Labrador Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠓⣶⣤⣀
⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣽⣿⣦
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣹⣿⣿⣿⣿⣿⡯⢘⣿⣿⣿⣝⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣤⡉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡤
⠀⠀⠀⠀⠀⠘⡏⢨⠝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣉⡁⣤⠀⠀⠉⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣤⡙⣿⠁⣸⠟⢅⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠃⠄⠘⠄⠉⢁⣿⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢸⣿⣧⠀⠰⢲⢲⠲⠀⠀⣾⣳⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢹⣿⠀⡄⢀⣇⢠⠀⢰⣿⠁⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣿⣿⣤⣡⠞⡻⣄⣀⢸⡟⣷⣯⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⣟⠂⢷⣾⣿⠦⠩⢛⣵⣿⡿⠈⣧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⡏⠀⠀⠀⣾⣿⣿⠇⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣀⣼⣷⡄⢠⣼⣿⣿⠏⠀⠀⠀⢠⡟⠀⠀⠀⣶⢀⡄⡀⠀⠀⠈⢿⣌⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣾⡇⢀⡇⠀⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢸⣿⣧⣼⡇⢀⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢠⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠉⠁⠀⠀⢶⡀⠀⠙⠋⠁⠀⠀⠀⠀⣠⡤⠀⠀⣿⣿⣿⣿⣧⡾⣿⣿⡿⠋⠻⣿⣄⣄⠀⢀⠈⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⢀⡶⠋⠉⠈⠻⣿⣷⣶⣶⣿⣿⣿⡿⠟⠁⠀⠀⢿⠿⠛⠋⠁⢀⠀⠀⣲⣦⠀⣨⠏⣻⡇⠀⠁⠈⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⡠⠞⠋⢁⣠⡴⠋⠀⠀⠀⠀⠀⠀⠉⢉⡽⠋⠉⠀⠀⣀⡴⠆⠀⠀⠀⠀⠀⠀⠆⠠⢼⣿⠟⠉⢠⠞⠋⠀⠀⠀⠀⠀⠙⠿⣿⣿
⠀⠀⠀⣀⠤⠞⠀⢠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢀⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠸⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠜⠉
⣏⣰⢤⣧⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠶⠄⠛

⠀⠀⠀⠀⠀⠀⡴⠚⠛⠒⠛⠛⡋⠉⠛⠛⠓⠒⠓⠲⢤⣀⣤⣄
⠀⠀⠀⣠⠖⢩⡾⠁⠀⠀⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⢀⡴⠊⠁⢀⡞⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠙⠢⡀
⢸⡁⠀⢀⡞⠀⠰⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⢈⣢⡀
⠈⢿⣀⠀⠐⠺⠀⠀⠀⠀⠀⠀⠐⣻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠈⢿⠃⡀⡀⠀⠀⠀⠀⠀⠤⠸⠟⠛⠛⠃⠀⠀⠀⠀⣤⣄⠀⢰⣤⣠⣿⣿⡿⠋
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⣸⡆⠀⠀⠶⠦⠄⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⡿⠋
⠀⢀⡄⣀⣀⡀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡿⠋⢣
⠀⠀⣐⣁⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢺⣿⣄⠀⣰⠇⠀⠀⣷⡀
⠀⠀⢹⣟⣿⣿⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⣽⡿⣿⣏⠀⠀⠀⠀⣷⡀
⠀⠀⠈⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⡠⠬⠷⢿⠟⠉⠀⣽⡟⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⣻⣟⠓⠿⣿⡄⠈⠀⠀⠀⠀⠰⠁⠀⠀⠀⣿⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⣻⣦⡀
⠀⠀⠀⠉⠙⠲⣷⢾⣿⣶⣤⣤⣤⡾⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⢀⣴⡿⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠉⢽⣏⠀⠀⠀⠀⠀⠀⢀⣀⣰⠏⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⢦⣀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠙⣿⣧⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠸⣿⢀⣶⡇⡀⠀⠀⢠⣿⡖⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠧⠤⠿⠿⠿⠦⠴⠾⠿⠥⠀⠤⠶⠿⠿⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⢀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠉⠀⠀⠙⠁⣀⣄⣀⠐⢆⠹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⡿⡿⢃⣤⣶⣶⣤⣰⣟⣻⣿⣭⣦⣀⡀⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⣼⣥⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿⣝⣶⣸⣿⣿⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢻⣿⣾⣛⠧⣿⢛⣿⣿⣿⣄⠀⠾⠿⣿⡓⣓⣻⣿⣹⣿⡿⠉⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠉⣿⣿⣿⣯⣥⠉⣻⣿⣿⡏⠀⢠⡀⣤⣿⣿⢉⣴⣿⣿⣿⡇⡄⢸⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⠄⠇⢠⣻⣿⣿⣿⣿⢃⡿⢫⣾⠏⠴⠛⣿⣬⡏⢋⣿⣿⣿⣿⣿⠇⡆⠈⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⡀⠈⡇⣿⣿⣿⠏⣿⣷⡟⢡⠴⠀⠀⠈⣿⡇⣼⣿⣿⣿⣿⣿⣷⢁⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠓⢹⣿⣿⣾⣿⣿⣴⣿⣶⣿⠿⢷⣿⣧⣹⣿⣿⣿⣿⣿⣿⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⣸⣿⣿⣿⣿⢿⣽⣷⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣟⣫⡻⣻⣿⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣷⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠽⢿⣿⣿⣿⣿⣿⣿⣿⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣆⠉⠻⢿⡿⠭⠿⢿⡟⣼⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠔⠋⢠⣿⣿⣿⠛⠛⠻⠿⣿⣿⣿⣯⡏⠛⢻⣿⣿⣿⣧⣐⢠⣽⣤⣤⡤⠀⡸⠟⣿
⠀⠀⠀⠀⠀⠀⠘⠗⠲⠖⠿⢥⣤⡟⠁⠀⠀⠀⠉⠻⢿⣿⠓⠀⠀⠀⠙⢿⣿⣋⣼⣿⡿⠿⠏⢠⠁⣼⠃⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠆⠀⠀⠀⠀⠀⠀⠀⠉⠣⠀⠀⠀⠀⠀⠀⠉⠹⠧⠀⠀⠀⠀⠀⠋⠠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⠔⠒⠈⠉⠉⠉⠉⠉⠉⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⣤⡀⠀⠀⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠄⠀⣀⣤⣀⠀⠀⠀⠀⠀⣀⡉⠀⠀⠐⠆⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢰⠂⠀⠓⠟⠊⠀⠀⠀⠀⠁⢝⡅⠀⠀⠀⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⣸⠀⠀⠀⠀⡎⠉⠁⠲⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠃⣿⠀⣀⠀⠀⠱⣄⣠⠖⠁⠀⠀⠀⣠⠀⠀⣶⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠛⣆⠘⣆⠄⣀⡤⠦⣤⣤⣤⠴⣺⠃⢀⡴⡟⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢧⡈⢦⡇⠀⠀⠀⢸⡤⠞⢡⠔⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠙⣼⠀⠀⠀⠀⠘⣅⡴⠁⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡄⠀⠀⠀⡇⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢧⠀⠀⠀⢧⡀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡇⠀⠘⢇⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢰⠃⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⡾⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡞⡇⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢰⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠁⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣰⠃⠸⡄⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣵⠀⠹⠀⠀⠀⠀⢻⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢠⠞⠁⠀⠀⠹⣄⠀⠀⠀⠀⡇
⢀⡴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡇⠀⠀⠀⣰⢿⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢸
⡜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢹⠀⠀⣰⠃⢸⡄⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⡆
⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠘⡆⠘⠁⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢹
⢷⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢳⣀⣠⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠈⣇
⠈⢧⡀⠈⠳⢤⣄⣀⣀⣠⡤⢿⡀⠀⠀⠀⡀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⣯⠤⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠸⣄
⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⢻⡔⠀⠀⠀⠉⠛⢇⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣷⠀⡀⠠⠨⣆⠀⠈⠐⣌⡆⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠛⠉⠉⢧⣀⣸⣀⣀⣆⣨⡇⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣿⠀⠀⢠⠀⠠⡝⢿⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠰⠄⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡳⣶⡾⣤⣠⣷⠞⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣒⣛⣓⣒
⣿⣿⡟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠖⠋⠁⠀⠀⠉⠙⣲⡤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠚⢠⠞⢁⣀⡴⠒⠀⠁⠀⠀⠀⠀⠈⠙⠢⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣤⣸⡇⢿⣰⡿⠋⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡹⠞⠋⠡⠞⠋⠀⠀⠀⣾⠿⣿⣌⠁⠀⠀⠀⠀⠀⠀⢸⡆⠈⠻⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠚⠂⠀⠀⣀⡠⠀⠀⠀⠠⢾⣥⡶⣄⠈⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡀⠀⠙⣷⣄
⠀⠀⠀⢀⣾⣿⡷⢌⣑⢲⣦⣤⠶⠋⠁⠀⠀⠀⠀⢀⠺⡻⢟⣛⢳⣄⠀⠀⠀⢀⡄⢠⠏⠀⠀⡇⠀⠀⢸⣿⣧
⠀⠀⠀⢰⠶⣿⢠⣿⠻⢲⡟⠛⢆⠀⠀⠀⠀⠀⠀⢸⡆⠈⠉⠒⠀⠘⣷⡿⠲⠋⢠⡟⠀⠀⣸⠀⠀⠀⢿⣿⣿⣧⡀
⠀⠀⠀⢸⣿⣷⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣷⣤⡤⠀⠀⠀⠀⢠⡏⠀⠀⠀⡇⠀⠀⠀⢾⡿⠛⠻⣷⡄
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⢀⡟⠀⠀⠀⣸⡇⠀⣀⠀⠀⠀⠀⣰⡟⣿
⠀⠀⠀⠀⠹⣿⣜⢿⣿⣿⣍⡓⠦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⣼⠁⠀⠀⢰⣿⣷⣾⣿⣷⣶⣶⣿⠟⢀⣿
⠀⠀⠀⠀⠀⠈⠙⠓⠛⢿⣦⡉⠐⠂⠦⠀⠀⠀⠀⠀⢀⠀⠀⠲⣤⣄⣤⣾⣿⠀⠀⠀⣸⣿⣿⣿⠿⠟⠛⣉⣡⡴⠟⠁
⠀⠀⠀⠀⢠⠞⢀⣴⠟⠛⠻⠿⠶⢤⣀⣀⣀⣀⣀⣀⣼⣷⣦⡀⠀⠙⢿⣿⣿⣆⠀⢸⣿⠿⠋⣁⣤⡶⠛⠉⠁
⠀⠀⠀⠀⢸⣤⣨⣀⣀⣠⠴⠶⢶⣶⣿⣿⣿⣿⣿⣿⡿⣿⣿⠇⠀⠀⣼⠋⠉⠻⢷⣶⡶⠶⢺⣿⣿⡇
⠀⠀⠀⠀⠰⣭⣛⠛⠛⠻⢶⣞⠛⠛⠿⢿⣶⣦⣭⣥⠾⠻⠏⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠈⠓⠶⣤⣤⣄⣀⣀⣀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠉⢻⣷⣤⣈⣉⡉⣩⡿⠀⠀⢠⣶⣤⣴⡶⠾⠿⠛⢻⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠀⠀⠀⣿⣇⠀⠉⠹⣇⠲⢶⣶⣿⠟⠋⠁⠀⠀⠀⠀⢈⣿⣿⣿⣦⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⡆⠀⠀⡿⣿⠀⠀⠀⠙⢷⣄⠙⢿⣿⣶⡄⠀⠀⠀⠀⠀⠛⣿⣿⣿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣼⠁⠀⠀⠀⢸⠀⠀⠀⠀⠈⣿⣿⢆⠹⣿⣿⡄⠀⠀⠀⠀⠀⣾⣿⣿⠀⠘⢷⡀
⠀⠀⠀⠀⠀⢀⣾⡟⠀⢠⡟⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣿⡿⠀⠀⢿⣿⢧⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠈⢿⣄
⠀⠀⠀⠀⢠⣿⡟⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠘⢿⣆⠁⠀⠀⠀⠀⠘⣿⣿⣿⣦⡀⠀⠹⡆
⠀⠀⠀⠀⣿⡿⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠙⠷⡄⠀⠀⠀⠀⠈⢿⡇⠈⠙⠓⢀⡏
⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠉⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⠎
⠀⠀⠀⠀⠈⠻⢷⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡀⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣂⣀⠂⢉⠀⠀⠈⠉⠛⢿⡖⠁⠀⠀⠀⠉⠣⠁⠐⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡾⢁⠈⡩⠉⠀⠀⠀⠀⠀⠀⢀⠀⠙⠀⡶⠀⡀⠀⠀⠀⠀⡄⡌⢧⡀
⠀⠀⠀⠀⠀⠀⣠⡞⠁⢈⣡⡏⢀⠁⠀⠀⠀⡀⠀⢀⣠⢀⡇⠀⠀⠐⢠⢀⣤⠀⡀⠀⢸⣾⠸⠙⠳⣄
⠀⠀⠀⠀⠀⣼⢹⣿⠀⣿⣿⡇⢸⠀⠀⢀⡾⣿⣶⣿⣧⢿⠀⠀⢀⠀⠀⠻⣿⣽⠦⠄⠈⣿⡀⠀⠀⠻⢷
⠀⠀⠀⠀⠀⠸⣆⣿⠀⠘⣿⡇⢸⡆⠈⠀⠀⠻⠏⣹⡿⠀⠀⠀⢸⡆⠀⠐⠙⢂⣀⠄⠀⣿⠁⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⢹⣷⢸⡗⠀⠘⢦⣄⠀⢾⠇⢀⣠⠴⠚⠋⢀⣀⠰⣟⠃⠀⠀⣿⡆⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠸⣿⢸⡇⠀⠀⠀⠟⠀⠘⢠⠏⠀⢠⢿⡛⠀⢹⡷⠘⡇⠀⠀⣿⣇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⢏⣾⣇⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢻⣡⣾⣮⡟⠀⠀⣀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡟⣿⣿⠀⠞⠰⣷⡀⠀⠀⠀⠀⡂⣾⣿⠏⣷⡤⠄⢰⡞⠀⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠸⡿⣆⠀⠀⠻⣷⣄⠀⠤⠄⠶⢾⡟⣾⡿⣧⣡⡼⡄⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⢃⠙⢦⡀⠀⠸⣿⣷⣶⡶⠊⡍⠙⠉⠀⢹⣩⠜⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠈⡆⠀⠈⠳⣄⢀⡁⠙⢷⣷⣍⣒⣀⣀⣠⡏⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠘⡀⠀⠀⢀⠀⠉⠓⠤⣤⣬⡭⠀⠐⠉⠀⠀⠀⢹⡟⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⣼⣿⡄⠀⠀⠀⠀⠀⡀⢸⠇⢸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢸⣿⠀⠀⠀⠀⠀⠘⣴⢀⠀⠀⠈⠀⠻⣿⠂⠀⠀⠀⠀⠀⠈⠀⠘⢀⣳
⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⢻⣤⡆⠀⠀⠀⠀⣿⣿⢀⡄⠀⠀⠀⠈⠂⠀⠀⠀⠀⢴⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣼⠀⠀⠀⠙⠻⡄⠀⠀⠀⠀⠘⢿⢷⣧⣠⡞⠷⣤⡤⠤⡀⠀⢪⡇⠀⠀⣸⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣾⢻⣿⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠈⠁⠀⠀⠀⠈⠆⠀⠐⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢦⡀⣀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠙⠷⣾⣦⣐⣦⡀⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠛⠛⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⣠⣴⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣷⣄⡀
⠀⢀⣠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣾⡟⢻⣿⣦⡀
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠟⣯⠟⢙⠹⣹⣿⣿⣿⣟⣻⣿⢿⣿⣿⣷⣯⡝⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⢎⣨⣴⣶⣦⡀⢾⣯⣿⣯⣽⣿⢿⣿⣿⣷⣿⣿⣿⣄⡀
⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣶⣿⣿⣿⣷⣦⣄⡀
⣿⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡿⠉⠻⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠘⣿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣧⣤⡤⠘⢛⠚⠛⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⣤⠊⠾⡟⢻⣿⢿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⣯⣴⣤⣶⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣷⣿⡷
⣿⣿⣦⣕⣿⣴⡈⠀⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣧⣌⠜⣿⣿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⡿⢿⣿⣿⡿⠛⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠁
⣿⣿⣿⣿⣿⣿⡿⣭⣿⣿⣿⣶⣿⣿⣷⣾⣿⣤⣈⢟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠉⠁
⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣟⢻⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣦⡹⣦⡼⠛⣹⠉⣿⢿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣮⣝⡳⠆⠹⣿⣌⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⢶⣶⠶⠒⠂⠈⠉⠉⠉⠉⠛⠲⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢋⣠⠆⠀⠀⠀⠀⠀⠑⠀⢾⣆⡤⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢲⡀⠐⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠔⠁⢠⣾⡏⠀⠀⠠⠶⠾⣷⣀⡀⠀⣿⠀⠀⠀⣴⢃⣞⣁⣀⠀⢠⠀⠀⠀⢀⠙⠆⢿⠐⠀⠀⠲⣄
⠀⠀⠀⠀⠀⣠⠊⠀⠀⣼⣿⠀⠀⢁⡤⣲⡦⠘⣿⡇⠀⠀⠀⠀⠘⡁⠋⢿⡳⢟⡷⠀⠁⡀⠀⢸⡆⠀⠈⠃⠀⠀⠀⠀⠑⢄
⠀⠀⠀⣠⠚⠁⠀⢀⣾⣿⡟⠀⢀⠀⠷⠭⢒⡴⠋⠀⠀⠀⠀⠀⠈⠙⠒⢄⠈⠉⠀⠀⡄⠉⠀⠈⣧⠸⣄⢀⠐⢄⠸⣄⠀⠀⡕
⠀⢀⢴⠋⠀⠀⠀⠸⠋⢸⣿⠀⠘⢷⣤⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡁⠙⢶⡖⠀⠀⢠⡀⣿⠀⢿⢸⡀⢸⣦⣼⣦⣰⣿
⠀⡱⠁⠀⠀⠀⠀⠀⠀⢸⡻⡄⠀⠈⡟⠀⡀⠠⠀⣠⡄⣠⣬⣷⡄⠀⢀⡀⠛⢶⡀⠁⠀⠀⣼⡇⣿⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣧⢱⠀⠀⠃⠀⡇⠀⣸⣿⣧⣿⣿⣿⣃⠀⠘⡟⠀⠈⢃⣆⠀⠀⣿⡷⢸⠀⢸⣿⢻⣿⣿⣿⣿⠿⠋
⠈⠢⣤⣠⣤⠤⠀⢀⣤⣿⣿⠈⠀⡀⠀⢰⠇⠀⢙⣙⢭⡼⢟⣿⣿⠀⠀⠂⠀⠚⠋⡟⠀⢀⣸⠃⢸⡄⢸⣿⣸⣿⡿⠋⠁
⠀⠀⠈⠣⡀⠀⠀⠘⠛⠿⠋⠀⣴⠇⠀⢸⡀⠀⠹⣿⣷⣶⣿⠿⠇⠀⠀⠀⠀⠀⠀⣀⡀⠸⠉⠀⣸⣧⠈⣿⣿⠋
⠀⠀⠀⠀⠘⢄⢄⡀⢀⠄⡠⠊⢹⠀⡄⠈⠁⠀⠀⠈⣡⣶⣦⣤⣀⠀⠀⠀⠀⣠⣾⣿⠃⠀⡄⢀⣿⣿⣦⠘⢧
⠀⠀⠀⠀⠀⠈⠒⢍⣁⠔⠁⠀⠈⡄⣷⠀⣄⠀⢀⠼⠿⣿⣿⡿⠿⠿⠖⣤⣾⣿⣿⢾⠀⢠⠀⣸⣿⠁⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠈⣯⣿⡄⠀⠀⠀⠀⠀⠄⢀⣿⠛⡷⢿⠏⠀⠊⠀⠛⠀⢀⣿⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⢀⣇⡴⠟⠲⡴⠦⠧⣀⣒⡍⣥⠰⠗⠁⣀⠀⠀⠀⠀⠀⣾⠃⠀⣯⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡞⣦⠀⠙⢦⡉⠛⠿⠿⠟⠛⠍⠊⠀⣠⡶⠚⠋⠀⠀⠀⠀⢸⡇⠀⠠⣿⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⢹⠘⢷⠘⢦⣽⣷⣦⣀⡀⢀⣠⣤⣾⣿⣄⣠⣴⡾⠟⠁⢀⠘⡇⠀⠀⣿⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠘⡆⠀⠳⡀⠙⠟⠛⠻⢿⣿⠟⠻⣿⠋⠛⠛⠃⠀⠀⠀⣼⠀⠀⠀⠀⣷⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠀⠀⠀⠀⢻⡀⠀⠹⣆⠀⠀⠀⠀⣿⠀⠀⢿⠃⠀⢰⣷⡀⠀⢀⠋⠀⠀⠀⠀⠡⠀⠀⠀⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣾⠀⠀⠀⠀⢸⣇⠀⠀⠘⢇⠀⠀⠀⠋⠀⢠⣧⠀⠀⠀⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⣀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣇⢰⠏⠀⠀⠀⠀⠀⣿⣆⠀⠀⠘⠀⠀⢸⡆⠀⠀⠻⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣆⠀⠀⢉⣄⡀⠈⢿⣧⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢷⣶⣿⣿⠿⢶⣤⣽⣧⣴⣦⣄⣺⣧⣤⣄⣴⣦⣀⣀⣠⣤⣤⣴⣤⣶⣶⣶⠶⠞⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣉⣙⣿⣏⣀⣉⣻⣋⣉⣋⣀⣉⣉⣉⣉⣉⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣯⣿⣽⣽⣯⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣯⣯⣯⣽⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣆⠉⠰⣄⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡬⠀⠀⠀⠀⢀⡟⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡸⣿⡇⠀⠀⠀⠸⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠚⠀⠀⠀⣀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠀⠀⠀⠀⢤⣿⣿⡀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣤⡞⣣⣤⢬⣹⣦⣷⣴⣄⣤⣿⣆⣴⣦⣶⣴⣞⣁⣉⠲⡆⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣾⣿⣾⣯⡛⣳⣿⣙⠿⠟⠛⠙⠛⠛⠛⢿⣿⢻⣿⠾⠟⣿⣦⣦⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢸⣿⢿⠛⠦⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢉⠑⠚⢁⣽⣿⣿⡟⢠⣧⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢧⢸⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣘⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⣿⣷⢸⣿⠇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠴⠤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢹⣿⡇⠘⣿⡀⠀⠀⠀⠄⣾⣶⠃
⠀⠀⠀⠀⠙⣦⡀⠀⠀⡀⣶⣿⢏⣼⣿⠀⠀⢀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢀⡀⠀⠘⣷⡆⠀⠀⠀⠀⠀⢸⣿⣷⡘⣿⣿⣿⢸⣴⣾⡾⠁
⠀⠀⠀⠀⠠⣴⠟⠷⣦⣅⣠⣤⣾⣿⣿⡷⣠⠏⠀⠀⠀⠀⠀⢆⣶⣿⣿⣿⣿⣶⡴⠀⠀⠀⠀⠀⠸⣄⣾⣿⣿⣿⣦⣈⣈⣠⡴⠯⠤
⠀⠀⠀⣠⣴⡇⠀⢰⣦⣿⣿⣟⠙⣿⣿⡿⠉⠀⠂⠀⢄⡤⠀⠀⢭⣵⣼⣿⣮⠌⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⢻⣿⣿⣧⡅⠀⠑⢀⣀⢀
⠀⠀⡴⠟⢻⣷⠗⠛⢿⣿⣿⢿⣷⣿⣿⠅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢻⡿⠃⠀⠐⠠⠄⣀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⠟⠒⣦⡾⠿⢧⡀
⠀⠀⠘⢶⣾⣿⣶⣶⣤⣿⣯⢴⣿⣿⣿⣧⡀⠀⠀⠀⠀⢐⡓⣃⣤⣴⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣦⣘⣿⣥⣴⣦⣿⣷⣶⠞
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢘⠈⠓⣂⠀⠈⣶⢚⡄⠀⢸⣿⡆⣿⡗⠈⠐⠪⢹⣿⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠈⢧⡸⢿⣦⠀⣿⡈⡇⠀⡞⢿⣿⣼⣾⣷⢄⠀⠈⢻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠀⠈⡇⠀⠻⣼⠁⢿⡖⣰⠃⠊⢻⣿⣶⣿⣶⣝⡀⢿⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠⣦⡈⠁⠀⠀⠀⣙⣧⡼⠿⠛⠋⠉⣉⣭⠜⣩⡽⢻⢿⣿⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠸⢾⣷⠥⠶⠶⠾⠯⠭⠤⠤⠤⠤⠴⠚⠛⠓⠛⠛⡐⠉⢾⡻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣤⣴⣤⣤⣥⣤⣤⣤⣤⣥⣄⣄⣠⣀⣁⣀⣀⡀⣸⡇⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠐⡇⠀⠀⠁⠈⠀⠀⠀⠈⠀⠏⠇⠀⠀⠀⠀⠀⠈⠀⢹⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢠⡄⠿⠜⢈⣧⡞⢷⠀⡦⠀⢠⡀⠀⠀⠐⠈⠀⠀⠀⢠⡇⣀⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠐⠛⠀⠀⣨⣭⣽⣟⠀⣿⣉⣀⣿⣤⣤⣴⡀⠀⠀⢤⡜⣷⣙⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⣧⣀⢹⡿⠁⠀⢸⠀⣿⠀⠀⢹⣴⠒⠛⢣⢶⠶⠒⠧⢌⠉⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢨⠄⡼⠈⣷⣒⣒⣾⡆⣷⠴⠄⡾⣣⣀⣀⣸⣿⣤⣤⣾⡟⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡀⠂⠀⡇⣥⡌⠸⠗⠹⠀⠀⠁⢸⠇⠀⢹⠈⡏⠉⡏⠏⠹⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠁⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣺⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠴⢀⠀⢸⣿⡇⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠀⠀⠀⣠⣀⣀⠔⠀⠘⠛⠀⠀⠁⠀⠀⠀⣹⣿⡄⠘⣿⣿⡆⢀
⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠞⠉⢹⣶⢖⣴⠴⠀⠀⠀⠀⠀⠀⣾⠈⠀⠀⠘⣿⣿⣎⠳⣦⡀⠀⠀⠀⠘⣧⣀
⢴⣿⣄⣰⡀⢀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠚⢛⣩⠴⠊⡀⢀⡄⣸⣧⡀⠈⠳⢀⡄⠀⠀⢹⡿⠷⠦⢙⠻⢷⣶⣄⣉⠛⠳⠦⡄⠀⠀⣦
⣿⣿⣿⣿⣧⣿⣼⡆⣸⣿⣿⣼⣀⠀⣠⣀⡄⠉⠁⣼⣦⣿⣸⣿⣿⣿⣷⣿⣄⣸⣿⢠⣤⣼⣧⣧⢠⣸⣇⣸⣧⣏⣿⣹⡄⠀⣰⣼⣾⣿⣷
⣿⠿⠛⠙⠛⠛⣛⠻⠿⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⣙⡛⠉⠉⣉⣙⠿⠿⠿⠿⢿⣿⣿⣿⣷⣿⣿⠿⢿⣿⣿
⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠀⠈⠉⠉⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣧⠀⠀⠀⠀⠀⠀⠇⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣼⢸⣿⣿⣧⡀⠀⠈⠄⠀⠀⠀⠀⣹⣿⢀⣤⣶⣾⣿⣿⣷⣶⣤⣀⢸⣿
⠀⠀⠀⠀⠘⠀⠀⠀⡌⠀⠀⠀⠀⠀⣠⣾⣧⢠⣶⣶⣶⣷⣄⠀⠐⡀⠀⠀⠀⢿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢀⣴⣿⣿⣿⢸⣿⣿⣿⣿⣿⢰⣄⡀⠂⢠⣾⡛⠙⠻⣿⣿⣶⡄⠈⠙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⡔⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⠏⠠⠠⣿⠋⣡⡀⠀⠙⢿⣿⡄⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡀⢼⣩⡅⣩⣭⣭⣭⣭⢈⣭⣭⣭⣭⣭⡈⠀⢀⠀⠀⠀⠄⠀⠀⢆⢠⠋⠀⠀⠀⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠸⢀⣠⢠⠤⣬⣿⣇⡿⠿⠟⠛⠛⠚⢛⡟⠛⢿⡿⣁⣀⠄⠀⠐⠀⢀⠀⢀⠀⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⠀⠀⠀⠐⢀⠈⢰⡏⠁⠐⡄⠀⠀⢀⠔⡁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡡⠄⠀⠀⠀⠙⠄⠀⠀⢉⡀⠈⡷⢖⣶⣞⣻⣄⣁⠔⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠘⠉⠭⣽⠖⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⠀⠀⠀⠀⠀⢀⠓⢒⡄⠀⠒⠦⡤⠤⠤⠴⠒⠋⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠒⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠐⢃⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣀⠀⠤⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠴⠀⠀⠀⠀⠀⠀⣿⣿⢿⠇

⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠟⠁⠀⠐⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡄
⠀⣰⣷⠀⠄⠠⠀⠀⠻⠶⠖⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡘⠅⢴⣶⣄
⠀⠈⠋⠀⠀⠀⠀⠃⠀⠀⠀⠈⠁⠀⠠⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠀⢼⣿⣻⣋⠉⣀⣀⣰⣀
⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠢⠀⠑⠦⠀⠈⠑⠦⠀⠀⠠⠶⠶⠚⠛⠛⠩⠽⠿⠟⠋⠽⠿⠿⠿⠿⠿⠿

⠀⠀⠠⡤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄
⠀⢠⡌⠫⠤⣤⣤⠸⢧⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣛⢛⣉⡟⣫⣄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⢙⣏⣀⣀⣀⣀⣬⣥⣈⣏⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⣛⣓⣲⣒⣶⣿⣿⣿⣾⣏⣉⣩⣽⣿⣿⠿⡿⠉
⠀⢸⣿⠿⠿⠟⠟⠟⠟⠿⠿⠿⠻⠻⠻⠶⠶⢤⡀⠀⠀⢿⠿⠿⠟⠟⠝⠝⠟⠯⠮⠮⠪⠫⠫⠫⠻⠳⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣶⡄⡀⠀⠀⠤⠈⠁⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣉⡀⠀⠒⢀⡀
⠘⢿⣧⣤⣤⣤⣤⣤⣴⡿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⠿⡿⣿⣦⣤⣤⣿⣶⡆
⠰⠿⠿⠿⢿⣿⡟⠛⠛⠓⣼⣿⡄⠀⣀⣀⣀⣤⣤⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠙⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢀⢠
⠀⣠⣾⣿⣦⣤⣴⣾⣏⣷⡄⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⣤⣦⣦⡄⠘⢺
⠀⠻⢿⣿⢟⣻⣻⣏⣿⣛⡇⣀⣤⣄⢰⣿⣿⡆⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⡶⢤⣤⣋⣼⣿⣿⣿⣿⣉⣿
⢠⣶⣴⣿⣾⣿⣿⣿⣿⠿⣻⣿⣿⣿⣄⣠⣄⣀⠀⠀⠀⠦⣀⣩⣷⢶⢶⢶⢶⢾⢿⢿⢿⢿⢿⣿⣿⣿⣿
⠈⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠋⠁⠁⠁⠁⠁⠁⠁⠁⠁⠁⠁⠁⠁⠁⠉⠙⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠉⠉⠉⠀⠀⠀⠀⠁⠄⠠⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠄⠀⠀⠀⠀⠐⠆⠀⠀⢣⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⡆⠀⠀⠀⠐⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠄⠀⠀⠀⢀⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠄⠈
⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠂⠀⠀⠐⠖⠀⠀⣀⡀⢀⠠⠐⠋⠀⠀⠀⠀⠀⠠⠀⠀⡔⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⡠⠤⣀⠂
⠀⠀⡐⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢐⡰⠒⠀⠁⠀⠄⠀⠀⠀⠢⢄⡈⠀⢀⣀⡠⠊⠀⠀⠀⢀⠈⠄⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠐⠒⡒⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⡠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠈⠉⠑⠒⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡤⠊⠀⠠⠄⠐⠂⢐⠆⢀⠠⠗⠀⠐⠒⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠄⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠙⠓⠂⠀⠀⠀⢀⡄⠀⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠉⠩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⠀⠤⠤⠤⠌⠀⠀⠤⠄⠀⠤⠤⠀⠂⠂⠩
⠀⠀⠸⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⠲⠶⠒⠋⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⢄⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢀⠀⠐⡀⢈⠀⢂⠔⢂⠔⠡
⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⢈⣤⠄⠂⠓⠶⠶⣤⡀⠀⠀⠀⠠⠊⠀⠔⠁⠀⢀⠀⠀⠀⠊⠀⠀⠐⢀⠊⣤⣁⠀⠘⠁⠀⠀⠀⢠⠋⠀⣮
⠀⠀⠀⠀⠸⣧⠀⣀⠠⠰⠴⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⡴⠤⣀⠀⠀⠀⠀⠀⠀⠤⣀⡠⠔⠒⠉⠀⠀⠉⠳⠶⠶⣠⠴⠴⠃⠠⠎
⠀⠀⠀⠀⠆⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⢤⡁⠓⠲⠦⠤⠄⢀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣀⣀⣁⢰⡄
⠀⠀⠀⢰⣦⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣠⠄⢤⠀⣤⣤⣈⣄⣀⠀⠀⠀⠀⠤⠐⠒⠈⠉⠀⠈⠐⣠⠃⢀⠞⠀⠀⢸⣾⠰⠋⠁
⠀⠀⠀⠀⣿⡄⢻⡇⠀⢀⠤⠄⠐⠒⠊⠉⠉⠉⠀⠀⡀⢀⡀⢀⡀⢠⠀⣠⡀⣀⢉⡉⣩⢱⣄⡀⢠⣄⣄⢀⡀⢠⡀⠀⢠⠂⠀⢀⠀⢻⠏
⠀⠀⠀⠀⢹⣷⠘⣿⣷⡀⠀⢠⡄⣠⠠⡄⢠⣤⣶⣄⣿⣼⣿⣸⣷⣾⣷⣟⣧⢽⣾⣿⣿⣿⣏⣿⣿⣿⣿⣼⣷⡾⠁⡰⠁⠀⢀⡏⢀⠎⠀⣰⠃
⠀⠀⠀⠀⠈⣿⡆⢿⣿⣧⠀⠸⣷⣿⣶⣷⣾⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⡄⠀⠀⠀⢈⢇⠎⠀⡀⢯
⠀⠀⠀⠀⢀⣿⡁⠘⣿⣿⡄⠈⠉⢉⠁⠁⠀⠀⠀⠀⠀⠃⠉⡙⠙⠻⣿⣛⡛⠛⠛⠛⣻⢛⡉⣉⠙⣉⡿⢋⠔⠉⠀⠀⠀⢀⠾⠊⠀⠀⠄⢸⡆
⠀⠀⠀⠀⠀⢹⣿⡸⣿⣿⣷⣄⠀⠀⢀⠀⣄⠀⣀⡠⠤⠤⠤⠝⠛⠛⠁⠛⠛⠋⠁⠀⠁⠈⢀⣀⣀⡜⡡⠁⠀⠀⠀⠀⡰⠋⠀⠀⠀⢀⡔⠙⠻⡆
⠀⠀⠀⠀⠀⣺⠿⠇⢿⣿⣿⣿⡄⠀⢠⣤⣾⡇⠀⠀⠀⠀⣀⠠⠤⠀⣀⠠⠄⠒⠒⠊⠛⠉⠋⢉⠏⠀⠅⠀⡀⠀⢀⠞⠁⠀⢀⡤⢤⡎⠀⠀⡴⠇
⠀⠀⠀⠀⠰⡇⠁⠀⡀⢻⣿⣿⡇⠀⠀⠞⢛⣣⣤⣀⠀⠈⠁⠀⠀⠚⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠰⠷⠿⠞⠁⠀⠀⡔⢡⠀⠘⠳⡦⢀⣄
⠀⠀⠀⠀⠒⣶⠀⠎⠁⣈⣿⡿⠟⢠⠀⣾⣿⣿⣿⣿⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠒⠒⠒⠒⠒⠂⠰⢤⣀⣀⣺⠣⠾⠶⣿⣼⣿
⠀⠀⠀⠀⣦⣜⠂⠘⣩⣥⣤⣴⣶⡾⢶⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠂
⠀⠀⠀⠀⣿⣿⣿⣅⠀⢀⡀⢤⣿⣇⠀⣬⣿⣿⣿⡟⠹⣿⣄⠀⠀⠀⠉⠉⠛⠓⠲⠶⡦⣤⡄⢰⡄⢀⡀⡀⠀⠉⣉⠁⠀⠀⢀⡢
⠀⠀⠀⠀⣿⣿⣿⣿⣦⠈⢷⡘⢻⣿⡀⠔⣸⣿⣿⣷⡀⠀⠙⠿⣇⠀⠀⠀⠀⣤⠀⡀⠀⠀⠀⠸⠛⣿⣿⡿⣷⣷⣿⡠⢀⡀⠀⠉⠀⠐⡦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡈⠹⣷⢿⠟⢳⣿⣿⣿⣿⣧⠀⠀⠀⠙⢿⣆⠀⠀⠁⢸⡇⠀⠀⠀⠀⡀⠉⠉⣿⣿⣿⠛⣓⡶⠈⠱⠀⡀⠀⠂⠤⣤⡀
⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣷⠀⣩⠏⠤⢶⡾⠿⣿⢻⣿⡄⠀⠀⠀⠀⢿⣷⣮⣲⣿⡇⠀⠀⠐⠆⠘⣺⢻⣤⡤⠛⠉⠕⠀⠀⠰⣁
⠀⠀⠀⠀⣿⡌⣿⣿⣿⣿⣿⣧⡯⣤⡖⢋⠀⠀⢹⣆⣾⡗⠀⠀⠀⠀⠀⠛⣿⣿⣿⣇⡇⠀⠀⠀⠣⡌⢷⣼⣧⠀⠀⠐⠤⣀⡀⠀⠉⠉⠛⠒⠒
⠀⠀⠀⠀⡃⠁⢻⣿⣿⣿⣿⣿⡛⢿⣤⡘⣷⠀⠘⣿⣿⠃⠀⠀⠐⠂⠀⠀⢿⣿⣿⢿⣇⠀⠀⠀⠀⡠⠲⢠⣽⡘⣷⢦⣄⢀⡌⠁
⠀⠀⠀⠀⡃⠀⠸⣿⣿⣿⢿⣿⣿⣌⠋⢿⣆⠑⠄⢸⣿⣆⠀⠀⠀⠀⠀⢹⣿⣿⣿⣄⢻⡀⢀⠄⠀⠐⡖⠢⠹⣧⡹⣞⣿⣛⣿⣶⣿⣄⣀⣀⡀
⠀⠀⠀⠀⠐⢶⣤⣙⡿⠿⢸⣿⣿⡘⣆⠸⣿⣧⠀⠈⣿⣿⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣆⣷⡄⠀⠀⠂⠀⠀⠀⠋⠹⡿⢿⣿⠟⣿⣿⣿⣿⣟⣿⡄
⠀⠀⠀⠀⠀⣠⣽⣿⣷⠤⠾⣿⣿⣧⠹⡄⢹⣿⣧⣀⣸⣟⠸⣧⠀⠀⠀⠜⣿⡿⠟⠻⣿⠋⠁⠀⠀⠰⠴⣤⣀⠀⠀⢠⣀⢠⣤⡍⠿⢿⣿⣷⣿⠁
⠀⠀⠀⠀⠸⠟⠋⠉⠉⠑⠦⠈⠛⢿⣄⠀⠘⣿⣿⣿⠉⠛⢿⣾⣇⡀⠀⢿⣿⠇⠀⣠⢟⠀⠀⠀⠀⠀⢀⣧⢉⡄⠀⢸⣏⢹⣿⣀⣿⣽⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⣿⣧⡄⢻⣿⣿⡇⠀⠀⠘⢿⣿⣆⣸⡟⠀⡐⢣⢹⠀⠀⠒⠁⠀⠘⠻⢈⣇⣼⣺⠿⠈⠋⠙⢻⠧⣻⣿⣿⡇
⠀⠀⠀⠀⣄⣠⣴⣾⣿⣿⣷⡄⢻⣿⣿⣿⣷⡈⠛⢿⣟⣀⣀⠀⢀⠉⠻⠿⣷⣖⠀⢐⣾⡇⠀⠀⠀⢺⡤⣠⡄⠎⠉⢷⠀⣾⡄⠘⣿⡶⡯⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣷⡶⠋⠉⠙⣿⣿⣷⣄⠀⠀⢾⣿⣷⡀⢿⡇⠀⠀⠀⠀⠀⠁⠋⠀⢣⣤⢤⣽⠛⡶⡟⣳⣤⣤⡿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠢⠂⠝⢿⣿⣿⣷⣶⣿⣿⣿⣷⣸⣇⣀⡀⠀⠀⠢⠀⠄⠀⠶⠋⠘⠈⠐⡶⣯⡅⢡⣉⢱⠇
⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠂⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠟⠿⠿⠿⢿⣷⣤⠀⠀⠀⠓⠀⠁⠁⢤⠁⠼⠰⠌⠒⠻⢿⡆

⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢣⡙⢄⠀⠈⢻⡛⠛⠛⠛⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠈⡀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⡃⣇⠀⠀⠘⣧⠀⣀⠄⠀⠀⠀⠀⠀⠀⠁⠑⠀⠤⠤⣤⣤⣴⣶⡿⠉⠀⣹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢌⣿⡅⡇⣠⣔⠒⠛⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠀⣠⣾⡿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡆⠀⢩⣞⣅⡀⡀⢤⠠⠀⠀⡀⢀⣴⣶⡿⠟⠁⠀⣠⣾⣿⣿⢣
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⠹⣿⠯⠐⠚⠛⠛⠟⠛⡷⠿⢷⣾⣿⡿⠛⠁⠀⠀⣠⣾⣿⣿⣿⡟⣾
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣏⠀⠀⣀⡀⣔⠀⠀⠀⠀⠀⠁⢀⢀⣀⢀⣴⣶⣿⣿⣿⣿⣿⠃⠨
⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣟⣒⠀⢀⠀⠀⢠⣯⣷⣿⣾⣿⣿⣿⣿⣿⣿⢏⢄⡇
⠀⠀⠈⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢛⣉⣙⠛⠛⠛⣛⠋⠉⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⢿⡼⠁
⠀⠀⠀⠼⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣷⡶⠦⠾⠿⠶⠷⠳⠤⠴⠤⢤⣤⣴⠷⠋⢵⡟
⠀⠀⠀⠘⠻⠻⠿⣿⣿⡟⣿⣿⣿⣿⣷⣄⡀⠹⣶⣤⣴⣤⣤⣄⣄⣠⣤⣤⣤⣤⠤⢤⣀⡀⣼⠛⠓⠚⠓⠒⠛⠛⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⢿⣿⣿⣦⣌⠛⠛⣟⠿⠿⠛⠿⠻⠿⢿⣶⣶⣤⣬⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠤⢠⠄⢠⣄⣀⣀⠀⢀⡁⣁⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠉⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⣀⢠⣤⣴⣶⣶⣶⣦⣄⣀⣀⣀⠠⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡴⠊⣇⠑⢄
⠀⠀⢀⠤⠒⢉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢆⠀⠀⠀⢠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢣⢂⣀⣁
⠀⠎⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⡄⣀⢈⣄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⠀⠀⠀⠠⠤⠤⠀⠀⢀
⢰⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡆⠀⠩⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠖⡀⠀⠠
⢸⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠁⠀⢀⠔⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠕⢈⡴⢨
⠀⡄⠀⠀⢹⣿⣿⣿⣿⣿⡈⠁⠀⠀⠀⠀⠀⠀⠀⢨⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣠⣶⠟⠃⠂
⠀⠘⡄⠀⠀⠙⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢊⣴⣾⠟⠁
⠀⠀⠈⠢⡄⠀⠀⠹⣿⣿⣿⣆⠀⠀⠈⠀⠀⠀⠀⡠⠃⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⣾⡿⠊
⠀⠀⠀⠀⠸⠒⠠⠀⠀⠉⢩⠍⠋⠖⡀⣶⡒⠒⡘⠁⠐⠀⠶⣓⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁
⠀⠀⠀⠀⠀⠀⢰⠀⢠⠄⡄⠀⠀⠀⡜⠀⠀⠩⠄⠀⢀⠀⠀⠀⠀⠉⠉⢀⠒⠄⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢰⠀⢸⣴⡾⡄⠀⠀⠹⣶⣦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠆⢁⡠⡆⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠻⡀⠀⠔⠛⢿⣿⣿⣿⣿⣿⣿⣿⡖⠒⢶⣻⣿⠿⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣈⠩⠀⠉⠉⠛⠛⠿⠿⡇⠀⡟⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⢀⡴⠂⠁⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡀
⠀⠀⠀⠀⠠⠈⠁⢀⠀⠀⠀⠀⠀⠄⠠⡤⠄⢤⡤⢠⡄⠶⠀⠛⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠡⣤⠀⠈⡄⠀⠀⠀⡧⠔⠒⠗⠒⠒⠳⠼⠧⠤⢀⣥⣿⣦⠀⢴⠆⠀⠀⢠⣶⣄⠀⠀⠿⠀⠀⠀⠀⠐⠠⠄⡀
⠀⠀⠀⠀⠀⡩⠋⣑⣷⡀⠀⠀⠁⠤⡴⠃⠂⠂⠁⠀⠀⠀⠀⠀⠉⠫⡀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⢠⣿⣷⠀⣶⠀⠈⠐⢄
⠀⠀⠀⠀⠠⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢶⠄⠀⠀⢰⡦⠀⠀⠈⠛⠋⠀⠀⠀⣠⣤⠀⠐⢄
⠀⠀⠀⠀⠸⠟⠐⣶⣾⠁⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⣿⠃⠀⡀⠁
⠀⠀⠀⠀⣀⣀⡠⠛⠋⠀⠑⠄⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⢀⣠⡜⠀⣿⣿⠂⢀⣄⠀⢠⣾⣦⠀⠀⠛⠁⠀⠀⢀⣄⠀⠀⠘⠋
⠀⢀⣒⠁⣶⠀⠀⠀⠀⠀⠀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠀⠈⣥⢀⡈⠁⠀⢈⡉⢀⠈⠿⠏⠀⣀⠀⠀⢀⣤⠀⠉⠐⠀⣤⠈⡀⠀⡀⢀
⠀⢿⣿⡇⠀⠀⠄⠀⠀⠀⠛⡤⠇⣀⣀⣀⠀⣀⣀⣀⡀⢀⠀⠀⢀⠀⠰⡎⠉⠁⠉⠀⠀⡀⠝⠒⠄⠀⠛⠀⠀⢿⣿⠇⣴⠁⠀⢈⡤⠚⢈⣵⣦⠀⣨⡄⢀
⡆⠈⠛⠡⣦⠀⠐⡀⠲⠀⠀⠈⠐⡆⠠⠄⢀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠒⠲⣤⠦⠀⢠⣾⠀⠀⢀⢀⡠⠀⠀⠠⢤⣥⣴⠁⠀⢀⠋⢴⠆⠘⢿⠟⠀⠈
⠇⠿⠂⢰⣿⣧⢠⡀⠀⠁⠀⠀⠀⠈⢠⠀⢠⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⢠⠟⠁⠰⢲⠟⠉⠁⣿⡄⠀⠀⢠⣤⡀⠀⣠⡄⠀⣰⣶⡄
⠀⢄⠀⠈⠛⠃⠈⢁⡀⢀⠀⠀⠀⠀⣨⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⣬⠏⠀⠀⠀⣿⣧⢀⠀⢻⣿⠇⠈⠀⡀⠀⠹⠿⠃
⠀⠀⠈⠐⠀⡄⠠⣿⣿⡆⠀⢀⡈⢁⣸⠠⢥⢀⠑⠢⣀⠀⠀⠀⠀⢠⠠⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢩⣿⣾⢰⡆⠀⠀⠀⣿⣿⡄⠀⣦⡠
⠀⠀⠀⠀⠀⢘⠀⣉⠉⠙⣦⡙⠁⠀⠊⠀⠉⢚⣴⣶⡄⠱⠆⠀⠀⢲⡗⢦⠅⣀⠀⠀⠀⠀⠀⣀⣀⡠⡧⠴⠐⠂⠈⣿⣿⣍⠀⣠⣤⠀⠀⠙⠋⡶⠈⠉
⠀⠀⠀⠀⠀⢸⠀⠛⠀⢸⣿⡇⠀⣶⢼⣶⠂⠀⠹⠿⠇⢠⣾⣷⣀⣀⡀⣰⣶⣦⠉⠿⠀⠉⠈⢿⡿⣀⠀⢰⣿⣷⠀⠛⠋⣶⠀⢿⣿⠇⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠉⣁⠚⠛⠃⠠⠤⠊⠈⠂⢺⡇⠀⠀⣴⣦⡀⠀⠟⠀⠘⠿⠋⠀⠀⠰⡷⠀⠀⠀⠛⠁⠈⠻⠋⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠄⠒⣁⠀⠀⢸⡏⣿⡀⡄⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠆⠜⠛⠛⠿⠥⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠢⠠⡁⠀⠀⠀⡠⠀⠒⠲⠶⢦⡄⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠄⠀⡈⠀⠀⠀⠀⠀⠈⠙⠱⡄⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠁⢈⠀⠄⣄⠠⠀⡐⠀⠀⡀⠈⡀⠃⠐⠠⡀⠀⢐⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠁⠴⢡⡄⣀⢠⡄⠲⠀⠀⠀⠀⠀⠄⡀⣀⣴⠇⢀⠀⠀⠉⠁⠐⠓⠄⡀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠃⠀⠘⢺⠟⠋⠙⠙⠿⡗⠃⠀⠀⠀⠀⠀⠀⠉⠁⢆⡭⠌⠀⠂⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠀⠂
⠀⠀⠂⠀⠀⠀⣎⠀⠀⠴⠀⠀⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⢎⡀⣀⠀⣀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠿⠓⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⣀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠄⣀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⡠⣍⠈⡤
⠀⠀⠀⠀⢠⠛⠁⠀⣆⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⡽⠷⢤⡑⠃
⠀⠀⠀⠐⠁⠀⠀⡴⢯⡰⠃⠀⠀⠀⠀⢠⠔⠂⠀⠀⠀⠀⠀⠠⠒⠒⣾⡀⡄⢈⠀⠀⠀⢸⣇⡀⠀⣇⣀⢀⡀
⠀⠀⠐⠀⠀⠀⣼⡁⠀⠇⢀⡀⠎⣀⢠⣞⢠⠊⠁⠀⠀⠀⢀⠀⠀⢸⠋⢹⡅⠈⠀⠀⠀⠈⣀⣀⠀⠡⣄⠁⠃⠀⠀⠀⠀⠀⠀⠀⠊⢉⣤
⠀⠀⠰⠀⠀⣴⢗⡇⠈⣨⢯⠗⢸⣁⡈⢹⠦⠃⠀⠀⠀⠀⠌⠀⢰⡏⡆⠈⡀⠀⠁⠒⠒⠿⡦⠽⠿⠒⠀⠫⠟⠀⠀⠀⠀⡔⠒⠾⣷⠯
⠀⠀⠀⠉⢓⠗⠲⢝⡒⠀⠀⠀⠆⠡⠁⢀⢀⡀⠀⠀⠀⠀⠀⣀⣸⠿⠛⣄⠀⠀⠀⠀⠀⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠁⠸⠇
⠀⠀⠀⠀⠀⠂⠀⢀⠀⡔⠀⠀⠈⠘⣂⠈⣹⣄⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⠃⠤⠀⠀⠀⣈⣣⡀⢀⣘⠀⠀⠀⠀⠀⠀⠸⠀⡌⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠼⠀⠇⠀⠀⠀⠀⠘⣭⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⢁⠌⠈⢇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠘⠆
⠀⠀⠀⠠⠎⠀⠀⠀⣆⡀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢹⠅⠁
⠀⢀⠔⣤⠂⠀⠀⡐⢡⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠂⠀⠀⠀⠀⠀⠀⠀⠔⠢⡀⠀⠀⠈⡖⠳⠄⡄⠀⠀⠀⡠⡔⠰
⠀⠀⠁⠄⠀⠀⠀⠉⠦⠈⠐⠣⡀⠀⠀⠀⠀⣍⡀⠀⠀⠀⠘⠆⠀⠀⠀⠧⠤⠶⠀⠈⠒⠒⠒⠁⠀⠀⠘⠚⠉⠀⠀⠀⠀⠠⠁⢀⡛⢄

