Crucified Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠀⠸⠷⣿⣀⣸⠈⠉⣉⠉⠘⣷⡟⣉⣙⡛⢛⣙⣻⣿⠿⢛⣫⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠂⠈⠉⠉⠉⠁⢰⡗⡚⠿⣄⣘⠉⠉⠉⠉⠈⡉⣁⡠⠞⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⢤⣀⣈⣳⣄⠀⠈⠉⠀⠀⣀⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠢
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣯⡀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⣆⡀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠉⢢⡄⠀⠀⠀⠀⠀⠸⡿⣷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⣇⠀⢀⡄⣠
⠀⠀⠀⠀⡀⠺⠿⠁⠱⢾⡃⠀⠀⠀⠀⠀⠀⠰⣿⣶⡶⠄⠀⠀⠆⠀⠀⠀⠀⠀⣠⣾⣃⡜⠁⠸⣾⣷⣿
⠀⠀⠀⢸⣧⠀⢠⠀⠀⠘⠃⠀⠤⠀⠀⠀⠰⢰⣿⢿⡆⠀⠀⡀⠀⠀⢀⣠⣴⣿⣏⣁⡭⢀⠀⠀⢹⣿⣿
⠀⠀⢀⣿⣿⡆⠘⣦⠸⣷⡆⠀⠲⣶⡄⠀⣬⡑⣻⣿⣇⠀⢠⣟⠃⢹⣿⣿⣿⣿⣿⣿⣧⡏⠀⠀⠈⡁⢛
⠀⠀⢸⣿⣿⣿⡀⠃⣰⣿⢷⣄⡀⠿⠷⣶⣿⣿⣿⣿⣿⡇⠸⣿⠆⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡄
⠀⠀⢸⣿⣿⣿⡿⠊⣼⡇⠀⠈⠃⠀⠀⢰⣛⡷⣿⢻⣿⡇⠀⣿⡀⢸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢸⠙
⠀⠀⢸⣿⣿⢟⠔⣸⣿⡇⠀⠀⠀⣿⡿⠿⣿⡿⠟⢘⣿⡇⢸⣿⣷⣄⢻⣿⣿⣿⣿⢦⣄⣰⣟⡆⠀⠀⣧
⠀⠀⢸⣿⠻⠏⡸⢻⣿⡇⠀⠀⢀⡟⠁⠀⠙⠷⣶⣻⣿⠃⣼⣿⡽⢋⠈⢻⣿⣿⣿⢿⣿⣿⢿⠠⠀⠀⣷
⠀⠀⢘⣿⣿⣿⠁⠸⣿⣇⠀⠀⠈⠃⠀⠀⢻⠀⡈⣼⣻⠂⣿⣟⢰⠂⠀⢸⣿⣿⡄⠀⢻⣇⠀⢃⢀⠀⠉
⠀⠀⢸⣿⣿⡏⡀⠀⣿⣧⠀⠀⠀⢀⣀⣴⠏⢸⠃⣿⣿⣿⣹⣿⡋⠀⠀⣿⣿⢿⡣⠀⠀⢻⡆⠊⢸⠀⢀
⠀⠀⢸⣿⣿⣿⠀⢠⣿⡟⡆⠈⢛⣏⡥⣀⠀⠛⠀⣻⣫⣌⣹⣿⣿⣿⢷⣿⣿⢸⣧⠀⠀⠀⠻⠀⢀⢸⣿
⠀⠀⢸⣿⣿⣿⣷⣾⣿⣿⡇⠸⣶⣿⣿⣥⡇⢘⠀⢩⣿⣭⡟⢛⣻⣯⣼⣿⣿⠸⣿⣆⠀⠀⣲⡇⢸⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⡿⢹⠟⢀⣿⣿⣿⡟⣠⡌⢷⣟⣿⣿⣯⡍⢙⣿⣿⣿⣿⣀⣿⣿⣧⡀⠹⣇⢸⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⢷⡏⡀⣼⣿⣿⣿⣷⣿⣿⣼⣿⣿⣿⡿⣿⣦⣆⠀⡿⠿⣷⣾⣿⣿⣷⣦⠹⣎⢿⣿
⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣷⣾⣟⣿⣿⣿⠟⠿⠿⠛⠉⠉⠙⠙⠿⠿⢷⣿⣿⣿⣿⣿⣿⣽⣿⣷⣭⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⠀⣀⢀⣶⣶⣿⣿⣻⣷⣶⣦⣄⡀
⠀⢘⣿⣿⣿⣿⡆⠀⠀⠄⠀⣿⣾⢋⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡉⣁⠔⠡⠾⠿⠞⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⣿⣯⣉⡉⠀⠀⠀⠀⠠⡤⣶⣿⣿⣿⡄⠀⠀⠀⠀⠀⠛⠀⠻⠟⢡⡞⠁⠀⠀⠀⢪⣿⣿⡻⣿⣿⣿⣿⣿⣿⢿
⠀⢴⣿⣿⣿⣿⣶⣦⡤⠒⠀⠓⣿⡟⢛⠛⠧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⣠⣾⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿
⠈⣻⣿⣿⣿⣿⢶⠤⢶⣶⣤⠀⠐⣵⣿⡟⠀⠃⣾⣅⡀⠀⠀⠀⠀⠀⠘⣷⠀⣀⣼⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⡿
⣸⣿⣿⣿⠿⣷⣶⣶⣼⣿⣿⣿⣶⣤⣿⡟⠀⢾⣿⣿⣿⣿⣶⣶⡆⠀⣶⣾⣀⣸⣿⣿⣿⣿⣿⢃⣤⣾⣿⣿⣿⣿⣿
⠈⣻⣿⡿⠳⣌⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⢲⡟⢻⣿⣿⣿⣿⣷⡄⣿⠻⣿⣿⣿⡻⣿⣿⣷⣜⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠐⠚⠛⠛⠛⠟⢿⡿⠿⣿⣿⣿⣻⣿⢗⡰⢧⠀⠛⣻⣿⣇⣻⣿⣌⣛⣻⣦⣿⣿⣯⣿⣿⣿⡟⠻⣿⣿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡍⠑⣾⢻⣿⣋⣠⣿⢿⣶⣶⣧⣹⣿⣷⣿⣿⣿⣿⢿⣿⣿⣿⣶⣿⡇⣿⣧⣄⣹⡅
⠀⠀⠀⠀⠀⣀⣀⣶⣶⣿⣿⡇⠀⣟⣿⣿⣿⣿⣿⠻⣿⣿⣿⡧⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣟⣿⡷⣸⡏⠀⣰
⠀⠀⣤⣶⣿⣻⣽⣿⢿⣿⣿⢯⣶⢹⣿⣿⣿⣿⣿⣷⢿⣿⣿⣷⡽⢺⡿⣟⠘⡿⠀⣿⣿⣿⣿⣿⣿⣿⡷⣿⡅⢠⣿
⠘⠉⠉⢫⣿⠞⣴⡏⣘⣿⣿⡛⣿⡟⣿⣿⣿⣿⣿⣿⠞⠛⠛⠛⠛⢿⣧⣿⣿⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿
⠻⣷⣻⣿⣤⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⢀⣠⣶⣀⡈⢽⣿⡝⡇⠊⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣉⡛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⣀⣾⣿⣟⣿⡛⠒⠒⠒⠡⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣈⣁⠑⠊⠙⠂⠀⠾⠿⠛⠛⠛⠋⠉⠃⠀⠀⠀⠀⠀⠈⢙⣵⡟⣿⣿⣹⣿⣿⣿⣿
⣉⣻⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣷⣶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣈⣈⣉⣩⣽⣿⣛⣉⣭⣽
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢠⡄⢠⡄⠠⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢘⣁⣂⠱⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⣤⣤⠠⠤⠤⠄⢤⣤⣤⠤⢤⢚⣁⣈⠙⣤⢠⡀⠀⠀⠀⣀⡀⠀⢀⢀⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣜⢻⣍⣀⣒⠒⠴⢦⣤⢄⠉⢰⣿⢻⡿⣿⣡⠈⠑⢂⡙⠤⠒⠛⡀⢬⠟⠃⠄⠂⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠈⠁⠉⠉⠒⠤⠬⣍⡙⠛⣿⣷⠐⣾⡟⢀⣉⠭⠒⠒⠉⠉⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠉⠁⠈⠉⢄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠃⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡤⠟⠁⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠒⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⣮⣤⠍⡃⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡍⣉⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣚⠃⢧⡿⢇⠍⠀⠀⠀⠀⢠⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢧⣮⡇⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢐⠙⢻⠿⠁⠈⠀⠀⠀⠀⠀⠀⠁⡃⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⠙⣿⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠆⠀⡕⠀⠀⢀⠀⠀⠀⠀⣦⣱⡅⠀⠀⠀⠀⠀⢀⠤⡀⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠠⠀⠀⠀⡁⠀⢀⠀⠀⠀⢸⠀⠉⣽⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠄⠈⡅⠠⠼⠀⠀⠀⠸⢰⠀⡗⠀⠀⢰⡆⠀⢀⡃⡀⠀⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠓⢿⠘⢸⡿⠀⠀⣧⠀⢱⠀⣶⠶⣤⡀⢰⠈⠰⠇⠀⡾⠤⠤⢶⡸⠘⡎⠱⠒⢉⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡄⢘⠁⠀⠀⣿⠀⠀⠀⡟⣠⣖⣂⡈⢈⠀⠘⡛⠒⠛⠃⠈⠁⠀⠁⠀⠰⠀⠈⠽⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠂⢻⡇⠘⠀⠀⠀⠿⠀⠀⠆⢀⢘⠁⠀⠘⠈⠀⠘⡇⠁⠀⠀⠀⠀⠀⠀⠀⠂⠣⠀⠀⠀⡟⠛⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠆⠀⣾⡇⠀⡌⠀⠀⠀⠀⠀⠀⠸⣿⣆⠈⠹⠀⠀⢸⣧⠀⠀⡊⠈⠙⡀⠀⠀⠀⠀⠀⢸⠦⡑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠂⡿⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠯⠁⠀⠰⢲⠀⠀⠀⠀⠀⠙⠛⠋⢹⠁⠀⢤⠐⡀⠀⠐⠶⠞⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣇⣶⣤⠁⠀⠀⢠⠀⢀⠀⣾⠶⡄⠐⠈⠀⡄⢸⣿⠀⠀⠒⠠⠤⢨⣦⡀⠈⠀⡛⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⢸⡚⢹⣿⠀⠀⠀⢸⠀⢬⣾⠟⠢⠙⠣⣀⠉⡀⠙⠗⠒⠟⠋⠐⠂⢰⣯⣷⣄⣽⣇⠉⠁⣉⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡚⠳⢾⡿⠀⢠⡦⣿⠀⠸⠋⠀⠀⠀⢀⣼⢒⢠⣀⡗⠐⠂⠰⠀⠀⠈⢓⣀⡦⠲⣀⣬⣤⣄⡡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⣒⠊⣡⡚⠛⠒⠀⢀⡠⠖⠛⠹⠧⠆⠠⠏⠦⢤⣤⣈⢆⠀⠉⠡⣧⠄⠸⠷⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠂⡈⠉⠉⢉⠑⠒⠲⠺⣯⣄⣈⢳⣄⠀⣀⣀⣀⣀⣀⣀⡀⠀⠉⡄⠀⠀⢴⡄⢀⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠂⠒⠚⠉⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠀⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠄⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣶⣤⣤⣤⣤⣶⣶⣶⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⡿⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⣀⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣩⣿⣿⡏⠁⠀⠀⢠⣼⡿⠁
⠀⠀⠀⠀⢼⣿⣿⡿⠿⠟⠓⢿⣯⣄⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣄⣀⠈⢿⣿⣿⣷⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡿⠀⠟⠿⠙⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⣴⣆⣭⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣶⣶⠶⣶⡄⣼⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠛⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢧⣽⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠙⣻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣉⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢬⣿⣿⣿⣿⣦
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣦⣤⡤⠤⠄⠸⠿⠿⠿⠛⠻⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠱⣿⣿⣿⣿
⠀⣿⣿⣿⣿⠿⢿⣷⣄⣀⣀⣀⠀⢀⡼⠛⣡⡞⣩⣿⡿⢿⣿⢿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣀⣤⣤⠤⠀⠀⠀⠀⣠⣷⣾⣿⣿⣿
⠀⣿⣿⣿⣿⣦⣄⠈⠛⠛⠿⣿⣿⣿⣷⣿⡷⢟⣿⣯⣴⣿⣶⣼⣿⣿⣿⣿⠿⢿⣻⣿⣿⢟⣉⠀⠀⢀⣴⣶⣾⣻⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠛⠛⠻⢾⣷⣯⣧⣴⣿⣿⣿⡿⣿⣿⠿⠀⠖⠛⠉⣀⣤⣾⣥⣶⡾⢟⡋⢛⣾⣻⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄⣀⡩⢿⣿⣏⣽⠟⠙⠃⠀⠉⠀⣀⣶⣾⣿⠛⠛⠉⡛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⡿⠛⢻⡇⠀⠀⠉⠁⠀⠀⠀⢀⣴⣾⣿⣯⡥⢶⣤⣤⣦⣿⣷⣶⣶⠶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣁⣰⣮⠅⠠⠤⠄⢘⣧⠀⠀⠀⠀⠀⠐⣶⣿⣿⣿⣿⣤⣁⣸⣭⣚⣷⣾⣿⣿⣟⣁⣨⣽⣿⣭⣽⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠇⠀⠀⠈⠉⠀⠀⠀⠀⠀⢾⣿⣿⣿⣏⣹⡿⠟⠋⡉⠁⠀⢀⣀⣀⣀⣉⣉⣉⣭⣭⢹⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⡄⠀⣀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⠛⠀⠠⢂⡀⠀⠑⠦⢤⣀⣉⡉⠉⢻⣿⣿⠿⣿⣿⡿
⠀⣿⣿⣿⣿⣿⡿⢿⡿⢃⣴⣳⣿⣏⣽⣟⣫⠀⠃⠀⠙⣷⣄⣽⣿⣿⡇⠀⠂⠒⠊⣥⣤⣤⣤⠔⠛⠛⠳⠿⢿⣿⠇⠀⢀⣿⣿
⠀⣿⣿⣿⣿⣿⣶⣶⣷⣾⣿⣿⣿⣧⡀⢀⢉⣀⢀⣠⠾⠧⣬⣿⣿⣿⣿⠋⢾⡃⠿⠉⡋⡙⣿⢷⣄⠀⣀⣤⣤⠻⣆⠐⣾⣿⣿
⠀⢿⣿⣿⣿⡿⣁⠀⠀⢸⣿⣿⣿⡿⠗⣾⡄⠉⠀⠀⡀⠀⠀⢨⣿⣿⣿⣷⣾⠟⡠⢈⠁⠙⣦⣜⣿⣧⢉⠀⣩⡟⣻⣿⣿⣿⣿
⠀⢻⣿⣾⣿⡟⠈⢳⣄⢸⡿⣿⣿⣧⠐⢻⡿⠀⡀⣿⡠⠀⠸⢧⣿⣿⣿⣉⣿⡧⢡⣀⠂⣀⢸⣿⣿⣿⡀⣋⣹⠿⣿⣿⣿⣿⣿
⠀⣼⣿⣿⣿⠟⠃⢸⣿⣋⣾⣿⣿⠟⣦⡼⠀⠀⠙⢫⡀⠀⠘⣾⣿⣿⡏⠈⠉⢹⡶⢿⡟⣩⡌⣿⡿⣿⣷⣟⣻⢴⣿⣿⣿⣿⣿
⠀⢿⣿⠛⠁⢸⣷⡘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⠁⠀⠀⣾⣿⣿⣷⣴⡾⠇⠀⠀⠁⠉⠀⠀⠀⠀⠈⠈⢷⣼⣿⣿⣿⣿⣿
⠀⣾⡇⠀⠀⠘⠏⢿⣿⣿⣿⣿⣿⢿⣿⡇⠀⠀⣸⣿⡄⠀⢘⣿⣿⣿⣿⠟⡇⠀⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⣸⣿⣿⣿⣿⣿⣿
⠀⣿⠇⠀⠀⠀⡀⠈⢿⣿⣿⠋⠁⣼⣿⣿⠀⢰⣿⣿⣿⠀⢸⣻⣿⣿⣿⣿⠁⠀⠀⣤⠀⡆⠀⠀⠀⠏⠀⠀⢅⣿⣿⣿⣿⣿⣿
⠀⣿⠂⠀⠀⠀⠀⣷⡌⢿⣿⣤⣼⣿⣻⡟⠀⠀⢸⣿⣿⣆⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⣄⢸⣆⢤⡞⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⣿⡆⡈⣄⠀⠀⢹⣿⣶⣿⣿⣿⣿⣿⣿⣿⣄⣼⣿⣿⣿⡇⢻⣿⣿⣁⣤⠀⠀⠀⠀⠙⣿⣷⠜⠁⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿
⠀⣿⠷⠞⣹⣦⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⢿⣿⠁⢸⣿⣿⠁⠘⣁⠶⠄⠀⠀⢈⠋⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿
⠀⡿⢸⣷⡾⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣧⠀⠀⢿⣧⣼⣿⣋⣾⠟⠻⣇⡀⠀⠀⠀⠀⠀⣼⡟⣻⠿⣿⣿⣿⣿⣿⣿⣿
⠀⣷⡿⢫⣼⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⢿⣿⣷⠀⢸⣿⣿⡿⢋⡤⢸⣿⠻⣍⡀⢀⡀⣶⣾⣿⠿⢱⣾⢿⣿⣿⣿⣿⣿⣿
⠐⣿⢁⣾⡟⠀⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣷⡾⣿⣿⣇⠀⣻⣿⡷⠉⠀⣼⣿⣿⣿⣿⣿⣿⢿⣿⠿⣾⣷⣿⣾⡟⣽⣿⣿⣿⣿
⠀⣿⣾⠇⣿⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⢿⣿⣿⣿⡆⠸⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⠣⣾⣿⣶⣌⢻⡿⢻⣿⣿⣿⣿⣿⣿
⠀⠉⣻⠘⠇⢀⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⢻⣿⣿⣇⢸⣿⠀⢀⣿⣿⣿⠛⣿⣿⠃⠀⠁⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⡇⣸⠃⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⢹⣿⣿⠀⣼⣿⣿⣿⠀⢻⡿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⠀⣼⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⡟⢰⣿⣿⣿⣷⣶⣿⣧⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣏⣸⣿⣿⣿
⠀⠀⠀⡇⢀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡇⣾⣿⣿⣿⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⢰⡇⢀⡿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣽⣿⣿⣿⣥⣿⣿⡇⢈⣿⣿⣿⣇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠁⢸⠃⠘⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡷⢿⣿⣿⡇⠀⣿⣿⣿⣿⣄⡀⢰⣤⣰⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠁⢺⠀⢠⣀⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⠁⠈⠹⣿⣿⣧⣿⣿⣿⣿⣿⣿⡇
⠀⡁⠿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⣿⠙⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣷⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠘⣽⡟⠀⠀⢻⣿⢸⣿⣀⡄⠀⢹⣿⠏⠉⠙⠛⣿⣿⣿⣿⡇
⠀⠁⢸⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⢹⣷⣴⣿⣷⡀⠀⣾⣿⢶⣿⣀⣅⠀⣿⣿⠀⠀⠀⣤⣿⣿⣿⣿⡇
⠀⣶⡟⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠁⠀⢰⡄⢻⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⠏⣀⣿⣿⣿⣿⣿⣿⣿⣏⠹⣿⠇
⠀⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣶⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⢰⣿⣿⣿⠿⠿⠓⠶⠿⠿⡄⣿⡀
⠀⠠⠥⠤⠆⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠫⠟⠿⠿⠿⠿⠿⠿⠦⠄⠀⠀⠀⠤⠶⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠉⠁⠉⠀⠁⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⣴⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⡶⠤⠤⠂⠈⠃⢀⣀⢠⡄⢠⣤⣤⣤⣤⣤⣤⠤⢤⠤⠤⠤⣴⣦
⠀⠀⠀⠀⠀⢹⣿⣧⣜⣁⠙⢿⣿⣽⣿⠛⠻⠖⠀⠀⠠⠀⠀⠀⣀⠀⠈⠙⢿⡋⠉⠀⣨⠀⠀⠀⢆⣀⢻⣿
⠀⠀⠀⠀⠀⠘⠿⠟⠻⠳⠆⠀⠙⠻⠟⠾⠶⠾⠒⠒⠀⠀⣾⡟⠋⡉⢳⡀⠠⠀⠄⠀⠊⠀⠀⠲⠽⠶⠿⠟
⠀⠀⠀⠀⠀⠀⠀⢾⠘⠀⠀⠐⢠⣀⠐⠂⠠⢀⠀⠀⠀⡴⣿⠀⢀⣀⠎⠁⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠲⢄⣀⡀⠀⢀⠷⠛⢧⡤⡏⠀⠀⠁⢀⡀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠘⠃⢠⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠠⠄⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠄⠀⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠀⠠⡄⠀⠀⢠⠴⠀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡥⠆⠠⢶⣤⣤⣀⠀⠀⣻⡏⢈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠂⠉⠉⡏⠉⠓⣄⢻⣅⠀⡁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⢀⠁⠀⠀⠈⠀⣿⠀⠙⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢧⣄⣼⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠆⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡀⠀⠘⠉⠁⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠠⠅⠀⠘⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠈⠁⠀⣤⣠⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢀⠀⢀⠀⠀⠀⢠⣄⠠⠌⡷⠆⠀⠀⠘⠀⠀⠙⠉⠉⠁⠀⠀⠀⠁⣈⡁⢤⠀⠄
⠀⠀⡀⠀⠘⠓⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⠀⢸⡉⠁⠀⠇⠀⠀⠀⠀⠂⠉⠀⠀⠤⠀⠀⠀⠀⠀⠀⡅⠸⠃⣐⢲⣒
⠀⠠⡉⠂⠤⡰⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠸⠃⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⢡⣊⠹⡟
⠀⠀⠀⠀⠀⠠⠀⠀⡠⣀⢀⡀⣀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠠⠀⠀⠀⠈⠀⠀⠰⢂⠀⠀⠲⠳⠈⠦⡰⠾⡃
⠀⣀⠀⠀⡀⠀⠀⢈⠀⠀⢴⣅⣧⠀⠜⠀⠀⠀⠠⠀⠄⠂⠠⠠⠀⢀⢤⠤⠀⠔⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠂⠐⠀⠡⠄⢿
⠀⣿⣶⣤⣁⠀⠀⠈⠀⠁⠂⠁⠸⠧⢧⠤⠀⠀⠀⠀⢀⠁⠀⠄⢀⣀⠂⠀⠀⠀⠀⠈⠀⠈⠀⠀⠰⠀⠄⢀⡄⢢⠄⠤⢵⣤⠿
⠀⣹⢿⣿⣮⡄⠀⠀⠀⡀⢤⠰⠣⠄⠟⣀⣄⣀⢀⣀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⣁⡠⠖⣒⠊⠀⠀⠀⠄⠀⠀⠭⠰⠃⠘⠛⠃
⠀⠁⠀⠉⠉⠋⠉⠀⠈⠁⠈⠀⠙⠛⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡿⠿⠿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣀⣀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣶⣌⠙⠛⠿⣿⣿⣿⡁⢰⡟⠛⠿⡇⠀⣿⣿⣿⣿⡿⠟⠋⣉⣴⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠂⠤⢀⠀⠈⠁⣸⣇⡫⣴⠿⠞⡛⠋⠉⢀⠠⠐⠊⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡇⠀⠈⠁⠀⠔⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣁⠑⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠤⠆⠀⠊⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⣁⡈⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠔⢊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡠⢤⡁⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢸⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⡇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣼⣷⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⠆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣦⣤⣾⣯⣦⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡅⣠⣄⠀⡄⣉⡉⠉⠁⢠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣾⣷⣿⡹⣿⣏⣿⢷⡟⢣⣾⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠼⠶⣿⣿⣭⡽⢟⣃⣿⣾⣿⠆⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣿⡟⣓⣈⣿⣱⣿⣿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⢛⣿⢿⣿⣩⣥⡛⢻⣽⡦⢻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⢿⣾⣿⣿⡟⣿⡿⣧⣸⣿⡈
⡀⣶⡖⣶⠒⠂⠐⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠒⠒⢴⣶⣆⢩⣞⣷⣞⣻⡎⣹⣿⠛⢻⣿⣿⣿⢿⣷⣾⣷⣶⣶⠄⠒⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣲
⠸⡿⣿⣿⣉⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣏⣾⣿⣨⣉⣉⣓⣿⣿⢿⣿⣿⣷⣿⡿⠋⠁⠀⠀⠀⠀⠀⡀⠀⣀⠀⠤⠤⣦⣴⣿⣿⣻
⠀⢿⣿⣿⣭⣍⣀⠀⠀⠉⠓⠂⠀⠂⠴⠆⣀⣀⣀⠀⢀⣻⣿⠿⣇⣼⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⣀⣴⣿⣿⠷⠚⠛⠛⠉⠀⠀⣠⣤⣽⣿⣿⣿
