Ivy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢀⣠⣄⣠⣄⣀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣾⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠒⡾⠁
⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⡇⢈⣵⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠛⠛⠿⠟⠛⣼⠁
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣴⡇⠾⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⣠⣷⣶⣤⣸⢏⠙⢿⣿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣰⣧⣷⣾⣇
⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⡷⣀⠹⠋⣉⠻⠿⡇⠀⠀⠀⠙⣿⣿⣿⣿⣷⡌⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠸⡿⠛⠙⢫⡇⠈⢳⣾⣿⣿⣶⠄⠀⠀⠀⣸⣿⣿⣿⣿⡿⠇⢱⣄⠀⣀⣤⡀⠀⠀⠀⠀⠀⢺⣿⣿⣯⡇⠙⠿
⠀⠀⠀⠀⠀⠀⣲⣶⣶⣅⡸⣀⣿⣿⣿⣿⣷⠀⠀⠀⠘⠛⠛⠉⠙⠟⠀⠀⣸⣶⣿⣿⣿⣷⣶⣄⠀⠀⠀⢸⣿⢿⡟⠹⢦⣶⣶⡆
⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⠗⢏⢑⣽⣿⣿⣻⠆⠀⠀⠀⠀⠀⠀⣶⣶⣶⣄⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣤⣤⠀⢀⠺⣿⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠸⡿⠏⣙⠀⠸⢿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⡿⠛⠉⠻⣿⡇⠀⠀⠀⠰⠿⣿⣿⠏⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡦⢀⠋⠙⢿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⠛⢻⣄⠀⣠⣬⡀⠀⠀⠀⠀⠀⠋⠁⢈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠀⠀⢼⣿⡆⠁⠀⠀⠀⠀⠀⠈⠉⠁⢀⣈⡁⠀⢸⢀⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⡄⢉⠈⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⢈⠀⠈⠁⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣁⣈⠋⠀⡀⣠⣾⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⠘⠁⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢻⡿⠛⡀⠀⠀⠀⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⣃⠀⠿⢿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠙

⠀⠀⠀⠀⢀⣠⣴⣦⣤⣀
⠀⠀⠀⠀⣻⡿⠿⢿⣿⣿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠒⠂
⠀⠀⠀⣴⡫⢤⣄⣚⠿⣿⣦⠀⠀⠀⠀⠀⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢠⠊⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⣰⠄⢁⠀⢻⣿⣷⣽⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠽⠃
⠀⢀⣡⣀⣈⣙⠚⠋⠁⠈⠙⢿⣆⠀⠀⢀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣶⣞⣺⣇⠀⠀⢀⠈⢀⡭⠓⢠
⠀⠈⠹⣿⡛⠛⠛⠀⠂⠀⠄⠐⣿⡀⠄⡐⠻⡋⣨⣍⣷⣶⣦⣄⠀⢀⠀⡀⢼⣏⠹⣿⣾⣷⣴⣷⣲⣠⣻⣿⣿⣆
⠀⢠⠂⠘⡹⠒⢤⣦⣤⣴⣶⣶⣾⣶⠺⢯⠹⣼⣈⣽⣿⣷⣿⣯⣽⣿⣿⣬⣿⣿⡿⣿⣿⣙⣿⠋⢯⡈⣿⣿⣿⣿
⠀⢚⣰⣾⣇⠠⢸⣿⣿⣿⣿⣻⣿⣭⣷⣄⣾⣿⣿⣿⣿⣿⣿⣶⣮⣭⠻⣿⠀⣯⠀⢠⣶⣤⣀⠈⢙⡛⠯⣍⠙⢿
⠀⠈⠛⠛⠿⣧⠙⢿⣾⣿⣷⡿⠟⠛⠁⠀⠛⢿⣻⣟⣯⣿⣿⡟⠛⠉⢦⣈⡀⠈⠛⢹⡈⣿⣿⣖⣺⣿⣦⠀⠉⠂
⠀⠀⢠⡴⢆⣘⡚⠒⠠⡉⠀⣖⣰⡀⢸⡆⠘⣧⣈⢻⣿⡷⠋⠀⠀⠀⢠⣧⡅⠀⡾⠁⢃⠈⠙⠋⠛⠛⠛⠷⡄
⠀⠀⣿⡿⢋⣧⣻⡶⣴⢢⡀⠙⢿⣷⣿⡟⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠀⠀⣤
⠀⠀⠹⣟⣻⣻⣿⢿⣧⡎⡀⠀⠈⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠈⠛⠿⣿⣼⣿⣿⠃⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠃

⣵⣾⣧⣀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠛⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⣾⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⢡⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿
⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣷⠀⡜⠻⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣛⣜⠀⠀⠀⠈⠻⣿⡟⠀⠀⠀⠀⠀⠀⢠⣤⣄⣠⢏⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠱⣶⡆⠀⠀⠀⠀⠘⠻⠿⠿⠿⢿⣿⡿⠀⠀⠀⠀⠀⢰⣶⣶⣴⣶⠀⠀⣨⠟⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⣀⠀⠀⢠⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⣤⣄⡀⢰⠋⣁⣀⡀⠀⠀⠀⢈⣿⣿⣿⡟⣠⠊⠁⠀⠀⠻⣿⣿⠟
⠀⠀⠀⣸⣿⣿⡇⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣋⣾⣿⣿⣧⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷
⠀⠰⣿⣿⣿⣧⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠛⠻⠛⢛⣿⣿⡿⠁
⠀⠀⠈⠉⢿⣿⡟⠀⠀⠀⢰⣷⣶⣦⣦⠀⢀⡴⠛⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⢀⡞⠀⠉⠁
⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣼⣿⣿⣿⠟⣠⠊⠀⠀⠻⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⠞
⠀⣤⣀⢀⡔⠁⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⣀⣀⣀⠀⠀⢠⣄⡀
⣼⣿⣿⣧⣾⣿⣿⠀⠈⠛⠻⠿⢿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣶⣤⣀⠀⡠⠞⠁⠀⠀⠀⠘⢿⣿⠿⡧⠄⠘⠟⠁
⠿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⡞⠀⠉⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡟⠋⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⢸⣿⣿⣿⠋⠁⠀⢀⣤⣤⡸⢡⣾⣷⡀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⣤⣤⠖⣭⣦⠀⠀⠀⠀⣸⣿⣶⡆⡰⠋
⠀⠀⠻⠟⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠇⠰⣿⣿⣿⣿⣿⣿⣿⣿⣶⡖⠒⢹⣿⣿⣿⣿⡿⠟⠀⢠⣾⣿⣿⣏⣝⠁
⠀⠀⠀⢀⣀⠀⠀⠀⠻⠿⣿⣿⣿⣿⡟⠁⠀⠀⠈⣹⠟⠛⢻⣿⣿⣿⠟⠃⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⠉⠙⢿⣿⠿
⠀⠀⠀⣿⣿⣷⣤⣤⠀⢠⠃⠹⣿⣿⠇⣴⣦⣔⢋⣤⣶⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⣠⣴⣶⣿⣿⣿⣿⡿⣠⠃⠀⣀⣠⣥⣾⣿⣿⣿⣾⣿⣿⣷⡄⠀⠀⠀⠀⢀⣶⣦⣤⡀⠀⠀⠀⠀⣰⣷⣞⣱⣾⣆
⣿⣿⣿⣿⣿⣯⣍⡼⠵⠚⠉⠁⠀⠙⠻⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⣠⣴⣿⣿⣿⣿⡏⢀⡠⠔⠚⠛⢿⣿⣿⡿⠛⠁⠀⢠
⠿⢿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠃⠀⠀⠀⠐⣿⣿⣿⣿⣿⣭⡖⠉⠀⠀⠀⠀⠈⠻⠟⠁
⣠⠖⠉⠹⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠛⠂⠀⠈⣿⣿⣿⣷⢦⣄⠀⠀⠀⠂⠀⠀⠀⣀⣠⡀⠀⢻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⢿⣿⣿⣿⣷⣮⣄⠀⢀⠀⠛⢿⣿⣿⣶⣤⠿⠿⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣿⣿⣿⣿⡇⢀⠉⠈⠁⣽⣿⡿⢿⣿⠁⠒⠄⠉⠀⢠⣤⣀⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠈⠁⣸⣿⣾⡇⢈⠀⠀⢈⣛⠀⣀⣀⠈⠀⠀⢻⣿⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡿⣿⠁⠀⣰⣾⣿⣿⣿⣿⡀⠀⠀⠿⠟⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⣿⣿⡿⢿⡇⠀⠀⠀⠀⠈⠁⠀⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣷⠀⣠
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣧⡄⠁
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⢏⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀
⠀⠀⠀⠀⠀⠀⣀⣠⣾⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣶⠆
⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠹⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠠⣶⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣴⣾⡗
⠀⠀⠀⠀⠀⢙⣿⣿⣿⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣀⠈⣉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡿⢿⡿⠉⠁
⠀⠀⠀⣴⣿⣿⣿⣁⠀⠃⠁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠈⢿⠿⡛⠉⢤⣶⣶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣯⠀⣠⣶⡀
⠀⠀⠀⠀⠀⢻⣷⣦⡄⠾⠿⢛⠻⣶⣿⣿⣧⣤⣄
⠀⠀⠀⣠⣾⣿⣿⣿⣧⡄⠀⢸⠀⣹⣿⣿⣿⠿⠋
⠀⠀⠀⠉⠉⢙⣿⣿⠿⠃⠀⠈⠀⠋⠙⠿⠏⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⣠⡄⠀⠀⠀⠀⠀⠀⣀⣼⣿⣶⣶⣶⠀⠀⠀⢶⣾⣿⣤⠀⠀⠀⣀⡀⠐⠃⠀⠉⠁⠲⡄
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠸⢿⣷⣿⣷⣤⡄⠀⣶⣶⡌⢻⣿⣿⣿⣿⠡⢤⡄⠆⠀⠻⠿⠉⠃⢨⣴⣿⣷⣶⣿⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣾⣿⣿⣿⡏⠀⢾⣿⣿⣿⣿⣿⠟⠻⠿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⢸⣿⣿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⢠⠃⡀⠸⡀
⠀⠀⠀⠀⢸⢠⠁⠀⠑⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠉⢖⠠⠤⠤⠤⠤⠤⠤⠤⠀⠒⠒⠒⠂⢳⡀⠀⠒⠒⠂⠀⠄⣀⡀⢀⠀⡰⢿⠿⠟
⠀⠀⠀⠀⠀⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⢀⡀⠀⠆⠉⠊⢥⡀⠀⠀⣀⣶⣶⣶⠟
⠀⠀⠀⠀⠀⠈⠂⡘⢡⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢀⣽⣿⣿⣾⣿⡆⠀⠀⠙⠦⡄⠊⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⡷⣐⠈⠒⢿⣷⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⢿⣷⣿⠟⠛⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠃⠀⠀⠀⠈⠒⣤⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣸⣷⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣠⣴⣶⣦⣄⡀⠀⠀⢻⣿⣿⣾⣿⣿⡂
⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡇⠀⠀⠈⠉⠀⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠺⡿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠉⠉⠻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟
⠀⠀⠀⠀⠀⠀⢀⢇⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢃⠘⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⡯⣿⣧⣄⣀⡤⠤⠔
⠀⠀⠀⠀⠀⢀⣰⣦⣀⠀⠈⣀⠔⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⠒⠋⡀⣛⢿⣿⡿⡀⠉⠀⠒⣂
⠀⠀⠀⠀⠀⣜⡮⠛⠋⠠⠀⠠⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢙⣆⡀⠀⠠⢇⣭⡽⢄⡴⠥⠀⢠⢶⢌⠁
⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠈⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠁⠀⠀⠘⢓⡋⡀⠂⠀⠀⣾⠵⠦⠀⣜⡐⠠⠀⡀
⠀⠀⠀⠀⠀⠘⣤⠁⠀⠀⡸⠍⢛⠋⠉⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠀⠀⠀⠀⠀⠀⠈⠻⣭⣬⣽⣑⣶⠾⡀⠀⠘⡁⠐⠊⠁
⠀⠀⠀⠀⠐⣠⡽⠀⠀⠀⠑⠚⠣⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣿⡶⢿⡾⠽⠃
⠀⠀⠀⠀⠀⢸⠀⠒⠀⢠⡄⠂⣀⠙⢶⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⣮⠔⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣲⠀⠈⠨⢿⡁⠰⠃⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⢸⠀⠀⠦⠚⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠓⣀⢠⡀⠀⡄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣄⠐⣧⡤⠀⠀⣿
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⢴⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⢦⡀⠘⢠⠸⠏⡁⠁⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⣦⣭⣴⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠙⢆⣀⠀⠀⠀⠀⡴⠶⡀
⠀⠀⠀⠀⣀⠂⠀⠈⠙⣯⣻⡐⠿⠶⠷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⣞⠉⡭⠀⠀⠀⡿⠠
⠀⠀⠐⠋⠄⢀⠻⠀⢄⣩⣴⠉⠉⠁⠙⠚⣃⡀⠀⠀⠀⠠⣶⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠋⠋⠁⠀⠀⠘⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠉⠀⠢⣿⠿⠀⢠⣴⠆⣹⡿⢇⡄⠀⠐⠛⡓⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠄⠀⠤⢀⣀⢠⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢁⡒⣦⣶⣾⢛⡚⢀⣀⡀⠀⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⠀⠸⠽⡶⠣
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠾⠶⣿⡿⢿⣽⠓⠉⠀⠀⠀⠀⠙⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠶⠂

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⣠⣶⣿⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⣟⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶
⠀⠈⢹⠿⣤⠀⠃⢠⠠⠆⢀⡀⠀⠀⠀⠰⠛⠝⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⣿⣿⣯
⠀⠀⠐⠄⠘⢀⡀⠀⠰⣿⠋⠁⣶⣄⠀⣀⠏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⡄⢀⡤⠀⣿⣿⣽⠏
⠀⠀⠀⢠⣷⠄⢻⠆⠀⠀⠠⠾⠿⠿⠙⠋⠛⢯⡀⢀⡿⠆⠀⠀⠀⣦⡀⠀⠀⠀⣾⣆⠀⠀⣼⣿⡇⣿⣿⣬⢀⣤⡀
⠀⠀⠀⠀⠙⣁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣄⡅⢊⡀⣀⠀⠀⠀⠻⣿⣄⣄⠀⠘⠿⠄⠀⡎⠉⢃⣠⠭⠉⠑
⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠉⠨⠞⠋⠉⠁⠻⣿⣼⡃⣴⣾⣆⠘⠛⠻⣆⠀⠀⣬⣶⠇⠞⣉⡍⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠛⢷⣔⣻⠀⠛⣿⣿⣢⠐⣞⠀⠀⢾⣷⣠⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠊⠀⣾⣿⡇⠨⠿⠿⠆⣄⣥⣤⡄⠘⢻⠛⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⣿⣿⠷⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣄⡀⢀⢀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⠀⢰⣿⣷⣾⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣯⣶⣾⣿⡿⢀⠀⢉⣿⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡹⣿⣽⠆⡵⠋⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⣠⣤⣤⣄⡀⠙⣿⣵⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⣿⣿⡿⢆⣿⣯⣽⣿⡋⠉⠈⠁
⠀⠀⣶⡠⠤⢤⣄⡀⠀⠀⠀⠠⠚⢁⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣼⣿⣷⠖⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢸⡷⣄⢀⠘⣿⣷⣴⣗⡄⠀⣼⡄⠀⠀⠀⠀⠀⠀⠈⠛⠚⢚⣿⣧⣤⣿⣇⣿⠿⢿⠿⠿
⠀⠀⠘⣧⣭⠈⠀⠈⠋⠻⡏⠃⣼⣿⣷⣦⠀⠀⠀⠒⢶⣦⣠⠞⡃⢻⣧⣼⣿⠟⣿
⠀⠀⠀⢸⡟⠆⠑⠀⠁⢤⣯⢘⣾⣟⣿⠃⠀⠠⠊⢝⣾⣿⣧⣾⣶⣦⠙⢿⣿⣧⡿
⠀⠀⠀⠀⢧⠀⠀⠈⠀⣙⢿⢼⣹⡿⢃⣀⡠⠤⣑⣿⣿⣧⣿⣿⣿⣿⡄⠈⠻⣿⡇
⠀⠀⠀⠀⠈⠋⣸⣦⣾⠿⢻⡌⠉⠉⠀⠀⢀⣀⡼⣿⣿⣽⣿⡿⠿⠿⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠠⣿⣾⣿⣶⣿⣟⣿⠀⣴⣾⣿⡿⠁⠸⡿⠟⠋
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣽⡆⣿⠻⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠻⠿⠛⠉⠁⠉⣛⡁⠙⣿⣿⣿⣿⣿⣿⣶⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣾⣿⣿⣗⣿⣿⣿⣿⣿⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠈⠰⠆
⠀⠀⠀⠀⠀⡀⠀⣸⠧⠀⢸⣿⢿⣿⣿⠹⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⡼⠷⠶
⠀⠀⠀⠀⠀⠙⠀⢻⣆⣀⠈⣿⣿⣿⣿⡖⣦⣤⣽⣽⣿⣿⣿⣷⣤⡤⠴⠚⡽
⠀⠀⠀⠀⠀⠐⢇⣼⣿⡋⠀⠘⠟⠋⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠽⣧⠀⠀⠀⠀⠘⢿⣿⣿⣿⡆⠈⠉⠛⠛⠛⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠛⢿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢞⣿⣶⡄⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣡⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⣬⣽⠇⣠⠟⢻⣄⣤⠶⠶⠛⠛⠛⠛⢓⡀⣭⣭⣭⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠦⠴⠞⠋⠀⠀⠻⣆⠀⠀⠀⠀⠀⢰⣿⣿⣌⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢧⡀⠀⠀⠀⠻⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠈⠻⣄⠀⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⣾⠁⢀⣀⣄⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⠦⠋⢾⣿⣿⣿⡀⠀⠀⠙⠷⣄⠀⠀⠀⣼⣿⣷⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣵⣿⢸⣷⣮⣿⣿⣿⡄⠀⠀⠀⠈⠛⠆⠐⠟⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣼⣿⣿⠿⠿⠛⠁⠀⠀⣠⣶⣿⣿⣡⣼⣻⣿⣯⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣼⣿⣷⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣤⡀⠀⠀⣴⡆⣀⠀⣀⡀⠀⠀⢴⣦⡀⢀⣠⣄⡀⣀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⣠⣴⣯⣿⠁⢿⣶⣿⢂⣬⣿⣳⣶⣾⣯⠹⠟⠿⣿⡿⠆⣶⣾⠿⠙⠿⠈⠙⠙⠿⠳⢿⠗⠀⣻⣷⣦⡀
⠀⠀⠀⠀⣘⣉⣁⡉⠀⠀⠁⠀⠈⣩⣵⠊⡙⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⣾⣿⣶⣷⣄⠀⠀⠀⠀⠀⠀⣈⣿⡏⠁
⠀⠀⠀⠀⠿⢻⡿⠿⠄⠀⠀⠀⠶⣿⠋⠀⠈⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠉⣥⡤⠀⢀⣴⣦⣮⡿⠋⠁
⠀⠀⠀⠀⠀⠹⠃⢰⣆⠀⣀⣰⣿⣆⠀⠀⠀⠼⠟⡿⠃⠀⠀⠀⠀⠀⠺⠏⣠⣶⠎⠉⠀⠀⣉⠋⢴⣶⡄
⠀⠀⠀⠀⠀⠀⣤⣼⠃⠈⢛⣏⡀⠀⠀⠀⠀⢴⠆⠛⢀⣄⠀⠀⢀⣶⡶⠀⠉⠁⠀⠀⠀⠀⢉⣠⠀⣿⡿⠂
⠀⠀⠀⠀⠠⣦⠉⠉⠁⠀⣼⠉⠁⠀⠀⠀⠰⠟⢀⠀⠈⢉⡀⠘⠋⠉⠀⠸⣿⠀⠲⣦⠀⠀⢘⠛⠀⠈⠐⡟
⠀⠀⠀⠀⠀⢀⣤⠀⢀⡄⡛⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠘⣤⡄⠀⠀⣀⣀⠈⠁⠀⠋⠀⠀⢴⣦⠀⣶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡗⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠂⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠛⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣶⣀⠀⠀⣰⡦⡀⠀⠀⠀⣵⡟⠟⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⢈⡟⣧⡋⠁⠘⠛⠁⢷⣦⠀⣠⢿⡏⠀⠀⢠⣿⣤⡦
⠀⠀⢠⡴⢾⣵⣿⠿⠀⣠⠋⠻⠃⠈⠂⡤⢠⣀⡌⢩⡽⠞⣳⡿⣶⣶⣆⢡⣤⡀⠀⠀⣀⣀
⠀⢀⡙⣋⠉⣽⣿⡀⠤⣿⣿⣷⣴⣎⡟⣷⣨⣿⢲⣏⣀⡮⣿⠅⠹⣿⣿⣰⣾⡻⣇⣨⡿⠿⢀
⠀⠀⣐⣏⡻⠮⣹⠷⡾⠲⠯⠉⠀⣧⣷⡄⣿⣶⢸⣿⣴⣮⣧⣶⡀⠈⠙⠿⡟⠛⣾⡟⠛⠾⠛⣇⣀
⠀⠶⠟⢫⡍⠄⣾⣿⣿⣆⠒⡲⠿⠉⣟⣿⣝⠿⠿⣿⠛⠻⣿⡿⠁⠀⠀⠀⠉⠘⢿⠀⣾⣧⢴⡋⣏
⠀⠱⡦⣾⠗⢀⣿⢷⣸⣇⠀⢩⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶⡟⢘⣇⣸⢟⣓⡄⢠⡀
⠀⠀⢠⣏⢰⠛⠋⠚⠈⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⢲⡿⣧⣬⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠘⠋
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠽⠋⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⠕⠉⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⡀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⢻⡀⠀⠀⢠⠃⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣴⣠⡀⠳⡀⠀⠸⡀⠀⠀⠀⠀⡠⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⢳⣷⠀⣠⢛⡩⠭⠭⣻⡶⣚⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠙⠤⠥⣰⠞⠁⠈⠉⠒⠦⣀⡀⠀⠀⠀⠀⢀⠎⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣊⡛⠀⠀⠀⠁⠀⠘⣆⠀⠀⠀⠀⠉⠑⠢⢄⡀⢜⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⠁⣩⡄⠀⠀⠀⠀⠀⠊⠳⣄⠀⠀⠀⠀⠀⠀⠉⠛⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡇⠀⠀⠀⠀⠀⠐⠀⠈⠑⠦⣀⠀⢀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⢒⡶⢶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⢀⡠⢮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣹⣍⣇⠀⠀⠀⠐⠉⠀⠀⡇⢀⠀⢀⢧⠋⣠⣾⢧⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠌⣑⠲⠤⣄⣴⠤⣤⢿⠋⡇⠾⠞⡄⠉⠀⠸⠒⠒⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠉⠲⣤⣈⢣⢀⠗⢧⣧⠀⢀⠟⢆⡀⢚⠉⠉⠲⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⠓⢬⡑⠿⣞⠤⠀⡛⣷⠾⡀⢒⣩⣣⠬⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠉⠢⣈⠙⠶⠗⠁⣠⠞⢢⢇⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠳⣄⣠⣞⠙⢖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⡟⠃⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠒⠶⠤⢄⣀⣀⡀⠀⠀⠀⣼⣄⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⠔⡡⠎⢉⠜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⣠⠤⣶⡥⠎⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⣤⣼⣁⣀⣀⣀⣠⣎⣀⢠⣾⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠛⠓⠈⠉⠉⠀⠀⠈⠉⠈⠛⠐⠀⠬⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠐⣀⣀⡠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠂⠉⠁⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⣀⣤⣤⠶⠶⠓⡿⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⢀⡀⠀⠨⠐⠀⠀⠀⠀⠉⠙⢋⣉⡠⠤⠴⠒⠛⠁⡀⠀⠀⠐⣀⠀⠀⢣⡄⠀⠀⠸⡀
⠀⠀⠀⠀⠁⠄⠀⠀⢠⡤⠶⠚⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠖⠁⢀⠀⠂⡐⣦⡈⢦⢨⣷⡀⡐⠀⢱
⠀⠀⠀⠀⠐⠈⣠⠠⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠄⢦⠔⢆⣤⠡⠙⣉⢤⠞⠛⠉⠀⠀⠀⠘⡂
⠀⠀⠀⠀⠀⣯⣆⢀⢾⡇⠀⠀⠤⠒⠋⡁⠀⢀⢋⠀⠩⣑⠘⠚⣀⠤⠒⠋⠁⠀⠀⠀⠠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⢈⢿⣆⠀⠫⠀⠀⡀⠂⠰⠀⠃⠀⠐⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠉⢻⡦⢼⡄⠀⠘⡂⣀⡁⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠠⢳⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⡌⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠔⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⢀⠀⣀⡀⣀⡄⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠁⠈⠛⠡⠯⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢆⡀⠀⠀⠀⢀⠤
⠀⠤⠠⠈⠈⠲⠇⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⢈⠁⠀⠀⢀⡤⠒⢉⠖⠰⠄
⠄⣁⠀⠩⠅⣀⡄⠣⠑⡀⠀⢀⢁⠀⠀⠐⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣴⡤⠤⣤⠀⣄⠀⣀⠀⣀⡰⢋⠀⣴⠁⠀⠰⠁⡀⠠⠘
⠐⣤⢠⠄⣰⠞⣷⣄⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠃⠹⠿⠟⠿⠶⠒⠀⠀⠃⠓⠉⠐⠠
⠀⠁⠀⠀⠠⠞⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⡐⠢⠀⠀⠀⠀⠀⢠⡀⢀⠀⠰
⠀⡀⠘⠀⡥⠘⠆⠀⠀⠀⠀⠀⠀⠒⠑⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠁⠸⠁⣤⠀⠀⠂⠀⠀⡁⠀⠂
⡎⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠡⠀⠠⠃⠀⡼⠀⠀⠀⢀⠆
⠀⠀⠀⢠⠙⢄⡈⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⢴⣏⡅⠠⠀⣀
⠚⠉⠁⠘⠂⠈⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡾⠺⡾⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠃⠀⠁⠀⢀⠐⠊⠠
⠀⠀⠀⠀⡀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠳⡄⠀⠈⣀⠀⠀⢶
⠀⡠⠂⠀⠈⡰⣾⠫⠀⠀⠀⠬⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣄⠀⠉⠂⠠⣄
⠒⠁⠀⠒⢁⠔⢉⡗⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⡀⠀⠀⠀⠀⠀⠈⠙
⠀⠠⠴⠛⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡦⠀⠂
⠆⠁⠀⢰⠋⠘⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠠⣀⡠⠔
⠐⠀⠀⢸⠖⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠈
⠀⠀⠀⠞⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡤⢀⡕⠛⠃⠀⠀⠉⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⠈⠁⠈
⢠⣾⠾⡏⠉⢀⠀⠀⠈⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣿⡀
⢠⡿⠈⠋⠉⠀⠀⠀⠃⠄⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⢀⠀⠀⢤⡆
⠏⠐⠒⠂⠀⠀⠀⠀⠀⠃⠉⠠⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠶⡶⡴⣥⠓⢠⡄⠀⠙⢲⠄
⠀⢄⠀⠠⢀⣀⡘⣶⠛⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤
⠀⠀⠠⠂⡼⠉⠀⠇⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⡀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡄⠀⠀⠀⡄⢀⡀⠀⠀⠀⠈⠉⢀⠀⡀⠀⠀⠈⠠⢤
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⢀⠀⠚⠁⣀⠀⠀⠀⠒⠀⠤⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠳⠀⠀⡤⠀⠀⠘⡂⢤⣀⣄⠀⠚⠀⠀⠐⣀⢀⠀⠈
⠚⠳⡶⠀⠀⠀⠐⠄⢀⠀⠀⠀⠀⠀⠤⡎⠒⠁⠀⠀⠀⠉⠑⢯⣄⣀⡀⣀⡀⡠⠀⠞⣁⠀⠀⠀⠀⠀⠣⠔⠀⠀⠀⡐⠁⡀⠈⡬⢿⢦⣄⡀⠘⠋⠀⠣⡀
⠀⠘⠓⢤⣀⠀⠀⠄⠀⠃⠀⠀⠛⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⣻⡅⠛⢇⠉⠀⠤⡼⣤⠄⠀⠀⠀⠀⠀⠰⠀⠀⣥⡄⠃⠀⡖⠺⢶⡒⠚⠙⠛⠃⠒⠥
⠀⠀⠀⠁⠐⠀⢘⡞⠀⠀⠀⠀⢴⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣻⠮⠁⠀⢀⠜⠙⢷⠤⢀⡐⢰⠃⠀⠀⠀⡀⠀⠁⠚⠩⠇⠀⠁⠨⠀⣤⠁⡄⠀⠀⠀⠀⠈
⠤⠀⠀⠀⠀⠄⠁⢦⠔⢀⣀⠀⠈⢤⣑⠀⠀⠀⠀⠀⠀⠀⣠⠏⢃⡀⠀⢠⠐⠁⠀⠀⠀⢉⠙⠀⠀⠀⠀⠀⢄⠰⠆⠐⠀⠀⠀⠀⠀⠀⠀⠉⠃⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡀⢀⡀⠀⠀⢀⠂⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠒⠊⠁
⠀⠀⠀⠀⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠈⠑⠀⠁⠀⠀⠘⠁⠈⠀⠛⢩⠋⠉⠆⠀⠀⡜⡿⠐⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠖⠀⠀⠐⠀⠀⠁⠈⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠰⢦⠍⠀⠀⠰⠶⢶⠆
⠀⠀⠀⢀⠀⠀⠠⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠃⠀⠂⠀⠀⢰⡈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⣾⠆⠀⠀⠄⠀⠀⠈⡀⠛⠀⠀⠄
⠀⠀⠐⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠿⠀⠀⠀⠀⠀⠀⢠⠄⠘⠃⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠃⠀⠠⢀⠀⠀⠀⠀⠀⠀⠛⠀⡆⠀⠀⠀⠁⠀⠀⣰⠈⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⡀⠀⢰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡜⠩⣰⢠⡄⠀⠀⠸⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠃⢀⠈⠀⣶
⠀⠀⣀⠀⠀⠀⠀⠀⡀⣀⢀⣀⠀⢀⡀⢀⣍⠀⠀⠀⠀⠀⠈⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⡀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠈⠀⡄
⠀⠀⠈⠙⠒⠀⢰⠀⢹⡇⢸⣇⠀⡾⢷⠀⣶⠁⠀⠀⡀⠀⠀⠰⠈⠀⢀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠈⠀⠐⠂⠀⠐⠒⠙⢂⠀⠀⠀⢀
⠀⠀⠀⢰⡆⠀⠏⠀⠀⠀⠀⠉⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢂⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠉
⠀⠸⢀⡈⠀⠀⠀⠀⠑⠲⠤⢴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠈⠑⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠠
⠀⠀⠈⠁⠀⠀⠐⠒⠒⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣄⡀⡠⠀⠉⠈⠀⠀⠀⠀⢀⡀⠀⠀⠀⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡆⠀⠀⡄⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⡀⠂⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠐⠀⠀⢓⠐⢀⡀⡀⠀⠀⠠⡜⠤⠀⢠⠀⠀⡦⠀⠃⠀⠀⠀⠀⠒⠀⠀⠀⡀⠀⠀⠀⡀⠘
⠀⠀⡅⢀⠀⢀⣀⣰⢀⣀⡅⠀⢠⠁⠀⠀⠨⠀⠀⡤⠐⠀⠒⠂⠀⠠⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠁⠀⠀⠒⠒⠂⠈⠭⠭⠃⠀⠀⢦⡼⠀⠀⠈⠦⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠑⠒⠶⠤⠤⣄⢠⡖⠉⣉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⣀⡤⣬⡇⡞⠁⣀⣀⣠⠞⢷⣀⡾⠉⠓⠦⣄⣼⣾⠉⠉⠉⠑⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠁⠀⢀⡿⠳⠚⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡴⢋⡤⠴⠒⠒⠲⢤⣀⢀⡴⠚⠉⠉⠉⠉⠓⠢⣄⡈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⠞⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⡾⢥⡏⠀⠀⢠⠖⠒⠲⣄⠀⢠⠟⠫⣷⠀⢀⡤⠖⠲⢤⡀⠀⠈⣇⣠⠽⣖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢩⣟⣹⠾⠯⠷⠶⠶⠶⢶⠖⠚⠲⣼⢦⡤⠿⠤⠬⠤⠤⠤⠤⠧⠤⠶⠿⠴⠶⣾⣷⡒⠲⠆⢀⡠⠴⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⡤⠬⢿⠀⠀⠀⡿⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⡞⡴⠒⠲⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠋⠀⢀⣀⡤⠤⡖⠒⠒⢛⡗⢒⡞⠉⠉⠙⠓⠒⠒⠒⠤⠤⢤⣀⣀⣀⠀⠀⠀⠉⠙⠒⠲⢧⣇⠳⣀⣀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⡤⣖⠉⠡⡄⠀⠈⠲⣄⡞⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠉⢉⠛⠒⣦⢤⣀⡀⠀⣧⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⠋⠀⠈⠳⠤⠞⠀⠀⠀⡾⠀⢀⡷⠤⠤⠤⢤⡤⠤⠴⠖⠚⠉⠀⠀⠀⢰⡃⢀⠜⠁⠀⠀⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠶⠆⠀⠀⠀⠀⠀⠈⠳⣇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠘⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠉⠒⠢⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠔⠚⠉⠁⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉





⣿⠿⠿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣛⣛⣿⣿⣿⣿⣿⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡄⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠾⠃⢤⡌⠁⠘⠀⡀⢄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠒⠐⠁⡶⡰⢤⣚⠉⠀⠀⠈⠃⠀⠁⡀⠀⠁⢠⣀⢀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠲⠄⠄⢸⣏⠀⠑⠎⠡⡀⠀⠀⠀⠈⠀⠀⠉⢀⠀⢀⡀⠉⠄⠀⠅⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⡀⠀⢀⣄⢠⡈⠉⠈⣀⠀⠀⠘⠃⠀⡘⠀⠀⢀⠀⠈⠓⢂⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣠⠀⡀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⠠⠀⠀⠀⠀⠀⠐⠀⠀⠙⢋⣀⣼⢶⡀⠀⡲
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣤⣀⡟⡳⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠿⣀⡆⠉⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠙⡏⡯⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣀⠀⠀⠀⠈⠙⣉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⢘⣎⣤⠀⠀⠀⠀⠁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⠀⠀⡄⠀⠀⢇⡿⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠀⣀⠀⣸⢠⡤
⠀⠀⠀⠀⠀⠀⡀⠀⡄⢀⡄⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢻⡉⠓⡦⢀⠐⠃⠀⢀
⠀⠀⠀⠀⠀⣠⣁⡴⠁⠈⠉⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣰⠐⡠⣄⡀⣀⡈⠗⡎⠠⠍
⠀⠀⠀⠀⠴⠟⠻⢶⡀⠑⠈⠀⠀⠐⠁⠈⠁⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀⡲⣴⡾⠖⡉⢻⣿⣧⢶⢷⣦
⠀⠀⠀⠈⠴⠀⢤⣼⢽⠀⠀⠀⢀⢀⡀⡄⠄⠀⢀⢸⠄⠀⠀⠀⠀⠀⠀⣜⠁⠀⢘⡗⠀⠀⠙⠻⠂⠋⡀⠎⠁
⠀⠀⠀⠀⠀⠀⡝⠁⠈⠀⠁⠤⠬⣈⠄⢀⠔⠀⣮⣼⠦⠀⠀⠀⠀⠀⠀⣝⠉⠉⠉⢳⣴⡆⠀⠉⠀⠰⠗⠂⣉⠁
⠀⠀⠀⠀⡄⠀⠂⠀⠀⢠⠖⠀⠀⠀⠀⢀⠀⠨⢿⣿⠑⠂⠀⠀⠀⠀⠀⠈⢆⠠⣴⢀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⣣⡀⠄
⠀⠀⠀⠁⠀⠀⠀⢁⠀⠾⠀⠀⠀⠀⠀⠘⡀⠠⠤⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡥⠀⠀⠀⠀⢀⣰⡃⠙⠷⠇⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁⠉⠈⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢛⡛⣻⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣾⣿⡷⢀⠠⢈⠀⠨⠂⠀⠀⠀⠀⠀⠀⠠⠄⣂⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡅⠁⣿⣿⣿⡟
⠛⢿⣷⣶⣶⣷⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠈⠛⢿⡇⢤⣿⣿⣿⣿⣶⣶⣾
⣡⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡄⠀⠀⠠⡂⠕⡂⠀⠀⠀⠀⠀⣠⡴⡢⠄⣤⡀⠀⠀⢱⣾⣿⣿⣟⠁⠀⠠⡉
⠿⠿⠟⠉⠙⣿⣿⣿⡿⠿⣿⣿⣧⣀⣀⣼⣷⠀⠁⠀⠀⠀⢠⣾⡟⠐⠀⢾⣿⣿⣷⣾⣿⣿⣿⣿⡇
⠈⠀⠀⠀⠀⢈⣿⣿⣧⠀⠀⠹⠏⣿⣿⡿⡋⠔⠒⠊⠀⠀⣿⣿⡇⠄⠀⠈⠠⢹⣿⠿⠿⠿⣿⣿⠃
⠀⠀⢾⣿⣿⣿⣿⠿⠿⠇⠀⠀⠘⠌⠉⠈⣀⣀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠐⣿⠁⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠘⣿⣿⣏⠀⠀⢀⢀⢀⠥⢀⠄⢀⣾⣿⣿⣿⣶⣶⣿⠿⠋⠀⠇⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⡇⠀⠈⠒⠁⡀⠀⠀⣸⣿⣿⣿⣿⣷⣶⣶⣿⣿⣶⣈⡀⠀⠀⠹⣿⡿⠂⠀⠀⠀⠀⣿⣏⢀⣿
⠀⠀⣼⣿⣿⣿⠃⠀⢀⣴⣿⣇⠀⠀⣿⣿⣿⠟⠛⠉⠉⠉⠉⠉⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡜⣿
⠶⠞⢋⠁⣿⣿⣷⣶⣾⡿⢿⣿⣦⣄⣽⣿⣧⣄⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⢄⠀⠀⠀⢀⣿⠣⣈
⡿⡀⠄⠀⢿⡿⠟⢻⣿⡿⠀⣿⠟⠁⠀⠂⣽⣿⣷⠀⢀⠠⠀⠙⠒⠚⠀⠀⠀⠀⢀⣤⣤⣀⡈⠉⠙⢍⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠙⠀⠀⠀⡀⣿⣿⡇⠀⠀⢀⣠⣤⣤⣀⣀⡀⠌⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡐⠄⠀⠀⢠⣶⡄⠀⠀⠀⠈⠀⠀⠀⠀⠂⠹⣿⣧⣀⣠⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣦⠇⠀⠀⣾⣿⡇⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠈⠻⠿⠿⣟⡛⠭⠝⠀⠈⢻⣿⣿⣿⡏⠛⢿⣿⣿⣿⠿⣋⣴⣾
⣿⠄⠀⢰⣿⣿⡇⠀⠠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⢀⢸⣿⣿⣿⡄⠀⠀⢸⠿⠛⡛⠑⠉⠀⠀⠉⢶⣾⣿⠃
⣿⢀⣠⣿⣿⣿⡇⠀⠀⣿⣿⣯⠀⠀⢾⣧⣍⣤⣤⣤⣾⣿⣿⠿⠃⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃
⣿⣷⣥⣿⣿⣿⣷⣄⣠⠟⢻⣿⣄⣠⣾⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢰⣿⣿⠏
⢿⣿⣿⣿⡟⠁⠝⢛⢡⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⣄⠀⢀⣴⣿⣿⠫⡉⢰⠀⠀⠹⠿⠋
⠀⠀⠙⢿⡁⡀⠀⠐⢘⠟⠁⠀⢹⣿⣿⣧⣀⠀⣠⠎⠴⠀⣾⣿⠀⠸⣿⣿⣿⢀⠄⠁
⠀⠀⠀⠀⠀⢐⠆⡀⠀⠀⠀⠀⠈⠛⠿⠿⢿⣿⣿⠐⠀⠀⣿⣿⣆⢀⣿⣿⡏⡌⠀⠀⠀⠀⠠⡴⠀⠁
⠀⣀⠀⠀⠀⠀⠠⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢴⣤⣤⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠔⡁⠀⠀⠄⠀⠔⠁

⡿⠿⠀⠀⠀⠀⢉⡛⣿⣿⣿⣿⣿⡿⠋⠀⠀⣩⣭⢹⣷⡄⠉⠀⠀⠸⣿⣿⣿⡿⣽⣿⣁⣀⠀⠀⠀⠀⢶⣿⣿⣿⣿
⠁⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⡏⢉⠁⠀⠀⠀⢿⡏⠀⠙⠓⠀⠀⠀⠀⣤⣶⣾⣿⢿⡏⣿⣟⠃⠀⠀⠀⣾⣿⣿⣿⣿
⢀⣤⢀⣴⠀⠈⠙⠉⣹⠿⠋⠁⢀⠀⠀⠀⠀⠈⢀⠀⢀⣤⣶⡇⠀⠀⠻⢿⠟⣱⣿⣿⠻⣿⠇⠀⠀⠀⠛⠛⠻⠟⢠
⣸⣿⣼⣿⣶⡾⠂⠀⠀⠀⠀⠀⢹⣿⣷⣦⠀⠠⣾⣷⣾⣿⣿⣷⣶⣴⣦⡄⠀⠻⠿⠟⠀⠀⣠⣤⡄⠀⠀⠀⢶⡄⣿
⠸⣧⣙⢿⣶⡄⠀⢠⣶⣶⣦⠄⠘⣿⣿⣿⠃⢠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⢠⣄⣀⠀⠀⠀⠿⠿⠁⠀⠰⣤⡘⣿⠸
⣷⣝⠻⢧⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⡠⠘⣿⣿⣿⣿⣿⣿⣤⡀⠀⢾⣿⣿⣷⡄⠀⠀⠀⠀⠙⠷⣝⢷⡘⣷
⠈⠙⠀⠀⣠⣶⠀⠀⠀⣾⣿⣶⢠⣶⣶⡄⠈⠀⠠⠟⢿⣿⣿⠿⡿⠛⠁⠀⠸⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠿⠈⢿⠘
⠀⢀⡀⠈⠛⠁⢠⣶⣦⡹⣿⣿⢹⣿⣟⣁⠀⠀⠀⠀⠘⠻⠏⠀⠀⠀⢠⣤⣀⠀⢀⠠⣄⠀⣾⡇⣴⣟⡀⠀⣤⣄⡀
⣿⣿⣷⡄⠀⠀⠘⠿⢿⡿⣮⣿⣾⣽⣿⡿⠃⠀⠀⣀⡄⢀⡄⠀⣀⠀⠈⠙⠋⠀⠘⠿⠟⡀⣨⣤⣌⠉⠁⢀⣿⣿⣿
⣿⣿⣿⡶⠃⠀⠀⢴⣿⡿⢿⡿⠙⠿⠷⠀⠀⠀⣰⠟⣴⡿⣡⣾⣿⣴⡖⢀⣀⠀⠀⢠⠋⠀⠙⠻⠄⢠⣴⣤⣙⣿⣿
⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⡀⠀⠀⢸⢇⣾⢅⣼⠛⣹⠿⣻⣾⣿⣥⠄⠀⠀⠀⠀⠀⠀⢀⣄⠈⠙⢻⣿⣿
⢀⣀⣀⠀⠀⡀⠀⣰⣄⠀⠀⣴⣼⣿⡇⠀⠀⠀⢸⠟⣼⣿⢸⣷⢠⣿⣵⣿⣷⣶⣶⡤⠀⠀⢶⣿⣾⣿⣤⠀⠈⠉⠉
⠿⣿⣿⣧⠀⠁⠀⣿⣿⣿⡄⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⠏⢸⣿⠀⢿⡇⠙⢿⡀⠀⠀⠀⣠⣶⣿⣿⣿⣷⡈
⠀⠀⠉⠉⠀⠀⢀⣈⣿⣿⣿⣿⣿⣯⣴⣶⣶⡤⠀⠀⠀⠀⠀⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⠿⡿
⣆⢠⣷⠀⢠⡄⠀⠙⢿⣿⠿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣤⣾⣀⣤⣀⡀⠀⠀⢠⣾⣿⣦⠀⠉⠈⠀⠀⠀⣀⣀⡀
⡿⢸⣿⢠⣿⡇⣠⣶⠀⠀⠆⠙⠻⢿⣦⠀⠀⠀⠀⠠⠀⢿⣿⣿⣿⣿⡿⠋⠀⠀⣾⠿⠿⠋⠀⡀⠀⠀⠀⠀⠈⠻⢿
⣷⡾⣁⣺⠟⣼⣿⢃⣤⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣥⣤⡀⠀⠀⠀⠀⠀⢀⣾⣿⣦⠀⠀⠀⠀⠘⣻
⣿⣿⣿⣥⡴⠛⣱⡿⢃⠀⠀⠀⢻⣿⣷⡀⠀⠀⢰⠿⢻⣿⣿⡻⣿⣿⣿⣷⠀⠀⠀⣿⣷⣼⣿⣿⣿⣠⣴⣶⠀⠀⠉
⠰⠿⠛⣫⣶⠿⣉⡴⠏⠀⠀⠸⢿⢿⡿⠇⠀⣀⣀⠀⠘⢿⣿⠇⠈⠋⠉⠉⠁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡿⠄
⠀⢀⠈⠉⠁⠞⠋⠀⠀⠀⠀⣀⠀⠋⠀⠀⠀⠙⠻⠃⠀⠈⠛⠀⠀⣰⣾⣶⣦⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⠀⠀⠀⢠⣤⣄⣄⠀⣰⣿⣆⠀⠀⠀⡀⠀⣤⣤⡀⠀⠀⠀⢸⡿⠟⠛⢁⣀⠀⠉⠉⠿⠛⠙⢻⡟⠙⠋⠀⠀⣤
⣿⡟⠀⠀⠀⢘⣿⣿⣿⣧⣹⣿⣿⢁⣀⠀⢻⣿⣮⡛⡛⢿⣆⠀⠈⠀⣠⣄⣸⣿⣧⣴⡄⠀⠀⠇⠀⠀⠀⠲⣿⣿⣿
⠀⠁⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠟⢿⣀⡀⢀⡀⢀⡠⠆⠸⡇⠀⠀⠀⢸⢶⣠⣄⠀⠀⢄⣷⣴⡆⠇⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡞⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡰⢾⣳⠀⠀⢴⣿⣿⣜⢿⣧⡰⡖⠂⠸⣦⣠⠀⠁⠘⢿⡟⢀⣶⡿⢿⠿⠇⠸⠄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠹⢆⢁⠤⠠⢈⠁
⠀⠀⠀⠀⡀⣀⣾⣧⡌⠉⠀⠀⠨⠿⠋⠛⠟⠛⢁⣿⡶⠆⠈⣷⢷⠿⠋⠁⣠⢿⠁⣀⣠⣄⠁⠀⠀⠈⠓⠂⠂⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⣨⠏
⠀⠀⠠⠤⠿⠿⡟⠉⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⢂⡄⠦⢱⣀⠠⢉⣯⣼⣿⣛⣃⠀⠀⣴⡀⣀⣀⠀⡀⢀⡉⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠆⠀⠀⠀⡀⢈⡠⣦⡀⠀⠀⠀⠐⣾⣿⣊⡀⠐⣼⣦⠠⠼⢿⣿⣿⣿⣿⣿⣿⣿⡤⠀⠺⣿⣿⠿⢓⣀⣸⣤⡃⠀⠀⠤⣸⠝⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠑⠀⠁⠠⣍⡓⠸⣿⣧⢤⠀⠀⠠⠈⠛⣿⣿⣿⠏⢡⡀⣌⣶⣦⣿⣿⣿⢿⣹⣻⣄⣤⣿⡇⣠⢬⡙⠿⠧⡴⢠⢠⣀⠀⠀⠴⢦⡀⠐⠀⠀⠄
⠀⠀⠀⠀⣀⡀⢠⣴⠦⠝⠃⠀⣥⡇⠘⣡⠄⢀⣈⠀⢨⡁⢌⢓⣺⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⡟⠁⢈⡛⢁⣠⣈⣻⣿⣿⣦⣶⣲⡛⡁⢀⡂⢢
⠀⠀⠀⠀⠈⣷⠀⠀⠤⠤⡄⠀⠀⣿⣻⣿⣿⣿⣾⡄⣠⣼⡅⣶⢽⣿⣿⣿⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠘⣿⡾⢿⠿⠽⣿⣿⡿⣼⣻⣴⠁⡸⣹⡏⡄⠂
⠀⠀⠀⠀⠘⠋⠀⡇⠀⠀⠄⠀⠀⣿⠁⠸⣿⣿⣿⠇⠋⠿⠿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣷⠀⣾⡷⠱⠜⢾⣿⣿⣿⠿⢿⠟⢿⣥⡌⣁⡀
⠀⠀⣧⠀⠐⠆⢰⡇⠀⠀⠀⠀⣶⣿⡦⠨⣯⠍⢈⣡⣔⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣸⣿⡗⠀⠀⣸⣟⣿⣯⣀⣾⣞⣯⣭⡽⢉⠁⠀⡀
⠀⠀⣾⠀⠲⠛⢸⡷⠀⢀⣤⢠⣽⣿⡗⠒⣷⣰⣿⣿⠟⣿⡶⢾⣿⣿⣿⣿⣻⢟⣩⣿⣿⣿⣿⣼⡻⣿⣿⣿⣾⣿⣷⣴⣿⣿⣿⠟⣻⣿⣿⣿⣿⠿⠃⠀⠀⠐
⠀⠀⣿⠀⠸⠗⢸⣿⣿⣿⠋⠿⠿⣿⣇⢦⣿⠿⣿⣿⡷⢟⡓⠞⣿⣿⣿⣿⣿⣾⣿⣫⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⢿⡟⢛⠛⢡
