Confirm Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠟⠛⠛⠛⠛⠛⠛⠛⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⢀⣤⠀⠰⠷⠀⠘⠿⠀⢰⣦⠀⠈⠛⢿⣦⡀⠀⢀⣤⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣴⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠉⢁⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣠⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠙⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠟⠁⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⠿⠀⢠⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡄⠀⠀⠹⣿⣿⣿⣿⣦⣄⠀⠀⢀⣶⣿⣿⣿⣿⡿⠋⠀⠐⠿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠁⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⢀⣤⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⠿⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠉⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣤⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠿⠁⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠉⢀⣄⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⣴⡆⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡈⠛⠁⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠆⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⡈⠁⠀⠾⠆⠀⣶⡆⠀⠿⠃⠀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢷⣶⣤⣤⣤⣤⣤⣤⣴⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⣀⣀⣀⣀⢠⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠂⠀⣤⠆
⠀⠀⠀⡎⣀⡀⠀⢠⡿⡇⠀⠀⠀⠀⠀⠀⢠⣿⠿⣿⡿⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⡿⠂⠀⠀⠀⠀⠀⠀⢠⡏⢀⡀⠀⣼⠏
⠀⠀⠀⡇⠙⢷⣠⡟⠁⡇⠀⠀⠀⠀⠀⠀⢸⣿⣦⡘⢁⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⢿⣾⠋⠀⡖
⠀⠀⠀⢣⣀⣀⣋⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⡴⠞⠁

⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡼⢁⡀⠀⢀⣾⡷⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⢀⣶⠂⡇⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⢧⠈⠻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢷⣾⠃⠀⡇⠀⠀⠀⠀⠀⠀⣇⠀⠘⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣠⣿⠋
⠀⠀⠀⠈⠳⢄⣙⣁⠤⠂⠀⠀⠀⠀⠀⠀⠘⢦⣀⣈⣁⣀⡠⠇⠀⠀⠀⠀⠀⠀⠈⠢⣄⣈⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁

⠀⠀⠀⡠⠄⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠄⣠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⡼⢁⡀⠀⢀⣾⢋⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⡇⠀⠱⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⢻⣤⡞⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠑⣦⣴⡟⠁
⠀⠀⠀⢇⣀⣀⣁⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣙⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠋

⠀⠀⠀⠀⢀⡤⠤⠤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠦⣤⠀⡀
⠀⠀⠀⡰⢡⡀⠀⢀⣼⢏⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠂⢦⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⢛⣿⠀⠀⠀⠀⠀⠀⢠⡟⢁⠀⢀⣴⡟⣥
⠀⠀⠀⢇⠈⢻⣤⡞⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⣼⠃⠀⣸⠀⠀⠀⠀⠀⠀⢹⣿⣄⠛⣠⣾⣿⠀⠀⠀⠀⠀⠀⢸⡇⠘⢷⣾⠟⠀⣿
⠀⠀⠀⠈⠢⣄⣉⣀⠴⠃⠀⠀⠀⠀⠀⠀⠈⠢⣀⠈⠃⣀⠴⠃⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣁⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⣰⠞⠋⠀⠀⠀⠈⠉⢀⣴⣶⡄⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠉⠻⣢⣾⣶⡄⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠈⠙⠷⢀⡴⠃
⠀⠀⠀⠀⣸⠏⠀⣠⡄⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⣸⠏⢀⠀⠀⠀⠀⠀⢀⣴⠟⡀
⠀⠀⠀⠀⣿⠀⠘⣿⣿⣄⣠⣾⣿⠟⢡⡟⠀⠀⠀⠀⢸⡇⠀⢶⣿⣦⣀⣴⣿⡿⠋⣴⠀⠀⠀⠀⠀⣿⠀⠈⢿⣦⡀⢀⣴⡿⠁⢀⡿
⠀⠀⠀⠀⠸⣇⠀⠘⢿⣿⣿⠟⠁⠀⣼⠇⠀⠀⠀⠀⠈⢷⡀⠈⠻⣿⣿⣿⠏⠀⣠⡟⠀⠀⠀⠀⠀⠸⣇⠀⠈⢿⣿⣿⠏⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠙⠷⣤⣀⣉⣁⣀⣴⠾⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣈⣉⣁⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⣀⣀⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⢀⡾⠛⠛⠛⠛⠛⠛⠋⠁⣠⣶⣄⠀⠀⠀⠀⣰⠟⠛⠛⠛⠛⠛⠛⠛⣡⣾⣷⡄⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠻⢀⡴⠁
⠀⠀⠀⠀⢸⡇⠀⢀⣄⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⢸⠁⢀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⢸⡇⠈⢿⣿⣧⢀⣴⣿⡿⢿⡁⠀⠀⠀⠀⠀⣿⠀⢾⣿⣦⣀⣴⣿⡿⢋⡤⠀⠀⠀⠀⠀⢸⠆⠘⣷⣄⠀⢀⣾⡿⠁⣾
⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⣿⣿⠋⠀⢸⡁⠀⠀⠀⠀⠀⣿⠀⠀⠻⣿⣿⡿⠋⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠘⢿⣿⣿⠋⠀⠀⣿
⠀⠀⠀⠀⠈⢷⣄⣀⣀⣈⣉⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣈⣉⣀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣁⣁⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠉⠛⠳⠆⣠⣿⡷⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠉⠉⢁⣴⣷⠄⠀⠀⠀⠀⠀⠤⠶⠛⠛⠶⠄⣴⣿⣷
⠀⠀⠀⠀⢠⡿⢁⣤⡀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠠⣾⣦⡀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⠀⠀⠠⣾⣷⡄⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⢸⡇⠘⢿⣷⡄⠀⣠⣿⡿⠃⡀⠀⠀⠀⠀⠀⣿⢿⣿⣄⠀⢀⣴⣿⠟⢩⡇⠀⠀⠀⠀⠀⢠⡙⢿⣿⣦⣾⣿⠟⢁⡶
⠀⠀⠀⠀⠘⣧⠀⠈⠻⣿⣿⠿⠋⢀⣾⠁⠀⠀⠀⠀⠀⣿⠀⠙⢿⣷⣿⠟⠁⠀⠸⡇⠀⠀⠀⠀⠀⠘⣷⡀⠻⣿⡿⠋⢀⣼⠃
⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠹⣦⣄⣀⣉⣁⣀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⢛⣩⣭⣭⣙⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢋⣾⣿⣿⣿⣿⣿⣿⣎⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡙⠿⣿⣿⣿⡿⠟⣡⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣶⣶⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⣿⣿⣿⣿⣿⡿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⣶⣶⣶⣶⣶⣶⡶⢠⣿⣿⣿⣿⣿⣿⢏⣴⡷⣨⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡿⢛⡻⣿⡿⣡⣾⠏⣴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣦⡻⣷⣌⣴⡿⢣⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠘⣿⣷⡜⢿⠟⣴⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉





⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠛⠒⠦⡄⠀⠀⣠⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⣠⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡴⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣋⡻⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⢰⠃⠀⠀⡴⠋⠀⠈⠳⣄⡠⠊⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢫⣾⣿⣿⣮⡻⠟⣡⣾⣿⣿⣿⢟⡁
⠀⠀⠀⠀⢸⡀⠀⠀⠙⢦⡀⠀⠀⠈⠀⠀⠀⢀⠔⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡙⢿⣿⣿⣿⣾⣿⣿⣿⢟⣵⣿⡇
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⠔⠁⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⢟⣵⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠙⢦⠔⠁⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⡙⢟⣵⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠰⡟⠛⠛⣷⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠋⠙⣷⠶⠾⠛⠀⠈⠻⠶⢶⡟⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣿⠆⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠾⣧⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠻⠟⠁⠀⢠⡶⠆⠀⠀⠀⠉⠉⠀⠀⢀⡀⠀⠈⠿⠋⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠀⢠⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠷⣤⣀⠀⠀⠀⢀⣀⣰⡏⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣬⣿⠀⠈⠙⣷⡄⠀⣿⠛⠛⠁⠀⠈⠁⠀⠀⢠⣄⠀⠀⠀⣠⣴⠞⠋⠀⠀⠀⣶⠀⠀⠘⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⢠⡟⣀⣴⠞⠋⠀⠀⢿⣤⣤⡄⠀⠀⡀⠀⠀⠀⠻⣦⡶⠛⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠈⠙⣷⡆
⠀⠀⠀⠀⢀⣀⡀⢠⡾⢷⡆⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⡾⢷⡀⠀⠀⠛⠶⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⣠⡾⢦⣴⠏
⠀⠀⠀⠰⣟⠙⠷⠟⠃⠘⠛⠾⠏⣹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢺⣏⠀⠀⠁
⠀⣴⠷⡾⠋⠀⢴⠶⠂⠀⠀⢤⣄⠙⢷⡾⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣰⡟⠛⠳⣶⠀⠠⣶⠛⠛⣷⣠⡴⠟
⠈⢻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢰⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⠷⠶⠶⠿⠀⠀⠈⠉
⣶⠾⠃⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠿⢶⡄
⠻⢶⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⢀⡶⠿⠃
⠠⣾⠟⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠹⠾⠻⣦⠈⠙⠀⠀⠀⠾⠛⠀⣠⡾⠳⠟⠁⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣾⠛⣷⣀⣤⣄⡀
⠀⠀⠀⠸⢯⣤⡷⢶⡆⢠⡶⠾⣦⣼⠗⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⡧⠘⠋⠉⠀⢉⡙⠃⢼⣇⣀⡀
⠀⠀⠀⠀⠀⠉⠀⠘⢻⡿⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡙⠋⣠⡾⠀⠀⠀⠈⠙⠃⠀⠙⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣤⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⠈⢷⠀⠀⠀⠀⠀⠀⠀⠈⠁⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡶⢷⣄⠀⠀⠀⢤⡶⠆⠀⣠⡾⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⠶⣦⠀⣴⠶⣦⣽⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣶⣄⡀⠀⢸⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣦⣿⣿⣿⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣏⠀⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣛⣋⣁⣻⣷⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⢹⡇
⠀⠀⠀⠀⠠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢸⡁⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠆⠘⠁⠀⠀⠀⠀⣠⡶⠋⠀⠈⠻⣦⡀⠀⠀⠀⠀⠘⠁⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⠀⠀⠀⢠⣶⣄⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⢸⡟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣄⣿⠀⠀⠀⠀⠙⢿⣷⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣇⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠈⠛⠶⣄⡀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠈⠙⠷⣤⣀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠏





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⣀⣀⣀⡀⢀⣄
⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠙⣣⣾⡿⠂⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠙⣧⣦⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⢉⣿⡟⠁
⠀⠀⠀⠀⡇⠀⣴⣦⡀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣤⡀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⠰⣶⣦⡀⢀⣴⣿⣯
⠀⠀⠀⠀⡇⠀⠈⠻⣿⣾⡿⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠻⣿⣦⣾⡿⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣿⣿⠟⠁⣿
⠀⠀⠀⠀⣇⣀⣀⣀⣈⣛⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣈⣛⣋⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣘⣋⣀⣀⣿
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣠⠄⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡤⢀⣴⠆⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡤⢀⣴⣦
⠀⠀⠀⠀⡇⢠⡀⠀⠀⠀⠀⢈⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⡇⢠⣤⣄⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⡇⠀⠙⣷⣄⣠⣶⡿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢿⣿⣆⣰⣿⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢿⣿⣦⣴⣿⡿⡇
⠀⠀⠀⠀⡇⠀⠀⠈⢻⡿⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠹⣿⣿⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠹⢿⡿⠋⠀⡇
⠀⠀⠀⠀⡧⠤⠤⠤⠤⠤⠤⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠬⠥⠤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠤⠤⠤⠤⠤⠴⠇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⢒⣴⣿⡧⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠚⠒⠚⣢⠴⠊⠀⠀⠀⠀⠀⠀⡖⠓⠒⠒⠒⠒⠒⢂⡴⠋⠁
⠀⠀⠀⠀⡇⢀⣶⣄⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠳⣦⡀⢀⣴⡾⡃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣦⣄⠀⠀⣴⠟⡄
⠀⠀⠀⠀⡇⠈⠻⣿⣷⣾⡟⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣿⣿⠟⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⣿⣧⣼⠏⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠈⠛⠛⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠿⠏⠀⠀⡇
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣴⣶⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⡄⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⣀⡴⠖⠚⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⣳⡀⣰⡿⢹⠃
⠀⠀⠀⠀⣴⢋⣤⣶⠿⣻⣷⠟⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠈⠀⢻⣿⠃⣼
⠀⠀⠀⢰⢧⣾⠏⣴⣿⠟⢡⡄⢇⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣏⣠⣿
⠀⠀⠰⠿⣿⣿⣼⣿⡁⠀⠀⢤⠊⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⡭⠴⢾⢸⡄⠀⠀⠀⠀⢀⡔⠁⣄⠀⡇⢈⠀⢹⡆
⠀⠀⠀⠀⠀⡸⣿⣍⣁⣆⠐⠋⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡏⠀⠢⠀⡞⣼⣁⠀⠀⠀⠀⠙⣆⠀⠀⠸⣵⢋⡇⣸⡇
⠀⠀⠀⠀⢰⡟⢦⠉⠉⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⣿⡁⠀⡤⠶⣿⣇⣽⠿⣛⢿⣷⡆⢿⠁⢀⣀⣠⠿⣴⣿⣇
⠀⠀⠀⠀⣼⠀⠈⣧⠀⢹⣧⠀⢀⣠⡴⠟⢛⢮⡻⢿⣷⡇⠉⠛⠁⠀⣿⣿⡏⢸⣿⡀⣿⣟⢻⣿⣿⠿⣓⡛⠿⣯⠿
⠀⠀⠀⠀⡇⠀⠀⠘⢧⣸⣿⠶⠋⢹⣷⣦⡺⣷⡿⠀⠘⣇⠀⠀⠀⠀⣿⠁⠻⢦⣿⠀⣿⣷⡿⣿⠃⣰⠿⢿⣷⡀
⠀⠀⠀⠀⣇⠀⠀⢀⠟⡏⠁⠀⢀⡾⡿⠉⢻⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⣤⣤⣤⣤⣴⣿⣿⢹⡟⢠⡏⠀⠀⣿⣿⡄
⠀⠀⠀⢠⣿⣦⣀⠈⠀⣁⣠⡴⠋⠀⣇⠀⢈⣿⡟⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠉⣾⢁⡾⠀⠀⢠⣿⣧⠹
⠀⠀⢀⡿⠉⠀⠉⠉⠋⠉⣿⡇⠀⠀⣿⠰⢣⠏⠛⢦⣼⣤⣴⣶⣆⠀⠀⠀⠀⠀⣀⠤⣿⣴⣇⣼⡁⠀⠀⣸⣿⣿
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⠇⠀⢰⣿⡄⠸⠀⠀⠀⣿⡇⣿⣒⣺⡇⠀⠀⠀⢨⣿⣽⡿⠀⣿⠉⢿⣦⢠⡟⣿⣿
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⣾⠏⢷⣀⠀⠀⢀⡿⣷⠛⠒⠚⠀⠀⠀⠀⢸⣿⠽⠥⢀⣿⠀⠘⣿⣿⠀⢸⣿
⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡏⠀⠀⠉⠛⠚⠋⠁⣿⠄⠀⠀⠀⠀⠀⠀⠈⢻⠿⣶⣿⣿⠀⢀⡿⠃⠀⠘⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⢡⣿⣿⣿⡴⣿⡇⠀⠀⠀⣿
⠐⣯⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡿⢨⣿⡇⠉⠀⣿⡇⠀⠀⠀⣿
⠀⠈⠛⠷⣶⣤⣦⣠⣶⠾⠟⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣇⢸⣿⡇⠀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠘⡏⠉⣹⠛⡏⠀⠀⠀⠙⠓⣶⣦⣤⣤⣤⣤⣤⠾⠇⠀⠀⠀⠀⠀⠀⢸⣏⣙⣿⣁⠀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⣧⠀⡿⢸⡇⠀⠀⠀⠀⠀⡇⠀⣸⠀⡏⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣿⣍⣉⢉⣿⠁⠀⠀⠀⣿
⠀⠀⠀⠀⣽⠀⢧⠘⠧⣤⣀⠀⠀⠀⣿⢀⣷⡀⣇⣐⣇⡀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣶⣿⡈⠉⠉⠉⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⢿⡶⢶⣿⣿⣿⡟⠋⠀⠀⣿⣿⣧⣬⣿⣯⣴⣿⡿⠶⠀⠀⠀⠀⠀⠀⠉⢹⣿⢉⣿⣷⣶⣶⡿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⠛⠛⠿⠛⠁

