Running race Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠔⠒⠉⠉⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢵⠤⠒⣒⡚⢳⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡠⠋⢸⠁⣀⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡐⠀⠀⠈⠙⠁⠒⢲⠀⠀⠀⠀⣀⣀⡀⠀⢠⡘⠍⣁⡠⢦⠞⠁⠀⠝⠃⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⡠⠊⠙⣤⠈⠅⢸⠀⠀⠀⠀⡟⠁⠀⠀⠉⠙⠋⠒⠂⢠⠊⠙⢯⣅⠀⡇⢸⠁⠀⠀⠀⡷⠊
⠀⠀⠀⠀⣀⠤⠤⠤⠤⠀⠀⠀⣀⡤⠊⣠⡤⠊⠀⠱⣘⠀⠙⢒⣊⡵⠁⡀⠀⠀⠀⠀⠀⠀⡴⢡⠒⠴⣁⠀⢱⠇⣠⠗⠒⠒⡋⡐⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠉⠁⡔⠉⠀⠑⡄⠀⠀⠈⠒⠒⢅⠀⣀⣰⠩⢠⡀⠀⠀⢰⢀⠀⡳⡀⣠⠊⠀⠈⠛⠁⡑⣄⠈⢢⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠐⣆⡇⠀⢠⠞⠀⠀⠀⠀⠀⠀⡼⡁⠑⡄⢀⡤⠃⠀⠀⠀⠙⠈⠀⡜⠁⠀⠀⠀⠀⢠⣇⣸⠲⠊⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⡖⠺⢅⡙⠚⠀⡌⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣰⠉⠉⠁⠈⠉⠑⠒⠁
⠀⠀⠤⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡉⠙⠉⠀⠀⣀⣀⣀⣀⠀⡰⠧⣄⡀⠴⠓⠠⠤⣤⠠⢤⡀
⠐⠂⠐⠀⠈⠁⢰⠒⠒⠚⠉⢹⠰⣄⣀⣀⣀⣀⠤⠴⠛⢳⠊⠉⢹⠀⠀⠀⠸⢥⣤⠤⢼⠞⠀⠀⠀⠀⠀⠀⠀⢠⠗⡢⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠉⡶⢶⢃⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡜⠀⠀⠀⠀⠀⡎⠀⠧⠲⣤⡀⠀⢠⠚⠢⠤⠎⡰⢁⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠮⠔⠳⡀⠀⠀⢀⡠⠤⠐⠒⢊⠇⣰⣠⠔⢢⠀⠀⠀⠑⠤⠴⠊⠀⠈⠑⠃⠀⠀⠀⢀⡟⠋⢁⡴⠃
⠀⢀⣀⣉⠩⠭⠭⠀⠸⣄⠤⠔⠊⠉⠉⠉⠁⠀⠀⠀⠀⢸⡾⠋⠀⡠⠊⠀⠀⠀⠀⠀⠀⣂⣉⡩⠤⠄⠀⠃⠺⠈⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠪⠝⡛⠿⣷⠤⠤⣀⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠲⠞⣓⠜⢓⠬⠛⠶⠓⠒⠀⠤⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡃⠀⠀⠀⠀⠀⠘⠿⣿⣶⡄⠀⠀⢸⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠰⣦⣀⠀⠀⠀⢸⣿⣿⣿⣦⠰⣿⣿⣿⣿⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠚⠛⠀⠀⠴⠟⠀⠀⣸⣿⣿⣿⣿⡟⠃⠈⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠉⠉⢿⣿⣿⣇⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡄⠀⠀⠀⠀⢀⣾⣿⣿⣿⡀⣴⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⢿⣷⣀⡀⠀⠀⣾⣿⡿⣿⣿⣿⡿⠋⠈⢿⡿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠙⠛⠻⠶⣼⣿⡏⠀⠈⢿⣿⣷⣾⣿⠟⠛⠘⠻⠿⠷⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⢸⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠆
⠀⠀⠀⠀⠀⠀⠠⠴⠆⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁⠲⠶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⢀⢹⣿⣿⣼⣿⣿⣿⣿⣿⣮⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡟⠀⠀⠀⠸⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣧⣀⠀⠀⠀⣧⢸⣿⣿⣿⣿⣿⣿⣿⣟⣿⣮⣽⣿⡿⠤⠤⠤⠄⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡦⣶⣿⠺⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⣿⣿⣿⣿⣿⡟⢟⣿⠟⠉⠃⠘⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⣿⣿⣿⣿⣿⣿⠿⣿⡀⠀⠄⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣰⡟⠁⠀⡀⣿⣿⣿⣿⣿⡇⠀⣿⠁⠄⣼⣿⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⢠⣿⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⠠⣿⠀⢷⣻⣿⣿⣿⣿⣿⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠋⠀⠀⣺⣿⣿⡿⢿⣿⣿⠇⠀⠉⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⢸⣿⡿⠀⠀⠀⠀⠀⠁⡸⣿⣿⣿⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⢼⣿⠃⠀⠀⠀⠀⠀⠀⠐⠹⣿⣿⡘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⣿⣿⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⢈⡷⣄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡀⠀⠀⠀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡤⠒⠁⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣠⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠛⠢⡺⣯⡆⠀⠀⠀⢰⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⠀⢠⠉⠈⣀⠙⣷⡀⠀⠀⢸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠐⢹⠀⣿⠀⢸⣿⡑⢾⡇⠀⠀⠀⣿⠃⠀⠐⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⡟⠉⠛⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢿⣀⣠⠄⠐⢸⣧⣄⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⠛⢛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠨⠀⠀⢸⡟⠉⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡿⠛⠛⠒⠚⠀⠀⠘⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠰⠁⠀⠀⢠⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡶⣛⠶⠥⠾⠶⢦⡴⡟⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣥⣼⣿⣿⡿⣿⣿⣷
⠀⢸⢹⢳⡒⣿⣥⣄⡴⠛⣁⠀⠀⠄⠤⠊⠀⡇⠀⡇⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠁⣸⣿⡇
⠀⢸⣏⠀⠀⢸⡟⣡⠂⢩⠃⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⣿⡟⠁⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠘⠋⠀⠀⢸⢸⡇⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠰⠀⠀⠀⠀⠀⠛⠀⠀⠸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠿⢯⣑⣠⡄⠀⠀⠀⠀⠀⠀⠀⢸⡧⠓⢒⡲⠀⠀⠀⠀⠀⠀⣽⣿⣿⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠃⠀⠀⠹⣿⣿⠄⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⣹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⢀⣤⡀⠀⠀⣀⣄⣿⣿⣿⣷⣦⡄⢠⣤⣿⣿⣶⣦⡀⠀⢼⣿⣿⣿⡇⠀⣴⣾⣿⣿⣤⣄⠀⣤⣶⣿⣿⣿⣿⣷⣷⣦⡀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⡿⢀⣼⣿⣿⣿⣿⣿⣿⣿⢷⡿⣿⠉⠉⠉⠁⢿⠀⠺⣿⣿⣿⠇⠀⠹⠛⠛⢻⣿⣿⣾⡿⠉⢹⠛⠛⠛⢻⣿⣿⣷⡀⠀⠀⣠⣶⣿⣧⣄⡀
⠀⣰⣶⣿⣧⣿⣿⣿⣿⣿⠉⠀⠀⢸⠛⠁⢸⣄⣀⡄⠀⠈⣄⣠⣽⣿⣿⣦⣤⡀⠀⠶⢾⣷⣿⣿⠃⠀⢰⠀⠀⠀⢸⣿⠙⣿⣷⠀⣠⣿⡛⠛⢻⣿⣧
⣰⣿⠍⠉⢹⣿⣿⣿⣿⣿⠿⠷⠀⢸⡇⢠⣿⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣇⠀⣯⠀⠀⠘⣶⣤⣴⣾⣟⠀⠈⠉⠐⠛⠉⡀⠀⢸⡇⢹
⠉⠀⣤⣤⣼⠀⠇⠀⣿⣿⣤⣤⣴⣾⡿⢸⣿⣿⣿⠛⢠⣿⡿⣿⡇⠀⠀⢾⣿⣿⣿⣿⣿⣛⠀⠋⠀⠀⢰⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⡇⠘
⠀⠀⣿⣿⣾⡆⠀⢀⣿⣿⣿⣿⣿⣿⡇⢸⡋⣿⡏⠀⢸⡇⢰⣿⡇⠀⠀⢰⠉⠛⢿⣿⣿⣏⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⡿⣿⡇
⠀⠀⣿⡟⣿⡀⠀⠈⣿⣿⣿⣿⡿⣿⡇⣾⡇⣿⠃⠀⣼⠇⢰⣿⣿⣾⣿⣿⠀⠀⣼⡟⣼⡇⠀⠀⠀⠀⠘⢻⣿⣿⣯⣿⣿⡇⠀⠀⠀⠀⢿⡆⠲⣧
⠀⠀⠸⣇⠈⠃⠀⠀⣿⣿⣿⣿⣿⠞⠀⢹⠀⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣽⠁⢹⣧⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣷⠀⠉
⠀⠀⠀⢻⡄⠀⠀⠀⣿⣿⣿⣿⣯⠀⠀⣿⠀⠈⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣿⡀⢨⡇⠀⠀⠀⠀⠀⠀⢸⣿⡯⢿⣿⡄⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠘⠃⠀⠀⠀⣾⣿⡇⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢹⣿⣿⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠸⣿⡇⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠏⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣦⣶⣶⣤⣿⣷⣴⣴⣶⣶⣶⣶⣶⣴⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣤⣶⣶⣶⣶
⣿⣿⣉⣬⣫⣤⣍⣽⣍⣿⣏⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⠙⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠁⢀⡀⠀⠹⣿⠟⠀⠀⠈⣿⣇⡀⠀⠀⢈⣿⣗⡄⠀⠀⠸⣿⣧
⠀⠀⠀⠰⢤⣠⣾⣿⡿⠳⣄⠀⠀⠀⢀⣀⣰⣿⣿⠷⣄⠀⠱⡴⢿⣿⡟⢳⡄⠘⣿⣿⢛⡶⣰⠻⣿⣿⡇⠀⠀⢸⣿⡟⢳⡄
⠀⠀⠀⠀⠀⠉⠈⣿⣇⣠⠃⠀⠀⠀⠀⠈⢉⣿⣿⣠⠋⠀⠀⠀⣸⣿⡷⠋⠰⠚⣿⣿⡏⠀⠏⢠⣿⣿⠁⠀⠒⢻⣿⡷⠋
⠀⠀⠀⠀⠀⢀⣴⡿⣿⣧⠀⠀⠀⢀⡀⠀⣼⡿⣿⠀⠀⠠⣶⡿⠛⣿⡇⠀⠀⢸⡿⣿⡧⠀⢀⣾⠿⣿⠀⠀⢠⣿⢿⡷
⠀⠀⠀⠀⢠⡟⠉⠀⠈⠙⠟⠛⠉⠉⠻⢼⡏⠀⠿⠶⡦⣤⡈⠙⢧⣼⣇⠀⠀⣿⠁⠘⠷⠶⢾⣿⠀⠻⠶⠶⣿⡇⠸⣷⣦⣤⣤
⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⠃⠀⠚⠁⠙⢧⡀⣿⠀⠀⠀⠀⣾⠈⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠛
⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠿⠀⠀⠐⠚⠛⠀⠀⠀⠰⠾⠇

⠀⠀⠀⠀⠀⠠⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠰⡄⠀⠀⢱⡷⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠫⢀⡔⠁
⠀⠀⠀⠀⠀⠈⠙⡖⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢲⣒⠗⠒⠒⠒⠂⡀
⣶⠖⠐⠀⠀⡰⢀⠀⠀⠀⠠⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡌⠀⠀⠉⡉⣹⠀⡇
⢿⣿⣸⠉⠉⠀⣸⠀⠀⠀⡀⠇⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡆⡏⠐⠁⠱⣦⠀⠀⢹⣆⢉⠁
⠀⠀⠀⠑⠒⠞⢩⠰⡀⠀⠷⣤⡔⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠂⠁⢸⠀⠀⢀⠽⠋
⠀⠀⠀⠀⠀⠀⠈⠢⠹⠤⠤⠤⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢀⡠⠀⠉⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠠⠁⠀⢀⡴⠛⢄⡀⠀⠀⠉⠉⠉⡷⢄⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡔⠁⠐⠤⡀⠀⠀⠀⠈⢑⢤⠤⣄
⠀⠀⠀⠀⢠⠃⠀⢀⠎⠀⠀⠀⠉⠢⠔⠒⠚⠶⣇⣨⠀⠀⠀⠀⠀⠀⢠⠃⠀⡜⠀⠀⠀⠀⠀⠉⠉⠙⠒⠚⢾⢀⣾
⠀⠀⢀⠴⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣶⢤⣄⢀⢇⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠢⢻⣟⠛⠲⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡤⣈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢰⢹⠀⠀⢀⣀⡄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡎⠸⣄⡇⠈⢧⡴⠁⠀⠻⡴⢇⢸⠉⠲⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠳⡀⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠸⣾⠏⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡘⣦⠉⠋⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠬⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣉⠉⡀⠀⠀⠀⢀⡤⠔⠋⠉⠁⠈⠉⢹⠖⠀⠀⠀⠀⠀⢴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢉⠏⡉⠉⠁⠀⢀⣠⣒⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡴⢦⠀⠀⢀⡏⠀⠈⣧⠀⠸⣠⡆⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⢸⠀⠀⢸⡷⠀⢀⡟⠀⡇⠉⢢⡖⠋⠙⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢟⡡⢃⡤⠆⠈⠓⠖⠋⢀⣼⡅⠀⠈⠀⠿⣢⠏⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⠀⠀⠐⠊⠙⣄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠑⢶⣤⣴⣶⣾⣿⠏⠁⢀⡟⠀⠀⣠⢶⢋⣉⣩⣁⣒⠤⡠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠻⣿⠟⠩⡏⣀⡴⠋⠀⢠⡞⠁⢏⠁⠈⡇⠀⡨⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⡹⠓⢢⠀⠈⠳⢦⣍⣙⣛⡉⡁⣀⣠⠴⠋⠀⠀⣈⡦⠴⠃⠐⣅⣀⠦⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠵⢾⢀⠆⣸⠀⠀⠀⠀⠀⠉⠉⣩⠟⠉⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠈⠣⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⣜⣫⠤⣄⠑⠤⣄⣀⣰⣊⠟⠁⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠓⠦⠤⠶⢲⠏⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⣀⡠⠷⠖⠒⠶⠦⠤⣄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇⠀⠀⠀⠀⣰⠿⠓⠺⠿⢷⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣴⡒⣧⠀⢀⡤⣄⠀⠀⢀⠀⠀⠈⠙⠲⡄⠀⣠⠷⡊⠲⡄⢀⠀⡹⢢⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⠃⠈⠛⠚⠉⠀⠈⠙⠛⢫⡀⠀⠀⢀⣼⠃⣠⠛⣱⡼⠚⢧⣋⠔⢁⡤⠋
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⢶⣶⠾⢧⡉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠣⣌⢙⡥⠚⠁⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠐⢏⠉⠉⠙⠓⠺⢭⣓⢮⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠈⠑⣼⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣒⣒⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠶⢤⡀⢀⠏⠀⠀⠀⠀⢀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⡠⣄⠀⠀⠙⢾⠰⠁⠒⠂⠁⠀⢣⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠋⡴⠈⠁⠀⠈⠱⢄⠀⡸⠃⠀⠀⠀⠀⠈⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣏⠀⡇⠀⢀⠀⠀⠀⠀⡇⠳⡄⢠⣀⠀⠀⢒⡂⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠖⢩⠀⢳⣬⣤⣀⣀⠆⣠⠃⠀⠱⣄⠙⠍⠉⠙⣀⠆⠀⠀⠀⠀⠀⢀⡤⠴⠶⠖⠛⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⣄⡠⢮⣂⠐⠁⣾⣥⣤⡀⢄⡬⢷⡄⠀⢰⠁⠀⠀⠀⠀⠀⠀⡎⢠⠤⣄⣀⣠⠞
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠴⡆⠇⠀⠀⣆⣀⣀⠙⣀⡤⠼⠇⢀⠸⠓⠉⠉⠙⢆⠀⠀⠈⡻⠊⡠⠴⠖⢶⡀
⢰⠋⠈⢝⠆⠀⠀⠀⠀⢠⠃⡼⡁⠘⣒⠊⠀⠨⣩⠟⠁⢀⠠⢀⡀⠀⠀⠐⣂⠀⠀⢣⠀⠀⣅⠎⠀⡀⠀⡀⠁
⠸⣄⠀⢊⠅⠀⠀⠀⠀⡏⢸⢰⠃⠀⠤⠄⢀⣾⠥⣄⡤⠂⠀⠈⣷⠀⠁⠠⠀⠐⡤⠊⢧⠀⢏⡀⢀⣀⠀⡀
⠀⠀⢳⠈⣆⡠⠒⠚⢺⠃⢸⡄⠀⠀⠀⠀⢰⠃⠈⠀⠀⢀⡀⣄⡼⠀⠀⠀⠀⠀⢧⡀⠀⢆⠀⠑⠜⠋⢙⡡⠃
⠀⠀⠈⣥⠜⠇⠀⢸⠋⢠⣾⡇⠀⠉⢚⡀⠻⢤⡤⠀⠂⡇⠀⠘⠀⠐⠀⠀⠀⠀⠂⠱⡄⠈⡀⣀⣤⠀⠸⣅
⠀⠀⢰⣃⡀⠈⠚⠓⠚⠁⢸⡆⠀⠀⠀⠀⠁⢸⠀⠀⠀⣰⠈⠉⠁⠀⠀⠀⠀⠀⢠⡤⢿⠀⣡⠆⠘⢦⠞⠁⣹⠀⠀⡠⢔⠦⡄
⠀⠀⠸⡉⠓⠳⢶⣤⣀⠀⣸⣇⠀⠀⠀⠀⢀⣸⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠉⠀⡄⠘⠀⠄⠀⠀⢛⠲⠒⠁⠘⣂⠽
⠀⠀⡇⡀⠀⠀⠀⠘⢻⡔⠓⢾⠉⠉⠉⠉⡁⠀⠱⣠⠓⠦⢤⣀⡀⠀⠀⠀⣀⣠⣼⠧⠀⠀⡇⢀⠠⠄⠀⠌⠈⡤⠖⠊⠁
⠀⠀⠱⠻⠟⠞⣀⠞⠉⠀⠀⠸⡀⠀⡠⠊⠀⠀⣰⠃⠀⠀⠀⠀⠉⠉⠉⡭⠁⠀⢸⡈⠂⢺⠁⠈⠉⠁⠀⠀⢰⠃
⠀⠀⠀⠈⠀⠈⠙⣆⠁⠰⣔⠈⠹⠜⠁⠀⠀⣠⠁⠀⠉⠓⠦⣀⡰⡖⠊⠀⠀⠀⠀⢧⠀⠸⣄⠀⠤⠤⠄⠀⢼
⠀⠀⠀⠀⠀⣠⠀⠘⠗⠋⠁⢀⠄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡜⠁⢱⠀⠀⠀⠀⠀⢈⡧⠀⡅⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⢹⣀⣀⢤⠒⠋⢹⠀⠀⠀⠀⢸⡇⠀⠀⠀⡠⠞⡇⠀⠈⢦⠤⠖⠒⢻⠀⠀⠀⡇⠀⠀⢠⢧⡀⡠⠂⠉⢂
