Money Symbol Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⢞⣛⣛⣯⣭⣭⣭⣭⣭⣭⣭⣛⣛⡳⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡵⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⢮⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣵⠞⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣮⣛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⠞⠋⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⢦⣄⡀⠀⠀⠀⠀⠙⠳⣍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⠟⠁⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠈⠳⣌⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⢫⡾⠁⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠘⢷⡝⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡾⣱⠟⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠻⣎⢷⡀
⠀⠀⠀⠀⠀⢀⡾⣱⠏⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠈⠉⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠹⣎⢷⡀
⠀⠀⠀⠀⠀⣾⢱⡏⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢹⡎⣷
⠀⠀⠀⠀⣸⢇⡿⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡞⠋⠉⠉⠙⠻⢦⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢿⡸⣇
⠀⠀⠀⠀⣿⢸⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⡇⣿
⠀⠀⠀⢰⡏⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠙⠛⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⢹⡆
⠀⠀⠀⢸⡇⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⢸⡇
⠀⠀⠀⢸⡇⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠸⣇⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣸⠇
⠀⠀⠀⠀⣿⢸⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣦⡀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡇⣿
⠀⠀⠀⠀⢹⡎⣷⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠈⠻⢦⣄⣀⢀⣀⣠⡾⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣾⢱⡏
⠀⠀⠀⠀⠀⢿⡸⣇⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣸⢇⡿
⠀⠀⠀⠀⠀⠈⢷⡹⣆⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣰⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡹⣦⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡇⠀⠀⠀⣾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⣴⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣜⢷⡀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⣣⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡙⢦⡀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⢀⣴⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⢦⣄⠀⠀⠀⠀⠈⠙⠳⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠞⠋⠁⠀⠀⠀⠀⣠⡴⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣭⡻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⡴⢟⣭⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⣝⡳⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⢞⣫⡵⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢮⣭⣭⣟⣛⣛⣛⣛⣛⣛⣻⣭⣭⠵⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⢤⡀⠀⠀⠀⢀⡤⠔⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠢⣄⠴⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠱⡄⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⣆⣀⡰⢦⣴⣁⣴⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠉⠀⠀⣀⣀⡀⠀⠈⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠋⠉⢀⠛⡅⠉⠓⠲⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⡴⠖⣿⢻⡷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⢸⡇⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⣻⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⣿⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣦⣼⣡⣯⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠴⠚⠢⢤⣀⣀⣀⣀⠤⠴⠮⠤⠤⠴⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠤⢶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢲⢃⣞⡆
⠀⠀⠀⠀⠀⠀⠀⠨⠀⢐⣋⣻⠀⠀⡞⢀⣖⡆⠀⠀⢸⠰⠯⡇⠀⠀⠀⠀⢘⡀⣟⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠖⢺⠀⠀⡇⢬⠵⠇⠀⠀⢸⠌⠽⡇⠘⠅⢀⣿⠀⡆⢴⡂⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⡀⠠⠅⢈⣉⣭⠀⠀⡓⠭⢽⠃⢀⠄⠀⡯⢭⡇⢐⡅⢐⣻⡇⡣⠽⣇⣀⠀⢸⡀⢽⡀
⠀⠀⠀⠀⠀⠏⢑⣺⠀⠒⠒⡛⠀⠀⢩⠹⢽⡀⢒⣚⣆⣮⣽⣆⣿⣧⣰⣾⣧⣬⣭⣥⣧⣰⣶⣿⣿⡁
⠀⠀⠀⠀⠀⢱⣐⣺⠄⣉⣩⠵⡀⠀⠈⠉⠙⣇⢀⣐⣷⣯⢈⣁⣀⡁⣉⡛⠋⢁⣩⣕⠾⡟⠛⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠂⠤⠔⠚⠠⠼⠋⡁⣀⡄⠈⠡⠿⣮⣋⣍⣉⣹⣭⣍⣀⣀⣏⣡⣈⢉⣉⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣤⣴⡀⠐⠒⠒⠁⠀⠀⠠⣀⠭⠍⠉⢉⠽⠯⠜⠋⠉⠁⠀⠈⠡⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠤⢤⣀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⠉⠀⠀⠀⡏⠛⠋⠀⠈⡳⢒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠇⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢀⠀⠀⠀⣄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⣠⣦⣤⣭⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⡼⠀⠘⠂⣸⡟⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠸⠃⠘⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢰⣿⣿⡿⢩⣿⣿⣿⠈⠻⣿⣿⣷⠀⠀⠀⠀⡀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣸⣿⣿⡇⠀⠀⠹⢿⡿⠁⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠄⠂⠁⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠈⠙⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀⠤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⢸⣿⣿⡇⠀⢀⣾⣿⣿⠇⠀⠀⠀⠀⡁⠄⠒⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠑⠄⠀⠀⠀⠀⠈⢿⣿⣿⣦⣸⣿⣿⣧⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠂⡉⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢧⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠐⢉⡀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠉⠀⠢⠄⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡄⠀⠢⣀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠒⠒⠒⠒⠒⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⣢⣤⣤⡄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠛⠁⠀⢰⡍⢏⣠⠖⣠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢖⡚⠛⠶⠲⢢⣄⡀⠀⠀⢀⣴⣶⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣥⣀⣀⠀⠀⠀⠀⠀⠀⠉⠒⠲⣿⣿⣿⣿⣿⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⠢⠒⠈⠐⡀⠀⠉⠭⠓⡲⢦⣤⣀⠀⡄⠐⣿⣿⣿⣟⢟⣀⣮⡇
⠀⠀⠀⠀⠀⠀⡀⠤⠐⣈⠗⠺⠆⠀⠀⠀⠀⠈⠓⠶⠶⠒⣂⢽⣵⣾⠈⢎⣞⣿⣟⠓⣠⢿⣿⢿⢻
⠀⠀⠀⠀⠀⣸⡦⣠⣉⡁⢆⣀⠒⠀⠀⠀⡀⠀⠀⣀⣤⣪⣶⣿⣿⣿⠀⣾⡾⠛⠒⠛⣿⣿⡿⣵⠋
⠀⠀⠀⠀⠀⢸⠉⠲⠤⣭⣛⠲⣭⡒⠂⣩⠗⣛⢬⣲⣽⣾⣿⣿⣿⡿⠞⠋⠁⠀⢀⡤⣻⣾⣼⡫⠵⠢⣤⣀⡀
⠀⠀⠀⠀⠀⠾⣀⠀⠉⣌⣒⠯⠝⠻⣷⣶⣯⣿⢿⣻⣿⣿⡟⠻⠋⠁⣐⡶⣶⣜⣮⢶⣿⣟⣭⠽⣇⠀⠈⠠⠬⣱⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢨⣝⣒⠶⢤⣐⢒⣿⣿⠿⠺⠏⠀⠒⠒⢉⢜⣾⣿⢩⢃⣰⠋⠁⠀⠆⠙⠫⢓⣶⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠲⠟⢂⠉⡁⠲⠤⡾⠻⠟⠀⠉⠀⠀⢀⡴⣵⣿⣿⣿⢨⡜⠃⣀⠤⠊⠀⠀⠀⠐⠂⠈⢹⠋⠓⠠⠤⣀
⠀⠀⠀⠀⠀⠀⣾⡃⠐⠥⠤⠒⣂⠠⠭⣒⡂⠀⢀⠄⢂⡲⢒⣰⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⣁⡂⡀⠠⡀⠑⣊⠴⣚⣄
⠀⠀⠀⠀⠀⠀⢻⡀⠬⠥⠉⣁⢐⣒⠀⠠⠌⠉⠑⠒⣒⠒⠿⣫⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⢑⣫⢕⣪⣵⡟
⠀⠀⠀⠀⠀⠀⠸⣖⡦⠩⠥⠤⠔⠒⣀⣀⡾⢏⣉⣘⣓⣒⣺⣿⣿⠟⠻⢿⣦⡉⠻⢄⡲⠤⠤⢀⣂⠕⣖⣬⣴⣾⣿⡿⠿⢛⡝
⠀⠀⠀⠀⠀⠀⠀⡼⣕⢍⡫⣛⠖⠖⡒⡂⢚⠞⠱⡳⠂⠖⠖⠿⣕⢤⠀⠀⠉⢾⡭⣒⣲⣶⣝⣓⣯⣿⣿⠿⠛⡋⠥⣐⣫⡵⠃
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠈⠑⠤⣤⣀⣁⡁⠈⣀⡁⢁⣡⣮⣼⣿⣄⡀⠀⠀⠙⠀⣾⣿⠿⣟⣋⢭⣔⠪⡑⠦⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⢿⣦⣔⠒⣋⣬⡿⣿⣿⣿⢿⡫⠛⠐⠤⡀⠀⣷⣾⡯⠾⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⡀⠀⠘⢴⠄⠙⣯⣿⢿⣿⠷⠋⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠄⣃⢸⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠢⢀⠄⡨⠀⠀⡀⢀⠀⠈⢀⠔⠁⢀⠄⡀⠊⠐⠀⠀⠀⠁⠈⠈⡀⢀⠔⡠⢊⢐⠁⡠⢂⠔⡡⡨⡪⢢⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⠕⠡⠊⠄⠨⠂⠕⠠⠊⠔⡁⢊⠔⠡⠪⢂⠀⠐⠠⠀⠔⠁⠁⠊⠐⠁⠊⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠲⢤⣞⠙⡗⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠃⠈⠉⠉⠁⠀⠓⠒⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠈⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⣴⡆⢸⡍⠲⣿⠀⡟⠲⣄⡀⠀⣠⠞⠁⠀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣄⡇⢸⠄⠀⣿⠀⡇⠀⣿⠉⠲⣅⠂⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠳⠇⢸⠄⠀⣿⠀⡇⠀⣿⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠀⢧⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠓⢶⣄⠀⡀⠀⠀⠀⠀⠀⠘⡎⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⢸⡁⠀⣿⠀⡏⣦⠀⠀⠀⠀⠀⢸⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠙⢗⢦⡀⠀⠀⡇⢸⡄⠀⣿⠀⡇⢸⠀⠀⠀⠀⠀⢸⠃⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠳⢵⣄⠀⡇⢸⠆⠀⣿⠀⣧⠏⠀⠀⠀⠀⠀⣸⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡐⠀⠀⠀⠀⠀⠀⠉⠛⠃⠸⠥⠤⠟⠀⠁⠀⠀⠀⠀⠀⢠⠋⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⢈⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢶⡄⢠⣤⡤⣄⠀⡶⣏⡁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠄⢙⣯⠀⡗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣷⣤⡏⠻⢽⣦⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠈⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⢀⠀⠙⢿⣷⣶⡾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣠⠞⠲⠶⠖⠋⣳⣶⡿⠛⠁⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⢠⡾⠁⠀⠀⣠⣴⡿⠟⠉⣀⡀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⡴⠋⠀⣀⣴⣿⠟⠉⢀⡴⠞⠉⠙⠓⠒⠚⢳⡀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⣁⣴⣾⠟⠋⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⣄⣤⣾⠿⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣤⣾⠿⠋⢁⡀⠀⠺⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣡⣶⡿⠛⠁⣠⡴⠋⠁⠀⢰⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠀⠀⣠⣼⡿
⠀⠀⠀⠀⠀⢠⣾⣿⣷⡿⠛⠁⣠⡴⠛⠁⠀⠀⠀⠀⠘⢿⣶⣶⠿⠿⣶⡄⠀⠀⠀⠀⠀⢀⡤⠞⠉⣀⣴⣿⠟⠉
⠀⠀⠀⢀⣴⣿⣿⡟⠁⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣤⠞⠋⢀⣴⣾⠟⠋
⠀⠀⠀⣾⡟⠀⢿⣧⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⡾⢿⡅⠀⠀⠉⢀⣤⣾⠿⠋
⠀⠀⠀⢿⣧⣄⠀⠻⣷⡄⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⢀⣠⣾⠿⠋⠁
⠀⠀⠀⠀⠙⠻⣷⣄⡙⣿⣆⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⢁⣠⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣮⣿⣷⡀⠘⢧⡤⠤⢤⣀⣠⡴⠛⠁⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄⠀⠀⠀⠀⠙⠁⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⣀⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠉⡍⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠁⠀⠀⠉⠋⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠉⡍⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠀⠀⠁⠁⠉⠱⠄⢀⡀⠤⠀⠀⠀⠀⣀⣀⣐⠋⣡⣴⣚⣀⡀⠀⠐⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠐⠲⢒⠈⢷⠈⡶⠍⠁⠐⠒⠉⠉⠀⠀⠀⠀⠀⣀⢀⠉⠩⢕⠢⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠀⠀⠤⣠⠤⢿⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⠲⠛⠻⠋⠁⠀⠠⠾⠶⠒⢤⣄⣀⠈⠑⡀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⡀⠤⠤⠀⣠⠀⡀⠀⡇⠀⡀⠀⠀⣂⣀⠼⠀⠀⢀⢀⣠⠀⣠⠭⢒⡩⠥⣆⣬⢆⣂⠤⣄⣐⠒⠀⡐⠦⠐⠒⡐⣀⠥⠔⠂⢛
⠀⠀⠀⠀⠀⠆⠀⠀⢀⣀⡇⡆⠀⣤⠀⠀⠀⠀⠀⠀⢠⠏⠙⠆⠠⣙⡿⢒⠈⠭⢒⡓⠶⣈⠉⠓⠛⠥⠒⠂⠔⠀⣒⠬⠳⠚⠓⠀⠀⠀⢠
⠀⠀⠀⠀⠠⣆⠀⠀⣠⣤⠶⢒⡲⡌⠀⠉⠙⠓⡚⡋⠄⣏⠀⠀⠃⢈⣇⣠⢀⣀⠀⠀⠀⣀⡈⢁⠈⠁⠀⠀⠁⠀⣀⢠⣢⠥⠲⠜⣀⠤⡾⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠉⠉⣏⠁⠀⡈⡈⣍⠀⠐⠂⠀⠐⠖⣨⣷⠆⣠⣌⠵⡋⢀⣉⣉⡊⠉⠙⠛⠚⠿⠿⢗⣢⠀⠀⢋⣭⠥⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠒⠓⠋⠑⠒⠒⠐⠖⠚⠉⠀⠀⠀⠀⡾⠩⠁⠂⠠⠤⡀⡄⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⣀⣀⢀⢂⠤⠇⠒

⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣶
⠀⠀⠀⣿⠀⠀⠀⠀⢰⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⡆⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⢀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣀⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣫⠵⠖⠒⠒⠶⢭⣝⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣵⠋⠁⠀⣠⡶⣦⣀⠀⠈⠳⣜⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⣼⠃⠀⠀⣾⠁⣾⣆⣹⡆⠀⠀⠹⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⠀⠀⠘⢦⣈⠙⢷⡀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⢿⠀⠀⠀⢿⠉⣿⡇⢠⡇⠀⠀⢠⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢷⡀⠀⠈⠳⣦⡶⠛⠀⠀⣠⢞⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡶⢤⣄⣀⣀⣤⡴⣞⡵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠾⠷⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⠶⠶⠶⠶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⠉⢸⣿⠈⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢸⣿⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⣸⣿⣀⣀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠶⠶⠶⠶⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⣀⣀⡀⣀⣀⣀⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⢀⠀⢀⡀⢙⣿⣇
⠀⠀⠀⠀⠀⢸⣿⠀⢀⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠁⠀⠀⠀⢰⡖⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⢰⡄⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠲⣆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⡄⢀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⡀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠁⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠂⢺⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣀⣀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠀⣿⡇
⠀⠀⠀⠀⠀⠘⢿⣷⣾⣿⡄⠀⠀⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣅⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠘⠃⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⣿⠀⠀⠀⠀⠀⠀⣾⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠳⠞⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠀⠀⠀⠀⠀⠀⠘⠷⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠲⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⣄⠀⠀⠀⠀⠀⢠⠶⠞⠻⠿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⢻⣿⣿⡟⠻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠀⢸⣿⣿⡇⠀⠘⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣦⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠻⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣇⠀⠀⢸⣿⣿⡇⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣀⢸⣿⣿⡇⢀⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⣿⣿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⢤⣤⠖⡛⠢⡤⣤⡖⢛⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠛⠻⢾⣽⣏⣬⣎⢈⣾⣡⢽⣝⣟⣝⣶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡒⣽⠁⠊⣿⠛⢳⡿⠀⢀⣠⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⠿⣇⣘⣾⠀⣼⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣿⣯⣥⣊⡽⠿⢻⢉⣳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣾⣹⣧⣬⣶⣿⣿⡟⠳⡦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠫⢻⠟⠁⠁⠁⠀⠣⠑⢄⠀⠁⢡⢩⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢁⠀⠐⠁⠀⠀⢀⣀⣀⠀⠀⠀⠁⠀⠀⠀⠊⠔⣠⠛⡲⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠔⢁⡄⠊⠁⠀⢀⢤⠄⠤⠼⢗⢼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢚⠔⡡⢓⢦⡀
⠀⠀⠀⠀⠀⡠⠊⠁⡘⠈⠀⠀⠀⣰⠕⢕⣵⣞⣝⢕⣵⢝⠗⣤⡀⠀⠀⠀⠀⠀⠀⠀⢪⠔⡡⢪⢗⡄
⠀⠀⠀⡠⠊⠀⠄⠂⠀⠀⠀⠀⠀⣗⢴⣿⠀⢸⢷⣽⡏⠓⢷⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⢩⠔⡡⢊⢜⢆
⠀⢀⠔⠀⡴⠘⠀⠀⠀⠀⠀⠀⠀⠹⣯⡾⡷⡾⡺⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠔⡡⢊⢎⢆
⢀⠎⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⡫⣪⣾⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢊⡴⡡⢃⢺⡄
⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⠿⣾⡫⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢈⠜⡐⡱⣷⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣿⣄⠀⢸⣵⣿⠀⠈⣿⢗⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡥⡪⠮⣮⣮⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡕⢹⡧⣼⣿⡻⣤⡴⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠪⢂⢔⣴⡝⡵⡇
⢹⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠵⣿⠉⣺⣫⣺⠯⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⡪⢔⡕⡩⡫⢊⣼⠁
⠀⢧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠔⣊⠔⡡⠊⢊⠔⣡⠆
⠀⠈⢢⠀⢐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⢀⠀⡀⡀⢠⢂⢄⣁⠜⡡⢫⢎⡕⡡⢪⡿⡉⠉⠉⢁⣠⣤⣀
⠀⠀⠀⠱⣌⠐⢠⡄⠀⡀⡀⠠⡠⠠⠡⢊⢌⠔⡡⢂⠕⢊⠔⡡⢊⢔⠔⡡⠣⡪⢃⠎⡔⡡⢩⣪⣔⣓⢒⣒⣒⣒⣒⣒⣒⣀⣀
⠀⠀⠀⠀⠀⡗⢤⣀⡁⠁⠁⠀⠁⣁⣁⣀⣈⣀⣁⡀⠤⠤⠬⠄⠁⠉⢊⢔⡕⡱⣡⢊⡼⠴⠯⠥⠤⠤⠤⠀⠀⠀⢀⣀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢋⡛⣛⣋⣈⠛⢻⣿⣿⠿⠿⢿⠿⠿⢃⣐⡲⠶⢷⣾⣾⣷⣭⣭⡭⠭⠭⠭⠭⠭⠭⠭⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋⢋⠀⠀⠒⢲⣖⣒⡒⣒⣒⣂⣒⠚⠛⠉⠋⠉⠛⠁⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⣦⡶⠟⠉⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠠⡄⠀⠀⣾⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣧⣀⣼⣇⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⢻⡟⠛⢿⡏⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠸⠁⠀⠈⣷⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⡟⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⠛⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡇⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⣀⣇⣀⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣀⣀⣀⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠻⠿⠿⢟⣙⠿⠟⣩⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢃⣾⣿⣷⣿⠟⣋⣴⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢋⣴⣿⣿⠿⢋⣴⣾⣿⡿⢟⣩⡙⠿⠿⠟⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⣡⣾⠿⢋⣥⣾⣿⡿⠟⣩⣴⣿⣿⣿⣿⣿⣿⣦⡹⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⢛⣥⣾⣿⣿⣿⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⢛⣡⣶⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠟⣩⣴⣿⣿⠿⢻⣿⣷⠌⠉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣩⣴⣿⣿⠿⢋⣥⣾⣿⣿⣏⠀⢾⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣹⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢀⣴⠟⣋⣴⣾⣿⡿⢛⣡⣾⣿⣿⣿⣿⣿⣿⣦⣀⣀⣤⣤⣄⠈⢿⣿⣿⣿⣿⡿⠟⣩⣶⣿⣿⠟⠋
⠀⠀⠀⠀⣰⣿⡏⣼⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠀⣸⣿⣿⠟⣫⣴⣿⣿⠟⠋
⠀⠀⠀⢸⣿⣿⣷⡘⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣠⠐⢿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠘⢿⣿⣿⣿⣌⢻⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⠹⣿⣿⣦⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣥⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡘⢿⣿⣷⡘⢛⣛⣛⠻⡿⢛⣡⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣆⠻⣿⣿⣿⣿⣿⣷⣶⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⡙⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠋⠉⠁⣀⣀⣀⣀⣀⣀⠈⠉⠙⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⣀⡤⠶⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠶⢤⣀⠈⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠛⠻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢹⣷
⠀⠀⠀⠀⠀⠀⢸⣿⠁⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣤⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠉⠓⠶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠶⠛⠉⢀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣄⣀⡀⠉⠉⠉⠉⠉⠉⢀⣀⣠⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠛⠉⠉⠉⠉⠉⠉⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣤⠴⠚⠋⣉⣉⣠⣤⣤⣄⣉⣉⠙⠓⠦⣤⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡤⠚⢁⣠⠖⠚⠉⠁⠀⣀⣀⣀⣀⠀⠈⠉⠓⠲⢄⡈⠓⢤⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢀⡴⠋⣠⠞⠉⠀⠀⠀⠀⣀⣠⠏⠉⠉⠹⣄⣀⠀⠀⠀⠀⠉⠳⣄⠙⢦⡀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⣰⠋⣠⠞⠁⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠈⠳⣄⠙⣆⠀⠀⠘⢆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⣴⠁⡰⠃⠀⠀⠀⠀⢠⠏⠀⠀⠀⡠⠶⠒⠲⠦⣄⠀⠀⠀⠙⣆⠀⠀⠀⠀⠘⣆⠈⣦⠀⠀⠘⣆
⠀⠀⠀⠀⢠⠇⠀⠀⣰⠃⣸⠁⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢧⣠⠤⠖⠋⠀⠀⠀⠀⠀⠈⣇⠘⣆⠀⠀⠸⡄
⠀⠀⠀⠀⣼⠀⠀⢀⡏⢠⠇⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢹⡀⠀⠀⣧
⠀⠀⠀⠀⡇⠀⠀⢸⠃⣸⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠉⠓⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⡇⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⢸⡄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⡏⢠⡇⠀⠀⢸
⠀⠀⠀⠀⢻⠀⠀⠈⣇⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢰⠃⣸⠁⠀⠀⡟
⠀⠀⠀⠀⠘⡆⠀⠀⠹⡄⢹⡀⠀⠀⠀⠀⢠⡴⠚⠉⠈⢧⡀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢀⡇⠀⠀⠀⠀⢀⡏⢠⠏⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠳⡀⠱⡄⠀⠀⠀⠀⢳⡀⠀⠀⠀⠙⠲⠦⠤⠶⠚⠁⠀⠀⠀⡼⠁⠀⠀⠀⢠⠎⢀⠞⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠹⣄⠙⢦⡀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⢀⡴⠋⣠⠏⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠈⠳⣄⠙⢦⣀⠀⠀⠀⠀⠉⠓⣆⣀⣀⣰⠒⠋⠉⠀⠀⠀⣀⡴⠋⣠⠞⠁⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠳⢤⡈⠙⠦⢤⣀⡀⠀⠉⠉⠉⠉⠀⢀⣀⡤⠴⠋⢁⡤⠞⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠈⠛⠲⢤⣄⣉⣉⠙⠛⠛⠉⣉⣉⣠⡤⠖⠛⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⣤⣀⣀⣀⣀⣀⣀⣤⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡤⠴⠒⠒⠒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⡏⠙⠉⠀⠀⠀⠀⣶⢲⠏⢹⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠈⠉⢉⣙⣿⠢⣄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠉⠓⠒⠒⡻⠛⠉⠉⠉⠉⠑⢮⣦⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⠤⠤⣄⡀⠀⢱⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡞⢁⡴⡶⠄⠹⡄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⣭⣯⠇⢠⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠀⠀⠈⠳⠤⠥⠴⠋⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⣀⣤⣠⢀⠀⣠⠄⢠⢄⣄⡤⣠⡄
⠀⠀⠀⠀⠀⠀⢠⣤⡀⣠⣀⡀⣀⢄⣄⣀⢠⠀⣈⠄⣀⣀⢠⢠⣠⣄⣀⡠⣠⠀⣀⢀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢰⣶⣶⣷⣶⣾⣶⣶⡆⠀⠀⠀⠠⣤⢀⣹⣿⣅⠠⣤⠀⠀⠀⠀⠀⣦⣤⣤⣀⣯⠗
⠀⠀⠀⠀⢸⣿⣿⣏⡲⢽⣿⣿⡇⠀⠀⠀⣾⣿⣾⣏⡩⣿⣷⣿⣦⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⢸⣿⣿⣷⣭⣼⣿⣿⡇⠀⠀⠀⠛⠛⣿⣷⣵⣿⣿⠛⠋⠀⠀⠈⠙⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⣤⣶⣳⣦⣤⣀⠀⠀⠀⠀⠀⣀⣈⡋⢀⣄⠀⠀⠀⠀⠀⠀⢀⣀⢠⣦⢘⡟
⠀⠀⠀⠀⠀⣿⠀⣿⢸⡇⢸⡇⠀⠀⠀⠀⠸⣿⣸⡇⠘⠚⠁⠀⠀⠀⠀⠀⢘⣟⢰⣷⢸⣿
⠀⠀⠀⠀⠀⣿⠀⣿⢸⡇⢸⡇⠀⠀⠀⠀⢶⣌⡉⠤⣶⣦⣀⡀⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿
⠀⠀⠀⠀⣴⣿⣶⣿⣾⣷⣾⣷⡄⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠇⠀⠀⠀⠀⠾⠿⠾⠿⠾⠷⠆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣽⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡈⠉⠛⠻⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⢀⣾⠛⠛⠉⠉⠉⠉⠉⢉⣉⣙⠛⠛⠛⠿⢿⣿⣶⣦⣄⡀⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⡀⣰⡿⣿⡀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠿⢶⣾⠟⠋⠛⠀⢀⣀⣠⣤⣤⣍⡙⠛⠻⢷⣦⣀⠀⠙⢷⣤
⠀⠀⠀⠀⢀⣠⣤⡶⠾⠟⠛⠛⠋⠉⣉⡛⣻⡇⠘⢿⣦⡀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠿⣷⣶⣶⠾⠟⠋⠉⠉⠉⠻⣿⣆⠀⠀⠉⢻⣧⠀⠀⠹⣧
⠀⢀⣤⣾⠟⠋⠁⠀⠀⣠⣶⡾⠟⠛⠛⠛⠻⣧⠀⡎⠉⠻⣶⣄⡀⠀⠀⠀⠈⠙⠛⠿⢶⣤⣤⣀⣀⠀⠀⠀⠀⠾⠟⠛⠻⠃⢀⣸⡟⠀⠀⠀⣿
⢀⣾⡟⠁⠀⠀⠀⢠⡾⠛⠁⠀⠀⢀⣤⣤⠀⠙⣿⣅⠀⠀⠀⡝⠻⢷⣤⣄⣀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠶⠶⠶⠾⠿⠛⠉⠀⠀⠀⢀⣿
⣼⣿⠃⠀⠀⠀⠀⣿⡃⠀⠀⣤⣤⣿⡏⠀⠀⠀⠈⢻⣷⣄⢠⠃⠀⠀⠀⠉⡟⠛⠷⢶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠟⣿
⣿⢿⣧⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠹⢷⣦⣤⣤⣶⠟⠋⠙⢿⣶⣤⣀⡀⢰⠃⠀⠀⠀⠀⠀⡟⠉⠉⠛⠛⠛⡛⠛⠛⠛⠛⠛⡟⠉⠉⢀⡇⣰⡟
⣿⡀⡟⣷⣄⠀⠀⠀⠈⠙⠿⣦⣤⣀⣀⡈⠉⠉⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣦⣤⣤⣀⣀⣇⣀⣀⠀⠀⢀⡇⠀⠀⠀⠀⢀⣃⣀⣤⣼⡿⠋
⢹⣷⡇⠈⠙⢷⣦⣄⡀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠷⠶⠶⠶⠶⠞⠛⠋⠉⠉⠉⠛⠛⠛⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠻⣿⣦⡀⢰⠉⠙⠛⠷⣶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠋⢹⢸⣷
⠀⠀⠈⠻⣿⣾⣄⡀⠀⠀⡇⠀⠈⠉⠙⠛⡟⠻⠷⠶⠶⢶⣶⣶⣶⡶⠶⠿⠛⡟⠋⠁⠀⢶⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣧⣀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣧⣀⣤⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣶⣷⣶⣤⣤⣤⣤⣤⣷⣴⣶⣶⣿⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠂
⠀⠀⠀⠀⢰⣶⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢤⡀⣠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⠀⢸⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⡇
⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠈⢠⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣷⣶⣤⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣯⣿⣿⣿⣿⣽⣿⣷⣿⡿⠃
⠀⠀⠀⠀⡀⠒⡒⢀⡀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⠿⢿⣿⡿⠛⠻⠛⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠰⠖⠞⠦⠢⠔⠀⠶⠂⠷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠺⠘⠃⠂⠃⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣍⠠⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠋⠉⠉⠀⠀⠉⠉⠙⠻⢿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠺⠿⠿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⠁⢸⡇⠙⣿⡷⠀⠀⠀⠀⠀⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣷⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⢰⣿⣧⠀⢸⡇⠀⣿⣿⠇⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣼⣧⣶⣿⠟⠀⠀⠀⠀⢀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠉⢹⡏⠉⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉
⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠿⢿⣿⣷⣦⣄⣀
⣴⣿⠛⠿⢿⣿⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣤⣶⣤⣀
⢻⣿⣤⣀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣷⣶⣦⣤⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⡛⢿
⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣷⣶⣶⣶⣶⣶⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠈⢁⣾
⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⡿⠿⠿⣿⣿⣿⠃⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣶⣤⣤⣶⣶⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠻⣿⣶⣤⣾⡿⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣥⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⡀
⠀⠀⠀⣀⣀⣀⠀⠀⣀⠀⠀⠀⢠⣤⣿⣿⣤⡄⠀⠀⠀⠀⠀⣰⡏⢹⣆⠀⠀⠀⠀⠀⠀⣀⣺⣗⣀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⠇
⠀⠰⣉⠀⠀⠀⣹⠶⡁⠀⠀⠀⠘⠟⣿⣿⠻⠃⠀⠀⠀⠀⠺⣤⡅⢨⣤⠗⠀⠀⠀⠀⠀⠉⢹⡏⠉⠀⠀⠀⠀⠀⠠⣄⣏⣨⠄
⠀⠀⠀⠉⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠀⠀⠀⠀⠀⠀⠀⣸⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣀⣼⣧⣀⡀⠀⠀⠀⠀⢠⡶⠋⠀⣀⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣀⣠⣔⣤⠀⠀⠀⠀⠀⢰⣶⢻⣟⣶⡆
⠀⠀⢈⣿⠀⠀⣿⡁⠀⠀⠀⠀⣿⡇⠀⠀⠙⠁⠀⠀⠀⠀⢘⣿⠛⠛⣿⡇⠀⠀⠀⠀⠀⠩⣼⡴⠋⣆⠀⠀⠀⠰⣿⢶⣯⣿⠶⡷⠆
⠀⠀⠉⠉⢻⡟⠉⠁⠀⠀⠀⠀⠘⠿⣦⣤⠴⠃⠀⠀⠀⠀⢸⡿⠀⠀⢻⡇⠀⠀⠀⠀⠘⢦⡼⠙⠾⠃⠀⠀⠀⠀⠸⠿⢼⡯⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⡀
⠀⠀⣠⣾⡟⢻⡗⣄⠀⠀⠀⠀⢀⢼⡧⣿⣄⡀⠀⠀⠀⠀⣠⡴⢻⡟⢦⣄⠀⠀⠀⠀⠀⠄⢸⡇⠤⠀⡇⠀⠀⠀⠀⠀⣀⣶⢠⡾⠂
⠀⠀⣿⣿⠟⠛⠀⢸⠀⠀⠀⠀⢸⠘⣋⡙⢺⠀⠀⠀⠀⠀⢷⡱⣼⣧⢎⡾⠀⠀⠀⠠⠶⠶⠾⡷⠶⠶⠇⠀⠀⠀⠿⣻⣿⣿⣏⡀
⠀⠀⠙⢧⣘⣃⡤⠋⠀⠀⠀⠀⠘⡸⣿⡿⡸⠀⠀⠀⠀⠀⠈⢻⣾⣷⠟⠁⠀⠀⠀⢰⠀⠒⠐⡇⠛⢀⠀⠀⠀⠘⠛⠋⠾⡗⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠐⡶⠜⠣⢶⠂⠀⠀⠀⠀⠀⠀⣯⣽⠀⠀⠀⠀⠀⢀⣀⠘⢻⡟⠃⣀⡀⠀⠀⢀⡔⠉⢸⡇⠉⢢⡀⠀⠀⢀⡔⢿⢤⡤⡿⢢
⠀⠀⢻⡅⠀⠀⢨⡟⠂⠀⠀⠀⠈⠉⢹⡏⠉⠁⠀⠀⠀⠸⣿⠖⢺⡗⠲⣿⠀⠀⠀⢸⠖⠒⢺⡗⠒⠲⡇⠀⠀⢸⣠⣾⣏⣹⣷⣄⡇
⠀⠀⠛⢻⡴⢦⡏⠛⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⢠⣼⣧⡄⠉⠁⠀⠀⠀⠣⣀⢸⡇⣀⠜⠀⠀⠀⠀⠣⣀⢹⡏⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠈⠉⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⣀⡀⠀⠀⠘⣞⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠈⠉⠺⣝⡢
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣾⠉⢉⠁⠀⠀⠀⢹⡒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣤⢤⡀⠀⠀⠈⢆⣡
⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠸⣿⠿⠿⠁⠀⠀⠀⢸⡭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣅⣧⣤⣄⠀⠀⠈⣆⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠙⠦⠤⠆⠀⠀⠀⣾⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⣄⣣⡼⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⠀⠀⠀⠀⠀⠀⢀⣼⡭⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠈⠉⠃⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠾⠛⠋⠀⠀⠀⠀⠀⠀⣀⠴⠒⠉⠉⠙⠉⠉⠓⢤⣀⠀⠀⠀⠈⠒⠤⢄⣀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡓⠀⠀⠀⠀⠀⣴⠶⠦⣄⠀⠀⠀⠘⡿⢧
⠀⠀⠀⠀⠀⠀⢀⠀⠒⠉⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠰⣾⣇⠀⠀⢿⠀⠀⠀⠀⣯⣻⡆
⠀⠀⠀⠀⠀⠀⠎⠀⣀⠀⣰⠀⠀⠘⡎⠆⠀⠀⠀⠀⢸⠇⠀⠀⠀⣀⣴⠟⠉⠃⠀⠀⠀⠀⠀⠀⡟⢾⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠈⡳⣿⣂⠀⠀⢸⢭⠀⠀⠀⠀⠸⡧⠀⠀⠀⠉⠙⢶⣄⡀⠀⠀⠀⠀⠀⣸⡋⡺
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠉⠙⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡵⢇⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⢀⠀⠀⠀⠀⠀⣀⡴⢯⣳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⢶⣟⡯⠵⠞⠋⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠉⠓⠫⣗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠘⣠⠤⠀⠀⡏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⣠⣶⡯⠑⢦⠀⠈⢫⡽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠲⢚⠱⣴⠀⠀⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⢀⠋⠸⣧⠀⠀⢢⠀⠀⢿⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠙⠋⠀⠂⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢈⠏⠀⠘⢤⢦⡞⠀⠀⢸⣖⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢳⣀⣀⠁⠀⢉⠆⠀⣸⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠦⠔⠁⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⠄⠴⠚⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠉⠀⠈⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⣿⣿⣧⣴⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠛⠛⠛⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣦⣤⣀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠋⠙⠁⢠⣿⣿⣿⡟⠀⠀⠀⣀⣤⣴⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣄⠀⠙⢿⣿⣿⣿⣄⣸⣿⣿⣿⡿⠋⠀⣠⣶⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠙⠛⠛⠛⠛⠋⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡀⢀⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⡇⢸⣿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡉⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡇⢸⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⠄

⡈⣠⣬⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⡉
⠀⣿⠀⡀⢀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⡇⢰⠰⡀⠀⠇⠀⡏⢱⠈⠁⣷⠀⠀⡁⠀⣤⠃⡃⠀⢸⢠⠃⢸⡜⠀⢀⡖⠀⠀⢱⣧⠸⠀⠀⢸⠀⠉⢸⠁⠀⣿
⠀⣿⠀⡇⢸⠀⠣⠀⡆⠀⡇⢸⢘⠀⡇⢡⠀⢰⠀⠄⡁⡅⢀⠧⡸⡀⢸⢃⠀⢈⢠⡏⠀⢸⠿⡇⢈⢁⢸⠀⢸⢸⠀⠀⣿
⡄⣿⠀⠈⠀⠀⠁⠀⠉⠁⠈⠁⠈⠁⠁⠈⠀⠁⠀⠉⠀⠉⠈⠀⠁⠉⠈⠈⠀⠀⠈⠁⠀⠈⠀⠁⠀⠀⠈⠁⠀⠈⠁⠀⣿
⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠂⠀⠀⣿
⠀⣿⠀⠀⠀⠖⠂⠀⠠⠄⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠐⠈⠀⠄⠀⠀⠊⠑⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⢀⢑⠓⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠐⠀⢀⡄⠀⠀⠀⠀⠀⠠⠐⠀⠀⠄⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠠⠄⠄⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣉⡉⠉⠀⠐⡄⠀⣿
⠀⣿⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⠂⠀⠀⠸⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠤⠆⠒⠁⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇⠀⠀⠜⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢁⣝⠛⠛⠛⠋⠉⠙⠛⠛⠛⠛⠛⠛⠙⠋⠙⠋⠛⠋⠋⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠋⠛⠙⠋⠋⠋⠛⠛⠛⠛⠛⠛⠋⡁

