Furniture Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⣉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠙⠪⣙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠳⢮⣑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣾⡒⠮⡛⢿⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠳⣝⡦⣌⡆⠙⠳⣭⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢶⣄⡀⠀⠀⠉⠉⠁⠀⠀⠀⠉⡗⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⢿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⡇⠀⢸⡇⠙⡦⣄⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠘⡇⠀⡇⠈⢳⣦⣀⠀⢸⢸
⠀⠀⢰⠚⣧⣤⣶⡮⠗⠚⠏⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⠀⠀⡇⠀⡇⠀⢸⠆⠉⡷⢾⡴
⠀⣀⠼⠞⠛⠉⠀⠀⠀⠀⢠⣀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣄⠀⣿⠀⡇⠀⠀⡇⢰⡇⠀⢸⠀⠀⡇
⢠⡅⠀⠈⠀⠀⠀⢀⡤⢶⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠖⠒⠒⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⠻⣤⡇⠀⠀⡇⠸⡇⠀⢸⠀⠀⡇
⠈⡇⠀⣈⡥⠔⠊⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣷⢸⡇⠀⢸⠀⢠⡇
⠀⠓⠈⠁⢸⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠦⣌⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣇⡀⢸⠀⢸⠃
⠀⠀⠀⡄⠘⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢻⢦⣝⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣤⣸⡀
⠀⠀⠀⡇⢠⠀⠀⠀⠀⢸⡠⠖⣨⢍⡉⣉⣉⠛⠛⠛⠛⠒⠒⢲⢶⢸⠀⠀⢸⡄⠈⠳⢤⡙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠤⠤⠤⠤⠽⡇
⠀⠀⠀⡇⢸⠀⢀⡤⠖⠋⠀⢮⣀⣤⢿⣤⣤⠇⠀⠀⠀⡠⢞⠡⠋⢸⠀⠀⢸⠁⠀⠀⠀⠉⠳⣌⢳⠒⠒⠒⠒⠋⠉⣉⣁⣀⣤⡤⠤⡴⠖⠒⢲⡞⠃
⠀⠀⢀⣄⠼⠚⠁⠀⠀⠀⠀⢼⣁⡼⠯⣀⡸⢂⣠⣔⠯⠂⢁⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢻⠗⠒⠒⣿⢻⠉⠉⢹⡀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⢰⣚⣒⣢⣶⣶⣖⣒⣒⣒⣒⣶⣶⣦⣶⡮⠙⣡⣜⡾⠀⡀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢹⢸⡄⠀⢸⡇⠀⠀⠀⢿⠀⠀⢸⡇
⠀⠀⠉⡇⢀⣇⠀⠀⢀⣀⢀⡇⣀⡇⠀⢹⣶⠿⠋⢹⠀⠀⡇⠀⠀⢸⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠘⡇⠀⢸⡇⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⡇⠓⠶⠒⠚⠻⠟⠛⡷⠺⡇⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⣸⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡇⠻⠶⠾⠃⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇⢀⡇⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⢿⣄⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡇
⠀⠀⠀⡇⠀⢸⠀⠀⠀⠉⠉⠀⠀⡇⠀⢸⠀⠀⠀⢌⣀⣠⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⡇
⠀⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡇
⠀⠀⠀⡇⢀⡇⠀⠀⠀⠀⠀⠀⠰⣇⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⠾⠃
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠚⠋

⠀⠀⠀⠀⠀⢰⡖⠶⠶⠶⠖⠲⠶⠶⠶⢲⠄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠈⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠐⠿⠿⠽⠿⠿
⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣄⣤⣤⣤⣤⣼⠂⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⢰⣿⣿⣿⣟⣻
⠀⠀⠀⠀⠀⢸⣇⣀⣗⣀⣸⣇⣀⣇⣀⣼⠆⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡟⠀⠀⠀⢈⣛⠛⠛⠛⣋
⠀⠀⠀⠀⠀⠀⠻⠗⠒⠒⠒⠒⠒⠒⠺⠏⠀⠀⠀⠀⠩⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠁⠀⠀⠀⠘⠏⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣶⣶⣿⣷⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠂⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠀⠀⢀⡜⣿⣿⣿⣿⣿⣿⣿⢧⡄⠀⠀⠀⣿⠉⠉⠉⠉⡉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠁⠸⠿⠿⠿⠿⠿⣿⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⠿⠿⢿⢟⠾⠀⠀⠀⠀⠿⠿⡿⠾⠿⠿⠿⢿⡷⡿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠴⠢⠠⠠⠠⠄⠦⠄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠸⣿⠤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⠀⢸⡶⠀⢿⠆⠘⢷⡶⠒⣤⡤⢼⣶⣄⣼⡇⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠄⣿⠀⠀⣤⢀⣸⠀⠀⠈⡇⠀⠛⠀⢸⠟⠉⢙⡃⠀⢠⣄⣀⠘⠟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠘⠀⠸⠟⠉⢿⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢸⡇⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⢺⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⡇⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣇⣸⣦⣀⣖⠀⠀⠀⠀⠀⠀⠀⢸⣏⠁⣩⡇⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⡎⠁⠀⠙⡇⠀⠀⣬⡇⣮⣙⣻⣧⡼⢻⣷⠀⡇⠀⠀⠀⠀⠀⠀⠈⢢⣄⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⣻⠀⣶⡄⢀⣴⠃⠀⢠⣿⣷⣿⣭⣿⣛⠁⠀⡟⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⣴⠻
⠀⠀⠀⠀⠀⠀⠀⠰⣧⣠⡾⢦⡧⣄⣤⡾⣆⣀⡼⠀⠰⠧⠀⠼⢿⣦⡴⠧⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠬⢉⣉⣉⠁⠀⠒⠒⠒⠠⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠁⠉⠁⠀⠁⠀⠀⡂⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⡘⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣈⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠤⣀⡀⠀⠙⠓⠢⢄⣀⠀⠀⠀⢀⠀⠀⢸⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠂⠀⡈⠓⠲⠤⣀⡀⠀⠉⠲⠤⣀⡀⢀⠇⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⡠⠈⠁⠒⠼⠄⣼⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⢀⠤⠊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠤⠈⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡀⠐⠁⢀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠓⢤⣀⠀⠀⠀⣀⠐⠁⠀⠀⠀⠀⠠⠊⠀⣠⠶⠁⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡰⢀⠃⢨⠁⠢⣄⡀⠀⠂⠌⡀⠐⠀⢀⠴⠋⠘⡀⠄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠁⠂⠀⡆⢠⠁⠆⠈⠒⢤⣀⢀⡠⠖⠁⠀⠀⠀⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⡰⠀⠀⠂⠌⠀⡸⠀⠀⠰⠀⠀⠀⢸⠀⠉⡁⠀⠀⠀⠀⠀⠁⠸⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠁⠀⠄⡐⠀⠀⣅⠀⠁⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⢰
⠀⠀⠀⠀⢠⠇⠀⡈⠐⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡜⠀⢰⠀⡆⠀⠀⠀⠀⠀⡄⢀⠀⠈⡀
⠀⠀⠀⠀⢎⡀⠐⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢨⠀⡇⠀⠀⠀⠀⠀⠠⣘⣀⠤⠃
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠈

⠀⠀⠀⠀⢠⡄⢤⠀⠔⠲⠴⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢰⠦⠶⣰⣠⢉⣐⠔⠂⠀⠊⠤⢴⡴⠀⠀⡀⠀⡀⠐⠄⠄⠐⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄
⢰⣚⣳⠘⠀⠈⠄⡤⡀⠀⠀⠀⡸⡇⠀⣄⠀⡌⠗⢁⡀⠀⠀⠁⠀⠀⣀⠀⠀⠀⢀⣴⠊⠀⠀⠂⠀⠀⠀⠀⠃⢠⢠⡁⠀⢀⢠
⢸⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠘⡥⠀⠀⠀⠀⠁⠁⠀⠀⢰⣾⠯⢸⡆⠀⢠⣭⣄⡄⢀⣀⣀⡀⢠⡈⣆⢨⣈⠁⡠⣶⠐
⠘⠰⠄⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⣇⠀⠁⠉⡁⠀⠀⠄⢀⢀⠀⠀⠀⠀⠁⠁⠄⠀⠸⠚⠶⠻⠿⢿⠀⢃⠁⠘⠔⠓⠀⠂⠉⠈
⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡄⠀⠀⢠⠀⠅⠀⠀⠀⣰⡀⢲⡦⠀⠤⠀⠀⠤⠄⠤⠠⠰⠀⢸⠀⡗⠇
⠀⠀⠈⢇⠺⠀⠅⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠀⠉⠉⠉⠈⠁⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⡋⠀⠀⠃⠘⢁⡘⢀⢃⣘⠁⡇⠀⠀⠀⠈⠀⠁⠀⠈⠯⠽⠨⠅⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⢈⣀⣀⣀⣀⣀⣀⣀⣁⣈⣁⣀⣀⣀⡀⠀⢀⢀⣀⣀⢀⡁⠀⠀⡶⣷⠰⠾⠷⠶⠶⠶⠶⠶⠶
⠀⠀⠠⢭⣿⡭⠭⢿⡀⢀⢤⠀⠀⠀⠐⣏⠀⠀⣼⠇⠚⠄⡂⠁⠀⢠⣘⡼⠒⠂⠒⠒⠐⠂⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠰⣐⢃⣀⣀⣀⣀⣛⣧⡠⠁⠠⠀⠀⢀⡷⡇⡼⠃⠀⠸⠐⠀⠀⢀⡾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⡀⠠
⠀⠈⠁⠀⠄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠑⠟⠁⠀⠀⠈⠀⠀⠀⠘⠛⠁⠀⠠⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠠
⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠁⡀⠀⠀⠀⠀⠀⣀⣀⡄⣀⡄⣀⡀⠀⣀⡠⠂⡖⠆⢲⣰
⠀⠀⠀⠀⠀⠖⠀⠄⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣶⡶⣶⣤⣿⣿⠆⠐⠀⠻⡇⠐⣿⠛⠙⠄⠀⠈⠙
⠀⢠⠦⠤⠤⠤⠤⠤⠤⠠⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠄⠦⠄⠀⠀⠀⣻⡷⢿⣿⢻⣿⣿⠀⠀⠀⠀⣿⣼⠇⠀⠠
⠀⢐⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣴⣿⣼⡧⢲⠀⠀⣀⠐⠂⠶⠀⠀⠀⠀⠯⠛⠈⠘⠘⠀⠘⠀⠀⠀⠀⠛⠿⠀⠰⠆
⠀⠈⠁⠀⠁⠀⠈⠈⠉⠁⠀⠚⠋⠈⠁⠉⠓⠀⠀⠛⠃⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣓
⠀⢸⡖⡆⡆⡄⢘⠘⣺⠆⠀⠀⡀⣀⠀⠂⠀⠀⣀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠈⠊⠘⠀⠂⠖⢀⣭⣹⡷
⠠⠼⠷⣤⣤⣤⣬⡽⠿⠀⠀⠀⠈⠄⠍⠆⠀⠀⠤⠤⠀⠀⠬⠆⠀⢆⣈⡀⡀⣀⣈⡈⡙⠀⠉⠂⠒⠚⠘⠛⠟⠩⠀⠀⠀⠰
⠈⡁⢀⠀⢠⠄⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠁⠈⠀⠉⠀⠀⠀⠀⠁⠉⠉⡁⠀⠀⠉⠉
⠀⠺⠌⠀⠃⠁⠁⢄⠀⠀⠈⠇⠀⡆⢱⡖⠻⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣀⠀⠀⠀⠀⡀⠀⠀⣠⠀⠰⠆⠀⡀⡀⡀⠆
⢠⠀⠀⠀⠀⡀⢀⠀⣃⡔⠂⠀⠤⡇⣬⠡⠐⡄⠀⠀⠠⠢⠏⠀⠐⠒⣕⢼⠛⠆⠀⣦⣤⡤⠤⣤⠸⠀⠘⠆⠅⠀⠄⠓⠠⠐
⢸⣀⣀⢀⣈⣀⣀⣀⠈⠀⠀⠂⠀⡆⠾⠀⡇⢶⠰⠀⠀⠍⠀⠀⠀⠀⠈⠋⠀⠀⠀⠜⠻⠿⠿⠻⢞⠀⠰⡄⠃⠂⠂⠀⠀⠈
⠠⠿⠟⠿⠿⠿⠿⠿⠠⠠⠀⠀⠀⠁⠀⠀⠛⠲⠾⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠋⠀⠐⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡤⠷⠶⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⡎⠀⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠘⠣⠶⠶⠤⠤⠤⠤⠤⠤⣤⣄⣀⣀⣰⠇⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⠊⠓⠒⠒⠒⠲⠶⠤⠤⠤⠤⣤⣤⣼⠀⡌⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡋⣲⠒⠒⠒⠶⠤⠤⠤⣤⣀⣀⣀⣀⡇⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡏⠉⠉⠛⠓⠒⠒⠶⠤⠤⣤⣀⣈⣹⡇⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⡏⠉⠑⠒⠒⠦⠤⢤⣀⣀⣀⠀⣸⠀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⢀⠇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡧⣘⡠⠅⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠶⠤⢤⣀⣀⡇⠈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠴⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣍⣀⠂⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢂⡩⠄⣶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣏⠉⠓⠲⠤⣍⣑⡒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠔⢊⣥⠴⠊⠁⢀⡀⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡟⢰⡤⢄⣀⠀⠈⠉⠓⠲⠦⣬⣉⡑⠒⠖⠒⣉⡤⠖⠚⠉⠀⣠⣴⣞⣉⡇⢘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠘⡇⠀⠀⠉⣓⣶⡤⢄⡀⠀⠀⢹⠛⢻⠋⡏⠀⣀⡴⠶⣉⣀⠀⠈⠉⡇⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠊⠓⠶⢶⣟⠉⠀⢀⡤⠜⠋⠓⢺⡀⢸⠀⣿⠉⠁⠀⠀⠀⠈⠉⠑⢲⣧⣤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠇⣀⠀⠀⠉⠉⠛⠳⢤⣤⣀⠀⢸⡇⢸⠀⣿⠀⠀⠀⠀⠀⣠⡴⠞⠋⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⣿⠉⠑⠢⢄⣀⠀⠀⠈⠙⠛⢻⡇⢸⠀⣿⠀⣀⣤⡶⠋⠁⠀⢀⡤⡆⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢻⠀⠀⠀⠀⠈⠉⠒⠦⣄⣀⠠⡇⠸⠀⣿⠞⠋⠁⠀⣀⠴⠚⠁⠀⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⠈⠀⣿⢀⣠⠔⠋⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⡆⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⣀⢀⠀⠀⠀⠄⢠⠀⠄

⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⢀⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡤⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⢀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⡘⢷⣮⣰⣬⣷⡞⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣠⣿⡇⣨⣽⣯⣭⣭⣭⣭⣭⣽⠇⠀⠀⠀⠀⠀⠈⠉⠚⢿⣿⣿⡟⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠛⣿⡆⠸⢷⠶⠶⠶⠶⠶⠶⠿⡄⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠿⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢈⣶⣟⡋⣘⣟⣛⣛⣛⣛⣛⣛⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⡃
⠀⠀⠀⠀⠀⢸⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠁⠈⠀⠀⠈⠉⠁⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠄
⠀⠀⠀⠀⠀⠀⢹⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣻⣿⣻⣿⣿⣟⣿⣗⣿⣿⣻⣿⣻⣟⣿⣟⣿⣟⡏⢹⠁⠀⡃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡏⢨⡇⢼⠁⢸⡇⣿⢸⢠⡏⢸⡇⣽⠀⢹⡅⡇⢸⠀⠀⡅
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡇⠀⠋⠙⠀⢸⠁⢻⠈⠈⡇⢸⠃⠙⠀⢸⠁⠃⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠸⠀⠀⠀⠸⠀⠀⠸⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠘⠓⠒⠒⠚⠂⠛⠒⠒⠓⠚⠃⠚⠒⠚⠒⠓⠚⢂⠀⡁
⠀⠀⠀⠀⠀⠘⠲⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢰⣦⣴⡤⣶⣤⡶⠀⢶⠂⠀⡁
⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣛⠛⢻⣛⡇⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠟⠀⢸⠒⡇⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⣺⣛⣛⣛⣛⣛⣛⣛⣾⢀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠅
⠀⠀⠀⠀⠀⢀⡸⠄⠠⣿⣯⠨⢽⣯⣭⣭⣭⣭⣭⣭⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄⠤⠄⠼⠀⠥⠀⢬⡀⠀⠆
⠀⠀⠀⠀⠀⠸⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠅
⠀⠀⠀⠀⠀⠀⢰⠀⠖⡶⠆⢶⠶⢶⡶⡶⠆⡶⠀⠶⢰⡶⠰⡶⠰⠶⠰⠦⠰⠆⠂⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢘⠀⠷⡇⠷⣿⠾⢾⡷⡇⠷⡇⠀⢸⢾⡷⠷⡿⢾⢸⢾⡶⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢻⠀⢸⡇⡇⠀⡏⠀⠀⢸⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢐⢀⠀⣇⠀⣸⠀⢸⡃⣇⡀⣇⠀⢀⢸⡇⢀⣇⢀⣐⢀⡇⠀⡂⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡅
⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣨⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⠀⠀⡃
⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⡁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠁
⠀⠀⠀⠀⠐⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣮⣭⣽⣿⣽⣿⣿⣿⣿⣽⣿⣷⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠿⠭⣽⣛⡷⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣭⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⢶⡶⠶⣶⠶⢶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⠀⠀⠀⠈⠉⠙⠛⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠦⣀⠀⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠻⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠖⠛⢿⣤⣼⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣄⣹⠀⢸⡇
⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣦⣾
⢀⣼⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⢊⣡⣿⡏
⠸⣷⣄⡀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠟⢋⡤⠖⣩⡿⠋
⠀⣿⠉⢻⣶⣄⡀⠀⠀⠀⠈⠙⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⢋⣡⡴⠞⣉⣴⡞⠙⢳⣄
⠀⣿⠀⢸⡇⠈⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣩⣴⠾⠋⣡⣴⠟⠉⠈⢻⣆⠀⢿⡄
⠀⣿⠀⢸⡇⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⣡⡴⠟⠉⣠⡴⠟⠉⠀⠀⠀⠀⠀⢻⡇⠸⣷
⠀⣿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⣠⠞⠁⣠⠞⢉⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿
⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⠀⠀⠀⠀⢀⡾⠋⣠⠞⣡⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡇
⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⣼⠃⣰⢏⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠸⣧
⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠶⠃⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣿
⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢹⡆
⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣼⣇
⢸⡷⠶⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⢹⡄
⣿⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠞⠁
⠛⠶⣶⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⣸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠉⠉⠉⠉⠒⢤⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠃⠀⠀⠀⠀⠠⠀⠀⢦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⠀⠀⠸⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⠙⠦⠶⠞⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠤⠤⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠲⣃⠀⠀⠀⣘⠶⠂⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠉⠀⠀⠀⠀⠀⠚⠛⠓⠀⠀⢠⠈⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠈⠛⠛⠛⠣⠤⠄⣠⡤⠐⠉⠀⠀⠀⠀⠙⠠⢄⡀⠀⠘⡀⠀⢀⡀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⢀⠤⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⢸⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⣀⣀⠠⠼⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠦⢀⡀⠀⠀⠀⠀⡤⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⣀⠀⠢⣄⠀⠔⠈⠁⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠓⠐⠂⠀⠀⠀⠄⢤⠤⠀⠀⠐⠂⠀⠈⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠸⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣨⠃⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⡍⠉⢉⣀⠀⣉⡈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠲⠖⠒⠂⢤⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠄⠀⠀⢀⡍⠢⠾⠆⢉⠉⠉⠉⠁⣶⠋⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡋⠀⠀⠀⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠠⡌⠀⠀⠀⠀⠀⢀⠦⠤⡤⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢠⡆⠀⠀⠀⡐⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⣀⠤⠛⠁⠀⠀⡄⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣾⣃⣹⣯⣹⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣛⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠦⠤⢤⣤⣤⣤⣤⣤⣤⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢰⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠶⠒⠒⠒⠒⠒⠛⠲⠾⠾⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡉⠐⠒⠒⠶⠦⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠐⢈⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠗⠐⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣮⣵⣬⣮⡓⢒⣫⠯⠉⠛⠛⣛⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠮⡭⠤⠀⠀⢠⢖⡪⢺⠆⠰⣍⣉⣛⡻⠷⠿⣶⡒⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠒⡼⠀⡠⢖⠕⡡⠖⠉⠉⠉⠉⠀⠀⠈⠉⠓⢲⠼⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢰⢗⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⡋⡩⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⠶⠋

