⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣹⠃⠀⠀⠀⣠⣤⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣚⠝⠁⡰⠃⣀⡠⢖⡿⢿⡟⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⣠⢿⠽⢿⡷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⠃⣠⡄⠀⢀⣀⡤⢶⡏⠑⠋⠁⣠⠾⠞⢉⡥⠞⠉⣠⠊⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣉⡉⠿⠛⢷⣄⠀⡤⢭⣷⣤⣀⠀⣰⠆⠀⠀⣀⡤⢤⣖⣻⡷⠿⣕⣊⡽⠗⣾⠉⠁⠀⣀⢧⣤⠖⣋⠥⠖⠊⠁⢀⣠⠞⢓⣾⢟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡯⠗⣲⣿⠷⠀⠀⠀⠹⣧⠀⠉⠉⠛⢿⡛⠁⠀⢠⠏⠁⢠⣹⣧⣤⣤⠞⠉⠀⣷⣞⣽⡷⠒⠋⢁⣀⣼⢁⣀⣀⣠⣴⣞⡩⠴⠚⢉⡴⠋⠀
⠀⠀⠀⠀⠀⠀⢠⣾⡉⠩⠒⠺⡏⠀⠀⣀⠀⠀⠀⣿⡆⠀⠀⠈⠙⣿⣦⣀⣾⣀⡆⣘⣿⠓⢲⠏⠀⠀⠐⠒⠶⢲⠿⠿⠿⠭⣽⣉⣉⣉⣉⣉⣉⣀⣠⡤⠶⢟⣲⣶⠀
⠀⠀⠀⠀⠀⠀⠘⢿⣶⠷⠒⠚⠛⠒⢺⠁⠀⠀⠀⢹⢳⡀⠀⠀⠀⠙⣟⢻⠿⠁⣸⣾⣏⠛⡟⠀⠀⠀⣌⣉⣛⣻⠦⠤⢤⣴⣇⣀⡒⠒⠚⠛⠛⠓⠒⠛⢉⣩⠴⠋⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⢿⣧⠀⠀⢢⡀⠚⢛⣲⣷⣦⡟⢦⣹⠃⠀⢠⣕⠶⠤⣬⣟⠒⠒⢶⢧⣀⡙⠻⠯⠭⣟⣚⣛⣉⣉⣹⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⣀⣀⡈⢻⣇⠀⠀⠉⠉⠙⠳⣌⣿⣿⠾⠃⠀⠀⣷⢮⣛⢲⢧⣌⠛⢲⡾⣄⡳⠭⣙⡓⠶⠤⣤⡤⠤⠖⠛⠁⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⡼⢩⡷⠖⠋⠉⠁⠀⠀⠙⠿⢿⡧⠴⠖⠒⠚⠛⠉⠉⠀⠀⢀⢠⡱⣎⡙⢻⡋⠳⣮⣩⢿⣄⠀⢿⣛⠦⣌⣉⣳⡶⠞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡴⣋⠤⢤⣈⠙⠲⠾⠁⠈⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⢠⢀⢀⢀⢠⣰⣎⢧⣙⡮⣟⢿⡙⠲⣤⣷⣀⠿⣷⣤⠬⣷⣦⣽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡞⢁⡞⠁⠀⠀⠙⢷⡀⠀⠀⠀⠀⣀⣠⣞⡀⢀⣀⣀⣤⣠⡄⣠⠆⡷⠿⡼⣿⣨⣯⣾⠿⡿⢏⢷⡈⢣⣹⡷⢻⣜⢮⡟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⣃⣿⠀⠀⠀⢀⣀⣀⣙⣦⠤⠞⠀⠉⠀⠀⠉⣡⠬⠓⠉⠙⢷⣏⣿⣧⣸⠁⡇⣿⢸⠸⡆⢳⡈⢦⣙⣶⡏⣷⣘⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⡴⠏⠀⠀⢠⠋⢉⣉⣀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠈⠻⠿⣏⡎⡿⠛⡿⢮⢦⣽⣄⡽⣶⢻⣤⡻⣜⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡇⢧⠀⠉⠉⠉⠛⠓⢶⣞⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠾⣿⣸⣾⣆⣾⣶⣜⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⠘⢧⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠙⢾⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠳⣜⢳⣦⠀⠀⠀⠀⠀⠀⠙⠿⣷⠦⣄⣀⠀⠀⠀⢠⣤⠖⠋⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣁⡞⠋⠉⠁⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠴⠋⠁⢰⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡟⢻⠲⠤⢴⡞⢿⣿⣧⣈⡙⠳⢦⣤⡶⠋⠀⣠⠖⠓⠲⣤⣀⣀⣀⣠⣾⡿⠃⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠾⠋⠳⠶⠛⠛⠓⠒⠚⠛⠛⠶⢤⣀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣷⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡞⠓⢲⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⣤⣤⣽⣿⣿⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣮⡟⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