Sports Emojis & Symbols

⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢗⢀⠃⠁⢐⠸⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣴⡰⣲⡄⠀⣤⡟⣥⡏⠐⢀⠀⠄⠂⣺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢹⠐⣿⠁⣼⡟⠫⣴⠀⢃⠄⠐⠈⡌⠎⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⡜⢳⡆⠀⡖⣿⠀⣞⣀⠢⠉⢐⠒⠠⠁⢀⡂⠠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⢳⡄⢰⡀⡇⢸⠀⣿⠂⡎⠃⠐⠂⠠⠀⠠⢂⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠻⡄⢳⣼⢸⡄⣸⣆⢣⠁⠀⠀⠄⠀⠄⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠹⡄⠹⣶⠇⠘⣷⡾⡊⠲⠆⢀⠰⢠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡄⢿⣷⠀⠘⡿⣵⣬⢁⣂⡲⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡿⣎⠌⢦⠀⠈⢮⠉⢉⠽⠊
⠀⠀⠀⠀⠀⠀⡇⠘⣇⠈⢳⡀⢸⠀⡌
⠀⠀⠀⠀⠀⠀⡇⠀⠻⣆⠀⠙⣯⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠒⠒⠖⠒⠤⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠷⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡰⠋⠑⠀⠀⠀⡀⠂⠀⠀⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⡁⡄⣷⡀⠀⠀⠈⢦⠀⠀⠀⢀⡞⠀⠀⠀⠈⡶⠊⠀⠀⠀⠀⢀⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⣿⣧⠀⠀⠀⠀⢣⠀⠀⡸⠀⠀⠠⠐⠈⠀⠀⠀⠀⡠⠂⠁⠀⠀⣀⡤⠞⠓⠒⠒⠒⡆⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⣧⣿⣿⢧⡀⠀⠀⠀⢳⡖⠉⠚⣅⠀⠀⠀⠀⠀⢀⠚⢄⠀⠀⣠⠞⠁⢣⡀⠀⠀⠀⠈⠙⠢⡈⢢⡁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣏⠁⠘⢷⡄⠀⠀⠀⠱⡀⠀⠘⣦⠄⠀⠀⠀⠂⠀⠀⠑⠴⠁⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⢸⡄⢀⡔⠛⠃⠷⣄⠀⠻⣄⠀⠀⠀⠰⡄⠀⠘⡄⠀⠀⡆⠀⠀⠀⢀⣧⡠⠔⠈⠻⣿⠿⠗⢀⠀⠀⡀⠄⠛⠿
⠀⠀⣠⠒⠉⠁⠙⣇⠬⠷⢀⠀⠀⠈⠳⠤⠈⢂⠀⠀⠀⠘⢄⡰⠃⡏⠈⠁⠂⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⡇⠀⠀⠀⠈⢧⡀⠀⠀⠉⠒⠤⠄⠀⠀⠘⢧⡀⠀⠀⠈⢣⣀⠘⢆⠀⠀⠐⠀⠀⠷⢄⠀⢀⣀⣄⠀⠀⠀⠀⡀
⠀⠀⢣⡀⠀⠀⠀⠎⠙⡇⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠹⡑⡄⠉⠈⠹⠤⠤⠒⡀⠙⣿⣿⣿⣷⡄⠀⠈⠁⠀⢄⣀⡰
⠀⠀⠀⠉⠀⠀⠒⠛⢄⣘⣦⡽⠂⠀⠀⠀⠀⠀⠀⠀⣈⣦⠀⢀⠀⣧⣸⠀⠔⠃⠀⠀⠀⠑⢤⠘⠻⠿⢏⠀⠀⠀⠀⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⠊⠑⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠈⠧⠄⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤
⠀⠀⠀⠀⠀⠀⡔⠶⢖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠐⠈⠱⡤⢈⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡆⢠⣻⣓⢲⣹⡇⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠘⠙⢻⠶⠊⡁⠀⠙⠃⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⠄⣿⠡⣤⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠴⠆
⠀⠀⠀⠀⠀⡿⠿⣶⣷⣾⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠿⠜⣿⣧
⠀⠀⠀⠀⠀⠌⠂⠐⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠚⠻⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⢀⠀⠺
⠀⠀⠀⠀⠀⠞⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⣭⣧⣀⠀⣀
⠀⠀⠀⢰⣤⣀⡈⣯⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣖⣐⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⡽⢉
⠀⠀⠀⠀⠉⠙⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣾⣇⣹⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⢤⣶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠙⠋⠉⢉⣿⣷⣴⣿⣦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢴⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⠁⠉⠉⠢⠀⢀⡐⢀⠀⠀⠀⠀⠀⠀⢠⡟⠘⠛⣿
⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠁⠀⠀⠀⠀⠘⠂⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⠆
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⢤⣦⡀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⣴⣥⠞⠀⠀⡠⢴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠚⠶⡀⠀⠀⠀⠀⠠⢿⣀⠸⣄⣁