⠰⠧⠼⣿⢿⠿⣿⢷⣶⢶⣤⣤⣤⣄⣀⣀⣀⣈⠉⠲⠛⠋⠑⠾⣿⡿⢿⣿⣶⣿⣿⣿⣶⣷⡶⢚⠻⠛⠛⠛⣛⣻⡁⣀⣀⣀⣠⣴⠿⠿⠿⠿⠿⠶⠿⠷⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣿⣿⣿⣿⣿⣷⣷⣄⠀⠀⠀⠀⠙⠙⠋⠀⠀⠉⠁⢠⣿⣷⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣧⡀⣀⠀⠀⠶⠶⠀⠀⠀⢀⣸⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣏⠀⣤⡄⢀⡐⠂⣠⠔⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠻⡶⢣⡴⠛⠛⢳⣮⣶⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⢱⡾⠓⠀⠀⠘⣿⡽⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡁⢘⠋⢐⢿⣌⢷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣨⣿⣿⣦⢾⢷⢿⣧⡼⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⠀⠘⢷⣄⢀⠙⠃⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠉⣿⡿⣷⣄⠀⠙⢿⣇⣠⡀⠑⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⣼⠿⣧⡈⠻⣷⣄⡔⢻⡿⢷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣵⣿⠄⠈⢳⣴⡛⣿⣿⣆⢘⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣿⣿⣇⡀⠀⠙⣿⡿⠟⢻⣾⣿⡝⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⣿⣿⣧⠀⠀⠈⢻⣿⣿⠿⠿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⢁⣼⣿⣿⣷⠄⠀⠘⣿⠇⠰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⣿⣿⣿⣤⢀⣼⣿⡆⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⡿⢿⣿⡿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣀⣼⣿⣿⣧⣠⣆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⣿⠛⠛⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣶⣷⣶⣦⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⢻⡏⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆⣸⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⠏⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣷⣄⣀⣀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣴⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢙⠋⠉⠉⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠗⠀⠂⠂⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣄⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⣷⡙⣿⣿⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢶⠟⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣬⠀⣾⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣿⣿⣿⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⣿⣿⣿⡏⠉⠉⠉
⠀⣦⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣦⣤⣿⣿⣿⣧⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠰⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣉⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣄⣀⣀⣀⡀⠘⣹⣿⣿⣿⣿⣇⣀⣀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠘⣳⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⡿⢾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇

⢰⡖
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⢸⠆⠀⠀⠀⠀⠀⠂⠐⠀⢀⡀⠀⠀⠀⠀⠀⢆⡜⠹⢿⣦⣀⣀⣀⠀⠀⠀⠀⠀⣠⡔⠃⢀⠀⠀⠀⠀⢰
⠘⢑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠐⠈⠋⠉⢀⣤⠤⠐⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌
⢠⣬⢻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣬
⢸⣾⠈⣯⡉⠁⠀⢺⡶⠶⠤⠄⣸⡗⠀⠀⠀⠀⢀⠀⠀⠃⢢⣿⠀⠀⠀⠠⡄⠀⣀⣀⠀⢲⠖⠻⣉⠀⣧
⢸⣟⡀⠸⡐⡜⢢⣾⡇⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡆⠠⠲⠄⠙⠛⠉⡅⠀⣵⠂⢠⡏⢠⡇
⢸⣭⣷⡀⢸⢺⣿⣿⣷⡴⠛⢥⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠸⡇⣙⠃⠄⠠⠄⠀⠀⠀⠈⠘⠋⣤⡞
⢸⣿⣿⣿⠀⠆⠛⢿⣿⣤⠀⠈⠙⣤⠀⡄⠀⠀⠘⡀⠀⠐⡀⠐⣧⠘⣡⠅⠆⠶⣴⡧⠀⠀⠀⢼⣾⣧⣰
⢸⠛⠛⣿⣆⠀⠀⢉⣿⠁⠀⠀⠟⠏⢠⣀⡀⢀⣤⠃⠀⢸⡇⢰⣿⣤⡏⢀⠀⠛⠙⡇⠀⠀⠤⣾⡿⠉⣹
⢸⠐⣾⢘⡏⠀⠀⢹⡏⠀⠂⣀⣤⠄⠉⠀⠘⢿⣿⡆⠀⣼⣯⡾⠁⠀⡧⠤⢶⠀⣿⠀⠀⠀⠰⣿⠃⢿⣿
⢸⢠⡿⢿⡇⠀⠀⢶⢿⠀⢾⣿⡏⠀⠀⠀⢰⣌⣿⠃⢰⣿⣿⠠⠀⠀⢃⠀⠀⠀⡧⠀⠀⢀⣾⣿⠀⢸⣿
⠰⠌⠀⠘⠀⠀⠀⢾⣿⡇⣴⢿⣷⠃⣶⡀⠤⣿⣿⡇⣾⣿⣿⠀⠐⠀⢸⡀⡶⠋⢠⣤⣷⣌⢿⣿⣄⡸⢿
⢀⣼⡀⢤⡴⠟⠂⠀⢻⠃⠈⢀⣉⢠⡍⠃⠀⠙⢿⠁⣿⢿⣿⠓⡶⠶⠂⣽⣿⣮⡜⢛⣽⣿⣿⠻⠿⠧⢠
⢸⣿⠁⠼⣁⡄⠨⡗⢸⣆⡔⢨⣿⠈⢈⣶⣶⣿⢈⣁⣿⣀⣿⣠⣦⣶⣋⣿⣿⣹⣿⣿⣿⣿⣯⠀⣶⣦⣀
⠰⠋⠀⠀⣮⣷⣾⡧⠀⠉⡴⢛⣿⠀⠺⣿⣾⣿⡈⢏⠋⠉⢹⡟⠉⢩⣝⢻⡿⣿⣿⣿⣿⣿⡿⠠⠺⣿⣿
⠀⠀⡀⠀⣲⣿⣿⠡⠀⠱⢰⣿⡿⣧⣄⣿⣿⣿⣧⠈⠓⠄⠰⠠⢠⢲⣿⣸⣿⣼⣿⣿⣿⣿⡴⣶⡘⠋⢀
⠀⢀⠳⠴⣿⣿⣿⣦⠣⢠⡾⣿⡇⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠛⣿⣿⣯⣿⠿⢿⣿⣿⣤⣽⠇⠘⣿
⠀⠃⢀⡘⠺⣟⠿⠿⣦⡀⠰⣸⡇⠈⣿⣿⣿⣿⣿⡇⠀⠀⢀⣻⣆⡀⠀⠈⢿⣿⣶⡏⣽⣿⣿⣿⣷⣿⣿
⠀⠂⠀⠉⠒⠙⠣⠐⣮⢻⠔⣹⣿⣧⣿⣿⣿⣿⣿⡇⠀⠀⢸⡟⢹⣿⠀⠀⠰⢿⣿⣧⣻⣿⣿⣿⡿⡿⣿
⠀⣷⠄⠄⠀⠀⠀⣠⡆⢤⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣧⣼⣿⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣿⣿
⠀⠟⠀⠀⣣⣤⡤⠼⢷⣶⣿⣿⣿⣿⣟⣿⣿⣿⣿⡇⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⢠⢀⡀⢸⡇⠀⠀⠀⠻⡿⢿⣿⣿⣿⢿⣿⣿⣿⣿⡃⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠞⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣾⣧⠸⣧⡀⡀⠐⠤⣂⠀⠈⠙⢿⣿⣿⣿⣿⣛⡀⠀⠙⠛⠷⠄⣀⠀⡄⠀⠀⠈⠻⠛⠛⠛⢻⣿⣿⣿
⠈⠁⠘⠂⠙⣦⡙⠂⠀⠀⠁⠀⠀⠀⢉⣿⡿⣿⣿⣤⡄⠀⠀⠒⠂⠐⠀⢨⠀⠀⢀⠀⠀⠀⣤⡄⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠀⠄⠄⠸⠿⠛⢛⡉⢭⣿⡿⠛⠙⡶⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠦⠒⠛⣋⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠄⠠
⠀⠄⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠉⠹⠇⠀⠐⠒⠛⠋⠉⠈⠹⠿⠿⠰⠀⣄⣤⣤⣤⣄⣀⠀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⢢⠠⡄⠀⠀⠀⢾⣿⣶⣴⡃⢀⣎⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠤⢤⡬⣄⡹⠦⢀⡚⣿⣿⠟⠝⣲⣿⣝⢢⠀⠀⠀⠀⣠⠦⢀⡀⢴⣦⣹⣇⠼⠇
⠀⠀⠀⠀⠀⣰⡾⢉⣽⣿⣷⣶⡾⣽⣴⡶⣿⣿⣧⣿⠿⠋⠀⠀⠀⣿⣶⣾⣿⡀⡛⠛⠿⣀
⠀⠀⠀⠀⠇⠙⠃⡐⠻⣿⠛⢻⣿⣏⡉⣥⢁⡐⣴⣅⠸⣧⣤⡤⠀⠙⣻⠿⠿⠷⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⣿⣿⣷⣥⣿⣿⣿⣿⣿⣿⣿⣕⣌⠙⠓⡠⣤⡀⠀⢠⡀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢯⣭⣥⣿⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⡿⣻⣿⣷⣦⢇⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⢠⡼⡯⣝⡿⠛⠉⣯⣿⣿⣿⣾⣽⡶⢳⣾⣿⢝⣿⣿⣿⣻⣻⠿⣷⡀⣀⣈⢠
⠀⠀⠀⠀⠀⠀⠀⡀⡤⢶⢠⣀⣤⣮⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⡂⣿⣿⣟⣟⣈⡁⠙⢿⣿⡽⠡⡀
⠀⠀⠀⠒⣫⡷⣾⢸⣋⣴⠾⢿⣿⠿⠛⢉⠛⣿⣿⣿⣿⣿⣿⡷⢦⣛⣯⠜⣿⠛⣽⣷⣀⣉⠁
⠀⠀⠀⠀⠴⠞⢳⠞⢛⡷⠗⠈⡟⠀⢁⣤⠉⠟⢿⣿⣿⣿⣟⠁⡈⠉⠀⠀⠋⠈⡿⠀⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡤⠀⠀⠀⠘⠀⠄⠄⠀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡲⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡷⡜⢀⠀⠀⠶⠃⠁⢠⠀⠀⠻⠆⠀⢀⠘⡴⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⠆⢀⠛⠄⠷⣠⠐⠀⢸⢀⠐⢠⡰⠇⡜⢃⠀⠾⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠤⡷⠀⠛⣢⡘⡂⠀⣌⠃⠀⡀⡀⣰⢼⡌⠔⡀⠀⠀⢋⡄⠀⡚⣠⡚⠃⠐⣦⠄
⠀⠀⠀⠀⠀⠀⠈⠛⣠⠄⢀⠡⣀⠀⡀⢙⣆⠀⠘⡃⠁⣸⠃⡀⡛⠀⢀⣞⠁⡀⢀⡠⢁⠀⢤⡘⠋
⠀⠀⠀⠀⠀⠀⣠⡄⢠⡤⠘⠛⠉⢇⠈⢲⡁⠹⢶⡇⠀⣿⠀⢠⣷⠾⠁⣱⠊⢀⠏⠙⠛⠠⣤⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠅⠀⠠⠀⢶⡖⠾⢧⣘⠋⠛⠢⢽⣆⣿⣠⣾⠥⠚⠋⢛⡠⠿⠖⣶⠆⠠⠀⠀⠍
⠀⠀⠀⠀⢀⡆⠀⠚⠢⡠⢔⡒⠳⠦⣤⢬⣽⣿⣶⣶⣿⣿⣿⣷⣶⣾⣿⣭⢤⡤⠶⠓⣒⠤⠠⠚⠂⠀⣄
⠀⠀⠀⠀⠈⠱⠶⡄⠀⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⡴⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠴⠶⠾⠿⠿⠿⠿⠿⠷⠶⠦⣤⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠈⣇⠀⠀⡅⠠⠄⠀⠆⠀⠀⠀⠀⢆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣆⡴⠃⠀⠀⠎⠀⠀⠀⠠⠀⠀⡏⠀⡠⢇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⡸⠀⠁⠀⠀⠀⠈⠒⢇⠀⠀⠈⠢⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣈⡇⠀⠀⠀⠀⠀⠀⢄⢸⡇⢸⣰⠃⢃⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⡀⠀⠀⠀⢸⡇⠀⣼⠅⠀⠀⠀⢀⠀⠀⠀⢸⡇⣸⠏⣀⡆⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠛⢻⣧⠞⠁⠀⠀⠀⠀⠀⢣⡀⠀⢸⣧⠟⠉⠀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢾⣿⠀⠀⠈⠠⣀⠀⠀⣼⡇⠀⣼⡟⠛⠛⠫⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⠀⢀⠀⠀⢸⣿⠀⠄⠀⠀⣸⠀⣠⣛⣇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠈⠙⠓⣶⠶⠶⣷⣄⣸⣿⠀⠈⢢⡴⠛⠛⠁⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠑⠦⣀⡨⠶⠋⠁⢀⠠⠈⠉⠻⣿⣆⢠⡿⠀⠀⠀⣠⣿⣿⣶⢶⣄⣀⠠⠄⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠈⣷⣿⠏⠀⠀⠀⠉⠛⠛⢶⣒⣒⢛⠓⠒⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⣼⣿⠏⠀⠀⢠⠀⠤⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⢀⡴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡥⠖⠋⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⢀⣀⣸⣄⠀⠀⠀⢀⣤⣤⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄⢀⡄⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠
⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⣶⣶⣶⣷⣾⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣴⡦⢠⣤⣔⣒⣲⣦⣴⣯⣭⣤⣤⣤⣤⡟⠙⣧⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢩⢤⢷⠶⢿⣿⣿⣯⡿⠟⠛⠙⣉⡉⠩⠭⠭⠭⠇⣸⠧⠤⣬⣅⣀⠀⢀⣁⡛⣷⣶⣤⣀⠀⣀
⠀⠀⠀⠀⠐⠸⢴⣊⢙⣩⢗⣛⠛⣋⣩⠤⠒⠚⣉⡩⠤⠄⠔⢒⣒⡂⣺⡁⠀⠈⠒⠒⠤⠭⣙⡒⠤⣝⠩⢝⡻⣭⣛⠍⠒⢀
⠀⠀⠀⠀⣠⠞⠃⠐⡐⡀⠀⢀⡊⠋⣀⡴⠞⠉⠁⢒⡈⠉⣉⣉⣉⣁⠃⢉⠉⠉⠑⠒⠢⢤⣄⠈⠁⠂⠄⠀⠈⠑⠉⠑⠢⣀
⠀⣰⢃⡞⠁⠀⠄⠀⠀⢀⡀⠀⢀⡾⠉⠀⡠⠀⠀⣴⣶⣾⠿⡿⠕⠁⠀⠀⠽⣿⣷⣶⡦⢄⡀⠓⢤⣠⣀⠀⠀⠀⠀⠐⡆⠈⠂⠀⠀⠀⠠⣄
⠰⠃⣸⠐⠀⠘⠀⡀⠀⡏⠈⠋⠿⠀⡂⢸⠀⠀⣶⢸⠇⠁⠀⠀⠀⣀⣀⣴⣮⣮⣿⠈⠂⠀⠀⠀⠀⣈⣿⣷⣤⠀⠀⠀⠈⢣⠀⠀⠀⠀⠛⡌⢦
⠘⠀⢀⢰⢀⢰⢡⢡⠀⣇⢀⣀⡀⠀⢇⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠉⠛⠛⠛⠉⢁⡠⠆⠀⠈⠀⠀⢨⣽⢿⣛⠄⠀⠀⢀⠈⡷⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠈⢎⢧⢻⢧⠱⡈⢮⡻⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠈⠉⠁⠀⣀⣀⡤⠤⠶⠀⢀⡾⠉⠀⣠⣾⠏⠀⠀⢀⡄⠀⠐⢸⢄⣷
⠀⠀⠀⠀⠈⢦⠳⣙⣴⣌⠂⣙⠪⣝⡷⣦⣀⠀⠀⠐⠢⠄⠀⡀⠀⢈⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⢠⠴⠋⢀⣤⠾⠋⣀⠴⢋⠤⠋⠀⠀⢀⠎⣼⠃
⠀⠀⠀⠀⠣⣀⠑⢌⠳⣗⠑⠢⣔⠠⣙⠻⢭⣟⡶⢤⣤⡀⠀⠀⠀⠀⠬⡭⠭⠭⠄⠀⣀⣀⣤⠤⠶⠚⣋⣉⠤⢖⣫⠵⢚⣡⠔⢠⠄⠀⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠑⠂⠈⠙⠲⢭⣒⡬⢽⡓⠲⠬⣭⣽⣛⠓⠚⠛⠛⣛⣛⣩⠭⠥⠖⣒⣛⡩⠥⠖⣊⡡⠄⠚⠁⣀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡁⠀⠀⠈⠻⢡⡀⠀⡄⠀⠀⠒⠒⢈⠉⠩⠥⠴⡒⠊⠉⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠈⠑⠈⠑⠊⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠒⠂⠀⠀⠰⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣤⡀⢀⣤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⠸⠿⠀⢿⣿⠺⢿⡶⢀⡄⣿⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣿⣾⢛⠹⠷⠞⠀⢰⣦⡀⣴⡇⢿⠇⠀⢸⠟⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠉⣤⡉⣿⠇⠀⣷⣦⢠⣝⠃⡙⠁⣠⣴⠆⣀⢼⡿⠇⢀⡄⢠⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣜⠃⣶⡠⢰⣬⣄⠈⣷⠄⡙⢀⡟⢣⣾⣶⠙⠋⣾⡿⢀⣴⣇⢻⠟⠛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⡉⠉⠀⠉⠁⠳⣍⠙⣇⣠⠊⣴⡄⠶⣶⢧⣸⣴⣦⠉⠀⠘⢿⡣⠔⠀⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣃⣙⣿⠗⢾⡛⠋⠻⢾⣿⡄⠛⠀⠀⣤⣤⣹⣇⣹⣴⠗⠫⣿⣶⠆⠀⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⡜⠛⣋⡀⢈⣭⣝⠲⣾⣿⣋⢻⣇⠙⠛⠢⣉⣹⡿⢛⣭⣽⣓⠒⢰⡾⠀⣶⠶⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⢀⡘⢿⣆⡭⠴⣾⣿⣍⠽⠿⣿⣷⣤⣤⣾⣿⣷⣮⣭⡭⠁⠀⣶⡿⠃⠀⠘⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠏⠠⡶⠀⢀⡀⢺⣷⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⠙⠍⠛⣻⣯⣅⠀⠘⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠈⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢰⣌⠙⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢻⣿⢿⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠖⣛⠿⢛⢿⡥⡴⠋⣹⡈⢯⡛⢤⣘⠿⣛⡛⠛⠛⠒⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠸⡀⠀⠃⠈⠂⠉⠑⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣁⣴⣶⣿⣿⣿⡇⠈⡆⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⣤⣾⡿⢿⠋⢹⣿⣿⠃⣸⠀⠀⠀⠀⠄⠤⠉⠀⠀⠙⢂⠒⢤⠀⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⣠⡞⠉⠛⡀⠀⡝⠐⣿⡟⢠⠇⠀⠀⣠⠀⢀⡀⠀⠀⠀⠐⢿⣷⣿⡇⠀⠀⢠⠊⢀⡄⢄⠠⡀
⠀⠀⠀⠀⠀⠀⣠⠎⣀⠀⠉⠀⠀⠀⢄⢀⡉⢡⣿⠂⡞⠀⠀⠀⣿⣴⠋⠀⢶⠂⢀⣀⠸⡆⣿⡁⠀⠀⠆⢠⠎⡁⢸⠆⢱
⠀⠀⠀⠀⠀⡔⠁⠈⠣⠄⠀⠀⢀⠀⡀⠀⡘⠺⣿⠀⡇⠀⠀⠐⢠⣾⠀⠀⠀⠀⠉⠟⢛⣿⣿⠁⠀⡘⠀⡎⡄⢹⠊⠤⠜⡆
⠀⠀⠀⠀⠈⠃⢤⠀⠀⠈⠀⠁⠀⡄⡈⠈⢳⠞⣿⠀⣇⠀⠀⠰⣼⠿⣄⠀⠶⠁⢀⣰⣿⠟⠁⠀⡰⢁⣶⡁⠟⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠈⢀⠀⣄⠀⡀⢉⣽⣻⣇⠘⢦⡀⠀⠠⠂⠘⢶⣶⣶⣿⡿⠿⠁⣀⠜⣰⡷⡟⠉⠈⠀⣀⠸⠁⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠘⠂⠀⠀⡀⢈⡀⠁⠄⡰⢷⣮⡦⠴⠀⠀⠀⠀⠀⠀⣩⣿⣀⣠⣤⣾⣥⢾⠟⡉⠰⠁⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡀⠈⠄⠀⠐⠉⠁⠀⠀⠀⠠⠂⠀⠘⠺⠟⡿⢿⣿⣿⣿⣿⢁⠰⠁⠀⠔⠀⠀⠠⠁⡜
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠔⠒⠶⠦⣔⠀⣈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠈⢻⣿⣿⡟⠀⠀⠘⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢳⠇⠀⠀⠀⣀⠤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢩⡄⡀⡁⠀⠀⠀⠀⠤⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣠⠖⠁⠀⢖⡠⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠁⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡤⡾⠁⠀⢀⣾⠿⠂⠀⠀⠀⠀⠀⠀⠀⣼⠹⠀⠀⠀⠀⠉⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠘⠁⢸⠁⠀⢀⣼⠿⠮⠀⠀⠀⠀⠀⠀⣀⡴⢻⠀⡇⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⢀⣾⣿⠋⠀⠀⠀⠀⣠⠔⠉⠁⠀⢸⠀⢍⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣴⣿⣿⠛⠂⠤⢀⣀⣘⣁⠀⠁⠀⠀⠘⣄⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣠⣤⠟⠛⠁⠀⠀⠉⠉⠙⠂⣰⠃⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡈⣠⡀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢾⡠⠄⠊⣡⠞⠁⠀⠈⠀⠀⠀⠀⠀⢧⡄⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣔⠀⠏⠀⢈⡶⠁⠀⠀⠀⠀⠀⠀⠐⠒⢦⠀⠀⢳
⠛⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣿⠁⠀⡸⠂⡼⣁⡀⠀⠀⠀⣴⠂⠀⠆⠀⢄⣠⡄⠸
⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠉⠁⠀⠀⢡⣴⣿⣿⠗⢀⠙⡄⠊⠉⠀⠀⠀⠀⠀⠀⠁
⣷⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⡂⠀⢫⣿⣷⠛⣣⡛⠈⠀⠀⠀⠀⠀⢀⣀⣆⠀⠈
⢼⠗⢄⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡾⠟⠀⠀⠀⠈⠁⠀⣿⣿⣶⢾⢽⣷⡇⠀⠀⠀⣀⣠⡟⠉⡈⠀⢠
⣾⣷⣾⣷⣤⣤⣄⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡹⠃⠀⡰⠂⢠⠀⠀⠀⢠⣿⡿⢡⡟⣼⣽⡇⠀⠀⠀⠀⠹⣧⢽⢣
⣿⣿⣿⣋⣿⡿⠿⠿⣧⣾⣿⣿⡗⠠⠀⠀⠀⠀⢀⣀⠀⠀⣠⣤⣤⠟⠀⠀⠀⠀⢰⣯⡇⠀⠀⠀⢹⣿⣿⣆⣽⡿⣿⡁⠀⠀⠀⠀⠀⠘⠶⢧⡀⡀
⣿⡿⠛⠉⠀⠀⢲⣶⣿⣿⣿⣿⡇⣀⡨⣿⢿⣶⣬⣯⢶⣾⡿⠋⢋⣀⣀⣀⣾⢟⣟⣮⡖⠀⠀⠀⠈⢿⣿⣿⣿⣧⡈⠷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣿⣿⣾⣿⣇⣀⡉⠻⠿⢿⡦⠰⢯⣉⡿⠿⡿⢋⣾⣿⣿⡇⠀⠀⠀⠀⠈⢻⣏⣻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣆⣿⣿⣿⣿⡿⣿⣇⡈⠁⠀⠤⣀⠒⠿⣞⠛⣾⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣧⣿⣿⣏⡳⢶⣶⠮⣯⣷⡴⠶⠏⣲⣿⠿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠙⠯⣭⣀⠈⠻⢷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠶⠿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣤⢞⣛⣡⣿⡳⠻⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣶⣷⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⢿⣿⣿⠿⠛⠫⠭⢭⣿⣿⣟⠚⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢿⣿⠟⠒⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡋⡳⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠻⠿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⢸⣷⠀⠀⠀⣤⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣬⣿⣤⠉⣾⢰⣆⠻⣚⣛⠁⠀⢴⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⣀⣻⣿⡆⣶⠉⠓⣶⢣⡘⢿⢢⡭⢁⡀⣿⣌⢱⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡘⣿⡇⢉⠩⣤⣤⡈⠀⣧⢉⡋⢻⡞⣰⣾⡿⢀⢿⣿⢈⡁⣰⡄⣶⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⣬⡀⠟⣶⠀⢭⣬⡶⣤⡸⢳⡎⣷⢛⡋⠀⢸⣷⠀⢛⡁⣿⣇⡉⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⣦⡀⠠⣭⣤⡤⡉⣾⠀⢹⡇⠨⣤⣪⡁⣿⠛⢡⣦⠷⢈⣀⡍⣁⣼⣋⣛⠛⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡗⠙⢹⡇⡀⠉⠁⢘⢦⡼⢻⣰⣷⠨⣥⡙⣿⢠⡞⢿⣴⣣⠴⢺⡅⠈⠛⠿⠓⠀⢸⣿⣀⣴⠂
⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠻⠃⠠⡼⠇⠸⠆⣿⡆⡙⢾⣜⠁⢶⣌⠁⣿⢈⣷⠿⣷⣤⡛⢂⠀⣜⢣⡄⢰⣦⡘⠟⣩⣵⠆
⠀⠀⠀⠀⠀⠸⣷⡄⣦⠀⣌⠛⠃⢹⡍⣛⣇⡪⣹⠇⣶⡙⣷⡜⠿⢶⣿⣿⠃⢠⡌⢉⠉⢸⣷⣉⠴⢛⢈⠿⡷⠘⣛⡋
⠀⠀⠀⠀⠀⠰⢾⡄⢉⠰⣿⡆⣷⣄⢷⡈⠉⢤⡬⠓⠾⣤⣘⣿⣆⣼⣿⡏⣀⠳⠕⣫⡄⣦⠙⡿⢚⣋⡘⢇⡜⢙⣋⠁
⠀⠀⠀⠀⠀⠰⠾⠿⠗⠑⢮⣄⣙⠿⠈⢳⣄⣀⣚⣛⡛⠀⠹⣿⣿⣿⣿⢣⣼⠷⡞⣩⡤⣉⡾⢳⡿⢟⣥⠟⣴⣿⢏⣾⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣃⣤⣭⡝⢛⣛⣛⢋⣹⠿⠿⠿⣶⣼⣿⣿⣿⠈⠿⣴⣧⣽⣿⣿⠷⠶⠶⠿⠧⠤⠽⢥⢠⣅⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠛⣠⣤⣉⠝⣭⠸⡆⠀⠀⠘⣿⣿⣿⣿⠀⣴⠟⠛⠉⢩⡍⣙⠲⣟⣧⡄⢾⣷⡌⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣶⡆⢿⡇⠁⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠛⢸⣿⢀⡓⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠇⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠀⠀⢄⠀⠀⡇⠀⡇⠀⡇⠀⢰⠀⠀⡞⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⣆⠈⢆⠀⠸⡀⠀⡇⠀⡇⢀⡇⠀⡇⠀⡼⠀⢠⠂⠀⡠⠋
⠀⠀⠀⠀⠠⡀⠀⠀⣄⠈⢣⡀⠈⢆⠘⡆⠀⢇⠀⢧⠀⡇⢸⠀⢸⠀⣰⠁⣰⠃⢀⡜⠁⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠈⠣⡀⠑⣄⠈⢆⠘⡄⠸⡀⢸⠀⡇⣸⠀⡇⢠⠃⡰⠃⣠⠊⢀⠔⠁⢀⡴⠊
⠀⢀⡀⠀⠀⠤⡀⠈⠳⢄⠈⢢⡈⢢⡈⢧⠹⡀⢇⢸⠀⡇⡇⢸⢀⠏⡰⠁⡴⢁⡴⠋⣀⠔⠉⢀⡤⠂
⠀⠀⠉⠒⠤⣀⠈⠑⢤⡀⠑⢤⡙⢦⠑⣌⢧⠱⠘⠈⠀⠁⠃⠃⠞⡰⢡⠎⡠⠋⡠⠞⢁⡤⠚⠁⣀⠤⠒⠉
⠀⠀⠠⠤⣀⡀⠉⠒⠤⣈⠓⢤⡙⠦⡑⠈⣠⠔⠚⠉⠉⠉⠉⠑⠲⢄⡁⠊⡴⢊⡤⠚⣁⡤⠖⠋⢁⣀⡤⠄
