Toffee Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠉⠛⠋⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡴⠀⠀⠀⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⡏⠸⠃⠛⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠙⠓⠷⣼⡥⠚⠉⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⡇⡟⢻⣄⠀⠀⣋⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡠⠴⢤⡤⠤⣄⠈⢳⡤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡞⢹⠀⡇⡇⢸⠈⡿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⡑⠋⠀⠀⠀⢈⡇⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⡇⡇⢸⠀⡇⡇⢸⠀⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠳⣄⠀⠀⣠⠞⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢹⢸⡇⡇⢸⠀⡇⡇⢸⠞⠁
⠀⠀⠀⠀⣠⠖⠛⠉⣉⣧⡀⠀⠈⠓⠚⠁⠀⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠬⠿⣸⡇⡇⢸⠀⣇⠟⠁
⠀⠀⠀⠸⡅⠀⠐⠋⣡⠊⠉⡷⠦⠤⠤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣩⠤⠀⠀⢸⠧⣇⣸⠞⠁
⠀⠀⠀⠀⡼⠀⠀⠀⠇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⣰⠃⡼
⠀⠀⠀⠀⠙⠦⠴⠦⣄⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡞⣀⠈⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⡼⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠭⠭⠽⢾⣵⡧⢞⣁⡬⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⣞⡛⠛⢦⡀⣀⡤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠓⠒⠒⠒⠒⠒⠛⢾⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠙⢦⡀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣖⡿⠒⠛⢦⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢳⡀⠀⣠⠞⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⣀⣀⣀⣀⣀⡀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠹⣄⠀⠀⠈⠛⠁⣠⠞⠁
⠀⠀⠀⢀⣀⠤⠖⣋⢽⡿⣭⠭⠭⠭⣭⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⣄⡈⠓⣦⣤⠴⠚⠁
⠀⠀⠀⠈⠙⣆⡈⠁⠋⡼⠉⠙⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣀⡤⠀⢠⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠹⣆⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠙⠻⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⣾⠃⢠⡶⠀⢀⡀⠀⢿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⣁⣀⣀⣀⣉⣻⣿⣇⣴⣟⣤⡾⠛⠁⢀⣠⣴⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣿⣟⣁⣤⡶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⢀⣠⣴⠶⠛⠋⣡⣼⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠋
⠀⠀⢴⣾⣛⠉⠀⢠⣴⠟⢋⣾⠋⣸⡿⣯⣍⣉⣉⣉⣉⣽⠿⠋
⠀⠀⠀⠈⠙⣷⠀⠀⠀⠀⠟⠁⣰⡟⠀⠀⠉⠉⠙⠉⠉
⠀⠀⠀⠀⠀⠙⠻⢶⣄⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡀⣀⢀⡖⠁⠘⡄⡀⠀⠀⠀⠀⠀⠀⢸⠷⠶⢶⣿⡦⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⣦⡀
⠀⠀⠀⢠⡏⢙⢱⡄⢀⣼⣶⠞⠁⠀⠀⢀⣆⣹⣤⣤⢄⣿⡃⣰⣿⣆⠄⠀⠀⠔⡒⠁⢪⠚⠛⣁⠟⡌
⠀⠀⠈⠀⠀⠘⠺⠁⠀⠈⠸⠀⠀⠀⠀⢠⣁⡈⠩⠉⠈⢉⣭⣿⠋⠠⠀⠀⠀⠈⠃⣧⢸⠓⠘⠛⠶⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠶⠲⠶⡾⣸⠛⠂⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⢀⠠⠚⠛⢇⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⢀⣾⠦⣀⣤⡴⣿⡇⠀⠀⠀⣀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣶⠋⡆
⠀⢠⣏⠀⠁⣜⣥⡶⠿⠃⠀⠴⠀⠀⠀⣯⣠⡀⠀⠀⠀⣀⠐⣀⡀⠀⣎⡁⠀⠀⠀⢂⠀⠶⠛⠉⣷⡊
⠀⠈⢳⡁⢸⠛⣸⠟⠃⠀⠀⠀⡐⠲⠄⠠⣞⡍⠀⠀⠠⠏⠆⡍⢹⣷⠿⠻⠄⠀⠀⠀⠹⢦⣴⠆⠁
⠀⠀⠉⠉⠑⠛⠁⠀⠀⠀⠀⠀⠃⠀⠠⠤⠏⠁⠀⠀⠀⠠⠬⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣠⣶⡆
⠀⠀⢠⣖⣈⣁⣄⣤⡴⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡟⠈⠉⠀⢢⡄⠀⠀⠀⠀⡀⠀⠛⠛⠐⣂⣀
⠀⠀⢎⠉⠉⠙⣿⠏⢀⣷⠤⠄⠀⠄⠀⠀⣶⠡⠀⠀⠀⣇⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢶⡶⣍⣾⠋⣿⠂⠄
⠀⠀⠀⠀⠀⠐⠚⠚⡋⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⣬⡿⠿⠴⠋⠁⠁⠀⠀⠀⠀⠃⠀⠠⢀⡿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⣠⠆⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⣿⡇⠀⣿⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠾⠿⠛⠛⠛⠛⠛⠻⢿⣧⠀⠘⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⢀⣀⣤⣤⣤⡀⠀⠙⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⢀⣤⠶⠛⠋⠉⠀⣠⡼⠃⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠤⠴⠖⠛⠉⢀⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣠⣤⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣿⠇
⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⣠⣾⠿⠛⠉⠁⠀⣀⣀⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢷⣦⣄⣀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⣠⣿⠟⠁⠀⢀⣤⠶⠛⠉⠁⠀⠀⣠⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⠟⠁
⣰⡟⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠁
⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⡶⠿⠟⠋⠁
⠀⠈⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢰⠇⠀⠀⢹⣿⠀⠉⠁
⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⢷⣶⣄⠀⠀⠀⠀⡟⠀⢀⡟⠀⢀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⣸⠇⠀⣸⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⣿⠀⢀⡿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⢿⣄⡾⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠉⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⢀⣾⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠠⠀⠀⠀⠀⣀⣀⡀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⢀⣀⠀⠐⠀⢀⣀⡀⠀⠀⠀⠘⠀⠀⠀⠀⢀⣠⠾⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂⠀⠀⠀⠐⠷⠚⠛⠀⠀⠂⠂⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⠂⠒⠒⠶⢆
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢠⠠⠤⠐⠒⠒⠚⠂⠀⣀
⠀⠀⠀⠀⠀⢄⠃⢈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠰⡼⠀⣸⠇⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠴⠒⠃⠉⠉⠀⠀⠀⠀⢠
⠀⠀⠀⣬⠎⢠⠊⡍⢠⢣⠃⡄⠀⠀⣀⣀⠀⣀⣀⣀⣉⣉⣓⣒⡀⠤⠀⠀⠀⠶⢾⣆⡀
⠀⠀⠀⢻⡄⢻⣤⣇⣼⣼⣼⠏⠀⠐⠀⠤⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠽⠙⠒⠁⠠⡀
⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠈⢀⡀⠀⠀⠀⣈⠀⡤
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⡦⣴⠈⠀⠀⢧⠀⡀
⠀⠀⠀⠀⢠⠊⠰⠾⠛⠀⢠⡟⠀⡀⠀⠠⠶⠶⠀⠀⣀⠀⠀⠀⣀⠀⣺⠀⠀⠈⡇⡇⠀⠀⡆⠀⠠⡇
⠀⠀⠀⠀⠸⡀⠄⠀⠂⠀⠔⠁⠀⠹⠀⠀⠀⠠⠐⢲⠏⠀⠀⠀⠀⠠⠠⠌⠐⠺⣇⣧⠤⠶⠧⠀⠀⠀⠒⢒⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠉⣉⣭⣿⣶⡦⢤⡀
⠀⠀⠀⠀⠀⢀⠔⠲⠤⣄⡀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢄
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠉⠢⣄⠀⠀⣠⠃⠀⠀⠀⢀⣼⣿⣿⡿⠋⠀⠀⠀⠀⢀⣤⣾⣧⠀⠀⠀⣀⢴⡺⠛⠀⠀⠉⢣
⠀⠀⠀⠀⢸⠃⠀⠀⠒⠢⠤⣀⡈⠑⢦⠃⠀⠀⢀⣴⣿⣿⣿⠋⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢧⠴⡪⠕⠉⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⢼⠀⢀⠀⢀⠀⠀⡀⠈⠁⣺⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⣠⣾⣿⣿⠿⠋⠁⠀⢸⡤⠐⠂⠉⠉⠁⠈⠁⠀⢧
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣀⡨⠭⢿⣾⣿⣿⡿⠋⠁⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⢸⠦⠤⠤⠤⠄⢀⣀⣀⠀⣸⠂
⠀⠀⠀⠀⠀⢸⠀⠀⠒⠈⠁⣀⡠⠤⢺⡛⠋⠁⠀⠀⠀⣀⣤⣾⣿⣿⠟⠁⠀⠀⢀⣠⣴⣶⡿⠒⠠⢄⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠤⠐⠊⠀⠀⢀⡴⢷⣶⣶⣶⣿⣿⣿⡿⠟⠋⠁⠀⢀⣠⣶⣿⣿⣿⢿⠛⠢⢄⣀⠈⠑⠢⢄⡀⢠⣻
⠀⠀⠀⠀⠀⠸⡀⠀⠀⣀⡤⠔⠊⠁⠀⠀⠻⡟⠛⠋⠉⠀⠀⠀⢀⣠⣶⣿⣿⡿⠛⠉⡰⠃⠀⠀⠀⠀⠉⠒⢄⡀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠉⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣤⣴⣶⣾⣿⣿⣿⠟⢁⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⣠⣿⣿⡿⠿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠰⣿⠟⠉⠀⠀⠀⠀⠈⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣰⣿⠏⠀⠀⠀⠀⢀⣀⣴⣾⠟⠉⠙⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣾⣶⣶⣿⣷⣰⣿⠇⠀⠀⠀⠀⢾⠿⠟⠋⠁⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⣤⣤⣀⡀⠉⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠉⠉⠉⠙⠛⠿⢿⣶⣄⡘⢿⣧⡀⠀⢀⣀⣀⣤⣴⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠿⠿⠿⣷⣶⣶⣤⣄⡀⠀⠀⠈⠻⢿⣾⣿⣿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣤⣤⣄⣀⠀⠀⠈⠉⠛⠿⣷⣦⣀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣋⣀⡀⠉⠉⠙⠛⠻⠿⣷⣦⣄⡀⠀⠀⠙⠻⣷⣦⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠛⠛⠻⠿⠿⣷⣶⣤⣀⠀⠀⠉⠛⢿⣷⣄⡀⠀⠈⠻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣤⣤⣄⣀⠀⠀⠀⠀⠉⠛⢿⣶⣤⡀⠀⠈⠻⣿⣦⡀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠉⠉⠉⠛⠻⠿⣷⣦⣄⠀⠀⠀⠈⠛⢿⣦⡄⠀⠈⣻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⡀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⠀⠀⠀⠙⢿⣦⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣼⣿⡛⠻⠿⣿⣶⣤⣄⡀⠀⠀⠀⠙⠿⣷⣄⠀⣠⣾⡿⠋
⠀⠀⠀⣀⣤⣴⣾⠿⠟⠛⠛⠉⠉⠉⠙⣻⣷⡄⠀⠀⠈⠙⠻⢿⣶⣄⡀⠀⠀⠘⣿⣿⡿⠋
⣠⣴⣾⠟⠋⠉⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠛⢿⣦⣀⠀⠀⠀⠀⠉⠻⣿⣦⣠⣾⡿⠋
⣿⣏⠀⠀⠀⠀⣠⣄⠀⢠⣶⡿⠛⠁⠀⠀⠀⠀⠙⠻⣿⣷⣶⣶⣶⣶⣾⠿⠛⠁
⠘⠿⣷⣤⣀⣾⡿⠁⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠈
⠀⠀⠈⠙⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠸⣿⣄⡀⠀⠀⣀⣠⣾⠟⠁⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⣿⡇⠀⠀⠀⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⢀⣴⣿⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⣿⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢡⠂⠸⠛⡄⠄⡔⠄⠀⠠⠂⡔⠀⠔⡤⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠈⠁⠈⠀⠀⠀⠁⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⡌⠕⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡑⢄
⠀⠀⠀⢀⠔⠈⠉⣷⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⣎⣐⡩⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⡆⢳⡶⠦
⠀⠀⢰⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡋⢞⡅⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⢁⠈⡘⣌⠃
⣄⡒⠚⡶⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⡠⠔⡲⢇⣢⡵⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⡀⢊⣄⠜⠁
⠘⠤⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⡴⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠌⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣀⣘⡀⡉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣔⠀⢓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡒⡲
⠀⠀⠀⣾⣿⣿⣿⣯⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠉⡸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠫⡪⡑⡀
⠀⡀⠸⣟⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠔⠁⡠⢊⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠙⢌⠢⡈⡾⠃
⠘⡄⠐⠝⡟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠻⡤⠪⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡬⣗⢕⢄⡱⠋
⣀⣓⣓⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣩⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣎⣜⣉⣉⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠈⠻⠿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣾⠏⠀⣠⣦⠀⠀⠀⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠛⠛⠛⠿⣶⣤⣼⡟⢀⣴⡿⠁⣠⣴⡿⠀⠀⠈⠛⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣦⣾⣯⣴⡿⠛⠁⢀⣀⣤⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣿⣁⣠⣴⡾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣤⣤⣤⣶⣴⣤⣶⣤⣶⣶⣴⣤⣦⣤⣶⣦⣶⣶⣶⣦⣴⣶⣦⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣀⢀⠀⠀⣀⠀⣀⣀⡀⠀⡀⠀⢀⠀⣀⢀⡀⢀⣀⠀⣀⠀⠀⡀⢀⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⢉⣹⣿⣧⡉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣡⣿⠏
⠀⠀⠀⣀⣤⣶⠿⠛⠋⠁⢀⣠⣶⠿⢫⣿⠟⢿⣿⣟⠻⠻⠻⠟⠛⠛⠻⢛⣿⡿⠛⠁
⠀⠀⠈⠻⢿⣶⣤⡀⠀⠰⠿⠋⠁⣰⡿⠃⢠⣿⠋⠻⢷⣶⣤⣤⣤⣴⡶⠿⠋
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠘⠛⠀⢀⣿⠃⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠻⠷⣶⣤⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢾⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢠⡿⠉⣻⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣴⠟⠀⢰⡟⣠⡟⠉⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠀⠀⠀⢀⣴⠟⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⣠⡾⠟⠷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⢶⣤⣿⠀⠀⠀⠀⠀⠀⠀⠐⠿⠟⠛⠁⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢻⣄⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢿⣄⠀⠀⠙⢷⡄⠀⠀⠀⠀⠈⠻⣦⠀⠀⣀⣀⣤⣤⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⠀⠀⠹⣦⡀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠹⣟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠙⣧⡀⠀⠈⢻⣆⠀⠀⠈⠻⢶⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠹⣦⡀⠀⠈⠻⣦⡀⠀⠙⢷⣄⠀⠀⠀⠙⢷⣤⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⣧⡀⠀⠈⢷⣄⠀⠀⠈⠻⣦⡀⠀⠉⠳⣦⡀⠀⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡀⠀⠈⢿⣄⠀⠀⠙⢷⣄⠀⠀⠈⠻⣦⡀⠀⠈⠛⢷⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠘⢷⣄⠀⠀⠙⢷⣄⠀⠀⠙⢷⣄⠀⠀⠈⠙⢷⣄⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠻⣦⡀⠀⠀⠙⢷⣄⠀⠀⠙⠷⣤⡀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣿⡄⠀⠀⠈⠻⣦⡀⠀⠀⠙⠳⣦⡀⠀⠈⣻⡿⠋
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡞⠛⠛⠉⠉⠉⠁⠈⢻⣆⠀⠀⠀⠈⠻⣦⣀⠀⠀⠈⠛⣷⡾⠋
⠀⠀⠀⢀⣴⠟⠋⢁⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠈⠙⢷⣤⣠⡾⠋
⠀⠀⠀⢻⣅⣠⡶⠛⠉⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠶⠶⠶⠞⠛⠉
⠀⠀⠀⠀⠉⠛⠳⣶⠾⢛⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⢹⡶⠛⠁⠀⠀⠀⣠⡶⢿⡇⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣠⡾⠋⢠⡟⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠾⠿⢶⣴⡟⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠛⠁

⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⠶⠶⠶⢶⣶⣶⠶⡶⠶⡶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⣀⡀⠀⡜⠉⠶⠄⢖⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⡰⠐⡄⡀⠀⣔⠓⠊⠀⠄⢤⠁⠋⠳⡐⣰⡆⡀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⢦⠀⡴⢶⡛⡉⠛⠛⡙⢟⢶⣸⠌⠋⡕
⠀⢱⠦⠑⠚⠫⠤⡳⣀⠀⠑⠊⡠⠤⡲⢿⠉⡀⠤⠜⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠠⠦⣬⠡⠀⢣⢡⠀⡇⠀⢸⢸⣧⡀⠀⢳
⡰⠋⠀⢀⣀⣠⡞⠉⠓⢀⢀⠀⠳⠴⠃⣼⠓⢦⠀⡞⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⡀⢠⢤⣸⢶⡁⡸⣸⢀⣷⣆⣼⡼⠙⢥⠋⠁
⠣⢄⡴⠛⠳⡀⠉⢳⡒⡜⣀⠜⡠⢊⡼⠁⠓⠤⠴⠃⠀⠀⢀⣤⠔⠒⢆⡁⣋⡵⠒⠒⠤⣄⠀⠈⢩⣇⡷⠈⠁⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠈⠓⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⣈⣀⠒⠋⠋⠙⠓⢌⢑⣾⡇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⡋⠉⠉⢩⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⠀⠀⡇⠀⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠆⠱⢒⡠⡀
⠀⠀⠀⠀⠀⡠⠤⠊⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠑⠄⢬⣀⣀⣀⣇⣀⣀⡸⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠄⠀⠀⠰⣁⡨⠊⢢
⠀⠀⠀⠀⢀⡇⠘⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠤⠃⠸⣀⡵⠀⠀⠰⡈⡂
⠀⠀⠀⠀⠀⡗⢄⠈⠳⢄⡀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠢⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⣄⢀⡀⠀⠤⡀⣆⣉⠇⠈⠁
⠀⠀⠀⠀⠀⠳⡀⠑⠢⠄⣈⠉⠐⠒⢉⣡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠀⠀⠀⠀⣠⠤⠉⠑⠒⠤⠤⣀⠀⠀⠘⢦⡇⠐⠒⠁⢀⣀⡾⢅⡀
⠀⠀⠀⠀⠀⠀⠈⡦⢄⡀⠀⠈⢉⡩⠟⠀⠀⠀⠀⠀⠀⠀⣀⡠⠐⠈⠀⠀⠀⠤⡀⠀⠈⠁⠀⠀⠀⢀⠠⠒⠉⡄⠀⠀⠀⠈⠁⠈⠲⣋⠃⡜⡖⠚
⠀⠀⠀⠀⠀⠀⢠⠃⢀⠏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⣉⣈⡋⠻⣀⠀⠀⠀⠀⠀⠀⠈⠙⢂⡠⠊⠁⠀⠀⡠⠃⠀⠀⠀⠀⠀⠈⠉⠀⠉⢃⠘⡆
⠀⠀⠀⠀⠀⢀⠞⢀⠇⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠉⠀⠀⠀⠀⢉⠝⡫⠦⣀⣀⣀⡠⢦⡐⠁⠸⡇⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠘⡄
⠀⠀⠀⠀⢀⡞⢀⠎⠀⠀⠀⠀⠀⠀⠀⠮⣁⠢⢀⠀⠀⠀⡀⣐⠑⠀⠀⠀⠀⠠⠘⠱⠄⠘⡀⡠⠗⠁⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠈⢦⣸
⠀⠀⠀⠀⡞⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢇⡈⠑⢢⠉⠉⠹⠊⠁⠈⢱⠊⠉⠙⠆⠀⠀⢆⣸⠇⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠊⠙⢦⡀
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠼⠆⠂⢼⠄⠒⠒⢾⠤⠒⠢⢤⠠⠒⠁⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⡠⠴⢤⣄⠠⢀⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢌⣠⣠⠋⣠⢤⠅⣺⠁⠐⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣇⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢇⠀⢈⡦⠔⢋⢂⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⢖⠛⣍⠉⠙⣇⣀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠈⠫⣔⠛⠲⣅⠂⠀⣸⠼⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠶⢊⠙⢄⠀⠀⠀⠱⡘⡄⢠⠿⢧⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠋⠉⠈⢂⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠛⢭⡀⢄⠀⠀⠐⠄⠇⠀⣀⣠⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠳⡀
⠀⠀⠀⢀⠁⠈⢏⠒⠤⠧⠄⠈⠳⠀⠙⡌⣆⣀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠙⢆
⠀⠀⠀⠈⡗⠒⠈⠱⠒⡦⠺⢅⣀⠤⠔⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠜⠃
⠀⠀⠀⠸⣄⠀⣾⠗⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠛⠂⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣆⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⣀⣀⣠⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣁⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠹⡿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠟⠁
⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡈⠙⠻⠿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡇⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣿⠀⣠⡟⠀⠀⠀⠀⠈⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠋⠉⠉⠙⠻⢧⣴⠋⢀⣤⠶⠛⠉⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠛⠳⢦⣄⡀⠀⠀⠙⢾⣋⣤⣤⠤⠤⠴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠙⠓⠶⣤⣀⠀⠀⠀⠀⠀⠀⠙⠳⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣏⣀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠉⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⠙⠷⣄⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠙⠳⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀⠀⠀⣠⡾⠋
⠀⠀⠀⣠⡴⠶⠶⠶⠚⠛⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡾⠋
⠀⠀⣸⠃⠀⠀⠀⣀⣤⠾⠋⢳⣤⡀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠘⠶⣤⡀⠋⠉⠀⠀⣠⠟⠀⣿⠛⠲⠶⠒⠛⠉
⠀⠀⠀⠀⠈⣧⠀⠀⠀⢠⡏⠀⠀⡟
⠀⠀⠀⠀⠀⠈⠛⠶⣤⠈⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⡴⠟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡠⡐⠈⠀⠀⠀⠀⠀⠀⣠⡜
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡴⠙⠋⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠆⠀⠃⠀⠘⠀⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢰⣀⣠⡖⠒⠀⠀⡀⠴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⡞⣉⣁⣀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠉⠉⠙⢂⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢱⣀⣉⣉⣉⣉⣉⡉⠀⠠⠴
⠀⠀⠀⠀⠀⠀⡌⠀⢌⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠄⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣢⠃⢀⣿⠀⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢀⡜⠁⡔⢩⠃⣰⢰⠁⣄⠀⠀⢠⣄⡀⢀⡀⣀⣀⣈⣉⣙⣓⡤⠤⠀⠀⠀⠰⠏⣻⣦⡄
⠀⠀⠀⠸⣇⠘⢧⣼⣀⣧⣯⡼⡟⠀⠐⠄⠠⠔⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠬⠏⠑⠚⠁⠀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠬⢍⡀⠀⠀⠀⠀⠈⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢱⣀⡴⠘⠁⠀⣆⠀⡂
⠀⠀⠀⠀⠀⡠⠊⣠⣶⠖⠀⠀⣷⠀⠀⠀⠀⣠⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠸⠀⡇⠀⠀⡆⠈⠀⡧
⠀⠀⠀⠀⠀⡇⠀⢀⣀⠀⠀⢈⠃⠀⠷⡄⠀⠀⠀⣀⣀⣤⡇⠀⠀⠀⠊⠀⠈⢓⠠⢤⣆⡆⡇⣀⣀⡇⠀⣀⣁⣀⡀⠀⠀⠤⠖
⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠊⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠛⠋⠉⠀⠀⠀⠄⠀⠀⠐⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣩⣝⣹⣝⣽⣻⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⠔⣠⠉⠀⠰⣦⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⣃⣠⣄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣴⠏⠈⣁⢠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠉⠁⠀⠀⠀⣀⠈⢀⣠⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡀⠊⠦⡀⠀⠀⠘⠉⡇⠀⠀⠀⠀⠐⠦⠟⠘⠀⠀⠇⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠉⠹⢋⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠚⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡈⠉⠙⠋⡉⠛⠆⠀⠀⠀⠌⠉⠋⠉⠉⠟⠀⠀⠀⢀⡀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⣰⡳⢼⠇⠀⠈⠈⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡄⠂⠀⣠⡀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠰⡞⢴⠛⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢠⠀⠈⠀⠀⢐⠒⢻⡇⠀⠀⠀⠀⠀⠀⠂⣡⠀⠀⠈⠉⠙⠛⠉⠋⠀⠀⠤⠂⠚⣇⠄⠀⠀⠀⠀⠀⠧⠀⠠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⠀⠇⠈⠀⢰⠀⣽⠃⠀⠀⠀⠀⠀⢙⡇⠀⢈⣇⠀⠀⠀⠀⠀⠀⠀⢐⠏⣶⣓⡈⢤⣄⢠⠀⠀⠈⠁⠐⠀⠀⠀⠀⠀⠂⢀⣀⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⠱⠍⠉⠠⡯⠁⠠⢀⠀⠀⠀⠀⠀⠂⣰⠁⣸⡁⠀⢀⢻⠄⢹⠐⠂⠀⠀⠀⠀⠀⠀⣬⣱⠎⠡⣤
⠀⠀⠀⠀⠀⠀⠀⠄⠹⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠃⠉⠉⠀⠀⠀⠀⠀⠀⠐⠓⠊⠁⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣶⣾⡄⣶⣖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠
⠀⠀⠀⠀⠀⢀⣶⣶⠘⠿⠋⣀⠉⢀⠈⠛⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⢐⡒⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠥⠤⠖⠀⠠
⠀⠀⠈⠀⠐⢌⠻⢿⡿⠂⢸⣿⣿⠿⠃⠀⠀⠀⠀⠁⠀⠀⢁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⢈⠐⡤⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡁⡂⠐⡀⢐⠀⣂⢡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⠀⠅⠤⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⢌⢂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢰⡶⠄
⠀⠀⠀⠀⠀⢂⢐⠐⠂⠐⠀⡃⣈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠙⠋⠈⣡⣴
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⡀⠁⠠⠤⠄⠀⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡿⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣀⠠⠆⠀⠤⡀⠣⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⢨⣤⠀⠀⠀⠀⠀⠀⠰⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡅⢠⣳⡶⠀⠁⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠤⡅⠛⣀⠈⢀⠁⠀⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠢⠀⠀⠀⠀⠆⠀⠀⠲⠒⠀⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣴⣶⠦⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢀⣄⣸⣶⡟⣹⡏⢿⡇⠀⠠⠃⠀⠀⠀⢀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠂⠀⢸⡶⢋⣿⠛⣿⡍⢛⠂⠛⠃⠊⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠿⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⡂⠰⠂⢄⡀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠆⢀
⠀⠀⠀⠀⠈⠑⠃⠠⠒⠋⠀⠀⠀⠀⠈⣂⠀⠀⠀⠀⠚⠊⠉⠀⠀⠐⡧⠀⠀⠀⠖⡦⠀⠀⠀⠈⢥⠾⠏⠀⠻⠉
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⣉⠉⣛⣫⡄⠀⠀⠀⠀⡀⠀⠀⠨⡖⣻⠇⠀⠀⠀⠁⠀⠀⠀⡄
⠀⠀⠀⠀⡄⢉⠉⢤⢛⡂⠀⠀⠒⠉⠉⠉⠛⠁⠀⠀⠤⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣉⡄
⠀⠀⠀⠸⠉⠟⠁⠈⣀⣽⠀⠀⠀⠀⠀⠀⡀⠀⠀⢤⠾⣬⣤⡶⠀⠀⢀⢸⡂⠀⠀⠀⠀⢹⡙⠋⠁
⠀⠀⠀⠸⠀⠆⠠⢀⠂⣾⠀⠀⠀⠀⠀⢤⠚⠡⡄⠀⠀⠀⠀⠀⠀⢨⢠⡤⢱⣀⢀⠀⠀⠸⠁⠀⠀⠀⠀⠠⠈⢀
⠀⠀⠀⠠⡀⠀⠀⠀⠀⡟⠀⠀⠀⢲⠂⠋⣴⠊⠁⡂⠀⠀⠀⠀⠀⡗⢘⡋⠀⠹⡯⢑⠠⠀⠀⠀⠀⠀⢠⣲⡼⢋⡅
⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠛⠁⠉⠀⠀⠀⠀⠀⠐⠒⠉⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢰⡄⠀⣤⠀⣰⣿⣀⡀
⠀⠀⠀⠀⠈⠳⠤⠝⢲⡿⠉⣀⠿⢂⡀
⠀⠀⠀⠀⠠⣾⡿⢠⣿⠃⠐⠋⢀⢽⡇⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠿⠄⠻⠧⠄⠀⠀⠣⠾⠃⠀⠸⣇⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣾⣿⠧⠀⠒⠒⠒⠋⠉⠓⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⣀⠀⠀⣈⣑⡒⠀⠘⠏⡇⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢣⠞⢁⣼⢶⣾⠟⢩⣿⣿⠂⣰⣫⣿⠆⣼⣿⠃⠀⣰⣿⠂⢀⣾⡆⠀⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢋⡞⣽⡟⣿⢏⡴⣿⡿⢃⡴⢳⣿⠃⣸⣿⠃⡠⣾⡿⠁⣠⣾⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠈⠛⠋⠀⠙⠋⠀⠙⠷⠋⠀⠈⠓⠊⠉⠛⠊⠀⠘⠃⢊⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣀⠤⢒⣾⠏⠒⢄⡀⠀⠀⢀⣤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣰⡿⠀⠀⣿⣠⣄⡀⡞⠁⣠⣾⠃⠀⠀⠀⠳⣄⠀⠸⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣠⣿⠓⢀⣿⠆⠀⠘⣃⡙⠛⠋⣵⡞⢓⠀⠀⠀⠘⢦⣀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡠⢾⣏⣴⣿⢋⣿⠗⣿⠰⣿⠒⡡⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠻⠃⠈⠉⣸⣏⠜⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⢠⡄⠀⣠⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠘⡿⠷⠀⠀⠀⠈⠁⠀⠉⠀⣶⠀⡟⠀⠆⡀⠀⠸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠈⠃⠀⠀⠀⠀⢀⡀⠒⢀⠀⠀⠀⡄⠀⠛⠀⠀⠀⠁⠁⠀⡤⠀⠀⠒⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⠆⠀⠀⠀⠀⠀⠀⢰⡆⠀⠈⢁⡒⠈⠀⠀⠀⠀⠀⠀⠀⣤⣼⣦⡄⠀⠀⡴⣠⠀⠰⠟⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠃⢴⣶⡶⠶⠂⠀⠀⠁⠀⠀⠈⠁⢰⣷⣰⣦⢠⡞⠀⠀⡼⠻⡏⠀⠀⠘⠱⠃⠀⠀⠀⠀⢀⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠟⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠘⣬⠃⠀⣠⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠘⠃⠀⠾⢧⡤⠀⠀⣸⣿⠟⠳⣆⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠲⠌⠀⠀⡀⠀⠀⣰⠟⠁⠀⢠⣿⡿⠀⣠⡟⠀⣾⠋⢻⠀⠿⠟⠛⢹⣿⠋⣩⡍⠉⠉⠀⣀⡀⠀⠀⠘⠃⠀⠀⢠⣾⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠁⠀⠀⠀⣼⣿⡿⠿⠋⠀⠀⠈⢹⠟⢛⣿⠆⢠⣿⡏⢠⣿⠃⠀⣠⢾⣿⠁⠀⣀⡀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣷⡤⠄⠀⠘⠃⠀⠀⣿⣿⠁⣴⢏⣿⠏⣰⠋⢠⣿⠏⢠⣿⣿⠀⢸⣯⠴⠋⣀⣿⣿⠶⠛⠋⠀⠀⠰⠀⠀⠉⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠟⠹⠏⠀⢀⣀⠀⠀⠀⢸⣿⣿⠀⠿⠋⠻⠞⠁⢠⣿⠏⠀⣾⣿⡇⠀⠀⢀⣴⠟⣻⣿⠃⠀⠀⢠⢀⠀⠀⠀⠀⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⡀⠰⠆⠀⢶⣿⡛⠉⠀⠀⠀⢸⣿⡏⠀⢚⣛⣛⡀⠀⣸⣿⠀⠀⣿⣿⠁⠀⢰⣟⠁⢠⣿⠏⠀⠀⠐⠂⡌⠀⠀⢹⣾⣷⠶
⠀⠀⠀⠀⠀⠈⡋⠰⠆⠀⣠⡾⠟⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠈⠁⠀⠀⠘⠿⠶⠿⠋⠀⠀⠀⢀⡀⠀⠀⠀⠘⠙⠇
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠉⠀⢀⠀⠀⠀⠙⠀⢀⣀⣤⣴⣶⡶⠶⠶⠶⢶⣦⠀⠀⠙⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣾⡤⢶⡇⠀⠀⠀⠐⠖⠒
⠀⠀⠀⠀⠀⠠⠶⠶⠓⠀⠀⠀⠈⡄⠀⠀⠀⡾⠟⠛⢉⣿⠇⢀⡄⠀⠀⢠⣶⠀⠀⣴⢻⣿⠇⢀⣤⣶⠶⡄⠀⠀⠀⠀⣾⣤⠞⠃⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⣈⠁⠀⣠⣿⣿⢂⡾⢁⣿⡟⣰⣿⡟⢁⡴⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠿⠀⠀⠀⠀⠒⠀⠂⠀⢠⣿⡇⢠⣿⢀⡞⣿⡿⢻⡿⠁⣸⣿⠃⣿⡿⠛⠋⠀⢀⡤⠀⠀⠀⠀⠂⣈
⠀⠀⠀⠀⠀⠀⠀⢤⠄⠴⠀⠀⣀⣴⠆⠀⠀⠀⠀⢸⣿⠀⠘⣿⠟⢰⣿⡇⠀⠀⢀⣿⣿⠀⣿⣧⡀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠘⠛⠓⣶⠆⠀⠀⠀⠸⠿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠈⠛⠛⠉⠀⠀⠀⠐⣾⣷⠂⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠸⠿⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠉⠁⠀⠀⠀⢠⡄⠀⠀⢀⠀⠁⠈⠀⣩⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⠆⠀⠀⡀⣀⠀⠀⠀⠖⠀⠀⣠⡄⠀⣀⣀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣴⣄⢀⡴⠀⠈⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⠞⠁⠷⠚⠁⣿⠀⢀⠀⠰⣶⣿⡤⠀⠸⠃⠹⠞⠀⠀⠰⣌⠳⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠆⠀⠀⠀⡰⠀⠤⠀⠘⠀⠀⠀⠀⠋⠻⠀⠀⡤⡐⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠉⣠⣦⡀⠀⠀⠀⢀⠀⡀⠀⢀⡀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠀⠀⠀⠀⠘⠀⠁⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⠛⠉⠉⢉⠟⢻⠒⠤⣀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⡔⢡⠃⠀⠀⢀⠎⠀⡎⠀⠀⠀⠑⢖⠋⠀⠀⠀⢀⡠⠴⠚⠀⠀⠈⠉⠉⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡎⠀⡎⠀⠀⠀⡞⠀⢰⠇⠀⠀⠀⠀⠈⢣⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡜⢀⣠⣇⠀⠀⠰⠀⠀⢰⣀⣄⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢀⠎⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⡻⠀⠀⠸⠀⣿⣿⣾⡇⠀⡇⠀⠀⢸⣿⣶⡗⠀⠀⠀⠀⢸⠒⠒⠒⠒⠂⠒⠒⠒⠂⠀⠀⣤⠤⠊
⠀⠀⠀⠈⡄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡇⠀⠀⡇⠀⠈⢹⠉⢀⣴⣷⣶⣦⠸⠏⠋⠁⠀⠀⠀⠀⡸⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⣀⠬⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠘⡀⠘⡿⠿⣿⣿⠀⠇⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠉⠒⠢⠤⢄⣀⣀⣠⠃
⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠀⠀⡇⠀⠀⠀⠇⠀⠹⢤⡿⠃⠀⠘⡄⠀⠀⠀⢀⡎
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠈⢆⠀⡇⠀⠀⠀⠰⡀⠀⠘⣄⠀⠀⠀⠘⡄⠀⣠⠊
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠔⠉⠣⣘⡄⠀⠀⠀⠱⡀⠀⠘⢦⡀⠀⢀⡼⠞⠁
⠀⠀⠀⠈⠑⠒⢻⠁⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠈⠙⠲⢤⣄⣀⣘⣄⣀⡀⠵⠚⠉
⠀⠀⠀⠀⠀⠀⠘⣄⠀⢀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠈⠁⠉⢹⠀⠀⠘⠀⠘⡇⠀⠀⣴⡄⣀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀
⠀⠀⠀⠀⠐⠀⡀⠤⠴⠛⣒⣻⠦⠅⠀⠀⠀⠀⢀⠠⠀⣤⡤⠶⠦⠀⠀⠀⢠⠀⣠⠆⠀⢲⢀⡰⠃⠀⠈⠁⢘⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡐⠁
⠀⠀⠀⠀⠀⠀⠙⢂⣄⣠⡿⠹⢿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⣄⠀⠀⠀⠀⠀⢸⡤⠁
⠀⠀⠀⠀⠀⢀⠈⠀⠉⠙⢄⠀⠖⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⠀⢀⡀⢠⢿⡀⠀⠀⡄⠀⠀⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⠀⠋⡾⢦⡀
⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡃⠀⠀⠀⠹⠋⠀⠑⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢚⣥⣰⠟⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠃⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⢿⠀⠀⠈⠘⢇⠘⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠈⡄⢁⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⢀⣤⣀⠳⢤⣎⢠⡄⠀⢀⣨⠶⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡡⠶⠂⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡀⢳⣀⡴⠀⠀⠀⠀⣀⣤⠖⢂⡀⠛⠷⠆⣠⡀
⠀⠀⠀⠀⠠⡀⠀⠐⠈⣽⠁⠂⢈⣹⡀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠈⠤⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⣠⡆⢀⣠⣄⣀⠾⢷⣤⣤⣾⣿⠋⠀⣀⡠⠄
⠀⠀⠀⠈⠀⠈⢳⡏⠉⢉⣰⣏⠉⢀⡄⡃⠀⠀⠀⠀⢀⠄⢠⡤⢠⠂⠓⢚⡀⠀⠀⠀⠀⠀⡀⠠⢦⣀⠀⠘⣿⣿⠅⠀⣰⠏⢀⣈⠙⣛⡛⠋⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡘⠣⠀⠀⠄⠁⠀⠀⠀⠀⠐⠶⠾⠗⣶⣶⠾⠿⠇⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈⠁⢉⡄⠀⠠⠬⠯⠐⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠄⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⠉⠁⠀⠄⢀⢤⡠⡤⣄⡀⠀⠉⠉⠩⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⡄⠀⠀⢰⣾⠃⠀⠀⠀⡙⢶⣄⣰⠀⣆⣀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⣷⣴⣴⢴⣼⢿⣭⣭⡭⡿⣷⣿⣿⣫⣿⣿⢿⡛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⣁⣦⣀⣄⢀⣆⣊⠀⢀⡆⣡⣿⣚⠋⢫⡙⣜⡄⢈⢇⠀⣳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠥⠒⢹⢫⠗⠉⠈⠉⠀⢈⠉⣉⡿⠯⢖⠻⠖⠉⠓⢻⡾⣶⣋⣴⠏
⠀⠀⠀⠀⠀⠀⢠⠊⠂⠀⠀⣀⡞⠉⠀⠀⠀⠀⠈⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠁⠳⣟⣄
⠀⠀⠀⠀⠀⠰⠃⡠⣠⡶⠚⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠃⣼⡼⠋⠀⠀⠀⢱⠀⠀⠀⠀⡎⢀⢤⣠⢤⢤⠠⣤⢤⢄⠋⠀⠀⠀⡈⠀⠀⠀⠀⠀⡔⠚⠋⠉⠁⢀⡠⣬⣉⠉⠃⠀⠂⠲⡄
⠀⠀⠀⠀⠈⡄⡟⠀⠀⠀⠀⢀⣨⣆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢇⢤⠀⠀⠶⡇⣀⣀⣈⣳⣤⣄⣠⣠⣇⣀
⠀⠀⠀⠀⠀⠙⢮⡙⠒⠉⠉⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⢀⢞⠛⠛⢛⠓⠛⠓⡒⢺⣻⠿⡿⢾⠛⢻⠉⢳⡀
⠀⠀⠀⢀⠠⠀⠄⡙⠲⢶⣖⡀⠀⠀⢙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⢀⠠⢿⢞⠗⠛⠊⠑⠒⢚⣷⢯⠷⡤⠳⠼⣧⣰⣇⡜⠁
⠀⠰⠚⠑⠒⡢⠚⠉⠁⠒⠠⢍⠉⠉⠉⠉⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠔⠁⠀⢀⡜⠃⠀⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠀⠳⢯⡉
⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠚⠛⠃⠘⠛⠃⠈⠉⠙⠓⠓⠓⠒⠒⠋⠉⠁⠀⠀⠀⠪⢔⡴⠛⠉⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⠋⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡒⠄⠐⢀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣙⠳⠯⣄⣀⣀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢉⠋⣀⡀⢀⣂⣤⡀⣠⡙⠢⢄⣠⣄⣀⣀⣠⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⢀⠔⠂⠀⠀⠀⠀⠺⣇⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢸⠠⡀⠀⠀⠀⠠⣺⠆⠿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠸⠀⠉⠛⠷⠿⠛⠁⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠐⠻⣦⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢶⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣐⠲⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⣰⠋⠃⠒⠂⠀⠀⠀⠀⠀⠀⢶⣤⣀⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⠁⢰⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠙⠛⠿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣐⣤⢀⣽⠃⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠸⠁⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⠓⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⣦⡄⠀⠀⠀⢀⣤⣄⠀⣀⣤⣿⣆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣙⠗⠛⣠⣄⣠⣶⡄⠀⢰⡟⠉⠀⢶⣄⣼⠇⣶⡄⠀⣿⡁⣿⠿⢿⡍⠀⠙⠛⣷
⠀⠀⠀⠀⠀⠀⣤⣶⠿⢻⡟⢋⣩⡿⢿⡇⠀⠈⠃⠀⠀⢿⡿⢋⣠⣿⣿⡶⣿⣿⣟⣀⣈⣿⡄⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⢛⣿⣩⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠈⢩⣿⠈⠿⠘⠿⠛⠋⣩⣌⠁⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⢿⣻⣹⡿⠀⠀⠀⠀⠀⠀⣀⣉⡀⠈⡋⠀⠀⢠⣄⠀⣾⠏⢁⣀⣤⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡙⣻⡇⠀⠀⣠⣴⡶⢿⣟⠉⣥⡀⢹⣆⣀⣼⣿⠀⣿⡾⠟⠛⢉⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠿⠃⠀⠀⠈⠀⠀⠈⣿⡀⢸⣧⠈⣿⠛⠉⠹⡧⠈⠻⠶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣿⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣶⢶⣄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠓⠀⠻⠋⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣶⡾⠟⠉

⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠴⢁⡎⠀⢀⣳⡄⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡀⢀⣁⣤⡄⠀⠀⠀⣠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⢠⣼⠃⢀⡤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣠⣈⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣘⡢⠀⠀⠀⠁⢝⠀⠀⠀⠀⠀⠒⠛⠊⠀⠀⠃⠀⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠤⢤⣀⣘⣩⣶⣂⠀⠀⠀⠀⢠⡀⠀⠀⠐⠏⠡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢠⡄⢀⠉⢠⣈⣙⠀⠀⠀⠒⠀⠉⠈⠀⠋⠀⠀⠀⠤⠄⠐⡄⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡄
⠀⠀⠀⠀⠾⠝⡟⠁⠀⠀⣈⣇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣰⣇⣁⣤⡷⠀⠀⠀⠀⣳⠀⠀⠀⠀⠀⠈⣟⠻⠋⠃⠀⠀⠀⠀⢈
⠀⠀⠀⠀⣦⢠⡁⢀⠀⠌⢩⣿⠀⠀⠀⠀⠀⢀⡁⠶⢀⡀⠀⠀⠈⠁⠀⠀⠀⢀⡉⣁⠙⡔⠀⠀⠀⠀⠀⡟⠀⠐⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠉⠘⠀⠀⠀⠇⢸⡏⠀⠀⠀⣄⡠⠠⠇⡠⠤⠗⡀⠀⠀⠀⠀⠀⠀⢸⠘⣿⠅⠐⢶⡜⠆⡀⡀⠀⠀⠀⠀⠀⠀⠠⣄⠢⠼⠂
⠀⠀⠀⠀⠘⠀⠀⠀⣄⢠⡿⠀⠀⠀⠀⠈⠃⠀⠊⠏⠀⢈⠐⠀⠀⠀⠠⠈⠀⠧⠤⠖⠀⠀⣼⣛⡠⠆⠁⠀⠀⠀⠀⢀⣀⡳⠟⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣍⣫⣬⣉⣿⣫⣿⣻⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠈⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡘⡀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠁⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠇⠠⡀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠸⠀⢁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⠰⠈⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠐⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⠀⠃⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⡀⠀⠀⠀⣸⠀⡀⠇⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠉⠉⠉⢻⡉⠙⠓⢦⣏⠀⢱⠀⢠⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠈⠳⢦⡀⣸⠁⠉⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠂⢀⣀⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢰⠋⠀⠙⢦⠀⠀⠀⠈⢻⠑⣆
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣠⠞⠉⠉⠹⡀⠀⠀⠀⠀⠈⡆⠀⠀⠈⢧⣠⠶⢄⠀⢇⠸⡆
⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣰⠃⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠈⠆⠘⡄⣷
⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠴⠞⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣯
⠀⠀⠀⠀⢀⣀⣠⡤⠟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠉⠦⠤⣀⡀
⠀⠀⠀⢼⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠄
⠀⠀⠀⠈⠙⠛⠒⠒⠒⠒⠒⠒⠶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⣻⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠌⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠂⠙⢄
⠀⠰⠀⠀⠀⠀⠉⠐⠒⠀⠒⠂⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠤⠤⠤⢄⡀⠀⠒⠠⠐⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⡠⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡶⠁
⠀⠀⠈⠿⡁⠲⢤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⣀⡤⠶⡎⠁⢰⡄
⠀⠀⠀⢸⠃⠀⢠⠂⠀⠈⠁⠐⠂⠤⠄⣀⠠⠀⠀⠀⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠤⡐⠂⠁⠀⠀⠀⡁
⠀⠀⠀⠸⣣⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠱⠄⠀⠀⢇⠀⠀⠀⠀⠀⡠⠂⠀⠂⠀⠀⠀⠀⠀⢹⡟⣼⡂
⠀⠀⠀⠛⠁⠀⠿⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠃⠀⠀⠀⠀⠈⢦⠀⠈⢦⠀⢀⡠⠞⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠉⠑⠠⠀⠀⠃⣀⠄⠲⠒⠲⠀⠷⣀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠡⠠⠏⢁⠔⠚⠂⠀⠒⠠⣍⠃⡀⢀⠄⠄⠀⠦⢄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡇⠰⠁⠀⠀⠀⠀⠠⠀⠀⢠⡗⠁⠀⠀⠀⠀⠀⢣⠀⢀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠓⠄⠀⠀⠀⠀⢠⢁⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠘⠂⠀⢶⡶⠶⠶⠶⠾⠁⣠⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠠⠀⣄⡐⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡑⠤⡀⠀⠀⠀⢀⠔⠉⠀⠀⠀⢀⣠⣞⠁⠀⠀⠀⠞⢋⡵⡋⠖⣇
⠀⠀⠀⠠⢀⠀⠀⠀⡄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠣⣀⠀⠋⢘⡣⢑⢦⠀⣀⠽⠢⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠲⠄⠁⠀⠀⠀⠘⢃⠠⡀⠀⠀⠴⠀⠁⠀⡀⠀⠀⢠⠉⠑⢄⢃⠉⠃⠈⠁⠀⠀⣸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢺⠀⡜⠉⠋⠉⠋⠓⠙⢹⠇⠐⠀⠠⠂⠀⠒⠀⠂⠐⠂⠈⢋⣠⣀⡀⣷⠒⢋⠠⡀⠠⠂⠉⢱⡠⣄
⠀⠀⠠⠀⠀⠀⠀⢸⡐⠅⠠⣂⠀⡠⠀⠀⠨⠜⠃⠀⠀⠑⠀⠀⠐⠃⠀⠰⠀⠀⠀⠉⠛⠁⢽⣉⠋⢝⣺⠛⠛⢦⠂⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡣⠈⠇⠀⡀⠉⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠈⢱⠀⠸⣆⠀⠮⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⡀⠒⢺⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠱⠆⠊⢩⠺⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠤⠀⠉⠂⠤⢀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡆⢀⠸⡀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠓⠦⣄⡀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠐⠠⢀⡉⠐⠂⠠⠤⠤⠤⠄⠒⠈⢀⣠⠔⠊⠀⠀⠀⠀⠉⠓⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇⠀⠀⠀⠀⢸⠉⠙⠓⠒⠒⠒⠒⠂⢩⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⠠⠄⠐⠒⠃⠃⠈⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠤⠄⠒⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⢠⠄⠀⠀⠼⠀⠀⠀⠐⢀⠀⠀⠄⠀⠀⠀⠑⠖⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢧⠀⠀⠀⠈⢂⠀⠀⠀⠀⠄
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠱⠤⠀⠤⠂⠀⠀⠙⠦⣤⡄⠞⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣾⣷⣿⣶⡷⢾⣾⣿⣿⣿⣿⣿⣷⣶⣿⣾⣾⣿⣿⣿⣿⣶⣶⣶⣿⣷⣿⣾⣷⣶⣶⣶⣾⢿⡶⠾⠿⢶⣶⡶⢿⣾⠿⢶

⠀⠀⠀⠀⣾⡷⢺⣾⣳⣦⣤⣦⡀⠀⠀⠀⠀⢀⣐⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃⠀⢴⣶⣶⣾⣿⣷
⠀⠀⠀⠀⢸⠃⣄⣀⣽⣿⣿⣿⡇⠀⠀⠀⣶⡘⣿⣿⡿⠇⠀⠀⢀⠤⣄⡤⣤⡶⠿⠛⢁⣤⣶⣶⡰⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⠈⢻⣿⣿⣿⣿⣿⣿⣳⣼⣷⣿⣿⡿⠃⠀⠀⡴⠀⠢⢤⢠⠢⣠⠛⠶⠃⢀⣀⣿⡿⢢⣿⣿⣿⡟
⠀⠀⠀⠀⢸⣤⠀⠸⣿⣿⣯⡉⣿⢻⣿⣿⣿⣿⠟⠁⠀⠀⣸⣷⣤⡀⠈⠋⠀⠁⠀⠀⢀⣈⣹⣿⡿⢪⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣇⠠⣤⠀⠙⠿⠿⠶⠿⠿⠟⠋⠁⣤⣀⣤⣴⣿⣿⣿⣷⣶⣀⣀⡀⣠⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣠⡙⠿⢶⡇⠀⠀⣤⠀⣠⠀⢠⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡟
⠀⠀⠀⠀⢸⣷⣾⠈⠓⠶⠞⣿⢠⡿⣤⡜⠻⠼⠋⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢼⣿⣿⣧
⠀⠀⠀⠀⢹⣿⣿⣿⣿⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⣤⣄⠹⠬⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⣿⣿⣿⡏
⠀⠀⠀⠀⢸⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣸⣦⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠻⣿⠿⠿⠻⠿⢻⡹⣯⣿⡿⣿⣧⠘⣿⣿⡅⠈⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡀⣴⣦⡀⢀⣤⣤⣼⣿⣿⣿⡟⢫⠞⠁⠀⠀⣿⣦⠾⠿⣟⣽⡿⠟⠀⢾⣿⣏⢥⣶⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿⣿⣿⠃⠰⣤⡼⣩⣥⣶⣿⠿⢿⣿⣿⡇⢼⣿⡟⢽⣿⡇
⠀⠀⠀⠀⠘⠋⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⣠⠋⣶⣿⣿⣿⡿⣷⣮⣍⠛⠃⢼⣿⡟⢲⣽⡇
⠀⠀⠀⠀⢺⣶⡿⣴⡿⢏⢿⠋⣿⢿⣿⣟⡜⢿⣿⡏⠋⠀⠀⢻⡿⢴⣿⣿⠿⠿⠃⠛⠿⠋⠀⠀⠀⣽⣇⠘⠘⠃
⠀⠀⠀⠀⢀⣉⣛⠻⠿⠿⣿⢿⣿⣾⣿⣾⣿⡼⣿⡇⣀⠀⢀⣼⣇⣼⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣇⣀
⠀⠀⠀⠀⢸⣿⣿⣷⣾⣧⣠⣦⣉⣉⠻⣿⣿⣿⣮⣤⣿⡇⠠⢿⣿⣉⣿⣻⣤⡀⠀⢀⠀⡀⠀⠠⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⣿⣿⣟⣀⣈⠛⣿⠍⢻⣿⣿⠀⠘⠻⣿⣬⡙⠶⢾⣦⣀⢚⢶⠎⠀⠀⣀⢀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠈⢛⣶⠀⠈⠉⠀⠀⠀⠿⡿⣿⠿⢿⣿⠿⠿⠓⠓⠒⠛
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠘⠗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⢀⠔⠀⠀⠐⠀⠐⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢆⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠳⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠉⠀⠀⠉⠀⠂⠤⡀⠀⠀⠈⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠠⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⣀⣀⡀⢀⠠⠄⠌⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⡀⠀⠀⠀⠀⠈⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠛⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣄⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠘⣿⣿⡄⠀⠀⠀⠀⠀⣿⣧⢀⣤⠤⢤⡈⠛⠛⠻⠷⠶⠶⢶⣿⣿⣦⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⣹⣿⠈⠛⠒⠒⢻⡖⠒⢶⣄⠀⠀⢸⣿⡟⠀⠀⠀⠉⠉⠉⠙⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢈⡇⠀⢸⣿⡇⠀⣾⣿⠁⠀⢀⣤⡄⠀⠀⠀⣀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⢸⠃⠀⢸⣿⠃⢠⣿⡟⠀⠀⣾⡟⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣠⡿⢃⣴⠗⣶⡿⠀⢸⠀⠀⣿⡿⠀⢸⣿⠇⠀⣸⡿⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⢀⡴⠟⢁⣾⠏⢠⣿⠇⠀⠺⠀⢰⣿⠇⠀⣿⣿⠀⠀⣿⠇⠀⢀⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡷⠋⠀⠀⣼⡏⠀⣼⣿⠀⠀⡇⠀⣼⣿⠀⠀⣿⣿⠀⠀⢿⡀⣠⢫⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⣿⠀⢰⠻⣿⠀⣸⠁⠀⣿⡇⠀⠀⠹⣿⠀⠀⠈⣉⣡⣿⡿⠤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⢿⣀⠆⠀⠙⠶⠃⠀⠀⠻⡇⠀⠀⢀⡠⠔⠊⠉⣠⣿⠏⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⢀⣀⣥⣴⣶⣤⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣾⡿⠟⠛⠉⠁⠀⠁⢸⣿⠀⠀⡎⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠁⠀⠀⢰⣶⠄⠀⠀⣿⣿⠀⠀⠙⠛⠉⠁⠀⠀⢠⣶⣿⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠉⣿⣿⣿⡆⠀⢀⣘⠃⠀⠀⢸⣿⡏⠀⣠⡀⢠⢺⣿⠂⣰⣿⠟⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠁⠀⠀⠀⢹⣿⣿⡇⠀⣾⣿⠇⠀⠀⣼⣿⠇⡞⣼⡵⢁⣿⠃⢰⣿⡯⠔⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⡟⠀⠀⢀⣿⣿⡾⠀⠿⠁⣼⡏⠀⣼⡿⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⣿⡇⠀⠀⡜⣿⣿⠃⠀⠀⠀⣿⠁⠀⠸⣷⡀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠸⣇⠀⡰⠀⠙⠿⠀⠀⠀⠀⠙⠁⠀⠀⠈⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠉⠛⠁⢀⣀⣠⣤⠴⠶⠒⠒⠛⠋⠋⠉⠉⠉⠉⠛⠛⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢁⡈⠑⡌⢢⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⢢⠀⠆⢙⣄
⠀⠀⠀⠀⠀⠴⡉⢆⠀⠀⡠⢳⡀⠀⠀⠀⠀⠀⢰⠁⢐⠡⠼⠀⡅⢸⠀⠀⠀⠀⠀⠀⠀⠘⠉⣉⢺⣮⡔⢉⣸
⠀⠀⠀⠀⢸⠀⠘⢆⡑⠶⣡⠊⡇⠀⠀⠀⠀⠀⠀⢃⠈⠢⠤⠚⢀⣎⠉⠒⠤⣀⠀⠀⠀⠸⡎⠀⠌⠑⢄⡠⠂
⠀⠀⠀⠀⠈⢇⡈⠥⠘⠒⠠⡽⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠐⠊⠁⠀⠈⠐⠠⠬⠂⠀⠀⠀⠈⠑⢺⣂⣺
⠀⠀⠀⠀⠀⠀⠁⠒⣖⣲⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠾
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣺⠀⠀⠀⠀⠀⢰⠌⢿⠉⠀⡁⠘⣄⣀⠟⡍⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠀⠀⠀⠀⠀⠸⠍⣅⠥⣈⡏⠁⢱⡠⢸⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠧⠄⠘⠚⠒⠓⠓⠚⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⢄⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢍⠉⠉⠒⢤⠡⡈⢖⢧
⠀⠀⠀⠀⠀⢠⠀⡡⣠⠒⠖⡖⣐⠖⡖⣒⠖⠤⡀⢀⠠⢤⠀⠀⠀⠀⠀⠀⠀⢰⣜⠦⠥⢾⢢⢄⡀⡗⠒⠚⠓
⠀⠀⠀⠀⠀⡇⠀⢨⣿⢠⠀⡏⣑⠀⡏⣻⢠⠀⡖⠵⠄⠘⠄⠀⠀⠀⢠⢤⠤⡼⡈⠑⠥⣞⢂⢻⣩⠃
⠀⠀⠀⠀⠰⣁⠬⠛⠹⣸⣀⣙⣻⣰⣉⣫⣸⣀⠟⢪⠁⢸⠀⠀⠀⠀⠐⣳⢧⡡⠛⠢⠄⠀⠭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⠉