⠀⠀⠀⠀⠀⢞⠀⠀⠀⢣⠀⡨⢧⠀⠀⠐⢣⢈⠖⠒⠋⡠⡸⠁⠀⠀⠸⡀⠀⠀⢸⣠⡤⢤⠋⠁⢐⠂⠀⠙⢦⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠈⣆⠀⠀⢻⠙⡄⠀⠣⡀⠀⠀⢿⠀⠀⢀⡴⠁⠀⠀⠀⠀⡇⠀⠈⠘⣧⡀⠈⠀⠀⠈⠀⠀⢀⠆⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠸⡄⠀⢸⠀⠙⠒⠊⠙⡤⡤⠘⡗⠒⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠃⢸⠦⠄⠀⠀⠀⣾⠹⠠⠏
⠀⠀⠀⠀⠀⠀⠀⢡⣄⢘⣆⠀⠀⢀⠜⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢸⠁⠀⠀⠀⠀⠀⠹⡄⠀⠡⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠈⢁⠜⠀⠀⣏⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡥⠒⠾⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠉
⠀⠀⠀⠀⠀⠀⠆⠀⠊⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠤⠄⢘⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣭⣭⣯⣮⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⠀⠀⠀⠀⢀⠊⣄⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⣄⠐⣂⠀⠄⡰⠂⠒⠊⠈⠿⠿⠖⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠠⠈⠀⠤⢀⠱⠁⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣷⣶⡶⡐⠀⠉⠂⡤⠀⡶⣄⠀⠀⠀⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠛⠿⢛⡴⡁⠀⠀⠀⠧⣀⠀⠈⠉⠀⠘⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⡇⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠐⠀⠋⠀⠀⠋⠀⡜⢀⠠⠃⠀⠈⢈⠗⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠤⠤⠚⡀⠁⠒⠀⠤⠀⠂⠀⢀⣀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠀⠀⠀⠀⢆⠉⠐⠀⠀⠔⠁⠀⠀⠠⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⣠⠀⠀⠚⠒⠄⠈⠢⣀⡠⠀⠀⠀⠀⠀⢠⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠛⠢⠄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⡠⠘⠀⠀⠀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠐⠉⠀⠀⠀⢀⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢠⢀⠊⠀⠀⠀⠀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⢀⠤⠄⡀⠈⠐⠤⣀⣀⠠⠤⠐⠀⠀⠀⠁⠀⠀⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠁⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⢉⠄⠀⠀⠀⠀⠀⠀⢐⠑⠁⢀⠄⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠔⠀⠈⠙⠄⠀⠀⠀⠒⠒⠒⠁⢀⣔⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣄⣰⣿⣀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠞⠋⠉⠀⠀⠉⠙⠳⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠃⠀⠀⠀⣿⣿⠀⠀⠀⠘⣧⣴⠄⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡇⠀⠀⢀⣤⣼⣧⣤⡀⠀⠀⢸⣇⡀⢀⣤⣀⣿⠷⠶⠿⠶⣾⣟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣸⠋⣡⣀⡽⠋⠀⠀⣠⣤⡀⠀⠈⠻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⣆⠀⠸⠿⠿⠿⠿⠇⠀⣰⠟⠿⠉⡿⠁⠀⠀⠀⠻⡿⠃⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢻⣦⣤⣤⣠⣤⣴⠛⠿⠆⠘⠻⡇⠀⠀⢠⣾⣿⣿⣿⡆⠀⠀⢈⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠙⢳⣦⠿⣣⡈⠛⠀⠀⠀⠀⣶⣿⡀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡴⠟⠛⠛⠛⠛⠿⣇⣠⣄⠀⠀⠀⠀⣠⣿⣤⡈⠋⠛⠛⠛⠃⣠⣾⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⠋⠀⠀⢠⣶⣦⠀⠀⠈⢻⣅⣀⠀⠀⠀⠈⠀⢸⡿⠛⣶⡶⠺⣿⡁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⣈⣛⣋⡀⠀⠀⠀⢻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⣸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢷⡀⠀⢸⣿⣿⣿⣿⡇⠀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢦⣈⡈⠉⠉⣈⣠⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠙⣿⠋⠹⠷⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠿⠿⠿⠷⠖⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣺⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣿⠿⣿⢿⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠀⠀⠀⢀⣤⣤⣶⣿⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣿⠟⠁⣠⣧⡀⠙⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣭⠶⠷⢮⣷⣄⡀⢨⣿⣿⠟⠉⢉⡟⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⣷⣶⣶⠸⢿⣿⡟⠛⣿⣿⡟⠛⣻⣿⠿⠁⣤⣀⣤⡄⠀⠘⣏⣟⠙⢦⣴⠋⢹⣼⢡⣾⣿⡏⠙⢲⣿⣄⣀⣿⣿⡿⠇
⠀⠀⠀⠀⠀⢀⣰⣿⠯⣤⣞⠹⣿⣦⢀⣿⣷⣦⣀⣇⣠⣶⣿⣇⢠⣾⡿⡛⣻⣿⣶⢄⣦⢷⣴⠋⠉⢢⣼⣹⡏⠹⣿⣿⣄⣼⠁⢀⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠙⣿⣦⣜⣉⣽⣿⠃⠈⣿⠋⠛⣿⣿⡟⠋⢻⡏⢸⣿⣤⠛⢣⣼⣿⠀⣤⠙⣭⡛⠟⣭⡟⣡⡄⠀⢾⡿⣿⣿⣿⣿⡿⠟⠟
⠀⠀⠀⠀⠀⠀⢶⡟⠛⢿⡿⠛⢹⡷⠀⠸⣇⠀⣨⣭⡀⢀⡾⠀⢶⡉⢿⠿⠿⠟⢀⡖⠹⣇⠈⣤⣤⡀⢀⡿⠀⠀⠐⣆⠀⠈⠛⠋⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠈⢧⢰⣿⣿⡄⡼⠁⠀⠀⠹⣆⠿⠿⢇⡾⠁⠀⠈⣧⢰⣿⣿⠄⡾⠁⠀⢹⡌⠿⠿⢃⡾⠁⠀⠀⠀⠹⣆⢼⣿⣷⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⠀⢻⣷⣶⣿⠃⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⢻⣷⣶⣿⠁⠀⠀⠀⠀⠀⢹⣬⣭⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠀⠸⣿⢿⡏⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⢁⡴⠖⠲⣄⠀⠖⠒⢦⡀⠐⣶⠒⢶⡖⠲⣆⠰⣖⠀⢲⡖⠀⣲⠆⣠⠖⠲⢦⡀⢲⡶⠖⠀⡇⣰⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣟⠚⠒⠛⢰⡏⠉⣻⡇⠀⡇⠀⢸⠇⠀⡿⠀⠹⣤⡟⢹⣰⠏⠀⣧⠀⠀⣸⠇⢸⡇⠀⠀⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠉⠛⠛⠁⠈⠙⠛⠛⠃⠘⠛⠀⠛⠃⠘⠛⠀⠀⠙⠀⠀⠋⠀⠀⠈⠛⠛⠉⠀⠛⠃⠀⠘⠛⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣤⣶⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣿⣿⣿⣯⡀⠀⠀⠀⢻⡗⠀⠀⠀⡀⠈⣻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣟⠀⠀⢾⡷⢾⣿⣶⣿⠚⠿⠚⢿⣿⣿⠟⠀⠀⠀⠀⣤⣶⣶⣄
⠀⠀⠀⣁⠀⠀⠀⣠⣴⣶⣤⣿⣿⡿⣿⠟⠻⠿⠃⠀⠀⠀⢴⣿⠿⣿⠆⠀⠐⢾⡟⠡⣿⣦⡀⠀⢸⠏⠉⠛⠻⠄
⠀⠀⠀⠙⣷⡀⢸⡿⠋⠉⠉⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠈⢷⣦⡻⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣴⣦⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣧⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣤⡄
⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⣤⡉⠁
⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣥⣤⣤⣤⣤⣼⣿⣧⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣬⣿⣿⣧
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠟⠛⠛⠓⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⡀
⠀⠀⠀⡔⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠠⠔⠒⡄
⠀⠀⠀⠳⠤⠤⠴⠄⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠤⠴⠃
⠀⡄⠂⢀⠀⠀⠀⠀⠀⠀⠴⠛⠻⠟⠛⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⣿⠷⢦⡀⠀⠀⠀⠀⡀⠀⣀⠀⠂⡄
⠀⠈⠉⣠⣴⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⠉
⠀⠀⠞⠉⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠙⠓⠶⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠤⠤⠤⣄⣀⣀⣀⣈⣉⡉⣉⣉⣉⣁⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢀⠔⠋⠉⠉⠙⠻⠍⠀⠀⢙⡷⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⢤⣀⣀⣀⣀⣤⡤⠶⠚⠋⠀⠘⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠿⣯⠉⢉⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣨⢷⡋⠁⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣷⣦⡴⢿⣿⣁⣀⣻⣾⣿⢡⣤⣼⣷⣴⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⣡⡜⢻⣿⡎⠉⠛⠻⠋⠋⢠⣶⡿⣫⣽⣭⡹⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⡄⣾⣿⣿⡠⢻⠃⣴⣿⠇⠀⠀⠀⠀⠀⢠⣿⡇⢿⣯⣽⡿⣹⣿⡁⣠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⣱⡿⢋⣿⣿⣿⡿⡟⣯⣴⣦⣴⣀⡀⠀⠀⢹⣿⣦⣭⣯⣷⣿⣿⣇⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⡿⠋⠀⠀⣀⣿⢻⣧⡶⣾⣿⣷⡆⠀⠀⠀⠛⠉⠻⠻⣿⣬⣿⣷⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢼⡇⢾⣏⠶⣩⣷⣻⣟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣆⠀⠀⠀⠀⠀⠐⢿⣾⡛⠻⣛⣷⣿⠉⢀⡀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠈⣿⠃⠀⣀⣀⣶⣤⣼⡍⠛⠻⠏⣛⣸⣷⣾⠿⢿⣿⣶⡆⠀⠀⢰⡿⠋⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⢰⡿⢠⣤⣿⣿⣿⣿⣿⣿⡟⠀⣀⣿⡿⠻⣷⡀⠀⣨⣿⣿⣿⠀⢸⣿⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⣶⣿⣶⣿⣤⡄⢠⣿⣿⣿⠁⠀⣿⣿⣿⠟⣹⣿⠁⠀⣈⣷⡾⠋⠁⢹⣿⠗⠈⠛⣠⣀⣌⢛
⠀⠀⠀⠀⠀⠀⠀⠠⣴⡿⠋⠀⠀⠀⠈⢻⣷⡆⣹⣿⣿⣷⣾⣿⣿⠿⠃⢙⣿⣶⡿⠋⠁⢻⣦⣀⣾⡿⠦⣤⡿⠛⠛⠛⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⡀⠀⠀⠀⠀⠀⣿⡷⠀⠈⠿⠛⣿⠛⠛⠀⠀⠈⠙⣿⣷⣦⣤⣤⣿⣿⠟⠇⢺⣿⠀⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⣴⣿⣿⣦⢿⣿⣇⠀⠀⠀⣀⣼⡿⠇⢻⣧⢀⣴⣶⣄⠀⣿⠇⠀⠀⠁⠈⠛⠙⣿⡏⢩⣴⣦⣝⣿⣄⢀⣄⠀⣠⣿⡋
⠀⠀⠀⠀⠻⢿⡿⢋⣼⣿⢻⣿⣿⡿⣿⠛⠋⠀⢸⣿⠸⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⡿⠈⢿⣿⠿⡿⠛⠃
⠀⠀⠀⠀⢠⣶⣶⣾⣿⣥⣾⡿⠋⠀⠀⠀⠀⠀⠈⠻⣷⣬⣭⣵⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣭⣭⣤⣶⡿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿
⠀⠀⠀⣼⡿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠙⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠘⠟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠿⣿⣶⣷⠄⠀⣠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠛⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⣡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠆⠀⠀⠊
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢻⣿⣿⣧⠀⢀⠘⢋⣿⣿⣿⣿⣿⠃⠀⢻⣧⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠈⡃
⠀⠀⠀⠀⠀⣸⣿⠿⣿⣿⣿⢸⣿⣿⣿⣾⡿⡷⠟⠛⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠸⠄⠀⢠⡃⠀⠠
⠀⠀⠀⠀⠀⣿⣿⣤⣿⣯⣤⠈⣿⣿⠙⠋⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠙⣋⣿⣿⣿⠀⡿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡯⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⡗⠿⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠸⡷⠾⠿⠿⢿⡇⠙⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠛⠛⣿⣷⣦⣤⣀⣈⠄⠇⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣤⣴⣶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⢿⣿⡿⠁⠈⠈⣗⣀⣤⣴⣾⣧⣤⣤⠤⠤⠀⠐⢲⠒⠂⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⠟⠁⣼⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡥⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣿⣿⠙⠿⣿⣻⡿⠟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡃⠀⠀⣿⡉⠀⠀⠀⠀⠿⠿⠿⠿⢿⡟⠋⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⡆⠀⡸⣿⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⣿⣴⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠇⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⢀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀
⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄
⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷
⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿
⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⡅⠀⠀⠀⠀⠀⠀⢨⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃
⠀⠀⠀⠀⠈⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣤⣽⡄⠀⠀⠀⠀⢠⣯⣤⣬⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢨⠃⠀⠀⠀⠀⠘⡅⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣄⠀⠀⠀⠣⣀⡤⠋⠀⠀⠀⠀⠀⠀⠙⢤⣀⠜⠀⠀⠀⣠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠉⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠤⢀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠔⢤⠊⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡠⠃⠌⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⢒⠄⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠁⠀⠀⠀⠀⠀⡆⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠀⠐⠀⠀⠀⢰⡀⠀⠀⠀⠀⡠⠊⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠋⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠀⢀⠤⠊⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠓⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠁⠀⠢⡀⠀⠀⠀⠀⡐⠀⠈⠢⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⡄⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⠁⠀⠤⢀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⢐⠂⠂⠠⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢆
⠀⠀⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠑⠠⠀⠀⡔⠁⠀⠀⠀⠀⠈⠐⠂⠄⡀⢀⡌⠀⠀⠀⠘
⠀⠀⠌⠀⠀⣖⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄
⢀⠀⠀⠀⠴⠁⠘⠂⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃
⡄⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠰
⠷⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⠃⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠂⠀⠠⠛
