Ecmascript Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠴⠶⠶⠶⣶⣶⠶⠦⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⠂⠀⠀⠀⠀⠀⠉⠙⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡉⠭⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠃⡀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠠⠤⠈⠀⠒⠹⠷⠄⠀⠀⠀⠀⠀⠑⠄
⠀⠀⢀⡔⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠰⣚⣲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⣺⣣⣄⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠈⢢
⠀⢀⡞⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⡀⠁⠀⠀⢙⡇⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣲⣤⣤⠂⠀⠁⠀⠀⠀⠈⠘⠐⠚⠿⠀⠀⠀⠂⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⣴⣭⡦⣤⡄⠑⠀⠀⠀⠀⠀⠀⢀⡠⠜⠑⠲⠤⣀⠀⠲⢤⣀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠘
⡾⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢌⠀⠀⠀⠀⠀⠁
⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠠⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄
⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⢷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰
⠀⢷⢸⠂⠀⠀⠀⠀⠀⠀⠀⠑⢤⣤⣄⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡞
⠀⠈⢿⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡆⠀⠀⠀⢀⡼
⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡞
⠀⠀⠀⠀⠙⣦⡀⠒⠂⠐⠀⠀⠰⠄⠉⠉⠉⢉⠉⠉⢂⠀⠠⢤⡄⢀⠀⡀⠀⠀⠀⠀⠔⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡠⠋
⠀⠀⠀⠀⠀⠈⠻⣶⡀⠀⢤⠀⠀⣀⠓⠀⣀⠆⠠⠗⠋⠛⠀⠈⠉⠉⠀⠩⠭⠿⠏⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠒⠦⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠂⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⣣⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢶⣶⣚⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠲⠤⠤⠤⠤⠤⠶⠶⠒⠒⠛⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠀⠠⠐⠀⠀⠄⠄⠄⠄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣠⣤⣴⣶⣶⣶⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢶⣻⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣤⠀⣔⣺⣭⣶⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⣀⣤⣾⣿⣿⠛⣊⣽⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⡀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⢋⣹⣫⣿⣿⣿⣿⣻⣿⣿⣿⣿⡿⢿⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⡈⠢⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠒⠺⣿⠿⢟⣩⣭⣶⣿⣿⡿⠟⢋⡁⣶⠚⠻⠀⠀⠀⣠⣶⠒⢆⠀⠀⠀⣿⠟⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⣭⣴⣿⣿⣿⠟⠛⣹⡷⠐⠻⠆⠀⠀⠀⠀⠀⠀⣿⡇⠠⠈⢧⣀⠞⣋⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠿⡋⠀⣶⠆⠉⠉⠀⠀⢠⣶⣿⣧⡀⠀⠀⣿⠇⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠉⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⡿⣻⡜⡇⠻⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠉⠻⣦⠖⢋⣠⣤⠴⠋⠀⠀⣀⣀⣀⣀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠛⠃⠀⠀⠀⣰⣿⢷⡄⠀⠀⢸⡿⠛⠃⠀⢀⡴⠋⠀⠀⣀⡴⠚⢋⡩⠤⢤⠀⢹⠄⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠻⣤⣠⡼⢁⣠⣄⣠⠟⠀⢀⡴⠚⠁⠀⡔⣡⣚⡿⢞⣠⠟⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⣿⠿⠒⠀⠀⣴⠏⠁⠈⠁⢀⠔⠋⠀⠀⠀⠀⠑⠚⠓⠚⠉⠁⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⣆⡀⣠⡿⠃⠀⠀⢠⡾⠃⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠀⠀⢀⡾⠃
⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡮⠉⠁⢀⡴⠖⠚⠋⠀⠀⣠⠖⠁⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡟⠀⠀⣠⠏
⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢸⠿⠀⠀⠀⢠⠟⠀⠀⠀⢀⡴⠚⠁⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⡴⠁⠀⢀⠉⢳⡄
⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣎⠳⢤⣤⠟⠀⠀⠀⣰⠏⠀⠀⢀⡴⢋⣠⣴⣾⣿⣿⣿⡿⠛⠋⠁⠁⠀⠙⠏⠀⢀⠞⠀⠒⠉⠉⠙⢮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠆⠀⢀⣤⠴⠞⠁⢀⣠⣞⣭⣶⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⠎⣠⠂⠀⠀⠀⠀⡄⢻
⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣄⠀⠀⣿⠟⠁⠀⣰⠋⠀⢀⣠⣴⣯⠾⣿⣿⡿⠿⠋⠉⠀⢀⣠⡤⠶⠖⠛⠋⠉⠉⠁⢀⠂⠔⠁⠀⠀⠀⠀⣰⠇⠈
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠙⠷⠾⠃⠀⠀⠀⠁⠀⢀⣤⡿⢻⠃⠄⠙⠛⠁⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠠⠊⡀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠈⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⣾⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⣴⠟⠉⠁⠀⠀⠀⣴⡿⠋⠀⠀⠀⢀⡤⠖⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⢀⣾⡇⡀⠀⠀⠀⢀⣾⠏⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⣾⣿⠇⠐⠀⠀⢠⣿⡁⠀⠀⠀⣰⠋⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⠄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⣼⣿⣿⠉⠳⠀⢀⣿⣿⣷⣦⣤⣰⢿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠧⠼⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⣤⡤⠀⢠⡿⠿⠿⠟⠛⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡠
⠀⠀⠀⠀⠠⠊⢀⣼⠫⠉⡘⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠵⣶⡉⠁⢀
⠀⠀⠀⠀⠐⢲⣿⣷⢄⡀⢠⠅⠈⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣹⣷⡀⠈⡰
⠀⠀⠰⠀⢠⣿⣿⡇⠀⢘⣍⠈⠀⠈⢙⡀⣀⠀⢠⣀⠀⠠⢰⡔⢃⠀⠀⢠⠀⢶⠀⣧⡄⠀⣿⣿⣿⠒⠡⢄⡀
⠀⠀⠀⠀⣽⣿⠹⣿⡆⣼⠏⢀⠀⢀⡾⠅⡇⠀⠀⠀⢛⢀⣹⠀⠀⡤⠛⠠⢀⠈⡷⣿⡇⢸⡿⢻⣿⠃⠀⠀⠁
⠀⠀⢀⠠⣿⡇⠀⢹⣿⣿⣧⣿⢠⣼⡇⢰⡀⢀⠀⢉⣟⣦⢿⡀⠛⠚⣘⠋⠉⢡⣿⢹⣿⡿⢁⢻⣻⡇
⠀⠀⠀⠀⢸⣇⠃⠸⣾⣿⣧⣹⡆⣿⡇⢸⡇⣧⢤⣼⣿⣿⡞⣿⣤⣄⠙⢀⣴⣾⣿⣾⣿⣿⠣⢈⣿⠁
⠀⠀⠀⠀⠈⠻⠒⠘⣿⡷⣤⣿⣿⣹⡇⣼⢳⣯⣼⣿⠛⣿⢿⣿⣿⡷⣤⣾⣿⣏⣼⡏⣿⡟⠀⣸⡿⡀
⠀⠀⠀⠀⠀⣀⢀⡀⣹⣗⢻⣿⠘⣿⣿⣿⣿⢿⣿⣧⡰⣿⠐⣿⣻⠟⣿⠃⣿⡇⣾⡇⣿⡗⠲⠓⠰⠁
⠀⠀⠀⠀⠈⢁⠀⠨⣽⣿⠸⣿⠞⣿⣿⢻⣿⠚⣧⣹⡇⣯⣴⣿⣇⣠⣿⣦⣿⠟⣿⢠⣿⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠈⢷⣄⡀⢻⣧⣻⣇⠁⠸⡾⣿⢠⠃⢻⣿⣿⠃⠟⣸⡟⠁⣼⡏⢬⢡⣼⡏⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⣿⣧⣿⣿⠺⢿⣿⡀⠀⠸⡿⠏⠀⢰⡿⡵⢶⡟⢶⣌⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠛⠛⠁⠸⠶⡄⠚⢷⠠⠄⠁⠀⠀⣿⠇⢀⠏⠀⠀⠙⢋⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⠀⠀⠀⠀⠰⠂⢂⠀⠀⠀⠀⠇⠀⠌⡐⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⠀⠀⠀⠁

⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⡇⠀⠀⠀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡇⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⡇⠀⠀⠀⢈⡏⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠁⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡁⠀⡘⡇⠀⢀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠛⠉⠁⠀⠉⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡾⠒⣒⣒⡒⠒⠒⠒⠒⠒⠶⠤⢴⡁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠠⠨⠭⠥⠐⠒⠲⠦⠿⠿⠅⢬⡧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠷⣤⣤⡋⣤⣤⡤⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⠀⠟⠰⢿⡄⠀⣿⠑⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠰⡿⠶⣄⣨⠟⢁⡼⠁⢰⡄⡀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⢦⣄⢀⣠⣼⣟⡴⠋⠀⠀⡞⠀⠈⠳⢄⣠⡴⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠙⣠⣤⣬⣭⣤⠾⠋⣀⣀⣀⣤⣧⣀⠀⣠⣤⠊⠳⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢞⣥⠛⠛⠛⠛⣿⢿⡿⢿⢿⡋⠁⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡜⢉⠞⠀⠀⣠⠎⣽⢸⠁⠀⡄⡇⠀⠀⢸⡁⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠤⣁⠀⣠⠞⠁⠀⢉⡭⠤⣀⠚⠁⠀⠀⠈⡍⠆⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣄⠈⢫⠁⠀⢀⡴⠋⠀⠀⣀⢳⣦⡠⠒⠚⣉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠷⣦⠶⠚⠛⢚⣁⣀⣄⠉⠀⢹⣿⣾⣧⡤⠾⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⡹⢠⠀⡄⢀⠠⡀⡀⣀⠈⠳⣄⠀⠀⠀⠘⣿⣿⣿⣿⠿⠟⠛⠆
⣋⣿⣿⢟⣿⣟⣋⣙⣛⣛⡟⠹⣿⣿⠇⡇⢨⠀⣥⠄⡄⠱⡌⢦⠓⣠⡞⠳⣄⠀⠀⠈⠉
⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠀⠐⠨⠀⠀⡆⠘⡄⠸⠄⠈⠄⠑⡁⣿⣿⣿⣛⠉⢳⠄⢠⠀⠀⠀⡇⠀⠀⡆
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡵⠚⡃⡄⠀⠀⠳⡀⠡⠀⠰⠀⠈⠀⢠⡈⡟⠩⡜⠀⣆⣷⣤⣤⡀⠀⠀⠀⢨⠃
⣿⣿⣿⣿⣿⣿⣿⡟⠘⢇⡴⠇⠁⠀⠀⠰⠀⡆⠀⠀⠀⠀⡆⠀⡀⢂⠀⠀⡌⠈⢿⣿⡿⡇⠀⠀⠀⠈⠁
⠛⠋⠛⢛⣿⠟⠹⡇⡄⠂⠀⡄⠁⠀⠀⢸⠀⡧⠘⠀⠀⠀⣄⠀⢇⠈⠀⠀⠀⡁⠸⢿⢧⣸⡀
⠀⣀⣴⣾⣿⣦⣬⠁⡆⡂⠀⡁⠀⠀⠀⢸⠀⣇⠀⠠⠀⠀⢃⢠⠛⠀⠀⠀⠀⢄⢸⢻⢸⣿⡿⠂
⣿⣿⣿⣿⣿⣿⡿⠀⢥⠃⠂⠀⠀⠐⠄⡀⢀⢫⠀⠀⠀⢀⠈⡴⠂⠀⠀⢈⠂⠀⡔⣯⣼⣿⣷
⣿⣿⣿⣿⣿⣿⡿⠈⡄⠀⠈⠀⠀⠀⠀⠇⠐⢇⢀⠠⢄⢂⡆⠡⠀⠀⠀⠈⠰⠀⠆⡧⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡀⠆⠀⠀⠀⢀⣀⠖⠒⠃⠙⠛⠃⠀⠀⠀⠀⠀⠈⠹⠗⠒⢪⡀⠁⢰⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⣀⠜⠁⠀⠀⠸⠟⠻⠟⠙⠧⠀⠀⠀⢀⢴⠲⣿⢿⣿⠿⣆⢉⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡇⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠉⠀⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣍⠏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠴⠦⢠⠴⢂⠇⠀⠀⠀⢠⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢷⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢠⡤⢤⣤⠤⠤⠀⠀⣾⣿⡇
⠛⠛⠛⠛⠛⠛⠛⠛⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠁
⣦⣶⣶⣦⠤⢤⣶⣾⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃
⡿⠟⠁⠀⠀⠀⠙⣿⣷⡀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠙⠲⠶⣤⣤⣤⣤⣤⡶⢿⣿⣿⡇⢦⡀⣀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠖⠒⠉⠉⠉⠙⢻⡋⠉⠙⠒⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⡈⠑⣢⠀⠀⠀⠀⠀⡇⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠈⠲⢄
⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠂⠀⠲⠅⠆⠀⠆⠀⠀⠆⠀⠀⠀⣠⡄⠀⠀⠀⠑⢄
⠀⠀⢠⢞⡁⠒⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⠀⠀⠉⠀⠀⠀⠀⡀⠎⠀⢀⠜⣨⡅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠳⡀
⠀⢠⠋⠀⣀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠐⠀⠖⡓⠂⠀⠀⡀⠉⠉⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡠⠰⠤⡄⠀⠉⠀⠀⠀⠈⢀⠹⠁⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡞⠒⠂⠀⠀⠀⠸⠴⡀⠒⠁⠉⠁⠢⠈⠑⠢⠀⠀⠀⠂⠐⠂
⡇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠀⠀⢂⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⢾
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠋⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁
⢇⢸⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠘⡌⡇⠀⠀⠀⠀⠈⠢⢀⡀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠹⡗⠒⠒⠒⠒⠒⠒⠂⠉⠉⠛⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢠⠏
⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣠⠋
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠃⣅⠀⠀⠶⣍⣉⡧⡲⠶⠄⠠⣤⣤⠴⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠵⠒⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢘⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠿⢭⣟⣁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣄⣠⠆⠀⠀⠠⢴⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠶⠆⠙⢉⣩⣤⣤⣤⣶⣶⣶⣶⣶⣦⣤⣭⣉⡛⠲⢷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠜⢋⣠⡴⠶⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠷⣶⣬⡙⠳⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠊⣁⠚⠛⠉⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠠⣴⣦⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡙⢿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⡠⠊⠀⠀⠀⠀⠀⠀⠐⢿⠿⠀⠀⠀⠀⠀⠀⠘⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡙⢿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠤⠆⠀⢲⢲⢦⠤⣄⣀⠀⠀⠀⢀⣿⣾⡀⠀⠀⠘⢿⣦⠙⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢃⣼⠟⠀⠀⢠⣠⡆⠀⠀⠀⣠⠴⠛⡏⡌⠸⢸⢁⠃⡇⡴⢸⢠⢁⡏⡟⡶⣄⠀⠈⠁⠀⣀⣤⣴⣾⣿⣷⡈⢿⣀
⠀⠀⠀⠀⠀⠀⠀⡿⠃⣾⠏⠀⠀⠀⠘⠻⠋⠀⣠⡞⡍⢸⢠⠃⠃⡆⠼⢸⢠⠁⡇⡇⠘⢸⢰⠀⣇⣮⣵⣶⣾⣿⣿⠿⠿⣿⣿⣿⣿⡌⢿⡀
⠀⠀⠀⠀⠀⠀⢾⠃⠴⠋⠀⠀⠀⠀⠀⠀⢀⣼⢁⠃⠃⡆⢸⢸⢠⠁⡇⡏⡘⢠⢰⣁⣧⣴⣾⣿⣿⣿⠿⠛⠛⢇⣠⠀⢶⣿⣿⣿⣿⣷⠈⣟
⠀⠀⠀⠀⠀⢰⡇⢀⡅⠀⠀⠀⠀⠀⠀⢀⡞⡇⢸⢸⢠⠃⡄⡜⡸⢰⣠⣥⣷⣾⣿⣿⡿⠟⠛⠻⡇⠘⣇⠀⣧⠀⢿⣆⠈⣿⣿⣿⣿⣿⡆⢸⡗
⠀⠀⠀⠀⠀⣼⠁⣭⠀⠀⠀⣠⣄⡀⠀⡼⢠⠃⡆⡘⢸⣠⣥⣷⣿⣿⣿⣿⣿⠏⠉⠙⢧⠀⢶⠀⢻⡀⢹⡆⠀⣠⣿⣿⡄⠘⣿⣿⣿⣿⣿⠀⣿⠂
⠀⠀⠀⠀⢀⣽⠀⣿⠀⠀⠈⠿⠻⠀⠀⡇⣸⣤⣷⣿⣿⣿⠿⠿⣿⡏⢀⡄⣙⠄⠸⣦⣾⣆⠀⣀⠈⢳⠀⢻⡀⢹⣿⣿⣿⣶⣿⣿⠿⠟⢿⡇⣿⡄
⠀⠀⠀⠀⠀⣿⠘⣿⠀⠀⢀⣀⣤⣴⣾⣿⣿⡿⠻⣿⠉⢿⡄⠀⠹⣷⣄⠈⠙⢧⡀⢻⡅⠹⡆⠘⣇⣈⣧⣼⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⣿⡇⣿⡄
⠀⠀⠀⠀⠈⢻⠀⣿⣶⣿⣿⣿⠟⢿⣿⠁⠈⢷⡄⠹⡆⢸⣇⠀⡄⠘⣿⠛⠦⠀⣷⣀⣀⣼⣟⣾⣿⣿⢿⢛⠋⡇⣼⠁⠀⣰⣾⠀⠀⠀⣿⠇⣿⠁
⠀⠀⠀⠀⠀⠻⡄⢿⣿⣿⣿⣿⣧⠈⢿⡄⢀⠈⢻⡄⠁⠀⣿⡀⢠⣄⣈⣶⣤⣾⣿⣿⡿⠿⢻⢉⠃⡇⡌⠸⢠⢁⡟⠀⠀⠁⠙⠀⠀⢠⣿⢀⡟⠁
⠀⠀⠀⠀⠀⠰⣧⠸⣿⣿⣿⣿⣿⡆⠘⣇⠈⢁⠀⢻⣄⣀⣼⣷⣾⣿⣿⡿⡟⢻⢹⠁⠇⡆⡬⢸⢀⢁⠇⡆⢸⡼⠁⠀⠀⠀⠀⠀⠀⣼⡏⣸⠟
⠀⠀⠀⠀⠀⠀⢼⣆⢻⣿⣿⣿⣿⡟⠀⣹⣀⣿⣷⣾⣿⣿⡿⡟⡛⢹⢰⠃⠇⡌⣼⢸⢀⠃⠇⡆⠸⢸⢠⢁⡟⠁⠀⠀⠀⠀⠀⠀⣰⡟⢠⠿
⠀⠀⠀⠀⠀⠀⠀⣽⣄⢻⣿⣿⣿⣷⣾⣿⡿⠿⠛⠻⣝⢰⠐⠇⡇⡌⣸⢸⠀⠃⡇⡆⠸⢸⢠⠁⡇⣌⡼⠋⠀⢀⣷⡔⠀⠀⠀⢰⡿⢡⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠻⣿⡿⠛⠉⠀⠀⣀⣄⠀⠈⠛⠚⠸⢀⠃⡇⡎⠰⢰⢡⠇⡇⣜⣸⡴⠛⠉⠀⠀⠀⠀⠋⠀⠀⠀⣀⡜⣡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡙⢿⡆⠀⠀⠀⠹⠻⠅⠀⠀⠀⠀⠈⠙⠓⠓⠓⠚⠚⠛⠋⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣴⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⡘⢷⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠀⠀⠀⠀⠀⠀⢹⣿⠶⠀⠀⠀⠀⢀⣤⡾⠋⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠦⣉⠻⢷⣤⣀⠀⠀⠀⠀⠀⠉⠟⠙⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣠⣶⠟⢋⡴⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⣦⣉⡛⠿⢶⣦⣤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⠿⠛⣋⣤⡞⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠗⢦⣤⣍⣉⣛⡛⠛⠛⠛⠛⠛⣛⣉⣩⣤⣴⠞⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠉⠙⠛⠙⠟⠉⠛⠋⠉⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠞⠛⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠁⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠉⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠶⢦⣄⡀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⢀⣤⣤⣀⣀⣀⣠⣤⠾⠋⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣀⣠⣤⣄
⠀⢀⡾⠁⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠹⣆
⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣆⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼
⠀⠈⠙⣦⡀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⣠⡴⠶⣤⡀⠀⠀⠀⣿⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠈⢷⡀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡏⠀⠀⠈⠛⠀⠀⠀⣿⡀⠀⠀⠀⠈⣷⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢷⣄⣀⡀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠙⢷⡀⠀⠀⢀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠙⣧⠀⠀⠀⣰⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⢀⡿⠁⠀⠀⠀⠘⣇⠀⠀⠀⠀⢸⡇⠀⠀⠸⣇⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⣄⠀⠀⢀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠹⣇
⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠉⠛⠷⠞⠛⠁⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⣿⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠙⢷⣄⡀
⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠈⢿⡄⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⣴⠏
⠀⠀⠈⠙⠋⠉⠉⠉⠉⠛⠷⣤⡀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⣠⡴⠟⠋⠉⠉⠉⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣤⣤⣀⣤⣤⣤⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠳⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠀⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⣀⠀⢀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠉⠀⠈⠁⠀⠁⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢿⠿⡟⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⠒⠒⠀⠀⠀⠀⠀⠀⠀⢠⣧⡄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⡞⣘⠛⡛⠛⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣾⠷⠤⢀⠀⠀⠀⠀⡀⢸⣿⢻⢠⢁⢴⢠⡏⣻⣧⣷⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⣿⠆⣠⢀⠄⠀⡄⠀⢠⢸⣿⣛⠀⠈⠈⠈⠀⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⣦⣤⣿⣤⣛⡙⠛⠛⠈⠓⠀⠃⠀⣼⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⢀⡀⠠⠾⠿⠟⠛⠛⠛⠟⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠟⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣒⣲⡿⠿⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⡴⢟⡫⠵⢒⣋⠁⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⡾⣫⢞⣩⠔⠚⠉⣁⣀⣀⠉⠉⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⡇⣴⣿⡾⠛⠉⣀⣤⣖⣫⠭⠶⠛⠀⠀⢶⣾⣿⡆
⠀⠀⠀⠀⠀⠀⢸⢡⠞⠁⣷⣠⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⢸⡿⡄
⠀⠀⠀⠀⠀⣠⣾⢦⡀⢸⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡼⡀
⠀⠀⠀⠀⠀⣿⣿⡜⣇⣼⡨⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠿⣽⢳
⠀⠀⠀⠀⠀⣿⢻⠃⣿⢿⡻⡗⢦⡀⢀⡴⠒⠒⢦⣀⢠⡴⠋⠉⡏⣰⢿⢮⠗
⠀⠀⠀⠀⢠⡟⣼⠀⣿⠀⡇⣧⠀⠙⠻⠦⣤⣤⡤⠼⢿⡟⢶⡖⢿⡥⠂⡾
⠀⠀⠀⠀⠀⠀⣿⠛⠚⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⢠⣀⡇⢸⣇⣰⡧⠞
⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠤⢻⠳⣄⠀⠀⠀⠀⠀⣀⠙⠷⢨⡼⢿⣿⣯⣝⡓⠲⢦⡀
⠀⠀⠀⣀⣤⠤⠴⣦⣄⡈⠐⠋⠀⠈⠳⢤⡀⠀⠀⢈⡙⣷⣼⡇⢸⢧⡀⠙⠿⣦⣄⠘⣆
⢀⡔⠋⣁⠀⢀⣤⣞⠛⢿⣧⣀⣀⣀⣀⣀⠉⠓⢤⡻⢿⣿⣯⠙⢶⣷⠻⢶⣤⣈⠻⣷⡘⡆
⢸⠀⢠⣿⡄⠿⣧⠈⢳⡀⠙⣿⣑⠤⣌⣉⣙⣛⣻⣿⡖⠓⠿⢦⣀⡽⠀⠀⠈⠙⢷⡈⢷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⢸⡀⠸⡟⢻⣧⠘⣧⡀⠹⡄⠸⡝⠻⢿⣟⡛⣭⣩⠏⢀⣀⣠⡤⠬⠤⠤⠤⠤⠤⠼⡿⢼⡏⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⢱⡄
⠈⡇⠀⣇⠈⢿⣧⡘⡷⡄⠹⣄⢻⢦⡀⣽⢈⡼⢃⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⢻⠀⢻⡄⠈⢿⡗⣄⠹⣆⠘⡄⢙⡟⣡⠎⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠘⡄⠀⠁⠀⠈⢻⡜⢦⠘⣆⠙⢉⡼⢱⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⡇⠀⠀⢰⡖⠢⣿⣼⣇⠈⣣⠋⢠⠞⣱⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡀⠀⠈⣇⠀⠈⢿⡼⣾⠁⡴⠋⣠⠟⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣘⣆
⠀⠀⠀⣇⠀⠀⢹⡀⠀⠘⣷⣨⢿⣴⠟⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣼⢉⡏
⠀⠀⠀⢸⠀⠀⠀⣧⠀⢠⡞⠷⠋⠙⣆⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⢹⠈⢻
⠀⠀⠀⠘⣆⠀⠀⠸⣦⠏⠀⠀⠀⠀⠘⣆⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⢸⡄⠘⡆
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡿⢸⢇⠀⡇
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣠⠞⠁⢾⣍⣉⣉⣉⣙⣛⣉⣉⣩⣭⣭⣭⣭⡭⠽⠿⠿⠿⠿⠛⠛⠛⠛⠻⠭⣍⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡇⢸⢸⠀⡇
⠀⣀⣀⡀⠀⠀⠘⢆⣀⣠⠴⠋⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣃⣀⣀⣀⠀⠀⠀⠀⠀⣸⠀⢸⠘⡆⣷
⠀⠀⠉⠉⠙⠛⠛⠛⠛⠒⠲⠶⠶⠶⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠛⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠄⠀⢠⠤⠤⠤⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠺⡀⠔⠛⠒⠒⠣⡀⠀⠀⠀⠤⡀⠈⠏⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠈⡀⠀⢀⡠⠁⢀⣀⡤⠴⢶⣶⡶⢶⣤⣤⣧⡤⢄⠀⠠⡈⠛⢢⣄
⠀⠀⠀⠀⠀⢠⣋⣖⠈⠉⠀⣨⣴⠟⠉⠀⠀⠀⢘⣏⣗⣚⡛⠛⠛⠙⠾⣗⡂⢆⠀⢎⠙⣦
⠀⠀⠀⠀⢀⡎⠀⠀⠀⣰⣾⡿⠁⠀⡀⠀⠀⡂⠭⠗⠂⠀⠀⠈⢐⠠⣴⣾⣷⡌⠒⠚⠀⠙⡆
⠀⠀⠀⠀⢘⠦⣀⡄⣰⡿⡡⠀⠀⠀⠀⠐⠁⢀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢮⡇⢹⡀⠸⠒⢶⢿
⠀⠀⠀⠀⢸⠀⠁⢠⡿⣾⠇⠀⠀⠀⠀⠀⠔⣡⠖⠛⠛⠛⢶⣄⠀⠀⠀⡈⣇⠀⣇⠀⠖⠺⢤⡇
⠀⠀⠀⠀⠸⡄⠀⢸⡗⢣⡀⠄⠤⠤⠀⠄⢠⡏⠀⠀⠀⠀⠀⢿⠀⣇⣀⡧⣿⠀⢿⠀⠢⡀⢈⡇
⠀⠀⠀⠀⠀⠈⠉⠚⠿⢭⡟⠉⠛⠉⡻⠛⠉⢁⠀⠀⠀⠀⢀⠼⠀⠈⠉⠃⡿⠿⢾⠀⡐⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠈⠤⢄⠀⣰⠗⡀⠀⢐⡗⠀⠀⠀⠀⣸⢻⣧⡿⣀⣈⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠳⣌⠪⣽⠃⠄⢀⠉⡉⢹⣆⢀⠔⣱⣿⣿⡿⠤⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠑⢤⣉⡓⠚⠒⠛⢉⣉⡴⠊⣿⡿⢛⠚⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣌⠙⡿⠿⠉⡯⢋⣽⢿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⢦⡤⣤⠔⡡⠻⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡟⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠛⠛⠛⢻⣿⣿⣴⣿⡗⣤⣤⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⢨⠅⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠘⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣶⣶⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⢹⣿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡔⣶⣿⣷⡆⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣙⣛⡁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠛⠀⠉⠉⠉⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⢰⣦⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠇



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠽⠏⠀⡄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⢠⣄⢠⡩⠓⢲⠶⠆⠀⠀⠀⢀⠒⠈⠁⠒⠁⠀⠀⠀⠀⡶⠲⡶⣶⢶⡂
⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⡔⢢⣶⠄⠀⠀⠀⠀⠠⡤⠀⠠⡤


⠀⠀⠀⠀⢙⠋⠀⠀⠀⠀⠀⠀⠀⡒⡂⢀⣀⠀⠀⠀⠀⠀⠀⢀⡠⢄
⠀⠀⠀⠀⠁⢘⠆⠀⠀⠀⠀⠀⢀⠐⢀⠬⠈⠀⠀⠀⠀⠀⠀⠉⣠⣮⣃⠀⠀⠀⠀⠀⠄⡀⠀⡈⡄
⠀⠀⠀⠡⢀⡈⠆⠀⠀⠀⠀⠀⠸⠴⠄⠀⠐⠊⠀⠀⠀⠀⠀⠀⠏⠂⠇⠀⠀⠀⠀⠀⠀⠁⡠⠀⠁


⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⢲⡆
⠀⠀⠀⠰⣍⣹⠂⠀⠀⠀⠀⢀⡰⢃⠘⠿⠃⠸⠀⠀⠀⠰⣶⠶⣢⣶⠶⢾⠀⠀⠀⠀⣸⠀⢦⡆⢸⡇
⠀⠀⠴⠆⠾⠷⠰⠦⠀⠀⠀⠀⠙⠙⠗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠻⠤⢼⡇⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⡤⣀⢶⠆⣤⣤⡀⠀⠀⠻⣥⡦⠶⠶⢶⡄⠀⠀⠀⠀⠀⣴⣦⣶⡄
⠀⠀⣿⣿⣿⡑⠀⠀⠀⠀⠀⠀⡔⣿⡇⠀⠘⢽⡇⠀⠀⠀⢸⠟⣤⠛⢠⢇⡀⠀⠀⠀⣘⣼⣛⣻⣌⡂
⠀⠀⢿⠿⢿⠇⠀⠀⠀⠀⠀⠐⠙⠚⠃⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⠿⠴⢷⡆⠀⠀⠀⠛⠛⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠐⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠰⠶⠶⠶⠶⠶⠶⠆⠀⠘⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠶⠶⠶⠶⠶⠶⠆⠀⠀⡇⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⢰⣤⡄⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠧⠤⠤⠀⠀⡃⠀⠀⠀⢀⡠⢴⣿⣶⣶⣿⣿⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⡠⠄⠒⡦⠤⠤⠴⠶⠷⠶⠶⠶⠆⠀⢀⠇⠀⠀⠀⣇⠠⠄⠼⡞⠉⠉⠙⢻⣽⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⢻⠋⢉⣉⣉⣙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⣩⣷⡶⠶⠿⣯⡉⠑⡆⠀⣿⣿⣤
⠀⠀⠀⠀⠀⢰⣶⣴⣶⡿⣧⣼⣀⣈⣉⣉⣉⣉⣑⣢⣤⣀⣀⣠⣤⣄⡀⠀⢣⣀⣀⣛⣹⠒⠀⠉⢙⣁⣀⣧⣼⣿⠏⠉
⠀⠀⠀⢀⣀⣾⣷⠟⠉⠉⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠇⠀⠀⠈⡀⠀⠀⡌⡀⠀⡿⠻⣿
⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡇⠀⢃⠃⠀⡇
⠀⠀⠀⠀⢈⣿⣻⣄⣀⣀⣠⣾⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⣼⠋⠀⠈⢳⡄⠘⢸⠀⢸⢸⠀⡇
⠀⠀⠀⠀⠙⠋⠛⢿⣯⡿⠿⢿⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠙⣦⡈⡇⠈⡎⡄⡃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢸⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢈⣧⢧⠀⢳⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⣠⠟⠘⡸⠀⠸⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠅⠆⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⠉⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢢⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻


⣏⠉⣹⣛⣛⣛⣛⣛⣟⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⠛⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣼⠏⠀⠙⠳⢶⣤⣤⣀⣀⣀⣤⣴⠶⣆⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣸⡏⢠⡶⠶⠶⣦⡀⠀⢩⡽⠷⠶⢦⡀⢹⣆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⣿⠀⠀⠀⢸⡟⠛⢻⡇⠀⠀⢸⡟⠛⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠘⠳⠶⠲⠛⠁⠀⠈⠳⠶⠶⠟⠁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⣤⡀⠀⠀⠀⣀⣤⠶⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠙⠛⠛⠛⠉⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠟⠙⢷⡄⠀⠀⠀⠀⠀⢠⡾⠋⠻⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠛⠛⠋⠉⠁⠹⣆⠀⠀⠙⢦⡀⠀⢀⡴⠋⠀⠀⣰⠏⠈⠉⠉⠛⠛⠶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢹⣶⡟⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠳⣄
⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣴⣿⣭⣿⣦⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡆
⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⠛⠛⠷⢦⡀
⠙⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠹
⣶⣶⣶⣶⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣦⣦⣤⣴⣶⣶⣶
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠘⠀⠁⠀⠁⠀⠀⠀⠀⢠⡞⣟⢷⡀⠀⠀⠀⠀⣠⠴⣶⡶⢤⡀⠀⠀⠀⢠⣶⣾⠒⢲⠀⠀⠀⠀⢀⣀⣀⠴⠦⡀
⠀⠀⠰⠴⠶⠶⠆⠀⠀⠀⠀⠈⢷⣶⣿⡇⠀⠀⠀⠀⠀⠫⠿⠯⠃⠀⠀⠀⠀⠘⡏⠙⣶⡾⠀⠀⠀⠀⠘⠛⠈⠢⢴⡃
⠀⠀⠛⠛⠀⠚⠛⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠤⢤⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠀⣸⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⢰⡶⠶⣶⣶⡆
⠀⠀⠻⠿⠿⠿⠟⠀⠀⠀⠀⠀⢸⣿⣽⠀⠀⠀⠀⠀⣽⣟⣿⣿⣿⡅⠀⠀⠀⠀⠸⠀⠝⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⣴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣾⠛⠻⣷
⠀⠀⠁⢰⠚⡆⠀⠀⠀⠀⠀⣰⠞⢾⣻⣢⠀⠀⠀⠀⢾⣿⣿⠆⡠⠁⠀⠀⠀⠀⢸⣸⣿⡇⠀⠀⠀⠀⠀⣷⣈⣤⣿
⠀⠀⠂⠀⠛⠐⠒⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠿⠶⠾⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣀⣀⢀⡀⡀⣀⢀⠀⠀⠠⠀⣀⢄⡀⠀⢀⠀⠀⠀⠀⢰⣶⣶⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇⠀⠀⠀⠀⢠⣤⡤⠞⠛⠃
⠀⠀⠁⠉⠉⠉⠈⠀⠀⠀⠈⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣷⡤⣤⡇⠀⠀⠀⠀⠈⠋⠓⢦⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠠⢤⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⣰⢰⣶⠀⠀⠀⠀⠒⠂⠀⢤⠀⠀⠐⠂⠀⠀⠀⠐⠀⠈⠉⠡⠈⠉⢠⡴⠶⠿⡀
⠀⠀⠀⠀⠀⠀⣿⣉⠇⠠⠤⠀⠐⠀⠲⠀⠈⠉⠂⠀⡀⠀⠄⠀⠃⠀⠈⠀⠀⠀⢀⣈⣤⡴⠖⠃
⠀⠀⠀⠀⠀⠀⢎⣿⠗⠸⡆⢀⡆⠀⠰⠀⠄⠀⠀⠰⠶⠀⠄⢐⣀⣠⠔⠲⠚⠋⠉⠀⠘⠃⠀⡀
⠀⠀⠠⠀⠀⠀⠈⠙⣾⣷⡀⡈⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⢛⡉⠹⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⡄⣲⡀⠄⢐⣒⡉⠙⠛⣏⠸⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠀⣳⡋⠉
⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡿⠟⣛⣋⣉⡭⣯⡭⣍⡽⣛⣻⠿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣟⢽⡂⣿⢹⡿⠋⣿⡿⣨⡾⠟⡾⠿⣿⣇⣶⣭⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⡫⣪⣾⡿⣧⢿⡸⢹⣿⡟⢱⣿⣿⡶⡏⣾⣥⢘⣛⣯⣿⢧⠝⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣽⡶⣛⣴⣼⣿⣌⣳⠜⣿⣏⣷⣭⡷⣻⢧⣿⣭⣿⣧⢉⣶⣷⡾⣤⡻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⣟⡚⢛⡿⢻⣥⡭⣷⣟⣻⢯⣽⣾⣥⣷⣶⣶⣶⣯⣽⣿⣯⣵⣶⢶⣦⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣝⣻⠾⢖⣻⡭⠶⢛⣭⣥⣿⣿⢹⣿⣿⢆⣭⣭⢛⣌⣻⡽⠿⣽⣃⣿⡃⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣝⢿⣿⣾⠟⡿⣏⣿⣿⢟⣭⡏⣭⣿⣭⣝⡻⣿⣿⣿⣿⢸⡟⣮⢿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⢰⣿⢏⣋⡿⣻⡽⣋⣻⠟⠿⢿⣬⣟⣋⣙⡾⢷⣝⢷⣮⢥⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⠿⣾⣿⣷⢽⠷⠿⠿⢿⣿⣟⣎⣻⣿⣿⠿⠿⠟⡼⣫⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣿⣿⢿⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣿⣿⣿⠿⣿⣷⡿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⣿⣿⣿⣿⣟⡉⠙⢛⠭⣈⢻⣷⢈⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⣿⣿⣿⡿⠉⠀⠀⠉⢹⣿⢯⠟⡽⠛⠉⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢺⣿⣀⠀⣤⣖⣂⠀⠀⣿⣼⣼⠁⣶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠘⠛⠋⠉⢠⣿⣿⣿⣷⣿⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⣠⠶⠀⠀⠀⠸⠿⢻⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢳⢆⠀⠀⠀⠀⢀⣼⢿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠀⣀⣤⣶⣿⠃⢘⡿⠟⠋⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⡿⠒⢉⡤⢖⣴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠛⠛⠁⢀⢰⣃⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣎⢿⠀⣠⠔⢁⣾⣽⣿⣿⣿⣏⣩⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⡏⣿⠧⠤⣽⣿⣿⣿⡿⣿⠟⠛⠉⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⣻⣿⣿⣿⣴⣪⣾⣿⣿⣿⣿⢋⠀⠁⠀⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢹⡿⠑⣿⣿⡿⢃⠞⣹⣿⠟⣿⠟⠁⠃⠀⠀⠀⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⢀⡿⠋⠀⠔⣰⠟⠃⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠰⡀⢠⡀⠀⢀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣤⡨⢉⣱⢀⠂⣀⠿⠶⠮⣑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣄⠀⠀⠘⡟⡏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣀⡇⠀⠀⠴⢇⡆⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠁⣼⣿⣿⡏⠀⢸⣿⣿⣿⣿⣇⣀⣘⣇⠀⠀⠀⠘⡇⠐⠚⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣰⡀⠀⣿⣿⣿⡇⠀⢸⣏⣿⣿⢿⡏⠁⠀⠀⣀⡤⠶⠾⠧⣄⠀⢸⡌⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⡀⠉⣛⣛⠿⢿⡇⠀⠘⣿⣿⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡇⢾⣇
⠀⠀⠀⠀⠀⠀⠀⣼⢿⡿⠉⠐⠀⢭⠓⠺⡇⠀⢸⣿⣿⡇⢸⣇⣀⣀⣤⣠⠀⠀⠀⠀⢀⣀⣼⠃⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣟⠟⢀⠄⠀⡀⠀⠈⠂⠈⢆⣸⣿⣿⡇⢸⡏⠀⠀⠀⠈⠀⠀⠰⠶⠾⠉⠀⠠⡏⠀⣷
⠀⠀⠀⠀⠀⠀⢀⡿⡧⠦⡄⠀⠈⠄⠀⠀⠀⢾⡄⠀⢹⡇⢸⡇⠀⠀⠀⢀⡀⣠⠇⠀⠀⣀⣀⠀⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⠎⠀⠀⡀⠀⠀⠈⣿⠀⠈⡇⢸⣇⠀⢀⣀⡰⠛⠁⠀⠀⠉⠁⠈⠙⡄⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠂⠉⠐⠀⠀⠀⠨⡄⠀⠀⢸⡟⠉⠉⠉⠻⢄⣀⠀⣀⣀⠆⠀⣸⠇⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢷⡇⠀⠀⠀⢰⢶⣺⡆⢐⣀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡄⠀⠈⠉⡉⠁⠀⢸⡏⠐⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢸⠿⣿⡇⠀⠁⠀⢀⠀⡀⢸⡇⠀⠀⣸⠋⠓⠚⠛⠙⠓⠒⠋⠙⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣘⠲⢼⠁⠀⠀⠀⣾⣾⣿⣾⡇⢀⣴⠋⠀⣠⠷⣤⠞⠉⠙⢦⡀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⡖⣦⣉⠞⢀⡾⠀⢰⡅⣹⣿⣿⡷⠛⠓⠶⠚⠁⠀⣸⠇⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣹⣶⡞⣥⣷⣿⣿⣿⣿⣿⡇⠀⣀⣰⣦⡴⠟⠋⠀⣄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣿⣽⣿⣿⣿⣿⠃⣟⠚⠋⣉⠀⢀⣴⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠈⠒⠶⠟⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠇⠀⠀⠀⢴⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⢀⢸⢸⣰⠀⠀⠀⣀⠀⢀⢀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⣀⠀⣰⣿⣦⡀⢠⠏⠁⠀⠀⠉⠀⠀⢁⠒⠾⠛⠻⣦⣉⠉
⠀⠀⠀⠀⠀⢠⣾⣿⣶⠀⠀⠀⡀⠀⠙⠳⣿⡿⢿⣭⣿⣿⣿⣿⠁⠀⠀⠀⢰⠀⠀⠀⣿⡇⠀⡆⠠⠘⠿⣿⠿⠙⡦
⠀⠀⠀⠀⠀⣬⢶⡶⢶⡆⠀⠀⠀⠀⠀⠀⣿⣧⣼⣿⣿⡅⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣿⣼⣷⣾⡇⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⡅
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢠⡆⠀⠀⣿⡟⢸⣿⣿⣿⣿⣿⠀⠀⢰⣿⣠⠄⠀⠀⠀⠀⠀⡀⠇⢰⣿⡇⣀⡀
⠀⠀⠀⠀⠠⠿⠿⠿⠿⠇⢰⡆⠘⠇⠀⠀⢿⠤⠸⣿⣿⣿⣿⣿⢀⠀⢸⣿⡏⠀⠀⠀⠀⠀⢸⣇⣀⠘⠛⠓⠺⠷
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣿⣿⡷⠀⠀⠀⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣟⣛⣿⣛⣛⣿⣭⢯⣽⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣀⣀⣠⡤⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢿⠿⠛⠁⠠⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣇⣀⣀⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⠿⢿⣿⣿⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠋⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⠇⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⣿⣤⣄⠀⢿⡿⠀⠀⠀⢸⡿



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⡴⠋⠁⢀⠀⠉⠳⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣼⠀⢠⡞⠉⠙⠂⣰⠛⠉⢀⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠲⢦⡀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢷⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢠⠄⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠟⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠸⣄⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠶⠶⠶⡄⠀⠀⢀⣠⠛⣦⡀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠈⠉⠀⠀⠀⠀⢸⠀⠀⢠⡄⠀⠀⣤⠀⡇⠀⠀⠈⠀⠀⠀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠀⢸⠀⢀⣟⣹⡄⠀⣿⠀⡇⠀⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡖⢲⣄⣀⡴⠛⠛⠀⢸⠀⠾⠁⠀⠳⠀⠿⠀⡇⠀⠛⠛⢦⣀⣠⡖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⣀⣀⠀⠸⠦⠤⠤⠤⠤⠤⠤⠴⠇⠀⣀⣀⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠉⠉⠀⢀⡄⠀⠀⢀⡀⠀⠀⢠⡀⠀⠉⠉⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⡿⠀⠀⠀⢀⣸⠇⠀⠀⢸⡇⠀⠀⠸⣇⡀⠀⠀⠀⢿⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⠀⠀⠀⢀⡼⢧⡀⠀⠀⠀⣿⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⣦⠀⠀⠈⠳⠞⠁⠀⠀⣴⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠖⠠⠀⡄⢀⣀⡀⠀⠀⠀⠀⠀⢸⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠠⠃⠀⠀⠀⠀⠀⡀⠀⠁⠈⠀⠁⠀⡠⡆⢠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡾⣷⡄⢰⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⢴⣶⡆⣀⠄⠄⠈
⠀⠀⠐⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⣱⡆⣼⠀⠀⠀⠀⠀⠀⠀⠀⡴⡟⠃⠶⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⠀⡀⢀⣧⡾⣩⡾⠿⢷⣾⣦⣿⢀⣀⣄⠀⠀⠠⡒⢁⣠⡞⣁⠈⠂⣰
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣷⠋⠁⠐⠢⣤⣤⣿⣶⣿⣿⠀⠀⠀⠀⣸⡟⣹⣿⡀⡀⣏
⠀⠀⠘⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢸⣻⡖⠒⢷⢶⣿⣿⡤⣭⣯⠇⠀⢀⠀⠀⣿⠁⣿⣿⠿⠃⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡆⠀⠀⠀⠀⠉⠀⠈⠁⠁⡦⠘⠢⡟⢻⣿⡿⠀⢀⠀⢠⡼⠁⠀⣠⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠐⢼⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⡗⡇⠀⠀⢀⣽⡗⠀⠎⠀⣾⣧⣿⣿⣟⢃⡼⠁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⡀⣰⣶⣶⣤⣭⠏⠀⠀⠀⠛⣙⡇⠸⠀⢸⣾⣿⣿⣿⣯⣡⡉⢀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠑⠀⠠⣌⠻⣿⣿⡅⣿⡟⣤⡄⠀⠀⠁⢋⡡⠃⠃⠀⠉⠉⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⡅⣝⡛⡻⠓⡿⢷⣦⣤⣄⣹⣿⣿⡅⢸⣶⣶⣿⣿⣿⣿⣟⡀
⠀⠀⠐⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣷⣭⣥⣴⣬⣯⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟⣻⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⣄⡈⢻⡏⢿⠻⠯⢭⡽⢿⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⡆
⠀⠀⠰⣤⣤⣤⣄⡀⠀⠀⢠⣤⠛⠇⠐⢠⣴⡇⠸⣷⣶⠥⠶⠶⠖⠺⠿⠛⠻⠾⠿⢾⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠛⠿⣆⡀⠁⠸⡦⠀⢸⣶⣿⡇⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣸⣿⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣷⣄⡀⠸⢿⣿⣷⣀⣀⣀⣀⣰⣶⣦⣬⠁⣁⣐⣶⣾⣿⣿⣿⠿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣵⣶⣯⣶⣶⣤⣤⣀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢠⣄⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⣄⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⢿⢿⣶⠄⠐⠒⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠐⠈⢀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠸⢇⠈⠁⠀⠐⠒⢲⠿⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⣀⠨⠅⠀⠈⢳
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣆
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢆
⠀⠀⠀⢠⡏⣀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢳⠄
⠀⠀⠀⠀⢹⠟⠛⠛⠶⣦⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡶⢶⠊⠁
⠀⠀⠀⢠⠃⠀⠀⠀⠀⡸⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠎⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠈⠀⣐⡀⠀⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⢼⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠨⠆⠀⠀⠄⠀⠀⠀⠀⠀⠏⠀⠸⠄
⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠈⠒⠂⠤⣤⠄⠘⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⣀⣀⣀⠀⠀⠀⠀⠈⠐⠂⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠹⡀⠀⠀⠀⠘⠓⠒⣂⣉⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠀⣠⡩⠏⢌⠙⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠀⠀⡠⡀⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡭⠟⢉⣁⠀⣠⠖⠦⣝⡎⠑⠢⡄⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⢢⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠐⠈⠲⣤⣯⣄⡀⠇⢀⣀⡘⠃⠀⠀⢹⡀⠒⠚⠒⠦⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠘⠀⠀⠀⢠⡷
⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠠⣄⠀⠀⠀⠀⠉⠛⢬⡉⠙⢳⣄⣀⣀⣀⠀⣸⠀⠀⡈⠀⠂⡀⠀⠀⡔⠐⠂⠄⢴⠀⠈⢠⡅⠀⣴⠟⠁
⠀⠀⠀⠀⠀⡠⠊⠢⢤⠀⠀⠘⣆⡤⠒⠀⠀⠀⠰⠋⠙⡆⢹⡄⠀⠈⠻⠇⢄⠀⣻⠔⠛⠃⠀⢠⠀⠀⠀⠀⠀⠂⠄⠒⠀⠈⠛⠠⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠤⣤⡀⠀⢀⡏⠀⡞⠀⠀⠀⠀⠒⢤⡀⠀⠇⠀⢀⠤⠤⠼⣇⠛⠓⢦⠀⠈⠆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢧⡀⠀⠀⡝⠀⠸⡅⠈⢧⡀⠀⠠⠚⠓⠄⠙⡄⠀⠀⣀⡀⠐⣄⠈⢳⡀⢈⣃⠀⠀⢀⠄⠈⠉⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢃⠀⠳⢄⡀⣧⠀⠀⠃⠀⠀⠀⠀⣰⠢⠀⠣⠀⡇⠀⣋⣀⡉⠁⢸⠇⢈⡇⠁⠘⠛⠈⠠⠄⠀⢀⠄⠁⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢁⠐⢦⡀⠘⢦⡀⠀⠀⠀⠈⡉⠁⢠⠗⢀⠈⠑⠛⠁⠀⠙⠚⠋⢉⣀⣤⡴⢊⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⠿⠛⠿⣦⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠹⠄⠀⠈⢉⣀⣀⡀⣇⡄⠺⢀⠀⠈⠐⠒⡖⠒⢀⣎⡀⢈⠀⠠⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡀⠈⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⢀⠀⠀⠀⠉⡹⠷⠐⠤⡈⠢⡀⠀⠙⢆⠀⠩⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠠⠀⡜⢂⠘⡂⠈⠀⠀⠀⠀⠀⠀⠀⢰⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠔⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣷⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠊⡠⠊⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣷⣶⣦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠈⣰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢄⠈⠀⢜⣦⡀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡂⢀⠀⠻⠓⢀⠀⠀⢀⡅⠀⠗⢶⣤⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢅⠂⠀⢠⠂⠀⢸⠈⠈⠀⠀⠉⠃⢀⠠⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⠀⠈⠀⠀⠉⠀⠀⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣝⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡋⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠒⠛⠿⣿⣌⣿⡓⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⣁⣴⣿⣦⡤⠄⠀⠈⠙⠻⢿⣧⡀⠈⠳⡀
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠴⠒⠀⠉⠠⢾⠟⠛⠋⠛⢾⠀⣄⠀⠀⠀⠀⠀⠙⠳⢤⡈
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣅⠀⠀⠀⠀⠀⠛⢿⣷⣠⡄⣄⠀⠀⠀⠙⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠙⠿⣿⠯⣬⣀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣶⡶⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠈⠓⠦⣍⣿⣦⣀⠀⠊⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣄⣤⡄⠀⠙⠟⢿⣦⣄⢀⡀⡀⠀⠀⠉⢻⣿⣷⣤⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⢐⣠⣠⡌⠙⠃⠻⣴⣼⢄⡀⠨⠏⠻⢾⣤⣤⣀⣀⡸⠁⠙⣷⣄
⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣷⣿⣧⣞⠻⣿⣧⣤⡀⠀⠹⢻⣿⣷⣿⣴⣚⣿⣿⣿⣿⣿⣿⠿⠛⠋⠙⢷⣄
⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣿⣇⣧⣀⣞⣛⣹⣻⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀
⠀⠀⠈⠉⠭⠭⠭⠭⠿⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠙⠛⠛⠉⠉⠁

⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠂⠀⠀⠀⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣁⣤⠔⣾⡷⠒⠒⣒⣲⡶⠶⢾⣶⣶⡒⢒⣲⡶⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢉⣙⣷⡶⠾⣟⣁⣠⠴⠚⢁⠨⢛⣉⠫⡀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢒⣿⠟⠋⣁⣤⣶⣛⣉⣐⣀⣿⣥⣬⣭⣤⣴⣮⣦⣄⡀⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⣋⣶⣿⡾⠿⠛⣛⣩⣿⣿⣿⡿⠷⠟⠋⠙⡩⣭⣟⠉⣠⠞⠁⠠⠂
⠀⠀⠀⠀⠀⠈⠲⠀⠀⣀⣤⣾⣿⠿⠟⠛⢹⣧⡴⠶⢿⣿⡿⠞⣋⠄⠈⠋⠀⠀⠠⢀⠙⠯⠿⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠐⣴⡿⠟⠋⣁⠤⣴⠞⠋⠉⠀⠀⠀⠈⠱⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⢈⡷⠖⠻⠯⠛⠛⠿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡯⡄⢠⠀⠙⢿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⡿⠷⠆⠀⠻⠿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⣟⣛⣛⣛⣛⡃⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢂⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠾⠷⠆⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⡅
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣶⣦⠀⠀⠀⠀⠠⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠴⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣿⣿⡇⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣯⣭⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣧⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠻⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠴⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠄⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠋⡀⢰⠂⠀⠀⣱⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣂⠘⠂⠀⢀⢞⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⡀⢥⣀⡃⢠⠁⠂⠀⠀⠀⠀⠀⠀⠀⣠⠇⠁⠀⠀⢀⠠⡀⡠⠀⠀⠌⠐⡊⢁⡐⠤⠂
⠀⠀⠀⠀⠈⢀⠀⠰⠂⠀⠀⠁⠂⠁⠀⠁⣤⠧⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡀⠀⠀⠀⡄⠀⠆⠐⠏⠐⠀⠀⢄⡄⠀⠔
⠀⠀⠀⠀⠢⠀⡇⠀⠐⠀⠀⠀⠀⠀⠄⠄⡄⢢⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⢌⠘⠁⣀⠀⠐⠀⠀⠀⠂⠁⠐⠁⠄
⠀⠀⠀⠀⠀⠢⠑⠀⠀⢈⠀⠆⠈⠉⠀⡈⠁⠔⠂⠀⠀⠀⠀⠀⠰⠛⠓⢦⠏⠀⠀⠀⠀⠄⠀⠀⠀⠁⠰⠀⡂⠀⠠⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠒⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠲⠦⣤⣄⣀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠






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

⠐⠿⣵⠤⠍⠁⠂⠲⢤⠤⣍⡉⠃⢤⣄⠀⡀
⠀⠀⢀⠀⠀⠀⠈⠳⢶⣤⡈⠻⣷⡦⢽⣿⡶⣴⣶⣦⣄
⡀⠀⠂⡀⠀⠀⠀⠈⢀⣿⣿⠞⠛⠛⠺⠿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠺⣿⡿⠗⠒⢠⣼⣿⣿⣀⠀⠀⠀⠀⠺⣟⣿⣞⣿⣿⣿⣦⡀
⠀⠀⠀⣈⠂⠀⣿⠿⣿⢩⡍⠁⣲⣶⣤⡤⠈⡿⠿⣿⣿⣦⠙⢏⠉⠲⣤⡀
⠰⣶⣤⣤⣏⠀⣰⣾⣿⢸⡇⠀⢻⣿⣿⡇⠀⢹⣏⠛⣏⡻⢧⣮⣿⠄⠸⡷⣍⣿⡗⠄
⠀⠉⠘⠫⠉⢹⣿⣿⣿⠞⣛⠓⣤⣥⣉⡁⠀⢸⣟⢺⡚⠇⠀⠊⣿⣧⡀⠘⣀⢿⠉⢳⡀
⡠⠵⠂⢀⣴⣿⣿⢸⣿⢈⣿⠀⢻⡇⢸⡏⢹⣞⣯⣟⣷⡀⠐⢄⢸⢿⣆⠀⠘⡄⢁⢠⣱⣄⡀
⣿⣿⣿⠟⢡⠀⠈⢿⡿⢛⡛⢱⠊⡉⠘⠃⠸⠇⠙⠛⠛⢷⣤⣄⡄⢄⣿⣦⡅⢴⢸⡅⠢⡻⠻⡀
⠟⣿⣿⡎⣿⣧⠀⢸⣧⣽⡿⢼⢰⡆⢰⣆⣰⠀⠓⠄⣀⠀⠈⠙⣻⣇⠻⣿⣷⡠⠄⠻⢰⣯⢷⣷⡀
⠈⠉⠛⠂⠸⠿⢄⣽⣿⣿⣿⣸⠀⠃⠘⠛⠨⠀⡀⠀⠀⠀⠐⣤⢿⣿⢲⣽⣿⣧⡖⠎⠳⡇⡨⢓⡀
⡶⠆⠀⠀⢠⣿⡆⢸⣿⣷⡦⣿⡧⡆⠐⣧⢰⣦⢠⡌⣄⣀⠀⡈⢪⣿⣧⡙⣾⣿⣿⠛⡄⢸⠇⢈⡁
⠀⠀⢀⡠⠘⠿⠧⣈⣻⣶⣆⣀⠀⠀⠀⠋⠈⡩⢘⡃⠛⠻⠀⠏⠘⣿⡟⢛⣻⡏⠛⡇⠧⡘⠇⠈⢳
⠀⠀⣉⣶⣶⣤⣄⣀⡈⠉⠙⠻⣧⡀⠀⢀⡀⣿⠁⡇⣿⠀⣠⠀⠨⣿⠋⠈⠹⣿⠀⠀⠦⠓⠘⡀⠘⡄
⠈⠛⢿⣿⠛⢿⣿⣿⡿⠶⢤⣤⠈⢿⣀⣼⣿⢿⣤⡇⠿⠀⠜⠀⢀⣿⣠⡀⣰⣿⠀⠀⠱⢧⡇⠙⣤⣜⢢⡀
⠀⠀⠀⠈⠲⠛⠻⠻⠳⣶⠀⠀⠀⠀⠻⡛⠛⠓⢴⣠⣾⣶⠄⠀⣈⠙⣿⣿⣿⣿⡻⡅⡄⠀⠀⠀⢻⡆⠦⢻⡀
⠀⠀⠀⠠⣤⣄⣠⣄⡀⠘⠯⠁⠀⠠⣀⣑⠐⡄⢾⣹⣿⣻⡄⠀⠁⠀⢟⡟⠻⣿⣿⣶⣷⡀⢀⠰⢿⠿⢄⠂⠨⡳⡄
⠀⠀⢀⣀⡀⠀⠠⠀⠙⠲⢤⣄⠈⠲⠻⣯⠃⢹⠀⠘⠻⣿⣿⠀⣷⠀⠀⣿⣶⢘⠿⣿⣿⠟⠀⠀⠻⠶⠚⢅⠢⡈⢿
⠀⠀⠀⠉⠀⣒⠛⢶⢦⡄⠰⠎⠙⣁⣶⣤⣀⠰⣷⣶⣴⣮⡛⠀⠟⢀⡠⠛⠛⠨⠀⠈⠨⣅⡑⠀⠦⡀⢒⡆⠀⠺⠾⣆
⠀⠀⠠⠾⠿⠇⢠⠁⠀⢰⡖⠃⢺⡫⢿⢿⣿⣄⣻⢿⣿⣿⣷⣤⣷⣮⣀⠀⠀⠀⠀⠠⠀⠐⠹⢒⢦⡐⠦⠐⣀⣰⡆⢊⣢
⠀⠘⣆⣀⠀⠀⠀⠤⠀⠀⣅⡀⣴⢡⣶⣿⣿⣿⣽⢟⠚⢻⣿⣿⣿⣿⣿⣅⣀⠀⣠⣀⡀⠀⠳⣦⣬⣾⠳⣆⠉⢻⣷⣿⠷
⢀⡀⣹⣿⠃⠀⠀⣠⣤⢰⣿⣟⣿⣾⣿⠿⡭⢹⡟⣎⢿⣶⡿⣿⣏⡉⠿⡿⢦⢿⣿⣿⣿⣧⠄⢀⠉⠛⣒⠺⢿⣿⣿⣿⣾
⡼⣻⣿⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⡷⣺⣅⣲⠀⠀⠀⠀⠚⠿⢿⣿⣷⠒⢶⣿⣾⣷⣍⠁⠉⣀⠀⠡⠀⠈⠑⠋⠁⠩⡻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣇⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣷⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠁⠛⠀⠄⠙⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠣⡀⠀⢻⡀⢠⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⣀⣀⣀⡀⠀⡄⠀⡄⠀⠀⠘⠂⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢱⣦⡀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⣾⣽⣿⣿⣿⡎⠀⢐⡃⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠘⠳⣦⠀⠀⠀⠀⡜
⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⢸⡇⠀⠀⣀⡆⠀⠀⠀⠀⠀⢀⠀⢠⡆⠂⠀⠀⠓⢀⡈⠂⠀⠦⠀⢰⠃
⠀⠀⠀⠈⢿⣿⣿⣿⠄⠀⠀⠧⡞⠉⠉⠁⠀⠀⠀⠀⠀⠘⠊⠛⣷⣆⠈⠂⠀⠀⠉⠠⠄⠀⢠⠇
⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠐⡀⠀⠀⠀⠁⣀⡀⠠⣤⠄⠀⠟⢻⣾⠃⠀⢠⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢹⣿⡿⢀⣀⡤⠄⣐⣓⣤⣦⣭⣭⣥⣴⠄⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⣠⠿⣛⣩⠶⠚⠋⠉⠤⠀⣸⣿⡉⠁⣀⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠓⣿⠉⣡⣶⣿⣿⣿⣿⢗⣿⣿⣿⣿⣷⣶⣧⣄⠀⠀⠻
⠀⠀⠀⠀⠀⢀⣿⠀⣹⡿⠿⣛⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⣿⣿⣷⡶⣾⣿⣿⣿⣿⣿⣿⡿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⢿⣿⡟⠋⠉⠉⣱⡋⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡴⠋⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠠⣼⢸⣶⡾⢢⣦⣦⣤⣠⢀⡀⢠⣙⣦⣀
⠀⠀⠀⠀⢾⡏⠙⠟⢀⣿⠃⠉⠡⠛⢸⣡⣼⡏⠻⡌⠶⢄
⠀⠀⠀⠀⠀⠀⢰⡞⢹⣿⣶⢂⣿⡇⢸⠋⣿⣇⡀⠀⠀⢨⢄
⠀⠀⠀⠀⢠⣆⣿⡟⣿⠟⣿⣿⢹⡇⠈⣡⠤⣿⣻⢂⣶⡧⡶⣷⡄
⠀⠀⠀⠀⣮⣿⢹⣧⣿⣤⠌⠉⠀⣴⣿⣧⢀⣿⠭⠉⣤⠀⡶⢰⡇
⠀⠀⠀⢸⠇⠉⠠⠬⣿⢩⣴⠶⣷⡻⣿⣿⡟⣿⣦⣠⣟⣰⣴⣾⣷⡛⢦⡀
⠀⠀⠀⠀⠀⢀⠤⢼⡟⢸⡯⠀⠈⠀⣿⠈⣧⣻⢸⣯⠸⣯⢶⣿⠋⣾⡦⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠴⠆⣧⢤⣶⣴⣆⡴⣿⣿⣿⣿⣿⣿⣦⣽⣿⣯⣼⣿⣶⠾⢷⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣴⣻⣿⠿⠿⣿⣿⡋⢻⣿⠋⠉⠉⡇⢸⡏⠀⠀⣤⡄⡇
⠀⠀⠀⠀⠀⣠⣠⣿⠟⠿⣿⣿⠀⠀⣿⠟⠵⠸⣷⡀⠀⠀⡇⠘⠀⠀⠀⠻⠇⠇⣄⠀⠀⢠⣼⡇
⠀⠀⠀⠀⠀⠋⠘⢻⡆⠀⣿⡿⠀⠀⢻⠀⠀⠀⣿⣇⠀⡀⣤⠀⡇⠀⠀⠀⢀⣿⣿⡆⠀⢺⡟⠃
⠀⠀⠀⠀⠀⠀⣾⣿⡇⢸⣿⣿⡄⠀⢹⡂⣀⢼⠟⢱⣖⢷⣦⣴⣿⣶⣶⣞⣻⣿⣿⣷⣿⡿⡓
⠀⠀⠀⠀⠀⠠⠿⠛⢃⣘⠋⠉⠋⠉⢫⣯⣵⣾⢿⣯⣾⣙⡚⡿⡟⣹⣳⣻⣿⣼⣽⣿⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠩⢇⠀⠀⠀⠉⠊⠑⠻⣿⣿⣿⣿⣿⣭⣿⣯⣶⣿⣿⣮⣿⣑⣾⡄⢤⡤
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠛⠛⠛⠋⠉⠟⢱⣏⣩⣿⡭⢿⣣⡤⢭⣻⡿⢿⣯⣭⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠙⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⢠⣀⣰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠰⠈⠁⠀⣐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠀⢠⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠀⠀⠇⠈⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡠⠋⠔⠀⠀⠀⠀⠀⠸⢀⣀⠀⠀⠀⠀⠀⠀⡲⣆⠀⠀⠀⠄⡄
⠀⠀⣤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠐⠀⠉⢀⢀
⠀⡎⠀⢡⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⣃⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠃⠀⠸⣸⠀⠈⡀⡄⠀⠀⠸⢀⡀⠀⠀⠦⣄⠀⠀⠛⠒⠵⣄⢀
⠀⢸⠀⠀⡏⡀⠀⠀⠹⣆⠀⠀⠹⡀⢀⠔⠠⠀⠑⠲⠠⠄⡠⣹⣬⠀⠈⠀⠐⡢⠄⡀⣂⣀⣂
⠀⠸⡀⠀⠁⠃⠀⢀⠀⠈⠂⠀⠀⡼⠏⢀⣀⠀⡀⠠⠤⠊⠀⠀⠌⠂⠀⠀⠀⠀⠠⠄⠀⢀⠨⠉⠉⠂
⠀⠀⢃⠀⢸⢸⢀⠏⠀⠁⠀⠱⢄⠃⠀⢀⣀⣀⣦⠠⠒⠂⠁⠌⠀⠀⣀⡠⠄⣀⠤⠒⠈⠀⠀⠀⢰
⠀⠀⠸⡄⠈⠸⡞⠠⠂⠀⠀⠀⠀⡏⠉⠉⠀⠀⠈⠐⠂⢀⠠⠀⠀⣑⠯⠒⠉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣷⣀⡀⡇⠀⠀⠒⠈⠒⠤⠁⠀⠉⠉⠓⠒⠲⠤⣌⣁⠀⠀⠀⠂⠀⢀⣀⣠⣤⠤⠴⠖⢒⡀
⠀⠀⠀⢡⡏⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠏⠉⠉⠀⠀⠀⠀⠀⠀⡌⠡
⠀⠀⠀⢸⠗⢻⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⢠⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⡸⣇⠹⢸⠦⣬⣄⣒⣒⣒⣒⣀⡠⢄⡀⢀⣀⠀⣀⠔⢡⠂⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡇⠋⠇⣨⣀⣀⠀⠈⠉⠉⠉⠉⠉⠀⠸⡤⠔⠋⢀⠔⠁⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢣⠀⠆⠀⠐⠁⠀⠀⡄⡇⡂⠀⣨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠦⠤⣤⣤⠤⠴⠆⡇⠃⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠗⣠⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⠀⠀⠀⠀⠀⠀⠀⣠⡄⢠⡄⢠⣀⣀⣀⣀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⢀⣸⡆⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⢧⠟⠸⠇⠈⣯⣉⣉⣉⣉⣩⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⠛⠉⠉⠁⠀⢀⡀⠀⠉⠉⠙⢻
⠀⠀⠀⣶⣸⣿⣿⢸⣿⣿⢠⣾⠂⠀⠀⠀⠀⡤⣄⡤⣄⠀⠘⢿⣭⡉⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⠏⠿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⢸
⠀⠀⣰⣟⣿⣿⡏⢼⣿⣿⢸⣿⠀⠀⠀⠀⠀⠷⠟⠣⠟⠀⠀⣠⡍⠻⢶⣄⠀⠀⠀⠀⠀⣴⡿⠿⠿⠿⠿⠿⠿⢷⡆⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸
⠀⢀⣽⣿⣽⣿⣷⣾⣿⣿⣼⣿⣨⠀⠀⠀⠀⢰⡄⡴⣦⠀⡾⠛⠒⠒⠚⠻⣆⠀⠀⠀⠀⠈⢁⠀⠈⠸⠀⠀⠐⡈⠀⠀⠀⠀⠿⠛⠋⠉⠉⢩⣿⣯⠉⠉⠙⠛⠻
⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⠘⠃⠳⠋⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠣⠤⣀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠉⠛⠳⠆
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⠄⢿⣿⠇⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢋⣍⠉⠳⢦
⠀⠀⠀⣞⡟⠉⣠⣿⣀⠉⢟⡦⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣌⢹⣄⠀⠈⢳⡀
⠀⠀⢴⠻⠀⣾⣟⢙⣿⡆⠠⠟⠄⠀⠀⠀⡴⠿⢷⣤⣀⣀⣀⣉⣧⣀⣀⡀⠀⠀⠀⠀⣀⣴⣤⣤⣤⣀⣤⣤⣾⠟⣛⣛⡢⠀⠀⠀⠘⣿⣿⣿⣿⣶⣝⣲⢶⢂⡇
⠀⠀⣴⣶⡾⠛⠻⠿⠛⠻⣶⣶⡆⠀⠀⢠⡾⣛⢷⡌⣿⣿⣿⣿⣿⣿⢛⣭⣓⠀⠀⠜⣩⠶⢮⡙⣛⣫⡙⠿⢡⣿⣫⣝⢿⡄⠀⠀⣀⢈⡻⢿⡿⢟⣫⣭⣭⣭⠅
⠀⠀⠘⠛⠡⡵⣴⢶⡴⡥⠙⠛⠁⠀⠀⠸⣷⣟⣽⠏⠿⠿⠿⠿⠿⠇⢿⣒⣽⠂⠀⠀⢷⣛⣽⠇⠛⠛⠛⠛⠘⢿⣼⣯⡿⠃⠀⠀⢇⡈⢛⠁⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠆⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣮⠛⣤⠀⠀⠀⣠⣶⣶⣦
⢀⡄⠀⠀⣤⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⣠⣾⡋⣹⠏⠀⠀⠀⠀⠀⠀⠀⣿⠓⢺⣿⣿⠀⠀⠀⠀⠀⠀⠉⢣⡝⢪⡲⢄⣤⡹⣿⣿⡿⠁
⠈⠁⠀⣼⣿⣿⣿⣿⣿⣿⡿⠇⣉⣽⠇⠀⠀⠀⠀⠀⠈⠛⣫⣾⣯⣨⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⢺⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣵⣿⣿⣿⠆⠁
⢠⡖⠀⠻⠿⣟⣛⣻⣭⣵⠶⠟⠋⠁⠀⠀⠀⠀⠀⠴⣾⣿⣿⣿⡿⣫⣾⠀⠀⠀⠀⠀⠀⠀⠀⡴⢛⠛⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢟⣵⡿⣦⡀
⠈⠙⠛⠛⢛⣻⣭⣭⣷⣶⡿⠃⢀⣴⣄⠀⠀⠀⢰⠁⠘⠿⣿⣿⠀⠻⡿⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠂⠀⠂⠂⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⠟⠁⠙⢫⣿⢏⣳⡆
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠙⠁⠀⠀⠀⠘⠂⠤⠆⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠶⠿⠁⠛⠤⠞⠁⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠙⠟⠉

⠀⠀⠀⠀⠀⢠⡞⠛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢴
⠀⠀⠀⠀⢀⣬⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣥⣄⡀⠀⢀⣹⣁⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⡶⣄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠙⢻⠛⢹⡇⢰⡟⢟⠟⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣛⣛⣛⣛⡃
⠀⠀⠀⠀⡯⢸⣿⣿⣿⠀⡟⠀⠀⠀⠀⠀⠀⡟⢰⡾⣷⠘⡇⢸⣅⡾⣦⣸⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣶⢸⣿⣿⣿⣿⣿⣿⢰⡆
⠀⠀⠀⠀⠻⢰⣿⣻⣿⠐⠇⠀⠀⠀⠀⠀⠀⠉⢹⡇⣿⠋⠁⠈⢹⡇⣿⠉⠀⠀⠀⠀⠀⠀⣀⣀⣐⠶⢖⣂⣀⡀⠀⠀⠀⠀⠀⠀⢿⠸⣿⣿⣿⣿⣿⡟⣸⠇
⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡆⣴⠀⠀⠀⠀⡅⣥⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠁⣿⣿⣿⣿⣿⡇⠉
⠀⠀⠀⠀⠀⢸⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣻⠀⠀⠀⠀⠇⠏⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠉⣽⡉⣹⡍⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠄⠀⠀⠿⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠦⠀⠆⢠⡄⠉⠃⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠶⢈⠴⠂⠁⠀⠀⠀⠉⢠⠀⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⢸⠀⠀⢀⠄⠂⠀⠀⣛⠰⠏⠄⠀⠀⠀⠙⢦⡀⠀⠤⠀⢷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠘⠄⠀⠀⠀⠀⡎⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⣠⠄⠀⠀⠔⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠚⠁⠀⠒⠀⣶⠀⠉
⠀⠀⠀⠀⠀⣀⠔⠊⠉⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠚⠀⣼⠃
⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⢀⣀⠔⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠉⠁⠀⠈⣳
⠀⠀⠀⢀⠴⠲⠄⡴⠚⠉⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠐⢦⡀⠋⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣄⡤⠤⠤⠄⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠱
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⣀⣀⣤⡦⠺⣉⡀⠀⠀⠀⠀⣀⡤⡀⠈⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⠲⣶⡿⡑⢀⠈⡛⢷⣶⣖⣺⢍⠅⠀⠈⣉⣀⣀⣀⣠⣤⣴⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠕⠦⠭⠛⠛⠊⠁⠀⠀⠀⠻⠛⠛⠛⠛⣋⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢀⣶⠾⢶⣦⠀⢀⣴⠾⠷⣦⡀⠀⣶⡶⠶⣶⣄⠀⠀⣶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇⠀⠀⣿⠀⢸⡏⠀⠀⢹⡇⠀⣿⡇⠀⠈⣿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⠀⠀⣸⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢽⡇⠀⣿⡇⠀⠀⣿⡇⠀⣿⣀⣀⡀⠀⢦⣄⡀
⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⢀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣸⡇⠀⣿⡇⠀⠀⣿⡇⠀⣿⠉⠉⠉⠀⠀⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣤⡀⢸⡇⠀⠀⣼⡇⠀⣿⡇⠀⠀⣿⠇⠀⣿⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠆⠀⣿⠀⠀⠀⢸⣧⡀⣀⣿⠀⢸⣧⡀⢀⣾⠇⠀⣿⣇⢀⣰⡿⠀⠀⣿⣀⣀⣀⠀⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠉⠛⠛⠉⠀⠀⠙⠛⠛⠉⠀⠀⠀⠛⠛⠛⠋








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⣀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⠄⡀⠀⠀⠀⠀⠰⣁⢠⢧⣸⣼⣧⠜⢫⡛⣶⢇⡄⠀⢣⢸⢸⡸⢇⣇⡹⢆⣸⢡⡶⡜⠀⡸⡼⢃⢧⠀⡇⣇⡀⡗⠏⠈⠳⡷⣁⡄⠀⠀⠀⠀⠀⠀⠄⢀
⠤⠭⠤⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠂






⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣄⣀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡄
⠀⠀⠀⠠⣿⣿⣿⡇⠀⠀⠀⢀⠄⠀⠀⠀⢠⣾⣿⣿⣿⣿⠛⠀⠀⣀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⢀⣴⡄⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣽⣿⣿⣁⠀⠀⡴⠁⠀⠀⠀⠀⣶⣿⣿⣿⣿⡅⠀⠀⠀⠛⠋⣹⣿⣟⣁⣠⣄⣠⣤⡄⠀⠀⠈⠛⣿⣿⣿⣷⡶⠋⠁⠀⠀⢠⣾⣿⣿⣿⣄⣤
⠀⠀⠀⢸⣿⣿⣯⠉⣽⠏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢰⣿⣿⣿⡉⠁⢀⣴⣶⡀⠀⠀⠀⢀⣻⣿⣿⣿⣷⣦⠀⠀⠀⠸⣿⣿⣿⡿⠉⢹
⠀⠀⠀⠈⡩⢹⣿⣾⠁⠀⠀⠀⠀⠤⠀⠀⠈⣹⣿⠟⣿⣆⡄⠀⠀⣴⣾⣿⣿⡿⢿⡤⢻⣿⣿⡗⠀⠠⣴⣿⣿⣿⣿⡿⠿⠟⠀⠀⠀⢴⡞⢋⠘⢷⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠉⠀⠀⠀⠙⠋⠉⠉⠁⠀⠀⠈⠙⠋⠁⠀⠀⠈⠉⠋⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡓⠀⠘⠺⠀⠀⣉⣠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠾⠚⠒⠛⠉⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⡆⠈⣿⠀⠀⠀⢿⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠠⠀⠀⣸⠇⠀⣿⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠓⠒⠐⠀⠀⠐⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣶⣴⣧⣴⣄⣠⣄⣼⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⡤⠀⠀⠠⢦⡆⠶⠄⢒⣦⣬⣽⣿⣿⣿⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠰⠚⡛⠛⡍⠛⢁⠈⢐⢾⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⡀⠀⢀⡻⠷⠈⠆⠀⠀⠀⠘⠃⢤⣿⢿⣿⣿⣿⣿⣉⣽⡿⢏⣿⣻⣦⡀
⠀⠀⠀⠀⠀⣀⣀⣁⣀⡁⡀⡈⠀⢩⣦⠀⠀⣶⠀⠀⠠⡄⠀⠀⠘⢈⠀⠈⣠⣿⣦⣿⣿⣿⡇⠀⠀⠀⠈⠀⢀⡀⠀⠀⠹⠿⣹⣶⣿⣯⡿⠿⣧⣀⡇⢼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⠠⠈⠿⠿⠦⣿⡀⠠⢀⡄⣤⠘⣿⢾⢇⣼⣿⣿⣿⣿⣿⣿⣿⠀⠶⡎⠀⠀⠐⠂⠁⠀⠉⠈⡋⠉⠋⠀⢡⣰⣿⣿⠇
⠀⠀⠀⠀⢐⡇⠀⠠⣧⠀⠈⡿⠀⠀⠀⠒⠀⠈⠉⣶⠀⠃⠸⠀⠛⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠛⠉⠃
⠀⠀⠀⠀⠰⠷⠾⠷⠾⠿⠛⠓⠒⠒⠒⠂⠒⠃⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢐⡄⢀⣀⠐⠒⠛⠛⠦⢄⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢸⡰⠋⠉⢳⠀⠙⠲⣄⡴⠊⠀⠐⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠠⢈⡤⠤⠄⠀⣏⠀⠀⠈⣁⡤⠤⢤⠴⠊⠉⠉⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢠⢦⡀⠀⡼⣧
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢠⠀⠈⣀⣀⣀⡾⠃⡀⢼⠁⠀⠀⠀⢤⣀⢸⠁⠀⠀⠴⠒⡀⠀⠀⠘⢦⣇⣠⡷⠃
⠀⠀⠀⠀⠀⠈⡀⢀⡆⠀⢘⢦⠞⠉⠀⢸⠃⢸⠁⠈⠂⠀⠀⠀⢠⡈⠻⡄⠀⠀⣠⠖⠈⢀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠐⠉⠉⣚⣇⢰⡁⣀⣀⠀⠀⠸⠀⢀⡔⠊⡀⠀⠀⠀⠀⡇⠀⢹⠀⠈⡇⠀⠀⢠⠆⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡞⠁⢀⠞⠉⡁⠀⠁⠀⠀⠀⡞⠀⢮⠀⠀⠀⠀⠊⠀⠀⠘⠀⠀⣿⢀⠴⠃⠠⡀⠀⠀⠀⠀⢀⠀⠤
⠀⠀⠀⠀⠀⢈⣀⡀⡟⠀⣼⠁⠔⠋⠙⠂⠀⣇⠀⠘⣆⠈⠲⠤⠤⠀⠀⠀⠀⣠⠏⢀⡴⠋⠐⢫⣉⣉⣡⣤⣤⣶⣶
⠀⠀⠀⠀⠀⡇⠀⠀⢧⠀⠘⠦⢠⠞⠉⠉⠓⠺⠆⠀⠈⣆⠀⠐⡄⠀⠀⠐⠊⠁⠀⠚⠀⠐⠀⠨⠋⠉⠉⢉⣭⡅⣀
⠀⠀⠀⠀⠀⠑⠀⠢⠤⠵⠄⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⡼⠀⢠⡧⠖⠛⠩⠴⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⡦⠄⠀⠀⢀⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠠⡀⠺⠇⠀⣸⠃⠀⠀⢀⠈⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⡬⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠠⠐⢒⡂⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀⠠⠈⢀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠆⠒⠒⢒⣤⠤⠴⠶⠦⣤⣤⣤⣄⣀⣀⣀⣉⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣕⣺⣏⣂⣿⣑⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⢟⣿⠿⠿⠿⠿⠿⢿⡿⠿⣿⣟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣭⣭⣭⡎⣿⣷⣜⣿⣛⣛⣛⣛⣻⣟⣛⣿⣻⣟⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠿⠿⠿⠿⢃⢹⣿⠟⣿⣿⣿⢻⣿⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡆⣸⣷⣜⣿⣿⣱⣿⣿⣣⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⢣⣿⣿⡿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣙⡍⣶⣿⣿⣯⡘⣿⡿⣣⣿⣿⢃⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣛⠿⣣⣦⣼⣍⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣚⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⡟⡛⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⢸⡆⠀⠀⠀⣿⣿⣿⡇⢤
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡘⣿⠀⠀⠀⣿⣿⣿⡇⠼
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⢀⣀⣤⣤⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⣼⠃⣠⣾⣿⣿⣿⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⠀⠀⠀⠀⢸⣧⣾⣿⡟⢀⣀⣀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢱⣟⣿⡆⠀⠂⢸⣿⣟⡁⠖⠻⢿⠛⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠛⠛⠁⢠⠆⢸⣿⣿⣿⣦⣠⡀⠸⢷⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣷⣶⣾⣿⣦⠾⣿⣿⣿⣿⣿⡿⢔⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢹⣿⣿⣿⣿⣷⣀⠈⠛⠿⣿⣿⣿⣦⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣴⣿⣿⣻⣿⣿⣿⣿⣿⣖⠠⠀⠀⣉⠉⢙⣉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⣭⣛⣿⡻⣟⢯⣻⣿⣿⣿⣶⣶⣶⣶⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠉⠙⠓⠮⠛⢯⣿⢿⣗⡿⣿⣿⣿⢿⣶⣾⣿⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢫⣤⡀⠀⠀⠀⠀⢦⣻⣦⢿⣿⣿⣾⡿⣿⣿⣿⠗⢻⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠃⠿⠿⠇⠀⠀⠀⠀⡈⢳⣻⣿⣥⠙⠛⠿⠆⠀⠀⠀⠘⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⡀⠀⠀⠀⠀⠀⠀⠘⢼⣧⢻⣿⢧⣄⠠⠀⠀⠘⡇⠨⣼⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠐⡈⣿⠀⢻⣷⣷⠀⣀⣀⠀⠻⠒⠉⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠛⠀⠘⣿⢿⠉⠙⠀⠀⠀⠀⠀⠁

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣷⣶⣶⣶⣶⣶⣦⡀
⠀⠀⣠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢃⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣀⣥⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢏⠁⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠋⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠁⠀⠘⠋⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⢘⠓⠄⠀⣠⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢠⣄⡐⠁⠙⡀⠈⠉⠉⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠁⡠⠐⢲⣿⡟⠀⠀⠀⠐⢤⣀⣠⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⣉⣽⡶⣶⣤⢴⣤⣎⠀⢠⡿⠃⠃⠀⠀⠀⣠⣾⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡝⠀⠀⣰⠇⠀⢰⠞⣉⠆⠀⠈⠻⣿⣽⠈⠠⠑⢢⣴⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⡟⢷⡀⠀⢿⠀⠀⠀⠀⠉⢀⠵⠀⠀⢹⣿⣧⣠⣴⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣉⡀⢨⣿⣦⣼⣷⣆⠀⠀⠀⠀⠰⣄⠀⣤⣙⣿⣿⣿⣿⣿⠋⠛⠓⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⡩⣩⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣿⣾⠛⠙⠉⠻⣿⣿⣿⣧⣄⡤⡁⢼⣿⠿⠿⠿⠛⠛⠛⢛⠉
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣤⣼⣧⡍⢸⠇⢾⣿⣿⣿⣇⠀⠀⢀⣾⡛⠁⠀⢀⣤⣴⣿⣿⡿⠛⠿⣷⣶⣯⣿⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣛⣻⠿⠳⣆⢸⣿⡇⢀⣼⠋⠉⠉⠉⢿⡇⠀⠀⡀⠙⠿⣿⣾⠂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠳⣮⣻⣿⣿⣾⡏⠀⠀⠀⠀⢸⣇⢀⣀⣀⣰⡄⠙⣿⡔⠀⣄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣄⣀⣨⣿⣿⣿⣿⣿⣦⣄⣠⣤⣿⣿⣿⡿⢿⣿⣿⣿⣿⣟⣀⣿⣷⣶⡂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⣈⣛⣛⣛⠻⢿⡿⠉⣀⢛⣿⣿⣿⣿⣿⣿⣾⣿⠟⠻⣿⣿⣿⣿⣟⠿⣧⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣀⠴⠾⣿⠿⠛⣭⡗⠈⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣾⣿⣿⣿⣟⠛⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣠⡴⠛⠁⠀⢢⣝⣻⣿⠿⢻⣿⣿⡿⠟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠢⠴⠛⠁⠀⠀⠀⢾⣆⣈⣭⣴⠾⠛⣉⣤⣶⣿⣿⡿⠟⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠡⠏⠔⠇⠀⠀⠀⠀⠹⠿⣿⣿⣛⣩⣴⣾⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣿⣿⣿⣿⣯
⠀⠀⢸⣿⣿⣻⣿⣟⣿⣾⣅⣀⢀⠀⣂⣀⣀⣠⣼⣾⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠉⠈⠙⠻⣿⡟⢿⢿
⠀⠀⠘⣿⣿⡿⢿⡿⢿⢯⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠋⠁⠀⠀⢀⠀⠈⠀⠈⠏⠑
⠀⠀⠀⠈⠙⠁⠈⢙⣛⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠠⠌⡽⠛⣿⡳⠾⠿⠛⠻⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢆⣐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⠟⣳⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⢿⣿⣷⣿⣧⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣄⠀⠙⠛⣹⣿⣿⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠁⠀⠀⠀⠘⣿⣿⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⡀⠔⠛⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⢀⣀⣀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠁⠀⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⢀⡀⠀⠀⢀⣄⡖⢤⠀⠀⠀⠀⢤⣈⢃⠀⠀⠀⢿⡿⠀⠀⠀⢾⡿⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⣷⡼⢦⠶⠒⠒⠿⠛⠋⢀⠀⣀⢸⡅⢀⠀⢠⣿⠃⠀⢾⣀⠈⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣒⡒⠐⠂⠀⠲⠤⠌⠁⠀⠀⠀⣨⠀⣯⠁⠉⠉⠡⠀⢀⣼⣿⣿⣷⣭⠈⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣀⣨⣿⣿⣤⣀⣄⠘⣀⣾⣿⡿⣿⣿⣿⣷⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⢶⠰⢶⡾⠟⠋⢽⣭⡄⣱⣾⣿⣿⠟⠁⠀⠉⠉⠉⠉⠉⠉⠐⠙⠈⠻⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠺⢼⢯⣴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠠⠂⠀⣼⣶⣾⣶⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢀⣷⣦⣾⣶⣾⣷⠎⠙⠢⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢾⣿⢻⣿⣿⢉⡁⠀⠀⠀⠀⠈⠐⠃⠰⠤⠒⠀⠀⠀⠀⠀⢾⣷⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⢘⣿⣿⣾⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⢿⣋⣴⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆⣹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⢈⣧⣾⠁⠙⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⣿⣿⣿⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠶⢷⣷⣶⣶⠀⠾⣤⣤⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⣿⣿⢲⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡿⢀⣼⣯⣿⡿⢰⠇⣿⡏⠀⠹⢄⠀⠀⠀⡃⠀⠀⠀⢀⢸⠺⣿⣧⣾⣿⣰⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣿⡿⣿⣿⠗⠋⣸⣿⣶⣤⣶⣦⣡⣦⣸⠁⠀⠀⢰⣿⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣀⣀⣰⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠟⠿⠿⣾⣿⣿⣿⣿⣼⣿⣿⠿⠿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠆⠐⠐⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣦⠀⣀⠀⠀⣀⠀⡀⠀⢤⣵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⢁⠀⠀⣀⣂⠘⣉⣁⣉⣁⣲⣀⣻⠟⢈⣯⢖⣠⣀⣀⡀⠀⠀⠀⠈⠋⢻⠟⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠄⠀⠀⠉⠈⠀⠀⠀⠀⠐⠚⠋⣍⢱⠣⣷⣿⣿⢿⡋⠟⠉⠉⠉⠀⠐⠣⠾⠻⢗⣶⣦⡀⠀⠘⠃⠈⠛⠉⠑
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠐⠖⠴⠒⢌⠍⠙⢩⢿⡿⠟⠊⠀⡀⢐⠀⠠⣾⣏⡞⠿⡟⠈⠛⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢺⣷⠀⠀⠀⠀⠀⠂⠀⠈⠃⠀⠁⠀⠀⠀⢔⡀⠂⠀⠔⠁⠿⡭⣖⡆⢰⠂⠙⠶⠟⢺⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣿⡄⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡄⠀⠤⠷⣯⣀⡈⡥⠀⠀⠀⢸⡿⡏⣿⣿⡧
⠀⠀⢀⠀⢀⠀⠀⠀⠰⠄⠀⣄⠀⣿⣀⣀⣀⣀⣈⠂⢸⡇⠈⡧⠀⣤⠀⠸⣧⠀⡤⢈⣿⣤⣀⣰⣿⣿⣷⣿⡀⡀⠀⢸⡟⠻⣟⠿⣇
⠀⠀⢀⣤⣸⣥⣤⣴⣦⢼⡈⣯⣹⣿⣿⣿⣿⣿⢸⣀⣈⣿⣆⣃⣀⣀⣀⣦⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣷⣤⣼⣿⣿⣿⣷
⠀⣠⣛⠛⣛⣛⣛⡻⠷⢾⢿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣟⣛⡛⣻⣿⡟⠉⠉⠉⠉⣿⣟⠉⠋⠉⠉⠙⣿⣟⣿⣟⣏⣹⣿⢽⣹⠁
⠀⠉⠀⠀⠀⠀⠀⠤⠿⠯⠯⠇⠂⠀⠐⠸⣿⣦⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠲⡶⣶⣿⣿⣦⠁⠀⠀⣀⣿⣯⣿⣿⠟⣿⣿⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡁⠀⢈⠀⣿⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡠⣤⡰⠀⠀⡀⠀⢀⠰⢺⡿⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣿⣿⣏⠀⠀⣿⣿⣿⣿⣿⣇⣿⣿⣿⡇
⠀⢀⣤⣤⠀⢾⣿⡟⣿⣿⣶⣦⡁⠀⠈⠀⣁⣱⣿⣿⣦⠀⠐⢦⣤⣀⠀⠀⠀⠈⠠⣿⣽⣿⣏⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣿⣿⣿⣿⢋⣉⠉⢉⡉⠉⠉⢉⣀⠀⣾⣿⣿⣿⣿⣿⣷⠤⠤⢽⣿⣿⣆⠀⢾⣿⣿⣿⣿⣿⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡀
⠀⠁⠀⠉⠉⡁⠀⠀⠀⠀⣴⣿⣿⣿⣀⡜⢛⣻⣿⣿⣷⣿⡀⠀⢸⣿⣯⠀⠀⠀⠉⣿⣯⣷⣿⠄⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡇
⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⣠⣴⣿⣿⣿⡟⠈⠻⢿⣿⣿⣿⠇⠀⢹⣿⡇⢰⣿⣿⡆⠨⠇⢙⣿⡀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠻⠛⢿⣿⡏⢰⣿⣷⠸⣿⠟⠁⣴⣤⠸⠋⠉⠈⠈⠈⠂⠈⠁⠚⣀⠀⠠⠀⢀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠋⠀⠓⠶⣆⣰⣶⣷⣷⣿⣿⣿⣶⡟⣢⣾⣿⠿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⢀⣠⡀⠡⠜⢿⣿⣿⣿⣿⣿⣷⠂⡐⠂
⠀⠀⢢⣷⣣⣄⡀⠜⠁⢠⣼⣿⣿⣿⣿⣿⠿⣋⣵⣾⣿⣿⣏⡀⠀⠀⠀⠓⠀⠄⡀⠐⠒⠚⠒⠈⠉⠵⢤⣤⣌⠙⠻⢿⣿⣿⣷⠰⣿⠟⡖
⠀⠀⣾⣿⣿⣿⣿⣤⣾⣷⣿⣿⢿⡛⢉⣤⣾⣿⣿⣿⣿⣿⣿⣧⣤⡦⣄⠀⠀⠈⠀⠀⠀⠀⠙⠶⣶⣤⣤⠀⠠⣭⣀⣀⣩⣷⣿⠯⠷⠆
⠀⠀⠀⠈⠹⠾⠽⢿⣿⠿⠊⠀⠤⠳⠯⢿⣟⣯⣹⣿⣿⣿⣿⣿⣿⣿⣻⣿⣾⡠⠤⠀⠀⠀⠠⠤⠼⠕⠂⠈⠘⠝⢈⠠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣠⠂⠀⠀⢀⣠⠿⠛⠛⠙⠛⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣾⣁⠀⠒⢻⠋⠁⠀⠀⠀⠀⠀⠀⢄⠸⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢶⡜⢿⡟⢿⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠈⣗⠊⣿⢸⡇⣫⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣰⣿⣧⣟⣼⠁⠴⣇⣁⠀⠀⢀⣈⣉⣓⣦⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⢄⠀⣯⡿⠀⠀⠘⠻⠯⠿⠇⠃⢠⣾⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣛⣳⣺⣾⣿⠗⠌⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡏⠹⡦⠀⠀⠀⠀⠀⠀⠀⠠⠤⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀⠀⠀⠀⠂⠀⠀⠀⣀⣤⣌⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⡋⢹⡿⣿⣷⣄⡴⠀⠀⠀⠀⠉⣭⣭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣾⡇⢈⣿⣿⣿⣶⣦⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣝⢿⣿⣿⡿⠋⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠉⠻⣿⠿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠿⠿⣷⣦⣀⣤⢙⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠦⣄⡀⠈⠻⠁⠙⢭⡙⢿⣷⣽⣿⣿⣿⣿⣿⣏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣠⡔⠈⠄⠀⠙⠆⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⡦⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠠⣄⠈⢿⣿⣿⣿⣿⣿⡿⠿⠾⢿⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣤⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡴⠟⠋⠉⠉⠉⠙⠛⠷⣦⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⡀⠀⣤⠀⠀⠀⠤⡄⢰⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠅⣆⣿⠀⣾⠀⠀⣷⢈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠲⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢹⡿⠀⢾⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⢸⡇⠀⢨⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⢸⡇⢰⣬⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣺⣥⣧⠾⢓⠚⠉⠻⢦⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠛⠁⢉⠅⠐⠂⠘⠂⠀⠀⠀⠈⠙⢲⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠲⣶⠀⢠⠞⣀⡒⣦⠀⠀⡁⠀⢀⠀⣀⢠⠋⢩⡏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣴⠿⢖⣛⣿⣦⣍⡉⣃⣀⣠⠶⠾⠶⣄⠼⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢿⣧⡀⠈⠉⠁⠄⠿⠛⠉⠉⠁⢀⠀⠀⠈⢀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣅⠤⠤⠌⢙⣧⡼⠿⠶⠶⣦⣄⣤⠴⠶⠖⠚⢛⡟⠓⠳⠶⢤⣴⡀⠙⠷⣤⡀⠀⠀⠠⠖⠦⣂⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣀⣀⣠⡤⠤⠐⠾⣿⠄⠄⠚⡲⣤⣤⠁⠖⠂⠒⠂⠀⠚⠒⠒⠂⠀⠁⠈⡻⣤⣄⠀⠀⠀⠀⠀⠀⣃⠀⠠
⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠐⠛⠛⣉⣁⣩⣄⡠⣤⠦⠀⠘⡷⠶⠚⠛⠆⠀⠀⠓⠋⠀⠁⠀⠈⠀⠀⠀⠀⠀⣤⣼⡇⠽⠛⠉
⠀⠠⢠⣾⡆⠀⠘⠛⠒⠀⠀⠀⠀⠀⠽⠝⠿⠛⠒⠊⠉⠀⠘⡇⢀⠀⠂⠒⠀⢤⠀⡄⠀⠀⠀⠀⡀⠠⠄⠠⠀⠈⢸⠇⠀⠀⠀⢴⠄⠴⠴⡀⠀⣰⣆
⠀⡄⢈⡽⣻⣦⡄⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⡀⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣴⠀⡇⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠓⠀⠘⠟⠋⢡
⠀⣿⢀⣬⣵⢨⣿⠒⢦⣄⣤⢀⣀⡀⢈⠁⠀⠈⠛⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣿⠀⠃⠀⠀⣠⠀⠀⣴⠀⠀⠀⠀⠀⡤⠀⢀⣤⣄⣩⣿
⡇⠻⠇⣠⣼⠟⠋⠈⠘⢯⣿⣿⣿⣷⣿⡆⠀⠀⠀⠀⢀⠀⣴⡇⢸⠀⠀⠀⠀⠸⠇⠀⠀⠀⠛⠀⠀⠀⠀⣿⠀⠀⢸⣤⣄⠀⠀⠀⠀⠒⠊⡙⠋⢹⣥
⣧⠓⠀⠀⠀⠶⡂⢀⣀⣾⠤⢠⠟⢻⣿⡇⠔⣶⡀⢠⠛⣿⢻⡉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠄⠀⠀⠉⠀⠀⢘⣿⣿⣿⣷⣶⣾⢶⡲⠻⡀⠈⢙
⡏⠀⠀⠀⠀⠀⠀⠎⠉⠉⠀⡞⣠⣿⠹⡇⢀⣿⡀⢠⡀⣿⠀⡇⠀⠀⢠⡀⠀⠀⠀⠀⡀⢤⢰⣧⡀⢄⡀⠀⠀⠀⠘⣿⣿⣿⣿⣭⠎⠀⠁⠀⠘⠛⣫
⠹⠀⠀⠠⠀⠘⢀⢈⣶⠀⠀⣀⢸⣿⣄⣅⣈⢈⣁⣠⣥⣠⡄⣉⣀⣄⣠⣧⣤⣤⣀⣀⡄⠀⠻⡟⠗⣾⡿⣧⡐⠀⠀⠉⢉⣼⣇⣹⡆⠰⣄⠀⣤⠠⣤
⠀⣀⣴⠄⠀⠂⠉⣿⣿⠂⠈⠙⠛⠛⠛⠛⠛⠹⠛⠋⠻⠋⠀⠈⠿⠏⣙⠋⠀⠉⠉⠁⠀⠀⣰⣶⠻⣝⡆⢿⡟⠷⣤⠦⠄⠁⣼⠟⣿⢴⣗⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠿⠷⠓⠀⠀⡠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠈⠋⠁⠀⠉⢃⣈⡁⢀⣀⡀⢠⡦⠀⠀⠁⠈⢻⡶⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠁⠟⣀⠀⠀⠀⠀⠄⡀⠤⠄⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⢀⣀⣉⣏⡀⠈⠍⠉⠀⡙⠿⣀⣐⠀⢀⣁⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣥⡀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⣀⠉⠙⢿⠋⣿⣽⢧⠀⡁⠀⠀⠛⠀⣼⡁⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠃⠀⠀⠃⠀⡄⠰⠀⠿⣀⢀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⣀⣰⡆⠀⢐⣈⡀⠛⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢹⡧⠠⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠐⠂⠀⣠⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡁⠰⡀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣋⣷⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠶⠟⠻⠬⣝⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠄⠀⠀⠀⠀⠀⠙⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣏⣀⠀⠀⠀⠀⠀⠀⣴⠚⣒⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠶⢒⠒⡛⢛⣉⡙⠷⣌⣾⣿⢛⡛⠿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠘⣿⣿⡇⢠⣿⡋⢻⠀⣿⣽⣿⣼⢰⣋⢺⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠐⣿⣿⡅⡟⢿⣿⡿⠀⠟⣿⣿⡿⣟⢟⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣒⣒⣿⣦⣠⣴⡿⠚⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠛⣷⡈⣿⡏⠈⣽⠛⣙⢳⣠⣤⣤⣴⣾⡾⣿⣽⠿⠿⠿⣻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣏⣽⣧⣿⣿⢸⣿⣿⣯⣾⣿⡿⣿⣷⠿⢋⣤⡶⠖⠒⠦⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣾⡿⣡⡶⠋⣠⣶⣾⣿⣿⣦⣬⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣻⣿⢯⣾⠿⢿⡆⣷⣾⣿⣿⢟⣾⠋⣠⣾⣿⣿⡟⠁⠀⠉⠉⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢱⡿⢡⣿⠃⢠⣿⠓⠛⠻⡿⣣⡿⠃⣼⣿⠿⣿⣿⡇⠀⡶⣡⡠⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣶⣿⣿⣂⣾⣃⣀⣀⣼⢳⡟⢁⣾⡿⠛⠠⣌⣿⣧⠘⣧⣫⣃⣸⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠹⣿⣿⣿⣿⣟⣛⣿⣧⣿⢀⣾⣿⣁⠀⠀⢢⣿⣿⣄⠈⠙⠚⠋⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⠈⣿⣿⣿⣿⣶⣴⣿⣿⣷⣾⣿⡿⠉⠨⣷⣦⣿⣿⣿⣷⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⢸⣿⣾⣍⣩⣿⣿⣿⣿⣿⣿⡏⠠⣄⠙⢻⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠛⠿⡿⠟⠋⠙⠛⠉⠱⠄⠈⡗⠼⡛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠶⠻⠛⠛⠓⢦⣀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⣾⣿⣿⣆⠀⠟⠀⠀⠀⠀⠀⠀⣭⡛⣻⣾⣿⣯⣮⠀⢀⡀⢀⣀⠀⠀⢠⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠶⠿⠿⠿⠿⠤⠤⠀⠀⠀⠀⠀⠰⣯⣭⣭⣿⠛⣫⢵⡏⠠⠀⣿⣿⣿⣣⡸⣿⡿
⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⣷⣾⣭⣿⠾⣿⣿⡟⢃⣴⣿⣟⣵⣿⣖⣿⡧
⠀⠀⠀⢂⡻⣞⣽⠗⣶⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣠⣭⠾⣧⣜⣿⣿⣭⣿⣿⣿⣿⡇
⠀⠀⢸⡉⠁⠽⢲⣻⣽⢻⣷⣤⠄⣀⠀⢀⣄⣸⣿⡇⠀⠀⠀⢀⣴⠟⠛⢦⣄⠀⠀⠀⢈⣹⣯⡁⠿⢻⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⣻⣼⠒⢴⣾⣽⣯⣼⣿⣿⣶⣿⣿⣿⣿⠃⢀⣴⡞⠉⠀⢠⣶⠀⢈⣳⣦⠖⢋⣡⣤⣿⣄⣘⡛⢿⣝⣿⣿⣿⣿⣿⡿⣿⣿⣷
⠀⠀⠀⣠⣿⠹⢦⢀⣸⣿⣿⠿⠿⠿⠿⢿⠿⢿⢿⣠⣾⣿⡀⠀⠀⠈⣩⣶⣷⢤⡁⡾⠋⠉⠁⠀⠀⠉⢻⣯⢙⠛⠿⢿⣟⣿⣵⣿⣿⡇
⠀⠀⠀⢠⣤⣴⡶⢶⣶⣶⠒⠒⣶⣶⣏⠋⠉⠉⠉⠉⠉⢹⣇⣤⣶⡿⡟⠛⠉⢸⡇⣛⠂⠀⠀⠀⠀⠀⠘⠹⡿⢿⣿⣶⡶⣭⣭⣙⣛⠻⠆
⠀⠀⠀⠀⠛⠛⠃⠚⠟⠻⠃⠘⠛⠛⠛⠀⠀⠀⠀⢠⣴⣿⡿⡛⡉⣶⡀⠀⠐⠘⡃⠌⢀⣄⠀⢠⡄⢠⡤⠀⣷⡀⡹⣇⣾⣷⣿⣿⡿
⠀⠀⠀⣰⣾⣶⡀⢠⣶⣷⡆⠀⣠⣶⣾⣿⣷⣶⣴⣿⣿⣿⣿⣿⡄⢈⠀⠀⡀⠀⣶⡄⠘⠋⠀⠈⠁⠈⠁⠀⢹⣉⣋⣿⣿⣟⣍⣿⣥⡀
⠀⠀⢠⣬⣭⣉⣡⡬⡭⠥⠵⠶⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⡏⠀⡟⡏⠁⠀⠀⠀⠀⠀⢀⠀⠘⠋⠉⠉⠉⠋⠉⠉⠉⠁
⠀⠀⠀⠄⣴⣿⡳⠄⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡰⠲⠶⠦⢤⡆⢠⡇⡠⠄⠀⠀⠀⠀⠀⢟⡀⠐⠒⠒⠒⠒⠒
⠀⠀⠀⠂⣿⣿⣿⡒⠂⠉⠉⠁⠀⠀⠀⠀⢠⠄⡄⠀⠀⠀⠊⠀⡮⢧⡀⠚⠃⢸⣏⡁⠚⠛⠀⠀⠀⠀⢯⠁⠀⠀⠀⠀⣤⣤⠀⠀⢠⣤⠄
⠀⠀⠀⠈⣿⣿⣿⠛⠛⠛⢀⢠⣶⣶⣀⣀⣄⣀⣀⣀⣿⣶⣄⣄⣉⣹⣿⣛⣇⣼⣃⣧⣠⣀⣜⣛⣃⣀⣋⣀⣄⣶⣀⡀⣀⡈⠁⠀⠀⠉⠙
⠀⠀⠀⠰⣿⣿⣿⣿⡿⠿⢿⣾⣿⣿⠟⡿⢿⣟⣿⣿⣿⣷⣿⠏⠿⢻⠏⠿⠛⣿⣿⣿⠃⢻⡿⣿⡿⣿⡟⢟⣿⣿⡿⣿⣯⣿⢏⣿⣯⠉⠉
⠀⠀⠀⣤⣿⣿⠿⠿⢷⡶⠶⠿⢿⢿⣶⢳⣾⣿⡙⢻⡛⠋⠉⠀⡀⣸⠉⠉⠛⠛⠛⠛⢰⡘⣛⢋⡙⢛⡙⠉⠉⠛⠛⡛⠛⠛⠛⠛⠛⣲⠒⠆
⠀⠀⠀⠀⣠⣤⡀⠘⠋⠁⠀⠀⠀⠛⠛⢻⣷⣿⣿⣼⣛⣀⣀⠛⠛⠛⠀⠀⠀⠀⠂⠀⠿⠿⠿⠿⠿⢿⠿⠷⠚⠑⣾⣇⣀⣠⣶⣶⣷⣿⡀
⠀⠀⠀⢀⣛⣛⣁⣀⣀⣀⣀⣉⣉⣉⣩⣽⣿⡭⠭⠥⠬⠠⠤⠀⠐⡀⠐⠀⠂⠒⣚⠛⠛⠛⡛⢛⡛⠛⠛⠻⠛⠛⠛⠿⠿⠟⣻⠛⢿⠿⠝⠃
⠀⠀⠀⠈⠙⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠐⠀⠐⠀⠀⠙⠃⠀⠈⠃⠀⠀⠈⠀⠉⠀⠀⠀⠀⠈⠉



⡿⢿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⢛⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣯⡿⢿⣿⣿⠻⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢛⢿⣿⣿⣾⣿⠟⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣸⠀⠛⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡌⠀⠀⠀⠀⠀⠐⢨⠟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠈⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠀⠀⠛⣄⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢃⠀⠐⠀⠀⠘⣷⣄⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠐⠤⣈⣋⠩⣍⡀
⠀⠀⠀⠀⠀⠀⣏⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡗⠄⡉⠄
⠀⠀⠀⠀⣠⡾⣋⣩⡄⠀⢌⠂⠀⠀⠀⠀⠀⠈⢹⣷⣄⠳⣄
⠀⠀⠀⢸⢿⡿⠟⠉⠁⠈⠒⡅⠀⡄⢠⠀⠀⠀⠈⣿⣿⡆⠙⣆⠀⠀⠀⠀⠘⣦
⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠹⣿⣿⡄⠸⡀⠀⠀⠀⠀⠈⠥⠀⡀
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⢿⣿⣷⡀⢱⠀⠀⠀⠀⠀⠈⣑⠺⡄⠀⠀⡤⢠
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇⣸⠀⠀⠀⠀⠘⣿⣿⣧⠈⡦⣄⠀⠀⠀⢠⣬⠑⠁⠀⠘⠀⣸⠂
⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠇⠀⠀⠀⠀⠀⢹⣿⠟⠋⠀⠈⢦⠠⠚⠁⠀⡁⢂⠀⠖⠀⠏
⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡈⠂⠀⣴⠇⠦⠈⣃⣴⠟⠁
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠖⢁⠴⠋⠀⠀⠀⢀⠀⢀⡜⠁⠀⠀⢿⡶⠛⠛⠁
⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢠⡤⠞⠁⠀⠀⠀⠀⠀⠀⣴⣟⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣣⠀⠂⢰⡇⡇⠀⠀⠀⠀⠀⢀⡜⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣥⣿⣆⡀⠘⠃⠀⠀⠀⠀⢀⡴⠋⠀⠈⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠉⠛⢻⡷⣄⣀⡀⠀⢠⡞⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠀⠙⠉⠙⠛⠛⠋⠀⠀⠀⣀⣀⣤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢷⣶⣶⣶⣶⣶⡆⣾⣿⡿⠿⠟⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠉⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡅⢠⡄⢤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣩⣤⣥⣬⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⢤⣤⣤⢤⣤⡤⢤⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠄⠤⣤⡄⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⢀⠔⠀⡖⠲⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣾⠀⠀⠀⡖⣿⣦⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⣿⠀⠀⠀⠙⢦⣸⠁⡴⠋⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⡇⠀⠀⠀⠀⠀⠀⠙⠓⠋⠀⣀⡀⠀⠣⣀⠼⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠼⠀⠀⠀⠀⠀⠀⢠⢖⣒⢢⡀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⠠⠤⡆⣦⢴⠠⠰⢤⠀⠀⢀⣄⣴⠶⣶⣠⣄⢻⡘⠿⣡⠇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡉⢻⣤⣟⢋⣹⣇⠈⠉⠁⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⡿⣿⢤⣽⣿⣿⣿⣧⣤⣤⣤⣼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠒⠛⠓⠀⠀⠀⠐⠚⠚⠚⠛⠓⠛⠛⠛⠓⠛⠛⠛⠛⠚⠛⠛⠓⠂⠀⠀⠀⠚⠒⠂

⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡀⠠⠄⡀⠀⠀⠀⠀⠰⣁⢠⢧⣸⣼⣧⠜⢫⡛⣮⢇⡤⠀⢣⢸⢺⡸⢇⣇⡹⢇⣸⢡⡴⡜⠀⡹⡼⢃⢷⠀⡇⣏⡀⡗⠇⠈⠳⡗⣁⠄⠀⠀⠀⠀⠀⠀⠄⢀
⠤⠭⠤⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠐⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⡀⠀⠁⣁⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⠀⠰⠂⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠉






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢠⣶⣶⣶⣤⢀⣀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⢣⠀⠰⡤⠤⣈⣀⣀⣤⡤⠤⠶⠂
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣟⠋⠀⠀⠀⢠⣿⣿⣆⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢠⣤⡄⠀⣠⣤⡀⠀⠀⠀⠀⣇⣴⣷⣄⣸⠀⠀⡇⢠⣤⣤⣤⣤⡤⠀⣦
⠀⠀⠀⠚⡟⠿⠿⠛⠛⢻⠃⠀⠀⣠⣿⣿⣿⣿⣆⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣿⣿⣷⣠⣿⣿⣿⠀⠀⢻⣄⣽⣿⣿⣿⡧⢦⠀⡇⠀⢿⣿⣿⣿⠃⠀⡇
⠀⠀⠀⣜⠀⣠⠤⠤⠔⠃⢀⢀⣴⣿⣿⣿⣿⣿⣿⣷⡀⡀⢸⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣹⣿⣿⣿⡀⠀⠘⢿⣿⣿⣿⣿⣷⠞⠀⡇⠀⢰⣿⣿⣴⠆⠀⠁
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠋⠙⠛⠛⠛⠛⠛⠙⠉⠉⠁⠈⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠈⠋⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢾⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡫⡎⢱⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣩⣮⣀⣇⣸⣀⣷⣝⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⢀⣸⣧⢸⣿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡤⠤⠤⠼⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡯⠥⠀⠬⠭⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡏⣡⣿⣷⣤⢩⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡭⢍⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⡣⠚⠁⠀⠙⢏⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣫⣎⡉⠀⠀⠈⠉⠉⠑⣌⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⡫⠞⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠑⢌⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⡫⠚⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢼⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡻⣋⠘⠑⠂⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠂⠀⠈⠀⣀⢠⠝⣍⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢋⠄⠥⠄⠀⠉⠀⠀⠀⠀⣠⡼⢛⡿⣧⣀⠀⠀⠀⠀⠤⠈⠈⠀⠑⢌⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢏⢴⠁⠀⠀⠀⠀⠀⢀⡀⣠⣾⢏⠟⠃⠀⢢⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣋⠟⠁⠀⠠⠀⠀⠀⠤⠤⣨⡾⢋⡄⠁⠀⠀⠀⠀⢈⠢⡙⢷⣄⠄⠀⠀⠀⠀⠀⠀⠀⠙⢍⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠀⠀⠀⠀⠐⠀⠀⢠⣾⢏⡴⠡⠄⠀⣀⣀⠀⠀⠀⠀⠈⣦⡿⣷⡄⠀⠀⠠⡀⠀⠀⠀⠃⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⢰⣿⣿⡆⠁⠉⢙⣿⠂⠒⠒⠀⠀⢀⣀⠀⠀⠐⣿⡋⢉⠀⠀⣨⣾⡍⡀⠀⠀⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢉⣷⣿⡏⠀⠀⢨⣿⠉⠀⣴⠤⠤⠴⠶⣤⠀⠀⣿⡅⠀⠀⠈⢽⣶⡇⡅⠀⠈⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠠⠴⠒⣿⣿⡧⠀⠀⢹⣿⠒⠀⣏⠁⢡⡄⠃⢹⠀⠀⣿⡆⠀⠀⠀⢾⣿⡟⠂⠀⠐⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠒⠂⠀⢀⣿⣿⡗⠠⠄⠸⣿⠂⠀⠇⡀⢸⡇⡁⢸⠀⠐⣿⡆⠀⠀⠐⣺⣿⡇⠆⠒⡆⢰⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠉⠀⠈⣹⣿⢿⡇⠀⠀⢈⣿⡁⠀⡃⡇⢸⡇⠐⢻⡀⠉⣿⡇⠀⠀⣈⣹⢿⣷⡃⠐⠂⢈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣼⣿⣛⣧⠀⠈⢸⣯⡀⠀⣅⠀⣀⢀⣂⢸⠀⠀⣿⡁⠀⠀⢠⡟⠻⣿⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⠄⠀⠠⠤⠤⢤⣤⠄⢨⢿⡤⠆⠿⠤⠤⠤⠥⠾⣤⣠⡿⡦⣀⡀⠤⠤⠤⠤⠤⠀⠀⣠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣿⣍⣉⣉⣭⣉⣉⣙⣿⣿⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠂⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣼⣿⣿⣾⣿⣟⣠⡤⠶⡏⠕⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠏⠁⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠟⠁⠀⠀⠀⢠⡠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⣿⠟⠁⠀⠀⠀⠀⣴⠞⢁⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣛⣷⣾⠫⠀⠀⣀⣠⣴⡿⠗⠚⢣⣠⣶⣿⡿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣶⣾⡏⠉⠉⠀⠀⣠⣿⣿⣿⣿⡿⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢳⡄⠀⣠⣾⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⠿⠿⣷⢷⡾⠿⠿⠛⡽⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡇⠀⠈⠀⣀⣴⠮⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡚⢢⡀⠀⣜⠡⠂
⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⢰⢃⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠁⠀⠁⠀⠀⣀⡴⠆
⠀⠀⠀⠰⠀⠀⢰⠁⣠⢴⠶⢛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⢀⡀⡰⢃⣴⡷⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡴⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠟
⠀⠀⠀⠀⠛⠻⠞⠛⠋⢃⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠏⠀⠀⠐⠈⠀⠀⠠⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣎⠀⠀⠀⢀⡠⠔⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢘⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠲⠀⠘⠇⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢀⠤⡐⢐⡐⢠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢐⢢⠋⠀⠘⢧⡄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠹⢆⡀⢀⡃⠀⠠⢰⡀⠀⠀⣠⠇⠀⠀⢀⢦⠀⢠⢦
⠀⠀⠀⠀⠀⠀⠁⢚⡍⢙⠒⠁⢠⡟⠠⢈⠁⠁⠨⠭⠷⢾⣄⠀⠈⠺⠦⡾⠋⠀⠀⠆⠿⠀⣴⠀⡀
⠀⠀⠀⠀⠀⡌⣤⠀⣹⠀⣠⠀⡠⠊⠱⣣⠞⠙⢦⠐⠒⢄⣠⠀⠀⠠⠀⠃⠀⠀⡦⠀⠶⣀⠈⠰⠁⣄
⠀⠀⠀⠀⠀⢧⠻⢊⡡⠤⠄⢀⡇⠀⠈⣙⡤⠤⢈⠔⠋⠉⠁⢀⠀⠀⠀⠀⠀⣀⠀⣠⠀⠈⢷⠀⠾⢀⡀
⠀⠀⠀⠀⠀⠀⠰⠉⠀⢠⠀⢀⣠⣤⣼⠋⡠⢼⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡈⠀⠙⠀⢀⡾⠀⢶⡈⠁
⠀⠀⠀⠀⢀⣀⡀⡆⠀⡈⠲⠉⠀⠀⡇⠘⠀⢀⠀⠀⠀⠐⢄⠈⢆⢤⣤⡤⠀⠁⢸⠆⠚⠉⣀⠸⣅⠉
⠀⠀⠀⠀⠀⠀⡤⠷⢸⡥⠤⠤⠀⠀⠃⢀⠎⢡⠀⠀⠀⢀⡼⠀⢸⠀⠈⡆⠀⣠⠏⠀⣶⠁⠽⠈⠁
⠀⠀⠀⠀⠀⠜⠀⢰⠋⢠⠄⣀⡤⣄⠀⢌⠀⢸⡀⢀⡀⠈⠀⠀⠀⠀⢀⡇⠐⣡⠄⠂
⠀⠀⠀⠀⢰⠒⠀⢿⠀⢻⡀⢀⠤⠤⠤⣸⡄⠀⢳⡀⠉⡉⠀⠀⠀⠤⠊⠀⡜⠁
⠀⠀⠀⠀⠘⠄⢤⣈⣂⡀⠀⠈⠀⠀⠀⠀⣁⠀⠀⡇⠀⣸⣠⠴⠒⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠂⠀⠉⢸⡇⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠈⠂⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠧⠅⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠇


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣴⡄⣠⣤⣤⣌⣭⣍⣩⡁⠀⠀⠀⠀⢸⣿⢣⣿⡿⣱⣿⡜⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣰⠖⠒⠒⠒⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣷⣿⣿⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣠⣴⣶⣶⣶⣶⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣻⣟⣛⣛⡋⠄⠀⠀⠀⠀⠀⠀⠀⢠⣴⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⣴⣿⢻⣽⣿⢗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠃⠀⠀⠀⢰⡟⢛⣦⠻⠓⠈⠁⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⣷⣿⢀⣀⡀⠀⠀⠘⠻⢿⣋⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠾⠶⠿⠿⠿⠆⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿⣿⡇⢤
⠀⠀⠀⠀⠀⠀⠠⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⠀⠀⢸⣿⣿⡇⠼
⠀⠀⠀⠀⠀⠀⠠⠭⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠥⠤⠤⠬⠭⠭⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⢀⣄⠀⡄⣀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠘⢢⡡⡇⣆⢿⢆⣇⡏⡶⣴⣠⢂⢀⡉⠉⠀⠒⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⠠⣇⠲⢰⣶⣌⢪⡳⣟⣾⢮⢰⣦⢦⡎⡅⣵⣵⢫⠾⣡⣶⣶⠶⢀⢀⣀⠐⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡄⣒⢹⣇⢿⣜⣓⣈⣁⠈⠛⠉⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⠈⠁⠲⠶⠾⣫⠍⡉⡉⣉⠩⠤⠖⢀
⠀⠀⠀⠀⠀⠀⡀⠒⢶⢘⣷⠘⣷⣭⣥⡤⠤⠔⠮⠙⠤⠤⠀⠀⢸⣿⣿⣿⡿⠀⠀⢘⡡⣤⠔⢒⣒⣒⡈⠛⠛⠛⠡⢤⣤⣤⡤
⠀⠀⠀⠀⠀⢤⡉⢁⠐⠌⠉⢓⢉⠉⡉⠢⢐⠤⠀⠐⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⣞⣊⢀⡨⡓⠙⢊⠭⠭⠭⠭⣛⠶⠄⠀⠀⠁
⠀⠀⠀⠀⠐⠂⡙⠮⣭⢤⢤⣤⣤⣤⣤⣬⡁⠐⣠⡶⣒⢀⣀⣀⣀⣤⣀⣄⣀⣀⣀⡠⣤⡘⠥⠠⠐⠌⠩⠭⢍⢭⣐⠭⠭⠭⢕⠲⠖⠶
⠀⠀⠀⠀⠈⠤⠂⢀⡄⣐⢶⣤⠤⣒⢖⣶⣶⣛⠵⣪⢕⢋⠻⡿⣿⣿⢹⣿⣿⡿⡻⣿⣜⣛⣛⣛⡲⠌⠛⢛⡪⢝⣈⡉⣍⡛⢦⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠐⢴⣿⡅⢿⣄⡿⢂⠵⣂⣘⣒⣑⢋⠠⣢⠕⣩⠾⣻⢇⣸⡀⣄⡳⠜⠖⠒⢒⣒⠮⣛⣛⢳⣦⢹⡆⢿⣇⢿⣿⠸⣷⣬⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⢫⢎⡔⣀⡀⠲⠕⣫⡴⢛⡵⢞⠝⣩⣿⠻⣎⢻⣶⣶⣶⢸⣿⡷⠌⣿⢸⣿⣦⢹⣦⢹⣶⣬⣍⠂⣍⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⢸⡟⢰⣶⠞⣡⡞⢩⣴⠇⣾⣿⣿⠇⣿⡘⢻⡟⣩⢸⣿⠑⣿⣿⠀⣿⣿⠈⣿⣆⠉⢛⡛⢷⣈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠁⢿⣿⠰⣿⡀⠾⣿⡇⠙⣿⡇⢸⣿⣧⣿⠁⡿⢈⠺⢿⣿⠿⠛⠿⠿⠐⠟⠿⠿⣿⠁⠸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠸⠿⠄⡹⡿⣄⠿⠟⠀⠋⠉⠉⠉⠉⠛⢾⡃⣹⡷⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠏⢀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⡒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⡁⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠄⢀⡾⠀⠁⢲⠄⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠦⠀⠀⣤⠀⠀⠈⣧⣄⡁⡺⠂⠈⢡⡄⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢂⣀⠀⣸⣿⣄⣀⣠⠙⠃⠀⠊⠀⠈⠘⠃⢦
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⢀⠠⢴⡄⠈⠙⢩⣿⣟⣭⠁⠰⠀⠀⠚⠑⠀⣴⣶⠀⠀⠤⣋⣤⡠⣦⢠⡄⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠋⠀⡜⠀⠂⠀⠉⣶⣦⢾⢺⢶⢨⢟⣿⡃⠀⠀⠀⠀⢠⡄⠀⠿⠿⠀⠀⠀⡹⠷⠓⠶⠾⠇⢨⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣆⠄⠃⢠⠀⠠⢄⣠⣴⣾⡏⠬⣼⣿⣿⡗⠂⠀⠂⠀⣩⣬⣈⣠⣤⣀⣤⡀⠠⢤⣠⣤⢤⣤⠤⠠⠌⠤⠄⣀
⠀⠀⠀⠀⢠⣄⠂⡏⠀⡆⢰⠀⠀⢠⣙⡿⠻⣵⠾⠿⡹⢿⡇⠀⠂⠀⠀⠉⠁⣦⢰⠎⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⣤⠀⠀⠃
⠀⠀⠀⠀⢻⡿⢱⣧⡆⣸⣬⠀⠀⠀⠿⠇⠀⠀⣀⣀⣀⠀⠁⠈⠀⠀⠈⠁⡄⠁⠈⢰⡄⠀⠀⠀⠀⠀⠀⠀⣆⠘⣿⡀⠀⡆
⠀⠀⠀⠀⣽⣿⠃⠟⠋⠉⠀⠀⠀⢠⡄⠀⠀⠀⠟⠩⢿⡇⠀⡄⠀⠀⠀⠀⠀⣿⣧⠙⠐⠆⠀⠀⠀⠀⠀⠂⡏⢀⣭⠀⠀⡍
⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣀⣀⡀⠀⣀⢀⡷⠀⠀⠀⠘⠁⠀⡇⢀⠀⠀⠀⢠⠟⠿⠴⠤⠤⠤⠤⠤⠤⠤⠴⠅⠘⠓⠖
⠀⠀⠀⠀⠀⠘⠛⠉⠉⠁⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠛⠀⣀⣾⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢿⣷⣶⣶⣽⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠉⠉⠉⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠱⠦⡕⠀⢘⠋⡙⠀⠀⠁⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠉⠁⠀⢀⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⡤⠀⠀⠀⣠⠞⠻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣾⣿⣧⣀⣀⣠⡴⠞⠁⠛⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⡈⠂⠀⠿⢧⠛⢿⠛⠉⠀⠀⠀⠀⠀⠘⠙⠲⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⡿⠀⠀⠀⠀⣴⠊⠀⠀⠀⢲⡄⠀⢠⠇⠀⠀⠈⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠾⠁⠀⠘⠃⠀⠀⠅⡀⠋⡀⡀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⡀⠀⣠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠰⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣥⣭⣑⠶⣯⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠈⠀⢀⠛⠻⢷⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⠆⠀⠀⠀⠁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣤⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⢀⡁⠢⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡦⠀⠀⣼⠃⠀⠀⠁⢠⠁⠄⣉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠓⠤⠿⠀⠄⠀⢀⡁⠀⠀⢹⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠘⡄⠀⠈⣥⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠁
⠀⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣁⣄⠁⠀⠀⡀⠀⠀⣀⣀⣀⡀⢀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⣡⣖⣫⣍⣀⣤⣴⣤⣤⣼⣿⣯⣽⠷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣦⠾⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢾⣶⠷⠾⠷⠖⠛⠛⠋⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣯⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢰⣤⣴⣦⡄⢀⠌⠃⠘⡄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣆⠀⠈⡙⠿⠦⠠⣠⠂⠀⠃⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⠀⠠⣏⢒⣸⠄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡰⣇⢻⣿⣿⣿⣿⣋⠜⠻⠶⣄⣸⣇⢈⠊⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⡵⠄⠈⠉⠉⠉⠀⠈⠙⠖⠶⠚⠉⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡭⣭⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠴⠶⠶⠶⠶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠷⠶⠶⣿⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡟⠃⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣓⣻⣆⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⡾⠁⢀⣀⣤⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠛⢀⡼⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡴⠟⠚⠛⠉⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣠⣞⣁⡀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢸⡇⠀⠀⠀⠀⠀⣠⠶⠛⠉⣉⣀⣈⣉⠙⠓⢦⣄⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠘⡇⠀⠀⠀⢠⠞⠁⣠⠖⠋⠉⠉⣉⣉⠛⠶⣄⠙⢷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣷⠀⠀⢠⠏⢀⡞⠁⠀⠀⠀⡾⠉⠉⢷⠀⠘⣧⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢹⡄⢀⣾⠀⣼⠁⠀⠀⠀⢀⡿⢦⣤⡟⠀⠀⠸⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣨⠷⠋⣿⠀⢿⠀⢠⠖⢲⡏⠀⠀⢈⡷⢤⡀⢠⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣠⡴⠛⠁⠀⠀⠸⡆⠘⣧⠘⠶⠞⠁⠀⠀⠈⠳⠞⢁⡾⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠚⠁⠀⠀⠀⠀⣀⣀⣹⣆⠈⠳⣄⡀⠀⠀⠀⠀⣀⡴⠟⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠈⠳⢤⣀⠉⠛⠛⠛⠋⢉⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠯⣆⠀⠀⠀⢠⣎⡅⠈⠁⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⢸⠀⠀⢀⣿⣀⡂⢠⠀⣰⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⢸⠃⠁⢦⠨⠉⠉⠉⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠠⠀⠀⠀⢀⡠⠋⣟⡐⠀⠐⣰⡍⠁⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡤⠧⠞⠂⠀⢰⣉⢑⠃⠉⠁⣶⠀⠁⠈⠀⢰⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⠀⠀⠀⠆⡀⢠⡇⢀⡀⢨⣇⣀⣟⠒⢸⠀⠀⠸⣿⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⢤⠀⠀⢀⠀⢈⡇⠀⠀⣶⠀⢸⠈⠀⠀⢚⣛⠃⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⡀⡀⠒⠚⡛⠐⠂⣀⠀⢠⠐⡶⡶⠉⢭⢰⢰⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠁⠀⠀⠠⠤⠀⠀⠛⠀⠤⠤⠏⠁⠴⠶⠬⠈⠁⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠈⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⢛⡻⣛⡛⣛⣛⣻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⡟⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣶⣿⣿⣿⣿⣿⡿⠻⠿⢿⣮⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠿⠿⣿⡙⠛⠟⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣦⡈⢻⣦⠀⠀⠀⡀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣸⡇⠀⣿⡆⠀⣤⡬⠙⠂⢠⣖⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣾⡏⣧⠀⠀⠀⠀⠀⠉⠘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⢿⣿⣿⣿⡿⠃⢻⠀⠀⠀⠀⠠⠄⠰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⢻⣿⣷⣄⢸⡀⠀⠀⠀⡠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣿⡏⠻⢿⣷⣤⣀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣄⡀⢸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣀⣘⣿⣿⣿⣿⣇⣩⣙⢾⣿⣟⡻⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠛⠭⣙⠻⣿⣿⣿⡏⢸⣿⣿⣧⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣼⣿⣷⣿⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣭⣿⣿⠿⢿⣿⣿⡐⠳⢸⣿⣿⣯⡿⣿⣏⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠐⠋⠉⢿⡍⢺⣿⣷⡹⣿⣽⣽⣿⡜⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠛⠛⠙⠋⠁⠉⠁⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⡾⣉⠺⣿⠟⠉⠑⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠋⠉⢀⡴⢋⣼⣿⣷⡖⣠⠆⡀⠀⠀⠀⠉⠑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠈⢧⡴⢋⣴⣿⣿⣿⣿⠿⠷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡼⢋⣴⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⡰⠁⣠⣠⠋⡴⠟⠁⠰⡰⡄⠀⠀⠀⠀⠀⡄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠂⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⢠⠃⢀⡽⠀⠻⡀⠀⠀⠀⢈⣥⣤⣶⣶⣾⣿⣿⠿⣵⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠎⣠⢾⣅⣀⣐⣡⣤⣶⣿⣿⣿⣿⣿⣿⠙⣿⣿⣷⣄⣀⠙⠓⢦⣤⡀⠀⠀⠀⣠⣆⣀⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠃
⠀⠀⠀⠀⣿⣷⡿⠉⣿⠛⠋⠛⢿⠿⣿⣿⣿⣿⣿⣆⣿⡿⣿⣿⣿⣿⣷⣶⣾⣷⣾⣴⡾⢻⣿⡇⠀⣸⣿⣿⣿⠀⢻⡏⣿⣿⢹⢸
⠀⠀⢀⣾⠿⠋⠀⠀⠙⠻⣄⠑⠌⡿⠛⠻⣿⡛⠒⠤⣍⠻⢦⣀⠘⠿⣿⣿⢿⣿⡛⠻⠍⢿⣿⠀⠀⣿⣿⣿⣿⣷⣾⣸⣿⣿⠘⠈
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠚⢧⡄⠀⠘⣧⣤⣤⠼⠀⠀⠉⠙⠛⠲⠖⠛⠛⠉⠛⠛⠚⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠀⢰⡟⠀⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢠⣶⡟⠋⠀⠀⠀⢠⡿⠀⠀⠀⠉⠛⣶⣤⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⡀⠀⣾⠃⠀⠀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⣿⣇⣀⢀⣉⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣶⠶⠶⠾⠷⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠸⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⠀⠀⠉⢻⣿⣿⣇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣷⣀⡀⠀⣀⣼⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⣿⣿⠏⠁⠈⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠂⠀⠤⠄⠠⣗⡃⠤⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡍⣴⠁⠒⠲⡀⠀⠂⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢄⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠄⡀⠀⠀⠛⢤⠁⢀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠳⣄⡠⠚⠀⠀⢣⠥⢤⠀⠈⠍⠔⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⡡⢄⠤⢠⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠚⠛⠒⠒⠀⠸⣀⣠⠃⠀⠀⠀⠈⠉⢁⣀⠖⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠀⠄⠠⠂⠀⠀⠀⠀⠀⡀⠀⠖⠒⠲⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠔⠀⠀⠤⠈⠀⠀⠐⠀⢀⣁⣀⣀⣏⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⢩⡝⢢⠀⠀⡀⠀⠀⡭⠭⠭⢭⠭⠭⠤⠼⠄⠀⠀⡀⠀⠀⠠⠄⠂⠐⣄⠈⠀⠐⠀⠀⣠⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣐⠁⠀⠀⠀⠀⠀⣠⠔⠋⠠⠤⡑⠂⠀⡀⠶⠐⠢⠐⠒⠄⠀⠠⠀⠈⠁⡄⠀⠀⠀⠀⠐⠛⠀⡀⠀⠀⢀⡀⠀⣀⣠⡀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠂⠈⡿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠛⢣⠸⠿⠖
⠀⠀⠀⠀⡔⡦⢴⠀⠀⠀⠀⠈⢿⠆⣤⣤⡀⠀⣤⣤⠀⠀⣤⣤⣄⠀⢠⡄⠀⠀⠀⢀⣤⣤⡀⠀⢀⣤⣤⣄⠀⢠⣄⠀⢠⡤⠀⠀⠘⢧⣉⡠⠖
⠀⠀⠀⠀⠈⠍⣻⠀⠀⠀⠀⠀⠀⠀⣿⣸⠇⠀⢸⡇⠀⣼⠃⠀⢻⡆⢸⡇⠀⠀⢠⣿⠀⠈⣷⠀⣿⠁⠀⣉⡀⠀⠹⣦⣾⠁⠀⠀⢀⠀⠀⠀⠀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣷⠀⢸⡇⠀⢿⡀⠀⣸⠇⢸⣇⠀⠀⠸⣧⠀⢀⣿⠀⣿⡀⠀⣹⠏⠀⠀⣿⠃⠀⠀⠠⠜⠃⢼⡁⠀⡦⢤
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠀⢀⠀⠀⠛⠛⠁⠀⠛⠛⠀⠈⠛⠛⠉⠀⠈⠛⠛⠛⠀⠙⠛⠛⠁⠀⠘⠛⠛⠋⠀⠀⠘⠋⠀⠀⠠⠀⠀⢄
⠀⠀⠀⠀⠀⠉⠀⣆⡀⢈⠂⠈⠀⢠⠚⠑⡆⠀⠀⣠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠚⢀⠀⣠⠉⠈
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⡨⣧⠖⠃⢀⠀⠑⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⣀⢠⡌⠀⠀⢠⠀⢄
⠀⠀⠀⠀⠀⠐⢈⠴⠉⠠⠈⠅⠀⠀⠁⠀⡠⠃⠀⠄⣀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠀⠀⠀⠴⣀⠈⠀⠀⠀⠀⣭⠆⠀⠀⠀⠀⠰⠂⠐⢂⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠠⠀⠀⠀⡿⢁⣀⠁⠂⠀⠀⠀⠀⠠⠀⠀⢀⠀⡀⠈⠀⠀⡀⠀⠙⠆⠀⠰⠀⠀⠀⣀⠀⠀⠒⠀⠣⠀⠠⠌⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⣈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠅⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠙⠉⠃⠈⠁⢀⢄⠀⠀⠀⢀⠀⠀⠠⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠠⢒⠙⠛⠀⠀⠀⠘⢻⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠂⠐⠒⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠂⠘⠆⠀⡆⠀⢪⡤⠀⡄⠀⢀⡅⠀⠒⣧⠀⠀⠀⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠐⠒⠢⡄⠀⠀⡉⠉⠸⠀⠤⠀⣠⠐⠃⠀⠀⠀⢸⡙⠲⠆⠀⠆⢀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣹⠀⠀⠁⠀⣰⣤⡁⣨⣤⠀⠀⠂⠀⠀⠀⠋⠓⠶⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠙⠃⠀⡀⠀⠉⠁⢄⠀⠈⠁⠀⣀⣀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⠀⠀⠀⠚⠤⠘⠧⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⠀⠀⣟⡧⣄⣾⠃⠀⠀⠀⡀⠀⠀⠀⢸⣠⠆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣰⠄⣄⡀⢠⡀⣆⣾⣇⢠⢾⠀⣰⣟⣟⣴⣿⢇⢤⠀⠈⡿⠀⠀⠰⢸⠯⠶⣼⣷⡀⢸⣿
⠀⠀⠀⠀⠀⣖⠀⢸⣿⣿⡆⢰⣦⣀⣿⣷⣾⣧⣼⣸⡿⣿⣧⣿⣼⢠⢿⣧⢳⣿⣿⡞⣿⣦⠀⣿⣠⣤⣾⣿⣠⠼⢿⣿⢧⣼⣿⠀⣤⣼⡄
⠀⠀⠀⠀⠀⣿⣤⢼⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⡿⢿⣾⠀⣼⣧⢸⣿⣿⣿⣿⣿⡌⣿⣿⣿⡟⢛⣹⣾⣽⣿⣿⣿⣿⣇⡝⣿
⠀⠀⠀⣄⣐⣿⣿⢸⣿⡿⣿⢿⣿⣿⣾⣿⣿⣿⣿⣸⡇⢠⣿⠻⢿⠖⣿⠓⢸⣿⣿⣿⣿⣿⣧⣟⡿⠎⢯⣿⣿⣿⣷⣿⣏⣿⣿⡿⠀⢸⣿⣦
⠀⠀⠾⣿⣿⣿⣿⢸⣿⣽⡿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣟⠋⠓⠚⣡⡄⠻⡆⠀⠋⣿⣿⣿⡿⠛⠉⠙⠇⠶⣛⣮⣿⣽⡍⠁⠻⣿⢷⣶⡾⠿⠿⡀
⢠⣶⣎⣿⣿⣿⣿⣿⣿⣭⣭⣯⣿⣭⣽⣿⣿⣿⣿⣤⣬⣤⣤⣤⣤⣥⣼⣍⢠⡈⠉⠉⠁⠁⢠⡴⠾⠴⢯⢿⠿⠉⢡⣤⢧⣆⣸⠭⣉⣽⡛⡉⠁
⠀⠁⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢁⠰⠆⠁⠀⠀⠀⢰⠂⣄⠀⠀⠀⠰⢡⠂⡄⠀⠀⠀⢀⠀⣀⠀⡀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⡈⢃⠀⠀⠀⠀⠀⣰⡗⢀⠀⠀⠀⠀⡰⠚⡄
⠀⠀⡀⠀⠀⠂⠀⠀⠀⠐⠀⠛⠄⠀⠀⠈⠬⠀⠁⠀⠀⠀⠈⠁⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡀⠀⠀⠀⠀⠿⠤⠜⠀⠀⠀⠀⠘⠙

⠀⠀⢀⠤⣄⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀⠤⢔⠉⡂⠀⠀⢀⠀⠆⡤⠂⠀⠀⠠⠘⠤⠃⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠛⠃⠀⠀⠀⠆⠂⠀⢀⠂
⠀⠀⣈⣓⣁⡀⠀⠀⢐⠀⡀⣀⣤⠀⠀⠀⠠⢞⠉⡂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⢠⠩⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢢⣤⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠠⣤⠀⢠⡄⠀⠀⠀⠀⡴⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⡤⠴⢄⠀⠀⠀⢀⡀⠀⠦
⠀⢠⠘⣇⡸⠦⠀⠀⠀⣄⢃⢀⠅⠀⠀⠀⠀⠢⠀⠤⠀⠀⠀⠀⠠⠉⠁⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠼⡋⢱⠀⠀⠀⢸⡃⢀⡤⠂⠀⠀⠈⠉⠀⡑
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠓⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠨⡡⠛⠄⠡⠀⠀⠀⠐⠀⠈⠀⠀⠀⢀⠰⠤⠦⡀⠀⠀⠀⡀⠀⣼⠄⠀⠀⠀⠫⢯⢿⡀⠀⠀⠀⠂⣀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠄⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠦⠈⠁⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠉

⠀⢀⠆⠀⡐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢐⠈⠀⠈⢒⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠛⢀⠀⠀⠀⠀⠀⠰⡖⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡐⢁⣄⡁⡀
⠀⠘⢀⡀⡀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠨⢙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠂⢀⠀⠀⠀⠠⠘⡛⠠

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣤⣶⣾⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠉⠉⠁⠀⣸⠃
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠼⣿⣿⡿⠿⠋⠙⠻⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠰⣤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⢀⣠⣶⣿⡆⠀⣀⣤⣶⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣠⣴⣿⣿⣿⣿⣿⣿⣅
⠀⣀⣤⣶⣿⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠙⠻⠿⠿⠿⢿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣄⣀⣀⣀⡀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡗⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠿⠋⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⣠⣾⡇⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⡿⠇⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⣦⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣧
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⢀⣴⣿⣃⣼⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣶⣶⣆⡀⣠⣄⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⡿⣷⣍⠙⠶⣿⣿⣿⣙⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣮⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠓⠶⠖⢶⣯⣝⣻⣟⢿⣿⣿⣿⣿⣿⣿⣿⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⢿⣿⣿⡿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠦⣙⠛⢿⡿⣿⣇⣻⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣮⢉⠲⣿⡜⣿⣿⣿⣿⣿⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠂⠘⢿⡜⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣻⣿⣿⣆⡙⣿⡜⣿⣿⢸⣿⣿⠟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⡻⢿⡜⣿⣿⡟⠋⣼⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⣿⣿⣿⣿⣶⣿⡝⣿⣵⡶⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠿⢿⣷⡼⣿⡇⠸⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡻⣷⣤⣤⣬⣤⣈⣷⡹⣿⡇⢸⡟⠛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⣛⣛⠛⢻⣿⣷⣹⣿⡿⠁⠀⣸⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡿⣿⣿⣿⣿⣿⠿⣿⣧⢹⣿⡗⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣶⣾⠟⠛⢶⣿⣧⢻⣷⣿⠏⠼⢿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣯⣅⣀⠻⢿⣧⢿⣯⡤⣶⣿⣿⡿⠃
⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⣠⡎⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣯⣙⢿⣯⣿⣷⠺⣿⡿⠛
⠀⠀⠀⣰⣿⣿⣷⣆⣄⢀⣠⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⣙⣿⣿⣾⡻⠁
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣟⣴⣦⣤⡀⠀⣸⡿⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠉⠙⢻⣏⣻⣟⣿⡅
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⡟⠁⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⡙⣿⠃
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⢰⣿⣿⣿⣿⠟⡛⢻⣿⣿⣿⣿⣿⣿⣾⣿⢇⢀⡟⠀⠀⠴⣄⡀⠀⠀⠀⠀⠀⢀⡀⠠⢄⡀
⠘⠿⢿⣿⣏⠀⢒⣠⡟⠡⣀⠙⣿⣿⠿⠉⢞⣼⠷⠀⠀⠀⠈⠙⠷⠆⠀⠀⠀⠪⡵⠐⠋⠀⠀⢀⡙⠦⢄⡀⠀⠀⠀⠁⡀
⠀⠀⠀⠠⠬⣙⣛⡻⠷⣤⣤⣼⣿⣿⣶⠞⠉⠀⠹⠓⢤⢀⠀⠀⠀⠀⠀⠀⠀⡴⠤⠄⠀⣀⠀⠀⠘⠣⣶⡀⠀⠀⢠⠀⠃
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⡝⣀⠃⠀⠀⠀⠀⠀⠀⠈⠑⠐⠢⠄⠀⠀⠈⠇⠀⠀⢀⡙⠁⠀⠀⠀⠈⠉⠶⣄⡀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠰⠀⠀⢀⣴⠀⠀⡸⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⢰⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣬⣧⣴⣧⣴⣦⣴⣿⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⣋⠉⠉⠉⢉⡛⢷⣦⡀⠀⠀⠀⠀⠀⣿⣉⣽⣏⣹⣏⣹⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣧⣀⣉⣀⣀⣀⣈⣁⣀⣿⢿⡄⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠁⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡇⠀⣴⠶⣦⠀⣠⡄⢀⣴⢶⣄⠀⣿⠟⣿⡇⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⡇⠀⠿⣶⠟⠀⢻⠇⠘⢷⡾⠟⠀⣿⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣇⢀⣤⣤⠀⣠⠀⢠⡄⢠⣤⣤⠀⣿⠉⠉⣀⣴⠶⠶⢶⣤⣀⣠⡶⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠁⠀⣠⣤⣤⣤⣤⡄⠀⠙⣿⡏⠀⢰⡟⠁⠀⠀⠀⠙⠿⠋⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣿⠉⠉⠉⢹⣇⣠⡾⠋⠀⠀⢹⡇⠀⠀⢠⣿⠛⣷⠀⣿⡇⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠷⠶⠶⠾⢿⡟⠀⠀⠀⠀⠈⠻⣦⡀⢸⣿⠻⣿⠀⣿⡇⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠿⠿⢿⣄⠀⠀⠀⠀⣼⡿⠶⠾⠶⠶⣦⡀⠈⠻⣦⡀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⢋⣴⢶⣄⣀⣀⣈⣿⣄⣀⣼⠿⣦⣀⣴⠟⠙⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⠿⣿⡶⠟⠙⠛⠛⣻⡟⠙⠉⠀⠈⣿⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⣷⣄⠸⠷⠶⠶⠶⠶⠶⠶⠿⢃⣴⡟⠛⠛⠛⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⠟⠀⠻⢷⣶⣶⣶⣶⣶⣶⣶⣶⡾⠛⠿⣦⣤⣤⣤⣤⣤⣤⣾⠿⣷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣿⠀⠀⣻⣇⠀⠈⠛⢷⣦⡀⠀⠉⠉⠉⠉⠉⠉⠻⠿⠟⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⣻⡷⠿⠛⠛⠉⠀⢾⣏⣻⡇⠀⠀⠀⠈⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⣉⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⣒⠿⠗⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠒⠒⠋⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠃⣠⣴⣤⡄⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠁⢈⣿⣿⠀⣿⡋⢙⣿⣾⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣿⣿⡀⠙⠛⠿⠋⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⣹⡍⠚⠋⣈⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣍⣿⣿⣿⣇⡉⢿⣷⣾⣩⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡆⠀⠀⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⡷⣿⡿⢋⠄⢘⠀⠀⢸⡟⢿⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⣿⣿⣷⣶⣦⣿⣿⣿⠿⢿⠿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠘⣶⣶⣾⣿⡈⡄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡇⢇⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣫⣭⣭⣭⣭⣭⣭⢙⠀⠀⠀⠀⢹⣿⣿⣿⡇⣿⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣸⣿⣿⣿⠿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡿⢸⠸⠯⠆⢀⢸⠿⣿⣿⣀⣿⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⣿⣿⡿⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡆⠀⠀⠀⠈⣾⠆⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢿⣿⡿⢀⣾⣿⣿⣿⣿⣿⣏⣛⣋⣉⣉⣛⣒⡀⠀⠀⢀⣾⣿⣷⡀⢿⣿⡟⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠠⢿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠻⠄⢀⣾⣿⣿⡯⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⠻⣿⣿⣿⣿⣿⣿⣷⣤⡔⠚⠒⠄⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⡀⢴⣶⢀⣮⣽⣾⣿⣿⣿⣿⣿⣻⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⠔⠂⠀⠀⠀⣸⣥⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⡀⢀⣌⣿⣻⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⣭⣯⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣧⡿⣏⡿⡛⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣶⣿⣿⡷⣨⣍⡙⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣦⣿⣿⠿⣿⣿⣫⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣆
⠀⠀⠀⠀⠀⠀⠠⠄⠊⠀⠀⠸⢻⡋⣙⣷⣥⣿⣿⠊⠧⣿⣿⡛⣛⣼⣿⣿⣿⣿⣿⣿⣿⣆⠙⠻⣿⣿⣩⣿⣿⣿⣿⣿⣿⠿⣿⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⡆⠴⣾⣿⣅⣿⣿⣻⣿⣿⡟⠿⡿⠇⠉⠴⣶⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣶⣿⣿⣿⣿⢿⣿⣿⣿⠋⠉⣠⣿⣿⣿⣿⣿⣊⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣦⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⡛⠁⠻⣿⡿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⡠⢀⢿⣾⣿⣿⣿⡿⣋⣽⣧⡀⠉⢁⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⣿⣧⣴⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣶⣌⣉⣡⣿⣿⣿⣿⠛⠯⠉⣁⣀⣾⣿⣿⡾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠿⠻⣻⣟⣧⣯⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⢿⣿⣿⣿⣿⣶⡟⣖⣛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⣿⣿⣶⣿⣿⣿⣷⢹⣿⣿⣿⣯⣾⣿⣿⣹⣷⣿⡟⠀⠉⠉⣥⣾⣿⣿⡿⠃⣦⠘⠋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢐⠻⠛⠻⠻⠿⢟⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣦⣠⡶⣈⡟⠉⢻⡿⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣿⣿⡇⢀⠀⣀⠿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣀⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣭⣝⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⣤⣿⣿⣿⡌⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣭⣦⣽⣏⣿⡿⢿⣿⣿⠿⢿⣿⣟⣿⣿⡿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⡏
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⣾⣯⣿⣿⣿⡃⢠⣾⣷⣼⢿⣿⣿⣿⣶⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⣤⣠⠟
⠀⣸⣆⢙⢻⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠚⠛⠙⠓⠻⠿⢿⣿⣿⣿⣿⠟⠻⢿⣿⣿⡿⠿⠿⠿⠿⡿⠁
⠀⢸⣿⣧⠤⠸⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢀⡉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁
⠀⢸⣿⣿⡇⠀⠀⢠⣶⡅⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢸⣿⡿⠃⣀⣀⠘⢻⣷⣤⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠖⠂⠰⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠤⠄⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠙⠂⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣲⠄⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠄⠈⢳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠌⠀⠀⠴⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠖⠂⠐⠚⠀⠶⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠐⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⡀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡁⠀⠀⠀⢠⣇⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⠀⣷⢹⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠄⡁⠀⠠⣦⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠰⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠋⠀⠐⠀⢀⠀⠀⠀⠀⠔⠁⢀⠜⠢⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⠄⠀⡔⠀⣁⡘⣁⠄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣤⠀⠀⠀⢉⢋⡤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠁⠨⣯⡠⠤⠞⠊⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣝⣿⣿⣿⣿⠛⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⣮⣛⣷⢿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⠃
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⡏⢉⣉⣉⠉⣿⣿⡿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠟⢁⣄⠉⣁⣼⣿⣿⣆⣈⢁⣤⡈⠻⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣅⠰⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿⠂⣼⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠺⠟⠀⣾⣿⡿⠋⣠⣶⣦⣄⠙⢿⣿⣷⡀⠛⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⣿⣿⣿⠁⣼⣿⣿⣿⣿⣧⠈⣿⣿⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣟⢙⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠇⡆⡔⣦⡔⡆⡖⡰⡄⡖⡖⡆⡖⡖⡆⣤⠖⣤⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠙⠃⠁⠉⠁⠁⠁⠁⠁⠁⠁⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠒⠒⠒⠒⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⣠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⢀⢤⡔⠂⠤⣄⠀⠀⠀⠉⠒⢤⡀⠀⠀⠳⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡼⣁⣀⠼⠂⠀⡤⣱⡀⠀⠀⠀⠀⠈⠲⡄⠀⠀⡤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⢀⡼⠤⠤⠥⠤⠤⠯⠤⠬⠧⡄⠀⠀⠀⠀⠀⠈⢦⣸⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡔⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠳⡙⠛⠉
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⠉⡇⠀⡔⠉⠲⣤⠶⠶⢆⡔⠋⠲⡄⢸⠉⡇⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠒⡇⠀⠳⠤⠔⢹⠀⠀⠨⠳⠤⠴⠁⢸⠖⠃⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡹⢯⠀⠀⡨⠯⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣑⣒⣒⣒⣒⣊⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢠⠇⣀⣸⠀⠀⠀⢀⢀⡀⣀⡀⠀⠀⢸⣀⡈⡇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⡞⡸⠁⢹⠀⠀⠀⣮⢧⢷⡻⠀⠀⠀⡸⠀⢣⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⡏⠀⡇⠀⠀⠑⠤⠤⠤⠤⠤⠤⠤⠤⠒⠁⠀⠸⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⡇⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⢸⠀⠀⠀⠀⠀⢠⢏⡙⡄⠀⠀⠋⢹⠀⠀⠀⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠳⡌⠙⠒⢄⣎⡎⣱⢱⡀⠀⣤⣼⠀⢀⡔⠋⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠟⠀⠀⡼⣨⣭⣥⡀⢧⠀⣿⣿⠀⠸⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠸⠷⠁⠀⠀⠘⠾⠇⠳⠟⠀⠀⠀⠀⠠⠚⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠦⠲⠶⠳⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠙⠃⣠⠾⡾⣋⢨⢈⣤⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⣘⣡⣄⣠⡄⠉⠉⠻⡅⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⢃⠀⠈⠙⠳⠃⠦⡀⠄⣀⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠰⠛⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡤⠀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠤⠒⠁⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠙⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠉⡀⠸⢄⣀⡴⠋⠁⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀⡴⠃⠀⠀⠐⠀⠀⠘⠁⠀⠀⠀⡠⠜⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⣀⡴⣶⠟⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢁⠃⠀⠀⠀⠀⢀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⢀⣠⠆⠑⣻⡿⠡⡀⠀⣜⠳⠒⠶⠶⠦⠤⠀⢀⣰⠇⢰⠀⠀⠀⠂⠀⠀⠈⠀⠀⠁⢠⡀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⣶⢡⡁⠀⠀⠁⠀⠀⠀⠙⢣⡤⣀⠀⠀⠁⠀⠀⠀⠉⢠⡇⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠟⢦⡀⠄⠀⠠⠀⢳
⣀⣤⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢠⡉⠢⡀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠲⢤⡀⠄⠁⣠⡾⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⢀⡏⠀⠀⠑⢄
⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠊⠉⢦⡈⢂⡀⠀⠀⠀⣀⠐⡏⠀⠀⠀⢈⣵⠿⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠈⡇⠀⠀⠐
⠀⢸⡟⠿⣦⡀⠀⠀⠀⢀⠀⠀⠀⠱⡄⠀⠱⣄⣳⣄⠴⣋⡤⠖⠁⣀⣤⠾⠋⣁⣴⠋⠈⠁⠀⠈⠀⠀⠈⠀⠀⠤⡴⠀⠀⠀⠀⠀⡼⠃
⠀⢸⡇⠀⠈⠛⢷⣄⠀⠘⡄⡆⠀⠀⠘⣆⠀⣀⠤⠒⠉⠁⣠⣴⠞⠛⠛⠉⠉⠉⠈⠳⣄⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠓⢀⠀⢈⡼
⠀⢸⡇⠀⠀⠀⠀⠙⠳⣦⡁⠸⠆⠀⠀⠈⠋⠁⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⣠⠶⠃⠀⠀⠙⠓⠦⣄⣀⡀⠀⠀⢀⡇⠀⡀⠠⢏⣀⠀⠀⣠⠞
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣀⣴⠾⠋⠁⠀⠀⡀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⣸⠟⠳⣏⢰⠆⠀⡸⠖⠋⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠟⠋⢹⠀⠀⠀⠀⠲⣼⣶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠁⠀⢀⣿⠁⢀⡴⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡀⠀⢀⣀⣤⣶⢮⣭⣤⣤⣤⣤⡤⢶⣿⠈⠉⠉
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠒⠼⡟⠋⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠸⡄⠀⣿⡇⠀⠀⠀⠀⠀⣿⠃⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢀⡏⠹⡄⠀⠀⠀⠀⠀⢳⠀⣿⡇⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠹⡄⠀⠀⠀⠀⠈⣧⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢀⣸⡇⠀⠀⠹⡄⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢀⣿⡀⠀⠘⣿
⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⣿⣀⣤⡴⠿⠛⠉⠉⢻⡏⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣀⣀⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⢀⣠⣤⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⢀⡼⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⠻⠿⣲⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠃⠀⠃⢸⡶⡇
⠀⠀⠀⢀⣠⠤⠤⠤⠤⠧⠤⣤⣀⣀⣾⣈⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠤
⠀⠀⡞⠉⡇⠀⣴⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣈⠉⢹⡟⠉⠉⠉⠉⠁⠒⠒⠒⠶⡴⠽⠋⠰⠚⠉⢉⣉⣉⣉⣉⣲⣤⡀
⠀⠀⡷⣾⣧⡀⢻⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⠇⡇⠀⡄⠀⠀⠀⠀⣤⠱⠞⡡⠄⠐⠊⠉⠓⡒⠋⠉⠙⠛⠳⠈⠉
⠀⢰⠀⠛⢠⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⢀⠃⠀⣌⣉⡉⠉⢉⣯⠄⠀⢡⠀⠀⠀⢀⡴⠃
⠴⣇⡀⢀⠾⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠺⠀⡇⠸⠀⢰⠀⠀⠀⠀⠠⢿⣇⠀⠸⡀⠀⠀⡏⠐
⠀⠀⠈⢋⢀⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡆⢸⠁⢸⠀⠀⠀⠀⠀⠀⠉⠉⠸⢳⣴⠃⢀⠔⠓⣤⠤⣄
⠀⠀⠀⢸⣿⢧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠃⡎⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣿⡆⢸⡀⠀⣸⣉⣈⣇⣀⣀⠠⠤⠤⠤⠤⠤⢄⡀
⠀⠀⠀⠘⠉⠁⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⢰⠃⠀⡇⠀⠀⠀⠀⠀⣀⣀⣰⡏⢩⡙⠇⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡸⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⢸⠀⠀⡇⠀⠀⠀⡴⠋⠁⠀⠀⠱⡈⢉⣠⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⣗⡖⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⠀⢰⠀⠀⠀⢸⢁⠀⠀⠀⠀⠀⠈⠉⠀⠱⡀⠀⠙⢏⠒⠠⠤⢤⣀⣀⣀⣀⠤⠤⠐⠉
⠀⠀⠀⠀⣿⣿⠀⠻⠛⠛⠛⠛⠛⠛⠿⠿⠿⠇⢻⠀⢸⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠘⠙⢻⠉⠉⠉⠉⠉⠙⠻⠛⠯⠟⣟⣋⠀⢸⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠞⠲⡄⠈⢆
⢰⢖⠒⠒⢞⢯⡙⢃⡲⣱⢆⡶⡰⢆⠶⣶⣶⡛⢝⠷⣾⠀⠶⠐⠷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠤⢛⣛⠉
⠈⠲⣷⣄⠀⠑⠿⠖⠛⠚⠛⠛⠛⠛⣻⠟⠇⠈⠓⢵⣿⣶⣶⣖⣉⡥⠷⡆⠀⠀⠀⠀⠀⠀⣠⡀⡿⠀⠸⣍⣁⢙⠆
⠀⠀⠈⠻⣍⣉⣉⣉⣉⣁⣀⣀⣐⠒⢺⠦⣸⠆⠀⠈⢻⢻⠛⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⡰⢇⣯⠇⠀⣾⠭⠉⠈⢹⡆
⠀⠀⠀⠀⢸⠿⠀⠀⠀⠀⠀⠀⢠⠚⠛⠢⡬⠂⠀⠀⢸⢸⠈⠁⠀⠀⠀⠁⠀⢀⠖⠒⠊⡇⠀⢏⢀⣴⢣⡤⡄⠀⠀⠁
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠇⠀⠀⠀⠘⢸⠀⠀⠁⠀⠀⡖⠒⢡⠀⠀⠐⠗⠛⣜⢷⡟⠉⠳⣸⣳⣀⡶⢦
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⠀⠀⡀⠈⠀⠀⠀⠈⢠⡦⢰⠎⡀⠀⠀⠀⠀⠈⠛⠀⢰⡤⣈⡇⠀⢸⠃
⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣅⣥⠴⠒⠊⠉⠉⣱⣆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡘⣧⡀⢈⠇
⠀⠸⣟⢕⣾⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⠈⠀⠀⠉⠉⠙⠛⢛⠒⠒⠒⠚⠋⠛⠁⠀⠈⠉⢱
⠀⠀⠈⠓⢝⠯⡑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠠⢤⣤⠶⠮⣤⡤⠄⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠑⢌⢢⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡄⠀⠀⠀⠀⠀⠒⢿⡷⡲⢶⠒⢺⠏⠓⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢴⣒⣊⣉⣉⣻⣗⡺⠿⡭⠥⠖⠤⠤⠈⢹⡍⣚⡱⢏⣀⣄⣐⣤⠤⠖⣶⠋⣉⣉⣐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠽⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⠛⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⢀⣤⡶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠾⠟⠛⠛⠻⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⢀⣾⠋⠀⠀⠀⠈⣹⣿⣆⠀⠀⠀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⢠⡿⠉⠻⣷⣄⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⠇⠀⠀⠀⠀⣼⡇⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⣿⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠘⣷⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠘⢿⣦⡀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣠⣴⡾⠛⠁⣴⠏⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡉⠙⠛⠛⠛⠛⠉⠁⣀⣤⠾⠃⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣶⣶⣶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣤⡀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⣿⡏⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠃⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣀⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣶⣤⣤⣤⣶⣶⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡗⠰⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢋⣼⣿⣿⡟⣰⣿⣮⡻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣜⢿⣿⡟⣰⣿⣿⡿⣣⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⢀⣀⣿⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣴⡄⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠟⣭⣭⣭⣭⣭⣍⣩⡅⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠰⣿⢛⡿⢾⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣠⣤⣤⣍⠻⣿⣧⣙⠿⠿⠞⣣⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⣼⣿⠌⢋⣽⡇⠹⠛⠉⠀⠀⠀⠻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⣦⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⡻⠿⠟⣡⣶⣶⣶⣤⠀⠀⠀⠀⠀⢬⣤⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣿⣿⣿⣿⣿⣿⣿⣿⣧⣍⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⣄⠀⠀⠀⠀⢰⣶⣶⣶⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢃⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⢿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣙⣈⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣁⣙⣁⣀⣀⣀⣀⣙⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⡖⠉⠙⢦⠀⠀⠀⠀⠀⠀⡴⠊⠉⠉⠲⡄⠀⠀⠀⠀⠀⠀⡤⠤⠞⠙⠤⢦⡀⠀⠀⠀⢀⣀⠘⣆⡞⣀⡀
⠀⠈⢧⣀⣀⡜⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⣠⠏⢠⠞⠓⣆⠘⢤⠀⠀⠀⣀⡴⠛⢯⠋⢣⣀
⢠⠖⠒⠬⠴⠒⢢⡀⠀⠀⠀⠀⠳⣄⣀⣀⡴⣇⠀⠀⠀⠀⠀⠈⣦⠘⠦⠤⠋⢰⠋⠀⠀⠀⠉⣇⠶⡼⠰⢠⠏
⠘⠦⣤⣤⣤⡤⠔⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠳⠄⠀⠀⠀⠀⠙⠛⠢⡴⠚⠚⠀⠀⠀⠀⠐⠏⠓⠲⠒⠉⠆

⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡤⡄⢠⡄⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⢧⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠁⠈⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⡴⢭⠇⠀⠀⢸⡇
⠀⢰⠖⠒⠲⡗⠒⠒⣦⠀⠀⠀⠀⠀⠀⠀⢶⠀⢸⠀⢸⡆⢸⣱⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⢀⡇⢄⢸⡇
⢠⠼⢦⠀⡴⠧⡄⢠⠿⡄⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⣠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢸⠁⠋⢸⡇
⠘⠓⠚⠀⠓⠒⠃⠘⠒⠃⠀⠀⠀⠀⠀⠀⠛⠀⠘⠀⠳⠋⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠚⠁

⠀⣠⡶⠒⠒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⢒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⡆
⡾⠷⣃⣀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣖⢸⢰⠄⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡆⠐⡆⠀⠀⡇
⡇⢼⣏⢹⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣋⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠇⠰⠃⠀⠀⡇
⡇⠈⠛⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣹⣿⣿⣉⠉⠉⠀⠀⠀⠀⠀⠀⠼⠭⠭⠭⠭⠭⠭⠭⠭⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠀⠀⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠀⢀⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⢸⣿⣿⠿⠛⠻⣿⣿⣿⣿⠟⠛⠿⣿⣿⡇⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⡏⢰⣿⣧⠸⣿⣿⠃⣼⣿⡆⢹⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠀⠀⢸⣿⡇⢸⣿⣿⢀⣿⣿⠀⣿⣿⡇⢸⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⣀⣀⣈⣉⣁⣈⣉⣉⣀⣉⣉⣀⣉⣉⣁⣈⣉⣁⣀⣀⣉⣉⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⢉⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⡉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⡿⠋⣉⣤⣤⣈⠙⢿⣿⣿⣿⣿⣿⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⡏⢠⣾⣿⣿⣿⣿⣷⡄⢹⣿⣿⣿⣿⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣉⣉⣀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣀⣉⣉⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠀⠈⠀⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣀⡀⢀⠀⠀⠀⡀⠀⢀⡀⢀⣀⠀⣀⡀⢀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠒⠁⢈⠀⡐⢸⠀⡄⠠⠭⡄⠁⢢⠀⡇⠀⡂⠂⠰⠠⠁⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡠⠒⠁⠉⠄⢂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠐⠄⠀⣀⠐⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠀⠀⠀⡐⠂⠀⢠⠤⠀⠠⠰⢦⠀⠔⡰⠃⠀⠄⢠⡠⢄⠀⠀⠀⠀⠀⡀⠀⠀⠈⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠞⢧⣄⠀⠊⠒⠂⠐⠁⠂⠀⠃⠓⠂⠰⠂⠴⠡⡀⢐⠩⠕⠂⠀⢱⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠒⢄⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠉⠃⠰⠶⠚⠂⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⡀⠀⠀⡠⠂⠤⠄⠀⠣⡀⠄⢀⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⢂⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⢄⣐⡀⠀⠤⣀⠤⠀⠀⠀⣶⠂⢸⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠈⢆⠀⠢⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠐⡀⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠁⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠊⠀⡀⠈⢀
⠀⠀⠀⠀⠀⠀⡀⠀⢠⣀⣀⣀⣀⠀⠀⠀⠀⠳⣄⠀⠀⠑⢤⣤⠀⢔⠠⡤⢀⣇⣀⣀⣠⣀⡀⠔⠈⠠⡀⠀⠀⣂⠅
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠉⠀⠀⠀⠀⠦⠤⠀⠓⠉⠉⢀⠀⡠⠆⠁⠀⠉
⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⡐⢀⢀⠀⠀⣀⢀⡀⠀⠀⠀⠄⡈⢒⠠⢀⠄⢀⠔⠀⠂⠔⣀⣡⡔⠊
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠒⠒⠂⠀⠀⠀⢸⠀⠔⡁⠔⡡⢒⠡⠂⢸⠉⠁⠀⡯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢊⠠⢊⠔⡁⠒⡁⢸⠀⠀⠀⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠐⡡⠊⡠⢊⠄⠈⡄⠄⠀⢽⠇
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢰⠀⢊⠔⡡⠔⡡⢂⠀⠀⠠⠀⣺⠅
⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠈⠤⠤⠤⠀⠤⠤⠀⠄⠀⠀⠀⠣⠀⢊⠔⢃⠔⡡⠀⠀⡀⠀⣺⡇
⠀⠀⠀⠀⡂⢁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠃⠠⢀⠁⠊⠐⡠⠀⠠⠀⣻⡇
⠀⠀⠀⠀⠘⡀⠠⠤⠤⠈⠡⣤⣤⣤⣭⣭⣥⣤⡄⡤⡠⠤⠤⢼⠀⠁⠂⠈⠀⠀⠀⠀⢻⣙⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠒⠒⠒⠒⠒⡖⠶⠚⠖⠒⠗⠓⠓⠟⠃⠀⢀⡠⠘⡄⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⡤⠤⠤⠤⠤⠧⢤⣄⣄⣁⣄⣄⣀⣍⣈⡀⠡⢂⠄⢘⡀⠁⡀⠈
⠀⠀⠀⠀⠀⠀⠴⡁⠀⠐⡒⣚⣶⣻⣿⣞⡥⠥⠫⠯⠽⡼⣽⣯⣫⡝⡀⠁⡊⢀⠁⠈⢀⠀⠈⡄
⠀⠀⠀⠀⠀⠸⠦⣄⢀⣒⣒⣒⣶⣶⣶⣲⣒⣖⣻⣻⣛⢟⣟⣟⣟⣿⣷⡀⠂⠔⠀⣧⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠈⠀⠉⠀⠁⢨⡅⠀⠀⠀⣩⡦⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠲⠛⠋⠙⠑⠂⠔⠊⠁⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢑⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢀⡼⢁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⢀⠎⠀⠋
⢀⡴⠂⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠒⠒⠒⠒⠲⣄⡴⠒⠒⠒⠒⠂⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠐⢦⡀⢀⡼⠁⠀⡠⠃
⢸⠀⣴⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠒⠂⠀⠐⣻⠓⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠂⠒⢲⢂⠀⣧⠞⠀⢠⡞
⢸⠀⡇⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠠⠆⢀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣼⢾⡴⠋⠀⣴⢋⡴
⢸⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⣒⡚⠋⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡀⣸⠞⠁⢀⡾⡱⠋
⢸⡄⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⠤⠤⠤⣴⠏⠀⣠⡏⠀⠋
⢸⡇⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⡾⠁⠀⡼⠋⠇
⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⡀⠀⠀⠀⠀⣠⠊⠀⢀⢼⡇⠀⡇
⢸⡇⠸⡇⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠅⠀⡠⠃⢸⠇⠀⡇
⢸⡇⢠⡁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢀⡶⠀⠀⣴⠄⠠⡇
⢸⡇⠠⣧⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠖⠋⠀⠀⠀⣻⡇⠘⠇
⢸⡇⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣿⠂⢠⡆
⢸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇
⢸⡃⠈⣦⡀⠀⠀⣀⣀⣀⣀⢀⣤⡄⠠⣤⠄⠠⠤⣼⣁⣤⠄⠠⣤⠀⠀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣿⠆⠈⠃
⠀⠳⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⡤⠎⠱⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠜




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