Frostbite Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠊⠉⠁⠂⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠔⠁⠉⢍⠁⠢⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⡆⡖⠀⠶⠀⠂⠰⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢢⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠉⠁⠀⠉⠉⠉⠁⠑⠀⠊⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠚⠀⠀⡳⠤⠒⠂⢸⠀⠉⠈⡏⠉⠀⠳⠆⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⠁⠀⠘⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⡇⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡸⠁⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡌⠀⠀⢇⠀⠀⡇⠀⠀⡅⠀⢠⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⡇⠀⠀⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⠀⠀⠈⠀⠀⡇⠀⠀⠃⠀⠀⡇⠀⢠⠀⠀⠀⢀⠀⠀⠀⠇⠀⢠⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠇⠀⠘⠀⠀⠀⢸⠀⠀⢰⠀⠀⠈⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡀⠀⠀⡇⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⠀⢀⡇⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆⠀⢰⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠄⠀⡇⠀⠈⡆⠀⢠⠀⠀⡸⠀⠀⡆⠀⠀⡇⠀⠀⡘⠀⠀⢠⠃⠀⡌⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣸⠀⠀⠃⠀⠠⠅⡄⠚⠖⠀⠁⠋⠈⠁⡏⠀⠁⠈⠀⠃⠶⠐⢣⡄⢀⡇⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠦⠔⠊⢉⡁⠀⣸⠀⡀⠤⠀⠠⠇⠠⠠⠦⠄⠤⠆⠄⠴⠇⠀⣴⣆⣀⣤⡏⢺⣶⣦⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⢋⣀⣰⣧⣶⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣿⣿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣾⣿⣟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣿⣟⣿⣿⣿⣿⠻⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠼⠚⣻⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄⡀⠿⠀⠀⠀⠀⠀⠀⢰⣤⣼⣿⣿⣿⣿⡟⣿⣿⡧⡈⠃⠀⡂
⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⣿⣿⣿⣿⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡻⣿⢿⣿⡿⣿⣷⢿⣿⣿⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⣻⣿⣿⡿⣏⣿⣾⣿⣿⠿⠶⣀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⡐⠁⠈⠁⠺⠆⠘⣷⠿⠹⢢⣿⣿⣿⠂⠐⠨⠅⠐⠂⠀⠤⣤⣤⣤⡤⠄⠐⣂⠩⠔⠨⠋⣿⡟⢿⢸⣾⠇⢻⠏⠉⠀⠀⠁⡄⠤
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡍⠻⣦⠀⠀⠀⠀⠉⠀⠀⠉⣿⡏⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠋⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠘⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟
⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀
⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠶⣼⣧
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣼⣯⣤⡀
⠀⢀⣾⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣤⣤⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⣼⠃⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠉⢻⣶⣄⡀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⢰⡟⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⣠⡾⠁⠈⠙⠳⣦⣤⣀
⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣀⣤⡾⠋⠀⠀⠀⠀⢠⡿⠉⠉⠛⠷⢶⣦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⣿⠁⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣼⠃⢹⣧⣤⣄⣀⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢿⡀⠀⢀⣠⣴⠿⢧⣤⣀⣀⣀⣤⡾⠁⠀⣿⠁⠀⠈⣭⣉⣙⡛⢷
⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣶⡶⠾⠿⠟⠛⠉⠀⠀⠀⠈⠉⢉⣽⠟⠀⠀⢰⡏⠀⠀⠀⣿⣿⣿⣿⠆
⠀⠙⢷⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠶⠶⠶⠾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣥⣀⣠⣴⡿⠀⠀⠀⠀⠹⣿⠟⣡⡾
⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠷⠶⠶⠶⣶⣶⣿⣋⣁⡀⠀⠈⠉⢁⣿⡁⠀⢀⣀⣀⣤⡴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣶⣶⣾⣋⣛⣛⣛⣉⣉⣁⣀⣀⣀⣀
⣿⣿⣋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢰⣿⠿⠛⢿⣷⣄⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢀⣿⡿⠀⠀⠀⠹⠿⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠿⢿⣶⣦⣴⣾⡿⠁⠸⠿⠀⠀⠀⢀⣀⣴⣾⠿⠁
⠀⠀⠀⠀⠀⣠⣾⠿⠿⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⢀⣀⠀⠈⠉⠉⠁⠀⠀⣠⣴⣶⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⢰⣿⠏⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠘⠛⠀⠐⠿⠃⣀⣴⣿⠟⠋⠁
⠀⠀⠀⠀⢺⣿⠀⢀⣾⡿⠛⠁⢀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠾⠇⢠⣾⡿⠋⠁
⠀⠀⠀⠀⠘⣿⣦⣼⣿⠁⠀⣰⣿⠟⠉⠀⣠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠈⠙⠛⢿⣷⣤⣿⣿⠀⠀⣾⡿⠋⠀⢀⣴⣾⡷⠀⠀⠀⠀⠀⠸⡿⠃⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣷⣶⣿⣇⠀⠀⣾⣿⠁⠀⠀⠀⠀⣀⣀⣀⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣶⣿⣿⣤⣀⣀⣠⣴⣿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⡀⠀⠀⠀⣿⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣾⣷⠀⠀⠀⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢹⡆⠀⠀⠀⣿⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣤⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢀⣴⠟⣩⣍⠻⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠈⠻⣆⠀⠀⠀⠀⠀⠙⠀⣠⡶⠟⠡⣾⡿⢋⣼⠟⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠈⣴⠟⠁⠀⠀⢀⣼⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣿⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣰⠏⠀⠀⢀⣴⠟⠉⠉⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸⡇⠀⢀⣴⠟⠁⠀⠀⢀⣤⠶⠞⠁⠀⢀⣴⠟⠁⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣸⣇⣴⠟⠁⠀⠀⠀⠀⠛⠁⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⣴⠟⠉⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣤⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠈⠻⣦⡀⢀⣠⡶⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠈⠻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢂⠴⡶⢤
⠀⠀⠀⠀⠐⢦⣆⠹⣁⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠔⠁⠐⠁⢈⡅
⠀⠀⠀⠀⢀⣤⠔⢛⠐⣤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠔⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈
⠀⠀⢀⠠⠌⠀⠀⢿⠂⠀⠁⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡈⢈⢞⢾⡁⠀⠀⠀⠀⠀⠀⡔⠲⡀⠀⠀⠀⠀⡠
⠀⠀⠈⠢⠠⠀⠀⠀⠒⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⡈⠢⡄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⡹⣁⣶⠆⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠑⡀⢆⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⢪⠓⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣈⡀⢆⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠄⡈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠃⢰⣴⢴⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂⠽⠀⠀⠀⢀⢅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠴⠂⣠⠁⠲⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠔⣃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣴⠀⠀⠀⠸⠿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠁⠤⢚⡴⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡊⢡⡀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣽⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⢸⠄⠀⣠⡀
⠀⠀⠀⢀⡀⠀⠀⢀⠀⢈⣩⣈⣫⡁⠀⠀⠀⠀⠀⡀⣀⡠⠊⠀⠙⢣⢔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡙⢑⢨⠑⠈⠉
⠀⠀⠀⡄⠘⠀⡌⠘⠀⠋⠀⣿⠀⠉⠁⠀⠀⠀⠀⣿⡇⢀⠠⠃⢀⠀⠀⢨⠉⠉⠥⠀⠀⠀⠀⠀⠀⡇⢸⠠⠞⠉⢰⡈⠙⠶⠄
⠀⠀⠀⠀⢰⠀⠦⡤⠀⠀⠀⠀⢠⠠⠀⠀⠀⠀⠀⣿⡏⠀⠀⡀⠂⠀⣐⢘⠂⣂⡒⠀⠀⠀⠀⠀⠀⡇⢸⢰⠀⠀⠸
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⠀⠁⠈⠈⠃⠀⢠⠄⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢸⣜⠄
⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠻⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣸⣠⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣼⡇⠀⢀⣻⣁⠀⠀⣿⣴⠆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣳⠞⣋⣍⠉⠛⢶⡟⠛⠛⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣄⣸⠇⣼⠉⠀⠀⠀⠈⣷⣀⣴⣋⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠉⠹⣆⠀⠀⠀⠀⠀⢀⡟⠉⠙⢯⠁⠀⠀⠠⣄⠀⠸⡇⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⠦⣤⣀⣠⡴⠿⣄⣀⣀⠀⠀⠀⠀⠀⠈⠳⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⢿⡇⠀⢀⣽⡀⠀⠀⣿⠻⡍⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠋⢹⠙⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠟⠳⣄⣠⡴⠛⢦⣀⣠⠞⠻⣦⣀⣴⠛⠳⣄⣀⡴⠛⠳⣄⣀⡴⠛⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣬⣥⣤⣤⣤⣬⣤⣤⣤⣤⣭⣤⣤⣤⣬⣭⣤⣤⣤⣬⣭⣤⣤⣤⡿





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣾⣇⣤
⠀⠀⠀⣀⠀⢀⣶⣶⣶⣶⣄⡀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⠟⠀⠀⠀⠘⠿⢶⣄⠀⠀⢀⡿⠀⢸⣿⣿⣿⣿⣠⡀⡽
⠀⢓⣜⡙⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣶⡇⠰⠶⠆⠀⠀⠀⠀⠛⣁⢠⠀⡀⣀⣀⣀⣀⣀⡿⠀⠀⢀⡿⣰⣾⣿⣿⣿⣿⣿⡇⡿
⣾⣿⣉⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣥⠉⠉⢉⠉⠉⠁⠀⠀⢈⡷⢹⣿⣿⣿⣿⣿⣿⠃⡿
⠀⠀⠀⠀⠀⠀⢶⡿⠟⠃⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⡇⠀⠀⠀⠀⠀⠀⠐⠋⠿⠙⠂⠀⢬⣿⣿⠄⠀⠀⠀⠈⠛⠈⠻⣿⣿⣿⣿⠟⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿

⠀⠀⡀⢀⣴⣶⣶⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⢀⡀⠀⠀⠀⠀⠀⣾⢛⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⡀⣤⣤⣤⡀
⠀⢾⠁⠸⣿⣿⣿⡿⠀⣳⠀⠀⠀⠀⠀⠀⠠⢠⡶⠛⠛⢷⣄⠤⠀⠀⠀⠀⣿⢸⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠆⣁⣀⣹⣇⣀⡀
⢐⡏⠀⣀⡙⠿⠟⣁⡀⠨⡇⠀⠀⠀⠀⠀⣠⣿⣤⣀⠀⠀⢹⡦⠄⠀⠀⣤⣿⣼⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠈⣜⣬⡼⠛⡛⠻⣿⣿⣿
⠘⢢⣿⣿⣿⣦⣾⣿⣿⣦⠛⠀⠀⢀⣴⡶⣿⢛⣛⢻⣶⣤⡟⠤⠀⠀⠀⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠀⠀⠀⠀⣿⣿⡇⠐⡿⠂⣽⣿⣿
⠀⣼⣿⣿⣿⣿⣿⣿⡿⣿⡄⠀⠀⠸⣿⣶⣿⣬⣽⣿⣿⡿⠃⠀⠀⠀⠀⠈⢧⣤⣤⣤⣤⣤⣤⣤⡼⠃⠀⠀⠀⠀⣿⣿⣿⣶⣶⣾⣿⣿⡿
⠀⠛⠃⠘⠛⠛⠛⠛⠃⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⡀
⠀⠀⠀⣀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⡴⣚⣃⡿⠃⠀⠿⣃⣓⢦⡀⠀⠀⠀⠀⠀⠀⢀⣆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣠
⠀⢠⣤⣬⣥⣯⣤⣄⠀⠀⠀⠀⠀⠀⡾⠛⠿⢿⡆⠀⣾⠿⠿⢿⡀⠀⠀⣶⠀⠀⢠⣼⣿⣿⡇⠀⠀⣶⠀⠀⠀⣀⠀⠀⠀⠹⣿⢿⠂
⠀⢸⣿⠛⡍⡝⢻⣿⠒⢦⠀⠀⠀⠀⢿⠀⠀⣸⠁⠀⣧⠀⠀⣼⠀⠀⠀⢿⣦⣲⣮⠻⣿⢟⣴⢆⣴⡿⠀⠀⢸⣿⡆⣤⣤⡄⠀⠀⠀⠀⣀
⠀⢸⡇⣽⡀⣸⡅⣿⣠⠞⠀⠀⠀⠀⢸⡆⠀⢻⡀⠀⡿⠀⢠⡏⠀⠀⠀⠀⠙⢿⣿⣿⢴⣿⣿⡿⠋⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿
⠀⢸⣿⣤⣅⣥⣾⣿⠁⠀⠀⠀⠀⠀⠈⣇⠀⣸⠇⢸⣇⠀⣸⠁⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠘⠿⠉⠉⠉⠉⠉⠉⠉⠉⠻
⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⢀⣠⣤⣤⣴⣾⣷⣦⣀⠀⠀⠀⠀⢰⣇⠀⠾⡥⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣿⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣬⣽⡱
⣠⠶⠾⠁⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢋⣤⣤⣥⣀⣘⠁⠀⠀⠀⠀⠀⠀⠀⠠⠟⣿⢿⠉⠀⠀⠀⠀⠀⠀⠀⢰⡄⣿⣿⣿⣿⠄⣦
⠻⣤⣤⣤⣤⣤⣭⣼⠏⠉⠀⠀⠀⠀⠐⣶⣶⣿⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⣶⣥⣶⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣿⣿⠘⣧
⠀⠀⠀⢀⣤⡀⠀⢰⣾⣷⡀⠀⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡀⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣸⣿⣿⣿⠘⣧
⠀⠀⠀⠘⠛⠃⠀⠘⠻⠛⠀⠀⠀⠀⠈⠉⠉⠩⣿⣽⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠻⠄⠻⣿⣿⠏⠘⠧
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠶⠐⠄⠂⠆⠄⠐⠀⠦⠶⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠠⠀⠶⠀⠆⠀⠤⠀⠆


⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⢃⣦
⠀⠀⠀⠀⠀⠀⠀⠰⠂⡰⠮⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢒⣿⣿⠲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠆⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠼⠄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡒⠒⢒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⡀⠁⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢉⡉⢛⡉⢉⡊⢉⡑⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢟⣩⣶⣶⡿⣿⣶⣬⡙⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⢡⣾⡟⠋⣷⡄⣴⣿⠙⢿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡟⢷⣷⡝⠃⠟⣵⣶⠞⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣆⣴⣶⢖⡀⣔⢶⣶⣄⣾⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣌⡁⣾⠇⢿⣷⢈⣽⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣷⣬⡻⢿⣷⣦⣶⣿⠿⣋⣴⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣶⣶⣶⣒⣒⣒⣒⣲⣶⣶⣒⣒⣒⣒⣺⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠿⠿⠿⠿⠭⠭⠭⠭⠭⠭⠭⠭⠽⠿⠯⠭⠭⠭⠭⠭⠭⠭⠭⠽⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢟⡛⣿⣿⠟⡛⢿⣾⠟⣛⢿⣿⡿⣛⠻⣿⣿⢟⡛⣿⣿⡟⣛⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣾⣿⣮⣥⣾⣿⣮⣭⣾⣿⣦⣉⣴⣿⣷⣍⣵⣿⣿⣮⣋⣼⣿⣦⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⢀⣤⣶⣾⠿⠿⠿⣶⣶⣤⣀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣰⣿⠛⠁⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢰⣿⠃⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣼⣿⣴⡿⠟⠛⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⣷⣦⡀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⣰⣿⠃⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣰⣿⠃⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠃⠀⠀⢰⣿⠃⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣋⣁⣀⣤⡀⠀⠘⠋⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠛⠻⠿⠛⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣤⣄⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠀⢀⣠⣤⣤⣀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢀⣀⣀⣴⣿⠟⠉⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠘⠛⠛⠋⠐⠿⢠⣦⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⡀⠘⢟⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡴⠶⢶⣄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡏⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⡿⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠲⢦⣄⠀⠀⢸⡇⠀⠈⠁⠀⠈⠁⠀⠀⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄⠀⠙⢧⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⠻⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣄⡀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣟⠀⠀⠀⡿⢠⣤⣤⣤⣤⣤⣤⠀⡇⠀⠀⢘⣻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣇⠸⠿⠿⠿⠿⠿⠿⠂⣿⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⢛⡿⠛⠛⠛⢻⡟⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣠⡴⣿⠀⠀⠀⢠⡞⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠉⠀⢿⠀⢀⣴⡏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣸⡷⠋⠀⠻⣦⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠾⣯⡁⠀⠀⠀⠀⠈⠙⠶⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠶⠿⠿⠷⠶⠶⠶⠶⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣼⣧⣤⠄⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣹⣇⣠⡤⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⠀⢀⠀⢦⣤⣈⣹⡏⣁⣤⠦⢀⡀⢀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⡀⣿⠀⣿⠀⠀⠀⠉⢻⡟⠉⠀⠀⢸⡇⠸⣇⣈⣷⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⣈⡽⠿⣦⣿⠀⠀⢀⣠⡾⢧⣄⡀⠀⢀⣷⡴⠿⣯⡈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣠⡴⠛⠉⠻⡾⠋⢁⣠⣄⡈⠛⣶⠛⠉⠳⣦⡈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠀⠀⠀⠀⡇⠀⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠙⠷⣄⣀⣴⣷⣄⠈⠛⠋⢁⣠⣿⣄⣀⣴⠞⠋⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠙⣷⣶⠟⣿⠁⠀⠉⠛⢶⡾⠋⠁⠀⠈⡿⠷⣶⣟⢉⣠⡄⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠉⣿⠀⣿⠀⠀⠀⣀⣼⣇⡀⠀⠀⢸⡇⢠⡏⢙⡿⢧⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢙⣀⡈⠀⠶⠛⢉⣹⣏⠙⠻⠶⠈⠁⠈⠁⠘⠃⠀⠀⠀⠀⠐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠶⠶⠛⠛⠉⠀⠀⠀⠞⠛⣹⡏⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⢻⡟⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠞⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠳⠶⠶⣤⣤⣤⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠷⠶⣤⣤⣤⣤⡤⠶⠞⠛⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⠶⠀⠀⠀⡇⠀⠀⡇⠀⠀⡤⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢱⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⣴⠃⠀⡕⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠈⡆⠀⠀⠰⠁⠀⢀⡇⠀⠀⢠⡏⠀⢸⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⡆⠀⠀⢸⠀⠀⣽⠀⠀⢰⠃⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢣⠀⠀⠀⠀⠀⢸⡇⠀⢀⡞⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢡⠀⠀⢸⠀⠀⡿⠀⢀⠇⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⡀⠀⠀⡄⠀⢸⠁⠀⡸⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⡀⠀⣸⠀⠀⡇⠀⡸⠀⠀⡼⠀⣠⠖⢦⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢇⠀⢰⠇⠀⢸⠀⢀⠃⠀⢀⠇⢀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢻⠀⡏⠀⠀⡇⢀⠇⠀⢰⠇⢰⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠸⡄⢸⠀⠀⢸⠀⡼⠀⠀⡿⢀⠎⠀⢰⠇