⠀⠀⣿⣠⡄⠀⢸⣇⠀⠀⠐⠶⠇⣿⣧⠄⣿⣳⣿⣩⣯⣶⣴⣙⣻⣿⣿⣿⠟⢻⣿⣿⠀⠀⢀⠙⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣟⣠⣿⡴⠋⠩⠓
⠀⠳⢻⣿⡇⡀⠘⣿⠀⠀⠀⠀⣤⣻⡿⣀⣿⠈⢀⢨⣟⠛⠉⠉⠛⣻⣾⣿⣿⣛⣻⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠤⠄
⠐⢤⣌⣿⡤⠋⠘⡝⠀⠀⠠⣄⠘⢻⣿⣻⣿⠁⠉⢹⡏⠀⢀⣀⣠⠉⢉⣈⣻⣿⣿⣍⡉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠙⣏⣠⡔⠠⠀⠀⠀⠀⠈⢡⣼⣿⣿⡿⠁⠀⢠⣿⠉⠉⠉⠀⠀⠀⠀⠉⠩⠟⠛⣛⣏⣉⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡀
⡀⣦⣸⣤⠋⠀⠀⢀⠀⠀⠀⠒⢾⣿⣋⡄⠀⡀⠀⡐⣿⡄⠀⠀⠀⠀⠀⠀⣀⣐⠲⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠈⠦
⠁⣈⣽⣏⠁⠀⡀⠀⠀⠀⠀⠀⠈⠙⣿⠀⠐⠀⢀⠀⡁⡀⠀⠀⠀⠀⣲⣶⡟⠉⠋⠙⠛⠻⠛⠻⠿⠿⠟⢻⠿⡿⠋⠻⠿⣿⣿⣇⠀⡄⠀⠀⠀⠀⠶⠰⢶⣦
⠈⢁⣀⠉⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⠁⠂⢄⠀⠋⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠁⢈⠒⠀⠀⠀⠐⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⢀⡠⠴⢂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⢆⠀⣴⡀⠁⠀⢠⡄⢀⠀⠉⢩⠀⠈⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⣉⠈⠀⠀⡇⠐⠤⠀⠀⠑⠉⠉⠁⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⣁⣉⠀⠀⠀⡀⣀⡠⠠⠖⠀⠀⠀⢀⠁⢀⠐⠉⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⠑⠆⠀⡴⣀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠃⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣆⠀⠀⠛⠨⠁⠀⠀⠀⡘⠂⠄⠀⠀⠙⠶⢞⠁⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣍⡠⣁⠶⠂⠀⠉⠀⠠⠂⠀⠀⢄⡠⣔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠎⠀⢙⡗⠒⢲⠶⠚⠛⠛⡖⠒⠓⣿⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠚⠃⢠⡞⠀⠀⣽⠀⠳⢀⣠⢙⠤⠔⠃⠑⠂⡴⢌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣀⢏⠀⠀⡠⠘⠃⠂⢡⠀⡖⠈⠉⠀⠤⠠⡄⠠⡀⠀⠀⡏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢰⠇⡀⠱⠞⠀⣤⢀⡠⠟⠠⠉⡀⠼⠀⡀⢀⢷⠀⢨⠉⠘⠣⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠟⠢⡀⡘⠀⠀⠀⠈⢠⠁⠸⣀⠀⠀⠀⢄⠀⢀⣄⠈⠠⠄⢠⡇⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⡏⠀⠀⠚⠁⠀⢀⠉⠂⢁⠀⠠⠉⠁⡈⠀⢀⠁⠀⢨⠀⢀⠀⠁⠈⡃⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡞⠁⠀⠀⡐⠀⠀⠤⣀⢠⠊⠀⡧⠒⠋⠁⠀⠀⠒⠒⢻⠀⠈⠒⠂⢠⡇⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⢰⠀⠠⠴⡎⠀⠀⢸⠀⢀⡌⠀⠀⠉⠂⢀⠄⠀⠀⠆⠀⠊⠀⠀⡀⠒⠁⢸⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⡼⢆⠀⡰⠁⠀⡀⠀⠈⠉⠀⠀⠀⠀⠂⠉⠀⠀⠸⡀⢀⠾⠀⠀⢑⠦⠀⣼⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡞⠀⠄⠚⠁⠀⠀⠧⠀⠤⠛⠀⠀⠜⠒⠀⡠⠀⠀⢠⠀⠁⢀⠆⠀⢨⠀⠋⠀⡆⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⢳⡀⠀⡠⠀⠀⠀⡄⠀⠀⡄⠀⢘⠀⠐⠊⠑⠀⠀⠘⠂⠒⠡⠀⠀⠀⠃⠒⢂⣯⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠉⠉⠛⠲⠦⣤⣄⣄⣀⣀⠀⠀⠀⢸⠢⠄⠀⢀⠂⠀⢀⡀⠀⢀⡀⢀⢀⣃⣠⠮⠜⠂⠀⠡⠀⠡⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⡇⠀⠀⠉⠹⠉⠛⠛⠛⠒⢓⠚⠛⠛⠛⠋⠉⠉⠉⡍⠀⠀⠀⡀⠀⢀⠀⠀⠀⠆⠀⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⢘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⡀⠀⠀⢠⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠸⠀⠀⡇⠀⠀⢘⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢤⠀⣄⢠⣀⣀⣀⣀⣆⣄⠀⠀⠀⠀⣄⢀⣀⠀⠀⠀⠸⠀⠆⠀⠀⠀⢀⣄⣠⠦⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠈⠀⠀⠈⠀⠁⠈⠛⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢄
⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠴⠒⠉
⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢹⣿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠈⠢⡀⠀⠀⢀⠀⠀⡌⠀⢠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⣷⠄⠈⠁⠀⠀⠀⠀⢠⣤⣤⠐⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠁⠀⢿⣄⡛⠉
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠄⠀⠀⠀⠀⢈⢿⣶⢷⣼⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀⠀⠣⢤⣀⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡈⢧⡈⠓⠶⠶⠄⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠹⡀⡔⣠⠀⡟⢂⠂⡄⢰⠴⣠⢦⢰⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠈⠀⠉⠀⠉⠁⠈⠀⠁⠈⠈⡄⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⣸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⢄⠀⠀⡇⠀⢀⡠⠊⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢠⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⣀⠱⣴⣡⠔⠉⠀⠀⠀⠑⠤⣀⠀⠀⢳⠀⠀⢠⠏⠀⠀⠀⢀⡤⠂⠁
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⠈⠻⡿⢉⠉⠁⠉⠀⠀⠀⠈⠙⢦⣘⡇⢠⠏⠀⢀⡤⠞⠁⠀⠀⢀
⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⣿⣿⣴⣞⡉
⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠁⠉⠉⡉⠑⠒⠤⡀
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡤⣐⣬⣶⣿⣿⣿⣿⣻⣷⢦⣖⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣬⣷⣿⣿⣿⣿⠱⣿⣿⡿⣟⣻⣿⣿⣛⣛⣿⣾⣽⣮⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣷⢟⣵⣿⡿⣟⣯⣿⣟⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣷⡄⠢
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣾⢯⣿⡿⣫⣾⣿⣿⡿⠟⠿⣭⣉⣭⡴⢶⣛⣛⣻⣷⣶⣯⣟⡿⣿⣆⢡
⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢿⣏⣿⣟⣽⣿⣿⣟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣶⣬⡻⣿⣮⣻⣆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⣿⢻⡟⣽⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣎⢿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠈⡿⢹⣾⢡⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡎⣿⠿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⢀⢂⠈⠳⣤⡀⠀⠀⠀⠀⠀⡇⢸⣿⣸⣿⠇⠀⠀⠀⣠⣴⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢸⣿⡇⢿⠀⡆⠀⠀⠀⠀⢀⡠⠄⠘⠃
⠀⠀⠂⠂⠀⠀⠈⠑⠒⠤⢀⢀⣇⢸⠛⠻⣿⡀⠀⠀⠀⣀⣀⣄⣈⣉⠛⠀⠀⠀⠰⠛⠛⠉⠉⠁⣿⣿⡇⡞⢰⠃⠂⠂⠀⠀⠀⢀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢾⢸⣸⡄⠀⡜⠃⠀⠀⠈⠉⣹⣿⡏⠻⠃⠀⠀⠀⠼⣿⣿⠛⡦⠀⣿⡟⢠⠃⠸⠄⠀⠀⠀⠠⡏⠀⠈⣱
⠀⠌⠌⠄⠄⠄⠠⠠⠀⠀⠌⠌⠎⣿⡇⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⢁⢎⣶⢸⠌⠄⢤⠤⢤⠹⠶⠶⠡⠌⠄⠄
⠀⠈⠈⠈⠀⠀⠁⠀⠀⠀⠈⠈⠜⡹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡎⣾⣿⡎⡌⠈⠱⠄⠴⠃⠀⠀⠉⠈⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡙⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠈⠀⠀⠀⠀⠀⣰⣟⢴⣿⣿⣿⣜⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⠳⡞⣧⠀⠀⠀⠀⠀⠀⢴⣤⣤⣤⣤⢀⣤⠄⠀⠀⢰⣿⣟⢾⣿⣿⣿⣹⣦⠶
⠀⠠⠡⠁⠀⠀⠀⠀⠀⠀⠀⣡⠅⣁⣹⠀⠡⢹⣷⡄⠀⠀⠀⠀⠀⠙⢿⠈⢿⠟⠋⠀⠀⣠⣿⡯⠿⠆⠙⣿⣿⣷⢿⡞⡁
⠀⠀⠡⠡⠀⠀⠀⠀⠀⠀⢠⠱⣏⠿⣫⡇⠐⡇⣿⠎⠀⢀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⡼⡿⣫⣾⣿⣿⣶⢪⣿⣿⡘⣱⠁⠀⠀⠀⠀⠀⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡰⣽⣽⣫⡞⣴⢧⡿⠀⠀⠀⠙⠲⣤⣀⣀⡀⠀⢠⣶⠋⣶⡁⢿⣿⡿⣿⠿⣨⠟⢋⣰⠁
⠀⢀⢂⢀⢀⠀⠀⠀⠀⠀⢀⢋⢢⣉⣹⣞⠋⠔⠁⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⢿⣧⠻⣿⣌⡉⢁⣐⣚⣭⠶⣻⢃⠀⠀⠀⡀⡀⡀⣀⢀⢀
⠀⠂⢂⠂⠀⣂⡠⠤⣤⣶⡖⠲⡎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⣽⢻⣮⣽⠛⣓⣦⠐⠚⠂⢂⠂⠂⠀⠀⠐⠀⠐⠀⠂⠂⠂
⠀⠀⣠⠔⠋⠁⠀⠀⣾⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⢈⣿⣿⣿⣿⣶⠂⢤⣄⡀
⢀⡞⠁⠀⠀⠀⠀⠀⣿⣤⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣤⣼⡟⠀⠀⠀⠉⢢⡀
⡜⠀⠀⠀⠀⠀⠀⢠⡛⣿⡟⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣿⣟⡇⠀⠀⠀⠀⠀⢑
⠃⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⠀⢀⡀⠀⠀⠀⠀⠀⢠⡾⢿⣿⡿⢿⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠐⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠐⠄⠀⠆⠀⠀⠀⠀⣷⣼⣿⣧⣾⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⣏⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠃⠀⠀⠀⠀⢸⣿⣉⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢿⣿⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠌⠡⠡⠁⠀⠈
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣶⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢰⣿⣯⣠⣿⣧⣨⣿⣅⣼⣿⣄⣽⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣅⣤⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠀⠀⠀⠸⡁
⠀⠀⠀⠀⠀⣾⡛⢻⣿⠛⢻⣿⠛⣿⡟⠛⣿⡟⢛⣿⡧⡀⠀⠀⠀⢀⡔⣻⡟⠛⣿⡟⢃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⡿⢿⣿⡿⢿⣿⡿⣿⣿⠿⣿⣿⢦⡀⣠⢿⣿⡿⢿⣿⡿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡷⠠⠀⠄⠄⠀⠀⠠
⠀⠀⠀⠀⠀⢷⣿⣷⣴⣿⣧⣴⣿⣦⣼⣿⣤⣾⣿⣤⣿⣷⣤⣿⣧⣴⣿⣧⣼⣿⣦⣼⣿⣤⣶⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣷⠀⠁⠈⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡄
⠀⠀⠀⠀⢰⣖⣴⢄⡀⠀⠈⠓⠶⠚⠁⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⣙⣻⣷⣎⡼⠆⠀⢀⡄⢠⣄⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⠃
⠀⠀⠀⠀⢩⣭⠿⣿⡴⢎⠾⣌⣷⣾⡿⢭⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠙⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠷⣿⡿⣯⣙⠓⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⠰⡷⡄
⠀⠀⠀⣀⢶⣘⣷⢰⡀⠀⠀⠘⠳⠆⠉⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠒⠒⠀⠛⠛⠃
⠀⠀⠈⠫⡟⣻⡿⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠁⠉⠀⠀⠀⢺⣷⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡮⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⢩⠇⠱⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠠⠂⠈⢼⠀⠠⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⢄⡀⠀⣸⡰⠁⠀⠙⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡀⠈⠑⢿⠣⣠⡹⢑⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡂⠂⠁⡿⢀⠤⣌⡡⢻⢦⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠒⡒⣒⠊⡉⠐⠒⢉⠢⡀⠀⠀⠀⠀⠀⡃⠄⢀⡠⠀⢾⢚⢰⠙⡡⣳⠁⠀⠀⠀⣸⠟⠢⢂⠄⢐⣒⠰⢒⠋⠁
⠀⠀⠀⠀⠀⣌⡷⡦⣄⠀⠈⢆⠈⠉⠧⡐⣌⠠⠇⠄⡎⠀⠀⣯⠚⡡⠆⡴⣩⣀⡔⣃⠂⡃⠘⢀⠜⠀⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠩⡀⠁⠀⠁⠒⠀⠀⣀⠈⠉⢷⠀⢠⢬⣨⡦⡾⡟⠔⣡⠈⠉⠓⢷⢀⡇⣀⡂⠷⠀⠒⠒⠂⠀⠐⠁
⠀⠀⠀⠀⠀⢀⣗⠦⠠⠀⠄⠠⠐⠉⠓⢢⣈⣇⠸⠃⡎⢼⠁⣟⠴⠁⡆⠀⠀⣸⠝⠁⣭⡄⠉⠀⡀⠀⠀⢇
⠀⠀⠀⠀⠀⠈⠀⠐⢤⠀⠀⠔⢀⡀⠀⣻⡬⡽⠳⢄⠱⣘⣆⢳⠂⠀⠀⢠⢚⢧⡂⢀⠐⠂⢀⠖⡾⢢⡤⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣈⠁⡌⠀⠘⢷⢒⠆⠳⢤⣈⠟⠀⣠⡞⠉⠁⠈⢿⡸⠆⠯⠷⠞⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⣱⣞⡠⣤⠄⡿⠷⡖⠷⡲⢿⡶⠶⡍⠂⡤⠆⢀⣄⠻⢆⣀⣤⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣠⢖⡡⠖⢺⡟⢸⠀⠆⠀⠈⢈⢠⢆⣡⠞⠀⠀⠈⠲⣀⣠⠊⣉⣀⢨⡟⢭⣱⡚⠵⠦⠄
⠀⠀⠀⠀⠀⠀⠰⠗⠚⠀⠢⠬⠤⢌⣑⠚⠂⠀⣝⡠⠛⠁⠀⢀⠀⠀⠀⠈⢂⣩⠵⠎⢄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⡋⠀⠀⠀⠀⠀⠀⢀⡼⠀⠈⠉⢻⡛⠶⢤⡈⠀⠀⡞⠁⢀⡏⠀⠀⣩⣅⠀⢀⡤⣤⠀⢿⡀⣀⣀⠀⢰⠋
⠉⢳⡀⠀⠀⢰⠟⠉⠀⣠⠞⣦⠘⢧⣄⠈⠛⠶⣄⠛⠶⠞⠁⠀⢸⡇⠈⢛⣯⣤⠞⠀⠀⠉⠉⠉⢧⡈⠓⢦⡀
⠠⣞⠀⠀⣠⡟⠀⣴⠚⠁⠀⢹⡀⢀⠈⢳⡄⠀⢸⠀⢰⡟⠛⣆⠀⠛⠦⠞⠁⢀⡴⠖⠛⠛⠲⣦⠈⠙⢷⠀⢳
⠀⠉⠛⠋⠉⠀⠀⠈⣷⠀⠀⠈⠉⠉⢳⠀⠙⠶⠟⢀⡼⠃⠀⠙⠦⡤⠤⣄⠀⠸⣇⠀⣠⣤⣤⠼⠀⠀⠈⠓⠋
⠋⠉⠛⠓⢦⡄⢰⡖⠋⠀⠀⢀⣀⣀⣈⡓⣲⠀⠀⠸⣇⣀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠁⢀⣀⣀⣠⠴⢤⡀
⣀⡀⠀⣠⠴⠇⠀⠙⠶⠶⠞⠋⠁⣀⣈⡉⠉⠀⣠⣄⡀⠉⠛⣆⠀⠀⠀⠈⠳⢶⡄⠀⣼⠋⠀⠀⠀⠀⠸⡇
⠈⢳⡀⣿⠀⣴⠚⠳⣤⣤⣄⠀⢸⡏⠈⠙⠲⠚⠁⢈⡷⠀⣠⠟⠀⠀⢀⣤⠴⠛⠀⠀⠹⣆⣀⣀⣀⡀⠀⢹⡆
⣀⡈⠓⠋⢀⡼⠀⠀⠀⠀⠻⣄⡀⠙⢶⡄⠀⢰⠟⠉⠀⢸⡇⠀⢀⣰⠏⠀⢀⣤⠶⢦⡀⠀⠉⠉⠀⠙⠶⠞⠁
⣭⡙⠒⢶⠘⢧⣤⠶⠶⣄⠀⠈⣹⠀⢸⠃⠀⢸⡆⠀⣀⣀⡙⠛⠋⠁⣠⡤⠞⠁⢠⡾⠁⠀⢀⡤⠶⠦⣄⡀
⢀⡷⠀⡞⠀⠀⣠⣤⡀⠙⠶⠶⠟⠀⠈⠳⠖⠋⠀⣰⠋⠉⠙⢷⡀⠸⣧⡀⠀⠀⠸⣆⡀⠀⣿⠀⠀⠀⠀⠙⣦
⠉⠀⠀⠻⣄⠘⢧⡈⠷⣤⠤⣄⠀⢰⠗⠦⣤⣤⠾⠃⢀⡶⠛⠛⠁⠀⠀⢿⣀⠀⠀⠈⢹⠄⠈⠛⠛⠛⢦⡀⠈
⠛⢳⠀⠀⣸⠀⣰⠇⢀⣤⣤⡼⠀⠘⢧⣤⡴⠖⠶⣄⣸⠇⠀⡴⠚⠲⢦⡀⠈⣷⠀⠀⣾⠀⣠⠖⢦⡀⠈⢷⣄
⠀⠻⣤⡴⠋⠀⠻⠴⠏⠀⠀⣠⠶⣤⡀⠀⠀⣀⣀⠀⠁⠀⣸⠇⠀⠀⣼⠃⠀⠙⠶⠞⠁⢰⠇⢀⡾⠃⢀⣀⡀
⠀⢀⣤⢤⣄⠀⠀⣀⡤⢤⡀⠹⣦⠀⠙⠒⠛⠁⠈⡷⠀⢸⡃⠀⠀⠀⠳⠶⠶⠶⠶⣄⠀⢸⡀⠸⡇⠠⡏⠈⣇
⠛⠉⠀⠀⠈⣧⠀⠿⣄⠀⢻⠀⣸⠇⠀⢀⣀⣀⠀⢧⡀⠈⠳⣄⠀⠀⠀⡟⠙⢷⣄⣼⠃⠀⣇⣰⠇⠀⢻⠀⠈
⡞⠋⠙⠶⠴⠋⠀⢀⡿⠀⢸⡄⠘⠲⠚⠉⠀⠙⡆⠀⣹⠀⢠⡟⣠⡤⣤⠹⡆⠀⣀⣄⠀⠀⠉⢁⣀⡴⠏
⣷⠀⢀⡤⣄⠀⢰⡏⠀⠀⠀⠳⠦⠴⠞⠛⢶⠀⠻⣤⠟⠀⠘⠶⠇⠀⠈⠛⠁⣰⠏⢈⡇⠀⣼⠋⢁⡴⠶⢦
⠁⢠⡏⠀⢸⡀⠘⢧⡀⠀⠀⠀⠀⣠⠴⠶⠞⠀⠀⢀⡴⠶⣄⣠⡴⣤⠀⢰⡛⠁⠀⠀⢷⡀⠈⠛⠋⠁⠀⠘⠧
⠀⠀⠻⣄⡀⢙⡆⠀⣧⣠⠖⠲⠞⠁⠀⡴⠖⡦⠀⠈⠳⡄⠀⢠⡴⠛⠀⠀⠙⣆⠀⠀⣀⡿⠀⠀⡞⠛⠲⠶⠶
⠀⣤⢤⡀⠉⠉⠀⠀⠀⠀⣠⢤⣄⠀⢸⠁⣾⠁⠀⣞⠛⠁⠀⠘⠧⠴⠳⡄⠀⡟⢠⠞⠉⣀⣤⣀⠙⠛⠛⠳⢦
⢸⡇⠀⠙⠲⠦⠤⠦⣄⠘⣇⡀⢻⡀⠈⢧⣌⣷⠀⠙⡇⠀⣠⣤⣄⣀⣰⠇⠀⠛⠋⠀⣼⠃⠀⠉⠓⠚⢳⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡌⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠀⠙⠛⠛⠛⠛⠛⠋⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣶⣶⣶⣶⣶⣶⣾⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢈⣃⣀⣉⣉⣉⣉⣉⣥⣶⣶⣶⣶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣼⣿⣶⣶⣶⣶⡶⠛⠉⠉⠉⠉⣿⣿⣿⣿
⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠠⣿⣿⣿⡿⣟⣻⣿⣿⡶⠞⣿⣏⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢀⣀⠠⠄⠐⠂⣩⣥⠀⢰⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣺⣿⣻⣧⡵⢿⣛⣿⣭⣷⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠈⠅⠀⠆⠘⣊⡀⠤⠐⠒⠈⠀⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣯⣭⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠐⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢎⡟⣄⡀⠀⠀⠀⠀⠀⢀⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡄⠀⠀⠀⠀⠀⠀⠀⣸⠛⣄⢨⢽⣻⠃⠀⠀⠀⢀⡴⠋⢠⠷⣤⣄⣤⡴⣋⣱⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢇⢳⣄⡀⠀⠀⠀⠀⢠⡟⠀⠨⢹⡼⡽⣇⠀⠀⡠⠫⠒⡫⠨⢍⠢⣢⣴⠾⢙⡼⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡴⠃⢸⡀⢻⡇⠀⠀⠀⠀⣾⢷⠀⠉⠀⣧⡎⠙⡷⡞⡡⠻⢷⣅⢎⣦⠿⣯⡤⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢢⠸⡇⢠⠹⡄⠀⠀⢸⢭⠂⠑⣄⠀⡇⣧⠅⢭⡚⡐⠉⣦⣾⠛⠁⠣⢈⡿⢯⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⢂⣿⡔⠀⢙⢦⣠⡏⠀⢆⠀⠈⢦⢻⡇⠉⠙⢇⣨⡶⠛⠍⠈⣷⠴⢿⣽⣢⡧⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡜⣀⣯⠀⠀⠀⢙⢣⠀⠀⢂⠀⠈⣶⠃⢀⣤⠟⡉⢊⢶⡊⢴⡟⣱⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠫⠎⠕⣻⡄⠀⡀⡞⠀⠇⠠⢤⡂⠀⢹⡧⠟⡁⠀⠈⡆⣸⠃⠤⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠀⣿⠚⠘⠀⢈⢧⡙⠁⠀⠀⣎⠀⠀⡸⠞⠛⠑⠿⠶⣥⣔⣲⠇⡠⣀⢛⠷⠒⢦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡗⡌⠓⠤⣀⡿⠀⠀⢠⡀⢸⣧⠀⢀⠼⣠⣥⣞⠓⡄⢢⡀⢀⠤⠚⠉⢩⠝⣃⠶⠟⠧⠶⣶⣶⣑⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⣻⡄⠀⢀⠁⡀⠀⢀⡇⠸⣯⣄⠈⠂⠌⠁⠈⠑⢾⡟⠊⢁⣀⣀⣱⡯⠎⠀⠀⠀⠀⠀⣬⡥⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢣⠀⠀⠀⢰⠀⣸⢁⠌⠸⣫⣄⣀⡀⠀⠀⠀⢠⠞⠉⠀⠀⢩⡯⠶⠒⠛⠋⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡽⠀⢀⠆⠀⠀⢰⣤⣯⡿⣀⠾⠿⣻⠞⢛⡓⠶⠶⣁⠀⠄⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠀⠈⠐⠦⢢⣾⡿⣩⠞⠋⠓⢀⣀⠀⠀⠈⡑⢦⡈⢑⠖⢦⣤⣄⣀⠀⠉⠛⠓⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠂⠀⠈⡿⣫⢶⣑⣂⠀⠀⠀⠀⢈⠑⠆⠀⠀⡠⠊⢠⠔⠻⡌⠉⠋⡐⠂⣀⣬⣷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢆⢂⠔⠓⠒⠒⠉⢭⣐⠀⠀⠀⠀⢠⡶⠶⠿⠒⠛⠛⠒⠚⠓⠶⢅⣜⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⣰⠃⠀⡅⠀⠐⠂⠁⠨⠉⢓⢦⣄⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠋⠉⠙⠓⠦⢄⣄⣀⠸⡷⢄⣛⣁⡌⠻⣦⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠙⠻⠞⠛⠷⠄
⠀⠀⠀⠀⠀⢀⠠⢐⠨⠒⠁
⠀⠀⠀⠀⢀⠕⠂⠁
⠀⠀⠀⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⡄⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠀⠀⠀⠀⣆⠒⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡰⠄⢄⡈⠐⠁⠆⠀⠀⠉⠋⠐⠐⢦⣀⠄⠀⠀⠀⡘⠶⣀⡔⠉⠀⠀⢩⡀⠄⢠⠔⠲⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⣀⢀⠑⢤⠀⢈⡀⠉⠖⠃⠀⠀⠀⠀⠆⠀⢀⠀⡉⠘⠁⠀⠐⡀⠁⢸⠁⠀⠀⠀⠀⡿⣯⠀⢀⡀⠈⡄⣀⠄
⠀⠀⠀⠀⠀⠀⢄⠈⠊⠄⠀⠂⠈⡃⠀⠀⠀⠀⠈⠢⡨⠄⠀⠹⠀⠀⠀⠀⠐⠁⣹⣠⣼⠀⢀⣠⡤⠞⣅⡇⠀⢨⠀⢸⠙⠁⠠
⠀⠀⠄⠄⠲⠤⠀⠙⠊⠁⡄⠀⠀⠳⠂⠀⠀⠀⠀⠀⠘⠂⠀⠐⠂⠑⠒⠐⠀⠀⠛⠟⠛⠻⠁⠀⠀⠀⠀⠑⠰⠂⠠⢆⠐⠚⠀⠀⠂⠴
⠀⠀⠀⠀⠐⣀⡀⠀⠀⠑⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠈⠀⠀⠈
⠀⠀⠀⢉⠁⠀⠈⢲⡋⡖⡁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⣆⢁⠠⠀⠄⠤
⠀⠀⠀⠈⢄⡀⠀⠀⢿⣺⡇⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠂⡠⢀⠄⡰⢠⢀⢆⡄⠔⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠘⣺⠞⠀⠀⠀⡆
⠀⠀⡜⠁⠀⠉⠛⠚⢾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⠌⠀⠀⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣏⣀⣀⡤⠆⠤⡀
⠀⠐⠀⠀⠀⠀⢀⣴⡋⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠜⠁⠁⢀⠈⡤⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⣿⣟⠁⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⢩⣃⠒⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⡁⠄⠁⠇⠐⠁⠏⠸⠀⠃⠺⠰⠇⠊⠇⠼⠀⠁⠀⠀⠀⠀⠀⠀⣽⢫⠀⠱⡄⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠐⠉⡈⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠊⠉⠇⠃⡈
⠀⠀⢀⣀⣁⠀⡀⢀⡀⢥⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡀⠈⢠⡌⠀⠠
⠀⠀⠀⠀⠀⠀⡠⠋⠁⡂⠀⠉⠀⠙⠦⣠⡤⠴⠞⠋⣿⠿⢽⠈⠁⢁⠉⡉⠀⡀⠀⠨⠏⢄⠀⠀⠀⠀⠀⠙⡀⢀⠀⠐⢁⠙⠠
