛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⣄
⠀⠀⠀⠇⠛⠒⠤⢤⡤⠤⠴⠒⠒⠒⠒⠒⠤⠤⠬⣦⣀⣤⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠬⠷⠄⠀⠀⠀⠀⣀⣠⢤⡀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⡩⠕⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠕⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⢸⠂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡒⣒⣒⣒⣒⣒⣛⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⠛⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣿⡿⣶⢶⡖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣺⣷⡏⠉⢸⢅⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣿⣯⡇⠀⠔⠶⣈⣁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⣿⡅⣿⡷⠀⠃⠀⠀⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⠀⣿⡇⣿⣷⣤⣠⠆⢿⡪⢈
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇⣿⣼⣿⣿⣿⡏⠀⣸⣦⡶
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣳⣿⣿⣿⣿⣷⣾⣿⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠈⢽⣯⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣦⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣦⣄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠻⢿⣿⣿⣿⡏⣾⣿⣿⡟⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⣿⠁⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⣧⠀⠀⠀⠀⢀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣿⣿⣧⣀⡀⠀⢸⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⣿⡿⠿⠇⠀⠛⠿⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠖
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⡄⢀⠂
⠀⠀⠀⠀⠀⠀⠑⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠂⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡐⡠⠁⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠡⡀⠆⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠠⠊⢀⠔⠉
⡔⠀⠑⢄⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⠔⡠⠊
⠀⠐⢄⠀⠢⡀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣐⠁⠈
⠀⠀⠀⠈⠢⢈⠢⡀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠲⢤⡀⠀⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠑⠰⣄⠀⠀⠀⠀⣀⠔⠁⠀⣀⣤⣤⡄⠀⠀⠀⢠⣾⣿⣷⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢀⡔⠁⠀⢠⡾⠛⠋⠉⠙⠀⠀⠀⠀⠀⠀⠉⠻⣇⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠟⢀⠄⠀⠀⠀⢀⠀⡠⠐⠂⠐⠠⡀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣀⣠⡁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸⠀⠀⢰⣳⡆⠀⢸⠀⠀⣶⣷⠀⠀⡁⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⠋⠀⠀⠙⠦⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⠀⠘⡀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⢀⠁⢀⡀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡀⠀⡸⠀⠀⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠋⠉⡀⠀⠀⠢⣀⢀⡀⠄⠀⠑⠄⣀⣀⠤⠁⠀⠈⠙⠓⠀⠀⢸⠁⠀⠀⠈⢆⠀⠈⠂⠑⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⠀⠀⣸⠀⠀⠀⠀⢀⠃⣄⠀⠀⠀⡈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⡏⠀⠀⢀⠇⠀⠀⢀⡴⢃⠌⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄⠀⠘⣿⣿⣷⣄⣀⣀⣄⡀⢀⣀⣄⣠⣤⣾⣿⣿⡿⠀⠀⢀⣾⡒⠒⠊⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣿⢦⡀⠹⣿⡿⠟⠛⠛⠻⠟⠛⠛⠛⠿⣿⣿⣿⡟⠁⠀⡠⠋⠛⠓⠒⠈
⠀⠀⠀⠀⣀⠀⢀⠎⢀⠔⠋⠀⠀⠹⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠀⣠⠊
⠀⠠⠊⠁⠀⠈⠉⢰⠃⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⢀⠖⠉⠉⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠎⠀⠀⢀⣴⡂⢠⠀⠀⢅⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠘⠢⠔⠿⠠⠁⠈⠦⡀⣠⠀⠀⠀⠀⠀⡇⠀⠈⠉⠀⠒⠒⠒⠀⠒⠀⣀⠀⠠⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠠⠀⠂⠀⠀⠀⣀⣀⡀⠤⠤⠄⠒⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠐⠒⠂⠈⣉⢁⣀⠀⠤⠤⠄⠒⠒⠚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠉⢉⣁⠀⠠⠤⠀⠒⠂⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠦⣤⣄⣀⣀⣀⣤⣴⠊

⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠳⠦
⠀⠀⠀⡴⠉⠀⢴⠄⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠈⠀⠀⠀⠸⡄⠛⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠂⢒⠚⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣒⣒⣒⣦⣀
⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⠀⣀⠈⡇
⠀⠀⠀⠑⢤⡀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠒⠒⠒⠒⠒⠒⠂⠀⠤⠤⠤⣄⠀⠀⠀⡇⠀⠂⠀⠀⠠⠅⠀⠂
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠇⢀⣁⣈⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⢦⠀⠀⠸⠀⢀⠀⠐⠀⠀⠀⠀⠀⢠
⠀⠀⣀⡀⣀⣀⣠⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⢀⢤⡀⡰⣤⠀⣤⢶⢀⣴⠀⠏⢡⠜⡷⠋⠀⠀⢠⠀⠀⡄⠈⠀⠉⠀⠀⠀⡀
⠀⡧⠈⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠠⢈⠀⠀⠀⣦⠌⠀⡾⢁⣎⣰⠇⠰⠤⠊⠃⠘⠉⠀⠀⠀⠀⢀⠖⠁⠀⠀⢸⠀⠀⠑⠠⠤⠶⠤⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⡂⠀⠈⠀⠳⠠⠊⠑⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠴⠮⠀⠀⠀⠀⢸
⠀⠈⠀⠒⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠡⠀⠀⠀⢸⣀⡀⠤⠀⣄
⠀⠀⠀⠀⠀⠀⢰⢧⢀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠔⠀⠀⠀⢸⠋⡠⠀⡆⠡⡱⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠶⠆⠠⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠂⠄⠀⢀⠓⠏
⠀⠀⠀⠀⠀⢠⠞⠋⠉⢳⠹⠄⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣈⣀⣀⣈⣴⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⡔⠈⠂⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠰⠒⠒⠠⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢆⣀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⡀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠒⠀⠈⠙⢢⣀⠄⢀⣀⠀⠀⠘⠢⠤⠤⠤⠚⢀⣀⣀
⠀⠀⣠⠞⠉⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⣈⣉⡉⡩⠭⠛⠋⠢⠀⠀⢀⠜⠁⠀⠀⠀⢠⠊⡤⢀⠀⠀⠈⣲⡀⠀⠀⠀⢰⠁⠀⠀⠀⡿⢝
⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⡔⢘⠈⡡⠫⡃⠈⡀⠄⠀⣐⠀⡂⠀⣆⠀⠀⠀⢀⠔⠁⠮⠀⠀⠑⢀⠔⠡⠋⠀⠀⠀⢸⠀⠀⠀⠀⡧⠘
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡄⠀⡇⣂⠊⢀⠄⠀⢁⡁⠐⠠⠅⠲⠁⢄⠛⠒⠒⠒⢻⢇⡒⠤⢀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠰⠁⠁
⠀⠹⡄⠀⠀⢀⣀⠀⠀⡏⠉⠈⠇⠀⠐⠀⠁⠀⠈⠁⠀⠀⢀⠔⠁⡠⠔⢂⠑⡀⠀⠀⠀⡐⠉⠁⢀⠐⠚⢳⠀⠀⡠⢆⣠⡏⠀⠙⠒⠚⠃
⠀⠀⠈⢦⣰⠉⡀⠱⡀⠘⡈⣀⡄⠀⠀⠀⠀⢀⠤⠒⠀⠀⠙⠄⠈⠔⡨⠐⢀⠐⡄⠀⠀⡑⠠⠀⢀⠕⢀⠎⠀⠌⠀⠀⠉⠁⠊
⠀⠀⠀⠀⠙⢄⠈⢄⢠⠀⠑⠉⠈⠂⠢⡀⠀⢃⠀⠀⠀⠀⠀⠈⢀⠈⠔⠈⢀⡤⠊⠀⠎⠁⠀⠀⠁⡔⠁⠀⠀⢆⡀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢢⠀⠀⠀⠀⠀⠈⢆⡀⠂⠁⠀⠀⡜⣫⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠡⠀⢀⠠⠐⠀⠉⠀⠀⠀⠀⢀⣡⠏⠐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⠔⡒⡀⢀⡆⠀⠀⢀⢀⡄⢀⠀⠀⠀⠀⠀⣠⠁⢸⡖
⠀⠀⠀⠀⠀⠀⠀⢀⣷⣷⣟⡴⠟⠛⠛⠓⠀⠀⡈⢿⡇⢸⣷⣠⠤⠖⣻⡿⢶⡟⠁⠀⢈
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠁⠀⠀⠀⠀⣀⣀⣈⣉⣸⣿⣌⠿⣿⣿⡿⠋⣠⠟⠈⠢⣄⣉⣤⡑⢐⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢀⣠⠴⠾⠛⠛⠛⢃⣉⣿⣿⣟⠻⢶⣦⣶⣿⣿⣤⣤⠞⠋⠉⠋⠖⢿⠴⠿⠉⠉
⠀⠀⠀⠀⠀⠀⢀⡝⢦⣴⡿⠲⢶⣿⣇⠴⠀⠖⠉⣽⣿⣷⣦⣿⣿⣿⣿⣿⣧⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣗
⠀⠀⠀⠀⠀⠀⢘⠷⣌⠻⣧⡄⢾⡧⣰⡶⠂⣠⣤⠴⢲⣿⡓⠛⠛⠉⠋⠋⠽⠿⢿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣴⣶⣾⡿⠟⠙