⠤⢤⣀⣀⠀⠉⠙⠒⠤⢄⣙⠲⢬⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⡡⠖⣋⡡⠤⠒⠊⢉⣀⣀⠤⠤
⠀⢀⣀⣀⣉⠉⠑⠒⠒⠤⠬⢍⠃⢰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢐⡩⠥⠴⠒⠚⠉⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠉⢉⣉⣑⣒⣲⣤⣿⣤⣤⣤⣿⣿⣿⣾⣶⣷⣴⣶⣤⣶⣦⣼⣿⣒⣒⡚⠉⠉⠉⠉⠉

⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠾⠋⠁
⠀⠀⠛⠉⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣴⣿⣟⢛⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⢻⣅⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣟⠹⠫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠉⠀⠀⠀⠀⢠⡄⣾⣿⣆⠀⠀⠀⠀⠀⠠⣿⣿⡿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡀⠀⠛⢿⣷⣾⣿⣿⢤⠀⠀⠀⠀⣠⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠱⣄⠀⠀⠁⠀⠤⠙⢶⣆⣴⣶⣾⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⡏⠻⡏⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡄⠀⠀⢰⣄⠔⠈⢱⣤⣄⣸⣏⠉⠀⠀⠰⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⢧⣀⠀⠈⠙⠀⣠⠏⠹⣿⠏⠙⠀⢀⣀⣠⣤⡶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠈⠙⣶⡒⠺⢁⣀⡼⠧⠴⠶⢾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⢀⡀⣼⣷⣤⠿⠟⠁⠀⠀⠀⠀⠈⠻⡏⠛⠻⣤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠀⠀⢰⣾⣷⣿⣿⣶⣿⣟⠁⠐⠆⠐⠀⠀⢀⡀⠀⢠⣬⣆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⢿⠟⠉⠙⠛⠋⠩⡄⠀⠀⠀⠀⣀⠀⢀⣸⣴⣿⣿⣿⣦⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⣀⡙⢻⣿⣿⣿⣿⣿⣿⢿⣿⣯⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠺⣿⣿⣿⣿⣿⣿⣁⠀⠀⠋⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⢿⣦⡀⠀⠈⢻⡿⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣧⡙⠁⠐⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠇⠀⣽⣿⣿⣿⣿⣿⡎⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⢀⠔⠀⠀⡤⠊⠁⠀⠠⠾⣖⣋⠉⠀⠀⠠⠂⠄⠸⡄
⠀⠀⠀⠀⠀⢄⡀⢠⣄⠲⢶⣶⡀⠁⠰⣶⠁⠔⢖⡋⠀⢶⣀⡀⠒⠒⠂⠒⠻⠫⣙⡃⠋⠠⡀⠒⠲⣄⠰⠒⠒⠒⠒⢲⡆⡀⡒⡆
⠀⠀⠀⠀⠀⠀⠑⠤⠀⠢⢀⣙⢣⠀⠀⠉⢧⠀⠈⠱⡄⠀⠀⠈⠁⠀⠀⠀⠀⠁⠏⡆⠀⠀⠀⠀⠉⠀⠉⠁⢠⠀⠀⠀⠐⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⣀⡀⠈⠈⠂⠀⠀⠀⠈⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⢀⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⠀⢀⠱⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠉⠀⠑⠄⠘⠡⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡶⠀⡂⠀⠙⠒⣠⡀⠈⠢⢈⡆⠀⠀⢀⣀⠀⠀⠀⠄⠀⢀⣀⡀⡀⠀⢠⡀⠀⠀⠀⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠙⠒⠒⠒⠒⠚⠛⠚⠛⠛⠀⠀⣿⣿⣧⠀⠀⠐⠚⠃⠤⠉⠛⣠⠝⡳⠆⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⡀⠀⠠⠁⠀⠀⠀⠐⠊⠀⠔⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⣿⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣶⠶⠿⠿⠿⠶⣾⣿⣿⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾⠛⣁⣤⣴⣶⣶⣶⣦⣤⡉⠙⢶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣶⣿⣿⣿⣤⣽⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣛⣛⣛⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⠍⠉⠉⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣭⣽⣯⣬⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣯⣿⣿⣭⣿⣯⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⢌⠱⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢀⠎⠈⠁⠈⣦⠀⠀⠀⠀⡿⣿⡄⢀⠁⠀⣀⣀⣤⣤⣄⣀⣀
⠀⣦⢶⠛⡿⠛⣦⣤⡀⠀⠀⠀⠀⠀⠈⠣⣄⢰⣃⡔⢀⢀⣀⣯⣽⣄⠀⣼⣽⠟⠀⠀⢀⡔⠁⠘⢯⣽⣉⣿⡿
⠀⠈⠺⠖⠛⠛⠁⠀⠙⢄⡀⠀⠂⠀⠀⠀⠈⠁⣾⠿⠄⢸⣿⣿⣿⡟⠃⣠⣿⣤⠤⣶⣿⡇⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⢶⡶⣢⠟⢆⠀⠉⢁⡼⠏⠀⣠⣿⣧⢰⠐⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⠿⠿⠀⠀⠁⠀⠀⠀⠐⣡⡻⡿⢿⠟⠀⠹⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣤⣔⣠⡴⠁⢀⣾⣿⡠⠑⠖⠀⡐⠀⢠⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⣻⡿⣚⢁⣴⣿⣿⡟⠛⠀⠀⠈⣠⡶⢻⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣧⢾⣿⢿⠿⠉⡇⣿⠉⣿⡄⠀⠀⣻⣾⣇⣘⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢨⠵⡿⠁⠀⡀⠀⠀⠀⡀⢿⣿⡀⠀⢹⠿⠉⡺⣿⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⢀⠀⠀⡇⢸⣿⣧⠀⠀⠶⠚⠏⣷⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⡀⢸⡇⢀⣿⡿⣀⢠⡦⠀⠹⢸⣿⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠞⢑⠀⠀⠀⠀⠀⢇⡾⡗⢸⣿⣿⡿⠀⠀⠀⠀⠀⣿⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡄⠀⠀⠤⠤⠖⠀⣀⣀⣠⣟⣡⠔⣫⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⠀⠀⠈⢻⡏⣧⢐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠖⠀⠀⢠⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠴⠀⠀⠀⢊⣾⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠀⢰⠟⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠚⠆⠄⠀⣀⠐⠂⢄⠠⠄⠀⠀⠀⠀⠌⠂⠀⠀⣢⣠⡤⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣴⣯⣶⡄⣷⣸⢿⣄⣤⣴⣷⠆⠊⣛⣍⠱⣿⣶⡘⡒⢢⢤⣄⡆⡤⠀⡀⢁⢺⣁⣴⣹⡇
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣊⣶⣷⣻⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣶⣿⣽⣿⣿⣿⣿⣯⣦⣤
⠀⠀⠀⠉⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠙⠋⠉⠁⠁
⠀⠀⠀⠀⠐⠀⠘⢁⣠⣴⣖⣀⣠⠀⣦⠀⠀⢠⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣧⣦⠈⣳⣴⡆⣯⣦⣶⢸⣯⡀⣤⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⠿⣿⣟⣛⣙⣿⣿⣿⠀⢀⣰⣦⣆⢀⣾⢿⡇⢰⢼⡄⠀⠀⠸⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠿⠉⢉⡛⠻⠆⠀⠀⠏⣿⣿⣿⣿⣷⡠⠼⠇⠀⠀⣆⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠀⠀⠀⠀⢳⡶⢖⠪⡿⠿⠶⠶⣿⣁⣈⣛⣻⣦⣤⠤⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢄⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠁⠉⠉⠉⠙⠛⠒⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⡜⠿⢷⠦⠴⣠⣴⣦⣶⣾⣿⣿⣦⠀⠈⠉⠉⠙⠛⠓⠒⠒⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠄⠹⣿⣿⣿⣿⣿⠿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠥⢄⡉⠂⡇⣑⣒⣆⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠹⠃⠳⠆⡿⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠘⠂⠖⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡴⠑⠦⠾⠟⠁⢀⣠⣶⣶⣾⣁⠀⠀⠀⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠤⠤⡄⠀⣠⣮⡙⠛⠉⠉⠙⠋⠀⠽⣾⠛⠋⠰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡙⣿⠟⠋⠀⢠⣦⡀⠀⢀⣨⡇⠀⣆⢀⠀⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡙⢻⣷⡶⠆⠀⠛⠉⠉⣶⣼⡏⠁⠀⢘⠿⠇
⠀⠀⠀⠀⠀⠀⣠⠈⠉⠀⠀⠁⠀⣴⣴⣿⣷⠐⢻⣿⣶⣶⠏⣸⣷⣄⢰⡿⠶⡆
⠀⠀⠀⠀⠀⠘⠋⠁⠀⣀⣶⠆⠸⡿⣯⣉⠻⣦⠘⣿⣿⡟⠴⣿⣿⠽⠒⠈⠀⠙⠲⣄
⠀⠀⠀⠀⢠⡾⢷⣦⠀⢼⠁⠀⢀⣀⠈⠻⣿⣿⠟⣛⠽⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠋⠠⠿⢿⡀⢀⣰⣶⣿⣿⡿⢿⡟⠀⠀⠑⢄⠀⠀⠀⠀⠀⠐⢶⣤⡤⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⣰⠟⣿⣿⣿⣿⣿⡏⠀⠸⡀⠀⠀⠀⠈⠱⢄⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⡴⠟⠛⢛⣩⣿⠉⠀⠀⠘⣖⠃⢤⣀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠖⠉⣉⡿⠃⢀⡀⢸⣿⡆⠀⣬⠆⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢔⣊⣥⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⠁⠀⢉⡁⠈⣿⣧⣶⡄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣀⡤⢖⣫⣵⠶⠛⠉⢥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⣀⣠⣌⠁⣤⣼⣿⣿⣿⡀⠄⠂⠀⠀⠀⠀⠓⠤⠖⣩⡵⠾⠛⠁⠀⠀⢀⣀⣄⣷⠾
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠙⠃⠐⠖⠉⠃⠛⠀⣨⣿⠿⠏⠙⠃⠀⣇⠀⢠⠀⠀⢀⡴⠛⠉⠀⠀⣀⣠⣴⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠾⠿⢿⣷⣄⡰⠄⣀⡶⠙⣦⣤⠄⢀⣾⠁⠀⢀⣸⣧⡙⢿⣿⣿⣒⠒⠒⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⡴⠿⠋⠀⠀⡿⢃⣠⠴⠾⢶⡸⣇⣾⣶⣿⣿⣿⣷⣌⣇⠉⠉⠉⠙⠒⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢯⢀⡟⠁⠀⠀⠀⠀⠛⠉⠈⠉⠛⢤⡉⠙⠻⢷⣖⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣀⠀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠳⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠿⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠙⠇⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠘⠋⢧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠐⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⢇⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⠤⠞⠉⠉⠉⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠯⢷⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠴⢟⡻⢦⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⣀⣤⠴⠞⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠑⢛⢀⡀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣯⠋⠁⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡷⣻⠓⠀⠀⠀⠀⠉⠓⠲⠶⠶⠒⠻⡄⠀⠀⠀⠉⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢡⣾⣿⠁⣠⡤⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠉⢨⣿⠖⠰⡶⠤⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢀⡀⠀⠀⣤⠀⠀⠀⣤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡴⢶⡽⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠄⢀⡀⠀⠀⠀⢀⣠⣠⣀⡀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⢀⣼⢇⡄
⠀⠀⠀⠀⠨⠵⢞⣳⡟⠻⠂⠀⠀⠀⣠⣀⣤⡀⠀⠀⠀⠈⢢⣄⣠⡄⠃⠀⠀⠀⣿⣿⡟⠁⠀⣀⣾⣿⢋⠈⠀⠀⠀⠀⠀⠀⠐⣯⠁⡄
⠀⠀⠀⠀⠀⠀⠁⢻⠀⠀⠀⠀⠀⠰⣿⠿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡇⠠⠔
⠀⠀⠀⠀⠀⠀⠀⡎⠰⠃⠀⠀⠀⠈⢾⣿⣽⡞⡗⠀⠀⠀⠀⢲⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⢀⠀⠀⠀⠀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠉⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⣤⠀⢘⣦⠾⠁⠀⠠⠐⣦⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⢠⣤⢞⠀⠀⢀⠈⡄⢠⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠔⠋⠀⠊⠀⠀⠀⠀⠀⠀⠅⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡟⠋⠉⠁⠀⠀⠈⠀⢑⡺⡍⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡇⠀⠀⠀⠀⠀⠈⠰⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⢀⡄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣨⠁⣉⠀⠀⠀⣶⡶⠀⠀⠀⡇⠀⡿⠋⠁⠀⠀⠀⠀⠀⠀⡀⢉⠀⠀⠀⠀⠆
⠀⠄⠀⠀⠀⠀⣀⠀⡦⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡃⡿⣃⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣿⠀⠀⠀⠾⡷⠄
⠀⠂⠀⠀⠀⠀⠈⢰⡖⠚⠀⠀⠸⠿⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠉⠳⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣾⠆⠄
⠀⠂⠀⠀⠀⠀⠀⠘⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉
⠀⡂
⠀⠁
⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⣶⣶⣶⣖⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢤⣀⣴⣿⣿⣿⣿⣿⣯⣿⣿⣿⣻⣿⣆⣄⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⠼⣿⣾⣿⣟⣿⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⢀⣠⣠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣶
⠀⠀⠀⢰⣾⠿⣿⢿⣿⣿⣞⣳⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣛⠿⠛
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣽⣿⢯⣯⣯⣿⣿⣩⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⣙⣛⣿⣿⣿⣶⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣿⣯⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⠟⠉
⠀⠀⠀⠀⢹⣾⣿⣿⢿⣿⣿⣿⡿⠿⣧⣿⡏⢻⣽⣿⣹⣿⡿⠋⢉⡽⣿⣟⣿⣿⣿⣾⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠻⣿⣾⣷⣿⣿⠝⠒⠚⠛⢷⣾⣻⡿⣹⣿⣽⠟⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠈⠏⠫⣷⣿⣿⡟⠀⠀⠀⠀⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢰⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠿⠿⠿⠧⠿⠿⠿⠿⠿⠶⠶⠦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠋⠁⠀⣀⣀⣤⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣏⢏⢷⣀⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⢿⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⡾⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠞⣛⣫⣭⡭⠭⠥⠆⠀⠀⠀⠀⠀⠚⠛⠛⠛⠿⠿⠿⠿⠿⠛⣿⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣯⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠲⠾⠿⠭⠭⠭⠭⠍⠙⠛⠛⠉⠩⠭⣭⣭⣭⣭⡭⠭⢩⡏⠀⢀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡾⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣾⠃⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡟⢛⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠲⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡤⠶⡾⠛⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣽⠁⠀⣸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠞⠋⠁⠀⣸⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣳⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⢖⣚⣩⡵⠞⠋
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⠾⠿⠿⠿⠿⠿⠿⠷⠒⠛⠛⠉⠉
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠇⠀⠀⠸⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡜⠀⠀⢠⠇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣰⠃⠀⠀⠁⠀⠘⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠁⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠚⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢠⡀⠀⠀⠉⠒⠷⣄⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠆⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣠⠂⠐⠁⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠓⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⣦⡄⡈⣢⡄⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠐⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⣀⣠⠀⠀⠀⠀⠀⠀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠶⠤⠚⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠙⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡁⢰⣿⣛⣿⠃⠀⣀⡀⠀⠀⠀⠀⠈⠉⢻⡟⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠙⠛⠋⠀⢸⡏⢙⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠛⠈⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠉⣳⢉⡵⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⢀⣠⡿⠀⠀⠀⠀⠀⢀⣀⡀⠛⠃⠰⠏⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠹⢶⡾⠟⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡇⠀⠀⠀⢰⣶⠀⢠⣤⣤⣾⣽⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⣿⠀⠀⠀⣾⡇⠀⠀⣻⡏⠀⠀⢈⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢹⣿⣿⠀⠀⠀⢻⣿⡀⠰⠷⠿⣶⢶⡾⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠀⠉⠛⠀⠘⣿⣿⠀⠀⠀⢸⣿⣷⣤⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⢀⣤⡀⠀⠀⢿⣿⠀⠀⠀⢸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠀⠀⠀⠉⠀⠀⠀⢸⡟⠀⠀⠀⠸⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁



⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣀⣶⠶⠺⡏⠉⠈⡉⠁⠠⠀⠀⣠⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⡿⢷⣶⣩⡅⣲⠔⡪⢙⢛⣷⡒⠠⣠⠀⣰⣿⣿⣿⣿⡆⢄
⠀⠀⠀⠀⠀⢀⠀⣰⠲⠿⣒⠩⠉⣈⠤⢒⡺⠷⡺⠝⣻⠗⠾⣲⢷⡿⣫⢿⢷⣰⠀⢹⣭⣿⣭⣭⠾⠋