⠀⠀⠀⣀⠀⡷⠟⠁⠀⠀⠀⠹⠯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠁⠠⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠈⠳⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡬⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠀⠈⠹⠾⠇
⠀⠀⠀⠀⠀⢻⣿⣶⣤⣀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⣇⡀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣮⠁⠀⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡟⠛⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣽⠟⢋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⠉⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠈⡹⣿⣴⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⣤⣧⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⢰⣥⣤⠆⠈⠙⠂⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⠶⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠘⠀⠜⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⢀⡀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣀⡘⣁⡀⢀⣾⣿⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠾⠟⠀⠀⣠⣾⣿⣿⣿⠟⣿⠿⢿⣿⣿⣧⣌⣉⣩⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡎⠉⣩⣤⣔⠀⠈⠀⢸⣿⣿⡟⠉⠀⠀⠀⠀⠈⠿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⡇⣿⣿⣩⣽⣦⠑⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠦⠀⠀⣨⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣮⣿⣿⠿⠿⣲⣄⠸⣿⣿⣿⡖⣀⡀⠀⢠⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣩⡽⠛⣋⣵⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⢻⣿⣿⠿⣿⣿⣿⣿⠋⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣿⣿⡿⠋⣿⣿⠋⠵⠚⠉⠁⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠻⠿⠧⠈⠩⠀⠀⢀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣅⠀⡰⠀⠀⢠⣠⣾⣿⣄⣴⠾⢋⠁⣁⣀⠀⠤⠛⣁⢀⢰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⣸⣤⣾⣿⣿⣿⢏⣁⠴⠊⠀⢹⡄⠡⠠⢾⣟⢢⣸⣿⠀⠀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢘⣭⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡿⠉⠂⢀⣐⣢⡶⣾⡇⢸⣿⣨⣿⣿⠿⠛⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⣿⠏⠁⢸⡏⠙⠛⠟⠛⠋⢀⣉⣠⣽⣿⣷⣦⣍⡤⠿⢿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠁⠀⠀⢤⣷⣤⡀⠀⠀⠀⠈⡻⢿⣿⣿⣿⣿⣿⡇⢠⣿⣬⠀⠄⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣭⣶⣾⣾⡇⣤⠀⠈⠋⠁⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⠀⢾⣿⠇⣀⣴⢀⡀⠀⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡛⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⢿⣦⣉⣀⣿⣿⣿⣷⣤⣶⣶⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠒⠠⠘⠟⠀⠀⣸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠹⠿⠿⠿⠿⠒⠾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡛⠻⠿⢿⡿⠿⠷⠄⠜⠛⠛⣿⣿⣶⡞⠃⠀⠘⣿⠿⣿⣋⣉⠚⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣿⣿⣦⡜⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⠀⡀⢭⠔⠀⠀⠉⠁⠀⢹⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣿⣿⡀⠀⢀⠴⢤⠄⢀⣴⣋⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠿⢿⣿⣶⣄⣤⠀⢀⣀⣼⣿⣿⡿⠛⠁⣠⢄⠱⣦⣄⣴⡿⢟⣛⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣶⣆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣾⣿⣾⡿⢟⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠙⠿⣿⣧⠉⠙⠛⠋⠺⠷⠿⠛⠻⣽⣴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⡀⠤