⠀⠀⠀⠀⠀⠀⡔⠂⠒⠒⠒⠐⠐⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⡀⠀⡀⢈⣉⣉⡁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣾⣾⣿
⠀⠀⠀⠀⡠⠊⡇⠐⠒⠒⠒⠒⠂⠀⡍⢢⡀⠔⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣭⣭⣭⣭⣽⣿⣿⣦⠀⣠⣄
⠀⠀⠀⢰⠤⣀⡇⠀⢀⠤⢢⠤⡀⢀⠷⠊⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⣿⡿⣻⣝⣻⢿⣿⣛⣵⣿⠟⠋
⠀⠀⠀⢸⠀⢀⡹⠒⠁⠀⠑⢄⠈⠁⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣎⣽⣾⣿⣮⡻⣷⣿⡿⣫⡇
⠀⠀⠀⢸⠖⠉⠀⠀⠀⠀⠀⠀⠙⠴⠉⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⣾⣿⣿⣿⣿⣿⣿⣮⣛⣵⣯⡇
⠀⠀⠀⠸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇


⣴⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣦⣦⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠺⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⠀⢀⣤⡄⠀⣤⡄⠀⠀⠀⣿⡇⠀⠀⠹⣿⡆⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠟⠉⠉⠙⢿⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣾⡿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⢺⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡗
⠀⠀⠀⢘⣿⣷⣤⣀⣤⣾⣿⠀⠀⠀⢸⣿⡄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷⣤⣀⣤⣾⣿⡃
⠀⠀⣰⣿⠟⠉⠛⠛⠋⠉⢻⣷⡄⠀⢸⣿⣿⡿⠋⠸⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣿⡇⠀⢠⣾⡟⠉⠙⠛⠛⠉⠻⣿⣆
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⣈⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣯⠿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢐⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⢲
⢰⠀⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠑⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡞
⠈⢆⠀⠀⠈⢦⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⠛⠛⠁
⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬



⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣠⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠤⡧⢤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢳
⠀⠀⡀⢴⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣡⠤⠧⢤⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠒⠚⠦⢊
⠀⠀⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤
⠀⠀⠀⠀⣠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣄⠀⠀⢀
⠀⠀⠀⠀⠉⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⠀⢀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⠀⠀⠣⣀⠜
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠙⠄⠀⠖⠉⠱
⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢻⣿⣿⣻⣷⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⠿⢿⡿⠇⠀⠀⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣽⡟⠀⢐⠂⠀⠀⠒⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡇⠀⠉⠀⠀⠐⢺⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣷⡚⢿⣿⢆⠀⢤⠀⡄⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣸⡿⠀⢀⡹⠙⢦⣀⠀⠠⠋⠉⠀⠀⠀⠀⠀⣠⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠛⡁⠄⠺⣷⣄⠀⣹⡷⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠻⣯⡿⠃⠀⠀⠀⠈⠀⢠⠞⠁⡀⣀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠬⠮⠀⠀⡠⠀⠀⡀⠀⠉⠀⠀⠀⣠⠂⠀⠘⠁⢀⡼⢍⢪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠱⣣⣼⣥⣾⣶⣤⣕⢄⡀⢀⣴⣿⣿⣦⣇⠀⢸⠖⠂⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⠝⡿⢿⣿⣿⣿⣿⣿⣾⡿⢿⣿⣿⣿⣿⣿⠀⠈⣦⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢳⣿⣠⣧⣉⣻⠿⣿⣿⣿⠟⠁⠺⣿⣿⣿⡿⠛⢆⠀⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⡿⢗⡀⠀⢒⣿⣿⠟⠀⠀⠈⢂⣠⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢹⣿⣿⡏⠀⠻⣿⣿⣯⣯⣭⠥⠄⠐⣿⣿⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⣿⡟⠀⠀⠀⢸⣿⣿⣷⣶⣷⠄⠀⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣸⣿⣿⣿⣿⠯⠀⢀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⠛⠋⠁⠺⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣭⣕⡀⠀⠀⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡋⠁⠀⠀⠀⢾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⠿⠟⠋⠅⠀⠀⠀⠰⠿⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣴⠶⣶⠏⠀⠛⠛⢳⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠿⠶⠿⠶⠶⠶⠶⠟⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣠⡶⠛⠛⠛⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠋⢰⡏⠀⠀⠀⢀⣤⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁⢸⡆⠀⠳⣶⠟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢈⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⡈⠻⣄⡀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠈⠉⢛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⢛⣿⣿⠻⢻⢟⣻⣿⡟⣿⡟⣿⡟⡟⡟⡟⣛⣿
⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡄⣿⢸⢨⣿⣿⡇⣾⡸⢿⡿⠇⣿⡇⢿⡇⣇⢁⡇⢼⣿⢠⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣏⣛⣛⣛⣛⣛⣛⣛⣓⠚⣛⣚⢛⣛⣚⢛⣛⣚⣛⣒⣛⣸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣿⣿⣿⣯⡀⢸⡏⣿⢸⣏⣿⢸⣿⣿⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡏⠁⢸⣇⣿⢸⡟⣿⢸⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⡍⣭⡭⠍⣠⣥⡄⣤⣭⡅⣬⠁⠈⣤⠭⠉⡭⢩⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡇⣿⣦⠀⣿⢸⡇⣿⣤⡇⣿⠀⠀⣿⡤⠀⠁⠹⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⢿⣤⡄⢿⣼⠇⣿⠀⠀⣿⣤⡄⣿⣤⠀⠀⢲⣬⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣽⣏⣤⣤⣠⣤⣄⣤⣤⣠⣤⣄⣰⣟⣤⣤⣀⣤⣤⣀⣤⣤⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠹⠏⠹⠏⠙⠻⠉⠛⠏⣹⡿⠿⠛⠟⠺⠿⠿⠾⠻⠟⠻⠍⠙⠿⠉
⠀⠀⠀⠀⠀⠀⠀⢻⣼⠇⣾⢻⡆⡇⢸⠀⠀⢿⣼⣶⡇⣿⢸⡆⠀⡇⠀⠀⠀⣿⢳⢸⡇⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⣸⡇⣇⣸⠀⠀⢸⣿⣿⠃⣿⢸⡇⠀⣇⠀⠀⠀⣿⣸⢸⡇⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣭⠄⣩⣍⣠⣤⢨⡅⣭⠀⠩⣬⠍⡅⡍⣭⠄⡄⣭⡍⠈⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠃⣿⠂⣿⣾⣿⣾⢸⣷⣿⠀⠀⣿⠀⡷⡇⡷⠂⡇⡷⡇⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣧⢳⣦⠈⠉⠉⢸⡾⠈⠉⠀⠀⠉⠀⣡⡥⢿⡇⠁⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⢿⣯⣵⣿⡿⣷⡿⣴⣿⡿⣶⡿⠿⣷⡿⣽⣟⣾⢯⣿⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠘⠛⠉⠣⣼⠿⠃⠙⠃⠀⠉⠃⠈⠙⠉⠛⠁⠙⠋⠉⠛⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔⠲⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⢎⡇
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠛⠀⠒⠒⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢊⠴⠁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠸⠃⠀⠈⡆⡔
⠀⠠⠀⠀⠀⠀⣰⣁⡍⢀⡴⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠓⠒⠒⠒⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠀⠀⠀⠀⠀⡠⠂⠉⠉⡇⠢⡀⠀⠀⠀⠀⠀⡇⡤
⠀⠀⠀⠙⠔⠉⠉⠉⠉⠁⡜⢣⠈⠉⠉⠉⠉⠑⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢘⠀⠀⢀⡴⢆⠀⢸⠀⠀⠀⠀⢀⠈⡀
⠀⠀⢠⠖⠉⠉⢹⠙⠢⡀⠑⠊⠀⢀⠔⠋⠉⠉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⢦⡀⠀⠈⢹⠈⠁⠀⠀⠸⠛⠀⠀⢀⠔⠁⠀⢈
⠀⢀⠁⠀⠀⢀⠾⡀⠀⢡⠀⠀⢀⠃⠀⠀⢀⠼⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠦⣠⡈⢆⡀⠀⣀⠞⢸⡤⠔⠁⠀⢀⠴⠁
⠀⠀⠙⠂⠉⠁⠀⠈⢙⠈⠀⠀⠈⠛⡆⠈⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡠⠒⠛⠂⠘⠓⠢⡀⠀⠀⠀⡤⠖⠛⠃⠘⠋⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡴⣔⢉⢩⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡔⠒⢄⠀⠈⠁⠒⢄
⠀⠀⠰⠢⢄⠀⠀⡰⠛⢤⡣⠄⠃⠈⢣⠀⢀⠀⢀⡆⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠢⡊⠗⢄⠀⠀⠈
⠀⠀⢐⠀⠠⠃⠀⠑⠀⠲⡀⠁⡛⠀⠘⠀⠀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠙⠦⠴⠃⡀⠜
⢀⠄⣈⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠑⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡀⠑⠄⣀⡠⠎⢠⠒⠄
⠣⠠⠆⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢦⠀⠀⠐⠒⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢬⣊⠀⠀⠀⠀⠀⠀⢈⡓⠅⢠⣄⡀
⠀⢠⠄⠤⡀⠀⠀⠠⣁⣀⡠⠚⠤⣀⡇⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢠⠀⠑⢄⠀⠀⠀⡰⠋⠀⠀⢠⠀⠉⢆
⠀⠸⡀⠀⠉⠢⡀⠀⢘⡀⠀⠀⠀⡇⠀⠀⡠⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠴⠋⠢⣀⣈⠀⠀⠀⣇⣀⣀⠤⠊⠢⢀⣈
⠀⠀⠈⠢⡀⠀⠈⠒⠤⠵⠦⠤⠾⠤⠐⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣀⣠⠃⠀⠀⠀⠀⡇⠑⣄⣀⣀⣠⠊
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠗⠉⠀⠀⠀⠀⠉⠲⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣤⣄⣀⣤⣤⣤⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣤⣤⣤⣤⣤⣤⣤⣤⣃⣀⣈⣀⣀⣀⣠⣄⣤⣤⣤⣃
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⡟⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⣧
⠀⠀⡏⢉⠔⠁⠀⠀⣤⠀⠀⢀⣤⡀⠀⣠⣄⡀⣤⣤⡄⢠⣤⣄⠀⣤⣤⣄⢠⣤⣤⠀⣤⣤⡀⠀⠑⢌⠙⡇
⠀⠀⡇⢸⠀⠀⠀⣸⣻⡆⢠⡇⠀⠀⢸⠁⠀⠀⣿⣤⠀⢸⣧⠼⠃⠀⣿⠀⢸⣧⡄⠀⡇⠀⣿⠀⠀⢸⠀⡇
⠀⠀⡇⠸⣀⠀⠠⠏⠉⠻⠄⠳⡤⠆⠘⠧⠴⠂⠿⠤⠤⠸⠇⠀⠀⠀⠿⠀⠸⠦⠤⠄⠧⠴⠏⠀⢀⡸⠀⡇
⠀⠀⠙⠲⡈⠣⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠊⡴⠚⠁
⠀⠀⠀⠀⠛⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⠿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⣀⣠⣄⡀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃⠀⣠⣾⠟⠛⠛⢿⡄⠀⠀⣰⠟⠉⠙⠻⣦⡀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⣷⡀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⢀⡠⠤⠤⣤⡀⠀⠀⠀⢀⡤⠚⠓⠒⢤⡀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢠⠋⠀⠀⣀⣀⣹⡄⠀⢠⡯⢠⣀⣀⣀⠀⢻⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠀⠀⠀⠀⠀⣼⡶⠾⣿⣿⠏⢈⡇⠀⠀⣇⠈⠿⣿⠏⠉⣻⠇⠀⠀⠀⠀⢸⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡴⠂⠀⠀⠀⠀⢀⣬⣦⣀⢀⣀⣠⣾⠁⠀⠀⣈⣷⣤⣀⣤⣴⡿⠶⠀⠀⠀⠀⠀⢲⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣉⣀⠀⠀⠀⣀⣤⡾⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣦⣄⠀⠀⣀⠘⣁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠉⠀⢠⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⢹⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠙⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⠟⠁⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣠⣄⠈⠙⠛⠿⠷⢶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠿⠟⠛⠉⣀⡀⠀⠀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡼⢻⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⣿⡟⢷⠄⢸⣿⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣿⡇⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⣀⡀⠀⣀⣀⣠⣤⡶⠛⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣦⣀⠀⠀⠀⠀⠀⠀⠠⣦⣄⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⣿⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠉⠻⢷⣦⣄⡀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠀⠀⢀⣀⣤⣶⡾⠟⠋⠀⢰⣿⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡇⠀⠀⠀⠀⠈⠙⠛⠿⠷⣶⣶⣶⣶⣶⣶⣶⡿⠿⠟⠛⠋⠁⠀⠀⠀⠀⢸⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿
⣶⠛⠻⢿⡟⠳⣄⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀
⠘⣧⠀⠀⠹⣦⣨⣷⠶⠶⠿⠃⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡄⠀⠀⠀⣠⠞⢋⣽⣧
⠀⠈⢳⡀⠀⠘⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⣧⣤⣄⣰⠏⡴⠋⠀⣰
⠀⠀⠀⠙⢦⣀⢀⣀⣀⣀⣀⣠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠻⠞⠁⠀⣰⠋
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⡀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠛⠛⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠻⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⣹⡏⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠾⠷⢀⣼⣧⣤⣀⡀⠀⠀⠀⠀⣀⣤⣾⡆⠀⣨⣿⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣦⣴⡾⠋⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣾⣿⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⢀⣠⣶⠶⢶⣄⣀⣀⣀⣀⣀⣤⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠓⠶⣦⣤⣀
⠀⠀⠀⢠⡶⠋⠁⠀⣠⣾⠿⠛⠋⠉⠉⠉⠉⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣀
⠀⠀⣴⠏⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⣾⠏⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⢸⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⣴⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠉⠛⢷⣾⣧⣴⠞⠛⠉⠉⠉⠁⠒⠀⠀⠀⠀⠀⠀⠹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠙⢿⡇
⢿⡄⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⣇⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢿⣆⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠻⣧⡀⠀⠘⢿⣤⡀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⢼⠁⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣇
⠀⠀⠀⠙⢿⣦⣀⠀⠹⣿⣷⣦⣄⡀⠀⠀⣸⠟⠻⣦⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠙⣿⡄
⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠉⠉⠙⢿⣶⣶⡟⠀⠀⠈⠑⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⣀⠀⠀⢀⣀⣠⣤⠶⠋⠁⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢰⡟⠙⢻⣆⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣄⣠⣤⣴⠶⠞⠋⠁⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⢸⡇⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣸⣷⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣧⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠞⠛⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣶⣤⣶⣖⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠉⠛⠛⠉⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢋⣠⣴⣶⢿⠿⡿⡿⣶⣦⣄⡙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢋⣴⣿⠛⠍⣃⣧⣤⣤⣥⣜⡜⠹⢿⣦⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣾⣿⣄⣱⣾⣿⣛⣯⣽⣛⣿⣷⣆⣴⣿⣷⡀⠿⣿⣿⣿⣿⣿⣿⣿⣽⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⠉⠁⢀⣀⢉⡉⢉⣉⢉⣈⡉⣉⠉⡉⣀⣀⣉⣁⡉⣈⢀⡀⠉⢉⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷⠀⢸⣩⢸⣿⢸⣿⣿⢺⣇⣏⣭⡇⣿⣿⢹⢸⣧⣿⣼⡇⠀⢾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣀⣀⠀⢁⣀⣁⣀⣀⣀⣀⣀⣉⣀⣁⣀⣀⣈⣀⣀⣀⡀⠁⣀⣀⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⢿⣿⠟⡙⢿⣿⣽⣯⣽⣯⣿⡿⠛⠻⣿⡿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠹⣿⣦⣄⡈⡉⠛⠛⡋⢡⣢⣤⡾⠋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⠉⠛⠿⠷⡶⠷⠿⠿⠛⠉⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣤⣴⣦⣤⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠖⠚⠛⠛⠒⠲⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⢀⡀⢀⡀⢀⡀⠀⡀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⢐⣦⡸⠋⠸⠉⠚⠃⠯⠽⢰⡴⣠⠄⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⢤⡀⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠓⠱⣋⠆⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⡐⢦⡨⠊⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠉⠉⠛⠲⢤⣀⠀⠀⠀⠘⠂⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⣐⠺⠆⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠐⣶⢂⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⣘⡴⠃⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠙⣃⡀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣼⠛⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠹⡄⠀⠈⣙⣇⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣝⠃⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⢻⠀⠀⢘⣿⡀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢈⣩⠆⠀⢰⡇⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢖⡃⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⠙⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢈⣓⡂⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠃⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡨⣉⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢠⠶⡄⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠑⠚⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⣉⣴⠄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠐⠂⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠐⡶⣀⠀⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡠⢷⡄⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠸⠥⢇⢄⠀⠀⠉⠓⠦⣤⣀⣀⣀⣀⣀⣀⣤⠴⠚⠉⠀⠀⢠⡠⠿⠆⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠛⢫⣤⠄⢀⡀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⢀⢤⡠⣷⡜⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠘⠟⢸⠞⠀⢠⡄⠀⡔⡆⢲⡆⡟⠜⠶⠁⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠦⢤⣤⣤⠤⠴⠶⠖⠛⠋⠁