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠠⠆⠐⠒⠒⠒⠂
⡇⠀⠀⡤⠀⠠⡄⠀⠀⠀⣀⣀⡀⡀
⡇⠀⠀⡇⠀⠀⣿⠉⠉⠙⡇⠀⠀⠉⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⢀
⡇⠀⠀⡧⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣶⠋⠉⢲⡊⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠁⠀⢹⠁⠀⠀⢸⡇
⡇⠀⠀⡇⠀⠀⢹⠀⠀⠀⢹⡀⠀⠀⠙⠆⠀⠀⢻⡀⠀⢂⠀⠀⠀⢹⠀⢸⠈⠉⢻⠀⠀⡖⠒⠆⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠂
⡇⠀⠀⡇⠀⠀⠸⠀⠀⠀⠸⡇⠀⠀⠀⠈⡄⠀⠀⢳⡀⠀⠀⠀⠀⢠⠀⠀⢰⠀⢸⠖⠲⡇⠀⠀⠀⠀⠀⠀⠀⡐⠀⢸⠃⠀⢸⠇⠀⢸
⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠐⡀⠀⠀⠙⡄⠀⠀⠀⠸⠀⠀⣴⠀⢸⡄⠈⡇⠀⠀⠀⠀⠀⠀⠐⠀⢀⡟⠀⠀⠈⠀⠀⢸
⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠱⠀⠀⠀⠘⡄⠀⠀⠀⡄⠀⠻⠀⢸⠀⠀⢰⠀⠀⠀⠀⠀⠠⠁⠀⡾⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠇⠀⠀⠀⠐⠀⠀⠸⠀⠀⠀⠀⠀⠃⠀⠠⠃⠀⠀⠀⠀⠀⠀⠨
⡇⠀⠀⠛⠒⠒⠒⠛⠒⠓⠒⠂⠛⠚⠒⠒⠒⠒⠒⠛⠓⠒⠂⠘⠓⠒⠓⠒⠚⠒⠚⠓⠒⠒⠛⠒⠂⠒⠐⠓⠂⠀⠛⠒⠀⠀⠀⠛⠒⠘⠒⠒⠐⠒⠂
⠇⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠤⠤⠤⠤⠄⠤
⡇⠀⢠⣄⣀⣀
⡇⠀⢸⠀⠀⢸⠄⠀⠀⢶⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤
⡇⠀⢸⠀⠀⢸⡆⠀⠀⢸⡄⠀⠀⢠⡤⠒⢢⡤⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢸⡇⠀⠀⠸⡇⠀⠀⠘⡆⠀⠀⢻⡄⠀⢢⢀⡠⠄⠀⠀⠀⠀⢀⣀⣀⣀⣀⠠⣄⡀⠴⢄⠀⠀⠉⠈⣡⠠⡀⠀⠀⠀⠀⠀⢠⠚⠓⡆⠀⠀⠰
⡇⠀⢸⠀⠀⠀⡇⠀⠀⠀⣧⠀⠀⠀⠀⢆⠀⠀⠳⡀⠀⠣⡀⠈⠂⠀⠀⠀⢸⠀⠀⡷⠀⠀⢸⡄⠀⠈⠄⠀⡀⠀⢻⠀⠃⠀⠜⠉⠹⠀⠈⠀⢸⠃
⡇⠀⠸⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠈⡄⠀⠀⠱⠄⠀⠳⡄⠀⠀⠀⠀⢸⠀⠀⠇⡀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠇⠘⠀⠂⠀⠀⠀⠆⠀⠈
⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⡀⠀⠀⠈⠆⠀⠙⣄⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢰
⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠄⠀⠈⢆⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⡆⠀⠀⡇
⡇⠀⠘⠆⠀⠀⠰⠄⠠⠤⠀⠸⠄⠤⠄⠀⠀⠀⠰⠤⠀⠀⠈⠤⠄⠀⠱⠤⠸⠦⠤⠀⠸⠤⠤⠄⠠⠦⠤⠀⠀⠀⠀⠀⠇⠀⠦⠀⠀⠣⠀⠀⠀⠐
⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⢀⡀⢀⠀⠀⠀⠀⠀⣀
⡇
⡇
⡇⠀⢰⠆⠤⠤⣤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⠤
⡇⠀⢸⠀⠀⠀⣿⠀⠀⢰⠐⠒⣦⠔⠂⠙⡄⠀⠀⠀⠀⠈⠀⠀⣶⠤⠤⣄⠀⡞⠁⠐⡶⠒⠒⠄⢀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⣀⡠⣄⠀⠀⠘⠉⠳
⡇⠀⢸⡄⣀⡀⡝⠀⠀⠘⡆⠀⠸⡆⠀⠀⠰⠀⠀⠀⠀⠇⠀⠀⡟⠀⠀⠀⢠⠀⠀⢰⠃⠀⠀⢿⠀⠈⢂⠀⠀⠀⠀⡏⠈⣿⠀⠀⠹⡅⠀⠈⠀⠀⢀⠀⢈
⡇⠀⢸⡇⠀⠀⡇⠀⠀⠀⢷⠀⠀⢻⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⡇⠀⠹⠀⠀⠀⢷⠀⠀⠀⠀⠈⠀⠈
⡇⠀⢸⠇⠀⠀⠃⠀⠀⠀⠘⠆⠀⠀⢧⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠈⡆⠀⠀⠀⠐
⡇⠀⠸⠃⠒⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⢠
⡇⠀⠠⣄⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣈⣤⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⡀⠀⣠⡀
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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢚⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠧⡄⠀⢀⣀⣀⣀⣀⡀⣀⣀⣸⠀⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠁⠒⠒⠒⠂⠤⠤⠤⠤⠬⣿⢠⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⠼⠧⠬⢉⣉⣁⣒⠒⠒⠒⢴⡇⡄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢺⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣺⠀⠐⠒⠠⠤⠄⣀⣀⠀⣸⢠⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠂⠂⠐⠐⠦⠤⣤⣀⣀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠇⡀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⡁⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠄⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁⠒⠒⠬⢄⣒⠂⠤⢀⡀⠀⠀⠀⣀⠤⠐⣀⠥⠒⠉⣀⡀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢈⢹⠒⠢⠤⣀⡈⠉⠐⠲⠤⣍⣉⣩⡤⠒⠉⠀⠀⠐⠺⢤⡇⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡤⢆⣀⡤⠶⠋⠁⣒⠤⢄⡄⠀⠀⡧⠔⠊⠉⠐⠂⠤⢀⡇⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠆⠀⠀⠁⠂⠶⡯⣁⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⣀⡤⠖⠉⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡟⠀⠂⠤⣀⠀⠀⠉⠓⢿⠀⠀⡇⢀⣠⠴⠊⠁⢀⡠⠆⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⡇⠀⠀⠀⠀⠈⠐⠢⢄⣸⠀⠀⠿⠋⠁⣀⠤⠚⠁⠀⢧⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⢻⣿⣯⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣺⣿⡾⢿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠨⣭⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣛⡋⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣄⡈⠿⣧⣉⣠⣶⣄⣀⣠⣤⠶⣤⠚⣿⢿⠟⡿⢿⡖⣄⠤⣤⡀⣀⣀⣤⣄⠛⣫⡿⠋⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⠳⣄⠙⣿⣿⠛⢹⣿⣟⠄⣤⡙⢇⠹⠂⠒⡐⠯⡰⢋⣤⡠⢿⣿⡟⠿⣿⣿⠏⢀⡞⣫⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣶⠙⡋⡀⠘⠡⢄⡈⢾⠇⠘⣮⠹⠦⠏⣱⠆⠘⡷⠋⣈⡈⢁⠀⠘⠃⣴⣟⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠘⠇⢴⣿⠷⠶⠀⠀⠃⡄⣴⣶⣶⣖⣶⣲⣦⣶⣦⠠⠀⠀⠀⡰⣶⣿⡆⢹⠻⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣾⣷⠞⣶⣦⣤⣶⣄⢨⡀⠀⡉⡝⣥⣾⠋⣿⣢⢩⠈⡀⢀⢀⣠⣶⣠⣽⠗⢶⡞⣦⡘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢑⠞⣻⡿⣿⣯⣉⡛⠻⢆⣤⠀⢜⢷⠷⢾⠣⠀⣠⡴⠟⢋⣩⣤⣶⢶⣿⢢⠘⣎⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣞⢠⡟⠀⣾⣹⣿⣿⣷⣆⠈⡓⢠⠈⠿⠁⡠⣞⣁⣠⣿⣿⣽⣿⣿⠆⢹⡄⡇⠐⡎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢧⠘⣿⡀⠘⠻⠿⠗⢙⣿⡿⠁⠀⠛⠀⠚⠀⠙⢿⣧⡋⠙⠿⠿⠋⠀⣼⠃⡆⢀⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢇⠈⢤⡘⠻⢷⣦⣤⣶⡟⠁⠸⢃⢄⡿⡂⢻⡠⡟⠀⠈⣻⣶⣤⣤⣴⠞⢃⡼⠀⡸⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣆⠀⠙⢯⣶⣌⡄⢳⠢⠴⠄⠛⣁⣞⢀⡎⠀⢸⠿⡇⠁⢦⡍⣲⢘⠛⠤⢤⠠⠚⣌⡠⣶⡽⠏⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣦⡀⠙⠛⠿⣯⣴⣶⢿⠿⠛⠉⠤⣴⣮⣿⠀⣿⣷⣦⠤⠙⠻⠿⣦⣼⣄⣏⡭⠿⠋⣥⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⣿⣿⡟⠀⠀⣀⠀⠀⠀⠀⠀⢂⢡⡶⣮⠻⢿⠀⡿⢟⡡⣦⢀⠂⠀⠀⠀⠁⠀⠀⠀⠘⣿⣿⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣳⣈⣦⠰⣤⣴⠆⢠⣏⣠⡄⢸⠃⢾⣿⡷⢸⡆⣀⣌⣧⠀⢤⣤⡆⢠⣎⣸⣻⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡎⠙⢛⢻⣷⣤⡈⠀⡰⠋⣙⡟⣿⠿⢿⣾⣏⢴⣷⢜⠟⣻⡉⠳⡀⢁⣠⣾⣿⡛⡛⠉⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⢌⣉⢛⠋⢱⢘⣓⠀⣠⣴⠃⠀⢈⠹⣽⠋⡁⠀⠩⣖⠃⠀⠟⢸⠉⠛⠛⣉⢴⣾⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠛⢫⠾⣵⣿⣇⡠⡁⠄⡁⢼⡇⠐⡳⢌⡒⠁⣌⡵⢟⠀⣹⡏⠐⡄⠚⣆⢘⠿⡝⢷⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠄⠀⣤⠀⣭⠸⢣⣭⣭⡴⠀⢙⡀⠙⣾⣀⠾⡉⠀⠉⣀⠉⠑⣧⠀⢰⠛⠀⣗⡠⣤⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡏⠀⢰⡿⣥⠈⠻⠞⠘⠁⠶⡘⣥⠄⣸⢣⣾⠛⡀⠨⣽⢀⠆⠘⠃⠼⠛⢠⣿⢿⡀⠀⢿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣰⣷⠈⠈⢯⡼⣖⡲⡿⢀⣀⣁⢉⣉⢁⡱⠄⢙⣷⡀⠁⣀⣠⡒⠷⢒⣴⢮⡿⠈⠁⣷⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⣿⣇⣥⣸⣮⢱⡛⢻⢃⡻⣟⡁⠰⣽⡀⠉⢀⣮⢠⡧⠀⣐⡹⣥⣨⢏⢻⡉⣿⣷⡀⣿⡧⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⣺⠛⢄⣨⣱⢯⣩⣦⣌⡁⣠⢿⣤⣈⣤⣮⣙⢝⣫⣃⠌⠛⢃⣼⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣿⡿⣿⣆⢛⣛⢷⣽⣿⢦⣠⣶⠽⡿⢷⣎⡩⢾⣿⣤⢻⡛⠊⣼⡿⣿⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡇⣿⣿⡟⣿⢹⣏⣿⢳⣾⢿⣖⣿⣾⣿⣶⢿⣯⣼⣋⣿⣻⠻⡇⣿⣧⡇⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣧⣼⣿⡏⢹⡧⣿⢿⣧⣼⣏⣿⣹⡧⢿⡿⣯⣼⠉⣿⢿⣴⣷⣿⣿⠇⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡇⢠⠀⣴⡀⣿⢻⣯⣿⣯⣿⣽⡧⣾⣟⡁⣰⡄⣠⢀⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣾⣧⣿⣷⣥⠉⣣⣺⡛⢿⠻⠇⣺⣍⡳⣿⣵⣿⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣀⠀⢀⣿⣿⣿⣿⣟⡋⢀⣾⠟⣼⠏⠀⠐⠚⠦⣝⠿⣮⣿⣶⣬⣍⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣏⠙⠀⣸⡟⣵⣿⣿⣛⣚⣩⣵⣾⠁⢀⣤⣤⣦⣄⠀⠀⠈⠙⢛⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣛⣿⣿⡙⣿⣿⣿⣿⣿⣿⣍⡃⠀⠀⠉⠁⠈⠙⠃⠀⠀⣰⣿⣿⣿⣿⣿⣿⢁⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣿⡿⠿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣌⣈⡿
⠀⣤⠖⠒⠒⠶⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢻⣷⣄⠀⠀⠸⣶⣭⣴⠞⠁⠀⣰⣿⣿⣮⣀⣉⣭⣸
⠀⣏⡁⠀⠀⠀⣀⣠⣤⡤⠤⣴⢶⣶⣿⣤⣤⣄⣀⣀⣀⠀⠀⢸⡇⠙⠓⢄⡀⠀⢉⣀⣤⣶⣿⡻⣿⣿⠽⣿⣿⡿⠟
⠀⠘⢧⣤⣠⡴⠋⠀⠀⠀⠀⠙⢦⣿⠿⠛⠿⢿⣿⣿⣭⣭⣴⣿⣿⠀⠀⣻⠻⣿⣿⣷⢻⣿⣿⣿⣮⠀⣦⠅⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⠴⠛⣿⣵⠟⠁⠉⣩⡇⠀⠟⢣⡀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠘⣧⡤⠖⠛⠁⠀⢀⡤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⡋⠀⠀⠹⣆⠀⠀⠁⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠛⠲⠦⣄⣈⠳⣤⣀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠹⣿⣿⣿⣿⣿⡗⠁⠀⠀⠀⠀⠀⣩⠽⠶⢭⡁
⠀⣀⣀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠘⣿⠶⣶⣶⣿⣿⣏⣀⠀⠀⢀⣀⣠⣿⣵⣶⠒⠄⣿
⣾⡓⠀⣿⣿⣿⠛⠛⣻⣷⣶⠶⢶⠓⣿⡄⠀⠀⠀⢀⡏⠛⠛⠿⠿⠿⣿⣿⣋⡉⠙⣷⠇
⠙⢿⢾⣿⣿⠇⢀⣾⣿⣿⣻⡀⢿⠀⣿⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⣸⣃⣀⠀⠉⣳⠏
⢠⣼⢸⣿⡟⠀⣼⣿⣟⠿⣿⠛⠛⠛⠛⠷⠾⠾⠟⠁⠀⠀⠀⠀⠀⡟⠀⠉⠉⣲⠇
⢻⣿⢸⣿⣧⡀⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣽⣻⡆
⠀⢸⣈⣽⣿⣿⣌⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⣿⣽⣟⣷
⠀⣿⡟⠉⢹⡏⠁⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠙⢷⣮⣿⣧⡀
⠀⠘⠳⠶⠛⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⡀⠉⠛⠛⠛⠓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣦⣤⣀⢀⣾⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⡝⠀⣹⣿⠛⣿⣭⣿⣿⣻⣿⡟⠛⢳⡀
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡒⣛⣳⣤⠖⢶⣤⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣹⣏⣷⣾⣿⣩⣟⣻⠟⣿⣦⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢿⠷⣾⣷⣒⣶⣾⣛⣶⣶⣿⣿⣿⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠴⢰⡟⣒⣛⢤⠂⠻⢗⠶⠶⣶⣴⣿⢤⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡧⢘⣿⡟⠉⠒⠱⡶⡿⠃⡠⢺⣷⣿⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡍⢀⡿⠔⢳⢚⡼⠳⠟⠪⣴⡿⣻⣿⣨⢼⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣌⢸⣇⣁⡻⣽⢣⠜⣍⢉⡸⡏⠿⡟⣫⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣲⣿⡋⢸⠏⣿⡟⡟⡀⠛⠙⠏⠴⣿⣭⡼⠇⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠟⠋⠿⠿⢿⣿⢭⣿⣇⡊⢻⣶⡔⠂⡤⣸⠟⢿⣶⣟⣭⣷
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡷⠴⠀⠂⠁⠀⠀⢸⣟⠂⣿⡿⣗⣿⡟⣿⠏⣸⣏⢾⣿⣟⣼⣯⡉
⠀⠀⠀⠀⠀⠀⠈⢿⣠⠝⣿⡇⠀⠀⠀⠀⠀⠀⢸⡧⡅⡿⠶⠶⠼⠾⠶⠶⠿⠷⠬⣋⣿⡟⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡥⢠⣏⣩⣩⣡⣁⣀⣁⣀⣀⣀⣹⣿⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢀⣀⣀⠤⣴⣚⢛⠟⠓⠛⣓⡶⠂⠐⠒⢲⣀⠀⣿⣿⡷⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣽⣭⣶⣛⣱⣶⣞⣂⣶⣢⡀⣈⣉⠀⣀⣀⣀⣈⣀⣀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⣽⣿⣿⣻⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣛⡛⠛⠛⢾⡏⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡏⠀⠀⠀⠀⠀⠀⠀⢛⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣷⡼⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡇⠀⠀⠀⠀⠀⠀⠁⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣟⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇⠀⠀⠀⠀⣀⣴⣾⠇⢨⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⣀⣹⠄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠽⣇⣀⣤⣶⡿⠛⠉⠘⠐⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⣱⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⢽⡟⠛⢛⣿⠿⠍⣤⠀⠒⠠⠤⠀⢀⣀⠀⠀⢀⢸⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠉⠛⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⣭⠁⠈⣩⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠐⠀⠀⠀⠀⠂⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠛⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣦⠀⠀⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠶⠖⠚⠛⠛⠋⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡞⠋⠛⠟⠉⢻⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣧⠀⠀⢧⣀⠀⠀⣠⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠈⠙⠻⠃⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠳⠶⠦⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣤⡴⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣙⣿⡿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣛⡳⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⠶⣚⣫⡽⠖⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠉⠛⠒⠶⢭⣍⣛⡓⠶⠖⣛⣭⠵⠖⣋⣩⣤⡶⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠀⠀⣿⠛⠶⠶⣤⣄⣉⠉⣿⢋⣩⡴⠶⠛⠉⠁⠘⡇⣿⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠀⠀⡿⠀⠀⠀⠀⠈⣿⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⣧⢿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢻⢸⡇⠀⠀⠀⠀⠀⠀⣿⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡇⠀⣼⠃⠀⠀⠀⠀⠀⣿⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⣿⢸⡆⢿
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣾⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠻⣾⣇⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣤⡾⠃⠀⠀⠀⠀⠀⠀⢻⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣼⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⣠⣶⠄⢠⣤⣤⣤⣤⣾⣿⣦⣀
⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣾⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠛⠛⠋⠛⠛⠛⠻⠟⠳⠶⠶⠾⣿⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠛⠉⠺⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠛⠉⠶⠶⠾⠿⠛⠻⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠙⢷⣤⡀
⠀⠀⠀⠀⠀⢻⣷⣄⣠⣤⣴⡶⠾⠟⠉⠀⠀⠀⣠⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢦⣤⠀⠀⠀⠉⠙⠿⣶⣤⣤⣄⣀⣠⣾⠏
⠀⣀⣤⣶⡶⠞⠋⠈⢹⡇⠀⠀⠀⣀⣀⣤⡴⠞⠋⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠉⠹⠷⢦⣤⣀⠀⠀⠀⠈⣿⠉⠁⠛⢶⣦⣤⣤⡀
⢶⣏⣁⢀⣠⣠⡄⠂⠘⠿⠶⠾⠚⠋⠉⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠛⠚⠛⠀⠀⠀⠀⠀⠀⢙⣻
⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣿⠟⢿⡆⠀⠀⠀⠀⢸⡟⠉⠛⠛⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⣤⣼⡇⠀⠀⠀⠀⢸⣷⢶⣶⣶⣆⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⣿⣿⠀⠀⠀⠀⠀⣿⠋⢻⡇⠀⠀⠀⠀⢸⣿⣿⠛⢻⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠆⠀⠀⠀⠀⣿⠀⢸⡷⠀⠀⠀⠀⢸⣿⣿⡄⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣧⠐⣿⣿⡆⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠠⣿⣿⡏⠀⠀⠀⠀⣿⠀⠀⣶⠀⠀⠀⠀⢸⣿⣿⠃⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⡇⠀⠀⠀⢈⣿⠀⠀⢻⠀⠀⠀⠀⣸⣿⣿⠀⠰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⢻⣿⡇⠀⠀⠀⢸⡟⠀⠀⣿⠀⠀⠀⠀⣿⠿⣿⠀⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢸⣿⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⣸⣿⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠸⣿⣟⠀⠀⠀⣸⡏⠀⠀⢹⡃⠀⠀⠀⣿⠁⣿⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⣽⠀⠀⠀⢿⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⠻⠷⠷⠶⢿⣇⠀⠀⠘⢿⠶⠶⠾⣿⠀⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⠀⠀⠀⠀⢈⣿⠀⢀⠀⠀⠀⠀⠀⣹⡇⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠛⠚⠛⠛⠛⠛⠙⠃⠈⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠁⠐⠈⠉⠀⠀⠀⠀⡏⠉⡅⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠘⢗⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠁⠀⡁⠀⠁⠠⠀⠀⠀⠀⠀⣀⣄⠀⠀⠰⡿⠿⠁⠀⠀⢳⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⡇⠀⠀⠂⠀⡀⠀⡔⠀⠒⠶⠶⠶⢶⣿⢾⠆⠀⠀⠀⠀⣤⠤⣀⠀⢀⠇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡗⠀⠀⠂⠀⡃⠀⠃⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠩⠀⠀⢀⠞⠀⢀
⠀⠀⠀⠀⡐⠀⠐⠀⡇⠀⡇⠂⠀⠂⠀⡃⠀⠁⠀⡇⠀⠀⠀⠉⠀⠈⠀⠀⠀⠘⠒⣶⠒⠊⠁
⠀⠀⠀⠀⡓⠒⠃⠀⠓⠒⠃⠀⠀⠂⠀⠃⠐⠒⠒⠓⠒⠒⠒⠓⠒⠒⠂⠀⠀⠒⠚⠛⠓⠒⠀⠀⠀⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⣴⠲⡆⠀⠂⠀⠀⠀⠘⣿⠁⢐⢠⠄
⠀⠀⠀⠀⣶⠚⢆⠀⠀⣆⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⠸⠘
⠀⠀⠀⠀⠆⢣⠈⢧⠀⠈⢂⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠳⡈⢳⡄⠈⠠⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⠀⡇⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠈⠀⢀⠀⠀⠱⠶⠳⠤⠀⠹⠏⠁⠤⠘⠀⠠⠀⠠⠀⠠⠀⠼⠤⠇⠀⠍⠻⠄⠀⠤⠦⠤⠀⠄
⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢸⡅⢨⣍⢉⣁⢀⣄⡈
⠀⠀⠀⠀⢸⠀⠐⠀⠀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⢄⠀⡄⠀⢀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⠆⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⢀⡠⠒⣉⡽⢋⠥⠒⠁⢀⠴⢂⡁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⡀⢸⣟⣟⣀⣀⣀⣟⣻⡀⣂⡀⣎⡡⢰⣿⠀⢉⢁⣀⠐⣾⡇⠀⣉⣀⣀⣀⣀⣀⣀⣀⣇⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣯⣮⣽⣭⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡛⢿⣦⠀⠀⠀⠀⠠⠽⠟⠁⠀⠀⢰⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⡤⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠘⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠻⣿⣿⣿⣿⣿⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⠉⠛⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⣶⣦⣤⣤⣤⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠗⠀⢠⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⠋⣠⠀⣼⡿⠋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⠀⠀⣰⣿⠀⠟⢠⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠁⠀⠀⠀⠀⠀⠿⠁⠀⠈⠛⠿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⠶⠶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠿⠷⠶⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣤⣄⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⣿⠿⠿⣿⠋⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡴⢿⠀⠀⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠀⠀⠀⠀⣿⡴⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⢰⠀⠠⠰⠀⠀⠀⠀⡒⢠⠄
⠠⠓⠇⠰⠀⠐⢴⣵⣤⡤⡤⠒⠛⠂⢠⡀⠀⠸⠀⠀⠀⠀⠀⠁⠈⠀⣠⠄⠰⠆⠀⠀⠀⠀⠀⠀⣅⠀⠔
⠀⠀⠀⡁⠁⠀⠅⠀⠁⢠⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢣⡀⡀⠿⢷⠃⠐⠷⣖⣢⣼⣏⠀⠀⠀⠈
⢀⢀⡀⡁⠀⠁⠀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠖⠀⢂⡁⠀⠐⠻⠤⠀⠀⠀⢀
⢀⠀⢁⣎⡄⠀⡐⡏⠀⣵⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⡀⠀⠀⡴⠰⢠⣴
⠘⠀⠀⠀⠀⡤⡁⠀⠀⢿⠀⠀⠀⠀⠀⠠⣄⠀⠐⠀⠐⠀⠀⠀⠀⠘⠛⢷⣦⣤⡀⠀⠀⢀⡼⠁⠀⠚⠑
⠀⠀⠀⠀⠀⢰⡅⠀⠀⢸⡇⠀⠀⠀⠀⠀⢺⠗⠀⠀⠀⠐⠒⠂⠀⠃⠀⠀⠈⠙⠃⠀⠀⠚
⠀⠄⠈⠀⠉⠈⠀⠀⠀⠉⠡⠀⠈⠈⠘⠄⠀⠀⠈⠀⠀⠀⠀⡤⡅⡤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⡀⢀
⠐⠤⠀⣀⣀⡀⢠⣤⡦⣀⠀⣤⠀⣤⠆⠀⠜⣦⢒⠀⠘⠃⢠⢌⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠚
⠀⠀⢰⣾⣻⣷⣾⣿⡇⠀⠀⢹⡄⡿⠀⠀⠀⠃⠀⠀⠀⠀⣜⠏⠁⠈⠀⠠⠀⠀⠀⠀⢀⠹⠉⠁
⠀⠀⢰⠋⠙⠻⠜⠛⠃⠀⠀⠸⣿⡇⢀⡀⠀⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠤⠄⠀⠀⠀⠠⠒
⠠⠀⠀⠀⠀⠀⠓⣦⠀⡄⢸⠀⠀⠀⠀⠀⠀⠘⠛⠀⢸⠀⠀⣰⣁⠀⠀⠁⢤⣤⣤⣴⠇⠀⠀⠀⢠⠂
⠀⠀⣠⠀⠀⢨⠰⢉⢂⠈⠀⠉⢮⠀⠄⠀⠀⠀⢀⠀⠀⠀⠘⣦⠡⠄⠀⠈⠉⠈⠙⡋⠀⡀⡀⠀⠉
⠀⠀⠀⠀⢅⣃⡆⠘⠈⠀⠀⠀⠀⢲⣶⣴⣷⣶⢶⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠈⠀⠘⠀⠀⢰
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠⠀⠀⠠⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⣀⢀⠀⡀
⠐⢆⠂⠠⠤⢀⢀⣀⡆⠠⡀⠐⠀⣀⣀⣂⢐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡠⠄⠀⠀⢀⠀⠡⠄⣦⣴
⠀⠀⠉⠂⠲⠀⠎⠈⡂⢀⠰⠂⢠⣿⠏⠀⠠⠀⢀⠀⠀⠀⠀⠐⠀⠂⠂⠆⠱⠆⠀⠀⢀⠦⠆⠘⠉⡿⣿
⠀⠀⠀⠀⠀⠅⠐⢂⡘⠨⠐⢣⣿⡟⠀⠀⢐⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠻⠂⠐⠂⠘⠙⠛⠀⠀⠀⠈⠀⠰
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⡇⠀⠀⠀⠀⣀⣀⣹⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢹⣿⠛⠛⠉⠉⣯⠉⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠸⣿⡀⠀⠀⠀⢸⠀⢳⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⡿⡇⠀⠀⠀⠸⡇⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⢛⠀⠀⠀⠀⡇⢸⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠸⡼⡀⠀⠀⠀⣿⣀⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡷⠷⠶⠛⠛⠉⠉⠇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢶⣀⣀⣠⣤⣤⠖⢻⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⣹⡉⠀⠀⠘⡇⢸⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⣏⣇⠀⠀⠀⡇⠈⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⢸⣀⣤⣤⣿⣠⣦⣤⣷⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠋⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠶⠤⠤⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⢶⠖⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣤⣤⣤⣤⣤⣤⣤⣄⣀⣈⡇⠀⢸⢉⣀⣠⣤⣤⡾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄⣧⠀⠀⠀⢸⡇⠀⢹⢻⡇⠀⢸⡏⠉⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡇⣿⠀⠀⠀⠀⡇⠀⢸⢸⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣿⠀⠀⠀⠀⡇⠀⢸⢸⡇⠀⢸⡇⠀⠀⠀⠀⣇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣿⠀⠀⠀⠀⡇⠀⢸⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡇⣿⠀⠀⠀⠀⡇⠀⢸⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃⡟⠀⠀⠀⠀⡇⠀⢸⣸⠃⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢰⣀⡇⠀⠀⠀⠘⠷⠶⠞⣿⠀⠀⢸⡇⠀⠀⠀⠀⣿⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠾⠁⠀⠀⠀⠀⠈⠉⠉