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠶⠤⠶⠶⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠈⠑⠦⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⢱⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠜⠁⠀⠀⠀⠀⠠⠤⠂⣡⠶⣄⠀⠀⠀⠀⠀⠀⡖⠲⡄⢀⢀⡀⠈⠧⢤⡆
⠀⠀⠀⠀⠀⠀⠈⠦⣄⣀⠀⠀⠀⡤⠠⠀⠔⠁⠀⠇⠀⠘⠀⠀⡏⠉⡏⠈⠀⠀⠓⢸⡡⠎⠀⢠⣎⡀
⠀⠀⠀⠀⠀⠀⠀⠰⢎⣁⣀⡀⠀⢦⣀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡷⢄⣵⠀⠀⠀⠀⠀⢱⠀⠀⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢒⡶⢀⣀⣀⡱⢄⡀⠀⠀⠀⠀⠀⢰⠈⣿⠀⠀⠀⠀⠀⢰⠗⠒⠒⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠀⠀⣩⠗⠦⣄⡀⠀⠀⠑⠂⠀⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠉⠀⠈⠉⠉⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄⠀⠀⠀⠀⠀⠀⡼⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣳⠀⠀⠀⠀⠀⢰⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⣏⢘⠄⠀⠀⠀⠀⢌⢙⡞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠀⠀⠀⠀⠀⠈⠉⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣤⣀⣤⣄⣀⣠⣄⣠⣄⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠇⠀⠀⠀⠀⠀⠀⠧⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠉⠿⠟⠃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠴⠄⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡴⠉⠁⠉⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣀⡀⠀⠀⠀⠀⠀⢀⡸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠒⠐⢶⠺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠘⢂⣴⠀⠀⠀⠀⠀⣀⣠⢄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠈⠀⠘⠣⠀⠀⠀⠀⡶⠦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠙⠃⠀⠀⠀⠀⠀⢴⣆⠠⠦⠄⣀⡀⠈⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣼⠆⠀⠀⠀⠀⢸⠟⢿⣶⡄⢠⠤⣶⠄⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠇⠘⣄⠀⠀⠀⠀⠳⡄⢸⡀⠀⣠⠉⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠰⡶⠀⠀⠈⠳⣄⡀⠀⠀⠀⠉⠉⠉⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⡛⠛⠛⠻⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⣀⣀⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡗⠲⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⠀⢀⡀⠀⠀⠀⠀⣟⠁
⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠠⢤⠄⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠚⠁
⠀⠀⠀⢀⠜⠀⠙⢦⡀⣀⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠁
⠀⠀⡰⠋⠀⢀⠴⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⡜⠁⠀⣴⠎⠀⠀⠀⢀⣈⡀⠀⠀⠀⠀⢀⣠⣼⣇⢀⣠⡤⠀⠀⠀⠀⡆
⡆⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠠⠈⠀⠉⠙⣿⣿⣷⠆⠀⠀⠀⠁
⠷⠠⠤⠠⠤⠤⠤⠀⠀⠰⡎⠉⠉⠉⠀⠀⠈⠁⠀⠼⠿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⢶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⢿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠠⠤⢤⡤⠤⠁⠸⠤⣤⠤⠤⠤⢤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⡀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⢀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢤⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣶⣾⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢷⣿⠿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡌⠓⣤⡀⠀⠀⠀⠀⠀⢰⣦⣀⣠⣾⣿⣿⠃⠀⢸⠓⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡔⠉⠙⢠⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⢶⣶⣎⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⡀⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠀⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⢻⣿⠧⡀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣄⠀⠀⣤⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣦⡀⠀⠻⣄⣠⣾⣿⣶⣤⡀⠀⢶⣿⣿⡷⣄⠀⣰⣾⣿⣿⡇⠀⠀⢀⣾⣿⡷⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠛⠞⠹⣿⣿⠀⢹⠇⠀⠀⠀⠐⠒⠶⢿⣿⣿⠈⣿⠀⠀⠙⠉⣿⣿⣇⡤⢗⣀⣼⣿⣿⡴⠋⣸⠁⢸⣿⡿⠃⠀⢀⣨⣿⣿⣀⡼⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⠏⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠞⠁⠀⠀⣀⣤⣿⣿⡏⠀⠈⠁⣸⣿⣿⡆⠀⠉⢀⣾⣿⣷⠀⠀⠈⢠⣿⣿⡅⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠻⣿⣆⣀⣀⣀⣠⣤⠀⣸⡿⢻⣿⠀⠀⠀⠿⣿⣏⠙⣿⡟⠀⠀⢀⣿⠟⣿⡗⠀⠀⣼⡟⢹⣿⠀⠀⢠⣿⠟⣿⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣸⡏⠁⠀⠀⠈⠛⠛⠉⠉⠈⠹⢷⣿⠁⠈⠿⠿⠶⢤⣆⠈⠙⣷⣿⣧⡀⠀⢸⡏⠀⠘⠟⠛⢻⣿⡆⠈⠛⠻⠷⣿⣏⠀⠻⠷⠶⠤⣤⠀⠀⢸
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠛⠀⠀⠟⠁⠙⢷⡀⢸⡇⠀⠀⠀⠀⢸⠇⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠙⠀⠀⢸
⠀⠀⠐⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣼⡇⠀⠀⠀⠶⠿⠇⠀⠀⠀⠤⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠂
⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡀⠂⠀⠁⠖⠒⠛⠛⠛⠟⠁
⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠐⠻⢦⠀⠀⠀⠀⠀⠀⠑⠂⠄
⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡤⠐⢤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⡻⠀⢀⡞⠀⡠⣶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢾⡄⡔⠀⢻⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡿⠃⠀⠀⠈⣧⢁⠀⠀⢻⡞⡆⠀⠀⣠⣤⣶⣦⣤⡤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢦⠀⠀⢀⡤⡞⠋⠀⠀⠀⠀⠀⠀⢸⡆⡆⠀⠀⣷⣷⣶⣾⠏⠙⢩⣿⠟⠟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠸⣶⡺⠟⠉⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣶⡶⣿⣿⡟⠿⠻⣷⠀⠈⠳⣌⢂⠀⠉⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⢸⣁⣀⢀⣀⠀⠀⠀⠀⣰⡿⠿⠊⠋⢻⣿⣿⢸⣿⢂⢀⢡⡀⠀⠀⠈⠈⢢⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠺⠛⠛⠉⠁⠀⢀⣤⣼⣯⡀⠀⠐⠀⡀⢻⣟⣾⠻⣧⠠⠀⠡⠡⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⠀⣆⠀⠀⠀⠀⢀⣼⣿⡿⣿⣷⣄⠀⠀⠀⠈⠉⢉⠣⡘⣧⠠⠠⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⠤⢤⣌⢳⠀⠀⠀⣾⣿⠉⠀⠉⢿⣝⣆⠀⠀⠀⠀⠀⠀⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⢠⢟⡆⠀⢰⣏⡏⠀⢀⣴⣿⡉⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⢀⡟⣸⠀⢠⡟⣸⢀⣶⡿⡟⡩⢷⡱
⠀⠀⠀⠀⠀⠀⣸⢣⠃⠀⠈⣹⣷⣾⣿⡄⠘⡓⡀⠧
⠀⠀⠀⠀⢀⣤⣿⣶⣶⣶⣿⣿⡿⠛⣿⠙⡄
⠀⠀⠀⢀⡾⣿⣿⣿⢂⠙⢝⣿⠣⠨⡿⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠉⢉⡓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣎⠉⠙⢻⠀⠙⣇⠀⠀⠑⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠳⣄⠈⢻⠙⠋⠀⠀⢸⡄⠀⢀⡀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠘⠰⣄⣸⠖⠛⠀⠀⢀⡔⠋⠀⣤⣈⠫⠑⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⠓⢦⡀⢀⣠⡤⠶⠶⠯⣅⠀⢠⠀⠀⡜⢀⠞⠙⠿⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣀⣀⡀⠀⢸⡉⠑⠒⠢⣄⣿⣋⡁⠀⠀⠀⠀⢸⠀⠀⢀⡾⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠶⠶⠺⡍⠉⢉⣙⣓⣦⡴⠋⠀⠀⣸⣓⠏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠙⠚⠁⠀⢀⡈⠉⠀⠀⠀⡿⠛⢉⣁⡉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠠⢀⡀⠀⡀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⢀⡼⢭⡟⠂⠀⠀⠀⠞⠋⠚⠉⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⠤⠤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⣠⡎⠑⣾⡇⠀⠀⠀⣶⠠⠴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡁⢈⣿⣿⡇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⡇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠫⣄⣸⠋⣿⣿⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢳⠀⠀⠀⠀⠀⠠⠚⠉⢳⡠⠞⠹⣀⠟⡿⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠏⡼⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡞⠁⠀⠀⠀⠀⠀⠈⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⡤⠴⠺⡇⠀⠀⠀⠀⠀⠀⠀⠰⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡰⣎⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠈⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠⣖⡛⣇⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠖⠚⠋⠁⢀⡠⠴⠒⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠⠜⠻⡄⣠⠞⡉⠉⠉⠉⠁⠀⠀⢀⣀⡤⠖⠊⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠞⢡⡖⠒⠺⢥⡀⢰⡀⢀⣀⡤⠔⠚⠉⣁⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠖⠛⠫⠽⠆⢹⠆⠉⠉⠀⠀⠰⡞⠉⠀⠀⠀⠀⠀⠀⠀⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢈⡿⠥⢛⣀⠀⠘⠠⠤⠤⠔⠒⠚⣿⣦⡀⠀⠀⠀⢠⠆⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠲⠤⣀⡘⠦⠖⠋⠈⣇⣀⠤⠤⠤⠤⡤⣤⡸⠿⠿⠦⣄⣠⠋⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⢆⠀⡞⢩⠟⠒⠲⠤⠤⣄⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣁⣀⡤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠋⠉⠀⠀⠈⢻⡁⡎⣀⣀⣠⣤⣤⠤⠤⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡤⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣖⠉⠉⢻⢳⠒⠤⣤⣤⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠘⠋⠉⠈⠀⠀⢀⠠⢽⡦⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠴⠴⠄⠤⠄⠛⠛⠉⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⢀⣀⣀⠀⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⣀⡻⣿⣿⠟⣁⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣤⣬⣛⣛⣥⣤⠀⠀⠀⣠⣬⣛⣟⣫⣤⠰⣿⣿⣿⡆⣠⣬⣻⣟⣫⣄
⠀⠀⠀⠀⣾⣿⣿⣶⣶⣿⣿⣷⠀⠀⠀⣰⣶⣭⣿⣵⣶⡽⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⢿⣵⣽⣛⣯⣶⣝⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⠿⠛⠁⠀⠀⠻⠿⣿⣿⠿⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠛⠉⠀⠀⠀⠙⠛⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⠀⠀⠀⣴⣿⣶⠀⠀⠀⢰⣾⣷
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣴⣽⣿⣵⡄⠀⢠⣮⣿⣯⣦⠀⢠⣬⣿⣯⣦⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⢠⣼⣿⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣺⣿⡛⠛⣻⣿⡛⠛⣻⣿⡀⠀⠀⠀⠀⠉⠉⣵⣾⣽⣷⣮⠉⢹⣾⣽⣷⣾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡗⠸⣿⣿⡿⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠀⠸⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⣦⣄⡀⢰⡟⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣭⡶⢷⣦⠉⠻⣾⡷⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⣦⣄⡀⢠⡿⠛⢻⡄⠀⠀⣴⠟⣁⣀⣀⡈⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣦⣉⠻⣾⡷⣦⠾⢃⡘⠷⠶⠾⠋⣉⣉⠻⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⡈⢻⡆⠀⠈⠻⠶⠶⠾⠛⠋⠙⠷⢶⣶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⣠⡾⢿⣄⠛⣛⣿⡇⠀⠀⠀⠀⣴⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⡇⠀⣿⠀⠀⠙⠛⢋⣉⣤⣤⣶⠶⣶⣿⣄⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣿⠆⠀⠀⢰⡟⠉⠁⣀⣀⣤⡙⣷⣭⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣾⠋⠀⠀⠀⠀⠈⠻⣦⣀⠙⢿⡝⢷⣮⣵⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢠⡶⠟⠛⠛⠷⣮⠙⣷⠈⣿⢀⣴⠾⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠹⡇⢿⠀⠿⢼⠇⠀⠀⠀⠀⠈⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣄⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⢡⣞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠
⠚⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⠦⡁⠀⠀⠀⠀⠀⠀⠹⡿⠀⢠⣶⣶⣶⡎⠀⠀⠀⠀⢀⡴⢻⡄⢀⠀⢀⡼⠋
⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠘⣏⡠⠁⠀⠀⠀⠀⠀⠀⠙⠶⣜⠀⠙⠛⠋⢢⠀⠃⠀⡠⠋⠀⢀⠷⠘⠀⠘⡄
⡟⢀⠶⣄⠀⠀⠀⠆⡀⠀⠸⢿⣛⣀⣠⣾⠃⠀⣀⢀⡀⠀⠀⣠⠞⣖⡤⠄⠲⠂⢀⡠⢤⣄⠈⢧⡀⢠⠃⠀⠀⡇⠀⢰⡾
⢀⠎⣀⡼⢃⠤⠀⣀⠉⠀⠀⠠⠥⣷⣶⣋⡀⢺⣹⡏⡋⠠⠤⢘⡷⠻⠁⠀⠏⠀⠑⣄⠖⠩⠄⠀⠻⣼⠀⠀⠀⢳⣰⠟⢣
⣼⠞⠁⠀⠀⠀⠀⠡⣤⠀⠀⠀⠘⡇⠙⠃⠒⠀⠻⠇⠷⠖⠀⠈⠀⡆⠱⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠳⡎⠛
⠀⠀⠀⣾⡄⣀⠀⠀⠠⢤⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠿⠛
⠀⠀⠀⠘⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⢀⠀⠂⠂⠀⠀⠀⠘⠀⠀⠀⢀⠘⠁⠀⡇⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠂⠀⠀⠀⠀⠐⠃⠐⠤⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡀⠀⠀⠀⠰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠄⡤⠀⠐⠂⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠉⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⡆
