⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣀⣀⣀⣴⡶⣿⣿⣟⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⠀⠹⡄⠀⠰⣢⣤⣬⣿⣿⣿⣦⣭⣿⣿⣿⡿⠿⣏⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠹⡔⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢀⡇⣠⣾⣿⢿⣿⣟⣾⣽⣇⣾⣈⣽⡈⣿⡏⣿⣿⡽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⣸⣿⣿⣿⣾⡾⣵⣿⣿⣿⣿⣿⣿⢃⣿⡟⢹⢿⢏⣿⣿⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡜⠀⢠⡟⢊⣿⣿⣿⣿⡟⣿⠟⢻⠋⣿⣡⣾⣟⣫⣟⣸⣸⣻⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢠⠃⠀⡼⠀⣾⢿⣿⣿⣿⡏⣯⣭⣽⢆⠻⠀⠛⣻⣇⣿⣿⣿⣿⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡜⠀⢰⠇⠀⠈⢸⢿⣷⣿⣷⠳⠿⠛⡸⠀⠀⠀⠀⢀⠟⣿⢟⡿⠽⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⠃⠀⡿⠀⠀⠀⠀⠀⠉⢻⣿⣆⠀⠀⠳⠶⠀⠀⠀⠁⠞⠁⣼⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡸⠀⢸⠃⠀⠀⠀⠀⠀⢰⠗⠉⠻⣦⠀⢰⣛⣓⠖⠀⠀⢀⣴⢹⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⢀⣠⣶⡿⠁⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⣸⠁⠀⠀⠀⠀⣀⣠⣶⠛⠙⡚⠻⠭⢝⠻⢭⣽⠛⠉⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⡇⢀⡏⠀⠀⣠⡾⢋⣁⣂⣈⡹⢿⣿⣦⡰⡀⠱⣤⡈⠓⢵⣀⢀⢸⣛⣻⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⠁⣸⣥⡴⣾⡿⠯⣙⡒⠦⣒⢂⡹⠷⡿⣿⣿⣦⡈⠻⣦⣤⣀⣹⣟⣋⠁⠰⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢾⠀⠀⡎⢀⢇⣴⣿⣧⣌⡉⠲⢌⢳⣿⢢⠶⢻⢿⣾⣿⣻⣧⣤⡙⠿⣟⢿⣞⣽⢻⣿⣷⡄⡼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡴⠟⠁⠀⡏⠀⢰⠃⢸⠉⠓⣶⣤⠈⠻⣦⣰⡝⣼⡃⠀⠀⡇⢹⣿⣯⣻⣯⠙⠶⣦⠀⣙⣿⠭⠛⠉⢸⣵⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡴⠋⠀⠀⣠⢾⠃⠀⠼⣀⣴⢟⣿⣾⣷⢷⡀⢸⣷⡼⢹⣧⠀⠀⡇⡎⢸⣿⢽⣿⢦⠤⠬⣙⠚⠋⠛⠷⠾⠿⣿⣞⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠟⠀⢠⣶⣮⣵⢻⠒⠚⣉⣼⣷⢿⣿⠿⣳⣾⣷⢸⢇⠇⢸⣷⠀⡠⣣⠃⣸⣿⡾⣾⣿⠇⠀⠀⠙⠒⠤⠔⠒⠒⣿⣿⣿⣺⡷⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣏⡼⠳⠾⠿⠾⢷⠿⢾⣩⣿⣿⢚⡻⣤⣿⣿⣿⣿⣞⢀⣰⢟⣥⢞⢽⢇⣠⢯⣏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⢻⣯⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡟⠈⠿⢭⢥⣴⣶⡒⣿⣿⣿⣿⠏⢠⣿⣿⣿⣿⣿⣿⢾⢟⣾⣻⡖⣧⠛⠋⣳⣿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠄⣷⢾⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⣸⣿⣿⣿⣿⣮⣫⣽⣧⣴⣿⣿⣿⣿⣿⣿⠟⣭⡾⣟⣿⣿⣬⢷⣾⣯⠪⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢳⠀⠀⠀⣿⣼⣁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⢿⣿⣿⣿⡿⠟⠽⡻⢛⣾⣾⡿⡟⣿⡏