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣴⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⢀⠤⡂⠀⠉⠙⡌⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠉⠙⡆
⢀⠔⠁⠀⠈⠓⣄⠀⠸⠀⠀⡜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠙⢗⠢⢌⣒⠒⠁⠀⣀⣀
⡄⠀⠀⠀⠀⠀⠈⣇⠀⡇⡼⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢠⣚⡉⣉⣥⣴⢾
⡧⢄⣀⠀⠀⠀⠀⢸⠀⣧⣃⡠⠞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡴⣩⠖⠊⠉
⢱⡀⠀⠈⠉⠐⠒⡞⢺⠛⡅⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⡴⠁
⠀⠑⢤⣀⠀⣠⠞⠀⡞⢀⡹⠖⠁⠀⠀⠀⠀⣠⠔⠐⠒⢔⠒⠦⢤⡀⠀⠘⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠟
⠀⠀⠀⠀⠉⠉⠒⠋⠉⠉⠀⠀⠀⠀⠀⢠⠎⢀⡉⢦⡀⠈⢂⠀⢡⠈⢣⡀⠘⠢⢝⡒⠶⠤⠤⠴⣒⡪⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡔⠁⢀⡠⢷⠀⢸⠀⠘⠀⢸⢳⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢀⣀⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⡠⠃⠀⣀⣄⢸⠀⢠⣀⣸⠼⡇⠀⠀⠀⠀⠀⠀⠀⡠⢚⣅⡢⠾⣿⠷⢦⡑⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠸⡇⠀⠁⢀⠾⢄⣀⠈⠉⠉⢀⣀⢴⠇⠀⠀⠀⠀⠀⢠⢊⣴⢷⡏⢨⠞⢄⡜⠃⢼⠸⡆
⠀⠀⠀⣠⠔⠛⢧⡔⠋⠀⠈⢟⠦⡀⠀⠱⣄⠀⢸⠤⣀⠈⠉⠉⢉⣁⣠⠞⠀⠀⠀⠀⠀⠠⢡⣳⠗⢅⡜⠑⠦⣎⠈⡰⢺⠀⡇
⠀⠀⡌⠀⠀⠀⢠⡀⠀⠀⠀⡘⠲⣝⣆⠀⠈⠑⠮⢷⣤⣬⣭⡭⠽⠛⠁⠀⠀⠀⠀⠀⢠⢣⡿⠋⡦⡂⢈⠖⢄⡠⠋⡲⡞⢰⡇
⠀⡜⣸⠦⠤⡖⠁⠙⠦⡠⠴⣄⠀⣸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣼⢉⠶⣀⡼⠓⣤⡊⢩⠾⡕⡡⢣⠁
⠀⣏⠀⠀⠀⠱⣄⣀⡜⠀⠀⢠⠋⠈⢦⡇⠀⠀⠀⠀⠀⢀⠤⠀⠲⠤⡀⠀⠀⠀⠀⠀⡇⡧⠉⣦⡎⠑⠴⣁⡜⡳⢊⠔⡱⠃
⠀⠘⣄⢀⣀⠊⠀⠀⠑⢢⠖⠣⣀⠀⡼⠁⠀⠀⠀⠀⢠⠊⠐⠢⡀⠀⢈⣆⠀⠀⠀⢐⢠⡘⠶⣁⣨⡳⠤⢛⡩⢔⡡⠊
⠀⠀⠈⠳⢄⣇⠀⢀⠠⠯⣄⣰⠧⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠆⠊⢸⠀⠀⠀⠌⣬⠶⢋⡠⠴⡒⠉⠁⠂⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣠⠔⠋⢉⡷⠃⠀⣀⡌⢈⡵⢊⠕⠊⠁⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠒⠒⠲⢤⠤⣀⡀⠀⠀⠈⠉⠉⠁⠀⢀⠾⠉⢍⢫⠈⠀⠀⠀⠀⢸⡀⣀⣸
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⣀⣠⢤⠴⠤⢦⠈⡍⡗⠀⠀⠀⠀⠀⡰⡉⢦⢱⢸⠎⠀⠀⠀⠀⢀⣼⣖⣗⣮⣄
⠀⠀⠀⠀⢀⠜⢄⠀⡠⠴⠎⠙⠉⠀⠀⠀⠀⠀⠇⢸⠃⠀⠀⠀⠀⡾⠤⡙⡼⡤⠃⠀⠀⠀⠀⠀⣾⢏⡏⢿⢙⢿⣄
⠀⠀⠀⠀⠊⠡⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠑⢄⣈⡷⠁⠀⠀⠀⠀⢀⣎⡟⣹⠓⢼⠉⣍⣷⢆
⠀⠀⠀⠀⠀⠀⢱⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠾⢁⣼⠑⣾⠁⣤⢹⠛⠧⡀
⠀⠀⠀⠈⠢⢄⣼⣠⠁⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠘⣅⠘⡹⠃⠫⣀⣨

⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠀⠀⣰⣾⠤⠀⠀⠀⠀⡠⠄
⠀⠈⢄⠀⢀⣀⢀⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⣀⠔⠉
⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⢀⠀⠀⠡⣀⠀⠀⠀⠔⠀⠀⠠⢂⠴⠊⠀⠀⠀⣠⠆
⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⡠⠈⠀⢀⠀⠀⠉⠈⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠰⠛⠂⠀⢴⣢⣄⠀⠀⠉⠁⠀⠈⠀⠀⠀⠙⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠁⠀⠀⠀⠈⢪⠘⣆⠀⠀⠀⠀⣦⠀⠀⠀⠠⠀⠀⢀⠔⠉⠀⠀⠀⠞⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣺⠟⠓⠲⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⡀
⠀⠀⠀⠐⠤⣤⠠⠔⠀⠀⠑⠤⠀⠀⢠⠏⠓⣾⠇⠀⢇⠀⠀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⡞⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢜⠁⠀⣠⠞⠀⠀⠀⠀⠀⡰⠀⠐⠂⠀⠀⠀⠀⠀⠃⠀⢠⡞⠀⠀⠀⠀⢐
⠀⠀⠀⡯⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠊⠀⠀⢠⣼⡦⠀⠀⢃⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⢤⡦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠈⠲⢄⡀⠈⠀⠀⠀⣀
⠀⢸⡀⠀⠀⠀⠀⣠⣾⡀⣀⣀⠀⠀⠀⠀⠈⠁⠘⠉⠉⠀⠀⠀⠀⠴⠠⢰⠄⠀⠀⠀⡠⠀⢀⡴⠶⠀⠀⠒⠁⠀⢠⠀⡄
⠀⠀⠳⢤⠀⠐⠛⠛⠻⠻⣛⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⢀⡀⠀⠀⡐⠂⠀⢀⣠⠤⢤⠀⢀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠣⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡈⠁⠀⢀⣿⠇⠰⣿⣿⡆⠃⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣶⣤⠄⡘⠁⠀⠀⠈⡉⣄⣀⠉⠩⣠⡤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠹⢿⣿⠂⡴⠿⠁⠀⠀⠀⡇