⠀⠀⠀⠈⠓⠒⠒⠒⠓⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠚⠁
⠀⢠⠒⠀⣀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠫⠿⠛⠛⠛⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠿⣟⠻⢦⡄⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⠒⡄
⠀⠈⠉⠉⣀⣤⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠉⠈⠁
⠀⠀⢠⡾⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠈⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠉⠛⠲⠦⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡤⠤⣄⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣠⠴⠞⠳⢦⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢀⡞⠉⠀⠀⠀⠀⠈⠳⠀⠀⠀⣠⡿⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣤⣄⣀⣀⣠⣤⡤⠶⠞⠛⠉⠀⠀⠉⠙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠈⠻⠿⠏⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⡏⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⢀⣠⣴⣶⣶⣶⣦⡘⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣛⠿⢿⣿⣿⣿⠿⢿⡇⣿⡿⣋⣭⣬⣭⡻⠟⣩⣥⣭⣝⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⡇⣿⢱⣿⣿⣿⣿⣷⡞⢿⣿⣿⣿⣇⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠛⠛⠉⣴⣿⢸⣿⣿⣿⣿⣿⢣⣎⠿⠿⢿⡏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⣿⣆⢳⣶⣶⡙⡟⣸⣿⣿⣿⡿⣱⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⣭⡝⣿⣷⣌⠻⣿⣤⣿⣿⠟⣫⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⢸⣿⢠⣤⣤⣤⣤⣤⡄⣿⡇⣿⡇⣛⢻⣿⣿⣮⡙⢟⣵⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⢸⣿⠈⠛⠛⠛⠛⠛⠁⣿⡇⣿⡇⣭⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⣛⣡⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⣀⣤⣀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⣸⣶⣿⡿⢷⣼⢿⣿⣿⠎⡷⣶⣵⢆⣙⡦⢛⠳⣮⡸⣄⡽⣼⣿⣿⣟⣿⣿⡿⣭⣙⣾⣷⣤⣿⠟⠸⣻⣿⣛⣿⡷⠿⣾⣷⠁
⠀⠀⠀⠀⠀⣸⣿⢿⡟⢂⣤⣆⣻⣿⣧⣀⡌⣹⣬⡞⢯⣦⣏⣠⣯⡗⣭⡙⢻⣿⣿⡟⠸⣿⣗⠈⢹⣿⡙⡯⡙⡦⢔⠱⡅⢻⣿⣟⢶⣿⠿⡆
⠀⠀⠀⠀⠀⢹⣾⢷⣝⠾⣧⣻⣏⣿⡝⡱⣿⢃⣾⡟⠟⢻⡝⣿⣻⣯⣎⣁⣿⣷⡏⠛⣶⡄⣈⠺⣿⣿⠄⠑⠣⠭⣘⣦⢰⡟⣺⣿⣈⡻⣤⡇
⠀⠀⠀⠀⠀⢺⣯⣶⢈⣝⢛⣽⣦⣿⠛⢿⣾⡏⠉⠀⡀⢸⣱⣿⣿⠛⠾⣯⣿⣯⢺⡇⠉⣴⡍⢛⠓⠺⣦⢼⢳⣤⣤⡈⠁⢸⣇⣯⠽⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣽⢫⡯⡿⣿⡷⠺⣿⣿⣀⠀⠀⣠⣿⠿⣿⣛⡂⡸⣿⢜⣯⣿⣶⣿⡓⣾⣦⣴⠻⣆⡵⣿⣿⡾⠄⠈⢿⡇⢤⡼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠿⢻⣿⣿⣧⣿⣿⣾⠾⠁⠈⠏⠀⢈⣡⡖⠉⢋⣁⣹⣋⣿⡏⠛⣹⠿⣧⡘⣫⠿⢻⣿⡟⢿⣼⡗⠾⡆⠉⠿⣮⣥⡜⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣿⣞⢹⣿⣷⣶⣷⣶⣶⣶⢿⣿⣧⡀⠀⢹⣻⡏⠉⣠⣴⡟⣓⡿⢸⡿⠀⡀⠙⢿⣿⠿⣏⢐⢳⣤⡀⠁⢏⣇⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣛⣻⣿⣻⣿⣿⣿⣿⣿⣿⣤⣤⣦⡀⠀⠀⢈⣿⣏⣹⣿⣼⣓⠩⠁⣿⡄⠀⠉⠀⠺⣿⣶⣿⠖⣽⢏⣿⡄⠘⢿⡅
⠀⠀⠀⠀⠀⣶⢻⣿⣿⣿⣿⣿⣯⣽⡻⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⣜⡿⡖⢮⠞⠛⢃⣾⣷⣦⣄⠀⣸⠿⠯⢋⣿⡏⣾⣿⠱⠀⣼⠇
⠀⠀⠀⠀⠀⣹⣿⢭⣽⣿⠿⣿⣛⣷⠁⠐⠀⠠⠀⠀⠈⠉⠉⠁⢸⡿⣿⣿⣏⣹⣌⣏⣛⠛⠉⠈⠈⠉⠀⠀⣇⣴⣶⣿⢿⣳⣿⣿⡿⢠⣿⣡
⠀⠀⠀⠀⢈⣿⢠⣎⣡⣿⣯⡓⡽⣄⣀⠤⠀⠰⠀⠀⠀⠀⠀⠀⠈⣿⣯⣯⣷⣌⡿⠟⠁⠀⣀⡉⠈⠁⠉⢹⡏⢀⢃⢹⣿⣷⣿⣿⡄⢿⣯⡏
⠀⠀⠀⠀⠀⠙⣪⢀⠹⣿⡛⢻⣽⠿⠟⢶⣴⣀⣀⣀⠀⡀⢐⡂⢀⣯⣉⣹⣉⣀⣀⣤⡴⠊⠉⠃⠀⠀⠀⡼⠀⡘⠁⠨⣟⠷⣿⡇⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠼⠛⠉⠁⠉⣉⡠⣶⣾⣞⣿⣿⣿⡇⠈⣿⡏⠉⢹⣿⣿⣿⣟⡿⣭⢟⣍⣿⣿⣷⠂⠄⠀⠊⠉⠀⢻⡄⠀⠙⢦⣬⡏⠐⢰⣯⡅
⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣇⠀⣻⣿⣾⣿⣿⣿⣷⢀⡿⢿⠀⠸⠿⠿⢿⡿⠿⠾⠳⠾⠿⠿⢿⠇⠀⠀⠀⠀⠀⢘⣟⢦⣀⣈⣯⣿⡘⡎⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣠⡽⠛⣿⡏⠉⢙⡻⢤⣷⠀⠞⢻⡄⡀⠠⢙⡀⠀⠠⠶⢶⣶⠎⠀⠀⢀⡀⠀⠀⢈⡏⠉⠛⠓⠉⢻⠃⣤⠘⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣤⠶⠾⣯⣄⢸⠟⠛⠛⠋⠀⢸⣿⣇⠀⡚⠛⢻⣆⠀⠀⢀⠃⠀⠀⠠⠊⠇⠀⠀⠸⠙⠳⠶⣀⠠⢼⣷⠋⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠈⠀⠀⠀⠀⠈⠙⠓⠶⢦⣄⠠⣀⡉⠁⣤⣣⣖⠁⠈⠀⠀⠞⠀⠰⠎⠀⠀⡄⠀⠀⠇⠀⠀⠘⠻⢦⣠⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠌⣼⣷⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡠⣇⢾⣿⢁⣴⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠚⠄⠟⠀⠴⢿⣏⣰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠂⠉⡔⠳⣶⣿⣿⡏⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣾⣿⣿⣥⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣎⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⠀⢈⢀⣙⣋⣩⣿⣿⣿⣿⠟⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢶⡛⠿⠊⠁⠈⣰⡿⠯⣤⣸⣿⣿⣿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣥⡴⠀⡂⠀⠉⣄⣀⣾⣿⣟⣉⣁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠈⣳⣊⣿⣿⣿⡿⢲⣴⢟⣿⣿⠟⠛⣯⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⡰⣯⣿⠀⣿⣹⣿⣿⣿⣿⣿⣿⠀⣠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠻⠿⣶⣿⣿⠿⠿⠛⠋⠁⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣤⣶⣶⣶⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠋⠙⣿⡇⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠆⠀⣿⠇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠢⣄⡀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠇⣀⠀⠀⠉⠛⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⠥⠒⠒⠒⠢⡉⢦⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡄⠈⠁⠂⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣠⡤⠰⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⣀⠰⠃⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⡀⠤⠔⠊⠉⠀⣠⠔⠒⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠸⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠁⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠉⠐⠢⠤⣀⠀⢀⣰⠄⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠓⠰⠆⠀⣀⠀⠀⠈⠀⠈⠀⠀⠸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣦⣤⣤⣤⣤⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡟⠛⠿⢿⣿⣿⣿⣿⡿⠀⠉⠁⠀⠉⠒⢄⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠙⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⢀⠎⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢡

⣀⡤⡴⠊⠣⡀⣀⡀⠤⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⡄
⢻⡄⠘⣶⡞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢡
⠀⢻⡄⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⡄
⠀⠀⠻⣆⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢡⡀
⠀⠀⠀⠹⣦⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠪⣳⣤⣴⠖⠉⢁⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠋⢉⣠⣴⠾⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣾⣋⣉⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⢈⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⡄⠙⣿⣿⣿⣷⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢿⣿⣿⣿⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⠁⠀⠘⣿⡇⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡋⠉⠀⠀⠀⠹⣿⣿⠿⠃⠀⠀⠀⢀⡟⠀⠀⠀⢻⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⢻⡄⠀⠀⠀⠰⢎⠁⠈⠒⠂⠀⢸⣧⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⡄⠀⢸⣷⡀⠀⠀⠀⠠⠐⠲⠄⠀⢠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⣼⣿⠃⠀⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠘⢿⡀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⠶⢦⣄⡘⣧⠀⢀⣾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⣠⣄⠀⠈⠙⠻⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠉⢻⣿⣦⡀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⣤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣄⡿⠿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠋⣹⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠸⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣰⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓
⣤⣤⣤⣤⣤⣴⣶⣶⣦⣈⣛⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⢳⠀⠀⡠⠄⢿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠸⡆⠉⣹⠀⡜⠀⠀⠈⡦⠊⠈⠱⡀⠀⢠⠊⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠻⠖⠁⠀⠈⢢⠀⢰⠀⠀⡀⠀⡇⠀⣊⠑⣠⣎
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢢⠀⡀⠀⠀⠀⣄⠀⠳⡏⢀⠀⡇⢠⠃⡔⠞⠉⠀⣁⠀⢀⠐⠂⠲⡄
⠀⠀⠀⠀⡠⠤⢤⣸⠤⡴⠁⠇⠁⢄⠀⠀⡏⡧⡀⠱⡟⢠⠇⡼⠀⡧⠂⢀⠀⡏⠀⢘⢄⣀⡀⡇
⠀⠀⠀⢸⠀⠀⢠⢻⢢⢨⠏⠀⣠⣀⠑⠲⢇⡇⠱⣀⡅⡘⢀⡧⠊⣣⢔⣋⣰⠥⠆⠀⣉⠈⣨⠁
⠀⠀⡔⠉⢂⣤⠎⣤⣀⡉⠀⠠⣇⠀⠑⠢⠴⣑⣤⣶⡿⠟⢍⠉⠭⠤⠳⠤⢤⠤⠄⣼⡀⠀⠈⡆
⠀⢸⠁⠀⠀⠀⠀⢀⡨⠐⠒⠛⠊⠀⠀⠈⢉⠽⠫⡀⠕⢶⡰⣉⠒⠧⢀⣀⡎⠦⣀⣡⡭⠦⢼⠃⠈⢆
⠀⠈⢇⠀⠀⠀⢀⡎⠀⠀⣼⡤⠤⣔⡲⠚⣣⠊⢠⢃⠘⡆⠉⠊⠑⢦⡀⠈⠉⠉⣵⠋⠐⢄⠀⠀⠀⡸
⠀⠀⠈⢆⠀⢰⢡⢣⠖⢉⠭⠔⠈⠁⣠⡞⡡⢤⠃⠈⡄⠘⡄⠀⠘⣄⣙⣯⣐⠒⣿⡄⢀⣸⡄⡀⣰⠁
⠀⠀⠀⠘⡗⢶⢸⡇⠀⠀⠀⠀⣠⢊⠎⡜⠀⠀⡇⠀⢱⠀⠘⢄⢰⠋⠀⠈⡆⠀⠀⠀⠀⣇⡇⣧⠇
⠀⠀⠀⠀⡇⠈⠄⢷⣄⠀⠀⠀⣧⡇⠀⢧⣀⣠⠇⠀⠈⡆⠀⠈⠛⣦⣀⡴⡃⠀⠀⠀⠀⠉⢀⡏
⠀⠀⠀⠀⡇⠀⠰⡆⢹⠓⠂⠉⢸⡇⠀⠀⠉⠒⠈⠐⢄⢸⡀⠀⠀⠈⠉⠀⠑⢦⠀⠀⠀⢹⠋⠇
⠀⠀⠀⠀⠃⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠸


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣀⣀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡄⠀⠀⠀⠀⢨⣯⣿⣿⣿⠀⠀⠀⠀⠀⢠⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣝⢿⣿⣷⡀⣠⡖⢲⣾⣿⣿⣿⣿⣷⡖⢲⣄⣰⡟⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠹⣎⢿⣿⡷⢻⠛⠛⠉⠱⠎⠀⠱⡉⠛⠛⠟⢉⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠛⢿⡟⢺⡷⢠⡥⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⡛⣽⡷⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡌⣁⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⣀⣀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⣆⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣽⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⢰⣰⣤⡖⢦⣤⣢⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠰⣏⣩⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠶⢶⣶⣶⣾⣻⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠰⣄⣰⠈⢃⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⢤⡾⠟⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡁⠁⠀⠀⠀⠀⠀⠀⠀⢾⡁⢀⣺⡇⣿⣿⢘⢘⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠈⢘⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢫⣿⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣄⠡⠉⠩⠈⡭⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣪⣿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣍⣽⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠩⠛⠷⠞⠛⠉⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⢐⣏⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠸⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠊⠉⠉⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢠⡌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠲⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠶⠿⡟⢘⡓⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⣥⡻⠷⢴⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣾⣿⡷⣼⡴⢦⣤⡰⢂⣀⡐⣥⡴⢴⣦⣈⢿⣿⣿⡄⠈⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⣾⣿⡿⠁⠙⠷⠾⢻⣿⣿⣿⣿⠟⠷⠾⠃⠙⣧⢻⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⠂⠀⠀⠀⠀⢘⣛⣿⣿⣿⠀⠀⠀⠀⠀⠘⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠉⠉⠡

⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣼⡇⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣿⠀⣠⡀⠀⠀⢠⡄⡆⢀⡄
⠸⡇⢿⡄⢸⣿⡄⠀⠀⠀⠀⠀⠸⡟⣿⢸⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣾⡏⢠⣿⡇⠀⠀⢸⢷⡇⢸⣿
⠰⣿⢾⣇⢸⡟⣿⡀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡿⠀⡾⢻⡇⠀⠀⠈⠘⠃⠟⠘
⠀⠙⠘⠟⠀⠃⠀⠀⣀⢠⡄⢀⠀⠀⠀⠀⠀⠐⣀⠤⠔⠗⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⠁⣼⡇⠀⠀⠀⠀⠊⠁⠄⠀⠀⣀⠠⠄⠒⠠⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣜⣧⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠇⡿⢠⡟⡧⠀⠀⠀⢀⡀⠀⣠⠐⣉⠀⠀⠉⠀⠂⠀⣀⠀⠀⠉⠐⠢⢄⡀⠀⠀⠀⠈⡏⢿⠀⡇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢅⠚⠰⠈⠀⠲⡄⠀⠀⠈⠒⠀⡀⠀⠀⠀⢀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠈⠛⣆⠟⠀⠀⠀⢀⣀⡀⠙⠠⡀⠈⠁⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠠⠄⠁⢀⠀⠀⠀⢉⣭⣍⡉⠀⠈⠢⠀⠐⡄⠈⢁⠛⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠑⠄⣀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠱⡀⠀⡀⠀⢣⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢠⠯⡁⠈⣄⡈⠨⠀⡇⠀⠀⠀⢠⣿⣷⣦⣤⣉⣉⣩⡽⠀⠀⠀⢀⢠⠔⡠⠸⠀⠇⠀⠀⡇⠀⢸⠄
