Stag Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣴⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣦⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣠⡾⡫⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⢷⣄⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⣴⣯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣶⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣶⠀⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⣶
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⢸⡇⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡇⡈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠘⡜⣇⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⢣⠇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⠷⠘⢄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⡠⠃⠾⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣄⣱⣿⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣿⣏⣠⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡹⣔⠤⣀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⠤⣢⢏⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣛⢒⠉⠒⠤⢠⡟⡀⠀⠀⠀⠀⢀⣏⡀⠄⠒⠉⢒⣛⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⠉⠙⠓⠿⣾⣷⣿⣄⠀⠀⢠⣼⣾⣶⠗⠒⠋⠉⣉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡌⠻⣿⡛⠉⢢⣌⣉⡽⠟⠛⠛⠹⢯⣉⣩⡔⠉⢛⣿⠟⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢧⣙⣿⣤⣸⣯⠀⢀⡀⠀⠀⢀⡀⠀⣽⣇⣤⣿⢏⡴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⠛⠛⢻⣟⡻⣾⠁⠀⠀⠈⣷⢟⣻⡟⠛⠛⠭⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠈⠀⠀⠀⠀⠂⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢠⣥⣄⣠⣤⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣿⢿⠀⠀⠀⠀⣾⣿⠏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⣤⠹⠄⠁⠀⠀⠀⠡⠏⣤⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢧⡀⠀⠀⠀⠀⢀⡼⢃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠙⠀⠀⠀⠀⠋⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⠀⢠⡀⠀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⢀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⣼⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠈⢿⣆⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⠀⠀⠀⡀⠁
⠀⠀⠀⠀⢶⣀⠀⠙⢿⠿⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠤⢀⡜⠁⠀⣴
⠀⠀⠀⠀⠀⠙⢷⣦⣤⣹⣤⡀⣬⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣀⡀⢠⠋⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢋⣴⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡯⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣯⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠙⢿⣯⣀⡀⠀⠀⠉⠉⠉⠑⠀⠉⠈⠿⠏⠀⢀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢾⠿⡿⣶⡆⠈⣿⣿⣿⣷⡀⠀⠀⠀⣴⣶⣶⣦⡾⢦⣼⣿⠋⠍⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠾⢻⣿⠿⣾⣿⡋⢹⣿⠿⠇⠘⠛⠛⠉⢀⠀⢿⡟⠍⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠨⣭⣵⣿⣿⡇⠀⠀⢀⣶⣶⣾⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠁⠀⠀⠀⠿⣿⣿⣇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣅⠀⠀⠀⠀⢸⣿⣿⣿⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣆⠀⣠⣤⣾⣿⣿⣿⠄⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡛⠛⠉⠘⣿⣿⣿⣯⣗⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣧⣝⣷⡄⠀⠑⢤⣀⣀⣀⣠⣤⣶⣾⣿⣿⡭⠀⠀⠀⠀⠀⠉⠁⣲⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠉⠉⠛⢻⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣤⣴⣿⣿⣿⣿⢿⣿⣮⡫⠝⠄⠀⠀⠀⠀⠀⢉⣭⣿⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⠙⢧⠝⠠⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢿⣯⠻⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡜⠄⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣄⣀⣠⣴⣾⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠯⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣟⠷⣝⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣟⡷

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡱⡄⠀⠠⡀⠀⠀⠀⠀⣰⠀⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⢶⡆⡀⢠⡖⣬⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⠄⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣇⠀⠀⣯⢀⣀⣠⣤⣤⡀⠀⠀⠈⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢸⠃⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⣤⣿⡟⠿⠛⠛⠛⣿⣷⡀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⢖⣿⡿⠋⠀⠀⠀⠀⠀⢻⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠸⠇⢸
⠀⠀⠤⠒⠒⠀⠖⠲⠶⠷⠶⠶⠶⠺⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠋⠉⠉⠃⠂⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡠⠄⠀⠀⠀⠠⣄⢂⠀⠀⠀⠀⡐⢀⠄⠀⠀⠀⠠⢄⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠁⠀⠀⠀⠀⠀⠸⣼⡄⠀⠀⢀⣧⠇⠀⠀⠀⠀⠀⠈⣧⡇
⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⢃⡆⢰⡈⣇⠀⠀⠀⠀⠀⠀⠀⢨⡇⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⣀⠀⡆⠀⣶⠀⣀⡿⠞⠀⠀⠳⢾⣆⠀⢰⠀⢰⠀⢀⣾⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍⣳⣧⣀⣿⠟⡉⠄⠀⠀⠀⠀⠠⣬⣻⣿⣄⣼⣞⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⢮⣹⡟⠇⠋⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣿⢛⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣌⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⢿⣷⣝⠆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⣀⣀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣍⠻⣧⢘⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢀⣿⡿⣹⣿⡿⠀⢿⣿⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⢻⣿⠏⠻⣿⣿⣿⣿⡇
⠀⠀⠀⣾⡟⠘⣿⡟⠀⠀⢸⡧⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠹⣿⡏⠙⣿⡀
⠀⠀⠀⣿⠀⠀⢹⡇⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⡇⠀⠀⢀⡟⠀⠀⢹
⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⢸⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⣼⠁⠀⠀⣼
⠀⠀⠀⢸⠀⠀⠀⠸⣧⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⠀⣰⡟⠀⠀⠀⣿
⠀⠀⠀⠘⠃⠀⠀⠀⠉⠁⠸⠇⠸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠇⠠⠧⠀⠉⠀⠀⠀⠐⠛

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣇⠀⢰⡀⠄⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⠇⠘⠧⠀⠀⠸⠶⠷⢤⣤⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠖⠛⠊⠙⠓⢯⣥⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡖⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣗⣙⣛⣿⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⣿⣿⣿⣿⠻⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣶⣭⣉⣓⡒⠂⠤⠄⢀⣠⣴⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⣿⣿⡫⠉⢩⣿⣿⣿⠋⠀⠀⠀⠀⠈⠉⠷⢃⣿⠙⠛⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠀⢻⢯⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⢿⣿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⡋⠀⠀⠀⠈⣦⠀⠀⠨⣿⣿⣿⣯⣀⣀⣀⣀⠀⣈⣿⡇⠀⠀⠀⣬⢹⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣆⡨⢏⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⢁⠀⠀⠀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣻⣵⡿⠃⠀⠈⠘⠻⠿⠿⠿⠿⠟⣿⣻⣦⠂⡼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⠿⢛⣷⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠺⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣶⠄⠰⣶⠦⣶⣶⡶⠆⠲⠄⠠⠤⠀⠤⡄⠤⠤⣦⣶⣶⣴⣤⣄⠱⣀⣠⣤⣤⣄⡄⣄⣤⠤⠦⠶⠶⠠⠂

⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡇⠀⠀⠀⢤⡀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢀
⠀⠀⢰⠀⠀⠀⠀⠀⢨⣶⣾⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⡀
⠀⠀⠘⣆⠀⠀⠀⢠⣾⡟⠉⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣠⣼⣟⢷⣶⣾⠁⠀⠀⠀⢰
⠀⠀⠀⠘⣷⣦⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠙⢿⣄⠀⠀⢠⣸⣇
⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⢦⡄⢸⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣴⡿⠋
⠀⠀⠀⠀⢠⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠸⠀⢠⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⢏
⠀⠀⠀⢠⡇⠀⠀⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠘⡆
⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⢹⡀
⠀⠀⠀⠸⣿⣦⣄⣀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠈⠛⠿⢿⡿⣷⣿⣿⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣤⢾⣿⣟⣀⣀⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠙⠻⣿⣿⣷⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⣆⣿⣿⡿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣀⣉⣻⣿⣿⣷⣦⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣰⣭⣶⠿⢟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠛⠛⠻⠿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⢸⣿⣿⣿⠷⠶⠾⣋⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⠓⠤⣄⡀⢾⣿⣿⣿⣿⣦⣀⠀⠀⢀⣠⣤⣟⠻⠟⣿⠃⢀⣠⡤⠴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠘⢷⣤⣨⣿⣮⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣴⣿⡁⢀⣤⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢭⣿⣿⣿⣿⣿⡿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠉⢈⢈⠛⡻⢻⣿⣿⣟⡿⢟⣻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣶⣶⣴⢦⣤⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣿⣅⣈⣀⣀⣹⣧⣬⣍⣁⣹⣍⣉⣹⣇⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣿⣿⣿⣻⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣴⢦⢛⣹⣷⣤⠀⣠⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⣦⣾⣿⣧⣽⡿⠿⠋⠈⠻⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣷⣼⣿⣻⣿⣿⣿⣷⣌⣁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣩⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⢾⣿⣿⣿⣿⣿⣿⣿⣷⣔⣿⣿⣿⣿⣿⣿⠿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣼⣧⢟⠘⡀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠄⠾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣇⣴⣿⡿⣿⠈⡗⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠁⠈⠉⠉⠉⠛⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢠⣿⠷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠮⣿⡄⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢀⣿⠃⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠘⣿⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⢸⣿⠀⢀⡴⠛⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠙⢦⡀⠀⣿⡇⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⣾⠋⠀⢀⡾⠁⠀⠀⢰⠀⠀⠀⠀⡆⠀⠀⠈⢷⡀⠀⠙⣳⣿⣷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣷⣦⣼⣇⠀⠀⢀⡾⠀⠀⠀⠀⣿⡀⠀⠀⣸⣧⣴⣾⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⠉⠛⠿⣿⣶⣿⡇⠀⠀⠀⠀⢸⣿⣶⣿⠿⠛⠉⠁⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣼⣗⣐⣴⣦⡈⢻⣿⣷⣤⣤⣤⣴⣾⣿⡟⢁⣴⣮⣁⣾⣷⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣘⣿⣿⣿⣿⣿⣿⠛⠿⢿⣿⣿⣼⢿⣿⣿⣿⣿⣿⣿⣿⣥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⢃⡀⠀⠀⠹⡟⢁⠀⠀⠘⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡛⠛⣄⡀⠀⠀⠀⣠⠚⢛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡰⢄⠳⠀⢰⠀⠸⢆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣏⠇⠾⣿⣿⠖⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣧⠀⠛⠋⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⡿⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⢿⣿⣿⣿⣷⣽⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢫⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⣶⣦⡀⠀⠀⠀⢀⣴⡷⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠛⣻⣿⣾⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣿⣿⣿⣿⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣴⣿⣿⡟⣻⡟⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣿⣿⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠙⣿⣿⣿⣿⣿⣿⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⢹⣿⣿⣿⠉⣿⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣼⣿⡙⠁⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠈⢿⣿⣧⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢿⡿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢸⡇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢸⢰⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠆⠀⠀⠀⠀⢸⢻⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠋⠀⠰⠟⡃

⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢣⣸⠁⠀⠀⠀⠆⠀⠀⠀⠀⡜⢀⡰⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⣤⠔⠀⠀⠀⢰⠋⠁⠐⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣞⠁⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⡙⢦⡄⣠⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢘⡇⠣⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⣀⣀⢰⣺⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⣽⣿⣿⡁⠐⠢⠀⠀⠀⠀⠀⠀⠀⠚⠉⠑⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⣀⣸⣷⣦⡆⠐⠀⢺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠝⣿⣟⣿⣿⡆⠀⢀⠀⠀⣿⣾⣿⣿⣷⡀⠀⠏⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣽⡀⠀⡉⢹⣿⣷⣴⣾⣶⣿⣿⣿⣛⢿⣿⡇⠀⠀⠃⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠆⠀⢨⣿⣿⣿⣿⠿⠛⢿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠃⢀⠛⠛⠋⠉⠀⠀⠀⠈⠻⣿⣿⣆⠹⣿⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠘⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡎⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠋⠀⠀⠈⢯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠃⠀⠀⠀⠀⠀⠀⠴⠿

⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⡠⠊⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠑⢄⠀⠀⠀⠀⠀⠀⠘⡀⢀
⠀⠀⠀⠀⢀⣿⠆⠁⠀⠀⠀⠀⡐⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢂⠀⠀⠀⠀⠘⢱⣿⡘
⠀⠀⠀⠈⣼⠏⡜⠀⠀⠀⠀⠠⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠄⠀⠀⠀⠀⢃⠹⣧⠃
⠀⠀⠘⠀⠏⢸⠁⠀⠀⠀⠰⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠆⠀⠀⠀⠈⡇⠹⠀⠂
⠀⠀⣷⠀⢠⠀⠀⠀⠀⠀⠂⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠐⠀⠀⠀⠀⠁⡄⠀⣶
⠀⢠⠃⢠⡟⠀⠀⠀⢀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠑⡀⠀⠀⠀⢻⡄⠙⡀
⠀⠈⠀⢸⠁⠀⠀⠀⡌⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢁⠀⠀⠀⠘⡇⠀⠁
⠈⠁⠀⣿⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠄⠀⠀⠀⣿⠀⠘
⣆⠀⠀⣿⠀⠀⠰⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⡄⠀⠀⣿
⣿⠆⠀⣿⠀⠀⡀⠀⠈⡟⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⢹⠇⠀⠀⠀⠀⣿⠀⠰
⣿⠀⣦⠘⡇⠀⡇⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠆⠀⠀⠀⠀⠀⠀⠀⠀⢼⣤⠀⠀⠈⠀⠀⢰⠀⢸⠃⣴
⢿⣿⣿⡄⠱⠀⠃⢀⣼⠀⢀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⣧⡀⢸⠀⠏⢠⣿⣾
⠀⢿⣿⣷⡀⠈⢰⣾⣿⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⣷⡎⠁⢀⣾⣿⡿
⠀⠘⢿⣆⠁⠀⠀⠹⣿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⠋⠀⠀⠈⣴⡿⠃