⣿⣟⢹⣟⠻⣿⢟⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⣤⣤⣄⣀⣠⣤⡴⠶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢸⡀⠀⠀⠀⠁⠀⠀⣷⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⣷⡶⠞⠛⠛⠛⡏⠹⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢻⣇⠀⠀⠀⠀⢷⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢸⡿⠀⠀⠀⠀⢸⠀⢰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡇⡇⠀⠀⠀⠈⡇⢸⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢷⢣⠀⠀⠀⠀⡇⠈⡆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣼⣤⣤⡶⠾⠟⠚⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⡇⠀⠀⠀⢀⣀⣤⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡷⣶⠛⠛⠉⢹⡄⢹⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢹⣸⡀⠀⠀⠸⡇⠸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡏⣇⠀⠀⢀⡇⢰⠄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣷⡾⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣫⠉⠉⠛⠛⠛⠒⠒⠶⠶⠶⡖⠒⢶⡤⠴⠖⠛⠋⠉⡩⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢺⡟⠛⠛⠻⢶⡶⠶⢶⣴⡇⠀⢨⣧⡤⠶⠶⠛⢻⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⠘⡇⠀⠀⠀⢸⡇⠀⢸⠅⡇⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⠀⠘⡇⠀⢸⠀⡇⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠀⡇⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⡇⠀⠀⠀⠀⢸⡆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠀⡇⠀⠀⠀⢰⡇⠀⢸⢀⡇⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠘⡇⠀⠀⠀⢸⡇⠀⢸⣸⠇⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⣰⡇⠀⠀⠀⠘⠓⠶⠛⣽⠀⠀⠀⡟⠀⠀⠀⠀⢸⣇⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠞⠃⠀⠀⠀⠀⠀⠁⠈⠁

⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⠶⠶⠾⠿⠿⠿⠿⠿⠶⢶⣦⣤⣤⣤⣀⣀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠋⠉⠛⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣠⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠁⠙⢿⣤⣤⣤⣤⣤⠶⢶⣶⠶⢶⣶⠶⠶⣶⡶⠶⣶⣶⣦⣤⣴⣾⠁⢸⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠀⠀⠀⣿⡇⠀⣿⡇⠀⢸⡟⠀⢸⣿⠀⢰⢿⠀⢸⡿⠀⣿⠇⣸⠏⢀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡇⠀⢻⡆⠀⣼⡇⠀⢸⣿⠀⣼⡇⠀⡌⡷⢰⡮⠀⣹⡆⢸⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⡄⠀⠀⣸⠃⠀⣿⡅⠀⣿⡃⠀⢸⡏⠀⢻⡇⠀⣿⠃⢸⡧⢸⡿⠃⣘⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣇⠀⠀⣿⠀⠀⣽⠀⠀⣿⠁⠀⣿⠇⠀⣿⠃⠀⢿⠀⣿⡇⢸⡷⠀⡿⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡟⠀⠀⣿⠀⠀⣿⠀⠠⣿⠀⠀⣿⠀⢸⣿⠀⢠⠘⢀⣿⠁⣿⠃⢰⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣷⠀⠀⣿⠀⠀⣿⠀⢸⣿⠀⢠⣽⠀⢸⣿⠀⢿⠀⢠⣿⠀⣿⠀⢼⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢈⣿⠀⠀⢿⡄⢸⣿⠀⢸⠉⠀⣸⡷⢠⠸⡇⢸⡆⠀⣿⠀⡌⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣹⠀⠈⡇⠀⠐⣿⠀⢸⣿⠀⢸⡆⠀⣿⡇⢸⡆⣿⢸⡇⢸⡿⢰⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⢸⡇⠀⢠⣼⠂⠈⣿⠀⠈⡇⠀⣿⡇⠸⡇⢻⡄⠁⣻⠀⢨⠁⣾⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣤⣤⣶⣶⣶⣶⣶⣶⣶⣾⣇⣸⡇⢸⡇⠀⢸⣿⠀⢠⣷⢈⠀⡇⠀⣿⠀⡆⣿⠈⣧⠀⠙⠀⣾⠀⣿
⠀⠀⠀⠀⣾⡿⠿⠿⠛⠛⠛⠛⠛⠉⠉⣉⠉⢉⣉⣉⣉⣀⣀⡀⢠⣤⣼⣿⡀⢸⡇⠀⢸⡎⠀⠘⡿⠈⠀⠇⠀⢹⠀⡇⢹⡆⠋⢰⣀⣤⡶⠼⣿⡀
⠀⠀⠀⠀⠹⠷⠶⠶⣿⣿⣿⠏⢻⣿⠁⠈⣿⡟⠉⠈⣿⡏⠀⠀⠀⢻⡿⢿⡇⠈⠀⠀⢸⡇⠀⣾⠁⠀⢠⠀⢸⣾⠿⠛⠻⠟⠿⠿⠿⠧⠤⠚⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⣿⡇⠀⢿⣇⠀⠀⢌⣿⠀⠀⠀⣸⡇⢸⡇⡀⢰⠀⢸⡇⠀⣸⠀⡀⣿⠀⠾⢻⣶⣶⠶⣶⢶⣾⡟⢷⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣧⠀⣼⣇⠀⢰⣿⠀⠀⢸⣿⠀⠀⠀⢻⡇⢸⡇⡏⢸⠀⣼⡇⠀⢿⠀⡇⢘⢀⡄⡇⠋⢸⣿⡏⠰⣿⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⡄⢨⣿⠀⠘⢿⡄⠀⢸⣿⡄⠀⠀⢸⡇⢸⡇⠇⢸⠀⢹⡇⠀⣸⠀⡇⢸⢸⠀⣿⠀⣸⠀⡇⡀⢾⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠈⢻⡄⣿⡄⠀⣸⡇⠀⢸⢼⡇⠀⠀⢸⡇⢸⡇⡀⢸⠀⡜⠃⢸⢸⡀⠇⠀⠈⠀⣿⠀⣿⠀⣿⣷⡈⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠘⣷⢹⡇⠀⣿⣇⠀⠀⢸⡇⠀⠀⢸⡇⢸⠃⣇⣸⠀⡄⠁⣸⠀⡇⢐⠀⡇⢰⠈⢠⡇⠀⣿⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢿⡘⣿⠀⠈⣿⠀⠀⣿⣇⠀⠀⢸⡇⢸⠀⣿⡏⠀⡇⠀⠈⠀⣿⢸⠀⡇⢸⡆⢸⡇⠀⣿⡿⡄⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢸⡇⣿⡄⠀⣿⡀⠀⣿⢿⠀⠀⢸⣧⣽⠀⣿⣿⠠⣟⠀⢈⡇⢹⡈⠄⢠⠘⡆⣼⡇⠀⣿⣷⡇⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠸⡇⢹⣇⠀⢻⡇⠀⣿⣾⠀⢀⣼⠟⢻⣤⠿⡇⢠⣿⠂⣿⡁⢸⣿⢾⣿⠀⣇⣿⡇⢠⣿⡋⠃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⣼⡏⠳⣾⣿⠶⢿⣿⡶⢻⣿⢷⣿⠿⠷⠾⠿⠶⠿⠿⢿⣷⣿⡿⣿⣤⣾⣽⣿⣉⣼⡇⣸⣿⡷⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠉⢁⠽⠿⠿⠟⠂⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⠇⢿⣬⣿⢸⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠐⠶⣤⣤⣀⣀⣀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣦⣾⡟⠓⠞⠋⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣶⣶⣶⣮⣭⣭⣭⣉⣉⣉⣉⡉⠛⠲⠶⠦⣤⣤⣤⣄⠀⣀⣀⣀⣤⣤⣤⣤⡤⠞⠃⠀⢉⣀⣀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣿⠃⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠷⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠿⠛⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢏⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠛⠛⠟⠿⠿⠿⠿⣿⡋⢙⣿⣶⣶⣤⣤⣤⣿⡏⢻
⠀⠀⠀⠀⠀⠀⠀⣰⡏⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣨⡿⠿⣯⡄⠉⠉⠉⠉⠙⢷⡸⣇⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⢀⣴⣿⣿⣧⢀⡟⠀⠀⠀⠀⠀⠀⢸⡇⢿⡀⠀⢀⡾⢻⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠁⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⠀⢀⣴⣿⣿⣿⡿⢿⠀⣧⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⢀⡟⢰⠇⣿
⠀⠀⠀⠀⠀⠀⣿⣀⣼⠏⠀⠀⠀⣀⣀⣠⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⡿⠋⠀⢘⠀⢿⡀⠀⠀⠀⠀⠀⠀⢿⣶⠿⣾⡇⡦⢸⡇
⠀⠀⠀⠀⢀⣼⠿⢿⣿⡷⣶⣾⣿⣯⣍⣁⣀⣤⡾⠻⠿⠿⠿⢋⡉⠛⠛⠛⠛⠻⠛⠛⠿⣏⣤⣀⣀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⢼⣅⣀⣾⣷⢀⣿
⠀⠀⠀⠀⢸⣧⣀⣠⡿⠛⠋⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⣤⣼⣿⣿⣿⣶⣿⣷⠿⠦⠤⣤⣤⣭⣽⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⣉⣸⡇⣼⠇
⠀⠀⠀⣴⣿⣉⣙⣿⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⣾⠿⠿⠛⠋⢉⣉⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠿⣷⠀⠀⠀⠀⠀⠀⠀⣹⣿⠟⣰⡏
⠀⢀⣼⡏⣼⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⢉⣀⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣿⠀⠀⠀⠀⠀⢠⡼⣿⠏⣰⡟
⠀⠘⠛⠿⠿⠿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡾⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⠴⣿⡀⠀⢀⣠⡴⣋⠜⠁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣿⣟⣿⣻⠧⠞⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣼⡏⠀⠀⢀⣀⣴⠾⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣛⣻⣿⣿⣿⣿⣿⣏⣡⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣃⣼⣯⣹⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣉⡉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⡇⠀⢸⠄⠀⢲⡆⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⣿⣿⡇⠀⢸⡇⠀⠈⡇⠀⠈⡖⠈⠱⡆⠁⠄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠉⠉⠃⠀⠀⢸⣿
⣿⣿⡇⠀⠘⠃⠀⠀⡇⠀⠀⠉⠄⠀⠹⡄⠀⠀⠀⠀⠀⠀⡆⠀⠅⠀⡐⠘⠀⠀⠀⠀⠀⡠⢄⡠⠀⢤⠀⢸⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⠐⡄⠀⠀⠀⠀⠀⢹⠀⢿⠀⣷⠀⠀⠀⠀⠀⠠⠁⢰⠃⠀⠚⠀⢸⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣶⣿⣿⣶⣶⣿⣷⣾⣿⣶⣶⣾⣿⣿⣿⣷⣿⣿⣷⣾⣿
⣿⣿⠛⠛⠙⠛⠋⠙⠛⠛⠛⠛⠉⠙⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⣿⣿⡋⠉⣇⠀⠠⣤⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿
⣿⣿⡄⠀⣿⠀⠀⢸⠀⠀⢁⡠⠐⣤⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠃⠀⢻⠀⠀⠸⡄⠀⠘⠂⠀⠘⣆⠀⢀⡄⠢⡀⠀⠀⠀⡤⢤⡄⠀⢤⠐⠒⠀⠀⠁⢨⡄⡄⠀⣀⣀⠀⢰⠉⢻⠀⠀⢸⣿
⣿⣿⡇⠀⢀⠀⠀⠀⠁⠀⠀⠀⢂⠀⠈⠆⠀⢣⡀⠀⠀⠀⠀⠀⠘⠇⠀⠘⡆⠀⠀⠀⠀⠀⠇⠠⠀⡃⠀⠀⡀⠀⠈⠀⠀⢸⣿
⣿⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠳⡄⠀⠀⠀⠀⠀⢀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣧⣤⣴⣶⣶⣶⣤⣶⣶⣶⣤⣤⣴⣶⣦⣤⣶⣦⣴⣶⣶⣶⣦⣤⣷⣶⣦⣴⣶⣴⣦⣤⣤⣶⣴⣦⣤⣴⣦⣤⣴⣶⣦⣼⣿
⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠄⠠⢤⡀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠸⡇⠀⢳⠂⢲⡔⠊⠑⠀⠀⠀⠀⠀⠀⡴⠠⠤⠀⠘⠁⢰⠒⠒⣀⠤⡀⠀⠀⠀⢀⠀⣀⠀⢀⡀⠠⠄⠀⠘⠉⢸⣿
⣿⣿⡆⠒⠒⠃⠀⠸⡄⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠠⠀⠀⠀⠈⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢇⠀⠈⠀⠰⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠇⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠈⡀⠀⠘⡀⠀⠀⠰⠀⢸⣿
⣿⣿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠃⠀⠀⠠⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⢸⣿
⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠈⠙⡄⡄⠀⠀⢠⠊⠁⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⡀⠀⠀⠈⡆⠀⠀⣃⠀⠀⠀⠀⡈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣘⠈⠰⠠⣤⠈⢡⡀⠀⠀⠀⠐⢄⡀⠀⢳⡞⠋⠙⢳⡤⠂⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠸⡇⠀⠀⠀⠘⡀⠀⠀⠀⠀⠠⠀⠀⠈⠉⠛⢻⡶⢤⣤⣿⣀⣀⡤⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠘⠀⠀⠙⢯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣠⣠⣤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⣭⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠟⠀⡀⢀⣠⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡠⠙⠁⢀⡔⣼⣷⠿⠶⠂
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⣦⠀⠤⠒⠲⠀⡐⠂⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠞⠁⠄⢀⣾⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣠⠎⢀⠴⠄⠀⠀⠀⠀⠀⠀⢸⢠⢆⣤⠖⠁⠀⠀⠀⣌⣴⣯⠴⠖
⠀⠀⠀⠀⠀⠀⠁⠢⡀⣤⣾⣦⡀⡰⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⠏⠀⠀⠀⡀⢀⠜⠁
⠀⠀⠀⠐⠁⠀⠠⠄⢼⣿⣀⣀⣿⡧⠤⠤⠤⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠦⠶⠶⠖⠊
⠀⠀⠀⠈⠒⠤⠐⠊⠀⠈⢉⠟⢅⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡀⠀⠀⢱⠀⠀⢠⠀⠀⠀⠀⠀⢰⢃⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠚⢶⣆⣀⣧⠀⢸⠀⠀⠀⠀⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣾⠀⠈⠀⢠⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇

⠀⠀⠀⠈⠻⢿⠻⠾⢻⠏⠁⠀⠀⠀⠀⠀⢸⡿⣶⠒⡚⠿⣗⠛⡹⡄⢦⠸⣆⠀⠀⠀⡄⠠⠀⢀⡴⠖⣛⢉⡉⠀⠀⣠⠀⠛⠀⢱⠀⡇⠀⡄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣿⣶⡯⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⣇⣦⢻⢰⠁⢻⢸⣆⠹⣆⠀⠀⣦⣤⠀⠀⠓⠦⣉⠛⣡⣔⡆⢉⠄⠱⡀⠘⠟⠀⠀⠉⢀⠀⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⣿⠛⡇⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⢠⢻⢿⡌⢸⡇⡇⢠⠻⣆⢹⡄⠀⠈⠋⠀⠀⠀⠀⠙⠶⢤⣬⣀⠈⢳⣦⡾⠆⠤⠀⠚⠀⡈⠣⠤⠦⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⣧⢸⠄⢸⡄⢸⡇⢻⡆⢻⡜⣧⠹⣆⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣤⣤⣴⡖⠒⠶⠦⠥⠔⠒⠒⠒⠖⠋
⠀⠀⠀⠀⠀⢸⡇⠀⣷⠀⠀⠀⠀⠀⢀⣿⣇⣾⢰⠘⣷⢸⠇⢨⢇⣸⡇⢈⣻⣮⡿⠃⠀⣠⡤⢤⡤⢤⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠈⡇⠀⠸⡆⠀⠀⠀⠀⠀⠙⣿⣯⣼⣤⣿⣾⣾⣷⣾⣿⠿⠛⠉⠉⠀⢠⠊⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠴⠛⠉⠉
⠀⠀⠀⠀⠀⠀⣷⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⢻⠻⡿⣇⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠁⠸⡄⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠂⠀⠀⢄⢰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣿⠀⠀⠀⠀⠈⣧⣀⠈⣹⡏⠀⠀⢀⡼⠃⠀⠀⠀⠀⣿⣿⡇⠰⣷⣦⣴⣤⡴⠿⠶⠿⠶⠃
⠀⠀⢀⢄⠀⠀⠀⠙⢆⡀⣹⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⢸⣶⣸⣿⣇⠀⠀⠀⠀⠈⠉⣹⡟⠒⠦⠤⠞⠁⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠐⠀⡬
⠀⠀⠃⠈⠀⠀⠀⠀⠀⠉⠉⠀⢀⡾⢿⡿⠳⡄⠀⠀⠀⠀⢸⡿⠋⠉⢻⡄⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠢⡀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡌⠳⣄⣤⠞⢽⡄⠀⠀⠀⢸⣄⣾⣧⣶⣿⡄⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠥⡠⠒⠚⠁⠀⠀⠀⠀⢀⡗⣀⠀⠂⠈⠀⣳⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⠀⣿⡿⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠓⠅⠀⠀⠀⠀⠀⣼⢥⠵⠋⠓⠒⠒⠉⣇⠀⠀⠸⣿⠿⠋⣨⣧⠘⡇⠀⢿⡇⠀⠀⠀⢀⣠⣾⣿⡟⡁⢠⡿⠁⠀⠀⠀⠀⣦⠀⡀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠒⠲⠄⠀⠀⣰⠁⠀⠀⠀⠈⠁⠀⠀⠈⣦⠀⠀⢿⢠⡴⣿⣿⣶⣿⠀⠈⠁⠀⢀⣴⣿⣿⡿⠋⠄⣰⡟⠁⠀⠀⢀⣠⣴⣿⣿⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠦⢶⣶⣶⡟⠉⠉⠀⠀⠀⠸⣿⣧⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⢋⡜⣠⡾⠋⠀⠀⢀⡰⠿⣿⠟⠹⠟⠛⣽⢷
⠀⠀⠀⠀⠀⣴⢶⡄⠀⠀⠀⠀⠀⠀⢸⠁⠀⡁⠀⠀⠀⠀⠀⠀⢹⣿⣄⣉⣁⡿⠀⠀⠀⠈⢿⣿⣿⣦⠾⠋⠀⠀⠀⠀⠈⠳⠤⢴⢤⠶⡤⣀⠀⠘⢳⡀
⠀⠀⠀⠀⠸⢿⠏⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⡁⢀⡎⠁⠈⠈⠁
⠀⠀⠀⠀⣀⠐⠁⠁⠰⡀⠀⠀⠀⠀⠀⡇⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⢀⡴⣶⡢⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⡏
⠀⠀⠀⢐⠛⠊⠁⠉⡉⣹⠄⠀⠀⠀⠀⢡⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣶⣝⣧⠩⢿⠇⣭⠻⠶⠦⡀⠀⠀⠀⠀⠀⠀⢸⣏⣾⠃⠀⠀⠀⢀⣀
⠀⠀⠀⠘⠓⠲⠾⡞⠋⠁⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠶⠤⣀⡰⢆⣈⣉⣀⣤⠤⡤⠤⠤⠒⠒⠀⠀⠀⠀⠀⢠⢾⠟⠊⠀⠀⣤⢶⣌⣛⣧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣏⣠⣄⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣤⣶⣾⣿⣿⡟⠋⠉⠁⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠶⢶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⢿⣿⣶⣄⣿⣿⣿⣿⣿⣿
⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿
⠀⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠸⣿⣿⣿⣿⣿⣿
⠀⣿⡇⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠋
⠀⠹⢧⣤⣀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠙⠿⣿⣶⣦⣄⣀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⣀⣠⣴⣶⣿⡿⠟⠛⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣦⣌⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⢀⣠⣴⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠻⢿⣿⣦⣌⡉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⣠⣴⣶⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠉⠛⠿⢿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣯⣤⣶⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇

⠀⢳⣄⠀⠈⠙⠻⢷⣶⣤⣤⣤⣀⣀⣀⣤⣤⣶⣾⣿⣿⣷⣦⣤⣤⣀⣀⣀⣤⣤⣤⣶⡾⠿⠛⠁⠀⣠⡞
⠰⡀⠙⣷⣤⡀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠋⠙⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⢀⣤⣾⠋⢀
⠀⠙⣆⠈⠛⢿⣷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣴⣾⡿⠛⠁⣰⠏
⠀⠀⠈⠳⣤⡀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⢀⣠⠞⠁
⠀⠀⠀⠀⠈⠻⢷⣤⣄⣀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣀⣀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣷⣶⣶⣶⣾⣿⣆⠀⠀⣰⣿⣶⣶⣶⣶⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⢙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡋
⠀⠀⠀⠀⠈⠳⣄⡉⠛⠓⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠚⠛⢉⣠⠞⠁
⠀⠀⠀⠀⠀⠑⢌⡛⠻⢷⣄⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⣠⡾⠟⢛⡡⠊
⠀⠀⠀⠀⠀⢶⣶⣭⡓⢦⣙⠳⠦⠤⠉⠓⠢⠀⠀⠀⠀⠀⠀⠖⠚⠉⣤⠴⠞⣋⡴⢚⣩⣶⡶
⠀⠀⠀⠀⠀⣼⣿⠿⣛⡴⢚⣉⡤⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⢤⣉⡓⢦⣛⠿⣿⣧
⠀⠀⠀⠀⠀⢀⠔⢋⣩⣴⡟⠉⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠈⠻⣦⣌⡙⠢⡀
⠀⠀⠀⠀⠀⣡⠞⠋⣉⣉⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣉⣉⠙⠳⣌
⠀⠀⠀⠀⠘⣡⠖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠲⣌⠃
⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣾⡟⠀⠀⢻⣿⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⢷⣦⣄
⠀⠀⠀⣠⠾⠋⠁⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣴⣦⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⠈⠙⠷⣄
⠀⢀⡾⠃⢀⣴⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠿⠿⠿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣦⣀⠈⢷⡀
⢠⠏⢀⣴⠿⠋⠁⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠈⠙⠿⣧⡀⠙
⠀⢠⠟⠁⠀⢀⣠⣴⣶⡿⠿⠿⠟⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠻⠿⠿⢿⣶⣦⣄⡀⠀⠈⠻⡄
⣶⣷⣶⢶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣾
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢤⣴⣦⡤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡸⠀⠈⠉⠉⠓⠚⠿⠿⢷⣖⣂⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡖⢫⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡎⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡟⡶⠦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢣⠇⠀⠀⠀⠈⠉⠓⠒⠦⢤⣀⣀⡞⢠⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡜⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⡞⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠒⠛⠉⠉⠒⠢⠤⣄⣀⡀⠀⠀⠀⢀⠇⢰⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠂⡞⠀⡏⡜
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢶⠞⡇
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢠⡇
⠀⠀⠀⠀⠀⢸⠉⠑⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡔⢡⠇
⠀⠀⠀⠀⠀⢸⠒⠢⡤⣀⡀⠀⠉⠉⠓⠲⠤⢤⣀⡀⠀⠀⠀⠀⢀⠔⠁⢀⠴⡃⠀⠸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⠉⠉⠓⠲⠤⣤⣀⣀⠀⠈⠉⠓⠒⡶⠃⢀⠔⠁⢨⡇⢠⢀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⡇⢸⠈⡏⠑⠒⡦⠤⣥⠜⡇⠀⠀⢸⡇⢸⢸⡃
⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⡇⠀⢰⠀⡇⠀⠀⢸⡇⢸⠘⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⠳⠼⠞⠁⠀⠀⡇⠀⢸⠀⡇⠀⠀⢸⡇⢸⢀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠘⠧⢼⠞⠁
⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠘⣇⠀⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣌⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⣤⣤⣤⣄⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣿⣿⠃⠀⠀⠀⢠⣾⣿⠀⠈⠻⣿⣿⣿⣿⡄⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⣿⣿⡿⠋⣁⠀⠀⢨⣽⣿⣿⠟⠁⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠹⣿⣿⣿⣿⣄⠀⠀⠀⠃⣿⣿⡟⣲⠄
⠀⠀⠀⠀⠀⣿⠟⠠⠷⠈⠀⣴⣿⣿⣿⣷⣶⣶⣶⣶⣤⣿⣿⠿⠋⣶⣄⡀⢀⡿⠟⠛⠛⠛⠓⠀⠀⠀⠈⢻⣿⣿⣤⣧⡀
⠀⠀⠀⠀⠀⣿⣠⠤⠀⣤⡴⣶⡌⠛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠚⢹⠟⢻⣟⠻⠗
⠀⠀⠀⠀⠀⣿⣿⣿⢰⣿⣿⣝⠻⣷⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣆⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⣀⡤⠃⠀⢸⡏⠁
⠀⠀⠀⠀⠀⣿⣿⣿⠘⣿⣿⣿⣿⣶⣭⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣿⣿⠀⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣾⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣽⡏⣻⠿⢠⡀⡶⠀⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⠛⣾⣧⣼⣩⣿⣿⠀⠀⠰⠀⠛⠀⣡⡄⠦⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣿⣿⣿⣯⣿⣍⣿⣻⣿⣿⣿⣿⣿⠟⣤⡀⠀⠀⠀⠀⠰⠄⠾⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⢾⡧⣾⣶⣿⣿⣿⣿⡿⠉⣁⣴⠉⠿⢶⣖⠀⠀⠀⠀⠈⠁⡛⢀⡁⢀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⡟⣿⣚⣿⣿⣿⣿⣿⢟⣋⣴⣾⣿⣿⠀⠀⠀⠉⠳⠦⣀⠀⠀⠀⠀⠨⠁⣤⢀⡀⣀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣹⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⡟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢆⠀⠀⠹⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⢩⣾⠈⢳⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣰⣿⣿⣿⡅
⠀⠀⠀⠀⠀⡀⠳⣄⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣾⣏⣾⡏⣦⠀⢧⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣿⡿⣫⣿⠇
⠀⠀⠀⠀⠀⠰⡀⢻⡄⠀⣿⣆⠘⣿⣿⣿⣿⣿⣿⣷⣿⣟⣾⡿⣰⠹⡄⠈⢣⠀⠁⠀⠀⠀⠀⠀⠀⣾⠟⢠⣿⡿⢰⡿⣹⡇
⠀⠀⠀⠀⠀⡀⠱⣼⣷⣤⠟⢸⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⡁⠻⣀⣿⣄⠀⠁⠀⠀⠀⠀⠀⠀⣼⡟⢸⣇⠛⠃⠚⣱⣿⠇
⠀⠀⠀⠀⠀⢱⠀⠸⡆⠀⣀⣈⢻⡞⠛⣿⣿⣿⣿⣿⣿⡏⣼⠹⣷⠁⣷⠸⡆⠀⠀⠀⠀⠀⢀⣴⣄⢤⡾⢿⣿⡏⣼⣿⢳⡇
⠀⠀⠀⠀⠀⣌⣧⣠⠇⢠⠇⠘⢧⠑⢶⡚⠻⣿⣿⣿⣿⣽⣿⢧⡿⠰⣏⠀⠹⠀⠀⠀⢀⣠⠦⢉⣴⠟⣸⣷⢸⡇⠿⠫⠾⠃
⠀⠀⠀⠀⠀⠀⠈⠧⠀⠻⠀⠀⠈⠳⣄⠉⠳⢮⠙⠿⢿⣿⣿⣾⣧⠈⠋⠀⠀⢀⣠⡴⢫⣴⡾⠟⣩⣾⣿⣿⠎⣿⠦⣾⣿⠇

⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⡿⣧⡈⢣
⡄⠀⠀⠀⠀⠀⠠⠄⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠃⡀⠀⠚⠄⠉⠀⠀⠀⠀⠐⢺⣿⣿⠿⠋⠀⠀⠀⠑⡄⢣
⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⢀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠐⢀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘
⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⢘⣀⡀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠛
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠉⠉⡏⠉⢩⡉⢠⣾⣿⡇⢴⡶⢦⡀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡉⣷⠶⠶⣿⣺⣿⣿⣿⣟⢻⣿⣾⡯⢸⡇⠀⠀⣀⣀⡀
⠛⠁⠀⠀⠀⠀⠠⠤⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡆⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⡿⠛⠓⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠒⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣠⣤⣿⢿⡿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⣿⠀⠀⠈⠈⠀⠀⠀⠀⠀⢴⠆⠛⠉⠉⠀⢸⠈⠃⢉⠠⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠈⢙⠳⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣤⠾⠛⠛⠾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⣄⣀⢀⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠉⠉⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣼⠃⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢹⡟⠉⠛⠉⢻⣿⡟⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⣿⠋⠛⠉⢻⡟⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⣟⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣇⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⣽⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁


⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⢾⣃⣸⣯⣹⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⣤⣤⣤⣤⣄⣀⣀⣀⣀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⣀⣀⡀⠀⠀⠀⠀⠀⢸⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠃⠉⠉⠉⠉⠉⠛⠓⠒⢺⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣀⣀⣀⣀⣀⡀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣿⣶⣶⣤⣤⣉⡉⠉⠉⢙⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣸⡇⠀⠀⠀⠀⠀⠉⠉⠉⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣾⣿⣿⣛⣓⠒⠒⠲⠲⠦⣾⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡿⠉⠉⠉⠉⠙⠛⠓⠒⠶⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⢠⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣿⣟⣛⣉⣙⡋⠉⠛⠙⠛⠛⣒⣒⣚⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣿⣤⣶⣿⣿⡉⠉⢙⠛⠒⠒⢾⠉⣽⢻⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡏⠈⣉⣽⢿⠿⣿⣿⣿⣶⣶⣾⠀⣿⣵⢻⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢘⡿⠟⠛⠯⢿⣾⣾⣯⣤⣄⣉⡿⢀⡏⣛⣾⡌⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣀⣤⣺⡿⠋⠁⠀⠈⠉⠉⡇⢸⠏⢹⢘⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⡞⣛⣩⠽⠋⠀⠀⠀⠀⠀⠀⠀⡇⣸⣠⣪⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⡿⣪⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠈⠉⠛⠛⠿⢦⣄⣀⡀⠈⠉⠀⠀⠀⠉⠛⣻⣿⣿⣿⣿⡏⠀⠀⠃⠀⠉⢛⢧⡀⠀⢦⠀⡄⠀⢀⣽⠆⠀⠀⠈⠢⠀⠀⠈⢂
⠀⠀⠀⠀⠤⣤⣤⣬⣭⣵⣤⣤⣄⣠⠐⠊⠉⠁⠈⢉⠿⡇⠀⠀⠀⠀⠀⠀⠣⠳⡜⣼⣿⡇⣠⣿⣷⣶⣶⣶⡶⠖⠆⠀⠀⠈⢳⡄⢰
⢀⠀⠀⠀⠈⠉⠉⠉⣉⣩⣽⡿⠛⠁⠀⠠⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⠛⠛⠛⠛⢻⣶⣶⣾⣿⡿⠟⠁⠀⠻⣿
⣼⣧⣴⡶⢗⠺⠓⠫⠛⠋⠉⠀⠀⠀⠀⠀⢠⣔⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠠⠐⠚⠉⢁⣴⣟⠃⢀⡀⠄⠀⠀⠹
⢛⣩⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠬⠿⠷⢿⣿⠿⠷⡄⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣠⣯⠀⠀⢀⣤⣶⣿⠿⠖⠋⠁⠀⠀⠀⠀⣠
⠀⠙⠿⠞⠛⠛⠛⠉⠙⠛⠛⠟⠋⠀⢀⡤⢒⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⣿⠿⠛⠁⢀⣠⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠃⠈⠀⠠⢿⢻⡇⠀⠀⠢⣀⠀⠀⠀⠀⢀⡍⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠏⡽⠁
⠘⠀⠀⠑⢄⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⢨⣿⣷⣤⣴⣾⣿⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⠃
⢸⢠⡃⠀⢀⠿⣬⣄⠀⡄⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢡⡾⠋⠀⠀⠀⢀⠎
⢸⣿⣥⡴⠏⠀⠘⣿⣶⣷⠀⠀⠀⠀⠀⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣤⡠⣠⣿⡽⠋⠀⠀⠀⠀⠀⡜
⣼⣿⣿⣡⡴⠖⠉⠀⠻⣿⡇⠀⠀⠀⠠⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⢠⣿⣷⡿⠋⠀⢀⣈⣀⣤⣄⣾⣿⠃
⣿⣿⣿⣥⡦⠐⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣷⣿⣿⣿⣷⣦⣸⡿⠏⠀⠀⢀⣀⣀⣩⣿⣿⣿⣿⣷⣦⣄
⠉⠉⠉⠉⠈⠉⠩⣿⡶⠶⣺⣷⡲⠶⣶⣶⣶⣶⣶⣶⡖⠒⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⣿⠳⣶⣶⡶⠶⠶⠾⠋⠉⠉⠉⠉⠉⢉⣭
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣰⣿⠄⠻⣆⠀⠈⠉⠛⠛⠛⠛⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⢀⣿⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠋
⣀⣤⣔⠒⠀⠀⠀⠠⢂⠟⣿⡀⠀⢹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⠏⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⡿⢟⢿⢁
⠈⠉⠙⢷⣶⣶⠄⠀⠈⠀⠘⠂⢐⣦⣿⡿⢆⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑⠶⣶⣴⣶⠇⠀⠀⣿⣀⣀⡤⠀⠀⠀⠀⠀⠀⢀⣾⡿⠅
⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⣤⣤⡿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢲⣿⣿⣤⣀⢸⡿⠟⠉⠀⠀⠀⠀⠀⠀⣰⡿⠿⣷⣶⣶
⠀⠀⠀⠻⢶⣶⣬⣁⡀⠀⠀⡟⣃⣀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢹⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠈⢡⣄⡀⠀⠀⠛⠻⠿⣿⣥⣀⣀⠀⠀⠀⠙⣷⣦⠀⠀⠀⠘⣛⣭⣇⣤⣀⣀⠀⠀⠀⠀⠀⠈⠁⠒⠬⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣨⣽⣿⡶⠤⣀⣀⣈⣿⣿⡛⠛⠉⠀⡀⠀⠈⠻⣄⣤⣶⣿⣿⣿⣿⣿⣿⣯⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢙⢻⣶
⠺⠿⣿⣉⠉⠁⠀⠀⠀⠀⠀⣯⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⡿⠿⡿⣟⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠛⠉⠅
⠀⠀⠀⠀⠀⠀⢀⣿⣶⣦⣤⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠋⠙⠃⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡐⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣩⣿⣿⡿⠃⠘⢹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣶⣽⣾⣷⣶⣼
⠀⠀⠀⠀⠀⠀⢀⣽⠏⠀⠀⠀⠈⣿⣿⠙⡀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⢻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠿⠿⠿⠿⠿⠿⠿⣿⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠉⣹⠉⠙⠛⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡎⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀⠀⠀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢢⣄⣃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠚⠧⡤⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢸⠃⣼
⠀⠀⠀⢀⣤⠴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣸⠀⣿
⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡏
⢰⡃⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡉⢴⡿⠁
⢈⡟⠲⣤⡀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣋⡥⢖⣩⠞⢯⡄
⢸⡇⢸⠇⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣭⠶⢋⣥⠖⠋⠉⢳⡄⢻⡀
⢸⡇⢸⠀⠀⠀⠀⠉⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣡⠞⢋⣤⠞⠋⠀⠀⠀⠀⠀⣷⠸⡇
⢸⠇⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⢠⠞⣠⢞⣥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣿
⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⡏⡰⣳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹
⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⡇
⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⣇
⣼⠒⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⢘⠄
⢿⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣤⣼⠂

⠀⠀⠀⡆⠠⠤⠤⠄⠤⠤⠤⠤⠤⠤⠠⠤⠤⠄⠄⠠⡄⠠⢖⠄
⠀⠀⠀⠡⠀⢿⡴⡦⠐⠲⠂⠔⠲⠶⠤⠰⠦⠦⠠⢶⡾⠒⢾⠊
⠀⠀⠀⠀⠀⠘⣇⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠐⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠀⢿⣰⡀⢄⣀⣀⢠⣀⣀⠠⣤⣦⠠⡾⡄⠀⠀⠰⠀⠡
⠀⠀⠀⠀⠀⠀⡀⠘⡿⠀⠈⠀⠃⠀⠃⠀⠀⠰⢺⠀⠁⢃⠀⠀⠀⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠀⢰⣃⠀⠂⠀⠀⠀⠈⠀⠀⠄⠀⠀⠘⡀⠀⠀⠀⢀⠀⢂
⠀⠀⠀⠀⠀⠀⠰⢱⡄⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⣿
⠀⠀⠀⠀⠀⠀⡄⢸⣧⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢸⡀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡛⠁⠀⢻⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠇⣷
⠀⠀⠀⠀⠀⠀⠰⡇⠈⡄⠀⣏⢂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⢀⡇⢠⠀⠡
⠀⠀⠀⠀⠀⠀⢹⠁⠀⠘⠀⠸⣎⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⢂⠀⢀
⠀⠀⠀⠀⠀⠆⣾⠀⠀⠀⢀⠀⢈⢧⢂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⣄⠀⠆
⠀⠀⠀⠀⢀⢀⡆⠀⠀⠀⠈⠄⠀⠈⢣⠀⠀⠀⢀⣀⠀⠀⢀⠀⢀⠀⢠⠀⠀⣤⠠⠤⠉⠶⠤⠀⠰⢶⣆⢠
⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠠⣀⠱⠐⠂⠀⣀⣁⣡⣤⠤⣤⢶⡶⠤⡶⠶⠖⠒⠒⢒⣻⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠂⢸⠀⠀⠀⢀⡀⠄⠆⠀⢣⣀⣤⡶⠾⠭⡤⢔⢒⣶⠾⠞⠋⣁⣠⠤⠖⠚⠋⠻⣯⢡
⠀⠀⠀⠀⠁⢸⡿⠛⠉⢁⣠⣴⣾⡄⠈⣿⣿⣧⣿⣳⣿⣿⣯⡴⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⢻⣧⠡
⠀⠀⠀⢸⣶⠿⠶⠖⠚⠛⠛⠛⠉⢃⠀⠸⠉⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠄⠐⠒⠂⠀⠀⠀⠈⠁⣱⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠃⠈⠁⠀⠀⠀⣀⣀⣀⣠⡤⠤⠴⠶⠖⠒⠛⠋⠉⠉⠹⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⣴⠚⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠤⠴⠖⠒⠒⠛⢻⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⢤⠲⠖⡖⠖⠒⡆⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⡇⠀⠀⡇⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⡇⡀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⢸⠀⠀⡇⡇⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⢸⠀⠀⡇⡇⠀⡄⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⡇⡇⠀⡇⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢰⠀⡇⠇⠀⠃⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⢸⠀⣇⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⠀⠋⡀⠀⢸⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢠⠀⡆⢸⡇⠀⢸⠀⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣸⣸⣄⣇⣸⣇⣀⢈⡀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⠉⠭⠙⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⠁⠒⠒⠀⠢⠤⠤⣀⠤⠖⠒⠂⠉⠁⣀⣀⢤⠴⠓⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠱⠒⠢⠤⢤⣀⣀⣨⣤⡔⠒⢻⡭⣭⣤⣄⣰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣸⠛⠒⠶⢼⠀⠀⠀⡟⢧⠀⡏⠀⠀⠀⠀⠈⢧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠈⢳⠀⢸⠁⢸⠀⣧⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⡀⢸⠀⢸⠀⡆⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠸⡇⢸⡄⠘⠧⠇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠼⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣷⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠟⠉⢹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡿⢿⡿⠿⠋⠉⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠁⠀⠀⠛⠃⠀⠀⠀⢹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡆⠀⠀⢰⡆⠀⠀⠀⠀⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢹⣄⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⢀⡀⠀⠀⠀⢀⠀⠀⠤⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡏⠉⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠛⠧⠄⠤⠤⠤⠘⠋⠙⠛⠂⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡵⠶⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⢸
⡶⠛⠢⡄⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠺⡇⠀⢀⡴⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠃⡀⠀⠈⠉⠀⠲⠶⠋⠸⠇⠀⠀⠀⠀⠀⣼⠉⠱⢆⣀⠀⠀⠀⣼⠇⢀⣼⠄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠘⢦⡤⠀⠀⠀⠉⡄⠀⠀⠀⠀⠚⠛⠋⠀⠟⠉⢤⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠈⠉⠀⠶⠶⠀⠀⢀⣀⠀⠀⠉⠉⠰⠖⢱⣦⣀⠀⠀⠀⠀⠀⠀⠈⠳⠾⢦⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⣤⠀⠀⠀⠀⠈⠉⠛⠛⠓⠀⠀⠠⣤⠀⠀⠀⠀⠙⠛⠠⣤⠄⠀⠀⠀⣠⠞⠀⠀⠀⠀⠳⡀⠀⠀⠀⢀⡴⠋⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠶⢀⡀⠀⠀⠀⠀⠉⠁⠊⠁⠀⠀⠀⠀⣀⠼⠛⢦⡀⠶⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⣤⠀⠀⢠⡴⠋⠀⠀⠀⠀⠙⣷⡦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠉⠁⠶⠆⣀⠀⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⠀⠐⠚⠛⠛⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠶⠆⠀⠀⠀⠀⠀⠈⣹⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠐⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢤⣤⣀⣀⣈⣉⣉⣀⣀⣀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠶⠶⠶⠶⠶⠾⠷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠙⠛⠉⠉⠉⠉⠉⠉⢻⠋⠉⠉⠉⠉⠉⠉⠛⠋⠉⠛⠋⠉⠉⠙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠷⣀⣀⣤⣀⠐⠶⠟⠀⠀⠀⢳⡄⣠⣀⠀⠀⠀⢀⣠⠞⠁⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣀⡴⠋⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⣄⡀⡀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⡷⠀⠾
⠀⠀⠀⠀⠀⢠⠌⠉⠀⠀⠀⠀⠀⠀⣠⠶⠒⠁⡠⠆⠉⠀⠀⠀⠀⠀⠀⢠⡄⠐⠆⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠰⣆⣀⣀⠀⣀⡀⠤⠚⠁⠀⠀⠰⣇⠀⠀⠀⠀⠀⢀⣠⠜⠃
⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣦⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⡿⠟⢻⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⡀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣞⣿⣶⣶⢒⣤⠶⣒⡲⢤⣤⡶⢀⢖⢆⡤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠻⡯⣧⣾⣯⣾⡞⢟⡿⡛⣺⢟⣵⣍⣾⣛⣽⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣽⣽⣷⣟⢻⡒⢕⣷⡽⣧⣾⠓⠽⡷⢿⡶⣾⣿⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⣾⠉⢹⡋⠐⡖⠛⣿⠛⢻⡟⠛⢻⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡇⠀⠀⡏⠀⢸⠀⢸⡇⠀⡿⠀⣸⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⡇⠀⣛⠀⢸⡇⠀⡇⠀⡟⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢀⡇⠀⡿⠀⢼⠀⢸⠃⢰⡇⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢰⡇⠀⡗⠀⣽⠀⣼⠀⣸⠇⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠶⠶⢶⡶⢶⢶⠶⡶⠶⠾⣿⠃⠀⢸⡇⠀⡇⠀⡇⠀⡏⠀⡟⠀⢠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢠⣆⡇⡇⠀⠀⢿⣀⣀⢸⠃⢰⡇⢰⡇⢰⡏⢰⡿⣦⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡌⣇⢧⣷⠀⠀⣾⠿⠋⢠⣾⠿⠃⠈⣥⣾⡷⠎⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣏⢏⣻⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡷⢳⣿⣼⣤⣤⣿⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠚⠛⠛⠳⠹⣿⣿⣿⣾⣿⣿⣿⣿⣽⣿⣽⣿⣿⣿⣽⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⣿⠟⠓⠶⠶⠶⢤⣤⣭⣉⣋⣙⣛⣛⠙⠉⠋⠉⠉⠛⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂⠯⠿⡿⠿⣟⣛⣾⣚⣛⣹⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣟⣀⣀⣀⣀⣠⣠⡤⢾⡇⠀⣠⣾⠟⠀⠀⢠⢿⠀⠉⢩⣿⢠⢦
⠀⠀⠀⠀⠀⠀⢠⣼⣇⣬⡽⠟⠛⠚⠻⠶⢼⣽⣾⣙⣋⣀⣀⣀⣸⢸⡄⣠⣜⣿⣏⡿
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣀⠔⣫⡿⠋⠀⠀⠀⠀⠈⢹⣸⣗⡿⠉⣽⣿⠇
⠀⠀⠀⠀⠀⠀⣾⠀⢀⣀⣠⠤⣖⣯⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣾⡋⠀⢠⣏⡟
⠀⠀⠀⠀⢠⢾⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢠⡟⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣶⣫⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⣒⣀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣧⣤⣤⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡩⠭⠤⠀⢷⠀⠀⢰⣋⣉⡉⢉⣳
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠉⠰⡏⠠⢰⡄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⡤⠤⠐⣂⣸⡄⠀⢸⠟⠛⠓⠚⢻⣆⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⢳⢠⣷⣴⠶⠶⣼⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠈⣧⠀⡟⢩⠅⠰⠀⣇⠀⢸⡤⠤⠤⠤⢬⡇⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⢸⢹⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣶⡾⠿⠓⠛⡄⢹⢸⠀⠀⡇⢸⠀⢸⠀⠀⠀⠀⠀⡿⠯⠭⠭⣿
⠀⠀⠀⠀⠀⠀⢰⠙⠒⠠⠞⡼⢸⣀⣠⢤⣼⠀⠀⢀⡞⠒⠢⢤⣿⢰⡖⠋⢳⢡⢸⠈⡄⠀⢰⢸⡀⢸⠃⠉⠉⢉⠹⣷⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⢰⢀⢠⠀⡇⠸⠒⠒⠒⢺⠀⠀⢸⢠⡆⢀⡄⡟⡆⡇⠀⢈⡞⡜⡇⢃⠀⠈⡄⡇⢸⡆⠄⣧⢸⠈⡟⠀⡇⢸⢸
⠀⠀⠀⠀⠀⠀⡆⡜⠸⢸⢀⡇⡀⠀⠀⠀⢸⠀⢀⡇⡼⡘⡘⣸⠀⢹⡘⠚⠉⠁⢳⢣⢸⡀⠀⣧⢳⢸⡇⢰⣿⢸⠀⡇⡇⡇⢸⣸
⠀⠀⠀⠀⠀⠀⡇⠇⠈⡌⢸⠇⣿⣙⣛⣉⣽⠀⣼⢠⢡⣷⢁⠇⠀⠈⣇⠀⠀⠀⠘⣿⡀⣇⢀⣸⢸⢸⡇⢸⣿⢸⠀⡇⡇⠗⠘⢺
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠇⢸⠀⡇⠀⠀⠀⢸⢠⢇⡆⣼⡃⡜⠀⠀⠀⠸⡀⣀⣠⣴⣿⣧⠩⠭⠔⢚⣾⡇⠘⠛⠀⠀⣧⣤⣤⣤⣼
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⣟⣒⣒⣒⣺⣾⠀⠀⠈⢰⠁⠀⠀⠀⠀⢿⠛⠉⠁⠀⣿⠒⠈⠉⠁⢹⡿⠶⠶⠶⢶⡏⠉⠉⠉⢻
⠀⠀⠀⣀⣀⣈⣓⣲⡦⢤⣞⡀⠳⠤⣤⣤⣼⠉⠙⢒⣶⣏⣀⣀⣀⣀⣀⣘⣦⣖⣒⣋⣙⣒⣒⣒⣊⣉⣓⣶⣶⣶⣶⣷⣶⣦⣴⡞⣀⣀⣠⡀
⠀⠀⠸⣇⣀⣀⣀⣀⣨⣍⣨⣭⣭⣭⣭⣍⣉⣉⣭⣭⣥⠤⠤⠤⠤⠤⠤⣤⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⣤⣄⣠⣤⣄⣠⣤⣤⣤⣼⠇







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⠉⠈⠙⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠡⠤⠀⠀⢤⡀⢹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠂⣠⠀⠆⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⢉⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢹⣧⣄⣉⣀⣠⠞⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⠀⠹⣆⠉⢉⡸⣿⠃⢩⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠈⠳⢦⣈⣿⣟⣴⣥⡾⡿⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠀⠀⠀⠀⠀⢹⠛⣿⠃⣼⢳⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡆⠀⠀⠀⠀⠀⢸⠀⣿⢠⠇⢸⠀⠀⠀⠀⠘⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠾⠴⠧⠤⠤⠤⢄⣀⣸⢠⡇⡸⠀⢸⡀⠀⠀⠀⠀⢻⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠖⠶⣄⠀⠀⠸⡏⢸⠇⡇⠀⢸⡇⠀⠀⠀⠀⠘⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡤⢾⡄⠀⢠⠾⠶⠶⢤⣧⠋⢀⣧⡀⣸⣇⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠉⠓⠛⠦⢀⣀⣠⢿⣹⠏⠉⠉⠉⠛⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⢺⡟⠛⠒⠋⠙⠟⠻⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⣤⣀⣀⡴⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢰⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡜⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠶⠜⢣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣞⣡⡴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠉⠛⠒⠶⠶⠦⠤⠤⣶⠶⠖⠛⡿⢧⣀⣹⣤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣶⣤⣤⣥⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⡴⠃⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠖⠛⠃⣲⣤⣦⣴⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠈⡆⠀⠀⠀⠀⠀⣴⠶⢶⠖⠛⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⣏⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢰⠁⠀⡇⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⠀⠀⢇⠀⠀⠀⠀⢨⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣤⣤⡞⠀⠀⠸⡇⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣠⡇⠀⠀⠀⢷⣀⣀⣀⣼⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠛⠁⠈⠁⢸⠀⠀⠀⠈⣏⠉⢈⣇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠉⠀⠀⠀⠀⠀⠀⢀⡜⢀⣠⠴⠞⠋⠉⠉⢹⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠋⠉⢙⡿⠛⠉⠁⠀⠀⠀⠀⠀⢀⣼⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠒⠒⠒⠉⠀⠛⠒⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣶⣛⣹⣯⣹⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠚⠐⠒⠒⠒⠒⠒⠒⠒⠏⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡤⠀⠀⠀⠀⣀⣀⣀⣀⡆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⠉⠀⠐⠒⠒⠒⠒⠶⢾⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠤⠄⠤⠠⠄⢀⣀⣀⣸⠐⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠉⠉⠀⠒⠒⠒⠠⠤⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠠⠤⢀⣀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠒⠈⠈⠈⠈⠒⠒⠢⠤⢀⣀⣘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢱⠂⠨⠄⣂⠀⠄⢀⠀⠀⠀⠀⠀⡀⠄⣂⠤⠒⢈⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⠒⠠⠄⣈⠉⠒⠢⢬⣀⣈⡥⠒⢉⡀⠐⠾⢽⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡖⠢⢤⡖⠋⢁⡰⠒⢤⠀⠀⡶⠊⠉⠁⠀⠂⢤⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⣤⢀⠈⠈⠓⠦⢤⣀⠀⠀⡇⠀⠀⣠⠴⠚⠁⠸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠂⠤⣀⠈⡀⡇⣷⠖⠋⢀⠠⠒⠉⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠂⠀⠀⠀⠀⠀⠀⠈⡇⡇⣧⠴⠊⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⡇



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

⠀⠀⠺⠟⠻⠿⠟⠷⢿⡿⠿⠿⠰⡿⣿⡿⢿⣿⣷⣀⠀⠀⠂⠐⠿⠿⠿⠿⠿⠿⠿⠿⠷⢰⡿⠃⠀⠀⠀⠀⠀⠀⠺⠂⣄⠀⠀⠀⠿⠿⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡆⠀⠀⠀⠀⠀⠀⢸⡏⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⠁⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠈⠉⠙⠋⠉⠁⠉⠉⠙⠛⠈⠉⠙⠿⠿⠿⢿⣿⠿⠿⠇⠿⠿⣿⡿⢿⢿⡿⡛⠛⠟⢸⣿⠛⠛⠛⠉⠋⠛⠟⠛⠃⠉⠉⠘⠃⡏⠋⠉⠉⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡆⣦⠀⢹⡇⠀⢸⠇⡆⠀⠀⢸⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠁
⠀⠀⢀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⣈⡀⠀⠸⠇⠻⠂⠀⡃⠀⠈⠐⠇⠀⠀⠈⡀⠀⠸⣀⢀⠀⠀⢀⣐⠃⡀⠘⣀⢁⠀⢀
⠀⠀⠈⠁⠈⠉⠉⠁⡟⠛⠛⠛⠈⠁⠉⠉⠈⠋⠉⠹⠟⢿⡇⠿⠿⠿⠿⣿⡿⠿⠇⠀⠀⠈⠁⠀⠉⠉⠉⠁⠈⠉⠉⠀⠉⠈⠉⠉⡟⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠘⡇⠀⠀⠀⠀⣿⠁⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣀⣓⢂⣀⣘⣀⣀⣀⣀⡀⠀⠀⠀⠀⣃⣀⣚⣃⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⢀⣀⣀⣚⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡷⠆⠀⠒⠈⠉⠀⠈⠉⠀⠖⣶⠙⠛⠃⠀⢀⠐⢰⡏⠉⠉⠉⠉⠉⠀⠀⢸⡏⠁⠈⠉⢹⡏⠉⠁⠉⠁⠀⠀⠉⠀⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠃⠀⠀⠀⢸⡆⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇⠀⠀⠸⠁⢰⡇⠀⠀⠀⢰⠀⠀⢰⡀⠀⠄
⠀⠀⣀⣀⣀⣀⣀⣀⣘⣋⣀⣀⠀⠀⠀⢀⣀⣈⣛⣀⠀⠀⠀⣀⣈⣃⣀⣀⢀⣀⡀⠀⠀⢀⣀⣀⢘⣃⣀⣀⣀⣀⣘⠁⠀⠀⠀⢘⣈⣀⣈⣃⣀⡀
⠀⠀⠉⠉⠈⠉⠁⠛⠃⠀⠉⠉⠀⠀⠀⢠⠄⠀⠘⢟⠀⠀⡀⠉⣿⠉⠉⠀⢘⠛⠁⠀⠀⢸⡯⠑⠛⠁⠀⠉⠁⠀⠁⠀⠐⠀⠐⢺⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣫⠀⠀⠁⠀⠃⠀⠀⠀⣿⠀⠀⠀⠈⢸⠃⠀⠀⠀⠀⡜⠀⠀⢠⡆⠀⠀⣿⠀⢿
⠀⠀⣠⣤⣤⣤⠤⣤⠄⠤⣤⣤⢀⣤⣤⣬⣤⣤⣤⢭⣤⣤⡄⠤⢤⣤⣤⣤⢭⣤⣤⣤⣬⢠⣤⣤⣤⣤⣬⠥⣤⣤⣬⡁⠤⠤⢭⡥⠤⢤⣤
⠀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⡛⠁⢚⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠐⡊⣿⠀⠀⣼⠋⢉⡁⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠛⠀⢀⡇⠀⠀⡇⠀⠀⠀⠸⠃⠀⢠⠁⠀⣸⡇⠀⠀⠀⠀⠀⢸⠀⡀⠀⢱⠀⠀⠹⠀⠀⠀⡇⡇⠀⠀⣥⠀⣸⡇
⠀⠀⣤⠀⠈⠠⣤⣤⣥⣤⣤⣤⠠⣤⣤⣤⣤⣤⡬⠥⠤⠠⠄⠀⠀⠈⠀⣤⣤⣬⣤⣤⣤⠠⠤⠤⠥⠤⣤⣤⣤⣤⣭⡄⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤
⠀⠀⠙⠀⠀⠀⠙⠛⠛⡉⠉⣿⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠁⠀⠀⠀⠉⣿⠋⡉⠉⠉⠀⠀⠀⠀⠀⣿⠁⠁⠁⠉⠁⠀⢠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡇⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⠀⢸⡄⠁
⠀⠀⢤⠄⠀⠀⠐⣶⣶⣶⣶⣶⠀⠆⠒⠒⠲⣶⣶⣶⡶⠔⠆⠀⠀⠀⠀⠀⣴⡀⠀⠀⠂⠠⠀⠀⠀⠀⢰⣶⣶⣶⣶⡄⠶⠶⢶⣶⣶⣶⡶⠶⣶⡦
⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⢠⡁⠈⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡁⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⡁⡀⠀⢸⠏⠉⢩⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢰⡇⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠈⠁⠀⠀⢸⠀⠀⠸⢸⠁
⠀⠀⠐⠒⠂⠒⠒⠒⠒⠒⢰⠖⢰⣶⠲⠶⠖⠒⠖⠲⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠲⢶⡖⠒⠂⠒⠂⠀⠀⠀⠀⢶⡶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⠀⠀⠆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼

⠀⠀⠰⠶⠲⢶⠖⠶⠶⣶⠶⠶⠆⣶⢶⣶⣶⣶⣶⣖⡒⠒⠒⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⢰⡶⠖⠀⠀⠀⠀⠀⠀⠰⠶⢠⡀⠀⠀⠰⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣰⠀⠀⠀⠀⠀⠀⠀⣿⠇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡿⠀⠀⠀⠀⠀⠀⠀⠟⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠘⠓⠚⠒⠓⠓⠘⠛⠒⠛⠂⠚⠐⢲⡶⣶⣶⣶⣶⣶⣶⡆⢶⣶⣶⣶⣶⣶⡶⣶⠶⠶⢰⣶⠒⠓⠶⠚⠳⠛⠶⠖⠷⠘⠛⠂⢲⢐⠐⠓⠚⠒⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢠⡄⡆⠀⢿⡅⠀⢸⡇⡅⠀⠀⢸⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢼⠇⣿⠀⠈⠃⠀⠘⠃⡇⠀⠀⠸⠃⠀⢠⠇⠀⠀⠀⠀⠀⠼⠀⠀⠆
⠀⠀⠘⠂⠀⠛⠛⠃⣸⠶⠶⠶⠆⠚⠐⠺⠂⠺⠞⠛⢷⣷⣶⡆⣶⣾⣶⣶⣶⣶⣶⡆⠀⠀⠐⠓⠀⠒⠚⠛⠒⠀⠒⠛⠓⠐⠂⠚⠛⢲⠷⠷⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣿⠃⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠏⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠐⠀⠀⠀⠀⢰⣶⣀⠀⣀⡀⠚⠚⠂⠲⠟⢂⣀⣀⠶⣶⠆⠀⠀⠠⢠⡶⠶⠶⠖⠛⠷⠀⠐⠐⡶⠖⠐⠶⠛⣶⠶⠖⠘⠛⠓⠂⠒⠒⠒⠖⠓⠲⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡏⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⠀⠀⠀⠀⢰⡀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⣤⠀⠀⠀⡟⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠠⠀⠀⠀⠀⠘⠀⣾⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠡⡿⠀⠀⠀⠁⠀⡿⠀⠀⠀⠀⠿⠀⠀⠸⠄⠀⠁
⠀⠀⠸⠿⠛⠿⠿⢿⡗⠛⠻⠿⠆⠈⠀⠀⠚⠛⠓⢲⡇⠀⠀⠀⠾⣷⠾⠿⠋⢹⣿⠏⠈⠁⢰⡷⠞⣧⠞⠛⠛⠗⠙⠻⠃⠈⠉⠉⣉⣼⣿⡞⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⡇⠀⠀⠆⠀⡏⠀⠀⠀⣰⠀⠀⠀⢠⢸⡟⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⡄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠠⠇⠀⠀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⠠⠘⠁⠀⠀⠀⠀⠼⠀⠀⠀⠿⠀⠀⠠⠷⠀⠃
⠀⠀⢸⡿⠛⠻⠛⠛⠃⠘⠛⠛⠃⣾⣿⣿⣿⣿⢛⣛⠛⠿⠿⠆⠛⠛⠟⠛⠛⠛⠛⠛⠻⣿⠐⠛⠛⡛⣿⡟⠛⠛⣿⡿⡷⠘⠛⠛⠻⠛⠛⢻⡷⠀⠀⡀
⠀⠀⢸⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠠⠀⠀⣴⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣤⢸⠁⠀⠸⠃⠀⣴⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠇⠀⠀⠾⠀⠀⠀⠀⠉⠀⠀⠆⠀⠀⠿⠇⠀⠀⠀⠀⠀⠸⠀⠆⠄⠸⠄⠀⠈⠀⠀⠀⠼⠸⠀⠀⠸⠄⠰⡿
⠀⠀⢸⠉⠉⠉⢿⣿⣿⡿⠿⣿⡇⠺⠿⠛⠿⠛⢻⠛⠛⠋⠙⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⠀⠉⠉⠉⠉⢻⡿⠿⠿⠿⡷⠸⠿⠿⣿⠿⠟⠻⠿⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⡇⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣄⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣀⠀⠀⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠃⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠿⠀⠀⠿
⠀⠀⠰⠆⠀⣀⡀⢻⣿⣿⣿⣿⡇⢘⠐⣉⣒⠻⣿⡿⠿⠛⠘⠂⠀⠀⠀⠀⠀⣾⣄⡀⠀⠀⠐⠀⠀⠀⠀⠀⠿⠿⠿⠿⢷⠘⠛⠛⣿⣿⣿⣿⠟⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡄⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣨⢨⠀⠀⢚⠀⠀⣶⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠧⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠉⠼
⠀⠀⠘⠉⠃⠙⠛⠛⠉⠙⠃⡟⠁⣾⡟⠛⠛⠋⠙⠛⠻⠟⠻⠆⠈⠉⠁⠁⠈⠀⠀⠀⠀⠁⠀⠈⠈⠁⠘⠛⣿⠛⠛⠈⠃⠀⠀⠀⡀⠐⢿⡟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠏⠀⠰⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠶⠶⠚⠛⠛⠛⠛⠒⠶⠶⣤⣀
⠀⠀⠀⢀⣠⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠈⢳⣄
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣤⡀⠀⠀⠀⠸⣇⢻⣷⡀⠀⠀⠀⠙⣦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢻⡻⣦⠀⠀⠀⠻⣆⢻⣿⣄⠀⠀⠀⠘⣇
⣿⠀⠀⠀⠀⣾⠻⣦⡀⠀⠀⠀⠈⢳⡄⠀⠹⣿⣇⠀⠀⠀⠘⢧⣹⣿⣆⠀⠀⠀⢻⡄
⢻⠀⠀⠀⠀⣿⠀⢪⣷⡀⠀⠀⠀⠀⠹⣦⠀⢹⣿⡄⠀⠀⠀⠈⢷⢿⣿⣆⠀⠀⠘⣇
⠸⡇⠀⠀⠀⠘⣧⡀⢿⢿⣄⠀⠀⠀⠀⠈⢷⡀⢿⣷⠀⠀⠀⠀⠘⣇⢻⣿⡀⠀⠀⢿
⠀⢻⡄⠀⠀⠀⠘⣧⠈⢷⣻⡆⠀⠀⠀⠀⠈⢷⠸⣿⣇⠀⠀⠀⠀⢿⢸⣿⡇⠀⠀⢸⡇
⠀⠈⣷⠀⠀⠀⠀⠘⣆⠈⢿⣿⡀⠀⠀⠀⠀⠸⡇⣿⣿⠀⠀⠀⠀⢸⡜⣿⡇⠀⠀⠈⣧
⠀⠀⢸⡇⠀⠀⠀⠀⢹⡄⠘⣿⣇⠀⠀⠀⠀⠀⣿⢹⣿⡆⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⢻
⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷⠀⢹⣿⡄⠀⠀⠀⠀⢿⡌⣿⡇⠀⠀⠀⢸⡇⣿⣿⠀⠀⠀⢸⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡄⠘⣿⣇⠀⠀⠀⠀⢸⡇⣿⣇⠀⠀⠀⠀⣿⢻⣿⠀⠀⠀⠈⡇
⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸⡇⠀⣿⣿⠀⠀⠀⠀⢸⡇⣿⣿⠀⠀⠀⠀⢻⣾⡿⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣧⠀⢸⣿⠀⠀⠀⠀⠸⣇⢹⣿⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⡇⠀⡆⠀⠀⠀⢹⣇⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡛⢷⣦⣄⡀⠉⠛⠶⣤⣄⡀
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠙⠶⣍⠻⢿⣶⣄⡀⠀⠈⠛⠳⢦⣄⡀
⠀⠀⠀⠀⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣅⠒⠭⣝⠳⣤⡀⠀⠀⠈⠑⠦⣝⠻⢿⡶⣄⡀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠰⡇⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠙⠷⣄⡈⠳⢦⣙⠳⣤⡀⠀⠀⠈⠛⢦⣍⣪⣿⡄⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⡇⠀⠀⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠠⢭⣟⡶⣤⣀⠀⠀⠈⠙⠶⣤⡉⠳⢮⡙⢦⡀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⡀⠄⠀⠈⢷⡀
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⠙⠛⠶⣤⣀⠀⠀⠀⠀⠈⠛⢦⡈⠙⠳⢯⣟⠶⣄⡀⠀⠈⠙⠶⢦⣍⣂⣹⡆⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁⠀⠀⠀⠸⣇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣄⠀⠀⠀⠈⠑⠦⣄⡈⠙⠪⣝⢶⣄⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠙⠷⣦⣌⣡⣹⡆⠀⠀⠀⠀⢀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡴⠶⠶⠶⢦⣤⣀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣄⡀⠀⠉⠉⠉⠀⠀⣀⣤⡶⠟⠉⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⡀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠀⠶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⣠⠾⠟⠉⠀⢰⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⣸⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⢻
⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⢻⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣧⠀⠀⠀⣀⣼
⠀⠀⠀⠀⠛⢦⣄⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠉⠛⠟⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⣿⢀⣠⡴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⣶⠛⣹⣿⣝⠛⢻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣦⠉⠰⢿⠁⠉⡩⢉⣹⠉⡛⢽⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠿⣿⡗⢺⣯⠐⠉⠛⠓⠚⠋⠻⣿⡦⠐⣿⣉⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⣺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣄⡀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣓⡛⠃⠀⠁⠀⠀⡀⢠⣃⣀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠿⠿⠿⢿⣿⠿⡿⠈⠛⢿⠟⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⢠⣤⠀⠀⢤⣀⣴⣿⣿⣿⣿⡧⠤⠶⠶⠾⠿⠤⠰⠆⠀⠈⠀⠀⣠⠄
⠀⠀⠀⠀⣠⡄⢀⣀⣠⣴⣿⣁⣠⣿⡤⠴⠿⠿⠛⠿⡄⢹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⠃⠿
⠀⠀⠀⠀⠠⢶⣴⣿⠉⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠹⠸⠇⠀⠀⠀⠀⣴⡅⡔⢀⣶⡘⢯⣇⡤
⠀⠀⠀⠀⠀⠀⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⠟⠃⢸⣿⡇
⠀⠀⠀⠀⠀⣴⣿⣟⣀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡟⠀⠀⠈⢻⣇
⠀⠀⠀⠀⢠⣿⣿⠿⠋⠀⠀⠒⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣼⣿⡏⠀⠀⣤⣾⣿
⠀⠀⠀⠠⣿⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣥⣾⣿⣿⣿⣿
⠀⠀⠀⡴⢻⣦⣁⣀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢘⣵⣿⣫⣿⠿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣶⣶⣶⣶⣴⣤⡴⣾⣽⡇⣷⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠘⢿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣾⢿⣿⣷⣿⣿⣿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⢤⡈⢿⡀⠀⠀⠀⠈⠛⠋⠛⠚⠈⠓⠛⠿⠅⣿⡿⠃⢀⣿⠈⣼⣿⠁⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠈⠃⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⢻⠂⣿⡏⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣟⠀⠀⠸⡇⣿⠇⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡿⠀⠀⠀⠀⠀⠀⠀⠑⢺⡇
⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣀⣀⡀