⠀⠀⠰⣿⢀⣄⠀⠀⠀⠀⠀⠈⣉⣹⣶⣦⣷⣤⣶⣿⡟⠉⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣶⣤⣄⣠⣴⣤⣤⣤⡀⣀⢀⡶⠛
⠀⠀⠀⠈⠉⠙⢷⣄⠠⠀⠀⠀⠀⢀⠛⠋⠁⠀⠀⠀⢀⣀⡤⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠻⠛⠛⠛⠛⠋⠛⢻⣿⠛⠛⠻⡿⠃
⠀⠀⠀⠀⠀⠀⠠⠞⠓⠀⡂⠰⠦⣈⡿⢧⠄⠀⠀⠀⠀⠉⠓⠦⢀⠀⠐⠛⠠⡄⠀⠀⠀⠀⠀⠠⣄⠀⠀⠒⠐⣤⢄⣠⣤⠀⡄⠀⡐⠸⣿⡄⣠⠤⠟⠀⠘⢇⡀
⠀⠀⢀⢀⣀⡖⠖⣀⢀⠀⠠⡐⢶⣄⢀⡾⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠳⠶⣔⠃⠀⠀⠀⠀⠰⠗⠈⣷⠀⠀⠸⠀⠀⢉⠇⠈⠁⢠⠇⢤⣿⣧⡏⢀⠀⣠⡄⠀⠛
⠀⠀⣠⠔⠁⠀⣸⡇⠈⣷⠂⠉⢀⡈⠛⢤⣅⣨⠀⢠⡀⢰⠀⢀⠄⠀⠀⡀⠈⢳⠀⠀⠀⠀⢀⣴⣿⣼⢧⠀⠚⠤⠰⠦⡀⠀⠀⣾⠰⣆⣰⢉⢀⡉⣠⠈⠳⣄⠤
⠀⠞⠁⠀⢠⣼⣿⡇⠀⢻⠀⡀⠈⢃⣦⠀⠙⠃⠀⣾⣿⡀⠀⠛⠀⠀⢠⡇⠀⢸⢀⣀⡈⠛⠛⡿⢻⡁⡌⣀⠀⠀⠀⠐⣱⠀⢺⡇⢀⢡⡏⢻⠀⠁⠙⠃⠀⠀⠤
⠀⢀⠀⠚⠈⡿⢿⡁⠀⠈⠀⠁⠀⠈⠆⢰⡄⠀⠈⡜⠻⠀⢀⡈⠀⠠⠊⠀⢰⣾⣿⣿⣏⡆⠀⠘⠈⣟⣠⡟⠐⡆⠠⡤⠟⠀⡿⢀⡈⣧⠀⠀⠰⢦⠀⠟⠙⠓⠖
⠀⢻⣿⠀⢰⠀⣼⠋⠀⢀⠀⠠⠀⠀⠈⣿⣦⣤⠀⢰⡄⠀⠋⠙⠒⠇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠘⡇⠉⠿⠟⢻⠶⡷⠀⢰⠀⣶⠞⠆⠀⠙⠳⣎⠙⠆⢰⠘⣥
⠀⢸⣿⡆⢸⠆⠙⠃⠀⠂⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠁⠀⠀⠀⠐⠉⣀⡈⠛⠿⠛⠿⠟⠓⢀⣤⠴⡷⠦⣀⠀⢠⡦⠀⢨⣤⣤⣥⠀⠀⠐⠲⠿⠊⠁⠀⠀⠈
⠀⠘⣿⡇⢸⠁⣠⡀⠀⣀⡀⢠⣄⢀⣤⣆⠀⢀⣀⡀⠚⠓⠠⡄⠲⣤⡋⣡⣤⠄⠀⠀⠀⢀⣀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢿⣿⡄⠀⢸⡄⢠⡭⠷⢶⣿⣿⣿⣷⠄⠠⠤⡉⢉⠐⠶⡌⠃⠙⠁⠛⣋⢀⠀⠤⣄⠉⠉⠁⠀⠀⠀⠀⢀⠄⠀⠢⡰⠄⠀⠀⠶⠀⠈⠁⠀⠀⠀⢠⠶⡴⠶
⠀⣿⣾⡷⠛⢠⣿⣡⠀⠒⠐⠂⠀⠶⠠⣤⢄⠀⠀⠸⠀⡈⣯⣄⠀⠀⢠⣌⡉⢶⡜⢶⡌⠀⠀⠈⠀⣀⣀⣘⡴⠄⠀⠂⠐⠲⠖⠉⠁⠠⠌⠉⠀⠀⠰⠏⠀⠀⠐
⠀⣿⣿⣷⡀⠸⠋⣀⡀⢠⣤⣄⢤⣄⣀⡀⠀⠉⠐⠓⠤⠀⣀⠀⠀⠀⠀⠈⠀⠀⠽⠀⠀⣀⣀⣀⠲⣿⠛⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠀⠭⢐
⠀⣿⣿⠉⠀⢶⡈⠉⠈⠁⠀⠁⠀⠉⠉⠀⢀⡀⠠⠤⠐⠈⠉⠀⠀⠀⠀⠀⠀⠛⠀⠤⣀⡉⠉⠙⠀⠈⠃⠀⠦⢀⣀⣀⡀⠀⢠⠖⠀⠀⡀⠀⣠⠄
⠀⢻⡏⠀⣸⠒⠛⢤⠀⠀⠀⠀⠀⠀⡠⠶⠨⠦⢀⠀⠀⠀⠀⢨⣨⣥⣤⣄⣷⠀⠀⠠⠙⣿⡇⠀⢀⣠⣄⢀⣀⠀⠀⠀⠈⠁⠘⠃⠲⠿⠀⢈⠃⢀⢠⠠⠘⣰⠆
⠀⠟⠙⠀⠹⠞⠁⠐⣿⡷⠖⣠⢤⠀⠀⠂⣀⣀⣀⣀⣉⣀⠰⠄⠀⠀⠀⠀⠀⠖⠀⠀⠉⠀⠀⠀⠈⠉⠋⠀⠛⠂⠀⢔⠒⠀⣀⣐⣦⣠⠀⠈⠉⠉⠓⠒⠂⠘⠂
⠀⣤⠄⢀⣀⣀⣀⠈⠠⠀⠒⠙⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠰⠀⠀⠠⠒⠳⠦⣀⡀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠉⢁⠀⠀⢀⠤⠄⠀⣠⣤⣂⠝
⠈⣿⠾⢛⣛⣉⣉⣀⡀⠀⠀⠀⠀⠀⢠⣄⣀⣙⢦⡌⠀⠀⠀⠀⠀⠀⠀⠾⠿⠶⢶⡖⢶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⢀⡀⢀⣉⣛⣉⡾
⠀⣿⣿⣿⣿⠿⠃⠶⢤⣄⠀⠈⠉⠑⣦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠦⡤⠀⠀⠀⠀⠙⢃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠄⠈⢁⣠⣤⣤⣤⣬
⠀⢿⠿⠏⠀⠀⠀⠀⠀⠈⠑⢦⣄⠘⠃⠚⠃⠀⠀⠀⠐⡀⠃⠀⠤⠀⠀⠀⠰⣆⣀⣀⣤⠴⠦⢈⡆⠀⠀⠀⠀⠀⠂⠠⣤⡄⠀⠀⡀⠂⣤⣶⣾⣿⣿⣿⠟⠻⠿
⠀⢈⠀⠀⠤⠤⠀⠠⠤⠴⠶⠄⠤⠂⠀⢀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⠁⢴⣀⠄⢸⡏⣀⣈⣶⡞⠛⠠⠾⣆⣿⡀⠸⠧⣼⡿⠛⠛⠉⣁⢔⣀⣀⡀
⠀⠈⠁⠀⠀⠀⣀⠀⠀⢀⠀⠐⠂⠉⠁⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢀⣄⣈⡀⣡⠀⡀⠀⣁⡀⠴⢟⠉⠋⠀⠷⠗⢚⣂⡄⠀⠈⣷⢠⠀⣀⣀⡀⠰⠇⡈⠁⠄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⢤⣤⡟⠓⠜⠛⣤⣀⣀⠀⠉⠉⠀⠰⣶⠀⠀⠀⣉⣤⡴⠋⠀⠀⠛⠛⠻⠿⢿⣷⣶⣶⣆
⠀⠀⣠⡊⠀⠀⠀⠀⠀⠀⠴⠁⠻⠏⠙⠳⠾⠒⠦⢠⡏⠀⠘⠣⠤⢄⣀⡀⠉⢸⠗⢛⠉⠉⠹⢶⡆⠀⠀⠀⢀⠤⠒⠊⡁⠂⠁⠀⠀⠀⠀⠘⠛⡷⠾⠿⣿⣿⣿
⠀⢀⣩⡿⠳⢤⣄⠀⠀⠀⠀⠀⣀⣀⣀⠰⠰⠄⠀⢸⡇⠀⡀⠀⠀⠀⠹⠁⠚⠛⠁⠈⠀⠀⣀⡀⠦⠄⢊⠀⠤⠐⠊⠀⠀⠂⠀⠀⡀⢤⠴⠞⠈⠁⢠⣦⣤⣼⣟
⠀⢀⡱⢶⣉⠙⠻⠂⢤⣤⣄⠀⠉⠁⠉⠃⠀⠀⠀⠀⠛⢶⣬⣙⠓⠀⠀⠀⠀⠰⠒⢈⣉⣉⠠⣤⣈⣁⣀⣤⡤⠄⠈⠉⠉⠙⠙⠛⠻⢶⡄⠂⠀⠀⣠⣴⣶⣿⣿
⠀⠈⠍⠉⠀⠀⢀⡀⠀⠀⠹⢤⡤⠐⠶⠀⠀⠀⠀⠐⠂⠁⠈⠛⠛⠛⠋⠋⠉⠉⠙⠁⠄⠋⠉⠉⠀⠀⠀⣄⠀⠀⠀⠉⠉⠘⠃⠦⡄⣀⠓⣀⣀⠀⡀⠚⠛⠻⠟
⠀⠀⠂⠀⠀⠀⠾⠷⠷⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠐⠀⠀⠠⠀⠀⢸⡇⠔⠉⠀⠀⠀⠀⢸⣆⠀⠀⢻⣿⣷⡞⠀⡀⠀⡄⠀⠀⠈⠘⠫⣏⣿⡟⠥⣐⠢
⠀⠀⠶⠈⢁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠆⣠⡀⠠⠆⠀⠀⠀⠀⠠⡞⠁⠀⠀⢰⡆⠀⠀⠘⣁⣠⠄⠀⠹⣏⢻⡄⠀⠈⣁⣀⠀⡇⠀⠀⠈⣿⠙⢦⡈⠻⡆
⠀⢀⠰⠉⠉⠉⠛⠛⠋⠢⣀⠀⠀⠀⢀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡀⠀⠙⠀⠀⠈⠛⠻⠆⠀⡀⢳⡄⢹⠀⠐⢉⠛⢡⡾⢢⡄⡀⣿⣆⠈⣿⢆
⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠈⠻⠉⠶⢈⡷⠀⠀⠀⢀⡈⠂⠀⠀⠉⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠈⢻⡀⢷⡀⠘⠀⢰⢁⠞⠀⡀⣿⣿⠇⡸
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⣤⠀⢠⣴⡛⠀⠀⠀⢀⡈⠤⠈⠀⣤⠐⢤⠀⠠⡌⢧⠘⣧⠄⠀⠘⠁⠀⠀⡟⢃⣸⠀⠙⠁
⠀⠀⠀⠀⠰⠶⣶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡌⠉⠁⠀⣤⠛⠉⠀⢰⡶⠛⣋⠀⠀⣀⠀⠋⠘⡄⠙⣦⠘⠇⢀⠀⠀⡇⠘⣿⢀⣀⡀
⠀⠴⠂⡀⠀⠀⠀⠀⠀⣉⠉⡁⠲⠦⢤⣴⡦⠴⢦⣄⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⢠⡴⠛⠉⠀⠀⠀⠒⠀⠀⠀⠀⠀⣿⠀⢿⡄⠀⠐⠦⣤⡇⠀⢟⠁⠘⠒
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠛⠻⢿⣿⣦⡀⢸⡆⢸⡇⠀⡈⣷⢈⠁⠀⠸⢫⣤⣀
⠀⠀⠀⠀⢀⡀⠀⢀⣀⣤⣤⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠒⠂⠁⠀⠠⠄⠲⠾⠟⠛⠛⠿⠿⠿⠿⠿⢿⣶⣌⣿⣷⣸⡇⠀⣷⠀⠄⠀⣀⡀⠐⠛⢣⣈⢏
⠛⠛⠛⠛⠀⠁⠀⠀⠈⠉⠀⠀⠀⠀⠖⣁⠀⠀⣤⣀⠀⠀⠀⠠⣤⡴⠆⠑⠀⢶⠶⣶⣶⣶⣶⣤⣀⣀⣠⣶⣾⣿⣿⣿⣿⡇⠀⢻⣀⡤⠀⠤⠄⣀⠐⢦⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡻⠀⠀⠀⠀⠀⠀⢀⣀⠐⠃⠀⠀⠀⠀⢠⣤⣬⣍⣛⣛⡻⠿⣿⣿⣯⣼⣯⣿⣿⡇⠀⠘⣯⣡⡖⠰⣂⡈⠋
⢀⡔⠂⠀⠘⠋⠀⠀⠀⣀⠀⠠⡚⠡⠘⠃⠀⠀⠀⢀⡤⠲⠹⠅⠀⣤⣤⡄⠄⣰⢆⣀⣩⡍⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠈⡉⠉⠒⠛⢃⣠⣤⣿⣦
⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠙⠋⠉⠁⠀⠉⠈⠉⠋⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠘⠃⠀⠛⠛⠻⠿⠿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡉⣡⣉⢿⡁⣸⠉⣉⣭⠋⣸⣽⠋⣅⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣉⠶⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠉⠇⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣭⠒⡋⢱⠀⠀⠀⠀⠘⠁⠀⡙⣥⠸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠀⢀⠼⠄⠀⠀⠀⠈⢦⡀⠀⠀⣀⣤⠤⠤⠤⣄⣤⠀⠀⠀⢠⠃⠀⢀⣘⠃⡌⢳⠀⠀⠀⡞⠁⠘⠟⠀⢒⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠯⠶⠼⠅⣅⣠⠋⠀⠈⠉⠃⠸⠀⠀⠈⠛⠀⠔⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⣀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⣛⢙⠉⠀⢻⡰⠂⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠸⠤⠄⠀⣠⠶⠋
⠀⢀⣀⣤⡼⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⡠⠔⡏⠨⠇⠀⠀⡖⢨⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⢡⠁⡄⠁⠀⠀⠀⠀⠀⠀⡀
⢀⣼⠀⠀⠀⠀⠀⢀⣀⣀⠀⠺⢤⡀⠀⣌⣀⣀⡂⠈⠅⠀⠀⡽⠋⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⢣⠀⢨⠆⠀⠀⠀⠀⠀⡼⠁
⠛⠢⠤⠖⠉⠙⠛⠃⠀⡠⠤⢤⠴⠛⠋⠉⠁⠈⠹⡄⢙⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠃⠀⠀⠀⢰⠋
⠀⠀⠀⠀⡀⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠆⠀⠀⠀⡜
⠀⠀⠀⢮⡁⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣠⠤⠤⣶⠫⠑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⢐⣊⠤⡀
⠀⠀⠀⠀⠻⠤⠶⠚⠛⠚⠋⠀⠉⠁⠀⠀⠀⠀⠈⠙⠣⢿⣄⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠉⠀⠀⠀⠀⠙⠚⢢⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠲⠖⠓⠲⠶⠤⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣳
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣲⣞⣛⣶⣴⣙⣓⣒⣋⣉⣉⣈⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⡖⠀⠀⠀⠀⠀⠀⣠⠞⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⣀⡀⠀⢀⣠⡤⠄
⠀⠰⠊⠁⢠⢀⠀⠀⠁⢠⣞⣁⢤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢚⣛⣒⣄⠀⢶⠎⠁⢸⠷⠟⠛⢻⡶⠟⠋
⠀⠠⠔⠀⠈⢋⡥⠂⣰⣿⣡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠞⠉⠀⣀⣼⣯⠀⠀⠀⣠⠔⡶⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⠁⣈⡁⡤⢸⡇⠈⢿⣄⠠⡤⠌⣬⣾⣶⣤⠀⠀⠀⠀⠀⠀⡠⠎⢀⣀⢚⣿⡅⠀⠀⠀⢀⠀⡀⠞⡁⠚⡁⠀⠀⡄⠀⠀⠀⠀⠀⣴⣿⣿
⠘⣳⢤⣍⡠⡷⠃⣸⠇⠀⠀⠉⠑⢦⠀⠀⠉⠟⠊⠹⠓⠀⠀⠀⠀⠀⠘⣧⠎⠈⠁⠤⠀⠠⡈⠁⠀⠀⡌⠈⠉⢗⢢⣿⠄⠀⠀⢀⣲⣿⣿⣿⣯
⠈⠛⢳⡙⣐⣸⣾⣧⠄⠀⠐⠉⠀⢼⡏⠁⠈⠢⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠁⠀⠉⠁⣰⣶⣿⣿⣿⣿⣿⣷
⠀⡀⢢⡙⠷⠾⣽⣝⠂⠀⠒⠗⠀⠈⢷⡀⠀⢀⠀⠁⠀⠠⠒⠉⠈⠁⠀⠀⠀⢠⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⢚⠀⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠐⢄⠀⠘⢦⣀⠈⠃⠀⠀⠰⠑⢄⠀⠀⠀⠑⣳⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠚⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⢿⣶⣄⡈⠁⢄⠀⠀⠀⠀⡘⣆⠀⠀⠀⠙⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠈⢀⡙⡉⠻⠶⢄⠉⠀⢄⠀⠡⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠⠃⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⣄⠀⠁⠈⠊⠙⠓⠍⠙⠪⠳⠀⠀⠀⠘⢁⠢⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠉⢀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠸⣯⡳⣄⠀⠀⢀⠀⠀⡤⠂⢀⡀⠠⠈⠀⠀⠀⠋⠲⢦⡤⡀⠠⠄⠚⡁⠈⠀⠀⠀⠀⠠⠀⢀⣀⣁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢀⠙⣿⣜⣖⢈⠀⠀⢸⡇⢆⣼⠀⠀⠀⠐⠂⣀⠂⠀⠀⠹⣤⡄⣤⠀⠀⡀⠀⡆⠀⠈⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠸⣷⡄⢙⣿⣗⣥⠴⠖⠿⣿⡿⢦⡀⠀⠀⠸⣿⣤⡄⠀⢀⣼⠑⢿⣷⣤⣷⣌⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠹⣷⣄⣝⠻⠁⠀⠀⠀⠈⠁⠀⢧⠠⡴⠛⠛⣉⡝⢹⡁⠉⠂⢰⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡟⠻⢿⣿⣿⣿⡇
⠀⣻⣿⣿⣷⣿⡀⠀⠂⢀⢰⣶⡧⠚⢻⠁⠀⠀⢉⣧⣸⣧⠀⠀⠺⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡠⢋⠞⡡⠀⠙⣿⣿⡇
⠀⠉⢀⠁⠈⢙⡁⠀⠀⠀⠙⢿⢁⣿⠓⡆⠀⠲⢴⡟⡧⡌⠁⠀⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⠉⡉⠉⠉⠛⠛⠋⠉⢙⠿⠛⡁⢮⣐⠋⢦⡙⣻⠃
⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⢰⢿⡸⠏⡸⠛⠀⠀⠹⠿⠀⡀⠀⠀⠴⠶⠾⠿⠷⠾⠿⠿⠿⠟⠛⠚⠓⠾⠿⠟⠷⠶⠆⣰⣦⣤⣄⡈⢷⣶⡟⣮⠃
⠀⣷⢀⣤⠀⠈⠉⢸⣷⣾⣿⣿⣵⡀⠱⢤⣤⣤⣴⣶⣦⠍⠀⢀⠁⠬⠀⠀⠀⠀⠀⠁⠢⠆⠦⣄⡀⠀⢾⡇⠀⢀⠾⠿⣿⠿⠿⠤⣽⢿⣯⣾⡷
⠀⣿⣿⣿⣧⠀⠀⢘⣿⢿⡿⡿⢹⠁⠀⠸⣿⡟⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠄⠀⠀⠀⠙⡄⠂⢠⣖⠃⠤⢠⢆⣶⣶⠂⣨⣷⣽⣿⣇
⠀⣿⣿⣿⣿⡉⠀⢨⡇⠨⠀⣿⡬⠀⠀⡔⠙⠁⢹⢿⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⢬⣄⠀⢸⣶⡟⠋⠀⠀⠀⠙⢻⣿⣾⣿⣿⣿⣿⠇
⠀⠻⠿⠟⠋⠀⠀⠸⢉⠅⠀⡧⢾⠀⠘⢑⠆⠀⠘⢸⡄⢤⠆⠀⡄⡤⣤⣀⠀⠀⠀⣀⠤⠴⠡⣾⣛⠀⢻⣟⠉⠒⠉⠘⠀⠀⠀⠙⠛⠟⠛⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⠀⠐⠀⠀⠀⠀⠐⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠤⠂⠊
⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⡤⠊⠀⠀⠴⠃
⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢰⡀⠀⠀⠀⠀⣀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀
⠉⠉⠲⣄⠳⠀⠀⠀⢠⣻⡯⢄⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⣀
⠀⠀⠀⠈⠳⣤⠀⢀⣿⣿⠀⠀⡁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⣿⡄⠀⡀⠀⠀⠑⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⣿⣿⣧⠏⠀⡀⠁⣀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡷⠀⠀⣴⣧⠀⠈⠉⠻⢄⠘⠦⠀⠀⠀⠰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣮⠀⣈⠴⠂⠀⣠⡄⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⢀⣾⡏⠁⠀⠀⠀⠠⠿⠇⠀⠀⠀⠀⢀⡔⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣿⣿⠁⠟⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟
⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢦⣤⣤⣀⣀
⠀⠀⢠⣿⠏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠦⣄⡉⠉⠛⠳⠶⣤⣤⣀⣄⣀⣀⣀
⠀⢠⣿⣿⣿⠇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠂⠀⠀⠀⠉⠉⠉
⣠⣿⣿⣿⣿⠲⠚⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠿⣿⣶⣦⣤⣀⣀
⠈⠈⠉⠋⠃⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢻⣿
⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠒⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⣀⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠲⣤⣀⣤⠴⠂⠐⠦⣤⣠⣤⠖⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣧⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⣼⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣩⡄⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠛⢻⠀⠉⠀⠀⠉⠀⡟⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠤⠤⢶⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡞⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⣷⠷⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⣂⣄⠀⠀⢀⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠈⢣⣆⣴⠁⠂⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣝⠆⠄⠀⢼⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠻⣧⡄⠘⡰⣒⣾⢒⠀⠀⣶⡒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠌⠉⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠦⠀⠀⢹⣷⣀⣹⠹⣿⡄⠱⣂⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣗⣹⠆⢻⣿⣈⣻⡿⣷⡰⢏⡿⠐⣁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⣶⢇⠀⠀⠀⠀⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣋⣿⡿⣿⣛⡿⣷⡯⡓⣻⣀⡯⣎⠂⠈⠂⠀⠸⡁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡠⠞⠑⠋⠙⠒⠂⠛⠃⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣌⢳⢡⡙⣷⡿⣿⣢⣽⠋⠁⠈⣿⢂⢆⡀⣸⣤⡄⠀⠀⣼⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⡌⠀⠀⠀⠀⠀⢀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠭⢿⣦⡝⠳⣯⣟⣨⣿⣨⣌⣻⡍⢛⣥⠓⠯⠀⣲⣴⠄⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠄⠀⠐⠀⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡔⠂⠀⠺⣷⣎⡟⣭⣦⠏⣫⡽⠃⠚⣠⠶⠙⢤⠀⢼⣻⣦⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⡆⠀⠀⠁⡏⠓⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⢨⠽⡲⢞⡽⣟⡁⠈⢀⠎⢁⠀⠀⠘⢤⣿⡃⠮⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡧⡂⣼⠖⢷⢼⡇⣁⣠⠟⢱⡆⣿⢰⡰⢰⣈⣿⠇⡆⣼⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠿⣏⢿⣧⣸⢾⣿⣿⣿⢿⣸⡇⠿⠼⠃⠘⠘⠉⠀⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠙⠸⠓⠈⠙⡛⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⠈⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢧⠀⠀⡄⠀⡏⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⣇⠀⠀⠀⢸⣖⡒
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡸⠟⢀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠠⣀⡠⠊⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒

⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⢐⣮⣿⣴⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣮⣿
⠠⣮⣻⣿⣫⢖⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡺⣟⣿
⠺⣿⣿⣿⣶⣿⡿⠀⠀⠀⠀⠠⣾⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣾⣿
⣛⠷⣽⣿⣿⣿⠿⡃⠀⢀⠠⣜⣿⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⣿⣿⣿
⣯⣷⣚⣿⣵⣶⡏⠀⢠⣿⣞⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢳⣶⣯⣿
⡝⣿⢿⣿⣹⠏⠁⣀⢿⣿⡿⡛⣻⡿⠿⣫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣾⣗⡌⠻⣍⣿
⠙⣿⡻⣿⢀⡁⢠⣿⣥⣿⣿⣷⣾⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⣤⡹⡿⣫⣦⠀⣿
⠄⡇⢳⣯⢘⣃⣶⣭⣽⣧⣿⡿⠹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡠⢧⣽⡿⣿⣾⣿⡓⣿
⠀⠁⠀⣟⢻⠉⠀⠻⢹⣿⣿⡗⢶⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣿⣷⡾⣿⠿⣿⣾⣿⠿⣿
⠀⠀⠀⡯⠹⠤⠀⠀⡀⡏⠘⡇⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣛⡇⠈⠛⣻⠘⠁⠗⣿
⠀⠀⠀⡇⠀⠀⢰⠀⠙⡏⠁⠃⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠁⢀⡂⣠⠄⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠏⢿⠀⠀⠀⣿
⣀⣤⣦⣧⣴⣄⣠⣄⣸⡇⠀⣀⣸⣇⣀⡀⢀⣀⣀⣀⣀⣤⣄⣀⣀⣴⣤⣤⣄⣿⣤⣄⡄⣀⣼⢀⣀⣀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠚⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡒⡂⠀⠀⠀⠣⠝⠀⠀⠀⢔⢲⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⢄⠀⠈⠀⢀⢤⠀⠀⠀⠀⠤⡀⠀⠁⠀⠤⡄⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠈⠈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⠤⠤⠤⠤⢤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⢸⡆
⠀⠀⠀⢀⣠⡼⠶⠚⢛⣉⣉⣭⣤⡤⠤⢶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠤⠤⣤⣭⣉⣉⡛⠓⠶⢧⣄⡀
⢀⣴⠞⣉⡤⢶⡞⠋⣉⠭⠋⠁⠀⣀⣀⠈⢆⠀⠀⢀⡀⠀⠀⠀⡇⣀⣀⠀⠉⠙⠬⣉⠙⢳⡶⢤⣍⡳⢦
⣾⡵⠋⠁⠀⢸⠗⠊⠀⠀⠀⠀⠋⠁⠀⠀⠀⠑⠦⡼⠉⠓⠲⠦⠇⠀⠈⠙⠀⠀⠀⠈⠑⠺⡇⠀⠈⠙⢮
⠹⣦⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣴
⠀⠈⠙⠶⢤⣼⠀⠀⠀⠀⠀⠸⠋⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠙⠇⠀⠀⠀⠀⠀⣧⡴⠖⠋
⠀⠀⠀⢀⡞⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢳
⠀⠀⠀⠀⠳⣄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⠞
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠘⠉⠓⠢⠤⠤⠔⠚⠉⠑⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏
⠀⠀⠀⠀⠀⡠⠒⠁⡇⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⢸⠈⠒⢄
⢀⡠⠤⠐⠊⠀⣠⠤⣇⢀⡄⠀⠙⢙⠓⠶⠤⠤⣤⣤⣤⡤⠤⠤⠶⠚⡉⠋⠀⢀⡀⣸⠤⣄⠀⠑⠂⠤⢄
⢎⣀⠔⣦⠀⢀⡵⠀⠀⢨⠃⠀⠀⠈⠙⠲⢤⠔⠃⡀⠀⠑⠦⠤⠖⠋⠀⠀⠀⠘⡅⠀⠀⢮⡀⠀⣴⠢⣀
⠀⠑⠊⠘⢤⣈⠆⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠰⣅⡤⠃⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠉⠁⠐⠒⠂⠀⠠⠀⠀⠄⠀⠐⠒⠂⠈⠉⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⠒⠒⠒⠶⠃⠘⠖⠒⠒⠒⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠤⠤⠤⡦⠀⠀⢤⠤⠤⠤⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⠀⠉⠓⠢⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣶⣶⣶⣟⣻⣿⣿⣏⣀⣀⣹⣿⣿⣟⣿⣶⣶⣶⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣫⠄⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⢛⡀⠀⠀⠀⢀⠀⣑⠠⠵⠕⠃⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠁⠁⠀⠀⠀⠀⠘⠘⢀⡀⠈⠁⠀⡄⠀⠈⠈⠀⠀⢸⠀⠀⠀⢠⢫
⠀⠀⠀⠀⠀⠀⠀⣰⣻⠀⡀⠀⡀⠀⠂⠀⠀⠀⠀⠈⠿⠢⠀⠀⡀⠀⠀⠀⣀⢼⣆⠉⠁⢸⡑
⠀⠀⠀⠀⠀⠀⣰⠿⢹⠉⣃⡁⠅⢀⡀⠈⠠⢠⣤⠄⠈⠈⠀⡴⠇⠀⠰⠊⠀⢸⠀⠀⢒⡞
⠀⠀⠀⠀⠀⣰⡝⠃⠀⠀⠀⠁⠀⠀⠉⢣⠀⠀⠀⠈⠂⠀⠀⢀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⡴⠿⣶⣷⣴⣤⣤⡄⡆
⠀⠀⠀⣀⣀⡀⢀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠂⣖⠀⠐⠞⠁⠆⠀⠀⠀⠰⠀⠈⠃⠣⡀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠈⠁⠉⠀⠀⠘⠁⠉⠉⠉⢻⠆
⠀⠀⠠⠤⠜⠗⢠⣄⢠⣐⣛⠻⠅⠀⠀⢠⠀⠀⠀⠀⠠⠷⠆
⠀⠀⠀⠀⠈⠉⠛⠙⠻⠉⣹⣿⣯⢭⣿⠩⢥⡦⠄⢦⠽⠿⠶⠶⠮⠷⠖⠒⠺⠥⠟⠿⠷⠮⠥⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣐⣪⡥⠶⠶⠒⠒⠒⠒⠶⠶⢬⣕⣂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣪⠴⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠛⢦⣕⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⢱⡄⠄⠀⠀⠀⠀⠈⠳⣮⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣡⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠖⠚⠁⠀⠀⠀⠀⠈⠳⣝⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣴⠋⠀⠀⠀⠀⢀⣴⠗⠢⡀⠀⢰⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣸⠃⠀⠀⠀⢀⡴⣪⠞⠀⢰⣿⣿⣯⠬⣹⣿⣿⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⡏⠀⠀⠀⣰⣯⡞⣿⠀⢰⣿⣟⣿⡴⢚⡿⣿⣿⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡈⣾⠀⣠⢤⣾⠟⠁⠀⠙⠀⢻⣿⣿⣿⣥⡾⠟⣿⣿⡇⢠⡟⠈⠐⠢⣄⠀⢠⠀⠀⠀⠀⠀⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠞⠁⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣥⡆⣿⣿⡇⠈⠟⠀⡀⠢⢌⠉⣿⠦⢤⣀⡀⢀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⣰⡂⠀⣶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⢸⣿⣿⡄⠀⠀⠈⠂⠀⣼⣿⣇⣼⡄⣯⣩⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣦⣽⡃⢘⣿⡄⠀⠀⣸⡘⣿⣿⣿⣿⣿⡿⢁⣿⢿⣷⠀⠀⣰⡄⣰⣿⣿⣿⣿⣿⣿⣿⡿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣿⣷⣾⣿⣿⠀⠀⣿⣿⣧⣄⣹⣿⣿⣶⣿⣷⡾⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣅⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⠟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⢟⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢝⠻⢿⣿⣿⣿⣿⠃⢠⣿⣿⣿⠀⢸⣿⣿⣿⣿⠟⡫⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠩⢝⡛⠻⠤⠾⠿⠿⠧⠠⠞⢛⠋⠍⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣅
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠉⠁
⠀⠀⣼⠷⢦⣄⡀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⡀⠀⠀⠀⠀⠀⢀⢤⡠⠄⠤⢄⠀⠀⢠⢊⠋
⠀⠀⠀⠀⠀⠈⠛⢗⢄⠀⠀⡇⠀⢱⠀⡀⠤⣀⠀⠀⠀⠀⠀⣏⣉⠉⡇⠀⠀⠀⠀⠰⡅⠀⠑⣴⡄⠀⠑⡴⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡕⢤⡃⡠⢼⠉⠀⠀⠀⢑⠖⠢⢄⣸⡀⣩⡧⠧⠒⠊⠉⠒⡎⠀⠀⢀⣩⠉⣶⠎⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠓⢌⡐⠪⢅⣢⡎⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠺⠵⣴⣝⣢⣁⠔⠁⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠑⠤⣈⡏⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢹⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠉⠐⠂⠀⠤⠦⠤⠤⠤⠤⠒⠂⠉⢛⠉⠉⢹⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠤⠤⠄⠐⠂⠉⡠⠃⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣀⣸
⠀⠒⠒⠂⠒⠂⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡉⣉⣶⢤⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠉⠁⠀⠀⠂⠀⠀⠉⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢠⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⢠⠴⠒⠶⠊⠉⠉⠀⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⠒⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣏⠙⠋⠓⠒⠶⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⣀⣠⣤⣴⣶⠾⠛⠋⠉⠙⠿⠿⠟⠛⠛⠉⠙⢿⣿⣦⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣀⣴⣾⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡦⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⠞⠉⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣁⠀⠀⢀⣈⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⣹⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠰⣏⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠘⠁⠀⠀⠈⠇⠀⠀⠀⠀⠠⢁⠈⠀⠐⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⣀⡄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⣄⣀⠀⠠⠒⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠀⠙⠅⠒⠢⠤⠤⠤⢤⣤⡐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡐⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⢦⠀⠀⠀⠀⢤⠀⠀⢀⡄⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠶⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⠀⠀⠈⠳⠀⡀⠂⠀⠀⠀⠁⠈⠣⡬⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠈⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⢀⠔⠁⠀⠀⠀⠈⠁⠀⠀⠘⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⠆⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣵⡀⠀⠀⠁⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀⠀⠈⢂⠀⠀⠇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠊⠙⢷⣄⠀⢀⠃⠀⢀⣉⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠈⡁⣯⠀⠀⠀⠀⠀⣠⣄⠀⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⣀⠠⠌⠀⠀⠀⡿⢛⠃⠀⠀⠀⡁⠘⠀⢀⠃⠀⠀⠀⠀⠀⢁⠀⠃⢀⠁⢰⡀⠀⢀⡤⠞⠁⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠞⠀⡆⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⡗⠉⠁⠀⠀⠀⠀⠀⠘
⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠑⠀⢀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠓⠀⣀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡀⠾⠀⠀⠀⠀⠈⠑⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⡔⠁
⠀⠀⠀⠀⠐⢄⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠂⠀⠀⠀⠀⠀⡇⠈⠉⠉⠛⠛⠛⠒⠒⠒⠒⠓⠒⠚⠋⠉⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⡀⠀⠉⠐⠂⠠⠤⠤⠠⠤⠴⢧⠀⠀⠐⠒⠂⠉⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠋⠒⢶⡀⠀⢎⠙⠋⠉⠉⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡈⠐⠒⠀⠁⠃⠰⠈⡀⠀⠀⠀⣐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠑⠴⠤⠆⠘⠁⢸⠀⠁⠈⠙⣀⠈⠁⢠⠄⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠒⠂⡁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠇⠤⠀⠀⠀⣀⣀⡀⣀⣀⠀⠠⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣦⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣶⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣓⡦⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⡴⠖⠚⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⠁⣠⡶⠉⠀⠙⠳⣝⡆⠀⠀⣠⡴⠋⠉⠙⠋
⠀⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⡟⠀⠀⣦⠀⠀⠀⠉⠲⣴⠋⠀⠀⠀⡼⠁
⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠈⠁⠀⠚⠁⠀⠀⠀⠀⠀⠘⣾⡄⠀⢸⠁⠀⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣟⠛⠃⢀⣴⠀⠲⠦⣀⠀⠀⠀⠀⣸⠀⠀⢹⣷⡀⠘⣇⠀⠳⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣿⠃⣀⡼⠀⠈⠓⠀⠀⣠⠏⠞⠀⢀⡇⠱⡄⠈⠳⢦⡙⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡀⢀⣀⣀⡀⠔⠁⠀⠀⢀⡼⣧⠀⠙⣆⠀⠀⠙⠲⣝⣦⡙⠦⣄⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣝⣒⣾⠋⠉⠛⠲⢾⣉⡁⠈⠳⢄⡈⠳⢤⡄⠀⢀⣹⣿⣷⣄⣙⣦⣤⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣥⣒⣒⣒⣒⣀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠷⠶⠶⢄⠁⠉⢙⡛⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⠀⠀⢀⣀⣀⣀⠹⣎⢻⡄⠈⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠉⠉⠉⠙⠇⠹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢉⠀⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢄⠀⣤⡞⠀⠀⠀⠠⡠⢔⣨⠄⠀⢀⠐⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢩⡿⠱⡼⠃⣈⠀⢈⡑⡧⠀⠄⡛⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⣙⢿⡈⣭⠘⢛⠁⠐⢼⣧⡄⡢
⠀⠀⠀⠀⠐⠄⠠⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠐⠀⠀⣀⠑⠆⠈⠘⢻⠑⠬⡆⣘⠃⠈⡻⡁⠪⣅⡸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣬⣄⠀⠐⠀⠀⠼⡀⠐⠁⡀⢢⣤⡟⠃⡉⢳⢤⡓⠀⠾⠂⠅⠒⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⠸⣗⠦⢀⣂⡀⢿⠄⠲⣯⣼⣴⣤⡦⠾⠹⣊⢿⣿⣷⣁⣢⣈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠠⠹⣆⠀⠘⠠⠁⢧⠙⣾⣸⠇⣿⣽⣿⣿⣿⣿⣼⣿⣿⡿⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⢸⣿⡐⠀⢹⣆⠀⠀⠀⠀⠣⣼⣿⠌⠉⠛⠙⠻⢿⣿⣾⠿⠀⢀⠘⠻⠁
⠀⠀⠀⠀⠀⠁⠈⠀⠒⢄⣀⣀⡀⡐⠀⠀⠀⠀⢸⣹⡀⠤⠀⢹⣦⠀⠀⠀⠀⠙⣀⠀⠲⡄⠀⠀⠈⠿⣇⠀⢰⠣⠤⠤⣄
⠀⠀⠀⠐⠐⠢⠖⣀⣠⣌⠉⠙⠑⢀⠆⠀⠀⠀⣈⣾⡇⠀⠀⠀⢻⣧⠂⠀⠀⠀⠈⢧⣠⠼⠖⡉⠰⢆⣀⣐⣏
⠀⠀⠀⠀⠀⠀⢠⣶⣥⣉⠙⣗⡀⠐⠀⠀⠀⠀⣿⣯⡇⠀⢀⠀⠀⢋⣶⡠⠀⠀⣀⣜⣳⣖⡀⠙⢦⠀⠀⡁⢀⡀⣀⣀⢅
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⢃⡨⠴⠓⠊⠚⠉⠡⠉⠃⠉⠉⠉⠙⠃⠀⠈⠱⠺⠋⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡇⣴
⠀⠀⠀⠀⣀⣄⣀⣾⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣤
⠀⡇⠀⢸⣿⣯⣩⣿⣿⣤⡇⣇⢀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡶
⣤⣧⢰⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣇⣿⣦⣦⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣠⣴⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣦⢀⠀⣤⣰⠀⠀⠘⣿⣿⠀⣿⡟⣸⣄⣴⣶⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇⠀⠀⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠟⠛⠉⣀⣀⣀⣀⣸⣿⡿⠁⣶⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠰⣿⣿⣿⣿⣿⡿⠟⠁⠀⣿⣿⣿⣧⡀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠻⣿⣿⣦⡀⢰⣿⣿⣿⣷
⠀⠀⠀⢠⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣰⣿⡟⠉⠁⣾⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠷⠄⢸⣿⣿⣿⣿⡟
⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡟⢩⣿⣦⣤⢠⣄⠀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣸⣿⣿⣿⣾⡏⣰⣿⣿⣿⣿⠇
⠀⠀⠀⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠏⣿⢧⣿⢡⣿⣿⣿⣿⠏
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠉⣸⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⣿⣿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢒⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠤⠤⠤⠀⣑⡉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⠀⠀⠀⢀⢀⣒⠒⠀⠩⠝⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡈⢀⡆⠀⠀⠀⠄⠒⠤⢌⡹⠐⠠⣀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠾⡟⠁⢀⠀⠀⠀⠐⠆⠀⠀⢣⡠⠦⡀⠑⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠈⠀⠀⠀⠀⢀⡀⠀⠈⢑⣁⠆⡠⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣄⠀⠀⠰⡶⠂⠉⡇⠀⣤⣡⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠓⠢⠤⣬⠭⠍⢀⣠⠖⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⡤⡔⣤⡠⡗⠋⢹⠿⠡⠤⢄⠀⠀⡿⡐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢮⠠⡢⣱⣱⠟⠈⠙⠉⠓⠬⠼⣁⠁⠢⡑⡜⠀⠘⡁⠂⢭⡲⡤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⢻⣦⢱⡞⠓⣬⡙⡉⢆⠀⠀⠀⠀⠀⢀⢴⣟⠀⠀⠀⣎⡱⠁⢰⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⡹⠞⣧⣀⢹⣧⠛⢾⣉⣉⣹⠟⠉⠁⢁⠶⠬⣔⣒⡤⣷⣀⠇⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⡲⠾⠌⢉⠉⠑⠢⣄⠀⠀⠀⠀⠁⠈⢹⠒⢇⣂⣀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠤⠤⣀⠀⠑⢄⠀⠀⠑⢦⣀⣀⣀⣀⡸⠶⢶⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⠊⢐⠢⡀⠱⡀⠈⣆⠀⠀⠀⣀⠤⠋⢄⠐⠁⣸⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⣀⠀⠉⢦⢄⣊⢠⣀⠌⢼⡴⠧⠊⠑⢌⠉⠀⠀⠀⡁⢸⣴⢾⣟⠓⡄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠳⢦⡈⠛⣧⡰⠃⡠⠊⠀⠀⠀⠀⠀⠉⠒⠒⠒⠧⠊⢹⠟⠉⠻⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠈⠻⣄⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⢀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⣉⣈⡳⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣤⡅⣦⡍⠆⠀⠀⠀⢂
⣠⠔⠀⠉⠈⠁⠀⡐⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠀⠉⠒⠀⠀⠀⠀⠈
⠣⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣥⣴⡆⠄⠀⠀⡜⠀⠄
⠀⠑⡀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠁⠙⠁⠀⠀⠈⠃
⠀⠀⠘⡄⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⢀⠔⠒⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠉⠓⡀⠀⠀⠸⠀⡀⡇⢄⢀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠄⠁⠈⢑⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠈⠀⠐
⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠤⠀⠀⠠⠄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠈⠐⡄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⢠⢈⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⣀⣀⡤⣊⡁
⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠈⠀⠀⠀⡁⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠈⠉⠉⠁⠈⠐⠂⠤⠤⠼
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣦⣶⣦⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢁⣀⣈⡉⣀⣶⢤⡐⠦⡤⠤⠤⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡟⠀⣨⣿⢭⠳⣤⡼⠀⢙⣷⠀⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠤⠀⠠⠒⠎⠀⠤⠤⠠⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⠁⠈⠐⠳⣌⣑⣦⠪⡾⠀⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⠀⠂⠀⠀⠀⢠⠀⠄⢸⠀⠫⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠀⠀⠈⡿⠀⡐⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⣁⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠠⠀⠤⠀⠀⠀⡠⡿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⢠⣀⣀⣀⣀⣀⣈⢸⠀⠰⢸⡇⠀⠀⠀⠀⢠⡤⣤⣤⠤⠖⠒⠂⠀⠀⠀⠀⠀⠀⢀⢀⡣⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣄⣶⡂⠐⠀⠀⠀⠀⠀⠘⠀⢀⣺⡇⠀⣠⠔⢈⢉⣁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣠⠿⠵⠶⠶⠇⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠒⠓⠛⠓⠒⣶⡴⢶⣶⠒⠒⠓⠛⠒⠛⠊⠀⠀⠀⠑⡍⠀⠀⠀⠀⠐⠀⠀⠀⢀⠴⠊⠁⠄⡀⠀⠀⠀⠀⡴⡀
⠀⠀⠀⠀⠀⠀⢼⢇⣰⣂⣀⣀⣘⣉⣉⣉⣉⣉⢓⣒⣲⣶⣶⣖⡆⠔⠲⡀⠀⡀⠀⠀⠀⠐⠂⠌⢀⡖⠁⠀⣀⡀⠀⠀⠆⠀⠀⡞⢈⡅
⠀⠀⠀⠀⢀⠀⡈⣆⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢄⡇⠙⢞⠁⠠⢣⡀⠀⠀⠀⣀⣂⣈⡁⠀⣊⡀⣱⠀⠀⠰⢀⠈⠀⠸⠁
⠀⠀⠀⠀⠹⡭⠟⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠔⠁⣦⡀⠀⠀⣇⠀⠉⠁⠈⠀⠀⠀⠈⠣⡀⠀⠉⠁⡀⠠⠚⠁⢀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⡇⠘⠛⠉⠉⠑⠒⠴⠤⠤⠤⠤⠤⠤⠾⠢⠤⠤⡶⠥⠟⠒⠋⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠀⠀⠀⢊⡅⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡃⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⣀⣄⣀⣴⣶⣦⣴⣶⠶⠲⡶⢲⣀⣸⣨⡗⣷⣷⣷⠤⢦⣤⠤⠤⢤⡬⢬⣤⣤⣤⣤⣀⣀⣥⣥⣼⠶⠶⢶⠾⠃⠈⠑⠂⠉⠀⠄⠒⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠎⠀⠀⠀⠉⠉⠉⠉⠙⠒⠲⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⣠⣶⣿⣿⣷⣈⣿⣝⢿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣀⣀⠀⠀⣠⣾⣿⣿⣟⠷⠞⠉⠉⠉⠀⠟⠿⢿⣿⣷⡄⠒⠚⢉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢸⣿⣯⣹⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠐⢶⣷⣦⣿⣿⠀⠀⠉⠉⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⡜⠁⢀⢀⣿⣿⣿⣿⣿⣏⣻⠀⠀⠀⣀⠈⠑⡄⠀⠀⠀⠀⢙⣿⣻⣧⡤⠖⠚⠉⠁
⠀⠀⠀⣠⠴⠚⠛⢛⣷⣶⣾⣅⡀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⡿⢿⠛⠉⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⣠⠟⠋⠁
⠀⠀⠀⢀⣐⠔⣀⠄⠉⠛⣿⣟⡿⣶⣄⠀⠀⠀⢸⠙⢿⡿⠻⠷⢋⠀⠀⠀⠀⠀⠀⠀⢀⠞⡀⠉⠣⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠉⢻⣷⣶⣤⣠⠄⠘⠿⣾⣿⣷⣄⢀⡼⡆⠠⢐⠀⠀⠸⡄⠀⠀⢸⣶⣄⠀⠆⠀⠙⢢⠖⡠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣾⣁⠄⠈⠿⣭⡻⣿⣼⠃⠄⠨⠀⠀⠀⠙⣆⠀⠀⠙⠛⣀⠤⣢⠖⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠘⢶⣤⡤⢀⡀⠙⠿⣿⣦⡀⢀⠈⠹⣯⣿⣶⣂⠎⠀⠀⣀⣤⣾⡷⠒⠒⠉⠁⣘⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠙⢿⣿⣔⢀⡀⠀⠘⠿⣷⡴⠂⠀⠈⠉⠛⠦⣀⠹⠿⠛⠁⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⡤⠀⢀⠀⠀⠉⠑⠲⢶⣄⣀⠄⠈⠙⠲⢤⣤⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠥⣖⠁⠀⠀⠀⠀⠉⢻⣿⣦⣴⠄⣀⠈⢙⣦⣀⠈⠑⠶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣴⣭⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠮⣁⠤⠀⠀⠹⢿⣿⣯⡁⠀⣀⠘⢿⣷⣄⠀⠀⠉⠓⢦⡀⠀⠀⢴⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁⠀⠀⠙⠻⠿⣌⠄⠀⠀⠹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⢤⣤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠐⣄⣰⣉⣷⡀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠁⢀⣀⣀⣀⣤⣤⠖⣛⣋⣿⣿⠧⣀⣀⣸⡟⢻⠶⠖⢺⣗⣚⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠉⠹⢿⠐⣿⣿⣿⣿⣟⡢⢄⠙⢿⡀⢣⡍⠀⠀⣌⣫⡙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣿⣄⠀⠀⠀⠀⠀⠎⠀⣿⠟⠁⠈⠈⠀⠀⠑⣄⣷⣀⡀⠀⠀⣨⠉⠻⣤⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠥⠀⣽⣿⣷⡴⠀⡼⢫⠂⠀⠀⠀⠀⠀⠠⣿⣿⣿⡧⣀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣁⣤⡤⠒⠊⠉⢵⢢⠁⣜⡷⠁⠈⢷⣄⠀⠀⠀⣴⡿⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡠⠚⣹⣿⠀⠀⠀⠀⠀⡎⠰⡎⠀⠀⠀⢀⣻⣦⡀⠀⠈⠳⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⡞⠁⡄⣿⡏⣀⠄⠀⠀⢰⣷⢳⣶⠞⢀⡰⢿⣝⠿⠆⠐⠲⣾⠿⡁
⠀⠀⠀⠀⠀⠀⠀⢰⡇⣈⠁⢀⢸⣿⠏⠁⣀⠀⣤⣼⢿⡿⠥⠒⠁⠀⠁⠉⠀⢰⣤⠀⢹⡄⢇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠘⣾⣿⠾⠋⠀⠀⠘⢿⣮⢻⡄⠀⠀⠀⢠⣦⠞⠁⠘⣧⠀⠀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⢀⠖⡁⠀⠀⠀⢹⢸⠗⠀⠀⠀⢻⡏⠀⠀⠀⠻⣷⡆⠀⠀⠘⠿⣦⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡸⣿⣿⣿⣀⡘⠁⡠⠋⠈⠀⠀⢤⣤⣸⣷⣄⠀⠀⠀⠉⠻⢄⣀⠀⠀⠈⠛⠷⣤⣤⠐⠧⠒⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢙⠻⢿⣟⣡⣾⡁⠀⠀⠀⠀⢠⣿⡏⠀⠙⠯⣕⠢⣤⣀⡀⠈⠉⠒⠢⠤⣄⣀⠙⠷⣶⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠘⠋⠉⠉⠉⣽⣭⣍⣩⡴⠎⠀⠀⠀⣸⠋⠉⠀⢙⣷⡄⠀⠀⠀⠀⠈⠙⠲⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠠⣿⣿⢻⣿⣇⠀⠀⣀⣴⠟⠀⠀⣀⡘⠏⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠐⠫⢻⡆⢻⣿⣧⣾⠟⣿⠺⢿⡛⠉⠉⢰⡏⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡁⡆⢘⠛⠿⠄⠁⠀⢠⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠁⠃⠘⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣤⣤⡴⠒⠀⡀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⣿⠁⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢘⣦⣘⣿⡏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⢠⡾⠛⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠱⣿⠷⣤⡾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣼⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣷⣄⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠐⠉⠩⠁⢀⣿⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⠀⢀⣀⣤⣶⣾⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠤⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢨⣿⣇⣿⠀⠀⠛⠻⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠁⢸⠀⠀⠀⠀⣨⠘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡎⠴⡶⢷⣾⡴⠾⠿⠿⠿⠸⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣾⣦⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⣉⡉⠍⠂⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠈⢡⡤⠴⠶⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⡉⣩⡦⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⡁⠋⠀⠀⠀⠰⡄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠠⠁⠀⠀⢀⣀⣀⠨⠤⠤⢀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠤⠘⠂⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠉⠀⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣉⣀⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢀⣀⣠⣤⠤⠶⠖⠚⠛⠉⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣠⠴⠖⠛⠉⠩⡄⠀⠀⠀⠀⠀⠀⠀⢀⠲⣦⣀⠀⣦⡈⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠀⠉⡽⠟⠾⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿⠁⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⣠⡄⠀⢀⣠⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣦⠖⠛⠈⠓⠦⡀⠐⠟⠀⠀⠠⠀⠙⠁⠐⠀⠛⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠐⢲⣄⠀⠉⠀⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⣀⣀⡠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣦⣄⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢿⣿⡿⠿⣿⡿⠿⢇⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠁⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠫⡓⠢⠤⣈⡀⠀⠀⠀⠀⠀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣀⠀⠈⠀⠀⠀⠀⠀⠀⠜⢡⠶⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡤⠚⠁⠀⠀⠀⠀⠈⢆⠀⠀⠠⠚⢃⠀⢠⠚⢧⡀⢸⠀⠀⠀⠀⠀⠀⠀⠔⠃⠘⠢⡀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠸⡀⠀⠀⠀⢀⠄⠀⠀⠀⠓⢴⠂⠀⠘⡤⠁⠀⠀⠙⣎⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡃⢀⡶⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠑⢄⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⡠⠀⠀⠑⠦⠐⠁⢪⡄⠤⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⠀⡜⠁⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⡆⠰⠀⠉⠀⠀⠀⡀⠀⠀⠀⠸⣅⢀⠔
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠱⣄⠀⠀⠀⠀⠌⠀⠀⠙⣷⣄⡜⠀⠀⠰⠤⢴⣄⠀⡸⠀⠀⠀⠀⠀⢣⠀⢀⡦⠠⠄⢡⠀⠀⠀⠀⢀⡼⠉⠢⠤⠤⠔⠉
⠀⠀⠀⠀⠐⠀⡀⠃⠀⠀⠀⠈⠣⡄⠐⠚⠀⠀⠀⣠⠏⢰⠁⠀⠀⠀⠡⢀⣀⠠⠃⠀⠀⠀⠀⠀⠀⠠⠀⡅⠄⠀⢸⣀⣀⡤⠚⠁
⡀⠀⠀⠁⠀⠀⠀⠀⠰⠀⢀⠔⠈⠠⠀⠀⠀⠀⠘⢃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠈⠀⢀⡄⠀⠂⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠂⠐⠀⠀⠀⠈⡄⠀⠈⠛⢄⣰⠀⠀⢸⠉⠂⠴⠹⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠙⢄⠀⠀⠀⠃⡄⠐⠀⠀⠀⠱⠒⠋⠀⠀⠀⠀⠃⠈⡿⠶⢦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠁⠀⣀⣀⡠⠤⠲⡏
⠀⠀⠀⠀⠀⠳⠀⡠⠀⠀⠀⠀⠀⠀⡈⠋⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀⠀⠀⠘⠀⠀⠀⢀⠌⠀⠀⠀⢰⠀⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣰⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢇⡀⠀⠀⢀⠌⠀⠀⠀⠀⠸⡀⠀⠀⠁⠐⠂⠠⠤⠤⠤⠀⠛⡆⠒⠒⠂⠀⠉⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣤⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡛⠛⠛⢻⢦⠀⠸⡉⠛⠉⠈⠩⢇⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡌⠁⠘⠂⠈⣰⠀⢦⣈⠀⡀⠀⢠⣜⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⠛⠀⠆⠂⠓⠉⠰⣇⠀⠉⠉⠏⡀⠀⠀⡢⠄⠒⠒⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠐⠢⡁⢀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠁⠎⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠠⠤⠤⠤⠤⠤⠀⠀⠂⠀⠒⠂⡀⣉⠀⠀⠃⠂⠠⠤⠀⠀⠄⠀⠤⠄⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡬⠩⠖⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⢀⣀⣀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡠⠀⠂⠒⠉⠁⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠂⠀⠀⢀⣀⣀⡀⠀⠤⠤⣤⣤⣬⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠤⠄⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣴⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠈⠁⠐⠨⠄⠠⡂⢄⠠⢄⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⡄⠀⠀⠀⣀⣀⠠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⣍⠛⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⡁⠀⠀⢀⠄⠀⠑⢤⠈⠀⢠⠂⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠈⠄⡤⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣇⠀⠐⠄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠠⠀⠈⠂⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠘⠻⣦⡀⡰⠀⢀⠁⠀⠀⡀⠀⠀⠀⠄⠀⢀⢀⢫⠀⠀⠀⢀⡴⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠂⠀⠀⡼⢫⠁⠀⠀⢀⠁⠠⠁⠀⠀⠀⠈⠀⠁⠀⠈⣄⣠⠔⠋
⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠈⠀⢀⠀⠀⠀⠀⠀⠄⢀⠀⠀⡓⢀⡰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⢄⠀⠀⠀⠀⠀⢉⢠⡜⠁⠀⠀⠊⢙⠂⠂⠤⠄⣀⣀⠀⠀⡀⣀⣀⠤⠔⡖⠁
⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠀⠀⠸⠀⠁⠂⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣀⣀⡀⠀⢸⣤⡠⡤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠣⠀⠀⠛⠄⡐⠈⠐⠐⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⠢⠀⠤⠘⠠⢉⠁⠒⢂⠙⠂⠤⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⠂⡠⠐⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠃⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠐⠂⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠬⢦⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠉⠀⡕⠾⠛⢹⡞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠔⠊⠀⠰⠖⠋⢹⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⡀⠀⠀⢀⡤⡀⠀⡇⠀⠀⠀⠀⠶⠋⠙⣯⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⢉⡟⠒⢼⠀⢈⠀⠇⠀⠀⠀⠀⠀⠰⠟⢀⡽⠛⡟⠟⠉⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈⠉⠉⡘⠛⢺⣂⠆⢸⠦⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠋⡀⢠⠄⠂⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠛⠓⣏⡙⠀⠈⣄⣀⣈⣱⣤⠤⢤⣤⣀⣴⣉⣀⣄⠠⡴⢺⠝⠁
⠀⠀⠀⠀⠀⠀⢀⠔⢃⠀⠀⠀⠀⠀⠒⠒⢲⣋⡇⠀⠀⣏⣭⡠⠤⠤⠐⠉⠙⠒⠒⠐⠂⠀⠉
⠀⠀⠀⠀⠀⢠⡾⣄⡀⠡⠀⠀⠀⠀⠴⡶⠾⣽⡇
⠀⠀⠀⠀⠀⠙⠲⢤⡉⠺⢦⡀⠀⠀⠀⢴⠶⠳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⢽⢦⡀⣀⠀⠀⠞⠙⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢝⢦⣈⠀⠂⠤⢀⡀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢗⣲⠤⢤⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉



⡟⠙⣿⣻⣻⣛⣻⣟⣟⣛⣿⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠚⠓⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⣟⠀⣄⠀⣀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠙⠦⠙⢦⡜⠳⢤⡄⣀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠛⡁⠀⠀⢀⡶⠀⠈⠁⠀⠀⠀⠀⠀⣹⢀⡞
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠏⠁⢀⠀⠈⡆⠀⠈⠁⠀⣀⠄⠁⣀⣀⠀⣈⣵⣮⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⠀⡇⠀⠀⠀⠀⣠⣏⠩⠥⠶⣰⣿⣿⣷⣮⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡟⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣟⣿⡻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠂⠤⡀⠬⠁⢀⠀⠀⠀⠀⣿⠛⣻⣉⣩⠻⠋⢹⣿⣿⣿⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠲⢤⣀⣑⢦⡀⢀⡫⢤⣙⠟⣉⣧⣴⣆⣽⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠦⣄⡀⠀⠀⠀⠉⠁⡄⢀⣠⣾⣧⣼⣶⣿⣯⣵⣾⣿⣿⣯⢻⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⠙⠛⠒⠒⠋⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠞⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠘⠁⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢤⣤⣤⢤⣶⣖⣯⣠⡆⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠰⠶⠶⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣠⠞⢋⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣇⡟⠿⢿⡟⡿⢿⠿⡿
⠀⠀⠀⢠⡞⠁⢀⣸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣛⡉⣠⠶⠋⠀⠀⣀⡤⠁
⠀⠀⠀⠘⠚⠋⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠉⠁⠈⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡏⡀
⠀⠀⠀⠀⠀⠀⢠⡗⠀⢠⣬⡷⢥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣍⣏⣶⡼⣶⣯⣖⣂⠀⠀⠀⠀⠀⡀⡀⠀⠄⠀⠀⢆⡀⠀⣀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣻⠷⠂⢰⣗
⠀⠀⠀⠀⢀⣉⣻⣿⣮⣿⡽⣷⣟⠿⡎⢁⣀⠐⠒⠚⢋⣉⣀⡀⢄⠀⠈⣹⡭⠀⠀⠨⠧⢄⠀⠀⠀⣀⡀⠀⢀⣨⣽⣯⡻⣷⣿⣷⣴⡇
⠀⠀⠀⠀⣉⡫⢞⣿⣿⣿⣹⣿⣿⣟⡛⠀⠀⠈⠀⠈⠁⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⢬⣿⣯⣿⣿⣿⣿⣿⣿⡟⢃⡀
⠀⠀⠐⠺⢟⣳⣭⣧⣾⣿⢻⡟⣿⡶⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⡀⣀⠄⠙⠒⠀⢀⣰⠶⠞⠛⠋⠈⠀⠀⠢⣤⣴⣇⣳⣽⣟⣻⡟⡻⣷⣽⡇
⠀⠀⠀⠀⠀⢀⣺⡇⢛⠿⢤⡇⠙⢿⠣⣀⣶⣤⣀⣀⣬⣭⡬⠿⠛⠃⠀⠀⠀⠈⠉⠀⢀⠀⠀⡀⠉⢀⠈⠲⢦⣤⡉⣿⣙⣟⣻⣿⡷⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣼⡷⠆⠀⠐⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠖⠲⠒⠒⠲⠶⣿⢭⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠤⣿⡏⣿⢁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠶⠿⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹⠩⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠈

⠀⠀⣀
⠀⡀⣿⢠⠀⢀⢠⣇⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⡇⣾⣼⠆⢸⣸⣿⠀⣦⣿⠀⠀⠀⠀⢀⠀⠀⠁
⠈⠻⢿⣿⡇⢸⣿⣿⢀⣿⣿⠀⣧⠀⣆⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⡄⡄
⠀⠀⠀⠀⠿⠙⠻⠿⢼⣿⣿⢴⣿⢀⣿⣸⡇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠘⠻⠗⠃⠃⠆
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠺⢷⣿⢠⡄⠀⠀⠀⠀⠀⠀⠁⠈⠈
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡂⠀⠀⠀⠀⢸⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⣿⠿⣄⡀⠀⠀⢸⠀⣧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠁⢰⣣⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⣀⠀⠀⠀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⠀⠀⠰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⡄⠀⢀⢂⡀⠸⠀⠀⠐
⠀⡇⡴⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⠉⢠⠠⠇⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⣨⡅⢀⠉⠅⢰⢢⡑
⠀⣅⢁⡤⡆⠌⠁⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⡆⡄⢖⠀⠉⡎⡞⣌⣟⡍⠅⠋
⠰⠏⠨⡿⡶⢶⢾⠃⡶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡄⠀⣀⣀⣟⣀⣯⣽⣶⣼⣟⣹⡷⠤⠸⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡛⠛⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠆⠀⠀⣀⣠⣀⣀⣀⠀⠀⠀⠉⠛⠛⠻⢧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠂⠀⣤⡀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⢀⡀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠾⠿⠛⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢉⣤⠖⠂⣀⡀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠶⠶⠖⠀⠀⠀⠲⣶⡶⠈⠀⠀⠀⠀⠉⠲⣆⡈⠙⠶⢀⡀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠎⠉⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠄⠀⠈⠙⠦⣀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠁⠀⢀⣀⠀⠀⠀⢀⡀⠴⠶⠟⠀⠀⠀⣀⡀⢶⣦⡀⠀⢀⠀⠿⣆⡀⠀⠀⠈⠁⠀⠘⡿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⣠⠶⠈⠉⠁⣀⠀⠀⠈⠉⠐⢶⣦⠉⠁⠀⣶⠀⢀⡠⠶⠉⠠⣄⡈⠉⠙⠛⠷⣦⠈⠣⡄⠀⠀⠀⠀⠐⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣰⠞⠁⣠⣴⠆⠈⠉⠙⠿⠟⠀⢦⡀⠉⠢⣀⠀⢰⡶⠈⠁⣠⣾⠷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠰⢆⠀⠀⠀⠀⠘⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⢀⡘⠁⢠⡄⠛⠃⢠⡞⠉⢀⣀⣀⣀⠀⠈⠣⡀⠉⣠⠀⠀⠀⠀⠛⠃⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⢃⡀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡆⠀⠸⠃⢀⡄⠀⠀⠀⠈⠁⣰⡿⠿⡿⠿⣷⣦⠀⠋⠀⠸⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠸⢿⡀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⠀⢸⡇⠀⠀⠀⠀⠀⠏⠀⡄⠀⣀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⠟⠋⠉⠁⠀⣰⣶⡟⢿⡆⠀⠀⠠⡇⠀⢃⡀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠈⠁⠀⠀⠆⠀⠀⠀⠠⣇⡈⠻⣿⡿⢿⡟⠀⢀⠀⠀⠀⠀⠠⠀⠀⢠⠈⢿⣿⣿⠇⣸⡇⠀⠀⠀⠃⠀⠈⢳⡤⠙⠋⠙⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡀⠘⢷⣆⠀⠀⠀⠀⠀⠘⠛⠀⠈⠁⡦⠁⢀⠛⠀⠀⠀⠀⠀⠀⠀⠛⠓⢠⣤⣄⠘⠋⠀⠀⠀⡀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡜⠛⠻⣷⣄⠀⠙⠃⣤⣄⠀⠀⠀⣠⣤⠐⠁⢀⣤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡄⠋⠀⣀⡞⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢀⡈⠀⠈⠙⠂⣀⣀⠙⠿⣦⣤⠀⠀⠀⠀⠀⠀⢀⣤⡴⠛⠋⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⡦⠀⠀⢀⠟
⠀⠀⠀⠀⣠⠞⠛⠓⠀⣀⠀⠀⠀⠙⢣⡀⠘⠏⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟
⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠻⢧⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡀⠐⢶⣿⣇⣀⣴⠃⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢀⣤⡴⠟⠁
⠀⠀⠀⢾⠁⠀⠀⠒⢀⣤⡀⠀⠀⢠⡶⠂⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⢀⣀⡀⠘⠒⠀⠀⢀⡹⠃⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠈⠁
⠀⠀⠀⠀⣶⣄⡀⠀⠀⠉⠳⢄⣀⣼⣧⣤⠾⠉⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠛⠁⢀⣀⡀⠶⠋⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⣀⡀⠀⢀⠿⣦⣀⠈⠉⠹⠏⠁⠀⠀⠀⠀⠉⠷⢦⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⣀⡴⠾⠋⠁
⠀⠀⠀⠀⣉⠓⠲⠿⠆⠉⠉⣀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠶⢄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠶⠊⠉⠁
⠀⠀⠀⢀⡟⠀⠀⠀⢀⡶⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠰⣶⠖⠒⠀⠈⠉⢻⠃
⠀⠀⠀⡿⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡶⠆⣀⠀⠀⠀⠀⠀⢿⣶⡶⠦⣀⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠶⢶⣾⠋⠀⠿⣀⠀⠿⠀⠀⠀⠀⡀⠼⠋⠀⠀⠉⣿⣇⡀
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠀⠉⠉⠀⠀⠀⠛⡀⠀⠀⠙⠿⠀⠀⢠⡴⠈⠉⠀⠀⠀⢠⠚⠁⠉⠉⠲⣦
⡀⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣴⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠙⠷⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠀⠙⢃⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣧⠀⣾⠀⠀⠀⠀⠺⠀⠀⢀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠙⣶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯
⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠙⠳⣤
⠀⠈⢷⣤⡀⢀⣤⣤⣤⡤⠔⠛⠛⠛⠻⠇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠘⠛⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⡘⠋⠀⠀⢀⡴⠛⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡼⠟⢀⣀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡴⠋⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⡀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃⢠⠞⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡀⠀⠀⣀⣀⠀⠀⠲⠶⠖⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠲⠶⣶⣄⠘⢧⡴⠃⠀⠀⠀⠀⢀⣀⠴⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⡊⠉⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⣏⠰⠎⠀⠀⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⠶⠶⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠰⢿⡟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠶⠂⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣶⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠺⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⢠⣿⠀⠀⣶⠀⠀⢀⣀⣠⣄⡀⠲⠾⢿⡟⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠈⠉⠁⠀⠙⡿⠁⠀⠁⠀⠀⠛⠓⠺⣿⠁⠉⠁⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⣇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⣿⠟⠁⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢾⣿⡷⢠⣤⠄⠈⠋⠀⣿⠀⠀⠀⠀⠀⠀⣤⡴⣿⡦⠤⠄⠀⠈⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠀⠀⢀⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢈⣤⠀⠀⠀⠀⢀⠘⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢘⣇⠀⢀⣤⡞⠛⢷⠀⠀⠀⠀⣴⠛⠛⠁⠀⠀⠀⠛⢃⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠻⠧⣤⣤⠀⠀⣘⠛⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⣠⣼⣇⡀⢠⣄⣴⣶⡀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⠉⠉⠉⠙⠓⠀⡀⠀⠀⠻⠿⠶⠟⠛⠛⠻⣿⠁⠀⠀⠀⠀⠀⠉⣿⠟⠓⠀⠀⠀⠀⠙⠃⢀⠙⠋⠁⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⣿⡄⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠒⠾⠛⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⣀⣀⣀⣤⣄⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠲⠖⠘⠛
⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠴⠶⠶⠖⠈⠉⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣀⡀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣀⣠⡿
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠆⠀⣀⣀⣀⣀⣀⣀⣀⡀⠰⠶⠖⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠂⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠂⠀⠈⠀⠠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠠⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⢹⠄⠤⢀⠄⢳⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠿⠄⠤⠀⠀⠓⢃⠀⣸⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠈⢆⠐⠂⠘⠚⢤⣤⠠⠤⠔⠁⠘⣧⠯⠀⢹⣯⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢺⣧⠘⣄⡀⠀⠀⣠⡿⠀⠀⡘⣿⢠
⠀⠀⠀⠀⠀⢀⣴⠻⣷⣄⠈⠻⢷⡘⣿⡼⢋⣭⡆⠀⢰⡇⣿⡄
⠀⠀⠀⠀⠀⣿⡹⣧⡙⣿⡟⠀⠀⠉⠀⠑⠿⣿⡇⠀⢼⣇⢸⡇⠆
⠀⠀⠀⠀⠀⢿⣱⡜⣷⡌⢿⣷⣄⠀⠀⠀⠀⠀⠁⠀⠘⢿⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄
⠀⠀⠀⠀⠀⠈⢿⣽⣮⢿⣆⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡈⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠻⣾⠷⣻⣧⡙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠆⠀⠀⠀⠈⠀⠀⠈⠀⠀⠒⠀⡖⠠⠠⢴⠐⠂⠈⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡟⣝⢷⣌⢻⡄⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⣀⡠⠔⠚⠉⠑⠤⣤⣤⣤⠼⢦⡀⠈⣤⣶⣆⣠⣞⣁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣻⣿⣦⠻⡀⠀⢀⡾⠁⠀⠀⢀⣤⣴⣮⣥⣤⣤⣤⣄⣀⡀⠀⠀⠀⠘⠛⠿⠀⣷⣦⣤⡜⣻⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣿⣷⣷⣠⣾⡿⢷⡶⢿⣿⣿⣿⣯⣭⣽⣿⣛⣛⡿⠿⠷⣶⣤⣬⣽⡛⢾⣿⠠⠛⠉⠉⠙⠒⠤⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠿⠾⠷⡾⣧⡼⠿⢼⣿⣸⣿⢸⣷⠉⣾⡏⣻⣟⣛⠷⣶⣶⠙⡿⠃⠈⠀⠀⠀⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡄⢰⡶⠂⠀⢰⡿⡵⢞⡪⠓⠋⠁⠀⠀⠀⠀⠀⡀⢉⠉⠓⠛⠣⠿⣇⣽⡏⡸⠟⠁⠀⠀⠀⠠⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⣸⡥⠊⡡⢤⠇⣢⡀⢀⣰⠀⠠⠙⠙⠉⠑⠀⠀⠀⠀⣸⡟⠉⠀⠀⢀⡠⢀⡥⠚⠁⠀⠀⡄⠀⠆
⠀⠀⠀⠀⢀⡸⠀⢀⡀⢶⠸⠟⠏⡾⢠⢗⡀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢌⠟⠀⠀⡀⢔⡡⠞⠁⠀⠀⠀⠀⠀⠊⠀⠁⠶⡄
⠀⠀⠀⢆⠈⠀⠙⠨⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⡶⣀⠐⣞⢀⡠⢊⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠛⠳⡌⠛⠛⠹⢸⡿⣫⣞⢡⡄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣩⠞⠋⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁

⣦⣤⣤⣦⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣀⣀⢀⡠⠉⠀⠸⠏⠀⠀⡀⠀⠀⠀⢀⡨⠕⠢⢦⠀⠀⠀⠀⠸⠀⠀⠻⠇⠀⠀⣂⣀⣀⣀⣀⣀
⠀⢠⣶⣿⡿⠿⠿⢿⠁⢀⡀⠀⠀⡄⠀⠀⠡⣀⡇⢀⡜⠁⠀⠀⠈⢧⡀⠂⢀⠀⠀⠀⠐⠀⠀⢀⡁⠘⡿⠿⠿⢿⣿⣦⡀
⢠⣿⡟⠁⠀⠀⠀⡘⡓⠀⠑⠄⢀⢣⡄⠤⣤⠗⠋⠉⠀⠀⠀⠀⠀⠀⠉⠛⠣⡤⠤⢴⡌⢀⠔⠁⠀⢞⠁⠀⠀⠀⠈⢿⣿
⢸⣿⡇⠀⠀⠠⠂⠉⢹⡀⠀⠀⡠⠀⠀⠠⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠅⠀⠀⢀⡏⠉⢰⠀⠀⠀⢸⣿
⠸⣿⣧⡀⠀⠀⢠⠖⠉⠁⠀⠊⠀⠀⣠⣤⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠒⠂⠈⠁⢲⡁⠀⠀⢀⣾⣿
⠀⠙⢿⣿⣷⣶⠁⠀⠀⢠⣠⠆⢠⡞⣱⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡌⢷⠄⠢⣀⡀⠀⠈⠄⣶⣾⣿⠿⠃
⠀⠀⠀⠀⠉⠉⠤⠂⠀⠀⠈⠀⢸⡇⢿⣿⣿⣻⣿⢿⣀⠤⠤⠠⠤⣀⡿⣿⣿⣿⣿⡿⢸⡇⠀⠀⠀⠀⠁⡤⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠘⠻⢿⣿⡧⠋⠐⣇⡀⢀⣽⠈⠑⣽⢿⡿⠿⠃⠚⠃⢠
⠀⠀⠀⠀⠀⠀⢂⢀⡀⠠⠆⠀⠀⠀⠀⠀⠀⠰⡇⠒⠦⡴⢿⠿⢤⠴⠂⢸⠀⠀⠀⠀⠀⠀⠀⡱⠄⢀⡀⡀⠁
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢫⣀⣀⡝⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠐
⠀⠀⠀⠠⠀⠀⠅⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠈⠓⠦⠶⡺⠁⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢠⢠⡤⠤⠤⠴⠦⢄⡀⡀⠀⠀⠈⠀⠀⠀⣠⠼⠦⠦⠤⢤⣄⡘
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⢀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⠄⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠉⣤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡷⠆⠀⣿⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⠃
⠀⠀⠰⠄⠀⠈⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⡀⠑⣾⠅⠀⠀⠸⣧⠖⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠈⠀⠀⠢
⠀⠂⠀⠀⣀⣤⡴⢶⣶⠶⣤⣄⡙⠲⠤⣤⣤⣤⠬⠿⠋⠀⠀⠀⠀⠉⠶⢯⣤⣤⣤⢤⠖⠋⣠⡤⢶⣶⡶⢦⣄⡀⠈
⠈⠀⢠⣶⣍⣿⠧⣾⡟⠑⣿⠻⣷⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢻⣾⠈⣻⣦⠾⣯⣙⣦⡀⠈
⠀⠀⣾⡟⠉⣧⣀⣿⠷⠞⠛⠓⠿⣿⡀⠀⠀⠀⢀⠤⠄⡔⠁⠀⢢⠤⠤⡀⠀⠀⠀⢠⣿⠟⠚⠛⠶⠿⣧⣀⣾⠉⣽⣧
⠂⠠⣿⡟⢾⠟⠁⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠁⠀⠀⢀⡀⢠⠀⠀⠀⠈⠀⠀⠀⣶⠃⠀⠀⠀⠀⠀⠀⠈⠻⡗⢻⣻⠄
⠀⠀⠸⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠂⠀⠁⢠⠂⠀⠀⠀⠀⠀⡄⠈⠁⠐⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠏
⠂⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠂⠐⢀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣤⣀⣤⠾⠋⠐⠀⠀⠁⠀⠘⠲⠤⠤⠔⠃⠐⠊⠀⠀⠊⠙⠦⣄⣀⣠⡤⠖⠋⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠊⠉⡆⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠵⡒⠁⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢦⣀⡼⠀⠀⠸⡯⠀⠁⠀⠀⢀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠓⠀⠀⠠⠄⠀⠈⠛⢇⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⣼⠁⠀⠁⠀⠀⢸⠀⠀⠘⡏⡇⠀⢀⣤⡈⠀⠑⠀⠀⠀⡀⠀⠀⠀⢱⠀⣸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠘⠻⢤⡀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠀⢀⠊⠈⠁⠀⠀⠃⠀⠀⠙⠿⠿⠿⠋⠀⠛⠉
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡖⠀⠀⡀⠀⠹⡆⠀⠀⡇⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⠇⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢣⠀⠀⠅⠀⣠⠇⠀⠀⡇⠀⠀⠀⠀⠀⣸⠘⣦⠀⠀⠀⠀⡄⠀⠈⢩⠉⠉⡇⠀⠀⢠⡮⡀⠀⠀⢘
⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⠈⠟⠛⠟⠛⠏⠀⠀⠀⡇⠀⠀⠀⠀⠉⢻⠀⠉⠻⠉⠉⠋⠁⠀⠀⢸⠀⣗⡇⠀⡖⠉⠀⠀⢀
⠀⠀⣠⣤⣿⣀⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣶⣶⣶⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣷⣧⣤⣷⣤⣤⣤⣼⣤⣬⣤⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠶⢲⡆⠀⠘⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣠⣤⣀⣤⠴⠦⣄⡀⠀⠀⠀⠀⠀⢁⣴⠋⡀⠀⣴⣏⣤⡄⠀⣴⣋⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠙⠏⠁⠚⠉⠀⠀⠀⢸⡇⣀⡀⠀⠀⠀⠛⠛⠋⠁⠀⠉⠉⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣠⠶⠶⣄⠀⠀⠀⠀⣴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠘⣧⣀⡀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠙⢳⡄⠀⠀⣠⡤⠴⠶⠶⢦
⠀⠀⠀⠀⠀⣽⠆⠀⠀⠈⢮⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⡶⢿⣤⣤⣤⡄⠀⠀⠀⠘⢿⣦⣼⠃⠀⠀⠀⠀⢘⡷⠶⠤⣤⣀
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⢸⡷⠶⠶⠞⠋⠁⢀⠀⠤⣄⡀⠈⠙⢦⡤⠀⠀⠀⣼⠃⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠈⢷⣶⠒⠒⠦⢤⣀
⠀⢀⣠⣾⣅⠀⠀⠀⠀⠀⠀⠀⢀⣙⣷⣄⣾⠁⢠⠶⢦⣄⠀⢀⣀⣠⡾⠁⠀⠀⠀⢻⡄⢀⣴⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠸⡏⠳⡄⠀⠀⠈⠉⠓⠦⣄
⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡇⠀⣿⠀⠀⣹⡦⠈⠉⠁⠀⠀⢢⡶⠃⠘⣷⡾⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⡄⠀⠀⠀⠀⠀⠈
⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠛⠶⠚⢿⠃⠀⠀⣰⠀⠀⢸⡇⠀⢰⡿⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢷
⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠘⠦⠤⠟⠁⠀⠀⣸⠇⢀⡾⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠸⡆
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣷⠀⠀⠀⢀⣀⡠⠋⠀⣾⠃⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⡉⠂⠛⠋⠉⠀⠀⣸⠏⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠋⡳⣶⣶⣤⣤⣴⡿⠀⠀⠀⠀⠀⢀⡍
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⣴⠾⠛⣃⠀⠉⠙⠛⣿⠃⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⣠⡶⠟⣉⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠛⢷⣿⣋⣴⠿⣫⣾⡿⠛⢾⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠦⣤⣄⣀⣀⣀⣈⣿⣿⡿⠿⠛⠁⠀⠀⢸⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠓⢄⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣯⠀⠀⠀⠀⢀⠜⢧⠀⠀⠈⠀⠀⠀⠀⠠⠁⢠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠈⠙⢦⠀⢧⠉⠀⠒⠈⠁⠀⢸⡇⠂⠀⠀⠖⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠰⣄⣀⣀⣨⡇⠘⣆⠀⠀⠀⠀⠀⡜⠀⠔⠀⢀⠀⠀⢀⠂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠚⠉⠉⠉⠉⠉⠙⠒⢼⣄⠀⠀⠀⢰⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⠤⠄⢀⠤⠒⠉⠉⠉⠓⢄⠙⢦⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⠴⠒⡟⠓⠢⣄
⠀⠀⠀⠀⠀⣠⠋⠀⠀⢁⠤⢴⡁⠀⠀⠀⠀⠀⠀⠈⣇⠈⣧⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢰⠇⠀⠘⡀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⢠⠃⢀⡀⢰⠃⠀⣸⠃⠀⠀⠀⠀⠀⢱⠀⢸⠀⢸⡖⠉⠉⠉⠳⢤⣀⣀⣀⣠⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢣⠀⠀⠀⠀⢳
⠀⠀⠀⡀⡾⠀⢎⠀⠘⠦⠖⠁⠀⠀⠀⠀⠀⠀⠈⡆⢸⠃⠈⡇⠀⠀⠀⠀⠀⠙⣤⠀⢸⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⢣⡀⠀⠀⠈
⣠⠞⠉⠉⣇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⠀⢸⠇⠀⠀⠀⠀⢀⠔⠁⢠⠟⠊⠉⠉⠉⠙⠓⠦⣄⢀⠞⠁⠀⠀⠀⠀⠀⠑⢤⣀⣰
⡏⠀⠈⠓⢾⡀⠀⠀⠀⣇⠀⠀⢠⡀⠀⠀⠀⣠⠞⣰⠇⢀⡞⠀⡀⠤⠤⢀⠣⣄⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠏
⢻⡀⠀⠀⠀⢳⡀⠀⠀⠘⢦⡀⠀⠉⠓⠒⠋⢁⡴⠃⣠⠞⡔⠁⠀⠀⡀⠁⡼⢲⠄⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⣠⠖⠋
⠀⠙⠲⠤⠶⠋⠳⣄⡀⠀⠀⠉⠒⠦⠤⠖⢚⣡⡴⠛⠁⡘⠀⠀⠀⡔⢀⠎⡡⠀⡀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠈⣧⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠦⢤⡶⠖⠚⠋⠁⠀⠀⢠⠁⠀⠀⠌⡠⡡⠒⣁⢔⠅⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠣⣀⠀⠀⠀⠀⠀⠘⡀⢀⠎⠔⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠈⠙⢲⠒⢤⡴⠀⠓⢁⡜⣰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⢀⣴⠋⠀⣀⡀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠈⠉⠢⡀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⣰⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⣄⠀⠀⠀⠀⠀⢀⡤⠊⢁⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠙⠓⠶⡶⠾⠷⠞⠛⠉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠐⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⢰⡇⠚⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠒⠒⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⡾⢣⡄⠀⠀⢰⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⣠⠞⠀⠀⢠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⣸⣀⡴⠞⠁⠀⠀⠀⣀⠇⠀⠺⣀⡈⠀⢨⠁⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⢄⣀⣤⡤⠖⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣠⣴⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢙⠶⠿⠋⠉⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣽⣏⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠈⠙⢦⡄⠀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⢀⣼⣷⠀⠀⠀⠀⠀⠀⠈⠈⢶⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⢠⣿⡿⠛⠀⠀⠀⠀⠀⣀⣀⡜⣧⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡟⢿⠀⣾⣿⣿⠠⠀⠀⠀⠀⢲⣼⠋⢀⣬⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢸⠀⣿⣿⣿⣤⠀⠀⠀⠀⠈⠹⣆⣿⡿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⠀⠀⠀⣿⣿⣿⠶⠀⠀⠀⠀⠀⢋⣤⣾⣿⣿⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣿⣼⠏⠀⠀⠀⣿⢸⡏⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣳⡀⠀⢸⡿⠋⠀⠀⠀⠀⡏⣼⠀⠀⠀⠈⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡍⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⣿⣿⣯⣠⣿⡿⣤⢀⠀⠀⢘⢇⡇⠠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⠿⣿⣿⣋⡭⠖⠲⠏⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢿⣿⣷⣿⡿⠛⠉⠩⠁⠀⣀⢢⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠿⠟⠻⠁⠀⠉⠀⠂⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠾⠿⠛⠛⠛⠿⠿⠷⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠋⠁⠀⠀⠠⠐⠆⠀⠀⠀⠠⠒⠄⠀⠉⠛⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠛⡙⠂⠀⠀⠀⠈⠀⠀⢀⡀⠀⠉⠀⢀⠔⢆⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠤⠀⠀⠀⠀⠔⣱⠆⠀⠀⠀⠠⠄⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠐⠀⢀⣀⣠⣤⣤⣤⣶⣶⡾⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⣿⣿⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⣀⣤⣴⡶⠾⠿⠛⠛⠉⠉⠁⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡈⠉⠛⠻⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠛⠋⠉⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣶⣤⣄⡉⠻⣷⡄
⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠈⠉⠻⢷⣌⣿⡆
⠀⠀⢀⣴⡿⠋⣁⣤⡶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⡿
⠀⢠⡿⠋⣠⡾⠟⠉⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣾⠇
⢠⣿⣡⡾⠋⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠀⠀⣴⡿⠃
⠸⣿⡟⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠂⢻⣏⣀
⠀⠹⣷⣄⡴⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⠀⣼⣿⣿⣧
⠀⣴⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⠉⠈⠙⠻⠟⠁
⠀⠘⠛⠛⠛⠓⢛⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢶⡀⠀⠀⠀⠀⣿⣿⣿⠟⠓⣄
⠀⠀⠀⠀⢠⡶⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣼⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⡟⠀⠀⠀⠀⣿⣿⣇⣀⣾⣿⣷
⠀⠀⢀⣶⣿⣧⠀⢹⡟⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢧⣀⣠⠇⠀⠀⠀⢀⣿⠃⠀⠙⠿⣿⡿
⠀⠀⠸⣿⣿⡿⠛⠋⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣤⡀⠀⠀⠀⠀⣠⣤⠞⠋⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠔⠛⠉⠛⠛⠿⢷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠿⠟⠻⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⡁⠀⠀⠀⠀⠀⠀⣤⣤⣀⡀⠉⠉⠉⢉⣹⡟⠉⠹⣧⣤⣤⣾⡇⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠁⠀⣸⡇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠐⠀⠐⠀⠀⠀⢻⣧⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⢀⣴⠞⠛⠉⠙⠛⢺⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡁⠉⢷⡄
⠀⠀⠀⠀⠀⠀⣰⠟⠀⣠⡾⠁⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⢀⡀⠀⡀⠀⠘⠀⠀⠀⢹⣟⠱⣦⠀⠙⠶⣄⣀
⠀⠀⠀⠀⣀⡶⠏⢀⣴⠟⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠠⠤⠠⠀⠤⠀⠤⠤⡄⠀⠀⢀⡄⠀⠀⠀⠈⣿⡄⠈⠛⠶⣄⣀⠉⠛⠦⣄⣀
⠀⠀⣴⠟⠉⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠈⠉⠳⢦⣤⡈⠉⠉⠛⠷⣦⡀
⣀⡼⠃⠀⡀⠀⢠⡷⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⡀⠂⠀⠀⠀⡁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠛⠶
⢻⣤⣤⡾⠻⣦⣼⠃⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⢠⡾⠿⣶⣶
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⠂⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠙⠛⠻⢿⣆⠀⠀⠀⠀⠀⢀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣧⣀⡀⠂⠠⠈⣿⣧⠀⠀⠠⠀⢀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠻⢷⣦⣤⣽⣿⣷⣀⣠⣾⠟⠋⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⢀⣼⠏⠙⠋⠀⠻⠟⠛⢷⡄⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠉⠁⢀⣀⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⡋⠉⠁⢀⣠⣴⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣶⣾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣶⣤⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⡀
⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡁⢀⠀⣁⣠⢀⡅⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠤⠀⠤⠤⠤⠤⠜⡀⡀
⠀⠌⠀⡄⢠⠆⢀⡀⢠⠀⠀⢹⠁⠀⠉⠈⠀⠀⠀⠀⠈⡇⠀⠆⠰⠀⠠⠤⠄⠂⠁⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁
⠀⠉⠀⠁⠈⡏⢈⠀⠠⡀⠀⣸⠀⣀⠀⠀⡀⠀⠀⠤⠀⠇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢀⡀⠠⣈⠃⠀⠀⡐⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⢀⠀⠀⠠⠀⠂⣀⠐
⠀⠀⠀⠀⠠⠆⠀⠀⠀⢀⡀⠀⢁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠒⠂⡑⡄⡅⠀⠀⠀⠐⠠⡤⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⣁⡒⠀⠠⠀⠒⠈⣀⠀⠐⠀⠀⠀⠀⠈⠀⠐⠄⠀⠀⠠⠄⠀⠆⡄⢀⠀⡆⠀⠑⠁⠀⠀⠀⢎⠀⠀⠀⣀⠘⠀⢀⡂⠶⡅
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠐⡱⠦⠤⠔⠊⠀⠀⠀⣇⢀⠙⣠⠀⢀⠠⠦⠀⢰⠀⠭⠃⢠⠠⡄⠀⠁⠀⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠔⠁⠀⠀⠐⠀⠀⠀⠀⠘⢦⡀⠈⠁⠉⠁⢂⡠⠬⠀⠀⠉⠀⠋⠀⠠⡀⠀⠀⠐⠈⠐⠉⠐
⠀⠂⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠂⠰⠀⠀⠀⠀⠀⠀⠉⡁⠒⠒⠊⠉⠀⢠⠀⠀⢰⠀⠑⠀⠠⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠔⠀⠈⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⡊⠐⠀⠄⠀⠘⠀⠑⠁⠦⠀⠀⢀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠂⠂⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⣤⣄⡀⢀⣀⣀⡀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⠈⠉⠉⠁⠁⠀⠀⠀⠈⠁⠀⠀⠀⡀⢸⣿⠀⠀⢸⠟⠉⢻⡇⣿⡇⢹⡇⠀⠀⠰⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠍⠀⠁
⠀⠀⠀⠀⢨⠈⡅⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠃⢸⣿⠀⠀⢸⣆⣀⣸⡇⢸⣧⣾⠇⠀⠀⢀⣀⣀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠆⠺⠀⠆⡀⠠⡄⠀⠀⠀⠀⠱⡀⢀⠔⠀⢸⣿⠀⠀⢸⡇⠀⢸⡇⢸⡏⣹⡇⠀⠀⢸⠉⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠘⠃⠀⠘⠁⠘⠛⠛⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠂
⢀⠀⠐⠉⠀⠀⠀⡆⠀⠂⠀⠀⠀⢠⢀⠀⠀⣀⠀⠀⢸⡛⡟⣻⠻⢿⠛⢟⣿⢻⡇⠀⠀⠀⡄⠀⠆⠐⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠈
⠀⠉⠀⠀⠒⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣷⣷⢾⡾⣾⡷⣷⡿⡾⡇⠀⠀⠀⠁⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠑⠀⠀⢸⠀⠠⠀⠀⠀⣀⠀⠀⠠⠀⠀⠐⢀⠀⠌⠐⢆⡃⠂⠀⠡⠠⢀⡐⠀⠀⠀⠀⠀⠠⡀⠠⠂
⠀⢀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠀⢀⣀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠵⠀⠀⠀⠀⠊⢔⠀⠀⠀⠀⠀⠙⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠀⠀⠁⠀⠂⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠂
⠀⠀⢰⡆⢰⡆⠀⠀⢠⡾⢷⡄⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠐⠁⠊⠁⡐⠀⠀⠀⠉
⠀⠀⢸⣧⣼⡇⡶⣦⠸⣧⢸⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠈⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣿⢸⡇⢀⡿⠀⣿⣸⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⡄⣡⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠈⠛⠘⠁⠙⠛⠀⠙⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠒⠒⠂⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢂⢀⣀⡈⠙⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠘⠁⠶⠾⣿⣿⣿⡆⠀⢠⣿⣿⣿⡿⠿⠓⠈⠉⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⣀⠈⠛⠛⠁⠒⠀⠙⠛⠛⠀⣠⣴⣾⣿⣿⣷⡄⠀⠀⣠⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⡀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡇⣠⣞⣩⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⠻⠿⠿⠿⠿⠿⠿⠛⠟⠛⠛⠛⠛⠀⠐⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣀⣀⣀⣀⡀⠠⡆⢀⣀⣀⣀⣀⡀⠀⠀⣿⠀⣀⣀⣀⡀⠀⠀⢠⢀⠀⣀⣄⣀⣀⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡍⠛⠉⢿⣟⣛⣋⠉⠉⠙⠛⠛⠛⠉⠉⠉⣿⡍⠙⠛⠚⠛⠯⡽⠟⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣯⠛⠻⠶⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠀⢷⡐⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡏⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣽⣦⠀⠀⠀⠀⠀⠈⠃⡀⠀⠀⠀⠀⡄⠙⢃⠀⠀⠀⠀⢠⠀⠋⡀
⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣦⡌⠀⠀⠀⠀⠀⢰⢠

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⣰⣶⣀⣀⠀⠀⠀⠀⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣀⠀⣴⢶⣠⣰⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⣾⣿⡗⣸⣿⣿⡏⠀⡯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⣻⣿⠿⣭⣿⣟⠀⣼⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⣿⣷⣿⣽⣿⡏⣿⣿⠃⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣿⣿⡿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⡿⠳⠆⢻⣆⠀⠀⠀⠀⠀⠀⡐⢡⠘⢶⣤⣹⡇
⠀⠀⠀⠀⠀⢸⣧⠀⠀⣠⣾⢲⠀⠀⠀⠀⠀⢠⢙⣿⣿⣿⣷⣀⠀⠁⠀⠀⠀⠀⡠⠊⢀⠏⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣴⣾⣿⠇⠀⠻⣄⠀⠀⢀⠆⢸⣿⣿⠏⠻⣿⣤⠀⠀⠀⢀⢎⡔⠀⠀⠀⣆⠀⢺⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠐⢄⠈⢳⣦⡈⠀⣼⣿⡟⠀⠀⣿⣏⠠⣠⠴⢩⡞⠀⠀⠀⢸⣿⣷⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⢷⣶⡤⠙⢦⣻⠳⣼⣿⡿⠁⠀⠀⣿⣗⡘⠁⠠⠋⠀⠁⠀⠀⡼⠋⣱⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣶⣍⢻⣿⣿⠿⠶⢿⣟⠁⠐⠀⠈⠩⣿⣇⠀⠀⣠⡾⠓⢋⡥⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⠶⣊⡙⠛⠿⣿⣿⣿⣿⣿⣶⣤⣤⠚⠉⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣿⣿⣿⣿⣶⡈⢿⣿⣷⣦⣤⣤⣬⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⢠⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⣄⡀⠁⠀⠀⠀⠀⠠⠀⠄⠀⠄⠀⠀⠀⠀⠀⠤⠤⠄⠐⠢⠀⠀⠡⢀⡄⠓⠄
⠀⠀⠀⠀⠀⢀⠄⢀⡀⠠⡆⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⡥⠤⠄⠀⡄
⠀⠀⠀⠀⠰⣅⡀⠀⠀⠀⠐⢄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⡠⠊⠀⠀⢠⠠⠬
⠀⠀⠀⢠⠊⠀⠀⠀⠀⣀⡤⠁⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠒⢀⢂⠙⠢⠄⠀⠀⠀⠀⢄
⠀⠀⠀⠀⢑⠒⠀⠀⢠⠎⠀⢀⣾⣿⡄⣦⢧⠀⠀⠀⠀⠀⠦⠀⠎⠂⠀⠀⣐⠥⣀⠀⠀⠀⠀⠀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⣠⣿⣿⣿⡇⠈⠇⠂⠀⠀⢀⠀⠅⠁⠀⢐⡁⣠⠄⠀⠤⠄⢀⡀⠀⢀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠄⣈⣘⢦⡀⠘⡌⠂⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⢀⣀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

