Paraffin Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠶⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠶⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣤⣤⣤⣴⣶⣶⡄⠻⢿⣿⣿⡿⠟⢠⣶⣶⣦⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⣷⣾⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⡿⠋⣉⣤⣤⣉⠙⢿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣾⣿⣿⣿⣿⣷⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣯⣿⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⡈⠻⣿⣿⣿⣿⠟⢁⣾⡿⣿⣿⣿⡟⢡⣶⡄⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦⣤⣉⣉⣤⣴⣿⣿⣷⣾⣿⣿⣷⣈⠛⣁⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡃⠜⢀⠀⠀⠀⠀⠀⠀⢰⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠂⢀⡿⡄⢀⣀⠀⠀⠀⢀⠑⠀⢤⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡼⠀⢳⠤⠤⠤⠤⠤⠤⣭⣠⠏⢱⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣋⠭⠥⠒⠒⠈⢧⣦⡞⠋⠉⠉⠉⠁⠀⠐⢿⢀⢸⠧⠬⠭⢍⣒⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡏⣔⣋⣡⠤⠤⠤⠤⠤⠶⠶⣿⠶⠶⠖⠀⠀⠲⠶⠶⠮⢿⠧⠤⠤⠤⣀⣀⣉⠑⢮⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠙⠮⣗⣦⢤⣀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣀⣠⠤⢴⣛⡥⢾⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠉⠛⠚⠻⠯⠭⢿⣗⣒⣒⣐⣀⣒⣐⣒⣒⣒⣻⣭⠭⠭⠖⠒⠛⠉⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠶⠶⠒⠒⠒⠛⠋⠉⠉⠀⣸⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠹⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠉⠙⠒⠒⠲⠦⠤⠤⠤⠤⠤⠴⠶⠒⠒⠛⠉⠉⣀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠰⠃⠚⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠴⠒⠋⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⣶⣤⣦⣤
⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣶⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢏⡙⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣴⣶⣿⣷⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠙⣠⡘⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢠⣄⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢀⣿⢿⡄⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⣿⠁⠘⣷⡀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣾⡇⠀⠀⠸⣧⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢰⡏⠀⠀⠀⠀⢻⣇⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢰⡿⠁⠀⠀⠀⠀⠀⣿⡄⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣦⣶⣾⣷⣤⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⢀⣤⡶⠶⠾⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠷⠶⣦⣄
⠀⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⣴⠶⠶⢦⡀⠀⠀⢀⡼⠃⠀⠈⢻⣆⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⢠⡾⠋⠉⢹⣇⡀⠀⢀⣾⠏⠀⠀⠀⢿⡆⣴⡟⠀⠀⠀⠀⠈⢻⣶⠟⠁⠹⢷⣤⣀⣀⡀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠁⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⠘⣷⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⣦⠀⣿
⣿⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣿
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢻
⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸
⣿⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈
⣿⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢻⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡾
⠀⣿⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⣇⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢹⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠈⠻⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡾⠏
⠀⠀⠈⠉⠉⠛⠛⠛⠛⢻⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⠿⠟⠛⠛⠛⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠁⠈⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣤⣤⣤⣴⣤⣶⣄⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠋⠉⣉⠉⠁⠀⠀⠀⠀⠉⣉⡉⠉⠛⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠷⠾⠛⠛⠛⠛⠛⢷⣄⠀⠀⠀⢀⣴⠟⠛⠛⠛⠛⠻⠶⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠰⠆⠀⣠⡶⠛⠛⠛⠻⣦⡀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⠀⠀⠀⠀⠀⠈⠃⠀⢻⡆⠀⠀⠀⠀⠀⠀⣾⠃⠀⠋⠀⠀⢀⣠⣄⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠘⠃⠀⠀⠀⠀⠀⠀⢠⡄⠈⢿⣦⣀⠀⢀⣠⣼⠏⠀⣤⠀⠀⠀⣾⡋⠉⣿⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠋⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣠⠤⠔⠛⢻⡟⠀⠀⢻⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠉⠁⠀⠀⠀⠀⠀⢨⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⢀⠀⣀⣱⠃⠘⠀⠀⠀⠀⠀⢀⠀⢀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⣀⣠⣤⣤⣤⣈⣀⣀⣛⣀⣀⣀⣀⣠⣀⣀⡆⠒⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣻⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠛⠛⠋⢹⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣉⠆⠀⠀⠀⠈⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣧⡋⠀⠀⠀⠀⢨⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠁⠀⠀⠀⠀⠈⢙⣷⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡇⢀⡀⠀⠀⠀⠠⢯⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣇⣆⠀⠀⣲⣠⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣖⣄⣶⣽⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⠠⣬⡍⢙⡛⡛⠛⠛⠛⠛⠛⠛⠛⠉⠉⢍⣵⣶⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠩⢅⡀⠀⢬⣿⠄⠄⠀⠀⠀⠀⠀⠀⢀⠀⠁⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁



⡿⢿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣒⣒⣒⣒⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣽⢿⣿⣛⣋⡉⠋⠙⢉⣛⣛⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠻⠿⣿⣿⠗⢀⡀⠹⣿⠿⠟⠛⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣠⣶⡶⠦⠶⠾⠷⠖⠶⠀⢶⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⣼⠃⠀⠂⠀⠀⠀⠀⠙⠛⠒⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⠀⡀⠄⠀⢰⠈⠨⣗⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⡖⠒⠢⢄⣀⠀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡏⠛⠁⠔⣣⣤⣽⡃⠀⠁⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡙⣷⣿⣿⢢⠉⠁⢿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⠃⠐⢠⠞⣿⡿⢿⣯⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⡿⢿⣿⡿⣄⠀⠈⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢰⠇⠀⠀⠉⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⢀⣀⢹⣿⢣⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⣸⠀⠙⠛⠛⢻⡟⠉⢹⡏⠉⠉⠹⡿⠀⠉⣽⣿⠛⠛⠛⠉⠘⣿⠼⡀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⣝⠀⠀⠀⠀⢨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠀⠀⠀⠀⠀⣿⡆⣧⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣴⠀⡿⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⠀⠀⡀⠀⠀⢸⡇⢿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠃⠀⠀⢾⣛⣩⣭⣦⣤⣤⡤⠤⠤⢤⡤⣤⡤⠌⣉⣙⣣⡄⠀⠸⣷⠘⡄⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⡇⠰⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⣷⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡿⢀⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣾⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⡇⠀⠃⠀⠨⠉⠙⠛⠛⠒⠒⠒⠲⠤⣤⣀⣠⠴⠒⠚⠛⠛⠛⠛⠛⠳⠆⢨⡗
⠀⠀⠀⠀⠀⠀⠀⠐⡹⣿⣿⣿⡏⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠿⣿⣧⢴⣿⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠢⡿⣿⠊⡏⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢸⠀⠀⢻⡇⠈⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢰⡇⠨⠀⡅⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠄⠀⠀⢸⣧⠀⡄⢿
⠀⠀⠀⠀⠀⠀⠀⣴⠀⡷⡸⠇⠐⢠⣇⡀⠢⣠⠞⠁⣀⣀⣀⠀⠀⠀⠀⣄⣀⣀⣀⡀⢘⣧⠈⠀⣠⡴⢶⣿⣠⡇⣸⡄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠟⡋⠀⠀⠊⡍⢹⣶⣿⣄⣠⡄⠀⠁⠀⠀⠀⠀⢠⠅⠀⠀⣀⣸⡯⢠⠾⠋⠀⢰⣿⣿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣸⡧⣽⠦⠀⠈⠀⣼⣉⣽⣟⣹⠻⠐⠛⠛⠻⠛⠛⠛⠛⠛⣏⠀⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠸⡷⢄⡀⠀⠈⠛⠋⢀⣈⣥⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⠀⠀⠀⣀⣀⡤⢛⠟⢈⣿⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⣈⠀⢤⣀⣊⡩⠟⢁⣈⣿⣿⠞⠉⠉⠉⠉⠛⢮⣿⠋⠀⠛⢿⡉⠉⠁⠐⢁⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⠜⠉⠀⠀⠀⣽⣿⣇⡀⠀⠀⠠⠔⢃⣼⣯⣷⣶⡄⠀⠈⠓⢶⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣅⠀⠀⢀⠤⣼⣭⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣭⡦⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣟⠳⠴⠤⢉⠙⢻⣛⣛⠛⠻⢻⣿⣿⣛⣻⣻⣭⠯⢽⡗⣊⣡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡗⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠀⠀⠤⠐⢲⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠉⠛⠿⠈⠥⠀⢀⡀⢀⣦⠀⢀⡄⣀⡀⠀⠀⠄⢦⠝⠙⠟⠛⠉⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣖⣀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡼⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠛⠛⢻⡟⠛⠛⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣰⣦⠀⠀⡀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡟⠀⣼⠙⠚⠁⢸⢠⡟⢳⣀⡀⣟⢳⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⡇⢀⡏⠀⠀⠀⡟⣸⠀⢸⡏⡇⢻⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠳⠞⠁⠀⠀⣸⠁⡟⠀⠀⠀⡇⢸⡜⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⢀⡇⠀⠀⠀⡇⠘⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢶⠀⡆⠀⠀⠳⠞⠁⠀⠀⠀⢷⢀⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢾⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢲⠀⡆⠀⠀⢳⢀⣶⡄⡇⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢼⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⢸⠀⣏⠛⠁⣧⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠋⠉⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⠀⠀⡏⠀⠀⠀⠉⠉⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢰⠚⠳⠆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠀⡇⠠⠖⠲⣄⣠⣄⡀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠀⠀⡴⠖⠛⠃⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⢸⠇⠀⠀⠀⠁⠀⠀⡿⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠳⣤⣀⣀⣀⠀⠀⠀⠙⢦⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠏⠀⠀⣀⣶⠒⠚⠋⠁⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡆⠀⠀⠀⠀⠳⠤⠶⠆⠀⠀⠀⠉⠉⠉⢉⠉⢉⡉⠉⢉⡄⠀⠒⠶⠖⠋⠁⠀⠀⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⡤⢤⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠉⣀⡀⠀⢀⣠⡖⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠙⠒⠒⠚⠛⠉⠓⠒⠒⠚⠋⠁⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⣿⣧⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣭⣭⣭⣤⠀⠀⠀⠀⣼⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣷⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣆⠀⠀⠀⠀⣛⣛⣛⣛⣛⣛⠀⠀⢰⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣿⣿⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣭⣭⣥⡄⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢨⣭⣭⣭⣭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣶⣶⣶⣶⣶⣶⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⡅⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⡇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠿⣿⠁⣶⡎⢻⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣄⡀⣾⠉⢿⡄⠲⢤⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⣺⣕⡦⠴⠛⠻⣿⣶⣦⡝⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠮⢽⠯⢟⣫⣤⣔⠊⠙⠛⢦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣽⡞⠀⢾⣷⠄⣸⠿⣟⡠⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⢩⣿⡥⠌⣿⣇⣀⡤⣿⡻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣖⣠⣤⣖⣮⣟⣳⣽⣳⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢈⡹⠿⡛⠯⢟⠮⠿⡷⠽
⣿⣿⣿⣿⣿⣿⣿⡿⢁⣤⣾⣿⠞⢠⣤⠤⠘⠳⣾⣧⣄⡹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⠉⣭⡿⢛⣻⣿⣍⠉⠙⠻⣧
⣿⣿⣿⣿⣿⣿⣿⠃⠛⢿⣿⣿⣄⠐⠿⠟⠀⣠⣿⣿⠟⠃⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣥⡀⠀⠈⢿⣯⣕⣤⣿⠟⠀⠀⣠⣼⡆
⣿⣿⣿⣿⣿⣿⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢚⣿⣟⣛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣋⣭⠤⢶⡺⠉⢠⣤⠈⠳⣌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣛⡩⠔⠚⠉⠀⠀⠀⣈⣇⠀⠈⣽⡄⠀⢿⠜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⠕⠊⠁⣀⠀⠄⠀⠀⢀⠀⠈⠘⣧⣦⣄⡐⠼⠋⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡥⠔⠉⠀⣀⠀⠀⠉⠀⠢⡄⠁⠰⠀⠄⠀⠀⠉⠁⠀⠀⠀⢃⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⠰⠂⠁⠀⠀⠀⠀⢠⠀⠰⣠⠀⠎⠸⡀⠊⡀⠈⠀⠀⠀⠀⠀⠀⠀⣘⣧⠘⣧
⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠶⠖⠛⡩⠄⠒⠈⠀⠀⠠⠐⠀⢀⠀⠀⠀⡇⠂⠀⢣⠐⡀⢣⠈⠘⡄⢀⠀⠀⠀⠀⠀⣠⠽⢿⡄⠹⣆
⠀⠀⢸⡇⢡⡥⠼⠒⠶⠦⣄⡈⠁⠀⢀⠀⠀⠀⢀⠀⢀⡁⠘⡘⡆⠄⠹⠘⡀⠈⣇⢡⠘⠶⠃⢉⡡⠀⠀⠀⠀⠀⡇⢸⣦⠹⡆⢹⡄
⠀⠀⠀⣷⠀⡇⠀⠲⣶⡆⠘⡇⠠⠀⠠⠀⠀⡄⠰⡀⣋⡀⠀⡇⢡⡌⠄⢃⣉⠕⠘⠋⢠⣀⣠⣌⠀⣀⣀⠤⠔⢒⣙⣲⣴⣞⣁⣈⣧⡀
⠀⠀⠀⠘⣇⠙⢄⠀⠼⠧⣠⣷⠂⠀⠘⡄⠠⠃⠐⣷⠀⠄⢀⡇⠀⠘⠶⠀⠀⠤⣀⠤⠛⢉⡠⢈⣈⣥⣴⠶⠿⣻⣛⣛⠛⠛⣯⠉⠁
⠀⠀⠀⢀⣘⣦⠀⠙⠒⠚⠟⠉⠀⠀⠀⢳⠀⠶⡀⢹⣇⢉⡠⠷⠊⠀⢀⡠⠔⠉⣀⠤⣊⣥⠶⠿⠿⠁⢲⣲⠟⢉⣭⡉⠳⡄⢹
⠀⠀⣠⢺⡇⢹⡎⠠⢱⠄⠀⠀⠀⠀⠀⠀⣇⡀⠱⠖⠁⠀⠀⢀⡠⠒⠁⡀⢔⣪⣷⡟⠫⠀⠀⠀⠀⠀⣈⡁⠀⢘⣻⡀⠀⣿⢸⡀
⢀⡔⠻⢸⡇⢨⢿⡀⠀⢾⡄⣀⣀⡀⠈⠀⠈⠠⠀⠀⢰⠦⠂⠁⠠⢐⣯⡶⠛⡡⠀⠐⠂⡤⠀⠀⠖⠄⠐⢷⣄⡀⡉⣄⡴⠋⠘⡇
⣾⢐⠀⠈⣧⠸⣌⣧⠀⡼⠉⠠⣤⠙⣦⠀⠀⠀⠀⣀⠓⢀⣰⣴⢾⠋⠀⢃⠰⠐⢲⠐⠊⢱⠀⣰⠃⠁⠀⠈⠙⠉⠉⠁⠀⢇⠀⣧
⢻⠘⡇⠀⢹⡀⠈⠛⣶⠀⠳⣄⠙⢂⡾⢠⠀⣂⣭⣶⠞⠋⠁⣼⠘⠂⡄⢸⠀⠀⢸⡆⡀⢸⠀⡴⡄⢰⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿
⢸⠀⡇⠀⠘⣇⠀⢘⡜⠶⠶⠶⠿⠿⠖⡟⠛⣏⠈⣷⠘⣁⣀⡇⠀⠃⠐⢸⠀⠂⡀⡇⠃⠸⠆⢃⣃⠈⠀⢀⣀⠀⠀⠀⢒⣿⡇⢺⡇
⢸⠀⢹⠀⠀⣿⠀⠀⣶⠀⠀⠀⠀⠆⠀⡧⡤⢸⢠⠟⠠⠤⠼⠧⠆⠈⠓⠈⠉⠉⠁⠙⢉⡴⠋⠁⠀⠀⠀⠀⠀⠉⠒⢎⣭⠙⢧⠘⡇
⢸⠀⢸⠀⡀⣿⡀⢀⡝⢉⣉⠑⣄⠀⠀⠀⠀⠀⠀⣄⣠⣀⣄⠀⡀⠠⠤⠤⠤⠤⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡬⠭⠥⣝⣰⠷
⢸⡀⠀⡀⠁⢸⣇⠈⣇⠀⢿⠀⣿⠀⠀⠀⢀⣀⠫⠉⠁⠀⠀⠀⠤⠤⠤⠤⢤⡄⢰⠁⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠸⡇⢣⠃⠸⠀⣷⣀⣈⣱⣰⣾⣥⣯⣤⠤⠴⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠉⠃⡆⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣧⠘⠈⠀⡶⠭⠭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠸⡄⣣⠿⢥⣄⡤⠤⠤⠶⠖⠒⠂⠀⠉⠉⠉⠉⠀⢀⣒⣒⣒⣀⣀⣀⣈⣥⠥⠽⣦⠀⠀⠀⢸⠁⠀⠀⠇⡄⠀⠀⠴⠀⠀⠘⠂⠘⠀⠄⠀⠸
⠀⠀⠱⠏⠠⠤⠤⠴⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⢸⠀⠄⠘⠀⡇⠀⣐⠀⠀⡀⠀⢰⠀⠀⡂⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡆⠀⠁⠈⠈⠁⠙⠀⠉⠈⠉⠈⠀⠁⠙⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣶⣶⣯⣥⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢸⠇⠀⠀⠀⠓⡀⠀⠀⠀⡄⠴⠄⠄⠀⠐⢠⣔⠂⠀⠀⣀⠀⡀
⠀⠀⠘⠠⠐⢀⠀⢠⣧⠀⢸⠇⠀⠀⠀⣠⣾⠂⠀⢾⣤⣀⡀⠀⠈⣅⠀⠀⠒⠀⠀⠀⠐⠀⠈⢸⣅⡀⠀⠀⠁⠄⠎
⠀⠀⠀⠀⠁⠀⠀⠀⠛⠋⠐⠀⠀⠈⠀⠍⠁⠀⠀⠀⠀⠉⠻⠂⠀⠉⠐⠀⠀⠉⠱⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠒
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢠⠿⠶⢿⠣⠀⠀⠀⠀⠀⠄⠀⠀⡀⠐⠄⠀⠀⠔⣂⡔⠀⠀⢸⣿⡿⠀⠀⠠⠆⢀⠀⠂⠀⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣤⠀⠀⠀⠠⢺⢏⠭⠀⣠⡌⠉⠀⠀⠀⠠⡁⠈⠀⠈⠄⠀⡶⠀⠀⠀⠂⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⣀⠀⠀⣀⣄⠄
⠀⠀⠠⢣⡁⠐⢂⡀⠀⣼⣷⡭⣣⡄⠀⣿⣅⣼⠂⢠⠃⠎⠤⠀⠀⠀⠀⠀⠠⠌⠀⣀⠑⡂⠀⠈⢿⠀⠀⢀
⠀⠀⠐⠋⠀⡍⠀⠀⠐⠹⢟⣘⠛⠁⠀⠿⠞⠋⠀⠀⠠⠏⠁⠀⠀⠀⠀⠀⠈⠹⠉⡖⡌⠂⠀⠳⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⣀⠀⠀⣀⣀⣀⣀⡤⣤⣠⢤⣄⣀⣀⣠⣤⣤⣤⠀⡔⡖⣿⣱⣽⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡼⠿⠘⠁⠀⠓⠈⠐⢛⠛⣫⠏⡇⢟⡕⠙⠟⠛⠉⠉⠉⠉⠉⠁⠉⠩⠤⠀⠉⣼⣯⣿⠅
⠀⠀⠀⠀⠀⠀⢀⢔⡕⠁⠀⢀⢤⡠⡀⠀⠀⡰⣡⣾⡟⡼⢱⢿⡇⠀⡆⠀⠀⢠⣠⣤⣤⡄⠀⠀⠀⠀⡇⢩⣏⣿⡄
⠀⠀⠀⠀⠀⡠⡣⠋⠀⠂⠀⠀⠂⠑⠁⠀⠀⣿⣿⣧⢊⡆⣿⣿⡇⢠⠃⠀⠀⠊⠛⠛⠛⠀⠀⢀⠀⡜⡀⣾⣿⣷⠃
⠀⠀⠀⠀⡔⣵⣅⣤⣤⢠⢂⡀⡴⠴⢂⣀⣀⣰⡿⢃⢪⡇⣿⣫⣄⣁⣀⣀⠃⣀⡀⢃⣐⣛⣫⠭⡜⣸⡞⣼⣿⠃
⠀⠀⠀⢬⣬⣭⣿⠿⣶⠶⣶⣮⣭⣥⣭⣍⣙⣻⣣⣇⣬⣇⣿⣭⣿⣿⣿⣿⣿⣿⣿⣯⡭⠼⠶⠶⡾⣻⣿⣿⠁
⠀⠀⠀⢸⢻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣗⣒⣶⠿⣿⣄⣀⣈⣭⣙⣶⣖⣶⣿⣿⠇
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⢯⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⢀⣤⣄⣀⣀⡴⢦⣀⣠⣿⡿⠛⠉⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⣠⠞⠁⠈⠉⢁⣠⣶⡿⠟⠉⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⡾⠁⠀⠀⣠⣴⣿⠟⠉⢀⣠⠶⣤⣀⣀⣠⡄⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⡴⠋⠀⣀⣴⣿⠟⠋⢀⣠⠶⠋⠁⠀⠀⠉⠉⠁⠹⣦⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⣀⣴⣾⠿⠋⠀⢠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣤⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⣤⣾⡿⠛⠁⣀⠀⠀⢻⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣠⣶⡿⠛⠁⢀⣴⠞⠉⠀⠀⣼⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡿⠟⠉⢀⣤⠞⠋⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⢠⣾⣿⣿⠟⠉⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢀⣠⠶⠋⠁⣠⣴⡿⠟⠉
⠀⠀⠀⣰⣿⠋⢸⣿⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣤⣾⡟⠀⠀⠀⠛⠁⣠⣴⣿⠟⠋
⠀⠀⠀⢿⣧⡀⠀⠹⣿⣄⠀⠠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠈⠻⢿⣦⣄⠘⢿⣧⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠀⢀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⠻⣷⣄⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠙⠛⠉⠙⠳⣦⠞⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣆⣀⣠⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡱⠁⣀⢠⡦⣴⣆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣹⡃⢴⠆⠈⣻⣥⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠰⣈⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⠀⠀⣬⣿⣶⣷⡌⢃⢨⣥⡀⠀⠘⢠⠇⠀⠀⠀⣠⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡁⠀⠀⠺⢛⡿⢿⡏⠛⠈⠀⠀⣧⡿⠀⠀⡔⢋⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⢘⠷⠻⣶⣷⣤⠆
⠀⠀⠀⠀⠀⣀⣤⣦⠀⠀⠠⠿⠃⠀⠀⢀⢤⣤⡀⠀⢀⠄⠠⡀⠀⠀⠀⠀⠀⠀⣸⣟⠃⠶⢰⣿⣿⠶
⠀⠀⠀⠐⠅⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠠⣈⡩⠃⠀⠸⢷⡅⠀⠀⠀⠀⠀⢀⣀⡀⣹⣄⠺⠃⣸⣿⠛⠂⠀⣈⣷⠆
⠀⠀⠀⠀⠀⢀⣠⠿⢲⣤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶⠀⣀⠀⠀⢨⡽⣧⠈⠈⠛⠹⠃⠀⠀⠀⠘⠴⠋
⠀⠀⠀⠀⠀⠒⡏⠑⢂⣿⣿⡁⠀⠀⠀⢰⣶⠀⠀⠀⢠⣜⡿⣿⣿⣿⣶⠂⠀⠑⠉⠀⠀⢠⣴⣂⣠⠶
⠀⠀⠀⠀⠀⠘⢻⣌⣁⢾⠿⠆⠀⢀⡀⢸⠏⡇⠀⠚⢟⣀⢤⠍⣉⠘⣟⠀⠀⢀⣄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠉⠀⠀⠀⢌⡾⠃⢨⢀⠇⠀⠀⠾⣧⣤⣈⣉⣴⠛⠁⡰⣁⠇⠀⠀⠀⠀⠴⢶⣤
⠀⠀⠀⠀⠀⠀⠘⠛⠂⠚⠃⠀⠀⠀⠀⠀⠁⣀⡀⠀⠀⠉⠀⠙⠁⠀⠀⠰⠿⠏⠀⠀⠀⢀⡀⣀⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠉⣁⡀⡀⠀⠀⠀⠀⠠⢶⢷⠀⠀⠀⢀⡙⢾⠿⢿⡆⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢀⣼⡉⠏⠹⣤⠀⠀⠀⠀⠐⠊⠀⠀⠀⢸⠟⢩⠻⡏⣻⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⢿⣿⣮⣨⡏⠀⠀⠠⠀⠀⠀⣶⢆⠀⠛⢱⣦⡤⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁⠀⢠⡴⠁⠀⠀⠀⠈⢣⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠿⣷⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣄⣀⣤⣤⣴⠶⠛⠋⣀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣁⣠⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣾⠿⠟⠛⠻⠿⣶⣤⡐⢶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡉⠙⠛⠻⠿⠿⠿⠏⣰⡿⢋⣤⣶⡿⠿⢿⣶⣄⠙⢿⡆⠹⠿⠿⠟⠛⠋⢁⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣤⡄⢸⡿⢡⣾⣿⡏⢰⣶⣦⣽⣿⣷⡀⣴⡄⢠⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⣿⡇⢸⣿⣿⣷⣬⣈⡙⢿⣿⣿⡇⠘⠃⢸⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣤⣄⢸⣷⡈⢿⣿⣏⠻⠿⠟⣸⣿⡿⢁⣿⠇⣀⣀⣤⣤⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣷⣌⠛⠿⣷⣤⣾⠿⠋⣠⣾⠏⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣧⡌⠛⠿⣶⣦⣤⣴⣶⠿⠛⠡⠾⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⢀⠤⠀⠀⠐⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠐⢤
⠀⠀⠀⢔⠀⣀⠐⠁⠈⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠱⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢔⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⡀⠔⠓⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠰⣁⠄⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢀⣀⡀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡠⠒⠘⠀⠓⠒⠜⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⢮⠤⣠⠂⡆⠀⠀⠀⠀⠀⢀⠀⠘⠀⠈⠐
⠀⠀⠨⢍⠁⠀⡴⠀⠀⠵⠄⠘⠢⠴⡀⠀⠀⠀⠀⠀⢞⠱⠎⢄⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡎⢖⠄⠀⠀⠀⠀⠀⡠⡄⠍⠀⠀⠀⠀⡤⢄⡠⠃⠀⠂⠀⠠⡠⠒⠃⠀⠀⠀⠀⠀⢄
⠀⠀⢠⣃⡀⠀⠀⠀⢀⠀⠀⠐⠂⡨⣸⠀⠀⠀⢨⠃⠀⠀⠀⠐⢠⠒⠽⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠁⠣⡀⡠⢜⠒⠁⠀⡔⠒⠁⠀⠀⠀⢀⠜⣈⠀⡐⢀⠤⣈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠘⠘⠈⠒⢲⣘⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠐⢦⠄⠀⠀⠘⠛⠁⠀⠀⣠⣤⡀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⠰⠖⠀⠀⠀⠀⡀⠀⠰⠂⣤⡀⠀⣀⡀⠀⠈⠁⠀⠀⠐⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠀⠀⢀⠀⠀⢐⡆⠀⡰⠋⠀⠠⠀⠀⠀⠀⠉⠀⣠⣶⡂⠠⡀⠀⠲⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠄⠊⠁⠀⠀⠈⠜⠀⠢⠌⠀⠀⢠⡄⠀⠀⣿⡄⠈⠓⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠊⢀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠠⡲⠄⠁⠀⠀⠀⠸⣳⡀⠀⠈⠳⢜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠂⢤⠀⢀⡀⠀⠀⠀⠐⠀⠀⢀⣄⠀⠀⢀⠀⡤⠀⠀⠀⠛⠦⡀⠀⢀⡁⠀⢤⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠀⠀⠀⠀⠀⠀⠀⠉⢲⣤⣉⠀⠈⠲⡄⠀⢻
⠀⠀⠀⠀⠀⠀⡁⠀⠛⠀⠀⠀⠹⠀⠘⠁⠀⠠⠆⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠘⠃⢠⡀⣦⠀⠀⠀⠀⠀⠍⠁⠈⠷
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠙⠃⣀⢀⡀⠀⠀⠀⠐⠂⠀⠀⠒⠀⡌⠀⠀⠐⠴⠤⠃⠉⡀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠤⠑⠀⢄⠀⠠⡄⠀⠉⠀⡀⠄⢞⠀⠀⠘⠒⠂⠀⠀⠀⠀⠄⠐⠐⠠⠂⠈⠀⠀⢠⡉
⠀⠀⠀⠀⠀⠈⠇⠀⢠⠠⠄⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⢘⠀⢠⡤⠀⠑⣤⠀⠒⠃⠀⠀⠀⣠⠨⠀⠀⠀⡇⠤⠀⠀⠂⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠂⡀⠆⠁⠔⢰⢰⡈⠀⢀⡈⠃⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡍⠂⠀⠁⠀⠘⠃⠀⠠⠄⠀⠈⠁⠀⢀⠢⠐⠂⠂⠠⢀⢀⠀⠀⠀⠘⠀⠈⡀⢀⠀⠀⠀⠀⠰⠒
⠀⠀⠀⠀⠀⠀⢤⣄⣉⠀⠠⣀⠀⠀⠀⢖⠀⠈⢁⡌⠦⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⢀⠀⢀⡀⠀⠀⠀⡆⠒
⠀⠀⠀⠀⠀⠀⠀⠋⠁⢠⣀⡀⠀⠀⠀⠸⠀⠀⠀⣀⠄⠀⠐⠃⠀⠘⠊⠀⣀⡀⠄⠀⠀⠀⢂⠀⠐⡂⢠⠄⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠶⠛⠃⠀⠆⠀⢀⡁⠀⠀⠁⠖⡄⠤⠒⠈⠈⠢⢄⣀⣀⠶⠀⠀⠀⠈⢠⡈⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⣄⠘⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠁⠀⠀⢀⠐⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣈⠁⠀⠀⣆⡀⠀⣀⠀⠀⠀⠰⠚⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠛⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⠇⠀⠛⠁⠀⠈⠂⣀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⡀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⢳⠀⠀⢤⢅⠄⡪⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⢀⠀⢠⡏⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⢦⠀⠀⠀⠀⣸⣄⠀⢄⠈⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⠇⠀⠀⠀⢻⠘⡳⣄⠈⠀⢜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⠉⠀⠀⠀⠀⠘⢧⢊⠄⠱⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡡⠺⡀⢀⡀⠀⠀⠀⠈⠳⡁⠂⢀⠝⢦⡀⠀⠀⠀⠀⠀⡰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠂⢐⠸⡇⡜⣇⠀⠀⠀⠀⠀⠈⢮⢐⠀⠠⢳⡄⠀⠀⠠⠊⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠑⡨⢐⣼⠁⠑⢌⠣⡘⣦⢄⡀⠀⠀⢳⠁⡨⡄⣷⠀⠀⣁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢂⠅⢊⠐⢹⡀⠀⠀⠉⠓⢸⡆⠱⡄⠀⣘⠈⡄⠀⢸⠀⠀⢡⡆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⡆⠀⢘⢢⡀⠳⣄⠀⠀⠀⡼⠐⠉⡇⣠⠏⠀⠃⠀⣚⠀⠠⡟⢇
⠀⠀⠀⠀⠀⠸⠓⠤⣀⠀⠀⠀⢣⠀⠘⡄⠈⢦⠈⢳⣀⡜⠁⡄⠀⠿⠁⡠⠁⡂⢨⡇⠀⢸⠇⢺⡀
⠀⠀⠀⠀⠀⠀⠣⢈⠈⢳⡄⠀⠀⢳⡀⠙⡄⠀⠑⡄⠛⢀⠎⠘⡀⢠⠎⠀⡰⢁⡜⠀⣠⡯⠂⠙⡇⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠠⢤⣈⡐⠇⠀⠀⠀⠘⣆⠘⠄⠀⠘⣀⠎⠀⢀⠖⠁⠀⡰⢁⣜⣠⠾⠁⠀⠀⢀⡇⠀⠀⢀⠜⢽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠳⢦⣀⠀⠀⢸⡄⠀⠀⠀⢻⠀⠀⠈⠀⠀⢀⠁⠚⠉⣀⠤⢒⠋⢀⣼⠃⠀⠰⢁⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢐⠙⢦⠀⢸⡇⠰⠀⠀⠀⠀⠈⠀⠀⠀⢰⢀⠔⠉⠀⡠⠃⢀⡘⠁⠀⠀⢋⡠⠖⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠣⠈⣇⡾⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢰⠁⢰⠟⢀⣴⠂⣰⠯⢐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⢰⠀⢃⠙⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡏⣰⢛⡏⣸⠅⡈⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠘⡀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢃⠀⡿⢊⠐⣧⡇⠜⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠃⠘⡆⠋⠸⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠈⢢⡅⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠅⠢⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠤⠀⠀⠀⠀⣀⣀⠀⠠⠌⠑⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠐⠈⠔⠈⠄⢂⠡⢀⠠⢐⠠⢐⠤⢂⠠⠐⠠⢐⠠⢐⠠⡰⠜⠢⠄⡂⠅⠢⠒⠄⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⢶⣿⡶⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣉⣉⠤⣉⣉⣛⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣤⣤⣤⣤⠄⠀⠀⢰⣿⡇⠀⠀⠠⢠⣤⣤⣤⠄⠀⠀⠀⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⡏⠀⢸⣿⣿⣿⡟⠛⠛⠛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⣿⣶⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠀⠉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣄⣹⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣄⣼⣿⡆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⡀
⠀⠀⢠⣦⣿⡟⠈⢹⣧⣦⠀⠀⠀⣶⣼⣿⢿⣿⣷⢀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣦⠀⠀⠀⠀⠀⣿⣿⡿⢿⣷⡄⠀⠀⠀⢀⠘⣿⣿⠻⣿⣿⡄
⠀⠀⠸⣿⠋⢀⣦⠘⢻⣿⠀⠀⠀⣿⣿⠇⠈⢻⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣦⣿⠟⠀⠸⢿⡧⠀⠀⠀⢸⣷⡟⠁⠀⢸⣿⡇
⠀⠀⠀⠻⣆⠻⠿⢃⡾⠃⠀⠀⠀⠹⣧⠀⠀⠀⣹⠟⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠹⣇⢠⣿⡆⠸⠃⠀⠀⠀⠘⢿⡇⠀⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⢀⠀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇
⠀⠀⠀⠀⢠⣾⣿⣿⠂⠀⠀⠀⠀⣀⣸⣧⣿⣿⣼⣇⠀⠀⠀⠀⠀⣠⣿⣿⣿⣦⠀⠀⠀⠀⠀⡀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆
⠀⠀⠀⢀⣿⣿⣿⡟⢠⡀⠀⠀⠀⢿⣿⣿⠿⣿⣿⣟⡄⠀⠀⠀⣰⣿⣿⠟⢿⣿⣇⠀⠀⠀⢸⣿⣾⣿⠉⢿⣷⣿⡆⠀⠀⢀⢰⣿⣾⣿⣿⣸⡇
⠀⠀⠀⣦⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⠋⢀⡀⢻⣿⡇⠀⠀⠀⣿⣿⠋⠀⢸⣿⣿⠀⠀⠀⠸⣿⡟⠏⢀⠸⢿⣿⠃⠀⠀⢸⣻⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁⠀⠀⠀⠈⠻⠀⣾⣷⠨⠏⠀⠀⠀⠀⠹⠿⠀⠀⠛⢹⠃⠀⠀⠀⠀⠙⠄⢰⣿⡆⠸⠋⠀⠀⠀⠈⠳⣿⣻⣿⣿⠿⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣄⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠀⠀⠀⠀⠀⠀⣴⣿⣇
⠀⠀⠀⣰⣦⣿⣿⣿⣷⡀⠀⠀⠀⢰⡄⠸⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⢰⣿⣿⣿⣠⣄
⠀⠀⢀⣿⣿⣿⠉⢿⣿⣧⠀⠀⠀⣾⣿⣶⣿⣿⢁⣴⠀⠀⠀⠀⣆⣻⡿⠀⠹⣿⣷⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⣄⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣄
⠀⠀⠈⣿⡏⠀⠀⠀⣿⡟⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⠁⣴⡆⢸⡿⠀⠀⠀⠀⣶⣿⢯⠈⣹⣿⡿⠀⠀⠀⢻⣿⣿⡏⠈⢻⣿⡿
⠀⠀⠀⠈⠻⠤⣀⠴⠛⠀⠀⠀⠀⠀⠛⠿⣿⠿⠟⠁⠀⠀⠀⠀⠈⠻⠦⣿⠿⠞⠁⠀⠀⠀⠀⠈⠻⠀⠀⠁⠟⠁⠀⠀⠀⠈⠻⠏⠃⠀⠀⠟⠁

⠀⠀⠀⠀⠠⠀⠀⢀
⠀⡀⠀⠂⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⢀⢶⡅⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡈⠂⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠘⢻⠃⠀⠀⠀⠀⠈⢶⣽⣦⣌⡁⢀⠀⠀⠀⠀⠀⠀⢀⠺⠁
⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠂⠀⠀⠀⢨⢢⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠋⠀⠀⠀⠀⠀⠈⠋⠁⠲
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⣇⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠂
⠀⠘⠉⢠⠀⢰⠆⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣄⣐⣀⣀⠀⠀⠀⢀⡟⠈⣹⡆⠀⠀⠀⠀⢀⣀
⠀⠀⡀⡀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣯⣩⡄⠀⠀⢸⡅⢀⡆⡀⠀⠀⡀⠄⢠⢼⡇
⠀⠀⠉⠛⠡⠒⠒⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠃⠀⠀⠈⢧⠾⠙⠀⠀⢴⣡⠄⣂⡿⠁
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠁
⠀⠀⡠⡄⠈⠉⠉⠙⠛⢦⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡠⠴⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⢀⡠
⠀⠀⠁⢰⠀⣠⡀⠀⠀⠀⠈⠀⠀⣒⡂⠀⠀⠀⠀⣀⢲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣨⣧⣴⣧⡽⠅⣤⡀
⢀⠤⢴⣼⣶⣡⠁⠀⠀⠀⠀⠀⣆⠇⣸⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⢸⢼⠿⣿⣻⡏⡡⠴⢴⡆
⠈⠓⠒⠻⡙⢾⢭⡑⣄⠀⠀⠸⢸⣼⡇⠀⠀⠀⠀⠈⠑⢍⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢱⡀⠟⠿⡛⢹⠎⡗
⠀⠀⠀⠀⠑⠚⢶⣻⡅⠀⠀⠀⠻⣟⣷⠀⠀⠀⠀⠀⠀⠋⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠥⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢀⢠⠄⠖⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠠⠂⠀⠂⠈⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠄
⠀⠀⠤⠞⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠠⢀⠂⠈⠒⠀⠀⢸⢁⡎⠱⠀⠀⠀⠀⠀⠀⢄⡸⠖⠯⠈
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⠐⠀⠒⠂⠰⠊⡜⠉⠣⠈⠀⠀⠀⠀⠀⠶⠇⢴⣿⠒⠂⣀
⠀⠀⣤⣔⠀⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠄⢢⠀⠀⠀⣀⡀⡀⠠⠀⠀⠀⠀⠀⠀⡘⠀⠀⠈
⠀⠀⠀⠀⠶⠀⡄⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢈⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦
⠀⠀⠀⠀⣾⣿⡇⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⣿⣟⣿⠟⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢹⣿⡇⠀⣿⣏⣴⠟⣡⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣏⣴⠞⣁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣏⣤⠞⢋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⡏⣡⠶⠋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⡟⢁⡴⠛⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⡿⢋⣤⠞⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⢸⣿⡇⠀⣿⡷⠋⣡⠞⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⢸⣷⣾⣵⣾⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣿⣿
⠀⠁⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠉⠛⠿⢿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⡿⠿⠛⠉⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⣿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⢀⡟⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢻⡀
⠀⠀⠀⠀⠀⣼⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⣧
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠸⣇
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⢀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⡀
⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠿⠿⠿⠿⠆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⠿⣶⣶⣶⣶⣶⣶⣶⣶⠿⠟⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡆
⠀⠀⠀⠀⣿⣿⠀⢰⣶⣶⣶⣶⣶⡄⢠⣶⣶⣶⣶⣶⡄⠀⣶⣶⣶⣶⣶⣦⠀⢴⣶⣶⣶⣶⣶⠀⢰⣶⣶⣶⣶⣶⡄⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⠀⢸⣿⢞⣥⠞⣿⡇⠰⣿⠞⣩⠞⣻⣗⠀⣿⡷⣫⡴⢛⣿⠀⣿⡷⢋⡵⢛⣿⡇⢸⣿⢞⣡⠞⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⠀⢸⣿⢋⡵⠞⣿⡇⠀⣿⠞⣡⠞⢻⣧⠀⣿⡟⣩⠶⢛⣿⠀⢸⡷⢋⡴⠛⣿⡇⢸⣿⢛⣵⠞⣿⡇⠀⢸⣿⣷⣶⣶⣄
⠀⠀⠀⠀⣿⣿⠀⢸⣿⢋⣤⠞⣻⡇⠀⣿⢋⣡⠞⢻⣯⠀⣿⡟⣡⡴⢛⣿⠀⢸⡟⢋⡴⠛⣿⡇⢸⣿⢋⣤⠞⣻⡇⠀⢸⣿⡏⠉⣿⣿
⠀⠀⠀⠀⣿⣿⠀⢸⣿⢋⣤⠞⣻⡇⠀⣿⠋⣡⠞⢻⣿⠀⣿⡟⣡⡴⢛⣿⠀⢸⡟⢉⡴⠟⣿⡇⢸⣿⢋⣠⠞⣻⡇⠀⢸⣿⡇⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⢸⣿⠋⣠⠞⣻⡇⠀⣿⠟⣡⡶⢻⣿⠀⣿⡟⢁⡴⢛⣿⠀⢸⡿⢋⣤⠞⣿⡇⢸⣿⠋⣡⠞⢻⡇⠀⢸⣿⣇⣀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⢸⣿⠛⣡⠞⣻⡇⠀⣿⠞⣡⡴⢻⣿⠀⣿⡟⢋⡴⢛⣿⠀⢸⡿⢋⣤⠞⣿⡇⢸⣿⠛⣡⠞⣻⡇⠀⢸⣿⡿⠿⠿⠟
⠀⠀⠀⠀⢿⣿⡄⢸⣿⠞⣡⠞⣿⡇⠀⣿⡞⣡⡾⣻⣿⠀⣿⡷⢋⡴⢋⣿⠀⢸⣿⢋⣴⢞⣿⡇⢸⣿⠞⣡⠞⣻⡇⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠈⣿⣾⣷⣿⣿⡇⠀⢿⣿⣷⣾⣿⣿⠀⢸⣷⣿⣾⣿⣿⠄⠸⣿⣿⣶⣿⣿⡇⠈⣿⣾⣷⣾⣿⡧⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⢀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣿⣿⣿⣿⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠀⠀⠀⣶⣶⣶⣶⣶⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⣿⠉⣿⠉⠉⠉⡏⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⣿⣀⣿⣀⣀⣠⣧⣸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠟
⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⠀⠀⠄⠀⠠⠀⠀⠄⠀⠠⠀⠀⠄⠀⠀⠆⠀⠀⠄⠀⠀⠀⠆
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠷⠿⠿⠿⠿⠿⠿⠿⠀⠀⠦⠤⠤⠠⠤⠄⠤⠤⠠⠤⠄⠤⠤⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠻⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⡿⣻⣿⣄⣠⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠿⠋⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠉⠻⣿⣿⣿⠃⠀⠀⣿⣿⡝⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⣹⣿⠿⡄⠀⠀⠙⠿⠁⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣾⣿⣿⡄⠻⠦⠀⠀⠀⠀⢸⡿⠀⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡴⠋⠀⢸⢸⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⡇⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⠁⠀⠀⢸⠘⢚⡥⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢸⠀⡇⡇⢸⠀⠀⠀⢀⢇⡀⠀⠀⣿⡀⢠⡀⠀⢣⠀⠀⠀⢹⠀⠀⢹⢀⣀⠀⠸⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⡇⢸⢸⠀⡇⡇⢸⠀⠀⠀⢸⢸⠁⠀⢠⡧⡵⣌⡏⢶⢬⣇⠀⢸⠈⡇⠀⠀⡏⠉⡇⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⣀⣧⣸⣇⣇⣸⣸⣀⣇⣇⣸⣀⠀⠀⢸⢸⠀⠀⡾⠰⠀⠈⠉⠀⠆⢹⠀⢸⠀⡇⠀⣠⠕⠸⢵⡒⠒⠚⠉⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡆⢸⡇⠀⠀⠠⠄⠀⠀⢸⡆⢸⢀⡇⠀⠛⠚⠒⠚⠃⠀⢀⣀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠳⡙⣾⠙⢤⡀⠻⠟⢀⡴⢋⡹⣼⡞⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⢤⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠸⠬⢷⠀⠀⠀⠀⠀⠈⠳⠞⢦⡏⠑⠊⢹⡤⠚⠓⠙⠀⠀⠀⠀⠀⠀⢠⢴⡏⠀⠀⠀⠹⡽
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢐⣒⣒⡂⢸⠀⠀⠀⠀⠀⣀⣠⡴⢻⠁⠀⠀⣈⣟⠦⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡞⠉⠀⠈⠙⡇⠈⠭⠭⠅⢸⠀⠀⡰⠋⠉⠁⢼⠀⡏⠀⠀⠈⠀⢸⠀⣹⠈⠉⠓⣄⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣤⡤⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠷⠶⢶⠶⠖⠚⠀⠀⡇⠀⠀⢀⡜⠁⡗⠒⠒⠒⠒⢻⠈⢣⡀⠀⠀⢸⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠛⠓⠲⡇
⠀⠀⠀⠀⢠⠜⠃⠀⣄⢸⢀⡄⠀⠸⢤⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡏⠀⡘⠀⠀⠀⠘⡃⠀⠀⡇⠀⠀⠠⡏⠉⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢏⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⢸⡀⢹⠀⠀⠀⢠⠃⣰⠃⠀⡆⠀⠀⡇⠀⠀⡇⠀⣀⣀⣉⣉⣓⣀⡀⠀⡇
⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⢧⠈⡇⠀⠀⡼⢠⠏⠀⠀⡗⠀⠀⢻⠀⠀⡇⠀⠭⠭⠭⠭⠭⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠘⡆⢻⠀⢰⠇⡞⠀⠀⢰⡇⠀⠀⢸⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⣸⡀⣀⣀⣸⣸⣆⣜⣼⢀⣀⣀⣸⡀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠉⠈⠁⠉⠉⠈⠁⠉⠁⠁⠉⠁⠉⠉⠉⠉⢹⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠯⠥⢜⣦⣠⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣿⡹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣶⠟⠁⠀⠀⢀⡧⣙⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠤⠤⠴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣾⣿⣁⡀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣾⣶⡿⠿⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⡆⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠙⠻⣟⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⠀⢀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⢩⣍⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣰⣿⣏⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⣿⣯⣿⣿⡄⠤⠀⠀⣀⡤⠤⣄⠀⠠⠤⠤⠄⣾⣿⣿⣿⠛⠿⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠛⣿⣿⡅⠀⣶⣦⠀⠀⠀⣠⣴⣾⣷⣦⡀⣿⡟⠙⢯⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⠒⣿⣿⠧⣼⣿⣧⣤⣤⣾⣿⣯⣍⣩⣿⣿⣿⣷⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠇⠈⢿⣿⣿⣿⠿⠁⠉⣉⣽⣿⠏⢽⣿⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣀⣀⣀⣉⣉⣡⣤⣤⣀⠈⣛⣁⣀⣿⣿⡄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠉⠁⠀⠀⠀⢸⣇⠀⠉⠛⢤⣀⠀⣹⢹⣧⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⡀⠀⣰⠞⠶⣀⢿⣄⠀⠀⠀⠈⠉⠻⢀⣽⣇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠴⠶⢾⣿⠇⠀⣿⣦⠀⠈⠃⠙⣷⣦⣀⠀⠀⠀⠀⠙⣿⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⣄⠀⠀⠈⠆⠘⠏⠳⢤⣀⣀⡄⠘⣷⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢷⣤⣤⣤⣿⣶⠀⠀⠀⠐⢶⣿⣾⣿⣻⣿⣦⠈⠇⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠁⠙⢏⡀⠠⣧⡀⠀⠀⠀⠙⢿⣿⣿⡙⢧⣄⠈⠀⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠁⠺⣿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣄⣽⣧⠀⢠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⢀⡀⢀⠈⠉⠛⢿⣷⡄⠐⠀⠀⠀⠀⠋⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢠⣾⣿⣧⣄⠀⠀⠀⠉⠑⠲⣆⡀⠀⠀⠀⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣾⠿⣷⠻⣿⢿⡄⠀⠀⠀⠀⢻⣅⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠀⢠⢜⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢲⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠿⢿⣿⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠠⠴⢬⣭⠷⢶⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠳⢦⣶⣦⣀⠂⠠⠍⢸⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠙⠓⠚⠂⠿⠿⠿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣄⣀⣀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣛⡿⢿⣿⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⢛⠹⠁⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡕⣃⢹⢰⣿⠷⠲⣦⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⣿⣅⠀⢿⣟⠀⢶⡏⣸⢛⡿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠌⣻⣶⠋⠉⢛⣾⡷⣷⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⢳⣦⡘⠦⢸⣇⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢋⣤⣠⣾⣟⠛⢿⣿⠉⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣴⣶⡤⣤⣞⡃⠀⠀⠀⠙⣿⡧⠾⡛⠛⠛⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣻⣿⣿⣿⣿⣿⡇⢿⣿⡃⠀⠀⢀⣼⠟⠳⣶⣿⣶⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⡇⢿⡋⢡⣄⡀⣀⠏⠀⣧⣉⣰⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠈⠉⣿⡃⢀⣾⢿⣿⡛⢻⠉⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣿⣿⣿⣿⣿⣿⣿⣇⢸⣇⣀⣴⣿⠂⠀⣼⡿⠈⣷⠈⢲⣶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣵⠃⠀⢰⣿⠃⠀⠉⠄⢡⢻⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⢃⠎⠀⡼⠂⠀⠀⠀⠀⠈⢆⢻⡍⢌⢷⣀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⡿⢡⠏⠀⣼⣤⡶⢾⡷⠶⣤⣄⠈⠊⢿⡌⢣⡻⣷⣄⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠛⢰⠋⠀⣴⣿⣿⡀⢸⡇⠀⠈⢻⠀⠀⠈⢻⡄⠀⢹⣾⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢻⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣹⡇⢀⣤⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡌⢿⣕⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣿⠀⣞⠇⡐⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣷⢀⠀⡄⠀⠀⠈⣿⣿⢞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢸⣿⣿⡃⠀⢠⠀⣄⠀⠀⠸⣿⠀⢻⣿⡟⠈⢀⣡⢀⠁⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣾⣿⣿⣧⠀⣞⡀⢿⣷⣤⣠⣿⣤⡿⠋⠀⠀⢿⡟⢸⣼⣈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢠⣿⡇⠀⠀⠀⠉⠛⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⣿⣿⢳⣼⣇⢠⢰⢠⣰⢠⣧⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣧⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠦⠤⠤⠈⢉⣉⣉⣉⣋⣙⣛⣛⣛⣋⣉⣉⣉⣉⠉⠠⠤⠶⠤⠤⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⡤⠖⠛⠊⣩⣭⣤⠔⠶⠶⠒⠒⠂⠉⠉⣩⣭⣩⣍⣉⣉⣩⣭⣭⣍⠉⠉⠙⠒⠒⠶⠶⠦⣬⣭⣉⠑⠛⠒⢤⣤⡀
⠀⢀⣠⠶⠛⣋⣠⠶⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠶⣄⣉⠛⠶⣄⡀
⣴⠟⠁⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠙⠒⠒⠖⠒⠒⠒⠒⠒⠒⠉⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⠈⠻
⣿⡀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣍⠱⢆⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠏⣩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⣿⠙⢦⡀⠙⠳⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠮⣭⣥⡶⠶⠒⠒⠿⠶⣬⣭⡴⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠋⢀⣠⠐
⣿⠀⠀⠙⠳⠶⣀⣈⠙⠓⠶⢦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠶⠶⠛⠉⣀⣠⠶⠋⠁
⣿⠀⠀⠀⠀⠀⠈⠉⠛⠓⢤⣤⣄⡉⠉⠙⠛⠛⠒⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠖⠚⠛⠛⠉⢉⣉⣤⣤⡴⠚⠛⠉⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠒⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠒⠋⠉⠋⠁
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⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⢹⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⡇⠀⠀⢸⣇⣀⣀⣀⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣬⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⣿⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⣿⡀⠀⠀⠈⠛⠳⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠶⠞⠛⠁⠀⠀⢀
⣿⠙⠷⣤⡀⠀⠀⠀⠀⠉⠉⠒⠶⠶⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠶⠶⠚⠉⠉⠀⠀⠀⠀⢀⣤⠞⠉
⣿⠀⠀⠀⠙⠛⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠶⠛⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⠖⢤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⠶⠞⠋⠉⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠁
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⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠤⠶⠴⠶⠴⠦⠴⠶⠦⠴⠶⠶⠶⠴⣦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠦⠴⠶⠶⠶⠶⠶⠶⠾
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠉⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠶⠶⠛⠋
⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⡉⠉⠙⠛⠛⢓⣒⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣒⡚⠛⠛⠛⠉⠉⣉⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠅⠀⠨⠁⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣴⣶⣦⣄⠘⣿⣿⣿⡄⠀⠀⠀⠀⠐⠀⠀⠀⡠⢖⠛⢷⡄⠀⠀⡤⡤⡙⠆⢀⣇⡜⡠⢠⣆⡱
⠀⠀⢀⣀⠈⠉⠉⣁⠀⠹⣿⣿⣿⡀⠀⠀⢠⡜⠀⡰⠫⠄⣤⢀⡼⠁⠀⠀⡔⡄⢀⣠⠺⡶⣠⢌⠁
⠀⠀⢿⣿⣿⣦⣄⠈⠓⠀⠹⣿⡿⠃⠀⠀⠀⣠⠻⠐⢋⠂⡠⠋⠀⠀⠀⠀⢿⣷⠧⠜⠉⠀⠿⠲⣏⡢
⠀⠀⠀⠉⠻⢿⣿⠇⢠⣿⠀⠀⠀⠀⠀⠀⢰⣇⣙⣠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠻⣑⠄⣮⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣦⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⡴⣫⠭⢭⣶⠀⠀⠀⢀⣼⣛⣯⣷⡎⠁⢰⠆⠀⠀⠀⣀⣄⠠⣿⣽⣻⣿⠀⣠⣶
⠀⠀⠀⠀⠴⠃⣀⣀⣀⢇⡇⠀⠀⠀⠀⠀⠀⢀⠿⣯⣿⡽⢇⡀⠀⠀⠀⠀⠈⠿⠏⠀⠉⠉⠁⢀⣀⡉⠉
⠀⠀⠀⡔⡫⠭⠿⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢩⡄⠀⢀⣠⣀⠀⠀⠀⠀⢰⡟⣾⣆⠀⢿⣽⣟⡷
⠀⠀⠀⢧⡧⠐⢗⣦⣴⡶⠶⠶⠆⠀⠀⠀⢺⣿⠇⠀⠀⠀⠀⢯⣿⣿⢁⠀⠀⠘⣿⣽⠛⣾⡀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠙⠙⠋
⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠎⠀⠀⡸⠀⡞⠉⡉⠉⠳⣄⠀⠀⠀⣸⡏⠀⠙⠳⠶⠦⢀⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧
⠀⠀⠀⡄⠀⠘⡄⠀⢹⠀⠄⠀⠂⢻⠀⠀⠀⢙⣤⡶⠿⠛⣃⣴⠟⠁⠀⠀⣠⢴⣶⣶⣦⣤⡉⠉⠉
⠀⠀⠀⠈⠑⠊⢀⡴⠋⠀⠀⠂⠁⣾⠀⠀⠀⠛⠡⠶⠾⠛⠋⣥⣤⡄⠀⠐⣿⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢾⡀⠈⠡⠈⠈⣰⠏⠀⠀⠰⢿⣷⣦⣤⢤⣤⣤⣄⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡄⠀⠀⠀⠀⠀⡞⠻⣽⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣶⣗⠀⠀⠀⠀⠀⢲⣿⠆⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡰⠋⠁⠀⠀⠉⣛⡼⢢⣀⠀⠩⣿⡄⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠐⢷⠙⢿⣯⡷⠶⣦⡁⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢒⣶⠶⠖⠉⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⡾⠿⢻⣢⡇⢸⡟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠸⠀⠀⠀⠀⠀⠀⡇⠀⢼⣴⣋⣽⣟⠮⠇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⢠⣾⢎⣱⠆⢿⣦⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠦⠤⠖⠀⠀⠶⣤⣤⣴⠄⣿⠸⠞⡋⣽⣸⢷⡄⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣼⡶⠟⣩⡥⢨⠃⠀⠀⠈⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡧⢹⡟⢿⣇⢈⠀⠀⠀⠀⢹⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡹⢷⡎⢳⣸⡏⣼⠁⠀⢤⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⡏⢺⣿⢸⠃⣶⢻⣤⡀⢀⣿⠟⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡟⣎⢻⣎⡷⠁⣯⢸⣹⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣷⣿⢀⡿⡁⡄⣿⣤⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣷⢻⡿⢷⣄⢷⠙⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⣸⣿⣼⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢈⣡⣿⠏⣿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣻⠛⠋⣻⠏⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣾⣿⣷⣾⣿⣏⢠⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣾⢻⡿⣰⣿⣏⣸⣿⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠉⠈⠁

⡿⢿⡿⠿⡿⠟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣿⣟⣿⣿⣿⣿⣿⣿

⠀⠴⣶⣦⡀
⠀⠀⠈⠛⠿⣆⠀⢠⣾⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀
⠀⠀⠀⠀⠀⠸⡄⢺⡟⢡⠀⣷⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠸⡇⠀⢀⣴
⠀⠀⠀⠀⠀⠀⢳⣸⡀⠈⣒⡁⣀⠀⢀⡀⠀⠀⠀⠉⢿⣿⣦⠀⠀⣴⣦⣼⣿⡄⠀⠀⠀⣀⣸⣷⣶⣷⣿⡁⠀⣇⣀⠠⠄⠚⠉
⠀⠀⠀⠀⢀⡠⠤⢿⣷⣤⣼⣿⣿⣷⣿⣧⠀⠀⠀⠰⣼⣿⡟⠀⠀⣿⣿⣿⠿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣣⣶⣾⠇⢀⡒⠄
⠀⠀⠀⢰⡁⢤⠀⠰⣾⣿⣿⣿⣿⡛⠉⠉⣷⣶⡄⠀⠈⠙⢧⠀⣿⡿⠋⢀⡀⠠⠶⢦⣭⣭⣽⣿⣼⣿⣿⣿⡇⠀⠀⡑⠤⠃
⠀⠀⠀⠀⣩⣼⣶⣾⣿⣿⣿⣿⣏⣿⡦⢀⣽⣿⣿⣦⡀⠀⣬⣿⣿⣤⣶⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⣧⣼⣿⣧⢀
⠀⠀⠀⢠⣿⣿⣟⣿⣿⣿⣿⣿⣟⠁⠀⠀⠉⠀⠘⠻⠿⢷⣿⡁⢙⣿⣿⣥⣀⡀⠀⠉⠀⠀⠉⠛⠛⠿⣤⣤⣼⣯⣿⣿⡏