⠀⠀⠀⠀⠀⠀⠘⢄⠙⢦⡈⠙⠾⢿⡋⡷⠿⠇⠀⠀⠀⠀⠙⠳⣖⠒⠒⠒⠛⠓⠒⠦⠭⣉⣛⠛⠛⠛⣛⣉⡉⠉⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠻⣶⣄⡀⠿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣽⣷⣶⣦⣧⣤⣤⣤⠚⠋⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠶⠭⠭⠛⠛⠛⠻⠟⠁⢀⡠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠐⠒⠂⠈⠁

⠀⠀⠀⠰⣶⣶⣶⣦⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⡄⡖⠀⠰⢶⡆⠀⢐⣲⡂⢤⡤⣤⣤⣤⠤⠤⡀
⠀⠀⠀⠘⣛⣭⣉⣛⣃⢀⡀⠀⠀⠠⠀⠖⢀⠠⠁⠄⢓⠀⠀⠀⢄⣀⣀⣉⠈⠁⠀⠘⠋⠂⡂⠁
⠀⠀⠀⢸⣿⣿⣿⡿⡉⠪⠭⢉⣙⠛⣻⣦⣤⠜⠲⠶⣾⠿⠿⠶⣾⣿⣿⡟⢻⣿⣿⠿⢽⣹⠁⠀⠂
⠀⠀⠀⢸⣯⣍⣽⡢⣛⠿⣷⣶⡤⣷⣏⡐⡂⣿⠐⢯⣀⣀⣛⡀⠉⢉⣉⣉⣀⣥⣴⣾⣿⠇⢀⣿⡃
⠀⠀⠀⢸⣿⣿⣇⣀⣫⡿⠀⠁⠀⣀⣻⡿⠧⢰⣴⣖⠛⠿⠿⠟⠛⠛⣛⣛⣛⣩⣭⣿⣴⣀⣾⠿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⡷⠆⠀⠀⠒⡯⠟⠫⠛⣯⣻⣿⡟⣷⡞⠀⠰⠈⠙⠛⠛⣿⣿⣿⣿⠉⣉⣙⡁
⠀⠀⠀⠸⣿⣿⣿⣿⠉⠁⠀⠀⢀⣬⠽⢃⡀⣿⣿⣇⣼⣿⡟⠉⠓⠂⣶⠻⠟⢿⣿⡿⣿⣯⣴⣻⡃
⠀⠀⠀⠈⣿⣿⡿⣛⡂⠖⢶⣤⣬⣵⣿⣿⣿⡿⣫⢷⣹⠿⠁⣀⡈⠉⠁⠀⠀⠈⢻⡿⠂⢤⡝⢿⡇
⠀⠀⠀⠀⣼⣿⣥⣿⣧⣠⣾⣿⣿⣿⣷⡟⢟⢸⡅⠰⢸⡀⡀⠉⠁⠀⠀⠀⠠⠄⢸⡇⠀⠀⣧⠘⡇
⠀⠀⠀⢰⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⢹⡇⠈⠁⢨⣽⣿⡄⠀⢀⠀⠀⠀⠀⣠⣾⢀⣠⣴⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢺⡅⠀⢀⣤⣿⢻⡇⠀⠨⣅⠀⣀⣨⢺⣿⣿⡛⠻⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣼⡿⠋⠙⣿⣯⣿⣆⠀⢸⣿⢚⣛⠿⣽⣿⣟⣆⠀⢸⣇⡄
⠀⠀⠀⢸⣷⣿⣿⡟⣿⠙⣶⣿⣟⣻⣽⣟⡇⠀⠀⣭⡿⠟⣿⣿⢏⣉⣈⢩⣤⣭⣭⣽⣟⠀⣄⣨⠇
⠀⠀⠀⠰⠬⠌⠛⠷⠽⡾⢿⡿⠷⠿⠿⠗⠛⠂⠀⠁⠠⠊⠉⠐⠂⠰⠂⠒⠲⠾⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢦⡀⠀⠀⢀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⡄⠀⠀⠀⢙⣦⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡣⡋⡄⠀⠀⠊⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠅⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠢⠰⠆⠀⠀⠀⠀⠐⠠⠿⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠒⠀⠀⢀⡀⢀⣀⣒⠀⠀⠀⠤⠀⠐⣀⢤⣤⠷⠟⠍⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠉⠓⠓⠶⢤⣠⠀⢀⠀⠀⠤⣝⡈⠛⠛⠠⣀⣀⣤⡶⠿⠛⠉⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠈⠑⠓⠒⠲⡘⠀⡼⠟⠛⠋⠁⠀⢂⠨⠄⠁⠢⠀⡠⠞⠰
⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠄⠒⣀⠄⠀⠀⠀⠀⠀⠃⠀⠐⠀⠀⠀⠀⢈⠁⠁⠶⢤⡄⠜⢸⣏⡀
⠀⠀⠀⠀⠀⡬⠐⢧⡀⢠⢊⠰⣢⠀⠀⢠⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣆⡀⢀⢄⠀⠸⡧⠆
⠀⠀⠀⠀⠀⠡⡈⣰⣿⠀⠀⠀⢂⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠓⠂⠀⠀⠳⡆
⠀⠀⠀⠀⠀⠀⠈⠚⠛⠀⠀⡤⠀⠀⠈⠫⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠛⣙⢰⡆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⢿⠂⠀⠀⠀⠀⠀⠀⠀⠘⠉⢉⣠⠖⠦⠉⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠈⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠠⠘⠀⠁⠀⠀⢀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠀⠀⠀⠀⠀⠀⡎⢰⠄⡆⠀⠀⠀⠀⡠⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠢⠤⠄⡹⢍⠄⠠⠖⠀⠉⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠒⠒⠀⠢⠤⣀⠀⠀⢀⠀⠂⠀⠉⠉⠀⠐⠠⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠞⢁⡠⠄⠉⠀⠀⠒⠐⠠⢀⢍⡔⠁⠁⣢⠀⡀⠀⠀⠀⠀⣀⠈⡠⡀
⠀⠀⠀⠀⠀⡐⠁⡈⠊⠀⢀⠠⠠⠀⠠⠄⢀⢀⡙⠐⠠⠘⠠⠂⣵⡄⠀⢀⠀⠀⠀⠀⠚⢀
⠀⠀⠀⠀⠀⠄⠀⠃⠀⡜⡡⠠⠒⠈⠐⠶⠒⣯⠟⠐⣂⠀⠀⠀⠈⡔⣷⢾⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠈⠃⢉⡄⢘⠀⢰⠀⠀⠀⠀⠁⢠⣶⠀⠶⣀⠄⡶⠀⠠⡀⢖⠹⣤⠂⠀⠀⢰⠀⡃
⠀⠀⠀⠀⠀⢃⠀⠇⠀⠢⣀⡤⡀⠀⠀⠀⠀⠉⠒⡃⠍⠀⠄⢠⠘⠻⠁⣆⠀⠀⢰⢶⣯⣅⠅
⠀⠀⠀⠀⠀⠀⢦⠚⣠⠄⠼⣗⣤⡤⠲⠖⣦⠀⠀⠀⠀⠀⠀⠀⣀⣂⠙⠁⠐⠀⠀⢸⠇⡏⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢈⣠⠻⠳⠝⠊⠀⠀⠀⠠⠄⡰⠂⣾⣟⢶⢐⢶⣻⡀⢨⣠⡭⡧⢀⠁
⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⣨⡗⠃⠀⠀⡀⠈⠁⠄⠀⠀⠀⠀⠈⢰⡆⢿⣾⣿⡿⢺⣲⢷⡉⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⣿⡟⠉⠁⠀⠉⠙⠳⣦⡀⠀⠀⠀⠀⠹⣸⠟⡩⢒⡍⠕⢀⠌⠐⢯⣻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠁⣿⡇⠀⠈⠂⡀⢾⣶⠌⠁⠉⠐⣋⡉⢰⣗⣤⠚⠋⠀⠀⠀⠀⣠⠔⢱⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡧⣠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢟⣟⡿⢋⣠⠤⣤⡄⠃⢀⣷⡄⠘⣿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⡟⡉⠓⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣼⣗⡆⢈⣤⡆⠀⠀⠀⠀⠉⠀⠀⣿⠇
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢋⠀⠐⡀⠀⠀⠀⠀⠢⠀⠁⡀⠀⢀⢾⣿⡇⣷⠀⠉⠁⠀⠀⠀⠀⠀⠀⠐⣿⡀
⠀⠀⠀⠀⢰⣿⡿⠛⠛⢣⡛⠃⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⠐⢣⣼⣿⡇⢸⣆⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠋
⠀⠀⠀⢰⣿⠋⠀⣘⠂⠶⢧⡀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠈⣶⣿⣿⠺⣏⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠄⡄⠰⠎⣛⠐⣶⡈⠀⠀⠈⠡⣛⣠⠝⠁⠎⠉⠐⣨⣵⣶⣦⣶⣾⣿⡗⠷⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢾⠴⣶⡶⠟⠋⣡⠛⠿⠟⠟⠙⠛⢾⣆⠀⠀⠀⠀⠀⠀⠉⣯⠙⢍⠗⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠆⠦⠰⣄⢀⣀⣀⣀⢴⡾⣪⣄⣀⠀⠂⠀⠀⠀⠁⢟⠶⢄⣀⠀⠀⠀⣝⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣬⠯⢉⠋⡙⠰⠁⠻⠿⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⣻⣵⢣⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠹⡀⠁⠈⠁⠀⠀⠀⠀⠀⣤⣀⠘⠟⡀⠀⡀⠀⠀⠀⠀⠀⠘⡿⣿⠖⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⢀⣿⣶⣦⡜⠉⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡐⠰⠊
⣾⣷⠾⢾⠾⠶⢿⠶⢶⣾⢿⣿⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣷⣷⣷⣷⣾⣾⣿⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⠒⠒⠒⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠲⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣦⣶⣶⣾⠿⠿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⣤⣀⣀⣠⣤⣤⣴⣶⣶⡶⠶⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣖⠼⠄
⠀⠀⠀⠀⠀⠀⠀⢐⣤⡀⠀⠀⠀⠈⠛⠲⢦⣄⡀⠀⠀⠀⢸⣿⠆⠀⠀⠈⠁⠀⠀⠀⣀⣀⣠⣤⠤⠴⢖⣚⣟⣛⣻⣭⣭⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠰⠋⠒⠬⣙⠲⢤⣀⠀⠀⠀⠀⠉⠙⠲⢤⣀⣿⣿⣶⣶⣶⢶⣒⣛⣯⣭⣭⣴⠶⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠉⠙⢻⣿⡟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⣷⡤
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣈⠑⠢⢄⡀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⢖⣚⣯⣭⣭⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠈⠐⠦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡈⠑⠠⢄⣻⣿⣯⣀⣀⣤⣤⠤⣶⣞⣿⣯⣿⡶⠾⠟⠛⠋⠉⠁⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡀⢈⣉⣩⣭⣿⠷⠾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⢸⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⢸⣿⢋⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠼⣿⣿⣄⣠⣤⣶⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠀⠀⠀⣦⣾⣷⣞⡀⠀⠀⠀⠀⠀⣰⣾⡿⣢⣴⣗⣡⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣦⠀⣠⣾⣿⣿⣿⣛⣁⠀⠀⣀⠄⢸⣿⣿⣿⣿⣿⡿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⣿⣿⣯⣵⣿⣿⡿⢻⣿⣿⣿⣿⣤⣾⢏⡠⢠⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣅⣾⣿⣿⣿⣿⣿⣷⣯⣼⣿⣿⣿⡿⣿⣿⠗⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⣅⣴⣾⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠊⠈⠀⠀⠀⠀⠀⠀⠰⠟⠉⠀⢛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⢿⠃⣠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠺⡟⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣤⣶⡀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠉⠛⠦⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣷⣦⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⠔⠒⢾⣷⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⢸⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⣤⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⢿⠹⠿⠿⢿⡊⠹⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣶⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠀⠉⠙⣿⣏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⢴⡆⠚⢳⣸⢀⡐⣔⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡁⢀⠘⢕⠋⠅⠰⠜⠟⢡⠟⣹⣫⣭⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠿⢭⣿⡄⠘⢷⢠⠸⡆⢲⠀⡐⡆⢐⣚⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⢂⠀⡄⢊⢀⢈⠀⠀⡀⠠⠈⡇⠀⠙⠤⠄⡒⠀⢄⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠞⠀⠀⠉⠀⠀⠢⠀⠀⠀⠀⠈⠀⠀⠀⠠⠌⠄⢠⠌⡼⡄