⠀⠀⠁⠀⣀⠀⢀⠀⠀⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⡀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠚⠁⠀⠀⠀⠀⠠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡠⠖⠋⠉⠈⠉⠒⢄⠀⠀⠀⣠⠔⠋⠉⠉⠉⠒⢤⠀⠀⠀⢀⠔⠚⠉⠁⠉⠓⢤⡀⠀⠀⢠⠖⠉⠒⠢⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⡰⠁⠘⢡⡄⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠑⢦
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡇⠴⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢣⡀⢿⠧⠀⠀⠀⠀⠀⡼⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠑⢄⠀⠀⠂⠀⠀⣀⠇
⠀⠀⠀⠑⠤⣀⣀⣀⣀⡤⠊⠀⠀⠀⠙⠦⣀⣀⣀⣀⡠⠚⠁⠀⠀⠈⠢⣀⣀⣀⣀⡠⠞⠁⠀⠀⠀⠀⠀⠙⠢⠤⠠⠔⠁
⠀⠀⢀⡀⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡤⠄⣀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣀
⠀⠀⡇⠀⠀⣠⠄⠉⠲⢄⠀⠀⠀⢀⣴⠁⢀⣴⣶⣄⠀⢱⣄⠀⢀⡔⠋⠀⠀⠀⠀⠀⠙⢆⠀⠀⡴⠋⠀⠀⠀⠀⡈⠙⢆
⠀⠀⢧⠀⠀⠀⠑⢕⠄⠀⠱⡀⠀⣾⠃⠀⠈⠿⠿⠏⠀⠀⠹⡄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠁⠀⠄⠀⢠⠀⡀⠂⠈⡆
⠀⠀⠈⢆⠀⠀⠀⠀⠈⡦⠀⢱⠀⠇⢠⣤⣄⠀⠀⠀⣠⣤⢀⠇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠐⠑⠁⠀⠇⠈⠌⠀⢂⡇
⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⢘⠀⠈⢾⣿⣿⠆⠀⠐⢿⣿⠏⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠳⡀⠀⠀⠀⡐⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠀⠀⠀⠉⠒⠢⠤⠔⠊⠁⠀⠀⠀⠀⠉⠒⠶⠤⠔⠚⠁⠀⠀⠀⠀⠉⠓⠲⠦⠖⠚⠁
⠀⠀⠀⢀⠴⠂⠉⠉⠑⠢⣄⠀⠀⠀⢀⡤⢆⣉⠉⠓⠢⣄⠀⠀⠀⢀⡴⠒⠻⠉⠓⠲⢄⠀⠀⠀⠀⠀⠀⡐⠳⢤
⠀⠀⡰⣡⡶⡻⣶⡄⠀⠀⠈⢣⠀⠰⠃⠀⠀⠀⠀⠀⠀⠈⢣⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢰⠄⠀⠀⠙⢄⡀
⠀⠀⡃⣿⠰⢡⣾⡇⠀⠀⠀⠈⠃⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⣿⣦⡀⠀⠀⠀⠑⣄
⠀⠀⢣⠈⠛⠟⠋⠀⠀⠀⠀⣰⠁⢣⠀⠀⢀⠀⠁⠀⠀⠀⣰⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⢴⣿⣿⣿⣿⣦⣄⠤⠔⠊
⠀⠀⠀⠳⢄⡀⠀⠀⠀⣀⠔⠁⠀⠀⠑⢄⡁⠀⠀⣀⣀⠔⠁⠀⠀⠑⢤⣀⠀⠀⠀⣠⠔⠁⠀⢀⠀⠙⡿⠛⠉
⠀⠀⠀⠀⠀⣉⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠁⠈⢀⣀⣀⣀⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠢⡀⠀⠀⡴⠋⠀⠘⣄⠀⠈⠲⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠓⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠑⣄
⠀⠀⡞⢰⡷⠀⠀⠀⠐⣷⠆⢹⠀⡜⠀⠀⠀⠀⠈⠱⢄⡀⢘⡄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡗⠒⠠⣀⠀⠀⠀⠀⠈⠉⠇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⣄⠀⠀⣠⣀⠀⠀⢀⠎⠀⠘⣄⠀⠈⠳⡄⠀⠀⢀⠜⠀⠘⢄⠀⠀⠀⠀⠀⠀⢀⡜⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠈⠑⠶⠬⠥⠤⠚⠁⠀⠀⠀⠈⠑⠢⠄⠼⠤⠒⠁⠀⠀⠀⠀⠑⠢⠤⠤⠤⠒⠉⠀⠀⠀⠀⠑⠢⠤⠀⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⠛⠉⠉⠉⠉⠉⠙⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣴⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⣡⠿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⢽⣒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣾⣣⡼⠖⠋⠉⢑⡜⠂⠀⠀⠀⠉⠉⡿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢙⣲⣤⠤⠤⠤⣿⠄⠀⣀⣀⠞⠁⠀⠀⠀⢀⡀⠀⢧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠘⠄⠀⠁⠀⠀⠘⠉⠉⠁⠉⠁⠀⠀⠀⣠⣏⠏⠁⠀⣁⡤⣤⣄⠹⡄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⢛⡶⠋⠀⠀⣀⣀⣉⣁⣠⠼⠁⠀⣠⠀⠀⣦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠉⠒⠦⠤⠤⠤⠤⠴⠒⠋⠉⠀⠘⢉⡴⠚⠑⠺⣄⠀⠀⠈⠑⠢⠤⠴⢻⣀⣠⠿⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⣀⣀⣀⣀⣠⠤⠤⠒⠚⠙⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡖⠒⢜⣯⣍⣠⣟⠔⠊⣉⠆⠀⠀⠀⠀⠀⠀⠀⣠⣞
⠀⠀⠀⠀⢀⣴⣾⠏⠁⠉⠹⣿⣦⣄⠀⠀⠀⠀⠀⠀⠸⣉⣱⢇⡠⠤⣽⠏⠇⠠⢐⠄⠁⠀⠀⠀⠈⣦⡠⢾⡿⢿
⢠⠞⡑⠠⡟⠋⠑⣄⣀⣀⡠⠋⠙⠻⣆⠀⠀⣀⠤⡄⠀⢉⣒⣊⣀⣴⠃⠀⠢⢈⡁⠔⠁⠀⠀⠀⠀⣧⢠⠼⠀⠘⠓⡄
⡄⠀⠇⣶⡅⠀⢠⣿⣿⣿⣇⠀⠀⠀⣻⡀⢈⡟⢠⢞⣆⣸⡄⠀⠀⠈⠙⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢻⠈⢆⣠⠤⠚⠁