⠀⠀⠀⠀⠀⠀⢀⠔⠱⡃⠀⠃⠀⠷⡼⠁⠀⣀⠀⠀⡇⠀⠹⠒⠁⢀⠐⠀⠀⠁⠀⢐⠂⠈⠃⠀⠀⢀⡀⠘⠆⠀⡦⡀⠈⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⣐⡠⠄⠪⡟⡄⠀⠀⠀⣸⠷⣼⡐⠀⠀⠈⢁⠳⣤⣌⡀⠀⠀⠀⠀⢀⠘⡈⠒⠢⢐⡔⠀⠈⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠈⢁⡤⡈⠁⠀⠀⠀⠀⠀⠌⠀⠀⠈⠘⠀⠀⠀⠐⠙⠢⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠀⠤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠁⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⠋⠉⠀⢀⡀⠀⠀⠘⠀⠠⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠋⠁⠀⣀⠀⠀⠀⢈
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠈⢴⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⠀⠀⠀⠁⠀⠠⠀⣶⡿
⣶⠉⠛⠛⠛⠉⠁⠀⠀⠀⠉⠍⠉⢿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⠙⠛⠛⠋⠉⠁⠀⠀⠀⠉⠉⠉⣿⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⣏⣈⣐⡀⢀⠀⠀⡀⠔⠀⡄⣠⣤⣬⣿⣿⣿⣿⣿⣶⣶⣦⣥⣴⣾⣿⣿⣿⣏⣀⣀⡀⠐⠀⠀⠠⠀⢀⠀⣤⣤⣼⣿⣿⣿⣿⣿⣶⣶⣦⣤⣶⣾⣿⣿
⠿⠿⠿⠋⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣾⣿⣿⣦⣤⣤⣤⣶⣦⣄⡀⠀⠀⠀⠀⠀⠙⠋⠁⢹⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣦⣤⣤⣴⣶⣦⣀⠀⠀⠀⠀⠀⠈⠙⠋⠈⢹⣿⣿⣿⣿⣿⣿
⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠀⠀⠀⠀⠀⣼⣿⡿⠛⠁⠀⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠁⠀⠀⠀⠀⢀⣿⣿⠿⠋⠁⠀⠉
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⣀⣠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⢀⣀⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠘⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠿⠿⠿
⣤⣽⣿⠟⠀⠀⠉⠉⣿⣿⣿⣦⣀⠀⢀⡠⠄⠀⠀⠂⠀⠀⠀⠀⢀⣤⣤⣤⣬⣝⣿⠇⠀⠀⠉⢙⣿⣿⣷⣦⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⣠⣤⣤
⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠃⢀⣤⣤⠀⢀⠀⠔⠀⠀⠂⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⠃⣀⣤⣄⠀⠀⢠⠐⠀⠐⠂⡻⢿⣿
⣿⠿⠛⠛⠿⣿⣿⣿⡿⠛⠋⠉⠉⠙⣿⠛⠛⢃⠈⢀⠆⠀⠀⠀⠐⢠⣼⣿⡿⠿⠛⠻⠿⣿⣿⣿⠟⠛⠋⠉⠉⢙⣿⠛⠛⢀⠀⢈⠀⠀⠀⠀⠐⣠⣾
⠀⠀⠀⠀⠀⠀⢯⣽⣶⣀⣸⣿⣦⣤⣿⡇⠀⠀⠉⠁⢀⣀⠀⠀⠀⠹⠟⠋⠀⠀⠀⠀⠀⠰⣯⣵⣦⣀⣻⣷⣦⣤⣿⡀⠀⠈⠈⠀⢀⡀⠀⠀⠀⠻⠛
⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⡿⠟⠁⠀⠀⢀⠰⠁⠀⠀⠀⠀⢀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⡿⠋⠀⠀⠀⢀⠙
⠁⠀⢀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠀⠀⠠⠀⠀⠀⠀⠄⢑⠊⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⠛⠉⠀⠀⠠⠀⠀⠀⠈⠀⠄
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠈⢆⠀⠠⢀⠾⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⠑⢀⠀⠀⠠⠾⠿
⣷⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⢿⣿⣄⠀⢀⠀⠀⠀⡀⠀⠀⠀⢀⣼⣿⣷⠉⠙⠋⠉⠁⠀⠀⠀⠀⠈⠁⠀⣿⣿⡄⠀⣀⠀⠀⠀⡀⡀⠀⠀⢀⣿
⣯⣀⣠⣄⠀⠀⠀⡀⠀⠀⠄⢤⣤⣴⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣇⣀⣠⡐⠀⠀⢀⠀⠀⠀⠤⣤⣤⣼⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿
⠿⠿⠟⡋⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢛⠉⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣿⣿⣿⣷⣤⣤⣴⣶⣶⣄⣀⣀⣀⡀⠀⠀⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣦⣤⣤⣶⣶⣦⣄⣀⣀⣀⠀⠀⠀⠙⠁⠀⢹⣿⣿⣿⣿⣿⣿
⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠉⠀⠀⠀⠀⠀⣾⣿⠿⠋⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠀⠀⠀⠀⢀⣿⣿⠟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⢀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢀⣠⣤⣤
⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠂⠘⠛⠿⠿⠿⢿⣿⣿⡿⡿⢿⣿⣿⣿⣟⡋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⡙⠻⠿⠿
⣦⣭⣟⡛⠀⠀⠀⢉⣿⣿⣿⣶⣄⡀⢀⠠⠀⠀⠀⠀⠀⠀⠐⠀⣠⣴⣶⣦⣦⣽⣛⠃⠀⠀⠈⢩⣿⣿⣿⣦⣀⠀⡀⠄⠀⠀⠐⠀⠀⠀⠀⠀⣡⣶⣶
⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠃⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⢁⣠⣤⣤⠀⠀⠀⠀⠀⠀⠂⡻⠿⣿
⡿⠟⠛⠛⠻⢿⣿⣿⠟⠋⠉⠉⠉⠉⣿⡛⠛⠣⠄⢀⠃⠀⠀⠀⠐⢠⣾⣿⡿⠛⠛⠛⠿⣿⣿⣿⠟⠉⠉⠉⠉⢉⣿⠛⠛⢠⠀⡐⠀⠀⠀⠀⠐⣤⣿
⠀⠀⠀⠀⠀⠀⢷⣴⣶⣤⣹⣿⣷⣤⣿⡇⠀⠀⠈⠀⣀⣤⠀⠀⠀⠹⠛⠁⠀⠀⠀⠀⠀⠰⢧⣶⣦⣄⣹⣿⣶⣴⣿⡀⠀⠀⠁⠀⣠⡄⠀⠀⠀⠻⠋
⣀⣀⣀⣀⣀⣄⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣋⣀⣀⣀⣠⣠⣀⣀⣀⣀⣀⣀⣀⣄⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣋⣀⣀⣀⣰⣨⣀
⣿⣏⠭⣟⣛⢛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠦⠼⠋⠁⠀⠀⠉⣠⡶⠆⠀⠀⠀⠀⠀⠀⢀⡀⣴⣠⠂⠠⠔⢀⣾⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣕⣯⢤⣄⠀⠀⠀⠘⠉⣀⠂⠀⡀⠀⠀⠀⡔⠘⣧⡻⠛⠀⠀⠄⠶⠀⣜⢽⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂⣙⡶⠽⢄⣎⠄⠀⠀⠀⠘⠃⠁⠀⠀⣄⣸⡽⡏⢂⠁⠒⠢⠀⠀⢀⣀⠀⠂⢐⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠈⡗⠉⠀⠀⠀⠀⠀⠀⠠⢈⠃⡆⣿⢯⡀⠸⢯⡔⠀⠀⢀⠤⣤⠀⡀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⢤⠒⠚⠀⠁⢿⣷⠀⠀⠈⠁⠀⠀⠀⠟⠚⠁⠀⡌⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢴⠀⠠⠈⡉⠀⠀⠀⣀⣀⠀⠀⠀⠀⡬⢠⡞
⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠄⠀⠁⢀⡠⠆⠀⢙⠄⢀⠷⠀⠀⠠⢇⠉
⠀⠀⡠⠖⠁⠀⠀⠀⠄⠀⠀⠀⢂⠀⠤⠊⠉⠉⠣⣀⡀⠀⢀⠃⠀⠀⠀⠈⢂⠀⠉⠁⠀⡀⠔⠣⠜⠀⠀⠄⠀⢀⡀
⠀⠊⠀⠀⠀⠀⠀⠀⠸⠀⠀⠡⠀⢸⣷⡄⠀⠀⡐⠂⠤⠤⠈⡀⣂⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠌⠀⠠⡺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢂⢀⣿⣄⢀⠔⠀⠀⠀⠀⠰⠁⠁⠀⠀⠀⣴⣢⣤⠅⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢾⣿⣿⣷⡀⠀⠀⠀⠐⠃⠀⠀⠀⠀⡠⠛⠛⢯⠁⠀⠸⡀⠀⠀⠠⠄⡎⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⢿⣿⣿⠁⠀⠀⠀⢀⠆⠀⡠⠔⠋⠀⠀⠀⢰⠀⠀⠀⠀⠒⠀⠀⠀⠠⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠉⠃⠀⠀⠀⢀⡮⠀⠀⠀⠀⠀⠀⢀⡠⠊⢆⠀⠀⢸⠀⢀⡠⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡠⠔⠉⠀⠀⠀⠣⠀⠘⡰⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠛⣁⣀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠀⢴⣁⠀⠀⡀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡈⠉⡃⠀⠉⠑⠓⠄⠀⠀⡀⠀⠰⠶
⠈⠛⠊⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⣤⠂⢀⠀⠀⠀⠐⠇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⢈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢠⠀⠀⠈⠊⠂⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠈⢤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⣀⣀⠀⠠⠄⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡀⠀⠈⠁⠤⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣈⣀⠈⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠆⠀⠀⠀⠤⠀⠀⠀⣀⣉⣉⣂⣄⠀⠠⠴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⣁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡄
⠀⢀⠤⠀⠀⠀⡰⠖⠋⠀⠀⠀⠈⠛⢉⡀⢀⠀⠀⠀⠀⠀⡀⢌⢀⡀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠀⠀⠀⠀⢸⠀⠀⠀⠈⢻⣿⠋
⠀⠀⠀⠀⠀⠁⠀⠠⠔⠂⠀⠰⡅⠀⠲⠇⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠚⠶⠤⣄⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠙⠐⠒⠤⢈⣢⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠇⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣀⠀⠀⡀⠀⠀⣀⠠⠀⠀⠀⣀⣀⡀⠀⢠⠏⢠⡀⡜
⠀⠀⠠⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠃⠘⠁⠀⠀⠀⠀⡃⠤⣀⡉⠢⣠⢸⠘⠿⠁⡠⠔⢒⡂
⠀⠴⢀⡄⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠐⣀⠀⠀⣠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣨⣙⣓⣼⣷⣴⣇⠠⠔⢀⡰⠁
⠀⠀⠀⠀⡀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⡘⠟⠈⣠⡾⢁⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⢠⠆⠘⠉⢉⠉⢛⣿⣿⣿⠟⠋⡉⠉⠉⠑
⠁⠀⠀⠀⠹⠒⢤⡄⠀⡠⠀⠀⠀⢀⠈⢁⣷⠟⣁⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⢠⣧⡴⠺⠋⠈⡴⠫⠟⠻⢍⢠⣄⠀⠈⠁⢀⠈⠂
⠀⠁⠀⠀⢠⠐⠢⠀⠀⠀⠀⠀⠀⢀⡶⠏⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠠⠖⠈⠀⠘⠀⡴⠘⣤⠌⢆⠉⠁
⢄⠀⡄⠒⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⢀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀
⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠁⠀⠐⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠘⡀⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⡠⢔⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢠⠂⢰⠀⠀⠇⡗⠀⡦⠊⠉⡁⠀⠀⠀⠀⠀⠠⠌⣡⡟⠀⠀⠀⠀⠠⠀⠄⢐⣾⣧⡀
⢀⠀⠀⠀⠀⡀⢀⠀⠀⠰⠅⠀⠹⠂⣁⠴⠋⠀⠀⠀⠀⣶⡞⠁⠁⠀⠀⠀⠀⠀⠀⢀⠁⠁⠈⠁
⡆⢸⠀⠀⢘⠀⠜⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠴⠄⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠇⣸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠴⠤⠆⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠀⠂
⠁⠠⠔⠁⣀⣄⣈⠉⠀⠂⠑⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⠃⠀⠈
⠀⠀⠀⠤⠲⢄⣀⣈⠉⠥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠀⣥⣿⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⡄
⠐⠲⠂⠒⠀⢀⠈⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠁⠀⠐⠀⢁⡀⠀⠀⠇⠀⠀⠀⠉⠒⢦⣀⠁⠀⢱⠆⣀⣧⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠁⠉⠀⠀⠀⠀⢠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⠩⢷⣤⡀⠠⠈⡉⠦⡃⡀
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠷⠆⠈⠙⠻⢧⣶⣀⠁⠡⡄
⣥⣷⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣐⣂⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣈⣐⣚⣁⣈⣉⣙⣂⣀⣀⣀⣀⣀⣀⣀⣀⣐
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⢀⡀⠄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⠆⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡀⢛⢉⡉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠙⢦⡀⠀⣙⠃⠉⠳⠆
⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠢⡀⠀⢙⡂⠉⠻⠦
⠀⠀⠀⢸⣿⣷⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢮⣄⠈⠛⠶⠄⠻⢶⡤⣮⡀
⠀⠀⠀⠘⣿⠟⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠈⠛⠦⠶⢦⡄⢶⣄⠈⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡦⣤⣀⠴⣤⠰⣇⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡄⠀⠀⠐⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠉⢳⡄⠀⢹⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⣿⢤⣾
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⢳⡆⠀⠘⣿⠁
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣤⣤⣄⣀⡀⠀⢤⣤⣶⣶⣶⣶⣶⣶⣤⠄⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠉
⠀⠀⠀⠙⠻⢿⣿⣿⣷⣦⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠀⠂⠀⠰⠀⠀⠈⠃⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠆⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⢿⠿⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⣆⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⡀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣤⣤⣴⣿⠂⠀⠀⠀⠀⠂⠀⢀⡈⠀⡉⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⠿⠿⡿⢻⠿⡿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡴⣀⣤⣤⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠑⣀⡁⠶⡉⣣⡜⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⣟⡄⡸⣾⠀⠀⠀⠐⠠⠀⡐⠂⡂⢢⡀⠀⠀⢀⠰⣄⠩⢤⠘⢣⣄⠹⣆⡙⣥⠚⡾⠋⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠚⠛⠘⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡈⢤⡈⢤⡀⠻⡤⠑⢢⡈⢵⠆⠓⠀⠈⠈⠀⠸⢡
⠀⠀⣤⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠱⠦⠉⠠⠚⠓⠂⠉⠀⠀⠀⠀⠀⠀⠉⠀⠖⠃
⠀⠀⠀⠀⠛⠃⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢠⡤⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⣀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣠⡤⠧⠀⢀⡤⠀⠀⠀⠀⠀⠀⡤⢀⣀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣀⡀⠛⠀⣠⡶⠀⠀⠀⠠⠇⠉⠉⠉⠉⠛⠉⠉⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁⢀⡤⠖⠀⠀⣀⣠⢠⣤⣤⠶⠶⠴⠁⠀⠄⠀⠀⠀⡄⠀⠀⠀⠀⣀⡀⠰⠁⠀⡀
⠀⠀⠀⠀⠀⢈⠛⠃⠀⠀⠀⠀⢲⣶⣾⡦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠘⠲⠆⠒⠊⠀⠀⠀⠀⠈⠉⠠⠈⠀⠁⠉⡗⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠓⠁⣀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠄⢤⠀⠀⠧⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠹⠁⠠⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠈⡦⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠰⠀⠘⠀⠀⠃⠀⠅⠀⠐⠀⠀⠀⠀⡶⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣐⢁⠺⠁
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠁⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠈⠀⠐⠀⠀⠀⢄⠀⠀⠀⠀⠀⠰⠀⠀⠀⢠⠤⠊
⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠠⢤⠄⠈⠀⠀⠀⠈⡀
⠀⡀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⢆⠴⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠉⠁⠉⠀⠀⠀⠂⠀⠠⠤⠄⠤⠔⠈
⠀⠀⠀⢀⡀⡧⠀⠴⠆⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⠀⠀⠀⢂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠙⠓⡼⠆⠀⣀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠐⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⠈⠁
⠀⠀⠀⠐⠒⢤⡄⠀⠈⠁⠀⠀⡀⠀⠛⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠈
⠀⠀⠀⢀⡀⠀⠀⠟⠀⠀⠴⠀⢁⢄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣚⠁
⠀⠀⠀⠀⠁⠀⠷⠀⢰⡀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠐⠆⠀⠀⠌⠀⠀⠀⠀⠀⡰⡀⠀⠀⠀⠀⠀⠀⠀⠒⠹⠀⠀⠀⠀⠀⡶⢠⡀⢒⠝⣇
⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠠⠈⠃⠀⠀⠀⣀⠀⠀⠀⠈⠇⡈⠕⠉⠀⠉⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⢁⠀⠃⠠⠄⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉
⣀⠀⠀⠀⠀⠀⣷⢰⠀⠀⢸⡀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣤⣤⡀
⣯⠛⠒⠒⠊⢰⣿⡸⡆⠀⠘⣇⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠁⡙⠷⣌⡀⠀⠉
⢸⡇⠀⠀⠀⢸⠸⠃⢻⣄⠀⢹⣆⠀⠀⠈⠶⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡇⠀⠀⠙⡆
⠈⢧⠀⠀⠀⣸⠀⢄⠀⣿⠀⠈⣯⠀⠀⠀⣼⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠈⡄⠀⠀⣇⠀⢠⡞
⠀⠀⡃⠀⠀⣿⠀⠘⣇⠸⣧⠀⠹⣆⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢰⡇⠀⠀⣿⠀⠈⢧⠀⡴⢤⣤⣄⣀
⠀⠀⢷⠀⠀⡿⠀⠀⢹⡆⠘⣷⠀⢹⡦⠀⣿⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⠀⠋⠀⠀⢠⠉⣠⠶⣞⣂⡀
⠀⠀⠘⢷⡾⠃⠀⠀⢸⡇⠀⣿⠀⢸⡀⠀⢿⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⠀⢸⠀⠀⠘⣠⡾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡏⠀⠸⣇⠀⢸⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⠁⠀⠀⢸⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠙⡆⠀⠘⠀⠈⠃⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠴⠚⠣⠀⠀⢼⣄⠀⠀⠣⣀
⠀⠀⠀⠀⣀⣀⡴⢟⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠳⣸⠆
⠀⠀⠀⣰⠀⠀⢀⡼⠁⢤⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠙⠓⠶⢤⣀⣀⣀⣀⣀⠀⢿
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠈⠛⠲⣝⡛⠛⠛⠛⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⢀⣀⣠⠞⠉⠁⠀⠀⠀⠘⢧⡀
⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⣄⡀
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠔⠁⡀⠙⢢⠀⠀⠀⠀⠀⢠⠃⣀⣤⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠇
⠀⠀⢨⠀⠠⠋⠓⠀⢣⠤⡄⢠⠒⣼⠘⠋⣤⡀⠀⢹⠀⠀⢀⡀⠤⢤⠐⠒⠒⣦⠀⠠⢠⣀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⡠⠀⠀⠀⠈
⠀⠀⢈⠀⢀⡾⠓⠀⢸⠀⣿⢨⠄⡟⠀⠠⣋⡉⠀⣼⠀⠀⡌⠀⡄⠀⠈⣦⠀⠀⠁⢄⠀⠀⡤⠃⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠘⡀⠀⣰⢦⡀⢸⠀⡇⠀⠇⡇⠀⠴⣫⡓⠀⣿⠀⠀⢰⠀⠈⢢⠂⠀⠑⡄⠁⠀⢣⠊⠈⢴⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡇⠀⡡⠶⠄⢸⡞⠀⠀⠈⣇⢀⣘⠥⢥⣀⣹⠀⠀⠀⠐⡄⠈⠑⣤⠊⠈⢢⡄⠈⠁⡠⡎⠀⠀⠰⠶⠦⠤⢤⣄⣀⡀⠀⠀⠀⠆
⠀⠀⢰⡗⢚⠃⠉⠉⠙⡇⠀⠀⠀⠸⡅⣠⢠⢠⡆⣸⠀⠀⠀⠃⠈⣦⠂⠀⢡⡀⠀⠈⢂⠈⠑⠇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠼⠤⡄
⠀⠀⠀⢿⣾⣼⣾⣇⣧⠇⠀⠀⠀⠀⠣⠽⠧⠼⠾⠟⠀⢀⣤⣧⡀⠀⠑⡤⠀⠉⢢⠄⠈⢣⣠⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣩⣀⣸⢧⣀⣀⣀⣀⣀⣀⠀⢠⣾⠉⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠄⠀⠀⢀⠀⠀⠀⠀⢽⠒⢻⡵⠊⠁⠀⠀⠉⠉⠉⠉⠛⣌⡿⠽⡇⠀⠀⠀⣴⣂⡤⢄⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠀⠀⠀⣜⣛⣉⣉⣉⣉⣛⣛⣒⣒⣒⡒⠒⠾⠷⠴⡇⠀⠀⣸⠇⣏⠑⡞⠉⣴⡂⢀⡀
⠀⠀⠀⢀⠐⠷⠶⠶⢤⣤⣤⣤⣤⣀⣤⠞⢀⣆⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⡏⠀⠀⣿⢠⢎⣩⠓⢠⠧⣸⠁⠲⢇⡴⢠⡄
