💜⢰⢢⣦⠀⠀⠀⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣯⡭⠹⠿⣶⣤⣀⠀⠀⣰⣿⢿⣿⣧⣄⣀⣀⡽⠟⠋⠉⢉⣭⠿⠓⢶⣤⠀
⠀⠌⣯⣭⣷⣄⣀⣀⣻⣷⣶⣤⠬⣍⣶⣄⠀⠀⠀⣀⣴⣿⠛⣉⣤⣶⣿⣿⣿⣿⣿⣯⣉⠓⠶⢼⣿⣿⣿⣦⣸⣏⣼⣻⣿⣿⣿⣿⡄⢀⣤⣾⣿⡁⠀⠀⠀⠈⠀
⠀⠀⠸⣎⡉⠉⠙⠁⣶⡉⢉⣘⣿⣯⣮⢿⣧⢀⣼⣯⣿⣿⠟⠁⣼⣿⣿⣿⣿⣿⡿⣿⠟⠳⣜⢀⢽⣿⣿⠿⠿⣿⣿⡟⠁⠀⠈⣛⣡⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀
⢀⠤⢶⣾⣿⣶⣦⣤⠉⣥⣾⡿⠋⠉⠻⣿⣿⣾⣿⣿⣿⠃⢀⡾⠋⠀⣿⣽⣏⢿⣷⠹⡄⣀⠘⣷⣴⠟⣿⠷⣄⣸⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀
⠁⠀⠀⠈⢻⣦⣄⡈⠀⠤⢀⣀⠀⣀⣤⣿⣿⣁⡟⣿⠘⣷⣿⡄⠀⣀⠋⢹⡍⣼⣿⡇⣹⣸⣶⡻⣿⣶⠛⢄⠈⢻⣿⣿⣄⢀⡀⠀⠩⣝⣿⠟⠁⠀⠀⢸⠇⠀⠀
⠀⠀⠀⠀⠀⢙⣻⣿⣿⣿⣿⡿⠿⠛⣿⣿⣿⡟⠀⠙⣿⣿⣧⢈⣻⣿⢶⣤⡧⢿⣿⣟⣿⣿⣿⣿⠉⠹⡆⠈⢷⡀⢻⣿⠀⠙⠻⠿⠭⠭⠵⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⡾⠛⠿⣿⣿⣟⠋⢉⣠⣾⡟⢹⣿⡅⢠⡎⣹⠉⠈⢿⡏⢉⣿⢿⡏⣿⣿⠉⡏⣿⣙⢎⢳⡀⢻⣦⠀⢷⠈⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠛⠀⠀⠀⠈⠙⠓⠾⠟⠛⠉⠀⣿⣿⡇⡿⠀⡇⠁⢀⡞⢀⣾⡟⣸⣷⡿⢹⠀⣷⣿⠈⡿⢿⣷⣌⣿⣦⣸⣧⢾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣁⣿⡀⢇⢠⣿⠁⡾⢱⢣⣿⡟⠀⡟⣸⢿⡿⠀⣘⣀⣻⣿⣿⢻⣿⣿⣯⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⡇⣿⠚⣼⡏⣸⢣⣾⣿⣴⡧⣼⣳⡿⢻⡿⣛⣉⣁⣸⣏⣿⣽⣿⣿⣿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⢻⣿⣼⣿⠸⡀⣿⢿⡏⣩⣿⣉⣱⣷⣿⠟⢡⡿⢻⣿⣿⣿⣿⡿⣿⣿⣾⣿⣿⣿⢟⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡹⣼⣿⣿⣟⣷⣇⣿⣾⡿⣿⣿⣿⣿⣿⣹⡈⠟⠁⠀⠻⣽⣭⣽⡴⢣⣿⣿⣿⣿⣿⣮⣿⣿⣦⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⡿⣿⣿⣿⣿⣿⡻⢦⣹⣯⣿⠖⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢈⡽⢳⢳⡾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⢻⣿⣟⠿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⢀⡿⣿⣽⣿⡞⡇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠋⣿⣿⣿⣾⣿⣏⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡆⢰⢃⡏⣼⠟⣿⣿⣿⣷⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⢳⣿⣿⣜⣆⡸⡍⠻⣄⠧⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣱⢿⢾⣿⣵⣾⡿⢋⣜⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣼⣿⣿⣷⣽⡟⠿⢷⡤⠼⠷⣄⠀⠀⠀⠉⠉⠛⠃⠀⠀⣠⣾⣿⣿⣾⣿⠟⢿⠋⠉⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠌⠙⢷⡙⠻⣿⣶⣜⣿⡿⡶⣾⣿⠲⠤⣀⡀⠀⠀⣠⡞⠋⢸⢿⢿⠛⠉⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡹⠉⠛⠛⠃⡼⢻⠀⠀⠀⠉⢳⡿⠋⠀⢀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡄⠀⠀⠈⠁⠀⡰⢻⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⣸⡄⠀⢀⡠⠞⠁⡞⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⣄⣀⣀⣀⣤⠤⢴⡖⠋⠁⠀⠀⠉⢙⠁⠀⠀⠀⢰⠃⠀⠀⠀⠀⠙⢿⠒⠤⢤⡤⡤⠤⠤⠤⠤⠤⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠁⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢿⠛⠒⠲⠤⠤⠤⠶⣷⣄⠀⢀⣿⠦⠒⠂⠀⠒⠚⢉⣷⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠐⢻⡟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⢹⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀ ⠀ ⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⡄⣼⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠈⢧⣷⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⡆⠀⣼⡆⣠⠎⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠳⠀⠈⣿⡆⢸⡇⠀⠀⠀⠀⠀ ⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣇⢀⣿⣿⠁⢀⣰⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣄⡀⠢⡘⢷⣸⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡿⣿⠿⠳⠶⠌⠁⠼⣷⡀⠀⠀⠀⠘⢆⠀⠀⢆⠀⠀⠀⢰⠀⠀⠀⠀⢀⣾⠁⠈⠛⢿⡛⠞⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢶⣷⠊⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠈⠶⠀⠸⡆⠀⠠⠆⠀⠀⠀⣠⣾⠿⠗⠂⠀⠀⠐⢿⣾⡄⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⢤⡀⠀⠀⠀⡇⠀⠀⣀⣠⣴⠾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡙⠻⣆⠒⠓⠒⡟⣱⠋⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠘⣆⠀⣼⡕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⣸⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡄⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢰⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡼⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡼⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⢸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠉⠳⣶⡤⣤⣤⣤⠤⠤⠀⠀⠀⠒⠰⠃⠀⠈⠉⣉⡫⠽⠶⣒⣢⠤⠤⣤⡴⣶⡾⠋⠀⠀⠀⠀⡎⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠘⣿⣿⡛⠋⠙⠒⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠠⠠⢘⡳⠿⠳⣯⡟⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠘⣿⣮⡿⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⠰⠉⣷⡿⡡⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡆⠀⠀⠀⠀⠘⢿⣷⡀⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⣿⠃⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⠀⠀⠀⠀⠀⠈⢿⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣽⠀⠀⠀⠀⠀⠀⡜⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠀⠀⠀⠀⠀⠀⠘⡏⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⣴⠇⠀⠀⠀⠀⠀⠀⢃⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⢹⡸⡴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⡝⣄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠞⠁⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡄⠘⢆⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣾⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢷⡀⠈⠆⠀⠀⣠⠞⠁⠀⠀⠀⢀⢸⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡙⡄⠀⠀⠀⠁⠀⠀⠀⠀⢀⢏⡇⠀⡞⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⡇⠻⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠏⡏⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡌⢿⡀⠹⡀⠀⠀⠀⠀⠀⠀⠘⡆⢘⡆⠀⠀⠀⠀⠀⠀⠀⡼⡹⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⡌⢠⠇⠈⢇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⢠⡹⡄⠱⣄⢳⡀⠀⠀⠀⠀⠀⠀⣇⠉⠀⠀⠀⠀⠀⠀⠀⡸⣱⠃⢀⢴⢻⠇⠀⠀⠀⠀⠀⣸⡵⠋⠀⠀⡞⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣞⣆⠉⠻⣄⠈⠛⢷⠀⠀⠀⠀⠀⠀⢹⢤⡀⠀⠀⠀⣀⡤⠶⠡⡧⠞⠁⢸⣼⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⣞⡇⢻⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣮⡹⣷⡄⠀⠙⠢⣌⣇⠀⠀⠀⠀⠀⢸⡦⢬⣉⡈⠉⠀⠀⠀⣀⡤⠖⠈⠹⡿⠀⠀⠀⠀⠀⡞⣒⠿⠛⠉⢀⣤⣫⣷⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⡁⢹⣿⣿⣆⡀⠀⠀⢸⡆⠀⠀⠀⠀⠈⡗⣄⣠⠭⠟⠒⠒⠢⠬⠭⣉⢁⢆⡇⠀⠀⠀⠀⢰⠋⠀⣀⣤⣺⣿⠿⣻⣿⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣶⣤⣿⡀⠀⠀⠀⠀⢿⣁⠠⠤⠤⠤⢄⣀⣀⣀⡠⠬⢼⠁⠀⠀⠀⢀⣿⣶⣿⣿⣿⣿⣥⣶⣿⣿⣧⡀💜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣃⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣆⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢱⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀❤️⠀⠀⠀⠀⢠⠇⣾⡀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠛⠋⠉⠻⣿⠛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣸⣿⣧⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠛⢿⣿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⢿⣿⡀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠉⠺⣿⣷⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣟⣿⡖⠋⣉⠙⢛⣻⡶⡄⠀⢀⡮⣿⣭⣥⣾⢿⡿⣿⣿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢿⡿⣿⣿⣿⣿⣿⣟⣿⡷⢿⣿⣿⣿⡟⠷⠈⠀⠀⠀⠙⠿⠟⠷⠟⢁⡟⡏⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢸⣇⠸⣿⣿⣿⣿⣿⣮⢿⡀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠚⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⣟⢆⠹⣿⣿⣿⣿⣿⣷⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡹⣆⠁⠈⣿⣿⣿⣿⣿⣿⣷⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣓⠂⢸⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠒⢣⠍⠉⠀⣠⡏⣸⠁⠀⠀⠀⠀⠀💙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⣀⠤⠚⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠤⣄⡀⢀⡤⠚⠁⡇⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠉⠉⠀⠀⠀⣯⡳⣜⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⡇⢀⠀⠀⠀⠀⢠⠀⣷⣷⠈⢳⣬⣙⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⣾⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⢈⣧⠀⠀⠀⡏⠀⠀⢹⣧⠀⠙⣎⠫⡀⠀⠈⠉⠉⠉⣲⢦⠹⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⠭⣌⣧⠀⢸⣠⠴⠒⡊⢯⢧⠀⠈⢣⡈⢦⠀⠀⠀⢀⡇⠘⡆⢳⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢃⣶⠊⠉⠙⠒⠺⢿⣿⣿⣿⣿⣿⢿⣿⡄⠈⠉⠓⠋⠁⠀⠀⠉⡘⡎⡆⠀⠀⠹⡄⢀⠀⠀⡘⠀⠀⢳⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡆⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⡄⠀⠀⠘⣎⡇⣰⣷⢻⠀⣸⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⢰⢹⣀⣴⣾⣿⣿⣿⣿⣿⣿⠛⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢱⠀⠀⠀⠘⢷⡟⡇⡞⢰⠃⢣⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣏⣤⣿⣿⣿⣿⠟⢻⢉⡇⡼⠀⠀⠀⠫⣕⣄⠀⠀⠀⡼⠁⠀⠀⠀⢸⣤⡆⣀⠀⠀⠈⢷⣽⢣⠃⠀⢸⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠸⣼⣿⣿⣿⣹⣦⡇⠘⢦⣧⠀⠀⠀⠀⠈⠻⢦⠀⡜⠁⠀⠀⠀⠀⣸⡼⠀⠀⠙⠯⣉⠉⢻⡏⠀⠀⠈⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⣿⣿⣧⠞⠁⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⣀⡜⠀⠀⢣⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇⠀⠠⣬⡏⠈⣑⣦⣄⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠟⠀⠀⠀⣠⠞⠁⠈⠑⠲⢼⡄⣴⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢧⠀⢰⠗⠋⠀⠀⠀⠈⢑⣦⣄⠀⠈⢦⡀⠀⠀⣸⠀⠀⢀⡴⠁⠀⣀⡴⠚⠒⠦⣄⣀⣀⣀⣀⣷⣏⡞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳⠈⢧⢸⠤⠤⠤⢤⠔⠊⠁⠀⠀⠙⠲⣄⠹⣦⠜⡽⡟⢤⠟⢀⡴⠚⠁⠀⠀⠀⠀⣞⡠⠤⠂⣸⣻⡟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⢤⠻⡗⠂⠀⠈⢇⠀⠀⠀⠀⢀⠀⠈⠑⢌⢿⣇⣽⢏⠔⠁⢀⣠⠄⠀⠀⠀⡜⠀⠀⠀⣠⡿⠛⠀⢸⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⡧⢹⣄⠀⠀⠘⢦⠀⠀⠀⠀⠉⠑⠒⠬⣕⣿⡿⠫⠔⠊⠉⠀⠀⠀⢠⠞⠀⠀⣠⠖⡉⠳⣄⠀⢸⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡼⠐⣫⠔⠋⠈⢳⣠⠤⢌⣳⣀⡤⠤⠚⠛⢉⣉⡼⠽⡷⢄⡋⠙⠋⠉⠭⠵⡯⠀⡶⣾⢧⠞⠁⠀⣀⠷⢸⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠸⣄⡘⠳⠤⡤⠄⣸⠈⠁⠉⢸⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠉⠁⠀⠑⠂⢸⣀⠀⠀