⠀⠀⠀⠀⠀⠋⠐⣈⠔⠈⠀⣠⠀⢻⡀⠀⠀⣿⣿⠿⢿⡿⣿⣿⣿⡇⠀⠀⠀⠈⠚⠠⠁⠨⡇⡆⠀⠀⡇⠀⠋⠂
⠀⠀⠀⠀⠀⠑⢅⠁⠀⠀⢴⡃⠀⠀⠳⡀⠀⠻⣿⣤⣀⣀⣼⣿⠟⠀⠀⠀⠀⠀⠀⡠⠒⠢⠈⡇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠣⠐⠈⠙⢄⠀⠀⠈⠢⡀⠈⠉⠙⠉⠉⠀⠀⠀⠀⣀⣀⡴⢄⣀⣀⣀⡤⠓⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⢳⠒⠒⠈⠉⠒⠲⠤⠀⠀⣀⠖⠚⠛⠩⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠂⠈⠉⡋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠤⠀⠀⠀⠀⠀⡰⠀⠀⠀⢀⢀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠰⠉⠉⠀⠉⢴⠂⠉⠀⠈⠀⠅⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠆⠀⠀⠀⣀⠀⠀⢀⣀⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢤⣀⣀⣀⣀⣀⣀⣀⣉⣀⡸

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣤⠈⠄
⠀⠀⠀⢔⠐⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡱⠀⠀⢠⣿⣿⠿⠛⠉⣇⠸⡀
⠀⠀⠀⣀⣽⣿⠦⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠱⠀⠀⠘⣿⣇⠀⠀⠀⡇⠀⠈⡄
⠀⠀⢠⣿⣿⡁⠀⠈⠢⡬⠒⢄⠀⠀⠀⠀⠀⠀⡄⢣⠀⠀⠈⢻⣷⡖⠈⢳⠐⢄⡁
⠀⠀⠀⠙⢿⣥⣀⣀⠠⠧⡄⠀⠵⢄⠀⠀⠀⠀⠡⠘⠁⠀⠀⠀⡜⠁⠈⠁⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠔⠁⠀⠀⠀⠈⠢⡀⠀⠀⠙⠢⠄⠐⠘⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠔⠁⠐⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠐⡀⢀⠤⠒⠐⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠀⠀⠀⠀⠀⠘⡳⠀⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠃⠂⠈⠉⠢⡀⠀⠀⢀⡠⠄⠂⠁⠀⢡⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠎⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⡘⠀⠀⠀⢀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡌⠀⠀⡠⠊⠉⢄⠀⠀⠃⠀⢀⠁⠀⡠⠂⠁⠈⠢⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠰⠢⠄⠀⠀⡘⠀⢀⠎⠀⠀⠀⠀⠡⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠈⠢⡀⠆
⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⡰⢀⠔⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠐⠦


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⢀⣀⣀⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣭⣯⣭⣭⣽⠷⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⡀⠀⠀⠀⣠⣾⡷⢶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠋⣿⠀⠀⢿⠛⠦⠴⠞⠉⣰⠇⢸⡇⠀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣿⠀⠀⠈⠳⣄⣀⣀⣴⠟⠀⠀⣧⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⣴⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣛⠛⣷⠀⠙⢷⣄
⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⡴⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣥⠀⢹⣦⣀⡀⠙⣦
⠀⠀⠀⠀⣾⠁⠀⠀⣰⠟⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠶⠄⣻⠀⠉⠙⠳⠸
⠀⠀⠀⢠⡏⠀⠀⣰⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠺⣇
⠀⠀⠀⣾⠁⠀⢠⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⣄
⠀⢀⣴⠏⠀⢠⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⠶⠾
⣰⠟⠁⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⣿⠀⠀⢀⣴⣈⡇⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠹⣦⡀⢾⡿⠿⠁⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣿⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡴⠛⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⡀⠀⠀⠀⢀⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠙⠛⠛⢳⡼⠃⠀⠻⣦⠴⠶⠞⠛⠛⠶⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢠⡏⢙⡆⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣴⠟⠀⣾⠁⠀⠀⠀⢸⡆⠀⠀⢰⡶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣿⠀⠀⠀⠀⢨⡗⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣿⠀⠀⠀⣰⠏⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡀⠀⢠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠋⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣠⡾⠋⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡾⠋⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣴⠟⠀⢀⣀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠀⣠⡾⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠣⠤⠔⢑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠒⠁⠀⢀⡀⠀⠈⠢⠤⣄⣀⠖⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡔⠁⠀⠱⠀⠀⡴⢪⡇⢸⡱⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⡀⠀⠘⠤⠤⠊⠀⠀⠳⣄⣘⠖⢤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⢈⡆⠀⣀⡀⠀⡴⢊⡶⠝⠀⠁⠉⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠕⠈⠉⠁⠘⠵⢮⠀⠈⠆⠂⢀⠅⠀⡎⢉⠆⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⡐⠉⠑⠀⠀⡎⠀⠀⠀⠀⠈⠑⡄⣈⡁⡠⠬⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠤⠀⢠⠃⠀⠐⠤⠔⠁⢀⢉⠆⠀⠀⠀⢀⡀⣍⠀⠑⠁⠀⠀⠀⢠⠤⠋⠑⠊⡆
⠀⠀⠀⠀⠀⡌⠜⡠⡾⡍⢑⠀⡤⠤⣀⡅⢔⢄⠀⠀⠀⠘⣃⡬⠀⠀⠀⠀⠀⠀⡴⠀⢎⢉⠆⢨⠃
⠀⠀⠀⠀⠀⢣⠑⢴⠁⠟⠀⠉⠀⠀⠀⠀⠈⠪⡒⠀⠐⠂⢸⡆⠒⠤⣀⣀⢔⡲⠀⢇⠤⣀⡔⠚⢠⢖⣄
⠀⠀⠀⠀⠀⠀⠱⡀⢸⢈⡑⠠⢐⡩⠃⠀⠀⠀⠀⠀⠘⠀⢸⠁⠘⠐⠒⠒⠁⠀⠀⠀⢔⠤⡀⠀⠳⣰⡃
⠀⠀⠀⠀⠀⠀⠀⢃⡼⢠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠆⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⠭⠥⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡳⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢐⠠⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⣉⡡⠂
⠀⠀⠀⠀⠀⠀⠀⡠⢃⠄⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⡄⢈⢹⠰⡄
⠀⠀⠀⠀⠀⠀⣎⠔⠁⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠘⡈⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠃⠀⠈⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠶⠿⠿⠟⠛⠛⠻⠿⠿⠶⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠉⠉⠉⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠉⠉⠛⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠿⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣼⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣤⣶⣶⣷⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠇⠀⠀⠀⢀⣠⡾⣿⡄⠀⡼⠟⠋⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠙⠻⢧⠀⠀⣤⣄⠀⠀⠀⠀⠀⠸
⣿⠀⠀⢀⣴⠟⠋⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠙⢷⣦⡀
⣿⡆⢠⣿⠁⠀⠀⠀⠸⣷⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡏⠀⠀⠈⠻⣦⡀⢰
⢹⡇⠘⣿⡀⠀⠀⠀⣰⡿⠀⢸⣟⠻⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠟⣿⠇⠀⣾⡇⠀⠀⠀⠀⢹⡇⢸
⠸⣷⠀⠈⠻⠶⣶⠾⠛⠁⠀⠈⣿⡀⠀⠀⠈⠉⠛⠛⠷⠶⠶⣶⣶⣤⣤⣤⣴⣶⣶⠶⠶⠾⠛⠛⠉⠁⠀⠀⢰⡿⠀⠀⠘⣷⣄⣀⣀⣠⡿⠃⣾
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠉⠛⠛⠉⠀⢰⡟
⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣴⠶⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣴⠟⠛⠻⢶⣦⡶⠟⠛⠻⣦⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⢶⣶⣤⣶⡶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⡻⢷⣶⣶⣶⣶⣶⣶⣶⣶⣾⢛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⠀⠒⠂⢈⠁
⠀⠔⠊⠀⠀⠀⠀⠀⠀⠁⠀⠐⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⡒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠁⠂⢀
⠀⠀⢰⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠺⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⢰⡟⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠻⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⡚
⠀⠑⢌⠀⠁⠀⠀⠀⢈⣉⣁⣈⡐⠀⠀⠀⡪⠈⠁⠀⠀⠀⠀⢨⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠊⠂⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢐⠀⢤⠤⠄⠀⡰⠛⠉⠀⠀⠀⠀⠀⢀⣤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣄
⠀⠀⠀⠀⠀⠀⠰⣁⠈⠙⠒⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⢃⠹⡀⠠⠀⠀⠀⠀⣒⠲⠶⠄⠀⠀⠀⡰⢁⠍⠉
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠄⠂⠄⣀⠈⠀⠁⢀⠠⢐⡨⠊⠁
⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⠬⣀⠧⠄⠒⠁
⠀⣯⢷⣀⠀⠀⡼⠷⠰⠦⠤⠄⠄⠐⡀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠘⠖⠀⢠⠄⠄⢀⡀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠐⠁⠀⠀⡠⠊⠑⡤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠤⠀⢰⡀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢄
⠀⠀⡀⠆⠣⣝⢎⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠂⡏⡟
⠀⠈⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠠⠄⠒⠊⠉⠉⠉⠉
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠤⡀⠐⢄⣠⢄
⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢎⠔⠋
⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠁⠋⠭⠕⢒⣂⣐⣒⡈⠥⠭⠐⠂⠙⠳⢄⣀⣀⣐
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⡿⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟⠀⠠⡀⠙⠿⠋⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⣀⢂⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⠑⠀⠁⡌⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠄⠀⣀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⣀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⡸⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢁⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡈⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠠⠆⣀⣀⡀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⣌⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢰⣿⡿⠁⠀⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡙⠻⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢸⣿⠃⠀⠀⢻⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣦⡉⠻⣿⣿⣿⣿⣿⡀
⠀⠀⣾⡏⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠙⣿⣿⣆⠘⣿⣿⣿⣿⡇
⠀⠀⣿⡇⠀⠀⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⡄⢹⣿⣿⣿⣷
⡆⠀⣿⡇⠀⢠⠓⣼⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠘⣿⣿⣷⠀⣿⣿⣿⣿⡆
⡇⠀⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⢸⣿⣿⣿⣷
⠇⣰⣿⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⣿⣿⣿⣿⣷⣄
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⢻⣿⣿⣿⣿⣿⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠸⣿⡿⠟⠉⠀⢀⠁⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣶⡿⠟⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠀⠀⠀⠠⢤⡤⣖⠂⠀⠀⠠⠄⠴⠶⠦⠤⠀⠀⣠⠀⠀⠀⠠⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⣀⣤⠄⠀⠑⠀⠃⢀⠁⢬⠀⠀⠉⠀⠀⣿⣿⣿⣿⡆⠀⠀⢀⣴⣶⣾⣃⡌⠀⠀⣤⣄⣾⣾⣷⡖⠐⢀⣀⢀⠀⠀⡀
⠀⠀⠀⠄⠀⠀⠈⠀⢠⡽⣷⣿⣷⡄⣄⠀⠀⠀⡄⡀⠀⠀⠐⢣⣐⣒⢛⡁⠈⠛⠫⡄⢀⣩⢿⡭⣿⣷⣄⡘⢷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠿⠛⠁
⠀⠀⡀⣀⡀⣂⡀⠀⠈⠉⠙⢻⣿⣷⣿⡑⠄⠀⠈⠐⣀⢒⡊⠳⢦⣬⣽⣲⣦⠀⠀⠀⠀⠙⠷⣾⠛⣿⣿⢥⣿⣿⣿⣿⣽⣿⣿⡿⡟⠛⠋⠁⠀⡀
⠀⠀⠁⠈⠙⢿⡗⢀⣀⠠⠺⣿⣿⡿⣿⣆⣐⡀⠚⣤⣞⢎⣻⠟⠋⠉⠉⠁⠀⠀⠀⣀⣀⠀⢈⠊⢻⣿⣿⣟⣛⣿⡿⢿⣏⢿⣿⢾⣿⣿⣵⣾⣶⡀⠀⠂
⠀⠀⠀⡀⣀⣠⢷⣈⣵⣾⣷⣿⣻⣿⣿⣿⣿⣽⡿⠿⣆⣠⣼⣶⣤⣤⣤⣤⣶⣶⣿⣿⣿⡆⠀⠀⡀⣿⣿⣾⣿⣿⣿⣿⣷⣿⣧⣿⣿⣿⣯⣿⣿⣷⣶⣶⣦⣄
⠀⠀⢁⠈⠵⡄⢘⣛⢘⣿⣿⡿⠯⢿⣿⡍⢉⢽⣆⣐⣋⣽⣮⢻⣮⣭⣶⡈⠉⠉⠉⠉⢿⣿⣶⡷⠛⠙⢿⣿⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢀⣀⣁⠉⠾⢿⣯⣭⣭⣯⣭⣍⢿⣆⣔⣀⣉⣁⣚⣳⣾⣷⡙⢿⡟⠳⠎⠉⠉⠀⠈⠛⢿⡷⣄⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⡻⣿⣛⣾⠿⢿⠟⠉
⠀⠐⠛⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠙⠛⠛⠛⠋⠉⢡⡿⢆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓
⠀⢀⣀⣀⣀⣘⣻⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣀⢑⠒⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠉⠓⠟⠛⠓⠐⠛⠾⠶⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣭⠹⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠒⠂⠂⠐⠂⠐⠀⠂⠀⠀⠀⠀⠐⠒⠀⠀⠂⠀⠒⠀⠀⠀⡂⡀⠓⢂⡐⠂⣀⣀⣂⡀⠂⠒⠐⠀⠂⠐⠒⠐⠒⠂⠐⠒⠀⠀⠒⠀⠒⠒⠒⠒⠒⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠉⠁⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡠⠚⠁⠀⡏⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢂⠘⡶⠉⠀⠀⠘⠃⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠑⣤⡟⠲⠀⠀⠀⠀⣀⣀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⠠⠤⠀⢤⣤⠁⠀⠀⠀⠀⡞⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠋⠁⠀⠀⠀⠘⠀⠹⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢩⠁⠒⠛⢛⠲⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⣀⡄⠐⠒⡇⢀⡴⠃⠀⠀⠀⠣⡀⠀⡸⠀⠲⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠏⠉⠡⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢃⠘⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠘⡄⠀⠑⢄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠣⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⡇⠀⠀⠀⠀⠀⠀⡰⠀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⠀⢰⠀⠀⠀⢀⡰⠊⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡄⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠰⡀⠀⠀⠀⡆⣠⠔⠉⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢢⠀⡘⠑⠂⠤⠄⢀⣀⣀⠀⠀⣠⠆⠀⠀⠱⠀⠀⠀⠉⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠱⣄⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠤⣬⡷⢀⣀⣀⣀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠋⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠈⠙⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠒⠒⢲⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢇⠀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠑⢄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠎
⠀⠀⠀⠀⢠⠏⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠾⠶⠻⡆
⠀⠀⣠⠒⣇⣀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠈⢦
⢠⡖⠉⠈⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢨⠴⠂⣀⠀⠀⠑⢤⡀
⠣⣝⠦⡀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠵⠶⣤⣭⠆
⠀⠀⠑⢌⣦⡀⠈⢆
⠀⠀⠀⠀⠉⠻⢦⣌⡧
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣿⣿⣿⡟⠋⠉⠉⠉⠉⠉⢿⣿⣿⣧⠀⠀⢠⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠠⠏⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣆⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡟⠉⠁⠀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣽⣿⣧⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠹⣿⣿⣷⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠛⠙⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⣠⣶⣴⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣹⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⠀⡀⠄⠐⠀⣰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠃⠀⠀⠀⠠⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠢⠄⡀⠀⠂⠀⠀⠀⢀⠠⠞⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⢀⠀⠀⠋⡏⠀⠀⠘⡍⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣂⠁⠀⠈⡈⠀⢥⠀⠀⠀⢠⠀⠀⢴⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡔⠁⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠈⠀⠀⠀⡀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣂⡴⠣⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⢀⠀⠈⠉⠀⠀⠀⠠⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⢀⠄⠀⠀⠈⣄⡠⠄⠊⠀⠀⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠢⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣀

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣧
⠀⠀⣿⣿⡿⣻⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⠽⠁⠀⠙⠺
⠀⠀⣿⡿⣿⣿⣿⣋⣙⣝⡫⣿⡭⠿⠟⠛⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⡟⣒⣀⣚⣠⡆⠤⢧⢨⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠿⢇
⠀⢀⣷⢻⣇⣹⣧⠻⣦⣦⠀⢀⣠⣄⡀⠀⠘⢛⠁⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠘⠋⠀⠀⠀⠀⢀
⠀⠼⢷⣊⠹⣖⣙⣭⢧⠁⣴⣿⣿⣿⣿⡆⠀⠀⠀⣭⣩⣤⣌⣨⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣄⡀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣉⡀
⢠⡶⠎⠵⠿⠯⠷⣵⠆⠘⢿⠿⢿⣿⡏⠁⠀⠀⠀⠋⠛⠛⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢰⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢹⣿⣯⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢠⣿⠀⠀⠀⠀⢸⣿⡟⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠰⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⡧⠀⠻⡿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⢠⣿⠋⠀⣼⡇⠀⠸⡿⠁⠀⢿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠤⠄⠀⠀⠀⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠙⠁⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠄⡀⠮⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠋⠉⠙⠛⠛⠒⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠍⠡⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⢤⡀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⡞⠉⠀⣀⠀⠀⡀⢀⡽⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢠⠏⠀⡔⠈⠀⠀⢀⣰⠏⢠⠏⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⣿⠀⠎⠀⠀⠀⢠⠁⣿⢠⡏⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⣀⣀⣁⣢⣀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠀⠀⢿
⠀⠀⠀⠀⢿⡘⠀⠀⠀⠀⠸⠀⠻⣾⠀⡠⠔⠁⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠉⠉⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢼⡆
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⡄⠀⣿⠊⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⣀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠖⣒⡛⠛⠛⠓⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡇⠀⢻⡀⡠⠐⠉⠀⢸⠇⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠈⣿⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡟
⠀⣀⡀⠀⠀⠀⢸⠇⠀⠀⢠⠁⠀⠀⡿⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣟⡼⠁
⡼⢻⡃⠀⠀⣠⠎⠀⠀⢀⠆⠀⠀⢸⠇⠘⡶⠖⠒⠶⢧⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠋⠁⠀⠀⢻⡁
⣧⠀⠉⠛⠋⠁⠀⡀⠔⠁⠀⠀⢠⡟⠀⢸⠁⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠘⢷⡀⠀⠀⠉⠁⠀⠀⠀⢀⡴⠋⠀⠀⠸⣆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⡇
⠀⠀⠙⠓⠶⠶⠶⠶⠖⠚⠉⠀⠀⠀⠀⠀⠈⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣤⣤⣶⣾⣿⣿⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⢿⡟⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠥⠤⠤⠒⠉⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⣾⠛⠖⣶⠤⠤⠤⣤⣤⡤⠔⠒⠓⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠉⠛⠒⠒⠚⠛⠛⠛⠉⢉⡀⠀⠀⢹⡄⠀⣇⠀⠀⣀⣀⣀⣀⠀⠸⢈⡰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡂⠤⢀⡶⠀⠀⠀⠀⠊⠉⠒⠀⢈⠀⠀⠀⢿⠒⠋⠉⠉⠉⠉⠉⠈⠙⠢⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⣶⡃⢀⡴⢳⣀⣥⠄⡀⢀⠎⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠈⠉⠋⠀⠈⠡⢼⡁⠈⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡟⠛⠓⠦⣄⠲⡖⠃⠀⢀⣀⣀⣀⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠠⣤⡤⠄⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠤⠤⢼⡧⠤⠤⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡟⠉⠉⢹⡏⠉⠉⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⢸⡇⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣶⣶⡶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⣟⣶⣶⣶⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⣿⣿⣄⠀⣠⢤⣠⣿⣿⣿⣿⣠⡤⣄⠀⣼⢫⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡈⢻⡝⣿⣿⡞⡿⠶⠛⢫⡝⠉⢩⠛⠶⢿⠟⣣⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⢾⣧⢨⡅⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠲⠉⠻⢻⡶⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡦⢥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢤⣶⣤⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⠶⠏⠳⢦⡄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⣞⠙⠉
⠀⠀⠀⠀⠀⠉⣽⣯⣭⣥⡍⠞⠀⠀⠀⠀⠀⠀⢀⡠⠀⢄⡄⣷⣾⠀⡥⠀⢠⠀⠀⠀⡄⠀⠀⢠⠶⢩⣭⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠿⠿⠿⣿⣇⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠃⠿⠿⠀⠆⠀⠘⠀⠀⠀⠃⠀⠀⠘⣠⣸⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⡚⢦⣤⣼⠴⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢧
⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⠿⠗⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠘⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠿⡟⢘⡃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⣬⡿⠷⢴⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⣿⢧⣷⠶⣤⣜⣃⣀⣘⣤⠶⣶⣦⡹⣿⣿⣆⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣵⣿⣿⠏⠀⠙⠚⠋⣿⣿⣿⣿⠙⠓⠋⠀⢻⡜⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⣯⢿⠿⠿⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠁⠈⢻⣿⡆⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⡿⡄⠀⠀⠹⠿⠿⠿⠿⠿⠤⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠒⠋⠁⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠠⣤⡀⠈⠙⠻⢿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⠛⠁⢀⠴⠦⣀⣀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋⠁⠘⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⡟⠁⠀⠀⠁⠀⢢⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠴⢿⣿⡁⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠗⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢆⠀⣈⠁⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⣀⠀⣀⣀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣡⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠢⣌⣳⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣡⠟⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣠⣦⡀⠀⠀⠀⠀⠀⠀⠘⡿⠛⢿⡿⠋⠙⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠀⠀⠀⠀⠹⣄⠀⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⢈⡓⠚⢉⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⣄⣀⠉⠉⠁⢀⣠⠤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠀⠠⠤⠀⠀⠴⠆⠐⠶⠒⠲⠖⠒⠒⠒⠋⠉⠙⡷⠄⢀⡼⢱⠄⡀