⠀⠀⠀⠀⠈⠎⢉⣀⠂⠅⣔⠒⠉⠀⠘⠉⠤⠬⢀⠊⠡⠨⠌⡵⠊⠻⠴⣫⠌⠹⣋⣬⢿⠿⣿⠃⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡘⡺⣫⣿⢿⣿⡷⠇⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠚⠴⣭⣡⠎⢀⠻⣠⠄⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⢑⠲⠲⠕⠀⠉⠈⣿⣟⠷⡷⣶⠲⠎⣩⣴⡛⠋⢹⣂⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠍⠀⠀⠀⠀⢀⣠⡤⠜⠱⢀⢀⡁⠀⠰⢛⡦⢤⠵⠒⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣯⡟⢁⠔⣠⣆⡌⠊⠙⣀⡹⡡⠀⠊⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⠀⡓⠈⠀⢻⠀⠈⠑⠋⠁⢀⢪⠕⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣡⠊⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡢⠢⡦⢁⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⣀⠀⠂⠤⣀⣠⠀⠀⠘⠋⢀⠠⠰⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠁⠀⠀⠐⠂⠲⠄⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠡⠠⡄⠒⠶⠒⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠁⠀⠒⠀⢁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⠀⢰⣦⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡙⠛⠃⠘⡿⠀⣾⣿⠃⠀⢀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠛⠁⠀⣶⡄⠈⢢⡉⠀⠀⠀⣟⡰⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣄⡀⠀⠸⡇⠀⠀⠀⠁⢢⠀⣧⡴⠚⠋⠀⠀⠰⣷⣞⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠐⢤⡃⠀⣴⡄⠀⠀⣷⠏⢠⡶⠀⠀⢀⣀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⢧⡀⢿⠇⠀⢸⡿⠊⠀⠀⢀⠰⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠗⠐⠊⣩⡍⠉⠛⣶⣤⡀⢸⡇⠀⢀⣠⠾⠒⢒⣀⠀⠴⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠂⣀⣉⣀⣀⠞⠁⠈⢻⣾⣿⣴⣯⠥⣤⠄⠈⠻⠷⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠁⠀⠀⢠⣄⠀⠀⢻⣿⣿⢀⣀⠀⠀⠀⣤⠲⠦⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠈⠀⠀⢸⣿⡇⠈⠉⣷⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷

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

⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠈⢧⠘⣧
⠀⠀⠀⠈⠢⡈⠃⠉⢣
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠣⡀⠀⠀⠀⠀⠀⠀⣀⠐⢀⠀⠄
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠙⢆⠀⠀⠀⠀⠀⡀⠐⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠲⢄⠑⠺⣆⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢌⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠚⠻⢷⣦⡀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠄⢉⣛⣛⣿⣦⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠐⠀⠈⠠⠀⠐⢚⠇⡀⠈⠛⠳⢤⠄⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⡁⠀⠀⠀⠁⠀⠉⠕⣂⠔⢠⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⣀⠠⠀⢀⠀⠀⡠⠈⠐⠾⣿⠆⠀⠀⢀⡇⠀⠀⠀⢀⠀⢠⣾⠋⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠐⠈⡀⠜⠁⠀⠀⠀⡼⠀⡀⣀⢠⡿⠐⣺⣏⣂⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⣤⣾⡀⣴⠃⣰⡟⠵⠂⠩⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣾⠏⣤⣼⡇⠚⣛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣾⠟⠣⢤⡽⣻⣿⣯⣥⣤⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣻⢾⣿⣿⡿⣰⡆⡸⠚⢩⡄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠡⠺⡿⢯⣿⣿⢏⣡⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⡟⣻⠟⣛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⣀⣭⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡔⠉⠀⠈⠑⣦⡖⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠲⣬⣢⣀⠔⠢⣀⣞⣵⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣤⣶⡿
⠀⠀⠀⠀⠈⠒⠤⠔⠊⠁⠉⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠀⠒⡺⠛⠱⣾⡷⠎⠻⣗⠂⠀⠀⠀⠀⠀⠀⠰⣶⣶⣾⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣧⣤⣤⣤⡀⠀⠀⠚⠒⠒⢲⢲⡖⡖⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣼⣿⣿⣷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠀⣰⢟⡻⣧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡗⠐⣷⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣴⣾⡦⠀⠀⣶⣷⣦⠀⠀⠀⠀⣿⡤⠀⣼⡇⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⣰⡟⠀⠁⢻⣦⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⢸⣿⢿⣷⡆⢶⣿⠿⣿⠀⠀⢰⣾⠿⠁⠀⠙⢿⣦⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⣿⢳⡆⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠻⠿⠛⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣄
⠀⠀⠀⠀⢹⣧⣾⣷⣼⡟⠀⠀⠀⠀⠀⢀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢹⡿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣀⣤⣶⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⡟⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⠇⠀⠀⢀⣤⣶⣿⣿⣿⣵⣶⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⡇⢸⣿⡇
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠻⢿⣿⣿⠿⠋⡡⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠠⣤⢀⡀⠉⠤⠈⡿⠴⢤⠀⠀⢄⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢐⣛⡣⠀⢋⠛⠈⣡⡄⡔⠒⠊⢠⣐⢎⡉⢁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣭⠀⣭⣉⢀⠣⠆⢀⣰⠀⠒⠁⠂⣦⡀⣚⣁⠜⠃⠾⠃⣦⣀
⠀⠀⠀⠀⠀⠀⢠⣤⣤⠆⡂⠩⢗⡉⠛⢠⡌⠻⢼⡩⠧⣽⠀⠘⠀⠿⠀⢤⡉⠈⡆⣙⠁⢷⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠹⢢⣅⢠⢨⠛⠖⡄⠰⠆⢺⠛⣶⣭⡓⣤⡃⣦⣻⠞⣁⠨⠶⢈⠷⠀⣄⣇⠤
⠀⠀⠀⠀⠀⠀⣤⡴⠘⢂⣑⠚⣄⠥⣤⢀⣤⢖⡻⣟⠰⡀⠀⣶⢶⡞⢱⣄⣿⣘⣦⣊⣒⠇⢠⡌⠔⠃
⠀⠀⠀⠀⠀⠠⠾⣤⠘⠿⢧⠇⢈⡱⢦⣌⡬⢻⡁⢻⣄⣴⠊⠷⣼⢠⡞⢭⢀⠖⣤⠐⢀⣵⡊⠁⡆⡒⠶
⠀⠀⠀⠀⠀⠀⢀⣀⠋⠴⠀⣂⡘⠁⣛⣉⢷⡖⠘⡙⣿⡑⠄⡽⣣⡟⡰⠦⣼⠳⠀⣶⢀⡘⣥⠰⠇
⠀⠀⠀⠀⠀⠀⠠⠝⢡⠤⣃⠤⣠⠴⠶⢤⣩⡟⢾⣅⢸⣧⣾⣵⠟⢸⣷⣎⡬⣕⠂⣽⣦⠉⡀⠠⠜⣤⡀
⠀⠀⠀⠀⠀⠀⠸⠇⣲⣀⠘⠂⣈⠋⣰⢿⣮⠳⣦⡙⣿⣿⣿⣷⡿⣿⡉⣱⡦⠝⢃⡄⢍⡚⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠨⠌⢁⠱⠀⣧⠡⠀⣀⠈⢻⣾⣿⣿⡏⠀⠈⠃⢀⡰⠒⠀⠠⡌⠑⢇⡐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠃⠀⣅⠷⠀⠰⡶⡉⠀⠀⢻⣿⣿⠀⠀⠀⠀⠨⠤⢀⣄⠃⠀⠠⠾⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠀⠀⠘⠁⠙⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠓⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠂⠈⠁⠀⠉⠁⣀⡀⠀⡁⠀⠑⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠤⠴⠀⠀⡀⣀⠹⠿⠄⠈⢀⣠⠀⣤⠤⠁⡙⠁⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢⢆⣀⡀⠀⢀⣤⠖⠀⠙⢚⣙⠀⠀⠄⠀⠠⣤⣀⡁⠀⣀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⣠⢤⣾⠁⠀⠁⠯⡿⠀⠀⠀⠀⠀⠲⢂⣾⣶⣤⠄⢠⣽⡀⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠀⠀⠀⢐⠘⠃⣹⢷⡄⠀⠀⠀⣻⣧⠀⡀⠀⠂⠉⠱⡀
⠀⠀⠀⡀⠀⠀⠀⣠⡀⡸⠮⠲⡇⠒⠼⣷⣿⠏⠀⠉⣰⣦⣺⢿⣯⢏⠛⠰⠆⠤⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⡀
⠀⠀⠠⠀⠀⠀⠀⡙⠈⢡⠄⠈⠈⡒⠄⢈⣻⠇⠀⠀⠊⠉⢼⣾⠟⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠆⡈⢹⢿⢿⡗
⠀⠀⠀⣀⠀⠚⠁⠙⠚⠣⠇⠀⡀⠷⠌⠋⢉⣇⠀⠀⠀⠀⡼⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⣤⠄⣀⡀⡀⠠
⠀⢀⣍⣻⣿⡏⢀⣓⣶⢷⣶⡏⣯⣤⣤⣶⠂⣤⢶⣾⣷⣾⣶⣀⣔⡀⠒⠀⣀⣀⣀⡀⠀⣀⣀⠀⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠈⠉⠁⠈
⠀⠸⡷⢶⠄⠻⠻⣿⣿⠏⡟⠿⠐⢚⡷⠶⣿⣟⠙⠛⠀⠛⠻⣟⣩⣐⠂⠀⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⠤⢄⣀
⠀⠀⠤⠤⠤⠥⠤⠭⠝⠛⠛⣁⣘⣥⠖⣖⣵⠀⠁⠀⠁⠀⢀⡘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠤⠤⠦⡄⢀⠀⠀⠀⠊⠁⠀⠀⠛⠠⠉⠁⠂⠀⠀⠀⠈⠃
⠀⠀⣀⡠⠎⠁⠀⢉⣤⡍⠀⠤⠢⣀⣀
⠀⢈⣍⠁⠚⠀⠀⠘⠓⢢⣤⡴⢠⣬⡍⢉⡂⠀⠄⠀⠀⠀⠈⠉⠁⢀⠄
⠀⠈⠁⠰⠄⠀⠢⠀⠈⠁⠀⠂⠀⠒⠘⠉⠀⠉⠁⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣶⣶⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢍⡱⠋⠉⠀⠀⠉⠒⢰⠆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢻⠻⠿⠓⢟⣀⠬⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⣳⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢿⣠⣤⡀⢼⣇⠀⢀⣞⠀⢲⣾⣿⣶⠛⠃⠀⣤⡙⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⣧⣶⡿⠊⣡⣤⣤⡐⢀⣀⡤⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣏⢹⠉⠁⣞⣵⣷⣤⡎⣿⣿⣿⣿⣿⠯⠃⠉
⠀⠀⠀⠠⠤⠤⡄⢸⠀⠀⣆⡖⠀⠁⠃⣸⠘⣿⢻⡋⣰⡏⠻⠛⡌⣽⣿⣿⣿⡅⠎
⠀⠀⠀⠀⠈⠂⠀⢸⣇⠀⣡⠁⠀⠀⠂⠘⠆⢻⣆⣿⣿⣧⢀⣰⢧⣻⣿⠯⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⠛⠗⠀⠀⠀⠀⠐⠂⠹⣦⣽⣿⣮⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠓⠀⡤⠊⠁⠀⠀⠀⠀⣄⣾⣿⣿⣿⠒⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢿⡧⠄⠀⠤⠀⠀⠀⠀⠀⠘⢛⣿⠛⣿⣽⣿⡿⠘⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣤⣠⣼⠷⠶⠓⠒⠀⠀⠀⠀⠀⢀⡟⠙⢯⣜⣥⣿⢥⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠂⠈⠉⠙⠙⠆
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠐⢀⠸⡇⠈⠑⢄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠳⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡈⠃⠀⠀⠤⠀⠀⠈⠀⢀⠀⠱⡀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄
⠀⠀⠀⢀⠐⠂⠐⠂⡃⢸⠀⠀⠈⠀⠀⢀⢀⡀⡀⠀⠀⠙⠢⣽⣀⡌⠀⣀⠶⡤⢤⣀⠀⠀⠀⣶⢰⠟
⠀⠀⠀⠀⡠⠐⠀⠀⠁⢨⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣣⠏⠀⡄⠀⠀⠀⠀⠤⣴⣯⣀⢀⡀⠀⢀⠶⡃
⠀⠀⠀⠀⠀⣠⠖⣃⠀⠀⢀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣯⣭⣭⠛⠀⠁
⠀⠀⠀⠀⠀⠨⠁⠙⡂⠀⠈⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠛⢿⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠂⠀⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠢⣀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠘⢦⡀⠈⢧⡀⢀⡀⢳⡀⢶⠀⡇⠺⢰⠇⡔⠀⡴⠃⠀⠀⣠⠞⠃⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠙⢦⡀⡑⣄⠱⡄⢣⠸⠀⡇⠆⡌⡼⢁⡞⢠⠜⣠⠞⠁⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠢⡙⠢⡈⠣⡀⠀⢀⡀⠇⡀⠈⣠⠋⠔⡡⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⣤⣀⠀⢬⡑⠬⡂⠀⠠⠂⣠⠤⠄⠢⠄⡁⠐⠄⢀⡠⠖⣉⠤⠒⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣤⣀⣀⣀⠀⠈⢉⡒⠈⠑⢈⠟⢠⠊⠀⠀⠀⠀⠀⠈⠳⡔⢅⠀⠪⠄⠒⠊⠁⢀⣀⣠⡀⣀⣀⣀⣴⠖⠶⠤⠤⠤⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠑⠒⠒⠤⠌⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⠀⠐⠒⠊⠉⢉⡝⠒⠛⠁⠈⠉
⠠⡤⠶⠒⠒⠒⠚⠋⠉⠀⠀⠀⠉⠉⠳⢄⣀⡉⠉⠉⠀⠐⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⡧⠬⠥⠤⠬⠭⣱⠖⠋⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⣄⣠⠤⠸⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⢀⣠⠼⠃⣀⠖⠋⠀⠀⠈⠉⢁⣠⠤⠤⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠐⠂⠁⠁⠐⢦⡀⣀⡀⠀⠀⠀⠤⡀⠀⡀⢠⣳⣖⡒⠒⠠⠤⠤⠤⠤⠤⣴⠞⠉⢉⡀⠐⠬⠈⠀⠴⡆⡀⠾⣉⣉⠁
⠀⠀⢀⣄⠀⣤⠄⢤⠄⠀⣠⠀⠤⠤⠬⠭⠥⠴⠦⠤⠤⠤⠴⠶⠿⠖⠂⠀⠀⠀⣤⠼⢃⢰⣀⠉⡤⡀⣀⠠⠀⣠⠔⢁⡽⠄⠈⢫⣉⣙⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡼⣃⣤⠾⠷⡼⠖⠛⠶⢅⣀⣀⣀⣜⣠⣤⣅⣐⣀⣈⣉⣄⣁⢇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡖⠒⣒⠲⠤⠶⠻⢛⣛⣋⢀⣤⣄⠤⠤⠤⠤⠤⠶⠤⠭⠭⠁⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣴⠶⠒⠒⠚⠛⠚⠛⠛⠒⣾⣉⡉⠀⠀⠀⠈⣡⣤⣵⣐⠒⣒⣒⣃⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⠤⠤⠔⠀⠀⠦⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠁⠀⠐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⢇⡴⣤⢴⠬⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⢤⣶⣾⡶⢠⠀⠀⠀⠀⠀⣹⣦⢤⠀⠠⣖⠀⣿⡆⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠡⢻⣿⣿⡗⠉⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⠋⢠⣿⡧⠀⠀⠀⠀⠀⠀⠙⣿⣛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠏⠀⣿⣯⡋⢀⠀⠀⠀⠀⢪⡉⠹⠿⠆⠋⢀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢥⡂⣻⣿⣟⠂⠈⢧⠐⢄⠄⠀⠀⠀⠈⢦⡀⢫⡚⡻⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣻⢃⣾⣙⢿⣾⣯⠆⢀⣹⣧⡀⠀⠀⠀⠀⠀⠐⢔⠀⢳⠞⢷⠀⠀⡀⠀⠀⠀⠀⠀⣾⣯⡦
⠀⠀⠀⠀⠀⠀⣴⠟⣯⣾⣏⢁⣾⣿⡟⠋⣀⢈⡿⡇⠢⣀⠀⠀⣴⠀⠀⠑⢮⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⢷⣀
⠀⠀⠀⠀⠀⢰⡟⠀⠐⢷⠿⡿⣈⣿⠀⣹⢻⣾⣵⣾⡀⠈⠑⠄⠋⠀⠀⠀⠀⠀⠀⢠⣀⡠⠀⡀⠀⠀⠀⠈⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣘⠅⢁⢈⢹⠛⠛⣸⣼⣿⣬⡀⠀⠀⠀⢠⢠⣄⠻⠀⠀⣼⡿⠁⠀⠀⢆⠀⣠⣠⣤⣤⡿
⠀⠀⠀⠀⠀⠐⠃⠀⠈⠀⢹⠂⠈⠐⣿⣶⣧⣿⣿⣿⣟⠀⡠⠤⢄⣾⡿⠇⠀⣠⣿⣿⣟⣓⢀⠀⠻⠋⣽⠟⠿⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈⠀⠀⢠⣼⡼⣿⣿⣹⠷⣿⠆⣤⡀⠀⠀⠈⠓⠿⢄⡿⣿⣿⡟⠁⠀⠀⣼⠏⣀⠂⠐⡦
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡆⢠⢇⠀⠈⢡⠟⠙⢿⣿⣿⣿⣿⣶⢤⡄⠀⠈⠙⠻⠏⢴⣦⢢⡆⡿⠀⠄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣴⡀⠀⢸⣿⣿⣿⣿⣧⣿⣻⣟⣠⣶⣧⣀⡀⠀⢨⣶⢀⡘⠐⠃⠁⠑⢶
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠹⠼⠁⠀⢰⠀⠙⠄⠀⠹⣞⠛⡛⢻⠀⣬⡿⠟⠛⠉⠙⢏⣤⣤⣉⣁⣡⣿⣾⣮⣌⢛⡀
⠀⠀⠀⢀⠀⢀⠠⠀⠀⠀⠀⡰⠁⠀⠀⠀⢘⠀⠀⠀⠀⠀⠻⡛⣾⣠⡄⠉⠀⠀⢠⣤⠀⣴⡿⡿⣿⣿⠋⠀⠀⠙⠃
⠀⠀⠀⠘⣷⢡⢂⠂⡌⢠⠛⠁⠀⠀⠀⠀⣸⡷⠂⠀⠀⠀⠀⠀⠙⣿⣿⣠⡾⣷⣿⠋⠘⠋⠁⠙⣞⠁⠀⣤⡀
⠀⠀⠀⢠⠤⢃⠎⡸⢠⠇⠀⠀⠀⠀⠀⠀⠸⢿⡄⠀⠀⠀⠀⠀⠀⠘⢿⣫⡞⠉⠁⠀⠀⢠⠀⣰⠟⠀⢠⡿⠀⢠⣤⣴
⠀⠀⠀⠘⠖⠋⡰⠳⠋⠀⠀⠀⠀⠀⠀⠀⢰⣶⡍⠁⠀⠀⠀⠀⠀⠀⠠⣤⠁⠀⠀⠀⢒⡃⢀⣿⠁⣰⠈⠁⢀⣼⣿⣿
⠀⠀⠀⠘⠧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⠀⠁⠀⠀⠀⠀⠀⢤⠆⠀⠀⠀⣠⡼⠁⠀⠀⠀⠁⢀⣴⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⡓⣒⡀⢀⢀⠀⢀⠎⠀⢀⠂⣠⡿⠁⣼⠀⠀⠀⣠⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣄⠀⠀⠈⠀⠂⠀⢀⠎⣸⡿⢁⣴⠟⢀⣠⣾⠏⣪⡻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⡌⠈⠁⠈⢁⣄⡤⠀⣠⠔⣡⠞⢋⣴⣿⣣⣴⣿⣿⣿⣾⡏⣻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠌⡱⣌⠂⠠⢿⠿⠟⢡⣼⣯⡤⠤⠶⠿⠟⠙⣋⡩⠉⠉⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣴⣿⣤⢤⣭⣭⠿⠶⠦⠒⠊⠠⠀⠀⡐⠀⠄⡔⣟⣀⡡⡤⣤⣤⢄⠠⠴⠒
⠀⠀⠀⠀⠀⠀⣀⠿⠶⠛⠛⠛⠉⠉⠠⠐⠀⣀⣀⣀⡠⠬⠄⠢⠶⡒⢑⠝⠉⠽⠟⠐⠛⠈⠉⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤⠶⠶⠖⠚⠛⠩⠉⠁⠀⠀⠀⠚⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⡇
⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⣀⡀⡿⠀⢸⠃
⠀⠀⠀⠀⢠⡏⠀⢻⠀⠀⠀⠀⢸⠉⠙⠇⠀⡾
⠀⠀⠀⠀⠀⣧⠀⢸⡄⠀⠀⣀⣸⡀⠀⠀⠀⣷
⠀⠀⠀⡴⠶⠾⡄⠀⢧⠀⠰⡏⠉⠃⠀⠀⠀⢸⡀
⠀⠀⠀⢳⡀⠀⠀⠀⠈⢧⠀⢳⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⢀⣳⡀⠀⠀⠀⠈⠳⣖⠻⠄⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⢀
⠀⠀⠀⣯⠁⠉⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⡘⢷⠛⠋
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠑⢄⠀⢠⠖⠈⠑⠲⡀⠀⢸⣧⠸⣇⣻⣔⠛⠋⠁
⠀⠀⠀⠀⢀⡼⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⠀⠈⠗⠁⠀⠀⠒⠀⠁⠀⣨⠷⢾⣭⣍⠛⠛
⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⠖⠀⣰⠃⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢓⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣤⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⡀⢀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠤⠒⢧⠤⢺⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣛⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠬⢭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣛⠳⠄⠀⠀⣤⣤⣾⣯⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠒⠒⠒⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⡖⠀⢰⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⠀⠀⢀⣭⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡾⠁⠀⠋⢠⠆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠆⠸⣇⠀⠶⠀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⢿⡆⠀⠀⠘⠈⠃⠀⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢞⣯⠉⠁⠀⠀⠀⣰⠃⠀⡆⠀⢀⣴⠀⠀⠀⠀⠀⢉⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣌⣋⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠁⠀⠀⢀⣀⣠⣤⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⡀⠀⠉⣉⣉⠐⠒⠓⠒⠒⠒⠒⠓⠛⠛⠉⠉⠉⠀⠀⠀⢸⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣀⣈⣻⠛⠃⠀⢰⡇⠠⡄⢠⣆⠀⠀⠀⠀⠼⢫⡿⠀⢸⡇⢠⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠏⠉⠙⢢⣤⠤⡄⠹⡄⢀⡄⠉⡇⠀⣠⠶⢶⡄⠀⠀⢸⣷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣨⠴⠶⣄⠀⠀⠐⠃⣠⠇⠸⠁⣼⢁⡾⠀⢈⣸⠇⠀⠀⢸⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠺⣧⡶⠂⠘⣷⠀⠀⠐⠋⠀⠀⠀⣉⡴⠃⠰⠟⠁⠀⣰⠆⢸⡏⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⣸⠏⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠰⠖⠛⠁⠀⢸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠆⠀⠀⠀⠀⣠⡾⠏⠀⠀⠈⠉⠁⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⣸⡆⢸⣿⡛⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⠛⠳⣦
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⡟⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣦⣤⣴⠏
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢸⣿⠁⠀⠘⡄
⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠤⠦⠞
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇
⠴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠈⠀⠒⠚⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠉⠙⠲⠶⢤⣤⣍⣉⣙⣛⣛⣀⣀⣬⣤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠜⠋⠁

⠀⠀⠀⣀⡠⠄⡀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠤⢍⣩⠩⣽⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣣⡄⠀⠀⠀⠀⣾⣷⣾⣂⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠨⠿⠞⠛⣟⣻⣿⣿⣧⡤⣀⡀⠀⠀⠒⣻⣿⡿⠅⠀⠀⣰⣾⣿⣿⣿⣿⣶⣦⠀⠀⠼⣿⣿⣟⣻⡿⣿⣿⡉⠃
⠀⠀⠀⢀⣀⣤⣴⣿⣿⣟⢿⣿⣿⣟⡿⢷⣤⣀⠈⠇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠁⢸⡏⠁⠉
⠀⠀⠐⠋⠉⠉⡬⣭⣿⠿⣟⡯⢿⣿⣷⣾⠛⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⢃⣀⡀⣐⣦⠤
⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠈⠻⠄⠀⠀⠀⠀⠀⢠⡄⢀⣀⠀⠉⠛⠉⠉⢹⠏⠉⠙⠋⠁⠀⠀⠀⠀⢠⣿⣿⣿⣧⣿⣯⣵⡆⡻⢶⡀
⠀⠰⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⣠⣤⣶⣿⢐⣬⠶⢉⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⣀⣴⣶⣦⡄⠀⠀⠋⠛⠛⢿⡻⡿⠿⠿⠃
⠀⠘⡿⣽⣿⣿⡿⠄⠀⠀⠀⢠⣿⣾⣿⣿⣏⣿⣿⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣜⢻⣿⣿⡟⠛⡃⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠘⠉⣉⣉⠁⠀⠀⠀⠀⠾⠘⠋⠿⢿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢯⣈⣻⣿⣿⣭⣤⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠠⣽⠦⠀⠀⠀⣠⣤⣤⣤⣤⣠⣀⡀⠈⠉⠱⡿⠉⠀⠑⠀⠠⣤⣶⣦⣤
⠀⣠⣴⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣏⣻⣇⠤⠄⠀⠀⠀⠀⠀⣾⡿⢿⣿⣿⣿⣷⣦⡄
⠀⣿⣯⠽⠿⠿⠾⣿⣿⣿⣿⡟⠿⣿⣿⣿⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣄⣀⠀⣴⢶⡾⣿⣧⣾⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠈⠀⠀⠀⠀⠿⠿⠯⠟⣿⠧⢴⣿⣿⣿⣿⣦⠀⠀⠉⠈⠉⠻⣻⣧⡿⠋⠉⠈⠉⠉⠁⠀⠀⠀⠈⠁⠈⢻⡿⢩⠋⠻⠟⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢺⠟⢟⠀⠤⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⡀⠒⠒⠄
⠀⠀⠀⠀⠀⠀⢀⠈⠂⠈⠆⠀⠀⠀⢀⠅⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡟⡉⠉⠂⢰⠤⠐⠓
⠀⠀⠀⠀⠀⠀⠈⠁⠸⢄⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠈⠃⠀⠠⠈⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⢀⠀⠀⠀⠠⡀
⠀⠉⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀
⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡄⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡠⠀⠤⠶⠈⠊⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⡆⢀⡀⠑⠠⠐⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⢦⠧⢄⠉⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠊⠑⠂⠉⠓⠒⠠⣀⣀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡠⠂⠀⠀⡠⠚⠁⠀⠐⢄
⠀⠀⠀⠀⢠⡟⠀⠀⠀⡠⠊⠀⢀⠄⠈⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⣀⡤⠤⠄⠀⠠⠄⠤⣀
⠀⠀⠀⣠⠞⠁⠀⡠⠊⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢾⡄
⠀⠀⠀⣏⢀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠢⣀⡲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠔⢦
⠀⠀⠀⡝⠁⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠔⠊⠉⢀⣉⠂
⠀⠀⠀⢷⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢀⠶⡉⠙⢅
⠀⠀⠀⡾⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠹⡁⡔⠘⠤⠎⠀⠀⠀⢠⢊⠉⠁⢒⡦⣀
⠀⠀⢀⡇⡨⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣢⠔⠊⠉⠉⢣⠀⠀⠀⠀⠀⠢⡀⡔⠁⠈⠙⠷⣄⡀
⠀⠀⡼⠁⠃⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢧⡠⣖⠘⢏⠁⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀⡐⠁⠉⠒⣼
⠀⡸⠁⠘⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⡤⠓⠀⣞⣉⠦⠤⢖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⣀⣴⠟
⠐⠁⢀⣡⡄⡀⠀⠀⠑⣄⠀⠀⠀⢠⠔⠂⠀⢶⡋⠀⢀⣀⣀⠆⢀⡤⣀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠧⠤⠼⠉⠉⠃⠀⠀⠀⠈⠳⢄⡀⡸⠀⠀⠀⠀⠙⠋⢳⣠⠀⠀⠘⡷⠁⢴⡁⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⡀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠳⠜⠒⠊⠀⢀⡠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⡸⠀⠀⠀⠀⠀⢀⠠⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡌⠀⠀⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠔⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠳⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠏⡇⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⣀⡀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⢺⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⡄⠀⠀⠀⠀⢀⣿⠃⠀⠀⢐⣽⠇⠑⢷⣄⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⢳⡻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃⠼⣧⠀⠀⠀⠀⢸⣿⠀⠀⡆⠈⠀⠀⢀⠘⢿⣤⠀⢸⣿⠀⠀⠀⠀⠀⣠⣿⠃⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⢛⣧⠀⠀⠀⠘⣿⡄⠠⠘⡆⠀⠀⠀⠀⠀⢳⡄⣸⡿⠀⠀⠀⠀⣴⣿⡟⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⢻⣿⣷⣄⡀⠀⠹⣿⣤⠀⠙⠤⠀⠠⠖⠀⠀⣷⡿⠁⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣦⣬⣿⣴⡁⠆⠀⠀⠁⠠⡆⠈⢠⣤⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⡘⣷⡄⣀⣀⣀⠀⣄⣀⠀⠩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⢰⠾⣇⠀⢀⠄⠀⠀⢠⠈⠙⣷⡅⠀⠀⠀⣿⠈⣿⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠈⠀⢾⣦⠋⠀⠀⣼⡇⠀⠀⢁⠐⢀⡿⢀⢹⠀⣷⠈⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡄⠀⣿⡇⠀⠀⠌⠴⢋⠄⣨⠀⠀⣿⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠘⠁⡟⠃⠀⣿⡇⠰⢚⣡⠞⣡⡾⢋⡀⢸⠻⣧⠑⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⢶⡀⠐⢁⡾⣀⠙⢒⡾⢋⣴⠟⢈⣅⣤⣤⢸⠀⣏⠑⢦⣱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡿⠗⠲⠿⡇⢻⡿⣫⠴⢛⣡⠾⠏⡇⢰⢹⣇⣇⢸⡄⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⢡⡟⢀⣤⠖⢛⠀⠈⠀⡇⢸⡹⣟⠎⠀⢛⣠⠊⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣾⠉⣿⠁⢸⠀⠘⣶⠄⢀⡇⢸⡇⢿⠈⠄⠈⢿⡁⢠⣈⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢛⠀⣿⠀⢸⠀⠀⠛⠀⣼⡇⢸⢿⠈⡆⠀⠀⠀⠙⠂⡽⢿⡄⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣼⠁⠀⠀⠈⠀⣿⠀⢸⠀⠀⠀⢰⣿⡃⢸⡌⠃⠸⡄⣰⣄⠀⣠⡆⠀⠀⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⡇⢸⣿⠀⢸⠀⠀⢠⡟⢸⠀⢸⢷⠀⠀⠹⡸⣿⣿⣿⣓⣷⣤⣿⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⡇⢸⣿⠀⢸⠀⠀⢻⡆⢸⠀⢸⠈⠀⠀⣶⢳⡙⠋⡁⢀⣤⠀⢸⡆⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⢰⠁⠀⠀⠀⠀⡇⣿⣿⠀⠸⡀⠀⠸⠃⣸⠀⢸⠀⠀⠀⠸⡄⢻⡾⣰⠏⢻⡆⠈⣷⢠⠀⠀⠀⢸⣷
⠀⠀⠀⠀⣾⡿⠀⠀⢀⣧⠎⠀⠀⠀⠀⠀⡄⠟⢸⠀⠀⡇⠀⠀⠀⣿⠀⢸⡆⠀⠀⠀⢹⢨⣤⠃⡔⢸⡇⡀⠘⢧⡄⠀⠀⠸⣿
⠀⠀⠀⢰⣿⡇⠀⠀⠸⠈⠀⠀⠀⠀⠀⠀⠙⣆⢸⠀⠀⣷⠀⠀⠀⣿⠀⢸⡇⠀⠀⢠⣾⠸⡏⣸⠀⠀⣷⠀⠀⠘⣇⠀⠀⠀⣿⡇
⠀⠀⢀⣿⡟⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠀⠀⢻⠀⠀⠀⢻⡀⠸⣷⠀⣦⠘⣿⡆⣿⠃⠀⠀⢿⢠⠀⠀⢻⠀⠀⠀⠸⣿
⠀⠀⣼⡟⠀⠀⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⢸⡆⠀⠀⢸⡇⠀⡿⡆⢸⡆⢻⡇⠇⠀⠀⠀⢸⡏⠀⠀⠈⡇⢀⠀⠀⢻⣇
⠀⣸⡟⠀⡄⡸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡧⠀⠀⣸⡇⠀⡇⠈⠀⢷⣼⣷⠀⠀⠀⠀⠘⣇⡇⠀⠀⢸⡘⡄⣄⠈⣿
⢰⣯⣤⡞⠰⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣧⠀⠀⢩⡇⠀⡇⢇⠠⡘⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢃⢻⡼⡷⣼
⠈⠁⡀⢠⠳⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡷⣄⠸⡇⠀⢧⠸⡀⢳⢹⣿⠀⠀⠀⠀⠀⢹⣼⠀⠀⠀⠈⠾⢷⣷⡄
⢀⡞⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠶⠶⣶⣿⠳⡈⠁⠀⠀⢸⡀⢳⡄⢸⣿⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⠀⠀⠀⠳⡙
⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠖⢻⠈⡟⠀⠀⠀⠀⠐⣌⡻⠄⠹⣿⠿⡄⠀⠀⠀⠀⠀⣷⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⠀⢸⠀⡇⠀⢸⡇⠲⣤⡈⠻⣶⠒⢶⣾⡇⠀⠀⠀⠀⠀⢿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣿⠀⢸⠀⡇⠀⢸⡇⠀⡟⠀⢲⣦⠅⣈⣩⠂⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡀⢸⠀⡇⠀⢸⡇⠀⡇⠀⠸⣿⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠉⠛⠖⠚⠓⠒⠧⠤⠞⠛⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠄⠀⠀⠀⢰⡀⣠⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⢠⢠⡀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡾⠁⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣴⡶⡀⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠷⠏⠀⠁⠀⠀⢀⣨⣴⢙⣿⡿⣿⣟⣿⣾⢿⣿⡇⠀⠑⠳⠽⠇⢲⡿⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⢯⠠⠑⢶⣶⣬⠭⣤⣥⣭⣤⣼⣿⣿⣣⣬⣛⢿⣿⣇⣐⣀⣖⣚⣿⣷⣾⣷⣿⣿⣦⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠶⣛⣽⣒⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡭⢙⣮⣡⣌⣿⣿⣿⣿⡯⠿⢿⣏⡷⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠁⠀⠈⠉⠋⠉⠋⠉⠚⠋⠉⠙⠛⠋⠋⠉⠁⠉⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠦⡀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠉⢠⣄⣀⠀⠀⠀⠀⢱⣼⣷⣾⠄⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⡀⢤⣚⣿⢿⡟⠀⠀⠀⠈⠀⠀⠀⠑⠄⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠑⠤⣄⡀⠀⠀⠀⠀⠀⢈⡄⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣿⣿⣻⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣤⡠⡀⠀⠀⠂⠄⠐⠰⢄⣀⢀⡠⠀⠠⠉⠀⠐⠀⠀⢹⢿⣿⡿⠄
⠀⠀⠀⠀⠀⠸⠅⠀⠀⢀⠄⠉⠀⠀⢰⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠏⠃⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⠁⠀⠀⠀⠸⡺⣄⠀⡀⠀⠀⢃⠀⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠤⠄⣀⠈⠻⢿⣾⣷⡤⠂⠁⠀⠀⠸⠁⠀⣤⣀⠀⢀⡀
⠀⠀⠀⠈⠀⠀⠀⡠⢮⡿⠁⡀⠀⠠⠬⠿⢻⣿⣿⢷⡦⠤⠖⠚⠁⠙⢷⡏⠿⠛⠀⠀⡁⠀⠐
⠀⠀⠀⠾⠁⠉⠀⠀⠈⠁⠀⠀⡄⢀⣨⠶⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠜⠃⢨⠀⣠⠀⠀⠂⠀⠀⠀⠀⠀⠀⣰⠌⠀⢀⡏⢓⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠠⣿⡃⠀⠀⠑⠒⠀⠒⠂⠁⠀⠀⠀⠩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠢
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⠀⣜⡄⡀⠀⢢⠂⠸⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡁⢀⣼⣿⣧⠀⠘⠀⠀⡇⣻⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣜⣡⣿⠏⢠⠆⠀⣰⢃⡆⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡿⣸⣿⣿⡟⠀⢟⠀⣰⢃⡬⢀⡼⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣯⣿⣿⠁⢀⡟⢐⣣⡾⣰⡿⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡟⢠⠾⣁⣾⢋⣽⠋⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⣿⣿⣿⠃⠙⠄⠸⢵⠟⢁⡶⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢸⣿⣿⣿⠋⢠⡴⠁⠀⣤⡴⢋⣠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣜⡻⠿⠶⠓⠘⢿⣿⠁⠀⢋⣴⠖⠁⢠⣾⠟⢋⡠⠖⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣷⠀⠀⠈⠄⠙⢷⡄⠈⣁⣤⠆⢨⣴⠿⠏⣁⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⡇⠀⣉⣠⣤⣴⠶⠾⡟⣉⣀⠐⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣸⡿⠁⢀⣁⣉⣙⣛⣛⣟⣋⣉⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡄⠈⠀⠀⣄⠲⠦⡤⢴⣭⣉⣛⣟⣁⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠈⢙⣄⠲⣶⣬⣉⣛⣛⡉⡁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⠀⠀⠘⢿⡿⢿⣥⢌⣁⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣀⡀⠀⠻⢶⣽⣓⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⡀⠀⠀⠙⠿⣿⣿⣿⠀⠀⠀⠙⠛⠛⠁⠆
⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⡘⠃⠙⢿⣦⢀⡰⣦⡀⠉⠻⣧⣀⢀⣀⣤⣄⣤⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣰⡖
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠹⢯⣑⣓⣢⣽⢏⣃⠘⣷⣄⣷⣜⢿⣿⣿⣿⣿⡿⠿⠉⠛⠿⣿⣿⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⠉⢳⣆⣈⣑⣺⣿⡙⢿⡆⢹⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣭⡁⠙⢺⣟⣘⣆⠸⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣵⠞⠋⠉⢟⣧⣤⣧⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢂⣀⣀⣙⣿⢿⣌⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣯⢀⣠⣴⠟⠉⠛⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠂⠚⠉⠘⠢⠔⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠊⠀⠀⠀⠀⠀⠀⣀⣴⣄⣀⡀⠀⣀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⡀⠀⠀⠀⠀⢀⣸⣯⢉⡙⠁⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠂
⠀⠀⠀⢀⡀⠤⠀⠒⠂⠚⠁⠹⠛⠆⢴⣏⣲⣿⣿⣿⣯⣧⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣲⣊⣄⢰⣿⡛⠁⠈⠉⠙⠐⢣⣤⡀⣄⢚⡃⠀⢠⠌⡧⠴⠛⠀⠀⠀⣤⠀⠀⡄
⠀⢀⣀⠀⣠⣀⣠⡀⢀⠀⠲⣼⣿⣿⣿⣿⡿⢆⣙⢶⣤⣆⡸⣷⣿⣯⣿⣦⠿⠷⣦⡄⣶⣬⣥⣀⣀⣾⣷⣊⡀⠂
⠀⠀⠁⠚⠒⠛⠿⣿⣿⣷⣿⣿⣿⣿⣿⣷⡶⠿⠿⣿⣿⣿⡿⢻⣿⣽⣥⣔⣒⡋⠁⠐⠋⠉⠋⠁⢛⣿⣿⠛⠁
⠀⠀⠀⠄⠠⠀⠂⠀⠒⠝⠛⢀⠛⠀⢜⢻⣅⢿⣯⣼⣿⣻⣷⣸⣿⢿⣷⣶⣷⣶⡆⢠⣄⣤⣄⣼⠛⠉⠁⠀⢀⠉⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢓⣦⣀⡚⠓⠙⠛⢛⣿⡿⢷⣇⡿⠋⠈⢿⡿⣿⠿⣿⡷⡿⠀⠉⠀⠀⠀⠀⣀⣈⡀⠀⠀⢀
⠀⢐⣂⡄⠀⢀⣀⡀⠀⠠⠉⢩⡭⢉⢽⡶⢦⣼⣿⡟⢁⡿⢋⡀⣸⣠⡏⢀⣤⣬⣟⣂⣄⠂⠀⠀⢠⣬⠣⠘⣫⠄⣤⣇⣩⡁
⠀⠈⢻⣶⢆⣍⠿⠖⣛⡉⣀⣮⡟⢥⣿⣧⣦⣿⣿⣧⣼⡧⣿⠛⢻⣿⢻⡏⠀⠸⠆⢴⠀⠀⠴⢡⣶⣷⠮⣿⣿⣾⣿⡶⣸⠿
⠀⠀⠀⠀⠉⠐⡿⣾⣿⣷⣶⣿⣿⣷⣾⣿⣿⣎⣿⣿⣿⣇⣿⣴⣿⣧⣾⣯⣤⣾⣧⣶⡭⠾⢗⣋⣿⣿⣿⣿⡿⣛⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠿⠛⣫⠿⣿⣿⣿⣿⠟⢿⣿⣿⣾⣿⠻⣿⢟⣿⣿⣿⣿⣷⣎⢻⡁⢠⡾⠉⠘⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⡿⠙⡃⠐⠀⠉⠛⠻⠟⠛⠛⠛⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⢀⣀⣤⣤⣴⠶⠿⠿⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⡄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠁⢠⠀⡀⢠⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠤⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠠⠃⠈⠑⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡶⠀⠀⣀⣤⠆⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡻⠻⠆⠀⠀⠀⠀⠀⠀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⠀⠀⢀⣠⣆⣠⠀⠀⠃⠀⢀⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⡿⠁⠀⢠⡋⠉⠹⢃⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠶⠆⠂⠀⠺⢿⢇⡀⡀⠀⡸⠿⠛⢀⡿⢶⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⢻⡃⢠⠇⠀⢠⠂⠀⠠⠀⠀⠀⣴⣿⠟⠲⣄⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⠏⠀⠀⡇⠀⢠⣿⠟⠻⠀⠀⢹⠆⠀⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢠⣄⠀⠘⢀⡄⠀⠃⠀⠉⠉⠀⡴⠞⠀⠀⠀⠀⢠⡆⠈⢢⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⠀⠀⠀⠞⢷⡛⠱⠄⠀⠀⡴⠀⠀⠀⠀⠀⠀⠐⠿⠃⣀⣸⡟⠦
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡄⢲⡆⠀⢠⣷⢀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⢸⡋⠁
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡎⠑⠇⠀⢘⠉⠘⡇⢢⣿⡶⠊⠀⠀⠀⠀⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠃⠀⠠⡘⢾⡄⠀⣏⡼⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⣷⡄⡟⢀⣶⣇⠀⠀⠀⠲⢀⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⡀⢠⡄⢱⠙⡄⢸⣿⢋⡀⣘⣀⠀⠘⠛⠛⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠱⣾⡇⢸⡄⠁⠀⡇⢈⠷⠛⡟⢃⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⣤⣾⣶⡷⠴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠀⠈⡇⣸⣷⣷⡿⠟⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⢳⡏⣿⠥⣶⣶⣽⣿⡴⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠠⣄⡀⠀⠀⠀⣸⠇⣿⣠⣨⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⠙⢦⡀⠀⣿⡔⡿⠃⠛⠛⢿⣼⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢿⡆⢿⡿⡇⠀⢀⣀⡀⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣿⠾⠃⠀⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲
⡰⠒⠢⡄⠀⠀⣀⠀⠀⠀⠀⠸⢱⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠒⡄⠧⠴
⠑⠤⠠⠎⠀⡘⠂⠃⠀⠀⠀⠀⠀⣤⡀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣩⠆⠰⠅⠜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢎⣠⠇
⠀⠀⠀⢠⡏⠉⠑⢦⡤⡀⠀⠀⠀⠂⠣⡀⠀⠀⠘⢍⢧⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⠁⠀⠀⠀⠀⢀⠴⠤⢔⡺⠛⢉⡭⠝⡩⠿⠽⢦⠀⡠⢤
⠀⢀⠀⠀⠙⠛⠛⠲⣄⠙⣄⣀⣀⡼⠆⣈⡉⡒⠤⠒⠃⠉⠁⠀⠈⠉⠉⠢⠴⠊⠑⢒⣁⡴⣤⠤⡝⣯⡛⠒⢒⣉⣰⡴⠋⠀⠀⡿⢀⡾⠔⠊⠉⠁⠀⠈⠉
⠀⠈⠀⠀⠀⢀⡴⠒⠈⢳⣉⠲⢷⡖⠲⣆⣏⣁⣠⠦⣀⣀⡤⠀⠀⠒⠲⠴⠂⠀⠀⠀⠠⠀⠀⡜⢵⠋⠙⠒⠋⠀⡜⠀⠀⠀⣠⠇⢸⠁
⢀⣀⠀⢠⠒⠉⡠⢄⠤⢌⡳⣧⡀⠉⠓⠏⠙⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⠢⠤⠦⠶⠤⠀⠀⣸⠁⠀⢸
⠣⠔⠁⠈⠁⠋⠀⠠⠀⠀⠳⢌⣻⣦⡀⠀⠀⠀⠀⠙⠓⢦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠦⣶⡦⣄⠀⠀⠀⠀⠙⠦⣤⡤⠤⣄⣀⣄⣠⠖⠶⠃⠀⠀⠀⠀⣀⡀⠀⠀⣸⡇⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⠀⠁⢿⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⠈⠓⠶⠛⠛⠋⢩⠉⠛⠉⠅⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠙⠶⠤⣤⠴⠋⠳⠦⠴⠒⠐⠦⠖⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠘⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⠀⠀⠁⠒⠂⠤⠤⢤⡀⠀⠀⠀⠀⠀⠤⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⣸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠂⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠂⠤⠤⠤⠤⠤⠤⠴⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⡟⠉⠉⢻⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡏⣼⠟⠁⢀⡀⠈⠻⣇⢹⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠱⣧⡀⢀⡀⢀⣴⠟⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢛⣭⣥⣄⠀⠀⢹⡇⢀⡀⢸⡏⠀⢀⣠⣬⣍⡛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠮⠟⠁⠀⠙⠿⠷⠿⠇⢈⠁⠸⠻⠷⠿⠋⠀⠈⠛⢵⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠀⠀⠐⠆⠰⠆⠰⠆⠰⢂⡰⠆⠰⠂⠰⠆⠰⠀⠀⠀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣄⣀⣴⣶⣲⣒⡆⢀⡁⢰⣖⡶⣲⣦⣀⣠⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣍⣉⠉⠀⠀⢸⡇⢈⡁⢸⡇⠀⠀⢉⣉⣩⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣷⣤⡀⢸⡇⠈⠁⢸⡇⢀⣤⣾⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡏⢸⡇⠈⠁⢸⡇⢹⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠁⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⠁⠈⠁⠈⢻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⡀⠈⠁⢀⣰⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣧⣀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣶⣾⣷⣿⣿⠶⠁⡺⣿⣿⢿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣹⣿⠿⢉⣉⣿⣿⣤⣬⢻⣿⡭⢄⣻⣿⣿⣿⣁⣤⡖⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣥⣼⣿⣯⣷⣦⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⣿⣿⣏⣩⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠠
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣭⣿⣯⣛⣛⣩⣴⣤⣀⡀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠈⣹⠟⠚⠛⣿⣟⢻⣲⣷⣿⣿⣿⣷⣿⣿⣿⣿⠆⣙⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣦
⠀⠀⠀⠀⠀⡀⣾⣿⣿⡟⠛⢉⠺⢛⡛⢻⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⢷⣼⡿
⠀⠀⠀⠀⠀⠚⠫⠋⣀⣠⣄⠴⣾⣿⣿⢀⡘⢯⣻⡟⢿⣿⣿⣿⣿⡿⢿⣿⡟⡫⣿⣿⡽⣿⣯⡳⢿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠁⢠⣽⣿⣿⣤⠀⠀⠙⢻⣦⣿⣿⡿⠉⠃⠀⠛⠀⠀⡓⠻⡆⠈⣿⠁⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠈⢻⠇⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣴⣀⣀⣠⣠⣤⣤⣴⣶⣿⣿⣿⣿⣶⣿⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣤⡶
⠀⠀⠀⠀⠂⡞⠿⡿⢿⠟⠛⢿⣿⡿⡿⡿⠿⠋⠉⠁⠠⠬⠭⠿⠿⡿⠿⠿⢿⡿⠻⠟⠻⠻⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠴⠤⠤⠤⠤⠄⠤⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠈⠀⣠⠔⠈⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⢂⠠⠤⠈⠀⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠒⡀⠊⠀⠁⠀⠃⢀⡴⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⠀⠀⢀⠘⢤⡀⠀⠢⠀⠈⢠⠀⠀⠀⠀⠰⡃⣀⠤⣤⣤⣶⡎⡁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠧⣀⡅⠀⢀⣅⠐⠠⠒⠣⠤⡰⠀⠀⠀⠞⢡⠿⠆⠛⣘⢋⣅⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠠⠁⠐⢀⣀⠀⡀⢺⢄⣤⡶⠃⠠⢀⠀⣀⣀⡀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢦⠀⠢⠄⠐⠒⠒⠛⠫⠛⠁⠀⠈⣉⡉⡡⠔⢁⠴⣿⣦⡥⠤⠅⠈⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠤
⠀⠴⠒⠀⠂⠂⠐⠀⠀⣀⠀⠀⠀⠴⠤⠀⣐⠒⠀⠰⠦⠶⠶⡴⠖⢚⠋⠁⠀⠀⠞⠂⠁⠙⠙⠃⠀⠀⠀⠐⠀⠐⠒⠐⠐⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠠⠂⠀⠀⠀⠀⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⡤⠒⠀⠀⠐⠢⣀
⠀⠀⠀⠀⠀⠈⠀⠰⠆⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⢤⡂⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠉⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠳⣤⣷⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⢀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣄⡀⠀⠀⢹⣇⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢷⡀⠀⣠⠒⠂⠀⠐⠒⠚⠻⣍⠛⠳⠾⢿⣀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠈⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠙⠓⠲⠶⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠈⠐⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠾⠇

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣷⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⢠⠀⡇⢸⠀⣄⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡀⢹⠀⡇⢤⠀⠀⠀⠀⠀⠀⠀⡀⣶⢰⡆⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡀⣴⢠⡆⣀
⠀⢸⡆⣧⢸⡆⣏⠀⠀⠀⠀⣿⡇⠀⠀⠀⢨⡇⣾⢠⡇⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⣿⣾⣧⣿
⠀⣤⠇⢻⠘⠇⠻⠀⠀⠀⠀⠛⠃⠀⠀⠀⠸⠃⠟⣸⠃⢧⡄⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣇⠀⠀⠈⠛⠀⠀⠀⣼⣿⣿⣿⣿⣯⡀
⠀⠘⢆⠀⠒⠶⠶⠶⠶⠶⠤⡀⡠⠴⠶⠶⠶⠶⠖⠀⢀⠎⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⣭⣭⣓⢦⢀⠔⣋⣭⣭⣽⣿⣿⣿⠟⠁
⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠇⠁⢾⣿⣿⣿⣿⠿⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⣀⣀⣀⠀⠈⠉⠉⠉⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠖⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠈⠙⢿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣸⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠀⢠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠃⢀⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠘⠃⠀⠤⠂⠀⢀⠀⢀⡀⠚⡀⠀⠁⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁⠀⠠⡀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠀⠂⠀⠀⠀⠒⠀⠸⡀⠄⠁⠀⠀⠀⠈⠀⠀⡀⢀⠀⠀⠀⠠⡆⠀⠀⡀⠊
⠀⠀⠀⠠⠄⠂⠀⠀⣂⠂⠀⠁⠁⢀⠤⠄⠀⠂⣀⠄⠀⠢⢄⠀⡴⡄⠀⠀⠀⠉⡉⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠨⠃⠀⠀⠂⠀⠒⠁⠀⠀⣂⠀⠀⠀⠀⠔⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡾⠛⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠤⣾⣏⣠⣤⣤⡄⠀⠀⠀⣀⠹⣷⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣽⣿⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠟⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠿⣿⡟⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⠿⠟⠉⠉⠉⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠦⠴⠶⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠉⠉⠉⠙⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⢐⣾⠂⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⢀⣼⣻⡀⠟⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣷⣼⣯⣿⣷⣶⣼⣦⣤⣶⣶⣴⣧⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠁⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⣠⠊⠙⢂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣀⣀⣀⡀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠠⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣷⣶⣄⣀⠀⠀⠀⠀⠴⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢻⣿⣿⣿⣶⡶⢤⡤⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠚⠿⠿⠿⠿⠟⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⣿⡇⠸⠁⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠋⠀⠒⠀⠉⢸⣿⣿
⣿⣿⣿⣿⣿⢣⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿
⣿⡟⠉⢻⠟⠀⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿
⠟⠁⠀⠀⠈⠀⢙⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣀⣀⣤⣤⣶⣄⣤⣾⣿⣿⠁⣴⣷⠀⡀⠰⠀⠀⣸⣿⣿⣿
⠈⠈⠃⠀⠀⠀⠈⢿⣿⣿⣿⣟⣿⣆⠀⠀⢸⣿⡏⣶⠿⣿⣿⣿⣿⣿⠏⠸⣿⣿⡀⡿⠀⠀⠀⣹⣿⣿⣿
⠀⠀⠀⠀⡆⠀⠀⠈⣿⣿⣿⣿⣿⣿⠗⠀⣤⠟⣴⣿⣷⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⡇⢸⣤⠀⠈⠿⣿⣿
⠀⠀⠀⠐⠇⠀⠀⠀⠙⢝⡛⠿⠝⠀⠀⠀⠀⠀⠻⠿⣿⣟⢻⣿⣿⡟⠀⠸⢯⣿⣿⡗⢸⠋⠀⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡟⠂⣀⣦⠄⠀⢀⠀⠒⠒⢾⣿⣿⠋⠀⠀⠀⠀⣿⣿⣿⡿⡆⠘⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡋⠈⣻⣿⠰⠞⠋⠰⠃⣴⣾⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⠇⢸⡇⢀⣾⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠚⠻⡟⠀⣀⣾⣧⣼⣿⣿⣿⣿⡀⠀⠀⠀⠘⠛⣿⣿⣦⢸⣿⣿⣿⣿⣿
⡗⢀⣶⣶⡀⣠⢀⣀⣠⣆⠸⣇⢠⣤⣤⣤⣿⣿⣿⣿⣟⣿⣿⣿⣅⠀⠀⠀⠀⢈⡿⠛⢀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⢈⣼⣿⣿⣿⣧⢀⣾⣿⣿⠟⢛⣟⢫⣭⣿⣸⣿⣿⣿⣦⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⡟⢿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣴⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣻⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣹⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣽⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⡿⣷⣿⣿⣿⣿⡿⠭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠘⠿⠏⢹⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⡀⠀⠀⣼⣾⢧⡠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠛⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠒⠛⠛⠒⠲⠦⣄⡀
⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⣠⠞⠉
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⠀⠀⠀⠀⠈⠡⠀⢷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣤⣦⣤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠢⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢏⡜⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠂⠈⠙⡆⠈⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡼⣄⣀⡂⠀⠀⠀⣇⠀⠀⠠⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣡⠤⠶⠚⣟⠁⠀⠀⢠⠝⠀⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠾⠵⠒⠶⣅⣀⢸⡆⡀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠁⠈⢣⡀⠀⢰⠀⠈⠉⠑⢢⡤⠂⠀⠀⠀⡤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢦⡤⠀⠀⢀⠀⡀⠈⠳⡀⠀⠀⢸⡆⠀⠀⢁⢀⠀⠀⠇⠀⠀⠀⡔⡠⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣠⠀⠀⠀⠀⣀⣀⡀⠐⠄⠀⠀⠙⠀⠐⢌⡴⢂⠀⠀⠀⠀⣐⣎⠋⠀⠉⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠈⠢⠀⠀⠸⡄⠀⠐⠀⠀⠈⢢⠹⢦⣾⠠⠂⠀⠀⠠⠀⠐⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠁⢀⡊⠀⠀⡄⠤⠆⠀⣤⡀⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠀⢠⡀⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠁⠠⠶⠂⠒⡻⢆⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠂⣷⡋⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠐⠚⠞⠁⠀⣈⣀⠤⠤⠬⢶⡀⠀⠀⠀⠀⣤⢤⡀⠀⠀⠀⠀⠆⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⣀⡤⠂⠉⠁⠀⡀⠠⠀⠐⠳⠆⣀⠀⠀⡇⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠈⢀⠄⡜⠱⠶⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢄⡄⠤⣠⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠖⠉⢶⠞⠁⠀⠀⠀⠀⠀⠀⠡⡀⠀⠙⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢨⠦⣴⡟⠠⠄
⠀⠀⠀⠀⢀⠀⠂⢚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠆⠀⠀⠁⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⡈⣤⠀⣤⣄⡂⠀⠀⠀⠀⠀⢀⢤⠔⠳⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⣀
⠀⠀⠀⠀⠀⠀⠠⠬⣉⣘⣻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠂⠤⠤⢬⣽⣿⣿⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡤⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠉⢀⣀⠠⠈⢛⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣷⡾⠇⢂⠠⠟⠀⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢀⡤⢅⣈⣻⡿⣿⢿⣿⣆⠀⠀⠀⠀⢠⣴⣿⡇⠀⠈⡾⢏⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⢿⣭⡉⠁⡌⠁⠀⢀⡀⡀⠀⠈⢻⠀⢀⣠⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠭⠿⠠⠄⠀⠀⣲⠶⠜⠒⠀⣴⡤⢨⡄⠈⢂⣇⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣤⡶⠀⠄⠀⠀⢺⣷⡟⠈⢃⣿⡟⠘⠀⠀⠲⠾⠷⣤⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠐⠀⢀⣀⣠⣼⣿⣖⢀⣴⣶⣶⣶⣶⢃⣼⣿⡆⠈⠏⣿⢿⣿⡛⣿⣿⢯⠄⣀
⠀⠀⠀⠶⠀⠀⣀⣽⣿⣿⣿⡿⣿⣼⣿⣟⠻⠍⠛⠻⠅⠔⠈⠀⠀⠠⣾⣷⣟⣛⣟⡿⢧⡄⠤⠐
⠀⠀⠀⠀⠀⠈⢉⣰⣟⣿⣿⣼⣿⣿⢙⣲⣀⣀⣀⠀⠀⡀⠀⠀⠀⠀⡀⠈⠉⠛⣿⣷⣦⢞⠷
⠀⠀⠀⠀⠀⠀⢹⣄⣸⣿⣭⣿⣿⣿⣿⣿⣿⡟⠹⡴⠐⣧⡀⠀⠀⠺⣿⣿⣧⣴⣶⣾⣿⠿⠆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠭⣿⣛⣻⣿⣿⣿⣆⠀⠀⠙⢿⣄⠀⠀⠙⢹⣿⣿⣽⣭⣬⣭⠀⠀⣀
⠀⠀⠀⠀⠀⡄⠀⠲⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⢨⠿⣄⠄⠐⢀⠙⢿⣿⣿⣿⡿⠠⠶⠁
⠀⢀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠻⡆⠀⠡⣦⠿⣇⢂⠱⡔⠘⠻⣿⣿⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠘⢿⣿⣿⠟⢳⢿⣿⣿⠟⠀⢀⣼⡀⣤⣿⣼⣿⣾⣆⢷⣴⣞⢈⠋⠀⠀⡌
⠀⠀⠀⠀⢈⠀⠀⠀⠠⠌⢻⣯⣾⣿⣽⡿⠀⡀⠀⣼⣿⣷⣬⣩⣏⣿⣟⣹⣿⣿⡟⠻⠃⠀⠀⠁
⠀⠀⠀⢀⠘⠀⠀⠀⠀⠐⠛⢫⠟⢿⣟⡔⣸⠁⣼⠿⣽⡿⡇⠠⢭⣿⣿⡷⠙⠁⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠐⣹⡽⠟⣸⡇⣧⢣⢻⣹⠀⠞⢯⠁⢄⠀⠀⠀⠠⠀⠀⢰⡷⠋
⠀⠀⠀⠀⠀⠀⢶⡦⠀⢡⠀⠀⣤⣼⠐⢃⠀⢻⢰⣿⣦⢳⣿⠄⡌⣦⠷⡀⠱⢪⠅
⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⢀⡂⠀⣿⠀⢸⠀⢠⡼⣿⣿⣧⠻⠇⠀⣸⡑⠈⢢⣴⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣤⣾⢿⠇⠘⠀⠻⣦⣼⣿⢿⣷⡄⣤⠷⡿⠲⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠓⠚⠦⣔⣰⣶⡿⣹⡄⢿⣷⣟⢀⠘⢆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠜⠳⣌⣿⣽⣴⣏⣇⣈⣯⣭⣥⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣯⡿⢿⣷⡌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠋⠃⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⠀⢴⣾⣿⣿⠀⠘⠈⠙⠓⠀⠀⠒⠀⠀⠀⠐⠁⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⡿⠀⡀
⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠘⠀⠀⠀⠄⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠄⠀⠀⠀⠀⠀⠀⠀⠋⡟⠁⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾
⢀⠔⠈⠁⠈⠁⣀⠀⠀⠀⠀⢠⣖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⢀⣾⣏
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠤⠀⡴⠊⠉⠉⠐⢄⠀⠀⠀⠁⢄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣀⡀⢀⣤⣿⣿⣟⡡
⠁⠉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠦⣄⡠⠲⠒⠖⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣋⣀
⡀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠑⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡻⣿⣿⣷
⠙⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠴⠊⠁⠀⠀⢀⣿⡟⠃⠀⠃⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠙⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣷⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡀
⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠀⠀⠀⠈⢆⠀⠉⠁⠀⢄⠵⠶⣦⠀⠀⠀⠀⠀⢀⠀⠀⢽⣿⣿⡇⠀⣾⠇⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⢀⣀⢀⡀⠉⠉⢻⣿⣿⣧⢠⣿⣷⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠻⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠳⡀⠀⣰⠁⠀⠠⡄
⠀⡄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⡀⠐⠃⠀⠁⠈
⠀⠐⢄⢿⡄⡀⠀⠀⠀⠀⠕⠀⠄⠀⠀⠀⠀⠀⠀⠒⢂⣩⡴⠋⡉⠀⠙⢷⡀⠈⠀⠀⠐⠒⠒⠄⠠⠄⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠆
⠀⠀⠀⠀⠙⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠋⠅⡔⠁⠀⠋⠀⠁⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣴⣶⣤⣬⡄⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠖⠀⠀⠀⠀⠀⠀⠁⠰⣀⠀⢀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠀⠀⠀⠈⠙⠙⠓⠚⠠⢄⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠉⠉⠒⠆⠒⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⣦⢠⠀⢀⣴⣿
⠀⠀⠀⠀⠇⠑⠂⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢣⡀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢺⣧⣠⣿⣿⣿⣿⠏
⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠸⠁⠀⠀⢀⣤⣞⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣟⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⢀⣀⠀⠀⠀⠈⢻⣿⣷⣀⢠⠀⠀⣤⣿⣿⠋⢀⠀⠠⠠⠀⠠⠄⠀⡸⠉⠐⠙⠧⣐⠀⠤⠤
⠲⢀⠈⠁⠀⠁⠀⠀⠠⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠈⠻⣿⣿⠛⠃⢸⣿⣿⡡⠊⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠙⠒⠒
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣷⣶⣦⣉⡀⠃⢌⣿⣟⡡⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎⠀⠀⠀⠐⠚⠉⠉⠉⠻⠛⠛⠛⢻⡿⢿⠁⣠⡆⢫⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⡀⣀⣀⠀⠀⢀⡀⠀⠀⡀⠀⠀⢀⡀⠀⢀⡀⠀⣀⣀⠀⢀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⢀⡛⠀⠈⠁⠈⠁⡉⠉⠉⠉⠉⠉⠃⠀⠈⠉⠀⡀⠉⠁⠀⠈⠉⡉⠉⠉⠀⠈⠉⠃⡀⠉⠈⠉⠉⠉⠉⠉⠁⠉⠉⠀⠉⠉⠉⠉⠉⢋⠀⣀⠘⡆
⠀⠀⣄⠀⠰⡶⠠⠒⠛⠐⠒⠚⠓⠚⠛⠛⠒⠐⠚⠑⠚⠷⠊⠒⠚⠋⠀⠀⠔⠋⠚⠒⠛⠐⠛⠂⠁⠈⠚⠋⠓⠓⠚⠓⠒⠋⠙⠒⠀⠚⣿⡇⢸⠇
⠀⠀⡍⠀⠠⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠘⡇
⠀⠀⢧⡀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⢉⣤⣁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠤⡄
⠀⠀⠆⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡁⠂⠀⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠟⢠⡟⢚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⡅
⠀⠀⡗⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⢇⡄⠠⠀⠈⠂⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⢘⡀⢤⡄⠋⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠸⡇
⠀⠀⣃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⡠⠁⢤⠀⢀⠈⠀⡀⡄⠀⠀⠐⠁⠈⢔⠗⠥⠀⠀⠐⠖⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⣾⠀⠠⡁
⠀⠀⣿⠂⢠⡇⠀⠀⠀⠀⠀⠀⠘⣶⡀⠀⠀⠄⢁⠀⠀⠈⡄⠀⠱⠁⠀⠀⠙⡆⠀⠀⠀⠀⠀⠐⠑⠦⠄⠀⠀⠘⠂⠀⢀⠀⠀⠀⠀⠀⣿⡂⢀⡆
⠀⠀⡴⠀⠈⡇⠀⠀⠀⠀⣄⠀⠀⠘⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢀⠠⠀⠀⠀⠀⢿⠄⠀⡁
⠀⠀⢯⠀⠐⡇⠀⠀⠀⠀⠋⠀⠀⢀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠃⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⣏⠀⠐⠇
⠀⠀⣾⠂⣈⡝⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⢀⡇
⠀⠀⣿⠀⠀⢱⠀⠀⠀⠀⢀⠌⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣉⢙⠧⡉⣉⠈⠃⠀⠐⠂⠀⠀⠀⠀⠀⠈⠀⠀⣘⠀⠀⠀⠀⠀⠀⢿⠄⠀⡀
⠀⠀⠟⠀⠐⡏⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢹⡇⠀⢁⣎⠁⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⣮⠀⠀⠃
⠀⠀⡟⠀⣀⡅⠀⠀⠀⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠊⢀⠀⠈⠙⠀⠀⠀⣀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠀⣳⠆⠀⡇
⠀⠀⣷⠀⠘⡇⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠦⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠀⠀⠀⠀⣿⡃⠈⠃
⠀⠀⡿⠀⠲⡇⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡁⠶⠀⡀⠀⢃⠀⠀⠀⠄⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡏⠀⢀⡆
⠀⠀⣸⠦⢀⡼⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢞⠃⠄⣠⣅⡠⠂⡸⠂⢀⢠⣀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣧⠀⠈⠃
⠀⠀⢹⡀⠈⣿⠀⠀⠀⠌⢀⠀⢁⡀⠀⠀⠸⣇⠀⠀⠐⠄⢠⢊⣵⡎⢻⡔⢧⠞⢁⣴⢦⡐⡄⢂⢀⣠⡏⠀⠀⠊⠀⠀⠀⣀⠁⠀⠀⠀⡿⠀⠰⡄
⠀⠀⣾⠁⠰⡞⠀⠀⢀⠄⠁⠀⠀⠁⠀⠀⠀⠛⢿⠛⣿⣳⡆⠛⠀⠈⣧⢷⡟⠠⠎⠀⠀⠈⠠⠮⡍⠛⠁⠀⠀⠀⡀⠀⠐⠃⠀⠀⠀⠀⣿⠀⢀⡇
⠀⠀⣰⡀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⢴⠷⠾⡃⠚⠂⠤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠀⠨⡇
⠀⠀⢻⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢁⠰⠆⠈⠀⢠⡄⠈⠀⠠⠦⢤⣀⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠠⡇
⠀⠀⡜⠑⠤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠉⠂⠭⠩⠀⠀⠆⣠⠀⠀⢀⡀⠀⠀⠐⠐⠀⠁⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢀⡃
⠀⠀⠰⠄⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⡋⠑⠆⡀⡀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⢀⠒⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⠊⡇
⠀⠀⢧⠀⠈⢻⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠘⠤⡀⠠⡄⠀⠈⠆⠀⠀⠂⠑⡀⠀⠀⠂⠰⠀⠀⠀⠊⠀⠂⠀⡀⠁⠀⠀⠀⠀⠀⠀⢀⡋⠀⠠⠃
⠀⠀⣞⠂⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⡇
⠀⠀⣱⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⡇
⠀⢠⣿⠀⠐⠿⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠁⠀⡠⠋⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⢘⠇
⠀⠀⣿⠄⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⣁⢲⣄⢰⣀⢀⢠⡀⣀⣠⢦⡄⠘⠋⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⠂⢨⡄
⠀⢀⣽⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡴⠞⠺⢥⠟⠷⢻⡏⢻⠿⠳⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⢘⡄
⠀⠘⣳⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⢸⡃
⠀⠰⣝⠃⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠩⠩⠍⠯⠍⢠⠄⠱⠀⠀⠀⠤⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⡆
⠀⢸⡧⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢠⠁
⠀⢸⠁⠀⣲⣿⠟⠛⠓⠛⠓⠐⠒⠛⠛⠘⠉⠙⠓⠲⠗⠒⠒⠏⠀⠒⠚⠛⠓⠐⡛⠙⠓⢾⠁⠀⠐⠃⠀⠐⠋⠉⠈⠓⡉⠀⠒⠆⠀⠚⢷⡆⠀⢁
⠀⠨⠁⠘⠏⠈⠙⠦⣀⠀⠤⠤⠤⠔⠂⠤⠄⠦⠄⣀⠰⠀⠀⠀⠆⠀⠀⠲⠤⠄⠈⠦⠄⠠⠦⠀⠀⠀⠦⠤⠀⠴⠤⣀⠶⠤⢄⣼⠦⠀⠈⠳⠖⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢠⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢘⠿⣦⡈⠳⣤⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠘⢿⣮⣿⣶⣤⡉⠓⠶⣤⣀⠀⢰⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⢿⣿⣿⣿⣿⣿⣷⣦⠀⠈⢳⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢿⣿⣿⣿⣿⣿⣿⣠⠴⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠻⢿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣿⣿⣿⡿⠛⠁⠀⠘⠃⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡘⠿⠿⠿⣿⣿⣿⣧⣤⣤⠀⠈⠉⠀⠀⠀⠀⠀⣴⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠐⢾⣿⢿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⣤⣩⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢒⣒⣒⣒⡒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣈⣩⣭⣭⠭⠍⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠌⣩⣤⠶⢚⡩⢔⠀⠀⠀⣾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡍⣴⡾⢋⡴⣫⢫⢢⡷⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣆⠴⢏⣼⢃⡾⢸⡇⢿⡆⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠧⠵⣮⣥⣮⣡⡶⠴⠋⠁
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣼⣶⣶⣶⣶⣷⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⡖⢢⠤⢀⡠⣴⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢡⡿⣇⠛⣛⣿⢿⣿⣿⠏⠻⢻⢷⣤⣀⣵⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣬⡫⠩⠝⠂⠍⣄⠘⣛⣦⣶⢶⣾⣿⣿⢻⠿⠿⢿⣾⣷⣄⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠐⠘⠀⠒⣶⠤⠤⣹⣾⣷⢾⡉⣾⡵⠻⢿⣤⢠⡄⢉⣽⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⣾⣿⠩⠉⢀⠀⠁⠐⠂⠳⡄⠠⠙⡃⠹⣿⡴⣦⡟⠻⡆⢳⣌⢽⣿⣿⣿⠥⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢴⣼⣤⠄⢿⣇⣆⢠⢈⠀⠃⠀⡀⣀⠀⡀⠤⠈⣧⠀⠀⢁⢰⣆⠞⣼⠼⣽⣿⣿⣷⣿⣗⡃
⠀⠀⠀⠀⠀⠀⠀⠰⢇⣻⣶⠾⠛⠻⡶⠂⠀⠤⠀⣠⣙⡷⢂⠀⡠⢄⡐⣄⣸⣛⡿⡻⠁⠐⠢⢻⣿⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⡂⠀⠀⣀⡆⠄⣒⣠⣾⣿⣿⣿⣴⣶⣿⣿⣿⣿⣿⣟⠳⢈⣷⣾⣿⡛⠿⣟⣻⣿⡇⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣾⣯⣠⣾⣷⣿⣯⢿⠿⠿⣿⣿⣿⣿⣿⣿⢿⣿⢻⣿⢿⢿⣿⣿⡿⠿⠷⣾⣽⣿⣿⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⣿⣿⣿⣿⣿⣛⠛⠻⣿⣄⢨⣿⣿⣥⣴⠟⣻⣿⣾⣿⣿⣿⣦⣧⣼⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⢰⠾⡟⠛⠿⠟⠛⠛⠛⠻⣿⣾⠋⣿⡏⢁⣴⣿⠟⢩⣽⣿⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢿⠟⣹⡿⠋⠀⢾⢿⣿⣿⣿⡿⣯⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⢀⠀⢰⣿⠁⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⢀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡸⢺⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣷⡜⠿⢿⢷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⠂
⠀⠀⠀⠀⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠐⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⠴⠀⠀⠀⠀⣀⠠⢀⣀⠀⣀⡀
⠀⠀⠠⡀⠀⠈⢄⠨⡒⢄⡠⢀⠃⢱⡀⠀⠀⠀⠀⠀⣀⠠⠤⠲⣀⣀⠤⠒⠩⠔⠁⠀⠂
⡤⠀⠀⠀⠀⢀⠈⢂⠑⠒⠠⠤⠐⠂⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠐⠒⠄⠃⢄⡀⢀⣀⣀⠄⠀⠀⠀⣀
⠀⠠⡌⢁⣀⡡⠤⠤⠗⠒⠒⠦⣄⣀⡳⠚⠃⢄⣄⡠⡆⢧⠀⠀⡀⠠⠠⢤⣀⡠⢀⢀⣭⡤⠄⠈⠙⠀⠐
⠀⠀⠈⠠⢅⣀⣀⠀⠀⠀⠀⠈⠀⠀⡀⠤⢴⣆⣀⠔⡁⠀⠀⠰⠔⠒⠀⠐⠂⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠐⠒⠒⠁⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠄⠀⠈⠙⢦⡀⠀⡴⣶⡀⢀⡴⠋⠁⠀⠠⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣿⣧⠏⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣣⣼⣠⣆⣶⣠⣄⣀⠴⠋⢀⡉⠙⠦⣀⣠⣄⣶⣰⣄⣷⣜⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣱⠟⣹⠟⢹⠏⠟⢁⢝⠁⠀⣠⡀⢀⣅⠀⠈⡫⡈⠻⠹⡏⠻⣏⠻⣌⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠁⠀⠠⠔⢚⠽⣻⣞⠴⠋⢰⣦⠙⠦⣳⣟⠯⡓⠢⠄⠀⠈⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠻⣦⠀⠀⢈⡡⠀⠀⠀⢸⣿⢸⡇⠀⠈⠁⣀⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⢀⣀⣀⡠⠄
⠀⠀⠀⠀⠀⠤⢄⣀⣀⣐⣒⣒⣉⣩⣤⣤⠶⠶⣦⡞⠃⣰⣞⣥⠤⡄⠑⠀⠀⢻⣼⣇⠀⠀⡾⢉⣤⠄⠀⠈⠉⠉⠉⣉⢉⠩⠝⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠼⠇⠸⠿⣿⢣⡎⠀⠈⡾⠄⠠⡙⢿⣆⠀⠃⠸⣇⡀⠀⠀⣀⣀⠤⠀⠒⢲⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢲⡌⢾⡀⠀⡴⢚⠉⠐⠜⢆⠙⢧⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠁⠘⢿⡾⢁⡞⣴⢇⢸⡌⢧⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠞⣡⣾⢠⡇⢸⡆⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣼⠫⠋⢼⣷⠾⢡⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠘⣧⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⣾⣿⣿⣇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠿⠛⡿⠿⠿⠿⠿⠟⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⠀⠐⠉⠘⠛⣿⣿⣿⡟⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⡹⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠒⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣦⡘⢃⣤⠄⠀⢀⡄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⣶⣄⠿⢃⡀⠀⠈⠛⠘⠉⢠⡄⠸⢟⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣙⣃⠀⡈⠉⠐⠛⠋⠀⠀⢠⠀⠀⣿⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠁⠀⠀⠙⠲⡀⠀⠀⢀⣿⠖⠉⠀⠀⠀⠀⠀⢠⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡻⠂⠀⠀⠀⢠⡼⣦⠀⣼⠁⠀⣿⣆⣴⠀⠀⠀⠈⠋⠒⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠸⠃⠀⠀⠸⠛⠀⠘⣷⡿⠘⠿⠮⠙⠋⠀⠀⠀⠀⠀⠀⢠⣶⠟
⠀⠀⠀⠀⠀⢀⣤⣀⠿⡆⠀⢸⡆⠀⠀⠀⢰⣿⢀⣀⠀⠀⣿⡇⠀⠀⢀⣀⠤⠒⠂⠈⠉⠁⠀⢴⡜⠿⠆
⠀⠀⠀⠀⡀⠈⠉⣽⠶⠂⠀⠀⠁⢀⡀⠻⢷⠅⠿⠃⠀⠀⣿⣷⠀⣠⠞⠁⠀⠀⣠⡄⠀⠀⠀⠈⢁⠀⣠⡄
⠀⢀⣤⣀⠿⡆⠀⣀⠀⠐⠈⠉⠉⠉⠛⠓⠶⢦⣤⡀⠀⠀⢻⣿⣼⠃⠀⠀⠀⢷⡝⡷⠆⠀⠀⠀⠻⢎⣯⡀⠀⠰⠿⣡⣤⡀
⠀⠀⠉⣩⡶⠀⣼⠟⠇⠀⠀⣀⣀⠀⠀⡀⠤⠶⠾⢿⣷⣄⠸⣿⡿⠀⠀⠀⠀⣠⣤⣤⣤⣤⡤⠤⠤⠬⠅⠀⠀⠀⢶⡌⠉⠁
⠀⠀⠀⠉⠀⠀⠈⠀⠀⠺⢶⡌⠉⠃⣼⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⢀⣴⡿⠋⠉⠀⠀⠀⠀⠀⣀⢾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⡎⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠠⢄⣤⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⣀⣀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⠠⠤⠤⠤⠀⠀⣀⠀⠈⠀⠋⠉⠉⠙⠛⠋⠛⠂⠙⠒⠒⠲⠶⠶⢶⣮⣉⣉⠉⠉
⣥⣤⣤⣄⣷⣤⣤⣶⣿⣿⣿⡿⠿⣛⣙⠃⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠋⠍⠉
⠭⠭⣽⣯⣭⡉⠿⠿⠿⡿⠯⠽⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⢀⡀⠀⣀⣀⣸⣶⣷⣶⣶⣶⣶⣶
⣶⣶⣾⣿⠿⢭⣭⣭⣭⣥⣀⣀⠀⠤⠤⠤⠤⠶⠶⠿⠛⠛⠓⠒⠚⠛⠛⠛⢽⠽⠿⠿⠷⠖⠲⠶⠶⣾⣿⣏⣉⡉⠡⠤⠤
⠛⠉⠓⠂⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣶⣖⣒⣉⣭⣤⣄⣠⣤⣴⣶⣶⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣀⡀⠀⠉⠭⣽⣿⣿⣟⣛⣛⣉⣉⣀⣀⣁⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⢛⡛⠿⠻
⠛⠛⠩⠭⠭⠥⣤⣤⡄⠀⠶⠶⠾⠿⠿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⣼⣤⠿⠭⠵⠶⠶
⢖⣶⣶⣿⣿⣿⡶⣶⠶⠶⠶⠒⠲⠶⠶⠶⢶⡖⢂⡀⠐⠂⠀⠀⣀⣀⣠⣤⡄⠁⠀⢀⣀⣉⣉⣉⣉⣙⡛⠓⠒⣒⣲⣶⣶
⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣶⠶⠶⠶⢾⣿⣿⣛⡛⠿⠿⣿⣿⣭⣍⣀⣀⠀⣀⣀⣀⣀⣈⣉⣉⣩⣤⣴⣶⣲⣶⣶⡶⠶
⣤⣤⡤⠤⠤⣴⣦⣤⣤⣤⠴⠶⠒⠒⠶⠶⠤⠀⠀⠐⠒⠒⠲⠶⠖⢺⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣶⣶
⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⢿⣷⠶⠶⠶⠶⠾⠛⠟⠛⠾⠿⠗⠒⠂⠀⠐⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠥⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠿⢟⣛⣛⣛⣛⣛⣿⣿⣿⡿⠯⠭⠽⠟⠛⠛⠛⠛⠉⠉⣉⣛⣉⣉⣀⣈⣉⣉⣀
⠿⠿⠿⠿⠿⠿⠾⠿⠿⠶⠾⠛⠛⠛⠋⠉⠉⢉⢁⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉
⣒⣒⣂⣠⡤⣄⣠⣤⡤⠤⠤⢴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠻⠲⠒⠒⠒⠒⠂⠛⠋⠀⠒⠀⠈⠙⠛⠓⠂
⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣤⣤⣤⣤⣤⣤⣀⣀
⣀⣀⣀⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣭⣭⣽⣿⣿⣭⣥⣬
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⡀⠀⠈⠉⢉⣩⣭⣭⠭⠭⠴⠶⠚⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠙⠉⠛⠛⢛⣛⣛⣛⣛⡛⠻⠿⠷⢶⣶⣶⡶⠶⠶⠶⠶⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣻⣿⣿⣿⣭⣭⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠟⠛⠛
⣀⣀⣀⣲⣀⣈⣭⣭⣭⣿⣿⣿⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣯⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠐⢲⣦⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣤⡖⠂⠀⠑⣲⣆⣀⢀⣀⣀⣀⡀
⠀⠀⣯⣉⣉⣛⠛⠿⡿⠿⢛⣛⣛⣛⣻⣯⣟⣻⣛⣯⣿⡷⠀⢠⣯⣉⣉⣛⠛⠿⡿⠿⣛⣟⣛⣛⣻⣭⣟⣛⣻⣽⣿⡇
⠀⠀⣭⣭⣭⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠛⠋⠉⠉⠉⢛⠛⠛⠛⠙⠛⢛⡉⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠉⠉⠉⠙⣟⠛⠛⠋⠉⠉⢙⠉⠛⠛⠙⠛⠛⠃
⠀⠀⠀⠀⠀⣦⠀⣠⣾⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣿⠀⠀⠀⠀⠀⣿⣄⢰⡇
⠀⠀⠀⠀⠀⣹⡞⣫⣾⣇⠠⠤⢤⡀⢸⣯⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⣡⣿⣄⠤⠤⣤⡄⢿⣯⣻⣧
⠀⠀⠀⢀⠔⢻⣾⣿⠧⠀⢴⡶⠾⠯⠤⢿⣿⣿⠳⣄⠀⠀⠀⠀⠀⡠⠞⣿⣾⡿⠅⠀⠤⡴⠯⠭⠌⢿⣿⣏⠳⣄
⠀⠀⠖⠁⣴⣿⡿⠁⠀⠀⠀⠠⠤⠤⠾⠯⣿⣿⣷⡎⠳⡄⠀⠠⠊⢠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⢯⣿⣿⣷⡌⢳⡄
⠀⠀⣴⣿⡿⣿⡉⠉⠉⣉⢉⡉⢉⢉⠉⣉⣛⣻⣿⣿⣇⡈⠀⠀⣶⣿⣟⣟⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣻⣿⣿⣧⡁
⠀⠀⣿⣟⣼⠩⠤⠤⠄⠀⢀⣀⣀⣀⣀⡈⣈⣸⣿⣿⣿⣷⠀⠀⣿⢏⡾⠥⢤⡤⠤⠄⢀⣀⡀⣤⡀⠀⣤⣼⡿⣿⣿⡇
⠀⠀⣟⡾⣉⣀⣀⣀⣀⠀⠀⠐⠀⠀⢀⣤⣄⡍⢭⣿⣿⡿⠀⠠⢏⡟⠁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠭⢭⣽⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠇⠀⢠⠀⠄⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⡜⢠⠀⠀⠊⠀⣴⠀⠀⠀⠀⠀⠀⠈⢡⣿⢷⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⡘⢠⠏⢀⠤⢤⡾⠋⠀⠀⠀⠀⠀⠀⠐⣻⣿⢻⣆⠀⠀⠀⠀⡇⠀⠀⢾⣡⣄
⠀⠀⠀⠀⢰⢀⡿⠀⠇⢠⡾⠃⠀⠀⠀⠀⠀⢀⠀⠐⠀⢿⠸⠙⢷⣄⠀⠀⠸⣦⠶⠀⠋⠙⣆
⠀⠀⠀⠀⡎⠈⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢧⠢⠙⢧⡀⠀⠈⢶⡙⡀⠀⠈
⠀⠀⠀⠀⡇⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⢀⣿⠀⢘⡇⠀⡀⣨⠷⣤⠈⠙⠃⠀⠐⡀
⠀⠀⠀⠀⠁⣾⣿⡇⠀⠀⠀⠀⡀⠰⢿⣤⣦⠀⢀⠀⢼⣿⡆⡌⠻⣿⣿⣷⡀⢸⡇⠀⠀⠈⣿⡗
⠀⠀⠀⠀⠁⠿⠛⠁⠀⢀⠀⢀⠀⠀⠀⢙⠟⠀⣸⠙⣿⣿⣇⠀⠒⠛⠻⣿⣷⣼⡇⠀⠀⠀⠛⠃
⠀⠀⠀⠀⡆⠀⠀⠀⠀⢈⣧⣏⠀⠀⠉⠛⠀⠀⢆⠀⠈⢋⣿⡇⢀⣵⡄⢸⣿⣿⡟
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⢸⣇⠀⠀⡸⠀⠀⠀⠠⠈⠠⠀⢹⠤⠤⠏⠀⠘⣿⣿⡏⠁
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⢁⠀⠀⠀⡀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡄⠀⠀⠀⢀⣏⡀⠰⠒⠿⠀⠀⣼⣿⢠⣿⣿⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⢸⡇⢀⠀⠈⠀⣄⣸⠏⠁⣤⣤⡴⠗⠚⣫⣡⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⠀⠀⢰⠀⠀⠉⠙⠓⠺⣭⣤⣤⡤⠾⠟⠟⠻⢿⣿⠀⠀⡗⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣧⡀⠈⠁⡸⠀⠀⢠⣔⠻⣿⣶⣖⠂⠁⣀⠀⠀⣿⠀⠀⠂⠀⢰⡧
⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⣸⣿⡇⠀⣧⡇⡀⠀⠈⢿⣦⠘⣻⣉⣤⡾⠃⠀⣿⣿⠃⠀⠀⠀⢸⡗
⠀⠀⠀⢠⠀⡀⠀⠀⣾⡎⢸⣻⡇⠠⣤⢇⡁⠀⠀⠘⣿⣷⣼⣿⠟⠀⠀⢸⣿⣿⡇⠀⠃⠀⢼⡇
⠀⠀⠀⠠⠀⡀⠀⢀⣿⣷⣾⢲⡇⠐⣿⢸⠀⡆⠀⠀⠹⣿⣿⡏⠀⠀⠀⣼⣿⣿⣧⣰⠀⠀⣿⡇
⠀⠀⠀⢰⠈⢣⠜⢻⣿⣿⣿⡾⢴⣤⡏⣿⣣⡿⡀⠀⠀⠹⢿⠁⠀⠀⢀⣿⣿⣿⣿⣿⡀⣄⣿⡇
⠀⠀⠀⢨⠀⠤⠂⠤⠿⠛⠛⠀⠀⠙⡇⠔⠉⠀⡀⠀⠐⠢⠌⢷⠒⡀⠚⠿⣿⡿⠟⠁⠉⠉⠙⠃
⠀⠀⠀⢸⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠓⠂⠤⠂⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣶⣦⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠰⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠈⠰⢧⡤⢣⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣋⠝⢃⠂⡀⠀⣀⣀⢴⡠⠀⠠
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠤⠤⠀⠁⠀⠈⣴⢤⡀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⡠⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠱⡛⠣⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣉⣤⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠊⠁
⠀⠀⠀⠀⢀⠈⠉⠀⣬⠧⠀⠀⢤⡄⠀⠀⣠⡄⠀⢀⡀⠀⡀⠀⢤⡄⡄
