Hesitant Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣿⡁⠀⠀⠀⠀⣀⣤⣶⣶⣤⣀⠀⠀⠀⠀⢀⣤⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⣰⡟⠁⠀⠀⠈⢻⣆⠀⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⣿⡀⠀⠀⠀⠀⢀⣿⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⢿⡆⢰⡾⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⣿⠁⠈⢿⡄⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⡟⠀⣾⢷⣤⣤⡾⣷⡀⢻⣿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢰⡿⠀⣼⠏⠀⠀⠀⠀⠹⣧⠀⢻⡆⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠿⠁⣰⡟⠀⠀⠀⠀⠀⠀⢻⣆⠈⠿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣀⣤⣄⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⢀⡾⠋⠀⠀⣉⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⣿⠄⠀⠀⣴⠋⠈⣁⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⢰⡇⠐⢦⡼⡄⠐⠂⣿⠀⠀⠉⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⢸⡇⠀⢸⡇⠀⠄⠀⢿⠀⠂⠠⢀⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⢸⡧⠀⢸⣇⠠⠀⠀⣾⡀⠚⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣅⠀⠁⠙⠶⠶⠟⠉⠁⠀⠐⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⢺⡉⠁⠈⠀⣀⠀⠀⠄⢠⣶⠄⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡀
⠀⢸⡇⠀⠋⠀⠉⠀⢦⠀⢀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢊⣩⡝⠲⣤⡀
⠀⢸⡇⠂⠀⠀⠀⡀⠀⠀⠛⠿⠀⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⡴⠋⠀⠹⢆⠈⢧
⠀⠸⣇⠀⠀⢤⣀⠁⢠⣀⡀⢠⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡼⠃⠀⠂⠀⣿⠀⠸
⠀⠀⣿⠐⢶⣄⠙⡿⠂⠙⠁⠈⠛⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⡆⠀⠀⠀⠀⢹
⠀⠀⠹⡧⢀⡀⠰⠷⠘⠀⢀⣄⠀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡠⠶⠀⣧⠀⠀⣀⡴⠆
⠀⠀⠀⢿⣾⠇⣤⠂⣠⡀⠀⣥⠘⠻⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠁⠀⠀⠈⠛⠛⠉⠁
⠀⠀⠀⢸⣇⠈⠉⠘⠛⣁⠀⢀⣄⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣠⠐⠾⠇⠰⠗⠀⠀⡀⢠⡀⠀⢸
⠀⠀⠀⠈⣿⡀⠀⡰⢦⠀⠀⠈⠋⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠿⠃⠀⠀⠀⠀⠄⠈⠀⠁⠀⠀⠀⣼
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⡀⢠⣤⠀⠀⢠⣤⡀⠀⢠⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣿⠆⠀⠰⠶⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣴⣎⠀⠈⠁⠀⣀⡀⠀⠀⡶
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠓⠀⢠⠀⢀⣴⠦⠤⢤⣿⣛⣦⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠿⠇⠀⠀⠁⠀⠀⠁⠀⢿⠗⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢻⡆⠈⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠉⠉⠳⠟⣿⣶⣖⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢀⣀⠀⠀⠰⠀⠶⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠈⢫⡀⠠⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣷⡦⢄⣀⣀⠀⢀⡼⠃⠀⠀⠺⠷⠀⠈⠀⠀⠘⠃⢰⡆⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣆⣀⣀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣈⠉⠝⠉⠁⠀⠀⠀⠠⠤⠺⠿⠃⠀⠠⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠂⠀⠀⠀⣄⠀⠀⠀⣀⡀⠶⠐⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⠤⠀⠀⠀⠀⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠿⠆⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⡈⠀⠀⠀⢀⣠⡴⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣯⣤⣌⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣧⣤⣤⣐⣛⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⡰⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠏⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⡴⠶⠋⠉
⠀⠀⠀⠀⠀⠀⢠⡶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠞⠛⠉⠀⠀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⠋⠉⠀⠀⠀⣀⣀⣤⡴⠾⠛⠋⠉⠁
⠀⠈⠉⠁⠉⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠟⠛⠋⠁⠀⠀⢀⣀⠤⠤
⡀⢈⠁⣀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣂
⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣠⢴⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢛⣡⢶⣍⠉⠙⠙⣿⡷
⠀⠀⢀⠀⠀⡀⡀⢸⡧⠤⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠈⠀⢸⠛⠶⣯⣄⣠⠾⠋
⠀⠀⣿⣿⣶⣶⡇⢸⡇⢠⣶⠆⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠈⠀⣜⠉⣙⣿⣳⠄
⠀⠀⣿⣿⣿⣿⡇⠀⡇⠀⣀⢠⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣤⡀⢰⣿⡝⠋⠙⠉
⠀⠀⣿⣿⢿⣿⡇⠀⡇⠀⠀⠀⠀⣿⡄⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣷⠤⢈⣠⡂
⠀⠀⣿⡿⢾⣿⡇⠀⠡⠴⠒⠚⠉⠉⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣷⡀
⠀⠀⠙⠓⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠲⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠂⠀⠈⣩⣤⣤⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡟⠀⠀⠀⠀⢀⣿⣿⡖⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠄
⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠈⡿⢋⣾⣿⣿⣿⣿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢤⣍⠛⣤⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⡀⠠⣄⡈⠀⣲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⠃⠲⠂⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠒⠶⣶⡶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⡀⢇⣾⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢼⣿⣿⣿⣿⣿⣿⣄⢸⣟⡴⢶⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡏⠛⢻⣿⣿⣿⡿⠻⣿⣿⣿⠮⠧⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠒⠛⠉⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣎⣁⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠚⠧⠴⢋⡞⠀⠀⠀⠀⠀⢢⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⠃⢀⠠⠀⠀⠉⠓⠲⠤⣤⣀⡀⠃⠀⠈⢆⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢺⣶⣶⠦⠴⣦⣀⣤⠤⠒⠲⣼⠉⢹⠖⠀⠈⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⠁⠀⠀⠰⠿⢿⣭⠟⠁⠀⠀⠀⣴⠃⢀⡏⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⣠⠤⠀⣀⣠⣔⣊⣡⡴⠚⠛⠒⠦⣬⣀⣂⡶⠃⠀⢿⣶⡞⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⣙⠦⢄⡀⠀⠀⠀⠈⠉⠉⠉⢉⣭⠉⠁⠈⢷⠲⠚⠉⠉⠀⢀⡞⠘⠀⠀⠀⠀⠀⠀⠠⡀⠸⡄
⡴⠞⠽⣤⡔⠊⠉⠑⠒⣶⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡓⣒⡒⣦⠀⠀⣸⠀⠀⣧⠤⠃
⠈⠳⣄⠉⠳⣄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⣩⣍⣡⠯⠤⠴⠋⠋⠉⠁
⠀⠀⠈⠳⣄⠈⠳⣄⠀⠈⣯⠙⠲⣄⡀⠀⠀⡤⠖⢢⠀⠀⠀⡜⠉⠉⠓⠊
⠀⠀⠀⠀⠈⠢⡀⠈⠳⣄⠘⣆⠀⠀⠙⢦⡜⡁⠀⣠⠃⠀⡜
⠀⠀⠀⠀⠀⠀⠈⢢⡀⠈⠳⣜⣆⠀⠀⠀⠙⢶⡋⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠈⠛⢦⠀⠀⠀⠀⠱⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢳⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢲⠀⠀⠀⡇⢎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠱⡄⠀⠀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣋⣩⣀⠀⠀⣧⠼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠹⡄⣠⠟⠚⠓⠒⠒⠒⠒⢲⠶⡶⠒⠒⠒⠒⢲⡖⠒⠋⡧⠽⣎⣧⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠋⠀⠀⠀⡶⠂⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡤⠖⠒⠙⠃⠀⠙⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠁⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠤⠤⠤⠤⠤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣘⠀⢡⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢯⡉⠭⠘⣦⠀⣧⣉⠀⠒⢢⠤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⢄⣤⣄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠁⠀⠀⣀⣨⣿⣿⣷⣤⣤⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠹⠿⠛⠛⣿⣿⡏⠙⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣫⣭⣤⣤⣬⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣝⠛⠳⡄
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡀⢁⠈⠹⣏⣿⣿⣿⣿⣿⣿⡇⣹⠇
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠙⠻⣿⣿⣿⣶⣄⢻⣜⣿⣿⣿⣿⣿⣳⣏⣀⣤⠃
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣰⣥⣦⣴⣴⣾⣿⣻⣿⠿⠿⠿⠿⠿⣿⣿⣿⡿⠿⡿⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢧⡟⠀⠀⠠⠖⠛⠛⠶⢦⣤⠀⠀⠀⢀⡴⠟
⠀⠀⠀⠀⠉⠐⠀⠤⠀⣀⠀⠀⢀⣿⡿⢿⣿⣯⡾⠁⠀⠀⠀⠀⢠⣒⡀⠀⠀⠀⠀⠀⣼⡔⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠋⠱⠋⠋⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⡏⠁⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡍
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠓⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠛⠀⢉⣈⣿⣦⣈⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡌⠀⠀⠀⠀⠀⠀⠰⠓⣴⣋⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣤⣶⣶⣤⣶⠀⡘⠀⠀⠀⠀⠀⠀⠀⣶⡊⢠⡈⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣦⣴⣶⡇⡰⠀⠀⠀⠀⣀⡀⣀⡀⠈⢿⡧⠈⠶⣀⢀⡠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠢⡀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠾⢃⠁⠀⠀⠀⠀⠈⠳⠀⠙⠇⠀⡀⠀⠈⣤⠉⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠿⣿⣿⠿⢿⠿⡿⢿⣿⠐⠀⠀⠁⠂⠤⠀⡀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠄⠁⠀⢠⣡⠌⠘⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣄⠘⡀⢀⣠⣴⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠒⠠⢤⡑⢤⠞⣄⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠈⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡿⠿⠿⠿⡟⠉⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠂⠢⣀⣀
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⢄⠈⠁⠐⠒⠤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠤⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠔⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡉⠙⣁⣀⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⠟⠀⠉⠉⠉⠛⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠰⠆⠀⠷⣄⠀⣠⡀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣾⣿⡇
⠀⠀⠀⡇⠀⠀⠀⠛⠃⠀⠀⢀⡙⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠙⠛
⠀⠠⠴⠷⠀⠠⠘⠱⣶⠀⠀⠚⠳⣤⣄⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠄
⣿⠀⠀⠀⠀⢶⣄⣀⡀⠃⠀⠀⣀⣀⠛⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠶⠆⠈⠉⠉⠀⠀⠀⠀⠀⠀⣠
⠙⠶⢠⡄⠀⠀⠉⠉⠁⠀⠐⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⣀⡀⠲⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⢸⣇⠀⠀⢠⣿⣦⣤⢤⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠈⠙⠛⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡀⠶⠂⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠦⣀⣀⣀⣀⣀⡠⠶⠶⠊⠉
⠀⠀⣤⣶⠀⠀⠺⣿⣶⣦⣀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠈⠁⠚⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢠⠀⠈⠉⠈⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠟⢡
⠀⠀⠀⠀⠀⠘⠷⢤⣤⣤⡀⠺⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⣾⡆⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢠⣿⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⢸⣿⣇⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠁⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡗⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠺⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣇⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣀⣀⣀⡙⠛⠀⠛⣃⣀⣀⣠⣶⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣟⠉⠁⠙⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣠⣤⣤⣤⣤⣤⣄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠃⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠙⠛⠿⠆⠀⠉⠉⠉⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡏⠀⢀⣀⡀⣭⣍⣁⠀⠀⡄⠀⠀⠀⣀⣤⡤⣀⣀⡀⠀⣸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠉⣥⠭⠤⠤⠤⠤⠿⠿⠬⢇⢰⠁⡠⠤⠮⠤⠿⠧⠤⠤⠤⢤⠍⠈⠙⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠐⠪⣦⡀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢃⠀⠀⠀⠀⠀⠀⠀⡴⠁⢠⠖⠂⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣀⡇⠉⠒⠒⠒⠒⠒⠒⢹⠀⠀⠈⠓⠒⠒⠒⠒⠒⠋⠀⠀⢸⣀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠴⢗⣚⣉⠝⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣄⣀⣀⣀⣀⣠⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠛⠻⠿⠿⠿⠿⠟⠛⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⣀⣀⣀⣀⣀⣀⡠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⠒⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠁⠀⢶⠿⡧⣤⠀⠀⢠⡀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠚⠦⡀⠙⣖⣿⣾⣇⡠⡜⣡⠐⡆⢀⠈⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡌⢠⣳⣼⣿⣿⣿⣿⢦⣟⠁⠴⣅⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣼⣿⣿⣿⣿⣿⣿⣿⡔⠁⣇⠰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⣦⣜⣿⣯⣿⣿⣿⣿⣿⢷⣼⠃⠔⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠞⠩⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠟⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠼⣾⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠀⠀⠀⠈⠿⠿⠟⠋⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡻⢛⡛⡛⣿⢻⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡂⡹⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢡⡍⢹⣡⠏⢀⠗⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⠁⢠⡃⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠠⢤⣀⠈⢦⠀⠀⢳⡀⢈⣥⣤⣀⡀⠀⠀⣠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠳⡜⢆⠀⠀⠹⡇⠀⠀⠀⠉⠓⢾⣵⠒⠒⠉⠉⠳⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡨⡄⠀⠀⢳⣰⣶⡆⠀⠀⠘⢯⡤⠄⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠙⠧⣄⡀⠀⠀⣠⠏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣸⣾⠀⠀⠀⠀⠀⠀⠉⠭⠝⢋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡤⢶⡶⠦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⣀⠤⠤⡄⠀⠀⠀⠀⠀⣀⢤⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀⠀⠄⠠⠀⠄⣀⡤⠮⢀⡉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠏⠀⢸⠀⠀⠀⠀⠀⣀⣀⡤⠤⠬⠉⢻⠛⠛⠋⠉⠀⠀⡰⢃⡴
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣄⢄⢸⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡴⣄⠞⣡⠋
⢿⠀⢸⠀⠀⠀⠀⠀⠀⣐⡄⠀⠀⠀⠀⠀⡤⢤⡤⠤⡇⠀⠀⢀⣠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠎⠀⠈⠻⡅
⢸⢠⠟⠃⠀⠀⢀⡤⠒⢒⠲⡄⠀⠀⠀⠀⠈⢻⡛⠒⠓⠠⠤⠍⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⣀⠀⢀⣀⠼
⢸⡿⣠⡀⢀⠤⣼⠷⢮⡝⡸⠁⠀⠀⠀⠀⢠⡤⠭⠭⣖⠢⠤⢄⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠒⠒⠒⠒⠒⠒⡶⠛⢾⡿⠋
⢸⡇⠈⡇⠙⠾⣖⠉⢣⠀⢙⡒⠒⠒⠒⠒⠒⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⢤⣀⣀⠀⠀⠀⠀⠀⡼⠉⠢⡼⢟⢆
⠸⠷⠖⠃⠀⠀⠈⠓⠦⠤⠏⠉⠉⠉⠉⠉⢏⢹⠀⢀⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠲⠞⠁⠀⠀⣣⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⣻⠀⠀⠀⠀⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⢤⢴⣇⡀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠉⠁⠀⠀⢸⢀⡏⠘⡉⢏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣇⣼⡤⡴⡇⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠛⢉⡏⢉⣁⣽⣥⣉⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⣀⣹⠿⠟⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⠉⠙⠛⠛⠓⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠟⠛⠛⠿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠳⣄⣠⠞⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢸⡷⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣹⢦⣄⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⣼⠃⠀⠈⠛⠶⠶⠛⠉⠀⠀⠀⠀⢀⣀⣠⣀⡀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣠⠶⣄⡀⠀⠀⠀⠀⠈⢷⡀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣀⣈⠙⢳⡄⠀⠈⠉⠙⠛⠲⣤⡀
⠀⠀⠀⢸⡇⠀⢀⣀⣠⡴⠛⠁⠀⠈⠙⠶⣄⣀⠀⠀⠸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢰⠏⠉⠙⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀
⠀⠀⠀⢸⡇⠀⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⡇⠀⣿⠀⠀⣠⠖⠲⢦⣴⠖⠲⣤⠀⠘⠓⠋⢀⣤⡴⠏⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢸⡷⠶⠇⠀⠀⢴⠆⠀⠀⠀⠠⡶⠀⠀⠸⠷⠶⣇⠀⢹⡄⠀⣧⠀⠀⠈⠁⠀⠀⣸⠂⠀⠀⠀⣿⠀⣠⡴⠟⠀⢀⡴⠖⠶⣤⡶⠒⢶⡄⠈⣧
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠈⢷⡀⠈⠳⣦⣀⣀⣤⠞⠁⠀⠀⠀⠀⣿⠀⢿⠀⠀⠀⢸⡇⠀⠀⠉⠀⠀⢀⡿⠀⣿
⠀⠀⠀⢸⡟⠛⡇⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢸⡟⠻⡇⠀⠀⠈⢳⣄⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠶⠟⠀⠀⠀⠀⠙⠶⣄⠀⣠⡴⠋⠀⢰⡏
⠀⠀⠀⢸⡇⠀⢧⡀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⣼⠁⠀⡇⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⢳⡄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣠⠟
⠀⠀⠀⢸⡇⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠶⢦⣤⣄⡀⠀⠀⠈⠷⣤⡼⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⣿⠛⠛⠛⢻⠋⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡠⢤⣄⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠉
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⠃⠀⠀⠈⢷⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⢀⣼⠷⠒⢲⡖⠒⠒⠛⠀⠀⠀⠘⠓⠒⢲⡖⠒⠾⢧⣀⠀⠀⠀⢹⣄⠀⠀⢀⡾⠀⠀⠀⠀⠉⠓⠶⢤⣤⣤⠴⠖⠋⠁
⠀⣴⠋⠀⠀⠀⠈⠳⣀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠙⢧⠀⠀⠀⠉⠛⠛⠋
⢰⡇⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠶⠶⠛⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⠶⢦⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡅⠀⢀⡷
⢈⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠙⠛⠋⠁
⢸⡇⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⢸⡇
⢸⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇
⠸⠇⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠘⠇

⡏⠉⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡏⠉
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⡔⠐⢦⠠⡄⢲⠂⢀⢶⠀⢰⠐⡄⢠⠀⠀⡆⠂⠀⢠⠂⠀⢰⠆⢠⡆⢢⢠⠆⠲⡄⣦⡠⠂⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠁⢇⢀⠼⠀⠅⢸⠀⡼⠒⠇⢸⡉⡇⢸⢀⠀⡏⡁⠀⠀⣉⠆⠸⠀⠸⠏⢣⠸⣄⣀⠇⡿⠱⡀⠏⠁
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠶⠒⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣋⣡⣤⠤⠤⠴⠒⠒⠒⠦⠄
⣀⣤⣴⣶⣾⣿⣿⣿⣗⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢟⡻⠟⠉
⠋⠉⠉⠈⠀⠀⠀⠉⠉⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣡⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠛⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠁⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠨⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⡀⠀⢠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⣰⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠔⠛⠳⣶⣿⣿⣷⠀⠀⠀⠀⠀⣤⢄⡐⢶⣤⣾⣿⣿⣇⣼⡇⠀⠀⠀⠀⣠⢺⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⢻⡟⠻⣿⡿⢿⡿⢿⡿⠛⠛⠛⣻⣿⣿⠆⠻⢿⣿⣦⡀⠀⡜⢱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⢸⣿⢆⠀⣀⣾⠀⠀⢣⣀⣴⡾⠛⠻⣿⣇⠀⠀⠉⣿⣿⡈⠁⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⣠⣿⠃⠀⣹⠟⠉⣳⠶⠉⠨⣿⡁⠀⡀⠹⣿⣷⣤⣴⣿⣿⠉⠙⢻⣦⡀⠀⠀⠀⠀⠀⠀⡠⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠋⠀⢠⠇⠀⣼⡷⠀⠉⠘⣿⡅⠀⠘⠀⠈⠛⣯⣍⣽⢧⠀⣀⣼⡏⢳⡀⠀⠀⠐⢤⣜⣣⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢯⡿⣤⣄⡆⠀⢸⣧⣿⠀⠀⠀⢸⣿⣤⠁⠀⢠⠞⠋⢻⣻⣧⣴⣿⠟⠁⠈⠀⠀⠀⠀⣐⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⢛⠋⠀⠘⠛⣷⠀⠈⡏⣹⡷⠚⠷⣟⠟⢻⡀⡰⠂⠀⠀⠸⣿⣿⠋⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠻⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⠀⠀⡠⠸⣿⡟⠿⢻⣿⣆⠀⠀⢰⠦⣾⣷⡖⠀⠀⠀⠀⣼⠿⣧⣀⣀⣀⣠⣤⠟⣿⣿⣿⡄⠀⣿⡇
⠀⠀⠀⠀⠀⢠⣿⡟⠻⣆⠀⠀⠀⢀⡄⠰⣿⣀⠀⢉⣿⣯⡤⠀⣠⡟⣠⠟⠱⢀⣤⡤⢾⡟⠀⠄⠉⠋⠩⡝⣣⣾⣿⣿⣿⠁⠀⣾⣿
⠀⠀⠀⠀⢀⣿⠏⠀⠀⠈⢻⣦⣴⡟⠀⠀⠹⡧⣼⣿⡟⢿⣿⡿⠋⠁⠘⣦⠶⠫⠁⠀⢸⡇⠘⠀⣴⣷⣷⣾⠟⠉⣰⣿⣿⠄⢀⣿⣿⠂
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⣿⡏⠀⠀⠟⣧⡀⢀⣼⠇⠀⠀⠀⠀⢸⡇⠀⠘⢋⣿⠋⠑⠖⠚⣹⡿⠙⠀⣸⣿⡟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⠀⠀⣿⠀⡀⠀⠀⢸⣿⢌⡁⢈⡤⠀⣀⡴⠻⣧⠀⣼⠋⠀⠀⠀⣠⡾⠋⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⢸⣷⣤⠶⠛⠉⠉⢻⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡇⠀⠀⡞⠻⣧⡿⠀⠀⢿⣌⠁⠀⠀⢰⡿⠋⠀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⢻⠀⠀⠀⠀⢻⣅⠀⠀⣧⡴⠪⡟⠲⡄⠈⣿⠀⠀⢠⡟⠀⣠⡾⠛⠉⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⢀⣿⠿⢿⣆⠀⠀⢸⠀⠀⠀⠀⠀⠙⠛⣬⣿⣆⣀⣾⡀⢹⡄⠘⣆⠀⢸⠇⣼⣿⣶⣶⣶⣦⣤⣞⣛⣻⣿⣽⣿
⠀⠀⣼⣿⡿⠋⢻⡆⠀⢀⣤⠞⠁⠀⠀⠹⣧⡀⠀⡆⠀⠀⠀⠀⢀⣴⠟⠁⣿⠛⠉⢉⣿⡇⠀⠙⡄⠈⠀⣿⣿⡿⣶⣤⣾⡉⠙⠛⠛⠛⠛⠉
⢀⡼⢿⣿⠁⠀⡘⢧⣶⣿⠃⠀⠀⠀⠀⠀⠈⢿⣄⣿⠀⠀⠀⠀⣼⠛⠳⣄⡿⠀⠀⢸⣿⡇⠀⠀⠘⣆⣀⠘⣿⣀⡼⠋⠙⢯⣷⣄⠀⠀⠀⢀
⣼⠁⠀⡇⠀⠀⢀⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢧⠀⠀⢰⣿⠀⠀⠀⠉⠳⣤⣿⣿⣿⡀⠀⠀⢿⣿⢀⣿⣯⣴⣶⣶⣦⠉⠛⠿⠿⠿⠋
⡇⠀⠀⠃⠀⠀⢿⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⠀⠈⣿⡀⠀⠀⠀⢀⣿⠟⢙⢻⣷⣤⣄⣤⡿⠿⠿⡿⣟⠀⠈⣿
⠀⠀⠀⢸⠀⠀⠸⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⢿⡇⠀⠀⢀⣾⠏⠀⠀⠑⠿⠷⢿⣧⡄⠀⠀⢈⣿⠀⠀⠏
⠀⠀⠀⠀⠁⠀⠀⣽⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⣇⢀⣠⣿⡿⠀⠀⠀⠀⠀⣠⣾⣿⣁⠀⣀⣼⡇
⠀⠀⠀⠀⠘⣤⡾⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢻⡇⠀⠀⠀⠀⡼⠏⢀⣼⢏⣸⣿⠋
⠀⢀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡇⠻⣆⣀⣀⣤⣼⠷⠞⠋⠑⣦⡿⠁
⠺⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⣶⠾⠛⠛⠛⠋⠉⠀⣷⠄⡀⢹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠿⠋⠑⠀⠀⠀⠀⠀⠀⠀⢠⠤⠐⠋⠀⠙⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⠿⡋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣼⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠙⡦⠄
⡞⠁⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠁⠀⠀⠀⠀⢀⠊⡠⠖⠚⠉⠉⣉⠂⠀⡥⢤⡤⠤⠤⡄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡭
⡂⣷⠤⠀⠀⣤⠘⡥⠀⠂⣥⠖⠁⡠⢾⣤⠂⢂⠄⢀⠈⣄⣶⢰⡟⢸⠁⠀⠄⠀⠀⠀⣠⢠⡀⠀⠂⠀⠀⠂⡈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰
⠛⠀⠀⠀⢀⣿⢢⠑⢀⡴⠋⣠⠊⠠⣄⠈⠦⢓⠀⣀⣿⠸⣿⢸⡇⣼⢾⣅⣀⠠⠊⠀⠀⠀⠑⠀⠀⢈⡉⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⣼⠁⣼⣷⠎⠠⡾⢏⠂⠀⡍⠀⣞⠶⠎⠀⠈⢀⠛⠘⢫⣹⣿⣲⢗⠾⠿⣿⡦⠀⠘⠀⣇⠈⠀⠀⠀⡘⢠⠀⠀⠀⠂⠘⠙⠎⠑⣦
⢀⣦⡀⢰⣿⢸⣿⡇⢠⠁⢱⡦⢉⠂⡍⠉⠁⡀⠀⠀⠀⡀⢥⠰⠙⠓⠘⠗⢩⣿⠿⠿⡇⢤⡠⡄⢸⣕⡋⢤⣡⠄⠈⡌⣀⠀⠀⠲⢀⠀⢸⣟⣷
⡉⠀⠀⢸⣿⡜⣿⠀⡆⠀⣄⣹⡾⠍⠀⠀⢀⣴⠝⠈⡐⢏⡌⠄⠸⠇⣴⢳⣤⠯⡄⣄⠚⡆⠳⣀⠋⡳⣴⠆⠑⠓⠀⠀⡭⠐⠂⠀⡈⠀⢀⠹⣾⡇
⠧⠄⠀⠈⢿⣷⡘⡆⠘⢄⠘⠁⠀⠀⢀⣴⢩⣱⣦⠀⠀⢄⠁⢄⠀⠀⡙⢮⣏⢑⠻⡃⢚⢡⡆⠻⠆⢹⣝⣿⣷⠟⡀⠠⠀⠀⠀⠀⠀⣠⣾⣷⣿⢻⡀
⡀⠀⠀⠀⠀⠙⢿⣬⣦⣀⠉⠁⢀⠀⢮⣶⣿⣿⣿⣦⠐⡈⢲⣦⡑⡀⡘⠉⠻⡬⠷⣀⣿⠇⣿⣿⣣⢘⡮⢈⡟⢎⢀⠀⠀⠀⢀⠀⡐⢿⣏⠂⣽⣿⠇
⠷⠀⠀⠀⠀⠀⣸⣿⡏⣭⠀⠀⡪⠶⢀⡉⠁⠋⠟⠟⢡⢨⠮⣧⣷⠋⠀⣾⠛⡆⣿⢮⣵⠀⢿⣌⣉⠀⢐⠟⢘⢄⡼⢀⣾⣿⡏⠀⠌⣠⣿⣤⣿⠃⣆
⠀⠀⠀⠀⠀⣰⣿⠛⠋⢓⡆⠀⠀⡷⢆⠉⢉⠛⡔⠒⠴⠟⠋⢠⠃⠀⣾⣫⣶⡧⣝⢶⣎⣄⢽⣟⣿⠀⢘⠓⣰⠃⠀⠘⠋⠉⠀⣡⡆⠈⡷⣸⡏⠀⠿
⠀⠀⠀⠀⢰⣿⣿⣆⠘⠿⢷⠀⠀⣏⣀⡈⠙⡳⠼⢷⣶⣯⠀⠎⠀⠀⣿⣽⣞⣎⠁⣐⢹⡄⣍⠁⢿⡆⢩⡜⢪⡀⢘⠆⠀⠀⣰⠿⠇⣌⢳⣟⠁
⠀⠀⣠⠀⠀⣿⣿⣿⡆⠀⢸⣇⢠⣬⣩⣵⣒⡀⠙⠛⠍⣿⡿⠀⠀⠘⢤⣿⣿⡇⢣⡉⠸⠇⢹⡗⢿⡷⠈⠖⠉⣠⣇⠴⢄⣴⣯⠴⢦⠤⠘⠈⠁
⣁⣴⠏⠀⠀⠹⢿⣿⣿⡄⠈⠋⣿⣇⣱⣥⡛⣥⠀⡗⣀⣩⣦⣼⣷⡦⢌⡻⣿⣵⢾⠉⠛⠶⢤⣤⣤⣤⡴⢶⠛⠁⠉⠳⣤⣈⣁⣀⣤⡔⠁
⣿⣃⣠⡾⢁⣼⠃⠈⣻⣷⢀⠀⠉⠁⠀⠀⣿⠦⣍⠛⠻⠯⠀⠚⠋⢰⣟⣛⡻⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠒⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀
⣎⡿⣿⢛⣾⣃⣃⢠⡟⣽⣧⣁⠩⠔⢚⣭⣶⡿⠛⢿⣶⣤⣴⡶⠗⠁⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⣿⣿⣿⣿⠃⢉⢱⡿⠳⠿⣿⣷⣾⣿⠟⠋⣁⠠⠐⠁
⣿⣿⣿⣷⣵⢾⠬⣧⣀⣀⣹⣿⡭⠴⠚⢋⡡⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣠⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣽⣿⣯⠛⣻⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣧⣤⣼⣧⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢀⡾⠋⠙⢷⡀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠘⣧⡀⢀⣸⠃⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⢸⡇⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢀⣤⠞⠉⢸⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⠓⢲⡖⠒⢲⡖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣸⣇⣀⣀⣸⡇
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⠷⠶⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⣸⠃⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣉⣉⣉⣉⣁⣉⣽⡇
⠀⠀⠀⠀⣿⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢧⡀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⣿⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣶⣶⣦⣶⣦⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣼⣃⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠊⠉⣉⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠐⠢⡕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⠿⠿⠿⠛⠛⠛⠋⣉⣉⡉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⢙⣛⠶⣦⣀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣴⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⡠⠴⠚⠋⠉⠉⠉⠉⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠐⠣⠀⠙⢷⡀
⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠆⡐⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠳⠤⠖⢊⣉⠉⠉⠲⣄⠀⠠⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠁⣰⣿⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠊⠉⠉⠉⠉⠲⣌⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⢀⣿⡏⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣄⣀⣀⣀⡠⠜⠃⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡆
⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠁⠀⠀⠀⢀⣠⠞⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⠇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠋⠛⣋⣉⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⡠⠂
⠀⢀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⡄⠂⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣋⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣴⠟
⠀⠈⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⠞⠁⢀
⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⣀⣤⡇⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠕⠊⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⣴⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠄⠈⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠷⠶⣦⣀⠀⠀⠀⠸⣆⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢷⣦⣀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠈⠛⢷⣦⣘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢈⡁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢁⠀⣄⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠐⠤⠀⠉⠻⢕⣀⣀⠀⠀⠂⢠⠆
⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣴⠞⣡⠆⠀⣁⠠⢖⠀⠠⠄⡛⢿⣷⣇⠀⠦⠀⠀⢀⠞⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣧⠋⢤⢖⡠⢄⠦⠠⣤⡶⠦⡉⠚⢻⣬⣳⣧⠔⠒⠌⢀⠀⠀⣠⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⢀⡾⢃⣠⠖⠀⠉⠛⠂⠁⠀⠀⠈⠀⡩⢴⣿⣮⣿⣦⣐⣀⡘⡀⠀⢀⡀⢀⡠⠰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⢸⢋⠀⠁⠀⢈⣤⣶⣶⣶⣃⠀⠀⡘⡴⠃⠙⣿⣷⣝⣿⠖⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠈⢦⠀⠀⢀⢴⡿⣊⣤⣡⣾⣿⡟⢹⠉⠉⠉⠛⠀⠴⠁⢄⠀⡞⣿⣜⣧⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⣸⡟⢛⣋⣈⢸⠟⡉⠀⠀⡀⠀⠀⢀⠂⢰⠆⠀⣠⢹⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠈⠁⠀⣠⣿⠛⠟⠛⠙⠉⠀⠃⠀⠀⠀⠀⠠⠁⠀⠋⠰⣰⢧⣿⣿⣳⣤⡄⡀⠀⠀⠀⠀⡊⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣡⣿⢿⢳⠀⠀⠀⠉⠀⠠⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠠⠀⠀⠀⠁⠀⠢⢀⠀⠀⣤⣷⡆⠀⠀⠀⠀⠀⣠⢾⠟⠀⣈⡸⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡿⣦⣧⠀⠀⠀⠀⢾⡟⢠⡇⣿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡀⠠⠄⠀⢀⠀⠀⠀⠙⢿⡟⣧⠀⠀⠀⠘⠆⣾⡏⢸⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⣎⠆⠀⠀⠄⢲⣽⡿⠘⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⣠⣜⠞⠀⠀⠀⠈⢿⠇⠀⠀⠁⢀⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠘⢿⠝⠻⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠙⢄⡃⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠚⠶⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠠⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣫⣟⣁⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⡠⠚⠁⠀⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⣭⣭⡷⠦
⠀⠀⠀⠀⠀⠀⠀⠐⠛⣓⡶⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⡙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠉⠉⣽⣣⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠬⠭⣯⡉⠛⠒⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠚⠓⢸⡀⠀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣠⠶⠚⠒⢆⣠⠞⠋⠉⠉⠙⢧⠤⢖⣒⣋⠭⣹⠇⠀⣿⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡞⠁⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣸⣈⠩⠤⠒⠒⠛⠓⠶⠋
⠀⠀⠀⢠⠞⠉⣉⠒⢤⣀⠞⠀⠀⣈⡓⢧⡀⠀⠀⡀⢰⠇⢀⠀⠀⢀⡴⢻⡀
⠀⠀⠀⢻⡀⠈⠑⠛⠆⠋⠀⣀⣼⠉⠉⢲⠙⢦⠤⠟⠋⠓⠿⠶⣶⣋⠀⠀⡇
⠀⠀⠀⠀⠉⠓⠒⠒⠒⢦⡀⠀⠉⠛⢶⡋⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠲⢤⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠸⡟⠲⣦⣤⣀⣀⣀⠀⢀⣀⣰⠚⠛⠛⠓⠒⠲⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⣄⠘⠶⠳⣀⡝⠳⠏⡙⢯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣻⢏⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢿⠀⠀⢀⠏⠹⡄⠀⣳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⢶⠋⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣤⣤⠴⠾⠀⠀⣸⡠⠤⢽⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⣲⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣧⢸⡄⠀⠀⢠⠿⡀⠉⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠻⣅⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡤⣼⡈⣧⡤⢤⡞⠀⠉⢹⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠏⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⡧⠞⠀⠈⠙⣦⣄⡈⠉⣩⠝⠁⢠⠏⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠁⠢⣀⠀⠀⣰⠛⢻⠙⠛⠁⠀⢠⠏
⠀⠀⠀⠀⢰⡋⠉⠛⣍⣓⣶⡤⢤⣄⣰⣇⠀⢸⣦⠴⠖⣚⠟⠛⠒⡦
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠳⠏⠀⠉⠙⠓⠒⠒⠋⠁

