Kahl Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠄⢦⠹⡄⢄⠁⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⡸⠃⠏⣸⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠠⠊⠐⠀⡄⢀⢀⢠⠞⠉⠀⠀⠄⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡇⠀⠀⠀⡠⠰⠀⠀⠸⠁⢀⠄⠀⠆⢸⠇⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠇⠀⠀⠰⣱⣦⠀⠀⠀⠀⡆⠀⠀⢀⡾⠀⡸⢀
⠀⠀⠀⠀⡆⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡒⢤⡐⠁⡿⠀⠀⠀⠀⠀⡇⠀⢠⠞⠀⢠⠃⢸⣦
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⡏⡘⠡⡄
⠀⠀⠀⠀⠀⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⠀⠀⠀⠀⣴⠏⠀⠁⢠⡏
⠀⠀⠀⠀⠈⢏⣰⣀⡆⠃⠀⠈⡀⠀⠀⠀⢀⠀⠀⢇⠀⠀⠈⠀⠀⠀⠀⢀⡤⠄⠺⣅⠀⠀⠀⠀⠴⠛⠁⠀⠀⢀⡼⠃⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣐⣄⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⡉⠀⠀⠀⠀⠀⢀⠀⠀⣠⡞⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣇⠁⠀⠀⠀⠁⠐⠠⣄⠀⠀⠀⠀⡰⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⡟⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠙⠀⠀⠀⠀⠀⠳⡀⠀⣰⡁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠎⠀⠀⠀⣼⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡀⠀⠀⠀⠀⠀⠀⡌⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⢀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠳⢤⣤⠤⠒⠁⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠐⣠⠞⠁⠀⣀⠴⠋⠀⠐⡇⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠇⠀⠀⣀⣀⡀⠀⠀⠀⢱⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠁⠀⠀⠀⠀⠀⠈⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠒⠂⠉⠉⠉⠀⠉⠀⠀⠀⠀⠁⠠

⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠠⢀⣠⣤⣶⣿⣿⡝⢿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣄
⠀⠀⠀⠀⢠⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣵⣾⣿⣿⣿⣿⣿⣿⣿⣧⡻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣀⣀⡀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣔⡄
⠀⠀⠀⣿⣿⠃⠈⠉⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠺⣿⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⢀⡏⠀⣠⣴⣟⣩⣭⣭⣽⣿⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⠻⢤⡀⠀⠀⢠⣾⡑⢾⢟⢻⣿⣿⣿⣿⣼⠏⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⢻⣦⠾⠀⢻⣿⣿⣷⠏⠀⠀⠀⠀⠘⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁⢀⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣄⣤⣀⣠⠤⠄⠀⠀⣠⣴⣿⣿⠟⠁⠀⢠⣾⠏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣠⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣤⣄⠀⢀⣠⣾⣿⠿⠋⠀⠀⠀⣰⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢾⠻⢛⣫⣥⢶⣄⠀⠀⣼⠟⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠸⣆⡾⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⢦⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠉⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣡⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣾⡿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠑⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⢁⣠⣄⠀⠀⠸⠟⡉⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢻⣿⣇⠀⠀⣡⡾⠛⠃⠀⢠⣾⣿⣄⠣⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠻⣿⠀⠀⠿⣡⣶⡶⠆⠈⠁⠈⠻⠇⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡰⣾⠟⠉⢀⣇⠰⠾⡟⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⢌⠀⢠⡼⠟⠁⠀⠀⠀⠀⢆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢙⣫⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠻⣷⣄⠀⠀⢀⣴⣶⠋⠁⠀⠀⠀⢀⠀⢠⣤⠤⠴⠿⣆⣀⡾⢋⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢳⣿⠟⠻⠷⠶⣟⣩⠜⠀⠀⠀⢀⣾⣿⠛⢻⣿⣷⠆⠀⠈⠉⠳⢾⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢏⠁⠀⢸⡏⢰⣿⣿⣿⡉⠙⢷⣄⠀⠀⢸⣿⡿⠀⣠⠿⣿⣧⠀⠀⠀⠀⠀⠹⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢷⡄⠈⢻⠋⣿⡋⠈⣷⣄⠈⢿⣷⠀⢸⡿⢡⠞⠁⠀⠈⠻⢧⡀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣿⣿⠄⣤⠀⠎⠑⢀⣿⣟⠻⢦⡝⣧⠘⣷⡏⠀⠀⠀⠀⠠⠄⠉⢷⣦⣤⡤⠀⠘⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⣿⣿⠂⠘⢷⡄⠀⣾⠋⠙⣦⢾⡄⠸⣆⠈⠃⠀⠀⠀⢠⡴⣄⠀⠀⠸⣷⣤⣤⣀⡇⢻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢻⣦⡟⠀⢀⠘⣆⠹⡄⠙⠀⠘⡿⠀⠀⠈⠶⠊⠁⠀⠀⠙⠟⢹⡟⠳⠀⠙⠻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⣿⠀⠀⠀⠈⣿⣷⡄⠀⢠⣼⠃⣧⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡀⠀⠀⠀⠀⠀⠙⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⣿⣿⡄⠛⠀⠀⣿⣿⣿⠀⠈⠉⠁⠀⠀⠀⠀⣧⣀⣀⡀⠀⠀⠀⠀⣠⡞⠘⣿⠓⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⣴⡎⠀⠹⡿⠃⠀⢀⣴⣾⠟⠁⣮⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠸⠋⠁⠀⠀⠘⠛⠛⠛⠁⠀⠀⠀⠀⠀⠹⣷⡀⠀⣩⡟⠉⢀⡼⠃⠀⢀⣿⣦⠀⠀⠀⣿⡀⢰⢃⡄⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⢾⠏⠀⣠⡟⠳⣄⣴⠿⢻⣻⣷⠀⠀⣿⣤⣾⣿⡷⠖⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠹⣦⠔⡛⢯⣠⠷⠯⡭⠤⢞⠉⠋⠉⢻⡄⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⠀⠀⠻⠶⠿⠊⠁⢀⣠⠄⠀⠀⠀⢀⡀⣸⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠀⠀⠀⠀⠀⠀⣯⣸⣦⠀⢀⣼⣥⣿⣧⡀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠂⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢨⡍⣿⠛⠛⠛⠉⡿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢰⣣⣄⡉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⠳⢀⡀⠀⠘⣧⡟⠀⠀⢀⣾⣟⠉⠀⠈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⢰⣶⡶⣿⣿⣿⣿⡇⠀⠀⢘⠿⣿⠻⣷⠶⢬⣛⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⡆⠀⠀⢀⡟⢡⣾⣿⢛⡟⠋⠀⠀⠀⣎⠑⡌⢳⡌⠷⣦⡈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡿⠿⠇⠀⢀⡞⣰⣿⣿⠟⠉⠀⠀⢀⡠⠊⣿⠀⠘⡀⠙⠀⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⠿⠛⠉⠀⣀⠄⢀⡠⠋⢀⡿⠛⠁⠀⢀⣀⠴⠋⠀⢀⣿⠀⠀⣧⠀⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⠁⠀⠀⡄⣁⠀⣀⡼⠳⠖⠉⠀⣠⠴⠚⠉⠀⠀⠀⢀⣾⡇⠀⠀⣿⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠉⠳⣄⢸⣄⢡⡜⠁⠤⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠋⠀⠀⠀⠹⠀⠙⠟⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠁⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠉⠀⠀⠀⠀⠀⢀⡴⡿⠿⠿⠿⠁⠀⠀⠸⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⣀⡀⡀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠵⠒⡈⠃⠈⠉⠝⠍⠂⢄⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⢇⢈⡉⠲⠀⠂⠀⣀⠵⠃⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠜⠛⠁⠀⡚⢡⠀⠁⠀⠊⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⣇⠀⠊⢰⠀⠸⢋⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣩⡈⠁⢀⠜⠄⠀⠀⢀⠀⠀⢧⡤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠢⢰⠀⠀⠀⠀⠰⢄
⠀⠀⠀⠀⣀⡤⠀⠂⣄⠀⠀⠀⠀⠀⠈⡰⢦⡀⠀⠀⠀⠈⠈⠀⡆⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠃⠤⠊⢠⡉⠃⡄⠀⠀⢀⡼⡹⠀⡌⠁⠤⠁⠀⠀⢰⠁⠀⠀⠀⠀⠀⠌⠀⠈⠒
⠀⠀⢰⠒⢠⡆⠐⣛⠁⠀⠇⠀⢠⡺⡤⠃⠀⠐⠴⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⠃⠿⠀⡆⡆⡄⠴⣳⠀⠰⡠⠀⠂⠀⢸⠃⠀⠀⠀⠀⠈⢣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠁⣰⠁⠚⠉⢩⠈⠁⡆⠀⡇⡄⠀⠀⡼⠰⠉⠉⠠⡁⠚⠁⠈⠂⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠙⢤⣈⠐⠒⠚⠉⡧⠀⢡⠁⡇⠀⣿⠒⠀⠀⢠⠅⠀⠀⠀⠀⠅⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⢐⠒⠀⠀⠀⠀⢃⠀⢸⠀⡇⠀⡇⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⡇⠀⠀⠀⠀⠀⡇⢠⠂⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠀⣽⡆⣵⢀⢄⠀⠀⠀⠀⠀⢠⡄⠈⠑⠀⠀⢀⣤⣠⣤⣤⣤⣁⣀⠀⠀⠁⠐⠀⠀⠀⠀⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡀⠤⢄⡰⣇⠘⠿⠀⠑⢀⠀⡠⠀⠠⡀⠻⠆⢀⣿⢉⠹⣿⣀⣤⡤⣈⠉⠉⠤⣌⠄⠀⠀⠂
⠀⠀⠀⠀⡈⠀⢀⠀⡰⠋⠀⠀⠀⠛⡷⠒⢖⡛⢶⢤⣼⠿⣥⣤⡔⠃⠀⢈⣽⣿⣤⣐⣽⠋⠀⠀⠁⠀⠤⣦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠠⠀⠀⢰⠀⢀⠈⢉⣧⣞⣀⠀⠀⠀⢀⣀⡀⠛⠀⠉⠙⠋⠱⠄⠀⠀⠀⠀⠢⠈⠻⡄⠱
⠀⠀⠀⠀⠀⠠⠀⣇⠀⠀⡇⠀⠠⠄⠃⠀⢀⠪⠀⣻⠟⠒⠞⠂⠚⠊⠀⠀⠀⠰⡄⠀⠀⠈⠳⠢⡀⠀⠀⠀⢀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢀⡆⠀⡈⠀⠀⠆⠀⣜⠃⢰⠁⠀⡄⠀⢀⡐⢦⣀⠀⠀⠀⠘⢄⠀⠀⠰⠀⢸⡦⠀⠀⠘⠀⠀⠹⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢀⠻⡄⣿⣧⠈⠁⣆⢺⠀⠀⠉⠓⠶⣆⠀⠈⣆⠀⠀⠃⢸⠙⠆⠀⠀⢡⡐⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣆⠐⠢⢤⠔⠀⠀⢧⡏⠀⡤⠀⠀⠑⠀⠀⠀⠀⢲⡿⠃⡄⢸⢃⠀⡀⢸⠀⠀⠀⠀⢨⠑⢸⡀⠀⠀⠀⠈
⠀⠀⠀⠘⠀⠁⠀⠀⢠⡀⠈⠀⠀⠀⠠⠀⠁⠈⠃⠸⣷⣿⡄⠀⠀⠀⢀⡁⠼⠁⠀⠘⠏⢀⣼⠓⡈⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⢫⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠜⢀⠞⠉⢀⠃⣄⠀⠀⠀⠀⢀⠂
⠀⠀⠀⢀⠀⠀⠁⠒⡀⠀⡄⠀⡄⠀⢸⠀⠀⠀⠀⠀⠈⠳⣄⠠⠆⠀⠊⠀⣀⠔⠁⣄⠂⠄⢀⡌⠀⣿⠀⣰⠁⡀⠋⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠂⠀⠀⠉⠀⠘⠐⢀⡇⢀⡁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠀⠐⠀⢸⡄⣾⠏⢠⡄⢸⠁⠀⠹⠀⢇⠞⠁⠀⣀⠀⠀⡄⠁⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠁⠀⠀⢀⠃⠀⠀⠀⠀⠀⢀⠄⠀⢠⡅⠂⢐⡂⢺⠁⠀⠀⠈⡉⠻⠆⠀⠀⡞⢘⠀⠀⠀⠹⠶⠀⠃⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠸⠀⠀⠀⠀⠀⡴⢉⡄⣸⠋⠑⠊⠉⠙⣡⠒⣄⠀⠀⢃⡀⠀⢸⠀⠅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠄⠀⠀⡄⠀⠠⠀⠀⢀⠀⡆⠀⠀⠀⣸⠧⠎⠉⠁⠀⠀⢂⡄⡀⢡⠀⠆⠙⠀⠈⢁⠀⠈⠀⠀⠀⠀⠁⠈⠃⢀⠤⡤⠒
⠀⠀⠀⠀⠀⠁⠀⠘⢀⠀⢂⡰⡀⠀⢠⠀⢀⡔⡵⣦⠀⠀⠀⠚⠐⠋⠀⠁⠀⢆⠀⠘⡇⠄⠈⠀⠀⢀⠀⢀⡠⣄⠐⠄⠀⠀⠀⢀
⠀⠀⠀⢆⠀⠁⠀⠚⠀⠀⠈⠈⠻⡄⠀⠰⣉⠎⢰⡧⠀⠠⡤⣄⡀⠀⠀⠀⠀⠸⠄⠀⠹⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠢
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠠⠁⣄⠰⠄⠀⠂⠹⠻⣄⠁⣦⣤⣤⡀⠈⠛⢠⠀⣧⠀⡆⠀⡸⠀⠀⢠⠈⠰⡇⠃⠘⠂⢹⠀⢂⡀⠁⢀⠀⠀⠀⢠
⠀⢀⠁⠀⠀⠀⠁⠀⠀⢥⠀⠁⠀⠀⡈⠀⢠⡇⠉⡟⠀⠀⠀⠉⠒⡀⠀⢈⡟⣰⠁⠀⡏⠀⠀⣼⠀⠜⢀⠀⠀⠀⠈⠀⠄⠁⢀⡆⠀⠀⠀⡂
⠀⠀⠁⠀⠆⠀⠀⠀⣀⣤⠀⡀⢸⡕⣁⠀⢴⡃⢸⢰⡆⠀⠀⠀⠀⢁⡟⠈⡇⢹⡀⢀⡇⠀⠀⠘⣆⠀⠀⡀⠀⠐⠤⠔⠶⠴⠋⣰⡤⠀⠚⠁
⢠⡀⠀⠀⠀⠐⠀⢀⡄⠀⠈⣉⡼⠟⠻⠀⠸⠇⢸⡀⣁⠀⠀⠀⠀⣸⠃⣸⠀⡞⡇⢸⠃⡀⢀⠀⠈⠐⠠⠬⠷⠶⣀⣠⣤⣆⡙⠓⠄
⠀⠀⠀⠟⠠⠂⡀⠉⠀⠀⠀⢩⠀⠀⢈⠆⠀⢻⡀⣧⢸⡀⠃⠀⠀⡟⠀⣍⠀⠁⢱⡜⢂⠘⢀⠑⣀⠀⢀⢀⡀⠐⢉⠁⠀⠉⣫⠀⠀⢷⡀
⠀⠀⠴⠀⠀⠀⠁⠇⠀⠂⠀⠈⠀⠀⠂⠀⠀⢨⡇⢀⠈⡗⠎⠀⠉⠅⡼⠁⠀⠀⠘⡇⠈⠀⠐⠋⡩⠒⠬⢀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⢧⣀⣀⡠⣐⠠⡤⠀⠄⠀⢀⠀⢀⠴⣃⡷⠘⠆⠁⠂⠀⠘⡤⠳⠀⠀⠀⣀⣳⠄⠀⢘⡆⠀⢤⡔⠀⠈⠁⠀⠄⡀⢠⠄⡀⠴⠂
⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⢨⢄⠁⠉⠑⠀⢰⡟⢡⠶⡎⠉⠁⡀⣀⠒⠞⢧⠈⠉⠀⠘⡇⠀⢀⠀⠄⢠⠉⡀⠁⠀⠀⠀⠀⠐⠆⠀⠁⠀⠁
⠀⠀⠀⠰⡀⠀⠠⠀⠠⠀⠀⠀⡇⠐⢀⡔⢋⡴⠃⢀⡇⠀⠀⡱⠀⠀⠀⠀⠆⠀⠀⠀⡗⠤⠈⠀⠀⠀⠀⠧⠀⠀⠀⢠⠀⠁
⠀⠠⠀⠂⠀⠀⢠⠀⠀⠐⠂⠞⠁⣴⣏⢴⠋⠀⠀⠀⢡⠀⠂⠀⠀⠀⠀⠀⠚⡄⠀⠀⢹⡂⠈⠠⣀⣀⠄⠀⠰⠋⠀⠰⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣠⠟⠐⠯⣀⡀⠁⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⣧⠀⠀⢸⠁⠀⠣⠙⢥⡠⠀⠢⠀⠀⠀⡴⠁⠐⡄⠂
⠀⢀⡐⠀⠀⠀⠀⢨⠀⠄⠉⠄⠀⠁⠀⠈⢻⡀⣇⠀⠀⠀⠐⠀⠀⠀⢤⡀⠀⢸⡇⠐⠈⣧⠀⠀⢁⠸⡇⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢧⣀⢀⠀⠀⠸⡏⠈⠛⠳⠤⠼⣦⡤⡄⠊⠁⠈⣿⠁⠀⠀⢀⣧⠴⠀⠂⠙⠒⠓⠀⠠⠂⠀⠀⠀⠀⠀⠂
⠀⠀⢀⠀⠀⠀⡀⠱⣀⠀⠀⠀⠛⡁⠠⣤⠘⣇⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠙⡄⠀⠀⢀⠛⢆⡀⠀⠀⡀⠁⠀⠀⠀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠁⢄⠐⠀⠀⣀⠔⠁⠀⢸⡁⠀⠀⠀⠇⢸⠛⣇⠀⠀⠀⠀⠙⢲⠀⠀⠂⠀⠁⠁⡐⠈⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢦⠀⢠⠋⠀⠀⠀⢸⡆⠀⢀⢰⢰⠀⠄⣿⠀⠀⠀⠀⢸⠀⢰⠀⠈⡤⠀⠀⠁⠀⣉⠰⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣸⠀⠸⠄⠀⠀⠀⠸⠇⣠⠘⠀⢸⠀⠃⣿⠀⠀⠀⠀⢸⠀⠀⢣⡼⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠤⠞
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⢻⠀⠀⢀⠀⠀⠀⠀⡇⠀⠰⠀⢸⠀⢸⡇⣷⠀⠀⠀⢸⡄⠀⡆⠀⠀⠀⠀⠀⠀⠰⡆⣣⢤⠤⠀⠁
⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠈⡆⠀⠘⡀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠸⠁⢻⡀⠀⠀⠀⢷⣰⠀⠀⠀⠀⠀⢸⠀⣴⠈⠠⠀⠀⠀⠠⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠃⠀⠀⠀⣷⠀⠀⠀⢸⡄⢀⠀⠘⡇⠀⠀⠀⠈⢧⡀⠀⠀⢀⠀⠃⣴⠇⠀⠀⠀⠀⢼⠇⠁⣤⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⡟⠀⠀⠈⠸⣷⠘⡀⠀⢻⠀⠀⠀⠀⠀⠹⣋⠉⢁⠀⠄⠁⠀⠀⠙⠖⠀⠀⠀⠀⠻⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⣯⠈⢣⠀⢸⡇⠀⠀⠀⠀⠀⠈⠳⡀⠀⢀⠄⠀⠀⡀⠀⠸⠇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⢄⠀⠀⣧⠀⠀⠀⠈⢹⡆⠘⡀⠈⢳⠀⠀⠀⠀⠀⠀⠀⡘⢦⠈⠀⠀⠀⡳⠂⠀⠀⠀⠞⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢠⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⡿⣫⣜⣮⣁⠲⣦⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣮⡥⢋⣿⣿⡏⠀⢴⡄⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⢠⣤⠀⠀⠀⣼⣿⣿⣿⣟⡩⣀⡹⣿⡿⣦⣾⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠠⣿⣦⣤⣾⣿⠿⠋⠹⣿⡿⠿⠟⠻⠷⣿⣿⣿⡿⠉⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣤⣤⣤⣀⠱⠛⠛⠛⠛⠛⠋⠛⡇⠀⢀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡾⢳⣾⠻⣿⣷⡆⠀⠀⠀⠀⠀⠀⣷⣀⢸⣉⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣽⣿⣿⣧⣸⣿⣿⣿⣯⡇⠀⠀⢀⣉⠉⠀⠙⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠉⠉⠛⢧⣷⣦⣭⣍⠥⠀⠁⠀⠀⠈⠛⠀⠀⠀⠀⠀⠉⠲⣽⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣠⣶⣶⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣽⠃⠀⠙⢷⣦⣤⣤⣤⣶⣦⣤⣤⣤⣤⠴⠖⠚⠛⠉⠉⠉⠛⠲⣤⣄⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠻⣦⡀⠀⠀⠻⠙⠛⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣛⠷⣦⣄⡀⠀⠀⠀⢻⣿⣿⣷⣶⣶⣤⣴⠶⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣝⣿⣷⣦⣄⠈⠛⠛⠛⠛⣉⣥⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⡟⠋⣉⠻⣿⣧⢤⠄⠀⢶⣿⡿⠛⠛⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢸⡏⠀⠤⣄⢹⣇⣦⣴⠟⠀⠀⠈⣶⠒⠂⢉⣙⣿⣿⡋⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣧⣀⢐⡈⢙⣿⣿⣿⣧⡀⠀⢀⣿⠛⠛⠛⠉⠁⠀⠙⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠋⣴⣇⠘⣿⣿⣬⣤⣾⠁⢰⠹⣧⡙⣛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⣡⣾⣿⣿⣧⠘⠻⠟⢋⡏⠀⢀⡄⠀⠉⠉⠀⠀⠀⢀⣀⠐⢦⡀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢘⣿⣿⣿⣿⠀⢻⣿⣿⣿⡏⠀⠀⠀⠈⢳⡀⠊⠀⠀⠀⠀⠀⠀⠀⠛⠿⣷⣄⡀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠈⣿⣿⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡀⣿⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠆⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⣿⣿⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣄⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⠛⢹⣿⣦⡙⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⣡⡶⢀⣿⣿⣿⣷⣌⣿⣿⣷⣤⣤⣤⣤⣤⣶⣶⣿⣿⠿⠿⠿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⣿⣿⣿⣿⣧⣾⣿⠀⢸⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⣿⡿⠟⠀⠀⠀⣸⠿⣿⣿⠿⠛
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣆⠸⣿⣿⣇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡀⠠⣄⣀⡜
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣧⣌⠻⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⢠⡾⠁⠀⣰⣷⠀⠀⣆⠀⠹⣦⡈⠉
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⠻⣿⣿⣿⣿⣿⣷⣤⣀⣾⣿⣠⣴⣾⣿⣿⠀⠀⣿⣦⣄⣀⣙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⣴⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣦⡀⢿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣖⣁⣠⡶⠃⠀⣾⠟⠉⢀⣠⠤⠂⠀⣀⡄⠀⠀⠀⣠⣄⠈⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠶⠛⠻⠶⠾⣿⣅⣤⡴⠾⠿⣧⡤⠶⠛⠉⠛⠻⠶⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⣀⣀⣤⢶⣾⣿⣿⣿⣷⣶⣶⣦⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⠿⠛⣉⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⡿⠋⣠⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡆⠘⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⣡⣾⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⢡⠞⠋⣡⠟⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⢄⣀⡠⠖⠚⠡⠊⠉⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⣠⢴⣾⣿⣿⣿⣻⣭⣍⡻⣛⠳⣦⡀⡀⢰⣾⡿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣦⡀
⠀⠀⠘⣷⣿⣿⣿⣿⣿⣿⣶⣿⣻⣿⣿⣾⣿⣾⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠙⣦
⠀⢠⣾⣿⣾⣿⣿⣿⣿⣿⣏⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢷⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣧⠀⠈⢷
⠀⠀⢹⣿⣿⣿⣿⡿⣻⣿⣟⣻⣿⠟⣻⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣾⣿⣿⣿⣧⣦⣶⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠂
⠀⠀⠈⠛⠻⢿⣿⠿⠛⠛⠛⠛⠋⠙⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠈⢿⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡟⢀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⠀⢸⡵⠛⠁

