Happy xmas Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⣾⠃⠻⡄⠀⢴⠄⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠁⠀⢀⠙⢄⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣿⡟⠀⠀⢸⠀⢰⠀⢀⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠍⠁⠀⠀⠀⢸⡄⠸⠀⢸⢡⠂⢢⠐⡄⣄⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢠⠖⠀⠰⡤⠐⠀⠈⠁⠀⠀⠘⠄⠀⠘⠀⠁⠀⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠘⡇⠀⠀⠀⠰⠆⠀⢀⠀⠀⠶⠀⠀⠐⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠵⢀⡇⢤⠐⢄⢀⡂⣆⣼⢠⡆⡄⡦⢠⢀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡤⠊⠋⠸⠀⠸⠈⠃⠸⠻⠚⠃⠃⠷⠉⠹⠆⠸⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠶⠒⠒⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⡀⢀⠀⢀⠀⠀⠀⠀⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡴⡟⠀⣴⠂⢀⡀⣶⠀⠐⣽⣀⢠⡖⣰⠂⠰⣧⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠀⠀⠀⠀⡼⢱⢇⢺⡇⢠⣯⢆⢞⡟⡰⣻⢃⠟⢠⡏⠀⠀⢀⡴⠋⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢸⠀⠀⡼⠁⢸⠇⢸⠀⠚⠠⠊⠘⠐⠁⠉⡡⠤⡼⠤⠒⠋⠁⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠘⠋⢀⡞⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡄⠐⢀⣠⣤⡄⠀⠀⠀⠀⣀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣐⠭⡀⠀⣠⢎⡵⠀⠀⢀⡄⡀⠠⢤⡾⠒⡋⢉⡁⣀⡀⢠⡶⣤⠆⢿⡤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠞⢁⡀⣸⣰⡧⠊⣶⠀⠀⣬⢁⣇⡠⡾⠁⣼⡟⣹⡟⡽⢡⡟⡰⡏⠜⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠺⠰⣿⡟⣸⢂⢏⡟⣼⢃⣾⢻⠄⣃⠌⠟⠀⠟⠈⠓⠁⠉⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠟⠐⠣⠊⠘⠊⠁⠁⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠚⠁⢤⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠱⠀⠀⠀⠀⢀⠐⠊⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠉⠐⠠⠀⠄⠠⡀⠄⠰⠄⠈⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣀⠀⠀⠀⣀⣀⣠⡤⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⠀⠀⢠⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢰⡏⠀⠀⠈⠛⢹⣿⠀⠀⠸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⣿⡇⠀⠀⢀⡤⠠⠸⠧⠶⣿⠀⠀⠀⣿⡀⠀⠀⢻⡆⠀⢹⡇⠀⠀⠴⢶⣦⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣿⡄⠀⢸⣧⠀⢠⣿⠀⠀⠃⠀⠀⣿⠀⠀⠀⠸⣧⣀⠀⠀⠛⠊⣸⠇⠀⠀⠀⢸⡟⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠷⠀⠀⢻⣇⠀⠘⣿⡀⢸⣿⠀⠀⠀⠀⠀⠸⣧⡠⠀⠀⠉⠁⠀⠀⠀⢀⠟⠀⠀⣠⠴⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠹⣷⡀⠻⣦⣀⡠⠊⢀⣤⡀⣤⠀⠀⢾⡁⠀⠀⠀⠀⠄⢒⡉⢠⡆⢰⠂⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⡆⠀⠈⢿⡀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠙⠀⣿⡄⠀⠀⠀⠀⢀⣀⠀⠀⣿⠁⢸⣇⠈⢷⣄⡀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠈⠀⠀⠀⣿⡇⠠⣶⠐⣶⠀⢻⡆⠀⣿⡀⠀⣿⠀⠠⠉⢿⡆⠸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡍⢀⠀⢰⡇⠀⠀⣀⠀⠀⠀⠀⢸⡷⠀⣐⠀⠀⠀⣿⡇⠀⣿⠀⢹⡄⢸⣧⠀⠻⣧⡠⠙⠷⠃⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠿⠇⢸⡇⠀⠀⣿⠀⣀⡀⠀⢠⣶⠀⠘⢷⣄⡀⢻⡇⠀⣿⠀⢸⡇⢸⣿⠀⠀⠈⠀⠀⠀⠘⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢸⣇⡠⣤⡙⠋⣿⡇⠀⢸⣿⠀⠀⠀⠘⣷⢸⡇⠀⠛⠀⠀⠀⠀⠻⠶⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡿⠀⢸⣧⠀⢻⣇⠀⠸⣿⠀⠐⢇⡴⠋⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣌⢧⡉⢘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⣧⠀⠀⣿⠀⠄⠛⠖⠀⢻⣆⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⡆⠻⠀⠀⢿⣄⡄⠀⠀⠀⠀⠉⠀⠀⠀⢰⠐⠀⡄⣤⣠⢤⣤⠀⢤⢀⣦⢤⢀⢰⠀⣰⢠⠀⣀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⡼⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚⣇⣃⠰⢸⠃⠙⠀⢸⣸⠰⠈⣾⡇⠀⢻⠲⢀⣿⢸⡋
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠛⠈⠛⠘⠀⠈⠀⠘⠛⠘⠄⠋⠃⠀⠘⠲⠘⠉⠋⠑

⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡩⡖⠙⣪⢉⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣛⠛⡃⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡔⠲⣌⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡌⠀⣾⠋⢹⣿⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⢁⡉⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡾⠀⣯⠉⡏⠀⣽⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠠⠀⢸⡇⡼⡇⠘⡡⣷⠀⠊⢹⠇⠈⣩⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢈⣬⣤⣤⣄⡘⠧⠀⡠⠋⢠⡄⣯⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠠⣄⠉⠤⡴⠂⣄⣭⠢⡀⢰⣷⠈⣳⣧⡴⠖⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠀⢀⡜⣿⣰⣦⣃⠀⡏⠙⡦⡥⢀⡟⣀⣙⠲⣇⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠞⡠⠋⠀⠋⢸⣿⠉⠀⡇⢸⡇⠀⣸⠁⣹⠯⠙⠒⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⠚⢶⡀⢰⡇⠀⢸⠹⢄⠐⠇⠸⠁⣰⢃⡶⠉⠳⣄⠚⠋⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠘⠁⡾⢀⡶⣤⢜⢓⣦⣄⣐⡊⣡⠞⠙⠶⣄⠈⠃⠀⠟⡿⣿⠀⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢧⡀⠀⠰⣻⠗⣾⢁⣿⠞⠀⠈⢽⠁⠀⣯⡴⠶⡄⠈⢠⠀⡄⣰⡄⡁⣄⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡳⠆⢠⡀⠉⠋⢡⡏⠀⡇⠸⠹⣆⡀⣀⣿⣀⡀⢀⣠⠞⠁⠀⣺⢻⣿⣳⠃⣷⣿⣿⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⢀⠠⡤⠀⠀⢀⠀⠀⠉⠛⠁⠈⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠁⠈⠈⠁⠀⠈⠀⢀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠈⠩⠅⠒⠉⠱⠔⠈⠉⡗⠢⠈⠲⠋⠀⠉⠢⠔⠒⠒⠤⣸⠀⠈⠑⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣦⣤⣄⠀⠀⢀⣤⣄⢠⠞⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⣧⢹⣷⡴⠁⠈⣿⡏⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡟⠘⣿⡇⠀⠀⣿⡇⠀⢸⡇⠀⠀⢀⡤⠄⠠⣿⡑⢀⣴⠂⣾⣡⣤⡿⠁⠀⣴⡶⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⠀⣿⡇⠀⢸⡇⠀⠀⣿⣇⠆⠀⡹⠓⣿⠃⠀⠘⢡⡿⠁⡠⣺⡿⠁⡰⣿⠇
⠀⠀⠀⠀⠀⠀⣠⡶⠛⠳⡀⠀⢰⡖⢆⠀⠀⣿⡇⠀⠀⢿⡇⠀⢸⣇⠀⠀⢻⣧⣠⠜⠁⠸⣏⡠⠊⠀⠸⡧⠔⠀⢿⡥⠊⣼⠏⠀⠀⢀⡷
⠀⠀⠀⠀⠀⢰⣿⠃⠀⢀⡇⠀⢿⣇⢸⠀⠀⣿⡇⠀⠀⠉⠁⠀⠀⠻⠦⠴⠃⠀⠠⠤⠶⣤⣤⣴⣆⣀⣠⠤⠒⣀⠀⢠⣾⠯⠤⠤⠶⠟⠁
⠀⠀⠀⠀⠀⣸⣿⠈⠷⠾⠁⠀⠸⣿⣸⠀⠀⠀⠀⢀⣀⡀⢠⣾⠀⠀⠰⠆⠀⣰⡟⡃⠀⠀⣹⡟⠉⠉⠀⢠⡊⣀⣤⠿⠃
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⣾⠁⣧⢾⣿⠀⠀⣶⡆⠀⢻⣿⠀⠀⢠⣿⠁⠀⣠⠀⣈⠉⢉⠁⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠈⣿⣆⠐⠙⣷⡘⠷⣿⢸⣿⠀⢀⣿⠃⠀⠌⢿⣇⡀⢼⡇⠀⢀⣿⠃⣼⠆⢹⡇⠀⣴⡟⢩⣿⠁⢸⣧⠆⠀⢦⡀
⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢹⣇⠀⢸⠈⣿⣀⠜⣿⣇⠜⣀⠼⣿⡆⢾⡇⣠⣾⡇⠀⣿⠀⣾⠁⢸⡿⠀⣼⡇⠀⡸⣿⡄⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⣿⣷⠀⠈⣿⡀⡜⠀⠀⠀⠀⢈⠽⠊⠁⠀⣿⣇⠈⠉⠁⠈⠀⠈⠃⠀⢿⠔⠻⠷⠊⠻⠦⢊⡠⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⡏⢿⠇⠀⠈⠋⠁⠀⠀⠀⡴⠁⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡛⠀⠀⣰⣔⡆
⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠶⠟⠀⠀⠐⠀⠀⣈⠤⠀⠀⠀⢠⠄⠸⠇⠀⠠⣿⣋⠽⠄
⠀⠀⠀⠀⢐⡟⠀⠀⠀⠀⡾⠎⠀⠀⠀⡓⣀⠀⣭⠀⠇⣀⠀⢸⢼⠀⠐⢀⣾⡟⢁⣠⢶⡀⢀
⠀⠀⠀⣼⣿⠤⠀⢒⠀⢠⡆⠀⢨⣃⢸⣇⡏⠺⢀⣠⣆⣸⡚⠉⠈⣉⣢⡄⢩⢶⡉⠀⡈⠁⡘⣀
⠀⢠⢰⣏⢥⠈⢁⠁⠁⡌⠀⠷⠌⠃⠜⠀⢷⡀⠸⠙⢠⣯⣉⣉⣉⣅⣤⣶⠀⢸⠖⢉⠀⢘⡁⢙⡀
⠀⠀⠳⣿⣮⡥⠤⢴⡄⠀⠀⢠⠀⠀⠀⠀⠈⣧⠀⠀⠀⢻⣞⣓⣛⣚⣙⡏⠀⠀⠰⠁⠀⣼⠄⠠⠷⡄
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢶⣿⣷⠒⡀⠀⢀⠇⠐⠀⠂⠀⠈⠙⠀⠘⠁⠀⠀⠀⠀⠀⠛⠛⠻⡟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠘⣄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠁⠄⠄⠄⠂⡄⠀⠀⠰⠂⠀⠀⠀⠂⠴⠆⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢮⣵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢀⣤⡙⠲⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣾⣶⣾⣿⣿⣿⣦⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣠⠞⠋⣠⣿⣿⣿⣝⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⠛⢥⢀⣴⣿⣅⡽⢿⣭⣻⣛⣦⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣥⣬⣹⡿⡟⣋⣥⣾⣁⠾⠟⠋⢀⡷⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠛⣹⠟⠋⣠⣶⣷⣤⣀⣩⣷⣾⣿⣥⣴⢋⣀⣤⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣤⠿⢻⡈⢁⢀⣰⣿⣿⣧⣄⣡⣱⣶⣤⡜⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣍⣀⡴⠂⢀⣤⣷⣼⣿⣿⣿⣿⣿⣏⠀⣹⣿⠛⡟⢯⣷⣦⣴⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣥⢶⣿⣿⣿⡿⠿⣿⣿⡟⣿⣷⣮⣄⣈⡉⠙⢲⡬⠁⠱⠚⣂⠲⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠚⣀⢨⠷⠛⠁⢀⣿⡏⢙⣟⣉⣀⣉⡻⣿⡭⠞⢫⣦⣶⣤⡾⢫⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢘⢉⡉⠀⠙⣆⠀⣴⡿⠛⢋⣩⡉⠻⣷⣤⡉⠁⠙⠷⣦⣠⣌⠉⠛⡛⢋⣋⣠⣤⣿
⠀⠀⠀⠀⡟⣸⣿⣶⠆⠀⠛⠾⢻⣿⢂⣵⡿⠱⢏⣀⣴⡿⢻⣷⡀⠈⠻⣿⣀⣀⢰⠋⢻⠻⣿⣶⣿⡾⠉⠁⠀⠁⠀⠀⢀
⠀⠀⠀⠀⣿⣿⣣⠏⠀⣀⣀⠀⠈⣹⣿⠏⢀⡀⣨⣿⣶⣏⢙⣿⣇⡀⠀⠹⣿⣿⣤⣭⣉⡀⠈⠛⠦⣄⡀⠀⢀⣠⣶⣸⣯
⠀⠀⠀⠰⠏⢀⣏⣦⢻⣿⣿⣷⠞⠋⢁⣴⣎⣿⣟⣿⣉⣉⢉⣿⡿⣧⣴⡆⢠⣌⣿⠷⣍⠉⣿⠶⢶⣴⣎⣛⢛⣿⡿⠿⣿⣀⡀
⠀⠀⠀⢰⡄⣼⠀⢻⣆⢿⣿⣿⠸⢶⡾⠛⣻⢾⣿⣿⣿⣿⠟⠛⠛⠛⣿⡇⣾⣿⣿⠁⡼⢃⣀⠀⠀⢘⡿⡿⡿⣿⠃⢰⠷⠦
⠀⠀⠀⢸⡇⠹⣖⢋⡟⠈⣁⣠⣴⣟⣶⣿⡿⠛⠻⣿⣿⠁⢀⣤⣤⣄⡟⠁⠻⢿⣯⣾⣷⠿⠛⢻⡆⢸⠤⣧⡼⢃⢄⡟
⠀⠀⡔⣿⣷⡀⠙⣟⣷⣿⣿⠉⠈⢿⣿⣿⠇⠀⠈⢧⢻⣤⡟⠁⠈⡿⠙⠓⠛⣾⣿⣿⣇⠀⡀⢸⣏⡾⣋⣩⣴⡯⠞⠶⠤
⠀⠀⠘⠻⠟⠑⣴⣙⡿⠋⠁⡀⢐⣿⣿⣷⣁⢀⡀⣿⡤⣿⣇⠠⠄⣿⢀⠀⠐⠷⣿⣅⢋⡀⢁⡜⢻⣯⣥⡾⠛⠛⢿⡶⠶⠐⠂
⠀⠀⠀⠀⣴⢶⠛⢟⠻⠿⣷⣪⡜⠉⣻⡉⠈⠑⠒⠋⠀⣏⢳⣥⣤⡾⠧⡠⡄⢶⣿⣾⣷⣬⣅⡴⢋⡈⠟⢂⡮⠄⠈
⠀⠀⠀⢰⣇⠘⣠⣈⣷⣄⣀⢩⣑⠛⠻⢷⣦⣄⣀⣤⣴⣿⣿⣿⣿⣄⣤⠋⣱⣤⣿⣿⡿⠿⠛⠋⠁⠀⠀⢸⣧⡄⠀⡀
⠀⠀⠀⢸⣣⡎⢰⣿⣿⠿⠋⢸⡯⣿⡆⠀⠀⠈⠋⠛⠛⠛⠛⢛⡛⡉⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠿⠃⢀⠝
⠀⠀⠀⣯⡾⡇⠈⣻⢈⠀⠀⠸⠧⣿⣇⡞⢲⠀⡤⢠⡀⡄⢸⠈⠁⡧⣄⣤⠤⡤⢄⣺⢰⣶⣿⣼⠍⠃⣸⠋⢑⡏
⠀⠀⠀⡟⠀⣹⣦⠇⠙⠒⠦⢤⣄⣀⣀⠁⠘⠀⠓⠘⠳⡇⠘⠀⠀⠗⠚⠻⠀⠃⠘⠉⠈⢉⣁⣀⡤⠔⢳⣀⠞
⠀⠀⠀⢻⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡠⢄⣀⣀⣀⣀⣀⣀⣀⠴⠶⠶⠖⠙⠈⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⢠⣿⡷⠀⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⣛⠏⣤⠹⠟⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠗⠛⢱⡿⠀⣦⢠⡆⠀⡀⢈⣠⢀⣴⢦⡆⡶⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⢀⣿⣾⠁⣎⠟⡿⠹⣾⠘⢺⣡⡧⠂⣨⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⡠⠂⠟⠸⠞⠃⠘⠃⠀⠁⠀⡴⡿⠀⠀⠀⣈⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⠀⠀⠀⠀⡤⢀⠀⣰⢀⣠⡈⡄⢠⢶⠀⣏⠀⠀⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡁⣾⣤⢠⣴⣦⡇⢸⣀⡏⣼⢹⢳⣇⣯⣾⣆⠿⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⣼⠖⢡⡇⣟⡾⠈⠹⠷⠸⠛⠛⠃⠈⠈⣁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠁⠀⠀⠀⡀⠤⠐⢂⠀⠩⠄⠂⣁⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠖⠉⠀⠈⠁⠀⢠⣶⡤⣶⣿⣿⠛⢀⣴⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⣴⣦⠉⠈⢉⣡⣅⣘⡛⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠙⠛⠘⠟⠀⠛⢿⣿⣿⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⢀⣰⣕⣠⣄⡀⢎⣄⣠⡢
⠀⠀⠀⠀⠀⠀⠈⣸⡇⠈⠙⣦⡾⠋⠀⢹⣷⣅⣀⣢⣠⣵⣀⣒⣀⣶⣀⣤⣀⡠⡄⣒
⠀⠀⠀⠀⠠⡀⢈⣸⡇⠀⠀⠈⠁⠀⠀⢸⡏⠈⣉⣹⠁⠠⠌⢻⠁⠠⠌⢻⡍⠹⠖⠋⣷⠔
⠀⠀⢆⣦⠞⠓⠾⣼⡇⠀⣦⡀⢀⣴⠀⣼⡇⠀⠰⢿⡇⠀⡐⢿⡇⠀⡐⢿⣿⡆⢰⢛⡵⢀⡀⠀⠀⣀
⠐⢽⡞⠁⢀⣀⢀⣹⡿⢾⡿⢿⡿⠿⠿⢿⡿⠶⠶⣾⠷⠶⢿⡿⠷⠶⣿⠿⢿⣷⡿⢶⡧⠶⢧⣖⣦⠷⠦⣮⠇
⠪⢿⡅⠀⠘⠟⠛⢻⡇⠈⠁⠈⣇⠀⠛⣠⣷⡄⢰⣿⢄⠐⠺⣶⠀⢰⢿⠀⠈⠋⠀⢸⡇⠠⠀⢳⡧⣀⠐⢿⣄
⠀⠜⠷⣤⣄⣀⣤⢾⣇⣸⣇⣸⣿⣀⣠⣨⣏⣁⣨⣧⣀⣀⡴⢻⣀⡿⢸⣀⣦⣠⣆⣸⣇⣴⢦⣨⣦⣀⣀⣼⢇
⠀⠀⠀⠊⠀⢠⡏⠈⠙⢳⡄⠁⠀⠀⢸⠁⠉⠉⢳⠀⠀⠀⠀⡾⠉⠓⠲⣄⠀⠀⠀⢸⠃⠙⠒⢦⠀⠫⠃⠀⠉
⠀⠀⠀⠀⠀⣸⠁⠀⠀⡼⠁⠀⠀⠀⢸⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⢠⡇⠀⠀⠀⢹⠀⠀⠀⡼
⠀⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⠀⠀⠀⢰⠃⠀⠀⡎⠀⠀⠀⠀⡼⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠙⢦⡀⢸⠄⠀⠀⠀⠀⠈⠓⢤⣀⣸⠆⠀⠀⠈⠢⣀⠀⢳⠀⠀⠀⠀⠙⢦⣀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⢀⠀⠀⠀⢈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⠖⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣾⠁⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⢧⡀⠀⠈⠳⢌⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⡀⢀⣇⢀⠀⢹⠓⠦⠤⠤⣧⡄⠀⢩⣿⠀⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠈⢂⠀⠙⢸⡏⣸⠁⣿⢀⡞⠁⠀⣿⠃⠀⢺⡇⠀⣿⢸⡇⠀⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣤⠀⠸⠀⠀⢘⡁⠉⢰⡇⠘⠧⠀⠀⢿⠀⠌⠘⢇⠰⠉⢸⡇⠘⣆⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢸⡏⢀⠃⠀⣰⠏⠀⠄⠈⠃⠀⠀⢀⡤⠠⠀⠀⡀⠀⠀⠀⡼⠁⣰⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠉⠀⠀⢠⡿⠀⣠⠤⡀⢀⡤⠖⢾⡇⠰⠆⣼⠃⠀⠀⠘⠁⠀⠀⠀⢀⡿⢁⡏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢀⡠⠀⠀⣼⡇⣼⡇⠀⠇⠈⠀⠀⢸⣇⢠⢠⣿⠁⠂⠄⡀⠀⠀⣀⠴⠛⠁⠀⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⡞⠀⠀⠠⣿⠀⠹⣧⢰⠀⠀⢀⠀⠀⢻⣇⢸⡇⠀⣀⠀⠀⠀⠀⠀⣠⠄⡀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠃⠀⠐⠀⣿⠀⡆⠈⢍⣾⠀⣤⠀⠀⠈⢻⣎⠷⠀⢸⡇⣄⢠⡀⠠⡟⣸⠃⠰⠁⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠠⠐⠀⠀⠀⠉⠘⣇⠀⠘⣇⠠⣧⠀⠄⠀⠈⣿⡀⠀⣿⢡⡇⣾⠀⢀⡁⠘⠂⠁⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠄⢤⡀⠀⢹⡇⠀⠁⢀⢰⡟⠀⠈⠙⣦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢸⡿⠀⢸⡇⠀⠰⠁⠘⡇⢀⠆⠀⣿⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡾⠃⢀⡿⠁⠀⠒⠀⠀⠈⠀⠀⣴⠏⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠴⠋⠀⠀⠀⠘⢄⠀⠀⠀⠈⠉⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠠⠔⠛⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⠀⢀⡄⠀⠀⠀⠀⠀⢼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠜⠓⠀⠀⠀⠀⢠⢾⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢈⣽⣅⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠒⢠⡤⡐⠀⠀⠀⢠⣾⡿⡄⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⢾⡣⠀⠀⠚⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠐⢉⣼⣿⡌⠂⠀⠀⢀⣾⣿⣧⠀⠀⠐⢨⡑⠀⠀⣼⣿⣿⠄⠀⠀⢀⣿⠀⠀⠀⠐⠾⣿⣿⡿⠒
⠀⠀⠀⠀⠀⢰⠖⠖⠂⠀⠀⠠⠋⣽⣿⡙⠦⡀⠀⣼⣿⣿⣿⣆⠀⢘⡱⣌⠂⠀⣰⣿⣷⣄⠀⠀⣼⣛⣇⠀⠀⠠⠾⣿⣿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣽⣯⠳⢄⠀⣸⣿⣿⣿⣿⣿⡄⠋⣸⣌⠃⠘⢛⣿⣏⠉⠀⢰⣿⠼⣿⡄⣀⣤⣾⣿⣿⣟⠒⠂
⠀⠀⠀⠒⠂⠀⠀⡆⠀⠀⠀⡠⠞⠉⡏⠳⠀⢠⣿⣿⣿⣿⣿⣿⣿⠈⢁⠈⠀⢠⣾⣿⣿⣆⠀⣾⣿⠱⣿⣧⠀⢀⣴⣿⣿⣿⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⠉⠉⠭⠉⠉⠘⠛⠛⠛⠛⠛⠀⠉⠉⠩⡏⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣄⠀⠀⠀⢀⠁⢀⡤⠀⡤⠌⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠄⡀⠀⠀⠀⠀⠐⡀⡀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⣄⣂⠀⠀⠀⠀⠀⠀⢠⠓⡄⠈⣶⢠⡄⡄⣤⣴⠖⣦⢹⣆⠀⢀
⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⠋⣿⠃⡷⠘⢦⡸⡏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄⠁⡀⠃⠲⠶⠃⠙⠀⠺⠃⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⠀⠀⢀⣾⣁⡟⣰⠟⠡⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠙⢀⠀⣾⡁⣷⡤⠀⠀⠘⠀⠀⣤⢠⠀⠀⠀⠀⢪⡵⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠈⠱⠀⠀⠀⠀⠈⠉⢀⠄⠀⠀⠀⠵⢖⠂⠀⠑⠒⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⠀⠀⠠⠀⠀⠈⠀⠈⠀⠀⡤⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢄⣦⡆⠀⠀⠀⠀⣾⠀⢀⡀⠁⠙⠓⠦⢦⢴⡤⠶⠃⠀⠀⠠⠃⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢀⣀⠀⠠⠈⢀⣈⠀⠄⠄⠀⠀⠠⠠⠤⠄⠀⢀⠀⢠⣦⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⢹⡇⢤⡄⠈⣿⠀⢀⠄⠀⠀⢢⠄⠂⣶⠂⢸⡇⢀⣿⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⡇⢸⡇⠀⣿⡀⢿⣄⡀⠠⡿⠀⠰⠷⠀⠛⠗⠀⣿⠀⠐⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠂⡀⠠⠀⢸⡇⠸⠗⠀⠙⠓⠀⠀⠀⢀⠀⢰⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠒
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠈⠋⢤⠀⠀⢀⡉⠉⠉⠲⡤⠒⢻⠛⠛⠛⠲⠦⣤⣿⣀⣿⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⢸⠀⠀⠘⠛⠀⠤⠀⣀⣀⢸⠀⠠⣤⢤⡽⣍⢠⡤⣄⢀⣀⡀⠀⠁⠐
⠀⠀⠀⡀⠀⠀⠀⠠⡂⠐⠀⠀⠀⢸⣀⡀⢠⡴⠶⢺⠘⢧⡉⢸⠀⠀⡇⢸⡇⢸⢠⠖⣻⠈⠳⣅⢀⣀⠠⠒⠚
⠀⠀⠀⠀⠐⠀⣨⡁⢰⣶⠁⠀⠄⢸⠁⢹⢸⡇⠀⢸⡀⣄⡽⠚⠶⠓⠓⠘⠃⠙⠈⠛⠙⠐⠳⠟⠘⠋⠀⠁
⠀⠀⠀⠀⠀⠂⠈⠁⠀⠛⠶⠖⠀⠾⠀⠘⠚⠃⢀⣀⣠⡤⠴⠖⠒⠙⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⢠⠁⠀⠈⠈⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡄⠀⠈⠁⢈⣄⣀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⣴⣿⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⢿⣆⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⣴⡿⠋⠀⠀⠈⢧⡻⡄⠀⠀⣤⡆⠀⠀⢀⣴⠟⠁⠀⠀⣠⣴⠿⡟⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⡾⠋⠀⠀⠀⠀⠀⠈⢧⢱⡀⠀⠛⠃⠀⣠⡾⠋⠀⢀⣴⡾⠋⠁⠀⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠞⣡⣤⣤⣤⣄⡀⠀⠀⠈⣧⢧⠀⢀⣠⣾⣿⣤⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢠⡾⠛⠋⠉⠉⠛⠻⣷⣄⠀⠈⢿⣷⢟⣛⣩⣭⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠇
⠀⠀⠀⢀⣀⣤⣤⣶⣶⠿⠟⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢸⠁⠿⠛⢛⣻⡷⠀⠀⣰⣶⣶⣤⣤⣴⣶⣶⣶⣶⣾⠿⣦⣄⣀
⠀⠀⠀⣿⠛⠻⠿⠿⠷⣶⣶⣾⣿⣷⣶⣶⣤⣤⣤⣤⣤⣴⣾⣿⣾⣿⣿⠟⣿⠛⠁⠀⠿⠿⠿⠛⠻⢿⣿⣍⣩⣭⣭⣥⣤⣶⣿⣿⣿⣿⣷
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⢉⣽⠿⠋⣡⡿⢫⡿⠋⠁⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⠛⠛⠛⠋⠉⠉⠉⠉⡏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⣼⠋⢠⡟⠀⠀⠀⠀⠀⣶⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡼⠃⢀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠘⠻⠿⣿⠿⣿⣿⣿⣶⣶⣶⠟⠁⠀⠀⢸⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣾⣿⣿⣿⣿⠇
⠀⠀⠀⣶⠀⢰⡇⠀⠀⠀⠀⠈⣹⠏⢀⣠⡆⠀⠀⠀⣸⢷⣶⣶⣦⣤⣄⣀⡀⢸⡇⢀⣠⣤⣾⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⢠⣇⣴⠿⠋⠁⠀⠀⢀⡏⠀⠀⠉⠉⠛⠛⠿⢿⣿⣿⠿⠟⠛⠉⢹⠀⠀⢠⣿⣦⠀⠀⡸⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡟⠁⠀⠀⢸⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡆⢀⡟⠀⠘⣇⣰⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢡⡾⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣶⣦⣤⣄⣀⣀⠀⠀⠀⣸⡇⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣶⣾⣧⣶⣶⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⢀⣐⡍⠑⣠⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⡠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢤⠛⠶⢤⠷⢸⡷⠞⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⣀⣾⢦⣷⠠⠂⢉⡳
⠀⠀⠀⠀⠀⠀⠓⡐⢲⡎⠋⣍⠉⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢸⡧⠓⢁⠋⢉⠉⠉⠉⠉
⠀⠀⠀⡀⠀⠀⠠⠀⢨⣇⢠⠈⡑⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⡂⡒⢄⢀⡠⠐⠂⣻⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢰⣁⡀⢸⢀⡀⢡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⡌⠈⠁⠀⠈⠃⠀⡁⠀⠀⠀⠀⠐⠀⠀⠊⢉⡹⢢⣍⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⠀⠀⠈⠀⠈⣡⣬⠈⠁⠉⠈⢃⣀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢁⡈⢈⠉⡇⠀⡦⠉⢀⡲⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠐⠀⠈⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠈⠀⠀⢡⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣀⠅⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠂⠚⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⡁⠀⠀⣀⡸⠀⢸⣀⣀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⣄⠀⠦⡀⠀⢀⠀⠟⡀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣀⠀⢀⣀⣀
⠀⠐⢄⠂⠈⠒⠁⠀⠀⣻⡆⠀⠞⠤⠊⠠⠖⠡⠂⠴⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠦⠶⣛⢉⡩⢒⡛⠛⠻⣟⡓⠁⡀⠀⠑
⠴⠂⠀⠒⠒⠰⠷⠾⠲⠿⠿⠿⠒⠒⠢⠔⢲⠄⠈⠰⠐⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⡇⠀⠈⡀⠀⠀⠀⢆⠀⠀⡄⢠⠊⠉⠀⠁⠈⢛⠂⠀⠀⢀⡀
⠐⠀⠂⠉⠀⡀⠀⠀⠀⠰⠀⡎⠐⠂⠀⠀⠀⠂⠲⠘⢳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠜⠈⠉⣷⡋⠀⠻⠈⠁⠉⠈⠾⡅⠈⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠀⣠⣄⡀⠀⠀⡀⠄⠀⠘⡆⠀⠀⠣⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡧⠸⣷⠟⠃⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠉⡻⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠩⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⣠⡆⠐⠃⠇⠤⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⡇⠀⢰⡶⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⢀⢤⠤⠤⢄
⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣑⣦⡋⣐⡠⠂
⠀⠀⠀⠀⡈⠉⠘⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡀⢀⠀⡀⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠉⠛⡿⠞⡋⠀⠀⠀⢀
⠀⠀⠀⠠⠇⠀⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠒⠒⠲⠶⠶⢧⡀⣅⡞⢁⠴⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠠⠖⠠⠄⣀⠀⢰⠂⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠴⠂⢤⠀⠀⠉⠁⠀⠀⢠⠄⠙⣀⠄⠀⠀⠈⠉⠈⠁⠓⠀⠤⠆⣀⠀⠇⠀⡨⠦⠛⠉⠁
⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⣀⠀⠈⠀⠐⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰
⠀⠀⢠⠹⣧⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢲⠖⠊⠀⡇⠀⣀⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⠀⠀⠈⠐⠁⢲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠸⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⢠⢀⠀⠴
⠀⠈⠀⠂⠀⠄⢻⣷⡀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠘⠻⣶⣦⣄⡀⠀⢻⣷⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄⢻⣷⡶⠿⠿⠿⠿⠿⢿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣤⣀⣀⣀⣀⣀⣀⣀⡙⣿⣧⣿⣷⣶⣶⣤⣤⡀⠀⢀⣨⣼⡿⠟⠻⠿⣶⣦⣤⣀⡀
⠀⠀⠀⣀⣀⣠⣤⣴⣶⣾⡿⠿⠟⠛⠛⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡿⠛⠉⠉⠙⠛⢿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⣿⣷⣦⣤⣀
⣶⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣿⣦⣬⣿⣿⡀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣶⣶⣶⣶⣶⣶⣶⣾⣿⣯⣁⡈⠉⠛⠿⣷⣦⣄⡀
⣿⣯⠙⠿⣷⣦⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡾⠿⠛⠛⠉⠉⠉⢛⣿⣷⣄⠀⠀⠀⠀⠀⢀⣿⣯⣀⣤⣤⣄⣀⠀⠀⠈⠉⠛⠿⣿⣦⡀⠀⠀⣉⣹⣿
⣿⣿⠀⠀⠈⠙⢿⣿⣶⣶⣶⠿⠻⣿⣉⡁⠀⠀⠀⠀⠀⠀⣀⣤⣾⠏⣹⣿⢿⣷⣶⣶⣾⣿⣿⡿⠛⠋⠉⠛⠻⣿⣦⡀⠀⠀⠀⠈⣿⣿⣾⠿⠛⠋⢹
⢿⣿⠀⠀⠀⠀⠀⢽⡿⣿⣶⣄⢸⣿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⢁⣴⠟⠁⠀⣠⣿⣏⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡷⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⢰⣿⠃⠈⠳⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠏⠀⠀⠀⣿⣿⣴⣶⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣦⣤⣶⣶⣿⠟⢋⣁⣼⣿⠀⠀⠀⢸⣿⡟⠛⠉⠀⢸⣿⠻⣿⣦⣄⡀⠀⠀⣀⣤⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡿⠟⠛⠋⠙⢿⣿⡿⠿⣿⣿⠀⠀⠀⠘⣿⣇⠀⠀⠀⢸⣿⠀⠉⠻⠿⣿⣿⢿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢿⣿⡀⠀⠀⠀⠹⣿⣆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⣧⡀⢀⣀⣀⣬⣿⣦⡄⠐⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⢿⣿⡿⠿⠛⠛⠉⠉⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⣿⡀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⢿⣧⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⢻⣷⡀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿
⠀⠀⠀⠀⠻⣿⣄⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢀⣠⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⢸⡇⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⢸⣿⡏⠀⠀⣀⣠⣴⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⣿⣷⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠟⠋⠁
⣿⣿⣿⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠤⠤⣄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠛⢄⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⡆⠈⢳⡀⠀⢹⡤⠴⠖⠒⠒⠒⠂⠂⢄⡜⠁⠀⠀⣀⣠⣦⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠱⡚⠉⠙⣄⠀⢇⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⣴⠋⠁⠀⠘⣆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠁⠀⠹⣦⡱⣄⠀⠘⣆⠘⡄⠀⠀⠀⠀⡴⠁⣠⠔⠃⣀⡽⢦⣀⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡳⣄⠘⡆⢹⣦⡀⢀⠞⣡⣞⡥⠖⠋⠁⠀⢀⣩⣷⣄⡇⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢻⡲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣿⡞⠉⢩⡿⠛⠉⢁⣀⡤⠴⠚⣋⣉⣀⣀⣼⠇⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣿⡀⠀⢸⠉⠒⠚⠛⠛⠛⠛⣉⣉⣩⡭⠥⠖⠋⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠉⠢⣄⣀⡤⠴⠖⠋⠉⠀⠀⢀⣨⠽⠻⡏⠁⠉⠉⠙⠯⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⢨⡇⠀⠀⠀⣀⣠⠴⠚⠋⠁⠀⠀⠈⢢⡀⠀⠀⠀⠈⠳⢦⡀⠀⢀⣀⠤⠒⠋⠉⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠢⣀⠀⢸⡇⠀⠀⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⣀⡤⠖⢻⡟⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠑⠼⣇⠀⠀⣿⠈⠑⢦⡀⠀⠀⠀⠀⠀⢀⠠⠄⠊⣯⠁⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠻⣤⡇⠀⠀⠀⠈⠢⢀⠠⠐⠈⠀⠀⠀⠀⢸⠄⠀⢀⣸⡧⠤⠶⠚⠉⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⢈⡗⢄⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⣀⡼⠓⠛⢹⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣸⠀⠀⣼⠁⠀⠈⠒⢦⡀⣿⠀⣀⣠⠴⠚⠉⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⠀⠀⡟⠀⢰⠇⠀⠀⠀⠀⠀⠉⡟⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣸⠇⠀⣾⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⣇⠀⢀⣀⣠⣤⠼⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⡤⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢸⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢸⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠚⠁
⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣴⠋⠀⠀⠀⣀⠀⠈⢲⡀⠈⣧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢸⠀⠀⡈⠉⠛⠻⡿⣶⣿⣌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣸⡄⠀⠳⣄⠀⣸⠇⢸⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣦⡼⣿⣯⣴⣟⣡⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡐⡆⢀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣉⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣛⣿⣿⣿⣾⣦⣶⣶⣾⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⠟⠋⠉⢿⣿⣿⣿⠃⠀⠀⠀⠀⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⣿⣿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠸⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣷⡆⠀⠈⣿⡿⠻⢿⣿⠟⠛⠛⠛⢻⣿⡟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠉⠛⠛⠉⢹⣿⡏⣠⣾⡿⠃⠀⠀⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⢻⣿⠀⠻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣇⠀⠀⠀⠀⢸⣿⡿⠋⠁⠀⠀⠀⠀⣿⠀⠀⠘⣿⠀⠀⠀⠀⢀⡾⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⢿⣄⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣿⠀⠀⠀⢀⡾⠁⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠘⠿⠆⠀⠀⠰⣿⡀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢠⣿⠀⠀⢠⣿⠃⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⢶⡄⠀⠀⠱⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⣀⣀⣀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⠤⣤⣙⣦⣄⠀⠙⣆⢣⠀⠀⠀⠀⠀⠀⡠⠃⢀⠔⠉⠀⠀⡘⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡈⠙⠻⠷⣤⣈⢙⣠⣤⣤⡀⡠⠜⡡⠔⢁⣀⣀⣀⣔⣁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠠⠀⡀⠀⠉⠛⠒⠒⠒⠊⢹⣀⠀⢡⡏⣐⠛⣉⣉⣁⣀⣀⣀⠀⠉⠳⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣄⣀⠠⢤⣤⡄⠁⠒⠒⠒⢒⠶⠾⣿⣆⣸⡇⠀⠙⣦⠨⢭⣉⣀⠉⠀⠁⣰⣧⠴⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠈⠁⠈⠒⢶⡎⣧⢀⣈⣉⣉⣁⠰⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⠤⢄⣀⣀⡤⠖⠇⢸⣦⡀⠀⠀⠉⢻⡇
⠀⠀⠀⠀⠀⢀⠁⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡀⡸⠀⠸⡇⠉⠛⠲⠒⠋
⠀⠀⠀⠀⠈⠓⠒⠛⡇⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⠀⠀⢸⡃⠀⢠⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡾⠉⠢⣸⠀⠀⢨⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠷⠒⠛⢆⢠⠇⠀⠀⢹⠀⠀⠸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⠀⠀⠀⢸⠀⠀⠀⠈⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠈⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡰⠀⡆⠀⠀⢸⠀⠀⠀⠘⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⠀⡇⠀⠀⢸⠀⠀⢀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣠⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⡀⠀⡇⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁
⣿⣏⢹⣟⠻⣿⣟⠟⡻⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠢⡀⠀⠈⠪⡉⠐⠂⠔⠁⠀⠀⡠⠊⡹
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⡲⠏⢉⡒⠈⢐⣄⣀⡈⣦⠎⣀⣀⡔⠪⠔⣊⠉⠲⢄⣀
⠀⠀⠀⡤⣒⠊⠉⠀⠀⠀⠀⠄⢀⣀⡀⠁⣂⣬⠭⡿⣤⠼⡯⣥⣄⠈⢁⣀⡀⠀⠜⠀⠉⠉⠒⡢⡄
⠀⠀⠀⡆⠀⠉⠑⠢⠤⣠⡄⣒⡩⠥⠒⠉⠁⠂⠉⡠⠋⠢⣀⠁⠀⠁⠐⠢⠍⡒⢤⡤⠔⠊⠁⠀⡆
⠀⠀⠀⡇⠀⠀⠀⠀⢀⠗⠋⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠈⠣⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠢⣄⣀⢀⡴⠁⠀⠀⠀⠀⢀⠔⠉⠉⠒⠢⢄⣀⣀⡤⠐⠊⠉⠢⡀⠀⠀⠀⠈⠂⠀⣀⡤⠂
⠀⠀⠀⠀⡇⠀⢋⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⡗⢄⠀⠀⢠⠛⠉⠀⡇
⠀⠀⠀⠀⡇⠀⠈⢆⠀⠀⢀⠜⠓⡧⢄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⡷⠊⠢⡴⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠈⠣⡤⠃⠀⠀⡇⠀⠈⠉⠒⠤⢄⣀⣃⡠⠔⠊⠁⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠧⢄⡀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠈⠑⠢⠧⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⢄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡠⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣀⢸⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣾⡀⣀⡖⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢿⡏⠓⢿⣀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡁⣾⠛⠻⠿⣟⢆⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣁⡴⠶⠦⢭⣿⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠈⢦⣿⡿⡄⠉⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡆⠀⠀⢯⣷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡇⠀⠀⢰⡏⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⠀⠀⠀⠀⠀⠀⠀⢈⣾⣧⡤⣠⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠻⢿⣦⣀⡀⠀⠀⢠⣺⡾⢋⣩⡾⠟⠋⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣵⣶⣌⣉⣙⣓⣪⡴⣫⣶⠟⠋⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣀⣤⣤⣤⣴⣶⣶⠶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡐⠐⠟⢨⣿⣿⣿⣧⣿⢗⣤⣀⢀⢀⣄⣠⣤⣶⣿⣿⣿⡿⠿⢛⣻⣿⣿⣿⣿⠏⠁⠂⠐⠲⠝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠛⣟⢛⣏⣸⣿⣈⠉⠛⠛⠿⠻⢿⡟⣿⠟⠙⠉⠘⡃⠸⣿⣿⣿⣿⣿⣷⡄⠀⠀⢰⣌⢸⡿⢂⣀
⠀⠀⠀⢠⡄⡀⢀⠀⠀⠀⠀⠀⠐⠒⠒⠒⡻⡻⠻⠟⢻⣿⣿⣿⣿⣶⣤⣤⠇⠀⠀⠀⣧⢀⠀⠀⣥⣾⣿⣿⣿⣿⣿⣿⣷⣔⢧⣾⣿⣿⡷⠿⠿⠿
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣀⡀⣠⣴⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣿⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⠿⠛⠉⠋⣧⡤⠆⠀⠰⣦⠀⠋⣹⡿⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠁⠀⠀⠀⠀⢶⣤⠿⠟⠛⠛⠋⢻⣿⣿⡿⣿⣿⠟⠛⠛⠛⠛⣿⣿⡆⠀⢻⣿⣇
⠀⠀⠀⠀⠀⢠⠀⠀⠴⠦⠤⣤⣤⣤⣤⣄⠀⠈⠂⣠⠸⡧⠤⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡏⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⠀⠉⠀⢹⠿⣦⣤⡶⣾⡑⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⡇⠀⢀⠀⠄⠀⠀⢸⣿⠃⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠰⣄⣷⡄⠀⢀⣀⠀⠀⢀⡏⠀⣀⣼⠑⠿⣇⠀⠀⠀⠀⠀⢃⡠⠠⣤⣬⣳⠟⢿⣿⡇⢸⡃⣼⣴⣷⣶⠧⠂⢸⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠠⢸⠟⠉⠙⠛⠺⢲⣿⣿⡿⣿⡿⠛⠳⣦⣨⠀⠀⠀⠛⠞⣷⣥⣾⣿⣿⣿⣤⣸⣿⢠⣼⠛⠉⠀⠀⠀⠀⠀⢻⠁⠀⠀⢀⣟⠠⠄⠀⢠⠄
⠀⠀⠀⠀⢠⠆⡇⠀⠀⠀⠀⠀⠌⠀⣿⡇⠈⠁⠀⠈⠀⠀⠄⣐⣥⣟⣆⣻⣿⠿⠛⠋⠉⠉⠈⣉⠀⢸⠀⠀⠀⢀⣀⣀⠀⣼⣤⣶⣶⠜⣿⢶⣶⣶⡆
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠘⠇⠀⠸⡇⢲⠐⣄⡤⣀⣀⣤⠘⠫⠙⠉⠁⠀⠀⠀⠀⠀⠀⣀⢻⠀⠀⣄⣀⣤⡾⣿⣥⣍⣵⣿⣿⣯⣍⣹⡿⠟⠋
⠀⠀⠀⢀⡃⡀⠀⣀⣤⣤⣄⠀⠰⢤⣤⠳⣼⣿⠿⠷⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠶⠿⠛⠓⠲⢯⣏⣉⣁⣀⣩⠟⠂⠉⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠉⠁⠈⠈⠀⠐⠂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⡔⠀⠀⢀⠀⠰⠀⠀⠐⠻⠿⠟⠋⠋⠉⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠱⣿⣄⣤⣀⡀⡀⠀⠖⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠚⠛⠛⠻⠿⣿⣿⣦⣦⣤⣠⣶⣶⣄⣀⣠⣄⣀⡛⢮⣀⡀⣟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⡀⠀⠀⠀⠉⠉⠛⠻⣿⣿⡿⠯⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠁⠈⠋⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠓⢤⠀⠀⠀⠀⢀⠴⠒⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠔⢲⣄⢳⠀⠀⣰⠃⢀⠴⠒⠦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠙⢎⢧⣠⠇⢰⣋⠴⠊⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠛⠒⠚⠛⠒⢺⠆⠀⠀⢯⠤⠤⠶⠥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠛⠛⢦⠀⠀⠉⠀⠀⡴⠉⠉⠳⣆⠀⢸⠁⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⡀⠀⠀⠀⡼⢁⡔⡂⠀⡧⠀⠄⠀⠀⡇⠠⠙⣦⡘⣆⡼⠀⡴⠋⠀⣀⣀⠙⡆
⠀⠀⠀⠀⠀⡜⢁⣠⣄⡀⠉⢦⡀⠀⡇⣾⠃⠀⡰⠃⣀⣄⣀⣀⢱⢄⡀⠘⢧⢳⣇⡜⢀⡴⢋⡠⠊⢀⡇
⠀⠀⠀⠀⠀⢣⡈⠑⠤⣉⣳⡀⠱⠶⢣⣡⠴⠞⠓⠉⠁⠈⣇⠈⠉⠉⠙⠓⠲⠆⠍⠁⣜⣉⣁⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠙⠢⢤⣤⠤⢼⠂⠈⠈⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠉⠉⠙⠓⢢
⠀⠀⠀⠀⠀⢰⠒⠉⠁⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀⣇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢡⠀⠀⠀⠆⠂⠈⠀⠀⠀⢩⠀⠀⠀⠙⠀⠀⠀⠗⠀⠠⠄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠤⠀⠀⠈⠈⠄⠀⠀⠰⡀⠀⠀⠀⠈⠙⡆⠀⠀⡄⠀⠀⢀⣀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⠠⠀⠚⠀⠀⠀⢃⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⡈⠀⠀⠁⠀⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⣆⣀⣀⠴⠴⠒⠉⠉⠒⠒⠲⠤⠤⠴⣄⣀⣀⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⠴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣾⠃⠀⢠⡖⢠⠀⣾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣾⡟⢀⡟⠀⣀⡘⢷⣘⡀⠻⣦⣀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣠⡟⣼⢣⠃⢸⡇⣾⠃⠂⠠⣿⠃⠠⣰⡏⣰⠇⣾⢣⡶⠂⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢂⡾⢇⠣⠠⠾⠣⡼⠀⠸⣇⠿⠤⠀⡀⠧⠔⠁⢻⡂⠛⢸⡏⠈⠁⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠘⠃⣪⠄⠄⠀⣴⢿⠀⠀⠈⠁⢀⣤⠀⠀⢠⠃⣰⠏⢀⡿⠀⢰⡏⠁⢱⠁
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠙⠄⣼⠇⠀⣼⡁⠉⠁⢠⡄⠀⠀⢸⡟⠀⠀⠚⠋⠁⠔⠋⠀⠀⠀⢉⠀⢠
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢸⡏⢀⡆⠘⢳⡶⢠⣶⢰⣿⠀⣾⠃⣰⡇⢰⡖⣰⡆⣴⠋⣾⠀⣿⢀⠆
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣾⠃⣼⢃⠀⣿⠁⣼⡇⠀⣿⡆⣿⢀⣿⢁⣿⢡⡿⢸⡇⢸⣇⠀⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠉⠀⢉⡥⠀⠉⠀⠈⠁⠎⢸⡇⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠐⠉⠀⠀⠀⠀⠀⠀⠁⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠁







⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠰⢁⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡀⢰⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠂⣀
⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠁⠉⠉
⠁⠒⠂⠁⠀⠃⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⣂⠥⣌⠂⠀⠤
⠀⠀⠀⠐⠀⠀⠊⠐⠄⣁⢀⠀⠀⠀⠒⠁⠀⠀⠀⢀⠔⠂⢀⠀⠀⣀⠤⠄⣀⡀⠤⠔⠊⠀⠀⠈⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢢⠆⠀⠀⠈⠈⠣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢇⠀⠀⡤⠀⡀⠀⠀⣆⠀⠉⠉⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⡇⠁⠘⠳⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⣀⠀⠀⣄⠀⠸⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡶⠁⠀⠀⡌⠀⠀⠀⠀⠀⠘⡳⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠁⠸⠞⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⢠⡇⠀⠀⠀⠀⠀⣀⢀⡁⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡄⠣⡀⠀⢠⠔⠋⠉⠀⠉⠑⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢃⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⡀⠀⢀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠢⣀⠀⠀⢠⠁⠀⣸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠈⢳⠒⢁⠀⠀⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⡄⠀⠀⢠⠀⠇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢁⠀⠀⢸⠀⢸⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⢸⡄⠀⠈⠀⢸⡄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡇⠀⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡧⠀⠀⠆⣀⠆⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠃⠀⠁⠘⣅⣰⠀⠈⠁⢸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣾⡛⣷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠉⠉⠛⠉⣿⢿⣍⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠳⣧⠀⠀⠀⠀⠧⠤⠟⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⡞⠀⠀⣷⣄⡀⠀⠀⠀⠀⠀⠀⢳⡘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢮⣀⣽⢷⢂⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠉⠛⣲⣄⣀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣹⡞⠲⢽⣦⣞⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣾⠈⣽⠀⢈⠑⣄⠀⢀⠠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠅⠲⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠒⠸⠏⢁⡠⢎⣀⡀⠀⠤⠤⢄⠀⠀⠀⠀⢤⣠⠄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣴⡉⠟⠢⣀⡋⠀⠀⠀⠀⠄⠀⠀⠀⠂⠠⠤⠀⠄⠁⠘⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⣀⠀⠀⣀⡀⣀⢀⣀⣤⢠⣤⡆⠸⠋⠑⣏⢉⡧⠀⠀⣉⣀⣀⣀⡀⠀⠀⠀⠉⠀⠀⠐⠒⠒⠂⠴⣧⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠃⠋⡹⢠⡿⢿⡐⠉⠉⠈⠁⠀⣿⠀⣻⠀⠀⠀⠉⠈⠁⢀⠜⠁⠀⠀⠀⠀⠁⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀⠙⠚⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⢴⣆⣴⢛⣁⣸⡦⡴⠇⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠄⠀⡤⠖⢡⠖⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⣡⣿⢇⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠯⠄⠀⡀⠀⠶⠀⠀⠀⠀⠀⠀⠀⡃⠀⢠⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠉⠙⢳⣷⣶⡱⣤⡔⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠐⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠠⠛⠃⠀⠀⠀⠀⠽⣤⡼⠀⠀⢠⠟⢻⠊⠛⢫⡷⢶⡰⠦⢆⡺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠐⠛⠒⡶⣵⠇⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠚⠀⠀⠸⣍⣩⠃⠀⠀⢰⠈⠐⠤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢄⡀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠤⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠗⣲⣤⡔⣠⢤⡄⣤⣤⠀⣤⣤⣂⡤⢔⡲⠴⢮⣴⠿⡥⢋⠝⠓⠠⠤⠄⠉⠉⣉⡉⣉⠤⠐⠒⠖⠀⠈⠁⠢⡀
⠀⠀⠀⠀⠀⠀⢠⡴⠃⢴⡾⣏⢈⣽⣶⡆⠀⠀⠀⣜⠋⢳⠀⠀⠀⠀⠀⠀⠷⢦⡞⠁⠀⠀⣶⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⢢⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣏⣹⣼⠤⣄⣀⣀⡈⠛⠉⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠡⡀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠀⠙⠀⢸⡏⠹⡖⠋⠛⠔⠒⠮⣱⣶⣆⠰⢴⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠐⡀⠀⠀⠈⡄
⠀⠀⠀⠀⢀⡀⠔⣁⡴⠆⠀⠀⠀⣀⢀⠘⣳⠞⡃⠀⣀⠀⢰⠆⣿⠀⣯⠀⡀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢠⠀⠀⠀⡇
⠀⠀⠀⠈⠉⠉⠡⠶⠑⢒⣀⠤⠪⠗⠹⠔⢫⠜⠓⠊⠀⠲⠁⠈⠉⠁⠙⠒⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡆⠀⠡⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒⠂⠁⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⠄⠀⠀⢀⠀⠠⠤⠐⠊⠁

⣶⣿⣾⣷⣶⣿⣾⣿⣶⣿⣶⣶⣿⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣿⣷⣶⣿⣿⣿⣷⣿⣶⣿⣶⣷⣷⣿⣿⣷⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣻⡟⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⡼⠃⠤⣤⣙⠉⠀⡴⠆⠀⠀⠀⠀⢠⡀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡴⠛⣀⠀⠀⠨⣥⡀⣆⠛⠀⠀⠀⢶⣦⢸⣇⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠀⡷⢛⡁⠀⠀⠁⠲⣄⣧⢚⣥⠄⠀⢢⡝⠞⣥⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣝⣷⠋⢐⣷⠘⣶⣤⠘⡇⣚⠅⢀⠸⣦⠙⣏⢉⣥⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡈⠉⠙⠳⠾⣇⡠⣤⣹⣴⡟⣡⠀⣾⠀⡀⢠⠷⠿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣧⡘⠖⠈⢛⣶⣭⣽⣦⣁⠀⢿⡄⢹⡿⠖⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡄⢩⣟⠛⣩⢅⡟⠁⢶⣭⣝⣿⣷⣾⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣼⠿⠃⠾⠋⠸⠁⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣤⣶⠿⡿⢀⡿⢛⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣴⢾⢟⣿⡛⢿⣿⡿⣿⢻⣶⣶⢶⠿⣻⢿⠇⣾⡽⠿⣾⣷⠿⠶⢿⣿⡟
⠀⠀⣴⣾⣿⣿⣟⣓⣛⠻⢭⡿⢿⡦⣧⣿⠧⠾⢛⣡⠾⣶⠏⡼⢡⢖⣛⢲⢬⣙⣛⣛⣿⠃
⠀⠸⠋⠉⢸⡿⢉⡵⣛⡓⢦⡹⣯⣿⠾⢷⣔⡛⠛⠒⣋⣵⣚⣷⡐⢜⡽⢢⡇⡌⣭⣿⠃
⠀⠀⠀⠀⢸⡿⡏⠸⡩⠿⢢⠇⣿⣿⠻⣳⣴⣶⡦⣤⣤⣔⠿⣿⡷⣔⢶⡟⣇⣿⣼⡇
⠀⠀⠀⠀⠘⣇⡇⢸⣬⢍⣭⣾⣿⣿⣿⣿⣼⣿⣿⡌⣧⡙⣷⣬⣿⢾⣇⣦⣿⣿⣿⣅
⠀⠀⠀⠀⢰⣿⢃⣇⣹⣶⣿⣿⣿⠃⠉⠙⠻⠿⠿⠿⠿⠿⠿⠛⢿⣿⣼⣿⣿⢿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠈⠉⠛⠛⢿⣿⡇
⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⡿⠃⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⢸⡟⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⣿⠃⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣶⡿⠃
⢸⣿⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠉
⠸⣿⡆⠀⠀⠀⠀⠀⠹⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⢠⣶⡄⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠁⠀⢹⣷⠀⣰⣿⣿⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⣸⡟⣰⢋⣿⠇⣰⢫⣿⡏⠀⢠⣤⠀⠀⣠⣄⠀⠀⠀⠀⠀⠚⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⣿⣿⠃⣸⣿⣴⢃⣿⡟⠀⣀⡘⢾⣇⡀⠙⣾⣄⠀⢰⡆⢀⣄⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⢠⣿⡿⠃⠀⣿⣿⠃⣾⣿⢡⣾⣫⡿⡼⢸⡷⢠⡏⣼⡇⣾⢃⣾⡟⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢠⣿⡿⠁⠀⠀⠙⠃⢸⣿⠇⠸⣯⣉⡼⠁⣿⠣⠞⢰⣿⠀⣿⠞⣿⣇⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⢿⣿⡇⠀⡛⠋⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠁⠀⠸⠿⠀⠀⠘⠿⠀⣠⢶⣿⠉
⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⣼⣿⠃⢰⣿⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⣾⠃⣼⡏
⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠻⠏⠀⣼⡟⠀⠀⢾⣳⠀⠀⠀⣦⠀⠀⠀⢠⣤⣿⣧⣤⣤⣤⣤⣤⠻⠶⠟⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⢀⡀⠀⠀⣿⠃⣠⡀⠀⣹⢶⣄⠀⠛⠀⣴⠖⠈⣿⡟⣀⠀⠀⣀⠀⠀⣠⡀⢀⣤⠀⠀⣿⠟⣿⠃
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠄⢸⣿⡞⣹⡇⢠⠏⣼⡏⢸⡷⣰⣿⡖⢠⣿⠃⣿⢠⢞⣿⣡⢺⣿⣳⡟⣡⡶⢠⢿⡄⠉
⠀⠀⠀⠀⠀⢻⣷⡀⠀⢀⣴⡿⠏⠀⣼⡿⠀⣿⠇⠋⢰⣿⠃⣾⡷⣿⠋⣷⢸⡿⢸⣿⠋⠸⠟⠁⣾⡏⠿⠟⣿⣡⠏⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠙⠃⠐⣿⠀⠀⠘⠛⠀⠙⠁⠛⠞⠋⣼⡇⠀⠁⠀⠀⠀⠸⣿⠀⠀⠀⠛⠁⠻⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⣀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣀⣼⡖⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡿⣯⣏⣽⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⡘⢑⡙⠎⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡤⠠⡤⠄⠀⠀⢀⡀⠀⣀⣀⣀⣀⣠⢀⣀⣤⣭⡍⠛⠊⠻⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⢹⠄⠉⡀⢉⡖⠶⡈⠈⣃⢰⣿⠂⢹⠠⣿⣺⣇⣽⣿⣷⣤⣤⣴⣿⣿
⠀⠀⠀⠀⣾⢛⡁⠈⠀⠀⢂⠀⢆⠓⢠⡀⠠⠰⠀⢙⡯⢼⡛⠉⣗⡑⠘⠿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣯⠘⣷⠀⠄⢀⠀⠀⠈⠃⠐⠃⠘⠃⢀⠉⠀⡉⠉⠀⠀⠃⢷⡀⠈⢻⣟⣿⡟
⠀⠀⠀⢸⡏⢠⠈⢷⡄⠀⠀⠀⠂⣸⣟⠂⠰⣯⣤⣍⣻⣧⣾⣶⣸⣷⣴⣷⡔⣻⣿⡿
⠀⠀⠀⠈⠓⠉⠀⠁⠈⠆⠀⢠⡿⠛⣿⣿⣾⡛⣷⡞⠓⢽⣿⣯⣷⠛⢳⣏⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⡴⢃⠀⢠⢒⠎⠀⠀⠉⠉⠛⠛⠓⠒⠒⠒⠛⠛⠙⣦⢻⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠂⡀⣠⠀⠐⢫⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠿⡆⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⣈⡼⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣧⢳⠀⠀⠀⠠⠖⠂⠀⠀⣰⠖
⠀⠀⠀⠐⢠⠃⠀⠀⠀⠀⠓⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠃⠈⠁⠀⠉⠀⠀⠀⠀⣅⡈
⠀⠀⠀⠘⠎⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡖⠐⠐⠋⠀⠉
⠀⠀⠠⠖⠀⠀⠀⠀⠠⠄⠀⠈⠁⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠿⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠇⠀⠜⠃⠀⠀⠀⠀⠀⠘⠲⠓⠉⠁⠀⠀⠁
⣶⣶⠶⠶⢶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⢶⣦⠀⢀⣀⠀⠀⠀⢀⠐⣉⠉⠉⠐⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⣼⢯⠜⠁⣿⢃⡤⣶⡌⠐⠋⠀⢀⡖⡄⠀⠀⢑⡂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢰⠄⣼⣏⠆⠀⢸⣧⠊⣰⡿⣡⡶⠒⠀⠀⠻⣷⣤⡀⣯⣭⡀⠀⣸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠀⣿⠾⠀⠀⣿⢃⣾⠟⢰⡿⠁⣨⠀⢀⣠⣤⠞⠁⠈⠙⡿⠠⣏⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⠀⠀⠹⠂⠀⠀⠁⣾⠋⠀⠀⢉⢤⡁⢠⡟⠁⡄⢠⡞⠉⢠⠄⠀⢈⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠞⠀⣀⣀⡀⠀⠀⠀⡏⠀⠀⠀⡃⠈⡡⢄⠑⠊⠀⠈⠓⠒⠉⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⢇⠀⢠⡾⠋⢀⡇⢀⣄⠀⢧⡀⠀⠀⠐⣀⡩⠊⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⢊⡙⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⢀⡘⠈⢱⣿⠥⠴⠋⠀⢿⣼⡀⠀⠈⠁⠈⠉⠀⠀⠁⠐⠒⠈⠉⢀⣤⠀⠀⠀⠀⠈⢉⡴⠃
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠘⠋⠀⠀⣼⡇⢀⠴⢶⡄⠀⠙⠛⣷⠰⠒⢫⡍⠉⣉⠉⠉⠛⠒⣶⡿⠻⠤⠀⠀⣐⠊⢁⣀⠀⠀⣀⣄⡀⠀⣠⠖⢢⠀⡄
⠀⠀⠀⠀⠀⠀⣾⣿⠀⡄⠐⡄⠀⣿⡷⠃⠀⣾⢃⣤⠖⠀⠁⢀⣴⡦⠀⣼⣁⠄⠀⠀⣼⡟⠀⢠⣾⡷⢊⡾⡔⣡⡿⢀⣾⠏⠰⠃⢰⣿⠘⠁⣠⠇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠃⠀⡇⠀⣿⠁⠀⡼⠁⠸⣇⢀⡴⠃⣾⠟⠀⠀⠘⢿⣦⣀⣴⡟⠀⢠⢻⡟⠀⣸⠏⣰⡟⠀⢸⣏⢀⡿⠀⠀⠻⣷⡘⢣⡀
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⡰⠀⠀⠛⠀⢸⠁⠀⠀⢀⠤⠄⢸⡟⢀⡄⢀⡤⠘⣿⡇⢸⠃⢠⠂⠈⠀⠀⠈⠀⢟⡰⠃⠈⠋⢸⡥⠚⢁⣠⠹⣿⠄⢻
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠊⠀⠀⢀⡴⣆⠘⢄⠀⠀⠈⣣⠎⠀⠁⠈⠀⠘⠤⠤⠟⠁⠘⣒⣡⠤⠄⠀⠀⠤⡀⠀⠀⡠⠒⠢⠀⠀⠠⣏⢀⣀⠟⠀⡼
⠀⠀⠀⠀⠀⠀⠰⣄⣀⣀⣤⠴⠋⠀⠹⣦⡀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⡤⢦⠀⠈⡀⠘⣇⠛⠊⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣀⣠⣤⣤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠍⠠⠞⠀⠀⠈⠳⢤⣀⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉






⠀⠉⠉⠁
⠀⠑⠒⠃

⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠲⡄
⠀⠀⠀⠀⠐⠀⣁⠃⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠉⠡⡀⡄⠠⠐⢀⠀⣀⠀⠀⠠⣄⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿
⠀⠀⠨⠀⠁⠀⠀⠄⠈⠃⠋⠀⠀⠀⠀⠐⠀⠀⢒⠀⢢⣦⣄⠱⣤⣠⠊⢁⣨⠴⠠⠀⠂⠀⠀⠀⠈⠉⠉⠉⠉⠋⠉⠙⠋⠁⠉⢉
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂⠀⠙⡿⠛⠓⢾⣃⣒⣚⣃⣀⣀⣀⠀⠀⠌⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠀⠈⠉⠉⠀⠀⠀⠀⠸⠁⠀⠀⠄⠀⠀⠿⠠⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠈⠀⠀⠐⠢⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠊⠐⡄⠀⠀⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠰⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠠⠃⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠀⢹⠂⡀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠜⠐⡇⠂⠀⠀⢰⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠒⠓⠀⠘⠒⠂⠀⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠠⢀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠘⢾⢀⡠⠞⠀⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡶⠀⠀⠀⠀⠀⠀⠀⢀⡁⠂⠀⣳⡤⠨⡭⠖⠐⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⠅⠀⠄
⠀⠀⠀⢀⡠⠄⢲⠁⠀⣀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠤⠖⠁⣀⠀⢄⣀⠠⣀⠀⠀⢠⣠
⠀⠀⠀⠀⠤⠠⡰⡀⡠⢊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠲⡈⠳⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡁⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠙⠀⠘
⠀⠀⠀⢀⣀⠀⢀⣿⢋⡀
⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡀⠈⠠⡀⣜⠄⢸⣀⣄⢬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢨⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠽⡶⢃⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸
⡄⠀⠤⠄⠀⠢⢼⠄⠸⠴⠤⠴⠶⠦⠤⠤⠄⠀⠀⠀⠀⡀⠐⠰⠁⠀⠀⣀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠠⠤⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢠⠃⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⡞⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠷⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣷⠀⠇⡁⠀⠈⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢰⠱⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⣴⢃⠀⠸⡃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠈⠓⡀⠀⠀⠀⠀⠀⠀⠀⡠⢘⠀⠀⠜⣶⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⢄⠘⡿⡄⠀⠙⢆⢦⡀⠀⠀⠀⠀⠀⠜⡄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⡼⡰⠃⢀⣾⣶⠃⡰
⠀⠀⠀⠀⠀⠠⠄⠀⠠⠀⠀⠤⠀⠑⢤⡌⠰⠄⠀⢠⡙⠆⠀⠀⠀⠀⠄⢧⠀⠀⠀⠀⠀⠀⠰⠘⠀⠀⢀⠀⠴⢋⠔⠀⣴⢟⣿⣿⡎
⠀⠀⠀⣀⠀⠀⡀⠀⢀⠀⠀⣀⠀⠀⡀⠙⢢⡴⢒⣒⠛⠓⠦⠌⠐⣤⣸⠀⠂⠀⠀⠀⠀⠠⠀⣃⣤⠂⢡⠴⠛⠛⢂⣲⣾⣿⣯⣾⣿⣄
⠀⠀⠀⠁⠀⠀⠁⠀⠈⠀⠀⠉⠀⠀⠁⠀⠈⠀⠀⠈⣌⣂⣀⡀⠢⣄⠈⢑⣦⠀⠀⠀⢀⣆⡈⢁⣠⠔⢀⣀⣂⣡⢰⣻⣿⣿⣿⣿⣯⣿⣆
⠀⠀⠀⠓⠀⠀⠃⠀⠘⠀⠀⠚⠀⠀⠃⠀⠐⠂⠀⠒⠠⠀⠀⠉⠳⠠⡙⡀⡀⢡⠀⣀⡄⠀⢀⠉⠠⠞⠉⠀⢀⡿⣹⣷⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢁⡜⢅⠀⠼⢠⡅⠦⢀⡀⢃⠌⠀⠀⢀⡾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢑⡉⠀⠁⠀⠀⢰⠀⠁⣊⣦⠐⢀⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠳⣐⠆⠚⠓⠰⣀⠞⢠⠂⢀⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⢃⠀⠀⠌⠹⠀⡏⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠆⠈⠛⠛⠂⠸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡆⠘⠃⠀⠀⠶⠃⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠜⢂⠠⠴⠦⠄⠺⠣⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠑⠐⠂⢊⠀⡠⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠭⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠛⠠⣀⣀⠄⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⣒⣒⣒⣒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠠⠬⠭⠭⠭⠭⠭⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠸⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡄⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⡶⠊⠉⠉⠙⠻⣕⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡰⠋⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠈⠉⠉⠒⠦⣤⠋⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢀⣴⠠⠤⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⠒⠒⠒⠒⠢⠤⣀⢀⠃⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠉⠀⠀⣀⡠⠤⠤⠤⠭⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣗⠀⠑⣄⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢠⠏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢻⠢⡀⠈⢆⠀⠀⠀⢀⡜⠀⠀⡠⢾⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⠀⢈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⡆⠑⡄⠀⢂⠀⠀⡼⠀⠀⡜⠁⡈⠀⢠⠁⠀⠀⠀⠀⠀⠀⣀⣀⠀⣸⡄⣃
⠀⠀⠀⠀⠀⠀⢸⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⠀⠘⡄⠘⡄⠈⡄⢠⠃⠀⡜⠀⡰⠁⠀⠃⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠲⢷⠦⣽⠀⢰⣼⠀⢸⠁⡰⠁⢀⠎⠀⠀⠀⡠⠊⠁⠀⣀⠤⠂⣺⠛⠂⡟⠒⠢⣌⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⡹⠆⠚⠁⠀⠛⠐⠦⣀⣠⡄⠀⡶⢄⡀⠈⢣⡈⢱⠀⢸⣴⣡⠴⠛⠻⠁⢀⠞⠁⢀⡴⠋⠁⢀⡴⠁⢀⠞⠀⠀⠀⠀⠙⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠟⠁⢀⣀⣀⠀⣀⠀⠀⢻⠁⠀⢳⠀⠉⢦⠀⢁⠀⡗⠁⠈⡵⢾⠀⡇⢠⠏⠀⡰⠋⠀⢀⡴⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⢀⡠⠄⣒⠶⠟⠋⣀⣠⠔⠋⠁⠉⢀⡈⣙⣂⣄⣣⡀⠘⢦⡀⢸⠀⢰⢆⡀⠀⡎⠀⡘⢠⣷⠇⣀⠞⢀⡠⠖⠁⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢰⠃⠀⠛⠠⠄⠀⠀⣭⠀⠀⠀⣀⠽⠃⠀⣠⠤⠘⠻⠿⢦⣀⠙⢺⣤⠺⣮⣃⢠⡣⠞⠁⣠⣃⠀⠙⠺⢭⣤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⢸⠀⠀⠀⣠⠔⠂⠉⠉⣉⢭⡟⠁⢀⣴⣊⡀⢀⣀⢀⡀⠤⠽⣻⣋⡵⠒⢿⣈⣿⣤⡴⠋⠀⠀⢙⠶⠀⠀⠈⢛⡁⠒⠂⠤⢄⣀⠀⠀⠀⠀⡠⠃⠀⣸
⠀⠀⠀⢳⠀⠀⡇⠀⠀⡰⠊⣴⡊⠀⣀⣀⠀⠀⠁⠁⠈⠁⠀⢀⣠⣼⣉⣷⣄⡼⢥⣯⡁⠈⣉⣉⠉⠉⠳⡄⠀⠀⠘⣥⣀⡀⠀⠀⠀⠉⠢⡤⠚⠁⠀⡠⠃
⠀⠀⠀⡜⠀⠀⠈⡆⡜⠁⠀⢀⡠⠊⡡⠊⠿⣁⣀⣀⣠⠔⡾⠊⡱⠊⡹⠧⠼⢶⠾⠉⠢⡉⠁⠀⠈⠳⢄⠘⣖⠀⠀⠘⣆⠈⢑⡦⢄⠀⠀⢱⠀⣠⠞⠁
⠀⡠⠊⠀⠀⢀⡜⢠⠁⠀⢒⡏⢀⠔⠀⠀⠀⠀⠀⡜⠁⣾⠀⠀⣰⢰⠀⠀⠀⢀⠄⠸⣦⢈⣉⡈⠉⣉⣉⣋⠘⠧⢄⠀⣼⠗⠂⠀⠠⢷⠀⠘⠁
⠾⠒⠙⢆⡠⠋⠀⢳⠂⢀⡜⢡⣌⣀⣀⣀⡀⠀⢀⡄⠀⡇⢀⣠⠋⡆⠀⠀⠀⣎⠀⠀⢹⡆⠀⠈⠙⠃⠈⢣⡉⠻⠊⢳⡈⢢⡀⠀⠀⢸⠀⠸⣄
⠀⠀⠀⠈⠁⠀⠀⠺⠀⠀⠀⢼⡔⢛⠶⠖⠛⢫⡭⡇⠀⠘⠁⢀⠎⢇⠀⠀⠀⠘⢷⠀⡟⠇⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠁⠀⢣⠀⠀⢀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠠⠛⣶⣀⠷⣇⣀⣶⡔⠉⣆⡟⠀⣠⠴⣟⠂⢼⣄⣤⡀⠀⠀⢣⡃⠰⠀⠀⠀⠀⠀⠀⠀⠈⢻⠑⠢⣄⠀⠀⠀⠈⠢⡀⠀⠀⣈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠟⣡⠞⠛⠛⠒⠚⠓⠠⠽⠏⣉⣻⡧⢴⡖⠉⡍⢻⣍⣧⣵⢄⠀⠈⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠙⢷⡄⠀⠀⠈⠢⣀⡁
⠀⠀⠀⠀⠀⢀⣤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠿⣙⢻⣍⣧⠹⠆⠀⠂⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡐⠋⠉⠁⠐⠒⠂⠀⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⣱⡾⠀⠀⠀⠱⠇⠀⠀⠀⠀⠀⠀⡠⢶⡤⢮⢀⣄⣼⣦⡀
⠀⠀⠀⡼⠛⠉⠉⠉⠉⠛⠒⠲⠦⣄⣀⡈⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣿⠶⣦⢀⣤⣠⣿⡹⠋⠙⠢⣄⡀
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣄⡈⠙⠲⢄⡀⠀⠀⢱⣀⡀⢀⣠⠄⢶⣏⣩⠟⠛⠷⢀⣄⣠⣟⠩⠋⠀⠀⠀⠀⠀⠉⡳⢤
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⠢⣄⠈⠓⢤⣘⣿⡥⣼⠟⢳⢎⠀⡙⠓⢶⣤⣬⠋⠟⠉⠀⠀⠀⠀⠀⠀⠀⠐⠁⣸⠃
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠙⢦⡀⠘⠿⢣⣏⢳⣮⣠⡼⢟⡾⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⢀⠴⠶⠃⠸⡀⠀⠀⠀⠀⠙⢆⡄⠈⡭⠿⠐⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠄⣀⣸⠃⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⡼⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡷⠲⠖⠚⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠴⠒⠚⠉⠀⢀⣠⠴⠚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠤⠤⠔⠲⣎⡁⠀⢀⡰⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢡⢠
⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠈
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠋⡍⠙⠂⢀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠄⠀⠀⠰⠤⠖⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠰⠦⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠆⠄⠀⠀⠀⠰⠰⠀⠀⠀⠀⠀⠀⠀⡰⢩⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠐⡀⠀⣍⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⣀⢀⢀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⢘⡧⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⡀⢧⣢⠘⠤⡀⠧⣄⠈⣆⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⡡⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⡄⠄⠀⡄⠀⠀⠀⠀⡀⢤⡀⠀⡄⠀⠀⢠⢀⠀⠀⡠⠀⠀⠀⣀⣿⡇⠀⠈⠫⣾⠍
⠀⠀⡤⢚⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⠄⠘⠇⠳⠀⠗⠄⠰⠀⠪⠄⠀⠷⠀⠧⠁⠇⠾⠀⠢⠐⠥⠀⠀⠠⠹⠻⠇⠀⠀⠀⣿
⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⡀⠀⣀⠀⢀⡄⠀⠶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠉⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣲⣲⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠈⠀⠉⠈⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⠂⡀⠀⠀⠀⠐⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠤⠂⠀⠀⠀⠆⠀⠆⢀⢀⣀⠀⠀⠀⠀⠤⡀⠀⠀⣤⠀⠀⠀⢸⡟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣬⣤⣦⣼⣮⣾⣭⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⢀⡀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠈⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠅⠀⠀⠠⠀⢈⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⣐⠀⡒⢀⠉⠀⠈⠀⠀⠀⠀⠸⠀⠀⠃⠘⠀⠄⠶⡀⠄⠘⠅⢠⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡐
⠀⠀⠂⠀⠈⠐⠢⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠐⠐⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠜⠠⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⢤⢠
⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢉⡮⡕⠆
⠀⠐⢢⣳⢥⠀⠀⠀⠀⠀⠀⠑⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠁⠘⠀⠀⢀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠑⠄
⠀⠀⡀⠀⠀⠀⠉⠀⠁⢀⠰⠞⠶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢰⠠⠠⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄⢐⢁⠂⢄⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⣦⡀⡠⠖⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠝⠜⠥⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢠⠠⢀⠀⠀⠀⠀⠘⠄⡺⠨⡒⠔⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠂⢁⠑⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠘⠀⠊⡀⠀⢀⠀⠀⠀⠈
⠀⠀⠌⠁⡠⣂⠈⡄⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠠⠄⣀⢀⠀⡄⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠄⠡⠠
⠀⠨⠀⠠⢀⡈⠀⠀⠒⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠈⠌⠀⠤⠂⠄⠀⠄⡀
⠀⠀⢀⠠⠂⡀⠆⠐⡂⠒⠀⡌⠀⠀⠀⠀⠀⠁⢈⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠠⠁⠄⠀⠡
⠀⠀⠘⠠⠀⠀⠈⠀⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢠⠁⠈⡀⠀⠀⠀⠀⡄⠀⠀⠂⢂⠀⠀⢔
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠠⠄⣀⠀⠀⠐⢀⠀⠃⡀⠠⠀⠁⠀⡈⠃⠐⢀⣀⡂⠀⠀⣀⠂⠀⠀⠀⠀⠀⢀⠁⠀⠀⣀
⠀⠐⡁⠀⠀⠀⠀⠀⠀⠠⠇⡐⠂⠀⠀⡀⠠⡆⠠⠄⠈⠀⠀⠀⠀⠀⡄⠃⡄⠘⠀⠚⠀⢑⡄⢀⠁
⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠉⠂⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁⠤⠁⠀⠂⠀⠐⠐⠂⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢠⠉⣤
⠀⠀⠀⠀⠐⠆⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⠀⢀⠿
⠀⠀⠀⠀⠇⢀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠪⠆⠒⢸⡀⠀⡀⠤⢄
⠀⠀⠀⠀⠐⢄⡡⢀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠡⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣁⡠⠆
⠀⠀⠀⠄⣀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡐⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡡⠞⠁⠀⠉⠉
⠀⠀⠦⠀⠤⠄⠁⠈⠢⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣋⡀⠓⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠈⢀⠶⡈⠑⠆⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⣉⡤⠚⠁⠈⠶⣄⠀⢀⡙
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠁⠀⠈⠉⠒⠒⠀⠈⠉⠉⠀⠀⠤⣄⠀⠀⠀⠀⠄⠀⢀⠐⢄⠀⠀⡤⠒⠀⠒⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠞⠁⠀⠀⠀⠀⠈⠲⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⢀⠀⠀⡠⠄⠀⠀⠀⠀⢀⡀⠒⠒⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣤⡀⠁⠀⣾⠆⠀⠀⠈⠥⠄⠀⠀⠀⢀⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠇⠀⠀⠀⢠⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢠⠀⠀⠀⢆⠀⠀⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠑⠀⠀⠀⠉⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠠⠴⡿⠦⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣻⢢⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⣀⡤⠄⠠⠦⡄⡀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠈⠣⡀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⡄⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⢑⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⠏⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⣄⡀⠀⠀⢠⠁⠀⠀⣸⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠛⠦⠒⢛⠀⠀⠀⠻⡄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢰⠀⢸⠀⠀⢰⠀⢡⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⠀⢸⠀⠀⢸⠃⢸⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢢⠸⠀⠀⠈⠀⠈⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⣄⠀⠀⠀⠀⠀⣦⠀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠘⠀⠀⢀⡇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠂⠀⡿⠀⣤⡄⢠⡠⠚⢁⠖⠀⠤⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⡇⠀⠀⠀⠀⢣⣀⡎⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠘⢷⣄⣀⣀⣀⣼⠟⠙⠢⠤⠤⠴⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⣙⣿⠛⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣠⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡄⠀⡜⠁⡏⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠸⡈⠳⡀⢀⠎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠲⠁⠸⡇⠀⡜⠁⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠁⠳⡀⠀⡇⠀⠳⠋⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠞⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠓⠚⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠓⢄⡀⠀⠹⡦⢄⣀⣰⣄⡄⠀⠀⢀⡴⠃⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠓⢤⡀⠀⠙⣄⠤⠷⣄⠀⠈⠉⡿⠤⣤⡏⠀⢀⠴⠊⠁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⠋⠉⠁⠀⡀⣉⠑⠢⣌⡶⠋⠀⠀⠀⠀⠉⠒⠊⠁⠀⠀⠉⢶⣡⠔⠊⡉⣀⠀⠈⠉⠙⠒⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠁⠀⠉⠙⠂⡞⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢳⡐⠛⠉⠀⠈⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣠⠴⠚⢹⠁⠔⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠣⠀⡏⠓⠦⣄⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⣿⣛⣷⡀⠀⠀⠀⠀⠀⣾⣿⣹⣧⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⡻⠟⡢⠁⠀⣀⡀⠀⠀⢿⠿⢟⡜⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡈⠉⠀⣠⣤⣶⣶⢦⣄⠀⠉⢁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣨⣇⡀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠦⢤⣀⣑⣊⣀⡠⠴⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢹⠀⠀⠀⠀⠀⠀⠀⠀⢫⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠏⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⢦⡇⠀⠀⡆⠀⠀⠀⠀⠀⢸⣧⠎⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⡇⠀⠀⣇⣀⣀⣸⠀⠀⢸⣷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡿⠃⠀⠀⠈⠿⣶⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⣀
⠀⠀⠀⡐⠒⣲⡔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣧⣧⢜⡆
⠀⠀⠀⣘⣬⡼⢦⠾⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⣀⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠉⡝⢉⠟⡇⠀⠀⠀⣀⣴⢶⠚⢒⣊⣳
⠀⠀⢀⠀⠀⡄⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠚⡀⡎⠀⡇⠀⠀⠈⡷⢧⡬⢶⣶⣮⣅⣄
⠀⠀⢸⡄⣠⠇⠸⠤⠤⠤⠤⠤⢼⠀⠀⠀⢸⠉⠉⠉⠋⠉⠉⣉⠉⠉⠍⡇⠀⠀⠀⡄⠀⣠⠃⡇⡴⡇⠀⠀⠀⣠⠊⢠⡎⢇⡱⣘⣾
⠀⠀⢸⠤⠤⡀⠀⡤⠤⠤⠤⠤⢼⠀⠀⠀⢸⠀⠁⠀⠈⢀⠀⠀⠀⠄⠀⢁⠀⠀⠀⡗⠊⠁⡠⢻⠀⡇⠀⠀⢸⣁⠜⠁⡳⠚⢀⡴⢻
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠐⠂⠀⡄⠀⠐⠀⠀⠆⣸⠀⠀⠀⡗⠒⠉⠀⡼⢀⡇⠀⠀⠀⡁⡠⠊⠀⡴⠋⡐⡻
⠀⠀⠈⠉⠀⠓⠚⠓⠒⠒⠒⠈⠋⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⢀⣀
⠀⠀⣀⣫⣄⣦⣠⣯⣖⡀⠀⠀⠀⠀⠠⡀⠐⡆⠀⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠤⢤⣤⣿⣧⣤⣤⠜⠀⠀⠀⠰⣒⢞⣛⣛⣯⡭⢀⣁⣱⠀⠀⠀⠀⠀⠀⠠⣈⣇⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⢄
⠀⠀⠀⠘⡀⢸⡇⢀⠇⠀⠀⠀⠀⢠⣡⠁⠀⠀⡀⠀⠗⠾⠂⠀⠀⠸⡁⠩⠍⠤⠤⠭⠯⠯⠭⠭⠭⠭⢽⠀⠀⠀⡏⠉⠉⠉⠉⠉⠷⡀⠀⠡⡀
⠀⠀⠀⠀⣇⢸⡇⣸⠀⠀⠀⢠⣔⠟⢹⢠⠀⠀⠇⠀⡄⠀⡅⠀⠀⠀⡄⢒⢐⡒⡂⢒⢖⠒⡒⣒⣒⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⣤⠊
⠀⠀⠀⠀⠘⠾⠷⠃⠀⠀⠀⠀⠈⠀⠸⠤⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠃⠠⠥⠬⠬⠤⠥⠤⠤⠤⠥⠤⠞⠀⠀⠀⠣⠠⠤⠤⠤⠤⠤⠦⠼

⠀⠀⠀⠀⢀⢆⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡠⠉⢢⡀
⠀⠀⠀⢠⢸⠴⣟⢰⣦⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠂⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣋⣄⣊⣁⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣤⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠄⠰⠛⠛⠂⠀⠤⠂⠀⠀⠀⡆⠀⠙⠛⠛⠛⠛⠋⠀⡂⠀⠀⠀⠀⠈⠀⠠⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠄⠀⡃
⠀⠀⠀⠸⣀⢀⢘⠀⡸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣃⠀⠀⠀⣀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⡀⡀⣈⣈⣀⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠖⠋⠇⢠⡤⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡄⠀⠀⠀⠈⠣⣀⠀⢀⣚
⠀⣤⣤⣿⣤⡄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⢸
⠀⠀⡿⠿⣯⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣜⢿⢹⣆⠀⠀⠀⠀⠀⠀⠘⡿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣿⣧⠀⢀⣠⡄
⠀⠀⣸⣿⡇⠀⠀⠀⠀⠺⣻⣿⣿⣾⣿⠛⠢⠀⠀⠀⠀⢸⠏⠈⠊⢫⠀⠀⠀⠀⠀⠀⣠⣖⣦⡀⠀⠀⠀⠀⠀⠰⢎⡟⠢⠀⠈⢻⠏⠀⠈⣿⠁
⠀⠀⣿⣿⣧⠀⠀⠀⠀⣾⣿⣿⣿⣿⣮⡓⠦⠀⠀⠀⠀⠎⣰⠘⡇⢧⢱⠀⠀⠀⠀⣼⣿⣿⣯⣿⠄⠀⠀⠀⠀⠒⣫⣏⡒⠆⠀⠀⠀⠀⣠⢿⣆
⠀⢰⣿⣿⣿⡆⠀⣤⣤⣶⣿⣿⣿⣿⣮⡛⠷⠶⠀⠀⢸⡴⠙⠴⠉⠋⡙⢇⠀⠀⢀⣽⣷⣿⣿⣶⣤⣀⠀⠀⣘⣯⠴⡧⣭⣁⠀⠀⠠⣾⣿⣿⣿
⠀⣾⣿⣿⣿⣿⠀⠰⣤⢾⣻⣿⡿⣿⣿⣿⣅⠀⠄⢠⢋⡆⢠⢺⡀⣧⠹⣌⢂⢾⣿⣿⡿⣿⣿⣿⡷⠟⠀⠀⠠⢤⢚⡟⠲⠤⠄⠀⠀⣼⣿⣿⣿
⢠⣿⣿⣿⣿⣿⡆⠴⣒⣿⣿⣿⢿⣿⣿⣷⣭⣉⠀⣮⠞⠥⠻⣨⠓⠎⠑⠋⢻⣠⡴⣾⣿⣿⣿⣿⣷⡦⣀⠀⢶⠚⣉⣏⣟⢒⣖⣠⣾⣷⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⠛⠿⢿⣻⢿⣾⣿⣿⣿⣿⣯⣿⢋⣴⢸⢀⠀⢰⢰⡘⣆⣿⣿⣿⣿⣷⣿⢿⣷⣿⣿⣟⣷⣒⣉⠥⡧⢌⣋⣻⣿⣿⣿⣿⣿⣽
⣿⣿⣿⣿⣿⣿⣿⡗⣿⣯⣷⣿⣿⣿⣿⡟⣯⣕⣗⢻⡝⠛⣼⢄⡞⣬⠑⠋⣁⣭⣾⣿⣽⣿⣿⢿⣾⣯⣭⡏⢤⡴⠒⡗⠲⠤⢤⣿⣿⣿⣿⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡛⣿⣿⣿⣍⡻⣦⣭⣉⣛⡓⠀⣀⣸⣇⣀⢰⣿⣿⣻⣿⣯⣿⣿⣻⣿⣿⣾⣿⣻⡦⠀⠀⣇⣠⣴⣿⣿⣿⣯⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣻⣿⢿⣿⡟⢿⣳⣯⣭⣍⣡⠆⣿⡿⢿⣿⠁⠠⣄⢤⠀⠀⠀⣿⠀⢀⣉⣉⡉⠁⠀⢈⣿⣯⣿⠿⢷⣿⣿⣿⣿⣿⣿
⠛⠛⠛⣛⠛⠛⠛⠛⠒⠛⠉⠀⢸⣿⣇⣀⣀⣭⣭⢥⣤⠀⢹⣷⣾⡏⠀⣿⣿⣿⡆⠀⠀⣿⠀⣸⣿⣿⣿⣿⠀⠘⣿⣿⡿⠀⠀⠀⠉⠙⠛⣿⡟
⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⠀⠈⠉⠉⠁⠀⠻⠿⠿⠃⠀⠀⠀⠀⠻⠿⠿⠿⠏⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁

⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⡀⣀⠀⠀⠀⡠⠀⠠⢀⠤⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⡀⡐⣀⠢⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠁⢠⠠⡴⠀⡢⠀⡂⢀⠀⠈⠀⠤⠁⠈⠀⠀⠈⠁⠀⡀⢠⢀⠤⠀⠀⠠⡀⠄
⠀⠀⠀⠄⡑⠡⡀⣀⠀⠒⡔⠀⠀⠀⠀⠐⠆⠁⢸⠀⡀⢠⠀⣇⢠⡀⠰⡀⢀⠀⢎⠀⠀⡄⠀⠀⠀⠀⠀⡲⠀⠀⡈⠈⠁⢀⡀
⠀⠀⠰⣑⢀⢊⠰⠄⢂⢁⠀⠁⠀⠀⠐⠐⠀⠀⠈⠀⠇⠀⠀⠀⠂⠁⠀⠂⠀⠑⠀⠁⠀⠃⠀⠀⠀⠀⠃⠀⢢⠉⡄⢑⠐⢑⣁
⠀⠀⠀⠂⡁⠐⠁⠊⠀⣄⢂⠁⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⣀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠡⢌⠁⠒⠄⠊⡈⠨⠄
⠀⠀⠀⠀⠁⠡⠒⠄⠐⠂⠀⠀⠀⠁⠀⠀⢀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠈⠄⢠⠰⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⡀
⠀⠀⠀⡰⠀⢀⠀⠀⠀⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⠀⠈⠀⠀⠀⠀⠒⠀⡀
⠀⠀⠙⡄⡀⣂⠀⠀⠀⠀⠐⡀⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠁⠀⠉⠂⠈
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡉⠀⠀⢀⠠⠐⠄⠀⢿⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠈⡀⠌
⠀⠀⠀⣄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄⠀⠀⠀⠀⢀⡂⠓⠀⠀⠀⢺⡍⠀⠀⠰⠈⠰⠂
⠀⠀⠶⢮⣹⠴⠄⠀⠀⠐⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠧⡀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠄
⠀⠀⠀⠃⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⢸⠀⠀⠠⡊⣠⠀⠀⢸⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁
⠀⠄⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡘⠀⠐⡀⠀⢸⠀⡀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠒⠒⠎⠀⠀⠀⠉⠒⠒⢺⠀⠀⠀⠀⠀⠀⠀⢸⠠⠄⠒⠲⠄⠀⠀⠸⠒⡒⠀⠀⠀⠰⡀⠡
⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠠⣀⠁⠀⠀⢀⠀⠀⠀⠈⠀⠢⡀⠀⠀⠀⠀⠈⡃⢀⠀⠀⠀⠀⠐⣰⠈⢐⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠰⠀⠈⠀⢀
⠀⢀⠤⠀⠀⠈⠃⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠨⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⡀
⠀⠈⠈⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠐⠀⠘
⠀⠀⢀⠀⢀⠀⠀⢀⠄⠀⠀⠆⠘⠀⠃⠘⠀⣀⠐⠏⢹⠀⠀⠠⡄⠀⠀⠀⡦⠠⠀⡀⠀⠄⠀⠂⠀⠀⡀⠀⠀⠀⡀
⠀⡐⢀⠐⠄⢑⠀⠂⠀⠂⠀⡆⢀⢀⡀⠨⠀⠒⠀⡄⢸⠀⠑⠚⠁⠁⠀⠉⠁⠀⠡⠀⡀⢰⠀⢐⠆⠐⠄⠐⠒⠎⠀⠀⣀⢸⣄⠖
⠀⠠⣀⠀⢁⠌⠀⠀⠀⠀⠀⠠⠼⠀⠄⢨⢀⠆⠈⢣⣸⠑⠚⢀⠄⠀⠀⠀⠀⢻⠀⠠⠇⠘⡀⠂⠠⠀⣊⠓⠒⠀⠀⠀⠀⡜⢺⠙⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀⣀⣉⣁⣉⣁⣈⣉⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠓⠀⠔⠉⠁⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡄⠀⠀⠀⠀⣏⠁⠀⠀⠳⡤⢰⠄⣼⠀⠀⠑⠂⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⡟⢰⡇⠀⡴⢒⠀⠨⢩⡟⠀⡀⠀⡟⢀⣏⠀⠀⠀⡎⠀⡸⠋⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⡤⠁⠀⢠⡇⢸⠀⠰⣗⠉⡰⠁⠘⠧⠀⠀⠀⠈⠐⡆⠀⠀⠀⠉⣩⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢷⣋⢁⡠⠂⠘⠇⠉⠃⠀⠈⢉⡀⠀⢸⠇⠀⠂⠒⠒⠛⠁⢀⡤⠀⠀⣿⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡽⢣⠀⠀⠀⠀⠀⠀⠤⠀⢸⡄⠀⣿⠀⢠⡔⢰⠂⡽⢠⡟⢠⡇⡠⢻⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⢞⢅⠀⢸⡇⠈⠀⠸⣁⡠⡦⢠⡆⠀⢸⣧⢘⡇⢀⡟⢀⡏⠸⣇⠈⠓⠈⠁⡰⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠨⠞⠀⢸⡇⢀⡗⠀⠌⣼⠃⢾⡁⢰⠃⣿⠈⠃⠈⠁⡤⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁
⠀⠀⠀⠀⠀⠠⠂⠀⠸⣇⠀⠀⡠⢿⡇⠸⠇⠊⠀⠉⠀⠀⠀⠘⠶⠋⠀⠀⠀⣼⠃⠁⠀⠀⠀⠀⢼⠃⡀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠒⠁⠀⠉⠃⠉⠀⠀⠀⢠⠔⠊⠁⠀⠀⠀⠀⠀⡀⠀⡀⣴⠂⣿⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠄⠀⣼⠁⡄⢰⡇⣼⢰⣇⠰⠇⠀⠀⠀⠀⢠⡎⠠⡌⠐⣄⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⠀⠀⢀⣈⠑⠈⠀⠀⠳⠔⡃⠘⠁⠈⠁⠀⠂⠀⢢⡀⠀⠀⠈⠓⠐⠃⠐⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢀⡶⢉⡀⠉⠂⠀⠀⣀⡜⢡⡆⠀⠀⠀⠀⠀⢀⢸⡇⢲⣄⡇⢸⡇⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢸⡁⢯⡀⠀⠂⢠⠎⢀⠃⣾⠀⡼⠉⡅⣼⠁⣸⣸⠃⣸⠻⡇⣼⠁⣀⠞⠁⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⢷⡀⠀⠀⠀⠈⢁⠈⢸⡇⠘⠇⠰⠇⣿⠐⠃⣿⠈⠁⠀⠀⡟⣊⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠀⠀⠀⠙⠒⠤⠤⠒⣁⠀⠀⠁⠀⠀⠀⠠⡇⠀⠀⠏⠀⢀⠐⠊⠰⣏⣀⣤⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⠀⣠⠶⡳⠄⠀⠀⠸⣇⣀⡄⠀⠀⠀⠀⣠⠀⡄⣴⠛⢴⠃⣰⠀⠀⠂⣰⠃⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⠳⠐⢠⡟⢠⡟⠀⣠⡄⢠⠀⣦⠀⠀⠀⠀⠀⠀⢠⡏⣸⠇⣏⠀⠛⠆⠛⠐⣁⣀⣬⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢾⡇⣼⠁⠀⠸⡄⠿⠀⠀⠀⠀⠀⠀⣨⡥⣿⠶⠤⠤⢤⣤⣔⡋⠁⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠘⠁⠙⢒⣁⣀⠀⠀⠀⠀⠀⠀⠀⠘⠅⠔⠁⣀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢄⣀⣀⣀⠀⠤⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠦⢄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⢶⣀⠐⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠈⠀⠀⠀⠀⠀⠀⠐⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢋⠃⠀⠀⠀⠀⠀⢠⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⢀⢔⡗⣉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣦⡤⠆⣀⠀⠀⠀⢀⣲⠿⠛⠁⠀⣀⡠⢤⢎⣼⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⣀⣴⠀⠉⠙⠛⠓⠛⠛⠛⢿⣷⣄⢀⣷⣐⠥⢆⢀⡾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣛⢿⣿⣿⣿⣋⢵⣿⡟
⠀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠦⣽⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⡆⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠘⢃⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡶⠿⠿⣿⣰⡿⢿⠃
⠀⠀⠀⠀⠀⣾⣿⣡⣴⠀⠤⠄⠐⠰⠤⡁⠀⠰⠶⢦⣀⣀⠀⠀⠀⠀⠀⠋⠹⠉⠁⠀⠀⠈⠛⠙⠋
⠀⠀⠀⢀⢸⣿⣿⠟⡽⣶⣮⣰⠷⠶⠶⠾⠀⠀⣰⣋⣩⣿⣆
⠀⠀⠀⠊⣼⡿⢁⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⣽⡏⠁⠀⠀⠉⠢⠄⠀⠀⠀⢀
⠀⠀⠀⡟⡟⢰⣋⣿⡿⠋⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⡂
⠀⠀⠀⢹⡇⠀⢶⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢺⡄
⠀⠀⠀⢺⡇⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠼⣧
⠀⠀⠀⠸⣇⢼⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠐⣿
⠀⠀⠀⢶⣿⠿⠀⢸⡻⠀⠀⠀⢰⡇⠀⠀⣸⣧⠀⠀⠀⠀⣄⣠⠀⠀⢸⡆
⠀⠀⠐⠿⠟⠀⠀⠸⣿⣷⡀⠀⠈⠉⠂⠰⢿⣿⠜⠓⠒⠂⠁⠀⠀⠀⣾⣿⢆⡀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠶⠿⠗⠀⠀⠀⠘⠻⡷⠶⠂⠀⠀⠀⢠⣐⠠⣶⣩⣤⡥⠀⢈⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠐⢠⠂⢀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠒⠬⢘⢭⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉
⠀⠄⠐⠂⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠄⠀⠀⠀⠀⠁⢀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘
⠠⠀⠌⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡤⠓⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠰⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠄⣖⡶⢔⡠⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠌⢘⣦⡀⠄⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠔⠚⠀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠀⠀⠀⠈⠀⠈⠄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡄⠀⠀⠀⠀⠀⢠⣾⣦⡀
⠀⠀⢀⣿⣷⡀⠀⠀⠀⢠⣿⣿⣷⡦⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⢀⣰⣿⣿⣅
⠀⠀⣼⣿⣿⣧⠀⠀⠀⣾⣷⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠦⠀⠀⠀⢈⣽⣿⣿⣏⠃
⠀⠐⠛⠻⠟⠻⠆⠀⠘⠛⠗⠿⠟⠛⠛⠀⠀⠀⠀⣠⠀⠀⠀⠀⠐⠛⠿⠛⠃⠀⠀⠐⠛⠛⠻⠿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠁⠀⢤⠤⡴⠲⣶⣦⡴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡀⢴⢾⡟⠙⣇⠀⡷⢶⣿⡆⠀⡀⠀⠀⠀⠀⢀⣄⣀⣤⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⡀⡙⠻⡎⠀⠘⠃⣈⠇⡙⢻⣿⣄⡀⠀⠀⠀⠀⢸⣿⣿⠟⠿⢿⣿⣿⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠘⠀⠰⡞⢉⠁⠦⢀⠿⠁⢐⠇⢠⡉⣿⣿⣿⣗⣀⡀⠀⣘⠂⢠⠈⣀⣀⣰⡟⠉⠥⢀⠁
⠀⠀⠀⠀⠀⢸⡦⠀⢠⡀⠀⠀⠃⠀⢀⡗⠀⢀⣿⣈⣑⣆⢈⠄⠨⡂⢀⡘⠹⠋⠁⠈⠉⠉⠛⠛⠿⠆⠀⢴⡏⠠⠉⣷⣔⡀⠈⠁
⠀⠀⠀⠀⠀⠸⠙⣄⠀⠉⡂⠰⣅⠀⣼⢿⠟⠉⠀⣠⠴⠚⣿⠃⢀⠝⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡷⡦⠀⠈⣛⠀⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠡⠘⢦⣀⣰⡄⠈⣸⢿⠀⠆⠠⠞⠁⢀⡞⠓⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢼⢿⣦⡀⢀⣀⡀⢀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⣥⠀⠉⣛⢻⣤⣿⣾⣶⣾⠧⣤⣤⣈⣀⡤⢶⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢾⣿⣿⣿⣅
⠀⠀⠀⢀⡔⢀⣀⣀⣭⢷⡶⠷⣿⣿⣿⣿⣿⡷⠿⠯⣍⣠⠴⢱⣾⣥⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⡀⠐⠻⠿⣿⣷⣦
⠀⠀⠀⣢⠒⠉⠈⠀⣀⠀⡠⡴⠋⡽⢻⣿⢿⠦⣤⣀⣀⡨⡟⠉⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡷⠶⣠⠋⣿⠉⠗⣠⡄
⠀⠀⠼⠉⠒⠖⢨⠀⣷⣊⢀⣤⣶⠃⠸⣿⣏⠳⡤⣭⡄⠀⢙⠉⠀⠀⠀⠀⠀⠀⢹⠎⡟⢹⢀⡀⣀⢀⣀⣀⣀⠀⠀⠀⠀⢸⣿⣟⡙⠙⢇⡀⠁⢰⠂⠢⠄
⠀⠀⠀⠀⠀⢇⣀⠸⣯⠉⢉⡿⠹⡀⠀⢸⡍⠀⡇⠀⠑⢄⠠⠀⠀⠀⠀⠀⠀⠀⠚⠐⠃⠙⠛⠚⠋⠙⠉⢻⡗⠀⠀⠀⠀⠘⣿⡁⡅⠙⠀⠉⢀⠉⠈⠷⣆
⠀⠀⠀⢀⠀⠙⠁⠖⢈⢀⡟⠀⠀⣱⡄⠸⡇⢠⠀⠀⠀⠈⠓⠤⢀⡔⣩⢆⣏⠀⡀⢠⠀⣠⣾⡀⣀⠀⣀⠙⡁⠀⠀⠀⠀⠀⢛⠃⠀⣙⣀⢇⠈⠦⠘⠂
⠀⠀⠀⠈⣀⠀⢻⣖⢿⡼⣀⠀⣰⣏⣘⣦⣧⣉⠁⡉⠀⠀⠀⠀⠸⢇⡤⠾⠻⠾⠙⠯⠾⠗⠧⠟⠟⠾⠿⠾⠷⠀⠔⠀⠀⠀⢈⠁⠁⠀⠉⠂⠐⢲⡆
⠀⠀⠀⠀⠘⠁⡀⠁⠘⢧⠉⣹⣿⣧⡛⠛⠫⠹⠇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡰⠆⠑⠈⠉⠳⢿⠀⠈⠃⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠁⠈⣠⠔⠒⠋⠛⡛⡛⠿⠴⠰⣖⢛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠻⣠⢰⣉⣛⢳⠊⢳⣄⠈⢰⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠫⣶⣻⣀⡶⠲⣤⣤⠶⢶⠟⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣣⠗⢄⣀⡻⢄⡾⠻⡟⠶⣀⢸⡄⠀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣽⠐⠣⠌⣳⣶⢶⣾⠓⣼⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣲⣄⢨⠉⠳⠆⣜⠁⣉⡟⠁⢠⠖⠄⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠲⢦⣤⣄⣀⡉⡿⣡⣼⡿⠻⢤⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⢀⣀⣠⣤⡠⠀⠈⣿⣿⣧⡀⠐⠆⠒⠤⢩⡭⢠⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣾⣿⣾⠿⠛⠹⠋⠉⠛⠙⢿⣿⣿⣿⣿⣷⣿⣿⡿⣿⣿⢧⣤⡀⡉⢹⡏⠀⠀⡇⢠⡄⠀⢢⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⢻⠁⠀⠀⠀⠀⡌⢀⣉⣰⣿⠿⠿⡟⠉⢻⣿⣿⠟⠃⠈⡯⠓⠀⠾⠇⢬⣾⡁⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⢄⣀⣤⣶⣿⠀⠀⢻⣿⠀⠠⢤⠈⠘⢿⢟⠀⠀⢨⣧⣤⡄⠄⢠⣿⣿⠏⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠿⠷⠿⠿⠿⠟⠟⠓⠖⠈⠀⠙⠟⠛⢦⡕⠋⢦⣡⣤⣿⡿⠃⠀⢠⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢷⡴⠶⠀⠀⠀⠀⣠⡾⠻⠛⠉⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⣵⣦⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣷⢶⡀⠀⠰⠿⠁⠀⡠⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢇⡀⣐⣤⠐⣿⠉⠉⠧⠀⠀⢸⣞⡁⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⢇⣱⠤⢄⣖⠰⠆⢴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⢷⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢶⣶⡛⣠⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠙⣿⣿⡿
⠀⠀⠀⠀⢀⣤⣶⣾⣯⣻⣵⣾⠩⢉⡁⠊⠸⣿⣿⣴⣾⣯⣉⣻⣚⣇⠜⢶⣟⣉⣟⡛⢹⣏⣽⣿⡃⠀⠠
⠀⠀⢀⠐⣾⠃⠞⣿⣿⠀⠀⠁⠀⠀⠀⠛⣶⢞⣿⡿⠟⣛⠘⠃⠿⠿⡟⣠⣤⡿⢿⣿⠛⣹⡾⢻⣿
⠀⠀⠻⢸⣿⠠⠀⢸⣿⠀⠀⠀⠀⢀⡀⠀⢰⠘⣷⣻⡟⠈⠙⠀⠂⠀⠘⠛⠻⣧⠿⣿⣽⢿⡛⣿⣿⠄⠀⠀⣠⣤⡆
⠀⠀⠀⢟⣿⡀⡀⢸⡟⠀⠀⠀⠀⠈⠉⠀⢀⠽⣏⣿⣌⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢛⣀⠠⣛⣵⣟⣠⠐⣾⣿⡟
⠀⠀⠀⠈⢻⣇⠡⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⣩⣿⠀⠀⠀⠀⠀⠀⠀⣰⣦⡾⠩⠄⢺⣏⣻⠿⣿⡟⡹⠋
⠀⠀⠀⠀⠀⣿⡄⠉⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀⢥⡄⠀⠀⠀⠀⠀⠀⠀⠀⡽⠟⠀⠀⠀⠉⠉⠏⠀⠙⡟⠁
⠀⠀⠀⠀⠀⠸⣿⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠁⠤⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠠⠄⠀⠀⠀⠀⠒⠒⢷
⠀⠀⠀⠀⠀⠀⢠⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣀
⠀⠀⠀⠀⠀⠀⣤⡀⠸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠰⣄⣼⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠟⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣾⡟⠋⠉⢿⣴⠶⢤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢣⠀⠀⡼⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣳⡦⣤⡷⢶⣷⠲⢿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣊⡴⠚⢉⣴⠞⢹⡇⢸⠈⠳⣄⠈⠻⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣁⠀⣰⠟⠁⠀⢸⡇⢸⠀⠀⠘⢧⡀⣽⡇⠃
⠀⠀⠀⠀⠀⢀⣀⣐⠒⣠⠤⠤⠶⠶⣮⡭⠿⣞⣹⣄⠀⠀⢸⡇⢸⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⣠⠟⠡⡍⢻⡇⠀⠀⠀⠀⡿⣸⠃⠈⠛⣷⡖⠒⢾⠀⢸⠲⠶⠶⢶⡟⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠐⣇⠀⠀⠙⣾⣇⠀⠀⠂⢸⣿⠏⠀⠀⣰⣿⠀⠀⢸⠀⢸⠀⢀⠀⢸⠇⠀⣤⠶⠶⠛⠛⠛⢷
⠀⠀⠀⠀⠈⢧⣤⠖⠚⠿⢤⣴⠤⠟⠛⢶⣡⣾⣋⣿⠀⠂⢸⡇⣿⠠⠀⠀⣿⣤⣼⡇⠀⠀⠀⠀⠀⢸⣷⠞⡛⠳⣦⡀
⠀⠀⣠⡤⠶⣾⠃⠀⠀⠀⣴⠛⣆⠀⠀⠈⢿⡉⠛⠛⢻⡆⢸⠃⢻⣠⡴⠾⠋⢰⡂⣇⠀⠀⠀⠛⠀⢸⡇⢠⠇⠀⠈⠙⣷
⠀⠀⡇⠀⢰⣏⣴⣦⠀⣼⠃⠀⢸⡆⠀⠂⠈⣧⠀⠀⢸⡗⢺⠀⢺⣹⡇⠀⠀⠀⢳⡸⡄⠀⠀⠀⠀⢸⣧⠏⠀⠀⠀⢀⡿⢰
⠀⡆⢿⣠⣼⣯⣤⣽⣴⡇⠀⠀⠀⢿⠀⠀⠀⢸⡶⢶⡾⠛⣽⠀⢸⠀⢳⡀⠀⠀⠀⠹⣷⠀⠀⠀⠀⣼⡋⠀⠀⠀⣠⡾⠁⠁
⠀⠘⡄⣿⠀⠀⠁⠀⠉⣧⠀⠀⠀⢸⡇⢰⠷⣬⡇⢸⣇⡀⢸⠀⣸⣀⣀⣿⣦⠔⠒⠒⠚⣳⡤⡖⠋⠉⠉⠳⣶⠾⠿⠶⠶⠶⠶⣦⡈
⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣷⡏⠀⠘⠁⢸⡇⠁⢸⣼⡏⠉⠁⢠⠏⠀⠀⠀⡼⠁⠀⠈⢦⡀⠀⠀⠘⢷⣀⣀⣀⣀⣀⣼⠃
⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠁⢸⡏⠀⠀⠀⠀⣸⠄⠀⢸⡝⣷⡤⢤⣿⡴⢶⡄⢸⠗⠖⠛⠛⠻⢷⡀⢠⣄⡈⢿⠉⠉⢉⡏⠁
⠀⠀⠀⢸⡇⠀⠈⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠠⠀⣾⠠⠀⢸⡇⣿⢹⡄⠁⠀⠀⠙⠟⠀⠀⠃⠀⠀⠈⣧⠟⠈⠙⠻⠆⠀⣾⠁
⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⡋⠀⢀⣀⣿⣤⣤⣼⣇⣿⣄⣱⣤⣤⣤⣤⣤⡤⣤⣤⣤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠋
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠶⠶⠶⠾⠋⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⡀⣆⠀⢰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠜⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂
⠀⠀⠀⢐⠀⠀⠈⡈⠉⠈⡀
⠀⠀⠀⢸⠀⠀⠀⡁⠀⠀⡁⠀⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢈
⠀⠀⠀⢸⡥⠤⠤⡿⢶⣏⡇⢠⢴⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣸⡀⢀⡦⣤⡀⢀⣯⠒⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡠⢖⡥⠁
⠀⠀⢠⠤⡬⡬⠴⠰⡠⡇⢸⠉⠀⠀⡍⡇⠀⠀⠀⠀⠀⢀⡠⠔⠈⡅⠐⠙⠿⠿⠯⠿⠛⠀⢀⡠⠥
⠀⠀⠠⢠⡜⠀⠪⠀⠄⡜⢼⠀⠂⠀⡃⠁⠀⠀⣀⠔⠊⠁⠀⠀⠈⠀⠀⠀⠋⠀⠀⠀⠈⠀⠀⠀⢀⡀
⠀⠀⢠⣨⣟⢄⣩⣨⣠⣌⣸⣀⣀⣀⣆⣇⠔⠊⠠⠇⡄⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠒⡆⠂⠀⠀⠀⠀⠙⡆⠀⠈⢤⡀⠀⠀⠁
⠀⠀⠆⠀⠀⠊⠁⠀⠀⠀⠐⠒⠀⠓⠂⠀⠒⠁

⣿⣿⣿⣿⣿⣿⣿⡇
⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢤⡄⢀⠈⣤⡈⡑⠃⣨⢐⡤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡤⢢⡤⠬⣿⣿⣿⣚⣉⡼⢙⣣⢖⣇⡒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠠⣄⡓⣎⡳⣷⣬⢻⣿⣽⣿⣿⢾⠟⢻⣆⡹⡁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⢸⠤⢧⢵⣿⢿⣳⣾⣿⣿⣿⣛⣿⣶⠾⢟⣼⣗⠶⢷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⠱⢴⢶⣼⣼⣿⠿⣿⣿⣯⣬⣿⣝⣿⣿⡿⣧⣭⢽⡄⣿⡥⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠞⣭⣿⢷⣿⣿⣿⣻⣿⣿⣽⣟⣿⣿⣿⣿⣿⣾⣷⣶⠏⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠛⣿⠻⣿⣏⣿⣯⣿⣿⣻⣯⣿⣽⣿⣟⣟⣷⡿⣿⣏⣩⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡘⣀⡏⣹⣿⣌⣼⣯⣽⢷⡾⣯⣻⣽⣿⣿⣯⢿⣷⢩⣻⡿⢪⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⠑⠙⣿⣾⣿⢭⣭⣻⣿⢿⣿⣧⢿⡿⣿⣽⣻⣾⣎⡱⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠃⠀⡛⢢⡟⢻⣽⣉⣿⡗⣷⣫⢷⣟⣏⡻⠎⣙⡍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡛⠒⢻⡦⣻⣯⣿⡝⣿⢋⡈⠽⠁⢹⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⡀⠈⢘⣺⢓⢣⠈⡳⠘⠉⠟⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡆⢠⣿⠃⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣸⣿⣠⡴⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣦⣀⠀⢸⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠉⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠻⣷⣼⡇⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣤⢿⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢠⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠌⢿⣶⣶⣤⣤⣀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢀⣀⣴⣶⣿⣿⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠘⠛⠻⠿⢾⣷⣄⠀⠀⠀⠀⠀⠀⢠⣿⣼⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⣆⠀⠀⠀⠀⢐⣼⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡀⢰⣞⠋⢹⣇⠀⠠⠊⠛⣿⣁⡀⠴⠶⠆⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣮⡿⠏⠁⠈⠉⠀⠈⢠⣿⡟⢡⣴⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠈⢿⡟⠁⠀⠀⠀⣄⣠⣄⠀⠘⠋⢀⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠃⠀⠀⠀⠐⠛⠛⠛⠁⠀⠀⠚⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣈⣿⣿⣿⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣭⣄⣀⣀⡀⠀⠀⣀⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠻⣿⣇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡏⠀⢻⣿⣷⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣿⣿⡋⠀⢙⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⣻⡯⠀⠀⠈⣿⣿⣿⠿⣻⣿⠃⣾⣿⡻⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡋⠀⠘⠿⠀⠀⠀⢿⣿⣥⣖⣺⣿⣵⣿⣿⣷⣮⣿⣇⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡃⠀⠈⡓⠀⠀⠀⠚⢻⣿⣿⣿⠟⢳⣿⣿⣿⡿⢻⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⣁⡀⣀⣹⣿⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⢿⣿⠇⠈⡟⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⡿⡟⣿⢻⣿⣿⡿⠀⢟⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⣿⣿⡿⠸⢿⠁⠀⠁⠀⠈⠁⣿⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣛⣿⣿⠿⠰⢷⣶⣦⣤⡀⠀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⣿⣿⣷⣶⣾⣮⣿⣿⣿⣶⣿⣿⣷⣶⣯⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢿⣿⣿⣿⣿⣾⣝⣛⡛⠿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⡆⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⣈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠟⠛⢉⠉⠉⢉⠻⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠀⠀⠘⠛⠀⣀⠘⣃⣠⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠰⢂⡀⠙⠛⠡⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡤⣤⣀⣤⣤⠀⣅⣁⣤⣦⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠓⠦⢤⣀⠀⠀⠀⣸⣿⡿⠿⠿⠿⠿⢷⣿⣾⣿⣦⠀⣀⣤⠶⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣐⠉⠀⠀⠀⠀⠀⠀⠀⢉⣻⣿⢻⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡉⠻⠒⠀⠀⠀⠀⠀⠀⠈⠁⢹⣧⢻⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠒⠊⠀⠀⠀⠀⠀⠀⠀⠶⠈⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠄⣀⣠⡀⢀⠀⠀⠀⠀⠛⠀⠈⡟⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠭⠛⠋⠉⠉⠉⠉⠈⠋⠙⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠠⣴⣶⣿⠶⣾⣮⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣒⡈⠠⣺⣿⣿⣿⡶⣬⣄⣂⣀⡀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠶⠞⠉⠐⠋


⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⣁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠳⣤⣴⢿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢧⠃⠂⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⢹⡷⡆⠄⣀⣀⠀⠀⢀⡻⠎⢀⠈
⠀⠀⠀⠠⢆⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠃⠰⠻⠁⠀⠀⠘⠛⠻⠺⢙
⠀⠀⠀⠀⠈⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣄⣀⣀⣀⣀⣲⠄
⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⢈⠛⠀⢀⣤⣤⢀⣀⣀⣠⣄⣴⡆⠀⠀⣀⣶⣿⣿⣿⣿⣿⣾⠟⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠘⣠⠀⠶⣬⣽⠿⠀⢻⣿⣿⣿⣿⢿⣿⣿⣷⢀⡄⡹⣿⣿⣿⡼⢿⣿⠋
⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠘⠙⡿⠁⠈⠀⡀
⠀⠀⠀⠀⠈⠞⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣈⢢⣄⠀⠀⠀⢄⠆⣀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⢨⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠲⣈⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠛⠉⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠋⠀⠸⣿⠆⠀⠀⢾⡿⠀⠈⢻⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣀⣤⣤⡴⣄⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⡇⠀⠈⠉⠛⠛⠷⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⡟⠀⠀⠀⠀⢧⠀⠀⠐⠦⠤⣤⡤⠤⠖⠂⠀⢠⠇⠀⠀⠀⠘⡇⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠿⣧⡀⠀⠀⢰⡾⡷⢶⣦⣤⣤⣤⣤⣤⣶⣶⠶⠿⣶⠀⠀⠀⣠⡿⠷⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢸⣯⣭⣭⣭⣇⣀⣀⣀⣀⣮⣭⣭⣭⣿⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠋⣿⣰⣽⡇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣿⢹⣼⡇⠀⢠⡶⣦⠀⠀⠀⠀⠀⠈⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿⣾⣿⡇⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣻⣻⡇⠀⢰⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠉⠉⠀⠀⠀⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠸⣧⡿⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠦⣄⣀⠀⠀⠀⠀⠀⢀⣀⡤⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⣾⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛
⠀⠀⠀⠀⠀⠀⢠⣿⣄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠚⢿⣿⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠲⢿⣿⣿⡿⠖⠂⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠰⠛⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠁⠀⠀⠼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣠⡴⠂⠀⠀⠀⠀⠀⠀⠘⠃⢰⡆⠀⠀⢀
⠀⠀⠀⠀⠀⣷⣤⣤⡤⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠹⠁⢠⡄⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⠀⠻⠁
⠀⠀⠀⢀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠰⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠠⣾⡄⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢴⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⣤⣤⡴⠀⠛⠃⠀⠀⠀⠀⣠⣶⣄⠀⠀⣰⣦⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⠀⡹⢿
⠀⠀⠀⠀⣀⣠⣾⣷⣤⣀⠀⠐⠷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠠⢾⣿⣿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠛⠿⠄
⠀⠀⠀⠀⠉⠛⢿⣿⠟⠉⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠰⠆
⠀⠀⠀⠀⠐⠆⠈⠁⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠘⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠗⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣴⠃
⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⢤⣤
⠀⠀⠀⠀⠀⠀⠻⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠿⣇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢨⣡⡄⠀⠀⠚⠛
⠀⠀⠀⠀⠀⣤⡀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠛⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠸⠛⠃⠀⠶
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁


⣿⣷⣶⣾⣶⡷⢾⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣷⡶⢷⠾⠿⠿⠿⣷⣶⠿⡶⠶⠶⠾⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⡀⠀⠀⠀⠀⣠⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⠙⠳⣦⠶⠛⡉⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠁⠐⠀⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠶⣦⠀⠀⠀⣰⣶⠶⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣀⠒⡐⠈⠻⣷⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡿⠛⠁⠖⠀⠀⠀⠀⠉⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡶⠿⠛⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⢿⠛⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣥⣤⡼⢆⠀⠀⠑⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠟⠁⠈⠣⡐⠄⠀⠀⠀⡉⠀⠀⠀⠀⠂⠀⡉⠢⠉⢛⡻⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠂⠰⠐⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡛⠛⠛⠟⠉⠷⡀⠀⠠⢴⣊⠀⠀⠀⠀⠀⠈⣠⠄⠀⠀⠀⠈⠐⠀⠀⠀⠙⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠿⠀⠀⠈⣠⠂⠵⡀⠀⠠⠕⠆⠀⠰⡦⠽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣏⠁⠀⠀⠀⠀⠐⢧⠀⠈⠢⡀⠀⠈⠦⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣦⡄⠀⠀⠀⠀⢣⡀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠛⠉⠁⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠉⠒⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢄⠛⠋⠹⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⡂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠀⣁⠤⠒⠉⠀⠙⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⣤⢀⡀⣀⢀⣀⡀⣀⢤⠰⠒⠉⠀⢐⣄⠀⠠⣤⣤⣿⠇
⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⢀⡔⠀⢀⡀⢩⠀⠀⢀⣴⣿⠟⢻⣦⣄
⠀⠀⠀⣾⣿⣛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠰⠎⠈⢁⡠⡺⢟⠝⠉⠀⠀⠈⠙⠿⣦⣄⡀
⣠⣤⣤⣤⣽⣿⠟⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⠈⣊⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡦
⠙⢿⣬⡉⠉⠀⢁⠢⠄⡀⠀⠈⠉⠿⢻⡦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠁⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⠇
⠀⠀⠉⢻⣷⠁⠀⠀⠐⠀⠉⠐⠂⠠⢄⣰⠖⠶⣄⠀⣠⠶⠶⣦⠀⠀⠀⠠⠄⠒⠂⠉⠀⠀⢀⣤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢿⣏⠉
⢀⣤⡾⠟⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⢨⠛⡅⠀⢀⣼⣯⣄⡀⠀⠀⠀⠀⣀⠤⠐⠉⠀⠀⣠⣤⡀⠀⢀⣤⣄⠀⠀⠀⠀⠀⢀⡈⠻⣷⣄
⠿⣿⣶⣶⣧⣤⣀⣬⣵⡦⢴⡟⠛⠛⠉⠙⡉⠋⠁⡘⡀⠉⠊⣀⡉⣉⣿⣥⠔⠂⣽⢤⡀⠀⣀⣀⣠⣏⢀⡙⠒⣋⣀⣸⣄⣀⣀⣀⣠⠟⣻⣆⠈⠻⠷⣶⣶
⢠⣾⣿⠿⠧⡼⡿⢦⠺⠷⣾⡇⠀⠀⠁⠐⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⢹⡇⣤⣼⡇⠈⣷⣤⣿⡉⠉⠁⠈⠁⠀⠈⠀⢬⣏⣹⣿⠤⣿⣿⡥⠻⣦⣶⡿⠟⠋
⠸⢿⡿⠶⢶⣁⢿⣈⡟⠛⣿⣷⠀⠐⠂⠐⠀⠀⡃⠐⠒⠀⠀⢐⠀⣀⣾⣿⣧⣀⠦⠠⢧⢤⡿⣾⡿⠛⢚⠇⢤⠀⡖⠒⠂⢸⣿⣗⣿⣿⠒⡆⢹⣿⡆
⠀⢸⠇⠄⠰⠛⠿⠟⠃⠐⣿⣿⠀⠀⠀⠘⣢⠀⠀⠀⠀⠈⠀⠟⠀⢻⡍⠋⡀⢻⠀⠀⢰⠀⠂⢸⣇⣄⠜⠋⠀⠲⠟⠀⢀⣼⣿⢿⣧⣿⠀⢱⡄⢻⡄
⠀⣼⠀⠀⠀⠀⣤⣤⠶⡾⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢦⡜⠉⠀⠀⢠⠈⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⡙⢧⣟⣸⣁⠾⣦⠀⣹⡈⢷⡀
⠀⣿⣀⣀⣀⣀⣿⣌⠀⣁⠋⣈⣽⣿⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢸⡷⠂⢀⠘⠀⠀⢀⡀⢃⢼⣇⠀⠀⠀⠤⣠⣤⡿⡽⠾⠟⠛⠟⠛⠋⠉⣻⢧⡘⣧
⠀⣿⡉⣉⣉⣉⣉⣍⢠⡉⠅⢉⠀⠀⡄⠠⢤⢴⣿⠀⠀⠀⠀⠀⠀⠸⣧⠘⡤⠉⡁⠀⢠⠘⠁⢸⡇⠀⠀⠀⠀⢸⣇⣀⡀⢰⠄⠀⠰⠦⠤⠠⣿⠄⢣⣻⡆
⠀⢸⡇⠈⠠⢀⠁⠠⠀⠁⠐⠋⠰⢀⠁⠠⠀⢸⣇⣠⣤⣀⣤⣤⣤⣴⣿⠁⢀⠀⠇⠀⠀⢠⠈⢸⡟⠒⠒⠒⠒⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠉
⠀⠘⠷⠶⠶⠾⠶⠶⠷⠶⠾⠶⠶⠶⠶⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣴⣥⣴⣩⣤⣼⣤⣅⣼⡇⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣴⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢧⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄⣓⣤⣀⢳⣀⡤⠚⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⡤⠠⡄⠀⠀⣿⠀⠀⠈⠉⠙⡟⠺⡷⢮⣭⣉⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡀⣈⠳⣄⣠⣼⣖⣙⣤⣯⣤⠀⠀⠀⢰⠀⠀⠇⠀⠀⠀⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣭⡿⡥⢞⠻⢍⡀⠀⠘⣧⣤⣄⡜⠀⢸⠀⠀⠀⠀⢀⡿⣀
⠀⠀⠀⠀⠀⠀⠀⣏⠉⠰⢯⡴⠇⠀⢳⡄⣾⠷⠶⣏⠀⠀⡇⠀⡼⢍⠙⢯⠁⣹⡇⠀⡱⠂
⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣸⡥⠾⡀⠀⢏⠃⠀⠀⠹⣆⠠⠇⣸⣀⣀⣙⠦⣧⣼⣴⣋⣠⣤⡀
⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠙⠓⠦⣧⣀⠈⡆⠀⠀⠀⠹⣼⠀⢸⣁⣨⡭⠿⠿⣿⢿⠍⠀⠈⣷⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢾⣄⠀⢠⡶⠛⠛⠉⠁⠀⠀⣀⣀⣽⡎⣶⠚⠋⢻⡀
⠀⠀⠀⢰⣧⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⢧⣠⣤⠴⠖⢛⡏⠉⠀⠼⠛⢇⠐⣂⣈⣷⣠⣄
⠀⠀⠀⣰⠃⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⢹⡄
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⡾⠀⠀⠀⡐⡗⣀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⠃⠀⠀⠿⡪⡻⣗⠓⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠚⠋⣧
⠀⣾⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠛⠛⡛⠛⢻⠛⠛⠉⠉⠁⠀⢠⡄⠀⠀⢹
⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⣀⠀⠀⠀⠤⣶⣶⠄⠀⠀⠀⢀⣇⠀⠈⠀⠀⠀⠀⠠⡮⣶⢕⠵⠆⢸
⠀⣇⠀⠀⠀⠀⠀⢀⢄⣹⣳⣿⣼⣏⡠⡀⠀⠀⠘⠀⠀⠀⠀⢰⠁⠀⣱⠔⠢⢄⠀⠀⠀⠈⠡⡇⠁⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠈⠉⣽⠟⣿⡻⣯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠴⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⣸⠤⠤⠤⢀⣀⠀⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠀⠀⠀⠑⠤⠤⠚⠁⠀⠀⢀⡀⠤⠒⠉⢧
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠉⠙⠛⠓⠒⠒⠶⠶⠦⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠊⣉⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠸⠀⠈⢪⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠐⢄⠀⢣⠀⠀⢿⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡀⠀⠐⢤⡑⠄⠱⣄⠈⣧⡠⠤⡒⠁⠀⠀⣀⡀⠤⠊⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠖⣌⣉⣁⡀⣉⣉⣐⡊⣑⣦⣬⡗⢻⢡⠀⠘⣍⠉⠁⠀⠀⠀⠀⣀⣨
⠀⠀⠀⠀⠀⢸⠉⠉⠁⠐⠒⢲⠐⠂⢉⠭⠽⠟⠉⣉⢁⡉⣲⠰⠀⡴⠮⢍⠉⠉⠛⠛⠂⠈⠉⠉⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠂⠀⢸⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣤⠄⢀⡀⢀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⢸⠀⠀⢇⡀⢀⡀⠠⡤⠆
⠀⠀⠀⠀⢀⠜⢸⠀⠀⠀⠀⢸⠀⠀⢸⠉⠉⠉⠁⠐⠚⠒⡇⠀⠀⢸⠉⢹⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠓⠂⢹⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢣⠔⠂⠤⠄⠘⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠂⠐⠧⣀⡠⠄⢀⡀⢀⣀⣀⣀⣀⡀⢀⡀⠠⠴⠀⠠⠔⠐⠒⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡄⠀⣀⣦
⠀⠀⠀⢀⡀⠳⠀⢴⠟⠙⠀⠀⢸⡏⠁
⠀⠀⠀⠈⠁⠀⠀⠘⣇⠀⢠⠀⢸⡇⠀⢀⠀⢀⣤⠀⣶⠀⠄⠀⡴⠌⠲⡄⠤⠐⢢⡄⠠⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡆⠀⠻⠀⣿⠀⠈⣓⡄⠈⠁⢸⡧⡖⠋⠀⠀⠀⠇⢀⣰⠇⠀⠀⢿⠀⠀⣿⠈⠀⠐⣏⠂⢠⣤
⠀⠀⠀⢀⠀⠀⣿⠀⠀⢰⡇⠀⢸⡏⠀⠀⠀⢰⡦⢇⡀⠀⠀⠀⡟⠈⢱⠀⠈⠀⣼⠄⢺⡁⠀⠀⡀⠙⣰⠋
⠀⠀⠀⠈⠋⠀⣿⠀⠀⣼⠀⠀⠘⠇⡀⠀⢀⣈⣃⣀⣻⡰⣤⣤⣱⠦⣬⡷⢀⣠⣙⣆⠀⢧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣯⠀⣀⡿⣀⣠⡦⠴⠟⠈⠉⠉⠉⠉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠀⠰⠶⢤⣱
⠀⠀⠀⠀⠀⣀⣈⣽⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⠞⠛⠃
⠀⠀⠀⠙⠚⠋⠉⠀⠀⠀⠀⣠⠶⠛⠉⢽⣓⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣰⠉⢰⡇⠀⢠⣤⠐⣦⠀⢠⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⢸⡇⢠⡶⣄⣷⣀⠀⠀⣤⠀⠀⡄⠀⠀⡤⠀⠀⢸⡇⠀⢰⠢⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣩⡄⢸⡇⠀⢨⣅⠰⡏⠀⢸⡇⠀⠳⣄⠈⠀⠀⣿⠉⠀⠀⢿⠀⠀⡇⠀⢀⠸⡇⠀⠈⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣿⠀⢸⣇⣄⣼⡇⠀⣧⠀⢸⡇⢀⡀⠈⢦⠀⠀⣿⠀⠀⡆⠀⡤⢀⡇⠀⣼⠀⣧⠀⢠⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢀⣀⡿⠴⠾⠃⠛⡿⠃⠒⠻⠲⠼⡇⡈⠂⢀⣾⠀⠀⡏⠀⠀⡏⢠⡇⢸⡇⠀⡶⠂⢸⡀⣈⣁⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠐⠦⠷⠀⠤⠇⠼⠠⠼⠧⠶⠇⠐⠚⠃⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢚⡠⠂
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⡀⠴⠀⠀⢀⡄⠈⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠆⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡠⠖⠊⠁⠀⠀⠀⠈⡦⢤⡀⠀⠀⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠂⠀⠀⠀⠀⢠⠂⠀⠂⠀⠀⠀⠀⣴⣋⣠⠤⠤⠤⣀⣀⢠⠞⢄⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠠⢀⡠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⣀⣀⣀⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠄⢳⡀⢠⢂⠁⠈⠀⠀⠀⡞⢠⠔⠋⠉⡁⠀⠀⠉⠙⠲⢄⡀⠈⡇⠀⠀⠐⠃
⠀⢀⡤⢤⡀⠀⠔⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⡁⠀⠀⠀⠐⠀⠀⢸⠃⠀⠠⠄⠁⡀⠀⡀⠉⠄⠀⢱⡴⠁⠤⠤⠒⠒⢢
⠀⠈⠄⠠⠃⢀⡖⠂⠐⠤⠤⠖⠤⠤⠞⠒⠤⣌⠈⠂⠀⠁⠀⠀⢨⠄⠀⠀⠀⠀⠐⠀⠊⠀⠀⠀⢠⠆⣸⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠈⠢⡀⢀⡤⠋
⠀⠀⠀⢠⠊⠀⡀⠀⠀⠂⠀⢠⠤⠴⠒⠒⠚⠂⠹⡑⡒⣩⠖⠋⠁⡈⢄⠀⢀⠀⠀⠀⠀⠀⣀⠖⠉⠑⢄⠀⢈⠅
⠀⠀⠀⠀⡼⠁⠀⠈⠐⠒⠉⢻⠀⠀⠀⡆⢰⣀⡜⠛⠻⠅⡀⢀⡴⠀⠀⠈⠈⠑⠶⠒⠋⠈⠁⠀⣀⠴⠊⠉⠁
⠀⠀⠀⡼⠁⣀⣀⣤⠤⠴⠖⠚⠛⣿⠈⠁⠀⡇⢣⣄⠀⠀⠀⣱⠃⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢰⠂⠢
⠀⢀⡼⠁⠸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠴⢶⡒⢿⣹⠀⠉⣱⡶⠯⣄⡀⠤⢘⠀⡉⡉⢀⠁⠁⣀⠠⢷⣀⠀⠀⠀⠉⠁
⠀⠘⠢⠤⢤⠁⠀⡄⠀⠀⠀⠀⠀⢺⠀⠀⠀⠸⡆⠒⠴⠹⡀⠀⠀⠉⠓⠒⠶⠯⠥⠤⠤⠴⠖⠒⠉⠀⣱
⠀⠀⢀⡀⠀⠰⠀⠄⠀⠀⣀⣀⣀⣸⣀⣀⣀⣀⡹⠈⠉⠉⠉⠱⠶⠤⣤⣤⣀⣀⣀⣀⣀⣤⣤⡴⠒⠋⠁⠀⠘⠕
⠀⠀⠈⠀⠀⠀⠐⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡠⠀⠘⠒⠀⠠⠅⠀⠀⠀⢍⠈⠀⠤⠼⢄
⠀⠀⠀⠀⠀⣖⢑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠤⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⢸⣠⠏⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⣟⠀⠀⠀⢰⠃⠘⣄⠀⢠⠇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢻⡀⠀⢠⡏⠀⣀⠈⢶⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠹⣦⣾⠛⠋⠁⠀⢸⡆⠀⢀⣼⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢶⠦⣝⠷⣴⠀⢀⣼⠷⣶⡛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠷⡼⠳⠞⠛⢀⢑⢋⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠋⠈⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⢧⣀⡀⢀⣀⣀⣾⡯
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠙⠷⠿⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⡄⠀⠙⡖⠤⠤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣾⢁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠰⣤⡀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡹⣿⡟⣇⣾⢳⡀⠀⠀⠀⡄⠀⠀⠀⠀⠘⣿⠀⢻⣷⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠸⠇⢻⣧⣿⣿⡈⢃⡀⠀⠀⣷⣠⠀⠀⠀⢸⡟⢸⣿⣿⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⠈⣧⠀⢸⣧⠟⠄⠀⢠⢟⣠⣿⢿⣿⣇⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⣸⣭⣤⣤⣶⡾⣿⣿⣷⣮⣿⣿⣧⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⢻⡇⢿⡇⡰⠉⠉⠉⠁⠀⠀⠈⠻⣿⣿⣿⣧⡙⠻⣷⣤⡈⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠇⢸⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⡻⣦⠀⠙⠫⣼⣟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢨⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢲⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⣿⣿⣿⣿⣿⣿⣹⣿⣫⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⣿⣿⢟⣽⣿⢿⣿⣿⣻⣿⣛⣿⣿⣿⣾⣿⣿⣿⣿⣿
⡋⠿⣍⡻⡭⣟⢻⣾⣯⣿⣿⣫⡼⢟⡻⣿⢤⢸⣉⣛⠿⡿⣟⣋⢿⡵⢿⡘⣿⢿⡃⣽⡟⢻⠻⣟⠟⠯⢻⣿⣿⣤⣽⡍⡙
⠙⣟⠿⢙⠖⠸⢭⡷⠛⠃⢛⠟⡷⠛⣧⠽⢲⣏⣧⣿⣇⠻⣭⠻⠚⠷⠧⠓⢃⡾⡻⣿⠌⠧⡇⡹⡛⠄⣜⣹⣷⣶⣿⣧⠛
⠀⠉⠂⠁⠄⠀⠘⠀⠀⠚⠉⠀⠀⠀⠡⠆⠀⠋⠀⠃⠀⠡⠎⠃⠉⢀⠀⠆⠀⠀⠒⠁⠀⡀⠁⠂⠋⠉⡀⠁⠀⠀⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣤
⠀⢀⠠⠀⡖⠀⠀⠄⠀⢠⢀⣠⠄⣰⡄⠀⡀⢀⠀⠀⢰⠈⡂⠀⠀⠀⠀⣀⢰⣄⠄⠀⢀⡅⡀⢄⡁⢀⠘⠀⠀⡀⠀⠀⡠
⢲⣼⣅⣤⡧⡤⠴⡷⣶⠌⢿⣣⡤⢥⣬⣡⡽⣍⡖⠻⣏⡷⣇⣶⠢⣄⡛⡃⠈⣶⣤⣇⢺⠟⡋⠋⣭⠔⢪⣊⢦⣶⣙⠓⢖
⡁⣒⣻⣷⣷⣾⣰⣿⣿⣿⣟⢿⣱⣺⣟⢿⣌⣍⡳⣮⡘⢿⣿⣛⠿⠟⢿⣈⣹⢈⣿⣥⣿⡌⡟⣿⣆⢶⣶⣧⣯⣼⣿⣴⡿
⣿⣼⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣳⣾⣿⣿⣯⣿⣿⣿⣿⣴⣿⣿⣯⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡟⢿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠋⠉⠉⠁⠀⠀⠀⠀⠈⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠐⡀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠠⠊⠱⡁⠂⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠤⠶⠦⠚⠒⠲⠶⠦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠢⠀⠀⣁⡤⠤⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠴⠚⠉⠀⠀⠀⠀⣀⣀⡤⠤⠤⠖⠒⠒⠚⠉⠉⠉⠻⡒⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⢀⣠⠤⠒⠊⠩⠥⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢱⡀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣤⠚⠉⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⢰⣖⠄⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠁⠸⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡓⠦⠤⣀⣀⣀⡀⣀⣀⣀⡠⠤⠖⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⠳⢤⣀⣀⡉⠉⡁⢀⣀⣀⠤⠲⡅⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡼⢩⣏⡏⠉⠉⠁⠀⠀⠀⠘⡄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢠⠀⡼⠀⡾⠰⡇⠠⠈⠀⠀⠀⠀⠀⢹⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⢠⡇⠀⣧⣀⣱⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠤⠋⢸⣠⣀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠉⠈⠙⠢⡀⠀⠀⢀⣀⠎⠁⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣄⣀⣀⡤⠏⠉⠉⠉⠙⠦⣤⣀⣠⠤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠘⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢰⡀⠀⢸⠀⠀⣴⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢠⠠⢀
⠀⠀⠀⠀⠀⠢⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠃⠀⢈⠀⠀⠁⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠠
⠀⠀⠀⠀⠈⠡⢠⠨⠁⠀⠀⠀⠀⠘⠀⢤⣄⠀⠀⢠⡀⠀⢿⠀⢀⣤⠀⠀⣀⣤⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠘⠠⠞⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠀⠙⠛⠉⡀⢀⢀⡉⠛⠛⠁⠈⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠖⠀⠀⢰⠟⠁⣼⠀⠙⢦⠀⠀⠒⠦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⠀⠀⠙⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢰⠃⠀⢸⠀⠀⢷⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢈⠀⠀⠀⠂
⠀⠀⠀⠀⢄⠈⠂⠘⠀⠊⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤
⠀⠀⠀⠢⠀⠰⢄⠸⢀⡴⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡘⠇⠸⠁⣀⡄
⠀⠀⠀⠒⠀⠒⢂⠀⢀⠒⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠭⠀⠀⣍⡁
⠀⠀⠀⠊⢀⠐⠃⠸⠀⠓⢀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠃⠳⠀⠁⠁
⠀⠀⠀⠀⠁⠠⠂⢰⠀⠢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⡀⢀⡀⠀⣀⢸⡇⠀⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠣⡀⠀⢸⢉⡗⢸⡇⣾⠁⠀⣽⠂⢸⡇⣾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢀⡟⠘⠃⢸⠇⠿⠀⠰⡏⠀⠸⠀⠙⢸⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣴⠀⠀⠀⢈⡟⠁⢀⡈⠃⠀⠀⣠⠀⠀⣴⠀⠀⡾⠈⠁⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠻⠀⠀⠀⡾⠀⠀⣿⠀⠀⠀⡂⢹⡄⢴⡟⠀⠈⠀⠠⠤⠼⠃⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠁⢰⡇⢸⠆⠈⣹⠃⢸⠃⠸⣧⢸⡇⢸⡇⡶⢰⢠⠖⡖⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⠀⢿⠀⠙⠀⠀⢹⡘⠄⠹⠑⠃⡟⠘⠘⠃⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⡐⢈⡧⠀⢸⠃⠀⠀⠀⠀⠉⠀⠀⣦⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘⠃⡠⠋⠀⠀⠀⠀⠀⠀⠀⠰⠇⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠟⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣯⣭⣭⣭⣭⣤⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⠿⠿⠛⠻⠭⠽⠯⠭⠟⠓⠀⠉⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣠⣴⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠿⣦⠀⠀⠀⠀⠀⢰⠟⠛⠣⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⠀⢀⣿⠀⠀⠀⠀⠀⢹⣆⣀⣰⠿⡄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣟⢙⡛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣷⠀⠀⠀⠸⡀
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣠⠼⠿⠿⠟⠛⢀⡄⠀⠀⠃⢀⣤
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠋⠀⠀⢀⠋⢀⣶⣠⣇⢀⣀⣀⣶⣴⣷⣶⡶⠟⢻⠏⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⣿⢿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠰⡃
⠀⠀⠈⠛⠛⠛⠛⠋⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡶⠟⠟⠋⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠃⢰⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⢣⡀⠀⠀⠀⠀⠙⠛⠷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠟⠋⠀⣠⠏⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⡶⠖⠶⠶⠶⢶⡶⠾⠿⠟⠛⠛⠉⠁⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣦⣤⣄⣀⣠⠀⠀⠀⠀⠀⣯⣤⣤⣤⣶⣶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⠀⠀⠀⠀⠀⣿⠉⠁⢸⣿⣷⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⢸⡿⠿⡿⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡜⠀⠀⠀⠀⠀⢻⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢰⡶⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⢸⡇⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠈⣇⠰⣿⣿⣷⠀⠀⠈⢣⣀⠀⠀⠀⠀⠀⠀⣀⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⡇⠀⠀⠀⠀⠀⠸⣧⠀⠀⢻⠀⠙⠛⠋⠀⠀⠀⠀⠉⠛⠒⠶⠶⠚⠛⠋⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠉⠛⠛⠻⠶⠶⠶⠶⠞⠛⠉⠀⣿⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠸⣄⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢹⡆⢀⣴⢿⣿⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣿⣧⣽⡾⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⠶⠶⠶⠿⢛⣋⣻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣀⣀⠀⠀⠀⠀⠀⠈⠻⠟⠿⠟⠿⠟⠛⠛⠁⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⢶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⠞⠛⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⡙⣿⠛⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠺⠀⠀⠀⠼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡠⠊⠀⠀⢠⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⠀⡀⠀⣰⣮⢖⣆⠆⠀⠐⠄⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠁⢀⣰⣿⠛⠉⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣬⡀⠀⠐⠂⠀⣸⡿⠁⡀⠀⠀⢀⠀⠰⠊⠁⠠⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠙⠓⠶⠶⣿⣿⣄⣘⣆⢨⣠⣤⣀⣀⠀⠀⠁⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⣿⣿⣿⣍⠙⠷⣄⣷⣄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⢿⣿⣶⣶⣦⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠦⠂⠀⠩⢽⣷⣄⠈⠻⣁⠖
⠀⠀⠀⠀⠀⠀⢠⡶⢿⣿⣿⣿⡷⢿⢟⣿⣿⣯⣿⠋⡉⠛⠿⠟⡿⢿⠛⠛⠓⠒⠒⠶⣤⣄⡀⠠⠞⠀⠀⠀⠀⠀⠀⣀⡈⣻⡷⢀
⠀⠀⠀⠀⠀⢰⡟⣰⣟⣿⡿⢿⠐⠀⡀⠰⢻⣿⠟⠀⠀⠀⠀⠀⠀⠹⠓⡀⠀⠀⠈⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⣵⣾⠁⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⣷⠘⣆⠈⣧⠀⠈⢹⣯⣃⣀⠀⠀⠀⠀⣠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⠘⠀⠙⣷⢀⣀⣀⣉⠉⠑⠀⠀⠀⠙⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⢿⣖⡷⢰⣅⡂⢉⣿⣿⣷⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠠⠀⠀⢁⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣉⠘⣿⣧⠀⢈⡿⠛⠲⠤⣌⡉⠉⠛⠛⣻⣿⣶⣷⣤⣦⣤⣴⣾⣇⢀⣄⡠⠶⠛⠁
⠀⠀⠀⠀⠀⢀⠐⢋⣤⠟⠁⡠⣳⢿⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢙⣿⣿⠛⠋⠁⠀⠀⠁
⠀⠀⠀⠀⡼⢋⡴⠋⠀⢀⣺⠏⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡜⣿⢻⡀
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠇⢸⣿⡇
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢻⡇
⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⣧
⠀⠀⣼⣿⣄⠀⠀⠀⠀⠀⠀⠸⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠛⠛⠒⠀⠀⠀⠀⠀⠘⠛⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⢤⣀⠛⠻⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠈⠀⠀⠀⠀⠀⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠊⢃⠢⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⡀⠠⠐⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠒⠈⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠔⠁⠀⣀⡀⠂⣄⠀⡑⠤⠈⠑
⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⡄⢁⣢⢎⡔⠂⣆⠀⠱⠄⠂⠈⠀⠀⠀⠀⡈⢄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠈⠀⠀⠠⠞⠁⠀⡉⠦⠇⠉⠠⢄⠀⢠⠐⢀⡠⠃⠀⠒⠕⢄
⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠤⠐⠁⠀⠀⠀⠐⠌⡀⠐⡀⢑⠮⢆⢀⠐⣎⡀⠀⡩⢥⢀⡠⠘⠣⠴⠥⡀⠀⠑⢠
⠀⠀⡆⠀⠀⡠⠐⠀⠁⠀⠀⠀⠀⢀⢄⣠⡄⠀⠀⠉⠙⡧⢀⡌⠉⡣⠤⣉⣉⣀⣀⣀⡀⠀⡄⠀⢀⡗⠀⠁
⠀⠀⢰⠔⠁⠀⠀⠀⠀⢀⢤⠔⠈⠀⣤⡄⠀⠀⢀⡠⠊⠀⠀⠈⠈⠃⠠⡎⠁⡄⢀⠀⠀⠐⠊⡈⠀⠈⠂
⠀⠀⠸⠀⠀⠀⢀⠔⠂⠑⠁⢸⡿⠂⠉⠉⠖⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⡞⠁⠂⡿⠤⡀⠬⡀⠀⠀⠀⠑⢄
⠀⠀⠀⠁⠤⠲⠁⠀⠀⠀⢀⠠⠓⠄⣤⠖⠋⠀⠀⠀⠀⢀⡀⠀⠌⠀⠑⠠⠇⠤⠜⠁⠈⠃⠀⡏⠈⠖⠒⣄⠄⡑⢄
⠀⠀⠀⠀⠐⡈⠑⠂⠄⠂⠀⠀⢄⠁⠀⠀⢀⠔⠐⣄⡸⠀⠈⡱⠒⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠐⢃⡀⣜⠁⠹⠂⡄
⠀⠀⠀⠀⠀⠇⠀⠈⠁⠀⠂⠈⠁⠈⠁⠒⠳⢄⣠⣄⠃⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢤⠂⠘⣄⠈⠛⠃⠉
⠀⠀⠀⠐⠢⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⢱⠀⠈⠁⠀⠀⠐⠒⠒⠂⠒⠀⣒⠈⢺⡀⢈⠤⠀⠁⠀⠀⠑⢄
⠀⠬⡀⠂⠀⠋⢂⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠁⠐⡞⠂⠁⡆⠀⠀⢀⠄⠄⠉⢺⢀⡸⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⡔⠒⠀⠀⠐⠀⠀⠱⠢⢄⡀⠠⠐⠊⠑⡤⠤⣀⠀⡸⠤⠤⠜⢖⠉⠀⢤⠜⡠⠓⠢⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠑⠠⡀
⠸⠁⠀⠤⣃⠀⣠⠁⠀⠃⠄⣀⠀⠑⠊⠃⠀⠧⣀⠰⠀⠜⠒⠈⠐⠒⠊⠈⠲⣀⡠⠔⢆⠁⠀⢣⠄⠈⠁⢄⣠⠂⠀⢁⠆⠀⠀⢀⡁⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠌⢁⣲⠆⠠⡤⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠙⠤⣔⡊⡀⠱⡶⠗⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠂⠸⡀⣸⠂⢢⡀⡀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⡀⠀⡅⠀⢀⠆⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠈⠒⠊⣀⡰⠒⢀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠩⠋⠉⠁⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢀⡠⠤⠚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠚⠀⢸⠊⢱⠄⢄⣀⡀⢐⠁⠉⡸⠒⠒⠀⠀⠀⣀⠀⠀⢀⠄⠠⠈⠑⢄
⠀⠀⠀⠀⠣⢀⠱⠀⢀⣱⠒⠁⠰⢄⣠⠒⢢⠀⢀⠄⢄⠀⢀⠤⠄⢀⠔⠐⢞⡤⠃⠀⢛⠇⠠⡔⠓⡄⠒⢔⠉⢪⡀⠹⢀⡨⠂⠄⠁⠈⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠂⠤⠑⣀⠂⠘⢀⣀⠤⠻⢄⠀⣈⠳⡅⠀⣈⠇⢄⡀⠖⠉⠑⠒⠁⠀⠈⠃⠄⠓⠤⠈⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⡆⠀⠠⠄⠀⠀⣀⣉⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢠⠤⠄⠀⠐⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⠘⠁⠉⠀⠀⠀⠀⠀⢒⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⢁⠀⠀⠀⠀⢠⣄⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡷⢦⡼⣧⣤⡶⣿⣿⣦⣠⣾⣆⠀⠠⠤⢬⠀⠠⠄⢄⡀⠀⠀⠀⠃⠀⠀⠀⠈⡀⠀⢀⠰⠿⠿⣷⠿⠿⢧⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢘⡿⡗⠁⢀⡇⢸⠁⣍⠉⢹⣏⠉⢉⠿⠿⢶⡶⡿⠶⡾⢿⣷⡄⣴⣈⡉⢲⡆⢁⣠⣇⠀⠘⠒⠒⠒⡟⣗⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢈⡏⡇⠢⠆⢠⢴⠀⠀⠀⠞⠸⠀⠀⡏⢸⠁⠀⡴⠀⠈⢫⠉⠩⠉⢿⡝⠳⠘⠫⡍⠉⠥⡮⣷⠤⠀⠁⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⣟⡇⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⡇⠈⢱⠲⠀⠑⠈⠱⠀⠀⠀⡌⠀⢰⠂⠀⡰⠀⠀⡇⢨⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠙⠀⠤⠤⠤⠤⠤⠤⣇⣀⣸⣀⣀⣀⣀⣈⣀⣀⣀⣸⣁⣋⣈⣁⣀⣀⣀⣇⠸⠠⠂⠀⠒⠒⠒⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠓⠂⠀⠈⠓⠤⣀⠀⠠⠐⢰⠀⠰
⠀⠀⠀⢂⣬⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠑⠤⡑⣆⡣⢆⡊⠐⠄
⠀⠀⠀⠀⠅⠘⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣠⡀⠀⠀⠢⠀⠀⠀⢀⠈⠳⢉⢏⠐⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠒⠛⠁⠀⣾⠇⠀⠀⠀⠀⠈⠖⢬⡐⢄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠑⠒⠒⢲⠂⠒⠀⡀⠀⠀⠀⠀⠀⡑⠂⠂⠀⠀⡀⠀⠀⠀⠀⠀⠆⠀⠈⠀⠤
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠂⠀⠘⠁⠉⠀⠀⢠⠀⠀⠀⠀⠤⠀⠐⠀⢀⠠⠐⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⠀⠀⠘⠀⠀⠀⠀⡁⣴⢦⡀⠀⠀⠀⠀⠀⠀⠠⢞⣿⡆⠀⠀⠀⡄
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠈⠀⠀⢀⡀⠙⠻⢟⠀⢠⣗⣶⣀⣈⣈⣄⠸⠛
⠀⠀⠀⡀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢴⡤⣦⣴⠾⠶⠶⢛⠀⠀⢤⠀⠀⠉⠛⠚⡿⠽⠫⠟⠀⠀⠿⣦⣀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠋⠀⠀⠀⠀⠈⠀⠠⣤⠀⠀⠀⠀⠀⣀⣀⣀⣤⡄⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⣀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡄⠀⠀⠀⠀⡧⠥⠗⠛⢃⠠⠀⠀⠀⠈⢷⣄⡀
⠀⠀⢠⠀⠀⠀⢀⢧⣸⢨⠀⠀⠃⡅⡇⡰⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⢠⡀⣀⣀⣠⡬⣆⠐⣄⠀⢰⠟⢿⠉
⠀⠀⠀⠀⠀⠀⢸⠀⠈⡜⠁⠀⠻⡀⡿⡀⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣯⡭⡭⣷⠼⢋⠑⠈⡀
⠀⠀⠀⣀⢀⠀⡘⢀⠘⢛⢂⡂⢃⠓⣃⠑⠀⡃⠀⠀⠀⢀⡀⠀⢐⠀⠀⣠⡀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢠⠋⠈⢹⠀⡇⡆⢘⢸⢸⠈⠂⠹⠀⡿⣠⡇⠀⣾⠀⢨⠈⠂⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⡆⠀⠀⢸⠒⡇⡷⡊⢸⠀⢢⠀⡄⠀⡇⠁⠃⢸⡠⡇⠈⢢⠀⠀⠀⠀⠀⠛⠂
⠀⠱⣄⡼⠸⠀⠳⡇⠱⡸⠳⣀⠇⠇⠀⠇⠀⢾⡇⠀⢸⠳⣀⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