⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢹⡇⠀
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢸⡇⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢠⡟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢠⡟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⡇⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⢠⡟⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣧⡶⠶⠛⠛⠓⠶⢶⣤⣀⣸⣼⣇⣀⣤⣾⠀⠀⠀⢠⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣻⡟⠛⢷⣯⡀⠀⣰⡟⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⡾⠋⣰⠤⠤⡤⣠⠀⠀⠀⠀⠀⠀⠠⢰⡏⠀⠀⠠⣬⣿⣶⣿⡀⠀⣀⣠⣙⣷⡀⠀⠀⠀⠀
⠀⠀⠀⣴⢟⣄⣸⢅⡴⢫⡿⣿⠈⠰⠄⠐⢆⡜⢀⣿⣀⣀⣀⣐⣻⡿⠁⢨⣿⣋⠉⠉⠉⣿⢷⡄⠀⠀⠀
⠀⢀⡾⣣⡾⠁⡏⠁⣰⣯⠶⣿⣄⢂⠀⢳⡘⠁⠈⠛⠉⢉⣉⣯⣙⢟⢛⠿⣧⡀⠀⠀⢠⡟⠀⢷⡀⠀⠀
⠀⣾⡽⡿⣁⢰⡏⢰⡿⠁⠀⣿⣿⣞⣧⡀⠹⣶⣶⣠⣶⢻⡟⣭⣙⢻⣏⡀⠫⠻⢷⣴⣿⠁⡀⠈⡇⠀⠀
⢸⡟⢹⠇⡿⢷⣧⣾⠁⠀⠀⣿⣿⣿⣽⣿⢦⣝⣾⡋⠙⢻⣷⣶⣿⣻⣿⣆⢀⡠⣄⣺⡾⠀⠉⡇⣿⠀⠀
⠉⠀⣾⡘⡇⠀⡟⣿⣄⠀⠀⣿⣿⣿⡟⠻⠦⣹⠛⠃⠀⢸⣿⣿⣿⣿⣿⣷⡷⠟⠉⡨⠇⢀⢃⣇⢸⠀⠀
⠀⠀⡇⠃⡇⡀⡇⢻⣍⣿⣶⡿⠿⠟⠃⠀⠀⠿⠶⣦⣤⣼⡿⣿⡿⠟⣿⠁⠀⠀⣶⠁⢠⣶⢸⣿⣼⠀⠀
⠀⠀⡇⢸⠇⢹⡗⡌⢷⡗⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡝⠁⠀⣿⠀⠀⠀⢁⠄⣞⣻⣾⡟⣿⠀⠀
⠀⢠⡇⢸⠀⢸⠇⠁⠀⡟⢦⣄⡀⠀⠀⠀⠀⣄⣀⣼⠇⠀⠀⠀⣀⣠⣿⠀⠀⡄⠀⡸⣸⡇⡇⡇⡟⠀⠀
⠀⠀⣇⡌⡇⢸⠀⠀⠀⡧⢸⠉⠛⠳⡶⣤⣤⣬⣭⣥⣤⣶⢾⠛⣿⠁⣿⠀⠀⡿⢀⣇⣷⢱⢻⣇⠀⠀⠀
⠀⠀⣿⢡⡃⢸⠀⠀⠀⡇⢸⠸⠀⠀⡇⢇⠀⢻⡟⠋⠁⣿⢸⡀⢹⠀⡿⡆⠀⡇⢸⢹⣾⣼⣼⠀⠀⠀⠀
⠀⠀⠘⣿⣧⢸⡆⠀⡆⡄⢸⢰⠀⠀⢸⢸⠰⠘⣷⠀⠀⢸⣾⠀⠸⣠⡇⡇⡇⡇⠸⡏⣏⠇⡏⠀⠀⠀⠀
⠀⠀⠀⢻⣿⠘⡇⠀⡇⡇⢸⠘⡆⠀⢸⡘⡆⠀⣿⣀⣀⣘⣿⡄⠀⢸⣷⠀⢱⣇⢸⠱⣿⠰⡇⠀⠀⠀⠀
⠀⠀⠀⠸⢿⡆⣟⡆⡇⡇⣼⠀⣇⣀⣬⣧⣷⣴⣿⣿⣿⣿⣿⢷⣄⣸⢿⠆⢸⢯⣿⢀⠇⠀⣇⠀⠀⠀⠀
⠀⠀⠀⠀⢸⡇⣻⠃⣷⢣⢻⢠⣿⡏⠁⡟⠀⣼⠋⠙⠛⠛⠉⠈⠛⣿⡸⠀⣿⠘⢟⢬⡜⡄⢿⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⠸⡇⢸⢘⣞⣾⢿⡇⢸⡇⢀⡏⠀⠀⠀⠀⠀⠀⢠⡟⠃⣸⠟⢷⡾⠸⠷⣾⡼⣇⠀⠀⠀
⠀⠀⠀⠀⢸⡟⡆⢻⡸⡿⡄⣿⠈⢻⣾⡇⢸⡇⠀⠀⠀⠀⠀⠀⣾⡟⢠⡟⠀⠀⣿⡼⡄⢻⣧⣻⡄⠀⠀
⠀⠀⠀⠀⣾⡇⣾⢪⢷⣿⣡⣿⡝⢦⣹⡿⣬⣇⠀⠀⠀⠀⠀⠀⣿⢀⡾⠁⠀⠀⣿⣷⢳⠀⣿⠻⢿⣆⡀
⠀⠀⠀⢀⡏⣽⣿⢸⢏⣿⢿⡏⠙⠦⣸⣷⣈⠙⠷⣤⡀⠀⠀⢸⣿⣿⠁⠀⠀⢠⡏⢹⣾⣄⢸⡄⠀⠉⠛
