Christmas Eve Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⡇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣶⣶⣿⣿⣿⠏⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣌⣰⣿⣿⣿⣛⣤⡿⢿⠹⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠛⠹⠫⣿⠉⠿⣿⣏⣹⣿⣥⢶⠟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣤⠂⢉⣽⣄⣻⣿⣿⡿⣿⣷⣯⣿⡽⢗⣖⠶
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣄⡀⠀⠀⢰⣾⠀⣼⡗⢛⣣⠘⡎⣿⣿⣿⣿⣿⣿⣛⠻⡿⢻⠛
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⢠⣿⣠⣿⡟⠉⠹⢤⣷⣿⣿⣿⣿⣏⠫⠏⠃⣀⠉
⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣷⣦⣶⣾⣿⣿⠟⠀⣿⡩⡿⢃⣽⣿⣿⣿⣿⣷⠶⣾⢧⣴⣿⡧⢤⠤
⠀⠀⠀⠀⠀⠀⠀⡽⢹⢿⣿⣿⣿⣿⣿⠟⠁⣠⠤⣭⣤⣾⢘⠉⣽⣿⣿⣿⣟⡿⠷⠫⠍⠉⠉⠻⠃
⠀⣠⣴⣿⣦⡀⠀⠀⠈⠉⣿⣿⣿⣿⣧⠀⡼⢿⣮⣁⢃⣽⣿⣾⣿⣿⣿⣿⣿⣶⣾⣾⣿⣖⢶⡖⠂
⠀⣿⣿⣿⣿⣿⡶⣷⠀⣀⡼⣿⣿⣿⣿⡠⢝⠾⠾⢿⣿⢿⣿⣿⣿⣿⣿⡟⠉⡝⠉⣵⠆⢀⣤⣤⣀⠀⢀
⠀⠘⠿⢿⣿⣿⣦⣸⣷⣿⢱⣿⣿⣿⣿⡗⠊⢹⠿⣦⣥⣤⣾⣿⣿⣿⣿⣷⣶⣷⢴⡷⢰⣿⣿⣿⣿⣦⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⡌⠁⣸⣿⣿⣛⣿⣿⡿⠛⣻⠛⢻⣿⣿⣯⣄⠸⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠸⣿⣿⣿⡏⠀⢹⣿⣿⣿⣿⣷⡟⣙⣵⣿⣹⣿⣿⣇⣀⣿⣀⣸⣿⣿⡯⠵⢂⠁⠀⣈⣡⣤⣾⣿⡟⠿⣿⣿⣿⣿⣷⡤
⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⢾⣿⣿⣿⣿⣿⣿⡿⢙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢽⠏⠂⠀⠎⢁⠉⣩⣿⡿⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⢺⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣏⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣼⡀⠀⣌⠑⡛⠛⢉⡉⠀⠀⢠⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠈⣿⣿⣿⠀⠰⣿⣿⣿⣿⠿⠋⢿⢱⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣽⣾⣷⣦⣽⣿⣷⣴⡟⣷⠀⠀⣸⣿⣿⣿⡿
⠀⠀⠀⠀⢸⣿⢿⣿⠀⠀⣿⡏⢻⣿⢠⢘⣿⡿⠟⢋⠟⠋⠉⣿⡿⢻⠛⣿⣿⣿⣿⡟⠉⠀⡜⠉⠁⠂⢲⠐⠀⢿⣿⡟⣿⣿⡀
⠀⠀⠀⠀⢸⣿⣾⡿⠀⣠⣿⠀⢨⣿⡾⠇⠐⠀⠀⢸⡀⡀⢰⣉⣁⠈⣶⠀⠀⢱⡞⠃⠀⠀⡇⠀⠀⠄⠸⠀⠀⣼⣿⡇⣿⣿⣧
⠀⠀⠀⠀⠸⡿⣿⣧⡀⠈⣿⡇⠈⢻⣗⠀⠈⠀⠀⢸⡇⠀⢸⠁⠀⠀⠿⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠃⠀⠀⢀⣤⣿⠄⣈⣽⣧
⢀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣈⣁⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣙⣉⣁⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⣤⠀⠀⠀⠀⠉⣤⣴⡿⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣴⣦⠀⠀⠀⠀⠀⠀⣶⣤⠀⠀⠀⠀⠀⣻⡿⣦⡀⠀⠀⠀⠉⣡⣬⣿⣤⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣾⣿⣼⡇⠀⠀⠀⠀⠛⠉⢻⡄⠀⠀⠀⢀⣤⣼⣿⣤⠀⠀⡴⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣿⠿⢿⣿⠛⠄⠀⢀⣠⠾⠛⠿⢿⠂⠀⠰⠋⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⣄⠀⣿⣿⣷⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣴⠀⠀⠀⢀⣶⣿⣶⣄⢰⡿⣿⢿⡿⡦⢰⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣴⣶⣶⣾⣦⡀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⣾⣿⠀⣤⢠⣿⣇⠀⠀⢸⣻⣿⣟⣻⢸⡷⣿⢾⡷⡏⢸⡷⣿⡇⣿⠉⣿⣇⣸⣀⣿⣸⣇⣿⢸⣿⣯⣿⣽⣿⡇⠀⣸⣿⣇⣼⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⢀⣸⣿⣯⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣾⣿⣧⣤⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤
⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣧⣿⣽⣽⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⢻⣿⣿⡿⣿⢹⠿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⣦⣿⣼⣼⣷⣮⣷⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠠⠀⢀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠁⠈⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢦⣀⠀⠀⠀⢀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⠈⡛⠶⠚⠉⠄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠯⢤⣥⣄⡀⠀⠀⠀⢀⣀⣤⣤⡽⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠆⠀⠠⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡿⠋⠤⠁⠀⠀⠙⠛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⠿⢻⠋⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⣶⡟⢣⠀⠀⠁⠂⠄⣀⠀⠀⠀⠀⠀⢀⡀⢼⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡟⠉⠀⠀⠑⢌⠀⠀⠀⠂⠀⠈⠉⠉⠀⠀⠀⣠⠔⠊⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⡄⠀⠀⠀⠀⠉⠒⠤⠄⢀⢀⠀⠤⠔⠂⠁⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠶⠶⢿⠛⣇⠀⠀⢀⡠⠂⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠂⠀⠀⠀⢶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⡆⠀⠀⢂⠀⡣⠀⠀⣜⢥⠀⠀⣄⣰⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⠁⠀⠀⠀⠈⢆⠀⠑⢄⠀⠀⠢⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⠀⠀⠀⠀⢢⡀⠀⠑⠄⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠛⠉⠉⠀⠀⠀⠀⠀⠙⢄⠀⠀⠈⠑⠂⠤⢀⣀⣀⣀⣀⣀⠀⠠⠄⠒⠊⢋⡩⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠰⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⠤⠤⠤⠀⠒⠊⠁⠀⡤⠤⠂⠀⣺⣶⣿⡃
⠀⠀⢀⣤⣤⣶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⣰⠢⠜⠀⣠⡾⡿⠋⠘⠻⢶⣄⡀
⠀⠀⠈⢛⣷⣦⠄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠊⠁⡠⠐⠝⡫⠊⠀⠀⠀⠀⠀⠉⠻⢷⣤⣤
⢿⣿⡛⠛⠋⠁⠄⡀⠀⠀⠒⢳⡦⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⢘⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠉⠻⣷⡔⠀⡀⠀⠁⠒⠠⠄⣀⢉⡴⢦⣄⠀⣠⠴⢦⡀⠀⠀⢀⡀⠤⠔⠂⠉⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣾⡛⠋⠁
⢀⣤⡾⠟⠑⠦⡀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢈⠛⡁⠀⣀⣿⣉⡁⠀⠀⠀⠀⣀⠄⠊⠁⠀⢀⣄⡀⠀⢀⣤⡀⠀⠀⠀⠀⢀⠘⢿⣄
⠻⣷⣶⣾⣤⣄⡤⢵⡆⢴⡛⠛⠋⠉⢉⠙⠁⡘⡀⠙⢋⣈⢉⣹⡧⠔⢂⡵⢤⠀⢀⣀⣀⣟⢀⡙⣚⣋⣀⣧⣀⣠⣤⣰⠏⣳⡄⠙⠻⣶⣶
⢰⣿⡿⠟⢶⢿⠳⣚⢛⣿⠃⠀⠈⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⡴⠾⠀⢸⡧⣼⡇⠈⡀⠀⡁⠀⠀⠐⠾⣭⣿⡔⣿⣷⠄⢻⣶⡿⠛⠉
⠘⢿⠛⠛⢧⣜⣦⡏⠉⣿⡇⠀⠈⢀⡁⠀⠁⠈⠉⡄⢀⡅⢠⣾⣷⠷⣄⠃⠘⠖⠾⠻⣿⠉⢉⡁⠂⠘⡁⠉⠐⣿⣯⡿⣿⠹⡄⢿⡷
⠀⣸⠁⠈⠉⠙⠋⢁⢈⣿⣇⠀⠀⠀⠑⠂⠀⠀⠀⠀⠘⠁⠸⡧⠁⠤⡹⠀⠀⠃⠀⠐⡿⠲⠄⠁⠀⠙⠁⣤⡴⢻⢻⡗⣿⠀⢹⡈⣷
⠀⡇⠀⠀⠀⢼⠛⠋⠏⡟⠻⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡈⠂⢠⠀⠀⠃⢀⠘⣇⠀⠀⠀⠀⠀⢰⣧⣘⣾⣼⣶⣽⠷⠴⣷⠘⣇
⠀⣷⠒⠓⠓⠚⠓⠂⠒⠔⠚⠛⠛⢛⣛⣙⣿⠀⠀⠀⠀⠀⠀⣿⠠⣀⠦⠀⠀⡂⠤⠈⣿⠀⠀⠀⠁⣿⠉⠉⠉⡁⠀⢀⣀⣀⣀⣏⢳⢻⡄
⠀⢻⡁⠋⡉⠙⢉⠘⠃⠰⠂⡄⠘⠀⡈⢰⡏⢀⣀⢀⣀⣀⣀⣿⠆⠁⠠⡆⠀⠋⡀⢪⣿⠤⠤⠤⠤⣼⡋⠉⠀⠁⠀⠀⠀⠀⢘⣧⠴⠟⠁
⠀⠸⠷⠦⠴⠦⠤⠦⠤⠼⠤⠴⠥⠤⠤⠼⠋⠉⠉⠉⠉⠉⠉⢻⣠⣊⣀⣂⣀⣢⣐⣈⡇⠀⠀⠀⠀⠘⣧⣤⣤⣤⣄⣤⣤⣄⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢰⣆⢤⣄⢀⡀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⢀⣉⣩⣧⡤⢭⣄⡁⠛⠀⠀⠀⠀⠀⠀⠀⠐⣦⣄⡀⡄
⠀⠀⠀⠀⠀⢈⠀⠈⠛⠵⡇⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⢠⣉⣿⡷⠚⣛⠛⠛⢧⡄⣹⡇⠀⠀⠀⠀⠀⠉⠐⠙⡿⠛⠢
⠀⠀⠀⠀⢠⠗⠦⢄⠀⠀⠀⠀⠀⠀⠀⣀⠀⣨⣁⠀⠀⠀⠀⠈⠁⠱⠤⣽⠤⠰⠇⠈⠙⡃⠀⠀⣀⣀⣀⣈⣂⣀⡰⣀⡤
⠀⠀⠀⢀⠗⠁⠁⠀⠈⠁⠀⠀⠀⠀⠄⢸⠒⠩⠍⢠⠀⠀⠀⠀⣀⠈⠀⢸⠀⠀⢀⣰⢰⠇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡌⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠈⡛⠤⠤⣜⣃⡀⠀⠀⠀⠙⠇⠀⢼⠀⠀⠐⠋⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠣⡈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠣⠤⡀⠀⠀⢒⠦⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠑⠬⠤⠄⠀⠀⠀⢀⣤⡄⠠⠄⠀⠉⠉⠬⡉⠀⠀⠀⠀⠀⠀⣤⠜⢋⣱⠆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠴⠀⠉⠀⠂⠤⠆⠀⠐⠂⠀⠀⣠⡜⠋⠀⠀⠈⠁⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠃
⠀⠀⢀⣀⡀⠀⡠⠀⡠⠀⠀⡀⠀⠈⠀⠀⣰⢖⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠈⠁⢀⡈⠀⣠⠂⠐⠧⠀⠀⠀⠀⠉⠋⠓⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠜⠛⠂⣹⠓⠒⣟⠐⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠁⠀⠂
⠀⠀⠉⡀⡺⠛⠀⠀⠘⠉⡀⢲⠁⠀⠀⠀⠀⠀⠚⠀⡀⠀⠀⠈⠚⠢⠀⠀⠂⠀⠀⠤
⠀⠀⡥⠊⢀⠄⠀⠀⠠⠀⠈⢀⠀⠀⠀⠐⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⠁⠀⡀
⠀⠀⢀⠐⠀⠈⠀⠀⠘⠁⡀⡸⠀⠀⠀⠀⠀⠀⠀⢀⢺⠁⠀⠀⠐⢂⠀⠀⢀⠀⠠⠐⠀⠀⢀⠒⠀⠀⠀⠐⠂⠒⠒⠰⣦
⠀⠀⠧⠁⠠⠾⠤⠠⠴⠀⠈⠀⠀⠀⠀⠀⠐⠐⠂⠀⠉⠀⠀⠈⠂⠀⠉⠀⠀⠁⠃⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⢂⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢑⣄⣼⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⡅⣿⣿⣿⣾⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠄⠻⣶⣿⣮⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠟⣷⡟⢻⣿⣽⣿⢻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⣶⣿⡏⢹⣟⣯⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣆⡼⢿⣿⣿⣿⣷⣾⠋⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢒⣀⣈⣯⣽⣻⣿⣿⢿⡀⠆⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⠯⣩⣝⣿⣇⣹⣿⣧⣾⠻⡷⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⠀⢶⠄⠈⠙⠑⢸⣿⡆⢹⠁⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣸⡇⠐⠷⠶⡶⣶⠄⠹⡇⢸⠷⠶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠨⢼⠷⢶⢾⣿⡏⢀⣤⡆⢸⣶⡤⣴⠇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢅⣠⡦⠤⣄⡾⠇⣯⣠⡘⣿⣧⣌⣟⡻⠙⠛⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢰⣀⡀⢀⣶⣽⣭⡿⣎⢻⣿⣿⣿⣶⣶⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⣷⠻⣌⠲⣶⣼⠿⠷⢾⣶⡟⢸⣿⡿⣿⣿⣝⣛⠛⡛⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣸⡏⠀⢃⢠⣶⣶⣶⣶⣿⣧⢠⣿⣿⠁⠻⢇⠉⠣⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣧⣾⣥⡴⠾⠛⠿⠿⠧⠽⢿⣻⣿⡿⠃⢼⠻⢿⡆⢄⣩⣀⡀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⠟⠙⠿⠿⠶⡶⣶⣝⣀⣠⢞⡛⢿⣿⣗⣿⣿⠯⠄⠸⣀⣍⣡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⢻⡌⠏⣥⣰⠏⡱⢾⡍⠉⣛⡻⣟⠳⠤⢀⣙⣻⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠣⠉⠀⠀⣾⠻⡄⠙⠃⣼⣶⣀⣀⠙⠛⢙⠻⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣿⠇⠐⠀⠀⠀⠈⠋⠑⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡀⣴⣦⠴⢢⠖⢠⠆⢀⣤⠇⢰⣿⠠⡄⢰⠀⠄⠤⠀⠤⠀⡴⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⢡⣄⠀⡁⠀⣸⣏⠀⠘⢩⢝⠃⠈⣠⣄⢡⡄⢬⠘⣁⣀⣼⣳⡿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠀⠻⠌⣾⠏⠉⢹⣷⡀⠐⣾⡀⣨⠋⠉⣝⠃⠋⡯⡹⠆⠃⠈⠛⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠈⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠉⠁⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⢀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣿⢰⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣶⣾⣿⣿⣿⣿⡿⣦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠃⠈⠱⢿⣿⣿⣷⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣠⣼⣿⣿⣿⣿⣽⡶⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⢿⣿⣿⣿⣿⣯⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣇⠀⣸⣿⣯⣬⣿⣿⣿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⢉⣴⣦⣿⣿⣿⣿⣿⣇⡹⢿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣶⣿⣶⣿⣿⣿⠟⣿⣿⣿⣿⣥⣍⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⠿⠟⢿⠿⠿⣿⣿⣻⣄⠛⠛⣿⣿⣿⡿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡀⣀⠛⣿⣿⣿⣿⣿⣿⣾⣿⣿⣽⣿⣷⣿⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣶⣛⣟⣧⣶⣶⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⡟⠛⠻⣍⣿⡟⢻⣿⣿⣿⣿⣿⣿⣬⣛⠻⠿⢶⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⣥⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣤⣀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣍⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢾⣿⣿⣿⣟⣇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⣼⣽⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⠛⢿⣿⣥⣿⣿⣦⣭⣽⣿⣟⣷⣿⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣏⣹⣿⠋⠙⠛⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿⣿⠿⠛⠻⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⡏⠑⠈⠁⣽⣿⣟⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣶⣴⣿⣿⣿⣿⣏⢉⣿⣿⣿⣿⣿⣅⣰⢀⣀⣿⣿⣿⣽⡙⠿⢿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⣿⣟⢿⣿⡿⠿⢿⣿⠁⠻⣿⣿⣿⡎⠛⠻⣿⡷⠈⠉⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠛⠃⠀⠈⠉⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⡀⠀⡀⠀⢀⠀⠀⠀⢀⠀⡀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠈⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠁⠀⠀⠀⠁⠁⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⡀
⠀⠀⠀⠒⠦⢀⡀⣠⠠⢄⡀⢀⠀⠐⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣤
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡼⢷⣴⣿⣠⡞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣔⣦⣼⣿⣍⣼⣿⣿⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠿⣾⡿⠯⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢽⣿⡄⠉⢀⡼⣷⣿⡏⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣾⣿⡷⠶⣻⣷⣻⣿⣇⡛⡳⠒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⡹⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣿⣿⣿⣿⣿⡿⢷⣏⣉⣏⣍⣉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⣾⣿⣟⣿⣿⣿⣿⣿⣦⣾⣟⠿⣿⡿⣀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣠⡌⠿⠿⣉⣿⡿⣿⣧⡷⣷⣿⣿⣿⣿⣿⣿⡷⠖⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠠⣅⣹⣶⣿⣤⣴⣿⣥⣴⣿⣳⣷⣜⣷⢢⡻⣿⣯⣤⡤⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⡟⣿⣿⡛⣛⣿⣟⣿⡿⣽⠻⣿⣿⢿⣷⣿⢷⣿⡿⡻⣿⣟⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⣯⣿⣿⣿⣿⣿⣛⣽⢳⣙⣷⠿⢿⣄⣷⡹⠋⠈⣱⣟⣿⣽⣿⣛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⣻⡿⣟⣻⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⢋⣏⠻⣾⣿⣻⣿⣯⣿⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠟⡛⠻⣿⣦⣭⣿⣿⡏⣝⣻⣿⣿⣿⣿⣷⣾⣿⠟⠛⠓⠾⠿⠷⠿⣿⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⢻⡉⠁⣛⣽⣧⢦⣸⣿⣿⣿⣿⣈⣿⣿⣶⣶⣷⠀⠀⠐⠏⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡇⠀⢾⡀⡀⢿⠩⢿⠘⡎⢿⠷⡀⠻⡿⣿⣿⠸⡿⣿⣆⡀⠀⠀⢸⡇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣦⣷⣶⣿⣿⡿⢾⠀⠈⠀⣧⡀⠀⣿⣦⣛⣿⣿⣶⣖⣿⣭⣷⣄⠤⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠁⠀⠼⠁⡄⠘⡣⠷⠀⠽⠻⢿⡿⠿⠿⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢰⡄⠀⢴⠀⢀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⡀⠸⠀⢰⠈⠊⠸⠠⠃⠸⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠠⡊⢈⡆⠱⠠⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠀⠀⢀⢎⠑⠂
⠀⠀⠀⠀⠀⠀⠈⡢⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠁⡰⠀⢀
⠀⠀⠀⢠⠀⠁⠪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⣡⠐⠄