⠀⠀⠀⠀⢠⠖⠲⣄⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠙⠋⠀⠀⠟⣠⠃⠀⣸⠃⠀⡶⠒⢦⠀⠀⠀⠀⣿⠀⠀⠉⠁⠀⠀⠈⠛⠁⠀⠸⢃⡞⠀⢠⡟
⠀⠀⠀⠀⠸⡀⠀⠘⣆⠀⠀⠀⢃⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠛⠁⠀⢰⠁⠀⠀⣇⠀⠀⢳⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢮
⠀⠀⠀⠀⠀⠳⡀⠀⠈⢦⡀⠀⡸⠀⠀⢀⡞⠀⠹⣆⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠘⣄⠀⠀⠱⣄⠀⢈⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠉⠛⠁⠀⢠⡟⢰⣶⣶⡝⣆⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⠛⠉⠀⠀⠀⠘⠃⣈⡀⠋⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⢠⠏⠐⠛⠾⢿⡓⠘⢆⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠈⢉⡠⠛⠃⡈⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢰⣏⣀⣀⣁⣀⣀⣉⣀⣈⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⠁⠚⠀⠉⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⠀⠀⠀⢸⠀⠀⢈⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⠀⠀⠀⠐⡇⠀⠀⡇⠀⠀⠀⢀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢱⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⣼⠃⠀⡵⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡆⠀⠀⢀⠀⠀⢀⡆⠀⠀⢠⠏⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⡆⠀⠀⢸⠀⠀⣼⠀⠀⣰⠃⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢱⠀⠀⠠⠀⠀⢸⡇⠀⢀⡏⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢱⡀⠀⢸⠀⠀⡟⠀⢀⠇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠸⣄⠀⢠⡆⠀⣸⡅⠀⣼⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⡅⠀⣸⠀⠀⡇⠀⡸⠀⠀⠸⠀⣠⠚⣷⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⡀⢸⡇⠀⣿⠁⢰⡇⠀⣸⠇⢠⡾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢻⡀⡏⠀⠀⡇⢠⠇⠀⢰⠇⡰⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⡧⠀⠸⠇⣾⠃⠀⣿⠀⡿⠀⢀⡿⢰⡟⠀⣸⡇
⠀⠀⠀⠀⢰⠛⠳⣄⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠙⠋⠀⠀⡟⣰⠃⠀⣼⠃⢠⡖⠀⣦⡀⠀⠀⠀⣴⠀⠀⠘⠃⠀⠀⠘⠛⠁⠀⣼⣣⡟⠁⢰⡟
⠀⠀⠀⠀⠸⡀⠀⠘⢆⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢰⠋⠀⠈⢇⠀⠈⢳⡀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠋⠀⠀⡏
⠀⠀⠀⠀⠀⠹⡄⠀⠈⢣⡀⢀⡼⠀⠀⣀⡀⠒⠒⢠⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠈⢆⠀⠀⠱⣄⠀⣠⠃⠀⠀⠀⡔⠁⠀⢀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠉⠉⠁⢀⣺⣿⣷⡀⢠⣿⣿⣧⠀⠀⠀⡎⠀⠀⠀⠀⠀⠈⢳⡀⠀⠈⠉⠁⠀⠀⠰⣦⠀⠀⠰⠏⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢨⠿⠿⠿⢿⡾⠿⠿⠟⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠘⢄⠀⣰⣿⣷⣄⠀⡜⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠑⠛⠻⠟⠛⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠁⠤⠤⠤⠂⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⣶⣶⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⠀⢸⡇⠀⢠⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠻⣦⣸⣇⣴⠟⠁⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠷⠶⠶⢾⣿⣿⡷⠶⠶⠾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⣴⠟⢹⡏⠻⣦⣀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠁⠀⣸⣇⠀⠈⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠛⠛⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡙⠿⣿⠿⢋⣉⡙⠻⢿⡿⠛⣉⣉⠛⢿⡿⠟⢋⣉⡙⠿⣿⠿⢋⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣻⣿⣟⠛⢛⣿⣿⣟⠛⢛⣿⣿⡛⠛⣻⣿⣿⡛⠛⣻⣿⣟⠛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣼⣿⣿⣤⣤⣼⣿⣿⣤⣤⣿⣿⣧⣤⣤⣿⣿⣧⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⣀⣀⣀⣤⣤⣶⡾⠿⠿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠉⠉⠙⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠛⠁
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢀⣤⡀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠙⠻⢷⣤⣤⣶⠟⠋⠁⠀⢠⣄⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢹⣯⠀⠀⠀⠀⠀⢸⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⣸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⢿⣦⣄⣀⣤⣾⣿⣦⣄⣠⣴⡾⠟⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠈⢹⣿⠋⠀⠀⠈⢿⡟⠁⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⢀⣾⣇⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣤⣶⠿⠛⠙⠻⢷⣾⠟⠋⠙⠻⣶⣤⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣸⣗⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⣴⡾⠟⠻⢷⣦⣄⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⣠⣄⡀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣤⡆⠀⢹⡏⠀⢰⣤⠄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡈⠉⠙⢦⣸⡇⣠⠞⠉⢉⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⣿⠀⢒⡿⠒⠒⠒⣿⣿⡓⠒⠒⢿⡒⠀⣿⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢹⡄⠀⢀⣀⣤⠞⢹⡇⠙⢦⣀⣀⠀⢠⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠳⣄⠀⠛⠇⢀⣼⣧⡀⠸⠛⠂⣠⠞⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠠⣔⢂⡄⠀⠀⠀⠀⠈⠳⢤⣀⡀⠈⠁⢀⣀⡤⠞⠁⠀⠀⠀⠀⢠⡐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠛⠛⣛⣛⣛⣛⣛⣛⠛⠛⠛⣛⣛⣛⣛⣛⣛
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢩⣭⣭⣭⢭⣭⣭⣭⣤⣤⣤⠌⠉⢩⣭⣭⣭⣤⣤⣤⣭⣭⣭⣭⠉⠁
⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⣠⠞⠳⣄⠀⣠⠞⠳⣄⠀⣠⠞⠳⣄⠀⢀⡾⠛⢦⠀⢀⡴⠛⢷⡀⠀⢠⠞⠳⡄
⠀⠀⠀⠀⠀⠐⠋⠀⠀⠈⠛⠁⠀⠀⠈⠛⠁⠀⠀⠈⠓⠋⠀⠀⠈⠓⠚⠁⠀⠀⠙⠒⠋⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠉⠈⠀⠁⠁⠈⠈⠈⠁⠁⠈⠈⠀⠁⠁⠈⠈⠀⠁⠁⠈⠈⠀⠁⠁⠈⠈⠀⠁⠁
⠀⠀⠀⠀⠀⣀⣈⣈⣈⣁⣁⣉⣈⣈⣁⣁⣈⣈⣀⣁⣁⣈⣈⣀⣁⣁⣈⣈⣀⣁⣁⣈⣈⣀⣁⣁⣀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣿⡫⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⡸⠓⣧⣎⠀⡀⢀⠁⡠⠈⠀⣨⠻⢈⣠⣀⡐⠫⠓⡦⢲⡖⣔⡀⠀⠀⡧⡤⢀⠪⢇⡠⠽⣿⢸⡸⠈
⠀⠀⢣⠠⢀⡇⠛⠠⢯⣧⣟⣉⣶⣴⡐⠚⡩⢤⠈⣙⡏⠙⢏⠁⠠⣬⠛⣫⢬⠅⠆⢹⣡⠀⢠⢂⡈⠧⢀⠉
⠀⠀⠠⠂⢸⠓⣌⠘⠳⡞⣽⡟⢒⠯⡇⠀⢠⡀⠈⠋⢹⠼⠘⡠⠔⠃⠀⠘⠰⣎⠀⠘⠂⠂⢉⣁⣀⡠⠴⠦⢴
⠀⠀⣠⡀⠆⠈⢀⠄⠀⠁⠉⠙⠚⠦⡼⠀⠲⡷⡤⢤⠬⢶⠛⠀⠀⠀⠀⠀⠀⠈⠡⣿⣿⡿⣿⡛⣥⣀⡂⠐⠨
⠀⡆⢩⠘⢠⢚⠀⢂⣐⡂⣐⠤⣄⠀⡘⡄⣟⣍⢍⣿⣿⡏⠀⡀⠴⠀⠁⠈⠀⠐⠀⢻⣈⣹⡇⣷⢏⠁⠸⡈
⠀⠓⠐⠋⣶⠀⣤⣭⣶⠀⠀⠀⣤⡑⡆⢹⢤⣿⠀⣿⣿⠱⠀⠿⠿⠇⠀⠀⠿⠉⠆⢸⡿⡿⣿⣝⠀⡁⠈⠀⠢
⠀⠀⠰⣦⢸⠟⠙⠘⠋⠃⠀⠺⠯⠿⠇⢫⠒⠛⠃⡿⠿⠀⣇⠠⠄⠀⠀⠀⠀⠠⢀⣬⣼⣥⣬⣧⣼⠁⢸⢅
⠀⢐⢾⣷⠀⠀⠓⢄⠀⠀⠀⠀⡠⠒⠀⠀⢀⠐⠀⢠⠄⠉⢘⠀⠀⠐⠀⠀⡞⠀⢽⢯⣤⣤⣀⡀⢉⣠⣨⡀
⠀⠘⢰⢸⡉⠀⠀⢀⠍⣲⢦⡀⠀⠀⠀⢸⠊⢀⣂⡉⢸⠀⣇⠤⢀⣠⣀⡀⠘⠄⡍⠐⠈⡗⠮⢤⢠⠀⢠⠀⢀
⠀⠀⠰⢸⠀⢠⡪⠑⠦⣤⣜⠟⠳⠀⢠⠆⢰⠀⢽⠏⢠⠊⠁⣦⠉⠀⠈⣰⠆⠀⠘⡆⣇⣁⣿⠀⢸⠀⠰⠤⠉
⠀⠀⢥⠿⡄⢠⠀⠸⡶⢣⢝⣲⢾⢰⡇⠘⠘⠂⠙⠀⢨⢠⠀⠁⠀⡀⠀⠘⠀⢠⠄⢿⡍⣉⣉⣉⣙⡂⡤
⢀⡀⢠⡄⠀⢺⠀⠱⢨⢋⠦⢱⡾⡟⠀⡀⠀⠀⠀⠀⠙⢿⢘⠀⠀⠀⠀⢀⠀⠈⠄⠗⠒⢰⣐⠀⠈⠃⠰
⠙⠋⠶⡄⢻⠸⠀⡖⠢⠍⠖⠸⠀⠓⡄⢠⢀⠰⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⢀⠀⣀⣀⣈⣂⠂
⠀⠀⢬⣀⡌⠁⡰⠀⠀⠈⡁⠀⠉⣁⢱⢸⡂⠀⠀⠀⠀⡸⠀⣀⠀⠀⠀⠈⠀⠀⠀⢀⠁⠲⠤⠽⣽⠀⠈
⠀⠀⠛⢉⡖⠐⠀⠀⠠⠀⠀⠸⠀⠘⡂⠃⠃⠀⠀⢀⣖⡃⠐⠁⠁⢀⡀⡀⠀⠤⠄⠈⣉⠛⢛⢛⢃⠀⠈⠉
⠀⠆⠽⢸⠀⡤⠀⠀⠀⠂⠀⠀⠀⠀⠇⠸⠷⠶⠦⠉⢯⡄⢀⠒⠄⠉⠀⠓⠀⢀⡄⠀⠂⠀⠀⠻⠾⠓⢾⢾⡀
⠀⠆⡽⢸⢧⡀⠀⠀⠀⠀⠀⣄⠀⠢⠄⢖⡁⠀⠀⠀⠀⡃⠉⠀⠀⠃⠀⠰⠠⡄⠀⣴⠠⠀⡀⠀⠘⠂⢨⠀⠈
⠀⡄⢠⡼⡾⠡⠄⣀⠀⠀⡔⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠈⡀⠙⠙⠁⠀⠆⠀⠀⢀
⠀⠉⢠⣐⠢⠲⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠿⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠄⠒⠀⠃⢀
⠀⠀⠘⢉⠈⠐⠲⣦⠀⠀⣠⠐⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠾⠀⢐⠀⢸
⠀⠀⠀⠈⠀⠀⠀⠉⡀⠈⢹⠘⡆⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠉⢆⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠉⠈⠉⠛⠶⣄⠀⠀⢀⣤⠶⠛⠋⠉⠉⠉⠉⠉⠙⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠉⠉⠉⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⣀⣠⣄⣀⣀
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠉⠀⠀⠈⠉⠛⣦
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠈⠙⢶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡁
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣛⣲⠖⣶⠶⢶⢿⣏⢩⣽⠇
⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣇⠀⠀⠀⠀⠀⠀⣀⡼⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣯⠉⠺⣧⠀⠀⠀⢹⡿⠷⣦
⠀⠀⠈⠳⢦⣤⣄⣀⣠⣤⣤⠴⠶⠛⠉⠀⠈⠛⠓⠲⠶⠒⣿⢿⡀⠀⠛⠻⣿⢶⡶⢤⣤⣤⡶⣾⠻⣾⢣⡞⠛⣻⡀⠀⠀⣴⢃⣤⡿⠁⣰⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠻⠂⠙⠶⠶⠶⠿⠂⠻⠶⠿⠈⠳⠾⠇⠀⠟⠒⠚⠉⢛⣓⡚⠁⣛⣛⣛⣛⡃⢘⣛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣃⡾⠋⣭⣽⢏⡴⠛⠙⠻⣴⢋⣿⢃⣦⡹⣏⣙⣦⡿⠉⠻⣎⢻⡉⠉⠉⠻⣮⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⣷⣀⣸⡅⠀⠀⠐⣿⡉⠁⣿⢨⡷⠈⣹⡏⠀⠀⠀⢹⡆⠙⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠟⢷⣤⣤⣤⡾⣡⠷⠛⠙⠻⢿⡜⢧⣤⣄⣠⢿⣷⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⡴⠶⢶⡶⡀⢿⡀⠀⠀⠀⣴⢋⣄⢶⡶⣤⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⡟⣴⢻⡌⠷⣶⡶⠾⢣⡞⠹⣎⢳⣌⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⣛⠋⢸⡇⠻⢶⡴⠟⢿⣠⣶⠟⠀⢙⡇⠉⣭⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣏⡼⣦⡿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠾⣵⣦⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣿⣿⣾⠟⠉⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠃⢿⡿⠛⠁⢠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠡⠙⠿⣦⣶⣤⡀⣠⡀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣤⠀⠀⢀⣹⣿⣿⣿⡿⣿⡿⠆⠀⠤⣄⠀⠤⡀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠼⠟⠋⠉⠉⠉⠀⠀⠀⠀⠈⠁⠀⢤⣀⣤⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⡃⢠⣤⣤⣤⡄⠀⠀⠶⣲⣷⠒⠒⠾⣿⡏⠀⢠⣿⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⢿⣿⡿⠑⡄⠀⠀⠀⠿⠿⠀⠀⠀⠉⠃⢠⣿⡟⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠋⠏⠀⠀⡴⠁
⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠾⣷⣦⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣴⣮⡄⠈⠁⠀⡸⠁
⠉⠁⠀⣀⠿⡇⠀⠐⣧⡀⠀⠀⠀⠀⠙⠿⠧⢀⠸⣿⣿⣦⣀⠀⠀⠶⠖⠀⠀⡠⣺⣿⣿⡇
⠀⠀⠀⠡⠤⠀⠀⠀⢈⣽⡗⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡆⠙⣶⠦⠄⠀⠀⢰⡈⠳⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠘⠋⠀⠀⣰⣆⣀⠀⠈⡡⠊⠉⠉⠉⠛⠛⠅⠀⠀⠀⠈⣿⡀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢤⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣶⣤⣄⡄⠀⢀⡀⣤⣴⣶⣤⣶⣄⣐⣥⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠷⢶⡶⠾⠿⠿⠛⠋⠻⢿⣿⣿⣿⣷⣾⣿⣶⣿⣿⣸⣿⣿⣿⣿⣿⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡀⢀⣴⡆⢀⠀⢀⣾⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠁⠘⣿⠃⠸⠀⢺⡟⢀⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡇⢰⣿⡟⠀⠀⣼⡿⠁⣼⣿⠟⠛⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢾⣿⠃⠀⢸⣿⠃⣸⣿⠏⣠⠆⢸⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⢠⡄⠹⣿⠇⢰⣾⡟⠀⢠⣴⡏⢠⣿⠃⣰⡟⢀⣿⡿⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⣸⣿⠀⣿⠀⣿⣿⠃⢀⣿⣿⠀⡾⠃⠰⣿⠁⣼⣿⠀⡄⠸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⡏⢠⣿⡋⠀⠁⢸⣿⣿⠀⣼⣿⡇⠀⢁⣴⣾⠃⣰⣿⡇⢠⡷⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢁⣼⣿⣇⠀⣿⣿⠀⢀⣿⠿⢿⣿⣿⣿⣄⣠⣾⡿⠃⣰⡿⠟⢁⣼⡇⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⠉⣽⡏⢀⣿⣯⣠⣿⣿⣶⣾⣿⣧⣤⣽⣿⣿⡁⠈⣁⣤⣦⣼⠟⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢁⣼⣿⡇⢀⣉⡙⠻⠿⠿⠉⣀⣉⡉⠛⠻⢿⣿⣿⣿⣿⣿⠟⢁⣠⣿⣿⣏⠉⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣾⣿⣿⣇⠘⠿⣿⣧⣠⣤⣀⠙⠿⣧⣴⣶⣤⣈⠉⠛⠟⢁⣴⣿⣿⣿⡿⠿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁⢤⣄⠙⠻⢿⣿⣷⡦⢀⡉⠛⢿⣧⣰⣷⣄⠘⢿⣿⣿⢿⣷⣦⣌⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠈⠻⣶⣶⣄⠙⠻⣿⣾⣿⣷⣤⣈⠛⣿⣿⣷⠀⣿⣷⣀⣹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣇⠘⢦⣈⠙⢿⣿⣀⣀⡙⠛⠿⢿⣿⣿⣿⡿⠃⠀⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠀⡈⠻⣷⡄⠈⠛⢿⣿⣷⣶⣼⣿⣿⠟⢁⣤⠀⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⢰⣿⣦⡈⠛⢿⣷⣦⣄⣽⣿⣿⡿⠋⣠⣿⣿⠄⠹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣄⡉⠻⢿⣿⣿⣏⣠⣾⣿⣿⠏⢠⡄⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⠄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣍⢺⣷⣶⣖⢲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣼⠆⢼⡃⠀⠇⢙⣷⠈⢱⣤⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⣿⠻⣾⣷⣆⣷⠟⠈⠓⢶⡒⢧⠀⠀⠀⠀⠐⣦⡜⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢿⡶⢻⣻⠉⠉⠀⠰⢣⣾⢏⡏⢡⠀⠀⠀⠀⢾⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠺⡿⣶⣯⠓⣒⣄⠀⠉⠁⢸⡏⡈⠀⠀⠀⠀⢀⣹⣿⣷⣜⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⠛⠻⢮⢿⣿⣟⠁⠁⢀⣠⠀⣧⠏⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣝⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠻⣿⢿⠷⢦⣬⣤⠂⠈⠄⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣝⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢰⡆⢚⣏⠐⣾⣟⠛⣷⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡝
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⣿⣷⠀⣿⡧⠤⠿⢧⣌⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠯⣀⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⢷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢒⡾⠟⢋⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⢁⢤⣾⡯⠖⢛⣩⠝⠛⠉⠉⠉⠉⠉⠉⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⠟⠁⠀⠀⠁⠾⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣀⡾⢋⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⠶⠾⠝⠻⠷⠶⠦⠽⣶⡀
⠀⠀⠀⠀⠀⠀⣴⣿⠋⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⢀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⡆⢿⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔⣨⠄⠒⠒⠒⠒⠢⢬⣁⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠡⠈⢧⡸⡀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠈⠉⠉⠉⠲⣄⣂⠤⢀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⢀⡼⣣⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠁⠈⠳⣷⡀⠀⠀⠀⠀⢸⡇⠀⣠⠖⠒⠲⢤⡀⢸⡇⠙⢦⡀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠶⠟⠛⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣤⡀⡞⠳⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢱⢠⠇⠀⠀⠙⢦⡀⠀⠀⠀⠀⣀⠀⠀⠀⡀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡎⣹⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡀⠀⠀⠀⠈⣝⠳⣄⠀⣀⣈⠁⠈⠉⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣧⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠈⢀⠀⠀⡀⠀⠀⠸⡆⢠⠞⢉⡉⢳⣀⣠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣷⠒⠒⠶⢦⣤⠀⠀⠰⠞⠋⠀⢄⣴⣾⣿⠛⣟⣿⠋⠀⠀⠁⠈⢀⠊⡀⢸⣏⢢⡘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣶⣯⣴⣀⠀⠀⠀⠀⠀⠀⣸⣏⢹⣿⣿⠇⡾⠃⠀⠀⠀⠀⣠⠀⡠⠁⡼⢻⡄⠱⡀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣏⠈⡉⣿⣷⣄⠀⠀⠀⠀⠛⢈⣲⣬⡵⠎⠀⠀⠀⠀⠀⠀⢿⠀⢀⡼⠁⠀⠹⣆⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣵⣬⠿⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⠙⠀⠀⠀⠘⢦⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⢣⡀⠀⠀⠀⠀⠳⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⡜⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣧⡀⠱⣄⠀⠀⠀⠀⢡⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠀⠀⠀⡀⠠⡶⠃⠀⠀⠀⠀⢀⣤⠞⠁⠟⢻⡏⠙⠷⣤⣅⣀⠀⠀⠀⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣍⠑⠆⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠈⣿⠀⠀⠈⠻⡙⢿⠶⣤⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⠀⢳⡀⠹⡇⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⢇⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣤⠤⠀⠀⠀⠀⢰⡇⠘⣷⡀⠀⠀⠀⠀⢸⠀⢹⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⣾⠁⢰⢸⠛⢦⣄⠀⠀⢸⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢸⠈⠀⠀⠙⣧⠀⠀⠀⣟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣧⠸⠀⠀⠀⠀⢸⣆⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⡠⢒⡽⢷⡀⠀⠀⠀⢸⣏⡴⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⢠⣿⢙⣁⣏⣹⠂⣐⠠⣀⣠⣴⡟⠀⠀⠐⠉⠀⠈⢷⠀⠀⠀⡼⠋⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⣿⠁⣀⠀⠀⢉⣼⠾⠛⠉⢁⣿⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⡜⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠋⠐⠐⠐⢺⡏⠀⠦⣠⡶⠋⠁⠀⠔⠃⣸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣷⣮⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⢀⣾⣤⠶⢻⡿⠁⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⡀⠀⢀⣠⠾⠛⠁⠀⠀⣾⠁⠀⠀⠀⠀⣠⡟⠁⠀⢀⣀⣤⡴⠞⢋⡼⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠁⢠⣴⠟⠁⠀⠀⠀⠀⣼⠇⠀⠀⠀⣠⠾⢋⣠⣶⣞⣫⣭⡴⠶⣿⠟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣤⡞⢁⣀⣀⣀⣠⣤⣞⣁⣀⡠⠶⠛⢛⣋⡽⠟⠉⠉⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⢀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⡿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⣿⣿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠋⠁⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠉⠻⠟
⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠗⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠺⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⢀⣴⣿⣿⢟⣵⣾⣿⠿⣁⡀⠀⠀⠀⠀⠀⠀⢀⣈⠻⣿⣷⣬⡻⣿⣿⣦⡀⠀⠀⢠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⢀⣴⣿⣿⢟⣵⣿⣿⠟⣡⣾⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⣷⣌⠻⣿⣿⣮⡻⣿⣿⣦⡀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣷⣿⣿⢟⣥⣾⣿⡿⣫⣄⠀⠀⠀⠀⠀⠀⣠⣝⢿⣿⣷⣬⡻⣿⣿⣾⣿⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠟⣩⣾⣿⠟⠀⠀⠀⠀⠀⠀⠻⣿⣷⣍⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠉

⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⡄⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡠⠂⠀⠀⠣⠖⠚⠋⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡰⠀⠘⣄⠀⠀⡀⢊⢻⡅⣀⣠⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠆⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠐⡄⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⢀⡐⠖⠒⠀⠀⠈⠀⠀⠀⠀⠈⠙⢧⠀⡄⠀⡠⠐⠉⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠉⠑⠖⠒⠆⠙⠁⠀⠀⠐⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠲⠂⠁⠀⠀⠀⠀⠀⠒⠐⠒⠒⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⡀⠀⠀⠀⣅⠀⠈⠀⠀⠀⠠⣶⡀⠀⠀⠀⠉⠡⠘⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⡘⠂⠀⢀⡀⠀⠀⠙⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⣌⣻⣶⣶⡄⠀⠀⠀⠀⣀⠚⠤⣀⠇⠀⠀⠈⠀⣀⣀⠀⠁⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣔⣋⣀⣀⡄⠋⡀⢲⡶⠛⠈⠉⠉⠀⢂⠄⡠⠞⠀⠀⠀⢈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠐⠠⠒⠀⠤⠐⠂⠀⠀⠀⡀⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣛⠛⠉⠀⠀⠀⢀⡀⠀⢠⡄⠀⠀⠐⢆⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢀⠀⠀⠀⠀⠀⡠⠄⠂⠠⠀⢲⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁⠀⠀⠈⠂⠉⠁⠀⠀⣴⠀⠀⠀⠈⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠀⠀⡾⠀⠀⢀⠂⠀⠀⠳⡀⠀⠀⠘⠷⠄⣀⡐⠠⠀⠀⠴⠀⠇⠀⠀⠀⠀⠀⠈⠲⡀⣼⠇⠀⠀⠀⠀⡇⠀⠀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠐⠄⠤⢶⠀⠀⠠⡏⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢉⡅⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣀⠠⠀⠀⢰⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⣄⠀⠀⠈⡄⠀⠓⠀⠐⠀⠀⢀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⡠⠂⣽⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠄⠒⠀⢧⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠦⠀⠀⢁⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠉⡔⠀⠀⡠⠊⠐⢀⡆⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠤⠶⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠶⠄⢀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠘⠆⢸⡅⠀⠄⣸⠃⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⢀⡠⠉⠀⠓
⠀⠀⠀⠀⠀⠀⠀⡆⠀⣼⠁⠀⡶⢋⠀⠀⠀⠀⠀⠀⠒⠀⠀⠤⣦⠤⣤⢤⣤⣤⣤⠀⠐⠛⠒⠈⠉⠉⠉⠉⠉⠉⠳⢾⡃⡟⠀⠉⠁⠀⠀⠀⢀⠦⠠
⠀⠀⠀⠀⠀⠀⠀⠐⡀⢈⣠⠀⡃⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⡀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣙⠷⣏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠸⠄⠈⠂⠈⠠⣤⢵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⡀⠈⠂⠤⠀⠀⠀⡄⠀⠈⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠹⣦⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣦⠀⠑⠄⠀⢰⡃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠀⠀⢀⠀⠀⠀⣄⡀⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠒⠀⠘⢦⡀⢀⠞⠉⠉⢰⠶⠆⠠⣤⢄⣰⡒⠀⢹⣄⠈⢢⠈⠓⠤⠤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡕⢤⡀⠀⠆⠀⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⢀⡆⢄⠁⣀⠴⠀⠄⠀⠹⡄⢀⡀⢀⡀⠛⠀⠀⠀⢦⠁⠁⠤⠉⠙⣦⠀⠱⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠬⢒⠠⠦⠄⠀⠀⠀⢠
⠀⠀⠀⠀⡐⠒⠣⠠⠔⠂⠀⠀⠀⠀⠀⠀⠈⠲⡙⠂⠑⠂⠀⢀⢀⠀⠱⣄⠀⠤⠀⠀⢳⣄⠀⠢⡀⠙⠦⠠⠐⣄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄⠐⠆⠤⠀⢀⡼
⠀⡀⠀⠀⠁⠉⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢢⡀⠀⣀⠺⠄⠀⠀⠈⢢⡀⠀⣀⣸⡜⢣⡀⠀⠄⡀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢰⠀⠀⠀⢀⠈⠙
⢈⡁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠙⢶⡁⠀⠀⠁⠔⠒⠀⠙⠦⠀⠬⠥⠽⠽⠶⠠⠘⠖⠐⠚⠓⠒⠋⠉⠁⠀⠀⠀⠀⠀⠸⠀⠐⠀⠀⠙⢤
⢸⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡁⠂⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠄⠀⠀⡀⠀⠀⢀⡀⠀⠀⠃⢠⡀⠀⠀⠀⢰
⠰⠃⠶⡤⠀⠀⠈⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⡕⢶⡀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠒⠀⠁⠐⠁⠀⢀⣂⡈⠓⠺⠛⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣳⠄⠁⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⣆⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣴⠶⠚⠛⠉⡉⢹⡇⡀⠀⢐⢀⡴⢀⠴⠛⢆⠀⠀⠀⠘
⠠⠀⠀⠀⢂⡑⣾⠃⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠀⠀⢘⣆⢠⡙⣢⣤⣴⣴⠖⠒⠉⢉⢛⣄⡀⢠⣀⡈⠀⢰⡂⣈⠙⣻⣰⡀⠁⠀⠀⠀⠈⢂⡀
⠀⠈⠀⠁⠈⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣆⡀⠀⠀⠀⠐⠟⠁⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠸⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠠⣾⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠋
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⠀⣤⣴⡄⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⠛⠀⠀⠀⠾⠋
⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠂⠀⠀⢠⣶⠿⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶
⠀⠀⠀⢠⣶⣿⠋⠉⣿⣇⣤⣶⠿⠿⣷⡄⠀⠀⠿⢷⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠘⠋
⠀⠀⠀⠀⢸⣿⣤⡶⠟⠋⠉⠀⣀⣀⣸⡇⠀⣴⢶⣦⣄⡀⠀⢹⣧⠀⠀⠀⠀⢠⣿⠃⠹⣧⡀
⢀⣠⣴⡾⠟⠋⠁⢀⣀⠀⠲⠟⠋⠙⢻⡇⠀⣿⡀⠈⠙⠻⠷⣾⣏⡀⠀⠀⢠⡿⠃⣀⣀⠹⣷⡀
⢻⣯⠀⢀⣠⣴⠾⠟⠋⠀⠀⠀⠀⠀⢸⣇⣠⣿⠛⠷⣦⣄⣀⠀⣹⡟⠀⢠⣿⣷⡾⠟⠙⠻⠿⣷⡀
⠀⠻⠿⢻⣿⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠙⠛⠁⠀⠀⠀⢹⡟⠿⠟⠀⠀⠘⢻⡷⠀⠀⠀⠠⣾⡟⠁
⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⣾⠏⠁⠈⠙⣷⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⢙⣷⣄
⣶⠿⠛⠛⠛⠛⠻⢷⣦⣴⠿⠿⠿⠿⠿⢿⣧⣄⠀⠀⢀⣸⣷⣶⠶⠿⠿⠿⠷⣶⣤⣤⡶⠿⠛⠛⠛⠛⢷
⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿
⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣠
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⢰⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣥⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠀⠸⣿⣿⣿⣿⡿⣱⣮⣝⣛⣛⡛⣫⣭⣶⣿⡿⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⣩⣭⢩⣵⣾⣿⣿⣿⣿⣿⣿⣶⣭⢉⣭⡁⠀⠨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣾⡀⠀⠘⣿⣇⠾⠇⠿⠇⠿⢆⣿⡟⠀⠀⠀⠀⢠⡀⢰⡆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠂⠈⣿⡀⠀⠘⠿⣿⣿⣿⣿⣿⡿⠟⠀⠀⣀⠀⣷⠈⠻⣾⡷⠋⢀⡿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡦⠈⠛⠇⠀⠀⢠⣭⣭⣭⣥⠀⠀⠀⠀⣉⣻⣿⣇⡀⢸⡇⢀⣼⣷⣟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀⢴⣾⣿⣿⣿⣿⣷⡄⣠⢈⣉⠉⠀⠈⣹⣿⣿⣋⠁⠈⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣦⣙⠿⣿⣿⠟⣡⣾⣿⡦⢉⣻⣿⡟⠉⢸⡇⠙⢻⣿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣧⣫⡅⡿⠀⣠⣾⣷⣄⠈⣷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠦⠘⠁⢸⡇⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣙⡻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢋⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠷⠌⢳⡜⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣼⣿⣶⣬⣭⣭⣭⣶⣶⣾⣿⣿⣿⣿⣿⣷⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣡⣿⣿⣿⣥⣦⣽⣿⣿⣿⣿⣯⣴⣬⣿⣿⣿⣌⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⣯⠹⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⢏⣽⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣜⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣫⣝⣛⣛⣫⣝⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣯⣽⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣛⣛⠛⠛⣛⣛⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣾⣷⣭⣛⣛⣛⣛⣭⣾⣷⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠏⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⢶⡆⡆⣶⣴⡇⣾⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠙⠁⠁⠁⠁⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⢉⠝⠻⢿⠑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⣠⠊⢁⡠⢀⢬⡖⡉⢢⠈⢢
⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠘⠉⢠⠋⢂⠀⠸⠃⠈⡆⡆⠀⢱
⠀⠀⠀⢀⡴⠋⠀⠀⠀⣀⣀⠀⢠⠔⠘⢦⣀⢱⠌⠀⠀⠀⠚⠆⢡⠀⠀⠇
⠀⠀⡴⠃⠀⢀⠔⠉⠀⢀⡌⢧⠴⡀⠀⠀⢇⡒⣀⠀⠐⠀⠂⢸⠘⠣⣀⠜⠋⠉⠒⠂⠤⠄⣀⣀
⠀⡜⠀⠀⢀⠎⠀⢆⣶⠿⣋⡌⠦⢬⣶⣄⡀⠀⠀⠑⢤⣀⠀⢘⣳⢧⠹⡈⠑⠒⠤⠄⠀⠀⠀⠀⠈⠁⠒⢤⡀
⢰⠁⡤⠊⢨⡄⠀⡎⡔⠀⠡⡱⣼⠀⢨⠳⡈⠉⠂⢤⠀⠣⣨⠛⡄⠀⠑⢆⠀⠀⣀⣀⠔⠁⠑⡄⠠⡤⠊⠁⡇
⠸⠀⣧⠞⢁⢥⡀⠰⡀⠀⠚⠡⠿⣷⢘⠀⢱⠀⠀⠈⢆⠀⠀⠀⠈⠢⡀⠀⢱⠎⠀⠈⡀⠀⠀⡇⠀⡄
⢠⠀⢤⠀⠘⠢⣽⢄⠱⡘⠒⠁⡧⠊⡘⠀⠀⢣⠀⠀⠈⢣⠀⠀⠀⠀⠈⢢⠌⠀⠀⠀⡶⠆⠀⢠⠀⡀⠀⢰
⢸⠀⠀⢻⠒⠤⣉⠻⠾⠟⠓⠚⠀⠀⠘⢢⠀⠈⡆⠀⠀⠀⠑⡄⠀⠀⠀⠎⠀⠀⡄⠀⠃⢠⠀⠸⠀⠁⣀⣸
⢰⠦⠒⠢⣧⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⢸⠀⠀⢰⡄⠀⠀⢀⠼⠶⣄⡸⠀⠀⡜⢁⠀⠸⣤⠀⠀⣧⢶⡍⣨⣷
⠇⠀⠀⠀⠈⠑⠢⣀⠀⠱⡀⠉⠀⠀⠀⣀⣥⣄⡰⠈⠁⠉⠀⠀⠀⠀⠉⠒⠚⢁⣼⠀⠀⢿⡆⡾⡘⣘⠋⣵⠈⢱⡠⢤⣀
⠘⠀⠀⠀⠀⠀⠀⠀⠑⠢⣵⡀⠀⡰⠋⠀⠀⠈⠉⠑⠒⠤⣀⠀⠀⠀⢀⡠⠖⠁⣿⡇⠀⠘⣿⠵⠞⣣⠌⠈⠢⡼⠀⠀⠈⠑⠢⢄⡀
⠀⡄⠀⠀⠀⠀⢄⠀⠀⠀⠀⠉⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠶⢶⣉⣀⣤⠖⠁⢸⡄⠠⠅⠠⠔⠋⢀⠄⠒⡗⠢⡀⠀⠀⠀⠀⠈⠑⠢⢀
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢣⡀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠀⠀⢀⣀⣤⠄⣖⣒⣒⣒⡒⠒⠋⠁⠀⡀⠧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠘⠈⠁⠒⢤⠄⣀⣀⣸⠖⠒⠂⠉⠀⠀⠀⠀⣠⠔⠉⠉⠀⠈⠁⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩
⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠰⡤⠔⠊⠁⡠⠞⠉⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⢀⠔⠋⠉⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠔
⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⡀⠐⠉⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡡⡞⠃
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠘⡀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⣀⠀⠀⠀⣿⣷⣠⣀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠃⠀⠀⠀⠀⢀⡠⢊⠵⠊⠀⣿⠁
⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⢀⡤⠚⠉⢳⢄⡙⠢⢀⠈⠣⢭⣍⠻⠶⣄⠀⠀⡠⠊⠀⠰⠇⠀⠀⣀⠔⣫⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⢀⠀⠀⢸⠀⠀⠀⣠⠔⠁⠀⠀⠀⠘⠀⠉⠑⠦⣉⠒⢄⡈⠛⢷⣦⠆⡏⢀⠔⠁⠀⠀⡠⠞⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠸⠀⠀⠀⣆⠴⠊⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠑⠢⣌⠐⠤⣈⠑⠛⠁⠀⢀⠴⢊⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⡂⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢌⠑⠢⣤⠔⣁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡴⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣤⣭⣴⣮⣽⣯⣴⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡽⢛⣽⠋⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣠⢀⣟⣴⣷⠆⠂
⠀⠀⠈⣵⣟⡇⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠃⠀⠀⠀⠀⠀⣀⣴⣾⠖⠀⠀⠋⠀⠀⠀⠜⢡⡢⢟⣿⠒⠋⠀⠀⠀⠠⠄
⠀⠀⠀⠐⣷⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⣀⣄⣠⣂⡘⢁⣠⠞⠁⢠⡄
⠀⣄⠀⠐⢉⡿⣿⣷⣄⣤⣄⣠⢄⠞⠋⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⢾⠟⠁⠀⢠⣇⠀⠀⠛⣿⢫⣉⠴⢶⠞⠁⠀⠀⠀⠁⠠⣦
⠀⠈⠀⠀⠀⠰⣻⣿⣫⠟⣿⣧⣼⣤⣀⡄⠀⢠⢀⡀⡤⡒⢂⠒⠀⠀⠁⠈⡠⢀⣦⣄⣑⡀⣀⣰⣵⣊⡛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠂
⠀⠀⠀⠀⠈⠀⠀⠌⠴⠚⣉⣔⣿⡿⣠⡐⠛⣌⢿⣿⣇⡇⡘⠦⢠⣶⣶⡈⢫⡾⣟⣭⢿⣿⡿⠛⣹⣶⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⠀⠀⠀⠀⢀⡀⠀⠀⠈⠙⠛⠉⢡⠈⠳⠛⢠⡿⡿⠇⠝⠩⢏⣄⠦⠾⣯⣡⣟⡫⠷⠚⠋⢀⣦⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡌⣠⠔⠊⢀⠐⠁⢤⡃⡴⣼⠋⠀⠀⠀⠚⠉⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣼⠂⡝⡡⠋⠻⠽⠹⡒⢄⠈⢤⣿⠀⠀⠀⠂⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠐⠋⢀⡄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢠⡄⠸⠇⠠⠃⠃⢨⠀⢸⡍⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠘⢀⠀⡑⢮⠀⡀⠑⡞⠁⣻⠔⠁
⠀⠀⠀⠀⠛⢄⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⣄⢧⠰⠶⠖⠶⢀⡎⠀⠼⡅⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣈⡧⠷⣃⡄⡗⠁⣀⠌⣄⠈⢮⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠚⠣⡜⣶⣂⠀⠀⠜⢢⠀⡶⠉⠀⠘⠛⠀⠀⠀⠀⠀⠀⡠⠾⠋⠉⠂⠙⠀⡇⢰⠋⡠⠅⣶⠋
⠈⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠃⡄⠁⠈⣻⡀⠓⣶⣼⣹⠀⠄⠀⠠⠠⡇⠀⠀⠀⠀⠄⠀⠀⠄⠀⡀⠀⠀⠀⣀⣀⢤⡤⢅⡦⣴⠀⠀⠰⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⣶⣿⣷⠀⢈⠋⢻⠛⠲⠃⠀⣑⢠⡶⢤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢣⣭⢀⠃⠀⠀⠀⢀
⠀⠀⠀⠉⠀⠀⠀⠀⣀⣤⡤⠀⠀⠀⠀⠀⠑⢪⠀⣀⡀⢀⠦⠊⠉⠉⠋⠉⠀⠀⠀⠀⢤⣄⣀⣀⣀⠀⠀⠀⠀⣤⣀⣘⣿⢀⠀⠀⠤⠀⠈⠁
⠀⠀⢀⣀⠀⠀⠀⠘⠋⠉⠉⠀⢰⣂⡀⠉⠀⠀⠄⠽⠥⠬⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⣺⣿⡿⠇⠀⠀⠠⠄⠀⠙⠛⠓⣺⣿⣿⠹⠤⠛⠒
⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠄⠈⠁⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠈⠛⠀⠈⠉⠁⠐⠉
⠀⠀⠀⠀⠀⠐⣀⠐⠂⠀⠀⠀⢀⡢⠀⠀⠀⠀⠀⠀⠑⠂⠀⠁⠀⠈⠉⠉⠁⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣄⣀⡀⠀⠀⠀⠢
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠚⢯⠧⠤⠄
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣦⣵⣶⣤⣾⣵⣿⣽⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⠇⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠉⠉⠲⣄⠀⠀⡞⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠁⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠠⠤⠀⣀⠐⠒⠂⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠿⢦⣀⠀⠧⠃⣀⡤⠊⠙⠒⠲⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡀⣠⠤⠤⠶⠲⡇⠉⠉⠉⠁⠀⢹⡉⠙⠒⠲⢤⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠘⠖⠓⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠴⠀⢀⡤⠒⠋⢀⣀⣀⣀⣀⣀⡀⠈⠉⠲⢄⠀⣲⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠈⡇⠀⢯⣀⠀⠘⠻⢥⡀⠀⠀⢸⡛⠛⠀⣠⠼⠃⣱⠄⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⣟⣃⣀⠈⢉⡇⠰⣏⠉⠀⢠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠉⠀⠀⠀⠉⠛⠚⠁

⠀⠀⠀⠀⡅⠢⠄⠀⠔⠁⠀⠀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⡀⢤⡝⣦⠉⠁⠀⠀⢀⡘⢧⡛⠂⠀⠀⠀⠀⠀⠈⠑⠒⠂
⠀⠀⠀⣸⡱⡀⣀⠎⠀⠀⠀⠀⠀⠀⠢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠃⠺⣂⢠⡄⢨⣥⠤
⠀⠀⡀⠹⣱⠡⠃⠀⠀⠀⠀⢀⠀⠀⠀⠨⣄⣷⠄⠒⠀⢲⢏⣄⠀⠀⠀⠀⠀⠀⣽⣍⠦⠀⠮⣶⠻⠀⠀⠀⣀⠀⣄⠀⠀⠀⠠
⠀⠀⡇⠀⠉⠀⠀⠀⠀⢀⠔⠁⠀⢀⠔⠉⠁⠀⠀⠀⢠⠊⢃⠿⠀⠀⠀⠀⠀⠀⠆⠉⠀⠀⠀⠀⠈⠀⠀⢀⡻⠋⢛⡶⠂⠀⡆⠀⢈
⠀⠀⠁⡀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⣾⠛⡆⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣶⠺⠧⠀⡘⠀⠀⣈
⠀⠀⠀⠩⡀⠀⠀⠠⠂⠀⠀⢠⢃⠤⠔⠒⠒⠁⠀⠀⠘⠿⠞⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣼
⠀⠀⠀⡇⠉⠁⠀⠀⠀⠀⣠⠋⣁⣴⣾⡟⠋⠲⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠠⠛
⠀⠀⠀⠙⣄⠀⠀⣀⣠⠞⠉⢀⠉⠁⠚⠒⠊⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠈⠑⠺⠇⠀⠠⢀⡈⣁⠀⠀⠀⠀⠀⠀⠀⠤⠔⠪⠁⠓⣒⠒⠢⡄⠀⠤⡀⠀⠀⠀⠀⢀⣀⠤⠐⠊⢁⣀⠀⣠⠀⡴⢩⠁⠀⢀⠀⢤
⠀⠀⠀⠀⠀⠀⠈⢄⡀⠆⠀⠀⠀⠉⠲⣤⣀⣀⡀⠀⣀⣀⡤⠂⠀⠀⠀⠀⠘⠀⠀⠘⢄⠔⠒⠽⠷⠃⠀⠀⠀⢠⠎⢂⣁⠴⠛⠁⠀⠀⡴⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠑⠤⣀⣀⠠⢤⠀⠐⠀⠀⠸⡛⢻⡋⠀⠐⠒⢊⣠⠅⠓⠘⠤⠤⠀⠀⠊⠴⢔⢋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⡀⠀⠀⡀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⢤⠤⠘⠀⠀⢃⣈⠭⠛⠁⠀⠀⠀⠄⠀⠀⠀⢀⢀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣄⣀⣘⠦⠄⠀⡇⠀⠀⠀⢀⠄⠂⠀⠀⠠⠀⠀⢸⠀⢧⠤⠤⡬⡥⠢⡀⠀⠀⠀⠀⡄⠀⠀⡨⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠀⠱⡄⢲⡆⠀⢸⠀⠀⠀⡰⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⡴⠊⠉⠀⠀⠀⠀⠀⠹⣻⣇⠀⡜⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣠⣔⠰⢠⠏⠀⠀⠀⠙⡄⠀⠀⠀⠀⢻⠈⢦⢀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠪⡄⠉⣰⡟⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⡹⠁⠀⢠⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡚⠂⠀⡰⢡⣅⠀⠀⠑⠲⠤⣀⠀⠀⠀⠀⢸⣧⠎⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠢⣀⠁⡠⣉⠀⢢⠀⢄⠀⠀⠉⠒⠠⠀⡸⠁⠀⠀⠐⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠙⠲⡥⣀⠀⠑⠀⠑⠀⡁⠤⠄⠀⢁⡀⠀⠠⢚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡃
⠀⠀⠀⠀⠀⠀⠒⠀⠉⠁⢠⠾⠛⢲⡄⠀⠀⠆⠀⠀⠀⠀⠉⢁⠆⠀⠀⠀⢀⡀⠋⠈⠋⠀⠘⠀⠚⠃⠘⠋⠀⠀⠞⣧⡣⠜
⠀⠈⠋⠀⠀⠀⠂⠀⠀⠀⠙⡀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠃
⠀⠀⠀⠀⠀⠀⠠⠤⠀⠐⠀⣻⡄⠠⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠀⢇⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠬⠀⠀⠀⠘⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠆⠀⠀⠀⠆⠐⠀⠐⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠷⡄⠄⠀⠐⠀⠀⠀⠀⢀⣴⠙⠋⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⢀⠀⠠⠀⠄⠀⠁⠈⠀⢁
⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⡂⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠈⠆⠀⢀
⠁⠈⠀⡒⠀⠠⠄⣠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠃⡄⠀⠀⠀⢀⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⢀⠈
⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠂⠂⠀⠁⠀⠈⠀⠀⠀⠀⠀⣀⢄⠀⠓⠘⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄⠈⠀⠐⠐⠒⠈⠉⠀⠇⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠄⠱⠄⠐⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⣹⡇⠀⠀⠠
⠀⠀⠀⠀⢀⡀⠤⠀⠈⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠲⠌⢷⣤⡼⠷⠆⡞⢀⡤⢤⡀
⠀⠀⠀⠉⢁⠀⡠⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⡃⠀⠀⢘⡀⡀
⠀⠀⠀⠀⠘⠁⠀⠀⢃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⡀⠘⠀⠑⠤⠠⠾⠷
⠀⠀⠀⠀⠠⠄⠛⠑⠁⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣐⠀⠀⠰⡆⡤⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡀⠀⡄⠀⠀⠀⢘⠀⠀⠀⡇⠀⠀⡇⠀⠘⢡⣿⣿⣿⣿⠀⠀⣃⡧⠈⠀⠀⠀⠀⠀⢀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⣀⠀⠀⠀⢹⠀⠀⠀⣇⠀⠀⠇⠀⢠⠜⠿⣿⣿⡿⠀⠀⢱⡷⠈⡁⠀⠀⠀⠀⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠈⠂⠀⠀⣁⣠⠄⣀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠈⠙⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠰⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠠⢵⡄⠀⠠⠀⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠠⠢⠄⢀⡀
⠐⠀⠤⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡒⠒⠶⠤⠤⢬⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠉⠛⠲⠖⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠠⠀⠀⠀⠘⡶⣄⠍⠁⠀⠀⠒⣄⠨⡉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡀⠠⡀⡀⠰⠋⣠⡄⢠⠀⠀⠀⠀⠋⠈⠁⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡗⢡⠈⣈⠏⠁⠀⠁⡂⠀⠎⠀⠀⢠⠄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡆⠀⠀⠉⠉⢈⡅⢀⠄⡠⠀⠀⠃⠀⢀⠔⡀⢀⡾⢦⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣆⠠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⡀⢤⡐⠆⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠙⠠⣀⡀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠀⣠⠀⠀⠈⠙⠒⠒⠲⡀⠀⠀⠀⠀⠀⢀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⡓⡠⡠⠒⠀⢠⠪⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠐⢹⠀⠀⠀⠀⢀⡎⠔⠀⡤⢤⢅⠤⣤⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠠⠐⠀⠀⡀⠀⠀⠠⠁⠀⠀⠀⣃⣒⠐⢂⠃⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠀⠐⠀⠀⠂⡀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠟⠛⠛⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⢀⣴⠟⠋⠉⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢀⣤⡾⠷⠶⢶⡿⠁⠀⠀⠀⠀⠀⠀⢹⣶⠶⠶⠶⣤⣀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢿⣄
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⠟⠉⠉⠉⠙⠻⣦⠀⠀⠀⠀⠹⣧
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠹⡷⠾⠿⠷⢶⣽⡇
⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀
⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀
⠹⣧⡀⠀⠠⣄⢰⣆⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⢿
⠀⠈⠛⠶⠶⣾⠿⠿⡷⠶⠶⠶⠶⠶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠠⠚⢳⠚⢦⠀⠀⠀⢦⡀⣸⣷⣀⠔⠀⠀⠀⠀⢦⣀⣸⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡝⠁⠈⣿⠿⠶⠶⠶⠶⠶⡿⠿⠿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠠⡀⠀⡆⠀⣠⢀⠜⢦⠔⠳⡀⠀⠀⠀⠀⠰⠛⢲⠚⠣⡀
⠀⠀⠀⠀⠀⠀⣀⣈⡖⠙⢺⣁⣀⠀⠘⠀⠀⢀⠀⢠⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢈⠷⡤⠾⣄⠀⠀⠀⠀⠀⠈⣳⠚⠓⣎⣀
⠀⢤⡀⢰⠀⠀⡀⠁⠀⠃⠀⠈⠀⠀⠀⠀⠉⠉⣹⢤⡤⢏⠉
⣀⣀⣹⠊⠙⣎⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠃⠀⠑
⠀⠀⡽⢦⡤⢯⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠛⠛⢿⣷⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠀⠀⠀⠈⠙⢿⣷⣄⣀⡀
⠀⠀⠀⠀⢀⣤⣾⣿⡏⠀⠀⠀⠀⠈⠛⠋⠙⢻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠿⣿⣷⡀
⠀⠀⠀⣰⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣧
⠀⠀⠀⢻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠋⠀⠀⠀⠀⣀⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⢠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠰⣿⣷⣶⣾⣿⣿⣶⣶⣿⡇⠀⠀⠉⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⢠⣤⣄⣀⣼⣿⣧⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠈⣿⣿⠁⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠟⠋⠻⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣦⣾⣿⣿⣷⡄⠀⠀⠀⠘⠛⠋
⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⢼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣦⣀⣤⣿⣿⣄⡀⠀⠀⠀⢀⣴⣿⠆⠀⠉⠁⠀⢹⣿⡏⠀⠀⠉⠁
⠀⠀⠛⠋⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠙⢻⣿⡟⠋⠛⠿⠃⠀⠀⠰⣿⠟⠁⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⣀⣀⠀⣀⣿⣿⡀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠛⠻⣿⣷⣄
⠀⠀⠀⠻⢿⣿⡿⠟⠿⣿⣿⠿⠋⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠘⠻⣿⣧
⠀⠀⠀⠀⣸⣿⣇⠀⢀⣿⣿⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⣿⣧⠀⠀⢀⣿⣿
⠀⠀⠀⢿⠿⠛⢿⣿⣿⠟⠻⢿⡷⠀⠀⣀⣤⣶⣶⣦⣄⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⢰⣿⣿⣿⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⣰⣿⡿⠿⠋⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⠟⠉⠙⢿⣿⣶⣦⡀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⢿⣿⣄⠀⠀⠀⠉⠙⢿⣿⡄⠀⠀⢰⣿⣿⣿⣿⡀⣸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄⡀⣀⣾⣿⠃⠀⠀⣾⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣠⣤⣴⣿⣿⣤⡀⠀⠀⢸⣿⡇⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣄⠀⠀⠀⢸⣿⣇⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣴⣿⡿⠛⠙⠛⢿⡿⠛⠛⠙⠛⢿⣿⣧⡀⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⠟⠛⠙⠻⢿⠿⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣴⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣴⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠛⣿⣿⡿⠟
⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠈⠛⣿⠛⠻⣿⡿⠋⠀⠈⢻⣿⡟⣻⠟⠛⠁⠀⠀⠙⢻⡟⠛⢿⠏⠙⠟⠉⢿⡿⠻⣿⠿⢿⣿⡿⣿⣿⣿⠋⠻⣿⣿⠿⣿⠛⡿⠛⢿⡿⠙⣿⠿⠟⠋
⠀⠀⠀⠀⠛⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠛⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠟⠀⠙⠋⠀⠀⠀⡿⠁⠀⣿⠀⠁⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠙⣿⠟⠻⣿⡿⠿⣿⣿⡿⠃⠀⠀⠘⢿⣿⡿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠻⠏⠀⠙⠋⠀⠘⣿⠏⠙⢿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠹⠀⢸⡏⠀⠙⢿⡇⠀⠀⠀⠀⠀⠉⠀⢹⡏⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠉⠁⠈⢻⡿⠋⠛⠿⠟⠛⣿⣿⠟⠻⠿⠟⠛⢻⣿⠟⠻⠿⠋⠁⠉⠉⠀⠀⠀⠀⠉⠉⠈⠹⡿⠛⣿⠟⢻⡏⠁⠉⠉⠀⠀⢹⣿⠋⠙⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⢘⣛⡛⠯⣪⢿⣿⣿⡿⡦⠹⠇⠺⣿⣐⣦⣼⡊⠞⡉⠊⡠⠐⠈⠈⡸⠃⠀⣶⠋⠀⠂⠀⠀⢈⠀⠲⠌⢀⢭⣴⣂⣿⠂⠻⢏⡼⣿⢿⣯⡯⣴⠟⣛⣛⡛
⠀⢸⡏⢩⡆⠰⣿⠿⠛⢻⡽⠎⠲⠶⣰⠈⡻⠒⡀⠁⠄⣠⠤⠴⠀⣊⣤⠴⡾⢬⣙⣒⣒⠀⠔⠤⣍⣑⠒⠶⣈⠓⠂⡒⢂⣒⣽⢿⡟⠛⠿⣿⢁⢮⣌⠈⡟
⠀⠘⠣⢙⠑⣡⡇⣬⡿⠋⢐⡿⡙⡢⢃⠔⠠⠂⣠⠖⢋⡤⣶⠛⣿⡩⡓⣏⠘⡾⡅⠿⣹⠏⣻⡶⢦⣬⣙⠲⣄⠑⢄⠑⠴⣥⡲⢦⢱⡿⣗⡨⣄⠀⠙⠐⠷
⠀⢸⠏⢸⣿⡿⠷⠭⣍⡛⡮⠍⠊⠔⣁⡴⢞⣷⢋⡾⠋⠻⣠⠂⣸⠆⡇⣘⣼⠶⠛⠲⠿⢾⣭⣧⡸⠰⠛⣳⣌⠁⠀⠀⠤⣤⣭⣭⢩⢉⠭⠶⠯⣽⣧⠙⡿
⠀⢸⢇⢘⠏⡶⣿⠗⠈⠃⡥⢋⠔⡁⢊⡴⢻⢛⡼⡈⡆⢆⢹⠰⡄⣱⠞⢋⠠⣤⠊⠉⡀⠀⣠⠌⠹⣼⢶⠋⡈⡷⣄⠀⠀⠈⠻⣿⣇⠘⠼⣛⠷⡌⡏⣰⣧
⠀⢸⡟⣽⠄⣒⠒⣪⢂⠞⠁⠀⠔⣠⣿⡘⣸⢻⡳⢁⠋⠜⠄⢎⣾⣇⠴⠃⣰⠃⡠⠊⡠⠃⠑⣔⢀⡟⡌⡎⣱⢁⠽⣶⡄⠀⠀⠈⢿⡌⢦⢛⢛⣀⣏⣿⣟
⠀⢸⣭⡟⠀⢤⠒⡴⠏⠠⠀⠌⣰⣿⣰⢶⣛⣿⣖⠵⠓⠋⣿⣛⣉⣁⣀⣠⡇⣊⣥⡬⠀⠰⠆⣌⣾⣃⡹⡸⠁⣁⣂⣯⡼⣄⠠⠀⠄⢻⡄⠲⡚⠉⠛⣽⡿
⠀⢸⢋⢔⣦⠆⠟⡐⠀⠁⢀⣴⡇⡿⣅⢀⣨⢣⣿⠀⢀⣔⡘⣋⣿⡍⠡⠈⢼⣿⠇⠀⠀⡐⣾⣹⣗⣿⡃⣷⣉⠀⣩⡎⣵⠜⣦⠀⡈⠀⢻⡄⠡⢷⢢⡜⢿
⠀⢠⠱⣷⣖⡡⠀⠀⠀⢀⣿⠃⡇⣹⡟⢫⣉⡎⢹⡀⢪⠳⠉⡜⢻⡘⠢⣄⠀⠉⠈⠀⠀⠁⠀⣰⢧⢡⢋⣩⠿⡜⢋⡜⣛⠁⠻⣧⠀⠀⠀⠛⢌⢲⡧⡸⣦
⠀⢐⠒⠆⡊⠀⠀⠐⠀⣎⠌⢢⡜⡇⣇⣼⠙⢘⣆⡌⡆⣆⡇⢳⠌⣷⣼⡈⠳⣤⢀⠐⢒⡠⣔⢧⣸⢞⡈⢋⢀⣿⠎⢞⢩⡄⢢⣽⢆⠀⠀⠀⠈⠔⢷⢬⡟
⠀⢸⣟⡄⠃⠀⢀⠁⣸⣯⠂⠈⣁⣿⢻⣇⣨⣣⣏⢌⣴⣭⣹⡌⢾⠟⢼⣛⣦⡈⢢⡼⠘⣧⡙⠘⠇⢜⣹⠀⢟⢀⣡⣽⡿⡠⠟⠍⣻⡄⠂⠀⠐⠘⣿⢸⣿
⠀⢰⠍⡔⣠⠁⠀⠀⡸⢻⣶⡷⠿⠿⠈⠻⣇⠝⢹⠯⠀⠀⣰⡿⢸⡞⢓⡏⠡⠀⣾⠇⠂⢀⠙⠲⣄⠠⢩⡼⠣⣡⣿⢿⡷⠿⣷⣻⠋⡇⠀⠀⠀⠃⢸⡷⣿
⠀⢸⠧⡔⠹⠀⠀⠀⠀⡾⣬⣝⣃⠀⠀⠀⢯⡞⠟⠛⣁⠛⢟⣀⣿⠁⣟⠀⠂⢸⡏⠀⢊⠁⡠⠂⠈⢷⣾⢂⢖⣿⠀⠀⠀⢛⣫⣥⣧⠀⠀⠄⠀⠀⠈⢣⠙
⠀⠀⠉⠁⠀⠄⠀⢀⢰⠟⡯⠥⢧⢤⡤⣤⣌⠳⣇⠨⠧⢊⠆⣿⠇⢸⠁⠈⢀⣿⡍⠀⠀⠀⢳⡄⠀⠀⠻⣆⣾⣁⣀⣠⣤⠬⠥⢊⢺⡄⣄⠂⡀⠀⠀⡈
⠀⢐⠒⠒⠒⠒⠒⠚⢸⡇⠉⠒⣸⢉⡅⠂⠙⢷⡌⠙⠲⢦⣔⡾⠀⠘⣇⠀⠈⣿⠇⠀⠀⠀⣼⢛⢦⡀⠀⠈⠻⡟⠉⣿⣫⠐⢧⣛⣿⡇⢙⢀⠈⠁⠉⠄⠤
⠀⢘⡀⢀⣀⣀⡀⢸⢸⣏⡁⢠⡟⢸⢾⠶⠲⠋⡽⢦⡀⠀⠈⠳⠀⠀⡼⣇⠀⢸⠀⠀⠀⢰⣏⡟⣸⠝⣦⡀⢸⠀⠀⣸⢟⠂⠀⡺⢏⡄⠀⢰⠀⠀⠀⠀⠥
⠀⢨⢶⠂⢕⡄⣩⢀⢸⡟⣡⠟⡘⡃⣎⠐⣡⠈⡷⣻⣼⠗⣦⣄⣡⡾⣡⢿⡄⠀⠀⠀⠀⣾⡟⣴⡅⣾⢹⣷⣬⣀⣴⢫⣎⢣⠱⠘⡈⡇⠀⢈⢩⡝⠉⣌⠛
⠀⢰⠂⢠⣸⢁⠛⢸⢸⡦⠳⠀⢇⠜⢈⡞⡠⡏⠁⡇⡝⠠⠷⣀⣭⢺⣼⠾⡇⢀⠀⠀⡀⢿⣇⢡⠋⡌⣧⣸⣮⢋⡸⢫⠟⡘⢸⢇⢣⡇⢠⢸⣆⢳⢀⠊⣶
⠀⠈⠀⣸⢃⠆⠀⢸⢸⡶⠖⠛⣌⣋⠒⠤⠤⣵⡳⠊⢲⠤⢲⣹⠾⣻⡇⢨⠀⠄⡠⠊⡠⠚⢿⠻⣤⡠⢷⡷⡥⠬⢴⡧⠾⠶⠓⠚⢢⡇⢰⢸⠈⠄⢿⠀⢹
⠀⢠⠀⣽⠘⡀⠀⠘⢸⡧⢀⣴⠏⣼⣅⡀⣀⣾⢡⢞⣦⣠⠞⠁⢰⠃⢡⢠⠰⢊⠔⠃⠀⠀⠘⡆⠈⠳⣼⢏⡀⢈⣍⣴⣯⡂⢀⠐⢋⡅⢸⠈⠀⡘⣸⠀⣨
⠀⠀⢠⠁⠱⠈⠳⠀⢸⢡⠡⢡⠶⠌⣸⢇⠹⣠⢏⣦⡾⠋⠀⢠⠏⠀⠈⡤⠞⠁⠀⠀⠀⠀⠀⢹⠀⠀⠈⢳⡟⡙⢇⡾⢫⠅⡰⠃⢣⡆⢸⠸⠃⠐⠁⢁⠈
⠀⠀⠃⡀⡱⡡⠀⢠⢸⡄⠇⠎⡤⣄⡣⢸⠗⣵⣿⠋⠀⠀⠀⠞⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠙⣷⡎⣷⣿⠀⡇⢃⠈⡃⢸⠀⠸⡕⠁⢈⠿
⠀⠀⠠⠋⠈⠁⠀⠈⠀⡵⢈⠂⠑⡄⠠⡟⣴⡟⠁⠀⠀⠀⠌⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢷⣷⠱⢠⢃⠼⡄⡆⠰⠀⠀⠁⠈⢣
⠀⠀⡆⢀⢂⠅⡆⠀⠀⣷⠘⢪⠖⢫⢋⣴⠏⠀⠀⠀⠀⠈⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⢻⡄⠱⡄⠒⣼⡆⢸⠀⡀⠀⣆⠀⡄
⠀⢰⢇⠸⣤⠀⠀⠀⠀⣿⡯⠴⢲⣿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢻⣇⠈⢩⡽⡇⢸⠀⠀⢀⡿⠰⢰
⠀⢠⡸⢆⠹⣆⡶⠀⠀⣿⢀⣎⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠈⡰⡇⢸⢠⡄⡾⢠⢧⠈
⠀⢠⡁⠘⣇⠹⣀⠀⠀⣿⣷⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠌⢷⣱⡇⢸⢈⡿⠰⠇⠀⡆
⠀⢨⠱⠾⠄⠲⢻⠀⢨⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⢸⡥⠁⠀⠎⣠
⠀⣠⣀⣀⣀⣀⡘⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⢀⡀⣀⣀⣁⣈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋⣶⣾⣷⣿⣿⣿⡟⢿⡿⠟⠀⠻⠿⠛⣿⣿⡚⠿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡏⠻⠏⢹⣿⣿⣿⣿⣿⠿⢿⣯⣤⣿⣿⣿⡿⣿⣿⣿⣿⠇⠘⢿⣿⡿⠿⣿⡏⣳⣤⣤⣶⣶⣶⣴⣾⣿⣿⣿⣿⣿⣷⣦
⣿⣿⣿⠿⢟⣯⣷⣶⣿⣿⣿⣇⣀⣀⣿⣿⣿⣿⣿⣿⠃⢿⣿⣿⣿⣿⣾⣧⣀⡚⣉⠁⠀⡀⠀⠈⠉⢀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿
⣿⣿⣿⡇⢸⣿⢿⣿⣿⣿⣿⢻⣿⣿⣿⣯⠉⠛⠛⠁⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⣿⣶⣿⣿
⠿⠿⠋⠀⠀⠉⠛⠿⠈⢸⣏⣼⣿⣿⣛⣿⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⣿⣿⣿⣿⣿
⠀⠀⢠⣀⠀⠀⠀⠀⢠⣿⣿⠅⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠭⠛⠛⠀⠀⠀⠙⠛⠀⣸
⣀⣀⣠⣀⣀⣀⣀⡀⢀⡽⠋⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⢇⡀⠀⢀⣶⣀⣤⣀⣄⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠰⠄⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⣁⣔⡹⣿⣿⣿⣿⣿⣿⣿⣿⣬⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠀⡤⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⢡⣶⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⣿⡿⠿⢛⣭⣴⣶⣿⣿⣿⡀⡀⣸⣿⣿⣿⣯⣝⣻⣿⣿⣿⣿⣿⣿⣷⣦⠀⢰⣿⠟⣛⣧⣾⣧⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣫⣭⣶⣾
⣿⣿⠀⣿⣿⣿⣷⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣯⣿⣿⠀⠀⠈⣿⣿⡏⣻⣾⣧⣿⣋⣽⣿⣿⣿⠋⢹⣿⣿⣿⣿⣿⣿⡗⠰⣿⣿⣿⣿
⠟⠁⠀⠈⠛⠾⠯⠉⢹⡏⣼⣿⠿⢿⣿⡏⠻⣿⡿⢿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠋⠻⠷⣬⣝⣫⣉⠉⠉⢹⣿⣿⠙⣸⣿⣿⣟⣿⠿⠟⠁⠀⠉⠳⢿⠏
⠀⣀⠀⢀⠀⠀⠀⢠⣿⣿⠉⣽⣾⣌⢿⡇⠀⠀⢥⣬⠽⠛⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⢸⡿⢡⣤⣿⣿⡇⠀⠀⠀⢀⡀⠀⡀
⠀⠉⠀⠀⠀⠀⠀⠈⡿⠟⠀⣿⡿⢿⡮⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⣠⣿⣵⣼⣿⣿⣿⣿⠆⠀⠀⠈⠁
⣿⣿⣿⣶⣶⣄⣠⠞⠀⢀⠀⠙⣷⠀⠳⠼⢷⡀⠀⠀⠀⠀⠀⠀⠀⣯⢠⡀⠀⠖⠂⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⠁⠉⠛⠂⠀⢀⣠⣶⣶⣶⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⣀⡀⠈⠳⠀⠀⠈⢗⠖⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣽⣿⣯⡰⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠟⢻⣿⣿⣿⡇⠀⠀⠉⠀⠀⠄⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣶⣂⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣵⣶⣿⡿⣿⣿⣿⣧⣄⣀⣀⣀⡀⢀⣀⣀⣀⣠⣠⡀⠀⠀⠀⠀⠀⢀⣹⡿⠿⠷⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣫⣶
⣿⡿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠈⠑⠒⠂⠀⠀⠀⢀⣠⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⣿⣿
⠉⠳⢿⡏⠉⢹⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⡇⠉⠂⠀⠀⠀⠀⣀⡀⠄⠚⠉⠈⣋⠉⢙⣽⣿⣛⠟⣹⣿⣿⣿⡟⢿⣿⣿⣿⡟⠻⠀⠏⠻
⢐⡀⠀⠀⢠⣾⣿⡟⢩⣿⣿⣿⣿⣿⣿⣟⢛⣵⣶⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣹⣾⠛⠿⣿⠿⢹⣿⣻⣿⣷⠀⠉⠀⠀⠀⡀⠠⢐⣷
⠁⠀⠀⠀⠛⣿⡿⠇⢸⣿⣿⣿⣿⣿⣿⡏⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣂⠀⠀⢀⣼⣿⡏⢹⣿⣦⠄⠀⠀⠀⠛⠑⠈⠁
⡀⠀⠀⢀⡴⠋⠁⠀⠸⣿⡇⠹⠟⠛⠋⠀⠀⢀⠉⠛⠁⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢿⣧⡀⠀⠈⢻⡿⠃⠘⠻⣿⡀⠀⠀⠀⢠⣤⣤
⠀⢀⢤⣿⠀⠀⠘⠃⠀⠙⠻⣀⠀⠀⠀⠆⠀⠐⠀⠀⠰⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⣀⡴⠃⢀⣀⠀⠀⠀⠀⣤⢀⠀⠘⠙
⠈⠀⠘⠀⠀⠀⠀⣴⠶⠀⠀⠋⠀⠀⠀⢀⣀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣁⡀⠀⠀⢀⣽⠀⠁⠂⠐⠂
⠀⠀⠀⣧⠀⠑⠀⠀⠀⠀⠀⠁⠀⠄⠀⠈⣩⠖⠁⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⣠⠚⢈⡄
⠀⠀⠀⠉⠶⠇⠀⠀⠀⠀⠀⢤⠀⣄⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄⠘⠇
⣶⡿⠀⠀⠀⠀⠈⠙⠆⢀⡀⠈⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠂⣀⣤⣶⣾
⣿⠁⠀⠁⠆⠀⠀⠀⠀⣿⠃⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣗⠀⢀⠘⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠉⠉⠉⠩⠉⠀⠉⠉⠉⠉⠉⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⣀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⡇⠀⠐⠒⠀⠀⠠⠀⡁⠀⠀⠀⠀⠀⡀⢀⣀⣀⠀⠀⢀⣈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⡇⠀⢀⢀⠀⣀⣀⠀⡅⠀⠀⠀⠠⠎⢁⠀⠠⠀⠀⠀⠐⠐⠨⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠈⠇⢀⡀⢐⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⡇⠀⠀⠀⠀⠘⠂⠀⡆⠀⠀⠀⠀⢃⠈⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠤⠄⡇⠀⠠⠤⠀⣠⡄⠀⡁⠀⠀⠀⠐⠀⠓⢄⡀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢀⠉⠉⠀⠀⠀⢀⣀⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠀⡅⠀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠛⠒⠋⣀⠈⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠐⠀⠀⠇⠀⠐⠀⠀⠤⠆⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡰⠊⠁⠅⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⡀⠀⢀⣀⠀⢀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢻⠀⡆⠀⠣⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡃⠀⠀⠀⠀⠉⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢡⡁⠠⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠃⠀⠀⠀⠅⠀⠀⠀⠀⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠠⠤⠄⠇⠀⠠⠤⠀⣠⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠈⠉⠁⡁⠀⠈⠉⠀⠛⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⣃⡀⢀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣽⡇⠀⠀⠀⠈⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣿⡇⠀⠐⠀⠀⠒⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣷⣤⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⠿⠿⢻⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣷⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠉⠀⠙⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠞⣶⡞⠛⠛⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠠⣄⣀⣉⠒⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠈⠓⠉⣙⣍⡉⠧⠄
⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⣠⠇⠀⠀⠈⠑⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⣶⢠⡄⢰⡌⠳⢟⡀⠀⠀⠀⠐⠒⠙⠀⠐⠖⠀⠈⠲⢄⡀
⠀⠀⢠⠀⠀⠀⢘⣁⡤⢄⡐⠚⢁⠀⠀⠠⠀⠀⠀⠘⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠩⡥⠾⠳⣦⠙⢛⡋⢁⣠⣀⠀⠀⣠⡤⠤⠤⠄⠀⠡⣀⠀⠙⣦
⠀⠀⠿⠀⠀⣀⠀⡡⠐⠀⠀⠀⠀⠈⠢⡀⠀⢆⠀⠀⡇⠀⠘⡀⠀⠀⠀⠀⠀⣀⠴⣤⠶⠶⠀⠀⠀⠙⠃⣼⡙⣯⡁⠂⠀⠀⠀⠀⠀⠰⠄⣀⣀⠀⠋
⠀⣤⡆⠈⠁⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⠀⡠⠁⠀⠰⠃⠀⠀⠀⠀⠄⣹⡿⣹⢇⣠⠀⠀⠀⠰⠿⠈⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘
⢠⣿⡒⣤⣤⠂⠀⠀⠂⠈⠉⠀⠀⠈⠐⠄⠀⣼⡂⠁⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⣨⣯⣤⠰⠷⣰⠶⣟⠂
⠃⢻⣝⠄⠀⠀⠁⠀⢀⠄⣀⠤⠀⠀⠉⢺⣦⣿⠗⠂⠀⠘⢻⢧⡴⠀⠀⠀⠀⠀⠀⠀⢴⢶⣼⠇⠀⠹⠿⡏⢋⠀⠀⠀⣄⡀⣠⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⣠⠊⢃⣹⠀⠀⠀⠀⠀⠀⠀⠰⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠟⠙⢷⣶⠄⠀⡀⠀⠀⣙
⡄⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠠⡞⠁⠀⠀⠀⠀⠀⠀⠠⢀⣴⠿⡇⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣧⡤⢾⡆⠀⢠⠁⠀⠀⠙
⠘⣶⠒⠈⠁⠀⣠⠖⠀⠀⠀⠀⠀⢀⣀⣠⠴⠎⠀⠀⠀⡾⣿⡶⠋⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⡌⠀⠀⠀⣿
⠀⠉⠤⠠⠄⠂⠁⠀⠀⠀⣼⠶⠋⢉⣠⣦⡤⣄⠀⠀⠈⡁⠋⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⡟
⢸⡆⠀⠀⠀⠀⠀⡐⢀⡾⠁⢤⣾⠟⣿⣟⣀⠈⠃⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠹⣦⣀⣀⣠⣈⡴⠛⡠⢣⠀⠀⠘⠉⠋⠀⠀⠀⠀⠀⠠⠀⢈⡁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠘⠀⠀⠀⢀⣰⠄
⠀⠀⠀⠉⠺⠷⠀⠀⢀⣀⠀⢉⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢦⡻⠀⡰⢦⣌⠑⠶⣄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠉⠁⢀⣀⠀⣠⠆⢠⠋⡉⠀⠀⢀
⠀⠀⠀⠀⠀⣄⠐⢠⠄⠀⠋⠁⠘⠂⢤⡀⠀⠀⠀⠀⠀⠀⣀⢀⠔⠁⠀⠀⠀⠀⠈⡀⠀⠀⠳⣄⡤⠲⠲⣿⠷⠁⠀⠀⠀⠄⣨⡍⣶⠋⣠⠿⠊⠁⠀⠐⣼
⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⡄⠀⠀⡀⠀⠉⣈⠉⠁⠀⠀⠉⠉⠣⣄⠀⠀⠀⠀⣀⡀⠠⢄⠀⠀⢹⡦⣶⡷⠂⠀⠤⠤⠴⠄⠚⠩⡆⠻⡉⢀⣀⠀⠠⠄⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⣧⠀⠀⠀⠇⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠒⠉⠀⠉⢄⡀⠀⠀⠀⣸⠇⠈⢧⠀⠤⢤⣴⠶⠛⠁⠀⢸⠀⠀⠁⠀⠀⠀⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡆⠀⠀⠠⡄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠘⣟⠙⡀⠀⠀⠈⣶⢖⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣐⣦⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⣤⡘⠒⠒⠂⠳⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⡟⠐⢷⡒⠟⠶⠾⡌⠐⢄⠀⠀⠀⠀⢀⠖⠀⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡠⠤⠃⠀⠀⠙⣧⢸⣷⠁⠀⢸⡀⠀⠀⠀⠌⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⠊⣀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⡀⠀⡜⠀⠀⢀⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣰⡔⣺⠋⠀⠃⢠⡘⠇⠀⠀⠀⠀⠀⢹⡇⠻⡄⠁⠀⠠⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣴⠛⠿⠉⣴⡇⠀⠀⠀⠁⠀⠻⠀⠀⠀⠀⠀⠈⣷⠀⠻⢞⠄⠁⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⢊⡄⠙⢃⠀⡐⢻⠍⠑⠢⣄⡀⠀⠀⠂⠀⠀⠀⠀⠀⣿⠀⡠⠊⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠀⠀⡔⠀⢷⡀⢀⠀⠀⠉⠐⠢⠆⣀⠀⠀⠀⠀⣿⠞⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠢⢼⠀⠰⣾⠀⠈⢢⡀⢤⡀⠀⠀⠈⠑⠂⠤⢰⠏⠀⠀⠀⠈⣁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠘⠳⣼⣄⠀⠀⠱⠀⠘⠂⠀⣁⡠⠤⠀⠃⠀⡀⠀⠠⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⠳⡉⠑⠲⠦⣠⣄⣁⡀⠠⠴⠖⠛⠁⠀⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⣼⠉⠁⠒⠒⠠⠤⣤⣥⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠲⣿⠀⠀⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡌⠀⢰⡇⠀⢀⠀⠀⢸⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠙⠿⣀⣠⣿⡇⠀⣸⠀⠀⣻⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢈⠉⠉⠉⠉⠈⠓⠂⢿⣇⡀
⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇
⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⢿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠟⣙⣥⡝⣿⡷⠀⣀⣤⡾⣿⣿⣿⡿⠛⠛⠿⠿⠟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠐⢿⠏⠉⠙⣿⣿⣾⣿⣿⣿⣿⡻⡿⠤⠟⠂⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⡿⠏⣠⣤⣄⢀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣀⡀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠚⠉⠹⠿⠿⠶⠂⠀⡀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄⠉⢑⣶⣦⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣿⠟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⣰⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⣿⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡇⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣾⣏⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠛⠿⣿⡿⠿⠿⢿⣿⣿⠏⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⡇⠀⠀⠀⠀⠁⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⣴⡀⢿⡇⠀⠀⠀⠀⣼⣷⡀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠙⠁⠀⠀⠀⣀⠀⠀⠛⠛⠁⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠰⣿⠇⠀⠀⠀⠀⢸⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⡿⠀⡶⠄⢠⡄⠠⠶⠚⠛⠟⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠠⠁⢠⠁⠠⠉⠁⠾⠷⠠⣶⠧⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡷⠀⠄⣤⠀⡂⠀⠀⠇⠀⠐⠀⠀⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⢀⡀⠀⠁⠐⠖⠣⡀⠐⡀⠀⠂⠈⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⢀⣿⣿⣆⡚⢸⣟⣷⠇⢀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣆⢿⣾⣿⡿⠟⠈⡿⢁⠀⠈⠀⠀⢰⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⣿⡇⠉⠉⠁⠀⢀⣤⣬⣴⣿⡦⣠⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿⣷⢂⣸⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡟⢿⢻⣿⣿⣍⣚⠏⠁⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠸⠋⠿⠿⠏⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⣚⣉⣭⠭⠭⠤⠶⠶⠶⠶⠶⠶⠶⠤⠬⠭⣭⣍⣛⣛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⡶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢦⣙⠲⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠛⠲⢦⣭⣍⣛⣓⣲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⣒⣚⣩⡵⠟⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠐⠒⣠⠖⠋⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢀⡴⠶⢦⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⢰⣯⣀⠀⠠⠤⣀⣀⣀⣀⣠⡤⢞⣿⠔⠋⢀⣀⣀⣈⢳⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡾⠈⠉⠛⠓⠲⢶⠶⠾⠧⣤⣤⡾⠧⡤⠤⠤⠴⠶⢒⡶⢿⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠳⣄⠀⠀⠀⠀⣼⢀⣠⠞⢻⡟⢧⡀⠀⠀⣀⡴⠞⠉⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠑⠦⣄⣀⣿⠟⠁⠀⣿⠃⠀⠙⢷⡞⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣠⠞⠉⠢⣀⠀⠸⡇⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣟⡠⠊⠁⠀⠀⠀⠈⠙⢶⡧⠔⠋⠀⠀⠤⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠻⢄⡀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠳⢤⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣠⠤⠴⠶⠖⠚⠛⠛⠛⠓⠂⠀⠀⠀⠀⠐⠲⠤⢤⣀⡀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠠⢄⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠈⠉⠙⠛⠲⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⠤⠴⠖⠒⠊⢁⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣀⣀⠘⠢⡀
⠀⠀⠀⠀⠀⠤⢤⣀⣑⣄⡀⠘⠂⠀⠀⣀⠀⠀⠀⠀⠉⠉⠙⠓⠬⠦⣀⠴⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⣲⣖⠦⠤⠈⠀⠀⡠⢂⣐⣂⣤⢄⡀⠀⠈⣳⠤⣀⣈⠠⢤⡀
⠐⢤⣀⡀⠐⠢⢄⡀⠀⠀⠈⠉⠫⠓⠦⡀⠀⢰⣿⢀⣄⣠⣦⠑⣌⡆⠀⢀⡀⠀⠀⠛⠷⣟⠲⠤⣄⣀
⠀⠀⠈⠛⢷⡦⢄⠀⠀⠀⠲⠤⣤⣀⠀⠚⠃⠸⡘⢌⠙⠃⠤⠄⣿⣧⡀⠀⠈⠉⠲⠦⢬⣉⣉⠓⠶⢶⡉
⠀⠀⠀⠀⠀⠈⠓⠦⣀⠐⠢⢄⡀⠉⢿⠷⡖⠀⡈⠢⢥⣤⣴⣿⣅⣔⣙⣄⠀⠀⠩⢶⣦⣤⣌⣉⣁⣂⣀⣈⣤⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⠢⢄⠀⡁⠒⠤⣄⡈⠋⠀⠈⢿⣽⣿⣷⣋⣻⣾⣗⢄⡀⠀⠀⠉⠛⠿⣓⠲⢤⣀
⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⠀⠀⠀⠈⠁⢢⣄⣀⠀⠀⠀⠈⠛⠋⠉⠹⣿⣿⣿⣶⣵⡀⠀⠈⠉⠉⠙⣛⣳⣶⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠶⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⣿⡏⢹⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣧⠀⢙⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⢀⣠⣼⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⣿⣿⣿⣿⠟⠁
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢸⣿⠟⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠋⠿⣿⠿⠋⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢛⣿⠿⠛⠛⠛⠛⠿⢿⣛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣠⣤⠀⠀⢀⣤⡄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠉⢀⣶⣌⡙⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠈⠛⠛⠿⠷⠄⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠉⠐⠃⠘⠀⠁⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⣬⣷⡄⠀⠀⠀⠀⠀⠀⠀⣠⠿⠂⠀⠀⠀⠀⠀⠀⠐⠻⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⣤⡴⠆
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠛⠳⠶⣤⣀⡀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢀⣀⣠⣤⡴⠶⠚⠛⠻⢧⣤⣀⡀
⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠷⢤⣄⣀⠀⠀⣠⣶⣶⡄⠀⠀⢠⠶⢿⡛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠉⠀⠀⠻⣿⣿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡶⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠿⣿⡿⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣤⣤⣤⣤⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⡜⡰⠃⡠⠊⠁⠀⠀⠀⣈⠩⠝⠉⢆⠀⠙⢎⠳⣥⠀⠀⢀⣠⢏⢦⠀⢠⡔⣞⣠⡤⡄⡠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡜⡰⢁⡜⣠⠔⠊⠉⠉⠁⠀⠀⠀⠀⠈⢆⠀⠀⢣⠸⡆⠀⠀⠀⠈⠈⠀⠀⠀⠉⠈⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⢠⠃⠎⡴⠷⠦⣤⡀⠀⠀⠀⠀⣠⠤⠶⠬⣄⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠇⡇⣰⢮⣴⣶⣶⣤⡉⠀⠀⠀⠀⣡⣴⢶⣦⣬⢆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⢸⢠⡑⡄⠈⣄⣻⣿⠛⠀⠀⠀⢀⠻⣿⢂⡽⠁⠠⢣⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡌⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡆⡸⠹⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⠈⢧⠀⠀⠠⣄⡀⠑⠀⠘⠀⣠⠄⠀⠀⡸⢠⠃⢸⠃
⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⡂⠀⡗⢄⠀⠈⠣⣁⠀⢂⠜⠁⠀⢀⠜⠗⠁⢀⠎
⠶⠈⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢣⠀⠇⠀⣑⢤⡀⠀⠉⠀⢀⡠⢺⠁⠀⢀⠴⠃
⣲⠒⠃⠀⠸⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠚⠣⣴⠊⠀⣰⠊⠉⠀⠈⠉⡅⠸⠒⠂⠁
⠹⠞⢲⠊⠁⠀⠈⠷⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⣇⠀⠀⡥⠔⣯⣃⠀⠀⠀⠀⠀⠈⡧⠤⠄⣀⣀
⠀⠀⠀⡇⠀⠀⠀⠈⡄⢑⡤⠔⠖⠈⠉⠀⠀⠀⡰⠉⢦⠎⠀⢀⠎⠀⠑⠂⠤⠄⠒⠉⠀⠀⠀⠀⠀⠈⠉⠒⠠⣀
⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⡧⠔⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⢆⠂⠂⢧⡀⠀⠀⠠⠛⠢⡘⠀⢀⠔⠁⡀⠀⠠⠜⣡⠆⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡹
⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⡟⠢⣀⠃⠀⠀⠙⣴⠃⠀⠠⠙⣦⠀⠘⡏⠀⠠⠍⠄⠀⠀⠀⠀⠀⢀⠞⠁⠀⠙⢦⡀
⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠣⠤⠌⣹⠦⠀⢰⢃⠀⣀⡢⣄⠀⠓⡤⠦⡞⠀⢀⣀⡄⠀⠀⠀⢀⠏⠳⡄⠀⠀⠀⠙⢦⡀
⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢃⢨⠀⠈⠝⠋⠉⠩⠇⠐⢟⠉⠉⠂⠉⠉⠉⠁⡟⠀⠀⠈⢦⡀⠀⠀⠀⠙⢄
⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⡀⠈⠾⡀⠀⠠⠶⠾⠁⢀⣅⠀⢻⡶⡆⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⠢⣀⢀⣀⠀⢧
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢈⠚⠀⡱⠀⠀⠀⠀⠀⠇⡹⠄⠀⠉⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⡸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠠⣀⡔⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠔⠁
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣻⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠀⠀⠀⣀⣴⠃⠀⠀⢀⡠⠒⠁
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⠁⣤⠏⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠩⡁⣀⠞⠁⢀⡱⠔⠊⠁
⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⡄⠘⡏⠹⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⡔⠃⠀⢀⠂
⠀⠀⠀⠀⠈⠑⠒⠂⠀⠐⠒⠉⠁⠀⠀⠀⠙⠶⠓⠁⠀⣠⠮⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠫⢄⣂⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⠈⠑⠦⡀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡶⠇
⠀⠀⠀⠀⠀⠀⠠⠤⠀⠒⠂⠀⣥⡶⢦⣄⠀⠀⠀⡀⠀⠄⠀⠀⢀⠒⠉⡀⠀⠀⠀⠀⠀⠸⠏⠰⠿⠀⢸⠆⠠⡶⠀⠰⡖⠀⠀⣤⣾⠠⣉⡃
⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰⢻⠀⠑⠀⠀⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠁⡀⠘⠄⠀⢠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠆⠐⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠴⠀⠀⠀⠐⠖⠘⠀⠘⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⣧⡀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡜⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠄⠠⠈⠀⢁⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠊⡄⠁⠐
⠋⠀⠀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠁⠀⢡⡼⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠀⠀⠠⠀⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢇⠈⠇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠠⡀⢠⡆⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⠚⠂⡘⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢸⡧⠄⠴⣿
⠀⠀⠀⠀⢀⠠⠤⠞⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠒⠂⠻⣦⡶⠻⠟⠰⠋⡠⠴⠤⣄
⠀⠀⠀⠉⠈⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢸⠁⠀⠀⠈⡆⢀
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⢀⠀⠂⠀⠓⠤⠤⢼⢷⠄
⠀⠀⠀⠀⠀⢀⠔⠖⠐⠈⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠠⠀⠂⠀⣎⢀⡀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⠀⠀⠀⠀⢰⠆⠀⠀⢰⠀⠀⠀⡆⠀⠠⠄⢰⣷⣿⣿⣦⠀⠀⠢⢸⡀⠂⠀⠀⠀⠀⠀⠀⠠⡄⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠢⠘⠃⠀⠀⠀⢸⠄⠀⠀⢸⡄⠀⠀⡇⠀⠀⠀⣿⣿⣿⣿⣿⠄⠀⠸⢼⡁⠰⠀⠀⠀⠀⠀⣌⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠇⠀⠀⠀⠘⣆⠀⠀⠨⠇⠀⠀⠉⠀⠀⠁⠈⠹⠛⠛⠃⢀⠀⠸⠾⣀⠐⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠒⢒⠢⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⣀⠐⡀⠀⠀⠀⠀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠲⣷⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⣀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣉⣉⡛⠒⠒⠶⠦⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠑⠋⠻⠝⢶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠐⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠟⠀⣀⣀⣀⣀⣀⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣀⣠⡴⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠐⠷⣤⡈⠙⠳⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣭⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⣀⣤⠶⠛⠋⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠉⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⣠⠖⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⣀⣠⡾⠋⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣍⠁⠀⠀⣠⠀⠀⠀⣠⡟⢀⣤⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣇⠀⠰⠟⢠⡟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⣠⡶⣠⡾⠃⠀⠀⢸⡆⠀⠀⣼⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄⠀⢀⡼⠋⠘⠋⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠛⠛⠛⠛⠳⢦
⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⢹⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠻⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁
⠀⠀⠀⠀⢀⣠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⡀⠀⠀⠀⣀⣀⣤⠶⠞⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠆⠀⠉⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⢿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⠉⠉
⠈⠙⠛⠒⠚⠛⠛⠋⠉⠉⠉⠉⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣄⣀⣠⣤⠴⠶⠛⠋⠉⠉⠛⠛⠛⠛⠉⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⣤⣀⡀
⠀⠀⠀⢶⣄⠀⢠⠞⠁⡡⠦⠼⢄⠀⢀⠤⠤⣄⠀⠰⣿⣿⣋⣽⠀⠀⠀⠀⠀⠀⣰⣿⣴⡿⡿⢿⣍⠁⠀⠀⢀⣾⠁⢿⡄⠈⢻⢯⣷⣄
⠀⢤⡿⣷⢫⣇⠺⣄⣸⠁⠀⠀⠐⡄⡇⠀⠀⢸⡇⣿⣟⣻⣟⡿⠇⢀⠄⠀⠀⡄⢿⣿⠻⡇⠡⠅⢻⣤⠀⠀⢸⣿⡦⡈⠳⠦⢿⣐⣬⣿
⠀⢺⣾⣎⣿⣿⠁⢾⣇⢁⡤⢰⡶⠁⢀⠀⢈⣨⠇⢹⣟⣿⣻⡟⡴⢿⠀⠀⠀⡿⠸⣿⢿⡿⢷⡞⣻⡟⠀⠀⠸⣿⣿⣷⣦⡤⣾⣿⣿⡏
⠀⠀⣿⣻⣺⡇⠀⠀⢿⣿⣿⡟⠀⠀⠘⣝⡿⡏⠀⠀⣿⣿⣿⠁⠱⣦⡤⠴⣶⠆⠀⠙⠿⣿⣾⡷⠟⠁⠀⠀⠀⣿⡷⠾⢽⡭⠿⣾⣿⠃
⠀⠀⢹⣭⡿⠀⠀⠀⠸⣿⡳⠆⠀⠀⠀⣷⠃⠀⠀⠀⢸⣯⡿⠀⠀⢻⡶⣶⡏⠀⠀⠀⣴⣿⠭⣦⡄⠀⠀⠀⠀⢹⣄⠀⠠⠄⠀⢸⡇
⠀⠀⠘⣷⠇⠀⠀⠀⠀⢯⣽⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⡗⡇⠀⠀⢸⣟⣻⡇⠀⠀⠀⠉⠉⠀⠉⣠⣴⠖⠒⠀⠘⢧⣀⡀⢐⣀⣼⠏
⠀⠀⠀⠉⠀⠀⠀⠀⣀⠈⠃⠀⣿⣯⣻⠆⢰⠟⢿⡄⠀⢛⣃⣀⠀⠀⠃⠛⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⠀⢀⡂⠀⠈⠉⠉⠉
⠀⠀⠀⠃⢰⠀⠀⣾⣿⣟⢦⠘⣿⣮⣿⠆⢸⠀⢸⡇⢰⣿⠛⣿⡷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣭⣁⠂⣺⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⡆⢸⣿⣿⣿⠹⢀⣿⠿⣿⡄⢸⠀⢸⡇⣿⡇⠀⢻⣷⣷⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⢈⣓⠿⠛⠁⠀⠀⠀⠀⠀⠉⠉⠛
⠀⣤⠀⠀⠀⣹⢸⣿⣿⣿⠰⠈⣿⣿⣿⡅⢸⠀⢸⡇⣿⣇⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⡿⣦⣾⠀⠀⡀
⠀⣯⣁⣀⣿⣿⢿⣿⣿⣿⠀⠀⣿⣟⣿⠇⢸⡀⣼⡇⠘⠿⣷⣾⠿⢃⣠⣤⣤⣖⣶⢤⠀⠀⠎⣿⠟⡿⢷⣿⡛⠓⠈⣧⠀⠀⠔⠴⢳
⠀⠀⠈⠉⠉⠀⠈⠛⠛⠛⠁⠀⠉⠛⠋⠀⠘⠓⠟⠃⠀⠀⠀⡁⠀⣼⣿⣤⣼⣿⣿⣿⠃⣇⣤⣇⣀⡀⣬⣿⠇⢸⡆⢹⣶⡿⣾⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢼⢿⣿⣿⣿⡿⠁⠀⠈⢙⣿⠷⣿⣿⠃⠀⠀⣿⣮⢾⡸⣿⠃⢏⠼⡀
⠀⡴⠊⠉⠑⣆⠀⢀⡤⠤⢤⣀⠀⢀⠤⠂⠤⡀⠀⣠⠴⠶⢤⡀⣷⣬⣼⣿⣏⣛⣧⡆⠀⠀⠘⠋⠀⣠⠏⠀⠀⠀⢿⣿⣷⣿⣿⣿⣿⣾⠇
⠀⣧⡀⠀⠀⣺⡀⣼⠀⠀⠀⢸⡆⣮⠀⠀⠀⢸⢸⡁⠀⠀⣠⡇⢸⡿⠉⠉⠛⣿⢻⠁⠀⠀⢠⢾⢇⣟⡀⠀⠀⠀⠈⢹⣿⣶⣿⣴⣿⠋
⠀⠛⠭⠴⠾⠟⠁⠙⠤⠠⠤⠟⠀⠻⠤⠤⠶⠛⠀⠛⠆⠴⠟⠃⠘⠳⠤⠀⠤⠿⠛⠀⠀⠀⠈⠒⠀⠛⠁⠀⠀⠀⠀⠘⠻⠽⢬⠷⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣦⡶⠶⣶⡴⠶⣦⣤⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠞⠛⠻⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⢙⣋⡴⠛⠛⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⣠⠤⢴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠰⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠿⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠁⠀⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡞⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠏⠉⠁⢸⡗⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡞⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⡶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡟⠉⠉⢙⣁⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡾⠉⠀⠈⣡⠶⠦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡾⠃
⠀⠀⠀⠀⢀⣠⣤⡾⠋⣩⠥⣤⠞⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡶⠶⠋
⠀⠀⣀⣴⠋⠁⢠⠖⠒⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢀⣀⣸⠟⠓⠛⠁
⠀⣼⠋⠉⣴⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠤⠴⠚⠁⠀⣤⣀⡿⠉⠁
⣴⠟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣄⣠⠞⠉⠉⠁⠀⢠⣤⣤⡼⠋⠉
⢻⡄⠀⠀⠀⠀⠀⢠⡀⢀⣤⡀⣀⡿⢄⣠⠏⠉⢡⣄⠀⢀⣼⢧⣤⣤⠞⠁
⢺⣅⣠⡄⠀⠀⢀⣄⠉⠉⣤⣉⠁⢀⣶⣄⣀⣤⠟⠉⠛⠋⠁
⠀⠀⠈⠛⠶⠞⠛⠙⠛⠛⠋⠉⠛⠋⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⡀⠄⠀⠀⠀⠐⠂⠀⠀⠒⠀⠀⠐⣦⠚⠹⡀⠀⠀⢈⡔⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡮⠶⢽⡄⠀⠂⠀⠀⢠⣔⣳⡀⠐⠀⢴⣁⡈⢙⡁⠀⢁⡞⣉⠉⢿
⠀⠀⠀⠀⠀⠠⠀⡻⠒⠀⢤⠇⠀⠀⠠⠀⣾⠤⢬⣧⠀⢴⣇⣀⣩⢹⠁⠀⣠⣋⠈⢩⢫⡁⠀⠂
⠀⠀⠀⠀⠐⠀⢘⠗⠒⠠⢤⠷⠀⠀⠀⢐⠗⠒⠢⡷⠴⠧⣀⣀⠈⢫⢁⢀⣞⣁⠀⠉⠻⡁⠀⠐
⠀⠀⠀⠀⠀⠀⠓⠒⡖⡤⠠⠧⠀⠁⠆⠉⠉⣛⠒⣚⣀⣀⣆⣌⣉⡉⠀⠀⠀⢸⢩⢉⣉⣁
⠀⠀⠀⢰⡗⠒⠒⠒⠓⠓⠛⠛⠛⠛⠓⠒⠚⠉⠉⠉⢉⣉⠀⢀⡉⠉⠉⠉⠛⠛⠉⠉⣁⣈⠙⢲⡄
⠀⠀⠀⢸⡇⠀⠑⠒⠀⠐⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠀⠀⠀⠀⠀⠀⠈⢸⡇
⠀⠀⠀⢸⡇⠀⠐⠤⣀⠠⠤⠄⠒⠂⠀⠀⠈⠉⠉⠒⠒⠒⠁⠀⠀⣀⠀⠈⠒⠒⠊⠉⠉⠁⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠤⠔⠒⠒⠀⠒⠒⠒⠢⠤⠀⠀⠀⠀⠉⠀⠉⠒⠤⠤⠤⠒⠀⠀⠀⢸⡇
⠀⠀⠀⠸⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠼⢧⡀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠠⠂⠀⠀⠀⠀⠈⠁⠀⠙⠢⡀⠀⠀⢀⡤⠴⠖⠒⠒⠦⢤⣀⡤⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⠞⡱⠀⢀⣀⡀⠀⠀⠀⠀⢀⡀⡀⠓⠘⣦⠚⠡⠀⠀⠀⠀⠀⠀⠀⠸⣌⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠅⡴⢁⠞⠁⠄⠈⡆⠀⠀⢠⠎⣀⠘⡄⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢀⢐⡽⠻⡌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇⠄⡼⢠⠊⠀⠀⠀⡀⢰⠀⠀⠏⠀⠀⡆⠱⡀⠀⡿⡄⠀⠀⡠⠦⡑⠄⠀⡜⠀⠀⠱⣾⢄
⠀⠀⠀⠀⠀⠀⢀⡜⠁⡎⢈⡜⢁⠎⠀⠈⠀⠀⡇⠸⠚⢨⠀⠀⠀⠈⢄⠙⠒⢃⡇⢀⢰⠁⠀⢰⠀⠠⡇⠀⠐⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⣾⠀⡞⠀⠀⠀⠀⠀⠑⡀⠇⠸⡀⠀⢠⠀⠀⠁⣀⠀⡷⠀⡟⠀⡀⠸⡈⣱⠁⠀⠀⠀⢀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠹⣝⠋⠀⠀⠀⡄⠀⠀⠀⠐⣘⠦⠃⠀⠀⠀⠀⠀⠁⠀⣇⡴⠃⠀⠀⠀⠉⠁⠀⠐⠀⠀⠀⣠⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠉⠓⠒⠦⠤⢄⣀⡘⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣼⠗⢢⡴⠚⠛⡆⠐⠀⣀⡠⠴⠖⠋⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠒⠒⠢⠤⠤⠤⠿⠦⠾⠽⠶⠚⠓⠛⠉⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣟⠲⠦⠤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠔⢀⠔⠀⠀⠉⠉⠉⠉⠋⠉⠙⠉⠛⠛⠛⠛⠛⠉⠋⠉⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠐⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣴⡁⣀⠀⠠⠂⣠⡮⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡴⠦⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⡒⢶⡲⣟⠻⠹⢟⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡊⠷⡹⣄⠙⠪⠅⣀⡨⢵⡽⢊⡿⢶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣔⣁⢊⠶⠍⣴⡒⡒⡈⡕⢎⠡⡈⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠒⠶⠦⠶⠶⠾⠖⠓⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠛⢦⡀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢦⠀⣾⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣀⠀⣿⠀⠀⠈⡇⢻⡀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢠⠏⢀⣤⢤⡀
⠀⠀⠀⠀⢰⠋⠉⢳⠘⠦⣤⠼⠁⠈⠳⢦⣤⡤⠞⠁⠀⠀⠀⠀⠀⠈⠓⠦⠴⠖⠋⣰⠋⠀⠀⣽
⠀⠀⣠⣤⡘⢦⣀⡼⠀⠀⠀⣀⣤⠤⠴⠶⠦⢤⡀⠀⠀⠀⠀⣠⠞⠛⠛⠲⢦⣀⠀⠹⣄⣠⠴⠃⣀
⠀⠀⣇⠀⣻⠀⠀⢀⡤⠖⠋⠁⠀⠠⠶⠶⢦⡀⠹⡆⠀⠀⡼⠁⠀⠚⠓⢦⡀⠈⠳⣄⠀⠀⢠⠟⠉⢹
⡴⢶⡈⠛⠉⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠸⢤⡴⠋⣀
⠻⠴⠇⠀⣴⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣰⠋⠉
⠀⠀⠀⣸⠃⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢹⡀⠙⠒⠚
⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⡇⠀⣴⢋
⠀⠀⠀⢿⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡇⠀⠈⠉
⠀⠀⠀⠘⣇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⣸⠃
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⢻⢀⣠⡀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⠀⠘⠉⠸⣧⠴⣆⢻⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣠⠞⢡⠞⢻⣄⡴⣆⠀⣠
⠀⠀⠀⠀⠀⠀⠿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡴⠃⠠⣤⠀⠀⠉⠀⠹⠟⠁
⠀⠀⢀⣀⢰⡟⠂⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⡞⠁⠀⣴⣃⡀
⠰⣤⣸⠉⠛⠃⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⣀⡴⠋⠀⠀⠀⠀⣼⠋
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⣀⣤⠴⠚⠁⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠂

⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⡀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠈⢉⡏⢹⠛⠋⠉⠁⠀⠀⠀⠀⠈⠛⠋⢹⠋⢹⠋⠛⠛⠉⠉⠀⠀⠀⠀⠀⠙⢻⣿⡿⠻⠛⢿⣿⡿⠿⠟⢻⠉⠉⠀⠀⠈⠛⠹⠿⣿⠿⠉
⠀⠀⠀⠀⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠈⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠈⠻⡿⢿⣿⡿⠻⠛⢿⣿⡿⠿⠟⠀⠘⠛⢻⠋⣟⠉⠛⠋⠿⠉⠛⠁⠀⠀⠉⠉⢻⠛⠙⡟⠙⡏⠻⠟⠁⠀⠀⠀⠉⣉⠻⠛⢉⠉⡟⡏⠋
⠀⠀⠀⠀⠸⢀⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⡇⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⠁⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠁

⠀⠘⠛⠻⠿⠛⢻⠛⢿⠛⠿⠟⢻⠟⢿⣿⣿⣿⣿⠿⢿⢿⣿⣿⣿⣿⡿⠃⠀⠙⢿⠿⡟⢻⠛⡟⢻⠛⡟⠀⠀⠀⠈⠛⠛⠻⡿⠿⡿⠛⠋⠙
⠀⠀⠀⠀⠀⠀⢸⡀⠺⠀⠀⠀⢸⡇⠀⠀⠈⠻⠟⠀⠀⠀⣿⢹⡄⠈⠀⠀⠀⠀⠀⢰⡧⢸⡆⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠰⠶⠲⡶⠶⠶⠒⡶⠶⡶⠶⠂⠀⠀⠒⠲⡶⢶⠒⡖⠲⠶⢲⠶⣶⠒⠶⠖⣶⠖⠶⠂⠐⠒⠲⡶⣶⢒⡖⠒⠀⠒⠒⠖⢲⡶⢶⡶⠶⠶⠒
⠀⠀⠀⠀⡅⠀⠀⠀⡇⢠⡄⠀⠀⠀⠀⠀⠀⠀⣾⡀⠉⠀⠀⢸⡇⣾⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⢈⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⣧⡀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⢀⣼⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣏⠙⣄⠀⠀⠀⠀⠀⢟⣆⠀⠀⠹⣾⣷⠏⠀⠀⣰⡿⠀⠀⠀⠀⠀⣠⠋⣹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣦⠤⡀⠀⠀⠘⡜⣆⠀⠀⢸⡇⠀⠀⣰⢣⠃⠀⠀⢀⠤⣴⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡻⡮⣝⢒⡳⢷⡘⣕⢦⣸⣇⡤⣪⢇⡾⢖⡒⣪⢵⢟⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢳⢙⠮⣣⣀⠤⢧⢹⠓⢼⡧⠚⡏⡼⠤⣀⣜⠵⡋⡎⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢷⣒⠦⢼⣎⠀⡜⠛⢄⠀⢣⣧⣸⣇⣼⡼⠀⡠⠛⢣⠀⣱⡧⠤⣒⡾⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⢬⣍⣓⠤⣄⠻⡦⣿⡜⢣⣿⢴⠟⣠⠤⣚⣩⡵⡲⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣳⣦⣀⣀⣈⣳⣙⣄⣉⣻⣛⠿⢮⡷⢾⡵⠿⣛⣟⣉⣠⣋⣞⣁⣀⣀⣴⣞⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡽⠟⠉⠉⢉⡽⣩⠋⣉⣽⣭⣷⢞⡷⢾⡳⣾⣭⣯⣉⠙⢍⢯⡉⠉⠉⠻⢯⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠮⢞⣋⡭⠒⠋⣴⠗⣿⢣⡞⣿⠺⣦⠙⠒⢭⣙⡳⠵⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡾⠭⠒⣺⢏⠀⢣⣤⠊⠀⡜⡟⢹⡏⢻⢳⠀⠑⣤⡜⠀⡹⣗⠒⠭⢷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡼⢨⢖⡝⠉⠒⡞⣸⡤⢺⡗⢤⣃⢳⠒⠉⢫⡲⣅⢧⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣵⢗⣫⠬⡵⡾⢡⡩⠞⢹⡏⠓⢝⡌⢷⠮⠥⣝⡺⣮⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠟⠒⠁⠀⠀⢰⢣⠏⠀⠀⢸⡇⠀⠀⠹⡜⡄⠀⠀⠈⠒⠻⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣏⣠⠋⠀⠀⠀⠀⠀⣾⠏⠀⠀⣰⢿⡿⣆⠀⠀⠹⣷⠀⠀⠀⠀⠀⠙⣄⣹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⡿⠁⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠀⠈⢿⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀⢀⡁⡀⠀⠁⢒⣀⡀⠀⠀⠐⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠆⠀⠀⠀⠠⠀⠀⡄⠀⠀⢀⠘⠀⠀⡠⠔⠁⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⠁⢠⡀⠀⠀⠚⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠤⠥⠤⠀⠤⠤⠬⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⢀⡀⠀⢀⠀⠀⠀⡀⣠⠂⠰⠂⠢⣀⡀⠀⠤⠀⠀⠘⠀⠀⢀⠈⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠀⠐⠂⠀⠈⠀⠀⠀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⢀⠀⠐⠀⠀⠈⠀⢀⡀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠆⠀⠀⠀⠠⠄⠀⠀⠤⠀⠀⠀⠤⠀⠀⠄⠀⠀⡀⠀⠀⠠⠀⠀⠀⠘⣀⡀⣀⢀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠐⠀⠀⢢⠀⠀⠀⠀⠐⠂⠀⠀⣀⠀⠀⠀⠁⠠⠀⢀⠠⠄⠐⠊⠀⠀⠀⠀⠘⠀⠢⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠠⠬⠡⠤⠠⠤⠤⠀⠤⠤⠤⠀⠀⠤⠬⠥⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡀⠀⠒⠀⠀⠀⠀⠰⠆⠀⡀⠀⠤⠀⠔⠈⠉⠐⢄⠐⠂⠀⠀⢀⠀⠀⠉⠀⠀⠀⠰⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⡀⠐⠘⠀⢀⠦⠀⠘⠄⡀⢀⡀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠰⠾⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠛⠁⠀⠘⠀⠀⠀⠠⠄⠀⢠⠊⠀⠀⠀⠀⡔⠁⠀⠑⢄⠀⠐⠠⢀⠀⠀⠉⠀⠀⠈⠉⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡄⠀⠀⠀⠰⠆⠀⠀⣀⠊⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠣⢀⠀⠈⠂⠒⠄⠰⠆⠀⠤⠀⠀⠀⠀⠀⠿⠇⠀⠉
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⡆⠀⠀⠀⠀⣤⠀⠀⠀⠎⠁⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠈⠁⠢⡀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⣄⠀⠀⠄⠀⠀⣀⠔⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⣦⡆⠀⠀⠒⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠈⠁⠀⢸⠀⠄⠀⠠⠀⣀⡀⡀⠠⠄⠀⠄⢸⠀⠀⠈⠐⢉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⡀⠀⠀⠘⠀⠀⠀⣀⣀⣸⢀⡤⠤⠬⣀⠖⠀⠉⠐⢄⣀⡀⠸⠀⠀⠀⠀⠈⢀⣀⠀⠈⣁⠠⠤⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠢⠂⠉⠁⠢⠔⠒⠂⢤⠀⠴⠁⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠒⠰⠦⠂⠀⠀⠺⠀⠀⠈⠋⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⡰⠖⠀⠂⢄⡠⠄⠠⢎⠀⠀⠀⠀⠈⠁⠀⠀⠑⠂⠈⠉⠐⠒⠀⠐⢤⠀⠠⡠⠀⠀⠂⢤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠤⠒⠀⠐⡤⠄⠠⢄⡀⡀⢠⠋⠀⠀⠀⠁⠀⠈⠀⠀⠈⠢⠂⠉⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠂⠒⠒⠔⠈⠉⠔⠀⠉⠐⠢⠐⠈⠀⠂⠔⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⢀⢀⡀⢀⣀⡀⠀⡀⠀⢀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢼⠀⡇⠀⠀⠀⡔⠁⢀⡜⠁⣰⢹⡀⣹⢠⠃⡇⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠓⠂⠀⠘⠒⠂⠚⠒⠂⠁⠀⠁⠉⠀⠃⠓⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⠈⠻⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⡿⠟⠉⢀⣤⣄⡀⠈⢿⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⠉⠭⡖⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⡆⠀⠉⠀⢠⡇⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⡄⠀⠀⠈⠓⠒⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⡀⠀⠖⠋⠉⠉⢀⣼⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣶⣤⣶⣾⣿⣿⣿⣿⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠙⠲⢀⡶⠶⢤⣼⠓⡄⠀⠀⠀⣠⠏⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢠⣤⠤⠈⡇⠀⠀⠈⠀⠙⠓⠚⠋⠁⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣠⠄⠙⠲⢤⣄⣀⠀⠀⠀⠀⢀⣠⠞⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠉⣿⠁
⠀⠀⠀⠀⠀⠈⠁⠀⢰⠃⠀⠀⠉⠉⠛⠓⠚⠋⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧⠤⠤⠼⠦⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⣠⣤⣤⣤⣀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠠⠿⠉⣁⣈⠉⢉⠀⠀⠈⠉⠉⣉⠉⠙⠷⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠇⠀⠀⠉⠙⠛⠈⢈⠀⠀⢁⠀⠉⠛⠋⠁⠀⠀⠘⣟⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⡃⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣇⠀⠀⠀⠀⠈⠲⠗⠦⠴⠛⠓⠁⠀⠀⠀⠀⠀⢸⠧⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣘⠀⠀⠀⢀⡤⠔⠒⠒⠒⠛⡒⠢⢄⠀⠀⠀⠀⣎⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⡉⠙⠛⠹⠠⠥⠥⠐⡻⠃⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⢷⡀⠀⠀⠐⠀⠂⠈⠀⢰⡆⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣧⣤⣀⣴⣶⣶⣤⣶⣿⣷⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣭⣿⡀
⠀⠀⠀⡘⠛⠻⠿⣿⣿⣤⣾⣽⣿⢏⣤⠖⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠂⠀⠀⠀⠨⣿⣿⣿⣆
⠀⠀⠀⠙⠻⠷⣦⣶⢟⣿⠿⠛⠋⠉⠀⠀⠀⠀⠠⢉⣍⣁⡒⠛⣀⣘⣂⡀⠀⠀⠀⠀⠀⠘⢼⡿⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠒⠄⠀⠐⠶⠤⠀⠀⣀⠒⠚⠂⠀⠬⠉⢛⣀⠀⠀⠀⢈⡖⣫⣎
⠀⠀⠀⢠⡞⠉⠀⠀⢽⣿⣿⣿⣷⡄⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢇
⠀⠀⠀⠘⡂⠀⠀⡀⠛⠋⠙⠋⣉⣇⠀⠀⠀⠀⠀⠀⢄⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢴⣾⢿⣆
⠀⠀⠀⠸⣿⣧⣴⣶⣷⣶⣿⣿⡿⠷⠀⢀⣀⣀⡀⢬⠐⢌⠀⢅⠠⣍⠑⢌⠁⣤⠳⠀⠀⣀⣀⠀⠀⠀⢹⠛
⠀⠀⠀⠀⠙⠯⣶⣦⣾⣿⣽⠿⠋⠀⠀⠀⠀⠀⠐⠂⠐⠄⠀⠀⠀⢀⡀⠀⠀⠈⠀⠬⠀⠋⠁⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢠⣆⠻⢷⣆⡈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠐⠒⠒⠦⠀⠀⠀⠀⠠⡴⠖⢒⡂⠤
⠀⠀⠀⠘⣿⣦⡈⠻⢿⣦⣄⡄⠉⠲⢤⡀⠀⠀⢀⣾⠟⠁⠠⣄⡠⠾⢷⣷⡄⠀⠀⠀⢸⡇⠀⠈⢻⣿⣶⠄
⠀⠀⠀⠀⠘⢿⣿⣦⣄⠉⠛⠿⣿⣷⣈⣻⠀⠀⠼⣿⣄⣀⡄⢈⣦⠀⠀⣽⡆⠀⠀⠀⠀⢻⡄⠀⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣶⣶⣬⣿⠛⠀⠀⠘⢮⣏⣩⠄⠸⢋⣠⡼⠋⠀⠀⠀⠀⠀⠐⠧⢈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⣀⡤⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⣄⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢸⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⢳⡄⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⢀⡿⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠤⠶⠶⠶⠿⠷⠀⢻⡀⠀⠀⠀⢸⡇
⠀⠀⠘⡇⠀⠀⠀⢸⠃⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⠃
⠀⠀⠀⣷⠀⠀⢀⣸⠀⠀⠀⠀⣠⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣄⠀⠀⠀⠀⣇⡀⠀⠀⣾
⠀⠀⠀⢿⡀⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣆⢀⡿
⠀⠀⠀⠘⡇⢿⠀⠸⣇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣸⠇⠀⡿⢸⠃
⠀⠀⠀⠀⢻⡜⢧⡀⠉⠗⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠺⠁⢀⡼⢣⡟
⠀⠀⠀⠀⠀⠻⣬⣙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠘⠋⠉⠛⠛⠛⠛⠉⠙⠃⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡙⠛⠲⠶⠶⠶⠶⠖⠛⢋⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠋⠁⠈⠛⠶⣤⣀⣀⣀⣀⣤⠴⠛⠁⠈⠙⠓⠶⠦⣤⣄⣀
⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄
⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣸⠁
⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢂⠥⠂⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠐⠠⢌⣀⠀⠀⠀⠀⠠⠤⠤⠔⠀⠀⢐⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠀⠉⠉⠑⠒⠒⠒⠒⠂⠀⠀⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⢀⠁⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠑⠒⠒⠀⠀⠀⠀⠀⠀⣸⠠⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣙⠉⠀⠐⠂⠀⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠒⠉⠉⠁⠀⠐⠄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⢀⣀⣠⠤⠤⠔⠒⠚⠉⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⣀⠜⠀⣀⡀⠰⡀⠳⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢐⢁⡤⠄⠀⠀⠀⠀⠉⣠⠴⠋⠁⠈⢧⠁⠀⠘⢦⠀⠀⠀⠀⠀⠀⠸⡀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠈⠀⠀⠀⢀⡠⠖⠋⠀⠠⠔⠊⠉⠀⢣⠀⠀⢀⠱⡀⠀⠀⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⠀⠀⣵⠀⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⣨⣠⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠰⠀⡇⠀⠀⠀⠀⠀⡇⠀⡄⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢣⠀⠀⡇
⠀⠀⠀⠀⢨⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⠀⠀⠀⠀⠀⠀⢧⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠞⠀⠀⠃
⠀⠀⠀⠀⠀⠱⣄⠈⢢⠀⠀⠀⠀⠠⠶⠤⠄⠀⠀⠀⠀⠀⠱⣌⠀⠀⠀⠀⠀⠀⠀⠉⠢⣴⠀⠰⠤⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠆
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡤⠤⠤⠖⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⠲⠂⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡧⠀⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⡿⠋⠀⠀⠀⠘⣿⣿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⣿⡇⠀⠀⠀⠀⠀⠻⠗⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣄⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⡀
⢀⣴⣶⣶⣄⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢶⣿⣇⠀⠀⠀⠀⠈⢻⣿⣿⣟
⠹⢿⣿⣿⣿⠁⠀⠀⠀⣿⣏⠀⠀⠀⠀⢸⣻⣷⠀⠀⠀⠀⠀⠰⣯⣿⠆⠀⠀⠀⢸⣿⠂⠀⢀⣀⣀⢸⣿⣿⣿
⠀⠈⣿⣿⣿⣄⣤⣶⣶⣾⣿⠷⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⡦⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠋⠉⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⢀⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠻⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡏⠀⠀⠀⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠆⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠟⠀⠀⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣤⣤⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⡇⠀⠀⠀⠀⢀⣀⠀⢀⣠⣾⠿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣶⣤⣤⣶⡿⠿⠿⠿⠛⠉⠀⠹⣿⣿⣿⣦⡀⠀⡀
⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣟⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠛⠛⠒⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⢀⣀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡀
⠀⠀⠀⣠⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣤⠶⠶⣤⡈⠉⠙⠳⣀⣤⠤⢤⣀
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣍⠷⣤⡴⢋⡴⠞⠳⣦⡙⠦⠴⠞⣩⡴⠒⢶⣽⠇⠀⢀⡀
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠳⠶⠶⠋⠀⠀⠀⠀⠙⠲⠖⠛⣹⠂⠀⠀⢀⣤⣀⣿⣿⣀⣤⡀
⠀⠀⠀⠙⠷⣤⣤⣤⣤⣤⣤⣤⣤⠀⣤⣤⡀⠀⣀⡴⢋⣉⣙⠷⣄⠀⢀⣠⠶⢛⣛⠳⣦⢠⡤⠞⠁⠀⠀⠀⠈⢳⣮⠋⠙⣵⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⡛⠚⣋⡴⠋⠀⠙⠷⣌⣛⣋⡥⠞⠋⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠰⣯⠷⣷⣾⠾⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⣀⠈⠉⠉⢀⡤⠶⠶⢤⡀⠀⠀⠀⣀⣤⠤⢤⣀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡟⡇⢀⣀⠀⢺⣍⠳⣤⡴⢋⡴⠛⠛⢦⡙⠶⠴⠞⣩⡶⠒⠶⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢭⣛⠇⢟⣫⠿⠀⠀⠙⠳⠶⠖⠋⠀⠀⠀⠀⠙⠓⠖⠛⠁⠀⣠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⣽⡆⣮⣝⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⡲⣿⢸⡶⣛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⣧⠇⠈⠉⠀⠀⠀⠀⠀⣀⡀⣾⢻⠀⣀⡀⠀⠀⠀⣬⢿⣣⢠⣛⢯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⡛⠘⣫⡽⠋⠀⠀⠈⠛⠋⣿⣸⠏⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣭⣶⢰⣮⣟⡦⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⡾⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣶⣶⣦⣤⣶⡾⠋⠀⠀⠀⠀⠀⠀⢠⡇⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢰⣿⡿⠿⢿⣿⠛⢛⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣴⣿⠃⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋⠀⠀⠠⠀⠒⠋⠁⠀⢀⠀⡈⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠃⠉⠀⠈⢿⣿⣿⠉⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢉⡀⡀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⣿⣿⠀⠉⠙⠛⠿⠿⠿⠎⠀⠻⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠈⣿⣇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⡄⠀⠀⠀⠀⣠⣴⠄⢠⣄⡀⠀⠀⠤⠀⠠⠀⠀⠀⠀⡰⠛⡿⠁⠀⠀⠙⣡⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣇⠀⠀⢀⢸⣿⣿⡀⠈⠛⠁⠀⠄⠠⣤⡄⣠⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣿⣿⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣶⣄⠀⢨⡑⠂⠀⠀⠀⠀⠀⠀⠀⠙⢢⡄⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⡇⢀⣤⠀⣈⣭⡙⢻⣿⣿⣿⣿⣧⣤⡐⠀⢀⣀⣀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠆⡀⣾⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠉⢰⣿⣿⣿⣾⣿⠋⢀⠇⣿⣿⣷⠀⠈⠉⠉⠀⠀⠀⣰⣿⣿⡟⡇⠀⠀⠀⠈⠀⠀⠁⠀⡠⡀⢈⠜⠀⠉⠀⣀⡀⢹⣿⣿⣿
⣿⣿⣿⣿⡆⠀⠈⠃⢸⣿⣿⠃⠀⠀⢠⠟⢰⣿⣀⣤⣦⡄⠀⠀⠀⢿⣿⠟⠠⠀⠈⠀⠀⠀⡘⡄⠀⠊⠦⠈⠀⣀⣤⣶⣿⠿⠓⠈⣿⣿⣿
⣿⣿⣿⣿⣧⠀⠀⠀⣼⣿⡟⠀⠀⠀⠘⠀⡿⢁⣿⣿⠿⡇⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⡀⢀⡂⠀⠀⣀⣡⣶⣿⠿⠛⠉⢀⠘⢒⠀⢹⣿⣿
⣿⣿⣿⣿⣿⡀⠀⠰⣿⣧⠀⠀⠀⠀⠈⠀⠃⣼⣿⣿⡇⣰⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠛⠄⢶⠀⠹⠛⠉⠀⠀⡀⡀⠺⢦⠉⠀⠈⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⢄⡺⡜⡠⠙⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⣤⣬⡀⠀⠀⠀⠀⠀⠸⠟⠋⠀⢀⡁⣰⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⣿⣿⣿⣿⣿⡆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⢀⣼⣿⡟⣿⣀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠁⠈⠁⠀⠹⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⠀⠀⠘⠛
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣤⣀⣀⣀⣠⡄⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡿⠁⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣄⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠙⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠶⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣇⠀⠉⢳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣴⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⠟⠉⠁⠈⠛⡻⠻⠀⠔⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⣿⣿⣿⣿⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠉⠀⠀⠀⠀⠀⢀⠀⢄⠒⠀⠂⠰⣿⣷⣿⣿⣿⣿⣿⡇⠀⠠⢀⣠⠄
⠀⠐⠠⠀⠀⠀⠘⢿⡿⢀⠃⠀⠀⣰⠎⣿⣿⢿⣿⣿⣿⣷⣄⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣧⡔⠀⣼⣷⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⣈⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣦⣿⡅⠀⠂⣴⣤⣦⣶⣤⢄⣾⠗⠁
⠀⠀⠀⠀⠀⠠⢎⣽⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢿⣿⣿⣿⣿⣧⣴⡇⠸⣿⣿⣿⣿⣿⣡⡄
⠀⠀⠀⠀⠀⠀⠘⢻⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⡟⠁⢠⣤⣭⣟⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣏⢠
⠀⠀⠀⠀⠀⠠⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣯⡉⣩⣽⣿⣿⣿⣿⠭⣾⠃
⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⢿⣿⡟⣶⠟
⠀⠀⠀⠀⠀⠸⣿⣿⣷⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡧⣶⣿⣿⣧⠀⠉⢘⣿⢻⣿⣿⣿⢸⣿⡇⠷
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠿⣿⢸⢻⣿⣿⣿⣿⣷⣿⣿⣿⣿⡆⠀⢸⣿⢰⣿⣿⠉⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠹⣋⣿⣷⣿⠀⣿⣿⠿⢿⣿⣿⡟⢿⣿⣿⡇⠀⣾⡏⣿⣿⣿⠨⣦⠟
⠀⠀⠀⠀⠀⢠⠠⠾⣧⣶⣾⣿⣿⣁⣷⡤⠹⠀⠀⢈⣉⣙⣃⡀⢹⠏⠀⢀⡿⠃⢸⡇⡿⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣍⣿⣗⠀⠀⢠⣿⣿⣿⣿⣿⣿⣄⠀⠀⣇⠀⢸⡷⡟⠀⡞
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢿⣿⡟⣯⠉⠀⠰⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⣿⡘⡟⠀⠱⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⣹⠻⣿⣷⣾⡄⠀⣠⡿⣯⣾⣿⣿⣿⣿⣿⣿⣇⣿⠃⡇⠀⡄⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣀⣼⣿⡿⠛⠙⣿⢋⣿⣿⠟⢿⢷⣿⡏⣿⣿⠙⢈⣧⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⡇⠀⠀⡏⢸⣿⣯⣀⣸⣸⣿⣤⣿⣟⣶⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡃⢸⣿⡏⠈⠉⢹⡿⢸⣿⡟⣏⢠⡆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⢸⣿⡇⠀⠀⢸⣷⠈⣿⡇⡿⠘⡇⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠁⢰⣿⠇⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠆⠀⢀⠀⠁⠀⠀⠀⢠⠀⠀⠉⠀⠀⠀⡇⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠀⠀⠀⠀⠀⠸⠀⠀⠀⠰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡷⠶⣿⣿⣿⣿⡶⢶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣯⣀⣀⣀⣀⣠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣾⣿⣿⣿⣿⣞⢿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣞⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⡉⠀⠀⠀⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢀⠀⠇⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠂⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡶⠂⠃⠘⠾⠟⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠾⠟⠛⠻⠿⠷⠶⠶⠶⢦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⢻⣿⠟⢉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⣿⣿⣿⡦⠾⠿⠿⣿⣿⠿⠿⢶⣤⣄⣀⣠⣤⣶⠶⠶⠛⠋⠁⠀⢀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣤⣴⣶⣶⣶⣶⣶⣤⣄⠀⠀⠉⢉⣉⣀⣠⣤⣤⣴⣶⣶⠿⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⡀⠀⠀⠘⢿⣿⣿⠏⠉⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠘⠿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⠉⠉⠉⠉⣁⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⠀⢶⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⡆⠈⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣀⣀⣠⣄⣀⡤⠴⠒⠋⠉⠉⠉⠉⠙⢿⣷⣤⣶⣤⣄⣀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠘⣿⣿⠀⠀⠀⠀⠀⣀⣠⣾⠯⢤⣤⣀⣀⣀
⠺⠤⠤⠤⠤⠤⢤⣤⠤⠤⠤⠤⢤⣀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠒⠻⠿⢿⣿⣿⣿⣿⣶⣿⣿⣦⣤⡤⠖⠛⠉⠉⠁⣀⣀⡤⠴⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢛⣛⣋⣉⣁⡤⠖⠚⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠛⠋⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⡠⠀⠀⠀⠀⢀⠄⠀⠤⣾⠔⣀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠
⠀⣠⣶⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⢛⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠉⠴⠂⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠈⠉⠃⠀⠀⠀⢠⠀⠐⠀⠀⠀⢾⠀⠈⠛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⢀⡖
⠂⠐⠐⠀⠐⠀⠀⠺⠬⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠅
⠄⣀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠁⠀⠉
⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠁⠀⠀⢀⠠⢀⠀⠀⠀⠐⠀⠃
⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣊⣘⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣤⣻⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⢿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠻⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠸⠏⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣤⣀⣀⡀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠐⠂⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶⣶⣿⣿⠁⠀⠀⠚⠛⠁
⠀⣀⣤⣄⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⢤⡀⠀⡀⠀⠀⡀
⣼⣿⣷⣿⢿⣽⡿⣯⣿⣿⣳⣶⣆⣤⣀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠿⠂⠠
⣯⣿⣯⣿⣿⣿⣿⣟⣿⣿⣻⣾⡿⣷⣿⢿⣶⡶⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢙⣷⣿⢿⣾⡿⣿⣿⣿⣽⣿⣯⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠈⢿⣿⣿⣟⣿⣿⣻⣿⣿⣷⣿⣿⣾⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⢿⣾⡿⣷⣿⢿⣽⣿⣯⣿⣿⣿⣿⣟⣿⣿⣻⣾⣟⣷⣿⣿⣾⢿⣯⣿
⠀⠀⠈⠫⠿⠿⣽⣿⣿⣿⠋⠀⠈⠻⣿⣿⣿⣽⢿⣯⡿⣿⣿⢿⣽⣿⣟⣿⣧⡀
⠀⠀⠀⠀⠐⡀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠙⠛⠿⠟⣿⣿⣻⣿⣟⣷⣿⣿⣾⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠠⣉⣿⣯⣿⣯⣿⣿⣿⣿⣿⠿⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣤⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠓⢤⡀⠀⠀⠹⡆
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠙⣄⠀⠀⢿
⠀⠀⠀⢰⡟⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⣀⣀⣤⡴⠶⠛⠻⠋⠀⠀⢀⠐⠀⠀⠀⢿⠀⠀⢸⡄
⠀⠀⠀⣿⠁⠀⠊⠀⠀⠀⠀⠐⢀⡤⠖⢋⡙⠉⠉⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⢀⣀⣀⣸⠀⠀⢸⡇
⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⣠⠥⡞⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢉⡁⠉⠛⠋⡄⠀⣼⠃
⠀⠀⠀⢻⣦⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠰⠞⠋⠁⢀⡄⠱⡄⠀⠀⠀⠀⠀⡀⠀⠰⠾⠇⠀⠰⠀⡇⠀⣿
⠀⠀⠀⠀⡿⠀⠂⠀⠀⠀⠀⢸⡄⠀⠀⠀⢤⣀⠈⠛⢃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⡇
⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⡀⠈⠃⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠈⣿⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠑⢄⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢤⠂⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠘⢷⡀⢀⠆⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠉⠙⠓⠶⠤⣄⣀⣀⣀⣴⡟⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠐⠠⠀⠠⠀⠀⢤⠤⠀⣀⡀⠀⠉⠁⠀⡗⠠⢀⡘⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢶⡶⠁⠀⠀⠉⠀⠀⠀⡦⠀⠀⠈⠑⠢⡀⠀⢷⠀⠀⠅⠀⠀⢻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢠⠜⠀⠀⠀⠀⠀⠀⠈⠃⣜⠰⠀⠐⡀⠀⢈⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠂⠄⠀⣀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠃⠠⠀⠈⠆⡈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⢀⠀⠀⠈⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠀⠙⢣⠀⠘⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠐⠀⠄⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠣⡀⠀⠑⠧⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⠠⢤⠄⠀⢸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⢣⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠙⢀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠤⠀⠃⠈⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠑⡦⢀⠀⠠⣽⡄
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⡀⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠇⠀⠉⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠈⠄⠀⠀⠀⠀⠀⢀⠈⡑⠊⠀⠀⠀⠀⠀⠀⠀⢈⠢⣖⠀⠈⡒⢄⡀⢀⣼⡂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠸⠉⠀⣷
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢰⠀⠀⠀⠀⠀⠢⠠⠆⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢤⠀⠀⠸⡀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⡸⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠔⠁⠀⠀⠀⢰⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣷⡄⢠⡟
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡇⢹⡞⠁
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠛⠉⠛⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⣠⡤⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠲⢦⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⢀⡴⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⡘⠛⠓⠦⣀⣤⣤⣀
⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠳⣤⡴⣫⡴⠒⢦⡙⠦⠴⢞⣩⠶⠲⣬⠇⠀⢀⡀
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠓⠶⠚⠉⠀⣀⠀⠙⠓⠚⢋⡿⠀⠀⢀⡤⣄⣿⣿⣤
⠀⠙⠶⠦⠤⠤⠤⠤⠤⠄⣴⢶⣄⢀⡴⢋⡥⢬⡑⢦⣀⣠⠶⣋⣭⣝⣳⠠⠶⠋⠀⠀⠀⠀⣹⡷⡁⣹⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⣭⠶⠋⠀⠀⠙⠶⠭⠴⠚⢁⣀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠓⠛⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢠⡶⣄⠀⣠⡶⣋⣭⣙⢦⣀⢀⣠⠖⣋⣙⠳⡄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⢼⣿⡧⣶⡆⠈⠳⣬⣛⣥⠞⠁⠀⠙⠳⣭⣭⠴⠋⠁⣈⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⣣⣀⣜⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⡏⣧⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⣿⠏⠙⠁⠀⠀⠀⠀⣀⠀⡞⡆⢀⡀⠀⠀⢀⣹⢷⡁⣱⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣝⠃⢛⣽⠟⠀⠀⠘⠛⠙⣇⡏⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠵⡆⡶⢭⡷⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠖⠒⠒⠒⠒⢒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠤⠶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠛⠀⠉⠉⠁⡇⠸⠈⠉⠁⠀⠁⠛⠀⡇⠀⠀⠀⠀⠀⣠⠈⡤⠤⠠⠄⠀⠀⠤⠤⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠀⠀⠀⠀⡇⠐⠀⠀⠄⠠⣤⣤⠀⡇⠀⠀⠀⠀⠘⠀⡀⠀⠄⠀⠒⠒⠒⠒⠒⠒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠁⢠⡀⢰⣔⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠛⠚⠂⡇⠈⠘⠒⠃⠈⠛⠃⠀⡇⠀⠀⠀⢀⠀⡆⠈⠋⠈⠉⠀⠀⠀⠀⠀⠀⡘⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⡇⢀⠀⡀⡀⢀⣀⠀⠀⡇⠀⠀⠀⠀⠇⠈⠦⣀⠒⠘⠋⠁⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠁⠀⠀⠀⠀⡇⠈⠀⠀⠀⠈⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠀⠈⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠒⠒⠂⡇⠐⠐⠒⠂⠀⠺⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠤⢀⣠⣤⣶⢤⡗⠊⠛⠁⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⡵⡖⡄⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠈⠀⡇⠈⠀⠀⠀⠘⠋⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠌⢳⡏⠸⠈⠂⠇⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠀⠤⠤⠀⡇⠀⠠⠤⠄⠠⣤⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢷⠘⡀⠀⠘⢶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢸⠀⢳⡇⠰⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⠙⠛⠁⡇⠸⠘⠋⠃⠐⠓⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢸⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣠⣠⠀⢀⠀⠀⡇⢠⠀⠀⠀⢠⣄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠶⠆⠀⠒⠒⠂⡇⠐⠐⠒⠒⠰⠶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢀⢀⠀⡇⢈⠀⣀⡀⠀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠁⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠶⠀⠠⠤⠀⣷⡆⠠⠤⠆⠀⠖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⢀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠉⠀⠀⠉⠀⣿⡇⠀⠀⠁⠀⠈⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⢠⡶⣿⣿⣶⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⢿⠿⠿⠿⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡠⠀⠀⠤⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠠⡄⠤⠀⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣯⣭⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣛⣻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣛⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⢻⣧⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⠉⠛⠻⢿⣿⣿⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢸⣿⣿⡿
⠰⢶⣶⣶⣤⣤⣀⠀⠹⣿⣿⡇⠀⢧⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠟⠁⠀⠀⢠⡆⢸⣿⡿⠁⠀⣀⣤⣴⣶⣶⠶
⠀⠀⠉⠻⣿⣮⣝⠳⣦⣿⣿⡇⢀⣾⡄⠀⠀⠀⠀⠀⠘⡿⠫⠟⠁⠀⠀⠀⠀⣸⣆⢸⣿⣧⡶⠛⣩⣾⣿⠋⠁
⠀⠀⠀⠀⠹⣿⣿⣷⡀⠙⣿⡇⢺⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠉⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢻⡟⣿⣿⡄⣿⡏⠀⠹⢿⣿⣿⣦⡀⠀⠀⡀⣠⠀⠀⣠⣴⣿⣿⠿⠁⢿⡇⣼⣿⡿⣼⠇
⠀⠀⠀⠀⠀⠸⣷⣌⠻⠋⣿⣄⠀⠀⢀⣙⠛⠿⣿⣷⣾⠗⢿⣶⣾⡿⠟⢛⣉⠀⠀⣸⡏⠛⢛⣥⡿
⠀⠀⠀⠀⠀⠀⠀⣹⣷⡀⢸⣿⣦⡀⠀⢉⣁⡤⠀⠀⠹⠀⢸⠁⠀⠠⣀⠉⠁⢀⣼⣿⠀⣰⣿⡁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠻⣿⣿⣿⣿⡿⣿⠉⢀⢴⣅⠀⠀⠀⢀⣵⢄⠈⢿⢿⣿⣿⣿⡿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⣾⢻⡏⠀⠘⣿⣇⢻⠀⡇⠠⠃⠀⠙⢷⣦⣴⠟⠁⠈⠃⢸⠸⠇⣿⡟⠀⠀⣿⢻⡆
⠀⠀⠀⠀⠀⣸⡏⣾⡇⠀⠀⠘⣿⣄⠀⡇⢀⣴⢚⣛⡲⠿⠿⣒⣻⢂⣴⣿⠀⣼⡟⠀⠀⠀⣿⡏⢿⡄
⠀⠀⠀⠀⣰⡟⣸⣿⠇⠀⠀⠀⠸⣿⣷⣷⡈⠉⠉⠠⠤⠤⠤⠤⠄⠈⢹⣿⣾⡿⠁⠀⠀⠀⢻⣿⡘⣷⡀
⠀⠀⠀⢠⡿⢠⣿⣿⣤⡀⠀⠀⠀⢻⣿⣿⣷⡀⢀⡤⠖⠒⠒⠦⣄⢀⣾⣿⣿⠃⠀⠀⠀⣠⣼⣿⣧⠸⣧
⠀⠀⠀⣿⠁⠛⠻⢿⣿⣿⣷⣄⡀⠄⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠄⣀⣴⣿⣿⣿⠿⠛⠃⠹⡇
⠀⠀⠀⠃⠀⠀⠀⠀⠉⠻⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢸⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠁⠀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠈⢿⣿⣿⣿⡟⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡘⣿⣿⡟⢠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⢤⡤⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡄⠀⠉⡠⢢⣾⣇⠀⠀⠀⠐⢬⣀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠠⠊⠀⡀⠐⢀⠐⣭⣝⢻⠀⠀⠀⠀⢠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡄⠀⣠⡖⠉⠉⢴⡁⠃⣍⡊⠄⠀⠀⡀⠀⠘⡄⠀⠀⠁⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⡥⠀⣀⣂⠑⠀⢴⠛⠎⢁⣗⢊⠌⠀⠀⠀⠀⢀⣀⢹⣴⡇⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢂⠠⡊⠀⠀⠀⠄⡠⠀⠐⢠⠀⠀⠀⠀⠀⡀⠠⠐⢉⡼⢿⠉⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠀⡰⠟⠊⣀⢴⡈⡁⠀⡀⠂⠈⠀⠀⠀⠀⠁⠈⢁⠀⠀⠈⠁⠈⠂⠁⠀⠀⠀⠁
⠀⠀⠀⠀⡠⢀⡔⠁⠊⡠⠚⠁⡠⠊⠁⢀⡴⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⠊⠁⠀⠠⡀⠀⠀⠀⠀⠰
⠀⠀⡀⠀⢴⠄⡡⠔⢀⠄⢲⠈⠣⡀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠠⠀⠀⠀⠐⠠
⢀⠜⣀⠘⠉⡊⡔⠎⠘⢀⡀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠢
⡆⣄⡤⣾⠙⢂⠠⠶⠐⠋
⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡶⠶⠚⠶⢦⣄⣠⣶⣦⣤⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⣹⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠃⣠⠉⠀⢀⠀⠀⠀⠀⣠⣾⣿⠏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣁⣠⠼⠁⣠⡶⠁⠀⠀⣠⠊⢙⣿⠛⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⣁⠈⠉⢁⣴⣯⡍⠀⠀⠀⣴⡛⠒⢉⡇⠀⠀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠘⠁⢀⣴⡿⢇⡿⠁⠀⢠⣾⣿⠛⢹⡟⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠋⢰⡿⠁⢀⣾⠛⣷⡾⠁⠀⠐⢿⣅⣭⠀⣿⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⢉⡷⠄⠈⢀⣴⠛⣥⠼⠋⠀⣠⣾⠷⣤⠉⠙⠋⠙⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠷⡿⠋⠀⢀⣴⣭⠝⠉⠀⢀⣤⣾⡿⠀⣠⠿⠀⣤⡀⣠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠟⢠⠞⠀⢀⡴⣯⣹⠃⠀⣠⣶⠟⠙⣦⣿⠏⠀⢀⣴⠋⢻⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢋⣠⡴⠁⢀⡔⣁⡀⠟⠁⢠⣾⡿⠟⢷⣶⠟⠁⠀⢠⣿⣿⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠟⣉⡼⠋⢀⡴⣿⠋⠁⠀⢀⣴⣿⣯⠀⢰⡟⠁⠀⠀⣠⠀⢨⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠟⠁⣰⢿⣤⡇⠀⠀⣠⡾⠛⠻⣿⣿⣿⠀⠀⢀⡀⢠⣶⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠏⢰⡞⠀⢠⣧⡴⠋⠀⢀⡾⠋⠀⣴⡞⠁⠀⠉⠀⠀⠋⢩⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣺⣿⣀⡾⠀⢀⣾⠟⠁⠀⣴⠋⠡⣤⠊⠉⠀⠀⢀⣠⣶⣴⣾⣿⣿⠇
⠀⠀⠀⠀⢠⣾⣿⠛⣿⠁⣠⡟⠁⠀⣠⣾⠉⢳⡤⠆⠀⠀⣠⣶⠟⢉⣿⣿⡿⠟⠁
⠀⠀⠀⠀⢈⣻⣿⡾⢁⣾⠋⠀⢀⣼⣿⣿⠶⠊⠀⠀⣠⡾⠛⣻⣶⣿⣿⠟⠁
⠀⠀⠀⠀⢹⣿⣿⠁⡾⠃⠀⡰⢿⡿⠋⠁⠀⠀⢀⣴⣿⣤⣼⣿⡿⠋
⠀⠀⠀⠀⣿⣿⠇⠸⠁⢠⣾⣤⠀⠁⠀⢀⠀⣺⣿⣿⣿⠟⠉⠋
⠀⠀⠀⠀⠙⣿⠀⢀⣴⠟⠋⠁⢀⣠⣦⣀⣾⢿⠏⠉⠁
⠀⠀⠀⠀⠀⣿⡷⠋⠀⣀⣤⣶⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠈⠳⠾⠿⠿⠟⠋⠁

⠀⠀⡀⠀⢀⠀⡀
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡈⢛
⣻⣿⡭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠏⢤⣴
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠰⠂⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⡇⠀⠀⠙
⣿⣿⣟⣋⡀⠀⠀⠀⠀⠀⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⡿⠤⠄⡀
⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠶⣶⣶
⣿⣿⣿⣶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠁⠀⠀⠀⢐⣛⣿⣿
⣿⣿⣿⣟⣋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢩⣽⣿⣿
⣿⣿⣿⠯⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣻⣿
⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠛
⣿⣿⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢾
⣭⣇⣐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡤⢤⡀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⠉⠉⠀⠀⠀⠀⣀⣹⡄⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⢴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠁⠀⠀⠀⠀⢀⣀⠀⠀⣀⡴⠛⠉⠁⠀⣠⣷⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⢰⢶⠀⠀⢸⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣄⣀⣀⠀⣀⣀⡴⠋⠉⠙⠋⠉⠀⢀⣠⣴⣾⣿⠿⠓⠚⢉⣩⠭⠟⠁⠀⠀⠀⠀⣿⠀⠀⢸⢸⠀⠁⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣷⠈⠉⠉⠉⠁⠀⠀⠀⢀⣀⣤⣶⣿⡿⠛⠉⠀⣠⠴⣚⠉⠀⠀⠀⠀⣰⡄⠀⠀⣿⠀⠀⢸⢸⠀⠂⠠⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣆⣀⣀⣤⣤⣶⣶⣿⣿⣿⡿⠛⠁⢀⣤⠖⠋⠁⠸⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢸⠀⠄⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⣿⡿⠿⢛⡹⠛⠁⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢸⢀⡀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠛⠁⢁⡤⢒⣉⡤⠖⠋⠁⠀⠀⠀⣶⠀⠀⠀⠀⠀⢠⡀⠈⣧⠀⠀⠀⠀⠀⣿⠀⠀⢸⢸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣀⣀⣀⣀⣴⠾⠟⠚⠉⠀⠀⠀⣠⠶⠶⣄⠀⢿⠇⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⠸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢀⣤⡿⠉⢉⡇⠀⠀⠀⣿⠀⠀⢸⢸⠀⠄⠀⡇⡇
⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⢧⠀⣀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠒⣟⠁⣀⣀⠀⠀⢹⠀⠀⢸⢸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⡈⣳⡘⠊⠃⠀⠀⠀⠀⠛⢦⣄⣴⠇⠀⠀⠀⠀⣦⣠⠤⡴⠋⠀⠀⠸⡄⠀⢸⠀⠀⢸⢠⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣾⠉⣉⣙⣓⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⡞⡀⠀⠀⠀⠀⢬⠀⢸⠀⠀⢸⣤⠀⠁⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⣿⣿⣿⣿⣿⣷⣾⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠱⡄⠀⠀⢀⠜⠀⢸⠀⠀⢸⣿⠀⠂⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⡻⡄⠀⠀⠀⠰⣶⠞⣡⠟⠀⠀⠀⠙⢠⡘⡯⠀⠀⢸⠀⠀⣾⣿⣷⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡓⠀⠀⣀⡴⠛⠋⠁⠀⠀⠀⠀⠀⢸⠇⠁⠀⠀⢸⠀⠀⠙⠛⠁⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢾⠁⢤⡀⠀⠀⠀⠘⣆⠀⠀⠸⠄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⢳⣠⠛⣆⠀⠀⠀⠙⣀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠠⣶⣄⣿
⠀⠀⠀⠀⣴⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡠⢻⡆⠀⠀⠀⠘⣦⠀⠀⠠⠶⠀⠀⠿⠾⣧⣦⣴⣿⠁⠉
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠻⣿⣿⣿⣿⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⠃⣾⠆
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠓⢦⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⡀⠀⠀⠈⠙⢦⠀⠀⢿⢳⣄⠀⠀⠀⠛⠋⠀⠀⠀⠀⢀⣿⠀⠀⣿⠇
⠀⠀⠀⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⠇⠀⢸⡆⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠉⠀⠀⠙⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠘⢳⡄⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⠆⡔⡠⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠾⢾⡴⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠙⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⢀⣴⣶⣦⡀⠀⣠⣶⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⠛⠋⠁⠀⣾⡟⠀⣿⡇⢰⣿⠃⢀⣿⠇⠀⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠀⠀⠀⠀⠀⣸⣿⡆⠀⣿⣇⣿⣷⠀⣼⡟⠀⠀⠀⠀⠉⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢸⣿⣿⡏⠀⣸⡿⠀⠀⣀⣤⡀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⣿⣿⡟⠀⢠⣿⠃⠀⣰⡿⢹⣿⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⣴⣶⣦⡀⠀⢰⣿⠟⠀⢸⣿⡿⠷⢀⣾⡏⠀⣼⡿⠁⣾⡏⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⢰⣿⠀⢹⣷⠀⣾⡏⠀⠀⣿⣿⠃⠀⣸⡿⢀⣼⡟⠀⣼⡿⠀⣰⣶⣶⡀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⣾⡏⠀⢘⣿⢰⣿⠀⠀⢰⣿⡏⠀⢠⣿⢣⣾⠿⠇⢰⣿⠁⠀⣿⡇⢻⣇⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⣠⡄⠀⠘⣿⡀⠘⢿⣿⣿⡏⠀⠀⠺⠟⠀⠀⣼⣿⣿⠋⠀⢠⣿⠇⠀⢠⣿⠃⣸⡟⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣼⡿⠁⠀⠀⣿⡇⠀⢘⣿⡿⠀⣤⣤⡀⠀⢀⣀⠛⠛⠁⠀⣰⣿⠃⣀⣴⣿⠏⢀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣾⣿⣶⣶⠆⠀⣿⠇⠀⠚⠏⠀⠀⠉⠉⠀⠀⠙⠻⠃⠀⠀⠰⣿⣷⠿⠛⠉⠋⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣰⣿⠃⠀⢸⣿⣷⣶⣤⣀⡀⠀⣴⡶⣶⣶⣦⣄⣀⠀⠀⠀⠈⠀⠀⠀⢀⣴⡿⠋⠀⠀⢰⣦⣤⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⠿⠁⠀⠀⠸⣿⡀⠀⠉⢻⡿⠿⣿⣤⡀⠀⠿⠟⠛⠿⣶⣦⣄⡀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⣀⠀⠀⠈⠙⠻⢷⣦⣄⡀⠀⢀⣿⠟⠻⣿⣏⠀⠀⠀⠀⠀⠺⢷⣦⡄⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⢸⣷⡀⠹⠿⢿⣦⣄⠀⠀⠺⠏⠙⠻⣷⣤⡈⠀⠀⠈⢻⣷⡀⠀⣴⣄⠀⠀⠉⠁⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⣿⣿⣦⣄⠀⠈⠻⢷⣦⣀⠀⠀⠀⠈⠙⠻⠿⠀⠀⢠⣿⡇⠀⠈⠛⠁⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⢻⣧⡉⠻⣷⣤⡀⠘⠟⠻⢿⣶⣤⣄⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⣼⡿⢿⣦⡀⢹⣿⣶⣤⣀⠀⠀⠉⠉⠀⠀⢀⣴⡿⠹⣿⡀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡆⢀⣿⠇⠈⠙⢿⣦⡀⠀⠙⠻⠿⠆⠀⠀⠀⣠⣿⠏⠀⠀⢻⣧⠀⠀⠀⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠀⠀⢠⣾⠟⠁⠀⠀⣠⣿⢿⣧⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⡇⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠀⠃⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⡀⠀⠀⢰⢦⡀⠀⢀⣤⠴⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣠⡄⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣡⣃⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡞⡼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⢙⡲⠤⣄⣸⢳⡧⠖⢋⣉⡉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠰⡏⠉⠛⠶⣾⣸⡶⠚⠉⠉⣹⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠦⣄⣀⣈⣉⣀⣀⡤⠖⠉⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⡆⠀⡆⠀⠀⠀⠀⠀⢺⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣀⢷⡰⠃⠀⠀⠀⠀⠀⢸⡀⢀⡇⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠈⠀⠀⠀⠀⡆⠀⣦⠀⠀⠓⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⠀⠀⠀⠳⠴⣏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⣀⠤⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⠀⠀⠀⢀⡼⠁⢀⡴⠲⣟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⢹⡀⠀⣠⣤⣞⠀⣠⠏⠀⠀⢸⡎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⣷⡦⢭⣉⣛⠓⠒⠒⠶⠶⠶⠖⠒⢚⣛⣉⣥⣶⠋⠙⠛⢁⠀⠈⢡⠏⠀⠀⠀⣸⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⢭⡂⠀⠀⣾⣓⡆⠀⠈⠉⠀⠀⠃⠀⠀⠀⠀⠉⠉⠠⠤⣾⡄⠀⠀⣚⡿⡄⢺⠀⣤⠤⠖⣣⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡘⠲⣤⣀⡀⠳⠌⢩⡉⠛⠂⠒⠀⠰⠶⠲⠆⠀⠀⠐⣚⠋⠥⠟⢀⣀⡭⠞⢁⡼⠁⠈⠙⠦⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⣉⡛⠓⠲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠒⠚⢋⣉⣥⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠛⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⣿⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⣭⣁⣀⣃⣀⣀⣀⣀⣀⣘⣀⣘⣛⣉⣭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣶⣶⣶⣶⣶⣷⢾⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣾⣿⣿⣿⣶⣶⣾⣷⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⠶⠶⠶⠾⢷⣶⡿⠶⠾⠶⠾⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⠤⣄⢀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠄⠀⠀⣿⠈⢧⡴⣃⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠀⠀⠀⣿⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⣿⠀⠀⠘⢧⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡟⢻⡆⠈⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇⢸⡇⢾⠀⠀⠀⢠⣠⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡞⠀⠀⢳⡌⢳⡄⠲⣄⢹⡇⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡏⠀⠀⠀⠀⢹⠀⣷⠤⢬⣿⣿⠥⢴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢳⣄⠀⠀⣀⡞⢠⡏⣠⠞⢹⡏⠳⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣌⣉⣉⣡⡴⠋⠀⠁⠠⠿⠳⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠



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