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠰⣮⠄⠀⠀⠀⠀⢀⠀⢀⢠⠠⠠⠔⠢⢀⠈⡱⣧⠤⠀⢨⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⠠⠀⠀⠀⠊⠀⠀⣀⢀⣀⣂⡼⣉⡇⠐⢧⡿⣿⠨⣹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⣁⢄⠐⢶⢳⠏⡻⣧⡸⡳⣝⣧⡉⠠⠤⣛⣈⠙⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡹⣆⢮⡩⢼⣲⣿⣿⢿⣚⡿⡤⢳⡆⢸⡀⣸⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠁⣴⠩⡭⡓⢈⡛⠍⣁⣤⢵⣬⡗⠲⠛⠙⠀⠀⠐⢔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣤⡾⠓⠬⡜⠘⢋⡡⠶⣛⣨⣴⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡷⢞⣒⣶⠺⠻⠹⣥⣫⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡟⣪⣋⡀⣀⣬⠵⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⠾⠵⢷⣞⠩⠙⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⢉⣉⣤⠴⠚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⣉⣉⡠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⢆⣤⣐⢒⠂⣠⣺⣟⢙⣷⣷⣶⢶⣤⠀⠀⠀⡀⣤⣴⣶⣞⡟⡟⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠲⠋⣻⡉⠁⠚⢀⠛⠀⠀⢦⣁⠛⢓⣼⣿⣯⣿⣷⡿⢿⣻⡍⠂⠀⣰⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠐⠛⢻⠶⣯⣯⡀⣼⡤⣶⣽⡿⣿⣷⡝⠻⣉⣵⣿⣟⢹⣧⡗⢡⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠃⠈⠓⠛⠛⠛⠀⢒⣾⠓⠖⠚⢿⠝⣪⠟⠛⠘⠷⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⠄⠂⠛⠀⠀⠀⠀⠀⠀⢀⡴⠋⣇⠀⠀⠠⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠈⠙⠆⠘⠉⢿⣿⠚⣖⣖
⠀⠀⠀⠀⠀⠀⠀⢉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃
⠀⠀⠀⠀⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠇⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠢⠿⠁⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁
⠀⡄⠀⢠⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠂⠐⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠄⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⡁⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠐⠀⠩⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⡜⠀⠀⠀⠀⢀⠀⡠⢀⡔⠀⠀⠀⠠⠀⡥⠂⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠃⠇⢰⡀⡄⠇⡀⠳⠏⠃⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠈⠁⠒⠒⠂⠁⠐
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠔⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠄⠄⠄⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠅⠀⠀⠐
⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠄⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠋⠓⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠆⠐⠒⠀⠐⠐⠐
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢾⠁⠀⠀⢈⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⢄⡀⠀⠙⢦⠔⠋⠀⠀⣠⠞⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⢄⡀⠙⢄⠉⠓⠒⠤⠴⠒⠒⠉⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠉⢢⠈⠓⢤⣀⠀⠀⠀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣀⠤⠒⠒⠒⠢⡀⠀⡇⠀⠀⠈⠉⠉⠉⢉⡭⠕⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠋⠀⠀⠀⠀⠀⠀⢹⠀⡞⠀⠀⠀⠀⠀⡰⠋⠀⢀⠴⠒⠒⢤⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠴⠃⣴⠁⠀⠀⠀⢀⡜⠁⠀⡰⠋⠀⠀⠀⠀⠱⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠉⠉⠀⢀⡤⠊⢿⢄⡀⠀⣠⠎⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⢣⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠉⠒⢌⡲⢍⣉⢀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢦⠀⠱⡄⠀⠁⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠀⠀⠀⠀⠀⡟⣄⠀⠀⠀⠀⠀⢀⠇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢸⠀⠀⠀⠀⠀⢇⠈⠦⣀⠀⢀⡠⠋⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠑⠤⣀⡀⠀⠀⠀⣀⠴⠋⠀⠀⣸⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⣀⠀⠀⣀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⡿⠻⠟⠋⠀⠙⠿⠻⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⠋⠈⠐⠀⠀⠀⠀⠀⠂⠀⠹⣿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠉⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠋⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⢀⠈⠉⠁⡀⠀⠀⠀⠀⠹⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣶⣿⣄⠀⣠⣷⣦⣤⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣞⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡛⠛⠛⣛⣛⣛⣛⡆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣿⣯⣭⣭⣭⣥⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⣀⡀⠈⠙⢻⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⠀⣴⣿⣿⣿⣷⠀⠀⢻⣿⣿⠟⠉⠀⢀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣄⠀⠘⠻⠿⠿⠋⠀⢀⣾⣿⠃⠀⢰⣾⣿⣿⣷⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⠀⣤⣶⣿⣿⣿⣧⡀⠈⠛⠿⠿⠟⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣦⢀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡏⢹⣿⣿⣿⣿⣿⡄⢸⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⣿⣿⣿⣿⢼⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣟⡹⣟⢿⣫⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⡿⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣀⠙⠛⠿⠛⠋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧

⠀⠒⠙⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣐⣷⣀⣀⢀
⠀⢰⡶⠚⣆⢠⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢠⣤⣿⣾⣿⣿⣿⣿⣷⣮⡲⣄⢀⡀⢀⡀⠶⠀⠀⠘⠁
⡘⢄⠳⣀⠻⣾⣁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣽⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣅⡀⠈⢆⡀⠀⣀⢠⡌⣠
⣿⣦⣕⣂⣉⣉⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣥⣧⣥⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣷⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣦⠿⣽⣿⣮⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⡻⢿
⠿⠋⠿⠘⠈⠚⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣯⣿⡟⣿⣶⠿⣿⣿⣟⠛⠻⠟⠻⠛⠛⠋⠉⠁⠀⢐⡶⣌⠑⢮⠳
⠀⢘⣠⡀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠛⠿⣟⣿⣟⠿⠟⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢈⡴⠚⠃
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⣾
⣘⣠⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⢶⢰⣠⢤⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣪⢙
⣿⣿⣿⣿⣯⣭⣥⣤⣦⣼⢢⡄⢀⣀⣀⣠⣴⣚⣥⣮⣿⣏⣾⢿⣿⣿⣷⣼⣿⣿⣦⡀⠀⠀⠀⠀⠀⢤⣠⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣾⣿⣿⣙⣿⣿⣿⣷⣿⣱⣆⣰⣶⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣁⡀⠠⣄⠠⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⢯⠛⣆⣠⠷⠀⠈⡟⠙⣛⠉⠧⠶⠤⡈⠉⠙⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠻⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠔⡸⠁⠀⠉⠀⠀⠀⣤⠐⠂⠈⠁⠙⠂⠙⠛⠿⠿⣿⡏⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠃

⠀⣀⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤
⠀⢣⠀⠀⠀⠀⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⡀
⠀⠀⢑⡆⠀⠀⠀⠠⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⢐⡊
⠀⠀⠈⢷⡀⠀⠉⠀⠈⠦⣀⣙⢦⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣶⠆⠀⠀⠀⠀⠰⣾⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⡴⣋⢀⠴⠁⠀⠉⠀⢀⡾⠁
⠀⠀⠀⠈⢿⠤⠀⠲⠦⣄⣘⠛⢣⡷⣄⠀⠀⠀⢠⠞⠀⠈⠉⠑⣤⣶⣶⣶⣶⡴⠊⠉⠁⠀⠳⡄⠀⠀⠀⣠⢾⡜⠻⣓⣠⠴⠖⠀⠠⡿⠁
⠀⠀⠀⠀⠘⣀⣀⠀⠀⠀⠀⠤⡆⠐⢌⢣⡀⠀⠸⣄⡀⠀⠀⣠⡿⣿⣿⣿⣿⢿⣄⠀⠀⢀⣠⠇⠀⢀⠜⡩⠂⢰⠤⠀⠀⠀⠀⣀⣄⠃
⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠋⠃⠀⠀⡀⠙⢦⡀⢸⣿⢀⣾⢽⣾⣿⣿⣾⣿⣷⡯⣷⡀⣿⡇⢀⡴⠋⢀⠀⠀⠈⠙⠂⠀⠀⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠙⣦⡹⢿⣿⣤⣿⣿⣿⣿⣿⣿⣤⣿⡿⣏⣴⠋⣠⠖⠁⠀⠁⠀⠀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠀⠴⠶⠿⢿⣷⣾⣿⡛⡿⠟⢛⣿⣿⣿⣿⡛⠻⠿⢛⣿⣷⣾⡿⠟⠶⠦⠀⠀⠀⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠔⠉⠛⣿⣿⡿⡀⠀⢘⣷⣿⣿⣾⡃⠀⢀⢿⣿⣿⠛⠉⠢⠄⠀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⢀⠈⣸⣿⡇⡀⠀⡚⣿⢿⡷⣿⢃⠀⢈⢸⣿⣎⠁⡀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠈⣸⢻⡇⠀⠠⠋⢻⣼⣇⡗⠙⠄⠀⢸⡟⣇⠑⢀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⣼⣿⣿⡇⠀⠀⠼⢻⣻⣟⡿⠇⠀⠀⢸⣿⣿⣧⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣷⡇⣿⣇⠂⢀⠀⡐⣿⣿⢂⠀⠀⠈⣸⣿⢸⣾⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢻⣿⣾⣄⠓⢿⣿⣿⡿⠚⣠⣷⣿⡟⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠉⠻⠶⠶⠟⠉⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢽⣿⣿⣿⣧⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⢿⣿⣿⣿⡿⠟⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⢷⢶⣶⣤⣤⣀⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⣟⣻⣯⡽⠿⡗⠻⡿⣦⡠⠐⠁⠀⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⡟⢻⡏⣹⣿⡄⣅⢹⡄⠘⡜⣷⡴⢶⡄⠀⠄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠄⢴⣇⣽⣿⣷⣄⣱⣿⣾⠷⠾⣧⣾⢷⢚⣦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣸⣧⣶⠼⣿⣛⣿⣯⣿⣿⣶⣿⣿⣿⡋⠌⠌⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣷⡟⢿⣿⢿⣿⣿⣻⣽⣿⡿⢹⢹⣇⢃⢁⢊⡇⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣇⠛⣿⣦⣿⡾⣿⣿⡟⠛⠙⡄⡈⣎⣿⣾⣦⣬⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢹⠈⣁⣇⣬⣿⣷⣾⡿⢿⣿⣿⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⢷⣾⣻⣟⣹⣭⠿⠶⢿⠛⠛⠛⠁⢸⣯⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡏⠙⠛⠋⠁⢳⣶⣾⣿⣄⣀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣆⠀⠀⣠⣿⣿⣿⡿⡿⠛⠃⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠉⠉⠛⠿⠟⠁⠀⠀⢀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⣠⣿⣿⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣷⣾⣶⣿⣿⣿⣿⡿⠀⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣤⣤⣴⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠻⣿⣿⣿⡟⠳⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⡀⠀⠀⢀⣀⣀⠀⠀⢠⠛⠀⠀⠀⢀⣼⣿⣿⣿⣀⠅⠀⠀⠙⣦⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣦⣴⣿⠟⠙⢿⣦⡹⢿⣷⣦⣼⣿⣿⣿⣿⣿⣿⣷⣶⣦⣶⠾⢁⡤⣶⠟⠻⠷⣶⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡈⣱⣿⠟⠛⠛⠛⠛⠻⠿⢿⣯⣁⣰⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣄⢀⣀⡀⠀⠀⣈⣳⠝⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⠿⠿⠿⠛⠛⠻⠿⣿⣿⣤⣏
⠀⠀⠀⠀⠠⠶⠟⠛⠛⠛⠻⣶⣄⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣴⣿⠿⢋⣽⡟⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⡀⠙⢿⣿⣿⣶⣄⣀⣤⣶⠟⠛⠉⠙⠳⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⣿⡏⢠⢸⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⣸⢏⣼⢧⠸⠈⢏⠻⣎⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠳⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⢿⢧⣿⣾⡿⣷⣬⣄⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⣼⠁⠰⠀⣾⣿⣿⣿⣿⣿⠿⠿⠿⢾⣯⣷⣿⣿⡈⢷⢰⠙⣿⣷⣌⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⢣⠞⠁⠀⡆⢸⣿⣿⣿⣿⣿⣿⣄⠀⢀⣀⣿⣿⣯⢿⡇⢸⡄⢇⠈⠻⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢃⡾⣷⠋⠀⠀⢰⠀⢿⣽⣿⠿⢿⣿⣿⣽⣿⣿⣿⡿⠟⠉⣈⣧⣼⡇⠈⢦⠀⠀⠙⣿⣷⡄
⠀⠀⣀⠀⠀⢀⣤⣤⣤⠴⠊⠛⠁⣾⢡⠇⠀⠇⠀⡇⠀⢸⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⡀⠈⠳⡀⠀⢸⠈⣿⣀⡀
⠀⠀⠙⠓⠛⠛⠛⠋⠀⠀⠀⠀⣼⠃⠾⠀⠸⠀⠸⠁⠀⢸⣿⣿⣿⣿⣿⢿⣿⡟⡿⠻⢻⡟⠛⠻⡟⠟⠀⠱⡀⠀⠙⢦⣸⡄⠘⣿⠛⠿⣶⣶⣦⣶⣄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣠⠃⢠⠃⠀⡇⠀⠀⠸⣿⢸⣿⣿⣿⠿⠸⠇⠃⠀⠀⢱⠘⠂⣷⠁⠀⠀⠡⠀⠀⠀⠙⢷⡀⢸⡇⠀⠀⠈⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡔⠁⢤⠃⠀⠀⠀⠀⠀⠀⢻⣾⣟⡿⠟⠀⠀⠀⠀⠀⠀⠸⣆⣰⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⢹⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠌⠀⠀⠀⡄⠀⠀⠀⠈⢻⣿⣿⣶⣤⣤⣤⣤⣤⣶⣾⣿⡟⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢳⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠊⠀⠀⠀⠀⠐⠀⠀⠀⡰⠈⠻⣿⣿⣿⣻⡧⠻⣿⡟⢿⡟⠙⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠻⣍⡁⠁⠀⢘⣠⠞⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠋⠁⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⡀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠘⠁



⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⢀⠠⠀⠴⠀⢸⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣄⣹⣿⣿⣳⡄⠀⠀⠀⠀⠉⠳⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣭⣌⠙⠿⢷⣮⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣾⣶⣿⡿⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠛⡗⡮⠙⠋⢀⣂⠖⠀⠀⠀⠀⠰⣆⣀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡚⠀⠀⠀⠑⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⢀⣀⡵⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣎⠀⠙⢿⢢⡀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠤⢟⣭⣥⣶⣟
⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢙⣦⣀⣳⡙⠢⠌⠁⠀⠤⠀⠀⣘⠯⠈⠙⠀⠉⠉
⠀⠀⠀⠀⠀⢀⠖⠁⠄⠀⠀⠤⣙⢦⡝⠹⣤⣻
⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠈⠑⣝⣆⠹⣟⡤⠀⠒⠒⢲⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢧⢹⡅⠀⠀⠀⢸⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⢄⣀
⠀⠀⡼⠃⠀⠀⢄⠀⠀⠀⠄⠀⣀⡀⢤⣹⠃⡈⡃⠀⠀⠀⠘⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠑⠤⡀
⠀⡰⠃⢀⠈⠢⢈⢢⡀⠀⠀⣀⣀⣭⣓⠚⢳⣁⣷⣄⠀⠀⠀⡇⠚⣡⣵⡶⢠⠀⠒⠘⢄⠀⠀⠀⠀⠀⠀⢀⠖⠁
⠞⠁⠀⠀⠙⠢⢤⣁⠻⣯⡉⠀⠀⠠⠤⠐⠚⠻⣷⠬⠀⠀⠀⡞⠂⠀⢁⠠⠄⠀⠀⠀⢀⠀⠀⠀⢀⢄⢴⣉⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢳⣄⠀⠀⠀⠀⠀⠀⢘⡇⣀⣀⣠⣁⣀⠡⣌⡃⠀⠀⠐⠀⠀⠀⠀⣀⠴⢧⣎⡀⢀⡀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢞⣰⡿⠀⢠⠀⠀⠀⠀⢫⠢⡄⠳⣄⠀⢀⣢⣛⠆⡀⠀⢠⣄⣤⢀
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠴⠚⠁⠎⣠⠔⠁⠀⠀⠀⠀⠈⡇⠁⠀⠈⠐⠋⠠⡙⠴⢀⠄⠔⣭⣭⠢
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠈⠳⢦⣀⠐⠊⣁⣀⣀⣀⣀⣤⡤⠤⠁⠀⠀⠀⠀⠐⠄⠈⠱⡄⠐⠂⠊⠁⠐
⠀⢾⡒⢒⠀⠀⠀⠀⠠⠠⠄⠀⢁⣀⣀⣀⡀⠙⢍⠣⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠢⡀
⢰⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠒⢯⡳⠄⠣⡈⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠁

⠀⠀⠈⡦⠀⠀⠌⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⡀⢠⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⢀⡀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⢂⠀⠀⠁⠀⠂⠀⠀⣀⠌⠀⢀
