⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡲⠜⠋⢉⣡⣴⠖⢻⣷⣶⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⡟⣩⣴⣷⣴⠟⠋⠉⢀⡎⡌⠻⣮⣪⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⣿⠪⡾⢃⣾⠟⢡⡾⣯⢋⡟⢺⣿⡱⣮⡎⢷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⡔⢾⣿⣿⣿⣿⣿⣿⠋⠀⡁⣼⣏⡴⡿⢯⣯⣞⣇⡿⠀⢀⡆⣷⣤⣡⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣷⣽⡻⣿⣿⣿⣿⢸⡀⢡⣿⣿⣶⣷⣿⣝⢀⣾⠃⣰⣾⣽⡏⡟⠃⡟⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⡭⣿⣿⣿⢾⣿⣿⣿⣿⣷⣎⣿⡁⢉⣙⡫⠃⠈⠫⠤⢾⣿⢉⣷⣇⣶⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡿⠿⣿⣿⣧⣿⢧⣽⡟⠻⠿⠯⠀⠈⢉⠀⠀⠳⠐⡻⣿⣿⣰⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⢏⡏⠀⡿⣶⣷⡌⠀⠀⠀⠀⠑⠒⠤⠄⠉⢲⣶⣿⣿⣿⣿⣿⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡀⠀⠀⣧⠞⡿⣏⣢⣀⠀⠀⠀⠀⠀⠀⢀⡼⢿⡿⢿⣛⡳⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡅⠀⠀⢇⣠⣿⣿⣿⣿⢷⣲⡿⣦⣶⣎⣁⣐⣋⣸⣉⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⡿⠛⠙⠻⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣦⣭⣕⣻⣿⣿⣿⣿⣿⣿⠿⠿⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠒⠒⣹⡿⠁⠀⠠⢈⢕⠟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣉⣉⣁⣈⣉⣁⣒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠦⣄⣰⡟⡹⠁⠀⠀⢲⡇⡎⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⠃⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢻⡉⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠁⠀⠀⠀⠉⢉⠙⠳⡶⢶⡚⠉⠉⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡿⠋⠀⢢⡀⠀⠀⢀⠴⠃⠹⡀⠱⣤⡀⣗⢖⠶⣨⣷⣦⣤⣈⣻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⠇⠀⠀⠀⠀⡠⢎⡠⠒⠁⢈⣿⢶⣀⣋⠀⠱⢄⠈⢦⢈⣧⡧⢓⣴⣟⡏⠉⠙⡟⢻⢮⡙⢟⣄⡀⠀⣠⣤⡲⢆⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠎⠀⠀⣠⢄⠮⠃⠁⠀⠀⣠⣿⣷⡿⢉⣭⡏⢶⡤⠷⠟⠛⠛⠻⣿⣿⣿⣽⡶⣖⢣⠤⣄⣙⣭⠉⠛⢋⡜⠁⣠⠏⠙⢣⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣜⠔⠊⠑⠉⠀⠀⠀⠀⠀⡴⣹⣋⣸⣾⠯⠓⠊⠁⠰⠄⠤⠀⠐⢦⡙⣿⡿⡍⠙⠳⢽⣠⣤⣴⣿⣬⣴⣿⣠⠎⣀⠔⢊⣀⢿⠃⠀⠀⠀⠀⠀⠀⠀⡠⠃⡠⠀
⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣜⡾⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⡿⢆⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠻⣦⣥⣔⣉⢀⠜⠀⠀⠀⠀⠀⠀⢀⠜⢀⠖⠁⠀
⠀⢣⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⢆⡼⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⢓⡀⠀⠀⠀⠀⠀⠞⢸⡇⠈⠻⣿⡏⢕⡃⠤⡀⠀⠀⠀⠀⠀⠋⠀⠃⠀⠀⠀
⠀⠀⢧⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠈⢎⣿⣧⣈⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠣⣀⡠⠖⣡⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢻⢆⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠈⢷⣿⣷⣿⣇⠀⠀⠀⠀⠠⣀⢆⠀⠀
⠀⠀⠀⠀⢀⢴⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⡀⢯⡆⠠⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⣝⣄⠀⠀⠀⠱⢦⢂⠀
⠀⠀⠀⠀⢸⣶⣮⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡁⠀⠋⠘⣧⣤⣶⣤⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡟⢸⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢘⢇⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⣆⡟⢻⣿⡿⣷⡇⢠⣢⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢜⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣇⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⣇⣾⡇⣰⠗⣡⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀
⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⢠⣿⣿⣷⠛⣴⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀
⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣼⣿⡿⠃⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀
⠀⠘⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣌⣿⣟⡕⠁⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⡜⠀⠀⠀⠀⠀⢀⣀⣤⢤⣶⣶⢶⡤⣍⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⠉⢃⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣷⠀⠀⣠⣴⣿⡿⣳⢯⠾⣿⣿⣻⣗⣿⣽⠿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⠀⡤⠊⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡉⢠⣾⣿⣯⡉⡣⢚⡥⠤⠤⠄⢿⠃⣞⣋⠸⣾⣯⣷⣄⠀⠀⠀⠀⠀⠀⠀⣷⠃⣰⠁⡆⡧⠃⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣧⣿⡿⢏⠖⠓⠊⠁⠀⠀⠀⠀⡌⢠⠇⠈⠦⡿⢿⣿⣾⢦⠀⠀⠀⠀⠀⢸⠿⡄⠃⠀⢱⠁⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡖⣶⡄⠺⣿⣿⣶⣦