⠀⠸⣿⣿⣿⣷⣴⡦⠀⠘⠿⠿⣿⣿⣟⣿⠇⣨⣿⡿⣼⣿⠣⣀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠐⢤⡈⠉⠁⠀⠀⠋⠲⣮⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢙⣯⣼⣏⣽⣇⢿⣂⣬⣉⡛⠛⢋⣍⠶⢧⠉⠁⣯⠁⠰⣉⠁⡆⠐⠂⠀⠀⠀⠀⠀⠀⠀⣠⡜⢣⠈⣠⠂⠀⠀⠀⠈⠲⠀⠀⢾⣿⡿⣿⣬⣿⣿⢿⣿⢿⣿
⠀⢀⣿⡟⣨⣿⠿⠋⠉⠉⠙⠟⠛⠹⣬⡿⠇⢠⠖⢲⠋⠈⢃⠀⡜⠃⢌⠀⠂⠠⠀⠀⠈⠁⠶⡑⢪⡄⣮⣰⢧⣤⠀⠀⠀⠀⠀⠀⠀⢿⡿⡷⢉⣉⣉⠻⢿⣿⠋
⠀⠀⢿⡿⠛⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠒⠛⡄⢰⢇⡰⢲⠀⠂⠀⠁⠀⠀⠀⠀⡞⠁⣿⡆⣿⣧⣟⣼⣟⠓⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⡘⡟
⠀⠀⣶⣶⡄⢰⠏⠀⢀⣀⠀⠀⠀⠀⠀⢂⣁⣾⡀⠀⠈⠁⣤⠴⠈⠀⠐⢂⠀⣠⡄⠀⣠⣼⣤⠿⢁⣈⣽⣉⠘⡿⠃⠘⢿⣦⠀⠀⠀⠀⢂⠘⠻⣿⣿⣿⡇⠙
⠀⢠⣿⣿⣇⣶⡀⢀⠈⣝⠓⠼⣿⣿⣿⣿⡿⠙⣋⠶⠑⠛⠄⠀⠈⠀⠀⠀⠜⠫⠀⠀⠉⣻⣉⠂⣙⠏⢸⡇⠀⣥⣈⣤⣏⢻⡇⢠⠀⠀⢸⡿⢺⣾⣟⠉⢾⡿
⠀⣸⣿⣿⣿⠛⠛⠁⠸⠈⠀⠀⠈⣿⠏⠉⠘⡠⢻⡄⠀⠀⠀⠀⠀⠀⠀⢁⡄⠀⠀⠖⢿⣁⠝⣳⠏⣘⠟⠁⠀⠁⠀⠿⢩⡘⡇⠈⡀⢀⣼⣷⣾⣿⣿⣿⡆⠁
⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⡀⠀⠀⣠⡴⠀⠀⠀⠀⠀⠀⠈⠀⠁⢀⡄⣀⣻⡧⡤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢢⣿⣿⣷⣿⣿⣿⣻⣷
⠀⣿⣿⣿⣧⡀⠀⠀⢀⣸⡿⠿⠟⠙⠛⠁⠀⠠⢛⡅⠀⠈⠛⠋⠉⢲⣤⣀⠀⢀⣀⡞⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡴⠉⠀⠀⠀⢸⣿⣿⡟⠹⢛⣿⣿⣿⡀
⠸⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⡖⠢⣀⠂⠎⣻⣆⠀⠀⢠⣤⣄⢀⠀⠃⣺⠋⢡⠖⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡄⠀⡀⠀⠘⣿⣿⣧⣦⣸⣿⣿⣿⣷
⠀⢹⣿⣿⣿⣿⣿⣏⠸⣿⡿⣿⣿⡿⠟⠻⠈⢡⠀⣿⡿⠀⠀⠀⠙⠛⠈⠀⠀⢹⠀⠀⠈⠠⠄⠁⠀⠀⠀⠀⢠⡰⣾⡇⢉⡷⠀⠳⠀⠂⠘⣽⣿⣿⣿⡟⢻⣿⣿
⠀⢸⣿⣿⣿⣿⡿⣇⠀⢿⣖⣏⣩⣶⡶⠃⠀⠂⢱⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⡦⠀⠀⠁⠀⠀⠈⠋⠀⠁⠀⠃⠀⠆⠀⢩⣿⣿⣇⣤⣾⣿⣿
⠀⣾⣿⣿⣿⠟⢴⣿⣧⡀⣙⠋⠉⠁⠀⠀⠀⠀⢉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠹⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⣿⣾⣿⡋⠀⠀⢹⣿⣿⣿⠃⡴⠂⢀⣀⣀⠠⢊⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠛⢥⠈⣛⠀⣄⠒⠀⠀⠀⠀⠀⠀⣀⠐⠁⠘⠀⠀⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣧⣀⣤⣾⣿⣿⡟⠀⡜⢰⣿⠟⠉⠀⠁⠐⠻⠁⠂⠀⠀⠀⠀⢀⣀⣠⡤⠀⠀⠀⠀⠈⢘⡥⢶⣄⡀⠀⢀⡀⠠⠴⢎⣩⡷⠀⠀⢧⢸⡌⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⠟⢛⣿⠇⢰⡆⠈⠁⠀⣀⣀⠠⣜⠂⠀⠄⠀⣀⣤⠚⠛⠏⡀⠀⠀⠀⠠⠀⠤⠀⠓⠀⢱⡁⠀⠀⠀⣤⡴⣿⠟⠀⣷⠀⠀⠻⣧⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣟⣿⣿⠀⢿⠏⠰⠏⠁⢀⣴⠞⣉⡴⠃⠀⣠⣆⢠⡞⠁⠀⣦⡄⠀⠀⠀⠀⠀⢁⠈⡀⠀⠀⣁⠠⠈⠃⠀⠀⠸⠟⣁⠀⡼⠋⡇⠀⠀⠸⢻⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⢌⣴⣿⢋⠴⣭⠆⠀⠰⠉⠁⢤⣉⡦⠀⠀⢤⡄⠀⠀⠀⡀⠈⢦⡕⠀⢰⠙⠓⠀⠀⠀⠀⠶⢦⠀⠶⡾⠀⣧⠀⠀⠀⣼⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⡀⠀⢀⠀⢸⣿⡇⠀⣿⡏⠀⠀⣠⠔⠁⠀⠁⢰⠦⠘⠀⠀⠀⠀⠀⠀⠀⠠⡆⢈⠏⡟⢋⠁⠠⠀⠀⢸⡄⠈⢣⠀⠀⠘⠀⠀⠀⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⡀⠀⠙⠳⠀⠏⢠⣀⢤⡍⠀⠲⠈⠁⠙⠀⠺⣷⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⣇⡀⠀⢀⠁⠸⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⢛⣿⡇
⠀⠘⣿⣿⣿⣿⣄⣿⣷⣄⠘⠂⠀⠀⠀⠀⠐⠋⠈⠁⠀⠀⠀⠀⢰⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠱⠊⠛⠣⡴⡇⠀⠀⢣⠀⠀⠀⢀⣿⣯⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠉⠉⠇⠀⠀⠀⢠⠀⠠⠿⠋⠰⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⡿⠻⣿⣧⣾
⠀⢀⣿⣿⣿⠿⣿⣿⠋⢹⣿⣿⠿⣿⣦⠄⢠⡆⣒⠀⠀⠀⢡⡄⠀⢰⠆⠠⢀⡀⢀⣀⠀⠀⠀⠸⠄⠦⠁⠊⠇⠈⠀⢀⡀⠸⣇⠀⠀⠸⠁⢰⠀⠠⠀⠀⣿⣿
⠀⢸⣿⣿⠛⠀⠈⠁⣀⡀⠈⠁⢰⣿⣿⠀⣼⠇⠉⠀⠀⠀⠌⣠⡆⢘⢶⠞⢇⠀⠈⠁⠀⠀⠀⠁⠀⣄⠓⠒⠐⠧⢀⠈⡇⠀⡏⠀⠀⠀⠀⠀⣤⣤⣤⣴⣿⣿
⠀⢸⣿⡿⡿⠀⢠⣿⣿⣿⣷⠀⠘⢿⣿⢠⣿⠈⠀⠤⢀⡀⢀⢼⡇⢁⠀⠀⠈⡂⠀⠀⠀⠀⠀⠂⠙⡻⠀⠀⠈⢀⠂⢙⣵⠀⠀⠀⠀⢠⡀⠀⣿⣿⣿⢿⣿⣿
⠀⢸⣿⣿⡇⠀⠘⢿⣿⣿⡿⠀⢀⣬⣿⣾⣿⠀⠀⠢⠀⡉⢸⣇⡵⠀⢀⡤⠸⡇⠀⠂⠀⠀⠀⢀⠈⠱⡆⠀⠀⠀⡇⢙⣹⠀⢰⡀⠀⠈⢀⡌⢻⡟⠀⠀⠈⣿
⠀⢸⣿⣿⣤⣶⣇⢀⡀⠀⠀⠀⠀⢹⣧⣿⡇⢸⠄⠐⠰⣎⡷⠀⠀⠠⠈⠀⣻⣷⠀⠀⠀⠀⠤⡀⣤⠀⠁⠶⠄⠀⣿⣷⠀⠀⠀⠀⠀⠄⠈⠁⠘⣱⣄⣀⣠⣿⡀
⠀⢸⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣾⣿⠿⠀⢸⡉⠂⢀⣿⣄⠏⠀⣦⠀⠁⠘⡏⠲⣀⠀⠀⠀⠀⠁⡄⠈⢠⠀⠀⣿⢠⡄⠀⠀⡀⠀⠀⠀⠀⢰⢈⣿⣿⣿⣿⡇
⠀⢼⠋⠀⠀⠀⠀⢀⣀⣤⣤⣴⡞⣻⡟⢁⣴⠗⠝⢀⠻⠛⢫⣄⣾⣿⡄⠀⠠⠈⠀⠇⠀⠠⠀⠀⠀⠀⠀⣤⠁⠀⢱⠸⡇⠀⠀⠛⡀⠀⠂⢀⠜⢠⣿⣿⣿⣿⡇
⠀⠘⠃⡀⢀⣤⣶⣿⣿⢻⣿⣿⡇⠀⠀⣾⠃⠙⢀⠘⣷⣯⣵⣿⣿⣿⡷⢃⠀⡄⠀⠂⠀⠀⠂⠀⠀⠐⡀⠀⠀⠀⢸⠗⣿⣧⠀⠀⠀⠀⠀⠋⢰⣿⣿⣿⣿⣿⡇
⠀⢠⣾⣷⣿⡿⠟⠻⣿⣿⣿⡟⠀⠀⢼⡇⠀⠠⠀⣾⣿⣿⣿⣿⣿⣿⡇⢿⣀⣀⣀⡄⠲⠞⠃⠀⠀⠀⢡⠐⡆⢈⠐⡆⠃⢀⠄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠸⣿⣿⣿⡁⠀⠀⣹⣿⠏⠀⠄⠀⠠⠄⠀⠀⣴⣿⣯⣹⣿⣿⣿⡿⢗⣟⣹⣿⠟⡀⠀⠀⠃⠐⡄⠚⠀⡄⠇⠀⠈⣷⣶⣿⡆⠀⠀⠀⠀⠀⠀⠜⣿⣼⣿⣿⣧
⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⡇⠆⠀⠀⠀⠰⠀⢰⠁⠹⠿⣏⣹⣿⣿⣴⠞⠉⢠⣴⡞⠁⠀⠀⢀⠀⠀⠘⣀⠇⢰⡀⠠⣿⣉⣿⠟⣦⣀⣼⣷⠀⠀⠀⠘⣿⣟⢿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣽⣿⡗⢀⡐⠀⢐⡂⠠⠌⠀⠀⠿⣿⣿⣿⣿⠉⣼⡃⣉⠉⠙⠓⢄⠐⣦⠈⠄⣀⡉⠰⢉⣤⢀⣿⣿⣿⣿⣿⣿⣭⣙⢆⠀⠀⠀⠙⢿⣿⠋
⠀⠀⢸⣿⣿⡿⣩⣍⣿⡿⠁⣀⡄⢙⡀⢀⠀⠀⠐⠀⢰⣿⣿⣿⣿⡀⡸⠃⠀⠀⠈⠀⠈⠀⠀⠛⣀⠀⠀⠀⣣⡄⠀⡿⢿⣿⣿⣿⣿⣿⠋⢙⣦⠀⠀⠀⢠⣤
⠀⠀⣽⣛⣿⢷⣿⣿⠟⡁⠒⠀⠁⠈⣉⠟⠉⢉⣻⢟⣿⣛⣿⣤⣿⠃⠁⠔⠀⠐⠀⢀⡌⠀⠀⢀⠻⡀⣰⠊⠀⢃⢀⡇⠙⣿⣿⣏⠙⢋⡾⢻⣿⠃⢀⣤⣾⣿⡀
⠀⢰⣿⣿⣿⣿⡿⢃⡞⠁⠀⠀⠀⠞⠁⠐⣠⣦⣽⣷⠙⠎⢾⣿⠁⠀⠀⠀⠀⠀⠀⡄⢀⡀⠀⡀⠀⠁⣍⢀⣸⠇⠠⠛⠓⠈⠛⣿⣶⣿⣷⣾⣿⣴⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣏⠀⠀⠘⢃⣀⣀⠀⠀⢰⣏⠀⠀⠹⣿⣿⣦⣠⣾⡿⠀⠀⠀⡀⠀⠰⠀⠙⠀⠀⠀⠙⠋⠠⠵⠚⣧⠀⠀⠀⠀⠀⡄⢻⣿⣿⣿⣟⠛⣿⣿⣿⣿⣿⣿
⠀⠸⣿⣿⡏⠀⠂⣴⡟⠁⠈⡇⠀⠸⣿⣦⡀⠀⠻⢿⣿⣿⢻⠀⠀⠀⠈⠙⠆⠐⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠟⡌⠛⠉⢻⣿⣿⡿⠿⠿⠿⣿⣿
⠀⠀⢿⣟⠃⠀⣼⣿⡇⠀⢸⣿⡀⠀⢿⣿⣿⣦⣀⠀⠘⢻⠈⠀⣀⡠⠦⠤⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢣⠉⠀⠤⠀⠀⣼⢿⡏⢶⣿⣶⣦⡈⠋
⠀⢀⣼⣿⣀⣾⣿⣿⣇⠀⣸⣿⣷⡀⠈⢻⣿⣿⣿⡿⠛⠂⠀⣸⣿⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣎⣀⠀⠰⠄⣸⣿⣦⣄⠘⢿⣿⡟⠃⣤
⠀⢸⣿⣿⣿⣿⣿⣿⣧⣁⣿⣿⣿⣿⣴⣿⣿⣿⣿⢃⠸⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⢀⠁⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⡄⠈⠀⠀⢀⣹⡏⠹⢿⣄⡸⠿⢟⣠⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠆⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⡉⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢃⠀⠀⠀⠸⣿⣿⠗⠀⠉⡻⣿⣿⣿⡏
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⡿⠀⠀⣿⡿⠀⠀⠀⣀⠀⠀⢀⣵⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⣜⠁⣾⣿⣷⡄⠘⣿⡿
⠀⠀⠀⠈⠉⠙⠋⠙⠛⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣸⣿⣧⠙⢛⡋⠀⣼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⢰⠁⡼⢠⠉⢀⠵⠀⣆⣠⣬⣧⠀⠀⠀⠀⠀⢹⠀⠐⡿⡿⣧⣸⠀⢸⡀⣀⣤⣤⠀⠈
⠀⠀⠀⢰⣤⣶⣄⡇⢠⠇⣌⣦⣈⠒⣼⠋⣀⠀⠈⢻⣦⠀⢠⠀⢸⣷⠈⣡⡼⠿⢾⡄⠀⡇⡿⠛⠁
⠀⠀⠄⡀⠈⠛⢻⡇⢸⢠⣿⣿⣍⣹⡇⠀⠀⢹⡆⠜⡟⠀⡸⢀⣿⣿⡴⠃⠀⡀⠀⢙⡴⠃⠃
⠀⠀⠀⠀⠀⣠⣼⠇⣾⣘⢹⢿⣿⣿⣿⡶⢶⡋⠀⠈⣀⣀⡵⢿⣿⣿⢁⡦⠤⠚⠿⣽⣶⠀⠐⣿⡆⠀⠈
⠀⠠⠶⢶⣟⣷⣿⠀⣿⣯⡷⠋⠀⠀⠀⠈⠻⣿⣟⠫⢍⡵⠚⠉⠁⠾⠺⢀⣠⣠⣄⣈⣿⣷⢺⣿⠇
⠀⡀⠀⠠⡛⢻⣿⠟⠠⣿⠀⠀⠀⠀⠰⠀⠀⢹⣿⣶⠟⠤⣀⡀⢀⡷⣤⠞⠥⢰⢀⠙⡌⠛⣦⠛⣇⠈
⠀⠀⠀⢸⣿⠲⡍⢀⣤⣿⡀⠀⠀⠰⠈⢀⣠⠞⠋⠉⠀⠐⠀⠠⣟⠼⠳⣄⠀⣙⣁⡜⠀⠊⢋⡁⠉⠁⡀
⠀⠀⠀⠸⠟⠘⢶⣄⡿⣿⣻⡄⠀⠈⠁⣞⡵⠋⠀⠀⠀⠉⠓⢤⡜⢀⣴⣂⣉⡭⠅⡒⠠⠙⠋⣹
⠀⠀⠀⠀⠀⠀⢰⡟⠓⠻⣇⣹⡿⢶⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⡧⠴⠋⠉⠐⠀⠁⠀⠀⠀⢠⡟⠳⠖
⠀⠀⢀⠀⠀⣠⠞⢀⣾⡆⠸⣿⡟⠻⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡤⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄
⠀⠂⠀⣠⡼⢿⡁⣸⠇⣱⡚⣿⠄⠀⠄⢹⡆⠀⠀⠀⠀⠀⠀⢈⣿⡙⠶⢶⢤⣤⣀⡀⠀⠀⢸
⠀⠀⠀⠉⠀⢸⡁⢍⡛⢋⡇⡻⣶⠻⠞⢓⠟⠲⢤⣄⣠⣤⣞⣧⣨⣟⣷⣇⠰⡽⠛⠻⢶⣄⣸⡇⠺⠓
⠀⠀⠀⢀⠀⣹⠏⣬⡈⠉⢀⣭⡯⢺⡀⡎⣤⢰⠒⠛⣉⣉⠀⠠⠐⣆⣘⣟⣩⡤⠀⠠⠀⠀⠀⡀⢠
⠀⠀⠀⢀⣠⣿⣶⣬⠡⠐⣿⠀⠀⢸⣲⡇⠻⠿⠆⠀⠸⠿⣧⠀⠀⠳⠀⢁⠀⠀⠀⣰⠂⠒⠀⠁⢆
⠀⠀⠀⠈⠉⠙⠟⠛⣶⡆⠟⣀⣄⠸⣼⡎⠀⢘⠀⠄⠀⡀⠉⠀⠀⢐⠖⠶⢶⣶⣾⠷⢼⡱⠀⠀⡸
⠀⠀⠀⡐⢀⣠⡇⠲⠭⠧⠾⠒⠳⣇⡸⢳⣾⣿⣿⣧⠀⠀⠐⠓⠀⣸⣠⣬⣺⢿⡁⠀⠀⣹⢶⣾⠁
⠀⠀⠀⢇⠘⠏⡑⢹⣦⣟⢩⣝⣻⠷⠄⣾⣝⠛⠛⠃⣀⠔⢁⡴⠚⠉⢀⡀⠌⠻⣗⢶⣶⣿⡇⠉⢄
⠀⠀⠀⣈⠳⣾⣴⣿⠀⠤⠤⠤⠤⠄⠀⢳⡈⠁⠘⠋⠅⣠⠏⠀⣡⣶⡽⠛⠳⡄⢸⣾⠯⠊⠁⠛⡿⠃
⠀⠀⠈⠉⠑⢦⣿⣿⡄⠀⠀⠀⠀⠀⡴⢊⠟⠶⠦⠄⡾⠁⡠⠊⠉⣻⣀⠛⢠⠇⢸⢿⣧⠄⠀⠀⠈⠁
⠀⠉⠀⢀⡈⠹⠿⡿⡇⠀⠀⠀⠀⠀⡜⠁⠀⠈⠉⠃⠁⣰⣥⣶⣿⡿⠿⠷⡏⠀⢸⠐⠳⣿⣤⣨⣄
⠀⠀⠛⣹⠗⠛⠋⠉⠉⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⠞⣩⣏⢣⡰⠀⢠⠃⢈⠛⢫⣰⠖⠾⠍⠋⠉⠉⠄
⠀⠋⠉⣇⠀⢠⡞⠙⢿⣿⣹⣿⣽⠃⢠⡆⠀⠀⠀⣈⡞⣿⣿⣷⡿⣤⠏⢀⡎⠀⠀⢳⡼⡦⠘⠂
⠀⠀⠀⢘⡦⣍⣑⣂⣸⡽⠿⠛⡟⠀⢀⠁⠀⠀⠀⠿⠀⠘⣿⣿⠂⠘⣦⡼⠠⠘⠃⡈⣷⠇⢐⣷⣤
⠀⠄⢀⡼⠟⠛⠉⢁⣀⣀⣀⣤⡇⠀⠿⠃⠀⠀⠀⠀⠀⠱⡜⢿⡃⣠⣘⣧⣌⣀⣠⣴⡫⠖⠋⠉⠄
⠀⢀⡎⠀⢀⡔⠛⠙⠫⣭⡷⢾⣇⠆⠰⡆⠀⠀⠀⠀⠀⠀⠈⠪⡳⣿⣟⣻⣟⣿⡿⢿⠄⢀⣤⣔⣀⡀
⠀⠈⣇⢰⡘⢧⡀⣀⣀⢿⠿⠛⠻⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡻⣟⠻⣴⣿⣟⡖⠛⠛⠛⠉⠁
⠀⠀⠸⠢⣅⡀⠉⠉⠈⡃⠀⢀⡤⣻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠳⢦⡿⣧⠙⠛⠂⠀⢀⠈
⠀⠀⠀⢀⣤⠽⠋⠉⠂⠁⠀⠀⠀⡇⠹⣷⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⠘⢶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠟⢏⡈⣉⠙⠛⢿⠟⠿⠛⠶⠶⠶⠶⠶⠺⣟⠋⠀⠀⠀⠙⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠉⠈⠁⠀⣰⣟⣷⣄⠀⠂⠐⠀⠀⣴⣿