⠀⠀⡀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠈⢆⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢠⠟⠲⠤⣍⠸⠁⠶⢥⣝⣉⠖⢀⡟
⠀⠀⢆⡀⠛⠛⠛⠛⠓⠒⠶⠤⠤⠄⠠⣶⣦⡄⠼⡄⠀⢻⢻⡟⠉⠁⠀⠀⠀⠀⠒⠒⠒⠶⢲⠋⠀⢰⡏⠘⠢⣤⣀⠛⢱⠆⣼⠈⡏⢘⡏
⠀⠀⠀⠈⠁⠒⠈⠉⠁⠀⠀⠀⠀⠀⠠⠤⠤⠀⠠⠔⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⢀⡴⠋⣤⠋⠰⢠⡍⢉⠓⠲⠤⣤⣁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⣟⡶⠒⠂⠒⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢤⣈⠑⠂⠠⠊⡨⠓⠦⢦⢨⠏
⠀⠀⠀⠀⡠⣶⣶⠯⠷⣻⣿⡿⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⠤⢔⢢⠆⠐⠒⠬⡈⠓⠒⠤⠧⣏⡐⠀⡌
⠀⠀⠠⠊⠙⢯⣿⣾⣻⣯⠟⠁⠀⠤⢀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠓⠌⡀⠴⠋⡀⠀⠀⠀⠈⠆⡀⠀⠀⠀⠙⠛
⠀⠐⠀⠀⠀⣀⠀⠉⠁⡠⢀⡾⡻⣷⣦⣄⣀⠀⠙⠺⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠠⠀⠀⣶⡟⣽⣷⡀⠈⠀⠈⠳⣕⣦⡟⠁⠈⠁⠉⠉⠁⠊⠷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠱⡀⣿⣿⣿⣿⢧⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⣀⠀⠀⡰⠃⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢄
⠀⠀⠑⣯⠉⠁⠀⠀⢧⠀⢦⡀⠀⠀⠀⠈⣉⠽⠪⣍⣀⠴⠊⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢌⠀⠀⠈⠻⠀⠀⠀⠈⢂
⠀⠀⠀⠙⠀⠀⠀⠀⠈⢧⠀⣈⠥⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⡀⠠⠄⠐⠂⠀⠀⠀⠀⠐⢤⠔⠁⠠⡀
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⠄⠀⠀⣀⡀⣀⣀⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊
⠀⠀⠀⠀⠈⢆⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠶⠤⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠈⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠠⠀⠀⣠⣤⣄⡀⢠⠔⠆⠀⣀⣀⠰⣇⠆⢰⠀⣦⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢰⣛⣯⣍⢹⣿⠷⠤⠞⠿⠷⠥⠤⠇⠀⠘⠶⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠈⠀⠀⠀⣰⠛⠂⢀⣄⣜⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠎⠛⠛⠛⠙⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⠀⠀⠀⢀⡀⡀⠀⣀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡄⠀⠀⠀⣧⣉⣭⣿⣟⣿⣉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⠉⣉⡉⣙⣻⠀⠀⡁⠀⠀⠀⠀⣁⠀⠀⠀⢰⣶⣶⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠈⠀⢰⠶⠓⠀⢰⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡖⣴⠾⣾⠏⣽⠐⣮⡀⣞⠚⠛⣧⣟⣶⡄⠀⠎⠉⠉⠉⠉⠉⠉⡆⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣇⣴⣖⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠚⣿⣇⣼⣄⠁⠈⠅⢈⡀⣁⠀⢄⡀⢀⣤⣤⣤⣤⣤⣤⣴⣧⣿⠷⣶⠀⠀⠀⣤⣀⣼⣯⣈⣉⣿⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣿⣿⡘⣿⣿⣿⣿⣿⣾⣿⣿⣤⣼⣿⣿⣿⣽⣽⣿⣿⣷⣤⣤⣄⣀⣠⣾⣥⣤⣿⣿⣭⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠀⠂⠀⠀⠀⠀⠀⡄⠀⢂⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠛⠀⢈⠉⠁⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⡇⠀⣀⣠⠀⠑⠀⠂⠀⠰⠂⠀⡀⣉⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈⠀⠈⠉⠉⠁⠈⠀⠉⠉⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠠⠤⠤⠀⠀⠀⠠⠀⠀⠀⠀⠤⠀⠠⠤⠀⠀⢠⣄⣀⣀⣀⠀⠀⠀⣀⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⡤⡠⣢⡪⢂⡠⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣰⡟⡱⣷⣫⠖⣡⣶⠥⠐⠀⠀⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡈⢦⠑⢌⠢⡦⢄
⠀⠀⣴⣶⣿⡱⣹⠟⡡⠞⠉⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡠⠀⠀⢳⡀⠈⡄⠱⡀⠳⡀⠱⡈⢆⢣
⠀⢸⠇⢸⡗⠡⢋⡼⠁⠀⢤⣤⠀⠀⠀⣴⣅⠀⠀⠀⠀⠉⠉⠀⠀⠹⣆⢸⠀⠱⡀⠙⢄⠱⠈⡆⠇
⠀⢸⣾⠿⠒⠀⠊⠁⠀⠘⠉⠛⠀⠀⠀⠈⠀⠠⠀⠀⠀⠘⠛⠀⠀⠀⠸⣿⠂⠀⠙⢄⡀⠂⡁⠰
⠀⠘⢼⠀⢄⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠀⠀⠀⠀⣿⠀⢀⢰⠀⠉⠂⣤⠤⡶⡄
⠀⠀⠀⠃⠀⠐⠃⠀⠀⠀⠀⠀⢀⣠⠤⢤⠀⠃⠆⠀⠘⡆⠀⠀⠀⠀⠀⡿⠌⠁⠂⠀⠑⠓⠀⠀⢉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠞⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⠄⠀⡄⢸⣄⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠁⠀⠀⠀⢸⠀⡿⠀⡰⢀⢿⡏⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢸⣧⠜⣠⢫⠛⠀⠆⠸
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠘⠁⠔⠱⢁⡀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠱⣦⣄⡀⠀⢠⣄⣿⣿⣏⣠⠄⠀⢀⣠⣶⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢻⣿⣧⣴⡄⢻⣿⣿⣿⠏⣦⣰⣿⣿⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⣿⣿⣽⣿⣿⣏⣀⡀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠰⠾⠿⠟⠻⠛⠉⠈⠙⠟⠻⠿⠿⠦⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⢻⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡏⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⢸⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠶⠦⠛⠴⠚⠰⠴⢦⠀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⠀⠀⠼⠷⢶⣶⣷⣼⣦⣤⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⢤⡄
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣄⠀⠀⠀⠂⠠⠰⣄⢢⠈⡙⢿⣵⡚⠯⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢤⠄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢤⡿⠋⣀⣬⠁⠀⠀⠀⢟⡃⠄⠐⠛⠣⣷⣹⠀⣤⣡⣂⣠⣂⣨⢉⡛⢷⡶⢶⣦⠄⠀⠀⠘⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠻⡯⠁⣺⢻⠃⠀⠀⠀⠀⣼⢷⡛⠦⠀⡨⠤⠙⠃⢌⠂⠀⠌⠑⠍⢃⣴⣶⠘⢣⡕⠂⠀⠀⠀⠘⢣⡄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⡀⢰⠄⠙⣄⠠⣈⣀⣤⣤⣤⣴⣿⢳⣁⡄⢠⣀⠤⠒⠢⠀⢠⡀⠠⢬⠹⣤⠁⣇⣼⠍⠀⢀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠠⠑⣸⣵⣿⣿⢯⣻⠿⠿⣿⣿⣿⡵⣾⣇⡤⠒⠂⠆⢤⢈⡷⣦⠄⡘⠿⣾⠓⡉⣤⡄⡄⠀⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣚⡠⠐⢄⠈⢻⣧⣬⣼⣼⣶⡿⢷⣦⣬⡝⢓⣯⡔⠄⠀⠆⡘⢀⠱⣿⠻⡿⡄⠟⣳⣯⣻⣗⣥⠀⠀⠀⠰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⡞⠲⠀⠑⢺⣞⠯⢿⣻⡏⣭⣻⣿⣿⣟⣽⡞⣇⡴⣬⠁⠐⡄⢺⠿⡟⢀⣺⢿⠀⢹⣟⠓⠋⠀⠀⣀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢿⠲⠆⠀⠐⣯⣻⣵⣷⢋⣼⣟⢸⡿⢱⡏⣱⣿⠿⠁⠃⢀⣸⠃⡓⣆⢘⣹⡧⡄⢨⣧⣠⣀⠴⠴⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⢸⠳⡆⠀⠀⠙⣿⡿⠄⢀⣿⣿⡟⣣⣿⣀⡿⡁⠌⠈⣤⡮⢾⡎⡁⠛⣄⠨⡇⠀⣼⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠈⠣⢤⡀⠀⠀⠘⢛⢀⠿⣿⡿⣼⢏⣽⣿⠥⠬⢰⡞⠒⠿⡁⡿⢶⣌⠺⡔⢠⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠖⠃⠰⣿⣿⣵⠟⠁⠀⠀⠀⠀⠂⠈⠉⣽⣬⡏⢼⣴⣿⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠀⠀⠠⠁⠀⠀⠀⠐⠉⠘⠻⡟⣿⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠔⠀⠀⠀⠀⠀⠀⠀⠀⠂⡂⠀⠰⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠑⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡉⠀⣸⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠑⠕⠙⣬⣇⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠶⡄⠀⠀⠀⠀⠀⠀⠀⠈⢛⣷⣄⠀⠘⣞⡟⠀⡴⠮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⣇⣄⣑⣒⣒⣒⣒⣒⣶⣥⣬⡵⠢⣾⣯⣽⣆⡇⠀⣅⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠅⠘⠒⠣⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠌⠚⠥⠤⠤⠤⠑⠚⠤⠵⡄
⠀⠀⠀⠀⠀⠀⠀⢧⠀⡠⠠⠤⠠⠄⠀⠠⠤⠄⠤⠤⠤⠀⡀⠠⠠⠤⠤⠤⠤⠤⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⢀⣀⡁⠀⡀⠀⢀⠀⡀⣀⠀⠀⣀⠀⠀⣀⢀⠀⢀⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠒⠢⢸⢒⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡇⠀⠀⠰⠒⢢⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠉⠀⠀⠀⠀⠀⢣⢱⠀⠀⠀⣧⡀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠀⠀⠀⠀⢀⡴⠃⣜⡔⠲⠀⡗⠷⠂⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠀⠤⠀⡔⠁⠀⠀⠻⠀⠰⡀⡇⠈⠁⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⢸⠀⠚⠀⢇⠀⠀⠀⠀⠀⢀⠇⡇⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠤⠜⢸⠀⠀⣠⣊⣑⣖⣉⣑⣖⣉⣢⡇⠀⢀⠐⠄⠚⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⠉⠉⠉⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣾⣶⣶⣶⣶⣶⣖⣲⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣶⣒⣶⣶⣶⣶⣶⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⣟⡿⣿⣿⠿⠿⠛⠛⠻⠟⠛⠛⠛⠛⠻⠿⠻⢿⢟⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠊⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠏⠁⠈⡄⠉⢳⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⢐⡀⠀⠀⣤⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡏⠀⠀⢀⣠⣀⡈⢷⡄⠀⠐⠀⠀⣰⠃⣀⣀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡃⢉⣿⠀⠙⠷⠶⠶⠚⣡⣏⡹⠈⠛⣦⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⢸⣙⣓⣿⠀⠀⠀⠀⠀⠐⣯⠢⡅⣢⠜⣸⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⢺⡇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢷⡤⡤⠴⠋⠀⠠⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣇⡀⢸⣒⣔⣶⣘⣆⡠⡤⠀⠀⠀⠀⣰⣶⣶⣼⡆⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⣀⣀⣒⣻⣿⣿⣻⣶⣒⣦⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠉⠉⠉⠹⠿⣛⢛⡛⠛⢛⣛⡛⣛⣛⣛⣛⣛⠛⠛⢻⠋⠋⠋⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡄⡇⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣉⣉⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⡉⠉⠉⢁⣀⣀⣀⣈⣉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠇
⠀⠀⠀⠀⠐⣶⡀⠀⣹⣦⠀⣴⠁⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡅
⠀⠀⠀⠀⠀⢻⣿⡷⣾⠗⣿⡿⡄⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠸⣿⣷⢸⢹⣿⠃⣇⣀⣀⣀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠀⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣮⣼⣿⠀⠠⠀⠀⡆⢰⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡀⡆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢻⣠⣤⣦⣤⣥⡼⠿⠻⠛⠻⢳⣶⣶⣶⣶⣶⣶⠶⠿⠾⠿⠿⢦⣷⣶⣦⣬⡄
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢻⣯⣤⣤⣤⣤⣤⠜⠛⠛⠛⠛⠻⠛⢤⣤⣤⣤⣤⣤⡿⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢟⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡐⠀⠀⠀⡠⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⡀⡀⠃
⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⢠⡄⡀⠀⠀⠀⠀⡠⠀⠀⠀⣈⠁⡄⠀⠀⠀⠘⠠⠈⠀⠀⠐⠿⢩⣝⡢
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⡀⠁⠀⠠⠄⠏⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠏⠮⣶⠀⢹⠅
⠀⠀⠀⠀⠀⠠⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⠢⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡅⡆⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡜⠠⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⠀⠀⠀⠀⡀⠀⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠁⠀⠰⠆⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣶⣈⣃⡀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⠀⠀⠀⠀⠀⢀⠛⠻⠟⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐
⠀⠀⠀⢡⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⡀
⠀⠀⠉⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⠀⠀⠔⠀⠀⠀⠀⠀⣠⢐⠐⠀⠄⠀⠀⡠⠤⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠁⠀⢀⠂
⠀⠀⠂⡙⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢐⠄⠒⣾⠪⠀⠀⠀⠀⠀⡀⠇⠀⠙⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡂⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⡀⣀⣀⡤⠤⠤⠤⠤⠤⠴⠁⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠹⡀⠀⠀⠀⠀⠀⠀⣔⠙⡀⠀⡘⡅⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⢀⡤⠟⠒⠈⠉⠁⠒⠲⡄⠂⠐⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠈⢍⠑⠂⠤⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠀⠄⡀⢀⠙⢄⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⡰⠁⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠈⢣⠀⠀⠀⠠⠈⠳⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠈⣹⣄⠀⠀⠀⠀⠀⣱⡄⠀⠈⢄⣁⡇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠒⠒⢄⠀⠸⠛⠒⠀⣠⡾⠃⠈⡗⠓⠊⠁⠸⠦⡄⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠣⡀⢀⠜⠀⠀⠀⣠⡾⠋⠀⠀⢰⠀⠀⠀⠀⠀⣀⠦⠒⠉⠀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠁⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⣠⠗⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⠀⠀⢀⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⣀⣀⣠⣤⠶⠛⠉⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠊⠉⠀⠀⠀⠀⣀⡤⠖⠉⠀⠈⡆
⠀⠀⠀⠀⠀⠀⣠⡴⠚⣩⡿⣿⠉⠣⠤⣀⣀⣀⣀⣀⡤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉⠁⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⢸⣇⣤⠞⡿⠁⣿⡇⠀⠦⠄⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⣀⡠⠔⠚⣇
⠀⠀⠀⠀⠀⠀⠉⠀⢸⡇⣰⢿⡇⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠉⠁⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⢸⡷⠀⠉⠁⠀⠀⠀⠠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⠇⢣⡀⠀⠙⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⣄⠠⣠⠀⠈⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠙⠣⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠰⠖⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣖⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⡶⠖⠀⢠⡤⠀⠐⠖⡖⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢸⠱⡄⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡌⠀⠑⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣄⠀⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡈⠧⡏⢇⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⡆⠀⠀⠀⠈⠀⡀⠀⠀⠀⠈⠀⠉⢂⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠁⠀⠀⢀⠘⢤⡀⠀⠀⠁⠂⠀⠀⠐⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠂⠀⡀⠀⠀⠀⠀⠁⠈⠀⠠⠀⠀⠀⠈⠂⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢂⢸⠖⠀⠐⠶⠂⠀⠈⠀⠂⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠂
⠀⠀⠀⠀⠐⠀⠺⠇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠠⠈⠁⠠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⢀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠒⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃
⠀⠀⠈⡀⢁⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠃
⠀⠀⡀⠁⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢐⠂⠀⣋⠀⢠⢠⠀⠀⠠⠀⠀⠐⠀⠀⠸⠣⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠉⠀⠀⠈⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠒⠄⡖
⠀⠀⠀⠀⢀⠚⠀⠂⠃⠀⠳⡤⠠⠀⡀⠀⠀⠈⠀⠀⠀⢠⡀⠀⠀⠀⠀⡠⠤⠠⠀⠀⣶⠂⠀⠁⠀⠒⠒⠀⡄⣗
⠀⠀⠠⣀⠄⠰⠀⠀⣀⠁⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⠀⠄⠁⢀⠀⠀⢀⢀⠀⠄⠀⠀⠉⢠⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠴
⠀⡄⠀⠀⠠⠀⡀⠀⠀⠂⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⠀⠀⠠⠀⠖⠀⠀⠀⠹⠀⠀⠀⠀⠀⡀⠀⢀⡠⢀
⠀⠀⠀⠀⠀⠐⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⠄⢀⡀⣠⡄⠀⡀⠸⠀⠰
⠀⢀⢐⡦⠀⢂⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣤⣄⣀⣁⣀⠀⠁⠀⠰
⢀⡀⠈⠀⠀⠾⠿⠿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⣩⢤⠀⢲⡴⠆⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠃⠒⠀⠠⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⢤⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠖⠶⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠈⠠⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣤⣤⣄⠀⠐⠀⠐⠀⠀⠐
⠠⡠⠀⠀⡀⠀⠀⠠⠠⣀⠠⠀⠂⠀⠀⣄⡄⠀⡤⡤⠀⢰⠄⠀⠘⠃⣀⠀⠀⠀⠀⠒⠢⠀⢲⠂⢀⡀⠈⠀⠀⠰⠁⠍⠨

⠙⢹⡏⠉⠏⠀⠀⠀⠀⠀⠙⠁⢰⣬⣿⣿⣿⣿⢿⣿⣻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⠃⠊⠉
⠀⣼⠁⠀⢀⡀⠀⠀⠀⠀⠀⢠⡼⠗⣹⣿⣿⣿⣿⣿⣻⣻⣧⣾⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣯⣹⣿⣏⠳⠰⠆
⠀⢹⡀⠀⠘⠙⢿⡶⠄⣴⢰⣯⣦⣿⣋⡼⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⠟⢶⣀⠁
⠀⠈⠁⢀⣶⣶⡏⢿⣻⢋⣉⣙⣿⣯⣿⣠⣾⣿⣿⣿⣷⣿⣇⣀⠙⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠂⣿⡍⡆
⠀⠀⠀⠙⢣⡙⠗⢽⡉⠇⠈⢻⠟⠛⣟⡿⡟⠛⠛⠛⠻⣿⢿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠉⠈⠹⣀⠑