⠀⠀⠀⠙⢷⣄⠀⠀⠈⠡⣄⣿⣶⠀⠀⠀⠀⠀⠀⠀⢠⠀⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠧⠀⡀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣠⠌⠁⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⢿⣶⣦⣀⠈⠻⣿⡄⠀⠀⠀⠀⠀⠀⠸⢸⡆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢠⡶⠇⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⣀⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⡈⠻⠄⣀⠀⠀⠀⠀⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⣀⠠⠛⣁⣴⣾⡿⠟⠁
⠀⠀⠀⠀⢰⣿⣿⣷⣶⣶⣿⣿⣿⣿⣦⣀⠈⠉⠀⣉⣄⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⣷⢠⣉⠀⠉⠀⣀⣴⣿⣿⣿⣿⣶⣶⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢌⡉⠛⢿⣿⣷⣶⣿⣿⣿⣬⣿⣶⠷⠶⠶⢿⡿⢿⡿⠶⠶⠾⣶⣿⣭⣿⣿⣽⣶⣿⣿⡿⠛⢉⡡⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⢡⣦⡀⠈⢻⣏⣻⣿⣿⣥⣶⣶⣶⣶⣶⣶⣄⣠⣶⣶⣾⣷⣶⣶⣬⣿⣿⣟⣹⡟⠁⢀⣴⠆⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⢠⣿⣤⡀⠘⠛⣿⣿⡟⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⢻⣿⣿⠛⠁⢀⣬⣿⡆⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠘⠧⠘⠿⠏⠀⢰⣿⠋⣾⣿⢶⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⡶⣿⣷⠙⣿⡆⠀⠹⠿⠃⠼⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⢀⣀⣾⣿⣆⡿⠿⠚⠋⢹⣿⣿⡿⠛⠛⢿⣿⣿⡏⠙⠓⠿⢿⣴⣿⣷⡀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⣧⣤⠀⠀⣾⠟⠁⢀⣴⣤⡀⠉⠻⣧⠀⠀⣤⣼⣿⣿⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣶⣾⣯⣄⣴⡟⠛⠛⢻⣦⣠⣽⣷⣶⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠘⣁⠀⣿⡝⣯⣿⣿⣿⣿⣽⢻⣷⠀⣈⠁⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣰⣾⣿⡄⣿⣇⣼⣿⣿⣿⣿⣧⣸⡿⢠⣿⣷⣆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰⣿⣿⣿⣿⣜⣿⡋⠀⠀⠀⠀⢙⣿⣡⣿⣿⣿⣿⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⣿⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢻⣿⣿⣿⡄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⣿⣿⣿⠁⠀⠈⢿⣿⠿⠛⠛⠿⣿⡿⠁⠀⠈⣿⣿⣿⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣥⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⡜⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⣠⠞⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠄⡰
⠀⠀⠀⠉⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⡐⢧⠀⠏⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠸⣧
⠀⠠⠇⠀⠀⠰⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢩
⠀⡀⠰⠀⠠⠂⠀⣀⣴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠙⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡀⢸
⠀⣆⠀⠀⠀⠈⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⠈⠀⠀⠀⠀⠀⢠
⠀⠹⣆⡠⣤⡀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⣼
⠀⠀⠈⠀⢀⡷⠆⠹⢃⠀⠀⠀⠀⠀⢀⡴⠀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⠀⠀⢀⣴⣿⡷
⠀⠀⠀⠘⠋⠙⠤⠀⠀⠀⠀⣀⠀⢠⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢀⡀⠀⠀⠠⠾⣿⡿⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⢀⡀⠀⠀⠀⢀⠴⠟⠁⠼⠿⠿⠟⠀⢀⣉⣀⣤⣴⡶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢰⣮⠋⢰⡶⣄⠀⠀⠀⠀⠀⠀⠀⡾⣁⠠⠄⠂⠀⠀⠀⠀⣠⡄⠂⠀⠑⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⠀⠀⠀⠘⢶⡿⠿⠷⣤⡄⠀⠀⠀⠀⠶⡤⠀⠉⠀⢀⣾⡷⠋⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠆⠀⣤⣸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠼⠆⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⣀⡾⠁⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠠⠐⢂⣤⣤⣤⣤⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣦⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠆⠀⠠⠆⠀⠀⢀⠀⠸⣿⣿⣿⣿⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡶⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣦⠹⣿⣋⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢸⣷⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣥⣼⣿⣿⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣰⣄⣤⠘⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣴⡶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠰⣀⡄⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⡀⠀⠀⠘⢦⡄⠀⢌⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⣿⠀⠀⠀⠘⠇⠀⠈⡟⠁⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣁⣌⣙⣯⣀⣼⣁⣩⣧⣉⣠⣾⣛⣸⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⢀⢀⠀⠀⡀
⠀⠀⣈⣶⣄⣷⠀⢀⠀⢠⣦⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣰⣆⣼⡾⠀⠾
⠀⣈⣉⠙⣾⣿⣿⡌⢸⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⢻⣷⣿⣷⡖⠋⠁
⠀⠈⠙⠓⠈⠿⣿⣇⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢸⢻⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠿⢸⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣃⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠹⣷⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣾⠟⣿⡃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡙⠻⣜⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⣧⣾⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣿⣿⣿⣄⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠀⠀⠀⠀⢉⣦⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣀⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣵⡾⣃⣀⣤⣾⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣍⠻⣿⣦⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣧⣠⣼⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠙⠲⢿⣿⣿⣿⡷⠆⠀⠀⠐⢦⣴⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣀⠈⢻⣿⣯⠄⠈⠁⠀⠂⠀⠸⣿⣧⣀⡤⠤⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠦⣾⣏⡙⣧⠀⠀⠂⠀⣁⡤⠙⣿⡍⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡟⠻⠿⣿⣇⠀⣰⡏⢾⢿⣿⡷⠵⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⣸⣦⣤⣿⠟⠀⠛⢀⣠⡎⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢸⡏⢻⣯⢰⡆⠤⢠⡼⣿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠘⠁⠈⠃⢤⣾⣿⣯⠀⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢠⠆⠀⠀⡄⠀⢀⣤⣁⡄⣠⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣼⡆⠀⠀⣤⣇⠲⣿⣿⢨⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⢀⠀⠘⠋⣿⣿⣿⣿⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣤⣤⡄⠉⠻⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⣼⣷⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠲⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡇⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠙⢷⣤⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠆⣼⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⣰⣿⡟⠁⢠⡎
⠀⠀⠀⠀⠘⣧⠀⠘⣿⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢠⡿⠋⠀⢀⡿
⠀⠀⠀⠀⠀⢹⡀⠀⠸⣷⡀⠷⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠟⠁⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠈⢷⠁⠀⠙⣷⣄⠠⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⠆⠀⠀⣠⡾⠏⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠈⠻⣶⣤⣀⠀⠀⠀⠘⢷⣀⠀⡀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⢀⣴⠏⠀⣠⣾⡟⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⣈⣉⠉⠁⡀⠀⠀⠀⠈⠻⢦⣄⣡⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡆⣠⣦⣶⠾⠋⢁⣴⣾⣿⡿⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠻⠿⠿⠿⢿⣶⣤⣍⡛⢿⣿⣶⣄⠀⠀⠀⠀⢰⣿⣿⡿⢋⣥⣾⡿⠟⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠉⠙⠛⠷⣿⣿⠿⠇⠀⠀⠀⠘⠛⢿⣷⣿⣯⠤⠶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡙⠛⠻⠿⠶⣾⡋⠀⠀⠀⠀⠐⠀⠀⢸⡿⠛⠁⢀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣹⠆⠀⠀⠀⠀⠀⠀⠘⠆⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢸⡇⠀⠀⣾⡇⣰⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⢀⡙⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⢠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢰⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⣶⢿⣝⣷⡎⣿⣿⣿⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣀⣿⣿⣿⣿⣃⣾⣿⣿⡷⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣽⣿⣯⣥⣭⣵⣿⣿⣿⣿⣿⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣨⣿⣿⠯⡻⢿⣿⡿⠛⠀⠘⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠙⠄⠁⢈⠶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣶⣆⠀⠘⠛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣄⢰⠆⠐⢆⠀⠀⠀⠠⠀⠰⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢃⠘⣾⡄⠀⠈⢱⣦⣀⠀⣧⠀⢳⠶⠄⠀⡇⢰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠓⠛⠿⠿⣷⣀⠉⠛⢯⠉⠙⠛⢄⠚⡇⠀⣣⣇⣼⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣷⣤⣀⣠⣾⣷⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⡏⣹⣿⠶⣦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⢿⣶⣤⣛⣛⣯⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣶⢿⣻⣿⡭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣵⣾⣴⣖⡀⢀⡦⠄⣿⠁⡀⠀⠸⣽⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣹⣿⣦⣐⣧⠦⠧⣀⠀⣽⣿⣿⣿⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣶⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣤⣶⣦⣿⣿⡋⣉⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠟⢻⣿⣿⡟⠉⠙⠛⠿⣿⣿⣿⡿⢿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⣾⣿⡟⠋⣠⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠉⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⣿⠋⠀⢰⣿⡿⠃⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⢻⡇⠀⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣼⡿
⠀⠀⠀⠀⠀⢸⣇⠀⣀⠀⢸⣷⢀⣁⣀⣀⣀⢀⠀⣿⡀⠀⠀⢀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣽⣿⣿⣿⣿⣷⣶⢾⡖⠀⠒⡤
⠀⠀⠀⠀⠀⠈⠋⠩⠉⠝⠛⠋⠏⠟⠛⠛⠛⠉⠙⠙⠛⠛⠃⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⣆⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⣳⣼⣀⠈⠳⠀⠀⠀⠀⠀⠀⡐⣣⠔⠭⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡽⢿⠆⣠⠤⢴⡋⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡄⠀⣨⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡤⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠐⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢠⣶⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⣇⠀⠀⠀⢻⠀⣀⣄⣠⣤⣶⣦⣆⡀⠀⠀⠂⠈⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢻⠃⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⡇⠋⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠁⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠁⢀⣴⣾⣿⡿⠉⠉⠉⠉⠉⠉⢹⣷⣿⡂⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⠄⣠⠞⣹⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠊⠁⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢸⡇
⠀⠀⠀⣠⣤⠤⠴⠒⢠⡤⢴⣦⣴⣧⣴⣦⣤⣴⡶⣿⣴⡶⠶⠾⠿⠿⠻⠾⣿⡿⢿⠾⢿⠿⠟⠛⠻⠛⠛⠒⠒⠒⠶⠲⠤⠤⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣩⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢀⡀
⠀⠀⠈⠀⠀⠀⠀⣠⠁⠀⢀⣀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⣀⠀⠀⢀⠀⠀⠀⠀⢈
⠀⣠⡆⠀⠀⠀⢰⣿⣶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣶⣾⣿⠀⠀⠀⠀⣦⡀
⣰⣿⠁⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⢹⣷
⣿⣿⡀⣴⣻⠋⠀⠀⣠⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⡀⠀⠈⢻⣳⡄⣸⣿
⢹⣿⣿⡿⠁⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠹⣿⣿⣿
⢸⣿⣿⠁⠀⠀⢸⣿⠀⠀⠘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⢸⣿⠀⠀⠀⢹⣿⣿
⢸⣿⡇⠀⠀⣀⣸⣿⣥⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣽⣿⣀⡀⠀⠀⣿⣿
⠸⣿⡇⣠⣾⡿⠛⠹⣿⣄⠀⣀⣼⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣄⡀⢀⣼⡿⠙⠻⣿⣦⡀⣿⡿
⠀⢿⣿⣿⠋⠀⠀⠀⢹⣿⣿⡿⠋⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠈⠻⣿⣿⣿⠁⠀⠀⠈⢻⣿⣿⠇
⠀⠘⣿⣿⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠙⢿⣶⣤⣄⣀⣤⣼⣿⣿⣷⣶⣾⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣶⣶⣿⣿⣿⣤⣄⣀⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠉⠙⢛⠛⠛⠻⠿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠿⠛⠛⢛⠛⠉⠉⠉⠙⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡤⢤⣄⠈⠻⣿⣿⣦⠀⠀⠀⠀⠀⢠⣾⣿⠿⠋⢁⣠⣤⣴⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⢞⡃⠈⢳⣄⠈⣷⣿⣿⣶⣶⣶⣶⣿⣿⡏⢀⣴⠋⠀⠘⣯⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠘⣿⡿⢻⣿⣿⣿⠟⢻⣿⣿⣾⡛⣿⡿⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣿⠇⢾⣿⣿⣿⣿⣿⣿⣿⣿⠇⢻⡷⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⣄⠹⣿⣿⣿⣿⣿⣿⢁⣴⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⣍⡀⣿⣿⣿⣿⣿⡇⣈⣍⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⣿⡟⣿⣿⣿⡟⣿⣿⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⢀⠼⢿⠿⢆⠸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡅⠸⣷⣶⣾⡿⠂⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⣷⣶⠚⣋⣛⢲⣶⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢫⣿⣿⣿⣾⣿⣿⡿⡏⢙⣿⣮⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⠀⠹⣿⣿⣿⣴⣴⣿⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⣿⣀⠀⠉⠛⣿⣿⣿⣿⢿⠃⣹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⢿⣿⣿⣤⡀⠀⠎⠛⣿⠀⢈⣧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡞⠟⢻⣿⣿⣧⡀⡀⣸⣿⣿⣿⡟⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡄⠈⠣⣿⣿⣿⣿⣿⣿⣿⡏⢡⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣥⠀⠀⠈⠘⢻⡿⠿⡇⢀⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣀⡀⠀⠀⠃⠀⣳⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣤⢀⣤⣴⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⢀⣿⠃⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣸⡿⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⢸⣷⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠋⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠻⣷⣤⣸⣿⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠈⠻⣿⣏⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⠹⣧⡀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣰⣿⢇⣰⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠟⣂⣻⣿⢿⣷⣶⣤⡄⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⢀⣄⣴⣾⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠛⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣿⣶⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⢿⣿⡆⠀⠀⠀⠀⠀⣼⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⠀⢰⣈⠉⢹⡇⠀⠐⠂⠙⢿⣏⣀⠠⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣷⣶⣦⣭⡿⠋⠀⠉⠉⠀⠈⢀⣶⡾⠃⣠⣶⣶⣶⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠈⠻⣿⠛⠁⠀⠁⠀⠀⣀⡀⠀⠈⠛⠀⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⡿⠀⠀⠀⠀⠴⠟⠿⠿⠂⠀⠀⠰⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣶⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢧⣶⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⠿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣵⣄⣷⣼⣿⣿⣿⣿⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣷⣶⣶⣶⣤⣄⣠⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢻⣿⣆⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠃⠘⢿⣿⣷⣤⣄⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⢿⣿⣟⠃⠀⢹⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣽⣿⣿⠇⠘⣿⡍⠀⠀⢈⣿⣿⣿⣿⠿⣿⡔⢰⣿⣟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠀⠀⢿⡓⠀⠀⢀⣿⣿⣋⣤⢥⣿⣖⣼⣿⣿⣤⣟⣿⡁⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡷⠀⠀⠰⠄⠀⠀⠀⢸⣿⣿⣿⣿⡿⢯⣽⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠗⠀⠀⠐⣶⣦⢠⣤⣴⣾⣿⣿⣿⣁⣶⣿⣿⣿⣿⠁⢹⡯⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣴⣶⣿⣿⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⡏⢹⡿⠀⠈⠃⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠛⠏⠇⠙⠘⠟⢿⣿⠁⠈⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⣼⣿⣭⣅⢈⣛⡀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣼⣿⣭⣁⡘⠛⢿⠷⣶⣤⣄⠲⣿⣇⣀⠀⢀⡀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⣿⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣾⣽⣿⣾⣿⣷⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣮⣭⣉⡛⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣯⡽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠢⢀⠀⢀⠴⢒⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢚⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠈⢁⡈⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡉⣩⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⣦⡶⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣷⣾⣿⣿⣏⣹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢚⠀⠀⠀⢲⢄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡹⢿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⡀⠘⠀⠛⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠓⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡄⠀⠄⠀⠀⠀⠀⠖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠀⢰⠁⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠁⠀⠀⠠⠈⢁⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⡐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⢀⠳