⠀⠀⠀⢀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡶⣦⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⢠⣯⠀⠀⠀⠀⠀⠀⠀⣮⢸⣿⣿⠀⠀⠀⠀⠀⠰⣿⣷⣄⡀⠀⠀⢸⣿⣿⣿⡟⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠳⠀⠀⠀⠀⠀⠀⠠⠻⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⢸⣿⣿⣿⡅⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡀⢸⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⠙⠁⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢤⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣾⣿⢠⡟⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⣹⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠸⡿⠘⢿⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣴⡄
⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠿⠇
⠀⠀⠀⠀⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢀⡰⠆
⠀⠀⠀⠀⣿⠊⠀⠀⠀⠀⠀⠀⠀⠠⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠈⠉⠿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠈⢷⣦⣤⣀⣀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢈⠟⠛⢛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧⣶⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⢿⣑⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⠟⠛⠋⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢧⠀⠀⡎⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⢤⣄⣀⣀⣠⠷⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠟⠛⠻⢶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣫⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡴⠁⣠⡾⣦⠀⢠⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⣸⡏⣠⠞⠁⠀⠀⠀⣠⠞⠀⣠⡼⣿⠀⣴⣋⣴⠞⠋⠀⠸⣧⢸⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⢀⡤⢞⣡⡴⠞⠋⢠⣿⠿⠛⠉⠀⠀⠀⠀⠀⠹⣧⡄⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⢸⡇⣤⣴⡾⠿⢛⣩⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣤⣄⠀⢸⣷⣄⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣷⢸⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠈⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⣴⣶⣶⡀⠀⢠⡇⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣻⠀⠀⠀⠸⣿⣿⣿⠃⠀⣾⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⢰⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⣿⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠹⢶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢀⡾⠛⣷⡀⣶⠛⣶⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠸⣧⣀⣾⠃⣿⠀⢸⣇⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠈⠉⠁⠀⢿⡄⠀⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⣀⣀⣀⡀⣀⣀⣸⣧⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⣹⡿⠛⠷⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡿⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢻⣦⣄⣀
⠀⠀⠀⠀⣀⣤⡤⠶⠖⠛⠛⠛⢻⣏⠀⠈⠻⣦⣄⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠉⠛⠶⢤⡀
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⣀⣼⠟⢿⣄⠀⠀⠀⠀⠀⠀⢀⣼⣷⡀⠀⠀⠈⢻⡄
⠀⠀⢀⣿⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠙⣷⠾⣿⠻⠶⠟⢳⣤⠾⠛⠁⠈⢿⡄⠠⠀⠀⢻⣆
⠀⠀⣸⡇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡆⠀⠀⠘⣿⠀⠀⣠⣴⠿⣷⡇⠀⠀⠀⢻⡄
⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣧⠀⠀⠀⠘⣷⠞⠋⠀⠀⠈⠻⣦⡀⠀⠀⢿⡀
⠀⢸⡏⠀⠀⢠⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠈⢻⣆⠀⠈⢷⡄
⠀⣿⠁⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡆⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠙⣷⡀⠈⣿⡀
⠀⣿⠀⠀⠀⠀⠀⣿⠀⢰⠖⠒⠒⠛⠛⠛⣷⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠈⢿⣄⠘⣷
⢰⡟⠀⠀⠀⠀⠀⣿⠀⠈⠓⠒⠒⠒⠚⠋⠉⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠻⣦⣹⡆
⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷
⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣿⡇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⢷⣄⠀⠀⠀⠀⠀⣼
⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠻⠶⠶⠶⠾⠏
⠻⣦⣄⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠙⠛⠛⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣄⣀⣀⣀⣸⣇⣀⣤⣤⣴⠶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⠿⠿⠿⣶⣦⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣥⡶⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⣤⠀⠀⠀⣠⠞⠁⣠⣴⠟⠻⣷⠀⠀⢰⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢀⣴⠞⠁⠀⠀⠀⠀⠀⣠⡶⠋⠀⢀⣴⢿⡿⠀⣠⡾⣋⣴⡾⠟⠁⠀⠀⢻⣧⠀⢿⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠁⠀⠀⠀⠀⣀⣴⠞⠋⣀⣴⡾⠟⠁⣸⣷⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⢿⣧⣼⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⣤⣶⣯⣵⡶⠿⠿⠿⣤⡄⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢴⣶⠿⢿⡿⠛⡋⢁⣄⣀⣤⡀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀⣀⠀⠀⣿⡿⢦⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡟⠀⠀⠻⠛⠛⠛⠉⠉⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⢾⡏⢉⡉⢉⡉⢉⠙⢷⡄⢻⡇⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⡇⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⣰⡆⠀⠀⠀⠈⠿⠾⠿⠿⠿⠾⠿⠟⠀⢸⣧⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡆⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠸⣿⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠁⠀⠀⠀⢸⣿⢡⣤⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡄⠀⠀⠀⢀⣀⣤⣤⣴⣶⠶⠿⠿⠿⠷⢶⣶⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⣼⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⠀⣠⣾⡟⠛⠉⢁⣀⠤⠤⠶⠖⠚⠛⠿⠿⠿⠿⣷⣍⡛⠛⢿⣶⣤⣀⠀⣿⡇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⢿⡾⠋⢸⡇⠀⣰⡿⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⠈⠻⣦⠀⠹⣟⠻⢿⣿⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠈⠀⠀⣿⠁⠀⣶⠿⠿⠿⣷⣶⣶⣾⡏⠀⢻⣷⢀⣿⠀⠀⠈⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⢠⣤⣭⣭⣿⡇⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢠⡏⠀⠀⠀⠀⠘⠿⣦⣤⣄⡈⠀⠀⠀⣹⣷⠀⠀⢹⣧⠀⠀⢀⠀⠀⠀⣿⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⢰⡄⠀⠀⢠⠀⠈⠙⠛⠛⠛⠛⠛⣿⡇⠀⠈⣿⣆⠀⠀⣷⠀⠀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣼⡇⠀⠀⣾⠀⠀⠀⠀⣤⠀⠀⠀⣸⣿⣄⠀⠘⣿⣄⢀⣽⣷⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠶⣶⣿⣄⣀⣀⣠⣿⣀⣠⣾⠟⠙⢿⡆⠀⠈⢿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⡏⠉⢹⡟⠛⢛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⡶⢿⣿⠀⠘⣧⠀⢸⡇⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠷⣦⣤⣄
⠀⠀⠀⠀⠀⢀⣤⣶⠾⠟⠛⠛⠋⠉⠀⠀⠀⠀⢸⣟⠀⠀⣿⠀⠈⣿⡀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠈⠙⠛⢷⣦
⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣄⢸⡏⠀⠀⣿⡇⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠈⠻⠿⣿⣦⣤⣿⣇⣸⣿⣿⣷⣦⣤⣴⣶⣆⠀⠀⠀⢀⣴⡿⢷⣦⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⠀⠉⠉⠉⠉⠉⠉⠁⣀⣼⠟⠁⠀⠙⣷⣶⣿⣿⡛⢳⣄⣿⡇⢠⡉⠀⠀⠀⠹⣷⡀
⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢿⣼⣷⡈⣷⣼⡿⠛⢿⣿⡁⠀⠀⠀⠀⠹⣷⡀
⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠛⠛⠉⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⢻⣷
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⢿⣧
⠀⢀⣿⠃⠀⠀⠀⠐⣷⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⢿⣧
⠀⢸⣿⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⢻⣧
⠀⢸⡿⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠈⣿⣆
⠀⣾⡇⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠸⣿⡀
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠶⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠈⠙⠿⣶⣤⣤⣤⣀⣀⣼⡿
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠉⠉⠉⠙⠛⠋
⠙⢿⣷⣤⣀⡀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠈⠛⠛⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⡾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⣸⠞⢼⣿⣮⡓⢀⠀⠀⢰⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⡞⣿⠻⡏⠙⢦⣉⣲⣿⣍⢻⣿⣿⣗⣹⣿⣿⣿⣿⣿⡟⠉⠙⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣇⠀⣀⢻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡄⡀
⠀⠤⡤⢤⣄⠀⡀⠀⠀⠈⠁⣘⣽⣿⣟⣯⣿⣿⣿⣿⡿⢿⣷⣿⣯⣹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠃⠤⣶⡿⣿⣿⣿⣿⣧⢽⣿⣿⣯⣿⣿⣿⣟⣿⣿⡀
⠀⠀⠀⠈⢳⣠⢠⣤⡈⠀⣤⡀⠀⢀⡐⢀⣈⡽⠛⠛⣿⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠈⠔⠁⠀⢊⠽⠿⢦⣴⣏⣙⣿⣧⡇⠴⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠈⠀⠀⠀⡠⠉⠀⠠⡀⢼⡷⠿⣩⣑⣠⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⣠⠤⠇⢠⡄⠀⠟⠋⠠⣠⣿⣿⣧⣿⣟⣿⣿⣿⣿⣿⣿⣿⣧⣿⣯⣿⠆
⠀⠀⠀⠀⠀⠘⠁⢸⣴⡥⠊⢁⣠⣤⣄⣤⣤⡵⣿⣿⣿⣿⢻⣿⡿⣿⡿⣿⣿⣿⣿⡏
⠀⠀⠀⠐⠀⠀⠴⡾⠋⠤⠌⠠⣶⣿⣭⡷⣬⢻⣿⠿⣿⣿⣿⣿⣲⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠣⡤⠀⠀⢶⣶⡾⡉⠐⠀⣌⣈⣰⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⣈⣉⣙⣩⡷⠶⢭⡋⢰⠐⣻⣿⡿⠻⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠳⡄⠀⣠⡿⠚⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣴⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⣀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⣣⣾⣿⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⣛⣛⡛⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠐⠛⠛⠛⠛⠀⠀⠀⢹⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣷⢹⣿⣿⠿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣸⣿⣿⡇⣿⣿⣿⢸⣿⡇⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇⣿⣿⢳⣿⣿⣿⡷⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⣿⣿⢸⣿⣿⡿⠁⠀⠀⣿⡇⣿⡆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢸⣿⣿⣇⣿⣿⣸⣿⣿⡇⠀⠀⡀⣿⣧⣿⣧⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⣛⣛⣛⣛⡛⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣻⣿⡇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣁⣀⣀⣀⡀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢚⣉⠤⡤⠤⣄⣈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢸⣷⠀⠀⠀⠈⡦⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⡞⠀⣎⣀⡜⠁⠀⠀⠈⢣⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠛⠒⠒⠒⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢆⠈⣿⠉⡉⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠦⠆⠀⢣⠀⠀⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠈⠀⠀⠤⠶⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠐⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠐⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣴⠲⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⢰⠲⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠢⠄⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⣛⡛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣛⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣤⣉⣵⣯⣿⣿⣧⠌⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⢤⡘⢩⡷⠠⣤⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⣻⡾⣿⢿⣿⡟⠺⠿⠆⠀⡇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢺⠀⢸⠀⢰⠋⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣹⠞⠛⠛⠑⠀⠀⡀⡀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⡄⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⣀⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⣀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠈⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠙⣄⠀⠀⠀⣠⠾⠋⣠⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡈⠢⣈⣷⣶⣿⠥⠤⠶⣤⣥⣀⡈⢳⡀⢀⣀⣤⠴⠒⠋⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣀⣀⣀⣸⢃⣠⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢯⡉⠀⡀⠄⠂⠘⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠤⠀⠀⢀⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢀⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡀⠀⠐⣟⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠂⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣿⢣⡀⠀⢀⡤⠞⠉⠉⠀⣈⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⣧⡈⠀⠀⣯⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠄⠀⢀⠀⢸⣄⠙⢦⠎⢀⣠⠶⠛⠛⠙⠛⠳⢦⣷⠀⠀⠀⠀⠀⠀⣿⠑⠀⢀⣀⠉⠓⠦⣤⣀⡀
⠀⠀⠀⢠⡴⠚⠋⠉⢀⡀⠠⠄⠐⠂⠀⠉⠳⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠉⠐⠤⠀⣠⠟⠃
⠀⠀⠀⠀⠙⢦⣀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣤⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⢀⣼⡇⠀⢀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⣼⠁⠰⡀⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠆⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠶⠶⠒⠚⣉⣤⠞⠁⠀⠀⠀⠀⠀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⢊⠵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡶⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠄⠘⣷⡀
⠀⠀⠀⠀⠀⠰⠯⣄⡈⠐⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⠤⠤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢁⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠙⣛⣿⡿⢯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠌⠀⠀⡀⠤⠀⠀⠂⠀⣼⠏⠀⠀⠀⢹⡄⠀⠀⠜⠑⠀⡀⠀⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠐⠂⢀⣀⣀⣀⡀⠀⠘⠀⣿⣄⠀⠈⢀⣼⡇⠀⠊⠀⣀⠀⠀⠑⠲⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠖⠚⠋⠉⠉⠁⠀⢻⡄⠀⠐⠙⣯⣝⣛⣭⢿⣌⠀⠀⣼⠋⠉⠙⠓⠲⠶⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣴⠟⠉⠀⠉⠀⠀⠉⠻⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⡄⠀⣴⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠻⣿⣿⣿⣿⡿⢋⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣦⣍⣉⣥⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣆
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⢋⡙⠿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣾⠿⢿⣶⣌⠻⣿⣿⣶⣶⣤⣤⣀
⠀⠀⠀⣸⣿⣿⣿⡿⠟⣋⣴⣿⣿⣶⣌⠙⠿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⡏⣠⣶⣦⡘⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣿⣿⠁⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⢹⣿⠀⠀⠀⢸⣿⣿⡿⢛⣛⠻⠿⢛⣛⠻⣿⣿⣄⣛⣠⣿⡿⠿⢁⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠛⠛⠀⣿⣿⡿⠻⣿⣿⣿⣿⡿⠻⣿⣿⡇⠘⠛⠀⠀⠀⠸⣿⡟⢰⣿⣿⣷⣴⣿⣿⣷⠘⣿⣿⣿⣿⠁⣶⣶⡿⠋⣰⣿⡿⠟⠛⠻⣿⠟⠛⠻⢿⣿
⠀⠀⢠⣾⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣆⠀⠀⠀⢻⣿⣄⠙⢿⣿⣿⣿⠟⢁⣼⣿⣿⣿⣿⠀⣿⡇⣰⣿⣿⣿⠀⣾⣿⣷⣄⣾⣿⣷⡌⣿
⠀⠀⠀⠛⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠃⠀⠀⠀⠀⢻⣿⣷⣦⣍⣉⣥⣶⣿⣿⣿⣿⣿⣿⡀⠿⠇⣽⣿⣿⣿⣆⠙⢿⣿⣿⣿⡿⠋⣠⣿
⠀⠀⠀⣾⣿⠘⣿⣿⣿⣿⣛⣛⣛⣛⣻⣿⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠾⣿⣿⣿⣿⣿⣿⣦⣍⡛⣉⣴⣾⣿⡟
⠀⠀⠀⣿⣿⣆⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣿⣿⣿⣦⣌⠛⠿⠿⠿⠿⠿⠿⠟⢋⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣿⣿⣿⣿⣦⣉⣡⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⣴⣶⣶⣶⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⢀⣨⣭⡍⢩⣭⣭⣤⣾⣿⣿⣿⣤⣬⣭⣭⢉⣭⣍⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉
⢀⣴⣿⣿⣿⣿⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣿⣷⣄
⣾⣿⣿⣿⣿⣿⣿⣷⣬⣉⣛⠛⠛⢛⣋⣡⣴⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠛⠋
⣿⣿⡟⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠹⣿⣿⡇
⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⡇
⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⣿⡇
⣿⣿⡟⠻⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠂⠈⠉⠉⠉⠀⠒⠢⢄⡀
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⣀⣤⣤⣀⠀⡶⠤⣀⠀⠈⠲⣄
⠀⠀⡠⠊⠀⠀⠀⣠⠔⠛⠉⠉⠉⠉⠙⠳⢯⣊⣓⣄⠀⠈⠢⡀
⠀⡜⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠙⢯⣾⢧⠀⠀⠑⡀
⣸⠃⠀⠀⢀⠃⠀⠀⠀⠀⠀⡠⢊⢔⡡⠀⠀⠀⠀⠙⣯⣣⠀⠀⢷
⣯⠀⠀⠀⠈⠀⠀⠀⠀⠀⢨⠔⠁⠉⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢸⡆
⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡢⠀⠄⠀⠀⣧⡇⠀⢸⡇
⢸⣥⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢊⠄⡠⠂⠀⢠⣷⠃⠀⢸
⠀⢳⡅⠀⠀⠐⡀⠀⠀⠀⠀⡠⠊⠊⠐⡡⠊⠀⠀⠀⣾⠋⠀⢀⡟
⠀⠀⠳⡵⣄⠀⠈⠢⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠞⠁⠀⢀⣾⣷⢄⡀
⠀⠀⠀⠙⢧⡧⡄⠀⠀⠑⠂⠤⠄⠤⠤⠤⠐⠊⠁⠀⣀⣴⣾⣿⣿⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠙⠻⢥⣅⣄⡦⠀⠀⡄⢀⠀⣤⣤⣶⡿⠟⠉⣟⣫⣃⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠚⠚⠛⠉⠁⠀⠀⠀⠘⢗⢵⠤⠄⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣖⠒⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⡫⡀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡴⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⡣⡂⡀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡢⣂⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡷⣠⡄⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⢳⡀
⠀⠀⠀⣠⠖⠋⠉⣉⡓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠁⠈⠑⣄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡤⠒⠀⠤⣀⠀⣠⡾⣋⣀⣠⡄⢧
⠀⠀⠀⣇⡀⠄⡄⠠⠋⠘⣒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⣠⣤⣤⣤⠈⢱⡀⠀⠀⡠⢊⠤⠉⠠⣱⢫⠞⠉⠉⠲⡌⠳⣿⠞⡉⠀⢀⠈⢾⡇
⠀⠀⢸⣁⣴⡞⠁⢠⣦⣛⣬⡆⠀⠀⠀⠀⠀⠀⠀⠀⢨⣴⡞⢙⠀⠀⡉⣿⣴⠃⠀⢸⣰⠃⢀⠀⠀⡍⣸⠀⠠⠀⠀⢉⢲⠇⠀⠀⢀⠀⠀⠘⣇
⠀⠀⠀⠑⢛⡁⠀⠀⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⠅⠀⢐⡂⢠⣶⡅⠀⢠⡋⠍⠀⠀⠀⠀⣧⠧⡀⠀⠀⠀⢸⣿⡄⠀⠀⠂⠂⠀⣰⠁
⠀⠀⠀⢀⡮⣷⣶⡶⠤⢤⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣈⡴⠃⠈⠀⠀⠀⠉⢦⣀⠀⠐⣰⣣⣧⣴⣦⡄⠀⢸⠟⠉⠂⣄⣀⣠⠔⠁
⠀⠀⣴⣋⢡⠆⠉⠛⢓⠲⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠞⢹⠈⡛⢹⠦⣄⠀⠀⠀⠀⢈⣅⣀⣞⠉⠛⠛⠛⢉⣄⡴⢃⡷⡲⠃⣈⠹⢀
⠀⢸⣦⡋⠀⠀⡜⠀⠉⠀⠤⠻⡄⠀⠀⠀⠀⢀⡠⢚⣽⢀⠆⣷⠐⡁⣾⠆⠠⠍⢢⡤⠞⠙⠀⠁⠐⠉⢲⠤⠞⠉⠋⣦⠗⠉⠑⠲⠰⠖⠁⠈⠑⡄
⣰⠋⠀⠉⣻⢚⠡⠐⡘⠁⠔⠀⠻⡄⠀⠀⣰⢋⠔⠉⢸⠈⢃⠋⠔⠁⣿⠀⠀⠀⡿⠀⠒⠀⠀⠀⢀⠸⠁⠀⢀⠙⢛⠗⠒⠂⠤⠤⢤⣇⣀⣄⠀⢧⡀
⢇⠀⢀⣾⡡⠉⠀⢀⣀⣰⠈⠀⡐⠹⣄⣠⣜⠢⣁⠀⠘⡄⠈⠃⠀⠀⢸⢀⣔⠀⠀⡀⠀⠰⡄⠀⠛⣄⠙⢋⡋⢢⠈⠳⢀⡀⠠⢤⣄⣀⣀⣙⢚⣃⡇
⠀⠁⢻⠁⠁⡐⠀⡠⠀⠗⠐⠊⢴⢲⢉⡤⠤⢍⡹⢷⢼⡇⠀⠀⠀⠀⣞⣽⠛⠊⠀⠀⠀⣀⠇⠀⠀⠈⠑⢄⠙⢿⢱⡀⠈⠋⠉⠁⠀⡏⠉⠙⠏⠉⠁
⠀⠀⢸⠄⠂⠀⢨⢄⡠⠒⠉⡁⠀⢸⡀⢀⣀⡀⠹⢈⢧⡇⣀⠀⠀⢠⡭⠈⠲⠖⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢱⡦⡙⡇⠀⢸⡤⡆⠀⣧⡄⣴
⠀⠀⣼⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠈⣇⠋⠙⢁⣨⠞⢸⠀⠀⠀⠀⠀⡇⠀⠀⢇⣀⣠⣤⣴⣀⣀⣀⠀⠄⠀⢺⠃⢣⡇⠀⢸⠀⠈⠁⠀⠀⢨⡀
⠀⠀⠈⢿⠛⠛⠿⠿⠶⠶⢦⡦⠤⠔⡻⠳⡞⠉⠀⠀⣾⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠉⣤⣤⣤⣤⡤⠴⠶⢺⠁⠀⢠⡇⠀⢸⠀⠀⢀⡀⠀⠘⡇
⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⢸⢀⣜⣥⠌⡁⠈⣖⠀⠀⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⠀⢸⠀⠀⣸⠀⠀⢸⠀⠀⠈⡇⠀⠀⡇
⠀⠀⠀⠸⡄⠀⠘⠀⠀⠀⡆⢺⣙⠢⠑⣦⠾⣉⠇⢀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⡇⠀⢸⠀⠀⡏⠀⠀⢸⠀⠀⠀⡇⠀⠀⣷
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⣐⡉⠛⢲⠌⢾⠉⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢸⠀⢰⠇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⠀⠀⠸⣝⣬⣤⡼⠀⠀⢸⠀⡉⠈⠙⠉⠈⠁⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⢸⠀⡼⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢹
⠀⠀⠀⠀⡜⠀⠀⠀⠀⣸⠀⠀⢠⠉⠀⠀⡇⠀⠀⠸⠤⢄⣀⣠⢀⣠⣤⠟⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⡸⠀⠙⠢⠤⣀⣾⠀⠀⠀⣧⠀⠀⢸
⠀⠀⠀⠀⣥⠤⠴⠤⠤⠿⣄⡀⠀⠀⠘⠀⢣⠀⠀⠀⠀⣼⣉⡥⢸⣉⣽⣄⠀⠀⠀⠀⡀⠼⣄⣀⣀⣿⠴⠓⠀⢄⡀⢀⡇⢸⠀⠀⠀⣿⠀⠀⢸
⠀⠠⢶⣶⣷⣀⣀⣀⣀⣀⣀⣸⡖⢓⡺⣏⣉⡹⢶⣖⣋⣤⠤⠟⠛⠣⠤⠄⢈⣱⠶⢿⣀⡤⠴⠞⠛⠛⠓⠒⠶⣾⡿⠿⣥⣴⠀⠀⠀⣿⣀⣀⡼⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠎⣁⣀⣀⣀⡠⣗⣀⣀⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⢉⡉⠉⠉⠉⠉⠉⠉⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣾⢼⣠⢃⣳⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣤⣤⣤⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣧⣤⣼⣤⣤⡼⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣯⣏⣩⣉⣯⡅⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢋⡉⠉⡙⠃⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣛⡧⠨⠅⢎⣩⠿⠛⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣆⠡⠔⠒⠳⢀⣤⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠈⠁⣿⣿⣿⣿⠁⠀⢀⡀⡀⢀⡀⢠⡉⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⣀⣀⣀⣀⠿⠤⠨⣿⣿⡿⠧⠜⢒⣓⡱⣦⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⣀⣀⣀⠠⠄⠒⠉⢙⡄⠀⠔⠁⠘⡎⠀⢰⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣀⠤⠄⣘⠀⠀⡀⠀⡰⠣⠤⢚⠆⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣼⠀⠒⠖⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠄⠊⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚⢅⠀⠀⠀⠀⠠⡀⠀⡠⠃⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠉⠦⣶⣤⠀⠀⠁⠀⠒⠒⢺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⡴⣶⡼⢲⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠰⠁⢈⢱⠀⠙⣥⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠤⣶⢶⡸⠃⠀⠀⠈⢆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⡜⠃⢸⠀⠒⠖⠆⠉⡍⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠈⠁⠀⢐⣠⠀⡐⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⣀⠀⠀⠙⡛⠁⣀⣅⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠺⡿⡦⠀⡈⠀⠐⡏⠡⢴⡶⠓⠤⣄⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠤⠶⠤⠤⠤⠄⠘⠓⠲⠺⠤⠶⠴⠼⠆

⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⢿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢿⣯⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⡿⠟⠋⠀⠀⠀⠀⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣥⡀⠀⠀⠀⠀⢀⣬⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠌⠁⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠄⠀⠀⠠⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⢰⡆⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡈⣹⣀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢥⡍⢹⣡⠋⢠⠟⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠉⠁⢴⡃⠀⠀⠸⡆⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠤⢤⣄⡈⢦⠀⠀⢻⡀⣉⡥⠤⣄⡀⠀⠀⣠⠴⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⢦⠀⠀⠹⡇⠀⠀⠀⠈⠓⢾⣵⢒⠂⠉⠉⠓⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡬⡆⠀⠀⢳⣴⣿⡆⠀⠀⠈⢧⡤⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠙⠧⣄⡀⠀⠀⣠⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣸⣾⠀⠀⠀⠀⠀⠀⠉⠭⠝⢋⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡤⢶⡶⢦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⡤⠤⡆⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⠈⠙⠦⣤⣈⡐⠄⠠⠀⠄⣀⡤⠾⢄⡈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠏⠀⢸⠀⠀⠀⠀⢀⣀⣀⡤⠤⠬⠉⢻⠛⠛⠉⠉⠀⠀⡰⢃⡴
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣄⢤⢸⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡴⣄⠞⣡⠏
⣻⠀⢸⠀⠀⠀⠀⠀⠀⣐⡆⠀⠀⠀⠀⠠⡤⢶⡤⠤⡇⠀⠀⢀⣠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠎⠀⠈⠻⡅
⢸⢰⠟⠇⠀⠀⢀⡤⠒⢒⠲⡄⠀⠀⠀⠀⠈⢻⡻⠒⠓⠤⠤⠍⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⣀⠀⢀⣀⠼
⢸⡟⣠⡀⢀⠤⣼⠶⢮⡽⡸⠁⠀⠀⠀⠀⢠⡤⠭⠽⣖⠢⠤⢄⣀⡀⠀⠈⠉⠉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⡶⠛⢾⡿⠋
⢸⡇⠈⡇⠙⠾⣖⠉⢣⠀⠛⠒⠒⠒⠒⠒⠒⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⠤⢤⣀⣀⠀⠀⠀⠀⠀⡼⠉⠲⡼⢟⣆
⠸⠧⠶⠃⠀⠀⠈⠓⠦⠤⠏⠉⠉⠉⠉⠉⢏⢻⠀⢀⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠲⠞⠁⠀⠀⣷⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⣻⠀⠀⠀⠈⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢤⣴⣏⡀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⠀⠀⢸⢁⡏⠸⡍⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣼⡠⡴⡇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠛⢋⡟⢉⣉⣽⣅⣙⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⣁⣹⠽⠝⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⠛⠛⠓⠣⠤⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣦⣶⣶⣶⣶⣶⣾⣿⡏⣴⠟⢷⡌⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣍⣭⣍⣭⣭⣽⣿⣿⣿⣧⡘⠳⠛⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠻⣿⠟⢛⣿⣿⡿⠟⢋⣼⣿⣿⣿⣿⣿⣿⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠰⣿⣭⣤⣴⣾⠟⢛⢛⢛⣛⠋⣉⣤⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣴⣿⣷⣬⣿⣿⠛⣉⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⢻⡇⢸⣿⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣅⠸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣤⠀⠀⠀⢠⣤⡄⠀⠉⠙⠻⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⣿⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠿⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⢆⡜⠱⠔⠶⠀⠀⠐⣿⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠐⠉⠀⠒⠀⣠⠞⠁⡴⡒⠢⠀⠀⠀⠇⣠⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠀⠐⣫⣀⠀⡦
⠀⠀⠀⠀⡠⠛⠲⠎⠁⠀⠀⠣⢀⠔⠀⠀⠀⠀⠁⠀⣠⠀⠀⠀⡴⠊⠑⡤⠀⡀⢀⡀⠀⠀⠀⠀⡠⢀⣟⣊⡁
⠀⢧⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠲⠛⠀⠀⢸⠇⠈⢰⠇⠈⠀⡼⢸⢃⡎⠀⣴⣷⣯⠛⠁
⠀⠀⠁⠀⠀⢤⡄⢀⠀⢼⠀⡖⠋⠸⡆⠳⠀⠀⠀⠾⡠⠀⠀⠘⢦⠀⣾⢰⡄⢀⠧⣿⢼⡀⠀⠙⠛⠁⠀⠀⠀⠋⠐⠆
⠀⠘⢶⡄⠋⠁⡇⠈⢣⢸⡄⠒⠃⢢⣇⡐⠒⠖⠃⠀⠀⠀⠀⡀⣀⠀⠀⠉⠉⠙⠚⠁⠦⠀⢠⡖⢙⠒⠦⠀⠀⠈⠉
⠀⠀⠀⡇⠀⠀⢹⠐⣈⠔⠧⠖⠚⠁⠈⠀⣠⣤⣄⠀⠀⠀⠘⠆⠤⠃⠀⠀⣠⢦⣄⣠⠦⡄⢾⡀⠄⠁⡸⠀⠀⠀⢀⡄⡀
⠀⠀⣀⣃⠤⠞⠘⠋⠁⠀⠀⣀⣀⡀⠀⠀⠛⠟⠛⣠⢶⣶⣦⡀⠀⠀⢀⣤⠯⢄⣠⣄⠀⠑⢤⡑⠒⠚⠀⠀⠠⢊⡅⢠⠟
⠀⠀⠉⠀⠀⠀⠀⠀⢀⠔⠈⠁⠈⢻⡿⠦⡀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠘⢇⠀⢸⡀⢀⡇⠀⡼⠇⠀⠀⡰⢾⠶⣫⠖⠁
⠀⠀⣰⡆⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⠛⠛⠛⠁⣀⡀⠀⠈⣗⠀⢉⣉⠀⢠⠁⠀⠀⠀⠛⠾⠋⠁
⠀⠀⠈⠿⣷⡄⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢰⡟⠉⠛⣦⠀⠈⠓⠁⢈⡙⡁⠀⠀⠀⠆⠀⠀⠰
⠀⠀⠀⠀⠉⠀⠀⠀⠈⢣⡀⠀⡆⠀⠀⡜⠁⠀⠀⠀⠀⠀⠈⠷⣤⣄⣼⠀⠀⠀⠀⠀⠀⠰⢐⡖⠂⠰⠐⡔⣦⣼⡃
⠀⢠⢤⡔⣖⡤⢠⣤⠄⠀⢧⠀⠳⢀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⣀⣀⡀⠀⠈⠀⠉⠈⢀⠈⠀⠁
⠀⠘⣎⢱⡇⡀⠨⡥⠀⠀⣾⡘⢸⣾⠁⠀⢀⣾⡿⠿⢿⣶⡄⠀⠀⣰⠿⠛⠛⠻⠿⠆⠀⠀⠀⢸⣦⣴⣴⣶⣷⣷⣶⣿⡦
⠀⢀⢘⠈⣇⠃⢀⠈⠃⠀⢹⣯⣿⣿⠃⠀⢸⣿⠀⠀⠀⢹⣿⡀⠀⢿⣤⣀⣀⡀⣸⡟⠀⠀⣀⣈⡉⢉⣡⣀⣀
⠀⠈⠁⠀⠉⠁⠈⠉⠀⠀⠀⠉⠉⠁⠀⠀⠈⠻⣷⣄⡀⠀⣿⡇⠀⠀⠙⠛⠛⣿⡿⢀⣴⠀⢇⢈⡩⠿⠁⣁⠘⠢⡀
⠀⢀⡰⢂⣻⠃⠀⠀⢀⣀⠀⠀⠀⡖⠀⠀⠀⢀⠈⠛⠿⢷⢿⠿⠗⠂⣀⡀⠀⠙⠀⣨⠐⠠⣨⠀⠀⣤⠲⠀⠀⠐⢙⡆
⠀⠹⠟⠉⢡⠄⣤⠤⠿⢨⠀⠀⠸⡧⠈⠒⡆⠀⠀⠀⠀⠀⠀⠀⢠⡴⠃⠐⠦⠀⢰⣇⡀⢀⢸⠁⣸⢡⡈⣨⠃⣨⡜⢰⡄
⠀⠀⠠⠠⠌⠀⠀⠀⠐⠀⠀⠀⠀⢳⡀⢰⡃⠀⠀⠀⡀⣠⢤⠀⠀⡧⠖⠀⠲⠀⠀⣄⠄⠈⢺⣤⢀⢫⠇⡀⠘⠀⠀⣜⠁
⠀⠀⢸⡀⠀⡆⣆⣆⢠⡀⠀⠀⠀⣎⠀⠠⢥⠀⠀⢈⠎⠁⠐⠖⠀⠁⠰⠀⠀⠀⠀⢹⡀⠐⠉⡀⠒⣻⡞⠁⠟⠀⡄⢸
⠀⠀⠨⡿⠠⡿⠽⣿⢼⣧⠀⠀⠀⠈⠉⠁⡾⠀⠀⠈⠐⢋⡠⠤⠀⠀⠀⠀⠀⠀⠀⠈⠛⢏⡙⣍⠪⠇⠱⣇⣠⣊⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠛⠓⠒⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠷⣦
⣠⡾⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⢀⣤⣤⡿⠁
⠛⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣤⡛⠁
⠀⠀⠀⢛⣀⠀⠀⠀⣀⡴⠚⠋⠉⠉⠀⠉⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠐⣦⠀⠀⠀⠀⠀⢻⡀⠀⠉⠁
⠀⠀⠀⠈⠉⠀⣠⠞⠁⢳⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠌⠛⠦⣄⣀⡀⠈⣇
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠙⢦⣀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠈⠁⠀⣿⣀
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣈⣳⣄⠀⠀⠀⠀⢻⠀⠀⠀⠀⡞⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⠙⠲⠦⠤⣼⡆⠀⠀⠀⠹⣄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⠜
⠀⠀⠀⣰⢺⡄⠀⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠓⡄⠀⠀⠀⠈⠉⠱⣄⠀⠀⠀⠀⠀⠰⣏⣉⡷⠀⠀⠀⠀⠀⢀⠞⠉⠁
⠀⠀⠀⢹⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⠇⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠁
⠀⠀⠀⠀⠉⠚⢳⡀⠀⠀⠀⠀⠀⠒⠋⠙⠂⠀⠀⠀⠀⣠⠞⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⣿⠻⣶⡶⠦⠤⢶⣶⠾⢻⡏⢹⣶⣄⠀⠀⠀⠀⠀⠀⣀⢀⠞⠹
⠀⠀⠀⠀⠀⢠⠏⠙⠦⣄⣀⣀⡀⢀⣠⠴⣶⠄⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠺⣧⣤⡼⡆⠀⠉⣳⣶⡉⠀⢀⡞⣤⣴⠿⠈⢷⠀⠀⠀⢀⡼⠽⠋⠀⢀
⠀⠀⠀⠀⠀⢾⠀⢀⡴⠚⣹⢆⠉⠋⠀⢺⠓⡿⠉⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣴⠟⠀⢻⡴⠛⣇⣸⡟⠢⣼⠁⠙⣧⡀⠀⠘⣇⠀⢠⡟⠛⢦⣄⡤⠊
⠀⠀⠀⠀⠀⠸⣤⠞⢀⠜⠁⠀⠳⣄⣤⡞⠋⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠻⣦⠀⠈⣷⠀⡏⢹⡇⣰⠇⠀⣠⡾⠁⠀⠀⢻⢀⡾⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⢀⡏⠐⠋⠀⠀⢀⡼⢹⣿⡇⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢰⡇⠀⣀⠀⠀⠙⢷⡄⠘⣿⡇⠘⣷⡟⢀⣼⠏⠀⠀⠀⣷⢸⣿⠁⠀⠀⢀⡟
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣠⠏⠀⢸⣿⡇⠀⠀⠀⠀⢠⠀⠀⢸⡄⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠻⣦⢹⣇⢠⡟⣠⡿⠁⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢠⡇⠀⠀⠀⡴⠃⠀⠀⢸⣿⡇⠀⠀⠀⠀⣼⠀⠀⢸⠃⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⠀⠙⢷⣿⣾⡱⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠣⣀⣀⡚⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢹⠀⣠⠏⠀⠀⠀⠈⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣄⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠈⡟⠁⠀⠀⠀⠀⠀⠘⣦⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⣀⡀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠘⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡇
⣶⣶⡶⠶⠶⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⢩⣃⡀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⣤⢡⣣⠏⢠⠟⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠀⠰⢇⠀⠀⠘⡆⠀⣀⡀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⠔⠲⡦⣄⢱⠀⠀⠹⣀⡤⠶⠦⠤⣀⠀⣠⠖⠒⢢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⠀⠀⠹⡄⠀⡀⠀⠈⠙⢷⠏⠍⠀⠀⠈⠲⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⡇⠀⠀⠹⣾⠿⠀⠀⠀⠘⢲⣈⣁⠀⠀⠠⠼⠤⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⠀⠀⠀⠈⠓⠦⠤⣄⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣒⣒⠚⠿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⣖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠙⠋⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡴⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⡇⠀⠀⠀⢀⡴⠊⣷⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢥⠬⠤⠥⠖⠋⠉⢱⠄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢾⠀⠀⣇⢀⣀⠤⠖⠒⠊⠉⠉⠉⠀⠘⡄⠀⠀⣀⠀⡴⢁⠞⠁
⣿⠓⢲⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡇⠀⠀⠀⢈⡣⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢀⡜⠉⠙⢴⠃
⢹⢠⠾⡄⠀⠀⠀⡀⠬⠵⡀⠀⠀⠀⠀⠑⢾⣧⠤⠧⣀⢀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⢈⡧
⢾⣾⢁⡀⢀⣀⣸⣤⣖⡇⡞⠀⠀⠀⠀⢀⣈⣓⣒⡤⢄⣀⡀⠈⠉⠁⠒⠒⠒⠒⠒⠒⠠⠤⠤⠤⠤⠤⣤⠗⣽⣶⠚⠉
⢸⡅⢹⡃⠫⢾⡕⠲⡌⠘⠢⠤⠤⠤⠤⠼⢧⠀⠀⠀⠀⠀⠈⠉⠁⠒⠢⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⣰⠓⢦⣸⡷⡄
⠸⡧⠴⠃⠀⠀⠑⠦⢄⡸⠉⠉⠉⠉⠉⢲⢻⠀⠀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠤⠴⠁⠀⠀⣇⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢒⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⣿⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢤⣴⣇⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠁⠀⠀⢸⢀⠏⢸⡙⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣇⣼⡤⡾⡇⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⣉⣏⣀⣤⡼⡤⠬⡋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣄⠀⢲⢀⣠⡶⢶⣿⠣⣽⣆⣮⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢰⠿⣄⡰⢣⢞⣽⣿⣿⣴⣿⣿⣡⡴⣣⠶⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢔⠫⣹⣮⡿⣷⣿⣿⣿⣤⣷⣿⣿⣿⣿⢟⣱⣦⡴⢤⣴⢟⣾⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣨⣿⣿⣻⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⢷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢨⡙⢮⣻⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⡵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢱⣼⡼⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣯⣶⡟⢁⡔⣠⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠋⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣭⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⠶⣾⣿⣿⣿⣿⡿⠻⢡⠛⠙⠟⢿⣿⣿⣿⣽⠿⠟⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⡴⠛⢻⣿⡽⣯⠁⠀⠀⠜⡀⠈⠈⣿⡟⣿⢿⣻⣦⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢒⡀⢹⣟⡇⠉⠀⠀⠀⢠⡀⠀⠀⠘⠃⡛⠒⠈⠙⠓⠵⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠭⠄⠱⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢤⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠂⠄⠀⡴⠂⠉⠀⠀⢣⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⡆⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠊⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⣄⣴⣿⣷⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⣣⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣦⣤⣄⠀⠀⠀⠀⢠⣮⣙⠿⢟⣩⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣫⡻⢿⣿⣿⣷⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⣩⠶⠶⢦⡙⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣫⣵⣾⣿⣿⣶⣭⣛⢻⡇⠀⢸⣿⡿⠿⢿⠿⠿⣿⣇⢛⣼⡿⢪⡷⣸⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣡⣿⣟⣻⣿⣿⣟⣿⣿⣈⡁⠀⠘⣿⡸⣿⣷⣿⡿⣸⣿⣿⡇⡾⢛⣵⡿⣫⣝⠛⣩⡙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⠻⠋⠀⠀⠹⣷⣮⣛⣫⣶⣿⣿⣿⡇⠇⣿⣿⣧⡹⢿⣿⡿⢋⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣭⣭⣿⣿⡟⣼⡇⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⢋⣍⢻⣿⣿⣿⣶⣥⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣮⣛⠻⠿⠿⢟⣫⣾⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣮⣭⣾⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⣿⡇⠀⠀⠀⢠⣶⣶⡄⠀⠀⠙⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⢰⣶⣾⣿⣿⣶⣶⢄⣶⣤⡀⠀⠀⠈⠻⠿⠃⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣭⣛⣛⣛⣛⣥⣾⣿⣿⣿⡆⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⡿⠛⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⠀⣠⣾⣿⣿⡿⠟⠋⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣦⡀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⣿⣿⣿⠀⢠⣄⠀⠙⢿⣿⣿⣆⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣷⠀⠀⢻⣿⣿⣆⠀⢻⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢿⣿⣿⡀⠘⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⠇⠀⣿⣿
⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣧⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡀⠀⢀⣿⣿⡿⠀⢰⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢻⣿⣿⣦⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠟⠁⢠⣾⣿⣿⠃⠀⠾⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠹⣿⣿⣷⣤⡀⠉⠉⠉⠉⠉⠉⠀⠈⠀⣠⣴⣿⣿⣿⣿⣿⣦⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠚⠛⠋⠉⠉⠉⠉⠛⠛⠛⠲⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⣼⠁⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠋⠀⠀⠀⠀⠈⣧
⠀⣸⠃⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠴⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣼⠁⠀⠀⠀⠀⢠⡶⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⠀⠀⠟⠀⠀⠻⠀⠀⠀⢠⡶⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⠀⠀⠀⠈
⡿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⡇
⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⣀
⢹⡀⠀⠀⠀⠀⠀⢰⡏⠀⠙⡇⠀⠀⠀⠀⠀⠀⠴⠞⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⣇⠀⠀⠀⠀⠀⡾⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢹⡄⠀⠀⠀⣸⠃⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⢻⡄⠀⣰⠏⠀⠀⠀⣟⣀⣠⣤⡤⠴⠶⠖⠛⠛⠉⠉⠁⠀⠀⠀⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⢻⡴⠋⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⣤⣤⡴⠶⠶⠒⠛⠛⠲⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠖⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣿⠁⠀⠔⠂⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡎⣹⣀⠌⣿⣻⣿⡄⠀⣰⠆⣤⠔⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡰⢹⣧⣼⣿⣿⣿⣯⡷⢋⣼⠒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣷⣾⣿⣿⣿⣿⣏⢮⡏⢶⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⠃⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⠑⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣉⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣭⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⣇⣀⠀⢀⣴⣿⣿⣿⣷⣄⠀⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣤⠀⠀⠀⠀⠀⢀⣰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣴⣿⣿⣿⣿⣿⣿⣿⣷⣌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣷⠀⠀⢙⡟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⢩⣿⣿⢿⡿⣿⢿⣿⣮⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠰⢻⡟⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣛⣛⣚⣛⣛⣛⣛⣛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣧⣤⡄⠀⠀⠀⠀⠀⢸⣿⣿⠸⣿⣿⣼⣧⣿⣽⣿⡿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠭⣭⣭⣭⡭⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⢰⣶⣶⢳⡞⣶⢳⣶⣶⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣠⡄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢰⣿⣿⡇⣿⢟⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⣧⡰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⢶⣶⠀⠀⠸⠁⠀⣘⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢘⢿⣿⣿⣿⣮⡛⠃⠀⠀⠀⠀⠀⠀⣾⢫⣭⣽⣿⡇⠿⠦⠤⠤⢤⠄⣭⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣷⣭⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢸⣿⡿⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠞⠛⠛⠛⠛⠛⠛⠳⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢿⡄⠀⠀⠀⢀⣾⢻⡆⠀⠀⠋⠉⠁⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⢸⡇⠀⠀⠀⣼⠃⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠘⣷⠀⠀⣸⠏⠀⢸⣏⣀⣤⣤⡶⠶⠞⠛⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠹⣇⣼⠋⠀⠀⠙⠉⠉⠁⠀⠀⣀⣤⡴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⣿⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠶⠶⠶⠶⠛⠛⠋⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠒⠶⣒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡟⠈⠀⢽⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣾⣿⠉⠉⠈⣹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡝⣿⣿⠐⠚⣳⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣿⡟⠀⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠄⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡀⣸⣿⡇⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡯⠁⠀⢀⡠⠐⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣼⣋⣉⣿⣿⢱⣶⣿⣟⣿⠿⠟⠿⢿⠛⢿⣿⣼⣃⠉⠒⠉⠀⠀⢀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣤⣻⣿⣿⡋⠀⡀⢀⠀⠀⠄⠜⠋⠁⠀⠀⣀⣤⡾⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⠓⠴⠐⠴⠆⠲⣚⠉⠀⠪⠥⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠛⣈⣉⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠋⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠭⠇⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠗⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠆⠀⠾⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠲⢾⠂⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠻⣿⣿⣿⣯⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢀⣿⣿⣿⣿⣿⠿⠛⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣫⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿
⣿⣿⠀⠀⠀⠀⠀⠳⠢⠄⠤⣀⠠⠤⠤⠤⠴⠾⠿⠿⠟⠁⠀⠛⠋⠙⠃⠖⠋⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠠⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣶⠞⠳⢶⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⢾⡇⠀⠀⠀⡿⠶⠀⠠⠶⠶⠶⠶⠶⠶⠶⠆⠀⣴⠛⢁⣀⣀⣀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠿⢦⡴⠾⢧⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠆⢸⠇⠀⠈⠁⠀⠉⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠶⠶⠆⠸⡇⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⠀⠹⣦⣀⡀⠀⣀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠉⠉⠉⠉⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡞⠉⠀⠀⠈⢳⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠟⠻⠛⠟⠿⠻⠟⠷⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⡴⠟⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠹⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣠⣤⣤⣤⣤⣤⣀⣄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⡖⣦⣰⡆⣟⢦⡄⣲⠆⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠉⠁⠁⠘⠁⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⡋⠁⠁⠠⠈⠉⠭⠑⠒⡛⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⣸⡃⠀⠀⠀⣌⢀⡤⣌⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⠃⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠰⠀⠀⠸⣇⢹⠀⠀⠀⠈⠁⠀⣀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣧⠼⠀⠀⠀⢜⣩⡤⠤⠔⠊
⠀⠀⠀⣠⠴⣒⠲⢄⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠠⠴⠃⠀⠀⠀⢸⠇⠀⠀⠀⠀⣬⡅
⠀⠀⠀⣇⣸⠉⣹⢀⠆⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⡼⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠈⢀⣞⡵⠋⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠁⣐⡒⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⢡⡦⠀⠀⠀⢀⣀⠤⠔⠒⠚⡳⡦⠀⣬⠀⢀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠇⢇⣀⠬⠀⠸⢸⠧⣀
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡄⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠈⢸⠀⠈⠙⠦⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⢦
⠀⠀⠀⠀⢀⣏⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡝⠀⠀⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⣇⢠⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢠⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢹⡌⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢸⠀⠀⠀⠻⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⠉⠉⠀⢘⣗⡿
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠉⠉⠙⠓⠲⠤⢄⣀⣀⣧⠀⠀⠀⠀⢧⠠⢤⣸⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⠘⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠉⠛⢳⡖⠒⠒⠒⠲⠶⠶⠤⠤⠤⠤⢿⡄⠀⠀⠀⢸⡥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠘⠂⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠛⠛⠳⢦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣫⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⡖⢀⣴⠿⣆⠀⣤⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣏⡤⠚⠁⠀⠀⢀⣠⠞⠀⣠⡾⣿⢀⣴⣫⡶⠛⠁⠀⢹⣆⢻⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢺⡏⠀⠀⠀⣠⣴⣋⣤⠶⠛⠁⠰⡿⠟⠋⠁⠀⠀⠀⠀⠀⠻⣶⡄⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⢤⡶⠿⠟⣛⣭⣦⣄⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⣶⡄⠀⢿⡷⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⢀⣼⣿⣷⡀⠀⣸⠇⠀⠀⠀⠀⣤⣶⣄⠀⠀⢸⡇⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⣿⡿⠁⢀⡿⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⢸⡇⢸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⢸⣧⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢰⡟⠉⣿⠀⡟⠉⣷⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⢷⣴⠟⠈⣷⠀⢹⡇⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣦⣤⣤⣤⣤⣤⣼⣷⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⢀⣾⠋⠈⠛⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠞⠛⣿⠟⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⢻⡗⠶⣤⣀⡀
⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠉⠙⢷⡄
⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠹⢧⣄⡀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠈⢻⡄
⠀⠀⠀⣼⠇⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠳⣄⠙⢷⣤⣀⠀⢀⣩⡿⠳⣦⡀⠀⠀⠀⠙⣷⡀⠀⠀⠀⢻⡆
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠙⠛⠛⠉⣤⠶⠈⢷⣄⠀⠀⠀⠈⢿⣦⠀⠀⠀⢻⡄
⠀⣸⡟⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠙⢷⣄⠀⠈⢿⡄
⢠⡟⠀⠀⠀⠀⠈⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠻⣦⡀⠈⢿⡄
⣿⡁⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠘⢷⡄⠈⣿⡀
⠈⠻⣶⣤⣄⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠈⠻⣦⠸⣇
⠀⢠⣿⠈⠉⠙⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠙⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠈⢿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠈
⠀⢸⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⣷⡀⠀⠀⠀⠀⢀
⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⠤⠶⠖⠚⠛⠛⠛⠛⠲⠶⠶⠤⠤⠄⠀⠀⠀⠀⠀⣿⡇⠘⢿⣦⣄⣀⣀⣼
⠀⢾⣇⡀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠙⠻⠷⢶⣶⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠉⠐⢢⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⢔⠑⢆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⡔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠠⢄
⠀⠀⠐⠀⡀⢀⠀⠄⠀⠀⠀⠈⠂⢩⠒⠉⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠂⢄⠀⡇
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠦⡐⣃⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠤⢚⡤⠞⣠⠇
⠀⠀⠀⠀⢠⠋⡢⠀⠀⠀⠀⠀⠀⠀⢨⣀⣨⠇⠀⣀⠑⠀⠀⠀⠀⠀⠀⠈⠐⢀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣄⠠⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠠⠄⠂⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡠⠄⠀⠀⠠⠁⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⣀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⠀⡐⢢⠈⠄⠀⠀⠀⠀⠐⢂⠀⠀⢀⠔⡀⠌⠂⠀⠀⠀⠈⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⢆⢁⠊⠀⠀⠀⠀⠀⠀⠈⠓⡀⠈⠢⠥⠀⠀⣠⡤⠄⠠⠄⠄⠁⠀⠀⠀⢀⠀⠀⡂⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠒⠊⠀⠘⠍⠧⡄⠀⠀⠀⠀⢀⡠⢠⡀⡓⠊⠀⠀⠀⠀⠀⠬⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠠⠀⠤⠀⠀⠀⠀⠀⠀⡐⠀⠀⢆⠀⠀⠀⠀⠈⠀⠱⣠⠉⢒⠁⠀⠀⠀⠀⠁⠤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠂⢀⠄⠀⠀⠀⠀⠊⠐⠲⠚⠃⠀⠀⠀⠀⠀⠀⡨⠃⠉⠀⠀⠀⠀⠀⠐⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣠⠈⠀⠀⠀⠀⠀⠀⠐⢀⠀⠠⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⡠⠀⠂⠄⠀⠀⠀⠀⠀⠀⢐⢀⡆⠀⠀⠀⠄⠀⠀⠰⠐⡤⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⢄⡀
⠀⠀⠀⠀⠀⢀⡐⢑⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡆⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢯⣰⠖⡆⢹
⠀⠀⠀⠀⠀⠀⠨⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠄⠒⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⢉⠕⠋
⠀⠀⠀⠀⠀⠀⠠⡉⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡆⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠖⠉⠡⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⢢⢈⠀⠀⠀⠀⠀⠀⣀⡂⠒⠂⢤⡠⠐⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢀⠊⠀⠀⠀⢠⠂⠁⠀⠀⠙⠄⢸⠇⠁⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡆⠀⠀⠀⢧⢀⣀⣀⡀⠀⠈⣩⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠉⠐⠀⠻⠟⠩⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⡿⠁
⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣐⣄⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠛⠛⠛⠛⣿⡏
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣂⣤⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠛⠛⢻⣿⣿⣿⣿⣿⣿⠇⠀⣀⠔⠁⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢛⠋⡀⢀⣬⣦⡀⠀⠀⣟⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⠀⠒⠂⠈⠀⠀⣷⡿⠋⠄⠀⠀⠀⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠁⠠⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠉⠁⠀⠀⠀⠀⠐⠈⠀⠀⢈⡄⠀⣀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣷⠀⢀⠀⣶⣦⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣜⣹⣿⢻⡇⠀⠀⠀⠀⣿⡿⠛⠻⣿⠀⠀⠀⠈⠁⠀⣦⣦⣾⡇
⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣤⣼⡇
⠀⢀⣤⣤⣤⣤⣤⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⣁⣀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⠄⠐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣀
⠀⠘⠿⠛⠛⠛⠃⠀⠀⠀⣿⣿⣿⣿⣿⣟⠁⠀⠠⠇⢲⣾⢿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⣀⣙⣛⣛⣛⢸⣿⡇⠀⠀⠀⣿⣿⣿⡇⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⠸⠿⠃⠀⠀⠀⣿⣿⣿⣧⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⠇⣿⣿⣿⡇⢀⣿⣿⠋⠉⠙⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣿⣿⣿⠴⣿⣿⣿⡿⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⠤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡃⠀⢹⣿⣿⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣦⠀⣿⣿⣷⣶⣤

⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⠀⠀⣠⣤⡤
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⣧⣾⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣽⡿⠋⠙⢿⣿⣉⣉⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⠟⠁⠀⠀⠀⠙⣷⣬⡿⢿⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⢴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠟⣵⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢻⣿⣿⡏⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⡌⢿⣿⢡⣿⣿⣿⣿
⣿⣿⡇⣿⣿⣿⣿⢹⢸⣿⣿⢸⣿⣿⣿⢩⠁⡀⡀⠈⠁⢀⡀⠈⠁⢀⢀⠈⡍⣿⣿⣿⡇⣿⣿⣷⡀⠿⢸⣿⣿⢸⣿
⠉⠉⠁⠉⠉⠉⠉⢈⣈⠉⠉⠈⡏⠉⠙⣩⠍⠉⠉⠁⠉⠩⠍⠉⠈⠉⠉⠩⣍⠋⠉⢹⠁⠉⠉⠉⠀⠀⠀⠈⠉⠈⠉
⣄⣀⣄⣤⣀⣀⣀⣀⣀⣠⣄⣀⣤⣀⣀⣠⣄⣀⣀⣀⣀⣠⣄⣄⣀⣀⣀⣀⣄⣄⣀⣀⣄⣀⣤⣀⣀⣀⣀⣀⣤⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠢⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠻⠿⢿⣿⠿⣿⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⢙⡁⣤⣶⣾⣾⣷⣶⣬⡄⠀⠀⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠈⢉⡽⠟⠛⠉⠁