⣿⣿⣿⣯⣭⣯⣶⣶⣶⣶⡶⠶⠶⠶⠶⠒⠒⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⠿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣠⣶⣶⣶⣶⣶⣖⡠⡆⠀⠀⠀⣼⡽⡿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⡈⠉⠙⠊⠻⢿⣽⣇⠃⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠝⣿⣿⣿⣿⣿⣿⠏⠃⠀⢳⣶⣦⣀⠸⠶⠞⠀⠀⠀⠀⣬⣶⣤⣤⡀⠀⠀⠿⠛⠋⠙⠿⣿⡇⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⣰⣋⡬⠋⢿⣿⠟⠛⠿⣿⣿⣿⣿
⡇⠀⣿⣿⡿⠿⠋⠡⣄⠀⠀⣽⣿⣁⠋⠃⠀⠀⠀⠀⠀⠸⣿⣿⡏⠉⢿⠀⠀⠰⣿⣿⣶⠀⠹⠃⠀⠀⣴⣿⣿⣿⣿⣿⣶⡀⠈⠉⠀⠀⠈⠁⠈⠃⠀⢸⣿⣿⣿
⠀⠀⠉⠀⠀⠀⣷⣄⣸⣿⣮⣿⡟⠁⠀⠀⠀⠀⣾⠆⠀⠀⣿⣿⢧⡀⡝⠀⠀⠀⢻⣽⣿⡀⠀⢠⠀⠀⠀⠉⠉⣩⣿⣿⣿⠃⠀⠀⠈⠁⠀⠀⠀⠀⢀⠰⠉⠻⡿
⠀⠀⠀⠀⠂⢶⠻⣿⣿⣿⣷⡼⠇⠀⠀⣀⣠⣤⣼⣷⣿⢰⣿⣷⣈⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣇⣶⣤⠙⣶⢾⡝⠟⢻⡇⠀⠀⠀⠀⠁⠀⢀⣆⣼⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢻⣴⢿⢿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣷⣀⠀⠀⣴⠀⠀⠋⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⡆⣸⣿⣷⣦⣀⠀⣀⠀⠈⠙⠋⠀⠀⡠⢖⣾
⠀⠀⠀⠀⠀⣼⣏⠘⠀⠻⣿⣷⣄⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣆⢸⠿⢀⣤⣰⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⣱⣿⣿⣿⣿⣿⣷⣿⣴⡀⠀⠀⠀⠘⣅⡈⢻
⢀⠤⠼⣧⣀⣾⣾⣯⣀⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣿⣿⣿⣾⠟⠁⠀⢙⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣇⢆⡁⠀⠁⠈⠀⠉⠁⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⢿⣿⣿⣠⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⡉⠃⠉⠀⡝⠠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠠⣿⣿
⠀⠀⣦⠤⣄⣬⣽⣿⣿⠿⠿⢿⠛⠁⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⣿⣿⣿⣿⣿⡇⠀⢀⣴⣿⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣶⣷⣤⣤⠔⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⣷⣶⣿⠿⠃⠈⠁⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿
⢤⡄⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠈⠉⣹⡽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠰⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢧⢻⣿⣿⣿⣿⣿⣿⣿⡿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⠃⠀⠀⠀⠰⣿⠋⠈⢻⣿⣿⣿⣿⣿⣿⣷⣤⣴⣦⣀
⢶⣶⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠻⠀⡠⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣿⠈
⠿⢹⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⡀⢿⣿⣿⣿⣿⣿⣿⣿⡧⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⠷⠃⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡇
⠀⢸⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣧⠈⣿⢿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠀⠀⡠⠀⣠⡾⠁⠀⠀⠀⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣼⡇
⠀⢸⡃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡟⠇⣿⡄⢹⠈⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣼⣿⣿⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⢀⠸⣿⠀⠀⠀⣄⠸⣿⣿⣿⣿⡿⢃⠀⣿⣿⡆⠀⢸⣿⣿⣿⡿⠄⣿⣿⣿⣿⣿⣿⢧⣿⣏⡄⠀⠀⠉⣷⣾⣿⠋⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⠿⠿⠀⠀⠀⠘⣦⣿⣿⣿⣿⠋⢻⠀⣿⣿⣷⠀⠀⢻⣿⣿⠃⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣬⡛⠛⠿⣷⠦⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣶⣶⠀⠀⠀⡀⠈⣿⣿⣿⣿⣀⣼⡂⣿⣿⢿⡇⠀⠈⣿⣿⣠⣾⣿⣿⡿⢿⣿⣿⣿⣿⡿⠁⣠⣾⣿⣿⣶⣴⡁⠂⠀⠀⢀⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠿⣿⢿⠿⠿⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢟⠋⠉⠉⠁⠀⠸⠉⠉⠓⠢⢍⡛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠲⣌⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠳⡈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠻⣑⢤⡀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⢣⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⡳⣄⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠨⡄⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢀⠝⢠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⡠⠋⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣤⡀⠠⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡵⢊⣤⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⣷⣖⣠⠤⠤⠤⠼⠤⠄⣒⣩⣥⣶⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠟⠛⠛⠛⠻⠿⠿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣸⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⢃⡤⣄⢀⣣⣃⠀⠠⠟⠁⠀⠀⠀⠀⠀⠀⠻⡆⠀⣿⡇⠘⠃⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀⠀⣸⠶⠛⠉⠉⠭⣭⡳⣄⠀⢠⡖⠲⡦⣄⣠⠴⠒⠒⠛⢿⣴⣛⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣠⠤⢶⡾⠃⠀⠉⣻⡷⣾⡁⠀⠀⠀⠀⠀⠈⠛⠧⡀⢠⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠀⣾⠀⢀⣠⠖⢻⣽⡶⠞⠓⠒⠛⠛⠻⠿⢯⣉⠉⠳⣄⠀⠀⠀⠀⠀⢢⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠯⣶⠞⠋⢀⣀⣤⡤⠤⠤⢤⣤⣀⠀⠈⠙⠦⣿⣿⣦⣄⡀⠀⢸⠂⡇⠐⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⣠⠞⠁⣠⡾⠟⠉⠀⠀⠀⠀⠚⠀⠈⠙⠲⣄⠀⠈⠳⡄⠀⠙⠳⣦⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡴⠃⣠⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠙⣆⠀⢆⠘⣆⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣰⢣⠏⠀⠀⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠘⣆⠘⡆⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⡇⡾⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢹⡀⢻⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⠶⠀⠀⠀⠀⠀⠈⡶⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡇⢸⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡀⣇⠀⠀⠀⠀⠀⠀⠀⣱⠶⠃⠀⠀⠀⠀⠀⠀⠀⠒⣿⠀⠀⡇⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣇⢸⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠃⡟⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣆⢳⡀⠀⠀⠀⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡞⡼⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢦⡙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣼⡽⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠢⣍⡒⠤⣀⡀⠶⠀⠀⠀⢀⣠⠖⠋⠀⢀⣾⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣤⣀⠉⠛⠒⠲⠶⠒⠒⠛⠉⠀⣀⣤⡖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁⠈⠙⠓⠶⠶⠶⠶⠶⠖⠚⠋⠙⣦⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠖⠒⠒⠂⠉⠉⠉⠉⠉⠑⠒⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⣀⣠⣤⡶⣲⠶⠻⢟⠟⠛⠒⢲⠶⠦⣤⣀⡀⠀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣀⡤⡾⢛⢹⣴⣿⠶⠿⠻⠿⠛⠿⠻⠳⠻⡿⢾⣵⣬⣝⡛⠲⢤⣀⡀⠀⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⡞⡙⣤⣷⠿⠋⠁⡀⣤⢰⡆⢰⣶⡄⢠⣦⠀⣠⡄⢀⡀⠊⠙⠫⢖⣬⣋⡳⢦⣤⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⡏⣨⣶⡟⢉⣥⡄⢶⡌⠟⠉⠀⠀⢸⣿⡇⠀⠁⠀⠉⠁⠘⠏⢴⡗⢀⣤⡈⠙⠿⣯⣭⢛⣶⣦⣄⣑⣤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⣴⠻⣠⣾⠏⠁⣴⡹⣧⢻⣆⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⣤⡄⠀⠽⢗⣴⣚⡻⣮⡝⡻⣦⡠
⠀⠀⠀⠀⡠⠃⢀⡞⢀⣾⠟⢁⣀⠻⠎⠀⠸⣧⠹⡧⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠉⠰⠿⢀⡀⠉⠿⢶⣛⡻⣦⡟⢷⣆⠒
⠀⠀⠀⡐⠁⢠⠏⢤⡿⠃⠀⣮⡙⠀⠀⠀⠀⠀⠡⠘⣆⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⢡⡶⠀⠙⢿⣯⣃⠙⣎⣿⣆⣐⡀
⠀⠀⡐⠀⣰⠏⣄⡞⠁⣰⣶⣄⡀⠀⠀⠀⠀⠀⠀⠁⡘⣆⠀⠀⠀⠇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣆⠀⠘⢿⠁⠈⢏⠹⣷⡤
⠀⠠⠀⢰⠿⠐⣾⠅⣀⠉⠻⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⡘⣧⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠟⠋⢡⣤⠀⠀⢂⠀⠘⣧⢸⣧⠄
⠀⡄⠀⡞⢂⣻⠇⢀⣙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⣆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⠀⠈⠄⠀⠸⣧⢻⡇⠀⠅
⢰⠀⢸⣏⣾⡛⢀⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⠐⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠸⠀⠀⢳⢿⡿⠂
⡈⠀⣬⢋⣿⠃⣬⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⡆⠀⢸⣱⣿⣭⡄
⡇⠀⣯⢪⣿⠂⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣷⠀⠁⠀⢸⢕⣿⡂
⡇⢰⡇⠐⣿⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⡛⠋⠀⠀⠀⢸⢺⣿⣒⡀
⠀⠐⢧⠪⣷⠀⢛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠆⠀⣞⢽⣿⠒
⢰⠀⠷⢌⣻⣆⠈⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠃⠀⡸⠀⢀⠗⣹⡿
⠈⡆⠘⣗⠨⢧⡀⠀⡶⠆⢀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣈⠋⠀⢠⠁⠀⣘⡩⣾⣯⡭⠩
⠀⢡⠀⢹⡬⢹⣧⡀⠰⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣶⣄⡈⠛⠀⠀⡌⠀⢠⠇⣺⡟⠉⠁⠄
⠀⠘⡄⠀⢻⣆⠻⣧⡀⠈⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠏⠀⠀⡜⠀⢠⢛⣭⣟⣃⠈
⠀⠀⠈⢆⠸⢳⡌⠹⣯⡄⠀⠺⠓⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⢀⠈⠀⡰⢋⣿⡟⠓⠒
⠀⠀⠀⠈⠣⣞⡻⣶⡞⢷⡢⡀⠈⠋⣴⡆⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀⠘⠷⠀⠀⠀⡠⠋⢀⡜⢭⣵⡟⠒⠒⠂
⠀⠀⠀⠀⠀⠙⣧⡟⢷⣔⣻⣷⡀⠀⠀⠰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⢴⣆⠛⠃⠀⠀⣠⠞⠀⣰⢓⣰⡾⠯⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣎⣹⢶⡀⠛⠦⣀⠀⠈⠁⠾⢂⣶⠀⣀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⠀⣤⡙⠿⠟⠈⠁⠀⣀⠔⠊⠀⣠⠾⢉⣰⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣅⠋⢦⣄⠈⠑⠢⢤⣀⠀⠁⠘⠛⠸⡷⠘⣿⣷⢰⣆⢸⡆⠙⠃⠉⠁⠀⣀⡤⠒⠊⠁⢀⣠⠾⢍⣽⠟⠓⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⢻⣧⡨⢱⠢⣄⡀⠀⠉⠐⠢⠤⠄⢀⣀⣈⣁⣀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠀⣀⣤⡶⣯⣥⡶⠿⠡⠬⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠽⠷⢧⣡⢉⢉⠱⡰⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢤⡔⠶⢙⣩⣤⡜⠛⠋⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⢀⣙⣛⡳⣶⡤⣬⣌⣏⣫⡹⡙⢝⠏⡫⡙⠬⢑⣬⣦⠶⠽⠛⠛⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣠⠖⠚⠛⠛⠛⣛⡛⠛⢛⣛⣛⣛⣛⡡⠬⠌⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠛⠉⠉⠀⣀⢀⡀⣀⠀⡉⠉⠛⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠛⠁⢠⡀⠆⠘⠀⠁⠈⡘⣃⠁⠈⠀⠃⠰⢀⡄⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢄⠙⠈⠓⡄⡄⠀⠀⠀⣇⣜⠁⠀⠀⠀⢠⠚⠀⠋⡠⠀⠙⢿⣷⣄
⠀⠀⠀⠀⣴⣿⠟⠀⣄⠑⠀⠀⠀⠀⠇⠇⠀⠀⣤⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠊⣠⠀⠻⣿⣦
⠀⠀⠀⣸⣿⠏⠀⠲⢦⢀⢀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡤⠖⠀⠹⣿⣇
⠀⠀⢠⣿⡟⠀⣈⠁⠀⢸⠻⡸⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠈⣁⠀⢻⣿⡄
⠀⠀⣾⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⣧
⠀⠀⣿⣿⠀⢘⣂⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣐⡃⠀⣿⣿
⠀⠀⣿⣿⠀⢨⠍⠀⠛⢀⣀⣠⣤⣤⡶⠶⠟⠛⠛⠻⠟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠩⡅⠀⣿⣿
⠀⠀⢿⣿⡄⠀⠂⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢠⣿⡿
⠀⠀⠸⣿⣧⠀⢉⡀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⣰⡆⠀⢀⡉⠀⣼⣿⠇
⠀⠀⠀⢹⣿⣆⠀⠴⠚⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠓⠦⠀⣰⣿⡏
⠀⠀⠀⠀⠻⣿⣦⠀⠋⡠⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⢄⠙⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠊⣠⠀⡤⠁⠀⠀⠀⠀⣼⣅⠀⠀⠀⠀⠐⢥⠀⣄⠑⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠘⠁⠆⢠⠀⡀⢀⢨⡅⡀⢀⠀⡄⠰⠈⠃⢀⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣤⣀⣀⠀⠉⠈⠁⠉⠀⣀⣀⣤⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣀⠤⠄⠒⣂⣀⣀⣐⡒⠠⠤⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⠔⡫⠔⠂⠉⠀⠀⠀⠀⠀⠀⠉⠐⠬⣑⢦⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⢋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢆⢳⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⠇⡎⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⡆⢇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢦⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⠞⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠳⣜⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⡡⠋⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠈⠓⢬⣓⠦⢤⣀⣀⣀⣀⣀⣀⡤⠔⣊⡵⠊⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠈⠉⠒⠲⠮⠭⠭⠴⠖⠚⠉⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡻⢿⢛⠿⢿⣿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠛⢷⣶⡾⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠛⠀⠈⡡⠤⡂⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠐⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠁⠀⠀⠈⠉⠀⠀⠈⠒⠁⢹⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⢠⣤⣤⣼⣧⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣾⣧⣤⣤⣤⠀⠀⠀⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡿⢿⣿⡿⠿⢿⣷⡄⢶⣿⣿⡆⢿⣿⠿⢘⣿⠿⠿⠿⠿⠿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣷⡿⠋⠀⣠⡾⠛⣿⣀⣉⡁⠘⠀⣠⡶⠾⠋⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡏⠀⣠⣾⣿⣶⣤⣾⣛⠛⣻⡿⠿⠟⣃⣠⣤⣶⣿⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⣷⡾⠋⠘⣿⠀⠉⠙⠛⠿⢿⣷⡾⠟⠛⠉⠁⢸⡟⠀⠀⢀⣿⠇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢹⣿⠀⣠⡾⣿⡆⠐⣶⣦⣤⡀⠀⣠⣤⣶⣶⠀⣾⡇⠀⠀⢸⣿⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⣿⡾⠋⠀⣹⡇⠀⣿⠈⢹⡇⠀⣿⠉⢸⡟⠀⣿⠁⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣿⡇⣠⡾⢻⣧⠀⣿⠀⢸⣷⣶⣿⠀⢸⡇⢠⣿⠀⠀⠀⣿⡇⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣿⡄⢹⣿⠋⠀⣸⣿⠀⢻⡆⢀⣀⣀⣀⠀⣼⡇⢸⡏⠀⣀⣰⣿⣥⣼⣇⡀
⠀⠀⠀⠀⠀⠀⠀⢻⠳⣄⣀⣸⣷⡀⢻⣧⣾⠟⣿⣶⣾⡇⢸⡟⡟⣿⣀⠿⢶⣚⣿⣭⣿⡷⣶⢶⠶⣶⢹
⠀⠀⠀⠀⠀⢠⣄⡈⢧⡳⢶⣶⣿⣷⡈⢻⣷⣄⣴⠟⣹⣧⠼⢟⣫⣭⡶⠿⣿⠹⢹⡏⡇⣬⢻⢠⠘⣿⢸
⠀⠀⠀⠀⠀⠀⠙⢬⣽⡷⢾⠋⠶⣿⣿⡤⢽⣟⣓⣯⣽⡶⣿⢛⡏⣿⠀⡰⣿⢸⣸⣧⣷⡴⠾⠾⠿⠿⠜⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⢸⣿⠛⠟⢻⡟⡛⣿⢉⢸⣇⢻⢸⣧⣹⣶⠷⣛⡻⢭⣷⠰⣿⣿⣿⣟⣛⣼⡟⢛⡮⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣿⠀⠼⡏⣇⠟⣼⣸⣼⡷⠿⣛⣭⣵⠶⠋⣁⣴⡿⢧⡺⠿⠿⠿⣛⣛⣛⣛⡳⠳⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣛⣛⣛⣛⣛⣿⡯⠽⠶⣾⣯⣌⣉⣤⣴⡿⠟⠉⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠉⠳⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⣿⣷⢻⣎⠻⣮⣿⣆⠳⣌⠛⠷⣶⣶⣽⣿⣿⣿⣾⣿⢃⣴⣿⡿⢟⣫⣾⣧⣾⣿⢟⣠⣿⣵⠏⢀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⡄⢻⣧⡀⠹⣿⣿⣿⣿⣶⣄⡈⠙⣿⡎⢻⣿⣿⣾⣿⣯⡾⣫⣭⣿⣿⣿⣿⣿⣿⠏⣠⣶⣿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀⢻⣷⠀⣿⢇⣤⣌⡙⠻⢿⣦⣹⣧⣾⣿⣿⣿⣿⣿⣴⣿⡿⠛⢉⣡⣤⣸⣿⣾⡿⠋⣠⡿⢣⣿⣹⣿
⣿⣿⣿⣿⢿⣷⡀⢻⣷⣿⣸⢳⣎⣙⢦⡀⠙⠿⠛⠋⠉⠉⠉⠙⠛⠿⠋⢀⡔⣋⣥⠸⡇⣿⡟⣠⣾⠟⠁⣼⢣⣿⣿
⣿⣯⢻⡟⣧⡹⣿⣌⣿⣏⣿⢸⣙⡾⣧⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡰⣿⣿⡃⡇⣿⣿⠟⠁⣠⣾⣿⡿⣿⣿
⣿⣿⣎⢿⣌⠻⣿⣿⣿⣿⣾⡘⣛⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⢣⣷⣿⣯⣴⣿⣿⢟⣩⣾⣿⣿
⣿⣷⣿⣾⣿⣦⣌⣻⣿⠿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢟⣿⣿⣯⣶⣿⢟⣱⣿⣿
⣿⠶⢈⣙⣻⣿⣯⡉⠉⠁⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠈⠉⣉⣽⣿⣿⣿⡿⠛⢻
⣿⣫⣭⣭⣿⣟⣛⣛⡶⡄⠀⠀⠀⠀⡞⠀⢰⡏⠳⡄⠀⠀⠀⢠⠞⣽⠀⠈⡧⠀⠀⠀⠀⠀⢾⣛⣿⣿⣿⣿⣿⣿⣻
⣿⡿⢟⣿⣿⣫⣭⠍⠁⠀⠀⠀⠀⠀⠸⡄⢸⡇⠀⡇⠀⠀⠀⢸⠀⢿⡇⣠⠃⠀⠀⠀⠀⠀⠈⠭⣭⣛⣿⣯⣟⡻⢿
⣿⣴⢟⣿⣿⠟⠁⢀⣀⣀⣀⠀⠀⠀⠀⠈⠚⠓⠊⠀⠀⡤⠀⠈⠑⠚⠛⠁⠀⠀⠀⠀⣀⣀⣀⡀⠈⠻⣿⣿⣷⣭⣻
⣿⣶⣿⣿⢇⡴⢋⣭⠿⠛⠉⣿⡷⢦⣄⡀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⢀⣠⡴⣾⣿⠉⠙⠻⣍⠛⢦⣹⣿⡿⣿⣿
⣿⣾⣿⣿⡟⢀⡾⠁⠀⢠⠾⢫⡇⠀⠈⠙⠳⠦⠤⣤⢰⣿⣂⡤⠤⠴⠚⠋⠁⠀⢸⡙⠧⡀⠀⠈⠳⠀⢿⣿⣿⣿⢻
⣿⣿⡇⢹⣧⠸⠁⠀⠀⠀⠀⢚⣇⠀⠀⠀⠀⠀⠀⢀⡏⡇⢯⡀⠀⠀⠀⠀⠀⢠⣼⡃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣾
⣿⣿⣧⠈⢿⡄⢤⠀⠀⣀⡤⢟⣿⠀⠀⠀⠀⠀⢀⡞⣠⣿⡌⢧⠀⠀⠀⠀⠀⢸⣿⠳⢤⡀⠀⠀⡄⢠⡟⠟⢸⣿⣿
⣿⣿⣿⣷⡈⠳⠸⡷⠤⣤⣴⡿⢻⣷⠀⠀⠀⢀⡞⣰⠃⡇⠹⣌⢧⠀⠀⠀⢸⣾⠟⢷⣤⣤⠴⣾⠇⠟⠀⣠⣿⣿⣿
⣿⣿⣿⡿⢿⣷⣶⣿⣷⡿⢯⣤⡀⢿⣴⠀⠀⣼⡼⠁⠀⡇⢀⠘⢮⣧⠀⠈⣾⡟⢀⣤⡽⣿⣾⣷⣶⣶⡾⢿⣿⣿⣿
⣿⣮⠻⣷⡀⢠⣾⣿⣿⣿⣶⣌⢻⡌⣿⡀⠀⣿⣇⠀⡀⠟⠀⠀⣼⡿⠀⢀⣿⢠⡟⣡⣾⣿⣿⣿⣷⡄⢠⣿⢏⣵⣿
⣿⣿⣧⠘⣷⣿⣿⡸⣿⣼⣿⢿⣾⣇⣿⠙⢆⢸⣿⠀⠀⣤⠀⢀⣿⠇⡴⠋⣿⣾⣾⡿⣿⣷⣿⢡⣿⣿⣾⠃⣾⣿⣿
⣿⣿⣿⠀⢸⣿⠹⣿⣿⢿⣿⣸⣿⣿⠇⠀⠈⢿⣿⡆⡦⠀⠄⢸⣿⡾⠀⠀⢻⣿⣿⣅⣿⡿⣿⣿⠫⣿⡇⢠⣿⣿⢻
⣿⢻⣿⣇⠈⣿⣷⡿⣃⠘⣭⣿⣿⣿⠀⠀⠀⠸⣿⡇⠀⠀⠀⣸⣿⠁⠀⠀⠀⣿⣿⣿⣭⠁⡛⣿⣾⡿⠀⣼⣿⠏⣸
⣿⣶⣿⣿⣷⣬⣝⣛⣻⠿⣿⣿⣿⣿⡀⠀⠀⠀⣿⣷⠀⠀⠀⣿⣿⠀⠀⠀⢠⣿⣿⣿⡿⠿⣛⣋⣭⣤⣾⣿⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠞⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣯⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣀⣀⣀⣀⣀⡀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠉⠉⣿⡏⠉⠁⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⣀⣠⣤⣤⣶⣿⣷⣤⣤⣤⣀⡀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⣿⣧⣴⣾⣿⡿⣿⡛⠿⠿⠿⠿⠿⠟⣻⠿⣿⣿⣶⣤⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠋⢁⣴⣿⡿⡿⠟⠂⠉⠀⠐⠀⠁⠸⠀⠁⠐⠀⠈⠁⠋⠟⡿⣿⣷⣄⠉⠻⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢀⣴⣿⣟⠟⡈⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠈⠙⢟⣿⣷⣄⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡝⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠷⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠟⠁⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢆⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠐⠦⢄⢼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⣢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡀⠠⣪⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣗⢄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⢔⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⣦⢜⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢘⢤⣪⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣶⣦⣤⣂⣠⣡⣬⢤⣥⣀⣲⣬⣶⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⠶⡛⢳⡀⠀⠀⠀⣀⣀⣠⠀⠀⠀⠀⠀⠀⣀⠤⣀⢀⡠⠖⠒⣄
⠀⠀⠀⠀⠀⠀⣠⡞⠑⠀⠁⠐⠊⣷⣚⣛⠋⠉⣿⠁⠀⠀⠀⠀⠀⢀⡇⠀⠴⣅⣱⡂⠀⡾⡀⠀⠀⠀⠀⠐⢈⠄
⠰⢶⣶⠛⠛⣩⣽⣆⠀⣠⣄⣶⡖⣿⠛⠛⠻⣇⢻⠀⠀⠀⠀⠀⡠⠊⠙⠦⠞⠁⠀⠉⠉⠀⢸⠀⠀⢀⣀⠀⠈⠩⢠
⠀⠀⢹⠘⣟⡿⢋⡽⢷⣽⣍⣠⡾⣁⣤⡆⠀⣿⢸⡇⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣀⣤⣄⢼⠀⠀⠘⠻⠾⡷⡀
⠀⠀⠸⡇⣿⣠⡾⣷⠺⠷⠾⠷⠛⣉⢸⡇⠀⢿⡘⡇⠀⠀⠀⠈⢇⠀⠀⢀⣴⣶⡜⢿⣿⠟⢨⠆
⠀⠀⠀⢷⠸⣏⣰⢿⡀⣿⢲⡀⣟⢹⡀⡇⠀⢸⠇⡇⠀⠀⠀⠀⠀⠉⠲⡙⠿⠟⠃⠂⠀⡞⠁
⠀⠀⠀⠘⣇⢻⣇⡼⣧⢹⡈⠛⣛⠸⡇⣧⣠⣾⣾⣷⣆⠀⠀⠀⠀⠀⠀⠑⠒⠉⠓⠒⠋
⠀⠀⢀⠀⠹⣆⢻⣤⠾⣤⣇⢻⣿⣶⣿⠿⣿⡯⣷⣼⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢢
⠀⠀⡘⢵⢶⣿⣦⡙⣶⣯⣿⢿⣻⣿⣼⣿⣿⣿⣿⣿⣷⣯⡿⡙⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣤⣤⢀⠀⠁⠔
⠀⠀⠉⠛⠻⣿⣿⢟⣿⣿⣽⣯⣿⣿⢟⣡⠞⠛⠛⠛⠉⠉⠉⠳⠀⠀⠀⠀⢠⡴⠟⠀⠀⠀⠀⠈⠈⠑⠛⠛
⠀⠀⠀⠀⠀⠸⣿⣿⣿⠾⠛⠛⠷⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠤⠠⠐⠒⢄⠀⠀⠀⢀⡀⠤⠀⢀⢀⣠⡀⣤⠈⡆
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⠈⠀⠂⠀⠀⠐⠦⠁⡶⣀⡀⠀⠅⠀⢀⡈⡥⡆⣾⠀⠿⠘⠑⢃⣉⣀⡇⣀
⠀⠀⠀⣀⣰⡶⠀⠀⠀⠀⠀⣠⣴⠆⠀⠀⠀⠀⠀⡇⢰⣶⢰⣃⡟⠇⠇⣃⠡⠄⠀⠀⢩⠄⠀⠀⠘⠁⢖⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⣆⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢧⣀⡯⣝⠀⣸⡁
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣟⠛⠛⠛⠛⣻⠁⠀⠀⠉⠀⠈⠉⢉⡟⠛⠛⠛⠛⢛⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣶⢾⡿⠿⣧⣀⣤⣶⠰⣿⡖⢸⡿⠿⠿⢶⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣷⠋⢀⡴⠋⣿⣬⣍⠈⣠⡶⠛⠀⡀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣯⡴⠛⡟⠛⠿⢤⣼⣏⣭⠴⠞⢛⡏⠀⠀⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⣿⢀⡴⣷⠀⣶⣤⡀⢁⣤⢴⡆⢸⡇⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿⠋⢀⣿⠀⣿⠀⣇⢸⡇⢸⡇⣼⠀⠀⢸⡇⣸⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⣇⢻⡶⠋⢸⡇⢻⠀⣉⣉⠁⢸⠁⡿⠀⣀⣿⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣦⣀⣿⡎⣿⡶⠛⢷⣾⠀⣿⣹⡧⢞⣚⣭⣭⣿⣿⡿⣿⢹
⠀⠀⠀⠀⠀⠀⠲⣚⣋⣿⢿⣿⣜⣿⣶⠯⣞⣻⣭⣶⡟⢻⣏⣿⣿⣟⣹⣷⣿⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣶⠿⢿⣶⣾⣟⣻⣽⣿⣗⣿⢿⣿⠿⣿⣿⣿⣿⣿⣼⣿⣟⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⡷⠞⣫⡴⠞⢿⣿⣿⠿⠿⠿⢿⣻⡌⠁⠠⡤⠄⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠛⠛⠉⠁⠈⠛⠷⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠉⠙⢻⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⡑⠶⢄⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢂⠀⠂⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣱⠛⡊⠤⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡘⣄⣓⡀⠃⣤⠀⠀⠀⠀⣠⠤⠄⣤⣼⣧⣤⠀⠀⠘⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣯⡮⣝⡶⡅⠢⠀⠀⠀⠈⠳⣦⣤⡀⢛⣿⣿⣿⡃⢀⣤⠄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⣫⡓⢨⡚⡷⠀⠀⠀⠀⠈⠉⠁⢀⠃⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣯⣤⣿⢇⠶⢲⠄⠀⠀⠀⢀⠠⡄⣀⠀⣄⢀⠄⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣯⣻⡗⣯⡀⠀⠀⠀⠀⠀⠐⠫⡆⢸⣀⡔⠀⣠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⠿⠟⠁⠀⠀⠀⣶⣆⠀⡀⢠⣾⡟⢀⣘⡟⠀⠀⠠⡒⠄
⠀⠀⠀⢠⣀⡀⠡⠀⣥⡀⣤⠀⠀⠀⠀⠀⠀⢦⠈⠋⠙⣇⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⣿⣿⣿⣥⠀⠴⠞⠻⢿⡿⠃
⠀⠀⠀⠈⣿⣿⣦⣒⡈⠼⠤⣱⠀⠀⡄⢀⣘⣷⣀⣤⡀⠛⠀⠀⢀⣤⣤⣤⣀⠜⢹⣿⣿⣿⣿⣿⣦⠀⢲⣄
⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⠿⣄⣾⡷⠛⠚⠻⣿⣿⣿⡗⠂⠀⣿⣿⣿⣾⣧⣴⠋⢀⡼⠛⢿⡿⠿⢧⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠰⢶⣶⡄⠈⡏⠙⠳⠀⣠⣿⣿⣿⣿⣿⣿⠰⠋⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠘⠻⢻⡇⠹⡄⠀⠘⠛⠛⢿⣿⣿⣿⣿⣷⣀⡢⢄⠀⠀⠀⠀⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣧⠀⣼⣗⣠⣧⣀⠀⠄⣀⡄⢻⣿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣷⣿⣿⣿⣿⡥⠾⠘⠉⠀⢠⣿⣿⣿⣿⣦⣲⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠁⠀⢈⣷⣶⣮⣿⣿⣿⣿⣤⡀⠀⠀⠀⢀⣸⡟⠛⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣾⣿⠛⠁⢳⠀⠉⠛⣿⠥⠤⢴⡾⠋⠀⠀⠀⠈⠉⠉⠀⢀⣀⣠⡤⡴⢼⠓⠣⢿⣐⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⡏⣩⠏⠁⠀⠀⠀⢣⠀⠀⢸⠀⠀⣨⡇⠀⠀⠀⠀⠀⠀⢀⠀⡞⠧⣞⢡⣇⣾⣯⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠉⠁⠀⠀⠀⠀⠀⠀⠳⡀⢺⠀⣰⢹⣿⣀⣤⠀⠀⠀⠀⣍⢷⣴⣺⣄⣳⣿⣿⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⢠⣇⡜⢸⣿⣿⠀⣠⡴⠚⠛⠿⠿⠿⠟⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡏⠀⣉⣉⣀⠀⠈
⠀⠀⠀⠀⠐⣦⣄⣀⣀⠀⠀⠀⠀⠀⢀⠈⣿⣷⣶⣤⣹⣶⣿⣾⣿⠟⠛⠉⢉⣠⣘⡀
⠀⠀⠀⠀⠀⠀⠑⠈⠛⢻⣿⡀⠀⣀⣉⣀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠗⣤⣼⣾⡿⢋⡿
⠀⠀⠠⢤⣠⣀⡔⡤⢠⣿⠟⠁⢾⡿⣻⣿⣿⢿⣿⣿⡿⠛⡿⠏⠀⢹⡉⣿⠑⠒⠉
⠀⠀⠀⠒⠛⠛⠶⠛⠋⠀⠀⠀⠀⠙⠛⠋⠁⣾⣿⣧⡀⠀⠁⠀⠀⢜⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⢀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⢣⡄⠀⠐⠈⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃⠂⠀⠀⠀⠀⠀⠀⢻⠟⢡⠏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠏⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡇⠀⢻⣿⢸⡇⣸⣿⣿⠀⠅⠀⢨⣿⣿⣇⢸⠀⣤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⣿⣾⣇⣯⣿⣿⠀⡀⢀⢸⣿⣿⣿⣾⢄⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣄⠘⣿⣿⡏⣿⣿⣿⣾⡇⢸⢺⣿⣿⣿⣿⡄⣿⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⣿⣿⣿⠀⣿⣼⡏⣿⣿⣿⣿⡇⢸⣸⣿⣿⣿⣿⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡷⠍⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡟⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣾⣭⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣿⣿⣿⣿⣟⠛⢓⣦⣤⠀⡆⣀⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⢿⣿⣷⣿⣋⣻⣿⣿⣿⡿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⡿⠻⢿⡿⠿⢿⣿⠀⢸⡟⢿⣿⣿⡿⠿⣿⣯⣾⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡀⠀⠀⠀⢸⣏⠀⠀⠻⣼⡽⣿⠀⣠⣿⣿⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣷⣷⡀⠀⠻⣿⣿⣿⠿⠉⠻⣿⣾⣿⣿⡋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⣶⣶⣿⣷⣶⣶⣶⣿⣿⣿⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣷⣶⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣉⣍⣉⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⣿⣿⣿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠆⢀⣤⣾⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡍⠀⠙⠿⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣷⡄⠀⢠⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⣿⣿⣇⣸⣿⠇⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⣾⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣤⣀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⢀⣤⣤⣶⣤⡄⣼⣿⣿⣿⣿⣛⣿⡏⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢾⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣧⠨⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣺⡅⠘⣿⣿⣿⣿⣿⣀⣤⣀⠀⢸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣿⡏⠁⣠⣾⣿⣿⡗⠀⣿⣿⣽⣿⣿⣿⣯⣵⠉⠿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⠃⠴⣿⣿⣿⣿⣵⣿⠛⣿⣿⣿⣿⣿⣿⣧⠀⠀⠘⠿⢿⣿⣿⠟⣿⣿⣿⡟⠺⠿⠿⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⠻⠟⠂⠀⠈⠉⠙⠉⠉⠉⠙⠛⠿⢯⣤⠉⠁⠁⡤⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠠⠀⢀⡀⠀⡀
⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠠⠋⠀⠀⢠⡀⠀⠉⠀⠀⠀⠀⢀⡄⠀⠀⠉⠀⠊⠀⠀⠀⠀⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⣗⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⠻⡉⠀⣸
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡄⠀⠄⠀⠀⠀⠀⠀⠁⠂⠀⠀⢶⠃⠀⡇⠀⢀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠐⢀⢐⠀⠀⠀⣴⡦⠀⠀⠸⠂⠀⠇⠀⠈⠀⠀⠀⢈⣁⠄⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡸⢬⠂⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠾⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⠟⠛⠛⢩⣿⡟⢁⣠⣤⣶⣾⣿⣿⡿⠿⣿⣿⣷⣦⣄⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢀⣴⣿⠿⠛⠉⠀⠀⠀⠀⠀⣾⣿⣶⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⠀⠈⢻⣿⣄
⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⡄⠀⠹⣿⣦
⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠙⣿⣧
⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⢸⣿⡆
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⣿⣷
⢸⣿⡇⠀⠀⠀⠀⢠⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣄⠀⣿⡿
⠸⣿⣇⠀⠀⠀⠀⠀⠙⢿⣶⣶⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣤⣤⣤⣶⠟⠁⠀⠀⠀⠀⣾⣿⢿⣿⣿⡇
⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠆⠀⠀⠀⠀⠞⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⣿⡿
⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣶⣄⠀⠀⠀⢀⣤⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⢿⠁
⠀⠀⠀⠻⣿⣧⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠘⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⢀⣿⡿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⡁
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠉⠛⠻⠿⠛⠛⠋⠁⣠⣶⡄⠈⠛⠿⠿⢿⣿⠿⠟⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠻⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣶⣶⣶⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠊⠉⠈⠁⠒⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⣀⡤⠴⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⣤⣀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠀⠀⠀⢈⠆⠀⢄⢀⠀⠠⡊⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡄⢊⠡⠀⠀⠀⠃⠀⠀⠄⠑⠠⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠒⡁⠔⢉⣠⡤⠖⠒⠒⡒⠒⠲⢤⣄⠈⠂⠈⠂⠄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡴⠋⠁⡆⠀⠀⡄⠱⠄⠀⠀⠌⠙⢦⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣠⠏⡀⠀⠀⠇⠸⠀⠁⠈⠀⠀⠇⠀⢀⠀⠹⣄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠇⠀⢹⢰⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠁⠹⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⣠⡀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⡀⢠⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠈⠀⠀⣠⣴⠾⠛⠋⠉⠉⠉⠉⠁⠘⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡆⠀⡈⠢⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⠀⢠⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠹⣆⠁⠀⠀⢨⠁⠀⢀⠀⠀⠰⡂⠀⠀⠈⣠⠏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠈⠳⣄⡐⠂⠀⠀⢛⡃⠀⠀⠰⢂⣠⠞⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢠⡈⠙⠲⠦⠤⠨⠄⠤⠴⠖⠋⢁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠤⠤⠀⠀⠂⠈⠀⠑⢈

