Weakling Emojis & Symbols

⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁⠀⣻⣮⣻⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⢃⣯⠁⠀⢸⣿⣿⡿⠇⡙⡯⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠀⢘⡛⠛⠩⢠⣼⣿⣷⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⠛⣹⡄⠀⠐⠖⢀⣠⡟⢶⣿⡯⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⠟⠁⢠⣿⡷⣤⡔⠙⣆⠀⠘⣿⣼⣧⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⠃⠀⠀⠘⠋⠙⡌⢻⡄⠈⣧⠀⢹⠘⣿⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣄⡂⠀⠀⢀⣀⣸⡄⢻⡄⠸⣀⡈⠀⡄⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡏⠋⠙⢾⡆⣷⢀⣵⣾⣿⣟⡉⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣧⣆⠀⠀⠓⠻⠯⣈⠉⠐⠤⣈⠃⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡿⠆⠄⢢⣁⣬⣀⠙⠲⣄⠈⣹⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠘⠛⠛⠉⠉⠛⠐⠛⠉⠓⠈⠓⠀⠈⠑⠋⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠴⠊⠀⠚⠓⡾⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡸⠁⠀⠀⠀⠀⢀⡜⠁⠀⠈⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠴⠀⠀⠀⠀⠁⠶⠀⠀⠀⠐⢁⡀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⣹⠋⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢠⡟⠀⠀⠀⠀⢠⠾⠉⢸⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣾⠀⠀⠀⠀⠀⠀⠀⢠⣏⡀⠀⠘⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⡘⠀⠄⠀⠀⠀⠀⠀⢼⣿⡷⠀⠀⠀⠀⠘⢿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠟⠁⠀⠀⠂⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⣠⠾⠋⠛⢆⡀⠀⠀⠀⠀⠀⢀⣤⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠛⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢀⡘⠉⠟⢲⠟⢢⣴⠋⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣴⠁⠀⠀⠀⣸⡇⣤⠀⡄⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠙⠛⢦⣄⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⣿⠀⠀⠀⣶⠉⠀⠸⣿⣋⣺⡏
⠀⠀⠀⠀⠀⠀⠀⢀⢁⡀⠀⠀⠀⠀⠈⠳⢦⣤⡀⠀⠀⢸⠀⠀⠀⠛⢧⠀⠸⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢦⡀⠀⠈⠉⢻⡁⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢀⠋⢠⡆⠀⠀⠀⠀⠀⠀⢀⣬⠋⠀⠀⠀⠦⠀⠀⠀⠈⠳⣄⠀⠀⠙⢣⣄⠀⢀⡀⠀⠀⠀⠀⢻⣄⣀⣀⣴⠟⠻⡆⠀⠀⢸⣅⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠚⠀⡼⠻⣦⡀⠀⡀⢀⣀⢸⠇⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠈⠓⠢⣄⡰⠉⠳⠞⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠷⠀⢰⠟⠻⣤⣤⣤⠾⠋
⠀⠀⠀⠀⠀⡘⠀⡶⠁⠀⠈⠙⠛⠳⠚⠛⠁⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⢠⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡏⠀⠀⠀⠁
⠀⠀⠀⠀⡰⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⢀⡀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⡦
⠀⠀⠀⣠⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢰⠁
⠀⠀⣰⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠿⠷⣦⣀⡜⠀⠀⢲⡄⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⣿
⠀⢰⡏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠋⠁⠀⠀⠀⠀⠈⠿⠃⠀⠀⢴⡟⠋⠉⠉⠙⠃⢄⣀⣀⡀⠀⠀⠀⠀⢀⣠⣿⠁⠀⢰⠇
⢀⡏⠀⠀⠀⣠⡀⠀⠀⣤⠤⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠉⠁⠁⠀⠉⠁⠀⠀⣾
⣿⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⣿⠀⠀⠀⠀⢀⣀⣤⣴⠂⠀⠀⠀⠀⣠⣤⠞⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⢉⡄⠀⠀⠀⢀⡞⠁⠀⠀⠻⣦⣼⡇
⣿⠀⣠⣴⠶⠉⠁⠀⠹⡇⠀⠀⠀⢰⡟⠁⠀⠀⣿⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠸⡇⠀⠀⠀⣼⡇
⣿⡠⠋⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢻⡇⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⡀⠁⠀⠀⠀⠙⣁
⠉⠁⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣶⠀⠀⠀⠀⡇⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⢠⡜⠁⣠⠇⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠹⣦⡀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣀⢀⣠⡾⠃⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠉⠻⠶⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠷⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠗⠀⠈⠉⠛⠛⠛⠛⠛⠛⠊⠉⠉⣠⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠉⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⡀⠀⣠⣀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠚⠉⠙⣿⠉⠉⠉⠉⡛⠉⠉⠉⡟⠉⠁⠀⣸⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇⠀⠀⠀⡀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢸⡇⠀⠀⠀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢰⠄⠀⠀⠀⠃⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡤⠀⡀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡏⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠂⠀⢼⣇⠀⡇⠀⠀⠀⠀⣤⣿⠄⠚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⢶⠶⠶⠾⡇⠀⠀⠰⠟⠻⣇⠀⠀⢀⠔⠛⠁⠀⢰⣎⢠⣤⡶⠶⠖⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⠀⠀⠀⠀⣤⣤⠌⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢦⣀⡀⠀⠀⢀⣀⣀⡀⠀⠐⠶⠶⠶⢿⠇⠀⢹⣄⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠷⢤⣈⣙⠛⠂⠀⠤⠤⠄⠀⠀⠒⠒⠒⠊⠀⠀⠾⠥⠤⠈⠉⠙⠻⠷⣆⣀⣐⡚⠣⣤⠞⠉

⠀⠀⠀⠀⣴⣶⣶⣤⡀⣠⠶⢻⡆
⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⠀⢸⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣧⡀⠀⠈⢿⡀⠀⠀⠀⠀⠈⠙⠛⢾⠁⠀⠒⠲⠦
⣴⣷⣦⣾⣿⣿⢿⣿⣿⣷⣄⠀⠈⢷⡀⠀⠀⢠⡶⠛⠛⢶⣀⣴⠞⠳⢦⡀
⢿⡟⠻⣿⣿⠇⢸⣿⣿⣿⣿⣦⠀⠀⢻⡄⢰⡏⠀⠀⠀⠀⣿⠁⠀⠀⠈⣷
⠀⠀⠀⠈⠁⠀⠀⠋⠁⠈⢻⣿⣷⡀⠀⠹⣾⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣀⣤⣿⣇⠀⠀⠀⣰⠏⠻⢦⣤⣼⣇⣀⣀⣀⣀⣀⣤⣤⣤⠴⠶⠶⠶⠚⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡉⠀⠙⠓⠒⠚⠁⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠲⣤⣼
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⣠⡴⠲⠿⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⣦⣤⣤⣤⣤⣥⣤⡷⠶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠃⠰⢾⡿⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⣤⠴⠶⢿⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣴⢟⢛⣿⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠸⣿⣿⢸⡏⠀⠀⢰⡆⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢀⣿⡿⠿⠛⠛⠛⢛⣷⢋⡤⣶⠾⠳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠈⠁⠀⠀⠀⠀⢠⡞⠁⠘⢻⣧⡐⢛⡦⠀⠉⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠘⠾⠟⠋⠙⠛⢻⣣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠇⠸⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡾⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣫⡥⣶⠆⠀⣿⠶⠶⠶⠒⠛⠿⠿⠿⠿⠭⣭⣭⣭⣥⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡶⠋⠀⠀⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⠋⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣸⠇⠀⠀⠀⢀⡿⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⡾⠋⠉⠛⠁⠿⣄⡀⠀⠀⡼⠁⠀⠀⠀⣤⠘⣧
⠀⠀⠀⠀⠀⢸⣅⡀⠀⠀⠀⣠⣌⣽⣶⣾⣁⠀⠀⠀⠀⠉⣀⣽⣿⡶
⠀⠀⠀⠀⠀⠀⠘⠓⢻⡄⢠⡏⠁⠀⢹⡾⣟⠛⠛⠛⠛⠛⠉⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠂⡀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠰⠶⠾⣾⠛⠃⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠀⠀⠠⠀⠀⣿⣧⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⢶⣿⣿⠋⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡿⠀⢸⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢱⡇⢠⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⢃⣿⣴⣯⣤⣉⣛⣒⣒⠒⠾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠈⠁⠀⠀⣀⠈⣉⢩⡉⢻⣯⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠂⠲⣌⣷⣘⡿⠟⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠙⠛⠓⠛⠀⠐⠛⠚⠛⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠙⠲⢄⡀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⣀⣀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⡏⣧
⠀⠀⠀⠀⠀⢀⣶⢣⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢸⡇⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣱⣶⣄
⠀⠀⠀⠀⠀⣾⢧⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢸⡇⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢹⣿⣿⣿⡆
⠀⠀⠀⠀⢰⣿⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣯⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣈⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⡇⠉⠉⠉⠉⠉⠉⠉⠉⠻⠿⠟⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠻⠿⠟⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⡼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠶⠶⠞⠛⠋⠉⠉⠉⠉⠛⠛⠲⠶⠦⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣷⡻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣎⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣠⣴⣶⣶⣦⡀

⠀⠀⠀⢀⡤⡄⠀⢀⣤⣞⣡⡄
⠀⠀⡴⣿⣿⣯⢞⡵⣛⠝⠋
⢠⢖⡫⢞⡿⠁⢀⣔⣥
⠸⣾⣿⣿⡟⣀⠜⠻⡀⠀⠀⠀⡀⠀⠀⠀⠐⠁⢀⣀⡀
⡇⢿⣿⣿⡿⣇⠀⠀⠱⣄⠀⠈⡤⢤⡀⠀⢀⡞⠁⠀⢸⠆
⠑⠓⣾⣅⠀⠀⠑⢢⡄⠀⠑⠲⣇⠀⠈⠳⣼⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⡏⢫⡷⣄⠀⢀⣭⡓⢤⡀⠈⢧⡀⠀⠈⢳⡯⠥⠤⠤⠤⠤⠴⠖⠒⠉⠁⠀⠀⠀⢈⡇
⠀⠀⠳⢿⡈⢮⡑⢮⡀⠈⠙⢿⣦⡀⠙⠦⣤⡠⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣂⡤⠚
⠀⠀⠀⠀⠉⠉⠈⠉⠉⢀⡠⠤⠻⠿⠆⠀⠀⠠⠃⠀⠀⢻⡁⠒⠒⠒⠒⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠐⠋⣋⠥⠒⢆⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠁⠀⠀⠈⢆⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢲⠫⡝⠛⢽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢣⠀⠀⠙⠿⠓⠲⡲⡄⢀⣠⣶⠿⠛⢁⣸⣬⡤⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢹⡩⠟⠋⠁⠀⠀⢧⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡦⣀⠀⠀⡼⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⡷⢄⢄⡘⣛⢟⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢀⣸⠀⠀⠱⠴⢳
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡤⢒⣩⠥⠐⠒⠒⠒⢲⡀⢣⠈⠁⠈⠳⣌⡲⢄⠀⠀⠀⠀⠀⠀⢑⣬⡇⠀⠀⠘⠦⡞⠒
⠀⠀⠀⠀⠀⠀⠘⢧⡈⠣⣄⣠⠖⡡⠚⠉⠀⠀⠀⠀⠀⢰⢫⡧⠼⠁⠀⠀⠀⠈⢏⠢⣝⠢⠴⢖⡢⠭⠕⠒⢻⣀⣀⣀⠠⠗⠒
⠀⠀⠀⠀⠀⣔⠒⠒⠮⢓⡀⠀⢈⣓⣒⠈⣹⠆⠀⠀⠀⠠⡗⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠈⠉⣊⠭⠖⢃⠜⠁⠀⠈⠉⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣀⣀⣀⣀⣀⣀⣀⣀⢤⣤⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠿⠯⡧⠘⠚⠯⡟⠚⣄⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⠏⠑⣷⣶⣿⠷⠶⣶⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⣶⠾⠾⠿⢿⣶⡶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⣰⢫⠏⠀⠀⢀⠞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣿⢉⠙⢟⣇⡀⠁⠀⣈⡆⣰⠊⠁⠈⠓⠒⣶⠦⠐⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣖⢿⡏⡻⢛⠝⠉⠀⠙⠭⢒⠲⢾⣟⠋⣝⣶⡽⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠽⠿⠊⠁⠀⠀⠀⠀⠀⠀⠈⠓⠪⢷⣿⡾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠂⠉⠉⠉⠉⠉⠉⠉⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡼⠁⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠶⢦⣤⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣬⣤⣴⠁⠀⠀⠀⠘⠀⠀⠀⠀⠤⠄⡀⠀⣀⠠⠤⢀⠈⠻⣆⠀⠀⠸⡇⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠦⠁⠀⠀⠀⠑⠄⠈⠀⠀⠀⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⡀⠀⢀⣠⡀⢀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡿⠿⣿⣿⣿⠇⠀⠀⠀⠀⢂⠀⠀⠰⣟⡇⠐⠃⠀⠸⠯⣇⠀⠀⠀⠀⢠⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠡⠖⠀⠐⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⣾⠁⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⠛⣦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢸⣄⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠊⠁⠀⡄⠀⠀⠀⠀⠀⠀⠉⠒⠀⠤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡿⠀⡠⡶⠛⠛⠛⠛⠛⠲⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠐⠂⠠⠄⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⣸⠃⢰⣙⢀⡖⠉⢀⡀⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢂⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠂⠁⠀⠀⠀⠀⠀⣰⠃⠀⠀⠹⣌⣧⡀⠘⠋⢉⡇⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠉⠃⢚⠉⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⠠⢄⡀⠀⠁⠒⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⣷⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢡⠊⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠟⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠒⠒⠒⠂⠉⠁⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⢠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣠⡶⠛⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠙⣤⣠⣤⠎
⠀⠀⠀⠀⠀⢀⡞⠉⠙⢦⠀⠀⠀⠀⣼⠃⠀⠀⢀⣴⠛⠁⠁⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⣀⡀⢠⢦⣼⠃⠀⢀⡞⠀⠀⠀⢰⠇⠀⠀⢠⠟⠀⠑⠦⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⢰⠏⢹⡏⣸⡏⠀⠀⣼⣃⣠⠔⠒⠺⠀⠀⢀⡏⠀⠀⠀⣤⠶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠛⠦⣾⢁⡿⠀⠀⢠⡿⠉⡇⠀⠀⠀⠀⠀⠈⠋⠘⢦⢰⠇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠛⢾⠇⠀⠀⡼⢷⠎⠀⠀⠀⠀⠀⠀⢀⠞⠳⢾⡏⠀⠀⢰⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⣻⠃
⠀⠀⠀⡎⠀⠀⢰⠁⠘⢦⣤⠀⠀⠀⣀⠀⢨⣀⡀⣼⠀⠀⢀⡟⢰⡧⣤⡤⣤⠤⠤⠤⠤⠤⠤⠔⠒⠒⠋⠉⠁⠀⠀⠀⡿
⠀⠀⠀⠙⠶⠶⠏⠀⠀⠀⠈⠲⠤⠴⠿⣀⣠⠟⢹⡇⠀⠀⣼⠁⣿⠀⣸⡄⠂⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⡿⠾⠏⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⡿⠀⢠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠁⠀⠀⠄⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣀⣀⡀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣤⣀⣘⢀⠃⠀⠂⡄⠀⠀⠀⠀⠀⣀⣠⠔⢿⡟⠋⠉⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⡿⠿⢶⣷⡶⠒⠛⠛⠉⠉⠀⠀⣀⣅⠀⠀⢦⣎⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣤⠀⠀⣌⣀⡀⠀⠀⠀⠀⠰⢏⡠⢧⣤⠴⠟⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⡉⠷⢄⣀⣹⠭⡹⠆⠀⠀⠀⡀⠌⣳⡀⠀⠀⢀⡦⠄⠉⠐⡒⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠈⣱⠀⠀⠀⠀⣼⠳⢤⡀⠀⢸⠁⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⡁⠘⠀⠀⠀⠈⠁⠒⠂⠉⠁⠀⠀⢹⡄⡿⣄⠀⠠⠐⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⡤⠄⡀⢀⡼⠃⠘⠤⣛⠶⣌⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⡑⢦⡜⡩⠒⠁⠀⠀⠀⠈⠁⠚⠋⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣤⠒⠍⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠠⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡳⠦⠀⠀⠀⠀⠤⢀⠐⠃⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠁⠀⠀⠌⠀⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠰⠀⠀⠀⠠⠀⠀⠀⢠⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡴⠲⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡿⠀⠀⠀⠀⠀⢈⡀⠐⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠄⠀⠀⠀⠀⠀⣾⠇⠀⠀⢀⡸⠀⠀⠀⠀⢀⣤⢤⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠰⠉⠀⢀⠒⠒⠢⢀⡤⠉⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠁⠀⠀⠼⠀⣀⡀⣼⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠐⣤⡀⠀⠘⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢀⠀⠉⠰⠃⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⡜⠀⠀⠀⡀⠀⠀⠠⠀⠀⠑⢄⠀⡀⠀⠀⠈⠦⠀⠤⠊⠗⠀⢀⡦⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠀⢤⠰⠆⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⡴⠆⠁⠀⠀⠈⠻⠀⠀⢼⠂⠉⠉⠂⣀⠀⠀⠀⠀⠀⢰⠏⠀⡤
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠦⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢀⣀⡄⠀⠀⠀⣀⠠⢶⣄⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠂⠀⠀⠀⡐⠀⠀⢆⣸⠁
⠀⠀⠀⠀⠀⠀⠘⢀⠔⠈⠁⠀⠡⠀⠀⢠⠁⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠒⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠈⠁⠂⠀⠀⠀⠐⠒⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢻⠉⠉⠙⠛⠉⢹⠋⠀⢠⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠘⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢀⡀⢀⡉⠀⢹⣤⡇⠀⠀⠸⠎⠉⣸⢀⣀⣠⣤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢰⠀⠀⠀⠡⠈⠉⠀⠀⠀⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠲⠀⠀⠀⠀⠀⠀⢉⣉⠙⠀⢠⠆⠠⠤⣤⣀⠀⠀⡀⠒⡿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣬⣶⣶⣶⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣷⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠾⠟⢦⣴⡟⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠛⠀⠈⣉⣀⡀⠀⠁⠀⠀⠀⠸⠟⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠛⠃⠀⠀⠀⠀⠿⠈⠁⢀⣤⡄⠀⠀⣠⣄⠀⠘⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣴⠶⢦⠀⢀⣶⣦⡘⠿⠃⠀⠾⠋⠉⠀⠐⠟⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣶⡿⠀⠀⠀⠘⠃⠘⠃⠀⣼⠛⠦⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⠷⣤⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⣦⠀⠀⣠⡶⠛⠛⠛⠳⢦⡀⠀⠀⢰⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⢿⣇⠀⠀⣠⣤⣬⡛⠛⠉⢠⡾⠛⠆⠀⠀⠻⢿⣦⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡦⣀⡿⠀⠀⠉⣀⠘⠃⠀⠀⠈⢁⣠⣤⡶⠂⠀⢸⣿⣠⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡰⠏⠳⣄⠀⠀⣹⣿⣿⡀⠀⠀⣸⣯⣽⡇⠀⠀⢾⡇⢉⡀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣴⣿⠀⠀⠻⠟⠉⠉⣤⠈⣡⡙⠛⠁⠀⠀⠈⣿⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢻⡄⠀⠀⠀⣰⢋⡀⣙⣿⠄⠀⠀⢀⣠⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⠀⠀⠈⠙⠿⠟⢉⣤⣄⠀⢺⡇⠈⠻⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⣀⣾⣶⣾⣿⡿⠃⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⢿⡀⠛⠀⠀⣴⡟⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢶⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠙⢷⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡟⠋⠀⠈⣹⣿⣿⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠈⢿⣿⣧⣀⣴⣿⡿⠃⠀⠀⠉⠛⠛⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠋⠀⠀⠀⠀⠻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣷⢀⡀⠀⠀⢀⣀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣿⣿⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡏⠹⣦⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠈⢷⡀⠀⠀⠀⣿⠘⠃⠀⠀⢸⡇⠀⢹⡆⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠈⣷⠀⢸⡇⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿⠀⢸⡇⠀⠀⠀⠘⣇⢠⠄⠀⠀⣿⡇⢸⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡇⢸⣇⠀⠀⠀⠐⣿⠀⠀⠀⠀⢸⣧⠀⣿⡄⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠃⢸⣿⠀⠀⠀⠀⢻⡄⢤⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⠀⠈⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡄⠈⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⣿⠀⢰⣿⣀⣀⣀⣀⣸⣏⣀⣀⣀⣠⣼⡷⠀⢿⡆⠀⣿⡇⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣽⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣶⣿⡇⠀⠀⠸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⣧⡀⣿⠀⢸⡇⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣧⡀⢀⣾⡿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡏⢉⠙⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⢿⣷⣦⡀⢀⣾⠟⠛⢿⡏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢿⣿⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠻⣿⣿⣿⠂⢠⡿⠁⠀⢸⡏⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⣿⡀⢀⣀⣠⣾⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤⣾⣇⠀⢀⣾⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣼⡇⠹⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⢹⣿⣿⣶⣶⣾⣿⣿⣷⣶⣾⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⡟⠁⠀⠀⠀⢹⡏⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢸⡇⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣧⠀⠀⠀⠀⢸⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡆⠀⠀⠀⢸⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢿⡇⠀⠀⠀⢸⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢸⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⡇⠀⠀⠀⢸⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⠁⠀⠀⠀⠀⠙⠛⠛⠛⠃