⠃⠀⠀⢻⡟⠀⠙⢿⣿⠿⠛⠒⢤⣴⣿⠁⠈⠻⡈⢧⠚⢇⣻⣄⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠚⠛⠦⡄⢡⣊⠙⢶⡄⠀⠀⢄
⠀⠀⠀⠈⢅⠀⠀⠀⡇⠀⠀⠀⢰⣿⠃⠀⠀⠸⢵⡌⠁⠛⠁⠈⠳⡀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⢺⠉⣇⠀⢣⠀⠈⡆⠱⠀⠀⠑
⠀⠀⠀⠀⠀⠑⠲⠾⢿⣶⡶⠖⠋⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠒⠊⠁⠀⠱⣄⠀⠀⠀⠀⢀⣴⣿⣁⡼⢾⡽⣇⢈⡿⠀⠁⠀⠄⠘⠃⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠤⠖⠉⠀⠀⠀⠀⠈⠫⣈⡼⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢠⠀⡄⢀⠀⡄⢀⠀⡄⠀⡀⠀⡀⠀⡀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠫⠉⢍⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⠍⠊⠀⠀⠀⠀⠈⠒⢍⡻⢿⣿⣿⣿⡿⠟⢋⠡⠐⠀⠀⠀⠀⠉⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄
⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦
⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠟⡁⠤⠈⢐⡀⠀⢀⡀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀⠄⠀⠒⡈⠈⠀⠙⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠟⡡⠊⠀⠀⠀⠀⠀⠁⠀⠒⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢩⠄⠒⠀⠈⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⢸⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⠁
⠀⠀⠀⠸⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠉⠉⠉⠉⠉⠉⠩⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠇
⠀⠀⠀⠀⢳⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡞
⠀⠀⠀⠀⠀⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠞
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⣤⣤⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⣤⣤⣶⣶⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⠬⠥⠤⠤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⣀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⡠⠊⢱⠤⠒⠒⠈⠙⢆⠀⠀⠀⣰⣿⡘⣯⣵⣶⣶⣶⡄⠀⠀⢀⢾⡁⠀⢀⡤⠤⣌⢢⡀⠀⢠⡸⣿⣿⠿⣛⣿⡛⢦
⠀⢠⣇⠀⠀⠣⣀⡠⠤⣄⣈⡆⠀⠐⢿⣿⣷⣝⠿⣟⣛⢛⡿⠀⠀⡎⠀⠉⠉⠁⠀⠀⠀⢣⠇⠀⣿⣿⣶⣶⣿⣿⣿⣿⡎⡇
⠀⠘⡏⠣⣄⢀⡎⠀⠀⠇⢀⠇⠀⠸⣮⡻⢿⡿⣸⣿⣿⣸⣿⠀⠀⢇⠀⣀⣀⡀⠀⠀⠀⡜⡆⠀⣿⣿⠿⠿⣿⣿⣿⣿⢇⡇
⠀⠀⠙⢄⣨⠏⠀⠀⣼⡠⠊⠀⠀⠀⠙⢿⢗⣵⣿⣿⢇⡿⠃⠀⠀⠈⠻⣀⠀⠈⠙⠒⣫⠜⠀⠀⠈⢵⣿⣿⣶⣭⣯⣥⠞
⠀⠀⠀⠀⠉⠑⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉
⠀⠀⠀⣠⡶⠓⠀⠺⣶⣤⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡙⠻⣦
⠀⠀⠈⠁⢀⣤⣤⡆⠀⠉⠀⠀⠀⢀⣾⣿⡟⠛⠛⣿⣿⣾⣧⠀⠀⠀⠀⠀⠀⠀⣠⡦⠙⢷⡄⠀⠀⢀⣾⣿⣿⠏⢙⣷⣌
⠀⢰⣦⠀⠸⢿⡿⠇⠀⣠⣶⡆⠀⠈⠻⣿⣅⡀⢀⣽⣿⠟⠙⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣾⣿⡿⠁⣰⣿⣿⠏
⠀⠀⠿⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⢦⣀⠙⠁⠀⠀⠀⠀⠀⠀⢈⠻⣿⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠐⢾⣿⡶⠁⠀⠀⠀⠀⠀⠀⠙⠏⠁⠈⠹⠟⠉⠀⠀⠀⠀⠀⠉⠳⠦⠀⠀⠀⠀⠀⠀⠘⢷⣦⡉⠻⠛⠁

⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣴⣶⣷⣶⣦⡀⠀⠀⠀⠀⢀⡤⠂⠈⠀⠒⢀⡀⠀⠀⠀⢀⢰⣶⣿⣷⣶⠄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠠⢁⠇⠀⠀⡴⠚⠉⠑⡀⠀⣰⡇⣾⣿⡿⢫⣴⣾⣷⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣿⣿⣿⣿⣿⣿⠀⠀⣧⡞⠀⠀⢸⠁⠀⠀⠀⡁⠀⠛⣼⣿⣿⠅⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣿⣿⣿⣿⠟⠀⠀⠸⡀⠀⠀⠸⡄⠀⠀⢠⠃⠀⠸⣿⣿⣿⡅⢿⣿⣿⡿⠁
⠀⠀⠀⠁⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠈⠂⠄⠀⠉⠶⠒⠁⠀⠀⠀⠈⠻⠿⢿⡦⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⣀⡠⠄⢀⡀⠀⠀⠀⢀⣿⣧⢀⣀⣀⣀
⠀⠀⠈⢿⢆⠀⠀⠀⠀⡀⠀⠀⠀⠙⣤⡀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠈⢀⠞⠁⠦⡀⠀⠈⢢⠀⠀⣾⣟⣵⣿⣹⢿⣿⣷⣄
⠀⠀⠀⠀⢂⠁⠒⠈⠁⣔⠀⠉⢐⡄⠘⣿⣶⣶⣿⡟⠽⣷⣶⣶⠦⠀⠀⢸⠀⠘⠂⠉⠀⠀⠀⡇⠀⢿⢹⣿⣧⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠢⠀⢀⣀⡁⠤⠀⠀⠀⠀⠘⠿⠿⠿⠷⠽⠟⠛⠛⠉⠀⠘⢌⣄⠀⠀⠀⠀⢀⡔⠀⠀⠘⠯⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁
⣟⣿⣻⣻⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣿⣿⣛⣛⣟⣿⣻⣿

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⡄⠀⢀⡀⢠⠀⠀⠀⠀⠀⠠⠖⡲⢦⡀⠀⠀⠀⠠⡊⢳⣿⣿⣿⡏⢑
⠀⠀⠀⠀⣤⣷⣿⠇⠀⠀⠀⠀⠀⠈⢃⠀⣾⡿⠟⠀⠀⠀⠀⣈⠂⠜⢠⠃⢽⠀⠀⠀⠀⠙⠺⣿⣿⣿⠗⠁
⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀⠠⣶⣾⡟⢀⣤⡀⠀⠀⠀⠀⠘⢕⣒⣣⡈⠎⠀⠀⠀⠀⠀⠀⢉⣿⠁
⠀⠀⠀⠀⠘⠿⠛⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛

⠀⠀⠀⠀⢀⠀⢀⣤⡄⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣧⡂⠀⠀⠀⠀⠀⢠⣤⡄⣀⡀
⠀⠀⠀⠀⢨⣿⣿⡿⠋⠀⠀⠘⢟⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣰⣿⡣⣹⣦⡄⠀⠀⠀⢻⣿⣧⣿⡿⣦
⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠻⡟⠉⠉⠉⠀⠀⠀⠀⠀⠰⢟⣱⠿⢛⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⡥
⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠞⠋⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⢀⡈⣶⣷⠆⢳⡀⠀⠀⠀⢀⡤⡴⠗⠳⠤⢤⠄⠀⠀⠀⢠⣿⢃⡼⣭⣤⠀⠀⠀⠀⠀⣼⡙⣎⣣⠶⣀
⠀⠀⠘⢇⣈⠉⢰⣶⠀⠀⠀⠀⠀⠙⣷⠀⠀⣴⠃⠀⠀⠀⠀⠰⣟⣶⣽⣷⡝⠀⠀⠀⠀⡼⠡⠗⢉⣤⠶⣉⡄
⠀⠀⠀⠈⠛⠃⠈⠁⠀⠀⠀⠀⠀⠰⠿⠛⠛⠾⠀⠀⠀⠀⠀⠀⠈⠛⠓⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⡿⣶⠄⠀⠀⠀⠀⠀⠘⣿⠇⣴⡿⠃⠀⠀⠀⢐⣴⣿⠟⢻⢹⣿
⠀⠠⠞⠋⡳⠤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⣶⣿⣿⡾⠋⠀⠀⠀⠀⠀⣾⢏⠘⡐⣼⡼⡋
⠀⠀⠀⠀⠁⠢⢈⠗⢦⡄⠀⠀⠀⠀⠛⠛⠟⠃⠀⠀⠀⠀⠀⢻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠿⠦⠾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣿⣿

⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠔⠁⠀⢀⠀⠉⢢⡀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⠔⣅⠀⠀⠀⡨⠢⡀⠀⠀⢀⣤⣻⣿⣿⣿⢟⣤⡀
⠀⢸⠀⠀⠀⠀⠀⠠⠄⡃⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡌⠀⢸⡄⠀⢠⡇⠀⠁⠀⠀⢸⣿⣧⣿⣿⣿⣼⣿⡇
⠀⠸⡀⢀⠀⠂⠀⠀⣠⠃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠰⡀⡸⠁⠀⠀⢇⢀⠆⠀⠀⠸⣿⢏⣿⣿⣿⡹⣿⠇
⠀⠀⠈⠲⠤⠠⠤⠞⠁⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⠣⠄⠀⠠⠜⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠗⠁

⠀⠀⢀⠄⠒⠀⠒⠠⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⡀⠀⠀⠀⠀⠠⢔⣂⣀⠐⠢⢄⠀⠀⠀⠀⣠⡴⠶⠿⣶⣦⣄
⠀⢰⠁⠀⠀⠀⠀⠀⠽⠆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣻⡄⠀⠀⣐⡝⠁⠀⠀⠙⡄⠈⠂⠀⠀⣰⢫⣾⣿⣿⣮⢻⣿⡆
⠀⢀⠀⠀⠀⠀⠀⠀⡅⡂⠀⠀⢸⣿⣿⣿⣿⣿⣿⢾⡇⠀⠀⠌⡇⠀⠀⠀⠀⠸⣄⠘⠀⠀⢿⢸⣿⣿⣿⣿⣧⢻⡿
⠀⠈⠢⡀⠀⠀⠀⣀⠜⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⢜⣦⡀⠀⠀⠀⡨⠁⠀⠀⠈⢷⡻⣿⣿⣿⣿⡷
⠀⠀⠀⠈⠁⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠂⠂⠁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠋⠁
⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⢀⢔⣿⣷⡆⠀⠉⠦⡀⠀⠀⢀⣴⠋⠉⢽⣿⣷⣤⡀⠀⠀⢀⠎⠁⠀⠐⠆⣌⠱⡀⠀⠀⢀⣾⣿⣿⣯⣹⠻⣷⡀
⠀⠈⠀⠙⠉⠀⠠⣴⣆⠁⠀⠀⢸⣿⣶⣾⣿⣿⠋⠹⡇⠀⠀⡂⠀⠀⢀⣄⠀⠀⠀⢠⠀⠀⣾⣿⣿⣿⠻⣿⣿⣿⣧
⠀⠸⣤⣶⣄⠀⠘⠻⢋⠇⠀⠀⠸⡟⠙⠿⣿⣯⣄⣴⠇⠀⠀⠡⡀⠀⠀⠉⠀⠀⢀⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠈⠛⠏⠤⠤⠖⠁⠀⠀⠀⠀⠈⠲⠾⠿⠛⠋⠁⠀⠀⠀⠀⠑⠢⠀⣀⠠⠔⠂⠀⠀⠀⠀⠘⠻⢿⣿⡿⠟⠋

⠀⠀⠀⠀⣀⠄⢤⡄⠤⠀⠀⠀⠀⠀⠀⣀⣤⡀⢤⣤⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⢠⠊⠀⢀⣤⠙⢶⠀⠀⠀⠀⢠⣾⣿⡿⢛⣷⡙⠀⠀⠀⠀⠀⡠⠊⠀⢠⠂⢘⠀⠀⠀⠀⠀⣠⣾⣿⣟⣿⣿
⠀⠀⣇⠀⣴⠟⠁⠀⡔⠀⠀⠀⠀⠻⣿⠋⣡⣾⣿⡟⠀⠀⠀⠀⠎⠀⣀⠔⠁⡠⠊⠀⠀⠀⠀⣼⣿⢿⣿⣿⡿⠃
⠀⠀⡉⢳⣄⣀⠤⠊⠀⠀⠀⠀⠈⣷⣝⠻⣿⠿⠋⠀⠀⠀⠀⠀⠳⣈⢀⡠⠊⠁⠀⠀⠀⠀⠀⠹⣿⣿⠿⠋

⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⡔⠉⡍⠁⠀⠑⡄⠀⠀⠀⡠⠚⠡⠊⠀⠉⢢⡀⠀⠀⠀⡴⠋⠀⢀⠀⡙⢆⠀⠀⠀⡠⠊⠁⣀⣀⠈⠳⡀
⠀⢸⠲⠀⠀⠐⠐⠐⢽⠀⠀⠀⡇⠈⠦⠀⠀⠐⠀⡇⠀⠀⢸⠀⠀⠀⠈⠠⠄⢸⠀⠀⠀⡃⢰⠋⣶⡆⢱⠀⠁
⠀⠘⣄⠅⠀⠣⡀⣀⡜⠀⠀⠀⠳⣁⡆⠀⠀⠀⣰⠃⠀⠀⠘⣄⠀⠀⠀⠀⢀⡜⠀⠀⠀⢣⡈⠢⠽⠥⠊⣠⠃
⠀⠀⠈⠑⠒⠒⠛⠁⠀⠀⠀⠀⠀⠈⠉⠓⠒⠉⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁

⠀⠀⡠⢚⡽⠋⠉⠲⡄⠀⠀⠀⢀⣴⠉⠉⠉⠑⣄⠀⠀⠀⠀⣠⠖⣩⡙⢉⡲⡀⠀⠀⠀⣀⡠⠤⠤⠤⢤⣀
⠀⢰⢱⠃⠀⠀⠀⠀⣸⠀⠀⠀⡏⢸⠀⠀⠀⠀⡇⡇⠀⠀⢸⠡⠈⢄⠈⠀⠐⢻⠀⣠⢚⠁⠀⠀⠀⠀⠀⢸⡓
⠀⠸⡈⢆⡀⠀⣠⠞⡸⠀⠀⠀⢧⠈⠲⠤⠤⠞⢠⠃⠀⠀⠸⡔⢢⡀⠡⠀⠤⡻⠀⠘⢿⣍⠉⠀⠀⠉⣉⡽⠟
⠀⠀⠙⠢⠬⠭⠕⠊⠀⠀⠀⠀⠀⠑⠦⠤⠤⠒⠁⠀⠀⠀⠀⠈⠳⠤⠦⠥⠊⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⢀⠤⡶⢲⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢤⢒⢢⠛⡿⠾⡆⠀⠀⠀⠀⣠⠴⡶⠖⡤⣀
⠀⢠⠃⠐⠀⠀⠀⠀⢳⠀⠀⢠⡖⠉⠁⠀⠀⠉⠑⢦⠀⠀⠀⢸⣿⢸⢰⢻⣸⠁⠀⠀⠀⡜⢡⠛⢊⡅⠂⡏⡆
⠀⠸⡀⠀⠀⠀⠀⠀⡿⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢻⣾⣾⡾⠃⠀⠀⠀⠀⣗⠰⠐⡆⠐⣩⠒⡇
