⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣒⣒⣶⣯⡿⡿⣉⡰⢶⣤⡛⠉⣀⡀⠤⣐⣒⣦⣴⣶⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⡷⡞⢫⣩⣽⣿⡟⠀⠈⠻⣛⣿⣯⣭⠽⠶⠺⣯⣍⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠙⠻⣒⠦⢤⡐⠒⠛⢛⣓⠺⢟⡴⡻⠽⣿⣿⡇⠀⢿⣄⠹⣿⣭⣝⡻⣿⡿⠿⠭⠭⢿⣍⣛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣉⣙⣶⠥⣖⡭⠟⢚⡳⡟⣽⣜⢿⣥⣾⢿⣷⣿⣿⣿⠇⣹⣶⣶⣶⣶⣶⣖⣦⠭⣵⣢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡦⣤⠤⠼⠓⢚⣷⣽⢣⣿⢫⣷⠶⢛⠿⣛⠩⣥⣶⢪⣿⡳⢯⣻⣿⡻⣯⡻⣿⣦⣝⠿⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⡾⡿⠳⢴⠒⠋⢀⡏⢰⣏⡿⢃⠔⠊⢁⠀⠀⣹⣷⣏⡳⣍⠢⡙⣯⡻⣮⢻⡜⢿⣿⣷⣬⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡠⣞⡵⠛⢩⣾⡾⠾⠟⣛⣊⣉⠴⣛⠜⠀⠀⠐⢉⡤⠔⡲⠋⠙⢻⣿⢸⠳⠙⡘⣧⢹⣇⣿⡘⣿⣿⡝⢿⣌⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠒⠛⠋⢉⢴⣺⠹⠦⠭⣿⠛⠳⠒⠍⠋⠀⠀⠐⠀⢀⡟⢀⠐⠀⠀⠀⣴⠏⣠⢾⡇⠃⢻⡀⣿⢹⣇⢡⢿⣷⠀⠻⣧⡙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⠗⠋⠀⠀⠀⠀⡏⣿⣦⣠⣇⡄⠠⠳⡶⡲⠛⠩⠄⢂⣠⠮⠒⢉⣴⠏⢎⡇⢰⢸⡇⣿⢸⣿⢸⡸⣿⡆⠀⢿⣷⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡀⠀⠀⢱⢻⣿⡍⠀⠀⠀⠈⠈⠀⡠⠰⡪⣋⠤⠒⠚⣿⠏⢈⣾⢃⡟⣼⡇⣿⢸⣿⢸⡇⣿⡇⠀⢸⣿⣇⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⠀⠀⢀⣜⡾⢸⢖⢲⠸⡀⠘⡐⠀⠠⠞⣊⣠⡶⢖⣫⣵⣺⠟⣡⣾⣳⣿⢷⡏⢸⣿⢸⢇⣿⠀⠀⣿⣿⣿⢸⠈⢳⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠊⠀⢀⡮⠋⠀⣸⠼⠼⢤⡮⢗⢄⠲⡒⢪⢶⠿⠞⣿⣟⣯⡶⢫⣿⡿⢛⣿⢸⣷⢸⣿⢸⢸⡏⠀⢰⣿⣿⣿⣸⠀⠀⢻⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡼⠊⠀⠀⣠⠈⠀⠀⠀⠹⡦⠀⠘⠐⠀⢀⠀⠼⢿⠁⠀⣸⠽⣚⣭⣶⠿⠟⢱⣿⣿⢸⣿⣾⣿⠇⣿⠁⠀⣾⣿⣽⣿⣿⡆⠀⠀⣷⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠇⠀⠀⠀⢸⠃⠀⠀⠀⠀⡄⠀⠀⠈⢯⡮⠷⠿⠿⠿⠃⠀⣸⣿⣡⣿⣾⣿⣿⠏⣼⠏⠀⣼⣿⢃⡟⣿⣿⡇⠀⠀⢸⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡴⠋⡠⠀⠀⣐⣮⡰⡀⠀⢠⣾⠖⠒⠒⠈⠁⠀⣠⣿⠟⢋⣹⣿⡿⢃⣼⠏⢀⣾⡿⢃⣾⢹⣿⡇⠱⠀⠀⢸⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠼⢥⠷⠋⠁⢀⠉⠋⠹⣾⣁⠄⣀⣤⣶⣾⡿⣩⣿⣿⣿⠿⢋⣠⠟⠁⣴⣿⠟⢁⣾⢃⣿⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⡄⠀⠀⣿⡀⠀⢰⣿⡇⠘⠿⠛⢁⣴⣿⣿⠟⠋⣡⠴⠋⢁⣤⣾⠟⣡⣴⡿⢃⣾⠋⠀⠀⢀⠆⠀⣸⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⠀⠈⢠⣾⡿⠟⠁⢀⣤⣶⡿⠛⢉⣠⡴⠛⠁⣠⣴⡿⢟⣥⡾⠟⢋⣴⠟⠁⠀⠀⠀⡜⠀⢠⡏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠒⣿⣉⣉⡉⢀⣠⡴⠟⢋⣡⣤⡶⠛⠁⣠⣴⣾⠟⣫⣶⠟⠉⣠⣴⠟⠁⠀⠀⠀⢀⡞⠀⢀⡞⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⠟⣋⣥⣶⣿⣿⠟⠉⣀⣴⣾⡿⠋⣡⣾⠟⢁⣴⡿⠋⠀⠀⠀⠀⢀⣴⠏⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣞⡭⠖⣛⡽⢋⡵⠟⢁⣴⣾⣿⠟⠁⣠⡾⠟⢀⣴⡿⠋⠀⠀⢀⠄⢀⣰⠟⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣡⡴⠞⠁⣰⠏⢀⣾⣿⠟⣽⠋⢀⣴⠟⠁⣴⡿⠋⠀⠀⢀⡔⠁⣠⡾⠃⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡳⢤⣤⣀⣤⡤⠴⠞⠋⣁⣤⡶⣿⠃⣰⣿⠟⠁⡼⠁⢠⣾⠋⢠⣾⠟⠀⠀⠀⡴⠋⠀⣰⡟⠁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠤⠶⠞⠋⠉⠀⢰⡏⣰⡿⠁⠀⡼⠁⢠⣿⠃⣰⣿⠏⠀⠀⣠⠞⠁⠀⢰⡟⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡟⠀⠀⣸⠃⢠⣿⠃⣰⣿⠏⠀⠀⣰⠋⠀⠀⢀⡿⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⢰⠏⢀⣾⠇⢰⣿⡟⠀⠀⣼⠃⠀⢀⣴⢻⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⡏⢀⣿⡟⠀⠀⣸⡏⠀⢀⡾⠁⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣾⣿⠁⣼⣿⠁⠀⢰⣿⠁⢠⡟⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⣾⡿⡏⢠⣿⡇⠀⢀⡏⣿⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣾⠟⠉⢰⡇⢸⣿⡇⠀⢸⠇⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡿⠟⠉⠀⠀⠀⢸⡇⢸⡿⣇⠀⢸⡀⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣧⢸⡇⢿⡀⠘⡇⠸⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⣷⠈⢷⡀⠹⡄⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠈⢿⣄⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠉⠳⢮⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