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⣄⠛⠀⠃⠀⡆⠐⣶⣀⠎⠤⠁⠀⢡⠀⣀⠆⠀⠀⠁⠀⠒⢁⠂⠈⡀⠛⠀⠃⠀⢢⠀⢶⣀⠼⠠⢀⠀⠀⡀⢀⠀⢀⠃⠁⠀⠒⢃⡀
⠀⠀⡄⠤⠇⠃⣀⡓⠤⠄⠀⡄⠀⡄⠀⠧⠤⠀⡄⠙⣀⠈⠟⠤⣆⢠⠀⡄⠡⠗⠃⢀⡛⠤⠤⠀⡄⠀⡀⠀⠬⠤⠀⢤⠛⣀⠀⠿⠤⣄⠀⡄
⠀⠀⢰⠀⣦⠀⠧⠴⠀⣶⠂⣅⠆⠁⡄⠂⠀⡆⠀⠠⠤⠀⢰⠀⠤⡜⠀⡕⠀⠀⠀⠧⠤⠀⢰⠂⢅⠆⠀⡔⠂⠀⡆⠀⠠⠤⠀⢰⠀⠤⠼
⠀⠀⠉⠉⠁⣦⠀⠠⠂⠏⠒⠀⠀⠐⠁⠃⢠⠁⠖⠀⠀⡄⠈⠂⡄⠐⠚⠛⠁⠀⢰⡄⠀⠖⠊⠓⠀⠀⠐⠙⠂⢠⠈⠘⠀⠀⡄⠀⠂⣄⠐⠒
⠀⠀⠝⠀⢸⠉⡀⠘⠒⠛⠀⢉⡆⢸⠀⠀⡟⠀⠚⠋⠐⡇⠘⠲⠋⢡⠀⣖⠀⡧⠈⢁⠀⠒⠛⠂⢉⡇⢸⠇⠀⣿⠂⠚⠋⠀⡇⠐⠒⠋⠁⠀⣧
⠀⠀⡊⠁⡆⠀⡀⡆⠀⠀⠉⠉⠀⠀⢠⡄⠀⠛⠓⠈⢙⡇⡆⠀⠙⡃⢰⡏⠉⠀⠀⢀⣶⠀⠂⠉⠉⠀⡄⠀⡄⠂⠓⠳⠊⠑⡇⣦⠀⠙⣋⠀⡄
⠀⠒⠘⠛⠒⣤⠀⢅⢠⢸⠀⠁⢘⡀⠸⠗⠁⢦⡘⡛⠋⠑⡇⣺⠇⢀⢽⡛⠛⠓⢠⠄⢯⢠⣸⠀⠁⠈⡇⠠⠟⠀⢢⡐⢛⠋⠙⣿⢸⠏⢀⠱⡇
⠀⢲⣀⣀⣶⣦⣀⡜⣪⠙⠂⠙⠋⠐⠄⠐⠖⢯⣀⣈⠀⡆⡇⠋⣘⡃⠀⣱⠀⢰⢰⣀⣠⢻⠙⠃⠛⠋⠑⡤⠘⠶⢻⣀⣈⠁⡦⣯⠋⣘⣇⠀⢨
⠀⢻⠁⠀⡉⠉⠉⠉⠉⡇⡦⠀⠀⣾⢠⠀⡔⡃⠀⢹⠚⡗⠓⡎⠀⢘⠛⠛⠲⠏⠉⠉⠉⠉⣧⣶⠀⠀⣾⢠⡄⠔⢊⠀⢈⡓⠻⠛⢸⠀⠈⡟⠛⠒⡄
⠠⠚⠓⠚⠚⢦⠀⠀⡇⠧⠗⠀⢀⡿⢼⣠⠗⠚⠒⠓⠢⣇⠔⠚⣒⡃⠀⢰⡆⢀⡇⠀⠀⡇⠷⢿⠀⠀⡽⢼⣇⠖⠚⠓⠋⠒⢦⡰⠚⢓⣚⠀⢠⡆⡇
⠆⣤⠀⡔⠉⠙⠆⠀⠧⠀⠀⢰⣏⠀⠀⣿⢠⠀⠀⢀⡄⣿⢠⡎⠀⠈⡃⢸⣇⣴⣇⡀⠀⠧⠄⠸⠐⣾⠀⠀⣹⢠⡄⠀⠀⣤⢸⠀⣼⠁⠀⣳⠸⣇⠇
⠁⣿⣀⣑⣤⣴⣷⣀⠀⠀⢠⠜⠂⠓⠚⠓⠾⡀⠀⢸⡇⢸⠼⠯⠶⠾⠧⢾⣸⠁⠀⢹⠀⠀⠀⢰⠜⠣⠓⠚⠛⠺⡇⠀⠀⣿⢸⡤⠯⠶⠶⠿⢤⡁
⠘⠙⢰⠋⠉⠳⡀⡬⠶⢤⣸⢀⡀⠀⠀⢠⡴⠧⣄⢠⡏⢹⢀⡀⢀⠤⠤⣤⠯⠷⠤⠾⢤⡤⠤⢼⢀⡄⠀⠀⢀⡄⣇⡤⠤⡏⢻⢀⡀⠀⡤⠤⢤⡇
⠀⠀⢹⣄⣀⡴⢻⡀⠀⠀⣿⢸⡇⠀⠀⢨⠀⠀⣸⠀⡇⢸⢸⡇⢇⠀⠀⡕⠀⠀⠀⠀⣸⠀⠀⢈⢿⡇⠀⠀⢸⡇⣏⠀⠀⢸⢸⢸⡇⢸⠀⠀⢀⡇
⢰⠋⠉⠉⢉⡭⠭⠵⠶⠮⠽⢾⡇⢠⠴⠚⠛⠚⠛⠲⢆⠸⣼⠓⠒⠛⠛⠚⠳⡄⢀⡤⠿⠷⠶⠿⠾⢧⠀⠀⢸⠷⠯⠷⠶⠧⠼⣼⡧⠤⠵⠶⠮⠧⢤
⣹⢠⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⡀⠀⠀⠀⣀⢸⠀⡂⢠⠀⠀⠀⢀⡀⡇⡎⠀⠀⠀⠀⠀⠀⠀⣷⡆⡇⠀⠀⠀⠀⠀⠀⢘
⣸⢸⡇⠀⢸⢸⡇⠀⠀⠀⢸⡇⡇⢸⢸⡇⠀⠀⠀⣿⢸⠀⠇⢸⠀⠀⠀⢸⡇⣇⡅⢸⡇⠀⠀⠀⣿⠀⣿⡇⡇⢸⡇⠀⠀⠀⣿⢸⢸⡇⠀⠀⠀⢸⡇
⠹⠼⡇⠀⢸⢸⡇⠀⠀⠀⢸⡇⡇⢸⡼⡇⠀⠀⠀⡿⡼⠀⢡⡾⠀⠀⠀⢸⡧⠿⡇⢸⡇⠀⠀⠀⣿⠀⣿⡇⢇⣸⠇⠀⠀⠀⣿⣸⢸⡇⠀⠀⠀⢸⡇
⠀⠀⡇⠀⣌⠋⡅⠀⠀⠀⢸⠙⠉⠁⠈⡇⢀⡄⠀⡇⠀⠀⠀⠙⠀⢠⡄⢸⡇⠀⣿⠋⡇⠀⠀⠀⡏⠛⠁⠈⠉⠉⡇⠀⡀⠀⡏⠉⠛⡇⠀⠀⠀⢸⠙
⠀⠀⡇⠀⣿⠀⡇⠀⣿⠀⢸⠀⠀⠀⠀⡇⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸⡇⢸⣇⣀⢿⣀⡇⠀⣷⠀⡇⠀⠀⠀⠀⠀⡇⢸⡇⠀⡇⠀⠀⡇⠀⣿⠀⢸
⠀⠀⡇⠀⣿⠀⡇⠀⣿⠀⢸⠀⠀⠀⠀⡇⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠈⡇⠀⣿⠀⡇⠀⠀⠀⠀⠀⡇⢸⡇⠀⡇⠀⠀⡇⠀⣿⠀⢸
⠀⠀⣇⣠⢿⣀⡇⠀⣿⠀⢸⠀⠀⠀⠀⠧⡸⠧⡠⠇⠀⠀⠀⠀⠦⠞⠣⠜⠀⠀⠀⠀⡇⠠⣿⠀⡇⠀⠀⠀⠀⠀⡇⢸⡇⠀⡇⠀⠀⡇⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠳⠴⠛⠦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠛⠤⠇⠀⠀⠀⠀⠀⠙⠋⠈⠋⠀⠀⠀⠳⠴⠛⠦⠞

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀⣀⡀
⠀⠀⠀⣠⠖⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠲⣄
⠀⠀⠀⣿⠲⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠖⢻
⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠛⠛⠓⠛⠒⠓⠛⠙⠙⠋⠉⢉⠉⠁⠁⠀⢀⠀⢸⣀
⠀⠀⠀⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⡼⢉⣳
⠀⠀⠀⠀⠈⢻⠓⠒⠶⠦⠤⡤⠤⣤⣤⣄⣀⣠⣀⣀⣤⣠⣤⣤⣤⡴⠦⠶⠶⠞⠟⠛⠛⠉⢸
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣤⢄⡀
⠀⠀⠀⠀⠀⠘⢦⣤⣄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⢚⡷⠞⠉⣇⠀⠝⠢⡀
⠀⠀⠀⠀⠀⠀⢸⠉⠒⠶⠯⠭⣭⣭⣛⣒⣚⣓⣒⣒⣒⣒⣚⣒⣒⣛⣚⣉⣩⣭⡥⢤⠶⢲⡏⠀⢀⠀⠚⡄⠀⠑⢘⢦⡀
⠀⠀⠀⠀⢀⣀⣼⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠈⠁⠁⠉⠀⠀⠅⠀⠀⠀⠀⠰⠀⠘⢷⠀⠀⠀⠀⢹⡔⡀⠀⠁⡙⣄
⠀⠀⠀⣞⠉⠠⠚⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⣠⡼⣆⡀⠔⠁⠀⠱⡀⠀⠀⠈⢈⢳⡀
⠀⠀⠀⡏⠉⠓⠒⠦⢬⠭⢭⣉⣛⣒⣒⣲⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⠖⠛⠛⠉⡏⠁⠀⠘⣆⠀⠀⡠⠀⠹⡄⠀⠀⠀⠡⡹⣄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠘⣖⠁⠀⢀⠄⠛⣄⢀⠀⠀⠀⠜⢦
⠀⠀⠀⠳⢤⣄⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⣋⣽⡄⠀⠀⠈⢶⠊⠀⠀⢀⠘⢦⠢⠀⠀⠈⠈⢣⡀
⠀⠀⠀⠀⢸⠀⠉⠙⠛⠛⠷⠶⠶⢶⠶⠶⡶⠶⢶⡶⠶⣶⠾⠷⡶⠿⠿⠿⠛⠛⠛⠉⠁⢸⡇⠀⠀⠀⠈⢳⡄⠂⠀⠀⡀⠷⡐⠀⠀⠀⠂⢳⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⠹⣄⠄⠈⠀⠀⠹⢎⠂⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠈⣷⡦⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣃⣀⣠⣤⠤⠖⣚⡇⠀⠀⠀⠀⠀⠀⠘⢦⡀⠄⠈⠀⠈⡳⡄⢄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⡇⠉⠙⠓⠒⠯⠭⠭⢭⣭⣙⣛⣛⣛⣛⣛⣛⣛⣉⣩⠭⠭⠭⠭⠲⠖⠛⠋⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠔⠈⠀⠈⢦⡀⠀⠀⢧
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠠⠐⠀⠀⡙⠦⣄⣸
⠀⠀⠀⠀⠀⠳⠦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣠⡴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠐⠁⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠓⠒⠛⠒⠒⠓⠒⠒⠓⠒⠚⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⢠⣤⣿⣤⣤⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿⣅⣿⣀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⠉⣿⢙⣷⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠛⠛⣿⠛⠃⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠉⠀⢀⣠⣴⠟⠁⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⠶⣦⣤⣀⠀⠀⠉⠛⠛⣿⠛⠛⠉⢀⣴⠟⠋⠉⠉⠉⠙⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⣿⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣿⠀⢠⣿⣤⡄⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠐⠻⣾⣷⡀⠀⣿⣴⠾⠿⢷⣤⣤⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠛⠋⠉⠛⠿⣿⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢶⣶⣶⣶⣶⣶⣶⠿⣶⠶⣶⣶⣶⡶⢶⢶⣤⡀
⠀⠀⢠⣶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢸⣿⠀⠀⠀⣠⣤⡀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡃
⠀⠀⠸⡿⠀⠀⢸⡏⠈⣿⠆⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢰⣧⠀⠀⠈⠛⠛⠋⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢷⣤⣤⣤⣤⣤⣾⡏
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢹⣧⣠⡶⠾⠷⣦⡄
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⡾⠟⠉⠀⠀⠀⠈⣿⡄
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠛⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠷⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠾⠿⠿⠿⠛⠛⠋⠁
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠛⠛⠛⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣀⣤⠤⠶⠶⠤⣤⣀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣴⠋⠀⢀⡴⠛⠁⠀⠀⣀⣀⣴⠂⠈⠛⣦⠀⠀⠹⡄
⠀⠀⠀⣸⠃⠀⢠⠏⠀⠀⠀⠀⢼⠋⠈⠙⢷⠀⠀⠈⣧⠀⠀⢻⡀
⠀⠀⢀⡏⠀⠀⡟⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠤⠤⠤⠤⢤⣤⣄⣀
⠀⠀⠘⡇⠀⠀⣧⠀⠀⠀⣀⠀⠀⠀⣻⠄⠀⠀⠀⢀⡟⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⢉⣁⣤⠤⠤⠶⠶⠶⠶⠤⢤⣤⣉⠙⠓⢦⡀
⠀⠀⠀⢻⡀⠀⠘⣦⠀⠀⠙⣷⠶⠞⠋⠀⠀⠀⣠⠞⠁⠀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣾⢋⣥⣴⡆⠀⢀⣴⠟⠛⠛⣳⠶⠎⠙⡆⠀⢹
⠀⠀⠀⠈⣷⣄⠀⠈⠛⠦⣤⣁⣀⣀⣀⣤⠴⠚⠁⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠙⠳⠦⣬⣝⣛⣉⣀⣀⣀⣀⣭⡤⠶⠚⠁⢀⣼
⠀⠀⠀⠀⠈⢿⣷⣤⣀⠀⠀⠈⠉⠉⠀⠀⢀⣠⣴⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣷⣦⣤⣀⣀⣀⣉⡉⢉⣁⣀⣀⣠⣤⣶⣾⣿⢿
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣯⣟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣻⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⣷⣝⡻⢿⣿⣷⣶⣾⣿⣿⣿⣿⣶⣶⣾⣿⠿⣟⣯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⣉⣉⣉⣀⣀⣄⣈⣉⣉⡉⠛⠲⢿⣮⣛⢿⣿⣶⣾⣭⣭⣭⣭⣭⣭⣭⣽⣷⣾⣿⢿⣫⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣴⠞⠋⣩⡍⠀⠀⣠⡴⠶⠶⣭⣭⡟⢶⡄⠈⢻⣿⣷⣯⣭⣽⣿⣻⣛⣛⣿⣿⣭⣭⣷⣾⣿⢿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠳⢬⣉⣙⠶⠶⠞⠁⠀⠀⠀⢛⣀⣠⠴⠃⠀⣸⣿⣭⣟⣻⣿⡿⠿⠿⠿⠿⢿⣿⣛⣿⣭⣶⡿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣀⣈⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⣀⣠⣴⣾⣿⣯⣟⣻⡿⠿⢿⣿⣿⣿⣿⠿⠿⢿⣟⣯⣵⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣛⠿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⢿⣫⣾⣿⣛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣛⣯⣶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⡻⣿⣶⣿⣭⣭⣿⣟⣛⣛⣿⣿⣭⣽⣷⣾⡿⣟⣽⣿⡿⠛⠉⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⢿⣻⣵⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⣝⢿⣷⣾⣭⣟⣛⣛⣛⣻⣟⣛⣛⣛⣯⣭⣷⣿⣿⣿⠋⠀⠀⣠⠶⠚⠛⠛⠓⠶⣤⡀⠀⠙⢿⣿⣫⣾
⠀⢀⣠⠴⠖⠛⢉⣉⣉⣉⣉⣉⣉⣉⡉⠛⠓⠶⣿⣿⣿⣾⣭⣟⣛⣿⠿⠿⠿⠿⠿⣿⣟⣛⣻⣭⣾⣿⠃⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠻⣟⣽
⣴⠋⠀⡴⠞⠋⣩⡍⠁⠀⣠⡴⠾⢯⣍⣙⡓⢦⡀⠙⢿⣝⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣿⡟⠀⠀⡟⠀⠀⣤⠀⠀⠀⠀⣀⣀⠀⠀⢻⡄⠀⢻⡿
⣿⡀⠈⠳⣜⣛⡛⠶⠶⠞⠁⠀⠀⠐⠋⣀⣠⡼⠃⠀⣸⣟⠿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⡿⣿⣿⣇⠀⠐⡇⠲⠾⣇⠀⠀⢀⡾⠋⠉⢷⣀⠀⣷⠀⠈⣧
⣿⣿⣦⣤⣀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⣀⣠⣴⣾⣿⡿⢿⣿⣶⣶⣿⣿⣭⣭⣿⣷⣶⣾⣿⢿⣿⣿⡀⠀⢻⡀⠀⠙⠦⡴⠟⠁⠀⢀⡾⠉⠁⣿⠀⠀⣿
⣿⣽⡻⢿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⢿⣫⣾⣿⣿⣶⣾⣯⣭⣿⣿⣿⣿⣭⣽⣷⣶⣿⣿⣿⣧⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣰⠇⠀⢀⡏
⣿⣝⢿⣿⣾⣯⣭⣿⣛⣛⣛⣛⣛⣿⣯⣭⣷⣾⡿⣟⣽⡏⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠈⢿⣿⣷⡄⠀⠙⠶⣄⡀⠀⠀⠀⢀⣠⡴⠋⠀⢀⡾⠁
⠙⢿⣿⣶⣭⣽⣛⣛⣿⣿⣿⣿⣿⣛⣛⣿⣭⣷⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⡀⠀⠉⠙⠛⠛⠉⠁⠀⢀⣠⠞
⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣦⣤⣤⣤⣤⣴⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣻⣽⣿⣿⣿⡷⠖⠛⠉⠉⠀⠀⠀⠈⠉⠙⠓⠶⣿⣿⣿⣻⣽⣻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠘⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⢠⣶⣿⣿⣿⣿⣧⠀⠀⠀⢠⣿⣿⣿⣿⣷⣤⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣴⣿⣿⠟⡻⣻⣿⡏⠀⠀⠀⠈⣿⣿⡻⡛⢿⣿⣷⡄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⣿⣿⣿⣸⣷⣿⣿⠁⠀⠀⠀⠀⢹⣿⣷⣿⣸⣿⣿⣿⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢻⣿⣿⣿⣾⣯⣵⠏⣴⣶⣶⣶⣶⡌⢷⣯⣽⣾⣿⣿⣿⠃⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠻⣿⣿⣿⡿⠋⠀⠈⠉⢻⠛⠉⠀⠈⠻⣿⣿⣿⡿⠋⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠉⠁⠀⠀⠀⠲⠴⠚⠲⠴⠂⠀⠀⠀⠉⠁⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⢤⣤⣄⣀⣀⣤⣤⡤⠴⠖⠛⠉





⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣻⣿⣻⣻⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠿⠁⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠁⣤⣤⣤⣤⣤⠀⢹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⠀⢹⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠁⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⠀⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⠀⢠⣤⣴⣿⣿⣿⣧⡄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠂⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠐⣟⡏⠙⡟⣿⠲⠒⠲⠶⣶⢶⠶⠶⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢷⠳⠶⢤⣤⡤⢤⣤⣤⣤⣶⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠳⠾⠤⢤⣤⣼⣇⣸⠟⠋⠁⠀⠈⣉⠙⠻⠶⢄
⠀⠀⠀⢀⣠⣤⣴⠶⠾⠶⠖⠲⣦⣤⣤⣄⣸⡟⡄⠀⢀⠄⠊⠈⠢⡀⠀⠀⢿⣿⠛⠷⠶⣦⣄⣀
⠀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣇⠀⠉⠂⠀⠀⠀⠀⢄⠀⠀⠙⣷⡀⠀⠀⠈⠙⠻⣦⡀
⠀⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡉⢿⣆⠀⠀⠁⢀⠀⠀⠀⠑⢄⠀⠸⣷⠀⠀⠀⠀⠀⠈⠋
⣾⡹⢿⣁⣾⠭⡷⠒⣀⣐⣶⣶⣦⣤⣶⡾⠛⢳⡈⢫⡣⡀⠀⠀⠑⠄⠀⠀⡠⠓⠀⣿⣿⣤⡶⠶⠽⣷⣾
⢿⣙⠷⠛⢿⣿⣷⣶⣤⣤⠤⠤⠶⣶⣗⣓⣊⣭⣿⣄⠙⢮⣲⢄⡀⠀⠆⠊⠀⠀⣠⣿⠿⠿⠤⠤⠴⠟⠛
⠀⢻⣿⡶⠶⢤⣤⣀⣀⣀⣉⣹⣉⣀⣀⣀⣽⣿⣿⣿⠙⠿⣟⢙⣾⣿⣤⣶⣶⣼⡿⠋
⠀⠈⠛⠳⢤⣀⣀⣀⠈⢉⠈⠉⠉⠉⣇⣀⣸⣿⠿⠟⠀⠀⠀⠉⠛⠛⠚⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣉⡉⠤⠤⠤⠄⣉⡉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣈⠻⣦⡀⠀⣀⣤⡴⠶⠶⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠊⠀⠀⠀⠀⠀⣀⡖⠒⣄⡀⠀⠀⠀⠈⠣⣈⣷⡾⢋⡠⠔⠒⠈⠁⠒⠢⢬⡙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⠁⠀⠀⠀⠀⢠⠊⠁⣀⣀⣀⢈⠇⠀⠀⢀⡜⣹⠏⡰⠋⠀⠀⢀⡀⠀⠀⠀⠀⠈⠳⢌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠇⠀⠀⠀⠀⠀⣟⠀⠸⢅⡀⠀⠉⠀⠀⠀⡼⢰⡏⢰⠁⠀⠀⠐⡏⠙⠂⢑⡀⠀⠀⠀⠈⠳⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠈⠙⠢⡀⠀⢀⠃⢸⡇⡌⠀⠀⠀⢸⠀⢠⠚⠉⠁⠀⠀⠀⠀⠀⠱⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡄⠀⠀⠀⠀⠀⢀⡀⠀⠉⠳⡄⠀⠃⠀⠸⠅⢸⡇⢡⠀⠀⠀⠈⢆⠈⠓⠂⠀⠒⢤⡀⠀⠀⠀⢣⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠱⡀⠀⠀⠀⠀⣏⠀⠉⠀⠉⠀⣠⠃⠀⠈⡇⠈⣷⠸⡀⠀⠀⠀⠈⠑⠒⠒⠒⢤⠀⢱⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠱⣄⠀⠀⠀⠀⠉⠑⣆⣠⠋⠁⠀⠀⠀⢻⡀⠹⣧⠱⡀⠀⠀⠀⠀⢀⡤⠤⠚⠀⡸⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣷⣈⣳⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠴⠊⣳⡀⠘⣧⡑⢄⠀⠀⠀⠀⠣⠤⠰⢤⠕⠀⠀⠀⡜⢸⠇
⠀⠀⠀⢀⡤⠒⡾⠛⠛⠋⠉⠉⠉⣭⣉⣽⣿⣿⣯⣭⡉⠉⠉⠙⠛⠓⢟⠳⢶⡟⠁⠱⣄⠈⠻⣬⡓⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠞⣠⠏
⠀⠀⠀⢸⠛⠶⢭⣷⣶⣦⣤⣤⣤⣭⣍⣙⣟⣛⣁⣭⣥⣤⣤⣶⣶⣾⠵⠶⠟⡇⠀⠀⠈⠳⣄⠈⠻⢦⣍⡓⠒⠤⠤⠤⠒⢊⣡⡶⠋
⠀⠀⠀⠘⠦⣤⣀⣀⣀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠁⠀⢀⣀⣀⡤⠶⠃⠀⠀⠀⠀⠈⠑⠢⢄⣍⣋⠛⡳⢖⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠚⠚⠚⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠛⠛⠛⠓⠲⠦⣄⡀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠚⠃⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⣠⣧⣀⠀⠀⠙⢦⠀⠀⠀⠀⡴⠋⠀⣴⠶⠶⠶⣤⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣟⠀⠀⠈⠇⠀⠀⠈⣧⠀⠀⣼⠁⠀⠛⣿⠛⠛⠛⠛⣿⠃⠀⢻⡀⢀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠓⠶⢦⡀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠉⣿⣉⣉⣉⡽⠋⠁⠀⢸⡇⢈
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⣇⠀⠀⢀⡿⠀⠀⢀⡟⠀⠀⢻⡀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⣼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⠙⡟⠉⠁⠀⣠⠞⠀⠀⠀⠈⠳⣄⠀⠻⠀⠀⠀⠀⠀⢀⡼⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠤⠤⠶⠚⠉⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⡈⠙⠲⠦⢤⣤⣤⣤⣤⡤⠴⠖⠋⢁⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠤⠀⢀⣀⣀⠤⠤⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⢠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢠⠇⢻⡀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⡏⠀⠀⢧⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡼⠀⠀⠀⠈⣇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢰⠃⠀⠀⠀⠀⢹⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠘⣇⣀⣀⠀⠀⣀⣀⣸⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡏⠀⠈⡇⣼⠁⠀⢹⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⠀⠀⡇⣿⠀⠀⢸⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠀⠀⢀⡅⢨⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠏⠀⠀⢸⠃⢸⡄⠀⠀⠧⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⣼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⣠⠖⠦⣄⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠴⠦⡄
⠀⠀⠀⡰⠃⠀⠀⠈⠙⠧⣄⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⡠⠞⠋⠀⠀⠀⠙⣄
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠘⢦⡀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠙⠫⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠋⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢰⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠀⢠⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡷⠀⠀⠀⢀
⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠋⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃
⠀⠀⠙⠟⢁⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠁⠀⠀⠀⠀⠀⠘⠁⢠⠔⣢⣤⠀⠀⠀⠀⠀⣠⣶⡠⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⢰⣏⣴⣿⣿⡆⠀⢀⣾⣿⣷⣙⣆⠀⠀⠀⠀⠀⠀⠀⢻⣽⣿⣿⡇⠀⠀⠀⠰⣿⣿⠿⡧⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠾⠟⠛⠋⠀⠀⠀⠙⠛⠿⠧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⡴⠚⣻⣶⡄⠀⠀⠀⠀⣰⣾⣟⡓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠇⠀⠀⠀⠀⠻⣿⣿⡽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠸⢿⡇
⠀⠀⠀⠀⢠⣿⡿⠇⠀⠀⠀⠘⠛⢿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢫⣶⣶⡄⠀⠀⢀⣴⡶⣆⠀⠀⠀⠀⠀⠀⣿⣫⡵⣤⣄⠀⠀⠀⠀⠀⠀⣠⣶⠮⡙⣿⠃⠀⠀⠀⠀⠀⠀⢀⡤⣤⡀⠀⠀⠀⢀⣤⠤⣀
⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠈⠛⠻⠃⠀⠀⠀⠀⠀⢀⣰⡷⣾⣿⣿⣧⠀⠀⠀⠀⣾⣿⣿⣷⢿⡉⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⠀⠀⠀⣾⣿⣧⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡿⢿⠿⣿⣿⣿⠃⢀⣤⣤⡀⠘⣿⠿⠿⠷⣿⣦⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠰⣿⣭⣽⡷⠀⠀⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠐⠀⠙⠿⠟⠁⠔⠀⢰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠙⠛⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠋⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⢀⣾⣿⣿⡄⠀⠀⠀⣼⣿⣿⣧⣀⣄⠀⠀⠀⠀⠈⠻⠿⠿⠶⠄⠀⠀⠀⠀⠠⠴⠿⠿⠟⠁⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⢶⣯⣼⡿⠟⠃⠀⠀⠈⠛⠿⠿⠦⠿⠛⠀⠀⠀⠀⠀⠀⡴⢂⣐⣦⡄⠀⠀⢠⣴⣶⡒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢻⣿⣿⡟⠀⠀⢸⣿⣿⠟⡻⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣶⣄⠀⠀⠀⢀⣴⣧⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⢰⣯⣿⠆⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⠤⠖⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒⠤⣄
⠀⠀⢸⡷⠤⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠼⡇
⠀⠀⢸⡇⠀⠀⠀⠉⠉⠉⠙⠉⠙⠛⠒⠓⠒⠛⠉⠙⠉⠉⡉⠉⠁⠀⠀⠀⣇⡀
⠀⠀⠘⠧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣴⣃⢽⠆
⠀⠀⠀⠀⢸⠉⠉⠋⠓⠒⠚⠒⠲⠒⠶⠖⠲⠖⠒⠶⠓⠒⠛⠛⠋⠉⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣆⣠⠴⡋⠲⢄
⠀⠀⠀⠀⠀⢸⠻⢿⣶⣒⣒⣤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⢒⣒⣚⣋⡭⡔⠋⠀⠀⢧⠈⠀⠱⣄
⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠈⠉⠉⠋⠉⠙⠉⠙⠋⠉⠋⠉⠙⠉⠉⠉⠀⠁⠀⢷⠀⠈⠀⠈⣆⠀⠀⠈⠳⡀
⠀⠀⢰⣋⠡⠚⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡬⢇⡀⠀⠀⠘⣆⠀⠀⠀⠙⢆
⠀⠀⢸⠈⠉⠒⠒⠶⠭⠭⠭⠭⢵⠷⠶⡶⠶⡶⠶⠶⠶⠶⠞⠒⠛⠛⠉⢹⠁⠀⠈⢦⠀⠠⠐⠉⢆⠀⠀⠀⠊⢣⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⡀⠀⠈⢧⠀⠀⠐⠈⢣⡀⠀⠀⠀⠱⡄
⠀⠀⠀⠙⡖⠶⢤⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣤⣤⣶⣶⡿⠥⠒⡇⠀⠀⠀⢣⡀⠀⠄⠈⠳⡔⠀⠀⠀⠘⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⡄⢀⠄⠀⠙⢆⠀⠀⠀⠘⣆
⠀⠀⠀⠀⣧⢤⣄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⣀⣨⠴⣧⠀⠀⠀⠀⠀⠘⢆⠀⠠⠀⠈⢳⡄⠀⠀⠘⡆
⠀⠀⠀⠀⡏⠉⠒⠒⠯⠭⠭⢭⡵⣚⣓⣒⣒⣒⣚⣒⣚⡭⠭⠭⠭⠵⠖⠒⠉⣿⠀⠀⠀⠀⠀⠀⠀⠳⣄⠠⠀⠁⠈⢦⡀⢀⠸⡄
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠐⠀⠀⠜⠢⣤⠇
⠀⠀⠀⠀⠈⠓⠲⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⠤⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠉⠉⠛⠳⡄⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡆⠀⣀⣀⣀⠤⠚⠁⠀⠀⠀⢀⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⠶⠿⠿⠿⠿⠿⠿⠶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠐⠿⢿⣯⣤⣤⣾⠟⠉⠀⣹⡿⠿⠂⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡟⢿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡿⢻⣿⣿
⠀⠀⠀⠀⠀⢹⣿⡇⢸⣿⠉⠛⣿⠿⠷⢶⣶⣶⣶⣶⣶⣶⡶⠾⠿⣿⠛⠉⣿⡇⢸⠿⠋
⠀⠀⠀⠀⠀⣿⣟⠻⣿⣿⣀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⣿⠀⣀⡿⠃⠀⣠⣴⡾⠿⣿⣿⡿⠿⠿⠿⠿⢶⣦⣀
⠀⠀⠀⠀⠀⣿⣿⣦⣀⠉⠛⠿⠿⣶⣦⣤⣿⣧⣤⣤⣼⣿⣤⣶⣾⠿⠿⠛⠁⣠⣾⣏⣁⣠⣾⠟⠁⣠⣴⣶⣶⣤⣄⠉⠻⣷⡄
⠀⠀⠀⠀⠀⣿⣿⡏⢻⣿⣶⣤⣄⣀⣀⠈⠉⠉⠉⠉⠉⠉⠁⣀⣀⣠⡄⠀⣼⡟⠉⠉⣽⡿⠁⣰⡿⠋⠁⠀⠀⠈⠻⣷⣄⠈⢿⣆
⠀⠀⠀⠀⠀⣸⣿⣇⣸⣿⠀⠉⣿⠛⠛⠻⣿⡿⠿⠿⢿⣿⠟⠛⠛⡟⠀⣸⣿⣶⣶⣼⡟⠀⣼⡟⠁⠀⠀⢠⡀⠀⠀⠈⢿⣆⠈⢿⣆
⠀⠀⠀⠀⠀⣿⣯⡙⠿⣿⣤⣠⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠐⠃⢠⣿⠃⠀⢠⣿⠁⢸⡿⠀⠀⠀⢀⣾⣧⡀⠀⠀⠈⣿⡆⠘⣿⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣦⣄⡉⠛⠛⠿⠿⠶⣿⣷⣶⣶⣾⡿⠶⠿⠿⠀⢸⣿⣤⣤⣼⡏⠀⣿⡇⠀⠀⠀⣿⡏⠉⠋⠀⠀⠀⢸⣷⠀⢿⡇
⠀⠀⠀⠀⠀⢻⣿⡇⢸⣿⠻⢷⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡤⠀⢸⡏⠀⠁⣿⡇⠀⣿⠀⠀⠀⠀⠘⢿⣤⠀⠀⠀⠀⠸⣿⠀⢸⣧
⠀⠀⠀⠀⠀⣾⡿⣷⣼⣿⠀⠀⣿⠀⠉⠉⣿⡏⠉⠉⢹⣿⠉⠉⠀⠀⢹⣇⣀⣀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢰⣿⠀⢸⡟
⠀⠀⠀⠀⠀⣿⣿⣌⠙⠻⢷⣶⣿⣄⣀⣀⣿⡇⠀⠀⢸⣿⣀⣀⣠⠀⢸⣿⠛⠛⢻⣇⠀⣿⡇⠀⠀⠀⢶⣤⣼⡿⠀⠀⠀⢸⡿⠀⣾⡇
⠀⠀⠀⠀⠀⣿⣿⡟⢿⣶⣤⣀⡉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠈⣿⡄⠀⠘⣿⡀⠸⣿⡀⠀⠀⠀⢹⡏⠀⠀⠀⢀⣿⠃⢰⣿
⠀⠀⠀⠀⠀⠹⣿⡇⢸⣿⠉⠛⣿⠿⠷⢶⣶⣶⣶⣶⣶⣶⡶⠾⠿⣷⠀⠹⣿⡛⠛⠻⣷⡀⠹⣷⡀⠀⠀⠀⠀⠀⠀⢠⣾⠏⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠹⣷⣤⣤⣽⣷⡄⠙⢿⣦⣀⠀⢀⣠⣴⡿⠃⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣶⣶⣤⣿⣧⣤⣤⣼⣿⣤⣴⣾⠿⠿⠃⠀⠙⢿⣯⣉⠘⢿⣦⣄⠉⠛⠛⠛⠛⠉⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢶⣿⣿⣿⣶⣶⣶⡾⠿⠛⠁



