⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠺⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⡀⠠⠀⠄⢂⡐⠠⡁⢌⠠⡁⠌⡄⣁⢂⢁⠂⡔⢀⢂⡐⠠⠀⠄⠠⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢓⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⢀⠠⠐⣈⠐⠤⠑⡌⡘⢄⠢⡑⢌⠢⡑⢌⡒⡰⢄⠪⣄⠣⡘⡄⠦⡐⡡⢉⠔⡁⢆⡐⠠⡁⠌⡠⠁⡌⢠⠁⡌⡐⠠⢀⠙⠦⣌⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⡀⠐⡈⢄⠢⡑⠤⡉⢆⠣⡔⡱⢊⡱⡘⣌⠣⡍⢦⠱⡑⢎⡱⢂⠧⡱⡘⢦⠱⡡⢍⢢⡑⠦⢨⡑⢌⠢⡑⢌⠰⡁⢎⠰⣁⢃⢆⢢⠐⡈⠻⣤⡀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠠⠐⠠⠌⡑⢌⠢⡑⣌⠲⡩⢌⠳⣌⡱⢣⠖⣍⠶⣙⡜⢦⢫⡝⣎⠵⣋⠶⣱⡙⢦⢳⢱⢪⡱⣌⢣⢣⠜⡬⢱⡘⢬⠱⣘⢌⢣⠜⡌⣆⠣⡍⢦⡑⠌⢷⡄⠀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠌⣀⠣⠑⡌⡒⢬⢡⠓⡬⡑⢥⣋⣶⣤⣷⣧⣞⣬⣣⢳⡜⣫⠶⣹⢬⡳⡭⡞⣥⢻⠬⣇⡏⢶⡱⢎⢧⣋⠞⡼⣡⢞⣡⢳⣉⢎⡞⣬⢓⡬⣓⡜⢦⡙⡜⡄⢻⣆⠀⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠠⢁⠒⡠⢡⠓⣌⢱⢊⠦⣹⣴⡿⠿⠛⠉⠉⠉⠉⠉⠉⠛⠻⢿⣶⣭⣓⢯⡳⣝⡳⣝⢮⡻⣜⡞⣧⡝⣮⢳⡜⣏⢶⡓⣮⠖⣧⠞⣮⡜⢦⡏⢶⡱⢎⡳⣜⢲⣉⠦⣹⣧⠀⠄⠀⠀⠙⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠄⡡⢊⠔⣡⢃⠞⣤⢓⠮⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⡿⣿⣶⡹⣞⡽⣎⡷⣝⡮⣝⡶⣹⣎⢷⡹⣎⢷⡹⣖⡻⣜⡻⢦⣛⢧⣛⢧⣏⡟⣴⣋⢶⡩⢖⡡⢽⣆⠌⡐⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠠⢁⡘⢄⠣⡜⢤⣋⡜⢦⣋⣾⡿⠗⠚⠉⠋⠙⠋⠛⠛⠛⠳⠶⢦⣤⣀⠀⠀⠉⢿⣷⡝⣾⡵⣛⣮⡽⣞⣵⡻⣼⢫⣷⢫⣗⣻⡼⣳⣭⢟⡧⣟⣮⢽⣺⢼⡹⢶⡹⢶⡹⣎⠵⣋⣽⡆⢰⠁⠄⡀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⢁⠢⡐⣊⠖⣩⠖⡼⣘⢧⢲⣿⠇⠀⠀⣤⡄⠀⠀⠀⣤⡀⠀⠀⠀⠈⠙⠻⣶⣦⣬⣿⣿⣱⢿⣹⣞⣵⣻⡼⣽⢳⡿⣼⢻⣼⣳⡽⣳⣞⢯⣞⡷⣞⣯⢞⣳⠿⣭⣛⣧⢻⡜⣧⠳⣜⣷⠘⣎⠐⡀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠄⢡⢂⡱⠰⢥⡚⢥⡛⡼⣙⢎⣳⡿⠀⠀⠀⠙⠁⠀⠀⠀⠛⠁⠀⠀⠀⣴⣦⠀⠈⣿⡈⢹⣿⡞⣯⢷⡞⣧⢷⣻⡵⣯⠷⣏⡿⢶⣏⣷⡻⣞⣻⢮⣟⡾⣵⢯⣟⣯⢷⣫⣞⢧⡟⣼⢳⡱⢾⡇⣷⢈⠤⠁⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣠⠖⠊⠔⠂⡌⢑⢢⣉⣦⣵⣮⣷⣾⡿⠷⠿⠶⠷⠾⡶⣶⣶⣶⣶⣤⣄⣀⣀⠈⠁⠀⢠⣿⠀⢸⣿⡽⣛⣾⡽⣯⣟⣶⡻⣵⣻⡽⣽⣛⣮⢷⣻⡽⣏⡿⣾⢽⣯⣟⡾⣽⣞⣷⣫⣟⢾⣱⢏⣳⢻⣷⢸⡎⡰⢁⠐⡀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠐⠀⠀⢌⣠⣵⣶⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠿⢷⣶⣼⣿⣳⡝⣮⢽⣳⣛⢾⡿⣷⣯⣞⡷⣛⡾⣝⣧⢿⣝⡿⣭⣟⡶⣯