Hojas Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣰⣿⣷⣶⠒⢶⣶⣄⠰⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠛⠂
⠀⠀⠀⢰⣿⣿⣿⣿⣷⣀⣹⣿⣇⢹⣿⣿⡿⢸⣶⣄⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣦⡘⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⡇⣸⣿⣿⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⣻⣿⣿⣿⣿⣷⣶⣶⣦⣤⣌⡛
⠀⠀⠀⣈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠸⣿⠏⣠⣾⣿⠛⣻⣿⣿⣿⣍⡛⠻⠿⣿⣿⡆
⠀⠀⠀⢻⣷⣶⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢋⣼⣿⣿⠃⣼⣿⡿⣿⣿⣿⣿⣷⣦⣄⠹⡇
⠀⠀⠀⠈⣿⠿⢿⣿⣿⠿⠿⢛⣻⣿⣿⡝⠿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⣼⣿⣿⠀⣿⣿⣿⣷⣌⠛⢿⣷⡀
⠀⠀⠀⠀⠈⢷⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣄⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣦⠸
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⢿⣿⣿⠿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣶⣫⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣠⠖⡠⠚⠁⠀⣠⠶⠃⠤⠄
⠀⠀⠀⠀⠀⠀⢀⡔⠻⠀⠀⣼⠋⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⢀⣀⣰⣿⠋⠀⢀⣀⣽⣡⣤⣶⡶⠚⠻⠂
⠀⠀⠀⠀⠀⣠⣯⡉⠢⠀⣾⣿⡓⢶⠀⠀⠀⡀⠁⠰⠀⠀⠀⢠⣴⠶⠷⠛⠛⠛⠛⠙⢿⣙⠻⢦⣄⣀⣀⠈⠁
⠀⠀⠀⠀⣾⢍⠉⢷⣤⣾⠿⣉⠘⠇⠀⠀⠀⠁⠀⠈⠆⢀⣤⠋⠁⠀⠀⠀⢀⡀⡀⠀⠀⠙⠷⢦⣄⠉⠉⠙⠁
⠀⠀⠀⡴⠟⠦⢌⣻⠏⠉⠀⠀⠉⠀⠀⠀⢡⠀⠀⠀⣿⠿⠤⣤⣀⠀⠀⠀⠀⠙⠚⠶⣤⡀⠀⠀⠨⠳⠆⠀⠀⠀⣀⡄
⠀⠀⣸⠇⢄⡀⢸⡿⢦⣄⠀⠀⠀⠀⠀⠀⠀⡄⠀⣷⠇⠀⠀⠀⠈⠉⢓⠤⠤⠄⠀⠈⠀⠁⠀⠀⠀⠀⠀⢀⣴⠾⠁⢣
⠀⠀⡁⠀⠀⣿⣿⠋⠀⠈⠇⠀⠀⠀⠀⠀⠀⠰⣠⣏⣤⠤⠴⠦⠤⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣹⠟⠀⠀⠀⠡
⠀⠀⣷⣂⣤⡿⠿⠷⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠧
⠀⠀⢻⠙⣿⡿⢷⠄⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠟⠃
⠀⠀⠀⠙⠻⠧⠤⠓⠀⠀⠀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣶⠛⡁⠀⠀⠀⠀⢰⠀⠀⠀⠘⢿⢧⣽⠉⢢⡀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⠑⢄⠀⠀⢸⠀⠀⠀⠀⠈⣿⠿⣆⢠⠉⠐⠀⠀⣸⡟⠀⠀⠀⠀⠀⠰⡢⠄
⠀⠀⠀⠀⠔⠀⠀⠀⣀⣼⠛⠓⠤⡀⡀⠈⡇⠀⠘⠆⠀⠀⠀⠠⠋⠁⠈⣿⠀⠀⠸⠀⣿⠃⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠘⠒⠂⠒⠚⣻⠻⢦⡀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡄⣿⠀⠀⠀⢠⡤⠒⠂
⠀⠀⠀⠔⠂⠋⠉⠉⣿⠂⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⢿⡧⠀⣸⣿⠀⠀⠐⣿⡀
⠀⠀⠀⡆⠀⠀⠀⢰⡇⠀⠀⠈⢐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡄⠸⣇⠀⣿⡯⠀⠀⠀⠉⣳⠄
⠀⠀⠀⡛⠀⠀⢀⣼⠳⢤⣤⣤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣀⢻⣴⣿⣯⠀⠀⣀⠼⠋
⠀⠀⠀⠼⠒⠋⠁⡎⠀⠀⠒⠒⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢻⣿⣿⢂⣼⠃
⠀⠀⠀⠠⠀⡀⣀⡇⠀⠀⠀⠀⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣽⣿⣿⠇
⠀⠀⠀⠀⠙⠉⠀⣵⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠚⠋⠳⡀⠀⠠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣧⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⢡⠸⣆⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⣯⡙⠛⠓⠲⠷⣿⣆⢿⡇⠀⠀⢀⡤⠔⠚⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠛⠛⠛⠋⠛⠿⣧⠀⠰⠁⡀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡈⠳⣅⣠⢀⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣷⣦⣪⣭⣿⣷⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠙⡄⠀⠀⢀⣤⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠀⠀⢡⠀⣼⣟⡵⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⢦⣀⠈⣷⣿⣿⠷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⣇⣹⣿⣿⠏⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣠⣴⣾⠟⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠞⡕⡽⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠞⣉⣠⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠊⠁⢠⡾⣋⠔⣭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣄⠀⠀⠀⢀⡘⢀⣿⣰⣡⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠘⣿⢮⢳⡄⠀⡸⠷⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠛⣆⠀⠀⠀⠀⣿⡆⠀⠀⠀⣿⣶⣄⠀⠀⠀⠀⡿⠘⣾⠃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⠀⠀⡆⢯⢻⠀⠀⢠⡟⠃⠉⠳⣄⠀⠀⣦⣾⣿⢊⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠃⠀⠀⠀⢸⡆⠀⢧⠀⠈⢇⠀⠘⣵⣀⠡⠂⣿⠀⠀⣽⡿⣿⣿⠚⠋⠁⠈⠉⢛⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢦⡿⠀⠀⠀⠳⠀⣎⠀⠀⠀⠉⢻⣿⣭⠶⠋⠁⠀⠀⠈⠉⠂⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠹⣶⡿⠁⠀⠀⠀⠀⠐⣿⠀⢀⣀⠤⠾⠯⣤⣶⣶⣶⣤⣄⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠲⢿⡷⠶⠶⠶⠶⠶⠶⣿⠞⠉⠀⠀⠀⠀⠀⠙⠻⠿⢽⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⣦⠀⠀⠀⠀⣸⢯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢄⡻⣇⠀⠀⣀⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣇⡟⠀⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠀⢷⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋
⠀⠀⠀⠀⠘⣇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠐⢿⠗⠀⠀⠀⢠⡀
⠀⠀⠒⠒⠒⠚⠏⠀⠀⣴⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠈⠉⠁⠀⣠⡄⠀⠀⠀⠀⠙⣦⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠘⠇⣀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢀⡀⠉⠁⠀⠀⠀⠀⠀⠘⣧⣼⠃⠀⠀⠀⢀⣴⣶⣶⣶⣶⠂
⠀⠀⠀⠀⠲⣄⡀⠀⡿⠁⠀⠀⠀⠰⠞⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠐⠛⢁⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⣠⣽⣟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⡾⢀⡦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡞⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⡿⠃⠈⠉⠉⠉⠁
⠀⠀⠀⣰⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⣀⣀⣤⡀
⠀⠀⠐⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⢠⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡏
⠀⠀⠀⠀⣀⠀⢀⣠⣤⣀⣀⠀⠀⠀⢤⣶⡶⠀⠸⠃⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀⠀⠻⠿⠟
⠀⠀⠀⣾⣿⣦⠙⠿⣿⣿⣿⡿⠀⠀⣤⣤⣤⡄⠀⠀⠈⢡⡆⠀⠀⠀⠀⣤⡀⠀⠀⠁⢀⣴⡶⠿⠛⠀⠀⠀⠀⠘⣿⣿⣆⣤⣾⠟
⠀⠀⠀⢻⣿⣿⠀⠀⠈⠀⠁⠀⠀⠀⠀⠉⠉⠀⢠⣴⠆⠈⠸⣧⣴⠂⠀⠈⠻⡆⢀⠴⠋⠁⠀⠀⠀⠀⠀⣶⣦⣄⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠛⢻⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠠⠿⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⢳⢀⡠⠖⠛⠉⠁⠀⠀⠀⠘⠻⢿⣤⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢠⡆⠀⠀⠀⠀⠀⠸⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠉⠀⠀⠈⢛
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⡂⠀⠌⠀⠓⢶⠂
⠀⠀⠀⠀⠀⠀⠀⢧⠉⠙⠒⠚⠳⣄⠀⠀⡞⠀⠀⠀⠀⠀⠄⠀⠀⡇⠀⢀⢠⣀⠤⠤⣤⠴⡠⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⠐⠀⠀⠈⠓⠦⠇⠀⠀⢀⠀⠀⠀⠀⠈⢧⡴⠋⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢠⡀⠁⠐⠂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠠⠄⠀⠠⠀⠀⠀⢄⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⢠⠀⠀⠀⠀⠄⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⣀⣀⠤⠞⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⢇⡀
⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠔⠈⠂⠀⠆⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠉⠑⣲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⢌⡁⠀⠂⠀⠈⢀⣀⡀⣀⠀⠈⠀⠀⠠⠐⠀⡁⢀⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣗⣀⡴⠴⠚⠉⠀⣿⠀⠑⠢⠤⣤⣸⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧


⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣟⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣡⠞⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣈⡴⠀⣙⠳⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠢⡈⠇⡰⠃⠀⡠⠈⠛⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠢⣈⢻⠁⢀⠞⠁⢠⠊⠀⡉⠈⢣⡼⢧⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⡀⠑⢄⠈⠙⡶⠁⢀⡔⠁⢀⡜⠁⠀⢀⠄⠀⠹⠟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⠈⠲⣄⠙⠲⣼⣤⠊⠀⢀⠞⠀⠀⣠⠋⠀⠀⣰⠃⠈⠷⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠁⠲⢄⠀⠀⠙⠲⢄⣻⡀⡴⠋⠀⢀⠜⠁⠀⠀⡰⠃⠀⠀⡖⠀⢹⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⢁⠀⠀⠉⠒⠤⣀⠀⠈⢿⠀⠀⡰⠋⠀⠀⢀⡼⠁⠀⢀⠞⠀⠀⢀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠀⠀⠉⠒⠤⣀⠀⠀⠉⠒⠼⣧⠚⠀⠀⠀⣠⠎⠀⠀⣠⠎⠀⠀⢠⠏⠀⠈⠷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠢⣤⣀⠀⠀⠀⠙⠲⢤⣀⡀⠹⡆⠀⢀⡔⠁⠀⢀⡴⠁⠀⠀⣠⠋⠀⠀⣰⠂⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⠲⠤⣀⠀⠀⠈⠉⠓⠻⡔⠉⠀⠀⡠⠊⠀⠀⢀⡜⠁⠀⠀⡴⠁⠀⠀⠲⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠑⠢⠄⣀⡀⠀⠀⠉⠙⠒⠤⠄⣀⢱⡀⣠⠞⠀⠀⠀⡴⠋⠀⠀⢠⠎⠀⠀⠀⡰⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⢀⡀⠀⠀⠈⠁⠒⠤⢄⣀⠀⠀⠀⠈⠹⡅⠀⠀⣠⠊⠀⠀⢀⠔⠁⠀⠀⢀⡼⠃⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠉⠒⠢⠤⢀⡀⠀⠀⠉⠉⠒⠒⠤⢽⣤⠞⠁⠀⢀⡔⠃⠀⠀⠀⣠⠋⠀⠀⢠⠆⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠀⠀⡀⠀⠀⠀⠀⠉⠒⠂⢤⣄⣀⡀⠀⠹⡄⢀⡴⠋⠀⠀⠀⣠⠞⠁⠀⠀⡠⠂⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⠀⠉⠒⠠⠄⣀⡀⠀⠀⠀⠀⠈⠉⠙⠛⠻⣏⠀⠀⠀⣠⠞⠁⠀⠀⢀⡾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⡀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⢀⣀⣀⠙⣦⣠⠞⠁⠀⠀⠀⣠⠏⠀⠀⣠⠱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠁⠲⠤⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⡀⠀⠀⢀⡴⠃⠀⠀⣰⢃⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠲⠠⣤⡤⢀⣀⣻⣄⡴⠋⠀⠀⢀⣼⠗⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠁⠐⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⢠⠞⠁⡠⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⠀⢀⣀⡀⠀⠈⠉⠁⠛⠒⠂⠤⠤⣤⣨⣧⡴⠋⢀⡴⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⡀⠈⢉⠱⠒⠒⠠⠤⢀⣀⣀⠀⠈⣷⣴⠎⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⣀⠀⠀⠤⠤⣀⣀⣀⣀⣉⣉⣉⣱⣴⢋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠂⠀⣠⡤⢤⣤⣄⣈⣷⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣥⢤⣤⣤⣤⣤⠴⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠙⣿⠇⠀⢀⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀⠈⠀⢠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠻⣿⣷⡄⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⡈⠙⠛⡿⠛⠉⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⠀⠀⠁⢀⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠙⣿⣿⣧⢀⣴⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⣄⠀⠀⠉⠛⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣆⠀⠀⡐⠀⠀⢀⣠⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄⠀⠀⠈⠻⣿⣿⣿⡄⠐⣠⣶⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣇⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠚⢿⣿⣿⡆⠀⢠⠊⠀⠀⢀⣀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄⠀⠀⠀⠈⠻⢿⣿⣄⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⡙⠛⠛⠻⠿⢿⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⣿⣿⣿⣧⠠⠁⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠐⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⢁⣤⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠟⠻⠿⠛⠻⠿⢿⠻⢻⣻⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣶⣿⣿⣶⣿⣷⣾⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡝⢦⡀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⢏⠛⢦⠀⠀⠀⡞⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣒⠬⣧⠀⢳⡀⢸⣡⡇⢹⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠤⢄⣀⡀⠙⢝⠢⢷⡀⢇⢸⡵⡇⣼⠀⢀⡞⡽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢝⡶⢤⣻⢻⡲⣕⡦⣵⡼⠸⡀⡟⣸⢠⣏⡷⣧⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡙⠳⢷⡉⠢⡙⣄⠱⡗⠁⡼⣼⢣⢏⢔⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⣖⣢⣬⣕⣒⠬⠵⠷⢼⣆⡇⠀⣇⡷⣣⢃⣞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⠑⠲⢤⣉⡑⠲⢄⠈⢣⠀⡞⢠⣧⡾⣒⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢿⣚⠉⠓⠢⣷⣀⠻⡀⠸⣜⡩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣭⢭⡥⣌⡉⠛⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣗⣶⡲⡖⠛⠛⠒⣺⠟⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠀⠀⠀⠀⠑⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⠘⣆⠀⠀⠀⠀⢠⡾⠁⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣰⠏⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣧⢀⡾⠃⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⣼⠁⠀⠀⢠⣄⠀⠀⠀⠀⠀⢠⣿⣿⠃⣠⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⢻⣿⣶⣤⡀⠘⢿⡇⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⣾⡇⢀⡏⠀⠀⠀⣸⣿⡆⠀⠀⠀⠀⢸⣿⠃⠼⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣄⢀⣼⣿⠇⢀⣤⠀⠀⠀⠀⢠⡟⠀⣼⣿⡟⠀⢸⡇⠀⠀⠀⢿⣿⣷⠀⠀⠀⡠⠞⠻⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣿⠋⠀⣼⣿⠀⠀⠀⠀⣾⠁⢸⠿⠋⠀⠀⠸⡇⠀⠀⠀⠸⣿⣿⢀⡴⠋⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢰⣿⣿⠀⠀⠀⢸⣇⠖⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢻⣧⣎⣤⣶⣶⣶⣶⣦⣤⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⣀⠱⣼⣿⠇⠀⠀⠀⢸⡏⠀⠀⠀⢠⡄⠀⠀⢹⡆⠀⠀⠀⣰⠋⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⣏⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⣷⠀⠀⠀⢷⠀⠀⢰⠇⠀⢀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⢤⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠸⡇⠀⠀⢸⣿⣿⠀⠀⠀⠘⣧⠀⣼⣴⣾⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⣛⠿⠖⠚⠛⣻⡟⠓⠺⠷⣄⠀⢿⡀⠀⢸⣿⡟⠀⠀⢀⡀⠘⣇⣿⠉⠉⠀⠀⠀⠀⢀⣾⡆
