⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⣤⠤⢦⡤⣴⣲⡴⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣻⢷⣯⠿⣽⣏⢲⣱⣞⣷⣻⢷⣻⢾⣽⣻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣯⣟⣯⠟⡿⣞⡷⣦⡽⣯⠷⢛⣺⣙⣯⠻⣟⡾⣽⣻⢾⣽⣻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣯⣟⡷⣋⣴⣻⢷⣮⣝⣷⣻⣥⣞⣯⢷⣻⣞⡿⣶⣝⡳⣯⣟⡾⣽⢯⣟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣽⣳⢏⣶⣻⢾⣽⣻⢾⡽⣞⣷⣻⢾⡽⣯⢷⣯⣟⡷⣯⢷⣝⢺⡽⣯⣟⣾⣳⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⣟⡷⢭⣞⡷⣯⣟⡾⣽⣏⢿⡽⣾⡝⣣⢿⣽⣻⢾⣽⣻⣽⣻⣞⣷⡹⢷⣻⢾⡹⣯⣟⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⣟⡾⣱⣟⡾⣽⣳⢯⣟⣷⣻⣎⢽⡳⣼⠯⢻⡾⣽⣻⣞⡷⣯⢷⣻⢾⡽⣎⢽⣯⢷⡹⣾⡽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡿⣾⢱⣟⡾⣽⣳⢯⣟⡞⢠⢙⠺⣧⡟⣉⠲⡁⠹⡌⢷⢯⣟⣮⢻⡽⣯⢿⡽⣆⢻⣯⢷⡹⣿⡽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣻⠇⣿⢾⣽⣳⣏⡿⡞⢀⢂⠆⣥⣬⡴⣶⣳⢾⣅⠹⡄⠹⢾⡽⣎⢽⣯⢿⣽⣻⡄⢿⣯⢷⣹⢿⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡾⡽⢸⢯⣟⣾⣳⢭⣿⡑⢌⣊⣾⣳⢯⣟⡷⣯⣟⣾⣳⠭⡐⠌⢿⣽⠘⣯⣟⣾⣳⣟⠨⣟⡿⡌⣿⢾⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⣽⣻⠕⣞⡿⣞⣷⣻⢺⡷⢐⢊⣾⣳⢯⣟⡾⣽⣳⣟⡾⣽⣳⠜⣧⠈⢿⡣⢹⡞⣷⣻⣞⣇⠸⣝⣿⢸⣟⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣟⡷⣏⢲⣯⢿⡽⣞⡷⢸⡓⣈⣾⣳⢯⣟⡾⣽⣳⣟⡾⣽⣳⣟⡆⢻⡿⣄⢳⠄⣻⢳⣟⡾⣽⡄⠻⡜⡆⣿⡽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⢯⡿⣅⢺⡽⣯⢿⡽⣇⠸⡅⢺⡷⣯⣟⡾⣽⣳⣟⡾⣽⣳⣟⣾⣳⢸⡿⣽⠌⠐⢨⢵⡾⣽⣳⢷⠈⣳⢁⢻⡽⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣯⡟⡄⣻⡽⣯⢿⣽⡁⢘⠀⢻⣽⣳⢯⣟⣷⣻⢾⣽⣳⣟⣾⣳⢿⠸⣋⡵⣖⡌⢆⣻⣽⢳⣯⣟⡆⡐⢣⢸⡿⣽⠂⠀⣀⣀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢺⣷⠣⣀⡿⣽⢯⣟⡾⣅⢈⠎⢶⣬⣙⠻⠞⣷⢯⣟⣾⣳⢟⠚