⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⣉⠙⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡡⠖⠋⠉⠉⠓⢦⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠿⠛⠚⠛⠓⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⡀⠀⠀⢀⡞⠀⠀⠀⣤⠀⠀⠀⠀⠀⢸⣄⠀⠐⣄⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼⠁⠀⠀⡼⠀⠀⠀⠀⡿⡆⠀⠀⠀⠀⠀⡏⢧⡀⠸⣆⠀⠀⢦⠀⠹⣆
⠀⠀⠀⠀⠀⣼⠃⠀⠀⢰⠃⠀⠀⢰⠇⠀⡖⠀⢸⠃⢧⠀⠀⠀⠀⠀⣇⡀⢳⡀⡿⡆⠀⠘⡆⠀⠙⣆
⠀⠀⠀⠀⢰⠇⠀⠀⠀⣿⠀⠀⠀⣾⠀⣰⡇⣀⡿⠖⢸⡀⣼⠀⠀⠀⡏⠉⠛⢳⡷⢿⡀⠀⣿⡆⠀⠹
⠀⠀⠀⠀⡿⠀⠀⠀⢠⡇⠀⠀⠀⣟⣴⠿⣟⡟⠀⠀⠈⡇⣿⡀⠀⢠⠇⠀⠀⠀⠃⠈⡇⠀⣿⢿⣄⠀⢷
⠀⠀⠀⣸⠇⠀⠀⠀⢸⡇⠀⠀⠀⡿⠃⠀⠋⠀⠀⠀⠀⣷⠋⡇⠀⡼⠀⣤⣶⣶⣶⣶⡇⠀⡟⢸⡿⡆⢸
⠀⠀⠀⣏⡴⠀⡆⠀⠈⡇⠀⠀⠀⣧⣠⣴⣶⣾⣿⣿⡇⠏⠀⡇⢰⠃⠈⠛⠛⠉⠉⠉⡿⣰⠃⢸⡇⢻
⠀⠀⠀⣻⠁⢀⡇⠀⠀⣷⠸⡄⠀⢹⠿⠛⠛⠉⠉⠀⠀⠀⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⡷⣿⠀⢸⡇⢸⣿
⠀⠀⠀⣿⢠⠟⡇⠀⠀⢹⡀⠙⣄⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⡇⢸⠟
⠀⠀⠀⠘⠟⢰⡇⠀⠀⠈⣇⠀⠈⠳⣼⣆⠀⠀⠀⠀⠀⡤⠀⠒⠒⠒⣒⡇⠀⠀⠀⠀⢀⡾⠁⢸⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣆⠀⠀⢻⡀⠀⠀⠀⠀⠀⠙⠐⠦⠴⠚⠋⠀⠀⢀⣠⠶⢻⡀⠀⡼
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠻⡄⠀⠀⡉⠁⠒⠠⠤⠤⠤⠤⠤⠤⠐⠒⠂⠉⠀⠀⠀⢠⣀⠜⠁
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⡇
⠀⠀⠀⢰⢁⡴⠻⢄⠀⢸⠙⠢⢄⡘⢟⠗
⠀⠀⠀⠘⠊⠀⠀⠀⠑⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠒⠒⠒⠒⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⢸⡆⠀⠀⠀⠀⠀⡀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣀⠀⠀⣼⡿⠀⠀⠀⠀⠀⡇⠈⠹⡀⣆⢀⠀⠀⢣⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡟⠀⢠⡿⠁⢠⠀⢀⠀⣸⢳⡆⠀⠐⣼⣏⣆⠀⢘⣆⡀⡈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣷⠀⢸⡇⢀⣼⢀⣼⠤⠃⠾⠋⠁⠀⠘⣿⣿⣿⣿⣿⡇⣷⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿⡤⠚⠓⠉⠉⠀⠀⠀⠀⢄⣤⣶⣾⣿⢷⣾⣼⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⢿⣿⡟⢻⣷⣾⡗⠀⠀⠀⠀⠟⢱⣿⣿⣷⡞⢻⡾⢻⣟⣿⠀⢻⡄⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣷⣼⡟⠀⣿⣿⣯⡄⠀⠀⠀⠀⠀⠈⠛⢻⣿⠃⣼⡇⢸⣿⡘⣷⣤⢷⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡏⠁⢰⠀⠈⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⣿⠀⣾⣿⡷⠙⠋⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⣿⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣰⣿⠃⢠⣿⣿⡀⠀⠀⠀⠈⠷⠄⠘⠀⠀⠀⠀⠀⠀ ⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡾⠇⠀⢠⠀⣹⣦⣀⡀⠀⠀⠀⠀⠉⠀⠀⣀⣬⣿⡿⢀⣾⣿⣿⣇⢸⡄⠀⠀⢰⠈⠆⢁⠀⠀⠀⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⣿⢀⣿⣿⣿⣿⣿⡿⠛⢺⡿⠋⠉⢻⠟⣻⠃⣼⣿⣿⣿⣿⣼⡇⠀⠀⠈⠀⠈⢾⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⢻⣼⣿⣿⣿⣿⡟⠀⢀⣼⠀⠀⢰⠃⣠⢃⡼⠋⢻⣿⣿⣿⣿⠀⠀⠀⠂⠀⠀⣼⡇⠀⠀⢸⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠈⢆⠘⢿⣿⣿⣿⣿⣷⣤⣾⢹⠟⢹⣿⡾⠟⠉⠀⠀⣼⣿⣿⣿⡇⠀⢀⡼⠀⢀⣼⣿⠃⠀⣤⣾⣿⠇⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣈⢢⡀⠙⢿⣿⣿⣿⡏⢹⡾⠀⢸⡼⠿⢿⣶⣶⣿⣿⣿⣿⠟⣠⣴⣟⣡⣶⣿⡿⠃⠀⠀⢸⣿⡿⢀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣾⣿⣿⣿⣧⠘⢇⢀⡾⠁⢠⠞⠀⠻⣿⠿⡿⢿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⢀⣻⣥⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠽⠛⠛⠋⠉⣽⠗⠚⢁⣀⣴⣋⣠⠴⠚⠓⣄⣷⡜⠁⠉⠻⣿⣷⣤⣠⡶⢂⣴⣶⣈⣭⣝⠋⠀
⠀⠀⠀⠀⠀⣄⡀⡀⢸⣿⣿⣿⣷⡶⠁⠀⠀⣠⠀⠀⠣⠤⢲⠟⠉⠉⠉⠀⠀⠀⠀⢸⣿⣷⡀⠀⢀⣨⣿⣛⡛⠇⠘⢿⠿⣿⣿⣿⣾⣷
⠀⠀⢰⣷⣾⣿⣿⣿⡘⣿⣿⣿⣿⠃⠀⣰⠟⠁⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⢂⡙⢿⣿⡟⣡⣴⣷⣶⣶⢠⣾⣿⣿⣿
⠀⠀⣀⣘⠻⡿⠿⢩⣤⣿⡿⢋⣉⠀⢼⠁⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⣠⡞⢻⣿⠟⢸⣷⠈⠛⠘⣫⣌⠻⠟⣵⣿⣿⣿⣿⡿
⢀⣤⣿⣿⣷⣾⣷⣾⣿⣿⣷⡜⣿⣀⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠟⠋⣀⣼⣿⣶⠿⠟⢠⣿⣿⣿⣿⣿⢇⣤⣭⣭⣭⡁⠀
⡄⢛⣿⣿⣿⣿⣝⠟⢡⡜⠋⣅⠈⠿⣿⣄⡀⣤⣀⣀⣀⠀⠀⠀⠀⢀⣠⣴⢟⣧⡟⢻⣿⣿⣇⠾⣿⣿⣿⣿⡿⢟⣡⣾⣿⣿⣿⣿⡇⣸
⡁⠿⠿⠿⢯⣍⣩⣥⡜⢿⣿⣿⣦⣤⠈⢛⣁⣈⠩⣭⡽⣿⡷⣰⣿⣯⢉⡇⢺⣿⣿⡌⠉⢁⣤⠤⣀⡭⢩⣤⣶⣿⣿⣿⣿⣿⡟⢋⣴⣿
⣩⣶⣶⣿⢸⣿⣾⣿⣿⣶⡙⠻⣿⣿⣦⠘⣿⣿⣶⣤⠘⠿⢿⣽⣿⣿⡈⣾⣿⡿⠏⢸⣿⠟⣥⣾⣿⢡⣿⣿⣿⣿⣿⣿⡿⠟⣡⣿⡿⠋