⢀⠀⣍⣽⡇⠀⠀⢠⣉⣓⠶⣶⡶⣟⡽⡀⢀⠀⠀⠀⣤⣧⣀⣉⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠦⣶⣾⡍⠤
⠀⠛⣿⣿⣿⣿⣧⡶⢦⠋⠹⣿⣿⣿⢀⣁⠀⠒⠂⠙⠛⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⢉⣸⡿⠉⠐⡀
⠀⠀⣇⢹⣷⣿⣿⣷⡆⢳⣿⣿⣷⣿⢟⡉⠤⣿⣖⣀⣀⣀⣬⣷⣿⣿⣿⢿⠚⣻⣿⣾⣿⣿⣯⣼⡷⡆⠀⠀⠁
⠀⡄⢽⢹⡇⢻⣿⣿⡑⠀⠑⡻⣿⣿⡟⢱⢴⢾⠿⠧⠀⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠷⠃⡅
⠀⠻⣿⢺⡧⣞⣿⣿⣅⣤⣶⣦⣿⣿⣻⣿⣻⣤⣴⣶⣶⣿⣿⣿⡿⢧⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⡀⡇
⠀⠀⢿⣿⣇⠈⠋⣿⡛⢋⣀⢀⡩⣍⠛⢟⣛⣿⣿⣥⣿⣿⣿⣋⣵⠾⠟⠛⡉⣛⣽⣿⣿⣿⣿⣽⠃
⠀⠀⠂⣼⣿⣿⢇⣾⣇⠀⠙⠻⠻⠛⠾⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠁⠲⠭⢍⠁⠌⠉⢉⣻⣭⣀⠄
⠀⠀⠀⠈⠋⠁⠀⠁⠈⠀⠄⠀⠀⠀⠀⠁⢠⣿⣿⣿⡿⠀⠀⢀⠀⠀⢀⡈⡉⠛⣏⣠⣄⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠙⠷⠧⢾⣯⣻⣿⡟⣷⢶⡄⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠊⠛⢋⣛⡯⣓⣹⣦⣄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠉⠛⠛⠻⣯⠁⠛⠃

⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⠀⠂
⣀⡀⠀⠈⠀⣀⡰⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⡀
⠀⠠⢀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⠤⠖⠀⢤⠀⠀⠀⠀⠀⡄⠤⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⢀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪
⠄⠄⠂⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢿⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⡐⠀⠀⠀⠀⠀⢄⢰⢠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢄⡋⠠
⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠰⠂⢀⣀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⣷⠈⠻⠀⠀⠀⠤⠀⣁⠤⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠴⠀⠀⠱⠀⠀⠹⠆⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠁⠀⠀⠊
⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠁⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠐

⣿⡄⣀⣤⣀⣴⣶⠲⣶⣦⡄⠀⠀⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣏⠉⠙⠛⠃⠀⠿⣿⠿⠿⢿⣿⣷⣦⣾
⠉⣹⠉⠛⣿⣿⡁⠀⠀⠉⠁⠀⠰⢿⡿⠿⠟⠋⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⣀⡀⢀⣀⡀⠰⢿⠉⠉⠉
⢰⣿⣇⡀⠘⠿⣿⣶⣶⣤⡀⠀⠀⢠⣤⣦⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡛⠛⠛⠉⠙⠛⠛⠁⣼⣿⣿⣿⣿⡟
⢸⣿⣿⣿⣦⡀⠀⠉⠛⠿⠋⠀⢀⣾⣿⣿⣦⡀⠀⠀⢾⣿⣿⡟⠙⣿⣿⣦⡀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⣀⣀⣤⡀
⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⠀⠉⢉⡇⠀⣾⣿⣿⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣴⣿⣿⡀⠹⠿⢿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⣿⣿⣷⠆
⠀⠈⠛⠻⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⠛⠿⠃⠠⣾⣿⣿⣿⣷⣤⣄⠀⠙⣶⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀
⣿⣷⡀⠀⠀⠀⠀⠀⠚⢛⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣧⠀⠙⢻⣯⣤⣴⣿⠄⠀⢀⣴⣶⣦⡀⠀⠀⢻
⣿⣿⣿⣶⣶⠀⠀⠀⠀⠹⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣶⣿⠟⠛⢿⣿⣿⠀⢠⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⡄
⠉⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠻⣿⣯⣀⣾⣴⣶⣄⠈⠻⠉⠁⠀⠀⠘⣿⠟⠀⢸⣿⣿⣿⣿⣿⣿⣶⣿⣿⠟⠙⠻⣿⣄
⠀⠙⠋⠁⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⣿⣿⣿⣿⡧⠀⠀⣀⠀⠀⠀⣁⣤⡀⠘⠿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠶⡠⠶⢿⣿⠟⠋⠁⣠⣶⠀⠀⠀⠀⠈⠉
⣤⣤⣴⣶⡖⢀⣼⣿⣿⣿⣿⣿⡟⠉⠛⠉⠁⠀⠀⣠⣶⣿⣿⣿⡿⠿⣿⡟⠁⠸⣿⣤⣤⣀⠀⢠⣿⣿⣿⣶⣶⣤⠀⠀⢀
⣿⠃⢻⣿⣿⣿⠿⠿⢿⣿⣿⣿⣷⣦⣴⣶⣦⣀⣤⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⢸
⡟⢠⣾⣿⣿⣷⣤⣀⠀⠀⠙⠿⠿⠿⠿⠿⠟⠛⠛⢉⣨⣿⣿⡇⣠⣾⣿⣿⣶⣤⡄⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⢀⣤⣿
⠁⠀⠛⠛⢛⣿⠿⡿⠟⠂⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣁⠀⠀⠀⠀⠀⣰⣿⡿⠋⢀⣴⣾⣿⣿⠋
⣿⠀⠀⠀⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠃⠀⠀⠀⠀⠸⠿⠛⠋⣿⣿⣿⣷⣄⠀⠀⠀⠋⠁⢀⣤⣾⡿⠿⠿⠇⢀
⣿⣄⠀⠀⠙⢿⣿⣿⣿⡄⠀⠀⢠⣶⠿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡄⠀⠀⣠⣴⣿⣿⣿⣿⡶⠀⠀⠘
⣿⣿⣿⣷⡄⠈⢿⣿⡟⠁⢀⣠⡀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣀⣤⣄⠀⠈⢿⣿⣿⣿⡟⠁⠀⠸⣿⣿⣿⣿⣿⡏⠀⣰⣤⡘
⠹⣿⣿⣿⣿⣀⠀⢉⣀⣴⣿⡿⣷⣦⡀⠀⠀⠀⠘⣿⡟⠻⣿⣿⣿⣧⣀⣨⣿⣿⠏⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣿⡟⠿
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠘⢿⣿⡆⢀⣤⣾⣿⣿⠆⠈⠉⠈⢹⣿⣿⣿⡇⠀⠀⠀⠠⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣦
⣾⡇⠀⠛⠉⠉⢻⣿⣿⠿⠃⠀⢰⣿⣿⠀⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠀⠀⢹⣿⣿⡟⠁⠀⠈⢿⣿⣿
⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⠈⠻⣿⠏⠀⠀⠀⠀⠀⢶⣿⣿⣿⣷⣦⣤⣀⡀⠈⢻⣿⣦⣴⣷⣄⠀⠛⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⠶⠤⢤⣐⣶⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⠿⠛⠒⠀⠈⠁⠀⠈⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⡟⠓⠒⢄⠀⠀⠉⣉⣙⢲⡀⠀⠀⠀⠀⠀⠀⣠⡖⠉⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡎⠀⠠⣄⠈⡋⠉⠁⠀⠀⠀⠀⠀⡎⠁⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⡠⢤⡂⠀⢸⠇⠀⡸⢡⠖⠒⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠹⣜⢳⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⡿⠂⠀⠘⣆⣸⠀⢠⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢣⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⡇⠀⠀⠀⠘⡇⠀⣸⣽⠀⠀⠀⠀⢠⠞⠉⠉⠒⢦⢀⡀⠤⠄⢤⣠⠎⠙⣦⣀⠀⢉⣉⣙⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢷⠀⠀⠀⢀⠇⠀⣯⠿⣂⡭⢑⡆⠀⠓⠤⠤⣶⠞⠉⠀⢀⡠⠊⠀⢀⡜⠁⠈⠀⠀⠉⢢⡉
⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠊⠉⣠⠌⠲⣤⡤⠊⠀⠀⠉⠊⣉⣠⠚⠉⠉⠉⠀⠘⣇⣀⡤⠖⠋⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢧
⣠⠤⠒⠂⠉⠁⠀⠀⠀⠀⠀⣠⠞⡹⠀⠀⠀⠀⠀⠀⠀⡎⠁⠈⠉⠉⠓⠒⠒⠒⡆⠀⠀⠀⠀⠀⠀⠘⢫⡭⠭⢑⠆⠀⠀⠀⠀⡇
⣇⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⡗⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠉⠀⠀⠀⠀⡜
⠈⠑⠒⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣔⠲⡖⢇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢏⠳⡘⠜⣄⠀⠀⠀⠀⢳⢄⣀⠠⡾⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⠀⠈⠿⠳⢤⡀⠘⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⠀⠀⠀⠀⠀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢦⠀⠀⣺⡓⠷⠀⢇⡠⣄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣈⡆⠀⠀⢸⡏⣰⠆⣰⠃⠀⠀⠀⠀⠀⠀⣬⢠⠋⠀⠙⣆⠀⠘⣠⠞⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠙⠃⠀⢀⡸⠟⠁⢺⠀⠀⠀⠀⠀⠀⠀⢀⠇⡜⠀⠀⣜⣁⣳⣞⠁⢠⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⣹⢀⣀⡤⠔⠚⠁⠀⣠⢶⡅⠀⠀⠀⠀⠀⠀⠀⡜⢸⠁⣠⠴⢊⠉⠁⠈⠓⠃⠀⢤⡉⠦⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣹⠏⠈⣆⠀⢀⠴⠊⠀⡀⠃⠀⠀⠀⠀⠀⠀⢰⠳⠟⠉⠀⠔⢺⡀⢆⠀⠀⠀⠀⠀⢻⠢⡈⠢⡀⠀⠠⡇⡇⣰⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠞⡗⠀⠀⠸⠖⠁⢰⠀⠀⢡⠶⢤⠀⠀⠀⠀⠀⠸⠒⠉⠁⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⠆⠈⠢⡘⢦⣸⢇⣥⢃⠜⣠
⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⣀⣸⡄⠛⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⡏⠙⠈⠁⢙⣊⠴
⠀⠀⠀⠀⠀⠀⢣⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⢿⣀⡠⠖⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠶⠦⢤⣤⣤⣤⣤⣤⡞⣠⠖⡡⠚⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⡗⢲⠶⠦⠶⠖⠒⠒⢛⠇⡎⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⠉⡙⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢇⢸⠀⠀⠀⠀⠈⠉⠚⠈⠁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠠⣸⣄⣇⠀⠀⠀⠀⣀⣀⠴⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡖⢲⣦⠄⠐⠃
⠀⠀⠀⠀⠀⢀⣀⣀⣬⣀⠈⠙⡛⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣀⣀⣀⣀⣘⡹⡀⢫⢡
⠀⠀⠀⠀⢎⣁⣀⡈⠿⠓⠋⠉⠉⠒⠢⠤⠤⠤⣤⣤⣴⠾⠓⠒⠢⢄⠀⠀⠀⠀⠙⠠⡤⠒⠒⠚⠛⠛⠓⢷⡒⠇⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠐⡆⠀⠀⠀⠀⢀⣰⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢡⠎⠀⠀⠀⠀⠀⣾⡏⢸⠁⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠗⠁⠀⠀⠀⠀⠀⢸⣿⢀⠇⢠⣾⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠀⣠⣤⡄⠀⠀⠀⣾⣿⡞⠀⣾⢇⠎⠀⠀⢀⣀⠀⡠⠚⠉⢀⡤⠖⠂⣩⠟⡿
⢀⣤⣄⡠⡀⠀⠀⠁⢹⣿⠞⣴⣿⠁⣀⡀⠀⢻⣿⠀⢸⣿⠏⠀⠀⡆⠙⣿⡿⠀⠀⡴⠃⠀⣠⠞⠁⢠⡇
⢠⠿⠣⣅⠓⢰⣀⣀⠈⠉⢰⣿⠇⣼⡏⡼⢀⡞⠻⡤⢾⣿⠀⠀⡼⢧⣠⡿⠃⠀⠀⣳⢶⠊⠀⠀⢀⡜
⠈⢹⠓⠈⠳⢌⡁⢠⡇⠀⢸⣿⠀⣿⣏⣠⠊⠀⠀⠀⢸⣿⣠⠞⠁⠀⠀⠀⠀⠠⠊⠁⠀⠑⠒⠛⣉
⠀⠀⢡⡏⠀⣨⣹⠲⢍⡀⠀⠉⠉⠈⠉⠁⢠⣤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒⡏⠁⠀⠳⣤⡀
⠀⠀⠀⠉⠛⠁⠁⠀⠀⢉⣀⣀⣀⣀⣀⣀⣿⡏⠀⠀⣀⣀⡤⠤⠈⠁⢠⣤⠊⢹⣿⠁⣿⡇⠀⠈⠒⠒⠋⠺⠃
⠀⢠⣶⠛⢢⡀⠀⠀⠐⠉⠀⠀⠈⠉⠉⣻⣿⠛⠋⠉⠉⠁⠀⠀⠀⠀⣾⡏⠀⣼⡇⢰⣿⠃⠀⣀⡀
⢠⣿⢁⣴⣿⠁⠀⣰⣦⠀⠀⣴⡆⠀⠀⣿⡏⠀⢀⣾⠂⠀⣠⡀⠀⠀⡿⠃⠀⠈⠀⢸⣿⠀⠀⣿⡟⠉⢉⣷
⠘⠟⠋⣽⡟⠀⣰⣿⠇⢀⣼⣿⠇⠀⡰⢻⣷⣠⢾⣿⠀⢰⣿⡇⠀⡼⠁⠀⠀⠀⠀⢸⣿⠀⢸⣿⠇⠀⣰⣿
⠀⠀⠀⢿⣇⡼⠉⢿⠴⠋⣼⣿⠀⡼⠁⠀⠉⠀⠘⢿⡴⠚⣿⠀⡼⠃⠀⠀⠀⠀⠀⠐⢿⣀⠼⠟⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠴

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⢀⠀⠀⠀⠀⢀⡀⢀⠀⠀⢀⡀⡀⠀⠀⢀⠀⠀⡀⡀
⡇⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⢭⡥⠬⢤⣬⡤⠤⠤⠥⠤⠤⠥⠬⠤⠤⠬⠤⠤⠬⡄⠁⠁
⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠁
⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢠⣀⡀⠀⠀⡇⠀⣆⣄⠂⠀⠀⠂
⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⣸⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⢸⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡇
⡇⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒⠂
⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⢸⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠙⠉⠉⠉⠛⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠙⠉⠉⠉⠉⠋⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⣯⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠘⣧⠀⠀⠀⠀⡇⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣦⠀⠈⢧⡀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⠚⠁⠀⠀⠈⢳⣄⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣤⣴⠆⠀⠀⠰⣶⣏⢉⣷⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢰⠟⢱⡆⠙⣯⡉⠀⡿⠀⣿⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⢸⠏⣧⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠈⠳⠛⠁⠀⠈⣻⣤⣟⠀⣿⠀⠀⠀⣼⠹⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣻⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡟⢻⣆⣀⣄⠀⠀⠀⠀⢠⣾⠉⣹⣽⠷⣿⠀⠀⠀⣟⠀⣿
⠀⠀⠀⠀⠀⣤⣬⣽⣥⣤⡀⠘⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⣩⣟⣥⡀⠀⠀⠀⠈⠻⣏⠉⣿⠀⣿⠀⠀⣤⣬⣯⣥⣤
⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⠀⢰⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⢯⣀⣿⠀⠀⠀⢠⡴⢬⡻⣯⡀⣿⠀⠀⡇⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⢸⡇⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⢶⣞⣉⣀⣀⣀⣀⣩⣤⠶⠶⣤⣼⣦⣼⣃⣀⣉⣻⡶⠴⡇⠀⠀⠀⣿
⠀⠀⣠⠶⠶⡏⠀⠀⠀⢸⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠶⣇⠀⠀⢸⠷⠾⠿⠿⠿⠿⠿⠶⠶⡇⠀⠀⠀⢹⠶⠶⣄
⣠⡞⣁⡀⣀⣙⠂⣛⣛⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣛⣙⣛⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣛⣓⡋⢀⣀⣈⣳
⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣟
⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹
⠀⠙⡟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠛⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡟⠙⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⣿⠋
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⢛⡟⠋⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⠀⠈⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⡟⠋
⠀⠀⠀⢻⠀⢀⡴⠛⠛⠻⢦⡀⢠⡿⠒⣶⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢀⡴⠛⠛⠳⣦⠀⠘⡇
⠀⠀⠀⣿⠀⢸⠁⠀⠀⠀⠘⡇⢸⣇⣀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠻⣶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢰⡏⠉⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠶⢶⡄⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢠⡟⠀⢀⣀⣀⡀⠀⠀⣿⡀⢸⡇⠀⣛⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣾⢁⡴⠋⢉⡾⠛⠃⠀⣿⠉⢹⡏⠉⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⢰⡇⣾⠁⠀⠈⣷⠀⠀⠀⣿⠀⠸⣧⣤⣿⠀⣿⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⣠⣤⣼⡇⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠈⢷⣏⠀⠀⠀⠸⣇⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⠘⡇
⠀⠀⠀⣼⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢹⡄⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢀⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠙⠛⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢺⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⠸⣆⡀⠀⢀⣼⠃⠸⡇⠀⠀⠀⢸⡇⢀⣴⠖⠛⠻⢦⣄⣠⠶⠛⠛⠶⣄⣠⡴⠞⠛⠲⣦⡀⣿⠀⠀⠀⠀⣿⠀⠹⣄⡀⠀⢀⣼⠃⢸⡇
⠀⠀⠀⣿⠀⠀⠈⠙⠛⠋⠁⠀⢐⡇⠀⠀⠀⢸⣇⣼⣁⣀⣀⣀⣀⣹⣇⣀⣀⣀⣀⣸⣏⣀⣀⣀⣀⣈⣧⣿⠀⠀⠀⠀⣿⠀⠀⠈⠙⠛⠋⠁⠀⢘⡇
⠀⠀⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣤⣤⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣶⣿⣿⣾⣿⣾⣿⣾⣿⣿⣿⣾⣶⣶⣶⣶⣷⣤
⣿⣿⣿⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠦⢤⣤⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠄⠚⠛⠁⠴⢶⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠈⠁⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠊⠀⠀⢠⠔⠀⠀⠀⠀⠒⠒⠄⠀⠀⠉⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠈⢳⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠘⡃⢤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠟⠀⢱⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠀⠀⠀⢸⠂⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⠀⠀⣠⠄⠀⠀⠈⢧⡀⠀⠁⠒⠠⠤⠤⠤⠤⠀⠐⠂⠉⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⡴⠖⠁⠀⠀⠀⠀⠀⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⠶⢶⠶⠶⠖⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠛⠛⠛⠉⠛⠋⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡂⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢣⡈⠙⠳⢤⣄⣀⡀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠤⠁⠀⠀⠀⠀⠇⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠱⣀⠀⠀⠀⠉⠉⠐⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⡤⠊⠀⠀⠀⠀⠀⡀⠀⠟⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠃⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠷
⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢾⡄⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⡆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠀⢰⡆⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠃⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠃⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠤⠇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣤⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠠
⡼⠓⠒⠒⠲⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠂
⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠉⠛⠛⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣴⣦⣴⣶⣶⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠿⡄⠀⢀⢀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢠⣤⣤⡤⣠⡄⠀⠘⠃⠀⠀⡁⠀⢀⠀⠀⠺⠿⠂⠀⢀⣄⣀⣄⡀⠀⠀⠀⠀⡄⠀⠈⠀⢠⡄
⠀⠀⠸⢿⣿⡽⠿⠓⠀⠀⠀⠀⠐⠛⠉⠙⠋⢢⡀⠀⠀⢀⣾⣶⣶⣾⣿⡆⠀⠀⠜⠱⢀⠀⢰⡵⡰
⠀⠀⠀⠀⠁⢻⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠄⠀⠈⠉⣿⣿⣿⣿⡆⠀⠀⠓⡁⢺⠦⠚⠧⠘
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢠⣤⠤⠀⠀⠁⠀⠀⠀⠀⠙⡿⠀⠘⠇⠀⠀⠀⠈⠀⠀⠀⣤⣄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⢠⣻⣼⣃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⣏⠋⠀⠀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡾⢀⣀⣀⢷⠀⠀⠀⡀⣤⢄⡐⠊⠉⡉⢿⠄⠀⢀⣄⣠⣤⣴⠀⢀