⣟⣷⣻⢮⡷⢯⣛⣮⢟⡼⣣⢿⡨⣗⢡⢊⡐⢀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⣴⣾⠿⠛⠉⠀⣀⣀⣠⣤⣄⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣷⣯⣶⡹⣎⢷⡹⣞⡻⢿⣯⣽⢯⣞⣟⡾⣽⣳⢯⣟⣷⣻⢾⡽⣯⣟⣯⣟⡾⣝⡾⣱⢿⡔⣯⢒⢢⠐⢂⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡟⠁⠀⠈⠉⠓⠒⠒⠒⠲⠤⣀⠀⠀⠀⢾⣅⣾⠟⠉⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠉⠙⠻⢿⣮⣷⡭⣝⠧⣏⡟⣿⢾⣽⣞⡷⣯⣟⣾⣳⢯⡿⣽⣳⣟⣾⣳⢯⣟⡼⢧⣿⡘⡽⢌⠦⡉⠆⡈⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⠶⣤⣸⣿⠃⠀⠀⠴⠟⠋⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠉⠛⠿⣾⣽⣎⡽⢎⡟⣞⠿⣿⣾⡽⣾⡽⣯⣟⣷⣻⣞⡷⣯⣟⢾⣹⡗⣾⡵⣙⢎⠲⣁⠎⠠⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⡁⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣶⣄⡀⠈⠻⢿⣿⣽⣚⣭⢻⡜⣯⢿⣷⣻⢷⣻⣞⡷⣯⣟⡷⣯⡟⣧⡟⣽⡗⡜⢮⡑⠦⡈⠅⢂⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣿⠐⢄⡀⠑⠢⠤⣄⣀⣀⣀⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣦⡀⠈⠸⠿⣿⣾⣧⣛⢾⣹⡞⣿⣿⣷⢯⣟⡷⣯⣟⣷⣻⣳⢯⣿⢧⡹⢦⡙⠦⡑⠌⠠⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⢀⡴⠃⠀⠀⠉⠲⣴⡁⠀⠀⠀⠀⠈⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣆⠀⠈⠰⢽⢻⣿⣷⡝⡾⢧⣟⡿⣿⣾⣽⣳⣟⣾⣳⢯⢷⣿⢆⣻⠲⣍⠲⡁⠎⡁⠐⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢣⠐⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⣰⣷⣭⣟⣶⢶⣶⣴⣶⣿⣿⠟⠛⡛⠻⢿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⠛⢼⢻⣿⣿⣝⡾⣽⣻⢿⣷⣟⣾⣳⢯⣟⣻⡿⢎⣧⢛⡴⢣⡑⢌⠠⠁⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠋⠒⢆⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⠀⠀⠀⣿⣿⡄⠘⣿⣿⠏⢩⣿⡟⠁⣴⣿⣿⣷⡄⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣻⣿⣷⣽⣹⢿⣞⣿⡾⣽⣻⣞⣿⣏⡻⣴⢋⠶⡡⢎⡐⢂⠡⠀⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣠⠏⠀⠈⠦⡑⠤⡀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⢸⣻⣭⢷⡀⠙⠁⣰⣿⡟⠀⣼⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣉⠛⠻