⠀⠀⠀⠀⣰⠖⠲⢦⡀⢀⡴⠚⢿
⠀⠀⠀⢰⡏⠀⠀⠀⣹⠏⠀⠀⢸
⠀⠀⠀⢸⡇⠀⠀⠰⣯⡀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠐⠛⠳⣾⠀⠀⠤⠴⢤
⣠⠶⣄⣼⠁⢀⡴⡆⠈⠻⣄⠀⠀⠘⣆⠀⠀⠀⣠⣴⠶⠶⣤⡀⣠⡶⠶⣤⡀
⣷⡘⢿⡉⢀⡟⠀⡇⢀⣀⡘⢧⡀⠀⠘⣦⠀⣰⠟⠀⠀⠀⠘⣿⠃⠀⠀⠈⢿
⠈⠁⠀⠙⠛⠁⠀⠻⠋⠉⠻⣌⠳⡄⠀⠈⢷⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⠻⡄⠀⢈⣿⡀⠀⠀⠀⢠⡟⢷⣀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣽⡾⠛⠙⠳⡤⠤⠴⠋⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠙⢿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⣀⣈⣟⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠐⢀⣐⣣⣬⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢀⡴⠋⢁⠤⣬⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡿⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠉⠙⢉⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⣿⠀⠀⠀⠀⠀⣠⡤⣤⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⡞⠁⠀⡿⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠘⡇⢠⢸⣇⣀⣀⣀⡆⣠⠏⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢠⠾⠛⠋⠉⠉⠉⢉⡽⢃⣴⣻⠛⠉⡙⠳⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢙⡿⣄⣉⣽⠀⢀⣀⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠈⠓⠛⠉⠀⠉⠁⡿⢸⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣴⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⣇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠃⠸⠦⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢊⣥⣤⡶⠀⠀⡷⠶⠶⠶⠖⠒⠲⠤⠤⠤⠤⢤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣡⠞⠉⠀⢀⡇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡴⠁⠀⠀⠀⣼⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣼⠁⠀⠀⠀⢠⡏⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣠⠞⠙⠓⠏⠸⣇⡀⠀⠀⢀⡞⠀⠀⠀⠀⣀⠘⣧
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⡈⠙⣦⣤⡾⠁⠀⠀⠀⠀⠛⠂⢻⡷⣦
⠀⠀⠀⠀⠀⠀⠙⠧⢤⡄⠀⣰⠟⠋⠑⣮⠙⠒⠒⠒⠒⠒⠒⠚⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣴⠏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢿⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⣀⣀⣀⣤⠼⠷⣶⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⣿⡇⠀⢸⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⡿⠀⣾⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣸⢃⡼⠃⠼⣀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠠⠿⠿⠶⠶⠂⠤⣤⣬⣄⡉⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣀⠀⠀⠀⠀⠀⢀⡀⢦⡐⣦⣨⡿⢤⡼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠁⠀⠈⡘⠁⠸⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠈⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠈⠋⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠛⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⠀⠀⠐⠀⠀⠀⢴⡆⠀⠀⠀⠐⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠃⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⣶⠀⠀⠀⠀⠀⣠⠞⠛⣆⠀⠀⠀⠀⠀⣠⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⡜⠁⠀⠀⠏⠘⠙⠋⢰⠎⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣠⠤⠤⢄⡀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢠⠁⠀⠀⠜⠀⢢⣤⢠⡎
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠳⢤⣀⠀⠈⢃⠀⠀⠸⢃⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⠀⠀⠘⢆⡀⠈⠑⠻⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⢡⡄⠀⠀⠀⠀⠀⡽⠀⠀⠈⢄⠀⠀⠈⠢⡀⠀⠁⠄⠀⠰⠀⠀⠀⠘⢦⣄⡠⠞⠓⡄⠀⣨⡆⠀⠀⢀⡾
⠀⠀⠀⠀⠐⢀⠊⠙⢤⢠⡄⣤⠄⠁⠀⠀⠀⠀⣆⠀⠀⠀⠈⠁⠘⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠋⠙⠶⠖⠋
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠃
⠀⠀⢠⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠘
⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠛⠛⢦⡼⠀⠀⢣⣀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡇
⢠⠏⠀⠀⡇⠀⠀⢀⣠⡤⠒⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠋⠀⠀⠀⠀⠱⠶⠂⠀⠀⠀⠐⠚⠃⠀⣼
⡌⠀⠀⠀⠓⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟
⣇⠀⠀⢀⣀⠴⣶⠀⠀⠀⢀⡤⠞⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠙⠀⠀⠀⢀⠃⠀⠀⠻⠤⠃
⡇⠰⠞⠉⠀⠀⠸⡆⠀⠀⣸⠀⠀⠀⡆⠑⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢿
⠛⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡌⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⣆⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠙⢦⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠧⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠂⠈⠁⠒⠒⠒⠒⠂⠈⠁⠠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠤⠤⠤⠀⠄⢶⠄⠦⢶⠇⠀⠀⡶⠠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⢸⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠘⠀⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠀⠀⢸⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⡀⠀⡇⠀⠀⢰⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡁⠀⠈⣧⣴⠀⠀⠀⡼⠛⠁⢠⠀⣀⣀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⢰⡋⠀⠀⠀⠈⡄⠀⠁⠀⠀⠀⠸⡆⠀⠉⠀⠀⠀⣉⠙⠋⠀⠀⠀⠀⢙⡄
⠀⠀⠀⠀⠀⠠⠛⠠⢄⣀⠀⠀⠤⠀⠀⠒⠒⠚⠇⠀⠰⠀⢀⣀⣤⡀⠀⠀⠀⡀⠤⣿

⠀⠀⠀⠀⠉⠃⢴⣏⠀⠙⠒⠠⠼⣀⣿⣀⡄⠀⠀⠀⠀⠘⣷⣦⣿⣿⣿⠉⠀⠀⠀⠐⠂⠀⠀⠀⠀⢻⣿⣴⡇⠆
⠁⠀⠀⠀⠀⠒⠈⠀⠉⠒⣠⣼⣼⣿⣿⣫⠘⠄⠀⠀⠀⠘⢹⣑⣿⣿⠠⢀⣴⣶⣶⣿⣿⣿⣷⣶⣦⣝⠿⣏⠇
⠀⠂⠀⠀⠀⣀⣄⣺⢿⣿⠿⢺⠯⠛⠫⣿⡆⠀⠀⠀⠀⠀⠈⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⠄
⠀⠀⠀⣤⣸⡿⣿⣿⣿⣿⣿⣼⠀⢠⠲⢻⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡗⣀⢠⡀⡼⣇⠈⠀⠀⠀⠤⠄⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⡋
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣟⣿⡟⢿⡆⠀⠀⠀⠀⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⣿⣿⣿⠁⢋⣻⣇⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣸⣿⣿⣿⣿⣿⣿⣿⠟⣰⣿⣿⣷⣤⣾⠀⠙⢳⠀⠀⢀⠀⠰⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠
⢿⢻⣿⣿⣿⣿⣿⡟⡄⠿⣿⣿⣿⣿⠋⠀⡀⠀⣾⣿⣿⣟⢻⡆⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠄⠼
⠨⠌⠉⠟⠉⠉⢳⣠⣡⣿⣿⣟⢿⣟⣁⣄⠁⢰⣿⣿⣶⣿⣭⠸⣦⠀⠀⠈⠛⠿⠿⣿⣿⣿⡿⠿⠛⠉⠠⢤⠀⡀⠐
⢀⣠⣀⣾⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢪⠏⠉⡉⠆⠾⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣩⣼
⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⡿⢡⠂⠀⠀⢁⢀⣿⣾⣟⡀⢀⠄⠀⠀⠠⠈⢀⣀⣤⣤⣼⣶⣿⣿⣭
⢹⣿⡿⣯⣭⢭⣿⢿⣿⣟⣽⣴⣈⣸⣿⣿⣿⡣⠁⠀⠀⠀⢀⣿⣿⣿⠟⠡⠀⠀⠀⠐⡏⠛⠛⠻⠉⡽⠛⠈⠉⠝⣿
⢨⡏⣨⣿⣥⣬⡿⢶⡛⣿⣿⣿⣿⣿⣿⣿⢿⠠⠀⠀⠰⠀⠿⢻⣿⣿⡇⠀⢀⠤⡷⠀⠀⠀⠀⠨⠿⣾⠀⠀⠄⠀⢻
⢸⣯⠿⡋⠉⢹⠀⣠⣷⣾⣿⣿⠟⣻⣿⡇⠸⠀⠀⢀⠂⢉⠃⠠⣹⣿⣏⣄⠀⠐⠁⠀⠁⢀⣐⣀⣙⡁⠀⠰⠆⢤⣼
⢸⣿⣾⢿⣄⢾⣤⡴⠄⢸⣟⣥⣴⣿⣿⣄⡸⠀⠀⠀⠀⠥⠀⠀⠉⠉⣿⣿⣿⣿⣢⣠⡴⠚⠛⠉⠁⠱⡄⠀⠀⢰⡾
⢸⣿⣿⢖⡉⠀⠀⣤⡖⠺⠷⠼⢿⣯⠰⢾⣿⣀⣨⣀⡌⠁⠀⠀⠀⠀⣿⣿⠷⡼⠋⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇
⠸⠷⠁⠿⢿⣷⡶⣀⣻⢏⣓⣒⠛⠉⠂⠀⠀⣉⠹⠛⠳⠀⢂⠀⠀⠀⠿⢿⡇⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣴⣿⡇
⢸⣠⣴⣤⣀⣨⣿⣿⢋⣘⣛⣿⣿⣷⣦⣾⣿⣯⣠⡄⠀⠰⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⡐⠛⢛⣙⣉⡉⣽⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⡿⡏⠛⠿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⠁⠀⣴⣄⠀⠀⠀⠀⠤⢤⡸⣿⣿⣿⣿⡇
⣸⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⡀⠀⠉⠛⢷⡆⣐⣂⣴⣾⣄⣠⣴⣾⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠠⠐⡦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⢠⡑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠁⠀⠀⣈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠄⠀⢐⠖⠌⢂⠀⠠⡤⡄⠀⠀⠀⠀⠀⠐⠁⢀⠀⡠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠈⠐⢤⢻⠀⠠⣉⡀⢸⠈⠃⠠⢤⠀⠀⠀⠠⠊⠀⠀⢁⠌⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢛⠂⠀⠈⢿⢉⢀⣈⡜⠀⠀⠀⠤⠀⡀⠀⠀⠀⢀⠀⠃⠪⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠈⠀⠀⢠⠑⠁⠀⠈⠂⠀⠀⠥⠐⠊⠀⡄⠆⠴⠁⠀⠀⠀⠈⡐
⠀⠀⠀⠀⢀⠄⠂⠁⠀⢀⣒⢠⠄⠒⢁⢂⠀⠔⡠⢀⠄⠄⠀⠀⡌⢃⠤⠊
⠀⠀⠀⠀⠐⢄⡀⠒⠍⠁⠁⠐⠁⡑⡁⡡⢊⠄⠔⡡⢊⠔⢁⠀⠂⠁
⠀⠀⠀⠀⠠⡑⠈⠉⠚⠒⠒⠂⠈⢈⢈⠔⡡⠂⢊⠔⡡⢊⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⡡⠊⢔⠕⠡⡪⠊⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⡩⠊⢐⢕⠄⠠⠂⠊⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡨⢀⠔⠁⠕⡠⠂⠄⢣⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠂⠑⠁⠁⠆⠨⡐⠅⡈⡆⠘
⠀⠀⠀⠀⠀⠀⢀⠐⠀⡁⠀⠠⢐⠡⠀⡞⠀⢈⠢⠕⡀⠐
⠀⠀⠀⠀⠀⠀⠈⠊⠀⠇⠀⠀⠥⡪⠀⢁⠀⢕⠨⠅⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠜⡪⣠⠈⠆⠐⠂⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠘⡆⠀⠀⠀⠀⠘⠻⡗⡋⠠⠐⠨⢐⠀⢐⠀⡐⠄⡀⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠓⠒⠖⢂⠄⠂⠄⠂⠀⠊⠐⠊⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⠿⠿⠿⠿⠿⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⠀⠀⣠⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣀⠀⠀⠈⠙⢷⡀⠀⢠⣾⠟⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⢻⣿⠙⠻⣦⡀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⢀⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⢰⡷⣶⠾⣧⣤⣄⠀⢀⣿⣄⠀⠀⢀⡴⢻⣇⣀⠀⣰⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣠⣾⠃⢸⣇⡿⢰⣿⠋⣿⠀⠈⣿⠀⠀⠀⠈⠀⠸⣿⣿⠆⢿⣏⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠛⠛⠋⠁⠀⠈⠛⠁⠀⠉⠈⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢠⣤⣄⣤⡆⠀⠀⣶⠀⠐⣿⡟⠃⠀⣴⡿⢿⣆⠀⠙⢻⣧⠀⢸⣿⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣼⡟⠻⣧⠀⠀⢀⣿⠀⣠⣹⣧⠀⢰⣿⠀⢸⣿⠀⣀⣀⣿⡇⢸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⣿⡀⠀⣿⠀⠀⢸⣿⠀⣿⡇⣿⡄⣸⣿⣠⣾⠏⣾⠏⠉⣿⡇⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢿⡇⢀⣿⠀⠀⢸⣿⢰⣿⡇⣿⡇⢹⣿⠉⠁⠀⣿⠀⠀⣿⡇⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠸⣷⣼⠇⠀⠀⠸⣿⣿⠻⣿⡿⠀⠘⣿⣤⣶⠆⠻⣧⣴⣿⡇⠸⡿⠈⠻⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠉⠙⠛⠶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡶⠿⣻⡟⠁⠀⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢿⡀⠀⠻⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢰⡟⠀⠀⠀⢻⣄⠀⢿⡀⣾⠋⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢿⣄⠀⠀⠀⣾⠃⠀⠀⢀⡾⢋⣄⠈⠛⠁⡾⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠋⠉⠉⠛⢦⡀⠉⠀⠀⢸⠇⠀⠀⣰⠟⠀⢸⡟⢷⣄⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢠⠟⠀⢀⡾⠁⢀⣤⠀⣿⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⠏⠀⢠⡟⣠⠟⠛⠋⠳⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⡀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠞⠁⠀⢀⣜⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠈⠉⠛⢶⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣠⣤⣄⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣯⣹⡧⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣬⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣤⡽⠿⠛⠉⢿⡆⣠⣤⡀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⡴⠋⢀⣠⠶⣶⡶⢀⣿⠁
⠀⢀⣀⣀⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣧⡴⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠚⠛⠛⠁⣠⡿⠁
⣴⠏⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠿⠷⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠛⠉⠉⠁⠈⢿⣯⣤⣤⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠛⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠃⠀⠀⠀⠉⠶⢄⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠉
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠻⣧⡀⠀⢤⠤⠀⢀⡠⠀⢀⣤⠀⠠⠀⠀⣀⣠⣴⠾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠈⠻⢶⣤⣤⣄⣈⣀⣀⣈⣤⣤⣶⠶⠿⠛⠉⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠾⠓⢤⣄⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⢦⣄⠙⢦⡈⢷⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⣤⡌⠻⣄⢳⣌⣿⡀⠀⠸⣧⠰⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠑⠰⡿⠛⠉⢁⡼⠀⠀⢹⣇⠀⠙⢿⣶⣄⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⢷⡤⠶⠋⠀⠀⠀⠀⢿⡄⠀⠈⢿⡜⢷⣄⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⠋⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠈⠿⠋⠁⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣇⠀⠀⠀⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⣦⣤⣤⠤⠴⠶⠶⠶⠛⠛⢻⡿⠀⠀⠀⠀⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠦⠤⠤⠤⠤⠤⠶⠖⠒⠛⠛⣿⠁⠀⠀⠀⢠⡄⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⣼⠀⠸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⠇⠀⠁⠀⠀⠀⠀⢸⠀⡆⠀⢹⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠿⠶⣤⣤⣤⣄⣠⣾⠀⣴⡴⠛⠉⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠸⢷⣄⣀⡀⠀⠙⣉⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣄⣀⣀⠀⠀⠀⠉⠉⠉⠁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠈⣿⡏⠙⠃⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣶⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢿⠉⠛⠻⣿⣿⡿⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠀⠀⠘⡟⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⠀⠀⠠⣇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡿⠀⢧⣄⠀⠀⣇⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣧⣴⠖⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⢷⣤⡟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣴⠟⣧⡄⠀⠀⢀⣀⣠⠴⠶⠚⠛⠛⠋⠉⠉⠀⠸⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡉⠻⢿⠿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣀⣤⠆⢠⣶⠶⠶⠶⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡐⠈⠁⠀⠈⡘⠃⠰⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠁⠀⡀⠀⠀⠀⢀⡀⠀⠈⢁⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡟⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠞⠀⠀⠀⢠⠟⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠿⠀⠀⠀⠀⠀⠀⣣⡄⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣠⠶⠦⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣈⠁⠀⠀⠀⠠⡞⠁⠀⠀⠟⠠⢠⠄⣀⡔⠉⠉⠹
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠖⠀⠀⢀⡶⢠⠀⡄⢀⡏
⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠓⠤⡀⠀⠀⠘⠓⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⠀⠀⠀⢎⠀⠀⠻⠧⣿
⠀⠀⠀⠀⠀⢠⠀⡂⠀⠀⠀⠀⠉⠛⠦⠀⠀⢀⠀⠀⠀⠢⡀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣶⡄⠀⠀⣿⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠄⠀⠷⣄⠀⠀⠀⠀⡘⠀⠀⠀⠈⢠⠀⠀⠀⠀⠢⠀⢀⠃⠤⠴⠀⠀⠀⠀⠙⠓⠒⠋⠀⢁⠀⢠⠿⣆⣀⣀⠴⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠈⠒⠛⠘⠛⠁⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠆⠀⠈⠉⠉
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⢄⣠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐
⠀⠀⡔⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢰
⠀⣰⠁⠀⡇⠀⠀⠀⠀⠀⠀⣀⡤⠒⠈⠉⠀⠉⢷⠇⠀⠀⣴⠀⠒⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⡈
⠐⠃⠀⠀⣇⠀⠀⠀⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠓⠐⠒⠒⠒⠀⠈⠉⠀⠀⡇
⡇⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁
⡇⠀⠀⣀⣤⠐⢶⡄⠀⠀⠀⣴⠖⠉⣿⢦⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠆⠀⠀⢀⡆⠀⠀⠈⠓⠚
⡇⠰⠊⠁⠀⠀⠀⠁⠀⠀⢠⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠉⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⢁⠀⠀⠀⡇⠀⠀⠐⠀⠀⠀⠀⠐⢀⠃⠀⠆⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⣦⡀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠘⠦⣤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡁⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠉⠶⠦⠀⠀⠠⠶⠂⠙⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⣆⣀⢀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠈⢃⠀⠀⠀⡏⠀⠀⠛⠀⠀⢠⡍⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡇⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠁⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⠀⠀⠀⠐⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠀⠀⠀⢸⣀⡀⢠
⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⢤⣤⠤⢄⠀⠀⠩⠖⣿⠀⠀⠀⠾⠋⠁⣐⠃⣠⣤⡤⠤⠀⣤⡄
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⡃⠀⠀⠀⠭⠈⠉⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⡛⠂⠤⠄⠀⠀⠀⠀⠀⠒⠒⠒⠚⠇⠀⣧⠀⣀⠀⣠⡀⠀⠀⠀⠠⠄⢺⠇
⣀⣀⣀⣀⣀⣀⣈⡒⣢⣴⣒⠒⢀⣀⣀⣉⣉⣁⣀⣀⣀⣁⣀⣀⣀⣈⡉⣶⣤⣤⣔⢂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⢿⡿⠿⠿⡿⢿⣿⣿⠿⣿⢿⣿⠿⠿⢿⣿⣿
⣿⣦⣼⣥⣬⣤⣾⣤⣽⣧⣤⣼⣥⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣾⣷⣶⣶⣶⣿⣿⣿⣶⣶⣶⣿⣶⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣧⠀⠀⠀⠀⠀⠀⠇⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣼⢸⣿⣿⣧⡀⠀⠈⠄⠀⠀⠀⠀⣹⣿⢀⣤⣶⣾⣿⣿⣷⣶⣤⣀⢸⣿
⠀⠀⠀⠀⠘⠀⠀⠀⡌⠀⠀⠀⠀⠀⣠⣾⣧⢠⣶⣶⣶⣷⣄⠀⠐⡀⠀⠀⠀⢿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢀⣴⣿⣿⣿⢸⣿⣿⣿⣿⣿⢰⣄⡀⠂⢠⣾⡛⠙⠻⣿⣿⣶⡄⠈⠙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⡔⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⠏⠠⠠⣿⠋⣡⡀⠀⠙⢿⣿⡄⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡀⢼⣩⡅⣩⣭⣭⣭⣭⢈⣭⣭⣭⣭⣭⡈⠀⢀⠀⠀⠀⠄⠀⠀⢆⢠⠋⠀⠀⠀⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠸⢀⣠⢠⠤⣬⣿⣇⡿⠿⠟⠛⠛⠚⢛⡟⠛⢿⡿⣁⣀⠄⠀⠐⠀⢀⠀⢀⠀⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⠀⠀⠀⠐⢀⠈⢰⡏⠁⠐⡄⠀⠀⢀⠔⡁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡡⠄⠀⠀⠀⠙⠄⠀⠀⢉⡀⠈⡷⢖⣶⣞⣻⣄⣁⠔⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠘⠉⠭⣽⠖⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⠀⠀⠀⠀⠀⢀⠓⢒⡄⠀⠒⠦⡤⠤⠤⠴⠒⠋⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠒⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠐⢃⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣀⠀⠤⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠴⠀⠀⠀⠀⠀⠀⣿⣿⢿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠖⠒⠈⠉⠉⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠀⠰
⠀⠀⠀⠀⠀⠀⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠁⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢰⠔⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⣀⣀⡀⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠸⡄⣤⠘⣷⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⠀⢀⠊⠉⠙⠒⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠱⣍⠀⠀⠈⠀⡀⠔⠊⠁⡠⠈⡠⠃⡐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⡀⠀⠀⠀⡘⠀⠈⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⢀⣈⠤⠊⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⡁⠒⢦⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇
⠀⠠⢐⡲⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡇
⢈⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠬⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣭⣭⣥⣶⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠟⢿⣯⢸⣿⣿⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⣀⣾⣿⣿⡿⠟⠉⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣀⣠⣿⣿⣿⣿⣷⣄⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠑⠊⠉⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⣀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⠿⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡐⠁⠀⠀⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠈⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⠈⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣿⣿⡟⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⣤⣴⣿⠟⠋⠙⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣥⡀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣟⠀⢀⣾⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣧⡘⠉
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃

⠀⠀⠀⠀⠀⠀⣀⢠⣤⣴⣶⣶⣶⣦⣄⣀⣀⣀⠠⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡴⠊⣇⠑⢄
⠀⠀⢀⠤⠒⢉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢆⠀⠀⠀⢠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢣⢂⣀⣁
⠀⠎⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⡄⣀⢈⣄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⠀⠀⠀⠠⠤⠤⠀⠀⢀
⢰⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡆⠀⠩⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠖⡀⠀⠠
⢸⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠁⠀⢀⠔⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠕⢈⡴⢨
⠀⡄⠀⠀⢹⣿⣿⣿⣿⣿⡈⠁⠀⠀⠀⠀⠀⠀⠀⢨⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣠⣶⠟⠃⠂
⠀⠘⡄⠀⠀⠙⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢊⣴⣾⠟⠁
⠀⠀⠈⠢⡄⠀⠀⠹⣿⣿⣿⣆⠀⠀⠈⠀⠀⠀⠀⡠⠃⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⣾⡿⠊
⠀⠀⠀⠀⠸⠒⠠⠀⠀⠉⢩⠍⠋⠖⡀⣶⡒⠒⡘⠁⠐⠀⠶⣓⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁
⠀⠀⠀⠀⠀⠀⢰⠀⢠⠄⡄⠀⠀⠀⡜⠀⠀⠩⠄⠀⢀⠀⠀⠀⠀⠉⠉⢀⠒⠄⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢰⠀⢸⣴⡾⡄⠀⠀⠹⣶⣦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠆⢁⡠⡆⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠻⡀⠀⠔⠛⢿⣿⣿⣿⣿⣿⣿⣿⡖⠒⢶⣻⣿⠿⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣈⠩⠀⠉⠉⠛⠛⠿⠿⡇⠀⡟⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠢⠠⡁⠀⠀⠀⡠⠀⠒⠲⠶⢦⡄⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠄⠀⡈⠀⠀⠀⠀⠀⠈⠙⠱⡄⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠁⢈⠀⠄⣄⠠⠀⡐⠀⠀⡀⠈⡀⠃⠐⠠⡀⠀⢐⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠁⠴⢡⡄⣀⢠⡄⠲⠀⠀⠀⠀⠀⠄⡀⣀⣴⠇⢀⠀⠀⠉⠁⠐⠓⠄⡀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠃⠀⠘⢺⠟⠋⠙⠙⠿⡗⠃⠀⠀⠀⠀⠀⠀⠉⠁⢆⡭⠌⠀⠂⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠀⠂
⠀⠀⠂⠀⠀⠀⣎⠀⠀⠴⠀⠀⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⢎⡀⣀⠀⣀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠿⠓⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⣀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠓⠒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣤⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⢠⠃⠉⠙⠲⣆⢸⠀⢣⠀⠑⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⠘⢶⠋⠋⠀⠠⢴⠂⠑⠾⠀⠈⣧⠀⠸⣾⠀⠀⠀⣠⠂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⡄⠈⢧⠳⣄⡠⠴⠦⡐⢠⠀⠀⢸⡀⠀⢹⡇⢀⣼⠇⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣼⠀⠀⠂⠈⠙⠛⠋⠁⡏⠀⠀⢸⠆⠀⠀⣷⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢀⡀⠀⣰⢃⠀⠀⠈⠀⢀⠀⣹⣿⣿⣿⣥⣤⡤⠖⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠆⠀⠀⠉⠉⢁⠮⠒⡉⠀⠀⠒⡜⢷⣿⣿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡀⢀⣀⣀⡤⠄⠁⠀⢈⣠⠀⠀⠀⢰⣾⣿⣿⣉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢠⣐⠒⡆⠀⠀⠀⢿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠂⠄⣀⢸⣜⡄⠀⠀⠀⠸⣿⣿⣆⠀⠀⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠲⠲⠒⠚⠉⠁⠀⢀⡤⣤⡟⠀⠹⠀⠀⠀⠀⠀⢇⠉⠻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⡐⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠉⠈⠉⠁⠀⠀⢀⣠⠤⠀⢀⠀⠀⠀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠁⠀⠀⠀⠈⢆⠀⠀⠀⡇⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡀⠠⢤⣤⡀⢀⡟⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢋⠀⠀⠀⢸⢠⠫⢠⠆⠀⠀⠀⠀⠀⡆⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠄⠜⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⣾⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠒⠒⠁