⠀⠀⠀⠀⠀⠈⠁⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠈⢁
⠀⠀⠀⠀⠀⠐⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠄⠀⠐⠈
⠀⠀⠀⠀⠀⠌⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠂⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠂⠀⣤⢀
⠀⠀⠀⠀⠀⠀⣦⠠⢽⣀⢀⣤⠠⠀⠀⠀⠀⠀⠀⢀⣨⡄⠐⠃⠐⠶⠦⠰⠎⠁⠀⠀⠀⠀⠀⠀⠠⡰⢻⣹⠖⠒⠇⢱⣿⠋
⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠈⠙⡟⠄⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⢿⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⠂⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡄⠀⠀⠊⠀⠀⠀⠀⠀⠠⡄⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡔⠸⡡⡄
⠀⢠⡀⢇⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢰⠐⣇⢂⠑⡇
⠀⠈⣰⡬⠁⠀⠲⠀⢠⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣴⠀⠀⣴⡆⠸⠀⠀⠀⠀⠀⠀⠈⡘⣄⠀⣰⣤⠍⠳⠏
⠀⠄⠉⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠉⠉⠀⠈
⠀⠁⠀⠘⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢰⠆⠄
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢠⠀⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣻⣦⡀
⠀⠀⠀⠀⠀⡠⣾⡷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⢀⣤⣿⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⡀⠀⠀⠀⠙⡹⣧
⠀⠀⢀⣿⢯⣿⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠀⢻⡆
⠀⢀⣾⠁⣼⡧⢀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⡈⢿⡀
⠀⣸⣏⠀⣩⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⡄
⠀⠾⢃⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⣿⡏⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠸⣿⣧⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⣿
⠀⠀⢻⣷⡈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⣾⣿⠏
⠀⠀⠀⠻⣿⣦⣉⢄⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣾⡃⠒⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣤⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⣤⣉⡀⠈⡀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣿⣿⣯⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠻⡿⣿⠻⣿⣿⣝⢶⣥⡢⡈⠻⠧⡄⠀⠀⠀⠀⠠⢦⣤⣿⡟⣡⣿⡿⢻⣭⣠⠞⢙⡟
⠀⠀⠀⠀⠀⠀⠀⢲⣿⣮⣻⣿⠹⣯⣿⣷⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿⠛⣤⣬⣛⣠⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣮⣹⣿⣿⡿⠀⠀⠀⠀⣦⡾⠟⠛⣿⣿⣿⣿⣤⣴⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡉⢉⠻⣯⠟⠁⠀⢄⢀⣤⡌⠀⣠⣴⣿⢿⣿⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠠⣶⣄⠀⠈⠉⢻⣿⣿⣽⣿⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠉⠉⠀⠀⢠⣼⣿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣄⢠⣼⡇⠀⠈⣹⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡌⢠⣤⡁⣀⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠈⠉⠀⠉⠛⠁⡛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠉⠉⠀⠀⠈⠉⠉⠻⢿⣿⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢣⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢠⣄⠠⣄⣄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⡏⠉⣹⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⡜⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⢹⠆
⠀⠀⠀⠀⠀⠀⡸⢰⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⠶⢶⠙⡣⠃
⠀⠀⠀⠀⠀⠀⠘⡄⢑⠊⠋⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⣄⣌⠜
⠀⠀⠀⠀⠀⠀⠀⠈⢎⢂⡀⠋⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣰⠂⠀⢸⠉⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣉⠱⡘⡦⡀⠀⢸⡀⠀⠀⠀⠀⠀⢰⡇⠀⡀⠿⣒⠪⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢉⣈⠑⠶⠸⠀⠀⠀⠀⠀⢘⠓⠋⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡖⢂⡄⠀⠑⠄⡸⠛⠉⠉⠟⠙⢅⠔⠉⠀⠒⠒⣺⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠁⠙⠦⠜⠀⢀⡀⠀⡤⠀⠈⠒⠱⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⢦⡄⣦⡀⢉⠀⠁⢐⣤⢤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠋⠃⡄⠇⢰⠚⠃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠄⠸⠁⠀⠈⠀⣄⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⡀⣻⣿⣿⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡛⠕⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠘⠂⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠸⡟⠲⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠸⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡞⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡘⢦⡀⠈⣷⠀⠀⠀⠀⢸⡄⠀⢀⡞⠁⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣦⣼⡇⠀⠀⠀⣸⣧⠞⠩⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⡏⣯⠹⣿⣶⣾⠇⢸⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣡⢹⣶⠂⠐⡞⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢉⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡈⠿⠏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣾⣧⢸⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⣿⣿⣼⣿⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣯⡿⠀⠛⠛⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

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

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡟⣇⠀⠀⠠⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⠄⠀⠀⣸⢻⡇
⠀⠀⠀⠀⠀⢻⡜⢆⠀⠀⢰⠃⠀⠀⡀⣀⣀⡀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢀⣀⣀⢀⠀⠀⠘⡆⠀⠀⡰⢣⡟
⠀⠀⠀⠀⠀⠀⠻⣄⠑⠤⠸⡀⠀⢀⡵⠋⠉⠉⠁⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠈⠉⠉⠙⢮⡀⠀⢀⠃⠤⠊⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠑⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⠱⡄⠀⠀⠀⠀⢠⠞⡇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠊⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠑⠦⢄⡀⠀⠀⠀⠈⡰⠁⠀⠀⠀⠀⠈⢆⠁⠀⠀⠀⢀⣠⠴⠊⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄⣈⠑⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠜⠀⠀⠐⠊⣁⣠⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠢⠄⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠠⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡙⠛⠓⠲⠄⠀⠀⠀⠐⠋⣷⣾⠙⠂⠀⠀⠀⠠⡖⠚⠛⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⣠⡀⠦⣤⡑⢦⠃⠘⡴⢊⣤⡴⢀⣄⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠷⢸⡇⢸⡇⠾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⠒⠛⠀⠀⠓⠒⡺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢶⡂⢀⡷⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠁⠈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⣄⣀⠰⣿⠀⢀⣷⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⠛⠾⣿⣇⠀⠀⠀⠀⠠⠀⠀⠈⣿⣄⡧⢰⣾⣿⣦⡄⢸⡇
⠀⠀⠀⠀⠀⠀⣀⠴⠞⠛⠁⠀⠀⠈⠙⢷⣄⠀⠐⡷⠤⠄⢀⣸⠦⠄⠀⡇⠀⠀⠀⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡇⠀⠀⠐⣻⠀⠈⠀⣿⠀⢀⢌⣼⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⢻⠀⠀⣣⡟⠀⣨⣿⠏⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣼⣦⣾⠿⠷⣺⣿⣷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡝⠋⠑⠒⠻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⡀⠀⠛⠛⠳⣿⡦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⡄⠀⢤⣀⡤⠞⢁⣈⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠀⠀⠀⠠⣤⣾⠝⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠘⣴⠺⣿⠉⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢰⠀⠀⠈⣿⠀⠸⠉⣿⣿⡒⠒⠖⠚⠉⠉⠉⠉⠉⠐⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠛⠀⠀⠀⣿⠀⠀⠀⠘⠉⠁⢤⡀⠀⠀⠀⠀⠀⢠⣤⠀⢾⣗⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠄⠀⠀⣴⠁⢀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣾⣿⠀⠈⠹⣷⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⢀⡾⢁⣴⡏⠀⠀⠀⠀⠀⡀⣠⣄⠐⠄⠀⠀⠈⠈⣆⠀⠀⠛⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡿⢠⣿⡿⠀⠀⠀⠀⠀⢠⣿⠟⢩⡄⠀⠀⠀⠀⠀⣿⣀⣰⡶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠸⡿⠁⠀⠀⠀⣰⡇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠘⣿⠀⢰⡏⠀⠉⢻⠏⣀⣀⡀⠀⠀⠀⢀⣠⣾⣿⣿⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠻⣾⣿⠷⣾⠀⣦⢰⠿⠿⠿⠿⠷⠶⢶⣾⣿⠿⣿⢻⣿⣧⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡿⠅⠀⢹⠀⣇⡄⠀⠀⠀⠀⠀⠀⠀⠻⣿⡆⢹⡀⠻⣿⣆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢇⡆⠀⢸⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣼⣷⣄⠙⣿⣎⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⠀⢸⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠆⠈⣿⡆⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⢹⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⣼⡆⠇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠗⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠐⠿⠷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⡆⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡴⠎
⠀⠀⠀⠀⠀⠀⠸⣮⡄⣆⠀⡀⠀⠀⢀⠄⢰⠀⠀⣘⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣶⣷⡄⠀⣸⣤⡾⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⡟⢸⡯⡃⢯⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⢉⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣮⣟⣿⣇⣤⣤⣴⣔⡲⣶⠀⡀⠀⠀⢀⣴⢐⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⣿⣿⣯⣿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠉⣾⠁⢰⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⢀⣹⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀⡟⠀⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⣿⣿⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⣽⣇⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⣳⣦⣤⣠⣴⣯⣀⡀⣀⣀⣴⣾⣿⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⣼⣿⠉⣹⠿⠿⠿⠿⣿⣿⣿⠀⠙⣿⡇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣰⣿⠀⠀⠀⠀⠈⢻⡿⢧⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠐⣿⠃⠀⠀⠀⠀⠀⠀⣧⡟⠀⠀⠈⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣿⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⡀⠀⠀⠀⠀⣰⡏⠀⣀⣀⡀⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡿⣀⢀⢸⡇⠤⢤⠤⢾⡿⠶⠞⠛⢛⠛⣡⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⢋⣼⠷⣖⣒⡐⠊⠈⠛⠁⠶⠾⢏⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠦⣄⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠹⣷⠀⠀⠀⠀⠀⣸⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣼⣦⣶⣶⣶⣤⣿⡨⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠋⠉⠟⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⣾⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢅⢰⠷⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠁⢠⣤⣴⣾⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⢀⣴⣿⣿⡏⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⠛⠉⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢻⠀⣿⣿⣿⣿⣿⡿⢿⣿⡇⣾⣽⣿⣿⣿⡿⡟⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⡠⣿⣦⣀⠉⠛⠿⡿⡿⠿⠿⠿⠿⢻⣿⣿⣿⠃⠀⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡼⢋⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣷⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⡜⣠⣿⣟⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠈⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡸⠀⠺⣯⣽⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡇⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⢿⣿⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠈⢿⣾⡄⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢸⡜⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣷⡻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠹⣷⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⣠⠾⠁⠀⣠⠖⢰⡄⠀⢸⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⡞⠸⣄⣰⠞⠁⠀⣸⣇⢀⣼⣏⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣫⡴⠚⡄⠀⣴⠀⣠⠏⣙⣋⣤⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢿⣤⢾⡇⢸⠷⢞⡷⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠈⠳⣤⣈⠳⣼⣇⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣉⣻⣶⣟⠁⠀⠉⠃⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠘⠛⠃⠲⢤⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠈⣁⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⡆⠘⠶⣶⡶⠚⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⢸⠃⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡾⠋⠀⠀⠀⣾⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠖⠓⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⡆⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⣰⡟⢁⡤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢰⡇⢰⡟
⠀⠀⠀⠀⠀⠀⢀⣿⢠⡞⠁⢠⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⡘⣰⠟
⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀⠀⢸⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⡼⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠙⠃⠹⣜⣧⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠓⠲⢤⣀⣀⣀⣀⣀⣀⣠⣼⡇⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣸⠃⣸⠟⣿⡄⠀⢀⣤⠀⠀⢸⡏⠉⠓⠲⠶⠤⣤⣭⣭⢭⡭⠔⢶⡇⠀⠀⠰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⢇⡾⠁⣰⠏⠀⢹⡇⢠⠏⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⢀⡿⡾⠀⣸⠃⠀⠀⣼⢁⡟⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⡼⢻⠀⣰⠃⠀⠀⡴⢃⡞⡀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⢸⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢼⠃⠙⣾⠃⠀⠀⣾⠁⠈⠛⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⡏⢸⠀⢀⡿
