Caramel Emojis & Symbols

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






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

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⡎⠲⠀⠀⠀⠀⢀⢤⣤⠤⠔⠓⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⠀⢸⡀⠀⠒⠀⠀⠀⠆⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠰⠀⠀⠁⠀⠈⠁⠤⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⢸⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⣠⡶⠒⠂⠀⠀⠀⡴⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠰⠾⢉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠉⠁⠀⠀⠀⠀⠘⠛⠉⠉⠁⠁⠀⠈⠉⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠉⠉⠭⠭⠍⠛⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢢⠠⠤⠤⠶⠒⠒⠚⠓⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢺⠃⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⡂⠈⠭⠉
⠀⠀⠀⠀⠀⢀⠁⡇⠀⢸⡎⢳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢑⣒⣒⠠⠤⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⢜⡼⠁⢠⢾⠇⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⢌⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡀
⠀⠀⠀⠀⣧⠏⢀⡎⢁⡏⢠⡎⡜⢀⣆⠀⠀⠠⡤⣄⡀⣀⡀⣀⣀⣤⣤⣬⣭⠭⠥⠤⠖⠀⠀⠀⠰⢓⣛⣷⢤
⠀⠀⠀⠀⢿⡄⠸⣧⣸⡇⢸⣰⣇⡼⡽⠀⠀⠦⠀⠤⠔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠿⠩⠔⠊⠀⠀⡀
⠀⠀⠀⠀⠈⠙⠒⠚⠋⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡓⠀⠀⠀⠀⠀⠀⠒⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠓⢢⡀⠀⡠⠴⠂⠀⡀⠊⢠
⠀⠀⠀⠀⠀⠀⠀⡠⠄⢀⣀⡀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⢀⡗⢺⠃⠀⠀⡀⠹⠆⢰
⠀⠀⠀⠀⠀⢠⠎⠀⠾⠿⠛⠁⠀⣠⠝⠀⢰⠀⠀⠐⠶⠶⠖⠀⠀⢠⡄⠀⠀⠀⢀⠀⠘⣸⡄⠀⠀⠈⠁⢸⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⡃⠀⠀⠤⠔⠂⢀⠤⠁⠀⠸⡳⠀⠀⠀⠀⢤⡤⠴⣾⠁⠀⠀⠀⠀⠀⣀⣀⣈⠐⠶⣧⣿⢸⣠⣤⣤⣏⠀⠠⠤⠤⠤⠄⠀⠀⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠒⠒⠂⠒⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡻⠿⢛⠻⠿⡿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢖⣹⡭⢦⢶⢑⡄
⠀⠀⠀⠀⠀⠀⠀⣆⣱⡄⢒⠉⣧⠵⡜⣏⣣⠘⣄
⠀⠀⠀⠀⠀⠀⢰⠻⣆⠴⡊⣣⠬⢖⠙⣉⣦⠷⡘⣆
⠀⠀⠀⠀⠀⠀⠸⢸⡟⣆⠙⣤⢷⠚⣯⡽⡞⢏⣁⠼⡆⠀⠀⠀⠀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⠂⠇⡏⣧⡨⢖⢫⣈⢥⠚⡏⣿⠀⠇⠀⠀⠀⣴⣑⠖⡶⣬⣉⡒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⢃⢻⢹⠒⠃⠃⢸⠀⠇⠟⣠⠀⠀⠀⡴⢠⢏⡝⡲⢧⣜⢉⠏⠠⢉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⠸⢸⠀⡆⠇⢘⡤⠶⠛⠁⠀⢀⡞⣰⠓⡼⣸⢉⠞⢤⣍⡰⠇⡜⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⠴⠒⠉⠁⠀⠀⠀⠀⠀⢸⢦⣉⡛⠶⢥⣟⢑⠦⣴⠉⡖⡇⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢈⢹⠒⠮⣀⠋⠖⣠⢾⢡⠹⢃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣘⠊⢸⠈⠀⠃⠙⢰⢹⢸⠘⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⡀⠇⢸⢸⠘⢈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⠗⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄
⠀⠀⠀⡀⠀⡀⢠⠖⠁⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣵⡤⢤⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣢⡀
⠀⠀⠘⠉⣸⠓⡟⣧⠀⠀⣱⣥⡶⠆⠀⠀⠀⢠⡄⢇⣀⣀⡈⣿⠏⢀⡾⣧⢀⠀⠀⢀⠤⠴⠂⢸⠡⠾⠏⣹⢆⠐
⠀⠀⠘⠀⠉⠀⠧⡿⠂⠈⡝⢙⡀⠀⠀⠀⠀⠙⠛⠛⡛⠛⠘⠛⢓⣾⡽⠃⡁⠀⠀⠀⠲⠅⣆⢸⣧⠀⣶⢧⡼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⢲⣂⣁⣀⣀⣴⢫⡿⡅⠀⠀⠀⠀⠀⠀⠀⠟⠚⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⢀⣤⣦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠈⠉⣫⢾⢣⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⡔⠉⠀⠈⢣⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢤⢀⣀⡄
⠀⠀⣠⠾⢋⠢⣴⠾⢛⣾⠃⠀⠀⠀⠤⠄⢠⡶⠂⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⣠⠀⠀⠀⠀⠀⠀⠀⣀⣤⢽⢏⡰⠄
⠀⢰⣧⠀⠀⣰⣥⣾⣛⠟⠁⠀⠘⠀⠀⠀⠸⡦⣦⡀⠀⠀⠀⣤⠸⢠⣆⢐⢸⣥⡀⠀⠀⠀⠱⣄⠻⠋⠀⢸⠯
⠀⠀⠹⣦⡐⠟⣫⡼⠋⠀⠀⠀⠂⣜⠃⠒⠐⢸⣧⡄⠀⠀⡐⠂⢀⢸⢈⡿⠿⠋⠳⠂⠀⠀⠀⠘⠷⡶⠞
⠀⠀⠉⠉⠈⠙⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠤⠼⠉⠀⠀⠀⠀⠰⠬⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣶
⠀⠀⠀⣔⣀⣉⣀⣀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣴⡾⠀⠉⠁⠤⣄⠀⠀⠀⠀⠁⠀⠀⠘⠛⠃⠖⢀⡀
⠀⠀⣤⠋⠻⠙⢿⣿⠛⠀⣧⡄⡀⠀⣀⠀⠀⢠⡖⠩⠉⠉⠉⡇⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠁⣤⣤⣜⣱⡾⢹⣷⢀
⠀⠀⠀⠁⠀⠄⠨⠷⢴⠚⠁⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠔⣿⣶⣂⡴⠛⠐⠀⠀⠀⠀⠀⠇⠈⠁⠈⢿⣦⠞⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠄⠶⣯⠙⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠄⡘⠁⣰⡄⠀⠀⠀⠸⡀⠀⠀⠀⢀⣢⣤⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠄⢀⡶⠈⡠⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠤⢤⠶⠀⠀⠀⠀⠀⠀⠀⢠⡨⣅⠀⠀⠁⠃⠀⠀⠀⠁⠉⠁⠀⠁
⠀⠀⠀⠀⠒⠒⣒⠶⠀⠀⠉⠒⠒⠛⠷⠀⠀⢀⠐⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠞⢺⠃⠈⢈⣦⠀⠀⠀⠀⠀⡀⠀⠀⣰⣆⣈⡤⠀⠀⠀⡆⠀⠀⠀⠀⣾⠞⠃⠀⠀⠀⢀
⠀⠀⠀⠆⠄⠀⢀⠀⣿⠀⠀⠀⠀⢤⠚⢠⠀⠀⠀⠀⠀⠀⡬⣤⠙⣀⢀⠀⠀⠎⠀⠀⠀⠀⠀⠘⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡋⠀⠀⠐⠆⠉⡶⠉⠐⠀⠀⠀⠀⢠⢃⡭⠁⢘⡯⣱⠂⠀⠀⠀⠀⠠⡵⠎⣥
⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠔⠒⠒⠒⠢⢄⠀⠀⠀⠀⠀⠀⡔⠉⠡⠀⠀⠠⣄
⠀⠀⠀⠀⠀⢀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡜⠁⣠⠀⠀⠀⠆⠀⠑⡄⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠯⠀⢄⣀⠀⠀⠀⠀⠀⠀⢸⠅⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢻⠋⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢀⡴⠋⠠⠄⠀⢄⡀⠱⣆⠀⠀⠀⠀⢸⡆⠀⢇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⣀⣀⣀⡴⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠈⠹⡶⣠⠀⠀⢠⠞⠀⠀⠀⠀⠀⢀⠃⢀⣿⣆⠀⠀⠀⠘⣿⣄⠈⠦⣀⣀⠀⠀⠁⢰⠇⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡈⠀⠀⠀⠀⠀⠀⠌⢀⠞⣿⠟⠀⠀⠀⠀⠈⠻⣦⣄⣀⡀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡅⠰⣠⠤⠤⠊⢁⡤⠊⣸⢦⠃⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣧⣤⣤⡄⠀⠚⢩⣄⣴⠟⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠒⠦⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠙⠷⠦⢄⣀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠄⢁⣀⣉⡀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣾⢦⡈⢫⠁⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⢂
⠀⠀⠀⠀⠀⣿⢹⡇⠀⠧⠄⡀⠀⠀⢀⡆⠀⡆⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣷
⠀⠀⠀⠀⠀⢹⡀⠁⠀⠀⠄⡀⠀⠀⢉⣀⣴⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡏
⠀⠀⠀⠀⠀⠀⠙⢤⣀⡀⠀⢀⢀⣉⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠙⠳⣆⡀⠀⠀⢀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉

⣿⣿⠿⠿⡿⠛⠿⢿⠿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⢂⡿⣿⣿⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⣿⠿⢹⡿⢃⠾⣋⣿⠋⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣯⡿⠁⣿⣵⣫⣒⣁⡠⢤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡠⣄⣿⣿⣋⣿⡅⠊⠀⠘⠛⠉⠉⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠈⢳⣨⢹⣿⢟⡉⠌⠻⣶⡶⡶⠾⠖⢚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⡿⣿⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠄⠀⣡⡿⢵⣶⣯⣓⢶⣶⣦⣤⠴⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠈⠿⣄⡘⢮⣀⠀⠀⠀⣂⣤⣦⠴⢏⣆⠀⠀⠀⠈⢉⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠐⠀⠀⠀⠀⣝⣶⣼⣿⣿⠛⠋⠉⠂⢰⣍⣿⠿⠾⠛⣻⣏⣴⣆⣄⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⢿⣉⠛⠛⢋⣁⠀⠿⢽⣿⣷⣆⣀
⠀⠀⠀⠀⠀⣀⣤⣴⣄⢙⣓⣧⣿⠶⣬⣁⠀⠀⡀⣀⣜⠁⠦⡔⠁⠀⢀⡄⠙⠁⠀⠀⢃⢝⣬⠂⠈⠙⠛⠁
⠀⠀⠀⢠⠚⣓⣢⠔⣫⠽⢣⣿⠏⢠⣾⣉⣽⠟⠛⠉⢿⠃⠘⠃⠃⠘⠋⠀⠀⠀⣀⢔⡼⠋⠼
⠀⠀⠀⠘⠡⠿⢱⠾⠿⠴⠛⠓⠈⠉⠉⠁⠀⠀⠀⠀⠈⠳⣦⡀⣄⣀⣀⣤⣤⠿⠟⠋⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠂⠒⠒⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠿⣿⡿⠿⠭⢟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠹⠄⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢷⣦⠄⠀⠀⠀⢰⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠼⠃⠨⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⣴⡀⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠀⠀⠀⣸⣟⣦⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⢠⣸⣿⡇⠀⠀⠀⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⢠⣷⡿⠉⠈⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠛⢳⣿⢀⣤⣾⣿⣿⣿⣿⣶⡤⠄⠚⠀⠰⠶⣾⣿⣿⣿⣿⣿⣾⡀⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⢠⣾⠟⣿⣿⣿⣿⣿⣿⣿⣿⣷⢣⣾⣿⣖⣾⣿⣿⣿⣿⣿⣿⣿⡆⢸⡆⠀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡜⠛⠀⢻⣿⣿⣿⣿⠿⠿⠣⢎⣼⣿⣿⣾⡜⢿⣿⣿⣿⣿⣿⡿⠁⠸⣷⣶⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠉⠉⠀⠁⠀⠀⠀⣾⣿⣿⣿⣿⣉⣄⠀⣀⡀⠉⠀⠄⠀⠀⢸⣍⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣀⣀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⣯⠏⠃⠁⠀⠠⠀⠀⠄⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠐⠀⠉⠺⡿⠁⠁⠀⠀⠀⠀⣠⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⢿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⡌⢿⣿⣷⣄⣀⣀⣀⣀⣀⣀⣠⣤⣠⣤⣽⣿⣿⡏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣆⠈⠻⣿⡏⣌⠃⡏⢸⡃⡏⣏⣙⣩⣿⣿⠿⠟⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⠀⠀⠈⠑⠓⠦⠶⠸⠿⣹⡤⣭⣴⠿⠃⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠈⠃⠀⠀⠀⠀⠈⠋⠉⠙⠁⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢆⠀⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠃⠀⠀⠠⢈⣿⠇⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣤⢠⢀⡄⣀⢠⣤⡀⣤⡄⢠⡤⠀⣤⣄⠀⠀⢠⣤⢠⢤⡀⣤⠤⢀⡤⣤⢠⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⡏⢸⢼⡇⣿⢸⣤⠇⣧⡿⢸⠶⠀⡇⢸⠀⠀⣿⠈⢸⣼⠃⣿⠆⢸⣧⣼⢸⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠃⢸⠘⠇⠿⠸⡇⠀⠯⠀⠸⠤⠄⣧⠞⠀⠀⢻⡴⠸⠘⠆⠿⠤⠸⠇⠸⠸⠇⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠆⠀⡆⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⠀⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡇⠀⠘⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⡻⠀⠀⠀⠁⠀⢰⠀⠀⢰⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠥⢀⡔⠁⠀⢀⠌⠀⠀⠀⠀⠀⢸⠀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⠚⠁⠀⢀⠠⠋⠀⠀⠀⠠⠀⢀⠌⢠⠃⠌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢀⠔⠁⠀⠀⠄⡀⠔⠁⡠⠂⡴⠃⢠⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠑⠀⠀⠀⢀⠠⠔⠊⣀⠄⠊⣠⠀⠀⠀⢠⠀⢰⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⢀⠂⠀⠈⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠊⠀⢀⠠⠐⣈⣉⣀⡠⠤⠔⠊⠁⠀⠀⠀⡀⠔⠁⠀⡠⠀⢀⣼⡅⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡥⠚⠉⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⠀⢀⡠⠊⡠⠞⠁⢸⡇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠉⠀⠀⠠⠒⠀⠒⠒⠒⠀⠉⠀⠀⠀⠠⠔⠊⡁⠀⠈⠀⠀⠀⡸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠄⠐⢒⡀⠀⡈⢉⣉⠥⠄⠒⠒⠒⠀⠁⠀⢀⠔⠀⢀⡴⠁⠀⠀⠀⣾⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠘⠃⢀⣠⠔⠚⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣠⠴⠋⠀⠀⠀⢀⡜⠁⢇⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠒⠒⠒⠈⠀⠀⢀⠤⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠖⢹⠀⠀⡸⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠒⠀⡀⠀⢠⠞⠉⠉⠓⠒⠢⠤⠤⠤⠔⠂⠁⠒⠁⠀⠀⢀⠀⠀⠁⠀⠀⠀⣠⠃⠀⣠⠃⢀⠂⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠐⠈⢙⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠁⠀⠀⢀⠤⠊⠀⠀⠀⠀⣀⡤⠚⠁⢀⠔⠁⡐⠁⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠁⢀⣠⠴⠒⠋⠉⠀⠀⣠⠆⠀⠀⠈⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⠀⡠⠒⠉⠉⠉⠉⠁⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣠⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣔⣋⡠⠴⢾⡉⠙⠂⠀⠛⠉⠀⠀⠀⠀⣠⣴⣒⣋⣉⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣄⣀⣀⣀⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢠⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⣇⠈⡇⡀⠀⠀⠀⠀⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠲⠚⢻⣿⣿⣿⠿⠶⠶⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⡆⢻⡜⡄⠀⢠⠀⠄⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠚⣡⡾⠉⠉⡡⠄⢀⣈⢩⣶⣾⢣⡀⠀⠀⠀⠀⠀⠀⠀⣦⡆⢧⠈⢿⣜⠀⢐⡄⢀⠈⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠘⠹⣦⠟⣿⣿⣷⠀⠀⠀⠀⢠⣶⣼⣿⣧⠈⢇⠈⢿⡂⠀⠀⣀⣠⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠚⣡⣿⣿⣿⣿⠀⠀⡀⠀⠈⠿⢿⣿⣿⣿⣎⠢⣀⠛⠲⣶⣝⣿⣦⣿⠇
⠀⠀⠀⠀⢀⣎⡀⠠⠃⠀⠀⠀⠀⠠⠐⢁⣴⣿⣿⣿⣿⠟⠀⠀⠀⣠⣤⣤⣀⣿⣯⣿⣿⣷⣬⠑⢄⡙⠿⣿⣿⡿
⠀⠀⠀⠀⢠⠟⡄⠁⠀⠀⠀⠐⢀⣴⣀⣨⣿⡿⠛⠃⠀⠀⠀⠾⡿⣿⣟⣻⡿⠛⠙⣿⡿⢹⣿⣿⣦⣬⣑⡢⣝⡗
⠀⠀⠀⠀⠀⣆⠀⢀⣀⢤⣔⠠⢚⣽⣿⡿⠋⠀⣠⠎⠀⠀⠁⣰⠟⠉⠉⠙⢿⣤⣴⡿⢁⣾⡿⠿⠿⠛⠛⠛⠺⠃
⠀⠀⠀⠀⠀⠘⠢⢽⣞⠯⣖⣤⣿⣿⣿⣷⢴⡶⠋⣀⣀⣤⣾⠏⠀⠀⣀⣤⡾⠃⣀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠀⣸⡀⠊⠀⠉⢙⣥⣄⡠⠴⠋⣀⣴⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣍⣙⣒⣒⢒⣁⣀⣠⣤⠖⠋⠁⢓⣿⣯⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⡟⣻⡟⠁⠀⠉⠀⣰⣯⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣽⣷⣍⣀⣠⠀⣀⣭⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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