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

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠛⠛⠛⠛⠛⠶⠦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠀⠒⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⠴⠖⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣤⣤⣀⡀⠀⠿⠀⠀⠀⢀⣀⣀⣀⣠⣤⡤⢶⣶⠶⠚⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢿⣛⣛⣶⣚⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸⠉⠉⠉⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣀⣤⣄⣸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⣯⣀⠀⠉⠁⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣦⣌⣁⡀⠀⠀⠀⣸⣧⡤⠴⣶⠶⢾⡟⠋⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⢤⣤⣤⠤⠞⠁⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣅⣼⣧⠤⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⣼⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣤⣤⣤⣤⣤⣤⣤⡤⢶⡶⠛⠋⠛⠻⣏⣧⠈⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣤⣀⣀⣀⣀⣀⣀⣀⣘⣶⣾⣙⣧⣤⡤⠶⢿⣏⣉⣭⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⢤⣬⣉⣉⣉⣉⣉⣉⣩⣭⣤⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⠀⢠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠲⠴⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢠⢿⡃⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢿⡠⠒⠋⣉⠍⠉⠈⢉⠓⢤⣀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢩⣄⠀⠀⠀⠈⠳⢤⡈⠓⢦⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⠲⡄⠀⠀⠀⠀⠀⡏⠀⠀⠉⣀⡀⠀⠀⠀⠙⠏⠀⠀⠀⠳⡄⢀⡾⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⠀⠀⠀⠀⠀⢧⠀⡰⠋⠈⠉⠉⠉⠉⠳⣄⠀⠀⠀⠀⠙⠍⡷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠈⢧⡁⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣀⣤⣠⡼⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠤⣼⠖⡆⠀⠀⠀⠀⠀⠉⠓⠦⡤⣤⣀⣀⣤⡥⠴⠚⠁⢀⣅⡀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢦⣘⡄⢧⠀⠀⠀⣠⢲⠎⠉⡟⠣⠤⠴⢳⠒⠦⣄⣀⠞⠁⢰⠓⠤⠦⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠢⣤⢯⠟⠈⢦⠀⡼⢁⡞⠀⠀⠓⠶⠦⠤⠾⠀⢠⢇⡏⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠓⠚⠈⠀⠀⢾⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠴⠲⣤⠞⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣷⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠒⠚⠛⠛⠛⠛⠛⠛⠓⠛⠧⠼⠓⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⢒⣚⠚⢲⠖⠚⠒⠛⠓⣶⢲⣶⠒⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠎⠥⠬⠂⠸⠦⣤⣄⣠⡤⠟⠒⠚⠁⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⡀

⠀⠀⢰⠓⢆⠀⠀⢂⢤⡶⢧⡀⡀⠀⠀⠀⡀⠀⠀⡆⢸⠀⢰⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠑⣄⣈⢁⠀⠐⠁⠀⠀⠀⡀⠠⡀⠀⠀⢡⠀⠀⠀⠘
⠀⠀⠸⡀⢸⠆⠀⢸⠸⣄⠠⠀⠱⠀⠀⠀⠘⢆⠀⢱⢸⠀⡘⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀⡀⠀⠀⠀⠈⡇⠀⠈⠈⡆⢸⡀⢠⣠
⡆⠀⠀⡇⠙⣄⠀⠈⠀⣰⠹⣄⠀⠀⠰⢄⠀⠈⠧⠈⢃⣁⠇⡘⢀⠔⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⠃⢁⠄⠐⠄⠐⣇⠀⠀⠘⡅⢸⠀⠘⡏⣆
⠀⡠⢤⠃⠀⡆⢱⣴⣆⠁⠀⡏⢷⠀⠀⠀⠙⠦⢠⠂⠀⠀⠱⡔⠁⠀⠠⠤⠀⠀⠀⠀⠠⠀⠻⠀⠀⠀⠸⠀⠀⡿⠀⠈⠓⠀⠊⠀⠈⠃⠘⠁
⠀⠈⢨⡄⠀⡇⠀⠀⠀⠘⡆⡎⠀⠀⠉⠉⠂⠂⢸⡀⠀⠀⠐⢀⡀⠀⠀⠀⠀⡀⡀⢀⠀⠀⠤⡀⡀⠀⠸⠄⠔⠁⠀⣀⠀⠀⠀⣤⠀⠀⢀⡀
⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⣠⠉⠀⠀⠠⡀⠈⠙⠢⠄⠀⠁⢘⡇⠘⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠁⠰⠁⠘⠃
⠀⠀⠀⢀⠀⠁⠀⠀⢀⢴⠀⠡⠄⠀⠀⠀⢀⠔⠁⠰⠀⡇⠀⠙⣄⠀⠀⠀⠀⠘⠀⠳⢀⡠⠊⠱⡀⠀⡠⢠⠀⠀⠀⠀⠜⠁⠐⢆⠀⠀⠀⠀⡀
⠀⠀⠀⢸⠀⢸⠀⠀⠈⠈⢠⡈⠀⡀⠀⠀⠁⠀⢠⠃⠀⡇⠀⠀⣀⣠⣦⣶⣦⣤⡀⠀⠀⠀⠀⠀⠈⢰⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈
⠀⡀⠀⢸⠀⢸⠀⠀⠀⡰⠈⠃⠘⠃⢠⡂⠀⠀⠀⠀⠀⠁⠀⣰⣿⠟⠋⠁⠁⠀⠙⢧⡄⠀⠀⠀⠀⢘⠀⡰⠁⠀⢢⠀⡇⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠈⠀⠀⡄⢸⠀⠀⢸⠀⢠⢀⠀⠠⠀⠈⣢⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⣀⠋⠁⢀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢡⠀⠀⠀⠀⠀⠀⠀⠱
⣆⠀⠀⠀⡇⢸⠀⠀⠸⡄⠀⠈⠓⡀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⡄⠘⠃⠈⠁⠀⠂⡸⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⡀⠀⠀⠀⠀⢀
⠀⢱⠂⠰⠦⠘⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢥⠀⣀⡀⠀⠀⠀⠉⠉⠂⠀⢀⠀⢄⠀⠙⢦⡀⠀⠀⠀⠀⢇⠀⣄⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠐
⢿⡈⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⢣⠀⠀⠙⢦⠀⠀⠀⠸⡄⢰⠀⠀⢳⠀⢁⠀⠀⠀⠐⠀⠀⠐
⣀⢧⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡀⠀⡤⡄⠀⠀⠀⠀⢣⡀⠀⠀⢳⡤⣄⠀⠳⠸⠃⠀⠘⠀⠘⠀⠀⢀
⠋⡁⠑⠢⠄⢀⡀⠀⠸⠄⠀⠀⠀⠀⠀⠀⢼⡀⠘⠀⠀⠀⡇⠘⣾⢁⣮⠄⢀⠀⠀⡘⢳⠀⠀⠈⠁⠋⠀⠀⠀⠀⢀⠠⠀⠀⢀⢂⠜⠁⢠⠀⠈
⢈⠃⠐⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⠀⢸⠋⠀⠀⠱⠔⢣⠀⠀⠉⠀⠐⠀⠀⠀⢀⠰⡀⢢⠀⠀⠀⠀⢀⢆⠂⠀⠀⠀⠉⢐
⢉⠘⠋⠀⠈⠒⠠⠄⠀⠀⠂⠀⡄⠀⠀⠁⠀⠀⠀⠃⠀⢠⡋⢧⡄⢤⠔⢻⠁⠀⠀⠂⠀⠀⠀⡀⠀⢠⠀⢈⠒⢃⠂⢤⡀⠀⠂⠀⠠⣀⠐⠂⣥
⠘⠡⣠⠙⠠⠀⣀⡀⢰⢠⠐⠈⠀⠀⠄⠂⠀⠑⠐⠀⠀⠀⢀⠎⠑⠃⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⣼⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠉⡰⠂
⠉⢔⠆⠘⠃⡠⠊⠁⣨⢌⠀⠀⠀⠈⠀⠀⠀⢀⢀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢄⡀⠐⠁⠐⠦⢐⠂⠁⣤
⠀⠀⢶⠀⠁⠀⠠⣀⡇⢠⠀⠀⠀⠀⡀⠀⢇⠈⠀⡅⠀⡇⠙⠲⠀⠀⡠⠒⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠐⠂⢤
⠀⠀⢸⠀⢠⠀⠀⠘⢇⡎⠀⡆⠙⠀⠱⠀⠈⠒⠊⠀⠀⠑⠄⠊⠀⢰⠇⠀⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠀⢰⠂⡄⠐⢶⡄⠀⢸
⠀⠀⠀⠠⠃⢶⡗⠄⠈⢧⠀⠱⢄⢀⠃⠀⠀⡠⠀⠤⠤⠤⠀⠠⢄⡀⠀⠀⠀⠘⠧⣘⡆⠀⠀⠀⠀⠀⢀⡀⠰⡆⢏⢈⠆⠀⠹⡇⡠⡄⢣⠀⠈
⡀⠀⡔⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠈⠀⠀⠎⠀⠀⢤⠀⡀⠀⠠⡄⠘⡄⠀⠀⠦⡀⠠⠁⠂⠰⡃⠀⠀⠈⢣⠀⠙⠈⠁⠀⠀⠀⢻⡀⠀⣼
⠘⢸⠀⠀⡀⢀⠀⡀⠀⠈⡇⠀⢀⠤⠤⠘⠈⠂⠀⠀⠁⠣⠄⠤⠃⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡆⡄⠀⠀⠀⡠⠊⢀⠀⠀⠀⡏⠀⠩⡄
⠀⢸⠀⠀⡇⢸⠀⠶⠀⠀⣥⠀⣾⠀⠀⠀⠄⠀⠢⡀⠀⠀⠀⢀⠀⢄⠀⠀⠉⠑⢄⠀⢀⠰⡀⠗⠈⠃⠁⢢⡀⠸⠀⠆⠀⠀⠀⠀⡇⠀⠀⢇
⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠨⠀⠈⠲⠤⢼⡀⠲⠀⡇⠀⠀⠀⡇⠰⠀⡇⠀⠀⠀⡸⠀⠈⡶⠁⠀⠀⢰⠰⠀⡇⠠⠌⠀⠀⠘⠀⠀⡇⠀⠀⠘⡄
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠠⠀⠀⠈⠒⠀⠉⠀⠁⠉⠀⠀⠀⠇⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢤⡀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⢒
⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⠤⠄⠰⠊⠁⡇⠀⠈⠉⠓⢲⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⢐⡀⣆⠈⢳⠀⣇⠀⠀⠀⢀⡾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠙⠢⡀⠈⣢⡭⢹⣗⠚⢢⡜⢝⠲⣠⠏⢰⣀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢠⠏⠀⠳⠾⢥⠠⢃⠄⠈⢑⣾⠿⢆⣠⢗⣊⠭⣤⠘⠂⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄
⠀⢀⡠⠞⠃⠀⠀⠀⠀⠘⠶⣧⡞⠁⣉⠀⠀⠀⢙⠀⠀⢀⠽⢏⣳⠀⠦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡠⠀⠈⠀⢁
⠀⠘⠒⠒⠉⠉⢹⠓⠢⠤⣴⡻⠁⠀⠈⠀⠀⠃⠛⠒⢂⠤⠒⠈⠁⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣵⣾⣿⠮
⠀⠀⠀⠀⠀⢠⣁⠤⣒⡴⢮⣝⡓⣤⠀⠐⠂⡀⠾⠂⠀⡀⠠⠶⠶⢖⣲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⢀⠜⡡⠚⢁⠖⢹⠀⠈⠉⠉⡘⢋⠀⣠⡔⠉⡴⠃⣀⣠⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡛⡅
⠀⠀⠀⠀⠸⢴⠐⢃⠾⢴⣸⣆⠀⠀⢰⣃⣈⣉⣀⣇⡞⡠⠊⠁⡉⢀⡼⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣱⡀
⠀⠀⠀⠀⠀⢨⠋⠴⠶⡜⠁⠈⠳⠖⡟⡟⢆⠀⠉⠙⠿⣧⠶⠋⡰⢏⡀⠀⠘⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡆
⠀⠀⠀⠀⠀⠈⠳⠗⣾⠁⠀⠀⠀⠀⠇⡇⠘⣆⠀⠀⠀⢰⢀⠞⠁⢀⣿⡄⠖⢀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢓⡀
⠀⠀⠀⠀⠀⠀⣠⢴⡇⠀⠀⠀⠀⢸⠀⡇⠀⠹⡀⠀⠀⢸⡏⠀⣠⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡇⠀⠁
⠀⠀⠀⢀⡠⠞⣡⢿⡇⠀⠀⠀⠀⡘⠀⡇⠀⠀⢳⡀⠀⢸⡧⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡛⠃
⠀⢀⠔⠉⣠⢚⡇⣿⠀⠀⠀⠀⢀⡇⢸⡇⠀⠀⠀⢧⠀⢨⣿⣿⣿⣿⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠈
⣰⠃⢀⠜⠁⠈⡇⢿⠀⠀⠀⠀⢸⠃⢸⡇⠀⠀⠀⠘⡆⢸⢿⣿⣿⠏⠙⠁⠀⠀⠈⢻⡠⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠃
⢇⠀⣏⠀⠀⠀⠙⣾⠀⠀⠀⠀⡼⠀⠈⡇⠀⠀⠀⠀⢣⡾⣿⣿⡇⢰⠀⠀⠀⠀⢰⡌⣳⡤⣀⣀⣀⣀⣀⣀⣀⠀⠠⠤⠤⠄⠠⠇
⠀⠉⠉⠀⠀⠀⠀⠸⡇⠀⠀⢀⡇⠀⠀⡇⠀⠀⠀⠀⢸⢧⣿⣿⣿⡎⣆⠀⢱⣄⢲⡵⢶⢾⡏⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠄⣷⡀⢀⣸⣅⣠⡴⢛⢷⡶⠒⠛⢻⢸⣿⣿⣡⣟⠻⡶⠋⠈⠉⣿⠃⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠂⠁⣎⢻⡏⠀⠀⢸⢂⢢⢿⡇⠀⠀⣸⢶⣿⠟⣡⠎⣩⠆⠀⠦⢰⠷⠦⢤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠂⣾⠻⠯⢱⡎⡀⠣⢘⣯⠄⢶⡏⣾⠏⡐⠁⠀⠁⠀⠀⡴⠁⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⡁⢻⠀⠀⠁⠘⣿⣅⣈⡅⠛⢶⣀⣠⣆⡀⠀⣸⠇⢣⠀⠀⠐⡆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠙⣋⡟⠃⠀⠀⠀⠠⢷⠶⠤⠇⠠⡝⡼⡟⠀⠉⠙⣿⡆⠯⣀⡀⣠⠟⠹⢿⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠤⠄⠚⠋⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠃⡇⠀⠀⠀⠈⠹⡟⠙⢯⡀⠀⠀⠀⠈⣿⢅⠈⠗
⠄⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⠀⡇⠀⠀⠀⠀⠀⠹⠀⠀⠘⣦⣀⡀⣀⣾⣎⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢽⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣧⠀⠳⣄⠀⠀⠀⠀⠀⣀⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠶⠶⠶⠶⠖⠒⠺⠿⠿⠖⠚⠛⠓⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⣰⠞⢩⣿⡛⢿⡷⣦
⠀⠀⢠⣏⣰⣿⣿⣿⣶⣿⡟⣷⣄
⠀⠀⢀⣟⣹⣯⡼⢻⣿⢻⣧⣿⢹
⠀⠀⢸⡟⢿⣟⣿⢿⣿⣾⣿⣷⡟⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠈⢷⡈⠻⢭⣽⣿⡿⠿⠿⠃⠀⠀⠀⣀⡴⠞⠋⠁⢀⣀⣤⣤⠀⣀⠀⣉⡙⠳⣤
⠀⠀⠀⠀⠉⠛⠛⠹⣇⠀⠀⠀⠀⠀⣠⠞⠉⠀⣠⣶⡿⠿⠛⠉⠉⢠⢚⣼⡿⡑⠄⣤⠘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⡾⠁⣀⣰⣿⠟⡁⢅⠂⡥⠌⣠⣴⠿⠋⢂⢃⣼⣿⣦⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠠⢤⣾⢟⡠⡔⢁⣐⣬⣶⡿⠛⢁⠌⠒⣡⣾⡟⠀⡙⣿⣤⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣱⡿⠃⣪⣴⡾⢟⡛⢉⠑⠐⠎⣣⣬⡾⠋⠀⠻⣶⣔⠈⢿⣦⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠸⠟⣠⡿⢋⡁⢒⣠⣤⣶⠶⠶⠟⠋⠁⠀⠀⠀⠀⠈⢻⣮⠘⢿⣆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣌⢇⣾⢟⡔⣢⡷⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⢲⡀⢀⡀⠀⢻⣯⠸⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡇⣼⡏⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⢻⡇⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣹⣟⣼⠋⠀⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⠀⠀⠘⣿⣿⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠙⣿⠁⠀⠀⢿⡀⠀⣿⠀⠀⠀⣠⠄⠀⠀⠘⣇⠀⢠⡟⠀⠀⠀⣿⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣿⠀⠀⠀⠈⠛⠛⠁⠀⠠⣾⣁⠀⠀⠀⠀⡈⠛⠋⠀⠀⠀⠀⢿⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⠀⠀⠀⣴⠟⠃⠀⠀⠈⠁⠀⠀⠈⠙⢶⡀⠀⠀⠀⠀⣾⣿⣭⣙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⡇⠀⠀⢠⡾⠛⠙⠓⣦⠀⠀⠀⡗⠀⠀⠀⢠⣿⠐⡩⡻⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⡻⢷⠶⣄⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠁⠈⠛⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⠛⠛⠛⡛⣛⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⣍⠙⠛⠁⠀⠈⠋⢋⣽⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠦⣤⡀⠀⠀⠀⡾⠀⠊⠀⠙⠷⣦⣀⣀⣠⣾⠛⠁⠈⣠⠈⣷⠀⠀⠀⣠⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣬⠿⠛⠀⠀⢹⡆⠀⢀⡇⢨⣄⠀⠀⠀⣿⠉⠉⠈⣿⠀⠀⠀⣠⡀⢹⡄⠀⢰⡃⠀⠀⠛⠻⣯⣀
⠀⠀⠀⠀⢠⡾⠋⢀⣠⣤⢶⡞⢛⢿⡄⢸⡇⢸⡏⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢿⡇⢸⡇⢠⣿⡛⢲⡶⢦⣄⡀⠙⢶⡀
⠀⠀⠀⠀⠻⠴⠞⠋⠁⠀⠀⢿⡀⠊⣿⡄⡇⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⢸⣇⣾⠣⠀⡾⠀⠀⠈⠙⠛⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠜⣿⡇⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢺⡇⢸⣿⠫⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠃⠘⣷⣸⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣾⣧⣾⢃⠇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠐⠀⠽⠛⠿⠄⠀⠐⢿⠀⠀⠀⡿⠀⠀⠂⠿⠛⠋⠐⠀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⣀⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠲⣤⣀⣤⠴⠂⠐⠦⣤⣠⣤⠖⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣧⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⣼⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣩⡄⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠛⢻⠀⠉⠀⠀⠉⠀⡟⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣷⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⡏⠀⠀⠀⠿⠿⠿⠿⠿⠈⠿⠿⠿⠿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣶⣤⠄⠀⠀⠀⠙⠉⠉⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⣉⣙⠻⢿⠀⠀⠀⠐⠒⣶⠖⢄⠀⠀⠆⢀⠤⣶⡖⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠠⠄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⣄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⣿⣶⣤⣀⣀⠀⠀⠀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⣀⣀⣀⣀⣀⠤⠊