⡡⠊⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⡈⢀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⡈⠀⠀⠁⠤⠈⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡤⢊⠽⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠄
⠀⠠⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠐⠃⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠓⢄⡀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣴⠆⠀⠠
⠀⠀⠀⣀⠤⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠠⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⡤⠴⠋⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢄⠘⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡐⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡂⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠊
⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠈⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠈⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡴⠻⣄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣟⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⣶⠀⡀⠈⠐⠋⠀⠀⠈⢷⡄⠀⠠⠀⡀⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠘⢶⣤⣄⡀⠀⡜⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⣀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⡀⠀⠀⠀⠉⠙⠛⠾⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠙⠒⠒⠋⠙⠃
⡀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠂⠀⠀⡄⠀⠃⠀⠀⢀⠀⠀⠀⢀⠀⡄⠀⠠⠐⠂⠄
⠀⠀⠀⠀⠐⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠠⠀⠈⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⢀⠀⠄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠠⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠔⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣈⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣔⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠠⠀⠄⡀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠨⠀⠀⠀⠈⠀⠠⠀⡀⠠⠔⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡄⠀⢀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣄⠀⠀⠀⠀⡀⠸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢇⠀⠤⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠈⠃⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠈⠁⠊⠀⠀⠀⠀⠄⠀⢀⣄⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢐⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠄⢀⠀⠀⠀⠀⠀⢠⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠋⠊⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠆⠀⠠⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⢁
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡀⠈⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠈⠁⠀⡇⠀⠀⠀⠄⠠⠀⠁⠀⠂⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠀⠀⠀⠀⢀⠈⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡀⠀⠀⢀⣀⣀⣀⣀⢠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠐⠐⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠔⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠟⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠐⠛⠀⣠⠤⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃


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

⠀⠀⠀⠀⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⢘⢯⣭⠿⠂⠀⠀⢀⣦⡀⠀⠀⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⢰⣶⠀⠀⠀⠰⣿⡇⠀⢠⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⠃⠀⠈⠻⠽⠟⠀⠀⠠⣿⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⣰⣶⣄⠀⠀⢦⣀⡀⠁⢀⠀⠀⣀⢀⣀⠀⠀⠎⡟⠀⠀⠀⣀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⢼⣿⡧⠀⠀⠀⢈⢏⠃⠀⠀⠀⠏⠀⠀⠀⠀⠀⠁⠠⠀⣰⢓⣶⡤⣄⠀⢸⣿⠀⠀⠀⠗
⠀⠀⠀⠈⠟⠁⠀⠀⠀⢸⣻⣴⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴⠏⣿⣿⣏⠂⠀⠘⠑⠀⠀⠀⣁
⠀⠸⣷⣆⠀⢀⢠⣶⡇⠀⠉⠁⢀⡀⠀⠀⠀⠠⡍⠄⠀⠀⠀⠉⠉⠉⠉⠁⠀⣤
⠀⠀⢹⣞⢻⣿⢻⣿⠀⠀⠀⢠⠻⡟⠄⠀⠀⠀⠉⠀⠀⢠⠀⡄⡀⠀⠀⠀⠘⠾⠃
⠀⠀⠈⠛⠁⠀⠛⠃⠀⠀⠀⠈⠺⠖⠁⠀⢀⣀⠀⠀⠀⠈⠒⡓⠂⠀⠀⠀⠹⠿⠁
⠀⠀⠐⣦⡄⠰⢤⣤⣤⣿⣤⣴⡶⠂⠀⠀⣾⣿⡀⠀⠀⠀⢸⣇⠀⠀⠀⣠⡄⠀⠀⡀⠀⠉⠐⠓⠀⠁
⠀⠀⠀⠐⡀⠈⠉⠉⠉⠉⠙⠛⠉⠁⠀⠈⣿⣟⡇⠀⠀⠀⢸⠁⠀⠀⠀⣿⡇⠀⠀⡇
⠀⠀⠀⠈⠨⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠃⠀⠀⠀⠻⠋
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠸⣿⡿⠀⠇⠇⠜⠆⠈⠂⠸⠬⠂⠀⠅⠃⠟⠅⠜⠆⠻⠇⠠⠳⠀⠀⠿⠀⠧⠁⠣⠄⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠂⠐⠛⠛⠛⠂⠰⠛⠛⠛⠃⠀⠛⠛⠛⠀⠘⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠴⠊⠉⠙⢢⡀⠀⠀⢠⠖⠈⠑⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠀⠉⠓⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⡾⠘⠊⠉⠉⠀⠈⠉⠉⠒⠔⢢⠀⠀⠀⠱
⠀⠀⠀⠀⠀⢠⠎⠀⣀⣀⣠⠖⠉⢷⠒⢒⣐⣐⡚⠳⣄⠀⠀⠀⠀⠀⢀⣘⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⠢⡀⠈⠃⠰⡀⠀⠘
⠀⠀⠀⠀⢀⠇⠰⡏⠀⠉⡤⠦⣀⣸⠉⠀⠀⠁⠀⠙⢍⢆⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⠈⡒⠠⢄⠱⡀
⠀⠀⠀⠀⢸⠀⠀⠙⢄⠜⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠆⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⢀⡀⢈⡡⠘⠛⠀⠀⡰⠁⡧⠔
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠑⠚⠒⢤⣀⠀⣀⠀⠀⠀⣔⠃⠀⠀⠀⠀⢸⠙⠉⠉⠉⠉⠁⠍⠀⠀⢸⠀⠑⠈⠀⠀⡇
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡞⠀⠀⠒⠀⠀⠀⠉⣉⠈⡙⠉⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⣀⠀⡰⠁
⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⠃⠀⠀⠂⠀⠀⠀⠠⠌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠠⣀⡠⠄⠀⠀⠀⠀⠉⡉⠜⠉
⠀⠀⠘⠣⡄⠀⠀⠀⣴⢧⣀⠀⠀⠐⠢⠠⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠌⣀⣀⡀⠀⢤⣔⡊
⠀⠀⠀⠀⠙⠚⢢⡼⠀⢀⠑⠭⠶⣦⣒⣂⡬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠳⠴⠋⡀⠀⠂
⠀⠀⠀⠀⠀⠀⡘⡄⠀⠘⠀⠀⠀⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⡆⠀⠀⠁⠰⢄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠄⠀⠘⠄⠀⢆⠀⠉⠂⠄⣗⠄⣀⠀⠑⠢⢄⣀⠂⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⢐⠄⠀⠀⠀⠄⠈⠀⢀⠄
⠀⠀⢨⠀⠀⠀⠈⠁⠀⠑⠢⠤⢀⡀⠉⠙⣢⡰⢄⣠⠏⠑⠦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⡀⠀⠐⠉
⠀⠀⢸⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠈⠑⠲⠋⠁⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠲⠤⠄⢀⠠⠆
⠀⠀⠈⡄⠀⠀⢸⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠
⠀⠀⠀⠣⠀⠀⠜⠀⠀⠀⠐⠄⠀⣀⡠⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠋⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⢂⠉⠄⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢀⡀⠁⣀⡠⠭⠒⠂⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⢿⣷⡄
⠀⠀⠀⢀⣠⣾⡿⠛⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠒⠲⠶⢶⡄
⠀⣠⣶⣿⠟⠉⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠶⠂⣠⠾⠃⠀⠀⣀⣤⣦
⣼⣿⠋⠀⠀⠀⣠⣴⠇⠀⠀⣼⣿⢀⣠⣤⣤⣄⣠⣤⣶⣶⣶⣾⣧⣤⡚⠛⢦⡀⠐⠿⠛⣿⠇⠀⠀⠀⠀⣀⣠⣄⡀
⢻⣿⣄⣠⣴⣿⣿⣿⠀⠀⠀⣿⣿⣿⠟⣹⣿⠿⠋⠉⠀⠀⠀⠀⠉⠙⠻⣷⣿⣶⡿⠿⢰⣿⡶⠟⠃⠀⣾⡿⠛⠛⠿⣷⣦⣄⡀
⠀⠉⠛⠛⠋⠁⣿⡟⠀⠀⢸⡿⠋⢀⣾⡟⠁⠀⠀⠀⠰⠶⣶⡄⠀⠀⠀⠈⠻⠁⠀⠀⠈⠁⠘⣿⡄⠀⣿⣧⡀⠀⠀⠈⠙⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣠⣴⣾⡯⠀⠀⠀⠀⠀⣴⣾⣉⡀⠀⠀⠀⠀⠀⠀⣈⡿⠖⠀⠀⢼⣿⣶⣬⡻⢿⣷⣦⣄⡀⠀⠀⠈⠙⠻⣿
⠀⠀⠀⠀⠀⢸⣿⠁⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠀⠈⠙⠻⣿⣿⠛⠋⠁⠀⠀⠀⠀⢀⣽
⠀⠀⠀⠀⠀⠘⢿⣷⣾⠟⠁⠀⣀⣀⣠⣤⣤⣶⣶⣆⠀⠀⠀⣠⣴⣾⣿⠁⠀⠀⠀⠀⣴⣶⣤⣤⣤⣤⣄⣀⠈⢿⣧⣀⣀⣤⣴⣶⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢺⣿⠿⠿⠟⣿⣿⡿⠁⠀⠀⠀⠉⠁⣾⣇⣠⣤⠀⠀⠀⠙⠛⠛⢛⣿⣿⣿⠟⠀⠀⢿⣿⣿⣛⠋⠉
