⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡉⢦⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢃⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢵⡤⠻⢧⢄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡾⠗⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡺⣿⣉⣉⣉⡓⠲⡖⣚⣉⡿⡇⣾⡿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⡩⠗⠛⠛⠹⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣹⣎⢎⠪⣿⡄⣨⣭⡭⠥⡿⡿⠁⣿⣿⡦⣙⠆⠤⠤⣄⠀⣀⣀⣾⠔⢊⡥⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡏⠉⣻⣆⠈⠙⢿⡿⠁⢀⣃⡧⢤⡜⣏⠑⠺⢍⠒⢲⡦⣽⠯⣍⣹⠾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠒⠲⠶⣄⣤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣲⢣⠎⡇⠙⢦⡀⡠⢷⢔⣯⣽⠥⢆⣷⢻⠀⢀⡬⠿⠛⣪⣳⠟⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠃⠉⠉⠓⠒⠭⠭⢙⣒⡒⢿⠶⢶⣶⡧⢾⣧⡾⠠⣇⣀⠤⣿⣧⡀⢹⠿⢀⣀⡀⢹⣿⡿⢯⠀⢀⣼⡋⢩⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣾⣾⣫⣍⡉⠉⢱⡖⢓⡶⠛⠛⠛⣶⣷⣜⣿⣧⣼⢋⣀⣴⣟⠉⠀⢹⣿⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠹⡟⢿⢷⣚⢢⢼⡄⠀⢸⣿⢿⣿⡿⣧⣽⣞⣏⠛⠻⣏⡗⠤⣾⣽⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠤⣷⢸⢄⡎⠙⢦⢿⣿⡶⢶⣾⠶⣿⣿⣿⡹⠿⠀⢀⡇⣹⠉⠙⠯⠷⢷⡶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡼⢀⠏⠘⣮⡤⠔⠚⢏⢁⣠⣾⢣⣿⣿⣞⣿⠛⢿⣿⣟⡽⠭⢿⢿⡿⣿⢾⡧⢾⡬⠭⣕⣒⡀⠤⢤⣀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢷⣸⣗⠿⢋⣀⣀⣠⡼⣻⢿⢧⣏⣀⢨⢯⣿⣧⣀⠿⣏⠀⢀⣧⣎⡜⠁⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⢻⡟⠫⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡗⠹⣭⣷⠿⣗⣩⣿⣾⡏⢀⣜⣷⡾⣿⡯⠭⠺⡝⣇⠀⠘⣷⠊⡎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢊⡡⣷⡞⣻⢩⡻⠒⠶⢾⠷⣝⠻⢏⡀⠰⠀⠝⣿⣀⣷⣬⢿⡤⣩⣻⢃⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣊⠴⠂⠁⠀⠙⣷⣡⠟⠀⠀⡠⠊⠀⠀⠑⠢⣝⡶⣂⣀⣉⣩⣾⣘⠒⣛⣿⣽⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⠯⠒⠉⠀⠀