Bag Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡏⢉⣹⣾⢋⡻⣦⡶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢾⣧⣴⡿⠻⠶⠟⠛⠷⠾⢿⣔⣶⡿⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣷⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢦⣷⡿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡩⡍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣥⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣔⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢠⠹⣏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠏⠆⣿
⠀⠀⠀⠀⠀⠀⢰⡏⠄⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣼⠏⠀⠰⢹⡆
⠀⠀⠀⠀⠀⠀⣼⠃⠄⠀⠀⠈⢷⣍⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣢⡾⠁⠀⠀⠀⠜⣧
⠀⠀⠀⠀⠀⠀⣿⠐⠀⠀⠀⠀⠀⠙⠷⣤⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣤⠾⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣮⣄⣂⡀⠄⣶⣿⣿⣷⠤⢀⣂⣨⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⢹⡇
⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣧⡑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣧⣤⣥⣤⣄⣐⣀⣀⣀⣒⣀⣐⣀⣀⣁⣀⣈⣤⣬⣥⡴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⠍⠲⡲⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡭⣿⣝⢟⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡠⠤⣄⡀⢸⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢟⣛⢿⢿⡧⡇⣿⣿⢸⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸⣿⣿⣿⣿⣿⣿⡏⢹⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣷⣿⡇⡏⠀⠀⠀⠀⠀⠀⢰⡞⣿⣿⣿⣿
⠀⠀⠀⢰⣶⣾⣿⣶⣾⣿⣿⢹⣿⣿⣿⣿⣷⠘⡀⠀⠀⠀⢸⣿⣿⣿⡏⠉⠁⠁⠉⠁⠁⠀⡆⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿⢹⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⠀⢣⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣾⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡆⢸⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⠿⠿⠿⠿⠧⠞⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣀⣀⣀⣛⣯⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⡡⠤⠐⠒⠀⠠⠤⣍⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⣠⣋⣀⡀⣀⠀⠀⠀⠀⠀⠙⡄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡟⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡠⠊⡼⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⢠⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⡆⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠋⠀⠀⡌⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⢆⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⡠⠊⢁⡠⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠲⢄⣹⡛⣻⣒⣒⣊⡡⠤⠒⠉⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠤⠒⠒⠒⠂⠤⠭⢉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⡀⠀⠀⠀⠀⠀⢸⠀⠀⡰⢂⡲⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⢸⠀⢰⢱⠁⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢸⠀⣌⡌⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⡟⠁⠀⠈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠱⣄⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⢸⠀⡇⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣑⠢⠤⣉⣒⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⢀⡆⢸⠀⢸⠀⡇⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⠬⠭⢑⣒⡠⠤⠬⢉⣑⡒⠒⠒⠉⣠⠏⠀⠘⡀⢱⡀⠀⡠⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠒⠒⠩⠭⢍⣄⣀⡀⠀⢣⠀⠈⢉⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠁

⠀⠀⠀⠀⠛⠛⠛⠓⠂
⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠈
⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠊⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣄
⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢧⠴⠋⠀⠀⠐⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠲⢦⢤⣄⣀⠀⣸⠁
⢠⠴⠒⠒⠒⠒⠚⠛⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⣆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⢄
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂
⡇⠀⠀⠀⡟⠆⠀⠤⠤⠀⠀⠐⠁⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⡇⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠂⠤⠤⢤⣀⠀⢀⠀⠀⢀
⡇⠀⠀⢠⣧⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠁
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⡠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⡏⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡠
⢀⠀⠀⢸⡟⠦⠤⣒⣒⣨⡥⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠔⠀⠀⠀⠀⠀⠉⠒⠲⠤⠔⠁⠸⣀⠀⠄⠀⠀⠀⡆⢧
⠀⠀⠀⠀⠙⠿⠛⠉⠉⠉⠀⣠⡤⠶⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠈⠉⠒⠒⠲⠾⣅⡼
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠿⠴⠟⢶⠗⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⢉⣉⣉⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢤⢄⠀⡀⢢⣀⣠⣿⠤⠤⠤⠒⢳⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢳⣡⠸⡀⠀⠻⠀⠀⠀⠐⠂⠈⠀⠈⡇⠀⠀⠀⢰⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⣇⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠒⠀⠁⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢀⡴⠛⢉⡻⣟⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠞⠛⠛⣯⠉⠉⠳⣏⠀⣴⠋⠀⠸⡆⠀⠸⣆
⠀⠀⠀⠀⠀⠀⢀⡴⠞⣩⠞⠉⠀⠀⠀⠀⢸⣆⣤⣤⡽⠶⠷⠶⠖⠒⠓⠒⠲⠾⢦⣄⣀
⠀⠀⠀⠀⢀⡴⠋⢀⡼⠃⠀⠀⢀⣤⠶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⢠⠞⠀⣠⠟⠀⠀⠀⣴⣋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⣰⠏⠀⣸⠏⠀⠀⠀⣼⡟⠉⠁⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⢠⡏⠀⣰⡏⠀⠀⠀⣸⡏⠀⠄⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⡾⠀⠀⣿⠀⠀⠀⢀⡿⠀⠈⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢰⡇⠀⣸⡇⠀⠀⠀⢸⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⣧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠀⠀⣿⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡏⠀⠀⣿⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣇
⡇⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠠⡀⠀⠀⠀⢠⠞⠋⠙⣧⠀⠀⠀⠀⣼⠃⢻⡄
⣷⠀⠀⢹⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠹⣄⠀⠀⠁⠀⠀⢸⡄⠀⠀⢹⠚⠛⠛⠋⠁⠀⠈⣧
⣿⠀⠀⢸⡆⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⠦⠤⠴⠞⠛⣷⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢹⡄
⢸⡀⠀⠸⣇⠀⢀⡟⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠛⠛⠀⢀⣀⣤⡤⠶⢤⡈⣧
⠘⡇⠀⠀⣿⠀⢸⠇⠈⠇⠈⠉⠉⠉⠁⠀⠀⠀⢸⠃⣿⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⡴⠶⠛⠋⠉⠀⠀⠀⠀⢳⢹
⠀⢻⠀⠀⢻⡆⣾⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣿⠀⠀⢹⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡇
⠀⠘⣇⠀⢘⣷⡇⠸⣶⠶⠶⠶⠦⢤⣤⣄⣀⣀⢀⡷⣿⠀⠀⢸⣇⠀⠀⠀⠀⠄⠀⢀⣀⣈⣤⣤⣤⡤⠶⠺⡇⠈⡇
⠀⠀⢻⡄⠀⢸⠃⠀⡟⠀⠀⠀⠀⠀⠀⠈⠉⢹⢻⠃⣿⠀⠀⣾⢧⣀⣤⡤⠶⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⢻⡀⡇
⠀⠀⠀⣧⠀⡿⠀⢠⡇⢀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⣿⠀⠀⣿⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢧⡇
⠀⠀⠀⠘⣦⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠀⣿⠀⣸⠃⠀⠀⢀⣀⣠⣤⡤⠶⠶⠶⠚⠛⠋⣡⡞
⠀⠀⠀⠀⢸⡇⠀⠘⢷⣄⣀⣀⡀⠀⠀⠀⢀⡿⢸⡇⡟⠀⠀⠉⠓⠿⠶⠶⠛⠋⠉⠀⠀⠀⣀⣀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠓⠋⣸⠇⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⣤⣤⣄⣀⣀⣠⣴⣯⣤⡤⠶⠖⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣤⣤⣠⣴⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣍⣿⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠻⣦⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠻⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠉⠛⠛⠳⠶⠶⢦⣤⣤⣤⣤⣀⣀⠀⠀⣼⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠟⢷⣄⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢰⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠇⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠁⠈⠉⠛⠛⠳⠶⠶⢶⣤⣤⣤⣤⡀⠀⠀⣿⠀⢠⡿⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡟⠀⢸⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⡿⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢸⡇⠀⠀⢠⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸⣧⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣼⠃⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⣦⣤⣄⣸⣇⣀⣀⣿⣤⣤⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⣂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡡⠊⠁⠀⠉⢆⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠈⠉⠉⠳⣍⠉⠒⣄⠀⠀⢸⠀⡇⠀⠀⠀⠀⢨⠀⠇⠀⠀⢀⠔⠚⣩⢝⡉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⣠⠖⠉⠀⠈⠆⠀⠘⡆⠀⢀⣃⣧⣀⣀⣀⣀⡎⡜⠀⢀⡔⠁⠀⡔⠁⠀⠈⢢⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⠜⠁⠀⠀⠀⠀⢸⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⢎⡀⠀⡜⠀⠀⠀⠀⠀⢱⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⠊⠀⠀⠀⠀⣠⠚⠁⠀⠀⢀⡠⠤⠐⠒⠒⠂⠤⣀⡀⠀⠀⠀⠉⢲⡁⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰
⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⠎⠀⠀⠀⠀⡔⠁⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠸⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡆⠀⠀⠀⡜⠀⠀⠀⠀⡜⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡆⠀⠀⠃
⠀⠀⠀⠀⢠⠁⠀⠀⢠⠁⠀⠀⠀⢰⠇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠸⡆⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠸⠀⠀⠀⡌⠀⠀⠀⠀⡞⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢻⠀⠀⠀⠀⠀⢡⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⢠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡆⠀⠀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡌⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢲⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⢠⠁⠀⠀⣀⠠⠤⠤⠤⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠄⢀⣀⡀⠀⠀⡆⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⡇⠀⢀⠴⠾⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠗⠒⡄⠀⡇⠀⠀⠸
⠀⠀⠀⠀⡇⠀⠀⡇⠀⡜⠀⡆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⠀⠁⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠃⠀⠀⠃⠀⡇⠀⡃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠄⢰⠀⠀⢰⠁
⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠁⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡇⠀⢸⠀⠇⡘⠀⠀⡸
⠀⠀⠀⠀⠈⡄⠀⠘⡄⡇⠀⠀⠀⢸⠀⠉⠉⠈⠉⠉⠁⠀⠀⠐⡖⠒⠒⢐⡆⠈⠉⠉⠉⠉⠉⠉⠀⠀⠃⠀⢸⠀⣷⠁⠀⢠⠃
⠀⠀⠀⠀⠀⢃⠀⠀⠱⡇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⣠⠃⠀⢀⠏
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠱⣄⡇⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⢃⠀⠀⠀⠉⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠤⠄⠒⠊⠁⠀⠀⡦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⣓⠲⢄⠈⠑⢤⣀⣀⣠⢤⣤⡴⢶⡶⠖⠲⣶⡦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡴⠊⠹⠀⠉⣳⠵⠚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⢙⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⢀⡷⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⢢
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠛⠒⠂⠤⣘⡴⢋⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇
⠀⠀⠀⠀⠀⣠⡫⠒⡖⠀⠀⠐⠒⠢⡬⠔⠒⠂⠛⠛⠒⠢⠭⣀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡄
⠀⠀⠀⠀⠠⡝⠀⡸⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧
⠀⠀⠀⢀⠃⠁⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢖⣦⠀⣀⡟
⠀⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡡⠀⠀⠀⠀⢴⣛⠽⢻⣧⠖⢒⠁⡨⢿
⠀⠀⠀⠰⠘⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡴⠒⠒⠛⠛⠲⡄⠀⠀⠀⠀⠀⡙⠦⣴⣒⣀⣠⣤⣄⣼⡽⣷⠛⠉⠀⠈⡇
⠀⠀⠀⠀⢁⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⡘⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⢠⠀⢰⠀⠀⠀⠀⠀⡇⠀⠀⢇⠀⠀⢶⠄⠀⠀⡿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⡤⠔⠒⠒⢶⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⡇⠘⡆⠀⠀⠀⠀⡇⠀⠀⠈⠓⠒⠂⠀⠠⢺⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⡁⠀⠀⠀⠀⣼⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢣⢹⠀⠱⡀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⠒⠒⠒⠉⢹⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⡈⡆⠀⠱⡀⠀⠀⢻⡀⠀⠀⠈⠙⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢀⡼⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⢣⢳⡀⠀⠘⢄⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠣⡑⢄⡀⠀⠉⠒⠒⣳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⠉⠉⠙⠛⠒⠉⠙⠲⠤⣀⣀⣀⠀⡀⢀⣀⡼⠥⠤⠄⠐⠒⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⢀⣴⠒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄
⠀⠀⠀⠀⠀⡀⠤⠥⠤⠬⣙⡆⠀⠀⠀⠀⠀⠀⣾⡇⢀⠩⠐⠶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠞⠉⠰⡄
⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⣰⡿⠀⠃⢀⡴⠒⣈⣉⢲⠀⠀⠀⠀⢠⡇⢀⠀⠤⠀⣠⣽⣷
⠀⠀⠀⣠⣄⣀⣀⣀⠀⠀⠀⢦⣇⠀⠀⠀⢰⠉⠀⢠⠀⣸⠀⠀⢈⣃⣸⠀⠀⠀⠀⢸⠀⢰⠖⠒⠐⠂⡟⢿⡄
⠀⠀⠀⢩⡖⠀⡠⠠⠄⠀⠀⠈⡟⡀⠀⠀⣿⣀⠈⠛⠂⡇⢠⠀⢠⠂⢀⢴⠀⠀⠀⠛⢠⢀⡄⠀⢠⠀⡅⠀⣿
⠀⠀⣏⣸⡇⠀⠑⠄⡀⠄⠀⠀⣇⠁⠀⠀⠉⠛⠀⠠⡆⠀⣄⡶⠂⠀⠙⢺⠀⠀⠀⠀⢀⢂⡀⠀⠠⠄⣃⣨⡽
⠀⠀⠈⠈⠃⠠⠀⠠⠀⠀⠒⠂⠉⠀⠀⠀⠀⠈⠓⠤⠇⣄⣏⠀⢈⡤⠴⠊⠀⠀⠀⠙⠉⠈⠀⠤⠤⠬⠥⠞⠁

⠀⠀⠀⠀⠀⣀⡤⠶⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣒⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣀⡀
⠀⠀⠀⢠⠏⢩⠖⠒⠲⣿⠿⣦⠀⠀⠀⠀⠀⢠⠎⠕⣒⣒⣒⠫⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⢸⠀⡇⠀⣀⠠⠉⡀⠬⣇⠀⠀⠀⠀⡘⢘⡏⠀⠀⠀⡿⡕⡇⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠨⡇⡇⡿⠄⠀⠀⠀⠀⡆⠀⠀⠀⠀⣧⡞⠁⠀⠀⠀⠀⢹⢿⠀⠀⠀⠀⢀⡿⠃⠀⠀⢸⢸⢀
⠀⠀⠀⢀⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠀⡀⠀⠀⠀⢠⡈⠺⡄⠀⠀⠀⢸⡇⠀⠀⠀⣸⢼⠈
⠀⠀⠀⠺⢻⡧⣇⠀⠀⠀⠀⠀⢠⠀⠀⠀⡼⠂⠐⠇⠀⠀⠀⠈⡇⠀⢱⠀⠀⠀⠸⡇⠀⠀⠀⠉⢸⠀⠠⠔⣎
⠀⠀⠀⠀⠀⢸⠸⠟⢀⣤⣤⢷⣾⠀⠀⠀⠇⡀⠀⠀⠀⠀⠠⠜⠁⠀⠸⠂⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⢀⡠⠛
⠀⠀⠀⠣⠤⢾⠀⠀⠀⠩⠛⠋⠁⠀⠀⠀⠘⠓⠶⠶⠤⠴⠶⠶⠖⠛⠀⠀⠀⠀⠀⠚⠁⠀⠠⠷⠒⠊⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⣿⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⣰⣭⣜⣦⠀⠀⠀⠀⢀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠻⡀⠰⠠⠀⠀⠀⠀⠐⠊⠄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠀⡉⢉⠉⠁⠘⠂⠶⡀⢀⠊⠀⠀⠈⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠕⠀⠀⠀⠂⠠⢀⠀⠀⠀⢀⠀⠈⠢⡀⠀⠀⠀⠠⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⡄⠀⠘⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠁⠀⠁⢻⠄⠀⠐⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠊⠉⠀⠓⠐⠠⠄⠠⠀⢀⠈⠗⠀⠠⠀⠈⢁⠀⠀⢁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⡀⠀⠀⠇⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡎⠰⡆⠁⠀⠀⠰⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢙⡇⠈⠀⠠⠉⠀⠁⡀⠀⢀⠊⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠁⢸⡀⠈⢀⠏
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡐⠂⠊⠀⠀⢠⣖⠀⠀⠀⢸⣷⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡔⠂⠀⣐⢫⡘⢣⠈⠹⡿⡘⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⣀⣀⣀⠀⠀⠔⠀⠀⠀⠀⠘⢫⡰⠂⣐⢓⢷⡁
⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣝⡠⠒⠠⢍⡽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢮⡷⢒⡤⢮⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⡠⠀⠀⠉⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠛⢛⣛⡒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⢋⣤⠔⠋⠉⠉⠙⢿⣧⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡃⢠⠞⠁⠀⠀⠀⠀⠀⠀⢻⣿⣆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣼⣿⣇⢹⣶⣶⣶⡶⠶⠶⠶⠶⢾⡿⢷⢡⡟⠓⠲⠢⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡍⠉⠉⠉⠉⡩⣽⢸⡆⡄⠀⠀⠀⠀⠀⠀⣿⢉⢣⡟⢡⠀⠀⠀⢳⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⣇⠀⠀⠀⠀⢹⣌⠉⡡⠞⠀⠀⠀⠀⠀⠀⢸⡟⢂⣚⡁⠀⠀⠀⠈⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣄⣼⡀⠀⠀⠀⠀⡇⣭⠤⡌⠀⠀⠀⠀⠀⠀⠈⣇⣇⠀⡁⠀⠀⠀⠀⢹⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣿⣿⡇⠀⠀⠀⠀⣧⠆⠀⡀⡇⠀⠀⠀⠀⠀⠀⢻⣭⣴⠗⠀⠀⠀⠀⠀⢧⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⣾⠟⢿⣿⠀⠀⠀⠀⢹⣴⢾⡔⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡆⠀⠀⠀⠀⠀⠘⣇⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡠⢤⡿⠃⠸⣿⡀⠀⠀⠀⢸⡆⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢻⠖⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⣿⠁⡀⠀⠙⡇⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠔⠈⢸⣦⡇⣷⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⡆⠀⡌⠀⠀⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢰⠃⡜⠀⠀⠀⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣼⢸⠁⢀⣀⣠⢻⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⢰⣿⣿⣿⣿⡿⠃⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣶⣽⣿⡇⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣿⣷⣿⣷⡷⠷⠶⢿⣿⠟⠁⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⢻⡇⠘⣝⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠀⠈⠉⠉⠉⠉⠈⠉⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠘⣧⠀⠙⠋⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢉⣡⠤⢔⣲⠒⠀⠲⣶⣦⣤⣤⣀⣀⣀⣀⣀⣠⣴⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠙⣷⣀⠀⠈⠀⠈⠉⠉⠛⠛⠫⠉⠀⢂⣠⠒⣉⠴⠚⠉⠁⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣀⣀⣀⣀⣀⣄⢄⣠⡽⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠟⠛⠛⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣫⡵⠶⠚⠛⠛⠛⠲⢭⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠵⠾⠷⠶⠦⠤⠤⠤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠓⣶⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⡶⠲⠶⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣾⣅⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢼⣥⣤⣀⣰⠃⠀⠀⣀⣀⡀⠀⠀⢠⡟⡼⠁⣽⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⢸⡇⠄⡟⠁⠀⢀⡼⠉⠉⠉⡿⢀⡾⣼⣧⣼⢳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⠿⠿⠷⣴⣶⣾⠣⢰⣧⣀⣀⣘⣓⣶⣶⣾⣡⣾⠃⠙⠾⢣⡟
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⣞⣿⣦⣾⡏⠉⠛⠛⠛⠛⠛⠛⣹⠃⠀⠀⢀⡞
⠀⠄⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣠⠏⠝⡿⠋⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡾⠁
⠀⠂⠀⠀⠀⠀⡾⠀⠀⠀⠀⢠⡾⠁⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠐⣯⣝⣒⢦⣄⣀⡙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣍⡉⠛⠓⠿⠿⠯⠭⠭⣿⣷⣶⣶⣶⠾⠁⣀⡼⠃⣰⠏
⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⢤⣄⣀⣀⣀⣀⣐⣓⣛⣛⣉⣤⠾⠃
⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⡤⢦⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⢦⣴⡁⣶⠋⠀⠀⠈⣿⠆⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣴⠟⠻⢷⣌⠹⡇⠀⠀⠀⠀⠘⣦⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⣤⠟⠁⠀⠀⡈⢸⣇⠹⡀⠀⠀⠀⠀⢻⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠦⣷⣶⢤⣤⣤⣧⣸⣾⠈⣇⠀⠀⠀⠀⢸⡣⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⡇⡐⣞⣟⣿⡎⣟⣿⣿⣿⣻⡆⢿⣤⣤⣀⡀⠀⡷⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡯⣽⣙⣛⢿⠡⣝⣿⣿⡾⣿⢹⣟⣿⣿⣿⡇⢸⡿⣿⡿⣿⣽⣷⣤⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠚⢁⡨⠟⡋⠀⢬⣿⣿⡏⠿⢿⣿⣷⣿⡟⡁⢸⣿⣯⣿⣿⢿⣎⣿⣿⣦⣯⣽⡷⣦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠔⠻⣀⠴⡇⠀⣟⠀⣰⡜⠙⢻⡶⣍⣙⡛⡇⢸⣷⣿⣿⣿⣯⡿⣷⣟⣿⣿⣿⡿⣯⡽⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢙⡄⠚⠙⣰⣆⢈⣿⠊⠁⣰⡔⠉⢃⣀⠌⠻⡇⢸⣿⣹⣛⠻⢯⣿⡏⢽⣘⠧⠿⠓⠉⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠠⡀⡀⠜⢯⣼⣇⠶⡋⠀⢈⡴⠊⠁⠀⢀⡇⠰⠙⢉⣩⠛⠾⠁⠀⠀⠀⠀⠀⢀⠊⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣦⠔⠻⡄⣠⠔⢏⠀⢀⡵⡞⠁⠘⣆⠀⠛⢴⡟⢲⡤⠈⠑⣄⠶⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⠔⠛⡄⢀⡴⢿⡁⠀⣨⡤⠊⠉⠃⣀⠼⢿⠉⢙⣦⠊⠁⣻⠀⢰⠏⠀⠀⠀⢸⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢇⠀⣠⡴⠚⢇⠀⣀⠷⡎⠁⠈⣤⠔⠊⠱⣀⡤⠻⡁⠈⣳⡞⠋⠀⠠⠃⠀⠀⠀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠜⢯⠀⢠⡤⠜⢫⡀⠀⡈⢖⠋⠀⣡⡤⠚⠉⠀⢀⡸⢾⡁⣈⠦⠀⢀⠄⠀⠀⠀⠀⢃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣦⠄⢞⠁⠈⣀⠤⠻⠉⠀⣀⢷⡈⠁⠘⣄⡀⠘⠫⣀⡤⠛⢇⣀⠀⢂⣾⠀⠀⠀⠀⣎⠂⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣈⣧⡤⠻⠋⠈⣦⡀⠘⢏⠀⢀⡱⡖⠉⠀⢁⣤⠂⠙⣄⡤⠊⢙⠀⢈⢡⠂⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣱⣤⠞⢏⠀⠱⣄⠴⠿⡁⠀⣈⢶⠊⠁⢈⢶⡊⠉⠱⣄⠛⠁⠀⢃⠀⠀⠀⠀⣴⢂⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢷⣄⠈⣦⡤⠛⢏⠀⣠⠳⡎⠁⣀⠽⢎⣁⡼⠳⣆⠀⠈⡄⠀⠀⠀⠁⠊⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠴⢎⡛⣄⡠⠞⢛⣀⠤⠊⢙⣤⡖⠙⠀⠀⢠⠂⠀⠀⠀⠀⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠶⣏⣁⡶⠚⠁⣀⠽⢮⣇⠀⠀⠀⣀⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠦⠤⠤⠿⠖⠂⠈




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⢶⣶⣶⣶⡶⠶⣤⣄⡀⠀⠀⣴⢻⡀⠀⠈⠉⠉⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⡾⠋⠉⠉⠙⢿⣦⠀⠈⠙⢦⣀⣿⣈⣧⣀⣀⣀⣀⠀⢸⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠹⣧⢀⡤⠚⠉⠁⠀⠀⠀⠀⠉⠉⠉⣉⣉⣩⣭⣍⣻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⡠⠔⣚⡭⠵⠒⠋⠉⠁⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠁⣾⠁⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⢀⠔⢊⡡⠖⠋⠁⠀⠀⠀⠀⢀⣠⠴⠒⠒⠉⠉⠉⠉⠙⠓⠶⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⣠⠞⣡⠔⠉⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⡴⠁⡴⠁⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣷⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⡼⠁⡼⠁⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣿⠀⠀⠀⢰⡏⠀⠀⠀⠀⢰⠁⡸⠁⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⣾⠁⠀⠀⠀⠀⡏⢰⠃⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⢰⡏⠀⠀⠀⠀⢰⠁⡏⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣠⡄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⣾⠃⠀⠀⠀⠀⡌⢰⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠾⠭⠭⠭⠭⠽⠶⠿⠟⠛⠛⠛⠛⠒⠛⢻⢿⠃⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠃⡸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡇⢰⡏⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡜⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣾⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⡿⠀⡇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⣧⠸⡇⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠙⠷⣇⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠤⠴⣶⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣧⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠈⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣤⠾⢤⣄⣀⠀⢀⣀⣠⣤⠤⠤⢤⣤⣤⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠚⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠀⢾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠚⠛⠻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠈⠉⢃
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⣠⡤⡼⠈⠳⣄
⢠⡙⠦⠈⠉⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠇⣠⠞⣍⡺⠿⠃⠀⠈⢳⡀
⠀⠙⠂⠀⠄⡀⠘⠙⣦⣀⠀⠀⠀⠀⠀⠀⢠⡟⣿⣾⡽⠛⠀⠀⠀⠙⠄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⢤⡀⠀⠀⠀⣾⡿⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠙⠦⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⠀⠀⠀⠀⡆⢰⠐⠀⠀⠀⠀⠀⢀⣠⠐⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣀⣠⠤⣀⠨⠟⡆⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠤⢴⣏⣛⠻⢆⡻⡟⣹⠷⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠿⢲⢿⠖⢿⡧⠿⡆⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣧⢲⣄⣠⣷⣄⢽⣄⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣄⢈⣭⡙⣻⡃⠀⣩⡆⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡝⣏⡙⣿⢟⣻⣿⣇⠫⠛⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡰⠂⠃⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠜⠀⡮⠀⠂⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠇⢸⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⣀⣤⠴⢶⣿⢻⣽⣽⣿⠻⣿⣋⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡤⣟⠈⢉⣰⣶⡏⡉⢋⠀⢩⣄⢛⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡠⠀⢀⡞⡁⠀⠁⠈⡆⢠⣶⡄⢿⣟⣿⣯⣾⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠜⠁⠴⠋⠀⠀⠀⠀⠀⣾⠉⠋⠀⠙⣿⣿⣿⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠁⢸⠃⠀⠀⠀⠀⣿⣷⡟⠀⠀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣤⠴⢤⣀⠀⠀⠀⠀⠀⠀⠄⠊⠂⠋⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠛⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢒⣊⣉⣍⣍⣭⣭⣙⣒⡰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡛⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢨⣀⣸⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠽⠋⠋⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⡈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠒⠐⠀⡀⢀⠀⠀⠂⢀⠔⠂⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠈⠀⠀⠠⠁⠀⢹⣿⣟⣽⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠰⠁⠀⠀⠔⠁⠄⠠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢼⢃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠂⠀⠘⡀⠀⢰⣽⠘⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢐⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄⠀⠊⠀⠀⠀⠀⠀⠀⠠⠀⠸⠀⠀⢸⣿⢈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠄⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⡷⠊
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⡇⢪⡀
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣟⠃⡏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⡶⠶⠲⠲⠶⠖⠖⠖⢿⠄⠀⢿⡶⡲⠆⠆⠠⠄⠦⠄⠄⠦⠦⠤⠤⢤⡴⢿⠭⠀⡇⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⠈⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⡀⣸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⢁⢸⡋⠀⣗⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣆⠀⡇⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣀⠄⠀⠠⢸⡵⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢸⠖⠀⡇⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠨⠈⡧⠀⡇⡔⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠟⠙⠛⠛⠃⠀⠀⠸⡧⠀⣿⢸⣿⡄⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢍⡀⠀⠀⡁⠠⣿⢸⣿⠅⣿⣿⣿⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡯⠀⣿⣃⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠈⠉⠁⠀⠒⠂⠠⠤⠤⠄⣀⡀⣀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⣻⡦⢯⠿⠙⣀
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡁⠐⠒⠠⠆⠀⠤⠀⡤⡤⠄⠀⠄⠈⠅⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢠⣾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠧⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⢉⠙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠄⠀⠀⠀⢀⠈⠛⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠀⠀⠀⡄⠳⣤⡀
⠖⢦⡀⠀⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⣠⣾⣛⣻⣠⠀⠈⠻⣆
⠸⣆⠙⠓⠀⠀⠈⠙⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢛⠃⣠⣞⡱⣦⡉⠿⢋⠀⠀⠀⠈⢧
⠀⠈⠗⠀⠀⠄⠀⠀⠛⠉⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⣿⣾⣡⠟⠉⠀⠀⠀⠀⠱⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠊⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⣸⣛⡿⠿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢀⣿⠟⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠈⠛⠒⢿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⡇⠸⡀⠃⠀⠀⠀⠀⠀⠀⢀⣤⠐⠁⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠤⡂⣈⠠⡻⢧⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢀⣠⣴⠺⠿⢿⣦⠾⢯⡶⠲⣿⡀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢻⣷⣷⡉⣴⣝⣨⣧⡙⢤⣅⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⢓⡠⠋⡟⠉⠿⣿⠙⡿⡄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⣷⢜⣷⣈⣠⣿⣦⡤⣽⣄⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡁⠀⢉⡁⡝⢻⡟⠠⠈⢉⣆⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⣿⠲⣿⣷⡶⢿⣿⣶⠠⢟⡽⠄⠂⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣷⠤⠬⣧⡴⠖⠛⠛⠈⠁⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠃⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡴⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠞⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⠀⢸⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡌⠀⢸⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣄⣤⣶⣾⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⣿⠉⢯⣹⣽⣿⡟⢻⡿⢏⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢉⠛⠃⣰⣤⡝⠛⠫⡏⠈⠙⣃⠀⠿⣿⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣼⠉⠙⠂⠀⢻⡉⠛⣃⠐⢠⣾⣤⣴⡿⠇⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣇⠀⠀⠀⠀⠀⡰⠁⠀⣠⠏⠸⠀⠀⠀⠀⠸⣆⣩⣿⡿⢋⣿⣥⣾⣷⡾⢿⣿⣽⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠜⠁⠀⠞⠁⠀⠀⠀⠀⠄⠀⢸⠿⠁⠉⠀⠀⠙⣿⣿⢿⡇⠀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣼⠇⠀⠀⢻⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣀⣈⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡠⠒⠤⠼⠃⠀⠀⠀⠀⠀⠾⠟⠛⠁⠀⠀⠀⠸⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠟⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠅⠀⢀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠃⠂⢀⣿⣿⣿⣷⣄⠀⠘⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠾⠿⠿⠿⢿⣟⣛⡓⠚⠿⠭⠭⣭⣽⡀⠀⡟⠙⠛⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠉⠙⠳⢤⣀⠀⠀⠈⠙⢷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠠⠤⠈⠙⢦⡀⠀⠀⠀⠀⠉⢳⡀⠀⠀⠈⢷⠀⠍⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢠⡤⠤⠶⠶⠶⠾⠛⠋⠁⠉⠑⠒⠒⠒⠀⠈⠉⠃⠀⠀⠙⢦⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⢇⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣿⣟⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠠⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣿⣯⠅⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠤⠼⢙⡛⠲⠖⣤⠄⣀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣿⣟⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠀⠈⠁⠊⢷⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠀⣿⣯⠅⠠⠀⠀⠀⡤⢤⠠⠄⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⣇⠀⠀⠀⠀⡤⢀⠀⠀⣿⡇⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠀⠀⠀⣯⣟⠂⠀⠀⠀⢀⡀⢈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⠜⠀⠐⠉⡂⠾⣿⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠛⣯⠅⠀⠀⢀⠉⠀⠀⢉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠊⠀⠀⠀⠀⠀⢠⣿⡆⠘⣷
⠀⠀⠀⡈⢐⠤⡀⢸⣿⠀⢸⠀⠔⠑⠀⢿⠂⠀⡐⠉⠀⡀⠉⠉⠁⡄⠀⠰⢀⠀⣰⠉⡍⠐⢲⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠁⠠⠀⠠⠀⣿⣇⠀⢿⡄
⠀⠀⠀⡆⠀⢀⠐⢿⠻⠀⠀⡌⠁⠀⠀⣸⡅⠀⡇⠀⠀⠀⠐⠀⠀⡇⠀⡆⡠⠦⠬⡤⠇⠉⠑⠠⡀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡐⠁⠀⢹⣿⠀⠸⣧
⢀⡴⢄⠱⠀⠁⠀⠈⢠⠀⠀⠁⠀⠀⢠⠋⠳⡀⡇⡾⢘⡄⠀⠀⠀⡇⢰⡇⠀⠀⢀⠀⢀⠀⠀⠀⠐⠀⠀⡜⠁⠀⠀⠀⢀⢀⠜⠀⠀⠀⢸⣿⡄⠀⣿
⠋⢄⠀⠉⢣⡀⠀⠀⠈⢃⡠⠀⠀⠀⢨⡇⠀⢣⡄⠅⠀⠇⠀⠀⠀⣃⠻⠄⠀⠀⠈⠀⠀⠀⠀⢀⠆⠀⡞⠀⢀⠴⣂⡠⣀⠆⠀⠀⠀⠀⠈⣿⡇⠀⢸
⠀⠀⠑⢠⣊⣈⣒⣄⣀⣀⣡⣀⣀⣄⣤⣥⣤⠾⠷⠶⠶⠶⠶⠖⠒⠓⠚⠚⠒⠶⠶⠶⠦⢤⣤⣼⣰⡆⡅⠀⣸⠁⠀⡡⠈⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⢀⣾⠿⠛⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠙⢳⣄⠈⠢⣤⡀⠀⠀⠀⠀⠀⠄⠒⠻⡇
⠀⢠⡿⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠉⢳⡔⠀⣇⡼⠢⡤⠂⡁⠀⠀⢸⡀⢀
⠀⣾⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⡞⠁⢡⢮⠔⠁⠈⠑⠒⢺⠀⠀⢸
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⠁⠀⠀⠀⠀⢻⣿⡇⠘⠀⠀⠀⠀⠀⠘⡆⢀⣾
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡟⠀⠀⠀⠀⠀⠈⣿⢠⢠⠀⠀⠀⠀⠀⠀⣧⡾⠃
⠘⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣸⠃⠀⠀⠀⠀⠀⠀⣿⠈⢠⠀⠀⠀⠀⠀⠀⣿
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠀⠀⠀⠀⠀⢠⣿
⠀⠸⣷⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠹⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠄⢀⣾⠋
⠀⠀⠀⠹⣧⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢠⣿⡾⠟⠁
⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣦⣤⣤⣤⣄⣤⣤⣤⣤⣴⣶⣶⣏⣩⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠾⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠠⠤⠤⠒⠶⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⠤⠤⠤⠤⠤⠤⣄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠀⠀⠀⠀⠒⠒⠒⠛⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠚⠒⠒⠒⠀⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡇
⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⡔⠐⠒⠀⠒⠒⠒⠒⡄⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⡱⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠰⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣰⣤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡠⠚⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠺⠃⠀⢀⡠⠖⠉⢇⣐⡁⢠⡁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠘⢧⡀⠀⢠⠋⠀⠨⢷⠋⠀⠀⢠⠇⠀⠀⣀⣀⣠⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢄⡀⠀⠀⠙⣄⢸⡀⠀⠀⢀⡦⠤⠴⠟⠒⠛⠉⠉⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⠀⠘⣆⠑⠦⠔⠋⠁⠀⠀⠒⠦⠤⠤⠴⢲⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣀⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡞⠵⠖⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⠀⠀⠙⠂⠐⠊⠁⠐⡀⠀⠀⠀⠠⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣈⠤⠂⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣸⠃⠀⠈⠂⢤⣀⠀⣠⣾⣽⡈⠁⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠓⠛⠀⠈⠉⠁⡗⠑⢇⢠⡎⠀⠀⢀⡤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠤⠈⡟⢾⠀⣎⡿⠐⡟⣉⢗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡅⠈⡃⡘⢣⠀⠭⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⢺⢁⣾⡎⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢱⣠⠺⢿⠸⠁⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⠬⡏⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣈⠀⠀⠒⠞⠉⠑⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢽⣋⠭⢁⣀⣒⣲⡒⠒⢛⣯⣉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠁⠀⠀⠀⠠⢤⣶⣿⣿⡉⠙⠛⠻⢿⡀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠉⠀⠐⠀⠀⢸⡿⢿⡏⠁⠀⠀⠀⢸⣧⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋⠀⠀⢀⣴⣾⣿⣷⣀⣴⡷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠠⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡟⣿⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠙⣻⠿⠿⣦⣄⠀⠉⡀⠀⠀⠀⠀⡬⠿⠛⢿⠿⠿⠂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⢶⣥⣄⡛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠷⠾⠷⠀⠀⠀⠀⠀⢀⡀⠀⢠⡴⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⣿⡿⣯⡤⠆⠀⠀⠀⠀⠐⠃⢀⠀⠀⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡼⣿⣽⣿⢻⣶⣄⣄⣀⠀⡀⠀⠀⢨⡄⠀⢴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠟⢃⠀⠙⠿⣿⣿⡏⠛⠿⣶⣥⣤⣠⣀⣤⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⢙⠛⠉⠰⢀⡀⡀⢠⠀⠵⠞⠀⠁⠀⠀⡆⠈⠙⠻⣿⣶⣄⣈⣿⣿⣿⠿⣿⣧
⠀⠀⠀⠀⣠⣶⣿⠃⢠⣤⣤⣤⣦⣆⣀⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠛⠻⠿⣿⣶⠏⠁
⠀⠀⠀⢀⣿⣇⢴⣲⡿⠛⠋⠉⠛⠣⢌⡡⡙⠿⠀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⢾⣧
⠀⠀⠀⠸⣿⣇⣸⠟⠀⠀⠀⠀⠀⠀⢈⣷⠉⢤⡸⡇⠀⠀⠀⠀⠀⠠⣄⠐⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠙⠛⠀⢀⣀⣀⣀⠀⠀⠀⣿⣦⠺⣇⠀⠀⠀⠙⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⣶⣷⣾⣿
⠀⠀⠀⠀⠀⣀⡖⠈⠃⠀⠀⠈⠉⠁⠀⠈⠙⠶⠳⠦⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⣟⣈⣟⢹⣿
⠀⠀⠀⣤⡾⠉⠀⠀⣀⣤⣤⣤⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣢⠦⠛⠛⢁⡟⠓⠋⠸⢻
⠀⢠⣾⣯⣀⠀⣠⠞⠁⠀⠀⠀⠸⠶⠤⢤⣀⠐⠂⠠⠆⠀⠀⠀⠘⠁⣤⢤⡄⢠⠄⠋⠀⠀⠀⠀⠘⡇
⠀⢺⣿⣿⠏⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣦⣀⣀⣀⡤⣬⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠻⡆
⠀⠀⠻⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⠛⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡶⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣥⣧⡄⠀⠒⢲⡆⣠⡤⢴⡆⢀⠻⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⣦⣅⣶⠆⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠗⣸⣇⢄⡀⠀⢀⡴⠀⠀⠀⢀⣀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⣼⢿⣿⣙⣿⣿⣿⣿⣿⣌⣧⣀⠀⠀⠀⠀⠀⠀⠉⠲⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠰⢗⣿⣿⣿⣿⣻⣿⡿⠿⣯⣠⣄⣀⣀⠔⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⡷⢞⣽⣟⣽⣿⣿⣿⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣡⣾⢋⣿⣿⣿⣿⣯⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠐⢤⠀⠀⠈⣷
⠀⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠒⠒⠲⠒⠂⠰⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠰⣆⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡋⠀⠀⣠⣄⣀⡀⣀⢀⠀⡰⠔⠃⠀⠀⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣯⡛⢿⣿⠖⢶⣞⣻⣿⣻⠆⠀⠀⠀⠘⠉⠀⠀⠐⠉⠀⠐⠀⠀⠀⣠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢁⣤⣼⣷⣾⣿⣭⣾⣟⣿⣶⣾⣷⣤⣠⣄⣀⡀⠠⢀⣀⣀⣴⡞⠙⡛⣦⣄⣀⣀⣀⣀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠛⢉⣀⡀⢈⣉⣉⢉⣻⣿⣿⡿⢿⣿⣹⡛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣹⣟⣋⣉⣉⣭⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⣡⡾⠂⠀⣀⡀⢀⠀⠀⠉⠀⠈⠛⠉⠛⠁⠈⣁⣹⣿⣿⠿⡿⢿⡟⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣴⣿⣿⣶⡶⢴⠶⠶⠶⠖⠚⠓⠛⠛⠋⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢡⣤⣴⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⠉⠀⠀⠈⠁⠀⠁⠀⠀⠐⡤⠀⢸⣿⣿⣫⡭⣿⣿⣷⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⣧⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⢸⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⠄⠹⣄⠀⢠⡄⠀⠀⠢⡖⠀⠀⢸⣿⣿⢸⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣾⣿⣷⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣻⣿⠛⠛⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⣠⣿⣿⣿⣀⣀⣀⡀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⣼⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢾⣿⣷⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣭⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣼⣿⣧⣀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⡿⠟⠿⠿⢿⣶⣦⣽⣽⣿⣆⠸⣿⣶⣿⣿⣿⣦⡀⠀⠛⣶⣤⣀⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠙⠁⡉⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣶⣄⣀⣰⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠁⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋⠙⡟⠛⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠃⠀⣸⣷⣤⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⣹⣿⣿⣿⣿⣿⠏⠀⠀⢀⣶⡆⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⣤⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⣠⣿⣿⣿⣿⡟⠁⠀⠀⣠⣿⣿⣷⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⢰⡆⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⡿⠀⠀⠀⡀⠀⠈⣿⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣄⣤⣿⣿⣿⡿⣿⣆⣀⣿⣿⣿⣿⣿⠀⠀⠀⠟⠁⠀⠀⠀⢻⣄⣳⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣇⠸⠿⠿⠉⠉⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢃⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢣⠀⠀⠀⠀⠀⡀⢀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⡀⠀⠀⠀⠈⢷⣘⣿⣿⣿⣿⣿⣿⠛⠉⠉⠻⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣰⣿⣿⣿⠻⠇⠀⠀⠀⠀⠀⢾⣿⣿⣿⠿⠃⠈⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠛⢿⣿⣿⣿⡆⢠⢠⠀⠀⠀⢤⡀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠘⢿⡌⣿⣿⣿⣷⠈⠀⠀⠀⠀⠈⢳⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⢰⡀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⢤⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⣸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠹⣦⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⢹⣿⣿⣿⡆⠀⠀⠀⠀⠘⣦⠈⢿⣿⣿⡆⠀⠀⠀⠀⠀⠠⠆⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⣾⡿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠈⣿⣿⣿⣷⡀⠀⠀⠀⠀⢹⣧⠀⢻⣿⣿⠶⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⠁⠀⠀⠀⠀⣾⡿⠁⢀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⢹⣿⣿⣿⣷⡄⠀⠀⢰⣄⠙⢷⣾⣿⣿⡄⠀⠀⠀⠀⣤⣹⣿⣿⣿⣿⣿⠀⢠⣶⣦⡾⠋⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⢈⣿⣿⣿⣿⣿⡄⠀⠀⢻⣷⡀⠙⣿⣿⡇⠀⠀⣀⢶⡈⢿⣿⣿⣿⣿⠃⢀⣼⣿⡏⠀⣠⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠸⣿⣿⣿⣿⣿⣷⡀⠲⣄⠙⠻⣶⣾⣿⣿⡀⠀⢻⣿⣿⣙⠻⣿⣿⣿⣾⣿⠟⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣷⡀⢹⣷⣤⣼⣿⣿⣿⣷⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣷⡄⣬⡍⠙⣿⣿⣿⣿⡿⠋⠁
⢀⣼⢿⣿⠏⡉⠛⠻⠿⣷⣶⣿⣿⠿⠋⣹⣿⣿⣿⣿⠀⣴⣦⠀⠀⠀⠀⠀⣀⠉⠉⠉⣡⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣷⣾⣿⣿⣿⣷⣶⣦⣤⣤⣍⣁⣴⣿⣿⣿⣿⣿⣿⣼⣯⣴⣿⣿⣤⣴⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣶⣶⣶⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⢸⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿
⠀⠀⠀⢰⣶⣦⣬⣍⣉⣛⠛⠛⠃⣿⣿⡟⠛⠛⣛⣉⣉⣥⣤⣶⡆⠀⠀⠀⢸⣿⣿⣿⣿⠉⠙⠛⠶⠶⠦⣤⣤⣼⠀⠀⢸⣤⣤⡤⠶⠶⠞⠛⠋⠉⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣛⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢸⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢀⣠⣤⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣾⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣿⣿⡿⠿⣿⣇⡿⣷⣯⣿⣿⣿⣿⣿⣿⣾⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣠⣿⣿⣿⣤⣾⣿⣤⣄⣼⠋⠉⣿⠿⢿⡿⢿⣿⠟⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠈⢻⣿⣿⣿⣾⣦⣥⣄⠀⠸⣿⣿⣼⣟⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡟⠀⠀⠠⣌⣽⠿⠿⣿⣿⢿⣿⣿⣾⣋⣹⣿⣿⣛⡉⠙⠛
⠀⠀⠀⠠⣾⣯⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡉⢁⡀⠸⣷⠈⠁⠀⠀⠀⠀⢀⣙⣿⣿⣏⡉⠛⢻⣿⡿⠿⢿
⠀⢀⣴⣶⣌⣹⡏⠙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢛⠛⠛⣿⡆⠀⠀⠰⠾⠟⢿⣆⠘⠿⢿⣿⡙⠻⠿⢿⣿⣇⣴⡾
⢰⣦⡙⢧⡙⠛⠃⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⣷⠀⠈⠀⠰⠶⢶⣶⣤⡀⢻⡆⠀⢠⣿⣿⣧⢀⣤⣉⣿⣿
⣾⠋⠙⠻⠇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢿⣭⡴⠃⠀⠀⠀⣀⣀⡀⠀⢹⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋⠁
⢻⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢠⣤⡀⢀⡀⢰⣿⡿⢻⡟⠷⠈⠛⠀⠀⢀⣼⡟⣛⣿⣿⣿⡇⣠⡄
⠸⣏⣹⣆⠀⠀⠀⠀⢀⣶⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⢠⡞⠋⠉⠻⣿⡁⠸⣏⣁⣼⠻⣆⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠈⠙⢿⣦⣀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢻⣿⣿⣷⢀⡟⠀⠀⠈⢠⡀⠰⣄⠙⠓⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⡟⠀⠀⠀⢸⣷⠀⠀⠰⣿⣿⠟⠛⣿⣻⣿⣿⣿⣉⣥⡄
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⢀⣀⣸⣇⠈⣻⣯⡿⢿⡟⠀⠀⠀⠚⣛⣠⣾⣦⣭⣿⣶⡿⢛⣿⣯⣽⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣦⣬⣁⣤⣾⣁⣀⡀⣀⣼⣿⡟⠉⠉⠉⠁⠀⠀⠈⠻⣿⡿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠃⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢧⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢱⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣼⢿⣤⣼⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⠋⢸⣿⠿⢿⣿⣶⣿⣷⣶⣦⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣾⠃⠀⠀⠀⠈⠉⠙⠛⠻⣿⡿⠶⠾⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣍⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠉⠙⢿⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣠⣿⠏⠀⠀⣀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠠⣿⣿⣿⡀⠀⠀⣰⣿⠋⠀⠀⣴⣿⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⠀⢀⣤⣾⣿⣿⡛⢿⠄⠀⠀⠀⠀⠀⣹⣿⣿⡇⠀⢰⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⣠⣶⠟⠋⢀⠙⣿⣿⣿⣦⠀⠀⠀⠀⠀⢸⣿⣿⡇⢠⣿⠏⠀⠀⢀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠸⣿⣿⡇⣀⣴⠞⠋⠀⠀⢠⣾⢦⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢿⣿⣷⣿⡏⠀⠀⠀⣿⣿⣿⣿⡏⢸⣿⠈⢿⣿⣿⣿⠁
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣾⣿⣿⡿⠋⠀⠀⠀⣰⠎⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣾⣿⣿⣏⠀⠀⠀⢰⣿⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⢸⡟⠀⠀⠀⠀⢂⣾⣿⡿⠀⠀⠀⠀⣼⣧⡼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⣾⠁⠀⠀⠀⠀⣸⣿⡙⠇⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣆⠀⠀⠀⠀⢻⣿⣿⣿⡀⣼⣿⡿⢋⣿⣟⣿⣟⣿⣿⣿⡟
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠈⠀⠀⢀⣤⣾⣿⣶⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠘⣿⡀⠀⠀⠀⠈⢿⣿⡏⢳⣿⣿⠃⠈⠻⠿⣷⣿⣿⣿⣿
⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⢠⣴⡿⠿⠛⠉⠀⠀⢈⣿⣿⣿⣿⣭⣿⠋⠀⠀⠘⣷⡀⠀⠀⠀⢸⡿⣧⣿⣿⣇⣀⣤⣤⣴⣾⣿⣿⣿⡃
⠀⠀⠀⠀⢻⣆⠀⠀⠀⢀⣾⣿⠋⠠⣀⣠⣤⣀⣤⣾⣿⣽⣿⣿⣿⣿⣾⣿⣿⣿⣻⣿⣄⠀⠀⠈⡹⣯⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠙⢷⣶⡶⠿⠛⠟⠋⠐⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⠛⠛⢿⣷⣶⣾⣿⣿⠿⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠈⠇⣠⣴⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⣴⣶⣭⣤⣁⣀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠇⡂⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠢⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⠠⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠉⠀⠉⠉⠁⠶⠶⢤⣄⣀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⣄⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢧⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠤⠤⠴⠶⠶⢿⣍⡻⢿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠘⡗⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡾⠁⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠒⠲⠛⠛⠉⠃⠀⠀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⣈⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣶⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⣛⣥⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⠿⠿⠛⠻⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⣥⠴⠶⠶⠦⣬⡛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⣿⣷⣞⣉⣀⣀⣀⣀⣀⣀⣙⣶⣾⣿⣇⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⡴⠋⠳⡄⠀⠀⠀⢸⠀⡇⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣤⣧⣼⣤⣤⣤⣤⣇⣠⣀⣧⣤⣤⣤⣼⣤⣧⣤⣤⣴⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠁⠀⢹⣻⣟⡏⠁⠈⠹⣏⣉⣉⡿⠁⠀⠙⣟⣟⣻⠉⠀⢹⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡟⢻⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠛⣿⠀⠀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠸⡇⢸⡇⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢿⠀⣿⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠘⢦⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⠞⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠁⠀⠀⠈⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡴⠒⠋⠙⠛⠒⠲⢤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣔⡴⠶⣶⠴⢦⣠⣴⣲⠶⠤⡤⣿⣦⡤⠤⠤⢤⣤⣤⣴⣧⣤⣤⡴⣆⣀⣀⢶⣶⡤⠤⢤⣤⣤⡤⠤⣄
⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⣠⠞⠛⠁⠠⠾⠋⠉⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠉⠁⢠⡾⠋⠡⡄⠀⠉⠁⠐⠢⡀⡆⢸⡆
⠀⠀⠀⠀⠀⢸⠑⢊⠔⠋⠁⠀⣇⠀⣀⣧⠀⠀⠀⠠⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⠀⢸⡇⠀⠀⠙⢦⡀⣈⡦⠀⣧⡇⢸⡇
⠀⠀⠀⠀⠀⢸⣀⠏⠀⠀⠀⠀⢧⠤⠤⣤⠀⠀⠁⠀⠓⢄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢤⡇⠀⠀⠀⠀⠙⢦⡀⠀⡿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢀⠠⠀⠀⠀⣿⠀⠀⠀⠑⢀⠈⠐⠉⠀⠀⠂⠂⠀⠀⠀⠀⡃⠀⠀⡼⠇⠀⠀⠀⠀⢀⢠⣙⠐⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡃⠀⠀⠑⠔⠉⠀⠀⠀⠀⠀⠐⣠⢦⠀⠀⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠠⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠰⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⢀⡀⡀⠀⠸⠀⠀⠀⠀⡀⠀⠀⠀⠄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘⠀⢫⡀⠀⢀⠀⠀⠅⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⢀⠀⡈⠃⠀⠀⠀⢰⠀⠈⠁⠂⠀⠤⠤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⡳⠠⠀⠀⠤⣤⠆⠀⡆⢀⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢡⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡆⠀⠀⡆⡀⠈⠑⠪⡥⢤⡌⠀⠀⠃⢸⠃
⠀⠀⠀⠀⠀⢠⠋⢢⠀⠀⠀⣡⡠⠀⠀⢸⠀⠀⠐⠀⠀⠀⠉⡀⠀⠀⠀⠈⠉⠀⠀⠀⡁⠁⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠀⠼⡄
⠀⠀⠀⠀⠀⢸⡀⢀⠑⣄⠉⠀⢸⠀⠀⢸⠀⠀⠀⢀⠀⠀⢀⠄⠀⠀⢀⠀⠂⡘⡄⡀⠑⡇⠀⠀⡆⠀⡘⠀⠀⠀⠀⠂⠈⢶⡄⢸⡇
⠀⠀⠀⠀⠀⠸⣅⣨⣤⣈⣷⠤⢬⣀⡀⣸⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠁⠁⠀⠀⡇⠀⠀⡗⠈⠀⠀⢀⠀⠁⡀⠀⢟⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠧⠤⠀⠷⠦⠠⠴⣁⣀⣐⣁⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⡄⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠟⠉⠁⠈⠉⢪⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡟⡤⢤⡀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⢤⣤⢞⣽⣶⣃⣉⡑⢿⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⢰⣯⠏⠀⠀⠉⠉⠉⠉⢻⣿⠒⠒⠛⠲⠶⠤⠤⢤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⣷⡷⡶⠾⠶⢶⣶⣶⣿⣯⣾⣿⣧⣼⣧⣤⣦⣴⣿⣄⣰⣤⡦⣤⠀⣄⡀⣴⠶⢟⢟⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢻⣋⡿⠃⣈⠀⠈⠫⠟⣿⢁⠀⠀⠀⣠⡤⣄⠉⠉⢹⣏⣿⣟⠛⠛⠛⣿⠿⣿⣧⠀⡫⢊⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣏⣻⠀⠀⠸⡭⠀⠀⠈⠸⡐⢔⡹⠀⠀⣜⡿⠩⠃⠐⠀⠘⢿⣢⠞⣿⠀⢐⠵⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡍⣷⠀⠂⢠⣴⠮⠏⢶⣄⠂⣠⡄⢄⠈⠁⠄⠁⠀⠈⠀⠀⢠⡀⢀⡀⠀⠀⢸⣿⡇⠀⡪⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣌⣱⣟⠀⢀⣿⠁⣶⢥⣤⠙⣷⠝⠶⢠⠃⠀⠀⡤⠤⠄⠀⢰⢡⢂⢲⢰⠈⠀⣠⡿⣿⠀⢒⠕⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠉⠉⠉⠉⢉⣽⡿⠿⢿⣿⡛⠛⠛⠛⠛⠛⠓⠒⠶⠾⠾⠾⣧⡄⠐⡈⠠⢵⠏⠀⠈⠹⡯⢼⡇⠀⡪⠚⣧
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣚⣫⣀⣀⢀⠰⣠⣿⠃⠰⠆⡀⢸⣷⡀⠰⠆⠀⠀⠀⠀⢀⣠⡴⢚⠙⡇⠀⠈⠉⠁⡀⠀⣠⣠⡿⠘⣿⠀⠐⡣⢻⡆
⠀⠀⠀⠀⠀⠀⢸⠏⠉⠉⠙⠛⠛⠛⠛⢿⡷⠶⠶⠶⠷⢿⣼⣷⣤⣼⣴⣤⣤⣤⣼⣆⠀⠑⡀⣻⠀⡀⠀⠀⠀⠀⡻⣿⣿⠄⢹⡇⠨⢆⠜⣷
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡈⢘⡄⣴⡮⠰⡄⠠⠈⠈⣯⢁⠀⣿⠀⠡⠊⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣔⠅⣿⡹⠳⢗⡟⠀⠀⠀⣿⢄⡀⢹⡆⠈⡀⠍⣿⡀
⠀⠀⠀⠀⠀⢀⣿⠖⠶⣤⡤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠄⡀⢿⠈⠉⠉⡄⠀⠀⠀⣿⠔⠠⠈⣇⠀⢤⠴⠘⣇
⠀⠀⠀⠀⠀⢸⡇⡸⢎⠀⠙⣤⠊⠉⠢⣠⠋⠙⢍⡩⠟⠛⡍⢉⠝⠛⡍⢉⠝⢛⡛⣻⣧⠀⠊⢀⠹⣀⡄⡁⠀⢠⠤⢤⣹⡈⠀⠀⢻⠀⠠⠊⠄⢹⡆
⠀⠀⠀⠀⠀⢸⣏⠀⠀⡱⣞⠀⠑⣤⠊⠉⠢⣠⠞⠑⣄⢀⡜⠣⡀⢀⠜⠣⡀⣠⠟⢽⣿⡄⠐⡁⢨⡿⠻⠃⠀⢹⡶⣶⢿⡇⠀⠀⣼⠷⠶⢷⣿⣿⠟
⠀⠀⠀⠀⠀⣸⡏⢣⡜⠀⠈⣢⡞⠀⠳⣠⠖⠉⢦⢀⡼⠣⡀⢀⡜⠣⡀⡠⠚⢦⠀⣸⣿⣧⠈⡪⠀⡧⠀⠀⠀⠀⡈⠡⠘⡇⠀⣸⠟⣨⣶⠿⠋
⠀⠀⠀⠀⠀⣿⡇⠁⠈⣦⠚⠁⠙⣤⠞⠉⠢⣀⡴⠛⢄⢀⠜⢯⠀⡠⠛⢤⠀⡠⠛⢼⣿⣿⡀⠰⠁⢿⣞⣕⡄⠀⠀⣴⠲⣗⣸⣿⡿⠛⠁
⠀⠀⠀⠀⠀⣿⠈⣢⡞⠁⠑⣤⠞⠁⠱⣄⠔⠉⢦⣀⠔⠳⡄⣠⠛⢄⠀⡰⠛⢄⢀⢼⣿⢾⣇⣀⡦⢸⠿⠿⠤⠥⠼⠿⢷⣿⠟⠉
⠀⠀⠀⠀⠀⡿⢎⠀⠈⢢⡔⠉⠣⣀⠔⠉⢢⣀⠔⠓⢄⡠⠞⢣⡀⡠⠻⡀⠀⡴⠻⣼⣯⣿⡏⡛⢻⣿⠄
⠀⠀⠀⠀⠘⠷⠤⣵⣶⣥⣌⣤⣔⣉⣢⣀⡔⠙⢄⡠⠊⠳⡀⡠⠛⢄⠀⡨⠫⡀⢀⢼⣿⡿⣈⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠚⠷⠶⠤⣼⣧⣀⣀⣿⡀⠀⡹⢇⢸⣿⣹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠶⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢀⣠⣄⣀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⡆⠀⡇⠀⠀⠉⠛⠶⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠉⡭⠭⠭⣍⡙⠛⠷⣤⡀⠀⠀⠀⠈⠙⢦⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡘⢦⡀⠀⠉⢻⠶⣄⡙⢶⣄⠀⠀⠀⠈⢳⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀⠀⠳⡄⠹⣦⣍⠳⣄⠀⠀⠈⢷⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠲⡄⠙⣦⠹⣏⠳⣌⠳⣦⡀⠈⢧⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣳⣄⠈⢦⠈⠳⣌⠳⣌⢧⡈⢷⡀⠈⢧⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠛⠛⠛⠛⠿⠷⠾⣦⣷⣌⣷⣀⣸⣇⣹⣀⣠⣤⣤⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢰⠁⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣾⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢀⡠⠺⠣⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡴⠛⠁⠀⠀⠈⠛⢦⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠋⠀⠀⠀⠀⠀⠀⠀⣀⣙⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠷⠶⢾⣿⣶⣶⣶⣶⡶⠿⠿⠛⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣠⡤⠶⠶⠶⠶⠤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠉⠉⠛⠛⠲⠶⣤⣀
⢰⡟⠁⠀⠀⠀⠀⢀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⣀⠀⠀⠀⠀⠈⠳⣄
⠸⣦⣠⣤⣤⣶⠞⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠉⢳⣤⣀⠀⠀⢀⡿
⠀⠀⠉⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠸⣧⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢻⡄⠀⠀⣼⠟⢁⣀⣀⣀⣀⡈⠛⢻⣦⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢻⣄⣸⣿⠞⠋⠉⠀⠀⠉⠙⠓⠦⣿⡇⣰⠇⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠦⣤⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⣖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡙⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣠⣾⣯⠙⠳⡄⠀⠀⠀⣴⣿⡍⠛⢳⡄⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⢿⣿⣿⣤⣴⡟⠀⠀⠀⣿⣿⣷⣶⣾⠇⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠀⠀⠀⠀⠈⠛⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⢀⣼⢋⣤⠀⠀⠀⣀⣠⣤⣤⠶⠶⠶⠶⠚⠛⠛⠛⠛⠛⠲⠶⠶⢦⣤⡀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⣼⠇⢼⡇⠀⢰⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢸⡿⠀⠸⣧⠀⠈⢳⣄⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣿⡇⠀⠀⠹⣧⠀⠀⠙⢷⣤⡀⠀⢸⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⢿⡇⠀⠀⢀⡼⢷⣄⠀⠀⠈⠛⠷⣿⣀⡀⠀⠀⠀⠸⡇⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠸⣿⡀⢠⡟⠀⡴⠛⠷⣤⡀⠀⠀⠀⠉⠛⠛⠷⠶⠾⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠙⢿⣿⠀⠘⠁⠀⠀⣼⠟⠓⠶⠦⣤⣤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠿⠛⠁
⠀⠀⠀⠀⠀⢘⣷⣄⣀⣀⣴⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠿⠛⠻⣿⣄
⠀⠀⠀⠀⢀⣾⠟⠉⠛⠛⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⢠⣿⠏⠀⠀⠀⠁⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⡀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠶⠶⠦⣤⠘⢿⡄
⠀⠀⢀⣾⠏⠀⠀⠤⠠⠀⣴⡶⠟⠛⠋⠉⠉⠁⠀⠀⠹⣆⠀⣼⠋⠁⠀⠀⠀⠀⢀⡀⠀⢸⡇⠈⢿⡆
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⠀⠀⠀⠸⣆⠀⢹⣾⡏⠀⢀⡟⠀⠀⠀⠘⣇⠀⠀⣷⠀⠈⣿⡄
⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢹⣦⡾⠿⢦⣤⣼⡇⠀⠀⠀⠀⣿⣤⣴⠟⠀⠀⠸⣧
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠿⢦⣤⠴⠾⣧⣤⣴⠶⠟⠃⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠟⠁⠀⠀⠀⠀⠀⢿
⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣸⡇
⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠛⠻⠷⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣾
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠷⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠾⠿⠿⠿⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⢉⡹⠟⠓⠲⠦⣀⠀⠀⠀⣀⡤⠤⠤⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠒⠛⠓⠒⠒⠛⠓⠚⠛⠛⠲⠾⠷⠤⢮⣅⣀⢀⡤⠞⣉⡥⠤⢬⣙⠲⣄
⠀⠀⠀⠀⠀⢀⡠⢖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡬⠭⠿⣦⠚⠉⠙⢦⡀⠈⠳⣌⠳⡄
⠀⠀⠀⠀⣠⠋⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠊⠳⢗⢦⠀⠈⠳⡄⠀⠀⠳⡄⠀⠘⢦⠹⡄
⠀⠀⠀⢰⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢧⠀⠀⢳⣳⡀⠀⢻⠀⠀⠀⠹⡄⠀⠈⣧⠹⡄
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠸⡆⠀⠀⢯⢷⠀⠘⡇⠀⠀⠀⢻⡀⠀⠸⡆⢳
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⢸⠀⠀⡇⠀⠀⠀⠈⡇⠀⠀⣷⢸⡆
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⢺⠀⠀⡇⠀⠀⠀⠀⢹⠀⠀⢸⠀⡇
⠀⠀⣸⠀⠀⠀⠀⠀⠀⢀⠴⠂⠠⠤⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⣼⠀⠀⣇⠀⠀⠀⠀⢸⡄⠀⢸⡆⣧
⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⠀⢈⣯⡄⠀⢻⠀⠀⠀⠀⢸⡇⠀⢸⡇⣿
⠀⠀⠹⣄⠀⠀⠀⠀⢀⠱⠤⣤⠤⣤⣄⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢻⠀⠀⢀⠀⠐⠀⠸⡆⠀⠀⠀⢸⡇⠀⠀⡇⢹
⠀⠀⠀⢸⡟⠒⠲⠶⠤⠤⠤⡇⠀⠀⣯⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠃⠀⠀⢸⡄⠀⠀⠮⠵⠀⠀⢷⠀⠀⠀⢸⡇⠀⠀⡇⣸
⠀⠀⣠⠼⠓⠒⠒⠒⠒⢒⠶⠇⢀⣨⡗⠒⠒⠶⠶⠶⠤⠤⢤⣄⣀⠀⠀⠀⠀⣇⢀⡴⡶⠒⠛⠛⠉⠛⠲⢦⢸⠁⠀⢸⡇⣿
⠀⢰⠇⠀⠀⠀⠀⠀⠀⡗⢎⣉⣁⡶⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢻⢸⣿⣿⡄⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⠃⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠛⠓⢒⠖⠋⠀⠀⠀⠀⠀⠀⠀⠘⢠⡟⣷⠀⠀⠀⢸⡜⡏⠉⣧⠀⠀⡄⢀⠀⠀⢹⠀⠀⣼⢠⡇
⠀⣿⠀⠀⡀⡀⠀⠀⠀⠀⠑⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⠀⠀⠀⠘⡇⣧⠀⢹⠤⠤⠤⡤⠶⠒⣯⠀⢠⡇⣼
⠀⢻⡀⠘⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⣱⠄⠀⠀⣼⠀⣿⠀⠀⠀⠀⡇⢻⠀⠸⠀⠀⠀⣵⠀⠀⢿⠀⡞⢠⠇
⠀⢸⡟⠒⠶⠤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠞⡇⠀⣿⠀⠀⠀⠀⡇⢸⠀⠀⡄⠀⠀⢸⠀⠀⢸⡞⣠⠏
⠀⠸⡇⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⢀⡇⢸⠀⠀⡇⠀⠀⢸⠀⢀⣟⡴⠃
⠀⠀⢷⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⡇⢀⡏⠀⠀⠀⣸⠁⠸⣤⣀⣇⣀⣀⣽⠴⠛⠉
⠀⠀⠀⠉⠻⣗⡒⠶⠦⠤⠤⣤⣤⣤⣀⣀⣀⣀⣸⣀⣀⣀⣸⣠⠞⠁⠀⠀⣰⣃⣠⡤⠤⠤⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠶⠦⠤⠤⢤⣤⣤⣄⣀⣈⣩⣥⡤⠤⠴⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣿⣿⡿⠿⠿⠛⠛⠛⠛⠻⠿⠿⣿⣿⢷⣶
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣶⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣾
⠀⠀⠀⠀⠀⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⣉⣉⣉⡙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠸⣿⣿⣿⣿⡇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣡⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠛⠛⠛⠛⠃⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣒⢒⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⠎⡔⠁⠀⠀⠈⢆⢱⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⢀⣉⢦⡀⠉⢢⠀⠀⠸⡀⡇⠀⠀⠀⠀⡜⢸⠀⠀⢀⠔⠊⢡⠔⠯⡁⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⡠⠚⠁⠀⠀⢱⠀⠀⣇⣀⠤⠷⠷⠤⠤⠤⠴⣥⡃⠀⡰⠃⠀⡰⠃⠀⠀⠈⢆⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠞⠀⠀⠀⠀⠀⡨⠗⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠣⣄⢠⠃⠀⠀⠀⠀⠈⢆⠀⠀⢳
⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⠎⠀⠀⠀⠀⡰⠋⠀⠂⢀⠤⠒⠊⠉⠉⠉⠉⠒⠢⢄⠀⠀⠂⡈⠻⡀⠀⠀⠀⠀⠀⠸⡄⠀⠈⡆
⠀⠀⠀⠀⠀⡏⠀⠀⠀⡎⠀⠀⠀⠀⡜⡀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠈⠂⠱⡀⠀⠀⠀⠀⠀⡇⠀⠀⢳
⠀⠀⠀⠀⢸⠀⠀⠀⣸⠁⠀⠀⠀⣸⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠁⣇⠀⠀⠀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⡟⠀⠀⢀⡇⠀⠀⠀⢠⡏⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⢹⡄⠀⠀⠀⠀⢸⡄⠀⠸
⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢇⠀⠀⠀⠀⠀⠃⠀⠀⡅
⠀⠀⠀⢠⠃⠀⠀⡌⠀⠀⠀⢀⡇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠈⠒⠢⠤⠤⠤⢀⣀⣤⣤⣤⣤⣄⡠⠤⠤⠤⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⢠⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⠀⠀⠀⠁
⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⣀⡇⠀⢠⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠑⡄⠀⢸⡤⡀⠀⢸⠀⠀⢸
⠀⠀⠀⢸⠀⠀⢸⠀⠀⡏⢸⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⡆⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠈⡄⠀⢸⠀⢸⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⡆⠀⠀⡟
⠀⠀⠀⠀⡇⠀⠈⡆⢸⠀⠸⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡇⢸⢠⠇⠀⢸⠇
⠀⠀⠀⠀⢷⠀⠀⢣⢸⠀⠀⠀⠀⡏⠓⠒⠒⠒⠂⠀⠤⠤⠤⡤⠤⠤⢤⠄⠀⠐⠒⠒⠚⠛⠓⠉⢹⠀⠀⡇⢸⡼⠀⠀⡞
⠀⠀⠀⠀⠘⡀⠀⠘⣾⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡇⡸⠁⠀⡸⠁
⠀⠀⠀⠀⠀⢳⡀⠀⠘⣄⢸⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡏⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⢻⠀⠀⠈⠓⠢⠤⠤⠤⠄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠚⠀⠀⢠⡥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠒⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠋⢹⡄⠀⣀⡤⠖⠋⠉⣉⠛⠷⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢀⣀⣀⠀⢷⣉⠀⣀⠴⠉⠊⠍⠀⠁⠀⠀⢀⣠⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣰⠞⠉⠀⢘⣧⠤⠾⠋⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠇⠀⢹⡠⠴⠚⠉⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⣸⠤⠴⠒⠋⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⣉⡤⠒⠺⢉⡍⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣰⣶⣏⣁⣀⡀⠼⠂⠡⠐⠊⠁⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠉⣟⠙⠛⠓⠶⢤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣇⠀⠀⠀⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠲⢿⣄⣀⠀⠀⣼⠀⣀⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢀⢻⣴⣖⣸⠦⡀⠀⠀⠀⠀⢀⠠⠤⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⡞⠸⢻⡀⠀⡇⢰⠀⠀⡠⠊⠀⢀⠔⠫⡀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⡤⠄⠘⠃⠒⠐⠃⠠⢥⡀⣆⠞⠀⠀⡰⠁⠀⠀⠘⡄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⢉⡈⠓⠒⠒⠀⠦⠀⡀⠀⠀⠀⠉⠢⣀⡰⠀⠀⠀⠀⠀⠰⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠒⠉⠁⠀⠀⠀⠉⠐⠢⢄⠀⠀⠀⠠⠄⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠇⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⠀⠀⠀⢵⡆⠀⠀⠈⢄⠀⠀⠀⠀⠸⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠘⠁⠀⠀⠈⠁⠐⠂⠄⢀⠀⠀⠀⠀⠑⡀⠘⡛⢻⣇⠀⠀⠈⡄⠀⠀⠀⠀⡆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⣄⠀⢠⡀⠠⠄⣀⠀⠀⠀⠀⠉⠶⣆⠀⠐⡀⠀⠘⢯⡇⠀⠀⢘⠀⠀⠀⠀⢁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠈⠀⠒⠨⢐⡢⠄⠄⠀⢋⠇⠀⠁⠀⠀⠀⢤⠀⠀⠀⡆⠀⠀⠀⠈⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠄⠀⡀⠀⠘⠀⠀⠀⠐⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⡆⠀⠀⠀⣇⠀⠀⠈⠀⠀⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠸⠐⠂⠖⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠁⠀⠀⠀⣠⠀⣀⠀⡇⠀⠀⠀⢄⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⢸⠀⠀⠀⣗⠒⢹⡇⢀⠀⠀⠀⢎⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⣀⠀⠀⠀⠀⠁⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⣻⡄⢸⠀⠀⡠⠉⠉⠈⠣⣸⠀⠀⠠⠉⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠒⠀⠨⠁⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡇⠈⠀⠀⡇⠀⠀⠐⠁⢹⠀⠀⡌⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠠⢁⡐⠂⠄⠀⢀⣀⣈⡤⠂⠀⡆⠀⠃⠀⠀⠀⠀⣿⣷⡼⠁⣼⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠉⠒⠠⠄⣀⠀⠀⠀⠀⣷⡷⠦⠄⠀⢀⠀⠼⠿⠇⢰⠿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠓⠀⠀⢿⡛⣧⡖⠺⣼⠁⠺⣷⢇⠞⠀⠳⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠁⠀⠀⠀⠘⠷⡏⢑⠴⡉⠲⠧⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠠⡆⡟⠣⢀⡘⠃⠖⢸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣥⢧⡔⠁⢈⠟⢂⡾⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠭⠅⠒⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢸⠀⠹⢦⡗⠋⣣⠴⢷⢷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠨⠄⡂⠤⠄⠠⠶⠊⠀⢘⠀⠠⠤⠛⠛⠛⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⣤⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⡙⡵⠛⠻⣿⡙⢮⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢱⠀⠀⠀⢣⢷⠀⠎⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⢸⢸⠀⠀⠀⠈⣸⠀⢠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⢸⢀⠀⠀⠀⢀⣻⡄⢈⢰⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⠴⠒⢀⠈⠉⠉⠉⠉⠉⢹⡼⠈⠉⠁⠀⠀⠀⠈⣨⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠉⣸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠺⢿⠗⠀⠀⠀⠀⢐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡊⢽⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠐⠀⡀⠀⠀⠀⠀⠀⢰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠂⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⢄⠔⡠⡀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠪⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠃⢊⢀⠔⠀⠀⠀⠕⡁⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢪⠂⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠐⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡐⡄
⠀⠀⠀⠀⠀⠀⠀⡯⠐⢽⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡈⡐⡄⣔
⠀⠀⠀⠀⠀⠀⠀⡧⠀⣹⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢂⠎⠈⢀⢌⠔⠌⡴⠣
⠀⠀⠀⠀⠀⠀⠀⡇⠂⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⡡⠡⠀⡀⠠⠪⠀⢊⠎⠸⠀⡐⠇⠰
⠀⠀⠀⠀⠀⠀⢰⡇⠨⣺⠀⠀⢀⠄⠀⠀⠀⠀⠀⠊⠌⠜⠐⠁⡠⠔⡑⡑⠑⠡⠂⡊⠠⠀⠌⢀
⠀⠀⠀⠀⠀⠀⢸⢇⠄⣺⠀⢘⠑⡁⡁⡠⠠⢀⠄⡀⡠⠀⠊⢊⢊⢊⠊⠌⡠⡑⡁⠀⠐⡀⠄⢀
⠀⠀⠀⠀⠀⠀⢸⢕⠬⣺⠀⢸⠨⡪⢊⠔⡰⠡⢊⠈⠔⡑⡑⡑⡡⠢⢂⠄⠘⠔⡐⡀⠐⠀⠈⢀⠊
⠀⠀⠀⠀⠀⠀⠸⢵⣾⠿⣆⢸⠐⡔⡡⢊⠔⡁⡡⢂⢊⢌⢜⡔⡑⡡⢪⢊⢎⠌⡄⡰⠀⠄⢂⠠⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⠒⠒⠒⠒⠂⠀⠁⠣⠤⠬⠬⠌⠔⠃⠃⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠄⠀⢀⠀⡠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠈⢂⠄⡠⢐⠁⡀⢄⠈⢀⠅⠠⡈⠀⡁⡊⠔⡡⢂⠄⡨⢂⠔⡀⢀⠔⡠⢂⠔⡠⢀⠁⡀⠪⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠀⠁⠊⠀⠁⠈⠀⠁⠈⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣒⡼⠿⡿⠿⢿⣒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣡⠞⠁⠊⠉⠉⠙⠲⡝⣦⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢰⡏⠀⠀⠀⠀⠀⠀⠀⠹⣾⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣞⣀⣀⢀⣀⣀⣀⣀⣀⡀⣹⠀⢸⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠁⠀⠀⠈⢉⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣌⠀⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠀⠀⠀⢠⠟⠉⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡁⠲⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⡖⣀⠐⢺⠀⠀⢠⡏⡶⠂⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠞⠤⢀⠀⠀⠉⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠰⠏⠀⠀⡼⢰⡇⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠰⠀⠀⠙⠉⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠁⠳⠤⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢀⠒⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣦⡤⠶⠚⢻⠁⠀⠒⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣠⠄⠀⠀⠀⠐⠚⠋⡄⠀⠀⠀⠈⠉⠁⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⡏⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠶⡇⠀⠀⠀⠀⠀⣄⠀⠈⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢟⠀⠀⢀⡇⠀⢸⠃⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣄⡀⠀⠀⠀⣿⠙⠛⠁⠀⠉⠉⠉⠉⠉⠉⠑⠒⠒⢻⡄⢿⠀⠀⢸⡇⠀⠸⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⠿⡇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣼⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠱⠄⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠊⣧⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡆⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠉⢉⡽⣛⣲⡄⠀⠀⠉⠙⡐⠀⠀⣠⡿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣾⡿⡿⠛⠁⠀⠀⠀⠀⠈⢹⡛⣿⠀⠀⣻⠦⣤⣀⣀⣀⡤⠖⠋⢡⠅⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⡀⠀⠀⠀⠀⢿⠁⡇⠐⠂⠒⠠⠤⠄⣀⣠⣧⢾⠀⠀⣿⠀⠀⠈⣉⡉⠄⠀⠀⢸⠁⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠙⡋⡟⠒⠛⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢹⠀⠀⠀⠘⠀⠀⠀⠀⢘⡀⠀⠐⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⡇⠀⠀⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣠⣴⡭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⢠⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠓⢺⢸⡄⣄⣀⣸⡴⢃⠀⠀⠀⠀⠀⠀⢸⡇⡼⠀⠀⢸⠀⠀⢀⡀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⣈⣀⣀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠚⠋⠀⠀⠀⣾⠀⡤⠞⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⣠⣿⣦⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⠖⢫⠗⠋⠉⠉⠻⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⡀⠙⢆⣠⠃⢀⠏⠀⠀⠀⠀⠀⠘⡎⢣⠀⣠⠞⠉⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢰⠙⡆⢠⣏⠠⠞⠀⠂⠀⠒⠂⠐⠠⣹⠈⡷⠁⢠⠞⡇⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⡸⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣀⠋⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢸⠎⠀⠀⠀⠀⢀⣀⠤⠤⠀⠀⠤⢀⡀⠀⠀⠈⢀⠀⠰⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠋⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠡⡤⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣠⠂⠀⠀⣰⠒⠒⠒⠒⠒⠒⠒⢦⠀⠀⠈⢆⠀⠱⡐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣰⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠈⢂⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡸⠀⠀⠀⠀⠀⣇⠐⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠰⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⢻⠷⠲⠶⢦⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣤⣤⣤⡤⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠆⢡⢤⣀⣀⣀⠀⠀⠈⠈⠉⠈⠉⠉⠁⠉⠈⠁⠀⠁⢈⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠓⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣫⣴⠾⠛⠛⠻⣿⣿⣤⠤⠴⠶⠤⢃⠈⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⠟⠁⠀⠀⠀⠀⠈⣏⢹⠦⣀⢀⠀⠠⠀⢙⣢⠀⠰⠊⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠋⠀⠀⠀⠀⠀⠀⠠⢈⣁⡠⢤⠳⠒⠃⠉⠻⣿⠠⠰⠰⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠤⠈⠁⠀⠀⠀⠀⣀⠁⠀⢐⣀⣠⣱⡶⢻⠛⢹⠸⠃⠀⠀⠀⠀⢀⠴⣿⠀⠀⢀⠴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⢀⠀⠄⢐⣢⣄⠀⢱⣶⢿⡛⠋⠉⣈⠡⠀⠀⠘⠀⠀⠀⣤⠀⠀⠈⢦⣻⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⢀⠔⢈⣥⣶⣿⣿⡿⡿⠐⠸⣏⠐⠁⠄⢈⣴⡶⠈⠀⠀⠀⠀⠀⠂⠑⠄⠀⠀⢹⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢽⠀⠀⠀⠎⢠⣿⣿⡿⣯⣿⣧⣿⠠⠀⡏⢀⠀⣰⣾⡿⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠸⢸⣿⣿⣟⡿⠋⢿⣷⠶⠚⡇⠀⠀⠟⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠸⠀⣿⡿⡿⠇⠀⠈⢿⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⠖⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⣇⣟⡠⠒⠦⢼⠆⠰⠃⠀⠀⣟⣃⡒⠤⢀⠀⠀⠀⠉⡅⠀⠀⠀⡀⠀⢘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⣿⣿⡜⢱⡁⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠑⠃⠀⠀⠓⠀⠢⢄⠘⠆⢸⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⣿⢻⡆⠮⡁⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⢸⠀⠀⠀⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⢻⣨⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⢈⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⡁⢸⡇⡀⡀⠀⠀⠠⠀⠀⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⣴⣿⣿⣟⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⢸⣿⢁⡇⠀⠀⣰⡆⠀⢺⠉⠢⡄⠠⠐⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠟⠏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⡀⠀⠀⢸⠈⣿⣿⣿⡶⠀⡿⡇⠀⢸⡄⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⡿⠛⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣶⣤⣀⠸⠅⣿⣻⣿⣇⡆⡇⣿⠀⢸⣷⣀⣠⣴⣶⣿⣿⣯⡛⠲⢭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⠛⠿⢿⣦⣿⣿⣿⣩⣛⣧⣽⣤⣾⣿⣿⢿⣿⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠍⠓⠦⢶⠝⠻⠿⣿⣿⣿⣿⣿⣿⡿⢿⡻⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⢁⠀⠀⠀⠉⠋⠛⠙⠢⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿
⢀⣴⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣦⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣸⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⢍⠁⠀⠈⠉⠀⠁⠀⠀⠘⡤⠀⢸⣿⣿⣫⡭⣿⣿⣿⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⢷⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⣿⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⢄⠹⣄⠀⢠⡄⠀⠀⠣⠖⠀⠀⢸⣿⣿⢹⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠤⠤⠴⠒⠒⠒⠒⠒⠲⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢩⠤⠤⠴⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠉⠛⠒⠒⢢⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⡆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣄⠧⠤⠤⠤⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠚⠋⠉⠉⠉⢹⠉⡇
⠀⠀⠀⠀⡏⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹
⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡸⡇
⠀⠀⠀⢀⡿⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⢠⠇⣧
⠀⠀⠀⢸⡄⡇⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⢿⣿⣀⣀⣀⣤⠤⠴⠒⠚⠉⣁⡤⠤⠚⠁⠀⢸
⠀⠀⠀⢸⡇⠙⢤⣀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⡇⠀⠀⣿⣇⣀⡤⠤⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⡇⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⣷⣤⣤⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿
⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⠏
⠀⠀⠀⠀⢻⣻⡟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠖⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠙⠳⠿⠷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠰⠒⠒⠒⠊⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢆⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠿⠷⠴⠴⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠛⠛⢿⣟⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢻⣿⡶⠟⠛⠓⠦⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢞⣩⣥⣶⣾⣿⣿⣶⣄⡈⠙⣳⣦⡾⠛⠃⠀⠀⠀⠀⠀⠛⢿⣴⣞⠋⠁⣠⣶⣿⣿⣿⣶⣤⣭⣛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠛⠿⣿⣿⣿⣿⣿⡏⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠹⣿⣿⣿⣿⣿⠿⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⣟⠋⠀⠀⠀⠀⢀⣀⣴⣏⠀⠀⠀⠀⠀⠈⣿⣦⣶⡀⠀⠀⠀⠀⠘⢻⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠹⣧⡀⠀⠀⠀⣉⣙⡛⢿⡄⠀⠀⠀⠀⣰⠟⢉⣀⣀⡀⠀⠀⠀⣴⠟⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⢿⡀⠀⣰⢿⣟⢻⣿⣦⠀⠀⠀⠀⠀⠉⣼⣿⣟⣹⡿⢦⠀⠀⣿⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⡿⣅⠐⠁⠠⠿⠟⠣⠏⠀⠀⠀⠀⠀⠀⠙⠌⠛⠿⠀⠈⠀⣈⠛⠻⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠸⣁⣄⠀⢠⣌⠇⠀⠀⢰⡟⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠉⠁⠾⠈⠉⠀⠀⣰⠏⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⢠⡀⠀⠀⠈⠛⠶⢶⡾⠟⠛⠻⣶⡶⠟⠃⠀⠀⠀⡠⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⢄⠙⣧⡀⢀⡀⠀⠀⠙⠁⠒⠛⠁⠀⠀⠀⠀⣰⠟⠴⠋⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⠀⠈⢷⢾⣿⣤⡀⢀⣄⠀⠀⠀⠀⣰⣿⣼⠃⠀⠀⠀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⣀⡀⢀⡹⠉⠹⣾⠿⣶⡾⢷⡾⠋⠈⠹⣀⣤⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠶⠶⠦⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢉⡡⠄⢒⣒⣒⣒⠂⠤⢍⡙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣡⢞⣡⣶⠿⠿⠿⢿⣿⣿⣷⣮⣙⢦⡉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡾⣱⡟⠋⠀⣀⣤⡶⠾⠿⣿⣿⠿⣿⣧⣻⡆⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡜⣱⠏⠀⢀⡾⠋⠀⠀⠀⠀⠀⠙⢦⡀⠙⢿⣿⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠈⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿⣴⣆⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣿⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢡⡘⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣴⡿⠀⣸⣿⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠳⣜⢿⣎⠻⢿⣷⣶⣶⣶⣾⡿⠛⢁⣴⡿⠃⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣷⣦⣌⣉⣉⣉⣁⣠⣴⡿⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣉⠛⠛⠛⠛⠛⠛⢋⣁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠾⠟⠛⠛⠛⠿⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣿⣷⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣇⣼⠇⠘⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠸⠟⠁⠀⠀⠀⠀⠀⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⢠⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⣀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣿⠀⠀⢀⡿⠀⠀⢀⣴⠟⠋⠁⠀⠉⠻⣶⡄
⠀⠀⢀⣤⠾⠋⣹⠇⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⢿⡀⠀⣉⠁⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠘⣿
⠀⣠⡿⠁⠀⠀⠉⠀⠀⣀⣤⡶⠟⠉⠀⠀⠀⠀⠀⣸⠿⠛⠉⠛⢷⣿⣛⣿⣧⠀⠀⠀⠀⠀⠀⣿⠁
⣰⡟⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⣦⡀⠀⠀⠀⠀⣸⡏⠀⠀⠀⢰⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⠶⠶⠶⠶
⢿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠀⣿⡇⠀⠀⠀⠀⢀⣼⠟⣀⣀⣤⣤⣴⠶⠾⠟⠛⠉⠉⠀⠀⠀⠀⠀⢀⣾
⠀⠉⠙⠛⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⢿⡄⠀⠀⠀⠀⠀⠀⣾⣷⣦⣤⡶⠾⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠘⣷⡄⠘⢷⣄⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠙⠷⠶⠾⠟⠋⢰⡏⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠶⠶⠶⠿⠟⠛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢠⣶⠟⠉⣁⣀⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⣿⠁⣰⠿⣯⣉⡿⠋⠁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠈⢉⣀⣴⠾⠋⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⣼⡟⠀⠀⣄⠀⠀⠀⠀⠀⠈⠙⠻⠟⠛⠉⠁⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢀⣴⣾⠉⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⢠⣿⠁⢹⣆⡀⠀⢠⡿⠁
⠀⠀⠀⠀⢹⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⢠⣟⣸⣷⠾⠛⠉⣰⡟⠁
⠀⠀⠀⠀⠘⣷⡀⠀⣿⣷⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⣆⠀⢸⡿⠋⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠘⢷⣄⣿⠻⣦⡀⠀⠀⠘⣿⠛⢶⣤⣀⠀⠀⠀⣀⣴⣿⣤⣤⣿⡄⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠿⠂⠈⠻⣦⣤⣀⣹⣧⠀⠈⠙⠛⠶⠾⠛⠁⣿⡀⢦⣹⡇⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⣯⠙⠂⠙⣧⡀⠀⣴⣯⠀⠀⠀⠀⠀⠀⢠⣴⠖⠚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠀⠀⠈⢷⣼⣧⣜⣷⡀⠀⣠⣶⣶⡟⢀⣴⢟⣿⠿⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠸⠷⠒⠀⠈⣿⠀⢹⣿⠁⠘⣿⠀⣿⠟⠛⢱⡏⠀⣀⣴⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⢿⠀⠀⣿⣦⠀⠻⣿⡛⠀⠀⠘⠛⠋⣻⣉⠉⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢀⠀⠈⠳⣄⠀⠀⢰⡏⢳⣄⠈⠻⡷⠀⢠⡾⢻⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠿⣄⠀⠀⠙⢷⣤⣶⠿⠀⣹⡇⣸⠃⠀⣼⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⡄⠀⠀⠀⠙⣷⡄⣰⣿⣿⡿⠀⢀⡿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⠀⠀⢀⣼⣿⣤⣟⠙⣿⡇⠀⣾⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⣠⡾⠛⠻⣿⡋⠻⠟⠀⢠⣿⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⢻⣷⣦⣤⣼⠇⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠾⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⣀⣠⠤⣾⠟⠦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀⣀⠀⠀⠀⠐⠊⠉⠠⠞⠁⠀⠀⠀⠈⠉⠙⠓⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⣋⣿⠟⠛⠓⠢⢤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⠿⠿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⢾⣏⣀⣤⠶⠦⣤⣤⣄⣀⣉⣉⣙⠛⠋⠉⠁⠀⠀⣈⣻⣆⢀⣀⣈⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⢀⣀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢠⢒⣐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢠⠕⢿⠓⠚⠑⡀⠀⠀⠠⠂⢉⣥⠛⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣀⠀⠿⠸⠨⠇⢀⡀⢁⠠⠂⠀⡠⠀⠀⠑⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠋⠙⠒⠒⠀⠤⠀⡀⠀⠈⠙⠣⣀⠰⠀⠀⠀⠀⠐⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠊⠁⠀⠀⠈⠁⠂⠤⡀⠀⠀⠠⠀⠀⠀⠱⡀⠀⠀⠀⠀⠃⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠢⡀⠀⣸⣶⡀⠀⠘⡄⠀⠀⠀⠘⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⣀⡀⠐⡀⠈⠈⡷⡀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠚⠂⠉⠃⠠⠀⡀⠀⣀⢀⠀⠉⠷⠀⠐⠀⠀⠙⠃⠀⠀⢀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠽⠀⠀⠀⠀⠆⠀⠀⠩⠀⠀⠈⡀⠀⠀⢠⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡆⠀⠀⠃⠀⠀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠃⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡤⠀⡀⠀⠀⠀⠈⣄⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢆⠀⠀⡄⠀⠀⠀⠀⠀⠀⣤⡄⠀⡄⠀⠸⣇⡈⡕⢰⠀⠀⠀⣇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣀⠀⠀⢀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⢿⡀⠇⠀⡔⠀⠀⣈⢢⠀⠀⡈⠀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠈⠀⠀⠂⠅⢀⠀⠀⠄⠀⠀⠀⠀⠀⠈⢘⠀⠀⠀⡄⠀⠀⠀⢸⣇⢀⠀⣠⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠐⠂⠠⢈⡀⠂⠊⠁⠀⠀⣤⣀⠀⠀⠀⢨⣿⠏⢰⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠆⠀⠀⡗⣇⠔⢖⡖⠲⣶⡆⠖⠐⠦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠑⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠷⡏⢢⠌⠑⠆⣼⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⣇⡏⠃⡤⡋⡐⢒⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢠⠳⣌⣀⠈⠁⠟⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠁⠲⢀⡀⠄⢀⠀⠀⠀⣀⠄⠀⠇⠈⠛⣭⣴⣞⣩⠏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠌⠁⠠⠄⠒⠈⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣹⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⠆⠀⢈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣟⣿⣿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢦⢀⠀⣈⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣦⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠀⠖⠀⠀⠀⠈⢛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣾⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⣰⠁
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠁⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂⠀⠀⠸⠋⠀⠊
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⢀⠰⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠤⠴⠿⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠲⠶⠤⣝⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⡤⠴⠂⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠿⠡⠄⠸⣷⠠
⠀⠀⠀⠀⠀⢰⠁⠀⠀⣀⣀⣀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⡸⠀⢀⣤⠶⠒⠛⢉⠓⣆⠀⠀⠀⠀⠀⠀⡞⠀⢀⣀⢀⣀⡀⣈⣽⣿⣇
⠀⠀⠀⠀⠀⠐⠀⠎⠁⠀⠀⠀⠀⠉⢲⡸⡇⠀⠀⠀⠀⠀⡘⠟⠁⠀⡅⠀⢸⠁⠀⠀⠻⠟⠀⣿⠀⠀⠀⠀⠀⢀⡇⠰⢩⣀⡀⠠⠤⠍⢱⢿⡟
⠀⠀⠀⠀⠾⡟⡞⠂⠒⠒⠒⠒⠀⠀⠀⢳⣷⠀⠀⠀⠀⢠⡇⠀⢤⣸⠆⠀⡿⠀⠀⠠⠠⢒⡒⠛⡄⠀⠀⠀⠀⣤⡇⡇⠸⠃⠀⠀⠀⠀⢸⠀⠛⣧
⠀⠀⠀⡈⠀⣿⠃⠀⡰⠒⠒⠊⠀⠀⠀⠘⣇⣁⠀⠀⠀⢸⣷⢸⠀⠈⠉⡇⠇⢠⠀⠈⠳⡍⠀⣤⢳⠀⠀⠀⠀⡏⡇⣧⢸⡶⠀⠀⠼⠀⢸⠀⡀⣿
⠀⠀⠀⡇⣭⣿⠀⠀⠑⢤⡀⢀⠄⠀⠀⠀⣷⠈⠂⠀⠀⠈⠘⡟⠄⢀⣴⠇⠀⣸⢸⡶⠂⠀⠉⠹⣼⠀⠀⠀⠰⠀⡆⢸⢈⡀⠐⠂⣠⡄⢻⡀⡏⣽
⠀⠀⠀⠑⠙⢧⡀⡀⠀⢀⣈⡁⢀⣀⣀⠴⠛⠂⠀⠀⠀⠀⠀⠳⣆⣀⣸⠀⠀⢻⠋⠀⣈⠀⢀⣀⠼⠀⠀⠀⠀⠳⠗⠚⠘⠁⠁⠀⠀⠒⠛⢉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠓⠛⠂⠈⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⣴⠚⢛⣡⡤⠤⢬⣿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣩⠭⠭⢭⣙⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠙⢉⡉⢤
⠀⠀⠀⠀⠀⡇⠀⡏⠀⠔⠈⠉⠵⠉⠙⠻⡄⠀⠀⠀⠀⠀⠀⡼⢡⣧⠖⠒⠒⠒⢢⡝⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡆⠐⠀⠠⠤
⠀⠀⠀⠀⠀⡇⢸⠀⣸⢠⠄⠂⠀⣆⠁⠒⢧⠀⠀⠀⠀⠀⢀⠇⢀⡟⠀⠀⠂⠁⠸⠻⡌⢿⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⡃⠀⠀⣀⣠⡤⠄
⠀⠀⠀⠀⠀⣧⢸⠀⣿⡇⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⢾⣤⡟⠀⠀⠀⠀⠀⠀⠀⢿⠿⡄⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⡇⣿⢠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣿⢸⠀⡇⠀⠀⠀⠀⠀⢁⠀⠁⠀⠀⠀⠀⠀⣀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣇⣽⠘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣰⣿⣸⠀⡇⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡤⠀⠀⠀⠀⠀⢻⠀⢰⢱⡀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢸⠏⣿⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠉⠃⡾⢛⢧⣄⡀⠂⠀⠀⢀⠀⢰⡇⠀⠀⠀⣶⠛⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠈⠀⣿⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠀⣿⠀⠔⠊⠉⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⡟⠇⢈⣤⣠⡼⠖⣿⠇⠀⠀⠀⢻⢀⠀⠀⠀⠀⠀⠀⠤⠴⠋⠀⠀⠀⡿⠀⠀⠀⠀⠀⢹⡚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛
⠀⠀⠀⠘⢦⡀⢀⣯⠀⠀⠀⠈⢉⣫⠵⠛⠉⠀⠀⠀⠀⠈⠿⣦⣤⣤⣄⣀⣤⣤⣤⣤⣤⠶⠂⠀⠀⠀⠀⠀⠠⢾⠧⠀⣀⣀⣴⣤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣟⣻⣻⣹⣻⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣽⣯⣿⣿⣯⣽⣿⣯⣭⣯⣽⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⢉⡩⠟⠋⠉⢦⣠⣀⡤⠴⠶⠄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣠⠖⠁⠀⣠⠔⠊⠉⠻⣿⠉⠁⠒⠂⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⣉⣮⠶⠋⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣦⠟⠙⣆⠀⠀⠀⣤⣸⡀⣠⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢽⠟⡁⠀⠀⢼⠂⠠⡎⠁⠀⢙⡃⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⠃⠉⠓⣾⡄⠀⡇⠀⢀⡜⢦⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠦⢾⡇⠀⠈⠚⠉⠀⢄⠀⠉⠓⠦⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠛⠦⠁⠀⠀⠀⠀⠀⢸⠳⡄⠀⠀⠀⠀⠉⠙⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠐⡆⠀⠀⠀⠀⠀⢸⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⠀⠀⠀⠈⡆⠀⠀⠑⢤⡀⠄⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⠉⠉⠉⠉⠉⠁⢸⠀⠀⠀⠀⢻⢦⡀⠀⠀⠉⠲⢤⣀⣀⠀⣀⡠⠞⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⢸⠀⠀⡠⢶⠉⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⢙⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸⡀⡠⢦⣘⣠⣔⣚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣡⠧⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⡹⢳⠁⠈⠁⠈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⡟⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣠⣼⢩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡄⠣⡄⠀⠀⠀⠀⣀⣼⢾⡅⠀⠀⠀⠀⠈⠃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡗⠀⠀⠀⠀⢹⡀⠀⠀⠀⢹⡀⠰⠶⠲⣋⠩⣭⣆⣀⡝⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠘⠀⠀⡇⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⣧⠀⠀⠀⠈⠉⢿⣁⣀⡜⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡇⠀⠀⢷⣄⣀⣀⣼⣧⣀⡀⠀⢀⣀⠸⡄⠀⠀⠀⠀⠘⣦⣼⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⢠⠃⠀⠀⠣⣠⠯⡴⢇⠸⣍⠉⠉⠙⠒⠛⠉⠑⠒⠒⠒⠂⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢸⣿⡀⠶⠚⡇⢀⠇⣸⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠣⠤⣤⣀⣀⠀⠘⣇⢧⠤⠚⠥⠞⠒⠃⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠉⠙⠶⠃⠀⡇⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢀⣄⠀⠀⢀⣈⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⣆⠀⠀⠀⢰⣤⡀⣀⣤⣤⣬⣶⣿⣿⣷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⢯⣭⣽⣿⠿⠛⢛⡇⡿⡄⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣿⣠⡀⡄⣤⠈⠑⢻⢹⡏⢰⣤⣼⡄⢣⣿⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠘⠛⠒⠛⠛⠋⠉⠙⠛⠛⠛⠉⠛⠛⠛⠴⠽⠯⠬⠍⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠅⠀⠉⣇⢢⣖⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⣶⠤⠤⠖⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠙⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢼⠿⠋⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣰⠜⠁⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢰⠁⢰⠋⠀⠀⣸⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠸⢀⠏⠀⠀⢰⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⣀⣠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠀⠀⢻⡀⠀⠀⠀⢀⠤⣲⠊⠡⠐⠒⠈⠉⠉⠁⠹
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⠀⠆⠀⠀⠑⠤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠒⠂⠡⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⡠⠄⠐⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣎⣇⠀⠀⠀⠀⣀⡤⠔⠋⠯⠝⠓⠉⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⣟⡉⠓⠀⠀⠐⠁⠁⠀⠀⠀⠀⠀⣀⠜⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡆⠀⠸⣇⡇⠀⠀⠀⠀⣀⣠⠤⠤⠒⠒⣋⡡⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⢧⠀⠀⠈⠛⠒⢒⣉⣩⠤⠤⠀⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠂⠒⠈⠉⠉