⢸⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⣭⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢀⡂⠤⠅⠑⠒⠀⠀⠀⠈⠉⠛⠐⠲⢄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣿⣿⢸⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⢮⡁⠀⠀⠀⡀⠔⡊⠅⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣸⣿⢯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⢣⠆⢀⠔⡨⠔⠈⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢤⠀⠀⠀⠀⠀⠀⠀⢧⣿⣿⣽⣿⣻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠼⢿⠖⡡⠊⠀⢆⠱⡀⢄⠤⠐⠌⢢⠐⢍⠉⠉⠉⢈⠑⢄⠀⠨⡂⠐⠄⠀⠀⢄⢢⠀⢄⠈⢳⣤⠤⠤⠤⠤⢠⣼⣿⣿⣿⣾⣯⣿⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⡶⠃⡰⢀⢰⠀⠈⢢⠱⡈⠒⠒⠈⠀⠱⡀⠉⠀⠉⠀⠉⠀⠀⠀⠈⠉⠁⠀⠀⠈⠃⠣⢄⢢⠀⠹⣿⣿⣻⢿⡿⣶⡷⣿⣻⢿⣽⣻⣟⣿⣻⢿⣟⡿⣿⣻⢿⣟⣿⢿⡿⣿⢿⣿⣿⣿
⢁⡣⠃⠁⢸⠀⠀⠀⢡⣱⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢪⣷⣯⢿⣽⣳⣟⡷⣯⣟⣾⣳⣟⣾⡽⣟⣾⢿⣽⣟⣯⣿⣾⢿⣻⣿⣻⣷⣯⣿
⠎⠀⠀⠀⢸⡀⠀⠀⠀⠱⣧⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢸⡜⣿⣿⣾⣷⣿⣿⣷⣿⣾⣷⣿⣾⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣷⣿⣷⣿⣿
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠄⢹⡆⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢘⡾⣹⣷⣻⣞⢷⣻⣞⡷⣯⣟⡾⣯⣟⣾⣽⡷⣟⣯⣿⣽⣟⣯⣿⣻⣟⣿⣻⣿
⠀⠀⠀⡄⣸⠊⢏⢆⠀⠈⢦⣳⠀⠀⠀⠀⠀⢘⣯⢆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠠⠀⢠⣧⣮⠗⡽⣷⣻⣞⣯⢷⣯⣟⡷⣯⢿⣽⣯⢿⣞⣿⣟⣯⣷⡿⣯⣿⢷⣿⣻⣽⣿⣽
⢠⠀⠀⡗⢿⠻⠋⠙⢿⢓⠢⣽⡆⠀⠀⠀⠀⠀⢻⣯⡆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣼⣻⣧⢻⠵⣟⣷⣻⣞⡿⣾⣽⣻⡽⣟⣾⡽⣟⣯⣷⢿⣯⣷⡿⣿⣽⡿⣯⣿⣻⣾⢿
⠀⣆⠀⢱⠈⡆⠀⠀⠀⠉⢹⢶⣷⠀⣀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⢀⠰⠀⢀⠀⡀⠀⣸⣏⣹⣏⠿⡎⣿⣷⢿⣾⣹⣷⣏⣷⣿⢿⣏⣿⣿⣹⣏⣿⡿⣾⣿⣏⣷⣿⡿⣷⣿⣏⣿
⣧⣹⣄⣈⣶⢻⣽⣿⡿⣿⣿⣿⡿⡍⠀⠀⠀⠀⠀⣧⣄⠹⡀⠀⠀⠀⠠⠀⠀⢀⠀⠀⣴⠿⣜⣻⣭⢻⢱⣿⣯⢿⣞⣿⢾⣽⡷⣿⣻⢿⣾⢷⡿⣯⣿⣽⡿⣾⣟⣯⣿⣟⣿⡷⣿⣿
⣯⣻⢟⣧⠀⡼⠟⢸⡷⣿⢿⡜⠀⡇⠀⠀⠀⠀⠀⢋⡉⠃⢱⠀⠀⠀⢰⠀⠀⠀⣀⡾⣏⠿⣜⡳⣎⢿⢸⣿⣽⡿⣯⣿⣟⣯⣿⣟⣿⡿⣯⣿⢿⣟⣷⣿⣻⣯⣿⣟⣾⡿⣽⣿⢿⣾
⣿⣽⣻⢿⣧⠀⠀⠈⢧⡹⠟⠀⠀⡇⠀⠀⠀⠀⠀⢸⡱⠀⣼⠂⠀⠀⡏⠀⣠⣾⡻⣝⢮⡻⣜⡳⢭⡎⣿⣿⢯⣿⣟⣷⣿⣻⣷⢿⣯⣿⢿⣽⣿⣻⣿⢾⣿⣽⣷⣿⣻⣽⣿⣻⣿⢿
⣿⢾⣽⡯⠏⠀⠀⠀⠂⢄⢠⠄⠀⢡⠀⠀⡄⠀⠀⡾⢀⣴⡇⠀⠀⢰⣇⣼⢻⣜⡳⣝⢮⣓⢧⡹⣹⢰⣿⣿⣿⣯⣿⣟⣾⡿⣽⣿⣻⣾⢿⣻⣾⢿⣽⣿⣻⣾⣟⣾⣿⣻⣽⣿⣯⣿
⡿⣯⢿⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⢸⠀⢰⠃⠀⢀⣿⣿⠿⠀⢰⣠⣿⡹⣎⠷⣎⢷⡹⣚⡼⢢⢳⢣⢿⣿⣿⣾⣟⣷⣿⣻⣽⡿⣷⡿⣯⣿⢿⣽⣿⣻⣾⣟⣷⣿⣻⣽⣿⣯⣷⣿⣿
⡿⣭⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣾⠀⣠⣻⠧⣿⠇⢠⡷⣻⢶⡹⢎⡟⣬⠳⣭⢓⠬⣃⣟⡾⣼⣿⣿⣷⣿⡿⣽⣿⣻⣽⡿⣽⡿⣽⣯⢿⣞⣯⣷⢿⣯⢿⣟⣿⣾⢿⣽⣷⣿
⠓⠯⢯⢷⡀⢳⡆⠀⠀⠀⠀⠀⠀⣆⡞⣧⠞⣯⣯⣊⡟⣠⢿⣱⡟⢶⢩⠳⡜⢢⣻⠇⣍⣲⣽⣞⢇⣿⣿⣿⣿⣾⣿⣿⣷⡿⣯⣿⣻⣽⣟⡾⠿⠽⠗⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⢀⣀⠰⠯⠵⣊⣽⠿⠋⡼⣜⣛⣾⡯⢝⡂⢣⢑⠬⣵⠟⣠⣾⣿⡟⣬⡿⢿⢿⣿⣿⣽⣿⡾⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⢟⡶⡶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠚⡗⢋⣡⣞⣑⣢⣼⡶⢿⣻⣫⠼⢋⣛⠟⠷⣿⠁⢀⣠⠔⠒⠉⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⡀⠀⢈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⢀⠠⠄⠒⠊⠉⠁⡠⠂⠉⠀⠀⠀⠈⢻⡋⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢷⠀⡠⠂⠅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⠸⡌⠀⠀⠀⠠⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢸⠇⠀⡇⡀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣦⣀⠀⠀⠈⡇⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⡞⠀⠀⡟⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣷⣍⡠⡇⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡏⠀⢰⠡⢀⠄⡀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣟⢤⡠⠘⡇⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⣇⢦⡟⡜⣢⠜⠈⠀⣀⢄⡂⠥⠀⠒⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠿⠟⠒⠒⠉⠉⠉⠁⠐⠒⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⢀⣀⠠⠬⠤⠒⠛⠺⠿⠽⢥⣎⣈⠉⢉⣁⣀⣀⣀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠤⣀⠀⠀⠠⠄⣲⠖⢊⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢅⡀⢠⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄⣀⠀⠈⠱⡄⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡄⠀⠀⢻⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠤⠸⡌⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⢃⢹⣄⠀⠀⠀⠀⠀⢀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣦⢖⡲⠭⠃⠀⠀⠀⠀
⠓⠒⠒⡶⢷⡶⢶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⢶⢿⡶⠒⠚⠛⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢱⠀⢹⣉⣿⣆⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠉⣹⡾⡏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⣾⣎⣷⡈⠱⢆⠀⠀⠀⠀
⠀⠀⠀⣸⣴⠟⠊⠁⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠉⠑⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠀⠀⠀⠈⠙⠓⢦⣬⣢⣀⠀
⣠⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠤⠒⠂⠈⠉⠉⠉