⠀⠀⠀⠠⣀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠿⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⢿⠛⣦⡀⠀⢀⡴⠋⡟⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠺⣿⣩⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣝⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⠀⠀⠀⠠⢴⣾⣿⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⣅⠀⢀⣀⣀⠀⢹⣿⣿⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⣿⣏⠀⢈⣿⡁⠀⢸⣿⣿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⢀⠀⠀⠀⠉⠉⠉⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣌⠳⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠾⢷⡿⢿⣿⣷⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠋⢁⡾⢿⣿⣾⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣅⢙⢴⣿⡿⠞⣩⠗⣷⡾⠁⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠛⢟⡿⠟⣨⣽⣍⡉⣛⣤⣶⣖⣡⠃⢀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠟⢖⠉⣀⣰⣿⣿⣤⣩⣵⣦⣤⣻⡿⣿⠿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⡴⢂⣠⣼⣷⣿⣿⣾⣿⣿⠄⠼⢯⣹⡻⢿⠷⡾⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠻⣿⡿⠟⢫⡿⢿⡽⠿⠷⣶⣦⣄⣸⡃⡀⢉⣭⣬⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠷⠀⠰⡌⠉⣠⣤⠿⢓⠾⣿⣒⠒⠊⢿⣭⡈⠛⠻⠿⠥⠷⢀⣠⡄
⠀⠀⠀⡰⣰⣶⣤⠀⠻⠴⢿⡦⣰⣿⠺⢋⣠⣶⢽⣦⠈⠻⣿⡀⠀⡜⢻⠿⣶⣼⣶⠟⠋⠉⠁
⠀⠀⠀⢻⠿⣱⠃⣀⣀⠀⢈⣿⠟⢁⣀⣽⣷⣟⣹⣿⣀⠀⠸⢿⣷⣬⣭⣄⠈⠛⠤⣀⠀⢀⣴⣄⣿
⠀⠀⠀⠏⢠⢳⣎⢿⣿⡗⠋⢁⣴⣦⢿⣿⣧⣤⣠⣿⣿⣶⡦⢰⣬⡟⠺⡁⠿⠛⠶⣦⣭⣽⣿⠛⡿⠤
⠀⠀⠀⣇⢺⣀⢿⡜⠟⠛⣘⣟⣩⣬⣿⣿⣿⡟⠉⠀⠀⢹⡇⣿⣿⣇⣞⣥⣦⣄⠀⡏⡞⣻⠇⢠⠛⠁
⠀⢀⣴⣿⠀⢳⣾⣴⣾⠛⢻⣾⣿⡏⠀⠸⣻⣇⡼⠛⠻⡖⠤⠤⣽⣿⣿⠁⠀⢻⣦⠧⢛⣫⣮⢏⣀
⠀⠀⠺⠟⠣⣼⣻⠟⠋⡀⢰⣿⣿⣄⢀⢰⡇⢽⣇⠠⢸⣆⡀⠐⢿⣿⠙⡀⢃⠛⣿⣯⡽⠛⠳⣦⢤⠐
⠀⠀⠀⣰⢲⠙⢟⠻⢿⣥⠏⣩⣏⠈⠑⠚⠁⣸⣱⣤⡴⠷⣠⡖⢾⣿⣷⣦⣅⢖⠁⠙⢂⠧⠄
⠀⠀⠀⡧⢈⢤⣼⣶⣤⢰⢬⣉⠛⠳⢦⣴⣶⣿⣿⣿⣿⣴⠅⣴⠾⠿⠟⠛⠉⠀⠀⠀⣸⣷⠄⢠
⠀⠀⢠⢻⡇⠸⡻⠏⠁⢸⣸⣿⢠⣀⠀⡀⠀⠉⢡⠰⢰⠀⠀⠀⢀⢠⣄⣠⡄⣤⢢⢀⡿⢡⠄⠊
⠀⠀⢸⠋⢳⣰⠑⠦⢤⣀⡈⠉⠃⠸⠀⠇⠯⡇⠸⠀⠸⠽⠼⠸⠏⠞⠒⠛⠋⢉⡀⢼⢀⡞
⠀⠀⠸⣆⡿⠁⠀⠀⠀⠀⠈⠉⠙⠒⠤⣀⢤⣀⣀⣀⣀⣀⡀⠤⠴⠤⠐⠊⠉⠀⠀⠀⠉
⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⣕⠮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠂⠁⢄⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⡀
⠀⠀⠄⠀⡀⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠁
⠀⠀⠦⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠒⠒⡀⠀⠀⢀⡀⠀⠀⠂⠀⠀⠠⠀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠀⠤⠀⠄⠀⠁⠄⠀⠀⠀⠀⠄⠀⠀⠠⠠⠄⠀⠴⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠐⠐⢀⠂⠀⣀⡄⠀⡀⠀⠀⠀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠴⠠⢀⠄⠂⡰⠄⠁⠀⡂⢠⢁⠆⡁⠁⢱⠦⠁⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠠⣸⢴⡂
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⡁⠀⠐⡈⡐⠀⠀⠁⠀⢀⠐⠀⠀⠀⠀⠀⠊⠃⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠄⠀⠀⠄⠀⠂⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⡉⠀⠐⠀⡆⢈⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠤⡷⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠁⠀⠀⠂
⠀⠐⠉⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠰⠨⠐⠪⠀⠀⠀⠀⠀⠠⠠⠁⠀⠀⢀⣀⣧⡴⠂⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⠀⠀⠀⢠⠤⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠁
⠀⡀⢥⠢⢈⠄⠂⠌⠈⠔⡁⠀⠀⠀⠀⠀⠀⠀⢤⠓⡍⠋⠀⠀⠀⠀⠐⢄⠐⠀⠔⠀⠀⠀⠀⠠⠠⢠⠈⠀⠀⠀⠀⡀
⠀⠀⡃⠔⢲⢈⣃⠢⠀⢈⡁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠰⠠⢐⠀⡀
⠀⠀⠠⠠⢁⡈⠀⠀⠌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⠐⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠐⣐⠂⠀⠔⠠⣀⠄⠬⡄
⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⡐⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠈⠀⠀⠀⠘⠀⠈⠡⠈⠃⠄⠀⢍⠄
⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⠀⠀⠠⠂⡀⠀⠐⠉⠁⠉⠁⠀⠑⠀⠀⠢⠀⠁⠀⠀⠀⠀⠀⠀⢁⢂⡄⠁⠑⡀⠄⠄
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠆⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠑⠀⠓⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠲⠄⠐⠂⢀⠀⡀⠄⢀⠁⠰⠀⠀⠀⣤⡄⠀⠀⠀⠀⠁⠀⠄⡂⢀⠤⠂⠂⠰⠂⠀⢠⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠐⢄⠀⠢⠆⠈⢀⠠⢐⠀⠀⠀⠁⠀⠀⠀⢐⠂⠄⡀⠀⠠⠂⢀⠄⠀⡠⠄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣹⠦⡄⠁⣤⡆⣠⠄⠀⢩⣠⠀⠀⠀⠀⠀⢠⣠⠀⠀⠠⣄⢢⡔⠁⣠⢴⣁⠀⠀⠀⠀⠀⠃
⠀⢐⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠄⠀⠤⠄⠀⢠⢠⡀⠀⡀⠀⠀⡀⡀⠀⠀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⢢⡄⣉⠀⠀⠃⣠⢀⡐⠲⠃⠆⢐⠁⠀⠌⡀⡔⠰⠄⢆⠐⠲⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠀⠐⠬⠅⠚⠀⠒⠂⠃⠬⠚⠍⠡⠀⠄⠄⠀⠧⠰⠇⠴⠂⠢⠼⠉⠡⠀⠰⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⠄⠀⠀⠀⡤⠀⠀⠀⢤⠄⠀⠀⠠⡄⠀⠀⠀⢤⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡀⢄
⠀⢀⠐⠀⠁⠁⠀⠀⢐⢢⠀⠈⠀⠀⠀⢸⠃⠀⠀⠐⠑⠀⠀⠀⠊⠂⠀⠀⠘⠁⠀⠀⠀⢹⠀⠀⠀⠊⠀⢠⣘⢧⠖⠀⠀⠀⠂⠠⠁
⠀⠀⠐⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢠⠂⠀⠀⠠⠘⠠⠀⠀⠀⢣⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠜⢲⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠉⠀⠀⠁⠀⠀⠀⠂⠂⠀⠀⠀⡁⠀⠈⠁⢀⠠⠁⠨⠑⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠈⠀⢐⠀⣁⣀⣑⠈⠀⠈⢈⠁⣀⠤⠄⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⡀⠄⢀
⠀⠀⠐⠄⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⠀⣀⠀⠆⠄⠘⠀⢸⠀⢤⠠⠃⠀⠀⠀⣤⠀⠀⠀⠀⠀⡎⠐⠐⡄⠁⠀⠀⠤
⠀⠀⠑⠂⠂⠁⠀⠀⠀⠀⢀⠀⠀⠀⠙⠁⠀⠀⠀⠄⠀⠀⠠⠈⠄⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠰⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⢀⡀⠀⠀⠀⠀⠀⠔⢉⢌⠑⡀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠤⡀
⠀⠈⣹⢽⠢⠆⠀⠀⢠⡄⠀⠀⠀⠀⠈⢁⡀⢰⠀⠀⠀⠀⠀⠀⢄⠁⠐⢀⠂⠀⣀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠟⠀⠀⠀⢄⠀⢀
⣀⣀⣁⣘⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⣿⣋⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣧⣿⣿⣯⠛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⣿⣅⣹⣷⡟⣷⡟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⣿⣷⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡟⢿⣿⣾⣿⣿⣿⣏⣉⣿⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⡰⣿⣿⣿⣿⣻⣿⣿⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡶⣿⣿⣟⣹⣿⣿⣧⣿⣨⣿⣿⡽⠧⣝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡼⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⢤⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣐⣶⣤⣿⡿⢿⣏⣿⣿⣿⣿⣿⣿⡟⢻⣷⣿⠷⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢋⣿⣿⣷⣾⣷⣿⣿⣽⣷⣾⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠁⣩⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⡟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣿⠿⣿⡟⢻⣿⣿⣟⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⡏⣯⣤⣤⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⣿⠐⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠿⠿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⣤⣾⣷⣾⣟⣿⡟⣿⣿⣿⣿⣿⣛⣿⣾⣿⣿⠿⣿⣿⣿⣥⡴⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠼⠾⠿⡿⢯⣤⣿⣿⣻⣿⣻⣇⣻⣿⣿⣿⣿⢁⣿⣿⣉⣿⣶⣿⣿⣿⣿⣿⠻⢏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠷⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⢿⣻⣿⡿⣧⣿⣿⣿⣿⣿⣿⡇⢀⣤⣈⣀⣠⡤
⠀⠀⠀⠀⠀⢀⣀⣳⣤⣿⣿⣿⢟⣷⣤⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡿⣿⣭⣿⣿⣿⣿⣿⠿⠿⣻⣷⣍⣯⠁
⠀⠀⠀⠀⠀⠈⠉⣻⠇⢹⣏⣙⠿⢿⣿⣿⣿⡿⣿⣻⡿⢿⣿⠉⣿⠷⠛⣯⣽⣿⣿⣿⡿⣭⣽⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⢿⣤⣿⡏⣅⠀⠺⣿⣶⣾⣿⡏⢠⣾⣿⣿⡂⣿⢠⡾⠏⢹⣿⡟⠛⣽⣀⢺⣿⣿⣿⠿⣷⣶⣷⣶
⠀⠀⠀⠀⢠⣤⣶⣶⠈⣿⣇⠀⡐⢃⡝⢿⣿⣿⡇⣾⢻⢣⣾⠛⣯⣾⡟⣁⢻⣿⢁⠰⣿⡿⢻⣋⣁⣤⣶⣾⣯⡼⢿⡄
⠀⠀⠀⠀⠸⣿⡟⠉⢉⣉⡉⢃⣾⠿⢟⠿⣻⣿⣧⡿⢸⣾⣼⣷⣾⣿⡇⠙⣼⣿⡌⠥⣿⢿⣾⣯⣽⣿⣿⣿⠿⠳⠋⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣴⣿⣿⡏⣑⠛⢱⣻⣿⢿⣿⢸⢻⣾⡧⡴⣶⢉⣙⣻⣿⠷⢻⣿⣽⣿⡾⠷⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠻⠷⣿⢿⡿⠟⠉⠛⠐⠚⠿⠇⢷⡏⢻⡿⣻⣧⠴⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠉
⣦⣶⣶⣶⣦⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣴⣦⣴⣶⣦⣤⣴⣶
⣿⣿⣏⣀⣁⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠟⣁⣀⣶⠐⠋⠙⠛⠛⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢦⢍⡝⠁⣼⠀⠞⢹⣿⣧⠀⠂⠀⠀⠀⠠⣷⣶⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠟⠁⡤⠳⠂⡈⠣⡔⠀⣰⢁⠂⢼⣿⣷⡄⠀⠀⠀⠄⠛⠛⠁⠁⢹⡟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠊⠀⢰⠀⠀⢱⠘⠠⡀⢠⠀⡀⠑⠈⢇⠟⠛⠙⠛⠲⠶⢾⡆⠈⣠⠟⢈⠻⣄⠄⠀⠒
⠀⠀⠀⠀⠀⠟⢆⠈⠓⠀⢤⡀⢀⡿⡶⠋⠉⢁⡬⠝⡿⠄⡸⢆⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡤⠀⠙⠏
⠀⠀⠀⠀⠀⠈⠈⢦⣀⣱⡀⠁⡾⡇⠀⠀⠜⠃⢀⠞⠃⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠠⡾⣷⣄⢀⣀⡀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⢠⡀⠈⢟⡳⢼⣷⣷⣾⢏⣤⣤⣈⡠⠴⠶⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠻⣿⣿⣿⣄
⠀⠀⠀⠠⢂⣀⡤⠴⠿⠶⠺⣿⣿⣿⣿⠷⠚⠿⠤⣴⣚⡺⠧⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣄⡀⠛⢻⣿⣷⣆
⠀⠀⢨⡆⠉⠀⡀⣤⢀⠔⠞⢡⠏⣿⣿⡟⠲⢶⠤⠚⣏⡠⠄⠁⠀⠀⠀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⣛⡾⢡⡏⠐⠱⠦
⠀⠀⠋⠀⢉⠑⠂⡾⠥⠤⣶⣟⠀⠙⡿⠈⣗⠲⢆⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠇⣇⣴⣴⢦⡶⣤⣆⠀⠀⠀⠀⣿⡿⠤⢈⠁⡄⠀⠎⢨⠙
⠀⠀⠀⠀⠨⡶⢊⠻⠀⣴⠃⠸⡀⠀⣇⠀⠊⠀⠈⠤⣌⠀⢀⡤⣄⣤⠀⠀⠀⠀⢀⡄⠀⠀⠀⢸⠏⠀⠀⠀⠀⡺⡧⠒⠤⠀⡄⢤⠀⣌⠹
⠀⠀⠀⠂⠀⢰⣤⣴⣐⠁⠀⢠⠛⢆⣍⠘⠂⠢⠀⠀⠀⠁⣞⠐⣱⢿⣰⠦⣆⣴⣸⣱⣶⣦⣶⣠⣆⠀⡀⠀⢀⠐⠆⠀⠒⠲⡅⠠⣁
⠀⠀⠀⠀⠗⠀⠙⠹⣇⠚⣶⣿⡻⠾⢟⡓⣖⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⠀⠔⠓⠦⣴⠀⠲⡄⢙⡆
⠀⠀⠀⠀⠀⠉⠀⢢⡴⠴⠛⠛⠻⢶⣦⠰⡦⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠘⠇⡀⣒⡳⢦⠜⢢⡀⠃⣄⢹
⠀⠀⠀⠀⠀⠀⠈⠫⣤⣓⣀⡶⠮⣤⣴⠶⡼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣂⠧⢀⣘⠇⣸⠿⣿⠦⣀⢼⡀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠫⠿⠘⠥⢈⡶⠶⣾⠗⣼⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡀⢌⠉⠲⢴⢉⣈⡝⢀⣠⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠘⠷⣶⣤⣤⣙⣏⡶⣟⠛⣣⣾⣿⣿⣿⣿⣿⣦⣄⣀⣀⡀⡀⣀⣠⣴⡆⠄⠀⣿⣿⣧⠀⢙⠂⠱⠠⠶⠠⠆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⡿⠛⠉⠘⠁⠀⢁⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣶⡆⠄⢹⡅⠀⣸⠀⠖⠀⢰⣾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠁⠀⠀⠀⣰⡦⠀⢶⣿⠟⠉⠋⠀⢹⣿⡟⠁⠀⣼⠉⠀⠛⠁⣿⣷⡄⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣶⣆⣴⣶⣿⣿⣷⣄⡀⠛⠢⣬⣽⡀⢄⠟⡢⠀⣠⣿⡿⠂⠁⣼⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⢀⣰⣀⣄⡀⠀⠉⠁⢀⣷⣿⠟⠋⠀⠀⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠉⠀⠀⠀⠐⠛⠁

⠀⠀⠀⠐⠀⠈⠁⠈⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣄⡁⠗⠀⠀⡘
⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣴⡤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⣤⠀⠀⠀⠀⠀⠀⣶⣿⣷⡄⠀⠀⠀⠀⣩⣁⠀⠀⠀⠀⠘⡛⠀⠀⠀⠀⠀⠀⠴⠄⠀⠘⠋⠀⣤
⢀⡀⣤⡀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀⠀⠀⢸⣿⣿⠓⠀⠀⠀⣼⣿⣦⡀⠀⠀⠀⢠⣾⣆⠀⠀⠀⠀⢩
⢠⣾⣿⣷⣌⠁⠀⠀⠉⣿⣿⣿⣋⠁⠀⢴⣿⣿⢿⣷⡄⠀⢈⣽⣿⣿⡛⠀⠀⢠⣿⠃⡷⡆⠀⠀⢀⣿⡄
⣻⣿⣿⣿⣿⣍⠠⠀⣞⣿⣿⣿⣿⣆⣰⡟⡟⣿⢿⣿⣷⡴⢻⣿⣿⣿⡿⠦⠀⣼⣿⠈⡟⢌⢢⠀⣼⣿⣇
⣿⣿⣿⣿⣿⣯⡀⠾⠛⣿⣿⣿⣿⢍⣿⣿⣶⡟⣦⣿⣿⣷⢿⣿⣿⣿⣿⣦⡜⢻⠏⣤⡗⢬⠫⣠⣿⣿⣿
⣿⣿⣿⣿⣿⡷⣬⣁⣾⣿⣿⣿⣿⣿⣋⢏⠏⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⡟⣰⡀⣇⢹⣌⢳⣾⣿⣿⣿
⣿⣿⣿⣿⡿⣿⣮⣼⣾⣿⣿⣿⣿⣿⣷⠀⠀⢱⠀⠡⠾⠟⠛⠉⢹⠉⡩⢋⠟⡆⢨⡀⠉⡄⢹⣿⣿⣿⣿
⣿⠟⣿⣿⣿⣮⣝⣛⠉⠉⢹⡇⣠⣦⣤⡄⢸⡛⡿⠀⢰⡟⢻⡆⢸⠀⠓⣡⣋⠐⠊⠱⠤⠈⠻⠛⠛⡟⠛
⠁⠀⠛⠀⠙⠋⠉⠐⠀⠀⠘⠁⣿⣿⣿⠇⠈⠛⠃⠀⠘⠛⠛⠃⠈⠀⠸⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⣧

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⣏⠿⢻⣌⣳⣶⡀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠓⠛⠦⠀⠀⢀⠀⡀⠉⠁⠙⠏⣡⣼⣤⠴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⣶⡂⠐⢂⢀⣰⣟⡶⠄⣀⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠸⣧⡼⣈⡿⠆⠿⣛⣬⡁⢾⣇⡈⢳⡽⠁⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢀⠍⠁⠈⠉⠁⡀⣀⠀⠉⢯⣍⠿⡇⠩⠛⣦⣼⠶⡄⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⠀⢀⡶⠎⠁⠘⠀⠀⠀⠀⠀⠙⠀⠠⡀⡀⠀⠉⠀⠉⠑⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠚⣿⣿⡤⠛⠁⢀⣀⣀⢄⣠⠀⠒⢤⣄⣀⠀⢀⡌⠖⠀⠒⠶⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡚⠓⠛⠛⠛⠁⠠⠆⠈⠁⢠⡒⡁⠨⣿⣿⡄⠀⠋⠄⠀⠁⠒⠰⠄⠑⢥⡤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠝⠒⠐⠲⣁⠑⢤⣀⠀⡀⠀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢱⠆⡀⠀⠀⠀⢢⡀⠂⡀⠀⠀⠀⠀⡁⢀⠀⠀⠀⠀⡴⠀⡶⢚⠐⢤⡀⠁⠀⡌⣿⣿⣭⣝⣓⡁
⠀⠀⠀⠀⠀⠐⠠⢄⠀⣦⡾⠐⠕⢄⡴⢶⠂⠀⠘⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠈⣳⠈⣂⡈⡀⠚⠙⡆⠈⢇⡀⠀⠀⠀⣀⠝
⠀⠀⠀⠀⠀⢀⠤⣾⡟⢻⡀⠀⠀⠘⡆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣟⠽⠆⠠⢴⣿⡄⠈⠹⢟⠙⡍⢯⠀⢦
⠀⠀⠀⠀⠀⣈⠲⣼⣇⣠⣴⡀⠀⠀⢹⡄⠈⢂⡀⠀⠀⠀⠀⠀⠀⠠⡀⠆⠀⠀⠳⣤⡉⠀⠀⣀⡜⣹⡀⠀⠶⠀⢶⣧⢸⡅⣼⡀
⠀⠀⠀⠀⠊⠀⠀⠀⢏⠁⠀⠁⠀⠂⠘⠏⢢⡀⠁⢄⠀⢀⠀⡀⠀⠀⢠⡄⠀⠀⠀⢙⣷⡆⠉⠁⠈⠀⠈⡁⢄⡈⠼⢿⣼⡷⣿⡧⡀⣀⣀
⠀⠀⠀⠀⡇⠐⠀⠀⣟⠀⠂⠄⣀⠀⡄⢲⡀⠹⣄⠘⠁⠀⠀⠃⠀⠀⠰⡇⠀⢀⡄⠈⠍⠁⠐⠂⠀⠙⠉⠁⠀⠁⠀⠖⠨⡇⢭⠀⢼⠧⡤⠐
⠀⠀⠀⠀⡇⢨⣆⠀⣿⣄⣠⡘⣉⢢⣈⠢⠀⠀⣿⠗⠀⣀⠲⣆⣀⡴⢖⡹⠀⠀⢀⣀⣰⡆⣰⡀⣠⣞⣁⣼⡤⣿⣧⣿⡤⣋⠴⣞⣿⣿⡞
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠁⠈⠁⠀⠁⠀⠀⠀⠉⠀⠈⠁⠀⠈⠉⠙⠛⠋⠉⠉⠉⠈⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣠⡿⠁⠈⢷⡀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠁⠀⢀⡄⠱⡄⢀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⠟⠀⠀⢸⡇⠈⡇⠀⠀⡗⠀⠀⠀⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣛⠛⠉⠀⠀⠀⢸⡇⠀⣿⠀⢰⠇⡄⠠⡀⠦⡀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⠤⠀⢄⢀⠀⠀⠈⠏⠀⠀⠀⠸⡀⠓⠀⠧⠀⠇⠙⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡼⠁⠀⠀⣸⡇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⢸⠃⠄⠀⡀⠀⠉⠀⠀⢠⠀⠀⠈⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠓⠃⢸⡇⢶⠀⠇⡄⣨⠰⣇⣸⠂⣶⢰⡄⡇⣰⢀⡄⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡴⠙⠃⠾⠈⠸⠇⠙⠂⡹⠻⠎⠛⠘⠸⠇⠉⠉⠧⠀⡽
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⢀⣘⣛⣂⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢁⡿⠄⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣹⢸⡟⣇⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡍⠰⠀⠀⠀⠀⠀⠴⣠⠄⡧⣠⠀⠀⠀⠀⣰⢻⡇⣿⢸⡇⢹⡄⠀⠀⠀⠐⠀⠀⠀⠆⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣬⠭⠵⣫⡴⣿⡂⠀⠀⣠⠇⡾⢸⣿⢸⣿⠈⢷⡀⠀⠀⠐⠀⠁⠀⠖⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠥⠞⠑⠃⠀⠀⠉⠈⠁⠉⠉⠈⠉⠀⠈⠁
⠀⠀⠀⢰
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠸
⠀⠀⠀⢸
⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⠃⠐⢩⢔⠀⠀⠀⠀⠀⠚⠥⠀⠀⠀⠀⠀⠀⢠⢋⡂⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⡔⠃⢒⡀⠀⠀⠀⠀⡈⠉⠭⠂⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠐⠠⠠
⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠃⠀⡈⠀⠰⠣⠀⠀⡀⠭⠭⠭⠙⠀⠀⠀⠀⢸⠗⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠄⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠋⠉⠌⠄⡠⠀⠀⢑⣒⣒⢒⣒⣲⠄⠀⠀⠍⠀⠀⠀⠀⡄⠀⠀⡄⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠌⠌⠀⠀⠀⠀⠀⠀⠸⠡⠤⠬⡭⠥⠤⠄⠀⠐⠦⠀⢀⠀⠀⠀⠀⠀⠀⠁⣀⡐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣨⣿⠋⣥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣿⣿⣿⣷⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡈⣻⣿⡟⠛⠫⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡾⢿⡏⢙⣧⣀⣴⠛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡀⢀⣿⠟⢿⡟⠛⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⡿⢿⣿⣾⣿⡿⠿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣏⠀⠀⣿⣄⣼⣇⠀⣀⡟⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⢹⣿⡿⣿⡿⠛⠻⣿⣿⠿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣦⣤⣼⣷⣴⣿⣇⠀⣀⣿⣧⣤⣾⣭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⠛⣿⣿⠛⠛⢻⣿⠟⠿⣿⣧⣽⣿⡏⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣴⣿⣿⣄⣀⣸⣿⣤⣤⣿⡟⠉⠹⣿⡿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⠛⢿⣿⡿⢻⣿⣿⣿⣧⣤⣶⠿⠿⢿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣼⣷⣤⣼⣿⣿⣿⣿⣧⣀⣿⣿⣏⡀⠀⣼⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⠋⢻⣿⢉⣻⡏⠀⠈⢻⣿⣿⠋⢹⣿⣿⣾⣿⠟⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣾⣿⣿⣿⣿⣿⣤⣾⣿⣦⣶⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣍⣀⣸⣿⠁⠀⢹⣿⠛⠛⣿⣿⣥⣿⡟⠁⠉⢿⣏⠁⠹⣟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⣿⣿⣿⣿⣦⣴⡿⢻⣶⣾⣿⠋⢹⣿⣧⣄⣠⣿⣿⣿⣿⣿⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⣛⣿⣿⠿⢿⣶⣾⠿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠉⠁⠀⠈⠉⠉⠉⠉⠁⠉⢫⠝⠛⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠀⠀⠀⠄⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⠀⠤⣄⠠⠀⠀⠉⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁

⣿⢿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⣰⣲⡄⢄⣛⣡⣴⣿⣶⣶⣾⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣉⣁⣀⣀⡈⠉⠛⠷⣤⣀
⠀⠀⠀⠈⠉⡀⠈⡹⣿⣟⣛⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣛⣉⣹⣿⣿⣿⣿⣿⣶⣦⣄⠙⠻⣶⣄⡀
⠀⠀⠀⠀⢰⠁⢾⡿⠂⠹⠋⠉⠙⠁⣰⣄⣀⠀⠀⣀⣤⡤⠎⠐⠿⠛⠛⠟⠛⠛⠉⠉⠻⠿⢿⣿⣿⣿⣿⣦⣈⠻⣿⣷⣶⣤⣀
⠀⠀⠀⠀⣼⣦⣀⠀⡈⠂⠀⢨⠻⣿⣦⣹⣿⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠻⢿⣿⣿⣶⡈⢿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⣼⣷⣝⢿⣿⣿⣷⣶⢰⠐⠋⠉⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣄⣀⠀⠀⣿⣷⣶⡘⢿⣿⣿⡄⠹⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠹⣦⣽⣿⡿⣿⣷⠷⠖⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⣿⡄⣈⠹⣿⣿⣿⡃⢿⣿⣿⡄⠘⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠛⠉⠙⢿⠏⠀⠀⠰⣦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣟⠀⠻⣷⣶⣾⠟⠡⣾⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢀⡠⠀⠴⠶⣶⣶⣶⣤⣤⣀⣀⣀⡀⠀⠀⠀⢘⣏⢠⣥⢶⣦⣌⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣦⠀⠀⠁⠀⠀⣤⣄⠀⠨⠟⠋⠛⠛⠛⠛⢿⣷⣦⣤⠈⠛⠘⠻⣿⣿⣿⢯⣭⡝⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⣠⣶⣦⡤⠤⢤⣝⡻⣶⡀⠀⠀⠀⠀⠀⢀⡈⣻⣿⣧⠀⠀⠀⠀⣰⣴⣿⡟⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣤⢤⣴⣿⣿⣶⡇⠲⣤⣤⠄⣀⣠⣤⣼⣍⢻⡇⠀⠀⢰⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣤⠀⠟⠃⠀⣬⣽⠟⣽⡷⢀⡔⠛⠋⠻⠿⣿⣷⣿⣃⢱⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣼⣿⠋⠀⠀⠀⢉⡉⠀⠰⠟⠁⠀⠉⠒⠠⢿⡾⢿⣟⣼⣯⠈⠀⠂⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⢀⡄⠀⣼⣿⡅⢤⣧⣶⣾⣿⡀⢼⣥⠔⠉⠉⣶⣄⠀⠀⠀⣿⣿⣝⢿⣷⡦⠀⢻⣿⠛⢿⡇⠀⢹⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢀⣀⡋⠀⢬⣴⠀⠀⣏⣽⣤⡀⠀⠈⠁⠘⠷⠾⠷⠞⠻⠟⠋⠘⠷⣶⣿⡿⠟⠙⢤⡞⠻⠷⠘⣿⡀⠀⠁⢀⣸⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠠⠚⠉⢀⣀⣤⡤⠤⠤⠒⠂⠹⠟⠙⠒⠧⣶⣶⣤⡀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⡤⠘⢶⣬⡻⣶⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣠⡤⠖⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣟⠙⠦⣄⣀⡠⢦⣤⣶⠿⠿⠯⣥⣄⣺⣿⣿⢾⡟⠶⠊⠉⢉⠉⠻⣷⣤⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠙⠋⠙⢿⣷⣦⡄⠀⠀⠀⠈⠁⠀⠀⢠⠈⣿⣿⠋⠁⠀⠁⠀⠀⠀⠀⠀⢲⣿⣿⣅
⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢏⢻⡿⠂⠀⢄⠀⠀⠀⠀⠀⢱⣼⡟⡀⠀⠀⠀⠀⠀⠀⣈⡶⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠘⠁⡆⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡼⠀⡇⠀⠀⡀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠻⠻⣿⣿⣿⣿⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⡄⠀⢿⠀⠴⢠⡎⠁⠀⠀⠀⠀⠀⠀⠁⠀⢷⠀⢰⡇⠰⣿⢸⣷⡈⢢⣄⠐⢤⣀⣀⡒⣿⣿⡿⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢈⠀⠛⡜⠋⣇⠀⢠⠀⠀⠀⡆⠀⣠⡀⠀⠀⢸⡄⠀⢡⣾⣿⣿⣿⣿⣿⣶⣿⣿⡿⡁⢿⣧⢻⣶⢷
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠘⠛⣼⠂⠀⣽⣠⠁⠀⠀⢸⠇⢰⣿⡄⣠⣇⠀⣷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⢸⣿⣷⡜⠻
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁⠀⠸⠇⢠⣍⠁⠀⠀⠀⢀⣞⢠⣆⡘⣷⣿⣾⢀⣿⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣶⡦
⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⢀⣡⠈⠊⠁⡧⠘⢠⡴⠛⠻⢸⣿⣿⣜⣿⣿⣸⠇⠀⣿⣿⣿⣻⣿⣿⣿⣏⡑⠾⣏⢰⣿⠿⠋⣰⣦
⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠀⠀⣬⠃⠜⠀⠀⡠⠊⠋⠁⢀⢠⡿⣿⣿⣿⣿⣿⡅⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣽⠟⠁⠻⣷⡏
⠀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠇⢀⠄⠀⠈⠀⠀⠀⢠⣿⠟⢰⣿⣿⣿⣿⡿⢄⣿⣿⣿⣿⣻⣿⣿⠟⠋⠀⣔⠊⠁⠂⣸⣿⡿⣣⡀
⠀⢧⡰⠇⠀⠀⠀⠀⣀⣀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣀⠄⣠⣿⠇⢠⣿⣿⣿⠓⢯⣶⣿⣿⡿⣿⣿⢺⣿⣯⣴⣯⣴⣤⣦⠀⢻⣿⢟⣼⡏
⠀⠀⠄⠀⠀⢀⠔⡝⠋⠉⠀⠀⠀⣀⣀⡀⠀⠀⣀⣤⣶⠾⢛⣥⣾⠟⣃⣴⣿⣿⡟⣧⡼⢿⣻⣿⠿⠛⡛⠩⠉⠓⠚⠛⠓⠒⠶⠧⠴⠿⠓⠚
⠀⠀⢧⡄⠀⠠⠊⢀⣴⠾⣿⣿⣿⣿⠏⠁⣤⣾⣿⠛⠁⣰⣿⣿⣿⣿⣿⣿⣿⣾⣿⣯⣿⠿⠕⠀⠈⠁⠀⠀⠒⠋⠉⠉⠙⠛⠓⠒⢤⣀
⠀⠀⠀⠁⡀⠀⠀⠉⠸⣷⣿⣿⣿⣅⠀⠀⢰⣿⡅⢂⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠄⠀⠿⠟⠃⢈⣩⣽⣿⣿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠒⠀⠀⣴⠀⠀⠀⠀⠀⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠐⣹⣏⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⣺⡃⠀⠀⣼⣿⠄⠀⠀⠀⠀⠀⠀⠓⠀⢀⣼⣄
⠀⠀⠀⠀⠀⠠⠟⠁⠀⠀⠀⠜⣹⣏⠢⡀⠀⠀⢀⣿⣧⠀⠀⢀⢼⢤⠀⠀⣼⣿⡄⠀⠀⠀⣤⠀⠀⠀⢀⣤⣿⣷⣤⡀
⠀⠀⠀⠀⠀⣠⢤⠤⠀⠀⢀⠴⣻⣟⢧⡀⠀⢀⣾⣿⣿⡆⠀⠠⢺⡣⠄⠈⣹⣿⡍⠀⠀⢰⡿⡆⠀⠀⣀⣴⣿⣿⣤⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠔⣻⣟⢦⡈⠀⣼⣿⣿⣿⣿⡄⠒⢹⡑⠆⠼⢿⣿⠛⠂⠀⣾⣩⣿⠀⢀⣠⣼⣿⣷⠤⠤
⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⢀⠼⠋⡟⢦⠉⣸⣿⣿⣿⣿⣿⣷⡐⢩⠙⠀⢠⣿⣿⣧⡀⣸⣿⡣⣿⣇⠉⢉⣽⣿⣿⡷⠒⠂
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠇⠀⠀⠉⠉⠉⢹⠉⠉⠉⠀⠸⠀⠀⠛⠉⡏⠉⠁⠛⠛⠛⠛⠛⠀⠛⠉⢹⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠄⠐⠒⠒⠒⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠓⠈⠘⠊⠉⠀⠀⠙⠦⠔⢦⠤⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⡼⠃⠀⠀⠀⠀⣀⣀⠠⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣼⠁⠀⠀⢀⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣸⠇⠀⢀⡠⠌⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢻⠿⠖⠛⠓⠉⠉⠁⠀⠰⠟⢢⢠⣄⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣧⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠠⡎⠛⠁⠀⠀⠀⢉⡆⠀⠀⠀⠀⢈⣀⡀⣀⠈⠙⣇⡀⠀⠘
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⡥⠖⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⢁⣀⣀⠈⠒⣿⠈⣧⠻⠂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⣼⠃⢹⠋⠀⡠⢞⣉⠉⠲⣄⠀⠀⠀⠀⢠⡶⢣⣤⡀⠑⣆⢀⢰⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢿⣀⠀⠀⠀⠀⠠⠁⢀⠎⠀⢸⠁⣹⣿⣷⠀⠈⣆⠀⠀⠀⣾⡓⠾⡿⠃⢀⣨⣰⡄⠳⢄
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⢽⡀⠀⢰⡿⢆⠀⠈⢧⡀⠥⢤⣙⣿⡥⢄⡠⠊⠀⠀⠀⠻⡉⠒⠒⠊⠁⠈⠉⠉⠐⠎
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⡴⢇⡏⢹⡦⣉⣠⡾⠁⠄⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠙⣆⠀⠚⠀⠢⠀⠀⠀⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠐⠒⠚⠁⢎⡤⠊⠁⠷⠀⠀⠀⢈⠉⠀⠀⢀⠠⠧⠤⢤⣀⣀⠤⠛⠒⠤⠤⠖⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠐⢄⣀⡴⠋⠀⠀⠠⡆⠀⠀⠀⠀⠐⠄⠐⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⣤⣴⠲⠿⢿⣿⣿⣾⡿⠒⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠃⠁⠀⠉⠳⢤⣀⣈⣻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⣠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⠀⠀⣠⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠓⠚⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⡀⠠⠎⠀⠑⠉⠀⠐⠈⢻⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡌
⢱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠈⢢⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠈⠲⠃⣄⢀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⠀⠀⠀⠀⠈⠺⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢧⡈
⠀⠀⠀⠀⠀⠘⠣⠐⠢⠄⠣⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡟⠦⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣴⡄⠀⣯⠗⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⣘⡆⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠲⢄⣀⠐⠂⢤⣤⡤⠄⠚⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠱⠎⡁⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣷⣶⠂⠀⠀⠀⠀⠀⠄⠀⡈⠴⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠁⠙⠻⢦⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠰⠀⡀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠶⠀⠀⠰⠄⢀⠈⠀⠀⠂⠀⠈⢀⠀⠰⠀⠀⠰⠆⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⢰⣷⡀⠀⠘⠀⠀⠀⠃⠀⠁⠘⠀⠀⢰⣷⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠈⠁⠄⠀⠀⢾⣿⡇⠀⠀⠐⠄⢀⢁⠀⠔⠀⠀⠀⣿⣿⡆⠀⠈⠀⠉⠀⠀⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠆⠄⠰⠐⠀⠆⠀⠆⠀⠀⠰⠰⠀⠀⠆⠂⠀⠆⠀⠂⠀⠀⠆⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⡀⠀⠀⡀⢄⠠⢤⠉⠆⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠎⢁⠴⢀⠄⣀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠼⢀⡵⢘⡆⣹⠀⡅⠐⠁⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠘⠀⠇⢰⡁⢞⠱⣅⠘⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⠈⠀⠁⠀⠀⢀⠆⠀⠀⢀⠀⠀⡆⢴⣶⠀⡆⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠈⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⠀⠀⢀⠁⠁⢀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠈⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠋⠀⠀⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⢄⠀⠉⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠰⠘⠀⠀⠀⠀⠘⠃⢰⣶⠀⠛⠀⠐⠀⠀⠈⠐⠠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠆⠄⠄⠂⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⣋⣴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣾⣿⣿⣿⡿⠋⣁⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢟⣩⣶⣿⠟⣉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣫⣾⣿⠞⣫⠿⣫⣴⡿⢛⣡⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⣉⣴⣿⣿⡿⢟⣭⣶⡿⠋⣀⣴⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⢉⣴⣾⣿⣿⣿⡿⢞⡿⠟⣩⣴⣿⠿⠋⣀⣴⣶⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠾⢛⣽⣾⠿⢛⣥⢞⣫⣶⣿⠿⢋⣡⣶⡿⠿⠿⠿⡿⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣤⣾⣿⡿⣿⣿⣿⣵⣾⠿⣋⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⣋⣵⣾⠟⣫⣿⣿⣵⡾⠛⠉⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⢉⣴⡾⠟⣩⣴⣿⣿⠿⠛⠁⣀⡴⠞⢋⣡⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⢁⣤⡾⠛⣉⣴⣾⣿⠿⠋⢁⣠⡶⠟⣉⣤⣾⡿⠟⣁⣤⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⢁⣴⣾⣿⠿⠋⣁⣴⠾⢛⣡⡴⢛⣽⢞⣫⣶⣿⣿⣿⣿⢛⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⣤⣾⣿⠿⢋⣡⣶⠟⣋⣥⠾⣋⣵⣾⣿⣿⣿⣿⣿⣿⣿⡟⠋⡛⠛⠃⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣋⣥⡾⠟⣫⡴⠞⣉⣴⣾⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣵⡾⢛⣭⠶⠛⣡⣴⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⣟⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣛⣵⠞⢋⣠⣶⣿⣿⣿⣿⣿⢟⣫⣵⣾⣿⣿⣿⣿⣿⢟⣫⣵⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⢉⣠⣶⣿⣿⣿⣿⣿⢟⣭⣶⣿⣿⣿⣿⣿⡿⢛⣩⣶⣿⣿⣿⡿⣿⣿⣿⣿⠿⣿⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⠟⣋⣵⣾⣿⣿⣿⠿⣛⣭⣾⣿⠿⢛⣥⢞⣭⠶⣿⣅
⠀⠀⠀⠀⠀⠀⠸⠟⠉⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣾⣿⡿⢛⣩⣴⣿⣿⣿⣿⠟⣫⣵⣾⣿⠟⢋⣡⣾⣿⣿⣿⣿⣿⡿⠛⠿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⣯⣷⣿⣿⠿⣋⣥⣶⣿⣿⣿⡿⢟⣫⣶⣿⡿⠟⢋⣤⣾⣿⣿⣿⣿⣿⣿⠿⠋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣫⣴⣾⣿⣿⣿⡿⣛⣽⣾⣿⡿⠛⣉⣴⣾⣿⣿⣿⣿⣿⣿⡿⣛⣥⣾⣿⣿⣿⣯⡀
⠀⠀⠀⢀⣠⡶⣛⣿⣿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⣿⣿⣿⡿⣻⣵⣾⣿⠿⢋⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⢋⣤⠀⣠⡄
⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⡿⠟⢋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣾⣯⣵⡾⣋⣤⡶
⣸⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⠿⠛⣁⣤⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣵⣾⣿⣿⠟⣫⣿⣿⣿⣿⣻⡷⠻⣆⣠
⠉⠛⠉⠛⠁⠉⠁⠀⠀⠘⠋⢩⣽⣿⣿⣿⡿⠟⢋⣡⣴⣿⣿⣿⣿⣿⣿⠿⢛⣩⣾⣿⣿⣿⣿⣿⣿⣿⠻⣿⣛⣠⣾⠟⣉⣴⣾⣿⣿⣿⢛⡿⠛⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟⠉⠀⠘⢋⡽⢿⣿⡿⠿⠛⢋⣡⣶⣿⣿⣿⠟⠉⠉⠋⠙⠛⠉⠃⠺⠏⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⣿⣿⠿⣿⠟⠁⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣡⣞⣥⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢋⡵⢋⣤⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠞⠋⣼⣷⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣻⣿⣿⡿⣿⣿⢻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡟⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠙⠀⠀⢹⡄⠀⣇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠠⡄⣀⠀⠀⢀⣀⠀⠀⢳⠀⢹⠀⠘⠀⠀⠀⠀⠀⣰⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣸⣿⠀⣀⣤⣤⣜⠀⣠⣤⣥⣤⡀⢠⣥⣤⣤⡀⢠⣄⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⢻⣿⠐⣹⣥⠾⣿⠀⣿⡏⠀⢻⡇⢸⣿⠁⢸⣿⠈⣿⣄⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⡘⠿⠀⠀⠸⠿⠀⠻⠷⠾⠿⠆⣿⡿⠶⠿⠃⢸⣿⠶⠾⠏⠈⠘⣿⡏⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠘⠟⠠⠤⠒⠰⠿⠛⠀⠀⠤⠴⠆⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣨⣉⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣱⣿⠏⠀⣼⣧⢸⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⢱⣿⠏⢀⣐⠸⠷⠸⠟⠿⠀⠀⠀⠀⠂⠀⠀⠀⢠⣐⠢⠤⣄⣈⠉⠉⠛⠒
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢠⣿⡟⣀⢸⣿⠀⠀⢀⠀⣶⡆⢰⡆⠀⠀⣶⡄⣴⡆⠈⠱⣶⡀⢉⣶⠆
⠀⠀⠀⠀⠰⠿⠀⠀⠀⢠⣿⣿⣬⣥⣼⣿⣤⠰⠃⠀⣿⡇⢸⡇⢰⠀⣿⡇⣿⡇⡀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡭⠉⢻⣿⠋⠸⣷⣤⣿⠃⠸⣿⣤⣴⡿⠃⣿⣧⣭⣤⡄⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⡖⠈⠉⠁⠀⠀⢨⠉⢭⠀⢢⠈⠉⠉⠉⠀⠈⠁⠙⠂⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⣿⠀⢸⠄⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠰⢀⠂⠐⠂⣆⢂⡶⠖⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠈⠉⠈⠛⠡⠌⠀⠀⣄⠀⠀⠀⠍⡲⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠁⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠴⠋⢀⡀⠀⠙⢤⠀⠠⠏⠀⠀⠁⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢀⡔⢄⣀⣤⠵⠶⠚⠛⠛⠛⠻⠷⢶⣦⣥⣐⠀⡀⠀⠀⠀⣀⠀⠬⠦⡀
⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⢀⠦⠠⢞⣭⡶⠛⠉⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠈⠙⢿⣶⣅⡠⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠠⠀⠒⢂⠒⠈⢁⢒⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣮⡢⣾⠆⠐⠤⢀⡵⡀