⠀⠀⣽⡻⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⠀⠀⠀⠘⢄⡀⢰⣼⣧⡀⠀⠀⠀⡘⠀⠙⣷⣄⡈⣿⢰⠀⢇⠀⠀⠀⠀⠀⠀⠀⣠⢴⣻⣷⣦⣿⣗⣼⡇⢀⣀⠉⠛⠛
⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⡄⢻⣿⣧⠀⠀⢰⠃⠀⠀⢿⠀⠈⠙⡷⠀⠸⠂⠀⠀⠀⠀⣰⣋⣿⣌⣛⡿⢟⣋⣈⠜⢻⡇⠀⠈⠁⠀
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣹⣿⡄⣰⡋⠀⡀⠀⡈⡇⠀⠀⢹⠀⠀⠀⠀⠀⢀⣾⡟⢲⡀⠀⢀⡖⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠿⠿⠇⠀⠀⣤⡟⡀⠀⢀⣇⠀⠀⠀⠀⣾⣿⡷⣸⢃⠒⠉⠀⠀⠀⠀⠀⠀⠀⢾⢦⠀⠀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠛⠂⠉⠉⠉⠈⢆⠀⠀⢰⣿⣋⡵⢋⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠇⠀
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢸⣿⡄⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣧⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣷⣦⣤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⠀⣸⣷⣶⣤⣤⣀⡀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⢻⣿⣿⣿⣿⣿⡿⢻⣟⡵⣡⠞⣡⡶⠛⣉⣉⢭⣭⣉⣒⠮⣕⡯⣿⢿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⡀⠈⢿⣿⣿⣿⣿⣿
⡌⢿⣿⣿⠋⠁⢠⡎⡞⡵⠁⣴⠋⣠⠞⠃⠀⠀⠀⠈⠙⠛⠓⢽⣻⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⡻⣿⣿⣧⢹⣿⣿⣿⣿⠆⠘⣿⣿⣿⣿⣿
⣇⣼⣿⡁⠀⠀⣾⡞⢰⠃⣼⠁⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢽⣿⣿⣿⣿⣟⣿⣿⣿⣿⡻⣿⠉⠛⢻⣧⣿⠀⠈⠀⠸⣿⣿⡆⢿⣿⣿⣿⡧⠀⠘⣿⣿⣿⣿
⡟⠟⠁⠹⣄⢸⢱⢣⠇⣰⠃⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⢽⡿⣿⣾⣿⣤⣀⠀⠁⠶⣬⠳⢄⣀⠀⠸⣿⣧⠀⢿⣿⣿⣷⡄⠀⠻⣿⣿⣿
⠁⠀⠀⢀⡞⡇⡎⡜⢠⡏⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⢽⣕⣒⠫⢽⣿⣿⣿⣷⣦⣄⠈⠲⣝⢦⠀⠹⣿⣧⣼⣿⡟⠛⠁⠀⠀⢹⣿⣿
⠀⠀⢠⠏⢰⢹⢡⣧⣸⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠒⠂⠤⣤⣀⠠⡄⠀⠈⣹⣿⣶⣽⣮⣻⣋⠙⢝⠳⣿⡈⠻⣷⡄⠹⣿⣿⣿⣧⠀⠀⠀⠀⠀⢻⣿
⠀⠀⣿⠀⢸⠸⢸⡲⣾⣾⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣖⡭⢒⡉⢀⣨⣥⣞⣛⣻⣷⣤⣶⣶⣏⣹⣿⢹⣿⣿⣯⡻⣎⢳⣜⢿⣄⠘⣞⢦⡙⣿⣿⣿⡆⠀⠀⢀⠀⠀⢿
⠀⠀⣿⠀⡿⣇⣤⡱⠝⠃⠀⠀⠀⡰⠃⠀⣴⣿⡿⢟⣥⣶⠽⣎⢩⣦⠙⢏⠹⣍⣼⣿⣿⣭⡙⠚⠿⠿⣿⣿⣿⣷⡉⢧⠹⣆⢻⣧⠈⣧⣽⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣅⡇⡇⢿⣟⣷⡄⠀⠀⡴⠁⠀⣺⠿⣳⣶⣿⡋⠳⣦⣪⣻⣿⠄⠈⢣⡻⣿⡟⣮⣿⣿⡂⢠⣶⢻⣿⣿⣿⡷⡄⠳⡈⢆⢻⣧⠈⣯⣿⣯⠽⢿⣤⠀⠀⠀⠀⠀