⠀⠀⠀⠀⠀⠀⠔⠂⠉⢷⡀⠢⠾⠷⠫⠞⠿⠅⣤⢛⠉⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⣄⣠⠴⠈⡈⠀⠄⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⠀⠈⠉⠉⠛⠒⠲⠤⠤⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠉⠉⠲⠤⣄⣀⠀⠀⠀⣠⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢌⡃⠀⡀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⠀⢧⡉⠒⢍⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⠒⢬⡲⢤⣭⠢⡍⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡐⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡛⡆⡈⠑⠯⣛⠾⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡈⠂⠄⡀⠀⠀⠀⠀⢉⢁⢢⢃⢠⣆⢨⠙⠢⣀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣁⠃⡀⠀⠀⠀⠸⡄⢸⡈⡀⡈⡄⡔⡈⡓⢤⡀⢤⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠯⠭⠳⢤⣙⠢⠀⠆⡇⠀⠇⠁⠁⢃⢠⢡⢱⢂⢍⠲⣌⠧⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠉⠤⠬⠭⠭⠓⢦⣀⠃⠀⠸⠀⣸⠸⡘⠈⡈⡘⠘⡼⢨⠑⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⣀⡀⠀⠨⠭⠄⣰⠟⠢⣅⠃⠅⡆⠇⠃⢡⠃⢃⠣⠇⣫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠐⠒⠒⣂⡀⣽⠋⠀⠀⠀⠙⠾⣵⢰⠸⠸⠐⠸⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠠⠤⠤⠤⢠⡖⠁⠀⠀⠀⠀⠀⠀⠈⠛⠦⣇⣆⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠤⠉⠀⢠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠄⠉⠉⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢒⡒⢪⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⠀⠀⠀⣀⣒⡒⠒⢒⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠐⠒⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠁⠀⠀⠀⠤⠀⠀⠩⣭⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠂⠀⠀⠀⣀⡁⢐⡀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢀⠈⠀⠀⠀⢀⠀⠲⠂⠐⢂⡾⠁
⠀⠀⠀⠀⠀⠀⡰⠁⠔⠀⠀⠀⠀⠠⠤⠤⠤⠤⢤⡟⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢈⣀⣒⣒⡂⣀⡞
⠀⠀⠀⠀⠀⠈⢦⡀⠀⢀⡐⠒⠒⠀⠀⠐⣐⠞
⠀⠀⠀⠀⠀⠀⠀⠑⢶⡀⠠⠤⠍⠥⠀⣒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢭⣭⣭⣴⠏
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⡠⢐⡡⢉⡤⣊⠕⡣⠔⡠⢒⡠⠊⡡⠒⡡⠐⡠⠔⡡⢖⡠⢖⡠⢂⡠⢂⡠⠄⡠⢀⡠⢀⡄⠀⠀⢀
⠀⠀⠀⠀⠈⠐⠀⠈⠔⢩⠔⣡⢔⡡⢊⠕⢋⠔⡩⠔⣁⠔⢉⠔⢋⠔⢊⠔⣋⠔⣩⠔⣉⠔⢉⠔⢉⠔⢊⠔⣉⢔⡡⣪⢞⡵⢋⠔⡡⢊⠔⡡⢀⢔⡠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠈⠀⠘⠁⠈⠁⠈⠁⠀⠁⠈⠁⠈⠁⠘⠁⠈⠁⠘⠁⠘⠁⠚⠁⠉⠘⠁⠉⠀⠁⠈⠀⠁⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⡄⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⢧⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⢳⡄⣤⡀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢣⡴⠋⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠶⠿⠼⠇⠟⠹⡄⠀⠠⠐⠓⢦⣄⠀⣠⢋⠔⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡀⠑⠬⠀⠃⣀⠀⣤⠀⠀⠘⣷⢡⠃⠀⠀⠀⠀⢈⡹⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠉⢁⡀⠀⠈⠡⢴⢀⣀⠀⠀⠀⠈⠀⠀⠤⠞⠉⠉⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠑⢦⡀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣙⣆⣀⣀⡤⠔⠉⠀⢁⣤⣥⡽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡐⢿⠗⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡆⣾⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⡇⣿⡿⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⣷⣶⣶⣾⣿⡇⣿⣿⣶⣶⣶⡾⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣣⣴⣴⣶⣶⣶⡆⣶⣶⣶⣶⣶⣦⡻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⠉⠉⣿⡇⣿⡏⠉⠉⠉⢹⠟⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣿⣶⣴⣶⣦⣤⣶⣶⣶⣶⣶⣶⣶⣿⡇⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡅⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⡐⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡇⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣷⣦⡀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⢗⣌⡻⢿⣶⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣙⠻⡇⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠻⢿⣷⡄⠙⢿⣷⣤⡀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣿⣶⣍⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡟⠿⣷⣦⡉⢾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠈⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⢇⠻⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢿⠗⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣴⢲⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⣬⣦⣤⢤⣍⣻⣿⣿⡶⣿⠾⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⠰⠦⣿⣉⠀⠐⣶⣶⣿⠳⠞⣿⣿⠿⠋⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⣻⣿⣧⡳⣾⣿⡞⢲⠛⡻⣿⣤⣶⡾⣤⣤⠖⠺⣿⣯⣣⣤⢠⠒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⣿⣿⣿⣿⣿⣿⣶⣬⠵⢿⣿⣧⣠⣃⣽⣶⠀⠀⢘⠛⠛⢲⠿⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢿⠻⠟⠁⡟⠻⣿⣿⣽⣷⣿⣿⣿⣿⣿⣷⣴⣾⣦⡄⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠠⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⡀⣤⣤⣥⣥⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣷⣿⣿⠛⣿⣿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣤⢀⣀⣤⡟⢿⡿⠷⡌⢤⡛⠻⢿⢿⣿⣿⣿⣿⣻⣿⣿⡿⢝⠻⠿⢣⣖⢾⢾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠛⣻⣿⣟⣿⣿⣷⣦⣾⣿⣶⣦⣼⣿⣿⡿⣷⣿⣿⠐⣀⣉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠇⠈⢋⣻⣿⠇⢿⡻⣿⣿⣿⢫⣾⣿⣿⣿⡟⠻⠉⠀⢴⣴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣬⣿⣿⡿⣧⢹⣿⣿⣿⣿⣵⡆⢀⣷⣼⣿⣶⣤⡐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣄⡀⢀⢀⡛⣿⣿⣿⠹⣿⢠⣿⢿⣿⣿⣿⣇⠀⢿⣯⡍⠍⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⢿⣿⣿⣮⣭⣯⣥⡉⠀⣿⣧⣶⣄⢋⣭⣿⣍⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠹⠷⠇⢡⣿⠉⠹⡟⠀⣿⣷⠛⠷⠻⣼⣿⣻⣿⣃⣤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⡉⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠒⠢⠀⡈⠙⠟⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣭⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠺⠛⠃⠃
⣿⣿⠿⠿⡿⢻⠿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⡀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⡇⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢷⣼⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇⠀⠙⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣄⠀⢹⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡟⢷⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠠⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⠀⠈⠁⠀⠚⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠁⠀⠅⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡀⣀⠀⣤⠦⠐⣄⠤⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠌⠳⡄⡀⠀⠀⡕⠊⠀⠀⠀⠀⠀⠀⠀⠐⡢⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠘⠃⠀⢀⠀⠀⠈⠛⡠⠀⠀⠀⢀⠄⠤⢈⠴⠫⢀⠔⠢⠤⠁⢀⠀⠀⠀⠰⠆⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⠀⠀⡀⠀⠀⠀⢠⠣⡈⠢⢀⢠⠐⠁⠀⠀⠀⠤⡀⠐⠃⠀⠲⢿⠁⠂⠄⡌⠀⠠⠒⡂⠑⠃⠀⡘⠤⠃⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠉⠌⢊⠢⡀⠀⠀⠀⠑⢄⢡⡀⡀⢀⡞⢞⣳⠀⠀⠀⠐⢄⠱⡀⡆⠨⢰⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠁⠜⢀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⢸⡀⠄⠀⠀⠂⠀⠀⠺⠓⠂⠉⠀⠢⡠⠴⠀⠀⡃⢀⠃⡝⢑⠂⡰⠈⠀⠐⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⢀⠰⣀⡐⠁⠀⡠⢠⡟⡏⠅⠀⠀⠠⢈⡤⡁⠂⠄⠀⠰⠖⡐⠋⠀⡀⠔⣡⡮⠀⡑⠀⠀⠀⠠⠄⠠⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡣⠄⠠⡄⠕⠐⣬⡂⡊⡤⠋⢈⠄⠨⡐⠪⡴⠛⣁⣤⣄⡀⢤⢲⠿⠤⠤⢤⣆⣁⣼⣀⢨⣥⠀⣉⠉⠠⠔⠄⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢜⢯⡑⠞⠫⠒⠀⠐⢁⣭⡈⢀⡙⢹⡆⠀⠈⠓⢓⠈⡀⡀⠀⠠⢌⠀⢨⡼⠀⠈⣣⣰⢁⣾⡷⢎⣡⠁⡰⠀⢂
⠀⠀⠀⠀⠆⠆⠀⠀⠀⠀⠑⠩⠑⢤⣀⠀⢠⡟⠄⡽⠿⣉⡉⠀⠀⣠⡤⠗⠛⠲⣤⡤⠋⠷⠔⡉⢳⠀⠉⠈⠡⠈⡋⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⣤⠀⣎⠆⠀⠆⠐⠂⠀⠰⡙⡖⠫⠉⢉⣤⠴⠃⠂⠐⠑⢀⡤⣐⡒⠖⠠⠤⠐⠒⠀⠂⠉⣀⠀⢄⠁⠉⠠⡩⠂⠀⠛⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⣀⡆⢠⠒⠀⠀⠀⠀⠀⠀⠀⢰⡈⠀⠘⠚⠑⠲⠓⠉⠙⣻⣮⢔⣲⡭⠁⢠⠒⢒⡂⢁⠀⠂⠐⠀⠡⠈⠀⠀⠂
⠀⠀⢀⠀⠀⠀⡄⣡⠴⢄⠀⠀⠂⡠⢀⠆⠀⠂⠀⠀⢀⡄⠰⢆⢬⡁⠉⠀⠠⢀⠀⢗⠀⡆⢀⢀⠀⠀⠀⠀⠐⠀⠀⠔
⠀⠀⠀⠁⡄⢠⣅⡨⡗⠀⠦⡭⣤⠴⠻⠄⠀⠀⠀⣄⠴⢀⡾⠞⠛⠁⠒⠒⠂⠁⡀⠀⢇⡱⠈⠊⠀⣤
⠀⠀⠀⠂⠁⠪⡉⢫⡀⠀⠀⠀⠠⢄⡁⠀⠀⠀⣊⠡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠈⠜⣆⠈⠀⠀⠑⠈⡠⠔⠂⠉⠀⠀⠀⠘⠀⣀
⠀⠀⠠⠀⠀⠀⠀⠠⣨⢗⡂⠀⠀⠀⡁⠀⠀⠀⠣⠃⠀⠀⢀⡀
⠀⠀⠀⠑⠀⠀⡠⠚⠉⠋⠐⠀⡀⠀⠐⠂⠠⠤⠎⠬⠄⢒⠄
⠀⠀⢀⠔⠰⠶⡇⠀⠒⠀⠒⠂⠀⠂⠠⠆⠒⠨⠄⠒⠠⢣
⠀⠀⠈⠉⠀⠀⠐⠈⠀⠀⠘⠄⠐⠐⠠⠄⠉⢀⣘⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠉⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠚⠁⠒⠀⢄⠀⠀⢀⡬⠀⠀⠀⢠⣠⣀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⣠⠄⠄⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠡⡀⠘⠀⠀⠀⠀⡘⠉⠃⠀⠀⠀⡴⠀⠀⠀⢀⡴⢋⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠴⠂⠀⠀⠀⠢⠀⠀⣷⠀⠀⠀⡀⣦⡄⢀⣻⣀⡑⠀⠿⠀⢐⠀⠀⠀⠀⠀⠀⣀⠃⡠⠀⠀⠈⠁⠀⠈⢢⡀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠰⠀⠠⢍⡩⠅⠈⠄⠉⠀⠑⠀⠀⢀⣀⢀⠀⣈⡀⠀⠡⠀⠀⠀⠀⠈⠋⠀⠈⠀⠃⠀⠆
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠀⠀⠄⠠⣦⡀⣰⠚⠆⠀⠀⠤⠀⠈⠀⡄⢁⠀⣨⠼⠋⠀⠀⠀⠀⠰⠀⠼⢦⡀⠀⠌⠀⠀⠤⠐⢠⠖
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠠⠈⠀⠘⢠⠖⠀⠀⠐⠀⠰⠂⡠⠀⢠⠆⠀⠀⠀⠀⠰⠃⠀⠈⢀⡀⠀⠑⠦⢦⡀⠠⠤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⣤⠋⡀⠀⠀⡀⢳⡆⠀⠀⠀⢦⠖⠀⠀⢡⡀⠂⠀⡟⠀⠀⠀⠁⠀⠀⠀⠀⠄⠈⠀⢀⠔⠉⡀⠁⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢹⠀⠀⠚⣀⠇⠀⠀⠀⠀⠇⠠⡆⠀⠀⢀⢀⠀⣰⢇⣤⠂⠀⠡⠄⠀⠄⠱⠀⢠⡲⢆⣤⣤⣁⡀⠀⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⠀⠀⠐⣿⠀⠀⠈⠀⠄⠙⠊⠀⠀⠀⠀⡇⠀⡟⠁⢠⢢⠄⠀⠀⠀⠂⢀⠐⢡⠊⠀⠄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⡃⠀⡀⠀⠀⠀⠀⠑⡇⠀⣆⠀⡟⠀⡇⠀⠄⣀⡀⠈⠀⠀⡰⠃⠠⠃⠀⠑⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠘⢢⡀⠀⠀⠀⠀⠆⡃⠀⠹⡀⢾⠂⠈⠡⠆⠈⠉⠉⠀⠀⢀⠞⠀⠀⠀⢀⡠⠈⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⠀⢈⠧⡀⠀⢰⠂⠂⠘⠆⣘⠀⣨⠀⠀⡸⠃⠀⠄⣠⠌⠀⠨⡀⣠⣴⣷⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠉⠈⠃⠤⣄⡆⠀⢹⣶⠁⠀⠀⣧⡠⠔⠂⣀⡀⠔⠐⢠⡏⠀⢸⡇⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠇⠠⠄⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠎⠁⠀⠀⠀⣾⠀⠀⣾⠁⠀⠀⠀⠀⠖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠰⢄⣀⡀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⡀⠀⠀⣀⠆⠀⠀⠀⠀⢠⡏⠀⡼⠃⠀⠀⠀⠂⠀⠠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⣀⠙⢆⠆⠀⠀⡀⠀⣶⣶⠶⡶⣶⣤⣤⡀⠐⠇⠀⠀⠂⢀⣤⠟⠀⠀⠙⢳⠀⠀⠀⠆⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⢠⡌⠁⢠⠀⠈⠑⠲⠞⠉⠉⢳⣀⠀⠀⠀⠈⢙⠻⡿⠒⠶⠖⠒⠋⠁⠀⡄⡀⠀⢾⣀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣬⠀⠀⠀⠀⠁⠂⠈⠀⠀⠀⠀⠀⠀⡄⠀⢹⡀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠋⢠⠙⠂⠀⢸⡀⠠⠀⠀⠀⠀⠀⠀⡞⠀⠀⠘⠃⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠐⠀⠀⠃⢀⣤⣴⣶⠟⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠘⠦⠘⠀⠀⠁⠀⠀⠁⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠞⣧⠀⠀⠇⠠⠄⠀⠀⠀⠀⠀⢠⡀⢀⠈⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠓⠈⠠⠂⠀⠀⠀⠀⠠⣤⠄⠀⢀⠀⠀⠂⠈⠂⠀⠀⡾⠂⠀⠰⢿⡶⠒⠚⣦⣠⣄⠐⣴⡆⠤⣶⣻⣦⡲⣐⠔⠢⣶⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠊⢁⠀⠀⠀⢀⣠⠃⠸⠀⠀⠀⠤⡀⠀⡄⠀⠀⡅⣄⢀⣀⣤⠀⠀⢀⣈⡁⠀⠀⠀⠀⠈⣹⢏⣾⢻⡜⢿⠓
⠀⠀⠀⠀⠀⠀⢀⠄⠼⠆⠈⠀⢀⡔⠈⠁⠠⠊⠀⠐⠂⠀⠐⠄⠀⠠⠐⢖⣀⣀⣈⣀⡄⢀⣀⣤⣀⠀⠀⣀⡀⠀⢡⠻⠟⠋⠻⢦⡑⣤⣦⣀⢀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢰⡏⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢷⡴⠿⠟⢛⠀⠤⠁⠀⠁⠀⠀⠈⠁⠀⠁⠀⠈⡋⢁⠀⠱⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢉⠛⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠄⠀⠀⠐⢂⠀⠰⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠒⣨⠀⠂⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠽⣇⠈⢷⡀⠆⠘⠆⠀⠰⢃⠤⠀⠀⠀⡀⠀⠀⠄⠈⢤⣀⣀⠤⠤⠀⢉⠀⠉⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠒⠋⠳⣤⠛⠾⢆⡀⠀⠐⠃⠴⠀⠀⠀⠃⠈⢁⠚⠀⠀⠀⡀⠀⠀⠀⠊⠈⠛⠉⠓⢄⠀⠀⢰⠛⡉⠑⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⠤⠄⢨⡧⠄⣀⡀⣔⡀⢀⠀⠀⢤⠀⠀⠈⠁⠚⠉⠀⠀⠀⠆⠀⡈⠀⠀⠀⡄⢀⡆⢀⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣈⠀⢠⠄⠁⠀⠀⠀⠀⠀⠉⡁⠀⠓⠀⠠⠘⠀⠑⡀⠀⠀⠠⠐⠒⠒⠄⠀⠁⠈⠂⠸⠀⠀⠙⠁⡀⠀⠀⠠⢉⡘⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣈⢀⠘⡁⠤⠌⠻⠀⠀⠉⠁⡈⠉⠀⡄⠀⠰⠃⢠⠀⢠⡐⠂⠀⠂⣄⠀⢠⡅⠀⠀⡜⠀⠃⠀⠀⠀⠀⠄⠀⡘⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠜⠁⠀⠀⠀⠀⢀⠀⠀⠐⠒⢢⠈⠉⠐⠒⠛⡄⣠⠕⠚⠉⠉⠓⠛⠁⠀⠀⡈⠉⠀⠀⠐⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠤⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠂⠉⠂⠁⠀⠤⠀⠀⠀⠤⠂⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠰⡒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⣽⣾⣷⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡄⠛⠍⢙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⡀⣰⢀⠁⠁⠘⢧⠴⢦⣀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⠟⣄⠠⠁⢠⠀⣠⠃⠀⠙⣴⣯⣾⣷⣡⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⣽⠷⠾⠓⠠⠼⣋⣁⣁⠠⠔⠉⠹⢿⠛⠭⡛⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⢀⠄⣀⡀⠀⣻⣗⣀⡈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠱⡐⠋⠉⠠⠆⠉⠿⢿⣃⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⢿⣀⠀⠈⣁⣠⡶⣶⣶⣏⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⡽⠿⠋⠻⣽⣟⣽⣠⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠲⣤⣶⠘⠟⣧⣷⣚⠁⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⡿⣭⣀⡀⠀⠀⠈⢻⡄⠆⢹⡟⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠄⣈⡁⠒⢦⣆⣻⠈⠀⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠛⣷⣰⣿⣷⠶⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣈⣽⣿⡽⠐⡏⠉⠃⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡻⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠁

⠀⣰⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⡀⠠⢠⠀⠀⠀⢀⠀⡤⠋⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢰⡿⠂⢀⣤⠀⡀⠠⢀⢀⡆⠊⢛⣿
⠀⠈⠂⠸⣃⠰⠋⠐⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⢦⣄⠈⠑⡼⣠⠰⠋⠀⢠⡼⠁
⠀⠀⠂⠀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠆⢑⠀⢦⡄⣰⡏
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢌⠛⡋⠤⠋
⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠
⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠆⢨⣀⠔⢉⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡤⡈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠
⠀⠀⠀⢠⡂⠐⠙⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⡴⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠠⡄⠀⠀⠀⠠⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⢠⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠶⢦⡀
⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⣿⠉⠛⠳⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠞⠛⠛⠉⠁⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢸⡇⣀⣠⡴⠶⠞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣤⣀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢿⡇
⠀⠀⠀⠀⠀⠘⡇⠈⠙⠳⠶⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠛⠶⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣬⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⡀⣀⡠⠄⠠⠄⠀⠀⠀⠀⠀⠒⠈⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠁⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠄⢀⡀⠀⢀⠀⡆⠀⠀⠀⠂⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⡀⠀⣤⠠⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣷⣦⢀⣴⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡿⣡⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢁⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁






⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏⢹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠉⣿⡇⢸⣿⠉⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣧⣤⣿⡇⢸⣿⣤⣼⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⣉⣉⣉⣉⡁⢈⣉⣉⣉⣉⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⡟⠛⣿⡇⢸⣿⠛⢻⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⣿⡇⢸⣿⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⡛⢿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡅⠸⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣦⡙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡇⢸⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁


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