⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⡉⡻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣄⣀⠀⠀⠀⠀⠀⠀⢰⡟⠛⢿⣤⡤⢿⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠍⢨⣷⠚⠉⣆⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢘⣬⣯⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⢲⠚⠉⠀⢸⣻⡄⠀⠀⠀⠀⠀⣧⣀⣤⡴⠟⠛⠋⠉⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣮⣄⣀⣀⣠⣷⠷⠤⣴⣖⣾⣯⣿⣼⢯⣭⠄⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⠉⠉⢩⣻⡿⡼⣿⣿⣽⡿⣶⡿⣿⡷⡾⠀⣀⡀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⢷⣺⣹⢭⣿⠀⠹⣶⣿⣯⣶⢮⣷⢖⡂⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣐⣚⣴⠶⠿⠿⠛⠀⠀⠈⢉⢀⣴⣟⣄⣀⣀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠨⠏⠉⣙⣲⣤⣐⠀⠀⣧⠀⠀⢻⣭⣿⣞⣵⠒⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠔⢒⡋⣿⣷⡟⠀⢀⣽⣤⣄⣀⠻⣗⢋⡓⠧⠄⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣁⠾⠛⢲⣧⣴⣛⠻⣏⣽⣿⣟⡿⢯⢅⡩⣅⣀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠭⠄⡿⠿⠽⡛⠓⠃⣿⣈⣤⡴⠒⢒⣩⣭⣥⡀⠘⠷⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡤⠼⠟⢓⣓⡋⠉⠙⢯⡳⣹⡞⠀⣼⣿⣭⡀⠉⢻⡆⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠀⣠⡾⠛⢉⡿⣿⡆⠑⢷⠛⡇⠀⢿⣇⠈⠛⠷⣼⡏⢠⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡁⣰⠟⠁⣸⡇⢀⢸⣧⠸⡄⠈⠛⠷⠶⠾⠋⣠⢎⡟⠁⠉⢓⣄⡀⢀⣤⡶⠿⠛⠛⠛⣿⠿⣷⠦⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠘⠿⣯⣤⡶⠟⢡⢋⣿⡿⡶⣽⣮⣔⣀⣀⣀⣤⣶⡟⠀⠀⢠⢞⢟⣿⠟⠉⠀⠀⠀⢠⡾⠁⠀⠈⢿⡿⣻⣦⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠓⢄⣀⠀⠀⢐⣂⣵⣿⣿⣳⣿⣏⢿⡿⢿⣛⣛⡻⠿⢷⣤⣴⢯⣫⣿⠋⠀⠀⠀⠀⣰⢿⠁⠀⡐⠆⢀⣴⣿⣿⣷⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⡴⣇⠀⠀⠀⢹⣯⣿⣿⣿⣿⠽⢻⣿⣿⣿⣾⡷⢤⣀⠀⠀⠀⢠⣟⣽⣿⣿⠏⠀⠀⠀⠀⣼⣃⡼⢳⡟⠁⢠⠞⣿⡏⠀⣿⣧⠀⠀
⠀⠀⠀⢀⣠⡶⠟⠛⠛⠻⣿⣿⣿⣾⣗⢽⣶⣤⣫⡿⠁⠀⠀⠀⡰⢫⢿⣩⣿⣿⠙⢦⡈⠑⠶⢾⣿⣿⣧⡀⠙⢷⣄⠀⢀⡼⠋⠉⣻⠏⠛⣲⠋⢠⢻⣿⠀⣿⣿
⠀⢀⣴⣿⣝⢦⣀⠀⠀⠀⠈⠻⣯⣿⣿⣿⣿⡿⠿⣿⣶⣦⣤⠞⢠⠏⠘⡏⠉⢹⠀⢦⣙⣦⡀⢾⡛⣿⢿⣻⣦⠀⢹⣶⣟⠀⠀⠀⠉⠀⣼⣧⣤⣲⣿⣿⣾⣿⣿
⢠⢿⣿⠋⠀⠈⠘⣷⢤⡀⠀⠀⢹⡜⠃⣸⠏⢀⣾⣿⡿⣿⢥⣴⣯⣤⣤⣙⣀⣮⠖⠋⠁⠈⣷⠘⣿⣿⠋⣿⣿⣿⣌⠉⢻⣧⣀⣀⢀⣼⣿⣿⣿⣿⣻⣿⣿
⣼⡷⢍⡀⠀⠶⣶⠿⣤⡽⠢⣤⣼⣿⣿⠋⢠⣿⡿⢻⣴⡏⣸⡇⠀⠈⠉⠙⢿⠃⠀⠀⠀⠀⢸⡅⠸⣿⠀⠯⣿⣿⡿⣷⡄⠘⣿⣃⣾⣿⢟⣿⣿⣿⣿⣿⣿⠟⠀
⣿⣇⠀⠙⠦⣀⠾⠒⢷⡀⠀⠈⠙⢿⠏⣠⣿⣿⣧⠸⢾⡇⢸⡂⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⢘⡷⣖⣿⢀⢠⢹⣿⣷⣜⣿⣦⡈⢹⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀
⣿⣿⣦⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⡞⠀⣾⣿⣿⣿⡀⢰⣇⣴⣗⠀⠀⣀⣴⡾⠷⠶⠶⠶⠶⠿⡟⠛⢻⣾⣯⡇⣿⣿⣿⣿⣿⣷⡶⣻⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀
⠸⣿⣾⣶⠐⠠⠴⣿⣿⣦⣄⠰⣾⠁⣸⡏⢻⣿⡍⠀⣀⣧⠶⠿⡿⠛⠋⠁⠀⣸⣀⡀⠀⠀⠀⢻⡀⠈⣿⣿⢁⢹⣿⣿⣿⣿⣿⣿⣷⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⠀⣼⣿⣇⢰⣻⣿⠀⠀⡏⠀⠀⣴⠟⠉⡇⠉⠙⢦⢠⡼⣷⠀⢻⢸⣾⣾⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣃⣸⡇⠀⣿⡟⣿⡀⠹⡿⡇⠀⣯⡐⢾⡃⢀⣠⣷⣦⣄⡈⡟⠀⣻⡀⢸⣼⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⣯⣿⣧⣸⣿⣧⠘⣧⠀⣧⢿⠀⢻⡋⠉⢧⠞⠋⢹⡀⠀⠉⠳⣀⣾⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣏⣧⡈⠀⣼⣾⣇⠀⢳⣀⡎⠀⢀⣼⡷⢦⣄⣀⡿⣻⠀⢸⣿⠿⣿⣿⣿⣿⣀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡌⣿⣿⢻⡄⠀⣳⠹⡾⠋⠀⢱⠀⠀⠈⢳⣸⠀⡼⢁⣼⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡎⠛⢻⣿⣿⣿⣷⣄⠈⣧⡇⢀⣴⠿⠿⠶⠤⣼⢻⠾⡅⣾⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢶⣦⣿⣿⣿⣿⣿⡆⠸⣾⠏⠁⠀⣄⠀⠀⢈⣾⣄⣹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣀⣻⣧⣤⣴⠿⣶⣾⣿⡿⠛⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢾⡍⠈⠉⠙⢿⣏⣻⣿⣿⠟⠀⠀⠀⢀⡈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠰⢯⢴⡄⠀⣐⡿⣸⡿⠟⠛⢷⣄⠀⣤⣹⣌⠛⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⣤⠖⠉⠀⠀⠀⢀⣠⣾⣷⣮⡵⠟⠉⠀⠀⠀⠀⠙⠿⣷⠝⠁⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡇⢸⢳⡀⢀⣠⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⢠⡯⣿⣿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⡴⣚⢛⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⢑⣻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡞⠁⠀⡹⣾⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣾⠻⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣯⣽⡟⢦⡉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠼⣣⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣞⡢⣙⢮⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠘⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣍⢪⢟⣆⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁⠀⣀⣀⡴⠟