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⣶⡄⣠⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣼⣿⣷⣿⡿⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⣀⠀⢀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⣿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠉⠉⠙⠛⠛⠛⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡇⠀⢨⣿⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡾⠿⡿⠛⠁⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣾⣷⣦⣀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⣿⣿⣄⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣶⣦⡌⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣾⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣶⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣧⣀⣠⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡰⠾⢿⣿⣷⣿⣿⣷⣾⣷⣶⣤⣤⣦⣄⠀⣀⢠⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⣤⣀⣀⣀⣀⣄⣀⡀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣄⠙⠻⣦⡳⣝⠄⠀⢄⠳⣽⣶⡐⢮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠉⢽⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣾⡛⢦⢀⠘⢿⣮⠂⡰⠀⠱⣼⡝⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣟⡿⣦⡁⠳⣄⠻⣿⡌⢦⡀⠱⠻⣶⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣽⣿⣿⣿⣿⣿⣶⣝⢿⡿⣷⢾⠟⣦⠘⠢⡘⢷⡜⢿⡦⡘⠙⢿⣎⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠣⣹⡿⣿⣿⣿⣿⣿⣧⡙⢬⣷⣵⣌⠳⣄⠈⠂⠻⣆⠹⣄⠀⠬⢻⣷⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣽⣿⣿⣿⣿⣿⣿⣧⣙⢿⣯⠳⡘⢧⡀⠁⠘⢦⡈⢷⡀⠑⢿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⡄⡀⠀⠀⢀⠀⠰⣿⣦⡭⡿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣜⠺⣿⣆⠀⠈⠻⡄⠳⠀⠀⠉⣹⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠠⡀⠛⢷⡄⠀⠙⣆⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠐⢶⣶⣶⣶⣶⣿⣿⣿⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⡄⠙⠷⡤⠈⢳⡀⠀⠀⠀⠙⡛⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡂⠈⠀⡀⠙⢦⡀⠀⡀⠈⠝⣧⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠻⣿⣿⣯⠽⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠈⠠⡈⠙⢆⠀⠀⠀⢻⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⣿⣧⡙⣿⣛⣿⣷⣿⣋⠻⠹⣿⡟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠂⠀⠀⠀⠀⠀⠹⣟⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣿⣿⣧⠈⢧⡙⠻⣿⣿⣦⠀⢼⣧⠘⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡐⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠑⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⢆⠀⠐⡀⠀⠀⠀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣸⣦⡀⠀⠀⠀⠀⠀⠀⠐⣤⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⠁⡀⢾⡀⠀⠀⠈⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢠⣸⣶⣦⣶⣾⣿⣿⣷⣦⡄⠀⢠⡀⣤⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠐⠌⣷⡀⠀⢧⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⢟⡿⣛⣛⣻⣻⢿⣿⣧⠹⠀⣦⢿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣣⠈⢸⣷⡀⠈⢄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠉⢩⡏⠉⡏⠈⠙⠛⠃⠀⠀⠀⠡⠿⣉⣷⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠠⡈⢷⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⣰⣿⣇⣚⣷⣶⢦⣤⠀⠀⠀⠀⠀⠀⢘⢻⡿⣾⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣆⠐⠘⣧⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣿⣿⣿⣿⣿⣿⠿⢿⣿⡷⣤⣶⣶⣤⣬⣯⡾⢿⣿⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠃⠹⣆⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣶⣿⣿⢿⣿⣼⣷⣾⣯⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡄⠀⢻⠀⠈⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠟⠻⣿⣄⣻⣻⣿⣯⣽⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢣⠀⠈⣇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣻⣾⣿⣥⣼⢿⣭⣿⣿⣿⣿⣿⣿⣿⣌⢿⣿⣿⣿⣿⣿⣿⣿⠘⣇⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⣿⣿⣿⣿⣿⣆⢿⡑⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⡜⣧⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⠈⠻⠿⣿⣿⣿⣿⣷⠸⣆⠀⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠉⠙⠙⠋⠋⠉⠀⠉⠉⠉⠉⠻⠿⠟⠛⠿⠿⠻⠿⠟⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠀⠈⠨⢷⠼⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣾⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢾⣿⣿⣿⣿⣽⣻⡟⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠷⣿⣿⣿⣿⣿⣿⣿⣷⠺⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠏⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃⠀⠀⠀⠻⣿⣿⣯⡿⠋⠀⠀⠀⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣤⣄⣀⡀⣰⣿⣿⣆⢀⣀⣤⣤⣾⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣯⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠁




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

⠀⠀⠀⠀⢀⣤⣶⣶⣷⣶⣶⣄⠀⠀⠀⠀⠀⢀⣤⢶⣶⣾⢶⣶⣄⡀⠀⠀⠀⠀⠀⣠⣴⣶⡾⠒⢶⣤⡀
⠀⠀⠀⣰⣿⡏⣸⣿⣿⡮⣿⣿⣧⠀⠀⠀⢰⣿⣿⣟⡿⣿⣿⣯⣿⣷⠀⠀⠀⢠⣾⡟⠋⢌⣤⣷⣾⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⠟⡃⠀⠀⠀⣿⣿⣿⣿⣧⣿⠿⠿⠟⠻⠀⠀⠀⣸⣿⡶⣒⣷⣿⠿⠿⠟⢻
⠀⠀⠀⣿⣿⡟⠛⠿⠁⠀⢠⠀⢡⠀⠀⠀⢻⣿⣿⠛⠻⠃⠀⢠⠄⠰⡀⠀⠀⢸⣿⣿⠛⠻⠏⠀⠀⡄⠘⡄
⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⡐⠃
⠀⠀⠀⠀⠀⡏⠀⠀⡤⠤⠤⠜⠀⠀⠀⠀⠀⠀⡉⠁⠀⢠⠤⠤⠴⠃⠀⠀⠀⠀⠀⢹⠁⠀⢠⠤⣤⡤⠃
⠀⠀⠀⣀⣴⣇⣀⣀⣷⣄⡀⠀⠀⠀⠀⠀⣀⣤⣇⣀⣀⣸⣄⡀⠀⠀⠀⠀⠀⢀⣀⣟⣀⣀⣸⣦⣀
⠀⠀⠚⠿⠿⠿⡿⠿⠿⠿⠛⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠟⠂⠀⠀⠀⠘⠻⠿⠿⢿⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣀⡀⠀⠀⠀⠀⣈⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⠀⠉⠉⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣠⠖⠛⠲⣄⠀⢀⡴⠞⠛⢶⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣰⡇⠀⠿⠿⢸⠆⢸⡅⠐⠿⠇⣿⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⡞⠁⠙⠦⣤⡴⠋⠀⠈⠳⢦⣤⠾⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢧⡀⠀⢀⡼⠁⠀⠀⠀⠀⠸⣆⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠉⠛⠉⠀⣀⣀⣀⣀⣀⣀⠈⢛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣤⣤⡤⠶⠚⠋





⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣦⠄⣀⣴⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⡿⢃⣾⡌⢿⣿⣿⣿⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⠛⠛⠻⠿⢿⣥⣿⠛⣿⣦⣿⠿⠿⠛⠛⢻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⡆⢿⡿⢿⣿⣿⣿⠀⣿⣿⣿⣿⢿⣿⢃⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣌⣿⣆⠻⣿⣿⣄⣿⣿⡿⢣⣾⣏⣼⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⢡⣾⣿⣦⠹⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧⡘⢿⣿⠟⣰⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣎⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣌⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣶⣦⢹⣿⣿⣿⣿⣿⣶⣤⣍⣉⣛⣛⣋⣉⣥⣴⣬⣙⠛⠻⠿⠿⠛⠛⠉
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⣛⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠿⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⡄⣤⠤⢠⠶⢠⡄⢦⡤⢠⣤⠠⣤⠄⣤⠦⠀⢠⡄⢠⡄⣤⢠⡄⣤⢠⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢙⡇⣿⣉⢨⣹⢆⡇⢸⡇⣾⠺⡆⣿⠀⣿⣉⠀⣸⢻⠏⡇⣿⢸⠙⣿⢸⣄⡾
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠱⣬⠿⠉⠙⢛⣷⠶⡟⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⡽⠛⠉⠀⠀⠀⠻⡌⠃⠉⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠁⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣸⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣻⡤⠶⠶⣤⣤⣤⣼⠃⠀⠀⠀⡟⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡼⠃⠀⠀⣴⠋⠀⠀⣿⠀⠀⠀⢰⠇⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⠀⠚⠀⢀⡆⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣧⠀⠀⠀⢷⡀⠀⠀⠀⣠⡞⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢻⡄⠀⠀⠈⠛⢶⡶⠚⠉⢳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠈⣧⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢹⡄⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡇⠀⠀⣸⠇⠀⠀⠀⠀⢨⣷⣤⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣠⣼⣏⣀⣠⣤⡴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠁


⡟⠉⣿⣛⣛⣟⣿⣛⣛⣟⡟⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣿⣿⣻⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣀
⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠁⠉⠛⠆⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢿⣻⣿⡂⠀⠀⠀⡰⠋⢁⣠⣤⣄⠈⠻⡄
⠀⠀⠀⠀⡜⠁⠀⡤⠶⠶⣄⠀⠀⢸⠀⠀⠀⠀⠀⢸⢿⡟⠁⠀⢹⣩⣹⠃⠀⠀⠀⠁⠠⡃⠀⠀⠈⠇⠀⠘
⠀⠀⠀⠀⡀⠀⡌⠀⠀⠀⣸⠀⠀⡸⠀⠀⠀⠀⠀⠸⣾⣷⢀⣴⣿⡿⠋⠀⠀⠀⠀⢦⣤⠵⠄⢀⣀⠞⠀⢠
⠀⠀⠀⠀⠙⠚⠁⣤⠖⠚⠁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢡⡻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢉⣁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⡇⢰⠚⠉⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⣿⠇⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⢸⠉⠁⠀⣀⣤⣄⡀
⠀⠰⠊⠉⠑⠄⠀⡇⢸⡀⠀⠀⠀⠀⠘⠉⠙⢧⠀⠀⠀⢹⣿⠦⠀⠀⠀⠀⠯⠈⠙⠄⠀⠸⢬⠆⠀⠸⠅⠀⠀⢙
⠀⠀⠀⠀⣠⠆⠀⠘⣒⣃⠀⠀⠀⠀⢀⡤⢤⡾⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⣀⠄⠀⠰⣋⠀⠀⠀⠀⢀⡤⠎
⠀⠀⠐⠉⠁⠀⠀⠀⣁⣨⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢈⣛⣁⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠘⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠰⠀⠀⠀⣠⠶⠛⠉⠉⠉⠛⠗⢦⡀⠀⠁⠀⣀⣤⣶⣶⣶⣶⣦⣄⠉
⠀⠀⠀⠀⢠⠞⠋⠉⠉⠉⠛⢷⠀⠀⠀⠀⠀⢸⠁⢠⣾⠛⠲⣤⡀⠀⠀⠁⠀⠀⣼⡿⡟⠛⠛⠛⢿⣏⠙⡆
⠀⠀⠀⠀⢫⣀⡴⠚⠛⢆⠀⢈⡁⠀⠀⠀⠀⠘⢤⣠⡜⠀⠀⠀⡇⠀⢀⡖⠀⠀⢼⣟⣿⠀⠀⠀⢸⣿⣴⠃
⠀⠀⣀⣄⡀⠀⠀⠀⢀⠎⣰⠟⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠃⣠⠞⠁⠀⠀⠀⠉⠁⠀⠀⣴⣿⠟⠁⢀⣀⡀
⠀⢨⣀⠀⠙⠀⠀⠀⡞⢸⠁⠀⠀⠀⠉⠉⣻⠀⠀⠀⢀⠴⢋⠔⠊⠁⠀⠀⠊⠙⠢⠀⠀⠀⢸⡟⠁⠀⠀⡁⠀⠈⡆
⠀⠀⢀⡤⠖⠀⠀⠀⢻⡞⠀⠀⠀⠀⢸⠉⠉⠀⠀⠀⠘⢶⠃⠀⠀⠀⠀⠈⢁⠤⠎⠀⠀⠀⣘⡁⠀⠀⠀⠀⠊⠉
⠀⠀⠠⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠸⣌⠇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠛⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣄⡖⣀⣰⣷⡆
⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡷⠲⣾⣧⣤⣤⣤⣤⣤⡤⠤⠶⠶⠶⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⣿⣷⣷⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣷⣤⣿⡟⠛⠙⢿⣿⣿⣿⣿⣻⠇⢀⠸⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠈⡿⢿⣿⠈⣿⠀⠈⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⣿⡁
⠀⠀⠀⠀⠈⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠈⢻⡷⠦⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⡀⠀⠀⠀⡈⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣤⣤⣤⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⢀⠈⠅⢼⠀⣰⠦⠈⠁⠀⢸⣿⣿⣿⣿⣿⣿⠉
⣠⣴⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⡏⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⢩⡇
⣿⣿⡟⡔⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣆⠀⠈⣷⣼⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⠀⠈⡁
⣿⣿⡀⣇⡜⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⢄⣀⣀⣤⣤⣶⡶⣶⣶⣶⣶⣶⣶⣾⣿⣷⡆⢹⠀⠀⠇
⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⠿⠏⠀⠀⠈⢹⡿⠿⠿⠿⠟⣿⡿⠳⠘⠳⠶⠂
⣿⣿⣿⣿⣧⢺⣦⣦⣇⠀⡀⢿⠀⠀⠀⠀⠀⠻⣧⣀⡿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠉⠑⣄⣸⣷⣦⣤⠤⠯⢥⡄⠀⠀⠀⠄
⠛⠛⠛⣿⣿⣾⣿⢧⣿⠀⣷⣘⢧⠄⠀⠀⠀⠀⢈⣿⣿⣿⣿⣷⣦⣀⣦⣀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣁⡀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢿⣿⣿⣿⣷⣿⡆⣷⣧⠘⣄⠀⠀⠀⠀⠘⠛⠉⢉⣿⣿⣿⣿⣿⣿⣶⣆⠀⣠⠂⣠⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣤⣤⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠙⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠿⠻⡿⠛⢻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠛⠛⠛⠿⠛⠛⠉⠉⠉⠉⠙⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡄⢤⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠀⠀⠀⣻⡦⢱⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠠⢢⠄⠀⡀⠀⠀⠀⠀⠀⠈⢧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⢀⠀⠀⠐⣦⣤⣤⣵⠄⡒⠈⠋⠀⠀⠀⠀⠀⡅⢰⠀⢠⠘⠆⠸⠃⠀⠰⠊⣠⠸⡲⠀⡁⠔
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠒⢀⣠⣀⠑⠜⠄⠀⠀⠀⠀⠀⠀⠀⢀⣅⢠⠀⢏⠀⠀⠀⠰⢟⣁⢰⠇⠀⡃⠂
⠀⠀⠠⠂⡀⡀⣷⣿⣂⠀⠀⠆⠂⠈⠀⠀⠀⠀⠀⠀⢀⠙⠻⣭⣀⠆⠀⠀⠈⠀⠀⠔⠀⠀⣿⣿⡶⠡⠈⣀⠀⣵⣿⣾⣷⣆⣤⡁⠠
⠄⠀⠀⠀⠈⠀⡏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣦⢨⠏⢀⡤⡶⠊⠀⠀⠀⡀⠀⠈⢟⠁⠀⣤⡟⠛⢍⠉⠉⠈⠑⠉⠿⠃
⠀⢀⠀⠀⠀⠀⣗⠠⠊⢀⠀⡀⠀⡴⠂⠖⠒⢀⣐⡻⢤⡸⠛⠈⠀⠀⠀⠈⠀⠐⢛⡛⠛⠠⠎⠚⢐⣵⣿⣿⣾⠆⠙⠲⢤⣀
⠠⢁⠈⠀⠔⠀⠈⠦⠙⠁⠀⠀⠄⠀⠀⠀⢙⡎⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⣛⡓⠀⠶⠶⠶⠋⠀⠀⠀⠀⠀⠀⣠⣶⣨⡓⢤⡀
⠀⠐⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠀⡠⠀⠻⣿⣁⠉⠐⡄⠀⠠⠽⠿⠟⠛⠛⠛⠂⠀⠀⠀⢀⠠⠐⡾⠿⠿⠛⠯⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣃⠀⠓⠥⣈⡀⠉⠍⠐⠀⠀⠀⠀⠀⢀⠀⠀⠦⠀⠒⠀⠀⠀⠀⡀⠀⠄⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⠑⠛⠓⠀⠀⠀⢀⣀⠀⠀⠀⡀⠀⠀⠤⠀⠀⠀⠀⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⣤⣴⡆⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⢿⢿⣧⢹⡾⢷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢩⣯⡾⣆⣡⣈⣅⣜⢛⣿⡟⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⢀⣀⣀⣀⡀⠀⠀⣀⡠⢼⡟⠙⠛⠛⠋⠁⠈⣿⡣⣼⠀⠀⠨⠄⠀⠤⠄⠤⠤⣤⠔⠊⠙⣄
⠀⠀⡞⠁⠙⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⠀⠀⡀⠀⠀⠘⢿⣿⣄⣀⣀⡀⠀⠀⢀⠀⠚⠀⠀⠀⠀⠈⠙⡆
⠀⠀⢇⠀⠀⠀⠙⠂⣀⡒⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠁⠀⠀⠀⠀⢈⡟⠰⣛⡫⣭⠒⠢⣀⠂⣡⠄⠀⡀⠀⠀⡠
⠀⠀⠀⠳⣄⠀⡠⠀⡈⡻⣷⣀⣀⡀⠀⢠⣾⣆⢀⣀⡀⡀⠀⠀⠀⣿⠇⢾⣫⣅⣿⣇⠔⠹⣎⠀⠀⡼⢀⢂⠎
⠀⠀⠀⠀⠈⠳⣄⠀⢇⠀⠀⠇⠇⠀⠃⠾⢿⡿⢆⠀⠋⠁⠀⡴⠐⣟⢐⣶⣿⣿⣿⠇⠀⠀⠀⠀⡜⠁⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠙⢮⣦⠀⠀⠸⠀⠀⠁⠈⠀⠀⠑⠂⠀⠀⠠⠒⠁⠉⠉⠉⠉⠈⠀⠀⠀⠀⠈⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠁⠀⠀⠀⠀⠀⠀⠁⡇⡇⠀⠀⠀⢀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⢀⠀⠀⠀⠁⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡈⢀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣰⠁⠎⡄⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠈⠀⠘⠡⢎⡴⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠇⠀⠀⢰⠀⠀⠀⠀⠐⠀⠲⠛⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠰⠀⠠⠤⣤⠤⠤⠄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠒⠋⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣅⠄⢀⣤⡖⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢤⠀⠙⠒⠋⣎⣀⠀⠀⠀⠈⢦⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⢀⠒⠀⠀⠀⠀⠉⢑⡒⢦⠤⣀⠀⠑⢖⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠙⣛⣭⡭⣽⠶⠚⠉⠉⢩⠇⢀⠏⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣋⡵⠤⢄⣀⠈⠍⢙⡣⠀⣤⣀⣴⠋⠀⡜⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢫⣍⠁⠉⠉⠈⠛⠓⠒⠒⠒⡚⠛⠋⢻⠤⠤⠞⠛⠁⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⣠⠴⢺⣅⠤⠛⠒⠢⢤⡤⠤⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠶⠠⢤⡀⠀⢰⠆⠈⣆⡸⠆
⠀⠀⠀⠀⠀⠈⠳⣖⠘⢦⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⡉⣉⡼⠥⠤⠊⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠙⢦⡀⠀⢻⡉⠒⢄⡀⠀⢠⠖⠒⡄⠀⠀⡜⠉⠉⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⢦⡀⢧⠀⠀⠙⠦⣇⠀⣠⠆⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⠢⣧⡀⠀⠀⠙⢯⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠳⡀⠀⠀⠀⠁⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠰⢷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠘⡄⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⠤⠇⠀⠀⢃⢬⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡘⣆⠀⢸⠷⠦⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⢿⣒⣏⢲⠀⢸⠉⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠸⠚⠁⠀⢠⠤⠤⠤⠤⠿⠚⠒⠒⠒⠒⠚⠳⣄⣈⡭⠼⠿⠀⠘⠦⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⠒⠠⠤⠔⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⡤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡇⠹⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣄⠒⠊⢱⡀⢡⡱⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠒⠂⠉⠁⠈⠛⠒⠶⠊⠀⠭⠆