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




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⢦⡴⠖⠚⠋⠉⠉⢉⣉⣉⣭⡽⡱⠊⢷⠤⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣿⠀⠀⡹⢤⡶⠖⣶⡿⢻⣶⠶⠊⣼⡁⠀⠈⣧⠀⠀⠘⢿⣿⡉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⣿⠀⠀⠀⡸⠁⢸⡟⠀⣾⠁⠀⠀⠀⠁⠀⠀⢸⡄⠀⠀⠘⣿⡇⠀⠀⠀⠙⣦⣄
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⠇⢀⣀⠀⠁⠀⠞⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⣿⠓⢄
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⡠⡞⠀⠈⠉⠓⠀⠀⠀⠀⠀⠠⠖⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢸⣿⠀⠀⠱⣄
⠀⠀⠀⠀⡴⠃⠀⠀⠀⡠⠊⠀⡇⠀⠀⠀⠀⠀⢺⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⣽⣦
⠀⠀⠀⣸⠃⠀⠀⢀⠈⠀⠀⢀⣷⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢠⣿⡏⢣
⠀⠀⢰⠇⠀⠀⠠⠁⠀⢀⡴⠋⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⢀⣾⠟⠀⠀⠇
⠀⠀⡿⠀⠀⠀⣴⣠⡠⠋⠀⠀⠀⠀⠑⢤⡀⠀⠀⢀⣀⣀⠀⠀⢀⣀⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⠀⣾
⠀⢰⡇⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢀⡽⠛⠉⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠁⠀⠀⠀⢀⣾⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⣀⠔⠂⠉⠉⠈⠉⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠁⡇
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣗⣄⣰⠊⠀⠈⠒⠄⠀⠀⠀⠀⠴⠶⠚⠛⠉⠁⠀⠀⠀⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠇⠈⠀⠀⠀⠀⠀⠀⣠⠴⠒⠤⠤⠐⠚⠉⠉⠀⠀⠉⠐⠢⢄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡀⠀⠈⠓⠶⢦⣤⣶⣾⡿⠿⠇
⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣿⣊⣼⣷⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⠲⢶⣤⣌⠁⠀⠀⢮
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢻⣟⠩⢿⣿⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠹⣄⡼⠀⠀⠀⠀⠀⠐⠿⠿⢿⣧⠀⣀⠹⡄
⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠈⢷⣢⣿⠿⠛⢄⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣸⠖⠁
⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠫⡀⠀⠀⢈⢦⡀⠀⢇⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠐⢤⣤⣾⠟⠉⠓⢼⡄⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠙⢦⡀⠀⢀⣼⣿⡄⠀⣠⡦⣤⣤⠄⠀⠀⢶⣶⣿⣯⣁⠠⠐⠉⠈⢀⣘⠫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠉⠓⠿⣿⡋⠀⠀⢸⡗⣿⡏⠀⠀⠀⢀⡿⠛⠉⠁⠀⠀⠀⣤⡀⠈⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠉⠀⠛⠛⠛⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⡈⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠙⠛⠿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠛⠀⣀⣴⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣀⠀⠐⠾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠙⣿⡆⠀⠀⠀⠀⠐⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⡿⠋⠀⣠⣴⣆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⣿⣿⣿⣿⡉⠉⠀⠀⠀⠉⠛⠛⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣿⣿⣷⣄⠈⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀
⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣾⣿⣯⡀⠀⠙⢿⣿⣿⡆⠀⠈⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦
⣾⡿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠻⣿⣿⣦⡀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⢀⣴⣿⣿⣿⣧
⠙⣿⣦⠀⠐⢶⣾⠂⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣦⣤⣴⡾⠟⠀⠀⠘⠛⠉⣀⣼⣿⡆
⠀⠈⠻⣷⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣧⣤⣤⣀⡀
⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠻⣿⣶⣤⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠘⡷⣤⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⣿⣿⠿⣿⣦
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣶⣤⣤⣶⠟⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠋⠀⠀⠸⠟⠁⢀⣾⣿⣷⡀
⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡁⠀⠀⠀⠀⠀⠀⠠⡿⠿⠛⣿⣷
⠀⠀⠀⠀⠀⠹⣿⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠋⠁⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⡀⠀⠈⠻⠄⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣏⠀⠈⠙⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⢻⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⠿⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠈⠁⠀⠀⠀⠀⠉⠉⠁⡀⠀⣿⣿⠃⠀⣠⡀⠀⠀⣠⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⣾⣿⣿⣿⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⣷⣤⣶⣿⣤⣾⣷⣴⣿⣿⣷⣾⣿⣿⣶⣾⣿⣷⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⠿⠿⠿⢿⣿⣿⡟⠋⠀⢈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⢿⣧⣤⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠙⠛⠛⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠟

⠀⢀
⠀⠈⠃
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠂
⠀⡀
⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣦⠆⠀⠐⠀⠀⠀⠤⠤⠀⡀
⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠈⠇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡃
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⡈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠄
⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣄⣿⠤⠇⠀⠑⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣬⡀
⠀⠨⠊⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠘⠀⠰⠂⠀⠀⠀⠀⠠⡤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⣰⣶⣀⣄⣤⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⢰⣿⣿⣾⣿⣦⣠⡄⠀⢲⠀⠒⠂⠀⠚⠳⠀⠀⠀⠀⠒⠂⠶⠆⠀⠀⠈⠈⠀⠀⠈⠉⠉⠉⠉⠀⠀⠉⠀⢨⣃⡀⠂
⠀⠀⣿⣿⠻⠁⣾⡿⠁⠀⢹⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠙⣿⡦⠐⡟⠀⠀⠀⢰⠀⠀⠀⠀⢀⣀⡤⠄⠈⡉⠁⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⠀⠠⠄
⠀⠀⠀⢟⡛⠛⢛⣳⠶⠶⠾⠆⠀⢀⠔⠁⠀⠀⠈⠉⠁⠀⡧⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⠈⠀⢀⠂⠀⠀⠀⠀⠀⠀⠤⠔⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠹⣿⣷⣦⣶⣤⣤⡖⠰⡟⠀⠀⠀⠀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⠀⢹⢿⣿⠄⠤⠤⠭⠹⣟⠀⠀⠀⠀⠺⣙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡟⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠠⠠⠤⠀⡤⠤⠀⠶⠤⠀⠤⠤⡤⠄⠤⠄⠄⠦⠤⠤⠀⠤⠤⠠⠦⠤⠰⠤⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠰⠲⠀⠂⠀⠆⠰⠂⠐⠂⠂⠰⠆⠀⠀⠖⠔⠀⠐⠀⠃⠤⠲⠂⠐
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡉⣽⠦⠤⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠡⠁⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⣁⣠⡤⠤⠐⠒⠒⠒⠒⠺⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠴⠊⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠉⠀⢀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣖⠒⠒⠒⠺⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣴⣶⡾⠛⠋⠉⠉⠛⠛⠛⠋⠉⠈⠻⣿⣆⠀⢀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣁⠴⠚⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⢙⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠠⣏⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣀⠙⠋⠀⠀⠀⠀⠀⠀⠘⠃⠈⠈⠋⠀⠀⠘⡆⠀⠀⠀⠀⡀⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⣷⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠁⠀⠀⠀⠀⣅⡀⠀⠀⣀⡀⠀⠀⠀⠀⣾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠹⠿⠿⠆⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣄⠀⠀⠀⠀⠈⠁⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠃⠈⠈⠑⠒⠢⠶⠶⢖⣋⠁⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠊⠀⠀⠀⠘⢄⠀⢀⠐⡆⠀⠔⢤⠀⡀⠀⠀⠀⠀⢀⠄⠐⠛⢟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢁⠀⠀⠀⡀⠀⠀⠈⠰⠆⠀⠰⠄⠀⠀⠹⡅⠀⠀⠀⠀⠆⠀⣄⣀⣨⠁⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠑⢤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⡀⠄⠣⠀⠈⠉⠁⠀⢈
⠀⠀⠀⠀⠠⠂⠀⠘⢆⠀⠀⠀⢀⠈⠙⢷⣄⢀⠂⠀⣨⢉⡀⠀⡒⠀⠀⠀⠐⡂⢀⡈⡩⢹⠀⠀⠀⠀⡤⡄⠀⢠⢀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠳⣄⠀⠂⠀⠀⣠⠋⠏⠀⠀⠀⠀⡃⠀⠁⠀⠀⠀⠀⠁⠈⠥⠀⢸⣀⣠⠔⠋
⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠈⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⡀⠃
⠈⢄⠀⠀⠀⠀⠀⠀⠅⡄⠀⠀⠀⡴⢄⡁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⡀⠀⠀⠀⡀⠀⠈⠉⢠⠴⠓⠀⠉⠘⠑⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡔⠂
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢁⠀⠉⠉⠉⠛⠛⠓⠒⠚⠚⠉⠉⠀⢠⠃
⠀⠀⠀⠀⠀⠈⠒⠄⢀⠀⠁⠀⠀⡐⠀⠀⠀⡆⠘⠄⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠠⠞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠳⢄⠀⠀⠈⠁⠀⠀⠈⢹⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠒⠲⢦⡀⠸⠳⢖⠚⢻⣋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠁⠐⠂⢉⡃⡦⢁⣀⢀⠠⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠈⠁⠁⠉⠉⠀⠣⡀⠀⠉⠄⡀⡑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠈⠢⡔⠈⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⡖⣲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠹⣟⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣾⣷⣽⢼
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⣶⣿⣿⣿⠃⠀⢀⣀⠀⠀⠀⠀⠀⢠⡏⠉⣾⣾⣳⣸
⢄⡀⠀⣤⣶⣦⡀⠀⠀⠀⠀⠀⡇⠀⢀⣼⣿⣿⣿⣿⡆⠀⢸⣹⢧⡀⠀⣠⢤⢾⡇⠾⣿⣿⣯⡼
⠀⠳⢄⣸⣶⣿⠛⠀⠀⠀⠀⠘⡟⠛⠉⣸⠛⠛⢻⡿⢿⣴⡋⣽⣇⡇⠀⠉⢀⡿⣭⠉⢻⡏⢹⠃
⠀⠀⠀⠙⣿⣿⡆⠀⠀⠀⠀⢀⡇⠀⢀⠇⢠⠈⠡⡇⠘⡻⠤⢧⡟⡇⠀⢀⡞⠁⢿⡀⢨⣏⠀⠁
⠀⠀⢀⣾⣡⡄⢛⡄⠀⠀⠀⢸⠀⠀⢾⣆⡸⣤⣤⠇⢠⠃⡆⢸⠋⠀⠀⢠⢟⠶⠟⢲⢶⠿
⢠⡀⡾⠁⠀⠈⠉⠻⣦⢤⠀⡞⠀⠀⣨⡞⠀⠀⢫⠃⠈⣿⢛⡟⢀⡀⢀⣞⠏⠀⠀⠀⣧⡄
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠳⢆⡗⢳⣶⡿⠀⠀⢀⣀⣿⠾⠿⠾⣧⡈⠫⠻⠏⠀⠀⢀⣤⣞⣧
⠐⠒⠚⠛⠉⠉⠻⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠛⠛⠛⢛⡏⢉⣉⠭⢭⣝⣛⠛⠓
⣿⣟⣛⣛⣛⣻⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠈⢻⣷⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠾⠀⠀⠸⣦⣿⣏⠐⣂⠴⣯⠛⣿⣄⣼⡟⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠀⠀⠀⠀⠹⠋⠁⣼⣏⠀⠙⣷⡉⡀⠉⠳⠻⣿⡻⠧⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣶⣶⠶⢤⣤⣶⣄⢀⠙⠃⠀⠢⠠⠑⠀⠀⠉⠀⠙⠛⠒⠀⣀
⠀⠀⠀⠁⡀⣤⣦⣶⣦⠾⠟⠛⠉⡉⠤⢀⣴⣄⣍⣉⣹⣿⣷⣴⣤⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣤
⠀⠀⠀⠀⠙⢉⡀⢡⣆⣴⣴⣦⡈⠠⣾⡿⣿⣯⣿⡻⠿⣿⣿⢷⣾⣍⡛⣛⠷⢶⣴⣷⣶⣤⣀⣀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠙⠓⠺⠟⠒⠉⠙⠺⠿⠿⠖⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠙⠓⠦⠴⡆



⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣟⢿⠿⢿⣿⠻⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣿⠁⠩⡿⠸⣷⢺⢴⣎⢿⡣⢶⡜⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⣻⡷⡇⣨⣤⣥⡖⣿⣷⣷⣿⣧⣵⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣬⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣯⣭⣭⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⣻⣿⣿⣯⣿⣿⠿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣯⣿⣿⡿⢿⣿⣿⣿⡏⠁⠀⠀⠀⡠⠤⣤⣽⣿⣿⣿⢧⡤⠤⠴⠶⠒⠒⠿⠛⠅⠀⠀⠘⡆⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡏⠀⢢⠈⢻⣿⡄⠀⠀⠀⠀⠠⢾⠉⠙⣽⡅⠠⠠⠀⠀⠀⠀⠀⠂⠒⣋⡑⠀⠂⠒⣟⡓
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⣿⣿⣿⣿⣷⡀⢘⡄⠀⠻⡷⠀⠀⠀⠀⠀⠈⠿⠟⠿⡷⠶⠶⣤⣤⣤⣤⣤⣶⣾⣰⣇⣀⣀⣀⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣌⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣈⠄⠀⢸⡇⠀⠈⠉⠉⠛⠛⠛⠛⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⡀⢀⡴⠋⠉⢋⣹⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠒⠶⠖⣻⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⣧⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⣿⣖⠤⢀⡀⠀⠀⠀⠀⢰⣤⣀⡀⣀⡠⠞⠀⠀⢀⠴⢺⣷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⡉⠛⣻⢦⣌⡁⠒⠒⠲⠼⢿⣤⠀⠀⠀⠀⠀⢠⠃⢹⣹⠏⣿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⣿⣿⣷⣾⣿⣷⣦⡀⣠⠞⠍⣇⠀⠀⠀⠀⣨⠀⠞⠁⢀⡿⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣯⣿⣿⢿⠿⢿⣿⣿⣿⣿⣿⢷⣄⡀⢺⡄⠀⠀⠀⡏⠀⠀⠀⣿⠁⠨⣺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⡿⡍⠿⡀⢈⢧⠣⢭⣛⣿⣿⡆⣿⡷⣦⣷⡀⢀⡀⣇⠀⠀⠀⠙⠖⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡛⣴⣷⡘⣿⣆⠃⡣⣙⢻⣿⣿⣿⡮⣹⣼⣿⣷⣿⣟⠻⣄⠀⠴⡆⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⡒⢌⢻⣿⡞⢿⣳⡔⣹⣿⢿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣷⣌⡟⡿⢷⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡛⡹⣭⣻⣩⣿⣎⡿⣽⣞⡹⣶⣿⣿⠏⣽⡿⣿⠿⣿⣿⣫⣻⣿⣷⣶⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡼⣏⣿⠿⣤⡹⣿⣿⣳⡬⢿⣿⣿⡿⣿⣿⣼⣿⡷⢝⢾⣿⡿⢳⡿⣛⡞⡿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣿⣁⢻⡽⣌⢿⣿⣣⡺⠳⣿⣿⡽⣿⣻⡻⣾⣫⣷⣽⣷⣯⣟⡟⡼⠁
⠀⠀⠀⠀⠀⠀⣿⢾⣿⢯⢿⢞⢾⢿⣻⣽⣇⢻⣿⡷⡴⡹⣽⣿⣿⣷⣷⠣⠛⣯⣻⣿⣜⣏⡜⠁
⠀⠀⠀⠀⠀⠀⣿⡽⣿⡿⢻⣧⢳⡠⣻⡻⣯⣷⣿⣿⣟⣵⡜⣿⣿⣿⣧⡆⡀⠙⢻⣿⣟⠞
⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠁⠁⠁⠈⠁⠁⠋⠛⠙⠛⠋⠉⠈⠈⠛⠛⠉⠙⠘⠂⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢨⠀⠀⠉⡅⠀⠀⠁⢤⠤⠤⠤⠤⠤⣤⠠⠀⠀⠀⠀⠀⢀⡀⠅⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠐⠀⠀⠀⠰⠸
⠀⠀⠀⠸⠀⠀⢈⠀⠀⠤⡀⠆⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⡄
⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⡂
⠀⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⢰⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣇⣀⡀⠀⠀⠀⡀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠆⠸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠠⠖⢠⠜⠉⠣⠀⠀⠉⠓⢤⠤⠥⢆⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡠⠊⢉⣰⠎⡀⠔⠉⠀⠀⠀⡀⠀⠈⠉⠐⢱⡆⡈⠉⠷⠒⠒⠛⠘⠤⠤⠄⠀⠄
⠀⠀⠀⠀⠀⣧⠔⠁⠀⡌⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠑⡼⠁⢉⠁⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠺⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⡌⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠃⠀⠀⣰⠠⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⢄⡀⢀⣀⣀⣀⠀⠀⠠⠊⠀⠀⠀⠀⠸⣇⠘⣖⣲⠆⠁⠈⢂
⠀⢀⠀⠀⠀⠀⣿⠄⠀⠤⠂⢁⡁⡀⢀⢀⡀⠀⢉⠐⠖⡀⠒⠐⣃⣤⡈⠉⠉⠀⠊⠉⠉⣨⣀⣀⣀⣀⣀⣀⡀
⠀⠈⠀⠀⠀⠀⡏⠤⡎⠀⣀⠿⢣⠄⠀⠀⠄⣉⠞⠵⡀⢘⠀⠀⠉⠘⠁⠉⠉⠉⠋⠉⢻⣿⣉⣉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠰⠄⠲⠚⠚⢁⠄⠊⠐⠄⠂⢀⠀⠁⠀⠉⣛⡛⢛⠂⠻⣃⠀⠂⠂⠀⣶⣼⢇⡀⡈⣧⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠃⠀⠄⠀⢀⠥⢠⠀⠈⠁⠉⠂⠁⢤⠤⢄⠁⠈⠡⠃⠀⠩⡉⠉⠉⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠁
⠀⠀⠀⠀⡞⠄⠡⠔⠂⠇⢀⠌⠈⠓⠒⠐⠒⠂⠈⢄⢀⠃⠐⠀⠀⠄⠈⠡⠀⠐⠠⢤⠔⠈⠀⠀⠀⢠
⠀⠀⢠⠘⠀⠀⠀⠀⠀⠉⠉⠈⠉⠀⠀⠀⠃⠀⠀⠈⠉⠀⠉⠉⠉⠁⠀⠀⠓⠀⠃⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠤⠤⠬⠥⠍⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠠⢀
⠀⠀⠀⠀⠀⢀⠆⠠⢤⠀⠠⠄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢄⣀⡄⠀⠤⡀
⠀⠀⢀⡀⢂⡈⠁⠀⠀⠀⠈⠉⠉⣉⣁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠎⠀⠀⠀⠀⢀⠀⠑⠈⢊⠂⠀⢂
⠀⠀⠋⠀⢸⠠⢯⡟⢴⢂⢤⡭⠟⠦⠈⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⡎⠄⠀⠀⠉⠉⠉⠀⠀⠸⣄⠀⠀⠆⠃⠀⡆
⠀⠀⢠⠀⠀⣾⠘⠿⠃⠀⠀⠶⠖⢠⡄⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠄⣲⡲⡀⢤⡐⣦⡜⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠑⠤⣟⡃⠀⣀⠀⠀⠀⢀⣟⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣯⠈⠘⢷⡷⣶⠈⢶⠃⡀⢠
⠀⠀⠀⠀⠀⠈⠙⢃⣄⠀⣠⠴⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠄⠀⠀⠀⠀⠀⠁⠀⢀⠺⠀⠁
⠀⠀⠀⠀⣰⠂⠐⠀⢇⠀⠚⠀⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠂⡀⠲⠖⢛⡠⠞⠃⠀⠀⠠
⠀⠀⠀⠔⠈⠀⠘⢆⡠⠳⡱⢧⣀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⡒⢈⠏⠀⠄⡀⡔⠉⠉⠂⢹⠟⣇⡀⢠⠀⠁⠀⡄
⠀⠀⢈⠀⠀⠀⢰⠀⢱⣀⣽⡧⠌⠆⠀⠀⡈⠢⡀⠀⠀⠀⢠⠌⠸⠀⣠⣤⠖⠂⠄⣈⡄⢂⠀⡁⠉⢈⡀⠱⠄⡀
⠀⠠⠀⢀⣢⡄⠀⡰⠋⠈⢸⣧⠀⠀⡔⠘⠄⠀⠈⡆⠀⢠⣵⠂⠀⠈⢏⢸⡒⠐⢄⠘⠀⠀⠃⠀⠀⠀⢳⠀⠐⠁
⠀⢃⠀⠀⠀⢹⢽⠀⢀⠶⢥⡜⠉⠀⠂⠐⠀⠀⡰⠀⠀⠀⠙⠳⠾⠀⠈⠁⡝⠛⠛⠀⠀⠀⠀⠆⢀⣐⠁⠢
⠀⠀⠀⠀⡅⠀⡍⠀⠀⠀⠀⠂⢀⡄⠙⠂⢀⠔⠀⠀⠀⠀⠀⠀⠐⢀⡀⢰⠁⠀⠀⠀⠀⠀⠰⠀⡠⠁⠀⡀
⠀⠀⠀⠀⢃⠀⢙⡒⠂⠐⡂⢻⠈⠗⠖⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠀⠀⠀⠀⡗⢀⠔⠊
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡘⡄⠀⠐⡊⠄⠀⠹⡆
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠒⠐⠁
⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⡐⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠠⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⣅⣀⡀⠁⠀⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠢⠠⠀⣀⢀⠄⣀⡀⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⡀⠀⠈⡀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠤⠤⡁⠀⠀⠀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢀
⠀⠀⠘⡖⠔⠂⠀⠀⠀⠠⠄⢠⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢰⠀⠀⠀⠀⢸⠆⠘⣦
⠀⠠⠐⠉⢐⠀⠀⠀⠀⢠⡀⢀⣀⢉⣉⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠧⡔⠈⠃⠀⠀⠀⠈⠓⠠⠓⢕⠄⡀
⢠⡥⠀⢄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠀⢀⡠⠢⠃⠀⠀⠀⠀⠀⠢⡀⠂⡠⠚
⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⢲⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⡀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣠⣆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⡆⠀⠄⠀⠀⠀⠀⠀⠀⠠⣴⡆⠠
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⢀⣀⣀⠀⣀⣀⣠⣼⣿⢁⣸⣿⣷⣀⣴⣶⣄⣤⡤⠀⠂⠀⠁
⠀⣾⣶⣦⣀⣠⣤⣤⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠙⠛⠿⣿⣿⣿⣿⣶⣮⣤⣶⣿⡄⢀⣤⣾⣷⡀⠀⣴⣶⣤⣠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⢠⣤⠄
⠀⢀⡀⠀⣤⣌⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠿⠇⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡄
⢀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⢛⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠋⠉⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⢠⡀⠀⠀⠀⠀⠙⠃
⠀⢾⡿⠿⢿⠿⢿⣿⢿⡿⣿⣗⠷⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠎⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⠶⠤⣴⣒⠒⢤⣼⢥⢡⢼⠬⢽⡄
⠀⠀⠀⠀⠀⠀⠀⣠⢴⡏⢄⣣⣹⠶⠾⠛⠾⠞⠊⠉⠉⠉⠙⠓⠤⡀
⠀⠀⠀⠀⠀⠀⡰⠁⢸⢵⠝⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣠⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣦
⠀⠀⠀⠀⢰⠁⠱⣤⠃⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢣
⠀⠀⠀⢀⠞⡄⠀⡟⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⢀⡀⠘⡄⠀⡦⡀⠀⠈⢧⡀⢇
⠀⠀⠀⣼⠀⠙⢼⠁⠀⠀⠀⢰⠁⠀⠀⢰⡆⠀⠀⠈⡑⡄⡇⠀⢸⠹⡀⡀⣸⢧⣸
⠀⠀⢰⠛⡄⢀⣸⠀⠀⠀⠀⡌⠀⠀⠀⡿⢇⠠⠤⠤⠃⠑⠛⠒⠉⠀⠉⠁⠀⠘⡈⡶⡄
⠀⠀⣠⠀⡸⠁⢸⠀⢀⣀⣰⠓⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⠀⡇⢡⠆
⠀⢰⠉⢣⢇⣀⡼⡄⡏⠀⡝⠀⡆⠀⠀⠀⠀⠞⠳⠀⠀⠀⠠⣄⠄⠀⢀⡠⡀⠀⡇⢸
⠀⠸⣴⠁⠀⣠⠃⠱⡕⠤⣆⠀⡇⠀⠀⠀⠀⠀⠀⡀⣀⢀⡠⠤⠐⠰⠯⢄⣳⢀⡇⡸
⡰⠑⠦⢭⠏⠀⠀⠀⠈⠒⢼⠀⡇⠀⠀⠀⠀⠀⢸⢀⠈⠁⠀⠀⠀⠀⠀⠀⠙⢇⣷⠃
⣧⠆⡇⡸⠀⠀⠀⠀⠀⠀⠈⡆⢸⠒⠢⢤⣤⣤⣀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⣧
⠈⠛⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⢮⡆⠀⡇⠀⠀⠑⢇⡀⠀⠀⠄⠀⠘⠀⠀⠀⣀⡷⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢧⠀⠀⠀⡠⣻⢆⣀⡠⠤⢄⡀⣠⠋⠀⢈⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⠣⡀⡜⡵⠁⠀⠈⠑⢆⠀⢸⠙⢤⡤⠾⠚⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡠⢔⡪⠋⠓⢦⣄⣀⣀⡠⠞⠚⠉⠀⠀⠀⠀⠀⢸⠹⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠖⠁⠀⠀⠀⠀⠈⠉⡞⢳⠤⠒⠲⡖⠐⡲⠋⠑⡌⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⢠⠃⡎⠀⠀⡰⠀⠀⠃⠀⠀⡗⠚⢾⠥⡖⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⣀⣀⠀⢀⠎⡜⠀⠀⡰⢇⣀⣀⣄⠀⡠⠓⠠⠴⠤⠧⠷⠒