⠀⠀⠑⠤⣄⣀⡤⠔⠁⠀⠀⠀⠈⠉⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⡿⠿⡇⠀⠀⠀⠀⠀⠘⠲⣥⣔⣷⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⡠⠤⠤⠤⣀⠀⠀⠀⠀⠀⢀⠤⠄⠒⡢⢄⡀⠀⠀⠀⠀⠀⢰⠉⡆⠀⠀⠀⠀⠀⠀⠈⢣⣈⣯⡰⠋
⠀⢠⠊⠀⠀⠀⠀⠈⢳⠀⠀⠐⢭⣁⣠⠀⣟⣭⠆⢱⠀⠀⠀⠀⠀⢸⠉⡇⠀⠀⠀⠀⠀⠀⡠⠚⠛⠙⠣⣄
⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⣲⣷⣖⣧⣜⣋⡇⢸⠀⠀⠀⠀⢀⣎⣀⣧⠀⠀⠀⠀⠀⡜⡐⢡⡀⠀⠈⡘⡄
⠀⠈⠣⣄⡀⢀⣀⠴⠃⠀⠀⢸⣞⣻⢽⣿⣁⠤⣤⠎⠀⠀⠀⠀⢸⠦⠢⢾⠀⠀⠀⠀⠀⢡⡀⠀⠀⢀⠠⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠃⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⣾⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⢴⣤⡓⠃⡀⠀⠃⠀⠀⠀⠀⢀⠐⠒⠶⣃⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠉⠀⠀⢘⣀⣴⣿⠃⠰⠀⠀⠀⠀⠀⠀⡁⠀⢠⣿⠟⠋⢀
⠀⠀⣦⡄⠀⠀⠀⠀⠀⠀⠀⣴⠶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣉⣩⣀⠀⠀⠀⠀⠀⠀⠀⠻⡶⣿⡟⠀⠀⣬⣷
⠀⠀⢹⣷⡀⠀⠤⠔⠋⠓⠿⠮⢕⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠂⠀⠀⣷⡀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠺⡼⠐⠔⠋⣷⠀⡀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⢠⣆⠀⢸⣿⠁⠀⠲⠂⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⣀⣶⠒⠉⠉⠉⠁⠘⠀⠘⠋⡁⢀
⠀⠀⠀⠀⣴⠖⠲⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢺⣁⣀⣀⠀⠀⠀⠀⠀⢢⠈⠀⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⣀⠀⠀⢀⠗⠶⠒
⠐⠶⢶⡌⠁⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠓⠓⠀⢿⣯⠭⡀⠐⠢⣦⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣄⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡅⠀⠀⠀⠀⠀⠀⢳⠾⠿⢭⣯⣶⣷⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠃
⠀⠀⠀⠀⢠⠶⣿⡆⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍
⠀⠀⠀⠀⠀⠀⠈⣓⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣚⡺⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠘⣿⣤⡀
⠀⠀⠀⠀⠀⠈⡱⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣿⣵⢾⡀
⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⠈⠛⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⢀⣾⣿⠟⠋⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⣿⠆⠘⠛
⣄⠠⠂⠀⠀⠀⠀⠀⠸⡇⠀⠀⣼⣿⠃⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⣬⣥
⠫⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠻⡟
⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣼⣯⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣤⢄⠤⣄⡀⠀⠀⠀⡠⢀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢀
⠀⠀⠔⡏⣠⣼⢦⡈⢏⢂⠀⠀⡧⢸⠊⢘⠰⠄⠀⢀⠀⠉⣑⠄⡀⠀⠀⡄⡙⠀⠀⠀⠊⠂⠔⠁⠀⡛⡠⡈⢄
⠀⠀⣾⠀⢸⠯⢼⠁⢀⢿⠀⠀⣇⡏⡰⢣⠖⠀⠀⡞⠕⠾⠁⢀⠃⠀⠠⠉⠙⡀⠀⠠⠰⠀⠀⠀⠈⠂⠈⠀⢨
⠀⠀⢛⠁⢱⣤⣤⠉⠉⠞⠀⠠⠁⢢⠕⠁⠸⠝⠀⠐⠀⠐⠃⠁⠀⠀⡇⠀⠀⡇⠀⠆⡀⠀⠑⢔⠄⠀⢀⠠⠁
⠀⠀⠀⠁⠛⠿⠿⠒⠈⠀⠀⠀⠉⠁⠀⢀⢴⠙⢉⠩⠱⢄⠀⠀⠀⠀⠇⡀⣀⠅⢠⠀⠁⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠠⠂⠐⠄⠀⠀⠠⠰⡆⢒⣄⠀⠀⡈⠀⡙⢤⠐⡂⠒⠂⣄⣐⣒⣂⡄⠀⠀⠀⠠⠈⠛⡖⡄⢀⠀⠠⠄
⠀⠀⠀⠀⢠⠀⠀⠀⡁⢠⠇⡞⢀⡄⠀⠱⡄⢈⡙⢀⠁⡴⠀⠙⠛⠋⠛⠃⠀⢀⠁⡄⣀⡑⠍⠅⢀⠤⠀⡀
⠀⠀⠀⢀⢸⢀⡠⡀⠐⠏⠀⠧⠞⠀⣀⢄⠈⠒⠐⠂⠈⠀⠀⠀⢀⠈⠈⠉⠉⠁⢀⠀⡨⠀⠄⠀⢀⠀⢀⠇
⠀⠀⢠⠈⠈⡄⠒⠃⠀⢀⠀⡤⠀⣀⠱⠰⠁⠀⣪⣶⣶⣕⠄⠀⠈⠠⠤⠀⠀⠀⠂⠀⠀⠀⠷⢀⣀⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠌⠀⠀⠈⡉⢄⠀⠀⣿⣿⢿⣿⠆⠀⠀⡠⠀⡠⡀⡠⢀⠀⠀⢀⣺⣩⣼⣌⠻⢂
⠀⠀⠀⣷⣶⠀⠀⠈⠈⣰⠀⡀⠔⠁⠀⢀⡀⠀⢜⠿⠿⡫⠂⢠⣀⣠⣆⣁⣸⢦⣴⣀⠀⢸⣇⣸⠛⢫⠠⣿
⠀⠀⠀⡿⢿⠀⠀⠐⠀⢸⠀⠐⠐⠈⠁⣠⠀⠀⠀⠁⠈⠀⠀⠘⠙⠋⠙⠋⠙⠛⠉⠛⠀⠀⠡⢼⣿⣧⠠⠁
⠀⠀⠀⠒⠒⠀⠀⠀⠑⠡⢩⣀⠀⡀⠜⠡⡒⡄⠀⣾⣽⠀⠐⠀⠲⠀⠉⠑⠢⠀⠀⡤⠔⠀⢄⢠⣄⠐⡒⢂⢠
⠀⣠⣴⠒⠒⠒⠒⣤⡀⠀⠀⣀⣀⠀⡀⠀⠁⠀⠀⠿⠻⠀⢨⣀⠠⣀⢀⡲⠤⠀⠰⠀⠀⠶⢈⠀⠺⣶⣓⣸⠋
⠀⠸⣤⠀⠀⠀⣠⣸⠁⠀⠀⡟⣗⣦⠀⠤⠀⠐⠀⠀⠲⢤⣤⣤⣤⣤⣤⣤⠴⠃⠀⠱⡄⠐⠁⠀⠀⠿⠞⠿
⠀⠀⠀⢁⡦⣆⠁⠀⠀⠠⢤⠐⠀⠐⡠⠤⡀⠀⠀⠀⡄⣸⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⣁⣤⣀⠀⠀⢠⡄
⠀⠀⠀⠒⠓⠚⠃⠀⠀⣠⣠⠀⠀⠸⠀⠀⡨⠀⠀⠀⣧⠀⢀⢄⡀⠀⠀⠀⠈⡀⠀⡾⠉⡁⠙⣇⠀⢈⡌⠔⠄
⠀⠀⡠⣖⢠⠀⠪⠽⠠⠁⠀⡂⠀⠀⠈⠩⠐⠂⡉⠀⠀⠀⠈⠚⣀⠀⡀⠀⠀⠀⠀⠳⣄⣀⣤⠏⠀⠀⠀⠛
⠀⣞⠱⣎⢫⢇⠀⠀⠀⠀⠀⡇⠀⠀⢰⡢⣠⠞⠀⡴⠀⠀⠀⠀⣹⢠⠏⡀⠀⢰⠉⡀⠀⠉⠀⠀⠀⢀⠁⡆