⠀⠤⠂⠉⠉⠉⠐⠢⢄
⡌⠀⠀⠄⠒⠠⢀⠀⠀⠑⡀
⢇⠀⠸⠀⠀⠀⠀⢡⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠌⠀⢀⠃
⠀⠀⠀⠀⡀⠊⠀⠀⡠⠊
⠀⠀⠀⠘⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡙⢄⠀⠀⡔⠄⠀⢀
⠀⠀⠀⢠⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠄⠁⠀⠀⠁⠀⠀⠀⠃⠈⠰⠁⢆⠰⠄
⠀⠀⠀⠈⢄⠀⢀⠂⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠨⠀⠢⠀⠀⠀⠀⠀⠀⠀⡠⡶⢠⠄⠀⠤⢀⠀⠀⠀⢀⡀⠠⢤⡀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠄⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢰⡄⠂⠀⠂⠀⠀⠐⠂⠐⠠⠀⠀⠀⠀⡠⢤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣸⢠⣖⠒⠢⡀⠀⢠⣄⠠⢄⠀⠀⠐⠎⡠⠂⡆⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⢄⡀⡁⢻⠘⠥⠀⠔⠇⠀⠫⣉⣀⠜⠀⠀⠀⠀⠑⠀⠇⣠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠀⢀⠈⠙⢸⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠸⠀⠀⡀⠀⠀⠁⠄⠤⠀⠀⠀⠀⠀⠀⠀⢘⡁⢀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠁⠀⠀⢃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⢠⠁⠐⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠑⠠⡀⣀⠀⠤⠀⠈⠀⠀⠀⠀⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠈⠐⠀⠤⠄⠤⢴⠀⠠⡀⠀⠀⠀⠀⠀⡠⠂⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⡰⠤⡒⠉⠀⠀⠀⢀⠄⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⠀⠀⠀⠀⠀⠐⡠⠚⢠⢀⡘⢆⠀⠠⠂⠁⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠂⠀⠀⠒⠀⠀⢀⠂⠑⢄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠊⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠚⠀⠢⡀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⡀⠀⠀⠇⠀⠀⠀⢈⠀⠈⢢⣤⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡘⠀⠀⠀⠀⠀⠂⠄⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠄⢤⠀⠀⢠⠀⠀⠀⠀⠀⠄⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠟⠋⢹⣷⡀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⡄⠀⠐⠻⠿⠿⠿⢿⣿⣿⣆⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⢠⡈⠉⠉⠉⣿⣿⣿⠀⠀⠰⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠆⡀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⡀⠀⠠⣿⡄⠀⠀⠀⢀⠠⠤⢸⠿⠿⠋⠀⠀⠀⠀⣰⠇⠀⠉⠂⠀⠀⠀⠀⢀⠞⠁⠈⠑
⠀⠀⠀⠀⠀⠔⠉⠈⠑⠄⣿⡇⠀⠀⢀⠃⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠀⠀⠀⢀⠃⠘⠁⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠸⠀⠀⠀⠀⠻⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠸⠀⠀⡀⠡⠀⠀⠀⠀⠀⢸⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠆⣀⠀⠄⠃⠀⠀⠀⠀⠠⢀⡀⠠⠘
⠀⠀⠀⠀⠘⠄⣀⢀⡠⠁⠀⠀⠀⠀⠀⠑⠠⠠⠊

⠀⢀⡤⠒⠚⠉⠉⠙⠒⢤⡀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠒⠒⠤⡀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠉⠓⠦⣄
⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⡇⠀⠰⡦⠀⠀⠀⠸⠇⠀⠀⣷⠀⡎⠀⠀⠀⢠⡀⢠⡄⠀⠀⠀⠀⡇⠀⢰⠃⠠⣤⠄⠀⠀⠀⠞⠂⠀⢳⡀⠀⡏⠀⠀⠀⠀⠀⠀⡀⠠⡄⠀⢳
⡇⠀⠄⠀⠀⠀⠀⠀⢀⡆⠀⡏⠀⡇⠀⣠⠀⠈⠁⠀⠀⠀⠒⡇⠀⡇⠀⢸⠀⢀⣀⣀⣀⣠⣤⡤⣤⡄⠈⡇⠀⣇⠀⢻⠀⣶⠆⠘⠛⢀⡇⠀⢸
⠸⡄⠈⠳⠤⣄⣤⠤⠋⠀⡰⠁⠀⠱⡀⠙⠐⠢⠴⠢⠴⠓⠊⠀⢠⠃⠀⠘⣆⠈⢿⡤⡧⢾⠒⣷⠟⠀⣸⠁⠀⠸⣄⠀⠳⢤⣄⣠⠴⠚⠀⣠⠏
⠀⠈⠲⢤⣀⣀⣀⣀⠤⠚⠁⠀⠀⠀⠈⠢⣀⡀⠀⠀⠀⢀⣀⠔⠁⠀⠀⠀⠈⠳⣄⣈⠉⠉⠉⣀⡠⠞⠁⠀⠀⠀⠈⠳⢤⣀⣀⣀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⢀⠤⠒⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠶⠶⢤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⠒⢦⡀
⢀⡔⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠈⢦⡀
⡼⠀⠀⠀⠤⠀⠰⠆⠀⠀⡀⢹⠀⢠⠏⠀⢠⡀⠀⠀⠠⢦⠀⠀⠘⡆⠀⢰⠁⠀⠀⠰⠀⠐⠀⠀⠀⠀⢹⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⡇⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠀⢸⠀⠀⠉⠁⠀⠀⣀⣀⣀⡀⠀⡇⠀⢾⠀⠀⠀⢀⣀⠤⠤⢎⣇⠀⠀⡇⠀⡇⠀⠰⠷⠀⠀⠀⠰⡶⠀⠀⣿
⢳⡘⢄⠀⠀⠀⠀⠀⠀⣠⢏⡞⠀⠘⡄⠘⢯⡭⠭⠭⠵⣲⠞⠀⣸⠃⠀⠸⡄⠠⠚⢯⠇⠀⠀⠀⠈⠁⣰⠁⠀⢳⡀⠀⠀⠐⠉⠳⠀⠀⠀⣠⠃
⠀⠑⢤⣉⠒⠢⠤⢒⣊⠵⠋⠀⠀⠀⠘⢦⣀⠉⠒⠒⠊⢁⣠⠔⠁⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠑⠦⣀⡀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⢀⡤⠖⠖⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⠶⢤⡀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠶⢤⡀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⠲⠤⣄⡀
⢀⡴⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦
⡼⠀⠀⢦⡄⠀⢐⣴⠀⠀⠀⢸⡄⠀⣸⠁⠀⢀⠀⢠⠀⢀⡀⠀⠘⡆⠀⢸⠁⠀⠠⠀⠀⠀⠐⠀⠀⠈⣇⠀⠀⡼⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈
⡇⠀⠀⠁⠈⠀⠈⠈⠁⣀⠀⢀⡇⠀⡇⠀⠘⠉⠉⠀⠀⠁⠉⠀⠀⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⣀⣀⣀⣀⣀⠤⠤⠆
⠹⡀⠀⠉⠉⠉⠉⠹⠽⠁⠀⡼⠁⠀⢳⠀⠀⠀⠀⢾⡷⠀⠀⠀⣸⠃⠀⠘⣆⠀⠀⠀⣠⠤⢄⠀⠀⢠⠇⠀⠀⢹⡀⠀⢹⡙⣿⠀⠀⠀⠀⠀⢠
⠀⠙⠢⣄⡀⠀⠀⠀⣀⡤⠞⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⣀⣠⠞⠁⠀⠀⠀⠈⠳⣄⡘⠁⠀⢀⣀⠴⠋⠀⠀⠀⠀⠙⠦⣄⣉⠁⠀⢀⣀⡤⠖⠋

⠀⠀⡐⢂⢄⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠑⠊⠉⠢⠤⣀
⠀⠀⠉⠀⠀⠈⠣⡄⡀⠤⠤⢀⠀⠀⡠⢁⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠀⠀⢈⠂⢀⣀⠔⠁⠀⠀⠑⡄
⠀⠈⠣⣀⣂⡠⠋⢸⡠⠒⠓⢄⡃⠘⡄⠑⠘⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⡠⠒⠤⣀⠈⠆⡎⠀⠀⠀⠀⠐⡩⢍⢢⠈⠈⠒⢄
⠀⡠⢤⡌⠈⣤⠤⠼⡄⠀⠈⢠⣵⠜⠀⠀⡠⢄⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⠶⠁⠤⠀⠠⠀⠣⠆⠆⢘⠙⠊⡁⠈⢰⢪⠊⡠⢤⠤⠄
⠈⠀⠀⠈⠉⠀⠀⢰⠈⡖⢒⠁⡘⡢⠀⠀⠑⠉⠠⡊⠀⠀⠀⠀⠀⠀⠀⠈⠒⡀⠀⠤⠀⠀⡖⡃⠈⢆⡉⠉⠀⠀⢘⡂⠀⠑⠤⠔⢁
⢀⢰⠀⠀⠀⠀⢠⠁⠙⠠⠄⠋⠈⡇⡔⡆⠀⡔⢢⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⠤⢤⠚⠀⡇⠀⠀⣈⡐⢢⠀⠘⠊⠀⠀⡤⠐⠁
⠈⠈⠀⠀⠀⠀⢈⢐⠀⠀⠀⠀⡂⡡⠌⢀⠄⠨⡥⠐⡎⠀⠀⠀⠀⠀⠀⡠⠆⠤⠘⠀⠐⢠⠄⠇⠀⠸⣀⡸⠀⠑⠠⠤⠄⠊
⠀⢸⠀⠀⠀⠀⠸⢉⡀⣀⣀⣀⡏⡇⢠⠐⠒⢰⢳⠒⠒⢄⠀⠀⠀⠀⢰⠀⠀⠈⠑⠒⠒⠉⠀⠀⠁⢠⠒⡀
⠣⢬⠀⠀⠄⠀⠘⠀⠀⠀⠀⠀⡇⢇⠸⡀⠀⢸⢸⠀⠀⡀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠃



⠀⠀⠀⢀⣀⡀
⠀⠀⢰⣁⡀⠜⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠉⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢸⠪⠀⠀⡇⠀⠀⠊⠉⠒⠉⢱⠀⢸⠈⠁⢢⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠠⠔⡊⠁⠀⠠⠒⠤⠒⡄⠀⣏⠀⠀⠀⠈⢢
⠀⠀⠈⠠⠤⡤⠃⠀⠀⠑⠄⣀⠤⠊⠀⠸⡂⢀⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠎⠀⠚⡬⠂⠀⣈⡀⢤⣔⣁⠀⢇⢏⠐⢲⡀⢘
⠀⠀⡠⠚⠤⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⢁⠦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⡍⠞⠁⢠⠜⢸
⠀⠀⠀⡌⠉⠢⢰⠀⢸⠉⠉⠀⠉⠉⡃⡆⢰⠑⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢐⠂⠁⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⡆⠀⢸
⠀⠀⠀⣇⡀⡀⢸⠀⠈⠢⠜⠀⠣⡴⠁⡄⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠤⠴⠒⢤⠀⠀⠀⠀⠀⠀⠀⢀⡨⠀⢧⣀⢸
⠀⠀⠀⡇⠀⠉⠈⠀⠀⠘⠀⠀⠀⠗⠒⠓⠊⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠂⠒⠄⠀⡇⠀⠀⠀⠀⠀⡐⠁⣈⠤⡉⠀⢹
⠀⠀⠄⠈⠉⢹⠋⠉⡒⠊⠉⡉⠉⢑⠊⠹⡉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡇⠀⢐⠀⢸
⠀⠈⠀⠀⠀⡌⠀⠀⡅⠀⠸⡀⠀⠪⠀⠀⢡⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠄⠀⠇⠀⠸⠀⠸
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡂⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⠀⣿⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣶⣶⣶⡄⢸⣿⣿⣿⠀⣿⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⢰⣶⣶⡦⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⣿⡇⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠘⣿⣿⡇⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⡟⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡈⠻⡯⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠿⠋⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣀⠛⠻⠿⠃⠸⠿⠛⢋⣀⣴⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⠿⠛⠉⠛⢁⣤⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠿⣿⣷⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣠⣿⡿⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣝⠻⣿⣿⣿⣿⣿⣿⣿⠿⢋⣵⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣦⣬⣍⣉⣉⣭⣴⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠿⢿⣿⣿