⠀⠀⠀⠀⠀⣾⠀⢸⡇⠀⠀⠀⠘⣷⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠃⠈⣇⢸⡇
⠀⠀⠀⠀⠀⢿⠀⣾⠁⠀⠀⠀⠀⠘⣧⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣽⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠘⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢹⠰⡇
⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠸⣆⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⠀⠀⢸⡄⣇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⣼⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⡇⢻⡄
⠀⠀⠀⠀⢠⣿⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⢿⡄⠀⢸⣷⡎⢷⡀
⠀⠀⠀⠀⠈⢹⣧⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣧⣬⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢺⣿⣆⠀⢩⡿⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁

⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡈⠀⠀⠈⠓⣄⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⢇⠀⠀⠀⣰⡿⠛⠛⠳⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣀⣿⢦⣤⡖⠀⠀⠀⡄
⠀⠀⠈⢳⣦⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠻⣧⠀⠀⢀⣹⡀
⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆⣸⡟⠁
⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⡾⠋
⠀⠀⠀⢰⠀⢈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⡆⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠈⡆
⠀⠀⠐⣷⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢸
⠀⠀⠀⠻⣷⣦⣤⣙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠉⠙⣿⠿⣿⣷⣤⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠤⣧⣿⣷⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣉⣛⣿⣷⣤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⣐⣤⠾⢟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠛⠛⠿⣿⡷⣦⡀⠀⠀⠀⠀⠀⢀⣴⣆⣺⡿⠿⠷⠶⢟⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣈⠐⢦⣄⢿⣿⣿⣿⣦⣤⣀⣤⣴⣾⣝⣩⡿⢀⡤⠒⠚⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⣼⣿⣾⣿⣷⣴⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣶⣿⣧⣴⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢏⣿⣿⣿⣿⡿⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡯⠉⠁⠹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣄⢠⡠⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣇⣿⠉⣿⣻⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⣿⣿⣩⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⣌⣣⣿⢤⠀⢴⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡃
⠀⠀⠀⠀⠀⠀⠞⣮⣿⣷⣟⣿⠁⠀⠛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣼⣿⣿⣶⣾⣿⣟⢷⡶⢺⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠨⣹⣿⣿⣿⣿⣿⣿⣉⣽⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠠⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢹⡖⡃⣁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠠⠦⠵⠾⠿⠿⣿⢿⡿⡿⣿⠿⣷⠶⢻⠓⠇⠙⠈⠿⠿⠿⢿⣿⣿⢿⣿⠿⢿⡿⢿⣿⠿⠿⣷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⣀⣀⣠⡀⠀⠀⠀⠀⢀⣾⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣾⡶⢀⣸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡐⢤⣼⣿⣿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣈⣿⣿⡷⣼⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡾⡿⠯⠹⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡛⢠⣷⣀⡠⠤⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣿⣿⣿⣷⣤⣾⠻⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠐⠶⠒⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣯⣶⣴⣄⣠⣀⣀⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣦⢻⡇⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣡⣾⡟⠁⠙⢽⣿⣿⣿⡿⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣯⣾⠋⣿⣿⣿⣿⣿⣿⣧⡁⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⠀⠀⢺⣿⣾⣿⡧⠀⠀⠀⠀⠰⣍⣿⣿⣿⣿⣿⣿⡟⠂⠹⣿⣿⣿⣿⣿⢿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡦⠀⢹⣿⣿⣿⣿⣶⣶⡀⣠⡴⣽⣿⣿⣿⣿⣿⣿⣧⢻⣄⠻⣿⣿⣿⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣷⢄⣾⣿⣿⣿⣿⣿⣬⣥⣙⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡯⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣤⣀⣤⣤⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠉⠀⠀⠀⠀⣿⣿⣿⠿⠋⠉⠉⠀⠀⠀⢻⣿⣤⣤⠀⠁⢬⣥⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠉⠀⠀⠀⠀⠀⢠⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠃⠀⠀⠀⠀⠀⠰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢠⡋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⡀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢿⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⣠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠆⠀⠀⠀⠀⠀⠈⢧⣰⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡆⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢈⡀⠀⠇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢻⡄⠀⠱⡄⠀⠀⠀⢀⠋⣼⡟⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⢸⣇⠀⠀⠁⢾⣤⣠⡜⢠⡟⠁⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠢⣽⣆⡀⠀⠈⠉⠀⠀⠀⠀⢈⡿⠁⠻⣿⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⡀⣀⣀⠐⠈⠀⠀⢤⣿⣤⠀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⢨⠌⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⣾⢠⡆⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠐⣿⡄⢷⡄⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⢿⣧⠸⣧⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢈⣿⡆⠻⣧⣌⢺⠇⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣉⣛⣧⣴⣿⣿⡿⣷⣄⣤⠀⠀⠀⣀⣰⡟⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⠇⠀⢿⣿⡟⢰⣾⠶⢿⣁⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢷⡀⣸⡟⠅⠀⠙⢿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⣀⠈⠠⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢡⣤⣤⡀⠀⢠⣀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠺⡿⠿⠯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣄⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⢠⡤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢿⣯⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡤⠀⢰⣄⡻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣄⢀⡀⠉⣑⣾⣶⣶⣮⣷⡦⠤⢍⣩⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣁⣈⣈⣁⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠀⡈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⢰⡖
⠀⠀⠀⠀⡇⠀⢠⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠈⠓
⠀⠀⠀⠀⠈⠁⠀⠀⣀⡤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠱⢤⣀⠀⠀⠃
⠀⠀⠀⡀⣀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⠙⣿⢷⣶⡄
⠀⠀⢰⣿⠏⢙⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⢀⠀⠀⢉⣴⣿⠟
⠀⠀⠙⠻⠶⣤⣄⣀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣠⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⣷⣶⣶⣦⣧⣐⣀⣠⡆⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡶⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⡉⢤⣌⠙⣿⣯⣍⡰⠻⢶⣤⣀⠀⠀⠀⠀⣀⣤⡛⠺⠟⢒⣶⣶⣶⠶⠞⣛⠛⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠷⠀⠈⣯⣟⣷⣦⣾⣿⣿⣷⣾⣦⣞⣿⣿⣤⢁⣴⡟⣩⠋⠁⠀⡈⠉⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠾⠊⣽⣿⣿⢏⢩⣌⡛⠿⠿⠛⣹⣿⣿⡟⠛⠇⢠⠴⠞⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠛⢷⣾⣿⡦⠶⢖⠰⣾⣿⣿⡿⠿⢀⣴⠃⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣠⣷⣤⣶⣿⣻⡿⣿⣏⣴⣷⠈⠉⠠⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣏⠉⠀⠰⢿⣿⡭⠀⢱⠛⢿⡄⠉⠹⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣯⠞⣿⠋⣀⣐⣛⣷⣄⣻⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡟⣹⣿⣿⣦⡏⣼⡟⣻⣏⣿⡿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢧⠚⠿⣿⣦⣤⣿⠥⠛⠛⠿⢏⣼⠟⣿⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡟⠀⠀⠙⠻⠟⢿⣶⣶⣦⣤⡾⠃⠀⣿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠐⠀⠈⠂⠀⢬⡴⠋⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔

⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⡷⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠄⠀⠀⠀⠀⢼⡄
⠀⠀⠀⡿⢡⠀⠀⠀⠠⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠄⠀⠀⠀⡌⢿
⠀⢀⡈⠀⠃⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠘⠀⢃⡀
⠀⡘⢡⡞⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⢳⡄⠃
⢠⠀⢸⠁⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⡇⠈
⡀⠀⢸⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⡇
⣷⠀⢸⡀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢸⡅⠀⠀⢀⡇
⣧⣰⠘⡇⠀⠀⠀⠀⠀⢼⠆⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠠⡣⠀⠀⠀⠀⠀⠀⠸⡧⠀⠀⠁⠀⠀⢸⢃⣆
⢸⣿⣧⠀⠆⣠⣿⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣆⠰⠀⣸⣿
⠀⠻⣯⠂⠀⠹⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⣿⠏⠀⠀⣹⠟
⠀⠀⠈⠳⣄⡀⠈⠠⣿⣷⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠁⢀⣠⠞⠉
⠀⠀⠀⠀⠈⠻⣶⣦⣈⠻⣆⠀⠀⠀⠀⠸⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⡞⠇⠀⠀⠀⠀⣰⠟⢁⣤⣶⠟⠉
⠀⠀⠀⢠⣶⣶⣤⣽⣿⣷⣦⣁⠀⠒⢐⣀⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⣀⠀⠂⠀⣀⣴⣾⣿⣯⣤⣶⣶⡄
⠀⠀⠀⠸⣇⠀⠀⠀⠨⣉⠛⢿⣿⣶⣿⣿⡿⠿⠟⠞⠻⠟⠻⠟⠶⠻⠿⢷⣿⣿⣶⣿⡿⠛⣉⠁⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠹⣆⠀⠀⠀⢈⣷⠀⠸⢿⣿⣿⠞⠿⣿⣿⣿⣷⣾⣿⣿⣿⠿⠳⣿⣿⡿⠇⠀⣾⡁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠘⢧⣀⠐⠎⠻⠗⠀⢸⡟⣵⣶⣄⢉⣿⣿⣿⣿⣿⣿⡉⣠⣶⣮⢿⡇⠀⠺⠟⠱⠂⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣠⣿⣯⡿⠛⠉⢹⣿⠿⠛⠛⠿⣿⡇⠉⠛⢿⣿⣿⣄⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣶⣠⣾⠁⣠⡾⢷⣄⠈⣷⣄⣴⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠙⠉⣹⡟⣻⣶⣶⣿⢻⡏⠉⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣴⣿⣧⢻⣧⣿⣿⣿⣿⣼⡟⣼⣷⣶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣿⣿⣿⣦⣿⣤⣀⣀⣤⣿⣾⣿⣿⣿⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⡟⠙⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⣿⣿⡏⠀⠀⠛⠛⠁⠉⠛⠛⠀⠀⢸⣿⣿⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣆⡀⢀⣠⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣠⣄⣬⣁⣤⣠⣄⣀⣤⣤⣀⣤⣤⣠⣄⣠⣤⣠⣤⣤⣤⣤⣤⣤
⣿⣛⣟⣻⣛⣻⣿⣛⣛⣿⡛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣻⣿⣿⣟⣛⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠈⣿⣼⣿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣷⣼⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⢹⣆⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢀⣾⠁⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢯⡻⢷⣤⣀⠻⣶⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡿⢃⣠⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣝⡛⠿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠮⣍⡻⣿⣿⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣭⣻⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡶⢶⣶⣦⣤⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣥⣤⣶⣶⠶⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣦⣬⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣩⣤⣶⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠘⣿⣿⣿⣿⣿⣿⣿⡿⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠸⣿⣿⣿⣿⣿⣿⠁⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⣈⣙⠛⢛⣉⣡⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⠀⣀⠀⠀⠀⠀⢀⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⣼⣵⠋⣭⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣾⡿⠿⢛⣻⣉⣛⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⣼⠳⣟⢛⣾⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣇⡈⣁⡭⡘⠿⢽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⢤⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣉⣉⡛⠻⠎⠺⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣰⢏⠁⠒⠆⠈⠃⣀⣁⣀⠠⠄⠼⠅⠀⢬⣗⡒⢛⡊⢹⣯⣼⣿⠙⣻⣿⣿⣅⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣰⢂⡀⠋⣆⠸⠃⠀⠐⠀⠀⢯⠀⠙⠆⠀⠐⣒⠚⠾⣭⣀⢘⣇⣉⣿⡶⣻⠜⡿⣿⢿⣿⣿⣿⣾⣏⣿⣿⠇
⠀⠀⠀⠀⠀⣸⡿⣈⡁⠀⠈⠀⠀⠈⡀⠈⢓⡈⠃⠤⣀⠀⠀⡛⠀⠈⣩⡥⠶⣏⡙⢐⣏⣁⠁⠈⠛⢾⣿⣿⣏⣿⣿⣿⡟
⠀⠀⠀⠀⢠⣿⡇⠹⣧⠀⠂⠀⢀⠀⠁⠀⠀⠳⠄⠴⡗⠂⠛⠂⠀⠘⢀⠐⠒⠉⠈⠉⠀⠚⠐⣷⠀⠃⠹⣿⣿⣻⣿⡿⠁
⠀⠀⠀⠀⢰⣿⠇⠀⠙⢷⣄⠀⠘⠁⠉⠠⡄⠀⣴⣤⡄⠀⣠⡒⠈⠶⠠⣿⠭⢹⣦⣠⢰⣦⡀⢸⣦⡀⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠀⡼⠀⢊⠻⣆⠀⠀⠘⢀⣈⣥⣿⣧⣌⡠⢽⣿⣿⠽⠶⠼⣷⣶⣽⣷⡾⢗⣛⡟⣿⣿⣠⡿⣿⡟⠁
⠀⠀⠀⠀⠀⠓⠊⠀⠀⠈⠀⠀⠚⠤⠄⣰⡿⠛⢫⣿⣷⣷⣾⣛⣻⣿⡙⠋⠾⠿⢿⡇⣿⡌⠉⢁⣿⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡓⢁⠀⠀⠀⢂⠎⠀⠀⠈⠉⠙⠛⠳⠶⠶⠶⠶⠶⠶⠚⠛⠓⠛⢧⡘⢿⣿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⢀⠂⠉⠉⢄⠁⢀⡾⣄⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘
⠀⠀⠀⠀⠀⢈⡤⢄⡴⠃⠀⠀⠸⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢧⡙⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣤⡛⠁⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⢷⡈⡇⠀⠀⠀⠀⡰⠊⠉⠀⡆⠀⢠⠾⠛⠁
⠀⠀⠀⠀⠀⠦⣨⠁⠀⠀⠀⠀⠀⠑⠲⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠈⠋⠉⠉⠉⠁⠀⠀⠀⠀⢰⡁⠀⠁
⠀⠀⠀⠀⠐⡀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡤⠜⠉⠓⠤
⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠿⠂⠂⠚⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠀⠀⠿
⠀⠀⠀⠠⠟⠁⠀⠀⠀⠀⠀⠔⠒⠀⠀⠈⠀⠀⢀⠖⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣀⠾⠿⠄⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⢸⡄⠀⢠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠐⠦⠴⠞⠋⠉⠀⠀⠀⠁

⠀⠀⠀⠀⢢⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠨⠳⣼⡀⣶⠋⠀⠀⠀⠁⠀⠀⠀⢠⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⢱⡄⢘⠀⢰⡄
⠀⠀⠀⠀⠀⢣⣘⠇⠘⢦⡀⠀⠸⣆⠀⠀⠀⢸⡇⠀⠀⢠⠇⠀⠀⠀⢀⣠⠞⣷⡧⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠿⢶⣦⣽⣆⠀⠀⢠⣷⣀⣴⣏⣀⣠⣤⣶⣿⡥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣮⣿⣷⣾⣿⠛⡛⠛⣻⡟⠉⢉⠀⠀⢀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡞⠣⠟⣿⣿⣿⣿⣧⣼⠿⠃⠀⠀⠱⣄⠈⠳⡄⠀⠁⠀⠀⠀⠀⠀⠀⣼⠀⣠⢀⠀⠂⢀⡀⢀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢻⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠂⣠⣠⣦⠴⠰⠀⠀⠠⠆⢀⣯⠀⣀⣨⣀⠂⢀⠤⠀⢠⠟⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣤⡔⠛⠀⠩⠴⠆⠀⢀⠛⢺⣽⣿⢼⣿⠿⠖⠀⠈⢀⠀⠀⢰⣏⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⠀⠘⠁⠣⣀⢀⠶⠖⠁⠈⠀⠀⣩⣿⡟⣻⣿⣶⠀⠀⠊⡼⠀⢠⢍⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣄⠀⠿⠖⠀⠀⠀⢠⢿⣿⣿⡿⣿⣷⡀⠀⡟⣰⠋⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⢩⠳⠈⣤⣄⣤⣲⣙⣿⣿⣿⣿⣪⣿⡅⣀⡴⠃⡠⠄⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⡀⠀⠀⠀⢡⡈⠻⣿⣿⣿⠿⢿⣿⣿⣻⢿⢿⡿⣁⡀⠊⠁⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠉⠀⡈⠷⣿⡗⢿⣿⣿⡙⠪⣠⢣⣽⣟⣀⠘⠀⠤⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠘⠓⣷⣾⣿⣿⣧⠜⢃⣿⣿⣿⣿⣧⠄⣤⡿
⠀⠀⠀⠀⠀⠁⣀⠀⠀⣀⠄⠀⠀⢿⣩⣍⣈⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣮⣿⣿⣿⣿⣿⣿⠀⠀⠀⠹⣿⠷⠉⣽⢆⣎⣸⣿⣿⣿⣿⣶⣿⠇
⠀⠀⠀⠀⠀⠀⢡⠄⡉⠁⠀⠀⡀⠈⢷⣿⣧⠆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣤⠶⠂⣀⣻⡏⣼⢶⣿⣿⣿⠿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠐⠂⠂⠠⡀⠐⠀⠀⠀⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠠⠤⣬⢛⣠⣿⠉⡃⣿⢺⢿⣟⣸⠛⢿⡗⠃⠀⠀⢀
⠀⠀⠀⠀⢄⠀⠀⠐⠐⢂⡄⡶⠀⠀⠀⠘⣿⣿⡏⣿⣿⣿⠛⠿⠿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣏⠀⡔⢁⡙⢋⣤⣒⣻⢻⣼⠞⣳⣽⣞⣏⠁⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠞⠐⡕⠲⡀⠀⢹⣿⣇⣽⣿⡇⠀⠀⠀⠈⢿⣿⣧⠀⠀⠈⠙⣿⣿⣿⣆⠒⠈⢋⢼⠑⣿⣿⣯⣿⣿⡷⣛⡿⠀⠲⡒⠂
⠀⠀⠀⠀⠀⠈⠁⠀⠐⢀⣠⣌⣣⣴⣠⡄⠈⣿⣿⢺⣿⠀⠀⢀⣀⡀⠀⠹⣿⣷⡄⠀⠄⡄⠉⠛⢿⣷⣶⡄⠈⣰⣿⣿⣿⣿⣻⣿⣁⣴⠀⢀⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢄⣀⠀⠀⠈⠠⠀⠈⠗⠀⠸⣿⣾⡇⡀⢰⣬⣟⢃⣄⢠⣿⠏⠀⢀⣤⣷⣶⣯⣠⡌⢿⣿⠶⣿⣿⣿⣿⣿⣿⣿⣋⣉⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠠⠄⠀⠘⠥⠀⠀⢀⡀⢠⡿⢿⣦⣯⢻⣇⠀⠀⠱⡏⢻⠛⣿⣷⣦⣶⣾⣿⣿⣿⣿⣿⢿⣴⢻⣶⣿⣿⣿⣿⣿⠻⡛⠻⣿⣋⠣⠤⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡈⢛⣶⣄⡦⡀⠂⠀⣿⢈⣿⡆⠠⡤⢷⣇⡔⠈⠷⢿⠿⢿⣿⠿⣿⡿⣿⠸⠁⢈⣿⣾⠿⠿⠿⢷⡾⠯⡀⠀⠀⠂⠀⡀⠨⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠧⣤⠌⠀⠀⣀⣸⠟⠁⢸⣷⣾⠕⡼⢿⡀⣀⣠⢰⣠⡄⣠⠀⠬⠩⢠⡶⠞⠛⠛⣓⠂⠰⠦⠀⠂⠀⠒⠀⢀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠐⣠⣿⠂⠉⠠⣠⠠⠐⢀⠀⢀⣬⣿⣿⣿⣿⣾⣷⣴⣿⣷⡴⠊⠉⣠⣶⡆⠓⠁⡀⠀⠀⠀⠸⡁⠲
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠁⠤⠁⣤⠂⠈⢹⣶⣤⢂⠀⡉⢉⡉⠛⠛⢿⣿⣿⣿⣋⡤⠎⣼⡿⠋⠀⠀⠘⠇⠐⡄⠀⠋⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠗⠊⣏⠀⠾⡆⠋⠀⠒⠘⠉⠉⠉⣸⠧⠀⠀⠀⠀⠘⡀⠂⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠘⠁⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡆⣰⠏⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⠎⠰⡀⠀⠀⡼⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠙⣤⡼⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣄⢀⣼⣤⠴⠚⠀⢸⡇⠀⠀⠀⡼⠁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣈⠻⢧⡀⢀⠀⠀⠸⣧⣀⣠⣾⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡙⣦⠙⣷⣄⠀⠿⣩⠭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠏⠘⠁⢠⣤⠠⡡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢢⠀⠀⠀⠁⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣌⢷⣤⠄⣀⣠⠤⢾⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠿⠖⠀⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⡄⠀⠙⡗⠦⠤⣤⣤⣤⣤⣤⣤⣤⣦⠶⠶⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣤⣾⢁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢠⣤⡀⠀⠀⠈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣈⢿⣿⡏⡇⣼⢷⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⠀⢻⣿⡀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⣿⣇⣿⣿⡘⠷⠀⠀⠀⢸⡄⡀⠀⠀⠀⢸⡿⢠⣿⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣧⠀⣷⠀⠀⣿⣹⢷⠀⠀⢀⡾⢁⣾⣿⣿⣇⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⣿⠀⠀⠿⠋⠀⣀⣠⣭⣶⢿⣟⣉⢿⣿⣦⡀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡻⣿⡟⣿⣿⢀⠼⠿⠿⠿⠛⠋⠉⠺⣿⣿⣿⣿⣿⠻⣿⣦⡀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⡇⢸⡏⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣟⢦⡀⠙⠻⣷⡆⢳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠡⢸⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣹⠆⠀⠈⠙⢿⣦⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢰⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠏⠀⠀⠀⠀⠀⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢨⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠳⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⢻⣿⣿⣿⣿⣿⣿⣾⣏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣽⣿⣿⣿⣿⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣦⢀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡀⠀⠀⠀⠀⣀⠸⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄⠀⠀⣸⡆⢙⣧⣿⣿⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢶⣶⣿⣷⣿⣿⠿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡄⠀⠀⠀⣿⣿⣷⣾⣿⣿⣿⣿⠏⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣀⣴⣿⣫⣾⣿⣿⡅⠀⠼⠀⢀⡤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣻⣿⣿⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⢻⠛⢿⣿⣿⣿⡿⣿⣿⣿⡇⣺⣦⣀⣁⣤⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡟⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣤⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠼⢿⣻⣿⣿⣿⣦⡏⠉⣽⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⠏⠀⣸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⠙⠛⣸⣿⡛⢻⣿⣿⣿⢿⣿⣿⡖⠀⢿⣿⣿⣿⣇⣿⠃⢿⢿⣿⣿⡏
⠀⠀⠀⠀⠀⢀⡙⠉⠁⠣⠀⠙⠿⣿⡁⢀⡤⠻⡇⠀⠉⠉⠁⠠⡀⠘⠛⣿⣿⢟⠄⠚⣟⣿⣿⠃
⠀⠀⠀⠀⠐⠉⠀⠁⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⡻⠋⠉⠀⠀⠹⠟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠐⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠈⠀⠀⣤⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⣸⡟⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠘⣷
⣿⡇⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠠⡇⠀⢹
⢿⣷⡀⠀⠀⣠⣿⠟⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⣿⠀⣸
⠘⣿⣿⣶⣾⣿⠋⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⣿⣤⣿
⠀⠘⣿⣿⣿⠃⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣶⣿⣿⠁
⠀⠀⢸⣿⡟⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆
⠀⠀⠸⣿⣇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢹⣿⣿⣿⠸⣿
⠀⠀⠀⢻⣿⡄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠀⢻
⠀⠀⠀⠀⠻⣇⣿⣧⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡄⠀⠀⣼⣿⣿⠇⠀⣿
⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⠀⠀⣰⡿⠟⢋⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣾⠏⠀⣀⣡⣴⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣶⣄⣻⣷⣄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢀⣴⣿⣵⣾⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠙⠻⢿⣿⣿⣿⣿⣿⡦⣀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣷⣿⣿⠿⠛⢉⣤⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠷⣄⡀⠈⠙⠻⣿⣿⣿⣦⡙⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⢁⣤⠾⠛⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠈⠑⠦⣄⠀⠀⠉⠛⢿⣿⣾⡷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠀⠐⠛⠁⢀⡤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⢢⣄⠀⠈⠀⢲⡄⠀⠀⢻⣿⣧⣈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠀⠁⠸⠿⡏⢀⡆⠀⠀⣴⠏⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠙⣿⣆⠀⠀⢻⡆⠀⠠⠟⠉⠁⠀⣤⣤⠀⣤⣴⣦⠀⠈⣳⣄⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⡀⠀⠀⠈⠉⡀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠻⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⢀⣾⡟⠀⢰⣾⠟⠂⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠉⠃⠀⢹⣿⣿⣿⣿⠛⠛⢿⣿⣿⣿⣿⠀⠀⣈⡁⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⢳⣴⣯⣀⣤⣦⣄⠀⠀⠀⢈⣿⣿⡿⠁⠀⠀⠀⠈⢿⣿⣿⠀⣾⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠸⣟⣿⠿⠇⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣆⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣷⣿⣿⣾⣿⣦⠜⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⢰⣶⣷⣶⠀⠀⡤⠀⠀⣀⠀⣸⣿⣿⣿⣿⣿⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠸⠟⠛⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠀⠀⠀⠀⣶⡄⠀⠀⣀⢀⡤⠀⠘⠻⡟⠛⠛⠛⠃⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣸⣄⠓⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⣀⣀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⣷⣆⣿⣿⣷⣦⣤⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣿⣿⡿⢣⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠉⠀⠉⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠂⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠑⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⣿⠆⠁⠀⠀⠀⠀⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢂⠀⠀⠀⠀⠘⠰⣷⡈
⠀⠀⠀⠈⣴⠋⡌⠀⠀⠀⠀⠀⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠄⠀⠀⠀⠀⢣⠹⣧
⠀⠀⠐⠀⠋⢸⠁⠀⠀⠀⠠⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠄⠀⠀⠀⠘⡇⠙⠠⠂
⠀⠀⣧⠀⢀⠀⠀⠀⠀⠀⠃⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠐⠀⠀⠀⠀⠁⡀⠀⣴
⠀⢠⠃⢠⡟⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⡀⠀⠀⠀⢻⡄⠙⡀
⠀⠈⠀⢸⠁⠀⠀⠀⡄⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢁⠀⠀⠀⠘⡇⠀⢁
⠈⠁⠀⣿⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⣿⠀⠘
⣆⠀⠀⣿⠀⠀⢰⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⡆⠀⠀⣿
⣿⠂⠀⣿⠀⠀⠀⠀⢨⡏⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢹⠃⠀⠀⠀⠀⣿⠀⠸
⣿⠀⣦⠘⡆⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠆⠀⠀⠀⠀⠀⠀⠀⠀⢼⡤⠀⠀⠀⠀⠀⢰⠀⢸⠃⣴
⢿⣿⣿⡄⠱⠀⡇⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣦⠀⠈⠀⠏⢠⣿⣿
⠀⢿⣿⣷⡀⠈⢱⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⣷⡄⠀⠀⣾⣿⡿
⠀⠘⢿⣆⠁⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠈⣴⡿⠃
⠀⠀⠀⠙⢷⣄⠀⠀⠈⠡⣄⣿⣶⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⠀⡄⠀⠀⠀⠀⠀⠀⠀⣶⣿⣠⠌⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⢿⣶⣦⣀⠈⠻⣿⡄⠀⠀⠀⠀⠀⠀⠸⢾⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡷⠇⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⣀⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⡈⠛⠄⣀⠀⠀⠀⠀⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⠀⠀⠀⣀⠠⠛⢁⣴⣾⣿⠟⠁
⠀⠀⠀⠀⢰⣿⣿⣷⣶⣶⣿⣿⣿⣿⣦⣀⠀⠉⠀⣉⡄⣸⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⣇⣠⡁⠀⠉⠁⣀⣴⣿⣿⣿⣿⣶⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢌⡉⠻⢿⣿⣷⣶⣿⣿⣿⣬⣿⣶⠷⠶⠶⢾⠿⠿⡷⠶⠶⠾⣶⣿⣡⣿⣿⣾⣶⣾⣿⡿⠛⢉⡡⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠰⣦⡀⠈⢻⣏⣻⣿⣿⣡⣶⣶⣶⣷⣶⣶⣄⣠⣶⣶⣾⣶⣶⣶⣬⣿⣿⣟⣹⡟⠁⢀⣴⠌⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⢰⣿⣄⡀⠘⠛⣿⣿⡟⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⢻⣿⣿⠛⠁⢀⣬⣿⡄⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠘⠇⠘⠿⠏⠀⢰⣿⠋⣿⣿⢶⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⡶⣶⣿⠙⣿⡆⠀⠹⠿⠃⠼⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⢀⣀⣾⣿⣇⡿⠿⠚⠋⢸⣿⣿⡿⠛⠛⢿⣿⣿⡇⠘⠓⠿⢿⣴⣿⣷⡀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⣧⣤⠀⠀⣼⠟⠁⢀⣤⣤⡀⠈⠻⣧⠀⠀⣤⣼⣿⣿⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣶⣾⣏⣄⣴⡟⠛⠛⢻⣦⣠⣿⣷⣶⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠈⣁⠀⣿⡝⣯⣿⣿⣿⣿⣽⢻⣿⠀⣈⠁⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣰⣾⣿⡄⢿⣇⣽⣿⣿⣿⣿⣧⣸⡿⢠⣿⣷⣆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⣿⣿⣿⣌⣿⡉⠀⠀⠀⠀⢉⣿⣡⣿⣿⣿⣿⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⣿⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢻⣿⣿⣿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⣿⣿⠁⠀⠈⣿⣿⠿⠛⠛⠿⣿⣿⠁⠀⠈⣿⣿⣿⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⢰⠁⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡀⣸⠃
⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣸⣿⠟⠁
⠀⠀⠀⠀⠀⠈⢻⣄⣶⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⣀⣼⣿⣥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣠⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⡶⢶⣼⣷⡀⠀⠀⠀⠀⣠⣿⣷⡾⠿⣛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣤⣩⠁⣴⣶⣶⡦⠈⢉⣀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⢮⣿⣿⣧⡄⠸⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡌⣹⣿⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠝⠻⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣰⣔⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⣴⣶⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⢿⣿⣿⣿⣝⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣯⣻⣞⣿⣿⡄⣻⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⡀⣼⣿⣠⣴⠟⠁⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠙⢷⣄⣹⣿⣀⡀
⠀⠀⠀⠀⣼⡟⣾⣿⡿⠁⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⢻⣿⣿⠻⣷⡀
⠀⠀⠀⠀⣿⡇⣿⣿⠇⠀⠀⠀⡎⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⡄⠀⠀⠀⣿⣿⡇⣏⡅
⠀⠀⠀⠀⣿⣇⡵⣿⡀⠀⠀⡜⡼⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠻⡰⡀⠀⠀⣿⡧⣷⣿⡇
⠀⠶⣒⣚⠸⣿⡇⣿⡇⠀⢰⢧⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⢧⠀⢀⣿⡇⣿⡿⢘⣒⡶⠆
⠀⠀⠀⠙⣷⣾⣧⣿⣷⣠⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣸⣿⣷⣿⣴⡟⠁
⠀⠀⠀⠀⢡⢿⣿⣿⢷⣿⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠜⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢻⣷⢿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⠟⣡⣿⣧⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣇⠀⠀⠀⠀⠀⠀⠀⢀⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣰⣿⣦⡙⢿⠃
⠀⠀⠀⠀⠀⡎⣼⣿⣽⣿⣀⠙⠦⢄⣀⣀⣀⡀⠀⠀⢸⣼⡟⠀⠀⠀⠀⠀⠀⠀⠸⣿⣸⠀⠀⢀⣀⣀⣀⣀⠤⠚⢁⣽⣿⣿⣿⠈⡂
⠀⠀⠀⠀⠀⠹⣿⣿⣷⣭⣍⣙⣛⣳⣶⣶⣦⣭⢽⣲⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣶⣺⢭⣤⣶⣶⣖⣛⣛⣉⣭⣽⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠦⣨⡿⣧⣸⣄⠀⠀⠀⠀⠀⢀⣾⣸⡿⣧⡠⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⡿⡨⠓⠒⢲⠒⠒⠫⡺⣿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⠀⠀⠀⢸⠀⠀⠀⢼⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠄⠀⢸⠀⠀⠄⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡓⣄⠀⠀⣸⡀⠀⣀⠜⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠗⢬⣷⠀⠘⠀⣴⡏⠚⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⢸⢻⠀⢸⠀⢛⡛⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⠈⠇⢸⢠⠃⣰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸⢀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⣏⣸⣘⣷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⠛⠋⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢼⠄⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡠⠀⠀⠀⠀⢤⡀⢣⠀⠀⠀⠀⢀⠂⣠⠄⠀⠀⠀⠠⣀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡾⠀⠀⠀⠀⠀⠀⠸⣜⡆⠀⠀⠀⣼⣼⠁⠀⠀⠀⠀⠀⠘⣆⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⡇⡄⠀⡄⢿⠁⠀⠀⠀⠀⠀⠀⠀⢘⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡆⠀⢀⡄⠀⡄⠀⠀⣾⡴⠃⠀⠻⣼⣧⠀⠀⡄⠀⢠⠀⢀⣾⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠳⣼⡇⠀⣧⡴⠛⡉⠀⠀⠀⠀⠀⣙⡳⣦⣇⠀⣼⡴⢟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠯⣻⡽⢻⡿⡆⠗⠁⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⠛⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡹⣧⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣾⣆⠛⢿⣮⡁⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⢿⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣹⣿⣿⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⡿⣱⣿⡿⠋⠈⢿⡟⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢿⣿⠀⠙⣿⣿⣿⣿⣷
⠀⠀⠀⣾⡟⠀⢻⣿⠁⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢸⡟⠀⠀⠈⣿⡟⠈⢿⡇
⠀⠀⠀⢸⠀⠀⠈⣿⠀⠀⠀⢸⠁⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢀⡿⠀⠀⢸⡇
⠀⠀⠀⢸⡀⠀⠀⢸⡆⠀⠀⣸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣼⠃⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⢿⡄⠀⣿⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⣸⡟⠀⠀⠀⣸
⠀⠀⠀⠘⠃⠀⠀⠀⠈⠉⠀⣿⠀⠹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⡇⠀⠛⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⡏⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⣧⠀⣿⠃⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀⠀⠀⣿⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⠶⠛⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⣦⣄⣿⠁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⣿⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠸⣷⣤⣄⣀⡀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢀⣤⣾⣿⣥⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠋⠛⠻⠿⢧⣷⣄⠀⠀⠀⠀⠀⠀⢸⣇⣶⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⣿⣦⠀⠀⠀⠀⢀⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⠀⣷⡛⠙⣧⠀⠠⠈⠻⣿⢁⡠⠴⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣽⢟⠁⠀⠉⠀⠀⢰⣿⠏⣤⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠈⣿⠉⠀⠀⠀⣤⣤⣤⡀⠈⠁⣸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠞⠛⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢉⣉⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣦⣠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠷⠀⠀⠀⠀⠀⠀⢀⡀⢀⣿⣴⣤⣿⣿⣿⣿⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣿⣧⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠀⢻⣿⣷⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢻⣿⡏⠀⢈⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡇⠈⣿⡂⠀⠀⣿⣿⡿⠛⢿⣟⢰⣿⣯⢿⣿⡿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡁⠀⣙⠃⠀⠀⢻⣿⣾⣯⣽⣿⣾⣿⣿⣿⣾⣷⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢨⣥⡀⣀⣈⣽⣿⣿⡟⢩⣿⣿⣿⡿⠹⣿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣶⢤⣶⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡇⠀⠚⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢹⣿⣿⡇⣿⡟⠋⠃⠋⠹⢻⣿⠃⠨⠆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢼⣿⣿⡄⣽⣅⣀⠀⠀⠀⣈⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣷⣶⣤⣬⣽⣟⣿⣷⣮⣝⣿⣤⣤⣴⣂⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢻⣿⣿⣿⣿⣿⣻⠿⠿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⣮⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣩⡄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣟⢿⣦⣿⣴⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣀⣾⠛⢃⢿⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠄⡙⠛⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣴⡟⢰⡿⢹⡧⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⣿⣿⣿⣿⣧⣟⣫⣽⣤⣤⣤⡶⣦⡀⠀⠀⠀⠀⢀⣠⣾⡿⠉⣷⣿⢷⣄⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⠛⢿⣿⣷⣿⣿⣿⣿⣏⡙⠛⠃⠀⠀⢠⣶⣾⣿⣿⠟⠁⠀⢻⣷⣤⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡃⠀⠉⠉⠀⣀⠀⠀⠀⠛⣻⡿⠋⠀⠀⠀⢸⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠉⠹⣿⣷⠤⠟⠀⠘⠁⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⣻⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠒⠛⠫⡅⠀⠈⠀⠃⠠⠂⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣿⣷⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣸⣿⣯⣰⡿⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⡿⠉⢀⣿⣿⣿⣿⣶⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⣾⣧⡴⢿⣿⣿⡿⣿⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡄⢹⣿⣾⡿⠋⠀⠈⣿⣿⣶⣬⠁⢤⡷⡄
⠀⠀⠀⠀⠀⠀⢰⡿⠿⠉⠟⠀⠀⠀⠀⠈⣿⣿⣷⣦⠈⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣉⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢛⠓⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⢠⡀⠈⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣄⠀⠀⠀⢰⣿⣿⡏⠀⣾⠀⠀⠉⠙⠿⠇
⠀⠀⠀⠀⠀⠀⠀⡸⠋⢀⠸⣿⣇⣀⣿⣿⣿⣷⣀⣹⣄⣀⢀⣇
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣶⣦⣴⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠏⠁⠀⠀⢹⣿⣿⣿⣷⡆⠀⣴⡆⢀⡈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⣿⣷⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠋⢀⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⡾⠀⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠃⠀⠀⢹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠀⢻⡏⠀⢀⣴⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠈⠀⣰⡟⠁⠈⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⢰⣿⡇⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⢸⣿⣿⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣧⠘⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⡀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣷⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⠟⠟⠓⢄⣀⠀⠀⢀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣀⣉⠻⢷⣿⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢘⣦⣤⣼⣿⡱⢿⣶⣍⣉⣭⣤⡀⠀⢀⠀⠈⠙⠛⠛⢿⣿⠟⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡴⢿⡿⠛⠉⠀⢀⣰⣿⣿⣿⣤⠃⠀⣿⣷⣦⣀⠀⠀⠀⠁⠀⠀⢀⣤⣤⡀⠀⣴⠄⠆
⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣧⣄⠀⠹⣿⣿⣿⣶⣶⣤⣶⢾⣿⠛⢿⠿⠟⢻⣿⣶⣆⡔⢢⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣠⣼⣿⣿⣿⣟⣹⡇⣈⣿⣶⠈⠦⠀⠀⡁⠀⠉⠛⠣⠄
⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣷⡿⣿⢧⣠⡿⠀⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡟⠉⠙⠙⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠔⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠑⢄⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⣼⡆⠐⠀⠀⠀⠀⠀⠂⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠂⠀⠀⠀⠀⠐⠂⣾⡄
⠀⠀⠀⠈⣰⠟⣀⠃⠀⠀⠀⢀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⢀⠀⠀⠀⠀⢣⡘⣿⡀
⠀⠀⠀⠄⠟⢰⠇⠀⠀⠀⠀⠄⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⡄⠀⠀⠀⠈⢧⠘⠇⡄
⠀⠀⣦⠀⠀⠈⠀⠀⠀⠀⠄⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠄⠀⠀⠀⠘⢀⠀⢠⡀
⠀⢀⠟⢀⣾⠀⠀⠀⠀⠐⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠰⠀⠀⠀⠀⢹⣆⠘⠇
⠀⠘⠀⢸⠇⠀⠀⠀⠀⠁⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⠁⠀⠀⠀⠀⣿⠀⠘
⠀⠇⠀⣸⠇⠀⠀⢀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⠀⠀⠀⢿⡀
⡄⠀⠀⣹⠀⠀⠀⡄⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠈⡄⠀⠀⢼⡇
⣿⡄⠀⢹⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢸⡟⠀⠀⠀⠀⣼⡇
⣿⠀⣄⠸⡇⠀⡄⠀⠀⠃⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠈⠁⠀⠀⡀⢀⡿⢁⡀
⣿⣤⣿⡀⢻⡀⠇⠀⢀⡀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣀⠀⠀⠃⣸⠃⣸⣧
⠈⣿⣿⣇⠀⠡⢇⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⠄⠀⢠⣿⣿
⠀⠹⣿⡍⠂⠀⠈⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠚⣿⡿
⠀⠀⠙⢿⣦⡀⠀⠀⠙⠇⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄⢠⠟⠁⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠙⠿⣦⣀⡀⠀⠓⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣇⣴⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⣄⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡖⠃⠀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣤⡈⠻⣿⡀⠀⠀⠀⠀⠀⠀⠉⣿⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡏⠁⠀⠀⠀⠀⠀⠀⣸⠟⠉⣠⣴⣿⠿⠋
⠀⠀⠀⠀⢀⣤⣤⣤⣀⣙⣿⣿⣶⣤⣉⠐⠂⠀⠄⠠⠜⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠄⠀⠀⠀⠐⢊⣡⣴⣾⣿⣛⣁⣠⣤⣤⡀
⠀⠀⠀⠀⢸⣿⠟⠛⠛⠛⠛⠿⠿⣿⣿⣷⣦⣀⣠⣠⣾⡔⢿⣾⣠⣀⣀⣀⣠⣤⣤⣀⣀⣀⣠⣹⣾⠷⣶⣦⣠⣀⣠⣴⣿⣿⡿⠿⠟⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠑⠢⠀⠙⠻⣿⣿⣿⣿⣿⡿⠞⢛⣋⣉⣉⣉⠛⠉⠙⢋⣙⣉⣙⣛⠛⠾⣿⣿⣿⣿⣿⡿⠋⠁⠰⠆⠁⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠈⣻⡶⠀⠀⣿⣶⣿⣿⡷⠾⢿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠟⠷⣿⣿⣷⣾⡇⠀⠰⣾⠃⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⣐⠿⣷⣦⠀⠀⣸⣿⣿⣁⣀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⢀⣀⣹⣿⣿⡀⠀⢠⣶⡿⢗⡀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠙⠀⠙⠋⠀⠀⣿⣏⢸⣿⣟⣿⡄⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⣴⣏⣿⣿⢈⣿⣇⠀⠈⠛⠁⠘⢁⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⣤⣠⣤⣾⣿⣿⣾⠉⠉⠀⠀⣿⣿⠿⠋⠁⠈⠻⢿⣿⡇⠀⠈⠉⢻⣾⣿⣿⣤⣤⣤⡴⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠹⢿⣿⣿⣶⡆⠀⣰⡿⠁⢀⣴⣾⣶⣄⠀⠹⣧⡀⠀⣶⣾⣿⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⠿⠿⣷⣦⣿⣍⣉⣉⣽⣧⣶⡿⠿⢿⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣠⡄⢰⣿⠈⣾⣿⣿⣿⣿⡞⣼⣧⠀⣤⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣿⣿⣿⡈⣿⣦⣿⡿⠿⠿⣿⣧⣾⡟⣸⣿⣿⣧⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣿⣿⣿⣿⣷⣸⣿⣀⠀⠀⠀⢀⣸⣿⣰⣿⣿⣿⣿⣇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣾⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢻⣿⣿⣿⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⡏⠀⠀⠸⠿⠿⠟⠋⠛⠿⠿⠿⠀⠀⠈⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣀⣼⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠹⣄⡀⠀⢀⣿⣄⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⣨⣉⣡⡬⠭⠯⢍⣓⡒⠀⠀⠸⣤⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠤⠤⠠⠤⠶⣶⡶⣖⣀⡀⠀⢻⣿⣷⠄⠀⢀⣠⠈⠉⠉⠒⠒⠂
⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠘⠉⠛⠻⢆⣸⠏⢀⣀⣴⣾⣋
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢀⡄⠀⠀⢸⡟⠀⣿⣿⣿⠫
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠸⣧⠀⠀⠘⡟⠀⠘⣿⠯
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣆⠀⠀⠀⠙⣦⣦⣄⠡⡀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠠⣤⡈⠻⣷⢿⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠘⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⢙⠿⣦⣷⠀⠀⠀⠀⠀⠀⠀⢺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣿⣧⣦⣠⡀⠂⢺⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠃⠃⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢌⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⣎⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⢸⢀⠅⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⣾⣸⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠦⠀⣀⠘⠧⡀
⠀⠀⠀⠀⠀⡷⣎⡇⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢠⠀⣄⣀⡕⣦⣌⠢⡄
⠀⠀⠀⠀⠀⠃⡇⣤⢏⡀⣀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢬⡋⠀⢠⢩⡝⣾⡄
⠀⠀⠀⠀⢰⣀⠟⠑⢀⠭⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠁⠀⢸⡆⡁⠸⠁
⠀⠀⠀⠀⢸⣿⡶⠈⡀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⡀⡔⢊⡴⠋
⠀⠀⠀⠀⠈⢇⠛⢦⣁⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢴⢈⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠙⣾⢅⡠⢀⠀⡀⠀⠀⢰⢁⠀⠀⠀⠀⠀⢳⡀⣀⠴⣺⠞⠉⡠
⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠘⢶⣌⠲⢄⡺⡋⠀⠀⠀⠀⠀⢰⢻⢄⡞⠁⣠⢊⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣗⠀⠑⣄⡀⠹⣿⡂⢝⢗⣘⠀⠀⠀⠀⣸⣤⠙⣠⡾⠁⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠘⠉⠢⠸⣷⣎⣯⣮⣴⣴⣶⣿⣿⣶⣤⠟⠁⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣀⠀⠀⠀⠈⡙⢻⡛⣭⣿⣿⣿⣿⣿⣿⣷⢤⣀⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⠦⣰⢯⣍⢻⢿⣟⢿⣷⣿⣿⣯⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢆⣀⠈⠃⠉⢳⣌⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣏⡦⢘⡷⠋⢀⣾⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠯⣝⡛⠁⠀⢿⠟⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⣟⣟⣿⣏⣉⣧⣈⣿⣿⣤⣭⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⣤⡱⢞⣷⣛⠠⠥⠀⠀⠉⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡣⠙⡥⣞⢻⣷⢄⠀⠀⠀⠀⠈⢿⠿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣛⣹⢞⠠⠥⠏⠀⠌⢁⣳⡲⢵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡫⡽⠽⠠⡎⠐⠁⠂⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠟⢴⣭⡾⢞⣳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠆⢠⠆⣀⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⡟⣷⣃⠾⠿⢶⡖⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⡇⢸⠰⣿⣿⠿⠛⠛⢾⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢰⠀⡇⠘⢧⣿⠃⠀⠀⠀⠊⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⢸⡘⠀⢹⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠘⣤⣀⣈⣳⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⣽⣿⣉⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣿⣿⣿⣷⣦⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠻⠿⣿⣿⣿⣿⣿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⣤⣶⣶⣾⣀⣼⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⢿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀
⠀⠀⠀⡄⠀⠀⣴⡃⣀⣴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⣀⢘⣆⠀⠀⢠
⠀⠀⣸⡇⠀⠀⣹⡿⠋⠁⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠙⢿⣏⠀⠀⢸⣇
⠀⢰⣿⠁⢀⣼⠏⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠹⣧⡀⠈⣿⡆
⠀⠘⣿⣦⣾⠃⠀⢰⡿⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢿⡆⠀⠘⣷⣴⣿⠃
⠀⠀⢹⣿⡟⠀⠀⢸⣇⣴⠏⠀⢰⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡆⠀⠹⣦⣸⡇⠀⠀⢹⣿⡏
⠀⠀⠀⣿⡇⠀⣠⣴⠿⣧⡀⠀⣸⡇⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⢸⣇⠀⢀⣼⠿⣦⣄⠀⢸⣿
⠀⠀⠀⢻⣷⣼⡿⠋⠀⠹⣿⣿⠟⠀⠐⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠂⠀⠻⣿⣿⠏⠀⠘⢿⣧⣾⡟
⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠘⣿⣦⣀⣠⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣀⣀⣴⣿⠃⠀⠀⠀⢨⣿⡟
⠀⠀⠀⠀⠀⠹⢿⣶⣶⣶⣾⠿⠿⠿⠛⠛⠛⠛⠛⠻⢿⣷⣦⡀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢀⣴⣾⠿⠟⠛⠛⠛⠛⠛⠿⠿⠿⣶⣶⣶⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣾⣿⣿⣿⣿⣿⣿⣷⣾⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⣠⠾⠁⠀⣰⠆⣴⡄⠀⣹⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⡞⠺⣄⣴⠞⠁⠀⣸⣇⢀⣼⣏⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣫⡴⢚⡅⢀⣴⠀⣰⢏⣙⣋⡤⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣜⢿⣤⢾⠇⢸⠷⢚⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠈⠷⣤⣈⠳⣼⣆⢺⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣉⣻⣶⡏⠁⠀⠉⠃⠛⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠘⠛⠃⠲⢤⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠈⣁⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⡆⠘⠶⣶⡶⠚⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⢸⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡾⠋⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠖⠓⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⡆⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⡦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢸⠇⣸⡏
⠀⠀⠀⠀⠀⠀⢠⣿⢠⡞⠁⢠⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⡜⣰⠟
⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀⠀⢸⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⡞⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠙⠃⠹⣜⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠓⠲⢤⣄⣀⣀⣀⣀⣀⣠⣼⡇⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣸⠃⣸⠟⣿⡄⠀⢀⡤⠀⠀⢸⡏⠉⠓⠲⠶⠤⣤⣭⣭⢭⡥⠔⢶⡇⠀⠀⠰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⢇⡾⠁⣰⠏⠀⢹⡇⢠⠏⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⣿⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⢀⡿⡾⠀⣼⠃⠀⠀⣾⢁⡟⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡅⠀⣛⢛⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⡼⣹⠁⣰⠃⠀⠀⡴⢃⡞⡀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠀⣛⢙⠀⠀⣸⡃
⠀⠀⠀⠀⠀⣼⠃⠙⣾⠃⠀⠀⣾⠁⠈⠛⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢐⡏⢹⠁⢀⡿⠁
⠀⠀⠀⠀⠀⣾⠀⢸⡇⠀⠀⠀⠘⣷⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⠁⠘⡇⢸⡇
⠀⠀⠀⠀⠀⢿⠀⣾⠁⠀⠀⠀⠀⠘⣧⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠘⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢹⠸⡇
⠀⠀⠀⠀⠀⢸⣧⡇⠀⠀⠀⠀⠀⠀⠀⠹⣆⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⠀⠀⢸⡄⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣧⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⣼⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⠀⠀⢸⡇⢻⡄
⠀⠀⠀⠀⢠⣿⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⢿⡄⠀⢸⣷⡎⣧⡀
⠀⠀⠀⠀⠈⢹⣧⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣧⣬⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢾⣿⡆⠀⢩⡿⣾⣷⡄
⠀⠀⠀⠀⠀⠈⠋⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣼⠇⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⣷⠀⠀⠀⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⣿⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⡄⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⣧⣸⣿⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⡀⠀⣿⣧⠀⠀⠀⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣽⣿⡄⠀⠀⣿⣄⠀⠀⣸⡟⠀⠀⣀⣠⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢪⣿⣿⣿⣷⣄⠀⠈⠙⣷⣾⣿⣷⣿⢻⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣷⡝⣿⣿⣿⣿⢿⣿⡿⣿⣏⣽⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠵⣿⢝⣿⠿⣯⡿⠷⢾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠟⣹⣿⣟⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠻⣟⣻⣿⣿⣤⣼⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠈⢿⡏⢻⡿⢏⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣽⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣧⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣯⣿⣷⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢿⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣯⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣾⠁
⠀⠀⠀⠀⢀⠀⣐⠃⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡄
⠀⠀⠀⠀⠈⢦⡀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠈⢠⣯⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⡇⠈⠂
⠀⠀⠀⠀⠀⠈⠻⣷⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣸⢷⡙⢷⣄⣠⣄⡀⣸⡀⡀⠀⣀⣠⣿⠈⠛⣿⣷⡶⢀⣀⢀⣄⣠⣴⣿⣇⣼⢿⣿⢁⡴⢁
⠀⠀⠀⠀⠀⠀⠀⠹⢧⣝⣶⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⠟⠓⢈⡏⢈⣷⣿⣿⣿⣾⣿⣿⣿⣿⣯⣮⡿⠿⠥⡌
⠀⠀⠀⠀⡴⢶⣶⡄⠀⠌⠙⠻⠶⣦⣙⠻⣿⣿⠛⢻⠗⣶⡀⣼⣧⣾⣿⣿⣿⣟⣿⣿⢟⡿⠿⠛⠉⠀⠘⡳⠓
⠀⠀⠀⠀⢀⡐⠋⣇⠀⠂⠀⠀⠀⠀⠙⢦⣿⣿⠀⠚⢪⣿⣷⢿⣿⣋⣩⣤⣭⣭⡭⠖⠋⠀⠀⠀⠀⠀⠀⠋⠙
⠀⠀⠠⣐⣛⣿⢿⣿⡇⣀⡀⠀⣀⡀⠀⢀⣹⣿⣷⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⡇
⠠⠴⣶⣿⣿⣿⣿⣿⡇⡛⠓⢖⠈⢙⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⠃
⢲⣄⣿⣟⣿⣿⣿⣿⡽⣷⠉⠈⠁⣶⢻⡿⣾⣿⠮⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⣻⣿⣿⣿⣻⣿⣿⣾⡦⢬⡶⠦⠀⣿⣿⣷⣾⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⣀⣀⣀⣀⡀⡀
⢸⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠠⣾⣿⣿⣿⣿⣯⣿⣿⢙⡋⣍⣟⣿⣿⣛⣟⣿⡏⠀⠀⠀⠀⠠⠄⠀⣠⡀⣀
⢻⣿⣿⣿⣿⣿⣿⢣⠶⠒⠀⠀⣼⣿⣿⣿⣟⣺⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡁⠀⠀⠀⠤⠀⣀⢀⡀
⠺⠟⠛⠛⠿⠋⠉⠟⠐⠁⠀⢰⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡛⠛⢛⣻⣭⠭⡾⠶
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢉⣈⡉⣠⠚
⣀⣤⣿⣯⣠⣴⣴⣏⣄⣠⣀⣉⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⢹⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣿⣷⣿⣦⣀⠈⣿
⣿⣿⡿⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣉⣙⣿⣽⣉⣿

⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡟⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣧
⠀⠀⠀⢲⣟⡄⣼⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢘⡇⡠⣿⡆
⠀⠀⠀⠈⣏⠸⡇⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡞⣼⢁⡟
⠀⠀⠀⠀⠈⢳⡄⢘⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠁⢠⠎
⠀⠀⠀⠀⠈⣆⠙⢾⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠋⡔
⠀⠀⠀⠀⠀⠘⠳⡈⢛⠦⣕⣶⡀⠀⠀⠀⣆⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡔⠀⠀⠀⢠⣶⣫⠞⡛⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠓⠦⠷⠬⣽⣿⢖⡀⠀⠐⣆⢀⠃⠀⠀⠀⠀⠀⠀⢈⠀⡢⠀⠀⢔⢾⣿⡯⠴⠮⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠾⠤⢤⣭⣿⡄⣆⣀⠀⠀⣀⣤⣰⣿⣥⡤⠬⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⢲⣦⣈⣿⠿⠟⠉⠉⠿⢿⣯⣀⣴⡒⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣿⣿⣿⣁⣦⡄⠀⠀⣠⢶⣸⣿⣿⣯⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠇⣳⠀⢠⡇⠾⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⣿⣄⣸⣧⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡰⢫⣭⣭⡟⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣿⣎⠙⠋⣴⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡝⠿⣿⣿⠟⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣈⣠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀
⠀⠀⡀⠀⠀⢠⢚⠏⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠈⢟⢢⠀⠀⠀⡀
⠀⣼⠀⠀⡴⢡⠃⠀⡰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⡀⠀⢣⠱⡄⠀⢹⡄
⡘⠙⠀⡜⢀⠏⠀⡴⢱⠁⠀⠀⠀⠀⠀⠀⠀⡠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⢱⠱⡄⠀⡇⠘⡄⠘⠙
⡇⡄⢰⡅⡘⠀⡸⢀⠇⠀⠀⠀⠀⠀⠀⡠⢊⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠢⡀⠀⠀⠀⠀⠀⠀⢃⠸⡀⠘⡀⣿⠀⡆
⡇⢠⣻⠇⡇⢠⠃⠼⠀⠀⠀⠀⢀⡴⢊⡴⠋⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠳⣌⠲⣄⠀⠀⠀⠀⠸⠀⢣⠀⡃⢿⣧⠁
⡇⠸⣿⢻⠃⢾⠀⡇⠀⢀⣤⠖⣁⠔⠋⠀⠀⣜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⠀⠈⠓⢄⡑⢦⣄⠀⠀⡇⢸⡆⡋⢻⡿⢀
⢰⠀⣙⢦⣆⣿⠧⣷⠾⢋⡤⠚⠁⠀⠀⠀⢰⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢱⠀⠀⠀⠀⠉⠢⣌⠻⢖⡧⢿⣆⣧⢞⡁⢸
⠈⡆⠈⠿⠿⢒⣫⡵⠊⠁⠀⠀⠀⠀⠀⠀⢸⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠉⠲⣭⣓⠺⠿⠏⠀⡎
⠀⢱⠀⢠⣾⣿⡏⢡⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⣿⣿⣦⠀⢰
⠀⠀⠳⣿⣿⣧⡀⠈⠑⠂⠤⠤⠤⠄⠀⡀⠜⠈⢧⡀⠀⠀⠀⣀⣤⢶⣦⡶⣤⣄⠀⠀⠀⠀⣠⠏⠘⢄⣀⠀⠤⠤⠤⠄⠒⠉⢀⣠⣿⣿⡷⠃
⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠳⠶⠶⣶⣤⣤⣠⡌⠛⠦⢤⣼⠍⠀⠀⠙⠀⠀⠉⣷⣤⠤⠞⠋⣠⣠⣤⣴⡶⠶⠶⠛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠱⡄⠀⢸⠀⢀⠄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠘⠄⠘⠀⠊⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠐⠀⠀⠀⠆⣴⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡆⠀⢠⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢷⠀⣼⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣗⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣼⡴⠃
⣿⡿⠿⠿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣀⣤⠄⠞⠀⠀⠀⠀⠀⠂
⠀⠀⣾⣿⣳⣶⣤⣷⡀⣀⢸⡟⠁⠀⣤⣀⢄⣴⠶⠀⠀⠀⡠⠂
⠀⢰⣾⣿⣿⣿⣿⣿⣮⣷⣄⣀⣁⡚⠛⠃⠺⠤⠀⠠⠄⠊
⠀⠈⠑⠀⠻⣿⣿⣿⣿⣿⣿⣧⣦⣴⠋⠀⠉⠁⠒⠒⠒⢜⠶⠂
⠀⠀⠀⠀⠠⡀⠛⠟⠿⣿⣿⣿⣿⣿⣯⣼⣤⣤⣀⣀⣀⡈⠑⠤⣸⠁
⠀⠀⠀⠀⠀⠙⠲⡀⣿⡿⣿⣿⣿⣿⣿⣿⣿⢀⣽⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣼⣷⣭⣿⣿⣿⣿⣿⣛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣷⣌⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⡀⢀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⣿⣿⣿⣯⡹⠿⠿⠟⠛⠛⣉⡀⠀⢈⠻⣿⣿⣿⣿⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣾⣿⣿⡿⠁⠈⠉⠛⠉⠉⠉⠉⠹⣿⣿⣿⡷⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⣀⠢⢈⠙⠻⢿⣿⣿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⡀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣴⠇⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⠀⠀⠀⠀⠀⢠⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠴⠂⠰⣄
⠀⠀⠀⠀⢠⡞⠁⠴⠿⠛⢃⡠⠔⠉⣆
⠀⠀⠀⠀⢸⠏⠉⠉⢹⣿⠩⡇⠂⠀⠘⢆
⠀⠀⠀⠀⢸⡇⠀⠀⠠⠤⢷⢢⡅⠄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠢⠏⠇⢁⠨⠀⢀⢀⠉⡐⠒⠒⠂⠉⠉⠉⠉⠉⠉⠁⠒⠒⠲⠠⠤⠤⠤⠖⠁⡝
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠂⠊⡂⠄⢲⢦⡄⠠⠄⠐⠿⠆⡰⠀⢰⠐⠀⠄⠀⠀⠀⠀⢀⠀⢀⠐⢂⡼⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠰⠁⡁⠔⠄⠉⢀⠀⣂⠠⠀⠁⠀⠀⠛⠗⠀⢀⠅⠀⢒⢀⡁⡀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢲⣦⠐⡀⠀⠲⠷⠄⢀⠀⠁⡄⠀⢀⣀⠀⠁⡉⠛⠓⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⢉⡠⠒⠰⠀⠒⠀⡀⢡⠀⠈⠚⠃⠀⠂⠈⠈⠀⠄⠀⡈
⠀⠀⠀⠀⠀⠀⠀⡵⣄⠀⠠⠀⠀⠀⠀⠀⠐⠀⠄⠠⢉⣤⠈⠀⠀⠐⢧⠀⠀⠀⠀⠀⢁⡀⠊⢤⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠇⠈⠳⣼⠀⠀⠀⠀⠀⢐⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠳⣄⠀⠀⠀⠘⡏⡐⠀⠸
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠸⡄⠀⠀⠀⠀⢸⠒⠒⠒⠒⠒⠒⠒⠲⡖⠶⠒⢪⡧⠀⠀⠘⠰⠁⢠⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⡗⠀⠀⠀⡘⢠⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡘⠘⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡌⢀⠐⠉⢢⡏⠀⠀⠀⢁⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⣇⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢠⠁⠀⠐⢀⠇⡇⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠂⡎⠀⢱⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠈⣇⣤⡎⠀⠀⠈⣇⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⠈⠁⠀⠀⡆⠀⡎
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⢷⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠈⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⢿⣾⠃
⠀⠀⠀⠀⠀⠈⠀⠀⣰⣿⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢿⣧⠤⠤⠤⠤
⠀⠀⠀⠀⠀⣠⣤⣄⣿⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣷⣠⣿⡷⠒⠦
⠀⠀⠀⠉⠉⠀⠀⠈⣿⣿⣶⣾⠟⠁⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⡠⣮⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡉⠂⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⠒⢄⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢸⣿⡟⣱⠟⠁⢀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠱⢦⣤⠄⢦⡀⢸⣇⠀⣀⣀⣄⣀⣀⣾⣿⣿⣿⢏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⣿⠆⠈⠷⣮⠿⢿⣿⡿⣿⡿⢿⣿⣿⠿⣃⣚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠿⠷⠄⠈⠀⠄⣸⣠⣿⣴⢞⣙⣿⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⠆⢻⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠰⢸⣿⣿⣿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡏⣠⣾⣷⣼⣿⣿⣿⡄⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠇⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠰⠿⡞⣋⡎⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣀⡀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⢾⡷⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⢿⢏⡽⠛⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣈⠢⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠦⠄⠀⠀⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⣠⠀⠀⠀⠈⡻⠟⠛⣻⣿⡯⠍⠉⠿⠟⠁⠀⢀⠀⠈⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⠂⢩⣿⣯⣰⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠻⣿⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢬⠍⠉⢩⣍⡭⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⡠⠂⠂⠀⠀⠀⠀⠊
⠀⠀⠀⡄⣢⡏⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠄⡀
⠀⣰⣢⢿⠃⠈⢀⡐⠊⠀⠀⠀⠀⠀⠀⠀⢐⠀⠃
⡆⠈⠛⢸⡴⡫⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⡇⠀⠘⠿⠟⠠⠐⣀⣀⣀⣀⣀⣀⣴⣶⣿⠃
⡇⡡⠈⠀⠀⠀⠀⠈⢫⣿⣿⣿⣝⡡⠚⠁
⠃⠀⠀⠀⠀⠀⠀⠀⢰⣽⣿⠙⡾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⢹⣿⣿⣧⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⢶⣶⣶⣶⣾⣿⣿⠶⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠱⠇⠈⠀⠹⣿⡟⢿⡏⠃⠠⠿⠉⢩⣿⣯⡶⠀⠀⢸⣷⣏⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠃⢻⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠻⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⢿⠑⠂⠀⠀⠀⠀⠀⣷⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡣⡄⠀⠀⠀⢠⠀⠀⣀⣂⠀⠉⣀⡀⣀⣿⣷⠄⡀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡂⠀⣿⣿⣿⣏⣄⣬⣷⠿⠿⠿⠿⣟⢁⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⢸⡭⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⢘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢆⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣄⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⣷⣄⣿⠄
⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⣾⠃⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡧⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⣿⣟⠋⠙⠛⠋⠁
⠀⠀⠀⢄⣀⠴⠖⠲⢼⣿⣷⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣾⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠋⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢼⣤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⢠⣾⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡈⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣸⡇⠀⣠⢔⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠒⢄⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡟⣡⡾⠋⠀⡀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠢⢮⣠⠄⢦⡀⠀⣿⠀⢀⣀⣀⣀⠀⢀⣿⣿⣿⣿⡟⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⣿⡦⠀⣿⣤⣻⡷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢼⣿⣷⡀⠈⠃⢀⠉⣟⣠⣏⣡⠼⠻⣿⠀⠉⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠹⣿⣿⣿⣯⣾⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⡀⢻⣿⣿⣿⣧⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠇⢋⣿⣟⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢰⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠀⠘⢻⡿⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠛⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠉⠻⢋⣼⣆⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣀⣈⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⠛⠁⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⢿⣿⣿⣿⣿⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⠿⢟⢫⡟⠋⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⠱⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⡌⠁⠀⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠳⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠋⠀⠀⢂⡀⠀⠈⠀⠛⣿⠿⠿⣿⣿⣟⠉⠙⣿⡿⠋⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢙⠤⠐⣙⣿⡏⢡⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⠷⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠛

⢠⠀⠀⢀⡤⠂
⣾⠀⣴⠟
⣿⣼⠏⠀⠀⠀⠀⠀⠀⣰⠃⢠⡀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀
⢹⣿⠀⠀⠀⠀⠀⠀⣴⠃⠀⠈⣧⡀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣾
⠘⣿⡀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠈⢳⣄⣠⡟⠁⠀⠀⠀⠀⠀⣿
⠀⠹⣧⡀⠀⠀⢀⣿⠀⠀⣀⣠⠖⠀⢹⣿⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⡀
⠀⠀⠙⢷⣄⣠⣾⡿⠿⠟⠋⠁⠀⠀⠸⣿⠀⠀⠀⠀⠀⣰⡟⠀⢀⣰⠃
⠀⠀⠀⡈⠙⢿⣧⣄⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢀⣼⣯⣤⡴⠟⠁
⠀⠀⠘⣿⠳⣦⡉⠻⣷⣶⡂⠀⠀⠀⣦⣿⡿⠶⠿⠛⠛⠉⠁
⠀⠀⠀⢻⣆⠀⢹⣦⠈⢹⣿⣦⡄⠘⠿⣡⡴⠚⢛⡿⠂
⠀⠀⠀⠈⠻⢷⣝⠋⠸⠋⠁⠀⡀⣀⠀⡈⢠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠿⠛⠀⠹⣎⠡⡀
⠀⠀⠀⠀⠀⣰⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣀⣤⡀
⠀⠀⠀⠀⢠⣿⡄⢻⣆⠀⠀⣀⡀⠀⠀⠀⠀⣴⢿⣟⣡
⠀⠀⠀⠀⣾⡿⣿⣦⣙⠛⠛⠁⠴⠒⣿⠛⠉⠛⠛⠋
⠀⠀⠀⢠⣿⠁⠈⠛⠛⠛⠋⠀⠀⢸⡏
⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⣀⠀⠈⠳⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣿⣿⣇⠀⠀⠀⢀⣴⠃⠀⠀⠀⢸⠁⠈⠉⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⠟⠉⠉⠉⠉⠉⠛⠲⢦⣄⡀
⠀⠀⠀⢻⣿⣿⣄⣀⣴⣿⡏⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⣴⣤⡀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠈⢿⣿⣿⣿⡿⣹⠁⣸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⣴⡈⢿⣿⣿⡇⣿⣠⣿⡇⣿⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⢀⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⣌⠇