⠋⠁⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⣴⠄⢀⣠⣤⣤⣤⣄⠀⠀⠀⢀⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠞⢁⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠘⠃⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⡘⠀⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠱⡶⠀⢀⡀⠀⣀⠀⠉⠍⠙⠿⣿⣿⣿⣿⡿⢛⣛⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢐⠀⠀⢤⡀⠐⡀⢡⠀⣿⣿⣿⣿⣿⣿⠃⠘⠛⢁⣄⠀⠈⠿⠃⠀⡉⠀⠦⠀⠈⠢⡈⢿⣿⣿⡄⡏⣼⣷⠀⠉⠉⠁⠉⢿⣿
⠇⠀⠀⠾⠒⣼⡏⠀⠀⠈⠢⣧⢹⣿⣿⣿⡟⠁⡶⠀⠀⠈⠁⠀⠐⠃⠀⠀⠁⠘⠀⢀⠈⠀⠀⣀⠙⢿⣿⣦⣤⣅⣀⠀⠀⠶⠀⠈⣻
⠦⠤⡴⢦⠀⠈⠁⠀⠀⠀⠀⠈⠁⠹⠟⠁⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⣋⡠⡈⠻⣿⣿⡟⣫⠤⣀⣤⣠⡎⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠂⣠⡤⡆⠈⠁⠒⠲⢶⠶⣄⡀⠀⠂⠩⢉⡲⢌⡻⣷⡁⠚⡸⢿⣿⣿⡌
⠀⠀⠀⠢⠀⢈⠁⠀⠀⠀⠀⠀⢀⡀⣤⢀⣠⠤⠘⠁⠠⢔⣦⡞⡅⠀⠀⠀⠀⠀⡟⠁⠀⠉⠒⠒⠲⢄⡈⠉⠛⠎⢻⣷⣦⣽⣿⣿⣿
⡆⠀⠀⠈⠀⠀⠀⠀⠋⠙⠖⠓⠂⢣⠄⠀⠀⠒⠀⠀⠀⠈⢹⠁⢰⡀⠀⠀⠀⢀⠃⠀⠒⠀⠀⠀⠀⠀⠈⠉⣻⠷⡶⢻⣯⡿⠿⣻⣿
⠀⠀⠀⠀⠀⠀⠴⣀⣁⠀⠠⠖⣶⠶⣄⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⣧⡀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠄⠀⠔⠒⠋⣪⣾⣿⣯⣇⡀⡭⣿
⠆⠀⠀⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⠀⠉⠀⠀⠘⠦⠠⠤⠀⣿⠀⠀⠘⣷⣀⣰⠇⢀⠀⠀⠀⠀⠐⣠⣶⣿⣿⣿⣿⠟⣋⡍⣿⠡⣿⠿
⡄⠀⠀⠀⠀⢀⣤⣤⡄⠀⢀⣠⣾⣷⣴⣶⡄⠀⠀⠀⠀⠀⣿⣀⠀⠀⠉⠁⡇⠀⠀⠀⠀⢀⣾⣮⣿⣏⢻⣿⣿⣿⣷⣿⣾⢟⣷⣶⣿
⡇⢀⣼⣿⣦⠺⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⡏⠬⣧⠀⢠⠔⠁⠀⠀⣀⠀⠀⢿⣿⣿⠃⣼⣿⣯⣽⣿⢯⣟⣏⣳⡟⣛
⡇⢨⣿⣿⣿⣇⣻⢿⣿⣆⠛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⠁⢠⡏⠀⣼⣤⠀⠀⢸⣿⣷⣦⢸⡿⢡⣾⣿⣿⣿⣿⣿⣼⣿⣿⢿⣷⣼
⠃⠈⠻⣿⣿⡏⠀⠀⢻⣿⣷⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⣿⠀⢸⠃⢰⢻⣿⣷⠀⣿⣿⣿⣯⣾⢱⣶⣼⣻⡿⢛⣫⣮⣵⡌⡿⠛⣿⣽
⠁⠀⡀⠀⣿⡇⠀⠀⠈⠻⣿⡇⠀⠸⣿⣿⣇⠀⠀⢰⠀⣿⢠⡏⢠⠃⣾⣿⡟⠁⣿⣿⣿⡏⠀⣸⣿⣿⡇⠐⢿⣿⣿⣿⣿⡿⠀⢸⣯
⡇⠀⣷⠀⢸⡇⠀⠀⠃⠀⠙⣷⠀⠀⠘⢿⣏⢆⢀⡿⣸⡯⡔⠀⠆⢠⣿⡿⢀⠀⣿⣿⡏⠀⢠⣿⣿⣿⠟⠀⢠⢹⣿⣿⠉⠀⢀⣿⣿
⣇⣰⢻⡆⢸⠀⡀⠀⠀⠀⠀⢹⠇⠀⢡⠈⢿⡘⣿⠁⢿⡏⢀⡌⠀⣽⠃⠀⢘⠐⠸⣿⡇⠀⠈⣿⣿⠇⠀⠠⠂⢸⣿⣿⠀⡄⣼⡿⣿
⣿⣿⡻⣟⢿⣿⣿⣿⢿⣿⢿⣿⣿⣷⣿⣶⣾⣷⣿⣶⡾⢀⣾⣤⣤⣯⣤⣧⣬⣦⣤⣹⣄⣠⣀⡛⠁⠰⠂⢠⠀⣸⠟⠙⡄⠛⠛⠙⢚
⣿⣿⣷⡈⡳⢿⣽⡏⠆⢻⡈⣿⠻⣿⢈⣿⡏⢸⣨⠿⠃⢼⠈⢿⣿⡟⢿⣿⢿⣿⣿⣿⣿⡻⣿⣿⣟⠻⣷⣶⣶⣷⣶⣶⣾⣷⣬⣧⣀
⣿⣿⣾⡻⡟⣦⠻⢿⢶⡼⠇⣿⢠⣿⢹⢾⡇⣘⡳⢤⡀⠀⠳⣄⠙⢷⠈⠻⡌⢿⣷⣼⡻⣷⡌⣙⠻⣷⣌⠛⢻⣻⡝⣿⣿⡏⢻⣿⡻
⣿⣿⣿⢳⡜⡎⣷⡜⡾⣿⢦⢇⣾⡼⢼⣯⢻⡇⣽⡷⣍⡲⢤⣬⣑⣦⣥⣀⡳⠀⠻⣿⣿⣮⠻⡌⢳⡘⢦⢾⣿⣷⣰⣮⢻⣇⣦⡹⣷
⣿⣿⣿⠀⢿⣆⠸⣷⡅⣿⢪⣾⣿⣇⢺⣿⣾⣷⡌⢧⠈⣽⣳⣶⣶⣭⣥⣉⣛⠂⠀⠹⣿⣿⣷⡍⠼⢿⣌⢧⢻⣯⢹⣿⣷⡋⢾⣿⣜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠋⠘⣢⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣯⢀⠠⡞⠁⠀⠀⠀⢠⢿⠀⠀⠀⢰⡥⠲⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠓⠿⠀⠀⠀⢻⢉⢀⡭⠶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠼⠊⠻⠃⠀⠐⠀⠰⠀⢀⡾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠋⠧⣤⡄⠐⠂⠀⢀⡤⠶⣛⣉⠉⠙⠛⠓⣦⡀⠀⠘⢻⢉⡀⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠶⣾⠁⢀⠀⠀⢠⠾⢤⣀⡀⠈⣉⡷⠖⠋⢡⣽⣆⡀⠾⠋⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⢠⣧⣄⣀⣨⠟⠋⢁⣠⠴⠚⠉⠐⠄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣠⠞⢉⣀⣴⣚⣉⣀⣠⣤⣤⣤⣭⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡀⡄⠄⣂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢇⣂⣀⣀⣀⣀⣠⣤⡤⠤⠶⠒⠛⠉⢩⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢲⡄⠉⠀⣠⣤⠆⠀⠀⠀⠐⣶⢦⡀⠀⠀⣩⠄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠛⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠈⠆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢲⠀⠀⠀⠀⠀⠐⠋⠓⠀⠀⠀⣴⠅⢀⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠘⠉⠀⠀⠀⠈⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠗⣲⠦⡤⢤⡴⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣴⢦⡜⠃⠈⠡⠞⡡⢋⠜⣡⣟⢲⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠻⣀⠌⢧⡙⢦⣀⠀⠈⠑⠋⠘⣁⡵⢋⡴⣣⠟⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⠻⣤⡛⢲⠮⣭⣙⣋⣉⣩⠤⣶⣫⠟⠁⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⠄⢀⠀⠀⠉⠙⠒⠓⠚⠷⠞⠓⠚⠉⠀⠀⠀⢀⠄⢠⠄⢀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣡⠔⢁⠔⢠⠆⢀⡤⢀⡔⢀⡴⠀⡠⠃⡴⠃⡴⠁⡴⠃⡴⢋⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⢁⡔⢁⠔⠁⡠⠋⠠⠋⡴⠋⣠⠎⣠⠞⢠⠞⢠⠞⢀⠞⣠⢋⡤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠈⠀⠁⠈⠐⠀⡇

