Soup can Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣧⣴⣶⣶⣦⣶⣀⣦⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠟⠛⢷⣶⣶⣶⣾⣭⣾⣶⣦⣄⠀⠒⢤⣀
⠀⠀⠀⠀⠀⠀⣸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣶⣶⠶⠶⠾⠿⠿⠿⠿⠶⡾⠿⣿⣿⣟⣛⠷⢶⣭⣭⣿⡷⢦⡈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠢⡙⢧⡉⠻⣆⢻⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⣇⣘⣧⣤⢞⡩⠞⠹⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠶⠤⢤⣤⣤⣤⣤⣤⠤⠤⠤⠶⠶⣞⣛⠛⣫⣭⠭⠷⢚⣭⣤⣶⡅
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣄⣀⣀⣙⣛⣛⣛⣛⣛⣛⣛⣛⣭⣭⣭⣤⣶⣶⣶⣿⣿⣿⠿⠋⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⡀⠀⢀⣀⣀⡀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⣯⠞⢫⣿⡿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣴⡶⢦⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣣⣿⠉⣀⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠃⣼⢰⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢰⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⣿⣿⣽⣯⠇⣸⢇⣿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⢻⡿⣱⡿⠃⣸⡿⢉⣿⠉⣿⠛⣻⡿⢁⣟⠛⣿⡟⣱⣾⠙⣎⠈⣿⡿⢰⠏⣾⣷⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠑⣿⠃⣴⠟⢁⣾⠃⣼⠃⣰⡿⢃⣾⡟⣰⡟⣸⣿⢃⡾⢫⠖⡟⠁⡟⠐⣛⣋⣨⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⣾⣏⣼⣯⣀⡿⢁⣼⣏⡴⢹⡇⣛⣥⡞⢧⣬⠟⣧⠶⠛⠚⠉⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣼⣿⣿⠟⠈⠛⠛⠉⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⢿⣿⣿⣿⣷⡾⢿⣿⣧⣴⣶⣶⣶⣶⣶⡾⣿⡟⡛⣏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣾⣿⣿⣷⣿⣙⣿⣿⡏⣿⣳⣿⣤⣿⠷⠶⠿⠋⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣾⣿⣿⣯⣅⠈⠀⠀⣼⣷⣾⣿⣶⣤⣤⣴⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣷⣾⣿⣿⣿⣿⠖⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣤⣼⣎⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⡿⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢻⡿⠃⣴⠟⢶⡄⠀⢠⣤⡀⢀⣤⡄⠀⢠⡤⢴⣀⣶⠒⣺⣦⣾⠷⣽⡄⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⣿⡀⢸⣷⠀⣾⣿⣧⣼⣏⣿⠀⡾⢶⡆⣧⠈⡇⢹⠸⣟⣶⡿⠁⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠀⠀⠘⠿⠾⠃⠀⣿⠟⣿⣿⢿⣿⢰⣧⠖⣦⣼⠌⠧⠜⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠲⣦⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣏⣰⣿⠿⡆⢠⣾⣿⣿⣷⣆⠀⢶⣾⣿⢿⣿⡧⢺⣿⡾⢃⣰⠇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢠⡿⣟⣿⣿⣆⢸⣿⣿⣟⣿⣿⠀⢸⣿⣿⣸⣿⡇⢸⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⡀⠀⠀⠀⠀⠈⠷⠿⢿⡿⠋⠈⠻⣿⣿⣿⠇⠀⠘⢿⣿⣿⡿⠃⠘⠟⠛⠁⠀⠀⠀⠀⠀⣠⣾⠟⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣍⠻⢿⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠿⢟⣥⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣬⣍⣛⠛⠛⠿⠶⠶⠶⢶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣴⠶⠾⠿⠛⢋⣉⣤⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣤⣥⣍⣉⣉⣭⣭⣭⣤⣤⣶⣶⣶⠶⢾⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠷⠾⣿⣦⣭⣭⣭⣭⣭⣭⣭⣥⣴⡶⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡀⠀⠘⠛⠃⠈⠛⠛⠛⠛⠃⠘⠻⠷⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠈⠡⠚⡄⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠡⠄⠦⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣨⣴⡶⠦⠀⠀⠀⠀⠀⠤⠤⠄⣀⣤⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣿⣛⣿⣷⣶⣶⣶⣶⣿⣟⣛⣻⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢫⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡿⣻⢻⠟⠛⠛⡿⣛⢻⢫⣜⡋⡿⡽⡿⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣾⣇⣏⣼⣼⡹⣱⣫⣞⣻⣦⣹⣧⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡿⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣾⣯⣴⡿⠛⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⢻⣿⣿⣿⡶⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠭⠛⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢰⢳⡄⣶⣰⡆⢰⣶⠘⡟⢸⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠁⠛⠛⠃⠚⠙⠃⠋⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⢠⡶⣶⡄⣶⠲⡆⢺⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⣀⠀⠀⠛⠟⠉⠿⠾⠃⠹⠿⠃⠘⠁⠀⢀⡠⠖⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡉⠓⠂⠤⠤⠤⠤⠤⠤⠤⠤⠖⠒⠋⡁⢰⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠐⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠟⠛⠋⠉⠉⢉⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣯⣭⣭⣟⣛⡿⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣫⣵⣶⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣽⣛⣛⣻⠿⣿⣿⣿⣿⣶⣭⡻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣭⣟⣛⡻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣛⠛⠋⠉⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣮⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣛⣵⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣶⣶⣾⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣶⣶⣦⣤⣴⣿⣿⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠏⠉⣉⣛⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⣛⡉⠛⢿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣧⠠⡙⠿⣦⣿⣿⠋⠀⣶⣦⠉⠻⣿⣿⡍⠉⢱⣿⣷⡆⣶⣿⣗⠀⢸⣿⡇⠃⣼⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠋⣷⣮⡠⡙⢿⡇⠇⠀⣿⣿⢀⡀⢹⣿⢇⡄⢸⣿⣿⡃⣻⣿⣇⠆⠈⣛⣥⣾⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣄⣙⠻⠟⠁⣸⣷⡀⠀⣿⣿⠀⠀⣼⣿⡇⠀⢸⣿⣿⡇⢿⣿⡿⠀⠸⢿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣭⣭⣤⣾⣿⣿⣷⣤⣈⣉⣭⣶⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣟⢻⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⢟⣫⣾⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣩⡾⠟⠑⠿⢿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⡁⠀⠀⢀⣠⣤⣶⣶⣶⡿⠿⠿⢿⣿⣿⣿⠿⠶⠶⢾⣭⣤⣀⠀⠀⠀⣹⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣷⣦⣭⣝⣛⣛⣻⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣛⣛⣛⣯⣥⣶⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢁⣴⣿⣿⣿⣿⠁⠀⠈⣿⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣝⡻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣛⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣯⣭⣭⣽⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣯⣭⣭⣽⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢶⣤⡀
⠀⠀⠀⠀⢰⣻⣛⣛⣻⣿⡆⠀⠀⠀⠀⠀⠀⢧⡘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢧⡀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⢸⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣀⣉⣈⣁⣀⠀⠀⠀⢀⣀⣀⣀⣈⣀⣁⣀⣀⡀⠀⠀⣷⣷⣶⣾⣿⣿⠟⢻⡇
⠀⠀⠀⠀⢸⠘⠻⣶⠙⠛⡇⠀⠀⠀⢸⣯⣤⣤⣤⣤⣤⣬⡷⠀⠀⠻⣼⢹⣿⣿⣿⣿⣿⡧⠇⠀⠀⡟⠿⣿⡿⢿⡇⠖⢺⠇
⠀⠀⠀⠀⢸⠐⠖⠴⠴⠂⡇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⡇⠄⠬⠠⢼⡗⠋⠉
⠀⠀⠀⠀⠈⠓⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠙⠒⠒⠒⠊


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣧⣤⣤⠀⠀⠀⠀⠀⣤⣤⣴⣧⣤⣤⡀
⠀⠀⠀⣴⡒⠛⠉⠉⠙⢳⣫⡄⠀⠀⢠⣖⠚⠛⠻⠋⡟⣒⣦⠀⠀⢴⣻⣿⣿⢿⡿⣿⣿⣟⡆⠀⢰⣻⣿⣿⣿⣿⣿⣿⣟⡦
⠀⠀⠀⢻⣻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⢹⣿⣇⠀⠀⣸⣿⡏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠑⠿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠘⠺⢿⣿⣿⠿⠋⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⠇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⣴⣊⣉⣉⣉⣉⣑⡦⠀⣠⣶⣛⣿⣛⢦⡄⠀⣤⣄⠀⠀⢀⠲⠚⠊⠑⠑⢣⡀⠀⠀⠀⠀⠀⣤⢀⣄⣠⡀
⠀⠀⣠⠤⠤⠼⠯⠽⠿⡻⣿⣿⠇⢰⣯⣿⣿⣿⣿⣷⣹⡄⢿⡟⠀⢀⡏⠀⠀⠙⠑⠀⠀⢳⠀⠀⠀⠀⠀⣧⠈⠻⠿⠃
⠀⠀⢻⣷⣶⣶⣶⣶⣾⡗⠛⠁⠀⠸⣟⣿⣻⣿⣿⡿⣹⠇⢸⠀⠀⠈⣇⠈⣷⣶⣶⣾⠃⡼⠀⠀⠰⣒⡂⡠⢄⢠⢀⡀⡤⣄
⠀⠀⠈⠻⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠙⠾⣽⣿⣭⠾⠋⠀⠘⠀⠀⠀⠈⠳⣬⣉⣉⡥⠞⠁⠀⠀⠐⠶⠃⠓⠚⠘⠞⠣⡗⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⠚⠛⠚⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠒⠒⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⢀⣄⡂⢽⣿⡿⢿⣿⣲⠶⠮⢟⣛⣛⣒⡀⠀⠀⠠⠤⠤⠤⠤⠬⠛⠛⢛⣋⣩⠤⠶⠛⣻⡿⠿⣿⣿⡷⠶⣠⢀
⠀⠀⠀⠀⢠⠏⣾⠗⣫⡶⠛⣋⣥⡶⠖⣂⣨⣭⠶⠶⠖⠚⠛⠛⠋⠙⠋⠛⠛⠒⠲⠶⠶⣤⣬⣉⠑⠲⣬⣙⠳⢤⡁⠒⢼⣷⡝⡄
⠀⠀⠀⠀⠸⣆⢿⡘⠋⠀⢰⠉⠋⢠⢺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢖⢠⠉⢣⣠⠙⢾⣆⢹⢏⡾
⠀⠀⠀⠀⠀⢹⣷⣮⣑⣂⠸⣆⢳⣄⢘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠸⢃⣊⣼⠤⠞⡋⢥⡟⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣬⣶⣶⣭⣉⣛⠛⠛⠶⠶⠆⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⠶⠶⠛⠁⣛⣩⣭⣤⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣍⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣩⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⡿⠟⣩⣿⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⡾⠿⢛⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣿⣿⣿⣿⠃⣾⢱⣧⣾⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡏⢀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⣫⣾⣿⣿⣿⠏⣼⢇⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠁⣼⣿⡌⣿⠟⣩⡝⢛⣿⡿⠿⠟⢿⡿⠿⣿⣿⠋⠿⠿⣿⣿⠋⡀⠻⣿⠏⣵⣦⡟⣰⡏⣸⡏⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡄⣿⡿⣸⠇⣼⡟⢀⣾⠟⢀⣶⠂⣴⡆⢠⡿⠃⣴⡶⢀⣾⢀⣾⡿⢀⡿⢆⣹⡿⢠⡿⢰⡿⡿⠸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣾⣿⡀⣛⡁⠸⡏⢀⣾⠇⣸⡏⢠⡿⠁⣼⡿⢃⣼⡇⣼⡿⣡⡟⠰⠿⣟⠀⢛⣁⣨⣤⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣶⣿⣤⣼⠃⣤⣭⣴⣾⣿⣧⣤⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⢿⣋⣝⣿⠻⠿⣿⣻⢿⡿⣻⣿⠿⢿⣿⠛⣿⡟⢻⡏⡭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣿⣭⣾⣧⣤⣿⣷⣶⣿⣭⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⢿⣷⡆⢠⣴⣦⡀⠀⠀⣀⡀⠀⢀⡀⠀⠀⢀⣀⡀⠀⣠⣤⣤⡀⣰⡾⣷⣆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠙⣿⡄⢸⣿⣷⢀⣿⣿⠀⠀⣾⡿⣿⠀⠛⣿⡏⢀⣿⠀⢸⡿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠇⠀⢻⣧⣴⣿⠀⣸⡟⣿⣿⡟⣿⠀⢰⣿⠷⣿⡇⠀⣿⡇⠈⢿⣶⠿⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠙⠁⠘⠛⠀⠛⠀⠚⠋⠀⠘⠋⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣦⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣤⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠘⢿⣿⣯⣟⠃⢰⣿⡿⣿⣿⣷⡄⢺⣿⣿⠛⣿⣿⠀⢹⣿⣏⣿⣿⠇⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⢸⡀⠆⢀⠀⠀⠀⠀⠰⣿⣍⣿⣿⣦⢸⣿⣦⡈⢻⣿⡇⠘⣿⣿⡀⣿⣿⠀⢸⣿⣿⠋⠁⠀⠀⠀⠀⠀⡀⠘⣨⡃
⠀⠀⠀⠀⢀⣼⠛⢦⣌⠀⢠⠄⠀⠀⠙⠛⠻⠛⠃⠈⠻⣿⣿⣿⠿⠃⠀⠿⣿⣿⡿⠇⠀⠿⠛⠛⠀⠀⠀⠀⠀⠆⢀⣥⡾⠛⣇
⠀⠀⠀⠀⠸⣏⣤⡀⠉⠛⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⣉⣤⣴⠾⠛⠁⢀⡴⣿⠆
⠀⠀⠀⠀⠀⠙⢿⣙⢦⣤⡀⠀⠉⠉⠛⠻⠶⠶⢤⣤⣤⣤⣀⣀⣀⣀⣀⣈⣀⣤⣤⣤⣤⡶⠶⠾⠛⠛⠉⠁⠀⣀⣤⣶⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣥⣟⠐⠂⠤⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⣴⠖⣉⣤⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⢤⣤⣤⣉⣙⠛⣛⠛⠛⠒⠚⠛⢛⣛⣛⣋⣉⣉⣭⣥⠴⠾⠛⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⢋⣉⣩⣤⣤⣤⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣴⠾⢋⣥⡶⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠾⠿⠷⢶⣦⣤⣤⡶⠟⣡⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⠳⣤⣀⣀⣀⣁⣤⠼⠃⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠘⠷⣦⣤⣤⣤⣤⣤⣴⠶⠟⠋⠁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⢸⠇
⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⢸⣏⢉⣀⢀⣀⠀⣀⡀⢀⣀⠀⣀⣀⢀⣀⠀⣀⡀⢀⡀⢀⣀⡀⢀⣀⠀⣀⡀⠀⡈⠉⠉⣿
⠀⠀⠀⢿⡌⠁⠀⠀⠀⠈⠀⠈⠁⠀⠉⠁⠈⠉⠀⠉⠁⠈⠁⠈⠉⠁⠈⠉⠀⠉⠁⠈⠉⠉⢱⡟
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡟⠛⠛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⣛⢠⣬⠭⣄⠒⠒⠂⠀⠀⠀⠀⠀⠀⠶⢤⣤⠠⠤⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣴⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⣶⣼⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠟⣫⢍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⡴⢿⡿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠋⣤⠿⠿⡸⡸⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⡊⡼⢰⠏⡄⡠⣰⢃⡖⡨⣱⠆⢅⡼⠳⠱⠇⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣾⣾⣧⢀⣬⣴⣧⣴⣿⣼⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣷⣼⣶⣷⣿⣾⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢫⣶⢓⣰⡮⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠟⡿⢿⣿⡿⣥⡇⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣻⡿⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢆⡴⢦⠀⣤⣀⣤⠀⣠⣄⢠⣤⢤⠖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⢧⣼⢠⡿⣿⢿⢀⡿⢿⡀⡿⠸⠶⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⣶⣦⢄⣠⣤⡀⣤⣤⣤⣤⣤⣤⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⡀⠁⠀⠾⠿⠞⣷⣽⠇⢿⣤⡇⢼⡟⠉⠐⠃⠁⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠄⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡴⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣄⣀⣀⣀⣠⣤⣤⢤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣦⣾⣿⡋⠉⠁⢀⣀⣿⠿⠿⠿⠷⠤⠼⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇⠉⢰⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⢀⠔⣈⠐⡿⢿⣛⣯⠭⠟⠓⠒⣤⡤⠥⠤⠀⢀⣀⣒⣒⣒⣀⣀⠀⠀⠈⠩⠤⠐⢒⡋⠉⠭⢛⣛⠻⠿⠀⠀⢀⡠⡀
⠀⠀⠀⠀⡎⢸⡟⢩⡶⠛⢉⣴⡾⠛⣩⡤⠴⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⢦⣄⡈⠑⠦⣬⠑⠂⠀⠀⠺⣿⡌⡄
⠀⠀⠀⠀⢳⣜⠢⡉⠀⠀⡇⢈⠀⠚⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠢⠀⡤⡆⠈⣾⣆⠙⣱⠇
⠀⠀⠀⠀⠀⣿⣷⣦⣄⢀⡱⠌⢧⣄⣽⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣃⣀⠐⠡⠇⢐⣉⢀⣼⠋
⠀⠀⠀⠀⠀⣟⠿⣿⣿⣷⣼⣷⣶⣶⠐⠉⠉⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠖⠛⣋⠉⢠⣠⣴⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣷⣬⣙⡛⠿⠿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣀⣀⣉⣉⣉⣉⣉⣉⣉⣁⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⡿⠿⣛⣥⣾
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⣤⣭⣭⣙⣛⣛⣛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⣛⣛⣋⣩⣭⣴⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⢩⣿⠟⢋⣵⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⡟⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⡎⠁⠰⠿⠛⣋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣸⣿⣿⣿⡿⢠⡟⣼⣷⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢟⣵⣿⡿⢿⣿⢁⡿⢰⡟⢻⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣷⢸⡿⢋⣤⠉⢹⣿⠛⠛⠛⠻⠟⢻⣿⡟⠁⠟⠻⣿⡿⢁⡀⢛⡿⠡⢶⣼⠃⣾⢃⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⢏⣾⢁⣿⠏⢠⡿⠃⢰⡿⢀⣶⠀⣼⡟⢀⣾⡇⢠⡖⢠⣿⠏⣠⠗⣠⣼⠏⣸⠏⣸⠳⠏⢻⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣧⣴⣾⣯⡘⣋⠀⢞⡅⢠⡿⢁⣾⠃⢼⠋⢀⣾⠟⢠⣿⠁⡿⠏⣴⡇⠸⢛⣅⢀⣩⣀⣴⣦⣾⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣾⣿⣤⡾⢠⣦⣤⣶⣿⣿⣦⣴⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⠶⣿⢫⡝⣿⠻⢻⡟⣛⢿⡟⢛⣿⠻⠟⣿⠛⣿⡏⣿⡏⢎⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣾⣷⣽⣾⣧⣬⣿⣾⣶⣿⣥⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣩⣭⠁⠩⣉⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⡶⣶⡼⠻⠦⢠⣿⣬⡷⢝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢹
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⠇⢹⣧⣾⣷⡦⣷⣴⣿⠿⣁⣬⡼⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⢠⠘⣿⣿⣷⣷⣶⣮⣿⣦⣦⣄⡟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡⣭⣽⣿⣾⣍⣤⣽⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡟⠿⠛⠛⢣⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⢿⣶⡆⢠⣤⣤⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⣤⣤⠀⣴⡿⣷⡄⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠹⣿⠀⢸⣿⣧⢠⣿⣿⠀⠀⣾⠿⣧⠀⠙⣿⡏⢠⣿⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠇⠀⢿⣆⣰⣿⠀⣸⡟⣿⣾⠟⣿⠀⢰⣿⠾⣿⡄⠀⣿⡇⠈⢿⣶⠟⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠛⠁⠙⠛⠀⠛⠀⠺⠃⠀⠙⠃⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣦
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡌⡘⠃⢠⣾⡿⣿⣿⣦⡀⢺⣿⣟⠛⣟⡿⠀⢻⣿⢈⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠠⣾⡉⣿⣿⣢⢸⣿⣆⠈⢻⣿⡇⠸⣿⣿⡀⣿⡗⠀⢰⣿⢿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⣸⠳⣦⡀⠀⠀⠀⠀⠀⠛⠻⠾⠿⠃⠘⢿⣿⣷⣾⡿⠃⠀⢿⣿⣿⣿⠇⠀⠿⠛⠛⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⡄
⠀⠀⠀⠀⠐⣟⢄⠈⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣠⣤⠶⠛⠁⢀⡴⣿
⠀⠀⠀⠀⠀⠹⣿⡓⣤⣀⠀⠈⠉⠛⠳⠶⠦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠋⠉⠀⢀⣠⣶⣿⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣉⣶⠂⠤⢀⣀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣀⣀⠤⣤⡖⢋⣡⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⢤⣤⣀⣀⣉⡀⠒⠒⠲⠶⠶⠶⠶⠶⠖⠒⠒⠚⠛⣋⣉⣩⣥⠶⠟⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣁⣐⣒⣒⣒⣒⣒⣚⣋⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠶⠶⠶⠶⠂⠀⠂⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠐⠶⠶⠶⠶⠤⠤⣀⣠⣀⡀
⣠⡖⠛⠋⣀⣠⠤⠤⠴⠶⠤⠤⠤⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠶⠶⠦⠤⠤⠀⣀⣀⡈⠙⠛⠲⠦
⢟⠳⢄⣻⣦⣄⣀⣀⣀⡀⠠⠤⠄⠀⠋⠁⢀⡀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⢈⣉⡛⠛⠓⠶⠶⢤⣄⣀⣀⣀⣀⡷⢀⣀⠠
⢸⡷⠤⢄⣈⡉⠉⠁⠤⠤⠤⠤⣀⣀⣀⣀⣈⣀⡀⠙⠂⠀⠠⠤⠤⠴⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣌⣙⣃⣀⣀⡠⠤⠤⠤⠌⠛⠋⣉⣁⠤⠴
⢸⡇⠀⠀⠀⠉⠙⠛⠓⠂⠤⠤⠤⠤⢄⣀⣈⣁⣀⣈⣉⣉⣉⣉⠒⠛⠛⠛⠛⠛⠛⠛⠛⣛⣋⣉⣉⣉⣉⣀⣀⣀⣀⣀⠠⠤⠤⠤⠤⠐⠛⠛⠉⠉⠀⠀⢠
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢸⡇⠀⠀⣠⡄⠀⠀⣠⡤⠒⠛⠢⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⣿⡏⣠⠞⠁⣠⠞⣹⠇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠀⠀⠀⠀⠀⣠⢤⣀⡤⠒⠳⡄⠀⢸
⢸⡇⠀⠀⢻⡟⠁⠀⠀⠀⠚⣁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡅⣠⠇⠀⠀⠀⣸⠃⣰⡟⢠⢶⣀⠇⠀⢸
⢸⡇⠀⠀⣼⠁⣤⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡨⢃⡵⠋⢀⡀⠀⢀⠏⢰⡟⢣⠏⣸⡇⠀⠀⢸
⢸⡇⠀⢠⠃⢰⣅⠖⢦⡀⠀⣠⠔⢢⣄⢀⡀⠀⣠⣄⣀⣠⡄⢀⣤⢤⡄⠀⡰⠋⢹⣆⠀⠀⠀⢀⡾⠁⠀⣴⠋⣠⠖⢉⡉⢲⡜⠀⣎⢻⡌⣴⠃⡄⠀⠀⢸
⢸⡇⠀⢸⠀⡿⢃⣾⢇⣷⠞⢡⡶⣦⠀⢸⡇⡾⠁⠈⡋⠀⠘⡋⠀⠀⣇⢰⠁⢀⣈⠉⠹⢦⣤⠞⢁⣤⠀⠈⣷⠁⠐⣿⡵⡿⢁⢿⡘⣦⡹⡆⢠⡄⠀⠀⢸
⢸⡇⠀⢸⠀⡇⠻⡝⣾⠏⢠⠏⣠⠋⢠⢇⡜⠀⢠⠾⡇⢀⣾⠟⠀⢶⡸⠃⣰⠏⣹⠃⢀⣾⠀⣰⠋⡜⠁⣠⡷⡶⠀⣭⡶⢁⡘⣬⢣⣾⢷⡈⢺⣄⠀⠀⢸
⢸⡇⠀⠘⣤⠉⢋⡴⠛⡄⠹⠞⠁⠀⣼⣾⠁⠠⣎⠶⢀⣿⠏⠀⠸⠷⠁⢰⠁⣴⠃⠀⣼⡇⢸⡃⣼⠃⣰⡿⢋⠞⣡⠞⠁⠘⡷⠁⢾⣧⠈⢉⣀⣨⠃⠀⢸
⢸⡇⠀⠀⠀⠉⠉⠀⠀⠱⠶⠞⢣⣠⠜⠻⣤⠖⢹⣀⡜⢿⠀⣠⣴⠂⠀⠈⠉⠀⣠⠞⢡⡀⠀⢉⡡⠞⠡⣅⣀⣉⠶⠻⣄⡤⠟⠲⠞⠉⠛⠉⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⠁⢀⠞⠉⠉⠉⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠀⣠⠶⣦⠀⣤⡀⣤⠀⣤⢤⣀⠀⣀⣀⡀⢀⡀⢠⡀⢀⣠⡄⢀⣤⣤⠄⣼⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠂⠹⠦⠾⠀⠏⠻⡷⠀⣯⣀⣽⠂⣿⣉⡁⢸⠛⣾⠅⢈⣛⣷⠌⢿⠴⠄⠻⠦⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠁⠈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠒⣓⠒⢒⣠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⢠⡞⣩⠝⠛⠛⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠀⠀⢌⣰⡁⢀⣤⣒⠶⣶⣤⡜⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴
⢸⡇⠀⠉⠉⠒⠒⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠻⣷⣤⠀⣀⣼⣿⣿⣧⡞⠉⠀⠀⠀⠀⢉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠔⠒⠊⠉⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠒⠒⠦⠠⣴⡇⠀⠈⠲⢩⣍⠿⢻⣿⠿⣭⡶⣀⠀⠀⠀⠀⣧⣤⣤⠤⠔⠒⠒⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⣀⢀⡴⢊⡇⠀⠀⠱⣯⡄⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠈⠛⠓⠒⠉⠀⠀⠀⠀⠻⠇⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠤⠤⠄⠰⠶⠞⠋
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢸⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⢸⡉⠙⠛⢳⡆⢀⡀⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⣠⠖⣛⠰⣄⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠙⡇⣾⠛⢡⠏⢠⠴⣄⠙⣆⠀⢸⠟⠛⠛⡆⢰⠛⠛⢻⡄⠀⠀⣾⠛⠛⢳⡄⢸⣇⡀⠀⣀⢽⡜⠁⡞⠁⣷⠘⡀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⣿⠀⠃⠀⢸⠀⢸⠀⢸⠀⢀⠀⣧⡚⠀⠀⢸⡇⠀⢠⠃⠸⣧⠈⣆⠀⠈⡅⠀⡏⠀⡇⢰⠀⠀⣧⢀⠃⠀⠀⠀⠀⠀⠀⡇
⠘⠀⠀⠀⠀⠀⠀⠀⠠⡇⣿⠀⢻⠀⠻⣀⡼⠀⢸⠀⢸⠀⣿⡇⠹⠇⣸⡇⢸⡇⠀⡼⠀⣛⣛⠀⢿⠀⠀⡇⠀⡇⠀⢳⡈⠶⠴⢋⡼⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠳⣄⣈⣀⡴⠃⠀⣇⣠⡇⢻⡀⣀⠛⣀⣸⡇⢸⣇⣰⠋⠙⣇⣀⡇⠀⣧⣠⡇⠀⠀⠙⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠁⠀⠉⠉⠀⠈⠉⠁⠈⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⢀⣤⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠲⢦⣄⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⣀⠀⣠⣄⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡷
⢸⣅⠁⠀⣀⠲⠀⠀⠀⠀⠀⠀⠘⣄⠀⢾⡔⢢⣬⠇⢠⠞⢉⠀⠉⠙⢦⡀⢺⡉⠉⠉⡿⢏⠉⢉⡿⢣⡀⢀⡴⢢⡄⠘⡃⠀⠀⠀⠀⠀⡀⣤⠀⠙⠂⠀⣇
⡿⢯⡑⢦⣄⠈⠀⠀⢎⡉⠀⠀⣠⠞⣷⢦⡈⠱⢦⠠⡇⠰⡏⠉⠳⡄⠈⣵⠀⠋⠀⢸⠀⢸⠀⢸⠃⠀⠃⠘⠳⠞⣁⡼⠃⠀⠀⠶⠀⠀⠱⠀⠀⠀⣠⣴⢿
⢱⡄⠙⠶⣌⡉⠲⠦⣄⡀⠀⠀⠈⢦⡀⠙⠁⠀⢸⠀⢣⡀⠘⠷⠠⠏⢀⡿⠀⢡⠀⢸⣄⣼⠀⡜⠀⢠⠇⠰⣏⠁⠀⠀⠀⠈⠁⠀⠀⠀⣠⡴⠖⣉⠾⠁⡼
⠀⠙⠶⣀⠀⠉⠓⠶⢄⣈⡉⠓⠲⠦⢌⣉⣍⡉⠉⠀⠀⠙⠶⠤⠠⠶⠊⠁⠀⠘⢤⣀⠀⣀⡼⠃⠈⠻⠶⠖⠘⠃⢀⣀⣀⡠⠶⠖⠋⣉⣠⠶⠚⠁⣠⠞⠁
⠀⠀⠀⠀⠙⠲⠦⣀⡀⠀⠀⠉⠛⠐⠦⠴⠤⢀⣉⣉⣉⡉⠀⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠛⣉⣉⣉⣀⠤⠶⠖⠚⠋⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠶⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⣀⠠⠶⠖⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣿⡿⢟⣻⠽⠿⣉⣙⣷⠶⠶⠶⠚⣉⣉⣉⣉⣉⣉⣙⡒⠲⠶⠖⠛⣉⣙⡻⠿⢿⣟⡻⠿⢉⠱⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣿⠟⣡⠞⢉⣴⡾⠛⣩⣴⠶⠛⠋⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠙⠒⠲⢦⣍⠛⠲⣦⡍⠻⣦⣍⢻⣿⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣝⠂⠁⠀⢺⢸⡁⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠂⠈⠿⡇⢸⣿⠦⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣬⢜⣓⡻⠾⢿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣵⠧⠾⢟⣓⣭⣔⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⢿⣿⣿⣿⣿⣿⣍⣀⡀⠈⠉⠉⠉⠁⠀⠰⠶⠶⠶⠶⠬⠙⠋⠉⠉⠉⢀⣴⣴⣾⣿⣿⣿⣿⣿⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⣭⣛⣻⠿⠿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠿⠿⣟⣻⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⢿⣶⣶⣶⣦⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡿⠋⣠⣾⠟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢫⡿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⣬⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⢏⣼⣿⣿⣿⠇⣾⢃⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣸⣿⡛⣿⡿⢛⡿⢿⣿⡿⣿⡿⣿⡿⣿⣿⡟⢻⣿⣿⣿⠏⠠⢴⣿⠟⣩⢙⠏⣼⠇⣼⠃⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⠏⢰⡿⢁⣾⠏⢀⣧⢀⣬⠀⣼⠟⢀⣤⠀⣹⠁⣠⡶⢀⣯⡄⣰⠏⣰⠏⣸⣿⠆⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣽⣵⣿⠀⢛⠁⠼⡫⢀⣾⢁⣾⠃⣼⡋⢠⣿⠏⣴⡏⢰⡿⢡⡾⢡⡾⡋⠀⣋⠀⣥⣁⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣧⣾⣿⣤⡟⢀⣌⣡⣾⣿⣦⣠⣴⣿⣧⣤⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣏⠴⣿⢩⢹⡟⠻⣿⢛⡻⣿⠛⣿⠟⠿⣿⠻⣿⠋⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣶⣾⣿⣾⣿⣾⣶⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢲⡆⢀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⣼⡏⢹⣟⡋⢠⡾⠗⠀⢠⣤⣤⡄⢀⣀⣀⣠⡄⢀⣤⡀⢠⣶⢶⣶⢸⡏⠀⢸⣿⠖⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡟⠀⢸⡟⠃⢸⣇⠴⣶⢸⣿⣭⡁⠙⠛⣿⠋⠁⣿⣻⣧⠀⣿⠿⢷⣸⣏⣀⣸⣿⡶⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠙⠋⠀⠻⠾⠃⠸⣿⣤⡄⠀⠀⣿⠀⢰⣿⠉⠻⡦⠿⠶⠟⠘⠛⠉⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠸⣿⣿⣛⠿⠀⣴⣿⣿⣿⣦⡀⠰⣶⣶⡶⣶⣷⠀⢻⣿⣟⣿⣿⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⣮⣙⣿⣷⠀⣿⣷⡀⠙⣿⣷⠀⣿⣿⠀⣿⣿⠀⣼⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢦⣀⠀⠀⠀⠀⠈⠛⠛⠻⠋⠀⠙⢿⣿⣾⡿⠃⠀⠻⣿⣿⣿⠏⠀⠿⠛⠋⠀⠀⠀⠀⠀⠀⢀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣄⠀⠉⠛⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣳⢦⣀⡀⠀⠉⠉⠛⠓⠶⠶⠦⢤⣤⣤⣤⣤⣀⣀⣠⣤⣤⣤⡤⠤⠴⠶⠒⠛⠋⠉⠀⠀⢀⣠⣶⣥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣽⣟⠲⠦⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢠⡤⠀⣀⡼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠤⠀⠀⣈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠠⠴⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠉⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠟⠿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠏⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠁⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏
⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⣴⣿⣿⠋
⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⢠⣾⣿⡟⠁
⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⣴⣿⡿⠋
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⢀⣾⣿⠟⠁
⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⢀⣾⡿⠋
⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⢀⣿⡿⠁
⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣦⣤⣦⣤⣤⣤⣤⣤⣾⣿⠃⠀⠀⢀⣿⡟
⣿⡆⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠏⠀⠀⢀⣾⣿⣷⣦⣤⣄⡀
⢿⣿⣦⣤⣴⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⢀⣼⡟⠁⠈⠉⠛⠻⣿⣷⣦⣀
⠀⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆
⠀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟
⠀⣿⣿⣿⣿⣿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⣰⣿⡏⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⡇
⠀⣿⣿⣿⡏⠈⠙⠛⠿⠿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣤⣴⣶⣿⣿⣷⣶⣶⣿⣿⣿⡿⠟⠛⠛⣿⣿⡇
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿
⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⢸⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿
⢸⣿⣿⣿⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠉⠉⠉⠀⠀⢸⣿⣿
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⢾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣾⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿
⠘⠿⣿⣿⣿⣿⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⠉
⣿⣿⣿⣶⣾⣿⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠻⡿⠿⠿⠿⠛⠿⠿⢿⡄⠛⠿⠻⡿⠿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣛⣟⠉⣩⣙⠛⣛⣯⣽⣿⣟⣋⡉⠙⠛⠛⣛⣛⠛⢛⣻⣿
⣿⣷⣶⣿⣶⣶⣶⣶⣿⣶⣾⣿⣶⣾⣶⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣀⣀⣴⣿⣿⣿⣷⣶⣿⣴⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣭⣿⠿⠟⣛⣛⣭⣭⣭⣶⣷⣶⣶⣶⣦⣬⣥⣒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣮⡩⣉⣵⣶⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⠿⣷⡌⣶⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⣿⣿⣿⣿⢱⣿⡟⢽⢒⣻⣍⣩⣝⠒⠛⢛⠒⠶⠶⠞⠓⢂⣈⣴⣿⣷⢹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⡰⣹⣿⣿⣿⣿⣧⢸⣿⣿⠠⢤⣶⣄⢀⣰⣆⡀⢀⣰⠬⠷⣲⠛⣛⣭⡉⣿⣿⢻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣰⣷⣿⢻⣿⣿⣿⣿⢸⣿⣿⢸⡟⣵⡧⠀⠀⡟⠃⢀⣀⢰⠟⣽⠵⢃⡼⡇⣿⣿⢸⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣾⢸⡟⣩⣽⢫⢉⢹⢯⢏⢡⠆⢥⠟⢋⣓⡇⣿⣿⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣏⠒⢻⠖⠶⠶⢲⢀⡶⠛⢳⠋⢺⣿⢹⠀⠠⣿⣿⡌⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⡏⡀⢨⡠⣶⣶⣾⣟⣖⣶⣺⣶⣾⢽⢿⡇⠀⢹⣿⡇⣿⣿⣿⣿⣿⣿⣂
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣧⠁⡐⠅⠀⠈⣉⡵⠟⠉⠱⣄⢸⡆⠈⠡⢄⣾⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⢲⣦⢤⣤⣀⡞⠁⠀⢀⠀⠘⡆⣁⡶⠊⠙⢘⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡸⠋⡀⠀⠈⠙⢦⡀⢀⣀⠜⠁⠀⣦⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇⣆⡥⣀⣀⡈⣈⣉⣋⢇⣀⣠⢤⣤⣾⡂⠆⣿⣇⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⡷⡇⠇⣾⢴⢹⣇⡿⡼⡿⡎⣿⣸⢿⡻⠀⠀⣿⣿⢹⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⢀⠪⠉⡷⠚⠛⠓⠓⠓⠉⢋⠁⠀⠀⠤⠂⣿⣿⠸⣿⣿⣿⣿⣇⡀⠠
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣷⢼⡇⠀⣞⣶⣦⣶⢦⡖⣖⣶⠳⡵⠀⠀⢀⢿⣿⡇⣿⣧⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⢸⣿⣯⣰⣶⢤⣿⣥⣟⣭⣜⣣⣭⣻⠾⠷⠶⢚⣩⣾⣿⠏⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⡿⢛⣥⣶⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣮⣭⣟⣛⡿⠿⠿⠿⡿⣟⣛⣛⣭⣭⣶⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠛⠛⠛⠛⠓⠲⢤
⠀⠀⠀⠀⢠⣤⠤⠤⣀⡀⠀⠀⣠⡤⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁
⠀⣀⡤⠖⠋⠉⠀⠀⠀⠉⠒⢄⡇⠀⠀⠀⠀⢀⠜⠦⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠖
⠙⠓⠦⢤⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠾⣲⣾⠛⠉⠹⠏⣉⡉⡉⣯⢩⠉⠹⠉⠉⠙⠓⠒⠶⠤⣴⡊⠁⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠉⠁
⠀⠀⠀⠘⠡⠤⠄⢀⡀⠀⠀⠀⠀⠀⡀⠀⠈⢸⣽⣦⡴⠔⠈⠙⠉⠉⠙⠛⠟⠷⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⢀⡤⠚⠉
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⠋⠉⠉⠉⠓⠄⡈⢿⣀⡀⠀⠀⠀⠀⡠⠐⠒⠂⠠⡙⠛⠛⠛⠛⠛⠛⠓⣲⣶⣤⣀⠀⠿⠫⡀
⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⣠⣤⡤⣀⣠⣬⣤⣿⡉⣉⡀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠊⠁⣈⠁⠂⠀⠀⠈⣉⠙⠛⠷⢴⡹⣆
⠀⠀⠛⠲⢶⣶⣾⣯⣤⠤⠴⠚⢛⠛⣉⣉⣉⣉⠉⠉⠉⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⣿⡀
⠀⠀⠀⢠⢸⣯⡿⠿⠭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⠿⢄⣤⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠤⠔⠂⠀⠀⣠⣿⠃
⠀⠀⠀⢸⣄⠙⢷⣄⡀⠠⠀⢀⠀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠤⢤⣤⠀⢀⣠⠴⢫⣿
⠀⠀⠀⠘⡏⡳⣄⠉⠙⠳⢦⣤⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠠⠀⠀⠀⡰⠈⠂⠀⠀⠀⢀⣀⣤⡴⠒⠉⣡⡴⠋⡟
⠀⠀⠀⠀⢻⠀⠁⠙⠶⣤⣀⡈⠉⠋⠛⠛⠓⠖⠲⠶⠤⠤⠤⠥⠤⣤⣤⡤⠤⣤⣤⡤⠶⠶⠖⠚⠉⠙⣉⣡⣤⠶⠛⠉⠀⢸⠇