⠀⠀⠀⠀⠁⠀⠀⠀⢒⡄⣰⣿⣿⡟⣥⣶⡿⠁⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣄⢾⠆⠀⠘⠃⠑⡀
⠀⠀⢀⠂⠀⠀⠀⠀⠌⣼⣿⣿⣿⣷⣿⡅⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠸⡆⠀⢿⣿⣿⣿⣆⠀⣈⡀⠀⠀⠰
⠀⠀⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⣻⣿⣿⣷⣴⣦⢀⣬⡇⠀⢀⡠⠖⠚⠉⠉⠹⠛⠛⢶⣿⣄⣸⣿⣿⣿⣿⣇⢛⡣⠈⠐⠀⠃
⠀⠀⠀⠀⢀⠂⠀⢸⣿⣿⣿⣷⣚⢿⣿⣿⣿⣿⣿⣿⣷⣾⠿⠒⠀⢀⠀⢰⡔⠒⠒⠀⢿⣙⢿⣿⣿⣿⣿⣿⡈⣁⠀⣄⡀⠘
⠀⢸⠀⢀⠀⠀⡗⣼⣿⣿⣿⣿⣿⣈⡿⣿⣿⣿⣿⣯⠈⢹⣤⠶⠛⢻⣧⡀⠑⢐⡀⠀⡖⡂⠉⠙⠹⣿⣿⣿⡇⠀⠀⠄⠙⠶⡆
⠀⠈⠀⠈⠀⢀⠃⣿⣿⣿⣿⣿⡿⡏⢉⠛⣿⣿⡿⣿⣶⣤⣄⠀⠀⢀⡝⢁⣀⡸⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠂⠀⠘⠀⠀⡅
⠀⢐⢀⠀⠀⠎⠀⣿⣿⣿⣧⠈⠄⠁⢀⣴⣿⣿⡇⠀⠙⠛⠁⢀⡒⠋⠙⠁⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠈⣿⡇⠀⠀⠠⡄⠀⡇
⠀⠀⠘⡄⠀⠀⠀⢸⣿⣿⣧⠏⠀⢠⣿⣿⣿⣿⡷⠀⠀⠀⠀⠈⠀⠀⣀⣀⣠⢤⠔⠖⠛⠛⠋⠀⠀⠀⠀⢹⢃⡥⠀⠀⠛⣦
⠀⠀⠀⠰⣄⠀⢀⣠⣿⡟⠀⠀⠀⠀⠀⣀⣹⣿⣿⣆⠀⠀⢀⢰⢾⣾⡟⠛⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⢧⡌
⠀⠀⠘⠀⢈⣴⣿⣿⡿⠃⠀⠀⠉⠀⠀⠀⡈⣿⡠⠙⣿⠀⢄⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣆⡀⠺⠁
⠀⠀⢠⣿⡿⢛⠄⠀⡁⠀⠀⠀⠀⠀⠀⡀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀
⠀⣠⣿⡟⢱⠃⡄⠐⠁⠀⠀⠀⢠⠀⣠⣤⣾⣿⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇
⢠⣿⡉⢀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣿⣿⣿⣿⣟⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⢸⡋⡇⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦
⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣟⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠋⠙⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠻⠟⠿⢿⣿⣷⡀⠰⡧⣠⠄⠀⠀⠠⠀⣶⣤⣾⣿⡿⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠆⠐⣿⠁⢶⣂⣴⣾⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⣿⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⢾⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣄⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣭⣷⣿⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠁⠀⠀⠀⠀⠀⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠛⣰⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠿⠛⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠀⢾⣿⡗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣧⣤⣀⠀⠀⠀⠀⣻⣿⣷⣤⣁⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣜⣿⡟⠁⠀⠀⠀⠀⣿⣿⡏⠉⢉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠙⠇⠀⠀⠀⠀⠀⣻⣿⡿⠗⣂⣬⣿⣿⠟⠻⢿⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡷⣤⣤⣤⣿⣿⣌⣷⣌⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⣬⣭⣭⣿⣿⠩⣤⣤⢙⣿⣿⣿⣿
⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⣿⣿⡟⣿⢋⡙⣿⣿⣶⣤⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠸⡿⢿⡧⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣆⠀⠀⠀⠀⠀⠀⠀⠠⣾⠟⠉⠙⣿⡆⠀⠀⠀⠀⣺⣿⣷⠤⠾⠧⣿⣿⢻⡟⣿⢻⣿⣿⣿⣿
⠀⠀⠀⢐⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣲⣇⣗⣿⡇⠀⠀⠀⠀⢼⣿⡷⠒⠶⠶⣿⣿⣈⣁⣙⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠂⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⡇⠛⠀⠚⣿⣿⡉⠩⢭⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣛⢛⣛⣿⣿⠟⠛⣠⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠟⠘⠛⣿⣿⠶⠶⠶⢾⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⢂⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⡛⣿⣿⠹⡿⠿⢾⣿⣿⣿⣿
⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡙⠛⣁⣿⣿⣀⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠀⣀⣼⣿⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⣷⠄⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣶⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢟⡛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⢀⠉⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠓⠉⠊⠑⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⡀⠰⡶⠀⣀⠀⠹⣿⣿⣿
⠀⠀⠀⣦⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣆⣤⣰⣶⡇⠀⠀⠀⠀⠀⠀⠴⢤⢘⠠⠰⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠘⠓⣶⢲⠚⠓⠀⠀⣿⣿⣿
⠀⠀⠀⠹⡿⠿⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣟⠃⠀⠀⠀⠀⠀⠀⢠⠴⢘⠱⢢⠄⠀⠀⠀⠀⢽⣿⣿⡆⠀⠘⠏⢨⣅⠙⠏⠀⢀⣿⣿⣿
⠀⠀⠁⠐⠃⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀⠀⠀⠁⠀⢀⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠈⠓⣆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⡐⠈⠙⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠁⠀⠐⠂⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⠁⠀⠀⢤⡄⠠⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣐⡥⠜⠃⠠⠘⠀⠀⢨⣷⠃⠀⠢
⠀⠀⠀⠀⠀⢀⢠⠠⠋⠀⠀⠀⠀⣀⠉⢂⡦⠰⠎⢣⣄⡈⠁⠠⣄⡀⡀⢠⡄⠶⠈⠉⠂⠀⠁⠀⠀⢫⠘⡃⠠⡔⣈⡼⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠁⠀⣠⡞⠋⠉⠑⡇⠀⠈⠃⠀⠀⠉⠈⠋⠁⠀⠀⠄⢀⠀⠀⠀⠀⠀⠈⠂⢡⠐⠺⢁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⡴⡏⠁⡤⣤⣀⣣⠀⠀⠀⠈⢀⠀⠀⠁⡀⠀⠀⡀⠓⠄⠃⠈⠀⠀⠀⠜⠀⠘⢄⡀⢒⠓⠈
⠀⢀⡀⣀⠒⡌⠙⠀⠠⢂⡁⣠⠄⠶⠀⠈⠙⠎⢆⠀⠀⠀⠀⠀⠈⠄⢀⠴⠘⡁⣀⠂⠀⠃⠉⠐⠀⠈⠀⢀⠤⠞⠁
⠀⠊⡉⠁⠈⠙⠀⠀⢔⠉⠋⠉⠀⠰⠟⠀⠾⢿⠐⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠁⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠠⠀⣒⠀⠠⢀⠋⠐⠀⠀⠨⠒⠀⢀⠀⢀⠀⠀⠀⠀⠀⠍⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⢄⣶⣁⡠⣀⣥⣦⡀⠄⢦⡀
⠀⠀⠌⠈⠉⠐⠀⠀⠀⠀⠀⠀⠐⠾⢤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠘⠛⠃⠸⣿⣟⣛⣫⡥⡗
⠀⠁⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣠⠀⠀⠀⠓⠒⠲⠤⡄⠠⠀⡀⠀⠀⢀⠤⣈⠀⢀⣀⠤⠞⠑⠀⠀⠀⠀⠀⠀⠛⠅⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠦⠐⠉⠽⠅⠠⠀⠀⠀⠀⠀⠀⠹⢶⣶⣞⣤⠀⢅⠀⠙⠶⠂⠀⠢⠴⡂⠈⠂⠀⠀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠉⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢧⡃⠄⠀⠀⠀⠊⠱⠉⠁⠘⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡀⠐⠀⡈⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣌⢁⠈⠁⢰⡏⠈⣥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠀⠀⠀⠀⠐⠂⠀⠐⠀⠀⠄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠀⢀⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⣀⠄⠀⢠⠀⠀⡠⠀⠈⠀⠀⠀⠀⠁⠀⠁⠀⠀⠠⠀⠂⢀
⠀⠀⠀⠀⠀⠄⣈⣁⠠⠀⠐⠀⠫⠛⠀⠒⠂⠀⠀⠴⢀⠀⣁⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⢘⡃
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠂⠠⠀⠀⢁⠀⠀⡾⠀⠀⠈⠀⠀⠈⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⢲⡄⠀⠀⢤⠄⠀⠐⠀⠁⠀⠀⠀⠄⠀⠂⠈⠀⠘⠀⠀⠁⠀⠀⠂⠀⢀⠀⠂
⠀⠀⠀⡀⠃⠀⠠⠀⠀⡀⠀⠠⠀⠀⠈⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠐
⠀⠀⢠⣻⠀⠀⠀⠀⠀⠁⠀⠑⡂⠀⠁⠀⠘⠀⠀⠀⡆⠀⠉⠀⠀⠀⠀⠀⡀⠢⡀⠴
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⡇⠀⠰⠀⠀⠀⠁⠵⠶⠐⢠⠘⠀⠀⠀⡀⢠⠀⠀⠐⠁⢀⡀
⠀⠀⠀⠀⠀⠁⠀⢃⠉⠁⠀⠈⠀⠌⢀⡀⠁⠀⠀⠐⠁⠀⠀⠀⠊⠀⠀⠰⢤⣀⠈⢁⡤⠞⠁
⠀⠀⠀⠀⠰⠆⠀⠂⠀⠐⠀⠘⠀⠀⠀⠀⠀⢀⠀⠀⢀⠄⠐⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠧⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠚⢃⠀⠈⠓⠀⠠⡤⠒⠔⠀⠛⠒⠀⠋
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⣀⢀⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⢀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠒⠛⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠡⠀⠀⠐⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⢖⠒⠒⠂⢒⠐⢒⡶⠆⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠀⢀⠀⠀⠈⠉⠀⠒⠻⣑⠭⠀⢀⣀⠀⡀⠀⠉⡒⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢁⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⠂⠀⠀⣰⠆⠀⠀⠉⠲⣄⠀⠀⢸⡗⠁⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⠆⢀⠁⠀⣛⣛⣬⣬⣭⣭⣭⣥⣼⣒⡀⠤⢈⡃⠀⠜⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠀⠀⢠⡴⢓⣨⣴⠾⠋⠉⠉⠀⠀⠀⠀⠀⠉⡉⠛⠛⠻⣶⣮⣔⡠⡀⠀⠀⠰⠀⠘⠉⠙⢦
⠀⠀⠀⠀⠀⠀⡠⡁⠀⠀⠀⣠⠞⢃⢜⣥⣾⡟⠉⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣦⣕⠄⡀⢀⡀⠄
⠀⠀⠀⠀⢀⠌⠀⠉⠩⡍⠀⢀⢜⣵⣿⣿⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣔⢾⠟⠀⠱⢆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠘⡤⣢⣾⣿⣿⡿⣤⣾⡿⠃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠹⣿⣿⣿⣧⡹⠿⡀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⡰⣱⣿⣿⣿⣿⣵⣿⡍⠀⠀⠀⠻⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢳⡀⠀⢿⣿⣿⣿⣷⡐⡀⢡
⠀⠀⢐⠀⠀⠀⠀⠀⠠⢱⣿⣿⣿⡿⣿⣿⣿⣷⣦⣶⡀⢀⣹⡆⠀⠀⢀⣤⠶⠚⠛⠉⢹⠿⠷⢶⣮⣇⡀⢸⣿⣿⣿⣿⣿⡘⠻
⠀⠀⡎⠀⠀⠠⡆⠀⢣⣿⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣷⣿⣿⣧⣴⣾⡥⠄⠀⠀⠀⢠⣠⡤⠤⠈⢻⡟⢿⣿⣿⣿⣿⣿⣿⣧⢹
⠀⢠⠀⠀⠀⠉⠀⣤⢸⣿⣿⣿⣿⣷⣗⢹⣿⣿⣿⣿⣿⣿⡟⠛⢿⠃⢀⡤⢤⣷⡀⠙⠇⠀⠀⠀⢈⣉⡉⠻⠿⢿⣿⣿⣿⣿⡈
⠀⠸⠃⠀⣦⠀⢀⠇⣿⣿⣿⣿⣿⣿⣿⡿⣧⣿⣿⣿⣿⣿⣷⡀⠘⠛⠉⠀⠀⠙⣿⡄⠀⠀⡆⠀⠊⠐⠀⠄⠀⠈⠙⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡌⠁⣿⣿⣿⣿⠿⠿⠫⠟⠐⠈⢺⣿⣿⡏⠻⢿⣿⡇⠢⠄⣀⡴⣁⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠸⠀⡀⠀⠘⠁⢠⢻⣿⣿⣿⣢⡐⠀⠀⣠⣶⣿⣿⣿⠃⠀⠀⠈⠀⠀⡬⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⠀⠀⠀⠀⢘⣿⠃
⠀⠀⡆⢇⠀⠀⠀⠀⠸⣿⣿⡟⡼⠁⠀⠰⠿⡿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢤⣤⠖⠛⠛⠉⠙⠁⠀⠀⠀⠀⠀⣿⣰
⠀⠀⢐⠀⢷⡀⠀⢀⣠⣿⣿⠋⠄⠂⠀⠀⠀⣀⢸⣿⣿⣷⡄⠀⠀⢀⢰⣿⣿⣿⡋⠻⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠈⠆⠀⢙⣴⣿⣿⣿⠟⠀⠀⠀⠉⠀⠀⠀⡀⢻⣇⢍⠟⣷⠀⡈⡀⠐⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣴⣾⡿⢟⠁⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⣴⣿⡟⣡⠏⠐⠀⠳⠀⠀⠀⠀⢀⠀⢀⣋⣠⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⣾⡟⠟⠀⡉⠀⠁⠀⠀⠀⠀⠀⠰⠀⣠⣾⣿⣿⣿⣧⣴⣿⠃⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣦⠀⡆⠙⠀⠀⠀⠀⢠⡴⠂⠀⠀⠙⣿⣿⣽⣿⣿⣿⣶⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⢺⠀⠀⠀⠀⠀⠀⠀⠌⠐⠊⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣿⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿

⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢒⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⣼⣟⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢒
⠀⠀⠀⠀⢐⣻⠵⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣛⣭⡿⠿⣟⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠗⠂
⠀⠀⠀⣠⣿⣓⣫⣭⠥⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢛⣋⣛⣉⣍⣁⡀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣊⣭⢄⣀
⠀⠀⠀⠴⠛⠉⠁⠀⠀⠀⠀⠉⠙⠒⠒⠂⠀⠀⠀⠀⠈⢋⡉⠉⠭⠤⠖⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠤⠤⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢤⠙⠂
⠀⠀⠀⠀⠀⠀⣨⣠⣶⣾⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⠶⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄
⠀⠀⠀⠀⠀⣘⣿⣿⣟⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⣠⠴⠂
⠀⠀⣀⣶⠶⠟⠋⠋⠉⠉⠉⠉⠁⠐⠒⠦⠤⠀⠀⠀⢠⠴⠞⠩⢭⣉⠈⠉⠓⠄⠀⠀⠀⠀⠀⠀⠀⢀⡁⠒⡈⠀⡀⢠⣀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⣱⠎⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢤⣿⣷⡶⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠞⠛⢁⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣶⣾⢹⡅⣠⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⡿⣧⣾⣷⣤⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⡋⣀⡿⠿⠛⢠⣼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢉⣁⣅⣠⠄⢾⠿⠾⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠏⠈⢻⣄⣘⣷⠈⠑⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠠⣤⠟⠁⡈⣠⡴⠋⠉⡾⠆⡷⠂⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢂⣾⣿⠋⡪⣴⣀⢱⡤⠶⢂⣤⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣄⠴⠲⠾⠿⢃⢠⣄⣼⠛⠋⡀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⢠⡶⢠⣀⣈⣉⠀⠴⠶⡅⠀⣀⡤⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠘⡻⠊⣸⡿⠃⠀⠘⠋⠀⠀⡈⠁⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀⣤⡂⠀⠈⠀⢸⡿⢁⠠⣀⠖⠀⠀⠀⠈⠓⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠐⣿⣾⠋⠀⠉⠀⢠⠀⠀⠈⠰⠄⣠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣄⠀⢀⠀⠄⢻⣿⣶⡀⠀⡰⢿⠀⢀⢀⠄⠂⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠀⠈⠁⠀⠀⡰⠃⠀⠀⢀⣧⡀⠀⠀⠦⠈⡀⠰⠊⠀⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠉⠒⠥⠰⠚⠃⠐⠒⠀⠚⠛⠟⢉⣠⠤⠊⡑⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠂⠐⠂⠀⠂⠠⣤⣦⠔⠀⠀⠹⠷⡰⠚⠓⠋⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⣤⣄⣀⠆⠚⢸⣿⠏⠀⠀⠀⣶⡆⠀⢈⡀⠀⠀⠒⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄⣨⡼⠏⠀⠀⡸⢡⠀⣀⣠⠁⠀⠀⠀⠀⠘⢆⠀⢀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠁⣿⣿⣿⡇⠀⢠⡈⠀⢀⠀⠈⠡⠀⠈⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⠝⣿⠁⠐⠈⠁⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⡕⠘⣖⣐⠐⠒⠒⡲⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠆⠀⠉⡢⢒⣴⡀⣠⠐⢀
⠀⠀⠀⠀⠦⢶⢂⡈⢥⠉⠳⡖⠁⠀⠀⠰⠆⠠⠀⠐⠒⠂⠀⠐⠐⠒⠂⠀⠀⠀⡘⢂⡑⣂⡰⣏⡌⡴⠋⠓⠘⢒
⠀⠀⠀⠀⣅⡧⠛⠛⠊⠙⠛⠋⠉⠐⠈⠙⠉⠉⠙⠉⢩⣓⠘⠚⠈⠊⠉⠉⠁⠈⠓⠈⠉⠉⠁⠉⠉⠉⠉⢉⡛⠁
⠀⠀⠀⠀⡀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡙⡁
⠀⠀⠀⠀⠉⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⡴⡸⢧⢤⠀⠀⢀⣈⡑
⠀⠀⠀⠀⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣇⢇⣈⣏⠀⠀⠰⢄⣀
⠀⠀⠀⠀⣁⣈⡀⠀⠀⠀⠀⠀⢠⣴⠳⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠁⠀⠀⠣⠢⠄
⠀⠀⠀⠀⣍⣈⡁⠀⠀⠀⠀⠀⠐⠛⠚⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⠒
⠀⠀⠀⠀⡀⣄⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠐
⠀⠀⠀⠀⣍⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠄
⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄
⠀⠀⠀⠀⣲⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢒⣀
⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠅⠀⠅⠨⠀⠅⠀⠄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡂⢀⡔⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡰⠆⠴⠀⢶⢰⢆⠴⢤⠶⢰⠆⠰⠄⠖⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠈⠉⠁⠉⠈⠉⠉⠈⠁⠁⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣆⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠑⠒⠀⠠⠤⠤⠀⠤⠤⠤⠒⠚⠂⠉⠁⠀⠀⢀⠠⠀⡄⠢⠀⢰⡄⢰⡀⠀⠐⠀⠀⡄⡀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢨⠀⠀⡗⠎⠀⠈⠇⢸⠠⡄⠀⠀⠀⡆⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⡰⠀⠀⠆⠰⢸⡀⠀⠀⠀⡇⠀⠀⠚⡆⠀⠀⠀⠀⠀⠇⠘⠀⠃⠐⠘⠀⠃⠐⠄⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠋⠀⠄⢰⡆⢨⠀⠀⠀⠀⡃⠀⠀⠀⠐⠀⠀⢀⣀⠠⠄⠒⠒⠒⢀⠊⠀⠐⠒⠢⠤⣀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⢀⡀⠀⠀⠈⠃⠀⠀⢀⣉⠀⠤⠐⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠉⠢⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠒⠶⠖⠶⠲⠶⠲⠶⠒⠖⠲⠒⠲⠒⠶⠒⠶⠒⠶⠒⠶⠲⠶⠲⠖⠒⠖⠲⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠂⠀⠀⠀⠂⠐⠀⠐⠀⠐⠀⠂⠀⠂⠀⠂⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⡀⠈⠀⠘⠀⠈⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠄⠀⠀⠀⠀⠈⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠂⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠸⠀⠀⠐⠒⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⣿⠿⠿⠿⡿⣿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣾⡛⣷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠉⠉⠛⠉⣿⢿⣍⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠳⣧⠀⠀⠀⠀⠧⠤⠟⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⡞⠀⠀⣷⣄⡀⠀⠀⠀⠀⠀⠀⢳⡘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢮⣀⣽⢷⢂⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠉⠛⣲⣄⣀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣹⡞⠲⢽⣦⣞⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣾⠈⣽⠀⢈⠑⣄⠀⢀⠠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠅⠲⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠒⠸⠏⢁⡠⢎⣀⡀⠀⠤⠤⢄⠀⠀⠀⠀⢤⣠⠄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣴⡉⠟⠢⣀⡋⠀⠀⠀⠀⠄⠀⠀⠀⠂⠠⠤⠀⠄⠁⠘⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⣀⠀⠀⣀⡀⣀⢀⣀⣤⢠⣤⡆⠸⠋⠑⣏⢉⡧⠀⠀⣉⣀⣀⣀⡀⠀⠀⠀⠉⠀⠀⠐⠒⠒⠂⠴⣧⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠃⠋⡹⢠⡿⢿⡐⠉⠉⠈⠁⠀⣿⠀⣻⠀⠀⠀⠉⠈⠁⢀⠜⠁⠀⠀⠀⠀⠁⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀⠙⠚⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⢴⣆⣴⢛⣁⣸⡦⡴⠇⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠄⠀⡤⠖⢡⠖⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⣡⣿⢇⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠯⠄⠀⡀⠀⠶⠀⠀⠀⠀⠀⠀⠀⡃⠀⢠⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠉⠙⢳⣷⣶⡱⣤⡔⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠐⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠠⠛⠃⠀⠀⠀⠀⠽⣤⡼⠀⠀⢠⠟⢻⠊⠛⢫⡷⢶⡰⠦⢆⡺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠐⠛⠒⡶⣵⠇⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠚⠀⠀⠸⣍⣩⠃⠀⠀⢰⠈⠐⠤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢄⡀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠤⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠗⣲⣤⡔⣠⢤⡄⣤⣤⠀⣤⣤⣂⡤⢔⡲⠴⢮⣴⠿⡥⢋⠝⠓⠠⠤⠄⠉⠉⣉⡉⣉⠤⠐⠒⠖⠀⠈⠁⠢⡀
⠀⠀⠀⠀⠀⠀⢠⡴⠃⢴⡾⣏⢈⣽⣶⡆⠀⠀⠀⣜⠋⢳⠀⠀⠀⠀⠀⠀⠷⢦⡞⠁⠀⠀⣶⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⢢⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣏⣹⣼⠤⣄⣀⣀⡈⠛⠉⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠡⡀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠀⠙⠀⢸⡏⠹⡖⠋⠛⠔⠒⠮⣱⣶⣆⠰⢴⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠐⡀⠀⠀⠈⡄
⠀⠀⠀⠀⢀⡀⠔⣁⡴⠆⠀⠀⠀⣀⢀⠘⣳⠞⡃⠀⣀⠀⢰⠆⣿⠀⣯⠀⡀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢠⠀⠀⠀⡇
⠀⠀⠀⠈⠉⠉⠡⠶⠑⢒⣀⠤⠪⠗⠹⠔⢫⠜⠓⠊⠀⠲⠁⠈⠉⠁⠙⠒⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡆⠀⠡⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒⠂⠁⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⠄⠀⠀⢀⠀⠠⠤⠐⠊⠁

⣶⣿⣾⣷⣶⣿⣾⣿⣶⣿⣶⣶⣿⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣿⣷⣶⣿⣿⣿⣷⣿⣶⣿⣶⣷⣷⣿⣿⣷⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣻⡟⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣻⠉⠉⠉⠉⠀⠉⠀⠀⠀⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⡄⢀⠀⠀⠀⠘⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⣯⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡤⠴⠖⠹⠒⠁⠙⠉⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢰⠃⠀⠀⣀⠀⠀⠀⠀⠀⡜⠷⠃⠀⠀⠀⠀⠀⠀⠐⠚⣣⠄⠀⠴⢋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣀⠀⠋⢠⠂⠀⠀⡶⠚⠀⡠⣀⡛⡐⠀⠀⠀⠀⠐⣉⢉⠉⢃⡐⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠨⡇⣀⠀⠘⠁⢠⡴⠃⢤⣜⠯⠥⠀⠀⠑⠐⠉⠲⢀⠀⠀⠀⡘⢁⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠻⢿⠀⠀⣄⣸⠀⠀⠀⠲⢀⡀⠀⠀⠀⠀⠀⠀⢀⢹⡀⠉⠀⢼⢌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⡴⣆⣿⠃⠀⠂⠀⣀⡀⠉⢉⡔⠀⠐⠦⠴⠋⠁⠈⠙⠢⠼⠀⡇⠻⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡔⠉⢸⠀⠰⡇⡎⠸⣀⡬⠋⠀⠀⢀⣠⠎⣑⠦⠖⠤⠤⠴⠊⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠆⠀⢰⠁⠰⣇⠀⠀⠀⠙⠦⢤⣄⣤⠖⠉⠁⠀⠀⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠾⣧⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢧⠀⠀⠀⠀⠀⠀⢠⠞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠴⣷⣀⣈⠧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⢧⣀⢀⡄⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡈⠁⠀⠀⠀⠘⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⣷⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠳⠦⠷⢤⣤⣄⣀⡀⣄⠈⡀⠀⠀⠘⢦⣘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠙⣄⠀⢀⡞⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢸⣆⡤⣤⠀⢀⣤⡀⠀⢀⡀⠀⠀⠀⢀⡀⡠⣄⠀⣴⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠛⠀⡟⢀⣏⢀⡗⠀⢸⢇⠞⢹⡦⢾⠏⢀⣼⠻⡇⠀⢸⡇⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠓⠊⠈⠀⠧⠄⢹⡏⣴⡽⠃⣿⠀⠧⠏⠀⣉⡀⢸⠧⠒⠉
⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡽⠀⠀⠀⠀⢀⡴⡄⠀⢸⠇⠀⠀⠀⠛⠀⠀⡴⠋⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠉⠀⣀⣀⠀⢀⣞⠜⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠳⠄⠤⠞⠁
⠀⠀⠀⠀⠀⡔⢹⡇⣠⣄⠘⣧⣌⣷⣼⠁⠀⠶⠀⣠⢤⡾⠀⢀⠴⢄⠀⢠⡄⠀⠀⢰⡋⠷⠀⣸⠂
⠀⠀⠀⠀⠀⠛⠀⠷⢀⡟⠀⣏⢀⠏⣿⠀⣰⠃⢰⣇⣰⡇⠀⣏⣠⡏⠀⣿⠀⣠⠏⡈⢷⡐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠄⠉⠁⠀⠈⠉⠘⠂⠊⠉⠘⢧⡤⠁⠀⠓⠀⠈⣹⠏⢾⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⢤⣤⣀⡀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠙⠛⢶⣤⣯⣒⣉⣉⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⠤⠤⠤⠖⠒⠈⢠⠋⣤⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⡷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣷⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⢏⡟⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⠸⡗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⠺⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣴⣧⠗⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⡁⣔
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡙⠺⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡤⣀⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢺⡿⢿⡿⠞⠿⢦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⢝⠫⠖⣀⢚⣛⡀
⠀⠀⠀⠀⠀⠀⡈⠦⠤⠌⣀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⡇⣉⣤⣾⣶⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⡓⣻⡇⢌⠉⠻⣭⡄
⠀⠀⠀⠀⠀⣀⠛⠲⠀⠤⠁⠐⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣷⡟⣿⣟⡻⣷⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⢉⣵⡖⢹⣄⠻⠆⠖
⠀⠀⠀⠀⢠⠈⠁⠈⠘⠃⠁⠈⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣽⣿⣷⣿⣿⣿⣿⠃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣋⣋⢉⣁⣠⠠⠀⢡⠚⠋
⠀⠀⠀⢀⡀⠙⠓⠶⠒⡄⠙⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⣻⣿⣿⣿⣿⣿⡿⣆⢀⡀⠀⠀⠀⠀⠀⠀⠈⠋⢶⢁⣉⠁⠐⠒⠳⠿⠓⢚⡁
⠀⠀⠀⢨⠇⠠⠄⠰⠶⠰⠆⠀⠀⠞⠀⡀⠀⠀⠀⠀⠀⠀⢤⣤⡤⣭⣿⠋⠵⣿⢩⣯⢹⣟⡟⢋⢠⠀⠀⠀⠀⠀⠀⣫⠖⣩⢿⣷⠞⣽⣄⠑⠎⠞⣁⢀⣀⡀
⠀⠀⠠⠀⠀⢠⢀⠀⠀⠀⠀⠐⠴⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢛⣷⣿⣶⣿⣸⣿⣶⣿⣻⡿⠉⠀⠀⠀⠀⠀⠰⠭⢙⠃⣘⡋⣀⡅⠉⢴⠒⣈⡄⠉
⠀⠀⢤⡀⠀⠀⠈⠁⠒⠀⠐⡀⠀⠄⠄⠂⡠⠀⠀⠀⠀⠘⢦⣢⡼⢟⣭⠟⢩⣿⡎⣿⣷⣿⣿⣤⣆⣤⡖⠀⠀⣈⣰⣠⡆⣀⡘⠉⠹⠗⠂⠀⠜⠣⣿
⠀⠰⢌⠀⠃⠒⠂⠰⠂⠀⠂⠉⠀⡀⠀⠀⠀⠓⠀⠀⠀⣀⣄⣈⣴⣿⣿⢿⣺⣿⣿⣉⣹⣿⠦⣟⡿⣍⠀⠀⠠⣌⣛⣻⡍⢩⣥⡀⣶⡇⢒⡶⣦⡀⣰⢶⣆⠠⠄
⠀⠀⠀⠀⠀⠉⠉⠐⠐⠒⠘⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣹⣿⣏⣤⣿⣿⠻⣿⣷⣲⣦⢼⡿⠒⠋⠀⠀⠀⠉⠋⠀⠃⠺⠞⠟⠃⠂⠀⠈⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠈⠙⠷⠜⠛⠛⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠐⠚⠛⠿⠿⣿⣿⣿⣻⢿⣿⣟⡲⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡠⠆⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣯⡻⣿⣌⡻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⡴⠋⠀⣠⡤⠀⢀⡀⠀⠀⠀⠀⠀⠠⠌⠑⠌⢻⣿⡹⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⣉⣤⣶⣿⣯⡽⠿⠒⠩⢭⣥⣖⣀⣠⠀⠈⢢⡀⠈⢻⡇⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡟⢀⣿⣿⠦⠄⠉⠁⠈⠩⠉⠛⠻⢿⣦⡀⠀⢻⡌⢊⠃⣾⡿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠱⠀⢺⢟⣠⠍⣛⣻⣿⣿⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⣴⣆⣠⣶⣷⣄⢀⠈⣿⡆⢠⡀⠀⢽⡆⠀⢀⡄⠈⠐⠋⠉⠉⣩⣿⠋⠉⠻⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠒⠫⠵⡻⠟⠋⠙⠆⣐⠐⠚⠉⠁⠀⠈⢧⡀⠂⠀⠀⠠⠀⠤⠾⠋⠁⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⢿⢸⣆⣤⠴⠛⠁⠈⠀⠀⠁⠉⠉⠢⣤⣤⡄⡆⠀⢈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠓⠌⣼⣿⠃⢠⠀⡁⣴⣿⣿⣦⡀⢄⡀⠈⠙⠓⠛⠂⠀⢃⡀⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⢠⠀⠀⣤⠻⠀⠁⠙⢿⡿⠋⠈⠀⠙⠒⠒⠂⠀⠀⠀⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠑⠀⠀⢀⠀⠊⣄⠀⠀⠦⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠑⠦⠤⢤⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣨⣦⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢿⢰⣾⣶⣤⡆⣷⣼⡖⠄
⠀⠀⠀⠀⢀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣾⣿⣿⣿⣿⣿⢛⣤⣶
⠀⠀⠀⠀⠘⣿⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣾⣟⠁
⠀⠀⠀⠀⠀⣽⣿⣿⠟⠂⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠢⠛⠛⣽⣿⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠙⢫⠃⠀⢀⣾⠓⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠳⣌⣉⣋⣤⣿⣿⣿⣿⣫⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⡋⢠⡄⢲⡖⢠⣾⣦⣹⣿⣿⣶⣶⣶⣾⣧⣴⣿⣿⣿⣿⣿⣿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠜⠠⢄⣠⠴⠿⠿⠈⠛⠻⠿⠿⠿⠿⠿⠟⠛⠉⠙⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠆⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣀⠱⡀⠂⠁⢀⠁⠂⠀⣰⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣠⠤⢤⠄⠀⢀⣀⣀⠀⠐⠐⢄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⠐⠦⠷⠶⠶⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⠒⠈⠀⠀⠀⠀⡀⠀⠀⠄⠁⠀⠀⠀⠀⢀⠌⠀⠀⠀⣓⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢘⠢⢀⠀⠀⠀⡠⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠤⠾⣁⠼⠀⠈⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠰⠂⢠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⠁⡠⠀⡊⠈⠂⠀⠀⢠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠠⠁⡀⠀⠀⠉⠒⠀⠠⠤⠀⢀⣀⣀⡠⠊⠀⠄⠅⠰⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠘⠀⠂⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⡜⠀⠀⠀⠀⠒⠠⠤⠤⠀⠀⣀⠊⠀⡔⠀⢀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⣀⠀⠀⠀⡔⠁⢀⠎⠀⠀⠀⠀⠀⠀⣀⡀⠠⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠒⠐⠃⠀⠀⠀⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠑⠀⣠⠀⠀⠀⣩⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠀⠀⠄⢠⠒
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠂⠒⣤⠘⢇⠉⠉⠁⠀⠐⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⣆⠀⠀⣄⣀⣤⣰⣀⣰⣠⢄⡠⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠃⢁⠀⠄⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⡀⠀⢀⡀⠳⢤⠔⢊⠴⠚⠀⠀⠉⠓⣮⡗⡶⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⢧⡀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠈⠇⡇⠀⡸
⠀⠀⠀⠁⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠤⠤⠤⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣨
⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠘⠀⡀⠀⠁⢀⡀⠁⣀⠂⠌⠀⠀⡀⠠⠀⠀⠀⠄⠀⠀⡀⠀⠁⠀⣠⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠈⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠶⣦⠈⠀⠊⠀⣠⢆⠈⡀⡂⠀⡀⢤⠐⣠⣤⢴⣄⣢⡢⡀⠠⠀⠓⠀⡐⢃⠐⠠⢠⠀⢠⠁⠂⢀⢀⢀⠀⠀⢉⡚⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄⠀⠙⣆⣀⡰⠦⠤⣄⣚⠀⡆⠘⢂⠀⣿⠉⢀⡀⠈⡎⡆⠈⠀⠐⠰⣢⣱⣎⠃⡁⠀⡀⠀⠀⠀⢸⠀⠈⠀⠀⠌⠾⠀⣿⠄
⠀⠀⠀⠀⠄⠀⠀⡐⠀⢂⠓⢀⡿⠃⡤⠀⡄⠀⠹⣦⣁⡉⣑⣰⣿⡀⠣⠂⣤⣿⡁⠀⠀⠂⢀⠀⠁⢀⠘⠹⣦⣁⣉⣠⣵⣿⡀⠐⠀⠐⠀⢠⣴⠿⠂
⠀⠀⠐⠀⠀⠁⢴⣤⣠⣠⣤⡾⡃⠀⠐⠂⡤⠛⢀⡿⠋⢁⠀⠀⠙⢿⠿⢟⡽⠗⠛⠻⣦⣄⡆⠀⠈⠀⠀⠀⢿⣧⠟⡉⠉⢉⠛⣷⡟⠷⠶⠛⠉
⠀⠀⠀⡀⠈⢀⡀⠈⢩⠍⢀⡐⠘⢦⣤⣄⣠⣤⣾⠁⠀⢁⠀⠛⠀⠀⢿⡏⠐⡄⠠⢀⡀⣮⢧⠠⠸⠠⠄⣴⢾⡇⠀⡱⢀⠄⠀⢸⡗⠀⠀⢀⠀⠀⢀
⠀⠀⠀⢠⠈⠄⠠⠄⡠⡀⠂⣀⠔⢣⣿⠟⠉⠙⠻⣦⡀⠃⠊⠋⠂⣀⣧⢣⡠⡃⠴⠁⢁⣏⡿⣯⣗⠺⠷⠇⣼⣿⣦⣐⢀⣀⣤⣯⢁
⠀⠀⠐⠀⠀⠰⢋⠆⠀⠉⠐⠒⠶⣿⠇⢚⡄⠈⡆⢸⣿⣷⣶⠶⠾⠫⠪⢷⣿⣶⣶⣶⢾⢿⠖⠀⠀⠀⠀⠀⠈⠛⢉⣿⡿⠋⡡⠀⠠⠑⢀⡀⠀⠘⡀
⠀⠀⠀⠀⠈⠐⠈⠀⢠⡐⢂⡄⠀⢹⣧⣁⠓⠐⣡⣾⡣⠋⠀⠀⠀⡀⠀⢀⠈⠪⡻⣄⣾⡎⠀⠃⠁⠀⠈⠐⠀⠀⠈⣋⡇⠀⡉⢀⡀⠁⠀⢱⣄⣀⠀⠉
⠀⠀⠀⠁⠠⠀⢢⠀⢄⠨⠀⡠⠀⣼⠽⠛⠛⠻⠟⣿⡇⠀⠘⠂⠀⠀⡀⢠⠠⠄⠈⣞⣷⣆⠀⣆⠐⢠⠀⠑⡀⠀⢀⠟⡗⢄⡣⠨⠀⠂⠀⠾⠋⠈⠳
⠀⠀⠈⠀⣠⠴⠦⠿⢦⣥⣬⡴⠊⣀⠠⠀⠠⠠⠀⢹⣇⠀⠀⠔⠄⠀⠒⠒⠠⠒⠀⠸⡿⢤⣦⡀⠀⠀⠀⠀⠀⣠⠚⡿⠁⣤⠘⠒⠒⠷⠀⢠⡄
⠀⠀⢠⡞⠁⢖⠂⢀⡀⠘⣿⠁⠀⠀⠀⠀⠀⠁⠀⢨⡿⣦⣀⠂⠂⠛⠁⠀⠉⠀⣠⣨⢇⡾⠭⢶⡾⣶⣰⣾⣤⣿⠿⠶⣦⡰⠑⠄⠀⠀⡀⠀⠂⢀
⠀⠀⠸⣧⠀⡠⠄⠊⠀⠁⣿⢇⡀⠠⠁⠈⠢⠄⢀⠖⡉⢈⠙⣿⣶⣶⣶⣶⣴⡷⠾⣡⠋⠀⠀⠠⠙⠾⣦⣾⠏⢀⠤⢄⠈⠻⣦⣄⡴⢭⣶⠠⠄
⠀⠀⠀⠙⢷⣤⣉⣀⣁⡼⠋⠲⠩⡦⣤⡤⣤⣶⣇⠀⡨⠄⢀⢿⠕⠃⠁⣉⠙⠻⣇⣿⠀⡀⠣⠃⠀⠴⠤⠇⠀⠠⠤⡄⡇⠀⢸⣿⠅⠀⠙
⠀⠀⠀⠠⠀⢈⠜⠉⡹⢄⠀⡋⢀⡞⢁⢀⠀⠈⢉⠱⣦⣴⣿⠁⠐⠀⠰⠒⡀⠀⢹⣎⠦⣄⣀⣥⠞⠳⡻⣷⡀⠺⡂⠂⠂⢀⣾⠗⠛⠄
⠀⠀⠀⠀⠀⠀⡤⠤⠁⠙⠀⠠⣞⠀⠈⠂⠀⠂⠀⠁⠈⣿⣷⠀⠠⠡⠀⠍⠇⢀⣿⠖⠚⠒⠒⠲⣫⡅⠉⢨⡝⠲⢤⠴⠰⠏⠠⡀
⠀⠀⠀⠀⠐⠀⠀⠀⡠⠘⠀⠠⠻⡀⢠⠂⢠⣀⠒⡀⣰⠇⠈⠻⣵⡄⣤⣤⣾⢿⠃⠐⠖⠀⠀⠀⠆⠠⡄⠋⠐⠀⠨⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠘⠂⠀⠙⢢⡀⢀⡤⢴⠞⣁⣠⡎⣄⠀⢀⠛⠀⠴⠸⠀⠀⡀⡀⢒⠀⠄⠀⡹⣠⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡐⠇⢨⣓⠅⣀⠁⠄⠁⠈⠓⠀⠠⠁⠀⠂⠀⠈⠐⢀⡠⠝⠛⠂⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠈⠺⠉⠈⠁⠄⠀⠀⠀⠀⠀⠀⠀⠉⠂⠂⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⢀⣦⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⠀⠀⠀⢀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⡄⢀⠀⣶⡧⠀⠀⢀⣿⣿⠟⢳⣄
⠀⠀⠀⠀⠀⠀⠐⡾⢁⠀⠀⠀⠀⠀⠀⢰⢋⠉⠀⠀⠀⢰⡀⡀⠀⠀⣤⠏⠓⡂⠀⠀⠀⠠⣗⢚⡄⢉⠃⡀⠐⢩⣿⣷⡖⠁⠛⢃
⠀⠀⠀⠀⠀⠀⢀⡷⢻⠀⠀⠀⠀⠀⣰⠿⠟⠀⠀⠀⠀⠤⠓⠃⡀⠀⢠⡄⠘⠟⢀⡄⠀⠀⣻⢼⡃⠀⠰⠀⡘⣿⡟⠉⠀⣀⡴⣿⡀⢀⡄
⠀⠀⠀⢀⢀⣰⣿⣃⣀⠁⢠⡄⠀⢀⣁⡄⠀⠀⠐⢶⠀⢼⠃⣿⠁⣶⠈⠁⢀⣀⠀⢳⡴⠞⠓⠐⠛⠦⣄⠀⠈⠛⣡⡶⠛⠁⠠⠸⠃⢀⡧⡀
⠀⠀⡰⠀⢼⣿⡿⢅⣠⡀⡨⢦⡀⠸⡿⠃⣀⡀⣿⡿⠀⣾⠶⣿⠀⠉⠠⣦⣾⡟⢲⣿⣤⣤⣤⡤⠤⠶⠞⣛⡀⢞⠙⡾⠀⣠⠶⠁⢀⣼⠄⢷⡀
⠀⠀⢧⠸⣿⣀⢶⡎⠀⢈⠃⠄⡇⡆⠈⠐⠿⠷⠈⠁⠸⠁⠀⠸⣧⠀⠠⠃⠘⠃⣾⣤⣤⣠⣤⣀⣤⢠⡶⣿⡧⠈⢢⡿⠋⢀⠄⠀⢀⣅⠀⢠⣅
⠀⠀⠘⡇⣾⣽⣔⢑⣄⣡⢈⣤⡠⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢻⣄⠀⠀⠀⠀⠹⣿⣼⡷⣦⡿⠴⠌⠿⣾⠃⠀⠀⠁⢴⠉⠀⠀⣸⣄⣀⣀⣸⣆
