⠀⠀⠀⠀⠀⣰⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⣦⣤⣠⡿⣧⢴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠙⣿⡱⣙⠦⣏⠽⣶⣶⡶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⡚⣏⠻⡝⡶⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠻⣟⡧⣏⠵⣫⣜⡳⡜⡽⣧⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣤⠾⣙⢦⢣⡝⣬⢳⡹⣜⢣⢮⡹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⣤⠴⣷⢭⢞⡱⡷⢯⡱⢇⢯⢿⣉⣁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣤⡾⣡⣿⠜⠋⠛⠈⠛⠲⣷⢌⡳⢎⡵⣋⢿⡀⠀⠀⠀⠀⠀⠀⠀
⠀⣀⢉⣹⣏⠮⡵⣙⠶⣙⢮⢣⣛⢷⠟⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣭⡶⠋⠀⠀⠀⠀⠀⠀⠀⠈⢧⡝⣎⠶⣩⠞⣧⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠙⠚⣿⡱⡝⡼⣩⠳⣎⢧⡹⣚⣿⡽⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⠼⣌⣳⡟⣻⡟⢷⡀⠀⠀⠀⠀⠀
⠀⠀⠒⠾⢽⣷⢭⢳⡱⢏⡼⣒⠧⣝⢺⣖⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡄⡀⠀⠀⠀⠀⢾⣀⠀⠀⠀⠀⠀⠀⠉⠻⡝⢧⣙⢾⡷⠶⣦⣀⠀⠀
⠀⠀⠀⠤⣴⡾⣯⠶⣙⢮⡱⡭⢞⡜⣣⢿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠠⠈⡏⣶⢶⠶⠲⣠⠷⡀⠀⠀⠀⣀⠔⠀⠀⢸⢣⢎⡳⣜⠳⣼⡹⣶⠀
⠀⠀⠀⠀⠀⠉⣹⣟⡬⢇⣳⣙⢮⣱⢓⠮⣿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢄⠀⠈⠙⠿⠖⠒⠁⠀⠉⠢⣄⡘⠘⠀⢀⣠⢎⡳⢎⡵⣊⣷⢣⠳⣼⠁
⠀⠀⠀⠀⠀⠉⠓⠻⣯⢞⡱⢎⣖⢣⢏⡳⢭⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠁⠀⠀⠀⢑⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡘⣮⢟⡹⣚⢭⠲⣍⡳⣽⠂
⠀⠀⠀⠀⠀⠀⠲⠖⢻⣯⠞⡭⢎⡝⢮⣱⢫⢽⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡠⠅⣀⢀⠠⠄⡂⠄⠁⠵⠊⠉⠉⠉⠉⠑⠢⣄⠀⠀⠀⢻⣿⡟⡵⢪⡵⣭⢮⡷⣼⡟⠋⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⣿⣱⢋⠾⡱⢎⢧⢫⣿⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠠⠠⡀⠀⢀⠀⠤⠤⣀⠀⠈⢧⠀⠀⠈⣿⣿⠼⣏⠳⡭⢶⣻⡉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣷⣫⢝⡱⣏⢎⡗⣺⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⠁⠢⠘⣦⣴⠿⣿⡿⠟⠫⣆⢸⠶⠂⠀⢸⣿⠳⣌⠷⣙⢮⡱⣛⢷⣀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢷⣫⠼⣜⣺⣜⡱⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣾⣇⠀⠀⡈⢾⣃⠀⠉⠈⠛⡳⣬⠳⣍⠶⡱⣍⢞⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢳⣏⠶⢧⠟⣼⣹⣯⣽⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢟⠫⠋⠀⠀⠀⠠⡀⠀⠀⠈⡌⡷⣌⠳⡕⣎⢮⡱⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⣽⣮⣿⣿⣿⢿⣻⣽⣟⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢀⣠⣤⢤⣤⣀⡀⣀⣁⠀⡔⡼⣙⠾⣟⣾⡘⢦⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⢿⣽⣯⣿⣻⠾⣿⣻⢿⣆⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢟⡱⢎⠶⣘⢦⡛⣴⢒⢮⡙⢶⡩⢞⡴⣙⢮⣷⠟⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⡅⣄⠈⠛⣯⣷⣻⣷⣿⣻⣟⣯⡾⣽⣄⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗⢮⡱⣋⢞⡱⢎⡵⣊⠟⣦⣟⣦⣽⡾⠶⠛⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⢐⠃⠀⠀⠀⠈⠘⣿⡿⣽⣾⣷⢯⡇⠋⢿⣻⣄⣀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⣠⢾⡛⣭⢻⣏⡷⣝⢮⡱⣋⠶⣩⢞⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡀⠀⠀⠀⠀⠈⣽⣿⣟⣷⡿⣟⣾⣦⢿⣿⣿⣿⡇⠠⡌⠀⠀⠀⠀⠀⠀⣾⣋⠶⣙⠦⣗⢎⡻⣋⢶⡟⣧⣛⡴⢫⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠓⠄⣀⠌⢀⢬⣿⣾⣿⣻⣿⢿⣻⣶⣼⣻⣿⣿⠃⢒⠀⠀⠀⠀⠀⠀⢸⡷⣌⠳⣍⢞⡯⠿⡴⡽⣞⡹⢲⣍⠻⡝⢦⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠴⢺⣿⡽⣷⣿⣯⣿⣿⣿⣾⡷⣿⣿⣿⢀⡌⠀⠀⠀⠀⠀⠀⠈⣿⢮⠷⢾⢫⢳⡙⣶⡱⢎⡵⢣⢎⡳⣭⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⢻⣿⡿⣯⣷⢿⣳⣯⣿⣽⣿⣿⡿⢀⠁⠀⠀⠀⠀⠀⠀⢹⡛⢷⣋⢎⢧⢣⡝⣜⣿⠻⠞⠷⠾⠟⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢆⢈⠉⢿⡿⣽⣯⣿⣿⠟⠋⣽⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣙⡎⣎⠳⣜⡾⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢿⣿⠟⡁⠡⠀⢀⣿⣿⣿⠘⠀⠀⠀⠀⢰⠀⠀⠀⡻⢧⠙⠻⠞⠿⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠊⠀⠀⠁⠂⡛⢿⣿⢘⠀⠀⠀⠀⠜⠀⠀⠀⡇⠀⠁⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⢝⠨⠀⠀⠀⡨⠂⠀⠀⠀⡇⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