⠀⠀⣴⣶⣶⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⡏⠈⠉⠻⡆⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⡆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣤⣀⠀⢀⣤⣶⠀⠀⠀⠀⡇⢀⠀⢀⠥⡀⠀⠀⠀⢸⠁⠀⣀⣣⠀⠀⡁
⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠧⠤⠤⣾⢶⡇⠀⠀⠀⢸⣀⣀⣿⣧⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⡖⠒⠒⠒⠒⠒⡄⠀⠀⠀⢀⡔⠒⠒⢲⡄⠀⠀⠀⠀⠀⢰⠚⠛⠛⠛⠂⠀⠀⠀⢠⣤⣤⣤⣤⡤⣤⡄
⠀⠀⡥⠤⠤⡴⠶⢮⠀⠀⠀⠀⢸⡇⠀⠀⢀⣘⠀⠀⠀⠀⠀⢸⢀⣏⣵⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣛⣷⡇
⠀⠀⠧⠥⠥⠧⣀⢼⡀⠀⠀⠀⢸⢇⠀⠀⣸⣤⠇⠀⠀⠀⠀⢸⠸⠿⠷⠀⠀⠀⠀⠀⠘⠟⢿⠷⠛⣿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⠒⠓⠂⠛⠛⠃⠀⠀⠀⠀⠘⠒⠒⠒⠒⠂

⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡖⠒⢦⡀⠀⠀⠀⠀⠰⠒⠒⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⣿⣿⠿⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⡇⠀⠀⡏⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠴⠞⠉⡁⠀⠘⡄
⠀⠀⣿⣿⣴⢛⣿⣿⣿⠀⠀⠀⠀⠣⡀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠘⠢⠤⣾⢾⡤⠔⠃
⠀⠀⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠰⠦⠤⠤⠴⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡇⠀⢠⣁⡄⠀⢸⠀⠀⠀⢀⣠⣀⣈⣙⡂⡀⠀⠀⠀⠀⢀⠿⠤⠤⢧⠀⠀⠀⠀⠀⡠⢴⢖⣲⠦⢄
⠀⠀⣇⣀⣘⣛⣀⣀⣸⠀⠀⠀⠈⣿⣿⣻⣿⣿⡇⠀⠀⠀⠀⢸⢰⣿⣾⠀⠁⠀⠀⠀⠈⠢⢼⣝⣛⡤⠔⠁
⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠿⠿⠷⠾⠿⠇⠀⠀⠀⠀⠘⢤⣉⣈⠔⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡶⢦⡀⡶⠶⠄⡶⠶⡄⢰⡄⣠⡆⣤⢀⡴⠦⠀⡴⠦⠀⡆⢠⠶⢦⢠⣦⢠⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡟⠋⠀⣏⣁⡀⡟⢻⡅⢾⠻⢿⡇⣿⢠⣍⣳⠀⣍⣹⠆⡇⠸⣄⡼⠸⡏⢻⡇⠀⠀⠀⠀⣿
⠀⠀⣀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀
⣴⣞⠋⠀⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣿⠀⢙⣳
⣿⠉⠛⠶⣿⣄⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⣠⣿⠶⠛⠉
⣿⠀⠀⠀⠀⠉⠛⠶⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⠶⠛⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡶⠟⠋⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣩⡿⠟⠋⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡿⠿⠿⠿⠿⢿⣿⡿⠟⠋⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣦⣴⡞⠋⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠉⠛⠷⣦⣄⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀
⣿⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⣀
⣿⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀
⣿⣶⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣶
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⡰⠀⠀⠀⠀⠀⠀⣀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠧⣿⠂⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠰⣶⡀⠀⠱⢠⣶⢶⣐⣌⣯⣴⡿⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠱⡄⠐⠀⠀⢀⠀⠀⠀⢢
⠀⠀⠀⠀⢠⣿⣷⠈⢊⠹⣿⢊⣉⢁⣬⠴⠚⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠳⠅⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⡀⢠
⠀⠀⠀⠀⣿⣿⣿⡇⢶⡴⣛⡀⢖⡿⠧⢀⣠⡾⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠙⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⠂
⠀⠀⠀⠀⣿⢿⢿⣧⢻⣷⣿⣃⢻⣯⣶⣿⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠠⠆
⠀⠀⠀⠀⠻⣇⠀⢿⢾⢿⢞⠻⣄⣼⣿⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡑⠂⡀⠀⠀⠀⠐⢡⠀⣧⡀⢀⠃⢐⡈
⠀⠀⠀⠀⠀⢹⣆⠀⠀⠺⡗⠗⠑⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠁⠄⠀⠀⠀⢸⢀⠋⢹⣆⠈⢩⢀⡃
⠀⠀⠀⠀⣴⣿⣯⡿⢦⣤⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⠄⠈⠀⣠⣿⣾⡧⡗⡟⢠⣇⠼⠁⡀
⠀⠀⠀⣾⣿⣿⣌⠙⠒⣅⣼⡄⢻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠈⠉⠻⣿⣿⣷⠄⡴⠿⠁⠴⠋
⠀⠀⠀⣿⣿⣿⣿⣦⡀⢿⣿⡇⢼⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠷⠶⠺⠊⠐⠀⠀⢀⢐⣠⡥⠂⠀⠀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣧⣸⣿⣷⣸⣿⣿⣿⣿⣿⣿⣟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣆⣴⣾⣙⡛⠛⢿⣿⡾⣠⡁⠀⢐
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⣌⠂⡘⠿⣿⣿⣿⠶⣦⣌⡀⠁⡀⢀⢳
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣧
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⣉⣿⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⢀⡴⠋⠀⠀⠀⠀⠀⠉⢉⣉⣩⣽⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣷⡞⠁⠀⠀⠀⠀⢀⣴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠟⠀⠀⠀⠀⢀⣠⠽⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣠⡴⠾⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢀⡾⠋⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠉⡀⠀⠰⡏⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⣾⠀⠀⠸⣏⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢻⠀⣀⡀⠹⣆⠀⠀⠀⣿⢀⣀⣤⣤⣤⣭⣥⣀⡀⠀⠀⢀⣠⣭⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⢳⡄⢹⡆⠀⢀⡿⠛⠉⠀⠀⠀⠀⠀⠉⠛⢶⡞⠋⠁⠀⠀⠀⠀⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣞⣷⠀⢾⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⣼⣟⢺⠟⠉⠉⠉⠉⢉⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡼⡇⠈⠳⣤⣟⣁⣀⣀⣠⣤⣴⡾⠋⠉⠛⠓⠶⠶⠖⠻⠿⠤⠶⠶⠶⠶⠚⠛⠛⠛⠛⠛⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢉⣉⣉⠛⢿⡄⠀⠀⠈⠉⠉⠭⠵⠞⠋⠀⠀⠀⠀⠻⣆⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠉⠑⠛⠓⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡶⠶⣦⣤⣬⣤⣈⣁⣈⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢼⠁⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢈⣧⣼⡦⠾⠗⠚⠁⠀⠀⠀⠀⠀⠀⡿⠲⠖⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢟⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⢙⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠛⠉⢀⣠⣵⠚⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠤⠤⠤⠤⠤⠖⠒⠒⠛⠀⠀⠀⠀⠀⠀⠉⠉⢹⡦⣄⣠⡤⠤⠶⠖⠚⠉⠁⠀⠀⢀⣠⠴⠚⠉⠀⣹
⠀⠀⢀⣿⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠤⠀⠀⢸⡇⣩⡟⠛⢦⡀⠀⠀⠀⣀⣤⠞⠋⠁⠀⠀⠀⠀⣿
⡴⠚⠉⣥⡬⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡐⠀⠀⠀⠀⣠⣤⣼⣟⣏⠀⠀⠀⠙⠲⣴⠞⠉⠀⢀⠀⠀⠀⠸⣗⠀⢿⣶⠄
⠳⠦⣤⣼⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠁⠀⠀⠀⠸⡇⣖⡆⠉⠉⣹⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⡀⠐⢫
⠀⠀⣿⠀⠐⢯⢗⡇⠀⠀⠀⠤⡠⡡⠀⠁⠈⠀⠀⠀⠀⠀⠀⢠⡞⣿⡉⠉⠉⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⣀⣀⣨⡧⠀⠘⣆
⠀⠀⣸⠟⢛⣿⣿⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣯⣥⣤⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡁⠀⠀⠀⢀⣿
⠀⠀⠙⠦⢼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡰⠳⠀⠘⣗⠒⠒⠒⠒⠒⠀⠤⠤⠤⠴⠇⢸⡧⠶⠖⠛⣿⠉
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢲⣾⡏⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠲⠾⠃⠀⠀⣸⠃
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣴⣟⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠖⠒⠒⠚⠛⠛⠛⠉⠙⣏⠙⢿⣏⡭⢉⡤⢙⠢⣄⡦⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠤⠬⠽⢿⣳⡬⣶⡟⢿⣦⣀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⣷⣤⣤⣤⣀⣀⣉⣰⣿⠁⣺⡿⠈⠉⠉⠁⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⢘⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣠⣬⣿⡶⠶⠛⠉⠉⠉⠙⠛⠉⠛⠲⠶⠦⢤⣤⣤⣠⣤⣤⣤⠴⢶⣞⣿⣻⠟⠉⠉⠛⢶⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⢀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⢃⣠⠖⠶⠶⠶⠖⠛⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠙⢻⣿⠉⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⣿⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠃⠀⠀⣿⠀⠀⢸⣇⣀⡟
⠀⠄⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣿⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠤⠴⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣶⣤⣴⣿⣿⣿⣿⣦⣀⣲⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡆⢾⣿⣿⣿⣿⠿⠿⠿⠿⠻⠿⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣑⠿⠿⠋⠁⠀⠀⡀⠠⡆⢠⠀⢘⣉⡅⠀⣴⣧⢠⠖⠆⣠⣀⠀⠀⠈⠙⠿⣿⣿⣛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣮⠝⠀⠀⠀⠠⠀⢯⠆⠀⠓⠈⠋⠁⠉⠀⠀⠀⠈⠈⠓⠀⢧⢀⡼⠲⢀⡀⠀⠈⠻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡿⠃⠀⠀⣴⠂⡘⠵⠊⠀⠀⠀⠀⢀⣀⣀⣀⣄⣀⣀⡀⠀⠀⠀⠈⠑⠰⠻⠝⢠⡀⠀⠈⢻⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠀⠀⢰⡂⢈⠛⠁⠀⠀⢀⣤⠶⠚⠉⠉⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀⠀⠀⠀⠰⠋⣡⢄⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠀⠀⢤⣠⠍⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠘⢎⡨⢤⠀⠀⠘⣿⣿⣄
⠀⠀⠀⠀⢠⡾⣿⣿⡟⠀⠀⠀⠀⠙⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠈⠓⠚⠀⠀⠀⠹⣿⣿⣷⣆
⠀⠀⠀⠴⠾⠇⠼⠿⠶⠴⠶⠶⠶⠶⠶⠦⠶⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠶⠦⠴⠶⠶⠾⠿⠿⠷⠶
⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣀⣤⣤⣤⡀⠀⢀⣤⣤⣤⡀⠀⣤⣤⣤⣤⡄⠀⣤⣤⣤⣤⣄⠀⢠⣤⣤⣤⣤⡄⢠⣤⣤⣤⣤⠀⢠⣤⣤⣤⣄
⠀⠀⠀⠀⢀⣾⠙⣷⡀⠀⢰⡟⠁⠀⠉⠁⢠⡾⠉⠀⠈⠉⠀⣿⣧⣤⡀⠀⠀⣿⣁⣀⣀⣿⠇⠀⠀⢸⡇⠀⠀⢸⣿⣤⣄⠀⠀⢸⡏⠀⠈⠹⣧
⠀⠀⠀⢀⣾⠷⠴⠿⣷⠀⠸⣧⣀⣀⣠⡄⠘⣿⣄⡀⣠⣄⠀⢻⣏⣉⣀⡀⠀⣿⠋⠉⠉⠁⠀⠀⠀⢸⡇⠀⠀⢸⣿⣉⣁⣀⠀⢸⣇⣀⣀⣴⡟
⠀⠀⠀⠘⠃⠀⠀⠀⠙⠃⠀⠈⠛⠛⠋⠀⠀⠈⠙⠛⠋⠁⠀⠛⠛⠛⠛⠋⠀⠙⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠛⠛⠛⠛⠀⠘⠛⠛⠛⠉
⠀⠀⠀⠛⢛⣏⣚⣛⡛⠛⠛⠛⠛⠛⠛⠛⠂⡛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⠟⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⡟⠛
⠀⠀⠀⠀⠘⢿⣿⣿⣧⠀⠀⠀⡴⠦⡀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⣤⡀⠀⠀⠀⣼⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⠙⡋⡑⡄⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠌⠁⠀⠀⣠⢾⡛⠛⠀⠀⠘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠙⢋⡠⠆⠀⠀⠀⠀⠙⠳⢦⣄⣀⣀⡀⠀⣀⣀⣠⡤⠾⠋⠁⠀⠀⢠⡤⣄⠠⠇⠀⢀⣼⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣷⣄⠀⠀⠃⢴⣢⠖⢤⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡠⠞⣦⡀⠞⠀⠀⢀⣾⣿⡿⠟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠁⠢⠾⢁⢙⡆⠤⡄⡀⠀⡀⠀⣀⡀⡤⡄⢰⠀⢰⣻⠄⠊⠀⠀⣠⣴⡿⠷⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡏⣀⢀⣀⠀⠀⠈⠉⠐⠶⠃⢸⡟⠀⠸⣅⡇⠠⠿⠐⠛⠀⠀⠀⢀⣤⣾⣿⣿⢷⡶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣾⣿⣷⣶⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡆⣸⣿⣯⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠿⣿⣰⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⠷⠾⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠛⠛⢿⣿⣿⡿⠟⠛⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⢸⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣤⡄⠀⠀⣿⡿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⠀⠍⢻⣧⠀⠀⣤⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠙⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠈⠉⠒⠠⣀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⡤⣄⠠⣄⠠⡄⢤⠒⠖⣶⠰⡖⢢⡀⣶⠙⠘⣷⡈⠋⠋⡏⠋⣿⠀⢸⡆⠈⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠈⢸⡇⢸⡆⠙⣆⠃⢸⡤⠀⣿⠀⣇⢸⡇⣿⠲⠀⠈⢷⡄⠀⡇⠀⣿⠀⠒⢿⠀⣇
⠀⠀⠀⠀⠀⠀⠘⣇⢀⠸⡇⢸⠇⣀⠸⡇⢸⡇⠀⣿⠀⡿⠼⠃⠿⠤⠆⠦⠈⠃⠀⠛⠀⠛⠚⠃⠘⠃⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠉⠂⠤⣀⡀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⣴⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠒⠒⠒⠒⠲⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⣠⡄⠀⠀⢰⣶⠀⠀⢀⣦⣤⣤⡀⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⢀⣴⣦⡀⠀⢻⣧⠀⠀⢸⣟⠀⠀⣸⡿⢶⡌⠀⣴⣷⡆⢀⡈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡀⠀⢸⣿⡼⠿⣶⡼⠿⠞⠛⠙⠛⠛⠋⠛⠳⠶⠆⣼⠟⣿⣇⣾⠏⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢀⡴⠋⣀⡀⢙⣿⣦⣸⡿⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⡶⢶⣤⣤⣀⠀⠀⠸⠿⢣⣾⠿⠛⢷⣄⠙⢦⡀
⠀⠀⠀⢠⠞⠀⠀⠙⣿⣿⠁⠙⠋⢀⣠⣶⡟⠋⢻⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣶⣄⠀⠘⢿⣄⣷⡄⠉⢀⡀⠳⡄
⠀⠀⢠⠏⢀⣾⠃⠀⢈⡻⠃⢀⣴⣿⣿⣿⣷⡀⠈⣿⣿⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⣿⠿⣦⠀⠙⢫⣴⣾⡟⣷⡀⠹⡄
⠀⢠⡏⠀⠈⢿⣦⣤⡾⠁⢠⣾⣿⣿⣿⣿⣿⣿⡄⣙⣿⡿⣿⣿⠀⣾⣿⣿⣿⣿⡿⠋⠀⣘⣷⡄⠈⢿⡌⠟⠈⠃⠀⢹⡀
⠀⣼⠀⠀⠀⠀⠉⠉⠀⢠⣿⣁⡈⠙⠟⡿⠽⠛⠉⠁⠀⠉⠺⣏⣸⠿⠛⠺⣝⠋⣀⣴⣾⣿⣿⣿⡄⠀⠁⠀⠀⠀⠀⠀⣧
⢠⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⣷⡄⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠙⠯⣿⢿⣿⣿⣿⣿⠀⠀⠀⢠⣤⡄⠀⢸
⢸⠁⠀⠀⠸⠇⠀⠀⢸⣿⣿⣿⣿⣿⡿⢺⣿⣦⠀⠀⠀⠀⠀⡄⢰⠀⠀⠀⠀⠀⠀⠈⡟⠛⠛⠋⢻⡇⠀⠀⠈⠛⠁⠀⠈
⢸⠀⠀⠀⠀⠀⠀⠀⢸⣇⣈⣭⣍⣩⣵⣻⣿⣿⡆⣿⢑⡆⠛⠀⠈⠀⡀⠐⠀⠀⠀⢤⣿⣶⣷⣿⣿⡇⠀⠀⠀⠰
⢸⡀⠀⠀⠀⠀⠀⠀⠸⠿⣻⣿⣿⣿⣿⣷⣿⣿⡇⠈⣉⣙⡀⠀⣘⠁⠀⠀⡀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀
⠘⡇⠀⠀⠀⠀⠀⢆⠈⢏⣉⣡⡟⠉⣻⣿⣿⣿⣿⠀⣛⠭⠒⠋⠁⠀⠀⠀⢿⠀⠀⠀⣻⣻⢟⡻⠟⠀⠀⠀⠰⠗⠀⠀⢸
⠀⢻⠀⠀⠀⣾⣦⠈⢲⠋⢸⣿⣄⡀⣹⡏⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠼⡁⠈⠢⡀⢄⡁⠀⠀⠀⠀⠀⠀⡟
⠀⠈⣇⠀⠀⠘⠉⠀⠀⢯⣀⡤⢼⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣄⠈⢢⡀⢐⢊⣴⣶⣦⡀⠀⠀⣸⠁
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢐⠄⠺⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠋⢻⡷⠄⠋⠠⣿⣥⡀⢹⡗⠀⣰⠃
⠀⠀⠀⠘⢦⠀⠠⣶⣶⣿⣻⡷⠀⠀⠙⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣷⠑⠋⠀⣠⡾⢃⡀⠙⠻⠟⢀⡼⠃
⠀⠀⠀⠀⠈⠳⣄⠀⠀⣹⡿⢡⣶⡶⣶⣄⠀⠉⠙⠛⠻⠶⠶⠶⠶⠞⠛⠋⠉⢀⣠⣴⠎⠻⣷⣟⣡⡶⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠁⢿⣏⠀⠀⢁⣶⡶⠶⡦⢠⣤⣤⣤⠀⣤⡶⣶⡜⠋⢻⣧⠀⠀⠘⠟⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠙⠛⠀⢸⣿⡀⠀⠀⢸⣿⣦⡄⠀⢻⡗⠟⠃⠀⠀⠿⠇⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⠉⠙⠃⠀⠺⠿⠶⠶⠀⠸⠗⠀⠀⠀⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠶⠤⠤⠤⠤⠶⠖⠒⠛⠉⠁

⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘⠅⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢠
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢆⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⢁
⠀⠀⠀⠀⠀⠔⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢲⠀⢂⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠡⠀⠃⡀⠆⠀⠀⠀⠀⢀⠎⢀⡈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠁⢄⠉⠂⠈⠀⢀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠈⠠⣀⠁⠐⡠⣀⠁⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠃⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠁⠊⢆⢄⠉⠂
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡁⡐⠁
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢜⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢘
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣭⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠒⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠢⣄⠀⠀⠀⠀⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣒⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠬⠅⠀⡇⠀⠀⠀⠀⠀⠀⢀⠔⠁⡠⡢⠈⡢
⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠯⠃⢠⠁⠀⠀⠀⠀⢀⠔⢁⢤⠑⠉⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠤⠞⠀⠀⠀⢀⠔⠀⡐⠌⠀⡠⠊
⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠛⢁⡤⠊
⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⡄⡈⠉⠑⠒⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⣈⡁⠑⠓⠞⠴⢀⠀⣄⡈⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠠⠤⢀⡈⠀⡰
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠠
⡠⠊⣀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⢀⢀
⢀⠀⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠂⠈⠀⠀⠀⠀⠀⠀⠀⡠
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠠⠐⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⢀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠰⠀⠐⠒⠉⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⣀⣀⣄⣀⣠⣀⠀⠐⠂⠂⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀⠀⠀⢈⠔⠒⠃⠈
⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⣠⡆⣀⣒⢀⠄⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⡀⠠⡐⠀⠀⠀⠈⠄⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠂⠀⢈⢠⠟⠀⠀⠉⠉⠓⠺⡄⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠔⠅⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠼⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠐⠀⡼⠴⠀⠠⢬⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⠀⠉⠀⠀⠈⠉⠁⡤⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠴⠦⠆⠀⠔⠋⠁⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⠠⠄⠀⡆⠠⢸⡀⠀⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⢄⡀⠀⠀⡈⠀⡠⠂⢀⡀⠀⠀⠀⠀⠀⠙⢄⡁⠂⠀⣀⡄⢁⠀⡀⠳⣄⠈⠀⠁⡠⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠢⢄⣀⢥⠚⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠆⠙⠒⠊⣩⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠀⠁⠂⣤⠐⠈⢀⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠒⠓⢤⡀⠀⡠⠓⣥⠤⣇⠸
⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠙⠒⠚⠑⠙⠒⠉⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠁⠂⠤⠤⠅⠀⠈⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⡇⠀⠒⠀⢰⠀⠀⡖⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡠⠂⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠊⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠶⣄⡀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣠⣀⣀⣀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡤⠖⠛⠛⠲⣼⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠻⢿⣷⡖⠋⠉⠙⠳⣄
⠀⠀⣾⠁⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢸⡆
⠀⠀⢿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⠳⢤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠁
⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢠⠄
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢠⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⠀⠀⢸
⠳⣤⣾⠀⠀⢸⠀⠀⠀⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⡇⠀⠀⢸⠀⠀⢸⡦⠴
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠹⣄⠀⢸⡀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣼⣀⣀⡼⠁
⣶⣶⠶⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣧⠀⠠⠂⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⣴⢾⣿⣿⣿⣿⣿⣿⣄⣸⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⡇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⣇⠀⠀⡀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠴⠕⠂⠁⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠸⠿⠿⢟⠛⠋⠉⠉⠀⣀⣀⣤⣶⢂⠀⠀⠀⡄⡀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⢀⡀⠀⡄⠀⠀⠀⣀⠀⣀⣠⣼⣤⣶⣶⣾⣿⣿⣿⣿⠃⣾⡅⠒⢸⡇⠀⠾⠿⢇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⣀⣧⡀⠒⢾⣃⡀⠀⠸⠦⠴⠒⡛⠦⢭⡿⠿⠛⢉⠛⠛⢟⡉⡀⠐⠻⠇⣈⣼⣿⡰⠎⠀⠈
⠀⠀⠀⠀⠀⠀⢸⣿⠠⠛⠉⠀⡠⡊⡈⠘⠃⠘⠀⠠⠧⠡⠒⠀⢀⠀⠰⠀⢸⡄⠀⠈⠘⠓⠢⣧⡀⠀⠈⢢⠔⠈
⠀⠀⠀⠀⠀⠀⠘⠁⠔⠀⠀⡀⠀⠀⣇⣰⠀⠒⠀⠀⠀⢠⡀⠊⠀⠄⢀⡇⣿⣇⠀⣁⡀⠀⠀⠀⢁⡄⠀⠰
⠀⠀⡄⠀⠀⠀⠀⠄⢀⡀⢀⣤⣤⣤⣝⠀⠀⠤⠀⣠⡄⣸⡇⠲⠥⠠⠚⠛⠛⠉⢁⠁⠂⠀⠀⠈
⠀⠀⠀⠉⠀⢀⡁⠀⡀⣀⣀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠻⠟⠀⠀⢠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠺⠖⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠉⢉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠉⠉⠉⠙⠛⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⣼⣷⡆⠀⣴⣿⡆⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠛⠁⠀⠙⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⣀⠠⠤⠀⠚⠛⠙⠛⠀⠀⠀⠰⠀⠀⠐⠒⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⣤⡄⣀⣀⡀⢀⣤⠀⣶⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠁⠀⠀⢰⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⠀⠀⣨⡀⠀⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠆⠁⠀⠀⠀⠲⢀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⠟⣀⠠⠁⢠⡄⠶⠠⡀⠙⢢⡀⠀⠀⠀⡀⢋⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠒⢨⡐⠦⡀⠀⠈⠉⠲⣄⠀⠀⢀⠜⠀⢸⡇⠀⠈⡇⣌⠁⠀⠀⠀⠀⠑⡈⠛⡀⢰⠎⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⠑⠬⣒⠄⡀⠀⠀⠙⠢⢍⡀⢠⣿⣷⠀⠀⢹⡏⠀⠀⠀⠀⠀⢠⣳⠀⡥⠼⠒⠡⠈
⠀⠀⠀⠀⠀⠀⠀⢀⢂⡼⢁⠄⠒⠂⠀⠀⠀⠈⠒⠤⡂⢄⡀⠀⠈⠳⢮⣟⠿⠀⣄⠁⠀⠀⠀⠀⣀⠤⠸⢀⠀⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠠⢃⠔⢙⢠⡄⠀⠀⠀⠀⠀⠈⠃⠜⡱⢄⡀⠀⠉⠛⠦⣀⠀⠀⠀⣠⢤⡉⠉⠉⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⡐⣠⠎⡘⢁⠂⠀⢀⠋⡀⢘⠀⣰⠎⠦⢀⠀⠀⠈⠑⢮⡔⣠⠀⠀⠀⠉⠲⣌⡏⠀⠙⣦⠀⠀⠀⠠⠁
⠀⠀⡀⢤⡔⡰⠋⠀⠁⠘⠊⣵⢂⠎⢀⠌⣼⠃⢰⡆⣠⠟⠒⠤⡀⠀⠈⠑⢬⡐⠤⣀⠀⠀⠑⠦⢄⠈⠳⣄⡠⠁
⠀⢴⠀⠀⢾⡁⠀⠀⠓⠒⠊⠁⠠⠔⢡⡞⣡⠆⠎⣰⠏⡔⠂⠴⢋⡦⠀⠀⠀⠈⠓⠤⣓⠤⡀⠀⠈⠑⠢⢌⡁
⡄⠙⠶⠈⠁⠈⠓⠤⣀⠀⠀⠉⠄⢀⡋⡐⠏⢀⡼⢃⡀⢘⡆⣠⠎⠠⡀⢀⠀⠀⡀⠀⠈⠓⢤⡁⠄⡀⠀⠀⠈⠠⢀
⠈⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠉⠐⢠⡞⠡⠊⢀⠏⣰⢋⠔⠠⢡⡞⣡⠄⣈⡶⢄⠀⠀⠈⠓⢤⡁⠢⣀⠀⠀⠘⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠈⠁⠀⣊⡼⠃⣎⠀⣠⠟⡀⠑⣠⠙⢀⡔⢊⠗⠠⠀⠀⠈⠑⠤⣑⠢⣄⠀⠀⠈⠒⢤⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠙⠢⢄⡀⠀⠈⠃⠤⠤⡚⠃⣎⠁⠺⢃⡀⡜⠁⢋⠇⢀⣨⠓⠄⠀⠀⠈⠓⠤⣙⠢⢀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠱⢄⣀⡾⢡⡟⠀⡐⠀⢈⠄⠠⠁⠀⠔⢈⠀⢀⡀⠀⠈⠓⢄⠓⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠀⠈⠁⠡⠀⠈⡐⠀⠂⠌⣠⠗⣠⠄⢂⠐⡡⠁⠀⠀⡰⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠠⣀⠀⠀⠀⠐⠀⠊⡼⢋⠀⠉⣼⠂⠐⠁⠀⢀⡜⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠓⠤⣀⠀⠀⠙⠂⠀⠁⡼⢃⠂⠀⠀⡀⢋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠙⠦⣀⡐⠀⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⠶⣄⠀⠀⢀⡞⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠒⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠭⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⠤⠤⠄⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣧⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢸⡟⠛⠛⢻⡇⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⡇⠀⠀⠀⣠⠴⣖⣷⠦⣀⠀⠀⠀⠀⠀⢤⠀⢀⣀⣀
⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⢻⢫⠝⡟⢹⠀⠀⠀⠀⠀⢿⠀⢀⣿⣽⡗
⠀⠀⠀⣧⣤⣤⣼⡇⠀⠀⠀⠘⠛⢛⣿⣿⡛⠛⠃⠀⠀⠀⠈⠛⢼⡤⠗⠊⠀⠀⠀⠀⠀⠉⠀⠀⠈⠗
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣾⣫⣟⣷⣄⠀⠀⠀⠀⠤⢒⣉⣉⡒⠤⠀⠀⠀⠀⠀⢘⣧⡴⢒⣆⠀⠀⠀⢰⠒⠒⡒⠒⣶⣶⡆
⠀⠀⡧⡧⠴⡧⢾⢾⠂⠀⠀⠀⠀⠡⢒⣒⠌⠁⠀⠀⠀⠀⢰⣫⣤⣿⣿⠏⠀⠀⠀⢸⠹⡇⡇⢦⣬⠍⡇
⠀⠀⠙⢿⣾⣷⣻⠟⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⡀⠀⠀⠀⠸⠤⠤⠥⠥⠶⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠠⢲⢶⣀⣶⡲⡄⠀⠀⢠⠒⠉⠀⠉⠉⠐⡄
⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⢠⢀⠵⠤⠤⢎⠙⠀⠀⠀⢰⣟⠋⣾⣿⡦⢏⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡏⣿⣿⡿⠁⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠀⠀⠀⠈⠹⡶⠾⠿⠷⡾⠀⠀⠀⠀⠉⠒⢦⠔⠀⠉
⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⡀⢀⢄
⠀⡔⠀⣲⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠄⢠⠀⠀⠀⠀⣦⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠑⠠⠿⢁⠔⡠⠤⣄⠀⠀⠀⡄⠀⠉⠉⠀⢸⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠮⠥
⠀⠀⠀⠀⠃⠚⢇⡀⠿⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠀⠀⠀⣼⣛⣿⣟⣟⡿⠀⠀⠀⠀⠀⢰⢲⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠘⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠚⠛⠛⠓⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢛⠉⣄⢰⡖⣆⢸⣎⡁⢼⣶⠄⢲⡤⠉⡛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣁⠐⢯⡤⠸⡆⢷⡼⠸⠧⠄⠏⢿⠀⡾⢀⡾⢁⣤⡌⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣀⠻⣾⢷⣌⠧⣠⠽⠒⠚⠋⠉⠋⠛⠓⠲⠇⣘⠵⢻⡟⣠⠖⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠸⣍⡳⣌⢓⡴⠋⠁⠀⠀⠀⣶⡄⠠⣤⠀⠀⠀⠈⠙⢮⡺⠅⠀⠀⠀⠈⣷⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣾⠡⣖⠂⢬⡉⣱⠋⠀⠀⡀⠘⠃⠀⠉⠁⠀⠈⠀⣉⣀⣠⣤⣤⣿⣶⣶⣾⣿⣿⡿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠈⠙⠛⣱⠁⢀⣀⣀⣡⣤⣤⣶⣶⣶⣿⣿⠿⡿⠿⡟⢛⠛⠉⣹⠉⣿⠀⠘⣿⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣀⣠⣿⣤⣤⣴⣶⣾⣿⣿⠿⡿⠟⠛⡟⢻⡏⢉⠙⣧⠀⣤⣷⠀⠁⠸⣷⠀⣿⡆⢹⡀⠆⢹⡆⢹⣿⣿⣿⡄
⠀⠀⢰⣿⣿⡿⠛⠻⣿⠉⡉⠻⡄⠸⡀⣧⠀⠶⣿⠀⣇⠸⡇⠸⠀⢀⣼⡄⠀⠀⣿⡄⢹⣇⠸⠇⢠⡀⣇⣀⣀⣿⣿⡇
⠀⠀⢸⣿⣿⡇⠸⣦⣽⠀⢿⠀⣇⠀⠀⢸⡀⢠⣾⡄⢻⠀⠇⢘⡇⠈⠉⣇⣰⣄⣽⣧⣼⣿⣶⣷⣿⠿⠿⣿⠟⠛⠛⠁
⠀⠀⠀⣿⣿⣧⠀⣟⠙⡆⠸⠀⣿⠀⣆⢈⣅⣸⣿⣧⣼⣦⣲⣾⣿⣿⡿⠿⠿⠛⠛⠛⢉⡟⠉⠀⠀⠀⢀⡟
⠀⠀⠀⢹⣿⣿⣦⣀⣴⣿⣦⣶⣿⣿⣿⠿⠿⠛⠛⠛⠋⠉⠉⠁⠀⠀⢀⡀⠐⠀⠀⢀⡜⡑⠳⠦⣤⠆⣼⠁
⠀⠀⠀⠈⠻⠛⠟⠻⣿⠋⠉⠉⠀⣀⣙⢦⠀⠀⠀⠃⠀⠶⠄⠰⣶⠄⠈⠁⠀⢀⡴⢫⣌⠻⣿⡶⠄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢠⡞⡁⠛⣤⣝⠳⢤⣀⣀⠀⠀⠀⢀⢀⣀⡤⠖⢩⠾⣧⠈⠳⡌⢁⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡉⠁⣾⣡⠟⣼⣦⠆⡭⠍⣭⢩⣭⡅⣶⣂⢻⣾⣦⠈⠳⢀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣬⡉⠰⢻⡟⢰⡟⠂⣿⢸⡇⣇⢸⣯⡄⠑⢉⣥⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣬⣄⣀⣉⣈⣉⣡⣤⡤⠶⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⡀
⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣠⠟⣩⣭⣭⣭⣭⣭⣭⣭⣭⠹⣆⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣴⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆
⠀⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠀⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣸⣿⡟⠀⠀⠀⠀⠀⠀⡾⢋⣭⡉⠁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠋⣭⣍⢳⡄⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⣾⣿⡇⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢸⣿⣿⠈⣇⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⣿⣿⡇⣤⣭⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣭⣤⣸⣿⣿⡆⣿⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⣿⣿⣷⣻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣹⣿⣿⠇⣿⠀⠀⠀⠀⠀⠀⢸⣿
⢹⣿⣧⠀⠀⠀⠀⠀⠘⣧⠹⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢰⡇⠀⠀⠀⠀⠀⠀⣼⣿
⠈⣿⣿⡄⠀⠀⠀⠀⠀⠘⢧⡙⢿⣿⣿⣿⣿⣿⣿⣷⠀⣼⣿⣿⣿⣿⣿⣿⡿⢋⣴⠏⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠙⢶⣍⠻⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡟⣋⡴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠸⣇⣛⣛⣛⣛⣛⣀⣙⣛⣛⣛⣛⣡⡿⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠻⣷⣄
⠀⠀⠀⠀⠀⣠⣶⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣦⡀⠀⢀⣴⣿⣟⠁⠀⠀⠀⣸⣿
⠀⠀⠀⠀⣾⡟⠁⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠹⣿⣶⡿⠋⠙⢿⣧⡀⣠⣶⣿⡃
⠀⠀⠀⠀⣿⡇⠀⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣦⣾⠿⠋⠀⠀⠀⠀⣹⣿⠟⠉⢹⣿
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢀⣴⣿⠟⠁⢀⣤⣿⠟
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⢠⣤⡀⠀⣴⣿⠋⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠋⠁⠀⠀⠀⣠⣴⡿⠋⠀⢀⣴⡿⠛⠁
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠻⣿⣾⠟⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⣹⣿⡿⠋⠀⠀⠀⠀⣠⣾⡟⠋⠀⣠⣾⡿⠋
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⢀⣴⣿⠟⣿⡇⠠⣾⠟⠁
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⢀⣀⠀⠀⣠⣾⠟⠀⠀⢀⠀⡀⣸⡿⢿⣷⡄⠀⣠⣴⡿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠘⢿⣷⣴⡿⠋⠀⠀⠘⠿⠿⣿⣿⠁⣀⣹⣿⣾⠿⠋⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠙⠋⠀⢀⡀⠀⠀⠀⢰⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⢀⣄⠀⠀⣠⣾⠟⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠈⠻⣷⣾⡿⠁⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣾⣿⣦⠀⠀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⠟⠈⠉⠁⠈⠿⠿⣷⡄⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⠆⣿⡇⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣧⠀⠀⠀⠀⠀⠀⢾⣟⠉⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⣿⣇⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣾⢿⣦⣴⡿⢿⣿⡿⠿⠿⠟⠁⢀⣿⠇
⠀⠀⠀⠀⠘⢿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠈⠙⠛⠁⠀⣿⣧⣤⣤⣤⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣀⣠⣴⣤⣄⣀⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠋⠉⠉⠛⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢀⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⡤⢛⠁⢸⣿⠟⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡁⣼⣿⠾⠁⡇⠀⢀⣀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣿⣿⣇⡘⠃⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⣿⣿⡇⠀⠀⠀⠉⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣤⣶⠷⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠻⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⢻⣿⣿⣿⣿⣿⠐⠀⠀⠀⠀⠀⠃⢠⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡆⣼⣿⣿⣿⡯⠛⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠈⣿⣿⣿⡇⠀⢠⡀⠀⠀⢀⡆⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⣷⣿⣿⡟⢀⠀⠀⢡⡀⠀⡜⣡⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⣶⡾⣾⣿⣴⣻⣿⠻⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠻⢿⣿⡏⢉⣿⣷⣿⣿⣿⣿⣿⣢⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⠀⠛⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⠯⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡖⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠻⠟⠃⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧⣼⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣀⡀⣤⣰⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣟⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢻⣿⣿⣟⣿⣿⡿⠟⢀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⣻⣿⣷⣿⣿⡟⢿⣿⣥⣄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣀⣀⡛⣻⣿⣿⡋⠀⠈⠻⡛⠋
⠀⠀⠀⠀⠀⢀⣾⣿⠿⠛⠋⠉⠛⠟⠉⠘⠇⠀⠀⠀⠈⠖⢤⣠⡇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠬⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣀⣴⣾⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⡿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⠿⣿⡿⠿⠿⠿⠿⣿⣿⣷⣦⣀
⠀⢠⣾⣿⠟⠉⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠉⠻⣿⣷⡄
⢠⣿⡿⠁⠀⠀⠀⣴⣿⠿⠿⣿⣷⠀⠀⢸⣿⡿⠿⣿⣷⡄⠀⠀⣿⣿⠿⠿⢿⣷⡄⠀⢸⣿⡿⠿⠿⣿⣦⠀⠀⠀⠈⢿⣿
⣿⣿⠁⠀⠀⠀⢸⣿⡇⠀⠀⠈⠉⠀⠀⢸⣿⡇⠀⠈⣿⣿⠀⠀⣿⣿⣀⣀⣸⣿⡇⠀⢸⣿⣇⣀⣀⣼⣿⠃⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⢠⣶⣶⡄⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠿⠿⠿⠋⠀⠀⢸⣿⡿⠿⢿⣿⣇⠀⠀⠀⠀⠀⣿
⢹⣿⣆⠀⠀⠀⠘⣿⣷⣄⣤⣽⣿⠇⠀⢸⣿⣇⣠⣴⣿⠟⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿⣿⡀⠀⠀⠀⣰⣿
⠀⠻⣿⣦⡀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠘⠛⠛⠛⠛⠉⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠘⠛⠁⠀⢀⣴⣿⠟
⠀⠀⠙⠻⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⠟⠉
⠀⠀⠀⠀⣠⣿⣿⠟⢻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡟⠻⣿⣯⣄
⠀⠀⠀⣾⣿⣟⠁⠀⢸⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⣠⣄⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢸⣿⡇⠀⠈⣻⣿⣷
⠀⠀⠀⣿⣿⢿⣿⣦⣸⣿⡇⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠙⠋⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠁⠀⢸⣿⣇⣴⣿⡿⣿⣿
⠀⠀⠀⣿⣿⠀⠉⠻⢿⣿⣧⡀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⢀⣼⣿⡿⠟⠉⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠙⠻⣿⣷⣄⡀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⣠⣾⣿⠟⠋⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⣶⣶⠀⠀⣶⡦⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⣿⣿
⠀⠀⠀⣿⣿⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⣿⣿
⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⡗⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⡁⡐⢄
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠁⠘
⠠⠀⡶⠉⢫⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡚⠛⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⢼⣉⠀⠀⢖
⡄⢻⡤⠔⠚⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠴⠊
⠘⢦⣈⣓⣂⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⣜⣂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠠



⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠄
⠀⠀⢸⣄⣬⡀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠒⡂⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⡆
⠀⠀⠸⣿⣇⠇⠀⠀⠀⠀⠀⠀⢧⣀⣈⣉⡁⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠳⡅⠀⠜⢀
⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠘⠛⣿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⠄



⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡍⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡘⢢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⠄⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⢔⠂⢹
⠀⠀⠧⠤⠭⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣒⣒⣒⣲⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⣞⠡⠨⠤⢌⣳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⡀⠀⠀⠀⠀⣠⡤⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣦⡀⣠⡾⠋⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣨⡿⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠋⠀⠈⠻⣦⡀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⣀⣀⣀⣀⣀⠈⠉⠀⣿
⠀⠀⠀⠀⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠏⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⠀⣰⣿⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠰⢿⣷⣄⣀⣼⣿⠏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠙⢿⣿⡿⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠽⠷⡖⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⣀⣠⡄⠀⠀⠀⠀⠀⠀⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢉⣤⠾⠖⠒⢂⠟⠉⠉⠓⠤⡀⠀⣀⡬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⢩⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢀⡞⠀⢀⠙⡆⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⠀⠀⣁⢴⡀⠀⠀⠸⢀⣀⠤⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⢀⣀⣑⡞⠁⠀⠹⠀⠀⣳⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣨⡇⠀⠀⢸⠇⢠⣿⠗⣊⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠘⠦⡔⠋⠀⠈⠁⠞⠋⣰⣯⣍⣉⡉⠉⠳⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣶⣿⣿⣿⣿⣿⠀⢸⡆⠀⠷⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⡻⢝⠿⢿⠀⢸⡇⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⣿⣿⣿⡿⢯⡰⢍⠲⣵⣿⠀⢸⡇⠀⢸⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢀⣾⣿⡿⠯⠘⢦⣭⣷⣿⣿⣿⠀⠀⡇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠒⠐⠉⡼⠁⠀⠀⠀⠀⣸⣿⣏⠠⣀⠄⣲⣾⣿⣿⣿⣿⠀⠀⡇⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⣟⠑⠪⣕⣭⣾⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⢸⡃⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢰⣯⣳⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⢸
⠀⠀⠀⣀⠀⢎⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⡞⣁⣼⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⢸
⠀⠀⢨⠀⠑⢬⣳⡈⢆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠴⡇⠀⠀⠀⣀⣠⡤⠴⠚⠉⣁⣤⢯⣭⣭⣭⣭⣍⣉⣉⣉⣉⠀⢰⣇⠴⠞⣤⣄⡀
⠀⠀⠀⠉⠒⠂⢤⡁⠀⠀⠱⠤⠤⠤⠄⠀⠀⣠⠊⠀⠉⠿⣥⢘⣤⠒⠋⠉⠁⠀⣀⠤⠒⠉⠀⠐⠂⠁⢪⢼⡤⢀⣉⣁⣉⣉⣉⣉⣁⢔⠞⠋⢀⠝
⠀⠰⡋⠉⠉⣉⡩⠤⢤⠀⡴⠂⠉⠉⠉⠁⡸⠁⠀⢀⠀⢠⣯⡰⢻⡄⣠⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀⣯⡈⢉⣁⡐⠒⠒⠢⠜⠉⠉⠁⠒⠿⠐⠒⢢
⠀⠀⠈⠉⠉⠁⠀⠀⡘⠈⡇⠀⠀⠀⠀⠀⠙⣂⠖⢁⡔⠃⢸⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢨⠁⠚⠒⠛⠛⠛⠳⡀⣠⠤⢄⠚⢧⣤⣐⠉⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠃⠀⠀⠀⣠⠖⠉⠁⣰⠋⠈⠉⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣟⡷⢞⢒⠒⠒⢒⢒⢒⠶⠃⣾⡶⠋⠙⠦⠤⠜⡀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⢲⠞⢑⣄⠀⡢⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡗⠃⠋⠋⠋⠛⠃⠋⢲⡶⠋⠀⠀⠀⠀⢀⡔⡜⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⡆⠀⢸⠈⢀⣠⡟⠛⠓⠚⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⢻⠭⠭⠿⠿⠿⠿⠿⠯⠤⠤⠄⣤⣤⣤⢎⡾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⢦⡟⣺⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⠤⣤⢤⣼⠊⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⢀⢀⡜⢑⡧⠇⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠹⡏⣈⠟⠛⡿⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠖⣒⣚⣳⣦⣀⣀⣀⣀⠀⠙⠁⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠉⠐⠒⠒⠀⠤⠤⠤⠤⠤⠠⠶⠶⠾⠭⠽⠤⠤⠬⠷⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦⣀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⠀⠉⢉⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⣀⠀⠘⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠛⠀⠀⠈⡛⠛⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠐⠐⠊⠀⠀⠀⠀⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣤⣴⣶⣶⡄⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡅⣿⣿⡿⠃⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣾⡀⠀⠀⠀⡑⠠⠻⠞⠋⠉⠐⡊⡁⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠈⠁⠀⠀⠈⠲⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠢⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣿⠀⠀⠀⠀⠀⠈⢩⣥⣥⣭⣐⣀⠂⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⡀⠀⢄⣢⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣶⣶⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢋⣡⣤⢰⣶⠆⢶⣶⢠⣤⣉⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢉⠰⣾⡇⠙⠿⠀⠛⢀⠘⠃⠸⠟⠁⣿⡶⢀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⡰⠿⣧⠈⡁⠰⣄⢸⣆⣼⣄⣾⢀⡴⠀⡉⢠⡿⠷⣀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠠⠿⠷⠄⠠⠀⠹⠶⠿⠿⠿⠿⠿⠿⠿⠷⠾⠃⠠⠀⠴⠿⠇⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⠶⠶⠶⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢠⣶⡄⢠⡶⠶⠄⣴⠶⠆⣶⠶⠆⣶⠶⣦⠴⣶⡶⢰⡶⠶⢰⡶⢶⡄⠀⠱
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⣹⣧⢸⡇⠀⢸⣿⠀⠀⣿⠶⠆⣿⡦⠿⠀⣿⡇⢸⡷⠖⢸⡇⢸⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠟⠉⠿⠈⠻⠶⠂⠻⠶⠆⠿⠷⠆⠿⠂⠀⠀⠻⠃⠸⠿⠶⠸⠷⠞⠃⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡀⢶⡶⠂⢐⠀⣴⠶⣶⣶⣶⣶⣶⣶⣶⡖⢶⡄⢐⠀⠲⣶⠆⣰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡌⠱⣾⡏⢀⡁⠰⠋⢸⠏⢻⠋⢿⠈⠳⠀⣁⠈⣿⡶⠉⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣈⠰⢿⡇⣠⣷⠀⣴⠈⢰⡄⢰⣧⡀⣿⠷⢈⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣌⣙⠛⠸⠿⠇⠿⠿⠘⢛⣉⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠶⠶⠶⠞⠛⠛⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠴⢖⣒⣂⠉⠩⠭⠴⠖⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠤⠶⠒⣒⣛⡉⠭⠅⠖⠒⣒⣋⡉⣉⠅⠀⣶⡆⠀⣿⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢖⣒⣂⣭⠭⠭⠴⠒⢒⣋⡉⠉⣠⡤⣦⠰⣶⡌⠃⠋⢿⡇⠃⢿⡇⠀⡛⣿⡄⢹⣧⢀⣸⡄
⠀⢲⠒⣋⡭⠬⠅⠒⠒⠀⣀⣈⠨⡄⢴⣦⠲⠄⣿⡀⢻⡇⢻⡆⢿⡧⡆⠀⡙⢿⣆⠀⢸⣿⠀⢸⣷⢠⡄⠹⠷⠘⠛⠚⡁⡇
⠀⠘⠀⢠⡆⢢⢀⣏⢘⣦⠈⢿⣧⡅⠘⣿⠴⠀⢿⡇⢸⣿⢸⡟⢸⣷⠠⠆⠧⠈⠛⠀⢘⣉⣡⡬⠤⢴⣒⣒⣺⠯⠭⠴⠂⠋
⠀⠀⡇⣾⡇⠀⠸⣿⠈⣿⠀⡆⠻⣷⠀⡿⠇⠀⠸⠛⣘⣋⣈⣤⠤⠤⣔⣒⣒⡯⠭⠭⠒⠒⠚⠋⠉⠉
⠀⠀⢃⠘⠗⠠⠃⠙⢃⢋⣀⡉⠤⠤⠴⣖⣒⣺⠭⠭⠝⠒⠒⠚⠉⠉⠁
⠀⠀⢈⣐⣒⣺⡯⠭⠴⠖⠒⠒⠉⠉⠁
⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠈⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡛⠁⠀⠀⠀⠀⠀⠛⣿⣟⠋
⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⡄⠀⠀⢀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣦⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢣⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣧
⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣭⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⠎⣀⣀⣀⠤⠤⠶⣚⡉⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠛⠛⠉⠽⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢁⣠⣴⡾⠿⠿⠿⠿⣶⣤⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠀⣴⡿⠋⢁⣤⣤⣶⣶⣤⡄⠁⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⣾⡟⢀⣴⡿⠛⠛⠻⣿⡟⠁⣴⡿⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⢸⣿⠀⣼⣿⡄⠸⣷⡀⠉⢀⣾⠟⢀⡄⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⣿⣿⣿⡄⠙⣿⣤⣿⠋⢠⣾⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠘⣿⡄⠸⣿⣿⣿⣄⠈⠛⠁⣰⣿⡿⠁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠘⢿⣄⠈⠻⢿⣿⣿⣶⣿⠿⠋⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⡈⠛⢿⣶⣤⣀⣀⣀⣠⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣈⠉⠙⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⢀⡀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠣⠈⠠⠈⠇⠘⠁⠈⠘⠅⠠

