⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣐⡤⠖⠋⠉⠀⠀⠇⢆⠀⠀⠀⢳⡀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠉⠙⠶⣄⠀⠀⠀⠀⢰⠏⣜⡹⣟⣿
⠀⠀⠀⢱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⣠⠴⠞⡋⠁⠉⠙⠒⠲⠦⡧⣤⣀⣄⣠⠤⠴⠒⢫⠁⠀⠀⠀⠀⠀⠀⠰⠸⡀⠀⠀⠘⡌⢦⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠙⠲⢄⣀⢸⣟⡼⣽⣻⢾
⠀⠀⠀⠈⠀⠙⠓⠒⠒⠒⠒⠒⠚⠛⡻⢹⠃⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡆⢣⠀⠀⠀⢣⠀⠉⠢⡀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⠄⣀⢀⡙⣯⣿⣳⣯⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⠸⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢷⡘⡄⠀⠀⠈⣦⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⡾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡀⢧⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣯⢻⠀⠀⠀⢋⢣⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⢼⠀⠀⠀⡈⡆⠀⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⠀⠀⡀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢸⡭⢧⣃⠀⠀⠘⡄⢳⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⡰⠁⡇⠀⠀⠀⠀⠀⠀⠸⡀⢳⡀⠀⠀⠀⢹⠀⠀⠸⡀⠀⠀⠀⠀⠀⢸⡆⢿⠙⣆⠀⠀⢱⠀⠹⡦⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠩⠥⠶⠶⢤⣤⠴⠒⢁⢼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⣿⠀⢠⠇⢵⢷⠀⢸⡀⠀⠀⠀⠀⢧⠈⣧⠀⠀⠀⠈⡇⠀⠀⡇⠀⠀⠀⠀⢠⠀⡇⠀⠱⣼⣆⠀⡀⢇⠀⠙⣝⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠸⡒⠒⠊⠉⠁⢀⣐⡶⢋⠜⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣿⠀⡞⠀⡿⣸⡀⠀⣇⠀⠀⠀⠀⠘⡆⠸⣆⠀⠀⠀⢹⠀⠀⡇⠀⠀⠉⠉⣹⠉⡏⠒⠂⠐⢽⣦⠐⢈⢆⠀⠈⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢇⠀⠀⠀⢀⣵⣫⠞⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢈⣿⣾⠃⡼⠀⣿⣇⡀⣸⡆⠀⠀⠀⠀⠘⡆⠘⣄⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠈⠻⣦⠀⠘⣦⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣤⠴⠞⠋⠉⠀⠀⠀⠀⢀⡤
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣿⡇⢸⣃⠜⢣⣿⠀⠘⣿⡀⠀⠀⠀⠀⠸⡄⠸⡄⠘⡀⢻⠀⠇⠀⠀⠀⢸⡿⢠⠇⠀⠀⠀⠀⠀⠣⣇⠀⠸⣤⣜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⡇⠀⠀⠀⢀⠀⣀⣠⢜⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⣇⣸⣿⣺⠃⠀⠀⠈⣿⡆⠀⢧⢷⡀⠀⠀⠀⠈⠹⣄⠹⡄⢣⠘⡆⢸⠀⠀⠀⣾⠃⢻⠂⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣧⡤⠖⠒⢾⡽⢿⣞⠁⠀⠀
⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⡇⠠⣿⣏⠐⡿⠁⠀⠀⠀⠀⠘⣷⠀⢸⡆⢳⡀⠀⠀⠀⠀⠘⢗⡼⣎⡆⢱⠸⡀⠀⡼⠃⢘⡘⠀⢀⣀⣀⣀⣀⣀⣀⣀⠙⢦⡈⢻⡀⠀⠀⠀⠀⢠⠄⠀⢸⠀⢸⡆⠸⣀⣴⢖⡺⣇⣯⠧⣤⡀⠀
⠐⠂⠀⠀⡂⠀⠀⠀⠀⠀⡇⠀⠿⢃⠘⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⣷⠙⢷⣄⠀⠐⢄⠀⠀⠱⣼⣿⡀⢇⡇⡼⠟⣻⣿⣧⣥⣴⣿⣶⣾⣿⣿⣿⣿⣿⣶⣽⣦⣙⣮⣤⡴⠞⠉⠀⠀⠈⡆⢸⣿⠀⢳⣈⣷⢻⡿⣞⣟⣨⠇⢀
⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⡇⠀⣰⠻⡄⣇⠀⠀⠀⢀⣚⣉⣉⡘⢆⠘⣇⠀⠘⢦⣄⠈⠐⢇⡀⠀⠉⢿⡾⣟⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣶⡀⠉⠉⢻⣿⠛⠷⣄⠀⠀⠀⠀⠀⡇⢸⣿⡆⠸⡈⣇⠟⣫⣟⠸⣧⠀⢠
⠠⠀⠀⡜⡁⠀⠀⠀⠀⠀⢱⠀⢸⠀⡇⣇⣒⣞⣯⣥⣦⣾⣭⡉⠈⠢⡹⡄⠀⠀⠉⠳⢤⣀⢙⠦⣀⠈⢿⣜⢏⠀⠐⣿⡿⣽⡟⠿⣿⠞⡙⣯⣟⣿⡇⠀⠀⣸⡏⠀⠀⠈⢙⠫⠗⠲⠶⡗⠈⣏⣿⡀⢷⡈⡉⢇⠙⢛⣅⡰⠃
⠀⠀⢰⠻⡄⢇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠻⣄⠀⠀⠀⠀⠈⠉⠑⠚⠓⠬⣫⡙⢦⡀⠹⣿⡽⣆⠡⠌⡒⢤⣟⠾⣼⡇⠀⢠⢿⠀⡠⠀⠀⢸⠀⠀⠀⠀⣇⠀⣿⠸⣇⢸⣖⢓⡒⢓⣸⠏⣀⠴
⠀⠀⢸⡀⡇⢸⠐⠦⣄⣤⣴⣾⣿⡟⠻⡿⣿⣿⢿⡿⢯⣿⠞⢳⡿⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠹⣿⢭⠳⣜⣰⡟⡜⣫⣼⣡⠴⠫⢼⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸⠀⢠⣀⣿⡄⣿⢥⣶⢞⣡⠞⠁⠀
⠀⢀⡀⠙⢇⠈⡇⠀⠀⠀⠀⢿⢿⣷⠀⠘⣟⢿⣻⣄⠢⢉⡐⣸⠻⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠾⠽⢶⠾⠛⠻⣋⣁⠔⠁⡞⢸⢸⠀⠀⡜⠀⠀⠀⠀⢸⠀⡸⣡⠺⡇⢹⣾⣿⠿⠛⡆⠀⠀
⠀⠀⠘⢲⣘⡄⢹⡀⠀⠀⠀⠸⣆⠙⡷⡄⠙⣌⢿⣯⢳⣆⣴⣃⣟⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠃⣾⢸⠀⠀⡇⠀⠀⠀⠀⣾⠀⢠⣼⠟⡇⢸⠛⠙⢀⠀⡇⠀⠠
⠀⠀⠀⠀⠉⢷⡄⢇⠀⠀⠀⠀⢻⣆⢱⡈⠲⢤⣭⣽⣽⡟⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⣻⢸⠀⢀⡇⠀⠀⠀⠀⡿⠀⢼⠀⡆⡇⡼⡇⠈⡞⠀⢱⠀⠀
⠀⠀⠀⠀⠀⠀⠳⡈⢦⠀⠀⡀⠀⠻⣦⢳⠈⠲⠯⠤⠘⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠏⢽⡎⠀⡼⢀⢠⠀⠀⣼⣻⠀⡟⠔⢸⣧⠃⡇⠀⢳⡇⢸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢹⢄⠳⡄⠂⠀⠀⠙⣧⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⠋⠀⣾⠇⣰⠇⡎⠨⠀⢠⣏⡏⣸⣋⣮⣿⠃⢠⣧⡄⠸⣇⠀⠇⠀
⠀⢠⡀⠀⠀⠀⢀⢸⠀⠑⢬⡲⣌⠓⢤⣈⠻⣕⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠊⠀⠀⠀⢸⢰⡟⣰⠁⠀⢠⠎⡿⣴⠛⡏⠁⢸⣰⠘⡛⠁⠀⢹⡄⠀⢈
⠀⠀⠻⣇⡒⢂⠏⢸⠀⠀⠰⢹⡙⠓⠦⣙⣷⣬⠷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⡴⠃⠀⣠⠋⣰⡟⠡⠐⢿⠄⠈⡟⠀⠀⠀⠀⠀⢃⠀⡀
