Washing Emojis & Symbols

⠀⠀⠀⠀⢀⠀⣠⡶⠶⠶⠶⠶⠶⠒⠒⠒⠚⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣒⠒⠒⠶⠶⠶⣤⡀⢀⡀
⠀⠀⠀⠀⠎⣼⠃⠀⣴⠶⠒⠛⠛⠛⣛⣛⠉⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⡉⢉⣩⣍⠉⣉⣉⡙⠛⣷⠀⠀⢹⡆⠓
⠀⠀⠀⠀⠀⣿⠀⠀⣿⢰⣟⣻⡇⢼⣏⣻⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠒⠚⠀⠻⠶⠇⠀⣿⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⠹⠟⠁⠈⠙⠿⠁⠘⠷⠶⢤⣤⣤⣤⣤⣤⠾⠃⠸⠭⠝⠀⠯⠭⠇⢰⡏⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⣓⣚⣒⡒⠲⠶⠶⠶⠶⠶⠶⠶⠛⠛⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣤⠶⠞⣛⣩⠭⠭⠭⠭⠭⠭⢭⣛⠳⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣠⠾⢋⠴⠒⠋⢁⣀⣠⣤⣤⣤⣤⣤⣄⡈⠙⠲⢬⡙⣿⡉⠉⠛⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠃⡴⠁⢠⡴⢛⣩⠭⠒⠚⠛⠛⠛⠒⠮⣝⢷⣄⠀⠙⣎⢻⡝⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇⡼⠀⢠⡟⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡹⣆⠀⠸⡀⢿⡀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠘⡇⢀⡶⢾⡇⡇⠀⢸⡅⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢻⡀⠀⣇⢸⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⣷⠘⣇⠸⡇⣇⠀⠸⣇⢧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⢸⠸⡇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢿⠀⣿⠀⢻⡸⡄⠀⢻⡌⢮⡻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⠁⠀⢸⢰⡇⠀⢠⡏
⠀⠀⠀⠀⠀⢠⢸⡇⠘⠳⠾⢿⡜⣆⠀⠙⣦⡙⠦⢌⣀⣀⣀⣀⣀⣀⡤⢞⣱⠟⠀⢠⢏⡾⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠘⠈⣇⠀⠀⠀⠀⢻⣌⠳⣄⡈⠙⠳⠶⣤⣭⣭⣭⣭⣤⠶⠛⠁⣀⠔⢁⡾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠳⣦⣙⠲⠤⣀⣀⣀⠀⠀⢀⣀⡤⠖⢊⣡⡴⠋⠀⠀⠀⢀⡟⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠐⠦⠉⠛⠶⠶⠤⣬⣭⣭⣥⠴⠶⠞⠋⠉⠀⠀⠀⠀⠀⢸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠉⠉⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠒⠶⠶⠶⠶⠧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟
⠀⡀
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⣤⣤⣠⡟⠁⡴⠃⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠋⠁
⠀⠀⢀⣀⠀⠀⠓⢺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⢶⣤⣀
⠀⠀⣏⠀⠴⢤⠦⣾⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠙⠿⠆⠀⢰⣿⢐⡇⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣸⠀⠀⢀⣤⣤⠞⠋
⠀⠀⣀⣌⣤⠆⣼⠃⢸⣷⠀⢸⡇⠀⢰⡀⠀⠀⠀⠀⣠⠄⠀⢠⣿⠀⠀⣀⡈⠁⢀⣴⠟⠉
⠀⢰⡏⠀⠁⠀⢹⡄⢹⣿⠀⢸⡇⠀⢸⣿⠀⠀⠀⣼⡿⠀⠀⢸⡏⢀⡀⣨⠟⠀⣼⡟
⠀⠀⣿⠂⣀⣀⡀⢻⡌⢻⡀⢸⡇⠀⢸⡟⠀⠀⣰⡏⣴⠀⠀⣸⡟⠈⣹⠁⠀⣼⣿⡁
⠀⠀⠈⢻⡋⠀⠁⢀⣽⠶⢿⠈⣧⠀⣿⡁⠀⠀⣿⠃⢿⠀⠀⣿⣁⡿⠋⢀⣾⠟⢉⣷
⠀⠀⠀⣾⣧⡾⠀⣾⠃⠀⡘⣧⣹⡄⢸⣇⠀⠀⢿⡀⠘⢧⣀⡟⠈⣠⡴⠛⠁⠀⠈⢿⡀
⠀⠀⠀⢿⠀⠀⠀⣿⢀⡾⢁⠀⠉⢳⣜⣿⡀⠀⢸⡇⠀⠀⢉⣴⠾⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⡷⠂⠀⢻⡾⣱⠏⠀⣴⡿⠉⠛⢻⡄⠈⣿⢀⣴⡾⠋⠀⠀⠀⠀⠀⢀⣷⠏
⠀⠀⠀⣼⠋⠀⠀⠸⢷⡟⢀⣸⡟⠁⢀⡖⠀⢻⣦⠟⠋⠁⠀⠀⡄⠀⠀⢀⣷⠾⠃
⠀⠀⠀⠘⣧⠀⠀⠀⠘⠛⠛⠋⠀⠀⠻⠶⠶⠛⡉⠀⠀⠐⢦⡖⠃⠀⣀⡾⠁
⠀⠀⠀⠸⣧⡀⠀⡀⠀⠀⠀⢧⣠⠀⠀⢀⣀⣠⠷⣄⣀⣠⡾⠃⠀⠉⣿
⠀⠀⠀⠀⠸⣧⣴⠏⠀⠀⠀⠀⢙⣻⣷⡚⠉⠁⠀⠸⣏⠉⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠖⠛⠋⣩⢙⠻⢷⣤⣀⠀⠈⠛⠒⠚⠋
⠀⠀⠀⢰⣞⣷⢸⣿⣄⡀⠀⠘⠶⠟⣠⠤⢤⡈⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠉⠁⢸⡇⠈⠛⢶⣄⡀⠀⠙⠚⣯⣤⣶⣶⠀⠙⠻⢶⣦⣄⠀⠀⠀⠀⣏⣹⠆
⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠉⠛⢶⣄⡀⠀⠀⠋⡵⢶⣻⠆⠀⠈⠙⢻⣶⡄⠀⠈⠉
⠀⠀⠀⠀⣴⣋⠙⢷⣄⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⣀⣤⣤⡶⠞⠋⠉⡇⢀⡠⢤⡀⠀⠀⠀⡶⠛⡆
⠀⠀⣀⣸⣇⠙⢷⣾⣉⠳⣤⡀⠀⠀⠀⠀⠀⠈⠙⣿⠟⠋⠉⠀⠀⠀⠀⠀⣇⠈⣀⣨⡿⠀⠀⠀⠀⠉⠁
⠀⠈⢯⡽⠙⠛⣿⠀⢈⣷⠈⠙⢶⣄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠁⣰⠛⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠈⠻⣦⡀⠀⠀⢺⠀⠀⠀⢀⣠⣴⠶⠛⠉⠀⠀⠀⠀⠻⠤⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣾⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⠿⡏⠉⠩⡈⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠀⠀⢀⠀⠁⠀⠀⠀⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⣶⢠⡀⠑⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⢠⡱⡆⠑⢤⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠙⡅⠀⠈⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢣⠀⠀⠀⡇⠄⠀⠀⡸⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⢀⢷⠀⡠⢿⠂⠀⠂⠀⠀⡀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠍⠉⠁⠈⠻⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⡀⠑⠀⠀⢀⠀⠄⣀
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⠀⠸⡄⠙⠒⠁⠀⠀⠈⠙⠐⠄⠀⢀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠄⠀⠀⠀⠘⠆⠈⠄⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠂⠠⢄⡀
⠀⠀⠀⠀⠀⣸⣿⠀⡀⢀⠠⠀⠀⠀⠁⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡀⠀⡆
⠀⠀⠀⠀⠀⠿⠘⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠤⠀⠀⠠⠁⠀⠀⡇
⠀⠀⠀⠀⠀⣷⠀⠁⠀⢃⠀⠀⠈⠀⠁⠀⠀⠈⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠧⠲⣄⠶⢏⣤⠐⠛⠁
⠀⠀⠀⠀⠀⡿⠀⠀⠀⢘⡷⠦⣠⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡴⠶⠶⠒⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢃⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠐⠛
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠋⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣿⣦⣤⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠿⡏⠙⠋⠙⠋⣿⡿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠖⠂⠀⠀⠀⠀⠋⠁⠠⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣦⡘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣀⡀
⠀⠀⠀⠀⠀⣴⠚⠉⠉⠉⠉⠉⠉⣹⠆⠀⠀⠀⠀⠀⠀⠸⣧⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⢟⠛⠳⣄
⠀⠀⠀⠀⣀⣹⡶⠤⠴⠖⣶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣧⣆⣤⡴⠚⢷⣠⢞⠕⠁⠀⠀⠈⢷⡄
⠀⠀⠀⢰⢱⠛⠓⠒⠒⠒⠛⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠑⢁⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⢸⡘⢦⡤⠤⠤⢤⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⣴⠏⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⣷⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⡅⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⠀⣠⡾⢁⣴⠋⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⢠⢎⣠⠼⠁⠀⠀⢀⣴⠟⠛⣦⡞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢻⡀
⡔⠒⢤⡀⠀⢀⣀⡀⠀⠀⢸⠀⠀⣇⣀⣉⠉⠀⠀⠀⣀⣀⣿⣀⣰⠟⠻⣦⡀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠙⠒⠤⢵⡄⠸⣄⡙⡄⠀⣼⠀⢀⣯⣍⠉⠙⠛⠋⠉⠉⠉⠈⣿⠁⠀⠀⠀⠙⢦⡀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣶⡏⠀⠻⣍⣉⠛⠳⠶⠶⠶⠶⠒⠛⠋⢳⣄⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠶⠖⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣷⣄⡀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠠⣏⣉⡩⠶⠀⢻⣿⠿⣛⣋⠀⠀⠀⠀⣠⡾⠛⠛⠛⠉⠉⠉⠀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣠⠀⠀⠀⠀⠀⡈⣿⣟⣭⣵⠞⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡔⠋⣠⠏⠀⠀⠀⡴⢚⡇⠀⠉⠈⠛⠋⠁⠀⢤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠾
⠈⠋⠁⠀⠀⠀⠸⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠁⠀⠀⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠈⠀⠀⠀⠀⠀⠉⠀⠀⠠⠖⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⡠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡞⣠⠖⠉⠀⢠⡞⠁⢀⡾⠁⢀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣚⣠⠗⠁⠀⢀⡠⣾⠁⠀⣾⠀⠀⡼⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣩⣤⡤⠴⠒⠉⠀⣿⠀⣸⠇⢀⡾⠁⢠⠞⢹⡇⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢿⣰⣯⡶⣋⠤⠚⠛⠳⠆⠹⣄⠀⡆⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⠀⡀⠀⠀⠀⠈⢦⠀⠀⠾⠋⠀⠀⠉⠉⠉⠀⠀⡠⠀⠐⢤⠀⠘⣶⡇⢪⡖⠀⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠁⠂⣤⠠⠙⠒⠀⢠⢩⡆⠀⣠⣬⣌⠑⡄⠀⠀⠀⢰⣿⣿⣶⡀⠁⠀⠛⠁⠈⡛⠠⠞
⣀⡀⠤⠄⣠⠂⠀⠀⠀⠁⠀⠀⠙⢄⡀⠄⠐⠈⠀⠀⠑⠻⠀⠹⢿⣿⡇⠄⡠⠖⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⡜⢀⠀⠄
⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠚⢄⠀⠀⠀⠀⠙⠒⣀⢰⡄⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠉⠀⠀⠀⠀⣀⡤⢲⠙⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⡀⠸⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢲⠶⡲⠖⠒⡉⠍⠀⠃⠀⠀⠀⢡⡀
⠀⠀⠀⡆⠀⠀⡀⠀⠀⠀⡇⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠑⡀
⠀⡇⠀⡇⠀⠀⠧⠆⠀⠀⡇⠀⣿⣀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⠈⠄⠀⠀⠀⠀⠀⣀⣠⡤
⠀⣷⠀⣧⠀⠀⠀⠐⠀⠐⠓⠂⠉⠀⠀⠀⠀⡀⠀⠠⠤⢤⣀⣀⡠⠔⠁⠀⢑⣄⡄⠀⠀⠀⠀⣀⠜⠐⠂⠠⡤⠀⠀⠀⠈⢄⣤⠴⠒⠋⠙⠢⢄
⠀⢹⠀⢸⡀⠤⠄⠀⠀⣀⠠⠄⠂⢀⡀⢀⣀⠀⠤⠀⠠⠗⣴⠂⠀⢀⠠⠖⠋⠡⢑⠦⠂⠐⠈⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈⣦⡀
⠠⠘⠉⠀⠀⠀⡠⠂⠁⡀⠀⠀⠐⠃⠀⠀⠀⢙⠁⠀⠹⢤⡙⡢⠈⠉⠒⠀⢴⡒⠈⡀⠀⢀⡀⠄⠐⠉⠀⠀⠀⠀⢀⠠⢖⠉⠄⣈⡱⠶⢖⡊⠉
⠀⠀⠀⠀⠀⠀⣄⠀⠣⡀⠀⠀⡈⠀⢠⠉⠑⠥⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⠆⣠⠊⠑⠠⣀⢀⡀⠄⠒⠈⠀⠀⠀⠂⠌⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠈⠐⢄⡈⠑⣤⠀⠀⠆⠀⠀⣠⠼⠆⠘⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠏⠲⣄⣀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢌⣂⢐⠦⡀⠁⠀⢀⡰⠶⢃⡀⠁⠒⠠⣀⠀⠀⠀⠀⢀⠠⠀⠊⢀⡤⢺⠐⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠈⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡈⠑⠤⣀⠻⣀⡉⡼⠀⠀⠀⢀⣀⠦⠐⠈⣀⡠⠔⠊⠁⠀⠸⣠⡇⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢀⣀⠤
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠉⠂⠤⠤⠄⠒⠈⢁⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⢠⠌⠙⠤⠄⠀⠐⢒⣉⡠⠤⠛⠻⢅⡀
⣀⣤⠴⠖⠋⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠀⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡄⠀⠀⢨⣅⠀⠀⠀⠀⠀⠈⠙
⠧⣄⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡐⠳⡀⠀⠈⡄⠉⣒⣤⣤⠖⠊⠉
⠀⠀⢉⣲⠤⣔⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠗⠂⠔⠒⠈⣱⠀⣈⠢⠀⠙⠲⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⣭⣿⠘⢯⣤⣤⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⠁⠀⠀⠀⠀⠀⠀⢰⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠛⠛⠳⠶⣤⣀⣀⣤⠤⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣴⠋⡴⠄⠀⠀⠴⠂⠈⡏⠀⠀⠀⢸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⣀⣀⣀⣿⠀⠀⢻⡝⢣⡟⠀⠀⣿⡀⠀⠀⢸⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡰⢋⡴⠖⠒⠻⢦⣐⣒⠈⠭⢐⣒⡿⠟⠛⠲⠶⠾⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢄⠀⢸⡆⠀⡇⢸⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⠒⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⡃⣞⠛⣋⣉⣉⣹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡹⢋⡞⢡⠄⠀⣄⢲⡹⣝⡆⠀⠀⠀⠀⠀⠀⠀⢠⡿⢛⡻⣿⠟⣛⠻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡈⠀⠈⠀⠸⢘⡣⠘⠄⠙⠀⠀⠀⢰⣲⠀⠀⠀⠀⣿⠿⣯⡀⣇⠸⣿⢇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠓⠞⠁⡠⠀⢠⠀⠀⠀⠀⣄⠀⢠⡀⠀⠀⠀⠀⡠⣄⠀⣹⣧⢭⣥⠿⢷⡶⠻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⣀⡀⠀⠰⠁⢀⡾⠛⠛⠳⣄⣸⣤⣄⢱⡞⢙⡇⠈⠂⠊⠀⣿⢅⡲⣿⢘⣻⡘⠟⣻⡽⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣉⣹⢆⡴⠚⠛⣷⠶⣤⡟⠻⣄⣠⢼⡇⠙⠋⠁⠀⠀⠀⠀⠻⣮⣥⣾⣶⣾⠿⠛⠉⣾⣿⡞⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⡁⢾⠀⣰⠶⣿⡀⠘⣷⠀⠙⣷⠀⠻⡟⠓⣦⠀⠀⠀⠀⠀⠈⠉⠋⣉⠀⠀⠀⠀⠉⠉⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⠴⢿⡾⢧⣽⡄⠈⢧⠀⠈⢧⠀⠘⣇⠀⠹⣆⠘⡇⠀⢰⣷⠀⢀⡀⠀⣿⠀⠀⠀⠀⢀⣴⣶⠸⣷⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⠀⢀⡿⣞⠁⢻⡄⠈⢧⠀⠈⢷⠀⠈⠀⠀⠘⣶⠇⠀⢸⣿⠀⣼⣧⠀⣿⣀⣶⣶⣦⠘⠿⣦⡄⢿⣿⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠙⣩⠖⠛⣰⠶⣿⡄⠈⠧⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⣿⣀⣿⠿⣷⣿⢿⣇⣾⣿⣄⣴⡾⠏⠘⠃⠰⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠁⠀⠀⢻⡄⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⠿⠟⣠⡉⢁⣈⠛⣋⣈⠛⣡⠀⣶⣾⣆⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣧⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢿⣇⣼⣿⣼⡟⢻⣿⣿⣀⣿⠻⣿⠏⠙⠁⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠈⠳⢤⣤⣤⣶⣋⣷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠙⢛⣿⠙⠿⠿⢃⣙⠛⣿⣠⣿⡶⣶⡜⢿⣦⣍
⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⢸⡁⢈⡇⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠙⣆⠀⢸⣆⠀⢿⣿⠋⣴⣶⡀⢸⣿⣧⣿⡇⣿⡆⢸⣿⠀⠈⢻⣧
⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠉⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠘⣧⠸⣿⠀⢘⣿⡀⣿⣛⣿⡌⣿⡝⣿⣿⢸⣷⣾⢿⣦⣤⣼⡟
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣠⠏⠀⢿⡿⠛⢿⣇⣿⠟⠙⠿⢿⡇⠈⠛⠀⠛⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⠤⠴⠚⠁⠀⠀⢸⣷⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⡧⠐⢿⠂⠀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠒⡉⠑⢾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢡⠃⠆⠠⣷⡄⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡹⠁⠀⣸⠷⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢼⠘⠿⣲⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣁⣐⣂⣁⣤⢤⠛⢷⡀⠀⠀⠀⠀⣀⣤⣶⣦⡴⢁⣎⠉⠁⠀⠀⠀⢀⣴⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⢸⠃⠈⠁⠀⠘⠊⠉⠪⡑⠦⢐⡨⠿⣿⡯⣿⣷⣼⣯⣷⡄⠀⠀⣠⠏⠈⢯⣦⣿⠋⢹⡄
⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠀⣿⣿⣞⣯⣿⡼⢏⠳⢔⣫⡟⣴⠀⣸⢹⡀⢸⡾⡇
⠀⠀⠀⣛⡭⠭⠝⠒⠒⠉⠉⠉⠛⠓⠒⠲⠤⠤⠤⠀⣀⣀⣀⠀⠯⠽⣿⣛⣿⣿⡥⠄⠀⠉⢩⢎⡾⢹⠀⢸⡄⢳⢈⣧⡙⢦⡀
⠀⠀⢰⠋⡿⠿⢿⣶⣶⣶⣦⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢹⢀⡾⠯⠝⢻⣷⠁⠀⠀⠀⣇⣾⢁⡾⠀⢈⡇⢸⠸⢸⡎⣇⣷
⠀⠀⢸⠀⡇⠀⠀⠀⣀⣈⠉⠉⠉⠙⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡟⣿⣶⣿⣇⠀⠀⠀⠈⠉⠻⢧⣀⣼⣤⡷⠤⠼⠃⠈⠁
⠀⠀⢸⠐⡇⢀⡔⣫⠭⠭⡙⢷⡄⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣽⣻⠿⢿⣿⡆
⠀⠀⢸⠐⡿⣿⢸⠃⠀⠀⠘⡌⣿⣤⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣥⣹⣷⣶⣯⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣀⣀
⠀⠀⢸⢸⡏⠹⡜⢧⣀⣀⡴⢠⡿⢭⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⣿⠉⠀⠀⠀⠀⠀⠀⢀⣜⣹⡭⢷⣿⠋⠁
⠀⠀⢸⢸⡇⠀⠈⠒⠭⠭⠔⠋⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡤⣤⠤⠿⠀⠀⣀⣀⣀⣀⣀⢎⣾⡿⢣⡿⠁
⠀⠀⢸⢸⡇⠤⠤⢀⣀⣀⣀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢇⣠⠔⠒⢉⣩⠭⠤⠒⠲⠏⢾⡷⢳⡿⠳⢤⡀
⠀⠀⢸⢸⡧⢤⣀⣀⣀⠀⠀⠉⠉⢹⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣧⡀⠔⠋⠁⢀⡀⠈⢀⡀⠤⠀⡉⠙⢂⣠⡏
⠀⠀⢸⢸⡗⠂⠤⠤⠄⣉⣉⠉⠓⢺⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢛⣏⠙⢛⣟⡲⠲⠶⠾⠗⠶⠚⠻⠿⣿⣏⠗
⠀⠀⢸⣸⡋⠙⠒⠂⠄⠤⠤⢈⣉⣹⡇⢸⠀⠀⠀⠀⢀⣤⣶⠋⠀⠀⠀⠟⠘⡆⠈⠻⢀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠛⠻⠿⢷⣶⣶⣤⣤⣄⣀⣸⡇⢸⠀⣀⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⠢⣀⣀⠀⠀⠀⠀⠀⣀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣲⡶⠶⠶⣶⣒⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⣤⣤⡇⠀⠀⠀⠀⠀⢠⡒⠒⠒⢲⠀⠀⠀⠀⠀⢀⡔⠒⠒⣎⠀⠙⠢⢄⣀⣠⠤⠚⠁⠘⡖⠒⠲⡄
⠀⠀⠀⠀⠀⡴⠒⠒⡟⠒⠒⠒⢳⡀⠀⠀⠀⠀⢸⠉⠉⠉⠉⡆⠀⠀⠀⢀⠎⠀⠀⠀⠈⠙⠲⢤⢴⠋⡦⠤⠤⠒⠋⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⡇⡎⡜⠀⠀⠀⠀⠀⢳⠀⠀⠀⣠⠞⠒⠒⠀⠚⢧⣀⣀⣠⣋⣀⡀⠀⠀⠀⠀⠀⠀⠈⡇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⣧⠟⠀⠀⠀⠀⠀⠀⠀⢣⢠⠎⠁⠀⣀⣀⣀⠀⢀⡈⢳⠖⠤⠤⠬⠭⠭⣉⣉⠭⠶⣆⠇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢰⠋⠁⠀⠈⡆⣿⢱⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣤⡿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⡇
⠀⠀⠀⠀⠀⠳⠦⠤⠤⠤⠤⠤⠤⠴⢲⡏⠀⡇⠀⠀⠀⠀⡇⣿⢸⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠯⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⡇
⠀⠀⠀⠀⠀⢰⠤⠔⠊⠉⠉⠉⠒⠒⣾⡇⠀⡇⠀⠀⠀⠀⡇⢿⣸⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣶⡟⠒⠒⠒⠒⠒⠒⠂⠀⠉⠉⠉⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⠀⠀⠀⠙⠒⢄⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠷⡿⠭⠭⠭⠭⠭⠭⠭⠭⠽⠿⠛⡆
⠀⠀⠀⠀⠀⣘⡒⠤⠤⠤⠤⠤⠤⠐⣻⡇⠀⡇⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠤⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⡅
⠀⠀⠀⠀⠀⡇⠈⠉⠉⠉⠉⠉⠉⠁⠀⣇⠀⠈⠉⠐⠒⠒⠂⠉⠁⠀⡟⠒⠒⠲⠦⠤⠤⣧⠤⠲⣟⡿⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⠇
⠀⠀⠀⠀⠀⠳⠤⠤⢀⣀⣀⣀⠤⠤⠔⠻⠤⣀⣀⣀⣀⣀⣀⣀⣀⠴⠋⠙⠛⠓⠒⠲⠶⠾⠖⠚⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠴⠶⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠴⠶⠖⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢸⠏⠁
⠀⠀⠀⠀⢤⣴⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠻⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠰⡇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣿⣿⣿⣿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⡇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⣠⣤⣧⣤⣾⡀⢀⣧⣤⣤⣤⣄⡀⠀⠈⠙⢷⣌⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣠⠖⠚⢯⣁⢀⣀⣀⡀⠉⠉⠀⢀⣀⣠⠬⣝⠳⣦⡀⠀⠛⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⡖⠁⠀⢀⡸⠋⠉⠉⠉⠉⢉⣻⠟⠉⠉⠁⠀⠈⣇⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⣰⡶⠚⠛⠛⠛⢣⣀⣠⢿⣤⣀⣀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠉⢹⠄⠈⢷⣄⣀⣀⣀⣀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣻⣿⡶⠶⠤⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠆⠘⢧⣀⣀⣤⠀⠀⠀⣠⣼⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⠴⢧⣀⡀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠲⠤⠖⠁⠈⠷⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠾⡇⠀⠀⠹⡛⢻⡁⢀⡿⠻⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣹⠆⢨⠿⡟⢿⠄⠀⢀⣧⣄⣉⣉⣀⣀⣀⣀⡤⠤⠔⠛⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠃⢪⡀⠀⢸⠋⢻⠛⡏⠉⡿⠛⠃⠀⠀⠀⠀⠠⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡍⠛⢿⡶⠾⠿⣶⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⣀⣤⡶⠛⠛⠛⠓⠒⠶⠶⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣁⠀⠈⢳⡒⣦⣀⣀⣀⠳⠤⠴⢶⡀⠀⢀⣖⣋⣀⣤⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠼⠀⠀⠀⠉⠁⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣾⣿⣿⣶⡦⡔⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣏⡉⢿⣿⣿⡜⡞⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣾⣿⣿⣿⣿⣿⣿⣿⣇⢠⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⣿⣿⣿⣿⣿⣿⡏⡈⣘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⢾⢿⣿⣿⣿⣿⣿⣿⡷⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠻⠿⠿⡇⠓⠦⠀⠀⠀⢐⣀⣀⣶⣤⠤⠔⣠⣴⠟⠂
⠀⣀⣀⣀⣀⣀⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠃⠀⠀⠀⠀⠠⠄⠀⠠⠀⠀⠈⠉⠉⠄
⠀⠈⠙⠛⠛⠛⠿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠷⠦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⢤⡤⣔⢲⣆⣲⠰⠯⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠸⢹
⠀⠀⠀⢀⠆⡃⡞⠙⠁⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠉⠡⠤⠠⠤⠐⠒⠒⠒⠢⠤⡀
⠀⠀⠀⠀⠀⢀⠃⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⡸⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⢀⠠⠤⠐⠒⠂⠀⠀⠒⠒⠤⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠇⢀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠐⠒⢲
⠀⠀⠀⠀⢰⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠄⢇⡀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⢸⠀⠠⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⣧⠠⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⡖⠉⠉⠀⠘⠀⠁⣢⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⠠⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡁⢘⠀⠀⠀⠀⢈⠀⠈⠉⠁⠈⠉⠉⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⣤⠀⠀⡀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠰⠀⠤⠀⡇⢰⠀⠀⠀⠀⠸⡀⢰⠁⠐⠒⠈⠂⢸⠀⠀⢀⠀⠀⠀⠈⠄⠈⡁⢀⠀⠠⠀⣊⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣧⠜⠀⠀⠀⢀⡔⠃⠈⠢⠨⠬⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⢌⠀⠠⠀⠀⠀⠊⠀⡀⠈
⠀⠀⠀⠀⠈⡗⠢⣤⠦⠒⠙⠀⠀⠀⢠⢊⡔⠊⠈⠉⠉⠁⠐⠒⢄⠀⠀⠠⠀⠀⠀⠀⠀⠀⡐⠉⠁⠀⠀⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠈⡄⠀⠀⠀⣀⣀⡀⠀⢘⢓⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⢬⢠⡈⡰⠂⠤⠀⡤⠠⠔⡨⡤⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⡜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠌⠡⠁⠀⢀⢂⣤⠀⠀⠀⠀⠀⠈⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠐⠒⠀⠉⠀⠀⣀⠠⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠤⠤⠔⠒⠋⢉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⢀⠄⣤⣤⣤⣤⣤⣤⣤⡄⠀⣠⣄⠀⠀⣠⣄⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠠⡀
⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⡧⣾⠉⠙⣧⣼⠉⠉⣷⢸⡯⣤⢤⡥⣤⢤⡤⣤⣼⡈⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡌⣦⣤⣤⣤⣤⣤⣤⠇⣙⠷⠾⡁⣘⠷⠾⡋⣸⡷⡾⡾⠷⡷⢾⡷⣾⠾⠇⡀⣠⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠇
⠀⠀⠀⠀⠀⢠⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠉⠀⠀⠉⠉⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣠⠤⠐⠒⠒⠂⠤⣄⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡾⠁⠀⢠⠔⠉⣀⡤⠶⠶⠶⠶⢤⣀⠉⠲⡄⠀⠈⢷⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡾⠀⠀⡰⠁⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⢆⠀⠀⢷⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠃⠀⢰⠃⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⡆⠀⠘⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⣧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠸⡀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⠇⠀⢀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣧⠀⠀⠣⡀⠙⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢀⠜⠀⠀⣼⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢧⡀⠀⠑⢄⡈⠙⠲⠤⣤⣤⠤⠖⠋⢁⡠⠊⠀⢀⡼⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠒⠢⠤⠄⠠⠤⠔⠒⠉⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀⡀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⢤⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠚⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠈⠹⠍⠉⠉⠩⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⠍⠉⠉⠩⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠽⠤⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠬⡀⡨⠀⠀⠀⠊⠀⡖⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢱⠀⠊⠁⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⢀⠙⠐⢁⠤⢊⠂⠀⠀⣈⠂⠤⠊⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⣉⡉⠙⠀⡇⠀⠀⠀⠀⠀⠀⠈⡠⠊⠀⠒⠓⠊⠀⠀⠒⣄⠀⡂⡂
⠀⠀⠀⠀⠀⠀⠠⠐⠊⠁⠉⣉⢀⡐⡲⠛⣈⠝⢀⠀⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⣞⠠⡀
⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⢄⣀⡠⠴⠮⠤⠤⠜⠀⡇⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠠⠤⢤⠐⠊⠥⣂⡈⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠃


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢐⠙⠀⠉⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⡁⠀⠀⠙⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠄⠀⠈⠀⠈⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⣉⣀⡀⠀⠀⠀⠀⠀⠂⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠒⠠⡀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠒⠚⠉⡉⡉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢲⢙⠠⢙⠃⠄⢡⡐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠰⠀⠐⣤⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠆⣀⠀⢳⡨⡀⠌⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⠢⣁⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠉⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢰⢲⡶⠶⠶⠶⠶⠶⢶⠒⣒⣶⣒⠒⢒⣶⣖⠒⣲⣶⣶⣶⣶⣶⣶⣶⣶⣶⡒⠒⢲
⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⢴⡏⠀⢹⡦⣟⠀⢘⡇⣿⠧⡤⢤⠤⡤⣤⠤⡤⢷⡇⠀⢸
⠀⠀⠀⠀⠀⢸⠼⠶⠶⠶⠶⠶⠶⠶⠤⠽⠿⠯⠤⠽⠿⠿⠥⠿⠶⠷⠾⠾⠷⠿⠶⠷⠾⠧⠤⢼
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠁⠀⠀⠀⠀⠉⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⣀⠤⠒⠒⠀⠒⠒⠢⢄⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⠏⠀⢀⡴⠋⢀⣤⠶⠚⠛⠓⠶⢦⣀⠉⠢⡀⠀⠘⢧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠇⠀⢀⠎⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀⠘⡄⠀⠈⣧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⡜⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢸⠀⠀⢸⡄⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠆⠀⢸⡇⠀⣀⣸⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢷⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣸⠀⠀⣸⠃⠸⣅⣀⣀⣼
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⣆⠀⠈⢦⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣠⠃⠀⢠⡿⠒⢲⠯⠭⠥⠿⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⣆⠀⠀⠑⢄⡈⠙⠲⠶⠤⠴⠶⠛⠁⣠⠖⠁⠀⣠⣿⢠⢒⠇⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠓⠢⠤⠤⠤⠤⠒⠉⠁⠀⣀⡴⠋⢸⡼⠋⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⣀⣀⣤⡴⠞⠉⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⡗⠒⠒⠀⣤⡔⠒⠒⢺⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠲⣷⠒⠊⠁⠀⠈⠉⠉⢺⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⢚⠯⢵⣶⣖⣒⣒⣶⡭⠽⡇
⠀⠀⠀⠀⠀⠀⠈⠫⠤⠤⠤⠼⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⠈⠁⠈⠯⢽⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠈⠉

⠀⠀⠀⠀⠀⢹⠛⠛⠛⠛⢻⣩⠿⣏⣩⠿⣏⣽⠿⠿⠿⠿⠿⢿⡏⠙⡆
⠀⠀⠀⠀⢠⣼⣤⣤⣤⣤⡼⠹⣦⠞⠹⣦⠟⢿⡽⣽⣽⣽⣽⣽⠇⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠐⠖⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠲⠶⠶⠶⠶⠶⠶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠲⠦⢤⣀⠀⠀⠀⠀⡠⠔⠓⠈⠁⣀⡄⠈⠍⠢⢄⠩⢥⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⠤⠄⠀⠠⠤⣀⠈⠳⣄⠀⢰⠓⠒⠀⠀⣻⠷⠤⠬⠭⠭⠟⢈⠉⠓⠓⠚⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠊⣠⠖⠚⠛⠓⠲⣄⠑⡄⠈⢷⠈⠀⠀⠀⢠⣋⣋⣉⣉⣉⣉⣉⣛⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠁⢠⠁⡞⠁⠀⠀⠀⠀⠀⠈⢳⠈⡄⠈⣿⠀⠀⠀⣎⣉⣉⣉⣉⣉⣉⣉⣉⣡⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠰⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⢃⠀⣰⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠘⡄⢳⡀⠀⠀⠀⠀⠀⢀⡞⢠⠃⢀⡏⢸⢀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⢑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠘⢆⡙⠲⠤⣤⠤⠖⢋⡠⠃⢀⡞⠀⠸⠈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠐⠒⠀⠒⠊⠁⣀⡴⠋⠀⠀⠀⠀⣉⣉⣁⣉⣉⣁⣉⣉⣁⣉⣈⡁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠶⠒⠋⠁⠀⠀⠀⠀⠈⠁⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⢠⡆⣈⣉⣁⣉⣉⣈⣈⣁⣉⣀⣀⡁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠸⡃⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⡀⡄
⠀⠀⠀⠀⠀⠑⠖⠒⠒⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠲⠣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡔⠈⠑⢆⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢱⣄⡀⣘⠀⠀⠀⡇⡠⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⠄⠤⠀⢊⠉⢻⠉⠀⠀⢰⠉⢩⠋⠛⡏⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⡎⢀⢀⠀⢈⡆⠊⠉⠒⠀⠒⠁⠙⠢⠐⠃⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⡋⠊⠴⣾⠿⣀⣀⣀⠀⠀⠀⠀⡠⠔⠊⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⢞⠆⠒⠉⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣢⠤⠤⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠉⢀⡠⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠤⠔⠊⠁⢀⡰⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⣀⠠⠐⠊⠁⣀⠴⠀⠀⠀⡀⠄⠊⠁⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠣⠠⠤⢲⠋⢀⣀⡤⠖⣛⠢⠬⣁⠒⢤⣀⠤⠊⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡒⠢⢔⠛⠥⡜⣀⠀⠉⠈⠉⠁⠀⡀⡡⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⠪⢕⠢⠬⠭⢿⡤⡁⠒⠀⢐⠬⠡⠔⠁⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠈⠉⠀⠀⠁⡠⠭⠙⠁⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂⠀⢎⠀⠀⣀⡀⠤⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠠⠤⠄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢃⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠎⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠒⠠⠤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠎⠀⠐⠋⠁⠈⠑⠒⠤⡤⢀⣀⣀⡀⠀⠈⣂⠀⠀⠈⠙⠀⠀⠀⠀⠹⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡸⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠄⠀⠀⠀⠀⠀⠀⢡⠀⠸
⠀⠀⠀⠀⠀⠀⢀⡤⢦⣀⠃⠀⠀⠀⠈⠄⡄⣱⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢰⡀⢦⡄⠁⠀⠀⠀⠀⠀⠈⣄⠧⠤⡀
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠙⠀⠀⠀⠀⠻⡜⢧⠟⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⢷⣟⡵⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠉⠉⠉⠉⠉⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⡛⠛⠛⠛⠛⣏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠥⣀⣀⣀⠠⠜⠙⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠑⡄
⠀⠀⠀⠀⠀⠀⢠⠖⣉⢉⡑⢄⠀⠀⢠⠏⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡆⢸⠁⠀⠱⠞⠀⠀⡞⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣀⣀⠀⢸⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⣅⣠⠤⠤⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣈⣉⣉⣉⣧⠀⠀⠀⠀⠸⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠸⠥⣴⠒⠊⠉⢳⡀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⣰⠃⠀⠀⡸⠁⠀⠀⠀⠀⢀⡠⠦⠼⣀
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠙⠢⣄⣀⠀⢠⣄⢀⣈⣀⡴⠁⠀⠀⡰⠁⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢀⢶⠀⠀⠀⠀⠀⡝⠲⢤⡀⢀⡀⠈⠈⢠⣍⠀⠉⠁⠐⣀⡴⢾⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣧⣸⣄⠀⠘⠖⠀⠀⠀⠀⠀⡇⠀⠄⠀⠉⠉⠉⠉⠰⠓⠄⠒⠂⠉⠀⠀⢾⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡠⠤⠶⠤⠤⠼⠇⠤⠤⠤⠤⠤⠤⠤⠧⠄⠤⠤⠤⠬⠶⠶⠦⠤⠶⠠⠾⠁⠤⠄⠼⠤⠤⠤⠤⠤⠤⠼⠄⠤⠤⠤⠤⠴⠦⢤⡀
⠀⠀⠀⠑⠐⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠘⠁
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠲⠀⠠⢤⣤⣤⡤⠤⠄⠒⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣶⣾⡿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠉⠀⠀⠉⠉⠛⠛⠶⢶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣶⣶⣿⠿⠿⠿⢿⣿⣿⣛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣒⣛⣛⣛⣻⣭⣭⣭⠭⠿⠶⣶⡶⠖⣛⠛⢻⡟⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠁⠀⠀⠘⣿⠐⣿⣿⡿⠿⠿⠿⠇⢸⡇⠀⣾⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠄⠆⠂⠐⢘⠀⢸⡇⠀⠙⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠰⣿⠟⠿⠿⠀⣿⠀⢀⣃⣁⣁⣬⣤⣤⡼⠷⠶⠖⠺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣶⣶⣿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠁⠀⠀⠀⢀⣠⣤⡶⣶⣶⡶⢶⣤⣀⠀⠀⠀⠀⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣶⡿⣫⣷⣿⣿⣿⣭⣛⠶⣝⢷⡄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢷⣻⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡇⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⣿⡿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠹⣿⣷⡽⢿⣿⣿⣿⡿⢟⣫⣾⣿⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠀⠀⠘⠿⣿⣷⣭⣽⣿⣿⣭⠿⠛⠁⠀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⢀⣀⣠⣤⣶⣶⣾⡿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⣤⣤⣴⣶⠾⠿⠛⠛⠉⣿⡇⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶⣾⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⣀⣴⣶⡷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠋⠉⠁⠀⠀⠀⣀⣀⣠⣄⣤⣤⡶⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣴⣶⣾⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠂⠉⠁⣤⠔⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠈⠁⠢⢀
⠀⠀⠀⠀⠀⢀⡠⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠤⠤⡤⠔⠈⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⢀⣠⠤⠒⠀⣈⡉⠍⠁⠀⢢⠀⠀⢀⠄⠐⠀⠀⢖⠒⠢⡴⠁⠀⠀⣀⠄⠐⠉⠀⠀⠀⠀⠀⠀⠁⠢⢽⡀
⠀⠀⠀⠀⠀⠋⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⢃⠔⢁⣠⠄⠀⠀⠀⠉⢹⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠃⠀⠀⠀⠒⠂⠀⠀⠠⢄⡀⠀⢸⣷⠟⠉⠀⠀⠀⠀⠠⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⠐⠒⠂⠠⠝⣂⣸⣧⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠈⠒⠒⡒⠢⣄⡉⠐⠢⢔⣋⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⣇⠀⠙⢦⣉⠀⠀⡸⡏⠉⠙⡏⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⡏⠀⠉⢻⣏⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠸⣷⣤⡀⠈⢉⠿⢀⠈⠀⠀⢳⣶⣽⣷⣦⣄⣀⣑⡒⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠛⠉⠀⠀⣠⡟⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠉⠁⠸⠢⢄⣀⠀⠀⠇⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠲⣤⠀⠀⠀⠀⣇⣀⡤⠤⠒⢛⣿⠁⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠉⠁⠀⣀⣀⠀⠠⣄⡀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠈⣇⠀⠀⠀⡟⠀⡐⠀⡆⠈⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢃⠸⠀⠀⠀⠀⠀⢇⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⢹⠀⠀⠀⡇⢀⠁⢠⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⠀⢗⠒⠂⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⢰⠃⡌⠀⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠂⠀⡧⢀⢀⠴⡃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠱⠐⠁⢀⡁⠐⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡀⠀⠀⠀⠀⠣⠀⠆⠀⠀⠀⠄⠀⠄⠀⠀⠀⠇⠄⠂⠀⠀⠀⢀⡀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡅⠑⡀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⢀⠊⢉⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢡⠀⢁⠀⠀⠁⠀⢸⠀⠀⠀⠉⠉⠁⠀⠀⢠⠀⠀⠀⠀⠘⠀⡌⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⡀⠈⡄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⠂⢀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠒⠁⠀⠘⡄⢀⠆⠀⠀⠀⠀⠀⠀⠀⢆⣀⠜⠀⠀⠑⢁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠑⠒⠃⠀⠀⠀⠀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠘⠐⠃⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣵⣶⣿⣿⣷⣶⣄⢢⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡫⠛⣿⣿⣿⣯⢳⡹⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣻⣿⣿⣿⣧⣆⣽⣿⣿⣿⡆⢳⢻⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡇⢸⢘⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣜⣸⡀⢀⠸⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⢿⣿⣿⣿⣿⣿⣿⣿⣟⣴⠃⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢿⣿⣿⣿⣟⠹⣯⣀⢀⠀⠀⠀⠤⠀⠀⣠⣄⠁⠀⡠⠄⠀⣉⣶⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢦⠤⡄⠀⠀⠀⠀⠐⠒⠒⠚⠋⠉⠉⠁⢚⠭⠷⠤
⠀⠲⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠍⠉⠈⠉⠉⠉⠐⠀⠀⡀⠀⠁
⠀⠀⠀⠉⠉⠉⠙⠛⠿⣿⣿⣤⣄⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⣀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣄⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣰⣄⠠⠆⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣥⣄⡀




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠰⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠤⠤⠤⠤⠤⠼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠽⠟⠻⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠤⠤⢞⠀⠀⠀⠀⠀⡠⠐⠒⠒⢲⠓⠂⠤⠤⠄⠒⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢸⠀⢀⠤⣀⡜⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣼⠀⡇⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⡠⠄⠒⠒⠒⢢⠀⠀⡴⠲⡄⠀⢽⠀⣃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢰⠀⢸⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢨⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⡇⢸⡋⠁⠒⠒⠒⠰⠦⠤⠴⠖⠐⠒⠒⠂⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠣⠤⠜⠀⢸⠂⣀⣀⣀⡠⠔⠒⠒⠒⠒⠢⠤⠄⠠⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⡀⠀⢸⣿⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢸⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⠒⠠⠤⠤⠤⠤⠤⠤⠤⠐⠚⠀⠀⢸⡀⣙⣒⣢⣤⣤⡤⠤⠤⠤⠤⠐⠒⠊⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠶⢶⣶⣶⠿⠿⠿⠟⠛⠛⠛⠻⣿⣿⣏⣛⠶⢶⠒⠒⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠒⠻⢶⠶⠶⠶⠛⢛⣛⣋⣉⣡⣭⣤⣤⣤⣼⣄⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⡇⠀⠀⠀⠀⠀⢸⢀⠚⠻⠭⢿⣿⣛⣉⣭⣭⣥⠭⠿⠿⠓⠚⢹⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⣄⣸⣨⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣠⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠼⠷⠖⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠶⠶⠶⠖⠒⠶⠦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣿⣿⣿⣷⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠛⠻⢽⣓⣶⡶⣶⣶⣒⣚⣛⣿⣿⣯⣭⡿⠿⣿⠿⣶⠚⠛⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢹⡟⠉⠉⠉⠀⠘⣿⠰⣶⠿⠿⠿⠿⠃⣿⠀⢰⣿⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡀⠀⠀⠀⠀⢸⠀⢀⣀⣀⣀⠀⣿⠀⠐⢐⢐⢈⠨⠀⣿⠀⠀⣉⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡀⠀⠀⠀⢸⡇⠘⠛⣉⣋⣀⣿⣤⣤⣤⡶⠶⠶⠚⠛⠛⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⡇⠀⠀⢰⢸⡿⠿⠟⠛⠛⠋⠉⠉⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⡇⠀⢸⢸⢸⡇⠀⠀⠀⠀⢀⡴⣚⣭⣶⣶⣖⠮⣝⢿⣦⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⣷⢸⢸⢸⠸⡇⠀⠀⠀⢀⣞⣼⣿⣿⣿⣿⣿⣿⣎⢳⡹⣷⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹⢸⢸⢸⢸⠀⣇⠀⠀⠀⣸⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣇⣿⡇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⢸⢸⢸⠀⣿⠀⠀⠀⢻⢻⣿⣿⣿⣿⣿⣿⣿⡿⢀⣇⣿⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⢸⢸⠸⠘⠀⣿⡇⠀⠀⠘⣞⢿⣿⣿⣿⣿⣿⠟⣡⢞⣾⠏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⠀⠀⠃⣿⡇⠀⠀⠀⠈⠢⣝⡻⠿⠿⣗⣫⣷⠟⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡄⣿⣧⡄⠀⠀⠀⠀⠀⠉⠙⠛⠉⣉⣀⣠⣤⣶⣶⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠃⣿⣿⡇⠀⠀⣀⣀⣤⣤⣶⠾⠿⠛⠛⢹⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢻⣿⣿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⣀⣸⣧⡴⠿⠗⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⢸⠀⢀⣀⣀⣤⣤⣦⡶⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⠒⠛⠋⣉⠉⠉⠉⡉⠉⠙⠛⠒⢒⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣤⡀⠀⠀⠀⢀⣠⣔⣯⣿⡟⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⠿⠿⠿⠯⠬⠿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠀⢰⣿⡎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣏⣩⣍⣤⣤⣴⡶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⣿⣧⢻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡯⠻⠛⠉⠉⠀⠁⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣟⣻⣿⣿⢹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⣀⡀⠠⠤⣤⣀⣒⣒⡒⠒⠺⠟⠿⠻⠿⠛⠛⠛⣛⣛⣛⣻⡿⠿⠿⢿⣿⣯⣿⣛⡋⣾⡿⣟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠑⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⢾⣫⣭⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣧⣿⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⠔⣫⣭⣑⢦⡀⠀⠀⠀⠀⢸⡞⣵⡿⠋⣙⣙⣯⣵⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢀⣀⣀⣸⡇⠀⣮⣾⠁⠀⢹⣦⡇⠀⠀⠀⠀⢸⣳⣿⡁⢸⠛⠛⢻⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⢠⣏⣠⣾⣿⣇⢸⠣⣹⢶⣶⢿⣿⠇⠀⠀⠀⠀⢸⣏⢿⣧⡀⠙⠛⣩⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠟⠀⠉⠛⠛⠛⠁⠀⢀⣀⣀⣀⣈⣈⣉⣙⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠉⠉⠷⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⣻⠋⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣭⠉⠍⠩⠍⠍⠩⠭⠭⠭⠭⠽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠶⠀⠰⣾⣿⣿⠀⠀⢀⡀
⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢤⣾⣻⡏⣧
⣰⣿⣿⣮⠉⣉⣉⣉⣛⣒⣒⣲⣶⣦⣤⣿⣿⣿⠿⣿⡿⠿⠿⢿⣿⣿⠋⢹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣯⣭⣷⣶⣶⣶⣾⣿⠉⠹⣿⣿⠃
⣿⡏⠙⢿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠻⣿⣿⢀⣾⣿⡿⡿⢹⡏⠹⠇⣿⣿⣿⣿⣿⣿⡛⠛⣿⣿⣿⠀⠀⡏⢿
⢹⣷⣤⠜⣰⣿⣿⣿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⣤⣀⡀⠀⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣿⠀⠀⣷⢸
⠀⠈⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣹⢏⣼⣿⣿⠀⣿⣿⣿⡿⢿⣿⣼⣿⣿⣿⣿⠀⠀⢸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⠉⠉⢉⣉⠉⠉⠉⠉⠉⠉⠵⠋⠸⣿⣿⠀⣿⣿⣿⣼⣿⡟⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣭⡻⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠘⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢿⣿⣿⣜⣿⣿⠿⢿⣇⡀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⠀⠀⠘⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣿⡿⣹⣿⣷⣶⣾⣿⡿⠀⠀⠀⢤⣿⣿⠀⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡷⠿⣿⣿⣿⣿⣿⣿⠥⠄⠀⠄⣾⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⣔⣒⣯⠭⠭⠶⠖⢛⣒⡀⠀⠀⣠⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀
⠀⠀⠀⠀⢠⡾⠋⠉⣉⣩⣭⣭⡍⣁⡀⣂⣤⣦⣽⣿⡟⠻⠿⠿⠿⠶⠶⢶⠶⠶⠿⢿⡿⠿⠷⢿⡿⠿⠿⣿⣟⠿⣿⣿⣻⣿⣿⣿⡿⣿
⠀⠀⠀⠀⣾⠀⡭⢚⠩⠍⠶⣐⡿⠿⠟⣛⣿⣿⡟⠛⠻⠶⠶⠶⠿⢭⣉⣉⣙⣛⠂⣠⣤⣀⣶⣤⣽⣶⣶⣶⣿⣿⣿⣿⠯⣟⠓⠂⢀⣿
⠀⠀⠀⠀⢻⣌⡖⠁⠰⣿⣿⣿⣿⣿⣿⣶⣆⠀⠈⠉⠉⠉⠭⢭⣽⣿⣮⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣛⣳⣬⣭⣴⣶⣶⣿⣏⣁
⠀⠀⠀⢀⣤⣬⣽⣷⣶⣶⣭⣭⣟⣛⣛⠛⠋⠀⠀⠀⠀⠀⠶⣶⣿⣿⡿⢻⣟⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⠿⠿⠶
⠀⠀⠀⠀⠉⠛⠛⠛⠻⠷⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠛⢩⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠉⠁⠀⣀⣠⡤⢼⣦⣄⡀⠈⠉⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠛⠉⠉⠀⣀⣠⡤⠶⠚⠛⠉⠁⠀⣼⠀⠀⠉⠓⠶⣤⣀⠀⠈⠉⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠚⠋⠉⠁⠀⣀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣠⣤⣤⣀⣀⡈⠉⠓⠦⣤⣀⠀⠉⠛⠳⣤⣀
⠀⢀⣠⡤⠖⠛⠉⠀⠀⣀⣠⣤⠴⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠦⣬⣷⠟⠁⠀⠀⠈⠉⠻⣄⡀⠀⠀⠈⠙⠲⢤⡀⠀⠉⠙⣳⢦
⠀⣿⠣⢤⣠⡤⠶⠚⠉⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠶⠤⣤⣤⠽⠿⠚⠋⢁⣼
⠀⢹⣦⣀⡈⠙⠲⢦⣄⡴⠒⠛⠷⠶⠛⠉⠉⠳⣄⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠒⣶⠊⠁⠀⢀⣠⡴⠞⠉⣿
⠀⠈⣧⠈⠉⠑⠦⣄⡈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠈⠢⣄⣤⡤⠤⠶⠦⣤⣤⣤⣀⡀⠙⠳⣦⡤⠔⠚⠉⠉⢧⠀⠀⠀⢸⡦⠖⠛⠉⠁⠀⠀⠀⡏
⠀⠀⢹⡄⠀⠀⠀⢠⡍⡗⠦⣄⡉⠛⠶⣤⡀⠀⠀⠀⠈⠓⠲⠶⠒⠀⠀⠀⢀⡩⠿⠚⠉⠁⠀⠀⠀⠀⠀⢸⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣧⠀⠀⠀⠘⣿⡅⠀⠈⠙⠲⢄⡈⠙⠛⠶⣦⣄⡀⠀⠀⠀⣀⡤⠶⠋⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⣷⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢻⡀⠀⠀⠀⠻⣕⢤⡀⠀⠀⠀⠙⠲⢦⣄⣀⠉⠛⢳⡖⠋⢁⣠⠴⠖⢻⡇⠀⠀⠀⠀⡄⠀⠀⠀⢸⡇⠀⣸⠀⢿⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠛⠾⢷⣶⣤⣀⣠⠟⠀⠈⠙⠲⢼⡗⠚⠉⠀⠀⠀⢸⠁⠀⠀⠀⡼⠀⡀⠀⠀⢸⡇⠀⠏⠀⢸⡆⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡇⢠⠃⠀⠀⢸⡇⠀⠀⣀⣸⡇⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡟⠀⠀⠀⠸⠀⡌⠀⠀⠀⠸⡷⠞⠋⠉⠁⠀⠀⠀⣀⡾⠁
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠀⣀⣴⠇⠀⠀⠀⢀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⣠⠶⠶⠖⠚⠋⠁⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧⠶⠋⠁⠀⠀⢀⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣤⡀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⢸⡏⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣼⣧⣤⠶⠚⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⠶⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⠲⣄⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠘⡇⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡧⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣧⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡯⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡤⣤⣤⣤⣤⣤⣤⡤⡤⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢠⠶⠶⠶⠶⠶⠶⠶⠶⠤⣄⠀⠀⠀⠀⡼⠋⠈⢹⡆⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢸⠃⠀⠀⢸⡇⠀⢸⣧⣤⣤⣤⣤⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⣠⠟⠈⢳⡄⠀⠀⢿⠀⠀⢻⠀⠀⠀⢸⡇⠀⢸⡷⠶⠶⠖⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⣰⠋⠀⠀⠀⠹⡄⠀⠘⣆⠀⠘⣇⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⣏⠀⠀⠀⠀⠀⡷⠀⠀⠘⣦⠀⠈⢧⣀⢸⡇⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠘⠦⣄⣀⣤⠞⠁⠀⠀⠀⢸⠀⠀⠀⠉⠛⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠒⠒⠒⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠈⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⢾⡁⣹⠆⠀⠀⠀⣿⠁⠉⠈⠉⠉⠁⠉⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⢛⣋⣉⣉⣛⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣥⠖⠛⠉⠉⠀⠈⠉⠙⠓⢦⣉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⠏⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡈⢷⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⡄⢷⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡿⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⡆⢷⠸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢸⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢷⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠃⡟⢸⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣇⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡟⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⢦⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣡⠟⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣍⠓⠦⢤⣤⣀⣀⣤⠤⠖⢋⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠤⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢜⣹⣷⡆⠀⠀⠀⠀⠀⠀⠦⠷⡆⠀⠸⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣻⣆⠀⠀⠀⠀⠀⡀⣸⡇⠀⠀⡀⠀⢀⣀⡠⡀⡘⢠⣿⣿⣿⣿⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⣀⣷⠜⢅⣸⣿⣿⣿⣿⣶⣭⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠛⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣦⣧⡀⠙⢿⣻⣿⣵⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡟⢿⣿⣿⣿⢣⡰⢇⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣛⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣰⣃⡘⣿⣯⣾⢟⣿⣿⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡿⡿⣠⣤⣤⣄⣀⣀⣀⣀⣀⡈⠉⠙⣿⠉⡉⢿⣯⣆⢺⣥⣽⣿⣿⡟
⠀⠀⠀⣠⣼⣿⣿⣶⡞⠀⣩⣾⣯⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⠿⣿⣼⣟⣿⣿⣿⠋⠉
⠀⠀⠐⣿⣿⣿⣿⣿⣶⣾⣿⣿⣽⣿⣿⣿⣟⣿⣿⣿⣿⣿⣯⣭⣽⣛⣛⣛⣟⣿⣿⣿⣿⣿⣄⡀
⢀⣤⣾⣿⣿⣟⣿⣿⡿⡿⣿⣻⣿⣿⣯⣿⣿⣟⣿⣿⡿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⢨⣿⣿⣿⡿⢛⣤⣚⣿⣭⣽⣿⣿⣿⢿⣿⣿⣷⣿⡛⠛⣿⡿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⠿⣃⠄
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠂⠛⠛⠋⠉⠙⠛⠛⠳⠿⠶⣠⣻⣿⡿⢿⣿⣾⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣤⣴⡹⠋⠹⣉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⢴⣶⣶⣶⣶⣶⣶⣶⣶⣼⣶⣿⣷⣦⣼⣷⣶⣿⣿⣿⣹⣏⢀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠧⡄⣈⠟⢻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⣻⣷⣤⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠈⢳⡀⣸⣄⠀⢹⣿⣯⣷⣶⣶⣶⣶⣶⣶⣶⣼⢻⣽⣽⣿⣿⣿⣿⠿⣏⡀⣀⣇⣀⡀⡄
⠀⠐⠒⠒⡶⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡶⠛⠉⠉⠉⠙⠛⠛⠛⠋⢸⣯⣯⡿⣿⢽⣿⡾⣴⣶⣶⣿⣶⣿⣧⣤⣄
⠀⠀⠀⠀⠱⡀⠀⣾⣿⣿⣯⣿⣿⣟⣿⣿⣿⣏⠀⣴⣶⣶⣶⣖⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇⢻⣿⣯⣿⣷⣿⡷⣿⣧⣤⣄⡀⡀
⠀⠀⠀⠀⠀⠐⠄⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣯⢿⣿⣿⣿⠿⠭⠩⢽⠭⠛⠛⣛⣷⣿⣷⣎⡉⣿⣹⡏⢻⣇⡼⠃
⠀⠀⠀⠀⠀⠀⢈⣿⣇⣼⣿⡿⠿⢿⡟⣿⣿⣿⢀⣤⣤⣴⣶⣄⣶⠶⠶⣶⣼⠿⡏⢱⣿⣟⣿⣷⣿⣿⣿⣿⣧⣾⡟
⠀⠀⠀⠀⠀⠀⠐⣿⣟⢩⡏⢳⣄⣸⣗⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣽⣿⣿⡇⣿⣿⣟⣛⣿⠉⠀⣹⡇
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠈⠛⣷⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠸⠿⠿⠛⠛⠛⠉⢉⣭⡭⠽⠥⠴⢚⣚⣛⣋⠩⠭⠽⠷⠚⠛⠛⠛⠁
⠀⠀⠲⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⣀⣀⣠⣤⣤⠶⡶⠒⠛⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣁⣈⣉⣉⣉⣉⣉⣉⣉⣋⣉⣉⣉⣉⣉⣉⣉⣁⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂
⠀⠀⠀⠀⠀⠐⠶⠒⠄⠀⠠⠀⣤⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣨⡡
⠀⠀⠀⠀⠀⡠⠠⠄⠀⠀⠠⣖⣒⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⢨⠿⠀⣀⠀⠀⠠⠠⠀⠀⣀⠠⣮⠁
⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠐⣒⣒⡂⠸⡇⣠⡄⠀⡀⠀⠀⠈⠉⠀⠉⠁⠉⠉⠁⠉⠀⠀⠀⠀⠆⠒⢻⠽
⠀⠀⠀⠀⠠⠖⠀⠁⠀⠀⠀⠒⠒⠂⠈⠁⠀⣀⠐⠀⣉⠤⢉⠉⠉⠙⠙⢒⠒⠲⡒⠒⠀⠀⠀⣀⠈⡛
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⡀⠈⠃⡦⠄⡐⠀⠒⠒⡆⠉⠀⢀⠈⣡⠶⠖⠺⠀⢠⠤⠐⠠⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠂⠐⠆⠀⠀⠀⠀⢤⠤⡅⠀⠀⠀⡇⠀⠐⠦⡤⠤⡴⠀⢀⠆⠀⠀⠀⠈⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⢢⠀⠤⠤⡇⠀⠰⠂⢀⣀⠘⠀⢄⠀⣄⡤⠀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⢔⡐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠨⠽⠈⠀⢤⣴⠀⠀⠘⠀⠸⠿⠆⠀⠀⠀⡈⣙⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⡎⢈⠀⢀⠰⡆⡄⠀⠀⠀⠠⠤⠭⠭⠤⠤⠤⠀⠠⠤⠀⠀⠠⠤⠬⠽⠖⠲⠲
⠀⠀⠀⠀⠀⠀⠁⠁⠉⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠁
⠀⠀⠀⢰⣰⠀⠀⢸⢾⡇⠀⠐⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠐⠀⠐⠀⠂⡰⠂
⠀⠀⠀⢸⢼⡀⠀⢸⢻⡄⠀⢸⣿⡿⣿⣿⣿⠇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⣳⣄⣀⣠⣜
⠀⠀⠀⡼⠘⡇⠀⠸⠻⠇⠀⢸⣿⣷⢾⣿⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠟⠘⠇⠀⠀⠀⠀⠀⢸⣿⣿⣾⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢿⣋⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⠾⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠉⠓⢦⣀⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⠀⠘⠓⠒⣄⣀⡀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠘⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠂⠒⠛⠓⠀⢰⣿⣿⢸⡀⠀⠀⠀⠀⢀⣶⠆⠀⣀⠁⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡄⠀⠘⣿⣿⣿⠁⢾⣦⠀⣀⠀⠁⠀⠸⠿⠀⠀⠘⣿⣿⣿⣿⣿⠇
⠀⠀⡀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⠀⠀⠁⠀⠻⠷⠀⠀⠀⡀⠀⠀⠀⢻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣀⣠⣤⣴⡄⠀⠀⠀⢈⣿⣿⣿⡃
⠀⠀⠀⠀⠀⣠⠖⢉⣉⣉⠑⢦⡀⠀⠀⠀⠀⠘⠿⠿⠿⠀⠀⢀⣼⣿⣡⣤⣤⣿⣷⠀⠀⢠⠎⠈⠛⠁⠱⡄
⠀⠀⠀⠀⣼⢁⣾⡿⣿⣿⣿⠂⢱⡀⠀⠀⠀⠀⣀⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠟⠉⠁⠐⠃⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⡇⠐⣯⠀⠀⠀⢹⠀⢸⡇⠀⠀⠀⢀⣿⡟⠶⣄
⠀⠀⠀⠀⠘⣄⠙⠷⣶⡶⠟⢀⡞⠀⠀⠀⠀⢸⣿⣿⣦⣬⣷⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠈⠑⠀⠤⠀⠒⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠆⠀⠸⢿⣿⣿⣿⠟⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣾⠉⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣀⡈⠉⠉⢳
⠀⠀⠀⠀⠀⠀⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡖⢲⢰⠊⡇⢰⠉⡇⠀⠀⢰⠧⠈⠁⠘⠦⠀⢰⡍⠉⠨⢷⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠙⠁⠈⠉⠁⠈⠈⠁⠀⠀⠘⢤⣆⣐⡶⠃⠀⠳⣤⡀⣠⡞⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣿⠤⠤⠭⠭⠭⠭⠭⠭⠭⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠿⠷⠶⠶⠾⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠋⠉⠉⠉⠉⠓⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⣀⣠⣤⠤⢤⠤⢤⣀⡀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⠴⡾⠉⡜⠁⣰⠃⢠⠎⠀⡜⠓⣦⡀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡴⢣⠞⣠⠎⢀⡴⠁⣰⠃⢀⡞⠀⣰⠃⢸⣆⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣼⡵⠃⡴⠃⢀⠞⠀⡴⠁⢀⠎⠀⡰⠃⣰⠋⢸⣆⠀⠀⠘⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠟⢀⡞⠁⣠⠋⢀⡞⠀⢠⠎⠀⡴⠁⣰⠃⣰⠋⣼⠀⠀⠀⢻⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⢠⠞⠀⡰⠃⢠⠎⠀⣠⠋⢀⡞⠁⡰⠃⡰⡥⠚⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠋⠀⡜⠁⡠⠃⠀⡴⠁⢠⠎⢀⡼⠁⡼⠑⣇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡄⠀⠀⢹⣀⠞⠀⡴⠁⢀⡞⠀⣠⠋⢠⡞⢀⡜⠀⠼⠉⡿⠲⡆⠀⢠⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⣏⢀⡜⠁⢠⠎⠀⡰⠁⢠⠏⢀⡞⠀⡼⢀⠖⠀⡶⠀⠀⠸⠁⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠈⢿⡀⣰⠃⠀⡼⠁⣰⠋⢀⡎⠀⠜⢁⡟⢠⡿⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠙⠧⣄⡞⠀⡴⠁⢀⠎⠀⠌⢀⣏⠴⠋⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠉⠚⠓⠒⠛⠒⠚⠋⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⣀⣀⣀⣀⣀⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠃

⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠱⠤⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢈⠤⠂⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠚⠀⠀⠀⠁⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠒
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⡇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀
⠀⠀⠀⢠⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠄⠃⠀⠀⠀⠀⠐⡀⠀⠁
⠀⠀⢀⠆⠀⠀⠀⠀⠀⠣⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢡
⠁⠀⠉⠒⠒⠶⠤⠤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠈⠁⡀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠠⠐⠂
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢸⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠁⠀⠈⢆
⢘⠃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⠀⢀⣂⣰⠀⠄⠀⡂⠀⢠⠀⠀⠒⠊⠀⠀⡀⠀⠀⡀⠀⡆⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⢂⠈⡁⠀⠠⢀⣤⠤⠤⠃⠉⠀⠀⠀⠠⠀⠴⠀⠅⢤⡄⠴⠶⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠈⠀⠐⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⠀⢀⠴⣠⣤
⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⠔⢊⠉⠑⣧⠞⢁⡸
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠴⠒⠋⠹⢓⠎⠁⠀⠉⠂⠀⠀⠀⠙⠀⠋⠘⠉⠿⣧⠤⠚⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠂⠉⢀⡀⠀⠀⠠⠄⢀⠀⠐⠐⠀⠠⠀⠀⠀⠀⠐⠂⠠
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠀⣀⠀⠀⠀⠉⠀⠂⠀⠀⡄⠀⠀⠀⠀⡁⠀⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⢀⡼⠢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡔⠀⠀⠀⠁⠈⠄⠀⠀⠀⡀⠀⠀⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠻⠄⠀⠀⠃⠀⠀⠀⢀⠀⠃⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠈⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠄⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⡀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠠
⠀⠉⠭⠩⠁⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠠⠇⠀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠉⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣚⠷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⣤⣴⡾⣿⣿⣛⢶⣄⠀⣾⢹⡷⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣦⣶⣮⡑⢤⣤⣤⣶⣶⣒⠚⠋⠉⠉⠉⠁⢀⣾⣿⢫⣾⣿⣿⣿⣿⣿⠀⠛⣿⣴⡆⢿⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⢿⡇⢿⣿⣿⣿⣿⣿⣿⣥⠼⣿⣤⡇⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠍⠛⠛⠛⠛⠛⠻⠟⠛⣿⣶⣶⣤⣌⡻⠎⠻⢿⣿⣿⣿⣿⠙⣆⣿⣄⡇⣼⣤⣸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣆⠀⢸⣿⣿⣿⣿⡇⢠⣷⡹⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⣿⣿⣧⢸⡇⠈⢦⣴⣿⠇⢹⣿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⢸⣷⣀⢀⣹⣷⠖⠒⠛⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢠⢸⣿⣿⣿⡞⡿⠟⠉⠙⠿⠀⠀⠀⢠⣗⣀⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⢸⣿⣿⡟⡛⠃⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣶⡦⠤⠶⢶⠶⠶⠶⠶⠶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣾⡿⠁⢸⢸⣿⣿⣇⠁⠙⠃⠈
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣥⣾⣿⢿⣿⣿⣿⣿⣶⣦⣤⣤⣶⣾⣿⣿⣿⣿⣾⣿⣩⣽⣛⣁⣀⣘⣘⣛⣿⣿⣷⣶⢎
⠀⠀⠀⠀⠀⢠⡾⠯⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠯⠏⠉⠩⠭⠭⠭⠭⠭⠭⠽⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡟⠋⠀⠂
⠀⠀⠀⣶⣶⣶⣶⡶⣶⣶⣶⣶⡟⡟⣿⣿⣿⢻⣿⣿⣿⣿⣶⣟⣷⣶⣶⣶⡶⣶⡶⣶⣶⣶⢶⡶⢲⣶⡶⢲⢲⠲⡇
⠀⠀⠀⡹⠊⠈⢻⠃⠏⠘⠟⢸⠃⠃⠀⠉⠇⠃⢸⢻⠿⠙⠿⠉⠈⢻⠙⠺⠀⠛⠃⠉⠋⠛⠘⠇⠀⠋⠛⠀⠀⠈⡇
⠀⠀⠀⠘⠶⠶⠶⠶⠶⠷⠶⠾⠶⠶⠶⠶⠾⠶⠾⠾⠶⠶⠶⠶⠶⠶⠷⠞⠛⠲⠶⠾⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠼⠋⠁⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠟⣉⣥⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠲⢾⣿⠯⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⡞⢻⡟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⢿⣷⣶⣶⣤⣤⣄⣀⣀⠈⠉⠀⠀⠀⠀⠂⠀⠀⠀⠀⣠⣽⣿⡇⡇⠘⡿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⣠⠴⢏⣡⣦⠀⣠⡀⢈⠉⠉⠛⠛⠻⠿⠶⠶⢤⣀⣀⡀⢀⣶⡿⡟⢿⢻⣇⣷⠞⡃
⠀⠀⠀⠀⠀⢀⡞⠁⢠⣈⣽⠟⣻⡿⢶⣾⣧⣤⣷⣂⣤⠤⢀⢀⢀⠈⠱⣤⣿⣿⠁⠁⠀⢼⠉⠀⠀⡇
⠀⠀⠀⠀⢠⡟⡄⢤⣤⣿⣛⣿⠿⣷⣾⣧⣼⣟⣩⡿⢛⣷⡿⣧⠔⣡⡾⢻⣿⡇⠀⠀⠀⠠⢀⣠⣾⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⣼⠀⠀⢠⣯⣭⣝⣛⡻⠿⢶⣾⣏⣽⡿⣻⡿⢿⠒⢀⣾⠟⠀⢸⣿⣇⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡓
⠀⠀⠀⠀⣿⠗⠦⠈⠉⡙⠛⠻⠿⢿⣶⣦⣤⣭⡛⠛⠓⠋⢠⣿⠋⠀⠀⡼⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠁
⠀⠀⠀⠀⢘⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠈⣉⡉⠁⠀⠀⢀⣼⡿⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⠿⠿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠀⠉⠙⠛⠿⠿⣿⣷⣶⣦⣤⣯⣙⣿⣿⡁⠸⣷⣶⣿⣿⣿⣿⣿⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣿⣿⣿⠿⠿⠛⠟⠋⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣭⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⠉⠀⠀⠀⢀⣈⡙⠛⢄⡀⠀⠉⠑⠒⠲⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⢀⣴⠋⣀⣀⣈⡉⠒⠈⢉⡢⠖⠊⢩⢳⠀⢹⠙⠤⢄
⠀⠀⠀⠀⠀⠀⠀⡼⠶⠶⠶⠒⠒⠒⠒⢚⣧⣲⣤⣀⣀⣀⣀⣉⣉⣡⣖⣉⡉⠉⠛⠒⠚⠒⠒⠶⢇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣏⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣸⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠯⠭⠭⠭⠭⠭⠭⠭⠭⠽⠿⠭⠭⠭⠭⢽⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡆⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡸⠓⠒⠛⠓⠒⠒⠒⠒⠚⠛⠛⠒⠒⠒⠒⠒⠒⠳⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢠⠧⠤⠤⠤⠤⡤⠤⠤⠤⡤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠵⣄⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢙⠶⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠶⠒⠒⠒⠒⠒⠶⠒⠒⠲⠖⠶⠶⢶⠖⠒⠒⠒⠒⠒⠒⠒⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠋⠀⠘⠃⠂⠘⠛⠀⠛⠁⠀⠛⠀⠀⠀⠀⠘⢣⡤⠿⡼⠉⠿⣂⠉⠉⠉⣑⠳⠋⢢⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⢄⣠⠤⠤⠤⠤⠤⠤⠤⣄⠤⠤⠤⠤⠤⡴⠋⠀⠀⠑⠢⣄⡀⣩⠶⣍⢁⣀⠤⠜⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠈⠹⡀⡏⠁⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣤⣐⣂⣴⣐⣀⣐⣰⣖⣀⣀⣐⣦⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠺⠭⠭⠭⠭⠭⠭⠭⠭⠽⠯⠭⠭⣷⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⣟⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡞⠒⠒⠒⠂⠀⠶⠶⠶⠶⠶⢾⣦⡷⠶⠶⠶⠶⠶⠆⠐⠒⠒⠒⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠒⢺⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⢺⡭⣷⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⡤⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠉⠀⠀⠐⠒⠒⠒⠒⢺⣷⡟⠒⠒⠒⠒⠒⠀⠀⠈⠉⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢻⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣧⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣾⣿⣿⣷⡀⠀⠀⠠⠤⠀⠀⠀⢸⠉⣹⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣽⣿⡀⠀⠀⠀⠀⠀⠀⢰⢀⣸⣿⡀⠀⠀⣿⡀⠀⠀⣀⣀⡤⢤⠀⡎⢀⣿⣿⣿⣿⣿⣿⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣅⠉⣷⣮⢿⡀⢸⣿⣿⣿⣿⣿⣿⣤⣿⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣛⣯⠀⠙⢿⣿⣿⣿⣿⡿⢟⣿⣻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣙⢻⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⢧⡸⡾⢇⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣆⢠⣾⣣⣬⣹⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣤⣿⣿⣿⣩⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠉⠛⢿⣿⡍⡉⢽⣿⣟⣹⡆⣿⣭⣭⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⠟⠻⠿⣻⣮⣷⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣾⣿⣦⣿⣿⣿⣷⣽⣿⣿⣿⡿⠻⠿⠃
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟⢀⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣯⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⢀⣠⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣀⠂
⠰⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣽⣯⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⣀⡴
⠠⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⢹⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠽⣿⢿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠙⠛⠻⠶⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⢤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠒⢒⡒⠲⠶⠤⠤⠤⢤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠔⠀⠀⠃⠀⠀⠀⠀⠀⠁⠉⠉⠉⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠘⠚⠁⠃⠀⠀⠨⠀⠀⠀⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠊⠀⠀⠀⣇⠀⠀⡅⢈⣉⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠊⢏⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠇⠃⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠆⠦⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠐⠀⠀⠀⠉⠓⠢⢄⡀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢉⠉⠀⢇⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⢧⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡌⠀⢰⡘⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢱⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣆⠀⢨⡃⠀⠀⠀⠀⠀⠇⡆⢀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⣿⡀⠸⠀⠀⠀⠀⠀⠀⢀⢧⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠷⡉⢀⠄⠀⠀⠀⠀⠀⠀⠺⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠏⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠈⠽⣆⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⢀⣠⣴⠟⢃⠀⠀⣔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠊⠃⣀⠀⠀⠀⠉⠉⠀⠚⠛⠛⠉⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡆⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠕⠦⣀⣀⠀⠀⠀⠀⠀⠀⢠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠻⡇⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠂⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠄⡀



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠤⠤⠤⠤⠤⠤⠤⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠒⠒⠒⠒⠒⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠉⠉⠉⠉⠉⠉⠉⠉⢧⡀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⢸⠋⠭⠭⠭⠭⠙⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠈⠳⡾⠠⠤⠤⠤⠤⠤⠔⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⡀⠀⡜⠀⠀⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡠⠖⠊⠉⠀⠀⠀⠀⠙⣄⠃⠀⡎⠀⢙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⢸⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢱⠤⠤⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠤⠤⠤⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠤⠄⠤⠔⠊⠉⠀⠀⠀⠀⠉⠙⠒⠒⠒⢢⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢹⠈⠦⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠞⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢲⡟⢹⡏⢹⡏⢻⡖⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣭⣈⣷⣾⡿⣿⣿⣛⣿⡿⣷⣾⣁⣭⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠱⣦⣾⡟⡿⣯⡿⣟⣿⠿⢿⣷⣿⠟⢿⣷⣴⠎⢣⡀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⣶⣿⣿⣷⣫⣽⣾⡿⠿⠿⢿⣿⣯⣶⣿⣯⡿⣶⠃⢳⠀⠀⠀⣧⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⢳⣿⣿⣿⣿⣿⣿⠃⠰⠾⠷⠆⠈⣿⣿⣿⣿⣿⣿⡞⠊⡆⠀⠀⠈⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⠀⡄⠤⠄⠤⠀⠠⠀⠀⠤⠀⠠⠀⠤⠠⠤⠀⠆⠘⠛⡇⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⡞⢨⣥⣭⣥⣍⢉⣭⣭⣥⣭⡁⣡⣀⣤⣭⡍⢳⠘⠛⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⠃⠈⠀⡀⣠⠘⠘⠀⠁⢈⡌⠃⠃⠀⠀⣀⠁⢸⠘⠛⣿⣶⠄⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⠇⠀⠈⢉⡇⠐⠀⠀⠉⣿⠀⠀⠅⠉⢹⠁⠀⢸⠘⠛⣿⣿⡤⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⡆⢰⣠⣉⣅⣸⢰⣄⣌⣡⣠⡆⣇⣨⣉⣄⡄⢸⠘⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⠇⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣦⢸⠘⠓⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⠲⠬⠍⠉⠉⠩⠭⠭⠍⠩⠭⠉⠍⠩⠉⠍⠅⠞⠘⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⢺⡇⠀⣿⠀⢰⣿⠀⢸⡇⠀⣿⡆⠀⣿⠀⢸⡇⠘⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚⠃⢸⡇⠀⣿⣆⢸⣿⡄⢸⣧⠀⣿⡇⠀⣿⠀⢸⡇⠘⠓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡒⣃⣸⣗⣀⣿⣿⣸⣿⣇⣸⣿⣀⣿⣇⣀⣿⣀⣸⣇⣘⣒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⡶⢶⢶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⡶⣶⡶⢶⣶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠸⠇⠿⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿⣿⡿⠿⠿⠿⢿⣷⣄⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⣡⣶⣿⣿⣿⣶⣍⠻⣷⡄⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢏⣾⣿⣿⣿⠩⣿⣿⣿⣷⡹⣿⡄⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣾⣿⣿⡻⣿⣿⣿⣿⣿⡏⢷⢹⣿⣤⣤⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣛⢡⣿⣿⣿⣷⣿⡿⣫⡛⣿⣿⢻⡜⣛⡻⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡸⣿⡿⢛⡛⠿⠿⣮⣴⣿⣿⢸⢃⣿⣿⣷⣮⣭⣵⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⢿⣇⠿⣿⣾⠇⣽⣿⣷⠝⡿⣸⣿⣿⣿⣿⣿⣿⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡜⢿⣇⠻⠿⣸⣿⣿⣷⡾⢣⣿⣿⣿⣿⣿⣿⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⡻⣿⣿⣿⣿⡿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣮⣭⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⡀
⠀⠐⢲⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀
⢠⣤⣾⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⢸⢸⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣬⣭⣿⣭⣀⠀⠀⠀⠀⠀⠀⡟⣿⡂
⢸⢸⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣾⡷⠟⡇⢸⡇
⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠉⠉⠛⢻⣿⣿⣿⣿⣿⣿⣀⡂⡇⠀⡇⣼⡇⢀⣀
⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⢇⡟⠛⡇⣿⡦⣾⣿
⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣇⡀⣿⣿⡆⡏⢻
⢸⢸⣿⣿⢫⢿⣿⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠙⠋⠛⣿⣿⡇⡇⣿⠀⣴⣿⠶⠿⢿⣿
⢸⢸⣿⣿⣾⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⡃⣹⣿⡾⠁
⢸⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⡏⣾⣿⣿⣤⣤⣤⣤⣿⣿⣿⣿⡿
⠘⠻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⣿⣿⣿⠿⣿⣾⣿⣿⡿⠋
⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣸⣿⣿⢇⣿⣿⣿⣿⣿⠁
⠀⠀⡗⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣰⣿⣿⣿⡎⣿⣿⣿⣿⣿⣆
⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣆
⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠑⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⡀⠀⠀⠀⠀⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣞⣼⣷⢽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⠦⣀⢀⣀⣿⣿⣅⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣦⡀⢰⣿⢻⣿⣯⣼⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣻⣿⠇⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣥⣼⣧⡌⣽⣿⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣷⣶⣶⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣧⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣵⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡋⢩⠙⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣫⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⡢⣼⣾⣿⠀⢿⢽⣿⣿⣿⢿⣿⣿⡟⠷⣄⠀⠀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⣿⣿⣛⣿⡄⣶⣶⣿⣿⣿⣿⠿⠿⠃⠀⠙⠿⠲⠀⠤⠤⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣽⣿⣿⠟⢧⣿⣿⠉⠁⠙⠁
⠀⠀⠀⠀⠀⣀⡀⣀⣀⡀⠀⠀⢀⣸⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣵⣸⡷⣿⣟⣀⣀⣀⣀⡀
⠀⠀⠀⢀⡾⠋⠀⠛⢉⠂⠀⠤⠌⣽⡍⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣉⢢⣄
⠀⢀⠔⠁⢀⠀⠄⢠⡊⠀⠀⠀⠈⠏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⡀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀
⡀⣀⣀⡀⠀⠀⢀⣈⣉⣉⣛⣛⣻⣿⣛⣉⣉⣉⣙⣋⣛⡋⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⣿⠿⠿⢿⣿⣿⣿⡛⠒⠃⠀⠀⠀⠀⠀⢀⡀⢈⣀⣙⣢⣀
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣾⣿⡾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⣿⣷⢷⣿
⢸⣿⣿⣿⣾⣿⣿⣿⣎⣉⣝⣙⣙⡫⣛⣛⣛⣛⣿⣿⣾⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣼⡇
⠻⣿⣛⣛⣛⣻⣻⣿⣿⣿⣿⣿⣯⣽⣭⣾⣿⣿⣿⣿⣿⣷⣛⣛⣛⣛⣐⣋⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠒⠒⠒⠒⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠾⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⡴⢺⡳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠦⠼⠤⠄⠤⢤⢀⡀⢠⡗⣞⠑⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢀⡀⣀⡀⢀⠔⢄⠈⣿⣷⠘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢱⣍⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠣⠄⠄⠊⣿⡟⣠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠇⠸⣧⠀⠀⠀⠀⢀⠀⡄⠇⢻⡽⢆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⡀⠀⠀⠀⠸⠀⠁⢀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡤⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⠀⡄⢰⠀⠀⢀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⡀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠁⡰⠁
⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠳⠒⠒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢊⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠒⠒⠲⣄
⠀⠀⠀⠀⣠⢊⣀⠀⠀⠀⠀⡀⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⠀⠀⠀⠀⠀⢀⣀⣀⣠⠀⠀⠀⢀⠀⠈⢦⡀
⠀⠀⠀⠀⡏⠀⠉⠀⠈⠀⠁⠉⠉⠁⠈⠈⠀⠀⠀⠀⠠⠤⠤⠤⠀⠠⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⠉⠉⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠉⠈⠁⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⢰⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡄⢀⡀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢰⡀
⠀⠈⣤⣤⣤⣤⣤⣬⣭⡍⠀⠀⠀⢸⣿⡆⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠈⠁⠠
⠀⠀⣤⣉⡉⡉⢉⣩⣍⣀⠀⠀⠀⣼⣿⡟
⠀⠀⠉⠉⠉⠙⠉⠉⠉⠽⠆⠰⣿⠿⠿⠃⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠐⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⢸⡟⠀⢠⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠈⠉⠛⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⢰⠛⣿⣿⣷⡀
⠀⠘⠁⠀⠟⠀⣀⢀⡄⠀⠀⠀⠸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⣾⣿⣿⠁
⠀⠀⠀⢀⠀⠀⠻⣤⣧⠀⠀⣀⡀⠙⠻⠿⢟⠁⠀⠀⠀⠀⠀⠀⠰⠐⠐⠇⠈⠀⠀⠉⠀⠀⠀⠔⠂⠈⢻⣿⡯
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⣤⡀
⠀⠀⣠⣤⣠⣬⣌⢁⠀⠀⢴⠀⠀⠆⠀⣠⣴⣦⣤⣄⣠⣀⣀⣄⣀⣤⣴⣶⣿⣿⣷⡄⠀⠀⢸⣿⣿⣧⡀⠀⣀⣠⣤
⠀⠀⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠘⣿⣿⣿⠇⢶⣿⣿⠿
⠀⠀⠀⠉⠉⠁⠀⠀⠀⣤⣤⡄⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⢀⠀⠈⠉⠁⠀⢈⣉⠉
⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠂⠈⠙⠿⠦
⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣦⣤⠀⠁⠀⣀⣀⣀⣀⣀⠀⠀⢰⣿⣿⣽⡟⢀⣠⣦
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠀⠀⠀⠉⠙⠉⠉⠉⠀⠀⠸⠟⠟⠋⠁⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⠘⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠃⢰⣷⣤⣤⣠⣿⠃⠩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢰⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠛⠛⠛⠛⠛⠛⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠞⠛⣦
⠀⠀⠀⠀⠀⠀⢸⢸⠀⡤⠤⣤⣤⢤⡤⣤⡤⢤⠤⠤⢤⣤⡤⢤⡀⠀⢀⣀⡀⣀⣀⢀⣀⣀⣀⡀⣀⡀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⢸⢸⠀⣷⡟⢉⣽⣿⠟⠉⢝⣽⣿⡿⠋⣩⣿⣿⡇⠀⢸⣧⡿⣿⣻⡍⣯⡿⣿⢻⢿⣇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢸⠀⣟⣶⣿⣿⣷⣧⣴⣿⣷⣿⣾⣾⣿⣿⣻⡇⠀⠀⠶⠆⠴⠰⠆⠶⠰⠰⠆⠶⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢸⠀⢀⣀⣀⡀⠀⣀⣀⣀⡀⠀⣀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢸⠀⡏⠉⢩⣽⢺⡏⠉⣩⢽⢸⠉⠉⣩⣷⣿⠉⢉⣽⡇⠸⠿⠗⠀⠰⠿⠗⠀⠸⠿⠗⠀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢸⠀⣇⣐⣛⣹⠸⣇⣞⣛⣾⢸⣀⣚⣋⡿⢿⣰⣛⣣⡇⢰⠶⠆⠀⠰⠶⠶⠀⠰⠶⠶⠀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢸⠀⡴⠶⠶⢦⣠⡶⠶⠶⣦⢠⠶⠶⠶⣄⣴⠶⠶⢶⡄⢠⣤⡤⢀⡴⠖⠦⡀⢠⣤⣤⠀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⢀⣴⢿⢸⠇⣠⣾⢿⢸⠀⣠⡶⣿⣿⢀⣴⡏⡇⢠⣤⡄⢸⡄⠀⢀⡗⢠⣤⣄⠀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢸⠀⢓⣛⣒⡚⠈⣓⣓⣒⡛⠘⣒⣓⣒⠋⢙⣚⣒⣚⠃⠨⠥⠥⠤⠭⠝⠭⠤⠬⠭⠅⠀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢸⠀⡏⠉⢩⣹⢺⡏⠉⣩⣽⢸⠉⠉⣩⣷⣿⠉⢉⣽⡇⠠⠂⡀⠠⠀⠂⡀⠠⠠⠂⡀⠀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢸⠀⣧⣐⣛⣹⠸⣇⣚⣛⣼⢸⣀⣚⣋⡿⢿⣐⣛⣫⡇⠀⣉⠀⠀⠀⡈⠀⠀⠀⣈⠀⠀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠘⠀⡴⠶⠶⢦⣠⡶⠶⠖⣦⢠⠶⠶⠶⣄⣴⠶⠶⢶⡄⠘⠠⠂⠐⠀⠠⠆⠐⠈⠡⠄⠀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢀⣼⢿⢸⠇⣠⣾⢿⢸⠀⣠⡶⣿⡿⢀⣴⡏⡇⢠⠐⠀⣀⠀⠔⠀⢀⢀⠄⠀⠀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠛⠓⠒⠚⠈⠓⠓⠒⠛⠘⠓⠒⠒⠋⠙⠚⠒⠚⠃⠀⠀⠁⠀⠀⠐⠁⠀⠀⠀⠁⢀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⢿⠿⡿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⢿⠿⢿⠿⠿⠿⢷⠶⣶⣶⡶⠶⢶⣶⡶⡶⢿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠚⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⢛⣦
⠀⠀⠀⠀⠀⢀⣠⣤⣖⣚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠉⠁⠈⡇
⠀⠀⠀⠀⠀⣿⢀⣀⣀⣀⣉⠉⠉⠉⠉⠙⠒⠒⠒⠲⠤⠶⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⢹⠈⠀⠀⢀⢘⠀⠀⠀⠀⣠⠀⢠⠚⡝⡆⠀⡇⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⢹⡀
⠀⠀⠀⠀⠀⢸⣘⣲⣤⣤⠔⠀⠀⠀⠀⠀⠀⠘⠦⠴⠃⠀⠃⠀⣀⡠⠔⠊⠉⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⡗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡤⣒⠭⠝⠯⢭⡓⢄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⢀⠎⠊⡴⢒⠒⠲⢤⡈⢆⢣⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⠀⠀⢸⡏⢸⠀⡏⠀⠀⠀⡵⠘⡈⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⡄⠀⠸⣷⠸⡀⠣⣀⠀⠀⢣⢠⢁⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠣⡣⣙⠲⠤⠤⢔⡣⢊⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠚⠭⠿⠭⠝⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⠤⢤⣄⣀⣀⣀⠀⠀⠀⠀⣇⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠒⠋⢉⣉⡀⠀⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠒⠛⠉⠉⠀⠀⠠⣴⣶⣾⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⠿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣀⣀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠒⠒⠋⠉⠉⠉⠁⠀⠒⣂⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣋⡤⠔⠒⠒⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢀⡞⠀⠀⢀⣤⣶⣶⣤⣀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⣿⣿⣿⡿⠛⣁⠥⠤⠤⠴⠶⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡼⠀⠀⣴⡿⢋⣴⣾⣿⣿⣧⡀⠀⠀⢿⣷⠀⠀⠀⠀⣿⣿⡏⣰⠋⠀⢀⣀⠤⠴⠒⠒⠒⠚⠛⠛⠒⠒⠲⢶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢢⡇⠀⣰⡟⢠⣿⣿⣿⣿⣿⣿⣷⠀⠀⠸⣿⡇⠀⠀⠀⣿⣿⢀⠇⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠤⣀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⢸⠀⠀⣿⠃⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣿⣷⠀⠀⠀⣿⣿⢈⡞⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⢸⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡇⢸⠀⣀⣤⣴⣶⣿⣿⡿⠿⠿⠿⠆⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⢸⢸⠀⠀⣿⡄⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣿⡿⠀⠀⠀⣿⣷⡌⢻⣿⣟⣯⣥⣶⡶⡶⠷⠿⠿⣆⠀⠀⠀⣾⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠜⡆⠀⢹⣧⠘⣿⣿⣿⣿⣿⣿⡿⠀⠀⢰⣿⡇⠀⠀⠀⣿⣿⡇⢸⣿⢛⣭⡤⣶⡖⣶⡇⢾⣿⠰⠀⠀⢠⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⢳⠀⠀⠻⣷⣈⠻⣿⣿⣿⡿⠁⠀⠀⣾⣿⠀⠀⠀⠀⣿⣿⣿⠈⣿⡟⣛⡁⣭⡄⣶⡆⣶⣖⢳⣶⣶⣾⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⢧⠀⠀⠈⠻⠿⠾⠿⠋⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⣿⣿⣿⡄⢿⡟⣻⣅⣭⣤⢴⣆⢰⡆⢰⡦⣼⣿⣽⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⢻⣋⢨⣥⢨⣦⢰⡦⢰⡆⣼⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠈⣿⣿⣿⢈⣭⢰⣧⢰⣦⣼⡧⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⣿⣿⣿⣉⣿⢨⣯⣸⣯⣼⡇⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣿⣿⣿⡇⢸⣿⣿⣏⣿⣭⣯⣼⣯⣽⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⡄⡇⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⡀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠙⢿⡆⣾⡟⠙⢻⣇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣴⡿⠃⠻⣷⣴⡾⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠿⠛⠛⠛⠛⠿⠷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⢋⣁⣤⣶⡶⠾⠿⠿⠶⣶⣦⣤⣈⠛⢿⣦⣄⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣰⡿⠋⣠⣴⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⡈⠻⣷⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣼⠟⢀⣾⠟⠁⣀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠘⢿⣆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⡏⢠⣿⠋⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠈⢿⡆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡿⠀⣿⠇⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣸⣿⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⢸⡿⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠋⠙⠛⠛⠛⣿⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡀⡀⠀⠀⣀⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣇⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡿⠛⣿⠟⠋⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⡄⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣼⡟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠹⣷⡀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⣰⡿⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⢿⣄⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⢁⣼⠟⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠻⣷⣄⡉⠻⠷⣶⣤⣄⣀⣀⣀⣀⣤⣴⣶⠿⠋⣡⣴⡿⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣉⣉⠛⠛⠛⢋⣉⣁⣤⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠟⠛⠛⠛⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⢿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣉⣉⡉⠉⠉⠉⠉⠛⠛⠛⠒⠒⠶⠶⠶⠤⠤⢤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠙⠛⠛⠒⠒⠶⠶⠶⠤⣤⣤⣤⣀⣤⠶⠛⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⢀⠀⡀⢀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠷⠻⠾⢻⣽⡅⣻⣄⡿⣄⣨⢠⣤⢀⣀⠀⠀⡇⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣄⣠⣀⠀⡀⢉⡌⠁⠉⠉⠋⠘⠘⣿⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠏⠁⠙⠉⠛⠛⠿⠻⡇⡷⠆⠀⢠⣴⡿⠁⠀⣷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠲⠦⢤⣤⣀⣀⡀⠀⠀⠀⠚⠡⠚⡴⢂⠀⠀⣿⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠦⠤⣤⣄⣀⡀⣸⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⣤⡤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⡠⠤⠤⠤⠧⠄⠀⠀⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠘⠀⡠⠀⡜⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡆⢰⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠤⠒⠉⠉⠉⠑⡀⢰⠉⢣⠀⠀⢸⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠃⠀⠀⠀⠀⠀⢀⠁⢸⠀⠀⡇⢸⠒⠒⠀⠠⠤⠤⠤⠤⠤⠀⠐⠒⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠈⠤⠔⠁⢸⢀⢀⣀⠤⠐⠂⠀⠐⠂⠤⠄⠠⡀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢂⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠀⣀⣀⣀⣀⣀⣀⣀⠠⠆⠀⢸⠘⠤⢀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣤⣴⣶⣾⠶⠿⠿⠿⢿⣿⣶⠒⣦⣤⠠⠐⠒⠣
⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠀⢺⠀⠈⠐⠛⢳⠶⠖⠒⢚⣛⣛⣩⣭⣥⣥⣤⣼⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠣⠀⠀⠀⠀⢸⠀⠋⠋⠝⠿⣭⡭⠭⠴⠶⠛⠛⠃⠉⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠚⠒⠤⢄⡀⣇⣀⡤⠤⠴⠒⠒⠋⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⣿⣿⢰⣶⣶⡄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⡇⠀⣿⡇⠀⣿⡷⠀⢾⡷⠀⢾⡷⠀⢸⣿⣿⢸⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣶⡆⠀⣶⡆⠀⣶⣦⠀⣶⣶⠀⣶⣶⠀⢸⣿⣿⢸⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⣿⣿⣬⣭⣭⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣶⡆⠀⣶⡆⠀⣶⣦⠀⣶⣶⠀⣶⣶⠀⢸⣿⣿⢛⣛⣛⡛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣤⡄⠀⣤⡄⠀⣤⡤⠀⣤⣤⠀⣤⣤⠀⢸⣿⣿⣬⣭⣭⣥⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠐⠛⠓⠒⠛⠛⠒⠛⠛⠒⠛⠛⠒⠛⠛⠂⢸⣿⣿⣿⣿⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣤⡄⠀⣤⡄⠀⣤⣤⠀⣤⣤⠀⣤⣤⠀⢸⣿⣿⣿⣿⠗⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣍⡅⠀⣩⡅⠀⣩⣅⠀⣩⣍⠀⣩⣍⠀⢸⣿⣿⣿⣿⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠰⠿⠷⠶⠿⠷⠶⠿⠷⠶⠿⠿⠶⠿⠿⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣤⡄⠀⣤⡄⠀⣤⣄⠀⣠⣄⠀⣠⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣛⡃⠀⣛⡃⠀⣛⡋⠀⣛⡋⠀⢛⡋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣿⣧⣤⣿⣧⣤⣿⣯⣤⣿⣿⣤⣿⣿⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡼⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠶⠶⠶⠶⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢉⣀⣀⣀⣀⣤⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣉⣉⠉⠉⠉⠉⡿⠁
⠀⠀⠀⠀⢰⡶⠒⠒⣲⡿⠂⠀⠸⡇⢸⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢙⡆⠀⣏⠉⢉⡇⠀⠀⡇⠀⢠⡴⢤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢻⣀⡴⠋⠀⠀⠀⠀⣇⠸⡀⠀⠀⠀⠀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣥⣤⣀⣈⣉⣉⠁⠀⢰⡇⠀⠀⢹⡄⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠶⠶⠦⠉⠀⠀⢸⡇⠀⠀⠀⠹⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠒⠦⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⡠⠒⠉⠉⠛⠦⡀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢹⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⡇⠀⠀⢰⡇⠀⠀⠀⠀⠀⣠⠶⠗⠀⠠⠀⢸⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⣇⠀⠀⠀⠀⠀⠙⠛⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢦⣄⡀⠀⠀⢀⣠⠏⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢱⡀⠀⠁⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣤⣄⣤⡤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠤⠤⠤⠤⠤⢐⣒⣒⣒⣛⣋⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠙⠛⠒⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⣀⣤⣤⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⢳⣄⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠤⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠉⢉⣀⣀⠀⠀⠀⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⢂⣈⣡⠤⠄⠀⠂⠀⣉⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣃⣀⡭⠀⠖⠀⠚⣉⠁⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠄⠀⠖⠀⣀⠉⠩⠀⠀⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⢉⣀⡤⠀⠀⠒⠂⢀⡈⠀⣤⠄⠠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡁⠠⠤⠒⠒⠉⣀⠈⠠⠤⠤⠒⠀⠈⠀⠀⢐⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⢀⣀⣀⣀⠀⠁⠀⠀⡤⠐⠒⠒⠚⣁⡀⠠⠀⠀⠐⠀⠀⠁⢰⠀⠘⢃⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⢠⢔⠂⢴⣾⠀⠀⡴⠶⣏⠀⠀⠀⣀⡤⡖⢦⣤⣤⣤⣴⣮⣒⣒⣖⣛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣆⣀⣥⣂⣼⣶⣾⠿⣟⠁⣡⠌⢟⣁⣼⣿⣟⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢿⣷⣾⣿⣶⣭⣭⣿⣿⣿⣿⡠⣼⣆⢘⣰⣿⡿⢃⣾⠇⣼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⠙⣿⣽⣿⣿⡿⠿⣿⣮⣻⠿⠿⠿⠿⠛⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣇⡘⣿⣿⣷⣶⣶⣿⣿⣿⣿⣷⣶⣶⣿⡿⢿⣟⢛⣿⣉⡿⢭⡟⢲⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢽⣿⣿⣧⡙⣿⣿⣿⡿⠿⣿⠛⣻⠍⡽⠧⣾⢓⡾⢭⣿⠠⡾⣁⠚⢡⣏⠰⣧⡬⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣌⠻⣿⣟⠺⠷⢸⣓⠰⠧⣘⠓⣾⣋⣾⡥⠟⠖⣛⣓⣬⣭⣽⢏⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣮⡛⢿⣦⣠⠤⢉⣒⡻⢋⣭⣤⠴⢶⣿⣿⣿⣿⣿⣿⠾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣝⣿⣿⣿⣿⣿⣶⠶⠿⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣴⠶⠶⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣸⠦⠤⢾⠀⠀⠀⠀⢰⠁⠀⢱⡀⠀⠀⠀⡠⠊⠁⠧⡀⠑⠢⣤⠐⠊⠁⡸⠈⠑⢄
⠀⠀⠀⠀⠊⣤⡏⠈⠉⠈⢆⠀⠀⠀⣠⠀⠀⠠⡇⠀⠀⡰⠁⠀⠀⠀⠀⠁⠚⡄⡃⠉⠁⠀⠀⠀⠈⢢
⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠈⠆⠠⠊⠀⠀⣀⠀⠈⠲⡭⠅⠀⢐⣒⣒⣒⡤⡄⠇⡇⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⢠⠊⠁⠀⡇⣿⡇⢠⠀⠀⠀⠀⠀⡇⠀⢿⣤⠧⠤⠤⠤⠤⠤⠤⠄⠀⡃
⠀⠀⠀⠀⡗⠐⢒⠒⢒⣒⣒⢺⠁⠘⠀⠀⠀⡇⣿⠀⢸⠀⠀⠀⠀⠀⡇⠀⣿⣛⣗⣒⣒⣒⣒⣒⣂⣂⡐⡇
⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠣⢙⡃⢸⠀⠀⠀⠀⠀⡇⠀⣿⣿⡧⠤⠤⠤⠤⣤⣤⣤⣤⡆
⠀⠀⠀⠀⣟⠤⠀⠀⠀⠠⠤⣻⡀⢠⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⡇⠀⣿⡤⡥⠤⠤⠤⠤⠤⠤⠤⢤⡇
⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠁⢸⡇⠈⠉⠀⠀⠀⠉⠀⢸⣁⡒⠒⠀⠠⠧⠐⣯⣿⣏⣉⣉⣉⣉⣉⣉⣉⣀⠇
⠀⠀⠀⠀⠁⠒⠒⠒⠒⠒⠒⠈⠑⠒⠀⠠⠤⠄⠐⠒⠁⠀⠉⠉⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣍⣍⣭⣍⣈⣀⡠⠤⣤⣤⣤⣭⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠈⡉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠶⣾⣿⣇⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣸⡷⣀⣀⣾⣿⣿⣧⡾⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⠟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⣿⠟⠛⠿⠰⠏⡈⣸⠁⠀⠩⣹⣮⡧⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠁⢹⠀⠀⠀⣿⣿⠰⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⢸⠀⠀⠀⠉⡏⠀⢀⡀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⢸⠀⠀⠀⠆⠡⠀⠘⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⣞⣒⣶⣖⣒⣂⣀⣀⣀⣀⣠⣦⣸⣷⡷⣼⠸⡗⠴⣾⠀⠀⠠⠾⡀⡷⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢹⣯⡟⠋⠉⠉⠉⠁⠀⢸⣯⢿⡿⠛⠛⢳⡗⠲⠼⠦⠤⠠⠚⠛⠛⠓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⠀⠀⢸⠿⠟⠋⠛⠋⠉⠋⠁⠈⠋⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⣿⠀⠀⢸⡷⢤⣤⣤⣤⣀⡤⠤⠴⢒⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⢸⠃⠁⠀⠀⠀⢰⠏⠀⢀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠀⠀⠸⡄⡆⠀⠀⡀⠘⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣤⣤⣤⣷⡄⠀⠈⠀⠀⠘⢠⠄⠠⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠋⢙⣷⣶⡄⠤⢠⣤⣤⣤⣿⣿⡿⠃⠀⠙⠻⣇⣘⢀⣀⣀⣀⣤⣠⠞⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⠀⠀⠀⠘⢛⣛⡛⠃⡀⠀⢚⡛⣀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣛⣶⣶⣶⣶⣶⣶⣖⣒⣒⣐⣂⣂⣀⣐⣂⣂⣀⣐⣰⣦⣦⣴⣤⣶⣶⣶⢒⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠤⠤⠤⠤⠤⢤⣄⣈⣛⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⣤
⠀⠀⠀⠘⠋⣤⡶⠚⠋⢉⣠⡤⠶⠒⠚⠓⠛⠒⢖⣶⣦⣤⣌⠉⠛⢷⠀⠀⠀⠀⠀⢠⣾⡿⠯⠉⣉⠩⣥⠴⠶⠶⠀⠶⣤⣀⡀⠀⢐⢶⣈⡀
⠀⠀⠀⠀⡜⠉⢂⣴⣾⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄⠈⠀⠀⠀⠀⡰⢫⣿⠃⠀⠾⢋⠠⢴⣚⣩⣽⣶⣶⣶⣭⡋⠀⠀⠻⣿⡇
⠀⠀⠀⠈⠁⣠⣿⠟⠋⠀⣀⣤⣶⣾⠿⠿⠿⣿⣷⣶⣄⡀⠈⠙⣿⣷⠀⠀⠀⠀⣇⣿⣿⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠊⢾⡝⣆⡀⠀⠘⡇
⠀⠀⠀⠀⢠⠇⠁⢀⣴⡾⣟⣉⣥⣤⣶⣾⣿⡟⣀⠉⠻⣿⣄⠀⠈⠿⠀⠀⠀⠀⣿⡟⡙⢠⢻⣿⠿⢻⣿⣿⣿⡷⣦⣄⠀⢰⣿⡿⣿⠀⠀⡇
⠀⠀⠀⢠⡏⠀⣠⣿⣿⣿⣿⣿⣿⣿⢛⣋⡉⠈⠛⣷⣴⡞⣿⡆⠀⠀⠀⠀⠀⠀⣿⠟⣿⡻⠍⠀⠀⣾⣿⣿⣀⣀⠈⢹⣇⣼⣿⠋⠐⣁⠀⡅
⠀⠀⠀⠸⠃⢠⣿⠃⠛⣭⢵⣿⣿⡏⠛⠙⠻⣶⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⣇⡃⠀⢰⣿⣿⣿⣿⣿⣷⣀⣿⣿⣿⣾⢙⣿⡷⡇
⠀⠀⠀⢸⠀⢸⡇⠀⢸⡇⣼⣿⣿⢡⣾⣷⡆⢹⣷⣿⢿⣷⣾⣿⡄⠀⠀⠀⠀⠀⣻⠀⣿⡇⡂⠀⣾⣿⣿⣷⣾⣿⣿⡿⣿⣿⠹⣿⣬⣷⠂⠁
⠀⠀⠀⢰⠀⢸⣧⠀⢸⣷⣿⣿⢇⣈⣿⣿⣿⢿⣿⡟⣿⣿⣻⡿⠁⠀⠀⠀⠀⠀⣿⠀⣿⠘⣗⢰⣿⣿⠧⣿⣿⠹⣿⣿⡿⢿⡄⢬⠋⠁⠀⣦
⠀⠀⠀⠈⡆⠸⣿⡄⠀⣿⣿⡿⢸⣿⢿⣿⣷⣾⡿⠓⠛⣻⡏⠀⣠⡇⠀⠀⠀⠀⢻⡀⣿⠀⠈⣾⣿⣿⢰⣿⡟⠀⣈⣁⣠⣄⡤⠃⠀⠀⠀⣿
⠀⠀⠀⠘⢳⠀⠙⢿⣾⣿⣿⠳⡿⠟⠼⠿⠽⠃⢀⣴⡾⠋⢀⣴⡿⠇⠀⠀⠀⠀⢸⠃⣿⡇⢰⣿⣿⠣⠻⠿⠿⠿⠿⠿⠛⠉⠀⠀⢀⡈⡡⢛
⠀⠀⠀⢀⡈⣦⡄⢀⣿⣿⡏⣶⣤⣤⣤⣤⣶⠾⠟⠉⣀⣶⡿⠋⢀⠀⠀⠀⠀⠀⢸⡆⢸⡛⣾⣿⡿⡆⠀⠀⠀⠀⠀⠀⢀⣀⠀⣚⡷⠋⣀⠘
⠀⠀⠀⠀⡇⠸⣷⣾⣿⡿⢁⠀⠉⠩⠍⠉⡀⣀⣤⣶⠿⠋⠀⣤⣾⠀⠀⠀⠀⠀⢸⡏⢸⢇⣌⡙⠲⢤⣿⣿⣛⣺⣲⣜⢭⠖⠒⠉⠠⠊⠁⢸
⠀⠀⠀⢰⠱⠀⠉⠋⠱⠶⣿⣿⣿⣷⣿⣿⡿⠟⠋⠁⣀⣴⣾⠿⢋⠀⠀⠀⠀⠀⢸⡧⣼⣧⣉⠛⠲⠦⣤⣨⣍⣅⠀⣤⡤⠶⠘⠋⠀⠀⠀⣸
⠀⠀⠀⠘⠆⢲⣧⣆⣀⣀⡀⡀⠀⠀⢀⣀⣀⣤⣶⣾⠿⢋⣁⣠⣼⠀⠀⠀⠀⠀⢸⡗⡿⠻⢾⡿⣶⣶⣶⣶⣶⣶⣶⠤⠶⠶⠛⠟⠛⣫⠟⠉
⠀⠀⠀⠀⠀⢈⠛⠻⠿⠿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠶⠖⠓⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢼⣄⠀⠀⠀⠀⠀⣈⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠙⢷⡄⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣶⣀⣀⠀⡀⠀⠠⠿⠷⠯⠤⠤⠤⠄⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠐⠈⠀⠈⠈⠉⠉⠉⠉⠁⡆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢠⡴⠀⠀⠀⠀⢀⠄⠂⠀⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢄⠀⠀⢠⠀⡉⡯⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠣⠔⠐⠀⠀⢠⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠃⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠣⢄⣶⡆⠤⠂⠀⠀⠀⠀⠘⠠⠘⠿⠤⠈⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠤⠄⠀⣀⣀⣀⣀⣀⣂⣐⠀⠀⠠⠠⠄⠀⠀⠄⠀⢚⡠⠇⠀⡀⠄⣦⣀⠒⡢⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠆⢀⡐⠀⠀⠤⢉⡀⠒⠂⠀⠀⠀⠠⡄⠀⣤⣤⣤⠄⠀⠒⠀⠒⠈⠉⢀⢀⣠⡪⣴⠍⠊⠀⠁⠢⢸
⠀⠀⠀⠀⠀⠀⡇⠈⠉⠙⠓⠒⠭⣀⡖⣀⠠⠭⢐⡂⠀⠤⢄⣚⠃⠘⠛⠛⠀⠤⢄⣀⣤⣴⠮⡻⠕⠋⠈⠀⠀⠀⠀⠀⠀⢘⣦⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⢀⠉⠀⠀⠒⠦⢍⣐⠂⣠⠉⠛⣀⢀⣶⠾⣛⠟⠉⠂⠁⡤⠠⠊⡒⠒⠣⢄⡀⠀⠀⣨⠏⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⣲⠐⠀⠀⠀⠀⠀⠀⠀⠀⡐⠚⠲⠒⠒⡀⡉⠀⠀⠀⢠⡤⢂⠖⠁⠀⠈⠢⠈⡆⡰⣎⠥⠂⢀
⠀⠀⠀⠀⡠⠴⡇⠀⠀⢠⠲⠀⠙⣌⠄⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠆⠀⠀⠀⠀⢀⠃⡎⠘⠛⠒⡦⠄⠇⠁⣇⣇⢠⢀⠌
⠀⠀⠀⠰⡀⠺⣇⠀⠀⢠⠍⠁⢠⠲⣄⠀⠀⢠⡘⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠣⠀⢀⠊⢀⠔⠀⠀⢻⣏⠤⠊
⠀⠀⠀⢀⠈⠢⢬⡝⣆⠈⠐⠃⠀⡍⠉⠀⠀⢖⠚⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣾⠃⠬⢅⠔⠀⠀⠀⢈⢸
⠀⠀⠀⠈⠢⡀⠀⣇⢸⠀⠀⠀⢇⡨⠦⠄⣀⣀⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠋⠁⡀⣴⠄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠈⡛⠛⠉⠀⠀⢀⠤⢄⠀⠀⠀⠑⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢀⠰⠂⠴⠀⠙⠋⢁⠐⡂
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠚⠀⠀⠀⢦⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠹⠾⠘⠛⠀⠤⢤⡀⠏⠛⠈⠂⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠁⢒⠀⠄⡀⠘⠉⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⠧⠉⠀⠉⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠀⠀⠀⠈⠐⠀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⢀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⡀⠄⢀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠠⢌⠐⣸⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⡁⢀⠀⡠⠀⢀⠄⠐⡠⠠
⠀⠀⠀⠀⠀⠀⠐⠈⠄⠐⡈⠀⠈⡁⠒⣦⠤⢄⣀⡀⠀⠀⠀⢳⡸⠃⠔⠸⢀⣀⠤⠠⠖⠈⢋⠔⡠⢊⠔⡡⠊⠔⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠉⠀⠐⠉⠄⢈⠭⠛⡠⠚⡒⠖⡂⠌⡉⠄⠈⠔⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⢀⡠⠀⠀⡀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⣀⠔⠁⠀⠠⠚⠁⠀⣨⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⣶⣶⡐⠒⠉⠁⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠷⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⣿⣿⣆⠀⠈⠀⠄⠒⠉⢀⠠⠀⠀⠀⠀⠀⠈⢀⣀⣠⡴⠖⠀⢉⡘⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⠉⢳⣄⠀⠀⠔⠊⠁⡀⠔⠈⠀⢤⠰⣟⢉⡉⠠⢆⣀⡐⢉⣛⠒⠀⠠⠏⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣴⠷⠄⠙⠰⡷⠉⠄⠠⠀⠒⣩⣤⣶⣷⠩⠸⠆⠀⣸⠇⢀⣾⣿⣿⡾⠏⠀⠏⠀⠀⠉⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⠀⠀⠄⠐⢉⠂⠀⠈⠉⢹⠉⠀⢄⡀⣀⡠⠊⠀⣼⣿⣗⣒⡒⠀⠈⠀⠀⣠⢷⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⢉⡄⢰⣦⠽⣧⣀⡀⠀⢀⣸⡶⢦⠀⡀⠀⣀⣶⣿⣾⡭⠂⠀⣀⢠⠞⢏⣥⣶⣬⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣌⢻⣢⡀⠀⣀⢶⡾⠟⠁⠀⠀⠀⠈⠉⠉⠀⠀⣠⠰⢎⣯⣶⣾⡛⢏⣫⣶⣾⠿⣻⡂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣾⣦⠀⠀⢠⠒⢊⣉⣷⢦⡶⣘⣣⣶⡿⢿⢏⣓⣴⣾⣿⠻⢧⡭⠿⡏⣓⣩⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡻⣿⣿⣿⣿⣿⣿⡆⠀⠐⣿⣻⣽⣫⣶⣾⡿⠿⢫⣋⣶⣾⢻⣅⣃⣴⡞⢿⢜⠓⢂⣩⡍⢻⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣮⡻⣿⣿⣿⣿⣿⡘⠀⠈⣿⣿⡻⠿⣬⣵⣿⡟⠽⢯⡍⠶⣘⡳⢶⡭⢛⣹⣶⠿⢋⡥⢊⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣷⡀⢾⣿⣿⣿⣿⠻⠿⣼⡃⢿⣈⠱⠦⡭⢛⣡⣶⡿⢛⡡⠖⡡⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣇⡘⣿⠿⢝⡛⢽⣷⢍⠐⠦⣭⣙⣤⣾⡿⠟⢁⠔⢋⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠀⠁⠈⠢⠀⠙⢮⣁⠋⠛⠀⠰⠟⠟⠉⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣷⣄⣝⡂⠙⠓⠛⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⢿⣿⣿⡿⠂⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠙⣆⡀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⠆⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⢠⠞⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣠⠄⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠋⠉⠛⠉⠙⠛⡇⠀⠀⠀⠀⠈⠁⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠⠄⠀⠀⠀⠀⢀⡞⠀⢀⡼⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠢⠀⠀⠀⠀⢀⡾⠀⠀⡼⠁⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠋⠉⣉⡇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⡼⠁⠀⢸⠀⠀⠀⠀⠀⣠⠞⠋⢀⣠⠴⠖⠛⠋⠉⢹⡀⠀⢀⣀⡾⠁⢣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡸⠁⠀⠀⢸⠀⠀⠀⢠⠞⠁⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠉⢳⣀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣼⠃⠀⠀⠀⢸⠀⠀⢀⠏⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⡀⠀⠀⠀⠀⠀⣼⠁⠀⣰⠃⠀⠀⠀⠀⡏⠀⠀⡿⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠚⠁⠀⠀⠀⠀⣼⠃⠀⢰⠋⠀⠀⠀⠀⣰⠁⠀⠘⠃⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⢧⡀⠀⠀⠀⣰⠁⠀⢠⠏⠀⠀⠀⠀⢰⠇⠀⠀⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢠⠇⠀⢀⡞⠀⠀⠀⠀⢠⡟⠀⠀⠀⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⠀⠀⠀⢀⡞⠀⠀⠀⢠⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣈⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠘⣆⠀⢀⡞⠀⠀⠀⠀⢸⡆⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠘⢷⠟⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⢦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣄⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



⣄⣤⣤⣠⣤⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣠⣠⣄⣠⣤⣀⣠⣤⣤
⣿⣿⣿⠟⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣬⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠴⠄⠀⠀⢀⡀⣸⢼⠁⢷
⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠄⠀⠔⠀⠙⣿⡎⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠴⣾⡇⠀⢠
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣒⠂⣖⠀⠀⠀⠀⠀⠀⠻⡾⡄⡼
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠁⠀⠀⠐⠁⠁⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠤⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⢀⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⡀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⢀⠎⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⡀
⠀⠀⠀⢀⡎⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡀⠘⢦
⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢸

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡅⡎⠉⠉⠉⠉⠉⠉⠉⠉⡆⣴⠞⠻⣦⡀⣴⠞⠻⣦⡀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣀⣀⣼⠃⣷⣀⣀⣼⠃⣿⠛⣿⢛⡟⢻⡛⣻⡛⣻⢛⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡧⠻⠳⠟⠛⠛⠛⠺⠒⠛⠣⠬⠛⠛⠡⠠⠬⠟⠛⠥⠠⠿⠛⠛⠛⠛⠛⠛⠻⠛⠿⠛⠛⠰⠀⢾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠴⠶⢤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⣀⡤⠤⠒⠒⠒⠒⠤⢤⣀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⡴⠋⠁⣀⣤⣤⣤⢤⣤⣤⣀⠈⠙⢦⡀⠀⠀⠹⣦⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⠃⠀⠀⡰⠋⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠙⢆⠀⠀⠘⣧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡏⠀⠀⣰⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⣆⠀⠀⢹⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠐⡅⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢨⠂⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⡀⠀⠀⣇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣸⠀⠀⢀⡿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣧⠀⠀⠘⣄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⣠⠃⠀⠀⣼⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣆⠀⠀⠈⢦⡀⠈⠳⢦⣄⣀⣀⣀⣀⣠⡴⠞⠁⢀⡴⠁⠀⠀⣰⠏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠙⠢⣄⣀⠈⠉⠉⠉⠉⠁⣀⣠⠔⠋⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠈⠉⠙⠒⠒⠋⠉⠁⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣤⣤⣤⣠⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣀⣀⣀⣀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠣⢤⡤⠤⠤⠤⠤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⠤⠤⠤⠤⢤⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⠠⠄⣀⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠤⠤⠤⠤⠤⠖⠒⠒⠒⠒⠛⠃⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠢⠤⣀⡀
⠀⠀⠀⡟⠓⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⢄⣀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠉⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠠⠤⠤⠤⠄⠒⢒⠒⠒⠂⡈⠉⠭⠭⠍⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠄⠒⠖⠒⠒⠒⠂⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠀⠀⠀⠐
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⣠⠒⠀⢶⡄⠀⠀⡔⠢⠀⡖⠐⠀⢏⠈⠀⡇
⠀⠀⠀⡏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡅⠀⠀⠀⠛⠀⠀⠧⠀⠀⠳⠄⠀⠙⠂⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠴⠋⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠤⣤⠴⠶⠶⠶⠚⠛⠒⠃
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢠⣤⠠⠤⠠⠤⠔⠒⠒⠒⠊⠉⠉⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠉⠈⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡒⠷⠚⠛⠛⠛⠓⠒⠤⢀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡡⠊⠀⠀⠀⠀⣠⡤⢒⠂⠀⠀⠒⠤⣀⠀⠀⠀⠑⣄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡰⠁⠀⠀⠀⡠⠊⠀⠀⠀⠈⠑⠢⡀⠀⠈⢳⣄⠀⠀⠈⢆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠈⢺⠁⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠹⠂⠀⠀⠘⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⡆⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⢡
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⡀⠀⠀⠀⠘⣕⠢⢀⣀⣀⡀⠄⠂⠀⠀⠀⡠⠁⠀⠀⠀⢀⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣣⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠀⠀⢀⠎
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠉⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⠤⠤⠤⠄⠒⠊⠁
⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠐⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠄⠒⠒⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⢀⣀⣀⣤⠤⠤⠔⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠉⠁⠈⠉⠁⠈⠉⣄⡤⣀⡄⠈⣠⢌⡉⠉⣉⠋⡋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⡀⢸⠀⠀⠀⢊⡧⣷⠇⠈⠷⠾⠟⠀⠳⠶⠿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠉⠉⠉⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠂⠒⢒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣆⢐⡀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢀⣴⠶⠟⠛⠛⠷⣦⣑⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠄⡀⠹⣀⡀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠄⢏⠈⢰⡇⠀⠀⠀⠀⠀⠀⠁⡈⣆⢸⢢⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠘⠓⠀⠀⠀⡤⣾⠿⠭⣡⠀⠎⠤⠵⠾⢇⣀⠀⠀⣀⣀⡴⠊⠀⢸⠄⠀⠀⢼⠆
⠀⠀⠀⠀⠀⡠⠀⠴⠴⠵⣤⣤⣦⡺⠿⠓⠶⠾⠿⠼⠃⣤⣀⡙⠊⣩⠍⢱⠒⠨⠆⠰⠇⠰⠠⡄⣀⣸⢠⠄⢀⢸⡀
⠀⠀⠀⠀⠈⣀⠴⠒⠒⠀⠀⠉⠛⠛⠓⠒⠉⢉⣿⣿⡡⠤⠤⠭⠥⠥⣶⡌⠫⣥⠒⠒⠺⣯⣐⣧⠚⠁⠊⠒⢺⡞⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠠⠁⡤⣤⢤⣤⣠⣀⣂⣀⡂⠀⢂⣾⠛⣛⣋⠀⢛⠛⡃⠀⡻⡇⠈⠈⢡⡶⢾⡇⢾⠶⠞⠋⢂⢴⠟⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠠⡀⣹⠈⢛⠛⠚⠛⠘⠋⠛⠛⠋⠘⠛⠛⠀⠛⠒⠃⠀⠛⢓⣦⡔⣸⣁⣀⡐⠽⣤⣤⡶⠗⠁⠀⢀⠤⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠶⠆⠳⠶⠖⠘⠿⠾⠘⠟⠒⠂⠚⠛⠟⠀⠛⠛⠛⢀⡇⢈⠉⠉⠓⠋⠁⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠻⠶⠆⠿⠿⠂⠷⠶⠆⠷⠽⠷⠸⠿⠟⠘⠿⠗⣡⣾⣶⣶⣶⣶⣶⢶⠷⠷⠶⠖⢶⢶⣶⣶⣇⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠓⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠈⠀⠈⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠐⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣤⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠉⠉⠉⢳
⠀⠀⠀⠀⣠⠤⠤⢤⠧⢤⣤⣤⡤⢿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠒⠒⠒⠒⠛⡆
⠀⠀⠀⡎⠀⣀⣀⡼⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⡇⢸⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⣀⣀⡀
⠀⠀⠀⡇⢸⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⡤⠄⢀⣈⣉⣉⣉⠉⠁⠐⠒⠒⠒⣒⡶⡄
⠀⠀⠀⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢸⠀⠀⠀⣠⠤⠒⠒⠐⢦⠀⢠⡖⢆⠀⢡⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠉⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠀⡎⠀⠀⠀⠀⠀⠈⡆⢸⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⢣⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⡎⡇⠀⢀⠃⠀⠀⠀⠀⠀⠀⡇⢸⡇⠘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⢀⡠⠤⠤⠤⢄⣀⣀⢀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⢸⡎⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠸⣇⡰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠑⠒⠤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⢸⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⣸⣈⣉⠒⠒⠒⠒⠒⠒⠒⠒⠊⣉⣀⣇⡇⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⡇⠀⠀⠈⠉⠙⠛⠀⠐⠛⠋⠉⠉⠀⠀⢸⠓⠒⠒⠂⠤⠤⠤⢄⣀⣀⣸⣀⡤⠤⡇
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠦⠤⢤⣤⣀⣀⣀⡀⠀⢸⠀⣀⣠⠇
⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠀⠈⠒⠒⠦⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⣉⣙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢷⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠔⠉⠀⠀⠈⠙⣦⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿⣿⣿⣿⣿⣿⠟⠃⣾⣴⣶⣾⣿⣿⣿⣿⣿⣟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠠⡿⠸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠋⠀⠀⢰⣽⣿⣿⣿⣿⣿⣿⣿⣿⣈⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣤⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⣸⣿⣿⣍⠉⠹⠿⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⢯⡿⣦⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣟⣻⡿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⣿⣿⣿⣿⡇⣿⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣬⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡜⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⢇⣿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠻⣿⡝⠂⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡟⠨⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⢄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⢏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⡇⠀⠈⠱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠻⠁⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣇⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⡇⢸⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠉⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⠇⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣷⡯⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⣸⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⣽⡿⠷⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠀⢿⣿⣶⣷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣷⠀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠁⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣏⠀⣿⣿⡟⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠹⣿⣿⣤⣿⣿⣷⣤⣼⣇⣀⣀⣠⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠿⣿⣿⣿⣿⣷⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣁⣼⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⠿⡿⠯⠥⠤⠤⠬⠭⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠴⠶⠒⠂⠀⠐⠛⠛⠒⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⣉⣉⠉⠉⢁⣀⣀⣒⣒⣒⣒⣂⣀⣀⠀⠀⠀⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⣀⣀⡀⠀⠀⠀⠸⠀⠀⣧⣀⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⣀⣀⣬⣥⣈⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠐⠂⠈⠉⠉⠐⠒⠭⢁⠀⠀⠐⠒⠶⠶⠢⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⠤⠤⠤⢄⣀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠋⠀⡀⢀⠔⠀⠀⠀⢠⠀⠈⠙⠢⡀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠃⠀⡞⡰⣡⠀⢠⢂⡤⠎⢀⠞⡆⠀⠈⢢⠀⠀⠀⢡⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡎⠀⡸⢰⠁⡇⠀⡞⠀⠤⠖⠁⠀⠘⠄⠀⠀⢃⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡇⢸⠀⡇⠀⢱⠀⠀⣀⠤⠔⠒⠤⣄⣠⢜⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢁⠸⡀⠰⡀⠀⢣⡀⠀⠒⠒⠒⠒⠤⡀⢀⠃⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠀⠀⡀⢳⡀⠙⢄⠀⠙⠢⢄⠀⠀⠀⠀⠀⡘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢂⠀⠈⢆⠙⢤⡀⠑⠢⠤⠄⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠡⡀⠀⠑⠢⣍⡒⠲⠦⠤⠶⣒⠔⠊⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠄⢀⡀⠀⠠⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠶⠦⠤⠤⢤⣀⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠲⠶⠤⠤⠤⠤⠀⠀⠠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠾⠛⠛⠛⠛⠿⠶⠶⠶⣦⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋⣉⣠⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣀⣉⣉⡉⠛⠛⠛⠷⠶⢶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⢀⣤⡶⠟⠛⣩⣴⠟⠿⣿⡟⠛⠩⣉⠉⠉⠛⠛⠻⠟⠛⢻⣧⣭⣭⣿⡿⣿⣿⡿⣷⣶⣶⣤⣤⣄⣩⣭⣟⢿⡄
⠀⠀⠀⠀⣼⠏⠀⢴⣿⣏⣀⣙⣂⡈⠻⣶⣶⣧⡶⠶⠾⠛⠛⠷⢶⣌⣿⡄⢰⠈⢢⣀⠹⡻⣿⣍⣽⣿⡛⠋⠉⢤⡙⢿⣇
⠀⠀⠀⠀⠹⣧⣀⣀⡀⠉⠉⠉⠛⠛⠻⠶⠿⣿⣶⣶⣔⣀⣒⣒⣈⣉⣻⣿⣼⠶⣦⣄⣤⡶⠿⠿⢭⡉⠙⢦⡀⠈⠁⠀⣿
⠀⠀⠀⠀⠀⠈⠹⣿⠛⠛⠻⠶⠶⢶⣤⣤⣤⣀⣀⣀⠈⠉⠙⠛⠛⠿⠿⢿⣶⣦⣤⣽⡿⢿⡷⣦⡀⠉⠀⠀⠀⢀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠷⠶⠶⣦⣤⣤⣄⣀⣀⣀⣠⣼⣷⠘⣷⠀⠹⠄⠀⢸⡇⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⢻⡆⠀⢰⠦⡄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠀⣿⠀⢿⡄⠀⠀⡆⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣧⣨⡇⠀⢸⡀⡇⠀⢸⠉⣿⠀⢸⠛⡶⠀⣠⣤⡄⠀⠀⠀⢰⣿⡇⢸⡇⠀⠀⠃⠀⠀⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢰⡶⣦⠀⢨⣭⡁⠀⠘⠶⠿⠀⢸⣤⣇⠀⣿⠀⡇⠀⠀⠀⠸⢾⡇⣿⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣇⣹⠀⢸⡇⣿⠀⢸⡏⢻⠀⢰⠶⣤⠀⢙⡛⠃⠀⠀⠀⢀⣸⣧⢻⣿⠀⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣬⢭⡄⠈⣛⡛⠀⠸⠧⣾⠀⢸⡀⣿⠀⣿⠉⡇⠀⠀⠀⢸⣯⣿⢸⡏⠀⣧⠀⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢸⡀⡇⠀⡏⢹⠀⢰⡶⣦⠀⢀⣉⣉⠀⠻⠶⠇⠀⠀⠀⠘⢻⡇⣸⡇⠀⢸⡆⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠘⠓⠇⠀⢿⣼⠀⢸⣇⣻⠀⢸⡇⢹⠀⢰⠶⣦⠀⠀⠀⢰⡟⣿⣿⣄⣤⣬⣇⣠⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠘⠷⠼⠂⢸⣀⡏⠀⠀⠀⠸⠷⢋⣼⠟⠋⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠳⠶⢶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠶⠶⣦⣤⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣉⣉⣉⣙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣿⣛⣛⣿⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠋⣿⡏⣿⡟⢻⣿
⠀⠀⠀⣤⣤⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣀⣿⣇⣼⣇⣸⣿
⠀⠀⠀⣿⣧⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛⠛⠛⠛⠻⣿⡄
⠀⠀⠀⠈⠉⠙⠛⣿⠀⠀⡀⠀⠀⠀⠀⢀⣤⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠄⣿⣿⣷⣴⣦⣶⣾⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⢀⣿⣼⡟⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣥⣿⣷⣶⣶⣶⣶⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠻⠟⠋⣼⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⡀⠀⠀⢀⠀⡀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⣼⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡁⠀⠀⢻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠄⡠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣃⡀⠆⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠦⠀⠀⠀⠶⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡀⠀⠀⠛⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣛⣢⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠛⢣⣄⠁⠀⠀⡀⣀⠻⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀
⠀⠀⠀⠀⠀⠀⡶⠊⠀⣤⣀⡀⠁⠉⠴⠆⠉⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠤⡈
⠀⠀⠀⠀⠀⢰⣷⠒⠒⠒⠒⠒⠒⢒⠒⠒⠒⠒⠒⠒⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠖⠲⣶⡆⠀⠀⠦⠄⠀⠤
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡟⠁⠀⣄⠀⠀⠉⠀⠀⠀⠀⠃⠀⡀⠈
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣤⣤⣶⣦⣄⡀⠀⠀⠀⣿⠀⠀⠀⢀⣤⣶⣶⣦⣤⡀⠀⠀⠀⡇⠐⠀⠸⠀⠀⠀⠀⠀⠀⢀⡀⣀⣅⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣇⠀⠀⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣿⠀⠀⢰⣿⡿⠿⣿⣿⣿⣿⡇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢿⣮⣿⣿⣿⡿⠅⠀⡀⠀⣿⠀⠀⠀⠻⣄⣾⣿⣿⣿⡿⠁⡀⠀⡃⠐⢂⠀⠀⠘⠁⣀⡦⠀⠀⠀⠀⣀⣀⣼⡖
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⠀⠀⠉⠉⠉⣀⡠⠞⠁⠀⣿⠀⠀⠀⠀⠈⠙⠛⠛⠉⣠⠔⠁⠀⣧⠀⣀⣤⣀⠀⠀⠀⡀⢀⣀⣤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠐⠶
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠿⠟⠛⠀⠀⠤⠤⠄⠀⠤⠤⠤⠤⠤⠴⠶
⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⠄⠀⠀⠀⠀⡤⠀⠀⠀⠄⠀⢠⠀⠀⡄⠀⠀⠄⣄⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⢤⣼⠀⢴⠋⠌⠠⠀⠀⢠⣤⢠⠄⠸⠁⠀⠈⠀⠀⠅⠀⠀⠀⠈⡄⠀⠠⠀⠹⠆⠀⠈⠂⠢⠉⠿⠆



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣽⣯⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣷⣷⣶⣿⣿⣷⣶⣷⣾

⠀⠀⠀⠀⠀⠀⠀⠀⡍⠉⠉⠉⠉⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃
⠀⠀⠀⠀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⠉⠉⠉⠉⠐⠒⠤⣀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀⠀⠈⠣⡀
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡰⠁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠙⡄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠁⠀⢀⠃⠀⠀⠀⠀⢀⠀⠐⠀⠀⠂⠄⠀⠀⠀⠀⠀⢡⠀⠀⠸
⠀⠀⠀⠀⠆⠀⠀⠀⠀⡎⠀⠀⡌⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠂⠀⠀⠀⠀⢇⠀⠀⢁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⡇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⡄⠀⠈⠄⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠜⠀⠀⡰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠱⡀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⣠⠃
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠈⠂⠄⡀⠀⠀⠀⠀⠀⠀⣀⠠⠊⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠀⣀⡀⠠⠤⠔⠒⠉
⠀⠀⠀⠀⡃
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