⠀⠀⠀⠀⢀⣀⣠⣤⣀⣠⠚⠁⠀⠀⠀⣰⣿⡇⠀⠀⠀⠈⠳⣌⣧⠀⠈⢿⠁⣠⣾⣿⠃⠀⠸⣿⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⣠⣾⣿⣿⣿⠟⢿⣿⡀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠈⠙⣧⠀⢰⣸⣿⡿⠁⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠚⠛⠛⠉⠉⢀⡄⢸⣿⣷⠀⠀⠀⠀⠟⠁⠀⠀⣤⣀⡀⠀⠀⠀⠈⢷⣸⡋⠁⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠘⣿⣿⠁⣠⣴⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⣰⣿⡇⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣄⠀⠀⠈⠻⣧⠀⠀⠀⠀⢀⣴⡇⠀⢻⡀⠀⠀⠀⢿⡧⠈⠛⠛⠛⠉⠁
⠀⠀⠀⠀⢠⣿⣿⠇⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠀⠀⠀⠙⣧⠀⠀⠀⣼⣿⡇⠀⠘⣇⠀⠀⣠⠎⣀⣀⣀⣀
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡤⠀⠈⣻⡦⢤⣀⡀⠘⢧⡀⠀⣿⡟⠀⠀⠀⢻⡀⣰⠛⠻⠿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠋⠀⣰⣿⡿⠀⠀⠀⠉⠳⢌⣷⠀⡍⠀⠀⠀⠀⢸⣧⠇⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠛⠋⠉⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠹⣷⡇⠀⠀⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⢿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢹⣿⣆⠀⠀⢸⠀⠀⠀⠀⣼⠁⠀⠀⣠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣄⠀⠀⢀⣿⡇⠀⠀⠀⠀⢻⣿⡀⠀⣾⠀⠀⠀⢀⡟⠀⠀⢠⣿⡿⠁⠀⣀⣤⣴⣶⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⢸⣿⡇⠀⠀⠀⠀⠀⠙⡃⢠⡏⠀⠀⠀⣾⠁⢀⣀⡼⠯⠀⠀⠺⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⢸⣿⠇⠀⠀⠀⠀⠀⠀⣇⡾⠁⠀⠀⣼⡧⠞⢿⣷⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣴⡟⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⢀⡼⠋⠀⠀⠀⠈⠙⠛⠿⠿⠿⠦

⠀⣿⠀⢸⣠⣞⣉⣉⠁⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⣤⣤⠖⡿⡛⠁⠋⠀⠀⠀⣈⣡⣤⠾⡇⢸⡄⠘⠀⠀⠀⠀⢳⡈⠷⠚⣉⡀
⠀⣿⠀⡜⠋⣉⠀⠀⠀⠀⠀⠀⠀⠀⠛⡉⢉⣡⢾⡟⠀⣾⠟⠁⠀⠀⠀⠀⠉⠁⣴⢹⡇⠁⠀⠁⠀⠀⠀⠘⣆⠀⡻⠖⠋⠉
⡆⢿⣾⠿⠛⠋⠉⠉⠁⠀⠀⠠⠶⠟⣩⣿⠏⣱⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⠃⠈⠃⠀⠀⠀⢀⠀⠰⣆⠘⡷⠗⠛⠛⠁⠀⢀
⣷⣧⣴⡶⠖⠒⠒⠀⠀⣠⣴⢶⣶⠞⣱⠏⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⣆⠈⣇⠀⢻⣾⣤⣶⡶⠆⠀⠀⢿⡀
⠃⢀⣀⣀⣀⣀⠀⠀⣠⣥⠔⢻⠇⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢲⡀⢹⡄⠘⣤⣯⣤⣤⠤⠀⠀⠀⠀⣼⡇
⠿⠋⠉⠉⠉⠀⠐⢛⠛⡇⠀⠉⠀⠀⠀⢷⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⠀⠘⣇⠀⡹⠾⠤⠤⠤⠤⠀⠀⢸⣇⠀⢿⡿
⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠙⢄⡇⣠⠀⣤⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⢷⣀⡼⠛⠲⠶⠶⠶⠤⠀⠀⠀⢸⣿⢀⡞⣁
⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣙⣛⣓⣿⠀⣧⠀⡇⠀⡄⠀⠀⠈⣧⣹⣤⣖⠉⠙⠛⠳⠶⢢⡄⠀⠀⠀⣶⡄⣿⡾⠞⠋
⠁⠀⠀⠈⠻⣿⣦⠀⠀⡾⠀⢀⠀⠀⠀⠀⠀⢀⣉⣉⣑⣿⡌⣧⠀⡏⠀⠐⠉⠉⠑⠲⠍⠙⠓⠒⠀⠀⠀⠀⠀⢠⡀⡏⣧⣾⣴⡶⠋
⠀⠀⠀⠤⣤⣄⣉⡠⣸⡇⢀⡧⠀⠀⠀⠀⠀⠀⠀⠩⠭⠥⠼⠿⢄⣻⡄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢘⡟⠉⣁⣠⡄
⣠⡖⠀⠀⠀⠈⠉⠉⠛⠣⣸⠇⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠶⠒⠶⠿⠧⣜⣇⢰⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣼⠖⠋⠁
⡭⠖⠀⠀⠀⠉⠛⠛⠛⠛⠛⢦⠀⣿⠀⢰⡆⠀⠀⡄⠀⢀⠀⠀⠀⠶⠶⠶⠿⠲⣿⡀⢆⠀⠀⠀⠀⠀⠀⠀⢺⠀⣽⠁⢀⠀⠀⠀⡘
⠒⠋⠀⠀⠀⠀⠀⠰⠶⠶⠶⠾⠟⢯⡀⢸⡷⠀⠀⡇⠀⢸⡃⠀⣀⠀⠀⠀⠙⢋⣈⣉⣻⣧⣄⣧⠀⠀⠀⠀⠸⣶⡷⠞⠉⠀⠀⣄⠙
⠉⠀⠀⠀⠀⠀⢰⢂⠆⠀⣀⣤⣤⣤⣼⠾⣇⡀⠀⡇⠀⢾⡇⠀⢿⡆⠀⢸⡇⠀⠈⠀⠀⠰⠶⠟⠓⠦⠀⠀⢠⡏⠀⠀⠀⡀⢲⡈⠟
⠁⠀⠀⠀⢀⡀⣿⠇⠀⠈⠛⠛⠋⠉⣀⣤⣤⣽⠶⢇⡀⣾⠃⠀⣹⠇⠀⢸⡇⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠸⡄⠣⣠⣛⣛
⠀⠀⠀⠀⢸⣣⣧⠔⠀⠀⠀⠀⠐⠛⠛⠋⢉⣠⡤⠶⠛⠛⠢⣄⣿⠀⠀⣾⠇⠀⣿⠃⢠⠆⠀⠀⠀⠀⠀⢀⠘⡆⠸⡄⠹⠬⠭⠭⠍
⠀⠀⢀⡇⣸⢟⡁⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠁⣀⣤⣴⠶⠟⠉⢈⣙⡶⠯⣤⣀⣿⣀⣻⣀⣀⣀⠀⠀⠀⡈⣇⠙⣦⣜⣑⡒⠒⠒
⠀⣰⠸⣧⣿⠟⠁⠀⠀⣀⣀⠀⢀⣀⠀⠀⠀⠾⠛⠋⢀⣠⣴⡾⠟⠉⢀⣠⣴⠞⣡⡼⠋⠉⠁⠀⠀⠀⠀⠽⢾⣛⠤⠤⠉⠉⠉
⠀⢹⢀⣟⣴⠃⠀⠉⠙⢗⠲⢾⢭⣥⡰⢖⣀⣤⠤⠀⠿⠀⠀⠀⠀⡔⠝⠋⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣶⠊
⡄⢹⡾⠋⠁⠀⠀⠀⠀⠈⠀⠀⠑⡈⢯⡙⣖⠢⣴⡶⠒⣛⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢛⣋⣽⠟⠁⠏
⣇⢸⣡⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠃⠙⣏⠹⡏⢵⡖⠚⣛⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣭⣩⢾⡇⠜
⢻⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠹⡈⢷⠹⡯⣴⠆⠂⠦⠀⠀⠀⠀⠀⠀⢀⣈⣁⡤⢺⠃⡟⠀⠃
⣬⠔⠁⠀⠀⢠⡆⠀⠀⢀⣄⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠁⢻⡙⢶⣄⠀⠀⠀⠀⣘⣋⣉⡽⣿⠀⠏⠀⠂
⠁⠀⠀⢰⠀⢸⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠉⠂⣀⣈⣉⣭⡝⣯⠀⡏⠀⠀⠀⠀⠀⢀⣴⠆
⠀⣀⠀⢸⠀⣿⣠⣋⣡⣤⡄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡤⢤⡤⢺⠋⠀⠀⠀⠈⣉⣩⡤⣻⠃⡿⠀⠻⠀⠀⠀⢦⠀⠞⢃⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠃⠀⢶⡆⠀⠠⠄⢡⡄⢠⠖⢁⠀⡀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢰⠳
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⡈⠀⠘⠂⠀⠀⠈⠀⡖⠀⠀⠠⡈⠇⠀⠀⠀⠀⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡈⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠐⠀⠀⠐⠦⠤⠾⣤⡀
⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠥⢐⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⡧⠀⠀⠈⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣤⡴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡴⠁⠀⡄⠀⠁⣀
⠀⡤⠖⠀⠀⠀⠠⢦⠀⢀⢀⡠⢀⡴⠊⢡⠏⠀⠀⠐⠤⠤⠀⠤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠇⠀⢰⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢇⣠⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠆⠀⠁⠀⠘
⠈⠀⢢⣈⣁⣀⣀⣀⣹⣀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠋⠁⠀⡼⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢨
⡀⠀⠤⡤⠤⠀⠾⡀⠈⠁⢹⠀⠀⠉⠁⠐⠂⡂⠀⠀⠀⠐⠺⠦⠄⠀⣀⠄⠁⠀⠀⠀⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⡤⠤⠤⢄
⣈⣖⠈⠀⠀⠀⠀⠧⠀⠀⢸⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠀⢀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁
⠀⠉⠳⡄⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⣞⠁⠀⠀⠉⢁⡶⠚⠀⣠⣄⡼⡀⡀⠠⠄⠀⠀⠃⠀⠀⠀⡠
⢄⡀⠀⠸⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⡤⠤⣀⠀⠀⠳⣀⡤⠖⠉⢻⣀⠀⠀⠀⠸⠀⡤⠚⠁⢀⠄⠂⠑⢴⠇⠀⠀⠀⠀⠀⠰⠆
⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠊⠁⠀⠁⠀⠢⠊⠀⠀⣠⠊⠀⠀⠀⠈⢿⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢄⠀⠀⠀⠀⠀⠀⢠⡦⠒⠒⠢⣄⠀⠀⠀⠰⠦⡴⠁⠀⠀⠀⠀⠀⠈⠃⠀⠘⠤⠖⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢣⡀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠁⠤⠀⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣀⣄⣲⠂
⢠⡄⠀⠠⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣆⠀⢠⣿⣇⣀⡀
⡎⠁⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⡸⠊⠉⠙⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⡟⡻⠛⠉⠁⠈⠙⠣
⠀⠀⠀⠄⠀⠀⠀⠀⠃⣠⠂⠊⠀⠀⠀⠀⡄⠠⣄⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⢠⠏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁⠈⢀⣀⡠⠤⣀⣀
⠉⠀⠀⠀⠂⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠴⠋⠉⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⣴⠀⠀⠉⠣⣤⠤⠀⠀⠀⠀⠐⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⡆⢀⣔⣠⡟
⠀⠀⠀⠀⢨⠁⠀⠀⠀⢸⡆⢑⡠⠞⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠔⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠈⠙⢲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⣴⣄⠀⢂⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡚⢦⡀⢀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣇⠀⠀⠀⠀⠀⢹⣿⣇⣽⠏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣙⠛⢿⣴⡇⣀⢛⠇⠀⠀⠀⠀⠙⠢⡀⠀⠙⠲⣶⣾⡿⠔⠊⠀⠀⢸⣼⣿⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠛⠛⢧⣄⣹⣿⣿⡾⠀⡠⠂⠀⠀⠀⠀⠀⠉⠸⢿⡿⠿⣿⡁⠀⠀⠀⢘⣿⣏⡰⠂⠙⠋
⠀⠀⠀⠀⠀⠀⢀⡦⠤⠔⠒⠒⠲⠿⠿⠿⢿⣧⡀⡏⠀⣀⠀⠀⠀⠀⡤⠴⠛⠀⠀⠈⢳⣄⠀⢰⣾⣿⡙⣤⣉⠰⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠠⠤⠴⠚⢄⣀⡈⣃⣿⣢⠖⠉⠙⠂⠀⠙⡄⠀⠀⠀⠀⠀⠉⠀⠀⠛⠛⠧⠄⠈⠻⠲⠳⢤
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣷⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢄⠀⡀⡄⠀⠆⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡇⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⡄⢣⠀⠀⠀⡞⠀⠆
⠀⠀⠀⠀⠀⠀⢀⡄⢠⡆⠠⠔⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⡄⡄⣿⣣⡀⠀⠔⠁
⠀⠀⠀⡠⠄⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠑⠀⠀⠀⣠⣆⣷⣿⡻⠕⠁⠠⠊
⠀⠀⠀⠀⢀⡠⠄⠀⢸⡄⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠐⡀⠰⡀⠀⠀⠀⠀⠀⠁⠂⣶⣴⣶⣿⣿⣿⡿⠿⠂⠀⠀⠁
⠀⠀⠀⠴⠃⠀⣀⠀⠉⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⣦⣷⠁⡠⡀⠀⠀⠀⠉⣀⠀⠚⠛⠋⢙⠛⠻⠇⡀⠐⠂⠀⠀⢀
⠀⠀⠀⠁⠀⠸⠈⠀⠈⠀⠀⢤⣀⡠⠬⠃⠀⠀⠀⠰⠀⡠⠙⢿⢄⣾⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⢄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⣿⡎⣾⣿⡝⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢇⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠈⠢⢀⢬⣿⡽⠈⢿⣷⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡜⢁⠀⠀⢀⠀⠀⠀⠀⠀⠒⠂⢁⣿⣅⢳⠀⢮⢆⠔⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢱⣰⢁⠄
⠀⠀⠀⢀⠀⠐⠷⢤⣠⡀⠀⡟⠁⣠⣀⠀⢀⠄⠀⠈⠀⠀⢨⠭⠄⢳⡐⡡⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⣾⣷⡞⣠⠂
⠀⠀⠀⠀⠀⢀⢹⡆⢹⠘⣷⠟⢉⡄⣀⡀⠀⡀⠀⠀⠀⠁⠄⢒⣩⣅⣻⡾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠳⠿⣿⣦⣽⣶⣟⠛⠋⠀⠀⠀⡢⠀⠀⠀⠀⠀⠀⠈⠑⠊⠀⠀⠂⠀⠀⠀⠀⠀⠒⠒⠘⠾⣿⡿
⠀⠀⠁⠀⠀⠀⠀⠞⠉⠉⣹⣿⣀⠀⢀⣈⡒⠂⠀⠀⠀⠀⡤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠒⢄⣉⣀⡉⢀⣻⣷⣀⡄
⠀⠀⠀⠀⠀⠀⠀⢑⡀⠀⢽⠠⠚⡿⢿⠿⡏⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⣏
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠸⠀⠘⣁⣀⠀⡀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠒⠄⠰⣦⡀⣴⠎⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠀⠆
⠀⠀⠀⢀⡌⠀⠀⠀⠲⣾⣯⡦⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⡀⢀⡶
⠀⠀⠀⠘⣋⠉⠀⠀⢀⣼⣿⣻⣀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠒⠛⠒⠤⠌⡐⢤⣄⣿⣾⡥⠔⠀⡀⠒
⠀⠀⠀⠀⣯⣥⣄⠀⢀⣠⣿⡏⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠆⠀⠀⠠⠤⣤⣶⣬⣿⣿⣿⡏⠛
⠀⠀⠀⠀⠘⣁⡠⢀⠠⢀⠞⢻⡲⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠛⠒⢀⣀⣴⡶⢿⠛⢿⠃⠆⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠻⡾⣁⡀⠀⠀⢻⣷⡂⠀⠀⠀⢐⠀⠀⠀⠀⠀⢁⠠⢄⣤⡫⠹⠃⡃⠈⠀⢠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠁⠻⣴⡀⠀⠁⠈⡷⣦⠀⠀⠆⠀⠀⠀⠀⠀⠈⢠⠞⠉⠁⠀⢠⠁⡀⠀⣸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⢷⢀⠀⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠒⠃⢠⢷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⡀⠀⠀⠀⣀⠀⠤⢄⣀⡀
⠀⠀⠀⢀⠀⢐⣾⣷⡤⠀⢀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠁⣴⠾⠋⢁⠀⠘⢠⣤⣀⡅⠀⠀⠲⠚⠿⡄
⠀⠀⠀⢸⡇⠐⠀⠉⣷⣶⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠁⠐⣻⠛⠀⠀⠛⠀⠀⠀⠚⣿⣷⡁⠐⠀⠀⢠⣙
⠀⠀⠀⠀⢡⡄⢠⠀⠁⠘⣷⡂⠀⠀⢦⠀⠀⠀⠀⡸⠉⠀⣼⠏⠁⠀⠀⢨⣦⠔⠃⠀⠩⣿⢿⣅⠀⠀⢀⡛
⠀⠀⠀⠀⠀⠙⠇⠀⣼⠀⠈⠳⡄⠀⠪⡁⠀⠀⠀⡋⡡⢈⡟⠂⠀⢠⠄⠁⠙⠄⠓⠰⠁⠀⠀⠙⠷⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠠⣤⡇⠸⠀⢑⠂⠀⠀⠀⠛⠄⠨⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⠁⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢠⠄⠀⠀⠀⣴⣢⣿⣯⠀⡀⣞⣿⣟⡀⣀⣼⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⠂⠀⢸⣦
⠀⠀⠀⠰⠾⠀⠀⠀⢛⣾⣵⡿⠇⠰⣼⣿⣿⡟⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡵⠂⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠋⠀⠀⠀⠘⠉⠀⠀⠉⠿⠛⠛⠉⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠹⠀⠀⠀⠸⡿⠁
⠀⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⣀⠸⣷⣰⠆⠀⠀⠀⢠⠖⠈⠀⠀⢀⠘⣿⡆⠀⣠⡀⠀⢠⣀⠀⠀⢀
⠀⢨⣿⣷⣆⠀⠀⠀⠀⠀⠀⢹⣟⡺⣏⡴⢋⠀⠀⠡⠀⠀⠀⠀⣻⣇⠀⠀⣐⣬⡷⠄⠈⢿⡆⣼⣿
⠀⠰⣿⣿⡇⢀⣄⢀⣶⢀⣠⠰⣶⣭⣿⠼⠛⠁⠀⠀⠀⠀⠀⠀⠴⠶⠀⠀⠒⣿⡤⢖⠀⠀⠀⠛⠁
⠀⠀⠻⠗⠀⣨⣿⣿⣿⣾⣋⠀⠀⠀⢰⠀⠀⠀⢐⠀⢠⠀⠀⠀⢨⣷⢀⡲⠤⢸⠾⢟
⠀⠐⣷⣄⡀⠈⠩⡿⠿⡯⢉⡀⢠⡄⠀⠀⠂⠀⠀⠉⠀⠈⠀⠀⠈⠃⠀⢭⣍⡒⣪⡵⠀⠀⠀⠀⢀⡀
⠀⠀⠻⣿⡇⠀⠀⠀⠀⠀⠠⠽⠿⠀⢀⠀⠀⢰⡆⢟⠴⠂⠀⠀⠀⠀⣠⡆⠀⠉⠀⠀⠘⠃⠀⡀
⠀⠀⠀⠀⢠⡀⡇⣇⣤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠛⠈⠁⠀⠀⠀⠀⢰⠟⢠⣄⠐⠟⠀⣶⠀⠛⠃⠀⢀⡠
⠀⠀⠀⣜⢦⣣⣷⠯⠉⢀⣠⠦⢺⣧⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣾⣿⠀⢀⣐⠶⣫⠄⠀⡠⣽⣷⣅
⠀⠀⣙⣾⣗⡾⠋⠀⠀⠊⣡⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⠀⠸⡿⠀⣺⠅⠀⠬⢿⣿⠅
⠀⠀⠚⠉⠓⠉⠀⠀⠀⠈⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠳⠚⠀⠀⠉⠙⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣛⣻⣛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⢸⣿⠀⣾⢋⣤⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⢹⡇⠀⠰⠂⠀⠀⡿⠀⠃⠈⠉⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⢀⠀⠟⠀⠀⠀⠀⠀⣀⠀⠀⢀⣤⡀⠀⣠⡶⢶⣿⠁⠀⠠⠄⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⣠⡾⢉⠇⣰⡿⢠⠇⣴⡟⢠⣿⠇⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣧⡼⣾⡏⢻⡾⢾⡟⣠⢿⡿⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣤⣴⣶⡄⣼⡏⢀⡞⢠⣿⣠⠏⠀⠸⡷⠋⣾⡇⠀⢀⡾⠁⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠉⢁⡼⠃⠹⠷⢋⣠⣬⡿⠷⠦⣄⠀⠀⠀⣿⡇⢀⡾⠁⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⡶⠛⠉⠀⣴⠂⠀⡼⠀⠀⠀⠘⠳⠛⡁⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠸⣿⣄⠀⢀⣠⡾⠋⡀⠀⣼⠏⠀⠀⠀⢰⡏⣠⠞⣱⠆⢀⣾⠀⣰⠟⣹⠇⠀⠀⠛⣥⣴⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠈⠛⠛⠛⠁⠀⠀⠁⢸⡏⠀⠀⠀⢀⣿⢿⡅⢠⣿⡴⢻⡏⣸⣿⠞⠁⣠⠄⣀⠀⠰⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠀⠀⠀⠀⣠⣴⡾⠿⠿⠿⣷⣄⠀⠀⣼⡇⠀⡟⠀⠉⢀⣾⠷⣿⠁⠀⣰⠏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠗⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠈⠛⠀⠀⣿⠀⡼⠁⢀⡴⢻⡟⠀⠻⣶⠾⠋⠀⠀⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠻⠟⠁⣠⠏⠀⣾⠁⢠⣶⡆⠀⠀⠀⢰⡿⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⠀⢠⡏⢀⣼⠃⢠⡞⣯⡀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠛⢁⡀⠀⠉⠀⢁⣀⣠⣿⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⣠⡞⢹⠄⠀⣴⠋⡷⠀⢀⣾⠟⢩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⣿⢃⣾⣿⠀⣜⣠⣾⢿⣴⣃⣠⣿⠁⣠⢿⡟⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⢀⣼⠇⣼⡇⠘⡿⠉⢸⡇⣴⠋⠉⢸⣇⡴⠃⢸⡇⠀⣠⡟⠀⠀⠛⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣀⣀⣀⣤⣿⣿⠀⣿⣠⡞⠁⠀⠸⠟⠁⠀⠀⠀⠉⠀⠀⣼⠛⠞⠋⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⢁⣾⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡆⣸⠇⢀⣤⠄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢠⣦⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⣼⠀⣰⡇⢀⣾⠃⠀⣴⣿⣾⠿⠋⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⣼⢇⣼⣿⠀⣼⠇⣠⢞⣿⣿⡁⠀⠀⠀⢀⣾⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣾⡿⠀⠀⠀⢀⣀⡀⣿⠟⢸⣧⡾⠻⠶⠃⣸⠇⠘⢷⣤⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⣿⠷⠶⠿⠿⠟⠛⠁⠀⠀⠀⢉⡀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠀⠀⢀⡾⢹⡇⠙⠀⠀⠀⠰⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠋


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠆⢾⣋⣉⣛⣋⣩⣉⣉⣋⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⢁⣤⣤⣶⣶⠦⡄⣼⣿⠋⢱⡆⣠⣾⡿⣀⠀⣰⣿⠏⢠⣿⡟
⠀⠀⠀⠀⠀⠀⢠⣿⡟⢼⣿⠟⣻⣿⠃⠀⢸⣿⠃⢀⣿⢰⣿⠏⢠⣿⢰⣿⠏⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣴⣿⠃⠀⠀⢿⡟⢀⣼⠃⣿⡿⠀⣾⠇⢸⣿⣠⡟⢹⣿⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠘⣿⣇⣠⣾⠟⠁⠀⠀⠀⠈⠛⠚⠁⠀⢿⣧⡼⠋⠀⠀⠉⢡⣾⡆⠙⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⢀⣾⣿⠀⢀⣀⠀⠀⣤⣾⣿⣧⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⣾⣿⡏⣠⣿⡟⢀⣾⡿⠫⣿⡟⣠⣾⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⢸⣿⣟⣴⣿⣿⠁⣼⣿⠁⠀⠈⠀⣼⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⢈⠻⠛⠁⣿⡇⠀⢿⣇⣠⡴⠂⢠⡿⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣠⠏⠀⠀⠀⠘⠓⠚⠁⠈⠉⠀⠀⠘⠃⠀⠹⣿⣧⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⣂⡄⡀⡔⠒⡀⠀⢀⠀⡤⠔⠒⠠⡀⠀⠀⢀⠀⡤⠔⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡈⠀⡨⣜⢰⢅⠃⡰⢁⠊⠅⡄⠂⠄⠑⠀⡰⢡⠊⠁⡄⠢⡀⠱⠀⠀⠀⠀⠀⠀⠀⢸⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠡⡀⡁⡘⢹⠁⢀⠃⡌⠀⠃⡈⣒⠽⠀⢠⠁⡀⠀⢆⡈⣒⢽⠀⠇⠀⠀⢀⠠⠤⢤⣃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠊⠀⠀⠆⢘⠀⡇⠀⠀⠀⠀⠀⠀⣸⠀⠇⠀⠀⠀⠀⢸⠀⡀⢀⢊⠔⠁⡑⣦⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠇⠀⠀⠀⠸⠀⠠⠀⠀⠀⠀⡘⠀⠿⠀⢠⠀⠀⠀⠀⡌⠀⠃⠁⢌⠀⠀⠇⠁⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⢀⠄⠄⠀⢣⠀⠑⠤⠤⠊⠀⡔⠀⢣⠀⠑⠤⠄⠊⠀⡌⠀⡀⠐⡀⠀⠈⢻⠧⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠤⠊⠀⠀⠀⠑⠠⠀⠠⠄⠊⠀⠀⠀⠑⠠⠀⠠⠔⠊⠀⠀⠑⠢⠄⠁⠩⠊⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡄⠀⠀⠀⠀⠀⢈⠀⢡⠀⠠⠪⢪⢢⠀⠀⠀⡠⡊⠥⢈⠑⢄⠀⠈⡀⡅⢀⠆⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢤⠀⠀⠈⡄⢸⠀⢀⢠⠌⡘⠀⠀⠌⠔⠀⠄⡂⠂⢘⠀⠀⢠⢠⡌⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠈⠀⠀⠀⠀⡈⠀⠈⠎⢀⠁⠀⡘⠘⠀⠀⠈⠢⠤⠊⠀⠀⢠⠌⡞⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠆⠁⠀⠌⠀⡌⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡐⡡⡆⠃⠈⢢⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠧⠊⢀⠌⠈⠀⠆⠁⠰⠀⠀⠀⠀⠀⠠⠀⢰⠀⡇⠀⠀⠀⢃⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡇⣗⠂⠠⠤⠤⢲⡆⠀⠀⢀⠌⠀⠀⠀⢰⠐⠄⠑⠄⣀⠀⢠⠖⠀⠠⣀⡣⠁⠀⠈⢌⠒⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠜⠀⠈⠁⠒⠂⠁⠑⠤⠐⠁⠀⠀⠀⠐⠊⠀⠀⠑⠒⠒⠂⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣀⣤⣤⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣝⡻⢿⣿⣿⣿⣿⠟⢋⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣙⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣵⣾⣿⣿⣿⣿⣿⣿⣿⢸⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣟⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⡿⠃
⠈⢿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢰⣿⡿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣾⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠚⢛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣖⡒⠶⠶⣤⣀⣾⢁⠀⡀⠀⠀⢉⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡒⠂⠨⣀⡢⢍⡻⣿⣆⠞⡴⢀⠀⠀⠔⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⠀⣀⡒⠭⣓⠯⣺⢎⡞⠐⢁⠄⡀⠐⡿⠀⣀⣤⣤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⠀⠀⠬⣕⡪⢝⠊⢳⠀⠀⠀⠈⣠⡾⣡⠞⡙⠀⠁⠐⠀⠋⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣤⣤⣙⢦⣤⣙⡒⠍⠀⠀⢠⡆⡔⠀⣠⠏⡴⠏⠰⢁⠄⡠⢀⠀⠀⠀⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⠁⠀⠂⠈⢢⡙⢷⣮⡙⠺⣄⡀⢀⢸⡏⡴⡴⢃⠞⠁⠀⠀⢈⠜⡡⢃⢎⡄⡀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⠢⣈⠢⠁⠀⠀⠉⡳⢤⡙⠦⣙⣏⣞⡵⠳⢂⣞⡰⠤⠤⣞⢼⣱⢿⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠐⠬⣑⠮⣑⠀⠀⠠⣑⣊⣑⣉⣓⣻⣿⣿⣖⠻⡛⠉⠀⠀⢤⡲⢝⠮⡓⢍⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣦⣽⣶⣹⢦⢟⢀⠀⠁⠉⠱⢫⠞⢩⠿⡽⢍⡓⢦⣀⢀⠢⡑⢌⠢⡑⢌⠢⠁⢈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⡵⡱⢣⢣⢎⠄⡄⡀⣴⠋⣰⢏⠜⣷⠒⠝⢦⣌⠳⣵⡘⠂⠁⠉⠀⠀⡐⣌⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⠰⢡⢣⠏⡎⡜⣰⡼⢡⢞⠁⠀⠀⠺⣏⠣⠀⠈⠳⣮⡙⠦⣷⣼⣤⣦⠽⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⢀⠁⠀⠈⠀⠀⢠⣟⣵⠟⣩⢖⡄⡠⠀⣐⢀⠀⣀⠒⠤⢉⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣮⣴⣠⣥⡷⠞⠉⣾⠅⢈⠔⣫⢞⠔⡱⣯⡢⣕⠢⣉⠀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠰⡷⠂⠀⢊⠵⣫⢎⢜⣵⢝⠮⡑⠀⠀⠀⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⣿⡥⢀⠁⢊⠔⠁⣾⠿⣧⠁⠀⠀⠀⣈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡟⠀⠀⠀⠀⠀⠘⢶⣥⣔⣡⡴⠞⠁⠀⠈⠛⠶⠾⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠰⡦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠘⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡜⠉⠀⠁⠀⠀⢠⣄⠀⠠⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⣦⡀⠀⠀⠁⠹⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠘⣿⣷⣀⣷⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡚⠿⣾⣁⣴⢀⢸⢿⠀⠀⠀⠀⠀⠰⢄⠀⠀⠘⠦⣤⣤⣾⣾⠤⠒⠀⠀⠱⣄⣿⣿⣿⣩⣈
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠾⣗⣀⠛⣿⣾⣯⡞⠀⢀⠄⠀⠀⠀⠀⠀⠐⠒⣲⣾⣿⣿⣷⠀⠀⠀⠀⠐⣿⡿⠉⡭⠛⠻⠃
⠀⠀⠀⠀⠀⠀⡔⣁⣀⣤⢤⣬⣽⣿⣿⣿⣿⡇⢰⡇⠀⠀⠀⠀⠂⠀⢀⣀⣠⠟⠁⠈⠻⢦⠀⠀⢀⣤⣿⡟⢏⠒⠆⣄
⠀⠀⠀⠀⠀⠘⠉⠥⢄⡀⠀⢀⣀⣠⣇⠀⠙⠻⣜⣧⠀⣺⠤⠠⡄⠀⠩⢌⠁⠀⠀⠀⠀⠘⠓⠀⠈⢿⣿⣷⡙⠛⠳⣄⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⠉⠀⠈⠑⠶⡴⣶⣿⡟⠁⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠙⠮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠆⠘⠀⠀⢈⡜⠀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠙⠃⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⡘⠃⢂⡆⠈⠀⠈⠠
⠀⠀⠀⠀⠀⠐⠆⠸⡇⠐⠋⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢈⠃⠀⢸⣷⡀⢀⠀⠁⡀
⠀⠠⠒⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡈⠀⠀⠀⠠⣷⣿⣿⣯⣑⡅⠀⠊
⠀⠀⠀⢠⠖⠀⠀⢸⣆⠀⣀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣈⠀⡀⡤⠀⠀⠀⠀⠀⢀⣈⠀⠿⢮⣿⣾⣿⣿⣿⣏⠛
⠀⠰⠎⠁⢀⢤⠀⢉⠻⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠚⣧⣷⠃⢠⣢⢀⠀⠀⠀⠀⡀⠀⠋⠙⠉⠈⠉⠋⠓⢤⠁⠀⠀⠀⠀⣠
⠀⠈⠀⠀⡷⠁⠀⠂⠀⠀⠢⣀⡠⠤⠉⠁⠀⠀⠀⢐⠀⡰⠈⢿⡔⣲⣿⡟⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⢄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⣿⡮⢸⣽⡿⢁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠈⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠑⠆⣀⢬⣿⡼⠌⢻⣿⡟⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⢠⠃
⠀⠀⠀⠀⠀⠄⠀⠀⡘⢁⠀⠀⠀⡀⠀⠀⠀⠀⠐⠢⠀⠨⣽⣇⠱⡈⢱⡵⢂⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡼⢀
⠀⠀⠀⠀⠰⡦⡁⣀⡁⠀⡶⠊⠀⣄⡀⠀⣀⠀⠀⠐⠂⠀⢂⣻⡃⠱⣎⢞⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢰⣿⣧⡞⢀⠔
⠀⠀⠉⠀⠀⢷⡉⢋⠱⣦⡷⠖⢈⠉⠁⠀⠁⠀⠀⠀⠀⠂⣁⠤⢖⣋⠹⣬⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣹⣿⡴⠁
⠀⠀⠀⠀⠱⣮⣷⣿⣄⣹⣥⣶⠿⡾⠋⠀⠠⠀⡄⠀⠀⠀⠀⠈⠉⠛⠛⠩⠄⠀⠀⠤⢀⡀⠀⠀⠀⠤⢤⠀⣬⣿⣿⣏
⠐⠀⠀⠀⠀⢀⡼⠟⠛⢻⣿⠃⠁⠀⠀⢠⣤⠐⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⢄⠀⠐⠂⠰⠄⢝⣿⡇
⠀⠀⠀⠀⠀⠐⣀⠀⠠⢾⣛⣻⣶⣶⣶⣷⡀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠶⠖⠻⢿⡖⠁
⠀⠀⠀⠀⠀⠀⢱⠖⠈⢸⠀⢠⠞⠙⠋⠙⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣶⣴⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤
⣿⣿⣥⣥⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⡑⠶⢄⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢂⠀⠂⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣱⠛⡊⠤⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡘⣄⣓⡀⠃⣤⠀⠀⠀⠀⣠⠤⠄⣤⣼⣧⣤⠀⠀⠘⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣯⡮⣝⡶⡅⠢⠀⠀⠀⠈⠳⣦⣤⡀⢛⣿⣿⣿⡃⢀⣤⠄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⣫⡓⢨⡚⡷⠀⠀⠀⠀⠈⠉⠁⢀⠃⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣯⣤⣿⢇⠶⢲⠄⠀⠀⠀⢀⠠⡄⣀⠀⣄⢀⠄⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣯⣻⡗⣯⡀⠀⠀⠀⠀⠀⠐⠫⡆⢸⣀⡔⠀⣠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⠿⠟⠁⠀⠀⠀⣶⣆⠀⡀⢠⣾⡟⢀⣘⡟⠀⠀⠠⡒⠄
⠀⠀⠀⢠⣀⡀⠡⠀⣥⡀⣤⠀⠀⠀⠀⠀⠀⢦⠈⠋⠙⣇⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⣿⣿⣿⣥⠀⠴⠞⠻⢿⡿⠃
⠀⠀⠀⠈⣿⣿⣦⣒⡈⠼⠤⣱⠀⠀⡄⢀⣘⣷⣀⣤⡀⠛⠀⠀⢀⣤⣤⣤⣀⠜⢹⣿⣿⣿⣿⣿⣦⠀⢲⣄
⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⠿⣄⣾⡷⠛⠚⠻⣿⣿⣿⡗⠂⠀⣿⣿⣿⣾⣧⣴⠋⢀⡼⠛⢿⡿⠿⢧⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠰⢶⣶⡄⠈⡏⠙⠳⠀⣠⣿⣿⣿⣿⣿⣿⠰⠋⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠘⠻⢻⡇⠹⡄⠀⠘⠛⠛⢿⣿⣿⣿⣿⣷⣀⡢⢄⠀⠀⠀⠀⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣧⠀⣼⣗⣠⣧⣀⠀⠄⣀⡄⢻⣿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣷⣿⣿⣿⣿⡥⠾⠘⠉⠀⢠⣿⣿⣿⣿⣦⣲⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠁⠀⢈⣷⣶⣮⣿⣿⣿⣿⣤⡀⠀⠀⠀⢀⣸⡟⠛⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣾⣿⠛⠁⢳⠀⠉⠛⣿⠥⠤⢴⡾⠋⠀⠀⠀⠈⠉⠉⠀⢀⣀⣠⡤⡴⢼⠓⠣⢿⣐⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⡏⣩⠏⠁⠀⠀⠀⢣⠀⠀⢸⠀⠀⣨⡇⠀⠀⠀⠀⠀⠀⢀⠀⡞⠧⣞⢡⣇⣾⣯⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠉⠁⠀⠀⠀⠀⠀⠀⠳⡀⢺⠀⣰⢹⣿⣀⣤⠀⠀⠀⠀⣍⢷⣴⣺⣄⣳⣿⣿⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⢠⣇⡜⢸⣿⣿⠀⣠⡴⠚⠛⠿⠿⠿⠟⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡏⠀⣉⣉⣀⠀⠈
⠀⠀⠀⠀⠐⣦⣄⣀⣀⠀⠀⠀⠀⠀⢀⠈⣿⣷⣶⣤⣹⣶⣿⣾⣿⠟⠛⠉⢉⣠⣘⡀
⠀⠀⠀⠀⠀⠀⠑⠈⠛⢻⣿⡀⠀⣀⣉⣀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠗⣤⣼⣾⡿⢋⡿
⠀⠀⠠⢤⣠⣀⡔⡤⢠⣿⠟⠁⢾⡿⣻⣿⣿⢿⣿⣿⡿⠛⡿⠏⠀⢹⡉⣿⠑⠒⠉
⠀⠀⠀⠒⠛⠛⠶⠛⠋⠀⠀⠀⠀⠙⠛⠋⠁⣾⣿⣧⡀⠀⠁⠀⠀⢜⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⢀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⢣⡄⠀⠐⠈⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃⠂⠀⠀⠀⠀⠀⠀⢻⠟⢡⠏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠏⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⢿⣟⠛⡜⠛⣻⣷⠀⠀⠀⠀⢠⣴⣴⣄⣆⡆⣰⣶⣤⣦⡄⠀⠀⠀⢠⣄⣤⣀⣶⣀⣤⣀⣤
⠀⠀⠀⠀⣯⠝⠀⣷⠀⣻⣿⠀⠀⠀⠀⢸⣧⢿⢿⣿⣷⣿⣽⡟⣯⡄⠀⠀⠨⣿⠀⠐⠒⣿⠒⠂⠀⣺⡇
⠀⠀⠀⠀⣿⣋⠁⣷⠈⣿⣿⠀⠀⠀⠀⢸⣷⡻⡳⣽⣿⣯⢪⢎⣾⠃⠀⠀⠀⣿⠈⠁⠤⣿⠏⠉⠁⣸⡇
⠀⠀⠀⠀⣻⣿⣄⣿⢰⣿⣿⠀⠀⠀⠀⢰⣱⡱⣝⣿⣿⣷⣷⢫⣮⡄⠀⠀⠨⣧⠀⠠⠬⣿⡧⠤⠀⢼⡇
⠀⠀⠀⠀⣿⠿⠀⣿⠘⣿⣿⠀⠀⠀⠀⢸⡟⣿⣮⡻⣿⢟⣵⡿⣻⡄⠀⠀⠰⣯⠁⠩⠈⣿⡭⠍⠈⣹⡇
⠀⠀⠀⠀⣿⡣⠀⡿⢰⣿⡟⠀⠀⠀⠀⠈⣯⢮⣙⣿⣿⣿⣫⢞⡽⡀⠀⠀⠐⣷⡒⢚⣓⣿⣛⣛⠛⣺⡇
⠀⠀⠀⠀⣳⡾⠀⡏⠀⢯⣿⠀⠀⠀⠀⠸⣌⣛⣻⣿⣿⣿⣗⣻⣵⠇⠀⠀⠠⡿⠀⠀⠀⣿⢂⣀⠀⢽⡅
⠀⠀⠀⠀⠛⠛⠛⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠠⠀⢀⡀⠀⡀
⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠠⠋⠀⠀⢠⡀⠀⠉⠀⠀⠀⠀⢀⡄⠀⠀⠉⠀⠊⠀⠀⠀⠀⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⣗⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⠻⡉⠀⣸
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡄⠀⠄⠀⠀⠀⠀⠀⠁⠂⠀⠀⢶⠃⠀⡇⠀⢀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠐⢀⢐⠀⠀⠀⣴⡦⠀⠀⠸⠂⠀⠇⠀⠈⠀⠀⠀⢈⣁⠄⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡸⢬⠂⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠾⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⡲⠀⠀⠀⢀⣄⡀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢤⠀⢰⣆⣄⠄⡖⢰⣦⠀⡴⠄⡖⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠘⠁⠀⠀⠀⠸⣿⣷⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠸⡤⠸⠇⠹⠀⠯⠸⠇⠀⢬⠇⢧⠘⠀⠀⠀⠀⠀⠰⠛⣿⣶⣤⡤⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⣶⣦⣌⣻⠈⠠⠂⠛⡇⠀⠀⠀⠀⠀⠧⠷⠷⠇⠸⠿⠸⠾⠾⠇⠆⠅⠇⠟⠇⠿⠉⠇⠀⠀⠀⠴⣶⣿⣿⣿⡷⢦⡴⠂⠀⡇
⠀⠀⠀⠀⠀⠉⠻⢅⡿⠟⢶⡶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⢹⡿⠁⠀⠇⠀⠀⠁
⠀⠀⠀⡄⠀⠀⠀⠄⠀⠀⠼⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⣄⠋⡀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⣀⡀⠙⣿⡇⣼⠟⠀⠀⠀⠀⣤⠒⠄
⠀⠀⠈⠘⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣹⣿⣦⡬⣿⣷⠚⠒⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠻⣧⣤⣶⣄⣀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⣿⣿⢁⠏⢳⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣷⣄⣾⠟⢻⡇⠀⠉⠋⠁⠀⠁⠊⠁⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠁⠀⣼⣿⡟⠃⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⣿⠣⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠀⠀⠀⠀⠀⠐⠿⠄⠀⠀⠀⠀⠀⢠⡀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠋⠽⣦⡀⠓⣆⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣦⣹⣿⣷⣀
⠀⠀⠀⠀⠀⠄⠀⣦⠀⣄⠀⠀⠀⠀⠀⠀⣸⠷⣾⠏⠀⢰⡾⠷⣄⠀⢰⡿⠿⠻⣶⠀⠀⠀⠀⠀⠀⠀⢀⣶⡏⠀⠀⠀⠀⣸⠉⠉⡿⢋⣿⣷⣄⡀
⠀⠀⢰⣾⣙⣦⣠⡟⣰⠿⣄⣀⣀⠀⠀⠀⠁⠀⠀⠀⠀⢘⣷⣤⣾⣷⣏⡒⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⡿⠉
⠀⠀⠈⠉⠉⠉⣙⣷⣷⣞⠋⠉⠁⠀⠀⠀⣀⠀⠀⠀⠀⠸⢯⠿⠋⢸⣯⠛⠻⡦⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣄⣈⡁
⠀⠀⠀⠀⠀⠘⢿⣿⡁⠀⠠⠀⠀⠀⠀⠈⠙⣷⠊⢿⠀⠀⠀⠀⠀⠀⠹⠆⠀⠁⠀⠀⠀⠘⣿⢀⡄⠀⠀⠀⢀⡁⠃⠀⠀⠀⠀⠀⡏⡷