⠀⠀⠀⣿⣧⠘⣿⣿⠁⣿⣿⣿⡇⠹⣿⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⢸⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⠛⠀⣸⣿⣷⣿⣿⣿⣷⡀⠀⣤⡀⠀⠀⠀⢰⡹⢇⣿⠀⠀⠀⠀⠀⢠⣿⠏⢀⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢹⣷⠀⠀⠀⣼⣧⣼⠛⣇⠀⠀⠀⢀⡾⠋⢀⣾⣿⡇⣿⣿⣿⡀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⠀⠀⢀⣿⡿⠋⠀⠀⠀⠀⢐⣋⣤⣾⣿⠿⠟⠀⢿⣿⣿⣷⡄⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⣥⣤⣤⣶⣶⣾⣿⣿⠿⢟⣩⣶⣿⣿⣿⣦⣝⢿⣿⣿⣦⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠻⣿⣏⠙⣿⣿⡿⠀⣰⠛⠿⠿⠛⠛⠋⠉⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⡌⠻⢿⣿⣦⣄⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠹⣿⠀⢸⣿⠇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟⢷⣄⠀⠙⠻⣿⣿⣶⣄⡘⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢹⡆⢸⣿⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣌⠻⣷⣄⠀⠀⠙⠿⡏⣡⣽⡎⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠸⡇⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣼⣿⠀⠀⠀⠀⠀⠙⢿⣿⣄⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⢠⣅⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⢸⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⢀⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣛⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣬⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣶⣶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢻⣇⣦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠊⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⣿⠿⠷⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢫⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣷⣮⣝⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⢟⣿⣿⣗⢦⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣄⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠀⣿⣿⠉⠁⢃⣸⠟⢿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⠦⣾⣧⡀⠀⠀⠃⠀⠀⠀⢸⡇⡠⠌⣿⡟⠀⣦⣠⡿⣜⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣈⢽⣦⢤⣿⠀⠀⠀⣸⡿⣁⣴⢿⣧⣴⢿⣯⡾⢿⣏⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢹⣷⣝⢷⣼⣦⠀⣰⣿⢟⣥⢟⣾⡷⣿⢻⣷⣿⠰⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡈⠻⣿⠇⣿⡿⠿⠿⡻⢿⣯⣿⣹⡿⣱⡿⣿⡟⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢡⣭⣦⢿⣷⣀⣤⠀⠌⢻⡿⠻⣕⣻⣾⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣷⣿⣿⡏⠂⣴⡿⣿⣶⣿⣿⡏⠙⣿⠀⠀⠀⡗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠻⢦⡀⢀⣽⠋⢹⣿⣷⡄⢻⡄⠀⠀⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⢀⣾⣿⡏⠀⠀⢸⡇⠘⠀⣸⣿⡅⠀⣿⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢻⣾⣿⠟⠁⠀⠀⠘⠁⠀⠀⠁⢸⠇⠀⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡿⢿⣿⣇⠀⠀⠀⢤⣄⣴⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣿⣿⣧⠀⠀⢸⣿⣿⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢿⣧⡙⠄⠀⢸⡟⠁⡐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠐⢠⡿⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⣿⠀⠀⡴⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⣿⠀⢰⡿⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⠀⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣼⡟⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣿⡇⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡽⣦⡀⠀⠀⣿⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⠟⠋⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠈⠻⣷⣄⣿⠀⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢿⣿⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡏⢻⣆⣀⣀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢠⣵⣾⣯⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠤⢿⡻⠿⢿⣿⣦⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣤⣴⣾⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡿⢷⡀⠀⠀⣀⣠⣿⡿⠋⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣬⣉⡒⠦⣼⡧⠀⠸⠇⠀⠀⠀⣀⣻⡆⢈⣙⣋⣀⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⢰⡿⠋⣰⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠈⣿⠁⠀⠀⠀⠐⣤⣶⣦⡀⠀⠀⣠⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠰⠚⠋⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⣉⢉⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣢⣄⣸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣄⣻⣿⣿⣿⡧⢧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣤⣄⣀⣀⣀⠀⠀⣤⣾⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠉⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⣿⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇⠀⣿⣿⣦⣄⣈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠘⣿⡏⠀⠀⠈⣿⣿⣿⡿⠿⢯⠔⣸⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇⠀⢿⡗⠀⠀⢀⣿⣿⣋⣤⢈⣿⣦⣾⣿⣯⣬⣿⣟⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⢨⡅⠀⠀⠀⠽⣿⣿⣷⣿⡿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⡧⠀⠀⠀⢶⣶⢠⣤⣴⣾⣿⣿⣿⣠⣾⣿⣿⣿⡟⠀⣿⠇⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢰⣶⣿⣿⢸⣿⣿⣿⡿⣿⣿⣿⣿⣿⠋⢿⠇⠀⠁⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢸⣿⣿⣿⢸⣿⠟⠘⠁⠁⠘⠟⣿⡿⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⢸⣿⣿⣥⢈⣏⣀⡀⠀⠀⠀⠀⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⣯⣭⣥⣈⣙⣛⡻⣶⣦⣤⣛⢿⣦⡀⣀⡤⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢻⣿⣿⣿⣿⣿⡻⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣷⣶⣤⡙⠿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠉


⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣏⣙⣋⣛⣉⣉⣻⣧⣉⣉⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣤⣤⣭⣥⣼⣿⣿⣭⣭⣭⣭⣿⣿⣭⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣽⣷⣿⣿
⠀⠀⠀⣀⠀⠀⠀⣶⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠙⣷⣶⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⡇
⠀⠀⠀⠀⠘⣿⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⠃
⠀⠀⠀⠀⠀⠛⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣦⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⢳⣄⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢠⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣤⣀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⣤⣶⣿⣿⢟⣡⣴⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣭⣟⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠛⣩⣅⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠈⣉⠙⠻⣯⣙⠿⣿⣷⣶⣿⣷⣶⣶⣿⣾⣿⣿⣿⣿⣵⣿⠋⣁⡀⢤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⠉⡄⢀⡿⢷⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⣯⣈⡿⢫⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣴⠿⣧⣽⡏⠁⠈⢹⣿⣿⣿⣿⣿⡏⠉⠁⠀⠈⣿⣀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣬⣿⣤⣤⢨⣀⣿⠻⣏⠉⠁⢀⠀⠀⠾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠋⠁⠙⣿⡇⠀⣨⣄⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡧⢀⡠⣤⡛⢻⣷⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠠⣾⠁⣾⣿⣿⣿⣿⣷⠀⡦⠰⣶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣦⣸⣿⣰⣿⣿⣿⣿⣿⣿⣶⣿⣆⢼⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣏⣿⣧⣿⣿⣿⣿⣋⣙⣍⣉⣩⣿⣿⡿⣿⣧⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⠿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣯⣽⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣶⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⠛⣿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡅

⠀⠀⠀⠀⠀⢠⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⡄
⠀⠀⠀⠀⢠⡏⢀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⣦⠀⢳
⠀⠀⠀⠀⣼⢁⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢸⡇
⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⢠⠏⢀⣀⠀⠀⠀⠀⠀⢀⣀⠈⢧
⠀⠀⣿⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢀⡇⠀⠀⡞⢰⠏⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡆
⠀⠀⠸⣧⡈⣿⡄⠀⠀⠀⢠⡀⠀⠀⢠⠀⠀⠀⠀⣰⡟⣠⣾⠃⠀⢰⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣷
⠀⠀⠀⠉⠻⢿⣽⣄⠀⠀⢸⡇⠀⠀⢸⡇⠀⢀⣠⣿⡿⠛⠁⠀⣶⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢰⡆
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣼⡇⠀⠀⣸⣧⣶⡿⠛⠁⠀⠀⠀⠀⠸⣦⣹⣆⠀⠀⠀⡄⠀⠀⡄⠀⠀⢀⣿⣡⡞
⠀⠀⠀⠀⠀⠀⣏⣓⡲⢽⡿⢷⣶⣶⡿⢟⡩⢒⠏⡽⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⣸⡇⠀⠀⣇⣀⣴⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠒⢠⣷⡟⠉⢸⣿⢻⡇⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣝⣿⣧⣤⣴⣿⣿⣥⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⢸⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡙⣱⣿⡹⢿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠺⠿⢀⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⢇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣡⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⣿⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⢃⣾⣿⣿⣿⣿⣿⡿⣿⣿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⢡⡿⢟⣻⣿⣿⣿⣿⣿⣦⡉⠻⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀⠈⣴⣿⢿⣿⣿⣿⣿⣿⣿⡝⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡷⠀⠻⢃⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠀⢻⡿⣿⣿⣿⡹⠟⢁⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠀⠀⡀⠘⢿⣿⣿⠀⢠⡾⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⣀⡀⠻⠀⠀⠉⠁⠀⠈⢁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⠿⠿⠿⠿⠿⡿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣤⣤⣶⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡄⠀⠀⢻⣷⣶⣶⣦⣤⣄⣀⣾⣿⡆⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣆⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣆⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣼⣿⣶⠀⠀⢰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠉⠙⠛⢿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢰⣿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⠀⣻⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣰⣾⣿⣿⠟⠀⠀⡐⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣀⣀⣼⣿⣿⠏⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⢀⡀⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⣤⡠⡄⠀⠀⢠⠀⠀⢀⡀⣀⣠⠐⠯⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⣴⡄⠀⢲⣜⣻⢻⠞⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢭⣿⡿⣛⠙⠉⠻⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡜⢻⣌⠋⣷
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⠶⠾⠻⣿⣷⣯⣿⠏⢸⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⢠⣶⣶⣾⣿⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠋⢀⣤⠀⠘⠛⠛⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⢿⣿⡟⠛⣿⡇⠀⠋⢿⡇⠀⠃⠃⠀⠀⠀⠀⠻⢿⣻⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⠟⠻⣄⣿⣿⣿⣴⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⢹⣿⣿⣿⣿⣿⣿⣶⣤⣴⣶⣴⣾⣿⣿⣿⣿⣿⡏⠊
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠁⡹⠋⠉⠛⠛⣿⣿⣿⣿⣿⡿⠿⠛⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⠁⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠎⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠳⡀
⠀⠀⠀⠀⠀⠀⢀⡿⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢿⡀
⠀⠀⢀⡠⠔⡲⣾⡁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢨⣷⢖⠢⢄⡀
⠀⡰⠋⡰⠋⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⠀⠙⢆⠙⢄
⢸⣇⡸⠁⠀⠀⣿⡇⢇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡘⣸⡿⠀⠀⠈⡇⡸
⣿⡏⡇⠀⠀⠀⢸⣿⡘⣄⠀⠀⠀⠀⠀⢀⢚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡀⠀⠀⠀⠀⠀⣰⢃⣿⡃⠀⠀⠀⢸⢻
⢿⣧⢱⠀⠀⠀⣠⡿⠃⣚⠳⣄⡀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⡇⠀⠀⢀⡠⢞⡁⠘⢿⣄⠀⠀⠀⡆⣾
⠘⣿⡌⠳⠒⣊⠠⠀⣠⠈⠙⣂⠈⣉⠒⠾⣧⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣾⠷⠒⣉⣁⢘⠛⠅⡀⠀⠄⣑⠲⠞⢱⣿
⠀⠈⣿⡆⢊⠴⠒⠉⠉⠉⠉⠉⠛⠺⢿⣷⣜⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡿⣣⣾⡯⠟⠛⠉⠉⠉⠉⠉⠒⢦⡙⢴⣿⠁
⠀⠀⢿⣧⠸⣄⡀⠀⠀⣀⠴⣪⠝⠁⠀⠙⢿⣷⡀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢨⣾⡿⠉⠀⠈⠫⣗⠤⣀⠀⠀⢀⡠⠇⣼⡿
⠀⠀⠈⠻⢷⣦⣬⣭⣥⠴⠚⠁⠀⠀⠀⠀⠀⢿⣁⠀⠛⢶⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⠂⣨⡿⠀⠀⠀⠀⠀⠈⠓⢤⣬⣭⣥⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣬⣴⠛⠻⠀⣠⠤⣤⣤⠤⣄⢀⠟⠛⣦⣥⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠙⠓⠀⠸⠇⠀⠾⠋⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣆⠀⠱⣄⠈⠀⣠⠎⠀⣰⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡎⠀⠁⠀⠈⠁⡈⠁⠀⠈⠀⢱⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣼⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⠿⠣⠀⡀⠀⠠⠀⠀⢀⡀⠘⠿⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠸⣟⠿⣀⠀⠀⣀⠿⡿⠇⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⡄⢙⠀⠀⡏⢠⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⣸⠀⠀⣆⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠉⠀⠀⠉⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⡆⢰⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡇⢸⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡇⢸⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢵⡼⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠘⠀⠀⠀⠀⠀⠀⠀⠈⢦⣧⠅
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡾⡀⠀⠀⠀⠀⠀⠀⠀⠐⢮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣤⣰⠀⠀⠀⢰⣦⡔⠑⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣦⣻⣷⣶⣶⣿⣿⠔⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⡝⣳⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⡎⢠⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡀⣠⠞⠟⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠛⠒⠲⠤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣟⣸⢏⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⡁⠀⣼⠇⠀⠀⠀⠀⢀⡾⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠻⣡⠀⠛⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢳⣸⣿⠀⠀⠀⠀⠀⢸⡇⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⢋⡀⢀⡠⢻⣧⡀⠴⡀⠀⣷⣤⠀⠀⠀⠀⠀⢘⣹⣿⣖⠃⠀⠀⢠⣿⠇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣯⠀⠴⠃⡀⠈⢿⣯⠧⠌⠀⣿⣷⣿⣷⣴⣶⣾⣭⣽⡿⢿⡆⣄⠀⣾⠃⣸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣉⠙⡞⠁⠀⠀⢸⡗⢠⠏⢹⡿⢹⡇⠀⣀⠀⠉⠀⣀⠠⠟⢮⡁⣯⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣡⣬⠙⢏⠀⠀⢀⣸⠃⣼⣤⠼⡇⢸⠋⠁⠀⣀⠈⡝⢃⠀⠀⠈⣷⢘⣃⠖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⢥⣅⠈⢢⡾⠛⠛⠿⢰⠃⠐⠒⢻⡸⡓⠚⠋⠁⠐⠛⠿⠗⠁⠀⡿⣼⢿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢱⣾⠿⠟⠾⠛⠁⢀⡄⣆⣼⠬⠀⠠⠌⣇⣧⡆⣸⡀⠀⢀⡀⣠⠀⣠⠷⡏⢸⢹⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡔⣦⣶⣴⡿⣿⠲⡶⠸⡧⣿⣟⣶⣿⣷⣤⣨⣁⡻⢶⡏⢼⣶⣿⡾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠋⠉⠘⠁⠀⠀⠈⠀⠁⠁⠀⠀⠁⠉⠁⠀⠉⠈⠉⠀⠈⠙⠈⠋⠉⠃⠒