⠀⠀⠀⠀⠈⣧⣀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠶⠶⠒⠚⠛⠉⠉⠁⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠈⠫⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠐⡑⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣽⣿⣿⣿⣷⣶⣶⣶⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣦⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⡾⠛⢶⠖⢶⡄⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢦⡀⠀⠀⠀⢸⣄⠀⠀⢀⣴⠇⠀⠀⠀⣠⡼⠋⠉⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⢿⡄⠀⠀⢸⡁⣀⠀⢈⣿⠀⠀⠀⠀⢿⡀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⢻⡄⠀⠈⠙⠻⢦⣄⡙⣷⠀⠀⠀⢸⢟⣠⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠈⢷⡀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢧⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⣄⠀⠘⣷⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠘⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣠⡤⠤⣤⠀⠀⣠⡤⣤⡀⠀⣀⠀⠀⣀⢀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢸⠃⢴⣚⣉⠀⡞⢁⣠⣄⠙⢾⠉⡇⢸⠙⣿⠀⢉⡭⡍⢧⠀⠀⠀⢀⣀⡀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠒⠶⢦⠉⢹⡇⠸⣆⣸⠃⢸⡄⢧⡼⠀⡏⠀⠘⠶⠇⡸⠀⠀⠀⢸⠋⠙⣷⡄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡤⠤⠞⢀⡼⠙⢤⣠⣤⠴⠛⠧⣄⣠⠼⡇⢰⠶⠖⠛⠁⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⠓⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢠⡟⢀⣠⡾⠁⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠐⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠻⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠂⢊⠀⢫⣟⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⣿⣷⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢈⣶⣿⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣀⣲⡶⠒⠒⠒⠒⠀⠙⢻⠇⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠹⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣼⠀⡸⢠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠨⠛⣹⣿⡆⡄⠀⠁⠇⠀⣰⣿⠃⣀⣀⡀⠀⠇⢸⠇⠀⢰⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠘⣿⣆⠐⣯⡃⠀⢠⣤⢾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⡅⠤⠈⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⣷⣬⠀⢀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠬⠄⢀⡀⠀⠀⠀⠔⠀⢀⣿⣿⢰⠃⠏⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⡀⠠⢤⠀⠀⠀⠀⠀⠀⠀⣠⣿⡷⠃⡸⠀⠀⡀⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢋⠿⠇⠀⠀⠀⠀⠀⢀⣠⣴⠾⠿⠋⠠⠞⣁⠀⣰⠁⢀⠄⠁⠂⠀⠸⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡟⢻⡟⠫⠆⠀⢀⠉⠀⠀⠊⠁⠊⠑⠀⠈⠜⡸⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠁⣰⣶⠖⠂⠉⢀⡀⠀⠀⠀⢅⠋⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⠋⠀⠀⠀⠀⢀⡠⠋⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠔⠋⠀⠀⠀⡀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠉⠁⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡻⣧⣼⠇⠀⠀⠀⢀⡴⠋⠉⠙⢷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⢛⡿⣿⡿⣿⡀⠀⠀⠀⠀⠀⣀⡤⠚⠃⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⣿⢇⣿⢿⠀⠀⠀⠀⣸⠃⣼⠛⢦⣀⣉⣉⡙⢷
⠀⠀⠀⠀⠀⢀⡶⢋⡵⠒⢉⣡⢾⣿⣧⠘⣷⠀⠀⠀⡰⠊⠀⠁⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⡾⡼⣸⡿⢸⡤⠀⠀⣰⠟⡴⠃⠀⠀⠉⠀⠈⡷⢸⡇
⠀⠀⠀⠀⠀⣿⠁⡼⠀⡔⢉⠔⣿⡏⡏⣇⠸⡇⠀⢰⣅⠀⠀⠀⠀⠀⠀⠀⠤⠴⣧⠀⠀⠀⢰⠇⡇⠹⠃⢿⡇⠀⢰⢣⡞⠀⠀⠀⠀⠀⢀⡼⢃⡞
⠀⠀⠀⠀⠀⣿⠀⢿⠘⡇⣿⢰⣳⢃⡇⡿⢀⡇⠀⢸⡀⠀⠐⠤⠀⠀⠀⠀⠀⣀⡟⠀⠀⠀⣼⢸⡀⠀⠀⣸⠁⠀⢸⡌⠳⠤⠶⣄⠀⠀⣸⢣⡏
⠀⠀⠀⠀⠀⠹⣧⡈⢧⢱⣿⣾⣫⢞⡴⢃⣼⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢈⡽⠃⠀⠀⠀⡏⠸⠇⠀⢠⠇⠀⠀⠀⠙⠛⠲⡆⠈⠓⠚⠋⣼⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣮⣿⣿⣟⣛⣉⣤⠞⠁⠀⠀⠀⠀⠈⠳⡄⣰⠖⠚⠛⠉⠀⠀⠀⠀⠀⣧⣀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠈⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣷⡀⠀⠀⠀⠀⠀⠀⡾⠀⠘⠓⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⢰⠃⠀⣤⠀⠀⡼⠀⠀⠀⠀⢀⡴⡖⠉⠉⡇⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣘⣧⡴⠋⠉⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⢀⣧⣀⠀⡀⠀⢀⠇⠀⠀⢀⡾⠋⠀⢿⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠁⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣼⠁⠈⢉⣩⣤⡞⠀⠀⠀⡟⠃⠀⠀⠀⠀⠀⠀⠀⢹⣄⡴⠃
⠀⠀⠀⠀⠀⡾⠀⠀⢀⠀⢠⣄⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢰⢟⡆⠀⠀⠀⠀⢠⡷⢤⣀⠀⠀⡜⠀⠀⠀⠈⣇⠀⠀⠆⠀⣄⠀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⢷⠀⠀⣾⠆⠘⠿⠀⠀⠀⣿⣿⠃⠀⠀⠀⢰⡟⢠⠃⠀⠀⠀⠀⣼⠀⠀⠠⢤⡾⠁⠀⠀⠀⠀⠙⢦⣀⡀⣀⣸⠟⠁
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⡿⢿⢸⠀⠀⠀⠀⢠⡇⠀⠠⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠹⣄⡀⣀⣤⣤⣴⣾⡿⠟⠀⠀⠀⠀⠀⣿⢸⡏⠀⠀⠀⠀⢸⠋⠂⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢞⣩⡤⡤⣄⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣤⣀⠀⠀⢸⡀⣠⠏⠀⠀⠀⢀⣤⠶⠒⠶⣄⠀⠀⠀⣼⣳⠏⡁⢸⡵⣀⡹⣆⢹⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡇⠀⠈⢣⡉⢧⡀⠈⠛⠁⠀⠀⠀⡴⢋⡴⠛⠛⢦⢹⡆⠀⠀⣿⣿⢺⡽⣠⣄⠳⠆⣿⢸⡇
⠀⠀⠀⠀⢠⡞⠉⢠⡇⠀⠀⢹⡿⢦⡀⠀⠀⢠⡏⠀⠸⠁⠀⠀⠀⠃⠈⡇⠀⠀⠀⠀⣀⡼⢡⡿⠀⠀⠀⣸⠀⣧⠀⠀⢻⡜⢦⡀⠳⠞⢀⣴⢃⡼⠁
⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠘⠀⠈⢻⡀⠀⢼⠀⢠⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡼⠋⡴⠋⠀⢀⡀⠀⠙⣦⠘⡆⠀⠀⠙⠶⣭⣙⣛⣩⡵⠞⠁
⠀⠀⠀⠀⠈⢻⣶⠀⢀⡴⠀⠀⢀⠀⣼⠁⠀⠘⣧⠘⣆⠀⠀⢸⣀⣠⡟⠁⠀⠀⢸⡃⠸⣇⣀⡴⠛⢷⡀⠀⣸⠀⡿⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠿⢧⣤⡤⠾⠞⠁⠀⠀⠀⠈⠳⠿⣦⣄⣼⣿⠉⠀⠀⠀⠀⠘⠷⣤⣬⣥⠴⠶⣌⠓⠚⢁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡾⠋⠉⠉⠉⠉⠻⣿⣿⣿⣇⠀⠀⠀⠀⣾⡟⠋⠉⠉⠉⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⢿⣿⠟⠛⠻⠿⣿⠀⠀⠀⠀⠀⠠⠴⠄⠠⠦⠤⠀⠸⣿⣿⣿⣧⠀⠀⠀⠻⣤⣤⠀⣤⣤⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠃⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠘⠃⠀⠘⠃⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣆⠀⠀⠀⠀⢸⣿⠿⠛⠛⠛⠛⠛⣿⣿⣷⠀⠀⠀⠀⢈⣿⡿⠟⠛⠛⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣯⣤⡄⠀⣤⣄⠀⢻⣿⣿⠀⠀⠀⠀⠈⢃⣀⣀⠀⣀⣀⠀⢻⣿⣿⠀⠀⠀⠀⠸⣿⠤⠴⠄⠠⠤⡤⠀⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⣀⣀⡴⠶⢶⣄⣀⣀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠈⢄⠠⠀⡀⠠⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠚⠀⠀⠀⠈⠀⠅⠀⠀⠀⠈⠀⡄
⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⢀⠄⠔⢁⠀⠀⠀⠐⠠⠀⠒⠉⠢⣀⡑⡀⡀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⣑⠀⠄⠐⡀⠀⠀⠠⠀⠀⢀⠤⠤⣄⡀⠀⠀⠑⠂⠠⡀⠰⡠⠀⠙⠢⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠋⠂⠀⡔
⠀⠀⣀⠀⡀⠀⣦⠀⢀⠀⠀⢈⠀⠀⠀⠈⠣⣀⠀⠄⠀⢈⠼⠁⠀⠀⠀⢲⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⢄⠀⠀⠥
⠀⠘⠁⠘⠀⠀⣸⡰⢠⠀⡈⢂⡀⠀⠀⠀⠀⠀⠑⣦⠊⠁⠀⠰⠊⠀⢀⠂⢂⡉⠃⠀⠀⣠⡬⠐⠀⠁⠀⡈
⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠹⣄⠈⢂⠀⠀⠀⠙⢄⡀⠰⠂⣀⠔⠁⠀⠀⠄⠠⠒⠋⠉⠀⠂⠀⠀⠁⠀⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠹⠂⠈⠉⠐⢂⠀⠉⠢⠴⢄⡀⠠⠀⠝⠚⠉⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄
⠀⡀⠂⠂⠀⠔⠁⠀⠀⠀⠀⠀⠀⠱⢀⠐⡀⠆⠀⡀⠀⠀⠡⠀⡀⠈⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠈
⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⢀⠀⠀⡉⠄⠂⠀⠀⠀⠀⠀⣀⠀⢀⠰⢀⢀⠠⠄⠀⠊⠄⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠑⠂⠊⠀⠱⡀⠀⠀⠀⠀⠀⠠⠁⢸⡆⢸⠀⠈⠀⠀⠀⠤⢴⡇⠀⠀⡇⠀⡏⠀⠀⠀⠀⠀⠁⠀⡀⠁⠀⠘⠠⠀⠀⠆⡀
⠀⠀⠀⠀⢀⡀⠈⠑⠒⠒⠢⢀⠀⠀⠘⡇⣼⣴⡶⢠⣶⣦⣾⢾⢠⣻⡆⡗⡦⡷⣾⠇⢿⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⢰⠀⠀⠀⠻⠇⠙⠒⠊⢻⡇⠉⣈⣂⣉⣣⣯⣅⣀⣈⡃⠚⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⡀⠀⠐⠂⠀⢄⣂⣀⣀⠈⠃⠠⠄⡀⠒⠒⠃⠐⠚⠀⠉⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠰⠣⢀⠘
⠀⠀⠀⠈⠀⠉⠀⠀⢠⢦⣗⢦⠁⠀⠀⠀⠀⠂⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡅⠀⠂
⠀⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠱⣄⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀⠀⡀⠀⠈⠉⠐⠒⠒⠆⠀⠀⠀⣀⣀⣀⣀⣐⡀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠂⠐⠒⠦⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⠂⠠⡀⠄⠀⠀⠔⠐⠀⠀⢔⠒⠁
⠀⠒⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢆⠀⠡⠀⠀⠀⠠⠀⠐⠠⠀⠠⠠⠈⠀⢠⠀⠀⠆⠀⠀⠀⠀⠀⠒⢠⠄
⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠄⠄⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⡀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠀⡠⠄⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠁⠢⠀⠴⠁⢒⠠⠈⢤⠀⠊
⠀⠀⠀⠀⠀⢀⡈⣀⠀⠀⢄⠀⠀⠀⠀⣀⢀⡀⠠⢈⠐⠀⠀⠀⠂⠀⠁⠈⠐⠀⠀⠀⠀⣀⣀⠀⠃⠠⠁⠀⠠⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠅⠀⠀⠘⠑⠀⠀⠈

⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⠀⡌⢳⠪⡿⣥⢯⠧⣘⠉⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠤⢄⠀⠀⠀⢀⡩⢵⣴⣦⠀⡉⢺⢉⢺⠙⡯⠭⠨⠼⠐⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠁⠸⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠰⣤⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢆⠀⠀⠀⠀⠀⠄
⠀⢀⣀⡠⠖⠋⠀⠀⠀⠐⢈⣿⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠿⣾⡁⠀⠀⠀⠀⠠⡀
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⠉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣝⣥⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠀⠀⠀⠀⠀⣀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠁⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⡀⠀⠀⠀⠀⠀⠀⣤⢀⡔⣹⣆⡴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⢀⡤⠀⠐⠒⠒⠀⠤⡀
⣵⣶⡄⠀⠀⠲⠶⣷⡈⣷⣶⠇⠢⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⢠⡀⣀⠀⠀⠈⠁⠀⠀⠒⠒⠈⠁⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠉⣹⡗⠀⠀⠀⠀⢈⡇⠛⠃⠀⠀⢀⢤⣖⡢⣀⣠⡀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢈⠙⠚⠃⠀⠀⡔⢁⡀⡀⠉⠀⢠⡿⣿⠾⡯⣕⣳⠵⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠠
⠈⠀⠀⠀⠀⠰⢀⣻⣯⢧⡏⣛⡷⣜⡧⡧⠝⢒⠓⢡⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡄
⡀⠀⠀⠀⠀⠀⢨⣾⢻⣅⣓⢹⣜⡯⣷⡕⡝⣶⣻⠍⢰⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠈⡄⠣⡟⡬⣗⢝⡝⠔⣳⢟⣏⢟⡺⢁⠃⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠈⠢⡙⢷⣡⣟⣊⢆⣿⡃⠵⣉⠔⠁⠀⠀⠀⠀⠑⢴⡀⡀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠰⡄⢠⠄⠀⠀⠁⠒⠀⠤⠤⠴⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠬⠈⠡⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⡳⣷⡞⠐⠀⠙⠈⠁⠀⠀⢀⣀⡀⠀⢰⡟⢁⡴⠀⣰⠂
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠁⠀⠀⠀⠀⡠⠖⣉⣠⣤⠄⠉⠉⢾⠿⡷⢞⣩⣄⠀⠀⠀⠀⠀⠀⠀⢈
⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡰⠣⣓⣿⡷⢺⠀⠐⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⠂⠄⠳
⠟⠩⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠙⠂⠈⠼⠓⡀⠀⠀⠀⠀⠀⠀⠸⢠⣒⣊⣝⣱⣭⣅⣢⢤⣠⢤⢠⢀⢂⠀⠀⠀⠀⠀⠀⢠⠹⣒
⢶⣾⠧⡰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠁⠀⠀⠀⠀⠀⠀⣸⡓⢻⢭⠷⣟⢼⢾⠾⢺⢿⣯⠏⠀⠄⠀⠀⠀⠀⠠⡀⢭⡿
⣾⡖⣃⠂⢈⣶⣤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⠘⢿⣓⡟⡞⣤⣳⠧⣝⣫⣧⡗⢮⣣⠀⠀⠀⠀⠀⠀⠐⢌⠁
⠂⠐⠅⠐⠘⣿⡿⠃⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡉⠵⡾⢞⣫⠾⣍⡻⡼⣼⡻⢩⠎⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡷⠉⠃⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠳⠿⠹⠣⠋⡋⠔⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⣀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⢒⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠
⠒⠈⠉⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢱⢤⢀⡻⢧⣺⣕⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠫⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠀⠀⠀⠀⠉⠀⠀⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠀⠀⠀⠉⠐⢳⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠰⠁⠀⠀⢈⠂⠀⠀⠀⠀⠴⠒⠀⠀⠀⠒⠀⢳⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠰⠆⠠⠀⠂⠒⠉⢸⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡠⢊⡠⠊⠁⠒⠤⠤⠤⠤⠤⠤⠤⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠆⠀⠀⠀⠀⠀⠐⠀⠠⠁⠀⡇⠀⠄⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠬⠙⡆⠀⠀⠀⠠⣤⣤⠤⠴⠋⡼⢄⡀⠀⡀⣀
⠀⠀⠀⠉⠉⠙⠁⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠰⠀⠆⢀⠏⠀⠈⠓⠄
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠕⡦⢀⠀⢀⣠⠆⠀⠀⢸⠀⠀⠀⡰⠈
⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡌⠀⣠⡈⠓⠢⢸⡀⠀⠰⠁⢱
⠀⠀⠌⠀⠀⠑⡄⠀⠀⠀⠜⠀⠀⠀⠀⠠⣀⠘⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⡸⠀⠀⠀⢦⡇⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠑⠢⢄⢰⡀⣀⠄⠀⣸
⠀⠁⠀⠀⠀⠤⣙⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢦⠀⠀⣀⣀⡴⠃⠀⢠⠃
⡄⠀⠀⠀⠀⢀⠘⠐⠂⠤⠀⠀⠀⠀⠀⠉⠁⢠⠀⠘⡀⠀⠀⠀⠀⢸⠀⢸
⢷⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⢀⡇⠀⢸
⠀⠳⡇⠀⠀⢸⡤⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⡾⠀⠀⢸
⠀⠀⠹⠄⠠⠒⠀⠐⡄⠀⠀⠈⠑⠢⠤⠤⠤⠜⠀⠦⡀⠀⠀⢠⠇⠤⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠁
⠀⣁⣀

⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⠿⠿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⠿⠿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⢀⣴⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣷⣄⠀⠀⣠⣾⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⡀
⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⡿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣽⣿⣿⣷⣿⣶⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⠀⢀⣀⠀⢀⡀⠀⢀⣀
⠀⠠⠁⠀⠲⡄⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠈⠉⠀⢫⡀⠀⢀⡈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫
⢀⠀⠀⠀⢀⠀⠉⠀⠂⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙
⡎⡀⠀⢀⠂⡀⠀⠀⠀⡌⠀⢸⡀⢠⠤⡴⡂⠉⠀⠀⠂⠀⠒⠀⢒⣀⣾⣷⡶⢠⣐⡀⠀⠀⠀⠀⠀⠂⠐⢻⡂⠀⠀⠀⠀⢧⠙⡄
⢠⠁⠠⠃⢠⠁⠀⠀⢀⠆⠀⠈⠀⣼⢀⠀⢉⠄⣀⠀⠀⣀⡀⣺⡟⡉⢡⢔⠀⠀⠈⡕⠒⢤⡆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠐
⠇⡰⠃⢠⡇⠀⠊⡁⠀⠀⠀⠀⠀⡄⠘⢀⢸⠂⣿⠴⢋⢹⢭⠉⠉⣡⠱⠛⠓⢤⣤⣄⠠⣠⣶⠀⠃⠀⠀⣴⡈⡀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢡⢀
⠐⠋⣥⡄⠀⠰⠍⠠⢄⠠⠄⠠⢈⣧⠟⣬⡇⠨⠄⠠⢘⣼⠖⣏⠚⣧⡟⢿⡉⢸⡆⡿⠒⢘⣿⣄⠀⡴⠀⠄⡡⠶⠀⠔⠀⡐⠀⠀⠀⠀⠀⡀
⠢⢄⠈⠑⠢⣄⠀⠑⠠⡐⠀⠑⢼⠉⡖⢦⢄⡑⠦⣀⢸⣛⠲⠦⣌⣧⠥⡤⠦⢤⠭⡿⠆⠹⣿⣧⣾⡢⠄⢠⠀⠀⢂⢀⣌⠄⠊⡀⠄⠂⡀
⢸⠀⠹⢢⣄⡀⠙⠢⣀⠈⠐⠄⡆⠀⡇⠸⠨⠘⡑⠠⢙⣎⣂⣁⣚⣛⡛⣶⣯⡴⣊⢀⡤⠤⣭⠿⡏⠀⠰⠈⡄⡈⠌⢋⡠⠐⢀⠠⠂⠁⣀⠴⠞
⠇⢀⠇⡸⢘⡏⢰⠤⠀⠙⠢⢀⠃⢀⠇⡆⠈⠦⠥⠀⠀⠉⠁⠦⠄⣀⠀⠀⠀⠞⣁⣩⠜⠋⠀⠀⠀⠀⠀⠀⠉⢡⠈⢌⡄⠊⢀⣠⠔⠊⢅⠘⡄
⠀⡼⢠⠇⢸⠁⢸⡂⠀⠁⠂⣼⣄⠈⠠⠄⠀⠀⠔⠀⠀⢄⠀⠠⠀⠈⠙⠒⠶⣶⠿⠀⢀⣤⣄⣠⡄⢀⣠⡄⢂⡔⠀⣪⡼⡖⠉⠀⠀⠀⠈⢢⠐
⠀⣇⣞⠀⠛⠀⠈⠁⠐⠂⠈⠉⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⣴⡿⢶⠐⢂⡁⠲⠀⠀⠀⠀⠀⣠⡴⠚⠉⠒⠒⠶⠶⠶⠶⠶⠶⠷
⣸⣿⣟⣉⣙⣑⣀⡒⠒⠂⠒⠒⠖⠖⠀⠈⠳⢦⣄⠀⠀⠀⠀⠀⠠⢀⠑⠦⠖⣊⠴⠊⠁⠀⠀⠀⢀⣠⡖⠛⠁
⣍⣁⣀⣀⣀⣉⣉⣉⣉⣙⣛⣛⠛⠒⠒⠒⠒⠖⠚⠻⠦⣄⡀⠁⠂⢀⠉⠒⠋⣀⡤⠊⠁⣀⣴⠾⠉⢁⣀⣀⣀⣀⣀⣀⡀⠀⠈⠁
⣿⣽⣯⣭⣭⣭⣭⣭⣭⣭⣙⣛⣛⣛⣛⡛⠛⠛⠓⠶⠶⠾⠿⢶⣄⡀⠑⠲⠟⠁⢀⣴⠾⠯⠤⠤⠤⠤⠤⠤⠄⠀⠠⠤⠤⠤⠤⠤⠄⠤
⣿⣭⣽⣯⣭⣯⣭⣭⣭⣩⣍⣩⣉⣉⣉⣛⣛⣛⣛⣛⣓⣶⣶⣶⣮⣿⣶⣤⣤⣾⡛⡒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠄
⣿⣯⡛⣿⣯⣝⣯⣿⢿⣭⡋⠩⠻⣿⣍⠉⠙⠻⣯⣍⡉⠛⢿⣍⠉⠉⠉⠙⠛⠉⠉⠉⣉⡿⠛⠉⠙⣻⡿⠛⠛⠛⣓⡿⠛⠒⢒⡶⠛⠒⠒⣲⡶
⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣖⢹⣿⢷⣌⡠⠐⠉⠻⣶⣄⡙⢷⣄⠀⠀⠀⢀⣤⠾⠋⠀⣀⣴⠟⠉⠀⢀⣤⠞⠉⠀⣠⠶⠋⠀⣀⡴⠛⠁
⣿⣿⣿⣽⣿⣿⣾⣿⣿⣿⣽⡯⣿⢷⣧⡶⣉⠻⣦⣔⠉⡈⠙⢿⣦⡙⢷⣶⡾⠛⠁⢀⣤⠞⠋⠀⢀⣤⠾⠋⠁⣀⣴⠟⠁⢀⣤⠞⠉⠀⠀⣠⠴
⣿⣻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⡻⣿⣿⣦⣗⡬⣻⢷⣤⡀⠀⠙⢻⣦⡉⠀⣀⣴⠟⠁⢀⣤⠾⠋⠁⣠⡴⠟⠉⢀⣠⡶⠋⠁⠀⣠⠴⠋⢁⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣮⣿⣷⣤⡷⣝⡿⣦⣄⠀⡈⢿⣿⡟⠁⢀⣴⠟⠁⣀⣴⠟⠉⠀⢀⣴⠟⠁⠀⢀⡴⠋⠁⣠⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠢⠄⠀⠀⠀⣀⠂⡀⠀⡄⠀⠀⠀⠀⢀⠎⠁
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠆⠀⠀⢀⡀⠤⠤⠠⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠄⠸⡿⠁⡀⠀⠠⠀⡈⠐⣄⣼
⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠚⠄⣀⡊⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠀⠀⠀⠀⠐⠦⣼⣧
⠀⠀⠀⢐⡀⠒⠄⠀⡔⠠⢶⡀⠀⠀⠙⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠸⠿⠆⡀⠀⠃⣄⡀⠀⠀⠀⠐⡢⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠀⠀⠁⢠⠶⠃⠉⠀⠀⠀⡀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠠⠀⠀⠘⢄⣠⣠⠆⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⢠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⠀⠉⠉⠒⠤⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠢⠢⡤