⠀⠀⠀⠀⠀⢀⡴⢒⣶⠶⣶⢦⡀
⠀⠀⠀⠀⠀⣾⣠⣿⣿⣿⣼⣿⢳⣦
⠀⠀⠀⠀⠀⣿⣹⣯⣞⣿⡟⣿⣸⢻⠆
⠀⠀⠀⠀⠀⣏⢻⣯⣟⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠙⢦⣝⣳⣾⠟⠛⠛⠁⠀⠀⢀⣤⠴⠖⠛⠛⠛⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⢀⡴⠋⢀⣠⣴⠶⠿⠶⢶⣤⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⣴⠟⠉⢀⣀⣤⣄⣀⠈⠻⣦⡈⠑⠛⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⡀⣀⡴⠃⠀⠈⠁⣠⡾⠛⠉⠉⠉⠉⠻⣶⡘⠃⢀⣤⣤⣀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⡉⠉⠀⠀⠀⠀⣴⠟⠀⠀⣠⣶⣶⣶⣤⡈⠛⠀⠈⠉⠈⠙⣷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠸⢶⣤⣤⣤⡾⠋⠀⠀⣰⠿⢦⣍⠳⡌⠻⠄⠶⠶⠶⣄⠀⢹⡇⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⣀⠈⠉⠀⠀⠀⢀⣼⠋⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠘⣧⠈⢿⡀⣧⣀⡤
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠃⢨⡟⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠀⢸⣆⡈⠳⠦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠁⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠘⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡼⠛⣿⠀⠀⠈⣇⠀⣸⠁⠀⢀⡴⠀⠀⠀⠸⡀⢀⡾⠀⠀⠀⣿⠛⢷⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣤⣿⠀⠀⠀⠈⠉⢁⣀⠀⠛⠶⠀⠀⠀⣀⡉⠉⠀⠀⠀⠀⣿⣤⡼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢿⡀⠀⠀⠀⣰⠋⠁⠀⢀⣀⣀⡀⠀⠈⠙⢷⠀⠀⠀⢀⣿⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡄⠀⠀⠘⠀⠀⠰⠏⠁⠀⠙⠦⠀⠀⠘⠀⠀⢀⡾⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⣶⡶⢶⣶⡖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢶⠾⠏⠉⠉⠉⠽⠒⡶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⢀⡞⠁⢀⠹⣧⡀⠀⠀⠀⣠⡾⠁⠆⠙⣧⠀⠀⠀⣠⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣽⠷⠂⠀⠙⣆⠀⢸⡇⣠⡥⠠⠌⠛⠛⠾⠛⠋⠐⡄⢠⡁⢹⠀⠀⣾⠁⠀⠐⠿⣯⡀
⠀⠀⠀⠀⢠⠞⠉⣀⣤⠶⣾⠛⢻⣆⢸⡇⣿⠁⢣⡄⠀⠀⠀⠀⠀⠀⢈⢸⡇⢸⡆⣰⡟⠛⡶⠶⣤⣀⠙⢦⡀
⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠘⣆⠀⢻⣼⡇⣿⡄⡆⡁⠀⠀⠀⠀⠀⠀⠈⢸⡇⢸⣷⡟⠀⣸⠃⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢿⡇⢿⡇⢁⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟⡐⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⠌⢿⢼⡇⢈⠀⣀⠀⣂⡒⠂⠀⢂⣸⣷⡟⠐⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢀⠐⡢⠙⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠛⠀⡒⢀
⠀⠀⡈⠀⣧⠀⢀⣬⣦⣴⡈⢩⢳⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠻⠌⡱⠤⣤⣬⣄⢀⢃⡇⠈⡀
⠀⠰⢿⣄⠁⡐⠈⠈⡆⠀⠀⢢⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠞⠀⠀⠀⡌⠀⠀⣀⢁⣼⠧
⠀⠀⠀⠣⠷⠼⠇⠴⠃⠀⠀⠈⡀⠘⣄⠀⠀⠀⠀⢠⡶⣶⣦⡀⠀⠀⠀⠀⡰⠁⢠⠀⠀⠀⠀⠳⠄⠾⠵⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⢲⡀⠀⠀⢸⠠⠀⢄⡇⠀⠀⢠⡎⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠠⠈⠀⠩⣂⠀⣅⠀⣬⢀⣴⠋⠀⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠁⠄⠀⡀⠀⠬⠅⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⠀⠀⣄⠀⠀⠀⠀⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢿⠶⢚⣭⣒⢶⡟⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠸⠃⡦⠘⡉⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠉⣯⠁⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢣⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠋⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⢂⠶⠖⠛⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⣂⠀⠀⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣁⠀⠀⠀⠀⠉⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠆⡠⢺⢃⠞⢧⠀⢠⢿⡀⠀⠘⣎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣏⡞⠁⠈⠋⣀⠌⢧⡞⡀⠱⣜⣆⢹⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢱⠋⣡⠤⢤⡀⠀⢠⢚⣿⠀⠀⠐⢦⡀⠀⠀⠀⠈⠃⠈⠙⢿⠀⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠞⠁⠀⠀⡙⢆⠹⣌⡇⠀⠀⠘⠉⠁⡄⠀⠀⢾⠇⠀⠀⢸⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠓⢦⡀⢘⣇⣿⡷⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢀⢰⡘⣆⣇⢷⡎⠀⠉⠛⣛⣧⡀⠀⠀⠀⠋⠲⠀⠀⢀⣇⠀⣸⢋⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣜⠮⢷⣿⣭⡡⠏⠹⡄⣰⣻⡻⠙⠝⡗⠦⠀⠀⠀⠀⠀⠈⠋⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣑⠂⠀⠀⠀⠀⠀⠘⡟⠒⣍⠀⠀⢸⡀⠀⠀⠀⢯⣉⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⠟⡄⠀⠀⢿⠒⠲⠒⢊⡏⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠈⢳⡀⠈⠓⠢⢄⣸⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠙⢦⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠖⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⣒⣲⠶⠴⠤⢤⣤⣤⣤⣤⣤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⠿⠬⠽⠭⣭⠯⣭⠽⠭⠭⠯⠭⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠚⠒⠚⠒⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠳⣄⠀⣴⠃⠀⠸⡆⠀⣀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠲⢦⣄⠀⣸⠀⠀⠀⠀⠘⣶⠃⠀⠀⠀⣷⠞⠁⠀⠀⢸⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠛⢿⠀⡰⠒⠢⡀⢀⠊⠉⢣⠀⣉⠔⠒⡄⠀⣾⠶⠛⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠠⠊⠉⠐⠠⣀⠃⠀⠀⠑⢃⣀⡀⠀⠖⠁⠀⠀⣆⡠⠔⠒⢢⠀⠀⣸⣃⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣤⡤⠬⠧⠀⢆⠀⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠙⠛⠲⢤⡀⠁⠀⠀⠀⡜⠀⠘⠋⠉⠉⢹⡆
⠀⠀⠀⠀⣼⠋⠀⠀⢠⠔⠒⠒⠚⠂⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀⠀⠘⠉⠉⢀⠅⣀⡴⠋
⠀⠀⠀⠀⠙⣦⡀⠀⠘⣄⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠴⢁⡀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⢙⣶⠄⠈⣑⠤⢰⡇⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣈⠇⠀⠀⠈⡇
⠀⠀⠀⠀⣠⠞⠉⠀⡔⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⢅⠀⢤⡤⠶⠚⠁
⠀⠀⠀⠀⣇⠀⠀⠀⠈⠢⠄⡀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢱⠀⠙⢦⡀
⠀⠀⠀⠀⠈⠓⠒⣶⠖⢠⠚⠁⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠺⡀⠉⠁⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠧⣀⠀⢤⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡤⣀⠀⢡⠀⠺⣶⠶⠖⠛
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠠⠃⠀⠀⢀⠙⠲⢤⣄⣀⣀⣀⣀⣀⣠⡤⠞⡉⠀⠀⡇⠀⣀⠁⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠈⠓⠒⠒⢻⡇⠈⠂⠐⠊⠁⠀⠀⠀⢀⢭⠉⠉⠉⡱⡀⠀⠀⡏⠒⠀⠃⠀⢹⡗⠶⠤⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡀⠢⣀⠔⠁⠀⢆⠀⡠⠁⠘⠢⠔⠀⢰⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠴⠞⠹⡇⠀⠀⠀⣠⣷⠀⠉⠀⠀⣶⡀⠀⠀⠀⡏⠳⠦⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣀⡴⠃⠸⣆⠀⠀⣸⠃⠙⢦⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⠦⠶⠃⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠉⠉⠉⠉⠉⠁⠐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢤⣄⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⣬⣻⣶⣒⡂⠳⠆⠈⡀⢝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⠒⠒⠤⠭⠷⠌⠁⠀⠀⠀⡇⠀⢇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣠⡷⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⢀⠀⣀⣠⣄⡀⠀⢠⠟⡄⠀⣠⠌⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⣰⢣⡞⢉⣀⢈⠙⢦⣿⠚⠙⣾⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢉⣾⠰⠋⠀⡀⠂⠈⢲⠶⠞⣹⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠉⡘⠛⠲⠿⠁⡔⠀⠖⠳⣚⠉⢳⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⢀⣤⣮⠵⠊⠁⠀⢈⠒⠒⠊⢁⡌⠀⠀⠈⠉⠉⢉⠴⠤⣄⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣠⡀⠀⠀⡼⠁⢠⠄⠀⠀⠀⠀⠀⠓⠲⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠲⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢦⠀⠀⠙⠛⠀⠀⡇⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢦⡀⠙⠲⠄⠀⠀⠀⠹⢄⣈⡙⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⢾⠉⠁⠉⢱⡆⠀⡇⠀⠀⠀⠀⠀⠀⠒⠂⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠸⠤⠚⠀⠀⠠⠜⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠘⠉⠳⣄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠞⠉⠀⢀⡠⠐⠐⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⢸⠀⠀⠀⢰⣆⠙⠢⠀⠀⠀⡰⠀⠀⠀⡼⠃⠀⠀⡠⠊⠀⠀⠠⠞⠃⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢸⣄⠀⠀⠀⣹⠀⠀⠀⠀⢠⠁⠀⠀⠸⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⢸⠀⠀⠀⠀⢮⠓⠒⠀⢠⡄⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⢸⠀⠀⠀⠀⠈⠃⠀⠀⣴⢱⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⣀⠴⠋⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢦⠀⠘⠀⠀⢀⣀⡤⠀⠀⣰⢿⡌⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠉⠊⠉⠁⠀⠠⠚⠁⠀⠑⢦⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡜⢀⠂⠀⠀⡀⠀⠀⠀⠀⠈⢧⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢼⠀⣜⠀⠀⠀⢱⡄⠀⠀⠀⠀⢸⠀⠀⠘⣆⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠳⡌⢦⡀⠀⢠⠇⠀⠀⠀⠀⡎⠀⠀⠀⠈⠙⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠹⣄⢧⣠⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠠⠤⠒⠒⠒⠒⠒⠒⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠠⠋⠈⠻⡀⠀⠒⠈⠉⠉⠉⠙⠲⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢦⡀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⠈⣀⡤⠀⣀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣠⠀⡐⡛⡟⢻⢻⢻⠻⠛⢫⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠚⠥⠟⠁⠒⠀⠠⠤⠔⠒⠒⠛⠛⠓⠛⠁⠙⠛⠓⠓⠓⠛⠚⠛⠛⠓⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢦⠤⠽⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠿⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠢⣀⡛⣄⠀⣀⠸⡓⠀⠀⡠⠔⠒⠃⠀⠀⠀⠀⢠⡤⠒⠋⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠾⠱⣿⡄⠀⠧⢤⣈⠀⠀⠀⠀⠐⠺⣜⣠⠤⠖⠛⠒⠛⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠶⣖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡴⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⣀⣀⡤⠔⠊⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣒⠿⠛⠉⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠠⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠈⡙⢲⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⡙⠁⠘⡆⠈⢶⢤⣀⠀⠰⡶⢄⠀⡆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢦⠀⢸⡀⠀⠇⠀⠘⣆⠈⠉⠐⠛⠈⢺⠥⠤⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡜⠀⠀⠃⠀⠈⠀⠀⠘⠆⠀⠀⠀⣰⡃⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡧⠤⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢾⣿⡀⢿⠿⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⣖⡀⠛⢁⡀⠳⡄⠰⠒⣻⠁⠀⠀⠙⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⣇⣠⣅⠀⢠⠇⠀⠀⠀⠀⠈⠒⡛⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠦⣀⠀⠀⠈⢑⣿⠉⠬⠤⠞⠢⠄⠀⠀⠀⠀⠘⢅⣠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠿⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⢀⠀⠀⠙⠻⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠋⠛⠛⠛⠿⠿⠦⣠⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⠶⠿⠷⠶⠶⠂⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣉⣽⣿⣿⠀⠀⠀⢴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⡿⠃⠀⢠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣿⣿⣿⣿⣶⣦⠀⢸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⠉⠈⠁⣠⣿⣿⣿⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣿⡿⢿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡇⠈⠁⠙⣿⣿⣿⡿⠛⠛⣿⣿⡿⣿⣿⣿⣿⣿⠿⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠿⠿⣿⠛⠙⠿⣿⡿⣿⡗⠀⠀⠀⠘⣿⣿⡃⠀⢀⣿⡿⠀⢹⠊⣿⣿⣇⠀⠀⠙⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠸⣿⣷⣤⣾⣿⡇⠀⠸⡄⠸⣿⣿⡄⠀⠀⠘⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠹⣿⠙⢿⣿⡇⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⢀⣴⣶⣶⣶⣾⣶⣿⣿⣿⣿⠋⠀⠀⠀⢰⣾⣿⣆⣀⡀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠸⢷⣈⣿⣧⡀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⣼⣿⣿⣿⡏⠉⢻⣿⣿⡟⠁⠀⠀⠀⢀⣿⣿⡿⢿⣿⣧⡀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠁⠈⢿⣿⣄⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠈⢿⣿⣿⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣿⣿⡇⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⢻⣿⣆⣀⠀⢀⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣿⠇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠉⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⣸⣿⣿⣿⡿⠿⣿⣿⡿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⠀⠘⠇⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⣾⣿⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⣼⣿⣿⣿⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⡆⢸⣿⣿⣿⠈⣿⣿⠀⠀⠀⣴⣿⢿⣿⣿⣿⠿⣿⣿⡄⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠰⡀⠘⢿⣧⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⣿⣿⣤⣤⣼⣿⣧⠀⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢿⣷⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⠏⠀⣿⣿⡇⠀⣹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠟⢹⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⡧⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁⠈⢿⣿⣿⡟⠀⣼⣿⣿⣿⣿⣿⣿⣧⣀⣿⣿⣧⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⢸⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣶⠀⢀⣰⣿⢓⣦⡀⢀⣼⡅⠀⠀⣰⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣤⣤⣴⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠿⠿⠟⠛⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⠇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢠⣿⣿⡏⠀⢸⣿⣿⣿⠿⠋⣤⡈⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣸⣿⣿⠃⠀⣾⣿⣿⡇⠀⣾⡿⠁⠀⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣷⣤⣉⠉⠉⣠⡘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣷⣾⣿⣿⣿⠀⠀⣿⣿⣿⣄⣰⣿⡇⠀⣾⡏⢻⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⡿⠀⣾⡟⠁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠋⠈⣿⣿⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⢿⣷⣾⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⠃⠀⠛⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⢠⣿⣿⠀⢈⣿⣿⣿⣿⠁⣠⠄⠀⢸⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡄⠀⣤⠀⠉⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⢠⣾⣿⡏⠀⢸⣿⣿⣿⣿⣏⠁⢀⡀⠸⢿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡀⠙⢠⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠆⠘⣷⣄⢸⣿⣿⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠛⠃⢸⣿⣿⠇⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⠇⢀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢻⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⣿⡏⠀⠀⢠⣿⣿⣿⣿⠏⢀⠛⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⡇⠀⠀⢸⣿⣿⣿⠃⣠⣿⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⡆⠀⢀⠀⠀⣀⣀⣤⣤⣀⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣿⣿⡇⠀⠀⢸⣿⣿⣿⣾⣿⠇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠈⠉⠉⠉⠉⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⢸⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⣿⡿⠁⠀⣀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠠⠶⠶⢶⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣀⣿⠆⢰⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠠⣶⡦⠤⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠁⠀⠀⠀⣸⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢀⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⡿⠉⠻⠂⠀⠀⢠⣴⣶⣦⣤⣤⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⢸⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⣿⣿⣧⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⡇⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠃⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣄⣤⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣧⡄⠀⠀⠐⠀⠀⠀⠀⠀⣠⠘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⣨⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠃⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⢾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠃⠠⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠃⠘⠁⠀⠀⠀⠀⢀⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣼⣿⣿⠀⠀⠀⠀⠀⠀⠘⠛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡁⠀⠀⠀⠀⠀⢠⠄⠀⠘⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣇
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⠛⠁⣀⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⣤⡀
⠀⢀⣴⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠙⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠟⠻⢿⣿⣿⣷⣄⡀
⠀⣾⣿⣿⣅⣠⣦⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣄
⢀⣿⣿⣿⣿⣿⠿⢿⣿⠿⢿⣷⣀⣀⣤⣤⣶⣤⣴⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⣶⣾⣷⣾⣿⣶⣽⣿
⣿⣿⣯⣿⣄⣼⣶⣾⣷⣤⣤⣿⣿⡿⠿⠿⠟⠛⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣤⣀⣀⣀⣀⣴⣶⣦⣴⣾⣿⣧⣽⣭⣠⣼⡋⢙⣿⣿⣿
⠉⠙⠛⣛⠛⣿⣿⣟⠛⣻⣿⠋⢁⣠⣶⣦⣀⣀⠈⠉⠉⣀⣀⣀⡀⢀⣀⣀⡀⢀⣀⠀⣀⣀⣈⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⣿⡟⢿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⢿⣿⠛⢿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡆⠈⢁⣄⠀⠸⣿⡀⠀⣰⣯⠀⠀⢠⣶⡄⠙⠀⣴⣶⡿⠃⣴⣆⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠷⣶⣄⠀⠀⠀⠀⠀⠀⣾⡏⠀⣀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈⠻⣷⡀⠀⠀⠀⣴⡟⣀⠾⠁⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠶⠆⣼⣿⣤⣄⣤⣿⡇⠛⠀⠀⠀⣠⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢹⡏⠉⠉⠙⣿⣦⣴⣿⣿⣏⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⣀⣠⣾⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠙⠿⣿⣝⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠈⠻⢿⣤⠻⣯⡙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⠁⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠙⣿⣆⠙⣦⡙⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠷⣶⣤⡈⠀⠀⠀⠈⢿⣷⡘⢿⣼⢿⣿⣿⣿⣿⣿⣿⡇
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⡈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⣀⡀⠀⠙⣿⡆⠀⠀⠀⠀⠿⣧⡼⢻⣼⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣁⡄⢸⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢼⣿⣏⣠⠀⢸⡧⠀⠀⠀⠀⠀⠻⣧⡼⢻⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡟⠿⣧⡀⠀⠈⠙⠋⢁⣼⠋⠀⢰⠆⠀⠀⠀⠀⣿⡄⠀⠈⠻⠟⠃⢀⣾⡇⠀⠀⠀⠀⠀⠀⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡇⠀⠙⠿⣦⣤⣤⣴⠿⠉⠀⠀⠸⢧⠀⠀⠀⠀⠈⠿⣧⣤⣤⣤⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠟⠛⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠛⠛⠿⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⠶⠿⠛⠛⠁⠀⠀⠀⢿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⡏⢙⡛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⠿⢯⠛⣦⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⢀⣤⣤⡤⠚⠻⡿⠛⢠⡜⠀⢸⣤⠘⠛⣥⠀⠀⠀⠀⠙⠛⠛⠓⠢⣤⣤⢀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣿⠟⠛⠛⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣴⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣦⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣏⠀⠃⠀⠞⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠃⠀⡄⠀⠀⠹⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⢀⡆⠀⠀⠀⢠⡄⠚⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡠⠏⠀⠀⠀⠀⠀⠀⢀⠈⢀⠤⠀⠠⠤⠀⠀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠀⠀⠀⠀⠀⠀⠈⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⢰⣿⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⠷⠶⠶⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢉⠉⠉⠉⢙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⢿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⢸⠃⠀⠀⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⢠⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠘⠁⠀⠀⢠⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⢀⡜⠃⠀⠀⠀⢸⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡄⠀⠀⠀⠀⠤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⢀⡁⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠑⢦⣄⠀⢸⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠈⠉⢶⣿⡁⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠇⣀⠀⠀⠈⣿⡇⠀⠀⠛⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⡔⠂⠀⢻⣧⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⢅⠉⢤⡤⠙⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣤⣤⣤⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠉⠀⠈⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣓⡦⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⡴⠖⠚⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⠁⣠⡶⠉⠀⠙⠳⣝⡆⠀⠀⣠⡴⠋⠉⠙⠋
⠀⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⡟⠀⠀⣦⠀⠀⠀⠉⠲⣴⠋⠀⠀⠀⡼⠁
⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠈⠁⠀⠚⠁⠀⠀⠀⠀⠀⠘⣾⡄⠀⢸⠁⠀⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣟⠛⠃⢀⣴⠀⠲⠦⣀⠀⠀⠀⠀⣸⠀⠀⢹⣷⡀⠘⣇⠀⠳⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣿⠃⣀⡼⠀⠈⠓⠀⠀⣠⠏⠞⠀⢀⡇⠱⡄⠈⠳⢦⡙⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡀⢀⣀⣀⡀⠔⠁⠀⠀⢀⡼⣧⠀⠙⣆⠀⠀⠙⠲⣝⣦⡙⠦⣄⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣝⣒⣾⠋⠉⠛⠲⢾⣉⡁⠈⠳⢄⡈⠳⢤⡄⠀⢀⣹⣿⣷⣄⣙⣦⣤⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣥⣒⣒⣒⣒⣀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠷⠶⠶⢄⠁⠉⢙⡛⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⠀⠀⢀⣀⣀⣀⠹⣎⢻⡄⠈⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠉⠉⠉⠙⠇⠹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠉⠛⠋⠳⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠴⠶⠀⠠⣶⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠁⠀⠀⠈⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀⠀⢤⠤⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣄⣤⠄⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡿⠀⠀⠀⠀⡈⠙⠂⠀⠀⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠚⠒⡀⠈⠁⠀⣤⡤⠀⠀⠠⠤⠄⠀⣀⡀⠈⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⣦⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⣀⢀⣾⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢹⣿⠿⣄⣀⣠⠴⢋⠘⠲⠤⠤⣤⠴⢿⠋⢸⡇⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣼⠹⡆⠀⠀⠤⠤⠀⠐⠶⠶⠒⠀⠀⠀⢠⡟⢻⣄⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⢠⡏⠸⡟⠀⠠⠤⠀⠸⠀⠀⣀⣠⠀⠀⣠⠏⠀⠀⠙⠃⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠋⠀⠀⡿⠂⡀⠀⠀⠶⠀⠉⠉⠀⡀⠀⢸⠀⠀⠀⠀⢰⡆⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡄⢀⠄⠀⠀⡇⠀⣷⠀⠀⣄⠀⠀⠀⣸⠃⠀⣾⠀⠀⠀⠀⠘⣧⡀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢁⡞⠀⢀⣼⠷⣤⣽⣄⣀⣹⣄⣀⣤⣿⣴⠶⢿⡆⠀⠀⠀⠀⠈⢷⡀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠏⠀⠀⠸⠋⢙⣠⠉⠩⢭⣿⣭⠭⠁⢠⣄⠀⠉⣿⠀⠀⠀⠀⠀⠈⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠀⠹⡆⠀⠀⠀⠀⠀⠘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠁⡆⠀⠀⠀⠀⠀⠀⢀⣀⡶⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢺⣀⠘⣇⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢠⠄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠪⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠚⠁
⠀⠁
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⠩⠿⣁⣩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣎⠒⠶⢤⠘⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⡨⢠⠈⢀⡰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⠁⠀⠈⠑⠢⢤⣠⡏⠀⠀⠁⠃⠀⠨⠓⢄
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⢴⡾⡋⠢⠤⠄⠀⠀⢀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⢀⣠⠖⠋⠁⣀⠐⢱⠀⠀⠴⠆⠑⠂⢄⡀⠀⠈⡌⠀⠀⠀⠀⠀⠀⠜⢀⡀⠀⠑⡀
⠀⠈⢻⣄⠀⠘⠄⠸⠃⠈⠢⡀⡀⢀⠤⡪⠀⣈⡴⢻⠁⠀⠀⠀⠀⠀⠘⠀⠀⠑⠀⠀⠈⠠⡀
⠀⠀⢸⠈⠓⢤⡀⠀⡀⠈⢆⣀⠍⠓⣨⠴⠚⠓⠂⠣⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⢄⡠⠣⡀
⠀⠀⢸⠀⠀⠀⠙⠢⡉⠀⢀⣠⠖⠋⠁⠀⢀⠴⠋⠀⠀⠉⠓⠂⢀⡴⡀⠀⠀⠀⠀⠀⠀⢑⡢⡈⡆
⠀⠀⢸⠀⠀⠀⠀⠀⢹⠚⠉⡇⠀⠀⢺⠚⠁⠀⠀⠀⢀⣠⠔⠚⢁⣀⡜⠀⠀⠀⠀⠀⠀⠐⠠⠬⡀
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⠀⢖⠈⣿⠀⠉⠉⣏⢸⡇⠀⠀⠀⠀⠀⠈⠒⠤⠥⠃
⠀⠀⢸⡀⠀⠀⠀⠀⢸⠀⠀⣧⠀⠀⡜⢆⠀⠀⠈⡄⣿⠀⠀⠀⡿⠈⡇⠀⠀⠀⢠⣶⣻⣛⡏⣻⢙⣓⡄
⠀⠀⠈⠳⡄⠀⠀⠀⢸⠀⠀⣿⠀⠀⡇⠈⢆⠀⠀⠸⡇⠀⠀⠀⡇⠀⣷⠀⠀⠀⠈⣿⣟⣏⣹⣉⣯⣿
⠀⠀⠀⠀⠙⢦⠀⠀⢸⠤⠒⠋⠉⢹⠀⠀⠈⢧⠀⠀⢣⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⡀⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⣄⢸⠖⠒⠋⠉⠁⠀⠀⠀⠈⢧⣀⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⠺⠇⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣰⡟⠉⠉⠉⠉⢻⣦⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⣤⣶⣶⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠛⠛⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠒⠒⣞⠀⠀⠀⢈⢷⡂⠒⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡆⠀⠀⠈⠉⠉⠀⢸⡇⢰⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⡃⣸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠛⠚⡆⠀⠀⠀⠀⢸⠓⠛⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⣡⣄⡼⠃⠀⠀⠀⠀⢘⢧⣄⣌⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⢁⢼⣿⣿⣧⠀⠀⠀⠀⠀⠘⣼⣿⣿⣧⡈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠋⠀⡠⠂⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠑⢄⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠔⠁⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠈⠢⣀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠓⠒⠒⠒⠒⠚⠛⠛⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠙⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⡠⠠⣦⠀⢠⠀⠐⠆⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⢁⣀⡀⠂⠃⠔⠘⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠁⠀⠀⠈⠠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣘⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠆⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡋⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠠⠚⣸⠂
⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⡀
⠀⠀⠀⠀⠀⢀⡆⣀⡀⠠⠀⣤⣤⡄⣠⢖⡙⣗⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡤⢀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⢀⠜⠱⠤⠃⠊⢁⡌⢩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⡟⠻⠗⠲⢊⡤⠚
⠀⠀⠀⠘⣦⣀⠎⠉⠒⠤⠎⠀⠀⠀⠀⢺⠉⠀⣙⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢰⠤⢧⠀⠀⠀⠨⡯⢒⠠⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠑⣄⡘⠶⠀⠀⠀⠀⢀⠄⠠⢄⠈⠳⠒⠉⠀⠀⠶⡩⣔⣂⠾⣀⡀⠉⠂⠭⣒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠳⠀⠀⣏⡙⡇⠈⠢⣀⢀⠀⠀⢀⠆⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠚⠥⠒⢶⡅⡉⠊⠷⠄⢀
⢇⡀⢀⣀⠀⠄⠐⠈⠀⠀⠀⠀⠑⣀⠀⢁⠇⠀⠀⢘⠚⢷⢞⠁⠀⠀⠀⠀⣀⠤⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂⠤⠁⡒⢴⢦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠒⣤⡤⠂⢀⠺⡆⠆⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠤⡀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⣖⠁⠀⢀⠈⡀⠀⠀⠀⠀⠈⠀⠁⠒⢪⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠑⠄⢈⠀⠰⡀⠀⠀⠀⠀⠀⠀⢀⡿⡻⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠀⠀⠀⠙⠢⡄⢵⣰⠗⠒⣲⠀⢀⡾⠋⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⣠⣴⠀⠀⠀⠀⠈⠛⢿⡴⡒⢰⡥⠃⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⠀⢴⣉⡸⣇⠠⢇⠀⠀⠠⡀⠈⢠⡤⠛⠋⠀⡀⠀⠀⠀⣀⣀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠻⠔⠒⣣⠤⢶⠞⢻⡁⠀⠀⠀⠀⢀⣠⠖⠀⠀⠠⠀⠛⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠶⠆⠀⢰⠁⠀⠀⠀⠚⠙⢆⠀⠀⠀⠈⠹⣀⠀⠀⠀⣀⠎⠀⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⣈⡆⠀⢰⡆⠘⡤⣀⠀⡤⢀⠀⠸⡄⠐⠀⠀⠠⠈⣱⣖⡊⠁⠂⠀⠀⠈⡀⠑⠄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠃⢀⡘⠂⡉⠁⠀⣙⠢⣀⠥⡐⢾⣄⣭⡴⣾⣽⣖⣤⢀⠀⠀⠀⣀⢴⠄⣳⠉⠐⠢⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠁⢧⠄⠀⢠⡄⠈⠐⠢⢄⡙⣶⠒⠂⠀⠒⣞⢑⣖⠂⠒⠒⠊⠉⡝⠣⢄⡐⠒⢄⠀⠀⠀⣈⡑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣴⠿⢧⠤⠴⠉⢏⠁⢾⠀⠀⠀⠀⠀⠀⠀⠢⠠⠃⠈⠂⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠺⢄⣀⣀⡠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢄⠈⠂⠴⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠘⢤⠋⠢⡀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄⠀⠀⠸⠆⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠎⠀⠀⠀⠉⠉⠉⠉⠙⠒⠲⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⣠⣶⣿⣿⣷⣈⣿⣝⢿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣀⣀⠀⠀⣠⣾⣿⣿⣟⠷⠞⠉⠉⠉⠀⠟⠿⢿⣿⣷⡄⠒⠚⢉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢸⣿⣯⣹⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠐⢶⣷⣦⣿⣿⠀⠀⠉⠉⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⡜⠁⢀⢀⣿⣿⣿⣿⣿⣏⣻⠀⠀⠀⣀⠈⠑⡄⠀⠀⠀⠀⢙⣿⣻⣧⡤⠖⠚⠉⠁
⠀⠀⠀⣠⠴⠚⠛⢛⣷⣶⣾⣅⡀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⡿⢿⠛⠉⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⣠⠟⠋⠁
⠀⠀⠀⢀⣐⠔⣀⠄⠉⠛⣿⣟⡿⣶⣄⠀⠀⠀⢸⠙⢿⡿⠻⠷⢋⠀⠀⠀⠀⠀⠀⠀⢀⠞⡀⠉⠣⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠉⢻⣷⣶⣤⣠⠄⠘⠿⣾⣿⣷⣄⢀⡼⡆⠠⢐⠀⠀⠸⡄⠀⠀⢸⣶⣄⠀⠆⠀⠙⢢⠖⡠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣾⣁⠄⠈⠿⣭⡻⣿⣼⠃⠄⠨⠀⠀⠀⠙⣆⠀⠀⠙⠛⣀⠤⣢⠖⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠘⢶⣤⡤⢀⡀⠙⠿⣿⣦⡀⢀⠈⠹⣯⣿⣶⣂⠎⠀⠀⣀⣤⣾⡷⠒⠒⠉⠁⣘⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠙⢿⣿⣔⢀⡀⠀⠘⠿⣷⡴⠂⠀⠈⠉⠛⠦⣀⠹⠿⠛⠁⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⡤⠀⢀⠀⠀⠉⠑⠲⢶⣄⣀⠄⠈⠙⠲⢤⣤⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠥⣖⠁⠀⠀⠀⠀⠉⢻⣿⣦⣴⠄⣀⠈⢙⣦⣀⠈⠑⠶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣴⣭⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠮⣁⠤⠀⠀⠹⢿⣿⣯⡁⠀⣀⠘⢿⣷⣄⠀⠀⠉⠓⢦⡀⠀⠀⢴⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁⠀⠀⠙⠻⠿⣌⠄⠀⠀⠹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠛⠛⠋⠉⠉⠉⠛⠛⠓⠲⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣍⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⡉⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⠟⠘⢛⣓⣒⣒⠓⠲⣖⣀⣀⠀⠀⢀⣀⣀⡤⠔⡾⠀⠀⠀⠀⠈⠻⢯⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⠖⠉⠀⠀⠈⢳⡄⠀⢀⠀⠈⠉⠀⠀⠀⠀⠀⡧⡶⠀⠀⠀⠀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠞⠁⠀⠀⠀⠀⠀⠀⢳⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠓⢦⡁⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⡇⠀⠀⠀⠀⠀⢀⡀⡼⠁⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢿⠀⢀⣏⣤⠤⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠹⣄⠀⠀⠀⢠⡏⢁⣇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⠓⠉⣡⠤⢦⣄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡴⠚⢿⠒⠲⠖⠛⠉⢹⡻⠤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠳⠶⠋⠈⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢧⣀⣈⠃⣤⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⢦⣄⠀⠀⠀⠒⠦⠀⠀⠀⠀⢀⣀⡴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠈⣣⠀⠀⠀⠈⠙⠓⠒⢶⠶⠶⠶⠒⠛⠉⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢛⡖⠚⠻⡄⠀⠀⠀⡎⢠⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⢀⡤⢤⣀⠠⣴⠊⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⠟⣷⠀⢅⠀⠀⡸⣥⡇⠀⠀⠀⠀⠀⠀⠰⡬⡇⠀⠀⢸⠀⡶⡞⢶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡽⠶⢤⡑⠞⣡⠏⡇⠀⠀⠀⠀⠀⠀⠀⣷⠸⠤⠚⣡⡜⣇⡸⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣾⠀⠙⠶⠃⢘⡇⠀⠀⠀⠀⠀⠀⠀⢾⠣⠔⠉⢘⣏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡮⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣋⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⡇⠀⠀⠀⠀⠻⡤⣤⣀⣀⣀⣀⣀⣀⡼⠀⠀⠀⠀⣚⡉⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣇⣀⣀⣀⣀⣀⣸⠀⠀⢸⣅⣏⠀⠀⠀⣇⠀⠀⠀⠀⠘⣟⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⢼⠿⠼⣿⡼⠿⣿⣤⣤⣍⣽⣉⣁⣀⣀⣼⣿⣉⣿⣑⣿⡋⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢠⠛⡷⣾⠏⠻⣿⡭⠻⡁⠀⠈⢳⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠉⠑⠋⠁⠀⠀⠈⠓⠾⠳⠦⠾⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢀⡾⢛⠉⢛⣿⡿⠛⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⢀⣶⢟⠋⢛⣿⡿⡛⢻⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⢸⠱⣷⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⠱⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⡄⠀⠛⠛⠋⠉⠀⠈⠉⠉⣿⡷⠻⣿⣇⠀⠀⠀⠀⡴⠀⠛⠛⠋⠉⠀⠈⠉⠉⣿⡷⠻⢿⣇
⠀⠀⠀⠀⢰⠃⣠⣀⠀⠀⢀⣀⣀⡀⠀⠿⣿⡆⣸⡿⠀⠀⠀⢰⠃⣠⣀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡆⣸⣿
⠀⠀⠀⠀⠸⡀⣠⡉⠀⠀⠛⣩⣍⠉⠀⠀⣿⡿⢿⠇⠀⠀⠀⠸⡄⣡⣍⠀⠀⠚⠛⠋⠓⠀⠀⣾⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠘⠋⣠⠀⠀⠹⠏⠈⠁⠀⠡⡎⢈⠇⠀⠀⠀⠀⡁⠘⠋⢰⠀⠈⠉⠉⠈⠀⠀⠩⡎⢀⠇
⠀⠀⠀⠀⠸⡆⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⣡⠊⠀⠀⠀⠀⠐⡇⠀⠰⡅⠀⠀⠀⢀⠀⠀⠀⠀⣡⠎
⠀⠀⠀⠀⠀⢳⠀⠠⠤⠤⠤⠐⠛⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⢳⡀⠲⣖⣒⣲⠖⠃⠁⠀⡴⠉⠁
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠉⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠭⠁⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡖⠒⠲⠛⠳⠖⠲⢶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠘⢄⠘⡀⠀⠀⠀⠈⡄⠀⠉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠑⢄⡀⠀⠀⠳⣄⠀⢣⠀⠘⣤⡤⠛⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⢄⡀⠀⠙⠢⣀⡀⢈⡱⠦⠛⠉⠀⠀⣀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣄⣀⣈⠴⠦⠒⠊⠉⠀⠀⢀⡠⠤⠒⠉⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣗⠒⠦⢤⣴⣦⣉⣁⣀⣀⣠⡴⠚⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣹⣿⣿⡟⣛⣭⣭⣍⠀⠀⢀⣤⣾⡿⠍⠉⢉⠿⢿⣿⣽⡿⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⢟⡁⠀⠘⡆⡧⣶⠀⠉⣿⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠲⠶⠦⡄⠙⣿⡿⠉⠀⠸⣿⡇⠀⠀⢡⡘⠟⠀⠀⣸⠓⠲⠶⢶⣎⢧⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠐⢀⠘⣆⢹⡇⠀⠀⠀⠛⠛⠀⠀⠀⠉⠢⠤⡴⠃⠀⠀⠀⠀⢸⡎⠻⡏
⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⠾⠛⠛⠷⣿⣼⣷⣄⡀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠲⠤⣤⣤⣤⡾⢃⡀⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠐⠒⠒⢸⣿⣿⣿⣟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏⠀⠀⠀⢣⣀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢶⣴⡶⠟⠋⠉⠛⢯⡳⢦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠶⣤⠈⠓⠋⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣤⣀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠙⢦⣉⠻⣦⣄⠀⠀⠀⠀⠈⠁⠀⠀⠈⠃⢀⣼⠋
⠀⠀⠀⠀⠀⣼⠁⢀⣿⠋⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠈⠻⣷⣄⠙⠿⡶⣄⡀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⢻⣤⣼⠃⠀⠀⠀⠀⠀⢀⣴⡿⠿⢷⣶⣦⣤⣝⢿⡕⢄⠘⠮⣙⠶⣤⣶⡿⠋⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢈⣿⡈⢻⣆⠙⢦⡈⠳⣌⠙⢷⣄⡀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⡟⣿⣤⣄⣀⣀⣀⣰⡿⠿⠶⣶⣶⣴⣶⡿⢛⣤⠦⣿⣦⠀⠙⠦⡈⠑⢦⡈⠛⢷⣖⠾⣿⣦
⠀⠀⠀⠀⠀⠀⡇⣿⠀⠈⠉⠉⠉⣿⠁⠀⠀⢰⡟⢀⣿⠃⢸⡇⠀⠘⣿⣷⡀⠀⠈⠢⡀⠙⠢⡀⠉⠳⢦⣉⠳⢦⣄
⠀⠀⠀⠀⠀⠀⡇⢿⣄⣀⣀⣤⣴⣿⡀⠀⠀⣼⠇⢸⡟⠀⣸⣷⢴⠞⠛⢋⠻⣄⠀⢀⠘⠂⠀⠀⠀⠀⠀⠙⠷⣤⣼⠇
⠀⠀⠀⠀⠀⠀⣷⡾⠋⠀⠀⠈⠉⠉⠻⣧⢠⡿⠀⣾⠃⣼⠋⠢⠀⠀⠀⠈⠳⠹⣦⠀⠑⣄⠀⠀⠀⠀⠲⡀⠀⣈⣿
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣰⡦⣿⣾⠃⢸⡏⢸⡏⠀⠀⠀⡄⠀⠀⠀⠀⠈⢷⡀⠈⢣⠀⠀⠀⢀⣷⣾⠟⠉
⠀⠀⠀⠀⠀⠘⣧⣀⡀⠀⠀⠀⠀⠻⣤⣾⠇⢠⣿⣀⣸⣧⡀⠠⠀⠐⢀⠀⠰⠀⢁⣈⣷⡀⠀⣣⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣤⣤⣦⣤⣤⣿⠃⢠⡿⠛⠛⠛⣿⠻⣶⣦⣀⣽⣄⣴⣷⡟⠛⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠸⡏⢿⣄⣼⠋⢠⡿⠁⠀⠀⠀⢹⣆⣸⣏⣉⣽⠋⠁⠈⢻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢿⣈⡿⠃⣰⠟⠁⠀⠀⠀⢰⡿⠿⠿⢧⡀⠀⣠⡶⠛⣉⣉⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⣠⠾⣿⡀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢹⡿⠋⠉⠁⣀⠠⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡞⢁⡶⢾⣿⣦⡀⠀⠀⠀⠈⢿⡄⠀⢸⡇⢀⡴⠊⣡⠔⠀⠀⢿⣦⣤⠶⠛⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣼⠃⣼⠁⢸⣇⠀⠙⠃⠀⠀⢀⣼⠇⠀⠀⢻⣏⢀⠔⠀⠀⠀⠀⣠⠉⠁⠀⡀⠀⠡⠈⢷
⠀⠀⠀⠀⠀⠀⠀⣼⠏⢰⡏⠀⠻⣦⡀⢻⣦⣀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠻⣏⠀⢠⠃⠀⠀⢻⠀⠸⡀⠸⣄⠀⢱⣿
⠀⠀⠀⠀⠀⢀⣠⡟⠀⣾⠁⠀⠀⢸⡇⠀⠈⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣼⠀⢀⡀⠘⡀⠀⠳⡄⢈⣶⠟⠁
⠀⠀⠀⠀⠀⢸⡛⠻⢾⣧⠀⠀⠀⠘⣧⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣸⠀⠀⢳⡀⣠⣾⠟⠁
⠀⠀⠀⠀⢠⡞⠁⠀⢴⠟⠁⠀⠀⠀⠘⠷⣤⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠦⠶⠟⠋
⠀⠀⠀⠀⠈⠙⠳⠦⡼⠃