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠤⢤⡀⣴⣿⣦⣊⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣦⡻⣿⢏⣶⣀⣴⡀⠀⠀⠀⠀⠀⠀⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⡅⢿⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠟⠚⣿⢟⣿⣿⢿⣟⡿⠃⠀⠀⠀⠀⠀⠈⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⢿⡿⠄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠷⠛⢿⠋⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠔⠻⢼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠐⠃⠀⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⢄⠀⢀⠎⠀⠀⠈⡤⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⣄⠀⢀⣾⣽⣶⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠱⡘⠀⠀⠐⠄⠀⠀⠎⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡰⠋⠉⠐⠀⠀⠀⠹⡄⠀⠀⠀⠀⣤⡤⠃⠀⠀⠀⠀⢠⣾⣻⡿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢧⠠⡀⠀⠀⠀⢈⣑⠴⡠⠔⠺⠿⠅⠤⢄⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⣿⣿⣯⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠛⠋⠉⣸⠓⠄⠀⠀⠀⠀⠄⠀⠘⡄⠀⠀⠀⠙⠛⠛⠛⠋⠉⣰⠟⣿⡻⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠠⡀⠀⠀⠀⢻⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⣿⣷⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢆⣂⣀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠂⠔⠉⠉⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣶⣦⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄⠀⠀⢤⠅⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⣿⣿⣿⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢀⢣⡀⠀⠁⠀⢰⡿⠖⠉⠀⠀⢱⠀⠀⠀⢀⣠⣤⣤⣘⣿⣿⣿⣷⣿⣿⣷⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡀⠀⠙⢒⢄⣐⠝⠀⠀⠤⠱⠶⣟⠀⠀⠀⣾⣵⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠈⢣⠔⠒⠀⠀⠉⢠⠴⠶⣤⡀⠀⠀⠀⠀⣽⠀⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⣠⣴⠟⠀⠀⠈⢿⣆⠄⢰⠾⠟⠀⠀⠀⠰⣷⣾⣿⣿⣿⣿⣿⣿⡻⣿⣿⡿⢿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⢋⠃⠀⠀⡀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢹⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢡⡀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠁⠀⠉⢵⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠕⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⢀⠀⢠⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠠⣀⡄
⠀⠀⠸⡆⢸⡆⠀⠀⢠⡇⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠈
⠀⠀⠀⢷⠸⡇⠑⠀⢸⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠰⠀⠄
⠀⠀⠀⠘⠀⠇⢀⡇⠈⢸⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣧⢲⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠛⠳⣀⠃
⠀⠀⠀⠀⡰⠀⠀⢠⠀⠸⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠈⡇⠈⠠⡆
⠀⠀⠀⡀⢃⠀⢰⡸⡄⢻⠇⠈⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠲⡿⢀⣀⠄
⠀⠀⠀⠇⡞⢰⠀⠷⣧⠋⡆⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⣳⡀
⠀⠀⠀⢸⠃⠸⠀⠀⣿⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠹⠉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⡿⠘⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣧⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣤⡰⠦⠀⣙⡛⠀⠀⠀⠀⠋⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠖⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣾⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢴⣦⣃⡀
⠂⡐⠀⠈⠀⠀⠈⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⣿⡛⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⠀⡈
⠀⠀⠀⠀⠀⢤⢠⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⡞⠅
⠀⠀⠀⠀⠀⠈⢀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠇⢼
⠀⠀⠀⠀⠀⠢⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡽⣾⣉⣈⣆
⠀⠀⠀⠀⠀⢀⡀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⡾⠟
⠀⠀⠀⠀⠀⠈⠁⠀⢡⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠆⠀⣿⣟
⠀⠀⠀⠀⠀⠀⠐⠢⣬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢾⣶⣤⢏⣶
⣤⣤⣤⣤⣤⣤⣤⣤⣴⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣬⣯⣤⣿⣬
⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣵⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⣿⣿⣿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⣶⣋⣿⣿⣯⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡟⣸⣺⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣻⣿⣿⣿⡿⠟⠉⠙⣿⠉⡅⡈⠋⠻⢭⣽⡆
⠀⢠⣿⣿⣿⣿⣿⣿⣥⣿⡿⡝⢿⣾⣯⢻⣯⠝⣦⣷⡽⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⠾⡴⠂⠂⠉⠊⠀⡡⢎⠡⠀⠈⠋⠄
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣉⣸⣇⣻⡎⢟⣷⠾⣿⠿⣿⣿⣿⢋⣾⣿⣛⣿⣁⣳⣋⡈⣀⢀⡠⠁⢁⠀⠀⠀⠄⠀⠈⠉⢠
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣟⣿⡿⣶⣷⣶⡶⠄⣄⠀⡈⣗
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣆⢺⣧⣽⣿⣿⣷⣤⠄⠀⢘⡇
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣛⡛⢻⡿⠛⠃⠀⢀⣾⠃
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣯⡟⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡉⠉⠉⠻⠋⠁⢁⣈⣈⣦⣤⠶⠛⠁
⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠛⠛⠋⠉⠁⢀⣴⣿⠋⣼⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡙⠁⠈⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⠟⠁⢀⣯⣿⡟⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣪⠞⠉⠀⠀⠀⣿⣿⣷⡿⠝⣿⣿⣿⡟⢭⣽⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⣻⣟⣦⣿⣟⣿⣿⣿⡿⠿⢿⣽⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢻⢟⢻⣿⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣻⣿⣻⣿⣾⣿⣿⣿⣷⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⢩⣪⣿⣿⣿⣿⣽⣿⣭⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣟⣿⣿⣿⣿⣽⣿⡿⠿⢿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⢟⣻⣙⣯⣍⠛⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⠋⠃⠀⠘⠛⣛⠛⠋⠀⠀⠀⣠⣄⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣿⣷⠄⣀⣴⣾⠿⠿⠿⠿⠿⣷⣦⣈⠻⣿⣆⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣾⣿⣯⣿⣿⢁⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣊⠁⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠚⣿⣿⣿⣻⣿⢯⡍⢉⣉⡉⠛⠒⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⢻⣿⡦⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡄⠀⠀⣿⣿⣷⣺⣇⣻⡆⣏⠀⣹⢰⠟⠳⡄⣠⣌⡉⠓⠶⢤⣀⡀⠀⣿⣯⣺⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣸⡿⠛⣿⣿⣷⣤⡤⣌⣉⣁⡘⠦⠾⢡⡏⣀⡗⠀⢠⠆⠈⠙⠳⢿⣟⣺⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣋⡀⠙⣿⠁⢠⣿⣟⡺⠿⢠⣿⡏⠉⠉⠛⠓⠦⣌⣉⡀⠠⢯⡀⢠⡟⢀⡆⢈⡙⠧⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⡴⠋⠀⢸⣿⡒⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⢤⣘⠲⠟⣴⠋⠙⠆⣈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⣾⣿⢭⣀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⣌⡓⠃⡾⣷⠒⠂⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣹⣿⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠧⣿⡟⢳⣤⣙⠀⠀⣰⠏⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⢾⣿⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠋⠁⠀⢠⣿⠃⠙⢳⡞⠁⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⢣⣅⠀⣠⣿⠃⠀⢀⣾⣤⣤⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣤⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⣲⣿⠏⣴⣿⠁⠀⠀⢸⡟⣫⡃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⠉⠀⠀⠀⠀⠀⢀⣴⣿⡿⠍⠈⠻⢃⣾⡿⠁⠀⠀⠀⠀⣿⠋⠉⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣯⡀⠀⠰⣿⣿⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠩⠙⠀⡸⢉⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡄⢀⠃⢠⠰⠀⠀⢠⠈⠀⠀⠀⠂⠀⠀⠀⢰⡄⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠞⠈⡆⠀⠒⡾⠤⠒⣁⡀⠀⠀⠀⠀⠀⠀⠘⢻⡏⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠃⠈⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠉⠢⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠄⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⣾⢐⣸⢄
⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠀⠤⠐⢄⠀⠀⠀⠀⠀⠀⠀⠙⡿⠈
⠀⠀⠀⠀⠀⠀⠈⠒⢠⠔⢠⠀⠀⠀⠀⠀⠈⠀⢀⣁⠐⠁⠂⠀⠂⠀⠈⠀⠀⠀⠰⣮⡸⢣⢊⡷⡦⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠈⠃⠀⠀⠈⠢⠄⠃⡴⠀⢧⠧⢵⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢢⡁⠅⡠⣺⠇⡆⡄⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠄⠘⢂⠚⢽⡠⠤⠴⠀⠀⠀⠀⠘⠠⠆⠹⣪⣤⡃⠀⠀⠬⡄⠀⠀⠀⠺⠃⠀⠁⠰⠆
⠀⠀⠀⠠⠀⠀⠀⠄⠅⡀⠀⠀⠄⡀⠀⠀⠈⠡⠖⠋⣈⠟⣿⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⡀⠀⠶
⠀⠀⠀⠀⠠⠀⠀⠘⠀⡉⠀⠀⠄⠁⠀⠀⠀⠀⠀⡒⠂⠀⡇⠈⡀⢍⠁⠀⠀⠀⠀⠀⠀⡤⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠎


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⠀⠀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠈⡇⠀⠀⢸⠀⠀⢸⠀⠀⣰⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠘⡆⠀⡄⢡⠀⢀⠈⢰⠀⡆⠰⠀⠇⠀⠀⢠⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠘⢆⠀⢦⠐⠀⠱⠈⠀⠸⠀⠸⠀⠁⠇⣠⣤⠚⠠⠃⢀⠀⣠⠎
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠢⡀⠑⣄⠂⠀⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⡟⢠⡇⠁⡰⢃⠜⠁⡀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠑⠦⣀⠈⠢⡈⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⣀⣿⡷⠛⠀⠈⠀⠀⡀⠊⡠⢚⡁⠀⠀⠀⡀
⠀⠀⠀⠀⣀⠀⠀⠀⢌⡑⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡇⠀⠀⠀⠀⢰⠏⠀⡠⢸⡟⠠⠖⠋
⠀⠀⠀⠀⠀⠉⠓⠢⠄⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣇⠀⠀⠀⣠⠟⠀⠉⠀⢸⡇⡠⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡞⢹⣿⢻⠟⢦⣠⠼⠋⠀⠀⠀⠂⢸⡇⠀⠂⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣤⠀⠀⢠⣶⣤⠞⣹⣿⠀⣿⠘⠻⠟⢰⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠒⠒⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⣿⡄⠀⣾⣿⠈⡇⠸⢿⠴⠋⠀⠀⠀⢸⣧⠀⣀⠀⠀⠀⠀⠀⢸⠇⣀⣈⠁
⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⣿⡇⢰⣿⣷⣼⢿⣿⣰⡇⠀⠀⠀⠀⢀⣠⡀⢸⣿⣼⣿⠃⠀⠀⠀⡆⢠⡀⠠⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠈⢿⣧⡞⠘⣿⣇⠀⢬⡉⠀⠀⠀⠀⠀⣾⡟⠷⣸⣿⣿⡃⠀⠀⠀⢸⡇⠘⠃⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣿⣿⠀⢸⣷⠀⣶⡄⢰⡀⣿⡇⢀⡟⣿⡏⢿⡄⠀⠀⡾⠠⢄⣀⠁⠒⠲⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⢠⡏⢸⣿⡆⢸⣿⠀⢿⣇⢸⡇⣼⠛⠊⠀⠈⠋⠀⠻⣆⣼⢃⠀⠀⠌⣉
⠀⠀⠀⠀⠀⠀⣀⡤⠒⡡⠖⠁⡾⢀⠈⣿⡇⠸⣿⡇⣼⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠈⠙⠲⠄
⠀⠀⠀⠀⠀⠊⠁⠀⠈⢀⡤⢰⡇⠀⢠⣿⠁⠀⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡑⢦⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⢸⡇⠆⣼⠃⠀⢠⠈⠛⠁⠀⠀⠀⠀⢀⠀⠀⢠⠀⢀⠱⡄⠘⢦⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠛⠚⠁⡰⢀⠇⠀⠀⡆⠀⠀⡆⠀⠀⡆⠀⡄⢧⠀⢣⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣰⠁⠀⠀⡜⠀⠁⢸⠀⠃⠀⡇⠁⠀⢱⡀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢰⠇⠀⠀⢸⠀⠀⠀⢷⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⣤⠸⠀⠀⠀⣶⣦⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠃⠀⠀⠀⠀⣠⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⢀⣠⣤⡖⣾⠐⠶⠦
⠀⢀⠊⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠛⠋⠁⢡⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒
⣰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠈⡠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠑⠂⠐⠂
⡿⠀⡇⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠂⠉⠉
⡀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠓
⠐⠀⢠⡄⠢⣀⠀⠋⠉⠀⠀⠐⠒⠲⠤⠖⠾⠁⠀⠀⠀⠀⡀⠀⢈⡀⢠⠆⠀⢠⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣬⣯⣔⠀⠉⣀⣀⣀⣒⣚⣻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠒⠙⠹
⠀⠀⠈⠂⢿⠿⣿⣿⣿⣿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡀⠀⠄⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⢀⠀⠀⠙⠻⣟⣙⠙⠕⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⠀⠀⠀⡁⠀⢀⠀⠹⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠴⠄⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡄⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⣴⣦⡐⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠁
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣤⠀⡀
⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣦⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢧⠈⢻⢣⡔
⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠠⡀⠀⠄⠀⠀⠀⠀⠿⠛⠃⠀⡠⠴⠈⠃⠀⠀⠀⠀⢄⠈⠄⡀⠀⠇⠀⠀⠀⠰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⣠⢾⠡⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠹⢸⠈⠀⠀⠀⠀⠈⠀⠆
⠀⠀⠀⠀⠀⠀⢀⠀⣯⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢼⣳⣶⣿⡿⠟⡳⠶⠶⠦⢀⣀⡀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢠⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡴⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠘⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⡶⠄⡀⠀⠀⠀⠀⠰
⠀⠈⠑⠠⢤⡴⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠵⣄⠁
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⡓⠀⠀⠀⠀⠀⠀⢀⠄⠔⠁⢀⠀⠐⡠⠞⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡀⠸⠷⠁⠀⠀⠀⠀⠀⠀⠌⠀⢠⢋⢆⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠒⠀⠀⠀⠀⢼⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠃⠎⠀⡔⢀