⠀⠀⣀
⢠⣼⣏⣉⡙⠓⠶⠤⢄⣀⡀
⠘⡇⠀⠈⠉⠉⠛⠒⠲⠤⢭⣙⡒⠲⠤⣤⣀⡀⠀⠀⠀⢀⣀⣀⣠⠤⣶⣶⣀⣀⡀
⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⠭⠉⠓⡶⢋⣥⡌⢉⣏⣭⡤⢤⡄⢸⡆
⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⢓⣋⣭⠥⠶⢾⡃⠘⠃
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂
⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢖⠰⣄⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣏⠑⢉⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹⡱⡋⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⢷⣋⡉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠆⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⣸⡻⣤⡤⢷⠈⣧⣀⣀⡀⠀⡀⠀⠀⣀⣀⠀⣀⠀⡄⣀⣀⣀⢀⣀⠀⣀⡀⢠⣀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣾⠁⠃⣾⠀⣇⢸⡿⠾⡀⢻⣈⠘⡇⠀⡅⠐⠀⣿⣈⠁⣿⣧⠁⠃⣿⠘⢸⡇⠀⢿⡇⢸⡏⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⠀⣿⠎⢠⠔⠃⢈⡟⠀⠇⠀⠃⠸⠀⣿⠉⠀⡇⢿⡆⠀⣿⠀⢸⡇⠠⢼⣧⢸⡇⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠓⠃⠘⠛⠻⡄⠸⡖⠚⠉⠀⠀⠁⠀⠁⠀⠐⠋⠀⠀⠃⠘⠃⠐⠛⠂⠘⠓⠛⠘⠛⠀⠁⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢰⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠶⠤⣭⣀⣹⣼⠏⢉⣧⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠀⡾⢤⠤⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢠⣧⠜⢊⣉⡉⠙⢶⠦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠀⣉⠁⠜⣉⠀⠀⠈⠗⠀⠈⢙⠲⠆⠀⡀⠠⡀⠀⠀⠀⡄⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠡⢂⠀⢳⡀⠀⢋⠟⡇⠀⠀⠉⠓⠦⣄⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡦⠬⢃⣀⠭⠖⢳⠀⢸⡀⠀⠀⠀⠀⠈⠙⠲⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡖⠋⠁⢠⡀⠀⠈⢷⢊⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⢗⡀⠀⠘⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣨⠥⠴⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠲⠤⠤⠤⠽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⣀⣠⣴⣦⣄⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⣶⣿⢹⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣼⣯⣿⣿⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⡿⣿⣿⣿⣿⣿⣿⢿⣥⣤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⣮⣟⡿⢿⣛⠇⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠾⣮⣽⠃⠸⣷⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠉⠉⠉⠉⠉⠉⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⣤⡰⠂⠖⠾⠰⡠⣄⢄⠀⠈⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢤⡂⠈⠀⢀⣀⣠⣤⣤⣄⣀⡀⠉⠘⠕⢀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡌⠂⢀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⠀⢠⡄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⣐⠜⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⢘⣂⠀⠸⡄
⠀⠀⠀⠀⠀⠀⡞⠀⠰⡏⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠘⡆⠀⢸⠄⠀⢳
⠀⠀⠀⠀⠀⠀⡇⠀⢘⡄⠀⣾⠀⠀⠀⠀⣀⡀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⣷⠀⠨⡅⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⢈⠁⠀⢿⠀⠀⠀⠀⠙⢿⣦⡀⣴⡟⠀⠀⠀⠀⠀⠀⡿⠀⠨⡅⠀⢸
⠀⠀⠀⠀⠀⠀⢧⠀⠀⡀⠀⠸⡄⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠀⠀⠀⢠⠇⠀⠆⠆⠀⡼
⠀⠀⠀⠀⠀⠀⠘⡆⠀⢉⡥⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠂⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠘⡡⡀⠈⠑⢤⣀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⣞⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⠸⢢⡄⠀⠈⠉⠙⠛⠛⠋⠉⠀⣀⢠⡄⠛⠁⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠈⠋⠛⠀⠤⠀⢖⠀⠆⠓⠑⠁⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣀⣀⣀⣀⣀⣀⡤⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠰⠩⠀⠠⠑⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⣾⣷⡆⠀⠀⠀⠀⠀⢠⣤⠶⡶⢶⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠐⠂⡒⠲
⠀⠀⠀⠀⡐⡄⣦⡄⠀⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⡇⠀⠀⠀⠀⠀⠈⢩⣿⠩⡯⣻⣎⠉⢱⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⠀⢸
⠀⠀⠀⠀⠡⠁⠈⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⢀⡇⠀⠀⠀⠀⠀⠀⠿⠿⠷⠾⠿⠿⠖⠃⠀⠀⠀⠀⠀⠀⡸⠅⠤⠤⠼⣀⠌
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣠⣶⣾⣿⢶⣤⢀⠀⠀⠀⠀⠀⠀⢰⠶⡐⠂⠒⢂⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡍⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⢿⣿⣭⣿⣽⣿⠃⢱⠀⠀⠀⠀⠀⢸⠀⠈⠲⠞⠁⣠⣾⣎⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇⣀⣭⣯⣄
⠀⠀⠀⠀⠀⠿⠟⠫⣡⠒⠁⠀⠀⠀⠀⠀⠘⠏⠀⠠⢀⡸⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⡸⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠆⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃



⠀⠀⠀⠀⣿⣿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⡄⣠⣦⡀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠠⠐⠒⣦⡀⠀⠀⠀⠀⠀⠀⡖⠂⠒⣀⠂⠒⢲
⠀⠀⠀⠀⣿⣿⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⣘⣟⣁⠀⢃⣙⣟⣁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⣽⡇⠀⠀⠀⠀⠀⢀⡥⡄⠠⠿⠄⠀⢸
⠀⠀⠀⠀⣍⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠺⠿⠿⠿⠂⠀⠀⠀⠀⠀⠿⠷⠶⢤⡤⠤⢼⠅⠀⠀⠀⠀⠀⠸⣿⠟⡿⢿⠿⠟⠛



⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠰⡶⠀⠼⠧⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⠀⠀⡛⢍⠉⠉⣉⠝⡇
⠀⠀⠀⠈⠉⠀⣈⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⢠⣤⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⡘⡐⢛⡘⢰⠀⠀⠀⠀⠀⠀⠀⠃⡀⠑⠚⢡⢖⡲⡄
⠀⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣭⣭⣼⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠀⠂⡛⠛⣰⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂⠂⢣⣛⡓⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡄
⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆
⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣿⠿⠶⠶⠶⠶⠾⠿⠿⠿⠶⠶⠶⠶⢿⡇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣿⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡇⠀⣿⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠐⡿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢰⡟⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢻⡇⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢧⣿⣆⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⣾⣵⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠙⠻⣷⡀⠈⢷⣄⠀⠀⠀⣠⡾⠋⢀⣴⡿⠋⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠈⠀⠀⠀⠙⣷⠀⢸⡏⠀⠀⠀⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣠⣿⠀⢸⣇⣀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⣀⣠⣤⣄⠄⠀⠀⠀⠀⠀⠾⠓⠀⠀⠀⠀⠀⢠⡄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢀⠀⠀⠘⢙⣿⣭⢯⣠⠆⠀⠀⠀⣀⣄⣀⣤⣄⣠⠀⠀⣾⡅⠀⢀⣻⣿⡟⠛⢿⣿⠻⡟⢿⣿⡇⠀⣠⣄⣤⠀⣠⣾
⠀⠈⢁⡤⠀⠤⠠⠆⠀⠀⠀⠀⠀⣾⣿⠛⣿⣿⣿⣿⠀⠀⣿⡷⡤⢼⣿⣿⣿⣦⡀⠙⢛⢙⠛⢿⡇⣠⣿⣿⣿⣾⠟⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣿⣿⡿⠗⠀⠀⣿⣧⢨⣿⣿⣿⣟⠛⠛⢦⡙⠉⣠⣿⡇⢿⣿⣷⣿⣧⣦⡶
⣿⡿⠛⣿⣿⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⡋⠉⢭⣼⣏⠻⣿⣿⡿⣿⡇⣿⣯⣵⣿⡇⠈⠿⠁⠀⠀⢀⠀⡀
⠋⣠⣼⢿⣿⣷⣦⡘⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⠀⢣⢠⠀⣾⢿⡿⠀⣿⣷⣇⠻⢿⡿⠋⣿⣿⡇⠀⢰⣄⣀⣀⣼⠟⠁
⠸⠤⠋⠸⣿⢿⡿⠷⢼⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣐⣾⣌⠈⠥⠼⠷⣿⣿⣿⣿⢳⣦⣴⣾⠹⣿⡯⢀⣿⢹⠛⣻⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠀⢻⡇⣼⣿⣤⣉⣁⣀⣀
⣿⣿⣿⠛⢟⢛⠿⡻⢛⠛⡻⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⣠⣴⣦⠈⣷⣿⠿⠿⠿⣿⠟⠁
⣿⣿⣿⠻⠿⠿⢿⡿⢿⣿⠿⠶⠿⠟⣿⡿⠻⡛⠿⠟⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡷⠝⢿⣿⡶⣯⠉⠀⣠⣞⣥⣀⣀
⣿⣿⣿⠶⢷⣶⣶⡷⣾⣿⡶⣾⣷⣶⣿⣷⣷⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠈⠉
⣿⣿⣿⣶⣶⣵⣶⣷⣾⣦⣶⣷⣴⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⡇⣿⠇⠀⠀⠀⠀⣠⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢫⣭⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⣨⣍⡸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢸⣿⣿⢸⣿⣿⣿⣿⡸⠿⠿⠿⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣛⠻⣿⠟⣛⢿⣿⢟⡛⢿⣿⢸⣿⣿⣜⣛⢿⣿⡿⢛⣛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣛⣣⣿⣜⣛⣹⣿⣘⣛⣼⣿⢸⣿⣿⣿⣿⣷⣭⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢡⣶⣍⡻⠟⣩⣶⣾⣿⠿⢛⣛⣛⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⣿⣿⣿⣷⣝⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢣⣿⡿⣛⡻⣿⣿⣿⣿⣿⡆⣾⣿⣷⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⡸⣿⠗⣼⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣭⣭⣽⣏⢿⣿⣶⣾⣿⣿⣿⣿⣿⠃⠉⠉⠉⠻⠿⠋⠀⠈⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠦⠙⠿⢿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣔⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠔⢉⠠⠤⢈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡃⠐⠁⠀⠀⠀⢱⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠅⠀⠀⠀⠀⠀⢸⠀⡇⠀⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⢀⠎⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡊⠁⠉⠀⠀⠀⠀⠈⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢠⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢈⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣀⠌⠘⠤⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⣀⣆⣀⣀⣀⣀⣀⣀⣀⣀⢀⣆⢀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠄⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⠀⠀⣴⠄⠀⣴⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠆⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣼⣿⣧⣼⣿⣧⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡩⠵⣤⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡴⠭⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣇⡇⠀⠈⢷⡀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣰⠏⠀⠀⣇⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⢯⡗⠒⠢⡀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡠⠒⠒⣮⠇⠈⠢⣀
⠀⠀⠀⠀⠀⢀⡠⠔⢺⠖⠊⠁⠀⠀⠀⠘⡞⡄⢀⡇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣇⠀⡼⡜⠀⠀⠀⠀⠉⠒⢾⠒⠤⣀
⠀⠀⠀⢠⢾⠁⠀⠀⠸⣼⡀⠀⠀⠀⠀⠀⠹⡝⡍⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⡝⡽⠁⠀⠀⠀⠀⠀⡸⡜⠀⠀⠀⢹⢦
⠀⠀⠀⠸⡈⡄⠀⠀⠀⠙⣷⣀⠤⠤⠒⠒⠒⠚⠮⣓⢒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⢒⡪⠞⠒⠒⠒⠢⠤⢄⣰⡗⠁⠀⠀⠀⡞⡸
⠀⠀⠀⠀⠱⣘⣄⣀⡠⢤⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢦⠤⣀⣀⣜⡰⠁
⠀⠀⠀⠀⠀⠀⠈⠈⢆⠈⢣⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠑⣌⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⡔⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⠳⣌⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡴⠋⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠘⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡜⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣴⡾⠿⠛⠻⠿⣶⣄⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⣱⡿⠋⠀⠀⠀⢀⣴⣾⠋⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⣿⠁⠀⢿⣿⣤⣿⡟⠁⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠐⣿⡄⠀⠀⢻⣿⡟⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⡹⣷⣄⠀⠀⠉⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠉⠉⠈⠉⠉⠉⠉⠁⠈⠛⠿⣶⣶⣶⡾⠟⡋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠠⠄⠐⠒⠒⠒⠒⠂⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⣄⣀⣀⣀⣀⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⠀⠀⠈⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠂⠠⠤⠤⠤⠤⠄⠀⠂⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⣿⡇
⠀⣄⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⢀⡄
⠀⠙⢷⠄⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⡿⠋⣡⣴⣶⣶⣶⣶⣶⣶⣤⡉⠻⣿⣿⣿⣿⣿⡇⣿⡇⠀⠀⠀⢴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⠁⡾⢋⣍⣙⠛⠛⠛⢛⣉⣍⠻⣆⢹⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⢸⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⢸⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⡿⠈⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠸⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠁⣿⣿⣿⣿⣇⠰⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃⡴⢀⣿⣿⣿⣿⡇⢹⠃
⠶⠶⠶⠀⠀⠀⠀⣿⡆⣿⣿⣿⣿⣿⣦⣤⠘⣿⣿⣿⣿⣿⣿⣿⡟⢠⣤⣾⣿⣿⣿⣿⡇⣾⡇⠀⠀⠀⠰⠶⠶
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣧⡘⢿⣿⣿⣿⣿⠟⣠⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣉⣁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⠟⢋⣉⣥⣬⣉⣩⣤⣭⠉⠛⢋⣉⠛⠿⢿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡏⢩⣤⣴⣶⠿⠛⠛⠛⢷⣶⣤⣤⠈⠃
⠀⢀⣴⠄⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇⢸⣿⡿⢁⣶⣿⣿⡿⣆⠹⣿⣿⠀⠀⠀⠀⠀⢶⣄
⠀⠛⠁⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⣇⠸⣟⠛⢋⣴⣿⢀⣿⡿⢰⡇⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣿⣦⡙⠳⠿⠛⣡⣾⡿⠁⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠦⠙⢿⣿⣿⣿⣿⣿⠟⠁⠄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⡶⠶⠶⣶⣶⣦⣉⠛⠛⢋⣡⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢾⠆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⠿⣒⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⢛⣙⢻⣿⠟⣉⡻⣿⡿⣋⡙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣌⣋⣼⣿⣮⣛⣵⣿⣧⣙⣣⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⣿⣿⣿⣿⣿⣶⡍⠛⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿

⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣾⣿⣿⣷⣶⣦⣄⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣷
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣿
⣿⣿⣿⢋⣥⣦⣬⣭⣭⣭⣭⣭⣭⣭⣥⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿
⢻⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣮⣭⣍⣛⣛⣛⡛⠿⣿⣿⣿⣿⠈⣿
⠈⢿⣷⣜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡝⢿⣿⢰⣿
⠀⠀⠹⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡎⠟⣼⣿
⠀⠀⠀⠀⠉⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢙⠿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⠿⢟⢱⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣾⠿⠸⢿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⣠⣶⣿⣿⣶⣮⣍⣛⠿⠏⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡘⠿⢛⣫⣭⣶⣿⣿⣷⣮⠉⠁
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡿
⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⡧
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⠀⣤⡀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣄⣴
⠀⠀⠀⢸⠀⢀⣀⠀⠀⠀⣨⣿⠟⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⠀⠀⢸⡇⣀⡀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⢸⠀⠙⢿⣷⣤⣾⢿⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⣷⣀⣴⣿⢟⠁⠀⠀⠀⠀⠀⠀⢈⡇⠙⢿⣶⣴⣿⠛⡇
⠀⠀⠀⢸⠀⠀⠀⠙⠟⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠿⠟⠁⢸⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠙⠟⠁⠀⡇
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠃



⠀⠀⠀⢰⠖⠒⠒⠒⠒⠒⢲⠀⡠⠂⠀⠀⠀⠀⠀⢰⠖⠒⠒⠒⠒⠒⢒⡴⠛⠀⠀⠀⠀⠀⢰⡖⠒⠒⠒⠒⠒⢒⣴⣿⠂
⠀⠀⠀⢸⠀⢢⣄⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢸⠀⣶⣤⡀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢸⠇⣾⣷⣄⢀⣴⣿⠟⠁
⠀⠀⠀⢸⠀⠀⠙⣿⣾⡟⢱⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢻⣷⣾⡏⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣿⣿⡿⢹⡆
⠀⠀⠀⢸⣀⣀⣀⣈⣋⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣛⣛⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣉⣉⣀⣸⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂
⠀⠀⠀⢸⠉⠉⠉⠉⠉⢉⣵⣿⠿⠀⠀⠀⠀⠀⠀⢸⠉⣉⠉⠉⠉⠉⣩⠖⠉⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⢉⣵⠋⠁
⠀⠀⠀⢸⠀⢾⣷⣄⣰⣿⢟⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠻⣦⣠⡿⢻⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⣷⣄⢀⡾⢹⡇
⠀⠀⠀⢸⠀⠀⠙⢿⡿⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⡟⠁⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡿⠀⢸⡇
⠀⠀⠀⠸⠦⠤⠤⠤⠤⠤⠾⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠤⠤⠤⠤⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠙⠁⠑⠀⠂
⠀⠀⢀⡼⠻⡒⠤⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣶⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠃⠀⢀⢤
⠀⢆⡞⠀⢀⣱⡤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⢆⠀⢀⡱
⢃⠨⠻⣍⡉⠈⠃⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡛⢍⠢⠑
⠀⠓⠤⣌⣉⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠐⣉⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠬⠂



⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢉⣉⣉⡉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣄
⠀⠀⠀⣷⣶⡗⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⣒⣒⠒⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢠⠒⠲⢣⠏
⠀⠀⠀⡻⣿⡷⠁⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⣄⠀⢸⣿⣿⣟⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠳⡄⠀⠋⠠
⠀⠀⢀⣠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠈⠙⣿⠟⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠿




⠀⠀⠀⢠⠀⠀⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⠀⠀⢀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤
⠀⠀⠀⠀⢠⣉⡡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠳⠄⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠦⣀⡤⢖⡉
⠀⠀⠰⠤⠬⠥⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣛⣛⣛⣉⡦⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⡤⠤⠤⣄⣙⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡿⠿⠛⠛⠛⠉⠉⠉⠉⠙⠛⠛⠿⢿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄
⠀⠀⠀⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦
⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⢶⠶⣶⣶⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⡶⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣤⣽⣉⠉⠉⠉⠉⠉⠋⠋⠉⠉⠉⠉⢉⣭⣤⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣶⣶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣶⣶⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠀⠀⠀⠀⠂⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠘
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⢨⣆⢳
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠋⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢸⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠮⠤⠐⠈
⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠀⠀⠉⢖⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡅⠀⠀⠀⠑⠢⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⣀⡠⢸⠤⡄⠀⠀⠀⡀⠀⠐⠒⢒⠒⠂⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠃
⠀⠀⠀⢁⠀⠀⠀⠀⠀⠑⡀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢢⠀⠀⢈⡡⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠑⢀⠀⠀⢀⠔⠁⠀⡠⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠈⠁⠀⠀⡐⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡔⢄⣀⡠⢊⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠜⢄⠀⠀⠈⠢⠬⠟⢛⣠⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡶⠤⠤⠵⠦⠤⠠⠤⠤⠾⠟⠒⠒⠒⠚⠿⠛⠛⠛⠛⠟⠻⠟⠛⠛⠛⠛⠿⠿⠟⠿⠛⠻⠛⠦⠄⠤⠄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠉⡏⠙⡋⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠃⠉⠊⠛⠉⠛⠉⠉⠉⠉⠉⠉⠈⠉⠉⠀⡏⠉⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠰⠄⠀⠐⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠰⠀⠀⠀⠀⢠⠀⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⢀⠁⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣛⣒⣒⣒⣒⣒⣒⣒⣒⣒⠄⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠃⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠍⢸⠀⠀⠅⠉⠀⠀⠀⠁⠉⢠⠏⢨⡏⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠀⡆⠃⠀⠀⠀⠀⢰⡧⢾⠸⠷⡶⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠐⠃⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢇⡐⠀⡆⣦⠀⠀⠀⠃⠤⣀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘⠉⠒⠓⠛⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠛⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⣩⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⢴⣿⣦⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠃⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢠⣿⣹⣿⡟⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢿⣿⣦⡀⠀⠀⠀⣰⣿⣷⣿⠟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠹⣻⣿⣦⣀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛



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

⠀⠀⣴⢦⣄⠀⠀⠀⠀⢰⡟⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⠛⠛⠛⠛⢷⡀
⠀⣾⠁⠀⠙⠷⠶⠒⠛⣋⠉⠀⠈⠳⡄⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠁⣀⡠⠤⠴⠲⢶⡀⠀⠀⠘⡇
⢸⡇⠀⠀⠀⠀⠐⢾⡋⠉⣻⠀⡿⠚⠁⣌⣼⠀⠀⠀⠀⠀⠀⣠⠤⢾⣁⠀⠀⠀⡞⢁⣤⠀⠀⢀⣤⠙⠛⠋⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠙⢶⡟⢸⡇⢀⣼⣳⡏⠀⠀⠀⠀⠀⣸⢁⠖⠉⠳⢴⡶⠿⠷⠟⠈⠛⠛⠋⠙⠷⠶⠦⣄
⠈⢿⡀⢀⣄⠀⠀⠀⠀⠀⠑⢿⣅⠈⣵⠟⠀⠀⠀⠀⠀⠀⠹⣌⠳⠤⠤⠿⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢷
⠀⠈⣷⠏⡈⢧⡀⠀⠀⠀⠒⣩⠽⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⣾⠛⠀⠀⠀⠙⠦⣄⣀⣤⠴⠃⠀⠀⠀⠘⡆
⠀⣼⠇⠀⠀⠀⠳⣤⣤⣤⣭⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⢠⡷
⠈⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡿⠋⠉⠉⢻⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣸⠁⠀⠀⠀⣠⣇⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣤⣄⠀⠀⢀⣤⣄⣀⡀⣼⡏
⠀⠀⠀⠀⠀⠀⢀⣤⠿⠀⠿⣤⡀⣰⠟⠑⠒⠲⣝⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠟⠛⣋⣉⣤⡀⢻⡄⠀⡿⢀⠤⢭⣉⡛⢳
⠀⠀⠀⠀⠀⢠⡟⠓⠒⠀⠀⠈⠻⡏⠀⠀⠀⠀⠈⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢟⠉⠀⠀⠃⠈⣇⢸⢃⣌⡀⠀⠀⠁⣼
⠀⠀⠀⠀⢠⣟⣀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡇⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢨⣴⠆⠰⠛⠃⢹⡿⠀⢩⣁⠘⠛⢠⡏
⠀⠀⠀⠀⠘⣯⠛⠀⠀⠀⠀⠀⠀⣼⣄⠀⠀⠠⢞⡼⠷⢎⢻⡄⠀⠀⠀⠀⠀⠀⠀⢷⠀⡤⢦⠒⠊⠁⠈⣧⠀⢀⠃⢪⠇⣼⠁
⠀⠀⠀⠀⠀⠘⢦⣤⣤⠄⠀⢀⣴⠃⠈⠙⠒⠛⠉⣴⠖⠋⣼⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆⢹⡭⠄⠀⠀⠀⢸⡄⠀⠀⠸⢠⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠻⣤⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠂⠀⢀⣀⣠⣬⠟⠛⠲⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠛⠋⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠄⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠻⣿⠿⢿⡇⢸⡿⢿⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣄⣹⣇⠸⣷⣾⠇⣸⣏⣠⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡉⠛⢷⣄⢙⣿⣿⣿⣿⣿⣿⣿⣿⡋⣠⡾⠛⢉⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠷⢾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⣻⣿⡷⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠷⢶⣤⣿⣿⠟⠻⣿⠟⠁⠀⣠⣾⣿⣿⣴⡶⠾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⠾⠟⣿⣷⣄⠀⠀⠀⣠⣾⣿⣿⣿⣿⠻⠷⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⢾⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⡷⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣁⣤⡾⠋⣨⣿⣿⣿⣿⣿⣿⣿⣿⣅⠙⢷⣤⣈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠋⣹⡏⢰⡿⢿⡄⢹⣏⠙⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⣿⣷⣾⡇⢸⣷⣾⣿⣦⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⢀⣀⡀⠀⢀⣀⡀⠀⢀⣀⣀⣀⣀⡀⠀⢸⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠛⠛⠋⠀⠙⠛⠛⠀⠙⠛⠛⠛⠛⠋⠀⢸⣿⠀⠀⠀⠈⠻⢿⣿⣿⠏⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⢸⣿⠀⢀⣼⣿⣶⣄⠉⠁⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⢶⣶⡦⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⡦⠀⢸⠏⢠⣾⣿⣿⡿⠃⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⣰⣿⣿⣿⡟⠁⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⣀⣀⣀⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⠀⢀⣼⣿⣿⣿⠏⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠉⠉⠉⠀⠉⠉⠉⠀⠉⠉⠉⠉⠉⢠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⠀⣼⢻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⣿⣷⣶⣶⣶⣦⡀⠀⢀⣀⣠⣿⢿⡿⠁⣸⣿⠟⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡇⠀⠈⠉⠉⠁⠈⠁⠀⠉⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