⠀⠀⢰⡧⠛⢡⢤⠘⠿⣧⠀⠘⢻⣯⣿⣯⡼⠃⠀⠰⠡⢷⣓⢀⣀⠀⠓⠋⠀⠀⠉⢻⣿⡿⣯⠔⠉
⠀⠀⠘⡧⣴⠚⢀⣌⣦⣾⠀⠀⢰⠍⠋⢹⠀⠀⠀⠀⠀⠩⠉⠉⠀⠀⠀⠀⡀⠀⠀⠐⠛⢻⠞⠇
⠀⠀⠀⠑⠈⠓⠿⠿⠛⠁⠀⠀⠐⠶⠿⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠴⠶⠶⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⣶⡶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⣦⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⣤⠾⠋⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠈⠻⣿⣍⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⣩⡽⠛⠁⠀⠀⠀⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠻⣦⡀⠀⠀⠀⠙⠻⢶⣤⣠⡤⠶⠦⣤⣠⠖⠲⠶⣤⣴⠞⠉⢀⡄⠀⢀⣠⠶⠛⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠉⠻⣄⠀⠀⠀⠀⢀⣀⣉⡙⣷⡦⠄⠙⢷⡄⠀⠀⠙⢷⣶⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠙⠛⢿⡄⠀⠀⠻⠀⠀⠀⠀⠹⣦⣤⠴⠆⠀⠀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣬⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣠⣤⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣘⣷⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠉⠉⢻⡟⠛⢻⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣾⠃⢀⡿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢰⡏⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠁⠀⣿⠁⠀⠀⣿⠀⠀⠀⠀⣆⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠀⠀⠀⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⠀⠀⣿⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢻⡀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢻⡇⠀⢸⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠃⠀⢹⣇⢀⣤⣤⣶⣦⣤⣤⣤⣤⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣠⣤⣶⠶⠿⠿⠿⠿⠷⢶⣦⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡚⠄⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠂⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠂⠀⠀⡀⢉⠉⡡⠆⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢰⠃⣀⠈⠢⡄
⠀⠀⠀⠀⣄⠢⢀⢸⣏⢛⠂⠤⠀⠀⠌⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠉⠀⠀⠀⠠⠀⠀⠀⠄⠀⠤⠆⠀⠀⠀⠘⠀⠀⠀⢀⡄⠀⠀⠀⠂⠀⠈⡀⠀⠁
⠀⠀⠀⠀⣈⠁⠈⠁⠀⠀⠓⡄⠀⠀⡀⠀⢀⠀⠀⠘⠃⠀⠀⢀⠀⠄⠁⠁⠠⠀⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⠀⠀⠀⠑⠢⠀⠀⢀⠀⠀⢁⠀⢰
⠀⠀⠄⠀⠀⠘⠀⠠⢀⠓⠀⠀⠁⡄⡄⠀⢀⡀⠀⠐⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠁⠀⢠⡄⠃
⡲⠍⢇⢲⡀⠈⢢⠄⠊⠀⠈⠁⠉⠙⠃⠀⠀⣀⠀⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡘⠁⠀⠐⠉⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠁
⠁⠁⠘⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠒⠀⠀⠀⠉⠀⠀⠠⠖⢰⠀⠀⠙⢄⣀⠀⠀⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠰⠂⠁⠀⠀⠀⠀⠀⠦⠀⡰⠀⠀⠄⡈⠀⠀⠀⠈⢉⠀⠀⠁
⠈⠁⠀⠀⡄⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠆⠀⠜⠀⠀⠀⠀⠀⠀⠀⠠⠆⠃⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠁⠠⠀⠺⠁⣀⠀⠤
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣀⡀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⢀⠄⠀⢦⡀⠀⢠⡄⠀⠀⠀⠀⠀⡀⠂⠀⠀⠈⠁⠈⠉⠀⡀⠀⠔⠀⠀⢉
⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⡀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡄⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠠⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠄⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠄⠀⢠⠀⠀⠰⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠈
⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠸⠀⠀⠸⠀⠀⠐⠈⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣀⡀⠀⠑⠂⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠠⠄⠀⡄
⠠⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠀⠀⠀⠄⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠴⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠂⠐⠆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡀⠀⢀⠆
⠀⢠⠀⠀⣀⠀⠀⠈⠀⠀⠺⠀⠀⠀⠂⠀⠀⠀⡄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠩⠿
⠀⠀⠺⡄⠁⠑⢤⠀⠘⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠃
⠀⠀⠀⡀⠀⠀⠀⠁⠒⠠⢅⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⢀⠀⠠⠀⠄⠀⠠⠀⡠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣐⣤⣀⠀⡠⠄⠀⡀⠀⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠆⠀⢠⢲⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢁⠏⢹⠍⠀⠁⠀⠒⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠈⠑⠂
⠀⠀⠀⣀⣀⡘⢦⢾⡙⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠩⠀⠁⠀⡌⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠉⠈⠁⠀⠀⠀⠀⠓
⠀⠀⠘⠳⠜⣷⢸⣪⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠔⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠄
⡀⢀⣀⢀⣀⣉⣀⣉⣀⣀⢀⢀⣀⣀⢀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⡀⣀⣌⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⣫⠟⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⣤⡀⠀⠀⠀⣴⣏⣽⠟⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠈⣷⡀⠀⠀⣻⣿⣯⡉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠘⣷⣤⣤⣿⠏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠶⣄⠀⠀⠀⠀⠀⢾⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠛⣶⠶⣶⣶⠶⠒⠒⣒⣶⠶⠲⢦⣤⣼⡟
⠀⠀⠀⢀⣾⣿⣡⣤⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡾⢋⣤⠖⠋⣻⣷⣶⠏⠁⣀⠴⠚⠁⢀⡬⠛⣶⡀
⠀⠀⣰⠋⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡰⠻⣿⣏⣀⣠⠞⠛⢉⣠⠴⠋⠁⢀⡠⠚⢁⣤⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢰⠋⠁⣼⠃⠈⢧⣀⠀⠀⠀⠀⠀⠀⢿⣄⡼⠁⡜⠈⡿⠻⠿⣶⣾⣿⣷⣦⣴⣾⣷⣶⣾⠿⢿⡟⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣉⡁
⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⣾⣧⣴⣧⠀⠀⣼⠀⠀⠈⡏⠉⠁⠀⢸⠀⠀⠈⡇⠀⢹⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢷⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡟⠉⣿⡟⠀⢀⣷⠀⠀⢰⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠸⡇⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠏⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠇⠀⣿⠃⠀⢸⣿⢀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⡇⢹⣿⣤⣄⣀⣀⣀⣠⡴⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣻⠆⣠⣏⠀⢀⣿⣿⣿⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⣿⣷⣤⣷⣿⣿⡏⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣶⣦⣄⠀⠀⠀⠀⢸⠀⣿⠀⠈⡟⠁⢸⣧⢆⣿⢀⣸⣷⣶⣦⠀⣾⠿⠂⠲⢿⡿⠉⣿⢹⣿⡿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠐⠋⣭⠟⢻⣧⠀⠀⠀⠘⡀⢻⣀⠀⣧⠀⠸⣿⣿⣿⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⢸⡄⠀⣿⣸⣿⡇⠈⠛⢷⣤⣀⣀⣤⡤⠶⠚⠛⠛⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢻⣄⣠⣤⣴⣷⣾⣿⠀⣿⠀⠀⣿⡇⠻⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⢸⠃⠀⣟⣿⣿⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⢿⣏⡉⠉⢏⠙⣿⡄⢿⠀⠀⢻⡇⠀⠀⠀⣷⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠉⠛⠻⠿⣦⣿⠀⢸⡆⠀⠈⡇⠀⠀⠀⣿⠀⠀⠀⢸⠁⠀⢀⡿⠀⣼⣿⣿⣧⣤⡀⠀⠀⠀⠀⣠⠶⠚⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⢻⣦⣀⣀⠀⠀⠀⠀⠀⠀⢳⡹⡷⠘⣇⣀⡀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⣸⣧⣴⣿⣿⡟⠉⠉⢻⡀⠀⠀⠀⠁⠀⠘⠧⠤
⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠳⣿⡄⢹⡍⠀⢻⡟⠃⠰⢿⠷⠆⢸⡟⠛⢠⠏⢡⣿⣿⠏⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠈⠚⡿⠻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣱⣄⠀⢧⠀⠀⠸⠀⢀⣾⢀⣰⣯⣾⣿⠟⠁⠀⠀⠀⠀⢻⣷⣶⠦
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣾⣷⣶⣾⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⠏⠓
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣉⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⡄⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠀⠖⡉⠑⠀⡀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠢⣰⠂⠈⠁⠋⢶⠀⠚⢰⡀⣊⣀⡬⠤⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠢⡄⠈⡆⠀⣏⠠⠂⠀⠀⠸⠀⠀⢀⡟⠉⠀⠀⡄⠀⠀⠀⢨⠏⠈⠆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠡⡀⡸⠀⣇⠀⠀⠀⠀⠀⣇⡴⢋⡶⢈⠁⢤⠀⠀⠀⣸⠃⢂⡆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠀⡴⢠⠀⠱⢇⠀⢹⠀⠀⠀⡦⢠⣏⡤⢋⠆⠸⠀⠀⠀⠀⠀⣅⠤⠚⠉⠩⠀⠑⠂⠀⠀⠊⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡆⠀⡀⠁⠀⢆⠀⠈⢧⠀⡇⣄⠐⣁⣾⡏⠀⠴⠀⠀⢇⠈⡀⣀⡴⠃⠀⠋⠀⠀⠆⠀⠆⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡹⠣⡀⢸⠀⢰⠈⢻⡇⠀⣠⠏⠈⣷⠀⠀⠀⠀⠈⡤⠴⠚⣤⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⠀⠙⢾⠀⠀⡀⠀⣿⡞⠁⠀⠀⢸⡆⠀⠀⣠⠚⠀⠁⠂⠠⠀⣰⡄⠀⢠⡄⠊⠀⠀⠀⠀⢀⠀⠀⠀⡅
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠒⠒⢾⡀⠀⠈⢧⠀⢡⠀⠸⣧⠀⠀⠀⠀⢻⣦⠚⠁⠉⠀⠀⠈⣀⠠⠢⣄⠀⢈⠁⠀⠀⠀⢀⠄⠛⠀⢦⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠳⡀⠀⠈⠓⣎⠀⠀⣿⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣰⠁⠠⠀⠀⣀⡜⠒⠒⠖⠒⠣⢄⣴⠡⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠃⣠⠙⠲⣤⡀⠸⡄⠀⠙⢷⠀⠀⠘⡛⠀⠁⠀⠀⠀⣸⢁⣠⡾⠛⠋⠉⠀⠀⠀⠀⠀⣤⠄⠀⠀⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⣀⠞⠚⠶⣿⣦⡀⡀⢀⠀⢰⣷⠀⠀⣀⣤⣦⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⠓⠁⠀⠀⠀⠀⠈⠙⠻⣿⢷⡸⣿⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠞⣹⣿⣿⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⣿⣽⣿⣟⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡇
⠀⠀⣠⠀⡀⣄⣠⣀⡀⢀⣀⣀⡀⢀⣠⣠⣾⡿⣿⢿⣿⣿⡿⣿⣿⣾⣾⣿⣿⣷⣶⣄⠉⣁⡀⠀⣀⢀⣀⣀⣀⣠⡀⠀⢠⣀⣀⣀⣀⠀⠃
⠀⠀⣿⣿⣷⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣷⣟⣽⣟⢸⣯⣽⣿⣯⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡎⣿⣿⣿⣞⣿⣷⣽⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠉⠈⠀⠀⠈⠈⠀⠀⠁⠀⠁⠈⠁⠀⠁⠈⠀⠈⠀⠀⠀⠉⠁⠀⠀⠉⠀⠉⠀⠀⠁⠉⠈⠁⠀⠀⠁⠀⠈⠀⠀⠈⠁⠀⠈⠀⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠄⣤⠀⢠⢤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡦⠟⠘⢿⠦⠄⣒⠨⣗⡨⢫⠾⠏⠁⢴⣖⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢴⠁⡌⢐⣲⠝⢀⣏⠿⡅⠤⡿⡠⣠⣀⠓⣄⣯⠹⡿⣳⣤⡀
⠀⠀⠀⠀⠀⠀⣀⡉⣫⡭⣭⣿⣛⠻⢺⡇⢵⣽⡌⠧⣶⠄⠍⠁⡋⠁⢥⠄⢹⣵⣤⠏⠉⠄
⠀⠀⠀⠀⠀⠜⣷⢮⢭⣷⠘⣳⡌⣩⣞⡀⠈⠉⢁⡦⠁⠗⠇⠶⠀⠀⠈⠖⠈⣈⠃⢾⠃⡙⢣
⠀⠀⠀⠀⢸⣞⡧⢸⢈⡼⣿⠳⣮⣸⣟⢪⠑⠢⠦⣾⣇⣶⣇⠺⣿⢿⠷⢤⠰⠶⣴⠈⢾⡿⡜⢣
⠀⠀⠀⢀⣬⡈⢸⢞⡤⠐⠪⢻⣥⡉⡟⢠⠩⡈⡅⣞⠎⠉⢉⠐⡏⢹⠞⣷⢬⣄⣮⣄⡀⡙⣡⡠⣄
⠀⠀⠀⡞⢋⢗⢨⡽⡍⢽⡷⠸⣿⢹⡍⠙⠀⠸⢷⡸⢷⡼⡹⠼⠞⠻⡻⣿⡽⣄⠘⢊⡀⢃⡘⠟⣿⡄
⠀⠀⢠⡖⣛⣅⡿⣋⢵⡤⠡⠟⡇⠹⠾⠆⣰⠢⢹⠒⡾⠶⣫⢶⣦⠔⢠⣐⢪⠁⠔⠥⣀⢸⡟⢰⡋⠅
⠀⠀⢸⣹⡛⢈⣺⣦⣦⡏⢐⢱⡦⡀⠀⣀⠧⢸⣘⣼⡓⢇⠀⣑⢄⣴⢸⡵⢀⢠⡷⠀⠳⠹⣹⣬⠅⡇
⠀⠀⠈⠿⣻⣢⡽⢳⡗⠙⠨⡀⠃⢨⠀⢃⠂⠈⠊⠵⠒⣌⢱⢶⢣⣐⠑⣦⠴⠛⠓⣊⠐⢈⡲⠭⠦
⠀⠀⠀⢸⣯⢿⣷⢠⡛⢶⡄⠀⣷⠺⠀⠙⠁⢀⢆⢀⣲⠟⢸⢘⠐⠝⠀⠉⢈⡄⠒⣞⣥⡔⠉⠲⡏
⠀⠀⠀⠈⢷⡶⡦⣿⠛⠋⠁⢲⡋⠦⠠⢄⣠⣿⠴⢢⡔⠹⠘⣈⣨⡦⠄⢀⠀⣤⡾⢩⣍⣂⣤⡞
⠀⠀⠀⠀⠈⢿⢫⣋⣸⢎⣶⣞⣷⡒⢸⡿⠋⠁⡗⠦⣌⡡⠶⠐⠉⢾⠙⢋⣧⢀⣳⣁⡬⠉⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣔⣀⣛⣰⡕⣳⣛⡁⠘⠀⢃⣶⣶⠅⠸⡚⠀⠔⠁⠺⣱⠀⢻⣽⣟
⠀⠀⠀⠀⠀⠀⠈⠠⡛⢵⡿⢋⣡⡻⣿⣶⠠⠄⡉⠢⡋⣀⠔⠰⢃⠀⠄⢴⠤⣆⡆⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠲⢏⡀⠉⣹⣴⠋⢛⡄⠸⠍⠐⢦⡃⢶⣶⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠧⠼⠈⠨⠬⠔⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⢴⣶⣾⠾⣷⡀⠀⠀⠀⠀⢀⡾⠐⣁⠀⡀⢠⢀⡤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡖
⠀⠀⠀⠀⠀⠀⠀⠄⢸⣿⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⣿⡤⠺⠡⣶⢿⣟⠒⢿⡟⢾⡟⣀⣶⣠⡄⡠⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠚⠿⠌⠁⠙⣁⣘⣈⣢⣇⡘⢌⣧⣽⡶⡟⢿⡀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠄⠾⠵⣻⡰⠞⣠⣴⣾⣿⣿⣿⠿⣿⣿⣿⣷⣿⣄⡻⠸⣊⣽⢷⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣆⣖⣽⡉⢩⣾⣿⡿⡾⢫⠇⠻⣤⡾⠘⣧⠙⢟⢿⣿⣷⡜⠛⠈⣹⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠋⡟⡏⢃⣿⢿⡏⠐⣷⠃⢦⣺⡋⡿⣩⠇⣙⠿⠁⢻⣿⣿⡄⠰⠟⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣦⣾⣾⣅⣼⣿⠟⢶⠲⢊⠗⠒⠟⠛⠛⢋⠿⠁⠚⢋⠛⣿⣿⡷⠤⠼⠚⢯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣱⠟⢃⣿⣿⣤⣦⣤⣯⣤⣾⣤⣾⣤⣷⣤⣶⣤⣼⣦⣾⣿⣷⠀⠓⠋⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡍⠘⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⣿⢻⣿⢿⣿⣇⣿⡆⣿⢏⣿⡟⢻⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣿⢼⣿⡾⣹⣿⣿⡇⣿⡆⣿⠃⣾⣧⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣻⣿⣾⣿⡷⢸⣿⣿⡇⣿⠁⣿⢱⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣼⣿⣿⣿⣟⢻⡇⣿⠃⣿⡇⢹⣆⢹⣯⣿⣿⣿⣿⠃⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⣏⣿⣟⣿⣿⣿⢻⡇⢸⡄⢼⣡⢸⣿⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠘⠃⠀⠀⢸⣿⣿⣿⢿⣿⣿⣎⣿⣜⣿⣾⣿⣿⣿⢘⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⡾⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠛⠛⠉⠉⠉⠉⠙⠋⠉⠉⠉⠛⠙⠋⠉⠉⠛⠛⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄⠀⣠⠀⣤⣤⠀⣠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢸⠗⠀⡿⠀⢻⡋⠀⣽⠂⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⢸⠀⠀⣷⠀⢸⠁⠀⢿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠟⠀⠀⠀⠀⠀⠀⠀⠃⠋⠘⠗⠁⠹⠂⠘⠊⠂⠸⠚⠀⠃⠀⠀⠀⠀⢘⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠠⡆⢰⣦⡄⢠⡆⡄⢠⡆⠀⣄⢀⡄⠀⠀⠀⠉⠛⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⢸⡏⠀⢸⡇⢿⢸⠃⠀⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠐⣿⠘⡇⠀⢸⠀⡼⢸⡄⠀⡇⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠋⠀⠃⠉⠘⠃⠀⠘⠛⠁⠃⠈

⠀⠀⣶⡶⠆⠀⠀⠀⠀⠀⠀⠸⣿⠿⠛⠃⢶⣿⣿⣿⡟⠁⠀⣤⣴⣿⡋⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⢈⡁⠀⣩⣷⣿⣿⣦⣴⣾⣿⡗
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠛⠁⠀⠀⢀⣍⣁⡈⠉⣀⠀⠈⠛⠛⠟⢛⣿⠿⠋⣹⣿⣿⡏⠉⠸⣿⠉⣿⠿⠿⣿⣿⣿⣿⣷⠀⠻⠿⠓
⣀⣠⡀⢠⣶⣤⣴⡿⠛⣷⣄⠀⠀⠀⠀⠀⣾⡿⠛⣹⣷⠟⠉⠀⠀⠀⠠⠿⣿⣶⠤⣾⣿⣿⣿⣦⣤⣤⠾⢶⣷⠀⠉⠉⠁⣼⣿⠀⣶⣦⣤⣤⣶
⣿⣧⣄⠀⠀⠉⠉⢠⣾⣿⣿⣦⡀⠀⠀⠀⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢀⣹⠿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⢿⣤⡀⢀⣶⣶⣿
⣿⣿⣿⣿⠟⣠⣾⣦⣤⣤⣽⣿⡷⠀⠀⠀⠀⠉⠁⣤⣴⣾⠟⠂⠀⠀⠀⠀⠀⠀⢉⣁⣀⠾⠿⣿⣿⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢴⣿⣿⣿⣿
⡿⠛⢿⡧⠀⠀⠈⠛⢉⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠻⠿⠀⢠⣤⣾⡿⠗⠀⠀⢈⣉⣁⠀⠀⣿⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠄⠻⠿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠠⣶⠈⠻⢿⣷⠄⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠙⠁⠀⠀⠰⡶⠀⠉⠿⠃⠸⠿⠗⠀⠀⠀⠀⣤⣀⠀⣀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⡀⣠⡄⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⢈⡏⠀⠀⠀⠀⢀⣀⣴⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠿⠿⣷⠶⠀⠀⠀⠀⠀⠙⣿⣿⣶
⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⡇⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⢴⣶⠖⢰⣿⣌⠉⠉⠳⣤⡤⠀⠀⢠⣾⣿⣿⣷⣶⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠻⠿⠟⠉⠛⢿⣦⣼⣿⣿⣧⣤⡶⠀⢀⣀⠘⠓⢲⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉
⣀⣀⡀⣠⣶⡶⠀⣠⣤⣄⠀⠀⢀⣠⠿⠛⠁⢴⣤⣀⣴⣦⣤⣴⣿⣿⣿⣥⠀⠛⠀⠀⠀⠻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⣶⣤⣄⠀⠀⠀⠀⠀⠀⢀⣀⣠
⣿⣿⣿⣷⣄⠀⠀⣹⣿⣿⣷⣶⣄⡀⠀⠀⠀⠈⠙⠛⠁⠀⠙⢿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣄⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠂⠀⠀⠀⠀⠀⠀⡸⠿
⣉⡹⠛⢙⣿⣧⠀⠈⠉⠉⠉⠈⢹⣿⣤⣴⣿⣿⣿⣶⣤⠀⠀⣰⣿⣿⣿⣿⣏⣁⡀⠈⠛⠁⢀⣉⠀⣤⣴⣿⣿⣿⣷⣦⣤⡀⠰⣾⠇⢀⡀⠀⠀⠀⢈⣁⣀
⠁⠀⢀⣿⡏⠁⠀⠀⠀⠀⠀⢠⣶⣾⣷⡀⣉⣴⣶⣿⣧⠀⠀⠈⢉⣿⡿⠋⠉⠉⠙⣿⣿⣿⣿⣿⣷⣾⠻⠄⠀⠿⣄⣽⣯⡀⢀⣠⣾⣏⠉⠀⠀⠀⠈⠉⠉
⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⣿⠅⠀⠀⠙⠿⠟⠻⠧⠀⠀⠿⢿⣿⣦⣤⣤⣤⡾⢿⣿⣿⡿⠿⢻⣯⠴⣶⣤⣾⣿⣿⣭⣿⣟⠛⣿⣧⣄
⠀⠀⠀⠀⣀⠀⢀⣴⣿⣟⠀⠀⠀⠀⠀⠀⣰⣶⡶⢤⣄⣶⠤⠀⠀⠀⠁⣴⣿⣉⢀⢺⣿⣿⣧⡀⠈⠛⣠⣿⣯⠉⠿⠿⠟⢻⣿⠀⣈⡉⠀⠀⣀
⣶⡞⠃⠘⠟⠿⠿⠋⣨⣿⣷⡀⠀⠀⠀⠀⣻⡉⠀⠚⠛⠁⠀⠀⠀⠀⠋⠉⠻⣿⠉⢿⣿⣿⣿⣿⣟⠛⠋⠙⠛⠀⠀⠀⠀⠛⠋⠀⠛⣿⣿⡛⠛⠀⢀⣀⣤
⣿⣿⣷⣤⣤⠀⣠⡘⠛⠛⣿⣿⣆⠀⠀⠀⠉⠿⠶⠀⢀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠴⠟⠉⣹⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠿⢂⣰⣿⣿⣿
⣿⣿⣿⡟⠁⠚⠻⢿⠿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠿⣿⣿⠁⠀⠀⣠⣤⣄⠀⠀⠰⠶⠞⠉⠉⣿⡿⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣽⣿⣿⣿⣿
