Incident Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠱⣰⢿⠀⢲⣽⣍⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢹⣹⡇⠀⠀⢀⡀⠠⢹⣿⡄⢸⣿⣿⣿⣇⠀⠀⣀⠀⠀⠠⠀⣰⣶⠄⠀⠈⣿⣿⣶⣤
⠀⠀⠀⠰⡀⠀⠰⣣⣴⠀⢐⡟⢿⡈⣦⡙⣿⠀⣹⣿⣿⣿⣧⡀⠘⣿⠀⠀⠀⣼⡥⣄⠒⠂⢿⣿⢀⣤⣤
⠀⠀⠀⠀⠐⣾⣿⣿⡿⢦⣿⣿⡏⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠁⠈⢳⡉⠁⣭⣹⡌⠉⢸⣿⣽⣾⣾⣿⣷⣶⣤⣤⣀
⠀⢰⣾⣦⣿⣿⡿⠛⠷⣾⣏⠙⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⣸⠿⢻⠀⢸⣟⡛⠛⠛⣹⡿⢋⣵⣾⣿⣷⣦⡀
⠰⣿⣿⣟⣿⣧⣦⣰⣦⣤⣍⣁⣽⠒⠦⠄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢿⣦⣤⢸⣇⠛⡇⣧⡴⢺⡇⢠⣤⢠⣾⠿⣿⢻⡿
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣉⡁⠀⢾⡀⠀⠀⠀⠀⠀⡴⠛⢛⣉⡙⢦⣤⣤⣤⣤⣹⣷⣿⣿⠀⣿⣿⣿⣿⣇⣾⢧⠸⣿⣿⣿⣸
⠀⠘⣿⣿⣿⣿⣿⣽⡓⠀⣳⠛⢿⣷⣶⣾⡏⣿⡟⠀⣰⡿⠋⢱⡌⡛⠛⠛⠛⠉⢿⣿⡇⠀⠀⠀⠀⠈⢻⣿⣦⣴⣭⡝⣽⡇
⠀⠀⣿⣿⣿⣿⣿⣣⡵⠄⠉⢇⠀⠀⢻⣿⣷⣝⡳⣄⠹⡘⠘⡻⢽⠇⠀⠀⠀⢀⣾⣟⡁⠀⢀⡀⠀⠀⢀⠙⣿⣿⣿⣿⣾⠃
⠀⢺⣿⣿⣿⣿⠿⠛⢷⣌⠱⣿⣦⡔⣹⣿⠉⠉⠀⠛⠛⣻⣵⣧⡀⢀⣤⠤⢴⢶⠛⠉⠉⣰⣿⣦⠐⢤⣤⣾⣿⣿⣿⠿⠁
⠀⢸⣿⣿⣿⣿⡄⢸⣤⠛⣃⣤⣽⣿⢿⣿⢶⣄⡘⠃⢠⠟⢘⣗⣰⣿⣦⣖⣂⣀⣀⣤⡾⠛⠉⠈⠳⣄⡈⠙⣿⠿⠁⠀⠀⠀⡀
⠀⣴⡿⠃⢘⣿⠇⠈⠙⠓⠹⣟⡟⢉⣸⣟⠏⢀⡝⢶⣾⣿⣏⣀⣽⣿⡷⢿⣿⠿⣯⡣⣀⣀⠀⠀⣴⣿⡷⠞⠁⠀⠀⠃⠀⢐⡇
⢠⣿⠀⠀⣼⠏⠈⠓⠀⠀⢀⣼⣥⣿⣿⡿⣿⣋⣄⢸⣿⣟⣿⡟⠿⣶⣄⠀⢹⡆⠈⠻⣎⠙⠦⠀⠙⠿⢿⣶⣶⠀⢀⣤⣄⠀⠁
⠘⠿⠶⢶⣿⣇⠀⢤⡤⠞⠋⠁⣿⠿⣏⠻⣿⡉⠙⣿⣿⣟⣗⠾⠐⢚⡉⣥⣾⣧⠀⣠⣜⣤⠀⠒⣓⣀⣛⣿⣯⣤⣭⣥⣄⡀⡀
⠀⠀⠒⠬⢍⣛⡷⣿⢳⠄⠀⠀⡽⠂⠾⠀⠘⢷⡀⠈⢻⡆⣼⣷⠻⡅⠺⣯⡀⢁⣀⣙⣉⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠒⠶⠛⠳⠙⠷⣆⣠⣤⣄⣀⣠⣤⣤⣾⣁⣾⣏⣿⣦⣵⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠋⠉⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣠⣤⣤⣤⣤⣤⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⢦⡄⣶⠔⠘⡶⢾⣓⣀⣿⣿⣿⣿⣇⣾⣿⣶⣶⡶⠀⡄⠀⢤⣼⣄⠀⣤⣿⣿
⠟⣞⢹⠿⠿⠿⠿⠿⢳⡋⡯⠿⠦⠔⡟⠀⠀⠀⠸⢻⡋⣻⠿⠿⠿⠿⠟⣞⢙⡿⠿⠆⠀⠀⣠⠻⣤⠀⠉⣿⣿
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⢰⢶⣒⢦⣤⣤⣤⣄⣼⣿⣷⣶⣼⣷⡄⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣷⡀⢠⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠉⠿⣨⠟⠛⠿⠿⠿⠟⡗⠲⣽⣿⡜⠁⠀⢀⣄⡀⠀⣼⣿⣿⣿⣿⣿⣿⡛⠉⠁⠀⠰⢀⣀⣰⡄⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣛⣛⢛⣛⣹⡀⠀⢠⡏⠀⠀⢿⣿⡿⢿⠿⣿⠏⣷⣶⣿⣾⣿⣿⢿⣻⠇⠀⠀⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠃⠀⠈⠋⠁⠀⠀⠉⠈⠀⠈⠘⠀⠀⠸⠇⠇⠀⠀⠻⠚⠛⠧⠏⠁⠀⠀⠀⠀⠀⠀⡳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢿⡛⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢊⠹⢧⣾⢥⠀⠈⢺⡯⠀⠐⠀⠀⠔⠀⠀⢰⢶⣯⣹⣶⣤⣤⣤⡴⢲⡄⣿⣿
⠀⠀⣀⣤⣶⣿⠀⠀⠀⠀⠀⠀⢘⣷⣦⣄⣀⠀⠀⠘⡟⠆⠀⠀⠛⠀⠀⠀⠈⠚⠛⠻⠿⢿⡿⠿⠿⢿⡇⣿⣿
⣤⣬⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣧⣤⣮⣤⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣼⣷⡾⠟⠈⠻⣷⣿⣆⠀⠀⠀⢠⣿⠋⠁⠉⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⣿⠀⢹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠺⣿⡄⠀⠀⠀⠀⠀⢸⡿⠆⠀⠀⠘⣿⣄⠀⢀⣼⡟⠀⠀⠀⠀⠀⢰⣶⣶⣾⣿⠀⢸⣿⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢴⣿⠃⠀⠀⠀⠀⠀⢺⣷⡆⠀⠀⠀⠈⠛⠿⠿⣿⣄⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⢸⣿⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⡿⢷⣦⢀⣴⡿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠀⠀⠀⠀⢸⣿⠀⣸⣿⠀⣸⣿⠀⣸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⣀⣀⣀⡀⢀⣀⣀⣈⡀⠀⠀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣁⠀⠀⢀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⠛⠛⠛⠛⠁⠈⠛⠙⠛⠁⠀⠀⠛⠛⠛⠛⠋⠀⠙⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠁⠀⠛⠛⠛⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠺⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡄⠀⠀⠀⠀⠀⠀⠺⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠛⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣀⠀⠀⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⣿⡇⠀⢤⠀⠈⠋⠀⠀⠀⠀⠈⠁⢰⡦⠀⠀⠀⢀⣸⣿⣤⣤⣼⣿⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠉⠉⠉⠙⠛⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣶⡶⢶⣶⣤⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣤⠀⠀⠀⠀⠀⠀⢠⡄⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⣤⣤⣿⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢐⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⠲⠆⠠⢊⠀⠈
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿
⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣄⡀⣀⣤⣶
⠀⠀⠀⢠⣿⠏⣿⣿⣿⣿⠿⠿⠿⠛⠁
⠀⠀⠀⢸⣿⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠶⠶⠶⠶⠶⠀⠀⣠⣶⡿⠛⠛⠛⢻⡇
⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⠏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠆⠀⣴⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⣿⣿⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠋⠀⠙⠋⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠀⠀⠀⠿⠿⠿⣿⡇⠀⠀⠀⣹⡿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠛⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣿⡷⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠀⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣰⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠉⠉⠉⠉⣿⠉⠉⠛⠛⠛⠛⠻⣷⡀⠀⠀⠀⠀⠀⠀⠰⣿⡁⣷⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠈⢃⣽⠃⠀⢸⣿⣿⣷
⠀⢀⣀⣠⣤⣤⣶⣿⣯⣤⣤⣄⣀⣀⣀⣀⣀⣠⣿⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣀⣴⣶⣄⣀⣤⡐⠛⠁⠀⠀⢸⣿⣛⢛
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣧⠀⢀⣿⣿⣿⡿
⠀⢸⣿⣿⣿⣿⣿⡿⢋⣥⣶⣦⣍⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⠇
⠀⢸⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⣿⣶⡈⠀⠛⠿⠃⣼⣿⣿⣿⡟
⠀⠈⠉⠉⠉⠉⠀⠘⣿⣿⣿⣿⣿⡿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⣿⣿⣿⣿⣿⡗⠀⠀⣤⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠁⠀⠘⠿⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁


⠀⡉⡉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡆⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣽⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣾⠃⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣼⡆⠀⠀⠀⢄⡀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⢃⢀⠹⣿⡄⠀⠀⠀⡼⢳⡀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡆⢳⠀⠀⢢⡞⠻⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⠀⠈⣇⣆⡝⠣⡀⣴⡎⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⣷⡈⠛⢦⣸⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠟⠋⠀⢀⠿⠿⠲⠀⢠⠻⣯⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡃⠀⡇⠀⠀⠀⣸⠀⢀⡿⡁
⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⡠⠖⠁⠀⠒⠐⠃⢈⣾⣿⠤⠶⠊⠀⠀⠀⠀⠀⠀⠀⣇⠘⠻⣆⣠⠴⠃⠀⣼⣡⠃
⠀⠀⠀⠀⠀⠀⠀⡋⠙⠚⠳⣄⣀⣐⣐⣂⣀⣿⣼⠇⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡌⣳⢦⣙⠲⢤⡴⠚⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡶⣄⣀⡈⣉⠹⣿⠯⣽⣿⣛⣠⠞⠶⣦⢄⠀⠀⠀⠀⢠⣋⣼⠁⠀⠈⢧⣴⣷⡄⠀⠀⡇
⠀⠀⠠⠤⠄⠀⠀⠰⣽⣟⠛⠟⠿⠾⠭⢄⡠⠶⠟⠁⠀⠀⠈⠛⢶⠒⠀⠀⠈⠙⠻⡄⠀⠀⢸⡇⠀⠀⠀⣰⠋⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⣀⣀⣀⣀⣀⣈⣻⣧⡀⠀⠀⠀⠀⢹⣦⣤⣤⡀⠀⠀⠀⣰⣤⣤⣤⣤⡤⣴⢷⠀⠀⠘⣇⣀⡴⠾⠥⠤⠤⠤⠤⠤⠤⠶⠶⠶⠆
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠙⣦⠀⠀⠀⡸⠀⢀⡞⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⢸⡈⣧⠀⠀⠹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⢀⡞⠁⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⡇⠈⢷⡟⢲⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣧⠋⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢷⣀⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡷⣤⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣀⣿⠇⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⢈⣧⠀⢹⡏⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣤⣿⣵⣶⣿⣥⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⢻⠋⠻⠿⡛⣷⣄⣈⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⡽⠏⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⣶⡆⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢴⣶⣤⣦⡀⠀⠐⠀⠀⠀⠀⠁⠈⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠘⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠉⠻⠄⡁⠀⠀⠠⠀⠄⠀⠒⠀⠀⠸⠿⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⢴⣈⠁⠙⠒⠁⠁⠀⢀⣤⠎
⠀⠀⠀⠀⠀⠀⠤⠦⠴⠺⠦⢤⣆⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡀⠴⠀⠀⠠⠄⠐⠀⠀⠀⠈⡀⠀⠀⠀⠀⢴⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⡀⠈⡗⠀⢀⠀⠀⠀⠀⠀⢘⡌⠁⠀⠀⠈⠉⠉⠀⠀⠀⢀⡀⠘⠻⣆⠀⠈⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⢀⡠⠀⠀⠀⠀⠀⢸⣧⣀⣄⠈⠶⠀⢐⠀⠀⠠⠋⠀⠀⠀⢹⠀⠀⠀⠀⠀⠠⠂⢸⡇
⠀⠀⠒⠠⠀⠀⠀⣀⣠⣤⠥⠤⠞⠉⠀⠀⠀⠀⠀⠀⢠⠈⠑⠀⠉⢖⣂⣉⠄⢢⠀⠀⠀⠀⠀⢸⡅⠀⠀⢀⣄⠴⠀⡐
⠀⠀⠀⠀⠀⢠⣤⣤⣶⡾⠷⠖⠉⠄⠄⡄⠀⠀⠀⠀⢨⠀⠀⠀⢦⣄⠉⠁⣺⠺⡆⡀⠀⠀⠀⠈⠁⠀⠔⠚⠤⠤⠙⠁
⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⠀⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⠄⠙⢆⡁⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠐⠒⠛⠋⠀⡇⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣇⡀⢀⣀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢤⡖⠚⣩⣥⠀⢻⡀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣶⣾⣿⣛⠋
⠀⠀⠠⠀⢤⡀⢾⣷⠀⣘⣭⠴⣚⢿⠀⠀⠈⣿⣶⡶⠚⠰⠀⠀⠀⢀⠀⠀⣀⠀⠀⣼⢿⡃⠠⢿⣿⠍
⠀⠀⠰⣶⢆⡿⠀⠘⣯⣁⡀⠘⣿⣾⣧⠀⠀⠸⣿⣧⣤⣄⠲⣿⣶⣿⣿⣾⡿⠂⠀⠈⢻⣿⠀⠘⠛⠿⠄⣄⣀
⠀⠰⣧⢶⣿⣧⠀⠀⢸⡿⠓⣿⡏⠀⠸⡆⠀⠀⠘⣷⠀⠹⠇⣿⣿⣿⣿⣿⡇⠀⠀⢠⡾⣿⡀⠀⠀⠀⢸⢿⢿
⠀⠀⠀⢀⣽⣿⡇⠀⢠⣿⡄⢈⣡⡤⠖⢻⡀⠀⢰⡟⠀⠀⠀⡏⠉⠉⠉⠉⡇⠀⠀⠘⣷⠉⠛⠛⠂⠀⢸⣷⣿
⠀⠀⠀⢸⡇⠘⢷⣄⡀⠙⢿⠉
⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠤⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠒⠀⠸⡿⠂⠈⠀⠀⠄⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⣀
⠀⠀⠀⣾⣇⣸⣂⣹⡶⠻⠶⣾⣀⠀⠀⠀⠀⠰⣟⣿⣻⠆⠀⠈⢠⣤⡀⠀⠀⢿⢇⣼⠂⣷⡄⢹⣷⣄
⠀⠀⢷⠴⣿⣷⣈⣿⡀⠀⠀⢼⡋⠀⠀⠀⠀⠀⣼⠿⢷⡦⠀⠀⠸⠿⠃⠀⠿⠶⣿⣷⣴⡿⢷⣿⠟⢿⣄⣷
⠀⠀⠀⠀⣼⣿⡏⢩⡧⢶⡶⢿⠉⠀⠀⠀⠁⠈⠻⣦⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠈⠋⠀⣸⠟⣿⠃
⠀⠀⠀⠸⣯⠈⠿⣦⡀⠈⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠄⠀⠀⠶⠶⠾⣷⠾⠿⣦⡶⠶⠶⠶⠶⠿⠶
⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⢀⣾⣿⣄⢀⡄⠀⠀⠀⠀⠀⢠⣳⣤⡻⠃⠀⠀⠀⠀⠀⣠⠖⢲⣾⠛⣿⠛⠛⠒⠲⢄
⠀⠀⠀⣸⣿⣿⣿⠈⠀⠀⠀⠾⠛⠻⣶⣿⠿⢅⣀⣀⣠⣤⣤⣾⣵⣦⣶⣷⣠⣿⣤⣤⣤⣤⣌⣷⣤⣤⣀⣀
⠀⠀⠀⢿⡀⠸⣧⠀⠀⠀⠀⠀⠀⢰⡟⠉⠓⣾⣿⡿⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⡿⣿⣿⡿
⠀⠐⠒⠚⠒⢶⣿⡶⠒⠒⠀⠀⠀⠸⠇⠀⠙⠛⠛⢾⣋⣿⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢾⣉⣿⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡶⠶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⢁⣠⣤⣤⣤⣄⡉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⣾⣿⣿⣿⣿⣿⣿⣿⣦⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠀⣸⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⣿⡆⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠸⣷⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢠⣿⠁⣀⡉⠛⠉⣈⡉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠻⣷⣀⠙⠿⣿⣿⣿⣿⣿⠟⠃⣠⡿⠃⠀⠙⠻⠃⢰⣿⣿⣷⣶⣤⣄⣉⠙⠛⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢀⠈⠻⣷⣤⣄⣈⣉⣁⣠⣴⡾⠛⢁⣴⣿⣿⣦⠀⠻⠋⢹⣿⣿⣿⣿⣿⣿⣷⣶⡄⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⣴⣿⣿⣦⡄⠉⠉⠛⠛⠛⠉⠁⣤⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⣈⠉⠛⠻⢿⠟⢻⣿⠃⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣿⣿⣿⡿⠁⠸⠿⢿⣿⣿⣷⠀⣿⣿⣯⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣈⠉⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⣿⣿⣶⣶⠂⣠⣿⣿⣿⡿⠀⣿⣿⣿⠉⣿⡏⢹⣿⠉⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠹⣿⣿⣿⡁⣴⣿⣿⣿⡿⠃⣰⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⡈⠛⠿⠿⠿⠿⠟⠋⣠⣼⣿⣿⣿⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠠⡂⠄⠀⠠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢊⠴⠋⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠰⠤⢄⣀⣑⠀⠀⠀⠀⡀⡠⠴⠖⠤⠤⠤⠖⣢⣤⠤⠴⠒⠛⠉⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠒⠂⠀⠀⠀⠀⠀⡠⠤⣈⣙⡄⠀⢀⡮⢀⣀⠤⠒⠒⠉⠁⠀⣀⣀⣠⠤⠤
⠀⠀⠀⠀⠀⠠⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⠀⢰⠰⣿⠶⣅⡀⠀⠀⠀⠒⠉⠉⣁⡀
⠀⠀⠀⠀⡔⢁⣤⣾⡿⢿⣿⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⡇⣲⢀⣴⡿⠃⠀⠀⣠⣶⣿⠿⠛⠿⢿⣷⣦⡀⠀⠀⢰
⠀⠀⠀⡘⢠⣿⣟⠁⠀⢸⢸⠀⠀⢙⣿⣧⣀⣀⣠⣤⣤⡴⠖⠉⠀⠀⣇⢞⠛⠛⠻⠿⣿⣿⡿⠉⢏⢆⠀⡠⣪⠛⢿⣿⡄⠀⠈
⠀⠀⠀⠀⣿⣿⠛⠻⢽⣿⣻⣶⠭⠟⢻⣿⣯⠙⠛⠋⠁⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢩⡗⣀⡨⣿⢻⣿⡁⠀⢸⣿⣇
⠀⠀⠀⠠⠟⣿⡄⢀⢞⡽⠻⡳⡀⠀⣼⡇⢸⠃⠀⠀⠀⠀⣀⣠⠴⠚⠁⣧⣄⣀⣀⣀⣰⢿⣿⠒⠚⢹⢻⠓⠺⢵⣾⡟⠙⠦⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠈⠿⣷⣏⣀⣀⣹⣽⡾⠋⠀⠘⠶⠶⠖⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠻⣷⣄⣸⣾⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠉⢻⣷⡄
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠉⠈⣸⣿⡿⠂
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⠻⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃⠀⠀⢹⠁
⠀⠀⠀⠀⠀⢀⣤⣤⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠸⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣠⣦⠀⢀⣦⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆⣼⣿⣿⡆⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢰⣶⣶⡀⢰⣶⣶⠀⠀⠀⠀⠀⠉⠉⢁⣽⣧⡉⠉⠃⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠸⠿⣿⠉⣿⠉⢻⠋⢹⠀⠀⠀⠀⠰⠿⣿⣿⠷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣶⣾⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⡟⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠙⣿⠛⠛⠓⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⢀⣠⣾⣩⣇⣀⣀⣀⣀⣠⣿⣀⣀⣀⣀⣀⣸⣳⣄⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠈⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠉
⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⣿⠇⠛⢻⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃
⠀⠀⠀⠀⠈⠉⠉⢧⠘⠃⣸⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣟⠰⠆⣹⠛⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛


⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⣶⠶⠦⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⡴⠶⠶⠶⠶⢤⣤⣤⣀
⠀⠀⠀⠀⢀⣴⠞⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢹⣦⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⣿⣿⣷⡄
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣿⣶⣦⣤⣄⠀⢀⣤⣤⣶⣾⣷⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠘⠛⠛⣟⠠⡆⢹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⢋⡉⢳⠿⠋⣿⡿⢋⡉⢻⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠡⡤⢹⡿⠿⠋
⠀⠀⠀⠀⠀⠈⠲⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡥⠞⠀⠀⠀⠹⢤⣥⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠊

⠀⠀⠀⠀⠀⣠⡶⣻⡿⡋⣉⢟⡽⠛⡹⢛⢛⢟⠟⡻⢛⣻⢟⢟⢿⡶⠒⠖⠒⠖⣛⣿⢟⠟⢛⢛⡿⠛⢳⠖⢶⢶⠶⠶⠶⠶⠶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⣿⣷⣿⡏⠉⠉⠉⠉⠙⠛⠛⠛⡛⡛⠛⠛⠉⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⢻⠛⠛⠓⠻⠾⠾⠯⣄⡼⣇
⠀⠀⠀⠀⠀⣿⣟⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠡⣙⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠈⠀⠀⠀⠀⣯⢨⣿
⠀⠀⠀⠀⠀⣿⣏⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⣿⡏⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⡇
⠀⠀⠀⠀⠀⢻⣗⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⣺⠃
⠀⠀⠀⠀⠀⢸⣿⡯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣤⣴⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣿
⠀⠀⠀⠀⠀⢸⣟⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡡⣿
⠀⠀⠀⠀⠀⠘⣿⢿⣾⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⣿⢘⡇
⠀⠀⠀⠀⠀⠀⣿⡾⢸⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⠈⠉⠉⠉⠛⠛⣛⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⣷⢾⡇
⠀⠀⠀⠀⠀⠀⣿⡱⢫⡇⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢼⡯⢺⠁
⠀⠀⠀⠀⠀⠀⣿⣥⢟⡇⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣧⡟⣷⠀⡀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣳⡟
⠀⠀⠀⠀⠀⠀⢹⡏⠄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡏⣴⣝⡳⣶⡼⡤⠤⣤⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⡤⣤⡼⡤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⣸⡇
⣠⡤⠤⠤⠤⠤⠼⠿⠿⠿⠾⠿⠮⠦⠾⢾⣿⣵⣾⣵⣥⣼⣥⣴⣴⣵⣯⣎⣼⣮⣬⣼⣯⣿⣳⣋⣔⣑⣴⣳⣚⣵⣿⢏⣴⢋⣌⣼⣃⣀⣀⣀⣀⣀⡀
⢻⣄⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣤⣀⣤⣤⣤⣤⢴⣿⣤⣤⣤⣤⣤⣶⣿⣉⣠⣴⢶⣿⣿⣿⣿⣷⣶⣶⣶⣤⣴⣴⣦⣦⣤⣬⣥⣤⣤⣀⣠⣠⣤⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠋⠁
⠀⠀⣐⣂⣰⣶⣆⢤⣀⠔⠰⢶⣦⡄⡤⣤⣤⣴⡀⠀⠦⠾⣭⣏⣛⡀⠀⠀⠀⠺⠿⠛⠿⠿⠿⠿⠷⠿⠂⠱⠮⠀⠀⠀⠀⠛⠋⠁⠉⠉⠈⠉⠈⠉⠉⡉⠁
⠀⠀⣿⡙⢷⡻⣿⣷⣶⣦⣶⣞⢛⣧⣤⣯⢿⢠⣖⣦⣑⣤⣶⣮⣽⣦⣴⡶⡐⣿⣟⠒⣶⣦⣤⣤⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠨⠹⣿
⠀⠀⣿⣿⣿⡿⣋⣽⣾⠗⢻⡟⣛⣛⡏⣦⣖⡻⠛⢛⢿⣟⠛⣯⢻⡚⢿⢉⣉⣻⡋⠛⠛⣷⢾⣷⣏⣟⡟⢩⣷⡼⣉⣴⠄⣀⠀⢀⠀⡀⠀⣀⣀⡀⢠⣶⣿
⠀⠀⣿⣯⣭⡴⢷⡶⣿⠏⠸⢿⣿⠿⣛⣿⡛⠛⡏⣟⣻⠽⠀⠉⠈⠁⠉⠈⠉⣼⡍⠀⠀⠉⠈⠁⠈⠀⠀⠈⠉⠀⠛⢻⡎⠉⠉⠈⠫⡄⠤⠬⠥⠵⢬⣯⣿
⠀⠀⣿⣿⣿⡁⣬⡅⢠⣶⣿⣫⣈⣀⣽⣿⣃⡀⢸⣟⣁⠂⣀⠀⠀⠀⠀⠀⠰⣾⡇⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠿⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣁
⠀⠀⣿⣿⣿⣿⣿⣶⣿⣏⠉⣅⣨⣉⣸⣿⣥⣤⣼⣿⣴⣶⡓⣂⢀⢀⡀⡀⣘⣟⣷⣦⣤⣸⠀⠀⡀⠀⠀⣠⣶⣷⣴⠏⠁⠀⠀⠀⠓⣁⠀⠀⠠⡎⠀⠉⠉
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣴⣞⣀⡀⣙⣛⣛⣛⣛⣛⣋⣋⣛⣉⣉⡉⡙⢿⣿⣿⣿⣿⣿⢹⣄⣈⡁⢠⣾⣿⣿⣛⣁⣀⣀⣀⠀⠀⠀⣿⢦⣀⣸⣀⣀⣀⣼
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⢿⠿⡿⠿⠿⠿⠟⠿⠿⠟⠻⠛⡗⠿⢾⣿⣿⣿⣿⣿⣿⣟⣿⣡⣿⡏⢹⣿⣿⡄⠀⣤⠀⠐⠙⠂⠛⠿⠖⠮⠭⣟⣿⣟
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⢿⠿⠿⠓⠻⠓⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣤⣿⡿⠃⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢲⣥⠂⠀⠤⢀⠰⡶
⠀⠀⣿⣿⣯⣿⣿⣿⣿⣿⣿⣟⣡⠶⠿⠶⠶⠷⠷⠿⢚⣿⣷⢾⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣛⣍⠀⣄⡀⢰⣿⡽⣿⣿⢿⣋⡄⣠⡀⢨⣿⡄⠀⠀⠺⢷⡄
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣛⠀⠻⠖⠛⠒⠛⡴⣾⣿⣿⣿⡟⠒⠛⠒⠛⠛⣉⣩⠿⣱⣘⡿⣿⢟⣇⠇⢨⣿⣿⣿⡟⣿⢿⢷⣿⠂⣾⣿⣿⡄⠀⠁⠺⠛
⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⡟⠉⣉⣾⣧⣤⣤⢞⠀⠛⢿⣿⣿⣿⣦⢴⠶⢶⣾⡿⢧⠜⣯⠾⠯⠽⡾⢠⡄⢸⣿⣿⣿⡹⠇⠎⠤⢣⣾⣿⣿⣿⢿⡄⠀⢠⡄
⠀⢀⣿⣿⣟⣿⣿⣿⣿⣿⠏⣡⣯⣭⣁⡼⠉⠀⣀⣀⠀⠻⣿⢟⢭⣽⡃⠉⣉⠦⢼⢿⠏⠀⠀⢠⠀⠈⠇⠈⠟⠛⠂⠁⣶⠀⠀⢻⡟⠛⠟⠛⠰⠀⠀⠉⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠟⢁⣾⣼⣿⣿⠃⠉⠉⣿⣿⢿⣧⣶⣍⠃⠔⢛⡀⢀⠀⠀⠈⠀⠀⠀⢠⠀⠀⢸⠁⠀⠀⢸⣿⠭⠀⠀⠀⣿⣶⣾⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠞⠾⠾⠇⢸⣽⡿⣆⠀⣀⢿⡿⠽⠏⡁⠀⠀⠀⠀⠀⠺⠀⠦⠠⡼⠖⢶⣾⣆⢠⠜⠓⠰⠶⠾⠧⠄⠡⠄⠰⠿⣿⠿⣿
⠀⠀⣿⣿⣿⣭⣍⣽⣯⣿⣯⡽⢋⡀⢀⡀⠖⠡⠼⠄⠈⠀⣀⠀⠵⣿⡱⠉⢤⠀⡀⠀⠀⡆⠀⠀⠀⠐⠶⢿⡿⣿⡞⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⠀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣴⣞⠉⢀⢾⣏⠰⣷⢈⣹⣿⣾⣿⣇⠐⠷⢤⣤⡄⢤⣴⢟⣧⣿⣿⣮⣿⣿⣿⣭⣿⣹⣟⢛⣿⣿⡇
⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣶⣷⣿⣧⣼⣦⣤⡀⢈⣦⣾⣷⣶⡋⠉⠛⠍⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢻⣿⡯⢻⣧⣿⠿⠿⠛⠛⢯⡿⣿⣿⣿⣏⣿⣿⠛⠉⠻⠛⠨⡽⠛⠛⠦⡙⠃⠉⠍⠀⠙⢹⡿⠟⠻⠄⠀⠀⣾⡏⠙⢉⣏⣩⠭⡿⠛⠩⠙⠛⠋⠃
⠀⠀⠀⠀⠉⠁⠀⠀⠛⠀⠀⠀⠀⠈⠁⠘⠛⠛⠉⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙

⠀⡀⣀⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⢀⡅⡏⠀⠀⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀
⠀⡆⠁⠃⠀⠀⢰⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢹⣿⢠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠁⠀⠿⢠⡆⣻⠀⡇⢠⠀⠀⠀⠀⠀⠀⢀⠀⢱⠀⠀⡰⠶⠖⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⡖⣶⣶⡆⣿⠀⠃
⠀⣀⡀⠀⠀⠁⠛⢰⣷⡔⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠇⠋⠘⠁⠀⠀⡀
⠀⣿⡇⣷⢰⣤⢈⠀⢌⣓⣠⣄⢀⠀⠀⠀⣠⢄⠀⠤⠀⢸⠿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡄⢰⡆⢸⡇
⠀⣿⡆⣿⢸⣿⣸⣼⣶⣼⣛⠫⢽⣤⢳⣤⣭⣾⠀⣠⡀⢃⣄⠀⢀⡀⢰⡄⠀⠌⠁⠀⠀⠀⠀⠤⠀⠀⠘⠸⠇⠘⠃⠈⠁
⠀⠿⠧⣿⢼⢿⢿⢿⣿⣿⣿⣮⣟⡙⠿⣿⣞⣣⠀⠿⠇⡸⠿⠀⢸⡷⠾⡁⠀⢠⡀⣤⡷⠀⠀⠀⠀⠀⢸⢀⡀⠀⡀⢀⡀⢀
⠀⠀⠀⠀⢀⢀⢀⣀⣲⣿⢱⡜⣿⣿⣿⣿⣿⣿⠀⠀⠚⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⣶⣧⢀⡀⠀⠀⣿⠸⢸⡇⢘⡇⢸⡇⢰
⢸⣿⡉⣇⣿⣿⢿⣿⣿⣿⣾⣿⣿⠃⠹⠛⠛⠛⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠂⠀⠁⠁⠀⠀⠃⠈⠃⠘⠣⠈
⢘⠿⠛⠋⣿⣛⢻⡿⡿⢿⣮⣋⣭⣴⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡤⢠
⢈⣶⣾⡾⠡⠈⣯⣿⣿⣿⣾⣿⣌⠀⠚⠛⠻⡿⠷⠀⠀⠀⠀⠀⠀⠲⠖⠀⢠⣀⣀⠀⣐⣛⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠘
⠘⠃⠘⠀⡀⠀⠀⠈⠿⠿⣾⠏⣉⣼⡃⠄⠀⠁⠀⠀⠀⠐⠒⠑⠒⠀⠐⠆⠘⠛⠋⠀⠙⠛⠉⠛⠛⠻⠟⠿⢻⣧⣀⢌⣿⠁
⠀⠀⠀⠰⣷⠔⠂⠀⠀⢶⡀⣤⣸⠈⠙⠳⡴⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⢐⣊⣈⣋⣉⣉⡀⣁⣹⣤⣀⢾⣮⢙⣧⡶⣤
⠀⠀⠀⡀⡉⢠⡄⠀⠀⠀⠀⠀⠁⠀⠂⠘⠦⠀⢹⠙⣤⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⢠⠄⠈⠉⠉⠀⠀⠀⠈
⠀⠜⠀⠦⣧⢘⠇⣤⡆⠀⡜⠐⠦⠀⠴⠂⡴⡄⠸⠚⠉⠀⢀⠀⢀⡀⣶⠄⠀⠀⠀⠀⠀⢀⠇⢸⠠⠀⠃⢀⠀⢃
⠀⢷⡀⠀⠀⠉⠂⠿⠃⢀⣠⣀⣠⣤⣤⣀⢈⠈⢀⠈⠒⢀⣘⠳⠈⢐⣲⣆⣦⢀⡀⠈⠘⠁⠀⠛⠀⢀⣠⠊⢀⡀⢀
⠀⠶⠴⣶⡶⠀⠀⠀⠀⢂⡉⠁⠀⠶⢾⣿⣷⣤⠠⠤⠤⠠⠬⠻⡄⠀⠉⠉⠡⠾⠃⣦⠄⣠⠀⢰⠀⠀⣬⠀⢀⡁
⠀⠀⠠⠿⠿⠂⣠⡀⢀⣀⠀⣴⣄⠐⠦⣄⡀⠀⣠⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠋⣀⣽⠧⠾⢿⡤⣼⡏⠀⣠⣿
⠀⣀⠀⠐⠴⣿⣟⣱⡀⠀⠉⠙⠛⠳⣾⣿⣟⣿⣋⠀⢰⠖⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠲⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠈⢹⠉⠉⣻⠛⠛⠛⡟⠛⠛⢿⠛⠛⢻⠛⠛⢻⠛⠛⢹⡏⠉⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⣼⣄⣀⣽⣀⣀⣠⣇⣀⣀⣼⣀⣀⣸⣀⣀⣸⣄⣀⣼⣄⣤⣧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⠤⢤⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣁⡠⠿⣆
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⠷⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⣶⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⠀⠀⠀⠀⣴⠟⠛⠃⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠀⣼⠋⠉⠉⠉⡏⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠟⠀⠀⢿⣤⣤⣤⣤⣿⣤⣤⣤⠾⠁
⠀⠀⠀⠀⢀⣤⠶⠶⢶⣤⣤⣭⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡴⠶⢶⣤⡀
⠀⠀⠀⠀⣟⠀⠀⠀⡼⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠛⠁⠈⢹⣧⠀⢀⡼⠛⢶⣄
⠀⠀⠀⠀⠙⢦⣄⣀⣳⡀⠀⠀⢀⣰⠟⢀⣀⠀⠀⣿⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠙⢦⣠⡾⠋⠳⣮⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢛⣻⣷⠞⠋⠉⢻⡄⠹⣦⡀⢠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢿⡀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠻⢦⣄⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠈⢧⠀⠀⣠⠟⠀⠈⠻⣧⡀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⣼⠷⠛⠁⠀⠀⠀⠀⠈⠻⢦⣀⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⣶⣶⣶⡶⠶⣶⣶⣶⣶⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡿⠿⢰⣇⠸⠿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣇⣛⢻⡿⢛⣃⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣘⣃⣸⣿⣿⣿⡇⠀⠀⣿⡴⠖⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣡⣴⣾⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⡴⠶⠶⠶⠶⣿⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠳⠶⠶⠦⠴⢿⣿⣿⣿⡿⠟



⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣿⣿⣟⣛

⠀⠀⠀⠀⠀⠙⠉⠓⠛⠓⠘⠓⠒⠒⠒⠒⠒⢒⠒⠒⠓⠒⡒⠒⠒⠒⠓⠒⠒⠒⠚⠛⠚⠓⠚⠓⠛⠛⠛⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⣷
⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣷⠐
⠀⠀⠀⢸⣿⣿⣦⣾⣶⣶⣤⡀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢴⣾⣧⣠⣾⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⢠⣦⣴⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣤⣤⣼⣿⣿⣽⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣤⣿⣿⣿⣻⣿⡿⣿⣿⣿⢽⣟
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣤⣴⣶⣦⣀⣿⣿⣿⣿⣿⣿⣏⣿⠿⠿⣼⠟
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠘⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣾
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠙⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⡿⢿⡇⣿⣻⣿⣿⣿
⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠛⣿⣯⣭⣧⠘⠇⢿⣻⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠁⠀⢡⣤⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⢠⠀⠀⣛⠻⣿⣿
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡤⠚⠭⠄⠀⢠⠶⢦⣬⣿⣄⠿⢿⡇⢿⣿⣿⡷⣶⡶⣶⣷⣄⣛⣿⣊⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⡟⣿⢷⣤⣤⣀⡓⠀⠛⠘⡚⢿⣿⣶⣶⢿⣿⣿⣿⣶⣿⣧⣽⣿⡝⢿⣿⣯⣧
⠀⠀⠀⠸⢻⣿⣿⣾⣿⣴⣝⣿⣿⣷⣿⣻⣿⣽⣿⣧⡄⠀⠸⢿⣄⡀⠀⠀⠀⠐⢿⣿⣿⣟⣯⣿⣽⡇⢸⣿⢿⡿
⠀⠀⠀⣿⣻⢿⣿⣿⣿⣿⣼⣿⣿⣟⠛⣱⣿⣿⣿⣁⣀⣀⣀⣸⣿⣿⡄⢀⣤⣶⣾⢿⣿⣟⣿⣿⣻⣷⣿⢹⣎⢳
⠀⠀⠀⣽⣿⣿⣿⠿⣿⣿⣿⠿⣿⣻⣿⣿⣶⣺⣧⣌⣻⣻⣿⠛⠛⠿⢿⣿⣿⣿⣟⣿⣿⢿⣿⣧⣿⣿⣿⣿⣿⣯
⠀⠀⠀⣿⣿⣿⣽⣿⣿⣛⣻⣳⣿⢻⣯⣛⠛⢉⣉⢉⣉⣭⡛⠀⠀⠀⠀⠐⠛⠻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣂⣀⣀⣀⣀⣠⣤⣄⣠⣤⣄⣟⣭⣭⣿⣿⣿⣿⣿
⠀⠀⢠⣻⣯⣿⣯⣽⣯⣿⣿⣏⣿⣯⣍⣉⡉⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣷⣿⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣾⣶
⠀⠀⢼⣿⣹⢟⠋⢧⣤⣿⣿⣿⣿⣿⣻⣿⣤⣷⣿⣾⣾⢿⣿⡷⣶⣿⣷⣾⣿⡤⣿⣟⡥⣿⡿⠿⣿⣿⣿⣿⣿⣏
⠀⠀⠸⠿⠟⠿⠉⠻⠿⠿⠿⠿⠿⠿⠿⠋⠁⠿⠿⠿⠿⠿⠛⠿⠿⠿⠿⠟⠿⠏⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠘⠀⠒⣒⣒⠂⣲⠛⣒⡒⣒⣂⣚⣟⡙⣓⣂⣒⢲⡂⣒⣒⣒⣒⢒⣒⣛⣒⣛⣂⣛⣒⣊⣋⠚⠋⣛⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⠿⠛⢉⣀⣤⣤⣤⣤⣤⣄⣈⠙⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⡿⠿⠿⠿⢿⣿⣿⣷⣦⡀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁
⣿⣿⣿⠋⢠⣾⣿⠿⣿⣿⡯⠀⣤⠀⢸⣿⣿⠿⢿⣿⣦⠈⢻⣿⣿⡇⠀⠀⠀⠀⠀⠠
⣿⣿⠇⢠⣿⣿⠋⠀⠀⠉⠃⣴⡟⠓⠘⠋⠁⠀⠈⢿⣿⣧⠈⣿⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠐
⣿⣿⠀⣼⣿⣧⣄⡀⠀⠀⠀⠈⢟⡆⠀⠀⠀⠀⣀⣴⣿⣿⡀⢸⣿⡇⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⢈
⣿⣿⠀⢿⣿⣿⡿⠟⠃⠀⠀⢰⡏⠀⠀⠀⠀⠻⢿⣿⣿⣿⠇⢸⣿⡇⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⡄⠸⣿⣯⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣨⣿⡿⠀⣼⣿⡇⠀⠀⠀⠠
⣿⣿⣷⡀⠻⣿⣧⣀⣠⣴⡇⠀⠀⠀⢸⣶⣤⡀⣰⣿⡿⠃⣰⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⢧⠀⠠
⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⡇⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⡀⠀⠐
⣿⣿⣿⣿⣿⣷⣄⡈⠛⠿⢿⣿⣿⣿⣿⠿⠟⠋⣀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠢⠈⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⡉⡉⡉⡉⠙⠛⠿⣶⣄
⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⡇⠀⣴⡄⠀⢸⣿⣿⡿⢿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⢸⣿⠿⣿⣿⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⣼⣿⠏⠀⠀⠉⠛⠇⣰⢾⠲⠆⠸⠛⠁⠀⠀⠹⣿⣧⠀⠀⠀⠀⢀⣾⠏⢀⣾⣷⣦⣀⢸⠟⢐⡛⣿⢀⣤⣶⣷⡄⠈⢿⣆
⠀⠀⣸⣿⣏⠀⠀⠀⠀⠀⠀⠙⢾⣤⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇⠀⠀⠀⣾⡏⢀⣾⣿⣿⣿⣿⣿⣌⠸⢿⣿⣿⣿⣿⣿⣿⡄⠘⣿⡄
⠀⠀⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⣸⠞⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⠀⠀⠀⣿⠃⠀⠈⠙⠿⣿⣿⣿⡿⢛⣸⣿⣿⣿⡿⠛⠉⠀⠀⣿⡇
⠀⠀⣿⣿⣿⠟⠋⠁⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡿⠀⠀⠀⣿⡆⠀⣠⣴⣾⣿⣿⣿⣧⡸⣿⣿⣿⣿⣿⣦⣄⡀⠀⣿⡇
⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⢻⣧⠈⢻⣿⣿⣿⣿⣿⣿⣱⣿⣿⣿⣿⣿⣿⡿⠃⢰⣿⠁
⠀⠀⠀⠹⣿⣧⡀⢀⣤⣶⡇⠀⠈⠀⠀⢸⣶⣤⡀⢀⣼⣿⠏⠀⠀⠀⠀⠀⢿⣧⡀⠻⠟⠋⠁⢸⣿⣿⣿⣿⠀⠙⠻⠿⠁⣠⣿⠃
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣶⣶⣶⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣤⣀⣠⣤⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠪⢒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠐⠒⠶⣶⣄⡀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠉⠻⢷⡂⠤⠒⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⢼⡇
⢀⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢸⣦⡀⠀⠀⠀⠀⢀⡠⠛⠲⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⡈
⡟⠁⠀⠀⠘⡄⠀⠠⠄⠀⠀⠀⠀⣻⠀⠈⠳⡀⠀⢠⠔⠁⠀⠀⠀⠀⠙⢄⢀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠁
⢹⡄⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢳⣤⣶⣷⣦⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁
⠀⢹⡄⢰⣄⢤⣼⡄⠀⠀⠀⠀⠀⣸⣽⡄⠀⣀⢀⠆⡶⣄⡀⢐⡤⠀⣀⣀⣨⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠃⠀⠀⠀⠀⠀⣰⠄
⠀⠀⢻⣿⣷⡈⢿⣿⡆⠀⠀⠀⣠⡿⠻⠃⠚⠉⠉⠈⠐⠋⠻⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠟⢋⣡⣤⣶⠄⠀⢀⣠⡴⠞⠋
⠀⠀⠀⢿⣿⣷⡈⢻⣿⡄⠈⠈⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠉⠣⠛⡄⠀⠀⠀⠀⠀⠘⣧⣴⣾⣿⣿⣿⡿⠷⠛⠉
⠀⠀⠀⠈⢿⣿⣷⣼⠛⡇⢀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⢸⡉⠀⠀⠀⠠⡙⣶⣤⡀⠀⠀⠀⣀⣀⡀⣠⡴⠔⠋⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⢢⣼⠿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣿⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣠⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡱⠂⠈⠛⢿⣿⣤⣌⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢆⠀⠙⠻⣿⣧⣄⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⢀⢂⢀
⠀⠀⠀⠀⢺⢆⡀⠀⠀⠀⠀⣀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠢⣍⣿⣷⣏⣍⣇⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠀⠈⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⣀⡤⠖⢒⣶⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠘⠁⢀⠤⠚⣛⡛⢟⣟⣻⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⡄⠀⠀⠀⠁⡰⠟⢛⣿⣮⣭⣧⣤⣤⣝⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⢱⣤⣆⡴⢊⠔⢺⣿⣏⡙⣿⣿⣿⣿⣿⣿⢦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠛⢿⡴⠁⠀⠀⢻⠀⠀⢹⣿⣿⣿⣿⣿⡇⡟⢶
⠀⠀⠀⠀⢀⣠⡤⠶⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠺⠤⠈⠷⡦⠾⠆⠀⣠⣿⣿⣿⣿⣟⡏⢱⣾
⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⠄⠀⠄⠀⠘⠻⣿⣿⣿⣿⡟⠀⣾⡟
⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣉⢿⣿⣀⠀⢀⠀⠀⡀⠀⠈⠙⣿⣿⣷⡦⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠆⠀⠀⠀⠀⠀⠀⠻⢆⣿⡿⠋⠉⠁⠉⠁⢀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢈⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠈⠳⣦⣌⡉⠙⠛⠛⠛⠛⠛⢉⡉⠤⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⡀⠀⠀⠀⠐⡆⠀⠀⠀⢹⡤⡤⠐⠁⠀⠀⠀⣆⡀⠀⠀⠀⢸⠤
⠀⠀⠈⠙⠒⠷⠠⠀⠠⠈⠄⠄⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⠀⢀⣧⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣦⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠐⢶⠶⠆⠀⠠⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠒⠚⠙⠉⠙⠙⠒⠲⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠁⠀⠀⠀⢤⡠⠤⠤⡠⠄⠀⠀⠀⠹⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⡄
⠀⠀⠀⠀⢠⣶⠟⠇⠀⠐⠊⠑⠒⠢⠀⠀⠀⠀⠀⠔⠒⠒⠉⠒⠀⠘⠟⢷⣄⠀⠀⠀⠀⠀⢰⢻⢸⣷
⠀⠀⢀⡤⣾⠷⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣦⡀⠀⠀⠀⢸⣾⢸⣿⡀
⠀⠀⠸⣿⣿⣄⣀⣀⣸⡄⠀⠀⠀⢸⡇⠀⠀⠀⢀⡇⠀⠀⠀⢀⣾⣀⣀⣠⣾⣷⡟⠀⠀⠀⢸⢿⢸⣿⡇
⠀⠀⠀⠈⢿⣿⢿⣿⣿⣿⣷⣶⣶⣿⣿⣶⣶⣶⣿⣿⣷⣶⣶⣿⣿⣿⣿⢿⣿⠋⠀⢀⣠⣶⣿⢸⣿⠀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠉⠙⠙⠻⣯⣿⠿⠿⣿⣿⣿⡿⣿⣿⣿⡿⠿⢿⣯⡿⠋⠛⠉⠁⠀⠀⠸⠟⠛⢻⠈⣿⠀⡟⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⣤⣤⠖⠛⠛⠛⠛⠛⠉⠉⠉⠹⠛⠛⠛⠛⠳⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⠤⠶⣷⣤⣴⣤⣤⣤⣴⡷⠦⠤⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠘⣿⡇⠁⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡇⢿⠀⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢸⡇⢀⣿⡇⢸⣇⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⡏⡟⠏⠉⠉
⠀⠀⠀⠀⠀⢀⡤⠔⠉⡩⡤⠖⠂⢸⡇⢸⣿⣧⢸⣿⠀⠒⢤⢌⡉⠳⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⣀⣤⣤⣿⣇⡾⣿⠿⣌⣿⣤⣤⣄⡀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⣆⢸⠀⢸⢸
⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⡟⠉⠀⠈⠛⠀⠀⠀⠘⠁⠀⠉⢻⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⢿⠘⣦⠏⣸
⠀⠀⠀⠀⠀⠘⢿⣿⣿⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⣧⠈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢶⡤⢷⣤⣴⢦⡴⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣦⣤⣦⣠⣄⢠⣄⣤⣔⢠⡆⠀⠀⠙⠿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣠⣶⣶⣶⣶⣶⡆⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣡⣾⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠰⡇⣀⡴⠛⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢀⣼⣧⣤⣬⣤⣬⣭⣭⣭⣭⡍⣉⣉⢨⣤⣤⣤⣤⣭⣭⣬⣭⣭⣬⣥⣭⣭⣤⣭⣭⣭⢬⣭⣥⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣻⣿⣷⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣿⣿⣮⣻⣧⠀⠀⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣿⣟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠰⠖⠖⠒⠋⠉⠉⠉⣉⢹⠉⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⠤⠤⠤⠤⠄⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⠀⠀⠘⠛⠀⠀⠀⠀⠸⣿⢸⠀⠀⢘⣿⣿⣶⣶⣦⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣬⣭⣥⣤⡦⡴⠶⠷⠖⠀⠀⠀⠀⢸⠀⠀⠘⠛⠉⠉⠉⠉⢠⡀⠟⣵⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠠⣤⣤⣬⠁⢀⣀⣀⣀⣀⠈⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⢹⣿⠇⠀⠀⠀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣿⣿⠋⣰⣿⣿⣿⣿⣿⡇⢸⡀⢸⠂⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⢻⣷⣄⠀⠀⠀⣾⣿⣷⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢠⡿⠁⢸⣿⠉⠉⠿⢿⣿⡇⠈⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠋⣧⡟⠙⠃⠀⠀⣼⣿⢏⢿⣷⡄⠀⠀⠀⣀⣸
⠀⠀⠀⠀⠀⢀⣀⡤⠤⢾⣯⣤⣀⣀⡀⠀⢀⣠⠞⢇⠀⠘⣿⠀⠀⢀⣾⣿⣿⠀⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣀⣀⣀⣼⣿⡏⠀⠀⢻⣿⣦⣿⠿⠟⢻
⠀⢀⡤⠒⠋⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠁⠀⢸⠀⠀⠉⣁⣀⣈⣉⣅⣤⣤⣶⣼⣧⣤⣤⣶⣶⣶⣶⣿⣿⣿⡿⠿⠿⢿⠟⠛⠿⠔⠀⠀⠀⠹⣿⡟⠈⠀⢸
⢠⣿⣿⣷⣦⣤⣀⠀⠀⠀⢀⣠⣤⣤⣤⣴⣶⣶⣶⣿⡿⠿⢿⣿⣛⣛⡻⠯⠭⠭⢿⣽⡿⠿⠯⠉⠙⠛⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡙⠀⠀⠀⢸
⢸⡏⠉⠛⠻⢿⣿⣧⣀⣚⣛⠿⠭⠭⠽⠛⠒⠒⠓⠘⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣉⠛⣷⠀⢈⣀⣠⠤⠞
⣿⣇⡲⣦⣤⣇⡀⠀⠀⠇⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⠀⠊⢭⠁⢹⣏⣽⣷⣶
⠸⣮⣉⠒⠏⢷⡍⠉⠈⠀⠀⣴⠟⠋⠉⠉⠉⠙⠻⡷⣄⣧⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡇⠀⢀⣋⣀⣤⠤⠴⠖⠒⠛⠉⠹⢦⣾⣿⣿⣰⣋⡾⠛⠛⠋⠉
⠀⠙⢿⣟⡶⢶⣿⣀⣀⣀⡾⠁⠀⡰⠚⢽⣽⡢⠀⠀⠈⠯⠤⠔⠒⠒⠛⠋⠉⣙⣻⣿⡟⢋⣉⣩⣤⣶⣶⣿⣿⠿⠿⠛⠛⠁⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠙⢿⣷⣦⣭⣥⡼⠃⣄⠀⣄⠀⢈⡍⠇⣤⡴⠒⢶⣶⣒⣒⣺⣭⣭⣽⣿⣷⡶⠿⠿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠲⢮⣭⣉⣉⡀⠠⣿⣿⣿⣷⡮⣀⣼⣿⢃⣾⣿⣿⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠞⠻⣿⣿⣿⡿⠛⠁⠉⠉
⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⣀⣀⢀⣀⣀
⠀⠠⠀⠋⠋⠛⠚⠛⠛⠓⠠⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠒⠉⠓⠀⠉⠀⠉⠒⠘⠀⠀⠀⠈⠀⠀⠀⠀⢀⢀⣙⠇⠀⢀⠢⡤⠀⠀⠀⠀⠀⠀⡀⠀⠠⠐⢀⡀⣀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄⠡⠀⠀⠈⠉⠀⠁⠁⣤⠈
⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠠⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢧⡄⠀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡈⠠⠌⠀⠠⡀⠀⠀⠀⠀⠦
⠀⠀⠀⣤⣀⣤⡀⣀⣀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡹⠟⠚⠛⠛⡦⠆⠀⠀⠠⠉⠀⠀⠀⠀⣴⣼⣳⣿⡶⣤⡄⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⢰⠂⠠⠤⠂
⠀⠀⠀⠀⠘⠀⠋⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⢀⡞⠀⠈⠀⠀⠀⣀⠁⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢁⠻⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⡀⠀⠀⠀⢀⣀⣠⡤⠶⠋⠀⠀⠀⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠢⢄⠀⠀⠀⠠⣀⢀⡀⡀⠀⢂⠤⣀⣠⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⠻⠟⠶⠓
⠀⠀⠀⠑⠒⠂⠀⠀⠀⠈⠀⠀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋


⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⢤⣤⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⠄
⠀⠀⢸⣿⣶⣶⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⡰⣰⣯⡿⠟⠁
⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣶⣼⣿⣶⠀⠀⠀⠀⠀⠀⢀⠹⠀⠀⠀⠀⠀⠀⠀⢀⣈⣩⣴⣶⢻⠏⠡⠀⠐
⠀⠀⠀⠀⡇⢰⠈⣉⢛⠻⠿⣿⣿⣿⣦⣀⣀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⡾⠿⡿⠀⠐⠀⠀⢀
⠀⠀⠀⠀⢁⣼⣶⣾⡀⣸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠆⠈⠀⠀⡄⠀⢸⠀⢸⣾⣿⣿⣿⣿⣟⠀⠀⠀⠀⠈
⠀⠀⠀⢰⡞⢻⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠏⠀⠀⠉⠃⠀⠉⠀⠘⠙⠻⠻⠋⣛⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠨⠀⠀⣾⣿⣿⣿⣌⠉⣤⣤⡄⣾⣇⣿⠟⠛⣠⣴⡆⣺⠷⠿⠲⠗⠛⠟⠺⠿⠟⠿⢿⡁⣀⣀⣀⠈⡇
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣇⣽⣿⣿⣼⣿⣿⣆⠀⢹⣿⣿⣿⣷⣶⠞⠁⠀⠀⠨⠍⢙⣾⣯⣿⣿⣿⣭⣷⡇
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡈⠋⣿⣟⣿⣿⣿⣶⣦⣤⣤⣶⣶⣶⡶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⣉⣬⣯⣹⣿⣿⣿⡇
⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⠛⠙⠇
⠀⠀⠀⢴⣶⡿⢯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣟⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⠿⠿⠻⠿⠛⠛⠻⠛⠿⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠻⢦⣤⣀⣉⣉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣁⣬⠟⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⠶⣶⣤⣬⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣶⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⢸⡇⢸⡏⢹⡏⢹⡏⢹⡏⢹⡏⠉⡏⠉⡇⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣄⣸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⠀⡇⢀⣇⣠⣧⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠙⠛⠛⠻⠿⠷⠾⠷⠾⠷⠾⠿⠿⠛⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡾⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⣦⣤⣤⣀⣀⣀⣀⣤⣤⣤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣤⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣉⣉⣉⣛⣛⡋⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠛⠿⢿⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⠿⠟⠛⢻⡇⠀⠀⠀⠀⢀⣴⣟⡱⠋⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠛⠻⡿⠷⠶⣦⡀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⢰⡿⠃⠈⠹⣦⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⣠⡇⠉⠛⢸⡇⠀⠀⢻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡇⠀⠸⣧⣀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣷⣾⠟⠁⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢟⣛⡛⣟⠛⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠉⣀⣿⣀⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠠⠤⠠⠤⠠⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⠂⠉⠀⠀⣀⠀⣀⢀⣀⠀⠀⠀⠙⠀⠤⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⣀⠠⠆⠉⠁⠁⠀⠀⠀⠀⠈⠉⠐⠆⠀⠀⣠⠞⢁⡾⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠉⠀⠠⠆⠁⢀⣠⠴⠒⠛⠉⠉⠉⠃⠀⠀⢀⡤⢤⡞⠁⡀⢸⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠚⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠚⣏⠙⢲⡉⠀⡴⠃⠀⠐⠦⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠞⠀⢠⠉⠀⣰⠋⠀⠀⠀⠀⠀⣠⢾⡙⢦⡈⠳⣄⠉⢺⡁⠀⢀⠀⠀⠇⠀⠘⣂
⠀⠀⠀⠀⠀⠘⠀⢀⡌⠀⣸⠃⠀⠀⠀⠀⣠⢾⠙⢆⠙⢦⡙⠂⠈⠓⢦⠇⠀⠘⣆⠀⠈⣃⠀⢙⡀
⠀⠀⠀⠀⠀⡋⠀⢰⠀⠀⠇⠀⠀⠀⡠⢎⠹⣄⠳⢄⠙⠂⠀⠀⢀⡴⠋⠀⠀⠀⢹⠀⠀⢙⠀⠈⡅
⠀⠀⠀⠀⠀⡃⠀⢰⠀⠀⠀⣄⡠⢾⠀⠈⠳⣌⠓⠀⠀⠀⢀⠄⠀⠀⢀⡀⠀⠀⢸⠀⠀⢘⠀⠀⡅
⠀⠀⠀⠀⠀⣃⠀⠰⠀⠀⠀⠘⢧⠀⠳⣌⠳⣄⠀⠀⢀⡴⢋⡴⠛⣉⣩⣉⡙⠲⣄⠀⠀⣘⠀⢀⡅
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡠⠚⣳⣄⠈⠳⢬⣧⠴⠋⢠⠏⢠⠋⠁⠀⠀⠙⢦⠘⣆⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢰⠲⣤⠊⣠⠞⢡⠜⠓⠦⣴⠋⠀⠀⢸⠀⢿⣠⢤⣄⡤⣄⣸⠀⢹⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡈⠳⣄⠜⠁⢠⣀⠀⠈⠁⠀⠀⠸⣆⠸⣭⡛⠵⢙⣫⡗⢀⡏⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣬⡗⠀⠀⠈⠙⠲⠦⠤⠤⠤⠘⠦⣄⣙⠛⢛⣀⡴⠋⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠠⠄⢀⡀⢀⠀⢀⡀⣀⠀⠉⠉⠉⠀⣀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⢠⣄⢀⠀⠀⠈⠁⠉⠁⠀⠀⢀⡄⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠃⠛⠀⠛⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠤⠤⠤⠦⠀⠒⠿⠻⣿⡏⠙⣿⣿⣓⠚⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡶⠦⠴⠤⠤⠄
⠀⠀⠀⠀⠀⠠⠔⠒⠒⠚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⠰⢦⣴⠿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡦⠤⣤⣤⣿⣏⣀⣀⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⡤⣤⣤⣽⡟⣻⣛⡛⠛⠲⠶⣤⣄⣀⣀⣀⡀⠀⢀⣾⡏⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⡿⠐⠀⡀⣀⣀⠀⢨⣡⣭⠙⣋⣻⣇⣏⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠔⠀⣿⣿⣿⣿⣿⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠁⣀⣬⡤⠴⠒⠒⢛⣧⣘⣋⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⠠⠟⠿⠝⠉⠹⠤⠀⠀⠠⢤⣤⠀⠒⣦⣤⡤⠃⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⣉⠉⠛⠛⠻⠿⠛⠓⢲⣶⠟⠻⠿⠁⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣠⡤⡄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢴⠆⠀⠸⢃⠈⡅⠻⣿⡿⣿⣿⣿⣿⣷⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣭⣭⣭⣉⣽⣧⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠗⠚⡼⠃⠀⣛⣿⡞⣿⣿⣿⣿⣿⣿⣷⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⣿⣿⣿⠻⠿⠿⡾⠟⠛⡐⣾⣶⡆⠀⠀⠀⠃⢀⠞⡟⠛⠁⠀⣤⣿⣿⣤⣍⣉⠉⠉⠉⢉⣩⣭⣭⣤⣤⣄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣷⢠⣤⣤⣤⣤⡆⣴⣿⣿⡿⠀⠀⠁⣀⠘⠀⠁⠀⠀⠀⠋⠰⡆⠀⣀⣉⠁⢰⣷⣾⠩⢿⣿⣿⣿⠟⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⢿⣿⣿⣯⣽⣿⣇⣈⣈⣩⣥⡤⢴⣿⣻⠏⠁⠀⢰⣾⣿⢿⣰⣷⣾⣶⡶⢦⣤⣤⣼⣿⣿⣷⣘⣛⣛⣂⣉⣭⣽⣿⣦⣥
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠻⠷⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡿⠋⠙⠛⠓⠚⠛⠛⠿⣏⠪⣿⣿⠿⠿⠿⠿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⣀⠠⠄⠤⠤⢤⣤⣠⣤⣤⡄⠠⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠃⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠗⠛⣾⣷⣘⣈⣿⣞⣭⣤⣤⣶⣶⣶⣶⣶⣴⣶⣤⡬⠉⠁⠀⠈⣴⣦⣴⣦⡔⢻⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⠀⠀⢸⣿⣿⣿⣿⠸⣿⡎⣹⣿⣯⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣦⠀⢸⣿⣿⣿⣿⣴⣾⡷⠒⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣭⣭⣭⣭⣬⣭⣭⣭⣭⣭⣬⣽⣷⣦⣼⣿⣤⣿⣤⣤⣤⣦⣭⣤⢺⣿⣏⠀⣿⣿⣿⣛⣻⣟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣾⣿⣯⠀⣿⢇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⣷⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⡿⠿⠿⣿⣿⣿⡇⣏⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣉⣉⣹⡋⣉⣩⡏⢾⡿⢻⣿⣿⣶⣾⣿⣿⣿⣤⠿⢿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣾⣤⣤⣤⣄⣉⣀⣨⣭⣍⡉⠉⠉⣩⡟⣩⣮⣿⣿⣿⣵⣿⡇⠰⣿⡟⠉⠉⠛⠃⠀⠉⠙⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⣿⣿⣿⠟⠋⠉⠉⠀⠀⠘⣿⣿⣷⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠁

⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠸⠦⠀⠀⠀⠀⠀⣀⠴⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠲⠀⢤⠀⢀⡀⢀⡀⠀⠀⠀⠀⢰⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠢⡀⠀⠄⠀⠀⠀⠀⠀⠃⠀⠆⠀⠤⠀⢠⠀⢈⠁⠘⠁⠘⠃⢐⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⠀⠀⢠⠃⠜⠀⠀⠀⠀⠀⠀⠋⠐⠂⠀⠦⠀⠤⠀⢨⡁⢈⡀⠘⠁⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠐⠊⠹⡖⠀⠀⠀⠀⠀⠠⣂⡀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠈⠁⠠⠀⢀⠶⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠐⠤⠂⣓⣀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠐⠒⢖⠠⠤⠀⣀⠀⠀⠀⣁⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡷⠀⠀⢸⠃⠸⣧⠀⠠⠀⠀⠀⠀⠀⠈⠀⣀⡀⠈⠁⠀⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠛⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠚⠀⠄⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤⠴⠒⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡲⠞⢺⠉⠁⠤⡀
⠀⠀⠀⠀⠀⠀⣔⣮⠇⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠂⠀⠓⠦⡀⠈
⠀⠀⠀⠀⠀⡐⠉⠃⠀⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠂⠀⠀⠂⠀⠀⠀⠈⠓⠂⠉⠉⢢⠄⠍⠉
⠀⠀⠀⠀⠀⠃⢀⣴⣤⢀⠀⠀⠀⠀⠀⢠⡤⢤⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠄⠖⠓⠒⠉
⠀⠀⠀⠀⠀⠃⠘⠩⠞⠂⠀⠀⠀⠀⠀⠈⠙⠊⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⣶⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠄⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣄⣀⣠⣾⣇⠀⢀⣠⡎
⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⢠⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣤⣴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣯⣿⣶⣿⣿⣿⣿⣿⣶⣿⣽⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣵⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉
⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⢸⣿⡄⣸⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠃⠞⠁⠟⠹⡆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⠀⠀⠂⠀⠀⠀⠀⠃⠉⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣶⢦⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠶⡡⢿⡞⡏⠀⠀⠀⠀⢰⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡂⢀⠟⡉⠀⠁⣘⢳⢿⡄⢸⠀⠀⢨⠻⠻⣤⣇⢀⣖⠦
⠀⠀⠀⠀⠀⠀⣴⣥⠄⡀⠀⣹⡗⣡⣾⣿⠃⣰⡇⡸⠀⣿⠿⣄⣠⣿⢣⡀⠈⠹⣾⠁⠀⠈
⠀⠀⠀⠀⠀⠠⠋⠉⠛⠳⢾⣯⣾⣻⢳⠃⢠⣿⠇⡇⠀⢿⣾⡘⡈⢻⣇⢻⣤⠬⢶⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⠉⠭⠙⠒⠋⠛⢻⣿⡄⠚⠛⠓⠓⠘⠻⡏⢁⡤⠔⠈⣷⣀⠄
⠀⠀⠀⠀⠀⠢⡀⠀⠀⡀⠘⠇⠀⠀⠀⠀⠀⠀⢸⢸⡇⠖⠂⠀⠀⠀⠀⣹⣀⠀⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠈⠑⠤⠷⠞⢀⣀⠀⠀⠀⠀⠀⡄⢸⠃⠀⠀⠀⠀⢀⠌⠀⢠⠋⠱⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠈⠉⠀⠉⠉⢉⡉⠉⢩⡀⠴⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⠀⠀⠀⢀⠀⢐⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⣝⡀⠀⢀⠁⡊⠀⠀⠈⢻⢦⠀⠴⢂⡿⡄⠀⠀⠹⡦⡐⠐
⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠈⠓⢍⠓⢤⣁⠀⠀⠀⠀⡟⢀⠀⢀⠞⠀⡟⠀⠀⢠⠀⠹⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⢧⠀⠈⠳⡀⠐⣦⡙⢮⠢⢸⠁⢠⠃⠀⡠⠋⠀⠈⡇⠀⢀
⠀⠀⠀⠀⠀⠀⢖⠈⣀⣀⡀⠀⠀⢸⡤⠀⠀⢧⢠⣇⠹⡌⢳⠘⣆⡿⢀⡞⠁⠀⠀⢀⠆⢀⠎⢣⠀⡀
⠀⠀⠀⠀⠀⠈⠀⣈⠉⢳⡎⢢⡀⢸⠀⠶⡀⠈⠻⠂⠀⢱⠀⠀⡿⢽⣼⠀⢠⠀⢀⡌⢀⠠⡀⡏⠐⠁
⠀⠀⠀⠀⠀⠀⠈⢹⠙⠸⡇⢀⠱⣾⡄⢁⠈⠢⡀⢠⡄⡞⠀⢀⠇⡄⠋⡠⢁⢣⠏⠀⢸⢣⢰⠇⢀⢦
⠀⠀⠀⠀⠀⢀⣆⢺⡊⠀⠹⣇⠰⠠⢄⠀⠂⠀⠰⠂⢠⠃⢀⡞⡐⢸⠐⠀⢠⠃⠀⡰⠃⣸⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠘⡏⢦⣀⠀⠀⠈⠳⣝⢤⡈⠲⠀⠀⡄⢸⢠⠟⠐⠀⢾⠀⠂⡇⣠⠏⠀⣀⡎⠀⣠⢀⡸
⠀⠀⠀⠀⠀⠀⠱⣄⠈⠑⠢⢤⡀⠘⡄⠱⠀⠀⠀⠆⠈⠿⠀⠀⠀⠀⠀⡆⢹⠇⡄⢠⠎⢀⡰⣻
⠀⠀⠀⠀⠈⢻⠆⠈⠲⢄⠀⣄⡙⠶⣗⠀⠁⠀⠀⠘⢄⠈⠂⠰⠀⠀⠀⠐⡤⠊⢄⣮⠖⢋⡜⠁⣦⡀
⠀⠀⠀⠀⠀⠘⠒⠠⠱⠀⠱⡀⠂⠑⠄⠀⡄⠀⠀⠀⠀⠉⠂⣤⠀⠀⠀⠀⠄⠐⠎⠥⢈⠏⡀⠀⠁⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠢⡀⡇⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢐⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠙⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠈⠨⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠓⢤⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠂⢀⠁⠀⠀⡀⠁⠀⠔⠀⡀⠀⡀⠀⠀⢐⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠐⠀⠀⠈⠈⠀⠀⠁⠀⠈⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⠀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⡀⣀⠀⠐⣀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⢀⣀⠈⠀⠀⠀⠀⠀⠰⡿⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞
⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠢⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠙⠚
⠀⢾⣿⣿⣇⠀⣤⣤⣤⡄⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠹⠠⣀
⠀⢹⣿⣿⣷⣦⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⣨⣭⣭⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣠⠀⣤⠒⣾⠐⠿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣻⣄⠀⠀⠀⠀⠀⠀⠒⠐⡗⠤⠠⠛⢩⣏⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣶⢲⢾⢰⠟⢹⠂⢋⡠⠕⠒⣉
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣁⡀⠀⠀⢀⡤⣤⣤⡞⠐⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⠶⢋⣀⡤⠖⢊⡅⣴⡆⠘⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣶⣶⣶⣂⣴⣿⣧⠈⠃⢠⡌⠿⣄⠻⠀⢀⠀⠀⠀⡄⠀⠀⣘⢵⣿⡋⠂⡉⡇⢾⡇⢸⡇⢿⡇⢸⠿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣿⣯⣿⡇⢰⣶⠀⣳⣶⠁⣾⡆⢸⠇⠀⠀⡘⡉⣰⠉⢩⠟⠻⢀⠀⣇⠘⠁⠈⠀⠀⠀⡇
⠀⠸⢛⢿⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠈⠉⠐⠉⣉⠃⠉⠀⠀⠀⠀⠰⠽⣿⣿⢀⢰⡎⠀⢀⡄⡏⢰⡄⢰⡄⣴⡆⢰⣿
⠀⢠⣿⣼⣯⣿⣿⣿⣿⣿⣿⡕⣶⠻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⡅⠘⠠⢀⠘⠃⠃⢹⡇⢸⡇⢸⡇⢸⣿
⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣯⣿⣼⣿⣿⣿⣏⣉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠍
⠀⢸⢿⣿⣿⠿⣿⣿⣿⣿⣿⢻⣿⣾⡟⣿⣿⣿⣇⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⢾⣶
⠀⢸⡟⣿⠏⢻⣾⠷⢿⡿⣿⣿⣿⣾⣙⡿⠉⠉⠉⠛⠀⠀⠀⠀⠀⡀⠀⠹⠏⠀⠰⣦⣤⠀⢠⣽⣠⣤⣀⣀⣀⠀⣀⣀⠀⠰⠶⣤⠘⢻
⠀⢨⡍⣽⣿⣿⡿⠷⠿⠿⠿⠿⢋⣼⣿⣧⡘⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⡾⠦⠀⠁⠙⠀⢀⣼⣁⠉⠉⠉⡟⠘⠋⡿⢥⣀⠤⢣⡄⠘
⠀⢸⣿⣹⣿⣿⠿⠟⠀⠙⢿⡤⠿⣿⠀⢹⢷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠋⠉⠑⢻⠴⠂⣮⣶⣤⡤⣰⡀⠈⢤⡸⢦⡀
⠀⢸⡟⣿⣿⣿⢸⠀⢦⠀⠀⣀⣀⡰⢀⣟⠈⢻⢿⣾⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⢰⣶⢿⣧⣈⢭⣽⠀⠀⢸⠇⢘⣧
⠀⢸⡿⣿⣿⣏⣼⢰⣾⡇⢠⣿⣟⣿⡎⠳⠋⢿⡟⣿⠟⠋⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⣀⣀⣿⢿⣷⣦⣾⡧⣄⣹⡄⣠⡏⠁⠘⢦
⠀⢸⣷⣼⣿⣘⣻⡈⣟⣷⣿⣿⣿⣿⣧⣤⡾⣾⡉⣻⣿⣦⣴⣶⣶⠶⢶⢾⣿⢋⣧⣤⣶⣟⢹⣿⢠⣿⡟⢽⣿⡷⠁⣌⣹⣿⠁⣿⣦⠄
⠀⢻⣿⣿⣿⣿⢋⣰⣾⣛⣃⣉⣭⠄⠼⠿⡗⣻⣷⠟⠿⠿⠿⠷⢿⣷⡄⠀⠀⠈⠉⠿⠛⣿⣞⣯⣼⣹⡇⣸⣿⡷⢼⣿⡏⢛⡃⢹⣿⡧
⠀⢸⣿⢿⣿⣿⣿⣿⣿⣭⣙⣡⣿⣆⠹⣶⣄⡀⠀⢀⡄⢀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⢛⣿⣟⣱⠿⣶⢾⣿⣷⣷⣾⣿⣿⣞
⠀⢸⣷⣶⣿⣿⣿⣿⣿⣿⡍⢙⣻⣿⣿⣾⣿⣿⣿⣿⣿⠟⡿⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠉⣉⣩⣿⣿⣿⣿⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠁⠀⠉⢉⠁⠀⠁⠀⠀⠈⢀⠈⠀⠀⠈⠀⠀⠀⠈⠉⠉⠉⢉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠂⠀⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⠴⠉⠎⠃⠦⠍⠦⠤⠔⠉⠀⠐⠈⠑⢤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣺⣭⣭⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠭⠍⠭⠭⠭⠭⠭⠍⠩⠍⠹⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡠⡴⠁⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⡐⢠⢡⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢈⢈⠀⠀⠀⣉⢈⠀⠀⠠⠤⠀⠈⠆
⠀⠀⠀⠀⠀⡐⢀⠇⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⠠⢀⣀⣀⣀⣀⣀⣀⣠⠀⠐⢅⣓⠔⠀⠰
⠀⠀⠀⢀⠴⠠⠌⠼⠿⡴⠿⠿⠿⠿⠿⠿⠤⠀⠻⠛⠛⠟⠿⠇⠠⠠⠠⠤⠤⡤⢋⡙⡌⠠⠀⠀⠀⠀⠀⠀⡤⠄
⠀⠀⢀⣁⡀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣠⡄⠀⠀⡀⣀⠀⢀⣄⡄⠀⠀⠀⠀⠀⠢⡘⢃⠜⠀⠀⠀⠀⠠⡤⠠⡇
⠀⠀⢸⣀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠓⠐⡇⠐
⠀⠀⠸⢩⡀⠀⠀⠀⢀⠀⣀⣤⣄⣐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡏⠋⠙⠉⠉⠉⢉⠙⢋⣉⡛⠉⡉⠉⠙⠉⢉⣩
⠀⢀⣰⣒⣃⣀⠀⠀⢠⣾⡿⠿⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣡⣾⠿⠿⢿⣷⡌⠀⠀⠂⠈⣚⡀
⠀⠨⠀⠀⠀⢈⡇⢢⢸⣿⢀⡌⣀⢸⣿⡇⡗⠚⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⣆⣿⣏⢠⣁⡄⣿⣿⣐⠒⠲⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣻⣷⣾⣶⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣶⣷⣾⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈

⠬⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠈⠀⠀⠹⡀⠹⠋⣳⡍⣿⡴⢿⣟⠉⡝⠀⠜⡩⠊⠰⢁⠞⠙⠁⡠⠞⠁
⢤⣄⠀⠀⠀⠀⠀⠁⠈⢢⣴⠦⠤⣤⣄⡀⠀⠀⠀⠀⢠⢖⠛⣧⣄⢸⢱⠟⠋⠀⠀⣻⡜⣁⣤⣀⣁⣀⣀⢄⣀⣤⡈⠀⠀⠀⠀⣠⣔
⣒⠠⠀⠒⠄⠀⠀⠀⠀⣟⡀⠠⢀⠤⣼⠋⠗⠓⢀⢴⢟⡀⠀⠈⠀⠚⠓⡶⠀⠀⠈⠋⠉⠉⠀⠁⠠⠏⠀⢀⠃⠀⢻⡆⡠⠐⠉
⡒⠀⠠⠤⣀⠀⠀⠀⠀⣽⠂⠀⡈⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠈⠀⣀⡾⠃⠀⠀⠀⠀⢠
⠉⠛⠂⠀⠀⠀⠀⠀⠀⠈⢲⡆⠁⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠈⠠⢿⠉⠉⡧⠀⠀⠀⢘⡆⠀⠀⠀⠀⠑⠉
⠤⠄⠀⠀⠀⠀⠀⢀⡠⠤⣴⢳⡤⠀⠀⠀⠀⠀⠁⠰⠐⠀⣀⠀⠀⢢⠈⡄⠊⠁⠀⠁⠀⠀⠀⠀⠘⠃⠀⡴⠚⣫⠄
⠴⠒⠦⠤⠀⢀⡜⠉⠛⠁⠘⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠛⠉⠉⠷⣶⢦⡄⢀
⡒⠖⠂⠀⠀⣸⡏⠁⣀⠔⠚⠀⠈⠒⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣠⠟⣙
⢛⢉⡴⠎⠉⠃⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⡷⠀⠀⠀⠀⠀⠀⠀⡀⠶
⠈⣿⣀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠒⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢤⣤
⡒⡚⠻⠷⢶⣟⠀⠈⠳⢴⣎⡠⠀⠀⠀⠀⠀⠈⠀⢠⠀⠄⠆⠀⠀⠀⢀⠀⠠⠄⠈⠈⠀⠀⠀⠀⠀⠁⠀⠄⢰⡄⠀⠀⣀⣀
⢀⣀⡀⠀⠀⠘⠳⣤⣄⣀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠂⢀⠀⠀⠈⠀⠀⠀⠈⠀⠐⠄⠀⠀⠀⠀⠀⢴⢼⠁⢤⠚⠋⠉⠂⢉⣉
⠿⠁⠀⠀⠀⠀⢀⠀⠍⠁⣻⠀⠀⠀⠀⠈⠙⠉⡉⠀⡀⠀⢀⣀⡁⠀⢆⠀⠀⠈⠁⠀⠈⠉⠀⡠⠶⣤⡤⠞⠋⢀⡖⠀⠈⠐⠒⠄⠠
⠈⠀⠀⠀⡠⠂⠁⠀⢀⣴⠿⣄⣄⣡⠴⡄⠀⣠⡷⢚⠂⠀⠀⣹⡍⠂⢢⣄⣀⣤⠟⠦⠴⢧⣀⣀⣶⠏⠓⠲⠞⠋⠁⠀⠈⠀⢄⠐⠠
⣖⠔⠊⠀⠀⡠⠔⠋⠙⠁⠀⠀⠀⠀⠀⠉⠋⣱⠃⠸⣏⣠⣴⠋⢷⡄⠀⠀⢡⠇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠁⠀⢀⡴⢋⡤⠊⢁⣀⣠⢂⡖⢠⢈⡠⠀⢰⢃⣴⣤⡌⢷⡱⢳⣧⣙⡃⢟⢉⠀⠀⠀⠁⠀⢰⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠳⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢠⠞⣧⠀⠀⢀⣀⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⣠⠋⠀⠘⠚⠉⠉⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠙⠲⢤⠞⠉⠉⠙⠣⡀⠀⠉⠳⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⣧⠀⠀⠀⠀⡰⣋⣞⠉⣱⣄⡀⢸⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠙⣄⠀⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⢠⡾⣽⠣⣼⠃⢌⣇⡜⡸⠀⢀⡤⠲⡈⠳⣄⠀⠀⠀⠀⠘⠧⠤⠤⢤⣤⣀⣀⣀
⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠒⠲⢧⡦⢴⠛⣬⣶⠁⡰⠛⢢⠰⢇⣠⠟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⢾⣠⣿⠏⠻⣅⣀⠼⣀⡠⠃⢱⠛⡆⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⣨⠆⠀⠀⢠⡇⠀⠀⠀⠀⠠⠁⠀⣼⠟⠁⠀⠀⠘⠷⣤⢼⠒⠒⡇⢀⠇⠀⠀⠀⠀⠀⠐⠛⠲⠤⣄⣀
⠀⢀⡠⠖⠋⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠳⠿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠽⠗
⠶⣉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉
⠀⠀⠉⠙⠒⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡛⢏⣙⡦⢵⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠉⢫⠈⣇⣸⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡏⣧⣞⡁⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⣀⣀⣈⣶⡄
⠀⠀⠀⣴⣁⣤⠤⠤⠶⠒⣶⠂⠀⠀⠀⡠⢿⡀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠉⠙⠲⢤⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⠔⠋⠀⠀⠙⠓⠒⠚⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠑⠢⢌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡠⠞⠁⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡽⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⢀⡴⠋⠸⡆

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⢀⣤⣿⣿⣤⡀⠀⠀⣶⡶⢲⣶⠂⣀⠀⠀⢸⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠇
⠀⠈⠛⣿⣿⠛⠁⠀⠀⢿⡿⠾⠿⢿⡿⠀⠀⠀⠹⣿⣿⠏⠀⠀⠀⠈⠐⠊⠀⠐⠆⠀⠀⠀⢠⣀⠎
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⢸⠓⠚⠇⠀⠀⠀⠀⢸⠓⠚⡇⠀⠀⠀⠀⠀⣶⣿⡿⠀⠀⠀⠀⢸⢀⣁⠀⠀⠀⠀⠀⠀⣰⣦⡖
⠀⠀⢘⢢⠴⠂⠀⠀⠀⠀⢘⢢⡴⠃⠀⠀⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⢨⠈⠛⠀⠀⠀⠀⠀⣸⣿⠏
⠀⠀⠘⠐⠒⠂⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢀⡴⢊⡑⢦⡀⠀⠀⠠⠔⢒⡒⠢⠄⠀⠀⢠⣤⣯⣽⣦⡄⠀⠀⠀⠀⣰⠉⠇⠀⠀⠀⠀⠘⡛
⠀⠢⠸⠤⠤⠇⠜⠀⠀⠱⠀⠘⠃⠠⠌⠀⠀⠸⣿⣆⣴⣿⠇⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠸⣿⡿⡻⣿⠇⠀⠀⠀⠀⡅⢨⠀⠀⠀⠀⠀⢰⠈⠁⡆⠀⠀⠀⢸⡏⢉⡉⢸⡆⠀⠀⢸⠢⣤⣤⣤⡆
⠀⠰⣿⣿⣾⣿⠆⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠘⢳⡞⠃⠀⠀⠀⢸⡆⠻⠟⢸⡇⠀⠀⢬⠉⠉⠉⠉⡥
⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁⠀⠀⠀⠈⠓⠒⠒⠛

⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⢀⠃⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠚⡆⠀⠀⠄⠓⠁⠑⠀⠐⠀⠠
⠀⠀⠄⠀⠀⠑⠀⠀⠀⠀⠀⡇⠀⠀⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⠆⠀⠀⠑⠒⠊⠀⠀⠀⢀⣦⠀⢢⢀⠢⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠎⠀⠺⠇⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡤⣄⡀⢀
⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠬⣁⠰⠄
⠀⠀⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠆⠀⠀⠀⠰⠇
⠀⠀⠈⠉⠈⠉⠉⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠊⠔⠁⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠂⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⡄⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠈⠀⠀⠈⠈⡀⠀⠀⠀⠠⠼
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⠘⠀⡃⠈⡀⠄⠀⡀⠁⠐⠀⠀⠠⠀⠀⢃⠀⠀⠀⡄⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠠⠄⠄⠈⠠⠃⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠊⠑⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠄⡰⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠢⣀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢨⠃⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⢐⠆⠀⠀⠀⡀⢂⠀⠀⠀⣮⣁⣠⣤⠆⠀⠀⠀⡆
⠀⠀⠀⠀⡄⢘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠈⠀⠀⠀⠈⢀⠀⠀⠀⠄⠀⠀⠀⠄⠁⠀⠀⠶⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⡉⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠆⠐⠚⠛⠓⠀⠀⠀⢈⡉⠉⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠁⠀⠄⠠⠂⠈⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⡀⠀⢀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⢀⠴⡁
⠀⠀⠀⢦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠐⠀⠀⢠⠀⠀⠀⠤⠴⠂⠀⠀⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠠⠤⠀⠀⠄⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⣱⠫⡖⡚⠃⣿⠀⠰⠂⢰⣆⢀⡀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠠⡀⠀⢀⡦⡐⠒⢐⠋⢹⠃⣿⠀⣿⠀⠈⠀⠀⠈⠀⠃⠐⠁⠀⠄⠀⠀⢰⠉⠁⣜⢋⠁⠈⡀
⠀⠀⠀⠀⠛⣯⠸⢻⣯⢷⡂⢸⠀⢸⠀⢻⠀⡿⠀⠀⣀⠀⠀⠀⠀⣰⣆⡚⠀⣠⠔⠀⠲⣶⣮⢈⡛⠢⠁
⠀⠀⠀⠀⢘⣋⣀⠀⣰⡀⢁⡘⢣⣮⣴⡗⣼⢟⠀⣀⠀⠀⠀⣠⣮⠿⢿⡡⠖⠃⠀⠀⠀⠉⣿⣿⠖⡁⢸⣦
⠀⠀⠀⠀⠈⠉⠻⣿⣿⡭⣞⣿⢏⠧⠉⠀⡀⠀⠘⢿⣧⠓⢀⣟⣿⣧⣄⠉⠙⠓⢒⠄⠱⡀⢸⠋⢁⣾⣸⣿⣧⣀⡀
⠀⠀⠀⣠⡆⢀⠀⠹⢿⣿⠉⡉⠈⠀⠀⠁⠁⠀⣽⣿⠟⠱⠾⣁⣭⣶⠐⣿⠃⠀⢸⡄⠘⢲⠀⣶⡷⠘⠊⠻⡗⢀⠉⡄
⠀⠀⠘⣿⡟⠌⠁⠈⠋⠛⠀⠒⢂⣤⡄⠀⠀⠀⢿⣿⣷⣤⣴⣿⣿⣿⢀⡿⣇⠀⠀⠙⢄⣤⣆⣻⣧⡀⠀⠑⢴⣄
⠀⠀⠀⠈⠙⠂⠚⢃⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠉⣯⢽⡏⣿⣟⠀⠭⣤⠤⠀⠀⣼⣿⣋⣀⠈⢣⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣽⣷⠶⠂⠀⠀⣠⣿⡟⢁⡤⠐⠳⣾⣷⣿⣿⣦⣴⣯⠐⣦⣀⠙⠀⢸⣿⣼⡿⣧⣀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠓⠀⠸⠀⢸⣯⣭⣤⣿⡷⡉⣳⣿⣿⠀⠘⠋⠹⣿⣿⣿⢊⡸⠶⣄⠉⠉⠉⠀⠯
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⣀⠰⣁⣴⣿⡇⠀⢿⣿⣧⣴⣿⣿⣿⣴⣿⡆⠗⠈⢿⡟⠀⠁⠀⠈⠓⣨⣄
⠀⠀⠀⢠⣼⣽⣿⣿⣯⣄⡉⢃⣴⣿⣿⣿⣷⠲⠸⡿⣿⡿⠻⡛⠿⣿⣷⣿⣀⣴⢈⡏⠀⠀⠀⠀⠈⠛⠣⠄⢴⣤⡀
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣷⣾⢻⠟⠛⠛⠻⢆⠀⠁⠙⡷⠀⡀⠀⠀⠨⣿⣭⡝⢛⣦⣄⡀⠀⢠⠂⠂
⠀⠀⠀⣿⣿⣿⣿⣵⣻⣻⣿⣷⠀⢀⠀⠀⠀⢀⡠⠶⡄⠈⠀⠉⠩⠥⠂⠄⠙⢿⡷⣍⣻⠷⣶⡖⢄
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡑⠄⠀⠀⠀⡞⡀⢱⣌⠳⣤⣤⣄⠀⠀⠀⠀⠉⠉⠈⠉⢉⠁⠒⠂⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣴⣚⣃⣀⡘⠋⠉⠈⠉⠉⠛⠂⠐⠂⠀⣀⣀⣀⣀⣀⠀⡤⠴⠶⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠀⠀⠉⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠢⣄⡈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⠤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠙⠳⡂⢌⡛⢶⠤⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠳⡃⠀⡤⠚⠉⠉⠲⣄⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠈⠲⢌⠉⠓⠊⠙⠳⡴⠚⠓⠶⠉⠈⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⡇⠀⣀⠤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⡽⠄⠀⠀⠀⠢⣄⠀⠀⠀⠀⠈⣇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠗⢋⣵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⢿⠄⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡙⠦⡼⠘⢤⠃⡇⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢳⡆⠀⠀⠀⢀⣠⣤⠿⠀⠀⠀⠀⠀⠉⢹⣃⣀⣤⣤⠶⠚⠉⠀⠀⠀⠀⠰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⢈⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣁⣀⠀⠀⠀⠀⠀⠀⣠⠤⠬⠅⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠆⠀⠀⠐⠒⠛⠻⠶⡄⠀⠀⠀⠀⠀⠀⠀⡤⢝⣷⣌⡉⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⢄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠉⠓⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠔⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⣴⣴⡀⣶⣾⡁⠚⠆⠀⠀⠀⠀⠀⠀⠀⢺⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠏⠘⠇⢷⡏⠃⠙⠀⠀⠀⠀⡔⠢⠤⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡯⠴⠒⣞⠉⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠙⠢⢤⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⡤⠖⠁⠀⠀⠰⣕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣖⠚⠁⠀⠀⠀⠀⠀⠀⠁⠠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⡏⠹⡄⠀⠀⠀⢀⡄⠀⠀⠀⡇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡘⠀⠀⠉⠓⠒⠚⠁⠘⠦⠤⠞⠁⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠂⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⠤⢤⣀⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠦⢤⣄⣀⡀
⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⢤⡀⠘⠀⠀⡰⠀⠀⠸⠁⠐⠒⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⢤⣄⣈⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠙⠀⠈⠀⠀⠀⣁⡀⠠⠤⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⣤⣄⣀
⠀⠀⠀⠈⠉⣁⡤⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠤
⠀⠀⡀⣸⢈⡿⠁⠑⠠⢀⡈⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⡆
⠀⠰⠿⠿⣿⠃⠀⠀⠀⠀⠈⠐⠢⣀⠈⠑⠢⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠩⠳⣆⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⡇
⠠⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀⠀⠉⠐⠄⡀⠈⠀⠀⠀⠀⠉⢐⣲⡭⢅⡂⢽⡀⠀⠀⠀⠀⠁⠀⠀⠀⡇⡇
⢦⣶⣄⢀⠈⠒⠄⡑⠠⠀⠀⠀⢽⡲⠄⠀⣀⠀⠁⠀⠄⠀⢀⣈⣁⠉⣴⣶⣿⢦⢻⡐⠲⢤⣀⡀⠀⠀⠀⡇⡇
⢸⣿⣿⣏⣢⣄⠀⠈⠐⠀⠑⠠⠀⠈⠁⣹⣿⡶⠀⠀⠀⢞⣾⣿⠂⠀⣿⣿⡟⢸⣜⡇⠀⠀⠀⠉⠲⢤⣄⡇⠇
⠀⠉⠉⠲⣭⡺⡝⠢⣄⠀⠀⠀⠐⠤⡀⠀⠛⠋⠀⠀⢸⣿⠿⢟⠁⣤⢻⠓⠉⢀⡿⣷⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣄
⠀⠀⠀⠀⠈⠙⢧⣀⣧⡁⠀⣀⡀⠀⠁⠀⠂⢀⣀⣀⣾⠛⠉⡩⠀⠖⠉⠁⡀⠀⠇⣿⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣦⡀⠀⠉⠘⠋⢹⠀⡀⣴⠤⠖⢚⡡⠔⠊⣁⡏⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣿⣿⣇⠀⠀⠀⠀⢸⣀⡀⠄⢒⡩⣦⣤⣶⡯⣀⡼⠃⠀⠐⠠⡈⠙⠦⣄⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣷⣤⣤⠀⠉⡀⠤⠔⢾⡿⢟⣫⠵⠊⠁⠠⣀⠀⠀⠀⠈⠑⢤⡈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢀⠀⠀⢀⡤⠔⠚⠉⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠈⠲⣄⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢤⡀⠑⢤⡀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠑⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡉⠲⣄⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀

⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠘⣇⠈⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣸⠀⠀⠀⠀⠀⠀⠀⠰⠋⡇⣦⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠰⡆⣨⢛⢥⡶⠀⠀⠀⠀⠀⠀⢸⢸⣄⣧⣠⠀⠀⠀⠀⠀⢀⠀⣲⠛⡏⠀⠀⠀⠀⠀⠀⡾⡿⠜⡄
⠀⠀⠀⣷⣼⢀⡾⣤⠦⡼⠀⠀⠀⠀⢸⣾⠈⡟⠙⢻⠀⠀⠀⠀⠀⠘⢟⢫⣿⣀⡀⠀⠀⠀⢀⠠⢇⠠⣤⢷⡄
⠀⠀⠀⠘⠦⠾⠇⠘⠃⠁⠀⠀⠀⠀⠀⠁⠰⠛⠖⠁⠀⠀⠀⠀⠀⠀⠀⠱⠐⢯⠃⠀⠀⠀⠘⣷⠛⠘⢆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠜⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀
⠀⠀⠀⠀⠘⡿⡏⡅⠀⠀⠀⠀⠀⢀⢀⠀⣿⢣⣀⡀⠀⠀⠀⠀⠀⠀⠎⣼⣿⢰⡀⠀⠀⠀⠀⠀⡄⣾⡼⣃⡋
⠀⠀⠀⠀⠸⣧⠛⣇⠀⠀⠀⠀⠀⠈⠉⢷⠾⡌⠛⢛⡁⠀⠀⠀⠀⢰⣜⣇⢹⡾⣧⡀⠀⠀⠀⠀⡁⢏⢁⢹⠠⠇
⠀⠀⠀⠸⣆⣿⢶⣼⠀⠀⠀⠀⠀⠀⣻⡛⠀⡿⣆⠛⠇⠀⠀⠀⠀⠸⣿⠹⢰⡀⠻⠀⠀⠀⠀⠠⣼⡾⢿⡘⢾
⠀⠀⠀⠀⠀⠰⠏⠋⠀⠀⠀⠀⠀⠀⠈⠃⠻⠅⠙⠋⠀⠀⠀⠀⠀⠀⠙⠘⠋⠃⠀⠀⠀⠀⠀⠀⠈⠃⡎⠻⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠰⣄⣀⠀⠀⢀⣠⣶⠾⠼⢶⣤⣤⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢄⣽⣷⣶⣾⣿⣟⣥⣄⠀⠀⠈⠛⠻⢷⣄⡀⢤⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣷⣽⠏⠁⡦⠊⠀⠀⠀⠀⠀⠀⠈⢲⣾⣿⣏⣴⡙⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⡄⠀⠀⠀⡠⠔⠚⠓⠒⠄⠀⠈⠙⠍⠙⢿⣧⣤⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠐⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⠁⠉⠲⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠻⡀⢠⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡦⠀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⣧⡑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣾⣷⣄⢀⡀⢤⣠⢤⡀⠀⠀⠀⡠⠀⠀⠐⢺⣿⠁⠈⠉⠓⠤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠈⢦⣙⠿⠿⣿⡛⣆⣈⣉⣙⣳⣶⣍⣂⣀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠁⢈⠛⠻⠿⠛⡋⠉⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠛⠛⠉⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠘⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⢺⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠻⣿⡄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⡀⣰⣿⡿⠃⠀⢠⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡼⣿⠁⠀⠀⠸⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠘⣿⣿⣷⠈⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣦⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⢤⡀⠀⢀⣿⣷⡀⠀⠀⣸⣿⣿⡟⠀⢤⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⢋⣼⣿⣿⣿⣿⡇⠀⠀⠀⢱
⠀⠀⠀⢤⡀⠀⣴⡿⠀⡀⠙⣿⣿⣿⣿⡆⠀⠀⢸⣷⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⠁⠀⠈⣿⣷⡄⠀⠀⠙⠻⣿⣿⡿⠁⢸⣿⣿⣿⣿⣿⣷⣤⣀⣤⣿
⠀⠀⠀⠈⣿⣶⠟⠀⠀⣿⣦⣾⣿⣿⣿⠇⢀⣴⣿⠟⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⣿⣿⡀⠀⠀⠀⠀⠉⣰⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⣇⠀⢀⣿⣿⠀⡀⠀⠘⢿⣿⣿⣿⢟⣴⣿⣿⠋⠀⢠⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⠀⣿⣷⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟
⢿⣿⣿⣿⣿⣇⣀⣽⣷⡀⠈⠙⠿⢋⢸⣿⣿⣿⣷⣤⣾⡇⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⣿⣌⣿⣿⣿⣿⣿⡿⢡⣾⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣠⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⢠⠀⠙⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⡟⠀⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣷⣶⣿⣿⣿⣿⡟⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣘⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⡟⠛⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣿⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⡖⠒⠛⠓⠲⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣻⣻⣻⣿⣻⣟⣟⡛⣿⢿⣿⣿⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣼⣿⣇⣀⣀⣀⣀⣀⣀⣹⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⠿⢿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣵⡿⠿⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣶⣿⣿⢯⣶⡿⢿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⢻⣿⡀⠀⣸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢿⣯⠀⢀⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣦⣤⡄⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢣⠏⠀⠀⠀⣿⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⢹⣿⠀⠀⠀⠔⠈⡏⠉⠋⠐⠄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣱⢣⡏⠀⠀⠀⠀⣿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡆⠀⠁⠀⠁⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⣰⢣⣿⡇⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⡁⠀⡆⠀⠒⠀⡘
⠀⠀⠀⠀⠀⢀⠀⠄⠒⠊⠁⢸⠀⠂⠿⠼⠧⠤⠤⠤⠤⠟⠀⠀⡿⠦⠤⠤⠤⠤⠤⠤⠼⢻⠀⠀⠀⠈⠒⠳⠤⠓⠊⠀⠀⠀⠀⠂
⠀⠀⠠⢂⢉⠠⠀⠀⠀⠀⠀⢸⠒⠀⠀⠀⠀⠀⠀⠺⠶⠖⠂⠀⡇⠀⠀⠀⠀⠀⠰⠶⠶⢺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂
⠀⡔⠛⢶⣤⣤⠀⠀⠀⠀⠀⢸⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈
⡄⠟⠛⠉⠉⠉⠁⡀⢐⠈⠤⠍⣑⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⢂⠩⠤⠌⣐⠠⡀⠀⡀⠉⠫
⢁⠈⠤⠀⠀⢀⠌⡔⢡⡶⠒⠲⣦⠑⡌⢧⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠊⠌⣠⠶⠒⠶⣄⠱⠐⡄
⠀⠑⠂⠀⠀⢖⢠⠀⣿⠀⠀⠀⢸⡇⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠚⠀⠀⡏⠀⠀⠀⢸⡆⣁⠐⠂⠴⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠣⡙⠶⣤⡴⠞⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⠦⣤⡴⠟⡠⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣶⣶⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣶⣶⣶⣬⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⣤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⢄⣺⡵⠞⠳⠞⠳⠾⢷⠉⠊⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⢾⣭⣵⣾⣿⣿⣿⣿⣻⡶⣂⠓⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠾⢋⣼⣿⣫⣯⣿⣿⣧⣿⣿⢿⣳⣿⣦⣅⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⢹⡧⣙⣙⣉⡣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠜⢡⠟⠉⠀⡀⢴⣾⣿⡟⣹⡾⢟⣿⡇⢣⣜⢿⠿⡿⣛⢄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣦⣄⣀⣾⠀⠀⠀⢀⣾⡃⠀⣿⠋⢰⠉⣰⣿⣿⠟⢀⣤⣴⣦⣌⣙⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣿⠷⠠⣴⣛⣿⠄⢰⡇⣠⣾⡌⣿⣯⠗⢠⡏⣾⠿⠛⠛⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⢃⠀⣠⡈⣉⣀⡆⢰⣶⣿⣿⠇⠘⠋⠀⠚⠉⢠⣶⡟⣾⣿⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡛⠏⠛⠉⠉⢰⣿⣿⣿⠟⠀⠀⠀⠀⠀⠐⠛⠛⠘⢿⡃⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡐⣶⡀⣠⣿⣿⣿⣧⣄⠀⠀⠀⠄⠀⠀⠐⠚⠛⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⣄⣶⡈⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡉⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣀⠀⠀⠀⢈⠉⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⣴⡟⢡⣖⣼⠃⠀⢸⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⡛
⠀⠀⠀⠀⢀⣾⢿⣄⠇⠀⠀⢿⣧⣀⣉⡁⠀⠀⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢱⢻⡆
⠀⠀⠀⠀⠈⣿⣄⠙⠳⢦⣤⣈⣉⣙⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⡾⠁
⠀⠀⠀⠀⠀⢈⡿⢿⣷⣶⣾⣭⣍⣉⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⡿⠋⣿⣿⣿⡿⠛⠿⠋
⠀⠀⠀⠀⠀⠈⠳⣤⣿⣤⣤⣤⣬⣉⣉⣩⣭⣾⣿⠿⢋⣽⣿⠟⣁⣾⣿⣿⠋⢠⣾⡟⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⡿⠿⠛⠛⠿⠿⢛⣩⣤⡾⠟⠛⠉⢁⣴⣿⠟⢁⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠛⠛⠿⠻⢯⣤⡴⠶⠿⠻⠟⣋⡴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠛⠛⠉⣉⣁⣤⣽⣿⣷⣶⠿⠿⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⣷⣶⡾⠿⠿⠛⠛⠋⢹⣿⣧⣤⣤⠶⠶⠞⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⠿⠿⠟⠛⠛⣿⣿⣁⣤⣤⡴⠶⠖⠛⠛⠉⢛⣁⣀⣤⣤⣴⣶⣾⠿⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⢀⣠⣤⠶⠶⠚⠛⠛⠛⣉⣀⣠⣤⣴⣶⣶⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣰⠟⠁⢀⣠⣴⣶⣾⠿⠿⠛⠛⠋⠉⠉⠀⣀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⡏⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⣁⣀⠈⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢸⡟⠉⣿⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣸⡇⠀⢿⠶⠶⢾⡆⢸⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⠇⢸⡟⠉⠉⠁⠀⢀⣀⣤⣼⠇⣸⠃⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡄⠸⠷⠶⢻⡇⠀⣿⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⣇⣠⣿⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣉⣁⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⢀⣀⣀⣤⣤⣶⣶⡾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⠿⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣶⣶⡾⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠉⠻⣿⣦⣄⣠⣤⣤⣴⣶⣶⣶⣾⡿⠿⠿⠿⠟⠛⠻⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⠻⣿⣯⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣽⣿⣷⣄⡀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣟⡛⠛⠋⠉⠉⠉⠙⠻⣿⣶⣤⣀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠀⠈⠙⠻⣿⣷⣤⣀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⠁
⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣿⠃
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⣶⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣿⣿⡿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣦⣄⣀⣠⣾⠟⣿⣿
⠀⠀⠀⠀⠘⣿⣧⡈⠙⠻⣿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣿⣿⠟⠋⢠⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢸⡇⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣀⣀⡀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⢸⣏⢉⣿⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣟⣛⣿⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡟⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠟⢩⣉⠛⣷⡀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⢻⡇⣾⠛⠛⠛⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⣶⣾⣿⣶⢸⡧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢷⣄⡘⠛⣠⡾⠃⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣸⡇⣿⣀⣀⣀⣀⣀⣀⣀⣈⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣭⣿⣿⣭⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣧⣀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠿⣯⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣶⣟⣠⣶⣾⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣷⣦⣄⣀⣸⣇
⠀⠀⠀⠀⠀⠀⣿⡏⠉⣿⣿⡿⠋⢉⡉⠛⢿⣿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⡿⠋⣉⡉⠻⣿⣿⣏⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⣿⠀⣾⣿⣿⡆⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠀⣾⣿⣿⡆⠘⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠙⠻⠟⠁⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠙⠿⠟⢁⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡋⣝⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢷⣤⡀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣴⣿⠛⠛⠓⠂⠈⠓⠀⠃⠀⠘⠃⠀⠀⠂
⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣤⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠹⣆⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⣷⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠰⡵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⢠⣦⣤⣤⣤⣤⣦⠀⠀⠀⠀⠀⣴⣶⣤⣶⣶⣶⡆⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣾⣿⣿⣿⣽⢿⢿⣿⣧⠀⠀⢀⣾⣿⡿⣯⠿⣿⡿⣿⣇⠀⠀⢠⣿⣿⣟⢻⣿⡻⣿⣿⣇⠀⠀⢠⡿⣿⣿⣿⡻⣿⠟⣿⡆
⠀⣸⣿⣿⣷⣽⣯⣻⣹⣿⣿⡇⠀⣼⣿⣿⣿⣿⣷⣬⣺⣿⣿⡀⠀⣼⣿⣿⣏⢼⣾⣿⣿⣿⣿⠀⢀⣿⣇⣿⣿⣿⣡⣯⣷⣸⣿⡄
⢠⣿⣿⣿⣿⠛⡙⢛⢿⣿⣿⣿⢀⣿⣿⣿⡿⢋⠉⡛⣿⣿⣿⡇⢸⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⡇⢼⣿⣿⣿⡿⠟⠿⣿⣿⣿⣿⣷
⢻⣿⣿⣿⣷⣄⣀⢀⣼⣿⣿⣿⠀⣿⡿⣿⣧⡀⠒⢠⣿⡿⣿⡇⠘⣿⡿⣿⣇⠈⠒⢁⣿⣿⣿⡇⢸⣿⣿⣿⡀⠐⠔⢸⣿⣿⣿⡇
⠀⣿⣟⣙⠻⢿⣿⡿⣟⣉⣿⣿⠀⢿⣷⣤⣭⡛⠿⢛⣭⣴⣾⠅⠀⠟⣷⣈⣛⠳⠖⣚⣩⣤⣿⠁⢸⣿⣭⣿⡛⠷⠖⣋⣥⣽⣿
⠀⣿⣟⣩⣭⣴⣤⣙⣛⠟⢿⡏⠀⢸⣟⣤⡴⣾⣿⣶⣮⣝⣻⠀⠀⢸⣟⣠⣶⣾⣷⣲⣤⣬⣿⠀⠀⢿⣤⣶⣿⣿⣿⣶⣦⣼⡟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠐⠀⣿⡛⣋⣭⣭⣭⣭⣛⡛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⣿⣿⣿⣿⣿⣿⣿⠻⢿⡇⠀⠀⢛⣧⢺⣿⣿⠉⠛⣿⠇⠀⠀⠀⡟⣿⣿⣿⣿⣿⣿⣷⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢨⣿⣧⣿⣷⠙⣤⣿⠀⠀⠀⠀⣿⣿⣟⣿⣿⣿⣿⣿⠁⠀⠀⠸⣿⣿⣿⣷⣾⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⠿⣿⣿⠀⠀⠀⠀⣿⣿⣏⢿⣶⣄⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣶⡟⡇⠀⠀⠀⠀⣿⣿⣿⡘⣿⣿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢸⡿⠿⢿⠼⡿⣿⠇⠀⠀⠀⠀⢹⣿⠿⠷⠿⢿⠇⠀⠀⠀⠀⠀⠿⠿⠿⠿⠞⠿⠃⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠇

⠀⠀⣀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢐⡭⣗⢦⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣦⡀
⠐⣩⡴⣶⣶⣶⡶⣬⡑⠀⠀⠀⠀⣿⠛⢍⠉⠉⢿⡇⠀⠀⠀⠀⠀⠈⢣⢸⣿⡇⠀⠀⠀⠀⢸⡟⠉⠉⠉⠹⣿
⠸⣅⡀⣿⣷⣼⡟⢀⡽⠂⠀⠀⠀⢟⠐⠆⠻⣷⣌⡇⠀⠀⠀⠀⠀⢠⠏⡸⡽⠀⠀⠀⠀⠀⢸⡃⠲⠀⠰⠂⣿
⠀⠀⠉⠛⠛⠛⠋⢡⡤⠀⠀⠀⠀⠈⢇⠀⠧⢀⡜⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣦⡆⠀⠀⠀⠀⠱⡀⠸⠄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠁⠉⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⢸⣹⣅⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣅⠀⠀⠀⠀⠀⠀⠀⣠⣬⣟⠁
⠀⠘⣿⣿⣙⡻⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣀⡀⠀⠀⠀⠀⠀⢸⢿⣿⠻⣆⠀⠀⠀⠀⠀⢸⢿⣿⡟⠷⣄⠀⠀⣀
⠀⠀⣽⣿⣿⡇⣽⣦⡄⠀⠀⠀⠀⣼⠀⠻⠿⢯⣿⡀⠀⠀⠀⠀⠀⢸⡿⣇⠈⢆⠀⠀⠀⠀⠀⢸⡿⣧⢀⣤⣤⣴⡏
⠀⢯⡟⠛⠛⠓⠛⢯⡗⠀⠀⠀⠀⠘⠦⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⡾⠁⢻⠄⠈⠦⠀⠀⠀⠀⠾⠁⡿⠉⠟⠉⠹⠇

⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⣴⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠂⠀⠀⠀⠀⠀⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡄
⠀⠀⠀⣿⣿⣏⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠈⠿⢿⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⣾⡟⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠸⠇⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠤

⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠
⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢤⣀⣸⣧⡜⠉⠀⠀⠀⠀⠀⣴⣿⣷⣤⣾⣿⣦⠀⣷⠀⠀⠀⠀⢈⣿⢿⡁⠀⡀
⠀⠿⣿⣿⣿⡷⠛⠁⠀⠀⠀⠀⠀⣀⣹⣿⣿⣿⠛⠒⠀⠀⠀⠀⢿⣿⢡⢻⢟⣛⣛⡁⣋⠀⠀⠀⠀⢸⡗⢙⣿⠀⠒
⠀⠀⠛⢛⡟⠀⠀⣤⡄⠀⠀⠀⠀⠀⢠⠿⢩⡇⠉⠀⠀⠀⠀⠀⠒⠲⣿⣆⣾⠟⠁⠀⠀⠀⠀⠀⠀⢀⣽⡾⣅
⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⠀⠀⠀⢀⠠⣠⠞⣩⣤⣬⣅⢠⡄
⠀⠀⠀⢰⠿⢿⢿⡌⠉⠀⢸⣍⡉⠉⠉⢉⣉⡤⠶⣇⣸⣿⠑⠈⢁⣩⡟⢤⠤⠤⠤⢒⣠⠤⠤⢤⣀
⠀⠀⠀⠸⡀⢀⣾⣇⠇⠀⢀⡽⠟⠿⠯⣍⣀⣴⢦⣿⣿⡏⣶⣞⣉⣹⣷⣴⡴⠶⠿⠽⠖⠒
⠀⠀⠀⠒⠀⢸⣿⣿⡄⡦⠚⠉⠀⠀⠀⣪⣙⠿⣹⣿⡟⢿⡰⠂⠈⠙⠫⠉⢛⠒
⠀⠀⠀⠀⠀⠈⣿⣿⡟⣠⠂⠀⠀⣠⠀⣼⣩⣿⠛⣿⠇⡘⣥⣞⣋⠉⠉⠉⠛⠲⢤⣀⠠
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣀⡄⡤⣭⡱⣿⣿⣿⣸⣿⡾⢁⣄⡀⠀⠀⢀⣀⣤⡞⠃⠈⠉⠉⠉⠁⢀⡤⠖⠒⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⢱⡿⢙⣿⠈⣻⣾⣿⣁⠀⠈⠉⠉⠛⠻⢯⣅⣸⣧⡲⠛⠛⢛⣳⠞⠡⠤⣄⣀⣀⠴⡞⠉⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⠿⣿⣯⣵⡆⠹⣿⡯⠽⠿⠶⠶⢶⣞⣉⡛⢻⡏⠙⠛⠛⠛⠋⢁⣀⣤⣄⣀⠉⠓⠒⠛⠛⠋⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⢾⣯⠾⢿⣿⡚⣅⣀⣀⣀⣀⣀⣤⠴⡾⠛⠛⠿⣦⣤⣤⣴⣊⣹⣁⠀⠀⠀⣉⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠯⠚⠓⠚⠛⠋⠉⠉⠙⠒⠲⠿⢦⣤⠤⠤⠄⠀⠀⠀⠈⠉⠙⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠉⠛⢳⠆
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⣄
⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠐⣎⢧⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⣀⣼⣾⣇⠀⠀⡾⠁⠀⣠⠞⢳⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⣿⣿⣿⣿⣿⡆⠐⣇⢀⡜⠁⠀⣸⠃
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠘⣿⣿⣿⣿⠇⠀⠈⠻⠤⠤⠖⠃⠀⠀⣀⣀
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠉⢳⡄
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⠞⠉⠉⣹⢦⡀⠀⠀⠀⠀⢀⣤⣾⣿⣦⠀⣠⠴⣰⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡞⠀⠀⣏⠀⢠⠞⠁⠈⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣸⠁⠀⠀⠙⢶⣋⣀⣠⠼⠁⠀⠀⠀⠀⠻⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣴⣇⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠙⠒⠶⠒⠋⠀⠀⢰⡇⠀⠀⠙⠒⠶⠒⠋⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠲⣤⣀⣠⡴⠂⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡤⣄⣀⣀⣀⣀⣀⣀⣤⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣾⣧⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣶⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢇⣤⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⢠⠟⠙⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⡀⢠⡆⠀⠀⠀⠈⣧⠄⠐⠸⠀⡤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠀⡰⠉⠷⠀⠐⠑⠀⠃⠀⣤⡠⠞⡷⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠸⢤⡇⠀⠄⠀⠀⠀⠀⠀⠀⠉⠀⠐⡁⠘⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠈⠁⡐⠂⡰⠁⠀⠀⠀⠤⠂⡳⠀⠇⠆⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣦⡱⡄⠲⠇⠘⠀⠀⠀⠀⠀⠀⢀⠀⠞⠀⠔⢁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠌⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠊⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠈⡤⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢥⡀⠀⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠮⠴⠖⠖⢤⣼⢶⡿⣷⣾⣷⠋⢻⣿⣷⡖⠦⢤⡄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣤⣤⣴⠞⠋⠀⣸⠁⢀⣿⢻⣆⠀⠙⢯⣟⠉⣷⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢹⣁⣀⣠⣶⡇⠀⠸⣿⠄⣻⡆⠀⠈⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠈⣿⠷⠏⠛⠉⠘⠶⠶⠞

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣠⡇⠀⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠋⠀⢀⣾⠟⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⢀⣴⣿⡏⠀⠀⣠⣴⡏⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⢳⣿⢀⠠⠾⢻⣿⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⣿⡿⣱⣿⡿⠃⠀⠀⣼⣿⣼⠠⠄⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣽⣿⣿⣿⣿⣆⣠⣾⣿⢿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢟⣿⡿⣿⣿⣿⣿⣿⢣⣼⣿⡎⢷⣇⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⢏⣾⡟⣸⣿⣿⣿⣿⡇⡾⣿⣿⢻⡌⠻⣆⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢀⢠⠁⣿⠟⣿⡿⣰⣿⣿⣿⣿⣿⡘⣇⢸⣿⣿⣿⠀⠙⣷⣄⠠⡱⣄
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣿⣾⣿⡇⣿⢸⣿⣿⣿⡿⠧⠀⣼⡈⣿⣿⡇⢢⠘⣿⣧⠙⣿⣦
⠀⠀⠀⠀⠀⠀⢸⡀⣼⢿⣿⣿⠀⣿⣼⠳⠏⢠⣷⢀⣼⣿⠃⣸⢹⡇⣸⡄⠸⣿⡇⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣧⢹⡜⣿⣿⡄⢹⣿⠀⠀⢸⣾⡈⠻⣿⢀⡟⣹⠇⣿⣥⠀⣿⣷⠀⢻⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠹⣿⣿⣿⣦⡿⣷⣄⡈⠻⠏⠀⡹⣾⣱⠏⣼⣿⣧⠀⣿⣿⠀⢸⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⣻⣯⣛⣿⣿⡝⢷⣂⠠⣼⣷⣿⢏⠼⠁⢸⡇⢰⣿⡟⠀⣸⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠀⠈⠋⣉⣿⣿⣿⣷⡙⠿⠿⠯⠁⠀⢀⠴⠋⣴⣿⠟⠁⣰⢃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⣿⣿⣿⣿⣷⣦⣤⣤⣤⣷⣼⣿⣿⠗⢀⡴⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣹⣿⣿⣿⣿⣿⣿⣿⠿⢛⣡⠴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣧⣴⣿⡭⠤⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠿⠟⢻⠟⢿⠿⢻⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠛⠛⠛⠛⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡾⠁⠀⢠⣂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⣸⠇⠀⢈⠏⢹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⣿⠀⠀⠺⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣾⠀⠀⠀⠀⣴⠞⠋⠉⠁⠀⠀⠈⠉⠉⠛⠶⣤⣀
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣧⡴⠇⠀⣀⣀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠳⢼⡇⠀⠀⠺⠤⠜⠘⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠹⣆
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠳⣼⠁⢀⣤⣤⣄⣀⣀⣹⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣦⢄⢻⡄
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠋⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⠸⡇
⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⢀⡏⠓⢾⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢸⠷⢤⡾⠀⠀⢀⡔⣶⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⡾⠀⠀⢀⠀⠀⣸⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠇⠀⠐⠭⠇⠀⣿⠃⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⣯⠤⠶⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣌⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉



⣶⣶⢶⠶⠶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⡶⣶⣶⣶⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⣸⡇⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠀⠀⣈⣁⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠗⢰⣿⡏⠀⠀⢹⣿⡆⠺⠿⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣼⣿⣧⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢁⣠⣴⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠛⠉⠁⣀⣀⣀⠀⠈⠙⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⢀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢀⣠⣴⣿⡿⠁⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠈⠉⠉⠉⢹⣿⣿⣿⣿⠋⠉⠉⠉⠁⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠈⠛⠛⠛⠉⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣶⣶⣶⣾⠿⠟⠉⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠉⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠉⠉⠙⠻⣿⣆⠀⠀⠀⣿
⢻⣿⣄⣀⣀⣀⣀⣠⣿⠋⠀⣴⣿⣿⣷⣄⠀⢻⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⠁⢠⣾⣿⣿⣶⡄⠘⣿⣇⣀⣠⣿
⠀⠙⠻⠿⠿⠿⠿⢿⣿⡀⠸⣿⣿⣿⣿⡿⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠉⠛⠛⠛⠁⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠙⠛⠛⠋⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣴⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣦⣶⣶⡿⠟⠁
⣶⣶⡶⢶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣮⢳⡐⡄⠀⠀⠀⠀⠀⠀⡄⣾⠈⢁⡀
⠀⠀⠀⠀⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣷⣧⢳⡀⢀⣤⣤⣤⣀⣷⡸⢀⣾⡇⠀⠀⢀⣶⣮
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣸⣿⣟⣷⣿⣿⢟⣛⣿⣿⣷⣼⣿⣷⣶⣷⣿⣿⣃⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢹⣶⣾⣿⣿⣿⡿⢿⣾⣿⣿⣷⣿⣿⣭⣿⡿⣹⣿⡟⠁⠀⢀⠍⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⣤⣄⠀⠄⢘⢓⣏⠀⣠⣿⣎⣻⣿⣿⣿⣿⣯⣭⣻⡿⣿⣿⣿⡿⠉⢰⣿⣏⣤⠀⠀⠈⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢿⣷⣷⡦⣈⠁⠀⣾⡟⣛⢛⣿⡿⡿⣿⣟⣭⣽⣷⣶⣾⣿⣡⠀⠈⢿⡿⣿⣾⣶⣶⣀⣀⠈⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠺⡦⠀⠀⠀⠀⠀⠀⠉⠳⢶⣦⣝⣲⣿⣿⣁⣿⣿⣿⣿⣯⣹⣿⣿⣽⠛⠛⢿⣧⣎⣴⣿⣷⣹⣿⣟⣿⣿⣿⣿⣾⣵⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⣿⣿⣿⣿⣽⣿⣿⣭⣿⠏⠀⢿⡛⠩⠀⠀⠘⠃⣸⣷⣍⢻⣟⣮⢻⣼⣿⠿⠛⢋⣉⣤⣴⠂
⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠁⣀⣿⣿⣿⣿⣿⣿⠎⠋⢐⡂⠈⠐⢶⣶⠄⠘⣧⣽⣭⠛⢻⡟⢛⣉⠭⠔⢒⡈⠉⣉⠡⠤⠴⠶⡂⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⡿⣽⣉⣿⣿⡿⡿⠆⣐⣿⠀⠀⠀⠀⠁⠀⢀⡿⢿⣿⠀⣠⢫⣧⣶⣾⣿⣯⣭⣶⣶⠖⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠀⣤⣾⢠⣿⢧⢸⣿⣿⣟⢷⣿⠀⠙⠁⠀⠀⢠⣾⣧⡤⠉⣠⣾⣷⣦⣭⡾⢿⡿⣿⣟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠀⠀⠁⢼⣿⣿⣎⣻⣿⣿⣾⣿⣦⡀⣀⠴⠆⣼⣿⣿⡠⠐⠋⠻⠻⢿⣟⣱⡏⣶⡏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⣽⣿⡿⠻⠛⠛⢉⣨⣿⢇⠈⠉⠁⣩⣿⣿⠿⠛⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡿⣦⣤⣠⣶⡿⣿⣿⣟⡷⣶⣿⣿⣏⡀⠀⠀⠓⠘⢿⣿⡿⠁⣸⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠟⠉⡈⣩⣴⢿⣿⣧⣽⢷⣼⡿⣿⣿⣤⣬⣽⣻⣿⣎⡓⡄⠀⢈⡻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠉⠁⠀⠉⠹⠿⢷⣽⠶⠾⠿⠿⠿⠿⣿⣿⣿⣿⡿⣆⠢⡹⣿⣧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠟⠀⠀⠀⠀⡈⠀⠀⠀⠙⣧⢻⣎⠻⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢻⣆⠈⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠥⠤⠤⣀⡀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠏⠀⠀⠀⠀⠀⠉⠓⣄⠀⢀⡼⠁
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡞
⢀⡴⢏⠉⠋⠉⠙⠓⠦⣄⠀⣷⣀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠒⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⣾⡇⠀⠑⢦⡀⠢⡀⠀⠈⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⣯⠱⡀⠀⠀⠙⢦⡈⠢⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠹⡄⠱⣄⠀⠄⣀⠙⠢⡘⢦⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠐⠀⢀⡾⠁
⠀⠙⣆⠈⠣⣄⠀⠀⠀⠈⠢⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠈⠛⢦⣀⠉⠒⠦⠤⠤⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠴⠶⠚⠉⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⢀⡠⠞⠋⣻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⢀⣴⠋⠁⠀⠈⠁⡠⠖⠉⣠⠔⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠈⠛⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⣼⠃⠀⠀⣠⠔⠁⢀⡴⠊⠡⠀⠀⢀⢼
⠀⠀⠀⠀⣠⡞⠻⢭⣀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⣴⠋⠀⣀⠔⠁⠀⢀⣀⠤⠞⢁⡾
⠀⠀⢀⡾⠉⠙⠲⢤⣈⠉⠒⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡉⠉⠉⠋⠉⣉⣉⣀⡤⠶⠋
⠀⠀⢸⣧⣀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠒⠤⣄⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢿⡈⠓⠦⢄⡀⠀⠀⠀⠈⠙⠲⢤⣀⣉⣳⠖⢻⡆
⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⠑⠒⠒⠒⠒⠚⠋⠉⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣶⣶⣶⣶⣾⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⣀⣀⣀⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠰⡶⡶⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⢠⣿⣿⣿⠋⠙⣿⣿⠉⠁⢄⠀⠀⠀⠀⢀⣴⣯⡀⢀⣀⣀⣉⣀⣀⡀⠀⠀⡿⠿⡿⡛⣿⠿⠿⣷⠟⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠙⠚⠀⠀⣼⠃
⢸⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣷⣶⡄⠀⢸⣿⡄⠛⠘⠭⠟⠛⢿⠿⠇⠀⠀⣿⣿⣿⣿⣬⣿⣿⣿⣷⣶⢿⣶⡄⠀⠀⣿⡹⠞⠿⠿⠿⠿⠾⣿⡀
⠛⠛⢿⡽⠛⠛⠛⠛⠛⠛⢻⣭⠟⠃⠠⡞⠀⢿⠀⠠⠴⠞⠉⠑⠦⠄⠀⠘⠛⢻⣽⠟⠛⠛⠛⠛⠛⠛⢯⡽⠛⠀⢰⠏⡧⠀⠀⠀⠀⠀⡼⠉⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣀
⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣢⠀⠀⠀⠀⣾⣿⣷⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⢸⠂
⢠⣦⣶⣶⣷⣶⣶⣶⣷⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣵⡏⠀⠉⠀⠀⠀⠀⣿⣿⡿⢀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⣠⠞
⢸⣿⣿⣿⢫⣴⢶⣌⢻⣿⣿⣿⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⡌⢀⡠⠊⠁⣴⠒⠚⠒⣦⠀⠀⠀⠀⢀⡤⢄⡤⣌⢳⡞⠁
⢸⣿⣿⣧⢻⣦⣠⣼⢢⣿⣿⣿⠀⠀⠀⠀⠀⢯⣾⢟⡿⠋⠀⠀⠀⠀⠀⠀⢸⢀⠊⠀⠀⠀⣾⣫⠉⣝⣿⠀⠀⠀⠀⡏⠀⠈⠻⠋⢰⠆
⢸⣿⣿⣿⣷⣬⣭⣴⣿⣿⣿⣿⠀⠀⠀⠘⢷⡇⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣦⣤⡤⠤⣿⣽⣻⣫⣿⠀⠀⠀⠀⠳⣄⡀⣀⣠⠞
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉

⠀⣰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣖⣲⣦⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆
⢰⡿⠙⢃⣀⡀⢀⡀⣀⠀⠀⠀⠀⠀⢻⡛⡛⣛⢛⡛⡛⣛⢻⡟⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⢀⢀⢸⡇⠀⠀⠀⠘⢿⣿⣿⣟⡶⣾⢶⣺⣿⣿⡷
⢸⡇⠀⢾⣅⣿⢸⡇⣿⠀⠀⠀⠀⠀⢸⠭⡭⠭⠭⠭⡭⠭⢽⡇⠀⠀⠀⠀⠀⠀⢸⡇⠯⣼⢾⣼⢸⡇⠀⠀⠀⠀⠀⠈⠁⠣⣹⠤⠯⠜⠈⠁
⢸⡇⠀⠰⣭⡿⢸⡇⡿⠀⠀⠀⠀⠀⢸⣓⣓⣚⣚⣓⣓⣛⣺⡇⡴⠲⣦⠀⠀⠀⢸⡇⠙⠋⠘⠙⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣡⠶⣚
⠈⣿⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣧⡴⡴⣦⢦⣴⣼⡇⠻⡶⠋⠀⠀⠀⢸⣿⣛⡗⢺⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢚⠅
⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠃⠉⠘⠛⠛⠓⠚⠛⠂⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡖⡒⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡆
⠀⠘⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣓⣚⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣟⠃⠀⣠⣦
⢰⠉⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⣀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠻⣿⣿⣿⣮⠉
⢸⡇⢀⠸⢿⣿⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠈⡿⠿⠿⠿⠿⢾⡫⢝⡇⠀⠀⠀⠀⠀⣈⡛⠟⣁⡀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣵⣿⣿⣾⡿⠁
⢸⠿⠿⠀⠻⢿⠀⠀⠀⠀⠀⠀⠀⠚⠛⠳⣾⣷⢠⣾⠿⠂⠀⠀⠙⠋⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠿⠋
⠸⣀⣀⣀⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠛⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠀⠀⠉⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⠀⠀⣴⣶⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣀⣀⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠃⢀⣾⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡀⢀⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠶⠦⣞⣿⣿⣿⣿⢇⣾⣿⣿⣿⣿⠃⢸⡇
⠀⠀⠀⠀⠀⢠⣦⠀⠀⣀⡤⠴⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠱⣿⣶⣶⣶⣿⣿⣿⣿⡿⢫⣿⣿⣿⣿⣿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⣿⡿⠋⢠⣿⣿⣿⣿⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠉⠻⠿⡿⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠳⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⡆⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢠⡾⠛⣿⡀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⣴⡟⠻⣧
⠀⠀⠀⠘⢷⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣴⠟



⣿⣿⣿⣻⣛⣟⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣟⣻⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠂⡆⢰⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣘⣀⣁⣃⣘⣘⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸⠉⢉⣉⣉⣉⣈⡀⠀⠀⠀⡴⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢨⡇⠀⢀⠎⠈⠢⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠘⣀⠤⡄⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠤⠤⠤⠤⠤⠤⡇⢠⠊⠀⠀⢱⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠣⢀⣀⣀⣀⠠⠔⠊





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣰⡆⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣤⣿⣿⣤⣶⣶⣶⣯⣴⣷⣭⣿⡆⠀⠛⢲⡀⢛⣒⣶
⠀⠀⢀⢸⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⡏⢿⣿⣿⠀⢀⢀⡐⢾⢈⣦⣠⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⣰⣶⠟⣤⢿⡏⢈⡷
⠀⢀⣾⣿⣿⣧⣷⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠋⠀⢀⡌⣦⣠⢷⣏⣿⣯⣯⣧⡐⠀⠀⠉⢈⣍⣉⣩⣍⣭⣥⣤⡄⠀⣜⣿⣶⣼⠟⢹⣏⣸⡅
⠀⣼⣿⠏⠀⢿⣿⣇⠀⠀⠸⣿⣿⢿⢿⣿⣿⠃⠀⠀⢀⣜⣾⣷⣿⣿⣿⣛⣿⣿⣼⣿⣿⣷⣦⡰⣼⣿⣍⠓⢺⣟⡋⠉⠉⠀⠘⠿⠞⢿⣶⣿⠿⠛⠁
⠀⠻⢧⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⣁⣸⣿⣿⡟⠛⠈⠑⠀⠀⠀⠀⡀⠀⠘⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⡀⣴⢠⡀⠀⠀⠀⠀⠀⠀⠆⡐⠂⡆⠐⠀⠀⠀⠀⢰⠀⠀⠀⣿⣿⣿⣿⠇⢀⠀⠀⠀⠀⣠⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⣸⢼⣠⢧⣟⣳⢼⡰⣅⣷⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠉⠁⡄⣑⣘⣄⠀⣀⣙⣦⡴⣃⠀⠀⠀⢠⡀
⠀⠀⠀⢸⡆⠀⠀⣯⣻⣾⣾⣿⣿⣿⣾⣷⣿⣷⣿⡄⠀⠀⠀⠀⠀⢀⣼⣧⣶⡤⣄⠀⠀⠀⠀⢀⠙⠐⠀⠑⠿⠟⠀⢠⣞⣿⡻⣞⡽⠁⠀⢠⣤⣽⠆
⠀⣰⢠⡿⢁⡀⠀⠈⠛⠛⢉⠉⠉⠉⠙⠛⠛⠛⠋⠀⠀⠀⠀⢠⣼⣽⣿⣿⣿⣿⣿⣤⠀⠀⣀⡴⠛⢦⡞⠀⠀⠀⠀⠀⣹⣟⣿⣏⠀⠀⢰⣛⣦⣹⣦
⠀⢿⡈⢷⣼⣄⠀⠀⠀⣠⢿⣆⣤⠀⠀⠀⠀⣦⣷⡄⢣⣀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣞⠀⡄⡟⢷⡜⣮⣷⣴⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⢈⣽⣿⠁
⠀⢸⡿⢸⣿⣿⠇⠀⣰⣯⢾⣿⣏⣦⠀⠀⠲⣬⡟⣷⣦⡙⣷⠀⠁⢸⣿⣿⣿⣿⡟⠃⠸⣧⣿⣿⣵⣿⣟⣿⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⣿⡟⠋
⠀⣾⣷⣾⣿⣿⠀⠀⣿⡇⠀⠋⢹⡿⠀⠈⣧⣿⣷⣿⣼⣷⣡⣧⠀⠻⣿⡟⢽⣿⡅⠀⠀⣽⣯⣿⣿⣿⣿⣻⡇⠀⠀⣀⣤⣴⣆⡀⠀⠀⠀⠉⢻⣿⣦
⠀⠈⠛⠛⠛⠋⠀⠀⠈⠛⠀⠐⠋⠁⠀⠀⠙⠛⠉⠋⠋⠙⠛⠋⠀⠀⠘⠀⠘⠋⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠃⠀⠾⠿⠿⠿⠿⠟⠛⠗⠀⠚⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠀⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⣨⢿⣿⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⣻⠀⣽⡄⡴⢻⡿⣦⣦⠀⠀⠀⠀⠀⠀⡠⢃⠀⠀⢠⠛⢋⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⣧⣽
⠀⢛⣷⣿⣸⠇⡿⢠⢉⣯⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣼⠋⣿⣿⠽⠟⠛⡡
⠀⢽⣿⣿⣿⣷⣦⣙⠟⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⠀⢰⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠙⠈⢀⡋⠄⢒⢋⣤
⠀⢺⣿⣿⣿⣿⣿⣿⣇⣶⣽⣜⠃⠄⠀⠀⠰⠄⢉⠀⠈⡉⠀⠋⠀⡄⠀⠀⠄⠀⠀⡘⣿⡇⠀⣀⣯⣶⡆⣿⢹⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠿⣶⣦⡸⢿⠰⣿⠀⣾⡇⠀⡿⠀⠟⠐⠆⠀⢀⣮⠠⠛⠋⠘⠋⣯⠙⠁⠉⠀⠀⡄
⠀⡺⠾⢿⣇⠴⡾⡆⣀⣷⣷⣮⣿⣿⣿⠺⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣸⣿⣠⡷⠀⡤⡄⡏⣶⠀⣾⢰⡿⢰
⠀⣿⣿⣿⣹⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⠘⠷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⡗⠂⠛⠃⠀⢹⠀⠿⠸⠟⠸
⠀⡿⠛⠁⣿⣉⡿⠟⠿⣿⣿⣯⡿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠀⠀⠀⠀⠀⠀⣀⣈⣣
⠀⣵⠆⠀⠀⡩⠤⣌⡛⢈⣿⣭⢀⡜⠉⠉⠟⠆⠀⠀⠀⠀⠀⠐⢶⠀⢀⣀⣀⠀⣐⠆⢁⠀⠀⠀⠀⠀⠀⠀⣈⠉⢸
⠀⠉⠀⣶⡆⠠⠀⠝⠧⠤⠏⢀⣜⣃⠀⠀⠀⠀⠀⠀⠒⠘⠒⠲⣶⣦⠘⠃⠉⠀⢉⡟⠙⠓⠚⡶⠰⠾⢶⡀⠉⡙
⠀⠄⠀⣿⣿⠓⠀⠀⠸⠆⠤⠘⠀⠉⠁⢴⢤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒⠚⠫⢥⣀⠄⣀⡀⠀⠐⣅⠑⢀
⠀⣠⠸⣦⡇⣿⠀⠀⠀⢀⣀⠀⠀⠀⡈⠂⠀⣿⡷⡤⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠰⣾⡀⣦⠈⢧⠁⠉⠙⠀⠸⡒⢸
⠀⣯⠀⠙⠓⡋⠈⠀⠀⠶⠌⢓⠁⠈⡀⠽⠄⣻⡁⠀⠀⠀⠀⣼⣟⣷⡀⠀⢀⣠⣠⡾⣓⣿⠂⢹⣤⡄⣁⢀⡆⠂
⠀⣝⣢⣤⣀⠅⠈⠂⠀⠰⠆⠐⢒⡗⢠⡄⠀⡀⡀⠹⠷⢞⠛⠛⠛⢻⣷⣦⣾⡇⣾⠇⣿⠟⢼⡿⡟⢠⣤⣿⡇⢾⣦
⠀⣿⡷⣿⡦⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠐⠉⠉⠉⠈⠉⠛⠃⠀⠀⠀⠈⠉⡭⠿⢻⣶⣿⣇⠼⣷⠃⣾⣼⣾⣇⣸⡿
⠀⢯⣠⣤⠴⠾⠂⠐⠂⠀⠚⠓⢄⣀⣤⣤⣴⡇⠀⢀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠓⠛⣹⠏⠾⠟⢿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⣀⣿⡟⢻⣿⣀⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⡏⠀⢸⣏⠉⠁⠈⠉⣹⡇⠀⢹⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⡀⠘⠛⣿⣇⣸⣿⠛⠁⢀⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢷⣄⠀⠉⠉⠉⠁⠀⣠⡾⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⡀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠛⠁⠀⠘⠛⠛⠃⠀⠈⠛⠛⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⣤⠄⠀⢠⣤⣤⡄⠀⠠⣤⣤⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⠀⠀⠘⠛⠛⠂⠀⠘⠟⠛⠂⠀⠐⠛⠛⠃⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⡀⠀⢀⣀⣀⡀⠀⢀⣀⣀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠦⠀⠰⠶⠶⠆⠀⣿⠀⠀⠈⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⣿⠀⠰⠶⠶⠆⠀⠴⠶⠶⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠦⠀⠰⠶⠶⠆⠀⣿⠀⢸⣿⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣈⣉⣿⡇⠀⣿⠀⠰⠶⠶⠆⠀⠴⠶⠶⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⣿⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⣿⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢶⣶⡦⠀⠰⣶⣶⠆⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠰⣶⣶⠆⠀⢴⣶⡶⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠟⣉⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢶⣶⣬⣛⠻⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠉⠙⠳⣌⠙⠂⠀⠀⠀⠀⠘⢁⣤⠞⠋⠉⠉⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣴⣀⡀⠀⠀⠀⢸⡇⢸⠇⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢿⣿⠏⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⠀⣀⣠⠞⢁⠀⠀⢈⠳⣄⣀⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣩⣤⠖⠁⠀⠀⠈⠳⢮⣭⣉⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡴⡖⠒⣦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣾⣄⣇⣠⣃⣿⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⣿⣿⣿⣿⣿⣿⡇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡟⢻⠋⠉⢹⠻⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⣆⣠⡧⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣠⣤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠈⠉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢖⠋⠁⠀⠘⢒⡲⠥⢄⠤⠠⢓⡄⠀⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢃⠃⢠⠄⠀⠀⠀⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠠⢀⡄⠤⠄⣀⣀⣀⡀⡈⢸⠀⠘⠀⠀⠀⠂⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠰⠃⠈⠈⠤⠘⠀⠀⠀⠠⠀⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠄⠀⠀⠀⠀⠀⠈⣂⣀⣀⡀⠈⠀⠁⠀⣸⠤⠤⠖⠚⢄⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⢄⠀⡀⠀⣀⣀⣀⣀⣀⣸⠀⠀⠈⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠊⠁⠀⠀⠀⠀⠈⠢⢤⣤⠔⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣬⣭⣬⣭⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠟⠛⢞⣛⣭⣭⣭⣶⣶⣮⣵⣶⣭⡭⢛⡻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣵⣯⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣶⣥⣘⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣡⣶⡟⠛⠉⣙⣛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢷⣮⣛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⣼⣿⣿⣿⣿⠾⠟⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⣿⡎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣼⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⢿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⡷⢸⠆
⠀⠀⠀⠀⠀⠀⠀⣠⠞⢣⣽⣿⣿⣯⣥⣤⣴⣶⣿⡿⠿⣿⢟⣛⠧⠝⠣⠂⠀⠀⠀⠀⠀⠀⣀⠔⢺⣿⢿⣧⢻⡄
⠀⠀⠀⠀⠀⠀⢰⡏⣮⡿⣿⣿⣿⣿⣏⠁⠀⠀⢠⡶⠚⠋⠈⢉⠄⠀⠀⠀⠀⠀⠄⠀⢀⠞⠁⠀⢸⣿⠘⣿⡇⢹
⠀⠀⠀⠀⠀⠀⣸⢣⣿⡇⢹⡏⢻⣿⣿⢆⣠⠖⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⢻⣦⣭⡇⢸⡇
⠀⠀⠀⠀⠀⠀⣇⢸⣿⣧⣼⣿⣾⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⠛⠛⠳⡄⠀⠀⠻⣿⡇⣸⣧⣤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⡏⠈⢿⣿⣿⣿⠟⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⣠⠛⠛⠂⠀⠀⠀⠀⢿⣆⣥⣶⠶⢶⣬⣙⢷⡀
⠀⠀⠀⠀⠀⠀⢹⡆⣼⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠛⢠⣤⠀⠀⠀⠀⠀⢸⠟⢁⣤⡒⢤⠀⢹⣇⢿
⠀⠀⠀⠀⠀⣠⠾⢓⣼⠋⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠁⡆⠀⠀⠀⠀⢬⡉⠀⠀⠀⠀⠀⠀⣰⠃⠀⠹⡆⠀⢸⣿⢸
⠀⠀⠀⠀⡼⢣⡾⠛⠉⠳⣌⠉⠉⠁⠀⠀⠀⠀⠀⣀⡠⠶⠛⣩⠞⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠐⢂⠙⢦⡀⠀⠀⠀⣼⢇⡟
⠀⠀⠀⢸⡇⣿⡁⠾⠉⡆⠈⠁⠒⠂⠐⠲⠶⠎⣩⣤⠔⠂⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴⠊⠋⢁⣛⣤⠾⠃⡴⠃⠀⠀⣼⢏⡾⠁
⠀⠀⠀⢸⡟⣿⠃⣶⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢢⡄⠀⠙⢦⠜⠋⠉⠀⠀⠀⠑⢀⣠⡾⢣⡞⠁
⠀⠀⠀⠈⢷⡹⣦⡀⠂⠈⠳⣆⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠴⠴⠤⠠⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠟⣫⠶⠋
⠀⠀⠀⠀⠈⠳⣌⠻⣦⣀⣀⠘⠀⠻⠀⣡⡄⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⡾⣣⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠦⣭⣛⣛⡛⢿⣷⡉⣻⠌⠳⣤⡞⢉⡦⠀⠀⠀⢠⣒⣈⣶⣤⠀⠀⠀⠀⠀⠀⣠⣾⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣎⢻⣄⠁⠀⠲⣄⡴⣏⠀⠀⠀⠘⠿⠿⠻⡿⠁⠀⠀⠀⠀⢀⣾⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡛⢶⣤⣀⡔⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢬⣛⡻⢤⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢏⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣝⢷⣤⣀⠀⠀⢀⣀⣀⣴⡿⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣾⠁⣤⡉⠉⠁⣼⠙⣧⡘⠷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣼⣇⠀⠈⣹⠖⠚⠓⣤⠹⣿⣿⣶⣭⣝⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⣴⣿⣿⠹⠀⣠⠻⣆⠀⠀⢻⡛⠿⣿⣿⣿⣿⣿⣷⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣩⡞⣿⣿⣿⣿⡀⣷⠉⠀⠹⣧⣀⣾⡇⠀⢹⣿⡿⣿⣿⣿⣷⡜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⣿⣿⡘⣿⣾⣿⡇⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⣿⢗⣽⣿⣿⣿⣿⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⣿⣿⣷⠈⠿⣿⣧⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⢸⣿⢻⣿⢻⣿⣷⡿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢻⢿⣿⣿⣧⠹⣷⣿⡀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⡟⣾⣿⢸⣿⣿⣶⢷⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣿⢀⣿⣿⣿⣆⠸⣿⣇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣽⣿⣿⢼⣿⣿⣿⣿⣎⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣏⣾⣿⣿⣿⣿⣆⠹⣿⡄⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣼⡿⠟⠋⠁⢻⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⣿⣿⣿⣿⣿⣆⢨⣗⠀⠀⠀⢸⣿⣿⣿⣤⣤⠿⠋⠁⠀⠀⠀⠀⠀⢸⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡆⠀⠉⠉⠁⠸⣿⣿⣿⣆⠻⣷⣀⡀⠼⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⣷⣦⣤⣤⣤⣤⣿⣿⠷⠀⠻⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⠅⢒⡀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⣛⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣀⣼⣦⡶⠶⠶⠶⣛⣛⣛⣯⠭⠷⠞⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⣛⣛⣫⣭⠴⠶⢮⣭⡵⠶⠞⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣷⡿⠿⣿⣿⣿⣿⠿⣻⣿⣯⣵⣾⣿⡆
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣯⣝⣻⠿⠏⠀⠀⠀⠉⠀⠈⠐⣦⠀⠹⣯⣶⠿⢟⣛⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⢲⣟⠿⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢲⣀⣀⣀⣀⣰⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣤⡼⠿⢷⣶⣶⣿⣿⠛⢻⣟⣋⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣾⠿⢻⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠠⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⢛⣛⣛⣛⣛⣿⣿⡇⠀⠀⢰⢸⣿⣿⣿⣿⣿⣛⡛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠚⠃⠀⡀⢸⣾⣿⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⣶⣿⣿⣾⣿⣿⣷⣀⣀⣀⡀
⠤⠤⠤⢤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⡤⠤⠤

⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⢀⣼⣇
⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣤⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⢰⣷⣿⠇⠀⢻⣷⡆⠀⠀⢰⣷⣿⠿⣿⣿⣆⡄⠀⠀⠀⠀⢻⣿⣿⣿⣶⣇⠀⠀⠀⠀⢸⣿⣿⠻⣿⣶⠀⠀⠀⠀⡄⣿⣿⠏⢿⣿⡆
⠀⠀⠸⣿⠁⣰⣦⠈⣿⡇⠀⠀⢸⣿⠿⠀⠘⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠐⣶⡿⠁⡀⠛⣿⠀⠀⠀⠀⣿⡿⠁⠀⢸⢿⡿
⠀⠀⠀⠙⠦⡙⢛⠴⠋⠀⠀⠀⠀⠻⠄⠀⠀⠠⠟⠀⠀⠀⠀⠈⠻⢿⣿⡿⠟⠁⠀⠀⠀⠀⠹⠄⢾⡿⠀⠋⠀⠀⠀⠀⠙⠧⠀⠀⠀⠜⠁

⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣿⣷⠀⠀⠀⠀⠀⠀⣆⢸⣿⡄⡄⠀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀
⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀⠀⠀⠀⢠⡄⣾⣿⢿⣿⣀⡄⠀⠀⠀⠀⣶⣻⣿⣷⢀⡄
⠀⠀⠀⢸⣿⣿⣿⣇⣷⠀⠀⠀⢈⣿⣿⠏⠻⣿⣿⣶⠀⠀⠀⣼⣿⣿⠋⢻⣿⣧⠀⠀⠀⣿⣿⣿⠟⠈⣿⣿⣿⠀⠀⢰⣦⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣷⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⡇⢀⣦⠀⣿⠇⠀⠀⠀⢻⣿⠁⠀⣼⢿⡟⠀⠀⠀⠸⣿⠉⢀⣆⠉⣿⠏⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⠛⠻⠟⠛⠁⠀⠀⠀⠀⠈⠁⠘⠟⠁⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠛⠟⠀⠉⠀⠀⠀⠀⠈⠙⠚⠛⠛⠋

⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⢀⢀⣼⣿⣷⡄⠀⠀⠀⠀⡀⠈⣿⣧⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⠀⡀
⠀⠀⢀⣿⣿⣿⢿⣿⣿⡄⠀⠀⢀⣿⣄⣽⣿⡟⢀⡀⠀⠀⠀⡀⢻⣿⠛⢿⣿⣆⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⡀⣿⣿⣿⣾⣿
⠀⠀⠸⣿⡟⠉⠀⠹⣿⡇⠀⠀⢸⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⣿⣾⠋⢠⡀⢻⣿⠀⠀⠀⢠⣸⣿⡏⠿⣷⣾⡆⠀⠀⢸⣿⣿⣿⠛⢿⣿⣿
⠀⠀⠀⠻⣧⡀⠀⣰⠟⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⢿⣄⣿⣧⡼⠃⠀⠀⠀⠀⠻⡏⠁⠐⢻⡟⠀⠀⠀⠈⢿⡿⠷⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⢦⠀⠄⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠠⡀⠀⠂⠀⠀⠀⠀⢀⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡔⢀⡀⠀⠀⡼⠀⠂⠀⠀⢀⠍⢻⠆
⠀⠀⢸⣿⣶⣤⣤⠀⠀⠟⠀⠀⢀⠔⠊⢶⡴⠋⠝⠀⠀⠀⣷⣦⣤⣤⣴⣾⠀⠀⠀⠪⠙⠆⠃⣰⠃⠀⠀⠀⠀⠀⠁⡞
⠀⠀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠈⡁⢀⡤⠞⠁⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠈⠁⠀⠀⢀⣣⡀⣠⠟⠉⠀⠀⠀⠘⠀⢃⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣮⣀⡀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢉⡉⠉⠀⠀⠀⠀⢹⣦⡀⠀⠈⠉⠛⠲⠦⣶⡿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡗⠀⣄⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠀⠀⠐
⠀⠀⠀⠠⠀⠀⠀⠀⠰⣿⣦⡀⠀⠀⠀⠀⠛⠃⠀⠀⠁⠄⠀⠀⠜⠀⠀⠀⠩⠵⠀⠀⠀⠻⠀⠰⠀⢢⠀⠀⠀⣴⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⣾⣿⢿⡆⠀⢠⠀⡈
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡂⠀⢰⢦⠴⣶⠙⢰⠳⡄⡏⡔⠱⢰⡆⠐⠀⠀⠀⢠⡄⠀⡠⠀⢐⡇⠀⠀⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⡞⠂⢸⠀⡇⡇⡇⢠⣸⢼⠀⠀⠀⠀⠰⠀⠁⠽⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⠒⠚⠋⠐⠓⠘⠊⠋⠈⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣖⣲⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠁⠀⠐⠈⠁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⢄⡴⠊⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⡜⡖⠤⠤⢤
⠀⠀⡀⠀⢀⡋⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠁⠀⠀⠀⠀⠀⠈⠑⢒⣂⠛
⠀⠀⠈⠒⢭⡰⢂⡌⠰⣀⠄⠀⠀⠀⠀⠀⠀⠀⣤⣀⣤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⡤⢈⡥⠊⠁
⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠐⠅⠈⠋⠁⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠰⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠁⠀⠀⠈⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⡖⣋⠐⠄
⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⣰⣄⠙⢦⠀⠀⠀⠀⠀⠱⣀⠠⠌⣿⠻⠗
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⢄⣀⡀⠀⠀⠶⠆⠀⠀⠀⠈⠙⠓⠀⡐⠀⠀⠀⠀⠀⠘⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀
⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣇
⠀⠀⠀⠀⣛⣿⣿⣷⣾⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⣿⡀⠀⠠⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⡿⠃
⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠾⢛⣷⣤⣾⣿⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⡆⣾⣿⡿⢟⣩⠟
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⣤⣤⣾⡿⢿⡀⢸⣿⡆⢀⠐⡀⠀⠀⠀⠀⠀⢀⢠⣿⣿⡿⢃⣿⠥⠚⡉⡄⢘
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡃⣈⣁⣼⣿⣧⣉⡁⣧⣀⣰⣶⡀⣀⣸⣿⣿⡿⠚⣹⠡⠐⠀⡇⠀⢀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇⢈⣿⣇⢸⡟⠛⠛⠛⣿⣻⣿⡿⣿⠛⠷⠿⣻⠉⠀⠀⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⠛⠃⠀⠀⠀⠀⠐⠿⣿⣿⡇⣿⡷⢀⣶⣽⡇⢀⠀⡄⡄⢸⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠧⣤⣦⠘⡿⢸⠇⠈⠀⣧⣥⣸⠄
⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠘⠛⠃⠀⠀⠀⢧⣬⣿⡇
⠀⠀⠀⠀⣿⢿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⠟⠻⢿⠷⠀⠀⠀⠀⠀⠀⣶⣄⠀⣠⡀⠀⠀⣰⠃⠀⠀⠀⠀⢤⡀⠀⢈⣀⠸⡇
⠀⠀⠀⠀⠛⣶⣷⣞⢩⣧⣿⡿⣿⡿⢻⣿⣯⡴⠆⠀⠀⠀⠀⠀⠶⠿⠶⣾⣷⡀⠹⠿⠋⠀⠹⠟⠛⠇⠶⠆⡤⢷⣄⠀⡹⠃⠄
⠀⠀⠀⠀⢤⡴⣿⣿⠜⢷⡎⣿⣦⣶⣾⡇⢛⡿⣤⣀⡀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠐⢀⣀⣀⣦⠀⠀⠛⠉⠻⡻⣠⠄
⠀⠀⠀⠀⣸⢷⣿⣿⣴⠹⡷⠄⢩⣉⣻⠀⠺⠜⠻⡈⣷⣤⡀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠸⡧⢸⡟⠀⣿⠂⠀⢰⣿⡏
⠀⠀⠀⠀⣻⣼⣿⢷⣟⣰⣿⣀⣾⣿⣿⡏⠹⢋⣼⣇⣼⣿⣀⠦⠄⠀⠀⣿⣿⠟⠀⠀⠀⠀⣀⣴⣷⢸⣏⣰⣿⡄⢆⢸⣿⡘⠂
⠀⠀⠀⠀⣷⣬⣹⣾⣧⣿⣿⠿⣹⡏⢻⡿⣶⡚⣛⠙⠛⠋⠠⣕⠛⠒⠛⣹⣿⣰⠦⣶⠋⠛⠛⡿⠛⣾⣿⣨⡿⣇⠈⠹⡿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⡡⢊⠹⠇⠈⢀⡿⣞⡻⠿⠯⠚⠛⠛⠓⠒⠒⠻⠏⠉⠉⠛⠚⠃⣰⡗⢰⠇⠀⣌⠙⣿⡇⢹⡇⠀⠁⢺
⠀⠀⠀⠀⣿⣛⣏⣽⣿⣾⣷⣶⣴⣿⣷⣝⣿⣶⣤⣤⡄⢀⣤⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠠⡴⠏⠐⣻⠆⠼⠇⠄⠴⠂
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣉⣽⣿⣿⣿⣿⣿⠛⠁⠋⢟⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠈⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣾⣿⣿⣟⣀⡀⠀⣀⡀⠀⠀⣴⣿⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴⣾⣿⣷⣶⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣾⣿⣿⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⡀⠀⠀⣀⣀⣀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠚⠀⠂⠀⠉⠱⠟⠃⠈⠉⠋⠀⠛⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣞⠉⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡔⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠐⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠁⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣋⡄⠀⠠⠆⠀⠀⠀⠀⠀⡂⣺⠦⠋⠄⠀⠄⠀⠉⢍⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⣀⣑⠂⠀⠀⠀⠀⠾⠟⠻⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣷⠲⠀⠀⠀⠀⠀⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⠂⠐⠒⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠐⠀⠀⠙⠁⡀⡀⠄⣰⠦⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠤⠖⢇⣠⣀⡤⣧⠀⠀⠀⠀⠀⢠⣅⣀⡤⠐⠛⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠑⠊⡀⢿⠃⡀⠀⠀⠀⣿⠉⠑⠋⠉⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣀⣀⡀⠀⠘⠋⠙⠆⢉⠒⠛⠐⠁⢀⡄⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠴⠴⡾⠹⠤⣀⣰⣠⠖⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣀⠀⠀⠀⣠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠤⢤⡖⠒⠖⡇⠀⢀⠀⣿⠈⠑⠒⡋⠁⠈⠉⡦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠕⠀⡠⠒⠊⠉⠀⠔⠓⡴⣉⡊⠁⠃⠀⠒⠀⠀⠀⠐⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠰⠀⠀⠔⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠴⠋⠁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠂⠒⠐⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠈⠒⠒⠁

⣇⠀⠀⠀⠀⠀⠈⢻⣿⢟⣽⣿⡿⠿⢿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣏⠀⣾⣿⣿⡄⠈⠋⢿⣿⣿⡃⠀⣴⣿⣿⣿⣿⣯⣿⣿⠃⠀⠀⠀⠀⠀⢹
⣿⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠫⡼⣿⣿⣷⠞⢻⣸⣿⣿⣿⢶⣷⣽⣿⣾⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸
⣿⠇⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⠃⠁⠉⠁⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⣾
⣿⠀⠀⠀⠀⠀⠒⠛⢿⣿⣿⣄⡀⢀⣀⣤⢰⡇⣤⢸⢹⣿⣿⣷⣦⣄⡀⢀⣤⣿⣿⣿⠛⠒⠀⠀⠀⠀⣻
⣿⠂⠀⠀⠀⠀⠀⡴⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⠟⠻⣦⠀⠀⠀⠀⠀⣹
⣿⠁⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⣻⠿⠿⣿⡿⠙⠛⠿⢿⣿⣿⣿⣿⠿⠿⡏⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⢾
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠋⠀⢀⣤⣄⠀⣹⣿⣿⣿⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣄⡻⠿⢟⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣾⢹⣃⡟⣿⣿⣿⣿⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣼⣾⣇⣿⣿⣿⣿⡀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⢿⣿⡟⣿⣿⣿⣿⢿⣿⣿⣿⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢸⣿⠈⣿⡇⢸⣿⢹⣿⢸⣿⣿⣾⣿⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢃⣾⣿⡀⣿⠇⣼⣿⣾⣿⢸⡇⣿⢹⣿⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠀⣿⣿⡇⣿⠀⣯⣿⣿⣿⣾⡇⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⣿⣿⣿⣿⣇⣽⣿⣿⣿⣾⣷⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠾⠿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠸
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣟⣻⡿⣿⠿⠿⠿⣿⡿⢿⣟⣻⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠶⢮⣭⣭⣭⣭⣭⣭⣭⣭⡵⠶⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡶⠛⠛⢶⡞⠉⠀⠈⠉⢉⣭⣭⣿⣿⡛⠉⠉⠉⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣼⣷⡀⠀⠀⠹⢦⣤⣤⣤⣴⣿⣿⣿⣿⣇⣀⣀⣤⣿⡿⣿⠟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠻⢶⣄⣠⣶⣽⣶⡉⠁⢻⣿⣿⣿⡟⠉⠉⠀⣨⣽⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣀⠀⠀⠀⠀⣩⣛⣿⣿⣇⣠⣴⠿⠿⢿⣻⣿⣶⣤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣙⠻⠶⣦⣤⣿⣿⣿⣿⡟⠉⠀⠀⠀⢻⣿⣿⣿⠇⠀⠉⣉⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣌⣉⡙⠛⠋⣄⣀⣠⣤⠶⠟⢛⣉⣀⣤⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⠀⠈⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⢀⠄⠀⡆⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠰⠳⣄⡎⠀⠀⠁⠠⠀⡜⢆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠈⠆⠀⠀⢀⠀⠰⠁⢸⠈⠰⠃⢰⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡏⠲⢦⠀⠀⠀⠀⠀⠈⢄⠁⠀⢸⠀⠀⠀⡌⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⡘⠘⡐⢚⡇⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⡀⠀⠳⠉⠀⠌⡰⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢤⣄⡀⠀⠀⠀⠀⠂⠄⠐⠂⠀⠀⠁⠄⠂⠀⠀⠤⠊⠉⠈
⠀⠀⠀⠀⠀⣼⠹⣭⢍⣠⢴⠚⢠⣀⣤⡤⡖⠢⣤⢼⣐⠒⣦⠀⢲⢢⡀⠘⠛⣻⣻⡄⢠
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠻⠼⠛⠛⠊⠳⣾⡴⠃⠰⣌⣠⡼⠷⣤⣸⡖⠲⠮⠯⠟⠛⠛⠃

⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢇⠠⡆
⠀⠀⠀⠀⣒⡶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⠀⠀⠀⠻⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠑⢸⡀
⠀⠀⠀⠀⢸⢛⣱⡆⣤⡀⠀⠀⠀⠀⠀⢼⣲⡀⣀⠠⢊⣹⡆⠀⠀⠀⠀⠀⠀⢲⣿⡖⠀⠀⠀⠀⠀⠀⠀⠁⠈⣴⣶⠀⢹⡆
⠀⡀⣾⣿⡿⣻⣻⣛⡀⠀⠀⠀⠀⠀⠀⠈⠉⢣⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⢿⠀⢹⢱
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⣀⠀⠀⠀⠀⠀⢀⠀⣘⢇⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⢀⠞⢿⡿⠀⠀⠀⠀⠀⠀⠋⢿⣿⠙⠃⠀⠀⠀⠀⠀⢀⡀⢞⣻⠇⠵⠒⠂⠀⠀⠀⠀⠀⠀⠀⣼⡃
⠄⠀⢀⠀⡸⠁⢠⠞⠂⣵⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠼⠧⠿⡿⡿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⣷⠈⣀⣠
⢠⣀⣙⣁⣛⣛⣋⣀⣀⣟⣗⡀⠀⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡄⠀⠀⣈⣋⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡶⠀⠀⢀⣛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠠⡖⠒⠊⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠙⠃⠐⠒⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣂⡀
⠀⠀⣯⣤⠀⢸⢿⠉⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⣾⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⣂⠀⠀⠀⠀⠀⢀⣷⣿⢦⡆
⠀⢿⣿⡟⣇⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠶⠶⠾⢀⡀⠀⠏⡇⠀⠀⠀⠀⣰⣷⣼⣿⣷⣧⣿⣿⡀⠀⠀⠀⠀⠸⣿⣿⠋⡓⠂
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡭⢉⣳⣤⡀
⣿⠻⠿⣟⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⠛⠛⠛⠛⠛⣋⣾⣿⣿⡿⢻⠛⠛⢳
⣿⢰⠖⠤⠤⣈⣉⡓⢲⡄⠀⠀⢀⡴⢚⠉⠈⠻⡍⠙⠲⢦⣤⣴⠙⠛⠀⠈⠑⣿⠀⠀⢸
⣿⢸⠀⠀⠀⠀⠈⣿⢸⡇⢀⡴⢋⡶⣯⣳⠀⠀⠀⠀⠀⠀⢹⠈⡆⢸⣧⣀⡼⠋⠀⠀⢸
⣿⢸⠀⠀⠀⠀⠀⣿⢸⣷⡋⠀⠘⠓⠛⠁⠀⠀⠀⠀⠀⠀⠈⣧⢿⣿⣧⡈⠀⠀⠀⠀⢸
⣿⢸⠀⠀⠀⠀⠀⣿⡼⡿⠁⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡝⣄⠀⠀⠀⢸
⣿⠘⠛⠿⠶⢶⣴⣿⣽⠃⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⡯⢗⣂⠈⣧⠀⠀⢸
⣿⠀⠀⣀⡤⣴⢲⢷⣿⣀⡀⠀⣾⡿⡸⢀⡀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣭⣴⣶⢸⠀⠀⢸
⣿⠀⠀⣒⣿⠏⣼⢸⣿⠖⠂⠀⣿⢣⡇⣸⠀⠀⠀⠀⠀⠀⣼⡿⠟⠉⠉⠛⠛⠾⢦⣀⣸
⣿⠀⠀⠀⠀⠀⠸⡎⣿⠀⠀⠀⣿⢸⠇⡟⢸⠀⠀⠀⢀⡼⠋⠀⠀⠀⣠⠒⠒⠦⡀⠀⠉⠑⠢⢄⡀
⣿⣿⢰⠞⠓⢠⡶⢿⡸⣆⠀⠀⢹⣼⡴⢗⣯⠟⢝⡤⠊⠀⠀⠀⣠⠾⠁⠘⠀⢠⣾⡆⠀⠀⠀⠀⠈⠑⠢⣀
⣿⠿⠘⢦⣀⢸⣟⡛⣧⣻⣷⣴⣾⣵⣾⣷⣿⣠⢎⣠⡖⠚⠓⠲⣽⣄⣀⠀⠐⣟⡿⠁⠀⠀⠀⠀⠀⠀⣠⡞
⡿⠀⠀⠀⠀⢀⣩⣵⣺⣵⣾⡇⠀⣿⣿⣿⣿⠗⣻⠿⢤⣄⡀⠐⢌⣉⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢀⣾⠏⢁
⡗⠀⣀⢤⡺⠟⠊⠁⢸⣿⣟⡥⠤⢿⣟⣡⠾⠋⠁⠀⠀⠈⠙⢦⡀⠀⠉⠉⠳⣄⠀⠀⠀⠀⠀⣴⡿⠁⠀⡼⢄
⣧⢾⠟⠁⠀⢴⣒⡟⣸⣿⣿⣆⠀⣰⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⣼⠃⠀⠀⣠⣾⠉⠀⢠⠞⣠⣾
⡇⠃⠀⠀⠨⠽⢞⡣⠟⣫⣾⣝⠋⠁⠀⠉⠓⢦⡀⠀⠀⠀⢀⡴⣿⣲⡄⢀⣴⡃⠀⣠⣾⠏⠁⢀⡴⣣⣾⠟⠁
⡇⠀⠀⢀⡠⠖⢉⡴⠛⠉⠀⠉⠙⠶⣄⠀⠀⠀⢳⡤⢔⣾⠗⢮⣁⠈⣾⡏⠀⣹⣾⠿⠉⠀⡰⢋⣴⡟⠁
⣇⡤⠖⢉⡠⢚⣡⠔⠉⠀⠀⠀⠀⠀⢈⣷⢴⣾⣵⣾⣟⣁⣤⣾⣯⢿⣿⣠⣾⢿⡇⠀⡠⢊⣴⡿⠋
⡏⣠⢖⣭⠖⣫⠔⢊⠀⠀⠀⣀⢴⣺⡿⠞⠋⠁⣈⣩⣽⣿⡿⠟⢁⣴⣿⡟⠋⢀⡷⠊⣠⡿⠋
⣿⣿⣫⠶⣋⠴⢊⠡⢀⢔⠽⣪⠕⠉⣠⡤⠶⠟⠻⣏⢉⣠⣴⣿⠻⠻⣿⣀⡴⠋⣠⣾⢿⠁
⣿⣫⡵⣋⡵⠚⠁⢰⡷⠁⣼⣿⣶⠞⠀⠀⢀⣠⣴⢿⠟⠃⠃⠀⣀⣴⠟⠉⢀⣾⠟⠁⢸
⣿⡵⢚⡥⢒⠁⠀⣸⡷⠊⠛⠛⠁⣠⣴⡞⠟⠋⠉⢸⣀⡤⠖⠋⠁⣀⣬⣿⠿⠃⣀⡴⢿
⣿⣚⡵⢚⡡⠔⠋⢹⣇⣀⡴⡶⠟⠙⠘⠀⢀⡠⠔⠋⠁⢀⣤⣶⡿⠛⠉⣠⡴⠛⠉⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠤⠤⠤⠤⢤⢤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⣀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠉⠉⠉⠉⠈⠹⢿⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠤⠤⠤⠤⠶⠾⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠏⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⡌⠐⢲⡶⠲⠂⠒⠒⢒⣾⢻⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⡔⠺⣾⡟⢀⣦⣀⡸⠟⡀⣸⠀⠀⠀⠀⠀⠀⠀⣥⣿⡆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⢈⣿⡋⢁⣼⡁⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠓⠲⣶⠶⣾⠟⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠃⠀⠀⢤⣧⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢙⡆⠀⠀⣞⠉⣷⠀⠀⣚⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠈⠋⠁⣀⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢸⡇⣼⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⢀⠀⡤⠠⡄⠀⠀⣠⠒⢠⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⠉⠉⠉⠉⠉⠉⢸⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠼⠃⠀⠀⠢⠴⠃⠀⠀⠙⠶⠞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣋⣸⡇⠀⠀⢨⣀⣵⠀⠀⠺⣤⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠐⠀⠀⢠⣤⣤⣤⣤⣤⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡄⠀⠀⣔⠀⣆⠀⠂⢠⡁⣰⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢹⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠙⠛⠃⠀⠀⠈⠛⠋⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣈⣀⣀⣼⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁

⣷⠛⠛⣶⢶⡶⣶⠶⠶⣾⠶⢶⠶⠶⣾⠷⠶⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⢶⡶⡶⣾⢿⣿⣿⠾⠿⠷⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⣠⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠉⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠀⠀⢿⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠙⢦⣌⢷⣄⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠙⣆⠙⠷⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢛⡓⠦⡀⠈⢷⡀⠀⠀⠀⣿⠀⠀⠘⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⡄⠀⠻⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⣰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡆⠀⣠⠟⠀⠀⠀⠹⣄⡀⢻⠀⠸⣇⠀⠀⠀⠀⠀⢰⡇⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⠟⢰⡏⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠘⢷⣄⠀⠀⠀⠀⠻⣄⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⡉⠀⠀⠻⢦⣄⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠉⠳⢦⣄⠀⠀⠈⠳⡄⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠀⠀⠈⠉⢻⠓⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠃⠀⠀⠀⠀⢠⣸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⢠⡞⠀⠀⠀⢀⣤⠟⣿⠃⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠀⠀⢹⡆⢀⡤⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢁⡖⠉⠀⣀⡴⠞⠋⠀⣰⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠘⡇⢸⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠘⢧⣠⠞⠁⠀⠀⠀⠘⠁⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢰⠇⢸⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠁⠀⠀⠀⠀⠀⢠⢀⠞⠁⠀⣰⠋⣠⡾⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠃⢀⣠⠄⣀⣴⠆⢀⠀⠀⠀⠞⠀⠀⠀⠀⠟⠉⠁⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠰⣶⣤⣛⠀⠈⢁⣾⣿⣷⣿⣿⠃⠀⠀⣀⠀⠖⢀⣀⡀⠀⠀⠀⠀⠀⣰⣯⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⣍⣻⣦⣄⡀⠀⣀⠐⣀⣈⠛⠻⣟⣻⣿⣿⡿⠟⣋⠀⠀⣠⣤⣤⣤⡶⠿⠛⠁⠀⠀⠀⣀⡼⠯⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠴⠶⠛⠻⢶⣤⣄⣈⣁⣰⣿⡿⣯⣥⣤⣀⡛⣿⣿⣶⣬⣍⠀⠻⠷⠶⢶⣆⣤⣴⣾⣿⣶⣶⣶⠤⠴⠶⠖⣶
⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⣤⡤⠤⠶⠶⠚⠋⠉⠉⠉⠉⠁⣀⡤⣿⣿⣿⣿⣿⣿⣿⡿⠿⢶⠲⠶⣿⣿⣭⢤⣭⣍⣛⡛⠛⠶⠦⢤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠋⢑⣻⡿⠟⠋⢁⣀⡈⠉⠉⠉⠋⠉⠛⠒⠾⠭⣅⣀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⣯⣉⣀⠤⠖⠚⠉⠉⠀⠠⠖⠛⢉⣀⡙⠚⠙⠛⠶⠶⢦⣄⣠⣤⡴⠶⠯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠋⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢖⡊⠀⢩⠟⠃⠀⠀⠀⠀⣶⠀⠀⠀⠀⣝⡠⣄⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠲⢼⡆⠀⠀⠐⣖⠚⠀⠳⣤⡄⠘⠋⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠂⠀⠀⣸⡴⢦⡏⠁⢀⡄⠀⣀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣄⠀⠀⠀⣄⠀⠁⠀⢀⣠⡤⠤⠭⠤⣬⣅⡀⠀⠀⠀⣤⡤⠞⠘⢓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡍⣀⢩⠟⠋⠀⠀⠀⢀⡞⠋⠙⠒⠶⢤⣄⣀⣠⠽⠷⣄⠀⢀⣏⠴⣜⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⠻⠀⠠⠄⠀⢠⠞⠉⠓⠦⣄⣠⠴⠋⠉⣀⣤⠷⡛⢦⡈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢀⡟⠒⣦⡤⠞⠉⣀⡤⠖⠋⠁⠀⠀⢉⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣞⣁⣤⡴⠿⠥⠤⠴⠶⠶⠶⠶⠶⠶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣒⣩⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⣄⣁⣤⣤⣤⣤⠤⠤⠤⠶⠖⠛⠉⠁⠀⢸⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠒⣤⠀⠀⢀⣤⣴⠂⠀⠀⠀⠈⢻⢲⣄⠀⠀⠀⡴⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠂⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠱⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠲⡆⠀⠀⠀⠀⠀⠐⠛⠓⠀⠀⠀⢠⡆⠀⢀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⠃⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠖⠛⠉⠉⠉⠉⠃⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢲⣤⣄⣀⣀⣀⣠⠞⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡌⠴⢋⡩⢋⡽⢋⠞⣹⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⠖⠛⢧⠻⣄⠀⠀⠈⠠⠎⡴⢣⠞⠵⣠⠏⣹⠓⡶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠳⣎⡌⠳⣌⡓⠦⢤⣄⣀⣤⠤⠶⢚⣡⢾⣡⠞⠁⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠙⠻⣤⣝⣛⡶⢲⡶⠶⡶⣚⣿⠵⠋⠁⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢠⠖⣀⠄⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⢀⠀⢀⠀⢀⠔⢀⡴⠀⡠⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣠⠞⢁⠔⢁⡔⠁⣠⠆⣠⠎⣠⠞⢁⡴⠋⡴⠃⡰⠋⣠⠊⣠⠎⣠⠆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠴⠃⡴⠋⠀⠞⠁⠞⢁⠞⠁⣠⠋⢠⠞⢀⡞⠁⡼⠁⡰⠃⡴⢁⠔⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠇

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⢠⣄⠀⠀⢸⣿⡇⢾⣿⣦⣰⣿⡿⠀⠀⡟⢻
⠀⠀⠀⢀⠀⡘⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⣤⢄⢿⣿⠇⠀⢸⣿⡇⠀⢹⣿⣿⡟⠁⠀⠀⡇⢸
⠀⠀⢀⣾⡞⠀⠀⠀⢀⠔⢲⠀⢀⠠⠶⠂⠀⠀⠸⢿⣿⣶⡄⠀⠀⢸⣿⡇⣰⣿⡿⢿⣿⣦⠀⠀⣧⣼
⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠀⠀⠸⢿⠇⠘⠛⠀⠀⠙⠋⠀⠀⢿⡿
⠀⠀⠁⠀⠀⠀⢀⡀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣠⣔⣂⠁⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠆⢠⡀⢸⡇
⠀⠀⢠⠤⠤⡄⠀⠁⠀⠀⠀⠀⠀⣴⡾⠛⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠁⠔⠀⢥⡴⢸⠁
⠀⠀⠈⠈⠉⠁⠀⢈⠓⠶⠚⠇⢸⠋⣠⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠐⢃⠠⣴⠌⠁
⠀⠀⠐⠢⠤⠀⡆⠀⠀⢱⣶⠂⠀⠀⠫⠊⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⣤⠲⠆⢽⠆⠁
⠀⢀⡅⠀⠀⠀⡇⢈⠀⢘⠦⠀⠀⠀⢰⣶⣤⣤⣤⣴⣾⠀⠠⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⢀⡀⠒⡀
⠀⠈⠢⠶⠒⠚⠁⠀⠒⠺⠕⠀⠀⠀⠈⠻⣿⣿⣿⠿⠃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳
⠀⡀⠐⢀⠀⡄⠀⠀⠀⠰⢄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠃⠀⠠⠒
⠀⠻⠶⢎⣠⡼⠀⠀⠙⠒⢫⡄⠐⠊⠀⠀⠀⠀⠀⠐⢄⠀⠶⣦⣶⣄⡊⠀⠀⠀⠂⠀⠉⠁⢀⠀⠁⠘⢸
⠀⠰⣤⣤⠁⡀⠀⢀⠀⠘⢒⠀⢤⠄⠀⢀⡘⠛⢀⣿⠈⠀⠐⣫⣿⣿⡿⠀⠀⠀⣀⣤⣴⣦⠀⠀⠀⠅⡇
⠀⠀⠀⠀⠀⠙⠛⠫⣄⠠⠿⡗⠀⠀⠀⠘⠻⠖⠈⠁⢀⡀⠀⠀⠚⠁⠀⠀⠻⣿⣏⣹⣿⡟⠀⠀⢨⣽⠇
⠀⠀⠀⠠⠀⠀⠀⠀⠈⠳⡄⠈⠲⣄⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⢰⣶⠆⠲⠿⠟⠙⠙⣷⣶⡄⣼⣿
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⢢⡖⠘⢷⣄⠀⡾⠟⠉⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡀⠀⠀⠈⠀⠇⢠⢐⡴⠮⠀⠋⢸⡏⠉⢹⠀⠀⡠⣿⣄⠀⠀⠸⣼⠃
⠀⠀⠀⠀⢄⠀⠀⠀⠀⢸⣏⣿⣹⡇⠀⠀⠰⠬⣕⠻⠦⣵⡚⠣⢾⡇⠀⢸⠀⢰⣵⣿⣿⠀⠀⢠⡏
⠀⠑⠒⠒⠙⠿⠶⠿⠃⠈⠁⠉⠈⠁⠀⠀⠀⠐⠊⠠⠄⠒⠀⠤⠘⠿⠿⠟⠀⠀⠙⠛⠋⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢋⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣭⣛⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣴⠞⠛⠉⠛⠳⣭⡙⠢⠄⠀⠀⠀⠀⠠⠊⢁⣴⠖⠛⠉⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢹⡄⢰⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠸⣿⣶⡆⠀⠀⠀⢸⡇⢸⠁⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⠿⠁⠀⠀⠀⢸⡇⢸⡀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣄⡤⠞⢋⡠⠀⠀⢠⡙⠷⣤⣤⣤⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠖⠋⠀⠀⠀⠀⠉⠓⠶⠦⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢠⠖⡏⠙⡷⢄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣿⣄⣧⣠⣇⣾⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⣿⣿⣿⣿⣿⣿⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⡟⢻⠉⠉⢹⠛⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢌⣆⣀⡧⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣿⣿⣿⣟⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⣛⣛⣛⣻