⠀⠀⠀⣏⠇⠗⠧⣠⢼⡟⠀⡰⠁⢀⣴⣡⡿⠛⣽⡜⣟⣦⣼⣿⡏⡘⢯⣇⠀⠹⡹⣳⢿⣿⡝⢻⣾⣷⣿⡏⢻⣿⣷⠙⣾⡳⡈⢏⣧⣧⠈⣯⣾⣆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣿⢸⢀⡴⣿⢦⠀⣰⠃⣠⣾⣞⠁⠀⠀⠘⣷⣼⣿⣿⣍⢿⣿⣾⣯⢣⠀⢻⢿⣮⣿⣷⡾⣻⡎⣿⢻⡌⢿⣧⠀⠈⣆⢱⡘⠈⣿⡆⠸⣸⣾⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠿⡿⡟⢸⠁⡏⡸⠀⢡⢾⢿⣯⢿⣄⣀⣴⡼⣿⢷⡾⣿⣿⣮⡟⢯⢿⣏⣆⠀⣿⣿⣿⣹⡿⣟⣿⣿⢿⣷⡈⢻⠀⠀⠘⡎⣷⣆⢸⣿⡄⣿⡟⣿⡀⠀⠀⠀⠀⠀
⠀⠀⢠⢇⢧⡏⢸⢃⣇⡴⣿⢻⣿⣽⣯⣿⡟⡚⠛⠛⢫⣇⣧⢋⣼⣷⣾⣿⡿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣽⡜⢹⡇⠀⠀⠀⠀⣿⢸⣯⣆⣯⣿⢳⡇⢸⡇⠀⠀⠀⠀⠀
⣤⢠⡿⡞⢸⢣⡿⣾⣦⠾⣹⡾⣇⣳⣿⣿⣿⣾⢆⡀⠈⢻⣽⠿⣛⡹⠿⠛⢙⣿⣴⣿⣿⡆⠀⠙⠛⠛⣿⣿⢿⢇⠀⠀⠀⠼⢻⡎⠘⢯⣿⠇⢸⣷⣌⠁⠀⢀⣠⡀⠀
⢃⡾⡾⣠⢛⣿⣿⣟⣧⢀⡿⣳⣻⣿⡿⣿⣿⣾⣿⣿⡤⠤⠟⠀⠀⠀⠈⠀⠞⣽⣿⣿⡏⢿⡄⠀⠀⠀⠸⣿⣯⡘⣆⠀⠀⠀⠛⠀⠀⢀⣿⠇⡟⢏⠙⠓⡷⠋⠀⡇⠀
⣾⢟⣴⣥⡿⠋⢹⣟⣞⣞⣷⣝⣿⣿⣦⣌⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠷⠘⣷⣀⣠⠤⡶⣟⢿⣿⡻⣍⡳⣄⠀⠀⠀⠛⣿⣴⠉⠻⢷⡞⠀⠀⡼⠀⠀
⠒⠿⠿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣟⠿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⡼⢘⣿⣍⠙⣿⡇⠀⣿⠃⠘⠳⠿⢿⣿⣿⣾⣷⡆⠀⣠⡷⠃⠀⢠⠎⠀⠀⡼⠀⠀⠀
⠀⠀⠀⠙⠳⣖⢼⣯⣿⣿⣿⣿⣿⣿⢾⡿⣷⠆⠀⠀⠀⠀⠤⠠⠐⠁⠀⢀⠞⢀⣾⣿⣿⣿⣿⣿⣁⠏⠀⠀⠀⠀⢀⣀⣈⡙⠿⣿⣌⡁⠀⠀⣠⡇⠀⢀⡼⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣯⣧⠹⣿⣿⠿⠘⣧⠀⠀⠈⠓⣦⣄⣀⣀⣀⣀⣀⣀⡴⣋⣴⣿⡟⣿⣿⣿⣿⡿⠻⣦⠀⠀⠀⠀⠘⡅⠀⠙⢦⡀⠹⣿⡄⣴⡟⠁⢀⡞⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⣿⣦⣹⣻⡄⠀⢸⣧⣤⠔⠚⠁⢹⣤⣼⣿⣿⣿⣿⣿⣻⠟⣯⢻⣿⢫⣿⣿⠇⠀⠈⠳⣄⠀⠀⠀⢹⣄⠀⠀⢃⠀⣸⡿⠋⠀⢠⠏⠀⠀⣀⡤⠖⠀
⠠⢤⣀⠀⠀⠀⠀⠘⢻⣿⣿⡿⠞⡩⠚⢿⣇⠀⠲⠤⣽⢿⣿⣿⣿⣿⣷⣟⠿⡟⣼⣾⣏⡴⠃⠀⠀⠀⠀⠙⢧⣀⣀⣸⣬⣦⡶⢾⠛⡟⠀⠀⠀⠱⡭⠟⠛⠚⠋⠁⠀
⣀⠀⠈⠑⠶⢄⡀⠀⠀⣷⣸⠁⠸⠁⠀⠘⡿⣷⠶⠄⣿⠀⠀⠉⠙⠛⢻⣿⣷⣿⣿⢿⡛⣷⣦⣀⣤⣴⣲⣶⣿⡿⢿⡻⣝⢧⠐⠁⠈⠀⠀⠀⠀⠀⠑⠦⠤⣴⣒⠊⠀
⠀⠙⠢⣄⠀⠘⠛⢶⣤⣹⡟⠀⢠⠀⠀⢠⠃⠈⠲⠤⡟⠤⣋⣀⣀⣠⣼⣿⡿⣿⣿⣿⣿⣿⣿⡿⣿⣟⠋⠉⠁⠀⠀⠛⢿⣞⣆⠀⠀⠀⠀⠀⠦⡀⠀⠀⢣⡈⠉⠻⠍
⠀⠀⠀⠀⠙⠢⣄⠀⠈⠻⠿⣦⣼⡶⠶⠾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣿⢻⠉⠀⠀⣰⣿⢈⣷⡄⠀⠀⠀⠀⠀⠻⣿⣗⢦⣄⡀⠀⠀⠀⡀⠀⢀⡍⣷⠢⠀
⠀⠀⠀⠀⠀⠀⣀⢿⠆⠀⠀⠈⠳⠏⠀⠀⢷⡀⠀⢀⡇⠀⠀⠀⢈⠟⢷⣄⠀⠀⠘⡄⣇⠀⢠⡿⣹⣿⠿⠁⠀⠀⠀⠀⠀⠀⠈⢏⡏⠊⠉⠢⡀⠀⢣⢰⠋⠀⠏⠀⠀
⠀⠀⠉⠉⠉⠭⠗⢩⠄⠀⠀⠀⠀⠀⠀⠀⠘⠙⣄⢸⠇⠀⠀⠀⣾⣀⣸⠙⢷⣄⣀⡽⠼⠎⢹⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢀⣀⣨⡵⠶⢿⡀⢠⣆⠀⠀
⠀⠀⠀⢀⣠⣤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿⣾⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⣷⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⢠⡀⠀⣀⣠⣤⣶⣿⠿⣟⠉⠀⠀⢀⠜⢧⠀⡞⣆⠀