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣯⣟⡾⣽⣿⣳⣟⣾⡿⣜⣳⢭⢞⡱⡱⢢⠘⠄⠂⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢰⠦⠰⢲⠋⠁⠀⠀⠀⠀⠉⢢⡈⠃⢤⣀⠀⢀⣀⠴⡿⠀⠀⡸⣽⢞⣻⡷⠀⣾⣿⣿⠁⢰⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⣿⣿⠁⢀⣀⣀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣴⣿⣿⣷⡄⠈⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠫⢿⣿⣾⣽⣻⣞⣷⣻⣿⢳⡽⣎⢯⢎⡵⣡⢃⠎⡘⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡼⠀⡀⡸⢆⠀⠀⠀⠀⠀⠀⠈⠑⣤⠈⠙⠒⠚⢉⡴⠃⠀⠀⡇⣿⢺⣽⡇⢀⣿⢯⣿⠀⢸⣿⣿⣿⣿⣿⣿⠇⣸⣿⣿⣿⣿⣶⣾⣿⣿⣶⣶⣤⣀⠰⣿⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣼⣿⣿⣿⣿⣷⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⣿⣯⡿⣽⢯⣿⣟⡾⣵⢻⡜⣮⠱⡍⢎⡰⢁⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢘⠳⡈⠱⣀⠀⠀⠀⠀⠀⠀⠈⢻⡄⢀⡶⠋⠀⠀⠀⢀⠇⣿⢣⣾⠃⢸⣯⣛⣿⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⢿⡿⠀⢰⣿⣿⣿⣿⣿⣿⡆⠈⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡙⣿⣿⣽⣿⣟⣾⣽⣞⢧⣛⠶⡹⡜⢢⡑⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢰⡀⠀⣀⠤⠊⠀⠘⢆⡈⠱⣄⠀⠀⠀⠀⠀⢠⡗⠉⠀⠀⠀⠀⠀⢸⡜⣯⣞⣿⠀⣸⢇⢯⣿⡄⠀⢿⣿⣿⣿⣿⠏⢠⣿⣿⣿⣿⣿⡿⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡿⣯⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣆⢻⣿⣟⠉⣉⠉⠉⠛⠛⠾⠵⣍⣆⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣸⠄⠀⣽⡀⠀⠀⠀⠈⢷⡀⠈⠳⣤⣀⣀⡴⠋⣽⠀⠀⠀⣀⣤⡴⠾⣇⠙⠦⢿⣀⡟⣬⡿⢯⣷⡀⠈⢻⣿⠿⠋⢠⣿⣿⣿⣿⡟⠁⠀⠀⣠⠴⣿⠌⠻⣿⣿⣿⣿⣿⡟⠀⣼⣿⣿⡿⣿⣿⣿⣿⡃⢰⣿⣿⣿⣿⣿⣿⣿⠁⢀⣿⣿⣿⣿⡟⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠆⡉⢿⣷⡌⠳⣄⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣿⡂⠀⠈⢷⡀⠀⠀⠀⠀⠻⣆⠀⣠⡴⠏⠀⢀⣈⡧⠔⠋⠁⣼⠁⠀⠈⠙⠒⠲⠿⠶⣯⣞⣧⣻⣿⣦⣤⣀⣤⣴⣿⣿⣿⣿⣿⣷⣷⣄⢸⡇⢰⣿⠂⠀⠹⣿⣿⣿⣿⡇⠀⣿⣟⣷⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⣿⡇⣸⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⢁⠂⠅⡂⠙⢿⣧⡈⠳⣄⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡆⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠉⢶⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠸⣧⠀⠀⣈⢷⣄⠀⠀⠀⠀⣿⡞⠉⠀⠀⢠⡿⠁⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡉⠛⠛⠿⠿⢿⣿⣿⠿⣼⣿⣿⣿⣿⣆⠿⣷⡿⠀⠀⢸⣿⣿⡟⢻⣧⣀⠉⠉⠉⠉⠛⣿⣿⣿⣿⡀⠙⣿⣿⣿⣿⡿⠁⢀⣿⣿⣿⣿⣿⠁⡞⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⡀⠠⠈⡐⠠⠁⠌⢹⣿⣧⠘⢳⡀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⢣⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢿⡂⣼⠃⠀⠉⠷⢤⣤⣾⡟⠁⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠈⢓⡶⠶⠦⣤⣤⣀⣀⣀⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠙⠃⠀⠀⠀⠻⢿⣷⣏⣞⡽⢻⣿⡿⣿⣿⣿⣿⢿⣿⣿⣤⡀⠉⠛⠉⠀⣠⣾⣿⣿⣿⣿⡿⢸⠃⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀⠐⠀⠡⢈⣐⣿⡿⡽⣷⣄⠻⣦⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⢀⣤⡴⣾⠋⠀⠀⠀⠀⠀⠀⠙⣦⡀⢀⡴⠖⠋⠁⠀⠀⢀⡆⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠈⠭⠛⢛⠿⠷⠿⠷⣿⢾⣽⣿⣯⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⡿⢁⡏⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⡐⣤⣿⡿⣽⣳⢏⡿⣧⡘⢷⡀⠀⠀⢀⣀⣈⣙⣦⣀⣀⣀⣀⣀⣀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠾⠛⠁⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠀⠀⠀⢻⣅⠀⠀⠀⠀⠀⠀⢀⠀⠂⠠⣐⣼⣿⣿⣟⣷⢯⣟⣷⣽⣿⣶⡷⠿⠟⠟⠛⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠛⠳⢾⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⡁⠈⠀⠌⢀⠐⣈⣶⣿⣿⣿⣿⣾⣿⣿⠿⠛⡍⠑⠠⠀⠂⠀⠄⠂⢀⠀⡀⠄⠐⠀⠠⠀⠀⢀⠀⢀⠀⠙⢿⣦⣄⡀⠀⠀⠀⢀⣼⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡠⠊⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡍⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢄⡈⡐⢨⣰⣾⣿⣿⣿⣿⡿⣟⠫⢃⠔⡈⠆⠄⢃⠄⡁⢂⠁⠠⠐⠀⠠⠀⡀⠂⢈⠀⡀⠁⡀⠠⠀⠀⠄⠀⠙⣧⠙⠻⠶⠞⠛⠁⠀⠘⡍⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⡞⠁⠀⢸⡷⢄⡀⠀⠀⠀⠀⠀⠁⡀⠀⠄⠁⠀⢦⣤⣤⣤⠶⣾⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣧⠀⣼⠉⠙⠻⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣬⣽⠿⠿⠿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠐⣢⣴⣽⣿⣿⣿⣿⡿⠟⡍⠒⡄⠣⢌⠢⡑⢌⡘⢄⢢⣐⣦⣼⣴⣷⣾⣷⣷⣾⣿⣾⣷⣷⣷⣾⣶⣷⣼⣤⣦⣄⡙⣷⡀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⢸⡇⠀⠀⣹⣆⠈⢹⡶⠶⠶⠲⠟⠛⠁⠁⠠⠈⠀⠀⠈⣀⣤⣴⢧⡀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⡀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡟⠛⠛⠛⠟⠾⠶⠶⠶⠶⠷⠾⠳⠟⠛⣿⠟⠛⠛⠉⠉⠉⠀⠀⢀⣰⣿⠟⠉⠁⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⣿⣿⣿⣿⣿⡟⡋⠬⡑⢌⠱⣈⠱⣈⢦⣱⢦⡿⣾⣻⢯⢿⣹⡛⢮⠳⡙⢎⠳⡙⠎⡝⠺⢍⠳⡙⢮⠹⣍⣛⢻⠻⣷⣿⡄⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣠⠏⠋⠉⠒⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠁⠀⠀⠀⠀⠀⠀
⠀⠈⣷⡀⠀⠘⣿⣆⠠⡙⠷⣦⣅⠠⢀⠐⠀⠄⣠⣴⣶⠿⠋⣽⡇⢂⠹⣄⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢻⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⢣⠘⡄⢣⠘⡄⢣⢤⡳⣝⡾⣭⢟⡳⠭⠓⡍⠦⠡⠜⣂⠱⡈⢆⠱⣈⠒⡌⠡⢌⡐⠌⡠⠁⢄⡀⢃⠱⠈⠽⣿⣄⠀⠀⠀⢀⣴⠟⣷⢀⡾⠁⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀
⠀⢀⡜⢷⡀⠀⠈⠻⣷⣧⡘⠠⠉⠗⠳⠞⢛⠛⠋⠉⠀⠀⣸⣿⡘⡄⢂⠘⡆⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⢋⠔⡡⢊⠰⣁⠶⣸⡙⣎⠷⡙⠦⡑⢊⠔⡡⢋⠔⡡⢍⠒⡤⢃⠜⣠⠃⢤⠃⣌⠑⠢⢌⠢⡁⠎⠤⡐⢂⢂⡉⡐⢈⢻⡿⠶⠻⠛⠥⠂⢹⡿⠀⠀⠀⠀⠀⠀⠀⣧⠀⠱⣤⠶⠋⠉⠉⠳⣄⠀⠸⡆⠀⠀⠀⠀⠀
⢀⡾⠀⠈⠻⣦⣀⠀⠈⠻⢿⣦⣅⠌⡰⢈⠄⢢⠐⣀⠂⢡⣿⡷⣹⠐⡂⠄⢹⡀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟⢡⠊⡐⠰⣀⠧⡜⢣⠣⢙⠠⢂⡑⢢⢁⠣⣘⠰⢡⢊⠔⣊⠱⡰⣉⠦⣡⢚⣤⠳⣤⢫⡕⣎⠶⣱⢎⡖⡥⢎⡔⣠⠑⡌⡸⣿⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣟⠀⣸⠏⠀⠀⠀⠀⠀⣯⢳⡀⢹⢤⠀⠀⠀⠀
⢾⡇⠀⠀⠀⠈⢿⡳⣄⡀⠀⠉⠻⢿⣶⣬⣘⡄⠣⡀⠎⣼⣿⡳⢥⡓⠌⠠⢸⠃⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣹⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠏⡐⠂⡔⢁⠦⡑⠎⢌⠡⡘⡀⢃⠆⡘⢄⠊⡔⢢⡑⣪⢤⣋⠶⣭⢳⡭⡞⣵⢫⣎⡟⡼⣓⡞⣭⢞⡵⢮⡝⣾⡹⢞⡵⣫⠖⣥⢻⣿⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡧⢠⡿⠀⠀⠀⠀⠀⠀⣿⠀⢳⢸⡄⠀⠀⠀⠀