⡧⢌⣙⠲⣾⠕⡿⢋⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⢸⣹⠀⢀⡤⠞⢹⢳⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞⣠⠟⢸⢳⣄⡴⠛⣛⡷⠿⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⣋⠀⠀⠈⡳⠄⢸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⠇⠀⢀⣀⠀⠠⠔⢋⠤⢀⢿⢁⡶⠈⠭⣰⣷⣺⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⡸⡤⠛⣉⡭⢡⡖⢮⠤⠴⠒⠂⠀⠀⠀⠀⠀⠁⢠⣗⣾⠖⠋⠁⠀⠀⠀⠀⠀⢀⡞⡸⢋⣈⠒⠒⢋⣉⣈⡆⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠎⡰⠿⠶⠶⠶⣙⡴⠊⢿⣌⠑⢦⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡝⠀⠀⠀⠀⠀⠀⠀⢀⠎⠠⡏⡴⠋⠀⠀⠀⠈⡟⢡⠀
⠀⠀⠀⠀⠀⠀⡰⢣⠞⠀⠀⣀⠴⠊⠁⠀⠀⠀⠈⠳⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⢠⠏⡄⢸⠏⠀⠀⠀⠀⠀⢀⡇⠀⠀
⠀⠀⠀⠀⠀⡜⣡⠋⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣬⡑⢤⡀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣀⠴⠂⡏⠀⠁⡏⠀⠀⠀⠀⣀⠴⢻⠃⠀⠀
⠀⠀⠀⠀⡜⠰⣣⢴⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠓⢮⣓⣦⡀⣇⣀⠤⠐⣚⠭⠄⢐⣦⠇⡜⣰⠃⣀⡤⠖⢊⣡⢤⠇⠀⠀⠀
⠀⠀⠀⡸⢡⡷⠃⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⢋⠗⠒⠚⠉⠀⢀⣰⠟⠈⡀⢠⢷⣋⠁⠒⠋⡡⢀⠞⠘⠀⠀⠀
⠀⠀⢠⢃⢿⣧⠀⠀⠀⠀⠀⠀⠙⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠋⠀⠀⠀⡇⡿⢦⡈⠑⠤⠀⠙⢾⠀⠀⠀⠀⠀
⠀⢀⡊⣜⡎⡌⡝⠒⠢⠤⣀⣠⢾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⢤⣀⠀⡀⠀⠀⠀⠀⠀
⢀⠎⣰⡿⠀⠋⠀⠀⠀⠀⠀⢀⢏⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡁⠀⠀⠀⠀⠈⠫⡹⠇⠀⠀⠀⠀⠀
⠁⢺⣗⣇⠀⠀⠀⠀⠀⠀⢀⡾⠀⠑⢌⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢛⡟⠀⠀⠀⠀⠀⠀⢀⡀⣸⠏⠛⠯⣗⣄⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀
⢀⢧⣿⣿⡄⠀⠀⠀⠀⢀⡞⠐⣤⡑⠦⣑⡝⢦⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⡴⠋⢰⢻⣄⡀⠀⠀⠉⠛⢦⣠⠃⠀⠀⠀⠀⠀⠀⠀
⢸⠀⢸⡏⠑⠲⢤⣀⠴⠚⠀⠀⠈⠻⣗⢤⡈⠢⡱⣄⠀⠀⠀⠀⡜⠀⠀⠀⢀⠤⠊⠁⠀⢂⡏⡾⡀⠉⠓⠢⣄⡀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀
⡞⠀⢸⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⠓⢤⡈⠳⣄⠀⡞⠀⣠⠴⠚⣁⡬⠴⠒⠯⣽⢡⠇⣷⠀⠀⠀⠀⣩⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠣⣄⣸⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠓⠬⣿⣶⠯⠐⣺⠝⠋⠁⠀⠀⠀⠸⢹⣦⡘⠃⢀⡤⡾⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡇⣼⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⠿⣞⢴⡏⠁⠀⠀⠀⠀⠀⠀⢹⠿⡒⠽⡴⢋⠜⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡇⠈⡆⢯⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⣞⣾⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣈⢦⡃⠀⡀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡇⠹⡇⢸⡀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣿⠘⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣇⣼⣵⣡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⡄⢻⣸⡇⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⣰⠃⡏⡇⠘⡄⠀⢀⣠⠴⠒⠒⠒⠒⠦⣄⡀⠈⠉⠉⠛⠋⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⢌⠻⣿⡄⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠰⠃⠀⢇⣿⠀⣤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⢰⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠳⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡸⢸⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⡈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