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

⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠠⡇⠑⠦⠀⣀⡀⣠⡀
⠀⠀⠀⠀⠀⠀⣴⡃⠙⢦⡄⠀⠀⠉⠙⠶⠶⠦⣄⣀⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠘⢷⣄⢀⠃⠀⠀⢀⡄⠈⠙⠱⣦⠼⢷⡀
⠀⠀⠀⢠⡾⢟⠁⠀⠄⢠⠤⠽⣦⡀⠀⢀⣈⠤⠄⠀⠀⠀⠀⠈⢷⣦
⠀⠀⢀⡸⠃⠀⠡⢀⠀⠀⠀⠀⠙⢿⣦⡟⠀⠀⠀⠀⠀⡰⠥⠄⠀⠙⢶⡀
⢀⣴⠿⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠂⢿⣷⡀⠀⢀⠄⠁⠀⠀⠀⠀⢀⠈⢷⣄
⠈⣏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠉⣷⡴⠋⠐⠐⠰⣔⠃⢀⠦⠄⠈⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⢀⣀⠀⠀⢀⡀
⢰⣿⠂⠀⠀⠉⠁⢂⡠⠤⠀⢄⣀⡀⡀⠱⣿⣷⠀⠀⠀⠀⣠⡴⣥⣀⡀⠀⢿⠄⠀⠀⣠⠶⠶⠖⠟⠃⠀⠉⠉⠙⠋⠉⠉⠹⠓⠂⠴⠢⠄⡰⠂
⢸⣇⠀⠀⠀⠂⠞⠀⠀⠀⡠⠄⠉⠉⠉⠻⣿⣿⡀⢀⡴⠋⢀⢴⣿⢏⡁⠖⢸⠆⣤⠞⠉⠀⡁⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⣀⣀⡤⠖⠁⢠⠃
⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢈⣿⡷⠋⡾⠋⢀⣾⣵⣾⡇⣿⣯⡼⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⣦⣴⠟⠋⠅⠀⠀⠀⡀
⠈⢧⣄⠀⠀⠀⠐⠂⠚⡙⣿⣆⡂⠠⠀⡂⢨⣿⣧⠞⣁⠴⠛⣱⡟⠉⣟⠻⢏⠀⣀⠀⠀⠀⠀⠀⠈⠑⡀⣀⣶⠿⠟⠛⠂⠀⠀⠀⠀⢰⡇
⠀⠈⢧⡀⠀⠀⢸⡶⢾⡿⠻⣌⠛⢓⢦⣼⡟⠉⠀⠈⠀⠀⣴⣿⡄⢠⠌⠀⢰⡆⠀⠀⢀⠀⠀⠀⠀⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠁⠠⠈⠁⠀⠈⠀⠋⡙⣲⣤⣽⣿⣿⠀⠀⠀⠀⠐⠁⠀⣴⡇⣰⣆⣸⣀⠀⠀⠀⢑⠄⣠⣾⠟⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢘⡷
⠀⠀⠀⠀⣤⠀⡄⠀⠀⣀⣠⠿⠗⠀⠀⢻⡟⡀⠀⠀⠀⠀⣠⣾⣿⣶⣿⢿⣿⡿⣀⣴⠾⣛⣿⣿⣿⡦⣄⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⢛⠊⠑⢀⡀⠀⢹⣆⣀⡀⠠⣼⣷⣆⠀⠀⠀⣈⣩⣭⣭⡿⢷⣾⣿⣿⢋⣴⠛⠉⠉⠀⠉⠁⠘⢿⡶⣤⠀⠀⠀⠀⠀⠀⢀⡷
⠠⠄⣀⠐⢼⣄⣷⣿⣟⣛⣛⢛⡛⣻⠿⣿⣟⢛⠛⣿⣿⠋⢀⡾⠋⠈⠀⠈⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⡛⠳⠆⠀⠀⠀⣠⡏
⠀⠀⠀⠋⠀⠹⢿⣿⠻⠿⢿⣿⣿⠁⢀⡟⠋⠙⢻⣿⣯⡖⠁⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⢀⡠⠖⠛
⠀⠀⠀⠠⠚⠀⠜⠋⠁⠲⢿⣿⣷⢀⡾⠓⠀⢀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣤⣤⣠⣴⣿⣿⣿⣋⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣇⠀⢀⣤⣿⡿⣿⣿⣼⣄⠀⠀⠀⢀⡔⡺⢿⣿⣿⣿⣿⡿⣫⣿⣿⡿⠏⠀⠈⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⢑⣿⠟⠛⠁⠈⠑⠉⢻⣿⣦⣤⠾⠒⣀⣀⡉⠉⠛⠛⣟⢿⣿⡇⠀⠀⠀⠀⠈⢸⡆⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡶⢛⠋⠁⠀⠂⣀⠀⠀⠀⢿⢉⡴⠛⠛⠛⠻⣍⢢⡀⠀⣿⡗⢸⣷⠄⠀⠀⠀⠀⣴⣿⣠⠀⠀⡐⠀⢀⣠⣶⡿⠋⠀⠀⠸
⠀⠀⠀⠀⣀⠄⠒⠂⠀⠈⠉⠒⢝⢻⣿⣧⣀⠀⠀⠀⠀⠀⠀⠰⢾⣿⠆⠀⠀⠀⠀⠈⠇⢹⡀⠸⣧⣾⣿⣶⣦⣤⣤⣼⣿⣾⣧⠅⠀⠁⠀⠈⠉⠀⠀⠀⠀⣰⣿
⠀⠈⡍⠁⢤⣴⣶⣶⣤⠤⠀⣀⣼⣾⣿⣿⣯⡀⠀⠀⢀⣀⣀⣤⣿⣿⣄⠀⠀⠀⠀⢄⠘⣺⡇⠀⠙⠿⣿⣿⣿⣭⣭⣿⠿⠋⠀⠀⠈⠠⡀⠀⠀⢀⣀⠔⣺⣿⠏
⠀⠀⠷⣀⠀⠀⠀⢀⣀⣤⣶⣿⠟⡇⠻⣿⣿⣿⣷⣿⣿⣿⡿⠛⠙⢿⣿⡄⠀⠀⣀⠀⣀⣿⠃⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠶⠞⠚⠉
⠀⠀⠐⡀⠈⠉⠉⠉⠻⠋⣹⣿⡾⠁⠀⠀⠉⠉⠛⠛⠋⠁⠀⠀⠀⠀⠙⠻⠿⠦⠼⠿⠛⠒
⠀⠀⠀⠈⠒⣶⣤⣦⣴⡞⠉⠉
⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⢠⠄⡄⠐⠄⠀⠀⠀⠀⠲⢾⣶⣤⣀⣀⡀⠑⠒⣤⣄⣰⠷⠀⠀⢀⣤⠀⡄⠀⠀⠀⠂⡆⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠚⠀⣁⡀⠀⠀⢀⡀⢤⣘⡛⠿⠯⠈⢷⠇⣀⠾⠋⣉⢀⣶⣭⣀⡏⠂⡁⢀⠀⠀⠂⠢⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⢠⠠⠄⠀⠀⢉⡆⠉⠓⣶⠀⠀⠡⠞⠁⠀⠀⠀⠀⠴⡶⢫⣧⠀⠅⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠐⠈⡆⡀⠙⠉⣩⠶⠛⠋⡉⢉⣿⣶⡟⣴⢿⠀⠎⡐⠊⣤⢠⡴⢤⣄⠀⢀⠉⠛⠐⢀⠀⠀⠀⡀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⢀⣴⡯⢂⡠⣴⣚⣿⣿⣭⣥⡾⢃⣿⢠⡤⣀⣀⣀⠈⢡⣤⣦⣤⣄⡄⢀⠀⠀⠀⠀⠀⡅⢄⠀⠀⠀⡇⡇
⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣴⡟⡡⠂⣷⣾⣋⣍⣀⣀⣀⠈⠀⠚⠿⢋⣤⣬⣭⣥⢔⠲⠮⡙⢿⠂⠀⠀⠘⢀⠀⠀⠀⠀⢸⠈⠀⠀⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢸⡏⡌⢰⣤⠞⠛⣛⠛⠛⠻⠿⢿⣶⣶⡖⠙⠯⣽⣿⣷⣯⡓⢦⡈⡌⠆⠀⢰⠀⠈⠀⠀⠀⠀⠐⠀⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⣷⡿⠁⠠⣤⣭⣽⣿⣶⣦⣤⣤⣶⠀⠀⡠⣦⢙⢟⠆⣻⡆⡇⢠⠀⠀⠂⠰⠀⠙⠀⠈⠀⠀⠀⠈⠀⠈⡀
⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠀⣿⣇⣠⡞⣽⠿⠷⠮⠉⡻⣮⢹⣿⠀⠀⢹⡿⣾⣡⣾⡿⢋⡀⠀⠀⡈⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⡦
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢇⣳⠙⠿⣿⠾⣣⠆⠰⠦⠀⡀⣽⡆⢧⠀⠀⡎⢣⡬⠋⡀⠐⠉⡡⠂⡴⠁⠀⠀⠘⠀⠤⠀⠀⠀⠀⠀⠀⠱⢇
⠀⠃⠀⠠⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⠚⢯⣀⠀⠀⢀⡴⠋⠀⠀⠐⠈⠚⢿⡲⢦⡀⠹⣠⡤⣎⡄⢁⣤⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⢸
⠀⢦⠀⠀⠃⠄⠀⠀⢸⠀⠀⢃⠀⠀⠀⠀⠀⠉⣉⣀⣀⣀⣒⣤⣶⠎⡠⠄⡉⠀⠘⢦⠙⣿⣣⣄⠃⠀⠀⠸⣯⡀⠀⠀⠀⠘⠂⢀⠀⢀⠁⠃
⠀⠘⠀⠀⠸⠑⠀⠀⠀⡂⢁⠘⡀⠀⢁⡇⢨⣇⡀⠈⣽⢋⡔⢿⠉⢰⢴⡃⠀⠄⠂⡈⢳⡘⠇⠹⣄⠐⠀⠀⠈⠉⠒⠆⠀⠀⣤⢼⠀⠀⠀⠈
⠀⠀⠂⠀⠀⠂⠀⠀⠀⠰⠀⠀⢇⠀⢢⠁⢄⡹⢋⠔⣡⢛⣀⣉⣢⣄⠷⢌⢢⡀⡀⠙⠈⣿⡞⣦⣍⢉⣙⠲⢦⣄⠀⠀⠡⠀⢉⠈⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⢧⠸⡄⢸⡇⠎⣴⣁⣤⠴⣯⢌⠭⣭⡉⢷⣿⠀⣿⣺⣮⠀⡥⡿⠄⢀⢷⠚⣿⢦⣿⣆⠀⠐⢆⠈⠀⠀⠀⠀⠀⠂
⢠⢠⠠⡀⠀⠰⠀⠀⠀⠀⠰⠘⡀⢻⠁⡄⠢⣭⠁⠈⠞⠑⣦⠚⠉⢸⣎⣿⠀⣛⣧⠠⢥⡇⠙⠂⠠⣳⣾⣿⣾⠻⣿⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⢀
⠸⠆⡆⢇⠁⠀⠃⠀⠀⠀⠄⢄⠁⠈⣦⣄⠃⠀⠀⢀⢼⣆⠀⡟⠳⡌⢷⡸⡄⠻⢮⡓⢻⢟⣁⡤⠚⠁⠀⠈⢻⣷⣽⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠃⢠⢰⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠈⠙⠓⢶⠖⣡⡄⠁⣼⠇⠀⠈⠀⠉⠂⠰⠶⠾⡛⠛⠉⣀⣤⣤⡀⠀⠀⡇⣸⠇⠀⢠⠀⠀⠀⢦⠀⠀⠁⠀⠲
⠠⠰⠀⠀⡇⠀⢠⡓⢤⡀⠠⠀⠀⠀⠀⠈⠱⠌⢫⣰⢧⣄⣿⣧⣠⠀⢀⡀⢀⡴⢂⢤⣭⣭⣉⠉⠙⣻⣿⣆⣀⣧⣿⠀⠀⢸⠀⠀⠆⠠
⠀⠀⠇⡀⢸⡈⠢⡁⠦⠉⣦⢁⠀⠀⠀⡇⢰⣾⠋⠉⠛⠠⡟⣸⣿⢧⣮⣗⣋⣐⣶⠶⠿⣿⡏⠻⢿⣯⣠⣭⣞⡿⣵⠀⠀⠸⣄⠀⠀⠐⡃
⠀⢠⠰⠸⠀⢳⠀⠃⠀⠀⠏⠀⠀⠀⠀⠸⣿⣌⣩⣭⣵⠀⠘⢻⣟⢠⣄⣻⣿⡷⢭⣶⡺⠛⣧⢠⣄⡀⠙⠷⠿⠟⡁⠀⠀⠀⠉⣦⠈⠁⠰
⠀⠈⡈⠃⠃⢸⠅⠀⠀⠀⠀⠀⢀⡀⣀⡀⠨⢉⡛⢳⣿⠃⡻⡾⣧⣄⠘⢿⡋⢭⡛⠻⣿⠈⡄⡀⠀⠸⡀⠀⠀⠂⠠⠀⠀⠀⠀⢸⠀⢠⠀⡀⠀⠈
⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⢀⡞⡉⠛⡵⣿⣷⣮⠶⠿⢑⣘⠈⠀⠋⣻⣻⣿⡇⠈⠃⣠⡟⠳⠿⠧⢦⡖⣇⡀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠠⠄⠁
⠀⠀⠐⠀⡀⠀⠆⠀⠀⠀⢸⠁⣱⠇⣘⠛⢓⡲⢂⡞⠋⣉⣀⣀⠽⢟⣽⣿⣄⠀⠀⠉⠁⠀⣀⣴⠞⠀⣨⡿⠀⠈⠀⠀⠀⠀⠀⠀⢳⠀⠲⠘
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡄⠘⣼⣟⠻⣿⣄⣘⣛⣋⣁⣤⣶⣛⣩⣿⣯⣿⡆⠈⢀⣄⣠⣤⣴⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠆
⠀⠀⠀⢡⠀⠐⠀⠀⠀⠀⠀⠀⠳⣿⡪⠙⡋⠛⠛⠛⢛⣛⣭⣭⠭⠭⠭⠤⡹⣷⣤⣀⠉⠛⠛⠿⡶⠴⠖⠚⠻⣦⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⢄
⠀⠀⠀⠈⠀⡀⠀⢀⠀⠀⠀⠀⠀⠙⢯⡦⡈⠉⠳⣦⣄⠉⠉⢿⣿⣿⣷⠀⠐⣌⡙⠛⠻⠿⣿⣿⣿⣿⣿⡀⠀⣛⠀⠀⠀⠀⠀⠠⠀⠸⠄⠀⠘⡆⠳
⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠡⠷⡄⠠⠆⢻⠿⠇⠀⠺⣾⡿⠏⠀⠀⠘⠻⠷⢦⠀⠀⠹⠏⠻⠸⡇⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⢠⣿⣧⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣌⣿⣿⡿⣀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢰⣶⠿⣸⣿⣿⣯⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢌⣱⡞⢿⠛⠟⢻⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠐⡚⠛⣋⠩⣴⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠛⢷⣶⣴⣾⠖⢠⣼⢹⡇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠐⢴⠃⢌⣭⡾⠷⢺⣛⣱⡿⠈⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⡕⠊⠉⠀⠀⢘⣠⡞⢁⣀⣤⡾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠉⠉⠙⠲⠖⠒⠉⠙⠛⠿⠛⠉⢶⣶⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠆⠐⠐⠄⢠⠀⢀⡀⠀⡀⠀⡠⣦⡔⢾⢙⣧⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣠⠀⣆⢔⣀⣀⡆⠀⡇⣰⡆⠹⠳⡿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⢇⢻⡀⣄⠙⣿⣀⡿⢁⡗⣾⢦⣷⣼⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢼⡾⡏⣭⠓⢿⣀⣧⣾⡧⣼⣾⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣧⣷⢻⣧⣼⡍⣿⣸⣏⣿⣹⣿⣿⡗⣥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢼⣧⣹⣟⣿⣻⣿⣿⣿⣿⡿⠽⣋⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⡗⠻⢭⠄⠊⠀⠈