⠀⠀⠀⠀⠀⠀⣤⣭⣥⣤⡄⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠸⠿⠛⠉⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠀⠀⠀⠀⢸⣿⠛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⣠⣿⣿⣋⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣭⣄⣀⣀⠀⠀⠀⢸⣿⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠾⠿⠷⠶⠆⠀⢰⣿⣿⣿⣿⠿⠿⠟⠛⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠿⠿⠿⠃⠀⢠⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⡶⠀⠀⠶⢾⡿⠀⠀⠀⠀⠀⣠⣤⣾⡿⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣀⣀⣀⣀⠀⠀⠀⣀⣴⡿⠟⠁⠀⠀⠀⢴⡿⠶⠆⠀⢀⣀⣈⣩⡅⣾⢃⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠛⠛⠿⣿⣄⠀⠈⠛⠿⠿⠷⡶⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡟⠋⢸⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠸⣿⣷⣤⣀⡀⠀⠀⠀⢀⣀⣼⣷⣦⣤⣤⣶⡿⠛⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⢾⣿⡄⠀⠀⠀⠀⣰⣿⠀⠉⠛⠻⠿⠿⠿⠟⠛⠉⠀⢉⣩⡿⠟⠃⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠘⣿⣶⣦⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠠⣶⣿⣉
⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⢀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⣄⣀⡈⣹⡇
⠀⠀⠀⠀⠀⣰⡿⠛⠻⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠹⣷⣤⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠂⢀⣴⣿⣅⠈⡏⠁⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⣦⠶⣻⡉⠠⠸⣟⠁⠀⠀⠀⠀⠀⢡⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⣿⠳⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣷⣆⡀⠉⠲⣴⡟⠀⠀⠀⠀⠀⠀⣴⠀⢀⠆⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡧⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠁⠁⠈⠙⢿⣷⣄⢈⢷⡄⠀⠀⠀⠠⠴⣟⡁⣇⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠘⠇⠀⠀⠀⠀⣐⡛⠛⠑⠰⡛⠀⠀⠀⠀⠀⢻⠀⠀⠡⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⠀⢀⢀⡀⡀⠀⠀⠀⢀⠿⠯⠑⢆⠀⠀⢡⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡴⠇⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠁⠖⠈⡽⠁⡀⠀⠁⠈⠐⠄⠀⠀⠀⢀⡴⣀⠀⠀⠀⠀⢸⡂⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣴⡂⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡄⣀⣩⣵⣏⡀⠀⡀⣀⡹⣄⣠⠐⣋⣴⠏⠀⠀⠀⠀⠁⡎⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡄⢈⣦⠀⠀⠀⠀⠀⢰⡀⠀⠀⠹⣟⠿⣾⢹⣟⠿⢹⡙⣧⢯⠆⠉⠛⢥⡀⠀⠀⠀⣸⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡜⠁⠀⠡⢄⠀⠀⠀⠆⠑⢄⠀⠀⠙⠘⠿⠀⡈⠿⢨⢣⣿⢃⠀⠀⠀⠈⡷⠀⠀⠠⠛⠳⣇
⠀⠀⠀⠀⠀⢠⣦⣦⠄⠀⠀⠀⠙⠀⠈⠄⠀⠀⠑⢄⣠⢁⣼⠀⣳⡀⠀⣆⣰⡟⠁⢠⢆⣟⠃⢸⠀⣤⣤⣀⡅⢀⠔⠁⠀⡈⢸
⠀⠀⠀⠀⠀⠀⣡⢟⠶⠀⠐⢤⣀⠀⠀⠈⢀⠀⠀⠈⠁⠀⡉⢠⢿⡇⠸⣢⡈⠀⢀⠏⣼⠇⠒⢹⡟⠛⠞⠛⢳⠎⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠤⣴⡏⠀⠀⠀⠈⠉⠘⠁⠀⡠⢁⣾⡟⠑⢤⣤⡐⠤⣞⣸⣃⣀⣦⠮⠦⠄⠀⠀⢸⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠠⣀⡀⠀⠀⣰⠁⡾⣹⠃⠀⠨⡍⠁⠀⢘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠤⢴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠈⠉⢉⣅⢨⠏⡏⢀⡀⠀⡼⠂⣶⡟⠙⠁⠒⢲⡆⢤⡤⠄⠒⠁⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⢀⠤⠤⠀⠒⠀⠀⠀⠀⠀⣀⠠⠤⠒⠀⢬⡿⠄⣧⣀⣀⡞⢁⣾⢏⣠⠔⠒⠛⠛⠁⠀⠉⠢⡀⠀⠀⠀⠀⠀⠈⠐⠤⣀
⠀⠀⠀⠀⠠⠃⠘⠀⠀⠀⠀⢀⡠⠂⠁⠀⠀⠀⠀⢠⡿⠑⠀⠈⠙⠇⢁⣿⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⡸⢰
⠀⠀⠀⠀⠎⠠⠁⠀⠀⠀⡠⠊⠀⠀⢀⡀⠄⢀⣬⣏⣀⠀⠀⠀⠀⣠⣾⠋⠀⠀⢠⠀⠀⣀⠤⠤⠤⠠⢀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠳⠠⡀
⠀⠀⢀⠈⣀⣉⠠⠀⠀⢘⠁⠀⠐⠈⠀⠀⠀⠐⣿⠀⠉⠉⠉⠉⣿⡿⢷⡎⠉⢲⡛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠡⠁⠠⠀⠀⠀⠀⠀⠱
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⢨⣽⡀⠀⠀⣰⣾⣋⣀⣠⡇⠀⠈⢇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠈⠀⠀⢄⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣀⠤⠤⠤⠤⠿⠷⢶⠿⠟⠉⠉⠉⠉⠉⠒⢤⠄⠑⡀⠉⠀⠀⠀⠒⢂⠴⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠀⠀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠒⠒⠀⠀⠀⠒⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢄⣼⡿⠟⠛⠛⠛⢻⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⢰⡃⣿⠁⠀⠀⣴⣧⣦⡸⠿⠧⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠘⢧⣈⠣⢄⡈⢿⣿⣿⡀⠀⠀⢀⣿⡿⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣥⣤⣭⣿⣶⣶⣿⣿⣷⠴⠚⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢛⣿⠟⣉⠄⣒⣏⡄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣠⣾⣇⣸⡇⢹⠿⠛⠛⠋⠉⠉⠉⠉⠉⠙⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡎⣴⣿⣿⠟⠩⡳⣄⠀⢀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡼⣸⣿⠟⡁⠀⠂⠀⠀⠱⢄⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢰⡇⣿⠏⠀⡀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⣷⠀⡀⣴⠶⠦⣤⠀⠀⠀⢀⣀⣿⣿⡿⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⡇⣿⠐⢢⠀⠀⠀⠀⠀⠀⠐⢄⡈⠳⢄⠈⢳⣷⡐⢤⣤⡿⣠⡴⠾⢛⣹⣷⣿⣒⠂⡘⢿⣿⣿⣿⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣧⠃⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⣨⣣⣄⠛⡙⠀⠀⣤⢁⣴⣯⣍⡀⠈⠁⠀⠀⠛⣌⢿⣿⣿⣷⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⡆⢠⢰⠀⠀⣀⣀⡀⠀⠀⣀⣠⣴⣿⣿⣿⣷⣦⣤⠼⢫⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⢫⡈⣿⣿⣿⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⡀⠘⣿⡄⠢⠄⠾⠉⢉⣿⣿⡛⣻⡿⠃⠈⠻⢿⣿⣿⣴⣊⡶⠣⠤⣴⡄⠀⠀⠀⠀⠀⠀⠈⠇⢹⣿⣿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⠘⢿⣄⠑⡀⠲⣿⣿⠿⣿⠆⠀⠀⠀⢠⣧⠈⠉⠉⢢⣀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠠⡇⢸⣿⡟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⡈⠻⣷⣄⠀⠉⠀⠀⠁⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠇⣾⡿⢡⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣄⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⠄⣰⠟⣱⠟⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡈⠙⠿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢤⢀⠀⠀⢈⣡⡾⠋⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⡈⠙⠛⠿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⣉⣡⣴⡾⠛⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣶⣤⣀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⡛⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠦⢤⣤⣄⣀⣀⣀⣀⣀⣀⣤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠙⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢶⣦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠿⢶⣦⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⢀⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠀⠀⠀⣄⠀⢿⣿⡿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⣀⣼⣿⣶⣦⡤⡴⠶⠿⣿⣿⣿⡿⢿⡿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡏⠂⢐⢹⣿⠁⣸⡿⠩⠅⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⡇⠀⠀⢸⣿⢀⣿⠃⠀⠠⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡄⠀⠀⠈⣇⢰⣿⠾⠉⠉⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⣧⠀⠰⠎⡹⠿⣿⠀⣴⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠙⠒⡋⠁⠀⣿⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡐⠀⠂⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠲⢦⡄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⣾⣿⣶⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣦⣄⣀⣣⣀⣁⣠⣤⣾⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣠⡤⠀⠈⢿⣿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢷⡆⠠⠙⢿⣷⠀⠀⠐⠚⠛⠋⠀⠀⠀⣀⣿⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠂⠈⠿⠧⣄⣀⠀⠀⠀⠀⣈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡋⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠟⠛⠛⠉⠙⢷⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣮⠛⠿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⣀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣶⣾⡿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣆⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⢀⣴⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣷⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⠋⢀⣠⣾⣶⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣾⣁⣠⣾⣿⣿⣿⣦⡀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠈⣻⡟⢉⣵⣿⣿⠟⠁⠀⠀⣼⡟⠀⠀⠀⣀⣠⣤⣶⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢻⣿⡿⠛⠉⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠻⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡷⠜⣷⠀⠀⠀⢰⣟⣀⣀⣴⣦⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⠿⠟⠋⠉⠀⠀⠀⢹⣧⠀⠀⠀⠛⠛⠋⠉⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⣴⣶⠿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⣋⣥⣶⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢛⣿⣿⣿⡿⠟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣿⠿⠛⢉⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠺⣿⣭⡿⡟⠛⠒⠒⠒⠲⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠉⠁⠀⠀⢀⣶⢴⢤⢤⢤⢴⣦⡄⠀⠀⠀⠀⠀⠈⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠰⡇⡦⢰⠞⣰⡞⠶⠾⠧⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠄⠐⠀⠀⠀⠀⠆⢠⡞⠒⠒⠀⠀⣿⡇⠀⠀⠐⢺⡀⠀⠀⠀⠔⠀⠀⢀⠀⠂⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⣠⣄⠘⠀⠀⠀⠀⣯⣕⠀⠀⠐⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡞⠀⠉⠁⠀⠀⠀⠐⣇⣨⠄⠀⠀⠀⠈⠻⣿⣷⣦⣄⡐⠀⠀⠀⠀⢀⠄⠀⠐⠀⠀⠙⣦
⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⣶⣿⡟⠁⠈⡀⠀⠠⠀⠀⢀⣿⣿⣦⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⠄⠀⠀⡀⠈⢳⣄
⠀⠀⠀⢠⠞⠀⠀⠠⠀⡈⠀⠀⠀⠀⣴⣾⣿⣿⣿⣇⠄⠐⠀⠀⠀⠀⣦⡞⠉⠀⢽⣶⡄⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⡀⠆⠀⠠⠀⠹⣦
⠀⠀⢠⡏⠀⠀⠐⠀⠠⠀⢀⣀⢴⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⠀⣤⡿⠀⠁⣤⠈⠹⣧⡄⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣷⠄⠤⠄⠀⢈⡀⠀⠀⠀⠸⣇
⢀⣀⣏⠀⠀⢒⡛⠈⠈⠉⠉⣉⠉⠛⠿⢿⣿⡇⠀⠀⠀⣀⢀⣀⣀⣼⣇⠐⢠⣤⠖⠀⣹⣁⠀⡀⠀⠀⠁⠀⣿⡿⠟⠋⠡⠐⠒⠂⠀⠀⠺⠶⢦⣀⠁⠀⠸⣆
⠈⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⡀⠉⠡⣀⣶⣞⡯⠧⠟⠓⠙⠁⠀⠈⢸⠀⠀⡻⢿⡽⢟⣳⣦⣀⠔⠉⢀⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⣽⣄
⢰⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣄⠀⠻⠞⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡀⢀⠀⠀⠀⠈⠁⠓⢻⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾
⣸⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⠀⠀⠀⠄⠀⠀⠀⠈⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢸
⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠸
⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠃
⣇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⢸⠀⠀⣤⣤⣤⣄⠀⠀⢨⠀⠀⠠⠀⠀⢀⣠⣶⡦⣤⠀⡌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡯
⣿⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠠⡜⢿⡿⢇⠀⠀⠀⠀⠀⠀⠀⠟⢛⣛⡰⠃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⢸
⢸⠀⠀⠉⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⠛⠂⠚⠀⠀⠀⠂⠀⠀⠀⠃⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⣸
⠸⡆⠀⠀⠀⢉⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠏⠀⠀⠀⠀⡟
⠀⢷⠀⠀⠂⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣣⡀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⣄⣼⢁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠑⠂⠐⠀⠀⣸⠃
⠀⠘⣇⠀⠀⡀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⠏⠘⣿⠃⠀⡄⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠘⣿⡇⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠠⡀⢀⠀⠀⣰⠏
⠀⠀⠹⣆⠀⠈⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠜⠁⢀⣼⡿⠀⠘⡇⠀⠀⠀⠤⠀⠀⠀⠀⢀⡗⠀⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⣠⠏
⠀⠀⠀⠘⣧⡀⠀⠀⣐⠃⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠃⠀⣰⡇⠀⠀⠠⡄⠀⠀⠀⠀⣼⣧⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠈⢳⣄⠜⠉⠀⠀⠀⠀⡀⠀⠒⠘⢿⡿⠟⠁⣠⣾⡿⠁⠀⠀⠤⠀⠀⠀⠀⣰⣿⣿⣷⣄⣸⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠯⣀⡀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠒⠻⣿⡿⠑⠀⠠⠆⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠈⠀⢀⡀⠀⠀⠀⠙⡄⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠑⠀⠀⠀⠀⡀⠀⠀⠐⠼⠁⠂⣤⠀⠄⠀⢀⣴⠿⣿⡛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠁⠂⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠈⠀⠀⠀⠂⠠⠀⡀⠄⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⢠⠀⠂⠀⠐⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠁⠀⢸⢁⠀⠘⠀⠀⡃⠀⠀⠀⣦⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⣘⡄⠀⠙⡀⠠⠓⠠⠄⠀⡈⠒⡘⠀⠀⠀⠀⠀⢀⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀⠉⠠⠆⢠⠆⢇⣉⣾⣤⣤⡶⠶⠚⠛⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⢽⠟⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⡟⢷⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⡨⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⡀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠔⠛⠂⠠⠤⢤⣀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡀⠀⢼⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡠⠴⠯⢠⠏⠀⠀⠀⠲⡄⠀⠈⢣⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡜⠁⠀⠀⠀⣏⠀⠀⠀⠀⠀⠸⡄⠀⠀⡇⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡸⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢀⠇⠀⡰⠃⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⠀⠀⠀⠀⠀⠀⠘⠦⢄⣀⡤⠚⠒⠋⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢸⠀⢰⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠓⠤⠤⠤⠔⠚⠉⡄⡞⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡶⣁⣢⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⢷⢊⣢⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⣿

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠀⢄⣤⢄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⡼⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠒⠁
⠀⠠⣇⡞⠀⠀⠸⠇⣀⡄⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⣦⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣑⢤⡵⣧⣖⣀⣠⠔⠁⠀⡀⢀⣀⠠⠴⣶⣾⣶⠀⠀⡡
⠀⠀⠀⣉⠀⠀⠀⠀⢡⣤⣀⠀⠈⠀⠀⠄⠀⢀⣽⢿⣼⣿⡄⠒⠀⢂⡀⠐⠀⠀⠀⢀⡤⠶⠾⢱⣄⡜⢇⠉⠀⢀⠀⠊⢀⠾⣃⡀⢰⠿⠉⠛⣇⠉
⠀⠀⠏⣹⡄⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠊⠃⠀⢣⢛⡟⢠⠃⠀⠀⠀⠀⠀⠀⠂⡐⠋⠀⢀⣼⡿⢡⡜⡎⠃⠒⠁⠀⠀⣦⠀⠉⢁⢁⣤⠀⣤⠀⢄
⠀⠀⠀⢿⡀⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⠀⠀⡠⢪⢹⡟⡈⠓⠀⠀⠀⠀⠀⠀⡄⠛⠀⠀⠾⣟⡤⠋⢹⠀⠀⠀⠀⠀⡴⠋⠘⠃⠀⠈⠉⠀⣉⠄⠠⠁