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

⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⠐⠂⠠⢀
⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠄⠀⠠⠀⠄⡀⠀⠀⠀⢀⡀⠠⠤⠤⠀⢀
⠀⠀⠀⠀⡠⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠀⠀⣀⠀⠀⠑⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠌⠁⠐⢄
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠠⠁⠃⠀⢐⠀⠀⠀⠀⠉⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⡌⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠁⠀⠀⠀⠀⠠⣀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡀
⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠁⣀⠠⢄⡀⢢⡀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠤⠤⠤⠴⠦⠄⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⢐⠀⢰⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣧
⠀⠀⠀⠀⠾⣇⡸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣂⡉⣁⣒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡴⠊⠁⠀⠀⠀⠀⠉⠢⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠆⠀⡠⠤⠀⠀⠀⠀⠤⡀⠀⡆⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⢦⡇⠀⠀⠀⠀⠀⠀⣧⢎⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣔⡺⠿⠜⠛⠓⠚⠓⠼⠿⣒⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣖⠽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢗⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠊⠁⠀⢀⡠⠴⠒⠊⠉⠉⠉⡍⠉⠉⠒⠲⠤⣀⠀⠀⠉⠪⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡫⠊⠀⠀⣠⠖⠉⠀⠀⠳⠀⠀⠀⠀⡇⠀⠀⠀⡼⠀⠈⠑⢦⡀⠀⠈⠪⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡝⠁⠀⢠⠎⠀⠲⣄⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣠⠘⢦⠀⠀⠙⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡟⠀⠀⡰⠃⠀⠀⠀⠈⠀⠀⡠⠔⠒⠉⠙⠹⠑⠒⠤⡀⠀⠀⠋⠀⠀⠀⠳⡀⠀⠘⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡹⠀⠀⢰⠃⠒⣄⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡠⠄⢳⠀⠀⠸⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠇⠀⠀⡌⠀⠀⠉⠛⢭⣉⠓⠒⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠈⡆⠀⠀⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⢀⣀⡀⠀⠐⣯⡵⠢⣀⠀⢸⠞⠁⠙⢆⡤⣀⠀⠀⢀⣇⣀⡀⢀⣤⡄⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠐⡟⠛⠀⠀⠉⠁⠢⣀⡠⢾⠀⠀⠙⠢⢌⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⢃⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠂⡤⠍⣲⣤⣀⠀⢀⠃⠀⠀⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡹⡀⠀⠘⡄⠚⠁⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠈⠁⡜⠀⠀⡸⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡳⠀⠀⠘⣄⠀⠀⠀⣠⠀⠀⠉⠒⠦⠤⠴⣧⠤⠖⠊⠁⠀⠀⢦⣀⠀⢀⡜⠀⠀⢰⡳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡡⡀⠀⠈⠢⡀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠈⡠⠊⠀⠀⡰⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢆⠀⠀⠈⠢⢄⠀⠀⣰⠀⠀⠀⠀⡇⠀⠀⠀⠈⠃⢀⠤⠊⠀⠀⢀⢞⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⡦⡀⠀⠀⠉⠒⠤⠤⠄⣀⣀⣁⡀⠤⠤⠖⠊⠁⠀⠀⡠⣶⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⢗⡤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣖⠯⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠹⠷⠖⢲⣶⣶⠒⠲⠽⠑⠊⠉

⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢠⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠸⣷⠀⡇⣽⠀⠀⢸⡆⠀⠀⣴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠀⡀⠀⢠⢰⣟⣿⡇⠀⠀⠀⠀⠀⠀⢠⡄⢸⡇⢀⠀⠀⠀⢀⣿⣼⣷⣴⣷⣾⣴⡔⣹⡇⢴⣧⢿⣾⡖⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣿⡇⡣⠀⣼⣼⣿⣿⣿⡃⠀⠀⠐⣢⣤⣼⣇⣸⣧⣼⡆⠀⠤⠬⢿⣿⡟⠉⢻⣿⣿⡇⣚⢻⣽⣿⣧⡿⢡⡛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣾⣿⣿⣿⣧⣨⡁⢻⣿⣿⡿⢿⠁⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⡿⠀⠀⠀⠙⣯⣥⢿⣾⣏⣺⣿⢕⣯⡄⠸⣷
⠀⠀⠀⠀⠀⠀⣀⣼⣾⣿⣿⣿⣿⣿⣫⡿⠀⠈⣿⣿⠇⠸⣀⡆⠀⠀⡌⠀⣻⣿⣿⠏⠁⠀⠛⡛⠓⣾⠃⠀⡀⡀⠐⠛⣥⡾⢻⣍⢹⣏⣴⠿⣿⠤⠿
⠀⠀⠀⢀⣷⣿⣿⡟⣿⣿⣿⣯⢿⡏⣼⠇⠀⠀⢹⡟⠄⠀⣿⡼⣷⡿⠄⠀⠂⣿⣿⣀⠀⠀⠈⠉⣠⠼⠃⠁⠀⠈⠒⠠⣿⣇⢈⣿⣿⡏⢿⣿⡉⡷⠄
⠀⠀⠀⣾⣿⣿⣿⣿⡿⠿⠿⠓⠛⠓⠏⠀⠀⠀⢸⠇⢀⡀⠘⣷⣽⢓⣀⣄⠠⠝⠿⠛⠳⠾⣿⢶⡇⠀⠀⢠⣠⣤⣀⣀⡜⢻⡿⣷⡿⡈⢼⣿⠳⢟
⠀⠀⠼⠿⠛⠉⣉⠁⠀⠀⢀⣤⡤⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠋⠉⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠛⠉⠙⠗⠞⠃⠛⢂⠈⢠⣹⠆⠀⠃
⠀⠀⠀⠀⠀⠈⠽⠀⠂⠤⣤⣤⣧⠠⢠⣤⣤⡀⣀⠀⠤⠤⢀⣢⣄⣀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠶⠎⢉⣺⡟⣂⣉⣽⣷⣄⣀⣉⣱⡖⠀⠀⠀⠀⠀⠠⢶⣶⠤⢴⣈⡉⠷⣄⠀⠈⢻⡆
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠐⠺⠶⠶⠦⠶⠴⠶⠒⠛⠀⠠⠾⠷⠒⠒⠋⠉
⠀⠈⠀⠈⠁⢦⣴⣾⣿⣁⣀⣄⠀⣷⡄⠀⠀⣴⡄⠀⠀⠀⣆⠀⡀⠀⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣷⣷⡄⠹⣿⣤⣧⢾⡏⣧⣷⢠⢼⣷⡀⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⣿⣿⣿⣿⣾⣿⡿⠿⠛⢿⣾⣷⣿⡇⠀⢁⠀⡆⡄⠀⡀⣼⣿⣿⠀⣆⣿⡆⠀⠀⢀⢸⡄⠀⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠐⡻⢿⣿⡟⣿⠿⢿⣭⣥⠀⢘⠂⣻⢻⣧⣷⣿⣿⣿⡄⢸⢸⡇⠀⠀⠈⡄⣿⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠛⠁⠀⠒⠒⠀⠓⠄⠊⠀⡥⠜⠛⢛⠻⣿⣿⡗⡚⠉⢉⡠⠄⢠⣧⣿⡿⠴⠏⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠘⣿⠛⠏⠑⢿⠛⢛⢛⡻⢿⣧⣀⣀⣉⣭⡷⢶⣖⣚⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣤⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉⠉⠙⠁⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡼⡓⢿⢷⣶⣴⣢⡄⠀⣠⣄⢠⣴⣶⣿⣿⣧⡄⠀⠐⠒⠶⠓⠲⠶⠶⠒⠲⠤⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡬⢝⠀⢁⠰⣼⣿⣿⣿⣯⣽⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠲⠘⠻⢿⠿⠿⡟⠛⣫⣁⢹⠷⢆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠉⢙⠓⠞⢦⣤⡛⣐⠃⡨⠁⣿⣦⣤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠈⠿⢘⠳⠖⢾⠿⠛⠁⠉

⠀⠀⠀⠀⡟⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⡇
⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣦⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⢀⠇⠀⠤⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠋⠛⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣇
⠀⢀⠞⠁⠀⢶⠆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⡁⠉⠈⠀⠀⠀⣠⣾⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠘⣆
⠀⣸⣤⣤⣄⡀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠦⢤⠴⠶⠤⣀⡼⣆
⠀⢸⣿⣿⣉⣿⣷⣶⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣤⡄⠈⠁⠀⣴⣦⠀⠘
⠀⠀⠙⢿⣿⣽⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣉⣁⣀⣀⣀⣙⣋⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠁⠁⠉⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⢻
⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠍⠩⠍⠉⠉⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⣿⣿⡗⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠙⠛⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣒⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣷⣶⢸
⠀⠀⠤⢤⣤⠄⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⡂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣧⣼⢸
⠀⠀⣠⣼⣿⣭⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡒⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣿⣷⢸
⠀⠀⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠤⠤⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⠕⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⡖⠒⠒⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠤⣴⠁⢘⣦⠴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠉⢳⡀
⠙⢦⡀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡤⣮⠽⠯⣵⢤⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⡙⣄⣐⣿⣁⡀
⠀⠀⠙⡆⠰⠉⠺⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠰⣁⢀⡷⡇⠰⠆⣸⢮⡀⣨⠶⠀⠀⠀⠀⠀⠀⠠⠬⡦⠤⣽⠁⠀⣘⡄
⠀⠀⠀⠇⡇⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⢟⡳⢞⡛⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢈⠦⠬⠟
⠀⠀⠀⠸⠇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠙⡄⣠⠋⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏⢎⠻⣿⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠈⠣⠈⠙⠂⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠇⠀⠀⠀⠠⠶⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⡄⠀⠀⠀⠉⠁⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠒⠂⠀⠀⠀⠹⣿⣿⡿⠃⠀⢶⡤⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣄⣀⠀⠤⠴⠾⠟⠛⠉⠐⠦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡙⢿⣷⣦⣄⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢹⡄⠙⢿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣸⣧⣀⣠⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⡿⠋⢅⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⢀⣴⣶⣤⠀⢀⣠⣤⣶⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡟⠀⠀⠸⡀⠀⠀⠀⢹
⠀⠀⠀⠀⣠⣿⠏⠉⠙⠿⠛⠉⠁⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿⣧⠀⠀⠀⠀⡇⠀⠀⠀⢳
⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠸⣿⡆⠀⠀⠀⢣⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠠⠄⣀⣀
⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠻⣿⡄⠀⠀⠈⠶⠶⢶⣤⡀⠈⠢⢄⡀⠀⠀⠀⣀⣀⣬⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣦⠘⠛⠁⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⢿⣿⠏⠀⠻⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣄⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣼⣶⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠃⠀⠀⠁
⡖⠒⢲⠖⠶⠶⠶⠲⡖⠲⠶⠶⠶⠶⠖⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⡶⣶⣶⣶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⡀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⢂⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡀⠠⣀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⡄
⠀⠀⠀⠀⣠⠆⡀⠀⢰⣣⣐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢐⠏⠦⠃⠀⣏⣿⣾⠙⠁⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⠀⢠⢿⡵⠀⠐⠈⠉⠩⠕⠒⠐⠒⠀⠂⠀⠀⠀⠀⠀⠠⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⢀⠄⠄⠒⡉⢍⠁⠈⠉⠛⠢⢄⠃⠀⠙⠁⠉⠁⠀⣀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠰⠀⠀⠀⠀⡇⠀⠄⠀⢀⡠⢀⠔⠁⠀⠀⠔⢀⣴⠶⢉⣭⣥⣭⣹⣿⣆⡴⢠⣄⠀⡩⠤⠶⠊⠙⠃⠀⠤⡔⣖⠲⠦⣉⡱⠦⣤⠛⠉⠁⠈⠉⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡰⠋⠀⠀⠀⣠⠂⡟⠃⡀⠈⢢⡴⠀⠲⢷⣍⣝⣵⠋⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠙⣷⠀⠈⠸⠇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⡜⠀⡀⢙⣦⠅⠀⠀⠄⠀⡀⠀⠉⠀⠁⠈⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠲⢼⡆⠀⠀⢀⢠⡀
⢀⠀⢸⠀⠀⠀⢸⡄⠀⠀⠀⢸⡆⠀⠀⣴⠁⠀⣃⠈⢤⡀⠀⠘⡀⠠⠃⠀⠀⠀⠀⠀⣧⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⢿⠀⡀⢸⢧⣿
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡌⣇⠁⠂⠻⢰⣬⣿⣷⡟⠁⠀⠀⠀⠀⢀⡀⢠⡄⠀⠀⢸⠒⢨⠤⠙⣦⢚⡛⠁⠀⠀⠀⠀⢲⡟⣨⣎⣟⠀⠆⠀⠠
⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠙⣄⠀⢳⠌⣻⠿⡧⠁⠀⡄⠈⡀⠀⠚⠀⡿⠘⠀⠀⣾⠘⠀⠀⡸⠟⠉⠉⠂⠀⠀⠀⠀⢸⡇⡟⣿⣿⣿⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠃⣬⡟⠻⣶⠿⠦⣌⡀⠶⠀⠀⠀⠀⢀⡄⠠⠾⠿⠥⠞⠉⠀⠀⠀⠀⠀⠀⢲⡄⢀⣼⡿⠣⣿⢸⠃⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⠀⠐⣀⠴⠀⠀⠀⠀⠉⠓⠒⠲⠴⢶⣥⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⢃⡌⠀⠋⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠉⠀⠒⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣀⣠⣤⡤⠤⠤⣲⣾⢻⠏⠀⠁⠁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠈⠉⠛⣼⠟⢡⣾⠇⠎⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢨⠎⠉⠀⠙⠒⠦⢤⣤⡤⠴⠛⠁⠀⠀⠀⡼⠋⠀⠸⠏⠈⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⠄⡲⠀⠀⢄⣟⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⡀⠀⠀⠀⣼⡃⠀⢀⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠉⠁⠀⠀⠀⠃⠀⠀⡾⠊

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣀⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣊⣉⣵⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠁⠁⠀⠀⠀⠢⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⠀⣀⡠⠤⠐⠒⠒⠒⠒⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⠔⠂⠉⠠⠁⠀⠀⠀⠀⠀⠀⠠⠠⠠⠦⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⠀⠀⠀⠃⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣴⡤⠶⠤⠤⠥⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣤⣤⣶⠾⠛⠛⠛⠿⠿⠿⠛⠛⠙⢿⣿⣄⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡤⢶⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠚⠁⠀⠸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠴⢿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠙⠿⠟⠀⠀⠀⠀⠀⠀⢴⠆⠐⠐⠷⠀⠀⠱⡄⠀⠀⠀⢀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⠂⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⡠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠉⠀⠀⠀⠀⢹⠀⠀⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣤⠀⠒⠀⠀⠀⠀⠀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⠉⠉⠠⠄⠀⠀⠀⢀⡜⠁⠀⠀⠁⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠶⠠⣀⡀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠘⡀⠄⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠈⣄⠀⠀⠀⢀⠉⠉⢙⡱⠒⠀⠀⠀⠀⠀⠀⠀⠐⣠⣤⣌⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠖⠁⠀⠀⠀⠀⠈⢆⠀⡐⠀⠇⠠⠊⠱⣄⠄⠀⠀⠀⠀⠠⠂⠀⠉⠁⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠀⠀⡠⠂⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠈⢆⠀⠀⠀⠀⡁⠀⠲⡤⠤⠂⢀⡆
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠁⠞⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣆⠀⠄⠂⠑⠀⠀⠂⠈⠈⠠⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠑⡀⠀⠀⠀⠄⠀⠙⢷⣤⡌⠀⠀⠦⢠⠀⢠⠁⠀⠀⠀⠈⡄⢀⡄⠔⢰⠀⠀⠀⢀⡔⢲⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡢⠀⠈⠀⠀⣠⠎⡰⠀⠀⠀⠐⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡦⠤⠚⠁⠀⠘
⠄⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠀⢈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠐
⠈⠢⠀⡀⠀⠀⢀⠀⠰⠤⠀⠀⠀⢐⠣⢀⡸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢄⠀⠀⠘⢀⠀⠀⠉⠱⠤⠞⠀⠀⠀⠉⠰⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⡖⠁
⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⡀⠀⠈⠉⠉⠉⠛⠛⠙⠉⠉⠉⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠑⠄⢀⡀⠁⠀⠀⢀⠃⠀⠀⢠⠀⠳⠄⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠈⠀⠀⠀⠈⢹⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠶⠶⣆⠀⢸⠲⠖⠚⢻⠋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡌⠁⠒⠐⠹⠠⡄⡃⠀⠀⢠⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠒⠒⠒⠊⠰⢁⡈⠉⠉⠄⡁⠐⠒⠒⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠈⠉⠁⣄⠄⠊⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠤⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠠⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⢀
⠀⠖⠲⠀⢸⠀⠀⠀⠀⠀⢀⢀⣀⣄⣨⣀⣄⣸⣀⡇⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⢀⠤⠀⠀⠀⠀⠀⢛⡀⡀⠀⠀⡀⢀
⠀⣍⢈⡄⡜⣄⣸⠉⡄⠀⠈⠉⠉⠉⢉⠉⠉⢩⠀⠁⡇⠀⠖⠰⠠⠄⠐⠒⠈⠀⠀⠤⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈
⠀⠁⢀⠀⠸⠁⡜⠀⠣⠀⠠⠄⠠⠀⠈⠠⠀⠈⠤⠀⠇⠀⢀⠀⠠⠄⠀⠀⢀⠎⠀⠀⠀⠀⠀⠠⠠⠤⣏⡁⠀⠀⠌⠁⠐⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠈⠁⠀⢀⠈⠀⠀⠀⠀⠀⢀⡀⠂⢀⠀⠀⠈⠁⠸⠀⠀⡨⠀⠀⡆⠈⠣⠀⠁⠰
⠀⠀⠀⡀⠰⠈⠀⠂⠀⡤⠀⠠⠄⠀⠀⠀⠠⡀⠀⠀⠀⠂⠀⢁⠑⠘⠦⠀⠀⠀⠀⣠⠲⠆⠦⠀⡂⠀⠀⠀⠀⠀⠀⠊⢢
⠀⠀⢀⡉⡁⠒⠒⠂⠁⠠⠄⠈⠀⠀⠀⠀⠀⠀⠈⠡⠀⡀⠀⠂⠀⢠⠆⠀⢠⠐⢈⡅⠀⠀⠀⢘⠂⠀⡄⣤⣈⠀⠀⠗⠳⡆⠀⡀
⠀⠀⠀⠀⡆⢀⠤⡀⠀⠀⠀⠀⢠⠄⢐⠀⠀⠀⠂⠀⢈⡞⠓⠒⠈⠀⠀⠀⠘⡆⠈⠱⣦⢠⣄⠛⠁⢰⠃⠒⢇⡸⣐⠂⠀⠀⠀⠀⠀⢠⠰
⠀⠀⠀⠀⠀⢁⠀⠁⠀⠀⠀⠀⢀⣀⡒⠀⠀⠀⠀⠠⠋⠀⠀⡀⠀⠐⠀⠀⠀⠈⠢⣄⣐⠀⣀⣡⠔⠚⠀⠀⠀⣐⢁⠀⠐⡤⠀⠂⡀⠀⠁⠀⡁
⠀⠁⠀⠀⠀⠈⡆⠀⠂⢐⠀⠀⠠⠠⠀⠀⠀⠀⠀⠔⠈⠉⡉⠀⠜⠠⠀⠀⠀⠀⠀⠢⠉⠉⠁⠀⠀⠠⠀⠀⠘⠀⠈⠀⠐⢠⡀
⠀⠀⠀⠀⠀⠔⠀⠀⡀⠈⠠⠀⠠⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠔⠀⢀⢀⠀⠀⡀⢦⠀⢄⠔⡙⠀⠄⠘
⠀⠀⠀⠀⠀⠀⠔⡈⠁⠐⠀⠀⢀⡀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠄⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⠀⠙⠁⠀⢠⠀⠀⠀⠢⠒
⠀⠀⠂⠀⠀⣀⣀⣀⣈⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⣰⣶⠶⣶⡆⣶⠶⣶⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠠⠁⠠
⠀⠀⡀⠀⠄⠳⠄⠀⢸⠀⠀⠀⠀⢐⠀⣀⣀⡢⢸⣿⠀⠀⢿⡅⠀⢻⡇⣿⡀⣼⠇⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡁
⠀⠀⢧⠀⡝⢻⠀⠀⢸⠀⠀⠀⠀⢠⡀⠀⢠⠅⢸⣿⠀⠀⢸⡗⠒⢻⡇⣿⠿⣿⡄⠀⠀⡶⠶⠒⠒⢶
⠀⠀⠚⠁⠍⠈⠀⠂⢘⠃⠒⠀⠀⢀⠑⠐⠁⠀⠸⣿⣶⡆⣽⠇⠀⠸⠇⠿⣴⠿⠁⠀⠀⠃⠀⠀⠀⢸⠀⠀⠰⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⢀
⠀⠀⢠⠀⠠⠀⠀⠃⠩⠁⠉⠀⠀⡂⠀⠀⢀⠀⠀⢠⡤⣤⣤⣶⣶⣶⣶⣶⣶⡄⠀⠀⢀⠀⢀⣐⡀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑
⠘⠒⠤⠠⠤⠔⠂⠁⠀⡀⠀⠀⠘⠉⠑⠒⠚⠀⠀⢸⣧⣧⣿⣦⣿⣤⣿⣧⣾⡇⠀⠀⠸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠉⠉⢉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠐⠀⠘⢘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⢀⠃⠄⠀⠀⠀⢀⢂⠀⣠⠀⠀⠀⠀⠀⣁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠖⠀⠀⠀⠀⠀⠁⡌⠀⠀⣑⡄⠂⠀⠐⠜⡡⠀⠀⠀⠀⠀⠂⠁⠀⠐⠁
⠀⠈⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⡐⠀⠀⢀⠈⠀⠈⠀⠀⠀⠀⠂⠑⠢⠁⡀⠀⠀⠀⠜⠁⠁⠀⠈⠀⠀⠄
⠀⠈⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠀⠃⠠⠄⠀⠀⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠄⠠⠀⠠⠀⠀⠀⠘⠀⠀⠀⠀⠆⠀⠀⠡
⠀⠀⢠⡄⣴⠀⠀⠀⢠⡶⣦⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⡀⠀⠀⠀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠆⠰⠀⢀⠠⢀⠒⠂⠐⠀⠀⠀⠉
⠀⠀⢸⣇⣿⢠⣶⡄⢸⡆⣿⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢂⠀⠱⠚⠀⠀⠁⠀⠀⡌⠀⠀⠘⠀⡀
⠀⠀⢸⡇⢹⠀⢰⡇⢸⣇⣿⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠂⠁⢁⢈⡀⡄⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢠⠀⠂⠀⠐
⠀⠀⠘⠃⠘⠀⠛⠓⠈⠛⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠈
⣿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⠄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⣀
⠀⠎⠉⠑⡆⠀⡇⠀⠀⠹⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠤⣤⢡⣖⣒⣛
⠈⠄⢀⡠⠃⠀⠈⠒⠀⠀⠀⠀⢀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠀⠄⢀⠃⠀⢈⠷⠠⣤⠤⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⡀⠀⢀⡠⠀⢂⡠⠂⠀⠔⢁⢀⣶⠪⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠘⣍⠉⢻⡿⠟⢒⡎⠀⠀⠂⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠻⠒⠋⠀⠀⡤⣜⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠀⠀⠀⠀⠀⠐⢋⡤⠤⠈⡀
⠀⠀⠀⣀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠂⠴⠀⢀⣀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⣈⣁⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⢀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠶⡎⢹⠴⣴⣩⢈⡇⠀⠬⠭⠥⠄⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠾⢀⠿⠽⣏⠓⠒⠒⠒⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠉⠂⢄⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠦⠙⠦⣀⠀⠀⠀⣠⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠉⠀⠀⠀⢈⠆⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⣶⣶⣶⠄⠛⠂⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⣀⣴⠂⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⠐⠂⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣤⡶⠟⠻⠃⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⢀⣠⣿⡿⠤⢄⣁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⡞⡟⢱⠞⣘⡆⠀⢸⠀⣠⡟⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢦⡉⠁⠉⠉⠉⠉⣉⣀⡠⠤⠴⢲⠞⠁
⠀⠀⠀⠀⠀⠀⢁⠈⠈⠁⢸⠅⠀⠒⠋⠈⠁⠈⠀⡼⢩⠖⠒⢒⠀⠠⠄⣀⠀⡼⠁⠀⠀⢨⠗⠒⠀⠉⠁⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠏⠀⠀⢀⠄⠠⠤⠀⠐⠒⠃⠒⠛⠃⠀⠀⠀⠀⣸⠃⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⢠⠀⠀⢀⡞⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⢀⠀⢀⠈⠀⢀⠀⠀⠒⠋⠑⢤⡀⠀⠀⠐⠂⠐⠛⠉⠉⠷⣏⡀⠀⢠⠃⠀⣀⡠⠤⠀⠐⠛⡽⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠁⠀⠀⠀⢫⣙⠂⠴⠋⠀⠓⠂⢉⣉⠉⠠⠄⠒⠁⣯⠗⠒⠚⠉⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠾⠗⠊⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠠⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⡉⠉⠉⠀⠒⠀⠂⢉⣭⠁⢀⠒⢒⣠⡈⠅⠀⠀⢒⠀⠀⠠⠀⠒⠂⠠⠬⠁⢐⠒⠠⠬⠁⢒⡀⠬⢉⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠈⠓⠘⠉⠛⠛⠒⠰⠾⠿⢶⠴⡖⠲⠑⢲⠁⠲⠀⠹⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠚⠁
⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣲⠦⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⠀⠰⢇⡀⠫⢦⡑⢄⠀⠀⣶⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢐⡋⢗⠚⣛⠛⣛⣷⣀⣣⣶⣾⣍⠄⠿⢵⣦⣳⡄⠰⢿⣿⣧⠀⣄⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢠⣶⣷⣎⣿⣇⣤⣜⣻⠌⣧⠻⡿⠿⣿⣇⠀⠜⡿⠃⢹⡀⠸⠿⣿⣶⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠙⣿⠋⢱⣿⠿⠿⠗⠀⢸⡆⠀⠀⠀⠉⣾⣿⣷⡄⢸⠁⠀⠀⡘⠉⠛
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢴⣿⣿⡎⠁⠀⠀⢠⠀⠘⣧⠀⠀⠀⠀⠙⠿⠋⠀⣮⣤⣶⣾⣑⣖
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠙⠛⠁⠀⠀⠀⢸⠀⠀⡿⣄⡀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠋⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⣨⣷⣿⣿⣿⣷⣾⠿⣿⡿⠿⠿⠋
⠀⢀⣀⡀⡀⢀⡀⢀⢤⣶⣿⣷⣶⣶⣶⣾⣿⣿⡿⠛⣻⣿⣿⣿⡿
⠀⣤⣤⣈⠀⣘⣭⣵⠟⠛⠛⠋⠉⠉⢿⢿⣿⣿⣧⠀⡿⠏⣰⠋
⠀⠉⠉⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⡟⠟⢘⡆⣇⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣘⡄