Blacksmith Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢶⣿⣟⣻⣟⣫⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⣴⡿⠟⢛⣿⠟⠻⣿⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⡾⠃⣾⣿⣠⣿⡿⣶⣟⣑⢂⣶⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠋⢰⡿⠀⣸⣿⣿⣇⣸⣷⣼⣿⣿⢿⣿⣿⠟⢙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣀⣀⡀⢀⣏⣠⣼⠇⢠⣿⣿⣿⣿⣿⡿⢿⣏⠉⠚⣿⣧⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠂⠉⠉⠻⣿⡟⠉⡉⠉⠹⣿⣿⠃⠀⠈⣿⣿⣿⣿⣿⡟⣼⣿⣤⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠉⠀⠀⠀⠀⠀⢰⣿⣇⠀⠃⠀⢀⣿⠋⠀⢀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⢷⡀⠘⣯⣀⣠⠈⣶⡐⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⣿⡏⠀⠀⠈⠠⡈⠻⣟⠀⠀⢛⠂⠈⠙⠛⢿⣿⣿⣿⣿⡿⠛⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡸⢫⡆⠀⠀⠀⠀⢓⡀⠉⣶⣤⡈⠠⢿⣦⣦⣤⣤⣤⣸⠟⢳⠆⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠐⡄⠀⠀⢻⣄⠀⠀⢠⣇⣾⠃⠀⠀⠀⠀⠈⢳⠀⣿⡟⠉⠒⢂⢤⡌⢻⣋⣿⣿⣃⣤⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡁⠀⠀⣻⣯⠀⢰⠾⠟⣿⡆⠀⠀⣼⠂⠀⠀⠂⢿⣿⡾⢦⣽⣦⣀⣠⣿⠉⠉⠉⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⢿⡿⠀⠸⢀⣼⣿⡇⠘⠈⠁⠀⠀⠀⠀⠈⠻⣿⣶⣦⣾⣿⡿⠃⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⢻⡇⠀⡠⢾⣿⣿⠁⠈⠰⠆⠠⠀⢀⣀⠀⠀⠈⠉⠻⣟⣿⠇⠀⠀⠀⠀⠀⣤⠿⣤⠃
⠀⠀⠀⠀⠀⡴⠋⠀⢠⠀⠀⠀⠀⠀⢘⠀⠀⠀⠸⠀⠀⠊⢀⣴⣩⠁⠚⠁⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣼⣿⢃⡰⢀⠀⠀⠀⣾⠋⣰⠿⠆
⠀⠀⠀⢀⠘⠀⠀⠀⠈⠁⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢀⣴⡟⠼⢇⣀⠰⠦⠈⡄⠀⢀⡠⠟⠁⠀⠀⣼⣿⠉⢆⣱⠆⠀⠀⠘⠋⠀⠛⠀⠰
⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⢋⠀⠛⠁⠸⢍⣀⠶⠂⢀⠀⠀⣀⠀⣀⠀⣸⣿⠁⠰⠞⠁⣤⠀⠀⠀⠀⠂
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠁⣴⠀⠈⣂⢤⣟⡐⠊⠁⢠⣤⡈⣀⡀⢉⡴⠀⣼⡿⠋⢠⡾⢧⣾⠏⠀⣤⠀⢀⠀⠀⠀⠀⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⢀⠘⠁⣤⣈⠩⠔⢋⡀⢠⡾⠋⠛⡁⠉⠹⣧⠀⢰⡿⠇⢰⡿⠑⠏⠁⠀⣸⠁⠀⡾⠀⠀⠀⠀⢰⣄
⡇⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⠿⢷⡋⠀⠸⣧⠰⠀⡖⠉⠠⠀⠈⠀⣯⣅⠀⠀⠻⣇⢰⣿⡇⠘⢣⡶⠋⠀⠀⢀⡀⢨⡏⠀⠘⠃⢀⡜⠀⢠⠀⠙
⠹⣆⠠⠄⠀⠀⠀⠀⠀⢀⡄⠀⠈⠙⢲⣄⡀⠀⠀⠸⠟⠀⢠⣠⣦⠙⢿⣄⡄⠀⠘⣿⣟⠻⣶⣇⠀⠀⠀⠀⠀⣧⡄⠀⠂⡄⠀⠀⠀⠀⠋⢀⣤
⠀⠙⢦⣀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠐⠆⠙⠉⠁⠀⣠⣿⣿⣷⠀⠈⢿⠀⠀⠛⢤⡀⠀⠀⠀⢹⠁⠰⠀⠀⠀⠀⠀⠀⢀⡘⣿
⠀⠀⠀⠙⠻⢠⡶⠏⠁⠀⠀⠀⠀⠀⠀⠀⡀⠤⠴⢀⣠⠴⠋⠁⠠⠊⢠⣿⠻⣷⡀⠠⠀⠃⠰⢤⢈⠃⠀⠀⠀⠀⣶⠀⠁⠀⠀⠀⠀⠀⠀⠑⣸
⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⢀⠉⠀⠀⢀⣀⠀⠛⢣⠀⠘⢿⣀⢀⣤⣶⡶⠋⠀⠀⠀⠀⠀⠀⡜⣡⡄⠀⢀⣀⣀⠀⠀⡏
⠀⠀⠀⠀⠀⡌⠙⣷⣐⣄⣀⡴⢈⡵⠲⣤⣦⠀⠀⠀⠀⠀⢠⡈⠻⣆⡀⠀⢿⣠⠂⡉⢙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠓⠒⠉⠉⠁⠀⠀⢰
⠀⠀⠀⠀⠀⡇⠀⠹⣯⠉⣥⣤⣹⣷⡴⢧⡜⠣⣤⡀⠀⢷⡈⠳⣄⠈⠁⠀⠘⠃⠀⢹⠋⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠒⠀⠙⠉⠉⡀⠊⢁⡔⠈⢻⡶⠀⠱⠖⠁⠀⠂⣤⡎⢡⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⣠⣦⣴⢶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⡞⢀⡴⠋⣰⣾⠋⠀⢀⣄⠀⠀⣀⠞⣣⣴⡏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⠋⠀⠀⢰⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡁⡀⢠⡄⠈⠵⠋⠀⣼⠛⠁⠀⠀⠀⠙⢀⣰⠟⢠⣿⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠱⠀⠀⢁⡄⠀⣠⡟⠁⠀⠀⠀⠀⠀⢠⠼⣧⡴⠟⢙⣠⣀⢰⡞⠃⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⣏⣠⣾⠋⠀⠀⠀⠀⠀⠀⠰⠃⠘⠟⣥⣟⣿⡿⠋⠠⢹⡀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⢠⡀⠛⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃⠀⠙⣿⡿⣷⣟⣶⠊⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢷⣤⠀⢹⠀⢠⣤⡁
⠀⠀⠀⠀⣤⠀⠀⠆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠛⠟⣛⢀⣾⠋⣾⣿⡟⣥⢋⡰⠲⠄⠀⠀⣷⠀⠀⠀⢀⣠⣴⣦⣾⣭⡆⢸⣂⣀⡙⠇
⠀⠀⠀⠀⡉⠀⠀⢰⡆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢭⠉⠀⠀⠉⠻⠿⠟⠀⠑⠂⢀⣠⡝⠒⠈⡻⠿⠿⠛⢿⣭⡽⠷⠚⠉⠉
⠀⠀⠀⢨⠁⠀⠀⠀⢱⠀⠀⠀⠀⢉⣤⡀⣀⣠⣤⣀⣀⣀⣀⣤⣼⣤⣤⡀⠀⠀⠀⣀⣀⣤⡙⣿⣷⡾⠟⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠘⣆⠀⣾⣿⣷⣭⣼⣍⣠⣍⣈⣉⣉⣉⣩⣤⣄⣉⣁⣠⣶⣦⣭⣿⣿⡿⠿⠟⢻⡆⠀⠀⠀⠒⠀⠀⠒⠐⠚⠂⠀⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠙⠋⠉⠉⠉⠛⠉⠛⠛⠋⠙⠋⢻⠶⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢆⡠⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠳⣄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠀⠀⠀⠀⠀⠀⢀⡀⢄⡴⢤⣚⠿⠋
⠀⠀⠀⠀⠀⡸⠇⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⣀⠀⣤⣶⣾⣾⣧⡽⠿⠛⠊⠁
⠀⠀⠄⠀⢠⡅⢰⠀⠁⠀⠀⠀⠄⠀⠉⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡿⠚⠿⠋⠉⠉
⠀⠀⢀⡀⠸⣷⡄⠐⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣤⡄⢠⣄⠛
⠀⢠⠀⠘⠆⠁⣿⠀⠀⠀⠐⣀⠇⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢶⡀⠀⠀⢰⡿⣼⣷⣷⣀⠻⣇
⠀⠸⠀⠀⢠⡆⠀⣹⣦⠀⠀⠈⡄⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⢸⣷⣿⡿⣿⢿⣶⣉⡙
⠀⢸⡆⠀⠀⣸⢧⡛⣷⣄⠀⢀⠀⠀⠀⠀⠀⠀⣤⠈⠀⠰⠆⠀⠀⠀⠀⠀⣰⠀⠀⠀⢻⣿⣾⠷⠚⠛⠋⠛⠀⣀
⠀⢸⠀⠀⠀⣠⡾⠁⠘⠿⡠⠈⠁⠀⠀⢀⣦⣜⣻⣧⣀⡰⢀⣴⣤⣀⣤⣈⣿⠀⠀⠀⠚⠋⢀⣤⣶⣶⠿⠟⠋⠉⠀⠂
⠀⢸⠇⠀⠀⠟⠳⡀⠀⠠⠃⠀⠀⢀⠻⣤⡈⠉⠉⠉⠉⠁⠈⠀⠉⠙⠛⠛⠛⠃⣴⡄⠀⠾⠿⠋⠉⠁⠀⣀⡀⣴⣿⣀⠀⡀
⠀⠈⠀⡀⠀⠀⠀⢡⡌⠀⠀⣶⡄⠺⠧⣞⠉⠓⠒⠲⠄⠀⠀⣤⣤⠤⣤⣤⠀⠀⠠⠶⠆⠠⠄⠀⠀⠈⠈⢁⡘⢿⣿⣿⣴⡇
⠀⢀⠐⢧⠀⠀⠀⠳⠀⢳⠄⠛⠀⣀⠀⠙⠀⠀⠐⠂⠀⢀⣀⠀⠀⣀⣠⣄⣄⣀⣀⠀⣀⠀⣠⣤⣤⣈⠑⢾⣷⣾⣿⡿⢧⡇
⠀⠃⠀⠀⢠⣄⠀⠀⠚⠋⠁⠀⠀⠻⠀⢠⣿⠐⣤⣦⣌⠈⢁⣤⠶⠟⠻⡉⢿⣉⣿⣿⣿⣡⠿⣿⣷⢿⡄⠀⢹⡏⣿⣎⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢿⣿⣿⣿⡿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣳⣎⣭⡿⢉⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡷⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠛⠿⣿⡿⣿⠱⣤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⡟⠓⠄⣀⡔⠉⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⠖⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⣭⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⣾⣿⡟⠁⠀⠀⣸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⢱⣿⣿⠁⠀⠀⢠⡿⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣬⣻⣏⠀⠀⠠⢏⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣀⢀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣿⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⣀⣠⣼⠿⠻⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⠛⠃⣀⣴⣿⣶⠦⠀⠐⠚⠛⠋⢁⣀⣀⣝⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⠿⣿⣿⣿⣿⢿⣿⡆⠸⠋⠉⠉⠀⠀⠀⠀⢀⠤⠒⠉⡝⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡞⣹⣿⣿⣿⣄⠈⠙⠼⠿⣦⣹⣧⡀⠀⠀⠀⣀⣤⠒⠉⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⠆⠁⠀⠀⠀⠈⠉⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣼⠀⡞⠻⠟⢿⣿⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⣧⡷⢨⣿⣿⣿⡏⠀⠀⠀⢀⣤⡾⣿⠀⠀⠀⢠⣤⣤⣤⢤⣀⣠⣤⠦⡄
⠀⠀⠀⠀⢄⡾⠏⡑⢺⣽⣶⣿⠁⠀⠀⢠⡾⠋⢰⡏⠀⠀⠀⠀⠉⠉⠁⠠⠿⠁⠛⠛⠛⠛⠛⠛⠿⠷⠶⠶⠦⠦⠤⠤⠄
⠀⠀⠀⠀⠁⠀⢸⠇⣙⣿⣿⠇⠀⠀⢀⠚⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣤⣤⣤⣤⣴⣾⣴⣤⡤
⠀⠀⠀⠀⠀⠀⠸⣀⣿⣿⡟⠀⠀⠀⠀⠀⠀⢯⣆⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠹⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣟⣔⣤⠄⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠈⠓⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣿⡃
⠀⠀⠀⠀⠀⠀⠀⢨⣬⡿⠀⠀⠀⠀⠀⠀⠀⡠⢿⣿⣾⣷⠶⡀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⣺⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⡟⣉⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢯⣼⣧⡀⠂⠀⣠⠎⠀⠀⠀⣺⢿⡿⢫⣿⣿⣿⡿⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡮⠛⠉
⠀⠀⠀⠀⠀⠀⣀⢾⠷⠿⡿⢷⣾⣋⡀⠀⢀⠤⠐⠂⠀⠈⣷⢿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣤⣄⣤⡤
⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣯⣾⣿⣿⡿⠿⠯⠥
⠀⠀⠀⢘⣁⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣄⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⣿⣿⡿⠛⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿⣿⣿⡆⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡠⠾⣷⣿⣿⣿⣿⣏⠑⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡟⠓⠎⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⣿⣿⣿⡟⢿⣿⡿⢾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢈⣿⣿⣿⣿⣿⣇⠈⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠠⣾⣿⣿⣿⣿⣿⣿⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⢦
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡛⠀⡵
⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠈⠋
⠀⠀⠀⠀⠀⠀⢸⠁⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣤⣤⣤⣄⣀⣠⣤⣤⣼⡟⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣮⡳⣄⠀⠀⠀⠀⠀⠀⠠⡿⠿⠿⠋
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢧⠀⠉⠛⠿⣿⣿⣷⣵⣦⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⢧⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣦⣤⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⣧⠀⠀⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣏⡉⢉⣉⣉⣱⠶⣿⣶⣾⣿⣿⣿⣶⡦
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣧⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠉⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⣸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⢀⣴⣾⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠿⣟⢹⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡙⢂⠀⠀⠀⠀⠀⣀⣈⣷⠀⠈⣸⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠓⢢⠑⢄⠀⠀⠀⠈⣹⠿⣿⠞⠁⠈⢈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢱⣤⠔⠋⠀⢺⠏⠀⠀⢀⡼⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢮⣿⣀⣤⣴⡟⠀⢠⣠⣿⣷⣄⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⡁⠀⢸⣿⣿⣿⣿⡆⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⢀⡔⠁⣀⠴⠟⠁⠸⣿⣿⣿⣷⣠⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⢊⡴⠊⠀⠀⠀⠀⢀⣴⣾⣿⡏⣿⠇⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢠⡎⠀⣸⡥⠀⠂⠀⠀⣠⣿⣿⣿⣿⢱⣿⣦⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡀⢀⡀⣀⣀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⢣⡿⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⡦⢬⣭⡝⣵⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⠞⣲⠀⠀⠀⠀⣴⣿⣿⣿⣿⢏⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋⠀⠀⠀⣼⣿⣿⣿⣿⢧⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠙⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⣼⣏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⢠⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡁⣿⡿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣆⠀⠑⢤⣀⠀⠀⠀⠀⢿⣿⣿⠿⣱⡇⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣱⣿⡿⠿⠛⠿⢿⣿⣿⣷⣤⣀⣠⠝⣳⠀⠀⠀⢸⣾⣿⣿⡇⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠓⠋⠀⠀⠀⠀⠸⢿⣿⣿⠇⠀⣶⣶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠀⠀⠀⠈⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


⡿⠿⠿⢿⠿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⣿⣿⢿⠿⣿⢿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⡄⣤⢤⣤⣤⢤⠀⠀⢤⣤⣤⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣾⣽⣼⣧⣭⣬⣤⣿⣼⣥⣧⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢹⣿⡉⡏⠀⢹⣿⡯⣶⣿⣾⣿⡩⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡟⠙⠋⣼⡏⠉⢉⣩⣉⣙⠛⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣻⣿⣿⠇⢀⣜⡿⠁⣴⣿⣿⣿⣿⣇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣿⢰⣿⡿⠇⢀⣹⣟⣁⡉⣿⣿⣴⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣾⣿⣰⣿⣿⣾⣿⣌⠁⣰⣧⣾⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣯⡄⠘⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢡⣿⣿⣇⠈⠛⠛⠋⠀⢿⣿⣿⣿⣏⣿⣿⣿⡶⣀⣘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠐⢀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠜⢋⣿⣯⣿⣾⣿⣤⣄⡀⠘⢻⣿⣿⣿⣿⣿⣿⡿⣀⣼⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⢨⣿⣿⣯⠡⢹⣧⣀⣸⣿⡟⠛⢉⣽⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⣾⡇⠈⣿⣿⣿⣶⣬⢻⣿⣿⣿⣷⣶⣿⣿⡧⣺⠋⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣿⣿⣭⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡇⡿⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡛⢿⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣽⡇⠂⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⢰⣀⣼⣿⣿⣿⣹⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣰⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⠿⠿⠿⢿⢿⢿⣿⠿⠒⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⢹⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⣿⣿⠂⣿⣿⢿⣿⢿⣠⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠋⡇⠀⣿⣿⣿⣿⣿⣹⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣿⣿⠿⡟⠿⣹⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⣿⣽⣿⣦⣤⡼⣿⣶⣧⣤⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠿⠟⢸⣿⣿⣹⣿⣿⣿⣿⣿⣿⣷⣿⢿⣿⠿⠛⡿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠘⠟⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠈⠁⠉⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⢿⣿⣿⣿⡟⠋⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣟⣶⣬⣻⠛⠟⢁⠠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠛⣹⡿⠀⠀⠙⠃⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠐⣿⡟⣆⠈⢑⡞⠉
⠀⠀⠀⠀⠀⣠⠖⠛⠁⠉⠒⢄⡀⠹⠳⣾⣻⣯⡿⠆
⠀⠀⠀⠀⣼⡃⠀⠄⠀⠈⠃⠄⡈⠐⠀⠈⣷⣤
⠀⠀⠀⠀⠿⢀⣀⠀⠀⠀⢀⡄⢻⡀⠀⠀⢿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠙⠢⠄
⠀⠀⠀⢠⠖⣉⣥⣤⣄⢻⡄⢙⡟⠃⠀⠀⢠⠃⠰⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⢾⡿⠓⠀⠀⠀⠀⢿⡦⠀⠀⠀⠀⠠⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠻⠿⣦⣄⡀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠳⢄⡀
⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣄⣀⠀⣈⠓⠄⠀⠀⠀⣠⢟⣿⠏
⠀⠀⠀⠀⠐⠋⢈⠟⣷⠢⢄⠀⠀⠀⠀⠀⠀⠀⠁⠲⠽⠄⠄⠓⢦⢞⣡⠟⠁
⠀⠀⠀⠀⠀⢀⣾⠀⠁⣷⡘⠙⠒⠦⠤⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣰⡏⠀⠂⠄
⠀⠀⠀⠀⠀⣼⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣛⠳⣦⠔⠋⠙⠦⣐⠶⠷⡆
⠀⠀⠀⠀⠀⢿⡿⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠁⠀⠀⠀⢾⡿⠿⠿⠛⢿⡿⠃⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠳⣶⢊⣛⡛⣛⣧⡔⢸⣤⠀⣿⠶⠶⠋
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠹⠎⢹⣧⠉⣠⡷⠈⠀⡀⡆
⠀⠀⠀⠀⠀⠘⠦⠤⣀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡇⢹⣁⠀⢰⣄⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠃⠀⠂⠴⠀⠼⠿⠢⡀⢻⣷⣄⣵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⠀⠧⠀⠀⠀⠀⠠⢤⡀⣀⠐⢦⣠⣟⣛⣙⣳⣴⣴⣦⣄⣤⣴⣴⣿⠾⣃⢩⢏⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣷⣿⣿⣉⣉⣉⣩⡝⣋⢩⣉⣿⣷⣾⡇⢘⣤⢸⣾⡇⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣛⠿⣿⣿⡄⠀⠀⠀⠀⠈⠙⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⢿⣿⣯⡛⠘⢸⢸⡿⠃⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣭⣥⣶⠶⣾⣧⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⢸⠛⢻⠀⣾⣿⣿⣿⣿⣧⠹⣷⡽⡇⠀⡀⠈⢠⠟⡇
⠀⠀⠀⠀⠀⠀⠸⣯⣤⣭⡉⢹⠻⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⡌⠀⠃⠛⢿⡿⣿⢆⢠⠀⢇⣴⡀⠇⢀⣼⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠘⡇⠠⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣄⢧⠇⠀⢰⠀⠈⠿⣎⠰⣹⣷⣿⢿⡇⠀⠠⡟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⡯⠥⢼⡇⣸⣿⣿⣿⣿⠀⠀⠀⠀⣀⡤⠄⢿⡘⣤⠀⠀⢀⢂⡀⠙⣆⠹⣿⣿⢆⡇⠀⠰⡇⢠⣿
⠀⠀⠀⠀⠀⢀⠄⣸⣿⣿⣼⣷⣍⣛⠻⠛⠟⠉⡙⢠⡅⠀⡀⠀⠀⠳⣹⠄⠀⠀⠀⢳⡄⠈⠀⢻⠸⡞⡇⡇⠘⢃⢸⡟
⠀⠀⠀⠀⠀⠀⢠⣿⡿⢿⣿⢿⡿⠼⢿⣷⣴⣶⣦⣤⡶⠖⠃⢤⡄⢠⠑⣄⠀⠀⠀⢁⢿⡄⡀⠘⢂⠘⡇⢡⠀⠂⠸⢸⡒⠖⠀⠄
⠀⠀⠀⠀⠀⠀⢿⣯⣄⣉⡀⠸⠇⠀⠰⠋⢀⢽⠗⠉⢳⣶⣾⡷⠠⣄⠣⠈⠷⠀⠀⠈⠀⠀⠰⡠⠈⠓⢰⠀⣀⢀⣾⣯⣿⣛⠃
⠀⠀⠀⠀⠀⠀⠀⣉⡬⠿⠛⢿⣷⣶⣶⡶⠷⠟⠁⠀⢠⣭⣿⠃⠙⣦⣖⣠⠀⠀⠤⢠⣤⣤⣤⡴⢷⣤⣤⡼⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠎⠉⠩⠉⠁⠀⠀⠀⣠⠾⠩⠿⠿⠿⣿⢈⡃⠘⠛⢛⡿⠷⠿⠿⠿⠿⠟⠛⠻⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⡞⠛⠀⠀⠀⣀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣓⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡇⠠⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣖⠰⠲⠒⠒⠒⠛⠋⠉⠉⠉⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡛⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣀⠙⢉⣿⠋⠀⠀⠀⢀⠀⠀⣴⠉⠆⠀⠙⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠙⢁⣀⣠⠤⠾⠾⠒⠒⣿⠀⠐⠀⠈⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⢠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠠⡆⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⢰⠘⢀⡾⢿⠛⠁⣷⠒⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠥⡜⡤⢻⣦⣌⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠄⠈⢻⣿⣶⠿⡥⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⢸⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡇⠘⣰⡏⢿⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠂⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣹⠀⣇⢲⡟⠀⣼⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠂⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⠿⣾⠁⠀⣿⢦⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⣸⠃⠀⠀⣿⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣸⣉⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣗⣾⠁⠆⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡀⢀⣯⣩⡿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⣾⠁⢀⡇⠀⠀⠀⠀⠀⠀⢀⣤⢤⣶⠟⠊⠉⠉⠁⠀⠉⠉⢃⣀⣾⣿⡟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⠙⠉⢰⡇⠀⠀⢀⠀⣀⣀⢻⣿⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⢿⣿⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠉⢈⣿⡀⠆⠀⠀⠀⣀⣀⣀⣀⡄⠀⠀⠸⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠆⢚⡅⣠⠟⠋⠙⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⣶⣞⣡⠾⠁⠀⠀⠀⠀⠙⠿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣫⣤⣀⣵⣛⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢄⣀⣀⣬⢟⣥⣘⠻⠄⣉⡉⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠋⢠⣿⣥⣙⠶⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠈⢦⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠐⢶⣾⣉⣉⣉⡉⠒⠤⣄⠀⢀⡠⠤⠤⢄⠀⠀⠀⠀⣀⡴⢟⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣆⠀⠀⠀⠀⠀⠄⠀⠀⠈⢆⠀⠀⠀⠉⠛⠲⡾⢋⡉⢱⠂⠙⠲⣑⣀⣴⣞⠁⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣉⣃⠀⠀⠀⠀⠀⠀⠀⠑⡀⢫⣭⣍⣉⠭⠿⢋⣨⣵⡾⢉⣉⣽⣿⠿⠿⠯⠉⠉⠉⠐⠒⠒⢖⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠉⠁⠀⠀⠀⠀⢀⡠⢀⠤⠀⠈⢆⠰⠤⣀⡤⠴⠚⠋⠁⢈⣿⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣆⡀⠤⠀⠀⡀⣸⠀⠛⠃⣸⠒⠒⠚⠒⠒⠒⣲⠶⠛⠓⠒⠒⠒⠂⠐⠂⠀⠀⠀⠀⠈⠉⠩⠉⣭⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣦⣦⣧⡾⢰⣄⢠⣿⣀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠹⠿⣷⣿⣿⣿⣿⣿⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⢀⡀⠴⢾⣲⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⢺⣷⣿⡄⢥⢠⠈⡀⠀⣤⢐⡀⡤⢈⡡⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡸⠿⢿⢿⡤⣬⣷⠟⠀⠛⠸⣷⣡⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⠗⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠙⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣿⡇⠀⢀⣤⣶⣾⡟⠙⠉⢹⡛⢻⣶⣶⣤⣂⠀⠈⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⡶⠶⣾⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣷⣦⣤⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⣿⠛⡿⠿⣿⠿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⣄⣦
⠀⠀⢸⡋⣼⠟⠀⠀⠀⠀⠀⠄
⠀⠀⣿⣇⣿⠁⠉⠉⠑⠳⣴⣮⣭⣅⣤
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣟⣙⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠛⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣷⣿⠏⠰⢤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢴⣿⠉⠉⠐⢤⣤⣴⣄⣤⠤⣞⣿⢟⣿⣿⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⣻⣿⡟⠻⠓⠈⠳⣜⢃⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⠀⢻⡔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣀⠀⠀⠀⢠⣾⡇⠀⠀⠀⢀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣿⣿⣿⣷⣴⣶⡿⢻⣷⣀⣀⣠⣿⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣟⠧⠀⠀⠒⠓⠀⠭⠉⢻⠋⣿⣆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⣿⡗⢀⣀⠚⠉⠀⢀⠀⠀⠀⠘⢿⣷⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢿⣬⡛⠀⠀⠐⠉⢻⣧⠃⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡲⢽⡁⠀⣨⣏⠁⢘⡏⠀⠀⠀⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣧⣬⣘⠃⠀⠀⣏⣡⢼⡇⠀⠀⠀⠀⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣿⣿⡿⠭⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠐⢦⡈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠘⣹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠸⠇⣾⢿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢨⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⠻⠷⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡟⡁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠋⡦
⠀⠀⢠⣶⣦⡤⠴⠖⠋⣩⠃⡀⠁⠀⠀⢀⣠⣤⠄
⠀⠀⠘⣿⣷⡀⠀⢀⡼⠃⣰⠀⠀⠀⣠⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠈⠉⠀⢀⣿⡅⠀⠇⣀⣀⣀⢘⣿⣿⠿⢷⡖
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣸⢹⡻⡿⣿⡿⢿⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⡁⠊⠁⣷⠈⠧⠈⢧⠩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣼⣟⣧⠀⠀⡄⠀⣇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣿⢷⡀⢻⠀⠹⣿⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣮⣳⡘⣇⠀⣹⣇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⢹⣷⣴⠞⠃⠙⢦⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢣⡟⠉⢀⠙⠈⡵⢤⣤⣷⣾⣂⠀⠀⠀⠀⠀⠀⢀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣼⣷⣼⢦⡁⠀⠁⠀⠀⠉⠉⢙⣛⠷⣶⡿⣿⣿⣯⣤⡤⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⡆⡀⠙⢧⡇⠀⠀⠀⠀⢈⡹⠿⣿⣿⣿⣿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⠌⢣⠀⠀⠃⠀⠀⣀⣠⣬⣶⣿⣿⠟⠀⠀⢀⣨⣝⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡌⠀⠈⠀⠀⢠⠀⠀⣿⣿⣾⣿⣿⠇⠒⣚⣛⣛⣛⣛⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡖⠀⠀⡸⠀⠀⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠅⠠⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣇⠀⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠹⣿⡄⠀⠳⠀⠀⢸⣿⣿⣿⡿⣿⠀⢰⣷
⠀⠀⠀⡀⢄⣀⣶⣿⠦⠂⠀⣿⣿⣿⡇⢠⣶⢽⣿⣤⡜⣀⣭⣼⣷⣿⣿⣷⡇⠀⢸⣿⣇⠀⠀⠀⠀⠀⢸⠀⠀⡀
⠀⠀⠀⠠⠐⠉⢭⣽⣿⣾⣾⣿⡿⣋⣧⠘⠿⣾⣿⢟⡌⢹⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⢿⡄⡁⠀⠀⠀⢸⠒⠊⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⣿⣶⣄⡀⢸⣿⣷⠙⠿⢿⣿⣿⣿⣿⣷⠀⠀⠈⡘⣧⣡⣤⡶⠤⠌⠈⠐⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠲⠒⢛⣛⣿⣿⡛⠛⠻⣿⣷⣦⠼⠏⢉⣛⢋⡉⢙⠉⣡⣟⡒⣶⠀⢀⣀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⣀⣀⣈⠀⠀⠠⠤⠄⠀⠚⠓⠉⠃⠀⠉⠙⠛⠛⠻⠿⠛⠓⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⣿⣻⣟⣻⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⡶⠐⡠⠶⠚⠛⠉⠉⠉⠉⠉⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠸⢷⢖⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⢀⠄⠀⠀⠀⠘⠋⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠰⢅⡀⠀⠀⠀⠀⢠⣿⣿⡿⠛⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢧⣀⠀⠀⠈⠢⣀⡀⠀⢸⠿⡈⠁⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠙⠷⣤⣴⡆⠀⢻⣿⠁⠀⢣⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⠉⠃⠀⠘⣿⡄⠀⠈⣷⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⡇⠀⢧⢠⢿⡇⠀⠀⣿⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡟⣼⣷⣾⣾⣿⣷⡄⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣸⣀⡠⢊⣼⣿⣿⣿⣿⣿⠙⢦⡀⠈⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⣿⠏⠨⣽⣿⣿⢟⡿⢡⠏⠀⠀⠉⠳⢤⣿⣿⢟⠦⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢠⣿⣶⣾⣿⣿⣾⣿⣶⡇⠀⠀⠀⠀⠀⠀⠉⠩⠟⢶⡷⠦⣖⣲⣶⣶⡶⢤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠐⠚⠉⣶⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠿⢟⠯⠓⠜⣧⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠈⣧⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣷⣦⣼⡄⢀⣀⣠⠴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡎⠁⠘⠉⢳⡀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⢰⣄⣉⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡏⠀⠉⠁⠀⣴⢿⣿⣻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢠⡇⠀⠀⠀⠀⢸⡘⠏⠀⢤⠤⠀⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠸⡧⢤⠶⠶⠾⠏⠉⠀⠀⣈⣄⣠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠉⠙⠛⠈⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡂⠔⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⡆⠒⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡞⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠙⢦⡀⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠙⢦⣼⢀⠄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⢣⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢷⣦⣤⣁⣀⣀⢀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣴⣷⣶⣅⡉⠉⠉⠛⠛⠛⠛⢻⡋⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢳⠔⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⢉⣩⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣰⢇⡾⠀⠀⣠⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣡⠏⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠗⢲⡄⠀⠰⣿⠔⢨⣡⡼⠓⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣡⠾⣿⣿⡿⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣌⠀⠻⣤⣤⡤⠟⠀⣸⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠖⠉⠀⣿⣿⡇⠠⠀⠀⠀⠀⣰⠟⠁⠀⠀⠙⠢⢤⣀⣀⣀⠤⠞⠁⠀⠈⠛⠶⠶⠶⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠄⠀⠀⠈⠛⠛⢛⠛⢆⠒⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠄⠐⠀⢀⠀⠀⠀⠈⠐⢈⣷⠀⠀⠀⠀⠀⠀⠀⢠⣶⡖⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣦⣤⣌⣀⣀⣀⣀⡔⠉⠉⣁⣀⣀⣀⣀⣀⣀⠇⠈⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⣿⣿⡟⠀⠀⠀⠀⠉⠈⠁⠁⠉⠈⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠰⣿⣿⡇⠀⠀⠀⠀⣀⡴⠖⠲⢦⡄⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⣾⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢣⣀⡀⠀⣀⣀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡋⠙⠻⠀⠙⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⠀⠀⠄⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠒⡀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠁⣀⣴⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢛⡏⠙⠣⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⡇⠀⣿⠻⣿⡇⠀⢺⣁⣠⠆⠀⢀⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣋⣵⠀⣈⡀⠈⠁⠀⠀⠀⠀⠀⢠⣿⣿⡿⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⣴⣿⣿⡀⠈⠀⠀⠀⠀⠀⠀⠀⣴⣏⣥⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠐⠖⠁⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣶⣦⣴⣶⣶⣴⣿⣿⣿⣿⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⡄⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⡉⠙⠉⠿⢿⠟⢿⣿⠟⢿⣿⣿⠆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣰⣿⣶⣶⣶⣾⣤⣀⣴⣦⣼⣿⣏⠀⣿⣇⣀⣀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⡀⠀⠇⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠛⣿⣿⣿⣿⣿⣿⡧⠘⠁⠀⠀⠀⠘⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣶⠖⢂⣥⣴⣷⠀⣿⣿⣿⣿⣿⣿⣿⠁⢲⣧⠀⠀⠈⢉⣿⣉⣉⡀⠀⠀⡀⠀⠀⠀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠋⢰⡏⠀⠀⠉⠀⣻⣿⣿⣿⣿⣿⣿⣿⠏⣻⣶⣶⣶⣾⣿⣿⣿⣿⡌⠁⠀⢰⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⢿⣦⡀⣠⡀⠀⠈⠿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣦⣤⣿⠋⡻⠇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⠃⠀⠉⢶⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠏⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣄⣀⣰⣿⣿⣿⣿⣼⣿⣿⣿⣿⠟⢛⡁⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣼⣿⣿⣿⣿⡿⣿⣿⣿⠃⠀⢹⣿⣿⣿⣿⣿⣿⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⣿⣿⣿⣿⣿⡿⠀⢿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢀⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⠀⠀⠀⠹⣿⣿⣿⣿⣿⣦⠀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠠⠈⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢸⣿⣿⡿⠿⠛⠙⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⡇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣆⡀⠀⠇⠀⠄⠀⠀⣿⣿⣿⡆⣠⣦⠘⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣥⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣶⣤⣖⠀⠀⠀⢸⣿⣿⡇⣿⣿⣷⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣩⣟⠛⢿⣿⣷⣦⣀⡼⢿⣿⡇⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣾⣿⣿⣿⣿⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣶⣾⣿⣿⣿⣿⣿⣶⣿⣿⠶⠿⠿⠿⣷⡆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠋⣉⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠋⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠉⠉⡛⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣿⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⢸⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⣠⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⢀⢀⣰⣿⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢸⣿⣿⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣣⣮⣝⠿⢟⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢋⣭⣭⣤⣤⣤⣤⣤⣾⣥⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣛⣿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣛⠛⠛⠸⣿⣿⣿⣿⣿⣿⣿⣯⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣽⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣂⣀⣼⣿⣋⣿⠛⠙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢢⣿⠋⣼⣿⡞⢻⣟⣿⣤⣯⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⣸⡇⢰⣿⣿⣷⣾⣿⣿⡻⠋⢛⡛⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠛⠻⣿⣶⠚⠛⠒⣿⣿⡏⠀⢸⣿⣿⣿⣿⣧⣼⣧⣠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠋⠁⠀⠀⠀⠀⣼⣧⢸⡇⠀⣼⡏⠀⠠⠘⢿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⢧⡀⠻⣤⣤⠐⣦⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠈⢦⠙⠧⡀⠐⠃⣀⣀⡉⠛⠛⠻⣿⣿⠓⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢀⠀⠀⣼⡀⠀⠀⢨⣷⡏⠀⠀⠀⠈⣆⠠⣿⡷⠈⠀⠛⢻⡿⢿⣷⡤⢘⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣸⠀⠀⢰⡟⠀⢠⣿⣿⡇⠀⠀⡄⠀⠈⠀⣿⣧⣤⣰⣏⠃⢘⡿⠛⠒⠋⠉⠀⡄
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢺⠟⠀⠀⢿⡿⠀⠦⣠⣿⡇⠂⠘⠁⠀⠀⠀⠘⢿⣷⣭⣽⣿⡿⠃⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠘⠀⠀⠀⢻⡇⠀⠰⣿⡿⢃⡀⠒⠠⠤⠀⠤⠀⠀⠉⠙⣻⣿⠃⠀⠀⠀⢀⣰⣏⡷
⠀⠀⠀⠀⠐⠉⠀⡶⠀⠀⠀⠀⣿⠀⠀⠸⠀⠀⢁⡀⢧⠉⠀⠀⢀⠀⠀⠀⣀⡀⠀⠀⣼⡿⠁⠒⠤⠁⢀⣼⠃⣼⠃⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠁⡴⠻⡍⠛⡤⠊⣃⠙⠀⠀⠐⠋⠀⠀⣸⡿⠃⢠⠞⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⢀⡀⢠⡀⣠⡄⡱⠆⢉⠀⠀⠀⢸⡀⢁⣰⣿⠁⣨⣇⣴⠇⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣄⠀⠀⠂⡠⠊⠁⣈⢇⠿⢃⢀⣴⠚⠿⠒⠲⣦⠁⢸⡿⠀⣼⠫⡜⠋⠀⡞⠁⡸⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠤⠶⠾⣿⡊⠀⠰⣄⠂⠸⠋⠩⠀⠋⢨⡯⠀⠘⢷⢠⣾⣇⠘⣧⠶⠉⠀⠀⠀⣸⠁⠀⠇⠀⡄⢀⡌⠙
⠱⣆⠆⠀⠀⠀⠀⠀⡠⠀⠀⠉⠲⣄⠀⠀⠼⠋⠀⣤⠾⠜⠻⣦⣀⠀⠻⣿⠛⣶⣇⠀⠀⠀⠀⣶⠀⠐⠦⠀⠀⠀⠈⠁⣠
⠀⠙⢦⣴⠄⢀⣴⠊⠀⠀⠀⠀⠀⠈⠻⠄⠀⠈⢀⡁⠀⠀⠴⣿⣯⠁⠀⠙⠂⠀⠛⣦⠀⠀⠀⢉⡀⢇⠀⠀⠀⠀⠀⠄⢻
⠀⠀⠀⠉⢱⡛⠁⠀⠀⠀⠀⠀⡀⠀⠀⠈⠐⠶⠋⠀⠀⠁⣴⡟⠙⣷⡄⠃⠀⣀⣦⠞⠀⠀⠀⠘⠇⢀⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠈⠿⣤⣆⠀⢀⠐⡇⣤⣀⡀⠀⠀⠀⠀⠀⠰⡄⠀⠀⢀⠈⠑⠶⠿⡿⠃⠀⠀⠀⠀⠀⠘⣴⣇⠠⠶⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠄⢻⡟⢃⣭⣰⣷⣴⣏⠳⣀⡀⠀⢆⠙⢆⠈⠀⠀⠸⠀⠀⠺⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣴⠀⠁⠂⠙⠉⢉⠠⠉⡴⠈⢹⡦⠈⡶⠊⠀⢂⣴⠃⣄⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣠⣦⡴⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠠⠊⣀⡾⠃⠀⠰⣀⠀⣠⠜⣷⡞⠂⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠘⢷⠋⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠶⠀⠌⠀⡰⠋⠁⠀⠀⠀⢙⣦⠏⣴⠟⠇⠀⢀⡠⠀⢸⣄⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠛⠀⠁⠀⣰⠋⣠⡟⠁⠀⠀⠀⠀⢠⠞⣿⡟⡱⣾⣶⢇⠼⠁⠀⠀⢻⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠾⠋⠀⠀⠀⠀⠀⠀⢀⡤⡀⠸⣿⣿⣵⣦⣴⠃⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⢤⣀⠘⢃⠀⡀
⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠿⡃⣰⠟⢼⣷⢛⠇⡀⢦⠀⠀⢸⠀⠀⠀⢀⣠⣤⣤⣿⡄⢸⠀⡇⡿
⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠊⠙⠏⠀⠀⠙⢪⠟⠁⠉⢀⣀⡌⣂⢈⡻⠿⠟⠻⡏⠙⠷⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⠠⡤⢠⡴⢶⡤⣤⣤⣶⡦⠶⣤⠔⠀⢀⣀⣠⣾⣿⣿⣿⡋⠉⠀⠀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠄⠀⠀⠸⠇⠘⠿⠿⠿⠷⠶⠶⠶⢶⡶⣤⡾⠶⠶⣶⡾⣿⣿⣿⣿⣿⡉⢉⠗⠀⠀⠈⠉⠀⠈⠉⠉⠀⠈⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⣴⠖⠋
⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⢀⣠⣴⣶⣖⡶⠿⠽⠋
⠀⠀⠇⠠⢆⡥⠈⠀⠀⠀⠠⠀⠀⠀⠀⠈⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠷⠾⠋⠙⠋⠉
⠀⢀⠀⢀⢿⣄⠐⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣧⡀⢴⡘
⠀⢸⠀⠉⠈⠷⣀⠀⠀⠀⠉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⣏⣼⢻⣷⣌⠛⣤
⠀⣼⠀⠀⢷⣀⣿⣆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠁⠀⠀⢿⣿⣟⣿⣿⡷⣦
⠀⢿⠀⠀⢈⣽⠇⢻⡖⡐⠂⠀⠀⢀⣀⢼⣆⠀⠀⢀⣀⠀⠀⢰⣷⠀⠀⠰⠿⠉⣁⣤⣤⣶⠖⠓
⠀⣿⠀⠀⠼⠓⡀⠀⢱⠀⠀⢀⡦⣈⠛⠋⠛⠛⠁⠊⠙⠛⠻⠻⠿⠠⡤⠀⠴⠾⠛⠋⠁⢀⡀⣠⣆⡀
⠀⠀⢠⠆⠀⠘⠧⡀⠀⣰⡄⠸⢶⡉⠛⠀⠴⠄⠀⠀⠄⠀⢤⠤⠀⠀⠒⠶⠤⠄⠀⠀⡙⠉⡘⢻⣿⣧⣟
⠀⣄⠘⠃⠰⠀⠳⣀⠆⠈⠁⡄⠀⢑⡀⠀⠀⠀⢠⡄⠀⣠⣤⣦⣤⣤⣀⣤⣀⣤⣦⣜⡅⠸⣿⣿⣿⡟⠄
⣤⣤⣤⣤⣧⣤⣤⣭⣤⣤⣤⣥⣤⣼⣧⣴⣷⣭⣤⣴⣮⣭⣭⣬⣵⣼⣿⣿⣦⣽⣿⣿⣤⣤⣼⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠛⠻⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠂⠁⠀⠀⠈⠀⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠈⠀⢀⠀⠀⠀⠀⠀⠁⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⡁⣁⠠⠀⡀⠀⠁⡀⠅⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣇⡡⠷⠿⠺⢶⡾⠬⣌⡇⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⢑⣖⢿⠙⢇⣄⣀⡀⢙⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⣾⣇⣸⣐⣀⢴⣶⣆⣴⣨⣀⣤⣤⡦⢲⠖⣰⣶⣴⣤⠀⠀⠅⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠛⠉⠉⠉⠀⠀⠚⠁⠀⠀⠂⠚⠉⠉⠉⠀⠀⠀⠒⠒⠀⠀⠀⠂⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢠⣶⣿⣿⣿⣿⣾⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠩⠛⠉⠉⠀⠀⠀⠀⠀⠀⢠⣾⡟⠉⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃

⠀⠀⠀⠀⠀⠀⢸⢹⣿⣶⣶⣶⣤⣶⣤⣼⣧⠀⠀⠀⢶⠦⣤⢤⣼⣷⣶⣶⣦⣤⣤⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠻⣿⡟⠻⠛⠛⠛⠁⠀⠀⢸⡦⠷⢒⣿⣿⠿⠛⠛⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⡟⠿⠛⠛⠛⠛⠛⠿⠻⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣀⣠⣄⣀⣀⣀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⢀⣼⡿⠃⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣶⣤⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣾⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠟⢛⣿⡿⠛⠛⠛⠛⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⣿⡟⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣷⣶⣶⣿⣤⣤⣴⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⢀⣿⣶⣶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣾⣀⣤⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⠈⠙⠻⠿⠛⠛⠛⠉⠉⠉⣿⡏⠉⠈⠉⣿⠁⠈⠙⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣁⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⡷⡆⠀⢸⣿⣗⠀⣺⣶⣶
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣶⠀⠀⣀⣤⣀⡀⢸⣼⣿⣿⡇⡇⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⢹⣽⣽⡇⡇⠀⢸⣿⣿⣿⣿⣧⣿⣶⣖⡂
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⢼⣾⣿⡇⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⡧⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣭⣭⣭⣭⣽⣿⣿⣿⣿⡷⠆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡧⡅
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡷⠆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣇⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣠⣄⣴
⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⡿⢿
⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣿⡀⠀⠰⡆⠀⠀⠀⠀⣠⠀⠙⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣇⠀⠀⠀⢀⠀⠀⠉⠈⣿⠀⠙⠛⠀⡷⢶⣿⠀⠀⣿⣿⣿⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠰⠇⠀⠀⡏⠈⠻⠁⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⣀⡀⣿⢀⣀⠘⠛⢻⡛⠛⠁⠀⢀⡇⠀⠄⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠐⢾⡗⠂⠀⠉⠙⡟⠉⠀⠉⠉⣿⠉⠉⠀⠀⢸⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠐⣿⠗⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⢠⡧⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢨⡧⠀⠀⢹⠁⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣦⣿⣶⣯⣖⢄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠒⢿⡿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿⣿⣿⣟⣿⣿⣿⣷⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⢀⣠⣧⣤⣶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⣿⣿⣷⣿⣿⡿⣿⣿⣿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⢸⡏⠉⠋⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢽⣿⣿⣾⣿⣾⣿⡿⠗⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠙⠟⠿⠿⠟⠟⠈⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⡇
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣇
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⠀⠀⠀⣿⡇
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⡠⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠠⠄⠉⣿⠉⠀⠀⠀⢸⠁⠀⠀⠀⣠⠶⢤⡀
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⢠⡏⠀⠀⣿
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⠆⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⢀⣾⣷⡀⠀⣿
⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⡟⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠉⡿⠿⠁⠀⣿

⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀
⠀⣠⣿⣭⣯⡄⠀⢠⠤⢤⣤⠤⢤⡄⠀⠀⠀⢰⣟⣛⣿⠀⠀⠀⠠⠶⠠⠠⠤⠤⠔⠖⠀⢸⡏⢿⡿⢿⣽
⠀⣷⣶⣻⣤⡇⠀⢸⠶⢾⣿⠶⢾⡇⠀⠀⠀⢠⣤⣤⣼⡀⠀⠀⢨⢼⢼⢼⡧⡧⡧⡧⠀⢸⡇⣻⣟⣻⢾
⠀⣶⣴⣻⣤⡇⠀⢸⢶⢾⣿⢶⢾⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢈⢸⢸⢸⡃⡇⡇⡇⠀⢸⡇⣽⣯⣽⢺
⠀⣷⣤⣻⣤⡇⠀⢸⢶⢾⣿⣴⣼⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⢲⢺⢺⡗⡗⡗⡗⠀⢸⡇⢾⡧⣾⣻
⠀⣷⣴⣻⣤⡇⠀⢸⣴⣼⣿⣤⣼⡇⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⢰⢰⢰⣰⣆⡆⡆⡂⠀⢸⡇⢾⡷⢾⣽
⠀⣧⣤⣻⣤⡇⠀⢸⣤⣼⣿⣤⣴⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠨⠹⠹⠹⠏⠏⠏⠍⠀⢸⡇⣻⣟⣻⣾
⠀⠧⠤⠿⠤⠇⠀⠸⠤⠼⠿⠤⠤⠇⠀⠸⠿⠿⠿⠿⠿⠿⠿⠀⠰⠼⠼⠼⠧⠧⠧⠥⠀⠸⣧⣽⣯⣽⣼
⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⣠⢴⢲⢄⡀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⢲⡀
⠀⠀⠀⡇⣁⠀⠀⠀⠀⢸⣳⣺⣞⣶⣻⠀⠀⠀⠀⢀⡾⠿⢧⡀⠀⠀⠀⢨⢵⡀⠀⠀⠀⠀⢰⡛⡇
⠀⢀⣾⣿⣿⣆⠀⠀⠀⢸⣺⣺⣗⣿⣺⠀⠀⢠⢾⢿⡿⡿⣿⢿⢷⠀⠀⢸⣃⣳⡀⠀⠀⣯⣽⣟⣯⣽⡇
⠀⣯⣭⣿⣯⣭⡁⠀⠀⢸⢾⢾⡷⣿⢼⠀⠀⢸⣻⣻⣿⣟⣿⣿⣻⠀⠀⢰⠖⢶⢷⡀⠀⣏⣏⡯⣙⣏⡇
⠀⡿⠾⣷⡿⠾⠇⠀⠀⢸⢽⢽⡯⣿⢽⠀⠀⢸⢾⢾⣷⡷⣿⢾⢾⠀⠀⢸⠭⠭⠭⡇⠀⣟⣟⣛⣛⣟⡇
⠀⣿⣻⣟⣿⣻⡇⠀⠀⢸⢽⣽⣯⣿⢽⠀⠀⢸⣽⣽⣿⣯⣿⣽⣽⠀⠀⢘⣓⣛⣛⡇⠀⡟⡟⠷⢛⡟⡇
⠀⣯⣭⣿⣯⣭⡁⠀⠀⢸⣺⣻⣗⣿⣸⠀⠀⢸⣺⣺⣿⣗⣿⣾⣺⠀⠀⢸⠶⠶⠖⡇⠀⡿⡿⣟⠿⡿⡇
⠀⡿⠿⢯⡿⠿⡇⠀⠀⢸⣺⣺⣗⣿⣺⠀⠀⢸⣽⢽⣯⡯⣿⣽⢽⠀⠀⢨⣭⣉⡭⡇⠀⡿⡿⠷⢿⡿⡇
⠀⠛⠛⠛⠛⠛⠃⠀⠀⠘⠚⠚⠓⠛⠚⠀⠀⠘⠛⠚⠛⠓⠛⠛⠚⠀⠀⠘⠛⠚⠒⠃⠀⠓⠓⠛⠚⠓⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⡫⣒⣾⣿⡟⠻⣿⣿⣿
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢨⣿⣷⠉⠉⠛⠷⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣇⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⢻⠀⢀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⡟
⠀⣠⡠⣦⣤⠴⢾⡿⠛⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠰⣿⣿⢰⣿⡆⣤⡀⠀⠀⢻⣿⣟⣿⣿⣷⠃
⠀⠈⠃⠉⠀⠀⢰⣵⡆⠀⠀⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠈⠁⢿⡿⠸⠿⠿⠀⠀⣼⣹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⢘⡧⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣻⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣛⣟⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢸⣚⣿⣷⣿⠽⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣟⡆⠀⠀⠀⠀⠀⠨⣇⠀⠀⠀⠀⣽⣿⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢠⣳⣿⣿⣿⠋⠈⠙⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⠀⠀⠀⠀⠀⢘⣷⠀⠀⠀⠀⣸⢿⡀⠀⠀⠀⡾⣿⡿⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⢠⡳⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡄⠀⠀⠀⠀⠠⢿⡄⠀⠀⠀⣹⣧⡀⠀⠀⡜⣽⠈⠁⠀⠀⠀⣼⠃⠀⠀⠀⠀⡠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⣧⠀⠀⠀⠀⠠⢸⡇⠀⠀⠀⡼⣻⠆⠀⢸⣵⠃⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⡉⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠰⠿⠆⠀⠸⡟⣦⠀⠀⠀⠀⣾⠁⠀⠀⠀⠠⣔⢯⡳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠾⡇⠀⠀⠀⠀⣸⠀⠀⠀⠀⢰⣿⡆⠀⠀⠈⠁⠀⠀⠀⢰⡏⠀⠀⠀⠀⣴⣽⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⣭⠅⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣾⣶⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢺⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢣⣀⣠⣼⠌⠘⠄⢀⠤⠤⢤⠀⠀⣠⠤⢖⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠉⢁⣀⠀⣸⠀⣸⣤⡼⠼⠒⠚⢋⣍⡦⣧⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠤⣝⢦⠀⠀⠀⣀⡩⢵⢶⡌⠢⢘⣥⣤⡀⢀⣀⣨⡴⣚⣿⠇⠿⢐⡇
⠀⠀⠀⠀⠀⣞⡉⠲⡀⠀⠀⠀⠀⣎⣠⠖⠻⢿⣒⠺⣩⣠⣠⣠⠸⢸⢮⡇⠉⠭⣼⡟⡉⣠⡏⢸⣿⣇⣸⣐⣷⣤
⠀⠀⠀⠀⠀⠈⢫⠖⢛⠯⡉⢻⡉⠀⡞⠄⠀⠀⠉⠊⠉⠒⢤⡈⠙⠫⡉⡇⠀⣓⣿⣃⢱⡿⣠⣦⣬⡍⣙⡛⢁⣿⠁
⢎⣙⢦⡀⠀⠀⢸⢸⠤⡷⠷⢌⣧⣡⠭⠿⢷⡀⠀⠰⡆⢤⣦⣈⠣⣄⣌⢇⠀⠀⢿⢻⣿⠃⠈⠛⠻⠿⠶⢋⣾⠛⢆
⠈⠣⣶⠙⠢⣄⢸⠀⠑⠁⡆⡑⡇⢸⠀⡭⡟⢧⣄⣈⠪⣶⡿⢉⣷⡘⣞⣷⣆⠥⣿⣅⠘⡄⣭⡵⠞⠛⠛⣿⠟⢆⠈⣆
⠀⠀⠉⠳⣦⣂⡽⠀⠀⠀⡆⡢⣇⠈⢶⣶⢋⡌⠠⢮⡟⣛⠟⢩⣿⣷⢹⡿⠿⣩⣿⡝⣺⣘⡤⠶⠄⠀⣼⣟⣦⠈⡦⣙⡆
⠀⠀⠀⠀⠈⣹⠃⠀⠀⠀⣇⠂⣿⣇⣹⠛⢶⠁⠨⣼⡿⣿⠰⣿⣿⡾⠋⣿⣶⣿⢿⣿⠃⣻⡽⠧⢶⣶⡿⠿⢲⢋⡇⢒⡇
⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠸⡀⠉⢿⠻⢿⡏⢀⡥⣿⣶⣿⢣⢄⣉⣡⣴⣿⡧⠿⣒⣋⢭⠗⡶⢫⢋⠎⡐⡱⡡⢣⡧⢽⠁
⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠻⣼⠀⢐⣸⣾⣿⣷⡤⢾⣛⡿⡿⣳⢋⠝⡡⢣⢊⠞⡰⡡⢃⢞⠜⡔⡡⢡⠏⢽
⠀⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢈⣦⢒⣛⡿⡿⡻⠫⢋⢎⢎⠔⡡⢋⠞⡔⡡⢃⢎⠜⡰⢡⢊⠞⡔⡱⣹⠤⢽⠢⣀⡀⠠⠠⠄⣀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠳⠀⢽⢉⢔⢎⠔⠔⡱⡱⢡⢊⠜⡰⢁⢊⠜⡰⡣⢊⠎⡰⠡⢂⢎⠜⡔⣼⠭⣿⡑⢐⠀⠂⣙⡂⠈⠣⠥
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢀⠭⡧⢁⢪⢊⢎⠌⡔⡡⢊⠎⡔⡡⢋⢜⡔⡡⢃⠜⡔⡡⢣⢊⢞⣴⡾⢑⣟⡈⠠⢍⣀⡒⠂⣠⠤⠭
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⡷⡑⡡⡣⢣⠎⡔⡱⢁⢎⠔⡡⢋⢊⠜⠠⢊⠌⣔⣱⣥⣿⡻⣽⠷⠞⠩⠭⠉⠐⣒⡐⣃⡈⠉⠉
⠀⠀⠃⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠨⢽⠎⢜⠔⡵⢡⢊⠎⡰⡡⢊⡔⡱⢡⢊⣴⢥⣻⡹⠽⠶⠿⠉⢙⢛⣋⣈⣭⣭⣤⠀⠐
⠀⠓⠤⣘⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢽⢨⢊⠎⡔⡡⢃⢎⡔⣡⣮⠼⣶⣻⠽⠾⠭⠭⠥⠭⠭⠭⠉⠁⠒⠒⠒⠂
⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⡀⠀⡀⠽⡡⢣⢎⣼⡵⡗⣞⣽⡽⡶⠏⠩⠭⠥⠦⠤⠭⠭⠭⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄⡀⠁⠓⡂⣽⣲⣯⣯⣷⣿⣛⣃⣤⣤⠶⢖⣀⣀⣀⠀⠌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠶⠿⠿⠟⢀⣐⣒⣒⣒⣛⠓⠓⠖⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠐⠀⠀⢸⡏⠀⠀⠀⠀⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣠⡀⠀⠀⠀⠀⢻⡔⠀⠀⢸⣧⠀⠀⠀⠀⡄⢼⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠲⠈⣿⡇⠀⢀⠀⣰⣾⣗⠀⠀⢸⣿⠀⠀⠀⠀⡇⡍⠉⠉⠉⣶⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⣿⣧⠀⢸⡅⣿⣿⡿⠀⠀⠈⢻⡄⠀⠀⠀⠁⢦⣄⡀⠀⠿⡄⠀⠀⢸⡄
⠀⠀⠀⠀⠄⠀⠈⢠⣤⡄⣿⠀⠀⠀⠀⢸⡇⣿⢹⣗⠀⠀⠀⠘⠓⠠⣿⣿⣿⢸⣿⡇⠀⠀⠈⣿⣤⣨⣧⣤
⣤⣤⣤⣤⣤⣤⣤⣼⣭⣽⣿⣤⣤⣤⣤⣬⣥⣿⣼⣯⣵⣦⣤⣤⣤⣤⣿⣬⣭⣼⣼⣷⣦⣤⣤⣭⣭⣬⣼⣧⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣉⣤⠒⠒⢲⠒⠠⢌⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠋⠐⣎⢦⠀⠘⡀⠀⣠⠜⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⠃⣼⣁⠀⡀⠈⣆⠻⠀⡦⢊⠴⠊⠀⠱⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⠂⠠⢤⣄⡉⠀⡀⠀⡋⠒⠒⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣁⠀⠀⣠⣞⡵⠞⣳⡌⢫⠉⠁⠀⢐⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠻⠙⢫⠋⠁⠈⠀⡇⢿⠉⢦⡱⡀⢀⠎⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠉⠀⡀⠀⢀⡸⠀⢀⡱⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠈⠈⣿⡉⢤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⠀⠀⠀⢀⣀⡀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠁⠀⠉⠉⠉⠁⠀⡀⠄⠂⠙⡗⠲⣤⣀⣀⣀⣤⠔⠚⠉⠀⠀⠀⠀⠀⠙⢆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⢀⠔⠈⠁⠀⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⡈⠀⠠⠂⠁⠑⢄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⣇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡔⢃⠀⠀⣀⣀⣀⣀⠀⢦⠀⠀⠀⠀⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠈⢠⡡⣀⠤⠒⠉⠀⡀⠀⠀⡏⠉⠉⠉⠀⠀⢱⠀⠀⠀⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⢈⠀⠀⢀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⡠⠉⠀⠉⠓⠴⢦⡂⠀⠀⠀⠀⠀⡕⠸⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡄⠀⠀⠀⠀⡰⠁⠀⢠⠧⢄⡀⠀⢩⠁⠒⠒⠂⠐⠈⠁⠀⠀⡶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣁⠀⠰⠃⠀⠀⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡀⠀⠀⠀⠈⠢⢀⡀⠀⢀⠄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣄⠙⠶⢤⣀⣀⣀⣀⡀⢀⣀⣀⡠⠴⠚⠋⣠⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⠲⠤⣤⣀⣀⣈⣉⣀⣀⣤⠤⠖⠒⠉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠊⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡄
⠀⠀⠈⠉⠿⠿⠿⠿⠿⠿⠿⣉⣉⠉⠉⣉⠉⠉⠉⠉⠉⠉⢹⣿⣶⣶⣶⣿⣿⣷⠀⠠⢀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠁⠉⠛⠻⣉⢋⣿⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷
⠀⠀⠀⢰⢣⠒⠌⣿⡆⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠢⠼⢄⣠⠄⠤⣀⡀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⢈⡏⠾⡀⣿⡇⠀⠀⠀⠀⠸⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⢀⣿⡇
⠀⠀⠀⢸⡇⠀⠇⣿⡇⠀⠀⠀⠀⠐⠀⡄⠀⠀⢀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠠⠤⠤⠤⢄⠀⢂⣀⡀
⠀⠀⠀⢸⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠋⠹⠿⠯⢭⣭⣭⣭⣭⣭⣿⣗⣶⣶⣧⣤⡬⠈⣞⡏
⠀⠀⠀⠀⡇⢀⠸⣿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠒⠛⠿⣾⣿⣿⣷⣿⠶⢤⣀
⠀⠀⠀⠀⠅⡀⠀⢻⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠼⢳⠟⠶⣄⠈⢳⡀
⠀⠀⠀⠀⡇⡇⠀⣸⡇⠀⠀⠀⠀⣂⠄⡇⠀⠀⢀⣤⣦⣤⣤⣤⡀⠀⠀⠀⣀⣀⣠⣶⣄⣠⠀⠈⠉⠁⠀⠀⢹⠀⠈⢷⡀
⠀⠀⠀⠀⡇⢇⠀⣿⠁⠀⠀⠀⠀⡁⠀⡇⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣧⣀⠀⠀⠀⠀⠀⠾⠶⠷⠮⠷
⠀⠀⠀⠀⣷⢸⡀⣿⠀⠀⠀⠀⠀⡏⢽⡇⠀⠀⠀⠀⠉⠉⠉⠉⠙⣏⠀⠝⠉⣯⠉⠉⢿⠉
⠀⠀⠀⠀⢻⡈⠁⣿⠀⠀⠀⠀⣖⡃⢸⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠋⢻⡆⠀⠈⣇⠀⠀⠀⠀⢀⢠⡾⠛⢳⡄
⠀⠀⠀⠀⢸⠣⡌⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠘⡆⠀⠀⠀⣿⣿⣧⣤⠾⢧⡀
⠀⠀⠀⠀⠸⠀⠠⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠿⠿⠇⠀⠀⢸⡿⠉⢸⣇⣀⣠⡟
⠀⠀⠀⠀⠀⡇⢘⡇⠀⠀⠀⠀⠀⣿⣙⣿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣧⠀⢰⡟⠛⠻⣇
⠀⠀⠀⠀⠀⣇⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡾⠷⣤⡶⠋
⠀⠀⠀⠀⠀⡇⢸⠇⠀⠀⠀⠀⠀⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣄⣀⣀⣄⠠⠀⠀⠀⠀⠀⠀⠀⠙⠹⣇⣀⣸⠇
⠀⠀⠀⠀⠀⡇⢸⡃⠀⠀⠀⠀⠐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠛⢻⡟⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠀⠙⢯⣷⣤⡀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣤⡄⡀
⠀⠀⠀⠀⢀⠇⢸⡇⠀⠀⠀⠀⢸⣀⣾⣿⠀⠀⠀⠀⢀⣤⣾⡿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠻⠋⠉⠉⠻⠟⠃
⠀⠀⠀⠀⠺⠶⠶⠟⠀⠀⠀⠀⠈⠌⠁⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⠛⢛⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣬⣉⠙⢀⡄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡧⠥⡬⠀⠥⠬⠼⢿⣄⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠊⠀⠰⠀⠀⠀⠈⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢰⠞⠉⠉⠉⢳⡀⠀⠀⢀⡞⣁⣀⣀⡠⠤⠤⠖⠒⠀⠒⠢⠐⠄⠳⢤⣙⣧
⠀⡴⠖⠛⠂⠀⠀⠀⣸⡃⠀⣴⠛⠏⠠⠀⠀⡆⠀⠈⠀⠀⢘⠀⢰⠀⢠⠀⠘⡀⠉⣧
⢸⡇⠀⠀⠀⠈⢢⠀⠈⢿⡀⡏⢠⠀⢈⠀⠀⠁⠀⠀⠂⠀⠈⠀⠀⡄⠀⠀⠀⡇⠘⣸
⠀⣿⠂⠀⠀⡀⠈⠀⠀⣸⡿⠛⠀⢤⣚⣩⣭⣩⣒⡉⠉⠀⠴⠖⠒⢆⣀⡐⣖⣳⠂⠉⢳⡄
⢀⣸⡇⠀⠀⠁⠀⣤⠞⢹⠃⠀⠀⢰⠀⢀⠊⣴⡿⢷⠀⠀⠀⡴⢷⣦⠑⡄⠈⡇⠀⠀⠈⣷
⡟⠀⠀⠀⣆⠀⠀⡹⠶⠛⠠⡀⢀⡸⢠⣬⣄⣛⣿⠞⣠⣤⣄⢻⣿⠟⡠⣁⢀⡃⠀⠀⠀⣿
⠻⢤⣠⣤⠁⣀⢀⣠⡶⠓⠲⣦⣠⣾⣻⠟⠟⠩⢋⠕⠛⣏⠙⢌⠫⠑⠟⢿⢙⣷⣤⡤⠞⠁
⠀⠀⠀⠈⠉⠉⠉⠁⢀⣴⠞⠋⠉⠙⡇⠀⠀⠀⠀⠀⠚⡍⠂⠀⠀⠀⠀⠀⡋⠉⠉⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⠤⠔⠊⠘⠀⠀⠀⠀⠀⠀⠜⠙⢆⠀⠀⠀⠀⠀⡈⠉⠑⠤⢤⡙⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣀⣀⡀⢳⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡸⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⠄⠊⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸⠒⠂⢲⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠤⢽⡇⠀⠐⠒⠀⠤⢼⠀⠀⠀⠀⠀⠀⡸⠄⠒⠊⣿⣤⡴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠁⠀⠐⠒⠂⢎⣰⣣⣆⣴⣰⣠⠗⠒⠈⠉⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣛⣦⣤⣀⣀⣀⣤⣐⣿⣾⣓⣠⣄⣀⣀⣠⣴⣞⣯⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠉⠀⠀⠀⠈⠀⣼⣿⡀⠈⠀⠀⠀⠈⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⢷⡳⢄⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⠶⢶⣄⣀⠀⠀⠀⠀⣀⡶⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠉⠙⠷⠲⢦⡼⢧⡀⠦⢄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⠔⠲⡤⢄⢀⠀⠀⢀⣈⢥⠀⢈⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠀⢀⠉⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⠒⠃⠀⠀⢠⠀⠀⠀⠀⢀⣄⣼⠃⠰⠚⠁⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⣀⡴⢿⣰⠀⢀⣘⣿⣄⣠⣄⠀⣀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⠗⠚⡅⠐⠂⢹⡆⣸⠋⠁⠀⠀⣬⡟⠉⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⠓⠋⠀⠀⢠⠄⠀⠀⠀⢻⣧⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⠀⠀⠀⠀⠫⠇⠀⠀⠀⠀⠀⠀⡀⢺⡧⠤⢤⡈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣸⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡆⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⣿⡗⠒⠚⢁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⠒⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡦⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣑⣑⣚⣘⣃⣄⣤⣤⣤⠤⠤⠬⠶⠮⢕⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣠⣤⣀⣀⣠⣤⣀⣀⣠⣦⡀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⠀⣀⡀⢀⣀⠀⣠⡄⠀⠀⠀⢀⣾⡄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⠃⠀⠀⠀⠀⡬⣁⢆⣶⣶⣾⡿⠚⠓⠒⠛⠒⠛⠓⠀⠀⠀⣸⣿⣷
⠀⠀⠀⠀⣸⣿⣿⣧⣘⣀⣼⠀⠀⠀⠀⠀⠀⠀⣾⣾⣾⡿⠿⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿
⠀⠀⠀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠤⠤⠤⠤⠴⠿⠧⠀⠀⠀⠘⣿⣷⣀⣀
⠀⠀⢰⡋⠉⠉⠉⠉⢹⣿⣿⣿⢿⡿⠿⣿⠿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠛⠁
⠀⠀⢀⡇⠀⠀⠀⠀⢨⣿⣿⡿⠄⠡⠠⠂⠀⠂⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠂⣷
⠀⠀⠊⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡂⣿⢴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡷⠀⠀⠀⠀⢠⡀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢋⣍⡓⢄⠀⠀⠀⠀⠀⢸⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢜⢿⣟⣼⡀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⢼⣿⣿⣿⢰⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢶⡂⡄⠶⠶⠶⠶⣾⠷⣶⠶⠶⠶⠶⠶⠶⠶⣲⣶⣄⠀⠀⠀⢻⣧⢼⣿⣿⡿⠈⡇⠀⠀⠀⢐⣿⡇
⠀⠀⠀⠛⠓⠓⠲⠆⠛⢣⡊⠁⠑⠚⠓⠚⠃⠐⠋⠁⠀⠀⠀⠀⠀⠈⠻⠾⠿⠿⠁⠀⠀⠀⠀⠀⢸⠏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⡇⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣿⡂⢠⣾⣷
⠀⣿⣟⡟⡇⠀⠀⠀⢸⡇⣾⠀⠀⠀⢰⣿⣿⡆⠀⣾⣷⡀⢸⣯⡿⡇⣼⣽⣆⠀⣀⠀⠀⡾⡿⣆⠀⠇⣿⠄⢰⣿⣿
⠀⣿⣯⡯⡇⢠⣤⡄⢸⠇⢿⢸⣿⡇⢸⣿⣿⡇⠀⡯⢿⡇⢸⡵⣟⡇⣷⣿⣿⢘⡋⡇⠀⣗⡃⣿⠀⡅⣿⠅⢘⣿⣿
⠀⣿⣿⣇⡇⢸⣶⡇⢸⡇⣿⢸⣿⡇⢸⣿⣿⡇⢠⣿⣿⡇⢸⣟⣽⡇⣿⣿⣿⢸⡃⡇⢸⠭⡭⢽⡇⡄⣿⡀⢰⣿⣿
⠀⠿⠿⠷⠇⠸⠶⠇⠸⠏⠿⠸⠿⠇⠸⠿⠿⠇⠸⠾⠻⠇⠸⠿⠿⠇⠿⠿⠿⠰⠶⠧⠸⠾⠱⠾⠇⠿⠻⠷⠸⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⣀⡀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⡇⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢠⣿⣷
⠀⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⣀⡀⢸⣿⣿⡇⠀⣿⣿⡄⢸⣿⣿⡇⣾⣿⣧⢀⣀⡀⠀⣾⣿⣧⠀⣿⣿⣿⢸⣿⣿
⠀⣿⣿⣿⡇⢠⣤⡄⢸⣿⣿⢸⣿⡇⢸⣿⣿⡇⠀⣿⣿⡇⢸⣿⣿⡇⣿⣿⣿⢸⣿⣿⢀⣿⣿⣿⠀⣿⣿⣿⢸⣿⣿
⠀⣿⣿⣿⡇⢸⣿⡇⢸⣿⣿⢸⣿⡇⢸⣿⣿⡇⢰⣿⣿⡇⢸⣿⣿⡇⣿⣿⣿⢸⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿
⠀⠿⠿⠿⠇⠸⠿⠇⠸⠿⠿⠘⠿⠇⠸⠿⠿⠇⠸⠿⠿⠇⠸⠿⠿⠃⠿⠿⠟⠸⠿⠟⠸⠿⠿⠿⠃⠻⠿⠟⠘⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢀⣀⣿⣿⣶⣄⠀⠀⠀⠀⢸⢺⢸⡆⠀⠀⠀⢀⣤⡀
⠀⣼⣿⣿⣆⣀⠀⠀⠀⢴⢷⣿⣿⠀⣄⡀⣴⣿⣿⣿⣵⣻⣿⠀⣠⣄⣠⣾⢹⢸⣃⠀⣀⣀⣾⣿⣽⡇⣀⣤⣤⣄⡀
⠀⣿⣿⣿⣿⣿⠀⣤⣄⣸⣤⣶⣿⣧⣿⣿⡿⡿⣿⣿⣻⣟⣿⣴⡟⡟⣿⣿⣸⢨⣼⣨⣿⣿⣿⣿⣻⡇⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣸⣿⣿⣯⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠼⣞⣿⣿⣗⡇⣿⣿⢸⢼⢼⣯⡯⢹⣿⣿⣿⣷⣿⣿⣿⣿⡇
⢰⣿⣿⣿⣿⣿⣾⣻⣾⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣹⣿⣿⣍⣏⣹⣿⢸⢰⢸⣿⡇⢸⣿⡿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⢿⣿⣿⣟⣿⣷⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣳⣾⣿⣶⣺⣾⣯⣧⣼⣿⣟⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠩⠤⠰⢒⣲⣲⣟⣉⣡⣆⣖⣶⡶⡤⢄⣉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⣿⡷⠴⠛⠻⠿⠿⢿⢟⢻⢛⡛⠛⠃⢠⣤⣾⠃⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠃⠂⠈⠑⢾⣿⢿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡤⠊⠀⡠⣿⡟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠁⠕⠶⣤⣀⡀⠙⠻⣿⣿⣿⣹⣿⣿⣿⣿⠿⠋⠀⣀⣬⣤⢟⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⡿⣿⣶⣤⣀⡉⠛⢷⣆⡈⠛⢿⣿⣽⡿⠛⢡⠴⠶⠛⢉⣩⣴⠞⠋⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⣴⣬⡿⠀⠀⠉⠛⠛⠿⢷⣬⣥⡶⢤⣽⢫⣴⢶⣀⣤⡴⠶⠛⠉⠀⠀⠈⠁⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣱⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⡷⠊⠋⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠠⠟⠻⢏⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢎⢽⡄
⠀⠀⠀⠀⠀⠀⣗⠰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠻⡀
⠀⠀⠀⠀⠀⢸⣶⠁⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡰⠀⠀⠈⠥⠀⣧
⠀⠀⠀⠀⠀⣿⠘⠄⠄⠈⠳⢘⠲⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⠋⢻⡤⠂⠐⠠⢰⣽
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠐⢮⢀⣦⢠⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⣀⢰⣿⣿⠞⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⡿⠃⠁⠀⠀⠀⠀⣴⣆⣿⡜⠇⢠⡐⡠⣠⠀⠘⢁⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢻⣾⡿⠄⡖⡶⠆⠀⠀⠐⣿
⠀⠀⠀⠀⠀⣯⡆⠀⠀⠀⠀⠚⠟⢻⡿⣟⣿⣒⣛⣛⣈⣀⢀⠏⢐⢄⡟⠁⡽⠋⠉⠱⣖⠶⠺⡗⡻⢿⡇⡇⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⢇⡗⠀⠐⠠⠶⢻⣿⡝⠫⠍⠀⢹⠟⠛⠇⡮⠀⠴⠮⠒⠀⠀⠀⠀⠈⠁⠀⠀⠫⠞⢹⡗⠦⠄⠀⠀⠀⠀⢰⡾
⠀⠀⠀⠀⠀⢸⠜⠉⠀⠀⠀⠀⠸⠋⠙⠀⠁⠀⠀⠀⠀⠀⢳⣦⣾⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣥⣽⢿⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢺
⠀⠀⠀⠀⠀⠀⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡑⢀⡻⣿⠛⣛⠩⣶⢤⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠆
⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⡄⡀⠀⠀⡀⠄⠀⢨⡴⣟⣫⣻⡿⡖⠱⠘⠈⠁⠹⠃⢽⡠⣢⣠⣔⠀⢀⡀⠀⠀⠀⠀⣧⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⡲⡶⣿⡳⠛⠟⡿⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢧⡙⣲⠤⢆⣠⣤⣶⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢩⠏⠁⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠉⠉⢡⢁⣞⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠋⠄⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠈⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠈⠀⠚⠛⠛⠒⠒⠒⠠⠒⠚⠟⠉⠉⠛⠊⠉⠁⠀⠀⠀⠀⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠶⠄⠄⠖⢦⠐⢶⡶⠶⣶⣠⢶⣶⣶⡶⠲⠹⠟⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒

⠀⠀⠀⠀⢰⣇⣤⠀⠀⠀⠀⠀⠀⣾⣠⡄⠀⠀⠀⠀⠀⠀⣰⠗⠀⠀⠀⠀⠀⣀⣸⣃⡄⠀⠀⠀⠀⠀⢀⣴⡠⠀⠀⠀⠀⠀⠀⣠⣆
⠀⠀⢀⣴⠟⠉⠁⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⢠⣤⠞⠀⠀⠀⠀⢀⡠⠞⠉⣸⠋⠀⠀⢀⡴⠚⠋⢩⡏⠀⠀⠀⢀⣤⠤⢲⠟⠁
⣠⣴⡟⠁⠀⠀⠀⠀⢀⠠⡾⠋⠀⠀⠀⠀⠀⢠⣶⡿⠉⠀⠀⠀⠀⠀⠉⠀⢀⠜⠁⠀⠀⠀⠈⠀⠀⢀⡞⠀⠀⠀⠀⠋⠀⢀⡾
⠉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⢀⡴⠋⠳⠄⠀⠀⠀⠀⠀⣠⣶⡖⠀⠀⠀⠀⠀⠲⢶⣄⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠐⣅⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⢀⣴⠏⠙⠗⠀⠀⠀⠀⢀⡼⠻⠿⠂⠀⠀⠀⢀⡾⠛⠿
⠀⢠⣦⠀⠀⢈⡗⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠠⣎⢻⣿⠏⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⡶⠋
⠐⠀⠀⠑⠴⠛⠁⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠹⠞⠁⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠐⠟
⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣀
⠀⢸⣇⣀⣰⡆⠀⠀⠀⢠⡠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⡍⠉⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⢀⣴⣾⣿
⠀⢸⡇⠀⢰⡆⠀⠀⢠⣿⠃⣾⣧⣄⣾⠇⠀⢀⡴⠋⣰⠎⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣠⣾⣿⠟⠋
⠀⢸⡇⠀⠘⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⢠⠞⠁⠀⠀⠀⠀⠀⣔⡿⠁⠀⠀⠀⠀⠀⢠⣾⠿⠟⠛⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠐⠦⣄⡀⢰⡆⠀⠀⢀⣾⣛⣛⣿⣷⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⣾⣿⣿⣶⠶⠆⠀⠀⠀⠀⠀⢀⣤⣿⡷⠀⠀⠀⠻⠿⠿⠿⢳
⢀⣤⣤⣬⣿⣿⡇⠀⠀⣸⣟⠛⣿⣿⣿⠀⠀⣸⡟⠛⠛⢸⡆⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠋⠀⠀⠀⠀⠀⢀⡄⠈
⢿⣶⣶⣶⣿⣿⡇⠀⠀⠉⠉⠉⠉⠉⡏⠀⠀⠻⠀⠀⠀⠸⠇⠀⠀⠀⠀⣿⣷⠆⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠖⠖⠒⠒⠒⠒⠀⠀⠀⣀⣀⠀⣰⠶⣦⠀⠀⠀⠀⠀⢀⣶⣶⡀⠀⠀⠀⠀⠀⣠⣶⠃⠀⠀⠀⣶⣶⣶⣶⡦
⠀⣰⣿⠿⣿⡆⠀⠀⠀⡦⠤⠤⠤⠤⠤⠀⠀⢸⣏⣹⡇⠙⠶⠛⠀⠀⣠⣶⣷⣾⡟⠟⠁⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠺⠿⣿⠿⠗
⢠⣼⣿⣿⣿⣧⡄⠀⠀⣤⣤⣠⣄⣤⣤⠀⠀⠀⠉⠉⢠⡟⢻⡄⠀⠀⣻⣿⣼⣿⡇⠀⠀⠀⠀⠸⣿⠥⠄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⡀
⣴⡾⠛⢷⡄⠀⠀⠀⢰⣿⠶⠶⠶⠶⠶⠆⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣤⣤⣯⣭⣧⣤⡄⠀⢀⣀⣤⣶⣾⡷⠁⠀⠀⠀⣴⣿⣾⣿⣷
⣿⣦⣡⣾⡃⣀⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠛⠚⠁⠀⣿⣭⣿⣿⣯⣽⡇⠀⠉⣙⣿⠿⢍⡁⠀⠀⢀⣴⣿⣿⣿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⢉⡙⡲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠠⢂⠕⢒⣠⠴⠷⠛⠓⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢠⡴⠚⢉⣀⣀⣀⣀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠋⡠⠞⠉⠁⠀⠈⠉⠙⠶⣄⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢰⣃⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠟⠁⠀⠀⠀⠀⠈⠀⠀⠀⠘⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠀⠀⢀⠀⠀⠀⠲⠶⠂⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣺⣆⠀⠀⠸⡶⠦⠤⠀⢤⣤⠞⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣦⡀⠀⠈⠒⠤⠤⠐⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠲⣝⡟⣧⠀⠀⠀⢀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠘⣟⠄⠀⠀⢠⣅⠈⠑⠤⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠁⠈⠑⡄⠀⣸⠈⢳⡤⡀⠸⠃⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠛⢆⡀⢀⠛⠣⣴⠃⡴⢬⠷⣈⠷⢀⣀⡠⢸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠙⠏⠀⠀⠘⡄⠀⠀⡀⠸⡗⡦⠙⠛⣏⡀⠈⠛⢦⠶⠛⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢈⠉⠙⠀⠈⠷⠀⠀⣿⠙⠳⣄⠀⠀⢀⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠁⢉⠀⠀⡔⠀⠀⢸⡀⠀⢺⡀⠀⠻⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠒⠀⣶⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠳⡀⠁⠀⠀⢸⡇⠀⠈⠳⣄⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢦⡀⢀⡼⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⣿⢀⠃⢐⢀⡂⠀⠃⠆⡆⢺⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠠⠆⠀⣈⠀⠀⠀⠀⠁⠀⠀⠁⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢰⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⡠⣶⠞⠁⠀⠀⢸⡄⠀⡉⠀⠀⠀⠀⠀⠛⡟⠃⠀⠀⣿
⢠⢦⠀⠤⠟⣅⠀⠀⣠⠶⣼⠇⢘⡋⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⡋
⠸⠀⠑⢄⠀⠈⠳⠄⡏⠀⣸⡀⢸⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡟
⡰⡔⡄⠀⠑⢄⠀⠀⠂⠠⠂⠙⠺⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⡇⠷⠃⢀⠤⠀⡙⠀⠀⠀⠒⠒⠒⠚⢳⠀⠀⠀⠀⠀⡇⠀⠀⠀⢣
⠙⢄⠀⠀⣰⠂⣇⣀⣀⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠑⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠳⣬⣀⣀⣀⣠⣤⣤⣤⣤⠁⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠉⠉⠉⠉⠁⢹⠓⠲⠶⠖⠒⠋⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⢀⡿⣂⠤⣀⡀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⡰⠶⠶⠛⠓⠋⠉⠀⠀⠉⠐⠒⠭⠤⣤⣀⣀⣂⡾

⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⢰
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢰⣧⡄⠠⠏⠀⠀⠀⠀⠀⢀⣦⠈⢃⡀⠀⠐
⠀⠀⠀⠀⢀⣼⠋⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠉⠐
⠀⠀⠀⢀⣾⠏⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⡗
⠀⠀⠀⣸⣿⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣿⣿⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸⣿⡄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⢀⣤⣿⣄⢸⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠸⣿⣣⡄⢀⡀⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⡀
⠀⠀⢰⣿⣷⣾⡿⠻⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⣷⡆⠀⣄⡀⠀⢸⣿⠀⠀⠀⠀⠀⣣⡁
⠀⠀⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣟⠛⠁⠈⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⢿⣻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⣷⡆
⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡄⢀⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣤⣤⣴⣾⡇⠀⠀⠀⠀⠀⢰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠒⠂⠈⠁⠉⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠤⠔⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠶⠄⠦⠴⠶⠿⠿⠂⠉⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⣿⠀⢀⠂⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⢸⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠞⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡐⠁⠄⠉⡀⢢⠄⠤⠚⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⢀⠶⠀⠀⣠⡘⠀⠸⠀⠲⡀⠡⣄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠘⡴⠂⠈⠀⠉⠁⠛⠒⠊⠳⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠈⣿⡗⣶⡶⠰⠄⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣀⡀⢀⣿⣧⠼⠡⠇⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠛⠛⠿⠿⣿⣿⣿⣶⣤⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⣠⡀⢀⡀⠀⠈⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠁⠀⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠂⠉⠀⠈⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠁⠀⠸⠇⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⣶⡶⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠶⠆⣿⠀⠈⣿⣿⡇⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⣠⢴⡂⠙⣻⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠨⣞⣯⠅⠀⣿⡇⠀⠠⣞⣟⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⢽⣺⡅⠀⣿⡇⠀⠨⢵⣲⢯⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢐⣯⢷⡂⠀⣿⡇⠀⢐⡯⠿⣟⡾⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠲⣿⣿⠀⢸⣚⡽⡆⠀⣿⡇⠀⠐⠚⠙⠳⢭⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢨⠷⠋⠁⠀⠛⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⣤
⠀⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⡿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠠⡘⠁⠀⠀⠀⠘⠁
⠀⠀⠐⠠⠀⠀⠈⠀⠀⠀⠀⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠁⠀⢀⠑⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡀⠀⠀⠀⠀⠒⠀⣀⠂
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠
⣈⠀⠀⠀⠀⠐⠆⢀⠐⠄⠀⠀⠀⠀⠀⠘⠖⣠⣀⠀⠀⠡⠄⠔⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠄⠀⠈⠉⠀⠄⠀⠀⠀⠀⠀⠀⠫⠚⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣤⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢈
⠀⠀⠀⠀⠘⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠄
⡠⢀⠄⠀⠀⠀⠀⣰⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠃⠀⠀⡀
⠣⢀⠜⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠰⣨⡀⠀⠈
⣶⣷⠶⠷⠿⣿⣿⣶⣶⣶⣿⣿⣶⣶⣶⣿⣿⣿⣾⣿⣶⣶⣾⣿⣿⣷⣿⣿⣿⣿⣷⣿⣾⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⢀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⠀⠀⠀⠀⢀⡄⠀⠀⢠⣄⠀⠀⡀
⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⣸⠀⠀⠀⢸⡇⠀⠀⠀⢾⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⡇⠀⡇⢸⣿⡆⠀⢿⣄
⠀⠀⢹⣿⠁⠀⠈⣿⠁⠀⠀⣿⡀⠀⠀⣾⣷⠀⠀⠀⣿⡄⠀⠀⢸⣟⠀⢸⠁⠈⣿⠀⠀⠙⠛⢻⡟⠛⠋⠁⠀⠀⢸⣿⢿⣶⣷⡾⣿⡇⠀⠀⢻⣧
⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⠈⠯⠀⢸⠀⠀⠟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⡟⠙⡏⢹⣿⡗⢸⡖⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢻⣿⠀⠀⠀⣿⠀⠀⢀⣀⡀⠀⢸⠀⠀⠀⠠⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⡇⠀⡇⢸⡿⠃⢸⠛⢻⣿⡇
⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡆⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⠤⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣄⠀⢠⡄⠀⢰⡄⠀⡇⠀⠀⠀⢸⠀⢸⡿⠁
⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢿⠀⠀⢸⠉⡇⠀⢸⠀⠀⠀⢶⡶⠀⠀⢸⡇⢸⣿⡟⢻⡟⢻⣿⡇⠀⡇⠀⠀⠀⢸⣠⠞⠁
⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠸⡇⠀⢀⣀⣀⡀⠀⠘⣶⠃⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠸⣿⠛⢻⡟⠛⢻⡇⠀⡇⠀⠀⠀⢸⠁
⠀⠀⠘⡿⠀⠀⠀⣿⠀⠀⠀⣿⠇⣀⠀⠀⠁⠈⠉⠉⠉⠙⣦⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠈⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠃⠀⠀⠀⠋⠀⢀⠀⠀⢸⣿⠀⠀⢠⠀⠀⠀⢀⣴⠟⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⣰⡄⠀⢠⣷⠀⠀⣼⡇⠀⢸⣿⠀⠀⢸⡆⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⡀
⠀⠀⣿⡇⠰⣼⣿⡴⠀⢿⡇⠀⢸⣿⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇⠀⢀⣀⣠⡄⣀⡀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⠸⣇
⠀⠀⢿⠇⠀⠀⡯⠀⠙⢻⠋⠁⠘⡿⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⡇⢸⡇⢰⣿⣿⣿⣿⣿⠀⠀⠀⠙
⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡿⠀⠀⠈⠀⠀⠀⠀⣷⠀⠀⣿⠀⠀⠀⠀⣀⣀⣀⠀⠀⢸⣿⣿⣿⣿⡇⢸⡇⢸⠟⠛⠉⠁⣿⠀⣀⣈⣿⣀⣰⡆
⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠄⣉⣹⣉⠠⠀⠘⠛⢻⡟⠛⠃⠀⠁⠀⠠⢶⡄⠀⣿⠈⠙⠛⣿⠛⢻⡇
⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⣀⣀⣀⠀⠀⠋⢻⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⣿⡇⠀⣿⠀⠀⠈⣿
⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⣏⣯⣿⠇⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣼⣿⡿⠀⠀⣿⠀⠀⠀⣿
⠀⠂⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⡆⢸⠀⠀⠛⡟⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣾⣿⡟⠁⠀⠀⣿⠀⠀⠀⣿
⠀⠄⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⡧⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⡿⠋⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠄⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠂⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⡀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠇⠀⠀⢀⠀⠀⠀⠀⣸⠀⠀⠀⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿

⠀⠀⠙⠢⣀⢀⢀⡴⢄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠀⠐⠀⢉⣉⣀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⡀⣤⠄⢠⢈⠉⠀⡠⠀⠵⠀⠄⠀⠀⠀⠀⠰⠄⠀⠃⠀⠀⣆⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠚⣋⠉⠁⠀⠑⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠂
⠀⠀⠠⠐⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⢰⣆⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠁⢀⡞⠁⠀⠈⠁⠀⠀⢀
⠀⠀⠀⠐⠤⠄⠀⠀⠀⠀⠀⠀⠀⠉⣷⠊⠁⠀⠀⠈⠉⠀⠀⠀⠀⠰⡾⠃⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⡇
⠺⠶⠶⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠒⠠⠤⠀⠍⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳
⡉⠁⠀⠀⠓⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣐⠦⠴⣆⣀⡀⡀⠀⠀⠈
⠁⠀⠈⠉⠁⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣅⣐⣊⠉⠙⠐
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠀⠀⠉⠉⠁
⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⠑⠄
⠓⢷⣜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠠⠒⠈⠉⠉
⠀⠀⢛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠂
⠷⠖⠋⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⢀⡄⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢶⡶⠰⠰
⠀⠔⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠀⠠⣪⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡄⠀⠀⠀⠈⠐⢲⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⣄⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠈⠙⠁⠀⡘⠀⠀⢀⡀⠀⠰⠉⠀⠀⠀⠠⣄⡀⠀⢀⠀⠀⢀⠀⠀⢨⠀⠀⠀⠆⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡠⠧⠤⠀⠈⣧⠐⠖
⠀⠀⠰⢦⣀⢄⠁⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠂⠂⠀⠀⢺⡷⢀⠀⠀⠪⣟⠆⠸⠆⠀⠀⠀⢰⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⣲⠶⢓⣠⡀
⣀⣀⣀⣀⣀⣙⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣸⣃⣴⣃⣀⣹⣾⣑⣀⣆⣀⣀⣀⣈⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣉⣉⣉⣁⣹⣀⣀⣀
⣿⢻⠿⡿⢿⢛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣛⣻⣿⣟⣛⣿⣻⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⢣⣦⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⣄⣶⣠⣤⣶⣻⡏⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠢⠌⣩⡍⢉⣭⠐⠒⠟⢻⣦⣤⣤⣔⠒⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣷⣾⣿⣿⣿⡟⠚⠓⢾⣓⣒⣲⣒⣒⣶⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⠟⠛⡉⢉⡉⠉⠉⠁⢴⡆⢸⢸⡇⠀⢸⡆⠀⠸⠂⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠁⢰⣀⢧⣼⡇⠀⠀⠀⢼⡇⠈⢸⣷⠈⢸⡿⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⢀⣾⣼⣿⠭⠽⠯⣿⡀⠀⠰⠀⠀⠀⣠⣤⣤⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢃⡀⠈⠉⣁⣀⣤⣤⣠⣧⣤⡄⠀⠀⠀⠰⣿⣿⣿⠇⠀⢸⡆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢄⠚⣿⢫⢺⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠟⠛⠛⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠰⡏⣉⠛⠌⠸⠀⠀⠘⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠐⣉⠁⠉⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⢨⣿⣿⣿⡇⠀⢨⡇⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠉⠀⣆⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠘⣿⣿⡇⠀⠀⠀⠸⠿⠿⠿⠇⠀⠸⠇⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡇⠃⠃⠀⠐⠀⠀⠀⠀⣆⣼⣿⣿⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡀⠀⡆⠠⠏⠀⠈⠀⠸⢿⣿⣿⡏⠈⣿⣿⡇⢠⡄⠀⠀⣿⣟⣻⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠟⠀⠉⠀⠀⠈⠀⣀⢠⣼⡿⠿⠇⠀⠿⢿⡇⠀⠀⠀⠀⠁⠉⠀⠀⢀
⠀⠀⠀⠂⠀⢠⠌⠃⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⢶⣶⣶⣶⣶⣶⣶⣿⠂⠀⠀⢰⣶⣶⣾⣶⣶⢆⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠘⠀⠀⢸⣿⣿⣿⡭⣯⢿⣿⠀⠀⠀⢨⡷⣯⣶⣢⣤⢠⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⢿⢻⣿⠁⠀⠈⠘⠿⢿⠏⠘⡿⠸⣿⠁⠒⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠁⡩⠀⢀⡠⠁⡀⠰⠀⠀⠄⢰⠆⠀⢀⠀⣉⠀⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠘⠃⠀⠂⠀⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐


⣿⣿⣷⢿⣿⠷⣾⣷⣷⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣿
⣿⣿⣧⣶⣶⣶⣦⣶⣦⣾⣯⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡤⢀⠴⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡾⠛⠉⠠⠤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠳⣤⣤⣤⡤⠴⠚⢁
⠀⠀⠀⢠⠁⠀⠈⠁⠀⠀⢀⣀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠟⠛⠀⠑⠒⠒⠒⠛⠋⠉⡠⣄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠒⠚⠉⠁⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠶⠶⠖⠒⠒⠒⠒⠶⠤⢬⣽⠓⠦⢤⣤⣀⣀⣀⣀⣀⣤⠴⠶⠛⡏
⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠀⣤⠀⠹⠓⠒⠶⠦⠤⠤⠤⠤⠶⠒⠚⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢠⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠶⠀⠀⠀⠀⠿⠀⠀⠀⠘⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠰⠆⠀⠠⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠛⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⢸⣿⡀⣀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣿⣿⡟⠁⠀⠀⠙⣿⣿⠟⠁⠀⠀⠀⣶⠀⠛⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣇⠀⠀⠀⢀⣿⣿⣷⣦⡀⠀⢰⡆⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⣿⣷⣶⣶⣿⣿⣇⠀⠀⠀⠀⠀⠀⠶⠀⠶⠀⠀⠀⠀⠠⠐⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣾⠟⠉⢻⣿⠁⠉⠻⠆⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⠀⠈⠋⠀⠀⠀⢀⣀⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠙⢢⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠀⠀⢿⣦⣀⣀⣤⣶⠿⠟⠛⢻⡟⠛⠻⢿⣶⣄⡀⣠⣾⠟⢰⡇⠀⠀⠀⡸⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠋⢀⡤⠔⠂⢸⡇⠒⠦⣄⠈⠙⢿⣿⡁⠀⠀⠳⣀⠀⠁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢀⣀⠀⠙⢦⠀⠀⠀⠀⣰⣿⠋⢀⡴⢋⠴⠂⠀⢸⡇⠈⠒⢤⡙⢦⠀⠹⣷⡄⠀⠀⠈⠉⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢰⠁⠀⠀⠀⢸⠆⠀⠀⢠⣿⠃⢠⡏⡴⢃⡴⢋⣠⣼⣧⣄⠙⢦⠹⡌⢧⠀⢹⣷⠀⠀⠀⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢄⡀⢀⣠⠎⣀⣀⣀⣼⡿⠀⣼⢰⡇⡾⢀⣿⠉⠀⠀⠙⣧⠘⡇⢻⠸⡆⠀⣿⣦⣀⣀⣀⠸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠈⠉⠀⠀⠙⠋⠉⢻⣧⠀⠹⠘⠀⠁⣈⣿⣀⠀⠀⣠⣿⡀⠀⠘⠀⠇⠀⣿⠟⠉⠛⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢀⣠⠶⠛⠉⢬⣙⡛⢛⣋⠄⠙⠳⢦⣄⠀⣸⣿⠀⠀⣠⡄⠘⠿⠇⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⠁⠢⣌⠑⠦⢬⣭⣭⡤⠖⢉⡤⠂⠈⣻⡿⠁⠀⠀⠙⠋⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠿⠃⣠⣄⠀⠀⠀⠀⣸⣿⣦⣄⠀⠉⠓⠒⠶⠶⠒⠋⠁⢀⣠⣾⣿⡁⠀⠀⢸⣿⠀⠀⠀⠀⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠀⠀⠀⠈⣁⡀⠀⠀⣾⠟⠁⠉⠛⠿⣶⣦⣤⣤⣤⣤⣶⣾⠿⠛⠁⠙⢿⣦⠀⠀⠀⣴⣶⠀⢰⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣀⠀⢀⡀⠛⠁⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠉⣿⡏⠉⠁⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠘⠋⠀⠀⢰⠋⡀⠀⠙⡆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠰⣦⣤⣿⣦⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠑⠤⠞⠁⢀⠆⠀⠀⠀⠀⠀⠠⣤⣴⣿⠟⠉⠉⢻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠒⠁⢀⣶⣿⣷⣆⠀⠀⢀⣽⣿⡄⠀⠀⣰⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⠀⠀⠉⠉⣹⡿⢿⣿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠋⠀⠈⠟⠀⠀⠁

⠀⣶⠛⣷⠀⠐⠀⡞⡇⢰⣶⠖⠂⠈⠁⠁⠈⠀⠒⠶⣶⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣧⢉⡅⠀⠀⢈⠈⠀⠈⣿⣛⡟⣿⢉⠀⠀⢈⠀⠀⠸⢳⠂
⣀⣿⣀⣿⣀⣀⣀⣇⡇⠈⠙⠉⠉⠉⠙⠙⠙⠉⠉⠋⠉⠀⠘⢇⢄⢆⣤⡠⣠⡠⡸⣏⠁⡇⠀⢠⣶⣴⣦⣀⣽⡿⢸⣿⣀⣠⣶⣴⡆⠀⢰⣸⣀
⠉⠉⠉⠉⠉⠉⠉⠉⠁⣤⣄⣀⢠⠀⠠⡆⠀⣠⢀⢀⣤⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉
⣻⣿⣻⣟⢛⣟⣳⠀⠀⣿⡏⠀⠘⡇⡀⠀⠀⡛⠀⢸⣿⠀⢀⣀⣀⣀⢀⣀⣀⡀⣀⢀⣀⣀⣀⣀⠀⢶⣶⣶⣶⡦⠶⢶⣶⣶⣶⡶⠀⠐⣛⢻⠛
⠭⠭⠭⠭⠄⠅⠝⠀⠀⣿⣧⣤⣤⠇⡂⠀⡀⣤⣤⣬⣿⠀⠈⣿⠿⠿⠶⠶⣶⣶⣶⣞⣛⣛⣛⡏⠀⢺⣿⣿⣿⢟⣶⣺⣿⣿⣿⡗⠀⠰⠈⠌⠤
⢸⣿⠐⡇⠳⢘⠆⠀⠀⣿⡧⠄⢤⡏⠙⠙⢹⡧⠤⠔⣿⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣾⣿⠆⠀⢰⣿⣿⣿⡇⠀⠀⡷⢘⠆
⠼⠿⠶⠧⠿⠼⠧⠀⠀⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠆⠀⠙⠿⠹⠏⠿⠹⠿⠿⠿⠽⠿⠿⠇⠀⡸⠿⠿⠿⢂⠀⡸⠻⠿⢿⠇⠀⠠⠿⠼⠧
⣀⠤⠴⠒⠛⠲⠦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠚⠚⠒⠒⠛⠓⠓⠓⠓
⡀⣠⣄⢀⣤⡀⣤⡄⢠⣭⡛⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫
⡇⡏⠉⢸⠉⠀⠁⡇⢸⣿⣿⠀⠀⠀⠀⣤⣿⠏⠋⢿⣿⣯⣿⣭⢭⣿⠀⠀⢰⣿⢶⠿⣷⠾⡿⢷⣾⠿⠶⣿⣾⣿⠿⠿⠿⠿⣿⡿⠿⣯⠀⢰⣿
⡇⣏⡡⢈⢩⠀⣍⣅⢸⣿⣿⠀⠀⠀⠀⣭⣿⠀⠀⢸⣏⣹⣿⣋⣸⣯⠀⠀⢸⣿⠾⠯⣿⠿⠟⢽⡿⠨⠥⣿⣿⣿⠘⠃⠘⠃⢰⡇⠀⣿⠀⢸⣿
⠀⠉⠀⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠗⠊⠐⠉⠒⠀⠀⠈⠉⠀⠀⠉⠉⠈⠈⢉⡈⠀⠉⠉⠉⠀⣀⠀⠐⠉⠉⠀⠉
⣀⣠⡴⠶⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠿⠾⠿⠿⠾⣿⡗
⡿⡷⠲⡿⠻⡶⠺⣿⠾⠶⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣟⡛⠛⡟⠛⢛⣻⣿⣟⡛⠛⠟⠛⠛⡷⠀⠀⠀⠀⠶
⡇⡇⠀⡇⠀⡇⠄⣿⢀⠀⠀⠀⠀⣿⣿⡏⠉⢹⣿⠉⠉⣿⡍⠉⢹⡏⠉⠉⣿⠀⠀⠀⠀⠀⠁⠀⠀⠁⣿⡏⣏⢻⡇⠁⠀⡁⢘
⣷⣇⣢⣇⣠⣇⣲⣿⣤⣀⠀⠀⢠⣿⣿⣇⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣤⣥⣬⡄⠁⣾⣿⠸⣿⡇⠁⣤⣥⣬⡄⠀⠀⠀⠀⣄
⣿⣏⢸⣇⢸⡇⢸⣿⠘⠃⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⡀
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⠀⠀⠀⣄⣤⠀⠀⠀⢠⢠⡄⠀⠀⢠⢠⡄⠀⠀⡤⣤⠀⠀⠀⠀⢠⣤⠤⢤⠤⠰⠶⠤⠤⢤⣤⣤⠀⠀⠀⠰⠿
⢰⡆⠠⠰⣷⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡀⠀⢸⡿⠛⠻⣿⡟⠻⣿⡿⠛⢿⣶
⢻⡇⠀⢈⣿⠛⠻⠛⠛⠛⠀⠀⢸⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣿⡇⠀⣿⠀⠀⢸⣿⠀⠘⠟⠟⠛
⢸⡇⢀⢐⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠀⠚⠓⠒⠒⠛⠓⠒⠛⠒⠒⠚⠛
⢸⡇⠀⠰⣿⠀⠐⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⡀⠀⠀⡀⢤⡄⣀⠀⠄⣀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠂⠂⠐
⣸⣇⣀⣨⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⣉⣉⣙⣿⡃⠀⠈⣭⣷⣶⣶⣶⣶⣶⣶⣶⣷⣾⣭⠀⠀⢸⣯⣿⣯⡽⡯⣭⣿⣯⣿⠀⠀⠀⢀
⣿⢹⣿⡏⢹⣻⣄⣦⣳⣼⠀⠀⠀⣿⡿⢿⣿⢿⣿⡇⠀⠀⣿⠁⠀⢸⣿⠀⢸⣿⠁⠀⢸⣿⠀⠀⢸⣿⣿⣿⡀⠂⢸⣿⣿⣿⠀⠀⠀⢸⣦⣷⣼
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢂⣀⣀⣉⣉⣉⣉⣉⣉⣁⠀⢀⣿⣀⣀⣸⣿⣀⣸⣿⣀⣀⣸⣿⠀⠀⠸⠷⠶⠾⠷⠷⠿⠶⠶⠿⠀⠀⠀⠛⠛⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⡀⠠⠾⢿⠿⢿⣭⣭⢭⠿⣿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⡴⡤⡤⡦⣼⡷⠄⠀⠀⠀⢀⣀⣀
⠉⢉⣉⣉⠉⠉⣹⣿⡇⠀⢈⢉⣤⢸⠂⠀⢸⢀⡉⣠⡇⠀⢶⣶⣶⣶⣶⣶⣖⣒⣒⣲⣶⣶⠀⢰⠿⠿⡿⠿⠶⣶⣶⡶⠿⢿⠿⠿⡧⠀⢸⣿⣏

⠀⠒⠒⠒⠒⠂⠒⠒⠐⠒⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠠⠄
⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠶⡖⣒⠩⡩⣽⠿⠿⣿⣷⣶⣶⣦⣤⣀⡀
⢸⠀⠀⠀⠀⠀⣀⠴⢚⣭⣥⠀⠋⢻⠁⣿⠆⠣⣻⡤⠀⢿⠘⣿⡿⣛⡻⣿⣿⣷⣦⣄
⢸⠀⠀⢀⡴⣋⣥⣾⣿⣿⣿⠀⠀⠈⣦⢸⡀⠀⡘⠉⠓⢹⠀⢿⣷⣿⣿⣿⡍⠙⢻⣿⣿⣦⣀
⢘⣠⠞⣥⠀⠈⠃⢸⠿⠿⢿⠀⠄⠀⢣⢸⠀⢀⡹⠦⠤⢼⠀⢹⣿⣿⣻⣿⡶⠾⠻⠿⠛⢿⣿⣷⣄⠀⠴
⢸⣧⡦⣹⠀⠀⠀⢸⠖⡻⢸⠀⢀⠀⠊⠐⠃⣸⠇⠄⠀⢼⠃⣙⡿⡿⣿⣿⠀⢀⣀⣀⣀⣄⣽⣿⣿⡏
⢸⡏⢑⡝⠀⠀⠀⢘⠺⣩⣿⠀⠈⠁⠀⠀⠠⡇⠀⢡⠀⢸⣀⣻⣿⣶⢻⣿⢘⣯⣯⣯⣤⣿⣷⣿⣿⡇
⢸⡇⣿⣄⠀⢀⠀⢸⣿⣿⣿⡤⠄⢀⣴⣴⣶⣻⣿⡁⣀⣞⣻⡭⠟⢻⣞⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠠
⢸⡇⣿⣿⡏⠻⠀⠄⠀⠀⠀⣿⣶⠿⠿⡿⢷⡶⠶⡷⢿⣿⣿⣿⣶⣶⣿⣻⣿⣿⣿⣿⡻⣿⣿⣿⣿⡇⠘
⢸⡇⠛⠛⠀⠐⠀⠀⠀⠈⠙⣿⠃⠒⠖⠺⠠⣤⣶⡦⠤⠿⣿⣿⣿⣿⣿⣿⣟⡛⠛⣛⣱⣿⣿⣿⣿⡇⢸
⢸⡇⣄⡴⢀⠄⠀⠀⠀⠀⠀⣹⣶⣆⠄⠀⠀⠀⡠⠂⠀⠀⠈⣹⣿⣟⣿⢻⡏⡇⠀⠀⠀⠈⠁⣿⣿⡇⢀
⢸⡇⠉⠀⠀⢠⡖⠀⠀⠀⢰⡄⢸⡿⢀⠀⠒⠈⡀⢄⣠⣤⣄⠤⠚⢉⡿⣤⢹⡃⠀⠀⡀⠐⠀⣿⣿⡇⢸
⢸⡇⠀⠀⠀⣙⣇⠀⠀⢠⡞⣷⡾⠁⠾⠀⠀⡞⠭⣼⣿⠣⣀⣷⣠⣈⡬⠳⢽⣷⣤⣄⠀⠀⠀⣿⣿⡇⠘
⢸⡇⠄⣄⠉⠉⠀⠐⠒⣆⢻⡿⠓⣦⣠⣄⡀⣿⣖⡄⠉⠙⡛⢿⡇⣈⣧⣔⠲⣾⡟⡛⣷⣦⣄⣿⣿⡇⢀
⢸⡇⣿⣧⣀⣆⢀⢀⡄⠘⡼⣷⠀⠸⠿⠛⠀⣿⣿⣇⣤⣴⡟⠘⢛⢛⣻⠛⡛⣟⢻⣟⢻⣿⣿⣿⣿⣿⢻
⢸⡇⠉⣳⡤⣿⡏⠉⠻⠿⠤⣿⡄⠀⠀⠀⢸⡇⠘⢉⡉⢽⣧⣛⣉⣿⣙⣫⣛⣻⣿⣿⣾⣿⠟⣿⣿⣿⣾
⢸⡇⠀⢿⠀⣈⢳⣾⠡⠈⠀⢿⣿⠀⡆⠀⠿⣿⡇⢎⠉⢻⣤⣌⡻⣿⢥⡀⠹⣿⣤⣄⠀⠀⠀⣹⣿⡇
⠘⡇⣷⡿⡁⡒⢀⣏⣀⢒⠳⠘⡿⡓⣿⡀⣠⡈⣻⣾⠒⡼⠉⢧⣘⠓⣄⠛⡒⠛⢹⢻⢤⣤⡄⣾⣿⡇
⢠⡇⡾⠤⠟⠛⠛⣈⣛⣓⣆⡽⠇⠉⠻⠻⠛⡛⠉⢈⡛⠇⠸⡐⠋⣻⣄⣠⠗⢿⡁⢈⠻⢿⡴⣿⣿⡇
⢘⡇⠶⠀⠀⠒⠶⠀⠀⠐⠛⠐⠒⠖⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠿⠷⠤⠴⠶⠶⠶⠶⠦⣤⣿⣿⡇
⢘⡇⡀⠀⠀⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⣀⢀⡀⣠⡿⢿⣇⣀
⢰⠃⠖⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠶⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⢀⣀⡀⠐⠶⠶⣦⣴⡖⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⡔⠁⠜⢋⣀⡠⠶⠾⠛⠛⠉⠉⠀⠙⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢁⣴⡷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣠⡿⠉⣀⣴⠾⠛⠛⠛⠛⠛⠻⢶⣦⣀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣰⠋⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⡸⢃⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⡷⠻⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠴⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠉⢡⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠆⠉⠀⠀⠀⠀⣠⡀⠀⠀⠀⠰⠶⠶⠆⠀⢀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⣴⡄⠀⠀⠀⠀⣶⣤⣀⣀⠉⠀⠈⠀⠀⢁⣸⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⡄⠀⠀⠀⠈⢿⡉⠉⠉⠁⠐⠛⢿⠿⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣛⣦⣀⠀⠀⠀⠘⠓⠀⠤⠄⠚⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢦⡙⠿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⠟⢣⠘⣿⠀⠀⠀⠀⠀⢴⠟⠫⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠈⠿⣟⠀⠀⠀⠀⠀⢠⣤⡀⠀⠉⠲⣤⡀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠃⠀⠙⢳⡄⠀⠀⢀⣎⠉⢻⣦⡀⠀⠀⠹⠿⠁⢹⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⣄⣹⡆⠀⣼⠟⠀⠀⣿⣍⠳⣤⣿⠀⠀⠈⠀⠉⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠁⠀⠙⣦⡀⠀⣴⠉⠛⢳⣶⠋⢠⡞⠳⣾⠛⢆⣼⠟⢤⡄⣠⡛⠄⣹⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠻⢾⠇⠀⠀⠀⠉⣦⠀⠀⠀⠀⠀⠈⣻⠀⣤⣈⠙⠿⢾⡉⠀⠀⠉⠳⣦⣀⣴⠶⠶⠶⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠠⡤⠀⠘⣧⡔⠉⠛⣼⡅⠀⠀⢰⡿⢧⣤⣀⠀⠈⠉⠁⠀⠀⠀⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⣀⣀⣀⠀⠀⠀⢹⡇⠀⠀⠘⣷⠀⠈⠙⢷⠀⠀⠀⠀⡰⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠈⡄⠀⠀⢸⡇⠀⠀⠀⢻⡆⠀⠀⢸⡇⠀⠀⠸⣧⣄⠈⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠈⢷⡀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⣤⠘⠃⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣦⣀⣀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢷⡀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠙⠻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣷⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠏⠀⣹⡧⠤⡤⠀⠤⠤⢤⡄⠀⠴⢆⡀⢀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣶⠀⠀⢻⡇⠄⠀⠀⠰⠠⣼⡅⠀⠀⠀⠄⠼⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⠉⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⣼⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣰⡞⡿⠟⠁⠀⠀⠀⠀⢿⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⢸
⠀⣤⡄⠀⡀⣰⡧⣛⠀⠀⠀⠀⢀⣀⣀⡀⣷⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢸
⠀⡏⠙⣤⠀⠸⡇⠈⣷⣄⠀⠀⣿⠉⠉⢻⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡆
⠀⠃⠀⠀⠑⢄⠀⠀⠀⠘⠷⠶⡟⠀⠀⣿⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⢠⣄⣤⡀⠀⠀⠑⢤⠀⠀⠀⠀⣤⣤⣴⠇⠉⠳⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⢸⡇
⡄⠘⢧⡈⠀⠀⠀⠀⠙⢤⡀⠀⣀⣀⣀⣀⣀⣀⣀⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⣧⠐⠟⠁⠀⢠⡶⠀⠀⡈⠁⠀⠀⠀⠉⠁⠉⠉⠉⠉⠈⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠉⢦⡀⠀⠀⠀⣸⡆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠙⣦⡀⠀⠛⠀⠈⠟⠀⠒⠂⠀⠂⠀⠀⠀⠉⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠛⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠉⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠶⠶⠶⠶⠶⠶⠶⠒⠲⠿⢦⣤⣤⣠⣤⣤⣤⡴⠿⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⣸⣿⣥⠒⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢀⣠⣤⣤⣴⣷⣶⡦⠞⠛⠁⠀⠀⠛⠓⢤⣴⣞⣛⣁⣀⣠⠤⠤⠤⠤⢤⣿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣦⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣶⣶⣶⣶⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⠃⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⢻⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⣿⣻⣶⣶⣾⣿⠂
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡟⢂⣿⡏⢸⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣿⣿⣿⠟⠃
⠀⢠⣦⣠⣿⣿⣿⣿⡟⠈⠛⠻⢿⣾⣯⣤⣾⣿⣿⣿⣿⣛⡻⢿⣶⣦⣄⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠉
⠀⠈⢿⣿⡏⠙⠻⣿⣿⡄⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣭⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⢠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⡆
⠀⠀⠈⠉⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⣈⣽⣿⣷⣿⣿⣏⠀⢀⣀⣤⣤⣴⣶⠾⢿⣿⣿⣿⣏
⠀⠀⠀⠐⠛⠻⠿⠛⠻⢿⣿⣶⣶⣶⡶⢿⣿⣿⡿⠿⠿⠟⠛⠉⠉⠛⠿⣿⣿⣿⡾⠿⠛⠉⣼⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠙⢿⣿⣿⣧⡀
⠀⠀⠀⠐⠲⠶⠶⢦⣴⡿⢿⣿⣿⡷⣦⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⠉⠉⠙⠛⠻⠿⢿⣷⣶⣤⣤⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣸⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡄⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠻⣿⣿⣿⣿⡿⠟⠉
⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠀⠀⠀⢀⣠⠶⠛⠙⠟⠉⠁
⠀⠀⢻⣏⣹⣿⣿⣿⡿⠿⠟⠛⠛⠋⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⣀⣤⠞⠋⠁
⠀⠀⠘⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⣤⣄⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠉⠈⢩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠽⢀⢀⡴⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠦⡌⢆⡀⠀⠀⠓⠂⠀⠈⠈⡇⢰⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⡚⠉⢱⡀⠀⠃⣄⡀⠀⠀⠀⠀⢀⡤⢲⣦⡀⠀⠀⠀⠀⠀⠀⠓⠚⢼⠁⠀⠙⢋⠰⠤⠇⢀⠀⡰⢃⠏⡀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠃⠀⠠⠟⠁⢺⠇⢀⡴⠟⠀⠸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⡚⡮⡖⢄⠀⢙⠃⢘⣋⠻⡁⢸⡴⢁⡀⠀⠀⠀⡀⢣⡀
⠀⠀⠀⠀⠀⣀⠀⡧⣙⣾⡀⠀⠀⠀⣄⣶⣄⣋⣠⢀⠀⠠⠿⢿⡿⠇⢦⢤⣄⡀⠀⠀⠧⠙⡆⠘⢆⠈⢄⣘⣿⠀⢧⣾⠔⠋⢧⠀⢀⡀⡇⠀⠱⡄
⠀⠀⠀⠀⠀⠸⣿⠉⠒⠊⢡⠐⠴⢩⣹⣟⢯⣬⣽⡛⡷⢿⣿⣻⣿⢿⣿⡿⠛⠋⠀⠀⠸⢘⣧⣄⠈⠉⠛⢴⣾⡀⠀⡏⠀⠀⠈⢣⣀⣿⡇⠀⠀⡇
⠀⠀⠀⠀⣄⣠⣜⣢⣀⡀⡼⠉⠉⠉⠛⣲⡾⠟⢊⣷⣻⣾⢴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠀⠳⡀⠀⠈⣆⢍⣦⣟⡠⢄⣠⠛⢛⡭⣤⢶⢦⠵
⠀⠀⠀⢀⡨⡁⠀⠀⢉⡿⠀⢀⣠⠲⢻⣁⠤⠊⠁⣿⣼⡾⢺⣿⣿⣽⡞⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⣥⢴⡤⡬⣫⠴⡄⠱⠄⠃⠹⣟⣪⢿⣼⡴⣧⣣⡀
⠀⣠⠶⠵⠶⠼⡭⢽⣧⣄⠪⢋⣩⡇⢀⣱⠀⠀⠀⡇⠟⡗⠺⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣾⣟⠙⢆⡑⠝⢸⡇⠀⠈⢆⠀⢀⠀⠹⠔⠠⣾⣟⠹⣼
⠀⡷⠶⠤⠤⠤⠤⠤⣤⡷⠚⠁⠁⠓⠚⠁⢰⣘⡛⠋⠐⠓⠒⢿⣷⣧⠾⠄⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠙⢢⢺⠀⠀⠀⢸⡇⢸⡄⠀⠘⡶⠫⠛⢗⢿
⠀⡇⠀⠶⣀⣠⠶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣣⣌⠉⠐⠒⠶⠯⠤⢤⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⡱⠤⠖⠁⢸⠀⠀⠀⢸⣶⣯⠆⠀⠀⠈⠁⣻⣁⣼⠶⢦
⠀⡇⠀⡐⠉⢏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣞⠁⠈⠛⣶⠦⠤⠤⣶⢻⠃⠀⠀⠀⠀⣠⠖⢋⣉⠽⢚⠆⢀⠞⠀⠀⠀⢀⡟⠂⠒⠠⠤⠒⢃⣊⠁⣀⢴⡿⡟
⡐⣇⠔⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⡴⢿⠀⠀⠀⠈⠛⠛⠛⠋⡘⠀⠀⠀⠀⠀⣈⠷⠆⠀⠶⠿⣋⠁⠁⢰⠀⠀⣾⡟⣿⡿⣿⠶⠲⡮⢿⡿⢞⣿⠟⣇
⠙⣇⣀⣀⣀⣀⣀⣀⣀⡁⣀⡠⠔⠊⠁⠀⠀⣇⠀⠀⠻⠬⠥⠭⢁⠇⠀⠀⠀⠀⠀⡏⠲⠒⠒⢒⣒⡞⠀⡀⠃⠀⠸⣻⡏⠹⠾⠿⠆⠀⣾⢻⠒⠉⠯⠅⠿
⠀⠉⠉⠉⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⢀⠠⡄⡺⢄⣀⠀⠀⠀⣀⣾⡀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⠈⡏⠈⠁⠀⠀⣠⠷⠇⠀⠀⠀⠀⠘⠯⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠓⠚⠓⠻⡏⠻⣏⡭⠕⠃⠨⣯⠉⠉⢁⣐⡾⢀⢀⣀⠀⣠⠓⠁⠀⠀⡼⠁⠀⠀⠀⢀⢀⣲⡆⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⣷⣆⣁⣠⣄⣀⡀⢨⣎⣲⡈⠀⠑⠒⠂⠒⠚⠚⠷⠦⠤⠋⠀⠀⢀⣠⣾⡿⣪⡉⣿⣍⣽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠁⠀⠘⡌⠁⠂⠀⠀⠠⠂⠀⡀⠀⠀⣠⣴⠾⢛⠟⠁⠀⣉⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡠⠔⢃⡀⠔⠚⠉⠀⠀⠛⠛⠛⠛⠛⠓⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠒⠉⠜⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⣶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣶⡾⣿⠲⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⠿⠟⢛⣭⣽⢂⠀⢠⡶⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢤⣶⣾⣶⠿⣟⣛⣭⣵⣶⣿⣷⣶⠿⠟⠋⢘⣴⣿⡗⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⡿⠿⠛⣋⣸⣥⣴⣶⠈⣿⣿⣿⣿⣿⣿⡀⠄⠀⠀⠘⢿⡈⠐⠀⠀⣟⣂
⠀⠀⢠⣤⣤⣤⣤⣔⡺⢿⡿⠿⣛⣫⣭⣴⣾⣾⣿⣿⣿⣿⣿⣿⠾⠟⠛⠉⠀⣸⣿⡇⠀⠀⠀⠀⣀⣠⡴⢢⠞⠉
⢰⣶⣿⣿⡿⢿⣛⣯⣽⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣥⣴⣶⣶⣶⣾⣿⣿⣿⣧⣤⣴⣶⣿⣿⠉⡆⠊
⣸⣿⣿⠖⢋⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⢛⣿⣙⣿⠿⠛⠀⡇
⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡇⠰⠋⢹⣿⣿⣿⢿⣿⠀⠀⠀⠀⢷
⢻⣿⣿⣧⠙⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠘⣿⣿⡷⠀⠀⣾⣿⣿⡷⣾⣿⠀⠀⠀⠀⢸⡀
⠈⢿⣿⣿⣦⣴⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣤⣾⣿⣿⣿⣿⣼⣿⠀⠀⠀⠀⠈⡇
⠀⠈⠻⢿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠿⠛⢉⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣤⡄⢀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⠈⢁⣀⣠⣤⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⣿⣿⣯⣾⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣷⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣾⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠈⠉⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣶⣤⣤⣄⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⠀⠀⠀⠀⠀⢿⣿⠋⣁⣩⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⡿⠛⠋⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣠⣿
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠉⢠⡄⠹⣦⠀⠀⠀⣿⣿⣿⣇⢿⣦⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣯⣿⣿⣿⣿⠏⠀⠀⣠⡆
⠀⢠⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⣆⠙⢷⡄⠀⠸⣿⣿⣿⡜⣿⡆⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣼⣿⣿⣿⣿⣿⣷⠀⢠⠞⠉
⠀⠀⠻⣿⡇⠈⢹⣿⣿⣿⣿⣿⠀⠀⠈⢿⣄⡰⣿⡀⠀⢻⣿⣿⣷⣾⣿⣇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⢠⣾⣿⢿⣿⣿⣿⣿⣿⡴⠃
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠈⠻⣷⣿⣧⠀⠈⢿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⠃⣤⣿⣿⠋⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠘⣿⠛⣷⣀⠸⣿⣿⡟⠛⠻⣷⣿⣿⣿⣿⣿⣿⣿⢣⣾⣿⡿⠃⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⢹⣧⡘⣿⣧⣹⡟⠻⢷⣦⣼⣿⣿⣿⣿⣿⣿⣵⣿⣿⡿⠁⢀⣶⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣧⠀⠀⠈⢿⣧⠈⢻⣿⡿⢦⣤⡤⣿⣿⣿⣿⣿⡿⣿⣿⣿⠟⢠⣶⣿⣿⡟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣶⣶⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣴⣶⣶⢸⣿⣿⣿⣿⣇⣀⣀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⠉⢹⣧⣤⣤⡀⠀⠀⣿⣿⡇⢠⣶⣦⠀⠀⠀⠀⢸⣿⣿⣶⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⡇⣾⣿⣿⣿⠀⠀⠀⢸⣿⡿⣿⣧⡄
⠀⠀⢰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣷⣶⣶⣿⣿⡇⣿⣿⣿⣿⣤⣤⡀⢸⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣿⣆⣸⣿⣤⣾⣇⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⣿⣷⣿⡟⣿⣿⣿⣿⣿⢿⣷⣴⡿⠁⠀⠀⠀⢀⠀⢠⡀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣹⣿⣿⣿⠟⢣⢻⣉⣩⣽⣧⣿⣿⣿⣿⣾⡟⠰⣦⣼⣷⣾⣷⣿⣶⣿⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣛⠿⣅⣤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣉⠉⢻⡟⠁⠈⢻⣿⣧⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣾⣿⡿⠟⠛⠉⠹⡍⠉⠉⠛⠻⣿⣿⣿⣯⣷⣾⣿⣦⣤⣾⠿⣿⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣰⣄⣶⣲⢽⣿⣫⣿⣿⠟⠉⢦⣀⣠⠤⢶⣷⠶⠦⣤⣞⠀⠈⠛⢿⣿⣾⣿⣿⡿⠀⠀⠈⣿⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠃⠀⣠⠞⠻⣀⠀⠠⡿⠄⠠⣯⠟⠓⣤⢀⣨⢿⣿⣿⣻⡾⢿⡟⢻⣜⠛⠂
⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠲⣴⣇⢀⡀⠋⠀⣀⣅⣄⠴⠉⠢⡀⠘⢻⡀⠀⢻⣿⣿⣷⣾⢷⡞⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣇⣀⣿⣿⣿⣿⡇⠀⢰⡏⠈⠛⠀⢠⣿⣿⡿⣇⣾⣦⠀⠀⠀⠈⣧⠀⠀⣿⣿⣿⣿⣿⣙⣿⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠒⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⢼⣗⣲⡆⠀⣿⣿⣿⣅⣿⣿⣿⠆⠀⠓⠂⣿⠉⠉⣿⣿⣿⡿⡜⡟⠛⠒⡾⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡽⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⡇⠀⣠⡀⠘⢿⣿⣾⣿⣿⠟⠀⣰⢦⣀⡟⠀⢀⠿⣿⡋⡠⢟⣙⣤⣘⡳⣇⠑⢄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣼⡔⠋⠻⢭⣻⣿⣿⣿⡄⠀⣹⣾⠟⠓⢀⡀⠈⠉⠉⠀⣠⡄⠀⢉⡟⠓⠦⣼⣿⠒⠽⣱⡟⠉⠉⠉⢻⣎⢶⡩⡵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣧⡀⢀⣸⣿⣿⣽⣿⣿⣎⠁⠈⠳⣤⣼⠋⠀⠘⠇⠀⠈⣭⡶⠋⠀⠀⣜⢫⣏⣀⡇⢿⡀⠀⠿⠀⢠⡿⢘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⠛⠉⢻⣿⣿⣿⣿⣿⣿⢿⣿⣷⣄⠀⡰⠉⠛⠒⠒⡗⠚⠉⠁⠘⢠⣬⡷⠿⢿⣭⣵⠛⢌⠻⣦⣤⣴⠟⡡⠯⣝⡳⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⣄⣀⣰⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣦⣤⣄⣀⣧⣤⣤⣶⠔⣶⣿⣇⣀⣸⣧⣈⣣⣴⢫⠖⠒⢶⢻⣠⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣹⣿⣿⠏⠀⠉⣿⣿⣾⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⢛⣿⡿⢧⢿⣿⠟⠻⣎⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣼⠿⣷⣤⣴⣿⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣿⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⢸⣿⣿⣿⣷⡒⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⣛⣿⣾⠂⠉⠉⠙⣿⣿⣿⠉⢹⣿⣿⡇⠈⠉⠒⢦⣄⡀⠀⠀⣰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣻⣿⣄⠀⢀⣰⣿⣿⣿⣷⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣷⣦⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢿⣿⣿⣿⡟⠋⠉⠻⣿⣿⠏⠁⠈⠙⣿⣿⣿⣿⣌⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⣿⣿⡀⠀⠀⠀⣿⣿⡀⠀⠀⢀⣿⣿⣿⠉⠙⠁⠙⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⡿⠿⣿⣦⣴⣾⣿⣿⣿⣷⣾⣿⣿⡉⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡿⠉⣿⡿⠻⣿⡟⢻⣿⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠻⠀⠀⠉

⠀⠀⣀⣀⣤⣤⣤⣀⣤⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣦⣀⣠⣶⠀⠀⠀⠀⠀⢀⡴⠶⠦⣄⠀⠀⠀⢤⡀⠀⢠⡄⠀⠀⠀⣾⢰⡷⣾⡄
⠀⠀⣿⣿⣿⡿⠿⠛⠋⣽⡿⠋⠹⢿⣿⡏⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠀⢶⠆⢹⠂⠀⠀⠈⢷⣴⡟⠀⠀⠀⠀⢿⢸⣇⣼⠁
⠀⠀⠋⠉⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣤⡴⠋⠀⠀⠀⣠⠞⠛⠦⡄⠀⠀⠀⢈⣸⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣾⠀⠀⠀⠰⡇⢰⣷⠂⣻⠀⠀⢠⡏⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠒⠂⠀⣀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠘⠀⠀⠀⠀⠙⠦⠤⠖⠁⠀⠀⠀⠳⣄⣠⠼⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠗⠓⠒⠓⠃
⠀⢸⣿⠿⠟⠛⠉⠁⠀⢀⣾⠟⠁⢻⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠸⣿⡧⠀⠀⠀⠀⠐⠆⠰⠂⠖⠀⠀⠀⢺⣿⡗⠀⠀⠀⠀⠀⠀⣿⣅⣈⣿⣯⣀⣽⣿⣁⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠇⠀⣷⣌⠀⣡⡾⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁
⠀⠀⠀⢤⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⢴⡙⢿⠟⣡⡆⠀⣮⡻⡿⢋⣵⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠻⣿⣿⣿⢿⠛⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢨⠻⣷⡾⢋⡄⠀⣝⢿⣶⠟⣡⠀⠀⠈⢭⣿⣿⣿⣿⣶⢲⢋⣭⢳⢰⣾⣿⣿⣿⣯⠍
⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠀⠘⢷⣮⣴⠟⠁⠀⠻⣷⣥⡾⠋⠀⠀⠀⠀⠰⠿⠿⠛⠋⠉⠲⠔⠊⠊⠛⠻⠿⠷
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠊⡠⢐⣓⠂⠌⠢⠀⠀⠙⠁⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠰⠰⠆⠀⠰⠰⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣿⣸⠡⠶⢹⣼⣷⣶⣶⣿⡿⠋⠁
⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⢄⠁⠌⡏⠄⠊⠐⠀⠀⠀⠀⠀⢠⣤⣤⣶⣤⣤⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠑⠒⠙⠛⠙⠉⠉⠁
⠀⠀⠀⢠⣿⣷⠀⠀⠀⠀⢠⡄⠀⠀⠁⠀⠆⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⢻⣿⡇
⠀⠀⠀⡥⡯⢽⠥⠀⠀⣀⣾⣷⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣾⣿⡆⢀⣀
⠀⠀⠀⣛⣟⣻⡛⠀⠀⣿⡿⣿⢾⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣶⡄⠀⠈⠙⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠈⠉⠋⠉⠀⠻⠂
⠀⠀⠀⠉⠿⠏⠁⠀⠀⠛⠿⠟⠃⠀⠀⠴⠿⠛⠛⠛⠿⠟⠛⠉⠀⢠⡤⠤⠤⠤⢤⡄⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⣴⣄
⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣄⣠⣠⣄⣠⣀⣀⣀⣀⣀⠀⠀⢸⣷⣶⣶⣶⣾⠃⢠⣿⣿⣿⣿⣿⣷⠀⠀⢦⣀⠀⣿⣿⣆
⠀⢸⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠀⠀⠈⣿⡏⠀⣹⣿⠀⢸⣿⣿⢿⣿⣿⣯⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡶⠶⠶⠦
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⠟⠀⠀⠻⢿⣿⣿⠿⠃⠀⠀⠞⠋⠁⣿⣿⠏
⠀⢰⡇⠀⣧⠀⣾⠀⢸⡄⢸⡇⢀⡇⠀⣿⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋
⠀⢸⣇⢠⣿⠀⣿⡀⣼⡇⢸⡇⢸⣷⠀⣿⠀⣿⡄⠀⣾⠀⢸⠀⢰⡆⠀⡇⠀⣷⠀⢸⠀⢰⡆⢀⡇
⠀⢸⣿⢸⣿⠀⣿⡇⣿⡇⢹⣿⢸⣿⢸⣿⡇⣿⡇⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣶⠀⢰⡆⢰⡆⠀⣶⠀⣶⠀⢰⡆⢰⡆⠀⣶
⠀⢸⣿⢸⣿⢀⣿⡇⣿⡇⣸⣿⢸⣿⢸⣿⡇⣿⡇⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⡀⢸⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿
⠀⠈⠉⠈⠉⠀⠉⠁⠉⠁⠈⠁⠈⠉⠀⠉⠀⠉⠁⠀⠉⠀⠉⠁⠈⠁⠈⠉⠀⠉⠀⠉⠁⠈⠁⠈⠉⠀⠉⠀⠈⠁⠈⠁⠀⠉⠀⠉⠀⠈⠁⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⠲⠤⢄⡤⠴⠒⠒⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉⠉⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⡆⠀⠀⠀⠀⠀⣠⢋⡜⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠀⡴⠒⢄⢀⣠⠗⠉⢀⣀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡄⠑⠒⣺⢡⠃⠀⢀⡎⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠘⢄⣀⡤⠊⠑⠤⠖⠁⠘⠦⠴⠋⠳⠔⠃⠘⠦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠀⠀⠀⠀⠀⠀⣠⠎
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⡟⠀⡠⡄⢠⣤⠀⠀⣠⡉⠀⠀⣠⡦⢦⠀⠈⣿⡔⠒⠆⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠁⠈⠀⠈⡏⠀⠀⢀⡷⠁⠀⠀⡸⠀⡼⠀⢀⠇⠘⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⠤⠚⠀⠣⠔⠂⠀⠣⠤⠴⠊⠳⠴⠚⠐⠃⠀⠘⠒⠊⠀⠀⠉⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠊⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠦⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠔⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠉⠑⡄⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡗⠠⠄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠉⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠈⠢⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠄⠀⠐⠁⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠑⡀⠀⠘
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⠈⠀⠀⠀⠀⠈⠀⠀⢠⠋⠀⢹⣷⣦⣤⣀⡀⡇⠀⠀⠀⠀⠀⢡
⠱⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢡⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆⠀⠀⡇⠀⠀⣿⣿⡇⠀⡔⢊⠃⠀⠚⠑⠘⢄⣰⠀⠀⠀⠀⢀⢄⣿⠾⠻⢷⣕
⠀⠀⢃⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⡇⠀⠀⡆⠀⠀⣿⣿⣧⢸⠀⢂⡀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⡌⣿⠁⠀⠀⠀⣿
⠀⠀⠀⢂⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⣿⣿⣿⢘⡀⠀⠉⠉⠁⠀⡰⠀⠀⠀⠀⠀⠀⡿⡙⢷⣤⣠⣴⢟
⠀⠀⠀⠈⢆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢹⣿⣿⠀⢙⠂⠄⠀⡤⠊⠀⠀⠀⠀⠀⠀⢀⠀⡀⠐⡬⠍⠒⠁
⠀⠀⠀⠀⠈⡄⠀⢰⡉⠑⠤⣀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠁⠀⠀⢸⣿⣿⠀⠈⠀⣤⡀⠆⠀⠀⠀⠀⠀⠀⢀⠎⣼⡟⡰⠁
⠀⠀⠀⠀⠀⠘⣀⠇⠈⠑⠢⠤⢈⡁⠒⠢⢄⡀⠙⡄⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⢰⠀⣿⠀⡄⠀⠀⠀⠀⠀⢀⢂⣾⠟⡐⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⢲⣿⣿⣷⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠘⢸⣿⠀⠀⠀⠀⠀⠀⢠⢃⣾⠏⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⡇⠀⠀⠀⠀⠀⢘⠀⠀⠀⠘⣿⣿⡇⠀⢸⡟⢸⠀⠀⠀⠀⡠⢁⣾⠏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠧⠧⠀⠀⠀⠀⠀⠼⠀⠀⠀⠠⠿⠿⠧⠤⠾⠧⠴⠀⠀⠀⠴⠥⠾⠏⠬⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣬⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠙⠪⡙⠓⠦⣄⣀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠀⠈⠲⣄⠀⠉⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠙⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⠀⠀⠀⠀⠙⠲⣄⠀⠈⠳⡄⢀⣀⣠⣤⡴⠾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⣄⠀⠀⣀⣈⡧⠤⢖⣛⡋⡩⣥⡀⠜⢉⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠾⠟⠛⣛⣛⣯⣭⣷⣶⣿⣿⣿⣿⡇⡁⣸⡾⠠⠾⣻⣁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⢀⡤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣃⣨⣭⠶⠶⣛⣛⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢣⢸⠀⢸⠿⢸⣿⡿⠿⣛⣛⣭⡽⠷⢞⣛⡫⠭⠐⠒⠉⢉⣁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣼⠿⢈⣓⣻⣥⠴⠶⣚⣫⡭⢥⠶⣾⢿⣽⣿⣠⣤⣶⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠰⠞⣛⡫⠭⠖⠒⠉⠉⢀⣇⣧⡿⠖⠛⣋⣭⣿⣿⣿⣿⣿⠟⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣤⣤⠶⣞⢫⠋⠛⠛⠛⠚⢛⣷⣶⣭⣉⡉⠁⢀⣀⣤⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠧⠾⠬⠽⢿⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⣠⣀⡀⠀⠀⠙⢷⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠋⢀⡼⢛⣶⣶⣭⣥⣄⣀⢡⣉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠸⣿⣿⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢍⣙⡛⠻⠿⣿⣿⣿⣿⠇⠀⣠⢀⡀⠈⠉⠉⠙⠻⣿⣿⣿⣷⣦⣿⣙⣾⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡬⢉⣓⠒⠼⢟⣁⣤⡶⢋⣨⡶⠚⠛⠀⠀⠀⠈⠉⠛⠻⠿⣿⢙⣽⣿⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠀⠀⠘⠓⠃⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣶⣭⣿⣵⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⠉⠉⠉⠛⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠋⠉⠉⠉⠛⠓⢲⢤⣄⡀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣀⠀⠀⠀⠀⢠⣦⠰⣿⠇⠀⣭⡓⠦⣄⡀⠀⠀⠀⠉⢳⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠥⠼⠃⠀⢸⣻⣮⠟⠛⠙⠲⢼⡟⠁⠀⡀⠉⠓⢦⡀⢀⡟⣦⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⡄⢀⣤⢀⣸⠃⠀⠀⠀⠀⠀⠓⢤⡿⠋⠀⠀⠀⣙⡾⠀⣇⣀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠖⠃⠀⢹⡏⠀⠽⠂⠀⠀⠠⢶⡆⠈⢣⡴⠶⠂⠈⠉⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣯⡿⠁⣠⣄⣠⣄⣀⣠⡀⢀⡀⢸⠀⠀⠀⣄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠳⡀⠀⠀⠁⠀⠉⠀⠈⠉⠀⢈⡷⠤⠼⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠤⢤⣤⣄⣠⡤⣶⡿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠋⠙⠲⢶⣶⠶⢾⣋⣉⠀⢸⣧⠳⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⣨⣧⣄⣠⣼⣧⣀⣀⣈⣉⣠⣼⡇⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⣿⢻⠀⢠⡄⠀⣄⠀⠀⠸⠀⠀⠀⣴⢃⡜⠉⠛⠒⣶⠶⢭⡉⠁⠀⢸⡇⠀⢳⡀⢘⡆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⡇⠘⡆⣸⢧⢠⢿⡀⠀⠀⠀⣠⠂⣿⢿⡁⠀⠀⢸⠁⠀⠀⠙⠒⠦⣼⡇⠀⠀⠙⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣳⠈⢠⡄⠙⠃⠘⠋⣸⠁⠀⠀⠐⠇⠀⢁⣸⠧⠤⠤⣼⡀⠀⠀⢰⣧⡖⠃⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⠀⡼⠙⣆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠙⠒⣾⣲⣶⣏⣙⣛⣻⣛⣲⠭⣟⡷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠁⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠰⣯⣥⣤⠤⣾⠿⠭⢞⡿⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣧⣀⣀⣠⣧⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢡⢾⠉⡏⢇⠙⣍⠪⣝⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠳⣴⠃⣰⠖⢾⡘⡶⠋⣳⠸⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠉⠒⠋⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢃⠶⡞⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⡄⠀⠀⡠⠊⡤⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣸⣤⣧⣿⣿⠀⠀⠄⠀⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠄⠀⢴⠁⠤⠭⠇⠈⣦⢀⠤⠐⠂⠐⠢⢄⠀⠀⠀⠀⠀⠀⢸⠻⢛⢙⠛⣛⡛⢛⠀⠀⠀⠄⠀⠐
⠀⠀⠀⠀⠀⠐⠮⠽⠭⠦⠀⠀⠄⠉⠀⠈⠀⠈⠀⠤⠠⠄⠀⠀⠠⢤⠈⣀⠀⣈⠀⣀⠈⠀⠀⠀⣀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠙⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠆⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⣶⡗⢹⠀⠀⠋⠁⠋⠉⠛⠁⠀⠀⠘⠁⠑⠀⠀⠤⠤⠀⠤⠄⠴⢄⡀⠰⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠐⠂⢀⣁⣃⠀⠠⠲⡂⠀⠀⠁⠀⠀⠀⠀⣓⣒⢛⣒⣚⠀⠀⣉⣉⣉⣉⣉⠁⠀⠤⠀⣶⠀⠀⢈⠁⠈⠉⣀⠉⠁⢈⡅⢰⠄⢀⠀⠀⡂
⠀⠀⠀⠀⠺⠂⣄⢐⢸⠀⠶⠀⠇⠬⣴⠀⣭⣷⠀⠀⡿⠠⢼⡍⠸⡁⢸⠀⠄⠀⠀⠀⢸⠇⠀⠆⡄⡤⡄⠀⠀⠀⠀⠛⠃⠀⠀⢀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⠤⣠⠤⣄⡅⣄⡤⣄⠀⠀⠒⠂⠓⠒⠀⠀⣉⣉⡉⣉⡙⠁⠈⠉⠁⠈⠁⠁⠚⠁⠀⡉⡉⡉⡁⠀⡎⠁⢀⡉⣂⣐⢢⣘⠠⠁⠈⠀⠈
⠀⠀⠀⢸⣀⠿⣀⣿⣃⣹⣒⢚⠐⠀⠀⠂⠀⡀⠊⠀⡇⠀⣿⡇⢚⡇⠰⠿⠆⠿⠇⡠⢢⡀⠀⣀⡀⣀⡀⠀⠂⠀⠒⣤⡴⢂⡠⢾⢠
⠀⠀⠀⠀⠀⠰⠉⣲⠋⣷⠞⢳⠀⠀⢘⠀⠀⡁⠀⠈⠉⠁⠉⠁⠘⠓⠀⠂⠐⠀⠼⠁⠀⠁⣀⠀⠀⡀⣁⠀⠀⣀⡀⠭⠅⠠⠦⢨⣀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⣀⣿⣀⣿⡇⣸⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠆⠶⠆⢅⢠⡄⣟⠀⠀⡇⠐⢀⣸⠒⢺⡉⢉⠉⠉⣽⡅⡇⠒⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠒⠒⠒⠀⠠⠄⠠⠄⠸⠼⠄⠋⠉⠉⠁⠀⠚⠰⠀⠸⠁⠀⠀⠀⠸⠇⠇⠙⠃⠐⠒

⠀⠀⠀⡄⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠠⣦⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⢠⠀⠀⠀⠀⢰⡆⠀⠁⠀⠀⠸⢿⠟⠀⠀⠀⢰⣿⡆⠀⠀⠀⠃
⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⢠⣶⡄⠀⢸⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⡇⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠘⠛⠃⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⢶⡶⠒⠀⠀⠀⣀⣀⡀⠀⠀⣶⣴⣦⡆⠀⠀⠀⢰⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢰
⠀⠀⢸⡇⠀⠀⠀⠉⠛⡛⠃⠀⠀⠁⢸⠁⠁⠀⠀⠀⣸⡀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣸⣀
⠀⠀⢸⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢹⠈⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⠀⠀⠀⢀⣧
⠀⠀⢸⡇⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈
⠀⠀⠸⡇⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸

⠀⢀⣶⣀⣶⡀⠀⠀⢀⣰⣦⠀⠀⠀⢠⣤⣶⡄⠀⠀⠀⠀⠀⢀⣀
⠀⠘⡿⡿⡿⠃⠀⠐⠉⡿⡿⠀⠀⠀⢸⢠⠟⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣯⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⣶⣿⣷⣦⠀⠀⣠⣶⠀⠀⠀⠀⣤⣶⣶⣦⡀⠀⠀⡀
⠀⠸⣿⣧⣿⣿⠀⠀⠛⢻⣭⡄⠀⠀⣿⣿⣿⣿⡇⠀⣾⣿⣷⢦⣦⣀⣀
⠀⠀⠉⠛⠋⠁⠀⠀⠈⢹⡿⠁⠀⠀⠙⠿⡿⠛⠀⠀⠿⠿⠿⣼⣏⠙⣯⡙⠓⠲⠦⣤⣀⡀
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣷⠀⠀⠙⢶⣄⠀⠀⠀⠉⠙⠻⣶⣦⠄
⠀⠀⠀⣸⣇⠀⠀⠀⢀⣀⡀⠀⠀⢰⣀⣀⣀⣀⡄⠀⠀⢀⡟⠀⣿⠀⠀⠀⠀⠙⢷⣄⠀⠀⣿⣿⠄
⠀⠀⢸⣽⣽⡇⠀⠀⢸⡿⣿⠀⠀⠀⢹⣿⣻⡏⠀⠀⠀⠸⢷⣾⣿⡶⠶⠶⠶⢶⣶⣿⢷⣶⣾⡿⠆
⠀⠀⠈⠿⠟⠀⠀⠀⠘⠿⠟⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠀⠀⣀⡠⠤⠔⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠀⠀⠉⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣊⡉⠉⠠⠠⡀⠀⡤⠀⠀⠉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣴⣾⡿⠛⢿⣿⣿⣷⣦⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠁⢀⠇⠀⢃⠈⠁⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠙⠉⠁⠀⡀⠀⠈⠉⠉⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠀⠁⠀⠀⡃⠀⠀⠈⠀⣿⣿⠀⠀⣐⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡔⠋⠀⡘⠄
⠀⢀⠔⢳⠀⠀⡗⢦⢀⣿⣿⡄⠀⠀⡄⣤⣄⠀⠀⣼⣿⣿⣆⠀⣼⠀⢎⠈⢩⡇⠀⠀⠀⠀⠀⠀⡠⠔⠋⠁⠀⠉⠁⠀⠀⠀⠀⠉⠈⠙⠲⡄⠈⠂⠁⡈
⠀⢼⠀⠀⠦⠔⠃⣸⢾⣿⣿⣿⣦⣄⠁⠀⠁⣴⣾⣿⣿⣿⡟⢰⣿⢀⠎⡜⠉⠀⠀⠀⠀⠀⡰⢋⣀⡀⢀⠔⢢⡀⠀⠀⠀⢠⠆⠀⠀⠀⠀⢨⠀⣘⠈
⠀⠈⠢⠤⢄⠀⠘⣁⠼⠟⠛⡋⠉⠉⠁⠂⠀⠀⠀⢸⠉⠉⠓⣾⣯⢎⠌⠀⠀⠀⠀⠀⠀⡼⠀⡾⠀⠈⠁⠀⠀⠈⠉⠉⡇⢸⠀⠀⠀⢸⠀⡌⢀⡋⠳⡀
⠀⠀⠀⠀⠈⡄⠀⢅⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣿⢃⠎⠀⠀⠀⠀⠀⠀⡜⠁⠀⠠⠸⡁⢀⠤⠠⡄⠢⠀⠱⢄⠉⠉⠉⢁⠾⢅⡼⠁⢢⢳
⠀⠀⠀⠀⠀⢡⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⠳⠥⠎⠱⡀⠀⠀⠀⠀⢰⢇⠀⢐⣈⠅⠀⠸⣀⣀⠌⠠⠀⣠⠚⠀⠀⠀⢸⠦⠄⠀⠀⠀⡌⡇
⠀⠀⠀⠀⠀⠸⣆⠔⢳⡀⠀⠧⢀⢀⠀⠀⠀⢀⡀⠸⠰⡉⠂⠀⠀⠀⡇⠀⠀⠀⠀⢸⠈⠢⠤⠞⠀⠀⠀⠀⠀⠀⠁⠀⢇⠀⠀⠀⠀⠈⢯⣂⠀⢀⡴⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠉⠀⢀⠜⠀⠀⠀⠀⠀⢰⠀⠀⠈⠓⠒⠒⠤⠀⢀⡤⠒⠲⠊⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠊⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠐⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡖⠐⠒⠒⠒⠌⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡦⠤⠤⠤⠄⣂⡀⡄
⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣷⣬⣤⣤⣤⣤⣤⣤⣤⣤⣍⠰
⠀⠀⠀⠀⠀⠘⢻⠟⠛⠻⡟⠛⠛⠉⠉⢀⠄⠀⠀⠈⢉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠀⠻⡈⠑⢆⠘⢤⠉⢩⡭⠈⠙⠛⠋⣭⠟⠛⠑⡆
⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⠀⠀⠈⡆⢸⠆⠐⠀⠈⠀⠠⡀⣰⠀⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠑⠒⠊⠀⠸⡀⠀⡆⠀⠀⠀⢀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⣍⡄⢀⣀⡀⠠⣇⠀⠀⠀⠀⠀⠀⣸⡀⠀⢀⠜⡇⢀⠋⣂⠀⠀⠀⠀⠀⠈⠑⠒⠐⡄⠀⠀⠘⡤⠘⢖⠂⠠⣤⡤⠤⠐⡮⠤⢄⣀
⠀⠀⠀⠀⠸⠉⠀⣿⠾⠂⡏⠉⠋⠀⠘⠉⠉⢹⠠⠆⢹⠣⠀⠈⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⢀⠄⢺⣄⠀⠙⠚⠁⠈⠲⠊⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⣿⡇⠀⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠃⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠘⠃⠀⢊⠜⡂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⡠⠟⠛⠒⣃⠀⠀⠧⠤⠤⠤⠤⠤⠤⠼⠀⠀⢸⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠸⠀⠀⠀⢈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢳
⠀⠀⠸⠀⠀⠀⣈⣭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠜⠐⠚⠢⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠲⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⡄
⠀⠀⠰⣀⠀⠀⢤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢍⣀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠀⠀⠀⠀⢸⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠈⠻⠟⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣀⠀⣀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡉⠁⢻⠀⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣿⣦⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡏⢹⢟⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠠⣶⣿⣿⣿⣿⣿⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣷⡀⠀⢹⣿⡿⠛⠻⠿⣿⣿⣧⡘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠀⢿⡇⠀⠈⣿⣿⡀⠀⠀⠸⢿⣏⠉⠀⢸⣖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣏⠀⠀⢸⣧⡴⠀⣿⣿⣿⣦⣄⡀⠸⣿⡆⠀⠈⣿⠀⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⢿⡌⢦⠈⣿⣷⢻⣻⣿⠉⠈⠙⠹⣶⡿⠷⠄⠶⣿⣀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⣠⣼⣿⣿⣧⣿⣹⣇⡙⠿⣷⣄⣀⡀⠈⠀⠈⠉⢠⣾⣧⣏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣯⣙⣿⣛⣿⠟⣿⣟⣿⣿⣿⡄⠀⢸⡿⢦⣤⠀⠀⠀⣠⡾⠀⠀⣹⣤⠖⠂⠄⠀⠀⠀⠀⣤⡐⢦⣎⢾⣉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⡷⢜⣻⣿⠋⠀⠘⣻⣿⣟⣿⣿⣶⠀⠀⠀⠄⠀⠀⠁⠈⠁⠀⠀⣿⡏⠀⠀⠀⢀⣨⡶⣿⣧⣜⢷⣼⡿⠻
⠀⠀⠀⠀⠀⠀⢠⣿⡏⢿⣿⡸⠉⣿⡆⠀⠀⣼⣿⡟⢿⣿⣿⣶⣠⣤⣤⣤⣄⣠⣆⠀⢸⡀⣿⣿⣧⣶⢪⣟⣧⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣨⣿⣿⣧⣿⡄⠀⠰⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⢈⣀⣿⣄⡿⡿⠟⣩⣿⣿⣿⣿⣄⠁⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡏⢹⣻⣿⣿⡿⠁⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣣⠔⠻⠿⣿⣿⣿⡿⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠾⡇⢹⡟⣧⠀⠀⣸⣿⣿⣿⡿⣿⣷⣿⣿⡖⣿⣾⣿⡿⠟⣩⣴⣿⣿⠀⠀⠀⠈⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣿⠋⠀⠀⣿⢿⣿⣿⣿⣾⣟⣿⣿⠿⢿⠟⣁⣴⣿⣿⣿⣿⣿⣄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⣸⣇⠀⠀⣿⡇⢹⣿⣿⡟⠋⣽⡟⣀⣸⣿⣿⣿⣧⣶⣾⢹⡿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣤⡟⣿⡆⠠⣿⠀⣿⣿⡟⢛⣫⣿⣿⣿⣿⣿⣿⣿⣿⢱⡷⢸⣿⣸⡅⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠰⣹⣿⠐⣿⣤⠿⠋⣁⣨⣿⠟⣿⣿⣿⣿⣿⣿⣿⣸⣧⣰⣿⣿⠃⠈⠦⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣿⣿⠿⠋⠀⢨⣴⣿⣿⠿⢦⣼⡿⢿⠿⢿⣿⠿⠿⠿⠿⠿⠏⠠⠔⠒⠶⣶⣶⣄⠀⢐⣤⣤⣤⠠⡤⣤⡀
⠀⠀⠀⠴⠶⠾⣻⡿⣿⣷⣿⡿⠟⠋⠄⠂⢀⡀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡤⠤⠴⠒⠒⠋⠀⠀⢀⣲⣴⣧⣾⣿⣤⣀⣿⣟⠛⠛⠀⠀⣠⠤⢤⡀
⠀⠀⠀⠠⣶⣻⢯⣾⠟⡿⠋⠀⠈⣀⣤⡂⣉⣿⣿⡶⣶⠶⠟⢉⣉⣁⡀⠀⠀⠀⠀⣩⣶⣤⣄⣀⣘⣛⣉⣿⣿⡿⠿⠿⠿⢿⠿⠷⠤⣴⣿⣿⡆
⠀⠀⠀⠀⠉⠁⣾⠋⠊⠀⢀⡠⢞⠻⠿⣯⣿⢟⠻⣲⣯⣛⠶⢨⣉⣁⠀⠀⠀⠀⠈⠉⠀⠈⠁⠀⣤⣤⣤⣤⣥⣭⣂⣐⣂⣀⣒⣒⣒⡒⠖⠾
⠀⠀⠀⠀⠀⣸⡏⠀⠀⡴⡋⠀⣦⡀⣀⣀⡓⠺⠷⠿⠛⠻⠖⠛⠛⠻⠷⠶⠤⠶⢄⡀⡀⠀⠀⠤⣤⣖⣒⣊⣉⣉⣭⣭⣬⠿⠿⠿⠶⠶⠇
⠀⠀⠀⠀⠀⠛⠀⠀⠈⠁⠉⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠖⠂⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠛⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠐⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠛⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠀⠀⠈
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⢳⠘⡏⠃⢹⠘⠈⠁⠃⠀⠀⠀⠐⠶⠶⠒⠂⠐
⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢇⠀⠀⡤⠀⠄⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣡⠬⠖⠒⡒⢛⡋⢉⠉⣉⠛⡛⠒⠶⢤⣄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⣩⣄⡆⣿⠠⡇⠸⠇⠸⠀⠿⠰⡇⣾⢰⢢⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⢟⠁⠥⠀⣐⣀⣒⣀⣛⣀⣘⣂⣀⣚⠀⠈⢸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⠀⠀⠀⠐⣀⠤⠒⠒⣉⠉⣭⡍⣴⡆⣴⡆⢡⣭⢩⣍⢉⠛⠒⠶⣄⡀⢸
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣊⣤⣶⢸⠄⣿⢄⣿⡇⣿⡇⣿⡇⣼⡿⢸⣿⢸⣧⣿⢰⣠⠁⢸
⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣿⣿⡟⣼⠀⠏⠸⠛⠀⠟⠁⡟⠃⠛⠇⠿⠟⠸⠻⠟⣾⣾⠀⢸
⠀⢹⠤⠐⠊⠠⠐⠒⠋⠉⠀⠀⠀⠀⠀⠀⠟⣃⣁⡤⠤⠔⠒⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡟⣴⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⠘⠀⠄⣐⠋⠥⣄⠰⣶⢤⡶⠲⠾⠲⠟⠋⢺⣯⡝⠋⠉⠛⣛⣋⣙⣉⣥⣾
⠀⣶⠀⠀⠀⠀⣀⣴⡾⠛⠁⠾⢗⠀⠀⠀⠠⠠⣿⠦⠴⠶⠶⠞⠛⠛⠉⠁⢽⣿⣤⣻⡛⠉⠉⠉⠀⠀⠀⡟
⠀⣻⡐⠂⣂⠀⠙⠁⠒⠘⠀⡁⠀⡀⢠⡀⢤⡄⢻⠀⣤⠀⣤⡄⣦⠀⢻⡆⢸⠟⢿⠏⠀⠠⡄⠀⠀⣧⠀⡇
⠀⣼⡙⣉⣉⢀⣴⡄⣷⢸⡆⣿⠀⣷⢸⡇⢸⡇⢸⠀⣿⡇⣿⡇⣿⡆⣾⡇⢸⠀⢸
⠀⣺⡇⢸⢿⢸⢻⠇⡇⢸⡇⣿⢸⡇⢸⠇⣾⠃⣿⠀⣿⠃⣿⠀⣿⠃⣿⠁⢸⠀⣿⠀⠀⠒⡆⠀⠀⣴⠀⡄
⠀⣸⡇⣿⠸⢼⢸⠀⠇⠘⠀⠏⠈⠁⠈⠀⠉⠀⠋⠀⡿⠀⠟⠀⠏⠀⠁⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠿⠀⠋
⠀⣾⠁⢡⠀⠀⠀⠀⠰⠄⠀⠒⠂⢠⣴⣶⣶⣤⠀⠂⠀⠂⠀⠂⠀⠛⠀⠙⢰⣾⣶⡄⠀⡀
⠀⠹⣤⣎⡈⠈⠁⠀⠐⠂⣤⣄⣦⣼⡿⣯⣝⠈⠀⠀⠀⢠⡆⠀⡖⠀⣶⠀⢸⣿⣿⣿⠀⠸⣧⠀⠀⣿⠀⢠
⠀⣼⣭⣤⣄⡀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠸⣿⠀⠀⠀⠀⠸⠇⠈⡷⠀⣿⠀⢸⣿⣿⣿⣖⣒⣏⡀⠀⠟⠀⢸
⠀⢸⣿⠿⠿⠷⠖⠲⣀⣀⣙⣉⣙⣛⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⢉⣉⣉⣀⡈⠉⠉⠉⠛⠛⠋⠉⠉⠙⠃⢸
⠀⠺⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣥⣬⣭⣍⣉⣅⣀⣀⣀⣉⣉⣛⣛⣛⣛⣛⣛⣿⣷⠶⡾
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠈⠉⠉⠉⠉⠉⠰⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢁⡄⢶⠋⠉⠉⣙⢻⠋⠑⠶⣤⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡾⠋⠹⣄⢳⡀⠀⣿⢸⠀⠀⠀⣠⠟⣦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢰⠏⠀⠀⠀⠙⣆⠙⣦⡁⠸⠀⡴⠚⣡⠞⠁⢻⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡤⠄⠀⠲⠶⠒⠒⠒⠒⠒⠒⠒⠲⠂⠀⠀⠄⢠⡏⠀⠷⠠⠤⢤⣤⣤⣬⣧⠘⠁⠘⢁⣀⠊⠁⠀⠀⠀⢳
⢀⣠⠔⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠗⠒⠲⠢⠤⣤⣤⡀⠀⡀⠀⠈⣉⠉⠐⠒⠒⠂⣾
⣿⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢀⣴⣋⣤⡾⢻⣆⠙⣏⠉⠉⠉⠉⠀⢶
⣿⠀⠀⠀⠉⠙⠓⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣵⣤⣵⡦⢲⡶⠛⡩⠛⠉⡄⢸⡟⢶⡈⢷⡀⠀⠀⢰⠇⢰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠘⢂⡀⠈⠀⠀⠀⡇⢸⡇⠀⠙⣎⡱⣄⣰⠋⢠⠟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠑⠤⣀⡀⠀⣧⢸⡇⠀⣀⣨⡷⠉⢁⡴⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢈⠉⠁⠘⢻⡟⠈⣉⣁⣤⠖⠉
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢺⠟⠀⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿
⠀⣷⢠⡞⠙⠛⠦⠴⠦⠤⠖⠛⠛⠉⠉⢉⣛⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠛⠋⠉⠙⠛⠶⣄⠀⢠⡏
⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠁⠀⠈⢣⠉⠛⠳⠶⠶⠶⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇
⠀⢻⠀⣧⠀⠀⠀⠀⠀⠀⡞⠋⠀⠀⠀⣠⠂⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠇
⠀⢸⡆⢿⠀⠀⠀⠀⠀⣾⠀⢀⣀⠴⠚⠉⢣⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿
⠀⠀⡇⢸⡄⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇
⠀⠀⣷⠈⡇⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡹⠶⠖⠒⠐⠀⠒⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⢹⠀⣧⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡧⠀⠀⢀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⡇⣼
⠀⠀⢸⡇⢻⠀⠀⠀⠀⠀⠈⢷⠘⢦⡀⠀⣀⠴⠞⠁⠀⡧⠀⠀⠀⡟⠉⠉⠉⠁⠀⠀⠈⡇⠀⠀⠀⠀⣼⠁⣿
⠀⠀⠸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠑⠶⠞⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⢠⡏
⠀⠀⠀⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⢟⠀⡀⠀⠁⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢀⡏⢸⡇
⠀⠀⠀⢻⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠉⠳⢮⣵⣏⠀⠀⠀⠀⠀⠀⠀⢸⠅⡇⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠘⡆⢿⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢐⣇⡀⠀⠀⠀⢹⡗⠶⠠⠤⠤⠀⠀⠀⠶⠁⠀⠀⠀⣼⢠⡇
⠀⠀⠀⠀⣇⠸⡇⠀⠀⠀⠀⠀⢰⠋⠀⠀⢠⠏⠀⠉⠳⢦⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠇
⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⢃⣦⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠀⠘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟
⠀⠀⠀⠀⠀⣧⢸⡇⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⡄⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠇
⠀⠀⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣄⣀⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡇
⠀⠀⠀⠀⠀⠀⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠁
⠀⠀⠀⠀⠀⠀⢻⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢙⡆⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠞⠋⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⢦⣀⠈⠙⠛⠛⠲⠶⠶⠶⠶⠶⠶⠖⠛⠛⠉⠁⣀⡤⠾⠋⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠉⠛⠳⠶⠦⢤⣤⣀⣀⣀⣀⣤⠤⠴⠶⠛⠛⠉⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠦⢀⣀⣀⣀⣀⣀⣀⣀⡤⠶⠶⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣀⣠⣿⠙⢶⡦⣤⠀⢀⣼⣷⣶⣶⣿⣦⡀⠀⠀⣠⣴⣶⣶⣤⣀⠀⠀⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃
⠀⠀⠀⢸⣿⣿⠟⠀⠘⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⡁⠀⣾⣿⣿⠟⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠔⠉⠀⢹⣿⡇⠈⠑⢄
⠀⠀⠀⠀⣿⠋⠀⠀⢰⣿⠃⠀⣿⣿⣿⣿⣿⣇⣿⡅⠈⣿⣿⣿⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢀⣿⡀⠀⠀⠀⠑
⠀⠀⠀⠀⠘⢧⣀⣴⡿⠃⠀⠀⠙⢿⣿⣿⣿⣿⠟⠀⠀⠹⠿⣿⣿⣿⣿⡿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠙⠋⠋⠉⠀⠀⠀⢸⡇⢀⡇⠀⣤⠀⣶⡆⠀⠀⠀⠀⠀⠀⢨⢉⠁⠀⠀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡤⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾
⠀⠀⠀⠀⢨⠀⠀⠀⡆⠀⠀⠀⠀⠀⢬⡅⠀⢠⣷⣴⠟⢉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠠⢤⡤⠀⢤⣤⠄⠀⣶⠀⠀⢸⠁⠀⠉⡟⠁⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣀⣀⣀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢸⠀⠀⠠⡇⠀⠀⠁⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣀⠀⡀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣴⣿⣿⣿⠀⠀⣤⣴⣤⡀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠀⠀⠈⠀⠀⠀⡇⠀⠀⣾⣿⣦⣿⡾⠋⢱⠄⣀⣀⠀⣀⣤⡄⠻⠿⢿⣿⣿⠯⠼⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡇⠀⠸⢿⣿⠋⣿⠁⣴⡟⠀⣽⡏⠀⠛⢁⡀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠛⣿⡞⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⢸⡿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠈⠛⠸⡿⠀⠋⠀⠀⢈⠁⠀⠰⠿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠤⣄⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠴⡶⠢⠀⠀⠀⡗⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢃⡄⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⢔
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁

⣶⡶⠲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣷⣶⣶⣶⣶⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⣒⣚⡛⠓⠒⠤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣛⣭⠭⠽⢿⣭⣭⣛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠴⢪⣥⣤⡀⡘⠿⠿⠋⢀⣠⣄⡈⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣫⡵⠚⠛⣿⣧⣀⣠⣼⡿⠿⠿⣷⢹⣿⣿⣿
⠀⠀⠀⠀⡔⣁⣀⠙⠛⠋⠀⣿⠀⠀⠀⠘⠿⠟⠁⣹⠀⠀⠀⢸⣿⣿⣿⣿⣫⡾⠿⣦⣤⣴⣿⠈⣿⣿⣿⣧⣀⣤⣾⠏⣿⣿⣿
⠀⠀⠀⡞⠸⠿⠿⠃⠸⣷⡀⣿⡀⠀⣠⣤⣄⠀⢠⡟⠀⠀⠀⢸⣿⣿⣿⢳⣇⣀⣀⣼⣇⠉⢿⠀⣿⣿⠿⠛⠻⣿⡿⢸⣿⣿⣿
⠀⠀⠀⢧⣤⣴⡶⠿⣷⠈⠀⢻⣧⡀⠛⠛⢋⣰⡟⠁⠀⠀⠀⢸⣿⣿⣿⡘⠛⠋⢉⣀⡙⣷⣿⡄⠸⢿⣤⣤⣴⠟⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⢸⠀⠘⠚⠛⢁⣠⣴⠿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⡇⣿⣧⣥⣤⣿⠟⠋⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⠿⢟⣫⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣭⣽⣧⣈⣉⣉⣁⡤⠶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠻⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣄⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡙⠀⠀⠀⢄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠔⣂⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⡇⠀⢀⣀⡈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⢠⣁⣨⠟⢙⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠐⠐⠻⢉⣹⢠⣾⣭⣽⠟⠱⣦⡀
⠈⠛⠛⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⣢⣤⠀⠉⠉⠉⠉⣽⢲⡞⠀⢠⡄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠙⢦⣀⡄⣠⡤⠶⠿⣿⡇⠀⠀⠀⠀⢨⣿⣿⡃⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠤⣍
⠀⠀⠀⠀⠀⠀⠈⠛⢆⠀⠀⠀⠀⠩⡅⠀⠠⠤⢤⣬⣙⣛⡂⠀⠨⣃⣀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠔⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡄⠐⠀⢸⠅⠀⠀⠀⠀⠩⣿⡗⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠸⠄⠀⠻⠀⣠⠇⠀⠀⠀⠄⠀⠠⠀⠀⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠌⡆⠀⠀⠀⠀⠩⡽⠁⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⠂⠀⠀⠀⠀⢈⣯⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣂⣦⣀⢤⣄⡀⣲⣀⣐⣦⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢨⡎⣁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢁⣀⡙⠋⠁
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢀⠒⠈⠃⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣇
⠀⠀⠀⠀⡚⠋⠀⠘⠁⠰⡄⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢘⠀⠀⠀⠀⠂⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠇⠀⠀⠀⠘⠀⢓⠀⠀⠀⠀⠀⠹⢀⠀⢘⡆⠐⠀⠀⠀⠀⠐⠀⢠⠞⠊
⠀⠠⢡⣿⠁⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠁⠀⡈⠉⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⡀⢰⡏⠀⠀⠀⠀⠀⠘⡰⣯⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠰⢀⣿⠁⠀⠀⠀⠀⠀⠀⠓⢻⡃⠀⠀⠀⠀⠀⡰⠘⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰
⡆⢨⡅⠀⠀⠀⠀⠀⠀⠀⠸⡌⢻⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⢀
⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⠀⠀⠘⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⣠⠀⠰⣄⠼⢛⠿⣿⣿⣿⢿⣟⢫⡟⡛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⣛⢻⡆⢠⣶⣶⡯⣻⣿⣿⡛⣻⣿⣟⢨⣷⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⢻⢠⡶⠶⠶⢾⡟⠛⣯⣭⣭⣿⣍⢉⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⣮⣥⣴⣦⡛⠉⠙⣃⡋⠉⠙⣧⣶⣤⣧
⠀⠀⠀⠀⠀⣸⣿⣤⣶⣴⣿⣿⣿⣿⣿⣿⡿⣿⡟⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠻⠮⠿⠛⠻⠧⠾⠋⠈⠉⠁
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⢿⠟⢻⣿⠿⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣤
⠀⠀⠀⣴⣿⣿⣿⣿⣿⠿⣿⡀⢿⡀⣠⡏⠀⠀⢀⡀⠒⢢⠐⠀⡐⠢⡀⠀⡤⠒⠢⡄⠀⠀⢸⣆⡀⠀⢀⢿⠧⣀⣀⠼⡿⡀⠀⢀⣰⡇
⠀⠀⣾⣿⣿⣿⣿⣿⠁⠀⠈⠉⠋⠉⠉⠀⠐⠤⢾⡗⠚⢺⠓⠒⡿⠤⡨⣟⣄⡀⢀⣹⠀⢀⣮⠷⢮⡓⣷⠉⠙⡏⢱⠋⠉⣿⢚⠥⠾⡱⡀
⠀⢰⣿⣿⣛⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⣦⣶⣤⠼⠧⠶⠼⠶⠦⡷⠤⣴⣤⢲⣜⣋⣉⡀⢸⡘⠦⠤⠗⣘⣁⣚⠁⠈⣑⣎⣁⠸⠤⢴⣃⠇
⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⢾⣿⣿⣸⣿⣇⣿⣿⣿⠟⠉⠉⠉⠉⠁⠀⠉⠛⢺⡋⣯⠕⠛⠒⠒⠛⠫⢍⢙⡟⠚⠉
⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⠹⠶⠦⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣃⣛⠒⠠⠤⠤⠄⠒⣋⣘⠆
⠀⠀⣰⠊⠉⠉⠉⠉⠉⠛⢉⠀⠀⠐⢒⣒⠒⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠀⣀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣁⢄
⠀⠀⠺⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⡛⠒⠒⠻⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠲⠶⠾⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠉⢹⠛⠀⢀⡀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⣠⣤⣔⣚⠉⠁⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢀⡠⢔⡞⠋⠁⠀⠀⠉⠓⣆⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡋⠀⠞⠀⠀⠀⣈⠲⣄⠀⠈⣧⠴⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡙⠃⠀⠀⠀⠀⠉⠉⠀⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠤⢤⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀⣀⣀⣀⡠⠒⢟⠀⠀⠀⠀⠀⠀⠀⠀⠂⣨⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠈⠣⡀⠙⢎⠢⡀⠀⠑⢄⡀⠀⠀⠈⡶⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠙⣄⣀⢡⡈⠢⡀⠀⠙⢦⡀⢠⡄
⠀⠀⠀⠀⠀⠀⡜⠀⠈⠑⠦⠀⠀⠀⠀⠀⠀⠘⡄⠉⠹⡄⠈⢦⣀⡞⠙⣾⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⢰⠃⠀⠐⠉⠀⠛⠀⢠⠀⠀⠀⠀⠸⡶⠶⢷⠀⠀⠉⠀⠀⡏⣶⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠁⣈⠄
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢃⠀⠈⡆⠀⠀⠀⠀⢱⡇⠘⣆⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⢰⡀⠀⠀⠀⠸⡇⠀⠫⠈⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢀⣤⣾⠯⠀⣠⣤
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠉⠒⠒⠤⢄⣸⠲⡇⠱⡀⠀⠀⠀⠀⠀⠀⣀⣴⠦⣠⣴⠟⠁⢸⢷⣿⣼⠟⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠀⠀⠀⣇⠀⠀⠘⠈⠉⠙⠒⠢⠄⡈⠀⣿⠀⢳⠀⠀⠀⠀⣠⣞⣓⠦⠝⠂⢹⣀⡀⢸⣸⡟⣧⠞⠁
⠀⠀⢀⠀⠀⠀⠤⠀⣤⣤⣤⣤⠏⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠈⠄⠀⠀⢰⣧⡈⠉⠀⠀⠀⠞⢹⡇⠈⢁⡴⠃
⠀⠀⢸⠀⠀⠛⠒⠉⠀⠀⠀⠘⠦⣀⡠⠴⠒⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢣⣀⣴⠾⢹⠇⠀⠀⠀⢠⠀⠸⠧⠖⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⡾⠛⠁⣀⠾⡀⠀⠀⢀⠇
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⢀⡤⠊⢹⠀⠈⠒⠤⣾
⠀⠀⠀⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⢉⡷⠦⠄⣀⠀⠀⠀⠀⢀⣴⠿⠋⠀⣠⠖⠉⠀⠀⠈⢣⣀⢀⡰⠁
⠀⠀⠀⠀⠀⠈⠉⠓⠢⢄⡀⠀⠀⠀⠠⠀⠀⠀⡴⠋⠀⠀⠀⠀⠉⠑⠒⢾⠟⠁⣀⡴⠚⡟⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡦⠤⠤⠼⠒⠲⠧⡀⠀⠀⠀⠀⠀⠀⢰⠘⣤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⠤⠐⠊⠉⠉⠀⠉⢓⣶⡄⠀⡄⣾⢸⢀⣏⠁⠒⠲⢤⡷⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⡶⡻⠋⢱⢠⣁⡷⠚⠁⠀⠀⠈⠙⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣼⣁⠄⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢿⣿⡧⠀⠀⠀⠀⠀⠀⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⣠⡟⠈⠁⠀⠀⠀⠀⠀⢀⡜⠙⠁⠀⠀⠀⠀⠀⢐⡾⠅⠀⠀⠀⠀⠀⠀⢤⡼⠃
⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠞⠉

⠀⠀⠀⠀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢉⣹⣆⠀⠀⠀⠀⠀⠀⠀⢹⡴⠆⠀⠀⠀⠠⣤⣸⣇⣤⠄
⠀⠀⠀⣶⡋⠀⠀⠀⠀⠀⠀⠀⢰⠾⠀⠀⠀⠀⠀⠀⠠⠖⠉⠀⣟⠀⠀⠀⠀⠙⠁⠀⢀⠇⠀⠀⠀⠀⠀⢰⠟⠳⡄
⠀⠀⠘⠓⠃⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣤⠂⠀⠀⠀⠀⠀⠀⣠⣷⣾⡄⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠂
⠀⠀⠀⠐⣁⠆⠀⠀⠀⠀⠀⠀⠀⡏⠙⠁⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠲⣯⡿⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠃⡈

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⢀⡾⠁⢱⡄⠀⠀⠀⠀⣠⣷⣶⣶⣶⡆⠀⠀⠀⠀⠀⣠⡁⠀⠀⠀⠀⢀⣤⣴⡶⣿⡿⠿
⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⡾⠃⠀⠈⠿⡀⠀⠀⠀⣿⠇⠉⠉⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠈⣽⣿⢿⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⢤⡀⠀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤
⠀⠀⠈⠀⠀⡠⠀⠀⠀⠀⠒⠂⠒⠂⠀⡛⠓⠀⠤⠀⣹⣅⠀⠀⠀⠀⠠⠁⠀⠀⠐⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀
⠀⠀⣄⠀⠘⠀⠀⠀⠀⠐⠀⠀⠀⠈⠉⢀⡄⠀⠀⠀⠀⠈⠉⠀⠐⢰⣥⠔⠄⠀⠀⠀⠀⠀⣠⡀⠀⠀⢄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠲⠀⠤⡀⠠⢀⣀⠀⠀⠀⠢⡀
⠀⠈⠀⠈⠁⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⢀⠀⠢⠇⠀⠀⠀⠃⠀⠃⢀⠀⠀⠠⢂⢀⠛⠤⠱⠂⠈⠀⠂⠀⠌
⠀⠀⠀⠸⠀⠀⡴⠀⠀⠀⠤⠄⠀⠀⠠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠄⠀⠀⠈⠑⠂⠀⠀⡇⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⡤⢂⠀⠀⠀⠀⠆⠀⡠⢀⠀⠐⠀⠒⠀⠀⠀⠀⣀⣠⣆⠐⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠁⠀⠉⠰⠀⠀⠀⠤⠄⠀⠀⠀⠀⡀⠒⢀⡀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠸
⠀⢀⠀⠀⢀⠀⠀⠀⠻⠀⠀⠸⠂⡐⠀⢠⠀⠆⠐⠤⠄⠐⠒⠀⠀⠰⡖⢀⣃⣀⣀⣀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠄
⠀⠀⢀⠀⠈⠀⢀⠤⠠⠀⠀⠂⠀⠅⠀⠄⠀⠀⠀⠄⠠⠄⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⠀⢘
⠀⠐⠀⠈⡉⠀⢄⡀⠀⠀⠀⢤⠀⠀⠀⠀⡀⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⡄⠀⠀⠀⡄⠈⢳⠄⠀⡠⠀⠀⣠⠀⢀⠀⠀⠀⠊⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⣛⠢⠀⠀⠀⡄⠀⠀⠀⡇⠀⠀⡇⠀⠰
⠀⠈⠀⠉⠁⠀⠣⢄⠀⠀⠀⠈⠀⠣⠀⠀⠀⠀⠉⠀⠀⠐⢰⠀⠀⠀⠀⠃⠀⠀⠃⠂⠠⠒⠂⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠤⣄⡀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣰⠉⠉⠓⠲⢤⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⡀⠀⠀⠀⢢⠀⠘⢶⡖⠢⢄⣀⠀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⣰⢿⣦⣠⠤⠬⣃⣀⡜⠀⣔⣉⡴⠷⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣳⠦⠤⠬⣿⡆⠀⠀⠀⣀⠝⠀⢰⣷⣏⣀⣀
⠀⠀⠀⠀⡜⣱⠋⣠⠶⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⠠⡤⠈⠓⠒⠒⠉⠁⠀⠀⡷⢋⣡⡤⡌⢳⠀⡇⢸
⠀⠀⠀⠀⠀⡇⢸⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⡴⠋⠠⢁⠃
⠀⠀⠀⠀⠀⠉⠀⠑⠦⣥⣈⣁⣈⣀⣤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡆⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡆⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠁⠇⠀⣀⠤⠒⠊⣿⠫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⢹⠛⢲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠖⣉⠀⠄⢀⣀⣸⡯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⡰⠧⠤⢄⣀⣙⣢⣾⣠⣏⣙⣆⣀⣀⣠⣤⠤⠤⠤⠞⣆⠿⢋⡀⠒⠒⠢⢄⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⢙⡇⠠⡄⡈⠑⣄⢀⣴⠿⣦
⠀⠀⠀⠀⡰⢃⠖⣠⠞⣁⣸⠧⠤⠿⠭⠭⠭⠭⢉⣉⡉⢉⡩⠟⠉⠀⠀⠀⣠⠞⡇⠰⠁⠇⠀⠘⢟⠗⠕⠀⠁
⠀⠀⠀⠀⠃⣏⢰⡁⢊⡔⠀⣀⣀⣠⣤⠤⠶⠒⠒⣲⡞⠋⠀⠀⠀⠀⣠⠞⠁⠀⢿⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡏⠉⢹⠀⠈⠓⠦⣀⣰⠁⡇⠀⢀⣠⠴⠋⠁⠀⠀⠀⢸⡀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠆⡼⣸⠀⠀⢸⠀⠀⠀⠀⠈⠙⠲⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⣜⠀⢇⠻⣄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠇⢘⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡿⠿⢷⡶⠒⠒⠒⠒⠲⣶⡶⠶⣶⠟⠋⠉⠉⠁⠀⠈⠉⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠤⢾⠀⠀⠀⠀⠀⠀⣧⡤⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⠭⠿⠀⢯⠀⠀⠀⢀⣔⠾⣕⡶⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠁⠀⠀⠛⠒⠩⠥⠤⠬⠇

⠀⠀⠀⠀⠄⣤⡄
⠀⠀⢠⡇⣸⠿⠀⠀⠀⠀⠀⢀⠤
⠀⠀⣾⡗⣿⡏⠉⠙⠓⠲⣤⣮⣜⣻⣠⡄
⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⢸⠟⡋⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡧⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⢢⣄⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣴⡟⠓⠲⢄⡀⢀⣀⣄⣀⣀⡼⣿⡿⣿⣿⣿⡇⣀⡈⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣧⠀⠀⠀⠙⣿⣿⣿⣿⡎⠹⣯⣵⡟⣹⣿⣿⣿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣶⣤⣴⣿⣿⡇⠈⠀⠀⠈⠉⠐⠻⣿⣿⡿⠏⣶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢙⣿⠀⠀⠈⠻⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⣿⣧⠀⠀⠀⠀⠀⠀⣨⡏⠀⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⣀⣤⣴⣿⣧⡀⠀⠀⣠⡎⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣿⣿⣿⠛⠛⣛⣯⣈⠿⣷⣶⣾⢿⣇⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣻⣿⣟⠀⠉⣡⣬⠆⠉⠂⠐⠃⠸⣿⣦⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⢯⣴⠖⠂⠀⡤⣴⡄⢨⠀⠀⠘⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣿⡧⠀⠤⣼⣄⢈⣿⠇⠀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢿⢿⡦⠠⠤⣯⡄⠀⡟⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣿⣶⣬⣅⡀⠀⠷⠶⠞⣇⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣿⣿⡟⠒⠂⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠙⠳⣴⡇⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠙⠾⡿⠿⢷⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠿⠗⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣶⣶⣶⣶⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠙⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠉⠛⠛⠿⠿⣿⣿⣿⣟⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⣯⣝⠁⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣭⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⡀
⠀⠀⠀⣀⣼⢱⣄⣿⣿⡒⠻⡻⠿⠿⠿⠯⠭⠭⠿⠿⠶⠶⠶⣶⣶⣶⣶⣶⣦⣤⣄⣀⢀⣀⣀⠀⠠⡀⠀⠀⠀⢶⣶⣶⡆
⠀⠀⢸⢿⣇⣶⣴⣶⣶⣾⣿⣿⡿⠛⠛⠛⠛⠟⠿⠿⠾⣿⣶⣿⡾⣄⣀⣀⣀⢀⣀⢙⣶⢷⣾⣷⠀⠉⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠈⢙⣾⠿⠿⠿⢿⡟⠛⠙⣿⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣧⠀⠀⠀⣀⠀⠀⢹⣿⡇
⠀⠀⠀⢻⣿⣴⣶⣶⣶⣿⣿⣿⣿⡿⠸⢸⢸⢸⠹⢹⢓⡿⠶⠶⠿⠟⠒⠒⣶⡄⣶⡂⣾⣿⣿⡿⠀⠀⠀⠉⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠈⠀⠀⠈⠉⠙⠛⠛⠒⠓⠒⠛⠋⠉⠉⠉⠁⠀⠀⠀⡄⠀⠀⠀⠈⠓⢮⢷⣿⣿⡁⠀⠀⢰⣦⣄⠀⠀⠀⢸⣇⡀
⠀⠀⣠⡄⣀⡀⠀⣀⣀⡀⠤⠀⠤⡤⡀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠘⠛⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠋⠻⢿⣿⣦⣤⣾⣿⢣⡖⣀⡀
⠀⠸⣿⣧⣤⣤⣡⣤⣤⣄⣀⣀⣀⡘⡎⠓⢄⡀⠀⣸⣷⣤⡀⠀⠀⠀⢰⣦⠀⠀⢀⣿⣿⣿⣿⠀⠀⣀⠀⠀⢸⣿⣿⣾⣿⢸⡗⠻⡇
⠀⠀⠀⠀⠉⠉⠛⢝⡳⠿⣿⣻⡿⣿⣧⣴⣤⣬⣛⣯⠛⠿⢿⣀⣀⣶⣾⣿⠀⠀⠈⠛⠿⠛⠁⠀⠀⠉⠀⠀⣸⣇⣸⣿⣿⢺⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡉⠻⣿⡿⠿⠿⠿⣿⣿⣿⣷⣶⣶⣤⣤⣏⡙⠒⠤⢄⣀⠀⣀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣹⣿⢹⡇⢸⠃
⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⣀⣀⣁⡀⠀⠀⣀⠀⠀⠀⠈⢿⣿⡯⢍⠿⢿⡟⢬⣷⣶⣦⣭⣝⣎⡆⢿⣿⠿⠿⠛⠻⠛⢻⣿⣸⡧⠼
⠀⠀⢠⣤⣶⡌⠁⢸⣿⣿⠿⡻⣿⣿⡇⠈⢡⣶⣤⡄⠀⠸⣿⡷⢸⠀⠀⣿⠁⡏⠉⠙⠿⣿⠿⠃⠈⠀⠀⠀⠀⠀⠀⢸⣿⣷⠂
⠀⢄⠨⠟⣋⡛⢐⡿⠿⠿⡃⢘⠿⠿⢿⠂⢛⣙⠻⠅⡀⠀⢸⡇⢹⠀⠀⣿⠀⡇⠀⠀⣀⠀⠀⠀⠠⠄⠀⢸⢠⠀⠀⢸⣿⡿
⠀⠸⠶⣆⠿⠿⡏⠁⠀⢠⣶⣶⡄⠀⠈⢹⠿⠿⣰⡶⠇⠀⢸⡇⢸⠀⠀⠉⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣾⢈⡀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠐⠿⠿⠟⠋⠙⠺⠾⠿⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣀⠀⠀⠀⢖⣖⠀⠀⠀⣾⠀⡄⠀⣿⠈⡇⠀⠘⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣒⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣿⠀⡇⠀⣿⠀⡇⠀⠀⢹⣿⡓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠭⡽⠿⠿⠿⠯⠽⠿⠯⠤⠤⠤⠤⠴⢇⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⣠⡇⠀⣿⣤⡇⠀⠀⢸⣏
⠀⠀⢰⣶⣶⣶⠶⠾⠿⢿⣿⣿⣯⣭⣭⡼⠿⠿⠿⠿⠿⠿⠿⠿⣿⠏⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢸⡇
⠀⠀⠘⠿⠿⠁⠀⠀⢀⣼⡟⠙⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠴⠿⠾⠛⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠝⠛⠓⠀⠀⠺⠛⠻⠛⠻⠛⠟⠛⠟⠩⠉⠀⠂⠘⠛⠛⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡷⡸⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠰⠂⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠶⣆⣀⠀⠀⠀⠀⠀⠁⠀⢈⡇⢨⡁⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢠⣶⣧⡶⠄⡟⢿⡀⠀⠀⠀⠐⣉⢁⢉⡙⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢤⠀⠀⠘⠀⠀⢸⣿⣿⣟⡃⠇⢸⣧⣤⠀⠀⡀⠥⠬⠠⠅⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣇⠀⠀⠀⠀⢸⣿⣿⣯⠄⠀⠀⣿⡇⠀⠂⠀⣶⣔⠐⡂⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠇⢸⡿⢿⣀⡀⠀⠀⢸⣿⣿⡷⠂⠀⠀⣿⡇⠀⠀⠀⣿⣿⠈⣁⣉⣛⣿⣇⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣿⠙⠃⠀⠀⠀⢸⣿⣷⢸⢳⢸⣿⣿⠀⠀⢸⣿⣿⠏⠱⠴⣦⣿⣯⠀⠀⠀⢾⣿⣤⠿⠿⠿⠿⠿⢿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⢸⣿⣿⢠⡄⠀⢲⡰⠀⠲⠡⠙⣿⣿⣦⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣧⣿⣀⣀⣀⣀⣀⣸⣤⣿⣘⣛⣸⣿⣿⣘⣃⣀⣚⣚⣂⣐⣚⣒⣹⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣧⣄⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠌⠒⠒⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣌⢸⡇⢀⣀⠠⠤⠐⠒⠒⠒⠒⠲⠤⢤⣀⠀⣿⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣃⡼⠛⠉⠀⠀⢀⣠⠀⠁⠀⠀⠀⢀⣀⠈⠙⢻⣅⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⠿⣄⡀⠀⠀⠀⢋⠄⠒⠒⢄⡀⠒⠒⠠⡹⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠦⢤⡀⠀⠀⠀⡞⠀⠀⠀⠈⠷⠤⠘⠐⠁⠀⣤⡀⠀⠀⣠⣄⠀⠐⠰⡦⠏⠻⡀
⣦⣄⡀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠉⠢⡀⢀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠐⢀⠉⢀⠀⠃⠚⢅⡠⠂⠀⠀⠄⠀⡇
⢹⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠙⠦⣀⣠⡴⠧⠙⠢⠄⠀⢀⣀⡀⠀⠀⠀⠀⣇⣠⡴⢦⡀⢀⠇
⢸⡆⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⢛⠛⠛⢿⡀⠀⣀⠤⢾⠁⠀⠀⠐⠠⠀⡈⠙⠻⠶⠶⠖⠊⠀⡀⠠⠀⣻⠁
⠀⣇⠀⣀⡠⣶⡿⠛⠁⡠⠀⠀⠀⠀⠀⣠⠝⠋⠀⠀⣸⣇⠠⠀⡀⠀⠀⠀⠈⠉⠁⠈⠉⠁⢀⠠⠐⠈⣸⠳⢤⡀
⠀⠉⠁⠀⠀⠘⡆⡠⠊⠀⠀⠀⢀⡤⠊⠁⠀⠀⢀⡾⠁⠈⠲⣄⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢧⡀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣠⠞⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⡖⠒⠲⠒⠚⠉⠉⠀⠀⠈⣷⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠶⣿⣷⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⡾⠓⠤⣀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠙⠻⠿⠿⢷⡚⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⣤⣶⠿⠋⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢠⡇⠀⠀⠀⢀⣠⠶⣦⠀⠀⠀⣿⣷⡈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠋⣷⡞⠉⠀⠀⡶⢤⣀
⠀⠀⠀⠀⠓⠶⠶⠖⠋⠀⠀⢻⡀⢀⣠⠇⠈⢳⣦⣀⡀⠀⠀⠉⠙⠛⠲⠲⠶⠶⠶⠶⠶⠶⠒⠛⠉⠉⠀⠀⣰⠟⢧⠀⠀⢀⡇⠀⠈⠓⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⡇⠈⠙⠲⠦⣤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠚⠋⡇⠀⠈⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠙⡓⠒⠈⠉⠉⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣰⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠓⠄⠀⠀⠀⠀⠀⠀⠀⠠⢳⠆⠀⠀⠀⠀⠠⠞⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠠⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡘⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠒⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣁⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠃⠂⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠨⠈⠏⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣽⣽⣽⣿⣿⣽⣿⣿⣤⣤⣄⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⢿⡿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⢰⣾⣿⣿⣿⢿⡿⣿⣿⣿⢿⣿⣿⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣤⣶
⠀⠀⠀⢀⣀⣀⣀⣰⡷⠶⣶⣾⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⡛⢛⣿⣿⣿⣶⣶⣰⣦
⠀⠀⠀⢸⣗⣚⣻⣿⣷⣶⣿⣿⣿⣿⣿⣿⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⡧⠤⢼⣿⣯⣭⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⡗⠒⢺⣿⡿⠿⣿⣿⣿⣿⣿⣿⢾⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣬⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⡇⣩⣽⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⢸⣗⣒⣺⣿⣟⣛⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⢸⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⢀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠋⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣿⣤⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⢸⠀⠀⢀⣿⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢻⠟⠉⣿⠉⠙⡿⠁⠀⠀⠀⠀⠀⠀⠀⢿⢿⠖⠀⠨⣿⠅⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣄⠀⡀⠀⣠⡀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣇⣀⠀⠀⠀⢿
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⠀⠀⠀⡟⠀⢠⣤⣿⣤⣀⠀⠀⠀⢹⣿⣿⣿⢏⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠋⠀⣿⠈⠙⠁⠀⢸⠀⠀⠀⡇⠀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠂⢺⡗⠂
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠺⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢨⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠾⠀⠀⠀⣀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠙⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠛⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⣴⡿⠋⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢿⡏⢿⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣀⠉
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⢻⣿⠿⠿⠿⠿⠿⠿⠿⠛⠿⠿⠗⠒⠓⠀⠀⠀⠀⠀⢹
⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⢠⣶⣤⣴⡄⠀⠀⠀⠀⠀⠀⢠⣶⣦⠄⠀⠀⠀⠀⠀⠀⡺⠄⢀⣀⠀⠀⠀⠀⠀⠀⣠⣴⠆⠄⡀
⠀⢀⣴⣯⣯⣍⣦⣆⠈⠀⠀⠀⠀⠘⠻⠿⢛⠁⠀⠀⠀⠀⠀⠀⠨⡀⢈⠁⠀⠀⠀⠀⠀⠸⢧⣄⣻⡿⠃⠀⠀⠀⠀⢰⣿⣱⣛⡆⢠⡄
⠀⠈⠻⠽⠛⠛⠯⠟⠀⠀⠀⠀⠀⠀⠀⢻⣾⠃⠀⠀⠀⠀⠀⢠⣶⣏⣽⣶⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⠀⠀⠀⠀⠀⠉⠄⢉⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⣠⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠒⢤⡀
⠀⠀⠀⠀⣏⣁⣀⡍⠀⠀⠀⠀⢐⣤⣀⠐⣠⣖⣇⠀⠀⠀⠀⠀⢠⣶⡁⡹⠀⠀⠀⠀⠀⠰⠛⠲⣿⠷⣾⡇⠀⠀⠀⠀⣿⣿⡿⣷⡄
⠀⠀⠀⠀⡻⣿⣿⡇⠀⠀⠀⠀⠸⣟⣻⠇⢿⣛⡿⠀⠀⠀⠀⢰⣿⡿⠛⠀⠀⠀⠀⠀⠀⢠⣬⡸⣷⣽⣾⡇⠀⠀⠀⠀⢰⡆⠑⢹⣧⡦
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⣩⣗⣦⣒⡉⢺⡀⠀⠀⠀⠈⢳⣿⠞⠑⢢⡀⠀⠀⠀⠀⣸⠞⣀⣀⠻⣇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣁
⠀⠀⠿⢿⡟⢻⠗⠈⠀⠀⠀⠀⠀⣤⢿⡿⠿⢯⠇⠀⠀⠀⢰⣿⣧⣛⢛⣼⣿⡄⠀⠀⠀⠀⡀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠂⣉⣠⣿⣿⡆
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠐⢦⡀⣤⢤⠀⠀⠀⠀⠀⣠⣬⣵⣤⣄⠀⠀⠀⠀⠀⠆⣀⠒⣿⣿⠆⠀⠀⠀⠀⠀⣴⢟⢛⡷⣆⠀⠀⠀⠀⠀⠀⢀⣠⣀⣾⠉
⠀⢰⠿⠳⣶⡿⠿⣿⣿⠀⠀⠀⠀⠘⠛⡻⡿⡛⠀⠀⠀⠀⠀⠐⠧⣀⣾⣷⡄⠀⠀⠀⠀⠸⣧⠆⣓⢖⣿⠀⠀⠀⠀⠠⣾⣿⠯⢸⣛⣿
⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠙⠛⠚⠛⠃⠀⠀⠀⠀⠐⠛⠚⠁⠀⠛⠁


⠀⠀⢠⣤⣁⣈⣥⣤⠀⠀⠀⠀⢀⣤⣲⣖⣢⣄⡀⠀⠀⠀⢀⡶⣟⢻⢶⡄⠀⠀⠀⠀⠀⠠⡄⠠⣿⣷⠀⠀⠀⠀⠀⠀⣠⣭⣄⢀⣠
⠀⠀⢸⣛⡟⣻⣟⢳⠀⠀⠀⠀⠸⢟⡿⠿⠿⣛⠟⠀⠀⠀⢸⣷⢂⡲⣶⣶⠶⠀⠀⠀⠀⢸⡷⣶⣿⣧⣄⠀⠀⠀⠀⠀⢩⣽⣷⣯⣤⡄
⠀⠀⠘⠻⠓⠋⠉⠚⠀⠀⠀⠀⠀⠀⠢⠤⠔⠂⠀⠀⠀⠀⠀⠙⠛⠓⠿⠃⠀⠀⠀⠀⠀⠀⠁⠸⠿⠿⠿⠀⠀⠀⠀⠀⠘⢿⣩⠟⠛⠃

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠘⠀⠀⢠⡄⠀⠈⠀⠀⠀⠀⠰⠛⠁⠀⢀⡀⠀⢰⠉⠙⢿⡧⠀⠀⠄⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠳⣄⠬⠀⠀⠂⠠⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠠⢀⠀⠀⠀⠀⠀⠠⠂⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡧⠁⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠐⠀⠀⠁⠈⠉⢂⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠈⠀⠀⠀⠀⠠⠀⠊⣀⠀⢠⣶⡄⠀⠀⠀⠀⢀⡠⡀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢄⠀⠀⠀⠻⠆⠀⠀⠀⠐⠉⠀⠁⠀⠀⠀⠂
⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡄⠀⠀⢰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠂⠄⠂⠀⢀⠀⠄⠠⠄⣀⠀⠀⠀⠈⠀⢀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊
⠀⠀⠀⢀⡠⠚⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠈⠁⢀⡀
⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⠀⠐
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⢠⠀⢀⡟⠁⠀⢠⡀⠀⠀⠀⠀⠀⡀⠘
⠀⢻⣀⣠⣄⠀⠤⠆⠀⠀⠀⠎⠀⠀⠀⣐⡤⠌⠀⠀⡇⠀⠈⠀⠀⠂⠀⠐⠄⠀⠀⠐⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢎⠀⠀⠀⠀⡠⠂⠀⠀⠐⠀⠁⠀⠀⠀⢰⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠴⣄⣄⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⢤⡄
⠀⠀⠠⠀⠠⡀⠀⠉⠀⠀⠀⠀⣰⡟⠁⠀⣠⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣼⠏⠀⠴⢞⣲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠘⠄⣀⡨⠃⠀⠰⠊⠲⢤⠀⠀⠀⠐⠀⢠⡄⠀⢀⠀⠀⠈⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠁⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢒⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢽⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠛⢐⣒⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣄⣶⣦⡯⢽⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡃⠀⢰⡒⠀⠀⢹⡇⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡯⣿⣿⡯⢽⣿⣿⣶⣿⡆
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠸⡇⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣟⣿⣿⣟⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⡟⠃⠀⢨⡅⠀⠀⡘⠃⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣯⣿⣿⡗⢾⣿⣿⢿⣿⡇
⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠚⠛⠛⠛⠛⠛⣛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⣤⡶⢒⣒⡒⢒⣒⠲⣾⣷⣼⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠸⠿⠇⠿⠿⠟⠿⠿⠇⣇⡽⣫⣋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⠬⠭⠭⠭⠭⠭⠭⠭⢭⡿⣯⣼⣅⣿⣇⣠⣶⡋⠓⢲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠛⠒⠒⠚⢋⡽⣿⣿⣧⢸⣿⣿⡴⠛⠉⠉⠉⣿⣶⣬⡟⣩⠿⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⣀⣀⣀⡴⠋⣀⣟⢛⠋⢸⠿⠥⠤⠤⠤⠤⢤⣀⣉⣛⣋⣡⠖⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢓⣛⡛⣫⣩⡉⢩⣭⣍⡇⣴⡆⣿⣿⣿⣿⣿⣿⣿⣗⣾⣷⣶⣦⣤⣤⢹⣯⣤⣶⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠿⠇⠿⢿⣃⠘⣛⣋⣿⢙⣇⣤⣸⣭⣿⣬⣿⣋⣈⣛⡻⠛⠿⣿⢫⢿⡿⠿⣿⠀⠀⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⢧⣾⣿⡾⣿⡿⠻⠿⢟⣹⢻⡋⣛⣹⢿⢿⣿⣿⢿⣿⣿⣿⣿⣿⡦⣾⣶⣴⣶⣋⣉⠛⠒⠦⣄⡀⢸⡄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣀⣘⣏⣭⣶⢰⣿⣷⣿⡿⠿⢿⣷⢿⣿⣼⣰⣶⣶⣴⣿⣭⣨⣝⡛⣽⠿⠿⢿⡿⣿⡿⢃⣴⣄⠀⠉⠲⣧
⠀⠀⠀⢠⣎⠤⠖⠚⢋⣩⣥⣄⢹⡛⣋⡄⢴⣶⣯⣭⣹⡏⣸⠉⠉⠉⠉⠉⢉⠽⡿⠟⠿⣿⢧⣿⣷⣿⣾⣶⣮⢀⡻⠿⣷⣷⡦⠖⠉
⠀⠀⠀⠙⢦⣰⣾⣿⡄⢿⣿⠿⠀⢧⠿⣿⠞⠉⠛⠛⠛⠋⠁⠀⠀⠀⣠⠔⣩⣾⡇⣾⣿⣶⣾⣭⣹⣿⠻⣿⢿⣿⣿⣤⡿⠋
⠀⠀⠀⠀⠀⠹⡟⠋⠁⠀⢀⣠⣤⠸⣤⡏⣩⣭⡉⣹⣿⣍⢹⣿⣿⢹⢿⣟⣿⢿⢡⣭⣍⢱⠟⠛⢿⣿⣶⣦⣤⠉⡹⠋
⠀⠀⠀⠀⠀⠀⠙⣤⣾⣿⡘⡿⢿⣄⢻⡇⠿⠿⠇⠿⠿⠏⠸⠿⠟⢸⣛⣃⣥⣿⠸⢿⡿⣼⣿⣿⡿⣙⡋⠿⠟⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡟⣋⣡⡐⣿⣿⡟⡇⣶⣾⡆⣾⣿⣿⢸⣿⣿⢸⣿⣿⡟⣿⣿⣶⡆⣧⣤⣼⢿⣿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣻⡿⠇⠈⣡⣴⡇⢛⣛⠃⢙⣛⣁⢈⣉⣉⢸⣭⣴⣾⣏⣭⣟⢸⡛⠛⣿⣶⣤⣬⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢶⣿⡆⠹⠿⣧⣿⣿⡇⢸⣿⣿⢻⣿⡿⢸⡿⠿⢻⢿⣿⡟⣸⣿⣿⣛⣋⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⣭⣦⠐⣾⣿⣨⣭⣥⢠⣷⣶⣄⣤⣦⣼⣴⣶⣿⣾⣿⡇⡿⢀⡼⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠛⢛⣤⣭⢹⠛⠿⠟⠘⠿⠛⠙⠛⠛⣿⢋⣽⡟⠚⠛⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠼⠿⠿⠾⡇⠀⣶⣶⣿⣿⣿⡄⠀⡿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠿⠿⠿⠟⠛⠓⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠉⠉⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢀⠞⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⡀⠀⣰⠇⣰⡇⠀⣰⣆⠀⠀⠀⣠⡶⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢺⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠰⡏⠀⢠⡇⡜⢱⣣⠎⠿⡟⠀⡠⠊⢰⠇⡴⠚⡇⡴⣿⠀⣠⡆⠀⠀⠀⠀⢳⡀⠘⡦⢘⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⢠⠇⠀⢸⠏⠀⠸⠃⠀⠀⣷⠊⠀⠀⢸⠏⠀⠀⠟⠀⣿⠞⢹⢃⣤⠀⠀⠀⠈⡇⠘⣇⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⣸⠀⠀⣾⠀⠀⠀⢀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠏⢹⢀⢼⡇⠀⣿⠀⢇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡼⠁⠀⢀⣿⣤⣴⠾⠟⠋⠉⠛⣆⠀⠀⠀⠀⣀⣽⣦⣄⡀⠀⠀⠀⠸⠋⢸⠃⠀⣿⠀⠘⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⠟⢸⠁⠀⠀⠀⠀⢀⢸⡆⠀⠀⡜⠁⠀⠈⠳⣝⢦⡀⠀⠀⠀⢸⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢻⠀⠀⠀⠀⠰⠿⣸⠁⠀⣸⠀⠀⠀⠀⠀⣈⠙⡿⠒⠀⠀⢸⣠⠼⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢦⡼⠁⠀⠀⠀⠈⠳⣤⣄⣀⡤⠞⢁⡀⠀⢿⡀⠀⠀⠀⠰⠟⣠⠇⠀⠀⠀⣿⣣⢚⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡤⠋⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠋⣭⠀⠀⠀⠉⠓⢦⣤⡤⠞⠁⠀⠀⠀⢸⠛⢋⡽⠋⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣗⠺⠭⣄⠤⠼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢳⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠦⣄⡀⠀⠸⣆⠀⣠⣄⣤⡴⠶⢻⠖⢦⠞⣯⠤⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢯⡀⠀⠀⠀⠀⣀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠶⠶⠯⢤⡶⠀⠈⡻⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⣦⣤⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢽⣭⢭⡭⠭⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠠⠮⠤⠖⠊⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣃⣠⠴⠋⠁⠀⣀⠀⢦⣤⣉⣓⣦⣤⠀⣀
⠀⠀⠀⠀⠰⡇⠀⠀⣻⠄⠀⠀⠀⠀⠀⣀⣀⣀⡈⠳⣄⠀⠉⠉⠛⠓⠒⠒⠒⠒⠒⢒⣒⡿⠿⣍⠹⡄⢠⣄⠀⢹⠓⢬⣦⡀⠉⣩⠗⢴⠋⢠⡄
⠀⠀⢀⠀⠀⡇⠀⠀⠙⣦⠀⠀⠀⠀⠚⠯⢤⣉⡉⠙⠛⠒⢶⣶⡴⢲⣄⡤⣄⠀⣜⠉⠀⠀⠀⠈⠳⡝⣼⠀⠳⣼⡀⠀⠈⠉⠀⣳⢀⡼⠁⣾
⠀⠀⠈⢣⡀⠈⣢⠀⠀⠷⢤⡄⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠈⡇⢧⠀⠀⠀⠙⠉⠙⠖⠚⠳⠒⠲⠖⠿⠈⠀⠀⠈⠁⠀⠀⠀⢼⠁⣨⠇⢶⠁
⠀⠀⠀⠀⡇⠀⠙⢦⠀⠀⠈⠳⢦⣤⡤⠶⡄⠀⢀⡀⠈⢦⣤⣇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠈⣧⠀⠼
⠀⠀⠀⠀⠘⡆⠀⢎⠀⠀⠀⠀⠘⣧⠀⠀⠙⠋⠉⠓⠒⠋⠉⠉⠓⢦⢤⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠋⠀⡴⠁
⠀⠀⠀⠀⢀⠃⠀⠈⣷⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠙⠚⠙⠲⠛⠶⠶⡤⢦⠖⢶⡿⠙⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠛⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣕⡣⣌⡻⣷⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠉⠓⠿⣾⣧⣄⣀
⠀⠀⠀⠀⠀⠀⢿⢤⣀⠀⠀⠀⠀⠈⠙⠛⠯⣷⡲⠤⣄⡀
⠀⠀⠀⠀⠀⢀⡎⠀⠈⠙⠓⣦⠀⠀⠀⠀⠀⠀⠉⠙⠲⢭⣑⣲⡄
⠀⠀⠀⠀⠀⢸⣄⡀⠀⠀⠀⡟⠑⠻⢶⣦⣄⠀⠀⠀⠀⠀⠈⠉⠓⠾⣴⣢⠄⣠⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⣼⡃⠀⠀⠀⠀⠉⠑⠢⠄⣀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣝⡲⠤⣀⡀⠀⠀⠀⠀⢩⠖⠰⣒⠢⢤⣀⠀⠀⠀⠈⡙⢓⠶⣭⣓⠢⢤⡀⢀⡠⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣍⠓⣦⠀⠀⠛⢤⣴⣿⡹⠓⢢⣽⡀⡀⠀⡇⡜⠀⡇⡎⡙⠲⠿⣿⣖⠺⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠙⠶⣤⣀⡀⠀⠉⠉⠓⠚⢀⡞⠉⠛⢥⣇⣠⣧⣇⣿⢸⢠⡇⡎⠉⠓⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠃⠈⠉⠙⠓⠲⣼⣆⠀⠉⠲⢄⣻⠍⢻⡀⠺⠿⢼⣸⡇⣧⠀⢰⣇⣀⣨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡟⠀⠀⠀⠀⢀⡜⠀⢈⣽⠀⠀⡞⠈⠀⠀⠙⠒⠲⠚⠛⠯⣟⡤⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣧⡀⠀⠀⠒⠛⠒⠚⠋⢹⡄⠘⣧⠀⠀⣀⡀⠀⠀⠀⠀⢀⣠⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢬⣷⣦⣄⣀⠀⠀⠀⠀⢸⡇⣸⡋⠀⣿⣋⣽⠀⠀⢀⡔⣫⣜⣁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣶⠦⣤⡾⠹⣭⠗⠀⠀⠉⠀⠀⠀⡞⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⡆⠀⣇⡀⠀⠀⠀⠀⠀⢸⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠇⠀⠀⠀⠀⠀⢸⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣼⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡏⠀⠀⠀⠀⠀⠀⢸⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⣴⢛⡷⣄⠀⢸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⡓⠿⣤⣀⠘⠿⠶⠃⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠾⢭⣟⣒⣶⣀⡼⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⠘⠒⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣆⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⢼⣆⠾⠶⠶⠾⢰⠧⠆⠀⠀⠀⠀⠀⠀⡌⠀⡀⠂⠀⢀⡀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⢠⠖⠒⡄⠀⠀⠀⠀⢠⠒⠲⡄⠀⠀⠀⠀⠀⡅⠀⠿⠛⠛⠿⠃⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠔⢃⢤⠆⠠⡤⡘⠢⠔⠃⠀⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠐⠊
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠘⠶⢿⣀⠀⢀⣨⠷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⠒⠓⢢⣔⠁
⠀⠀⠀⠀⠀⠀⢶⢰⣖⣠⢠⣶⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠉⠉⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣤⣤⣤⣤⣼⣦
⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⠠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣦⠀⠐⢠⣾⣄
⠀⠀⠀⠀⠀⠘⠯⠇⠀⠀⠀⠀⠫⠝⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠘⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠈⠉⠉⠁⠈⠻⠇

⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⡖⣿⣶⣖⠀⠀⠀⠀⠀⠀⠀⣬⣷⠀⠀⠀⠀⣾⣥⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⣬⣅⡄⣤⢀⢌⣥⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣿⣿⣿⣅⣀⠀⠀⠀⠀⠀⠀⠠⡢⠀⠀⠀⠀⠀⠠⡲
⠀⠀⠀⠀⠠⣤⣭⠁⠁⡉⢈⡈⢩⣤⡤⠀⠀⠀⠀⠀⠀⠀⢻⣿⠻⠟⣿⡟
⠀⠀⠀⠀⠀⠈⠙⢶⣦⣄⣬⣶⠾⠉⠀⠀⠀⠀⠀⠀⠀⣖⡾⠁⠉⠉⠉⢷⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠲⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠻⠶⢲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠄⠛⢤⠀⠀⢀⣴⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠇⠀⡞⠓⠞⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢰⠃⠀⠀⠀⠸⡍⠿⠉⢙⣋⢙⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⣤⣤⠤⠤⢞⠛⠀⠀⢈⡀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠈⡛⠙⠋⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⠖⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡆⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⠃⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⡄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠙⠦⣀⢀⡄⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠂⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠈⠳⣖⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣼⡋⠛⠛⠻⠶⠿⠿⠿⠟⢻⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠹⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⣋⣵⣿⣿⡇⠀⠀⠄⠀⠀⠀⠀⠉⣿⠀⠀⠀⠀⠀⣀⣀⣰⠃⡆⠀⠀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢉⠝⣱⣿⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠖⣦⠀⠀⣾⠵⢊⣡⠼⠳⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡜⣿⣿⡿⡀⠀⠀⠀⠀⢀⣀⣀⣀⡇⠘⠧⠤⠤⠞⠀⣸⢯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⣿⣿⠇⠀⠀⠀⠀⣠⠏⠀⠀⠀⠈⠒⠦⠤⠤⠔⠊⠁⠀⠀⠙⠓⠒⠒⠒⠒⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⢀⠀⠀⠘⣇⣀⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⠈⠱⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣄⣐⣀⠀⠀⠀⠀⠀⢠⠴⠳⠀⠀⠀⠀⠀⠀⡜⠛⠧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠉⠛⢳⣶⣶⠃⠀⠀⠐⠒⠒⠒⠒⠒⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⣾⣿⡇⠀⠀⠀⠀⣀⡤⠤⣤⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠚⠋⠀⠀⠀⠙⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠹⠿⠿⠟⠛⣿⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠝⠛⠀⠰⡄⠰⠆⢼⠀⠀⠈⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠈⢆⠀⠀⢠⡄⠀⡞⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⠐⠂⣠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡏⢿⡿⣿⠟⠿⢛⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⡴⣾⡟⣿⠹⡗⠘⣶⠋⢀⢀⠟⠁⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁⣿⠁⢿⣿⣷⣼⣿⠲⠾⡘⡀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⣰⠟⠁⡰⣿⠀⠘⠛⠛⠛⠛⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢠⣾⡿⠛⠓⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⠆⠘⠛⠛⠓⠾⣧⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⡷⢶⣤⣄⣄⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡆⣠⡴⢦⣤⡉⠻⣇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⢿⡟⢿⠀⠀⣠⡤⣄⠀⠀⠀⡀⢻⡟⢡⠀⠀⡞⠻⣶⠇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣇⠈⠀⠀⠀⡇⣾⣧⡀⢸⣇⠈⢣⠀⡄⢻⣾⠁⠈⠀⢰⠇⢀⡏⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⣆⠀⠀⢰⠇⣿⡄⠙⠾⠋⠀⠈⡆⠸⣾⣿⠀⠀⠀⡟⠀⣼⠁⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢠⠞⠉⣉⠁⠘⣆⠹⣷⣀⠀⠀⢀⠸⣇⣷⣽⡟⠀⠀⠘⢀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠷⠤⢤⡔⣹⡆⢘⠎⡛⢣⢼⠀⠀⣽⣿⣷⡀⠡⣩⠉⠉⠁⠀⠀⣠⣾⡁⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠐⣶⠯⠴⠋⠿⠯⣼⣡⣧⣿⣃⠞⣸⣿⢉⠓⠶⠋⠀⠀⠐⠀⣴⣿⣿⣷⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡾⠁⠳⣄⣀⣀⣱⣄⣬⣿⣿⡏⠀⣿⣧⣬⣗⠒⠂⡀⠀⠀⣾⣿⣿⣿⣿⣇⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠈⢠⣾⣿⣿⣯⣿⣿⣿⡟⠀⢠⣿⣿⣿⣿⣯⠀⠀⣠⠾⣿⣿⣿⣿⣿⣿⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⣠⣿⣿⣿⣿⡿⠿⠿⣿⣀⠀⠸⣿⣿⣿⣿⣿⣿⠋⠀⣀⠈⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠛⠓⠿⠿⠿⠿⢿⣿⠀⢸⣿⢠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⡤⠈⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⣿⡿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⠿⠟⠋⠁⠀⣿⡇⠀⠉⠛⠿⠿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣤⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⣿⡇⠀⢠⣾⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠉⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⡆⠀⣿⡇⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⠇⠀⣿⡇⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⢷⣶⣤⣼⣿⠀⢀⣀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⢸⣿⠀⣰⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠁⣾⣷⠀⠀⠉⢻⣿⠀⢸⣿⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⢸⣿⣼⡿⠁⠉⠙⠛⠻⠿⢿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠋⠁⠀⢀⣠⣾⣿⠀⠀⠀⢸⣿⠀⠘⠛⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⢸⣿⠟⠀⠀⣤⡄⠀⢀⣤⡀⢸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣤⣶⡿⠿⠛⣿⣟⠀⠀⠀⢸⣿⠀⠀⠀⣴⡄⠀⣿⡇⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⣿⡇⠀⠘⣿⠇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠋⠁⠀⠀⣀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⣿⡇⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⣤⣴⣶⡿⠟⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠈⠀⠀⣿⡇⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⣾⡆⠀⢠⣶⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠋⠉⠀⠀⠀⣻⡯⠀⠀⠀⠘⠛⠀⢸⣿⠀⠀⠀⣿⡇⠀⢸⣿⡃⠀⢸⣿⠀⠀⠀⠻⠃⠀⠈⠛⠁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⣤⣴⣶⣶⡿⣿⣿⣠⣴⣦⣄⡀⠀⠸⠿⠀⠀⠀⣿⡇⠀⠘⠿⠂⠀⢸⣿⠀⠀⠀⣀⡀⠀⠀⣀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠉⠀⠀⠀⣺⣿⠟⠉⠉⠻⣿⣄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⢸⣿⡇⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣴⣶⣶⣿⠇⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠉⠁⠀⠀⢉⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⣴⡄⠀⢀⣾⡟⠁⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⣿⣷⣄⣀⣀⣠⣾⡟⠉⠉⢉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⣿⡇⠀⢿⠇⠀⠘⣿⣇⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣻⣟⠛⣿⣿⠛⠉⠀⠀⠀⣿⡿⠿⠿⠿⠿⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⠿⣷⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⣀⣀⣀⣀⣀⣽⣿⠀⣿⣿⠀⢀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠙⠋⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠤⠶⢒⠶⠢⡤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠇⠀⠀⠀⢀⠂⠀⠱⠂⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡤⠛⠀⠀⡇⠀⠀⠀⠀⡀⠀⡈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣶⠀⠀⠀⠀⡎⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣷⢼⡠⠀⠘⠀⠐⠗⢀⠁⣠⢃⣰⣹⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠠⠋⡹⢦⣑⡤⣀⣠⡆⢠⣌⡴⢏⣾⡿⢿⠙⡄⠀⠀⠀⠀⠀⠀⠠⠀⢿⠀⠀⠀⠀⢃⠘⠃⠀⠀⠀⣠⠚⠁⠈⣿⢫⡏
⠀⠀⠀⠀⢀⢃⡞⠁⠀⠉⠉⠛⠿⡷⠾⠿⠟⠛⠁⠀⠀⢣⣸⠀⠀⠀⠀⠀⠀⢘⠀⢺⠀⠀⠀⠀⠈⢦⠀⠀⠠⡼⠁⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡆⠀⠀⠀⠀⠄⠐⠚⠉⠠⠤⠀⠀⠀⠀⠑⠄⠠⠂⠀⠀⠀⠀⢿⢸⡆
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠐⠅⠀⣠⠴⠴⠦⣄⣀⣠⠗⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢸⠘⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠈⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣇
⠀⠀⠀⠀⣿⡆⠁⣄⣤⣧⣠⣴⣀⠀⠀⢀⣤⣤⣠⣄⢀⣀⣸⣿⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸
⠀⠀⠀⠀⢹⣇⠀⠀⢻⡿⣿⠿⠟⠀⠀⠈⠻⠙⠋⠙⠛⠁⢸⣽⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡆
⠀⠀⠀⠀⢈⡧⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡃
⠀⠀⠀⠀⠀⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠺⠀⠀⠀⠀⠀⠀⡀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⠆⠀⠀⠀⢸⠀⢻
⠀⠀⠀⠀⠀⡇⠛⣶⣤⣄⣀⣠⠴⠄⠰⠦⣀⣀⣠⣤⣶⠋⠸⠀⠀⠀⠀⠀⠀⠀⢁⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡎⠀⠀⠀⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠁⠀⡇⣾⡏⠈⣠⣤⣤⣦⣤⣤⡈⠹⡧⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠆⡆⠀⠀⠀⠀⢼⠀⠈
⠀⠀⠀⠀⠘⢇⠀⡇⢸⠀⢰⠿⣒⣤⣤⢖⣻⣧⠀⡇⢀⠀⡸⠃⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⡝⠀⣘
⠀⠀⠀⠀⠀⢸⠀⠀⠸⢑⠃⠀⠫⢾⡉⠈⠋⢹⡩⠅⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⢀⠇⠀⣲
⠀⠀⠀⠀⠀⠈⡀⠀⢈⡡⠆⠀⠀⡆⠁⠀⠀⠰⢎⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠊⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠰⡄⢸⢎⡁⠀⠀⠃⠀⡆⠀⢀⡳⡆⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⠀⠀⠀⠀⠀⠱⣸⠘⠆⠀⠀⠘⠀⠀⠀⢰⠃⣇⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡼⠀⠀⡄⠀⠀⢀⢠⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡈⠀⠀⠀⣶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢾⡁⢠⡶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢷⡀⢀⢀⣠⣾⣦⡇⠀⠀⢀⡗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠒⢧⣾⣅⣹⠛⠓⠒⠋⡀⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠃⠀⠀⠀⠀⠀⠀⠀⠙⠚⠓⠙⠓⠶⣬⣽⢃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠶⠶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⢀⣀⣀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣅⣀⡀⣀⢠⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠉⠃⠘⢸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⣿⡄⣀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠰⠃⣀⠐⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣦⣑⠠⡀⠀⠀⠀⡠⢾⣿⣿⣀⠀⠈⣠⣿⣏⠙⠂⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⠿⢷⣤⡙⠯⣷⣌⠲⢖⢘⣅⡘⣿⣿⣿⡟⢨⡏⠹⣿⣿⡆⡜⠡⢂⣀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣤⠴⠟⠛⠷⢤⣉⠷⠊⠉⠀⠙⣬⣿⣿⣿⠟⡀⠀⣿⣿⣷⣷⣦⡉⠉⠢⡱⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣤⣞⣉⣤⣤⠤⠴⡶⠚⠁⠀⠀⠀⢀⠀⠈⢿⣿⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⣱⣿⡉⢛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠀⠀⠀⠀⠸⡇⠀⣀⣀⣤⢾⣉⠉⠉⠘⠣⣾⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣷⣾⣿⣿⣃⣀⡀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⡀⠒⢷⣿⠉⠀⠀⠈⠻⡷⣄⢰⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠠⢤⣤⣴⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⡟⢮⡻⡿⠟⠳⢌⠻⣿⣿⣥⣼⣏⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠞⠛⠉⠁⢸⡇⠀⠀⢀⡠⠐⠁⠈⣿⣌⠢⡀⠀⠙⠁⠈⠳⡄⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠠⠏⢿⣷⣌⠁⠀⠀⣀⠀⢹⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡀⠀⡀⠀⠀⠀⡼⠁⢻⡆⡆⢰⠁⣷⡾⠦⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇⣸⠇⣰⠀⠀⠀⠀⣼⡇⠧⠈⢶⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣠⣿⢠⡿⠀⠀⣾⣄⣿⣿⣄⠀⢀⠙⠛⢛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠿⠿⠿⣧⣤⣾⣿⣿⣿⣿⣿⣷⣦⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣴⣴⣤⡄⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠟⠿⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⣤⡶⣶⣦⡾⠟⢿⣦⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⣾⠟⠿⣦⣴⠶⣦⣄⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⢠⣿⠁⠀⠀⣉⣤⣶⠾⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢷⣦⣌⡁⠀⠈⢻⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⠿⠛⢋⣠⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡈⠛⠻⢶⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⣄⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠘⢷⣄⠀⠀⣿⠁⠀⠀⠀⠀⢀⡀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠙⠃⢀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣴⠟
⠀⠀⣸⡟⠀⢠⣿⠀⠀⠀⠀⠀⣾⠃⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⢻⡆⠀⠀⠀⠀⢸⡗⠀⠘⣿⡀
⢠⣿⠋⠀⠀⢨⣿⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡗⠀⠀⠈⢻⣧
⠈⠻⣦⡀⠀⠈⣿⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⢹⡄⠀⢸⡇⠀⢸⡇⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢸⡇⠀⠀⣠⡾⠋
⠀⢀⣾⠃⠀⠀⣻⣧⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⣤⣤⡴⠾⠋⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠈⠻⢶⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⣠⣠⣿⡃⠀⠀⢻⣆
⣴⡟⠁⠀⠀⣼⠟⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠙⢿⡄⠀⠀⠙⣷
⠙⢷⣄⠀⠀⣿⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣸⡇⠀⠀⣴⠟
⠀⣸⡟⠀⠀⢻⣆⠀⠀⠀⢻⡇⠀⣠⠾⠛⢛⣿⠇⠰⣶⣶⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣶⣶⡶⠀⢿⣟⠛⠻⢦⡄⠀⣾⠇⠀⠀⢀⣿⠃⠀⠘⣿⡀
⠀⣿⡁⠀⠀⠀⠻⣦⡀⠀⢸⣧⢸⣇⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⣿⠀⣿⠀⠀⣠⡿⠁⠀⠀⠀⣹⡇
⠀⠈⠻⢷⣦⠀⠀⠈⠻⠷⢾⣧⠀⠙⠷⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠁⢈⣿⠾⠟⠋⠁⠀⢠⣶⠟⠛
⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠈⣿⠀⠀⠀⢠⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⡁
⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⢻⣧⡀⠀⣿⣀⣿⠀⠀⣠⡶⢦⣄⣀⣤⣄⡀⠀⠀⣀⣤⣄⣀⣤⠶⣦⡀⠀⢸⣇⣸⡇⠀⣀⣿⠁⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣦⣌⡉⠁⠀⠀⠉⠀⠀⠙⠋⠁⠈⠛⠛⠛⠋⠀⠉⠙⠁⠀⠈⠁⠀⠀⠉⣉⣤⡾⢿⣥⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠉⠛⣷⣶⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⡿⠋⠁⠀⠀⢹⡇⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠛⢿⣦⠀⠀⢀⣿⠀⠀⠀⢀⣾⣿⡛⠿⠟⠉⢻⡿⠛⠿⠛⢷⣶⣶⠿⠿⠛⠛⢛⣿⠋⠉⠀⠛⣿⣦⠀⠀⠀⢿⣇⠀⠀⢠⡾⠛⠋
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢸⣏⠀⠀⠀⣿⣯⠉⠁⠀⢀⣴⡾⠿⠿⠶⠾⣿⣯⣄⣀⠀⠀⠀⣼⡟⠀⠀⠀⠸⣿⡀⠀⠀⠀⢸⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠛⣷⡀⠘⣿⣉⠀⢠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢶⣿⣧⣄⣀⠀⣾⡟⠁⠀⠀⣰⡟⠁⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⠀⠀⣿⠁⠀⢼⣯⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠃⠀⠀⠀⣿⠇⠀⠀⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠹⣧⠀⠀⠉⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠉⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶⣦⡀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⣴⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠀⠈⠙⠻⠷⢶⣦⣦⣴⣶⠾⠛⠉⠙⠻⠿⠶⣶⣤⣤⣄⣀⣀⣀⣠⡾⠋⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡒⡒⠚⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠬⠬⡥⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⣴⡆⣠⢟⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⢾⢿⡟⠿⠂⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⣄⡞⠿⠰⢦⣀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣷⡟⠀⠀⠀⠠⠹⣧⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠴⡏⠀⠀⠀⠀⠀⢡⣾⠿⢤⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⡭⣧⠀⠀⠀⡄⢠⢞⠧⢇⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡇⣈⠃⡠⢠⣀⣛⡧⠟⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠿⣸⣠⠴⠿⠯⣠⠤⠶⠥⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣯⡤⠸⠧⢸⠇⠿⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣃⡀⠁⡘⣿⣏⡠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⢛⣏⢹⣿⣼⣷⡿⠯⠦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡀⢿⠈⣆⢨⣿⠁⣿⡾⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣧⠀⠀⠈⣿⡇⢠⠟⢳⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⡻⣿⠀⠄⠀⢸⠈⢿⣄⢸⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣸⣧⠀⠀⠀⡆⡀⣿⣿⡤⠠⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⡖⠓⡤⠛⠇⣙⣷⣄⣟⣿⣟⣀⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⡿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣸⣿⣇⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡉⢻⠿⢿⣿⡿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⢻⣿⠛⠛⣿⡇⢸⣤⣼⣿⣧⣤⡍⣭⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣼⣿⣤⣤⣿⡇⠸⠿⠿⠿⠿⠿⠇⠻⠏⠉⢹⣿⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⢻⣿⠛⠛⣿⡇⢹⣿⠿⢿⣿⡿⠿⣿⣿⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣼⣿⣦⣴⣿⡇⢹⣿⣀⣠⣿⣇⣀⣿⣿⠀⣸⣿⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⢻⣿⠛⠛⣿⡇⢸⣿⠿⠿⣿⡿⠿⣿⣿⠸⢿⣿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣾⣿⣶⣶⣿⡇⢸⣿⣀⣠⣿⣇⣀⣿⣿⢀⣼⣿⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⢻⣿⠋⠙⣿⡇⢸⣿⠿⠿⣿⡿⠿⣿⣿⠘⢻⣿⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠶⠾⠛⠒⣒⣛⣃⣘⣋⣀⣀⣛⣁⣀⣙⣛⠀⠺⠿⠤⢼⣿
⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠾⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠷⢶⣶⣤⣄⣀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣰⠛⠙⠳⡄⠀⠀⣀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⠉⡄
⠀⠀⠀⢻⣀⠀⣠⠇⠀⠀⢿⣄⣀⣀⣼⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⢸⢻⠀⠀⠀⠀⠸⣀⠇
⠀⠀⠀⠐⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⠏⠀⠀⢰⣄⣀⡀⡀⠉⠙⠃⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠈⣿⠙⣿⠀⠀⠀⠀⠹⣿⣿⡋⠙⢳⣦⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠹⣿⠙⠛⣻⡇⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢀⣼⣼⡄⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⠀⠀⠀⢸⡇⢈⣿⠀⠀⠀⣿⠛⣿
⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⣧⣼⡇⠀⠀⠀⣿⣀⣿
⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⣏⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣸⡟⠻⣧⠀⠀⠀⣿⠿⣿
⠀⠀⠀⢀⣿⣤⣿⠀⠀⠀⠀⣰⣿⣶⣿⡀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠁⠀⢿⡆⠀⢰⡏⠀⢹⡆
⠀⠀⢀⣼⣿⣿⣿⣧⠀⠀⣰⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⣸⡇⠀⣸⡇⠀⢸⣇
⠀⠀⢸⣿⠛⠉⠙⣿⡄⠀⢿⣿⠋⠀⠀⠙⣿⠀⠀⠀⠀⣿⠾⠳⣾⡇⠀⠀⠸⣿⡷⠾⣿⡇⠀⢹⣷⠶⣾⡏
⠀⠀⠈⢿⠀⠀⠀⠸⠁⠀⠈⢿⡄⠀⠀⠀⠉⠀⠀⠀⠀⢿⣀⢀⣼⠇⠀⠀⠀⢿⣦⣠⣿⠃⠀⠸⣧⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠉⠁
⣶⢖⣶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠾⠿⠿⠿⠷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣀⡤⠶⠶⠶⡶⠛⠛⠛⠻⠶⠛⠛⠛⠶⣌⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠙⢳⣄⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣾⡏⠀⡤⠖⠛⠛⠓⠀⠀⠀⠀⠈⠉⠉⠉⠙⠲⠄⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢠⡿⠀⠘⠶⠶⠶⠚⠁⠀⠀⠀⠀⠘⠓⠲⠒⠛⠀⠘⣧⠀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⢸⡇⠀⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⣼⡇⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣠⡤⠀⠀⠀⠀⠀⢠⣿⢁⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣬⣿⣄⣀⣀⣀⡀⠀⠀⠀⠉⣉⣁⣀⣀⣀⣠⣤⣴⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠛⠉⠉⢙⣷⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠟⠁⠀⠀⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⠀⣠⡴⠟⠋⠁⠀⣦⣤⡾⠿⢿⡟⠷⢶⣶⣶⣶⣿⣿⡟⢛⣛⣩⣥⣤⣴⠶⠶⣤⣄⡀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⢀⣾⠏⠀⢀⡴⠟⢛⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢻⣆⠀⠈⠉⠛⠛⠉⠁⠀⠸⣇⠀⠀⠀⠈⠙⠳⣦⡀⠹⣿⡄
⠀⠀⠀⠀⣾⣣⣴⠿⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠹⣆⢻⣷⡀
⠀⠀⢀⣼⡿⠋⠁⠀⠸⣇⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣄⣀⣀⣼⡃⣿⣇
⠀⣠⡾⠋⠀⠀⠀⠀⠀⠹⣆⠀⠈⢷⣄⡀⠀⣀⣀⣀⣀⣀⣀⣀⣼⣧⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢷⣼⣿
⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⢈⣿⡏⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