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

⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠉⣿⡏⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠋⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠋⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣧⣀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡟⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⣰⣿⣷⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⣼⡿⠃⠈⠙⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣶⣦⣄⣀⠀⣼⡿⠁⠀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠈⠉⠛⠿⢿⡿⠁⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠃⠀⠀⢀⣿⡟⠀⢀⣴⣿⣷⣶⣤⣄⣀⣼⡿⠁⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣾⡟⢀⣴⡿⠟⠁⠀⠉⠙⠛⠿⠟⠁⣠⣴⣿⠟⠋⠉⠁⣀⣀⠀⠈⠉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⣿⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⣾⡿⢻⣷⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⣴⣾⠿⠃⠘⠿⣿⣦⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠸⠿⠋⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⢸⣿⠃⢠⣿⡿⠀⠀⠁⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⢀⣶⡿⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⣿⣦⡈⠻⠿⣷⣶⣄⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⣹⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⡄⠙⣿⣇⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠘⢿⣷⣤⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⣀⡀⠀⢾⡿⠃⢀⣿⡏⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠙⢿⣷⡆⢰⣾⡿⠋⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⢻⣷⣾⡿⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣀⣀⡈⠁⣀⣀⣠⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⣿⣿⣿⡇⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⢹⣿⣿⡇⢀⣿⣿⣿⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣸⣿⣿⣷⣼⣿⣿⣿⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢿⣿⡏⠈⣿⣿⡿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⣿⡇⠀⣿⣿⠁⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⢿⣿⠀⢹⡇⠀⢻⡏⠀⣼⣿⣿⡇⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠈⢿⡆⠀⡷⠀⢸⠃⠀⣿⣿⣿⣷⣄⣠⣶⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠁⠀⠁⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠋⠉⢉⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡤⡄⣠⢤⡀⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⢤⠄⠻⡤⠇⠏⠻⠇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠄⠘⠇⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠛⠟⠋⢠⣄⠀⠀⠀⠀⣠⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠘⠋⠁⠀⠀⠀⠀⠉⢠⣴⣶⣦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡶⠄⠀⠀⠀⠈⠉⠁⠀⠀⢀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢀⣤⣤⣄⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣀⠀⠀⠀⠀⣤⣤⣤⡀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠋⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣤⣤⡄⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⠄⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠛⠉⠙⠛⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⢿⣿⣿⠟⠋⠉⠉⠉⠉⠳⣶⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠻⢿⣀⡀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⣄⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⡄⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠉⠒⠒⠲⠶⠒⠒⠉⠀⠀⠀⠀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⢀⠀⠐⠊⡽⢿⣝⢓⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠼⠤⢶⠾⠃⠉⠛⠀⢀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠛⠻⠶⠶⠶⠶⠶⠾⠿⠟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠰⠤⣤⣤⣤⣶⠆⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⢠⣄⣀⣀⣀⣤⣤⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠒⣳⣿⡶⠲⣤⡀⠀⠈⠉⠉⠉⠉⠁⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡚⠛⠟⣃⣀⣀⣤⣾⣷⠈⠛⠛⠛⠻⠿⠛⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠉⠉⠛⠛⠛⢛⣋⠉⣿⣿⠓⠶⠶⠶⠶⠶⠖⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣉⣉⣉⣉⡉⠀⣿⣿⠀⠶⠤⢤⣶⡶⠶⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⣀⣉⣉⣩⡅⠀⣿⡏⠀⠦⣤⣤⣤⣤⣶⠄⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠠⢄⣀⣉⣠⡄⠀⣿⣧⠀⣀⣀⣀⣀⣀⣀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣄⣀⣀⣠⡤⠀⣿⣿⠀⠈⠉⠉⠙⠛⢉⣀⣿⣧⣤⠴⠆⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣀⣀⣠⣤⣀⣻⣿⣤⠴⢶⣶⡛⠛⠉⠉⠁⠀⣀⡰⠤⠄⢀⣉⡛⠷⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣀⣤⠤⠴⣾⡛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠶⣶⣬⣁⠀⠉⠛⠒⠛⠀⠤⣐⡀⣈⣩⡛⠻⠷⣶⣤⣤⣀
⠀⠀⠤⠤⠶⠒⠚⢋⠩⢥⡀⠀⢒⣊⡭⠽⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣦⣄⣀⠀⠀⢀⣙⣦⣭⡽⠆⠀⣀⣀⡉⠉⣅
⢰⡆⠠⢄⡀⠀⠰⣌⠓⣲⣟⣯⡁⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⣾⣩⣤⠴⠶⠒⣛⣩⡭⠴⠒⢚⣿
⢰⣇⠀⠀⠙⠦⣄⠈⠙⠛⠻⠥⣌⣵⠶⠶⣤⣀⢀⣀⡤⠄⠈⠑⠢⠄⠀⠀⠀⢀⣀⣤⡤⠶⠞⠛⠉⢹⡅⠀⠒⣉⡩⠥⠶⠒⠊⢉⣼⣿
⠀⡏⠓⢦⣀⠀⠈⠙⠦⣄⡀⠀⠈⣅⠻⣲⣆⢻⡏⣠⠴⠋⠉⣁⣠⣤⡶⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣠⡤⠶⠖⠛⠉⢁⣿⡏
⠀⣗⠠⣀⠉⠳⣤⡀⠀⠀⠙⠲⢤⣀⠙⠛⠛⠋⣉⣥⣶⠾⠟⠛⠉⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠉⢉⣀⣤⣤⠶⠶⠛⢛⣿⡇
⠀⣿⢤⡈⠳⢤⡀⠙⠶⣄⠀⠀⠀⠈⠁⣴⠾⣛⣩⠥⠖⢂⣁⣠⠤⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⣉⣠⣤⠶⠶⠛⠛⣿⣷⣤
⠀⠉⣀⠙⠲⣄⠉⠲⢄⠈⠙⠦⣄⠀⠀⣿⣭⡥⠶⠖⢛⣩⡥⠶⠖⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣚⠫⠭⣔⣒⣬⣵⣾⣿⡿⠟⠋
⠀⠀⠈⠳⣦⡀⠁⠀⠀⠀⠀⠀⠈⠙⠲⣿⣤⡶⠶⠛⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣶⣾⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣀⣤⣴⡶⠞⠁⢸⡇⢀⣀⣤⣴⣶⣿⡿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⣿⡧⠖⠚⠽⠛⣋⠁⢀⣠⣤⣾⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⡀⠀⣿⣇⣤⠴⢞⣫⣵⡾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣿⣩⠴⠞⠋⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣈⡉⠙⠻⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢰⡟⠉⠉⠁⣀⣀⣀⣤⣭⠉⠉⠙⠛⠿⢿⣶⣤⣄⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣼⡏⢷⡄⠀⠀⠀⠻⣦⣄⡀⠉⢻⣯⣀⣀⣠⣴⠶⠶⠶⣶⣄⠉⠙⠻⣦⡀⠈⢻⣆
⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠉⢁⣠⣤⣴⠶⠶⢾⡇⢀⠛⢷⣄⡀⠀⠀⠉⠛⠷⢦⣭⣛⡋⠉⠀⠀⠀⠀⣼⡿⢶⡤⠀⢸⡇⠀⠀⢻⡄
⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⣴⠿⠋⠁⠀⢀⣤⠈⠻⣾⠀⠀⠉⡿⠶⣤⣀⡀⠀⠀⠀⠉⠙⠛⠻⠶⠶⠶⠶⠶⠶⠶⠾⠛⠁⠀⠀⣼⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣏⠀⠀⣤⣴⣿⠁⠀⠀⠙⣷⣤⣰⠃⠀⠀⠉⡻⠛⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⠾⢻⠃
⠀⠀⠀⢰⡟⢷⣄⠀⠀⠀⠻⣦⣀⠀⠀⠻⠷⣶⡶⠿⠋⠉⠻⢷⣦⣤⣰⠃⠀⠀⠀⠀⡌⠈⠉⠉⠉⠹⠉⠉⠉⠉⡏⠉⠀⢘⣠⡏
⠀⠀⠀⠘⣧⡈⠻⣦⣄⠀⠀⠈⠙⠻⠶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣈⣽⡿⠿⢿⣷⣶⣶⣧⣤⣤⣤⣤⣧⣤⣤⣤⣤⣧⣶⡾⠟⠉
⠀⠀⠀⠀⠹⣷⣄⠀⢹⠻⢶⣤⣀⣀⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣈⡽⢛⢹⡏⠙⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠘⢿⣷⣾⡀⠀⠀⡍⠉⠛⠛⠷⡶⠶⢶⣤⣤⣤⣤⣤⣤⡶⠶⡶⠛⠋⢹⢀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣧⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣇⣠⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣷⣶⣶⣶⣦⣤⣷⣶⣶⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣛⢛⣙⢛⡛⣛⡿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠂⠀⠀⠀⠀⠀⠀⠀⠤⣭⣶⡾⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⠀⠀⢀⣤⣴⣶⣶⣶⢦⣤⣤⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣏⣳⣰⣿⣷⣿⣿⣷⣿⣿⡿⠁⠀⠀⠀⠀⠀⡘⣡⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⠿⠿⠿⠿⠯⣿⢻⡇⠀⣠⣤⣴⣶⣶⣴⣭⣩⣹⣍⣱⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⢮⣿⣁⠀⠀⠀⠀⠀⠻⣾⠇⠀⠙⠋⠹⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁
⠀⠀⠀⠁⠀⠰⠄⠀⠘⠃⠀⠁⠀⠀⠀⠀⠀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣄⢀⣠⣄⢀⣤⣶⣿⣿⠁
⠀⠀⢀⣴⣾⣷⣄⢲⣰⢀⣴⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠈⠉⡟⠛⠛⠛⣷⣿⠛⠛⠟⡟⠁
⠀⠐⠛⣿⢿⣿⣿⣿⣿⣿⣿⡿⠛⠿⠛⠀⠀⠀⠀⠀⠀⣦⣀⠀⢀⣿⢆⣴⣦⠰⣿⡇⠀⠀⠉⠑⠀⠀⢻⡇⠀⠀⠋
⠀⠀⠀⠀⠀⠉⠉⠙⠿⠏⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠘⣿⣷⣿⠿⡊⠛⠉⠘⠓⠂⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⢀⢘⠀⠀⠀⠀⢀⡌⠁⠀⠀⠀⠈⠛⠿⠟⢍⠙⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⡐
⠀⠀⠀⠀⠱⡄⠀⠀⠀⢀⡀⡊⠀⠀⠀⠀⠈⣿⠶⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠉
⠀⠀⠀⠈⠀⠰⠆⢀⡀⢲⠆⠀⠀⠀⠀⢀⠀⠈⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⢠⣴⡇
⠀⠀⠀⠐⠀⠀⢄⣀⣤⣤⠖⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣉⠁⠀⠀⠀⢀⠀⠀⠀⠘⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠳⠴⠶⠶⠴⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⡀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠐⠒⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢧⣄⣸⣁⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠄⠘⠀⢀⣠⣿⢻⣷⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠃⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⢿⣿⣿⣶⣖⠚⠁⠈⢿⣿⣜⣿⡧⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⣶⡏⠀⢻⣿⡜⢿⣆⠀⠀⢸⡿⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠉⠛⠂⠛⠛⠶⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⢀⣼⣿⠟⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠛⢿⣿⣿⠟⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠋⠉⠙⠋⠉⢻⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢹⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⣶⣶⣶⣾⣶⣦⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣷⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢼⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣿⣿⠀⢸⣿⣷⣶⣶⣶⣶⣾⣿⡇⠀⣿⣿⣶⣶⣶⣶⣶⣿⣿⠀⠀⣿⣷⣶⣶⣶⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣠⣯⣥⡀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⣷⡤⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣠⣴⡀⠀⠀⠀⠀⠀⢀⣠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠦⢤⡀
⠀⠀⠀⠀⣿⢹⠿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠰⠏⣿⣿⣾⣿⠛⠃⠀⠀⠀⠀⠀⠟⢃⣤⡍⣹⡄⠀⠀⠀⠀⠀⠀⣀⣷⣙⣼⠁
⠀⠀⠀⢠⡿⣤⡼⣧⠀⠀⠀⠀⠀⠀⢰⠻⡦⢴⠟⡆⠀⠀⠀⠀⠀⠀⣤⡛⠁⠈⠻⣠⡄⠀⠀⠀⠀⠀⢷⡎⣙⣁⡹⠁⠀⠀⠀⠀⠀⢟⢷⢌⣾⢫⠛⡇
⠀⠀⠀⢸⣙⣻⣓⣹⠀⠀⠀⠀⠀⠀⢸⣀⣈⣁⣀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠙⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⣿⡿⠃⠀⠀⠀⠀⠀⢰⣿⡆⠀⠰⣿⡆⠀⠀⠀⠀⠀⢀⣀⣀⣄⣴⣤
⠀⠀⣴⣦⣻⣾⣶⣿⣆⠀⠀⠀⠀⠀⢀⣀⣻⣿⣿⣦⡀⠀⠀⠀⠀⠰⣿⡿⠳⠚⢿⣿⠂⠀⠀⠀⠀⠸⠭⠇⠀⠸⠥⠇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧
⠀⠀⠙⠙⠿⠃⠿⠻⠋⠃⠀⠀⠀⠀⢿⣿⣿⠏⠻⠿⠃⠀⠀⠀⠀⠀⡞⠛⣄⣰⠛⢳⠀⠀⠀⠀⠀⣿⣿⡿⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠛⢿⣿⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠘⠒⠋⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢰⣆⣤⣤⣀⣀⣶⠀⠀⠀⠀⠀⢰⠒⠚⠛⠒⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠰⣦⣸⣇⣨⡁⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⣿⡇
⠀⠀⢸⡧⠷⠾⠤⠤⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠡⡾⢿⢿⣭⠀⠀⠀⠀⠀⠠⣿⡷⠶⠶⢶⣿⡇
⠀⠀⠸⠗⠟⠒⠚⠗⠿⠀⠀⠀⠀⠀⠘⣲⣖⣒⣶⡚⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠠⠉⠀⠀⠀⠀⠀⠈⣿⣇⣀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⡆
⠀⠀⠀⠀⢰⡿⠿⠿⡇⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡢⢛⣁⠀⠀⠀⠀⠀⠀⢠⠊⠀⠁⠑⠢⡀⠀⠀⠀⠀⠀⠀⢀⣤⡛⢀⡄
⠀⠀⠀⠀⢸⡇⣠⣴⡇⠀⠀⠀⠀⢸⠀⡑⢄⣀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⡇⠀⠀⠀⠀⠀⠀⣵⣦⣄⣤⣾⡆⠀⠀⠀⠀⠀⠠⣼⣍⣙⡷⠂
⠀⠀⠀⢿⣿⠇⠙⠛⠁⠀⠀⠀⠀⠀⠑⠤⣁⣒⣀⣡⠜⠁⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠝⠋⠀⠀⠀⠀⠀⠀⠓⢮⣙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⢀⣠⠴⠖⠒⢚⣛⣋⣉⣉⣉⣉⣙⣓⣒⠦⢤⣀
⣴⠋⢀⡴⠚⠋⠩⣭⡴⠆⠀⢀⣀⣤⣬⡉⠙⠲⢬⡙⢦⡀
⣿⡀⠘⢦⣀⠀⠀⠛⠓⠒⠛⠉⢀⣀⣴⣿⡀⢀⣨⠇⠀⣿⡀
⣧⠛⢦⣀⠈⠙⠒⠶⠤⠤⠤⠤⠤⠴⠶⠖⠚⠉⠁⣀⡴⢻⡇
⢸⣧⣄⠈⠛⠶⠤⣤⣤⣀⣀⣀⣀⣀⣤⡤⠴⠶⠛⢉⣤⠞⢳⡀
⢸⡌⠙⠿⣶⣦⣀⣀⣀⠀⠀⢨⠀⠀⢀⣆⣤⡴⠞⣋⡥⠶⢻⡇
⠈⢻⣦⣀⡆⠈⠉⡟⠛⠻⠿⡿⠿⠿⠿⠿⠗⠛⡏⠁⡇⣠⣾⣧⡀
⠀⢸⡈⠙⠻⡶⣤⣇⣀⠀⠀⡇⠀⠀⣘⣀⣀⣠⣧⡶⠛⢋⣅⠴⠷⠖⠒⢒⠒⠲⠶⠤⢤⣀⣀
⠀⠀⠙⢦⣤⡃⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠁⠀⣴⠞⢉⡴⠞⠛⢛⣩⠉⠉⠉⠛⡛⠶⠶⣭⣝⠲⣄
⠀⠀⠀⠀⠉⠛⠛⠶⠶⣦⣤⣼⣤⣤⣤⣤⣷⣶⢾⣿⠀⠘⣇⠀⠉⢻⣧⣠⡴⠞⠉⠙⣷⣄⠀⠉⢻⡌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢸⡟⠳⣄⠈⠙⠲⠦⢤⣤⣀⣀⣀⣙⣉⣤⠤⠖⠋⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡨⠙⠲⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠞⢉⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣄⣈⡀⠀⢉⡁⣀⣀⣠⣤⣤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⣰⣶⡄
⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠉⣿⣿⡍⠛⣿⣿⡆⠀⠀⠀⠀⢰⣿⣷⣼⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⡀⣿⣿⡄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃
⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡟⠿⣿⣿⣆
⠀⠀⢠⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣶⣶⣆⠀⠀⣿⣿⡅⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⢀⣿⣿⣿⣿⡟⢿⣿⠇⠀⠀⠀⠀⠘⣿⣿⣦⣄⣿⣿⣄⣤⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠘⠿⠏⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣤⢀⡀⣶⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣶⣾⣿⣿⣿⣿⣶⣾⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⡀⠉⠛⣿⣿⣿⡿⠛⠉⣀⣽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷⠂⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠐⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣼⣿⠏⢀⣤⣤⣤⣐⢄⠀⠀⠀⠀⡠⣀⣤⣤⣤⡀⠹⣿⣧⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡿⠋⠭⣽⣶⣿⣿⡾⠛⠋⠉⠛⠻⢷⣷⣤⣤⣾⡾⠟⠛⠉⠙⠛⢷⣿⣿⣶⣯⠭⠙⣿⡄
⠀⠀⠀⠀⠀⢸⡇⠘⠻⣿⣿⣿⠏⠀⢀⣤⡶⢶⣤⡀⠙⣿⣿⠋⢀⣤⡶⢶⣤⡀⠀⠹⣿⣿⣿⠟⠃⢸⡇
⠀⠀⠀⠀⠀⠘⣇⠀⡞⢛⣿⣿⠀⢠⡟⣡⡀⣳⣦⢹⡀⢸⡇⢀⡟⣅⢘⣷⣎⢻⡄⠀⣿⣿⡛⢳⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠙⣦⣇⠈⢾⣿⣆⠘⣇⠻⣟⣿⡿⢡⠁⡸⢇⠈⡆⢿⢻⣿⡟⣸⠇⣰⣿⡷⠁⣸⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣤⣿⣿⠷⡌⠳⠬⠭⢖⣡⠾⠗⠺⠷⣌⡲⠭⠥⠖⢁⠼⣿⣿⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣧⣄⣀⣀⡤⠋⠄⣀⠀⠀⣀⠠⠙⢦⣀⣀⣠⣴⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡟⠀⠀⠀⠉⠃⠘⠉⠀⠀⠀⢹⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡉⠓⠒⠒⠒⠒⠚⢉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣤⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⠶⠴⠶⠶⠶⠶⠶⠶⠶⢦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠉⢁⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣌⣉⣙⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⢀⣤⣴⠶⠟⠛⣋⣉⣉⣡⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢶⣤⣈⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⡟⠉⠀⠀⠀⠀⠈⣿⣿⡋⠉⠁⠀⣀⣤⣶⠿⠿⠿⠿⢷⣦⠀⠀⠀⠈⠛⣧⡀⠈⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠈⢿⣦⣀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠋⠁⠀⠀⠀⢀⣠⣾⣿⣀⠀⠀⠀⣀⣽⠇⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⢾⠙⣷⣄⠀⠀⠈⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⢉⣉⣉⣤⣴⠾⠛⠁⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⢸⣧⡇⠙⢷⣦⡀⠀⠀⠀⠉⠙⠛⠻⠷⠶⠶⠶⠶⠶⠶⠾⠿⠟⠛⠛⠋⠉⠉⠀⠀⠀⢀⣠⣴⠟⡇⣿
⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⢸⠈⠛⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⡁⢀⣷⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠻⣿⣿⣤⣀⠀⢸⡉⠉⠛⠛⠻⣿⠶⠶⠶⢶⣶⡶⠶⠶⠶⠿⠿⡟⠛⠋⠉⢸⠀⢀⣠⣾⡿⠋⠀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡄⡏⠛⠿⣿⣷⣾⣇⣀⠀⠀⠀⡧⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⡇⢀⣠⣤⣾⣿⠿⠛⢁⣠⣴⣿⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣷⡇⠀⠀⠀⡙⠛⠻⠿⣿⣿⣶⣿⣶⣶⣶⣶⣾⣷⣤⣴⣶⣶⣾⡿⠿⢛⣋⣭⣤⡶⠾⢻⠉⠀⣼⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣄⡀⡇⠀⠀⠀⠀⣸⠉⠛⠛⠛⠿⠿⣿⠿⠿⠿⠿⠿⢿⠻⠛⠛⠉⢩⡇⠀⠀⢸⠀⣀⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠛⢿⣿⣦⣄⡀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⣀⣠⣼⣿⡿⢿⠋⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⠀⠀⠈⠙⡟⠿⣷⣷⣤⣤⣀⣀⣀⣀⣇⣀⡀⣀⣀⣀⣿⣤⣤⣴⣶⣾⡿⠟⡟⠉⠀⠀⢸⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⢰⡇⠀⠀⠈⠉⣟⠛⠛⠛⠿⠿⣿⠿⠿⠿⠿⠿⠛⣿⠋⠉⠀⠀⠀⡇⠀⠀⠀⣸⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣤⣄⣀⠀⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣧⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣿⣶⣶⣶⣶⣦⣿⣤⣤⣤⣤⣤⣤⣿⣶⣾⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉




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

⠀⠀⠀⠀⢀⣠⣤⣤⡶⠶⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⢤⣤⣀⡀
⠀⣀⣴⠞⠋⠁⢀⣀⣤⣤⠴⠶⠶⠶⠒⠒⠒⠒⠒⠒⠓⠶⠶⢶⣤⣌⡙⠻⢶⣤⡀
⢸⡏⠀⠀⣴⠞⠛⠉⠁⠐⢶⣶⡶⠟⠓⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⠉⠛⢷⣄⠈⠛⢦⡄
⣿⡆⠀⠀⢿⡀⠀⠀⠀⠀⠘⢿⣷⣶⣶⠶⠞⠋⠉⠁⠀⢀⣹⣿⠀⠀⠀⠀⢹⡇⠀⠀⣿⡆
⣟⢻⣄⠀⠀⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠟⠛⠛⢃⣀⣤⡾⠛⠁⠀⢀⣿⣇
⣿⣸⠙⢷⣤⡀⠀⠀⠉⠙⠛⠷⠶⠶⣶⣶⣶⠶⠶⠶⠶⠾⠛⠛⠋⠉⠁⠀⠀⢀⣤⠾⡃⣇
⠘⣿⣄⢸⠉⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠞⠋⠁⢀⣷⢿⣄
⠀⡟⢿⣿⣦⣄⡀⢸⠈⠉⠙⠛⢻⠛⠶⠶⠶⢶⠶⠶⠞⠛⠛⡛⠉⠉⠀⡇⣀⣤⣾⠟⠁⣀⣿⡆
⠀⣧⣰⠈⠛⠿⣿⣿⣤⣄⣀⣀⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢐⣇⣤⣴⣾⠿⠛⢉⣡⣴⠾⡏⢸⡇
⠀⠻⣿⣄⠀⠀⡄⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣭⣥⡶⣶⠛⠋⣩⣀⣀⣧⣾⣅⣀⣀⣀⣀⣀⣀
⠀⠀⣿⠻⣷⣦⣇⡀⠀⠀⢰⠁⠀⠀⠉⠉⢹⡏⠉⠉⠉⠉⣩⠉⠀⠀⢀⣿⠾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠶⣦⣤⣀
⠀⠀⣿⣸⠀⠉⠛⢿⣶⣦⣼⣀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⢀⣀⣰⡟⠁⠀⠀⠀⣠⣶⠶⠿⠿⠿⠿⠿⠿⠿⣿⣿⣶⣦⣤⣄⣈⠉⠛⠷⣦⣄⡀
⠀⠀⠈⢿⣦⣀⠀⢸⠀⠈⠉⠛⢻⠿⠿⠶⣾⣷⣶⣶⣾⣿⡿⠿⣿⣿⡇⠀⠀⠀⠸⣯⡀⠀⠀⢠⣤⣤⡶⠶⠶⠀⠀⠀⠉⠙⠛⠿⠿⣶⣤⡀⠉⠻⣦⣄
⠀⠀⠀⠀⠉⠻⢿⣾⣤⣀⡀⠀⢸⠀⣀⣠⣤⡴⠷⠶⠶⠶⠶⠶⣿⠘⢷⣄⠀⠀⠀⠙⠻⣦⣄⡀⠈⢿⣦⣤⣤⣴⠾⠟⠛⠛⠿⣷⡄⠀⠉⠻⣶⡀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⠿⠋⠉⠀⢀⣤⣴⠶⠿⠿⠿⣿⡀⣸⠙⢷⣤⣀⠀⠀⠀⠉⠛⠻⢶⣭⣍⣁⣀⠀⠀⠀⠀⣴⡿⠿⠶⠄⢀⣼⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⣴⠟⠉⠀⠀⢀⣤⡄⠈⢷⣇⠀⠀⠈⡟⠳⣦⣄⣀⡀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠷⠶⠶⠶⠶⠿⠟⠋⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢸⣏⠀⠀⣤⣤⣿⠁⠀⠀⠀⠙⣷⣄⣰⠃⠀⠀⠈⢹⠋⠧⠦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣦⠀⠀⠀⠀⠻⣦⡀⠀⠀⠙⢷⣶⣶⡶⠟⠉⠙⠻⣶⣦⣄⣀⡏⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⢛⠛⠛⠛⠛⢻⠋⠉⠀⣜⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⡟⢷⣄⡀⠀⠀⠈⠛⠷⢶⣤⣤⣀⣀⣀⣀⡀⠀⠀⢀⣉⣛⣿⡿⣷⣶⣶⣤⣼⣥⣤⣤⣄⣀⣼⣀⣀⣀⣀⣼⣤⣴⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠉⠻⢶⣤⣀⡀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠉⠉⠉⠀⠀⠀⠈⠉⠉⢉⣹⠟⢻⡟⠛⠛⠿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡇⠀⠈⠉⠻⠛⠷⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠛⠋⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣀⡆⠀⠀⠀⠀⢸⠁⠉⠉⠉⠉⠙⡟⠋⠉⠉⠁⢸⠀⠀⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⣶⣦⣤⣼⣄⣀⣀⣀⣀⣀⣇⣀⣠⣤⣴⣾⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠛⡁⢈⣙⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡘⠃⠘⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠇⠸⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣆⣰⣾⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣀⣀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣦⣀⠀⠀⠀⠈⢹⡏⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢸⣇⣠⣼⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠛⠷⣦⣼⡟⠉⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⢸⣿⣿⡟⠉⠉⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠐⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⢸⣿⣿⣷⣤⣴⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣉⣉⣉⣉⣉⣉⣉⡙⠛⠟⠿⠟⠿⠋
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠐⣶⣶⣶⣆⠀⠀⢀⣀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠿⠟⠛⢋⣠⣶⣿⣿⣿⣧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠷⡶⠶⠶⠾⠿⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠿⠗⠛⣩⠍⠀⠀⠀⠊⠁⠀⠀⡤⠀⠠⠄⠀⠀⢦⡄⠀⠀⢠⡀⠀⠉⠙⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⡋⡁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⣤⠀⠀⠐⠦⠀⠀⠾⠉⠳⢦⣄
⠀⠀⠀⠀⠀⣴⡶⠛⠁⠀⠈⠂⠀⠀⣴⠄⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⡀⠀⠀⢤⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⠦⠙⢳⡄
⠀⠀⠀⠀⣴⠋⠤⠀⠀⡠⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠔⠀⠀⠀⠀⠶⠙⢦⡀
⠀⠀⢀⣾⢃⠈⠁⠀⠀⠈⠀⠀⠘⠁⠀⠀⢀⠴⢄⠀⡀⡀⠀⡀⠀⠠⠠⢀⣀⠠⠂⠀⠂⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⢀⡀⠀⠈⢷
⠀⠀⢸⡧⡃⢄⠄⠀⠀⠀⠀⠀⢀⠠⡀⠠⠊⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⢄⡀⠀⢀⠀⠀⠀⠉⠀⠀⠘⡇
⠀⠀⣸⣏⣾⣦⢀⡤⢤⡄⠀⠀⢀⣴⢠⠀⣮⣧⣧⣦⣼⣬⣼⣿⣼⣾⣶⣾⣶⣶⣴⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⣿
⠀⠀⢸⣿⣿⠿⢿⢳⣰⣿⣿⣣⣰⣾⣾⣿⣿⣿⠿⠿⠛⠛⠛⢋⡉⠉⣉⣉⣉⡉⠉⠉⠉⠉⣩⣯⣽⣛⠛⠛⢳⣶⠶⠶⢦⣤⣤⣤⣤⣤⣀⣀⣤⠏
⠀⠀⠘⣿⣿⣷⣿⣷⣿⣿⣿⣿⠿⠛⠋⠉⢀⣤⠶⠛⠛⠓⠲⠋⠙⠛⠉⠉⠈⠉⠙⠛⠲⠚⠁⠀⠀⠉⠉⠙⠉⠉⡓⠛⠶⣄⡀⣀⣈⣉⡉⢉⣉⠳⣄
⠀⠀⠀⢈⡽⠿⣟⣛⣽⣥⣤⣤⣤⣄⣀⡴⠋⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠻⠏⠉⠙⠲⢿⢻⡞⠛
⠀⠲⠛⠋⣶⣿⠿⠁⠀⣔⠀⠀⢀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⢀⡤⠀⠀⣿
⠀⠀⠀⠀⣿⢀⣨⡀⠀⠈⠀⠀⠙⠀⠀⣠⡾⣛⣛⣛⣯⣤⣤⣤⣤⣤⣤⣤⠀⢀⡀⠀⠀⢀⣀⢀⣶⣿⣷⢶⣶⣦⡄⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠘⠻⣿⣿⣷⠀⣴⢦⠾⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⡄⢠⣿⣿⣿⣁⣀⡀⠀⠈⣻⣧⠀⣴⣦⡾⠛⠟⠻⠶⢶⣾⠋
⠀⠀⠀⠀⠀⠘⣿⣻⣿⣠⠇⠀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠶⠶⠶⠶⠒⠚⠛⠛⠛⠿⠿⠞⠛⠋⠀⠀⠈⠉⠉⠙⠛⠛⠚⠷⠦⠤⠤⣄⣀⣀⣠⣟
⠀⠀⠀⣠⣴⠿⢿⡿⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⣼⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⠴⠄⠀⠀⠀⠀⢀⣠⣴⠶⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⣦⣠⣶⣤⠀⠀⠀⠀⢠⣼⣧⡀
⠀⢨⡟⣉⠩⠷⣶⠖⠉⠈⠙⠛⢻⣶⣤⡀⠀⠀⠀⠸⣇⡄⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠾⣇⠀⢰⢦⡍⠀⠹⣷
⠀⠘⣷⣦⣤⣼⣷⠆⠀⢠⡀⠀⠀⠈⠙⠻⣷⣄⡀⠀⠉⠀⠀⠀⣠⣼⠏⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠟⠀⢀⡀⢠⡇
⠀⠀⠈⢻⣿⣧⣥⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣿⣶⡄⢀⣾⠋⠁⠀⠀⣈⣳⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⣉⣀⣀⣀⠀⢀⣀⣰⣦⣀⣀⣀⣀⣴⠟
⠀⠀⠀⣿⣿⣿⡛⠙⢹⡿⣟⣿⠟⠋⠻⣿⠽⠿⣿⣿⣿⣿⣾⡿⠛⣿⡟⠛⠉⢉⠉⠉⢹⡏⠁⣽⠋⠉⢻⠟⢿⣍⣛⣻⠟⠛⣿⠋⠻⠛⣩⡿⣯⣽⠇
⠀⠀⠀⠙⢹⣿⣷⣤⠏⠀⠈⠀⠀⠀⠀⢻⣄⣠⠋⠉⠛⠛⠋⠀⠀⢸⡇⠀⢸⠛⢷⡴⠛⠀⠀⣻⣦⣤⠏⠀⠀⠉⠉⠁⠀⠀⣿⡆⠀⢀⡟⠀⠉⣿
⠀⠀⠀⠀⢸⣿⣽⣥⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠞⠁⠀⠀⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣧⣬⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠳⠶⠶⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠶⠖⠒⠛⠋⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⢿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⢤⣀⡤⢤⢠⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠷⡤⠘⢧⠼⠸⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠉⠻⣶⣄⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⣾⠿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠙⢷⣶⢶⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⢀⣀⣀⣀⡀⠀⠀⠘⢿⣴⡿⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣦⠀⠀⣿⡇⠀⢸⣿⠀⠀⣴⡶⠶⣶⡆⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⠶⠾⠿⣶⣶⡿⠷⠶⢾⣿⡀⠀⣿⣧⣴⡾⠿⠶⣶⣿⡀⠀⢸⣧⣴⠶⠿⠶⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢀⣾⠟⠁⣀⣴⣦⣄⡈⠻⣦⣿⠋⠀⣠⣴⣶⣄⠉⢻⣦⡿⠋⣀⡀⠀⢀⣄⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠛⣿⡇⠀⠰⣿⣹⣏⡉⠀⠀⢹⡇⠠⣾⣿⡦⠀⠉⠁⠀⢻⡇⠀⠈⣿⣤⣿⠃⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣄⣿⡇⠀⠀⠉⣻⣟⣿⠄⠀⣸⡇⠰⢿⣷⠶⠀⣀⠀⠀⣼⡇⠀⠀⣹⣿⣏⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠉⠉⠉⢿⣄⠘⠛⠻⠟⠋⢀⣰⡿⢷⣄⠈⠛⠷⠾⠟⢁⣴⠿⣷⡀⠀⠉⠿⠉⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⣶⣶⣶⣿⣿⣶⣶⣶⠶⠟⠋⠀⠈⠙⠷⢶⣶⣶⠶⠟⠁⠀⠈⠛⠷⢶⣶⡶⠾⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡏⠉⠉⠉⠉⠉⠁⠀⠈⠀⠒⠤⢤⣄⠤⠄⣐⠒⠀⢨⡈⠍⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⡳⡄⠀⠈⠆⢠
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡇⠀⠤⠴⠤⣀⠀⠀⠀⠀⠣⡤⠤⠤⣤⢄⠀⠈⠻⡀⢀⢞⠘
⠀⠀⠀⠀⢸⡿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠉⠒⢤⣀⡀⠈⠢⡤⠨⠟⣛⣢⣄⠈⢣⠬⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣈⡛⠓⠦⠯⠹⠾⠷⠞⠀⢉⠕⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢍⣒⠨⢿⣍⣡⠶⠒⠀⢲⠂⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠭⢐⡒⠂⡍⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢆⣀⣘⣄⡔⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠍⡙⠃⠀⠀⠋⡩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⣱⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠎⢀⡠⠤⢤⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⠀⡏⠀⢎⡀⢹⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠳⣄⣒⣁⡼⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⡀⠀⠈⠈⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠚⠉⠒⠒⠒⠒⠒⠊



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠤⠴⠶⡶⠶⣶⠶⠶⠶⢶⡶⠦⠤⣄⣴⣚⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣤⣉⣡⣆⣀⣀⣀⣤⡄⣷⠪⡄⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⣏⣏⣉⣉⣍⣍⣈⣸⣠⣤⢾⣅⠢⣧⢀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣹⣉⣍⣭⣨⣠⣥⣥⡄⡰⠷⢅⢺⣆⠀⠐⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣏⣹⣹⣍⣯⣽⣬⡤⢃⢪⢷⢍⣿⣆⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢹⣙⣋⣉⣹⣉⣁⣅⣄⣶⢺⢓⢌⣻⣆⠀⠐⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⣙⣹⣉⣋⣟⣹⣩⢧⢇⡾⡎⠀⢳⠬⡺⢦⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣛⣹⣉⣋⣛⣙⣉⣉⣅⡭⠔⡲⡅⠀⠀⠑⡬⡳⢕
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠛⠛⠛⠛⡛⢻⢻⣋⣋⣿⡨⠥⠓⠁⠀⠀⠀⠈⠪⠔⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣯⣭⣿⣯⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣄⣿⣿⣴⣦⣿⣣⣜⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠟⠛⠿⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣛⠛⠓⠂⠀⠈⢻⣿⣿⡟⠁⠀⠐⠚⠛⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣄⠀⣾⣿⢃⣠⣽⣿⣿⣶⣌⢦⡀⠀⠀⢀⡴⣡⣶⣿⣿⣯⣄⡹⣿⣷⠀⣠⣴⣶⣤
⠀⠀⠀⠀⠀⠀⣾⠏⢠⣶⣮⣿⣿⣿⡿⠋⠉⠀⠉⠉⠙⠷⣿⣴⣧⣿⠾⠛⠉⠉⠀⠉⠙⠻⣿⣿⣿⣷⣶⡄⠹⣧
⠀⠀⠀⠀⠀⠀⣿⠀⠉⢛⣿⣿⣿⠏⠀⢀⣴⡶⠷⠶⣤⡀⠘⣿⣿⠋⢀⣤⠶⠶⢶⣦⡀⠀⠹⣿⣿⣿⡟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡄⢀⠟⢙⣿⣿⠀⠀⣿⢣⣆⢙⣿⣮⢳⠀⢸⡇⠀⡞⣡⠈⣿⣷⡌⣿⡀⠀⣿⣿⡋⠻⡄⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣬⡆⠈⢿⣿⣇⠀⣿⠸⣿⠿⣿⡿⢸⠀⡼⢧⠀⣇⢻⡟⣿⣿⠇⣾⠁⣰⣿⣿⠁⢠⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⣿⣿⣷⣌⠳⢬⣍⡭⠔⣡⡾⠵⠮⢷⣌⠢⢬⣉⡥⠞⣁⣼⣿⣿⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣦⡀⠀⠀⢀⡰⠋⡁⠀⠀⠀⠀⢈⠙⢆⡀⠀⠀⢀⣴⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠏⠀⠀⠈⠻⠆⠰⠟⠋⠀⠀⠹⣷⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠓⠤⣄⣀⣀⣀⣀⣠⠤⠚⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⢀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣶⣶⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⡀⠄⠂⠈⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠐⠀⡠⠲⠛⣹⣿⣿⡿⠶⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢁⣀⣤⣀⣄⣀⡀⠐⠠
⠀⠀⠀⠀⠀⠀⣏⠐⠒⠿⠽⠧⢶⢚⡇⠀⡀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⠷⢉⣸⣿⠿⢾⣟⣳⡄⠉⠄
⠀⠀⠀⠀⠀⠀⣎⠁⠒⠖⠛⠒⡟⣻⠇⡀⣠⠤⠶⢂⣰⣒⢀⠄⠁⠂⠀⠀⢘⢦⣄⣨⣭⣭⡭⠥⢿⠁
⠀⠀⠀⠀⠀⠀⡏⠒⠒⠳⢷⢾⠿⣽⣷⠣⢄⣊⣛⣛⣛⣋⣀⣵⡇⠀⠀⠀⠘⣀⠀⡉⢙⢻⣟⢹⣭⠂⠀⠀⠀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⡟⠲⢶⣶⣾⣾⡶⣿⣄⣧⡤⠤⠬⠭⠿⢷⣞⣿⣷⠞⠻⣿⣾⣀⠀⠉⠙⢿⣿⣙⣧⣇⣠⠀⣀⡄⢀⣀⠈⠤
⠀⠀⠀⠀⠀⠀⡟⠂⠾⠾⢦⠶⠏⣹⣿⡧⠌⡉⢙⣻⣫⣽⣭⡾⣿⣓⠒⣿⣿⣉⠛⠺⢿⠿⠭⡽⣾⡋⠘⣿⣻⡿⠉⣡⡇⠐
⠀⠀⠀⠀⠀⢠⠷⣤⣧⣼⣯⡉⠀⠻⣷⡶⠢⠤⠴⠶⠿⠗⡐⢚⣿⠭⠈⠿⢿⡽⢛⡒⢻⡦⣶⡟⣿⣝⠛⠉⠉⣙⣭⣿⠁
⠀⠀⠀⠀⠀⠘⣄⣨⣉⣛⣫⡉⠙⢻⣻⡗⠀⠤⠶⡬⠽⡧⠲⢻⡇⣸⢿⠗⣋⣀⣼⣧⡼⢅⡤⠴⣿⡏⠛⠈⠉⣿⣇⣾⣀⡀
⠀⠀⠀⠀⠀⠈⣄⠀⣏⢹⡏⠛⣜⣛⣓⣇⠉⠢⠤⢴⣶⠷⢶⣩⣿⣄⢀⣀⣯⣤⣾⣥⡦⢠⡶⠶⣿⣍⡙⠁⠉⢿⣹⣿⣭⣭⠿⠦⠤
⠀⠀⠀⠀⠀⠠⣄⢈⠙⢻⣟⠒⣶⣦⣽⢧⠙⠛⠻⠿⢿⡿⡿⢷⣷⣏⠀⢠⣃⡼⠿⣿⡦⢼⣷⠞⣿⡍⠛⠶⠾⠛⠻⠗⠳⣷⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⡿⠳⣿⣾⣦⣄⠈⠑⡞⣷⠖⡟⠟⢻⣿⣟⠉⠁⡖⣚⣿⣿⣇⣼⡾⠾⠛⠢⢤⣤⣤⣠⣀⣄⣠⠀⠈⠉⠉⠭⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⣭⣿⣏⠛⠭⣗⣛⣲⣾⡿⢿⣏⣹⠉⢉⣟⣫⣿⣿⣍⣽⡭⠭⢽⠋⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣏⠓⢶⣶⣶⣶⣶⣖⣯⣿⡙⠋⣫⣭⣭⣤⣤⢛⡋⢓⣚⣢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⣾⣿⣿⣴⢾⣾⣛⣂⣴⣒⣂⣩⣄⣠⣀⣀⡛⡙⠻⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠾⠛⠇⠈⠁⠉⠀⠟⡻⠖⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡚⠤⣀⠀⠀⠀⠀⠀⠀⡠⠼⢚⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣾⢷⠯⣭⡑⠄⠳⡀⠀⠀⢠⠊⡠⢊⡭⢗⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠋⠙⠄⠻⠿⣦⣀⣇⡀⢀⣨⣠⡴⠿⠋⠸⠉⠛⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣆⢄⡈⠉⠀⠀⠉⠉⠀⠀⠀⠀⠉⡉⠀⠈⠀⣉⢄⡾⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⣷⡚⢷⣄⠀⢤⢀⠀⠠⠀⠀⣈⠄⢀⣴⡞⢳⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢿⣻⣷⡀⠹⣷⣌⠀⠁⠀⠀⠀⠀⣡⢿⠋⣠⣾⣻⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⣿⣿⣆⠙⣟⣧⢀⣀⡀⡀⡴⣻⠏⣰⣿⡿⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⢯⣽⣦⡈⠚⠺⠺⠧⠓⠓⠁⣰⣯⡝⢡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠛⠿⣿⣦⣤⣤⣤⣤⣴⡾⢟⣫⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠻⠿⠿⠿⠿⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠦⠕⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠻⣿⣿⣿⣦⡀⠀⠀⠀⣾⣿⡟⣡⣙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⢠⣤⡈⢻⣿⣿⣷⡀⠀⠀⣿⣿⡷⢦⠌⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⣀⣉⠙⢻⣿⣿⣿⡇⠀⠀⠙⢿⣿⣦⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⠙⠿⠃⣠⣿⣿⣿⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣷⣠⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣄⣀⠀⠉⠁⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢛⣭⣵⣶⣶⣶⣮⣭⡛⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⣰⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀
⠀⠀⠀⢰⣿⣿⡟⣴⣿⣿⣿⣿⢟⣭⢀⣭⠻⣿⣿⣿⣿⣧⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣾⣿⣿⢱⣿⣿⣿⣿⣿⡸⣿⢌⣿⣷⣿⣿⣿⣿⣿⡎⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣶⢠⣭⠻⣿⣿⣿⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⢻⣿⣿⡜⣿⣿⣿⣿⣿⡹⣿⣔⣿⢇⣿⣿⣿⣿⣿⢡⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠈⢿⣿⣿⡜⢿⣿⣿⣿⣿⣶⣀⣶⣿⣿⣿⣿⡿⢣⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠈⢿⣿⣿⣦⣙⠿⣿⣿⣿⣿⣿⣿⣿⠿⣋⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣧⠤⠤⣶
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣭⣭⣭⣭⣭⣶⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠏
⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⢀⡞⠁⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣶⣄
⠀⠀⢠⡯⠤⠤⡶⠒⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⣫⣴⣾⣿⣿⣿⣿⣷⣮⣝⠿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣡⣿⣿⣿⣿⠿⣛⣛⡻⢿⣿⣿⣷⣝⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣰⣿⣿⣿⡿⢡⣾⣿⣿⣿⣦⣿⣿⣿⣿⡎⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⣶⠆⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⡿⠂⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡹⣿⣿⣿⣷⣜⢿⣿⣿⣿⢟⣽⣿⣿⣿⢃⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡝⢿⣿⣿⣿⣷⣶⣵⣶⣿⣿⣿⡿⣣⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⣝⡻⠿⣿⣿⣿⡿⠿⢛⣭⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠶⠆⠀⠀⠀⠙⠻⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠟⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠙⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠙⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣦⣴⡿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠙⣿⣿⠆⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣇⡼⣿⣿⡇⠀⣶⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⡇⠀⠀⢀⢸⢃⠀⠀⢸⣿⠙⣿⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣞⣿⣿⣿⢹⢃⣿⣿⣿⣴⣿⣶⠀⠀⠀⢸⣿⣿⣿⠁⠀⠈⢳⠀⠀⢸⡆⡞⠀⠀⠼⠋⠀⠙⣿⣿⣿
⠀⠀⢠⣿⣿⣿⡄⣿⣿⣿⠀⠀⣿⣿⢉⣿⣿⡿⡀⠀⠀⢸⣿⣿⡟⠀⠀⠀⢻⠀⠀⢸⣿⣿⣀⢠⣷⣶⠀⣶⣿⣿⣿
⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⠀⠀⢿⡿⠀⢸⣿⡇⠀⠀⠀⢸⣿⣿⣿⡗⣤⣽⣽⠀⡄⢨⣽⣿⣤⢸⣿⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠸⠃⠇⠀⠟⠉⠳⠀⠀⠞⠳⠀⠘⠛⠇⠀⠀⠀⢸⣿⣿⣿⣧⣿⣿⣿⣤⣷⣼⣿⣿⣵⣌⣿⣧⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⡆⠄⠀⣤⢢⠰⡂⢰⢲⠂⢠⠖⡄⡖⠆⢸⠀⠀⠀⠀⡉⠔⠢⡔⠒⣤⡖⡄⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠁⠈⠁⠈⠊⠈⠊⠀⠈⠀⠈⠐⠁⠁⠀⠈⠁⠀⠀⠀⠁⠈⠊⠈⠊⠀⠁⠁⠑⠁