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠂⢄⡤⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠒⠤⣀
⠀⠀⠀⡰⠋⢠⣦⠀⣶⡄⢰⡀⠀⠀⠀⠀⢀⣀⣠⣤⡤⣀⠀⠀⠀⠀⣻⣡⣰⠛⠈⡋⣀⢸⡛⠀⠀⢠⡾⠁⢠⣦⠀⠀⣶⠀⠑⢄
⠀⠀⠀⢣⠀⠘⠋⡤⠉⠁⣠⠇⠀⠀⢀⠔⠉⣀⠀⠀⡀⠈⠱⡄⠀⠀⢿⠀⡄⠐⠐⠓⡦⠈⡇⠀⠀⣾⠁⠀⠘⠟⠀⠀⠛⠀⠀⠀⣷
⠀⠀⠀⠀⠑⠀⠀⠀⡴⠚⠁⠀⠀⠀⠇⠀⠀⠛⠀⠘⠇⠀⠀⢰⠀⠀⠘⢦⣑⢤⣠⢜⡡⠞⠁⠀⠀⣇⠀⢰⠊⠉⠉⠉⠉⢹⠇⠀⣼
⠀⠀⠀⡴⡄⠢⠺⠿⠁⢀⣀⠀⠀⠀⡄⠰⣄⠀⠀⠀⢀⠖⠀⠄⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠸⣆⡈⠣⣀⡀⣀⣴⠏⢀⣴⠏
⠀⠀⠠⡃⣩⣭⠭⠭⠭⣍⣿⠀⠀⠀⠘⢦⣈⠁⠀⠀⣀⡤⠂⠀⠀⠀⠀⠀⢀⣤⠖⠒⠒⠒⠒⠢⣤⡀⠈⠛⠢⠬⡭⠭⠴⠚⠁
⠀⠀⠀⠉⠁⢀⣀⣀⣀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⣤⣤⣤⣤⣤⣤⣤⣈⣻⣄
⠀⠀⠀⡴⠚⠉⠀⠀⠈⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⡗⢢⡀⠀⠀⠀⠀⡟⠻⠿⠿⡿⠟⠙⣿⣿⣿⡿⣿
⠀⠀⡜⠁⠤⢤⡄⠠⠤⢤⣤⠹⡄⠀⠀⠀⠀⠀⠀⠀⠁⢀⠇⡀⠀⠀⠀⣧⠀⢦⡀⠀⠀⠀⠀⡬⠉⢠⡿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢆⡀⠀⠀⠁⠀⡼⡀⠀⠀⠘⢆⠀⠙⠂⠠⠶⠚⢁⣴⠟⠁
⠀⠈⢧⠀⠀⠀⢀⣀⣤⠾⠀⡰⠀⠀⠀⣼⠀⠀⠁⠀⢀⠔⠉⠀⣸⠀⠀⠀⠀⠀⠀⠰⠶⠶⠒⠋⠁
⠀⠀⠈⠓⠦⢄⣀⣀⣤⡴⠞⠁⠀⠀⢠⠘⠢⠄⠀⠀⠀⢀⡠⠚⠁⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⡴⠀⠀⢰⡇⠀⠀⣶⠀⠈⢢
⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠉⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠦⣄⣀⣤⠶⠶⠶⠶⢶⣏⣸⠀⠀⠀⡇⠀⠀⠘⠃⠀⠀⠋
⠀⠀⠀⢰⠃⠀⢺⠆⠀⢿⠂⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡼⠋⠀⠀⠀⠀⢀⣀⠈⢿⡄⠀⠀⡇⠀⢶⡀⠀⠀⠀⠀⣰⠂
⠀⠀⠀⣯⠀⢀⣄⠀⢀⣀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡤⠶⣦⡀⢰⡏⠀⠀⠈⣿⠀⠀⠹⣄⠀⠑⠦⠤⠴⠚⠁⠀⡐
⠀⠀⠀⢻⡄⠘⠈⠓⠞⠀⠙⠛⣠⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⣰⠀⢠⡏⠀⠀⠀⠈⠓⠦⠤⣤⣤⠤⠔⠊
⠀⠀⠀⠀⠙⠳⠤⢤⣤⣤⠴⠞⠁⢀⣴⠞⠋⠉⠉⠉⠒⢄⡀⠀⠱⡀⠀⠓⠄⣠⣤⠴⢃⣴⠟⠀⠀⠀⣀⠤⠔⠢⠤⣄
⠀⠀⠀⠀⢀⣀⣀⢀⠀⠀⠀⠀⢀⡾⠃⠀⣾⡆⠀⢸⡇⠀⠙⢆⠀⠈⠐⠀⠤⠤⠀⠀⠉⠀⠀⠀⢀⡞⠁⣤⠀⢠⡄⠈⠳⡀
⠀⠀⠠⠊⢠⠀⢠⠀⠑⠀⠀⠀⢸⠇⠀⢀⣙⣁⣀⣈⣁⣀⠀⠘⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠉⠀⣈⣁⣀
⠀⠀⠀⢀⠀⠀⠈⡠⠀⠀⠀⠀⠸⡆⠀⢏⠀⠀⠀⠀⢠⡟⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡔⠏⠇⢰⠐⣆⡇⢸
⠀⠀⠀⠄⣁⣀⣀⡠⠂⠀⠀⠀⠀⠹⣦⡈⠢⢤⣤⠴⠋⣀⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠉⠉⠉⠉⣁⡰⠃
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⠀⠉⠉⠁⠒⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢁⠂⠀⠀⢀⣀⣀⡀⠀⠀⠀⢈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠁⣾⢉⡤⣬⡹⠀⠀⠀⠀⢀⡷⠤
⠀⠀⠀⡠⠐⠈⠨⠁⠉⠉⠀⠉⠈⠐⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡩⠓⠉⠉⠉⠀⠀⠀⠀⠀⣠⣾⡗⠛⠂
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢀⣠⡤⣤⡤⠤⠒⣡⡞⠁
⡼⠀⠀⠀⠀⠀⠀⢠⠖⠒⢀⠒⠤⡀⠀⠀⠀⠀⠈⢦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣐⡓⠂⡤⣂⡴⢡⣴⡴⠶⠶⠦⠟⠉⠈⠀⢀⡀⠄⠐⠒⠒⠒⠠⠤⣀
⡇⠀⠀⠀⠀⠀⠀⠹⡖⠚⠶⣥⣄⠈⠂⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡠⠂⠁⠀⠈⢁⢳⣴⣴⠯⠁⠀⠀⠀⠀⠀⢀⡔⢪⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢈⡿⠉⠀⠀⠙⡆⠀⠀⠀⠀⢠⣿⠄⠀⠀⠀⠈⠓⠤⢀⡀⠀⣠⢿⡯⡅⠁⠀⠀⠀⠀⠀⠀⠀⡈⠐⠤⣀⠠⠤⠆⢤⠀⠀⠀⠀⠨⠈⡇
⢀⠀⠢⡢⡤⠐⡀⠖⠉⡟⠀⠀⠀⠐⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠐⠄⣀⠀⢈⠙⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠑⢢⣖⣤⣾⣺⠶⠁⠀⠀⠀⠀⢠⡯⠉
⠈⠢⠀⠙⢢⡊⠁⣈⡼⠃⠈⠁⠀⠙⢄⠀⠀⠀⠀⠨⠻⣤⠀⠀⠀⠀⠀⠀⠑⠒⠼⠛⠂⠀⠀⠀⠀⠀⠀⣠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⠇
⠀⠀⠑⢄⣽⡴⠞⠋⠐⠀⠀⠀⠀⢠⠀⠑⢄⠀⠀⠀⣡⢺⣇⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⢌⠲⠀⠀⠀⢀⡤⠴⠖⠢⠤⠤⠤⠤⠰⠚⢉⣴⠏
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠉⢶⢋⡵⠛⠁⠀⠈⠑⠲⣄⠀⠀⠀⠀⠀⢀⡠⠂⠁⠉⠢⣈⡒⢤⠞⢅⣴⣥⡶⢶⣶⣦⣤⣤⡶⠾⠯⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠟⣻⣼⠢⡀⣀⡀⠀⣀⠴⣿⡁⠀⠀⠀⢰⠯⡀⠀⡀⠀⠀⢠⣿⣼⣾⡫⠉⠉⠉⠀⠀⠀⠐⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⡀⠈⠢⡤⢞⠝⣸⠇⠀⠀⠀⠀⠰⡀⠈⠑⠢⣡⡞⣵⣿⠘⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣋⣼⣞⣓⠒⠀⠀⠀⠀⠀⠈⠑⠀⣀⣮⠼⠋

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⣤⣤⣤⣀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⢁⣀⣤⣤⣤⣀⡈⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣴⠟⠋⠉⠀⠀⠀⠉⠙⠳⣦⡀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠾⠛⣿⠏
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⢻⡄⠀⠀⠀⠀⠀⠀⢶⡟⠛⠉⠁⠀⠀⢀⣼⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡿⠀⣸⠇⠀⠀⣀⡴⢶⣤⣀⠀⠀⠀⠀⠀⠀⠹⣇⠀⢿⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣠⡾⠋⠀⠀⠀⠉⠛⠷⢦⣤⣀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣀⣴⠞⠛⢷⡄⢠⡟⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⡿⠀⣸⠇⠀⢀⣴⠾⠋⠀⠀⠀⠀⠻⠟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣿⡀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⡿⠛⠾⠋
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠟⠉⢷⡀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⢠⣾⣅
⠀⠀⠀⠀⣿⠀⠀⣠⡶⠋⠁⠀⠀⠀⠻⣦⡀⠉⠻⠶⣤⣤⣀⣤⣤⠶⠟⠉⣀⣴⠟⠀⠙⢷⣤⣄
⠀⠀⠀⠀⣿⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⣀⣀⣀⣤⡴⠾⢿⡄⠀⢀⣴⠟⠁⠙⣷
⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠙⣶⠟⠁⠀⣠⡾⠛⢷⣄
⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣷⣤⣾⠋⠀⠀⠀⠙⢷⣄⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⣄⠀⠀⣠⡾⠋⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡼⠋

⣿⢿⠿⠿⡿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣿⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣤⡄⠀⣤⠀⣤⡄⠀⣠⡄⢠⣄⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢠⣿⣇⢸⣿⠀⣿⡇⠀⣿⡇⠘⣿⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣸⡟⣿⣸⡏⠀⣿⡿⠿⣿⡇⠀⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⢿⣿⠃⠀⣿⡇⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⠉⠁⠀⠉⠁⠀⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⣴⣶⠀⣶⣶⠶⠆⢠⣶⠶⡦⠀⣶⡆⢶⣶⣶⣶⠆⢰⣶⣆⠰⣶⣶⣶⡶⢰⣶⡶⠶⠀⠀⠰⠶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣽⣿⠀⣿⣇⣀⡀⢻⣯⣀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣼⡟⣿⠀⠀⣿⡇⠀⢸⣿⣀⣀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣻⣿⠀⣿⡟⠛⠁⠀⠙⢿⣷⠀⣿⡇⠀⢸⣿⠀⢠⣿⣤⣿⡆⠀⣿⡇⠀⢸⣿⠛⠋⠀⠀⠀⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢽⣿⠀⣿⣷⣴⡄⣴⣦⣾⠟⠀⣿⡇⠀⢸⣿⠀⣼⡿⠉⢹⣧⠀⣿⡇⠀⢸⣿⣦⣶⠀⠀⠀⣶⡆





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣿⣿⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢯⣿⣿⡿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣳⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡶⠀⠀⠀⢀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⡴⠋⡁⠀⠀⠈⠙⢦⡀⠀⠀⠀⢠⡟⢠⣿⠃⠀⠀⠀⢸⡄⠀⠀⠀⢸⡇⠀⠀⠀⣰⠁⠀⠀⢀⡞⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣤⣀⣀⣀⠻⣄⠀⢀⡿⣠⣿⡟⠀⠀⠀⠀⠘⠃⠀⠀⠀⣿⡇⠀⠀⣰⡏⠀⠀⢠⠎⠀⠀⢀⡜⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣯⣥⣙⣦⣼⡿⣃⣨⣷⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀⠀⠀⠉⠁⠀⠀⠋⠀⠀⣠⠊⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡌⠙⢛⣿⣭⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣶⣄⡀⣀⣀⠀⠀⠀⠚⠁⠀⠀⣠⡶⠋⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡶⢋⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢬⠉⠳⡌⠻⣿⡙⣎⢫⡙⣆⠀⠀⢀⣠⠾⠋⠀⠀⢀⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⢡⣿⢃⡤⠞⣛⠋⢉⡶⠀⠀⠀⠃⢀⡀⠀⢧⠀⣿⠀⣿⣇⣿⢸⡏⣻⣄⡀⠘⠁⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⣿⠀⠼⠟⠒⠉⢀⣠⠴⠚⢸⡏⢹⠄⣸⣰⠏⢠⣿⣿⣿⣾⣧⡿⣿⠉⢦⠀⠘⠉⠀⠀⠀⢀⣀⡤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣿⢏⣈⣓⣲⣶⠶⣿⠋⠀⠀⠀⠈⢷⣦⡴⠛⠡⢶⣿⣿⠿⢛⣵⣿⣽⡿⠀⣾⠀⢀⣠⡤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⣿⡞⢻⣦⣭⣭⣥⡶⠟⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠋⠙⠶⣖⢋⡵⠿⣛⣤⣾⡃⠀⠀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢉⣼⣟⠳⠶⠾⠿⣛⡩⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⡿⠛⠉⠀⠀⣤⠤⠶⠖⠚⠉
⠀⠀⠀⠀⣠⣶⠟⢋⣠⠞⣱⠏⠈⢿⣿⣿⣿⣭⣶⣯⠶⠋⠀⠀⣀⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣴⠟⠋⢠⠔⢋⠴⠚⠁⠀⠀⢸⡏⠻⡟⠛⠉⠀⠀⠀⠐⠋⢁⣀⣀⣀⣉⠉⠀⠀⠀⠀⠘⠳⣦⡀⡾⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠋⠉
⢀⣾⠋⠀⡐⠁⠀⣠⠴⠦⢤⣀⠀⠸⣧⠀⠟⠀⠀⠀⠀⠀⠀⠰⣟⡿⠉⠛⠿⠛⠀⠀⠀⣴⣤⣤⣀⣹⠁
⣸⡇⠀⠀⠀⠀⡾⣏⠀⢀⣠⣿⡇⢀⣿⣧⣤⡤⠀⠀⠀⠀⠀⠀⠈⠓⠖⠚⠁⠀⠀⠀⠀⣿⡏⡉⠛⡿⠂⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒
⣿⠀⠀⠀⠀⠀⢷⣈⣉⣽⡿⠛⡟⣄⠘⢿⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠉⠁⠀⡇⠀⠀⠀⠰⠤⣄⣀⡀
⠸⣆⠀⠀⠈⠲⠤⢬⣿⠟⢀⣼⡇⢸⡆⢸⣷⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠀⡼⠂⠀⣸⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠂
⠀⢸⣷⣦⠤⠴⠖⠋⢁⣠⣾⡾⠁⣸⣧⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠙⠲⢄⡀
⠀⠀⠉⠻⣷⡶⠾⢛⣿⣿⠋⣠⣴⣿⣿⣿⢏⣾⠟⣦⠀⠀⠀⠀⠀⠀⠒⣒⠻⠿⣷⣶⠀⢀⡞⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠉⠳⠤⣀
⠀⠀⠀⠀⠛⠻⠿⢿⣷⣞⣻⣭⣴⣞⣫⣴⠿⠋⠀⠈⢷⣄⠀⠀⠀⠀⠀⠈⠳⠶⠛⢮⡙⠛⠦⢤⣀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣯⣿⣧⠙⡄⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⡤⠈⠛⠲⣄⡈⠓⢦⡀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣦⣤⣀⣤⡎⠀⠀⠀⠀⠈⠻⣄⠀⠙⠲⢤⣄⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠹⣦⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⢀⡴⠛⠉⠙⠛⠛⠛⠛⣓⣤⡍⠓⠦⣄⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⢳⡄⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣧⠔⠋⣀⡴⠖⠒⠒⢒⡳⠚⠁⢀⣤⠄⠀⠀⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢻⡄⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⠖⠋⠁⠀⠀⠀⡜⠁⣠⡴⠚⠁⢀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡼⠁⣼⠟⢀⠶⠛⠛⠲⢦⣀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣷⣄⢿⣿⡿⠋⠙⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣴⢁⣼⣏⡴⠋⠀⠀⠀⠀⠀⠉⠓⠶⣶⣶⠖⠀⠀⠸
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠈⢻⡾⡟⠁⠀⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠼⠿⠋⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢻⡀
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁⠿⠀⠀⠀⠀⠙⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⢠⢶⠈⠄
⣠⡷⡀⠀⢙⠀⠼⠄
⠘⢶⡥⠊⢱⠆⠂⠀⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠘⢇⠀⠉⢀⡴⡿⠈⠙⠀⠠⠤⠐⡒⠀⠀⠀⠴⠦⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⡶⣊⠐⣤⣐⣈⣉⣤⠴⠶⠓⢐⠦⢤⣤⣤⡄⢀⣀⡠⠄
⠀⢄⠚⢧⣴⢯⣠⠀⠐⠦⠀⠀⠺⡱⣤⠢⡀⠀⠀⠀⠐⠀⢢⣀⡤⢄⠒⠀⠁⠀⠀⠀⠀⠁⢠⡷⠄⣘⣛⠛⠛⡀⢈⠁⠀⣴⠱⡄⠀⠀⠀⡈⢋
⠀⠸⡎⢔⠉⢽⣕⠀⠀⠀⠠⢤⡀⠈⠪⣷⡈⡀⠀⠀⠀⠀⠀⠱⡊⡁⠀⠀⡠⠖⠀⢥⣭⡈⠈⢕⢿⣿⣟⠛⠛⠛⠓⠂⠀⠀⢀⣿⡌⠛⠁⢠⠈⣆
⠀⠀⢓⠁⠐⠕⠈⠂⠀⠰⣤⡀⠛⢧⣄⡈⠻⣰⠀⠀⠀⠀⠀⠀⠱⡀⠁⠠⠵⠒⠒⠓⣆⠤⢠⠀⠀⢹⣿⣥⡅⠉⠌⠀⠐⠐⢻⣿⢃⣄⣀⢠⠀⡾⠅
⠀⠀⢸⠀⠀⠀⠈⠀⠙⢦⡀⠙⣦⣀⢈⣳⣄⡠⣢⠀⠀⠀⠀⠀⠀⠁⠀⠄⠢⠀⠀⠀⢹⡉⠄⠀⠀⠀⠹⣿⣻⡚⠒⠀⢰⣶⠞⣷⠚⠛⠛⠛
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠛⠛⠉⠻⡢⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠈⠩⠙⠗⠐⢾⣛⣿⣿⣄
⠀⠀⠀⢻⠀⠀⠠⠈⢻⢿⣶⣶⡀⠘⣟⣛⣛⣀⣤⠑⠧⠀⠀⠀⢀⠀⠀⢂⡠⠄⠀⠂⠀⠀⠀⢀⣠⣄⡀⠀⠀⠠⢐⣀⡀⣹⣼⣿⣿⣶⣡⡀
⠀⠀⠀⢸⡄⠀⠁⠁⠀⠋⠣⠶⡇⠀⢋⡭⢭⠭⢤⣶⢀⡆⠀⠀⠀⠃⠀⠈⢖⢀⣢⠖⠩⠂⠀⠀⠘⠋⠉⣠⣤⣶⣾⣿⣿⠿⠿⠖⠛⠉⠁
⠀⠀⠀⠀⣧⠀⠈⠢⠰⠀⠁⠐⠁⠀⠘⠋⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣠⣤⡴⠶⠚⠛⠉⠙⠉⠉⠉⠁
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡸⠁
⠀⠀⠀⠀⢸⡻⣄⣀⣠⣤⣤⣤⣤⠤⠤⡤⠤⠴⠶⠒⠖⠒⠛⠛⠉⡉⠡⠀⠀⠈
⠀⠀⠀⠀⠀⡏⠁⠉⠀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢿⡆⢀⠈⠩⠀⠂