⠀⠀⠀⠀⠡⡀⠀⠀⠇⠀⣀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣶⣤⣤⣤⣴⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠰⣄⢒⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⣿⣿⣿⣿⣿⣿⡋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡄⣠⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣶⣿⣾⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠻⠟⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣜⣾⣿⡷⠧⣽⡟⣿⣿⡶⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠠⣷⡿⡟⠉⡃⠀⢰⠈⢺⢻⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⡀⠀⠒⠀⠇
⣤⡀⠀⠀⠀⠀⡷⠶⠆⠇⡀⠆⠀⢸⠀⡀⡌⣿⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢆⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠻⡇⠀⠀⠀⠸⠯⠭⢭⠀⠁⢃⣀⣸⠀⠀⢠⣙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠒⠀⢢⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠋⠀⠒⠈
⣿⣧⠀⠀⠀⠀⣿⠭⠽⡀⠀⠘⠿⠟⠀⠀⠈⡹⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢣⣀⣤⠀⠀⠀⠀⠀⠀⠀⣾⣶⣶⣶⡄⠀⣴
⣿⣿⠀⠀⠀⠀⠙⠽⣶⣣⢀⣀⣀⣀⣀⠤⣬⣿⡣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠓⠂⠻
⢻⠇⠀⠀⠀⠀⠀⠚⠻⣻⢿⣿⢿⣧⣾⣶⣽⠝⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠋⠁⠀⠄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣃⣀⣠⣤⣄⠀⣤⣤⣤⣌⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡏⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⣀⢀⡀⠀⣀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠀⠀⠈⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣦⡀⢸⠵⠷⢣⣎⡜
⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⣶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠁⠈⠙⣏⣶⣤⣄
⠀⠀⠀⠀⠀⢠⠀⣀⡀⠠⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠀⠀⠀⠤⢿⢻⢭⠭
⠀⠀⠀⠀⠀⠀⡜⢹⠙⡄⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣒⠀⠀⠀⢒⣻⠸⠭⣿
⠀⠀⢀⠀⠀⠀⠃⠀⡰⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣶⣶⣶⣾⣿⡯⡭⠭
⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⡀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡤⢤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⣿⠛
⠀⠀⠀⠀⢃⠀⢡⠠⠐⠀⢠⠃⠈⠁⠀⠀⠀⠀⠀⣰⢮⣶⠊⠀⠀⠆⠮⠀⠈⠹⣯⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠛⠋
⠀⠀⠈⠀⠈⠂⠁⠀⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢡⡀⡀⠀⢈⣁⠀⠀⠀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢠⣿⣿⣧⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣈⠙⠋⣉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⢠⣶⣦⡀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢶⠦⣀⣤⣤⡤⠾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⠙⠛⠛⠛⢹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⡿⣿⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣉
⠀⠀⠀⠀⡄⠂⠈⠋⠿⠁⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠈⣷⠠⣿⠀⢀⣀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⣿⠀⠀⠀⠃⠘⣿⣿⣿⠟⠁
⠀⠀⠀⠀⡟⠋⢲⣶⣶⡇⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠐⠈⠀⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣇⠀⠸⣿⠇⠀⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣴⣿⡿⢷⣿⣾⢿⣿⣧⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⣷⣿⢺⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣿⣷⣿⣷⣾⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⣿⣷⣿⣛⣿⣿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⢰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⣿⡿⢀⣿⣿⣿⣷⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣌⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣯⣿⣽⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣻⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣿⣿⣿⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣾⣆⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣷⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠾⠇⠙⠃⠙⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠋⠘⠋⠸⠷⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⠀⠈⠳⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠞⠁⠀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠅⠀⢀⣀⣀⠈⠀⢀⣴⠶⠞⠳⠶⣦⡀⠀⠁⣀⣀⡀⠀⠨⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠉⠀⣴⠏⠀⠀⠀⠀⠀⠀⠹⣦⠀⠉⠻⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠆⢸⡇⠀⣀⣀⡀⢀⣀⣀⠀⢸⡇⠰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢻⡾⠋⠉⢻⡟⠉⠙⢷⡟⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣴⠟⠁⠀⢀⣼⣧⡀⠀⠈⠻⣦⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡾⠁⠀⠀⣠⡞⠁⠈⢳⣄⠀⠀⠈⢷⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠼⠋⠀⠀⢀⡾⠋⠀⠀⠀⠀⠙⢷⡀⠀⠀⠙⠧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣧⣤⣤⣤⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣼⣧⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠃⠠⠊⠐⠗⣪⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠲⢤⡈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⠁⠀⠀⠀⢀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠄⠀⠀⠀⣠⣭⡈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡸⠁⠀⢺⣌⢱⠀⢴⣁⠇⠀⢣⢠⣄⠘⠛⠛⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⠐⢀⠞⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠒⠁⠀⠇⠘⠳⠎⢸⡀⠀⢄⣀⣀⠀⢸⠀⠑⠃⠐⠶⠆⠀⠀⠀⠀⡠⠊⠀⠀⠘⠈⢠⠏⡀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢺⠺⠘⠀⠀⠉⢿⡊⠁⠀⠈⢳⢸⠀⠀⠀⠀⣀⣀⠄⠀⣠⠊⠀⢀⠰⠀⠀⠔⠁⢊⡠⠀⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠔⣡⠋⠉⠑⠎⠇⠀⠀⠠⠀⠤⢂⡤⠊⠁⢠⢸⠘⣠⠖⠀⠀⠀⠉⢀⡌⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠉⠉⠀⠀⠀⠀⡎⠈⠁⠀⠀⠀⠀⠐⠂⠉⠐⠒⠀⠁⢀⡀⠸⠸⢀⠜⠁⠀⠀⠀⠀⠀⠈⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⢒⣋⣤⠤⠒⡹⢀⠂⡰⠃⡔⢀⠔⢀⠔⢠⠆⠀⢠⢂⠧⠇⣁⠔⠁
⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⢄⠖⡺⢉⢝⡵⡣⢂⠜⡰⢃⠞⣠⠊⡠⢋⡴⢋⡴⠁⡠⠂⡀⠨⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠠⣙⠲⠤⣁⣊⡔⠁⠊⠊⠐⠃⠞⡰⢃⠞⣡⠞⡠⠊⡠⠋⡠⠊⡠⢰⠁
⠀⠀⠀⠀⠀⠀⠲⠬⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⢈⠔⣡⠜⣡⠎⣠⠊⡠⠊⡠⠊⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢈⠔⡡⠞⣡⠞⡡⠊⡠⣊⠔⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡡⠚⡡⢊⡵⢋⠔⡫⢚⡁⢠⠏⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢈⡴⢊⠔⢁⢔⡵⢋⠔⢋⠀⡇⠀⢸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣩⠔⣡⠔⠁⢬⠐⡡⠒⡡⠄⢸⡀⠸⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠎⠀⠀⢚⡡⢚⠵⠂⡇⢸⠈⠰⣊⠔⢊⠀⢧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠄⢠⠀⠀⠀⡨⠒⣁⠔⠀⡇⡈⠀⡨⠔⣊⠡⠂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⡄⢸⠀⠀⠀⠴⢊⡥⠒⠀⡿⠀⠀⢒⡩⠔⢊⡁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠁⠸⡆⠀⠀⠚⣡⠔⣃⠄⢱⠀⢈⡥⠖⣊⠥⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢧⠀⠀⠈⠴⣊⠔⡂⠈⢇⠀⢒⣉⠤⣒⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠊⠔⠪⠶⠀⠘⠀⠀⠀⠈⢀⡰⠳⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡀⠤⠒⡀⠤⢳⠀⠀⠀⠀⠀⠀⠀⠙⠻⡭⢛⠡⢂⠤⠒⡩⠔⢊⡡⠔⣂⠠⢒⡢⠔⡠⠤⣠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠔⠊⠥⢒⠶⢦⡤⠤⢤⣤⠴⢐⡣⠐⣉⠤⠒⠉⠔⢊⡡⠖⠊⠀⠊⠁⠀⠉⠀⠈⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠶⢲⡆⠀⠘⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣠⣤⣀⣤⠴⠦⣄⡀⠀⠀⠀⠀⠀⢁⣴⠋⡀⠀⣴⣏⣤⡄⠀⣴⣋⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠙⠏⠁⠚⠉⠀⠀⠀⢸⡇⣀⡀⠀⠀⠀⠛⠛⠋⠁⠀⠉⠉⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣠⠶⠶⣄⠀⠀⠀⠀⣴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠘⣧⣀⡀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠙⢳⡄⠀⠀⣠⡤⠴⠶⠶⢦
⠀⠀⠀⠀⠀⣽⠆⠀⠀⠈⢮⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⡶⢿⣤⣤⣤⡄⠀⠀⠀⠘⢿⣦⣼⠃⠀⠀⠀⠀⢘⡷⠶⠤⣤⣀
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⢸⡷⠶⠶⠞⠋⠁⢀⠀⠤⣄⡀⠈⠙⢦⡤⠀⠀⠀⣼⠃⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠈⢷⣶⠒⠒⠦⢤⣀
⠀⢀⣠⣾⣅⠀⠀⠀⠀⠀⠀⠀⢀⣙⣷⣄⣾⠁⢠⠶⢦⣄⠀⢀⣀⣠⡾⠁⠀⠀⠀⢻⡄⢀⣴⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠸⡏⠳⡄⠀⠀⠈⠉⠓⠦⣄
⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡇⠀⣿⠀⠀⣹⡦⠈⠉⠁⠀⠀⢢⡶⠃⠘⣷⡾⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⡄⠀⠀⠀⠀⠀⠈
⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠛⠶⠚⢿⠃⠀⠀⣰⠀⠀⢸⡇⠀⢰⡿⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢷
⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠘⠦⠤⠟⠁⠀⠀⣸⠇⢀⡾⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠸⡆
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣷⠀⠀⠀⢀⣀⡠⠋⠀⣾⠃⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⡉⠂⠛⠋⠉⠀⠀⣸⠏⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠋⡳⣶⣶⣤⣤⣴⡿⠀⠀⠀⠀⠀⢀⡍
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⣴⠾⠛⣃⠀⠉⠙⠛⣿⠃⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⣠⡶⠟⣉⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠛⢷⣿⣋⣴⠿⣫⣾⡿⠛⢾⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠦⣤⣄⣀⣀⣀⣈⣿⣿⡿⠿⠛⠁⠀⠀⢸⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠓⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡀⠀⠀⠀⠀⣠⠀⣀⣠⠖⡄⠄⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣠⡾⠵⢎⠹⠛⣉⠤⠄⡙⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⠛⢚⣈⠉⢀⠀⠘⠛⠛⠁⢯⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⢿⡿⠀⡆⠀⠀⣾⡷⠀⢸⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣀⣠⣼⣤⣤⣄⣀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠺⢡⢶⡛⠛⠛⠛⣿⡙⠓⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠄⠘⠢⣝⣲⣶⡾⠞⠁⠀⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣠⣀⣀⢀⣀⣰⣤⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢶⠎⢩⠉⢏⠉⠁⢸⠙⡖⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢀⠀⠀⢸⠀⠈⢦⡀⠀⣠⠏⠀⡇⠀⠀⡠⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⠂⠀⠈⠖⠺⣀⠑⠞⠁⡖⠒⠁⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢠⠄⠤⠤⢤⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣐⠢⢄⣠⠀⠉⠉⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠠⠴⢚⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠓⢺⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠒⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠺⠦⢤⣤⣤⡤⢤⣄⣇⣀⣠⣤⡶⣒⠋⠉⠁⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠉⠉⠉⠉⢛⣿⡀⠀⢀⡤⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢤⣄⣠⣷⣿⡿⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣶⣿⣶⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⢿⣿⢿⣿⣿⣿⡿⣿⢿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣻⣿⣥⣿⣿⣿⣿⣧⣿⣿⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⢿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣰⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⣤⣄⡀
⢀⣀⣤⣤⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⢸⣿⡇⠀⣀⣀⣀⡀⠀⠀⠘⠛⠿⢷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⡿⠿⠿⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠋⠀⠀⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢠⣿⡁⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠶⠚⠻⡇⠈⠙⠛⠻⣶⠀⢠⣿⠋⡔⣀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⠃⢀⣿⠃⠀⠉⠁⠀⠀⡇⡇⣿⢸⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⣼⠏⠀⠠⠤⠴⠃⢀⡇⡇⣿⢸⠀⠦⣀⣀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠞⠁⠀⠀⠀⣦⠀⣰⠆⠀⠀⠉⠈⢠⠀⠀⡀⠀⡀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠛⠋⠉⠁⣼⠟⣷⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠘⠷⠶⠃⠸⣥⠆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢰⡟⢸⡇⠀⠀⠀⠀⠀⣠⠶⠚⠛⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣿⠁⠀⠀⠀⠀⠀⠻⠶⠶⠶⢦⣼⠇⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢰⣾⠛⠛⠻⣦⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⢻⡆⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠈⣿⢀⣠⣼⠏⠙⠻⣶⣄⣠⣤⠶⠟⢋⣡⡴⠟⣱⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣸⣿⡿⠉⠀⠀⠀⠀⠈⣿⣭⣤⡶⠾⠛⠉⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣴⠟⠘⣷⡀⠀⠀⢀⣠⡶⠿⠁⠀⠀⠀⢀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⣀⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠟⠉⠛⠻⢶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠀⠤⣀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠂⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢀⣀⡠⠐⠂⠀⠀⢀⣤⡄⠀⠀⠀⠀⢀⡴⠛⠛⢧⠀⠀⠀⠠⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⣞⠉⣄⡀⢀⣨⣟⣉⣛⢿⡀⠀⣀⣉⣽⣽⣿⣿⡾⣿⣿⣦⡄⢀⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⡏⢀⡶⠛⠛⣎⣻⣾⣿⡿⢉⠙⠛⠻⡛⠻⣆⣩⡽⠛⠉⡻⣷⣴⣆⡸⠀⠀⠙⢷⣼⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⢦⣾⣱⣏⣁⣼⢋⡟⣹⠛⣿⠃⢸⡆⢑⠀⣿⡶⠿⣿⣉⣧⡨⣀⣷⣾⣿⣷⢶⡾⣧⣼⢀⣿⡁⣠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢣⣼⣼⠃⣿⠿⣹⣷⣾⠋⠛⡢⣷⠀⣾⠗⢻⡿⣶⡶⢶⣿⡉⣾⣷⣿⣷⢤⠬⡿⣷⣤⣿⣟⣙⣽⢛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡔⢿⡍⢹⣧⣿⡉⣻⣿⣟⢴⣀⣶⠞⠉⣿⡴⠓⣠⣴⣔⣊⣹⡃⠤⢿⣡⣱⡄⣰⣷⣟⣉⣛⣿⢿⣷⡶⣯
⠀⠀⠀⠀⠀⠀⠀⠧⢼⣴⣦⢴⣿⡎⢙⢿⡿⢿⣿⣶⣿⠉⣷⢶⣿⣧⡜⠋⠉⠹⣭⢿⣷⣦⠰⡿⠙⣟⢻⣿⠃⠙⢻⣿⢎⣹⡧⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⢧⣾⣭⣿⣽⣇⡆⢸⡇⡈⢣⢻⡏⠀⢉⡼⠷⣆⣠⡞⢱⢿⣘⣾⣷⣿⡖⢛⣁⣾⣤⡾⠋⢁⢻⠿⡇⢂⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠆⢩⡉⣿⣿⢾⣷⣿⡋⢹⣦⡶⢿⣴⣏⢐⡓⣭⣟⣱⣶⠦⣾⣯⣾⣿⣿⢿⡼⣿⡋⠙⠒⣼⠑⠴⣿⣀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠷⡿⠯⣘⣿⡿⠒⣾⣯⣷⣤⣭⣿⣿⣽⡿⢿⣶⣤⣠⡿⠛⣼⠋⠰⢌⣉⢙⣭⢳⡦⣾⢶⡾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠩⠹⣿⠋⠉⠁⠈⣥⡀⠀⠀⠉⡄⠀⠈⠟⠿⡳⠾⠃⢀⠙⡀⠉⠀⠀⠘⢆⠀⢸⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠄⠊⠈⠲⡀⠀⠀⠉⠷⠠⡤⠴⠃⠀⠀⠀⠀⠀⠀⠀⢸⡄⡀⢀⠀⠀⠀⠀⠑⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠃⡘⢃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠰⢆⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⢅⠀⢸⡇⢨⢠⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣤⠀⠈⡄⠀⠈⠀⠀⠀⠀⠀⠀⠱⢀⠀⠀⠱⠚⠃⢰⠈⠀⡄⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⡿⣠⡼⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⢀⡀⢸⠀⠀⠁⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠁⠈⠀⠀⠀⠘⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠢⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⣇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣿⣾⣹⣿⣿⣧⠤⠆⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢹⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣂⣀⣴⣂⣾⣿⣿⣿⣿⣿⣿⣿⣶⣾⡁⣐⣦⣄⣀⣶⣠⣖⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣋⣴⣭⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⡤⣀⣈⠑⠀⠀⢠⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠷⡤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡎⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣿⣿⣿⡿⠟⠉⠛⠛⠛⢿⣿⣿⣿⣿⣿⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⣿⣷⣾⣦⣄⣀⠀⢠⣀⣀⡀⢀
⠀⠀⠀⠀⠰⠦⠀⠀⢚⣿⠻⢻⡟⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆⠀⠁⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣥⣷⣾⣿⣿⣿⣟⣿⣷⣿⣦⣕⢦
⠀⠀⠀⠀⠀⠈⠁⠀⠈⢓⣤⣼⠄⠀⠀⢰⣶⣦⡀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣠⣄⣳⣄⣰⣶⣦⣴⣦⣤⣄⣀⣀⣠⣿⣳⣶⣾⣿⣿⣿⣯⣿⡿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣶⣦⡀⢨⣿⣿⣿⣦⡀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠙
⠀⢀⣀⣀⣠⣤⣤⣴⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣦⣶⣤⣶⣿⣿⡿⠿⠿⠛⠋⢁⠀⠀⠁
⠀⠈⠉⠉⠉⠀⣿⣿⣿⣿⣿⣟⠈⡆⠐⡟⢹⣿⣿⣿⣿⣿⣿⣷⣄⠀⣙⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠉⢀⣀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⢳⣀⣾⢿⣿⡿⣿⡿⠟⠛⠻⣿⣷⣿⡞⠿⢷⡀⢀⡀⠀⠀⠀⠀⠀⠈⠈⣵⣿⣯⣤⣼⡟⠙⠓⠚⠋
⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⣿⡿⢿⠉⠈⠛⠻⠟⣿⣷⣶⡄⠀⠀⠀⠈⠿⢷⣄⣉⣙⠻⠆⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠈⠉⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣀⣭⣉⡉⠠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠛⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠛⠛⠛⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠀⠀⠂⠂⠃⠂⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠂⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣯⣿⣿⣿⣯⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣿⣿⣯⣿⣿⣿⣿⣿⣾⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⣟⡻⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣿⣿⣻⣿⣿⣻⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠿⢟⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⣋⣙⡛⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠠⠄⠀⠈⠉⠤⣄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠅⠀⠀⠉⠀⠸⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠐⠀⠐⠀⣀⠀⠥⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡠⠊⠉⠧⢤⠤⠊⠁⠀⠀⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⠈⠠⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⣠⣤⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡴⠋⠀⠀⠀⠈⠀⠀⠐⠚⠉⠙⠛⠛⠋⢹⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠀⠀⠘⠀⠀⠑⠢⠀⡔⠀⠠⠤⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠤⠀⡜⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢣
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢶⡀⠀⠀⢰⠁⠀⠈⠐⠢⠄⠀⠀⠙⢶⠤⠄⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⡆⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⢄⠈⠃⠒⠤⠤⠤⡄⠤⠤⠔⠚⠉⠘⣇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠑⡀⠀⢀⣀⠀⠀⠀⠤⠊⣀⠀⠀⠘⢆⡠
⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠀⠀⠀⠉⠒⠢⣀⠀⠀⠀⠀⠀⠠⠊⣁⢄⡴⠊⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣝⠲⠤⢤⡀⠀⠈⠑⠀⢀⣀⢀⡀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠄⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⡀⢀⠀⠀⠀⠀⢀⠀⠈⠇⠀⠀⠠⠒⠁⠀⣠⣂⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⡀⢂⢰⠜⠀⠄⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠛⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⡤⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣏⣀⣿⣽⣭⣭⣽⣯⣯⣽⣿⣍⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣿⣽⣽⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠒⠂⠉⠉⠉⠉⠉⠉⠉⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⣠⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡰⠃⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⡄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠆⠀⠀⢳⣠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢰⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⡿⠉⠀⠀⠀⠄⠀⠀⢴⡷⢀⠆⠠⠼⣟⡂⠄⠀⠀⠀⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠰⠤⠀⠂⠁⠀⠀⠀⠀⠙⢦⠀⠀⣰⠃⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠘⠻⠦⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⢹⡀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⡀⠀⣀⡴⠃⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠀⠀⢸⡧⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⢴⠋⡉⠍⠁⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡰⠁⠈⢦⠰⠁⠀⣰⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠘⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠙⠛⠦⠤⡤⠋⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢳⡄⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠚⠉⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣏⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠞⠉⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡖⠒⠒⠀⠉⠁⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣠⠴⢋⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⠢⠤⠜⠁
⠀⠀⠀⠀⠀⡼⠁⠈⢱⠀⠀⠀⢰⠃⠀⠀⣠⠾⣅⣀⡀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⢠⡶⢤⡟⣿⠃⠀⢰⠇⠀⣀⣀⡟⠀⠀⡼⠃⠀⠈⢉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⣾⣀⣾⢰⡏⠀⠀⣾⠲⡎⠀⠀⠁⠀⠀⠷⠠⡄⢰⠋⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠇
⠀⠸⢧⣾⠁⠀⢸⣧⡔⠁⠀⠀⠀⠀⠀⡠⠤⣼⡎⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡏
⠀⠀⢰⡇⠀⢀⡟⠐⣄⣠⠄⠀⠀⠀⢸⣀⠀⣸⠁⠀⢰⠏⣿⢤⣀⣀⢀⣀⣀⣤⠤⠤⠤⠐⠒⠊⠉⠀⠀⣸⠃
⠀⠀⠈⠳⠤⠞⠁⠀⠀⠙⠦⠀⠴⢀⣀⠝⢹⡇⠀⢀⡿⣸⡇⢰⡇⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡸⠓⠿⢉⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⡴⠃⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⠤⣄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢦⣤⣠⣁⣄⣀⣀⣀⣤⡤⠖⠋⠉⠏⠀⠀⣉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠉⠻⠋⠀⠀⠀⠀⠀⡤⢒⡛⠀⢠⡝⢓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢻⣅⠀⢸⡒⢲⡄⠀⠀⠀⡈⢦⡈⠉⠀⠀⣎⠐⠒⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢹⡂⠈⠀⠈⣱⢏⣀⠀⢠⡎⠀⠈⠙⠓⠚⠋⠁⠀⠀⠈⠀⠅⢈⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠑⠀⠀⠈⠉⠒⠒⠒⠉⠀⠈⢧⠘⣧⡀⠀⠀⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠐⠂⠀⠀⠀⠀⠀⠀⢀⢀⡤⠀⠀⢀⢾⠀⠀⠙⠷⣦⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠧⡚⠅⠀⠀⠀⠀⠀⠀⠈⠉⠔⠂⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣲⣐⣶⣤⣌⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⢿⣿⡿⠿⣿⣿⡿⣿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣶⣬⣦⣶⣦⣦⣷⣤⣾⣯⣼⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣤⣶⢦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡎⠠⠽⠛⣋⡀⣉⣴⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⠄⠀⠀⣾⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡔⣾⠃⠀⠈⠻⠉⣟⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠤⣄⠀⠃⠀⠀⣩⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡥⠬⠀⠀⠀⠸⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣉⣀⣴⡄⠀⠀⣠⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⡏⠀⠀⡞⣭⢻⣿⡿⢿⣿⣶⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢻⠀⠰⣶⣾⣻⣟⣾⢀⣒⡒⠄⡈⣩⢿⣙⣘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣾⣾⢸⣤⣗⣼⠛⢟⣸⡇⢾⡿⠁⠀⠀⡅⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⠏⠀⣨⣾⠛⠀⣿⣿⠃⣿⡏⠀⠁⢀⡿⢿⣿⠿⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠛⣿⠀⢠⣥⡆⠀⢀⡂⣬⠈⢉⠁⠀⠀⣸⢡⡘⠉⠲⠀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣾⡷⠀⢾⣿⡇⠶⣸⣿⡇⣴⣿⡄⠀⣶⡇⢴⣷⣾⣶⡖⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⣻⡿⠁⢹⣿⡏⢐⣧⣤⢻⣽⣾⣖⡝⣿⣧⢹⣯⣿⣽⡁⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡿⣯⠠⣾⣿⠇⢴⡻⣟⢿⠿⣿⣿⡏⢻⡿⢾⡗⠿⡿⠃⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠿⡇⣿⣰⣿⣿⡀⣹⢆⡀⢿⣷⡻⠀⣷⣾⡇⣸⣒⢠⣸⡆⢰⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡻⣧⣞⠹⣿⣿⠇⠘⣿⢿⠌⢿⡏⠻⢿⡟⡃⢹⣻⠻⢿⠧⠿⠁
⠀⠀⠀⠀⠀⠀⢀⡏⢾⡥⠽⢿⡏⢠⣿⣿⣶⣿⣿⡘⢰⣿⡿⠿⢶⣀⠷⣾⠿⢿⣿⠩⢅⡀
⠀⠀⠀⠀⠀⠀⢀⣳⣿⣤⡀⢐⠁⢸⣿⣇⠀⢸⠀⣁⠸⣿⠤⢐⢻⡌⢳⣌⠳⢾⠿⢧⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡧⢙⠛⠃⠈⠉⠉⠉⣿⣧⡼⣷⣴⡶⠐⣿⣿⢹⣿⡿⣿⣷⣤⣿⠇
⠀⠀⠀⠀⠀⠀⢰⡏⣤⡞⣡⠄⡤⠠⡔⠂⢀⣿⣹⣃⣻⡇⠤⠤⣷⣖⡙⢿⠷⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠟⡽⡄⠙⠻⢷⢦⡴⠚⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡜⠀⢀⡄⠀⠀⡇⠀⡀⠿⠛⠉⣸⠻⠿⠿⠛⢛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡜⣠⠔⢁⡾⠀⢸⠀⢠⠁⠀⠀⠀⣿⠀⠀⠀⢴⡟⠙⡃
⠀⠀⠀⠀⠀⠀⢸⠀⣀⡴⠿⠁⢠⡏⠀⠎⠀⠀⠀⢸⣄⠀⠀⠀⠀⠁⠸⠇
⠀⠀⠀⠀⠀⠀⠸⠀⠋⠀⠶⠆⠘⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠟⠛⠛⠛⠛⠛⠛⠻⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣀⡀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢿⣍⣉⣉⣉⣽⡟⠁⠀⠀⠀⠀⠀⠻⣯⣉⣉⣉⣩⣿⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠛⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⡶⠶⠿⠿⠶⢶⣦⣽⣷⡀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⢶⣦⣤⣤⣤⣤⣴⡶⠾⠟⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣾⢿⣟⣤⡿⣿⢿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⠀⣾⡟⠉⠉⠛⠿⣿⣿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣠⠿⠃⠀⠀⠀⠀⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣿⠃⠀⠀⠰⣦⣤⣤⣬⣛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇⠘⠀⠀⠀⠀⠙⡟⠃⢾⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⡆⠀⠀⣨⢀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⣲⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠃⠀⠈⠂⣄⠀⠀⠐⠒⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣇⠀⠀⠃⣠⢨⣿⣶⣶⢿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣼⣿⡏⣿⡄⠀⠀⠈⠻⣿⠋⠀⢸⣷⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣰⣿⣿⡀⠀⢠⣿⠟⣷⡄⢸⡇⣿⣞⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⣿⡿⣿⣿⣧⠀⠋⠙⣿⠟⠉⠸⠃⢸⣿⣿⣿⣿⣿⣄⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⡄⠀⠀⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⡄⢠⣿⣿⣧⠀⢠⢸⣿⣿⣟⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡸⣿⣿⣿⣿⣿⣟⢿⣿⣿⢿⡆⠀⢸⣿⣿⣿⣧⢹⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣎⠻⣿⣿⣿⣽⡌⢻⣿⣿⣷⡇⣿⣿⣿⣿⡏⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣷⡜⢿⣿⣇⣷⠀⠻⣿⡿⠀⣿⣿⣿⣿⣿⣿⡿⣾⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣦⠹⣿⠸⣇⠀⢹⠃⢸⣿⣿⣿⣿⣿⣿⡇⢷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢿⢿⡿⠟⢿⣿⣷⡌⠁⢿⣤⠇⠀⢸⣿⣿⣿⣿⣿⠟⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠀⠀⠈⠉⠉⠉⠁⠀⠈⠀⠀⠀⠀⠈⠉⠉⠀⠀⠙⠀⠀⠙⠙⠛⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠋⠉⠀⠀⠀⣩⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠰⠟⠃⣠⠀⠀⠀⠁⣶⠄⠈⢳⣄⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠃⠀⠀⠀⢾⡿⢻⡆⠀⠀⣿⠀⠀⠀⠀⠹⣯⢸⡿
⠀⠀⠀⠀⠀⠀⠀⣼⡇⢿⡀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡿⠀⠀⠀⠀⠀⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀⠀⠀⠀⠀⣀⣀⣤⡶⠿⣿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠛⠛⠛⢛⣋⣉⡁⠀⠀⣿⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⡄⠀⠺⣿⣿⣿⣃⡀⢀⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣉⣿⣷⣾⢿⣥⣼⣿⠿⣿⠿⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣿⢰⡏⠀⠀⠀⢻⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠻⠿⠓⠀⠐⠿⠟⠉⠛⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠙⠛⠿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠛⠀⣀⣴⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣀⠀⠐⠾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠙⣿⡆⠀⠀⠀⠀⠐⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⡿⠋⠀⣠⣴⣆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⣿⣿⣿⣿⡉⠉⠀⠀⠀⠉⠛⠛⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣿⣿⣷⣄⠈⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀
⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣾⣿⣯⡀⠀⠙⢿⣿⣿⡆⠀⠈⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦
⣾⡿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠻⣿⣿⣦⡀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⢀⣴⣿⣿⣿⣧
⠙⣿⣦⠀⠐⢶⣾⠂⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣦⣤⣴⡾⠟⠀⠀⠘⠛⠉⣀⣼⣿⡆
⠀⠈⠻⣷⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣧⣤⣤⣀⡀
⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠻⣿⣶⣤⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠘⡷⣤⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⣿⣿⠿⣿⣦
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣶⣤⣤⣶⠟⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠋⠀⠀⠸⠟⠁⢀⣾⣿⣷⡀
⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡁⠀⠀⠀⠀⠀⠀⠠⡿⠿⠛⣿⣷
⠀⠀⠀⠀⠀⠹⣿⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠋⠁⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⡀⠀⠈⠻⠄⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣏⠀⠈⠙⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⢻⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⠿⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠈⠁⠀⠀⠀⠀⠉⠉⠁⡀⠀⣿⣿⠃⠀⣠⡀⠀⠀⣠⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⣾⣿⣿⣿⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⣷⣤⣶⣿⣤⣾⣷⣴⣿⣿⣷⣾⣿⣿⣶⣾⣿⣷⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⠿⠿⠿⢿⣿⣿⡟⠋⠀⢈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⢿⣧⣤⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠙⠛⠛⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡛⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⣿⣧⣬⠇⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⣟⢺⣿⣿⣿⡀⢿⠇⠀⠀⠀⢀⣾⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⣿⠟⠀⠀⠀⠀⠀⢠⣿⠸⣿⣿⣿⣷⡞⠀⠀⠀⠀⠈⠻⣿⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣾⠟⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡆⣿⣿⢟⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣧⡀
⠀⠀⠀⠀⠀⣠⣿⡟⢸⠃⠀⣠⣶⣿⣿⣿⣷⠾⠿⠿⠿⠼⣿⣿⣿⡾⠯⠿⠿⢿⣷⣾⣿⣿⣿⣶⣄⠀⠘⠊⣿⣿⣄
⠀⠀⠀⠀⣴⣿⣿⠁⢸⣆⣀⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠈⣇⣴⣶⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣀⣠⢸⡼⣿⣿⣆
⠀⠀⠀⢰⣿⣿⣿⠀⢸⡿⠋⠁⠀⢨⣿⣿⠻⣿⣷⣤⣀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣫⣿⢛⣿⣿⣿⣿⣼⣧⣿⣿⣿⡄
⠀⠀⠀⠸⣿⣿⣿⠀⣸⣇⣤⣤⡾⣻⣿⣿⣷⣬⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣡⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢀⣿⠿⣿⣴⠿⠿⠿⢗⣿⠏⢹⣿⣟⢿⣿⡟⠛⠋⣉⣤⡄⣈⣭⣙⣋⣿⣿⢿⣿⡇⠹⣿⣶⣿⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠘⠿⢷⠶⠶⠀⠀⠚⠋⠁⠀⠀⢻⣿⢿⡿⢽⣶⠾⢋⣭⡏⣉⣛⣿⣿⡩⣿⣿⡿⠁⠀⠈⠙⠛⠓⠚⠿⠿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡙⣾⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣾⣿⣷⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣹⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⠀⠀⠀⢰⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢟⣫⡄⠀⠀⢠⣝⡻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⢟⣩⣶⣿⣿⠇⠀⠀⣼⣿⣿⣶⣍⡻⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡆⣿⣿⢸⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇⣿⣿⢰⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⢿⣿⢸⣿⣿⣿⣿⣿⡄⠀⢿⣿⣿⣿⣿⣿⣿⡇⣿⡿⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡸⣿⡎⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⢱⣿⢇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⢻⣿⡜⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⢣⣿⡟⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡻⣿⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⢟⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣷⣝⠿⣿⣿⣿⣿⠿⣫⣾⡿⣣⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⠿⣷⣮⣙⣋⣵⣾⠿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣙⠿⠿⣋⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⢤⣭⣭⣭⣭⣭⣭⣭⣭⡤⣤⡤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⢰⣿⣿⡸⣿⡇⣾⣿⡇⣿⣿⢸⣿⣷⢸⣿⢇⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠶⠴⠾⠿⠿⠷⠦⠾⠿⠿⠿⠦⠶⠿⠿⠿⠷⠴⠾⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢟⣭⠶⠶⠦⣝⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢡⠟⣴⣿⣿⣷⣌⢷⡹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⠻⣿⣿⣿⣇⣿⣼⣿⣿⣿⣿⣿⡜⡇⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⡶⢠⡿⣷⡰⣶⣶⣶⣶⣶⣶⣶⣶⡝⣿⡇⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⢸⣮⣷⡙⣿⣿⣿⣿⣿⣿⣿⡇⣿⢡⡇⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⢸⢹⣿⣿⡜⢿⣿⣿⣿⣿⣿⡇⣷⡟⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⣼⣿⣿⣿⣌⣼⣿⣿⣿⣎⢿⣿⣿⣿⣿⡇⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡐⣸⣿⣿⣿⣿⣎⢻⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣻⣛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣘⣛⣛⣛⣛⣂⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠒⠛⢿⣿⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡀⡈⣄⣄⢾⠷⠰⢯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡊⠿⠛⠋⠋⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⠅⠀⠀⠀⠀⠀⠀⣀⣤⣽⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠚⠿⠟⠀⠘⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⠀⠀⢠⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢉⠁⠄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢏⠀⠀⠀⣄⠀⢀⣠⣿⢿⠒⠤⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⡞⢡⠀⠈⠐⠒⠛⣿⣟⣻⣅⠈⡇⢠⠗⠀⠐⢄
⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠀⠀⢳⠄⠀⠀⠐⢙⣿⣿⡀⠀⡇⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⡀
⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠠⣄⣠⠳⡀⠀⠀⢸⣬⣭⣽⡄⠇⢀⡶⠀⠀⠀⢈⡀⠀⠀⠀⠁⠑⠄
⠀⠀⠇⡀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠁⠀⠙⢆⠀⢸⣬⣭⡭⠿⢤⣯⣀⣠⡀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⣄⠀⠀⠈⠣⠘⠉⠀⠀⠀⠀⠀⠀⠚⢧⡀⠀⠘⡇⠀⡀⠀⠀⠀⠀⠁⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠐⠠⡀⠈⢢⠀⠀⡈⠛⢻⡃⠀⠀⠀⠀⠀⠀⢀⣠⣤⡦⢀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⡐
⠀⠀⡆⢀⠠⠄⠤⠀⠀⠀⣼⡤⠊⠀⠀⢘⠁⡀⠀⣠⡔⢶⡞⠛⠟⢇⠠⠂⠀⠀⠀⠃⢸⠀⠀⠀⣀⠀⠀⠀⠐⠁
⠀⠀⢂⠀⠀⠀⢀⠀⡠⠒⠁⠀⠀⠀⠀⠈⢆⠈⣺⢿⠃⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣴⣠⣴⣶⣦⡍⢬⡁
⠀⢀⣊⡀⠠⡠⠂⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠉⠉⠉⠛⡟⠼⡇
⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⣬⣽⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡀⠀⣀⠀⠀⠀⢼⡂
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⡼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠈⠑⠀⠙⢃⡀⣱⠤⠙
⠀⠀⠠⡄⠀⠀⠀⠀⢀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠈
⠀⠀⠀⠳⣤⣀⣤⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠈⠙⠛⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⡿⠿⠿⠿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠆⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⡴⠊⠉⠲⡄⠀⠀⠸⠁⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣇⠀⠀⠀⠀⠀⠰⡇⠀⠀⢶⡿⠀⠀⠹⣄⣀⣹⠟⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⣠⠍⠁⠀⠀⠀⠀⠀⠙⢢⣴⡾⠗⠀⠀⠀⠉⠛⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠶⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠷⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⠀⠀⠀⠀⠀⣀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣴⠄⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠙⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠠⠴⠾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠰⢶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⢶⣤⣄⣀⣀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⣀⣠⣾⣿⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠉⠙⠛⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠉⠁⠀⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⣀⣠⣴⣿⠟⠁⠀⢀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠠⡏⢳⣤⠶⠾⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⣧⠀⠹⣷⣄⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⢀⣀⣤⡦⡤⠤⠿⠋⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢷⣄⠈⠛⢷⣤⣀⣀⣼⡷⠿⠶⠶⠶⠞⠋⢉⣿⣦⣀⣠⣤⡖⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠙⢷⣤⣀⠈⠉⠛⠁⠀⠀⢀⣀⣀⣠⡴⠟⠁⠈⠉⠁⡜⢀⡎⢀⡀⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠙⠛⠿⣆⠀⠀⠀⣀⠉⠻⣅⠀⠀⠀⠀⠀⠀⠷⢼⣇⢸⣇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⡀⠈⢻⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢯⣁⢀⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⡿⠿⣿⡿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠟⠛⣿⠉⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⡆⠀⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠀⠼⠿⠶⣶⣦⣤⣤⣄⣀⣴⡶⠿⠛⠛⠋⠉⠁⠀⠉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀⠀⢀⣀⣀⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠟⠛⠛⠛⠋⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣯⡿⠙⣿⠛⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢉⡉⠑⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⣘⠊⠳⣀⠈⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⡎⠉⠤⡀⠉⠓⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⢀⡀⠒⠚⢀⠶⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠐⡀⠛⠀⢐⡾⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠹⠀⠙⠢⠄⣈⡱⠋⠉⠑⠒⠦⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠋⢉⣽⢀⡵⠶⠶⠶⠶⠶⠦⣤⡀⠀⠀⠛⠋⠉⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⣰⠁⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⡆⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠐⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⢤⣄⣀⣤⣀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠣⣄⠀⠀⣀⠧⡀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⣅⠀⠀⠀⠀⠹⡄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠱⡅⠀⡤⠒⠐⠢⡉⠉⠉⠉⠉⠉⠉⠀⠀⢀⣸⣆⠀⠀⠀⠀⡹⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣰⠃⢣⡘⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⣠⡞⠁⠈⠛⠲⣦⠀⠁⢠⡀⠀⣻
⠀⠀⢸⡇⣆⠀⠀⠀⣠⠞⠋⠐⠒⠚⠁⠀⠀⣹⣦⡀⡀⠀⠀⠙⣆⠀⠀⠀⠀⠘⠋⠀⠀⠀⣠⠞⠱⡄⢀⡿⠀⠀⢹
⣰⠏⢉⡷⠈⢣⡤⠞⠁⠀⣀⣀⡤⠀⠀⣀⡞⠁⠈⣷⣧⠀⠀⠀⠸⠗⢤⡀⠀⠀⣀⣤⡴⠞⠁⠀⠀⠙⣏⡷
⢱⡖⢺⡇⠀⢀⠀⠀⠀⠠⠟⠁⠀⣀⡼⠉⠀⠀⠀⣻⢹⠂⠀⣠⠟⠀⠀⠀⠀⠀⣡⠟⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡼
⠀⠙⠧⠿⠔⠚⠒⠚⠙⠐⠒⠒⠊⠉⠀⠀⠀⠀⠀⣿⠈⠀⠰⠋⠀⣀⣠⣄⣤⠚⠁⠀⠀⠀⠀⣀⡤⠤⠞⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣞⣡⡄⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⣩⠴⡁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠋⠉⠀⠀⠙⢿⡆⠀⠀⠾⠕⠛⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⡿⠛⠁⠀⠀⠀⡴⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠎⠀⠀⠀⠀⠀⠀⡇⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢠⠇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡞⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢀⣠⠟⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡴⠛⠉⠀⣠⡞⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠈⠀⠀⠀⣴⠋⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢷⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢷⡀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠙⢦⡀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣾⠻⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡇⠀⢣⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⣷⠀⠘⣧⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠚⢁⣠⡤⢤⣀⡿⠀⢰⠋⣀⡼⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⡈⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠖⣖⣩⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⣀⡀⢶⣤⣬⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⣿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣿⡏⠀⠀⠐⣸⣯⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠗⠀⣈⣴⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣫⣠⠾⢿⣿⣿⠗⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠘⡿⢃⠐⣻⣿⣿⠁⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⣼⣯⣥⣳⠀⢀⣼⣿⣿⣿⡁⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⡀⠈⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣟⣤⣿⣿⣿⣿⣿⡷⠃⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢻⠇⠀⠀⠰⣾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣽⣿⣿⣿⣿⣿⢿⣿⡷⠟⠈⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠁⠀⠀⠐⠀⠻⠓⠀⠀⠀⠀⠀⠀⠀⢸⣿⣎⠛⣿⣿⣿⣿⡿⡿⠛⠉⣆⡹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⡏⢉⠧⠴⠚⢿⣿⣿⠟⠁⠸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⣴⡄⠉⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⣄⠀⠀⠈⠀⠀⡱⠾⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⢲⠀⡘⣦⠘⠀⠀⠀⠘⣆⠁
⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣧⠈⣿⡇⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢰⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠒⠀⠳⠾⡾⢿⠓⠰⠂⣀⠀⣾⣷⢴⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢤⣾⣶⣵⣦⣤⣤⢸⡟⢶⠙⣿⣏⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠄⢻⣿⡿⣿⣿⡝⠋⣷⠈⠁⠘⣛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣽⣿⣿⡇⠀⠸⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣺⣿⠛⠟⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣿⣶⠀⠀⠀⠀⠈⠋⠸⣯⠌⠠⠐⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠗⠀⠀⠈⠓⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡂⠀⠀⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠛⠛⢡⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠖⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⡈⠙⠿⣿⡿⠃⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⡦⠀⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢼⣿⡄⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣧⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⡆⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣷⠀⢹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⡆⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣷⣸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠙⢿⣿⡿⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣯⣤⣤⣴⣦⣤⣤⣤⣤⣽⣴⣶⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠛⠛⠛⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⡿⠗