⠀⠀⠀⠘⡿⠋⢻⡎⢻⣿⣼⡗⢦⣄⢸⡿⣏⠙⠢⢄⡉⠳⠦⣼⣿⠇⠀⠀⠀⣾⣏⣼⣿⣿⣸⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢷⣸⡏⢻⡟⢦⣉⢻⡇⢹⣶⢤⣀⠈⠑⠒⣾⣿⠛⠛⠶⢷⣿⠛⢁⣼⣿⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⢸⣇⠀⠈⢹⠁⠀⠹⣦⣌⡉⠓⠚⢻⠗⠒⠒⢲⣿⣋⣉⠭⠖⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⢧⡗⠳⣼⠀⠀⠀⠹⣆⠈⠉⢒⣿⠙⠒⠤⣾⢟⣀⠤⠤⠤⢾⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣿⠀⠀⠀⠀⠹⣗⠒⣺⠗⠒⠒⣺⣿⣀⡀⠀⠀⢀⠤⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠲⢦⣿⠀⠀⠀⠀⠀⢻⣎⡿⠉⠓⢲⣿⣃⡀⣀⠀⣀⠠⠤⢿⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣻⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣡⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣯⢷⣹⢿⣧⣤⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣣⢿⣯⣧⣤⡤⠴⣶⣖⣒⡚⣛⡻⢯⡭⣭⡱⣶⣾⢯⣳⢭⣿⡟⡯⡽⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⣿⡷⣭⡻⣽⢿⣷⣹⣛⢶⢺⡴⢫⡳⣝⢯⡳⣇⡟⡽⢎⡷⡹⣞⢯⡝⣳⡝⡶⣭⣛⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣵⣒⢾⡹⡽⣿⢶⡹⢧⡿⣟⢶⡹⣎⢷⡹⢧⡻⣜⢧⡻⣜⡽⢺⡝⡼⢳⡭⢶⡹⢣⡟⣵⢣⡟⢮⡳⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⣿⣾⣱⢻⣬⢳⡝⣧⢻⣬⢳⡝⣮⢳⡝⣧⢻⡜⣧⠻⣜⣞⣣⡯⠽⠷⣞⣟⡛⠉⠉⠉⠉⠘⣷⡹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣎⢷⡻⢿⡳⣎⣷⣽⠞⠓⠊⠓⡹⠬⠍⣉⠉⠉⠉⠉⠁⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⡳⢿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⡮⣝⣷⡝⣮⡗⠕⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢰⡽⣏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⢷⣹⢺⡽⣲⡇⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⢹⣆⣴⠊⠛⠋⠄⠀⠀⠀⠉⢦⠀⠀⠀⠀ ⢸⣽⣏⣿⣏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢮⣳⢭⣻⡗⣧⢧⠀⠀⠀⣀⡀⢠⠂⠀⠀⠀⠀⠈⠫⠀⠀⢠⢁⣴⡄⠀⠀⠀⠈⠄⠀⢠⡄ ⢸⣿⠼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣛⣮⢳⡿⣜⣿⠀⢀⣾⠃⠀⡆⠀⠿⠟⠀⠀⠀⠀⢥⠀⢿⠈⠉⠀⠀⠀⠀⢰⡇⠀⣠⠇ ⢸⣿⣻⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣻⠼⣧⣟⡾⣼⡆⠀⣿⠀⠀⠱⡀⠀⠀⠀⠀⠀⢀⠂⠀⠘⣄⠀⠀⠀⢀⣠⣾⠃⢸⡇⠀ ⢸⣿⣽⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠈⣿⢷⡿⢷⣿⣹⢷⡇⠀⠸⡆⠀⠀⢹⣆⡀⠀⠀⣰⡿⠀⠀⠀⠀⠉⠶⠾⠿⠏⠁⠀⣾⠁⠀ ⣾⡿⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡟⡽⣧⡀⠀⠀⠀⠀⠀⢻⣿⣽⣛⣾⢯⣟⣷⠀⠀⢹⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀ ⣾⣟⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣾⣟⣱⢫⢷⣄⠀⠀⠀⠀⠘⣿⡾⣽⢯⣿⢾⣽⡄⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀ ⣿⣿⣿⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⣝⣾⡽⣾⣽⣷⣄⠀⠀⠀⢻⣿⡽⣯⣿⣟⡾⣧⠀⠀⠀⠉⠉⠉⠳⢦⣄⣀⣀⣠⠤⠔⠒⠒⠒⠉⠀⠀⠀⠀⠀⢨⣿⣟⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣻⣿⠉⠻⣮⡝⣿⢿⣷⣤⣀⠈⠛⠿⢷⣟⡾⣽⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣴⣿⣟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣟⠅⢉⣫⣙⠛⢛⠶⠭⢞⣿⣿⠇⠀ ⠸⣿⣟⣳⣟⡿⣤⡤⣤⣴⢶⡶⣶⡶⣶⣶⡶⣿⢿⣟⡿⣿⣟⣿⣻⣟⣿⣿⣽⠾⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⡟⠆⣠⡿⡛⠉⠙⠓⠯⣤⣼⠿⠋⠀⠀⠀⠀⠉⠛⠛⢻⣿⣿⣛⠿⣞⣯⣟⡷⣿⢷⣿⣿⣿⣿⣾⣿⣗⠋⢙⣉⣩⣥⣁⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⣇⡠⣿⣳⠀⠀⠀⠀⠀⠀⢐⣠⡤⠤⠤⠴⠖⢒⣒⣒⣈⣩⣼⣳⣿⣻⡾⣝⡿⣽⣻⢾⣷⣽⢯⣿⣿⣿⣿⣿⣏⣷⣭⣯⣿⣟⣿⡽⣿⣼⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠨⣇⡄⡙⢿⣤⣀⡀⠀⠀⠀⢻⣇⠲⣤⣉⡽⢺⣭⣿⡟⢋⠛⢋⣫⣿⣟⡿⠿⢛⣽⣟⣿⣿⣾⢷⣲⡶⣿⠿⢽⣽⡟⣒⢓⣋⡸⣧⢟⡽⣯⢿⣿⣷⣦⣄⠀⠀⠀⠀
⠀⠿⣴⣄⡂⠀⠉⠉⠍⠛⠛⠛⠋⢃⡈⠙⣿⣿⣼⣯⢿⣿⡟⢯⣙⣉⣉⣿⣿⣿⣍⣭⣼⣿⣯⣟⡿⣷⢶⡶⢿⡳⣝⢧⡻⣜⡳⣝⢮⣳⣯⣿⣾⣽⣿⣿⡗⠀⠀⠀
⠀⢀⣠⢭⣛⠷⣦⣤⣀⣦⣴⠶⠾⠋⠙⠿⣿⣼⣻⣟⣿⣯⣿⡿⣽⢯⣟⡷⣯⣟⡾⣯⣟⣾⣳⣯⣿⣽⣛⣾⣷⣽⣮⣷⡿⣶⢿⡟⣿⣟⢿⣻⣟⣯⣟⣿⣷⠀⠀
⢰⡻⣺⣑⠫⣗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣿⣟⣿⣛⡿⣾⣽⣳⢯⣟⣷⣻⢾⣽⣻⣟⣿⣻⣟⣾⡽⣳⣝⣮⣳⣮⡵⠷⠾⠛⠛⠛⠋⠋⠁⠀⠀⠀⠀⠀
⢫⢗⣿⣤⢷⣏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢯⠷⡿⣼⢯⡿⠾⢷⠿⣿⢾⠷⠻⠶⠟⠯⠿⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠋⠷⠺⠽⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