⣉⠡⢞⠹⠋⠛⠈⠀⠘⠋⣙⡾⣽⡆⣡⣃⡬⠵⠒⣋⢍⣥⠛⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢮⠑⢸⢱⣯⡟⡾⡽⠂⠘⠀⠉⠘⠋⠫⠔⢢⡹⣞⣷⣻⣞⣷⡰⠈⡀⠠⢀⠀⡀⡶⠀⣴⡏⣿⣳⢯⠑⡩⢀⣇⡱⣼⠞⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⡤⣤⢤⡭⢦⣭⣘⢸⣷⣻⢷⣦⠀⠰⣎⡄⠀⢀⡀⢠⠘⣿⡽⣞⣷⣻⣞⣧⣟⣷⡠⠯⢧⣵⣳⢸⢳⡏⣾⣽⣻⡀⠇⣠⡴⣛⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⠦⣒⠴⠠⡄⠇⣸⢷⣯⣟⢾⣇⠲⣯⢳⡌⣓⣫⣼⣻⢷⣻⡽⣞⣷⣻⢾⡽⣾⣽⣻⢷⣯⢿⠈⣿⡘⣷⣏⡷⡃⢬⡱⣾⡽⣞⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⢮⣛⠶⢭⡐⢽⣻⢾⡽⣎⢽⡌⢿⣯⢿⣽⣳⢯⣟⣯⢷⡻⢽⡾⣽⢯⣟⡷⣯⣟⡿⣞⡏⢂⠧⣙⢾⡱⠏⡇⢆⡳⢯⡿⣽⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⡜⣸⢯⡓⢦⠨⡜⣯⢿⣽⡳⢆⠳⡘⣯⣟⡾⣽⣻⢾⡽⣯⢷⣸⢿⣽⣻⢾⣽⣳⢯⡿⡽⣰⠉⠰⣩⣞⡱⢉⡇⢎⡜⣫⢿⣽⠂⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⢰⠁⣯⡟⡼⡡⠇⣸⢹⣏⢾⡽⣎⠀⠱⡘⣯⣟⡷⣯⢿⡽⣯⢿⣽⣻⢾⣽⣻⢾⡽⢯⣛⣵⠋⡴⢈⡷⢎⡱⢀⠆⢣⠎⡵⣻⣞⡇⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⠆⢠⢿⡙⢶⡡⠇⢸⠸⣏⢎⢿⡎⠔⢢⠱⣌⡻⣽⢯⡿⣽⢯⣟⡾⣽⣻⢾⡽⣯⣟⣷⣛⣪⠟⠀⣜⠫⢆⡱⠀⠂⢡⠛⡴⢹⣞⡇⠀⠆⠀⠀⠀⠀⠀⠀
⠀⢠⣀⡄⠀⢨⣟⡹⢦⡑⢣⢈⠒⣽⢊⢮⡹⡄⢃⠦⡙⢿⡽⣯⢿⣽⣻⢾⣽⡳⣭⢿⣽⣳⣟⡾⡽⠃⡘⠰⡨⢃⠎⣀⠎⠁⠂⣝⢢⠇⣿⣳⠀⠈⡀⠀⠀⠀⠀⠀
⠀⠈⠇⡁⠀⢼⡧⢓⠦⡙⠆⡌⡂⢘⠧⡚⡴⠡⡘⢢⡑⢢⢉⠛⢿⡾⣽⣻⢾⡽⣯⣟⣾⣳⠯⢋⡔⠠⠁⡁⠰⡍⠀⣎⠸⢀⠁⢜⡢⢝⢲⣻⡀⢀⠰⡀⠀⠀⠀⠀
⠀⠈⠀⢠⠄⣿⠱⣋⠼⡁⡇⢠⢡⠈⢪⠱⡸⡁⠤⣁⠚⠠⢋⠆⠀⠉⠓⠯⣿⣽⣳⡟⠞⢁⠰⡌⠆⠀⠎⢠⢃⠰⢁⠆⣼⢀⡃⠈⡖⣩⠆⡽⣃⠌⠂⠁⡴⠀⠀⠀
⠀⠀⠀⠘⣠⡏⡵⣉⠖⢁⢃⠲⠘⡤⢣⡁⠇⡑⠂⡅⠆⡘⠥⡘⠀⠀⠁⠀⠀⡉⠁⠀⠐⠈⠀⠀⠀⠐⠡⠄⡂⢅⣢⡟⣸⠆⡩⠄⢘⠡⡚⢥⢳⠈⠄⠀⠘⠀⠀⠀
⠀⠀⠀⠀⣾⢑⣎⡱⠊⡔⡊⠄⢑⣿⢠⡙⠲⠤⡐⠈⠀⡍⢢⠱⡁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡉⢎⡁⠴⣯⠷⣠⢹⡇⢡⢋⠄⢣⠙⢦⠹⡌⠀⠀⠀⠀⠀⠀
⠀⢀⠂⣸⠇⢮⡰⢁⠲⣡⠓⡄⢈⢾⡸⣯⣃⠇⡜⠘⠐⠈⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⣻⣙⣶⣻⠆⡇⠸⣌⢒⡈⢖⡈⢧⠹⡀⠀⠀⠀⠀⠀
⠀⠌⠀⠋⠌⠀⠀⠀⠀⠀⠁⠈⡼⣆⡣⢿⡽⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀Ａｌｐｈａ⠀⠀⠀⠀⠀⠀⠀⡷⣹⢾⣽⡂⠙⠰⠈⠖⡰⠈⠖⡌⢣⢱⡈⠀⠀⠀⠀