⣿⣟⣻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⣿⣾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⠨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⢌⡀⠃⠈⠱⠀⠈⠒⡀⠊⡀⠉⢀⠂⠀⠀⠀⠀⠀⢠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⡌⠠⠆⠈⠠⠈⠡⢆⢁⠆⣀⠐⢀⡘⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠻⣿⣿⣿⠟⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⢸⠉⣿⣿⣿⠈⡏⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⠀⣿⣿⣿⠀⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⠄⠤⠄⠀⠀⠀⠀⠀⠠⠄⠄⠤⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡗⠊⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡁⠀⢠⢸⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⡳⠀⠘⠉⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠢⠀⠉⣉⣀⠱⢀⠴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠘⣻⣿⡛⠁⡀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈⡼⣻⢷⠟⠃⠈⠠⣑⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠊⠠⢪⢎⣭⣢⢠⢤⠐⠀⢰⡟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢤⣀⣀⣥⣿⣿⣿⡃⢬⡀⢺⣿⡟⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣠⣾⣿⠋⣿⣿⣿⣿⣿⣼⣷⣟⠛⠃⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡹⠏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢰⡇⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣧⣤⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠃⠘⠀⢸⣿⣿⢸⣿⣿⣿⡇⣿⢸⡃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣠⡄⢿⣿⣿⢸⣿⣿⣿⡇⣿⠈⡿⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡃⢸⡏⣿⢸⣿⣿⣿⡇⣿⢠⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⣷⣿⣼⣿⣿⣬⣥⣿⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣼⡿⣿⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣿⠻⣽⠃⠃⢨⣿⣼⣿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠉⠉⠉⠓⠔⠉⠙⠁

⠀⠀⠀⠀⢫⠐⠂⠤⠄⣀
⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠈⡩⠒⠂⠠⢀
⠀⠀⠀⠀⠀⠀⢆⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠈⢆⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣲⠄⠤⠄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠦⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⣀⠠⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⠃⠂⠤⠄⣀⠀⠀⠀⡠⢋⠤⢰⠖⠉⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠠⠤⢀⡀⠀⠀⠀⠀⠀⠉⢸⠁⢀⠔⠁⠀⠀⠀⠀⠀⡜⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠠⠠⢀⢘⡔⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠴⢮⠀⠀⠘⠒⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠤⢄
⠀⠀⠀⢠⠤⠤⠔⠁⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀
⠀⠀⠪⣀⡀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠀⠉⠲⣄⠀⠀⢀⡀⠤⠀⠒⠈⠉
⠀⠀⠀⠀⢈⣦⡤⢤⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉
⠀⠀⠀⢠⠊⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⡀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡠⠔⠂⠉⠉⠀⠀⠀⠀⠙⢦⢀⡀⠀⢀⡠⠖⠁
⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠪⠤⠄⣀⠀⠀⠀⠀⠀⠀⢀⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⡌
⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢰⠁
⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⢃⡐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠁⠀⠀⠀⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡀
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠹⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⡤⠴⠒⠋⠁⠀⠀⣀⡤⢾⡇
⠀⠀⠀⠀⠀⢸⠒⠚⠉⠉⠀⠀⠀⢀⣠⡤⠖⠋⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠈⢦⣠⣤⠤⠔⠒⠋⠉⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠴⢶⠛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢝⢢⣀⡠⠴⢲⠶⠦⣄⡀⢧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⡅⠉⠀⣰⠋⠀⣰⠃⠙⣎⣇⠹⣄⣀⣠⣄⣀⡀⠀⠀⣠⠴⠶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣰⠒⠀⠀⢀⡏⠀⢰⠇⠀⠀⠘⢻⡶⢫⠇⠀⣾⠀⠉⢲⠞⠔⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢼⣘⣿⣠⠴⢦⠀⢸⠀⠀⣾⡋⠉⠭⣛⠿⠀⢸⡀⠀⢹⠀⠀⠀⠐⣃⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡄⠀⡏⢳⡄⠀⠒⠀⠀⠀⣇⠀⠸⡆⠀⠀⢀⠀⠩⢤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⡟⠀⡪⠀⣄⠀⠸⣄⠀⠹⡄⣠⠏⠙⠲⠯⠥⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠾⣢⠞⠉⠙⠲⠾⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢯⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⡿⣿⡷⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣷⠾⣷⣿⣷⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⣶⣿⣿⣿⣿⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡏⢸⣿⣿⣿⣿⡧⡯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡼⠿⡟⢻⡿⠉⢿⠉⢉⣻⣟⠛⠿⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣭⡀⣰⡆⢸⡇⠀⠃⠀⠸⡆⠸⣿⣿⣷⠀⡆⢀⡍⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣻⡇⢹⣿⠸⡇⢸⡀⢣⠀⡇⢀⣤⡟⢻⠀⣇⢸⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⢸⡿⠀⢻⡀⣷⣼⡄⢷⠈⠛⠳⣾⡀⣿⣄⠿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⢛⡛⠿⣿⠛⠛⢿⡿⠾⠷⠖⣛⣥⣽⢿⣿⣷⠾⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠿⠃⢀⡟⠀⠀⠈⣇⠀⣰⡆⢠⡄⢀⣦⡄⠛⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⢲⡾⢿⡇⠀⠛⠀⢿⠀⠛⢀⣼⡇⢸⣿⣿⡆⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⡋⣰⠿⠧⣜⠅⢤⠸⠀⣄⢺⣿⡇⠈⣿⣿⣧⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠶⠞⠛⠷⠟⠳⠼⠷⢶⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⠶⠶⠶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⠴⠒⠉⠉⠉⠁⠀⠈⠉⠉⠛⠲⢤⣀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⣀⡤⠖⠒⠚⠒⠲⠦⣄⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢠⡏⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠘⣆⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣠⠤⠤⢤⡀⠀⠀⠈⢧⠀⠀⠀⠀⢸⡄⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⣼⠁⠀⠀⠀⠙⡆⠀⠀⠈⡇⠀⠀⠀⠀⣇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⠀⠀⠀⠀⢿⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⡿⠀⠀⠀⡿⠀⠀⠀⠀⡟⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⢠⡇⠀⠀⠀⢠⡇⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠈⠛⠲⠦⠤⠤⠶⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⣼⠁⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⣰⠃⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⣤⣤⠤⠴⠖⠛⠁⠀⠀⠀⠀⢀⡴⠃⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢉⣠⣴⡾⠋⠀⠀⠀⠀⡠⠐⠂⠀⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢶⣦⣤⣴⣶⣶⣶⣯⣭⣴⣶⡿⠛⣿⡅⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠐⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠹⣧⠀⠘⣿⡀⠀⠀⡔⠀⠀⠀⠀⠀⠀⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠹⣷⢀⠎⠀⠀⠀⠀⠀⡠⠊⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠉⠉⠁⠒⠢⠤⣀⢻⣆⠀⢹⣯⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣟⠒⠊⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠤⠤⠤⠐⠒⠊⠉⠀⠀⠈⣆⠀⢀⣽⣤⢀⡀⠈⠀⠀⠀⠀⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣿⡀⠈⣿⡄⠀⠈⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠹⣷⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⣠⠔⠃⠀⠀⠀⠀⢹⣧⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣸⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣴⠚⠛⢝⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠜⢡⡁⠀⠀⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠋⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⣏⠀⢸⠀⠀⠀⠀⢠⠥⣻⠆⠀⠀⠀⠀⢀⣴⠟⡉⠀⠊⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠇⠀⡞⠀⠀⠀⠀⢛⡵⠋⠀⠀⠀⢀⡴⠚⠁⠀⠀⢀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢡⠸⠉⠀⠀⢟⡴⠋⠀⠀⠀⠀⠀⠘⢦⣦⡔⠀⠤⢄⠀⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠰⡀⠀⢫⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢳⡎⠈⠉⠙⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠐⠒⢊⠘⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⢈⠵⠀⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠂⠀⠀⠀⠀⠀⠀⠐⣥⠀⠀⡈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⣄⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣱⠀⠀⠻⢩⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡾⢀⢀⡀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠰⢻⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠺⣅⠈⠗⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⠀⠀⠀⠀⠐⣆⡤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣃⡂⠀⠄⠆⠊⢤⠤⠀⠀⠃⠍⡠⠃⢚⣃⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠈⣀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠓⠲⠶⠶⠶⠚⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣥⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠓⢦⡄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢆⡀⡀⢹⠀⢀⡴⠛⠋⠙⠋⠉⠛⠛⠒⠲⠦⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⡨⢑⠘⠶⢫⠀⠀⠀⠈⠠⠀⠀⠂⠀⠴⡀⠈⠀⠊⠻⡄⠀⢠⠶⠋⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⢀⣀⣈⣀⡀⢁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⢻⣤⡟⠰⠀⠀⡂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠀⠀⠀⣠⠟⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠀⠏⠐⠂⠀⠎⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣦⠵⠚⠁⠀⠸⣇⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣦⣦⠐⠄⢢⢀⣥⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣿⣏⣈⡦⣀⡐⠀⢠⡀⠈⡀⣡⠟⠁⠸⣆⠀⠀⠈⢣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠓⠒⠒⠛⠁⠀⠀⠀⠙⠦⣔⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⣰⣿⠟⠁⠀⣿⣿⡆⠀⣠⣴⡶⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣴⡿⠃⠀⠀⢀⣿⣿⠁⢰⣿⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⢀⣼⣿⠇⠀⢺⣿⡇⠀⢰⣿⢀⣤⠄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢇⣀⠀⣠⣾⡿⠃⠀⠀⠘⣿⣷⡀⣾⠇⠸⢿⣧⣤⣿⣷⠾⠛⠋⠉⠁⠈⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠉⠛⣩⣵⡾⠛⠛⠇⠀⠈⠛⣿⡿⣤⣤⡄⠀⣿⣿⠇⠀⢀⣠⡄⠀⢀⡀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣿⣿⠀⠀⢰⣿⡟⠀⢀⣾⣿⠀⠀⣰⡟⣰⣿⠏⠀⢸⣿⡿⠀⠀⣾⣿⠃⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⢿⣿⠀⣿⣿⠀⠀⢸⣿⣇⣀⡼⣿⣿⣀⣴⠟⢰⣿⡟⠀⠀⣿⣿⡇⠀⢠⣿⡟⠀⢀⣿⣿⠀⠈⣿⣿⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠿⠋⠀⠻⠿⠃⠀⠀⠉⠉⠉⠀⠀⠉⠉⠁⠀⢸⣿⡇⠀⣸⣿⣿⠀⠀⢸⠿⠷⠤⠚⣿⣿⢀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⡀⠀⠀⠀⢀⣀⣀⣤⣤⡶⠶⠶⠶⠶⠶⢶⣦⣄⠘⢿⣷⠾⠛⣿⣿⠀⠀⡾⢀⡴⠖⢺⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⣿⣿⣿⠋⠁⠀⠀⠀⣀⣀⡀⠀⠈⣿⣿⡇⠀⠀⠀⠀⢿⣿⣄⡼⠁⣿⣁⣴⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠸⠿⠿⠃⠀⠘⠛⠋⣠⠞⣿⣷⡄⠈⠙⠋⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠀⠀⠀⣤⣶⡆⠀⠀⣀⣀⠀⣼⠋⠀⣿⣿⡇⠀⣀⣀⠀⠀⣴⣾⣿⠟⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⣼⣿⡿⠀⠀⣰⣿⣿⢸⠇⠀⢸⣿⣿⣠⠟⣽⣿⡇⣼⣿⣿⠃⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⠇⠀⠀⣿⣿⣯⡟⠀⠀⣾⣿⡿⠁⢰⣿⣿⠀⣿⣿⡧⠤⠚⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⡄⣀⡴⠛⠿⠟⠀⠀⠀⠻⠟⠃⠀⣾⣿⡇⠀⠹⣿⣿⣄⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⢀⡼⠃⠉⠉⠉⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⣿⣿⣧⠤⣤⡀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠤⠴⠋⠀⠀⠀⠀⠀⢸⣿⠃⣼⡿⠀⠀⠀⢀⣠⠴⠻⣿⣷⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣤⣤⡤⠴⠞⠋⠁⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⢉⠁⠀⠀⣄⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⠁⠊⡱⢆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⢀⡀⠀⠱⣀⡠⠚⡄⠃⡠⠤⡴⠃⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠐⢀⠀⠂⠀⠰⠀⠉⠐⠊⠇⠈⠉⠐⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠀⠀⣀⠸⠀⢛⠀⣀⠃⠉⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡈⠁⠈
⠀⠀⠈⠀⠀⠦⠀⠀⠈⠤⠁⠀⠘⠻⠄⠀⠠⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠐⠀⠉⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠉⢂⠑⠀⠀⡀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡓⠀⠀⠉⠤⡴⠇⠀⠀⠉
⠀⠀⠀⠀⡜⣳⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢈⡠⠔⠀⠀⣀⡤⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢠⠛⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠠⣄⣀⠀⠆⠀⠀⠙⢳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⠿⢴⡆⠁⠄⠘⢋⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠁⠀⠀⠀⠶⠀⠀⠀⠈⠙⠠⠜⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠰⡶⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⠉⠀⠀⣀⠀⢀⣠⠄⠀⡜⠀⠀⠀⠀⠀⠀⢐⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢣⠈⠈⠛⣴⠄⢼⢃⡀⠀⠀⠀⣠⠀⠘⠁⠀⠀⠄⢀⠀⠀⠐⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠐⠀⢸⣶⠒⠆⠉⣠⡞⠠⡦⡀⠀⠀⢉⠀⠈⠐⣀⠀⠀⡈⠀⠀⢠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡀⠀⡀⠀⠃⠀⣈⠀⠨⢥⠭⡏⠀⢀⡀⠃⠄⠀⠀⠉⠀⠀⣈⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⠁⠢⡁⠈⠂⠀⠢⠂⠰⠀⠀⠀⠀⠘⠁⠀⠑⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠁⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠤⢴⠒⠀⣀⠀⠀⠁⠀⠀⠀⠀⡄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠂⠀⠀⠀⠈⠉⠁⠀⠀⡀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠘⠆⠀⠀⠀⠀⠈⠢⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⠄⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠄⠀⣀⡁⠀⠀⠀⠐
⠀⢀⡠⠀⠀⠀⠀⠀⡸⢀⡠⠀⣠⠏⣼⠄⢀⡀⠀⢠⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠀⠀⢦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢑⠀⠀⠀⠀⠀⠠⠀⡀⠀⡃⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⠀⠁⠊⠐⠊⠀⠀⠀⠀⢀⢉⣀⠁⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠒⠶⠮⠉⠁⠁⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠃⢔⠀⠂
⠀⠀⠀⠄⠀⠀⠈⢠⠾⣤⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⠘⣧⢀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⢯⠀⠈⠙⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠔⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠠⠀⠄⠈⠣⠀⠡⠆⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠄⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⣄⡠⠀⠀⠰⠀⠂⢠⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡈⠉⡄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠐⠻⢙⣃⣀⠈⢠⠀⠀⠉⠶⠒⡃⠐⠒
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢀⠐⢈⠁⡀⠑⠀⠀⠈⠀⣤⡏⠻⣿⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⢀⠰⠄⠀⠀⠘⠂⠈⡄⠳⠒⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠈⠋⠔⠈⠀⡀⠀⠀⠀⠁⠂⠖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠠⣤⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣰⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠹⢦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠶⣿⠀⠃⣰⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠖⠋⣁⡤⠴⠒⡦⠀⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠒⢋⣡⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠻⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢲⡶⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⡄⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠃⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⣰⠟⠁⢀⠴⠚⠉⢉⡤⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣶⡶⠋
⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⣤⣀⣀⣠⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀⣀⡀⠀⠀⢠⠇⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣦⠀⠀⠀⡏⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠐⣇⡼⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠞⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁⠀⠀⠀⠈⠙⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠁⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢶⡆⠀⠀⣀⡀⠀⠺⠛⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣤⠀⠀⠀⠀⣀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠙⠻⣶⣄⡀⠀⠛⠃⠀⠀⠀⠉⠀⣀⣴⡾⠛⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠙⠻⢶⣤⡀⠀⠀⣀⣴⡾⠛⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⣶⡆⠀⠀⠀⠙⣷⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢠⣶⠟⠋⣿⡇⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠿⣶⣄
⠀⠀⠀⠀⢠⣿⠁⠀⠀⣿⡇⠀⠛⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⢿⡆
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⡾⠟⠁

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⡀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡼⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠹⢷⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⠶⣦⣼⡇⠸⠃⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡋⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠋⠀⠀⣠⡴⠖⠁⢀⣀⣤⢤⡄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠔⠚⠉⢁⡠⠖⠃⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣀⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠋⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠺⣆⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠾⠷⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⣀⣤⠤⠖⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣄⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠚⠉⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⡽⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢸⣶⠶⠶⠶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣤⣤⣄⡀⠀⠀⢸⠃⢠⠇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡄⠀⠀⠀⣾⠀⣸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⣏⢠⡇⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠙⠋⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠀⠀⠀⣾⣟⣼⠄⠀⠀⠀⠀⢠⣴⣦⡄⠀⠀⠀⠀⣠⡐⡦⣀⠀⠀⠀⠀⠀⠀⠂⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⢀⣾⣿⣿⣿⡄⠀⠀⠘⣷⢣⢠⡍⠀⠀⠀⠀⡀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄⠀⠀⠀⠀⡰⣟⡲⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠀⠀⠀⠀⣀⢸⡶⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⢠⣿⣿⡗⠀⠀⠀⠀⠀⠿⠮⠙⠁⠀⠀⠀⢀⣿⣿⣿⣿⡆⠀⠀⠀⠰⣬⡳⠁⠀⠀⠀⠀⠀⠘⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡿⠃⠀⠀⠠⣿⠊⠁

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠉⢙⡇⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⢠⣴⣿⠃⠀⠀⠀⠀⢠⣿⣷⡆⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠻⢿⡿⠏⠀⠀⠀⠀⣸⣷⡿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠾⣟⡀⠀⠀⠀⠀⣴⢿⣿⣷⡄⠀⠀⠀⠀⣀⣾⣿⣦⠀⠀⠀⠀⢾⣿⣿⠇⠀⠀⠀⠀⣶⣤⣴⣦⣶
⠀⠀⠀⠀⠀⠠⣤⣤⠀⠀⠀⠀⠘⠁⣾⣿⣿⡿⠀⠀⠀⠀⠉⠛⣛⣽⠃⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄⠀⠀⠀⠀⠀⢠⡴⠀⠀⠀⠀⠀⢀⠄⢼⠷⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠆
⠀⠀⠀⠀⠀⢠⣾⣿⡿⠀⠀⠀⠀⢀⠮⠁⠀⠀⠀⠀⠀⠐⡃⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⠀⠀⢀⣯⡀
⠀⠀⠀⠀⠠⣾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠂⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⢰⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⢀⣴⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⡦⢙⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡋⣴⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠎⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡀
⢠⣿⡟⠁⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠈⢻⣿
⣾⣿⠁⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⣿⡷⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⠈⣿
⣿⣿⡄⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢼⡿⠃⠀⠀⠀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠀⠀⢠⣿
⢹⣿⣷⡄⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠸⠿⢛⣤⣤⣴⣶⣶⣶⡿⠟⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢠⣾⣿
⠀⣿⣿⣿⣶⣦⣤⣀⣀⣀⣀⣀⣠⣴⣾⣿⠏⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣦⣄⣀⣀⣀⣀⣀⣤⣴⣶⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⠿⠿⣿⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⣼⡟⠁⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠘⠛⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠈⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃
⠀⠀⣙⣛⣻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⣛⣋
⠀⢰⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡆
⠀⣿⣿⣿⠏⠙⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⠹⣿⣿⣿
⢸⣿⣿⣿⠀⠀⠀⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠀⠀⠀⣿⣿⣿
⠸⣿⣿⣇⠀⠀⠀⠀⠈⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⠀⠀⠀⠀⣸⣿⣿
⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⠙⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⢾⣿⡿⠋⠀⠀⠀⠀⠀⠀⢿⣿⡟
⡀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⠟⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠢⡀
⠀⠀⠪⡡
⠀⠀⠀⠀⢂⠀⠀⠀⢀⠀⠀⠀⠀⠡⠀⢠
⠀⠀⠀⠀⠀⠡⠐⠀⠀⠑⠀⠀⠀⠀⠙⢄⠈⠐⠂
⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠐⠄⠀⠀⠀⠑⠄⡀⠀⠀⠁
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠄⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠠⠐⠈⠁⠀⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠰⣆⣠⠀⢢⠀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠐⠡⠀⠀⠀⠀⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⢆⠀⡀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠢⣈⠁
⠀⠀⠀⠀⠀⠈⢌⠀⠧⠤⠀⠀⠀⠈⠀⠀⠀⢀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠅⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠅⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠈⢕⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⢈⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠖⠉⠄⠀⠀⠈⠐⠈⠐⡁⠀⢀⠀⠄⠂⠀⠀⠈⠄⠀⠀⠀⠈⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠈⠈⠀⠀⠀⠀⠀⢀⠔⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⢀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠁⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣟⣛⣛⣾⣿⣿⣶⣶⣶⣦⣾⣿⣿⣶⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣻⣻⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣥⣄⣄⣀⢂⣸⢿⣿⣿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣥⣬⣿⡛⠋⠉⠙⠛⢿⣿⣵⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⢿⣿⣿⣿⣿⣍⣼⣿⣿⣿⣿⣿⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⣿⣟⣛⣯⠤⢖⣺⣷⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣅⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣶⠟⣉⡻⠟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠟⢛⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣷⣤⣉⣉⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣾⣿⣽⣿⣯⡭⢿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⣿⣻⣽⣿⡷⣬⣯⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣷⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠙⠛⠛⠛⠉⠛⠁⠉⠉⠙⠛⠉⠉⠛⠛⠙⠛⠛⠛⠛⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠙⠉⠉⠉⠙⠛⠛⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣴⣶⣴⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣦⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣤⣶⣦⣶⣶⣶⣶⣶⣦⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠐⠐⠀⠀⠒⠀⠀⠀⠒⠒⠒⠖⠒⠲

⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⡤⠤⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⢀⣾⠟⢁⢿⣏⠀⠀⠀⠀⣿⠂⠀⠈⠙⠓⠦⢤⣤⣤⡤⠤⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣼⣿⠀⠘⠾⠋⠀⣾⠟⣴⡍⣶⡆⣾⠛⣴⠆⢀⣤⣄⣠⣄⠀⠐⢄⣀⣾⡟⡻
⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⢸⠯⠴⣿⢹⡏⡜⠓⠺⣏⣼⡮⡿⢫⡿⣣⡶⣦⣠⡿⠋⠉
⠀⠀⠀⠀⢿⣎⢻⣇⠀⠀⢠⠇⠀⠀⣿⡈⠋⠀⣀⡤⠔⠒⠦⣄⣿⡵⢿⡉⣵⡿⠁
⠀⠀⠀⠀⠈⠻⣧⡈⠙⠋⠁⠀⠀⠀⠘⣷⡴⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠒⠻⣤⡀⠀⣿⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠤⠤⠤⠒⠋⠉⠻⣦⣀⠀⠀⢀⡠⠏⠀⠀⠀⠀⠀⠈⠻⣦⡀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡵⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⢸⠟⠉⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠤⠒⣒⣒⡩⠕⠒⡒⠔⠒⠢⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡇⠀⠀⢠⣮⠝⢢⡀⠀⠀⠀⠀⣼⠉⠑⠒⠲⠤⠤⣉⣉⡀⠀⠖⢊⣉⣀⠤⠤⠖⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⠙⢄⠈⠑⢆⢀⣀⡠⠴⠊⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠋⠉⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⣔⠱⣄⠳⣄⠑⢄⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⡔⠳⡈⠢⡈⠳⣌⠳⢄⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⢸⠙⢦⡈⠢⡈⠢⣌⡱⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢀⠤⣶⠒⠊⠉⠣⡀⠙⠦⣈⡳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠓⢮⣵⠖⠀⠀⠀⢸⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⠤⢤⣀⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⡠⠤⠖⠒
⠀⠈⣇⣠⣞⠟⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠼⠆⠒⠊⠉⠁
⠀⠀⠀⠀⠘⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠳⢤
⢀⠤⠤⢄⣀⠤⠴⢒⣒⣒⡒⠢⢤⡠⠤⠖⠲⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⢀⡜⠀⣀⠳⣄
⣇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢉⠁⠀⢀⣀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠫⢤⣀⠉⠫⣔⣋⣤⡤⠎
⠸⢸⠀⠙⠲⠤⠒⠉⠙⠒⢄⠀⠀⢀⡔⠉⠉⠉⠁⢸⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠧⢄⣀⠀⠈⠙⠢⣵
⢦⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠤⠊⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⠤⣀⠀⠈⠙⠢⣄⢀⠎
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠤⣄⡀⠀⠉⠒⢄⣀⠜⠁
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠉⠓⢤⣀⠔⠁
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣎⠀⣠⠴⢛⢦⣀⣀⣀⠔⠋
⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠎⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣤⣞⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⣉⣥⡄⠰⣶⣸⡏⠀⠀⠀⠛⣿⠿⠂⣠⡆⠈⣙⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢋⣤⣀⠘⠛⣿⡄⠀⠙⣿⡅⠀⠀⠀⢰⣷⠀⢠⣿⠁⣠⣿⡟⢠⣬⡙⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠁⠘⣿⣍⣿⣄⠀⠸⣧⡀⠀⠻⠇⣀⣀⣀⡸⠟⠀⢼⡏⣰⡟⢿⣶⣿⡿⠁⢠⣌⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠠⣶⣦⣄⠈⢿⣏⠹⠷⣀⣭⣶⣾⡿⠿⠿⠟⠿⠿⠿⣿⣷⣶⣭⡐⠗⣡⡟⢁⣴⣿⡛⠷⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢁⣤⣀⠹⣧⣽⠿⡶⣋⣴⣿⠟⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⡐⢿⣏⠙⠃⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢠⣿⣉⣿⠇⠘⠿⢠⣾⡿⠋⠀⠀⠀⣀⡀⠀⢴⣿⣿⠂⢰⣾⡦⠀⠀⠀⠀⠀⠙⢿⣷⡍⠀⠀⠀⠀⠀⠀⠈⣻⣷⣄⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠈⠙⠿⣦⠄⣴⣿⠋⠀⠀⠀⠀⠘⠿⠓⠀⠘⠉⠙⠀⠈⠉⠁⢀⣀⣀⣀⣠⣤⣤⣽⣿⣶⣶⣶⣾⣿⠿⠿⠿⠟⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣿⣿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⢉⣁⣀⠀⠀⣤⣤⣄⠀⣶⣶⣿⢿⡇
⠀⠀⠀⠀⢀⣠⣿⢇⣤⣄⣤⣤⣶⣶⣾⣿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠉⣀⣀⠀⠀⠀⣤⣶⣶⣶⣶⡄⣿⣿⠀⢹⣿⣿⣆⢠⣿⣿⣿⠀⢻⣿⣇⣀⣀
⠀⠀⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠁⣀⣀⣤⣤⣄⠀⢶⣶⣶⣾⣿⡟⣿⣧⢸⣿⡏⠀⠀⠀⠉⠙⣿⣿⠀⠀⢻⣿⡆⢸⣿⣿⣿⣸⣿⢻⣿⡀⢸⣿⡿⠿⠟
⠀⠀⢠⣤⣴⣶⣦⡄⠀⣾⣿⣷⡀⠀⢻⣿⡟⢻⣿⡇⠈⠉⣿⣿⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⣿⣇⢻⣿⣿⢸⣿⡇⠈⣿⣷⣀⣤⡄
⠀⠀⢸⣿⡏⠉⣿⣿⠀⣿⡟⣿⣇⠀⢸⣿⣧⣼⣿⡃⠀⠀⣿⣿⡀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠈⣿⣷⠀⣿⣿⠈⣿⣿⠈⣿⡧⠀⠿⠿⠿⠛⠃
⠀⠀⠀⣿⣿⣶⣿⠟⠀⣿⣇⣸⣿⡆⠈⣿⣿⠉⣻⣿⠀⠀⢸⣿⡇⠀⠀⠀⣿⡿⡀⠀⠀⠀⠀⠀⠸⠷⠇⠀⠀⠛⠛⠀⠉⠉⣀⠈⣀⣀⣤⣤⣤⣴⣶⣶⣶⣷
⠀⠀⠀⣿⣿⡍⠀⠀⢀⣿⡿⠟⢿⣿⡀⣿⣿⠀⢹⣯⡇⠀⠸⠿⠇⠀⠀⠀⠙⠉⠁⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⡿⠿⠟⠛⠛⠛⠋⢹⣿⠉
⠀⠀⠀⢸⣿⡇⠀⠀⠘⠿⠃⠀⠘⠋⠁⢉⣉⣀⣀⣤⣤⣤⣤⣶⣶⣶⣾⣿⡿⠿⠿⠿⠋⠛⠛⠛⠉⠉⠉⠉⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⣤⣤⣤⣤⣶⣶⣶⣶⣿⡿⠿⠿⠿⣿⣟⠛⠛⠉⠉⠉⠉⠀⢀⣀⡀⠀⣄⣠⡄⠀⢴⣶⠄⠀⠀⠀⠀⣴⣿⠏⣰⡗⢠⡀⠀⠀⠀⣾⡿
⠀⠀⠀⠐⠛⠋⠉⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠺⠿⠇⠠⢿⣿⠶⠀⠉⠉⠀⠀⢀⣤⣾⡟⢁⣌⠛⢷⣿⣁⣴⢆⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⢀⣴⡿⣦⡈⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣀⣤⣶⡿⠟⣁⣄⡹⣿⣷⣄⠈⠛⢫⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⣴⡿⠻⣶⠿⢡⣶⣦⠉⠛⠿⠿⣿⣶⣶⣶⣶⣶⣶⣿⠿⠟⢛⣉⢰⣦⠙⢿⣿⢾⣷⡹⢗⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡀⠀⢀⣴⣟⣿⠃⢰⡿⢳⣦⢠⣤⣬⣭⣭⢀⣤⠀⠀⠿⢿⣏⠀⠻⣧⡀⠻⣷⢉⣵⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣝⠉⢽⡏⢀⣿⠿⣿⠋⠀⣾⡏⠀⢻⣿⡏⠀⠀⠀⠈⣿⡆⠀⠹⢗⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣶⣤⣜⡛⠰⠿⠀⠀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⢘⡋⣴⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣷⣶⣶⣭⣤⣤⣤⣬⣵⣶⣶⣾⡿⠏⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠔⠒⠒⠒⠒⠂⠠⠤⢄⣀
⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡉⠲⣄
⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠘⣷⡄
⢰⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠀⢸⢿⡆
⣾⡇⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⡡⠊⣠⠟⢸⣿
⢻⡇⠀⠈⠛⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠬⠤⠔⣒⡩⠴⠛⠁⢠⢸⡿
⠘⢷⣄⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠒⠉⠉⠁⠀⠀⢀⡠⢃⡾⢃⣠⡴⠶⠒⠚⠉⠋⠉⠉⠉⠉⠉⠉⠡⢊⠉⠙⠛⠒⠶⠦⣤⡀
⠀⠀⢿⡓⠦⣄⣀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠤⠀⢀⣠⠵⢚⡿⣴⠏⠀⠀⣠⡤⠴⠖⠒⠒⠂⠂⠀⠀⠂⠐⠓⠒⠲⠶⠤⣄⡀⠀⠙⣷⡀
⠀⠀⠀⠳⣄⠀⠀⠉⠀⠀⠀⠐⠂⠒⠂⠀⠀⠀⠉⣀⠔⣠⠞⢸⣿⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣿⠄⠀⣹⣧
⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣫⠵⠚⠁⠀⣿⢻⣧⠀⠙⠷⣦⣤⣀⣀⣀⡀⠀⠀⠠⠤⣄⣀⣀⣀⣤⣥⠾⠋⠀⣰⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠒⠉⠉⠀⠀⠀⠀⠀⣿⠀⠙⠳⣦⣤⣀⡀⠀⠀⠉⠉⠁⠈⠁⠉⠉⠉⠀⠁⠀⣀⣢⣤⠿⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⠁⠀⠀⢠⢂⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠒⠒⠒⠀⠀⠀⠀⠐⠒⠒⠒⠒⠊⠉⠉⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⡰⢋⡵⢒⠒⢌⠙⡌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⣺⢟⡿⠋⢉⡟⣩⠏⠁⢀⡴⣫⠟⠉⠙⢦⡀⠀⠀⠀⢀⣀
⠀⠀⢠⠃⣾⢰⡯⣹⢀⡆⡩⠘⣦⣤⣄⡀⠀⠀⠀⠀⠀⣰⠛⣽⣫⠏⠀⣰⣣⣎⣀⣤⣯⡴⠷⠶⠖⣿⡿⠿⣖⡓⢶⡼⠃⠀⠀⠀⠎⠀⠛⠑⠖⠢⣀⣀
⠀⠀⠸⠀⠘⢦⣑⣒⣫⢴⡗⣋⣭⡭⢍⡙⢷⡄⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⣿⠗⠎⠣⢱⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠰⠶⣤⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠺⡷⠭⢜⡴⢸⢇⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠊⠠⠂⡛⡄
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠙⠦⣀⣈⣉⣴⠷⠋⢀⠆⠀⠀⠀⠀⠀⣀⡠⠤⠖⣒⣉⡭⠭⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣏⠛⠓⠒⠒⠒⠒⠒⠚⠋⠁⢀⡠⠊⠀⠀⠀⢀⡴⠋⠓⢞⠉⠁⠀⠀⢆⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⡀⢀⠀⡼⠁
⠀⠀⠀⠀⠘⠶⠦⠤⣤⣤⣤⣤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠸⡀⠀⠀⠘⡀⢀⣀⣠⠼⡖⠒⠂⠈⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠻⢧⣥⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠤⠔⠒⢻⠉⠁⠀⠀⢳⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⣆⣀⡤⠤⠺⡗⢋⣉⣁⣷⡆⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⠀⠀⠀⠀⢳⠒⠊⠉⠹⣀⠖⠲⣄⠜⠁⠀⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠘⠀⠀⠀⠀⠠⠊⠉⠒⠚⠋⠁⠀⠀⢀⡤⠤⠔⠒⠠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠛⠲⢤⣀⠀⠀⠀⡆⠀⠀⠀⢀⣀⣠⠤⠤⠖⠋⠉⠉⠁⠀⠀⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠈⠑⠢⣄⠇⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⡀⢠⡀⣀⠰⠐⠀⠧⠘⡹⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠔⠈⠠⠤⣍⠂⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣷⠄⠖⠓⠉⠈⠁⠀⠀⠀⠀⠀⠀⢇⠂⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡌⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠰⠋⡀
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠛⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⠴⠖⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡠⠚⠁
⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⡇⠀⠀⠀⠂⠄⡀⣀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⠀⠀⠀⢁
⠀⠀⠀⠇⠀⠀⠀⠀⠄⠐⠏⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⣴⠖⠚⠋⠉⠀⠑⠆⡇⠈⣦⡔⠁⠀⠀⠀⠀⡇
⠀⠀⠀⢀⠀⠀⠈⠁⠒⠐⠋⠀⠀⠀⠀⡠⠤⣤⣀⡀⢀⡞⢉⢹⠀⢰⠆⠀⠀⠀⠨⠗⠀⠀⠇⢃⠀⢻⡇⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢹⡇⠈⢸⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⢸⠘⡄⢸⡟⠁⠒⠒⠠⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠈⣷⠀⠏⡆⠀⠀⠀⠉⠀⠀⠀⠪⠂⢀⣣⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣀⢽⣄⣰⣐⡙⢁⣀⠠⠴⠗⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠔⠒⠉⠀⠀⠈⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣁⠠⠤⠤⠖⠒⠒⠒⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⡄⠀⠀⣞⢳⡀⠀⠀⠀⠂⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆⠀⣠⡶⢲⡄⠀⢠⣶⣶⣦⣄⣤⡀⠈⢸⠧⣀⡀⠀⣰⣿⣿⠃⠀⢀⣤⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡼⠋⣀⣼⡿⣻⣿⠟⢀⣴⣿⣿⠃⢠⡟⣴⣿⠇⢰⣿⡿⠁⠀⣠⣾⡿⠃⠀⣠⣿⡟⠀⠀⠀⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⣠⠞⣹⣿⢳⣿⠃⡴⢫⣿⡿⠃⣠⠟⣼⣿⠋⢀⣿⣿⠁⢀⢼⣿⡟⠁⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠹⠾⠿⠃⠘⠿⠚⠁⠘⢿⣥⠾⠃⠀⠹⠧⠴⠋⠻⠧⠔⠁⠘⢿⣄⠴⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⢏⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⢀⡴⠚⠉⣴⣿⠃⠀⠈⠳⣄⠀⠀⠀⢠⡾⠛⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢘⣻⡥⠴⢤⣄⠀⠠⡏⠀⢀⣾⡿⠁⠀⠀⠀⠀⠘⢧⡀⠀⠙⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⠀⢀⣿⣿⣡⠤⠒⣿⣥⠀⠀⠀⠠⣿⠇⠀⠙⠛⠛⠉⢀⣴⣶⠖⠲⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠖⣾⣿⠃⠀⠀⣼⣿⡟⠀⣠⣤⣀⡴⣶⣦⢀⣀⡀⢀⣾⣿⠃⢀⠜⠁⠀⠀⠀⠈⠙⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⣠⠞⣿⡟⣀⢼⣿⣿⠋⣸⣿⡿⢫⣿⡟⠸⣿⡿⠊⢁⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡤⠞⠁⠀⠈⠉⠁⣾⣿⠃⠀⠻⠟⠁⣼⣿⠁⡰⠻⠧⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢰⣿⣯⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⢂⣀⠤⢖⣒⣤⠤⣐⠒⠒⠂⠞⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡔⠉⡰⢃⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠤⡼⠥⢼⡤⠤⠤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣟⠶⢆⠘⡅⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡼⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠙⣣⡴⠃⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⡇⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡔⠀⠈⠢⠤⠔⠊⠁⠀⠀⢧⠀⠀⢠⡀⠀⡼⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢠⠛⡆⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⡏⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⢀⠆⠀⠸⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼⠀⠱⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠘⠞⠀⠀⠀⠉⠀⣀⡤⠤⠴⠶⠐⠂⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢸⠀⡇⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠠⠤⠶⠶⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠈⠛⠳⠛⠛⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣤⣿⣄⡀⠀⠀⠀⢀⣤⡶⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⣿⣁⡽⢀⣀⠈⠛⣦⡴⠞⠋⠁⠀⢀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⣀⠀⠀⠠⡏⣹⠇⢀⡼⣧⣠⣤⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣧⠀⣏⣈⡷⠀⠀⠉⠁⠀⠈⠓⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⢋⠀⠀⠉⡴⢤⡀⠠⡞⢳⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠰⣏⣹⠀⠀⠙⠦⠇⠀⠉⠉⣀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀⠈⠁⠀⠀⠀⣀⡤⣄⠀⢰⣣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⡤⠇⠀⣞⣹⠆⠐⢧⡤⠏⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⣸⣷⡀⠀⠀⢁⣤⣄⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⣠⡾⠋⠀⠙⠷⢦⣼⡧⣾⠟⠋⠁
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⢸⡇⠀⢹⡄
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⢦⣄⠀⠀⠀⠀⠀⠸⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡴⠛⠉⠻⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⣿⣿⣿⣿⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣿⣿⣛⣟⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢠⡄⠀⣀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠺⡿⠧⠀⠀⠀⠘⠁⠀⠉⠀⣴⠀⣿⠀⠤⣀⠀⢐⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠛⠀⠀⠀⠀⠁⢀⠀⠦⢀⠀⠀⠀⡄⠀⠿⠀⠁⠀⠃⠉⠀⢀⡈⠀⠐⠂⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⡴⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠛⠀⠆⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣆⣀⠀⠀⢠⢄⡄⠀⣴⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠷⢀⣤⣤⡴⡦⠀⠀⠘⠃⠀⠀⠘⠃⢰⡆⢠⣦⢀⣴⠂⠀⣸⠿⣿⠉⠀⠀⠋⠾⠁⠀⠀⠀⠀⢁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠉⣻⠇⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡿⠋⠹⠞⠁⠀⠀⠈⠀⠈⠀⠰⠆⠀⠀⠀⢆⡒⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠶⠀⢠⣾⣁⡀⠀⠀⣶⣶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠟⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣍⠃⠀⠀⠀⠀⠀⣴⠟⠁⠀⢰⣿⡟⠀⢸⡗⠀⣴⠶⣦⠀⣾⣿⠿⢿⣿⡟⢛⣛⠛⠻⠃⠀⠀⠀⠀⠀⠴⠀⠀⠀⢠⣞⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠘⠃⠀⠀⠀⣿⣿⣧⣶⠟⠁⠀⠻⢶⡾⠤⣤⡄⢀⣾⡿⠀⣾⡟⠀⠀⣠⣾⡗⠀⠀⠀⠀⠀⠀⠀⠉⣹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣾⣀⣀⠀⠀⡤⠀⠀⢰⣿⣿⠉⣠⢶⣶⠆⢠⠞⢀⣾⡟⠁⣼⣿⠃⢸⣿⣁⡴⠚⢱⣿⣧⣴⠶⠗⠀⠀⣀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠿⣿⠋⠁⠀⠀⠀⠀⠀⣸⣿⡏⢰⣿⠞⣿⡴⠋⢀⣾⡟⠀⣸⣿⡟⠀⠈⠉⢉⣤⣶⣿⡿⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠲⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢉⣤⣴⠶⠆⠀⠀⠀⣿⣿⡇⠀⣤⠤⠬⠀⠀⣸⣿⠁⠀⣿⣿⠇⠀⠀⣴⠟⠁⣸⣿⠃⠀⠀⣈⠁⠖⠀⠠⣦⣴⣆⣀
⠀⠀⠀⠀⠀⠀⠶⢈⣁⠀⠈⢛⣷⣦⠀⠀⠀⠀⠛⠛⠃⠀⠒⠛⠛⠂⠀⢿⡿⠀⠀⠛⠛⠀⠀⠐⣿⣄⣴⣿⠃⠀⠀⠀⠀⠙⠀⠀⠀⡾⣿⠋⠁
⠀⠀⠀⠀⠀⠀⡦⠈⠁⠀⠺⠟⠉⠀⠀⠀⠀⡆⠀⠀⠀⠀⣀⣀⣄⣠⣄⣀⣀⡀⠀⠀⢴⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢠⡿⢀⣠⠀⠀⠀⠈⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣶⡾⠿⣿⡟⠉⠉⠉⠉⠙⣛⠀⠀⢀⣴⣿⡿⠀⠀⣀⣀⡀⠀⠀⠘⠛⠉⡟⢀⣤⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢀⠀⠀⠘⠀⠀⠀⠘⠋⠁⠀⣼⡿⠀⢶⠆⠀⠀⣼⣿⡀⢠⡞⣹⣿⠃⣰⣾⡟⠉⣿⠀⠀⠀⠀⠘⠟⠋⠁⠀⢨⣽⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢠⣦⠀⣠⣾⣿⣿⣥⡟⢠⣿⡏⣸⣿⣏⣤⠞⠁⠀⠀⠘⠀⠀⢀⡀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠀⠀⠀⢀⠈⠁⠈⠀⠀⣼⣿⠀⢸⣟⣴⢋⣿⡇⠘⠋⠀⣼⣿⠁⣿⣿⠁⠀⠀⣠⡾⠀⠀⠀⠀⠈⠀⠆⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢃⡘⠃⠀⣠⣾⣋⡁⠀⠀⠀⠀⣿⡟⠀⠈⠛⠁⢸⣿⠃⠀⠀⢀⣿⣿⠀⠻⣿⣤⣤⠾⠋⠀⠀⣀⣤⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⣹⣏⣠⡄⠀⠀⠛⠃⠀⠀⠀⠀⠺⠿⠀⠀⠀⠸⠿⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⢹⠿⠏⢠⣾⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠄⠀⠀⠀⢠⣞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢤⣤⣄⠀⠀⠘⠃⠀⣠⠶⡆⣀⡤⣆⠀⠀⠀⠀⠀⣀⠀⠀⢀⡾⢷⢀⡾⠀⠀⠀⣀⠀⠀⢀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠻⠋⠀⠀⠀⠀⠘⠃⠐⠛⠁⠀⣿⠀⠠⠀⠘⣿⣿⠶⠀⠘⠃⠈⠛⠀⠀⠀⠳⠌⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⠀⢀⡋⠐⠆⠀⠈⠀⠀⠀⠀⠉⠙⠀⠀⠶⣈⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣴⣦⠄⠀⠀⠀⢠⠀⡄⠀⢠⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀
⠀⠀⡤⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠶⠶⠾⠷
⠀⢠⠿⠄⡘⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠊⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠆⠴⠒⠚⠂⠀⣐⣒⠛⠁⠂
⠀⣸⣢⣄⣈⡻⡇⠀⠀⠀⠀⠰⣷⣯⡭⠭⠟⠦⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⣠⡄⠀⠀⠀⠐⠒⠛⠫⠭⠁⣠⡶⡀
⠀⠘⠿⣬⡙⣿⠃⠀⠀⠀⠀⠀⣿⣦⠁⠄⠀⠀⠀⡄⠀⠀⠐⠬⣿⢓⠂⠈⠈⠀⠠⠒⠶⠖⣢⣶⣾⣩⠿⠟⢋⡔
⠀⠀⠀⠁⡏⠁⠀⠀⠀⠀⠀⠃⢣⠘⣄⣸⠸⠀⠀⠁⠀⠀⠈⠀⠑⠊⠁⠀⠀⠈⠁⠐⠂⠀⠩⠤⠤⠤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢂⠈⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠺⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠠⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡷⠆⠀⠀⠀⠀⠀⣀⠀⠀⠤⠀⠒⠀⠀⠀⠀⠀⠀⠀⣄⠤⡁⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠐⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⢐⠒⣋⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠐⣿⣦⠷⡀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⢰⠤⠄⢤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⢿⠿⠍⠁⡴
⠀⠀⠀⡄⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⢲⣄⡀⠀⠀⠀⠈⠁⠀⠀⠱⣰
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢄⡃⢜⠤⡀⠀⠀⠀⠀⠀⢀⡗⠠
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠈⢏⠉⠇⠀⠈⠑⢷⡀⠀⠀⠀⠀⠀⣾⢧⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢠⡀⢯⠲⡀⠁⢄⡑⢅⣠⠈⢷⢦⠀⠠⢤⠀⢷⠀⣹⡖⡁⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⠀⠈⢄⠡⡄⠀⡆⠀⠙⠧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢴⡀⢆⡀
⠀⠀⠀⢀⣠⠀⢐⢙⡀⠲⢄⠀⠉⠁⠓⠺⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢙⣆⠣⠆⣭⣄
⠀⠀⢠⡎⠉⠀⠀⠀⠀⠐⠈⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⣠⠀⠀⠀⠀⠀⠀⡜⢏⠻⢦⠈⠑⢾⠧⣡⣵
⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⢰⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠢⢍⣿⣽⠀⠀⠀⠀⠀⠀⠛⠮⠑⠦⢾⠿⠎⠁⠿⠉
⠀⠀⠀⢲⣤⠀⠀⢠⠠⣸⣿⡏⠀⠀⠀⠀⢸⣦⡀⣘⣿⣿⣥⣮⠽⠃⡀⣐⡦⡾⠀⠀⣜⠂⠀⠉⠉⠉⠩⠤⢖⡥
⠀⠀⠒⠂⢈⠛⠿⢾⣿⠟⠫⠃⠀⠀⠀⠀⠀⠈⠻⠶⠋⠉⠛⠡⠀⠸⣿⡿⠃⠀⠀⠀⠙⠅⠀⠀⠀⠀⠀⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠐⠈⠁
⣿⣿⠿⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠲⡄
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⡸⢭⣙⣳⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⢢⡀⠀⠀⠀⠀⠀⢸⠤⠖⠤⢾
⠀⠀⠀⠀⠀⠀⠀⢻⢮⣑⡏⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠘⡇⠀⠀⠀⠀⢸⡀⣀⠀⢀⡀⢈⠆⠀⠀⠀⠀⣏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣷⡼⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⢹⠁⠀⠀⠀⠀⠀⠙⣄⣍⣡⡸⠁⠀⠀⠀⠀⠀⠈⠓⡖⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⣴⡁⣹⡀⠀⠀⠀⠀⠀⢠⠞⠓⠛⢤⠗⠀⠀⠀⠀⠀⠀⣀⣟⣆⡀
⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠙⠦⡀⠀⠀⢠⠞⠓⠒⠛⠓⢛⡆⠀⠀⠀⠀⢮⣀⣀⣀⣼⠄⠀⠀⠀⠀⠀⢸⠶⠤⠤⢧
⠀⠀⠀⠀⠈⠻⣍⣑⣉⣉⣊⣩⠗⠁⠀⠀⢸⠟⠛⠛⠛⣛⣛⡇⠀⠀⠀⠀⠀⢣⣶⣶⠃⠀⠀⠀⠀⠀⠺⢍⣛⣛⣛⡩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡇⣃⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀⠀⠀⠀⠀⠀⢀⢔⡒⣒⠢⡀⠀⠀⠀⠀⠀⢀⡤⠒⠢⢤⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⢠⢊⡩⣍⠲⡄⠘⡄⠀⠀⠀⠀⡇⡏⣞⠶⡱⣱⠀⠀⠀⠀⠀⣏⢀⡀⢀⠀⣱⠀⠀⠀⠀⠀⣺⣿⣽⣽⣿⡇
⠀⠀⠀⠀⠀⢤⢸⠰⣀⡧⢸⣀⠇⠀⠀⠀⠀⠱⣕⣮⣭⠷⠋⠀⠀⠀⠀⠸⡍⣿⢹⡏⡿⢸⠇⠀⠀⠀⠀⣿⣿⣯⣭⣽⡇
⠀⠀⠀⠀⠀⠈⠣⣳⣬⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⢳⣻⣸⣧⣇⡟⠀⠀⠀⠀⠀⣗⣀⣈⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⡉⢢⠀⠀⠀⠀⠀⠀⠀⣠⡼⣝⢢⡀⠀⠀⠀⠀⠀⢺⢀⣀⡅⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⢸⠁⠏⠀⢀⠀⠈⡆⠀⠀⠀⠀⢠⠞⡄⢣⢸⠋⠁⠀⠀⠀⠀⢎⡉⠀⠈⢋⡱⠀⠀⠀⢀⡾⡀⢣⠀⣰⢃⡼⡆
⠀⠀⠀⠀⠀⠘⢄⣀⠀⢀⣁⡼⠁⠀⠀⠀⠘⣍⠢⡵⠜⠁⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠻⡀⠀⠀⠀⠀⠳⢼⠍⠉⠉⣽⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠋⠙⠦⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⢤⣤⠿⠗⠶⠖⢺⢧⣤⡄⠀⢤⣬⣿⠛⠒⠚⢻⣧⣤⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠙⠶⣄⠀⢀⣠⠞⠉⠀⠠⢦⣿⠆⠀⠀⠀⠠⣾⡷⠆⠀⢸⡿⠈⠳⣄⣠⠖⠛⠛⠲⣄⣠⠞⠙⢿⡇⠀⠀⣿⠏⠘⢦⣰⠶⠶⠶⠶⠶⣆⡴⠋⠸⣿
⠀⠀⠀⠀⣨⡿⣿⡁⠀⠀⠀⠀⠀⠈⠙⣲⣤⣖⠋⠁⠀⠀⠀⠀⡧⠀⠀⢈⡇⠀⠀⠀⠀⢹⡁⠀⠀⣾⠀⠀⠀⢸⡆⠀⠀⢹⠂⠀⠀⠀⠀⣏⠀⠀⠀⡇
⠀⠀⢀⡾⠋⠀⠀⠙⢦⠀⠀⠀⠀⢀⡾⠋⠀⠉⠳⡄⠀⠀⠀⢸⣧⠀⣠⠞⠹⣤⣀⣀⡴⠏⠳⣄⠠⣼⡆⠀⠀⣾⡤⢀⡴⠻⠦⣤⣤⢤⡴⠟⢦⡀⢠⣿
⠀⠀⠸⣧⣀⣀⣀⣀⣼⠇⠀⠀⠀⢾⣀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠿⠛⠁⠀⠀⠀⠈⠁⠀⠀⠀⠈⠛⠿⠀⠀⠀⠸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇
⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⣾⠋⠉⠳⢦⣀⡴⠛⠉⠉⠀⠉⠉⠻⢦⣀⡴⠞⠉⠙⣶⠀⠀⠀⢶⠟⠉⠓⢦⣼⠏⠉⠉⠉⠉⠉⠉⠉⠙⣷⡤⠞⠋⠙⣷⠀⠀⠀⢀⡤⠶⠶⢤⡀
⠀⠀⣾⣄⢀⣠⠶⠛⢷⣄⡀⠀⠀⠀⢀⣠⡾⠛⠶⣄⡀⣠⣷⠀⠀⠀⢾⣄⢀⣠⠴⢿⣄⠀⠀⠀⠀⠀⠀⠀⣀⡿⠲⣄⡀⢠⣿⠀⠀⢀⡟⠀⠀⠀⠀⠹⡆
⠀⠀⠘⠛⠋⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁⠀⠀⠀⠈⠙⠛⠂⠀⠀⠀⠐⠛⠋⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠛⠃⠀⠀⠈⣇⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠶⠲⢞⡁
⠀⠀⠰⡶⢤⣀⠀⠀⠀⣀⡤⠤⠤⠤⣤⣀⠀⠀⠀⣀⡤⣶⠆⠀⠀⠀⢰⣶⢤⣀⠀⠀⠀⣀⣤⠤⠤⠤⢤⣄⠀⠀⠀⢀⡤⢶⡆⠀⠀⠰⢾⡥⣤⢠⣤⡿⠆
⠀⠀⠀⣿⠀⠉⠓⣶⡟⠉⠀⠀⠀⠀⠀⠈⢻⣶⠚⠁⠀⣿⠀⠀⠀⠀⢻⠉⠀⠈⠓⣶⡞⠉⠀⠀⠀⠀⠀⠈⢻⣶⡞⠉⠀⠈⣿⠀⠀⠀⠀⠀⣿⢸
⠀⠀⢀⣿⣠⠔⠋⠉⠛⠦⣤⣀⣀⣀⣤⠴⠛⠉⠙⠦⣄⣿⡀⠀⠀⠀⠛⣶⣠⠴⠋⠉⠛⠶⣤⣀⣀⣀⣠⡴⠛⠉⠙⠲⣤⣰⡟⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠘⣟⠳⢤⣀⠀⣠⡴⠞⠛⠛⠛⠳⢦⣄⠀⣀⡴⠞⣿⠃⠀⠀⠀⠈⣿⠓⢦⣀⣠⡶⠒⠒⠒⠒⠒⠲⠒⠶⣤⣀⡤⠖⣻⠃⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⣿⠀⠀⣨⢿⣏⠀⠀⠀⠀⠀⠀⠀⣹⡿⣅⠀⠀⣯⠀⠀⠀⠀⠀⣸⠂⠀⣨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⡀⠀⣿⠀⠀⠀⠀⠀⠀⠻⠾
⠀⠀⢠⡷⠖⠋⠁⠀⠈⠛⠶⠶⠶⠶⠶⠛⠁⠀⠈⠙⠲⢿⡄⠀⠀⠀⠠⡿⠖⠋⠁⠈⠓⠒⠒⠒⠒⠒⠒⠒⠚⠋⠈⠙⠲⢾⡆

⠀⠀⣶⢦⣄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣠⢴⡶⠀⠀⠀⠀⠀⢶⡦⣄⡀⢠⡤⠤⠤⠤⠤⣤⠀⣀⡤⢶⠆
⠀⠀⢸⡆⠈⠙⣶⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣷⡞⠋⠀⢸⠃⠀⠀⠀⠀⠀⠘⡇⠀⠙⢺⡇⠀⠀⠀⠀⣿⠚⠁⠀⣿
⠀⠀⣼⣡⠴⠚⠁⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠓⢦⣼⣇⠀⠀⠀⠀⠀⣸⣇⡴⠚⠹⣧⣀⣀⣀⣠⣿⠙⠲⣤⣿⡄
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣶⠦⣄⠀⠀⣠⠖⠒⠲⣄⠀⢀⣠⠴⣶⠀⠀⠀⠀⠀⠀⣶⠦⣄⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣠⠴⣶
⠀⠀⠀⠀⠀⢹⡆⠈⠙⣾⠃⠀⠀⠀⠘⣷⠋⠁⢰⡇⠀⠀⠀⠀⠀⢸⡏⠀⠈⠙⣶⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣶⠋⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣾⣥⠴⠋⠙⢧⣀⠀⣀⡼⠋⠛⠦⣬⣷⠀⠀⠀⠀⠀⠘⣷⣤⠴⠛⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠛⠦⣤⣾⠃
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⠶⡶⠶⣶⡶⠶⡶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡉⣆⡀
⠀⠀⠀⠀⠀⠀⠀⡠⣤⠤⣴⣷⣏⣴⢗⡄⠀⠀⠀⣀⠤⠤⠤⠇⠀⠀⠐⢢⠀⠀⠀⢀⣠⠤⢤⢼⠑⠀⠨⢳⡄
⠀⠀⠀⠀⠀⣴⠋⡔⢁⠞⢀⠼⡛⠓⠉⠀⠀⢠⠊⠀⠀⠀⠀⠀⡗⠒⠂⠁⠀⢀⡔⠫⢐⠠⢀⠀⢸⠒⠛⠈
⠀⠀⠀⠀⣸⠃⡜⡰⠃⡠⢋⡼⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⣜⠀⠈⠐⠨⢐⠀⡾
⠀⠠⢔⣶⢿⣜⣰⠁⣔⣡⠟⠁⠀⢀⠄⠒⠚⠂⡀⠀⠀⣀⡴⠋⠀⠀⣠⠰⢖⠳⡨⢀⢐⢨⡰⠟
⠀⠙⢵⠁⡜⡿⠉⠉⠉⠀⠀⠀⠀⠈⠢⠀⠀⢀⠏⠉⠉⠀⠀⠀⠀⠀⠓⣌⠐⡐⡎⠉⠉⠁
⠀⠀⠈⠙⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠁

⠀⠀⠻⣿⣿⣆⠀⠘⣿⣿⡄⠀⢻⣿⡇⠀⢹⣿⡇⠀⣿⣿⠁⠀⣿⣿⠃⠀⣼⣿⡟⠀⢀⣼⣿⡿⠃⠀⢀⣴⣿⣿⡿⠋⠀⠀⢀
⣷⣄⠀⠘⢿⣿⣦⠀⠘⣿⣿⡀⠈⣿⣷⠀⢸⣿⡇⠀⣿⣿⠀⢰⣿⡏⠀⣰⣿⡟⠀⢠⣾⣿⠟⠀⢀⣴⣿⣿⠟⠉⠀⠀⣠⣴⣿
⢿⣿⣧⡀⠀⣹⣿⣷⣦⡈⢿⣷⠀⠸⣿⡄⠀⣿⡇⠀⣿⡏⠀⣾⡿⠀⢰⣿⠏⠀⣰⣿⡿⡁⠀⣠⣿⣿⣿⠇⠀⢀⣴⣾⣿⣿⠿
⠀⠙⢿⣿⣦⣹⣿⣿⣿⣿⡈⢿⣧⠀⢻⣧⠀⢿⡇⠀⣿⡇⢠⣿⠃⢠⣿⣏⣠⣶⣿⣿⣦⣷⣾⣿⡟⣿⣿⣧⣶⣿⣿⣿⣿⣷⣶
⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣷⠎⢿⣆⠈⣿⡀⢸⡇⠀⣿⠀⣼⡏⢀⣾⠏⢈⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣦⠈⢿⡆⢸⣇⠸⣇⢠⣿⢀⣿⠁⣾⠏⢠⣾⠛⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣷⣾⣿⣤⣿⠀⣿⢸⡇⣸⠇⣼⠏⣰⡟⢁⣴⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣀
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢳⠈⡇⠛⢸⠁⡟⣰⢋⡼⢋⣴⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠁⠀⠀⠀⠀⠀⠁⠞⡡⠞⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁
⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣯⣤⣤⣿⣿⣿⡿⢻⣿⣿⣿⡀
⣀⣴⣿⣿⣿⣿⣻⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠩⢯⣩⣉⣉⣽⣿⣿⣿⣿⡟⠀⠈⣿⣿⣿⣷⡀
⠿⣿⣿⣿⠟⠉⣉⣙⣿⣿⣿⣷⣦⡀⠈⠙⠛⠻⠉⠀⠀⠀⠀⠀⠀⠀⠒⠶⣤⣬⣉⣿⣿⣿⠿⠻⠿⠛⠛⠛⠿⢿⡄⠻⣆
⣤⣿⣷⣿⠿⠟⠛⢉⣈⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢤⣙⠳⢦⣤⣉⣿⣿⠻⢿⣶⣦⣤⣀⣀⠀⠈⠃⠀⠙⣷⣄
⠛⠉⣽⣁⣤⣶⠿⠛⠉⣸⠿⠛⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢦⣌⠻⢦⣌⠙⠻⢷⣦⣤⣀⠉⠉⠙⠛⠛⠿⢷⣖⣦⣌⣿
⡶⢾⣟⣉⡥⠀⣠⣾⠆⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠀⢠⠐⣆⠉⠀⠙⠷⣄⠙⠿⢶⣄⠈⠙⠻⢿⣶⣦⣄⣀⡀⡀⠀⠉⠙⠛
⣰⣿⡿⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠘⠁⠙⠀⠀⠀⠀⠙⠀⠀⠀⠀⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣶⣄⣰
⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣄⣤⣤⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⢹⣧⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣿⣦⣈⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⡀⢀⠀⣹⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿
⣿⣏⣁⡁⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠉⠁⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠂⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⡿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⡿⢿⢿⠿⢿⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⠃⠀⠀⠉⠛⠻⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠿⠋⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠋⠀⠀⠀⠀⠉⠀⠀⠀⠘⢿⣿
⢀⣀⠀⠀⠀⠀⣀⠤⠤⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢏⣉⡇⠀⣠⠞⠁⠀⣀⣀⠀⠈⠑⢆
⠈⠉⠀⢠⠃⠀⣴⠛⠉⠉⠙⣦⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⢀⡀⢠⡄
⠀⠀⠀⢸⠀⠀⣯⠀⠀⠀⠀⣸⠇⠀⡇⠀⠀⠀⠀⠈⣅⠗⠀⢦⠰⡂⢻⠀⢢⠆⠀⠸⠀⠇⠉⠠⠣⠘⠆
⠀⠀⠀⠀⠳⡀⠙⠷⣤⣤⡴⠋⢀⡜⠀⠀⠀⠀⠀⠀⠁⠀⠙⠈⠀⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠉⠢⠤⣀⣀⠠⠔⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠒⠋⠉⠉⠉⠉⠓⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣄⡀
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⣦⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠶⣄
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠈⣷⡀⠀⠀⢠⠞⠀⠀⠀⠀⠀⢠⡶⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠙⢷⡀
⠀⢀⡞⠀⠀⠀⠀⢀⣀⣤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣰⠏⠀⠀⠀⠀⠀⠀⣾⠀⠀⣼⠀⠀⠀⠀⠀⠀⣤⠖⢦⡀⠀⠀⠀⠀⠀⠘⣆⠀⠹⣄
⠀⢸⣇⠤⠤⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡟⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⡏⠀⠀⠀⠀⠀⢸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡄⠀⠹⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⡼⠀⠀⣸⠁⠀⠀⠀⠀⠀⣿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢻
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⡇⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠀⣾⠀⠀⣼⠁⠀⠀⠀⠀⠀⡼⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⡝⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⡟⠀⠀⠀⠀⠀⢀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠉⠀⠀⣀⣼⠇⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠇⠀⠀⠀⠀⠀⣼⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡿
⠀⣿⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠊⠉⠀⠀⢀⣠⡴⠚⠉⣾⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⠀⠀⠀⠀⠀⢠⡏⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢰⡇
⠀⣿⣀⡤⠤⠖⠛⠋⠉⠀⠀⠀⠀⣀⡤⠞⠋⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣾
⠀⡏⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡟⠀⠀⡜⠀⠀⠀⠀⠀⢀⡟⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠⠏
⠀⣇⣀⣤⡤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⠇⠀⠀⠀⠀⠀⢸⠃⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣸
⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⡏⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡇
⢰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡇⠀⠀⠀⠀⠀⣰⠃⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣴⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣾⠀⠀⣸⠁⠀⠀⠀⠀⠀⡿⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⡟
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⡏⠀⠀⠀⠀⠀⢸⠇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣸⠃
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⢻⠃⠀⠀⠀⠀⠀⠀⡸⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡿
⢸⡇⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠚⠋⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡎⠀⠀⠀⠀⠀⢠⡇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⠃
⢸⣇⣀⣤⣤⠴⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⢰⠋⠒⢦⣀⠀⠀⠀⠘⢧⣤⠼⠁⠀⠀⠀⠀⠀⣼⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡞
⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠉⢳⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠳⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢨⠇
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠳⠶⠶⠶⠛⠋⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿
⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣀⣀⣤⡤⠶⠤⠶⠋⠳⢦⡀⠀⢀⡏⠀⠀⢰⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⠀⠀⠀⣼
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⢙⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸
⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢠⡇
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⣍⡛⠲⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣸⠃
⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣙⡷⠶⠶⠦⢟⣯⣷⠶⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢠⡟
⠀⠀⠀⠀⠈⠉⠛⠒⠒⢶⠖⠒⠒⠒⠲⣶⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠉⣿⠻⢭⣭⣵⡒⢒⡒⠒⠋⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢨⡿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣶⣾⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⠐⢰⠀⠀⠀⠀⠀⠂⠒⠰⢶⣫⠛⡠⠤⢀⣀⡴⠚⠃⠀⠀⠀⠀⠀⢀⣤⠄⠠⣀⣴⠾⣿⠆
⠀⠀⠐⠃⠀⠀⢀⣶⢾⣌⣀⣨⠀⠀⢀⡀⠀⠃⢀⣀⠸⠿⠞⠀⠀⠀⠈⣿⣧⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣴⡯⣤⡦
⠀⠀⠀⠀⠀⠈⠁⠀⠘⠉⠀⠀⠀⠀⠀⠉⠙⠀⢟⣇⠀⠐⠢⣼⣿⠤⠤⡭⣀⣤⠖⠁⠀⠀⠱⣄⣀⣠⠤⠾⡿⠿⠓⠈⠂
⠀⠀⠆⠀⠀⠀⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠿⠉⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠸⣏⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠖⠂⠀⠀⠠⢰⣿⠋⠁⠈⠀⠀⠀⠀⠀⠀⠉⠙⠦⢛⡈⠀⣝⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢼⡆
⠀⠀⠀⠀⢠⠀⠀⠀⣀⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢏⣰⣿⣋⠀⠀⣦⠀⠀⠀⠆⠀⠀⠐⣡⣤⣀⡀⠿⠂⠀⣰
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⢻⠙⠻⠛⠇⠿⠀⠀⠀⠈⠶⠶⠒⠛⠏⢉⡶⠶⠒⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠶⠶⠦⠀⡕⠈⠀⠀⠀⠀⠂
⠀⠀⠀⡀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠢⣀⣤⣤⣶⣶⣾⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠂
⠀⠀⠀⠀⠀⢩⡿⠋⠈⣁⣿⣷⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⢀⣀⡄
⠀⠀⠀⠀⢀⠀⢒⣽⡿⡋⠀⠀⠀⠀⠈⣷⣄⢀⣤⣶⠾⢿⡟⠀⠀⠀⠘⠉⠉⠉⠉⠻⠻⣶⠖⠀⠀⠀⠉⠂⠄⡘⣋⡟⠃⡐
⠀⠀⠀⠀⠀⠈⠘⡯⠀⠀⠀⠀⠀⠀⠀⠱⠘⣿⣿⣯⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⢰⣾⠁⢠⡞⠁⠘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠘⠏⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⢀⡔⠁⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⠏⢡⣤⠈⠂⠠⢦⡄⢀⣤⠄⠀⠀⠀⢺⡆⢤⡄⢸⡇⢠⣾⠉⢰⡟
⠀⠀⠀⠀⠀⠀⡀⠸⣷⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠁⣾⠇⢸⡏⠀⠀⢸⡇⢸⡇⣾⠇⣼⡇⠀⣼⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡇⠀⠻⣆⠀⠀⡐⢿⡀⢸⡇⠀⡀⢸⡟⠀⣿⠀⣿⡇⡀⠀⢾⠇⠺⠃⣿⠀⠹⢇⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⠢⠔⣁⣈⠑⠀⠙⠀⠀⠘⣇⠀⠙⠂⠈⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢽⡇⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣾⠃⠊⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡈⠀⠀⠀⠀⠳⠄⠔⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣴⠋⠆⠀⢠⠞⢁⡠⠀⠠⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠻⠀⠀⠀⣿⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠉











⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣤⡴⢶⣶⣶⣤⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠂⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠈⠙⢿⣧⡀⠀⠀⠹⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⢀⡾⠁⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⣀⣴⠾⠟⠛⠻⠷⣦⣄⠀⠀⠀⠘⢷⡀⠀⠀⣸⠁⠀⢀⣴⠟⠋⠁⠀⠀⠀⠈⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⣇⠀⠁⠇⠀⢠⠞⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡀⠀⠸⡀⠀⠀⠀⠊⠀⢀⣴⡾⠿⠿⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠷⠶⣤⡀⠀⠘⠄⠀⠃⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠐⠂⠀⠀⠉⠲⠄⠀⠀⠀⣼⣾⣀⠞⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⣰⣶⡀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠴⠚⠉⠙⠛⢿⣦⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠃⠀⠀⠀⡼⠁⠀⠀⠀⠀⣠⣿⠟⠉⠀⠀⠉⠀⠀⠀⠸⠧⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠹⣧⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠸⠁⢠⣤⠀⠀⢰⡿⠁⠀⠀⠀⢠⣶⠶⠆⠀⠀⠀⠀⠈⠁⠈⠙⣷⡀⠀⠀⠀⠀⢹⡆⠀⢠⣿⣿⡄⠀⠀⠆
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⠁⠀⠀⠀⢠⡟⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠼⡿⠆⠀⠀⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠁⠀⠀⠀⠀⠁⠀⠀⣠⡄⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠚⠉⠒⠉⢉⡋⠉⠓⠊⠉⠒⠉⠓⠋⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣠⠶⢻⡆⢰⡏⠙⣆⢀⣠⣤⣤⣷⠄⠀⠀⠒⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠀⣴⠋⠀⣼⠇⠈⠷⣄⣼⡋⠁⢠⣿⠁⢠⡆⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠃⢀⣼⠏⣠⢤⡀⢀⠏⣹⠃⣾⠇⢠⡿⠀⡠⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠶⠛⢁⡾⢁⣾⢁⡞⢰⠇⢸⡟⡴⠻⡧⢞⣠⣼⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃⠸⣧⠋⠹⠊⠀⠘⠃⠈⠛⠀⠀⢰⠋⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠄⠀⠈⠛⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠛⠛⣿⠛⠉⠀⣴⠀⠀⣤⠀⢠⣦⠀⣰⡆⢀⣾⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡴⠂⢰⡿⠀⠀⣸⣿⠀⣼⣿⣰⢻⡟⣼⣿⠀⣾⣯⠏⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⡜⢹⡏⡸⢹⣧⠃⣾⡿⢹⣇⣼⣭⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣁⠞⠀⠸⡷⠃⠈⠋⠀⠈⠁⠈⠉⠸⣿⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⡈⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠁⠈⠁⠈⠁⠈⠁⠀⠁⠈⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠁⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⠋⠉⠙⠛⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠛⠋⠉⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢷⣄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣴⣿⡿⠛⠛⠛⠿⠿⢿⣶⣦⣄⣀⠀⢀⣴⣿⠏⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠋⠁⢀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⡏⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣹⣿⡿⠶⢶⣶⣶⣶⣶⣶⠾⠿⠛⠁
⠀⠀⠀⠘⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⡿⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⣰⣿⠋
⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣀⣠⣴⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣋⣁⣀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛
⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣟⠛⠋⠉⠁⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠿⢿⣿⣿⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣤⣤⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉




⣿⣿⣿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⠰⢄⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⠂⠀⠀⠀⠀⠀⣠⠏⡐⠶⠃⠀⠀⠀⢀⡐⠂⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⢀⣞⠉⠳⠞⠀⠀⠀⠀⠀⠀⠀⠡⡌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⡂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢦⡵⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠸⠃⠀⠀⠀⠀⠀⠀⠀⣾⠤⣌⠆⠀⠀⠀⢰⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢥⠊⠀⠀⢔⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣤⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⢠⡏⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢢⣀⡀⠀⠀⠀⢤⡐⠀⠁⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⡀⠀⠈⠢⡤⠀⢄⠀⠀⠀⠀⠀⠀⠠⣎⢂⢈⡰⠂⠁⠀⠀⠀⠀⢈⡤⠂⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡐⠉⢆⠀⠀⢄⡀⣠⠀⠀⠀⠀⡖⢢⠀⣩⠜⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡐⠀⠀⠀⠑⡄⠀⠀⠀⠑⢤⡈⢆⢙⡸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠⠒⠣⣰⣁⠁⠑⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡐⠁⠀⠀⡠⠊⠈⠢⡀⠀⠀⡠⣅⡜⠉⠀⠀⠀⢀⠊⠈⠲⠀⠀⠀⠀⢀⣠⢉⠀⠐⠗⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠃⡜⠀⠀⢀⠔⠁⠀⠀⢀⠜⠓⢄⠑⠉⠈⢢⠀⠀⠀⠀⠢⠄⠘⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠃⡜⠀⠀⡰⠃⠀⠀⡠⠖⠁⠀⠀⠀⣁⠦⢀⡀⠡
⠀⠀⠀⠀⠀⠀⡌⠌⠀⡠⠊⠀⢀⠤⠊⠀⠀⢀⠠⠒⠉⠀⣀⠠⠘⠋
⠀⠀⠀⠀⠀⡐⠌⢀⠔⠁⡠⠐⠁⢀⡠⠐⠈⣀⠠⠐⠊⠉
⠀⠀⠀⠀⢰⠏⡠⢃⠴⠊⡀⠄⢂⠡⠔⠒⠉
⠀⠀⠀⢠⣏⣨⢔⣡⠔⠍⠒⠉
⠀⠀⢠⡿⠟⠛⠉


⣿⢿⠿⡿⡿⠛⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣀
⠀⢀⠂⠀⠀⢱⡀⠀⡐⠀⠀⠀⠀⠀⢠⠈⠁⢀⠀⠀⠀⠀⢀⠁⢀⠀⠀⡀⠀⠀⡎⠀⣀⠀⠇
⠀⠈⢦⣀⣀⡼⠁⠀⢳⡀⠀⢀⡀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⣠⠃⠀⠀⠀⢤⣀⠤⠂⠀⢀⡀⡀
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⠙⢫⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡸⠉⠉⠀⠀⠀⠀⠀⡎⠀⠀⠀⣀⣙⣧⡼⠆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⠢⢤⡔⠀⠀⠀⣰⠁⠠⠀⠂⠄⠀⠀⠘⠀⣀⣄⡀⠹⣡⣏⢳⠟
⠀⠀⢀⡀⠁⠀⠀⠀⡠⠐⠠⡀⠈⠀⠀⠀⠀⠀⠇⠀⢀⠄⠠⡀⢀⠐⠗⠀⡆⠀⠀⢸⣿⣿⣿⠀⠀⠉⡇
⠀⠼⠥⠭⠦⠀⠀⠈⠀⠀⠀⠰⠀⣠⠢⠂⣠⠀⠸⡀⠀⠀⠀⢀⠀⠑⠲⠾⡁⠀⠀⢸⣿⣿⢻⠀⠀⠀⢸
⠀⢐⡒⢻⣋⡄⠀⢠⠀⠀⠀⠰⠀⢻⣤⠀⡤⠇⠀⠀⠇⠀⠀⠈⠀⠀⠀⠀⠙⣄⠀⠸⣿⣾⣿⠇⠀⠀⠈
⠀⠈⢫⣥⢤⠄⠀⠀⠀⢤⠄⠁⠀⢀⠟⠛⠋⠀⠀⠀⠀⠀⠀⠰⠀⡄⠒⠒⠦⠈⠀⠀⠈⡟⠉⢀⡤⠐⠀⠄
⠀⠀⠀⢹⠁⢀⠤⢀⠀⢸⠀⠀⠀⡜⠀⢠⣲⢛⠷⡄⠀⠑⡂⠀⢸⠀⠀⠀⢐⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠘
⠀⠀⠀⢸⠀⡏⠀⠀⠀⠸⠀⠀⠀⣁⠀⢇⠃⣍⣶⠃⠀⠀⡇⠀⠀⠓⠒⠺⡁⠀⠀⠀⠀⠸⠀⠈⠳⡤⠄⠂
⠀⠀⠀⠈⠆⡇⠀⠀⠈⠀⠀⢰⡛⢡⣁⠈⠉⢹⠀⠀⠀⠀⠂⠊⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠐⢦⣀⠜⠀⠀⡾⠁⠀⡨⠀⠀⠸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⣿⣀⣦⠇⠀⠀⠀⠃⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠓⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠒⠙⢄
⠀⡀⠀⠀⠀⠈⠁⠈⠉⠐⠒⠀⠤⠤⠤⠄⠀⠀⠀⠤⠤⠤⠤⠄⠒⠒⠂⠈⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠘⢦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⡏
⠀⠀⠀⢃⠈⢓⠖⠤⢄⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠁⠀⠀⢀⠤⠒⠚⡋⠀⢰
⠀⠀⠀⢸⢠⡈⠀⠀⠀⠀⠈⠁⠐⠂⠀⢠⠀⠀⠢⣄⠀⠀⢨⠀⠀⠀⠀⠀⠀⠜⢓⠉⠀⠂⠀⠀⢅⣤⢸
⠀⠀⠀⡶⠃⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠈⢄⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠘⠏⠻⠆
⠀⠀⠀⠀⠀⠈⠀⠀⢀⠤⠀⠠⢄⠀⠀⡆⠀⢀⢠⡀⣄⢱⡄⠈⢠⠒⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠑⠀⣧⠒⡁⠄⠀⠀⠠⡉⠣⡈⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⢁⠊⠀⠀⠀⠀⠀⠈⠢⣱⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠊⠌⠀⠀⠀⠀⠂⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢼⠁⠢⡀⢲⡦⠤⠤⠶⡟⢀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠸⠀⣠⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠢⠀⠀⠀⢀⡠⠊⢀⠀⠀⣀⣤⢊⣀⠀⠀⠔⠋⠱⡛⠖⠄
⡀⠀⠀⠀⢄⠀⠀⣀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢦⣄⡘⠘⡓⢒⡺⣀⡤⠗⠦⡆⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠙⠂⠀⠀⡀⣀⢉⣀⢢⣀⡀⠐⠀⠀⠀⠀⠀⠀⢌⠈⠲⣩⠀⠁⠈⠀⢀⡀⠃⠀⠀⢀⡀⠘
⠀⠀⡀⠀⠀⠀⢸⠀⠘⠀⠁⠈⠈⠁⠡⢇⡨⠐⠔⡐⠀⠈⢀⠀⠁⡀⠁⠦⢤⣄⠧⣹⣤⣢⣀⣅⣀⠰⡒⠤⠠
⠀⠀⠀⠀⠀⠀⠀⣤⡃⠸⡄⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⢱⠀⠻⢤⠒⠾⣀⠉⡷⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠫⢄⡁⠐⠂⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⢀⣠⠀⠈⠵⡔⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠤⢐⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⠢⠄⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⡇⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠸⢄⡀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠈⠀⠈⠑⠂⠠⠄⢀⣀⣀⠀⠤⠐⠊⢁⡠⠚⠀⠀⠈⠐⠢⡄⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣸⠀⠀⠀⠈⢹⠒⠒⠲⠶⠶⠦⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⠀⠤⠐⠒⠂⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠤⠐⠀⠁⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⣶⠀⠀⠰⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠐⠒⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠙⢄⠀⠀⠀⢀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒⠐⠊⠀⠀⠀⠑⠂⠐⠋⠀⠀⠀⠀⠀⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠶⠚⠛⠛⣻⠿⠿⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢻⡏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⣇⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣠⠾⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⡼⠁⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⢻⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢠⡟⠓⠛⠛⠛⠛⠛⠛⠳⢶⣤⡀⠀⢻⡄⠀⠀⣼⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⡇⠀⣠⠃⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⣤⣶⠶⠶⢤⣤⣤⣀⣸⣶⣧⣴⣃⡴⠞⠋⠀⠀⠀⠀⣠⡶⠛⠙⠳⣿
⠀⠀⠀⠀⢸⡇⠀⢀⡾⠋⠁⠀⠀⠀⠀⢀⣀⣤⣬⣿⣿⢭⣁⣀⣀⣀⣀⣠⡴⠞⠁⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠸⣇⣀⡿⠁⠀⠀⠀⢀⣤⠾⠋⠉⣠⠟⠉⡿⣦⡈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⢻⡋⠀⠀⠀⠀⢠⡿⠁⠀⠀⢰⡏⠀⢸⡇⠈⠻⢦⣀⠀⠀⠀⠀⣀⡀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⣿⠁⠀⠀⠀⢸⡇⠀⠈⣷⠀⠀⠀⠉⠙⠛⠛⠋⠉⠉⢳⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣿⠀⠀⠀⠀⠀⢿⡄⠀⠀⠈⠙⠻⢶⣄⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣤⣀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⣹⡷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⢾⡷⠤⠤⠶⠒⠋⠁


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⠶⠒⠛⠉⠉⠙⠓⠲⢤⣀
⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⡄
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⠀⢹⡄
⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠁⠀⠀⠀⠀⣀⣴⡇
⣧⣀⣤⡤⠴⠖⠛⠉⠉⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⢸⡇
⢯⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⣼
⠘⢧⣀⣤⠤⠶⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠙⠳⢤⣄⣀⣀⣀⣀⣀⡤⢶⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⣧⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢷
⠀⠀⠀⠀⢀⡶⠒⠦⣄⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠘⣆⠘⣧
⠀⠀⠀⠀⢸⡃⠀⠲⣌⢳⣠⡴⠛⠉⡽⠋⠉⢳⣦⣀⢹⡄⠸⡆
⠀⠀⠀⠀⠈⡧⠀⠀⢬⡁⠉⠀⢀⡾⠁⠀⢠⡟⠀⠙⣦⢻⡀⢻⣀⣀⣠⣤⣄⣀⡀⠀⠀⠀⣠⡤⠖⢲⡄
⠀⠀⠀⣰⠞⠁⠤⠤⠄⠀⠀⠀⣼⠁⠀⢀⡞⠀⠀⠀⠸⠞⣧⡴⠛⡟⠀⠀⡏⠈⠉⠳⣤⡞⣡⠖⠀⠈⡇
⠀⠀⢠⡏⣴⣒⣒⠀⣠⡄⠀⠀⡇⠀⠀⣸⡷⠖⠒⠦⣤⣴⡟⠀⢸⡇⠀⠀⣧⠀⠀⠀⠈⢈⡥⠀⠀⣼⠁
⠀⠀⠘⠶⠶⠷⠖⠋⠉⢷⡀⢸⡇⠀⠀⡿⣇⠀⠈⠛⠲⠙⠁⠀⠘⡇⠀⠀⢻⠀⠀⠀⠀⠀⠠⠤⠄⠘⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣧⣀⠀⣇⣘⣶⡀⠀⠉⣉⠀⠀⠀⢿⠀⠀⠘⣇⠀⠀⠀⢤⡀⢐⣒⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣸⠃⠀⢚⠅⢀⡄⠀⠈⢧⠀⠀⠹⣆⠀⣰⠃⠉⠓⠮⠭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣞⡥⢢⠞⠛⠳⢤⣌⣳⣄⣤⠽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡭⠞⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣾⣿⠶⠶⠶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⣿⣿⠀⢀⣀⣀⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠶⠶⠶⠶⢦⣤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⣿⠿⠿⠻⠿⠗⠖⢲⣶⠶⠶⠤⠤⠤⣤⣤⣄⣀⣀⠀⢀⣀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠓⠒⠶⢶⡄
⠀⠀⠀⣿⣿⠀⢸⣿⣿⡵⡷⠄⠀⠀⠀⠘⠃⠀⠈⠀⣀⣀⣤⣤⣾⣿⣾⣷⣿⣶⡘⠇⠀⠀⠉⠉⠻⠾⣿⣶⠲⡶⢶⣶⣶⡆⢸⡇
⠀⠀⠀⣿⣿⠀⢸⡿⠿⠁⠀⠀⠀⠀⠀⠀⣀⢔⡺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠐⢦⣄⡀⠀⠀⠈⠿⣁⠀⢻⣿⡇⢸⡇
⠀⠀⠀⣿⡿⠀⣼⣦⠖⠀⠀⠀⠀⠀⡠⢺⣄⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣵⣼⣶⣦⡀⠀⠙⢦⡘⣿⡇⢸⡇
⠀⠀⠀⣿⣿⠀⢹⣿⡄⠀⠀⠀⢁⢀⣴⣿⣿⣿⠟⠉⠉⠙⠛⠛⠛⠛⠛⠛⢿⣍⠻⣿⠿⢻⣿⣿⣿⣿⣿⠟⡆⠤⠀⠛⣿⡇⢸⡇
⠀⠀⠀⣿⡟⠀⣼⣯⡀⠀⠀⠠⢡⣾⣿⣿⣿⣣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⢹⣶⣾⣿⣿⣿⣿⣿⣼⡇⢂⠀⠁⣹⡇⢸⡇
⠀⠀⠀⣿⡇⠀⣿⣿⣿⣄⣰⣦⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣤⣈⣱⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠲⠀⠀⢹⡇⢸⡇
⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⣀⢈⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⡇⢸⡇
⠀⠀⠀⣿⡏⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡈⠉⠀⠀⢴⡆⢀⣾⣶⢸⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⠀⡀⠀⡇⢸⡇
⠀⠀⠀⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀⠀⢀⢘⣷⢸⣿⣿⣿⣿⣿⢿⠋⠁⠘⢁⣿⣿⣿⣿⣿⠿⢿⡏⢸⠀⠀⡇⢸⡇
⠀⠀⠀⣿⣷⠀⢸⣿⣿⣿⠯⠍⠛⢿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣿⣿⠿⣿⣿⣿⣿⣷⣤⣴⣍⣽⣫⣙⠉⢁⣴⣛⡇⠾⠐⠈⡇⢸⡇
⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⠏⠀⠀⠀⠉⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠋⠙⠋⠋⠁⠈⠀⠀⡇⢸⡇
⠀⠀⠀⣿⣿⠀⢸⣿⡯⠃⠀⠀⠀⠀⠀⢸⠇⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣾⣿⣇⠀⡀⠀⠀⣘⡀⣀⠀⣰⡇⢸⡇
⠀⠀⠀⣿⣿⠀⢸⣿⣟⠀⠀⠀⠀⠀⢀⠀⠄⠀⠐⠀⠀⣄⣦⣄⣨⣭⣷⠛⢻⠿⠿⠿⠿⠿⠟⠛⠒⠛⠛⠉⠉⠉⠉⢉⣀⣀⣸⠇
⠀⠀⠀⣿⣿⠀⢸⣿⣿⣧⡴⠀⠠⠴⠶⠶⠛⠛⠛⠚⠛⠉⠉⠉⠉⠁⣀⣀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠶⠶⠛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⣿⣿⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣴⡶⠶⠶⠶⠟⠛⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢻⣭⣵⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣷⣶⡷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢠⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⢸⢄⡀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠨⠁⠄⠡⠀⢻⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⣶⣤
⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⣏⣤⣭⣭⣤⣾⣆⠀⠀⠀⠀⠀⠀⣧⣠⣤⣀⠀⠃⠈⢸⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢸⠉⠙⠛⠻⠿⠿⠿⢋⡞
⠀⠀⠀⠀⠈⠈⢹⡄⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⢤⣤⡤⠏
⠀⠀⠀⠀⠀⠀⠸⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠻⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠿⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣻⣿⣿⣶⣤⡄
⠀⠀⢸⣯⣭⣗⡛⣵⣖⠯⢱⣄⠀⠀⠀⠀⠀⠀⢴⣟⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠰⠒⢦⠉⠙⢦⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠛⠛⠉⡇
⠀⠀⠹⣿⣿⣵⣾⣷⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠐⣭⣭⣿⣭⣭⡖⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣶⣶⣶⣶⣿⠀⠀⠀⠀⢰⠛⠉⢁⣀⠤⠤⠖⠒⣊⡇
⠀⠀⠀⠈⢿⣿⣿⡿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⢿⢻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠻⣝⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣊⣩⠥⠤⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉

⠀⠀⠀⠀⢠⣴⣦⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⠶⡶⣢⡀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠿⠯⠭⣼⠷⠆⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠩⠛⠲⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣶⣬⣙⢛⣛⣉⣠⡆⠀⠀⠀⠀⠀⠀⠈⣴⣶⣶⣦⣿⣶⠁⠀⠀⠀⠀⠀⢠⡟⠀⣴⠛⠉⠻⣆⠀⢿
⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢠⡶⠙⠛⠛⠛⠛⠛⠛⣡⣤⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⡆⠀⢷⣀⠀⣠⡟⠀⣼⠁
⠀⠀⠀⠀⢩⢩⡉⠈⡉⢩⢁⠀⠀⠀⠀⠀⠸⣿⣶⣰⣿⣿⣷⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠻⣀⡀⠉⠉⠉⢀⣠⠋
⠀⠀⠀⠀⢸⣸⠇⠀⢧⡼⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣭⣭⣭⣝⠲⣄⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⢀⣤⡞⣿⣿⣄⠠
⠀⠀⢘⠿⣿⣿⣿⡿⠿⢿⣿⣿⡆⠀⠀⠀⢀⡞⣴⣿⡿⣿⡿⣿⣷⡜⡆⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⣿⣷⣶⣶
⠀⠀⢸⣿⣿⣿⣿⣿⠦⠤⢼⣿⡇⠀⠀⠀⠸⡇⣿⣿⡿⣾⡾⣿⣿⡇⡿⠀⠀⠀⠀⠰⣦⣯⣭⣭⣭⣭⣭⣶⡆⠀⠀⠀⠀⠸⣿⣬⣿⣿⣾⣿⣿⣩⣿
⠀⠀⢸⣿⣿⣿⣿⣿⠶⠦⢼⣿⡇⠀⠀⠀⠀⢳⡹⣿⣿⣿⣿⣿⡟⣱⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢻⣿⣿⢻⣿⣯⣹⣿⠏
⠀⠀⠈⠙⠛⠛⠛⠛⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠙⠶⢭⣭⣭⠵⠞⠁⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⠀⣀⡤⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢤⢀⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠠⢂⠸⡭⡬⠤⣽⣿⡆
⠀⢀⡿⡎⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠐⣦⣹⠄⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣄⣤⠶⠶⠤⠀⠄⠴⠶⠶⠤
⠀⢸⣤⡑⠘⠲⣿⡄⠀⠀⠀⠀⢣⣦⣼⣟⣛⣻⣧⡈⡀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⢀⡌⠀⠀⠤⠤⠤⢿⠿⣛⣛⡀⣡⣄
⠀⠸⣿⣟⠳⣶⣾⠇⠀⠀⠀⠀⠀⣻⣙⠫⣄⠀⠐⠁⢨⡀⠀⠀⢀⣛⣿⣷⣄⠀⠸⠻⣁⣠⣤⣤⡭⠭⣥⣤⣶⣶⣾⡿⠞⣦
⠀⠀⠈⠿⣳⠞⠃⠀⠀⠀⠀⢸⡤⣿⠻⡇⠉⡶⣿⠐⢺⠇⠀⠀⠢⠍⢟⣡⠗⠀⠀⠑⠲⠭⣁⣀⣘⠻⠛⠛⠛⠋⣉⡠⠚⠁
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠻⡶⠁⠁⠀⢈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣆⣁⡄⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⣉⠃⠈⠴⠇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡀⠀⢀⠉⠁⠀⠰⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠿⠛⠋⠀⣀⣀⠀⠠⠤⠀⠀⠐⠈⠉⠁⠀⠀⠀⢸⠀⠀⢀⣧⣕⢄⣀⡀⣀⢀⣪⣦
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⢰⡇⠀⢸⠛⠶⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⠀⠀⠀⠈⠀⠀⠀⠙⠥⣀⣀⣈⣻⣻⡷⢶
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⣿⣶⠿⣀⡈⣀⣠⠃⠀⠀⠀⠀⠀⠀⠀⢰⠶⠖⠤⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⠿⡿⡯⠌⠀⡿
⠀⠀⠀⠠⢘⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⢦⣤⡄⠀⠀⠀⠀⠉⠀⠀⠀⢃⣸⠀⠀⠀⠐⠁⠀⠀⠀⢀⡤⠔⠂
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡎⠡⣈⡃⢼⢅⡀⠀⠀⠀⠀⠀⠀⢸⠏⠄⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠄⢻⡚⠻⡇⠀⠉⠻⣦⡉⠀⠀⠀⠀⢀⢀⣿⣆⡀⡈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢿⡳⡄⠛⢀⡘⢮⡂⠄⠘⣷⣤⡀⠀⣠⡼⢸⡇⠙⢿⣤⠃⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣁⣸⠻⡘⢃⠹⣀⠀⢹⡄⠈⢻⣆⠉⠓⠁⠀⠀⠉⠈⠙⠐⠋⠉⠉⠀⠀⣠⡄⢤⡀⠀⠉
⠀⠀⠀⠀⠀⣀⠤⠤⡦⠤⣀⡀⠘⠻⠆⠣⣨⡇⠘⢆⡀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢾⡏⢿⣄⡛⠆
⠀⠀⠀⣠⠾⢅⠀⠒⠒⠂⢀⡹⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡆⠂⠀⠀⠀⠀⠀⠀⠀⢠⡾⣷⡀⠻⢦⣉⣿⠿⠗⣄
⠀⠀⢰⡏⠀⢠⠤⠠⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⢼⣿⣿⡇⠀⠀⠀⠀⠀⠀⣯⡳⣌⡹⣶⣦⣬⡯⢓⣾⢯
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣀⣿⣿⠀⠀⠀⠀⠀⡄⠐⠀⣠⣤⣫⡕⣳⢼⡿⠃⢀⠀⢐⡄⠀⠀⣈⡛⠀⡭⠭⠯⢭⣤⢜⣫⡌
⠀⠀⠀⠘⢿⣆⣠⣴⣶⣹⣿⣿⣇⠀⠀⠀⠀⠀⠻⣶⣬⢾⡿⣿⣿⡿⠛⠁⣴⣵⣿⡿⠃⠀⠀⢶⣁⠀⠀⠀⠀⠀⠒⢚⣵⡇
⠀⠀⠉⠙⠢⠍⠹⠛⠻⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠉⠰⠀⠀⠿⠿⠏⠀⠀⠀⠀⠘⠏⡁⡀⡄⠀⡂⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⡀⢀⠀⡀