⠀⠀⠀⠈⠻⢿⣿⠿⠤⠀⠉⠞⠁⠀⠀⢀⣼⣤⣀⣀⡄⠀⠀⠀⢸⡟⢀⠀⠀⢀⠄⠙⠳⣶⠶⠶⠖⣲⡞⠋⠀⠀⠀⠀⠀⠀⣀⣼⣯⣥⣤⣤⣤⣽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⠉⢡⡀⠀⠀⣼⠁⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠟⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠋⠙⠒⠀⢿⡀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢸⢐⠃⠀⢀⠠⡀⠄⡀⡀⠀⢠⠀⠀⡄⡄⡀⢀⠀⡄⠠⢀⣠⣀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠰⠸⠁⠘⠀⠃⠘⢘⠁⠀⠠⠀⠀⠁⠀⠁⠈⠀⠅⠆⠘⠈⠀⠐⠸⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⠀⢈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡋⣳⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠰⡦⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣟⠛⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆⣤⡄⠀⡀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡏⢹⣧⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⡷⠁⠤⠀⠀⠀⢰⣿⠎⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠴⠁⢅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠒⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⡀⢀⠀⠀⣀⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠔⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠐⠄⠀⠀⠰⠂
⠂⠀⠀⠀⠈⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠤⠐⠁
⡎⠀⠀⠀⠀⠀⠀⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠂⠠⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⢀⢀⡀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠒
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠠⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀
⣶⣶⣶⢾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠐⠀⠀⠀⠂
⠀⠀⢀⠨⠄⢠⠀⠀⠀⠆⠀⠀⠀⠄⠉⠀⠀⠀⠐⠀⠐⡂⠀⠢⡀⠀⠁⡀⠂⠀⠠⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠂⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠈⠠⠀⢀⠘⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠕⠀⠠
⠀⠀⠐⠀⠀⠄⠁⠈⠀⠁⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⢐⣤⣰⢄⣤⢐⣂⣤⣤⣄⠤⣀⣠⣄⣀⢀⡄⣀⣠⣴⣤⣦⠖⣰⣶⣦⡤⣂⡴⣶⢦⢀⠂⣤⣬⣤⣆⣀⠤⣀⢀⡀
⠀⠹⠟⡵⢫⣴⡿⠛⠻⠯⠠⠾⠛⠿⡟⡿⢋⣾⠾⠟⠛⠉⠋⠚⠉⠍⠙⠾⠾⢿⠛⠟⠣⠾⠍⠯⠷⠿⡟⠾⣿⠼⠛⠒⠂
⠀⠈⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠀⠀⠀⢠⠀⠀⠀⠃⡀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠘⠀⠐⠀⠀⠀⢠⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠨⠂⠀⠃⠀⠰⠆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣀⣿⣏⠀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡈⢿⣿⣿⣶⣤⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣹⣷⣽⣿⣯⣴⢛⢀⡀⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣴⡾⠿⣿⣿⣗⠉⠉⠛⠛⠉⢩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣍⣯⠀⢀⢽⣿⡿⠦⠄⣁⣀⠀⠛⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⣤⡄⠽⢻⣿⡷⣾⠟⠑⢰⣼⣿⣛⣤⡶⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠓⠓⢓⡋⠰⢿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⣠⣄⠐⠏⣿⣿⡅⠀⠙⣿⣍⢁⣁⣈⣀⣤⢶⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠒⢿⡿⠿⢷⣶⣽⡳⢦⠾⠿⢏⣿⣿⣭⣻⠍⠁⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢂⣀⣩⣿⠀⠀⢰⣟⣿⣿⣿⣿⣷⣿⡛⠙⠻⠏⠈⣤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⠛⠿⠿⠲⣿⣭⣿⣿⣿⢿⣝⣿⣟⣶⡴⠾⣻⠿⢷⠶⡦
⠀⠀⠀⠀⠀⠀⠰⢽⣦⣶⣶⣾⣴⣶⡿⠟⢿⣿⢀⣟⣶⣿⣿⣿⠁⠀⠀⡸⣦⠴
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡤⡀⣉⡐⣤⣥⡼⣟⣯⣿⣏⣳⣮⣿⣷⣀⣈⣽⣧⣶⣯⣭⣀
⠀⠀⠀⠀⠙⠗⣶⣦⣄⣉⡉⢁⣭⣍⣛⣷⣿⣿⣿⣬⣋⠻⡟⣿⣯⣹⣿⡷⣶⣤⣬⣍⠙⠓
⠀⠀⠀⠀⠈⠀⠀⣈⠹⠙⢙⣿⢿⣿⠋⠉⠉⢿⣿⣷⣿⣟⣧⣸⣽⣿⠿⢿⣞⣂⣉⣉⡀
⠀⠀⠀⠀⢀⣠⣤⡤⠶⠚⠋⢀⣽⣿⠀⢠⢄⣹⣿⣿⣿⣿⣯⣀⣼⡁⣀⠀⣿⣿⠿⢿⣢⣦⣤⡀
⠀⠀⠀⠈⠉⠀⠀⠀⢀⣠⣶⣿⣍⣿⡿⣛⣿⠻⣿⡿⡏⣿⣏⠿⣼⡿⢶⠾⠻⡷⠄⡀⠈⠉⠉⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠙⠋⠉⠀⣉⣽⠿⠛⠁⣰⠟⠈⣷⡜⢿⣆⠀⠛⢷⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⢹⡧⠀⠈⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠊⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⣈⣏⣹⣋⣏⣫⣙⣏⣙⣉⣋⣟⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣭⣿⣿⣯⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⡶⠊⠉⠉⠙⠻⣕⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡰⠋⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠈⠉⠉⠒⠦⣤⠋⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢀⣴⠠⠤⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⠒⠒⠒⠒⠢⠤⣀⢀⠃⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠉⠀⠀⣀⡠⠤⠤⠤⠭⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣗⠀⠑⣄⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢠⠏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢻⠢⡀⠈⢆⠀⠀⠀⢀⡜⠀⠀⡠⢾⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⠀⢈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⡆⠑⡄⠀⢂⠀⠀⡼⠀⠀⡜⠁⡈⠀⢠⠁⠀⠀⠀⠀⠀⠀⣀⣀⠀⣸⡄⣃
⠀⠀⠀⠀⠀⠀⢸⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⠀⠘⡄⠘⡄⠈⡄⢠⠃⠀⡜⠀⡰⠁⠀⠃⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠲⢷⠦⣽⠀⢰⣼⠀⢸⠁⡰⠁⢀⠎⠀⠀⠀⡠⠊⠁⠀⣀⠤⠂⣺⠛⠂⡟⠒⠢⣌⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⡹⠆⠚⠁⠀⠛⠐⠦⣀⣠⡄⠀⡶⢄⡀⠈⢣⡈⢱⠀⢸⣴⣡⠴⠛⠻⠁⢀⠞⠁⢀⡴⠋⠁⢀⡴⠁⢀⠞⠀⠀⠀⠀⠙⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠟⠁⢀⣀⣀⠀⣀⠀⠀⢻⠁⠀⢳⠀⠉⢦⠀⢁⠀⡗⠁⠈⡵⢾⠀⡇⢠⠏⠀⡰⠋⠀⢀⡴⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⢀⡠⠄⣒⠶⠟⠋⣀⣠⠔⠋⠁⠉⢀⡈⣙⣂⣄⣣⡀⠘⢦⡀⢸⠀⢰⢆⡀⠀⡎⠀⡘⢠⣷⠇⣀⠞⢀⡠⠖⠁⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢰⠃⠀⠛⠠⠄⠀⠀⣭⠀⠀⠀⣀⠽⠃⠀⣠⠤⠘⠻⠿⢦⣀⠙⢺⣤⠺⣮⣃⢠⡣⠞⠁⣠⣃⠀⠙⠺⢭⣤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⢸⠀⠀⠀⣠⠔⠂⠉⠉⣉⢭⡟⠁⢀⣴⣊⡀⢀⣀⢀⡀⠤⠽⣻⣋⡵⠒⢿⣈⣿⣤⡴⠋⠀⠀⢙⠶⠀⠀⠈⢛⡁⠒⠂⠤⢄⣀⠀⠀⠀⠀⡠⠃⠀⣸
⠀⠀⠀⢳⠀⠀⡇⠀⠀⡰⠊⣴⡊⠀⣀⣀⠀⠀⠁⠁⠈⠁⠀⢀⣠⣼⣉⣷⣄⡼⢥⣯⡁⠈⣉⣉⠉⠉⠳⡄⠀⠀⠘⣥⣀⡀⠀⠀⠀⠉⠢⡤⠚⠁⠀⡠⠃
⠀⠀⠀⡜⠀⠀⠈⡆⡜⠁⠀⢀⡠⠊⡡⠊⠿⣁⣀⣀⣠⠔⡾⠊⡱⠊⡹⠧⠼⢶⠾⠉⠢⡉⠁⠀⠈⠳⢄⠘⣖⠀⠀⠘⣆⠈⢑⡦⢄⠀⠀⢱⠀⣠⠞⠁
⠀⡠⠊⠀⠀⢀⡜⢠⠁⠀⢒⡏⢀⠔⠀⠀⠀⠀⠀⡜⠁⣾⠀⠀⣰⢰⠀⠀⠀⢀⠄⠸⣦⢈⣉⡈⠉⣉⣉⣋⠘⠧⢄⠀⣼⠗⠂⠀⠠⢷⠀⠘⠁
⠾⠒⠙⢆⡠⠋⠀⢳⠂⢀⡜⢡⣌⣀⣀⣀⡀⠀⢀⡄⠀⡇⢀⣠⠋⡆⠀⠀⠀⣎⠀⠀⢹⡆⠀⠈⠙⠃⠈⢣⡉⠻⠊⢳⡈⢢⡀⠀⠀⢸⠀⠸⣄
⠀⠀⠀⠈⠁⠀⠀⠺⠀⠀⠀⢼⡔⢛⠶⠖⠛⢫⡭⡇⠀⠘⠁⢀⠎⢇⠀⠀⠀⠘⢷⠀⡟⠇⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠁⠀⢣⠀⠀⢀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠠⠛⣶⣀⠷⣇⣀⣶⡔⠉⣆⡟⠀⣠⠴⣟⠂⢼⣄⣤⡀⠀⠀⢣⡃⠰⠀⠀⠀⠀⠀⠀⠀⠈⢻⠑⠢⣄⠀⠀⠀⠈⠢⡀⠀⠀⣈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠟⣡⠞⠛⠛⠒⠚⠓⠠⠽⠏⣉⣻⡧⢴⡖⠉⡍⢻⣍⣧⣵⢄⠀⠈⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠙⢷⡄⠀⠀⠈⠢⣀⡁
⠀⠀⠀⠀⠀⢀⣤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠿⣙⢻⣍⣧⠹⠆⠀⠂⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡐⠋⠉⠁⠐⠒⠂⠀⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⣱⡾⠀⠀⠀⠱⠇⠀⠀⠀⠀⠀⠀⡠⢶⡤⢮⢀⣄⣼⣦⡀
⠀⠀⠀⡼⠛⠉⠉⠉⠉⠛⠒⠲⠦⣄⣀⡈⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣿⠶⣦⢀⣤⣠⣿⡹⠋⠙⠢⣄⡀
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣄⡈⠙⠲⢄⡀⠀⠀⢱⣀⡀⢀⣠⠄⢶⣏⣩⠟⠛⠷⢀⣄⣠⣟⠩⠋⠀⠀⠀⠀⠀⠉⡳⢤
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⠢⣄⠈⠓⢤⣘⣿⡥⣼⠟⢳⢎⠀⡙⠓⢶⣤⣬⠋⠟⠉⠀⠀⠀⠀⠀⠀⠀⠐⠁⣸⠃
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠙⢦⡀⠘⠿⢣⣏⢳⣮⣠⡼⢟⡾⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⢀⠴⠶⠃⠸⡀⠀⠀⠀⠀⠙⢆⡄⠈⡭⠿⠐⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠄⣀⣸⠃⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⡼⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡷⠲⠖⠚⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠴⠒⠚⠉⠀⢀⣠⠴⠚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠤⠤⠔⠲⣎⡁⠀⢀⡰⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡶⣤⣤⣴⣤⡾⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠀⠀⠀⠀⣼⣋⣻⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⡤⠉⣗⡆⠀⠀⠈⠉⠉⠀⠀⢤⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⡶⢠⡾⠿⣷⣀⡀⠀⠀⠀⠀⢞⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠛⠛⠉⠛⢷⡤⡄⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⢦⣴⣄⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠏⠀⣽⠿⢿⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⡧⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡤⢸⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣖⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⣀⡀⣀⡤⣤⣴⣲⠖⠻⣾⠿⢯⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⠗⣾⣽⣶⣿⠽⠛⠓⠋⠉⢹⣯⣿⡆⠀⠀⠙⠛⠋⠀⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣎⣥⣆⣴⢞⣁⣻⣭⣽⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣀⣈⠳⣤⣶⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⣩⡿⣅⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠉⠉⠛⣽⣶⣖⣦⣠⢤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⢿⠇⠀⠀⠀⠘⣿⣭⡏⠀⢈⣿⣻⡷⠛⠻⣽⣶⣟⠦⢴⣲⠤⣖⠶⠾⡿⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢲⣾⣵⣧⡶⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠃⠀⠘⣯⣭⠇⠀⠀⠀⠀⠙⣦⣴⣾⣷⠚⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡤⣀⣴⣺⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢟⣶⣶⣖⣦⣤⢲⣤⣴⣢⣤⣖⣦⠴⣲⠶⠾⣽⡿⣷⠛⠉⠛⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠗⢲⠿⣿⣪⣿⣾⠇⠉⠀⠨⣿⣙⡧⠀⠀⠀⠀⠀⠻⣶⠿⠀⠀⠀⠀⠐⢿⡾⢿⡖⠽⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣯⣿⢿⣲⣤⣠⢤⡙⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠉⠀⠋⠀⣽⡯⣿⠊⠉⠓⠛⠛⢯⡿⣾⡽⠶⢾⡷⠶⢏⣶⣶⡭⠖⠺⠽⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣊⢥⡶⠃⠀⠀⢀⡴⣤⢈⡟⣻⠀⣠⡄⢠⠿⣸⣯⣿⡁⣠⣄⢰⡄⣴⣯⣬⣧⡀⠀⠀⠙⢶⠬⣳⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠰⠞⠒⢏⡤⠞⠹⠋⠱⠊⠷⠛⠋⠁⠈⡇⠀⠀⠀⢸⠋⠁⠈⠛⠙⠞⠻⠏⠻⠏⠲⢄⡟⠚⠳⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢡⢠
⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠈
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠋⡍⠙⠂⢀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠄⠀⠀⠰⠤⠖⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠰⠦⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠆⠄⠀⠀⠀⠰⠰⠀⠀⠀⠀⠀⠀⠀⡰⢩⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠐⡀⠀⣍⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⣀⢀⢀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⢘⡧⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⡀⢧⣢⠘⠤⡀⠧⣄⠈⣆⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⡡⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⡄⠄⠀⡄⠀⠀⠀⠀⡀⢤⡀⠀⡄⠀⠀⢠⢀⠀⠀⡠⠀⠀⠀⣀⣿⡇⠀⠈⠫⣾⠍
⠀⠀⡤⢚⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⠄⠘⠇⠳⠀⠗⠄⠰⠀⠪⠄⠀⠷⠀⠧⠁⠇⠾⠀⠢⠐⠥⠀⠀⠠⠹⠻⠇⠀⠀⠀⣿
⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⡀⠀⣀⠀⢀⡄⠀⠶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠉⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣲⣲⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠈⠀⠉⠈⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⠂⡀⠀⠀⠀⠐⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠤⠂⠀⠀⠀⠆⠀⠆⢀⢀⣀⠀⠀⠀⠀⠤⡀⠀⠀⣤⠀⠀⠀⢸⡟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣬⣤⣦⣼⣮⣾⣭⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠛⠉⠉⠉⠛⠛⠒⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣦⡀
⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠙⣦⡀
⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠠⠬⠷⣄
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠻⢷⣶⣶⣾⠂
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⠤⠞⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⣼⡟
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⢀⣴⠶⠋⠀⠀⠀⠀⠲⠀⢰⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⣠⠞⠉⠙⢦⣠⠞⠉⠑⢦⡀⠀⣀⣀⣤⣽⣷
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠠⠀⠀⠀⠀⣠⠟⠁⠀⠀⡼⠁⢠⣶⣤⠈⠁⠀⣤⣶⡄⢳⡀⠘⠻⠿⢿⣥⡀
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⢶⡇⠀⠀⣴⠏⠀⠀⠀⢸⠇⠀⠈⠛⠁⠀⠀⠀⠈⠋⠀⠀⢷⠀⠀⠀⠀⠙⢿⣦
⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠻⢦⣴⣿⣶⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠘⢧⣀⠀⠀⠀⣀⡴⠛⠉⠀⣾⣿⠿⠾⠟⠛⠁⠀⠀⠀⠈⢧⡀⠀⠀⠀⣠⢦⡀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠻⢿⣿⣿
⠀⠀⠀⠉⠙⠛⠋⠉⠀⠀⠀⢀⣈⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠚⠁⠀⠙⠲⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⡿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡿⠃
⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠟⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄
⠀⠀⠀⠀⢿⣿⣛⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠓⠞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⡀
⠀⣠⣤⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢫⡧⢤⠄⠀⠀⠀⠀⠀⠀⠀⣤⢬⣏⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⡆
⠀⠘⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢧⡀⠀⠀⠀⠀⠀⠀⠀⡨⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⡿⠃
⠀⠀⠀⠈⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠙⡄⠀⠀⠀⠀⠀⡞⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⢠⣶⠿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⠶⣸⠉⠙⠒⠃⠀⠀⠀⠀⠀⠓⠊⠉⣇⡴⡆⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠙⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿
⠀⢠⣿⣋⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠋⠉
⠀⠈⠛⠛⠛⠛⠛⠛⠛⣿⣷⠆⢀⣠⣤⣤⣶⠶⠶⠶⣶⣤⣀⣀⣀⡀⠀⢀⡀⠀⢀⣤⣶⣶⣦⣄⠀⠀⢀⣀⣀⣠⣤⣦⣤⣶⣶⣶⣶⣶⣶⡶⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠿⠛⠋⠉⠛⠁⠀⠀⣀⣽⠉⠉⠉⠙⠿⣦⣿⡿⣿⠿⠃⠀⠀⠈⠻⣷⣶⡟⠋⠙⠛⠃⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠈⢹⡇⣿⠀⠀⠀⠀⠀⠀⠈⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠓⠒⠒⠚⠃⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠦⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠀⠈⢃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄
⠀⢰⢀⠀⠀⣠⠀⠑⡄⠀⡠⠒⠁⠀⠀⠀⠀⠄⢀⣇⠐⣹⠀⠔⡡⠂⠢⡑⢄⠀⠀⠀⠀⠀⡴⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠨⢀⠱⡊⠀⣠⠂⢁⡜⠀⡠⠤⠭⣄⠀⠀⠰⠁⠈⢁⠔⡡⠊⠠⠀⠀⠈⠂⡑⢄⠀⢠⠞⠄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐
⠉⠉⠣⡈⡔⠈⢀⡳⠺⣠⠊⠉⠉⠉⠙⢂⠀⠋⢉⠝⡁⠊⠀⠂⠀⠀⠀⠀⠀⠈⠢⡙⢇⡀⠀⠀⣠⡏⠉⠉⠉⠉⠉⠉⠉⠉⣙⠌⠉⠉⠉
⠀⠀⠀⠙⠮⡐⠉⠠⠇⠁⠀⠀⠀⠀⠀⠀⠂⠜⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠄⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⢌⠊
⠀⢀⠀⠀⠀⠈⢲⠐⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣥⡅⠀⠀⠀⠀⠀⢀⡢⠁⠀⠀⡀