⠀⠀⡲⠀⠀⠀⢀⣄⡀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢤⠀⢰⣆⣄⠄⡖⢰⣦⠀⡴⠄⡖⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠘⠁⠀⠀⠀⠸⣿⣷⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠸⡤⠸⠇⠹⠀⠯⠸⠇⠀⢬⠇⢧⠘⠀⠀⠀⠀⠀⠰⠛⣿⣶⣤⡤⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⣶⣦⣌⣻⠈⠠⠂⠛⡇⠀⠀⠀⠀⠀⠧⠷⠷⠇⠸⠿⠸⠾⠾⠇⠆⠅⠇⠟⠇⠿⠉⠇⠀⠀⠀⠴⣶⣿⣿⣿⡷⢦⡴⠂⠀⡇
⠀⠀⠀⠀⠀⠉⠻⢅⡿⠟⢶⡶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⢹⡿⠁⠀⠇⠀⠀⠁
⠀⠀⠀⡄⠀⠀⠀⠄⠀⠀⠼⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⣄⠋⡀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⣀⡀⠙⣿⡇⣼⠟⠀⠀⠀⠀⣤⠒⠄
⠀⠀⠈⠘⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣹⣿⣦⡬⣿⣷⠚⠒⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠻⣧⣤⣶⣄⣀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⣿⣿⢁⠏⢳⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣷⣄⣾⠟⢻⡇⠀⠉⠋⠁⠀⠁⠊⠁⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠁⠀⣼⣿⡟⠃⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⣿⠣⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠀⠀⠀⠀⠀⠐⠿⠄⠀⠀⠀⠀⠀⢠⡀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠋⠽⣦⡀⠓⣆⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣦⣹⣿⣷⣀
⠀⠀⠀⠀⠀⠄⠀⣦⠀⣄⠀⠀⠀⠀⠀⠀⣸⠷⣾⠏⠀⢰⡾⠷⣄⠀⢰⡿⠿⠻⣶⠀⠀⠀⠀⠀⠀⠀⢀⣶⡏⠀⠀⠀⠀⣸⠉⠉⡿⢋⣿⣷⣄⡀
⠀⠀⢰⣾⣙⣦⣠⡟⣰⠿⣄⣀⣀⠀⠀⠀⠁⠀⠀⠀⠀⢘⣷⣤⣾⣷⣏⡒⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⡿⠉
⠀⠀⠈⠉⠉⠉⣙⣷⣷⣞⠋⠉⠁⠀⠀⠀⣀⠀⠀⠀⠀⠸⢯⠿⠋⢸⣯⠛⠻⡦⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣄⣈⡁
⠀⠀⠀⠀⠀⠘⢿⣿⡁⠀⠠⠀⠀⠀⠀⠈⠙⣷⠊⢿⠀⠀⠀⠀⠀⠀⠹⠆⠀⠁⠀⠀⠀⠘⣿⢀⡄⠀⠀⠀⢀⡁⠃⠀⠀⠀⠀⠀⡏⡷
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠤⢤⡀⣴⣿⣦⣊⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣦⡻⣿⢏⣶⣀⣴⡀⠀⠀⠀⠀⠀⠀⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⡅⢿⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠟⠚⣿⢟⣿⣿⢿⣟⡿⠃⠀⠀⠀⠀⠀⠈⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⢿⡿⠄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠷⠛⢿⠋⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠔⠻⢼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠐⠃⠀⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⣠⠂
⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⢀⣠⡄⠀⡼⠐⠚⠁⠀⡀⠀⠀⠀⠀⠀⠐⠤⣤⡀
⠀⠀⠀⠀⠀⠈⢋⠀⢰⢣⠂⠀⡇⡄⢈⡄⡴⡰⢡⢦⠆⠀⡞⢡⣰⢀⡎⡼⢠⠄
⠀⠀⠀⠀⠀⣄⣀⠀⠈⠀⠀⡸⠑⠃⡞⡸⢱⠃⠁⡟⠀⠈⠀⢡⠃⡜⣰⢁⠎⠀⢒⠒⢦
⠀⠀⠀⠀⠀⣰⠏⠀⠠⢤⠄⠁⠀⠀⠁⠁⠉⠀⢦⡀⠀⠁⡠⠈⠀⢱⠃⠁⢀⠀⠀⠀⠈⡠⠄
⠀⠀⠀⠀⠀⠁⠀⠄⢄⣀⠀⠀⣶⠂⠀⠀⣾⠀⠀⢉⣤⣤⣧⢥⣤⡥⣴⢶⣿⡀⠠⡄⡓⠋
⠀⠀⠀⠀⠀⠀⢃⡶⡶⢦⣠⠀⢻⡀⠐⠚⠃⠈⣆⢸⣷⣽⣬⣼⣼⠧⠿⠚⢋⡁⠓⠀⣷⠀⡀⠌
⠀⠀⠀⢈⢤⠈⣌⠀⠉⠈⠀⢰⣟⣥⠀⠀⣠⣴⠏⠘⠋⠀⣀⣀⣀⣠⣬⣿⡏⠐⠛⠚⢻⣤⠌⠁
⠀⠀⠀⠀⠀⠐⢮⣄⣧⠀⠀⢸⡇⠉⠀⠀⠘⠅⡀⢰⣿⣭⣿⣛⣿⣹⣿⠢⠔⡙⠂⢄⠠⡌
⠀⠀⠀⠀⠀⠠⠤⠘⠿⢶⢀⣘⠇⠀⣠⣤⣄⠁⣄⠘⠿⠻⢟⠙⠛⠻⡅⠀⢐⠡⣦⢠⡁⢃⠀⣀⣀⡀
⠀⠀⠀⠲⣶⣼⡀⣤⣀⣿⠄⠿⠛⣛⣛⠺⠏⠉⣅⢀⠀⢀⠤⢽⠗⢠⠋⢤⡊⠀⠃⢘⠂⡗⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠩⢿⡿⢟⡛⢈⡉⠗⢰⣾⢸⡆⡇⣿⢸⢣⣿⠈⣿⠀⣷⠀⡆⡍⣉⣼⣰⠆⢠⠶⠄
⠀⠀⠀⠀⠀⡰⠃⡠⠏⣽⠈⡇⣿⡘⣿⠸⢃⡇⣿⠘⣸⡏⠆⣿⠄⣿⠀⡇⡇⡇⢰⡍⣧⠸⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠁⣿⡇⢹⢸⣿⡇⠿⢸⠘⢿⡇⢹⠀⣿⢀⡇⡇⡗⡈⠁⠃⠀⠈⠙
⠀⠀⠀⠀⠐⡄⠀⡆⠀⣿⢸⣆⣷⣶⡾⠾⠋⢀⣀⣠⣤⣤⣀⣀⠈⣿⣶⣧⣁⡇⣿⠀⢠⠂⠀⠄
⠀⠀⠀⠀⠀⠈⠀⣘⠠⣿⠛⠯⢉⢿⠀⠰⠀⠉⠉⠉⠉⠉⠉⠁⠀⠛⣿⡝⡅⢛⡿⠄⣘⣠⠊
⠀⠀⠀⠀⠀⠂⠬⣽⢦⡄⠸⠆⡀⠀⠑⠾⣇⠀⠀⠀⠀⠀⠀⠀⢀⣦⡿⡓⡀⠖⢠⡴⠿⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠴⠾⠉⠆⣶⠘⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡸⠁⠰⠃⣤⠈⠂⠟⠲⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⢶⢦⣤⣅⣄⠀⠈⠁⠘⠣⣄⣠⣄⠀⠀⠀⣀⣰⠤⣶⢶⣠⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠉⠙⣆⡠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢿⣀⣀⡟⠀⢈⡇⠀⠀⠀⠀⣀⡤⠖⢛⡭⠋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠐⢏⣀⣯⣀⣠⠇⠀⠀⣠⠞⠁⢀⣔⠁⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠔⢺⠃⠈⠙⠖⠒⠲⠺⠧⠄⠀⣞⠁⠀⠀⠀⠀⣿
⠀⠀⣠⣄⡤⠶⠶⠦⢤⣤⡴⠚⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠘⢦⡀⠀⢀⡜⣽
⠀⠀⣄⠀⠠⠵⠶⢤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠀⠀⠀⠉⠓⠯⠤⠿⣤⣀⣀⡀
⠀⠀⢸⡀⠀⠀⠀⡟⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⠈⠉⢉⡼
⠀⠀⠀⣧⠀⠀⢸⠃⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣴⣿⣧⠀⠀⠀⣀⣀⠀⠀⠀⢻⠇
⠀⠀⠀⠘⢷⢠⡏⠀⢾⠛⢛⣷⡄⠀⣀⠀⠀⠀⠡⣀⣹⠿⠗⠒⠉⠁⠀⠀⠀⠀⠀⡜
⠀⠀⠀⢀⣴⡏⠀⠀⠈⢂⣹⣿⡷⢼⠓⠉⢮⢀⣒⣉⠀⢐⣀⣁⣁⣀⣀⠀⠀⢀⡾⠶⠖⢲⡄
⠀⠀⣴⠚⠉⠀⢀⡀⠤⠐⠒⠀⢴⡈⠒⢴⣁⣀⡼⠉⠁⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠰⡇
⠀⠀⠈⠳⡄⠈⢁⡠⠒⡁⠄⠀⠀⠈⠉⢣⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⢶⣥⣠⠊⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⠀⠀⣀⣠⡴⠛⠓⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠸⡇⠈⠙⠒⢤⣤⣠⣀⣀⣀⣀⣀⣤⠤⢴⠟⠁⠉⠓⣄⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⣲⡀⠀⠀⠀⣀⣀⠤⠎⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⢶⠎⠀⠉⠉⠉⠉⠁⠀⢙⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⡓⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⢸⠈⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡰⠀⠀⠀⠀⠀⠀⠀⠐⡷⠄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡄⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢆⠀⠀⡇⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠈⠳⣜⡇⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠘⡗⠒⠒⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⢸⠀⠀⠀⠀⢀⡞⠙⢦⣠⠤⠖⠛⠉⠉⠉⠁⠙⠦⣄⣷⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠦⣼⠀⠀⢸⠀⠀⠀⢀⡞⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡚⠉⠁⠀⠀⢸⠀⠀⠀⠘⠷⣦⣉⡁⢩⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣨⠀⠀⣀⣼⡀⠀⠀⢠⡀⣺⡏⠉⠻⣬⣇⣸⣉⣛⠶⠶⠴⠶⠴⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠙⠒⠶⠾⠛⠋