⠀⠀⠀⠀⠀⠀⡀⠐⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣿⣗⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠤⣤⣼⣷⠄⠀⠀⠀⠀⠀⢠⣶⢳⣦⠀⢶⣦
⠀⠀⠀⠀⠀⠀⠿⠀⠿⠘⠟⠀⠀⠀⠀⠀⠀⠀⣾⣾⢿⡻⣫⣷⢻⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣶⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠿⠼⠏⠀⠀⢸⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣝⢫⣽⡵⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣽⠆⠀⡦⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠻⣦⡀⠀⠀⠀⠀⠀⠰⣿⠶⠶⠶⠶⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⢿⣽⠿⣅⣯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣼⣧⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣾⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠻⠟⠺⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡗⢶⣦⡀
⠀⠀⠀⠀⣿⠛⠋⢸⡷⠙⠛⢿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣥⣿⣿⠁⠀⠀⠀⠀⠀⠀⠲⣦⡄⣴⡾⢿⣶⠀⠀⠀⠀⠀⠀⠀⠖⠛⠋⣵⠂⠈⠛⠂
⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠟⠻⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠐⠒⠲⠒⠛⠓⠒⠂⠆

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣠⡀
⠀⠀⠀⠀⠀⠀⢠⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⣴⣶⣄⠘⠁⠀⠀⠀⠀⠀⠀⣶⢀⢀⣿⣉⡇⠀⡀⡆⠀⠀⠀⠀⠀⠀⠰⠗⠮⣭⠧⠾⠄
⠀⠀⠀⠀⠀⠀⣾⡇⣿⣿⢶⢢⠀⠀⠀⠀⠀⠀⠤⢠⣿⡟⠿⣿⣷⠠⠤⠀⠀⠀⠀⠀⣿⣙⣹⡿⠿⠿⡿⡷⣷⠀⠀⠀⠀⠀⠀⠸⢷⣻⣿⣿⣾⡃
⠀⠀⠀⠀⣀⣤⣿⣧⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠤⢼⣿⠶⠷⠶⡶⠞⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣿⣿
⠀⠀⠀⠀⣧⣼⠿⡿⢿⢿⣠⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⡼⣿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⣭⢽⡀
⠀⠀⠀⠀⢀⣴⣧⡒⢿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣾⣋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⣄⣎⣀⣇⢰⣲⣄⠀⠀⠀⠀⠀⠀⠀⠈⢮⣶⡵⠋
⠀⠀⠀⢠⣾⣿⣿⡇⣠⣴⣇⡻⡀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠟⠛⣷⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣾⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣶⡔⢂⢶⣶⡄
⠀⠀⠀⠘⠟⠝⠶⠭⢿⣿⣿⣯⠏⠀⠀⠀⠀⠀⠀⢸⣷⠏⠈⢿⣿⠀⠀⠀⠀⠀⠀⠘⠛⠛⢸⣿⣿⣿⡏⠛⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿⣍⣾⣾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡼⠙⣄⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⢤⣀⣿⣿⣦⠁⠀⠸⣾⣿⣿⢠⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠈⠙⣿⣿⡇⠀⢠⣿⣿⡟⠀⠀⣸⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⠀⠘⣿⠧⠤⠾⢿⡟⠀⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠛⠻⠿⣿⣷⠊⠀⠀⠀⠀⠀⠙⢾⣿⣿⠿⠋⠐⠒⢲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠤⠀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠟⠉⠀⣀⣼⣦⣀⠀⠀⠀⣀⣤⣯⡀⠀⠉⠉⢩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣀⠀⣼⣿⣿⡟⠁⢹⣿⡏⠁⠹⣿⣿⣶⣤⢄⣠⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⣿⡿⡟⠀⠀⣾⣿⣿⡀⠀⠙⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢧⠀⡠⠸⣿⣿⠳⣄⠀⡇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠀⠙⠏⠀⠀⠙⠃





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠋⠉⢀⠀⣠⣂⣸⣦⣴⡼⢧⠤⠠⣤⣦⣤⣆⡄⠀⢈⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⣀⣴⡶⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢦⣄⣠⠦⠌⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣀⠀⣤⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣼⠂⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢁⠀⣠⣾⠟⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠃⠀⠀⠀⠀⠀⠀⡄⠤⢰⡆⠀⠈⠛⢧⡔⠀⡙⠷⣄
⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⣨⡿⠋⠀⠀⠀⠀⠀⢀⣀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⢶⠛⠆⠈⢳⣄
⠀⠀⠀⠀⠀⡴⠋⠀⢠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⢰⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡈⠉⠉⠀⠀⠀⠀⠀⠀⠁⢦⡷⠀⠙⣧
⠀⠀⠀⢀⡾⠁⢀⣰⠟⠀⠀⠀⠀⣀⠀⠀⠀⢀⡴⠀⠀⠀⢠⡏⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⢀⣀⡛⢦⠀⣀⠀⠀⠀⠀⢠⠀⠀⠳⡋⠀⠈⢷⡀
⠀⠀⢀⡞⠁⠁⣼⠇⠀⠀⠀⢠⣬⡧⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⠀⠁⠈⣿⡄⠀⠐⠃⠀⠘⡆⠘⠇⣠⠈⣷⡀
⠀⠀⡾⠁⠀⣼⠏⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡴⢧⡷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠁⠁⠀⠀⠸⡀⠀⠁⠀⢀⣌⡷⠀⠀⣨⣧⠘⣧
⠀⢸⠃⠈⢰⡟⠀⠀⠀⠀⠀⠲⠖⡀⠀⠒⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠸⠤⠇⠀⠉⠀⠀⠀⠀⠀⠖⢦⢹⣆
⠀⡟⢠⠒⣾⠀⠀⠀⠀⠀⠘⠃⠁⣻⡶⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡄⣿
⢸⠇⠈⢸⡟⠀⠀⠀⠠⣇⠀⠀⠀⠉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⠀⠛⡇⢸
⣼⠀⠘⣿⠁⠀⢀⣀⣴⠊⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠟⠁⠀⠠⠀⢢⡀⠀⢏⣱
⣿⠀⠀⣹⠄⠈⠁⠉⠃⠀⡸⠂⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠀⡘⢀⡀⠀⠀⠁⢄⣹⠆⢈⣽
⣿⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂⠙⠛⠁⠀⣠⡶⢾⠁⠀⠈⣿
⣿⠀⢀⣸⡇⠀⠀⠀⡀⠀⠀⡀⠈⠈⢻⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠇⢀⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸
⢸⡆⢈⢸⡇⠐⠀⠂⠹⢄⡀⠙⠦⣄⢀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠃⠀⠀⠀⠀⠀⠀⠀⢻⡏⣸
⠈⣧⠀⠘⢿⠀⠀⠀⠀⠘⡇⠀⡀⠈⠀⠡⠼⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣥⣤⣿⠀⠀⠀⠀⠙⠚⢀⠀⢐⡛⠁⠀⣄⡀⠀⠀⠀⠀⠘⣾⢃⣿
⠀⢹⡆⢠⡘⣧⠀⠴⠋⠉⠁⠀⠓⠊⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣸⢿⡞⠃⠀⠀⠈⠡⠿⡄⠀⠀⢠⡟⣸⡟
⠀⠀⢻⡀⢳⢙⣆⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢠⡄⠀⠀⠊⢳⠀⠀⠀⢐⣡⡾⠁⠀⠈⣠⡄⢰⡶⠟⠀⢭⣾⢷⡿⠁
⠀⠀⠈⢷⡈⠃⣼⣆⠀⠀⠀⠀⠀⠀⠀⠙⠻⠋⠓⠀⠀⢀⡀⠀⠳⣦⣀⣀⠀⠉⢉⣉⡽⠛⠁⠀⠀⠀⠀⠀⠀⢐⣿⠉⠀⠀⠀⠀⠉⠀⠘⠁⠀⢦⡼⢫⣿⠃
⠀⠀⠀⠈⢷⡀⠈⠙⢦⡄⠀⠀⠀⠀⢀⣴⠀⠙⠶⠶⠚⠋⠙⠛⢃⣴⡿⢉⡴⢀⣀⣀⡐⠀⠠⣶⡛⠛⠳⠶⠟⠋⢁⢠⡀⠀⠀⠀⠀⠀⠀⡀⣰⡞⢡⡿⠁
⠀⠀⠀⠀⠈⢳⣄⠀⡈⠳⣄⠀⢀⡐⢨⡽⠂⠀⠀⠀⠀⣴⠧⢀⠞⣀⣀⡞⠀⠈⠉⠉⢹⣆⡀⠀⠉⣛⡟⠀⠀⠀⠀⠸⠃⣠⠀⠀⠀⠀⢀⣽⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢧⡌⠢⠈⠳⣦⡈⠈⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣠⢤⠀⠀⠘⠢⠀⢀⣼⣏⠀⠀⢀⣴⠟⢋⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠐⠉⢱⠆⡀⠀⠀⠀⠀⠘⠀⠀⠈⣣⠄⠀⠀⠀⠀⠀⠀⡠⠉⠀⣤⠁⠀⠀⠀⠀⠀⠛⠋⠀⣀⣈⡶⠟⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠢⣏⣳⠶⣄⡀⠀⠲⣀⡤⠿⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣹⡄⠀⠀⠀⠀⢀⠀⣰⣴⠿⠋⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣉⠀⠋⢩⡒⢶⠥⣤⣤⣀⠀⠀⠀⠀⠀⠘⠋⠁⠚⡁⠀⣠⠀⣀⣤⠶⠛⣉⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣉⠀⠂⠀⠃⠼⠇⠘⠁⠲⠀⣀⣈⡹⠗⠂⠐⣊⣩⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⣦⣤⣤⣤⣤⣤⣤⣶⠶⠶⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠒⠒⠒⠈⠉⠉⠉⠉⠉⠉⠛⠛⠲⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣹⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣞⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⣯⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠞⠛⠛⢻⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⡟⢻⢱⣣⠟⠁
⠀⠀⠀⠀⠀⡜⡑⡝⠁⠀⠀⠀⠀⢸⣜⢔⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⡟⢛⠹⡀⣇⡰⠜⠋
⠀⠀⠀⠀⢠⣷⠱⠣⠀⠀⠀⠀⠀⠀⢿⣿⢻⡿⡿⢿⠷⡶⢶⢶⡶⡶⢾⠟⡟⢛⠻⠉⠏⢹⢈⣄⡧⠼⠒⠉⠁
⠀⠀⠀⠀⢸⣷⢳⢔⣶⣄⠀⠀⠀⠀⠀⠉⠛⠾⠤⢮⣄⣧⣸⣀⣇⣰⣨⠤⠧⠼⠒⠓⠊⠉⠉⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠻⣌⠎⢟⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠈⠻⢜⡘⠀⠟⢻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⠾⢄⡆⢰⠉⡙⠻⡷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡻
⠀⠀⠀⣠⣤⠶⠛⠉⠁⠀⠀⠀⠈⠙⠲⢇⣸⠀⠎⢨⠋⡻⠻⡶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠪⢜⣼
⢀⢴⢟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠺⠤⣇⣸⠀⡎⢠⠋⡹⠙⡟⢻⠿⡿⠻⡿⢿⢿⢿⢿⡷⣺⣿⣿⢢⡀
⡏⡎⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠚⠲⠧⠼⠄⣎⣠⣃⣸⣀⣎⣌⣌⡮⠷⠛⠉⡇⡇⡇
⣧⡀⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢞⠔⣱⡇
⣿⠟⡦⣕⡻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣫⢵⢻⣷⡇
⠈⠧⣇⡇⡍⡟⡟⡿⢿⣷⣶⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣶⡾⢿⠟⢻⠙⠉⠀⢸⡸⠋
⠀⠀⠈⠑⠣⢧⣇⡇⠀⠀⢸⠀⢹⠉⡟⠙⡟⢻⠛⢻⠛⠻⠛⡟⠛⡟⢻⠋⢻⠉⢹⠁⡇⠀⡇⠸⠀⣸⡠⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠤⠼⠠⡇⣀⣇⣈⣀⣸⠀⣘⠀⡇⢀⣇⣸⣄⣸⢀⠼⠤⠷⠒⠓⠊⠉
⣿⣿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢸⡇⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⢀⠎⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⡘⢃⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⠀⠹⡀
⠀⠀⡞⠀⠀⠀⢀⠊⢀⠄⠀⠀⠀⠀⢀⡀⠤⢴⡏⠁⣤⣄⠀⠀⠐⠁⠈⠢⠀⠀⣠⡤⠈⢹⡦⠤⢀⡀⠀⠀⠀⠈⠠⡀⠑⠀⠀⠀⠀⢳
⠀⢰⠃⠀⠀⠀⠄⢠⠃⠀⠀⠀⢀⡴⠋⠀⠀⠠⢿⡄⠀⠋⠉⠀⠀⠀⠀⠀⠀⠉⠚⠀⢀⡿⠄⠀⠀⠙⢦⡀⠀⠀⠀⠘⡄⠠⠀⠀⠀⠘⡆
⠀⢸⠀⠀⠀⢰⠀⡇⠀⠀⢀⣀⠯⠤⠤⠤⠤⠒⢻⣇⠀⠀⠀⣀⠔⠀⠀⠢⣀⠀⠀⠀⣸⡟⠒⠤⠤⠤⠤⠽⣀⡀⠀⠀⢸⠀⡆⠀⠀⠀⡇
⠀⢸⡀⠀⠀⠸⠀⡆⠀⠀⠀⠉⠻⢷⢄⠀⠀⠀⠀⠉⠦⡀⣾⠡⠄⠀⠀⠠⠌⣷⢀⠴⠁⠀⠀⠀⠀⡠⡾⠟⠉⠀⠀⠀⢰⠀⡇⠀⠀⢀⠇
⠀⠀⣇⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⢧⢳⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⡞⣼⠁⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣸
⠀⠀⠘⣆⠀⠀⠀⠀⢣⡀⠀⠀⠀⣠⠏⢸⡇⠀⠀⠀⠀⠀⠘⣍⡁⠀⠀⢈⣩⠃⠀⠀⠀⠀⠀⢸⡇⠹⣄⠀⠀⠀⢀⡜⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠉⠒⠒⠚⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠓⠒⠒⠉⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⣿⣿⣿⣇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠘⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀⣹⣿⣿⣧⣼⣿⣿⣿⣇⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠟⠿⢿⣿⣿⣿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠋⠀⣀⣴⠟⠁⢀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡋⢀⣴⣾⡟⠁⢀⣴⠋⠀⣤⠀⢀⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠋⠀⣰⣿⠃⢀⣼⡏⠀⣼⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡁⣠⣾⡿⠃⢠⣾⣿⠃⢰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣴⣿⣿⣿⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠋⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠋⣡⠀⣤⣬⣽⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡀⠀⠀⠀⢀⣀⣿⣿⣿⣷⣄⡉⠀⠻⠿⣿⣿⣿⣿⡇⠀⠀⢀⠶⡀⠀⠀⠀⡎⠀⠀⠘⡄
⠒⠒⠒⠒⣄⡴⠒⠒⠒⠋⠀⠘⣄⡴⠒⠋⠈⣿⣿⣿⣿⣿⣿⠀⣶⡄⠈⣿⣿⣿⡟⢆⢀⠎⠀⠙⠒⡄⡸⠀⠀⠀⠀⠘⠒⠒⠒
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⣿⣿⣧⣭⣍⠀⣩⣤⣶⣿⣿⡟⠀⠘⠎⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣇⣀⣀⡀
⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠛⠛⠛⠉⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡉⠉⠛⠛⠋
⠀⠀⠀⠀⣠⣾⣿⣿⠿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢀⣾⣿⣿⡟⠁⣀⡈⢙⣿⣿⣿⡄⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⡀⠙⢿⡿⠁⣨⣿⣿⣆
⣴⡄⢸⣿⣿⣿⣧⣀⠉⠛⠿⣿⣿⣿⣷⢠⣤⡀⠀⠀⣾⣿⣿⡇⠀⠀⣠⣤⢰⣿⣿⡟⠛⠀⠈⠀⠐⠛⢿⣿⣿⠀⣤
⣿⣿⠸⣿⣿⣿⡋⠛⠿⠇⠀⣿⣿⣿⡟⢸⣿⠃⠀⠀⣿⣿⣿⡇⠀⠀⢻⣿⠈⣿⣿⡷⠶⠶⠀⠰⠶⠶⢿⣿⡿⢰⣿
⠘⣿⣆⠹⣿⣿⣷⣦⡀⣠⣾⣿⣿⡟⣡⣿⡏⠀⠀⠀⣿⣿⣿⡇⠀⠀⠈⣿⣧⡙⣿⣷⣶⣶⡀⢠⣶⣶⣾⡿⢁⣾⡿
⠀⠘⢿⣷⣌⡛⠿⠿⣿⡿⠿⠛⣩⣴⣿⠋⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠈⢻⣷⣬⡙⠻⠿⣿⡿⠿⠟⣋⣴⣿⠟
⠀⠀⠀⠙⠻⢿⣷⣶⣶⣶⣾⡿⠿⠋⠀⠀⣠⣤⣤⣤⣿⣿⣿⣧⣤⣤⣤⡀⠀⠈⠛⠿⣷⣶⣶⣶⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠀⢀⣄⣂⣩⡆⠀⠀⠀⠀⠀⣀⣀⠤⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠓⠶⠤⠶⠶⠶⠚⡇⠀⠀⢀⢀⢸⡀⠀⠐⢲⠒⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠁⣠⣬⠃⠄⠀⠤⠤⠴⢾⣏⣋⣁⣀⣀⣄⢸⡆⠀⠺⠛⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠓⠀⠴⢬⡷⠤⠤⣤⣤⣴⡾⡧⠀⠀⢠⣠⣤⢾⡗⠀⠚⢛⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠦⢿⠳⠤⢀⣄⠒⠒⣺⣯⠄⢀⣠⣴⣴⣾⣉⠀⡙⢛⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠠⣤⣿⠆⠀⢀⣀⣠⣡⡴⣿⠄⠀⣀⣬⣤⣮⡍⠈⠉⢋⣿⠦⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠴⠦⠴⢶⡤⠷⠤⡦⠿⠧⣀⣀⣠⣀⣀⣴⡿⠄⠀⣠⣤⢼⣛⠩⠀⠉⢏⣻⡗⠋⡛⣲⡇
⠀⠀⠀⠀⠀⠀⣶⢤⣕⣘⣯⠐⣚⠆⠲⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠬⠿⢦⣀⣤⣴⣾⣏⡍⠀⠛⠛⣰⠟⠀⣟⣿⠃⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠙⣻⡇⠀⠀⠀⠀⠠⠀⠀⠀⣤⣴⣒⣛⣉⡉⠄⠊⠠⠔⠀⠀⣈⣶⡶⠼⠻⠖⠛⠿⠿⢿⢛⣯⣿⣬⣷⣀⣰⣿⣶⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢁⡇⠀⡀⠀⠀⠀⡀⠀⠸⡿⣻⡷⣶⣤⣄⣹⠦⠵⣊⣛⣭⠬⢾⡶⠒⣚⠖⠒⢒⣾⣿⣉⣿⣿⣿⣿⣄⣭⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠹⢗⣢⠿⣾⣷⣧⠐⠀⠀⠀⠀⠈⠀⠀⡇⣂⠀⠀⠀⠁⠀⠉⡯⠾⠍⠺⠭⠛⠟⠛⣻⠿⠾⠿⠯⣟⠙⠀⡀⢤⢼⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⢌⣁⡀⠨⠠⠀⣷⣀⡉⠒⠂⠀⠀⠀⡧⠄⠀⠐⠀⠀⣠⣀⣽⣃⠀⠁⠉⢙⣢⣤⣶⣾⡞⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠐⠒⠹⠿⢟⡷⢶⣤⣤⣄⣧⢥⣤⣵⣾⡯⠿⠶⠛⠈⠉⠑⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠊⠓⠂⠩⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⢢⠀⠀⢄⣔⣤⢰⠂⠴⢦⢂⢠⣒⡄⡴⠆⢸⠄⠀⠀⠀⠈⡠⠒⢤⠒⢄⣴⢢⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠈⠈⠀⠉⠀⠡⠀⠐⠁⠀⠈⠨⠈⠢⠀⠁⠀⠈⠃⠀⠀⠀⠀⠈⠤⠉⠠⠀⠉⠈⠐⠌

⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢶⣶⣶
⠀⠀⠀⠀⠀⠀⣴⣾⣯⡃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣨⣿⠁⠀⠀⠀⠀⠀⠀⠰⣿⡦⢻⣿⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣤⣜⣿⣷⢘⣿⣟⣤⡀
⠀⠀⠀⠀⠀⠀⠛⣿⡟⡃⣹⣇⠀⠀⠀⠀⠀⠀⠀⡴⣶⣿⠈⡏⢹⣷⡶⡄⠀⠀⠀⠀⠀⠙⠻⠟⠻⣶⡋⢁⡾⠗⠀⠀⠀⠀⠀⠛⣿⢽⣿⣿⣿⢹⡟⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣟⠆⣻⡿⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠀⡇⢸⣿⠃⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡜⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⠿⠿⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠛⠙⠋⠋⠛⠙⠋⠙⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣄
⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⢿⡆⣀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠐⠒⠶⣆⣀⣠⣤⣴⣳⠄
⠀⠀⠀⠀⢀⣟⠁⢿⣿⣿⣿⠇⢹⣇⠀⠀⠀⠀⠀⠀⠛⠛⣠⣤⡘⠛⠃⠀⠀⠀⠀⠀⢀⣀⣤⡟⠛⠈⢠⣾⡏⠀⠀⠀⠀⠀⠀⣸⣿⣻⣀⣄⣸⣻⣿
⠀⠀⠀⠀⠯⠈⠇⠘⠿⡿⠟⠀⠇⠨⠇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⣷⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣮⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣥⣿⣤⡀⠀⠀⠀⠀⠀⠀⠈⠋⢙⡟⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢶⣶⡶⣶⣶⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⡦⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢠⡄
⠀⠀⠀⠀⠀⣀⣤⣤⣄⢠⡄⢠⣄⠀⠀⠀⠀⠀⠀⢸⣿⣼⣯⢿⢣⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣛⡿⠃⠀⣴⢶⠀⠀⠀⠀⠀⠀⣰⣿⡿⣻⣵⣶⠒⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠾⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣧⣽⣿⢿⣹⡿⣘⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠾⠀⠰⠗⠀⠺⠆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠻⠟⠀⠀⠀⠀⠀⠀⠻⣿⢟⣿⡟⢯⣾⠏
⠀⠀⠀⠀⠀⠷⠿⠿⠟⠙⠿⠿⠿⠇⠀⠀⠀⠀⠀⠸⠿⠗⠾⠿⠒⠿⠿⠀⠀⠀⠀⠀⠀⠸⠶⠾⠶⠗⠀⠺⠷⠀⠀⠀⠀⠀⠀⠀⠸⠚⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⢺⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠃⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⢬⣿⠂⠀⠀⠀⠀⠀⠀⠀⣰⢖⣧⣾⢻⣦⡓⢶⡀⠀⠀⠀⠀⠀⠀⠸⣿⣍⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣬⣟⠛⣩⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣝⠿⠨⣟⣵⣿⡇⠀⠀⠀⠀⠀⣤⣀⠈⠉⢁⣠⣤⣄⡀⠀⠀⠀⠀⠀⠉⠀⢿⣿⣨⣿⠇⠈⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣆⢦⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⢿⣯⣾⣿⣮⣿⣿⡇⠀⠀⠀⠀⠀⠈⠛⢿⣜⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠖⠀⣿⣿⡇⠐⠄
⠀⠀⠀⠀⠀⠀⠙⠋⠙⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⣆⣠⣠⣀⣀⣰⡆⠀⠀⠀⢸⢩⣭⣭⣭⡍⡆⠀⠀⠀⢰⠚⠚⠚⢳⣬⣿⠀⠀⠀⠀⠘⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡯⢭⢭⢭⣷
⠀⠀⡧⢼⣽⡥⠤⢼⡇⠀⠀⠀⢸⠠⠠⠠⢠⠄⡇⠀⠀⠀⢸⠀⠭⠭⢸⡇⣿⠀⠀⠀⠀⢠⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡇⠈⠘⠚⣿
⠀⠀⡗⠺⠖⠒⠶⢺⡇⠀⠀⠀⢸⠨⠬⠭⠸⠇⡇⠀⠀⠀⢸⡀⠒⠒⢸⠟⡛⠀⠀⠀⠀⣮⣤⠀⠀⣴⣼⡀⠀⠀⠀⠀⡇⠀⠄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⣠⠶⠶⢤⡀⠀⠀⠀⠀⠀⣠⠴⠶⠦⡀⠀⠀⠀⠀⠀⢀⡤⠶⠦⣄⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⠀⠀⠀⣠⠴⠶⢦⡀
⠀⠀⣼⠁⣤⣤⡄⢻⠀⠀⠀⠀⢰⠇⣤⣿⣄⢸⡆⠀⠀⠀⠀⡏⠐⣦⡖⠈⡇⠀⠀⠀⠀⣾⢀⣼⣧⡀⣷⠀⠀⠀⠀⢸⠃⠶⠶⠆⢹⡄
⠀⠀⠘⣆⣀⢀⣠⠟⠀⠀⠀⠀⠈⢧⣀⠛⣀⡼⠁⠀⠀⠀⠀⠳⣌⡁⣉⡴⠃⠀⠀⠀⠀⠹⣄⡐⢂⣠⠏⠀⠀⠀⠀⠘⢦⣉⢉⣡⠞
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⣣⣼⠀⠀⠀⠀⠀⠀⢸⠛⢛⣟⡛⠛⡁
⠀⠀⢸⣇⣙⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⣈⠁⠾⣇⠀⠀⠀⠀⠀⣹⠀⠿⣬⠇⢸⡇
⠀⠀⠘⠉⠉⠋⠋⠃⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢠⠤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡜⢝⠂⡇⢩⢱⠀⠀⠀⠀⠀⢣⡞⠐⣆⣰⠀⠀⠀⠀⠀⠠⣿⠄⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡄⣠⣞⠀⠀⠀⠀⢀⣀⠀⡟⠉⠉
⠀⠀⢆⢠⠧⡧⢁⢨⠀⠀⠀⠀⠐⠋⠙⠀⣸⠇⠀⠀⠀⠀⠀⠀⣿⠀⢦⡔⠀⠀⠀⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠃
⠀⠀⠀⠛⠤⠧⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠋⠓⠀⠀⠀⠀⠀⠘⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢠⡤⠀⠤⣤⣤⣤⡤⡤⠀⠀⠀⠀⢀⣠⣤⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠉⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⢀⡴⢋⠉⢉⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⠤⣀⣀⡜⢁⠂⠉⠋⠀⢸⠀⢸⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡰⠋⡆⠀⢩⠀⠸⠄⠀⡇⠀⢸⠀⠘⣠⠴⠒⢒⠒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡜⠁⠀⡖⠚⠒⡆⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢻⠀⠀⠀⠧⠤⢄⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⡄⠀⠀⡇⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠘⠦⣀⠉⠹⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠
⠀⠀⢰⠈⠉⠉⠉⢉⡍⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠂⠀⠀⠀⡖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⠀⠀⠐⠈
⠀⠀⢸⠀⠀⠀⠀⠊⠘⠆⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠃⠈⠠⠤⠤⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠤⠤⠤⠬⠄⠀⠤⠀⠀⠶
⠀⠀⢸⢀⡌⠉⠉⠉⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⠇⡄
⠀⠀⢸⠀⡆⢀⠴⠀⠦⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠤⢀⠀⠀⠀⣤⣀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠘⢆⡀⠀⢀⠶⡀
⠀⠀⠸⠀⠀⠁⠀⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣈⣉⣉⣁⣀⣉⣤⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠉⠁⠀⠈⠂
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠟⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒⠒⠒⠒⠒⡒⡂⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠁⠀⠀⠀⠁⠈⠁⠘⠓⢠
⠀⠀⠀⢅⢹⠉⠛⠉⠀⠀⢸⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡈⠉⠉⠁⠀⠀⢸⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠂⠀⠀⢠⠆⠀⡼
⠀⠀⠀⠓⠀⠀⠐⠒⠒⠒⢚⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠄⠀⡄⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⣀⣸⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⠉⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠐⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠑⠠⢁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡡⠒⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⢀⠠⢀⠤⣠⣶⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢷⣄⠀⠀⠀⠀⣀⡀⠼⡧⢐⡒⠈⠂⠈⠀⠁⠉⠉⠛⠦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡒⡀⡼⠛⠓⠚⠉⠉⠉⠔⠁⡣⢌⠙⣌⣐⣄⠐⣠⠀⠖⠀⠰⢓⡩⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⢀⠺⣄⠫⠀⠀⠀⡀⠄⠦⣀⡀⡩⠚⠈⡉⠝⠀⣀⡢⠅⣒⣩⢥⡶⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣄⢳⠠⡹⣇⡠⠶⠦⢀⣀⡤⣤⡮⠍⠭⢗⡶⠮⠽⡆⠒⠏⣊⣥⣶⣶⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠱⢤⡈⠺⢟⢺⡯⠭⠶⠒⣒⡋⠡⣈⡥⢔⣨⣕⣶⡿⢟⣿⡿⣿⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠈⠀⠙⢦⠉⣶⣦⣤⣖⣺⠭⣟⣫⣽⣻⣝⣫⣾⠟⢋⣥⣶⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡂⢴⡆⠩⣭⣔⡒⠊⣩⣽⣯⣵⣾⣿⣾⣥⣴⡾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢗⣿⠂⠑⣲⣾⣭⣥⡿⡷⠿⠟⠛⢋⠁⡀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠀⠀⡈⠉⠩⢈⠁⠈⠀⠀⠐⠠⡂⢀⠁⡨⢀⠅⡈⢐⠁⠁⡀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⡖⣾⢻⣻⢹⡟⡗⡖⣦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣿⣎⠧⠗⣛⣋⠉⠭⠭⠉⣙⣓⠣⢾⣾⣹⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⣳⡽⢊⠥⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠬⡑⢯⣞⡵⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⣙⡾⡡⠊⠁⠀⠀⠀⠀⠀⠀⡏⠈⡇⠀⠀⠀⠀⠀⠈⠑⢌⢶⣋⡵⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢯⣳⠏⡔⠁⠀⠀⠀⠀⠀⣠⡴⠒⠁⠀⠁⠐⠂⠄⠀⠀⠀⠀⠈⢢⠹⣞⡽⡄
⠀⠀⠀⠀⠀⠀⢀⣟⣓⠏⡌⠀⠀⠀⠀⠀⠀⢰⠋⠀⠔⠉⠀⠉⠙⠚⠀⠀⠀⠀⠀⠀⠀⢡⠹⣚⣽⡀
⠀⠀⠀⠀⠀⠀⣸⠤⡽⢸⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢷⠶⣇
⠀⠀⠀⠀⠀⠀⣽⣽⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠈⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠭⢭
⠀⠀⠀⠀⠀⠀⣗⣒⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⣄⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠘⢸⣟⣟
⠀⠀⠀⠀⠀⠀⢹⠶⢷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⣞⠒⡏
⠀⠀⠀⠀⠀⠀⠈⣟⡭⣆⢃⠀⠀⠀⠀⠀⠀⢰⠦⠄⣀⣀⣀⡴⠃⢀⠌⠀⠀⠀⠀⠀⠀⡘⣰⢭⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣞⡽⢆⠣⡀⠀⠀⠀⠀⠙⠢⠤⢄⡀⠀⠤⠔⠉⠀⠀⠀⠀⠀⢀⠜⣰⢯⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⣩⠷⡑⢄⡀⠀⠀⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡼⣍⡳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢞⡽⣳⢌⡒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢒⡡⣞⢯⡱⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⣏⡿⡷⢢⢭⣍⣀⣒⣒⣀⣩⣭⢴⢺⡹⣿⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠼⢼⣽⣇⣯⣇⡿⠼⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡀⠀⠀⣀⣤⠴⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠿⠋⠁⣠⠞⠋⠀⠀⣀⣀⣀⡀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⣴⠟⠋⠉⠉⠛⠆⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠴⣾⡷⠶⠶⠶⠆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢻⣟⠛⠛⠛⠃⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠙⢷⣤⣤⣤⡶⠆⠀⣠⡟
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⢀⣠⡾⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⣠⠾⠋⠁⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⢰⡟⠛⠛⢷⠄⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣸⣇⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⣿⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣼⣯⣤⣄⣀⣤⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠉⠁⠈⠉⠉⠁⣠⡴⠋⢀⣠⣶⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠖⠛⠉⠀⠀⠈⠛⠻

⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣤⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠙⢿⣿⣿⣄⠀⠹⣿⣿⣿⡿⠛⠋⠉
⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⣆⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡏⠙⠻⣿⣷⣄⠀⠙⢿⣦⠀⠀⠻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣄⠈⠙⠻⣷⡀⠀⠙⠇⠀⠀⠙⢿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⢿⣿⣷⣤⡀⠈⠙⠆⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄⡀⠉⠙⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣇⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⡤⠶⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠦⣄⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣴⣶⣦⡀⢷⠀⠀⠀⠀⠀⡾⢀⣤⣶⣤⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⣿⣿⣿⣷⢸⠀⠀⠀⠀⠀⡇⣾⣿⣿⣿⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢿⣿⡿⢫⠇⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⠇⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠹⢬⣉⣁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣶⠶⠦⢤⡤⠤⢤⡤⠴⠶⡶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⣇⣀⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣿⠛⠛⣿⠿⠿⢿⠿⠿⢿⡿⠿⠿⡟⠛⠛⣿⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠹⢤⣀⣿⣀⣀⣸⣀⣀⣸⣇⣀⣀⣇⣀⡴⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠛⠋⠉⠉