⠀⠀⠀⠚⠻⣿⣿⣿⠿⠛⢻⣿⣿⣿⣶⡤⣾⣶⣶⡶⠂⠽⢿⣿⣿⡟⠉⠙⢿⣷⣶⡿⠂⠠⠀⡽⠀⠀⠉⠙⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠁⠘⢧⠾⠿⢿⣭⠀⠀⠀⢀⣾⣿⠁⢤⡀⠀⠀⠻⣿⣿⠆⠀⣘⣥⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡷⠻⠁⠀⠀⣠⣶⣿⣿⠉⠀⢸⣿⠢⣄⠀⠀⠈⠛⣝⠻⣅
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠊⠄⡞⣀⠤⡀⠀⠀⢻⣿⢿⣿⠄⠀⢿⣿⣆⠈⠀⠀⡀⢤⠈⡆⠈⠳⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠙⠻⠿⠛⠉⠀⢰⡆⠳⣘⣦⡏⠀⠀⢸⢃⠜⠃⠀⠀⠀⠉⠻⠻⠀⠀⢇⣀⣠⢃⣀⠀⠈⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⣽⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⠉⠉⠉⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠙⠉⠉⠉⠉⠉⠋⠋⠙⠉⠉⠉⠉⠉⠋⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢳⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣋⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡨⠭⢄⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠊⠀⠀⠀⠱⡠⠊⠉⠐⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⡀⠀⠀⢤⠠⡇⢤⠀⠀⡠⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠣⣀⣀⣠⠞⠱⣀⣀⣠⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⡀⠀⡈⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠒⠒⠐⠀⠀⠀⠈⠐⠒⠂⢄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⣇⣀⡀
⠀⠀⠀⠀⡠⠒⢉⣁⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢄⣉⠑⠢⡀
⠀⠀⠀⡞⠀⡰⠋⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠈⠳⡀⠘⡆
⠀⠀⣸⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⡇⠀⢸⡀
⢀⠞⠁⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠀⠀⠀⠙⢆
⡎⠀⡜⢟⠢⣀⡼⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣄⡠⢚⠟⡄⠈
⠑⠒⠓⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠓⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⣩⣽⣭⡉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠹⢶⠾⠁⠀⠀⠀⠀⠀⠀⠀⣸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⣀⣼⣧⣷⠀⠀⣴⣿⣿⡄⠀⠀⠀⢀⢀⠀⠀⠀⠀⢲⣿⣿⣶⣿⣿⣿⣧⣿⣿⣿⣷⠀⠙⣷⢈⣣⡽⡂
⠀⠀⣤⣿⣆⢀⠀⠀⠀⢠⣿⣿⣿⣿⣶⠀⠏⢻⡿⠃⠀⠄⠆⡘⣠⣷⣆⣄⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⢸⣿⠡⢾⣿⣤⣿⡠
⠀⣰⣿⡟⢻⣿⡇⠀⠀⢾⣿⣿⣿⣿⣿⡆⠀⠀⠁⢀⣼⣸⣶⣿⣧⣯⣿⣻⢷⣥⢀⢀⢰⣦⢶⣒⣶⣶⠷⡶⠄⢲⣿⣻⣟⣡⣿⣾⡗
⠀⣿⡏⠀⠈⣿⡿⠀⠀⠈⠛⠋⠉⠛⠛⠁⠀⠀⢼⣾⡿⢿⡿⣿⠿⠿⠿⠿⠿⠿⣿⠇⢻⣿⣧⣤⡟⠶⡀⠀⠀⠀⠉⠁⡿⠿⠋⠁
⠀⠈⠉⠀⠈⠉⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⡀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣶⣿⣿⣿⡃⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣨⣮⡞⣥⡇⣆⢰⣠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠛⠉⡔⡅⢆⠀⢀⠻⣄⡴⠂⠀⠀⠀⡀
⠀⠀⠀⣧⠀⠀⣼⣻⣼⣽⣾⣿⣷⣷⣾⣿⣿⡄⠀⠀⠀⠀⢀⣴⣧⣦⣤⡀⠀⠀⠀⠀⠘⠀⠀⠺⠿⠏⠀⣜⣿⡻⢞⡿⠀⠀⣠⣹⠆
⠀⡴⢰⠏⡀⠀⠈⠙⠛⣉⠉⠉⠉⠛⠛⠛⠋⠀⡀⠀⠀⣴⣯⣿⣿⣿⣿⣷⡄⠀⢀⣠⢛⢦⡞⠀⠀⠀⠀⣨⣿⣿⣏⠀⠀⣶⣳⣹⣷
⠀⢷⡜⣧⣷⡀⠀⠀⣰⢿⣦⣆⠀⠀⠀⢶⣽⡌⢡⣀⠀⠻⢿⣿⣿⣿⣿⣿⡇⣰⣼⣹⡎⣿⣷⡎⠀⠀⠋⠉⠉⠉⠉⠉⠁⢀⣽⣿⡅
⠀⣸⡇⣿⣿⠇⠀⣼⡟⢻⡿⣾⡆⠀⡘⢶⡟⣿⢣⣹⢇⠀⢐⣿⣿⣿⣿⡏⠀⢻⣾⣿⣾⣿⣯⣧⠀⠀⠀⠀⠄⠀⠀⠀⠀⢿⣿⣁⡀
⠀⢿⣿⣿⣿⠇⠀⠻⣇⠀⢀⡿⠃⠀⢿⣿⣿⣿⣿⣿⣾⠆⠈⢻⡇⢹⡿⠂⠀⣸⣷⣿⣿⣿⣿⣽⠀⢀⣤⣶⣾⣧⣤⣤⡀⢀⣤⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢤⣯⠀⠀⢠⠀⠀⢀⠀⠀⠜⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡰⠋⠀⠀⢠⣼⡿⠃⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠞⠁⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠀⣠⣾⣿⣿⠁⠀⠀⣠⣦⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣾⡿⢡⣿⡇⠀⢀⡼⠟⢻⣿⠀⠀⠀⢀⡀⡄⠀⠀⠐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡗⢠⣾⣿⡿⣁⣾⣿⡶⠃⠀⠀⠀⣸⣿⡀⠄⣀⠈⠛⠀⠠⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠘⣿⣟⣴⣿⣿⣿⣁⠀⠀⢀⣿⣏⣿⣿⣆⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣬⣿⣯⣿⣿⣿⣿⣄⣴⣟⣿⣟⢳⡿⡟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢫⣿⣿⢫⣿⣿⣿⣿⣿⣽⢇⣼⣿⣿⡈⢿⣜⢌⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⣿⡿⣱⣿⡟⢠⣿⡿⣿⣿⡿⣿⡇⣼⣿⣿⣿⣷⡈⠿⣏⠠⠙⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣼⣿⡿⣰⣿⡟⢰⡞⣿⣧⣿⣿⣇⡿⢰⡏⢹⣿⣿⣸⣿⡀⠙⣧⣄⡈⢂⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⢠⣿⡟⢳⣿⣿⠀⣿⢳⣿⡿⣿⣿⣿⣇⠘⣇⠀⣿⣿⣿⣿⣇⢀⠈⢿⣷⣀⠰⣌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠅⠈⣿⣇⣼⣿⡧⢾⡟⢼⣿⣷⣿⣿⣿⠿⡇⡁⢰⣇⠻⣿⣿⣿⠀⣦⠈⢿⣿⣦⠘⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⢿⣿⣿⣿⠃⢸⡇⢸⡇⣿⡟⠉⢿⠀⠀⢰⣾⣿⡆⣿⣿⣿⠀⣿⡄⠈⣿⣿⣧⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣿⢸⣿⣿⣿⠀⢸⣿⣾⡀⠛⠀⣸⢺⡇⢰⣿⣿⠃⠀⣾⢼⣿⠀⣿⡇⠀⠸⣿⣿⠠⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⢻⣎⢿⣿⣿⣶⠀⣿⣿⡇⠀⠀⢿⣾⣇⢀⠹⣿⡀⢠⡟⢸⡿⢠⣿⣷⡄⠀⣿⣿⠀⠀⢻⡟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⢹⠿⣿⣿⣿⣶⣸⣿⣧⣄⠀⠈⢿⣿⠇⠀⢹⣇⣿⢡⡿⠁⣾⣿⣿⡀⠀⣿⣿⡃⠀⢸⡇⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠙⢿⣿⡿⢿⣿⣠⣟⠿⣿⣆⣈⠉⠀⢠⡏⣹⣿⡿⢃⡼⠛⢻⣿⠁⢰⣿⣿⠁⠀⣸⡇⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣝⣿⣦⣼⣿⣯⣷⣌⢻⣦⣀⣐⣯⣿⣿⠟⡠⠞⠁⠀⣿⡏⢀⣿⣿⡟⠀⢀⣿⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠀⠉⢀⡹⣿⣿⣿⣟⣷⡉⠻⠿⠿⠿⠁⠀⠀⢀⡠⠾⠋⣴⣿⣿⠟⠁⠀⣼⢃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠻⣟⣿⣿⣿⣽⣷⣦⣄⣀⣀⠀⢀⣩⣁⣾⣾⣿⣿⡅⠀⢀⡼⢡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠋⢀⣴⠋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣬⣿⣿⣿⣿⣿⣶⣾⣿⡿⠟⠋⣁⣤⠶⢋⠠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣶⣶⣿⣿⣤⡤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠿⡟⠿⢿⠟⠛⡿⠿⢿⢿⠛⡿⣿⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣶⣶⣾⣦⣾⣧⣦⣿⣧⣾⣿⣦⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣠⡄⠀⠀⠀⠀⠀⠀⠠⣀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣤⣿⣤⡠⠉⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠶⢾⣿⣿⣿⡷⠶⠆⠀⣀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡁⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠰⠊⢻⠿⢟⠑⢠⣤⠀⠀⠀⠀⠀⠀⠈⠛⠟⠏⠁⠀⠀⠀⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠰⡾⠀⠀⡶⠀⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⣿⣿⠀⠰⠷⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣷⡄⠀⠀⠀⠛⠃⠀⢠⣿⡿⠁⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠿⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⢿⡇⠀⠘⠀⠀⠀⠀⢶⡀⣤
⠀⠀⠀⠘⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠠⣷⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢠⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠈⡁
⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣤⠀⠀⢀⠀⠀⢀⣼⠃⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⢀⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠛⣶⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⠀⠀⠘⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠉⠉⣀⠀⠀⠀⠀⠀⢸⣿⡯⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣴⡶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠘⡷⠶⠶⠟⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⣠⣶⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣦⣤⡀⢾⣧⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣾⣿⣿⣿⣴⠾⠷⠤⣄⣀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣨⣿⣁
⠀⠀⠀⢀⣿⣿⢻⣿⣿⣿⣿⠻⣿⣿⠇⠀⠀⠀⠈⠃⠀⠀⠀⢸⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⠉⠉⠓⠦
⠀⠀⠀⢸⣿⠃⢈⣟⣛⣛⣏⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢻⣿⣿⣿⡇⠻⠿⠃
⠀⠀⠀⠘⠛⠀⣼⣿⣿⠛⠛⠂⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⣿⡇⢀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⣴⠀⠀⣿⣿⣿⣷⣶⣿⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢻⡇⢸⣿⣦⣤⣴⣿⡄⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠈⠀⢠⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠐⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⢛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢷⡈⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣌⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⠸⣄⡄⠂⠀⠀⠀⠀⠀⠀⡾⡇⢠⠇⠀⠀⠀⠀⠀⠀⠈⣉⣰⠀⠀⠀⠀⠀⠀⣴⡴⡀
⠀⠀⢠⡘⡰⢪⡕⠚⢀⠀⠀⠀⠀⣄⡇⢷⡾⣴⠀⠀⠀⠀⠀⣄⣠⣏⡍⠀⠀⠀⠀⠀⠐⠛⠛⣥
⠀⠀⠀⣷⢃⣾⠳⣉⠛⠀⠀⠀⠀⠻⠃⣰⣄⠹⠀⠀⠀⠀⠀⠈⢢⡻⣧⠔⠀⠀⠀⢰⣟⡆⣳⠛⠟
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⠀⠀⠈⠋⠠⠈⠂

⠀⠀⠀⠀⠀⢘⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⢾⣿⢊⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⡀⠀⠀⠀⠀⠀⠠⢂⣿⡄⡄⠀⠀⠀⠀⢀⢠⣆⡿⡸
⠀⠀⠀⠀⢾⡘⢶⠀⠀⠀⠀⠀⠘⠹⡶⢆⠛⠓⡃⠀⠀⠀⠀⣦⣺⠙⣷⣷⡀⠀⠀⠀⠀⠹⡉⢘⠃⡆
⠀⠀⠀⢲⣸⡷⣾⡇⠀⠀⠀⠀⠀⣟⠃⣼⢆⡙⡇⠀⠀⠀⠀⢻⡏⢣⡅⠻⠁⠀⠀⠀⢠⣷⠿⡏⠲⡄
⠀⠀⠀⠀⠀⠾⠙⠀⠀⠀⠀⠀⠀⠉⠘⠋⠈⠋⠀⠀⠀⠀⠀⠀⠁⠋⠉⠀⠀⠀⠀⠀⠀⠘⠰⠙⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠚⠲⠶⠶⠶⠶⠾⠷⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⣤⣤⣤⣤⣤⣤⣄⣤⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⢕⡩⠊⣡⠞⢁⡴⣋⢼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣷⣯⣤⣮⣤⣶⣥⣾⣥⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣶⡶⣶⠶⣶⠶⣶⠶⡶⢦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡯⢊⡥⢊⡡⢚⡥⢊⡴⣻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠻⠷⠶⠿⠶⠿⠶⠷⠾⠟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣟⠿⣻⠟⣛⠟⣻⠟⣛⢿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡗⣫⠔⣋⡵⢊⡡⠞⣡⢾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣉⣉⣉⣉⣙⣋⣙⣋⣉⣉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣟⡵⣫⠜⣩⠞⣩⢞⡩⢿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣷⣪⣴⣋⣴⣋⣴⣫⣴⣻⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⠘⠋⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣐⡈⢷⣤⣀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠟⠿⠉⠛⠷⣬⡿⣿⣷⣿⣦⡷⣶⣄⣈⣿⡁⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⢤⠚⡟⠋⠙⠻⣷⡌⠉⠙⠋⠀⢫⣼⣿⡿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢠⣤⣽⣶⣤⡶⠊⠉⠁⠀⣀⡀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡷⣄⠀⣸⣿⣷⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡷⠈⠲⢿⡇⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣦⡉⠁⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⢀⣿⣿⣼⣿⣿⡿⠯⢇⠀⠀⠀⠀⠹⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠆⠀⢀⠀⣸⠀⠀⠀⠀⠀⣸⣿⣿⣿⣟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠻⣧⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣀⣧⣀⣀⣸⣿⡿⢷⣿⣶⣶⠾⣷⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠻⣿⣿⠟⠉⣿⠃⠚⠻⠟⠁⠀⢿⡀⣼⡿⠙⣿⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠈⠁⠀⠘⢿⣤⠀⠀⠀⠀⠀⠘⢷⡿⠁⠀⣿⠀⢻⣿⡆⠀⠀⠰⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣿⡇⠀⢻⣧⠀⠀⠀⠀⠀⠀⣦⣄⢀⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠉⠀⠲⣾⣿⣷⠀⠀⠻⣆⠀⡇⠀⠀⠀⠀⠙⢟⣉⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠘⣿⣿⣿⣧⠀⠀⠈⠲⣧⡀⠀⠀⠀⣠⣿⠉⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣸⣿⣿⣿⣇⠀⠀⠀⠀⠈⠓⠄⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠹⣿⣿⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢹⣿⣿⣿⣿⣿⡎⢷⡀⠀⠀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠸⢿⣿⣿⣿⣿⣷⠈⣿⡄⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢹⣿⣿⣿⣿⡆⣿⣷⠟⣼⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢿⣿⣿⣿⣷⣽⡏⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣻⣼⣿⣿⣿⣿⡟⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢃⣼⢿⣿⣿⣿⣿⡟⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⠟⠡⠾⠍⠿⣿⠟⣶⢹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠄⠤⠶⢒⣶⡟⣼⡿⣿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⢤⣴⢁⣿⢟⣼⣿⠃⠙⠲⣇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠱⣾⣏⠚⠛⠋⠀⠀⠀⠀⠳⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⣶⣶⣶⣦⣴⣶⣶⣶⣤⣶⣶⣴⣦⣴⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣶⣤⣶⣶
⣿⣿⣿⡻⠛⠛⡿⠋⢛⠛⠛⡛⠟⣛⡿⢟⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣿⣿⣾⣿⣶⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡍⠠⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⠔⠢⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠤⡐⠊⠐⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⡠⠀⠀⠀⠐⠤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠆⠀⠸⠃⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠈⠢⡀⠀⠀⠀⠀⠐⠾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣑⣀⠈⠢⢀⠄⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⢠⠔⠀⠀⡀⠀⡀⠀⠼⢠⠐⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠠⠀⠀⠀⠀⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠑⠄⠀⠄⠀⠀⠀⠴⠀⠢⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠤⡀⠀⠀⠀⠀⠀⠁⠀⠬⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠚⠁⠘⠡⢀⠀⠀⠀⠠⠀⡆⠠⠀⠀⠀⢀⡠⠚⠀⠉⠒⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠖⠒⠚⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂⠀⠀⠁⠀⠀⠠⣀⠀⠀⢀⠉⠀⠀⠐⡀⠄⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠐⠂⠀⠀⢀⠁⠢⡄⠁⠀⠂⠀⠀⠀⠈⠤⠆⢈⠀⠀⠀⠐⠀⠀⠈⠳⣱⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠠⠃⠀⣀⠀⠀⠀⠀⠀⠈⠀⠁⠀⢀⠀⠀⠀⠀⢀⠀⠘⠠⠈⠀⠀⠀⠀⡐⠀⡀⠀⡻
⠀⠀⠀⠀⠂⡀⠀⠀⠀⢈⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⠠⠄⠁⠀⠁⠁⠀⠀⣿⠆
⠀⠀⠀⠀⠘⢠⡄⠀⠀⠠⠂⠊⠀⠁⢀⠀⡀⠀⠐⡀⠀⣴⠀⠀⡀⠀⠀⠀⠀⠐⠰⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠈⠂⠈⢀⠀⠄⠀⡠⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡄⡆⠃
⠀⠀⠀⠀⠀⠀⡈⣁⠀⠀⠀⠀⠀⡱⠊⠂⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⢉⣃
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠄⠀⠀⠀⠀⢀⠁⡀⠀⠉⠀⠂⢀⡀⠀⠀⠁⠀⠐⠒⠀⠐⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⡀⠂⠀⠀⠀⠊⢛⣄⠘⠸⠀⣬⠛⠃⠀⠀⠀⠀⠠⠀⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⢀⠔⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠐⠀⠈⠢⠀⠀⠀⢠⣀⣀⣀⠀⠀⠀⠈⠀⠀⠐⠢⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠐⠀⡀⠀⠀⠣⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠄⠀⠀⠈⠀⠈⠄⠁⠘⠀⠀⠀⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡨⠉⠀⠔⠀⠎⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⠔⢀⠐⠁⠀⠀⠀⠀⠀⢶⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠑⠀⠀⠐⠀⠐⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠒⠡⠄⢠⠀⠀⠀⡠⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⠢⠔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣋⠐⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠉⠀⠀⠀⠈⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣀⣀⣀⣀⣀⣀⣀
⠀⠀⣀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣀⣀
⣴⣿⣿⣿⣿⣛⡛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⢛⣿⣿⣿⣷
⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⢿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣷⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣾⣿
⣿⣿⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠻⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿
⠀⠈⠛⠻⠿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⣠⡴⠞⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠷⢏⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠗⠚⠀⢵⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠶⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⠛⠶⠲⣤⣤⣤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣥⣀⠶⣀⣴⠿⠟⠉⠀⣴⠋⠀⢀⡴⣀⣤⣤⣤⣀⣀⠉⠈⠁⣼⣿⡿⣄⡉⢛⣛⣋⣁⣈⡀⣀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⠘⣷⠀⠀⠘⢤⣯⡍⠉⠉⠉⠉⠀⠀⠀⣿⣿⠀⣽⠁⠈⠉⠉⠉⠉⠁⣬⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷⠀⠶⣶⡶⠞⠁⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣋⣿⣋⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣿⣿⣿⣷⡄⠀⣼⣶⣾⡟⢻⡶⠶⣶⡉⢹⡄⠀⠀⠀⠀⣾⠃⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣰⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣿⣿⣯⠙⠿⣤⣀⣀⠀⢀⣙⣃⣀⣸⣿⡇⠸⠇⣸⠋⣿⠛⢿⣦⠄⠀⣴⣿⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢰⠆⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠈⣿⡟⣷⠀⠈⠙⠻⢿⣿⠏⠉⢉⣴⡿⠁⢦⡸⠇⢀⣿⣤⠀⢿⡀⠀⠈⠙⠃⠀⢀⡄⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⡗⡇⠀⠉⠀⠀⠀⠀⠴⠶⢿⣿⠁⣰⣿⠃⠀⣼⣿⣶⣦⣴⣧⣀⣀⣀⣀⣀⣼⠃⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢠⣿⡏⣀⣠⣏⣉⠉⢉⣉⣹⣏⡉⠉⠉⠉⠁⠀⠀⠀⠀⣠⣟⣀⣀⣀⣀⣀⣀⣀⣘⣧
⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⢰⣿⣇⠀⠀⢀⣶⠀⠀⢀⣼⠏⠀⢠⡿⠿⣿⣿⣿⡟⠛⠛⠛⠛⠋⠙⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡟⠛⠛⠛⠛⣋⡹
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡟⢀⣤⡿⠁⣠⣴⠟⠁⢀⣴⣿⠃⠀⠀⠿⠛⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠃
⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣿⣿⣿⣿⣯⣴⡾⠟⠁⣠⡴⠋⣽⠁⠀⢀⠇⢰⡇⠙
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢰⣿⡿⠟⠋⣉⣁⠴⠖⠛⠁⢀⡼⠃⠀⣠⡟⠀⡆⢹⠄
⠀⠀⠀⠀⠀⢠⣿⠛⠃⠀⣸⣿⣤⣴⣿⠋⠁⠀⣀⣠⣴⣟⣁⣤⣾⣏⣤⣿⡼⠃
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢻⡟⠋⠉⠉⠉⢙⡟⠙⠋⠛
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿⣿⣟⣿⣿⡟⣠⡟⠀⠀⠀⠀⢀⣿⠀⠀⠀⣠
⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⣿⣿⣿⣿⣯⣼⡿⠀⠀⠀⠀⢠⣾⠃⠀⠇⠀⣿⡆
⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⣴⢿⡏⢀⣶⠀⢠⣿⡇
⣴⢓⣀⡀⢀⣿⡇⠀⣼⣿⣿⣿⣿⣿⠋⠀⠀⣀⣴⣿⣯⡆⢀⣾⠃⠀⣾⣿⡇
⠙⠿⣿⣿⣾⣿⣧⠸⣿⡅⠙⠛⠉⠀⣀⣤⣾⣿⣿⣿⠟⢀⣾⡟⠀⢠⣿⣿⠁
⠀⠀⠘⢿⣿⣿⣿⠀⠙⢿⣶⠶⣾⣿⣿⣿⣿⣿⣿⠋⣠⣾⣿⠁⠀⣸⡟⠃⢰⠃
⠀⠀⠀⠈⣿⣿⡏⠀⣰⣾⠟⣀⣤⣿⣿⣿⣿⠟⢁⣴⣿⣿⠇⠀⢸⣿⡇⠀⡂⢀
⠀⠀⠀⠀⠙⠿⣷⡤⢟⠁⠘⢿⣿⣿⣿⠟⢁⣴⣿⣿⠿⡏⠀⠀⣿⣿⠇⠀⡇⢸
⠀⠀⠀⠀⠀⢀⠛⢡⣾⣷⣄⠀⠙⢿⣁⣴⣿⡿⢃⠘⣦⠀⠀⣾⣿⣿⠀⢸⣧⠀⣶
⠀⠀⠀⠀⠀⠈⢰⣾⣿⣿⣿⣷⣄⠀⠙⣿⡏⠡⡎⢱⠁⠀⣸⣿⣿⡇⠀⢸⣿⠀⢻
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⡀⠈⠛⣧⣠⠇⠀⣰⣿⣿⣿⡇⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠋⠀⠛⢦⡄⠈⠻⣧⣄⢻⣾⣿⠘⠇⠀⠀⢿⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣶⠟⠀⣰⠃⢙⣦⡀⠈⠻⣿⣡⠙⣤⠀⠀⠀⢸⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣁⣀⡾⠁⢠⣿⠋⡹⣧⡀⠈⢿⣦⡟⠀⠀⠀⣸⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠀⣰⣿⠋⡀⠙⢠⡿⢦⡀⠙⢷⡀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⣀⣾⣿⠁⣠⠉⡄⣿⠁⣾⠛⢦⡀⠙⣷⣀⣿⠀⠀⠀⠈⣅
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⡿⠁⣰⣏⣿⢱⡏⢀⠋⠀⢼⠙⢦⡀⠹⢿⡀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠟⢀⣾⣿⣿⠇⣸⠃⣿⠀⠀⡻⠀⠀⠙⢦⡈⠛⣦⡀⢸⣿⡄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⡿⢠⡟⢀⠀⠀⣼⡇⠀⠀⠀⢸⠻⢦⡈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢻⣿⠟⣁⣴⡟⣿⣿⣿⣿⡇⣸⠃⢾⠀⢀⡿⠀⠀⠀⠀⣿⠀⠀⠙⢦⡘⢻⣷
⠀⠀⠀⠀⠀⠀⢾⣯⣾⣿⠟⣸⣿⣿⢹⡟⠀⣿⠀⢸⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⢨⡿⢦⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣿⣿⠃⣿⠃⣼⡏⢀⡏⠀⢸⠃⠀⠀⠀⢀⠃⠀⠀⠀⢸⡇⠸⣿⣦⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⣼⡟⠀⣿⠀⢸⡇⠀⡎⠀⠀⠀⠀⣸⠀⠀⠀⠀⣿⢱⠀⣿⣏⠙⢠⡙⢧⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⢰⣿⠁⢸⡟⠀⡘⠁⢠⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⢸⠇⣿⢸⡄⠀⠙⢦⡙⠂
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣠⣿⡏⠀⣿⡇⢠⡇⠀⣸⡏⠀⠀⠀⠀⢻⠀⠀⠀⠀⡇⢸⠀⡇⢸⡇⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠁⢸⡿⠀⠈⠇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⡇⢸⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇⠀⣾⡇⠀⣴⠀⠀⣿⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢸⡇⠀⡇⠈⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⡇⠀⣿⠀⠀⣿⠀⠀⠀⠀⢀⣿⠀⠀⠀⣰⠸⡇⠀⠁⠀⠹
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠻⣿⣆⢸⣧⠀⣿⠀⠀⠀⠀⠘⢣⣄⠀⠀⠻⣄⠰⣆⠀⢇⠀⢿⡀
⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠀⢼⣿⣿⣯⣀⣼⡆⠀⠀⠀⠀⢀⣼⣷⣀⣀⣙⣶⣿⣆⣸⣧⣄⣻⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣶⣶⣶⣶⣶⣤⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣬⣥⣬⣭⣭⣤⣬⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠟⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡯⣀⠀⠀⠀⠀⠀⠀⡼⠷⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⢷⣦⡀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⣿⣿⣿⣾⣿⣇⠘⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠴⣻⢿⢿⠿⡟⡏⠁⠉⠛⠛⠛⠛⠻⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⣯⡟⡼⠈⡀⠐⠂⠀⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠠⠤⠐⠈⠀⠀⠀⠀⠀⠀⢘⡟⠠⠁⢸⣧⣀⡰⠼⠟⠓⠓⠓⠛⠃
⠀⠀⠀⣠⣦⣾⣷⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠀⠀⢀⡉⠉⠀⠀⣸⠀⠀⠀⠀⣠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⢀⣤⣿⣷⣤⣥⠶⠛⠛⠛⠛⠛⢣
⠀⠀⠸⠿⠁⠋⠛⡉⣿⢹⠀⢸⠛⠁⠀⠀⢠⣴⣿⠏⠀⠀⠀⢼⣿⢹⡥⡍⠀⠀⣹⣀⣀⣀⣀⣄⡇
⠀⠀⣤⠀⣦⣄⣦⣷⣿⢸⠀⠈⣿⡿⣷⣚⣻⣿⡏⠀⠀⠀⠀⠘⢿⣾⣇⣿⠷⠾⠟⠛⠛⠛⠛⠛⢇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢸⠀⠈⣟⠀⠀⢸⠇⠀⠀⠀⠀⠀⣤⣤⢸⡇⢹⠉⣿⡆⠀⠀⠀⠀⢀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡆⣇⡀⠘⠀⠀⢸⣳⣄⠀⠀⠀⠀⠘⢿⣾⣧⣼⣶⣿⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⣀⣿⣿⣷⣦⡄⠀⠀⠀⠹⠏⠻⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠙⠉⠉⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣮⡾⣿⢣⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣇⣦⡀
⠀⠀⠀⢀⣠⣀⡤⣮⡾⣫⠞⢡⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⣾⢏⣿⢠⡀
⠀⢀⣴⣿⣿⡗⣺⢫⡾⠋⣰⡿⣻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⡤⣄⣴⣾⣿⣿⣿⣷⣿⣼⣇⣾⡿⡇⡀
⣴⣿⣿⣿⣿⣗⡿⣸⡇⣾⡟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣷⣧⢻⡆⣤
⣿⣿⣿⣿⣿⣿⡇⣿⢱⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢷⣝⠷⣝⡻⣿⣿⣿⣿⣿⣏⣾⣿⢻⣮⣻⣿⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣧⡣⡻⣿⣿⣿⣿⣿⣿⣿⣿⣯⡝⢷⣍⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣙⣿⣿⣶

⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢸⣤⢤⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⢻⣽⣆⠀⠀⠀⠀⠀⢰⢻⠇⠀⠀⠀⠀⠀⠀⠀⢀⣄⡤⡀⠀⠀⠀⠀⠀⠀⣾⡧
⠀⠀⠀⢠⣾⣧⣿⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣷⣶⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⣟⠛⠛⢏⠞⠀⠀⠀⠾⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠒⠒⠒⠃
⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣻⠃⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⢻⣷⠃⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⢢⠇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣮⣶⣿⣿⣿⡗⠀⠀⠀⠀⣠⣟⣍⣴⣋⠀⠀⠀⠀⠀⣾⡼⣧⣳⠄
⠀⠀⠀⢲⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠘⡋⢟⠛⠿⠋⠀⠀⠀⢠⣿⣿⣿⣿⡧⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠈⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⢠⡠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠺⠥⣠⠄⠀⠀⠀⠀⠰⠄⠀⠐⠁⡀⠀⠀⠀⠀⠀⠠⠔⠬⠁⢄⠀⠀⠀⠀⠀⠀⢐⡀⠆⠔⡀
⠀⠀⠀⠀⠠⠄⠨⠄⡀⠰⠀⠂⠀⠀⠀⠀⣾⠠⠹⠊⣩⣄⠀⠀⠀⠀⢂⣀⠀⠁⠀⢀⠂⠀⠀⠀⠀⢈⡂⣛⠀⠦⠀⠀⠀⠀⠀⢒⢬⠉⠈⠱⢁⡂
⠀⠀⠀⠀⠐⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠼⠅⠠⠯⠀⢼⠂⠀⠀⠀⠐⠂⡀⡐⠀⠁⠨⠀⠀⠀⠀⠀⠐⡀⠒⡀⢒⠀⠀⠀⠀⠀⠨⢧⣒⠠⣈⡌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠐⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈

⠀⠀⠀⠀⠀⠐⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠐⢆⡀⠁⡴⠂⠀⠀⠀⠀⠀⢀⠁⠄⠅⢁⠚⡁⠤⠀⠀⠀⠀⢄⠄⢠⠁⡀⡀⠀⠀⠀⠀⡤⠤⠀⠀⡊⠀⠀⠀⠀⠀⠂⠀⣀⢀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⡁⠀⠀⠀⠀⠠⠤⢬⠐⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠐⡃⣰⠊⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠈⣀⠠⠄⢠⠀⠀⠀⠀⠀⠀⠀⢁⠰⢠⠰⠐⠄⠁⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⢀⠒⡜⠒⠚⠀⢢⠀⠀⠀⠐⠀⠐⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙

⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢎⢿⠀⢟⣦⡀⠀⠀⠀⠀⠀⠀⡅⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⠓⡠⠉⠄⠀⠀⠀⠀⠀⢀⣔⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⢤⠘⠒
⠀⠀⠀⠀⠀⡎⣱⠀⢼⢸⡆⠀⠀⠀⠀⠊⡓⣩⠴⢯⠄⠀⠀⠀⠀⠀⠀⠁⠐⠀⠌⠀⠀⠀⠀⠀⠀⢰⠠⠀⠀⠠⣖⠀⠀⠀⠀⠀⠈⢀⣂⣘⠊
⠀⠀⠀⠀⠀⢻⠇⠀⣏⢠⠃⠀⠀⠀⠀⠂⠠⠄⠀⠀⡐⠀⠀⠀⠀⠀⠀⠂⠂⠢⠂⠀⠀⠀⠀⠄⠀⠀⠑⠄⡐⢂⠂⠀⠀⠀⠠⢊⠩⢆⠀⡑⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠨⣫⠅⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠁⢃⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⠁⠀⠀⠀⠀⠀⠉⠆⠀⠘⢠⢁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⡐⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢄
⠀⠀⠀⠀⠀⠘⠘⠘⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠠⠐⠰⠙⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⣀⣀⠀⠀⠀⠀⠈⡎⠀⢵
⠀⠀⠀⠀⠀⠠⡤⠶⠤⡆⠀⠀⠀⠀⠀⠀⠐⣧⣰⠆⠀⠀⠀⠀⠀⢐⡡⢈⠠⢀⠠⢀⠠⠀⠀⠀⠀⠈⡀⠀⠀⢈⠀⠸⠂⠀⠀⠀⠠⠇⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⢀⡀⠈⠁⠀⠀⠐⠁⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠨⡀⠀⢇
⠀⠀⠀⠀⠀⠀⢰⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠘⠁⠆⠀⠀⠀⠀⠀⠣⠀⠀⢒⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠿⣿⣿⠁⠀⠀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⢀⣀⠀⠀⠀⢸⣿⣆⣀⣀⡴⠲⢤⣀⠀⠉⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠉⠁⢀⣀⣀⣤⡤⠤⠶⠖⠚⠛⠋⠉⠙⢦⣀⡀⠀⣿⣟⠉⠁⠀⠀⠀⠈⠙⠓⠦⣄⣀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⡀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢲⠋⠀⠀⠀⠀⠀⢠⡞⠛⢷⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡆⠀⠀⠀⢠⡞⠉⢳⡄⠀⠀⠀⠀⠸⣇⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡆⠀⠀⠀⠀⠀⠘⠧⡤⠞⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣇⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠀⠀⠀⢹⡇⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⠀⢻⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣄⡀⠀⠀⢸⡄⠘⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡞⠀⣸⣿⣧⣀
⠀⠀⠀⢀⣠⣶⡿⠟⠉⢹⣿⡀⠸⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⠀⠈⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡾⠁⣰⣿⠋⠙⠿⣷⣦⣀
⠀⣠⣶⡿⠟⠉⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⣾⣿⡿⠋⣿⣿⠁⢀⣀⠈⢿⣿⣇⠀⢿⠀⢹⣿⠋⣀⡀⠙⣿⣿⡇⠀⠀⣼⠁⢠⣿⠏⠀⠀⠀⠈⠙⠿⣷⣦⡀
⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⠀⢸⣿⡿⠁⠰⢟⡋⠀⣭⣭⠀⢰⣿⣿⠀⢸⡄⠘⣿⣷⣭⡃⠀⣿⣿⡇⠀⣸⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡇⠈⣿⣿⣆⠈⠉⠁⢠⣿⣇⣠⣾⣿⡟⠀⠈⣇⠀⣿⣿⣀⣠⣾⣾⡿⠀⣰⠃⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿
⣿⡏⠙⠻⢿⣶⣄⡀⠀⠀⠈⣿⡇⠀⣧⠀⠘⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⢻⠀⢸⣿⣿⣿⡿⠋⠀⣰⠏⢀⣾⡟⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⣿
⣿⡇⠀⠀⠀⠉⠻⢿⣶⣄⡀⢻⣿⠀⢹⡀⠀⠀⠉⠻⠿⢿⣿⠿⠿⠛⠁⠀⠀⠀⠀⢸⡆⠘⣿⡏⠁⠀⠀⢠⠏⠀⣾⡿⠁⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣷⠀⠀⢠⡏⠀⣼⡿⠁⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⢠⡴⠲⣤⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿⡀⢀⡞⠀⣼⣿⣷⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⡀⠀⠀⠀⠀⠀⠸⣄⣀⡼⠀⠀⠀⠀⠀⠀⣸⡆⠈⣿⣇⡾⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢿⣿⠁⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠛⢦⡀⣀⣀⣠⣤⠤⠴⠶⠒⠚⠛⠉⠁⠀⢀⣀⣾⡿⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠉⠉⢀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⡿⠛⠋⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣶⣿⠿⠿⠿⠛⠛⠛⠉⠙⠿⠟⠉⠙⠻⢿⣷⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠙⢿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣦⣤⣄⣀⣠⣾⡿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣧⣌⡉⠙⠛⠿⣿⣿⣀⠀⠈⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠙⠻⢿⣷⣦⣄⡙⠻⠿⣿⣶⣴⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣶⣤⣈⡙⠋⠉⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣦⣤⣬⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠙⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣶⣦⣤⣀⡀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁
⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠈⠻⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣶⣦⣤⣄⣀⣀⡀⢀⣸⣿⠃

⠀⠀⠠⡀⠀⠀⢤⡀⠀⠂⢦⠀⠀⠚⠀⠀⠀⠀⢹⡆⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠄
⠀⠀⠀⠈⠢⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇⠀⠒⠰⠻⡧⢀⠀⠰⠖⠓⠀⢠⣀⠁⠈⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠒⠄⠀⠈⠁⢈⠴⠚⠁⠀⠀⣀⠁⠀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠰⠁⠀⠀⠀⠉⠠⠐⠊⢀⠄⠈⠩⠵⠂⢀⣄⠄
⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⢸⣶⣆⠀⢆⠀⠀⠈⠀⠉⢁⣀⡤⠀⠒⢉⠜⡠⢀⠃⠐⠂
⠀⠐⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⢸⣿⡏⠀⠀⠁⠀⠀⠀⠀⠠⡶⠿⠷⠗⠚⢊⣴⣧⡀
⠀⠀⠀⠀⡁⠀⠀⡁⢘⠀⠀⠀⠀⣿⠀⠀⠀⠀⢺⣿⡇⠐⠀⠀⠀⢠⡆⠀⠀⠈⠓⠶⠠⠶⠻⠀⠙⠛⣠⠄
⠀⠀⠀⠀⡄⠀⠀⠁⢈⠀⠀⠀⠀⠸⠀⠀⣀⠀⠼⠛⠛⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠹⣿⣄⣀
⠀⠀⠀⠀⡀⠀⣀⠆⠀⢀⠤⠀⠄⠉⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⢸⡇⢈⢀⡤⠐⠂⠀⠀⠁⠀⠈⢘⣿⠟⠋
⠀⠂⠄⠲⠀⠀⠀⠀⠐⠭⠿⠿⡖⢂⡀⠀⠀⠀⢠⣄⡀⠅⠀⠀⢨⣿⡏⢠⡠⠧⠖⠀⠀⠰⠏⡀⠀⠼⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠄⠩⢈⣀⣋⣉⣁⡄⡀⠀⠀⠈⣿⠧⠴⠓⣺⣉⠲⣄⠉⢳⠀⠀⠀⠤⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠟⣻⣥⢬⣴⣿⣽⣷⠂⠀⠀⢘⣿⣿⣅⠀⠀⠀⣨⠟⢀⣀⣤⡴⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠡⠖⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣅⠀⡀⠀⣰⠍⠩⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣦⣶⣶⣴⣦⣴⣿⣴⣷⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠀⢷⡀⣄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⣿⣤⡆⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡤⣸⣷⡀
⠀⠀⠀⠀⠀⠀⢀⢰⣧⣤⣾⢠⠘⣿⣿⣧⢠⣀⠀⢀⣼⡏⢀⣀⣇⣸⣿⣿⠃⣿⡌⣷⡀⢀⠀⠀⠀⢀⣾⣷⣿⣿⢻⣤⣷⡀
⠀⠀⠀⢠⠀⣠⣾⠟⣿⣿⣷⣿⣠⣿⢟⣿⣾⣿⣦⣿⣿⣧⣾⢋⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡀⣆⢱⣼⣿⣿⡿⣿⣿⣿⣿⣿⣀⡄
⠀⠀⠀⢾⣷⣿⢿⣾⡟⠟⢿⣿⣿⡏⢾⣿⡟⣿⣯⣿⡟⢿⢿⣿⣿⢻⠏⢻⠹⣿⠋⢼⣿⢻⣿⢿⣿⠿⢿⣿⠰⣿⣿⠇⠿⢻⣿⡷
⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⡀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⠟⠃⠈⠓⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡒⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉⠀⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠓⢤⠀⠀⠀⠀⡀⢠⠀⠀⠜⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⡀⠀⠓⠚⠉⠀⠁⠁⠀⠁⠂⢀⡤⠀⠀⡔⠢⣀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⡀⠀⠠⠔⠚⠀⠀⠄⠫⠤⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⢀⠀⠁⠀⠀⠀⠀⣰⠶⠡⠀⠀⠀⠀⠀⡀⠀⠶⠦⠈⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠅⢐⡠⣌⠀⠤⡀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠠⢭⠐⠂⠀⠀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠁⡌⠙⣆⣆⡤⠞⣓⠒⠢⢄⠀⠀⣀⠀⢠⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣉⢉⢉⣩⠞⠁⢰⠒⣏⠑⡢⣠⣇⡎⣡⠤⠶⣐⠒⡒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠁⡴⣴⠋⡜⣡⢋⢧⠀⠸⡯⠤⠴⣂⠑⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⢠⡿⣌⣛⡻⢢⠤⠻⠂⠂⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡈⠳⣄⢀⣠⣾⣷⣦⠀⠀⠀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠏⢰⣈⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣬⣁⣤⡈⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢉⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⠁⠙⠇

⠀⠀⣼⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢛⣿⡿⠁⠁⢸⣿⣿⡄
⠀⠀⢿⣿⣿⣿⣿⡿⠤⠾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢡⣾⡟⠀⠀⠀⠀⣿⣿
⠀⠀⠸⣿⣿⣿⣿⢹⣦⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣵⣿⣟⡄⠀⠀⠀⢸⣿⣿⡀
⠀⠀⢀⣼⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣯⣾⣿⣿⣿⠀⣾⣿⣦⣟⣿⣿⡇
⠀⠐⣿⣿⣿⣿⣿⣿⣿⣦⣘⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⢽⣿⣿⡿⢿⣿⣿⡆
⢀⣄⣽⣿⣿⣿⣿⡿⢳⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⢾⣿⣿⣿⣿⣿⣿⡿⠋⠀⢹⡟⠻⠾⠛⠀⠙⣿⣿⠻⠃
⠀⢹⣿⣿⣿⣿⣿⡇⣾⣿⡟⣼⡟⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⣿⣦⣤⡄
⢠⣿⣿⣿⣿⣿⡿⠧⢸⣿⣷⣿⡇⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⣼⠈⡆⠀⠀⠀⠀⠀⠈⠈⢿⣷⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⢹⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡆⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣰⣦⡀
⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠏
⣿⣿⣿⡗⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠁⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣹⣿⣿⣿⣿⣿⡿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⡇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⡇
⢿⣿⣿⣿⣿⣿⣿⣿⣟⡼⢿⣿⣿⣷⣼⣧⣿⣿⣿⣿⣹⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣤⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⡏⠀⢻⡀
⠀⢹⣿⣿⣿⣿⣿⣿⡿⢇⣼⣿⣿⣿⣿⣿⣿⣯⣿⠛⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢹⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢰⠇⠀⢸⣇
⢀⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠸⠀⠀⠀⣿⡄
⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡆⠀⠀⠀⢙⣧
⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢰⠁⠀⠀⠀⢸⡼
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣾⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡼⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠇⠀⠀⠀⠀⠀⢹⠀⡀
⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣬⣰⡀⠀⠀⠀⠀⠀⢸⡆⢁
⠑⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⣦⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡇⠘⡆
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢊⣽⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣱⠀⠧
⢶⣟⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠸⣿⣿⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⡤⠂⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⣠⠄
⠀⠀⠀⠠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⡆⠀⠀⠁⠆⡀⠀⠁
⠀⠀⢼⠈⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠇⢠⡇⠀⠀⠀⠀⠀⠠⠄⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠆⡀
⠀⠀⠀⠈⠀⡀⣀⠴⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠤⠖⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠄⠀⠀⠄⠀⠀⠉⠀⢠⣶
⠀⠀⠀⠐⠀⠤⠀⠄⠚⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⢻⡀⠀⠀⠀⠀⠀⢀⡤⠒⠀⡅⠉⠓⠂⠤⢄⠀⠈⠀⠀⠈⠱⠄⢀⣀⡀⠀⠘⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠼⠁⠀⠀⠀⢀⡔⠁⠀⡅⠀⠁⡄⠀⠀⠁⠪⠳⣄⠀⠀⠈⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⢀⡌⠁⠀⠀⠀⠄⠀⠐⠂⠀⠀⠂⠠⠈⡄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⢸⠁⠆⠀⢀⡀⠀⠀⠂⢈⠀⡐⠀⠀⠂⢒⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠐⠁⠐⢂⠀⠀⠀⠀⠈⠻⠶⠶⠼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠉⠀⠁⣠⠀⠈⠄⡸⠀⠀⠀⢘⠀⠀⠀⠠⠀⢀⠁
⠀⠀⠁⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠂⠀⡁⠀⠈⠀⢠⠄⢈⠀⢰⡠⠃⠀⠀⠀⠈⠀⠀⠐⠀⠂⠘⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠑⢄⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠠⠀⠀⢀⠀⠄⠈⣀⠔⠁⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠈⠉⠁⣀
⠀⠀⠀⠀⠀⠀⢩⠀⠀⠛⠂⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠐⠂⠀⠠⠔⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠀⠀⠀⠀⠀⠑⢖⠀⠀⠀⠀⠀⡠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠈⠀⠃⠀⠀⠘

⠀⠀⠀⢀⢀⡀⠀⠀⠊⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⣀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⡀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢠⢀⠀⠁⠘⠀⠀⠀⠔⠀⠒⢠⡕⡀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠐⠉⣄⠈⠀⠀⠀⠀⠀⠀⠊⠨⠀⠁⠀⠀⢠⢗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠤⠀⠀⠀⠈⢄⠀⠀⢀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠖⠀⢀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠈
⠀⠀⠀⠈⠓⠀⠘⠖⠉⠁⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⡛⠋⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠐⠐⠒⠒⠶⠖⠒⠀⠈⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠐⠆⠆⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⣀⣄⣀⣀⡈⠤⠶⠤⠤⣄⣀⣀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⠀⠀⠀⢸⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣒⣛⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠁⠀⠐⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠷⠚⠛⠛⠛⠛⣧⣀⠀⠀⣀⣼⠛⠛⠛⠛⠓⠾⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢈⡁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠐⠂⢀⡴⠛⠋⠙⠛⢦⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠠⠏⠀⠀⠀⠀⠀⣄⠀⣾⠁⠀⠀⠀⠀⠈⣷⠀⣠⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⣰⠶⢶⡄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠙⠳⠶⠶⠞⠋⠀⠘⠃⠀⠀⠻⣤⡼⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣄⣀⠀⢀⣠⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣾⠀⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠙⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡴⣛⣭⣭⣮⣭⣭⡓⢦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⢫⣾⣿⡿⠛⠈⠻⣿⣿⣦⡹⡎⢻⣿⡿⠿⣿⣿⠿⣿⣟⣯⣵⣿⣿⣿⠁⠙⣿⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⡅⠺⢸⣧⣼⣿⣿⣷⢹⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣧⣀⠈⠀⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡿⣷⢠⡉⢻⣿⣿⣿⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⢷⡹⣿⣿⣧⣈⠈⣁⣼⣿⣿⢣⡟⣴⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠳⣬⡻⠿⣿⣾⡿⠿⣛⡵⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠛⠉⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢷⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⢀⡀⢸⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⢸⠀⣀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣿⡇⠀⣼⢇⢸⢸⠇⣷⠀⠀⠀⠀⢀⡄⢩⢻⡀⠀⠀⢰⡇⣿⢸⢠⡟⡄⠀⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⡞⢠⡟⣧⠸⣥⢊⡄⢸⠀⣽⠀⠀⠀⠀⢸⣿⣌⢾⡇⠀⠀⢸⡀⢸⠈⡆⢡⡴⢠⡿⣇⠙⣦⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢠⣿⡼⣼⠃⣿⠷⣦⠾⣷⡜⣷⢺⡇⠀⠀⣠⣿⡿⡿⣦⡀⠀⠀⣿⣿⡜⣰⠛⢦⡴⢿⠀⢿⡸⣼⣧⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣿⡇⢱⢯⡗⢸⣀⢹⣶⡾⣹⡟⣿⠁⢀⠜⣿⣿⠀⡷⢻⡟⣄⠀⢹⡟⣷⠹⣶⣾⣁⣼⢸⣇⢧⠛⣿⡟⢹
⠀⠀⠀⠀⠀⣶⠀⠀⢸⢶⣼⣿⢏⢿⡃⣾⣿⡏⢻⡇⣯⠃⣿⠀⡜⣤⡏⣿⠀⣿⣚⢡⡼⡄⢸⡇⢯⡇⣿⠘⣿⣿⡀⣻⢎⢿⣿⣷⣼⠁⠀⢰⡆
⠀⠀⠀⢀⣾⠇⠀⠀⡼⣏⣹⡏⢆⡎⠀⣿⢸⢧⡌⢸⡏⡀⣿⠀⣿⢹⡇⢹⡄⡸⣧⣸⡷⡇⢸⡇⡜⡇⠛⣠⣿⢹⡇⠙⡌⡌⣿⣸⣿⡄⠀⠀⢿⣄
⠀⠀⢀⡞⢹⡄⠀⣴⠃⣿⣿⢸⡞⣸⠇⣿⠈⡞⢷⡄⠁⣏⢻⣄⢧⣼⢰⡇⣷⠃⣿⣿⣴⣇⣾⢱⡇⢁⣼⠛⡇⢸⡇⣿⡸⣶⢸⣿⡇⢳⡄⠀⣸⠙⣇
⠀⠀⢸⡇⡄⢻⡟⣡⡼⢀⡿⡾⢣⢿⡀⣿⢀⣷⡈⠻⠄⣿⠸⡏⣿⡏⣸⣿⢸⠈⢸⡇⢻⠏⡿⢸⡇⡾⠃⣼⢧⢸⡇⢻⡇⠿⣸⣇⠸⣠⡝⣿⢃⡄⣿
⠀⢀⣸⡇⣿⠈⣧⡴⣱⣼⣧⣇⠸⣾⠀⣿⢸⢸⣷⣸⡾⡈⢸⣿⡟⣸⡇⡟⢸⡇⣼⡇⣟⢷⣿⠻⢰⡶⣼⣿⢸⢸⡇⣾⡇⢠⣧⣿⡰⡱⣴⡏⣘⠃⣿⣀
⠀⣼⢿⣧⠟⣼⠏⠈⢱⢻⣿⡟⣆⣿⡄⡿⠈⠘⢹⡇⣧⢣⡟⣹⢱⢻⣄⡗⢸⣿⣿⢣⡿⡀⠸⣇⢧⡇⣏⠹⠈⢸⡇⣿⢡⠟⣿⢿⣧⠉⠘⢷⡽⢰⡿⢿
⢰⣟⣸⣯⡾⣣⢰⡏⣧⠏⣸⣇⢼⡏⢸⡇⣴⡀⢸⠇⢣⡟⢰⣿⣏⡏⠿⠁⡿⢁⡟⣾⣹⠸⣄⠹⣯⠃⣿⠀⢸⡀⣿⠈⡾⠆⣿⡈⣿⡇⣷⢬⡻⣼⣷⣘
⡿⢧⣼⡿⣱⡏⣸⢳⢱⢷⡟⣈⢠⣷⣿⢁⢻⡟⣽⣰⡿⣀⢸⡏⢋⠀⡤⣼⢃⢸⢡⡇⡟⡄⠉⣷⢹⣧⣽⠸⣿⢃⢹⣧⣧⢈⡘⣯⢷⢹⢹⠈⣷⠹⣿⣼
⣗⣸⢿⡇⣇⡷⢸⠎⣰⠏⣰⠛⣼⣿⠇⣼⠟⡇⣿⠋⡇⢿⡈⣄⣿⠸⠇⣿⣼⡾⡍⣷⢱⡿⢾⠇⣿⡏⢻⠄⡟⢾⡀⢿⣿⡜⢳⡈⢧⡘⣾⠸⣇⡇⣿⢿
⢹⣿⠘⣇⢛⣽⢀⡞⣡⠸⢋⡾⠋⡿⢸⡟⣄⡇⢀⠀⡇⠨⠀⠃⢸⠀⡄⢸⢸⠃⢳⣀⡄⠁⣤⡀⢈⡇⢀⠀⡇⣼⣷⠸⡏⠻⣮⠛⢤⡹⣆⢸⣽⢣⡇⢸
⠀⠻⣆⠙⣎⢻⣿⣼⢯⣶⡟⣡⢰⠇⡈⢧⣸⢠⣿⠀⣿⠀⠇⠀⠀⠏⠁⠘⠀⠀⡈⠋⢻⡀⢁⠁⣰⡇⢸⣦⢹⣠⢏⡀⣇⢠⡹⣷⣎⢷⣽⡾⢋⠞⢀⡿
⠀⢠⣽⣆⠘⣿⣇⣱⠏⣿⢡⣿⢸⣴⣿⡜⠇⣼⠁⠀⡿⠆⠀⠰⣇⠸⣆⠀⠆⡀⠛⠀⠸⡇⠘⢠⠟⡇⠀⢹⡈⠏⣼⣧⡽⢸⣧⢻⡏⢷⣉⣿⡏⢀⣿⣥
⠀⢸⣿⡏⡐⣿⢿⡇⡴⣯⢸⣽⡄⣿⠇⢿⠀⣟⡶⣸⠇⢸⡆⡿⢿⢸⡟⣷⡆⢿⡆⢸⠃⢿⡄⣿⠀⢷⠀⣜⡇⢸⠃⣿⠇⣮⣿⢸⡇⡆⣿⢿⡀⣜⣿⣿
⠀⢨⡟⢻⣰⣏⣸⡇⠇⡿⣸⡏⡇⢈⣤⢸⡆⡿⢠⡟⢰⡜⣷⣧⡘⣼⠇⠿⡸⢸⣿⢸⠃⡘⣿⡟⣦⠘⣧⡹⡇⣿⢠⡌⠀⣏⣿⠸⡇⢇⡇⣘⣧⣼⢹⣿
⠀⢸⡷⣌⠋⡾⢸⣿⣦⡏⣿⡿⣡⠞⡟⢸⣇⣴⠟⣠⣿⢰⡏⣾⢃⣿⡀⣠⢣⠏⣿⣤⢰⡇⢸⣧⠛⣆⠘⣷⣄⣿⠸⡟⢦⠱⣿⠂⣧⢾⣧⢹⡎⢣⣼⣿
⠀⠸⣧⠏⡆⢁⣾⣿⠈⠃⣿⢱⠏⡼⢃⡿⣻⢋⣼⡟⠸⣸⡇⢋⣾⢿⢃⣾⢏⡆⢻⡛⣸⡇⢸⣿⣠⠹⣷⣌⢿⡻⡆⠳⡬⣧⢻⡇⠋⢸⣿⣆⠁⣏⢸⡿
⠀⠀⣿⢸⡇⣼⢡⡇⠸⡆⠛⡘⣸⡟⢠⡷⢡⡏⣽⢱⢇⡟⢠⡟⡟⡼⣸⡏⢘⡄⡞⢱⢿⢀⣿⠘⣇⡦⢻⠙⣆⠱⣧⠸⣷⣸⠘⢃⡾⠈⣇⢻⡄⡿⣸⡇
⠀⠀⠘⣧⡀⣿⢸⡇⢳⣸⡆⢿⢻⠀⡿⠁⡜⣧⢿⣿⢸⣥⣿⢀⢻⡀⢻⡅⡞⡘⡀⢧⢢⡾⢿⣧⣽⢸⣆⢷⣹⠀⢹⡆⢹⢻⠇⣾⢰⠂⣿⢸⡇⣰⠟
⠀⠀⠀⠈⠻⣿⠀⢿⡘⣯⣷⣜⠎⣸⢧⠂⢱⠁⠘⣿⢸⠏⣿⢸⣎⢷⡘⢧⣡⢣⡜⣷⠟⣥⢺⡏⡿⢸⡇⠈⢹⠀⣷⢿⠘⢎⣶⣯⠏⣼⠇⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠻⣌⢿⣸⠀⡿⣸⣷⡜⣿⡄⣿⠈⣀⢹⡆⣾⠆⠹⣦⡙⢀⣼⢋⡞⡇⣿⢠⠀⣾⠁⣾⡏⣧⣾⢸⡇⠸⣸⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⣧⢷⢸⡇⢹⡟⡇⠈⡇⣿⠀⣿⠈⣷⠸⣻⠀⠹⣿⡿⣁⢨⣿⣸⡇⢾⡆⢿⠀⡇⠀⡿⣝⠁⣿⢺⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢱⡜⣇⢸⡇⢸⠁⣧⡆⢰⣿⢸⡇⣆⢻⡄⢻⡇⡀⣿⡇⢻⣤⠃⣿⢸⡇⡇⢸⡇⢡⣦⠁⢿⠀⣿⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⢹⣿⡇⣿⣼⢹⢿⡈⢿⠈⣧⡿⣼⡏⢀⣸⡇⣾⣧⠸⡇⠂⣿⢸⣧⡇⣼⠃⡼⣿⢸⣷⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢧⢻⡸⡆⢸⠀⣧⢰⣄⠻⠇⣿⠇⣼⣿⣄⣿⠹⣇⢻⢹⣿⡆⠿⢀⣵⢸⡇⢹⢀⡟⣸⢱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡹⣶⡁⣿⢸⢹⢧⠀⣿⡆⠏⣿⢸⡇⡀⢻⣦⡾⢹⡇⠰⠏⡗⢸⡇⣼⠞⣱⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⡓⢿⠀⣿⡀⡆⢻⡇⣇⡟⣾⢁⡟⣸⣿⡃⣾⢣⡇⣸⠇⣼⠟⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣈⡓⠷⠘⣿⡜⠃⣿⢸⢣⡿⢓⣡⡟⠸⠟⣋⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣾⣿⣦⣿⣠⡟⣴⢫⣿⡶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣥⡶⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⢿⣀⣮⣤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠛⠛⢻⠋⣷⣄⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢣⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣄⠙⣿⣦⡄
⠀⠶⣿⣼⣿⣿⣿⠟⢀⣠⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣿⣿⣿⣿⣿⠁
⣶⣶⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠿⠛⢯⣿⣿⣿⣿⣿⣿⣧⠀⣦⣾
⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠏
⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣬⣉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣤⣀⣀⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣽⣿⣿⣿⣶⣶⢈⠉⠉⠙⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋⠉⣉⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣿⣿⣿⣿⣿⣿⣧⡸⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠿⢿⣿⣿⣿⡿⠋⢻⣿⣿⣿⣿⣷⡌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣴⣿⣿⣿⠏⠀⠀⠈⣿⣿⣿⠈⢿⣿
⠟⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠈⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣿⠀⠀⢿
⠀⣾⣿⣿⠟⣿⣿⡿⢿⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡆⠀⠸
⠸⡿⠻⣿⢰⣿⢰⡇⠈⢿⠀⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⣸⡇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⢰⣿⣿⣆⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⢻⡀
⠀⠀⠀⢀⣿⠘⣻⣿⣿⣿⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢦⣾⣿⣿⣥⣄⢸⡇
⠀⠀⠀⢸⡏⢀⢼⣿⢿⣿⡿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣯⡥⢼⡇
⠀⠀⠀⣾⠇⠀⠈⠁⢸⡏⠀⠈⠃⢀⣤⣿⣿⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⡄⠀⠰⡛⠛⢹⢿⡏⠙⠿⠀⣷
⠀⠀⢀⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⡾⢿⡿⠶⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⠀⢴⢾⣿⣿⡅⠀⠁⠀⠚⠈⠁⠀⠀⠀⣿
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣼⣿⡗⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠂⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⣀⣀⣿⡇⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⣸⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⣾⡆⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⠀⣄⠀⠀⠀⠈⠿⢸⣿⡆⠻⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⠙⣠⡟⢰⡃⠀⠐⢦⣷⣴⣾⣿⡿⠀⢀⠀⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠘⠎⢧⣿⣇⠀⠁⢄⣀⠀⣹⣿⣿⡟⠁⠀⢾⣾⡘⢇⠀⢰⠃⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠠⡀⠀⢰⡝⡇⠀⠀⢸⣿⣿⣧⡀⢈⣿⣿⡏⢸⣿⣷⡄⠀⠘⣿⣿⡆⠀⣳⣆⠀⠚⣦
⠀⠀⠀⠀⠀⠀⣰⠇⠀⣼⣷⠀⢀⣴⣿⡟⣿⣿⡥⠞⠁⠛⠀⠈⢿⡿⣿⣦⠀⣿⣿⣃⣴⣿⡟⠀⠀⠹⣷⡜⣆
⠀⠀⠀⠀⢠⠀⠃⢠⣾⣿⡇⣼⣿⡟⠉⠀⣿⡟⠀⠀⠀⠀⠀⠀⠸⢃⣿⣿⣶⣿⣿⣿⠿⣿⡃⠀⢦⡀⣿⠇⠋⠀⣦
⠀⠀⠀⢠⣿⠀⢰⣿⡏⢿⣧⣿⣿⠀⠀⠀⢻⠀⠀⣹⠀⠀⠀⠀⢠⠾⠟⠛⠿⠋⠸⡏⠀⢻⣷⣤⣾⣿⣿⡀⣰⡄⢨⢳
⠀⠀⠀⢸⣿⣧⢻⡟⢧⠈⠻⣿⣿⡄⠀⢸⠀⠀⢰⣿⠀⠀⢠⡶⠀⠀⠀⢀⣤⠀⠀⠁⠀⠀⡿⠏⢹⣿⡿⣿⣿⣷⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⡄⢄⣀⡀⠁⠀⠀⠸⣧⡀⠈⢿⣆⠀⠻⣇⠀⠀⣾⠿⠁⠀⠀⢀⡷⠀⠀⠀⠚⠋⢀⡾⠟⣿⣿⠇