⠀⢈⡃⣀⠀⠀⢘⡀⠈⣄⠀⠀⣀⡠⢊⠔⠁⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⢌⠹⣦⡀⣀⣀⡠⣑⢁⣀⣀⠚⡈⠂⢄
⠀⠀⢫⡹⡀⠀⠈⢱⠀⢻⠀⡠⢉⠔⠁⠀⠃⠈⠀⠃⠈⠓⠀⠀⡏⠦⢄⠀⡜⠠⠠⡀⠈⠋⠳⠃⠀⠑⢌⠻⣈⠛⣵⡅⠀⡨⢃⠌⠀⢀
⠀⠀⠀⢻⢷⡀⠀⠘⡅⡠⠋⠔⠀⠀⠀⠀⠈⠅⠀⠀⠀⠭⠀⠀⡇⠐⠈⠈⠃⠐⠊⠁⠁⠀⠀⠀⠀⠀⠀⠐⠈⠻⡇⠀⡐⢠⠂⢠⡶⠂
⠀⠀⠀⠀⢳⢱⡀⡠⠋⠄⠀⠀⠀⠀⠀⠊⠃⠀⠅⠿⠄⠉⠢⠴⠄⠄⠀⠀⠠⠠⠀⠀⠐⠈⠂⠀⠀⠀⠀⠀⠀⠐⢈⠪⡐⠁⠀⡐
⠀⠀⠀⠀⠘⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠜
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠠⢈⠀⠀⠀⠦⠠⠀⠀⡍⠤⢠⠑⠀⠀⠀⠹⠗⠈⠂⠈⠁⡠⠤⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⢰⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠠⢘⠀⠀⠀⢠⢆⠀⠀⠁⠘⡃⠘⢄⠈⠀⠐⠀⣃⠀⣷⠀⠒⠂⢁⡰⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐
⠀⠀⠀⠑⢌⠢⡀⠀⠀⠀⠀⠀⢘⠀⠘⠒⢂⡄⢈⣈⠀⢂⢰⠸⠀⠼⠀⠑⡄⠈⠒⠁⠀⢋⠀⠰⢄⠈⡄⠀⠀⠀⠀⠀⠀⡠⢊⠔⠁
⠀⠀⠀⠀⠀⠑⢌⠢⡀⠀⠀⠀⠀⠀⠈⠀⠘⠑⠁⠀⠙⠊⠈⠁⠀⠓⠋⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠐⠋⠀⠀⠀⠀⡠⢊⠔⠁
⠀⠀⠈⢠⠈⠈⠈⠀⢈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠼⠉⠁⠐⠢⡀⠀⡾⡀⠀⠀⠢⡀⢆⠒⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠉⠁⠉⠀⠉⠉
⠀⢠⠀⠈⠃⠀⠀⠀⠀⢠⡈⠂⠀⠀⠀⠀⠀⠀⢠⠀⠇⠐⡀⣱⠠⠀⠄⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠋⠢⢀⢀⠌⢤⣌⡑⠄
⠀⠀⠉⢀⣠⠤⣀⣀⠔⠁⡨⠢⡀⠀⠀⠀⠀⠀⢸⠀⠒⠊⠀⡜⠀⠰⠼⠀⢆⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠈⠂⠤⡌⠰⢤⣁⠁⠂
⠀⠠⠖⠉⢄⠢⡀⠀⠠⠚⠀⠀⠘⠢⡑⢄⠀⠀⠘⠀⠈⠀⠉⠀⠃⠊⠁⠑⠘⠚⠀⠃⠀⠠⠄⢀⠔⡠⠚⠙⠒⢦⠄⠀⠠⠁⠄⠀⠀⠸⠒
⠐⠀⠀⠤⠀⠢⡈⠀⡈⡄⠀⢀⠔⠉⠈⠢⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⡊⠀⠀⠀⠀⠀⠙⣀⡂⠆⠀⠀⠀⠆
⠀⠈⠢⡄⠘⠂⠈⢓⠎⠀⣰⢄⡘⠃⡈⠐⢄⢂⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠡⠘⢄⠙⡆⠀⠀⠀⠀⠀⠈⠩⠄⠂⡄⡜
⠀⠐⠀⣌⠢⠀⠘⠡⢐⠠⡐⠀⠀⠑⢄⡀⠒⠀⣉⢤⠑⠀⠀⠀⠀⠀⠀⣀⠀⠀⠔⣁⠀⠀⠀⠈⠆⢹⠀⠀⠀⠀⠀⢠⠒⠠⠀⡔⠁
⠐⠙⠑⡋⠖⣂⠤⠊⠑⠀⠀⠀⢀⣀⠤⣄⠬⠼⠃⠂⠛⢄⠑⠀⠀⠀⠀⠀⠀⢀⠶⢅⠑⠒⠢⢀⠘⣀⣗⡀⠀⠀⠀⣇⠀⠀⠼⠃
⠀⠔⡠⠀⠈⠀⠀⠀⢀⠀⣀⠬⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠒⠀⠤⡁⢀⣈⣦⡀⠀⠣⣀⠴⠃
⠃⠁⠀⠀⢀⡠⠤⠙⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⡉⡑⢿⡄⣸⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠈⠁⢹⢀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⠀⠠⠒⠱⣄⡞⠀⠀⠨⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⢤⡀⢀⣿⠁⠀⠀⠀⠙⣌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⡀⠀⡀⠀⠀⠘⡆⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠱⡙⠄⠈⠑⠤⡀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠈⠳⡄⠀⣍⠎⠀⠀⠀⠈⢦⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⣀⡽⠀⡏⢢⣀⠄⠀⠀⠀⢳⡀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⣴⠊⢡⣈⡠⠊⠀⠀⠳⡀⠀⠀⡀⠀⠀⠀⢳⡀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⡱⢠⢀⢱⠀⠀⠀⠀⢷⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⡅⢸⠈⡇⢧⠀⠀⠀⠘⣧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠘⡆⠀⠀⠀⠀⣧⢸⠀⢸⣸⡄⠀⠀⠀⣿⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡟⠀⠀⠀⡇⢧⠀⠀⠀⠀⢸⣼⠀⠀⡇⣧⠀⠀⠀⣿⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢻⠀⠀⢀⣁⡘⠆⠀⠀⠀⢾⡟⠀⠀⢹⢸⡄⠀⠀⣿⠃⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⢆⢠⡎⠀⠈⢦⠀⠀⠀⡼⠁⠀⠀⠘⡎⡇⠀⢰⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⢦⡳⢤⠤⠀⠀⣑⠖⠁⢄⠀⠀⠀⣧⢻⠀⣾⠃⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣄⠀⠀⠀⠱⢤⡑⡌⠉⠀⠠⠔⠀⠈⢢⠀⢸⢸⣼⠏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⢦⣄⠀⠀⠀⠉⠛⠶⣤⢤⣀⣀⠤⠓⠀⢈⣼⠏⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠛⠁⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠂⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⢀⣠⡀⠀⣠⣤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⣿⣿⣿⣤⣶⣾⣽⣿⣶⣾⠅⣂⡠⠼⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠟⠉⢶⣿⣷⣿⣟⣿⡿⢿⣯⣿⣿⣥⡟⢺⡤⣗⠒⠒⠐⠂⠛⣤
⠀⠀⠀⠀⠀⠀⠀⢰⢃⣜⣢⣼⡿⣯⢿⣿⣏⣹⣇⣿⡿⣟⢯⠻⣿⣿⣤⣿⣷⣶⣀⣤⣞⢿⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠸⣯⣼⣿⣿⣿⣿⣿⡿⠿⠉⠙⣿⣿⣜⡎⢃⠣⢸⣻⣿⢿⣽⣿⣷⣼⠘⡏⣷⡄
⠀⠀⠀⠀⠀⣴⣦⣜⣿⣿⣿⣿⣿⢛⣂⣇⠀⢀⠼⠛⢿⣿⣿⠛⠒⠓⢿⣿⣿⣿⣿⣻⣿⣧⣇⣲⣿
⠀⠀⠀⠀⠀⡯⣷⢻⢷⣿⣿⣿⡟⡛⠉⢙⠖⠉⠀⢀⠔⠉⢳⣄⠀⠀⠀⣿⣿⠏⣸⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⡗⣿⢼⢊⣿⣅⣝⢳⠄⠀⣠⠃⠀⠀⠸⠀⠀⢸⡿⠀⠀⠀⢿⡿⣾⣏⣿⣿⣯⣿⣿⣿⣿⠷
⠀⠀⠀⠀⣰⠖⠛⢾⣸⣿⣿⣯⣨⣶⣾⣯⠀⠀⣠⡃⠀⠀⠘⣇⢣⡀⠀⠀⠛⠻⣽⣷⢻⣯⣅⡼⣌⠙⡄
⠀⠀⠀⠀⣽⢂⠀⠀⠈⢯⣿⣿⣿⣽⡟⠁⠀⢰⣯⣤⣤⠄⣭⣿⣯⡵⠀⠀⠀⢰⡟⠃⠁⢺⡤⢶⠃⠀⠨⡂
⠀⠀⠀⠀⠘⡄⢰⡀⡀⠸⣷⣿⣿⡿⢟⣧⡀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⣀⣤⣾⣟⠀⢀⣉⣷⣤⣁⠀⢀⠇
⠀⠀⠀⠀⣤⠛⠋⠉⠃⣢⣼⣿⣿⣏⡙⠲⢷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣽⣿⣿⢿⣿⣿⣽⡏⠃
⠀⠀⠀⠀⠘⢶⡂⢀⣀⠯⣨⣿⣹⣿⣿⣏⣺⣿⣻⣷⣦⣀⠀⢀⣤⣺⣿⣟⢙⣿⡿⣿⣿⡿⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠈⠱⣿⣾⣿⣿⣷⣿⢿⣿⣽⣁⣽⣿⣿⣿⣭⣵⣶⣾⣿⣿⣾⣯⣿⣿⣛⠄⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⡟⠋⠉⠙⢿⣽⣻⣿⣿⣿⣿⡟⣠⠿⣿⣾⣿⣿⣿⣿⣿⡆⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣆⠀⣄⣘⢼⣹⣍⣿⣿⣿⣿⣿⣿⡽⡋⣟⣺⣿⣾⣽⡗⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⢿⣿⣿⣿⣝⡻⢭⣩⠿⠬⠤⣥⡽⢻⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠉⠹⡿⠿⠛⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠉⠉⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠉⠉⠉⠉⠉⠉⢛⡳⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠀⣀⠤⠟⠓⠒⡶
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠚⠋⠀⠀⠉⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡃
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢠⣄⣠⣀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠋⢀⠄⠀⠐⠉⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠒⠉⠉⠰⠶⢄⣤⡙⣆
⠀⠀⠀⠀⢠⠞⢷⠀⣀⠴⠃⠈⣉⣉⣉⠶⠲⡶⢦⡤⠚⠃⠀⠀⠀⢀⡤⠶⢛⡉⠁⠀⠀⠀⠀⠀⢠⠤⠠⠦⡀⣿⠋
⠀⠀⠀⢀⡏⠀⠈⢷⠉⢢⣀⣀⡀⢀⡸⠊⠛⢢⣀⠇⠀⠀⣀⠤⣪⠕⠚⠉⠉⠁⠀⠑⢦⠀⠀⢠⠁⠀⠀⠀⠈⣿⡀
⠀⠀⠀⡜⠀⠀⠀⠈⠁⠈⠁⠀⠈⠋⠰⣄⣠⠜⠁⢀⠴⡎⠁⠀⠉⠉⠙⠃⠒⠦⠄⠴⠃⠀⠀⠘⣠⠒⠂⡀⠀⡇⠛⠋⠉⠛⠲⢤⡀⣀
⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡠⠚⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠘⡄⢷⣼⣇⣧⣤⠷⠓⢤⡀⠀⠀⠀⠻⣼⡷⣀⡀
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠊⠀⠀⠀⢀⣇⣀⠀⠀⠖⠀⠀⠐⢒⣼⠗⠒⠲⣤⡽⠚⠛⠛⠻⣇⠀⠀⢀⡇⠀⠀⠀⠀⢹⠃⠹⡍⠙⠲⣄⡀
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⡷⠾⠛⠲⠦⣄⣠⠴⠋⠀⠀⣷⠀⠀⠠⣵⣄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠌⢷⣤⡴⢚⠁⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠈⠳⣄
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠈⠑⠁⠀⠲⠤⣤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡷⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡿⡀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠋
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⢣⡙⠢⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣤⣀⠀⢀⣀⠀⠤⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠃⠰⢇⡨⡍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠉⠲⠤⣀⣀⣀⣀⣀⠠⠔⠚⠒⠛⠒⢯⡑⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢀⣀⣹⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⠏
⠀⢀⡴⠉⠁⠀⠉⠛⠒⣦⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⢀⡾⠋⠀⠀⠀⠀⠀⠀⠈⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠈⠓⠠⣶⣤⣤⠔⠻⠞⠁⠀⠀⠀⠈⠛⠦⢤⣄⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⢀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠛⠉⠲⢄⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠛⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⢰⡇

⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⣰⣣⡄⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠘⣣⠒⠀⠀⠀⠀⢾⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⠀⣴⣶
⠀⠀⠀⢠⡯⠏⠁⡀⠀⠀⠸⢿⡿⠇⠀⠀⠀⠠⠰⠀⠀⠀⠀⠴⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⡂⠀⠀⢀⣾⣿⣿⣧⡀⠀⠀⠀⣷⣿⣆
⠀⠀⠀⠚⠉⠙⠛⠛⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠘⠉⠉⠉⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⢀⣼⣿⣧⠀⠀⠀⢀⠡⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⢾⣿⣿⡂⠀⠀⠀⢀⣽⣿⣄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣴⣿⣿⡄
⠀⠀⠀⣠⣾⣿⣷⣄⠀⠀⠈⠐⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠰⣿⣿⣿⡦⠀⠀⠀⣠⣾⣿⣧⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⣿⣿⣿⡷
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠙⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠖⠋⠇⢠⡤⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡄⠀⠀⠀⠈⠣⣀⠀⢀⣚
⠀⣤⣤⣿⣤⡄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⢸
⠀⠀⡿⠿⣯⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣜⢿⢹⣆⠀⠀⠀⠀⠀⠀⠘⡿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣿⣧⠀⢀⣠⡄
⠀⠀⣸⣿⡇⠀⠀⠀⠀⠺⣻⣿⣿⣾⣿⠛⠢⠀⠀⠀⠀⢸⠏⠈⠊⢫⠀⠀⠀⠀⠀⠀⣠⣖⣦⡀⠀⠀⠀⠀⠀⠰⢎⡟⠢⠀⠈⢻⠏⠀⠈⣿⠁
⠀⠀⣿⣿⣧⠀⠀⠀⠀⣾⣿⣿⣿⣿⣮⡓⠦⠀⠀⠀⠀⠎⣰⠘⡇⢧⢱⠀⠀⠀⠀⣼⣿⣿⣯⣿⠄⠀⠀⠀⠀⠒⣫⣏⡒⠆⠀⠀⠀⠀⣠⢿⣆
⠀⢰⣿⣿⣿⡆⠀⣤⣤⣶⣿⣿⣿⣿⣮⡛⠷⠶⠀⠀⢸⡴⠙⠴⠉⠋⡙⢇⠀⠀⢀⣽⣷⣿⣿⣶⣤⣀⠀⠀⣘⣯⠴⡧⣭⣁⠀⠀⠠⣾⣿⣿⣿
⠀⣾⣿⣿⣿⣿⠀⠰⣤⢾⣻⣿⡿⣿⣿⣿⣅⠀⠄⢠⢋⡆⢠⢺⡀⣧⠹⣌⢂⢾⣿⣿⡿⣿⣿⣿⡷⠟⠀⠀⠠⢤⢚⡟⠲⠤⠄⠀⠀⣼⣿⣿⣿
⢠⣿⣿⣿⣿⣿⡆⠴⣒⣿⣿⣿⢿⣿⣿⣷⣭⣉⠀⣮⠞⠥⠻⣨⠓⠎⠑⠋⢻⣠⡴⣾⣿⣿⣿⣿⣷⡦⣀⠀⢶⠚⣉⣏⣟⢒⣖⣠⣾⣷⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⠛⠿⢿⣻⢿⣾⣿⣿⣿⣿⣯⣿⢋⣴⢸⢀⠀⢰⢰⡘⣆⣿⣿⣿⣿⣷⣿⢿⣷⣿⣿⣟⣷⣒⣉⠥⡧⢌⣋⣻⣿⣿⣿⣿⣿⣽
⣿⣿⣿⣿⣿⣿⣿⡗⣿⣯⣷⣿⣿⣿⣿⡟⣯⣕⣗⢻⡝⠛⣼⢄⡞⣬⠑⠋⣁⣭⣾⣿⣽⣿⣿⢿⣾⣯⣭⡏⢤⡴⠒⡗⠲⠤⢤⣿⣿⣿⣿⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡛⣿⣿⣿⣍⡻⣦⣭⣉⣛⡓⠀⣀⣸⣇⣀⢰⣿⣿⣻⣿⣯⣿⣿⣻⣿⣿⣾⣿⣻⡦⠀⠀⣇⣠⣴⣿⣿⣿⣯⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣻⣿⢿⣿⡟⢿⣳⣯⣭⣍⣡⠆⣿⡿⢿⣿⠁⠠⣄⢤⠀⠀⠀⣿⠀⢀⣉⣉⡉⠁⠀⢈⣿⣯⣿⠿⢷⣿⣿⣿⣿⣿⣿
⠛⠛⠛⣛⠛⠛⠛⠛⠒⠛⠉⠀⢸⣿⣇⣀⣀⣭⣭⢥⣤⠀⢹⣷⣾⡏⠀⣿⣿⣿⡆⠀⠀⣿⠀⣸⣿⣿⣿⣿⠀⠘⣿⣿⡿⠀⠀⠀⠉⠙⠛⣿⡟
⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⠀⠈⠉⠉⠁⠀⠻⠿⠿⠃⠀⠀⠀⠀⠻⠿⠿⠿⠏⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣇⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡭⢫⠄
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢿⣷⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡓⠾⢿⡖
⠀⠀⠀⠀⠀⡒⠢⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣛⣞⠀⠀⠀⠀⠀⠀⠀⠀⣸⣮⠑⠦⣽⠅
⠀⠀⠀⠀⠲⠌⠁⠂⢁⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣷⣿⡿⢿⡥⠀⠀⠀⠀⠀⠀⠀⠐⠒⣻⠴⣄⢯⡥
⠀⠀⠀⢄⣁⠀⠁⢀⠉⠀⠀⠀⠀⠀⠀⠀⠠⢤⡼⣿⣾⣿⣿⣃⠄⠀⠀⠀⠀⠀⠀⣉⡭⢩⠤⢄⣈⣟⢉
⠀⠀⢰⡆⡀⢩⠁⡀⠀⠠⠀⠀⠀⠀⠀⢀⣀⣸⣷⠟⣻⢿⡟⣿⡷⠂⠀⠀⠀⠀⠠⠠⣧⣶⢴⡍⢬⡭⢐
⠀⠀⢄⠠⢠⡀⠀⠀⠀⠂⠈⠀⠀⠀⠀⣀⠙⢛⣿⣶⣾⣼⣷⣿⡿⠋⠀⠀⠀⣀⠮⢙⢁⣋⣬⠉⠔⣊⣅⢉
⠀⡘⠀⡀⠀⢀⠁⠀⠄⠀⠀⠐⡄⠀⠀⠘⠺⣫⣿⣥⣾⣧⠿⣿⣿⣿⡞⠁⠀⣸⠿⠷⢦⠀⣩⢁⣈⡉⢛⣀⢀
⠀⠈⠁⠀⠂⠀⠂⠐⠐⠀⠀⠀⠀⠀⠈⠙⠛⣻⣿⡭⣿⡻⣿⣭⡤⠿⠟⠂⠀⠉⠛⠑⠙⠿⠛⠀⠉⠁⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠑⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡄⠀⠀⠀⠀⠀⢠⣾⣦⡀
⠀⠀⢀⣿⣷⡀⠀⠀⠀⢠⣿⣿⣷⡦⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⢀⣰⣿⣿⣅
⠀⠀⣼⣿⣿⣧⠀⠀⠀⣾⣷⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠦⠀⠀⠀⢈⣽⣿⣿⣏⠃
⠀⠐⠛⠻⠟⠻⠆⠀⠘⠛⠗⠿⠟⠛⠛⠀⠀⠀⠀⣠⠀⠀⠀⠀⠐⠛⠿⠛⠃⠀⠀⠐⠛⠛⠻⠿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⠀⠀⠀⠀⠀⠀⢀⣾⣆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣾⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⢸⣿⢻⡟⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⡼⢷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢾⣷⠀⡀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⢀⣀⣾⣆⣀⠀⠀⠀⢸⣿⣥⣂⠺⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣽⣿⣇⠀⠀⠀⠀⠀⠀⢹⣿⣿⡃⠀⠀⠀⢸⣿⣉⡩⢭⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢸⣿⣉⣑⣚⣿⡋⣍⠋⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢈⣛⣻⣿⠟⣭⡝⣿⣿⣿
⠀⠀⢀⣶⣶⢆⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⠀⢸⣿⠹⢋⡝⣿⣷⣶⣶⣿⣿⣿
⠀⠀⠈⢋⠛⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⠀⢸⣏⣤⣌⣿⠀⠀⠀⢸⣿⣓⣛⣻⣿⡍⠏⠇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⢸⣿⠰⠈⠶⣿⡛⠛⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⢉⣭⣿⡟⠁⠴⣿⣿⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢸⣿⣭⣬⣽⣿⣯⣭⣭⣿⣿⣿
⠀⠀⠀⣼⡅⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⡶⠄⠀⠀⠀⠀⠀⠀⣾⡀⡇⠀⠀⠀⢸⣭⣌⣛⣱⣿⣧⣭⣭⣿⣿⣿
⠀⠀⠠⠿⠧⠄⠀⠀⠀⠀⠀⠀⠀⠿⠙⠷⠀⠀⠀⠀⠀⠀⠴⠿⠶⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠝⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠠⠤⠼⢿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⣿⠋⢀⠠⡤⢀⠀⠹⣿⣿
⠀⠀⢶⣀⣼⣿⡄⠀⠀⠀⠀⠀⢸⣶⣴⣴⣿⠀⠀⠀⠀⠀⠢⢘⠢⠂⠀⠀⠀⢸⣿⡇⠀⢈⡱⡳⣈⡁⠀⣿⣿
⠀⠠⠈⠟⠛⠏⠁⠀⠀⠀⠀⠀⠈⠻⠟⠿⠋⠀⠀⠀⠀⠀⠋⠠⠈⠃⠀⠀⠀⢸⣿⣿⣄⠈⠀⠓⠈⠁⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣴⣶⣿⣿⣿⣿

⠀⠀⣀
⠀⠀⠉⠁⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠃
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠲⠀⢴⠆⠀⡄⠀⣴⠄⠠⠤⠀⠺⠇⠀⠉⠀⠀⠀⠙⠀⠤⠀⣀⡀⢀⡀⢠⣤⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⢤⣠⠗⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣮⣉⣩⣉⣭⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣤⣶⣤⣾⡿⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣄⢉⣁⡈⣉⣤⢒⡹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢛⡛⢋⣭⡥⠂⠀⠀⠀⠀⠀⠀⠀⠈⢉⡋⠳⠛⠚⣁⣠⣤⠂⠀⠀⠀⣠⠄⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠋⠁⠀⠀⠀⠀⠠⠀⠄⢀⣄
⠀⠀⠀⠀⠘⠋⠙⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢻⠶⠒⠒⣋⠩⠄⡀
⠀⠀⠌⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣤⣀⣈⣀⣠⠴⢟⡁
⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠄⠨⠄⠐⠈⠁
⠀⠘⠂⠀⠀⠀⠀⠀⠀⡹⠁⠀⠀⠀⠀⠀⠀⣀⠉⠀⠙⠁⡈⢁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠈⠙⠢⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⢷⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣧⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⡿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢯⣄⠈⢷⣜⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢫⣿⣾⠁⢳⣄⠻⡷⣝⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣱⣿⣿⡟⢲⡀⠻⣧⡈⢮⡳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢃⣼⣻⣿⡟⣇⠀⠹⣦⡈⢯⡢⣙⢦⡑⢽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⡽⢋⣴⡟⣱⣿⠟⣴⡏⢷⣄⠈⠻⣦⡙⢮⡓⢭⣢⣌⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠖⢋⣴⣿⢋⣼⣿⢋⣾⠳⡇⠀⠹⣷⢄⠙⢿⡳⣽⣦⡈⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠵⠚⣩⠞⣡⣾⠟⣡⣿⠃⢠⡟⢦⡀⠈⠳⡙⢦⣙⢦⣉⠻⢶⣄⡀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⢾⡿⣫⡾⡿⠃⣠⢿⣦⡀⠻⣦⡀⠙⢦⡈⠓⢬⡳⢤⡈⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠿⠛⣱⣟⡴⣫⠞⢁⣼⠟⢸⡏⠻⣦⡈⠻⣷⣄⡙⠦⣀⠈⠑⠻⢷⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢋⡴⠃⣴⣿⠋⣰⢻⠷⣄⠙⢿⣷⣌⠻⣝⠲⣬⡳⢤⡀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢁⡴⠋⣠⣾⠟⢡⣾⠏⣸⠧⣜⢷⣄⡙⠻⣟⢮⣝⣦⣍⠓⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⡴⣋⡴⣚⡿⢁⡴⣫⢏⡼⢻⠀⠈⠳⣝⢿⡦⣌⠙⠮⣙⠻⢿⣶⣤⣉⡙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⣡⠖⢋⡴⢋⡜⣡⠞⠁⢸⠀⠀⠀⠈⠓⢿⣮⣝⠲⠮⣽⣶⣄⡉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣡⠖⢋⡴⢋⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡈⠉⠙⠛⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⠿⠚⠉⢀⡴⢋⡴⠋⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣫⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⡇⠀⠀⣠⣤⣤⠀⠀⠀⣤⣤⠀⠀⠀⢀⡄⠀⠀⠀⣶⣶⣶⣦⡄⠀⢰⣶⣶⣶⣄⠠⣤⣄⠀⠀⢠⣶⡆