⠀⠀⠀⣼⣿⡟⢷⡒⢋⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠘⡆⠘⡙⣣⡗⢳⣤⠛⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣠⡶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⠟⢧⠈⠳⢼⣖⡀⠀⠀⣀⠀⠈⠀⠀⠀⢀⣸⢱⡄⣷⡏⠀⠐⠋⠳⡌⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⡤⠖⠋⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠀⢸⢷⡜⡮⢻⢦⡀⠀⠀⠉⠁⠀⠀⢀⡴⢾⢪⣷⢷⣃⣸⣀⠀⠀⠈⢦⣣⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡇⠀⡸⠀⢽⣷⣼⠀⠙⠦⣀⣀⣠⠴⣾⣻⡷⠛⣧⠟⣃⡀⢀⣈⡑⠦⡀⣔⣋⣠⡧⠤⡜⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠃⣰⠁⡄⠉⣻⠻⡤⣀⠀⠀⢀⡏⠀⣿⣯⣧⣞⠁⡴⡿⡠⠊⠉⠉⠫⣠⠒⠉⠁⠙⢆⢈⣦⣈⣣⣀⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠐⠁⠀⡡⣾⣽⣷⣧⡙⠒⡷⣳⠒⣺⠟⠓⢦⡉⠙⠛⢿⣅⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣯⡁⢣⣤⠖⠋⠀⢹⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⣠⢴⣲⣿⠶⣧⡈⣟⡻⣟⡾⣷⡷⠞⠁⠀⢦⣄⠉⠲⡄⣼⡬⠦⡀⠀⠀⠀⠀⣠⣾⠟⢉⡏⣧⢸⡄⢱⣴⢺⠈⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢼⡾⡵⢪⡿⢛⢿⣃⣬⣿⠋⣹⣾⠋⡌⠀⠀⠀⢀⣧⡘⢫⣜⡴⠋⣠⠞⢯⣵⣄⣠⢊⡼⢥⣴⡻⡀⢳⣺⣷⣚⣻⡇⠀⢱⣄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣷⢦⣿⡵⢷⣿⣱⣳⠚⢿⣾⣇⣴⡟⠁⠀⢱⡀⠀⣠⡿⠉⠉⠉⢉⣩⡾⠟⢗⣦⣍⣙⡷⠯⠖⠛⢼⣿⡱⣄⠁⠈⠫⣥⠀⠀⠈⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠀⣼⣮⣼⡳⠀⠀⢙⢯⣗⢤⣰⣿⣎⣽⡀⢀⣀⡠⢷⣾⣋⣀⣠⠤⢾⣿⣿⣷⣤⡈⢏⢿⣹⠤⣀⠀⠀⠀⣟⡆⢻⠛⠦⣄⣿⣄⣀⡰⡃⡇⠀⠀⠀⠀⢀⡠⠔⠋⡎⢈⠓⣯⡘⣼⣽⣄⣀⣾⠳⠋⠈⣿⠿⡽⠉⢿⠉⠉⠉⢿⣇⡀⢸⣀⣴⢿⣿⣝⣿⣿⣿⣾⢸⠗⠀⠀⠉⠒⠦⡇⢹⢮⠤⠖⠛⢻⡿⣿⡧⢵⡟
⠀⠀⠀⢰⠛⡢⣄⡜⠠⠋⡰⠛⡿⣾⠿⠛⣿⣅⠀⠀⢠⢾⢰⠃⠀⠀⠦⠀⠀⠸⣿⢾⡟⣏⣹⣿⣯⣻⣮⠙⣿⣿⣇⠀⠔⢄⠀⠀⠀⣯⠺⡈⢦⣷⠈⣁⣴⣟⡡⢻⠇
⠀⠀⠀⠘⢇⣀⣈⠽⣄⠼⠁⠘⡄⢸⢧⡜⠀⠙⢿⢶⣿⣸⠇⠀⠀⠀⢀⢹⣶⣒⠛⣿⣷⠻⣇⣿⡞⣿⢿⣾⡟⢣⡈⠻⣦⡈⢆⠀⢰⠳⣄⡹⡄⠈⡟⠛⠋⠀⢀⡼⠀
⠀⠀⠀⠀⠀⢸⣇⠀⢨⣿⣄⣰⠁⡇⠀⢷⠀⠀⠀⠘⢉⠏⠀⠀⠀⠀⡞⣿⣿⣘⣯⣿⢻⣦⣻⣜⣿⣾⠿⠛⢷⣌⡟⢦⠈⠙⢾⣇⣸⣳⡈⢛⡇⢸⠁⠀⠀⠀⢸⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠉⠉⢻⣿⣧⣸⠀⠀⠇⣣⡀⠀⢀⠎⠀⠀⠀⠀⣰⣸⢿⡿⠿⠿⣿⡜⣿⡟⠋⠙⣽⡀⠀⠸⣾⣳⣄⠑⢄⠀⢈⡟⣶⣽⣮⢠⡇⠀⠀⠀⠀⣾⠁⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⢀⢀⢸⣿⣿⣿⡄⣰⠀⡇⢣⣰⠟⠀⠀⠀⠀⢀⠇⣿⣾⡇⠀⠀⠈⢳⣿⠀⠀⠀⠘⡳⡀⠀⠹⣯⢻⣿⣮⣇⣸⣽⢾⣄⢸⡼⠀⠀⠀⠀⠘⡏