⢯⢿⣧⣱⠀⠀⣠⣶⡄⠠⠴⠋⠀⣰⡄⢧⠀⡀⣸⣾⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢧⡟⡿⠹⣿⣟⡿⠁⠈⢱⣦⣖⢋⡴⠚⣫⢤⣮⣷⣟⢅⢼⣼⣿⡘⣞⣿⣯⣧⠖⠉⢈⣥⡀⠀⣀⠜⠃⠸⡄⠓⣽⣿⣿⣯⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣷⡷⣾⣽⢿⣿⠲⢦⡿⠟⠕⠁⢀⡾⡢⠋⢴⣢⠔⠛⢟⣿⣿⣯⣽⣬⣿⣾⣧⠔⠋⢰⠏⠉⠁⠀⠀⠀⠘⢺⣿⠋⡏⡆⣯⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠹⣆⠈⠓⠚⠋⠀⠀⠈⠳⣄⠰⢛⣴⣦⡖⡋⣱⠂⡢⢊⡾⢱⣽⣶⠆⡈⢿⣆⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢙⣖⢯⠝⣶⣤⡈⠙⠷⣄⠀⠀⠀
⠘⣆⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢸⢠⣟⣕⣿⣏⣠⢃⣖⣯⠏⣼⣿⣯⣾⣧⢱⣿⣿⡌⠁⠀⠀⠀⠀⠀⢀⠆⢰⣿⢚⡶⣫⢾⠺⠽⢝⢾⣯⢳⣄⠈⢷⡄⠀
⠀⠘⢷⣄⣀⡀⠈⣷⠀⠀⠀⠀⠀⠀⢸⡸⠋⣾⣿⣿⢯⣿⣿⣵⢿⣳⣿⣿⣿⣿⡧⢹⣿⡋⠀⠀⠀⠀⠀⠀⢼⢀⠟⣯⡭⠭⢓⣚⣦⣀⠈⠳⣍⣛⣬⣷⣄⣹⡄
⠀⠀⠀⠀⠀⠙⢷⣼⡆⠀⠀⠀⠀⠀⠀⠳⣜⣟⢿⣾⣿⢳⡟⣟⣿⣿⣟⣿⡿⢋⣼⣿⠯⢔⡲⣄⠀⠀⠀⠈⢙⢎⠑⣷⣴⣾⣿⠟⠁⠈⣷⣄⠘⣯⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠚⠚⠛⠉⢀⣼⣿⢯⣤⣿⣿⡡⣔⠺⣁⠀⠉⠁⠀⠈⠁⣀⡕⠂⠽⣮⢟⠁⠀⢀⣼⣾⡆⣸⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣾⠇⠀⣿⠟⠁⠀⠛⠀⠈⠉⠒⠒⠒⠒⠊⠁⠙⠦⣀⣹⣿⡆⢠⣼⡾⣽⠾⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⠈⠀⠀⠀⠀⠀⢀⣤⣦⢦⣄⠀⠀⢹⣿⣿⡘⣷⠟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡻⠉⠀⣸⡄⠉⢷⠛⢿⣿⣶⣤⡀⠀⢀⢾⣿⣌⠃⠛⣧⠀⠈⢿⣯⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⠁⢀⣴⡃⣷⡀⠀⠣⡀⢯⠙⠺⠻⣷⠞⠀⠻⣿⣶⣷⣿⣥⣴⣿⣿⣏⢷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢠⣾⡷⠀⢈⢧⠀⠀⠘⢎⠃⡀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⣿⣻⠾⠇⡿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⣾⣿⠁⠀⠀⢸⡀⠀⠀⠀⢣⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⠀⠀⠀⠀⣿⠗⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⢿⣦⣀⣰⠋⠀⠀⠀⠀⣏⠀⢀⣀⣀⣀⠀⢇⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣼⣟⠭⠭⠤⠤⡤⠤⡬⢭⣛⠿⣿⣿⣿⣷⡼⣄⠀⠈⠑⢤⡀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢷⠎⠁⠚⠁⠀⠀⠀⠈⠉⠈⢱⢄⠘⣾⡛⠁⠙⡄⠀⣀⣀⡑⣄⠀⠀⠀⠀⣣⠀⣠⠺⣿⣿⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢑⣵⣢⣤⣽⣿⣷⣿⣿⡿⡀⠀⠀⣀⣠⡴⠛⢦⣸⠏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡇⠀⠀⠀⠀⢀⣀⣀⡀⣀⠀⠀⠀⠀⠹⣿⣿⣿⣾⣿⡯⠾⣊⣡⠤⠝⡻⠯⢽⣳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⣚⡽⠿⠭⢭⣭⣭⣿⣻⣷⡀⠀⠀⣰⣿⣛⠛⠲⠢⠭⠋⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠒⠚⠉⠉⠉⢿⠟⠽⣿⣿⠷⣴⡛⠚⡇⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⠟⣧⢤⡧⠤⣖⡶⠯⠭⠵⣶⡶⢫⠾⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