⠏⠀⠙⠋⠀⠀⠀⣀⠰⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⢿⠟⠋⠁⢀⣀⠐⠶⣶⡄⢠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠉⠛⠛⠿
⠀⠀⠀⠀⠀⣀⡈⠛⠀⠈⠙⠛⠁⠀⠀⠀⠀⣠⣶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠉⠀⠈⠉⠉⠀⠀⠀⠀⠻⣶⣤⣶⡄
⠀⡆⠀⠀⠼⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⠠⡶⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⢶⣶⣦⣀
⣤⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣷⣴⣧⣤⡄⠀⠀⢶⣶⣿⣟⠀⠀⠀⠀⠀⢀⣀⣀⠀⣤⠐⠾⢿⣟⠉⠉⠉⠉⠀⣠⣤⣤⣄⣀⣿⣿⣿
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣿⣿⣿⣿⠿⣧⣤⡀⢠⣿⣿⡛⠀⢀⠀⠀⠀⢹⣟⣁⣼⠟⠗⠀⠀⠈⠛⠋⠀⠀⠘⠛⠿⠿⠛⠛⠿⠿⠛
⠉⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⢋⡀⠈⢉⡉⠀⢠⣽⣿⣿⠟⠛⣿⠛⠋⠀⣴⣶⣤⣤⡼⠛⠓⠀⠀⠀⠀⢀⣀
⠀⡆⢠⡞⠛⠋⠀⠾⣿⣷⣤⣀⠐⠚⠉⠀⠀⠹⢿⣶⠟⠛⢿⣿⣿⣿⣿⣷⣤⣤⣤⣄⣀⠈⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀
⠀⣧⠀⠀⠀⠀⠀⠸⠿⠿⠿⠻⢷⣦⠀⢀⣤⣤⣤⣀⡀⠀⠀⢙⣿⣿⣿⣿⡿⠿⠛⢻⣿⠗⠒⠿⠖⠀⣀⣤⣤⣤⣄⡀⠀⠀⠈⣩⡄
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣟⠻⠿⢟⣻⣿⡟⠀⠀⠿⠿⣿⣿⡿⠷⠶⢦⣄⣠⣤⣴⣶⣄⣻⣿⠛⠛⣿⠛⢿⡿⠂⠈⠛⣡⡴⠦

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠈⠙⠲⡄
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡎⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡇⠀⡀⠀⠀⠀⠋⢀⢀⢀⠀⠀⠈⡗⠋⠉⠙⢆⠀⠀⡴⠋⠀⠉⢺⠁⠀⠀⡀⠀⡀⠽⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢧⠀⠀⠘⠍⣘⡃⢪⣤⠄⠀⠀⠀⣧⠀⠀⢀⡞⠀⠀⢳⡀⠀⠀⡾⠀⠀⠀⠠⣤⠕⢘⣁⠉⠂⠀⠀⡼
⠀⠀⠀⢸⠀⠀⠀⠐⠀⡔⠝⠆⠮⠦⢀⠀⢸⠀⡠⠊⠀⠀⠀⠀⠙⢄⠀⡇⠀⠀⠴⠴⠰⠋⢦⠈⠤⠀⠀⢀⡇
⠀⠀⠀⠈⡇⠀⠈⠀⠀⠀⢝⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⡎⠀⠀⠀⠑⠀⢸
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⡇⠀⠀⣀⣀⣀⣀⠤⠤⠤⠴⠒⠓⢤⠀⠀⠀⠀⠀⢀⠴⠛⠒⠦⠤⠤⢤⣀⣀⣀⡀⠀⠀⣸
⠀⠀⠀⠀⡔⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣄
⠀⠀⠀⠀⢇⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠤⠖⠒⠋⠀⠀⠀⠀⠀⠀⠑⠒⠲⠤⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⣒⣒⣒⣒⣀⣀⣀⣒⣀⡀⠒⠒⠒⢒⣒⣒⣚⣓⣒⣒⣒⣒⣖⡒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⡉⠉⠉⠛⠛⠋⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⢹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⣴⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢠⣿⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⢂⡀⡀⣿⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⢸⠟⠳⠁⠞⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⢀⠀⡐⠀⠸⠤⠤⠤⠤⠤⠿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢘⣦⣭⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡄⣴⠆⠀⠐⠹⠃⠉⠀⠀⢀⣠⣀⡀⠻⠁⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⣀⡡⣽⡆⠀⠀⡆⠀⠀⢀⣴⠎⣹⣿⢰⠉⡆⠀⣆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠥⠨⠯⠴⠷⠿⣼⡤⠿⠶⠦⣾⣷⠶⠿⣿⡾⠦⠷⠾⠷⠶⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠉⠉⠙⠛⠛⠛⣛⠛⠛⠛⠛⠛⠛⠛⣛⣛⠛⠛⠛⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣀⣀⡀⢀⣈⣉⣉⣉⣁⣉⠉⠉⠉⠀⢀⠀⢀⠀⢠⣤⣼⡄⠀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠙⣤⡤⠖⠋⠻⠙⠛⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣻⣷⡀⠀⠘⠆⠀⠀⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⣠⠿⠛⠻⢦⣀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣶⠀⣿⣿⣿⢿⡿⣿⣿⣿⡿⣿⣻⣉⣀⣼⣦⣀⣤⡾⠃⠀⠀⠀⢀⣬⡉⠁⠂⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣀⣀⣀⡿⠀⣿⣿⣽⣉⣇⣶⣿⣌⣠⣷⠉⣿⣿⣿⡿⠿⢿⣱⡀⠀⠀⣀⣼⣿⣿⡄⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣟⣿⣟⣁⡀⠀⠀⠈⠙⠻⣿⣿⡇⣿⣿⣿⠊⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⡤⠿⠷⡾⠓⠒⡶⠶⠶⠶⠒⢶⣿⣿⠿⢿⣿⣿⣿⣶⣶⣶⣶⣿⣷⣿⣿⣿⣴⢿⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠎⠀⠠⠄⠀⠈⠭⠀⠠⠤⠂⠀⢿⣿⣿⣿⣿⣿⣷⣷⣶⣿⣿⣿⣿⣿⣿⣿⡏⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠫⠿⠭⠉⢉⣉⡙⠙⢋⡽⢻⣿⢟⣏⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁

⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⣿⠀⠀⡀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⢿⣿⣤⣼⣿⠀⢸⣿⡿⣿⣶⣶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⣿⣦⠀⠀⠀⠈⠙⠛⠛⠁⠀⢸⣿⣇⣀⠀⠉⠉⠛⠛⠿⢿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣤⣤⡀⠀⣴⣿⠟⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠟⠀⠀⠀⠀⠀⠀⢼⡿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣄⠀⣿⡟⢻⣿⡆⠀⠀⠀⠀⢀⣠⣶⣿⢿⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣄⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⢠⣿⡇⠺⠟⠁⣀⠀⠀⣴⣿⠟⠋⣀⣴⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡏⠛⠛⠻⠿⠇⠀⣼⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠿⠿⠟⠁⣠⣴⣾⣿⣷⡀⠹⣿⣶⣾⠿⠋⢁⣤⣶⣦⡀⠀⠀⠀⢸⣿⣿⣶⣶⣤⣄⠀⢠⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣴⣾⠿⣿⣧⠈⠻⡿⠀⠈⠉⢁⣤⣾⣿⣟⠉⣿⣷⠀⠀⠀⢸⣿⣇⠈⠉⠙⠉⠀⣼⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠻⣿⣆⠈⠁⠀⠀⣀⣤⣀⠀⠻⠟⠉⠙⠿⠿⠿⠃⠀⠀⠀⢸⣿⡿⠿⣿⣶⠀⢠⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⠛⠀⣠⣴⣿⠿⠛⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⢁⡀⠸⠿⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣄⡀⢠⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⢀⣠⣴⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠙⠻⢿⣿⣿⡅
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⢿⣿⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣠⣄⠈⢻⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⣿⡟⠛⠛⢁⣠⣤⡄⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢀⣠⣶⣿⡟⠻⣿⣆⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠉⠿⠗⠀⠙⠋⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⢿⣷⡀⠀⠈⠿⠃⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢿⣿⡄⠀⠀⠀⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠁⠀⠉⠁⠀⠀⠀⠀⣿⡿⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⣿
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣞⠀⠀⢺⡆⢺⠁⡆⠀⣎⣡⠀⢺⣄⣾⣧⠘⠃⢈⡁⠀⠀⢿⣤⡇⠀⢸⠃⢰⣀⣖⠀⠀⠀⠈⠻⢦⠘⠁⠈⠃⢰⠀⠀⢛⠀⣟
⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠿⠺⡀⠃⠀⢀⠉⠀⠘⠁⠰⠟⢀⡀⢈⡀⠀⠀⠻⠹⡇⠀⠰⠀⠸⠀⢿⣀⡀⠀⠘⠁⠘⢠⡀⠀⠀⠸⣠⠄⠻⠀⢏⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢶⠶⠶⠾⠿⠛⡿
⠀⠀⠀⠀⠀⣤⣄⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠲⠀⠀⠀⣀⢀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⡿⠧⠿⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠄⠈⠑⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠴⠶⠿⡿⠟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠤⢀⡀⠀⠀⠀⠀⠀⢀⡶⢀⡴⠋⣽⣴⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠀⣸⠟⣠⠀⠉⠂⢀⠀⢴⡟⢻⠴⠛⠃⠀⠀⢀⣤⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠈⣛⡴⠃⠀⠀⢠⠀⠀⠀⠀⢀⠀⣤⡠⠀⠀⢽⠇⠀⢡⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣥⠬⠴⠃⠀⣀⡴⠛⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠠⣤⣤⣀⣠⣤⡄⠶⠟⠋⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠻⣿⣇⠀⣠⠑⣮⠃⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⣠⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⣤⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠂⣠⣾⣹⣷⣦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠛⠶⣯⣧⣼⣿⣠⡿⠋⠀⠀⠃⠈⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⢉⡈⢣⠀⢠⠄⠀⣠⣶⠀⠀⠑⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠰⠁⠀⠀⠀⢦⠀⠀⢤⡀⢃⡀⣃⣀⣀⣤⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣠⣄⣀⣀⣤⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠙⠟⠋⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠆⠀⠀⢘⠉⠐⠛⠿⣩⣿⢿⣿⣷⣾⡇⡄⠀⠀⠰⠀⠤⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠋⠀⢈⣡⡄⣠⠀⠀⡉⠋⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠑⠆⠀⢠⡀⠀⠀⠀⣸⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢦⣀⣀⠑⠈⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢶⡭⢢⡿⢁⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⣀⡴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠦⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠂⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠐⠂⡀⠀⠀⠀⡤⠀⠀⠠⠀⠀⠠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⡀⠀⠀⠀⠈⢠⡈⠀⠀⠀⠀⠀⠀⠰⠆⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠚⠀⠀⠤⠤⠀⣀⠀⠀⠀⠀⠀⢠⠀⠘⠧⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⠀⠁⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠐⠀⣀
⠀⠀⠀⠀⠠⠀⡀⠀⠀⠐⢠⠂⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠐⡀⠀⠀⣀
⠀⠀⠀⠀⠀⢻⣛⣲⣄⡀⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐
⠀⠀⠀⠀⠐⠛⠧⠀⠉⣻⣦⠙⠳⣴⡦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉⠉⠙⠓⠈⠙⠟⠓⠀⠾⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠂⢀⠆⠀⠠⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠠⠀⠀⠀⢰⣀⣤⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⢤⠽⣦⢹⣍⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠲⠤⠄
⠀⠀⠀⠀⠘⢷⣦⣤⡀⠀⠀⠀⠉⠈⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠁
⠀⠀⠀⠀⠀⣤⠀⠀⠀⣽⣿⣿⠟⠻⠿⠿⠉⠁
⠀⠀⠀⠀⠀⠱⠆⠀⠘⠙⠛⠁⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠠⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⢿⡿⡿⠿⠿⠿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿
⣿⣦⣿⣶⣴⣤⣾⣤⣿⣧⣤⣼⣧⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣶⣿⣶⣶⣶⣿⣿⣶⣶⣾⣿⣷⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠤⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠈⢀⠀⠀⠀⠀⠀⡄⠠⡄
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡐⢤⡄⠈⠈
⠀⠀⠀⠀⠸⣄⡀⡀⣀⠀⡀⠤⠤⠄⠀⢸⣄⡀⢜⣄⢸⣄⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡭⠀⠀⡍⣒⠀⠀⠀⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⡁⢀⠶⣤⡍⠄⠀⠀⠀⠸⠰⠀⠀⠀⡀⠄⠤⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢔⠀⠀⡱⠁⠀⠈⢶⣄⠤⣖⠐⠢⢠⠐⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠕⠼⡄⠀⢳⡇⡆⡳⣦⠀⢸⢛⡿⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠰⢠⡎⠄⢸⡇⠏⠀⢘⡏⠉⠐⢰⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢠⠖⣦⡀⣨⠿⠀⠰⡎⠸⣠⢶⡆⠀⠀⣤⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠥⠈⠛⡇⢬⡍⠀⠘⠐⠃⡆⠻⢀⠀⠻⣿⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣀⢧⣯⠀⠁⠈⠀⢀⠄⠀⠀⡴⠋⡄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢠⢄⡄⠈⠇⠀⣷⠁⠚⠀⣁⠸⠀⠠⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠛⠿⠻⡴⡶⣿⣤⣤⣄⣀⣁⢣⣴⡾⠇⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣴⠳⠀⢣⠀⠀⣨⢈⠁⠓⠛⢭⡁⠀⠀⢒⠒⠂⡚⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠜⠨⠀⠀⢘⣀⠀⣉⡊⠀⢾⠤⠀⠃⢠⡀⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠙⠀⠀⠀⠀⠉⠁⠀⠀⠀⠢⠀⠈⠁⠀⠄⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠆
⠀⠀⠀⠀⠀⠀⠂

⣾⣷⣀⣀⠀⡀⠀⠀⠀⠀⢀⡄⡘⠴⢟⠻⣿⣿⣿⠿⠿⣿⣷⡅⣿⣿⣿⡀⠀⣠⣾⣿⣀⣀⠠⠈⠀⠀⠀⠀⢀⠤⠤⠼⠟⠿⢿⣿⣿⠿⠿⣿⣷⡅⣾⣿⣿⡀
⢈⣿⣿⣿⣷⣶⣄⣠⣄⠀⠐⠕⢄⠀⠀⠀⠀⢉⡁⠀⣀⠀⠙⢧⣿⣿⠟⠛⢛⠻⢉⣿⣿⣿⣷⣦⣄⣠⣄⠀⠈⠖⣀⠀⠀⠀⠀⢉⣁⡀⣀⠀⠙⢧⣿⣿⡟⠛⠻
⡿⠿⠋⠻⣿⣿⠛⢿⣿⣄⣠⣾⠠⡀⠀⠀⣴⡟⡵⠸⢿⣷⣶⣿⣿⣿⠀⠀⠀⢂⡿⠿⠋⠻⣿⣿⠛⢿⣿⣄⣠⣾⠐⢄⠀⠀⣴⡟⡀⠸⢿⣷⣶⣿⣿⣿
⠀⢀⣤⣤⣿⣿⣦⠀⠉⡼⠿⠋⠁⠀⠀⠐⣿⣗⠀⠀⢠⡿⠛⠉⠛⠋⠀⣀⡀⠀⠀⢀⣤⣤⣿⣿⣦⠀⠉⡸⠿⠋⠀⠀⠀⠀⣿⡗⠀⠀⢠⡿⠛⠙⠛⠋⠀⣀⡀
⠀⠈⣿⣿⠉⠁⠀⡀⠊⡀⢠⣾⣷⣦⣤⣾⠟⢹⠀⠀⠸⣷⣄⠀⠀⠀⢸⣿⣿⡆⠀⠈⣿⣿⠉⠁⠀⠀⠎⡀⢠⣾⣷⣶⣤⣾⠟⠹⠀⠀⠸⣷⣄⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⡇⠀⠉⣠⡀⠀⣾⣿⣿⣿⡶⠿⠿⣷⣥⠀⠀⠻⠟⠀⠀⠀⢸⣿⢸⡇⠀⠀⣿⣿⡇⠀⠉⣠⡀⠀⣼⣿⣿⣿⡾⠾⠿⣷⣤⠀⠀⠻⠟⠀⠀⠀⢸⣿⢸
⣤⡾⢻⣿⣧⣤⣾⣿⣧⣀⣿⣿⡁⠀⢀⣀⠀⠀⠁⣠⠀⠀⡀⠀⠀⠀⠀⣿⠞⢷⣤⡾⢻⣿⣧⣤⣾⣿⣧⣀⣿⣿⡁⠀⢀⣀⡀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⣿⠘
⠷⠘⠈⠿⠟⢻⣿⠈⡿⠋⠉⢸⣿⡆⢠⠿⠛⠛⠛⠁⢀⢎⣼⣤⣅⣒⢊⣵⣿⣿⠷⠜⠈⠿⠟⢻⣿⠈⡿⠋⠉⢹⣿⡆⢰⡿⠛⠻⠟⠀⢀⢖⣤⣮⣔⣀⣚⣽⣿
⡀⠀⠀⢀⡀⠀⠁⠐⠀⠀⢠⢸⣿⡀⢀⣴⣶⣶⣦⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⠿⡀⠀⠀⢀⡀⠀⠁⠐⠂⠀⢀⢸⣿⡁⠀⣴⣶⣶⣦⣶⣪⣿⣿⣿⣿⣿⣿⣿⣿
⣜⠀⠀⣿⡇⠀⢀⣤⡀⠀⠀⠁⠙⡿⢿⣟⣛⠟⠉⢻⣿⣿⡟⡿⣿⣿⡿⣻⣵⣶⣼⠀⠀⣿⡇⠀⢀⣤⡀⠀⠀⠀⠛⠿⠿⣟⡛⠛⠉⢻⣿⣿⡟⢿⣿⣿⡿⣻⣵
⣿⠀⣸⣿⡇⠀⢸⣿⡇⠀⣀⢀⠀⡨⣸⣿⣿⡆⠀⢸⡟⠟⠁⠀⠀⢈⣿⡟⠀⠘⣿⠀⣸⣿⡇⠀⢸⣿⡇⠀⣀⠀⢀⠰⣸⣿⣿⡆⠀⢸⣟⠟⠉⠀⠀⢈⣿⡟
⣿⣔⣿⣿⣧⡀⣸⢿⣇⢀⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⣿⣔⣿⣿⣧⡀⣸⢿⣇⢀⣿⣿⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠿⣿⣿⠏⠙⠋⣴⡿⢿⣿⣿⠏⠁⠀⠀⠀⣤⠀⣠⣾⣿⢐⡂⠀⠘⠛⠁⠀⠀⠐⠿⣿⣿⠏⠉⠋⣵⡿⢿⣿⣿⠏⠁⠀⠀⠀⣤⠀⣠⣾⣿⠀⡆⠀⠘⠛⠁
⠀⠀⠙⠀⡀⠀⠋⠀⠀⢿⣿⣷⣤⡾⠑⠀⣿⡄⢹⣿⡿⠁⠀⠀⠀⢀⢀⢀⡀⠀⠀⠀⠙⢠⠀⠈⠋⠀⠀⢿⣿⣷⣤⣾⠉⠀⣿⡀⢹⣿⣿⠈⠀⠀⠀⢀⣀⣄
⢀⡀⠀⠀⠹⢤⠀⠀⠀⠀⠀⠀⠙⡏⣤⣠⣿⣿⣾⣿⣧⡀⢀⡄⢚⣁⢈⠔⠁⠀⢀⡀⠀⠀⠸⢢⠀⠀⠀⠀⠀⠀⠹⡏⣤⣠⣿⣿⣾⣿⣧⡀⢀⠴⢂⣀⢀⡑⠁
⣾⣿⣀⣰⡑⡈⠀⠀⠀⠀⢀⠐⡊⠴⠛⠻⢿⣿⣿⠟⠻⢿⣷⡆⣿⣿⣿⡀⢀⣠⣿⣿⣂⣐⡣⠇⠀⠀⠀⠀⢠⠴⠢⠌⠿⠻⢿⣿⣿⠟⠻⢿⣷⡄⣿⣿⣿⡀⢀
⣩⣿⣿⣿⣷⣶⣄⣤⣄⠀⠘⠬⢂⠀⠀⠀⠀⢈⡁⡀⣀⠀⠙⢣⣿⣿⠟⠛⠻⠟⣩⣿⣿⣿⣷⣶⣤⣤⣄⠀⠐⠠⢢⠀⠀⠀⠀⢈⡡⡀⣀⠀⠙⢧⣿⣿⠟⠛⢛
⡿⠟⠋⠻⣿⣿⠛⢿⣿⣤⣤⣾⢘⣀⠀⠀⣼⡟⡌⠸⢿⣷⣶⣿⣿⣿⠀⠀⠀⢽⠿⠟⠋⠻⣿⣿⠛⢿⣿⣤⣤⣾⢈⣀⠀⠀⣼⡟⡐⠼⢿⣷⣶⣿⣿⣿
⠀⢀⣤⣤⣿⣿⣦⠀⢉⠸⠟⠋⠀⠀⠀⢀⣿⡇⠀⠀⢠⡟⠉⠉⠛⠋⢀⣀⣀⠀⠀⢀⣤⣤⣿⣿⣦⠀⠉⡸⠿⠋⠈⠀⠀⢀⣿⣷⠀⠀⢠⡟⠋⠉⠛⠋⢀⣀⣀
⠀⠈⣿⣿⡉⠀⠀⡀⠪⠄⢠⣾⣿⣶⣶⣾⡟⠈⠀⠀⠸⣷⣄⠀⠀⠀⢸⣿⣿⡇⠀⠈⣿⣿⡉⠀⠀⡀⠻⠄⢠⣾⣿⣶⣶⡾⠟⠉⠀⠀⠸⣷⣄⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⡇⠀⠁⣠⡀⠀⣾⣿⣿⡿⠿⠿⠿⢶⣧⠀⠀⠛⠟⠀⠀⠀⢸⣿⢸⡇⠀⠀⣿⣿⡇⠀⠉⣠⡀⠀⣾⣿⣿⡿⠿⠿⠿⣷⣧⠀⠀⠛⠟⠀⠀⠀⢸⣿⢸
⣤⠾⢻⣿⣧⣤⣾⣿⣧⣀⣿⣿⣁⠀⠀⡀⠀⠀⠁⡀⠀⠀⡤⣀⠀⠀⠀⣿⠊⡷⣤⠾⢻⣿⣧⣤⣾⣿⣧⣀⣿⣿⡁⠀⠀⠀⠀⠀⡁⠀⠀⠀⠠⣀⠀⠀⠀⣿⢘
⠷⠚⠀⠿⠛⢻⣿⠈⡟⠉⠈⢸⣿⡆⠀⠊⠁⠀⠀⠀⢀⢬⣴⣦⣵⣂⣊⣽⣿⣿⠷⠜⠀⠿⠛⢻⣿⠀⠟⠉⠉⢼⣿⡆⠀⠂⠂⠙⠀⠀⢀⢂⣴⣦⣤⣓⣊⣷⣿
⡀⠀⠀⢀⡀⠀⠁⠐⠀⠀⢠⢸⣿⡀⢀⣴⣶⣶⣶⣶⣡⣿⣿⣿⣿⣿⣿⣿⣿⠿⡀⠀⠀⢀⡀⠀⠁⠐⠁⠀⢠⢸⣿⡀⢀⣴⣶⣶⣶⣶⣠⣿⣿⣿⣿⣿⣿⣿⣿
⣼⠀⢀⣿⡇⠀⢀⣤⡀⠀⠀⠈⠙⠿⠿⣟⣛⠏⠉⢹⣿⣿⡟⠻⣿⣿⠿⣫⣷⣶⣼⠀⢀⣿⡇⠀⢠⣤⡀⠀⠀⠀⠙⠿⢿⣟⣛⠏⠉⢹⣿⣿⡟⢻⣿⣿⡿⣫⣷
⣿⠀⣸⣿⡇⠀⢸⣿⡇⠀⣠⢀⡀⢈⣸⣿⣿⡆⠀⠸⡯⠙⠉⠀⠀⢈⣿⡟⠀⠘⣿⠀⣸⣿⡇⠀⢸⣿⡇⠀⣠⠀⡀⠀⣹⣿⣿⡆⠀⠸⡟⠋⠉⠀⠀⢈⣿⡟
⣿⣄⣿⣿⣧⣀⣸⢿⣇⣀⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⣿⣄⣿⣿⣧⣀⣸⢿⣇⣀⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿
⠻⢿⣿⠏⠙⠏⣵⡿⢿⣿⣿⠋⠁⠀⡀⠀⣤⠀⣴⣾⣿⠁⡆⠀⠘⠛⠁⠀⠀⠐⠻⢿⣿⢋⠉⢋⣵⡿⢿⣿⣿⠋⠁⠀⡀⠀⣤⠀⣴⣾⣿⢑⡆⠀⠘⠛⠁
⠀⠀⠙⠪⡀⠀⠋⠀⠀⠿⣿⣷⣴⡾⠀⠀⣿⡄⢸⣿⡯⠁⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠙⠢⡀⠀⠋⠀⠀⠿⣿⣷⣴⣞⠌⠀⣿⡄⢸⣿⡿⠀⠀⠀⠀⠀⢀⠠⡀
⣀⡀⡀⢀⣙⣢⡀⠀⠀⠀⠀⠀⣙⣇⣤⣤⣿⣿⣿⣿⣧⣀⣀⣔⣊⣀⣐⣊⠀⠀⣀⡀⠀⢀⣸⣢⡀⠀⠀⠀⠀⠀⢙⣏⣤⣴⣿⣿⣿⣿⣧⣀⣀⣀⣀⣔⣠⡒
⣿⡟⣉⣻⠟⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠐⠀⠀⠒⠬⡑⠢⢀⡀⠀⢀⣀⠠⠤⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡔⠁⠀⠀⠀⠀⠀⠀⠑⠂⠀⠤⠀⠀⠐⠀⠉⠉⠁⢂⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⢀⠔⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⠉⠈⠉⠁⠐⠀⠬⢐⣒⣒⠨⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠄⠄⢀⡀⠀⠀⢀⠔⣊⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠮⡀
⠀⠀⠀⢀⡠⠤⠀⠀⠤⠤⠀⣀⡀⠀⠀⣀⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⠔⡡⠒⠀⠉⠉⠁⠐⠒⠀⠀⠈⠁⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠒⠀⠤⢀⡀⠀⠀⠀⠀⢀⠆
⠀⡌⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠎⠀⠀⠀⠀⠀⠀⠈⠉⠐⡂⠁
⠀⢃⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢠⠊
⠀⠈⢄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⡄
⠀⠀⠀⠑⠬⠁⣒⣒⠀⠠⣀⡠⠐⠈⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠈⠣⡂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠑⠐⢺⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢀⡠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠈⢦⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘⣈⠱⠒⠦⠤⣄⣀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠢⡀⠀⠀⠀⠀⠀⡄⠠⡩⢄⢋⡙⠖⠲⡤⢄⣉⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠈⢢⠀⠀⠀⠀⡇⠀⠀⠈⠁⠘⠈⠳⠒⠦⠄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⢀⠂⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠤⠜⢁⣁⠄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⡰⠠⠋⡉⡰⠒⠃⡤⠴⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠥⠖⡚⣠⠆⠎⠉⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠒⠁⠁⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠈