⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠰⠀⠀⢐⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣵⣤⣂⣚⣾⡧⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣲⣾⣿⣻⣸⣿⣿⣆⢍⣹⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡝⣿⣿⡿⣱⣟⢿⣿⣟⡿⡄⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⡿⡟⡿⣏⠀⠚⠓⠋⠉⣷⠘⢫⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣟⣗⣃⡾⠡⠈⠐⠀⠀⢻⣥⠙⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⢶⢰⡇⡖⢂⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠽⣼⣲⣿⠔⠀⠁⠀⠀⡸⠜⡠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⢪⡿⣯⡷⡏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠴⣽⣿⢶⡽⡁⡀⢢⣶⣷⣿⣽⣯⠆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡽⠛⢠⣯⣇⣧⣧⡖⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⡿⢁⣤⠁⢼⠻⡉⢁⣼⣦⣼⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠻⡾⣡⣷⣿⣿⣯⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢵⠸⢿⢆⡂⠡⠃⣻⡿⠝⢸⣦⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠟⣿⡿⢡⣧⡤⢤⣄⡀⣀⠆⢀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⠉⣿⠴⠛⢿⢃⢹⣿⠸⣃⠸⢁⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠸⠏⣰⣿⠱⠿⠿⣿⠿⠇⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⣸⢿⠷⡶⠇⠀⣶⡇⣈⠀⡏⣷⣉⣿⠉⢷⠹⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢈⠴⣿⡟⡁⠀⠀⢦⡀⠠⠀⡘⢀⠼⣿⣵⠄⠀⣴⢀⣿⣿⣯⣷⣧⣤⠞⢧⡘⠽⠀⠳⣿⡩⢿⣇⠤⢱⡰⡆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⠷⣄⡸⣽⣷⡀⢀⢀⣽⣧⠄⠚⠀⠘⡹⠯⠇⣤⣽⣿⣿⣿⣿⣿⣿⠟⠁⢠⡞⠃⢁⠀⠀⢙⢷⣌⣻⢿⠿⣀⢹⡌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⢉⢖⣯⣟⣿⣍⣉⣚⣺⠁⡀⠀⠀⠜⡁⢀⢀⢶⣿⣿⣟⣿⣿⣿⣿⣧⣀⣦⢋⡽⣎⠺⠸⠓⡄⢿⣿⡈⢿⣜⢹⠿⢿⡀⢧⡄⠀⠄⠶⠲⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢞⢎⡾⣳⣿⣿⠏⡷⢃⣰⡞⣷⣾⣿⣶⢦⣐⢊⣾⣿⢯⣿⣽⣿⣿⣿⣿⣛⣧⣽⣷⠿⠀⢫⠀⠘⢮⣌⣻⣮⣿⣿⡏⢦⢱⠀⠈⢒⠤⡠⠴⠟⠀⠑⠂⢀⠀
⠀⠀⠀⠀⠀⠜⣺⡟⣞⣷⣿⠋⢰⠁⠈⢫⠗⣿⣿⢟⣿⣿⣿⣳⣟⣯⢿⣽⣷⠿⢫⣍⠃⢉⣾⣿⡏⠀⠀⢸⣠⣠⣾⣏⠉⡹⣯⢝⠺⣮⣶⣧⠀⣢⡀⠂⠄⡘⠀⠀⠀⠈⢢
⠀⠀⠀⢠⣬⠔⣯⣝⡳⣎⡏⢀⠇⠀⠀⠀⠀⢻⠇⣼⣿⣿⠿⢟⢻⣿⣫⢟⡥⢚⠁⠀⠀⠞⣉⠣⠀⢀⠀⠼⣿⣿⡿⠁⢂⠄⢹⣿⣦⣹⣷⣾⠀⣵⠹⡓⡞⡁⠀⠀⠀⠀⢸
⠀⠀⢀⢷⠻⣧⡗⢮⡑⠫⣠⠞⠀⠀⠀⢀⣠⣿⣿⣿⣻⡑⣎⣜⡶⣾⣿⣿⣷⣷⣾⣦⣴⣿⣿⣩⡯⣼⡤⠀⠘⢿⣷⠀⠸⡲⣼⣿⣿⡟⣛⢓⠀⢸⣡⠀⣇⢣⣀⡀⣀⣠⣟
⠀⠀⡎⠘⣆⢹⡟⣆⠣⡀⠹⡀⣇⣤⣾⣿⣿⣿⡿⣯⢳⣹⣾⣛⣿⢷⢟⣿⣿⣾⢷⣯⣿⣽⢿⣿⣗⣦⡑⡄⠀⠈⠛⢦⣁⣷⣟⡻⢍⠳⢼⣄⠀⣼⣿⠡⡙⣖⣻⢿⣟⣯⠏
⠀⡐⠀⠀⡇⢸⠷⣈⠒⠈⠐⠀⣷⣿⣿⣿⣿⣿⡟⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣟⣧⣽⣿⡿⡿⣿⣿⣿⣾⣫⠀⠀⠀⢀⣶⡿⠛⠻⣎⣣⢻⣇⣴⣿⣧⣳⡱⣎⢿⣿⡿⠋⠀
⢀⡇⠀⣴⣡⣸⠳⢠⠉⠆⠌⡸⠈⠓⠫⢝⢿⣿⣿⣿⣿⣟⣟⣻⣿⣗⣾⣼