⠀⠀⣠⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀
⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿⣿⣿⠀⣠⣤⣄⡀
⠀⠈⣿⣿⣿⠃⠀⠀⠀⠀⣾⣿⣷⡌⣿⣿⣿⠃⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⣴⣿⣿⣿⡈⣿⣿⡏⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⣤⣴⣿⣿⣿⡆⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣷⣬⣿⣿⣿⡇⢀⣤⣄⡀⠀⣴⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣄⡀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠀⢹⣿⣿
⣿⣿⣿⣿⡿⠟⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣶⣤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣠⣴⣿⣿
⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠉⠈⠙⣿⣶⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠉⢿⣿⣿⣿⣿⣿
⣇⠀⠀⠀⣹⣿⡇⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿
⣿⠆⠀⠀⣿⠋⠁⢀⣠⣀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿
⢠⣾⣿⣿⣷⠀⣰⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⠆⢹⣿⣿⣿⡟⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣀⣴⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⣴⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢨⠀⠀⠉⡅⠀⠀⠁⢤⠤⠤⠤⠤⠤⣤⠠⠀⠀⠀⠀⠀⢀⡀⠅⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠐⠀⠀⠀⠰⠸
⠀⠀⠀⠸⠀⠀⢈⠀⠀⠤⡀⠆⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⡄
⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⡂
⠀⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⢰⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣇⣀⡀⠀⠀⠀⡀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠆⠸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠠⠖⢠⠜⠉⠣⠀⠀⠉⠓⢤⠤⠥⢆⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡠⠊⢉⣰⠎⡀⠔⠉⠀⠀⠀⡀⠀⠈⠉⠐⢱⡆⡈⠉⠷⠒⠒⠛⠘⠤⠤⠄⠀⠄
⠀⠀⠀⠀⠀⣧⠔⠁⠀⡌⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠑⡼⠁⢉⠁⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠺⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⡌⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠃⠀⠀⣰⠠⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⢄⡀⢀⣀⣀⣀⠀⠀⠠⠊⠀⠀⠀⠀⠸⣇⠘⣖⣲⠆⠁⠈⢂
⠀⢀⠀⠀⠀⠀⣿⠄⠀⠤⠂⢁⡁⡀⢀⢀⡀⠀⢉⠐⠖⡀⠒⠐⣃⣤⡈⠉⠉⠀⠊⠉⠉⣨⣀⣀⣀⣀⣀⣀⡀
⠀⠈⠀⠀⠀⠀⡏⠤⡎⠀⣀⠿⢣⠄⠀⠀⠄⣉⠞⠵⡀⢘⠀⠀⠉⠘⠁⠉⠉⠉⠋⠉⢻⣿⣉⣉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠰⠄⠲⠚⠚⢁⠄⠊⠐⠄⠂⢀⠀⠁⠀⠉⣛⡛⢛⠂⠻⣃⠀⠂⠂⠀⣶⣼⢇⡀⡈⣧⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠃⠀⠄⠀⢀⠥⢠⠀⠈⠁⠉⠂⠁⢤⠤⢄⠁⠈⠡⠃⠀⠩⡉⠉⠉⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠁
⠀⠀⠀⠀⡞⠄⠡⠔⠂⠇⢀⠌⠈⠓⠒⠐⠒⠂⠈⢄⢀⠃⠐⠀⠀⠄⠈⠡⠀⠐⠠⢤⠔⠈⠀⠀⠀⢠
⠀⠀⢠⠘⠀⠀⠀⠀⠀⠉⠉⠈⠉⠀⠀⠀⠃⠀⠀⠈⠉⠀⠉⠉⠉⠁⠀⠀⠓⠀⠃⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠤⠤⠬⠥⠍⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠟⢷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⣤⡀⠀⣀⣸⠇⠀⡀⠰⠶⢶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠛⢛⡏⠀⠀⢀⠀⠀⠄⢀⡈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⡌⠀⠀⠀⠀⠴⠄⠀⠀⠰⠀⠈⢹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢠⢦⣤⠸⣷⣤⣤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠴⠢⠿⠟⠛⢿⣿⣿⣧⣀⣠⣆⡐⠒⠒⢦⣤⠶⢦⣬⢻⡆⠀⠈⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡄⣠⠀⠀⠀⠀⠀⠙⢿⣿⠟⠁⠀⠀⠀⠀⠈⠁⠀⢀⣘⣿⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣀⣤⣞⣿⣳⣷⣾⣦⣤⠄⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⣀⡀⠀⠀⢿⣿⠷⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠿⢿⣿⣍⣀⣀⣀⢀⣀⣿⣿⣦⠀⠀⣀⠞⣉⡵⠤⠄⠀⠠⣀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⠛⠟⠀⠀⠀⢿⣾⢿⡿⠛⠛⠛⠋⠀⠉⢻⡿⠙⢧⣤⠄⠀⠀⠀⠀⢀⡹⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣶⣶⣿⠀⡆⠀⠃⢸⣦⠀⠀⣠⡤⠀⣴⢤⣴⠆⠀⠀⠀⠙⠀⠺⠶⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣧⠈⢻⣿⠇⢸⡟⢛⡁⣼⣿⣷⣾⡿⠐⠀⠳⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⢇⣿⠀⠈⢾⣷⡈⠋⠉⣅⠀⠀⠲⠒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⢰⡈⠘⠷⣄⠀⠹⣿⣧⣀⣿⣦⡄⢀⣀⣾⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡛⢾⡔⢫⢧⡀⢠⡤⠍⠉⡉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⢳⡘⣧⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠷⡤⣙⣿⣿⡌⠣⠀⠀⢶⣢⠴⣋⠄⣠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⣿⣟⢷⡀⠀⠀⠈⣵⢞⡡⠾⠛⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⢿⣆⢁⢠⠀⢀⠀⠀⠀⢀⡤⠉⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣅⣀⣀⣀⣀⡀⢠⣾⡛⠻⣟⠈⠨⢀⡘⣥⡆⢠⣾⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣻⣿⠛⠛⠓⣠⢀⣽⡉⡁⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠈⠙⣯⡗⠉⠃⠀⠀⠁⠈⠉⢹⣿⡟⣿⡗⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣷⡦⡄⠀⢠⡤⠀⠀⠙⢿⡟⠘⠻⠿⣿⣿⣶⣶⣿⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠀⢻⡿⠋⠀⠀⠀⠀⣠⡤⠄⢀⠀⠀⠀⠉⠡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣠⡛⣿⡷⠁⠐⠁⠀⠀⠰⠏⠀⠀⠈⣧⠹⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠘⢃⣿⡄⠀⠀⠀⠀⠀⠐⣧⣀⣤⣾⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡄⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⣿⣀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣼⣿⡘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣧⠀⣿⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣠⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠛⠛⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠻⠿⢿⣯⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣷⣤⣨⣭⠖⢀⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠓⠒⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⡟⠿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣛⠛⠃⢰⣾⣿⠧⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠁⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⣀⡀⣀⣾⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠸⠟⠳⢖⣛⣉⣉⣭⣷⣮⣭⣭⣭⣭⡭⠝⣓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢛⣵⣣⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣧⣖⡝⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢛⣡⣶⠿⠟⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣮⡛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠫⣠⣾⣿⣿⣶⣶⡾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠈⢹⣧⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣘⣿⣿⣿⣛⣉⣉⣤⣤⣶⣿⣿⣿⡿⠿⣏⡳⢎⠅⠀⠀⠀⠀⠀⠀⠀⣀⠤⣾⣿⣿⡰⣏
⠀⠀⠀⠀⠀⠀⢀⡞⣅⣶⣿⣿⣿⣿⣿⠛⠋⠉⡉⢉⣭⠼⠟⠘⠛⡈⠀⠀⠀⠀⠀⡀⠀⠀⣠⠞⠁⠀⣿⠉⢿⣷⠙⡆
⠀⠀⠀⠀⠀⠀⢸⡇⣿⡏⠸⣟⢻⣿⣿⣇⠀⢀⡴⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢿⣧⠘⢿⠀⣷
⠀⠀⠀⠀⠀⠀⣿⢸⣿⣇⢘⣿⣤⣿⣿⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠶⣶⣤⡀⠀⠀⢹⣿⣿⠀⣿⢀⡀
⠀⠀⠀⠀⠀⠀⣿⠘⣿⣿⣿⣿⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡐⠁⠀⣤⢤⡄⠉⠀⠀⠀⢹⣏⠸⣛⣫⣭⣙⡳⢦⣀
⠀⠀⠀⠀⠀⠀⢻⡄⢸⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣺⢀⠀⠀⠀⠀⠀⠀⠈⣿⠾⠛⢋⣉⠙⠻⣦⡹⡆
⠀⠀⠀⠀⠀⠀⢸⣧⢸⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⡁⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠉⣠⠚⠳⢄⠁⠀⢸⡇⣿
⠀⠀⠀⠀⢀⡾⣫⣴⠾⣧⠒⠶⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣹⠇⠀⢃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣀⠰⣇⠀⠀⠘⠀⠀⣸⠇⡟
⠀⠀⠀⢠⡟⣰⡟⣠⣤⡀⠙⢤⢀⣀⠀⢀⣀⣀⢤⠤⠒⢊⡡⡤⠚⠁⠀⠀⠸⠇⠀⠀⠀⠀⠀⣀⣤⠶⢫⡀⢀⣦⠈⣳⠆⠐⠀⣰⡟⣼⠃
⠀⠀⠀⢸⡇⣿⡇⢙⠀⢹⡀⠀⠈⠉⠀⠉⠉⠙⠓⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠰⣀⡀⠀⠀⢾⡁⠀⣐⠶⠖⠋⠁⠰⡃⠀⠀⣴⠟⣼⠋
⠀⠀⠀⠸⣇⢿⡇⠘⡀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠙⠋⠁⠀⠀⠀⠀⢀⣡⣴⡾⣋⡾⠁
⠀⠀⠀⠀⠻⣌⠻⣦⡀⠀⠀⢹⡆⢀⡤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠋⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⡵⠞⠋
⠀⠀⠀⠀⠀⠈⠳⣌⡻⢦⣤⣄⣁⣈⣡⢠⢿⡐⢦⡀⠀⣀⣤⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣠⡾⣫⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢮⣭⣭⡙⣿⡧⢠⠗⠀⠹⢶⠉⣠⠗⠀⠀⢀⣴⣬⣴⣾⡷⠄⠀⠀⠀⠀⠀⣠⡾⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠻⣤⡀⠀⠳⠴⠛⢧⠀⠀⠀⠈⠛⠛⠙⠟⠀⠀⠀⠀⠀⢀⣾⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣙⠷⣦⣴⡰⢮⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣭⡛⢦⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢏⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡌⣷⣦⣄⠀⣀⣀⣠⣤⣴⡟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⡿⠐⢦⡀⠁⠀⡸⠉⢷⣜⡛⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⣇⠀⠀⣹⠛⠛⠛⣦⡈⣿⣿⣿⣦⣭⣛⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣾⣿⣷⠻⠀⣼⠻⣦⠀⠀⢹⡟⠿⣿⣟⣿⣿⣿⣿⣎⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣵⡞⣿⣿⣿⣿⠀⣷⠁⠀⠹⣧⣀⣼⡇⠀⠸⣿⡽⣿⣿⣿⣿⣦⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣧⢹⣿⣿⣿⡇⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⢹⡟⣬⣿⣿⣿⣿⣇⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣾⣿⣿⣆⠸⢿⣿⣇⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠈⣿⡿⣿⡿⣿⣿⣾⢧⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣿⣿⣿⣿⡄⢻⣦⣟⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⣿⢳⣿⡇⣿⣿⣷⡼⡇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡇⢸⣿⣿⣿⡄⢹⣿⡇⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⢹⣿⣿⣇⣿⣿⣿⣿⣷⡸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⢣⣿⣿⣿⣿⣿⡌⢻⣿⡀⠀⠀⠀⢸⣿⣿⣿⡄⠀⣾⣿⣿⣇⣿⡿⠟⠋⠙⣇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣿⡄⢙⡇⠀⠀⠀⢸⣿⣿⣿⣇⣀⡽⠟⠛⠉⠁⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⠈⠉⠛⠛⠋⢻⣿⣿⣿⡌⢿⣦⡀⠀⣈⡿⠿⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡾⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⢳⣤⣀⠀⣀⣀⣨⣿⣿⣭⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣟⡋⠤⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠶⣛⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣯⣤⠒⡀⠈⣡⣀⣀⣀⣀⣤⣤⡤⠶⠶⢟⣛⣫⡭⠥⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣙⢿⣿⣿⠿⠟⣫⣝⡛⠿⢛⣛⣩⡭⠽⠶⠶⠖⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠛⠋⠁⠀⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⡐⠀⠒⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⠀⠀⠀⠀⠀⠀⠐⠊⠐⣂⠂⢄⡀
⠀⠀⠀⠀⣀⣰⠟⢋⢴⣿⠗⣆⣀⡀⡀⠀⠀⠀⠀⣿⡿⣭⡽⣿⡇⠀⠀⠀⠀⣀⣁⣀⡖⠻⣷⢌⡛⢷⣄⡀
⠀⠀⣠⡞⠉⠀⢠⣧⡾⠁⠀⠀⠉⠛⠁⠀⠀⠀⠠⣌⠙⠛⠛⢉⡴⠀⠀⠀⠀⠙⠋⠁⠀⠀⠹⣧⣧⠀⠈⠙⢦⡀
⠀⠀⣿⡀⢀⣀⣨⡿⠗⠀⠀⣀⠀⠀⠠⢄⠀⠀⢰⣎⠑⠨⠔⢈⣴⠀⠀⢀⡠⠀⠀⢀⡀⠀⠐⠿⣯⣀⣀⠀⣸⡧
⠀⠐⢾⡍⠀⠠⢽⣤⠀⠀⠀⢻⣶⣄⠀⠘⠓⠶⠼⢙⣳⣾⣷⣟⡹⠶⠖⠛⠀⢀⣴⣾⠏⠀⠀⠀⣼⠯⠀⠈⣹⠖
⠀⠀⠀⠉⢿⡶⠍⠻⣶⣤⣤⣾⠿⣿⡆⠀⠀⠀⠀⠈⣻⣿⣿⡋⠀⠀⠀⠀⠀⣾⡿⠿⣦⣤⣤⡾⠋⠵⣾⠟⠁
⠀⠀⠀⠀⠀⠙⢦⣀⣨⣿⣿⡿⢛⠻⣿⣶⣤⣀⣀⣀⣺⣿⣿⣆⣀⣀⣠⣴⣾⡿⠛⠻⣿⣿⣯⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢹⣠⣿⠀⢹⢿⣿⣿⣿⠿⠿⣻⡻⠿⢿⣿⣿⣿⢿⡁⢈⣧⣸⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣴⣇⡟⢉⣏⣀⣄⣀⣿⣇⣀⣄⣈⣿⠉⣧⣷⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣼⠁⣾⠋⠁⠀⠈⣿⡏⠀⠀⠉⢻⡇⢸⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡁⢼⣶⠶⠖⠾⣷⡿⠗⠶⢶⣾⡄⣸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡏⠀⠀⠀⣿⡇⠀⠀⠀⡏⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢻⡦⠴⠾⢿⠿⠶⠤⣾⠇⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣸⡄⠀⠀⠀⠀⠀⠀⣸⣀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠦⢤⠴⣤⠤⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡻⣿⣷⣌⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣇⢹⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠟⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⢡⣾⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣋⣥⣿⣿⣿⣿⣿⣷⣬⣍⣛⡛⠻⠿⠿⠿⠿⠟⢃⣬⣭⣭⣍⠻⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣙⠿⣿⣿⣏⣴⣶⣬⣹⣿⣿⣿⣿⣿⣿⣩⣶⣶⣍⣿⣿⡿⢟⣡⣾⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠃
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿
⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⢿⣿⣿⣿⣿⣿⡏⣴⡌⣿⣿⣿⣿⣿⣿⠟⣼⣿⣿⣿⡟
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣷⣬⣴⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣧
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣭⣛⠛⠻⠿⠿⠛⢛⣫⡅⢺⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠟⣛⣭⣭⣴⣦⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⢠⣶⣮⣭⣙⡛⠿⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⣿⣿⢋⡛⣋⢻⣿⠟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡁⣾⡇⣿⡆⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢋⣽⣿⣿⣿⣿⣿⣿⣿⢡⣿⡇⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣍⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⡇⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⡇⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⡿⠿⠛⢛⣋⣭⢸⣿⣿⡇⣿⣿⣿⢨⣍⣛⠛⠻⠿⢿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠛⣛⣋⣉⣉⣭⣭⣭⣭⣤⣶⣶⣶⣿⣿⣿⣿⣿⠘⣿⣿⡇⣿⣿⡏⢸⣿⣿⣿⣿⣷⣶⣶⣦⣬⣭⣭⣭⣉⣉⣉⣛⡛⠇
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⣿⣿⡇⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠛⣋⣴⣌⡛⠃⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⠐⠂⠠⢀
⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠄⠀⠠⠀⠄⡀⠀⠀⠀⢀⡀⠠⠤⠤⠀⢀
⠀⠀⠀⠀⡠⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠀⠀⣀⠀⠀⠑⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠌⠁⠐⢄
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠠⠁⠃⠀⢐⠀⠀⠀⠀⠉⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⡌⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠁⠀⠀⠀⠀⠠⣀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡀
⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠁⣀⠠⢄⡀⢢⡀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠤⠤⠤⠴⠦⠄⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⢐⠀⢰⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣧
⠀⠀⠀⠀⠾⣇⡸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠉⠉

⢻⡙⢦⡀⠀⠀⠀⠀⠀⠀⠰⣏⠺⢭⣑⣪⠭⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠁
⠥⢧⠀⠈⠢⣀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣂⣹⡄⠀⠀⣠⠵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧
⠤⠔⣷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡩
⣀⣀⣸⣄⣠⣤⣤⣶⣶⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⡟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⠿⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⡤⠤⢤⣤⣀⣸⣂⣉⣁⣀⣈⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠈⢯⣭⣭⣍⣙⣛⣛⣛⣓⣒⣒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠲⠶⣶⣶⣤⣬⣭⣭⠟⠁⠀⠀⢀⣿⢤⡀⠀⠀⠀⠀⠀⠀⠀⢁⡤⣲⠶⢿⡻⠭⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⠸⡆⠉⠢⡀⠀⠀⠀⠀⠀⠈⠳⢬⠽⠶⠚⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣂⡩⢿⠀⠀⠈⠒⢄⡀
⠀⠀⠀⢀⡤⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠭⠄⠒⢚⣇⣠⠴⠚⠉⠁
⠀⠀⣀⠁⠀⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠉⠉⠁⢻⠀⠀⠀⠀⣀⣀⡄
⠀⠀⠻⣿⣶⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣾⣷⣾⣿⣿⣿⣿⠇⠀⠀⣧⡀
⠀⠀⠀⠀⠉⠛⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋⠀⠀⠀⠃⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡯⠭⠽⠿⢷⣾⣖⣒⣒⣒⣒⣦
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⣤⠶⣖⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣛⣓⣒⡒⠒⠲⠶⠦⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠛⠛⠛⠛⠛⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣻⣯⠤⠴⠶⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢯⣍⡀
⠀⠀⠀⠀⠀⢀⡴⠋⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣦⡀
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⢷⡀
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢷⡀
⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣧
⠀⠀⢿⠀⠀⢀⠀⠀⠀⠀⣴⠏⠀⠀⣤⣄⣀⡀⠉⣷⣄⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿
⠀⢀⣸⡆⣀⣼⡄⠀⢀⡾⢷⣠⠴⠋⠉⠈⠉⠛⠻⠇⠈⢷⡀⠀⢀⡞⢹⡄⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠋⠉⢹⠃⠘⣧⠀⣼⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⡾⠁⠀⠿⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣿
⢿⣄⣀⣸⡆⠀⠘⢧⣿⠀⠀⢀⣤⣶⠶⠄⠀⠀⠀⠀⠀⠀⠘⠋⠀⠐⠛⠿⣶⣄⠹⡆⠀⢤⣀⠀⠀⠀⠀⢸⠷⠷⢶
⠀⠙⠛⠛⣇⠀⠀⠀⠁⠀⠀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠉⠉⠉⠉⠳⣄⠀⠀⣿⠀⠀⢀
⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⢀⣤⣶⣶⣶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⢷⣸⣿⡶⠾⠛
⠀⠀⠀⡼⠙⣧⠀⠀⠀⠀⠀⢠⡾⢻⣿⣿⡏⠉⢳⡄⠀⢀⣾⣿⣿⣿⢻⣀⣸⣿⡄⠀⣠⣶⠶⠆⠀⢈⡿
⠀⠀⣸⠁⠀⠘⣧⠀⠀⠀⠀⣾⠀⠘⢿⣿⣾⣶⣿⣿⠀⢸⡟⠿⣿⣿⣿⣿⠟⢁⡇⠀⠁⢸⡆⠀⢠⣾⡁
⠀⠀⡿⠀⣸⠀⠘⣧⡀⠀⠀⢹⡄⠀⠀⠙⠛⠛⢋⡟⠀⠈⣧⡀⠀⠈⠀⠀⠀⣼⠇⠀⠀⢸⠇⣠⠟⠀⢻⡀
⠀⠀⣇⡀⣻⠀⠀⠈⠳⣄⠀⠀⠙⢶⣤⣤⣤⠶⠋⠀⠀⠀⠈⠻⢦⣤⣤⡴⠞⠁⠀⠀⠀⣼⡴⢋⣀⣀⣨⡇
⠀⠀⠘⡏⠻⣇⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⣼⠉⢩⡏
⠀⠀⠀⠱⡄⠹⣆⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣤⠆⠀⠀⠀⠀⠀⢹⣷⣤⣤⠶⠾⠛⠉⠀⠀⣰⠇⣠⠟
⠀⠀⠀⠀⠙⠳⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣤⣤⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢾⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢘⡧
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠉⠛⠶⣦⣤⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡶⠟⠋⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢤⣀⡀⠀⠉⠉⠙⣻⣿⠛⠛⠛⢿⣏⣉⠉⠉⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠉⠙⠛⠛⠛⠛⢻⡇⠀⠀⠀⣾⠀⠉⠉⠛⠛⠉⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣇⠀⠀⠀⠀⢀⣴⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠀⢀⠈⣂⣄⠲⡀⠀⡰⢊⢅⣀⠉⠀⠘⠃⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠿⠀⠐⠬⢦⡥⠁⠀⠁⠡⠼⠃⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⢒⣀⢒⣀⣀⣖⠽⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡰⢚⠈⡤⣀
⠀⠀⠀⠀⠀⢠⠊⠔⢊⠍⢒⠄⠀⠀⠐⠖⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡑⠣⢶⠛⣐⠲⠤⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠶⢚⠪⠑⠌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠋


⣶⣶⠶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⠋⢀⣀⣀⣀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠛⢯⡍⠀⠀⠀⠀⢠⣌⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⣿⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⡼⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠘⠲⠖⠃⠀⠀⢀⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⡀⠀⠀⢀⣀⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏⠐⠀⢻⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠉⠙⠋⠛⠛⠉⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⡄⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣿⠈⢿⣤⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡟⠀⣿⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢀⣴⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠸⣧⡀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠠⡄⢠⡞⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⣷⡄⢠⠄⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠷⠶⣅⣼⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣧⡴⠶⠿⣇
⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠈⣹⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠁⠀⠀⠈⣷⣄
⠀⠀⠀⠀⠀⠀⢻⡍⣳⡆⢶⣾⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢹⣷⠦⣰⣏⡰⠟
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠉⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠙⠛⠉⠙⣿⠁⠈⣿⠛⠉⠉⠉⠉⠙⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠶⠶⠖⠋⠉⠉⠁⠀⠀⠈⠉⠉⠙⠓⠶⠶⠶⠶⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢴⡖⠃⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⢀⡀⠙⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢰⣧⠹⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠰⡦⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⠟⣿⣷⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⣿⣿⡇⢹⣿⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡟⢸⡇⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠘⠃⠈⠃⠉⠀⠀⣠⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢀⣶⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣼⡟⢱⣾⡟⢻⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠛⠷⣮⣭⣦⣸⡹⢟⣼⡷⣼⣿⣿⢟⢷⣶⣤⣄⣘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢉⠙⠛⠿⠿⣷⣿⣼⣇⠻⣡⣿⡀⠿⣻⣿⠆
⠀⠀⠀⡀⠀⠀⠀⠀⠗⠹⡗⣰⣆⠀⠶⢈⣉⠛⠛⠿⢷⣿⣶⣿⡿
⠀⠀⠀⠁⠀⠀⠀⠀⣄⡀⠀⡀⠙⠘⠀⠺⣤⢀⣧⣴⠆⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢁⡎⡙⢀⠀⣤⢀⡤⣀⠀⠈⠀⠀⠀⢺⠉⣐⠂
⠀⠀⠁⠀⠀⠀⡤⠀⡀⢀⣉⡀⠘⠚⠁⠸⢶⡛⠀⢀⡀⠀⠀⠈⠑⠉⠀⣀⠠⢀
⠀⠐⠀⢀⡀⠐⠛⠻⠃⡾⠶⠀⢠⣦⠀⣀⡀⠁⠀⡆⢈⠂⠁⠀⠀⠀⠰⠂⡀⠴
⠁⠀⠀⠀⠀⠀⠉⠐⠂⠉⠉⠐⠋⠻⠄⣜⣹⡆⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠶⣌⠉⢀⠀⠀⠐⠆⠀⠀⠃⠀⠀⠀⠀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡀⣤⠓⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡤⠲⣄⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠁⣠⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠲⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠐⠂⢨⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠖⠧⠠⠀⠤⠘⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢀⣀⣀⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