⠀⠀⠀⠀⠐⢢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠌⠠⢃⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⢀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠠⡏⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠁⠀⠐⠇
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠀⡴⠺⠓⠆
⠀⠠⡇⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠺⣀⡁
⠀⠀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⢱⣶⣶⡾⢿⠷⢶⣶⣤⡀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠐⠀⠐
⣆⠠⠤⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠋⢀⣄⣀⠀⠈⠛⠁⠀⠀⠐
⣯⠙⠲⡄⡈⠀⠀⡆⠀⠁⠀⠀⠀⠀⠀⢀⡄⡀⢀⠀⠀⠉⠀⠀⠀⠸⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡔
⠙⡤⠀⠀⠀⠀⠀⡇⠰⠄⠀⠀⠀⠀⠀⠘⢿⡿⡇⠀⢀⠀⢀⢠⠂⠀⠀⠀⠘⠀⣠⠀⠀⠀⠂⠀⠈⡚⠀⠀⡄
⠄⠗⠓⡄⢀⡄⠈⡗⢠⡌⠂⠀⠀⠀⠀⠀⠀⢿⠁⢂⠏⢠⠀⠈⠀⣠⠄⢃⠄⠳⠟⠂⠀⠀⠀⠑⣌⠱⠚⠂⢀
⠀⣰⠲⡄⠀⡅⢀⡟⢾⣇⠀⠀⠀⠀⠀⠈⠒⠠⠃⠤⠆⢰⠂⣤⡰⠇⢸⠀⡄⣸⡞⠀⡀⠀⠀⠀⠀⠙⠖⠀⠈
⠀⠁⠀⢱⢀⣤⢨⡇⢸⢻⣎⠁⠀⠀⡄⣼⠆⡷⠀⠀⢌⢙⠂⠃⢍⡁⠻⣤⠳⠝⠃⠀⠑⣤⢀⠀⢀⠀⡀⠁⠆
⠔⠄⠈⠒⠋⢗⣠⡇⢂⠉⣿⠳⠀⠄⠀⡏⠀⠈⠳⣔⡀⢈⣀⡀⠀⠀⢾⣾⡧⢤⠀⠐⣤⠒⡖⣆⡓⠀⠈
⠀⢾⠾⠔⣶⣟⢙⡟⠋⡐⠿⠀⠇⢐⢟⡟⢠⣄⡒⠊⠭⣹⠀⠀⡴⢀⠈⣽⣦⡔⡀⢠⣤⢰⡿⣄⡁⡀⠀⠘⢣
⣥⢰⣿⡦⡰⠿⢾⣷⠄⠀⢣⠐⣌⣩⣿⣤⣜⣀⠀⠀⣀⡽⢒⣟⢠⣽⠈⡄⠤⢼⠃⠆⠀⠀⠀⠀⢳⠏⣔⣿⣠
⡤⢺⢿⣷⠟⠻⡖⠃⠀⢀⡈⢸⣫⣿⣿⣿⡿⠉⢄⡀⠹⡳⡷⠆⠻⣿⣿⠗⠂⡬⠜⢓⠠⢀⣯⣾⣁⣴⣣⣄⠈
⠁⢼⣗⣤⣄⢀⣥⣺⣶⡦⣦⣻⣉⢡⣶⠚⣗⣀⡴⠀⣤⣟⣆⡁⡘⠻⡞⠂⠀⢃⠰⢺⠐⠿⡝⣜⣧⢿⢷⠻⠆

⠀⠈⠄⠀⠀⠀⠉⠀⣀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣈⠙⠲⠆⣨⠐⣀⣄⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠀⠀⠀⠈⠐⢤⡀⠁⠄⠀⠁⡀⠀⠀⠈
⠘⠀⠀⠀⠀⠀⢀⡀⠉⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⡙⠒⠠⠀⠀⠉⣉⣿⣾⣆⡀⢠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠉⢀⡀⠙
⢀⠀⠀⢠⡀⢲⠀⠱⡄⢀⣛⡿⠿⢿⣿⣿⣿⣿⢿⣿⣷⣟⣻⠶⠶⠤⠀⠀⠉⠰⢾⣿⣾⣇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⢀⠐⣆
⡀⠀⠰⠈⠁⠈⣄⠀⢀⣈⠉⠛⠢⠀⠈⠆⠙⠿⣿⡉⣿⣿⡙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣳⣮⣿⡀⢠⡀⠀⠀⠀⠀⣀⡀⠀⡀⠀⡄⠀⠆⠘⠀⢹
⠁⢀⠀⠠⠀⢀⣁⣠⣬⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⠇⢸⠹⡇⢹⣿⣿⣿⣀⠀⠉⠓⠀⢀⣀⡈⠩⠾⠿⣿⡇⠀⡀⠀⠚⠋⠁⠀⠁⠀⠁⡀⠀⠀⠀⢈
⣀⣉⣀⣄⣠⣾⣿⣭⣤⣤⣤⡤⠀⠀⠀⠀⠰⢆⠀⠀⠂⡐⠁⡸⣿⠛⣿⣿⣷⣀⠀⠀⢉⣉⣠⠤⠐⠃⣀⣤⠾⠃⣤⣤⣄⣀⣀⣀⢀⣀⣁⣠⣄⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣋⣁⣀⡄⠀⠀⠀⠀⠀⠀⠈⢒⠀⢀⠁⣰⠃⠛⢠⣿⣿⣿⣿⣿⣋⣉⣤⣤⣤⣶⡾⠋⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⣋⣡⡤⠤⠀⠐⠀⠀⠀⠀⠈⣁⠸⠰⠇⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣴⣿⣿⣿⣿⠿⠟⢛⡿⢿⣿⣿⣿⣿⣿⣿⣿
⠛⠉⣉⣿⣿⣿⣿⣟⣩⣤⠶⠄⠀⠀⠀⠀⠀⠀⠀⢈⣉⠃⠀⠀⠀⣰⠿⠿⣟⠻⣿⣿⣿⣿⣿⣿⣇⣾⣿⣿⣿⣿⡇⠀⡀⠈⠁⠀⠉⠛⠁⢉⣻⣿⣿
⠀⠀⠈⠀⠈⢿⣿⣿⣥⣤⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⣴⡿⠿⢛⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⠃⠐⠁⠐⠁⡀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠋⠙⣿⣿⣶⣾⣿⡿⠛⠂⠀⢀⣀⣠⣐⡀⠠⠀⠐⠋⢀⣠⡶⠿⣻⣿⣿⣆⠈⠻⢿⣿⣿⣿⣿⣿⡇⠀⠀⢀⠈
⡀⠀⠀⠀⠀⠀⠀⠶⠿⣿⣿⡿⢉⣀⣤⠴⠊⠀⣉⣤⣽⡅⠀⠂⠀⠠⠛⢉⣀⡘⠛⠛⠛⠻⠷⣤⣀⠈⠋⠈⠉⢀⡙⠂⠀⠀⢀
⠀⠤⠂⠀⠀⠀⠀⠀⠀⢺⣿⣶⣿⠋⢁⣠⡶⠊⣁⣤⣦⡀⣠⠊⣀⣤⠀⠉⠉⣉⡉⠉⠑⠂⠠⣀⣉⠻⡆⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠛⢿⣿⣡⣶⡿⠃⣠⣾⣿⣿⣿⣏⣥⣾⠟⠉⣀⣀⣉⢉⡉⢉⣀⢲⣾⢶⡿⠻⠃⠀⣠⠄⠀⠄⠀⠀⠀⠀⠠⠀⠠⠄⠀⠁
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⢁⣴⣿⣿⣿⡿⠛⠉⠋⠙⠃⠀⠉⠁⠀⢠⣤⡼⡟⣿⠇⠈⠀⠀⠀⠘⠁⠰⠂⠀⠈⠀⠂⢀⠀⢀
⢨⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣛⣥⣤⠄⠀⠀⠀⠶⣾⡿⣿⠁⠀⠈⠀⠀⠀⠀⠀⠀⢀⠃⠀⡜⠀⡸⠃⠈⠀⢠⠀⠐⢀
⢸⡇⠀⢸⡆⠀⠳⠌⡁⠀⢀⣿⣿⣿⣿⣿⣿⣿⠑⠟⣉⠁⠀⠀⠐⢚⡿⣿⠉⠀⠉⠀⠀⠀⠀⠀⢰⡆⠀⡀⢙⡀⠀⠁⠀⠀⠀⢀⠀⠘⡄⠀⠹⡄
⠀⠀⠀⠀⠙⣦⣄⡀⢙⣿⣿⣿⣿⣿⣿⣿⡿⣷⠖⠋⢁⣀⠤⢐⠿⠙⡇⠈⠀⠀⠀⠀⡄⠀⠀⠀⢸⣇⠀⣿⣾⣧⠀⠀⡀⠀⠀⢸⡆⠀⠁⠀⠀⠹⣦
⣀⠈⢶⣶⣄⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣤⡴⠛⣡⡴⢂⠀⣠⠀⡇⠀⠀⠀⠀⠀⢣⠀⡀⠀⢸⣿⡀⣿⣿⣿⣿⣿⣿⣶⣤⣄⣡⡄⠈⢶⣶⣄⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⣡⣴⡟⢁⣴⠃⢰⣿⠀⡇⠀⠀⢰⡄⠠⠸⡆⢿⡀⠄⢻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠋⣠⣾⠃⢰⠿⠟⠀⠿⠀⢶⡘⣷⣄⠀⢿⣾⣿⣴⣮⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠏⣿⡿⢩⡿⢫⣿⠟⢹⣿⣿⣿⣿⣿⣧⣽⠃⣼⠟⠉⠀⠢⢤⡀⠀⠳⣤⠀⠙⠾⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⣷⡘⢿⣿⣇⠹⣿⣿⣿⠉⣿⡟⢸⡿
⣿⠀⣿⠃⠘⢡⡿⠁⠀⣾⢣⡟⠙⢁⣿⣿⣿⠜⠛⠷⠄⠀⠀⠀⠹⣆⠀⠹⠃⠀⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⡄⠙⢿⡄⠹⣿⣿⠀⣿⠁⠈⢰
⢿⡆⢸⠀⠀⠸⠃⠀⠀⡇⠸⠁⠀⣼⡿⠉⠛⠛⠀⠀⠀⠀⠘⣦⠀⠙⠄⠐⠀⠘⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⠿⣦⡈⠙⠄⠘⣿⠀⣟⠀⠀⠸
⡈⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠐⠒⠦⠀⠀⠐⢦⡀⠈⠳⠀⠐⠀⠀⣠⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠻⠦⣤⣉⠓⠤⠀⠈⠄⠉
⠴⣦⣾⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠑⠶⠄⠈⠂⡀⠒⠀⠀⠀⠀⠠⢤⣈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣴⢦⣄⣉⠑⠒⠀⠴⢴⣧⡀⡆
⠀⠤⠶⢿⡿⣴⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠴⠀⠁⠀⠀⠀⢀⠐⠄⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢠⣍⡁⠀⠀⠀⠤⠶⣿⢷⣴
⠒⠀⠀⠀⠛⠛⢻⣿⡄⣀⢰⡀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠁⠀⠀⠀⠀⠀⡄⠘⠀⠈⠳⠀⢶⣮⡉⠛⢻⡟⢿⣯⣙⣿⣿⣿⢿⣝⠛⠓⠀⠀⠀⠀⠛⠟
⣠⡄⠀⣀⣀⣀⣈⠉⠈⠿⠿⣷⢀⣦⠀⠀⠀⢀⣠⡤⠀⠰⠂⠰⠀⠈⠀⠀⠀⡷⠀⠠⢤⡄⠀⠉⠀⠀⠙⠄⠙⠻⣧⠸⡿⣧⠨⣄⣀⣠⠄⠀⣀⣀⣀
⣿⣷⣤⡀⠀⣀⠀⠤⠤⠀⠘⢉⡛⠿⠆⣤⡄⠈⠉⠀⠀⠀⠀⠀⠠⠀⠠⠀⢀⣶⣶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡇⢹⠀⣿⣿⣿⣦⣄⠀⢀⣀
⣿⠉⣿⣷⣤⣤⣤⡤⠶⠖⠉⢀⣠⡶⠟⠉⣠⣾⣿⣦⣤⣦⣤⣤⣦⣤⣶⣶⣿⣿⣶⣶⠶⠶⠶⠖⠀⠀⠀⠉⠷⣄⠀⡀⠁⠟⢀⢻⣟⢹⣿⣿⣷⣤⣤
⡈⠀⣿⣿⣿⣷⣶⣶⣶⣶⣿⡿⠉⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⡤⠄⠒⠀⠀⠀⠀⠀⠈⢄⣣⠈⠀⡾⢀⠋⢸⣿⣿⣿⣿⣶
⠀⠄⢹⣿⣿⣿⣿⣿⣿⣿⣿⠁⣴⣿⣿⣿⣿⠟⠛⠉⠽⠛⠿⣿⣿⠿⣿⣿⣿⣿⣿⣫⡥⠶⠒⠂⠀⠀⠀⠀⠀⠀⠀⢤⡇⡸⠁⠀⠰⠋⣿⣿⣿⣿⣿
⠀⡰⠟⢛⣯⣝⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠀⡀⠀⢀⠀⠀⠉⠀⠀⠼⠿⣿⣿⣿⣯⡶⠶⠒⠈⠉⠁⠀⠀⠀⠀⠀⠠⣴⠀⠀⠀⠀⠾⠛⣻⣭⣻⣿
⠀⢀⡴⠟⠛⣉⣽⣇⠘⣿⣿⣿⣿⣿⣿⣿⣏⠀⡋⠀⠁⠤⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣶⡶⠟⢋⣁⣀⣀⣀⠀⠀⠀⠀⠲⠀⠀⠀⠀⣠⠾⠛⢛⣩⣿
⠀⠈⢀⣴⠿⠛⣻⣿⣆⠈⠙⠿⣿⡿⢿⡿⢿⣄⠀⠀⠆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⠟⠋⠁⢀⣀⠤⠀⢐⡃⠀⠀⠀⠀⢠⡾⠟⢛⣿
⠀⠀⠀⠀⣤⣤⣀⠉⠙⠻⢶⣤⣀⠀⢀⡀⠲⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⡟⣡⣤⡶⠚⠁⠀⣤⠶⠾⢆⠀⠀⠀⠀⠁⢀⣤⣤⣤
⠀⠀⠤⠤⠤⠄⢈⣉⣐⣂⣠⣬⣝⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡿⢁⣠⡴⠊⢁⣤⣾⣿⣆⠔⣀⣴⠄⠠⠤⠤⠤⠤
⠠⠤⠤⠤⠄⠀⢤⣤⣽⠏⣿⠋⠛⠀⡠⠖⢀⠐⠀⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠙⣻⣿⣵⣿⡟⢀⣴⣿⣿⣿⣿⣧⣾⣏⠀⠠⠤⠤⠦⠤
⣤⣤⣤⣤⣶⣷⣾⣽⣯⣤⣤⣤⣤⣤⣥⣤⣯⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣴⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿
⣿⣿⢫⣤⣽⠟⠛⠛⠿⠟⠛⠻⠟⠻⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠎⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣿⣿⠟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠎⠁⠙⢻⣷⣤⣀⣂⠀⡄⠘⠀⣶⡇⢸⠀⠀⠀⠀⠀⣠⢀⣦⢞⣽⠟⠉⠎⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠧⠁⠀⠀⠀⠨⠿⠿⣿⣾⣷⣠⣧⣿⡇⢸⠀⢠⠀⣤⣠⣷⣾⣵⡿⠁⠀⠀⠂⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠐⡄⠀⠀⠀⣌⣉⡭⠾⢿⣿⣿⣇⣸⣿⣼⣿⣿⠿⠿⠃⡄⠀⢀⢀⢰⣥⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢭⣦⣤⣴⣾⣿⠷⠷⠛⠛⣻⣿⣿⣿⣿⣯⣭⣭⣬⣮⣤⣴⣿⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⢒⣿⣿⣿⡏⠻⣆⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⣋⠩⢴⣿⣿⢿⣧⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⣾⠹⣰⣿⡆⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⣿⢿⠔⠏⢻⡇⠀⠀⠘⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡏⠘⠀⠀⣿⡇⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠈⣠⣾⡿⠉⠁⢀⣴⣿⣿⠃⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣾⣷⣿⢟⠙⢁⠖⢺⣫⣿⡿⠃⠀⠀⠀⠀⠀⡇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣴⣯⣾⡽⠟⠋⠁⠀⠀⠀⠀⠀⠀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢧⠀⡄⠘⠶⠒⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢣⠀⠈⢇⢳⡤⠤⠖⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⣄⡀⠀⠘⡇⢀⠀⢣⡀⢸⠞⠒⠒⠒⠂⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠡⣀⠀⠐⡄⠈⠳⡄⢳⠈⢆⠀⢳⠯⠐⠒⠒⠒⠂⣰⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⢤⡈⠳⡄⢹⡀⢀⠘⣎⡆⠈⢣⠯⠴⠒⢂⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠙⢄⠘⠄⢣⠘⡄⡌⠻⠀⢀⣤⣶⠯⠽⢒⣒⠛⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠀⢈⠖⢫⠋⡩⢋⢟⣠⡀⠈⣀⡠⠔⠊⣉⡤⠤⠖⠂⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣠⠎⢠⠃⡰⠁⢋⢾⢧⡇⠀⢨⠙⡶⣉⣀⡠⠤⠒⠒⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠊⠁⠠⠃⠀⢁⡴⠋⣼⡏⢳⠀⡾⠀⡇⠈⡅⠀⢈⣉⣡⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠤⠴⠚⠉⠀⣸⡟⠀⠘⣆⠃⢀⡇⢠⡇⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⢆⠈⡇⠈⠇⠳⡀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⠓⢤⣀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠒⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠉⠉⠉⠙⠻⣷⣄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⢠⡾⠛⣷⠀⠁⠀⠀⢠⣿⠇⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⡀⠀⠀⠀⣾⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⣿⠃⠀⡟⠀⣀⠀⢀⣾⣿⣴⣾⠿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢠⣿⠁⢀⣠⡀⠀⠀⣿⣇⣼⠁⠀⠀⢀⣼⣿⠟⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠤⠀⠀⠀⠀⢀⣿⡏⠀⣠⡿⠃⢠⡟⢁⡟⠀⠀⢻⣿⡁⠀⢀⡴⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣥⡾⠛⠁⢀⣿⣡⠞⠁⠀⢠⡾⣿⣧⣴⠏⠀⣿⣿⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⢠⣿⡟⠙⢳⣄⣀⣼⡿⠁⠀⢀⣴⠋⢀⣿⣿⠁⠀⢸⣿⡟⠀⠰⠂⠀⠀⢰⡆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠙⠿⣷⣦⣤⠆⣸⣿⠇⠀⠀⣿⠁⢸⣧⣀⣴⠟⠁⣠⡿⢻⣿⠀⠀⣼⣿⡇⠀⠀⠀⠀⢠⡿⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⣿⠀⠀⠙⢋⣁⠀⣴⠏⠀⣼⣿⠀⠀⢿⣿⡇⠀⠀⠀⣴⡿⠁⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣿⣿⠀⠀⣼⠇⠀⠀⢀⣾⠏⣼⠃⠀⢀⣿⡏⠀⠀⠘⣿⣧⣤⣴⣾⠟⠀⠀⠉⠴⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⠀⢻⣿⣀⣼⠟⠀⡤⣄⡼⠁⢰⡏⠀⢀⣾⠟⠀⠀⡀⠀⠈⠙⢛⣿⠃⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠛⠛⠉⠀⣼⣳⢯⣧⠀⠈⠷⠶⠟⠉⠀⠀⠀⠁⠀⠀⠀⣼⡏⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠗⠀⠀⠀⢰⡏⠀⠛⠁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⢰⣿⠁⣠⠏⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣠⡄⠀⠀⢀⣿⠇⠀⠀⠀⢿⠃⠀⠀⠀⠉⠀⠀⠀⢠⣾⠟⠙⢃⣿⡏⣰⢃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣿⠀⢠⣿⠋⠀⠀⣹⡛⢳⡻⠟⠁⠀⢰⡆⠀⠀⢠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠾⠇⠀⢀⡾⠀⢠⣿⡇⢀⣾⡏⠀⣠⣾⣻⣻⡛⠁⠀⠀⠀⠀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⣾⠇⢠⢿⣿⢀⣾⣿⠁⣴⠏⢸⣿⡟⢿⣦⣀⠀⠀⢀⣠⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⣰⣿⣠⡟⢸⣿⡾⠁⠻⠟⠁⠀⢸⣿⠁⠀⠙⠻⠿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⢀⣤⣶⠿⠛⣿⠟⠀⠈⠛⠁⠀⠐⠟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣠⣾⠿⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠸⠂⠀⠈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠙⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⣼⣿⠀⠀⠀⠀⢳⡄
⠀⠀⠀⢀⡀⠀⠀⠀⠠⢤⣀⠀⠀⣿⡟⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠈⠑⢞⡶⢦⣄⠉⠁⠀⠘⠁⠀⠀⣠⠀⠐⣿⡇⠀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠙⠷⠄⠀⠀⠀⠀⠀⠟⠀⠀⠹⡇⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⢦⣄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠹⠆⠀⠠⡀⢀⡇⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣶⠶⠖⠀⠀⢀⡤⠀⠀⠀⠀⠈⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠁⠄⠀⠙⢿⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠒⠛⠛⠻⠷⠶⠤⢀⠀⠂⠀⠁⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠈⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠐⠀⠀⠀⠀⡀⢠⠂⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠼⠀⠄⠀⠀⠠⡀⢸⡅⠷⠒⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⣠⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠁⢀⡸⠀⠀⠀⠀⣠⠐⡇⡸⠤⠤⠀⡠⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠻⠃⠈⢛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⠀⠀⢸⣄⠸⣅⡠⠤⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⢠⡤⠀⠀⠀⠀⠙⠉⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠦⠀⠀⠀⠀⠀⠂
⠐⠒⠦⢀⡀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⢣⠀⡀⡠⠊⠀⠀⠀⠀⠰⣷⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⣼⡇⠀⢀⠀⠀⠀⣄⢌⢇⠝⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⠆⠀⡆⠶⠁⠀⠀⠀⠀⢀⣴⣿⡁⠐⠀⠀⠀⠀⠘⣦⡌⣀⡤⠀⠀⠀⠐⠀⢤
⠀⠀⠀⠀⢀⡀⠛⢚⠁⠀⠀⢂⡆⠹⠃⠀⠀⠀⠀⠐⢿⡿⠟⠁⠀⠀⠀⠀⠰⡌⣸⠛⠁⠀⠀⠀⠀⠀⠀⠸⣇⠀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠈⠑⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠖⠒⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢾⡄⢠⡀⠐⠊⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠇⢻⡇⢸⠆⠀⡌⠀⣀⡀⠀⢀⣠⠄⢀⠂⠀⠠⢾⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢺⣟⠴⠾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢡⣦⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⣀⡄⠀⠂⠀⢀⡴⠊
⠐⣄⣄⣀⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠉⠲⢠⣆⢋⠀⠄