⠀⠀⠀⠀⠙⢿⣯⡛⠛⠂⠈⠉⠻⣆⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠹⢦⠀⠀⠀⢿⠃⠀⠀⣴⠞⢀⣠⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠀⠙⠀⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢼⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠈⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢠⠄⠃⠀⠆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣰⠃⡸⠀⠀⡄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠁⢠⣿⢸⠀⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇⠸⡟⢸⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⠀⣿⣸⠆⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣁⠀⢻⡇⡴⠎⠁⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣩⠯⠽⠂⣼⣧⣂⣠⡤⠀⢻⠌⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢺⣉⣖⣢⣴⠹⠟⠋⢉⠗⠪⡉⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣤⣌⠁⣠⠴⢤⠰⠃⠀⠀⢱⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⠁⢸⠂⡅⠀⣸⢰⡇⠀⠀⢸⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣿⠀⢸⠠⡏⠘⠀⠀⠀⠸⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠹⣶⠏⠀⢷⠀⠠⠀⠀⣇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣸⠀⠀⠀⠀⢀⠇⠰⠆⠀⢸⣯⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠄⣿⠀⠀⠀⠀⢸⠶⠁⠀⠀⠘⡷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣿⠂⠀⠀⠀⢈⡀⡇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠸⡿⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠽⣶⣶⣆⣤⣄⣀⣤⠴⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⡛⣿⣟⣛⣿⣿⡿⢿⡿⠿⠿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⢄⡠⠤⢀⠤⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡆⠀⠀⠀⠢⠀⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠖⢇⠈⠙⢀⠒⠀⠂⠚⠛⠛⠓⠒⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠈⠉⠉⠉⠀⠀⢀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣈⣶⣶⣞⣁⣀⣀⣙⡶⣶⣎⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣤⡴⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⣆⠀⠀⣣⠀⠀⠈⠉⢠⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣴⠒⢺⣟⣿⠓⢒⡄⠀⠀⠀⠁⠂⠰⠄⠀⠀⠀⠀⠤⠆⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠉⠙⠋⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⢲⣶⣖⢢⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠄⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠁⡀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠭⠴⠛⠉⠉⡩⢲⠎⢩⡏⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⣉⣀⡀⠀⠀⠀⠀⢛⠈⣢⣉⣴⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠺⠑⠒⠀⠈⠉⠒⠚⠳⠦⣀⡀⣠⣾⣿⣫⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⢟⠒⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡿⠧⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣒⠭⠒⠁⠀⠈⢄⡀⠀⠐⠭⠩⢛⣶⡦⣄⢀⢆⢆⣿⣿⣿⡿⠏⣏⣠⣷⣿
⠀⠀⠀⠀⠀⠀⣀⠠⠔⣊⣥⢖⠙⠓⡀⠀⠀⠀⠀⠀⠉⠒⠒⠚⢈⡡⣪⣾⣿⠀⢫⣾⣿⡿⠟⢁⣶⢿⣿⢻⢫⡇
⠀⠀⠀⠀⠀⢀⣿⢴⣢⣉⡁⢦⣀⠑⠒⡀⠀⠀⠀⠀⠀⢀⡠⣒⢵⣾⣾⣿⣿⠁⣿⣿⡟⠟⠚⠻⣿⣕⡕⣵⠋
⠀⠀⠀⠀⠀⠀⡏⠑⢶⡬⣉⡓⠶⣭⣕⣲⠊⣩⡽⠟⡩⢵⣺⣵⣿⣿⣿⣿⣯⣖⠛⠈⠀⠀⣀⡄⣿⣮⣾⠋⠱⠶⢤⣀
⠀⠀⠀⠀⠀⠠⡏⠀⠀⠙⡑⠪⢝⣒⢿⣾⣽⣓⣞⣿⣾⢿⣿⣿⣿⠿⠿⠏⠓⠒⣦⣤⣄⢦⡛⣄⣿⡿⣙⣑⣦⠀⠩⠃⣩⡕⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢭⣛⡲⠬⣅⡂⠉⢘⠛⣿⣶⣿⣧⣟⠛⠈⠤⠤⠔⡪⣻⣯⡟⢺⠁⣹⠟⠋⠉⡀⠺⢕⡶⢄⣠⣶⣷⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠳⠦⢌⣙⡋⣼⡿⠟⠋⠬⠍⠀⠀⠀⢀⢞⣵⢿⣿⡏⢃⡽⠓⠀⢀⠔⠁⠀⠀⠈⣉⡚⠻⢿⠷⢇⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⢉⠴⠉⢰⡁⠒⠂⢬⣙⣋⠉⠁⠀⠀⣀⣀⣀⢔⣕⣟⡷⣿⣿⣦⠯⠄⠀⠒⠁⠀⠀⠀⠀⠤⠀⠀⠀⡜⠀⢄⢨⠉⣱⡆
⠀⠀⠀⠀⠀⠀⢐⡯⠬⠥⠉⣀⣀⡒⠊⠭⠭⠶⣢⡤⠤⠊⢁⠖⣃⢴⡿⡯⣾⣾⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⠂⣸⢄⣠⠴⢊⠥⢻⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠤⢂⣐⡒⠒⠀⠭⠭⠤⠐⢀⣀⣀⡨⡭⠍⠐⣫⣺⣿⣿⣏⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⡚⠭⢒⡩⣴⣫⣵⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢅⣒⣒⣒⡰⠯⠭⠭⠥⠔⢛⣒⣐⡤⠭⡍⡭⣿⣿⠿⡟⠙⠿⣿⢤⣘⠓⢬⠑⠲⣔⡢⠭⣐⣻⣵⣶⣿⢿⣿⠿⠛⡋⢬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢕⠬⡛⢌⡙⠩⢋⢟⡹⡋⢞⠑⡋⠛⠝⡒⢛⢷⣕⡢⠀⠀⠀⠹⢦⢽⣒⣺⣽⣕⣛⣭⣶⣿⡯⠟⢛⠩⢔⣊⣥⠔⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠉⠢⢤⣄⡀⢀⣁⣈⣀⡁⠈⣈⣰⣥⡿⣻⣄⠀⠀⠀⠈⠃⢰⣿⣿⠿⢟⢚⣉⣥⡒⠥⢒⠮⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠨⣛⢥⡀⠴⠚⣋⣽⢿⣿⣿⡿⣿⣟⠝⠒⢄⡀⠀⠐⣗⣖⣯⣽⠿⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠸⡩⡉⠹⢿⣻⣷⣟⣯⡿⠟⠊⠀⠀⠀⠀⠈⠁⠊⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⡀⠈⡆⢐⣿⡿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢐⠀⠂⠐⡠⠂⠔⡠⠂⡀⠀⠄⡠⢀⠀⠀⠀⠠⢀⠄⠀⠀⡠⢂⠐⠀⠀⠠⢀⠄⠀⢀⠔⡠⡠⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠊⠔⠥⢂⠔⢈⢔⡠⢊⠔⢊⠔⡠⢂⠑⡨⠐⠡⢊⠐⡡⢂⠀⡠⡠⢊⠄⠁⢒⡑⡠⢂⡔⢃⠔⡡⢪⠊⠔⠑⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠘⠁⠊⠁⠊⠐⠁⠊⠐⠡⠂⡠⢊⠔⠡⠪⠊⠀⠁⠊⠀⠋⠀⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠿⠿⠟⠻⠿⠿⠿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⢁⣀⣤⠤⠶⠶⠶⠶⠤⣤⣀⡈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿
⠀⠀⠀⠀⠀⠀⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡈⠉⠛⠒⠶⠶⠶⠶⠒⠛⠉⢁⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣶⣦⣤⣤⣶⣶⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀⢈⠂⠀⠂⠀⠢⡑⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢶⠀⠁⠢⢄⠀⠀⠀⠀⠈⠢⡀⠈⠢⠊⠃⢂⠤⠊⠁
⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠴⡄⢀⠇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠙⠦⣀⡠⠖⠁⠀⠀⠀⠠⠐
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠒⠈⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⡐⠤⠚⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠇⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⢀⡰⠆⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⡤⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠉⠀⠀⠀⠀⠲⣂⠀⠀⠀⠀⠀⠠
⠀⢂⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⠁⠉⠉⢓⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠾⠒⠙⠐⣶⠀⠉⠶⠄⠀⠀⠀⠘⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⢀⠬⢝⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠣⠖⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠄⠀⡲⠆⠂⡁⠈⠣⡀⠀⠀⠀⠀⠩⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⢄⡈⢒⠚⠠⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢀⣀⢠⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠀⠲⠌⠐
⡴⠋⡀⢠⡄⠀⠉⢦⠀⠀⠀⠀⠀⠀⠂⠁⠀⠐⠆⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⣌⠀⠀⠀⠸
⠀⠀⠀⠀⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠐⢀⠒⠂
⡘⠃⠀⠐⠴⡥⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠁⠰⠠⡆⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠜⠡⠀⣠
⠙⢦⣀⡀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠘⡡⠂⠀⠢⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠄⠊
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡇⠾⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠈⠂⠄⢒⡡⡂⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠤
⠈⠁⠀⠀⠆⠐⢀⠈⠀⠈⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠂⠄⡈⢑⡴⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠏⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠱⠢⢀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠆⠀⠀⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠓⠀⡀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠐⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⡠⠊⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠸⠁⣀⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡈⠀⠗⠁⢀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⡀⠈⠷⠄⠀⢀⡠⠊⠀⠀⠀⢀⣤⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢃⠌⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⠀⢀⡠⠔⠉⠀⠀⠀⠀⡰⠏⠀⠲⠒⡄⠀⠀⠀⠀⠀⠒⠒⠀⢤⠄⠊⡀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⡁⠜⠁⠀⠀⠀⠀⠀⠁⠉⠔⠋⡶⠚⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄
⠀⠀⣖⢊⡙⢢⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠴⠁⠈⢁⡜⠀⠀⠀⠀⠅⠠⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⡀⠀⠱⡄
⠀⣀⢉⠤⠽⠎⠧⠀⠊⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠈⠁⠀⠀⠀⠀⠀⢄⠀⠂⠀⠀⠀⠑⠀⠀⠀⡀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠡⢀⠀⠌⠀⠀⣀⠈⠄⠀⠀⠀⢨
⠀⠀⠀⠠⠀⠀⠀⠀⠙⡀⠀⠀⠀⡭⠊⠀⠀⠀⠘⠂⠀⠘⡀⠊⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⢀⠀⠀⠀⠀⠀⠀⠐⡀⠉⠀⡀⠀⠀⠈⠀⡘⠀⢀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣤⡤⢤⣾⣦⠤⣤⣤⡤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢈⣏⠀⠀⣸⡀⠀⢈⣏⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠙⢻⡟⠋⠛⣿⠛⠙⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⣤⣠⡼⢧⣄⣤⠿⣤⣠⡼⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢈⣏⠀⠀⣹⡁⠀⢈⣏⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⠟⠛⢷⡾⠋⠻⣶⠛⠛⢷⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣼⣧⣀⣠⣿⣄⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀
⢤⡀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠈⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠂⠀⢀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣀⣀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠃⠠⢻⡧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀
⠉⠙⠛⠈⠉⢧⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠼⠁⠀⠀⠀⠀⠀⠀⢻⣥⠤⠖⠶⠚⠋⠉⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣠⣒⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠖⠒⠒⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⢀⠞⢱⣤⣀⠀⠈⠁⠀⠀⠀⠀⠀⣯⠀⠀⠀⠠⣿⣦⣄⡀
⠀⣀⣀⠠⠤⠀⠂⠈⠁⠀⠀⠀⡰⠁⠘⣿⣿⣿⡆⠀⢀⠃⠀⠀⠀⠀⠙⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣼⣿⣿⣿⣿⡄⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠹⣿⣷⡂⠀⢸⡄⠀⠀⡀⣀⠀⠀⣿⣿⣿⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢸⣿⣿⣇⠀⣽⡿⠀⠀⠈⡇⠀⠀⢱⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⣿⣾⣿⠇⠀⠀⠀⡇⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣤⣶⣶⣶⣿⣿⣿⣿⡿⠿⠛⠛⠋⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣷⣶⣶⣦⠀⣀
⠀⣠⡾⠿⣿⣿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⢀⣀⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⣿⣿⠿⣷⣄
⣾⠋⢹⣶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠋⠉⠙⠛⠛⠛⠛⠉⠉⠙⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣶⡎⠹
⣿⣧⡌⠛⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⢁⣼
⣿⡋⠹⣿⠏⣉⣿⣶⠶⢀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⣀⣀⡀⢰⣾⣿⣍⣉⣷⡿⢻
⣿⡇⠀⠀⠘⠛⠛⢡⣶⣾⣿⠿⠿⣃⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣘⠛⠿⣿⣿⣷⣤⡜⠛⠛⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠿⠿⠿⠿⠿⠛⠛⢿⣿⣿⡟⠋⢻⣿⣿⣿⠟⠛⠛⠿⠿⠿⠿⠿⠗⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠙⠛⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣰⣿⠀⠀⠀⠀⣿⣅⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠛⠛⠿⢿⣷⠀⠀⠀⠀⣾⡿⠻⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣿⡟⠻⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠟⢻
⣿⡇⠀⠀⠙⠛⠛⢦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⡶⠛⠛⠋⠀⠀⢸
⣿⣿⣄⣀⠀⠀⠀⠀⠉⠉⠉⠙⠿⠿⠿⡿⠷⠶⠤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣴⠶⠶⠾⢿⡿⠿⠗⠈⠉⠉⠁⠀⠀⠀⠀⣀⣴⣿
⣿⡇⠘⠛⢳⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡾⠛⠃⢸
⣿⡇⠀⠀⠀⠉⠉⠙⠿⠿⠿⢦⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⡴⠿⠿⠿⠛⠉⠉⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠓⠶⠶⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠓⠀⠚⠟⠒⠲⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠚⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼
⠀⠈⠛⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⠟⠁
⠀⠀⠀⠀⠈⠙⠿⠿⢷⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⠼⠿⠿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⡀⠈⠉⠉⣉⣉⣡⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣈⣉⠉⠉⢁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠻⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣯⣀⣠⣄⣀⣀⣀⣤⣀⣠⣤⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⢠⡶⠶⠿⠿⠿⠷⠶⢶⡄⠀⠀⡴⠶⠾⠿⠿⠿⠶⠶⡄⠀⠀⡴⠶⠾⠿⠿⠿⠶⠶⡄⠀⠀⣴⠶⠶⠿⠿⠿⠷⠶⣦
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢸⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢈⠛⢿⣿⣿⣿⡇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣿⣷⣤⡀⠀⠀⠀⡇⠀⠀⣿⢸⣷⣦⡈⠻⢿⡇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣿⣿⣿⣿⣷⣄⡀⡇⠀⠀⣿⢸⣿⣿⣿⣷⣤⡁⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠻⣿⣿⣿⣿⣿⡇⡇⠀⠀⣿⠸⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⣦⣀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣤⣌⠙⠿⣿⣿⡇⡇⠀⠀⣿⢰⣄⡙⠻⣿⣿⡇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⣿⣿⣷⣦⣀⠀⠀⡇⠀⠀⡇⣿⣿⣿⣦⣌⠙⠇⡇⠀⠀⣿⢸⣿⣿⣶⣌⠙⠇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⢿⣿⣿⣿⣿⣷⠄⡇⠀⠀⡇⢿⣿⣿⣿⣿⣷⠆⡇⠀⠀⣿⢸⣿⣿⣿⣿⣿⡆⣿
⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠛




⣿⣿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠻⡄⠀⣠⠴⠂⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠴⣾⠁⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⡿⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣼⢃⡴⠚⠙⠲⣤⡀⣾⡟⠀⠀⠀⠀⠀⠹⣆⠀⠚⠉⡉⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠈⠙⢯⡿⢦⡀⠀⠀⠀⠀⠙⢧⡀⡿⠙⡟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣿⡄⠀⠀⠀⢀⣀⣭⡇⢀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠦⢤⣽⣷⣿⠤⠖⠚⠉⠉⠀⠙⠛⠓⠞⠷⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠴⠶⠶⠶⢦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢦⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠄⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠻⠶⢾⣷⣤⣬⣵⣤⣤⣵⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⣙⢦⡀⠀⣠⡟⠀⠀⠶⠖⣻⣿⠟⠓⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠁⣿⠂⢙⡾⠋⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡶⠟⠁⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾⠀⠀⠸⣧⠶⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣄⣤⠶⡿⢲⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣽⣷⣭⡙⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⣈⡝⢿⡀⠙⢿⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⣇⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣠⣶⣶⠾⠷⠿⠿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡯⠀⣿⠷⢼⡇⢰⠜⠃⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣄⢹⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠠⣾⣥⣤⣤⣤⡴⠛⣻⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢳⠀⠸⡇⠈⡇⢸⠀⢸⡿⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢷⣜⣧⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⢷⠀⣿⢸⠇⢸⣇⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⣄⡈⠻⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠸⡇⢸⡿⠀⠈⣿⢹⣆⣿⣤⢴⣶⡤⠶⣶⣆⢹⣷⠤⢀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠛⠋⠛⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢿⡀⠀⠸⡇⠀⣿⣸⣧⣤⠶⠛⠉⠉⠀⠀⢰⣿⠀⠀⣿⡽⠦⠿⠄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠛⠶⠖⠲⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠃⠀⠀⢿⡤⠼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣀⢹⡗⣶⠶⢆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣒⣚⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠈⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⡚⣣⡇⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠈⠙⠓⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠸⣿⡃⠀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⡟⠉⠉⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⡏⠙⣿⠉⢻⣏⣉⣠⡤⠶⠚⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⢿⣆⣠⣿⠀⠘⣏⠉⠀⠀⠀⠀⣿⣤⠤⠤⠴⠶⠶⠶⠖⠚⠛⠛⠛