⡇⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⠀⠀⢀⣾⣿⡄⠀⠀⣿⣿⠀⣹⣿⠀⢸⣿⡇⢸⣿⠆⠙⢿⣧⢠⣿⠟
⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⡿⢹⣿⡄⠀⣿⣿⠿⠿⠟⠀⢸⣿⡿⠿⠋⠀⠀⠈⢻⣿⡏⠀⡀
⡇⠀⠀⣿⣿⡏⠉⠉⢹⣿⣿⠀⣸⣿⣷⣾⣿⣷⡀⣿⣿⠀⣾⢳⡄⠈⣿⡇⠀⢀⢰⡀⠀⢸⣿⡇⠰⣿⠂
⡇⠀⠀⣿⣿⡇⠀⠀⠸⣿⣿⢰⣿⡟⠀⠀⢻⣿⣧⢹⣿⠀⠘⠋⠀⠀⣿⡇⠀⢸⣿⢟⠂⢸⣿⡇⢘⣿⠃
⡇⠀⢠⣴⣆⣴⣦⣆⣤⣦⣠⣠⣀⣠⣠⣠⣠⣠⡀⠘⠋⠀⠀⠸⡷⠀⠉⠁⠀⠀⠈⠀⠀⠀⠉⠁⠚⣿⠃
⡇⠀⠈⠙⠻⠙⠻⠋⠻⠙⠻⡟⠋⠉⠝⠛⠙⠛⢡⣄⢸⣿⣧⠀⠀⠀⣿⠀⢶⣦⢠⣶⠆⣠⣤⣤⣄⠉
⡇⠀⠀⣿⡇⢸⣿⠀⠘⠏⣓⣟⡹⠃⠀⣿⡇⠀⢨⣭⠀⣙⣟⡊⠀⢠⣿⡇⠈⢿⣿⠏⢰⣿⠏⠙⣿⡇
⡇⠀⠀⣿⣷⣼⣿⢠⣾⣿⣿⣿⣿⣷⠀⣿⣇⠀⢸⣿⠀⣿⡟⣿⡄⢸⣿⣷⠀⠀⣿⠀⠈⣿⣇⡀⠙⠃
⡇⠀⠀⣿⡏⢹⣿⠸⣿⣿⣿⣿⣿⣿⠆⣿⣿⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⡆⠀⣿⠀⣀⠈⠛⢿⣷⡄
⡇⠀⠀⣿⡇⢸⣿⠀⠙⠻⢿⡿⠿⠋⠀⣿⣿⣤⣤⠀⠀⣿⣇⣿⠇⣿⣿⣿⣷⠀⣿⠀⣿⣇⠀⣠⣿⡇
⡇⠀⠀⠿⠇⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⠘⠛⠋⠀⠀⠻⠇⠀⠀⠛⠀⠈⠛⠛⠛⠋
⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⣿⠟⠉⠙⠛⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢽⣭⡟⠉⢙
⠛⠂⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠛⠛⠛⠛⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡠⠲⣤⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⢻⢣⣠⠏⡟⢶⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣦⡴⣲⢆⢰⣄⡏⣲⣣⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⡿⠉⠋⠘⠃⠃⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣠⡀⣄⣠⣤⣤⣠⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠷⠿⠷⡿⠿⠸⠇⠿⠧⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡀⣤⣤⣀⡄⢠⣠⣤⡤⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠈⠉⠁⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠴⠄⠎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠀⠒⠀⠆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠴⠏⠒⠢⠤⢀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡀⢀⠤⢀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⠀⡁⠱⡄⠀⠀⠀⠀⠀⠀⠀⡎⠲⢉⣉⠬⢋⠀⠉⠐⠲⡐⢄⠀⢰⠀⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠈⢉⣉⢁⠀⣈⠦⠤⣀⠀⠀⠀⠴⠀⠀⠀⠑⠠⣝⠓⠢⢠⡄⠈⢺⠀⠈⠆⢀⡰⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⢀⠌⠁⠀⡀⢌⣱⢠⠊⠀⠑⠢⠄⠀⠀⣠⡅⠀⠒⠀⠄⠓⠒⠀⡶⠚⠈⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⡔⠁⠀⠈⢻⠀⠀⠀⠀⠀⠀⠠⠈⠙⠦⠤⠀⠀⠀⠤⠊⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⡀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠈⠒⠢⠄⠀⠀⠤⠤⠤⠒⢍⡁⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠊⠉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢦⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠇⠀⠀⠀⠑⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⠑⠢⡔⠁⠣⡀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢈⠶⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣖⢉⡁⠀⢆⠀⠀⠈⠢⠄⠠⠲⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢸⠘⠉⠉⠁⢒⠈⡇⠀⠈⠉⠉⠉⠀⠀⠘⠀⠀⠄⡈⠃⣄⠀⠀⠀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⢔⣤⣀⣀⣠⢀⠃⠀⠀⠀⠀⠀⣀⣀⠠⠤⠖⠊⠉⠉⠀⡍⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠲⡄⠀⢨⠀⠀⠀⠉⢉⠁⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡃⢘⠃⠘⢰⠉⢄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣤⡙⠨⠀⡘⠀⠣⡀⠉⠂⠀⠚⣧⢀⣀⠀⠀⠀⠀⣀⣀⠠⠤⠔⠒⠈⠁⠀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡑⢷⣈⣷⠧⣤⠀⠈⢲⠄⠤⠤⡛⢄⠀⠈⠉⠉⠁⠀⠀⠀⣀⡀⠤⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠒⢁⠜⢚⣶⠦⡀⢲⠒⢲⠊⠁⠀⠀⠡⠀⠉⠉⠀⠀⠘⡉⠉⠁⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⡄⠀⢸⠈⡀⢸⠀⠀⠀⠀⠀⡤⣀⠀⠀⠀⣀⠙⠠⠤⠤⠄⠄⠒⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠸⢀⣸⠀⡇⢁⠀⢀⣠⠷⠊⢀⡠⠤⡄⠰⢄⡀⠉⠉⠁⣈⢱⠄⠀⠀⠀⠀⠒⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡌⢖⠉⠀⠀⢠⠃⠀⠀⠈⠂⠀⠀⠉⠉⠁⠀⠀⡄⠀⠀⠀⠀⠀⠄⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣑⣄⠀⢘⠄⡀⠀⠀⢀⠀⠀⢀⣀⠀⠤⢰⢆⠀⠀⠀⠀⠀⠀⠀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠀⠈⠢⠄⣉⣉⣉⣉⣁⠠⠤⠤⠒⠚⡈⠄⣀⣀⣀⡠⠔⠊⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⡠⠄⠂⠁

⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⣹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡄
⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣟⠦
⠀⠀⠀⠀⣸⣯⣿⣧⠀⠀⠀⠀⠀⠀⢴⣶⣾⣶⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠐⠂⢁⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣿⣟⣆⡀
⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣶⣷⣶⣷⣶⣿⣿⡆⠀⠀⠀⠀⠀⠂⠠⢀⡀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⣿⠛⠄
⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣬⣟⣛⣛⣛⣫⣭⣾⣿⡆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣤⣿⣾⣿⣿⣿⣿⣇⡀
⠀⠀⠈⠉⠉⠹⠿⠉⠛⠃⠀⠀⠀⠉⠈⠛⠋⠿⠋⠈⠉⠈⠁⠀⠀⠈⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠏⠉⠉⠀⠀⠀⠀⠈⠉⠈⠉⠹⠟⠛⠿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⡤⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⢠⢄⡠⡄⠀⠀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⣡⢿⡈⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠃⠀⠀⠀⢩⠄⡠⠹⠄⠀⠀⠀⠀⠀⠨⠥⠀⠠⠬⠀⠀⠈⠁
⠀⠀⠀⠉⣺⠈⠒⠀⠀⠀⣠⣾⣿⡀⠀⠠⠂⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⢸⠿⣿⣿⠿⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠁⠂⠁⠀⡤⠀⠀⠀⢠⠄⠀⣼⡇⠀⠀⠀⠀⢸⠇⠀⠀⠀⣲⣆
⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⣀⣀⣀⠀⠀⣴⣶⠀⠀⣤⣤⣤⣤⣤⡄⢀⣿⣿⠀⠀⠀⠀⠸⠂⣠⣤⣤⣌⠁
⠀⠀⠀⣴⣿⣿⣧⠀⠀⠀⠈⠉⡧⠀⠀⠈⠀⠀⠀⠻⢿⣿⣿⣿⠀⣸⣿⣿⡇⠐⣞⣳⠂⠘⠋⣉⢉⠙⢧
⠀⠀⠈⣉⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠀⠀⠉⠉⠀⠀⠙⣀⠂
⠀⠀⠀⠈⠠⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⣠⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⢿⠿⠆⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠴⠿⢿⡇⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣶⠆⠀⠈⠛⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢀⠃
⠀⠀⠐⠿⠾⠿⠂⠈⠀⣀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠘
⠀⠀⠀⠸⢆⡀⠀⠀⠀⠉⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠸⣶⣀⢀⡀⠀⢠⡄⠀⠀⣠⣶⣿⣿⡆
⠀⠀⠀⢰⡌⠃⠀⠀⣠⠀⣭⠀⠀⢀⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠀⠀⠘⢷⣤⣴⣿⣿⣿⠟
⠀⠀⠀⠘⠃⠀⠀⠀⠉⠀⠛⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⠃⠀⠐⠀⠋⠙⠉⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⠀⢀⣾⠏⠘⣷⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠈⢳⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣥⠀⠀⠀⡇⢀⢙⠀⠌⠀⡇⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⣿⡿⠃⠀⠀⠀⡇⠀⢸⡀⠀⢰⡇⠀⠀⠀⠀⠶⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠯⠍⠁⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⣾⢠⡆⠘⡄⠘⣄⢰⡠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⣀⠴⠒⠐⣆⢀⠤⠀⠀⠛⠀⠀⠀⠀⢻⣀⣉⡀⠗⠁⠓⠈⡡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠃⠀⠀⢀⣿⠇⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢀⡇⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡄⠈⢸⠀⠌⠀⡀⠀⠈⠁⠀⠀⣠⠀⠀⠈⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠸⠺⠁⣿⠂⢤⠀⡧⣀⢈⡁⣾⠀⣿⠀⣶⢠⡄⣷⢠⡆⣀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⡴⠹⠇⣼⡐⠀⣯⠘⠇⡘⡿⢿⠜⠿⠸⠁⣧⠈⠋⢻⡄⠐⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢾⣱⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠉⠉⠗⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠐⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠖⠃
⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⣠⠒⠞⠢⢄⡀⠀⠀⡀⣀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠁⢠⣕⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠈⠦⠄⡄⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⣶⣶⡖
⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠤⠄⠀⢀⠄⣀⠀⠀⠀⠁⠀⣹⠉⠀⠀⠀⢀⠀⠀⠀⠄⠀⠉⠀⠀⠋
⠀⢂⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠀⠀⠀⠀⠉⠀⠉⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠁⠀⠀⠔⠓⠤⢀⣀
⠀⠀⠘⠀⡞⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢀⠈⠀⠇⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡄
⢇⠈⠀⢠⡇⠀⠀⠀⠀⠀⠈⢆⠀⢀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠈⠁⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠑⠤⣢⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠁⠀⠃
⠀⠀⠀⠀⠀⠃⠒⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀
⠀⠀⠀⠈⠂⠒⠒⠒⠀⠐⠉⠆⠚⠒⠚⠒⠚⠂⠒⠂⠁⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠠⠄⢀⠀⠔⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣉⣲⣤⣴⣊⣧⣤⣀⣀⣀⣤⣼⣢⣤⣴⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠉⠁⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠒⠒⠀⠀⠀⠀⠠⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠋⠉⠉⠉⠉⠒⠂⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢠⠞⠁⠀⠀⢀⠀⠀⠀⣠⠒⠉⠁⠈⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⣁⠄⠀⠀⠀⠀⠀⠂⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠁⠀⠉⠀⠀⣀⠀⠀⠀⠀⠀⣭⡬⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣀⣀⠀⠡⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⣠⠖⠋⠉⠉⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠆⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠉⠂⢾⠁⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⠿⣌⠀⠑⠀⠀⠀⠀⠀⢀⡠⠔⠀⣤⢤⡤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⣤⠈⠃⠀⠀⠀⠄⠀⠀⠀⠀⣠⠾⣁⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⣀⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢸⠹⡇⠘⠂⠀⠀⠀⠀⠀⠤⠴⠒⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠧⣼⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢺⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡄⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠁⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢠⠇⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣠⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⢀⡀⠀⠀⠀⢈⣀⡤⣀⠀⠁⠐⠒⠀⠀⠴⠚⠁⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠀⠈⠳⢤⣀⣀⠀⠀⠀⢀⣀⣤⠖⠋
⣀⣀⣀⢀⡀⡀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⣤⣤⡴⣶⣶⣤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣽⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠉⠉⠉⠉⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⠶⠤⣤⠀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠲⠄⣶⡄⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣭⣯⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠁⠴⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢬⡧⡄⠀⠀⠀⠀⢹⣿⣿⣿⣿⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⠴⢀⣭⣩⠛⢶⣤⠀⠀⠠⣆⣀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠙⣼⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠁⠨⢺⠋⠀⠀⠀⠋⠙⣭⡯⣤⣀⠨⣝⠲⠄⠀⠀⠸⣿⠂⢤⠄⢸⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⣿⢳⣤⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡄⠀⠀⠀⠀⣠⣈⣴⣀⣀⣀⣤⣀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣧⡐⢻⣿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢆⡀⠐⣀⣀⠤⠞⠋⠉⠉⠉⠉⠙⠛⠻⢦⠾⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹⡿⠟⠁⠈⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⣄⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⡄⠀⠈⡆⠁⠐⠀⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠶⣿⣿⣿⣞⡒⣚⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠲⢦⠞⠋⢀⡀⠀⣿⣶⠒⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠠⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠉⠈⠙⠛⠙⠉⠛⠙⡷⣶⣦⣀⣀⣀⠀⢁⣴⡾⠁⠀⢹⠇⡆⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢰⡏⢠⠇⠀⡠⠾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠈⠀⣟⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠈⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀
⠀⠀⠀⠀⠀⠀⠀⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠹⢶⣤⣅⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠈⣦⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⣠⣧⠃⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡾⠋⠀⠀⠀⢀⡰⡇⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠁⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢃⠸⠁⠀⠀⠀⡀⢸⠃⢣⠀⠐⠶⠌⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢰⡤⠾⡁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣴⡞⠀⡞⢀⠀⠙⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣠⠄⣴⢀⠀⢀⣼⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣼⠃⠐⠀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣥⠞⣿⡄⠠⢾⡏⠀⠀⠀⠀⠀⠀⢿⡀⠀⢠⡄⠀⠀⠀⠀⣴⠿⣤⣤⢠⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢠⣽⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢸⠃⠀⠀⠀⡼⠃⠀⢠⡔⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣠⡞⠀⠀⢀⣶⠀⠀⠀⠀⢦⡀⠀⠸⠀⠀⣠⠞⠳⠶⠖⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡼⠋⠀⣹⣇⡴⠖⠁⢿⡄⠀⠀⠀⠀⣀⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠀⣠⡼⢣⡜⠀⠀⢀⣾⡿⢲⣤⣴⡞⢁⣴⠛⠀⢾
⠀⠀⠀⠀⠀⠀⣀⡀⠤⠔⠋⠁⠀⣸⠀⠀⠀⢠⠛⣇⣼⠁⢸⣧⡘⢳⣄⠀⣾⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠇⠀⠀⡠⠋⠀⢻⡏⠀⠈⣿⠀⠀⠉⠀⢸⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠘⠉⠀⠀⠀⠈⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