⠀⠀⠀⠀⠉⠀⠈⠑⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠈⠀⠠⠄⠀⠀⠀⠀⠀⠀⠄⠰⡆⠀⢀⣀⠟⠀⠀⠀⠀⢀⡆⠀⠂⢠⡀⢛⡃⠐⠍
⠀⠀⠀⠀⣀⡠⢤⡀⢶⢈⡀⠀⠀⠀⢀⠠⠀⠀⠀⢘⡃⠀⠀⢀⠀⠑⠤⡏⢻⣐⣁⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄
⠀⠀⠀⠀⢀⣤⠄⠁⣼⡉⠀⣒⠀⣨⡁⠀⠀⠀⠀⢸⡆⠀⠇⡀⢀⠐⢠⠾⢾⡛⢍⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢀⡀⠁⡂⠀⡠⠊⠄⣰
⠀⠀⢠⠎⠁⠞⠀⢸⣨⠝⢁⣌⡉⠀⠉⠂⠀⠀⠀⢨⡇⠀⠀⠈⠁⠀⢸⢤⠀⠳⠀⠣⠈⠂⠀⠀⠀⠀⡠⡄⢀⠀⠀⠢⣀⠇⠈⢁⢨⡄⠈⠀⢀⡠⠂
⠀⠀⡎⠀⠀⢀⡀⣀⣠⣴⠋⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⡇⠀⠀⠀⢀⣤⣰⣀⣀⠄⠄⠀⠀⠐⠑⠒⡆⢧⠀⠠⣛⡯⠁⠈⢠
⠀⠀⠀⠀⠀⠀⠈⠀⢀⡟⠈⠁⠀⠀⠀⠘⡃⠀⢀⢀⣠⢄⠀⣠⠦⠀⠀⣠⠤⡀⢀⡴⠀⠀⣾⠀⢠⡤⡀⠐⠢⠔⠂⠠⢈⣷⣴⡶⠟⣛⡦⣀⠁
⠀⠀⠀⠀⠀⠀⠒⠚⠉⠀⣠⠄⠀⠀⠀⢸⡇⠀⢸⡟⠁⣾⠀⠙⢤⡠⣾⡧⠴⠃⣾⠀⠀⢰⡏⠀⠘⠦⡁⠀⠀⠉⠂⢰⠏⢿⡶⠶⠟⣋⠱⡌⠆
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠐⠀⠀⠀⠀⡅⢸⣧⠔⣿⠀⠀⢯⡔⢦⣠⡿⠻⠧⠤⠖⠻⢦⠴⠻⠧⠴⢄⣠⡿⠀⠀⠀⠀⡂⢸⠀⠶⠖⣛⡏⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢠⣆⠀⠀⠤⠁⠀⠀⠀⠀⡀⣳⠀⢀⠀⠀⢀⠀⠀⢀⢸⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⡙⠉⠀⡀
⠀⠀⠀⠦⣄⣀⣀⡠⢜⠰⢬⡁⠤⠄⠴⠂⠀⠀⢸⠀⣹⢰⡁⢸⢸⡁⢸⢰⡃⢸⠀⢸⠰⡇⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⣄⣄⠀⡔
⠀⢤⣀⡀⠤⠄⡶⠁⡄⠁⢙⡧⣀⣀⣀⠀⠀⢄⠀⠁⠈⠀⠁⠁⠀⠁⠁⠀⠉⠈⠁⠉⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠙⢤⣧⣫⠞
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠠⠊⠣⡀⠀⠉⠀⠀⠀⠤⠒⠞⢳⣋⡡⢔⠢⢄⠀⠀⢀⠀⡄⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠄⢸⣷⣿⢀⡒⠀⠂
⠀⢀⡠⠔⠊⠁⠀⠀⠀⢠⠀⠀⠙⠀⠀⠀⠀⠐⠀⠀⠐⠊⢬⢥⣀⣨⣽⠇⠀⠈⠸⠛⣶⠖⠷⢦⣀⣤⠤⠤⣀⠀⠀⠀⠀⠘⢡⢚⣒⣚⠷⡀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⢞⠁⡀⠈⠓⣌⠉⢳⠄⠀⠀⠀⠤⠋⠀⠀⠀⠈⠁⠂⢄⠀⠉⠉⠐⠀⣠⠃⢨⣭⡥⢖⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠄⠈⠀⠀⠐⠀⡀⠈⠛⠁⠀⠀⠈⠀⠀⠀⣀⠀⠢⠀⣀⣀⠔⢀⠄⠀⠀⠀⠁⠀⢀⠀⣬⡁⠀⠀⢖⣛⠃⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠐⠈⠀⠀⢀⡴⣳⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠳⣄⣾⡷⣿⣦⠇⠀⠀⠀⠀⠀⣀⡘⣟⠿⣌⠲⢦⡄⠁⠤⡄
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠡⠄⠰⡶⠀⢸⠁⠃⠀⠀⠀⢀⣤⡶⠀⠀⢀⠀⠀⠨⡅⠀⠀⢼⢀⡀⠄⡀⠺⢁⠤⢤⠸⡄⠘⢷⣦⣁⣌⣈⣄⠀⠀⠎
⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠈⠁⠀⠈⠳⣄⡘⣯⠕⠀⠀⠁⠈⠑⣦⣧⠀⠀⣼⠌⠄⠀⠀⠀⠀⠀⠀⢠⡿⠀⡤⠋⠈⠀⠻⠈⠓
⠀⠀⠀⠤⣄⠀⠀⠐⠀⠀⠲⠤⡀⠀⠀⠀⠀⠀⠀⢀⠾⣿⣿⠗⠢⢄⠀⠀⠀⡥⠠⠅⠀⠀⠘⡆⡁⠐⢀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡰
⠀⠀⠀⠀⠈⠉⠓⠶⠄⠒⠤⢄⠚⠦⢀⠀⠀⠀⠀⢂⠊⣹⡯⠃⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⠐⠉⡀⠐⠁⠀⠀⠀⢀⢀⠀⠉⠸⡽⠯⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣴⢊⠀⠈⠓⠀⠈⠚⠋⠀⠀⠀⠀⠨⠀⣩⠴⠀⠈⢄⢀⠂⠀⠀⠐⠀⠀⠀⠈⠉⠉⠀⢀⡄⠈⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠒⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡯⣭⣿⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⣟⣀⣉⣛⣲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⡿⠿⠶⠶⠶⠶⣶⣤⣤⣀⡹⣆⠀⠀⠀⠀⠀⣠⠟⢁⣀⣉⣥⣤⣭⣭⣽⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣤⠖⠋⠉⠉⠉⠉⠙⠻⣧⡀⠀⠀⣰⠿⠛⠉⠉⠉⠉⠙⠳⢦⡀⠀⠀⠀⠙⢿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢻⣿⣷⡄⣴⡟⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⣼⠃⠀⠀⢀⣀⣀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⠀⠀⠀⢹⡄
⠀⠀⢀⣴⠿⠶⣶⣞⣩⣅⠉⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠈⢁⣭⣙⣶⡶⠶⢷⣄
⠀⡴⠋⠀⠀⠀⠉⣡⠞⣋⡀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⣈⡛⢦⡉⠁⠀⠀⠈⠳⡄
⣸⠃⠀⠀⠀⠀⣰⡷⠞⢹⢇⡴⠞⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠘⢶⣄⢿⠙⢶⣷⡀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠉⠀⠀⢻⢿⡇⠀⠀⠀⠀⠈⠙⠳⠶⠤⠤⠶⣖⣫⣿⡄⠀⢀⡾⣽⣷⣶⠤⠤⠴⠶⠛⠁⠀⠀⠀⠀⠀⣿⢿⠃⠀⠈⠁⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠙⢦⠞⠁⠀⠀⠉⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠿⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠈⢧⡀⠀⠀⠸⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠆⠀⠀⣠⠏⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣌⣷⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢰⣏⣤⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠈⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣄⣀⠙⢦⣀⣠⣀⠀⠀⠀⠀⣀⣤⣀⡴⢋⣀⣤⣴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⣿⠛⠛⠻⡶⠾⢿⡷⢶⡿⠷⢶⠟⠛⠉⣿⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣇⠀⠀⠀⣿⠀⢸⡇⠸⡇⠀⢿⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣻⣄⣀⣴⣇⣀⡾⠀⠀⢳⣄⣘⣧⣀⣠⣟⣀⡼⠃

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡏⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⠀⡴⠶⠶⠶⠶⠶⢦⡈⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⠀⡇⠀⡴⠒⢦⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠂⠒⠒⠒⠀⠀⠀⢸⠀⡇⢀⣷⣤⣼⡀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠉⠀⠉⠉⢉⠉⠉⠉⠁⢸⠀⣇⣯⣀⣀⣀⣹⣼⠇⡇
⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢸⠀⣤⣀⠀⠀⠀⠀⢀⣀⣁⣀⡀
⠀⠀⠀⠀⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠛⠀⠋⠁⠀⠐⠛⢻⡭⢥⣀⡬⢭⠟⢳
⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⢯⠉⠉⠉⠉⠉⠉⠁⢸⡧⠤⡤⠤⢼⠀⠯⠇
⠀⠀⠀⠀⠀⠀⠓⠶⠶⠶⠶⠶⠶⠶⢶⠶⠾⠿⠶⢶⠶⠶⠶⠶⠆⠘⢇⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⣉⣉⣉⣉⠉⠉⢱⡦⠤⠾⠦⠄⠀⠀⢀⢤⡾⣟⠛⣹⢶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠤⠤⠤⠤⠤⠤⢸⡟⠒⠤⣄⡀⢀⠜⠁⠀⠑⣌⠉⡡⠋⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠶⠶⢦⡀⠀⢸⡧⣄⠀⠀⠉⠁⠀⢰⡆⠀⢸⠙⡇⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⣀⠰⡃⣎⢙⡆⡷⢀⡸⠃⠀⠀⠲⢄⣀⡤⢺⡇⠀⠀⠁⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⢮⣭⠖⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⣾⢸⡶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⡶⠛⠀⠀⠨⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠞⠛⠁⠀⢀⣴⡾⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠐⢿⣯⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣷⣦⣄⠀⠀⠀⠀⠀⠙⣷⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⠀⠀⠀⠀⠀⠿⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢠⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⣿⠁⠀⠀⠀⠀⠀⣼⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⢀⣼⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢸⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⢴⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⠤⠶⠶⣾⣿⣷⣦⢸⣷⣴⣦⣤⣤⣄⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣟⡉⠉⣉⠉⠀⠀⠉⣛⡻⣿⣿⣿⠉⠉⠙⠛⠛⢿⡿⠿⢿⣿⠿⣿⣶⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⣿⣿⣿⣿⣿⣿⣿⣷⣬⣭⣀⣀⣀⣀⣩⣿⣿⣟⣁⣀⡀⣀⣀⣀⣉⣉⣹⣦⣥⣶⣾⣿⣿⠛⣻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⡛⣿⣿⣿⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣝⢋⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⡟⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⡇⣼⡟⠉⠙⠻⣿⣿⠛⠋⠩⠝⢿⣿⠿⠋⠉⠉⠙⢻⣿⡄⠀⡀⢘⣷⣯⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣿⣿⣿⣧⣹⣆⣴⣿⡇⢽⣧⣠⡄⣿⣿⡄⢺⣤⡄⠀⢸⣿⣷⣮⣤⣿⡿⢿⡏⠉⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⣾⣏⠉⠙⢻⣿⣷⣾⣿⣿⣿⣼⣧⣿⣴⣿⣿⣷⣦⣽⣤⣴⣿⣿⣿⣿⣿⣏⠁⠀⢹⣄⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣿⣿⠆⣀⣾⣿⠙⠛⠉⠙⣿⠿⣿⣿⠿⠟⠋⠙⢻⡿⠛⠋⠙⣿⠋⠁⠀⠸⣧⣤⣼⡟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣀⣿⣧⣾⣧⡀⠀⠀⠀⣸⣷⣀⣀⣴⣿⣦⣠⣤⣺⣿⣧⣿⣧⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣿⣿⠿⡿⠿⠟⠿⠛⠛⠹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⢁⠘⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