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠁⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠂⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣧⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣤⠤⣤⠼⠟
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠛⢛⣻⣿⣿⣿⠿⠧⠤⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠒⠒⠂⣉⡿⠋⠀⣮⠈⠛⠓⠶⠶⢠⡿
⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠁
⢀⣀⣀⣤⣤⣤⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⣤⠴⠶⢿⡷⠻⠛⠒⠒⢠⡄
⠻⣤⣀⣀⣈⠹⣦⣹⣯⣭⡿⠿⠛⠛⠋⠉⠁⠀⠀⠉⠉⠐⠶⠾⠶⠧⠶⠶⠾⠋⠁

⠀⡀⠀⣄⠀⠆⢀⡄⠀⠀⠀⢀⡰⠆⠲⠤⣀⡀⠀⠀⠀⠀⡀⠀⡀⡀⠀⠀⠀⠀⠀⠄⣠⡀⣄⡀⠀⠀⠀⠀⠀⠄⡠⡀⢁⢀⠀⠀⠀⠀⠀⠀⠲⠀⡀
⠀⠁⠄⠈⠀⠀⠈⠰⠘⠀⠀⢸⡀⢀⡀⠀⠀⠉⠁⠂⠒⠀⠁⠉⠀⠀⠉⢆⠀⠠⠠⠟⠉⠀⠈⢩⠆⠀⠀⠀⣒⠄⠋⠁⠈⠚⢂⡀⠀⠀⢘⢰⡀⢤⠑⠀⠂
⠈⠒⠀⠀⠀⠀⠀⠀⡋⠁⠀⠀⠉⠉⠈⠉⠦⣀⡀⢀⡀⠤⠰⠆⡤⠤⠴⠂⠀⠀⢝⠆⠁⠈⠀⢠⡕⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⡂⠀⠀⠀⠎⠀⠈⠙
⠐⠊⢠⠀⡀⢀⣀⠈⠀⠃⠀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠙⠗⠰⠲⢘⠀⠀⠀⡨⠂⣠⡀⠀⠀⠀⠍⠄⠁⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠸⠀⠈⠃⠀⠀⠀⠁⠀⠍⢠⠒⡠⡀⠀⠀⠀⠀⢀⠀⠀⡉⠀⠠⠀⠀⠀⠀⠀⠀⢀⡀⣤⠀⠀⠀⠀⠐⡑⠈⠈⠁⠐⠀⠀⠐⠂⠀⠀⣸⡄
⠈⠁⠀⡀⡀⠀⠀⠀⠀⠐⠀⠒⠀⠂⠀⠈⠠⠀⠀⠀⢀⠈⠈⡀⠀⠁⠀⡀⠀⡀⠀⠄⠀⠰⠎⠈⠀⢘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⢈⢈⠀⠄⠉⠀⠀⠠⠈⠈⠡⠈⠂⠀⠁⠀⠄⠁⠀⠀⠀⠀⠀⠚⢠⠄⠆⠀⠐⣦⡀⠈⠓⠤⡄⡀⠀⠀⠀⠁⠀⠀⠈⠄
⠀⢐⠀⠈⠀⠆⠀⠉⠀⠀⠂⠘⠐⠒⠈⠀⠀⠐⢐⢀⠀⠀⢠⡂⠈⠁⢀⠀⠀⠀⠀⠀⣀⠀⠄⡀⠀⠀⠀⠀⠈⠃⡆⠀⠀⠉⠙⡣⠀⠀⠀⢠⠄⠀⠀⡀
⠀⡄⠀⠄⡐⠀⠀⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠚⠀⠚⠀⠉⠈⠀⠀⠀⠈⢆⠀⠀⢀⠰⠀⢣⡀⠀⠀⡂⠈⠁⠀⠀⠀⠀⠆
⠀⢇⢀⠰⠁⠀⡈⠀⠀⠨⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠠⡑⠁⠀⠘⠀⠀⢧⠁⠀⠀⠀⡀⠀⠀⠠⢀
⠀⢸⠁⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⡀⢀⣀⠄⠖⠂⠉⠀⠀⠀⠐⠉⠀⠀⡀⣀⡠⣴⠏⠀⡀⠀⠀⣄⣀⠀⠉⠒⠃
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠒⠠⠀⠀⠀⠈⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⡀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠀⡀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⡄⠈⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⢀⣤⣴⣶⣶⢶⣶⣶⣶⡶⠶⢾⣿⣿⠿⡛⢿⣿⣷⣄⠀⠀⠀⠀⢀⣀⡀⠐⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠄⠀⠀⠠⠂⢀⡀⠀⠀⣰⡟⣻⢁⠀⠀⠒⢠⡌⢦⡀⢤⣭⣍⠻⣷⡤⣋⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠐⠐⠀⠀⠀⠀⠉⠀⠀⢠⠃⠑⢠⠀⠀⠀⠀⢠⣿⡇⠃⢸⡆⢃⢀⣌⣙⣄⣁⣸⣧⣤⣤⡿⢿⡟⢿⠀⠘⣷⠀⠀⠀⠀⠤⠠⠀⡀⠀⡀⠂⢀⢁
⠀⠀⠀⠴⠀⠀⠀⠀⠀⠤⠀⠂⠉⠦⠤⠪⡀⠙⠃⠀⢸⣿⣷⣤⣿⠟⢿⣿⢉⣙⣯⡉⢹⡉⠤⠾⠄⠒⢠⠈⠀⣆⣿⡁⠀⠀⡈⠀⠁⠀⠌⠂⠜⠀⠀⠐
⠀⢀⡠⠤⢄⡀⠀⠀⠀⠀⠀⠠⠆⠀⢀⠀⠈⠂⠀⠀⠀⢿⡄⢂⠹⡄⢠⠙⠌⢿⠿⢿⡜⣧⠀⠶⠿⡀⡈⠃⢀⣿⣿⠁⠀⠀⠀⠀⠀⢃⠀⢄⠐⠍⡀⠀⡠
⢠⠏⠀⠀⠀⠙⡄⠀⠀⠀⢄⠀⠀⢀⠉⢀⠀⠀⠀⠀⠀⠀⠙⢮⣁⣠⣼⣷⣼⣦⣤⣾⣷⣼⣧⣤⣤⣷⣾⣾⣿⠿⠁⠀⠀⢀⠤⠀⠀⠀⠠⠄⠃
⠘⣆⠀⠀⠀⣨⠃⠀⠘⠀⠀⠀⠄⡈⣀⡘⠀⠘⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠀⠀⢀⣀⡀⡒⢢⣄⠇
⠀⠈⠙⠂⠋⠀⢀⣀⡀⠀⠠⠄⠀⠼⡉⠈⡇⠒⠃⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐⠴⡉⠟⠀⠞⠁⠃⠀⡀⠠⠄⠀⠂
⢠⠀⠀⡄⢀⠞⠁⠀⠈⠳⡄⠀⠤⠀⠙⠈⢀⠀⠐⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠗⢨⡀⠀⠀⠀⢠⡀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠂⠀⢀
⠈⠢⠔⠁⢸⠀⠀⠀⠀⠀⠇⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠈⢀⡔⠉⠑⡄⠀⠁⠄⠀⠧⣀⡰⠃⠀⠀⢀⣀⠈⠀⡠⢤⡔⢴⣶⠲⡀⠈⠀⠈⠀⠂⡀⢀⠐⠁
⠀⡀⠀⠀⠈⠦⣀⣀⡀⡔⠂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢣⣀⣠⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠘⢻⡏⢷⣼⡧⠚⠷⠀⠁⠀⠀⢀
⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠋⠀⠓⠀⠁⠀⠀⠀⠐⠀⠀⠀⢀⠤⠤⠄⡀
⠀⠀⠄⠠⡄⠃⡄⢂⠒⠒⣀⣰⡀⢃⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⢀⠠⢤⠀⡀⠀⠀⠀⢀⢊⠐⡀⠀⢀⡌⢂
⣄⠙⠠⠒⠃⠂⠒⠀⠈⠁⠀⠒⠈⢧⣐⠀⠀⠀⠀⣠⠔⠚⠋⢲⡀⠁⠤⠀⢀⠤⠤⣀⠠⠀⠀⠀⠐⠀⠀⠃⢔⠉⠀⠀⠀⠌⡢⠅⠀⠆⡄⢀⠀⢀
⣐⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢕⡚⠀⢀⡴⠋⠀⣈⣠⡠⠞⠀⠀⢀⡰⠁⠀⠀⡘⠀⠀⠠⠄⠀⠀⠀⠀⢊⡀⠀⠀⠀⢈⡢⠀⠀⠡⡰⢄⠀⠀⣀⢜⠔
⠉⠖⠱⠴⣀⢀⣀⠀⠱⡐⢀⠤⠑⡄⠁⠀⣎⣠⠖⠉⠀⠀⠀⢠⠒⠊⠁⢀⢀⠤⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠹⡠⠾⠐⠁⠀⠀⠀⠀⠈⠂⠈⠁⠐⠁
⠀⠉⠀⠃⡄⢠⠘⠀⠒⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠙⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⢻⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣼⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⢻⡟⡟⣿⢹⡟⡟⣿⢻⡟⡟⣿⢻⡟
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⠷
⠀⠀⢸⢹⡏⣿⢹⢹⡏⣿⣽⡿⢟⣛⣭⣭⣭⣭⣛⡻⢶⣼⣏⡏⣿⢹⡏⡏⣿⢹⡏⣿⣿⢹⡏⣿⣿⢹⡇
⠀⣀⣸⣾⣷⣿⣾⣾⣿⢟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡻⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣅
⠀⣿⢹⣿⡏⣿⢹⡿⣡⣿⣿⣿⣿⠿⠛⠀⠀⠛⠻⢿⣿⣿⣷⡌⢿⣿⢹⡿⡏⣿⢸⣿⡏⣿⢹⡏⡏⣿⢸
⠀⠙⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⡏⠀⠀⢰⣦⣤⣄⣾⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛
⠀⠀⣿⣸⢸⣿⡏⣾⣿⣿⣿⣿⣷⣄⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣧⢹⡇⣿⣸⢸⡇⣿⣸⢸⡇⣿⢸
⣤⣤⢿⣿⣿⣿⣇⢿⣿⣿⣿⣿⡿⢿⣿⣶⣦⠀⠀⠘⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⢿⣿⣿⣿⢿⡿⣿⣤
⣇⣿⣸⣇⣇⣿⣿⡜⣿⣿⣿⣿⣅⡀⠀⠉⠁⠀⣀⣼⣿⣿⣿⣿⢣⣿⣸⣇⣇⣿⣸⣇⣇⣿⣸⣇⣿⣿
⠉⡿⣿⢿⣿⡿⣿⢷⡜⢿⣿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⡿⣣⣿⣿⢿⡿⣿⣿⢿⡿⣿⢿⢿⡿⡏⠉
⠀⣇⣿⣸⣿⣇⣿⣸⣿⣦⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣾⣟⣿⣿⣸⣇⣿⣿⣸⣇⣿⣼⣸⣇⡇
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⠶⣮⣭⣭⣭⣭⣵⠶⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢻⣿⣿⣿⡇
⠀⠀⠀⠁⠒⠂⠐⣀⣴⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⢿⣿⣿⠇⠂⠄⠄⠠⠀⠀⠠
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠉⠁⠈⡁⣁⣀⡁⢈⣁⡀⣀⣀⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣿⣿⣿⣿⠀⢀⡀⢀⡀⠀⡀⢀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠤⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⠹⣿⣿⡇
⠐⠀⠂⣠⣶⣾⣿⣶⣦⣀⠀⠀⠘⠷⠶⣶⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣥⣤⣤⣤⣤⣴⣶⠶⠶⠶⠶⠂⠘⠿⠿⣿⣿⣿⠀⢿⡟
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⢰⣤⣤⣀⣀⣀⣀⣈⡉⠉⠉⠉⣉⣉⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣶
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂
⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣁⣉⡁⡉⠈⢈⠁⠁⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⡀⠀⠀⠀⠀⠒⠒⡀⠐⠂⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠈⢀⣤⣶⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠒⠐⠐⠒⠀⠐⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣿⣿⣿⣿⠛⣿
⠀⠀⠉⠉⠁⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠠⠄⠄⠀⠀⠀⠠⠀⠀⠄⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣈⡀⢀⢁⡉⠀⢉⠀⡈⠈⡀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣤⣤⡶⠶⠶⠶⣶⡟⠙⡟⠳⣶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⢯⣿⣧⣀⠝⠀⠀⠀⠀⠀⠀⠉⠛⢳⠋⠑⣤⣿⠋⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⣨⠿⣤⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⢛⠉⠉⠛⠛⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⡏⠙⢡⡌⢉⡅⠀⠀⠀⠀⠀⢶⡀⠀⢀⠀⠀⠉⠁⢀⣀⠀⠉⠀⠀⠀⠀⠈⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠷⠞⠁⠙⠋⠙⠛⠛⠛⠁⠀⣀⣄⠀⢹⡿⠏⣠⣴⡞⢠⡁⢈⡿⠃⡀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢻⠀⢀⣀⣴⡀⢀⡀⠀⠀⢶⣅⣈⣿⣷⡿⠤⣨⠁⠀⠢⠞⢟⢋⣠⠞⢁⣴⣃⣀⣀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇⡀⣣⠱⣤⣤⣉⢛⡇⣤⣤⣤⠟⢛⠁⢈⣦⣠⠟⠀⠀⠀⠀⠀⠀⠈⠛⠁⢰⣤⠌⠉⠉⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⠀⠉⠁⠀⣄⡈⠉⠉⠀⠀⠀⣨⠀⠀⠙⠛⡻⠗⢤⣼⣦⣤⡄⢀⠀⠀⠘⠲⠚⢿⠷⠲⠆⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡏⠀⠀⠶⠾⠛⠙⠚⠀⠀⠘⠛⠛⠀⢀⡄⠀⠳⢤⡴⠟⢁⣴⢣⡈⢣⠀⠄⣠⠀⣯⡄⣀⠀⠀⢠⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢾⣁⣸⣄⢀⣀⣀⣴⣶⣄⠀⠀⢀⠀⢸⠷⠿⡿⢿⣧⠀⢦⣤⡋⠁⢈⣷⠈⡧⠚⠿⠀⡏⠁⠀⠀⠀⠈⠳⢶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠀⣹⣯⣉⣉⡉⠛⣿⠶⠾⡇⢸⣄⣀⣾⠀⠈⢱⡈⠋⠓⠶⠚⠛⠲⣧⣀⡼⠊⢉⣀⣀⠀⠀⠀⠀⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠯⣉⣤⣤⡟⠁⠉⣿⣆⡀⣨⡦⠟⠉⠈⠣⠤⠞⠋⠀⠀⠀⠀⠑⠄⠈⠃⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣼⠏⠀⠑⠤⠶⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠢⠤⠜⠀⠀⠀⠀⠀⢤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣴⣆⣀⣴⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡇⠀⠀⠀⠀⠀⢄⣠⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠞⠒⣶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣾⣦⡄⠀⠀⡉⠙⠓⠋⠀⠀⠀⡀⠀⠀⡀⠀⢨⣟⢉⣹⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠿⣤⣀⣀⣀⣀⣠⣷⣶⣾⠿⠐⠛⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠉⠉⠉⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⣀⣤⠾⣦⣀⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣨⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠯⠶⠀⠰⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡴⠀⢀⠶⢴⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣿⣁⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡈⢧⡀⠺⣂⣼⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡥⣚⡉⠄⠂⣿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣸⠛⠶⢿⣿⠺⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡜⣿⡏⠀⠀⠀⢧⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣿⠋⠀⣀⣨⡿⠗⠚⣷⣶⠖⠋⠉⠉⠉⢉⣉⣤⣴⣿⡿⠀⠀⠀⠀⠘⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠛⣿⡛⠦⣭⣥⡴⠶⢟⣽⣏⣴⣿⣿⣿⠿⠿⠟⢻⢛⠁⠀⠂⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣠⡇⠹⣮⡳⣤⡉⠀⠀⠚⠁⣿⡿⠋⠁⠀⠀⠀⠀⠀⠘⠀⠀⠈⠴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠁⣰⡿⠁⠀⣌⠙⠦⣀⣤⣶⡀⠀⡟⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣾⠉⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⣧⣿⣄⢨⡳⣦⣙⢿⣿⣧⢸⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⢰⡏⣸⡟⢹⣿⣿⣆⢳⣬⣛⠷⣽⣿⣿
⠀⠀⠀⠀⠀⠀⡾⢡⣿⠀⠘⡇⢻⣿⡜⢿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⣼⢱⣿⠇⠀⠀⣧⠈⢿⣿⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⡇⢸⡿⠀⠀⣸⡟⡄⠢⡉⠉⠉⣩⣾⣿⠟⣇
⠀⣴⣶⡄⣿⢀⣾⠃⠀⠀⣿⣷⡜⠀⠙⣶⣾⣿⠟⢡⢸⣿⡆
⠀⠙⣿⢠⠇⣼⠃⠀⠀⢸⡏⠘⢿⡄⠀⠈⠉⠀⣴⡏⣼⣿⡀
⠀⠀⢸⠇⣼⠃⠀⠀⠀⣾⣿⣦⡀⢙⡁⠀⢠⣾⡟⢰⣿⣿⡇
⠀⠠⣋⣼⣿⠀⠀⠀⠀⣿⠋⠙⠿⠋⢀⣴⣿⠟⣰⣿⡿⣿⡇
⠀⠀⢻⣿⣿⣀⣠⣶⡀⣿⠀⠀⠀⣠⣾⡿⠃⣰⣿⣿⢣⣿⣿
⢷⠾⠿⢿⣿⡏⠉⠈⠁⣟⠀⢀⣼⣿⠟⢠⣾⣿⣿⡏⣼⣿⣿⡇
⠀⠀⠀⠘⣿⣧⠀⠀⢸⣿⣴⣿⡿⢃⣴⣿⣿⣿⡿⢠⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⡄⠀⢸⣿⣿⡟⣵⠟⣹⣿⣿⣿⢃⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣧⠀⢸⣿⡟⢸⡏⢠⣿⣿⣿⠇⣾⣿⣿⣿⢻⣿
⠀⠀⠀⠀⠀⢹⡜⡄⢸⣿⣇⣿⠀⣾⣿⣿⠏⣼⣿⣿⣿⣿⠀⣿⡄
⠀⠀⠀⠀⠀⠈⣷⢁⢸⣿⣿⡏⢰⣿⣿⡟⣰⣿⣿⠿⣿⡟⠀⢿⡇
⠀⠀⠀⠀⠀⠀⢹⡎⣸⣿⣿⢃⣿⣿⡿⣱⣿⣿⡿⠀⣿⡇⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠈⣷⢿⣿⣿⣼⣿⠏⣰⣿⣿⣿⠇⠀⢹⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡞⣿⣿⣿⡟⣴⣿⣿⣿⡟⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢻⣿⣿⣾⢿⣿⣿⣿⣡⠀⣧⢠⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣿⡿⢃⣿⣿⣿⣷⡏⢸⣿⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢹⢀⣾⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⢸⣘⡯⠯⣭⡟⠋⠁⠀⠀⠀⠈⠑⢻⣭⠽⢽⡋⡇⠀⠀⠀⠀⠀⠀⢸⢘⡯⠯⣽⡟⠊⠁⠀⠀⠀⠈⠑⢻⣯⠿⢽⡃⡇
⣀⣀⡀⠈⠽⣧⣽⠦⠉⠀⡀⣠⣿⡀⢀⠀⠉⠴⣯⢼⠿⠁⢀⣀⣀⣀⣀⡀⠈⠿⡧⣽⠦⠉⠀⣀⣠⣟⡄⢀⠀⠉⠴⣯⣼⠯⠁⢀⣀
⡶⡼⠾⡧⡅⠉⠈⠀⣀⣦⡹⡞⣽⢹⢏⣴⣀⠀⠁⠉⢨⣼⠿⢦⢶⡷⡴⠿⣧⡅⠉⠈⠀⣀⣦⡽⡏⣯⢻⢏⣴⣀⠀⠁⠉⢨⢼⠿⢧
⠟⣧⠳⡟⠂⠀⠀⠀⢮⣧⣥⢮⣿⡽⣬⣽⡵⢀⠀⠀⠐⢻⠿⣴⣸⣅⣦⠳⡟⠂⠀⠀⠀⠮⣧⣥⢧⣿⡵⣬⣼⡵⠀⠀⠀⠐⢻⠞⣼
⡿⠉⠃⠀⠤⣼⠮⣿⡅⠉⠓⠒⠚⠒⠚⠉⠨⣼⠷⣧⠄⠀⠘⠑⢟⡻⠈⠃⠀⠠⣼⠾⣧⠅⠉⠓⠒⠛⠒⠚⠉⢨⣼⠵⣧⠤⠀⠘⠋
⠁⠀⠀⣈⣳⠮⡥⣟⢹⠀⠀⠀⠀⠀⠀⠀⡇⣻⢌⠵⣾⣅⠀⠀⠈⠁⠀⠀⣨⣷⠯⡡⣟⢸⠀⠀⠀⠀⠀⠀⠀⡏⣻⢌⠵⣞⣅
⠀⠀⠛⠾⢗⣛⣛⡿⢺⠀⠀⠀⠀⠀⠀⠀⡇⢿⣻⣛⣻⠷⠛⠀⠀⠀⠀⠛⠼⣟⣛⣛⡿⢸⠀⠀⠀⠀⠀⠀⠀⡟⢿⣛⣛⣺⠗⠛
⣆⠀⠀⠐⢉⣵⣂⣿⡞⠀⠀⠀⠀⠀⠀⠀⢳⣿⣰⣮⡙⠃⠀⠀⡰⢆⠀⠀⠘⢋⣵⣆⣿⡞⠀⠀⠀⠀⠀⠀⠀⢳⣿⣰⣾⡙⠃
⡟⣶⢃⡄⠀⠘⠑⠃⣂⣦⣭⣩⣭⣍⣽⣶⣐⠘⠋⠃⠀⢠⣘⣯⢿⡿⣵⢃⡄⠀⠘⠙⠃⣂⣶⣯⣩⣭⣍⣭⣶⣐⠙⠊⠃⠀⢠⡘⣶
⣟⣅⣋⣟⡅⠀⠀⠀⠵⣧⣆⡕⣿⢺⡰⣾⠯⠀⠀⠀⢨⣿⣛⣸⣺⣗⣅⣛⣿⡅⠀⠀⠀⠵⣷⢆⡇⣿⢪⡰⣼⠯⠀⠀⠀⢨⣻⣙⣸
⠭⠬⠟⠃⣂⣶⣴⣀⠀⠁⠞⢿⢾⡞⠷⠈⠀⣀⣦⣶⣐⠘⠻⠥⠭⠯⠬⠟⠃⣂⣶⣴⣀⠀⠁⠞⢱⠷⡿⠳⠈⠀⣀⣦⣶⣐⠘⠻⠥
⠀⠀⠀⢰⠻⣇⢛⣧⢖⠀⠀⠀⠛⠀⠀⠀⡲⣼⡛⣸⠟⡆⠀⠀⠀⠀⠀⠀⢰⠻⣇⢛⣥⢖⠀⠀⠀⠛⠀⠀⠀⣢⣬⡓⣸⠟⡆
⠀⠀⠀⢸⢼⡷⡷⢞⣿⡱⠆⠀⠀⠀⠰⢎⣿⡳⢿⢾⡧⡇⠀⠀⠀⠀⠀⠀⢸⢸⡷⡷⢾⣿⡷⠆⠀⠀⠀⠰⢎⣿⡰⢾⢾⡗⡇
⠀⠀⠀⠸⣴⡏⢬⡟⠮⠀⠀⠀⣤⠀⠀⠀⠵⢻⣥⢹⣦⠇⠀⠀⠀⠀⠀⠀⠸⣴⡏⣬⡟⠯⠀⠀⠀⣤⠀⠀⠀⠴⢻⡥⢹⣦⠇
⣒⢒⣦⡄⠍⠿⠻⠉⠀⡀⢦⣾⣾⢧⡶⢀⠀⠉⠟⠿⠩⢠⣴⡒⣒⣒⢒⣦⡄⠍⠿⠻⠉⠀⡀⢢⡼⡶⣷⡴⢀⠀⠉⠟⠿⠩⢠⣰⡒
⣯⡏⣍⣯⡃⠀⠀⠀⣲⡟⠎⡣⣿⢼⠱⢿⣖⠀⠀⠀⢘⣿⣭⢙⢽⡯⡋⣭⣿⡃⠀⠀⠀⡲⡿⠎⡇⣿⢜⠹⢻⣖⠀⠀⠀⢘⣽⣩⢙
⣷⠿⡌⠃⠀⢠⡠⣄⠍⠟⣛⣙⣛⣋⣻⠻⠩⢠⢄⡄⠀⠘⢩⢿⣾⣷⡻⡍⠃⠀⢠⡠⡄⠍⠿⣛⣙⣛⣋⣻⠿⠭⣠⢄⡄⠀⠘⢡⢿
⠏⠀⠀⢠⣌⡻⠍⣿⢧⠀⠀⠀⠀⠀⠀⠀⡞⣿⠹⢟⣡⡄⠀⠀⠹⠏⠀⠀⢠⣌⡻⠏⣿⢧⠀⠀⠀⠀⠀⠀⠀⡼⣿⠹⢿⣡⡄
⠀⠀⣤⢶⣯⣭⣯⣷⣼⠀⠀⠀⠀⠀⠀⠀⡇⣾⣽⣭⣵⡖⣤⠀⠀⠀⠀⣤⢶⣯⣭⣯⣷⢸⠀⠀⠀⠀⠀⠀⠀⣧⣾⣭⣭⣵⡶⣤
⡀⠀⠀⢉⡽⢞⡑⣯⣸⠀⠀⠀⠀⠀⠀⠀⡇⣽⢊⣲⢿⡋⠀⠀⢀⡀⠀⠀⢉⡿⣖⡑⣯⢸⠀⠀⠀⠀⠀⠀⠀⣇⣽⢊⡳⢿⡋
⣵⣀⡄⠀⠒⢻⢶⡿⡃⣀⡤⠤⣤⠤⢤⣀⢐⢻⡶⡟⠂⠀⠠⡠⣮⣵⢀⡄⠀⠐⢻⢶⡟⡃⣀⡤⠤⡤⠤⢤⣀⢐⢻⡲⡟⠒⠀⢠⣀
⣧⡟⡴⣧⠄⠀⠀⠀⢞⡟⡛⢞⣿⣳⢛⣻⡳⠀⠀⠀⠠⣼⢶⠻⣹⣎⠟⣶⣧⠄⠀⠀⠀⢞⣟⡛⡞⣿⡳⢛⢻⡳⠀⠀⠀⠠⣼⢦⢻
⠷⠳⡶⡟⡃⣀⢀⠀⠉⠟⣱⢧⣻⣸⣞⠻⠉⠀⡀⣀⢘⢺⣶⠞⢾⡷⠳⣶⡟⡃⣀⢀⠀⠉⠟⣳⣏⣿⡽⣎⠻⠉⠀⡀⣀⢘⢺⣶⠞
⠉⠉⠁⢀⣶⡗⣻⠖⣀⠀⠁⠘⣽⠁⠉⠀⣀⠲⣟⢺⣶⡀⠈⠉⠉⠉⠉⠁⢀⣲⡗⣻⠖⣀⠀⠉⠘⣯⠋⠈⠀⣀⠲⣟⢻⣶⡀⠈⠉
⠀⠀⠀⢸⣨⣗⣖⡛⣧⢄⡀⠀⠀⠀⢀⣠⣼⣛⣲⣺⡅⡇⠀⠀⠀⠀⠀⠀⢸⢨⣗⣔⣻⣯⢄⡀⠀⠀⠀⢀⡠⣾⣟⣶⣺⡅⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣄⡀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠈⠒⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠙⡭⠹⢿⣧⡀⠀⠀⢀⠂⠀⠈⠁⠁⢂
⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢆⡺⢀⢛⣘⣿⡃⠀⠀⠁
⠀⠀⠀⠁⠄⠠⠈⠦⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣥⣽⣮⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠹⠛⠋⠁⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⣠⣤⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⣠⣿⣹⣾⣧⣿⣿⢧⡀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠀⠃⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠐⣛⣾⠾⣷⣿⣿⣿⣪⡅⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⠈⠀⠀⠀⠀⠆
⠀⠀⠩⣿⣿⣻⣿⣽⡿⡽⠀⠀⠀⠀⠁⠀⢀⣀⣠⠤⠤⢤⣀⣀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠉
⠀⠀⠀⠈⠉⠋⠛⠉⠓⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣄⡴⠉⠈⠁⠂⢤⣂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⣀
⠀⢠⠄⠀⠀⠀⠀⠀⠀⠲⠀⠀⢰⠔⠀⠀⠀⠀⠀⣐⣟⡀⠀⠀⠀⠐⢻⣶⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⡄⢀⠀⠀⣀⢴⣿⠂⠀⠀⠀⠙⠶⢴⣤⣤⣀⢀⣀⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡷⠂⠦⠙⢜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⢀
⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣤⣤⣾⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⡀⠀⠄⠀⠀⠀⠀⠀⠀⠨⢀⣴⣿⣿⣿⣿⠛⣿⣿⣦⠀⢨⡶⣾⣷⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠠⠄⣡⣾⣿⣦⡀
⣿⣿⣿⣷⡀⠐⢀⣴⣶⣦⣤⣶⣿⣿⣿⠟⠻⢿⠀⠈⠻⣿⣿⣿⡇⢈⣿⣦⣀⠀⠢⠀⡀⠀⠀⠀⠀⢀⣀⣾⣿⣷⡄⠈⠀⠴⣿⣿⣿⡿⠃
⣿⣿⣿⣿⣿⣄⠈⠉⠻⣿⣿⡿⢿⣿⣿⣄⠀⣠⣴⣿⣶⣝⠻⣿⣿⣿⣿⢯⠙⢿⣷⣌⠀⠀⢄⣠⣤⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣆⠀⠀⡙⢿⣿⣦⣉⣻⢿⣾⣿⣿⣿⠟⣿⢿⣷⣈⣻⣿⣾⣤⣾⣿⣿⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀
⠈⠻⣿⣿⣿⣿⣿⣷⣾⣿⣆⠛⠛⢿⣿⣦⣝⣿⣋⣤⣾⣿⣷⣬⡋⠉⠉⢻⢿⡿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣠⣦⣤⣾⣿
⣿⣶⣍⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣶⣤⣉⡋⠛⠻⣿⣿⣿⣿⣿⠟⠁⠀⢠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣦⣩⡿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡙⣿⠟⢁⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣮⣿⢀⡟⡯⢸⠻⢸⠃⡏⠿⠻⠁⠀⠛⠙⠻⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣡⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄⡒⠁⠄⠂⢁⠀⠠⠀⠀⠀⠀⠁⠸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⣤⣴⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