⠀⠀⠀⠀⠉⠁⠀⢸⠀⠀⠸⡀⡇⢣⠀⠀⠩⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⠠⠔⣪⣵⠞⢁⣠⠞⣁⡼⠋⡇⢂⠁⠸⡆⠀⠇⠀⡶⡄⠀⠀⠈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢹⢸⠀⠀⠀⢣⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⣩⠍⡾⢉⡴⡏⢀⠀⠀⠀⠀⡇⡐⠀⠀⣷⠀⢰⡇⢇⢣⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠘⡆⡇⠀⠈⡿⣼⡈⠗⣦⣄⡀⠀⠀⠀⠀⡠⠖⢉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣟⣍⣠⢞⡵⠋⠀⡇⠘⠀⠀⠀⠀⡇⠄⡁⠀⡿⣆⠈⣧⢸⠀⣆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣷⠇⠀⠀⡇⡧⡁⠀⠈⠙⡩⡷⣴⠴⠊⠀⠒⢩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⠙⠛⣛⡟⠀⠀⠀⡗⠀⠀⠀⠀⠀⡇⡐⠀⠀⡇⢹⡀⢹⠈⡆⠸⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣼⢸⡌⠀⠀⢱⢹⠄⠀⢀⠔⢣⠞⠁⠀⠀⢀⡴⣅⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡖⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⣀⡇⠀⠀⠀⠀⢀⡇⢀⠂⠀⡇⠈⢧⠈⡇⢣⠀⢱⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠈⡇⠀⠀⢸⢸⠂⢢⢪⡞⠁⠀⠀⣤⠚⣽⡖⠈⣭⠛⢲⣤⣤⣤⣤⡖⠚⢩⢡⠊⣿⠀⠀⠀⠀⠀⠀⠀⣴⣾⠋⠀⢠⣤⠒⠋⢡⡇⢠⠀⠀⠀⢰⠃⡄⠀⠀⡇⠀⠘⡆⢹⠘⡆⠀⢳⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡸⢀⡇⠈⡇⠀⠀⢸⢸⢂⡴⠋⠀⠀⢀⣾⠡⠀⢸⡇⠈⠀⢂⡁⢻⠩⢋⡟⠐⡈⡾⢈⠄⣿⣀⣀⣀⣀⣀⣐⡶⢯⠤⠒⠛⡉⠄⠣⠘⡀⡇⢸⠀⠀⠀⢸⠃⠀⠀⠀⣇⠀⠀⢳⡘⡆⢳⠀⠀⢧⠀
⠀⠀⠀⠀⠀⠀⢠⠀⡎⠀⣿⢸⠁⢰⡏⠀⠀⣼⣸⠋⠀⠀⢀⡴⢃⢸⠀⠀⢘⡇⠀⠀⠠⠁⢸⠀⡾⠐⢂⣐⡇⣆⢸⢻⡭⢯⡝⢯⡟⢣⠑⠌⣀⠃⡡⠘⣈⣴⣱⠴⡗⠸⠀⠀⠀⣞⠘⡀⠀⠀⢹⠀⠀⠀⢷⢳⠈⡆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢸⢀⠇⢸⢳⡞⠀⡞⢸⠀⢀⡿⠁⠀⠀⣠⠏⠃⠑⢸⠀⠀⢸⠁⠀⠀⠀⢆⣺⢰⠇⣉⣤⣾⣼⣶⣿⣿⡧⢳⡜⠯⡘⢄⣊⣴⣤⠦⠵⠚⠋⠉⠀⠀⠄⡏⠀⠀⠀⢸⣷⣶⣶⣤⣼⣀⣀⠀⠈⢿⡆⠹⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡇⡎⠀⡘⣸⠃⣸⠁⢸⡠⠋⠀⠀⢀⡴⠃⠀⠀⠀⢸⠀⠀⣼⠀⢀⣀⣭⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣤⣆⣀⠀
⠀⠀⠀⠀⠀⢠⢳⠀⢀⢇⡏⡰⢃⠔⠋⠀⠀⠀⠀⠚⠦⠤⠤⢤⣤⣸⣀⣤⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡼⠀⠀⢸⡜⡼⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡋⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡇⢄⣰⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⠄⠀⢀⣀⢀⣄⣀⠀⠀⠀⠀⠙⠿⣿⠿⠟⠛⡉⣡⣐⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⠎⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣸⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡐⠀⠀⠀⠈⠙⢗⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡾⠀⠈⠑⠢⢄⣉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣀⠀⠀⠑⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣶⣿⣿⣿⣿⣷⠦⣄⡀⠀⠀⠀⠈⠑⢤⣀⣠⣙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