⠀⠀⠀⠀⠀⣸⢺⢦⠀⠀⢠⠀⠀⠀⡠⠾⢿⣄⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠚⠂⢈⣻⣦⣤⣤⣤⣴⡷⠟⠛⠛⠉⠁⠀⢸⠀⠀⠉⠓⠒⠁⠀⠀⠑⢍⡈⠓
⠀⠀⠀⠀⣰⢣⠏⢸⠀⠀⠈⡆⠀⢰⠃⠀⠀⠘⡇⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡴⣿⢖⣛⣋⣻⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢀⡴⢡⠎⡰⡿⢲⣦⠀⠘⣠⣃⣠⣤⣤⣴⣷⣷⡶⠶⠾⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⢡⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⠋⠀⣨⠎⠀⡇⠀⣿⠒⠯⣉⠉⠉⠀⢀⠏⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠙⠦⣤⣀⣀⠀⣀⣀⣤⠼⡾⠀⠀⠀⠻⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠤⠞⠁⠀⠀⡇⠈⡿⠀⠀⠈⠉⠒⠒⠋⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠈⠁⠀⠀⠀⢰⠏⣰⠃⠀⠀⠀⠀⠙⠙⢆⠀⠀⣠⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⡇⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠦⣀⡀⠀⣀⣀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢀⠄⠀⠈⠳⡿⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠉⠉⢻⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⡿⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠞⠲⠤⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⡿⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠿⢷⣄⡀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⣶⡴⠃⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠻
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠙⣿⠇⠀⣠⠔⠋⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠞⠉⠀⠀⠀⣀⣀⣀⣤⢀⣀⡈⠉⠒⠦⡀⢹⢠⡞⠁⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠢⣄⠀⠀⠀⣸⡴⠟
⠀⠀⠀⠀⠀⣸⡥⠖⠂⠀⠀⢀⣀⣀⣀⣲⣶⣶⣶⣧⠀⠁⠀⠀⣠⣴⣿⣿⣿⡍⠉⣽⣿⣿⣷⡦⣄⠙⢾⠈⢀⡴⢿⣿⡾⢿⣿⡿⣿⠟⣿⣟⣧⣄⢻⡷⡄⠀⣿⡄⠀
⠀⠀⠀⠀⣰⢋⠴⠂⠀⠀⠀⠉⠉⠉⠛⠛⠻⠿⣿⣿⡄⠀⠀⡴⣿⣧⣟⣿⣏⣿⡿⠿⠽⣻⣿⣧⣏⠳⡜⣧⠏⣤⣿⠜⠙⠋⠁⠚⠁⠉⠉⠛⠻⣿⣭⡿⢻⣰⣏⠀⠀
⠀⠀⠀⡰⡱⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣶⣶⣭⣇⠀⡼⠱⠭⢻⠟⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠷⡚⣿⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡷⢻⡿⠋⠀⠀