⠀⠀⢀⠀⢠⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠠⣀⡄
⠀⠀⠸⡆⢸⡆⠀⠀⢠⡇⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠈
⠀⠀⠀⢷⠸⡇⠑⠀⢸⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠰⠀⠄
⠀⠀⠀⠘⠀⠇⢀⡇⠈⢸⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣧⢲⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠛⠳⣀⠃
⠀⠀⠀⠀⡰⠀⠀⢠⠀⠸⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠈⡇⠈⠠⡆
⠀⠀⠀⡀⢃⠀⢰⡸⡄⢻⠇⠈⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠲⡿⢀⣀⠄
⠀⠀⠀⠇⡞⢰⠀⠷⣧⠋⡆⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⣳⡀
⠀⠀⠀⢸⠃⠸⠀⠀⣿⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠹⠉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⡿⠘⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣧⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣤⡰⠦⠀⣙⡛⠀⠀⠀⠀⠋⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠖⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣾⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢴⣦⣃⡀
⠂⡐⠀⠈⠀⠀⠈⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⣿⡛⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⠀⡈
⠀⠀⠀⠀⠀⢤⢠⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⡞⠅
⠀⠀⠀⠀⠀⠈⢀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠇⢼
⠀⠀⠀⠀⠀⠢⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡽⣾⣉⣈⣆
⠀⠀⠀⠀⠀⢀⡀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⡾⠟
⠀⠀⠀⠀⠀⠈⠁⠀⢡⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠆⠀⣿⣟
⠀⠀⠀⠀⠀⠀⠐⠢⣬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢾⣶⣤⢏⣶
⣤⣤⣤⣤⣤⣤⣤⣤⣴⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣬⣯⣤⣿⣬
⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣉⢉⣿⣿⣿⣿⣿⡟⣸⠟⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⢄⣀⡠⡄
⣿⣿⡿⠿⠀⣽⣿⣿⣿⣿⠋⢠⣶⣾⣟⠛⢛⡽⢿⡟⢣⣹⡿⠃⠀⠂⢨⣿⣿⠀⠀⠀⠀⠀⠀⢠⠐⠛⠘⠃⢲
⣿⣿⠂⠀⠀⠹⠿⣿⣽⣿⣷⡿⢿⣿⠾⣄⢠⣞⣻⣳⣼⡿⣣⡦⢶⣾⣮⣿⣿⠀⠀⠀⠀⠀⠀⣿⣤⣄⠀⣴⣼⣧
⣿⣿⣿⣶⠀⢀⡠⣤⣬⠙⢿⣗⠻⣿⡛⢉⣀⠁⢀⣼⣿⡾⠋⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣄⠁⣀⡈⣡⣿⡀
⣿⣿⣿⡛⣘⣙⠷⡶⠾⠗⠀⠻⣿⣟⣿⣦⠉⠴⢾⣿⠏⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣯⣽⣷⣾⣿⣿⣽⣇
⣿⣿⣯⣄⠀⢈⣭⣤⡤⠀⠀⢀⣈⣹⣿⢃⡄⠰⣿⣇⣀⣼⠿⢯⡙⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣩⣿⡉⢻⣿⣿
⣿⣿⡿⠋⠀⠀⠀⠁⠀⢀⣴⣿⣿⣿⣿⣿⣦⣀⣿⣿⡏⢻⢀⠠⡙⢿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣧⣿⣿⣿⣿⣿⣿
⣿⣿⡷⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡇⣿⢸⡀⣷⡈⣿⣿⣿⠀⠀⠀⢠⣿⠟⣼⡷⣿⣿⣿⣿⡿⠋⠁
⣿⣿⣿⣶⠤⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⠇⣸⣧⣿⣿⣿⣿⣿⠀⠀⠀⢸⢏⣼⣿⠗⠟⢿⣿⠋⣶⡇
⣿⣿⣿⣿⣯⣽⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢈⣾⣿⣿⣿⣿⣿⣿⣽⢼⣷
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⡿⠷⡷⠺⠿⣿⣻⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣟⢻⣿⡟⠛⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣫⡄⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⡀⢹⠃⢰⣿⣿⣿
⣿⣿⣿⣿⡇⠸⠿⣿⣺⣞⣿⣿⣿⣿⣯⠀⢹⣿⣿⣿⡿⢋⣵⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⠇⢸⡆⢸⣿⣿⣿⡇
⣿⣿⣿⣿⠇⣀⣧⣿⣿⣿⣥⠺⣿⣿⣿⣇⠀⣿⢋⣥⣤⣫⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⠀⢸⣷⠀⠹⣿⣿⡇
⢻⣿⣿⢿⣿⣿⣿⣿⡷⣶⣦⣉⣿⢿⣻⣿⡧⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⠀⢸⣿⡄⠀⣿⣿⠁
⢸⣿⣿⣾⣿⡙⣻⡟⢡⡿⢹⡟⠀⢸⣧⠈⠃⢸⡀⠁⠻⣿⣿⣿⡟⢹⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⡄⠘⣿⣧⠀⣿⡿
⢸⣿⣿⣟⣼⣿⠏⢠⡄⢰⡿⠀⠟⢸⡟⠀⡄⢸⡇⠸⣆⣹⣧⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠛⠃⠀⣿⣿⠄⠘
⢸⣿⣿⣿⣿⣿⣶⣾⣥⣼⣇⣰⣦⣼⣧⣤⣧⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠂⠀⣠⠀⠀⢰⡖⠂⠀⠘⠛⡆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⣰⠉⠀⠀⢸⣆⣀⡀⢠⣤⡇
⠘⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠻⠿⠗⠘⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡉⠉⢁⡺⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠤⣀
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡠⠈⠀⠒⠒⠬⣢⣿⡄⠀⠀⠀⠀⠀⢠⢞⡡⠒⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠌⡐⠀⠀⠀⡠⠒⠛⠑⠉⠁⠐⠐⠤⠶⢇⣩⠤⠀⠀⠀⠀⢀⠐⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢠⠂⢀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢸⡷⣎⠁⠐⠠⡀⠀⠁⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠢⣠⣁⡠⠆⡘⠀⠀⡄⠀⠐⠀⠘⠇⠈⢆⠀⠀⠈⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⣷⡴⠉⢁⡲⢾⣇⡘⠈⡆⢀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢸⡐⡀⠀⠀⠀⠀⠷⠚⠏⠀⢈⣲⣿⣿⡟⣤⡇⢀⡏⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡀⡜
⠀⡀⠈⠈⡙⢦⡀⡇⢡⠀⠂⡀⠀⣀⠀⠂⠀⠀⠈⠉⠅⢾⣥⠵⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⢇⠀⠆⢼⡂⠓⠀⠀⠀⠀⡇⠀⠩⡀⢓⡲⠂⠀⢀⠴⣿⣥⡀⠀⠀⠀⢀⡏⠀⡀⠀⠀⠀⢠⠞
⠀⠘⣆⠀⠙⢷⣋⠞⠠⠀⡄⣇⠞⢓⠄⠐⣒⣆⠉⠣⢠⡤⡠⠰⠄⠀⠀⡜⠀⡐⠀⠀⠀⣠⠋⣀⣤⣀
⠀⠀⠈⠢⡀⠒⠤⠴⠁⡜⢀⠁⠤⠚⢠⠊⠇⠸⣒⠇⠀⠀⠀⠀⠀⠠⣸⠁⢀⠀⠀⠘⢠⡗⠋⣀⠀⠈⢷
⠀⠀⠀⠀⠈⠒⠦⠤⠤⣴⠏⠴⠁⠀⠀⠀⠰⠊⠘⠈⠍⠀⠄⠀⠀⠀⣧⠀⢸⡀⠀⠸⡰⣄⣀⣷⠇⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠙⠆⡀⠀⠀⣼⡀⠀⠐⠀⣁⣼⠑⠢⡀⠘⠦⣀⠣⡀⠀⠈⠀⠈⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠊⠀⠈⠻⠂⠀⠘⠉⠀⠀⠐⣀⠐⡌⠙⠚⠳⠶⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈




⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠤⠀⠤⠄⡄⠀⠤⠤⡄⠀⠀⠤⠀⠠

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠶⠶⠶⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⢀⣀⣀⣤⣤⣤⣄⣀⣀⡀⠉⠙⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⣀⡤⠖⠛⠉⠉⠀⠀⣄⢀⡀⠀⠉⠉⠛⠲⢤⣀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⡴⠚⠁⢀⢀⠀⠀⠀⠀⠀⠟⠰⠧⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠉⢻⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⢀⡴⠋⠀⠀⠀⠛⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⢦⡀⠙⢷⡄
⠀⠀⠀⠀⣰⡿⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⢿⡄
⠀⠀⠀⢠⡿⠁⢠⠏⢠⢠⣶⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⢻⡄⠈⣿⡄
⠀⠀⠀⣾⠃⢀⡟⠀⠈⠀⠉⠁⠀⠀⠀⠀⠘⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢷⠀⠸⣷
⠀⠀⢰⡿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⡄
⠀⠀⢸⡇⠀⣾⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡇⠀⢹⡇
⠀⠀⢸⡇⠀⢻⠀⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠾⠃⡇⠀⢸⡇
⠀⠀⠸⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⠀⠀⠀⠀⠀⢰⡇⠀⣿⠃
⠀⠀⠀⢿⡆⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⡾⠀⢰⡟
⠀⠀⠀⠘⣿⡀⠘⣧⠀⢘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡇⠀⣼⠃⢀⣿⠃
⠀⠀⠀⠀⠘⣷⡀⠘⢧⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣾⠃
⠀⠀⠀⠀⠀⠘⢿⣄⠈⠳⣄⠀⠀⠀⣲⡖⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⢀⣰⠟⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠈⠻⢦⣀⠙⠁⠀⠀⠀⠀⢼⣧⡀⠀⠀⠀⠀⠈⠟⠀⣀⡴⠛⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠉⠛⠶⢤⣄⣀⡀⠈⠉⠀⠀⢀⣀⣠⡤⠶⠛⠉⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋⠉⠀⣴⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⣄⠀⠀⠉⠙⣶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠙⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣏⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢉⣽⠃⠀⠀⠀⣰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⡶⠶⠦⠘⠛⢿⣿⣦⣿⠶⠶⣶⣶⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⢠⡶⢿⣿⣿⣿⣿⣷⣤⣂⣀⣀⣐⣾⣿⣛⢀⠀⠀⠐⣛⣢⣭⣭⣿⣷⢶⡦
⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⣿⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⣟⢻⡿⢿⣿⣿⣿⠿⢿⣿⣿⡿⠿⢿⣿⡟⠙⠻⣇⢼⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣾⣔⣄⣸⣿⢰⣆⡀⣿⡇⢤⣀⠀⣽⣧⣄⣴⡿⣟⠓⢤
⠀⠀⠀⠀⠀⠀⢀⡞⢾⣏⠉⢻⣿⣾⣿⣿⣾⣾⣷⣿⣿⣮⣧⣶⣿⣿⠿⢯⡀⠘⣦⣠⠃
⠀⠀⠀⠀⠀⠀⠈⠿⣾⣿⣶⣿⣷⣤⣀⣹⣍⣿⡉⠀⠀⣹⡅⢀⣼⣄⣀⢠⣿⣿⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⢿⣿⣿⣿⡿⣾⣿⠿⠿⠿⠟⠟⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠲⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⡄⠈⠓⡄⢠⠖⢤⡀⠀⠀⣿⠀⠈⠣⣍⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢰⣎⡀⠀⢱⠀⢀⣿⡄⠀⠀⠈⢣⡀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢘⢏⣀⡽⣤⣀⣀⠆⠘⡎⢁⠀⠀⠀⠀⠱⡄⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢈⣧⠀⠈⠳⡤⠶⠚⠇⠈⡄⠀⠀⠀⠀⠹⡀⠀⠀⠘⣧
⠀⠀⠀⠀⢀⣀⣀⣀⣀⡈⠒⠋⣰⠄⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⡀⠀⠀⠃⠀⠀⠀⢹⡄
⠀⣴⠮⠍⣁⡀⠀⠀⠀⠈⣹⠞⠁⠀⠀⠀⠀⠀⡠⠤⠄⠀⠀⠈⠙⠲⢤⣬⣀⣀⣀⣘⣇⢀⣀⡤⡆
⠀⠈⢦⡀⠈⠙⠳⢦⣀⡞⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠜⠛⠉⣰⠃
⠀⠀⠈⢧⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻
⠀⠀⠀⠈⢧⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡧
⠀⠀⠀⠀⠀⠳⣄⢸⣥⣦⡄⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠻⠉⠄⠘⢷⣤⠢⠄⠤⡀⠀⠄⠂⡁⠤⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠚⠁⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠘⠂⣜⠁⠀⠆⠀⠈⠒⢶⠄⠲⠋⠉⠛⣭⡁⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠄⠀⢀⣼⠃⠀⠀⠰⣌⠀⢚⡀⢠⣤⠴⢛⠀⠸⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠻⢵⣀⢀⡠⠔⠈⠀⠉⠈⠠⡀⢠⠊⡅⠀⠘⠇⠀⠀⠀⠀⠈⠳⣄⣀⣰⡏⢀⡀⠄⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠙⠻⣤⠄⠀⠀⠀⠀⠀⠴⠄⠊⠀⠀⠀⣺⠀⠀⠀⠉⠙⠲⣼⡄⠀⣻⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠠⠤⠀⣠⠶⠛⢉⣧⠀⠀⠀⠀⠀⢀⡵⢀⡟⠀⠛⠢⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⢤⠊⠁⠀⢀⡠⠼⣆⠀⠀⠀⠀⠈⠀⣟⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣠⠖⠉⠀⠀⠘⣆⠀⠀⠀⠀⠀⡏⠀⠀⠉⢳⣤⣀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⡈⠙⠓⠲⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⣦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣾⠁⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠾⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⢀⡤⠦⠷⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡔⠒⠂⠀⠁⠀⢀⣠⠾⠛⠒⠂⠀⠀⠈⠉⠙⠒⠉⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⡀⠀⠼⢭⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠆⣤⠖⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠀⠽⠷⡢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⢄⣀⡀⠠⠤⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣬⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠘⣿⠯⣾⠇⠀⠀⡀⠀⣠⠀⠀⣦⡙⠑⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⡄⠠⠺⠛⠋⠁⠐⢦⡄⠈⠚⠛⠂⠐⠀⠐⣦⣀⡀⢀⣴⣿⢿⣤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣃⠜⠛⠳⠃⠈⣀⣀⡤⠀⠘⠁⠬⣦⣤⠲⣾⡖⠀⠀⠀⠈⢿⣿⡭⠾⣋⣾⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⢀⠀⣿⣧⣀⣀⣸⠏⠀⣴⠶⠀⢐⠸⣯⠙⣿⣷⣄⣼⣆⡀⡀⢀⠰⢾⣿⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⢀⣀⠈⢿⡿⣿⣷⣄⢀⠠⠆⠀⠀⠙⢞⣿⣿⣧⡿⣀⣰⣯⣽⣬⣶⣬⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⡀⢺⣿⡆⢸⠟⣰⣿⣿⡏⠀⠀⠀⣤⡀⢘⣿⡟⣿⡏⠀⠉⠉⠁⠋⣸⡏⢙⣷⡮⠉⠖⠉⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢶⣨⣧⣄⣩⠥⡤⡶⠹⢿⣿⣅⡀⠤⠼⢿⣧⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢱⣾⣿⡀⢠⣰⠒⣩⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠁⠻⣁⠓⢓⣶⣤⣀⠒⣜⡯⠀⠀⠀⠀⠀⠀⠀⡁⠀⣠⣶⣷⡆⠀⠀⠀⠺⣿⠏⠛⠉⠉⢈⣭⣤⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢠⣾⣿⠿⠝⠋⠋⠛⠀⠀⢀⣶⡤⠁⠀⡔⣿⣷⣿⢿⣿⣿⠿⠶⢦⢀⠿⠂⠰⠤⣦⠈⠙⢿⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⢀⠁⠒⠀⠀⠉⠉⠀⠂⠈⠀⠀⣸⠀⠉⠈⢃⡉⠁⠈⠈⡳⣄⠀⡈⠂⣀⠀⠀⠁⠁⠄⠸⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⣠⣶⠟⠓⠻⠶⣶⣦⣤⢲⣄⣀⣀⡀⠉⠉⠀⠀⠠⠀⢻⠄⠁⠸⠻⠛⠃⠠⢿⣿⣿⠦⠤⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠉⢀⠂⠀⠀⠀⠉⠛⣿⠏⣠⣿⠿⠻⣿⣷⣄⣇⠀⠀⠉⣁⠀⠀⢀⣠⣤⡲⣾⣿⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠉⣀⣀⠤⠀⠀⠠⢈⠜⠋⠉⢛⣿⣦⢻⠿⡅⠈⠉⠉⣤⡀⠀⢠⣴⡌⠀⠹⣧⣦⣤⡔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣤⡬⢹⡇⢸⠃⠀⠘⠀⠐⡿⢿⡽⠯⠄⠒⠆⢤⣴⣿⣤⡠⠀⠀⠉⣠⠄⠀⠈⢩⣭⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣻⣥⣴⣟⡁⣰⠍⠁⣠⠞⠉⠀⢸⣿⠶⢬⠛⠛⣀⣾⣶⡞⠻⠟⠀⠈⣴⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣻⣿⣿⣟⡻⠾⣿⣿⡚⡅⠀⠈⡁⢀⡞⣀⠘⠻⢾⣿⣄⠀⣤⢀⠸⠏⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣾⣯⡉⢁⡀⣿⣿⡭⠁⢈⡄⠈⡾⠀⢿⣿⢦⠀⠘⠛⠃⣸⢁⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣅⣚⠀⠀⠎⣿⡌⠛⠀⠀⠀⠀⠀⠁⠀⣢⣿⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⣿⣿⡟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⡶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡁⠾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⣹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⢿⣿⣿⣿⣶⣤⣤⣬⣭⣭⣭⣤⣤⣴⣾⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢿⣽⣟⣟⣿⣛⣛⣟⣻⣻⣟⣿⣹⣏⡿⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉

⠀⡀⢤⠀⣴⣶⣦⠀⠀⢀⡀
⠀⠙⠈⠀⠙⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠃⢸⠀⠀⠀⠀⠀⠀⡠⠊⠌⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⡖⠀⠙⠀⠀⡇⠀⠀⠀⢠⠞⠁⡜⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⠼⠂⠉⣄⡐⣁⡀⠀⡐⠃⠀⠀⠃⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠂⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⢼⡀⠀⠀⠀⠀⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠊⢀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⢀⠂⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢈⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣆⣀⠃
⠀⠀⠀⠀⣀⡴⠒⠀⠈⠐⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⣤⠤⢶
⠀⠀⣠⢊⣀⡀⠀⠀⠀⠀⠀⠀⠀⠱⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠱⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠉⠀⠉⠉⠠⡀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡌⠰⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢤
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⢈⠁⠌⠀⠐⠀⠀⠉⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠒⣘⡁⠀⠀⠀⠀⠀⠈⠢⡀⢈⠄⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⡀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠚⠁⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣁⡀⣠⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⡍⠉⠐⢦⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⡂⠀⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⠔⠋⠉⠀⠀⠀⠀⠀⠉⠉⠑⠒⠀⠀⠀⠘⠃⠉⠀⠀⠈⢌⡁⠉⠂⠤⠤⠀⠠⠤⠄⠒⠁⢀⠜⡇⠒⠤⢀⣀⡀⠀⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⡀⠀⠀⠀⣀⠠⠔⠁⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡤⠚⡹
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡤⠊⠁⠀⢠⠁
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠅⣀⠀⠙⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⢸⠉⠉⠐⠒⠒⠒⢚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡰⠃
⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠤⣄⠙⢆⠀⠀⠀⠀⠀⠀⠀⢱⠀⠰⠁
⠀⠀⠀⢿⠀⠀⠀⠀⠀⣠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⡀⠈⡆⠈⠳⡀⠀⠀⠀⠀⠀⠀⣦⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠡⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⣀⣥⠴⠋⠀⠀⠙⣄⠀⠀⠀⠀⠀⠘⠤⠀⢤⡀
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⢀⠀⠀⡵
⠀⠀⠀⠀⠓⢤⢀⣀⣈⣤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⣇⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢧⠀⡄⠘⠶⠒⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢣⠀⠈⢇⢳⡤⠤⠖⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⣄⡀⠀⠘⡇⢀⠀⢣⡀⢸⠞⠒⠒⠒⠂⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠡⣀⠀⠐⡄⠈⠳⡄⢳⠈⢆⠀⢳⠯⠐⠒⠒⠒⠂⣰⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⢤⡈⠳⡄⢹⡀⢀⠘⣎⡆⠈⢣⠯⠴⠒⢂⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠙⢄⠘⠄⢣⠘⡄⡌⠻⠀⢀⣤⣶⠯⠽⢒⣒⠛⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠀⢈⠖⢫⠋⡩⢋⢟⣠⡀⠈⣀⡠⠔⠊⣉⡤⠤⠖⠂⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣠⠎⢠⠃⡰⠁⢋⢾⢧⡇⠀⢨⠙⡶⣉⣀⡠⠤⠒⠒⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠊⠁⠠⠃⠀⢁⡴⠋⣼⡏⢳⠀⡾⠀⡇⠈⡅⠀⢈⣉⣡⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠤⠴⠚⠉⠀⣸⡟⠀⠘⣆⠃⢀⡇⢠⡇⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⢆⠈⡇⠈⠇⠳⡀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⠓⢤⣀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠰⠣⡀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣂⠄⠀⠀⡠⠄⠤⣄⠀⢠⣄⣀⣀⢀⠀⡀
⠀⠀⠀⠀⢀⣠⠤⠤⢤⡤⠰⠒⠀⠀⠀⠀⠠⠤⣄⠠⠒⠢⢌⠠⠮⠤⡀⠜⠁⠀⣾⡿⠁⠉⠈⠙⠷⡤⠐⡆⠀⡀⠠⠤⢀
⠠⢤⠀⠘⢋⠀⠀⠀⠀⠘⢧⡀⠇⠀⠀⢰⡤⠊⣅⠀⠀⢀⣀⠀⢆⠀⠇⢀⠴⢊⠵⠽⢀⡰⠦⠀⠀⡆⠀⢹⡈⠀⠀⡠⠐
⠰⡄⠹⠄⠀⣒⡄⡀⡎⠁⠀⠉⠲⣄⠸⣁⠀⠚⠿⠂⢰⡟⠂⠀⠀⠉⢲⠜⡀⠀⠙⠖⠞⠉⠁⠀⠀⠑⢤⠘⠁⠀⢀⠁
⠀⠈⠞⠀⠀⠈⠑⠔⡲⠋⠀⡀⠔⠚⠀⠈⠦⣀⠀⠈⠉⠀⠀⠀⡀⠔⣏⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢠⡜⠳⠄⣀⠄
⠀⢹⣀⣄⡀⠀⠀⠀⠀⡠⣎⠀⠀⠀⠀⠀⠐⢤⡝⠐⠀⠀⠚⠻⡂⠀⠀⢠⡄⠀⠉⡖⢐⠂⠈⠁⠲⠁⠀⠀⠀⠈⢀
⠀⢀⠆⠀⠀⣉⠤⡾⠩⠛⠃⡂⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠇⠀⠀⢰⠑⢄⠀⡈⠈⠳⡀⠀⠀⡇⠀⠀⠀⠀⢨⠇
⠀⠈⡀⢠⠀⢱⠄⡄⠀⠐⠀⡞⠀⠀⠀⠀⠀⠀⠈⠰⢄⠀⠠⠂⠀⢀⡠⡏⠀⠀⢸⡄⠀⠀⠀⢑⠈⠀⠀⠀⠀⡀⡘
⠀⠀⢣⠀⢂⠀⠀⠀⡀⠚⢁⠀⠀⠀⠀⠀⠀⠀⠀⡘⡄⠉⡃⠀⢀⠋⠀⢅⠀⠀⠀⠀⠦⢄⠀⡈⠀⠀⠀⡀⠂⠀⡀
⠀⠀⢀⠂⠈⣆⡀⠈⡅⢠⣌⣡⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠇⠰⠆⡑⠄⠤⠇⠀⠀⢀⠈⠀⠀⡇⠀⢀⠜⠀⡀⠀⢀⡀
⠀⠀⠘⠶⠧⣀⡀⠜⠀⠠⠤⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠻⠄⠀⠠⠇⠀⠉⢁⠄⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣶⡂⡀⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣤⣸⣏⣠⣄⣿⣧⣠⣤⣰⡟⢿⢇⣤⣄⣿⡧⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣛⣛
⠀⡀⠒⠒⠂⡀⠀⠀⠀⠀⠀⢸⡟⣿⣸⡏⢹⣧⣿⣷⣼⡀⠛⣹⡿⣧⣤⣼⣟⣿⣿⣶⣮⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣛⣛
⢰⣤⣤⠠⣤⣴⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠁⠈⠛⠛⠁⠀⠈⠁⠀⠈⠉⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣟
⠀⠻⠁⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣁⣤⣸⣿⣯⣴⣿⣧⣶⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡄⢒⡀⣀⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⡿⠿⠿⠛⠛⠛⠷⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⡟⠂⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⣿⡇⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿⣛⡃⠀⠀⠤⠄⣹⣿⣏⣹⣿⣷⣦⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠘⣷⠀⠀⠰⢾⡗⠿⠛⢶⣶⢂⣄⡀⠀⠀⢠⠈⡄⠀⢻⣿⠀⠀⠀⠙⠛⠻⣿⠧⠜⠿⠃⠘⢻⣽⣯⠈⡶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⠀⣿⡀⠀⠀⠀⠀⠀⣀⡀⠉⠑⠈⠁⠀⠀⢨⠈⡁⠀⢸⣇⠀⠀⠀⠀⠀⠀⠛⠄⠀⠀⠠⠂⠈⠻⠓⠿⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣆⢸⣷⣶⣆⠒⠀⢈⣭⣿⡁⠀⠀⠀⠀⠀⠸⡔⢰⡀⠐⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣅⣶⣶⡄⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠹⣾⣿⣿⣷⣷⡿⢯⣉⠁⠒⠀⠀⠀⠀⠀⢘⡚⠾⣇⠀⣿⣿⣿⣿⢶⡚⢼⠗⠀⠀⣀⣠⣾⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠹⣿⣿⣿⣿⣷⣤⣿⡁⠀⠀⠀⠀⠀⠀⠰⣂⠏⢻⣶⣾⣿⣿⣿⣿⣇⣯⣤⣶⣾⠿⠿⠿⠄⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣜⢿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠠⢾⣦⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠻⣷⣶⣄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣦⣄⠀⠘⣿⣆⢻⣿⡟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢾⣆⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣆⣨⣽⡟⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⠀⠉⠻⣧⢿⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢹⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠡⠹⣿⣿⣌⢻⣿⣿⣿⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣧⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⡿⣿⣷⡆⢻⣿⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⣇⠀⢀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⡂⠀⠀⠈⠽⢿⣿⡄⢻⣿⠀⠀⣿⡄