⠀⠀⠀⠀⠈⠛⠂⠉⠙⠓⠂⢤⣌⠳⡀⠀⠀⢀⠀⠀⡀⠠⠀⠀⠀⠄⠀⡀⠀⠀⠱⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣗⣄⠀⠸⠀⠀⠃⠀⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⢠⠠⠀⠀⡀⠀⠁⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠍⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⡀⡀
⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣀⠀⠀⠀⠀⠀⠀⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠥⠤⠀⠀⠀⠀⠀⡀⠀⡀⠀⡠⠀⠐⠀⠀⠘⠛⠐⠀⣀⠔
⠀⠀⠀⡐⠠⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⡤⠀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⠀⢢⠞⠁⠀⡐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⡀⠀⠘⠀⠀⡁⠀⢱⠖⡘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠁
⠀⠀⡔⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠇⠀⠀⠀⠀⡁⠀⠃⠀⠀⠀⠀⠀⠀⠐⠈⠈
⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣆⠀⣀⠂⠀⢀
⠀⠀⠀⠣⢠⡀⠀⠀⠀⠀⢁⠠⠄⠀⡀⠠⡄⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⠂⠈⡆⠫⠀⠐⠁⢰⠶
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣴⡶⠟⠁⠀⠀⠀⠘⢄⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢀⠄⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠔⠁⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⠀⠀⠁⠐⡆⠀⠀⠀⠀⠘⠄⠀⠀⠀⠕⠀⡀⠀⠀⠀⠥⢥⡐⠂
⠀⢀⢂⠂⠀⠐⠄⠀⢀⠴⠈⠢⠀⠀⠀⢸⠀⠀⠀⡔⠀⠉⠐⠢⡀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⡐⠀⠀⠀⠚
⠀⠈⠂⠀⠀⠀⡜⢹⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠄⠄⠋⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⠤⠄
⠀⠀⠀⠀⠀⠀⣀⡀⠄⣀⡀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠑⠲⢄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⢤⣦⠀⠀⢀⠄⠃⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⣈⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡪⠂⢀⠎⠀⠀⠘⠍⠁⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⢄
⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⢆⠄⠄⡕⠁⡰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⡁⠡⠀⢊⣔⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⠐⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⡀⠈⠀⠀⠀⢈⡀⠀⠀⠀⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠊⢑⡦⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠊⠀⠐⠂⠀⠀⡀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠁
⠀⠑⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡡⠄
⠀⠀⠀⠈⠁⠀⠐⠁⠀⠀⠈⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠁⠂⠀⠐⠀⢱
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢁⠐⠀⣀⠠
⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠄⠀⠠⡜
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠖⠄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⢀⣤⣶⣦⣄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⢰⢿⣷⣷⣿⣷⣝⢿⣿⣿⡿⣣⢞⣿⣿⣝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣏⡿⣿⣿⣿⣿⣿⣿⣳⣾⣷⣻⣽⣿⣿⣿⣿⣼⡆⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣲⡶⣭⣿⣻⣭⣞⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣝⠿⠿⢿⣂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⣿⣿⣿⡿⣱⣿⣿⢹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣽⣿⣯⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣧⢹⣿⣿⢰⡟⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠧⠌⣸⣿⣿⢸⡏⣱⠃⣼⣿⠿⠛⣛⣛⣛⣛⣛⢿⣿⣿⣿⣿⡿⣿⣹⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣛⣯⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣈⣡⡾⢋⣴⣾⠟⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣷⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣾⣿⣿⠟⣡⣤⣄⠈⢿⣿⣿⣿⡿⠋⠀⣨⣿⣿⣿⡟⣰⠿⠛⠹⠞⣿⣿⠟⢋⣴⣿⣿⣿⣿⣿⣿⣯⣿⠮⣽⣅⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⡄⠘⣿⣿⡿⠁⣠⣾⣿⣿⣿⣿⡇⠀⠛⠓⠛⠛⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣦
⠀⠀⠀⠀⠀⠀⣙⢯⡻⣿⣿⣿⠀⢿⣿⡿⣿⡇⠀⣿⡿⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟
⠀⠀⠀⠀⢀⣾⣿⣷⣻⣽⣿⣿⣧⣀⣉⣀⣿⡇⢠⡿⠁⣼⡿⠟⠛⠻⣿⣿⠿⠛⠛⠿⢿⣿⣿⠟⠛⠻⠟⣿⣿⠟⣻⠿⠟⢿⣿⣿⣿⣿⡿⣋⡀
⠀⠀⠀⠀⠘⣿⣿⡿⣻⢿⣿⣿⣿⣿⣿⣿⣿⠁⢸⠃⣼⠋⢠⣿⡇⢀⡿⠁⣴⣿⡇⠀⣼⠏⢀⣾⣿⠀⣰⣿⠃⠀⣡⠆⢠⣿⣿⣿⣷⡟⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠈⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣸⣿⠀⠙⣋⣤⡾⠃⠰⣿⡟⠀⣸⠟⠀⢾⣿⠃⢠⣿⠏⢀⣾⡏⠀⣾⢋⣿⣿⣟⣧⣻⣿⡿⠁
⠀⠀⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⣿⣿⣦⣤⣭⣥⣾⣷⣤⣤⠆⢀⣴⣾⣦⣬⣵⣤⣬⣴⣤⣾⣿⣧⣤⣴⣿⣿⣿⣿⣯⡳⣌
⠀⠀⠀⠀⠀⠹⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡿⠋⡡⠀⣼⣿⠟⢛⣛⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢉⣟⡳⣿⣻⣿⣿⣿⣿⣿⠟⣩⢶⡟⢫⠄⣽⣿⡟⠁⣾⡇⢠⣿⣿⡄⢦⡉⣟⣻⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣯⠾⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⢷⣿⣿⣿⣿⢯⣾⣿⣾⠵⢋⣼⣿⣿⣇⠀⠿⢁⣾⣿⣿⣿⣤⡙⠻⠿⣿⣷⣽⣿⣿⣿⣿⣿⢿⡿⣻⣭⣧
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡟⣾⣿⣿⣿⣿⣿⣛⣭⣥⣶⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡜⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣻⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⣶⣶⣶⣝⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⢟⣯⣿⣛⠾⠿⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠾⡿⣿⣿⣿⣿⢿⣯⠿⡿⢯⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⣿⣿⡵⢫⣵⣿⣿⣷⣝⢿⣿⢿⣿⣻⠏⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠙⠻⠿⠛⠁⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⡄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⡀
⠀⠀⠀⠉⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠢⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠑⠄⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠂⠀⠀⠁⠀⠘⠐⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑
⠖⢺⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠠⢀⠆⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠐⡁
⠀⠐⠅⠀⡄⠐⠈⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⡍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠠⣄⣄⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠲⠋⢀⣀⣤⡀⠀⠀⠾⠁⠀⠀⠉⠉⠈⠒⠒⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠃⢠⠀⠀⠀⠙⠛⠛⡡⠀⠀⢰⣶⡴⠶⠀⢀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣀⠰⠂⠐⠈⡀⠠⠜⠁⠀⠀⠀⠀⠤⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡞⠉⠀⡠⠐⠒⠀⠀⠀⣀⣀⠄⣀⡔⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⡀⠀⣇⠀⠀⠀⠀⠀⠙⠿⠷⢀⣀⣀⣀⣀⣠⠴⠖⠋⠉⠫⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠉⠑⠒⠒⠒⡉⠉⠀⠀⠉⠁⠀⠀⣀⠀⠐⠄⠀⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠤⠔⠀⢀⠀⠀⠉⠉⠀⠠⠾⣿⡶⠀⠀⠀⠀⢡⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠋⠁⠴⠿⠗⠠⠖⣶⠂⠀⠀⠀⠀⠀⠀⠈⠉⠱⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡍⠉⠙⠒⠦⠤⠤⣤⡤⢄⡁⠀⠈⠀⡀⠠⣶⠶⠆⠀⢀⣀⣼⡒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡠⠚⠓⠀⡀⠀⠐⠻⠽⠿⣀⠈⠒⠲⣒⡛⠢⠤⠒⡫⠝⠋⠀⣠⣍⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡀⠀⠀⢠⠖⡋⠁⠀⢀⠄⠘⠛⠦⠀⢉⠉⣠⣶⣽⣷⠀⠀⠉⠀⠈⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠳⢤⢦⣀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠂⠸⠲⡌⢙⠓⠶⣿⣿⠷⣄⡀⢀⣀⣴⣀⣠⠤⣄⣀⣄⣀⡤⠶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⢀⠸⠷⠿⠃⠀⠄⠀⢀⣒⠒⠾⠋⠉⠤⠚⠁⠀⠀⠀⠈⡀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡶⢯⡀⠀⠀⠀⢂⡁⠈⢹⠁⠈⠐⢄⠀⠀⠈⠳⡠⠁⠀⠀⣿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡀⠀⠉⠓⠶⠶⡾⠗⣶⣤⣀⣀⣠⡾⢓⣶⡦⠴⠷⠴⠶⠚⠁⠀⠍⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⠁⠀⢾⠁⣀⠈⠃⠠⣙⠆⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢰⣿⡇⠀⠘⣿⣳⡈⠁⢀⣤⣤⡤⠀⠀⠀⠐⠤⡄⠉⠛⠛⠃⠀⡸⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠄⠀⠀⠒⠲⠶⠬⣄⡀⠀⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⣔⣀⣇⣀⡀⠴⠚⠉⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⣬⠉⠁⠛⠛⠓⠲⠖⢒⠋⠀⠀⠚⠉⠁⣀⡀⠀⠀⢀⣀⠂⣠⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⡀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡈⠉⠀⠀⠀⠺⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠙⠒⠊⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣴⠖⢲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⣿⡇⠀⠀⣿⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢙⠆⠈⣿⣿⠀⠀⣿⠁⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⣿⣿⠀⠀⢸⡇⠀⣿⣿⠀⢸⡇⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢹⣿⡄⠀⢸⡇⠀⢻⣿⣆⣾⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡠⠈⣿⣿⡄⠈⡇⠀⢸⣟⣿⡇⢀⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠘⠸⣿⣷⡴⠷⠆⣸⣿⣏⠀⠘⠟⠉⠃⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡀⠀⠀⠘⣧⡀⠸⠿⠃⠀⠀⠀⠈⠁⠀⣀⡆⢠⣤⣀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠿⣿⣷⣿⡀⠀⢀⡿⠈⠀⠀⠀⢀⣤⣾⣮⣿⡷⣿⠿⠊⣿⠿⣿⣦⣄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⡇⢹⣿⣿⡇⢀⡞⠁⠀⢀⣤⣶⣿⠟⣿⣿⢉⡄⠨⣅⣈⣀⣐⡉⢻⣿⣄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⢸⣿⣿⣷⡟⠀⠀⠀⠠⣿⣿⡛⣾⣿⣿⣿⣿⣷⢻⣿⣿⣿⣷⣷⣉⣻⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢘⣿⣿⡟⢀⠀⠀⠀⣠⣿⠇⣜⣶⣿⣿⣛⣿⠟⠠⠿⠿⠟⠛⠛⠛⠛⠓⠾⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⢨⣿⣿⢱⡇⠀⠀⢠⣿⠏⠱⣿⡿⢿⣿⡟⠃⠀⠀⠀⠀⠀⣠⣤⣤⣤⡤⢀⡀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡀⠘⢻⡟⣾⠀⠀⠀⣾⡿⣦⣼⣿⠀⣾⡟⠀⢀⠐⣂⣷⡞⠇⠉⢉⣩⣟⡳⠆⠙⢃⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⠀⠙⠛⠾⣷⡯⠀⠀⣾⣿⣿⣿⣿⣿⡻⣿⠇⠀⢈⣰⣿⣟⠋⣀⣤⣿⣷⣾⣿⣿⣷⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢃⠰⠋⠀⠀⠀⠀⢸⣿⡀⣠⣿⣿⣿⣿⣿⡏⣾⡿⠀⢠⣞⣿⣿⡟⣠⣿⣿⣏⢻⣿⣿⣛⣻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣀⠿⠂⠀⠀⠀⠀⠀⣼⣿⣿⣾⣿⣿⣶⣿⣿⢱⡿⠆⠀⢸⣿⣿⡋⠀⣿⡿⠉⢻⡿⠛⣩⣿⡛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣤⣽⣿⣤⣴⣾⣦⣤⣼⣿⣿⣿⣿⣿⣿⠏⢩⣶⣟⠉⠀⢀⣽⣿⣿⣿⠀⣿⣿⣤⡟⠃⣼⣿⣭⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢰⣿⣿⣿⡿⣿⠁⠋⢹⣿⣿⣿⣿⡿⠛⠁⠐⠈⠉⠁⠰⢦⣿⣿⣿⣿⣿⡗⣿⣿⠛⢷⣦⠙⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣼⣿⡿⠋⠀⠈⠎⣠⡞⣿⣿⠏⣠⡄⠀⣠⣄⣀⣤⣠⣤⣤⣬⣍⢙⠛⠿⠛⢿⣿⣦⡈⠛⠂⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⣿⠃⠠⠀⢲⣄⣽⣿⣿⠏⣸⣿⣦⡀⠳⠛⣿⡿⠿⣿⣯⣻⣟⣭⣷⣿⣧⠈⣿⣿⣿⣾⠛⠿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⢠⣿⣿⣿⡶⢤⣬⣿⣿⢟⢁⣴⣁⢈⠙⠛⠀⠀⠁⢣⣾⣷⠿⢟⠿⣹⣿⢻⡙⠣⣙⢿⡿⠛⠻⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠰⡙⠟⣻⣶⣮⣽⡿⢏⠀⢠⠟⣉⠀⢀⠀⠀⢄⠀⠄⠀⠛⠀⠈⣶⠜⠀⢻⣿⡀⠈⠀⣀⡤⢒⣈⡉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⣿⣿⣿⠿⠋⠀⠀⠀⡌⠋⣀⣴⣦⣶⣦⣾⣃⣴⠋⠒⣀⠀⡁⢀⠀⡸⣿⠓⢠⡞⣻⣿⣿⣿⣿⣧⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⣿⠋⠡⠀⠀⢀⣠⣾⣇⣼⡿⣿⡹⢻⣿⣿⣿⣇⡙⢷⠁⡆⠛⠀⠰⠟⣿⣠⣿⠀⣿⣿⢿⣿⣿⣿⣦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⡄⠀⠀⠀⢸⡟⢡⡦⠙⣵⣌⠰⣜⣿⣟⣹⣾⣳⠄⠳⡀⢶⠋⠀⢰⡟⠛⠃⣰⡇⠈⣿⣿⣿⣿⣿⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⠀⡝⠀⠀⠻⡄⠉⢠⣶⡙⢻⣧⣼⣿⣾⢦⣾⣳⡾⠋⠑⣆⡵⠊⣀⣰⠟⣱⢿⣴⣛⣻⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⢰⡏⠀⠀⠹⠀⢴⢸⣿⣧⠘⡏⢬⣹⡔⠂⡸⢍⢰⠆⠚⣉⣤⡿⣿⠁⢰⡟⢻⣿⣛⢋⣷⣾⣿⣏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⢸⣿⡄⠀⠘⣀⣆⣸⣿⣿⣿⠋⠂⡘⢏⡠⠁⣸⠇⠔⠊⠙⢧⠙⠁⣶⠎⢁⠞⢹⣧⡝⣳⢾⣿⡏⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡾⣿⣿⣷⡄⢿⡟⣿⣿⣿⣿⡀⠂⠀⠂⠠⠀⠡⣶⠄⠀⢠⣾⠀⣾⢋⣴⡟⢠⠟⣻⠰⣍⡾⣩⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡌⢿⣿⣞⠳⣝⠻⡿⢹⠿⡇⠠⠒⢀⣠⣏⡐⢀⡠⠲⡄⠁⣼⢃⡽⠁⣼⡿⢠⡿⠓⡈⠻⠈⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣮⣿⢯⣷⠜⣁⣄⣀⠀⡧⠀⣤⣟⣥⣿⣿⢟⠁⣴⠀⢐⣃⣜⣱⣾⢛⣿⡟⢁⡖⠀⢂⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣻⣿⢼⣯⡧⣀⣿⣌⢿⡿⢿⣿⡶⠂⡀⢀⡀⢏⣹⣟⠿⣧⣴⠟⢳⣬⠃⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡙⢿⣷⣧⣼⣿⣿⠛⠻⠋⠀⠀⢀⠎⠁⢰⡏⢠⡾⢿⡅⣨⠳⣄⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣭⠓⣏⢉⡱⣟⢀⢠⡔⣰⣁⣌⡁⢲⣤⡉⢱⣿⣬⠟⢶⠆⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣡⣬⡵⠏⢰⠋⣀⡞⠓⠾⣟⣁⣿⣠⣷⢾⠉⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣈⣷⠶⣻⣿⣿⠇⣿⡿⢁⣅⢹⢃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣍⣿⣧⣠⡽⣧⡾⠿⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣷⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⡃⠠⠃⠰⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣬⡿⣷⣶⡌⠁⠀⠀⣠⣶⣶⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠿⣿⣧⣿⠃⠀⠀⣰⣿⣿⣋⡛⢻⠿⠛⠉⣻⣿⣧⣶⡾⠆⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⣿⣾⠁⠀⠀⢰⣿⣿⠏⠉⠈⠁⠀⢠⣾⣿⣿⣿⠟⠱⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿⣏⢀⠀⢠⣿⣿⢿⢀⠀⠀⠀⢀⣾⡟⠋⠉⠀⠀⠐⣻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⢀⣸⠿⢿⣿⡴⢻⣿⣿⣾⠘⠁⠀⠀⣸⣇⠀⠀⠀⣠⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠁⠀⠀⢸⣿⣾⣿⣿⣟⠅⠀⠀⠀⣿⣣⣴⣾⣿⠟⠂⠐⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⢇⠀⡀⢼⡻⣿⡟⠁⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣀⣠⡟⠁⠹⣿⣿⣿⣷⣧⣚⠡⠞⢿⣤⣄⡀⢀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢰⣽⡿⠁⠀⠀⠐⠋⢿⠉⠀⠁⠀⠂⠀⢙⠻⠟⠳⠾⠷⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⢀⣼⣿⡁⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣬⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⠀⠆⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡍⠀⠀⠀⠀⠂⠐⢒⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠾⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡀⠀⠀⠀⠀⠴⠊⠀⠆⠀⠀⢀⠄⠀⠀⠀⠈⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡗⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣾⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣎⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⢠⠀⠠⠃⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣶⣞⠿⠟⣠⡧⢀⣷⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣮⣟⣦⣼⣧⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢦⡀⣴⡆⣿⡛⠀⣷⣴⣷⠀⣴⡆⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢰⡄⣿⠉⢹⠸⡟⢹⡇⢿⣥⠀⡏⠁⡇⣸⢿⠁⣸⠁⣷⣠⠗⠦⠀⠀⠀⠀⠀⠀⢀
⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠻⡌⠛⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠚⢱⣏⠃⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⣴⡆⣴⡰⠂⡶⠆⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⢶⠗⡇⢿⠹⡄⠯⡅⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠂⠀⠂⠐⠒⠒⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⣴⣶⢶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⢶⣶⣶⣦⣠⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⣠⠀⠠⠀⠀⢀⠀⠀⠀⣼⣧⣽⢾⠋⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠐⡆
⠀⢠⣿⠀⢃⢸⡀⣆⠀⣦⠀⣟⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣀⠨⣽⣿⣿⣷⣿⠂⠀⠀⠀⠀⠀⠀⢀⣿⣇⢀⡇
⠀⠀⠛⠻⠊⠻⠿⣿⠶⠿⠶⣿⡟⠛⢿⣶⣶⣾⣭⣡⣾⣿⣃⣌⣠⣄⣬⣬⣿⣧⣼⣴⣿⣿⣿⣿⠏⢿⠀⢰⠀⢠⡄⢰⣆⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠻⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⢿⠇⠀⠈⠀⠻⠆⠘⠃⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠴⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠠⢄⠀⠑⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣸⣿
⢀⡰⣃⠀⠄⠀⠨⠀⣀⣼⣿⢇⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠙⠒⠾⠛⠁
⠈⠛⠿⣿⣄⣠⣴⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢂⡀⠠
⠀⠀⠀⠹⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣼⡿⠀⠀⢠⠀⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⠃⠀⠀⣀⣴
⣦⣤⣤⣦⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣧
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈⠉⣁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠈⠁⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠀⢄⠔⠒⠢⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠈⠉⠂⢀⠀⠀⠈⠂⠀⣀⡀⣀⠤⠄⠊⠀⠀⠀⠀⡀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄
⣤⡒⠁⠁⠀⠈⠈⠝⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠚⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡎
⠑⠈⠦⣄⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠠⠁⠀⠈⠀⠀⡤⠄⠚⢡
⠀⠀⠀⠈⠢⡀⠉⡀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠈⠁⠙⡄⠀⠀⠙⠄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠄⠠⠀⠰⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠈⠐⢲⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠁⠒⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⢄⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠁⠺⠀⠀⠱⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠑⢄⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⣀⣄⠐⠂⠀⠠⢄⠀⢤⠃⠀⠀⠀⠀⠠⠄⠒⠂⠒⡠⠴⢤⡀⠀⠀⠂⠀⠀⠁⠢⡀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⢏⠙⢳⣻⠁⠀⠀⠀⠀⠀⠀⣀⡘⠀⠀⠀⠀⠁⠀⠀⠀⡠⢡⡀⠤⢈⠀⠀⠀⠰⡀⠀⠀⠈⠑⠠⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡋⠀⠄⡀⢐⣠⢜⠤⠉⠀⠀⠀⢄⠀⢠⡄⠀⠀⠁⠀⣀⣀⠚⠁⠀⠀⠀⠈⠐⠢⢀⡀⠀⠀⠈⢛⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠍⢐⡀⠀⠀⢄⠀⠈⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠁⠁⠀⠄⢛⣒⣐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⢀⢀⡀⠀⠀⠘⠁
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠂⠁⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣄⡀⢀⠀⣿⠃⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⠋⠙⢿⡄⠀⠀⠸⠃
⠀⠀⠀⠀⠀⢴⣾⣣⣂⠀⠀⡀⠀⠀⣀⣴⣿⣿⡟⠁⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣧⠀⠀⣿⣿⠟⠉⠀⠀⣀⣄⣠⡏⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠀⠀⠀⠁⠀⢀⣴⣾⣫⣽⡿⠖⠀⠀⠀⠀⣴⣶⠟⠋⠁⠀⠁⢦⣤⣶⡿⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣰⣿⣿⣿⣿⣏⡴⠀⠀⠀⢠⣾⣿⠁⠀⠀⠀⣷⠿⣟⠿⠋⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠆⠀⠀⢰⣿⣿⣿⣿⣿⡏⠸⠂⠀⠀⢸⣿⣿⠀⠀⠀⠼⣽⠞⠁⠀⠀⠀⡄⠀⠀⠉
⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⣾⣿⡿⣄⠀⠀⠀⠀⠀⠀⠀⢀⡾⣇
⢀⡠⢐⠒⣿⣽⣿⣿⣿⣿⣿⣿⡏⠐⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⢀⠀⣾⣿⣷⡴⠀⠀⠄⠀⠀⠀⠀⢸⣿⣿⡃
⣞⣿⡋⠽⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⢸⣿⣿⣿⣿⣏⡂⠀⠀⡎⠃⣘⣿⣿⣿⣎⡥⢀⡀⠀⠀⠀⢼⣿⣿⣿⣦⣀
⢹⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣻⣷⡄⠸⣿⣿⣿⣿⣿⣿⠳⣄⣤⣾⣿⣿⣿⣿⣿⣿⠉⡆⠀⠀⠐⢸⣿⣿⣿⣿⠿⠛⠋
⠀⠙⠛⠛⠉⠀⠀⠀⣼⣿⣿⣿⣿⠬⢽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡾⠁⠀⠀⠀⠈⣿⣿⣿⣿⣷⡄
⠀⠀⣠⣤⡴⣶⣶⣾⣿⣿⣿⣿⣿⡇⡀⠈⢡⡀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⣿⡃⠀⠀⡀⠀⠸⣿⣿⣇⡙⢷⣿⣷⣠
⠀⠀⢰⣯⢵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⣿⣇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠟⣡⠶⠒⠉⢄⠀⠀⠈⠀⢿⣿⣿⣤⠌⢿⡄
⠀⠀⠸⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣫⣿⣿⡷⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⢀⢾⠻⢷⣦⣼⣀⡴⣄⠀⠀⠀⣿⣿⣷⡄⠈⣧
⠀⠀⢠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⢟⣿⠁⠀⢀⣾⣿⣿⣿⣿⣿⣿⡟⠿⣇⠀⢤⡰⠉⢿⢿⣿⡚⠳⢄⣴⠋⢿⣿⣧⡰⡏
⠀⠀⢘⣬⣴⣿⣿⣿⣿⣿⣿⣿⣿⣻⣝⣷⣮⣷⣿⣿⣿⡿⢻⣿⣿⣿⢣⡀⣈⠲⢄⠴⢃⢰⢧⣿⣷⣿⣶⠋⠀⠀⢹⡛⣿⠁
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠟⢁⣾⣿⡿⢣⡈⠙⢿⣷⣬⣤⠀⣪⣷⣿⣿⣿⣧⠄⢐⠀⠀⢧⠘
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⡏⠀⣼⣿⣿⣿⠋⣢⢐⣶⣿⣿⣿⣮⠙⠦⣿⣧⣼⣯⠀⢀⠐⣜⠢⡄
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⣯⣠⡈⢿⣿⣿⢿⣉⣷⢾⡟⣷⣼⢿⣷⣿⣾⣟⡿⠏⠀⢰⠄⠈
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠉⣿⣿⣿⣿⣿⢳⡌⠻⣿⣧⣴⣿⣿⣿⣿⡿⢫⢼⣿⣿⣿⣿⡆⠀⠋⠀⠀⢀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣇⣽⡧⣾⣏⡿⢉⣿⣴⣦⣻⣿⣿⣿⠛⠉⠹⠀⢈⣿⣿⣿⣿⠀⠐⣄⠀⠀⠃⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠁⣾⣯⡿⣿⣿⣿⠟⠉⠀⠐⡈⣿⣧⣿⡿⠀⡉⠸⣷⡄⣶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣹⣟⠀⣿⣿⡟⣼⡿⠿⡄⠀⢠⣄⣀⣛⣿⣿⡇⠸⠁⠀⢻⡛⣹⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣶⣯⣸⣿⣿⣦⣀⣈⡐⢂⠈⠙⠻⣿⣯⡄⣷⢀⠀⢾⣇⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣻⣿⣗⣢⠌⢹⣿⣿⣿⢹⢾⡖⠈⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣟⣿⡟⢤⢿⣿⣿⣇⠿⠾⣅⡦⡿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣯⣴⠿⣽⣿⠀⣶⢾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡛⣰⣿⠉⣆⠺⢻⡿⡣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣍⡿⣿⢧⣶⣼⣿⢏⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⡿⠿⣿⣿⠟⣿⡿⣿⣿⠟⣿⡟⣿⣿⢿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⢸⣿⣷⢸⣿⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣻⣿⣦⣿⣧⣿⣿⣮⣿⣵⣿⣿⣿⣏⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠈⠉⠻⣧⣨⣿⣞⠿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⡿⢟⣿⣧⣠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⣀⣤⣤⣶⣶⣆⣿⡿⢀⣤⣀⠀⠀⠀⠀⠀⣀⣠⣀⣀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠨⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠉⠙⠛⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⢀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣭⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⡤⠤⠟⠛⠛⠛⢉⠉⠛⠛⠻⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢂⣠⣤⣀⣨⣾⣿⣶⠶⠤⠾⢿⣛⣻⣛⣩⣿⣿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⣤⣤⠔⠊⢀⣤⠿⠛⠋⠉⠀⠀⠀⠀⢀⢼⣯⣟⣽⣿⣿⣿⣷⣼⣃⣸⣎⣿⣻⡿⣿⣿⣿⣟⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠄⠚⠋⠁⠀⠀⠀⠀⠀⢀⣴⣾⣿⡘⣭⣿⣿⣿⣿⣿⣻⣿⡿⢿⡿⠎⠀⠀⣷⡛⠉⠉⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢿⣛⣫⠽⢿⣿⣿⣿⣿⣿⣿⣿⣓⠣⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠦⠦⣴⣶⣿⡿⠛⠊⢡⣾⣿⣷⣾⣯⣽⣿⠟⠁⢹⣿⣿⣿⣯⣶⣿⣾⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠘⠁⠀⠀⡟⣙⣿⣿⣴⣽⣿⣿⣷⠀⠀⠻⢿⣿⣿⣿⠛⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠀⠛⣾⣿⡿⢣⢿⠀⠀⠀⠀⠉⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⠄⠀⠀⣴⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣠⣴⣾⣏⣙⣿⣿⣿⣿⣿⣥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⣿⣿⣿⣿⣏⣀⣿⣶⣄⡀
⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⡏⣽⣿⣿⢻⣿⣿⣷⣶⣶⣶⣶⣤⣤⣶⣦⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⡇⣿⣿⡟⣼⣿⢣⣿⣿⣿⢹⣿⣿⢯⣿⣿⡛⣿⣯⣿⣹⣷⢻⣿⣦⢻⣿⣷⣍⢻⣿⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣷⣿⣿⣱⣿⣿⣸⣿⢩⡿⣼⣿⣿⢺⣿⣿⣿⣹⣿⢸⡟⣿⢸⣿⣿⣧⣹⣿⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣷⣿⣾⣿⣿⣿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠀⠂⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠊⠀⠁⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡼⠒⠒⠠⢩⣶⠘⠟⠒⠈⠤⠒⠀⠖⠆⡶⠂⠧⠖⠗⠴⠐⠀⠂⠆⠃⢠⣮⠡⠐⠒⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠁⠈⠆⠀⠀⠀⠀⠀⠁⠶⠀⣀⡀⠀⠀⠀⢮⣧⠞⠀⢬⡷⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠠⠀⠠⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠐⣿⠆⠀⠐⢷⠖
⠀⠀⠈⠀⣿⠦⠀⠀⠀⢀⠄⠒⠇⡈⠃⠀⠀⠀⠀⠀⠛⠀⠀⠀⠚⢀⠀⠀⢀⣀⠀⠀⠀⣀⢠⣤⣤⡤
⠀⠀⠄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠈⠀⡀⠀⠀⡄⣠⣄⠀⠀⠙⠀⠀⠁⠠
⠀⠀⠀⠀⠰⠀⠀⠀⣀⡀⠀⠀⠀⠤⠉⠰⠄⢂⠀⠈⠀⠀⠀⠸⠀⠂⠀⢰⠇⠋⢸⠀⠀⠀⠖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠀⠀⠐⠀⠀⠀⠀⠘⠀⠈⠄⠀⠀⠀⠄⠁⠀⢸⠁⠀⠘⠀⠀⠀⠀⠘⠷
⠀⠐⡾⠚⠀⠀⠓⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠈⠂⠠⠤⠄⠀⠈⠀⣰⠓⠀⠀⠀⠀⠀⠀⠚⠶⢷⠚⠃⢐⡄⠀⠀⠀⡸⠀⠀⠁⠀⠀⠀⢰⡷⠝⣇
⠀⢠⣄⡀⠀⠀⠀⠐⠟⠛⠖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⡠⠄⠀⡀⠤⡁⠀⠀⠈⠁
⠀⠀⣠⡷⠻⠃⠀⠀⠀⠠⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⡴⠔⠀⠀⢆⠀⠀⡄⠀⠀⠀⠀⠀⢠
⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠁⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠰⠈⠃⠀⠀⠀⠂⢠⠀⠠⠀⠀⢀⣀⣴⣴⠀⠀⢰⣿⡀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⣤⠀⠀⠁⠀⣿⣿⣿⠀⠀⠸⣿⠃⠀⠀⠠⠀⠀⠉⠀⠀⠀⠀⠘⠀⡘
⠀⠀⠀⡡⠂⣢⢬⡳⡄⠀⠀⠸⠟⠀⠀⠀⠉⣷⣿⣿⡤⠀⠀⢻⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠉
⠀⠀⣄⠣⠀⠠⠄⠂⠇⡖⠀⣈⣡⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠛⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠠
⠀⠀⠈⠀⠀⠀⠀⣀⠜⠀⠀⠈⠃⠀⠀⠀⠀⣿⣿⣽⠃⠀⠢⣤⠄⠀⠀⠁⠤⠤⣤⣄⠀⠀⠀⠤⠖
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⣶⣤⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣸⣿⠃⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢦⡀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡿⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠙⢶⡀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⢀⣀⣴⠇⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣤⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠞⠃⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⢤⣄⠀⠀⠀⢰⡇⠀⠀⢰⠏
⠀⠀⠀⠀⠀⢀⣤⡤⠤⠶⠶⠶⠶⢶⡞⠁⠀⠀⠀⠙⢷⠒⠒⠒⠒⠚⢳⡟⠁⠀⠀⠀⠈⢷⡖⠒⣿⠀⠀⢠⡿⠶⠶⠶⠶⠤⣤⣤⡀
⠀⠀⠀⠀⠀⣿⠉⠛⠛⠛⠒⠒⠲⠾⠶⠶⠶⠀⠶⠶⠾⠷⠶⠶⠶⠶⠾⠶⠶⠶⠶⠶⠶⠾⠷⠶⠷⠶⠶⠛⠒⠒⠚⠛⠛⠛⠛⠉⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢹⠀⢀⡾⠋⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⢳⡄⢠⡇
⠀⠀⠀⠀⠀⢸⡇⠘⣇⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣰⠇⣸⠃
⠀⠀⠀⠀⠀⠈⣧⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠈⠳⠞⠁⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⢀⡟
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢀⡶⠛⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⣦⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⢧⣄⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⠞⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠘⣧⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣸⠇⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣂⠈⡉⣱⠦⠤⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⡤⠜⠉⠉⠀⠉⠉⠓⢦⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⢺⣋⡵
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢠⡾⠁⠀⠀⡴⠋⠉⠉⠁⠂⠀⣴⠏⠉
⠀⠀⠀⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠁⠀⠀⠸⡏⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⢐⣲⠆⣀⠀⣰⠶⠚⢹⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣠⠴⣦⠴⠦⣬⡑⠚⠋⠉⠒⠮⡙⠍⢂⡀⢸⠀⠀⠀⠀⠀⠀⠈⠟⢌⣀⣀⣠⡾⠁
⠠⠚⣅⡀⠁⠀⠀⠈⢻⡄⠀⠀⠀⠀⠘⢆⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣴⠓⢄
⡂⢦⠏⡋⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⢣⡀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⡆⢱⡤⢤⡀
⣀⠚⠉⠃⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠋⡷⠊⠙⠺⢿⠛⠙⠲
⠘⣇⠀⠀⠒⠤⠄⣠⠴⠦⠽⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀⣏⡐⢲⣦⣄⡀⠀⠀⠀⣼⠀⢰⠃⠀⠀⠀⠀⠇
⠀⠈⠳⠤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠵⢒⣿⣿⠴⠋⠀⠈⠹⣆⠀⠀⣿⠀⡜
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇⢠⠃⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢀⣴⠾⢯⣍⠉⢧⡉⠒⢦⡀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⡇⠸⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⢀⡴⠋⠂⠀⠁⠳⠈⠀⠙⣆⠐⠿⣦⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠋
⢠⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣇⠀⠀⠙⠢⠤⢄⣠⣤⠴⠞⠁⠀⠀⠀⠙⠲⠤⣤⣀⣠⣇⠴⠃
⢸⠀⠛⡆⠀⠀⠀⠀⠖⢠⡊⠰⠤⣤⣼⠎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⢀
⣿⠀⠀⠳⠦⣀⣰⠧⠬⠀⠀⢀⡎⢀⠇⡆⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠴⠃
⠹⣄⡇⠈⠲⢶⠶⣄⠀⠀⢀⢞⡔⠁⠀⡇⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠊⠉⠉⠀⠴⠛⠁⠚⠋⠉⠉
⠀⠘⢧⠆⠀⠨⣄⠈⠙⢯⣴⣂⠀⢀⡞⢁⣸⠃⠀⠀⣀⡤⠴⠒⠋⠁⠀⠀⠀⢀⠀⢀⣶⠀⢦⡉⠉⠙⠒⠤
⠀⠀⠘⢧⣀⣀⣈⠓⠙⢓⣺⡿⣻⠋⡰⠋⠀⢠⣶⣉⣀⣠⣀⣠⣤⣤⡶⠴⠶⠾⠛⠛⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⣄⠀⠀⠀⠀⠀⣠⡖⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠁⠉⠉⠉⠉⠉⠊⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢽⣄⠀⠤⠄⠀⣤⡀⠀⠀⡼⠑⠒⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢍⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⣳⡄⢆⣰⣆⡠⠀⣒⡾⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡌⢢⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⣠⠓⢤⣀⣀⣠⠴⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢱⡀
⠀⠀⠀⠫⡉⠈⠀⠀⠀⠀⢹⡀⠀⠈⠉⠀⠀⠀⠀⠀⡔⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⣠⣷⡀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠈⡇⠀⠀⠀⣞⠒⠒⠛⠒⠒⠲⠤⠤⠤⠈⢧⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠳⣄
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠻⠦⠴⠒⠒⠒⠒⠒⠒⠒⠒⠮⠷⣄⡀⠀⠉⠒⠦⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠄⠐⠒⠒⠀⠠⠤⠤⠤⠤⠄⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠔⠒⠊⠉⠉⠉⠉⠒⠒⠲⠤⠤⠤⠖⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠤⠶⠀⠀⠀⠀⠂⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠊⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡊⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⣤⣧⣴⣶⣶⡶⠷⠶⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢟⡿⢏⣠⣶⣿⠟⠁⠀⠀⠀⠀⠀⢀⣠⣾⣇⠜⠉⢂⡇
⠀⠀⠀⢠⠀⠀⠀⠀⠈⠐⠋⢽⢶⢿⠿⠛⠁⡀⠀⣀⡁⠀⠄⠀⠉⠛⠋⠹⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠭⢅⠀⠀⠀⣠⣀⢲⠢⣄⣄⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⢀⠤⠊⠰⠆⠀⠀⠀⠀⠈⠄⢀⡀⠳⠷⠀⠉⠉⠡⠤⠋⠋⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠀⠦⠀⠢⢀
⠀⠈⠒⠄⠀⠀⠀⠀⠀⡇⠀⢠⠀⢀⠀⠀⡉⠉⠀⠀⠀⠈⢉⢁⠀⡀⠠⡀⠰⢄⠀⠀⠀⠀⢀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⢅⠀⠀⠀⠗⠀⡜⠀⡏⢀⠀⡇⠀⢨⠀⠀⢇⠈⡈⡀⡇⠀⠱⡄⠀⠓⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡈⠀⠁⠀⠘⠀⠀⠘⠀⠃⠁⡀⠀⠀⠄⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠛⠉⠉⠉⠉⠉⠙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣶⣦⣄⣀⠀⣿⣿⣷⣾⣿⣿⠿⠿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄
⠀⠀⠀⠀⠀⣾⣿⠋⢀⣿⠉⠛⠻⢿⣿⣿⣿⡿⠛⠁⣀⣤⣤⣄⣈⠙⠻⢿⣿⣶⣄⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀
⠀⠀⠀⠀⠀⢻⣿⣾⣏⡁⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠿⠟⠛⠻⢿⣷⣦⡀⠙⠿⣿⣷⣄⠀⠀⠀⠀⠀⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣦⡀⣠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠈⢻⣿⣷⡀
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠹⣿⣿⣄
⠀⠀⠀⣠⣾⣿⡿⠛⠉⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄⠀⠀⠀⠈⢿⣿⣷⡀
⠀⢀⣾⣿⡿⠉⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠙⢿⣿⣦⡀
⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠙⢿⣿⣦⡀
⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠻⣿⣿⣄
⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠈⢿⣿⣧⡀
⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⡀⠀⠀⢀⣠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⣄⣀⣀⣀⣀⣀⣠⣤⣶⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠋⠀⠈⠃⠀⠀⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠘⡄⠀⠀⠀⣀⣀⣀⠀⠂⠀⡰⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠈⠢⣄⡀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠉⣭⡉⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⡟⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⠀⠀⠀⢤⡀⢀⠀⢨⡄⠀⠀⢂⢀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡇⠀⠆⢸⠁⠀⠀⠰⣿⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠐⠒⠤⣐⢕⠠⡀
⠀⠀⢠⠠⠤⠤⠤⠤⠤⠄⠀⠐⠒⠀⠒⠒⠒⡄⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀⠈⠑⢕⡌⢂
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢻⣿⣿⣿⡄⠀⠀⠀⠀⠁⠀⠀⠈⢫⠀⠃
⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢁⠀⠀⢀⣀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠈⠐⠢⠤⣀⡀
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⠜⠀⠀⣖⡄⠁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠠⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢀⣀⡀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢎⠆⠀⠀⠀⠉⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⠉⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⣿⠿⠋⠀⠀⠀⢀⠀⠒⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢠⣴⣿⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠲⣄⢀⠀⢀⣾⣿⡿⠏⠈⠀⠀⢀⡠⠄⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣧⣿⣿⡗⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⠛⢁⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣆⢠
⠀⠀⠀⠀⢸⣿⣿⣿⣏⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠿⠛⠛⣿⣿⡿
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣶⡤⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠰⠞⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠾⠝⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠘⣮⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠒⠣⡀
⠀⠀⠀⠀⠘⢿⠹⡌⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⡔
⠀⠀⠀⠀⠀⠘⡅⠁⠄⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠛⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠙⠢⢤⣭⣤⣤⣶⣶⣿⣿⠉⠀⠀⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣷⣠⡢⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠻⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠟⠋⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⣶⣾⡧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠛⠋⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣇⠀⠀⠀⠀⢀⣠⣶⣿⣟⠉⠀⠀⠀⠀⠀⣠⣾⠟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⣻⠂⠀⠲⣿⣿⣿⣿⣅⡀⠀⣀⣠⣶⣾⢿⠁⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⡾⠛⠻⡛⢁⣀⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡒⡳⠓⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠅⠚⠙⠣⠴⣶⠪⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢗⠉⠑⠿⠤⣄⢈⢀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠸⠛⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠂⢀⡔⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⡠⠞⠉⡅⠀⣠⠞⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⠀⠀⠀⠈⠙⠒⠤⣀⣀⡾⠀⠀⠀⠈⠙⠁⠀⠀⡠⠕⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⣠⣶⡟⠓⢄⡀⠀⠀⠀⠀⠀⠀⡍⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⢄
⠀⠀⠀⠀⣀⣤⠶⠋⠁⢀⢤⠂⠀⠙⠋⠐⠠⡎⠈⠉⠉⠁⠀⠀⠾⢁⠃⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠈⢦
⢀⣠⠴⠚⠉⠀⠀⠀⠀⠈⢈⠢⡀⠀⠀⠀⠀⠈⢣⡒⠤⣀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠑⢌
⠻⣧⡀⠀⠀⠀⠀⠀⠀⠠⡊⠑⢌⠢⡀⠀⠀⡀⢸⡁⠀⠈⣡⠾⢻⢀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡇
⠀⣿⠙⠷⣄⠀⠀⢠⠀⠀⠘⣄⠈⠢⠟⣖⠭⠒⢉⣠⣶⣋⡥⠴⡇⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⡸⠁
⠀⣿⠀⠀⠈⠻⢦⡈⠘⡄⠀⠈⠢⠖⠉⣀⡤⠞⠋⠁⠀⠀⢀⡄⠈⠒⠤⣀⠀⠀⠀⠀⢀⠀⠀⢀⠞⠁⠐⢀⠞
⠀⣿⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠀⣰⠟⢴⢡⠄⠈⠧⠚⠁
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠶⠋⢹⡆⠀⠀⠐⢼⡶⠋⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠋⠁⢀⡸⣃⡠⠎
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣸⠀⠀⠀⢀⣤⡤⠖⣾⠛⠒⠒⠒⢶⠚⢻⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⢹⡀⠀⠀⠀⠙⡆⠀⣿⠀⠀⠀⠀⢸⡇⠸⣿
⠀⢹⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⡇⠀⠀⠀⡏⢳⡀⠀⠀⠀⠹⡀⣿⠀⠀⠀⠀⢸⡆⠀⣿
⠀⠘⢧⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⢳⡀⠀⠀⠀⢳⡟⠀⠀⠀⠀⢸⠁⠀⣿
⠀⠀⠀⠻⣄⠀⠀⠀⠀⢸⠀⣀⣠⡷⠤⠴⡖⠃⠀⠀⢳⡀⠀⠀⠈⡇⠀⠀⠀⠀⠚⠂⠀⢹⡄
⠀⠀⠀⠀⠈⢷⡀⠀⠀⢸⠋⠉⠀⣀⣀⣀⡗⠀⠀⠀⠀⢳⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠸⠷
⠀⠀⠀⠀⠀⠀⠙⣆⠀⢸⡛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⠤⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣰⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣶⣶⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠠⢌⡑⠂⢄⠀⠀⠀⣀⣤⣤⣶⣿⣿
⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⣂⡐⠲⠿⣓⡀⢄⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢿⣾⡿⣻⣡⣿⣾⣿⣿⣿⣿⣟⠿⣏⠟⠻⠛⠛
⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠂⠀⠐⠀⠀⣠⠀⠀⠀⠀⠀⠀⢺⢷⡶⠿⠻⢛⠟⣹⣟⢿⡻⣌⢣⠙⣄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠅⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢠⣾⣿⣿⢦⠳⡈⢀⢡⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠠⠀⠀⠀⠰⣟⡻⠯⡹⠽⠖⢱⡀⠀⠃⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣐⠀⢀⡀⠄⠀⠀⠀⠂⢀⠀⠀⠀⠀⠒⠉⠉⠁⠂⠀⠀⠀⠀⠀⠚⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣨⣷⣄⡀⢤⡁⠀⠁⠀⠀⢈⢲⣴⣋⠀⣀⠀⢀⠀⣀⠀⣀⣀⣀⣀⡀⡉⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠠⠾⠻⠙⠋⠦⠈⠑⠈⠙⠭⡙⠽⡿⣿⢯⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠈⠑⠀⠀⠀⠀⢸⢦⡀⠀⠑⠤⠑⢌⠛⡾⠁⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⢀⡀⠠⠄⡒⠈⠁⠀⠀⡀⠀⠀⠀⠀⠏⡙⢦⣀⠠⠈⠢⠈⠃⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠌⠷⣶⡾⠁⠀⠀⠀⠔⠀⢞⠀⠀⠀⠀⠁⠈⢻⣷⣄⠐⠄⠐⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠋⠀⠀⠀⢀⠄⡠⢁⣾⠀⠀⠀⢸⠀⠀⣛⣷⡍⠳⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢁⡴⠠⣿⣿⠁⠀⠀⠈⠀⠛⡡⢋⠔⢁⠄⠉⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠁⠄⠀⠒⠀⠈⠒⠀⠉⠑⠀⠀⠀⠀⠘⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠃⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠄⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⡀⣠⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠃⠀⣠⠀⣀⣀⠀⠀⠘⠀⠀⠨⠀⠊⠉⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡤⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠠⣀⢀⠀⠀⠀⠀⠀⠀⢀⣴⠁⠀⣀⠠⣤⣄⠀⠶⡄
⠀⠀⠀⠈⣇⣠⡉⡾⠛⢧⣦⡄⠀⠀⠀⠀⠀⣠⠀⠠⢀⢄⠀⠈⠛⡘⢷⣄⡀
⠀⠀⠀⠏⠛⠑⠀⠁⠀⠸⡏⢰⣤⣇⠁⢴⣦⢾⣾⠁⠀⠈⢁⣠⣀⡀⢀⠻⠿⡏⠑⠂⠀⠀⠓⠭⠒⠤⢀⡀
⠀⠀⢧⠀⡠⠀⠐⠀⠀⠀⢸⣛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⣮⢶⣄⠈⠙⠢
⠀⠀⠐⢄⡀⣀⣀⣠⢴⣾⣿⣿⣿⣿⣯⣿⣿⠛⠑⢷⠄⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⡆⠙⢎⠳⣽⠷⡆
⠀⠀⠀⠈⠛⠉⠸⠗⠛⠛⠻⠿⣿⣿⣿⡟⠁⠀⣸⣿⡖⠀⠀⢢⣠⠀⣞⠉⠁⢀⠀⢀⠚⠛⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⡏⠀⠀⢀⠀⠀⢀⠀⠚⢻⡄⠙⠃⠴⢛⡅⠝⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠂⠀⠀⠀⣀⠁⠀⣀⠀⠐⠉⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⣀⣀⡤⠶⠂⢤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣘⣧⡞⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡤⠒⠒⠲⣶⣠⣤⣤⡤⣤⠤⢤⣀⡀
⣦⡀⠈⠹⡍⠛⠶⣤⡀⠉⠀⠀⠀⠀⠀⣠⠤⠤⠄⠤⣄⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠉⠉⠉⠉⠁⠘⠛⠛⣿⣿
⢻⡷⠀⠀⠑⢀⣀⠀⢙⣷⣄⢀⠀⢀⡞⠋⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢽⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⡜⠳⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⡀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡈⠽⠿⠶⠥⣔⠄⠀⠀⠙⢿⣿⣷⣤⣠⣤⣀
⣶⠇⠀⠀⠀⠀⢠⣤⠶⠞⠂⠂⠠⣀⠀⢠⣠⣴⡯⢉⠹⢿⡛⠀⠀⠀⠀⡰⢡⠊⠀⠀⠀⠀⠀⡸⠿⠲⣦⠀⠀⠑⢿⣿⣿⣿⣿⣿⣶⣀⣴
⠋⠀⠀⠀⡠⠞⢋⢔⣶⠶⠞⢏⢀⣻⡿⠓⠉⠉⠛⢿⢤⡈⠳⡄⠀⠀⠠⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡆⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠁⠀⢀⣽⠏⠀⢀⠞⠉⠀⢸⡀⠀⠀⠀⠀⠱⣵⠀⡷⠀⠀⠀⣸⠀⠀⠀⣴⡿⢀⣤⠀⠀⠈⣧⡅⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⢀⣤⣀⣰⠋⣴⠇⠀⠀⠁⠀⢀⣤⣘⡢⢄⠀⠀⠀⠀⢿⣦⡃⠀⠀⠀⡷⠀⠀⠀⠘⠎⠛⠁⠀⠀⠀⡏⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠁⠀⢀⣿⠀⠀⠀⠀⠀⡼⠋⠈⣽⣮⣆⠀⠀⠀⠀⠀⠁⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠬⠭⠷⣦⣤⣤⣤⠗⡒⠉
⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⢀⠁⠀⠀⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠦⠒⠶⣼⣿⢿⡏⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠂⠀⠸⢿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⢣⣴⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⢀⣀⣠⡤⠞⠋⡁⠈⠀⠀⣧⡀
⣲⡦⠀⢠⣀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⣿⣿⣷⠡⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣉⡉⣀⣄⡤⠔⠀⠀⠀⠀⠙⠯
⠀⠉⠈⠉⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠛⣿⣿⣆⣇⠀⠀⠀⠚⠉⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠀⠀⢀⣼⣄⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠙⢋⣀⣨⣉⣓⣠⠄
⣀⣆⣀⡀⡀⠀⠀⠀⠘⣗⡀⠀⠀⠀⠀⠀⠈⢀⣿⢋⠇⠀⠀⠀⣿⣿⠀⠛⠙⢟⠲⠶⣴⠤⠀⠸⣡⣘⣷⠯⠒⠊⠙⠛⠃
⠛⠋⠉⠉⠀⠐⠀⠀⠀⠙⢜⢶⡲⠴⣶⠾⢾⡿⣧⠊⠀⠀⡄⠈⡿⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠉⠵⠂⠀⠀⠀⠀⠀⢣⠀⢿⠀⠈⡃⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⣶⢿⣿⣿⣿⣿⣿⣶⣶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡟⣻⣯⠊⠀⢙⢙⣛⣻⡟⢋⣭⣿⣿⣿⣿⣶⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠛⢷⣿⣏⣼⢜⣵⣯⣤⡼⣿⣿⣿⠿⣯⣕⡛⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢯⡙⢦⣄⠙⣿⣧⣼⣿⣿⡿⢽⣿⣿⣛⠿⢦⣌⣿⣶⣿⣷⣿⣔
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⢿⠳⣝⠢⣌⠳⣼⢿⢿⣿⠿⢿⣼⣏⣩⠬⠭⣳⣬⣿⣿⣯⡍⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡄⡙⠪⠿⠷⣿⠃⠔⡅⢀⣰⢻⢫⢞⡿⣶⣤⣼⡿⠭⠭⠭⢝⠔⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⣬⣙⢒⠢⣴⡇⠈⠘⠁⣴⢃⣳⡿⢣⢊⢴⡿⢿⠿⠍⠉⠉⠵⣦⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣻⣯⣤⣸⣿⣿⣿⡀⠀⢀⢿⡎⣿⠧⡳⣣⠉⢀⠟⠙⡲⢶⡤⡬⣑⢽⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠟⠻⠿⣿⣿⣿⣶⣿⣼⣧⣿⢠⢣⠃⢀⣞⣀⣀⣯⡵⣾⣮⣾⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣟⠲⠶⢤⣤⣙⣿⡟⣻⡟⣿⣿⣾⣿⣤⣾⡅⠉⠻⣗⣝⢞⢊⣦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡷⣮⣔⢤⣹⣿⢠⣿⣿⡏⢹⣿⡟⡿⣿⣿⠆⠀⠈⡉⡀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠺⣵⣾⣿⢨⡇⠸⡇⣾⡿⢣⣇⢊⣿⡁⢀⠀⣯⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣿⣿⠾⣧⣦⣇⣿⣷⢸⢉⢸⣿⣿⣼⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⡦⣭⣿⣿⣿⣷⣿⡿⠻⠟⠙⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣛⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠨⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡀⠀⠀⠀⠀⠀⠀⡂⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠃⡀⠦⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⠤⠂⠉⣍⠙⠉⠒⠠⡘⠧⣀⣀⣠⡟⠲⠊⠻⢳⠋⠀⠀⠈⠂⠉⠊⠀⠀⠀⡀⣀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⣠⠃⠀⠀⠀⣿⣿⠾⠋⠀⠈⢦⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⠠⠚⡥⢄⠀⠀⣉⠠⠔⠋⣥⣶⣦⡀⠁
⠀⠀⠀⠀⠃⠀⠴⠟⠿⣿⡿⠛⠋⠃⠀⠈⡄⠀⠀⠨⠆⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⡆
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠷⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⠟
⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣸⠟⡿⠈
⠀⠀⠀⠀⠀⠈⠒⢤⣀⡀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠁⠟⡲
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠡⠜⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⡠⠂⢄⡎⢠⡀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣲⣆⠄⢤⡝⢃
⠀⠀⠀⢀⠀⠀⠀⠈⠻⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢭⢜⢣⠿⠏
⠀⠀⠀⠀⢁⠖⢠⣄⠀⠀⠀⠀⡐⠀⠀⠀⠀⣀⠔⠚⠓⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⡿⢿⠧
⠀⠀⠀⠀⢨⠀⠉⠁⠀⠀⠀⡰⠁⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠩⠉⠀⠤⡕⡢⠀⠀⠔⠁⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠢⠄⡀⠠⠔⠈⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠠⠂⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠐⠁⠀⢀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⢀⣀⣢⠤⠒⠉⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊

⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠠⠄⠀⠀⢀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠄⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠄⠂⣠⣦⡀⠠⠀⠀⢀⠀⠀⠀⠀⠂⢠⣤⢴⠵⡀
⠀⠘⣄⠀⠄⠀⡿⠀⠀⠀⠀⠠⡀⠀⠀⠀⡊⠀⠀⠀⢄⠙⠮⣙⠒⠂⠀⠀⠀⠀⠈⠐⢺⠋⠑⠀⠑⠂⠀⠒⠉⠀⠀⠀⠀⣤⣲⡭⣆⣼⣽⣄
⠀⠀⠈⠓⠺⠀⠘⠆⠀⠀⣀⡴⠃⠀⠀⠀⣧⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠈⠃⠻⠜⡰⢡⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⡀
⠀⠀⠀⠘⠠⠟⠄⠐⢂⣠⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠤⠬⣳⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠈⠄⠀⠀⢠⠀⡜⢀⡴⠒⠀⣄⠀⠀⠀⢰⠀⠃⣀⣸⢼⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⠩⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠈⢠⠀⠐⠄⢠⠈⠀⠀⠀⠈⠦⡴⠒⠁⠈⠣⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⢰⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢸⡄⣀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠉⠓⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠉⠁⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⢀⢭⠘⠶
⠀⠻⠶⠄⠀⠁⠒⠦⢄⣀⠀⠠⠀⠀⠀⠀⢀⣀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠑⢀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠉⠓⠒⠂⠀⠀⠰⢄⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⣤⡤⣦⡤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠖⠤⢺⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠐⠒⠛⠃⠈⠀⠀⠀⠀⠤⠀⠠⠜⠀⠃

⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⡄⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣥⣖⢦⡀⠠⠄⡂⣤⡇⠂⠀⠀⠀⠀⢀⡠⠄⠂⠀⠀⠀⠀⡠
⠀⠐⠄⠑⡀⠐⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠀⠀⠀⡿⣹⠃⠀⠀⢀⠴⠋⠀⠠⡔⠀⠀⠀⠈⠁⠠⠂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠒⡈⠆⠐⡀⠀⠛⢠⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠁⠁⠃⠀⠀⠐⡀⠀⠀⠀⠀⠀⡠⠴⠁⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠈⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢧⡀⣀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠃⠀⠀⠀⡄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⣀⣰⣆⣼⣷⡦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢀⡤⠖⠚⠋⢉⡽⠛⠛⢿⠒⠖⡒⢠⠔⠄⢐⠀⠒⣤⠐⠂⠉⠁⠐⠢⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠠⠃⢀⡶⠋⠀⢀⡴⡾⠁⠀⠀⠀⠀⣳⠀⡌⢅⣠⣠⣄⠲⠒⠧⢤⡀⠂⠁⠀⠀⢈⢡⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⢧⣣⠀⢠⡀⣺⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⣶⡧⡘⠚⠁⠀⠐⠀⠀⠀⠹⣶⡾⢿⠛⠒⠿⣷⣶⡬⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢻⣧⣱⣮⣟⡛⠧⣀⢈⢥⠆⠀⠀⠀⠀⠠⡄⠀⠸⣿⡺⡿⠀⠤⡀⢠⠀⢀⣠⡸⣧⣄⡑⢤⣀⠈⢙⠇⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠛⢲⣿⣏⣻⡏⠑⠶⣤⡈⢺⢾⠀⠀⠀⠀⠀⠀⡶⢰⡀⠘⠓⢞⣷⣤⣅⢤⣶⢧⡿⡏⢋⠁⠈⠓⢶⣤⣬⣷⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⠀⠉⠉⠀⠀⠹⡅⠘⣿⠀⠀⠀⠀⠀⠀⠨⢸⡇⠀⠖⣪⡇⡯⢙⣻⣻⡏⢭⣽⠀⡐⠀⠀⠀⠛⢇⠛⠛⣷⠄
⠀⠀⠀⠀⠰⢿⠎⠀⠀⠘⣴⡴⣿⣇⡀⠀⠀⢀⢀⡀⠀⣿⡀⣿⣿⠀⠀⠀⠀⠀⢀⣿⡇⠀⢠⣿⡏⢠⣾⡿⡽⢿⣆⢀⠀⡄⠈⢠⣀⢀⣷⠀⠀⠘⡆
⠀⠀⠀⠀⡧⠈⣂⣀⢄⣤⠿⠤⠄⠙⡓⢒⣒⠆⣱⣗⣓⡚⢿⣿⣟⣧⠀⠀⠀⢀⡾⣿⣱⣣⣿⢋⣾⡯⠊⣀⡧⠀⠳⣦⣴⢷⣿⣢⠽⠾⣿⣂⣙⡶⠃
⠀⠀⠀⠀⣱⡀⣋⠩⡀⢎⠈⡧⣝⢞⣂⣩⣥⠶⠷⠿⠛⠷⠦⣿⣿⡿⠧⣄⠀⠁⡠⠿⢷⢫⡵⣟⡽⠞⠣⠍⡙⢄⡀⣹⠿⠉⢈⠀⣂⠝⠙⡓⠫⠆
⠀⠀⠀⢈⢯⣯⣶⣷⣿⣿⣟⡭⠟⠋⠉⢀⠠⢄⣠⡶⠶⠶⠲⡚⠝⢃⢖⣴⢇⢜⡑⣆⡉⣧⣐⣟⡱⠶⢲⠶⢍⡑⢡⣧⠀⣶⢊⢠⡷⠶⡶⣿⠴⠃⡀
⠀⠀⠐⠁⠈⠈⠛⢛⡲⠟⠉⠀⠀⠀⣠⠔⠚⠉⢀⡠⠠⠒⡄⣇⣾⢿⡟⣽⣾⢧⡇⠸⣟⠗⡐⡸⢝⣇⡛⢶⠀⢿⡈⣿⢿⡟⠛⢩⡠⠖⠃⢄⣀⢤⠁
⠀⠀⠀⠀⠠⠤⣤⣬⠄⠈⢁⡰⠖⠉⣀⡄⢔⢁⠃⢆⣸⣿⡶⣫⣽⡯⢕⣬⣿⣿⣻⡫⢻⣏⡔⢀⡗⠜⣮⣻⡄⠈⣧⠿⢤⣥⢴⣫⡠⠶⣦⣌⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢁⢏⠀⠄⠚⢤⡄⢆⣣⣰⣶⡶⠛⢉⣉⡀⢽⣶⠿⣽⣿⣿⡏⠿⠢⣿⣮⡉⢉⣵⣇⠈⢻⣿⡅⡤⣼⣧⢞⠵⠃⠀⠀⠈⠳⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠛⣒⣉⠤⠖⣁⣧⣵⡟⢁⣴⣾⣋⡭⠿⣾⣿⣷⣿⢉⠿⠀⠀⠑⡩⢷⡿⠞⠷⢬⣑⣸⡇⠹⠇⢛⣥⠁⠀⠀⠀⠀⠀⠸⢃
⠀⠀⠀⠀⠀⠀⠀⠔⠥⢿⣷⣶⠴⠏⢹⢙⡐⢠⠋⠀⠀⡜⠁⣠⣼⣿⠿⠒⠂⠒⠶⣖⡋⠀⣼⣏⢤⡀⠀⠣⡎⣤⡄⠀⢶⣺⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠸⠐⠉⠀⠀⠀⢸⠀⠀⢸⠁⠃⠀⠀⠸⠁⣾⣯⣙⣷⣿⣿⢶⣄⡀⢠⣷⠀⠹⡓⢼⣄⠀⠐⠻⣄⣉⢣⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⣄⣣⠀⠀⠀⠈⠦⠽⠟⢏⣛⠿⠃⣰⢞⣻⣿⡟⡆⠀⠈⠙⠿⣿⣦⠀⠴⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡌⢿⣿⠿⢾⡈⠁⠀⠀⠀⠀⠈⠛⣾⡄⠙⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠗⠠⢹⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⡶⠶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢰⠋⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⡾⠥⠽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⠞⣳⠀⠀⠀⠀⠀⠀⢀⡾⠂⠳⣄
⠀⢸⡑⠦⣄⣀⣠⠴⠋⠉⢻⠁⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⢳⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣠⣖⡿⢈⢂⡟⠀⠀⠀⠀⠀⡴⡫⠀⢠⠀⢊⠑⣄
⠀⠀⠙⠦⣀⣀⠀⣀⣠⠔⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢹⡔⠙⡇⠋⢣⠁⣴⠟⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⠛⠲⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠘⠦⣧⣀⣠⣤⡠⠧⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣀⠀⠀⣀⣰⠞⢛⢶
⠀⠀⠀⠀⠀⣀⣤⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠯⡏⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣮⣷⠶⡒⠦⣄⠀⠀⢰⣉⢊⣠⠌⣳⣴⢣⣊⣰⠒⠊
⠀⠀⠀⢠⠞⡽⢪⡽⢧⠙⣄⠀⠀⠀⠀⠀⠀⢀⣯⣭⡯⣿⣇⠀⠀⠀⠀⠀⢀⡞⢁⠊⠀⠂⠀⢂⠀⠱⡈⣇⠀⠀⠈⠉⠘⠦⠚⢿⡷⠃
⠀⠀⠀⣜⠀⠀⠈⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠿⣯⣿⠛⣿⣇⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠈⠀⠀⡇⣿⠀⠀⠀⠀⠀⣰⢞⠉⠑⢢⡀
⠀⠀⠀⠘⢆⣀⣀⣀⣀⡤⠋⠀⠀⠀⠀⠀⠀⠘⡇⣿⣿⣇⠸⣟⠂⠀⠀⠀⠘⣆⠰⡀⠸⠀⠀⢀⠀⠰⣠⠇⠀⠀⠀⠀⠀⢣⡀⠀⢀⣸⠃
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⡏⠉⠛⠁⠀⠀⠀⠀⠈⠓⠚⠒⠓⠒⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣽⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⠒⠋⠙⠲⢤
⠀⠀⠀⠀⠀⢠⠟⠁⠉⣯⠀⠀⠀⠀⠀⠀⠀⣀⡴⣿⡻⣗⢤⡀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⣼⠀⠢⢠⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⢠⠃⠀⠀⢰⠇⠀⠀⠀⠀⠀⢠⣼⢛⠹⣻⡻⡛⣗⠝⡄⠀⠀⠀⠀⣾⠀⣀⣀⣀⠀⢤⣤⣄⣹⠀⠀⠀⠸⣇⣑⡄⣀⠀⠂⠠⠴⡤⠃
⠀⠀⠀⠀⣼⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⢸⡇⢃⠰⠸⣏⢸⢸⢰⣹⡀⠀⠀⠀⠘⠶⠯⢿⡇⠀⠸⡟⠋⠁⠀⠀⠀⠀⠀⢸⠃⡀⠀⠀⠀⣾⠁
⠀⠀⠀⢠⡇⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢷⡐⠵⢬⣿⣃⡯⣂⡿⠁⠀⠀⠀⠀⠀⠀⠐⣧⣄⣠⠇⠀⠀⠀⠀⠀⠀⠀⠸⣀⡇⢀⣀⢀⡟
⠀⠀⠀⠘⠧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⠀⣠⠤⠖⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⣋⣩⢤⢤⣽⣄⡀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠲⠦⣄
⠀⢀⡞⠁⠀⠀⣀⡠⠷⠒⠒⠒⢦⡀⠀⠀⠀⡴⣫⠝⠿⢍⡑⢢⠀⠀⠀⠀⢸⡥⠒⠋⠉⠙⡆⠀⠀⠀⠙⡆⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⠳⣀
⠀⡟⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⡀⠀⠀⠀⡄⢸⡇⠀⠀⠀⢺⠀⠀⠀⣀⣾⣳⣤⣤⣄⣀⡇⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⡇⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢻⣠⠀⠀⢠⠁⡸⠀⠀⠀⠀⠈⠓⠒⠺⡻⠃⠁⠈⣼⠀⠉⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⣠⠎
⠀⠙⠢⠤⠼⠦⢄⣄⠤⠤⠖⠊⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠄⠀⣻⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣾⣷⣿⣾⣥⣾⣿⣷⣞⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣾⣿⣷⣾⣿⣿⡿⠿⠟⠛⠛⠛⠻⠿⣿⣯⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢟⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠻⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣥⡾⣻⣿⠟⠁⠀⠀⡀⠀⠀⠀⡀⠐⠋⠉⠀⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡿⠃⠀⠀⠐⠋⡉⢁⡀⠋⠉⠀⢠⠟⠁⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠁⠀⠀⠀⠀⠠⠛⠾⠿⠤⠴⠞⠉⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⠉⠀⠀⠀⠀⣠⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣄⡀⠰⠋⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⡄⠫⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣴⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⣀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠒⠚⠋⢰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⠀⠀⠀⠦⣝⣳⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⠶⠶⣬⣓⣶⠋⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⢌⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠙⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠿⠆⠀⠀⣶⡇⠀⢿⣧⡤⠴⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣁⠀⠀⠀⠀⢀⣸⣿⡇⠀⠀⠀⢀⣀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠙⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣠⣤⠠⠤⣤⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠞⠛⢁⣬⡄⠀⠤⣄⣀⣀⣀⣈⠉⠉⠷⡿⠬⢝⣂⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣶⣼⣿⠿⠥⣴⠊⠉⠀⠁⠀⢠⣏⡽⠋⠀⠠⠿⠦⠀⠀⣠⠞⠛⠳⠀⠐⢄
⠀⠀⠀⠀⠀⣰⣿⣿⣼⣀⡀⠀⠈⠟⠛⠓⢢⠄⠀⠛⠁⠀⣼⣦⣠⠀⢀⠀⢻⡾⠛⢲⣤⣴⣨⠷⡀
⠀⠀⠀⠀⢀⠻⣿⡿⠃⢀⠾⠛⠏⠀⣀⣾⣅⣀⣤⣤⣄⠀⠀⡄⠁⠀⡸⠛⠈⠁⠀⢿⣭⣿⣿⣸⡇
⠀⠀⠀⠀⠈⢦⡙⢤⠰⣅⡄⢀⣀⠀⠙⠋⠟⠁⠀⠀⡿⠟⠛⠀⠀⠸⢧⣤⢤⣄⣀⣶⡿⢏⣴⡏
⠀⠀⠀⠀⠀⠈⠻⣦⢑⡚⠶⡄⣹⣳⣶⣀⠀⠀⣰⣾⣶⣾⣶⣦⣄⠀⣤⣜⡶⠋⢁⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣭⣅⡒⠂⠌⠉⠥⡄⠈⠉⠚⣉⣉⣉⡋⡉⠀⠀⣀⣒⣭⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠏⢻⠓⠖⡶⢶⢤⣤⣤⣤⣦⣶⣶⣶⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢹⠀⠀⠀⠀⠀⠈⠀⠏⢹⡟⡟⢻⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠍⠣⠘⠏⠛⠙⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣭⣿⣹⣿⣽⣿⣾⣿⣽⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⡀⠸⢡⠀⠀⠀⢦⡀⣴⠂⠀⣠⠲⣈⠖⠂⠀⠀⠀⢀⡴⣾⡿⡿⠳⠄⡀⢀⣤⠀⠛⢳⠇
⠀⠀⠀⠘⣄⣷⡴⡰⠀⠀⢈⡧⡿⠂⠸⡁⠀⠁⠀⠀⢀⠀⠀⠊⠉⠈⠀⠀⢀⠎⠀⣼⠈⡀⠐⠠⢰
⠀⠀⠀⠀⣈⣻⡏⡀⠀⢠⡿⠀⢼⠀⠀⠁⠀⠠⠠⠔⠃⠀⠀⠀⠀⠀⣀⠐⠋⠀⠀⢹⣤⠰⣈⠄
⠀⠀⠀⡎⠁⠀⠀⠀⠀⢸⣽⢀⠇⠀⠀⠀⡀⢤⣠⣢⠀⠀⠀⢠⣾⡻⣽⠟⡆⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠈⠒⠠⡄⠀⠀⢸⣏⠌⠀⠀⠀⣾⢆⡀⠘⢰⡄⢰⠀⠩⠐⠷⠹⠋⠇⠀⠀⣠⠄⣾⠀⠠
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⢠⡏⣼⡿⠀⣸⠇⡜⠀⠀⠀⠀⠤⢥⡁⠀⣞⣿⣄⠉⠀⠀⠀⡆
⠀⠀⠀⠀⢀⠰⡛⡵⠃⢪⣷⠛⠞⡀⠈⠧⣿⣯⣴⣯⠜⠀⠀⡔⣩⣀⡄⢠⣷⠀⢿⡁⠙⠓⠄⡀⣠⠃
⠀⠀⠀⡠⠺⠅⡄⠀⠀⠈⣞⣷⣴⠂⠀⠀⠀⠉⠈⠁⠀⠀⠸⡇⠛⠟⢃⡞⠄⠀⠀⠉⠐⠶⠞⠉
⠀⠀⢀⡿⠀⠀⢣⠀⠀⠀⠁⠉⠈⢀⣴⢖⡛⣁⡀⠀⣠⢾⠁⠙⠀⠚⠉⠀⢠⡖⠊⠉⠉⠀⡄
⠀⢀⣾⡇⠀⠀⠀⢱⠀⠀⢀⠠⡤⠽⠿⠟⠛⠉⣁⡺⠋⠌⠀⠀⠀⠀⢠⠀⠻⣛⢄⡤⣂⠀⠀⠀⠀⡀
⠀⠘⣟⢱⣦⡀⠀⡸⠀⢰⢿⢰⣥⠀⠀⠀⠂⠀⠈⠑⢄⠀⠀⠀⢀⡔⣸⡀⢂⠀⠈⠃⠛⡭⠂⡔⣠⠃
⠀⠀⠈⠙⠶⠟⠋⠀⠀⢫⢫⢸⠀⠄⠀⢀⡆⠀⠄⢀⢸⣙⣈⠆⢸⡇⣿⡄⠈⠀⢀⣀⠀⣤⣩⡁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⠫⠧⠤⠴⠊⠀⠳⠦⠜⠀⠉⠁⠀⠀⢧⣽⣵⡖⠀⣀⣹⣮⣿⣼⡁
⠀⠀⠀⢀⡔⣾⠿⡆⠀⠀⠀⡠⢠⡖⠊⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡷⠛⣀⠈⡀
⠀⠀⠀⣞⢿⠛⡦⠷⠀⢀⡞⡆⠈⣷⡴⠡⠐⢈⠑⠀⣰⠾⠟⠈⠀⢆⡀⠀⠀⣴⡟⢱⠜⠚⣬⠘⡄
⠀⠀⠀⠛⢿⣇⡇⢠⠀⠸⡾⣷⠀⡜⠄⡀⠅⣸⢠⠃⣽⠆⠀⢀⠀⠀⠈⠀⠀⡟⣿⠟⢀⡈⠘⢀⡇
⠀⠀⠀⠀⡀⡗⠀⠀⠀⠀⠳⣻⣴⡧⡴⠁⠔⡡⠂⠀⠀⠀⠀⠐⡑⠀⠁⠀⠀⢳⢫⠿⣿⣧⣹⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠤⢁⡀⠈⠁
⠀⠀⠀⠃⠀⠀⢀⡀⠀⣄⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠈⠠⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠠⠆⠉⡵⠀⠆⠒⡲
⣤⡄⠀⠀⠆⠀⠈⠆⠰⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⡏⠁⠀⠐⣆⠠⣞⣤
⠘⡷⠂⠀⠀⠀⢀⡉⠀⠀⢀⠀⢐⡂⡀⢀⡤⠀⠀⠀⠀⠁⠀⠑⡀⠃⠀⠠⠀⠀⠀⠐⡀⠀⢀⢀⠀⠀⠂⠀⠀⠀⠀⡇⢤⠀⠴⠹⠀⠍⠉
⠀⠱⣄⠀⠀⠀⠀⠳⡴⠋⠁⠀⠀⠀⠄⠆⠀⠀⠉⠀⠀⠀⠐⡀⢠⡤⡀⠀⣀⡀⠀⠐⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠐⠠⠕
⠀⠀⠈⠑⠂⢰⠀⠈⠀⣤⣄⠠⠞⠀⠀⠂⠀⠀⣀⠀⠀⠀⠀⢠⠈⠰⠶⡀⢤⡇⢀⠎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⣿⠆⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⢀⠀⠀⠀⠰⢃⠈⢩⠀⠀⠀⠭⠿⡀⢉⠀⠙⠋⡄⡎⠙⠀⠀⠀⢄⡁⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⡇⠀⠙⠒⠜⠁⠁⠀⠀⠀⠁⢀⠀⠙⠂⣀⠀⠀⡤⠤⠞⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠹⣮⡵⠀⢀⠄⠀⠀⣄⠀⠀⠀⠀⠀⡄⠀⢁⡀⠀⠀⠻⠥⠔⠁⠀⡠⠢⡄⠀⢀⣤⡊
⠀⠀⠀⠀⠀⠀⢀⡖⠀⠐⢖⠊⠀⠈⣲⠖⠁⠂⠀⢖⠁⠀⠀⢀⡴⠋⠉⠁⠲⣄⠁⠘⢄⠀⠂⠄⠀⢉⡀⠄⠊⠀⢼⡁⠀⠁⠀⢱
⠀⠀⠀⠀⠀⣠⣬⣵⡄⢠⣞⣀⡀⣀⣳⣤⠀⠀⢠⣞⣀⡀⢀⣀⣙⣦⠀⢀⣴⣁⣈⠛⠛⠁⠀⠀⠀⣀⣼⡂⠀⢀⣀⣙⣦⠀⢀⣞⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⡀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣊⡘⡁⠀⠀⠀⢹⣧⠀⠀⠀⠉⠀⠀⣼⡿⠀⠀⠀⠀⣠⡾⠛⠾⢿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠻⢷⣤⣠⡁⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣠⡾⠋⡠⠞⠚⠛⢿⣿⠟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⢸⣿⠀⠀⣴⠟⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟⣳⣦⠀⠀⣸⡏⠀⣰⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⣸⣿⣕⠀⠀⣿⣷⣤⠛⠉⠀⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠇⢀⣿⣿⣿⣤⣾⡿⠃⠀⠀⠀⠀⠀⣄⣀⢀⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⢟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⣟⠀⠀⠸⣿⣼⣿⠿⠉⠀⠀⢀⣴⣤⣶⣿⣿⣿⡟⣿⣿⣿⣦⣴⣶⡶⠿⠿⠛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣠⣾⣿⣿⡿⠃⠀⠀⢠⣾⣿⣧⣤⣤⡿⡯⠼⠟⠛⠉⠀⠀⠀⠀⠀⠀⢀⣿⣇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣿⡿⠀⠀⠀⢰⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡃⠀⠀⠀⢾⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣾⠿⠟⠛⢻⠿⣿⣿⣏⣉⡙⣿⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣏⡠⢿⣿⣿⡅⠀⠀⢿⣿⣿⣿⠟⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢟⣾⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢾⣿⣿⣿⣀⣀⣿⣿⠟⡟⠀⠀⠀⠀⢀⡿⠀⠀⠀⣀⡀⠀⠀⢸⣧⠀⠀⢉⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣽⣿⡿⢷⣾⣿⣿⣿⣿⣿⠏⠁⣾⡵⠦⠤⠀⢀⣾⠃⢀⣠⣴⡟⠻⢷⡀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠿⣧⣾⠛⠉⠉⠹⣿⣻⢋⣿⡟⠁⠀⠀⠀⣾⣿⣾⣿⣯⠋⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣯⣿⣿⢿⣿⠁⠀⠀⠀⠀⠈⠉⢻⣿⡀⢤⠀⠀⠀⣿⣿⣿⡟⢠⡄⠀⠀⠀⠀⢿⡇⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢟⣼⣿⠄⠀⠀⠀⠀⠀⠀⣼⠁⠙⢾⠢⠤⠀⠸⣿⣿⣇⠸⣃⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢻⡏⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣷⣾⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⡟⣾⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠈⠛⠿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢷⡀⠀⠀⠈⠉⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⠂⣤⡴⠶⢾⣿⠀⠀⢻⡆⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⡀⠀⡤⠞⠁⠀⢀⣿⡏⠀⠀⠀⢹⡄⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠸⡇⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣼⠟⢹⡄⠀⠀⠀⠀⠈⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣼⠏⠀⠀⠗⠀⠀⠀⢀⣠⣶⣿⡟⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣦⠀⣀⣾⠏⠀⠀⠀⠀⠀⠀⢠⣞⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠒⠖⠒⢶⡞⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⢒⣿⡟⠀⠀⠤⠀⠂⠀⠀⠀⣼⣿⢿⣦⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢀⡜⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠠⠀⠀⠀⠀⣴⣶⣶⣦⣠⣬⣿⣄⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⢰⡿⡿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠆⠀⠀⠀⠘⣯⣝⣻⣿⣿⣿⡛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣴⠟⡷⠁
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⣿⣿⣿⣿⢿⣙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣹⣦⠞⠁
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠟⠉⠘⠻⠿⠿⣷⣤⣄⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⠄⠘⣭⣶⠏⠉
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠈⠛⠛⣻⣿⣿⣿⣿⣿⣿⣯⣀⡤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠴⢿⣿⠟⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣦⣤⣤⣀⣠⣤⣤⣤⡖⠋⣁⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⡀⠀⡀⢀⠀⢀⡀⢀⠀⢀⠀⠀⣀⣤⣤⣄⡀⢀⣀⣀⣀⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠐⣧⠀⣧⠀⡇⠀⠁⢸⠀⢸⠀⡼⠋⠀⠀⠀⠉⢉⠁⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⢠⠀⠀⠀⡔⠙
⠀⠀⢸⠀⠀⠙⣠⠇⢸⠄⣸⠀⡌⠀⠀⡀⠀⠀⠀⠁⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⢀⢀⠀⠀⠀⠀⠀⡏⠉⢹
⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⢀⠀⠠⡀⠀⠀⠀⣄⠀⠀⣄⠀⠈⠀⠀⢀⡀⣀⠀⡀⠀⣀⢀⣀⠔⡁⠀⠀⠀⡉⠂⠀⠰⠋⠉⠉⠴
⠀⢰⠇⠀⢈⣀⠀⢀⠀⠀⠈⠢⠀⠁⠺⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠻⠄⠿⠵⠛⠖⠋⠈⠇⠀⠁⠂⠁⠃⠩⠀⠈⠀⡀⢀⠀⠀⠉
⠀⠸⠀⠨⠉⠁⠀⠈⠨⠅⠂⠂⠐⠀⠠⠄⠀⠤⠄⠠⠠⠀⠤⠀⠀⡂⠀⠀⠀⠂⠂⠐⠐⢁⡀⠀⠲⠤⠖⠀⣠⠙⠀⢰⡂⠀⡀⢠
⠀⠀⠀⠀⠀⢙⢚⡁⠂⠀⠀⠀⠀⠀⠄⠀⡀⢰⠆⠰⠇⡀⢀⡤⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠅⠉⠀⣁⡔⠂⠀⠇⠀⠉⢉
⠀⠀⠀⠂⠀⠘⠐⠀⠂⠈⠀⠄⠀⠀⠀⢠⡄⠀⣀⡀⠀⠁⠈⠀⠀⣤⣤⣤⣬⣤⣥⡄⠀⠐⠄⠀⠂⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠁⢀⡸⡄⠐⢈⣆⡠⠀⠀⠠⠠⠀⠀⣹⠀⠿⠆⡤⠤⣄⣠⠤⣀⢠⡤⠤⢤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠄⠀⠄⠰⠀⠀⠀⢰
⠀⠀⢀⠀⢨⡀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⣿⠀⠀⠀⡇⣤⡈⠁⣤⢸⡆⢹⠀⣿⣶⢰⡟⠁⣠⢶⣄⣴⣦⠀⠀⠀⠀⠀⠀⠀⠒⠘
⠀⠀⠁⠀⠰⠂⠀⠀⠂⠀⠀⢃⣀⣀⢈⡇⠀⢷⠀⠀⡇⣼⡇⠸⡿⢸⠇⢸⠀⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢸⡁⢀⠀⠸⠀⠁⠰⡀⠀⢸⠀⡂⢸⠀⠀⠀⣿⡀⣇⣩⠴⣤⣠⠞⢰⣏⣀⣹⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⣀⡉⢠⠉⢀⢈⢀⠉⠀⠉⠉⠉⢈⡀⣀⣀⣬⢥⣄⣀⣀⣛⣠⣤⣀⠀⣀⣀⣈⣋⣀⣀⡀⣀⣈⣀⡀⠁⠁⡀⠀⠀⠀⠀⠨⠀⠨
⠀⠒⠈⠤⠤⠤⢹⡀⠰⠂⠀⡄⠐⠀⢯⢹⡏⢠⢀⣤⡀⠙⣿⠀⠀⣶⡌⠛⣥⡙⢠⣶⣿⢸⡇⡇⠲⢾⡇⠀⠀⠀⠠⢀⠀⠀⠠⡀
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠈⢧⠁⣿⢸⣿⠃⠀⣿⠀⠀⡉⢶⣧⣟⢿⡄⣭⣿⢸⡇⡇⣿⠙⠃⠐⠀⠀⠂⠘⠂⡝⢽⡇
⠀⠀⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣄⣉⣴⡀⠛⢀⡀⣿⡈⣇⠛⢀⡄⣛⣿⣘⣶⣇⣻⠠⠮⠿⠧⠀⣠⠀⢳⡿⠾⠟
⠀⠀⢠⡴⠀⠀⠀⡀⡀⠀⠸⣿⣿⡇⠀⠘⠛⠃⠈⠉⠀⠙⠛⠉⠛⠻⠉⠛⡟⠋⢉⡩⢉⠉⠉⠙⠙⠀⢰⣀⠀⠰⠇⠀⠀⠀⠠⣀⠄
⠀⠀⠀⠁⠀⠀⠐⠒⠒⠀⠀⠀⡀⠀⠀⠀⠘⠈⠉⠉⠁⠀⡀⠀⠀⠀⠀⠀⠁⠀⢬⠀⡠⠃⢀⣀⠀⠀⠀⠟⠀⢶⣶⣶⡖⠀⠀⡯
⠀⠀⢁⢨⡤⠀⠄⠤⠤⠤⢬⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠓⠆⠀⣀⢱⠎⠀⠀⠁⣤⣾⣿⣿⣿⣿⠀⠀⠀⠈⠸⠏⡁⡄⠀⠟
⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⡂⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡠⠀⢀⠁⠂⠈⠁⡀⠀⠠⡄
⠀⠀⠀⠀⠀⠟⡁⠀⠉⠃⠀⠃⠀⠀⡀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠮⠆⠭⠂⠀⠀⠂⠈⡇⠀⠁⠐⠀⠀⠄⠤⠈⠀⠁⡀
⠀⠀⠀⢠⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⡅⠈⠯⠀⠀⠀⠀⠀⠀⠀⢰⡂
⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠄⠀⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡟⠀⠀⠀⠀⠀⠀⠓⠀⠀⠴⠛⠂⠀⠛⠳⠆⠀⠁
⠀⠀⠐⠐⠃⠈⠀⠀⠀⠉⢙⠠⠄⠀⠸⠃⠀⡀⠐⠀⠰⠀⠀⠀⠸⠈⠀⠀⠀⠀⡇⢰⡋⢸⠀⢸⡅⣶⠚⠛⢲⠰⣤⠂⡖⠛⠋⢷
⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠁⠀⠀⠀⠈⠀⠉⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠤⠂⠀⠀⡀⢠⠒⠤⠠⡀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠄⠀⠴⠀⠀⠀⠀⠀⠠⠀⡀⢐⠀⠀⠠⠀⠀⠀⠀⠀⡐⠘⠉⠙⠐⠀⠀⠀⠀⠰⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠁⡄⠀⠀⠂⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠃⢠⠈⠀⣀⠀⠀⠡⠀⠀⠀⠐⠀⢀⠀⣜⠀⣎⠄
⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⢀⠈⠂⠀⡐⠐⠀⠰⠈⢠⠠⠄⠁⠀⠀⠈⠣⠃⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠇⠠⠃
⠀⠀⠀⠹⠄⠀⠀⠈⠢⠀⠀⠀⠃⠀⠀⠀⠈⠂⢄⡀⡐⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⡷⠋⠂⢄⠀⠠⠀⠀⠀⠂⡠⠾⢟⡃⠀⠀⣸⠂
⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡃⠈⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉⠀⠀⠈⠇⠀⠀⠁⠐⠛⢂⠤⠐⠂
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⢀⠀⠁⠂⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠄⠀⠄⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠄⠀⣠⡾⡇⠀⠀⡀⡀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀
⠀⠀⠰⠎⠀⠀⠀⠀⠀⠒⠁⠄⠀⠀⠀⠀⠀⠄⠀⠄⡀⠀⠀⠀⠙⠀⣠⠣⢈⡆⠀⠀⠀⠸⣡⢀⢀⠀⠄⠈⠀⠀⢀⣀⢀⣀⡀⠀⠀⣠⠀⠀⠀⢀⠅
⠀⠟⠳⠦⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠭⢙⡚⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⣾⡉⣠⣊⢔⠝⡄
⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠿⠉⡅⠀⠀⣞⢮⣱⠆⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢀⡀⠀⣈⢁⠌⠙⠛⠛⠁⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠠⠟⠀⠀⠀⠀⣠⣞⠗⠁⠀⠀⠒⠀⠀⠘⢓⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠛⢦⠤⢬⠃⠀⠀⠁
⠀⠀⠀⠐⠀⠤⠈⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠂⠀⠀⠀⠀⠂



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡀⢀⡠⠔⠒⠂⠈⠉⠉⠉⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⠂⠀⠘⠀⠛⠛⠻⢾⣶⣶⠄⣴⠀⠀⡀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠘⢖⠐⠁⠺⠆⠐⠀⠀⣠⡀⠀⢁⡴⡜⠁⠀⠀⠒⠀⠐⠛⠀⠀⠀⠀⡰⠋⠂⠀⠛⠁⠀⠀⠠⠤⠀⠑⢤⡀⡀
⠀⠀⠀⡠⠊⠀⠀⢁⠅⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⡐⠶⠀⠀⠀⠐⠛⠂⠀⣄⠀⠀⣷⡈⠙⠦⣀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⡏⠀⠐⠘⠀⠀⠶⠀⠀⠀⠀⠀⠁⠁⠀⢉⡽⡓⠀⠀⠀⠀⠀⠀⢠⣄⠋⠀⠂⠂⠀⠀⠀⠀⠀⠀⠌⠁⠀⢻⡇⠀⠀⠈⠢⡀
⢠⡏⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠘⠃⠀⠙⠃⠈⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠇⠀⠀⠀⠀⠈⢦⡀
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⢳
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⣆⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⠿⢿⣁
⣟⣠⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⢀⡉⠉⠓⠛⠐⠶⠦⠤⢿⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⢀⢠⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠐
⠸⣿⣶⠀⠀⠀⣠⡼⠋⠀⠀⠀⠋⠁⠀⠀⠀⠀⣀⠀⠀⠀⠉⠓⠦⣤⣴⣄⣰⣀⡄⠤⠤⠔⠂⠂⠈⠉⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢀
⠀⠈⠻⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡀⣸
⠀⠀⠀⠀⠈⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣀⡀⣼⣿⠋
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡠⠟
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣘⣆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠶⢄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣌⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠧⠄⢤⡴⠤⠶⠤⠐⠊⠁⠀⠈⠓⠴⢦⣄⣤⣀⣤⣤⡤⠴⠒⠚⠉
⣿⣿⡿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣤⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣦⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⡞⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⣾⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢿⡄⠀⠀⣰⣇⠐⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⣼⣟⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⣴⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⢈⣿⣿⡿⠉⠀⣠⣞⠉⠙⠻⣷⣴⣿⠾⠗⠛⢋⣉⣬⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠛⣤⣿⣿⡟⠀⠀⣼⣿⣿⣿⡿⠛⠉⠁⢀⣀⣴⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠁⠀⣿⣿⣿⠃⠀⣸⣿⣿⣿⣿⡇⠀⠀⢀⣼⠟⠛⠙⣻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠐⠋⠸⣿⣿⠀⠀⣿⣿⣿⣿⣿⡁⠀⠀⣿⣿⠀⠀⠀⠉⠘⢻⣏⠉
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠐⠒⣿⣿⣇⢠⣿⣿⣿⣿⣿⡶⢦⣾⣿⣿⠀⠀⠀⢸⣟⠻⠿⠷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣀⣶⣴⣟⡛⠉⠈⠻⠿⣿⣿⣿⠁⢰⣿⣿⣿⣤⣀⣤⣼⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣤⠀⠀⠀⠀⠀⢹⠿⣄⠈⣿⣿⣿⣿⣿⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⢿⣾⡷⠂⠀⠀⠀⠀⢢⠀⠈⠛⢊⠹⣿⣿⣿⡄⠸⣿⣆
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡻⡖⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⣿⣼⣆⣾⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣾⣷⠿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣵⣆⡀⠀⠀⢀⣆⣀⣀⣠⣴⣿⠛⢷⣔⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⡀⠀⣠⠞⠋⠉⢠⣿⠃⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡗⣨⠃⠀⠀⠀⣾⠏⠀⣬⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣃⣄⠀⠀⣴⠋⢀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣯⣴⣼⡿⠀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⡆⣀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⡆⣴⢳⣴⣦⣶⢰⡶⢰⣦⣶⢰⣶⢰⡖⡆⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠉⠈⠁⠈⠈⠉⠈⠀⠉⠉⠈⠉⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠰⣍⡷⢾⣹⢾⣏⡷


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⣉⢷⢊⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⠖⠁⢱⠟⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡇⠀⢀⠞⠁⢀⠔⠁⢨⠀⡧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⢧⣸⡳⢀⠔⢡⡀⠀⠀⢠⠁⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⢾⠴⠯⠥⠬⢁⣈⡠⠔⠁⠀⠀⠀⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠘⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⣀⠞⣡⠔⠁⠀⠀⠀⢀⡠⠤⠤⠛⠛⠠⠬⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢾⡼⠋⢁⣎⡥⣤⠀⠀⠀⣤⡏⠁⠈⢉⡩⠍⡉⠉⠉⢹⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣭⣭⣭⡍⠉⢙⣆⠀⠀⠀⡇⠀⢰⢡⠴⡒⡌⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⠉⠉⠉⠉⠉⢉⡝⠈⠳⡀⠀⡇⠀⠘⢜⠚⠿⣣⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⣒⣒⣒⣊⡹⠀⠀⠀⠘⠋⠷⣶⣶⣶⣿⣿⣶⣶⣆⠼

⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠂⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢂⠀⠀⡠⢀⢀⠀⠀⠀⠈⠂⠠⠀⠒⠀⠒
⠀⠀⠀⠀⠀⠸⢀⠀⠄⠈⠁⢈⠀⠀⠀⠀⠀⠀⠄⠤⠄⡀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢐⠀⠀⠈⠑⠄⢀⠀⠁⠀⠀⠀⠂⠠
⠀⠀⠀⠱⠀⠀⢠⠀⠄⠀⠀⠁⠀⠀⠈⠁⠀⠐⠚⠀⢀⢀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠅⠀⠀⢠⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀⢳⡴⣎⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠸⠏⢁⠀⠀⠀⠊⠀⠀⣀⡁⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠠⠀⠀⠀⠀⠀⢈⣧⣂⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⢀⠄⠀⢀⠀⠀⠀⠀⠂⠀⣜⠉⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⣀⠀⠂⠡⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠎⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⠄⠀⠒⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠁⠀⠂⠀⠄⠀⠃⠨⠌⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡠⠈
⠀⢀⡀⠦⠀⠀⠀⡐⠠⠀⠠⠀⠀⠈⠀⢰⠀⠀⠀⠀⢃⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⡀⠁⡀⢒⠀⡈⠀⡀⢆⠈⡀⠀⠀⠀⠘⠀⢘⠁⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⢁⠐⠄⡠⠀⡀⠢⠀⡌⠀⠀⠀⠀⠳⠆⢀⠘⢀⠆⡠⠞⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⢀⠴⠺⠀⠀⢀
⠀⠀⠁⠄⠀⠀⠈⠡⠠⠀⠀⠀⠀⠀⠀⠲⠆⡴⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠉⠂
⠀⠀⠇⠁⡄⠀⠸⠈⢠⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢡⡜⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠠⠀⢐⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢄⠰⠀⠀⠉⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠴⠀⢈⡴⡀⠛⡤⠀⢄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠠⣁⠆⠀⠀⠀⡀⠀⡀⠀⠁⠀⠀⢀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠠⡄⠀⠀⠀⠀⠐⠢⢘⠀⠀⠀⠈⠀⠀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠃⠀⠀⠀⠠⡀⠂⠂⠀⠀⠀⠀⡀
⠀⠀⢸⠄⠀⠰⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠐⣄⠀⠐⣁⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⡤⣄⡤⣤⡄⠀⣤⢠⣄⣤⠀⣤⣄⣤⣠⣠⠤⠀⡤⣄⢤⡄⣤⠀⣤⣄⡤⡄⣤⠀⡄⣤⣄⡤
⠀⠀⠀⠐⠋⠘⠃⠙⠋⠀⠓⠘⠀⠚⠁⠙⠀⠃⠛⠙⠛⠀⠛⠁⠋⠃⠋⠀⠋⠀⠛⠁⠃⠠⠇⠋⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠦⣅⠀⠀⢠⠁⢠⠖⠐⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣷⠀⢨⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢰⣿⡶⠋⢀⣀⣤⣤⣶⣀⣀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣾⡟⠁⢠⣾⡗⠚⠙⠉⢀⣀⣠⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⣻⣧⠀⢹⣿⠇⠀⢠⠚⠉⠉⢻⠧⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣽⣿⣶⡟⣩⠀⢀⡎⣠⡴⢄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⠀⠙⢻⣁⠀⢸⣿⢃⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⢦⣶⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣰⠳⡀⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠊⢀⡟⠀⢰⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⠀⠀⢀⠞⠡⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⡎⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠀⠒⠒⠒⠠⠒⠒⠤⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠖⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠈⡉⠉⠲⠤⢀
⠀⠀⠀⠀⢀⡤⢚⠁⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠂⠑⢦⡀
⠀⠀⠀⢠⠎⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠠⢀⠀⠀⠀⢀⠀⠀⠀⢇
⠀⠀⠀⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠊⠀⢀⡀⠘⠀⠤⠀⠀⠀⠀⠐⠀⡄⠁⠀⢸
⠀⠀⠀⠀⠣⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⡀⠁⠀⠀⠀⠀⠈⣠⠂
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⠤⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣴⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⠚⠓⡓⠶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡓⣨⠰⠏⠟⢥⠄⣴⠎⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠔⠛⠶⢧⣄⡠⢈⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠀⠹⠶⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⠀⢠⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⢤⠐⠁⠀⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠛⠲⢤⣀⠀⠀⠀⢀⡔⠉⠀⠈⠢⢠⠎⠁⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⣀⣤⠆⠀⠀⠀⠀⠀⠈⠙⠲⢶⠎⠇⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠋⠀⢀⠴⣮⠟⠁⠑⣄⣠⣀⣀⠀⠀⠀⠀⡌⠰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⣤⢒⠊⠀⠐⠉⠈⠑⠺⣁⠀⠀⠀⠀⠀⠀⠛⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣄⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠈⢨⡑⠄⠀⠀⠀⠀⠀⠈⢮⠑⠢⣄⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠈⠳⣄⠂⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠛⣦⣄⠀⠀⠀⠀⠀⠀⠀⢰⠉⠑⢄⠠⡀⠀⠀⢀⠀⢺⠀⠀⢀⣡⡾⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠷⣄⡀⠀⠀⡄⠀⠀⠑⡄⠈⠳⠼⠦⣚⠥⠒⢁⣠⣶⣛⣁⠴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠲⠃⠀⠀⠀⢀⡏⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⢦⡀⠰⢰⡀⠀⠘⢦⠴⠂⠁⣀⣤⠾⠋⠉⠁⠀⠀⡠⠈⠳⢤⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡠⠋⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢷⣄⠁⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠙⠒⠒⠀⢠⡾⣠⠒⣈⠁⢲⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⠶⠛⡇⠀⠀⠀⢀⣦⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠋⠀⢸⠗⠁⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣉⣁⣀⣠⡤⢴⡏⠙⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠒⢿⡋⠁⣸⡇⠀⠀⠀⠀⣿⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢷⠀⣿⡇⠀⠀⠀⠀⣿⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠀⢳⡀⠀⠀⠀⠈⡇⣿⠁⠀⠀⠀⠀⣿⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⢳⡀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣿⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣿⠀⣀⣴⠿⠒⠒⢶⠛⠀⠀⠀⢳⡀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣿⠋⠁⠀⣀⣀⣀⣼⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⣿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⣠⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⡶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣤⣤⠴⠛

⣶⣶⣶⣶⣶⡾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⠶⢶⡶⠶⠶⢶⣶⡶⠶⠶⠶⢶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⣎⠀⢀⡠⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢤⠀⠀⠀⠀⠕⠀⡀⢽⠀⠀⠀⡀⠀⣠⠌⢂⡔
⠀⠀⠊⠳⠘⠈⠀⠀⣀⠴⡆⠠⢄⠀⠐⠀⠀⠀⡀⠗⣤⣄⠂⠀⠀⠀⠀⠀⡞⢅⡻⣦⢰⡾⠇⠀⠈
⠀⠀⠀⠠⠀⠀⢀⠄⠀⠀⠈⣧⠈⡧⠀⠀⠈⠢⠆⠺⠇⠩⠀⠀⠀⠀⠀⠀⡏⠸⠈⠠⠀⡇⠀⠀⠀⢰⠧
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠙⠠⠀⠀⠀⠀⠀⠀⠀⢸⢈
⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠁⠀⠀⠤⡶⠄⠀⠨⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠲⠴⠂⠀⠀⠢⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠸⡀⡀⣼⠀⡦⠀⡔⠁⠀⢀⠄⠁⡠⠀⠈⠁⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⣀⠤⠂⠴⠆⡀⠀⠈⠻⣤⢿⣬⢡⡞⠁⠂⠀⠈⢆⠀⢗⣻⠂⢠⡀⠀⠀⠀⢀⠀⠀⠁⠀⡀⠊
⠀⠀⠀⢈⡁⠒⢅⢐⠘⡄⠀⠀⠀⠈⠙⣧⡋⠀⠀⠀⠀⠀⢠⢰⢌⠽⠛⠁⠀⠀⠀⠀⠀⢉⠀⠉⡶⠁⠘⡧
⠀⠀⠀⠀⢈⡁⡁⠂⡇⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠰⢿⡟⠉⡆⠘⠀⠀⠀⠀⡠⢠⠄⠀⠁⠀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠃⠀⠀⠀⠀⠀⠀⠰⠛⠁⠱⡀⢄⠀⢀⡆
⠀⠀⠀⠋⠀⠀⠀⢀⣠⡀⣀⣀⠀⠀⢀⡶⠀⠁⠀⣤⣶⡄⠀⠀⠀⠀⠀⣲⠆⢀⣀⠐⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠸⠿⠅⠀⠀⠀⠉⠉⠁⠒⠓⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⢠⡀⢠⠀⡠⠄
⠀⠀⠀⠘⠀⠀⢧⡈⠀⠀⣤⣤⠤⣀⠀⠀⠀⢀⣀
⠀⠀⢀⡤⢤⠀⢸⡇⠀⠠⡎⠀⣈⠙⠧⡆⠀⣈⠙⠈⡶⠂⠀⠠⡤⠶⠹⢶
⠠⠌⢉⠠⠈⠀⠀⠃⠀⡴⠁⢠⣤⠶⡆⡃⠰⣤⠄⠀⡶⠾⠃⠀⡇⠰⢦⠀⣄
⡸⡤⠁⠀⠀⢀⠀⠀⠔⠋⠀⠀⠠⠠⠉⡑⡤⠠⡄⠀⡄⠾⠁⢰⡦⠇⢸⢰⠄⠑⡄⠀⡀
⠁⢃⡀⠁⠀⠄⠀⠀⠀⠀⡐⠀⠀⠀⠀⠠⠃⠙⠂⠘⡉⠐⠑⠈⠀⠀⠄⠀⠀⠀⢄⠀⡇
⠐⡀⠙⢄⠂⠈⠁⢀⠀⠈⠴⡒⠑⠓⠀⠀⠀⠀⠂⠀⠀⠀⡀⠂⠀⠁⡀⡰⠀⠀⢈⡸⠃
⠀⠀⠄⠀⠙⠢⡑⠄⠏⡷⠀⠀⠀⠀⠀⡰⢒⠲⠶⣤⠤⠊⢁⡘⣢⡀⠔⠉⢀⠔⠉⠐⠀⠂⠄⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠂⠠⠀⠐⠒⠂⠈⠀⠘⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠐⠂⠀⠄⠀⠀⠁⠀⠀⠠⠐⠂⠤⡀⡀⢀⡈⣐⣢⠄⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠩⠑⠢⢄⣀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠍⣐⣀