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

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

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

⠀⠀⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣘⠃
⠈⠀⠀⠘⠄⠀⠀⠀⠀⣀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠌⠀⠔
⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⡀⠀⡆⠀⣂⠀⠀⠀⠀⠀⠈⠱⠄⠀⠀⠀⠗⠀⠀⠈⠓
⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡠⢃⠀⠁⠀⠻⣤⠀⠀⠀⠀⠠⠀⠀⠰⠄⠀⠰⣄⠀⠀⠀⠘
⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠉⠁⠀⠀⠀⠀⠉⠳⢄⣀⡀⠀⠀⠀⠒⠂⠀⠀⠈⠽⠠⠄
⢀⡔⠢⠄⢤⣀⠀⠀⠀⠙⠁⣀⡤⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠂
⠋⠀⠀⠀⢀⣿⠆⠀⡴⠚⠉⠀⠀⢠⠏⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠐⠄⠀⠀⠈
⢷⣤⣤⣤⠾⠋⢀⡎⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠂⠀⠀⡐⠀⠄
⠀⠀⠀⢀⣤⠖⠛⠁⠀⠀⠀⢀⡤⠒⠀⠀⠀⠀⠀⢠⠶⠛⠛⠒⠠⠂⠠⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⢰⠋⠁⠀⠀⠀⠀⠐⠀⡸⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠘⠆⠐⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠸⣄⡠⠄⠂⠀⠀⠀⠀⠀⠒⠀⠀⠀⢼⠁⠀⠀⠀⠀⠄⣀⡀⠔⠁⠀⠀⠀⠂⠀⠀⠌⠁⠀⠈⠀⡀⠈⠂
⠤⠄⠀⠀⠀⢈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠐⠀⠀⠀⠀⠀⠁⠀⢂⠄⠀⣢⡷⠀⠐⠏⠀⢀⣠⢄⠔⠀⠀⠀⠀⠀⠠⠔⠂⠀⠈⠀⠀⠀⠠⠄⠀⠀⢰⡇⠈⠙⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠖⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⣀⠔⠋⠈⢷⡀
⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠙⢦⡀⢀⠀⠂
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⡁
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠀⠈⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠳⣄
⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤
⢁⡀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠅⠀⠀⠀⠠⠂⠠⠍⠁⠀⠚⠛⠃
⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠈⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠈⠈
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠊
⠀⠈⣆⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠛⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠉⠁⠀⠀⠹⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡤⣤⣄⣄⠀⢠⠀⣀⡼⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⣄⣀⡀⡀⢀⢠⢀⣼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠉⠉⠛⠋⠉⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠉⠙⠛⠚⠛⠋⠀⠚⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠛⠓⠶⠶⠴⠤⣤⣦⣦⡤⠼⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⠷⢶⣴⣴⣴⣠⣀⣆⣄⣶⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢸⣸⣠⢀⢀⡃⣀⡰⠈⠉⠉⠩⠅⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠯⠍⠉⠉⠛⠛⠛⠳⠶⡖⣴⣴⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⢸⣯⡝⠛⠛⠛⠿⠿⠿⢿⣷⣶⣶⣶⣶⠏
⠀⠀⠀⠀⠀⠀⠀⢼⣁⠀⢀⡀⠀⠀⠀⠀⠸⣅⣸⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣾⠍⠀⢀⣀⠀⠀⠀⠀⢸⣇⣿⠀⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣉⡽⢿⡚⠳⢿⡵⣦⣤⣄⣒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠛⠛⠻⠿⠿⠿⣶⣶⣯⣁⣀⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢙⣄⡤⠷⡖⠛⠻⣅⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⢀⠤⠄⠀⠢⠈⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⢉⣧⡤⠞⢾⠉⣙⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠉⢀⠀⠐⠔⠀⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⡉⢀⣳⣤⠴⠷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⢀⡀⠀⢤⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠛⢯⠁⣘⣦⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠒⠀⠁⠀⣐⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡖⠚⠻⣅⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠢⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⣦⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠐⢀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣤⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢹⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⠿⠿⠛⠿⣿⣿⣿⠋⠉⢹⣿⣿⡿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⢻⣿⡿⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⠀⠀⢹⣿⣿⣿⣿
⠿⠛⠛⠛⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⣼⣿⡇⠀⠀⠈⠛⠟⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠐⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠉⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢿⣿⣿⠏⠁⠀⠀⠈⠙⣿⡿⠁
⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⣿⠟⠛⠻⣿⣿⠁⠀⠈⣿⣿⣿⣿⠛⠉⠙⠻⣿⣿⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⣹⣿⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⣼⣿⣿
⠀⠀⢠⣿⣿⡇⠀⢰⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠹⠿⠟
⠀⠀⣸⣿⣿⡇⠀⠘⠻⠟⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⣿⣿⣿⡿⠃
⠀⠀⠘⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡛⠘⢏⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠡⠀⠈⠙⠢⢤⣀⠰⠜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢱⣄⠀⠀⠀⠙⣷⣾⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⣠⡆⢀⠤⠀⣐⣈⡀⠐⢿⡇⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠰⠖⠉⠒⠓⠳⢰⠋⠉⣩⡶⠛⠁⠙⢦⠀⠀⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠃⠀⠀⠀⠀⢨⣷⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠞⠃⠀⣼⠀⠀⠀⠀⠀⡸⠃⢸⡥⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠞⢃⣀⣀⣤⠶⠛⠁⠀⣠⣵⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣴⠖⠲⠤⠔⠒⠒⠒⠐⠂⣀⡤⠖⠛⠉⠉⠁⠀⢀⣀⣠⣤⣿⣷⣟⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠉⠢⣄⡀⠀⠀⠀⣠⣴⠟⠁⠀⠀⢀⣀⣤⣴⣾⡿⢟⡭⠏⠹⡋⣋⢱⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠻⠟⠗⠖⠻⠟⠻⠋⠉⠁⢀⠈⢰⣞⣴⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⡦⣤⣄⣠⡄⠀⣀⠀⣀⢀⣀⣀⣁⣘⢠⣂⣠⣤⣼⣾⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢣⢹⡄⢹⡟⠈⢿⠛⣿⠛⢻⡟⢻⠟⢻⡟⢻⡏⢩⣇⣸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡎⣧⠀⣿⠀⢸⡀⢿⠀⣸⡁⢸⠃⢼⠀⢺⠇⢺⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⣸⡆⢿⠂⠸⡇⢸⢢⣼⡇⢸⠂⣿⡀⣿⢓⣿⣿⣿⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣿⣇⢺⣧⠀⣷⢾⡧⣿⣧⣿⢲⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⡸⣿⣿⡧⣿⡎⣿⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣿⣿⣿⣧⣿⣿⣷⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣶⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣾⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢟⡛⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠿⠟⠻⠛⠛⠙⠉⠉⠉⠁



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

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

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

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

⠀⠠⣤⢤⣤⣤⢠⡄⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⢠⣤⣤⣤⣤⣤
⠀⠀⠘⢠⡇⠁⠀⠀⠀⠀⠀⠀⠈⠁⣯⢀⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⣹⢻⠉⠁⠹⠿⠉⠉⢨⡇⠀⠀⠀⠈⠀⠉⣿⠉
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠉⠻⠿⡿⡟⠛⠛⣿⡿⠻⠟⠀⠘⠉⡏⢹⠉⠛⠙⠏⠉⠁⠀⠀⠉⠉⣛⠉⡛⠙⡋⠛⠋⠀⠀⠀⠉⡍⠋⠉⡏⡟⡏⠃
⠀⠀⠀⠀⠃⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠐⠇

⠀⠲⠶⢶⡶⢶⠶⣶⢶⡶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠖⠀⠲⣶⢶⠶⡶⡶⠶⢶⠆⠀⠀⠐⠶⠶⢶⡶⣶⠶⠖⠖
⠀⠀⠀⠀⠀⢸⡀⠿⠀⠀⢀⡇⠈⠁⠙⠿⠁⠀⠀⡏⣹⠈⠁⠀⠀⠀⠀⣾⠀⡇⠃⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠈⠁⢙⠉⠉⢀⡏⢙⠈⠁⠀⠀⠀⠈⢩⡇⠿⠈⠁⣹⢸⡅⠉⠁⠛⠀⠁⠀⠀⠈⠁⡇⠺⠀⠀⠀⠀⠀⠛⠉⡏⠈⠁
⠀⠀⠀⠈⠀⠀⠈⠁⠛⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠢⠤⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢰⣤⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣿⡏⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛⠉⢲⠇⣼⢸⠿⡂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡩⢷⡰⠀⠂⢐⣿⢸⡆⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣼⠁⠀⡹⠁⠀⢾⣿⠈⡆⠀⠠⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢐⢺⡇⠀⠀⡇⠀⠀⠘⣿⠀⢿⣄⠀⠊⠉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠄⠐⣄⠀⠳⣄⠀⠙⠹⣧⡈⢿⣼⣄⣀⡼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⢠⣶⠶⢈⡉⠒⠺⣟⣶⣶⠶⢷⣆⡈⠃⠒⠒⠺⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠘⠿⣼⠁⠀⠀⢀⢀⡀⠈⠉⠛⠁⠠⠤⢀⡀⠀⠀⠒⠊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢘⢧⣄⠈⠢⠤⡔⠋⠀⠙⠲⢄⡀⠀⠀⠀⠀⠉⠲⣏⣿⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⢀⠉⠙⠲⠤⢤⣤⣀⣈⢙⡛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⣀⣀⠙⠓⠾⣦⣠⡆⡀⠈⢉⠑⡒⠂⠢⠤⠈⠉⢉⣡⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠛⣿⡦⠤⣤⣭⣭⣿⣷⣦⣤⣤⣤⡴⠶⠞⣮⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⡇⠀⡇⠀⣿⠀⣾⠉⢹⠇⠀⡿⠀⢸⡇⣼⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⣿⠀⢁⠀⣿⠀⣿⠀⢸⠀⢸⡇⢀⣿⢠⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹⡇⠘⠀⢹⠀⡿⠀⡼⠀⣼⡅⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⢧⠀⠀⢸⠀⣷⢠⣧⣰⣿⢡⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣻⡆⠓⢺⣶⡿⢸⡟⣿⣿⣿⡿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⠋⠓⠚⠚⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⠃⢀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠁⠀⢀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠋⠀⠀⡀⠈⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠘⠂⠀⣤⡀⠈⠁⠀⢀⣀⠀⡗⢄
⠀⠀⠀⠀⠀⢠⠎⠀⠑⠄⣀⣀⣀⣀⡀⠀⠄⣀⡀⣉⠰⣡⢀⠦⡀
⠀⠀⠀⠀⠰⠁⢄⠐⠃⠀⠀⠀⠒⠀⠀⠠⠤⠀⠀⠀⢀⠤⢊⣀⠈⢆
⠀⠀⠀⠀⡆⠀⢀⠉⠒⠀⠤⠀⠀⠀⠀⠤⠄⠀⠒⠈⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠈⠁⠀⠀⠋⠀⠀⠀⠐⠒⠀⠀⠘⠁⠀⠉⢁⡀⡴⠁
⠀⠀⠀⠀⠀⠸⠈⠉⠀⠀⠉⠉⠀⠉⠉⠀⠉⠁⠀⠉⠁⠉⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠩⠶⠦⠀⠀⠀⠀⠤⠶⠭⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠢⠴⠃⠀⠀⣀⢄⢶⠋⢉⠔⠖⢄⣀⡀⠤⠐⠈⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡤⠠⠤⠄⠀⢀⠔⢠⢂⡆⢀⠊⠘⠀⠱⠱⡤⠄⠒⠁⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣀⠧⠀⠀⠀⠤⡜⠀⠂⢻⠃⣬⡀⠿⠆⠀⠀⡇⠠⠤⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⢄⢰⠀⠸⠈⣻⠏⢀⠀⡜⡰⠁⠒⠠⢄⣀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠈⢀⡠⠘⢢⡀⠀⠄⠰⡀⠈⣎⠔⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠒⠁⠀⠀⠀⠈⠑⠚⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢢⠀⠀⠀⢀⣠⡴⣖⠒⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢣⣀⡴⠉⠈⠲⠚⠀⠀⡀⠀⠀⡬⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠉⠁⠀⠀⣏⣄⠀⠀⠀⠀⠀⣏⢈⡇⠀⠳⠤⠏⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⣇⣨⠃⠀⡠⠤⡀⠀⠉⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⢼⠀⠀⠀⠀⢧⣀⡜⠀⠀⣀⣀⠀⠀⣀⣀⣇⣀⠤⠖⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣤⣄⣀⠀⠀⠀⠀⠀⡰⡆⠀⠀⠀⢧⡔⢲⠀⠀⠀⠀⠀⠰⡇⠀⡇⢰⡁⣸⠇⠀⠀⠰⡃
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠧⣀⠽⠈⠑⢦⡀⠀⡴⠁⡇⠀⠀⠀⠈⠳⡚⠀⠀⠀⠀⢀⡀⠈⠉⠀⠀⢉⡟⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⢀⡼⠓⡆⠀⣠⠒⢢⠀⠀⠀⠀⠀⢀⣱⡞⠀⠀⠈⠒⢹⠂⠀⠀⠉⠢⣄⡀⠐⣅⢈⠆⢀⣠⠔⠋⠙⠢⣀⠀⢸⠉
⠀⠀⠀⠀⠀⡞⠈⠋⠀⠀⠘⠒⠋⠀⠀⠀⠀⠀⡏⠈⣧⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠓⠼⠂
⠸⡓⠤⣀⣰⠃⠀⠀⠀⢀⡤⢤⡀⠀⢠⠎⠉⡆⠈⠉⢹⠤⠤⢤⣀⠬⠇
⠀⠙⡄⠀⠘⣶⠙⡆⠀⠸⣄⣠⠃⠀⠀⠓⠒⠁⠀⢀⠇
⡴⠚⠀⠀⠀⠹⡒⠃⠀⠀⠀⠀⠀⠀⠀⡠⠶⡄⢀⠎⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠒⠒⢦⣄⡀
⠈⠙⡆⠀⢀⡠⠛⢦⣰⡋⢳⠀⠀⠀⠀⠳⣤⠟⣁⠀⠀⠀⠀⢠⢾⠋⡲⠀⠀⣀⡀⠐⢇⣈⡏⠢⡀⠀⠀⠀⠀⢀⡀
⠀⠀⡧⠔⠋⠀⠀⠀⠈⠙⠓⠦⠤⠒⠒⠉⠀⠀⢹⠑⢄⠀⣰⠃⠀⠉⠀⠀⢸⣀⡹⠀⠀⠀⠀⠀⠘⡄⠀⢀⠔⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠙⣇⣄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⣹⠞⠁⠀⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⠁⠀⠀⠀⣿⢀⠇⠀⢸⠁⢙⠄⠀⢠⠖⠲⡄⢸⣀⣼⠀⠀⠀⠀⢉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⣧⠉⠀⠀⠈⠉⠉⠀⠀⠘⠦⠴⠃⠀⠀⣾⣀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠒⠋⠁⠈⠣⣴⠉⡆⠀⠀⠀⠀⡠⢤⠀⠀⢀⠞⠀⠀⠉⠓⠲⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢅⣀⠀⠀⠘⢦⣠⡧⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⠁
⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀
⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉⠉⠈⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣦
⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠋⠙⠛⠓⠦⣄⡀
⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⣠⡟⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣰⡏⠀⢀⡎⠉⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠈⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⣿⣷⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣿⣿⡇⠀⣸⡟⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠀⣰⡟⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⣼⠏⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠤⠤⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⢸⠇⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⢸⡄⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⠒⠀⢀⠀⠀⠆⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠂⠀⢀⡟⠳⡄⠀⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡴⢻⠆⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣻⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠒⠒⠦⣄⣀⢰⢷⣃⡤⠖⠒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡞⠛⠒⠭⣽⢸⡥⠖⠛⠛⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⠦⣀⡀⠉⠉⢁⣀⡤⠞⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢸⠀⡦⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢠⡜⠶⠃⠀⠀⠀⠀⠀⢳⡀⣸⠐⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡤⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⠉⠁⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⢋⠆⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣠⠞⠉⣉⣹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡁⠀⠀⠀⡰⠃⢀⠞⠁⠘⣎⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠷⣌⣓⠒⠦⠤⢤⣄⣀⣀⣀⣀⡤⠤⠴⠒⣋⡴⠳⠤⠚⠉⠳⢠⠏⠀⠀⠀⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⠄⠀⠀⣿⢭⠙⠛⠒⠂⠀⠆⠀⠀⠀⠀⠚⠛⢉⣹⡇⠀⠀⠸⣶⡀⣾⠀⣀⣀⠼⣡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠙⢮⣀⠀⠰⣍⡛⣒⠒⠆⠰⠀⠠⠤⠤⠄⠀⠀⠴⠖⢚⡽⠀⠀⣚⡵⢃⡼⠀⠓⢦⣸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣉⡓⠒⠶⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⡤⠬⠥⠶⠒⢚⣉⡥⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡗⠒⠒⠒⠶⠤⠤⠤⠤⠤⠶⠒⠒⢲⣋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠭⢭⣀⣀⣀⣀⣀⣀⣀⣘⣀⣛⣉⡭⠽⠃

⣿⠿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⣿⣟⣻⣿

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


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

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

⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠙⠛⠛⠁⠀⢀⠀⠀⠀⠀⠘⠑⠛⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠘⠙⠛
⠀⡀⣀⢂⠀⠀⠀⠁⠀⠀⠀⠀⢠⣠⠄⢙⡇⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢂⠀⠀⠈⠁⠀⠀⠀⠀⢠⣀⠄⢙⡇⠀⠀⠀⠐⣶
⠀⠱⠂⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡄⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠈⠓⠇⠀⠀⠀⠻⠲⡰⠦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠓⠇⠀⠀⠀⠻⠲⡰⠶⠀⠀⠀⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃⠢⠤⠴⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠦⠤⠴
⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠂⠈⠁⠀⢀⡀⠠⠀⠀⠀⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠂⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠂⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢤⠤⠤⣄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠁⡁⠀⠀⠀⠀⠀⠀⢐⠀⠀⡀⠀⢀⢤⠤⠤⣄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠋⠁⠀⡋⢠⠂⠂⠀⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠁⠀⡋⢠⠒⠂⠀⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡠⠃
⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢠⢚⠶⢀⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢰⠾⠶
⠀⠀⠀⠸⣦⣀⢀⣡⠟⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠠⠀⠐⠀⠈⠁⠀⢸⠁⠀⠀⠸⣦⣀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠠⠀⠂⠀⠈⠁
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠘⣿⣧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠀⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠉⠁⠀⠐⠀⠀⠸⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⠤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠘⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⣤⢤⣼⠁
⠀⠀⠀⡀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠡⠆⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⣀
⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣡⠰⠇⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣡⡰⠆⠀⠀⠀⠀⠟
⠀⠀⠀⠑⠀⠀⠀⠠⣄⡄⠀⠀⠀⡯⣑⠀⣘⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⢀⣄⡄⠀⠀⠀⣯⣁⠀⣘⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠐⢦⣀⡀⠁⠀⠀⠀⠈⠈⠘⠁⠀⠀⠤⡄⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡀⠁⠀⠀⠀⠈⠈⠊⠉⠀⠀⠤⡄⠀⠃⢀
⠶⠀⠄⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⢀⠄⠀⠀⠃⡀⠠⠍⠉⠁⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠠⠉⠉⠁
⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈
⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⠤⠆⠀⡴⠉⣉⡉⠃⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣀⠤⠇⠀⡴⠉⢉⡉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠄⠀⡅
⡆⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⢶⡽⠒⠀⠀⠀⠀⠘⠀⠤⠰⠊⠀⣠⣀⠀⠂⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⠶⠽⠖⠀⠀⠀⠀⠈⠀⠤⠰⠊⠀⣠⣀
⠃⠀⠀⢸⡀⠃⠀⠃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⣀⠀⠸⡄⠉⢰⠇⠀⠀⢸⡀⠃⠀⢃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠸⡔⠉
⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⢰⣿⡇⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⠀⣿⡇⠀⠀⠀⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢿⠟
⠀⠀⠀⠤⠄⠀⢀⣀⠀⢸⣶⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠂⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⢰⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⡿⢤⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⢧⡄
⠀⠀⠀⠀⠀⢀⡠⢲⠖⡲⠛⣯⣪⣫⠇⠀⠀⢀⠤⠒⠒⠒⠃⡀⠀⣈⠆⠀⠀⢀⠤⠖⠒⠺⢪⣀⣀⣹⠆
⠀⠀⠀⠀⢠⠏⡰⢁⠞⢀⠔⡇⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡔⠍⡀⠀⠀⠀⢰⠇
⠀⠀⠀⣀⡟⠰⢡⠋⡠⢃⡾⠁⠀⠀⠀⢀⡁⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢐⡃⠅⡂⠀⠀⢠⡜
⠀⢬⢕⠟⡹⡧⠦⠴⠕⠋⠀⠀⠀⡎⠁⠀⠁⡤⠤⠤⠔⠋⠀⠀⠀⡎⠉⠀⢑⠤⠦⠤⠒⠉
⠀⠀⠣⣴⣱⠇⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠧⣔⡸
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠃⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⢾⣄
⠀⠀⠀⠀⠀⠀⠀⣰⢻⠻⡐⠀⠀⠀⢠⡟⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠰⣟⣲⠿⣋⣩⡝
⠀⠀⠀⠀⠀⠀⠀⠉⣥⣀⣤⡀⣤⣤⣠⠁⠀⣠⡮⣭⡁⠀⠘⡆⠀⢸⣿⣿⣫⣭⣽⡿⣻⡆
⠀⠀⡠⠒⠚⠓⠶⣄⠉⣿⣏⠉⣉⣿⠟⢠⠞⠉⠀⣠⠼⠶⠴⢧⡀⠐⣿⣷⣶⣶⣶⣿⣉⠀⠀⣸⢷⣤⡀
⠀⢸⡁⠀⠀⠀⠀⣽⡆⢻⣿⠿⣿⣿⠀⣮⠁⠀⣴⠃⠀⠀⠀⠀⠨⡀⢻⣞⣽⣟⣩⣿⠇⠀⢸⡼⣊⡿⢽⣄
⠀⠀⠇⠀⠀⠀⠖⢾⡀⢸⣿⡟⣿⡟⠀⠀⡏⣶⠻⡄⠀⠀⠀⠀⢸⡇⠀⣿⣯⣭⣿⡏⠀⢲⡟⣓⢻⡆⣽⣷⡆
⠀⠀⢰⣠⢀⣶⣤⡞⠁⠈⣷⡟⣿⠇⠀⠈⠳⣿⣤⢈⣖⠒⢘⣿⡿⠀⠀⢸⣟⠟⣿⡇⠀⠙⣷⣷⣦⣭⣽⣶⠃
⠀⠀⠘⣭⢻⣽⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠋⠀⠀⠀⠘⣯⡿⣿⠀⠀⠀⢻⣿⡿⣷⠺⡟
⠀⠀⠀⢆⣹⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣉⡏⠀⠀⠀⠀⠀⢿⠿⣿⠀⠀⠀⠸⣾⡷⢿⢿⠇
⠀⠀⠀⢸⠷⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣶⢛⠃⠀⠀⠀⠀⠀⢸⣶⡏⠀⠀⠀⠀⢹⣿⢾⡟
⠀⠀⠀⠀⢂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⡄⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠂⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡀⠀⢀⠀⠀⠁⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⠆⢠⠸⡄⠠⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⣿⠀⠀⢸⠀⠀⠀⢱⠀⠀⡀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⣀⠙⠉⠁⠞⠓⠲⡶⠧⢤⡾⠗⠋⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣉⣉⣉⠛⠛⠛⠒⠒⠒⠲⠶⠖⠚⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠞⣋⡡⠔⠒⠚⠿⠓⠲⢄⡀⠀⠀⠀⠀⢰⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠰⠚⠉⠁⠔⠂⠀⠀⠀⠀⡄⠈⠻⣄⠀⡆⢀⠏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠥⠚⠁⠀⠀⠀⠀⠴⠊⠉⠑⠒⠒⠊⠀⢀⡄⢹⡞⣠⠏⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠐⠉⠉⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⢀⣾⠿⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡉⠛⠒⠂⠀⠀⣀⡤⠔⠈⠉⠉⠙⠒⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣦⣤⣤⣤⡴⠶⠛⠛⠛⠛⠛⠋⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠤⢶⡶⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠟⠿⠿⢿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠉⠀⠀⠀⠀⠉⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢧⡀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠛⠛⠛⠛⠓⠶⣤⡀
⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⣰⡟⠁⠀⣠⠒⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣴⡟⠀⠀⢠⣧⣠⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⢹⣿⣿⡆⠀⠀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣿⣿⣿⠇⠀⣼⡟⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠏⠀⣼⡟⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣠⣾⠏⠀⠀⠀⠀⠀⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⢶⣤⣤⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣶⣤⣤⣴⣶⠾⠟⠋
⣿⣿⠿⠿⡿⠿⠿⡿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀
⠀⠀⠀⠀⠀⠀⡏⡇⠈⠙⣒⠤⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⡇⠑⠖⠋⠉⡷⠀⠀⠉⣉⡒⠲⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠀⠣⣤⠴⠋⠀⠱⣀⡠⠔⠒⢦⠉⠉⠙⣒⣶⠦⠤⠤⣄⣑⣄
⠀⠀⠀⠀⠀⠀⢧⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠑⠒⠉⠀⠀⠙⠒⠶⠤⠖⠛⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⢾⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠚⠉⠉⠉⠉⠉⠉⢑⣲⠋⠈⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠒⠢⠠⣄⠀⣠⠞⠁⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠉⠁⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠃⠤⣄⠀⠀⣠⣞⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠱⡀
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢏⠉⠏⠙⠛⠲⣀⠀⠀⠀⠀⠀⠀⣷⢦⡘⡄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣴⣖⠲⠖⠒⡶⢺⣷⡄⢧⡂⢀⡠⣴⣭⡱⣦⠀⠀⠀⣸⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠐⡅⢀⠼⠁⠚⠒⢄⠈⢔⣒⠟⠁⠈⠉⠢⡝⠉⣼⢓⣚⣦⠀⣄⣿⣴⣤⣤⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠿⡤⣄⣳⠀⢨⣗⡁⡀⢄⣤⣮⣴⡞⠀⠀⢿⣾⠻⣍⠓⠛⠛⠋⣼⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢩⡅⣼⣿⠀⠄⢿⣿⠤⣀⡘⣿⠟⠁⠰⠀⢸⠛⢋⡽⡇⠀⠀⣠⠿⢁⡾
⠀⠀⠀⠀⠀⠀⠀⠄⢍⢣⡘⣿⣤⣤⣦⣿⣿⣯⠁⡙⢧⣢⢔⠀⡎⢰⡿⣄⣧⢀⣴⠋⠀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡄⠙⠙⢛⠁⡄⣷⣹⣧⠈⠈⠉⠙⠾⠲⠋⠀⠈⢟⣛⠁⠀⡆⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⢰⣷⠛⠃⠘⠿⠹⠟⠀⠰⠿⠷⢶⣦⣵⣆⣜⡢⠞⢰⣆⣿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⡄⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⣹⢻⣿⣷⣶⣤⡟⢹⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡃⠸⣧⣁⣤⣨⣰⣅⡆⢡⢢⢠⣰⣇⣾⣿⣿⠏⡙⡇⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⣿⢻⡿⠿⡿⠛⢻⣿⣼⣿⢹⢿⠇⣧⠾⠁⢇⡇⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⡇⠀⡇⠀⢸⠹⢿⡟⠘⢘⠘⠟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡇⠀⢿⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⣀⠀⠀⡔⠋⠱⠤⢶⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢠⠒⣄⠀⠀⢀⠚⠠⠂⢀⠤⣄⠓⠛⢢⡑⢄⣆⠀⡀⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠑⣄⠀⡴⢶⠲⡒⠒⡖⡚⢲⠲⣆⣠⠙⠚⡗
⠀⠱⡤⡑⠼⠟⢍⡑⣄⡀⠘⠂⠊⠀⢤⡠⣵⠋⠁⢌⣀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢠⣍⣘⢃⠀⢣⢸⠀⢰⢀⠀⡇⣿⣁⠀⠘⢆
⢀⠞⠀⠀⠀⠀⣠⠞⠳⠥⠀⠀⠀⢇⣠⠇⢸⡧⣵⠀⢠⠒⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⢀⡀⢹⣤⠂⢸⢀⠀⢸⡄⢠⢇⡿⢹⣱⠒⠋
⢇⠀⢀⡴⢾⡉⠙⠢⣄⣔⡇⠈⡂⢀⠤⣤⠋⠧⣈⣁⡜⠀⠀⠀⠀⢀⡀⠤⡤⠤⠤⢤⣤⣤⣀⡀⠀⠈⠓⢻⠃⣧⠜⠋⠈⠒⠞⠚⠒⠛⠛⠉⠉⠀⠀⠉
⠀⠉⠉⠀⠀⠑⠤⠴⠛⠾⣌⣉⣠⡧⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⡎⠑⠠⠀⣌⡶⡷⠷⢶⡀⠤⣍⣣⠀⠀⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⡒⠲⠤⠤⡶⠤⠤⠼⠖⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⠀⠀⠀⡇⠀⠀⢀⡄⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠒⡩⠤⢕⢤
⠀⠀⠀⠀⠀⡰⠲⡋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠠⠤⠤⠧⠤⠤⠚⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢡⠀⡠⢀⠑⠤⠚⠀⣣
⠀⠀⠀⠀⠀⣧⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠒⠁⠀⠢⠴⠃⢀⣀⠸⣄⡆
⠀⠀⠀⠀⠀⢏⠳⣄⠈⠓⠤⣀⡀⠀⠀⢀⣰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⡄⠀⠐⢄⠸⣄⡨⠃⡠⠁
⠀⠀⠀⠀⠀⠘⢦⠀⠑⠢⠤⣀⣉⡉⢉⢉⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⣀⡴⠂⠈⠉⠑⠢⠤⣀⠀⠀⠈⠣⣆⠀⠁⡁⠄⣀⣰⠻⠥⣀
⠀⠀⠀⠀⠀⠀⠀⢹⠦⣀⡀⠀⠈⣉⡡⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠂⠁⠀⠒⠀⠤⣠⠀⠈⠁⠀⠀⠀⠀⣀⠄⠊⠉⡄⠀⠀⠀⠀⠈⠈⡑⢟⡉⢠⠻⡖⠊
⠀⠀⠀⠀⠀⠀⠀⡼⠀⡰⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣎⣁⡘⠊⢇⡀⠀⠀⠀⠂⠀⠀⠀⠉⠓⢂⠔⠊⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠉⠁⠈⠙⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣰⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠈⠉⢀⠿⣓⣦⣀⠀⠀⢀⣤⣀⠂⠀⢸⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠱⡀
⠀⠀⠀⠀⠀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⣞⡋⠤⡀⠀⠀⠀⠀⠀⠔⠅⠈⠀⠀⠀⢉⡹⠓⡄⠉⡆⠀⣸⠗⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠱⣄⢑
⠀⠀⠀⠀⢰⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⢬⠑⠒⠒⢮⠒⠉⠓⢦⠒⠒⠲⣤⠐⠀⢠⠀⡾⠊⠀⠀⠀⠀⠀⣠⠔⠛⠉⡡⢈⠩⠛⢦⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣆⠠⣀⣔⠠⠤⢄⣍⣀⠀⣀⣸⠀⡠⠞⠉⠁⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⣀⣀⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⢏⠠⡀⢀⡴⢋⢀⢔⢹⡏⠩⢑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠚⡟⠀⠰⡉⢁⣼⢅⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠤⣼⠗⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠈⢷⡤⢀⣷⣀⠴⡻⠉⠈⠎⣆⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠚⠉⠙⠢⡈⢆⠀⠈⣷⡖⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣄⣙⣺⣤⡤⠴⡟⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠐⠚⠉⠁⢄⠙⢆⠀⠀⠀⢹⠈⣄⡴⠙⠛⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⢆
⠀⠀⠀⠀⡀⠠⣀⡀⠀⠀⠔⠉⢍⡉⠲⣌⠢⠀⠀⠀⠑⠘⣀⣠⠤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠳⡄
⠀⠀⠀⠐⡀⠀⠸⡈⠑⠚⠒⢀⡀⠱⠀⠈⢆⣷⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠘⢆
⠀⠀⠀⠀⡝⠉⠀⡨⠛⢳⠔⠣⠤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠘
⠀⠀⠀⠀⢇⠀⣸⠽⠛⠉

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡤⠒⣈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠏⢸⠀⠀⢱⢄⠀⠀⠘⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡂⠀⡟⠀⢸⠀⠀⠈⠚⠝⡲⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣆⣇⣂⡇⠀⠸⠁⠀⠉⠑⠄⠘⡌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⣇⣆⣲⣿⡿⠉⠀⠀⠀⠀⠀⡀⢠⠀⠀⡇⢸⣶⠀⢀⠀⢀⠀⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⡶⠒⡟⣷⣿⡟⠀⠀⣀⠤⠀⠀⠀⡠⠋⠀⠜⠀⡿⣒⣾⠒⢀⣰⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣿⣷⣶⣿⣿⡇⣠⠞⠁⣀⣠⠔⠊⠁⠐⠋⠐⠚⡇⢻⣿⣶⣸⠘⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠛⣿⣧⣄⣿⡉⠉⠉⠉⠁⠀⠀⠀⣠⠀⠀⠀⠀⡰⠁⢀⣿⣿⣿⣺⡛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡧⢄⣿⣟⣀⣀⣠⣤⡴⠖⣋⠠⠀⠀⣠⠞⠁⢠⢿⠹⣾⣾⠻⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠍⠟⢿⡁⠀⠀⠀⠉⠀⠀⠀⠀⢀⣠⠞⠁⢀⡴⠃⣸⠀⣹⠻⢿⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⣥⣿⣃⠀⠀⠀⠀⢀⣀⠤⠖⠋⣀⣤⠖⠋⢀⡴⠋⢀⣿⣦⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⡌⠉⠙⠛⠛⠻⠶⠶⠒⠚⠋⠉⣀⡤⠖⠋⠀⢀⡞⠀⡏⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢽⣇⠀⣀⠀⠀⢀⣀⣀⡤⠖⠋⠉⡁⠀⠀⢀⡠⠋⢀⡼⠁⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣿⠻⢭⣝⣲⠶⠖⠒⠚⠈⣀⣠⣄⣒⡯⠶⠚⠁⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⡄⠀⠀⠉⠉⠉⠉⠙⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣓⠶⠤⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠉⠁⡀⠒⠂⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠂⣀⠐⢀⡀⠐⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠐⠀⣀⠐⢂⣀⠐⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⢁⣀⠒⢂⣀⠉⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣁⠤⠒⢂⠤⡉⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣁⣉⣉⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⣿⣿⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⠿⢿⠿⢿⣿⠿⠿⡿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠑⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⠁⠢⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠈⠐⢔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠈⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠂⠀⠀⠀⠈⠒⠀⢄⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠤⢀⠀⡰⢀⠀⢀⠄⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠰⠄⣀⠌⠠⢀⠔⠁⠀⢀⠅⠊⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠲⣃⠀⠁⠀⡠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠉⠠⠑⢣⠀⠀⡠⠊⠁⠀⠀⡠⠉⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠔⠉
⠀⠀⠀⠀⠀⠀⠀⡠⠮⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢊⠀⠀⠀⠀⡐⠑⠒⢄⠀⠈⠂⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⢠⢁⡀⠀⠉⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⡀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠃⠀⠈⠑⢄⡀⠀⠀⠉⠉⠉⠐⠠⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⡀⠀⠐⢄⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠤⡀⠀⠐⠄⣀⠀⠀⠀⠀⡇⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠉⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠉⠐⢢⠃⠀⠀⠱⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⢄⡀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠠⡀⠌⠀⠀⠀⠀⡰⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠊
⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⢀⠠⡴⠛⠀⢀⠕⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⢀⡴⠉⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⢎⡁⠈⠀⠀⠊⠁⠔⢠⠃⠁⡒⠀⠀⠀⠀⠒⠈
⠀⠀⠀⢀⠠⠲⠋⠀⡠⠐⢠⠋⠖⠀⠤⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠈⠁⢢⠀⠀⠀⠀⠸⠀⢠⠁
⠀⠀⠀⠐⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⢇⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠰⡒⠀⢹⠉⠁⠐⠂⠀⢀⠀⠈⠒⠐⢄⡇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⢄⠚⠀⠀⢀⠐⠁⠀⠀⢡⠀⠀⡆⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⢔⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡐⡁⠀⠀⠀⠀⢸⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠑⣄⠄⠊⠀⠁⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢣⠀⠈⢂⠀⠀⠀⠀⠀⠀⠐⠸⣀⡀⠤⠂⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡂⠐⠠⢀⣀⣀⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠱⠀⠀⠀⠀⠀⠀⠂⢡⠀⠂⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢀⣀⠀⢀⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡇⠀⠀⠀⠀⠀⠰⠙⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⢐⡏⠀⠀⠀⠠⠄⢦⠀⢄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⠀⣀⠎⠀⠀⠐⠒⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡠⠂⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⢂⠀⠀⠀⠀⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⡠⠔⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠡⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⠗⠉⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠀⠀⡇⠀⡀⠆⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁

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

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

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

⠀⠀⠀⠀⠀⠀⢀⠖⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⣰⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀⣞⡅⠀⠀⠀⠀⠀⠀⠀⡴⠂⠒⢦
⠀⠀⠀⠀⠀⠀⢮⣀⣄⣄⡼⠀⠀⠀⠀⠀⠀⡰⠣⠭⢽⡀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠦⡀⠀⠀⠀⠀⢀⠷⠔⠒⢺⡄
⠀⠀⠀⠀⠀⠀⠈⢟⠢⡱⠁⠀⠀⠀⠀⠀⠀⣗⢒⠒⢒⡇⠀⠀⠀⠀⠪⣀⢤⣀⡠⣀⡼⠀⠀⠀⠀⠸⡀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢸⠁⠀⠀⠀⠀⠀⠸⣴⣤⣦⠇⠀⠀⠀⠀⠀⠀⠈⢹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣱⠦⡀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠢⡀⠀⠀⠀⠀⢀⡠⠚⣧⣼⣦⠀⠀⠀⠀⠀⡰⠁⠉⠉⡖⠁⠀⠀⠀⠀⠀⡴⠷⠞⢄
⠀⠀⠀⠀⢰⣫⢤⣀⠤⣄⡠⢌⣲⠀⠀⠀⣏⣩⣭⣤⣤⣤⡇⠀⠀⠀⠀⠳⡒⠒⢲⠞⠀⠀⠀⠀⢀⣠⣏⣙⣉⣹⣄⡀
⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠯⠭⠭⠭⠤⠴⠃⠀⠀⠀⠀⠀⢹⢻⠏⠀⠀⠀⠀⠀⠈⠑⠒⣢⣖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠽⠦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠤⠶⠭⡉⠑⣄⠀⠀⠀⠀⢰⠣⢢⢔⢮⢣⠀⠀⠀⠀⠀⡔⠉⠀⠈⠱⡄⠀⠀⠀⠀⢸⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⡏⡞⡉⠳⡈⡆⢸⠀⠀⠀⠀⠸⡰⣐⣾⣮⡾⠀⠀⠀⠀⢰⠳⡖⢶⢲⡖⣷⠀⠀⠀⠀⢸⣿⣿⡛⢻⡇
⠀⠀⠀⠀⠀⠱⣱⢌⣒⣁⠷⠊⠀⠀⠀⠀⠀⠉⢲⣿⠃⠀⠀⠀⠀⠀⠈⣇⣧⢸⢸⢧⡏⠀⠀⠀⠀⢸⠋⠛⠻⠿⣯
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡔⠋⠀⢙⠲⡀⠀⠀⠀⠀⠀⠀⢀⣠⡓⢤⠀⠀⠀⠀⠀⠀⡞⡁⡙⡆⠀⠀⠀⠀⠀⠀⣀⣄⣤⣀
⠀⠀⠀⠀⠀⡜⠡⠆⠀⠈⠀⢳⠀⠀⠀⠀⠀⡴⡏⢣⠈⡞⠁⠀⠀⠀⠀⡖⠚⠈⠘⠒⡆⠀⠀⠀⣰⡎⢱⡀⢀⠎⢑⡆
⠀⠀⠀⠀⠀⢸⣀⠀⠀⠙⢀⠎⠀⠀⠀⠐⣮⠣⣸⡤⠋⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢏⠀⠀⠀⠀⠙⢌⡏⠙⠋⢹⣡⠃
⠀⠀⠀⠀⠀⠀⠈⠙⣾⠉⠁⠀⠀⠀⠀⠀⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠋⠳⠬⠇

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⢄⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⠢⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣦⡀⢀⣤⡀⠀⠀⠈⠉⠃⠂⠀⠀⠀⠀⢄⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⢱⣶⣷⡆⠀⠀⠂⠒⣸
⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⣿⣿⣿⣿⣿⡄⣀⣀⠀⠀⣼⣿⣿⣷⠀⣤⣤⣄⠀⡀⡀⠀⠀⣾⣿⣿⣿⡇⣿⣿⣿⣇⠀⢸⣶⣤⣿
⣿⣿⡁⠘⣿⣿⡇⠀⠀⠘⠠⠔⢸⣿⣿⡇⢻⣿⣿⣿⣿⣿⠀⠈⠁⠰⣿⣿⣿⣿⠀⣿⣿⣿⣧⢀⣴⣶⡄⣿⣿⣿⣿⠀⢸⣿⣿⣟⣀⣼⣿⣿⣿
⣿⣿⣇⣠⣿⣿⡇⠀⠀⠀⢠⣤⠀⣿⣿⡀⠸⣿⣿⣿⣿⣿⠀⣾⣇⠀⣿⣿⣿⡏⠀⢿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⢠⣿⡀⠀⣿⠀⢻⣿⣇⣰⣿⣿⣿⣿⡟⠀⢹⣿⣀⣿⣿⣿⡇⠀⢸⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⢸⣿⣦⣴⡟⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⠀⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣇⢄⢀⣿⣿⣿⣿⣿⣿⣿⡝⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣼⢱⠘⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠾⡄⠉⠆⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢳⢧⠄⢰⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡞⡞⡄⢌⣆⠀⠀⠡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢹⡑⡑⡀⣍⠂⠀⠀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢭⣹⡱⡔⣌⢦⡀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠋⣀⡠⠀⠠⣄⡀⠀⠀⠸⢧
⠀⣠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⣮⡣⣝⠶⣌⡂⠀⠀⠀⢀⣀⠂⢈⠈⠤⠀⠀⠀⠀⠀⠠⠀⠀⠘
⢀⠁⡠⣐⡪⢥⣐⣒⣒⣒⡆⠤⠄⡀⠀⠀⠀⠀⠀⠀⠐⢌⠳⣮⣕⢮⣍⣛⠒⠤⠤⠤⠤⠤⠄⠐
⠀⠀⠈⠒⠍⢯⠙⠤⠣⠬⠭⣏⡒⠀⠑⠄⠀⠀⠀⠀⠀⠀⠁⠈⢛⣻⢾⣭⣛⡓⠒⠲⠒⠒⠒⠃⡉⡠⠀⠂⡀⠀⠀⠀⢀
⢠⠆⠡⠤⠄⠌⠉⠉⠉⣉⠁⠚⢼⡑⠄⠀⠀⠀⠀⠀⠀⠀⡆⣇⢢⢹⣿⣦⣯⣉⣉⡉⠋⠉⠉⣉⡀⠄⠒⢀⡀⢀⠀⡰⠂
⠀⣦⠀⠀⠀⢀⣒⢒⠤⠤⢈⠑⠤⠉⠀⣠⠖⠂⠀⠀⠀⠀⡇⢸⢸⠀⣏⡓⠺⠭⢭⣉⣉⣉⢉⡁⠤⠄⠒⣀⠔⠁⠊
⠀⠈⠳⠀⠀⠒⠒⠬⠉⠀⠀⠀⠁⠀⡠⠃⠈⠀⠀⡄⡐⢠⠃⠆⠸⢠⡷⠦⠍⣒⠒⠢⠤⠠⠤⠀⠄⢐⣂⡤⠖⠁
⠀⠀⠀⠀⠀⠐⠒⢲⣦⣤⣀⡀⠀⠀⠀⠀⠀⠡⠈⠐⠁⠀⠀⠠⠃⢆⡇⠀⠈⠉⠛⠛⠿⠶⠾⠓⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⠄⣀⠤⣀⡴⣢⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢋⢞⠵⣫⣔⡎⢗⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢯⠞⠓⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠤⠶⠧⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠹⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡀⠙⢷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢩⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣡⣦
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⢀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⢿⡀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⡀⠀⠛⢦⣀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢶⣤⣄⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⣠⡀⠀⢸⣆
⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣶⡏⠁⠀⠀⠈⠙⠿⠶⠿⠿⠋⠉⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠀⠹⢿⣄⠀⠀⣼⡏⠿⣇⣸⡿
⠀⠀⢻⣧⣴⠾⢷⣦⠀⠀⠀⣼⡟⢳⣆⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠈⢻⣧
⠀⠀⣼⡟⠁⠀⢸⡿⠀⠀⣾⡏⠀⢸⣿⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⢀⣿⡇⠀⠀⢸⡟
⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⣿⡇⠀⢸⡟⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠘⢷⡀
⠀⠀⣿⡇⠀⢠⣿⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣧
⠀⢠⡿⠁⠀⢸⣿⠀⠀⠀⣿⡇⠀⠈⠻⢷⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢨⡅⠀⠀⢸⣿⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠈⢻⣆⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⡿
⠀⣿⡇⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣠⠁⠀⠀⠀⠀⠀⢠⣿⠁
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣿⠿⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣿⠀⠻⢇⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⡀
⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⣴⠿⣿⣶⠀⢀⣼⠟⠉⠉⠹⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⣇⣀⡀⠀⠀⠀⣷⠀⢸⣿⠂⢸⡇⠀⠀⠀⠀⠀⢿⣿⣀⣠⣶⠶⠀⠶⣶⣶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣶⣿⠛⣿⣶⣤⣄⠀⠀⢀⣄⣀⣨
⣿⠛⠛⣿⣄⠀⠀⣿⠀⠸⣿⣀⣾⠇⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⣄⠀⣼⡟⠉⠀⠉⠉⠙⢿⣧⡀⠙⠋⠉⠙
⠟⠀⠀⢸⣿⠀⠀⣿⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃
⠀⡀⠀⠸⣿⣄⣰⣿
⠀⢻⠀⠀⠉⠛⠉⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿
⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠈⢠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⢰⡇⠀⠀⠀⠀⣶⠾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡶⠾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⣷⡄⠀⠀⠀⢀⡟
⠀⠀⠀⢱⡄⠀⠀⢀⣿⠀⣸⡇⠀⠀⢠⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⠀⠀⠀⠀⡄⠀⣿⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠘⠿⣤⣤⡾⠁⠀⣿⡇⠀⠀⡀⠀⠀⠀⠙⠛⠛⠛⢿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⢿⡿⠛⠛⠛⠋⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⣿⡄⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠻⣇⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⠀⠀⠀⠀⢀⣠⣷⣶⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣿⣿⣿⣿⣿⣿⣦⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⡔⣀
⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠨⣙⡂⠡⡇⢀
⠀⠀⠀⠀⠀⠛⠂⠀⠦⠀⠀⢀⠀⠀⠠⣄⠳⠿⣆⣣⣠⣤⡄⡀⠚⠳⠈⢱⡦⣀
⠀⠀⠀⠀⠀⠀⠋⠀⢀⢐⡟⣨⡀⠖⠲⠆⠖⠲⡄⠠⢭⠈⠄⠶⠊⡠⣥⠚⣽⣆⣳
⠀⠀⠀⠀⠀⠀⠀⠂⠊⡟⡬⡷⢑⣛⡻⠚⠛⢿⣆⡶⠶⢦⡒⢋⣻⣠⣴⢛⣴⢿⣿⣷⡄⠠⠌⠤⡌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠳⢿⣟⣓⢀⣹⣠⣷⠉⢐⣀⡉⢥⡟⡻⣿⣿⣼⣿⣳⣾⣿⠧⠖⡶⢒⣦⠶⣶⣦⡄
⠀⠀⠀⠀⠀⣠⣤⠶⠖⠒⠒⠲⢭⣭⣭⣿⣿⢠⣯⣬⣷⣶⣶⣇⢹⣿⣿⣿⣿⣿⣿⣿⣾⣯⣰⣭⣴⡿⠟
⠀⠀⠀⠰⡿⠉⠀⠀⠀⠀⠀⠀⠀⠈⠿⣫⣵⣾⣿⣿⣿⣿⣿⣿⣈⡩⠽⠟⠛⠛⠥⢄
⠀⠀⠀⢱⣬⣲⢤⣤⣀⣀⣤⣴⣤⣴⣾⢟⣿⣿⣿⣿⣾⣿⡿⠿⠿⣦⣄⠀⠀⠀⢀⣠⣷⠆
⠀⠀⠀⠈⢛⣻⣿⣿⣿⣿⣻⠿⢖⡻⠟⠛⠙⠛⢻⠟⠉⠀⠀⠀⠀⠀⠉⠻⣷⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠉⠙⠺⠷⠈⠐⠈⠉⠀⠀⠀⠀⠀⢸⣤⣤⣀⠀⠀⠀⠀⠀⢀⡹⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣧⣠⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣰⠂⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢠⡏⠀⠀⠀⠀⠀⠻⣦⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣻⡇⢸⡇⠀⡀⠀⠀⠀⠀⠈⠙⠛⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠛⠛⠛⠻⠇⠈⠀⣸⠃⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠴⠚⠉⢀⣠⠴⠖⠚⢻⡆⠀⢈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠠⠴⠚⠋⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⢀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡾⢷⠄⠀⠀⠀⠀⠀⠈⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⢀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣀⣀⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠠⠞⠋⠁⠀⠀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣦⣤⣤⣤⣤⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⡟⠀⡰⠀⠀⢻⡇⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⣦⡀⠀⠀⠁⢠⡇⠀⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠀⢰⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡀⠀⠻⣤⠏⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡾⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⢴⡄⠀⠀⠀⣀⢔⣒⣳⠀⠀⠀⠀⡠⢚⣩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠋⠀⠀⠀⣠⠾⠋⠀⠀⠀⣰⠛⠁⠀⠀⠀⠀⠀⠀⠷⠉⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠁⠀⡠⠚⢹⠇⠀⠀⠀⢀⡴⢺⠆⠀⠀⠀⡰⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡠⣒⣥⠔⠁⠀⠀⠀⢀⡴⢫⠇⠀⢀⠤⠴⢊⡼⠀⠀⠀⢀⠞⢀⡴⠋⢀⣔⣊⣑⣉⡠⠎⠀⠀⠀⢰⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠰⠏⠀⠀⠀⠀⠀⠠⠾⠦⠒⠉⠀⠰⠗⠒⠒⠉⡀⣀⠀⢠⠏⡔⠁⠀⠀⠞⠁⠀⠈⠀⠀⠀⠀⠀⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠹⠀⠀⠀⢀⡠⠖⢛⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠒⠁⡠⢃⠞⣠⡀⠀⠀⠀⠀⡠⢦⠀⠀⢀⡠⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣌⠙⡇⠀⠀⠀⡠⢋⠄⠈⣡⠤⠀⠀⣀⠤⠒⢩⡍⠉⠒⠦⠤⠤⠤⠊⢀⠎⢠⡴⠀⢶⠀⢀⠎⡀⠁⣠⠊⢭⠄⠀⠀⢀⠔⡆⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠀⣱⠃⠀⢸⠀⡇⣀⠀⠄⣰⠃⠰⠊⡀⢤⡴⠋⣡⠴⠒⣺⠆⠀⢀⣤⡤⢤⠤⠒⠁⠀⠀⡠⢢⠋⠀⡎⢰⠁⠀⣷⠁⠀⠀⠀⠀⢬⡘⠀⡰⢉⠔⠋
⠀⠀⠀⠀⠀⠀⠉⠀⠀⡸⢸⠁⠇⡌⢰⡇⠀⡞⠀⡰⠋⡠⢺⢁⠤⠀⠁⢀⠴⡟⠁⠀⠴⠊⠀⡠⠔⠊⠀⡁⠀⠀⡇⢸⠀⠀⠛⠀⢀⡔⡄⠀⠀⠀⡌⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠃⠀⡰⠀⣾⠃⢰⠁⡼⠁⡠⢇⠀⡱⢠⠇⡴⠋⢀⡴⠒⠉⡐⠹⣪⠂⠀⠔⡜⠀⠀⢠⣇⢸⠀⢋⡆⠀⠘⡇⠋⠀⠀⢰⠁⡎⠀⡠⢲
⠀⠀⠀⠀⠀⠐⠂⠀⣴⢷⠇⢠⠇⣠⣿⡶⣧⢞⠁⠾⠒⡻⠟⣥⢀⣾⣤⣀⣮⣤⣀⣰⠃⢰⡟⠁⠀⠀⠁⠀⠀⠀⡇⢸⠀⠟⠀⠀⠀⢧⡄⠀⠀⡏⢸⠃⢸⡵⠋
⠀⠀⠀⠀⠀⢀⡄⣼⠃⢸⣴⡧⠾⣻⠋⠈⠁⣉⣠⠀⣠⠴⠚⡡⠛⠯⠉⠁⠀⠀⠈⢋⡀⢸⡇⠀⠀⣰⠦⠺⠁⠸⠁⡎⠀⢀⡴⠂⠀⣸⡇⠀⠀⣧⢸⡀⠾
⠀⠀⠀⠀⠀⢘⡁⢻⣴⠋⠀⠀⠀⠁⠀⢀⣀⣤⠿⠾⠧⠀⠀⠀⠂⠀⠀⣠⠦⠄⠀⠸⠃⡼⡟⠻⢯⡀⠀⡄⠐⠁⢰⠁⢀⠏⠀⠀⢠⢣⠇⠀⠀⢸⠈⡇⠀⠠⡄
⠀⠀⠀⠀⢸⢸⣇⣼⡇⠀⠀⠀⠀⢀⡞⠉⠀⢀⡤⠤⠠⠀⠀⠀⠀⠙⢿⠁⠰⢄⠀⠀⠜⠀⠃⠀⠀⠛⣋⣥⣄⣠⡏⢀⡎⠀⠀⡠⢃⡞⠀⠀⠠⢫⠀⠁⠀⡔
⠀⠀⠀⠀⡸⣼⠿⠻⠀⠀⣠⠴⣦⣼⡀⠀⢀⣀⠈⢁⡄⠲⡋⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⡶⢀⡼⠛⠛⢿⡠⢺⠁⡠⠞⣠⠎⠀⣀⠔⢡⠂⠀⡄
⠀⠀⠀⣰⢱⠇⠀⠀⠀⠈⣠⠖⠋⠉⠉⠉⠻⠁⣌⠭⠐⢀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠀⠀⠀⠀⢠⡟⠋⠁⠀⠀⢸⡧⠞⠉⠀⣴⠥⠒⠞⢁⡴⠃⠀⢠⠃⠀⣰⢛
⠀⠀⢰⣧⣀⣀⡀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣾⣀⡀⠀⠀⠀⠀⠀⠈⠹⣆⣀⣀⠀⠀⠆⠀⢠⠀⠀⠀⣶⣦⠀⡟⠁⠀⣬⠴⢊⡴⢻⡇⡾⢠⠔⣣⠞
⢀⡴⠋⡉⠉⠀⠀⠀⣼⠧⠦⠀⠀⠀⣀⡄⠀⠀⠀⡠⠴⠋⠁⠈⠀⠀⠀⡶⠆⠀⠀⠈⢯⠉⠀⠀⠀⠀⢸⣄⠀⠀⠉⢀⣀⡙⠚⠋⠁⣠⠊⠀⡿⣸⢡⡷⠊⠁
⡼⠁⠂⠀⠔⠇⢠⡞⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠒⠲⢿⡄⠀⠀⠀⠚⠉⠉⠃⠀⠀⠀⣿⡿⠃⠀⣰⠟⠀⢰⠟⠁⠈
⣧⠀⠁⠆⠀⢸⣼⡁⠀⠀⣆⣀⠀⠀⠀⢰⠀⠀⠀⡀⠀⠈⠙⢲⣄⡤⠄⠀⠀⠀⣠⠠⢄⠉⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣾⠃⠀⠂⠀⠀⢀⡠⡀
⣹⠀⠃⠂⠀⠀⢻⡿⠚⠋⠉⠀⠀⠀⠀⠘⢦⣴⠟⣡⠒⠖⢤⠀⡏⠀⠀⠀⡤⠩⠤⠔⠆⣧⠠⠖⠀⠀⠀⢸⡄⠀⣶⠖⠛⠩⠖⣁⠀⠀⠘⠻⠭⠭⠔⠁
⡿⠀⠀⠲⣞⠀⢸⡇⠀⠀⢄⠀⢀⡀⠀⠀⢸⠇⠀⣧⠒⠓⠒⠀⠁⠀⠀⠤⣄⠀⠀⠠⡀⠀⠀⣠⠀⠀⠀⣼⠷⠛⠛⢂⡡⢚⠥⠖⠁
⢷⠀⠀⠀⠈⢦⣘⣇⡀⠀⢈⣷⢿⡇⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠘⡗⠀⢠⠟⠛⠉⢁⢀⡤⠾⠥⠴⠁⡴⢋⠔⠁⠀⢀⣀
⠈⢷⣤⣦⠀⠀⠨⣿⡇⠀⠘⠁⠀⠙⢄⠀⠀⠀⠀⣠⡬⠷⠦⣤⣤⡴⠂⠀⡰⠀⢠⠏⠀⠀⠚⢋⣙⣒⣒⡒⢂⣡⠔⠁⠀⠞⠀⠒⠚
⠀⠀⠈⠛⢧⣀⢀⣘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⣠⠾⠷⠶⠞⠋⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠛⠻⠿⠷⢦⣤⣤⣼⣦⣄⣀⣀⣈⣽⣦⣀⣀⡴⠷⠤⠤⠶⠚⠁
⣶⣶⣶⢶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣦⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉⠊⠑⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠋⣁⡠⠶⠖⠳⣄⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠒⠊⠉⠀⠀⠀⠀⠀⠣⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢡⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡀⠀⠄⠉⠀⠀⠈⠳⡖⠒⢦⣄⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⠀⣠⠞⠋⠙⠳⣄⣧⣴⠤⢤⣀⠀⠀⠀⡇⠀⠀⣨⣅⡀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⠀⠀⣸⢁⣀⣤⠤⠤⠤⣄⠀⠀⠀⠈⠉⠛⠳⣇⡴⠛⠁⠀⠙⢦⣤⣬⣇
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠠⣿⣤⡀⣠⠟⠋⠁⠀⠀⠀⠀⠈⠳⣤⣤⠴⠶⣏⢉⡽⠛⠋⠉⠉⠛⠳⢦⡀⠸⡇
⠀⠀⠀⢀⡴⠛⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⣷⠒⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⠇
⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣤⡾⢧⣄⣀⣀⣀⣀⣀⣈⣹⣆⣀⣏⣀⣀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘
⠀⣠⠞⠉⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣦⣤⣴⠚⢉⠉⢻⡄
⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠇⠀⠀⠀⢰
⣠⠟⠀⠀⠀⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠰⣾⡁⢠⡄⠀⠀⢀⣤⠀⠀⠀⢠⣄⠀⠀⠀⢀⣤⠀⠀⢠⣿⠀⠀⠀⠀⣼
⣏⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠈⢷⢾⡇⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⢸⣿⣆⠀⠀⠀⢹
⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⡇⠀⠀⢸⡿⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⠀⠀⢸⡷⠈⢳⡄⠀⠘
⠀⠈⢹⡶⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⢺⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⣠⢶⡄⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠄⠀⠀⢠⡆⢹⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠳⣄⣠⠾⠁⠀⠹⣶⠶⠶⠓⢦⣀⣤⣄⠀⠀⣠⣤⡶⠋⣧⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⡜⢸⠀⢠
⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠢⡀⢀⡠⠌⠃⢾⠀⣾⠋⣼⠁⠀⣿⠘⣧⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⡇⢸⣤⠞
⠀⠀⠀⠀⠸⡇⠀⠀⢢⡀⠀⠒⠉⠃⠄⠀⢠⡟⢠⡇⣼⠃⠀⠀⢸⡆⢻⡀⠈⣷⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣼⠃⢸⠁⣿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠐⢂⣀⡀⠠⠈⠉⣹⠃⢸⣷⠏⠀⠀⠀⠈⣷⠘⣧⠀⢻⡀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⠀⢀⡟⢠⡏⢸⡟
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠉⠢⠀⠀⣰⠏⢀⣼⡏⠀⠀⠀⠀⠀⠸⣇⠸⣇⠈⢣⠀⠀⠸⡆⠀⠀⢸⠇⠀⠀⠀⡈⠁⠀⡘⢀⡾⢠⡿⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠐⢤⡀⠀⠶⠈⠉⠙⠓⢛⡿⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⣦⠈⢣⠀⠀⠙⡀⠀⠀⠀⠀⠀⠰⠁⠀⡜⢁⡾⢡⡿⠁
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠑⠠⡀⡀⠄⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡘⢷⡄⠑⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠊⣠⠏⣠⡿⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠘⠢⠀⠈⠉⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢁⣴⠏
⠀⠀⠀⠀⠀⠀⢸⡆⠐⢀⠘⠓⠂⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣈⠉⠓⠒⠶⠖⠚⠛⢉⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠸⡇⠀⡀⠑⠶⠆⠈⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⠶⠶⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠦⠤⠀⢁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠢⢤⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢤⣀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢠⠀⠀⢀⡀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⣴⠋⠀⠀⠈⢧⠀⠀⠀⠀⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⠀⠀⠚⠁⠀⠀⠀⠀⠈⠣⠀⠀⠀⢀⣀⣤⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣳⣦⣤⣶⣤⣤⣶⣶⣶⣚⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠂⠀⠀⠀⠀⠈⠀⠐⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠁⠂⡀⠀⠀⣀⢀⠠⠤⡴⠄⠀⢀⡀⠀⠨⣐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⡀⠂⠈⠀⢁⢀⡙⣞⠤⡆⠀⠀⠀⢈⠈⢆⠄⣦⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡄⢆⠀⢶⡀⣀⠘⠟⢁⣶⡆⠸⡀⠀⠀⠀⠐⠁⠁⠁⠀⠈⠜⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡵⢁⡌⠉⢨⢚⠉⠀⠍⢦⠀⢠⠎⠀⠀⠀⣀⠌⠂⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⡄⠀⠀⠈⠓⠚⠁⣏⠂⠀⡀⠀⠀⢈⠲⠂⠀⠀⢠⠆⠁⢀⣼⣁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⠀⠀⠀⠀⡀⢻⠀⠌⠀⠌⡀⠀⠀⠀⠀⢐⡁⠈⣠⠞⠁⠈⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠝⡄⠁⠀⣀⠀⢀⡉⠀⠀⠀⠈⠂⡀⠀⠁⠀⢈⠀⡐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠐⠦⡀⠃⠡⡨⢀⠅⡐⠀⠈⢊⠈⠀⠀⠠⠂⠐⠚⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠉⠃⠉⠂⣄⡀⡁⠀⠰⢂⠄⠀⠨⠂⠀⠀⢀⠀⠀⢀⠀⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠰⠀⠀⠀⠄⠈⠈⢂⠀⠀⠈⠉⠣⠒⠒⠶⡅⠈⠀⠁⠀⠔⠡⠀⣀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠐⢱⡀⠀⠀⠀⠀⠠⠀⠀⢹⡀⢂⢀⡵⠀⠀⢠⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⠭⡄⠀⠀⠀⠀⠀⢠⠞⡨⠒⠁⠢⠐⠐⠣⡪⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠤⠀⡀⠀⠈⡇⠀⠀⠀⠂⡴⠓⠊⠁⠂⡀⠒⠠⠀⢈⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠀⠸⠀⠆⠐⠀⠐⢬⠀⢀⠀⠘⠀⠐⠠⠀⠀⠀⢘⠔⠀⣈⠐⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⢂⢴⠀⠀⠃⠀⠂⡁⡄⠀⠄⣤⠂⢀⠁⠀⠀⠀⠀⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠧⠀⠀⠀⠀⠀⢐⢰⠀⠀⠉⠀⢠⠀⠁⠀⢘⠀⠀⠀⠀⠀⠰⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣠⡰⠀⠀⠀⠂⡀⡄⠀⠀⡀⠀⠀⠀⠀⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⢀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠐⠁⠂⠀⠈⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠁⠂⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⠿⣿⣝⠛⠋⣩⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠱⡞⠛⢾⣷⣶⣿⣿⡿⡿⠵⠆⠈⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣛⣤⣤⣀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣶⣦⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠈⠉⠉⡄⠰⢤⣌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣿⠛⠻⢿⣿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠘⢲⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⡄⠀⠀⢰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⢀⣼⣿⣿⣿⣿⣿⡇⠀⢠⣾⣹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠙⣿⠋⠙⠻⠟⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⡀⣿⠀⠀⠀⣰⣆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠁⠀⠀⣤⠀⠙⠋⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠛⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠉⠉⠄⠉⠑⢄
⠀⠀⠀⠀⢠⠖⠉⠀⣀⠠⠂⠁⠀⠄⠐⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂
⠀⠀⠀⠀⠇⠀⠈⠀⠈⠀⠀⠂⠀⡀⠀⠰⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣬⡀
⠀⠀⠀⠀⢇⠈⢁⠀⠉⠀⠋⡀⠀⠀⠀⠀⠀⡠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⡎⠐⠾⠓⢹
⠀⠀⠀⠀⠈⡆⠀⠀⠀⣀⠀⠁⢠⠋⠉⢆⠀⠰⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠠⠀⠀⡧⣀⣀⣀⠜⢦⡀
⠀⠀⠀⠀⠀⠸⡌⠀⣾⠉⠙⠒⠋⠀⠀⠈⠂⠐⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠐⠀⡁⠐⠀⠬⠀⠈⠀⠠⠙⡄
⠀⠀⠀⠀⠀⠀⢣⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠈⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡠⡶⡻⠃⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⣩⠊⡴⠁⠀⣇⠑⠤⣆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡰⠁⢰⠁⠀⠀⠈⢦⡀⠀⠉⠳⣽⢦⡔⢢⢀⣤⡀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣀⣠⡎⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⣦⣹⠃⢸⠤⠻⢖⠚⠉⡦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠔⠒⡏⠀⢸⣇⠀⠀⡇⠀⠈⢆⠀⠀⠀⠀⠀⡇⠀⠀⣠⠣⣈⠒⢛⣦⣀⠠⢵⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣱⢄⣾⠀⠑⠒⠚⠢⠄⣈⠷⠤⠠⡤⠚⠉⠉⠉⠀⠏⠉⠣⡀⠀⠑⢦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠓⠂⠉⢸⡁⠒⠪⠕⣤⠊⠉⠢⠔⠉⠑⠤⠊⠉⠲⠊⠙⢦⠖⠻⢶⣀⣉⠯⠽⠥⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠏⠀⡏⠒⢶⠐⠸⡆⠀⠐⡄⠀⠀⡆⠀⠀⡛⠀⠀⣇⡤⠚⡉⠀⠠⠀⠀⠂⠀⠠⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠰⠀⠘⡄⠀⢣⠀⠀⠇⠀⠀⡇⠀⠀⡇⠀⡼⠃⠀⠂⠀⠀⣠⢄⠴⠛⢤⣀⠀⢁⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢣⠀⢱⣀⡈⡆⠀⢸⠀⠀⠁⠀⢰⠁⡜⠀⠈⡁⠉⢔⡲⠗⠂⠉⠉⠐⢦⠇⠄⠐⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠲⣧⠄⠠⢔⠂⠘⠿⠁⠀⠈⢷⡀⠀⡆⠀⠀⠀⡸⢠⠅⠈⠀⠀⠂⡈⠣⢄⣀⣀⡀⠤⠊⡀⠀⢠⠲⢄⡨
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠘⡂⠀⠀⠀⠀⠀⢸⠥⠄⠁⠀⠀⠤⠅⠤⠵⡞⠦⠚⢢⠀⠀⠀⠁⠄⠐⢀⠁⢀⠤⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡸⣁⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠓⠒⠉⠑⠤⠔⠉⠁⠉⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⠶⠖⠊⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣀⣀⣀⣀⡀⠤⠔⠊⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡈⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠄⠀⠀⠀⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡐⠐⠂⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣤⣠⣀⣄⣀⡀⡄⣤⠆⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⡐⠀⠀⠈⠁
⢀⠄⠀⠀⠠⠢⠀⠀⠀⠀⠀⠀⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈
⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⠀⠀⠀⠀⠈⠱⠶⠴⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀
⠀⠳⣄⠀⢀⠠⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠰⠃⠉⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⢀⡀⠤⠈⠁⠰⠀⠀⠀⠀⠀⡀⠙⠓⡆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠏⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠂⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⡠⢀⣠⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣮⣵⣦⣴⣧⣼⣯⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

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











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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⣹⢣⢎⣿⣝⢯⣝⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢖⡌⠀⢛⡠⡿⠋⢻⠘⣯⠈⣷⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⣧⣦⡿⣤⣤⠴⢋⣰⣟⣠⡿⠀⡿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠋⢛⣡⠤⡶⠖⠓⠺⠯⠉⣀⡀⠡⠚⠗⣠⣿⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠤⠊⠁⠀⣀⣤⣤⣶⣶⠿⠷⠶⢦⣶⠶⣾⠟⣿⣿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣤⠶⠛⠋⠉⣟⢡⠤⠤⠒⢻⣋⡡⠾⡟⢆⣿⣿⡇⣿⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣡⡼⠋⠁⣠⠖⠋⠉⣀⡤⠖⠚⠉⠁⠀⣀⣘⣠⠾⣻⣿⢃⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢇⣿⠀⣰⠋⠀⢀⡴⠊⠁⢀⡠⠴⠒⠈⠉⣁⣠⡴⠞⣉⠴⠞⠛⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⢿⡄⣧⠀⠀⡎⠀⢀⡖⠉⢀⣠⡴⠚⠋⢉⣠⡖⠃⠀⣀⣠⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠦⣄⣀⠈⠚⠳⣤⣇⣀⣿⢁⣴⣿⣃⣀⣤⣴⠿⣿⣏⣻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠈⠉⠑⠒⠲⠬⣬⣭⣭⣯⣹⣯⣿⣷⣿⡶⣿⢿⢻⡏⡿⢹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠈⠄⠇⠸⠄⢸⠠⡇⢸⡇⣿⡞⢸⢡⣇⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡆⢸⠸⡇⢸⢠⡇⣇⣿⣾⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠐⡈⠂⠀⠀⠀⠀⠀⣧⢸⢠⠀⡟⢸⠰⢼⣇⡏⣿⣿⣟⣿⣷⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣦⣯⢪⢺⠈⠀⢢⠁⢸⢸⣸⢰⣇⣿⣿⣿⢸⣿⣿⣿⣿⣯⣿⢿⣏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠺⣏⣏⣿⣧⢱⣸⢸⣼⣿⣿⣼⣿⣷⣿⣟⣿⣿⣿⣿⣿⢿⡿⢯⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠾⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠹⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠛⠿⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣤⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⠛⠥⢀⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⢀⣠⣤⣴⣶⣶⣶⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣤⣴⣿⡿⠟⠁
⠀⣾⣿⣿⠛⠛⠉⠉⠉⠙⠛⣿⣿⣷⣶⣤⣤⣤⣴⣶⣾⣿⠿⠟⠉
⠀⠉⠻⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⡍⠙⠛⠛⠛⠛⠉⠉
⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠘⢿⣿⣶⣶⣦⣄⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃

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

⢠⡄⠀⠀⠀⠀⠀⠀⠘⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⡀⠄⠠⠄⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣀⡀⠔⡆⠀⢀⣤⡀⠀⠀⠀⢀⡤⠒⠒⢦⠀⠈⠁⠀⠀⠀⠀⠀⡼⢤
⠀⠀⠀⠀⠶⠀⠀⢀⡠⠔⡶⠔⠉⡆⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣗⢚⣄⣠⡾⠉⠁⠀⠀⠀⠰⢁⡀⢀⠀⠇⠀⠀⠀⠀⠀⢀⠔⠋⠉⠁⠒⡔⠃
⠀⠀⠀⢀⣤⣐⣈⣁⣀⣀⡁⠈⠭⠤⠄⠐⠒⢈⡆⢀⠀⠀⠀⠀⠀⢀⠤⠾⢽⣯⠗⢳⢄⠀⠀⢀⠀⠀⡶⠆⡆⡏⠦⣼⠀⠀⠀⠀⠀⠾⠠⠤⣠⠦⠜⠣
⠀⠀⠀⢸⣄⡂⠄⠀⠒⣒⣒⣖⣒⠂⠤⠘⠛⢀⠀⠀⢱⠀⠀⠀⢀⡟⣶⠚⠓⣿⣙⡏⠭⡄⠀⠈⠁⠀⡇⠀⣇⠃⠀⣿⠀⠀⠀⠃⠀⢂⠱⠀⠇⠀⡇
⠀⠀⡤⠚⠦⠄⣀⣀⡁⠀⠠⠤⠤⠤⠀⠒⠒⠉⢀⡠⠊⠀⠀⠀⠈⡄⡓⠄⢰⠢⠀⠔⢳⠃⠀⠀⠀⠀⢇⠀⠙⠀⠀⣹⠀⠀⠀⠀⢀⠔⢆⠁⠐⠀⢀⠁⢈
⠀⢨⡦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠒⠉⠀⠀⠀⠛⠀⠀⠘⠪⠬⢈⠀⠈⠔⠋⠀⠀⠀⠀⠀⠈⠈⠩⠈⠃⠀⠀⠀⠀⠀⠘⠤⢀⣁⡐⠠⠥⠒⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠀⠀⠀⢠⡀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠤⠤⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⢠⣄⠤⠐⠒⠀⣐⠢⡀⠀⠀⠀⠀⢛⡢⠌⢰⡈⠁⠀⠀⠈⠉⠀⠀⢨⠑⠞⠄⢀⠌⠀⠀⠀⠀⠀⠈⠃
⠀⠰⠆⠀⢀⠎⢰⠊⠓⡊⢯⠱⣈⢢⠀⠀⠀⠀⡤⠊⣁⡠⠊⠒⠈⢀⠂⢰⠀⠀⠛⢄⡁⠈⠀⠀⠥⡀⠀⠀⠀⠀⢠⠒⣏⠢⢄⠀⠓⢤⡀⠀⠀⠈⠃
⠀⠀⠀⠀⡆⡾⠅⠠⠋⠉⠘⣷⠌⡙⠇⠀⢠⠊⢀⡌⠀⠀⠀⠀⠀⠸⡀⡞⠀⠀⠀⠀⠈⠁⡏⠀⠀⠈⡆⠀⠀⠀⠈⠢⢌⠙⠲⠏⠉⢈⣀⠀⠀⣀
⠀⠀⠀⠀⡗⠢⡸⠣⡀⠀⣀⠟⣗⡇⡄⠀⡆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠠⡐⢄⠀⠀⣀⠀⠀⠱⠄⢠⠖⠀⡉⢑⣲⠄⠀⠀⠉⠊⢠⠆⡵⠲⡤⠒⠒⡍⠢⡀
⠀⠀⠀⠀⠘⢄⠣⠥⢎⡉⡉⡮⢚⠜⠀⠀⠘⣦⣈⠑⢄⠀⠀⠀⠀⠀⠀⠘⡌⠢⡀⠉⠀⠀⠀⠀⠈⢦⢠⠌⣉⣢⠀⠀⠀⠀⢀⢡⠒⠁⡠⠄⠦⣄⠉⠒⣷
⠀⠀⣀⠀⠀⠀⠑⠢⠤⠥⠯⠒⠩⠄⠀⠀⠀⠀⠉⠳⣄⢣⠀⠀⠀⠀⡠⠔⢁⡴⠃⠀⠀⠀⠀⠀⠀⠨⡼⠀⠉⠁⠀⠀⠀⠀⢸⢎⡀⠰⡀⠀⠀⢨⠀⢰⢸
⠀⢀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡉⠐⠒⢉⣠⡾⠋⠀⠀⣀⠤⡐⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢸⠀⠈⠒⠂⠁⣀⢜⠌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢔⠒⢲⡆⠒⢄⠀⠀⠓⠀⠀⠈⠻⣶⡢⠛⠉⠀⣤⢀⡊⠀⠑⡻⠖⠚⠀⠀⠀⢀⡀⠀⠀⠀⠋⠀⠀⠀⠑⠭⣈⡓⣤⡊⠔⠋
⠀⠀⠀⠀⠀⡠⢖⡛⠄⠈⢲⠀⠀⠃⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠘⠋⠑⣶⠠⣤⠀⢤⣬⠥⠤⣤⡤⣤⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⢎⣣⠀⠐⡆⠀⠈⠀⢀⠄⡠⠂⠀⠀⠀⠀⠀⡠⣔⠚⠑⠒⠦⡀⠀⠀⠈⠑⠠⠄⠿⠤⠽⠤⠤⠽⠤⠀⠤⠥⠨⠧⠤⠃⠀⠀⠀⠀⣠⠄⠀⠀⠉
⠀⠀⠐⠀⠀⠘⠇⠀⠀⢀⡠⢔⡡⠚⠁⢀⡀⠀⠀⠀⢌⡴⠋⢠⡇⠘⢷⡌⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⢀⡀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⠻⡀
⠀⠀⠠⣀⠀⠀⢀⢒⡪⠅⠊⠁⠀⠀⠀⠈⠁⠀⡠⢄⠀⠀⠀⠀⠁⠀⠀⠀⡈⢢⠶⠀⠀⠀⢀⠔⠁⢀⣀⡀⣀⡈⠳⡄⠀⠀⠀⠁⠀⠀⡔⠁⠀⠘⡀
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠑⡆⠉⠓⠒⠊⠒⠒⠉⡹⠁⠀⠀⠀⠀⡎⠀⡔⠁⢀⠤⢄⡀⠀⠘⡄⠰⠆⠀⠀⡰⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⡤⠒⠒⠠⡊⠁⠀⠑⢎⠁⠈⡆⠀⠀⠈⠉⠀⠉⠁⠘⠈⠉⠀⠀⠀⠀⠀⠀⡇⠈⡀⠀⢇⠄⠀⡱⠀⠀⡇⠀⠀⠀⠸⡀⠀⠉⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⡀⠀⠀⠘⡤⠚⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠱⡀⠑⢄⣀⣀⠴⠁⠀⡜⠀⣀⠀⠀⠀⢩⠶⣔⢶⣲⠞⠁
⠀⠶⠀⠀⢈⠦⢀⠀⠀⠀⠀⠰⣀⡠⠞⠀⠀⢀⠔⡩⠰⠒⡐⡶⣕⢆⠀⠀⠀⠀⢀⡀⠀⠀⢠⢞⠦⠤⠀⠀⠤⠔⠊⠀⠀⠉⠀⠀⠀⢸⢲⡋⣹⢥⠆
⠀⠀⠀⠀⡆⠀⠀⠑⠢⢄⣀⠔⠁⠀⠀⠀⡐⡅⢺⠣⡀⢣⠱⡐⡘⡆⠆⠀⡤⠀⠈⠀⢀⠔⡱⠃⠀⢀⡀⠀⠀⢠⠔⣠⡄⠀⠀⠀⠀⠈⠉⣯⡝⢰⠀⢀⡀
⠀⠀⠀⠀⡱⢄⠀⠀⡠⠂⠀⠠⡄⠀⠀⢀⠀⠦⡀⢣⠱⡀⢣⠱⣁⡀⠐⠀⠀⠀⢀⡔⡡⠊⠀⠀⠰⡁⠈⡃⠀⢏⣩⠕⢂⠀⠀⠀⠀⠀⡧⢦⢼⡎⠀⠈⠁
⠀⠀⠀⠀⢇⣀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⢆⡘⢄⢣⠑⡌⠆⣸⢃⠃⠀⠀⠠⢊⠜⠀⠀⠀⠀⠀⠉⠉⠉⠒⢄⡀⠀⠉⠀⠀⠀⠀⠀⢰⡵⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠈⠢⢌⠒⠷⠧⠜⢋⡥⠋⠀⠀⠀⠓⠁⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠁⠰⠆
⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋⠃⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡋⠻⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⣻⣿⣷⠏⠁⢶⡷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣸⡂⢀⣤⣾⣷⢠⣤⢤⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⣾⣿⣿⣿⡷⣌⣁⣉⣼⠟⠛⢩⢛⣳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡀⠈⠉⠉⠁⠈⠻⠏⠀⠀⠀⢠⣜⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠿⠟⠻⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢦⣄⡀⠀⠠⠀⠀⠀⣄⣠⣴⣾⠿⠟⠉⠉⠁⠀⠀⢤⣄⠀⢰⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠙⠙⠛⠚⠘⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠏⠆⠚⠋⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⡟⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠉⠁⠐⠒⠐⠦⠤⢦⡤⣤⣤⣤⢶⠶⠶⠞⠛⣋⠉⠳⣤⣧⢉⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄⣄⡀⢀⡀⠀⢀⠀⠀⢀⡀⠄⠉⣤⢸⣷⣦⣄⢇⡉⣰⣖⣼⣿⣶⠻⢿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⡯⠵⠛⣯⣫⠾⠛⢦⣤⢶⠄⣰⠟⣿⢿⠋⠛⢾⣿⣿⡀⠙⠻⣿⠆⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠁⠀⡞⠚⠁⣼⠒⠁⠀⠈⠛⣶⠀⠀⠉⣿⡇⠀⣰⣿⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣷⠀⢰⡆⠀⠀⠀⠀⠀⣿⠀⣤⡀⠀⢰⣿⠐⠀⢸⣿⡧⠀⣿⡏⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠸⣷⠀⠀⣷⠀⠀⣿⣸⣷⣵⠀⢸⣿⣣⠀⣾⣿⠇⣀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⡆⢡⣼⣀⢠⣿⣿⣿⡿⠀⣿⣿⡏⢀⣿⣿⡇⢸⡿⣿⣿⣷⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣾⣇⣸⣿⣿⡸⣿⢮⣿⣧⢈⣿⣿⡃⢰⣿⣿⡆⣿⣻⣿⣿⡧⣞⣶⣾⠭⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⣿⣿⢰⣿⣿⡆⢿⣿⣿⣿⣾⣿⣿⡇⢹⣿⣿⡁⣷⣿⣿⣿⣷⣶⣴⢶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠤⠿⣿⣅⣍⣽⣷⣏⣘⣿⣾⣧⣽⣿⠽⣿⣿⣿⡿⣿⣷⡿⣿⣷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⢿⠿⠿⢿⡿⠿⠷⠛⠒⠛⠛⠁⠤




⣿⣿⡿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