⠀⠀⠺⠿⠅⣁⠐⢤⡀⠀⠀⠘⠀⣠⡶⣤⠀⠀⠀⠀⠀⢻⣶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⠀⠐⠈⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠲⢝⣆⠀⢦⠀⠸⢷⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢂⠾⠶⠒⠶⠲⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠸⠀⠄⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋⠛⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⢀⡴⠟⠋⠉⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⡚⠻⠏⠀⠢⠁⡇⠄⡄⠀⠀⠈⣿⠀⢰⡏⠀⠀⠀⠀⠀⢀⠘⢠⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣁⡒⡀⠀⠀⠀⠑⢄⠡⠄⡄⡶⠀⡐⢻⡇⣿⠀⠀⠃⠀⠀⢀⠸⢘⡠⠆⠉⠉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠬⠀⠤⠀⠀⠀⠀⠀⠑⢄⠡⠄⡆⡀⢸⣿⣏⠀⠁⡀⠃⠗⣉⡴⠋⠀⠀⠀⠀⠨⠁⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡔⠒⣐⠒⠀⠀⠀⠀⠀⠀⠀⠱⣌⠡⠀⢸⡏⣿⠈⡁⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⣤⠘⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠩⡄⠤⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⣿⠀⣿⠈⣡⠞⠁⠀⠀⠀⠀⠀⠈⢁⠐⠒⠆⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣳⣤⣃⡤⠅⠀⠀⠐⣀⣀⣀⣁⡀⠻⣶⠯⠞⠁⠲⠄⠀⢡⡀⠈⡤⠘⠒⣂⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⠩⠙⠖⢒⣛⠩⠉⢙⡛⠚⣻⢇⣤⠺⠿⣿⣝⣛⣛⣓⣒⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡅⠉⠀⠀⠀⠈⠀⠠⠀⢠⠙⣢⠴⠋⢀⡟⢹⡇⠑⢄⡈⠉⠉⠉⠉⠙⣛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠦⠀⠀⠀⠀⠀⠀⠤⢈⡤⠊⠁⠀⠀⢸⡇⣸⠗⠆⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⣨⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⣃⠀⠄⠀⠠⠌⣡⠞⠉⠀⠀⠀⠀⠀⢸⡇⣿⠠⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢁⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣐⠒⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠂⣸⢿⡇⠖⠀⠀⠀⠀⠀⠈⢣⡀⢀⠀⠀⠐⠀⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣷⠀⠄⠆⠀⠈⠀⣀⠐⠂⢆⡿⢸⡧⠆⠀⠀⠀⠀⠀⠀⠀⠹⠌⠤⠈⠤⠭⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠆⡄⢙⢸⠀⡛⠰⢇⣾⠃⠘⣷⠚⣰⠂⠆⠠⠀⠀⠀⠄⣠⣤⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣉⠘⠘⠀⣋⣤⢾⡇⠀⠀⠹⣧⡀⠞⡴⢀⠋⡴⠰⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⣿⢸⣇⠀⠀⠀⠈⠻⢦⣥⣉⣀⣥⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⣿⡄⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡌⠠⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⢀⣠⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⣉⣉⠉⠻⡿⠋⣀⣀⣈⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠉⠉⠻⠃⣼⣷⡄⠛⠉⣀⡀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣄⠙⠋⠠⡄⠹⣿⠁⠤⠈⠋⠀⠶⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠠⢴⣶⣦⡤⠀⠿⠠⠴⣶⠿⠖⠂⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⢤⡄⠁⢠⠀⣶⠀⠖⠀⢠⠆⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠈⢿⣦⡀⠁⠻⠆⠠⣿⡆⠐⠛⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣤⡾⠿⠛⠉⡉⠛⠟⠿⣷⣄⠙⢿⣧⡀⢲⡄⢻⠀⡿⠀⣼⡿⠋⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣾⠏⢠⠄⢰⡄⠉⣰⡆⢀⡀⠙⢷⣄⣸⣷⣄⡀⠘⠀⣠⣾⠏⠀⣠⣴⣾⠿⠛⠛⠛⠛⠛⠻⣿⣦⡀
⠀⠀⠀⠀⠀⣿⠀⢿⣆⠘⢉⡀⢻⡷⠈⢉⣀⠈⠛⢿⣿⣿⣿⣶⣾⣿⣷⣴⣿⠿⠋⠀⣴⠂⣰⣆⠘⢁⡄⠀⠙⣿⡆
⠀⠀⠀⠀⠀⢿⣦⣈⣠⣶⣄⡁⠸⠁⠜⠀⢉⠀⠷⠄⠙⢿⣿⣿⣿⣿⠟⢉⣠⡄⠰⠖⢀⠰⣿⠃⡀⠛⠁⣸⡇⢸⡿
⠀⠀⠀⠀⢠⡿⠋⣉⠛⠿⠛⠋⣛⠀⠛⠻⠿⠿⠒⠒⠀⠈⣿⣿⣿⠁⠤⠤⣤⣤⣦⣄⣉⠀⣋⢀⣉⣤⡈⠛⢀⣼⠃
⠀⠀⠀⠀⢸⡇⢰⡿⠁⣀⠙⢀⣿⡄⠋⣠⡄⠰⠟⢁⣴⠾⣿⣿⡿⣷⡀⠰⣤⠈⠉⢉⡍⢀⡋⠙⠻⣿⠿⠓⢿⣇
⠀⠀⠀⠀⠸⣷⡈⠃⠰⠟⠀⠺⡟⠀⠶⠀⢀⣴⣾⠟⠁⠀⣿⣿⠀⠘⢿⣦⣀⠐⠛⢀⠀⣾⡇⠘⢂⠀⢶⡆⠈⣿
⠀⠀⠀⠀⠀⠙⣿⣶⣄⣈⡉⣀⣁⣀⣤⣶⣿⠟⠁⠀⠀⠀⣿⣿⠀⠀⠀⠹⣿⣧⡀⠙⠀⢹⠓⠀⢻⡆⠸⠃⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠈⠙⢿⣶⣤⣄⡀⠓⢀⣠⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡤⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢻⣷⠈⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⢺⡀⠀⠀⣿⠟⠁⠘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⢸⡇⠀⣀⠀⠀⡀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣀⠀⠀⠠⣿⣿⣿⡿⠀⠸⢇⣀⣿⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⢿⣿⣿⣧⢰⡿⠞⠛⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣶⠀⢸⣿⣿⣇⠘⠁⣠⠖⠈⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⣿⡤⠀⠻⣿⣿⣶⣶⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣴⣿⣿⣿⣧⣿⣸⣤⣾⠿⠋⠉⠁⣾⠿⠿⠃⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣼⡟⠃⣿⠇⠐⠟⠃⠀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠈⢻⣿⣷⣾⣄⢁⡀⠲⠀⠿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣇⠀⣱⡟⠃⠉⠉⢙⣤⠜⢺⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡿⠷⠾⣷⣆⣹⣧⣼⣿⣿⣿⡿⠿⠟⠛⠃
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠘⠟⠙⢣⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⢸⡇⢠⣾⡅
⠀⠳⣄⡀⢸⣿⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⢸⣿⣾⠟⠁⢀⡄⠀⣠⡆
⠀⠀⣿⡇⠘⣿⡁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣀⣽⣧⡴⠞⠋⢀⣶⡟
⠀⠀⠉⢁⡴⠈⠉⠀⠀⠁⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠻⣿⣿⣿⣿⡟⠉⠁⣠⠶⣠⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠑⣄⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣆⠈⠻⣿⣿⣷⣮⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⠃⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⣿⡇⠀⣹⠿⠟⠋⠉⠀⠀⡀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠘⠃⠀⠀⣶⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡜⢷⢀⣿⠀⠀⢠⡤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⢁⣴⡗⠉⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡀⠉⠀⠀⠀⠠⣤⡀⠀⢸⣿⣿⠉⢿⣿⣯⡘⠻⣧⣴⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⡿⣨⢻⣶⣌⣿⠃⠀⠀⠹⠁⣠⡾⢛⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣿⣿⣠⣆⠀⠀⠿⠄⢹⣿⡏⠀⠆⠀⠀⣴⣟⣵⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣆⡀⠀⣤⡀⢻⣿⠀⢺⡆⠀⣼⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠻⣿⠉⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⡟⠉⠉⠉⠉⠀⠹⣿⡾⣿⣇⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⢰⣦⣤⣄⠀⠈⠁⢸⣿⣤⣿⠇⠀⠀⢀⣤⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⣰⠁⣀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⢻⣿⣿⣷⣤⡀⠈⠻⣿⠇⢀⡀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡇⢰⡿⠻⣿⣿⡿⢡⠀⣿⢠⡜⢣⣾⠏⠀⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠺⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠜⠁⠀⠘⢿⠇⢸⡄⠸⣿⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⠈⠁⠀⣴⠁⠈⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄⠀⠀⢠⠆⠀⢀⣤⡀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡌⠙⢧⡀⢰⣈⣷⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⣾⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣶⣼⣿⠊⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠟⠀⠀⠀⠺⣿⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣏⠈⣿⣿⣇⣀⣤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡀⠈⣤⠀⢀⣴⡀⣿⣧⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣿⡀⢃⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠛⢷⣄⡹⡀⢸⣿⠇⢻⣿⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠘⡷⠗⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⣤⠀⠀⠀⣉⠛⣿⠈⠻⠀⢈⡟⠀⠀⠀⠀⠿⠃⢠⠀⣀⠀⠀⠾⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣤⣘⠿⣤⡄⠀⠀⠸⠟⠀⠀⠀⠀⠀⡂⠁⢠⣿⡆⠘⣀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠿⣿⣷⣽⣿⣴⡀⢰⡆⠀⠀⠀⢠⣾⣿⣿⠀⢻⣿⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠈⠻⣿⣿⣿⠀⢸⡇⢠⡆⢠⡿⠋⠈⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠈⠻⣿⣤⣤⡆⢀⣶⠏⠀⣶⣾⣿⡀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠈⠙⣿⠁⠈⣡⡄⠘⣿⡇⢀⠁⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠸⡀⢀⣿⡇⢸⣿⣇⡈⣡⡭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⢰⣧⠘⣿⡇⠈⢻⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣄⠀⢸⣿⠀⠸⠇⠀⠏⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⣿⣿⣿⢈⣿⠀⠀⠀⠀⢸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⢻⣿⣇⠈⠸⠿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⢹⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣼⣿⡀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠀⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣆⣦⡿⣦⠇⠘⡽⠻⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢤⠀⠀⠀⡀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠙⠛⢳⣭⡴⣃⡤⢄⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿⣧⡁⢤⡘⠓⠀⢀⡁⣿⢦⣴⠀⠀⠀⠀⠘⢦⡄⠀⢀⣿⣿⣟⡉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⣿⣿⣿⣿⣿⠹⠃⠀⠰⢸⡿⣏⢙⣎⣻⠃⠀⠀⠀⠙⢛⣫⣿⠟⡘⣟⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⠟⠸⠿⠃⡀⠀⠄⢀⠋⡁⢁⡶⠀⠙⣶⠊⠀⠀⠀⠺⣿⣿⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⢰⣤⡤⠀⠂⠶⣶⠆⢠⠀⠉⠄⢠⣴⣻⣠⠄⠀⠀⠀⠀⠉⢩⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣴⣀⣠⣀⣁⡀⢀⣩⣶⡅⠈⠀⠀⠀⠀⠀⠀⢀⡹⠀⠀⢀⢀⡠⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣿⣽⣿⣿⣗⣿⣿⡿⠛⣹⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⣤⣁⣤⣾⣾⣿⣵⣿⠀⠀⡘⢎⣥⣶⣷⠤
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⡀⠀⠿⡿⣿⣿⠘⣏⠻⢇⣠⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⢿⣿⣿⣿⣿⠀⠀⢰⣸⡿⣻⣷⣶⠃
⠀⠀⠀⠀⠀⠀⣼⣩⣿⣿⣿⣁⠀⠀⠉⠀⠀⠹⣄⠀⠈⠉⠙⠉⡇⠀⣠⣆⣤⣠⣿⣯⠶⠀⡟⠋⢁⣀⠀⠈⢿⣿⡿⠀⠀⣞⠩⢤⡉⠽⣯⣴
⠀⠀⠀⠀⠀⠀⢯⣿⣿⢿⣿⡟⣷⡹⠋⠀⠀⠀⠹⣦⠀⠀⠀⠀⡇⠀⢟⣿⣿⣿⡿⠋⢀⣾⣤⣾⣿⣿⣇⠀⠈⠋⠀⣤⠋⠁⠀⢰⣾⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠻⣽⢸⡽⠟⠋⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⡇⠀⠀⠙⠺⡿⣿⣀⣸⢿⣿⢏⣿⣿⣿⣥⠀⢰⣾⡁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⡇⠀⠀⠀⠀⠀⠙⢋⠗⠊⠀⡼⠻⠛⠋⠉⣰⣿⣛⡛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⢀⡏⠀⠀⣼⠃⠀⠲⢶⣾⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣇⠀⠀⠀⠀⡼⠀⠀⣼⠃⠀⠀⢠⠞⠈⠉⠈⠘⠟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣧⠀⠀⢠⠇⠀⣰⠃⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣆⠀⡻⠀⢀⡏⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⣶⡇⠀⡼⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢿⡄⢀⠃⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡸⣷⡜⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡇⣸⠃⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠃⣽⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢺⣵⡏⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⣧⠀⢻⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⢠⣯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡠⠒⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡤⠀⡠⢀⠤⠀⣀⠄⣢⠄⢀⡠⠂⡾⠁⠀⠀⣼⠁⣠⠄⠀⣠⠄⢀⡤⡰⠄⣰⣏⠤⣶⠄
⠀⠀⠀⠀⠀⠻⢷⣤⣀⠠⢴⡞⠁⢾⡤⠜⠓⠸⠧⠤⠛⠂⠻⠖⠈⠃⠀⠀⣰⠇⠐⠿⠖⠈⠻⠄⠺⠧⠄⠸⠋⠐⠮⠅
⠀⠀⠀⠀⠤⠤⠤⠤⠤⣤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣴⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠄
⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢀⠤⠒⠚⠛⠓⠒⠶⢤⣄⡀⠀⢀⣾⠋
⠀⠀⠀⠐⠒⠚⠉⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⠿⢧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⢀⣀⣀⣠⡤⠶⠛⠉⠀⠀⠀⠈⠉⠛⠓⠲⠶⠒⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡰⡶⡖⡶⠦⡄⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡆⣀⣒⣶⡀⢀
⠀⠀⢀⡀⣈⡟⠲⣄⣆⠏⠙⠛⢦⡐⢂⠤⣢⠀⢀⠄⢀⠀⣧⠀⣠⠀⠀⠀⠀⠀⠀⠀⡺⠧⢤⣤⣊⠈⠈⢙⢢
⠀⠀⢀⢹⣧⡡⡀⠀⢙⣣⣄⠀⣊⠊⠀⣾⢏⠠⠀⠀⣬⠀⡅⣾⠃⠀⠀⠀⠀⢀⢳⣆⡱⡀⠀⢉⡵⣤⣒⡽⠁
⠀⠀⠀⠀⠻⡿⣭⣵⡏⠚⣫⡟⠀⠈⠑⡵⡷⠟⣧⠀⢿⠀⢱⣿⠀⣀⣀⢀⠴⠊⠀⢻⡿⢯⣤⡟⠀⢩⠞
⠀⠀⢈⣧⠈⣁⠀⡹⣿⣶⣜⠀⠀⠀⠀⠂⠐⠛⣻⡇⠈⠀⠘⠁⠾⠿⣇⣤⣆⣘⣦⡈⢀⠈⠙⢿⣷⣮⡀
⠀⠀⢈⡿⣇⣈⠛⠉⢈⢿⠏⠙⠀⠀⠘⣓⣛⣓⡛⢿⠀⠐⠐⢆⡜⠣⣄⠀⠈⠉⣿⣧⣈⡛⠉⠁⠿⣏⠉⠁
⠀⠀⠘⠀⠈⠀⠀⠀⡀⠀⠀⠡⠀⠀⣈⣩⣿⣿⣿⠆⠁⡀⡆⣠⢦⣤⡤⠟⢄⠐⠁⠈⠁⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠂⠉⠉⠉⠉⠁⠀⠀⢉⠐⠓⠘⢆⠀⠄⣠⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⡀⠁⠀⠂⠀⠀⠁⠀⢄⣠⣾⣷⣏⣡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⣵⣦⡂⢤⣀⡤⠝⠒⠤⣐⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⢀⣤⢂⠀⠀⠀⠄⠀⠀⢠⣟⠀⠒⢀⠂⡀⠹⣷⠀⣹⣄⠀⡠⠀⠀⠀⠀⠀⣠⠖⠀⠀⢀⠄⠀⠀⢠⣵
⠀⠀⢸⠇⠈⠓⠄⠀⠀⠀⠀⣀⢿⠀⣿⠂⢰⣛⠛⠾⠉⠀⠀⢻⡏⠙⢖⣤⣰⣾⠏⠈⠑⠀⡀⠀⠀⠀⢠⢹⡁
⠀⠀⠀⣀⣠⠊⢰⣤⡓⠒⠶⠈⣸⠀⠏⢠⣿⣙⠒⠀⠀⠀⠀⠀⣡⣴⡿⠋⠀⠙⢀⣠⠞⢱⣦⡗⠒⠲⠈⣸⡇
⠀⠀⢰⣿⣿⣿⢻⠤⠿⢄⠆⡴⠀⠀⠀⣼⡿⣛⣼⣉⡁⠁⠀⠉⡉⡟⠐⠲⠞⠀⣾⣿⣿⡛⠤⠽⠆⠸⡠⠃
⠀⠀⣸⠟⠅⠘⠆⣤⣄⠩⠴⢄⠰⢀⡱⠋⣼⣿⣯⣉⣉⠠⠀⠄⠀⠀⠀⠀⠉⣻⡟⠉⠈⢇⢵⣄⡨⠴⢦⡂⢀
⠀⠀⠀⠀⠀⢀⣶⡉⣆⠈⠉⠋⡓⡈⠀⠀⠈⠉⠉⠉⠉⠀⠴⠀⠀⠀⠁⠀⣰⠋⠀⠀⢀⣴⣭⢦⡈⠁⠉⣑⣈
⠀⠀⠀⠀⠀⠈⠁⠉⠀⠁⠀⠈⠉⠉⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠁⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠠⡙⢦⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠉⠑⢤⡀⠀⢰⠁⠀⠀⠀⠀⠉⠒⠂⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑⣄⢸⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⣠⠃⠀⠀⠀⠀⠀⠘⣾⠀⠀⣰⠃⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠊⠁⠀⠀⠉⠒⠤⣀⠀⠘⡇⢠⠃⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠴⠚⠛⠓⠒⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⠤⠤⢤⡴⠶⠚⡀⠀⠀⠀⠒⠢⢤⣀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⢀⡔⠁⠀⠸⡄⠀⠀⠀⠈⠙⠢⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢠⠎⠀⠀⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⡌⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⣧⠘⣆⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⠠⢄⡀⠀⠀⠀⢠⠟⡆⢹⠑⠦⣄⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣆⠑⠀⠀⢀⠞⠀⢱⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⢰⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢈⠃⠀⠰
⠀⠀⠀⠀⠀⢠⠀⣠⣴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⣿⠛⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠄⢺⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡴⠊⠁
⠀⠀⠀⠀⣰⠏⠀⠆⠀⠀⠀⠀⠀⠀⠀⣠⠾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣤⡿⠏
⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠨⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⢡⣴⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠰⣀⣿⠿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣦⣿⣯⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⢷⣦⣄⡁
⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣧
⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠘⠓⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡤⣶⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣤⡀
⠀⠀⢠⡆⠀⢠⡶⠀⠀⠀⣦⡄⠀⢀⣀⠀⠀⠸⡄⡏⡄⠀⠀⠀⠿⣿⣤⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠘⢷⡀⣼⠟
⠀⢀⠘⡇⡴⢋⡀⠀⣀⡀⠹⠿⣾⡿⠟⠀⢀⠳⣜⢊⡴⢀⠀⢀⣠⡛⠁⠀⠀⠀⣴⣿⣷⣦⡄⠀⠀⠀⠀⠈⠳⢁⣴
⠀⠘⣧⡈⢠⠾⠁⡄⠻⣷⣄⠀⣠⣴⡦⠀⣬⢷⡈⡡⠴⢋⡄⠘⠿⣿⣿⠀⢠⣷⣹⣯⣿⣾⣽⣷⡄⠀⠀⣈⢣⡿⠋
⠀⠀⢨⠛⠆⣠⡾⠃⠀⣨⡍⡀⠛⢋⠀⢀⡙⢇⢹⣠⠶⢟⠀⠀⢀⠀⡀⠀⣼⡛⣿⣿⣿⣿⣿⣻⣥⠀⠈⣿⢿⠁
⠀⠀⠸⣦⡀⠋⠀⠀⠀⠘⠿⣧⣾⡿⠃⠀⡳⡌⣯⣶⠞⠁⠀⢾⡿⣿⣿⠀⣻⠿⣯⣿⣿⣿⣿⢿⣿⢃⣀⠈⢿⢋⣤⡄
⠀⠀⠀⠈⠳⣠⡄⠀⠀⢿⣦⣏⣴⣶⠀⠀⠳⣘⣡⠆⠀⠀⠀⢀⣀⠀⣀⠀⢿⣋⣿⡟⢻⣏⣹⣿⣦⠈⠿⡀⡸⠚⡋
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠙⠟⠛⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⢿⠿⠿⠂⠀⠉⠉⠀⠀⠻⠿⠿⠃⠀⠀⠰⢡⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢏
⠀⠀⠀⣶⣄⠀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣆⢀⣤⣤⠀⠀⠀⠀⠀⠀⣿⢀⡄⠀⢸⣧⠀⠀⡀
⠀⠀⠀⢻⣿⠆⠃⣀⡀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠈⢁⣈⠛⠋⠀⠀⠀⠺⣦⡀⢹⡿⠃⠀⠀⠙⠀⣾⠇
⠀⠀⠀⠀⠉⣥⣾⣿⠇⠀⠀⠀⠘⢿⡇⣠⣴⣶⠀⣤⡄⠸⡟⢷⡶⠀⢀⣤⣤⡬⢥⡄⣀⣀⣄⠻⣶⡤⡄⠀⣰⣿⠂
⠀⠀⣀⣤⣀⠘⠋⠀⣀⣀⠀⢀⣄⠀⡀⣿⡿⠃⠀⢙⣥⡄⠀⢰⣶⡆⠈⠛⣡⡀⠀⠱⠀⢉⡋⠀⠀⠀⠬⣀⠛⢁
⠀⠀⠻⣿⡿⠿⡇⣾⣿⡿⠂⠸⣿⣷⠸⡇⢀⣴⡆⠈⠻⠓⢤⠘⠛⡁⠀⠘⢿⣧⠀⢆⣾⡿⠃⠀⢀⣤⣀⢸⠐⠁
⠀⠀⠀⠀⠀⣀⣸⠈⠉⠀⠀⠀⠉⠻⡄⣶⣿⣿⠟⠀⠀⠐⣄⢀⠞⠀⠀⠀⠀⠉⢠⠋⠁⠀⠀⠀⠀⠙⢻⠘
⠀⠀⠀⠰⣾⣿⡿⣠⣴⣆⢸⣿⣷⡄⢸⠛⠋⠁⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⡄⡞
⠀⠀⠀⠀⠈⠁⠀⠟⠛⠁⠀⠙⠛⠓⠷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⢀⠏⠀⠀⠀⠀⠀⢠⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠀⠀⠈
⡈⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⢠⡡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣧⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣆⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢮⣛⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣌⠳⣻⡵⡃⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠐
⠀⠦⡆⠘⢄⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠈⠯⢳⡵⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢰⡄⠙⠠⠂⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢰⠀⠀⠁⠘⢃⠉⣀⡀⠛⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⣨⠃⠀⢲⣁⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠁⡴⠂⠈⠈⣌⠁⢋⣠⠆⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⢐⠀⠱⠀⠈⢑⠀⠈⠉⠀⠀⠠⠖⡄⠀⠞⠀⠀⠀⠀⠈⣄⠩⠖⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠾
⠠⠀⡀⠀⠀⣨⠆⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⣛⣭⣿⣿⣿⠏
⠀⠀⠀⢠⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⠛⠛⠛⠛⠛⠛⠻⠷⣶⣤⣤⣤⣤⣠⣤⣤⣴⡶⠿⠛⣩⢖⣴⠞⣹⡿⠋⠉
⠀⠀⠀⢸⣿⣝⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⢉⡴⠂⠀⠀⠀⠀⠀⠀⣀⣠⠤⠂⠀⡉⠉⠉⠉⢉⡭⠀⠀⠀⣼⡵⠋⠀⣰⡿⠁
⠀⠀⠀⠀⠈⠙⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⣠⣴⡿⠋⣠⠞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⣠⡾⠋⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣶⣴⣶⠿⠛⣁⠴⠊⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⣀⡴⠿⠧⠦⠔⠒⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⠳⠶⢶⣮⣅⣀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⣀⠴⠋⢀⣀⣤⡴⠞⠉⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠈⢫⡙⠓⠶⠾⣭⣤⣤⣤⡤⠴⠶⠖⠚⠛⠛⠛⠉⠙⠓⠦⣄⡀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠠⡀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⢀⣀⣠⣭⣦⣤⣤⣤⣴⡾⠟⠋
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⡀⠀⠀⠀⠀⠉⠓⢀⣠⣶⠾⠟⠛⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠶⠶⠿⠿⠛⠉


⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣄⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠈⠹⣿⠟⠁⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠛⠿⢷⣦⣄⡈⢻⡄⠀⠀⠀⠀⠀⠀⣴⠏⢀⣤⡶⠿⠟⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠈⠛⠶⣽⣄⠀⠀⠀⢀⣼⡧⠞⠋⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⡠⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠁⠀⠀⠹⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣀⣀⣠⣴⠶⣻⣿⠏⠀⠀⠀⠀⠀⠘⣧⡙⠳⣦⣤⣀⣀⣀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⢁⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢾⡇⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠘⢿⣦⣤⣴⡾⠋⠻⣷⣤⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣆⡀⠀⠀⠱⣦⣴⣿⣿⣿⣤⣤⠆⠀⠀⣀⣤⣾⡿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣄⣼⡀⢹⣿⣿⣿⣿⣿⠃⣰⣀⣼⣿⣿⣟⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⣿⣷⣼⣿⣿⣿⣿⣯⣰⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣡⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣼⣿⣿⣷⣴⣿⣿⣿⣯⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⢿⡿⠛⠉⠈⠙⠻⡿⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠈⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣼⣿⣿⣿⣿⣧⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣿⣿⣿⣿⣿⣿⢙⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣝⢸⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⣿⡇⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣾⠣⢟⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠶⣄⡔⠋⠙⢦⡴⠊⠉⠙⠶⠁⠀⠀⣾⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠈⠀⢀⣤⣌⠀⢠⡶⣄⠀⢠⡶⡄⠀⣴⠿⣶⡿⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⠆⠁⠀⢠⣿⠋⢹⡄⠹⣿⣸⣠⡾⣧⣇⣼⡏⠀⣿⡇⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣾⡇⢀⢸⣧⢾⣏⡝⠸⣇⡜⠉⠹⠔⠁⢿⡇⠀⣼⠃⠀⣠⠟⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣷⠀⠘⠛⢁⣼⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠾⠋⠀⢊⡥⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⡟⠀⠀⠀⡞⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣼⣿⠞⠀⠀⠀⠀⠳⠋⠀⠀⠀⠀⣠⣤⡀⠀⢰⡿⣠⠖⣳⠄⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣀⣤⣿⡇⠀⠀⠀⢀⡄⠀⠀⣼⠀⠀⣼⡏⠘⠃⢀⣿⣿⣡⠾⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢤⠀⠈⠉⠉⠁⣿⡇⠀⠀⠀⢸⡇⠀⣼⣿⠀⢠⣿⠀⠀⢠⠎⣿⢿⣧⡀⠀⠀⠀⠀⠀⢀⣾⠏⢠⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⡆⠀⢻⡇⠀⠀⡰⢻⣇⡼⠸⣷⣠⠏⠻⣦⠴⠋⠀⠿⠀⠙⠿⣶⣦⣤⣤⣶⠿⠋⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⠻⠶⠞⠁⠈⠉⠀⠀⢈⡁⠀⠀⠀⣤⠀⠀⠀⣠⣷⡀⠀⠈⠉⠉⠀⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣤⠞⠷⣤⣠⣤⠞⠙⠶⠶⠚⠉⠓⠒⠚⠉⠈⠙⠶⢦⣤⣤⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⠀⠀⣠⣶⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠟⠁⠀⣠⣾⣿⣿⣏⣉⣀⣀⣠⣴⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢥⣄⢵⠄⠀⠈⠠
⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⢸⣿⡴⠚⠉⠉⢿⣿⣶⣦⣄⠉⠙⠻⢿⣿⡿⠿⠛⠉⠀⠀⢀⠀⠀⠀⠀⠀⢁⠀⠀⠬⢩⣇⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⠞⠻⣷⣶⣶⡄⠀⠙⠻⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠶⠄⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣦⣠⠏⠀⠀⠘⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⣛⡄⠀⠀⠀⠀⠊⡀
⠀⠀⠀⠰⣶⣤⣄⠀⠈⡿⠀⠀⢀⣤⣤⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠤⠼⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠘⣿⣿⣄⠀⢧⢀⣴⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠈⢛⡀⠀⠀⠀⠀⠈⡠
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⡟⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠉⡥⠀⠀⠀⠀⠊⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣬⣷⣤⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⠓⠦⠤⠄⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡀⡀⢡⡼⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤⠀⠀⠀⠀⠀⠀⠀⠈⠢⣘⣄⣷⣀⠰⣡⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⢿⡇⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠊⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠊⡁⠢⡈⣿⠡⠋⡠⠁⠀⠀⠀⠀⠀⠀⠻⣷⡄⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢻⡄⠀⢠⡇⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⢠⠒⠨⡁⠚⠿⢀⠥⠒⡂⠀⠀⠀⠀⠀⠀⠀⠙⢿⡌⣇⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢻⠀⢸⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠂⠈⠲⣿⠦⢄⠊⢤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⣿⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣼⢀⡿⠋⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣢⡀⠑⢀⣿⠔⠁⠠⠚⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⢹⡇⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⢀⡶⣄⠀⠀⠹⣤⣧⢋⡤⠚⠉⠉⢉⡽⠋⠀⠀⠀⠀⠀⠑⠄⠀⠠⣀⣯⡠⠒⢋⠠⠋⠀⠀⠀⠀⠀⠀⠙⢧⡀⢸⡇⠀⣿⠟
⠀⠀⠀⠀⠀⠀⢸⠀⠘⢧⡀⠀⣸⠿⠿⠤⠤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⡂⠀⠐⠠⢈⣿⡀⠤⠐⠘⢳⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⡇⣼⠋
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⣷⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠠⣀⠈⣿⠁⠐⣀⠠⢀⠀⠀⠀⠀⠀⢺⣦⡀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⣿⣡⠖⠋⠉⠉⢛⡿⠆⠀⠀⠀⠀⠀⠀⠀⢀⠱⠤⠀⠀⠙⠟⠈⠁⠀⠀⠜⠀⠀⠀⠀⠀⠈⠻⣧⡀⢸⡇⠀⢀⣴⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⠷⠶⠶⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠈⠒⢤⣥⣀⠀⠀⠒⡢⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⠀⣠⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠤⢀⣿⣀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⣿⠉⠀⠒⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠂⠄⡀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⢤⣀⠈⠐⠤⣀⣼⠁
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠠⠃⢀⣀⠀⠀⡞⠀⠀⠹⡆⠀⠀⠀⣿⠈⠁
⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠢⠤⠀⠀⠀⢰⡁⠈⣧⠀⢷⡄⠀⠀⡇⢀⡔⠛⣿⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣠⡄⠀⠀⠀⢸⣷⢠⡟⠀⣾⡛⠶⣴⠗⠺⣧⡼⠻⣆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢘⣿⣀⣀⠔⠁⣉⣉⡀⠀⠘⠳⠒⠁⠀⠀⠀⣠⠀⠉⠓⠦⠄⠊
⠀⠀⡠⠒⠉⠉⠉⠉⠉⠉⢹⡏⠁⠀⢠⠏⠀⠀⢹⠀⠀⠀⠀⣠⠤⡀⢰⡇⠰⣦
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢄⣿⣀⢀⠀⠊⠀⣄⠀⣼⠃⠀⠀⣸⣇⡴⠃⠀⠀⡠⠔⠒⢦⡀
⠀⠸⣆⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢸⣏⠀⠀⢸⠇⢰⣏⠀⣿⣆⣀⠔⠹⠏⠉⠓⡶⣮⣀⣀⣀⡼⠁
⠀⠀⠙⠳⠦⠤⠤⠖⠋⠀⠀⠀⠀⠀⢿⣄⣀⠾⠧⠜⠛⠊⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⡀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠜⠁⣠⠤⡔⠀⢨⠀⡤⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠁⠂⠀⠻⣦⠊⠁⢸⣶⣶⠿⢦⣃⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡰⠀⠀⠈⡄⠀⠊⠉⠀⠀⣿⣧⠴⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⣷⢤⣀⡟⠀⠀⠀⠀⠐⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢛⡿⠻⣾⣥⡤⠄⣀⠀⠴⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢩⡂⠀⠢⣠⠀⠀⢹⢀⣠⠄⠀⡄⠀⠀⢠⠋⠀⠀⠀⠉⠛⠶⠦⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⣠⣴⣴⡿⢶⣤⡶⠃⠙⠀⡰⠀⠀⡄⠁⠀⣀⡀⠠⠒⡄
⠀⠀⠀⠀⠐⠈⠀⠘⠂⠉⠑⠾⠈⠻⡇⠀⠿⠁⠀⣴⠞⠁⠀⡾⠀⠀⡔⢸⡀⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠀⠀⣃⢄⡇⠀⣀⣠⠞⠉⠀⠀⢺⠁⠀⢨⡇⣼⡁⠁⢀⡇
⠀⠀⠀⢀⡤⠶⠒⠛⠢⠄⠀⣄⣨⡽⠗⢺⡗⠒⠀⠀⠀⠘⠁⠀⠀⠀⣿⣿⣧⣴⡇⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠤⠤⠢⠖⠁⠀⠀⠈⢻⡄⠀⠀⠀⡈⠀⡠⠈⢀⡿⠁⠀⠁⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢠⠁⠀⠀⢠⠟⠐⢆⣣⣤⣅⡐⠃⠀⢀⡁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣼⠠⠀⢀⡎⠀⠀⠼⢮⢙⣟⠟⢋⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠱⣜⡀⠼⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠉⠐⠦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠱⣼⠉⠫⣄⣀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣿⠀⠀⢠⡿⠉⠓⠒⠤⠤⢤⣀⢀⣀⠀⣀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠉⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠴⠒⠊⠉⠉⠉⠍⣉⠉⠉⠂⠒⠈⠉⡀⠀⠀⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠉⠅⠒⠒⠒⠒⠒⠲⠤⢀⡀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠈⠂⠒⢄⣀⣀⠑⢂⡀⠮⠭⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠠⠤⠤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠒⠄⡀⠀⠀⠈⢓⡤⢤⠖⠊⡩⠋⠁⡜⠁
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⣀⠀⠀⠀⠀⢀⡨⠕⡲⠉⠁⢀⠎⠀⢠⠁⠀⡼⠁
⠀⠀⠀⠀⠀⡔⣁⡤⠤⠤⠤⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣩⠝⡶⠊⠁⠀⢀⠁⠀⢀⠎⠀⢠⠃⢀⠞
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡠⣴⠚⠉⠀⢠⠁⠀⠀⠀⡎⠀⠀⡎⠀⠀⠁⡠⠋
⠀⠀⢠⢃⠀⠠⠄⠀⠠⠤⠤⢀⡀⠀⠀⣀⡤⠚⠁⢠⠁⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⡰⠁⠀⢡⠎
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠋⡝⠀⠀⠀⡘⠀⠀⠀⢰⠃⠀⠀⢀⠇⠀⠠⠁⠀⡰⠁
⠀⡜⠀⠀⠀⠀⠀⠄⠠⣤⢴⠋⠀⠀⢸⠁⠀⠀⢠⠃⠀⠀⠀⡎⠀⠀⠀⠎⠀⠠⠃⢠⠞⠁
⡜⠀⠀⠀⠀⠀⢀⠴⠊⢠⠇⠀⠀⠀⡌⠀⠀⠀⡎⠀⠀⠀⡜⠀⠀⢀⠎⠀⠐⣀⠔⠁
⡇⠒⠉⢉⠩⠟⠁⠀⠀⠎⠀⠀⡠⠊⠀⠀⠀⡼⠀⠀⢀⠎⠀⠀⠐⢁⡤⠖⠉
⡇⡠⠂⠁⡄⠀⠀⣠⠎⠀⠀⡔⠁⠀⢀⠤⠊⢀⡠⢖⣁⣠⠤⠖⠊⠁
⠛⠒⠲⠤⠤⠤⠬⣅⣀⣀⣈⣀⣀⣠⠥⠤⠖⠊⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛
⠀⠀⠀⠀⠀⠀⢀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠴⣦⢴⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠂⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠋⠄⡿⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠂⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢁⠀⠀⠀⠀⡈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⡀⢠⠀⠘⡀⠀⠀⢠⠁⠀⡀⢀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠁⠀⠀⠱⡀⢀⠆⠀⠀⠈⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡀⠀⠀⠀⠙⠃⠀⠀⠀⢀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠴⠠⠄⡀⡀⠀⠀⡌⠑⣄⠀⠀⠀⠀⠀⢸⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡇⠀⠀⠀⠀⠀⡠⠊⢁⠀⠀⢀⢀⠄⠄⠦
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠠⢄⡉⠑⠘⠀⠀⡀⠑⢄⡀⠀⠀⢸⠀⠀⠀⠑⡄⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⢀⡤⠊⢀⠀⠈⠓⠃⢁⡠⠄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠉⠢⣀⠀⠀⠰⠀⠀⠈⠐⠂⠊⠀⠀⠀⠀⠘⡄⠀⠀⡠⠃⠀⠀⠀⠀⠑⠐⠂⠁⠀⠀⠂⠀⠀⡠⠔⠉⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⠓⢄⠀⡆⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⢦⡔⠁⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⢐⠀⡠⠊⠁⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠃⠀⠀⠀⠀⠈⠱⡅⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⠀⠀⠀⢰⠎⠀⠀⠀⠀⠀⠘⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠐⠠⢀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢡⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡌⠀⠀⡔⠁⠀⠀⠀⠀⢀⡀⠔⠂⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠈⠉⠐⠂⠤⠄⣀⣱⡄⠘⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠃⢠⣎⣀⠠⠤⠐⠂⠉⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⠃⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠈⢦⡔⠁⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⠆⠀⠉⠀⠒⠒⠒⠀⠠⠤⠵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠤⠄⠀⠒⠒⠒⠈⠁⠀⠰⠅⢀
⠀⠀⠀⠀⠀⢀⡀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢀⡀
⠀⠀⠀⠀⠻⢄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠛
⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠉⠒⠒⠂⠹⢤⡤⠤⠄⠀⠀⢀⣀⣀⣀⣀⣀⣑⣤⣤⣊⣀⣀⣀⣀⣀⡀⠀⠀⠠⠤⢤⡴⠃⠒⠒⠂⠉⠀⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢉⡉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⠀⠀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠔⠋⡟⢢⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⡴⠋⣀⠔⠁⠀⠀⡇⠀⠉⠢⣀⠑⢤⡀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠔⡡⠔⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⠢⢍⢢⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⢀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣌⣑⡲⡄⠊⣁⣤⣤⣤⣤⡄⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠁⠀⠀⠀⠀⠈⠈⣵⣿⣿⣿⣿⣿⣿⡇⠀⠁⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⠇⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠒⢦⣄⠀⠀⠀⠀⠀⣿⣟⠁⢀⣠⠴⠛⠉⠀⠀⣐⣠⣤⣤⣀⣐⠠⢀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠛⠉⠉⠉⠙⠒⠀⠀⠀⠈⠙⠲⡄⠀⠀⣿⣣⡶⠋⠁⠀⣀⠤⠐⠛⠋⠈⠁⠙⠙⠉⠹⠦⠀⠂
⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠏⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠋⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⠟⠑⠀⠀⠀⢸⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣶⣿⣿⣿⡟⢠⠀⠀⠀⠀⠀⢸⣿⣿⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣼⣿⣿⣿⣿⢫⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡌⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡧⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⣿⣿⠃⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⡜⢡⡏⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠲⠶⠿⠛⠉⠀⡐⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢠⣄⠀⠀⠀⠀⢀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠋⢀⣀⡙⢛⣗⡒⢒⣺⡏⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣰⣗⣿⣙⡿⣿⣓⡷⣏⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⣏⣀⣍⠀⠈⠁⠀⣼⢦⠁⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣿⠁⢋⡙⣷⠶⣶⢾⡟⣁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣟⠀⢸⣇⣏⣾⡉⢹⡷⣏⠀⢛
⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⢮⣍⣉⠈⠉⠈⠁⠹⣧⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣤⡤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣛⣻⣛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡄⠈⢧⠀⠀⠀⠀⣀⢀⠀⢀⣤⠘⠀⠀⠀⠀⢹⣧⠀⡂⣠⡀⠚⠓⡼⠳⠄⠒⡢⠰⡷⠛⠀⢹⢿⣿⠏⠈⣯⡀⡀⠀⠀⠈⠒⠤
⠀⠀⠀⠣⡀⡀⡦⢄⠛⢳⣿⡗⠁⠀⠀⠀⠀⢸⡯⠀⠁⢀⣶⡄⠀⠁⠶⠚⠛⠃⠁⠀⠀⠠⠖⢻⡭⣣⠀⠁⠘⣮⣶⣄⠀⡠⠤
⡀⠀⣆⠀⠁⠀⢻⡀⠙⠢⡄⠁⠠⠀⠂⠀⠀⠚⠀⠀⠀⠈⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⠀⠀⠀⠈⠉⠀⠀⠠
⡇⠀⠈⣆⠀⡄⠸⡿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠗⠀⠀⠘⠂⠁⠀⣀⠀⠉⠲⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤
⢀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣦⠀⠀⠠⠀⠡⠁
⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣷⣄⡀⠙⢷
⠀⠀⠀⠂⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡛⠳⠦
⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⡶
⣡⡀⠀⢘⢴⣗⣰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦
⣮⣷⠀⣼⣿⣿⠀⣦⠠⡞⠿⠛⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠁⠀⠀⠀⠒⠤⢀
⣿⢿⣆⣟⢹⣿⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⢳⣾⡝⢻⣿⣥⣠⣤⢶⠀⢴⢶⡄⠄⡀⠀⠀⠀⢠⠊⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠠⢀⣶⡋
⠀⣿⣿⣦⡙⠋⠉⠀⠀⠠⠀⠤⠶⢆⠀⠀⠀⠀⠆⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠘⠿⠄⠀⡀⣀⠀⡀⠉
⠀⡻⡀⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡘⠀⠔⢀⠋⠀⠀⠴⠊⠉⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠛⠉⠀⠁⠂

⠀⠀⡀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢿⣆⠀⢸⣆⠘⠋⣀⠀⠀⠀⣷⡆⢀⠀⢠⣄⠀⠰⠀⠀⠀⢀⠘⠇⠀⠀⠀⢠⣄⠁⣀⠀⠀⠠⣤⣁⣼⠂⠀⠀⢀⠛⢀⠂
⠀⠀⠈⠻⠆⠈⡛⠀⠾⠋⠀⠀⠀⠈⠁⡿⠁⠀⠁⠲⠀⠀⠀⢀⡘⢧⡶⠂⠀⠀⠀⣉⠾⠋⠀⠀⠀⣬⡙⠁⣀⠀⠀⠈⣈⠺⠈
⠀⠀⢶⣦⡀⣰⠆⢀⣴⡶⠀⠀⠀⠀⠃⠀⠀⠀⠰⡀⢀⡀⠀⠈⣿⣆⣴⡇⠀⠀⠈⢻⣧⡶⠀⠀⠀⠉⠛⣿⡟⠀⠀⠀⠉⠲
⠀⠀⠀⠉⠀⡿⠀⠈⠁⠀⠀⠀⠀⠶⠶⠂⠀⠀⠀⡹⠈⠁⠀⠀⠈⠉⠋⠀⠀⠀⠸⣇⣡⡄⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢈⡙⡔⠗
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⠰⣶⡄⢠⡄⠀⠰⢾⣶⢄⢿⠄⠀⠀⠸⣶⣾⣯⡔⠀⠀⢀⡀⠀⡄⣧⡀⠀⢰⣿⠀⠀⡀⠀⠀⢠⡄⠀⠀⡀
⠀⠀⢈⡄⠈⠀⠃⣀⠀⠀⢠⣌⠃⢸⠇⢀⡀⠀⠺⠃⠙⢇⣴⡆⠀⠀⠁⠀⠀⠀⠰⣌⢰⣧⡿⠆⠀⢠⣜⣡⠀⠀⠙⠳⠀⠘⠃⠀⢚⡁
⠀⠀⢀⠀⡄⠀⠀⣨⠀⢰⣬⡛⠀⡆⢰⣿⠃⠀⠀⠀⠶⢌⣾⠀⠀⠀⣠⠀⠀⠀⢳⣶⡧⠍⠀⠀⠀⢦⡟⢋⠀⠀⢠⣤⣦⣶⣤⠛⠉⠁
⠀⠀⠉⠀⠀⠐⠀⠀⠀⠘⠿⣷⠀⠄⠈⣁⣀⠀⠀⠀⠀⠀⠁⠀⠐⢶⣿⣶⠆⠀⢸⠃⠀⠀⠀⠀⠠⣤⣷⣿⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠐⠛⠛⠛⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠁⠀⠉⠛⢦⡀⠀⣠⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠈⢷⡾⠋⠁⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠳⡘⢦⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⢀⡴⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠉⠙⠳⢦⡀⠀⠙⢎⠳⡀⠀⢠⠇⢀⠔⣀⣠⠤⣖⣭⡾⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠱⡙⠀⣾⣾⡿⣛⣭⡶⠟⠋⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠘⢀⠟⣡⠾⠋⠁⣠⠴⣛⡭⠤⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣹⣿⣦⣄⣀⠚⠁⠤⢒⡩⠔⠋⠁⠀⠀⠀⠀⠀⡘⣧
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⣀⡬⠿⣿⣿⠟⢣⡟⢽⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣸⢣⡟
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠈⠉⠁⣠⣾⠟⠁⠀⣼⠃⠈⢿⢿⣷⣍⠑⠂⠀⠀⠀⣀⣪⣴⠟
⠀⠀⠀⠀⠀⠀⢸⣇⢦⡀⠀⢀⡠⣢⣾⠟⠁⠀⠀⢀⣿⠀⠀⠈⣿⣿⣿⣷⣄⠀⠀⠈⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠹⢷⣯⣭⣽⡾⠟⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡼⣿⣯⡙⠷⠄⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢳⡹⣿⣿⣦⣀⣀⣠⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠳⣭⣻⣿⣿⣛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⣀⣀⣤⣤⠤⣤⣤⣀⣀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠉⠉⠁⠀⠀⢌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠔⣡⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⡞⠁⠀⠀⠀⠀⠀⠀⣰⢲⡔⡄⠑⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⢀⡠⢂⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇⠀⠀⠱⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠃⣀⡀⠀⠀⠎⣰⡟⠁⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⢀⡄⢸⣇⡠⠤⢄⣀⣀⡀⠀⢐⢠⡀
⠀⠀⠀⠀⠀⠀⣼⠇⠀⢈⣉⠓⠀⢠⡿⠀⠀⠀⠿⠀⠀⣀⣀⢤⠀⢀⣐⡒⡄⠀⠀⠀⢀⣇⡾⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡑⢝⠂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⣉⡉⠁⣼⡇⠀⠀⠀⠀⠀⣸⠧⣌⣧⣀⣿⢧⣹⡇⣀⡴⠚⠉⣿⠃⢀⣀⠐⠚⠋⣁⡀⠀⠀⠀⠀⠤⠀⠀⠈⢻⡜⠂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⡠⠀⣿⡀⠈⠉⠉⢳⡆⢿⠀⢠⡏⠈⣏⠀⣸⠏⢹⡁⠀⣰⢿⣠⣟⡽⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠚⠀⠀⠘⣧⡀⠀⠀⣸⡇⠈⠑⠩⠄⠀⢙⠚⡩⠀⠈⢑⣊⠅⠀⠉⠻⠤⠄⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠪⣙⠒⣚⠏⠀⣠⠖⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⢀⠄⠀⠀⣀⣠⣄⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣾⢸⠇⠀⠀⠀⠀⠀⠀⠻⠟⠃⠀⠀⢀⡿
⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⣿⠀⣠⠟⠁⠀⠀⣸⠃⠀⣾⣷⢠⡟⠉⠓⠀⣿⡿⢖⡦⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⢀⠄⠀⠉⠓⠒⢿⡛⠁⠀⠀⠀⣴⢿⠀⠀⢷⡿⢸⣇⠀⠀⢀⣿⣏⠁⠀⢀⣄⡀⠀⠀⠀⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠘⣧⠈⢆⠀⠀⠀⠀⠀⠀⠃⠆⠀⠘⣎⠳⢤⣀⣠⠼⠋⠈⠓⠤⠚⠓⠋⢉⡓⠚⠍⠃⠈⠙⠢⢿⡞⠁⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡙⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⣀⠤⠖⠋⠀⠈⠳⣄⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⣌⣉⠉⠛⠛⠋⢉⣩⠴⠂⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡢⢊⣴⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣄⣀⣀⣀⣀⣀⣀⠤⠖⠋⡡⢞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣚⣚⣫⣶⣿⣿⣿⣷⣮⢳⡄⢀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢟⣫⣶⣶⣮⣝⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣾⣿⣿⣿⣿⣿⣷⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣇⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣮⡻⣿⣿⣿⣷⢻⣿⣿⢧⣿⣿⠿⣟⣿⣿⣿⣿⣿⣿⣷⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣵⣶⣶⣶⣶⣯⣤⣉⡻⢿⣧⢻⡏⡾⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢷⣮⡁⢐⣛⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡟⣼⡏⣿⣿⣿⣶⣦⣌⡩⣭⣿⣭⡽⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⡸⣿⣿⣿⣿⣿⣿⣶⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⣿⣿⣿⣿⣿⣿⡟⣼⢸⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⡎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡻⣿⣿⣿⣿⣿⠟⣸⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣽⣛⣭⣴⢧⣿⢡⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⢸⣮⡻⣿⣿⣿⣿⠿⣡⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⣿⢸⠈⠙⠲⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣛⣛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢀⠁⠐⠊⠉⠑⠸⢍⢱⠀⠀⠀⠀⢠⠤⢄⡼⠀⠈⠦⡤⠔⠛⡆⠀⠀⠀⠀⢇⢢⠈⠉⠙⠛⠋⠉⠉⢡⠂⡅
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡹⠒⢄⠀⠀⡗⠀⡸⠁⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠈⣺⠀⠀⠀⠀⠀⠀⠀⠘⡋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠑⠒⠺⠀⢀⠇⠀⣇⠀⠀⠀⠂⠀⠀⠀⡟⠉⢹⠀⢰⠋⣈⡇⠀⠀⠀⠀⠀⠀⣮⡈⢱
⠀⢀⡰⠃⠀⠀⠀⢀⠀⠀⠀⠀⢠⠃⠀⠘⡀⠀⠈⠀⠠⢀⣀⣀⡀⠴⠃⠀⢸⠀⠀⢻⡀⠘⡀⠀⠀⠂⠀⡸⠀⣨⠃
⠀⣦⣀⡔⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠙⢤⠀⠀⠀⠀⠀⠀⠀⠁⣀⡠⠃⠀⠀⠀⢳⠄⠈⠀⠈⠐⠊⠀⢴⠃
⠀⢇⠈⢩⡤⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠘⡄⢢⡶⠆⠀⠀⠀⡀⠀⠠⣸⠀⠀⠀⠀⢻⠀⠠⡀⠀⠀⢀⠄⠀⢸⠀⠀⠀⠀⠀⢣⠀⠓⡀⠀⠀⡰⠃⢠⠃
⠀⠀⠘⠢⠤⠒⠉⠉⠉⠳⠤⠴⠃⠀⠀⠀⠀⠀⠣⠤⠕⠒⠒⠮⠤⠔⠁⠀⠀⠀⠀⠀⠈⠦⠤⠎⠉⠙⠦⠤⠎
⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢈⡾⠷⠀⠾⢶⡡⠀⠀⠀⠀⠀⠠⠒⠢⠤⠠⠔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠑⠒⠶⠒⠊⢠
⠀⠀⣎⣡⠃⠀⠈⠈⠀⠀⠙⡀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠤⡀⠀⢀⠤⠀⢃
⠀⠀⡈⠋⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠸⠀⠀⠄⠀⠀⠀⠀⠘⡧⠈⢂⠀⠀⡊⠈⠑⠁⢁⠀⠀⠀⠐⠀⢠⣊⡉⡆
⠀⢠⠁⠁⠀⠀⠀⢀⠀⠀⠀⢈⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⡠⠃⠀⡈⡠⠤⡏⢀⣴⣿⣷⣿⣦⡀⢀⠐⠺⢁⣀⡇
⠀⠀⢢⠄⠀⠀⡨⠶⣏⠀⠀⠀⡃⠀⢀⠊⠀⠁⠀⢀⠠⠀⠈⠁⢀⠜⠀⠃⡄⢰⡏⣷⡆⠀⣴⡎⢷⡒⡆⢀⠬⠻⠁
⠀⠀⠘⠀⠀⢰⢁⣽⡏⠹⠉⠈⠇⠀⠘⡤⠄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠄⣼⣗⠮⠄⠀⠨⠔⣸⣧⢰⠂⠰⠎⠄
⠀⠀⠰⡀⠀⢴⠾⠷⢅⠆⠀⢀⠇⠀⠀⢃⠀⣀⠀⠀⠀⠀⣀⠀⢠⠀⠀⠀⠀⠸⣮⣿⡞⠛⣿⣿⣷⠩⣿⣆⣦⠈
⠀⠀⠀⠈⠁⠈⠁⠀⠉⠀⠉⠁⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⣤⣀⠀⠀⣀⣠⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡯⠀⣠⠤⣤⣩⠳⠞⣍⣠⠤⣄⠀⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣂⣀⣰⢂⠏⠠⢳⡞⢄⠸⣖⣆⣑⣐⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢀⢖⢁⡌⠠⠨⠔⣡⣌⡁⠁⠄⢡⡈⡲⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠗⢯⠥⣴⠶⡟⠸⡁⢻⠶⣦⠤⠭⡻⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡧⢞⠓⣉⣁⣾⣷⣈⣉⠎⡳⢼⣿⠏
⠀⠀⠀⠀⠀⣀⡤⠴⠖⢒⢲⣦⣤⣌⣷⡊⣰⠖⢉⢹⡟⡨⠺⣦⠑⣾⣣⣤⣴⡖⡒⠒⠦⢤⣀
⠀⠀⠀⢠⣾⠡⢰⠐⠈⣏⢎⢫⡻⠼⢨⡿⢷⣶⡈⡟⢳⠃⣴⣾⢻⡁⠧⢽⡝⢱⢩⠃⠆⡄⠌⢳⡄
⠀⠀⠀⡏⠘⡨⢨⣣⡐⡁⣺⡯⣕⠲⡽⢧⣨⡭⡽⣷⣾⢯⣭⣉⡤⢯⡖⢊⢽⣇⢺⣂⢜⡁⢤⠋⢸
⠀⠀⠀⢗⠀⢷⣩⠞⢤⡩⢽⢟⣊⣴⣷⣶⢞⡯⠗⣿⣿⠲⢽⡲⣶⣾⣦⣘⡫⡇⠝⢣⠳⢌⡼⠀⣸
⠀⠀⠀⠈⠷⣄⠧⠴⣵⡃⠾⣗⡊⣉⠋⠳⣬⣀⢺⡏⢸⡗⢀⡥⢞⠙⣉⣑⣚⠿⢈⣯⠶⠼⣩⡼⠃
⠀⠀⠀⠀⠀⢈⡏⡴⠓⠊⠀⠈⠹⣦⣁⣈⡤⣉⡿⢸⡇⢻⣉⠤⣁⣌⣴⠏⢁⠀⠕⠚⢣⢹⡅
⠀⠀⠀⠀⠀⢸⡁⢺⡓⣙⣢⠈⢤⡙⣌⣽⣶⠟⠀⢸⡇⠀⠻⣶⣯⣡⢏⡤⠅⢤⣋⠚⡇⢨⡇
⠀⠀⠀⠀⠀⠈⢧⠀⠑⠪⠘⡆⢄⡳⢓⡽⠃⠀⠀⢸⣿⠀⠀⠈⢯⡒⢾⡠⢰⠃⠄⠊⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠮⢧⣭⡬⠤⠚⠁⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠈⠓⠤⢥⣬⡴⠷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⢷