⢸⣧⠀⠀⠀⠀⠈⠻⣦⡙⠲⢄⡀⠀⠈⠙⠛⠿⣷⣾⣷⣾⣿⣱⣣⣮⠵⠞⠋⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠷⠷⠶⠶⠶⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠋⡐⠠⢡⠐⡊⠔⠡⡈⠄⠒⡠⢁⠆⡂⢥⠊⠖⡡⢃⠝⡰⢃⡍⢳⡉⢮⠱⡛⡜⡳⢮⡙⠷⡭⢞⡱⢮⡝⣮⢳⢧⡻⣝⢮⡵⣛⢦⣋⢿⣷⣦⣶⡾⣿⡇⠀⠀⠀⠀⠀⠀⣸⠇⣼⡇⠀⠀⠀⠀⠀⠀⣿⠓⠶⣸⡇⠀⠀⠀⠀
⠀⢻⣧⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠑⠂⠀⠀⠀⠀⠀⠈⠁⣉⣀⣁⣀⣀⣀⣤⣶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣿⣿⣿⣿⣿⣿⠃⢂⠁⢆⠡⢊⠡⢈⠐⠠⠑⢂⠁⠆⡘⡐⢢⠉⢆⡱⢌⡎⣔⡣⣬⣡⢚⡤⢣⠱⣌⠱⢢⡙⢬⡑⣩⢉⠧⡙⡜⣋⠞⡵⢫⠞⣶⡹⣎⢧⣻⣗⠢⢆⡰⢹⣷⡀⠀⠀⠀⢀⣰⡿⠀⣿⡃⠀⠀⠀⠀⠀⢰⡏⠀⢠⡿⣇⠀⠀⠀⠀
⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⠈⠻⣷⣄⡂⠐⠠⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢀⡿⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠛⡶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠛⠡⠐⣿⣿⣿⣿⣿⠇⡈⠄⢌⠢⠑⡀⢂⠂⠡⠌⡐⡈⢌⠰⡁⣜⣤⣿⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣷⣿⣶⣿⣦⣯⣶⣱⡜⣤⢛⡜⣣⢟⡲⣝⢮⣳⢻⣿⡇⡎⠴⣉⢿⣿⣷⡶⠶⠟⠋⢀⣀⣿⡇⠀⠀⠀⠀⢠⣿⢁⣴⣿⠇⠀⠙⠲⠤⠄
⠀⠀⠈⢷⡻⣷⣄⠀⠀⠀⠀⠀⠈⠹⢿⣦⣤⣠⣡⣤⣤⡶⠶⠒⠉⠀⠀⢀⣾⠃⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢷⣌⠙⠛⠿⣶⢶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⡾⠿⠛⡉⠤⢐⡁⠠⢁⢺⣿⣿⣿⡏⠐⠠⢌⠂⡁⢂⠐⠄⠨⠐⡀⠆⣐⣢⣷⣿⣿⢿⣟⡿⣯⣟⣷⢿⣾⣿⣷⡿⢿⡛⢯⡹⣙⢮⡱⢫⡝⣭⠻⣝⡻⢿⢿⣿⣾⣷⣯⣷⣯⣞⣧⣟⣿⡷⣙⢦⡑⣎⢩⠛⢿⣿⣿⣿⣿⡿⣿⢿⣶⣤⣤⣴⠿⣡⣾⡿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⢿⣎⠻⣷⣤⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣹⢧⠀⢀⠀⠹⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣶⣶⣶⣿⣿⣿⣿⠛⢩⠉⡍⢠⠐⠠⢁⠐⡀⣺⠁⠀⠀⢨⣿⣿⡿⢀⠑⡈⠆⢂⠐⠠⠌⣀⠃⢌⠰⣼⣾⣿⢿⡽⣾⣻⣾⢿⣽⣯⣿⣿⡿⡛⢥⡙⢦⡙⢆⠳⡩⠖⡭⢳⡜⣥⠻⣌⠷⣩⣾⠋⠀⠀⠈⠙⠻⠿⣷⣿⣾⣷⣍⡶⣉⢦⢣⣝⣢⢍⡻⢻⠞⡛⠔⡈⠿⣿⡿⡿⢿⡻⠛⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠻⣷⣌⣛⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⢷⡀⠀⠄⠘⢿⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢂⠱⠈⠄⡈⠐⠀⢀⠐⢃⣀⣤⣾⠿⣻⣿⠃⠄⠊⡔⠁⠂⢌⡐⠤⣀⣜⣴⣿⣿⣻⣞⣯⡿⣟⠿⣞⢯⣳⣿⠟⡋⠔⡉⢂⠜⠠⡘⠨⣁⠣⡙⢔⠣⡜⢢⡛⣬⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⢧⣞⡳⣯⣷⣫⢞⡵⢾⡶⣧⣦⣽⣩⣃⣅⣄⡐⠀⠈⠀⠀⠀⠀⠀