⠀⠀⠀⠰⣋⣉⣡⣤⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠋⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠯⣙⠚⠀⣸⠀⠒⠠⠄⣀⠀⠀⠀⡄⠀⡇⠀⢠⠂⠀⠀⠠⠤⢤⣄⣀⠀⠀⠐⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠒⢧⡠⢧⣴⠡⠔⠒⠉⠉⠉⠉⠉⠉⠉⠂⠀⠈⢄⡠⠂⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠞⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⣄⣤⣄⡀⠘⡄⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠀⠀⠀⠀⠀⣀⠤⠞⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣵⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠀⠀⢀⡤⠒⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡞⠣⣀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⢀⠤⠎⠁⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⢿⣿⡀⠀⢣⡀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠁⠀⠈⣆⣀⣀⣀⣀⢠⣆⡠⠤⠒⠂⠉⠉⠉⠉⠉⠉⠛⠒⠢⠤⣸⡇⠀⠈⢿⣧⠀⢸⣁⣀⡀⢄⣤⡭⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢈⣿⡴⠋⠀⠀⠉⠑⠪⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⢠⣤⣀⡀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢠⣤⣀⡀⢀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣿⠀⠙⠗⣾⠋⠉⣿⣿⣬⣯⣈⣀⣀⡀⠀⠂⣠⡿⠀⠙⠓⣿⠉⠉⣿⣿⣼⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠿⣄⣀⣀⣴⡟⠀⢰⣧⣰⡷⢈⣽⣿⣿⣿⡿⠟⠯⣄⣀⣀⣴⡏⠀⢸⣧⣸⡇⣩⣽⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⣹⣯⣽⡿⢷⣿⣿⣿⣿⣿⣟⣹⡿⠛⠁⠀⠀⣈⣹⣯⣽⡿⢷⣿⣿⣿⣿⣿⣟⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡍⠀⠀⠀⠀⣿⣿⣿⣿⣍⡛⠿⣄⡀⠀⣠⣴⣿⠉⠀⠀⠀⠘⣿⣿⣿⣿⣍⡻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣅⣸⣇⠀⠀⠀⠀⠘⣿⠉⠋⢉⡟⢷⣬⣹⣶⣿⣁⣸⡇⠀⠀⠀⠀⠹⣿⠉⠋⢹⡟⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡟⢹⡇⠀⣀⣀⣠⣤⣼⠇⠀⠈⠻⣿⠀⠀⠀⢸⡏⣹⡇⢀⣀⣀⣠⣤⣼⠇⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⠿⣿⣛⣩⣭⣥⣾⠛⢻⣄⠈⢀⣠⣿⣳⣶⣶⣾⠷⣟⣛⣩⣭⣤⣾⠛⣿⣀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡉⠙⠿⣇⣀⣿⡃⠈⠉⠹⣿⣿⠿⠯⢿⣿⢻⣿⡉⠛⢿⣅⣀⣿⠁⠉⠙⢹⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⣄⠈⢻⣿⣇⣄⢀⣤⣿⢁⠀⠀⠀⠀⠀⠈⣽⣦⣀⠙⢿⣿⣄⣀⣠⣼⣿⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⡿⠀⠙⠛⣿⠃⠉⣿⣿⡴⣧⣌⣀⣀⣀⠀⠀⣠⡟⠀⠙⠛⣿⠁⢉⣿⣿⡼⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠻⣤⣀⣠⣴⡏⠀⢰⣷⣼⡷⣨⣽⣿⣿⣿⡿⠛⠻⣄⣀⣠⣼⡇⠀⣸⣧⣼⡇⣨⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣽⣧⣼⠿⠿⣿⣿⣿⣿⣿⣿⣹⡟⠛⠀⠀⠀⣀⣽⣧⣽⠿⢿⣿⣿⣿⣿⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣯⣙⠿⣤⡀⠀⣠⣾⣿⠀⠀⠀⠀⠈⣿⣿⣿⣿⣬⡛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣸⡧⠀⠀⠀⠀⠘⣿⡉⠃⢨⡟⢳⡦⣽⣿⣿⣀⣸⡇⠀⠀⠀⠀⠹⣿⠉⠃⢸⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢹⡇⢀⣀⣤⣤⣤⣼⠇⠠⠀⠻⣿⠁⠈⠀⢰⡏⣹⡃⣀⣀⣠⣤⣤⣼⠃⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⡿⣿⣩⣭⣥⠶⣾⡋⢻⣄⣈⣀⣴⣿⣳⣶⣶⣾⢿⣟⣩⣭⡵⢶⣾⠉⣻⣄⣀⣀⣴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣅⠙⠻⣇⣀⣿⡃⠈⠉⠙⣿⡿⠛⠫⠿⢿⣻⣿⡈⠙⠿⣆⣠⣿⠁⠉⠉⢹⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⣄⡈⢻⣿⣧⣄⣀⣴⣿⢀⠀⠀⠀⠀⠀⠀⣽⢦⣄⡈⢻⣿⣤⣄⣠⣴⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⢂⣠⡿⠀⠈⠛⣿⠃⠀⣿⢿⡶⣷⣤⣤⣤⣀⣈⢁⣰⠟⠀⠉⠛⣿⠁⢈⡿⢷⡾⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠻⢤⣀⣤⣴⣏⠀⢸⣷⣼⣿⣠⣾⣿⣿⣿⡿⠛⠳⢤⣀⣤⣼⣇⠀⣸⣷⣼⣇⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡷⠾⠿⠻⣿⣿⣿⣿⣿⣿⣵⠟⠋⠀⠀⠀⣠⣽⡧⠾⠟⢻⣿⣿⣿⣿⣿⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⡄⠀⠀⠀⠀⢿⣿⣿⣿⣦⣙⠻⢦⣀⠀⣠⣾⣿⡀⠀⠀⠀⠈⣿⣿⣿⣿⣮⣙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⣼⡧⠀⠀⠀⠀⠘⢿⡁⠁⢨⣟⣻⡶⢽⣿⣿⣄⣼⡇⠀⠀⠀⠀⠙⣿⡀⠁⢸⣟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⠃⠀⠀⠀⠀⠀⠺⠃⠀⠀⠙⠟⠀⠀⠀⠘⠋⠘⠃⠀⠀⠀⠠⠀⠚⠃⠀⠀⠛⠟

⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⣬⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠩⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⢐⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠠⠐⠈⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠠⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠰⠁⠀⠐⠊⠀⠈⠀⠈⠁⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠠⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⡠⠊
⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣽⣽⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣷⣾⣿⣷⣶⣶⣶⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣄
⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠃⣈
⠀⠀⠀⠀⠀⠀⠙⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠿⠿⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣄⠀⠈⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣶⣶⣶⣶⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡀⠀⠀⠀⠀⠀⣠⠤⠔⠛⠁⣠⡾⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠠⢀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠒⠉⠁⠀⣰⣀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⢶⣒⣶⣶⣶⣖⣒⣛⠉⠻⣿⣿⣿⣟⣓⣲⣶⣾⠃
⠀⠀⠀⠀⠀⠘⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠟⠋⡉⠉⠉⠉⠉⠙⠛⠛⢻⣿⡅⠉⠉⠉⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⣿⣿⣿⠿⠿⠟⠛⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣶⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣀⣈⣽⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠿⣿⡟⠉⠻⠧⣄⣀⢠⣀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠉⠉⠙⠻⢷⣄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⢀⣀⢀⣼⠷⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠲⠶⠖⠒⣉⣑⣂⣀⣀⣀⠀⣀⣠⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠛⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠤⠖⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠉⠉⠉⠁⠒⢤⡤⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣠⣤⣶⠶⠶⠶⠶⣶⣤⡈⢿⡆⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⡁⠤⠤⢀⡀⠀⠀⠈⢻⣌⠻⠦⡀⠀⠀⠑⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⣁⡀⠀⠀⠤⣄⠀⠈⢆⠂⠀⡆⠆⠈⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⣦⠀⠀⠶⠀⠀⠀⠘⢤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⡤⠄⠊⠅⠀⠀⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡀⠉⠀⠀⣀⠔⠋⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣩⠀⠀⠀⣒⠖⠂⠁⠀⠸⠉⠉⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠀⠀⠊⢠⠀⠀⠀⠀⡧⠀⠀⠠⢄⠀⠈⠢⡀
⡔⠀⠚⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠀⠀⠀⠀⠀⠀⠁⠂⠀⡞⠀⠀⠀⠀⠀⠑⢄⠀⠑⡀
⠁⠀⢠⣿⠉⠐⠀⠠⢀⡀⠀⠀⢀⠄⠈⠑⢪⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⠐⡀
⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠈⠉⠀⠀⡠⠔⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠐⠂⠁⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⡰⠉⠆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠒⢄⠀⠀⣀⠤⠊⠀⠀⠈⢆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢏⠁⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠎⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣿⠃⠀⣱⠆⠀⠀⠀⠀⠀⠀⢂⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣶⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠁⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢏⡴⡚⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢖⠻⠊
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠃⠀⠈⠉⠀⠐⠒⠂⠀⠀⠒⠒⠂⠈⠉⠁⠸⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡠⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⠤⡀⠀⣰⣿⣶⠀⣠⣶⣦⡀⢀⠔⢢⠀⡼⠁⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠢⡫⣓⠾⠿⠛⠒⠛⠿⠿⢖⣱⡶⠁⡔⠀⢠⠚⡄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣙⢎⠀⠁⠀⠀⠀⠈⠀⣰⢋⠤⢊⡠⠒⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⡠⠛⡷⣄⠀⠀⠀⢀⠔⠋⢱⠊⠁⠀⠀⠀⠀⡇⠀⢀⡾⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡌⠀⠀⠈⠪⢑⡒⠚⠁⠀⡠⡏⠀⠀⠀⠀⠀⠀⢧⣰⠏⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡇⠀⠀⠀⠀⡰⠉⠹⢨⠋⢰⠁⠀⠀⠀⠀⠀⠀⢘⢆⠠⡈⠢⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠒⠁⠀⠠⠁⢀⠔⠂⢉⡀⠔⠁⢠⡀⠈⠢⢄⠀⠀⠀⠀⠀⠈⠂⠓⢍⣢⠜
⠀⠀⠀⠀⢀⠄⢺⡁⠀⠀⠀⠀⢀⠔⠁⠀⠘⢤⣀⡬⡶⠤⣀⠀⠈⢲⠀⠀⢑
⠀⠀⠀⡰⠁⠀⠀⠻⣤⠀⠀⡠⠋⠀⠀⠀⡮⢛⠕⠂⡀⠤⠤⠇⣴⣁⣀⣀⡜
⠀⠀⢀⠃⠀⡠⠊⠀⠈⣳⠊⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⡰⠁⣀⠀⠈⡁⢔⡄
⠀⠀⢸⠒⠁⢀⡠⠊⡠⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠑⢄⣸⢈⡼
⠀⠀⠈⠑⠒⠓⠂⠉⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠂⠤⠁⠀⠘⠀⠀⠀⠀⠀⠀⣠⣮⣆⡀⠷⢧⣂⡰⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠠⠊⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢫⣀⠄⣈⡯⠖⠋⠉⠀⠡⠡⠀⣀⡀⡢⠺⢏⡴⠈⡩⠿⣟⠉⠳⣦
⠀⠀⠀⠀⢠⠆⠀⠀⣀⣀⣠⠒⣀⠠⠙⠛⡄⠀⠀⠀⠀⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠁⠷⠊⡜⣕⠀⡠⢄⠃⣘⢀⠻⢀⣀⣬⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠘⡄⠁⠀⢀⡀⢀⠐⠿⣀⡀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⡸⠐⠍⠕⠀⠈⠁⠀⠉⠃⠁⠠⠐⠾⠊⡆
⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠠⠈⠁⠈⠀⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⠳⡢⡀⣠⡳⠁
⠀⠀⠀⠀⠀⠈⢰⠂⠐⠃⠀⢀⣤⣤⣤⣶⣾⣿⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⡐⡈⠊⠰⣇⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠀⢦⣀⠈⢁⡤⡌⠭⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡈⠆⠣⡿⣰⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡡⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠐⠇⡗⡰⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠒⠒⠒⠁⠉⠉⠉⠉⠀⠀⠀⠀⠱⡄⠤⠠⠛⡴⣂⡐⠨⠊⣀⣀⠀⢠⡴⢒⠆⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣧⣈⡩⢋⠉⡉⠱⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠈⠁⠀⠀⡀⠤⣰⠘⢹⡇⢉⠀⡰⣜⠕⢉⠿
⠀⠀⠀⠀⠀⠈⠒⠒⠒⢧⡉⡰⢁⡴⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠘⠐⠉⠈⡔⢬⡆⢸⣕⣿⢱⡰⠵⠟⢛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣠⠎⠀⠐⡀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣂⢐⢴⡘⠤⡤⢞⠗⠀⠉⠀⠀⠛⡜⠓⣚⠋⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠐⢆⢀⠷⡀⠘⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠹⣮⣐⡨⢓⠊⠁⠀⠀⠀⠀⠀⢰⢟⠵⠖⠘⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡰⡁⢠⡔⠀⢘⡤⠴⠊⠁⠀⠀⠙⡉⠉⠀⠙⣭⠓⠦⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⢠⣣⣤⣤⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠌⠱⡐⠊⠀⣰⠏⠀⠀⢠⡄⠀⠀⠀⢡⠀⣀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣁⣬⢬⣥⢀
⠀⠀⠀⢀⢄⡐⠂⠁⠀⠀⢠⠊⠀⠀⠱⣀⠔⢅⠀⠀⠀⠀⡇⠀⠀⠀⢸⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡺⠭⢭⣶⠡⠤⠄⠀⣀
⠀⠀⠀⠈⡥⠊⡠⠄⠒⠠⠤⠝⠀⠀⠀⠈⠉⠈⠉⠒⠒⠈⣇⠀⠀⢀⡨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⢀⠀⠀⠀⠀⠀⡠⢝⠢⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠄⢀⠀⣀⢀⣍⢑⡆⠏⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡈⠉⠃
⠀⠀⠀⠀⠀⢀⢀⠁⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠸⠿⠿⣿⣼⠛⠿⠃⢸⣿⠟⠦⠾⠿⠃⠀⠀⢀⣿⣿⣧⠀⠀⠀⢸⣿⣿⡀⠀⣸⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣤⣤⣤⣤⠀⠀⠀⣾⡿⠘⣿⣆⠀⠀⢸⣿⢻⣧⢀⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⡆⠀⣡⣿⠀⠀⠀⢸⣿⠉⠉⠉⠉⠀⠀⣼⣿⣷⣤⣿⣿⡄⠀⢸⣿⠘⣿⣼⡿⢸⣿⣿⣏⠘⠃
⠀⠀⠀⠀⠸⡿⠀⣿⣿⢀⡤⠂⢸⣿⣶⣶⡶⣶⡆⣰⣿⢏⢱⣤⠬⢻⣿⡀⢹⣿⠀⢿⣭⡇⢸⣿⡿⠇
⠀⠀⡀⠀⠀⠀⠀⠀⢠⠞⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⢠⣦⠀⠀⢠⣤⠄⠀⠀⣠⣤⠀⠀⣠⣴⣶⣦⣤⡀⠀⠀⣤⣤⣤⣤⣤⣄⠀⠀⣠⠈⣇⡀⢀⣤⣤
⠀⠀⠀⠸⣿⡇⠀⡾⠁⢀⡀⢀⣿⡏⢀⣾⡿⠉⠁⠉⠋⠹⡄⢀⣿⡏⣩⣭⠙⣿⡇⠀⣿⣶⢉⣴⣿⠟⠁
⠀⠀⠀⠀⢿⣷⢸⣶⠀⢿⣧⢸⣿⠁⢸⣿⡇⠀⠀⠑⢄⣾⡇⠀⣿⣷⣾⣿⣿⠟⠃⠀⣿⣿⣿⢿⣷⡀
⠀⠀⠀⠀⠸⣿⣿⡏⠀⠸⣿⣿⡏⠀⠈⢿⣷⣀⣀⣀⣼⣿⠃⠀⣿⡇⠀⠙⢿⣷⡀⠀⣿⣿⠁⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠛⠛⡡⠄⠀⠛⠛⠁⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠛⠃⠀⠀⠈⠛⠛⠀⠛⠛⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⠀⠐⠉⠴

⠀⠀⠀⢀
⠀⠉⠈⠉⠀⠁⠀⠁⠀⠁⠀⠁⠀⠀⠀⠉⠛⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠑⠦⠂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢀⠀⠀⠀⠀⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⡀⢀⣠⠔⠒⠋⢉⠁⠀⠀⠀⢈⣁⠀⠚⢄⡀⠀⢀⠜
⠀⠀⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⢄⠄⠀⠀⠘⠀⠀⠈⠁⠀⠉⠀⠄⡀
⠀⠀⡇⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠔⠱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣀
⠀⠀⠃⠀⠀⠢⠄⡀⠀⠀⠀⠀⣀⠠⠐⠣⠤⠤⠤⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⢀⡀⠀⠀⠀⠆⠀⠀⠃⠀⠆⠈⠀⠀⢀⡀⢀⠂⣣⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠁⠀⠈⠋⡀⠀⠀⠀⠤⣀⠀⠀⠄⠀⠀⡁⠘⠁⠈⠁⠴⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠤⠠⠂⠀⠀⠎⢰⣛⠆⠀⠀⠠⠭⠀⢀⡀⠀⢀⠇⠀⠀⠀⠀⠐⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⢀⡉⣀⠐⠁⠀⠀⡠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⡂⠁⠄⠀⠀⠀⠀⠀⠈⠂⠺⡀⠀⠀⠀⠀⠈⠳⠄⠤⠈⠀⡠⠊⠀⠀⠢⡀⢀⠔⠁
⠀⠀⠀⠀⠀⢠⠤⠧⢄⠀⠉⠀⠀⠰⢆⡀⠀⠤⠀⠀⠀⣈⠲⠀⣀⠀⠀⠀⠁⠁⡀⠖⠁⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠈⠁⡢⠄⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢉⡁⠀⡉⠀⠀⠈⠂⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠘⠀⡤⠀⢀⣀⠠⠔⠠⢀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⡀⠂⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⡹⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡴⠆⠁
⠀⠀⢰⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠒⠠⠋⠀⠄
⠀⠀⢸⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠆⠀⠀⢀⠀⢰⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⠀⢉⠱⡄
⠀⠀⢸⡀⠀⠀⠈⠁⠠⠤⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⡀⣀⡔⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⢤⢂⡀⢤⠃⠀⠠⠀⠀⢠
⠀⢀⠘⠁⠐⠈⠀⠀⢠⠊⠣⠄⠐⠆⠑⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⡠⠀⠚⠻⠀⠘⠀⠀⠀⠀⠀⡀
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⡀⣠⡀⠠⠊⠁⡠⠋⠀⠀⠤⠀⠘⡎⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠈⠄⠑⠄⠀⠀⠀⠽⠋⢃⠀⠀⡠⠁⢀⠀⠀⠀⠀⢀⠃⡀⠃⠈⠹⠁⠐⢆⢈⡀
⠀⠀⠀⠀⠁⠐⠇⠀⠤⢀⠠⠀⠀⡤⡄⠀⠀⠐⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⡠⠀⠀⠀⢀⠔⠉⠐⠧⡠⠐⠀⠀⠀⠈⠢⠀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡹⠀⠠⠜⠆⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠆⠠⠊⠀⢀⡤⠂⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠇⣀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⡤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠉⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠉⠀⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠁⠀⠀⠀⠀⠀⠀⢱⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠈⠀⠀⣀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⠒⠉⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⢢⠘⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠈⠀⠁⠀⠛⠀⢀⠃⠊⠀⠉⢀⠰⢄⠤⠀⠚⠠⠴⠀⢀⡀⠀⢀⠀⠇⠡⡠⢄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠑⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠟⠋⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣁⠀⣀⡀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢙⣇⣉⠉⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⢀⡠⢤⣴⣿⡀⠭⠄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⢞⣵⣾⡿⠉⢻⣷⠘⣡⣿⣿⣏⣁⢴⡇⠉⢋⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣟⣶⣿⣿⣏⣀⠀⢨⣿⠀⡟⠛⠿⢿⠁⡸⠧⠄⣨⣶⡟⠙⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⡿⢋⣴⡿⠛⢿⣿⣤⣴⡟⢻⠀⠑⠒⠤⠬⠿⠿⠾⠃⣿⢿⣧⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣷⡿⠁⣿⣿⡤⠴⠊⢘⣿⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣧⣠⣤⣄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢎⣿⠿⠃⡰⠀⠀⠀⠀⠀⠸⣻⣿⣷⣿⣷⣤⣀⠀⠀⢀⡀⠀⢀⣨⣿⣿⣿⠛⠛⠋⢃
⠀⠀⠀⠀⠀⠀⠀⢸⣾⠋⡠⠊⠁⠀⠀⠀⠀⠀⠀⢩⣿⣿⡟⠿⠿⠿⠛⠁⠈⠙⠛⢻⡟⠁⢸⣿⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣾⢃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⢾⣿⡶⠠⡎
⠀⠀⠀⠀⠀⣠⣾⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⡞⠀⡀⢀⠜⠉⠀⠀⣱
⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀⠀⠀⠀⠀⠠⢿⣿⣷⣀⣀⣀⣤⣴⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡧⠤⠤⠤⠤⠤⠄⠚⠛⠿⡿⠋⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⣦⡀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣯⣤⣄⣽⣿⣦⣠⣶⡖⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣿⣿⣿⡏⠀⠈⢻⣿⣿⣿⣧⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⣁⣀⣀⣠⣿⣿⣿⣿⡿⠂⢀⡀⣀⢥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠉⠀⢐⠿⡿⢻⣿⣷⡞⠉⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⣾⣰⣿⣿⣿⣿⣿⠀⢠⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⣿⡀⢀⣼⠋⠻⡟⢃⡻⣿⣧⣌⠳⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠁⠀⠀⠈⠉⠻⣼⣿⣿⣄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⣿⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠿⠾⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⠀⠌

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣟⣻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⠶⠾⠉⠉⠉⠷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠿⣿⠖⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⣠⠴⠿⢤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠏⠀⠀⢘⣇⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠘⠁⠀⠀⡞⠁⠀⣷⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⠞⢋⡍⠛⢦⡀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠺⣇⠀⠀⠛⠒⠂⣸
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⠃⠀⢨⣇⣀⡀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣧⡀⠀⠉⠉⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣶⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢛⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠘⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠛⣿⣿⡿⠟⣛⣉⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣼⣿⡟⣴⣿⣿⠛⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡿⢋⣥⣶⣶⣦⣄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡀⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢱⣿⣿⣿⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⢸⣿⣿⣿⣬⣭⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⠻⠛⡟⣻⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠾⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⠅⣸⡀⣷⢼⠅⣿
⠀⠀⠀⠀⠀⠀⡏⠀⠙⠁⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⢉⡁⠈⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠙⣆⡀⢾⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡰⠟⠋
⠀⠀⠀⠀⢀⡠⣀⠀⠙⣀⡁⢀⡠
⠀⠀⠀⠀⢸⠀⡿⡁⣧⠏⣇⢸⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣄⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⠆⠙⠚⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⢹⠛⣽⡏⣿⢿⡟⣟⡿⢿⢟⢿⠉⣿⡿⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⣜⣛⣴⣜⣮⣤⣧⣤⣼⣮⣾⣰⣽⣦⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡰⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠙⠛⠛⠛⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠀⣿⡈⠉⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⢟⣳⠶⣞⣻⣿⣿⡿⠂
⠀⠀⠀⡄⣠⢀⣀⢩⡅⡄⣀⢩⡁⣀⡄⢠⠔⠖⢀⣄⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠻⣿⣼⣿⣿⣿⣷⣿⠟
⠀⠀⠀⠃⠉⠈⠛⠂⠙⠃⠋⠐⠙⠉⠃⠊⠃⠃⠋⠈⠁⠙⠀⠀⠀⠀⠀⣀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢉⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⡴⢰⡀⡆⢠⡶⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣏⣹⡉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠁⠈⠁⠁⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⡀⢄⠠⢄⣀⢠⡄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠰⠁⠏⠘⠉⠚⠸⠣⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤
⣿⣯⣬⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⢀⣴⣿⡿⠿⣿⣦⡀⣾⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣻⣻⣟⣿⢸⣿⣿⡶⢃⣿⣿⡇⣿⣻⣟⣿⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⣿⣿⣿⣿⠘⢿⣿⣿⣽⣿⡿⠃⣿⣿⣿⣿⣿⣿⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⠀⠋⠉⠙⠛⠉⠁⠀⠀⠉⢻⡟⠉⠀⠀⠈⠙⠛⠋⠉⠙⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣿⣿⣿⣿⣿⡟⠻⣿⣷⣄⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⣠⣾⣿⠟⢹⣿⣿⣿⣿⣿⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⣿⣿⣿⣿⣿⡇⠀⠈⠻⠿⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠛⠛⠁⠀⢸⣿⣿⣿⣿⣿⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣴⣿⡿⠋⢹⣿⣿⣿⣿⡏⠙⢿⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⡇⠀⠀⠀⠙⠋⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠙⠋⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠸⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠘⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠒⣶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠥⠀⠀⣈⡹⣿⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⣿⢱⢀⠇⣠⠠⡉⠂⢶⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⢁⣀⣀⡁⠀⠙⠑⠁⣾⣴⣷⣿⣃⡤⠖⠲⢶⣶⣦⣤⡀⠀⠀⢼⣹⣿⣏⢹⠷⢦⣶⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢹⣬⣭⣵⣶⠖⠀⠙⡿⣷⠭⡉⢁⠀⠀⣀⠘⠛⠻⢿⣦⠖⠛⢏⠉⠩⠷⠦⠀⣿⣛⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠼⡇⠀⢻⣀⣩⠜⠁⠀⢠⠞⠁⠹⠶⡄⠂⠀⣀⣝⢳⡄⠀⣰⢣⣎⠱⢸⠇⢠⡒⢢⠀⣻⢛⡟⣏⣿⡃
⠀⠀⠀⠀⠀⠀⠀⣇⠀⣳⡀⠀⠀⠀⠀⠀⡰⠃⢸⡒⠢⣀⠁⠀⠘⢟⠸⠀⠁⢾⠇⠈⠉⡖⡁⠀⠘⠓⠁⠀⢹⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠓⢦⡀⠀⣰⠁⢠⠿⢿⣷⣌⠳⣀⣦⠀⠁⠀⣠⣾⠐⣯⠤⠤⠤⠤⢬⡶⠶⠂⠈⠋⡿⠛⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠂⡏⠀⢺⣀⣉⣉⡽⠛⠋⠉⠁⠀⠀⠙⢿⠀⠸⣿⡿⢛⣻⠏⠀⠀⠀⠀⢠⡏⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⢦⢻⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⠄⠰⠋⣸⣆⠀⠳⡭⠔⠁⠀⠀⠀⠀⣠⡟⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢀⡼⠷⠖⠋⢹⣹⢦⡀⠀⠀⠀⠀⠀⠀⡼⠁⠧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⡠⠤⢚⠀⠀⠀⠀⠈⠁⠀⠉⠳⢦⡀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣫⣾⡅⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣤⣤⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⠟⢷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠗⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠂⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⡠⢀⣀⠤⣀⣀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀⡀⠤⠤⠤⠤⢄⣠⠤⠤⠒⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠇⠀⠀⡤⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢠⡀
⠀⠀⠀⠀⠀⠀⢘⠀⡎⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠂⠑⠀⠀⠀⠀⠀⠰⠊⠑⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠘⠄⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀⠀⠘⠟⠉⢉⠇⠀⠀⠠⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⡤⠤⠤⠒⠂⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠔⠊⠁⠙⠦⠊⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠰⣿⠙⠄⣀⠀⡰⠃⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢆⠀⣀⠠⠌⠙⠁⠀⣿⡋⠣⣀⡀⢀⠞⠤⡀⠀⠀⠀⠀⠀⢀⡀⠸
⠀⠀⠀⠀⠀⠉⠉⠐⠄⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠢⠀⠁⠀⠀⡇⠀⠀⣿⡋⠳⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣉⣀⣁⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠒⠒⠚⢳⠑⠀⠀⢀⣀⡀⠀⠠⠤⠄⠀⠀⠀⣀⣀⣀⡀⠀⢀⡠⠔⠓⠒⠚⠻⡎⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⡆⠀⠀⠀⠒⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢡⠀⠀⠀⠰⠔⣇
⠀⠀⠀⠀⠀⡀⠀⠂⠉⣹⢰⠁⠀⠀⠀⠀⠀⠱⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠆⠀⠀⠀⠀⠀⠈⡆⠀⠂
⠀⠀⢀⡠⠊⠀⠀⣎⣋⣭⣧⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⣭⣽⠄⠀⠀⠀⠀⠀⠀⢁⢸
⣔⠊⠁⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣸
⢸⠀⠀⠀⠀⠀⠀⠢⣀⡀⠄⠀⠀⠀⠀⠀⠀⠿⣯⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⠠⠀⠀⠀⠀⠀⠀⠸⢿⣹⣧
⠘⢦⡤⢄⣀⡀
⠀⠀⠈⠉⠑⠒⠊⠭⠯⢐⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠯⠥⠤⠒⣒⣒⣒⣂⣐⣀⣀⣒⣒⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠶⢶⡶⠶⠀⣶⠶⢶⣦⠀⢰⣶⠀⠀⢠⣶⣆⠀⠰⠶⣶⡶⠆⢰⡆⠀⠀⢰⡆⠀⣶⡆⠀⠀⣠⣶⠾⠷⣶⣄⠀⢰⣶⣄⠀⠀⣶⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣤⣼⠟⠀⢸⣿⠀⢀⣾⠏⢿⡆⠀⠀⣿⡆⠀⢸⣿⣶⣶⣾⡇⠀⣿⡇⠀⢠⣿⠁⠀⠀⠈⣿⡆⢸⣿⠻⣧⡀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠉⢿⣆⠀⢸⣿⠀⣼⡿⠓⠻⣿⡄⠀⣿⡇⠀⢸⡇⠀⠀⣸⡇⠀⣿⣇⣀⡀⢻⣧⣄⣠⣴⡿⠀⢸⣿⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠉⠁⠈⠉⠈⠉⠀⠀⠀⠈⠁⠀⠉⠀⠀⠈⠁⠀⠀⠈⠁⠀⠉⠉⠉⠁⠀⠈⠉⠉⠁⠀⠀⠈⠉⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠉⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠉⠉⠉⠉⠛⠲⣤⡀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠉⠉⠙⠛⠦⣄
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⢀⣀⠀⠈⠻⣆⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢳⡄
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠈⣷⠀⢰⠇⠀⠀⠀⠀⠀⣀⣤⣜⠛⠀⠀⠀⠸⡆⠀⣾⠁⠀⠀⠀⣠⡤⢤⣄⡛⠁⠀⡀⠀⢻⡀
⠀⠀⠀⢸⡇⠀⠈⠀⠁⠀⣁⣈⣤⣿⠀⠿⠂⠁⢸⡄⣿⠀⠀⠀⠀⠀⢾⡟⣥⠉⠓⠀⠀⠀⠀⣿⠠⡇⠀⠀⠀⠀⠃⣠⣿⠟⠙⠒⠋⠀⠀⢸⡇
⠀⠀⠀⠸⣇⠀⠀⠀⠈⠉⠁⠈⠉⠉⠀⠀⠀⠀⣼⠀⢻⡀⠀⡰⣿⣾⡂⢻⠉⢴⣿⣮⡂⠀⢀⡟⠀⣧⠀⠀⠀⢀⡴⠛⣻⠆⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠈⢷⡀⠀⠙⠟⠃⠀⠀⠐⠙⠟⠁⢀⡾⠁⠀⠸⣆⠀⣀⠊⠀⣈⣁⣀⣀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠻⢦⣀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠶⠖⠚⠋⠁