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠁⠀⠐⠇
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠀⡴⠺⠓⠆
⠀⠠⡇⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠺⣀⡁
⠀⠀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⢱⣶⣶⡾⢿⠷⢶⣶⣤⡀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠐⠀⠐
⣆⠠⠤⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠋⢀⣄⣀⠀⠈⠛⠁⠀⠀⠐
⣯⠙⠲⡄⡈⠀⠀⡆⠀⠁⠀⠀⠀⠀⠀⢀⡄⡀⢀⠀⠀⠉⠀⠀⠀⠸⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡔
⠙⡤⠀⠀⠀⠀⠀⡇⠰⠄⠀⠀⠀⠀⠀⠘⢿⡿⡇⠀⢀⠀⢀⢠⠂⠀⠀⠀⠘⠀⣠⠀⠀⠀⠂⠀⠈⡚⠀⠀⡄
⠄⠗⠓⡄⢀⡄⠈⡗⢠⡌⠂⠀⠀⠀⠀⠀⠀⢿⠁⢂⠏⢠⠀⠈⠀⣠⠄⢃⠄⠳⠟⠂⠀⠀⠀⠑⣌⠱⠚⠂⢀
⠀⣰⠲⡄⠀⡅⢀⡟⢾⣇⠀⠀⠀⠀⠀⠈⠒⠠⠃⠤⠆⢰⠂⣤⡰⠇⢸⠀⡄⣸⡞⠀⡀⠀⠀⠀⠀⠙⠖⠀⠈
⠀⠁⠀⢱⢀⣤⢨⡇⢸⢻⣎⠁⠀⠀⡄⣼⠆⡷⠀⠀⢌⢙⠂⠃⢍⡁⠻⣤⠳⠝⠃⠀⠑⣤⢀⠀⢀⠀⡀⠁⠆
⠔⠄⠈⠒⠋⢗⣠⡇⢂⠉⣿⠳⠀⠄⠀⡏⠀⠈⠳⣔⡀⢈⣀⡀⠀⠀⢾⣾⡧⢤⠀⠐⣤⠒⡖⣆⡓⠀⠈
⠀⢾⠾⠔⣶⣟⢙⡟⠋⡐⠿⠀⠇⢐⢟⡟⢠⣄⡒⠊⠭⣹⠀⠀⡴⢀⠈⣽⣦⡔⡀⢠⣤⢰⡿⣄⡁⡀⠀⠘⢣
⣥⢰⣿⡦⡰⠿⢾⣷⠄⠀⢣⠐⣌⣩⣿⣤⣜⣀⠀⠀⣀⡽⢒⣟⢠⣽⠈⡄⠤⢼⠃⠆⠀⠀⠀⠀⢳⠏⣔⣿⣠
⡤⢺⢿⣷⠟⠻⡖⠃⠀⢀⡈⢸⣫⣿⣿⣿⡿⠉⢄⡀⠹⡳⡷⠆⠻⣿⣿⠗⠂⡬⠜⢓⠠⢀⣯⣾⣁⣴⣣⣄⠈
⠁⢼⣗⣤⣄⢀⣥⣺⣶⡦⣦⣻⣉⢡⣶⠚⣗⣀⡴⠀⣤⣟⣆⡁⡘⠻⡞⠂⠀⢃⠰⢺⠐⠿⡝⣜⣧⢿⢷⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⠿⠿⠿⢿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠿⠛⠉⠀⠀⠀⣿⢸⠀⠀⠈⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠞⠁⠀⠲⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠖⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠤⠤⠀⠀⠀⣠⣤⣤⣤⣴⡟⣦⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢀⣎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⢎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠎⠀⠀⠀⢠⡀⠀⠀⠀⠈⠁⢀⡔⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣤⣀⡀⠀⠘⠃⠀⣀⣠⣴⣾⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠙⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⣼⣿⠀⠀⠀⠀⢳⡄
⠀⠀⠀⢀⡀⠀⠀⠀⠠⢤⣀⠀⠀⣿⡟⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠈⠑⢞⡶⢦⣄⠉⠁⠀⠘⠁⠀⠀⣠⠀⠐⣿⡇⠀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠙⠷⠄⠀⠀⠀⠀⠀⠟⠀⠀⠹⡇⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⢦⣄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠹⠆⠀⠠⡀⢀⡇⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣶⠶⠖⠀⠀⢀⡤⠀⠀⠀⠀⠈⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠁⠄⠀⠙⢿⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠒⠛⠛⠻⠷⠶⠤⢀⠀⠂⠀⠁⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠈⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠉⣀⡴⠐⡖⠮⣄⡀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠶⡿⠛⢀⣀⣉⣀⡘⠫⡓⠦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⠊⡁⠀⠀⠀⣩⠶⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠁⠀⠙⢦⣤⠞⠉⠁⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡀⠀⠀⠈⠉⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠑⢄⠀⠀⠀⠀⢂⠄⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⣦⣌⡁⣀⡉⢄⣶⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉⠁⠀⠀⠑⠃

⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢤⠄⢠⡎⠙⢙⢳⡆⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⣀⣄⣀⣀⡀
⠀⠀⠈⣿⣿⣿⣬⣭⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⣿⡟⢿⣿⠿⢿⣿⣿⣷⣄
⠀⠀⠀⢻⣿⡖⢪⡔⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣴⣿⣾⡶⠶⢶⣿⠋⣸⣿⣇⣼⣧⣶⣿⣿⣿⣷⣝⢷⡀
⠀⠀⢠⣼⣿⣧⣴⣦⣼⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⣤⣤⣤⣤⡈⢿⣛⡙⠶⠍⣣⣀⠀⠀⠀⠹⣿⣎⢻⡄
⠀⠀⠈⢿⣹⣭⣿⣗⡻⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⢀⣀⣀⣈⣽⠿⠿⣿⣷⡼⠛⠛⠳⢾⣿⣿⣷⣤⣼⣷⣾⣯⠈⣷
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣵⣿⢿⠉⠁⠀⠀⠀⠀⠙⠛⢂⣀⣤⡾⢏⠠⠂⠈⠉⢉⣷⠙⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣄⠀⠀⠀⣀⡄⠀⠀⠀⠘⠛⣉⣤⣾⠀⠀⠀⡀⢞⣿⣧⢱⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠉⠀⠀⢀⣤⣤⣄⣤⠀⠀⣿⢿⣿⡟⠑⠀⠀⢹⣿⣿⣿⢿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠻⡿⠟⠛⢀⣶⣿⠟⣰⣷⡀⠀⠘⢿⣿⡼⡏⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠄⠄⠀⡀⠀⠀⠀⠀⠀⢀⠞⣋⣥⣾⣿⣿⣷⡀⠀⠀⢻⣷⡹⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣿⡇⢣⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠰⠀⠀⠖⣴⣾⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⣿⣿⣸⣆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⡿⠀⠈⣻⡿⠟⠛⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢟⠉⣿⠀⡿⢹⠉⣿⡿⣿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢋⣾⣿⣿⣿⣿⠏⡼⢠⡟⠀⠃⠀⢰⣿⡇⢸⡞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠐⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⣿⣿⣿⣿⠏⣼⠃⢸⡇⠀⠀⠀⣾⣿⠃⣾⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠉⠀⢀⢀⡄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⢏⣾⡏⠀⢸⣷⠀⠀⢠⣿⡟⣰⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢶⡦⠦⠠⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⢹⣿⣿⣿⣏⣾⣿⠁⠀⣿⢻⡆⠀⢸⣿⣷⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⣐⠁⠀⠀⢀⣀⠀⢀⣁⠀⡀⠀⠀⠀⠀⠂⠄⠀⢠⠀⠄⠐⠂⠀⢿⣿⣿⣸⣿⡏⠀⠀⣿⠈⣿⡀⢸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡈⠻⢀⣤⠄⠀⠙⠻⣿⣿⠂⣄⠢⡀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠙⢧⣿⣿⠇⠀⠀⢹⡇⢻⣧⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣮⣼⡀⠀⠀⠀⢸⠇⠀⠀⠁⢌⠐⠀⠀⠀⠀⠀⠀⠁⠀⠻⠂⠀⠀⠈⢿⣿⠀⠀⢰⡈⣧⠈⢿⣎⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣦⡀⣰⠋⠀⠀⠀⠀⠀⠑⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢸⣧⡘⣧⡈⠻⣯⣻⢧⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣯⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⡀⠀⠻⢷⣌⣷⣦⣈⣙⣩⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠙⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣣⣀⣤⡟⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣷⣶⣾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⢀⣤⣄⠠⢊⠀⢀⠼⣿⣿⣿⡇⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣯⠉⠃⠀⠈⢻⡿⠀⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣆⣆⠀⠀⠀⠃⢸⠇
⠀⠀⠀⠀⠀⠉⠹⠿⠿⣿⣽⡀⠀⠘⢀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⠀⠋⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⢀⡴⠟⠋⠉⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⡚⠻⠏⠀⠢⠁⡇⠄⡄⠀⠀⠈⣿⠀⢰⡏⠀⠀⠀⠀⠀⢀⠘⢠⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣁⡒⡀⠀⠀⠀⠑⢄⠡⠄⡄⡶⠀⡐⢻⡇⣿⠀⠀⠃⠀⠀⢀⠸⢘⡠⠆⠉⠉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠬⠀⠤⠀⠀⠀⠀⠀⠑⢄⠡⠄⡆⡀⢸⣿⣏⠀⠁⡀⠃⠗⣉⡴⠋⠀⠀⠀⠀⠨⠁⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡔⠒⣐⠒⠀⠀⠀⠀⠀⠀⠀⠱⣌⠡⠀⢸⡏⣿⠈⡁⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⣤⠘⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠩⡄⠤⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⣿⠀⣿⠈⣡⠞⠁⠀⠀⠀⠀⠀⠈⢁⠐⠒⠆⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣳⣤⣃⡤⠅⠀⠀⠐⣀⣀⣀⣁⡀⠻⣶⠯⠞⠁⠲⠄⠀⢡⡀⠈⡤⠘⠒⣂⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⠩⠙⠖⢒⣛⠩⠉⢙⡛⠚⣻⢇⣤⠺⠿⣿⣝⣛⣛⣓⣒⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡅⠉⠀⠀⠀⠈⠀⠠⠀⢠⠙⣢⠴⠋⢀⡟⢹⡇⠑⢄⡈⠉⠉⠉⠉⠙⣛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠦⠀⠀⠀⠀⠀⠀⠤⢈⡤⠊⠁⠀⠀⢸⡇⣸⠗⠆⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⣨⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⣃⠀⠄⠀⠠⠌⣡⠞⠉⠀⠀⠀⠀⠀⢸⡇⣿⠠⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢁⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣐⠒⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠂⣸⢿⡇⠖⠀⠀⠀⠀⠀⠈⢣⡀⢀⠀⠀⠐⠀⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣷⠀⠄⠆⠀⠈⠀⣀⠐⠂⢆⡿⢸⡧⠆⠀⠀⠀⠀⠀⠀⠀⠹⠌⠤⠈⠤⠭⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠆⡄⢙⢸⠀⡛⠰⢇⣾⠃⠘⣷⠚⣰⠂⠆⠠⠀⠀⠀⠄⣠⣤⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣉⠘⠘⠀⣋⣤⢾⡇⠀⠀⠹⣧⡀⠞⡴⢀⠋⡴⠰⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⣿⢸⣇⠀⠀⠀⠈⠻⢦⣥⣉⣀⣥⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⣿⡄⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡌⠠⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠋

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠉⠉⠉⠛⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣦⣤⣄⣀⣀⣤⣤⣶⣾⠿⠟⠛⠋⠉⠉⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⡀⣰⣿⠋⠉⠉⠉⠉⠙⢿⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋⠁⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣥⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣶⣾⠿⢿⣶⣶⣶⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁
⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⢀⣤⣾⣿⣏
⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣤⣴⣾⠿⠋⠁⢹⣿⡄
⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣿⣇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠹⣿⣄⠀⠀⠀⠰⣶⣤⡀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⢀⣀⣤⣾⠇⠀⠀⢠⣿⡟
⠀⠀⠀⠘⢿⣧⡀⠀⣄⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⡀⠀⣴⣿⠏
⠀⠀⠀⠀⠈⢻⣷⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢁⣾⡿⠃
⠀⠀⠀⠀⠀⠀⢿⣧⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⠿⠟⠋⠀⠀⣠⣤⣤⣀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡟
⠀⠀⠀⠀⠀⠘⢿⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣶⡄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣴⣾⠿⠿⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣄⣀⣀⣀⣤⣼⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣥⣄⣀⡐⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⠟⠋⠉⠉⠉⠉⠛⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠃⠀⠀⡀⠀⠐⢶⣶⡆⠤⣄⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⢄⠀⢀⢸⣹⡇⣶⣬⣇⠀⢨⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠀⢂⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡀⠀⣿⠛⣿⠛⢻⣿⣿⡏⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣻⠀⣇⠀⢸⣿⣿⣇⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⢀⡋⣀⣐⣛⣟⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠛⠻⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠤⠐⠶⠖⠾⠷⠷⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣄
⠀⠀⠀⠀⠀⢀⢤⡶⠂⠀⠀⠀⢀⠤⠒⡉⠍⠄⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠳⣧⣤⣀⡀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⢀⠀⠀⠀⠀⠀⠸⠀⢿⠀⠀⠀⠀⠀⠀⡀⠀⠛⢿⣇⢄⡀⠀⠀⠙⢿⣄
⠀⠀⠀⢠⠟⠀⠀⠀⣠⠀⡠⢊⠐⠀⠈⢦⡳⡀⠀⠀⠀⠀⣆⣸⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠙⢷⣕⣦⠀⠀⠈⢻⣦
⠀⠀⢀⠋⠀⠀⠀⣴⢃⠞⠁⠀⠀⠀⠀⠀⠑⡷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠄⠙⣿⣿⣄⡀⠈⢻⣧
⠀⠰⠏⠀⠀⢀⡞⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣠⣤⠀⠘⣿⣿⣿⣷⡈⢿⣆
⠀⡀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⡴⠟⠋⠀⠀⠀⠘⣿⣿⣿⡷⠸⣿
⠈⠃⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⠀⣻
⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⠁⢸
⠀⠀⠀⠄⣿⠀⠄⢠⣒⡒⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⣸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠿⢷⠀⠀⣿⡇⢹⣇⢨
⠀⠀⠀⢻⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⣟⡘⠇⢸
⢠⠀⠀⠈⣮⣇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣸⣿⠷⡏⠀⢸
⠀⡆⠀⠀⢿⠟⡄⠀⠀⣤⡲⡫⢵⠆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠒⠄⠀⠀⠀⠀⠀⠰⢿⣷⣦⢦⡄⠀⢠⣿⡿⠾⠋⢀⣾
⠀⠈⠀⠀⢹⣧⢹⣆⣀⠙⣫⠔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠉⠒⠙⠁⣠⡟⡟⢡⠋⢀⣾⠃
⠀⠀⠀⢀⠀⣻⣦⡛⡗⢀⠉⠀⠀⠀⠀⢀⣾⡃⠀⠀⣀⠀⣷⣤⠀⠠⠄⢀⢠⡀⠀⠀⠀⠀⠀⠀⣴⣏⠟⡴⠃⠀⣼⠋
⠀⠀⠀⠀⣦⠘⢿⣷⣌⢻⣷⣄⠂⠀⠠⡾⢉⠁⠀⠀⠉⠀⠉⠁⠀⠀⠀⡀⠀⠙⢆⠀⠀⢚⣤⣿⡷⣣⡾⠀⢀⡜⠁
⠀⠀⠀⠀⠈⠃⢈⠻⣿⣷⣍⡻⣷⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠈⠀⠀⠀⠀⣠⣼⣿⠿⣫⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢦⣼⣿⣿⣿⣧⣝⡻⠳⠲⠤⠀⣀⡀⠀⠀⠅⣀⣀⣀⣤⣴⣖⣭⠮⣋⣡⣾⠛⠀⣠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢻⣿⣿⣿⣿⣾⣯⣤⣄⣀⡀⠀⠽⠛⠛⠛⣛⣛⣋⣩⣶⣾⠿⠋⣁⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣭⡵⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠀⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣆⣦⡿⣦⠇⠘⡽⠻⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢤⠀⠀⠀⡀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠙⠛⢳⣭⡴⣃⡤⢄⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿⣧⡁⢤⡘⠓⠀⢀⡁⣿⢦⣴⠀⠀⠀⠀⠘⢦⡄⠀⢀⣿⣿⣟⡉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⣿⣿⣿⣿⣿⠹⠃⠀⠰⢸⡿⣏⢙⣎⣻⠃⠀⠀⠀⠙⢛⣫⣿⠟⡘⣟⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⠟⠸⠿⠃⡀⠀⠄⢀⠋⡁⢁⡶⠀⠙⣶⠊⠀⠀⠀⠺⣿⣿⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⢰⣤⡤⠀⠂⠶⣶⠆⢠⠀⠉⠄⢠⣴⣻⣠⠄⠀⠀⠀⠀⠉⢩⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣴⣀⣠⣀⣁⡀⢀⣩⣶⡅⠈⠀⠀⠀⠀⠀⠀⢀⡹⠀⠀⢀⢀⡠⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣿⣽⣿⣿⣗⣿⣿⡿⠛⣹⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⣤⣁⣤⣾⣾⣿⣵⣿⠀⠀⡘⢎⣥⣶⣷⠤
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⡀⠀⠿⡿⣿⣿⠘⣏⠻⢇⣠⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⢿⣿⣿⣿⣿⠀⠀⢰⣸⡿⣻⣷⣶⠃
⠀⠀⠀⠀⠀⠀⣼⣩⣿⣿⣿⣁⠀⠀⠉⠀⠀⠹⣄⠀⠈⠉⠙⠉⡇⠀⣠⣆⣤⣠⣿⣯⠶⠀⡟⠋⢁⣀⠀⠈⢿⣿⡿⠀⠀⣞⠩⢤⡉⠽⣯⣴
⠀⠀⠀⠀⠀⠀⢯⣿⣿⢿⣿⡟⣷⡹⠋⠀⠀⠀⠹⣦⠀⠀⠀⠀⡇⠀⢟⣿⣿⣿⡿⠋⢀⣾⣤⣾⣿⣿⣇⠀⠈⠋⠀⣤⠋⠁⠀⢰⣾⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠻⣽⢸⡽⠟⠋⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⡇⠀⠀⠙⠺⡿⣿⣀⣸⢿⣿⢏⣿⣿⣿⣥⠀⢰⣾⡁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⡇⠀⠀⠀⠀⠀⠙⢋⠗⠊⠀⡼⠻⠛⠋⠉⣰⣿⣛⡛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⢀⡏⠀⠀⣼⠃⠀⠲⢶⣾⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣇⠀⠀⠀⠀⡼⠀⠀⣼⠃⠀⠀⢠⠞⠈⠉⠈⠘⠟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣧⠀⠀⢠⠇⠀⣰⠃⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣆⠀⡻⠀⢀⡏⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⣶⡇⠀⡼⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢿⡄⢀⠃⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡸⣷⡜⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡇⣸⠃⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠃⣽⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢺⣵⡏⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⣧⠀⢻⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠹

⣤⣤⣤⣤⣤⣤⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⡇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⢀⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠙⠳⠦⠴⠞⠋⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣴⣾⣛⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣛⣉⣉⣉⣹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠛⠉⠉⠙⠻⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⡠⢤⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⡄⠀⠺⠁⠀⠈⢻⠀⣿⣿⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⢿⣷⣀⠐⠤⢤⡠⢀⣼⣿⠏⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠈⠻⢿⣷⣶⣶⣾⣿⠟⠋⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢻⣿⣿⣿⣷⣦⣄⣀⡉⢹⣁⣀⣤⣴⣾⣿⣿⡿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⢿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠿⣿⣿⣿⣇⣸⣿⠿⢿⣿⠟⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡠⠊⠉⠉⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⡄⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣴⡿⠃⠀⠀⢀⠶⠶⠶⢾⣿⡄⠀⠀⠀⠀⠑⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⠦⠤⠒⢚⡿⠀⠀⠠⠀⣿⣿⣦⡀⠀⠀⠀⠈⠂⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠉⠛⠿⢿⣦⣄⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠠⠂⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠦⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠇⠀⣰⡿⠛⠛⠻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣴⡟⠀⠀⡀⠀⠀⠀⠉⠛⠒⠂⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⡟⢀⣴⠿⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡴⠋⠀⠀⠀⠀⠈⠙⠳⣤⣀⠀⠀⠀⠀⠀⢀⣼⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡆⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣀⠀⠀⠀⠆⠸⡄⢻⣿⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢷⠍⣉⣉⣛⣻⣷⣤⣻⡇⠀⢀⠴⠖⠛⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠙⠻⣇⠀⠃⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⣤⣤⣌⣳⣦⣆⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣷⣾⡿⠟⠻⡟⠁⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠱⠀⠀⣠⡶⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⣀⠀⠀⣇⣰⣯⣔⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢮⡳⣦⣼⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠛⣧⠀⠀⣀⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣟⠟⡉⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣣⡾⠚⣀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠟⠉⣡⡴⢛⡽⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠁⡆⢰⢏⠜⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢻⣮⡳⣄⠀⠀⣿⣤⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠏⠹⣆⠀⠀⠀⠀⣾⡇⠀⠀⠀⣷⣄⠀⠀⠀⠀⢸⡇⢭⣿⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢂⠀⠀⣠⢿⢿⠀⠀⢠⡿⠏⠳⢦⡀⠀⢸⣠⣾⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠘⡆⠀⢩⠈⠘⡇⠀⠸⣣⠀⠂⠌⣷⠀⠈⣿⣿⣷⡿⠟⠛⠉⠛⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢠⣴⠃⠀⠈⠣⠀⣜⠀⠀⠉⠛⢶⣷⣣⡴⠚⠉⠀⠉⠉⠒⠤⠤⠤⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣾⠏⠀⠀⠀⠀⠀⣿⠀⠀⢀⡠⠼⢯⣥⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠲⢾⣷⠤⠤⠤⠤⠤⠤⣾⠖⠊⠁⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣦⡀⠀⠀⠀⢰⢏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡹⣧⠀⠀⣀⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣧⡹⠀⠘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣉⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡤⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢻⣷⠈⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⢺⡀⠀⠀⣿⠟⠁⠘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⢸⡇⠀⣀⠀⠀⡀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣀⠀⠀⠠⣿⣿⣿⡿⠀⠸⢇⣀⣿⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⢿⣿⣿⣧⢰⡿⠞⠛⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣶⠀⢸⣿⣿⣇⠘⠁⣠⠖⠈⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⣿⡤⠀⠻⣿⣿⣶⣶⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣴⣿⣿⣿⣧⣿⣸⣤⣾⠿⠋⠉⠁⣾⠿⠿⠃⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣼⡟⠃⣿⠇⠐⠟⠃⠀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠈⢻⣿⣷⣾⣄⢁⡀⠲⠀⠿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣇⠀⣱⡟⠃⠉⠉⢙⣤⠜⢺⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡿⠷⠾⣷⣆⣹⣧⣼⣿⣿⣿⡿⠿⠟⠛⠃
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠘⠟⠙⢣⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⢸⡇⢠⣾⡅
⠀⠳⣄⡀⢸⣿⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⢸⣿⣾⠟⠁⢀⡄⠀⣠⡆
⠀⠀⣿⡇⠘⣿⡁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣀⣽⣧⡴⠞⠋⢀⣶⡟
⠀⠀⠉⢁⡴⠈⠉⠀⠀⠁⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠻⣿⣿⣿⣿⡟⠉⠁⣠⠶⣠⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠑⣄⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣆⠈⠻⣿⣿⣷⣮⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⠃⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⣿⡇⠀⣹⠿⠟⠋⠉⠀⠀⡀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠘⠃⠀⠀⣶⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡜⢷⢀⣿⠀⠀⢠⡤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⢁⣴⡗⠉⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡀⠉⠀⠀⠀⠠⣤⡀⠀⢸⣿⣿⠉⢿⣿⣯⡘⠻⣧⣴⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⡿⣨⢻⣶⣌⣿⠃⠀⠀⠹⠁⣠⡾⢛⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣿⣿⣠⣆⠀⠀⠿⠄⢹⣿⡏⠀⠆⠀⠀⣴⣟⣵⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣆⡀⠀⣤⡀⢻⣿⠀⢺⡆⠀⣼⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠻⣿⠉⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⡟⠉⠉⠉⠉⠀⠹⣿⡾⣿⣇⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⢰⣦⣤⣄⠀⠈⠁⢸⣿⣤⣿⠇⠀⠀⢀⣤⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⣰⠁⣀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⢻⣿⣿⣷⣤⡀⠈⠻⣿⠇⢀⡀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡇⢰⡿⠻⣿⣿⡿⢡⠀⣿⢠⡜⢣⣾⠏⠀⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠺⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠜⠁⠀⠘⢿⠇⢸⡄⠸⣿⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⠈⠁⠀⣴⠁⠈⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄⠀⠀⢠⠆⠀⢀⣤⡀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡌⠙⢧⡀⢰⣈⣷⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⣾⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣶⣼⣿⠊⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠟⠀⠀⠀⠺⣿⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣏⠈⣿⣿⣇⣀⣤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡀⠈⣤⠀⢀⣴⡀⣿⣧⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣿⡀⢃⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠛⢷⣄⡹⡀⢸⣿⠇⢻⣿⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠘⡷⠗⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⣤⠀⠀⠀⣉⠛⣿⠈⠻⠀⢈⡟⠀⠀⠀⠀⠿⠃⢠⠀⣀⠀⠀⠾⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣤⣘⠿⣤⡄⠀⠀⠸⠟⠀⠀⠀⠀⠀⡂⠁⢠⣿⡆⠘⣀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠿⣿⣷⣽⣿⣴⡀⢰⡆⠀⠀⠀⢠⣾⣿⣿⠀⢻⣿⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠈⠻⣿⣿⣿⠀⢸⡇⢠⡆⢠⡿⠋⠈⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠈⠻⣿⣤⣤⡆⢀⣶⠏⠀⣶⣾⣿⡀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠈⠙⣿⠁⠈⣡⡄⠘⣿⡇⢀⠁⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠸⡀⢀⣿⡇⢸⣿⣇⡈⣡⡭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⢰⣧⠘⣿⡇⠈⢻⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣄⠀⢸⣿⠀⠸⠇⠀⠏⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⣿⣿⣿⢈⣿⠀⠀⠀⠀⢸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⢻⣿⣇⠈⠸⠿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⢹⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣼⣿⡀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠛⢿⣶⡾⠟⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢣⠀⠈⣣⣄⡒⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡟⠁⠀⠀⠉⠋⠀⠀⠘⠲⠞⠙⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡌⠉⠉⠁⣉⣉⣁⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣉⣉⣈⠉⠉⠉⠁⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣶⣶⣾⣿⣿⣿⣿⣧⡀⢀⣠⣶⣦⡀⣾⣿⣿⠆⣿⣿⣿⣿⣿⣿⣶⣶⣶⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣁⣼⠿⠉⠉⣩⣿⣿⡎⠻⠿⠟⢡⠝⠛⢉⣠⣿⠏⠀⠉⠀⠀⠀⣽⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡟⠃⠀⣠⣾⠟⠁⢻⣷⣶⣶⣤⣀⣀⣴⡿⠋⠉⠀⢀⣀⠀⠀⠀⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡆⣠⣾⢿⣿⡿⣷⣿⣯⣄⣀⣿⡟⢛⣉⣤⣴⣶⠿⣿⡟⠀⠀⠀⣿⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⠟⠁⢈⣿⡇⠀⠈⠉⠛⠻⠿⡿⠿⠛⠋⠁⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠘⣿⡇⢀⣴⡿⢿⡇⠀⣿⣶⣶⣤⡀⠀⣤⣴⣶⣿⠆⢀⣿⠃⠀⠀⠀⣿⡇⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣿⣷⡿⠋⠀⢹⣷⠀⢻⡇⠉⣿⡇⠀⣿⠋⢨⣿⠀⢸⣿⠀⠀⠀⢸⣿⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢹⣿⠀⢀⣴⢿⣿⠀⢸⡇⠀⣿⣧⣤⣿⠀⢸⣿⠀⣸⡟⠀⠀⠀⣼⡿⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⣇⠈⣿⣷⠟⠁⠀⣿⡀⢸⣷⠀⠈⠉⠉⠉⠀⢸⣿⠀⣿⡇⠀⠀⢠⣿⡇⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠘⣿⡄⠹⣿⣄⣠⡾⣿⣇⣸⣿⠀⢸⣿⣿⣿⠀⢸⣇⣠⠿⠧⢖⣒⣛⣭⣭⣭⣍⡓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⣝⣒⣒⣻⣿⡀⠹⣿⣟⠁⢙⣿⣿⣿⠀⢸⣇⡷⠿⣒⣩⣥⡶⣾⢿⢿⡟⠩⣿⠙⠋⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠻⣶⠤⠽⠎⣙⣿⠿⣿⣿⡄⠙⣿⣶⣿⣫⡴⠿⣚⣯⣵⡶⢿⡟⠉⣿⡇⣉⢸⡼⡇⠷⢸⣁⣇⣿⣷⣽
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣟⣛⡇⣘⣛⣛⣛⣒⣯⣭⣽⣶⡾⣿⢻⣏⢱⡆⣽⡇⣆⣻⣧⣿⣼⠷⢿⣛⣛⣛⣽⣛⡛⢸⣓⣲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⣏⢙⢋⢹⡏⣭⢹⣄⡄⣿⠌⣿⣸⣧⣽⠷⢟⣛⣭⠷⣶⡟⢻⣛⣛⣛⣿⣿⣥⣾⣋⢩⡵⢬⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣷⢸⢸⠘⣇⠛⣸⣯⣷⣿⠾⣟⣫⣭⣶⠾⠋⢁⣴⣾⠿⣧⡻⠿⠿⣿⣛⣛⣛⣛⣛⡳⠻⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣜⠻⠿⠿⣛⣛⣛⣫⠽⠷⣶⣿⣭⣈⢉⣠⣴⣾⠿⠛⠁⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠙⠻⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠊⡩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢱⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠑⠣⣀⠤⡜⠀⠘⡄⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⡖⠒⠂⠤⣀⠀⠀⠸⠀⠀⠐⡃⠀⣸⠀⠀⡇⢀⡠⠔⢊⡉⠉⠹
⠀⠀⠀⠇⠀⠀⢱⡀⠉⠢⣄⠳⣠⠴⠋⠉⠉⠫⢨⣔⠉⠀⠠⢋⠀⠀⢸⡇
⠀⠀⢸⠀⠀⠀⠀⢱⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢯⡀⠀⠀⠘⠇
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠽⣆⣠⠃⡆
⠀⠀⠸⡎⡄⠀⠀⡔⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠣⡸⢀⡀
⠀⠀⠀⠳⡈⣢⠎⠀⠀⠀⠐⠾⢺⣶⣶⡀⠀⠀⠀⠀⠀⣰⣿⡍⡏⠀⠙⠅⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀
⠀⠀⢀⡠⠛⠁⠀⠀⠀⠀⠀⠠⡹⠿⠟⢡⠀⠀⠀⠀⠀⠟⠛⡡⠁⠀⠀⠀⠀⢛⡦⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡉⠉⠉⠛⠓⢦⡀
⢲⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⢀⡄⠀⡄⠶⠾⢦⠄⠒⠈⠁⠀⢠⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠙⡄
⠀⠑⢄⡀⠀⠀⠀⠀⠀⡀⠤⠄⠒⠒⠒⠒⢖⡂⠀⠑⢤⠈⡸⠡⠤⢀⣀⣠⠚⢉⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢹
⠀⠀⠀⠿⣂⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠁⠀⠁⠒⠂⠉⠉⠀⠀⠀⡠⠊⠁⠁⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⡏⠒⠤⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡴⠊⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠰⡀⠀⠈⠻⡖⠦⢄⣀⣀⣀⣀⣀⣀⣤⡶⡿⠛⠣⣄⣀⣀⠴⠁⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠙⢄⠀⡐⠙⠦⣈⠀⠀⠀⠀⣀⡠⠽⠊⠀⠀⠀⠈⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠻⡆⠀⠀⠀⠉⠉⠉⠉⢭⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡒⠒⠐⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠢⣄⡸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡼⠁⠀⠀⠀⠀⠀⠀⢸⠉⠉⠀⠉⠁⠀⠈⠉⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡤⠟⠤⣤⠄⢀⣠⡤⠤⠀⠚⠉⡶⢄⡀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⣠⠔⠛⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃⠀⠙⠦⡀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⢸⡀⠀⡴⠒⠐⠃⠀⠀⠀⡰⠃⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡕⠀⡀⠀⠀⢀⡤⢺⢁⠀⡎⠀⠀⣸⠁⠐⡀⠁⠀⠀⠀⣠⠞⠀⠀⠀⠀⢀⠖⡡⠀⡀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢠⣄⣠⠴⠋⠀⠀⠙⠂⠓⠶⠚⠁⠀⠀⠙⠓⠒⠒⠉⠀⠀⠀⠀⠀⠀⠈⢦⣇⠀⡇⢀⡠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢈⠃⠀⠰
⠀⠀⠀⠀⠀⢠⠀⣠⣴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⣿⠛⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠄⢺⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡴⠊⠁
⠀⠀⠀⠀⣰⠏⠀⠆⠀⠀⠀⠀⠀⠀⠀⣠⠾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣤⡿⠏
⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠨⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⢡⣴⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠰⣀⣿⠿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣦⣿⣯⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⢷⣦⣄⡁
⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣧
⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠘⠓⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡤⣶⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⣶⣪⠃⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡉⢁⣀⣀⣼⠏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⠭⣅⢀⣴⠾⣧⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡜⢡⣴⡄⠛⢸⡟⢀⡾⠀⠀⢩⢿⣿⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡷⢾⡟⠀⠀⣠⠝⠏⠓⠲⣾⡗⠚⠀⢹⣦⡀⣠⠖⢉⣠⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⠛⣿⣟⣷⣌⠁⣠⡴⡆⠀⠀⠙⣦⣶⣻⠟⣷⡏⠀⠃⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠑⢾⣿⢱⣿⣏⣹⠋⠶⠽⠞⠁⠀⠀⠀⠉⢧⠀⣀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡈⣋⣛⣣⣼⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠿⢿⡏⠃⡀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠻⣟⢋⡟⢠⡮⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⡿⡀⠈⠳⢦⣤⣀⣀⣤⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠽⠃⢸⣸⠇⠀⠁⠀⠀⠀⠀⠀⠀⢠⡶⢀⣻⢯⣤⠿⢇⠁⠀⠀⢀⠀⡠⢻⣦⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠳⠦⡀⠀⠀⠀⠀⠀⠀⠈⢙⠟⣫⣻⣶⡖⣠⣶⡿⠿⠀⠐⠣⢾⡛⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣷⣦⡈⠀⠀⠀⠀⠀⠀⠀⢸⡀⠛⢛⣁⡉⠟⣇⠤⠤⠶⠖⠚⠛⢉⣩⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠋⠑⠶⠉⠙⠒⠿⣿⡄⠀⢀⠀⠀⠀⠀⠀⠙⠻⢯⣹⠿⠃⠛⠛⠀⠀⠀⠀⠀⠻⠿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⢀⡄⠀⠇⠀⠀⠀⠀⠘⢧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠿⠞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠘⠞⠀⠀⣶⣆⡴⠾⠗⠉⠈⠀⡅⠀⠀⠀⠈⠣⡀⠀⠀⠀⣠⣤⠀⢱⣤⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⢴⠟⢿⠀⠀⠀⢠⡄⢠⠃⢀⠃⠀⢴⠶⢷⠛⠀⠈⠉⠉⢿⡟⣃⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠸⣷⠀⠀⠈⣡⠃⠀⠀⠀⠀⠀⠂⢈⢣⠀⠀⠸⢻⣧⣿⣙⣛⡤⠀⠀⠀⠀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⣧⡀⠻⠀⢠⠀⠀⡀⠈⠀⣣⠞⠁⢀⣼⡀⠠⡀⠀⡷⢸⠠⢀⣠⠞⠋⢀⡟⢉⣽⣋⠳⡆⠀⠘⠿⢇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠈⠳⣄⣀⠀⠀⢀⣣⣴⡾⡁⢀⣴⣿⣿⣿⣶⡄⣷⣾⡏⠸⣇⣶⠄⢤⣶⣿⣟⣻⣿⣿⣥⠤⡴⠚⢉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⣀⠈⢩⣛⡋⠯⢿⣽⣡⣾⣿⣿⣿⣿⣿⣬⠧⣿⡿⡱⠁⠘⠿⣟⠉⡺⢿⣤⣀⣄⣄⡀⣾⢺⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠁⣰⠿⡷⣹⠿⣧⣼⠟⠋⠉⠉⠀⠉⣈⠁⣌⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠓⠊⠁⠀⢸⣏⠀⠀⠀⠀⣴⢋⣀⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣷⣌⣋⡽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣈⠓⣠⠏⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⣶⣴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣝⣧⡶⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠠⡙⢦⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠉⠑⢤⡀⠀⢰⠁⠀⠀⠀⠀⠉⠒⠂⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑⣄⢸⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⣠⠃⠀⠀⠀⠀⠀⠘⣾⠀⠀⣰⠃⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠊⠁⠀⠀⠉⠒⠤⣀⠀⠘⡇⢠⠃⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠴⠚⠛⠓⠒⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⠤⠤⢤⡴⠶⠚⡀⠀⠀⠀⠒⠢⢤⣀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⢀⡔⠁⠀⠸⡄⠀⠀⠀⠈⠙⠢⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢠⠎⠀⠀⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⡌⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⣧⠘⣆⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⠠⢄⡀⠀⠀⠀⢠⠟⡆⢹⠑⠦⣄⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣆⠑⠀⠀⢀⠞⠀⢱⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⢰⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃

⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠚⠢⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⡤⠀⠀⣀⠄⠀⢠⡀⠀⢤⣠⣄⣀⡀⡀
⠀⠀⠀⢀⡠⠤⠒⠒⠴⡂⠊⠀⠀⠀⢀⠐⢒⡠⠐⠃⠓⠠⠐⠐⠂⠐⠃⠀⣸⣟⠓⠀⠀⠉⠓⠖⠈⡆⢀⠄⠐⠂⢀
⠀⠒⡀⠈⠀⠀⠀⠀⣀⠙⢦⡊⠀⢀⡸⠖⢉⣦⠀⠀⣤⠄⠐⢀⡘⠀⡔⠋⢔⡘⢠⣔⠓⠀⠀⡆⠀⢣⠀⠀⠀⠂⠉
⠘⠆⣹⠀⠨⠭⠠⠰⢀⠀⠀⠈⠣⡈⢢⡀⠈⠛⢁⡸⠏⠀⠀⠀⢈⠗⠁⠀⢀⠙⠉⠀⠀⠀⠀⠈⢲⠁
⠀⣠⠁⠀⠀⠀⠁⠀⠁⠀⡠⠂⠁⠀⠀⠙⠢⢄⡀⠀⠀⣀⡄⠐⠩⢂⠀⡈⢢⣀⠀⠀⣀⢀⣀⡰⠊⠉⠂⠔
⠀⠈⡒⠒⠀⠄⢀⣀⢰⣛⠀⠀⠀⠀⠀⠀⠐⡏⠀⠀⠈⠁⠘⡀⠀⠀⡲⠀⠀⠨⡈⣅⠀⠀⠘⠀⠀⠀⠀⠀⢢
⠀⢰⠀⠀⠀⡎⠑⠁⠁⠈⢸⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠌⠀⠀⢠⠇⠐⠢⡄⠀⠈⠃⢄⠠⠀⠀⠀⠀⡀⡘
⠀⠀⡆⠰⠀⠛⠀⠀⢀⠤⠘⠀⠀⠀⠀⠀⠀⠀⢨⠐⠤⠀⠀⢀⠆⠘⠀⠀⠀⠓⡀⠀⠀⠰⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠡⠀⠀⠶⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⢢⠆⠀⠀⣀⡘⠀⠈⣢⠀⠀⠀⠘⡉⠂⡆⠀⠀⡠⢋⠀⠈
⠀⠀⢰⣠⠒⠓⢀⠜⠀⠉⡳⠀⠀⠀⠀⠀⠀⠀⠓⠂⠐⠀⠠⠸⠀⠉⠁⠀⠀⢰⡀⠀⠀⡇⠀⠒⠀⠈⠂⠠⠘
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⢉⣉⡱⡖⢀⡀⣶⣀⢀⡀⣴⡾⣶⣀⣀⣰⣆⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠦⠶
⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⣿⢿⣸⡟⠛⣏⣻⣯⣿⠁⠟⣻⣿⣀⣁⣽⢿⣿⣧⣜⣿⣍⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿
⢠⣡⣤⢠⣌⣡⠀⠀⠀⠀⠈⠁⠈⠉⠁⠈⠉⠉⠉⠙⠷⠟⠉⠀⠉⠁⠉⠉⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣛⣻
⠈⠿⠁⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣻⣻
⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⢀⠀⣀⣀⣀⠀⢀⠀⡀⠀⣀⣀⣀⡀⠀⡀⣀⡀⢀⣀⣀⣀⣀⠀⢀⠀⣀⣀⣀⣀⣈⣉⣉
⣿⣿⣿⣿⣿⣿⣿⡂⢀⣀⣄⢂⣿⣿⣿⣀⡄⠐⡀⢤⣿⣿⣿⡇⣠⢀⡀⡀⢈⣿⣿⣟⢃⢀⣄⠀⣸⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣀⣀⣍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠁⠀⠀⢈⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⠤⣖⣲⡭⢿⣛⣋⣁⣀⣠⡶⠦⣤⠀⠈⢙⣁⣀⣀⣀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⣰⠀⢸⣶⡭⠷⢚⣻⠿⠟⠛⠉⠉⠉⠉⠉⠉⠛⠓⠿⣄⡀⠸⣏⢆⠀⠉⠳⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡴⠞⠉⢀⡠⠤⣒⣺⣭⡭⢭⣭⣙⣒⠤⣄⠀⠙⠳⣜⢯⣣⡀⠀⠈⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠋⠀⡠⣖⡽⠞⡋⠁⠀⠀⡆⠀⢀⡆⢭⡛⢦⣑⢄⠀⠈⢿⣷⡵⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡠⣪⠞⡉⠀⠀⠛⠀⠀⠀⠀⠀⣨⡕⠾⠃⠀⠙⢦⡳⡀⠀⢻⡽⣞⢆⠀⣼⠰⡆
⠀⠀⠀⠀⠀⠀⢠⡏⠀⡜⡴⢃⣺⣿⠂⠀⠀⠀⠀⠀⠀⣼⢫⠇⠀⠀⠀⢠⠌⢳⡙⡄⠀⢷⠘⢮⡾⠃
⠀⠀⠀⠀⠀⢀⡟⠀⡜⡼⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⣰⣷⠏⠀⠀⠀⠀⠀⠀⠀⢷⢱⠀⠸⡇
⠀⠀⠀⠀⠀⣸⠃⢰⣿⡇⠀⠀⠀⠀⠀⠀⣠⠦⢤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⡇
⠀⠀⠀⠀⠀⢹⠀⠘⣿⡇⠚⠁⠀⠀⠀⠘⢧⡴⠞⠛⠻⡀⠀⠀⠀⠀⠀⠀⠘⠃⣸⢣⠃⢀⡇⢰
⠀⠀⠀⠀⠀⠘⣇⠀⢳⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢠⡏⡜⠀⣼⠁⠏
⠀⠀⠀⠀⠘⠄⢹⡄⠀⢿⣧⠀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣀⡾⢣⣞⡜⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⠻⣷⣄⠀⠀⣴⣶⠆⠀⠀⠀⠀⢀⠀⠀⠀⢉⡵⣫⠎⠀⣰⠃
⠀⠀⠀⠀⠀⠀⣠⠾⣝⣦⡀⠈⠻⣷⣤⣛⡛⠀⠀⠋⠀⠀⢈⣁⡤⢖⡫⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠻⠴⠞⠉⠙⢦⣄⠈⠙⠻⠭⠭⠽⣛⣛⣛⠭⠕⠚⠉⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣀⣀⣀⣀⣀⣤⣤⣶⡟⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣄⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠈⠹⣿⠟⠁⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠛⠿⢷⣦⣄⡈⢻⡄⠀⠀⠀⠀⠀⠀⣴⠏⢀⣤⡶⠿⠟⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠈⠛⠶⣽⣄⠀⠀⠀⢀⣼⡧⠞⠋⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⡠⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠁⠀⠀⠹⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣀⣀⣠⣴⠶⣻⣿⠏⠀⠀⠀⠀⠀⠘⣧⡙⠳⣦⣤⣀⣀⣀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⢁⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢾⡇⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠘⢿⣦⣤⣴⡾⠋⠻⣷⣤⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸



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

⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠀⣀⡀
⠀⠀⡇⠙⠒⠋⢸⠓⠒⠒⡟⠦⡖⢲⠒⢒⡖⠒⢲⠒⠒⡖⠒⠒⢻⢀⠔⠺⣿⢿⣿⣿⢼⠈⣿⠁⡗⢲⠒⠒⣲⠒⣶⠉⡇
⠀⠀⡇⢸⠀⡇⢸⠀⣿⠀⡇⢠⡀⢸⡆⢹⣿⠀⣿⠀⣉⡇⠈⠁⡞⣇⠀⠀⠀⠀⠉⢹⢾⠀⣉⠀⡇⢸⠀⡉⠙⠀⣉⠀⡇
⠀⠀⡇⢸⠀⡇⢸⠀⠿⠀⡇⢸⡇⢸⠷⠈⣿⠀⣿⠀⠿⡇⢸⡀⢧⠘⡖⠿⢗⠿⢗⡏⢸⠀⣿⠀⡇⢸⠀⠇⠠⠀⣿⠀⡇
⠀⠀⠓⠚⠒⠓⠚⠓⠶⠚⠓⠚⠓⠚⠒⠊⠙⠒⠋⠓⠒⠓⠚⠓⠚⠀⠈⠑⠒⠚⠉⠀⠘⠒⠛⠒⠓⠚⠒⠖⠲⠖⠛⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠈⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠿⠿⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠿⠛⣿⡇⠀⠀⠀⠀⠘⠻⣷⣶⣾⠋⠉⠉⠁⠈⠙⣿⠋⠉⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⢀⣿⣤⣤⣄⡀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣄⣀⣀⣀⣼⠟⠋⠉⠉⠁⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⣤⣴⣾⠿⠟⠛⠉⠛⠛⠻⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⢠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁
⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⢀⣠⣾⠟⢿⣿⡆
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁⠀⢸⣿⣧
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⣤⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⡤⠀⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⢤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣡⡄⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠟⠋⠀⣠⣤⠈⠛⠿⢿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⡎⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣮⢲⣋⡀
⠀⠀⠀⢀⠄⠀⠂⠀⢠⠬⠀⠀⠀⠈⢉⣁⣠⣄⠀⠀⡰⠟⠉⠀⠁⠀⠉⠻⢦⡀
⠀⠀⠀⠀⢀⠀⠐⠾⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⠈⢀⠈⠁⠠⠄⡀
⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠰⡀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⢈⣡⠀⠀⣀⣈⢣⡀
⠀⢅⡀⠀⠀⠀⣦⠀⠠⠀⠂⠀⠀⠈⠀⠀⠉⠢⠄⠀⠀⠀⠠⡌⢀⠀⠀⠈⠓⠉⣴⠞⠋
⠀⠀⠁⠲⠦⠂⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠀⠀⠀⠫⠢⡴⠿⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠰⠖⡠⠀⠐⠀⠈⠁⠀⠀⣠⡀⠀⢀⠀⣀⡴⠊⠈⠒⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠠⡄⣠⠔⠚⠉⠙⠁⡍⠁⠀⠁⠢⡀⠀⢀⡰⠀⢤⣄⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡈⠶⠀⠀⠀⠈⠣⣀⡀⠀⠀⠀⠘⠉⠀⠀⠀⠷⢄⡘⣧⠈⠆⠀⠔⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⢄⠀⠀⠄⠀⣧⠀⠀⠀⣤⡀⣀⠀⢂⠀⢩⠜⠢⠌⢠⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠢⠀⠠⠔⠋⠀⠀⠀⡄⠈⠠⡉⠒⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⠉⠈⠙⠓⢤⡀⠀⠀⠀⠀⠤⡄⠀⠀⠀⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣙⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢧⣔⠤⠐⠂⠉⠉⠉⠩⠀⠀⠘⣀⠀⢀⠃⠀⠀⠀⠀⠀⠲⠂⠀⠀⣆⡽⠀⢁⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠠⡔⡋⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⢈⡶⠃⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠈⠘⣧⣼⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⣟⣁⠀⠀⠀⠀⠠⠀⠈⠠⠀⠹⢦⣴⠂⡬⠁⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠘⠛⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⣭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠙⣯⣚⣓⠒⣺⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠄⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢉⡼⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠂⠀⣴⠀⠀⠀⠀⠀⠀⠼⠋⠁⠀⠀⢆⠀⢀⡠⢄⡀
⠀⠠⣀⠀⠠⠤⠐⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠚⢦⢄⠀⠀⠀⠀⠀⠀⣀⠤⡶⠂⠘⢦⡀⠀⠀⠀⠀⠘⠖⠠⣀⡀⠀⠀⠁⠀⡈⠁
⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠶⠖⠶⠾⠣⢶⣤⠁⠀⠀⠀⣤⣄⠀⠀⠀⠈⠒⣤⣄⠁⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠐⢦⡀⠀⠀⠉⠑⠠⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠈⣱⢀⡠⠔⠒⠙⠀⠒⠶⣄⠀⢂⣠⠎⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠠⡵⣆⠀⠀⠀⠀⠀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⢀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⠁⠀⠀⠀⠀⠀⠀⠀⠤⢦⡀⠀⡠⣆⠈
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⣣⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠃⠀⠷⣍⢲⡀
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢇⡀⠀⠀⠀⡀⠀⡀⠀⢶⣯⠀⠀⠟⣼⠃
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠈⢳⣄⠀⠀⠘⠎⠀⠀⠉⠉⣀⠄⠈
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⡀⢁⠀⠀⠀⣏⠶⠛⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⠀⠀⠠⣿⠀⠀⠀⠀⣠⣴⣥⡴⠆⠀⡰⢧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⣃⠀⠀⠀⣤⠁⠉⡹⠧⢀⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠁⠈⣦⠰⠂⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀
⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⡒⡼⠦⢴⡖⠉⠉⠀⠀⠀⠀⠀⢀⠴⠖⠋⠀⠀⠀⠈⣉⣉⣐⣄⡀
⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⢨⡷⠄⢾⣀⠀⢨⠀⠀⠀⠀⠔⠁⠀⠀⣀⠀⠀⠀⣩⡋⠁⠀⠀⠀⠀⠀⠀⣀⡀
⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠉⠉⠉⢌⠁⠀⣠⠤⠒⠉⠁⠀⠀⠈⠉⠉⠁⣀⠠⠔⠚⠃⢉⠉⠉⠈⡆⠀⢈⠝⠁⠀⠀⠀⠀⠀⠀⠀⡅
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⣤⢷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⢀⠔⡁⠀⠀⠀⣿⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⣠⠴⠋⠀⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠐⡟⠀⠀⠀⡠⣻⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠒⠉⠀⠀⣀⠀⢠⡹⠿⠷⡀⠀⠀⠀⠀⡴⠿⠟⢘⠃⠀⠀⠀⠈⠑⠦⣁⡴⠃⠀⡅⠀⠀⠀⠀⠀⠀⣀⣶⣶⣾⣿⣿
⣤⠄⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡟⠫⡭⠓⢂⣄⠐⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠾⢧⡤⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡤⡀⠀⠀⠁⠀⠀⠘⠁⠤⣤⣀⢀⡽⠳⠠⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠛⠁⠀⡼⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠉⢻⡗⠶⠤⢤⣄⣀⣀⡠⠤⢶⡶⠿⣯⢤⣀⣀⠀⠀⡰⠃⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⣶⣾⠋⠈⠓⠤⠤⠤⠤⠴⠖⠚⠁⠀⠀⠈⢷⡀⠈⠉⠉⠁⠀⠀⠀⠈⣆
⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠁⠀⠀⠀⠀⠀⠀⠀⣼
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢠⢠⠇
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⣤⡀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡞
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠉⠣⢄⡀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⣰⢃⡤⠊⢸⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠸⡆⠀⠀⠉⠓⠺⠀⠀⠀⠀⠀⠀⠀⣰⠛⠉⠀⠀⢼
⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣇⠀⠀⠀⠀⠘⠦⣀
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡃⠒⠇⠀⠀⠀⢠⠀⠀⠀⠀⠀⣰⠃⠈⢆⠀⠀⠀⠀⢄⠐⣕
⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡔⠀⠀⠀⠀⠀⢸⠀⠀⠐⠒⠛⠛⠀⢄⠀⠑⠦⢄⣀⣼⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠼⣀⠀⣀⠠⠶⣀⠀⠀⠀⠈⠀⢈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠊⠓⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠜⠁⣠⠤⡔⠀⢨⠀⡤⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠁⠂⠀⠻⣦⠊⠁⢸⣶⣶⠿⢦⣃⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡰⠀⠀⠈⡄⠀⠊⠉⠀⠀⣿⣧⠴⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⣷⢤⣀⡟⠀⠀⠀⠀⠐⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢛⡿⠻⣾⣥⡤⠄⣀⠀⠴⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢩⡂⠀⠢⣠⠀⠀⢹⢀⣠⠄⠀⡄⠀⠀⢠⠋⠀⠀⠀⠉⠛⠶⠦⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⣠⣴⣴⡿⢶⣤⡶⠃⠙⠀⡰⠀⠀⡄⠁⠀⣀⡀⠠⠒⡄
⠀⠀⠀⠀⠐⠈⠀⠘⠂⠉⠑⠾⠈⠻⡇⠀⠿⠁⠀⣴⠞⠁⠀⡾⠀⠀⡔⢸⡀⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠀⠀⣃⢄⡇⠀⣀⣠⠞⠉⠀⠀⢺⠁⠀⢨⡇⣼⡁⠁⢀⡇
⠀⠀⠀⢀⡤⠶⠒⠛⠢⠄⠀⣄⣨⡽⠗⢺⡗⠒⠀⠀⠀⠘⠁⠀⠀⠀⣿⣿⣧⣴⡇⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠤⠤⠢⠖⠁⠀⠀⠈⢻⡄⠀⠀⠀⡈⠀⡠⠈⢀⡿⠁⠀⠁⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢠⠁⠀⠀⢠⠟⠐⢆⣣⣤⣅⡐⠃⠀⢀⡁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣼⠠⠀⢀⡎⠀⠀⠼⢮⢙⣟⠟⢋⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠱⣜⡀⠼⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠉⠐⠦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠱⣼⠉⠫⣄⣀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣿⠀⠀⢠⡿⠉⠓⠒⠤⠤⢤⣀⢀⣀⠀⣀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠉⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢿⡷⠊⠑⠢⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀⠔⠊⠁⢾⡿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣠⣀⠀⣁⠤⠓⠂⠉⠁⠀⠀⠈⠱⣤⠎⢠⣈⠀⢀⣄⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢪⣎⠔⠋⠁⠄⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠈⠑⢶⡕⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢊⠝⠁⠀⠄⠀⠀⠀⠀⠀⠀⡰⠋⡴⠁⣇⠀⠀⠀⠈⠁⠀⠉⠑⠢⡀
⠀⠀⠀⠀⡦⠄⠴⠃⠀⠎⠀⠀⠀⢀⣀⠤⠀⠐⠒⠚⣁⢾⡡⠊⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠘⠦⠠⢴
⠀⠀⠀⠀⠑⠄⢠⠆⢨⠀⠀⢤⣔⣑⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⣁⣠⡤⠀⠀⠀⠰⡄⠄⠊
⠀⠀⡕⠤⠤⠒⠁⠀⠈⡄⠀⠀⠈⢛⠀⠀⢰⣶⣄⠀⠀⠀⠀⢠⣶⣶⠀⠀⡟⠁⠀⠀⠀⠀⠀⠈⠒⠤⠤⢊
⠀⠀⠘⠢⡤⠀⠀⢀⡄⠠⠀⠀⢚⡿⢃⡀⠲⡿⠏⠄⠀⠀⠠⠠⣿⡿⢀⡘⠛⠓⠂⠀⠀⡀⠀⠀⠀⢤⠔⠃
⠀⠀⠀⠀⡇⠀⠄⠉⠂⠂⠀⣠⠋⠀⠋⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠂⠀⠀⠀⠀⠀⠀⢰⠀⢸
⠀⠀⠀⠀⢇⠀⠀⠑⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁
⠀⠀⠀⠀⠀⢀⡄⠉⠐⠢⢄⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⡠⠖⠂⠉⢠⡀
⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠸⠈⠢⣀⣀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⣀⣀⠔⠁⠆
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢲⢦⣤⣤⡴⡖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⠖⠉⢀⣀⣀⠠⠔⠉⠛⠚⠷⠾⠓⠛⠉⠢⠄⣀⣀⡀⠉⠲⣆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⠄

⠀⠀⠀⠀⣀⡴⠖⠛⠋⠉⠙⠛⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢀⡾⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⢻⠛⠛⠛⡟⠋⠉⠻⠛⠉⠙⣿⡇
⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⡇⢹⠀⢸⠈⣹⠂⡧⡎⢹⠤⢰⠉⡧⣽⡇
⠀⣸⠃⠀⠀⠀⠀⠀⢀⣬⡀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣿⡇⠈⠀⣸⠘⠓⢀⣇⠙⠊⢀⡈⠓⠁⣾⡇
⠀⢹⡄⠀⠀⠀⠀⠀⡸⠛⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⣧⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣼⠟⠡⠀⠐⠀⠠⠉⠻⣆⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣾⠁⠂⠀⠀⠀⡔⠀⠀⠂⠘⣇
⠀⠀⠀⠀⠙⠳⢦⣤⣄⣀⣠⣤⡴⠞⠋⠀⠀⠀⠀⣿⠐⠀⠀⠀⠚⠁⠀⠀⠐⠂⣿⠇⠀⠀⢀⡴⠚⠋⠉⠉⠉⠉⠛⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢠⡀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀
⠀⠀⠀⠀⠀⠀⡴⠚⠛⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣌⡀⠀⠀⣈⣠⣾⠋⠀⠀⣨⣿⣿⣿⣷⣦⠀⢤⡤⠀⣠⣶⣶⣿⣾⡃
⠀⠀⠀⠀⠀⠀⠳⢴⣿⣿⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⢟⣩⣤⣼⣧⣤⣝⡻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣠⣤⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠟⠋⣡⣿⠟⡋⠉⠉⠉⠉⢙⠿⣿⣍⠻⢿⠇
⠀⠀⠀⢠⣾⠟⠋⠁⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠇⠀⠀⠀⠀⢀⣸⡛⣅⠀⠉⠀⠀⢀⡴⠋⠀⣘⣿⣧
⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⡾⠁⠀⠈⠁⠘⣿⡇
⠀⠀⣿⠇⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠉⠁⠐⠒⠒⠚⠁⠀⠀⠀⠈⠁⣿⡇
⠀⠀⢿⡇⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⣸⡏⠀⣴⣶⣶⢰⡶⣶⡄⣶⢶⣦⣴⡶⣶⠀⠀⠘⣿⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⡿
⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣿⣾⣿⢸⣿⣼⡇⡗⣿⣿⣟⣿⢺⠀⠀⠀⠘⢿⣦⣀⠂⠀⢠⠀⠀⢣⣠⣾⠟⠁
⠀⠀⠀⠈⠛⢷⣦⣄⣀⣀⣤⣴⡾⠋⠀⠀⠀⠿⠿⠿⠸⠶⠾⠃⠿⠾⠟⠻⠷⠿⠀⠀⠀⢀⡼⠋⠻⠿⢶⣶⣶⡾⠿⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛
⠀⠀⠀⠀⠀⠀⢀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠴⣦⢴⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠂⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠋⠄⡿⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠂⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⢺⠒⢺⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣾⣇⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⡇
⠀⠈⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⣿⡿⣿⣍⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠟⠛⠻⢛⠛⠛⠃
⠀⠀⠀⠸⡆⣷⣾⡶⡶⣶⣶⣶⠈⣬⣽⠛⢀⣼⣯⣠⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠶⠄⠁⠁
⠀⠀⠀⠘⠃⣿⣿⣄⠂⠙⠿⠿⣷⣿⣿⣤⣼⣿⣿⣷⣾⣿⣤⣭⣽⣿⣿⣿⣿⣏⣈⠁⣀⣀⣀⣰⣤
⠀⠀⠀⠀⢐⣟⣘⡿⠆⠀⠀⢸⣿⣿⣿⣠⣿⡛⠻⣿⣿⠟⠛⠛⠉⠉⠉⠙⠟⠿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠏⠁⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⠗⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠙⠛⢻⣿
⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⢈⠻⠿⠋⢙⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣾⡇⢸⡇
⠀⠀⠀⢠⣾⠀⡄⠄⠀⡴⠀⢀⠕⠒⠀⢛⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣷⣝⠓⢾⠇
⠀⠀⠐⠛⠀⠀⣧⠀⢸⡷⢾⡀⠀⠀⢀⣸⣰⠀⠀⠀⣄⠀⢀⠀⠀⢀⣀⠀⢀⠠⢶⣄⣶⣶⣿⣦⣤⣿⣿⣇⣸
⠀⠀⠀⠀⠀⢀⡧⡀⠀⠈⢦⡱⡄⠠⣾⣿⣿⠻⢨⡤⢮⡡⠤⠿⠤⠺⠗⠂⠀⠂⠁⠈⠉⠉⠈⠉⠝⠉⠈⠻⠿⠃
⠀⠀⠀⠠⠀⠠⠇⠀⠀⠀⠀⢷⣝⣯⣹⣿⠃
⠀⠀⠀⢀⠀⠀⠀⠀⠰⠾⠿⠋⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⡀⠀⠀⠀⠀⠈⠁⢰⠀⠀⡀⠀⠐⠼⠁⠀⠀⠀⠀⢀⠂
⠀⠈⠄⠀⠰⡄⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⢸⡄⡞⠀⠀⠀⠀⢆⢤⠀⠐⠀⠆⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣦⠘⢄⠒⠲⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡇⡄⠀⠀⠀⠊⠁⠔⠂⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⡀⠸⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⡀⠠⠀⠀⡀⠀⢀⠖⠀⠊⢀⠁
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠄⠀⠀⠀⠀⡦⠃⠀⢠⠀⡇⢀⠀⢰⡁⠀⢸⠀⡰⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠱⠠⠀⠀⠘⠁
⠀⠀⠤⠀⡤⠤⠒⡠⠔⠀⠀⠀⠀⠂⠀⡀⢀⡒⢦⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠀⠤⠠⠠⣄
⠐⢀⣠⡀⠄⠀⠀⡀⠀⠀⣀⠐⠁⢀⠔⠁⠠⠉⠋⠁⠀⠀⠀⠀⠀⢀⣻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠰⠲
⠀⠘⠛⠉⠀⠈⠀⠀⠉⠁⠙⠶⠛⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠢⣄
⠀⠀⠀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠀⠀⠀⠁
⠀⡄⢠⠀⠀⠀⢐⡀⣠⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠋
⠀⠻⠶⠘⡆⠀⠀⡼⠘⠛⠾⠧⠢⠀⠠⠄⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠄⠀⠘⠂⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢿⣆⠀⢸⣆⠘⠋⣀⠀⠀⠀⣷⡆⢀⠀⢠⣄⠀⠰⠀⠀⠀⢀⠘⠇⠀⠀⠀⢠⣄⠁⣀⠀⠀⠠⣤⣁⣼⠂⠀⠀⢀⠛⢀⠂
⠀⠀⠈⠻⠆⠈⡛⠀⠾⠋⠀⠀⠀⠈⠁⡿⠁⠀⠁⠲⠀⠀⠀⢀⡘⢧⡶⠂⠀⠀⠀⣉⠾⠋⠀⠀⠀⣬⡙⠁⣀⠀⠀⠈⣈⠺⠈
⠀⠀⢶⣦⡀⣰⠆⢀⣴⡶⠀⠀⠀⠀⠃⠀⠀⠀⠰⡀⢀⡀⠀⠈⣿⣆⣴⡇⠀⠀⠈⢻⣧⡶⠀⠀⠀⠉⠛⣿⡟⠀⠀⠀⠉⠲
⠀⠀⠀⠉⠀⡿⠀⠈⠁⠀⠀⠀⠀⠶⠶⠂⠀⠀⠀⡹⠈⠁⠀⠀⠈⠉⠋⠀⠀⠀⠸⣇⣡⡄⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢈⡙⡔⠗
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⠰⣶⡄⢠⡄⠀⠰⢾⣶⢄⢿⠄⠀⠀⠸⣶⣾⣯⡔⠀⠀⢀⡀⠀⡄⣧⡀⠀⢰⣿⠀⠀⡀⠀⠀⢠⡄⠀⠀⡀
⠀⠀⢈⡄⠈⠀⠃⣀⠀⠀⢠⣌⠃⢸⠇⢀⡀⠀⠺⠃⠙⢇⣴⡆⠀⠀⠁⠀⠀⠀⠰⣌⢰⣧⡿⠆⠀⢠⣜⣡⠀⠀⠙⠳⠀⠘⠃⠀⢚⡁
⠀⠀⢀⠀⡄⠀⠀⣨⠀⢰⣬⡛⠀⡆⢰⣿⠃⠀⠀⠀⠶⢌⣾⠀⠀⠀⣠⠀⠀⠀⢳⣶⡧⠍⠀⠀⠀⢦⡟⢋⠀⠀⢠⣤⣦⣶⣤⠛⠉⠁
⠀⠀⠉⠀⠀⠐⠀⠀⠀⠘⠿⣷⠀⠄⠈⣁⣀⠀⠀⠀⠀⠀⠁⠀⠐⢶⣿⣶⠆⠀⢸⠃⠀⠀⠀⠀⠠⣤⣷⣿⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠐⠛⠛⠛⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠉⠻⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣤⣤⣄⡀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠠⣶⣦⣤⣠⣾⠿⠛⠉⠉⠙⣿⡟⠋⠉⠉⠉⠉⠉⠉⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠻⣿⡄⠀⠀⠀⠀⠀⠈⠉⢻⣿⠀⠀⠀⠀⠀⣿⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣠⣿⠟⠛⠛⠛⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣤⣤⣶⡾⠿⠛⠿⠶⠾⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⣀⣠⣾⠟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣤⣄⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⢶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⡶⠂⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⢂⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠀⠀⠈⠙⠿⠿⠿⠿⠛⠋⠁⣠⣾⣿⡆⠈⠙⠻⠿⠿⠿⠿⠛⠁⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠃⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⣀⠀⠀⢀⣀⣀⣼⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⡴⠊⠋⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⠑⠉⢉⠳⣾⡅⡀⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢐⠭⣺⢿⣟⠎⢔⠀⡀⢺⠁⣠⣤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡀⠐⡋⠕⠁⠙⡀⠀⠈⡴⣣⠏⠁⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠋⠲⣽⡳⢦⡁⠀⢸⡗⣀⠞⡕⠁⠀⢀⢔⢄⠄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠠⡀⠄⠀⠈⠲⣌⡳⣝⣯⡯⢾⣠⣢⠤⠤⣯⣿⢿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢐⣬⣂⣅⠀⠨⠷⢽⢾⡿⣿⡿⣝⡉⠀⠀⡢⣟⢝⢕⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⡻⡻⡕⢄⢀⠀⣠⢋⡜⣹⡭⠦⣝⠢⣬⠪⠪⠂⠁⠀⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠐⠝⠌⠎⢂⡾⣡⡎⠀⠀⢿⠂⠀⠑⢮⣓⠴⠴⠴⠼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣂⠀⠀⣀⣠⢞⡟⠩⡪⢂⣄⣸⣀⢄⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡟⠉⠀⢸⠅⠈⢒⢕⢕⢟⣾⢕⠅⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠸⡅⡀⠀⠁⢡⠟⢯⡀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡇⠀⠀⠀⠀⠙⠲⠗⠚⠁⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣻⣿⣿⣿⣿⣋⡁⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢞⠘⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠿⣿⡿⠿⠿⠟⠛⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠷⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣽⡷⣾⣟⣻⣻⣿⣿⡽⣿⢶⣄⠀⣀⠔⠋⠀⠀⠐⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⡿⣶⠗⢿⣿⡏⠡⠎⢳⡀⠈⢦⠹⣿⡀⢀⣀⠀⠚⠁⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣻⠅⠀⢿⣀⣼⣿⣿⠰⡖⡈⢿⡑⡈⣧⢻⣿⠉⢻⣷⠀⢘⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠤⢸⣹⣇⣾⣿⣿⣷⣆⣘⣿⣷⣷⠿⢺⠻⣦⣿⠿⡚⡟⣧⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀⣿⣖⣸⣆⣿⣿⡿⢿⣿⣏⣿⣭⣿⣴⡾⣿⣿⣿⠛⡄⣃⢡⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢷⣻⣏⣿⣧⣼⣿⣷⣾⣿⣿⢿⡟⣾⣿⣿⢿⡛⣿⡇⢡⠸⠈⡜⡗⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢺⠻⣿⡧⣈⠛⣿⢹⣿⣿⣿⣶⣿⣿⣿⠛⠀⣇⢻⣷⠈⡆⠆⢡⢻⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⡇⠉⢿⣿⣷⣿⡿⢿⣿⣾⡟⠉⠁⢸⣆⣀⣸⣨⣿⣿⣿⣼⡬⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢹⠀⠸⡄⠉⠀⣷⣀⣽⣷⣳⣶⣿⡿⠿⣻⢿⣿⣮⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⣏⣦⣤⣷⣾⠿⢿⡟⠛⣿⣏⣽⣿⣶⣾⣿⠟⠁⢸⣿⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣽⣿⣷⣿⣷⠾⢿⠛⣉⣉⣥⡇⠀⠀⠀⠀⠀⢸⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣆⠀⠉⠉⠀⠀⠈⣻⣿⣿⣿⣷⣦⣤⣀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⣦⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣟⠛⠟⠀⠈⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠘⠛⠻⢯⣽⣷⡾⠁⠀⠀⠀⢀⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣾⣿⡇⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⠀⢀⡀⠀⣀⣀⣴⣿⣿⣿⠇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⡿⠀⠈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠉⠑⠒⢦⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⣷⠜⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠓⠾⣀⣀⣠⣯⠖⠀⢸⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⣠⠖⠁⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠐⢻⡀⢐⣯⣀⣀⣸⠀⠀⠀⠀⢠⠞⠁⠀⢀⡴⠊⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⡴⣿⠉⠛⠳⢤⡠⣤⣤⣤⣴⣃⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠿⡶
⢠⠶⣒⡛⠛⠛⠛⠛⠲⢦⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠓⢄⠀⠀⢀⡼⢰⡇
⢸⠀⠀⠈⠉⠐⠦⡀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⠂⠀⠀⠀⠀⠈⠐⠂⠤⠼⢧⡤⠀⢀⣀
⠘⡇⠀⠀⠀⠀⠀⡜⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠑⠛⢉⡌
⠀⢿⠀⠀⠀⠀⢰⢠⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣛⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺
⠀⠘⣧⡀⠀⠀⡆⠀⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢰⡟⣿⣿⠃⠀⢀⣀⣠⠤⠤⠀⠀⠀⠀⠀⡏
⠀⠀⠈⢏⠒⡠⠀⠐⢿⡟⠛⣹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠸⢧⣄⣻⡿⠿⠚⠛⠉⣁⣀⣀⣀⡀⠀⠀⠀⡰⠁
⠀⠀⠀⣠⠟⠁⠀⠀⠀⢄⠐⣿⣿⣿⡆⢠⡒⠶⠤⡄⠀⡴⢞⣫⠥⠶⠚⠛⠋⠉⠉⠉⠉⠉⠉⠀⢀⣼⣡⡤⠴⢤⡀
⠀⢀⡨⠟⠁⠀⠀⠀⠀⢀⣉⣉⠭⢭⡤⠻⣄⠀⣠⠎⠐⢚⣿⠷⠖⠒⠛⠛⠛⠛⠛⠛⠓⠒⠀⣠⠞⠁⠀⠀⠀⠀⣇
⠻⣀⠀⠀⢀⣠⠴⠚⠋⣁⡤⠖⣊⣽⠿⣆⣈⣙⣧⣤⣴⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⡿
⠀⠈⠣⡀⠀⠀⣠⠴⠋⣡⡴⠚⠉⠀⠀⠈⠉⠹⣿⣫⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠶⣎⡁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠛⠛⠛⠒⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⡇⠘⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⣴⠚⠉⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠉⠛⠻⣟⠑⠒⠒⠒⠒⠒⠚⠛⠋⠉⣿⠃⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠀⠀⢀⣀⣤⠶⣾⠃⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠘⢶⣶⠎⠁⠀⠈⠉⠉⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢾⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠓⠶⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠂⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣰⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠳⣄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣇⠀⠘⢦⡀⢸⠁⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢻⠀⠀⠈⠓⢸⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡐⠢⣀⠀⠀⠘⣷⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣀⡠⠐⠶⠛⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠧⡄⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡜⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢀⠤⠤⠘⠁⠀⠀⠀⡇⠀⠀⠀⠀⢸⠳⣄⠀⠓⠒⠲⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⢙⣿⠶⠤⢤⡎⢠⠞⢫⡶⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⣈⣆⠀⣀⣠⡴⢧⡀⠀⠀⠀⠐⠀⢸⡸⠃⠀⠀⠳⢼⣤⣤⣥⣤⠤⠞⠛⠛⠛⠛⠚⠛⠋⠉