⠀⠀⠉⠐⠣⣨⠄⠀⠐⠄⠀⠄⠀⠀⠠⣈⣨⠤⠐⠀⠀⠀⠀⢤⠃⢸⡠⠃⠀⢴⠀⠃⠀⠀⠀⠀⠀⠘⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠠⡄
⠀⠀⣤⣈⠃⢹⣀⡀⠀⠀⠀⠀⠀⠠⣶⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⡀
⠀⠀⠘⠃⠀⠈⣿⡁⠀⠀⠀⠀⠀⢀⣤⠟⢉⣿⠧⠀⠀⠀⠀⠀⠰⠤⠼⣿⡿⢧⠀⠀⠀⠀⢀⡜⣱⣮⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⢀⣤⣼⢆⢺⠁⣣⣤⡀⠀⠀⠀⠀⠀⢀⣿⠇⠊⠀⠀⠀⢠⡿⠛⣩⠟⠋⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠸⣿⡌⠀⠀⠀⢿⡸⣛⠂⠘⠛⣗⢿⣸⠀⠀⠀⣤⡄⣼⠙⣦⠀⠀⠀⠐⠋⣀⣾⡥⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠮⢳⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠋⠑⠁⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠚⠉⠙⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⣴⡿⣷⣾⣷⡄
⠀⠀⠀⠀⠀⠉⣻⣷⠀⠀⢠⢤⣀⡀⢀⣀⣠⣴⡿⠏⠑⠀⠀⠀⠈⢱⡿⢿⡀⠀⠀⠀⣴⢲⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⡟⠻⣄⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⠟⠀⢸⠇⠀⠀⠘⠋⠀⣹⡟⠛⠋⡟⠦
⠀⠀⠀⠀⠀⢠⠇⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢳⡂⠒⠑⠁
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠀⢀⣄⠀⡀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⢀⣼⣷⠦⠤⠴⠆⠀⠀⠀⠀⠑⣤⣾⣷⠀⠀⠀⠀⠀⣠⣄⣿⣀⣀⣀⣀⠀⠀⠈⠟⠀⢳⣀⡄⣸⠁⠸⠇
⠀⠀⢠⠏⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡇⠀⠀⠀⠀⠀⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃
⠀⠀⠀⣰⠏⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠈⢷⠀⠀⠀⠀⠀⢸⠿⡆⠀⠐⠀⠀⠀⠀⠀⠀⠀⣾⢿⡀
⠀⠀⣞⠁⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢸⡀⠀⠀⠀⠀⡿⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠈⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠈⠀⠀⢠⡆⠀⠀⠀⠀⠁⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣀⣼⣧⣤⠀⠀⠀⢀⠀⠀⠀⢘⣇⣀⣔⡈⠀⠀⠀⠀⣀⡤⠤⣤⣿⡃⣀
⠀⠀⠀⠀⢀⣾⣀⠞⠀⠀⠀⠀⠀⢰⡟⣿⣿⡟⢳⡀⠠⠋⠀⠀⠐⢸⣿⠋⠁⠁⠀⠀⠀⠀⠀⠀⣰⣿⡟⠓⠁
⠀⠀⠀⠀⣼⣿⠛⠛⠘⢂⡀⠀⠀⠀⠃⠹⡿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⡀⠀⠀⠀⠀⠀⢀⡴⠟⠛⣻⠇
⠀⠀⠰⠶⠾⠿⠿⠶⠄⠀⠃⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠀⠀⠀⢀⡼⠁⠀⡇⠀⠀⠀⠀⣴⠋⠀⠀⠐⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⡀⢠⢦⠤⣀⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠
⠀⠀⠀⠠⢂⠠⣦⣝⣣⣀⢌⢳⡙⣟⢏⢮⠲⣤⣤⠶⣔⡲⢶⡒⢶⢰⣦⣤⣄⡀⠀⠀⠀⢨⣽⡁⠀⢀⣀⠀⡀
⠀⠀⠀⠉⢻⣙⣬⠙⡟⡏⣾⠇⣇⣽⢸⢮⠳⡝⢮⣑⣄⣝⢷⡙⣮⢿⠁⢧⣹⡝⢯⣹⡗⣨⣁⠸⠇⠘⠛⠚⢏⣀⡀
⠀⠀⠀⠶⣬⣿⣿⠉⠚⣱⣣⣘⠉⣫⣾⠋⣸⠟⣦⢻⢿⢻⢺⡃⢹⣨⣧⣚⢨⡿⣿⣇⠀⠙⠠⣽⠆⢄⣀⣀⣠⠛⠃
⠀⠀⠀⠀⠙⠀⠜⣽⢭⡑⡔⢽⣁⣻⡘⣦⢵⠋⣤⡻⡈⠜⠐⢁⣼⣭⠮⣁⣍⢘⣙⡟⢆⠀⢀⠀⠑⠛⠇⠨⠟⠃
⠀⠀⠀⠀⠀⠀⣼⠖⢂⠔⣋⣵⣭⠟⠋⠁⠒⢒⡟⢉⡇⠠⡖⠉⠀⢳⠄⠈⠙⠢⣭⣥⡏⠉⡛⠑⡀
⠀⠀⠀⠀⢀⡚⠓⢠⠧⠚⠉⠠⠀⠐⠀⠋⠉⠉⠉⠀⠀⠠⠚⠀⡠⠜⠁⠀⠀⠀⠀⠉⠉⠉⡇⠀⠰⡀
⢀⠀⠀⠀⠉⣀⡸⠁⠀⡀⠐⠀⠀⢰⠁⠀⠀⠀⠀⠤⠴⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣄⣄⠠
⠈⠒⠂⠂⠂⠁⠀⠀⠀⠛⠐⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⢘⣿⡄⡀
⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⢀⠀⠀⠀⠀⣀⣤⠀⢠⣀⠀⡄⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⢿⣿⠟⠇⠟⣦⣀
⠀⠀⠀⠀⠀⠀⢘⠛⠛⢰⣿⣿⣷⣦⡀⠉⠁⢠⣾⡛⠊⡀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠀⠟⠃⠀⠀⢀⣸⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠁⣤⢼⣿⠟⠀⠉⠻⢿⣷⡤⠸⣿⣧⣾⠏⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⢀⣹⠶⡗⢫⡉⢿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣦⡀⠤⠐⠊⠉⠀⠠⢠⣥⣤⣤⠰⣞⡂⠀⠀⣀⣀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠰⠃⠻⠟⢠⣾⠃⠈⣇⠈⠋⢻⣿⣧
⠀⠀⠀⠀⠀⠀⣭⡻⠟⠋⠸⣷⣌⠳⣄⠀⠀⢀⠎⠀⠈⢛⣶⡘⢿⣟⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠸⡀⠀⠈⠛⢿⣧
⠀⠀⠀⢆⣛⣴⠟⠀⠀⠀⠀⠘⢿⣧⠹⣆⢀⠸⠤⠖⠊⠁⠀⠀⢀⣿⣷⡄⠈⣯⣍⡙⠲⣄⠀⠀⠀⢀⠀⠀⠀⢀⣤⢤⣶⡟⡁⠀⠀⠀⠀⣧⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⣿⣤⣶⡇⠀⢀⣾⣀⣀⣀⣀⣘⣩⣭⣤⣶⣿⣿⣿⣷⡈⠃⠀⠀⠂⠀⠀⠔⢋⣤⡾⢋⣮⣤⣤⠶⢁⣼⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣠⡄⠀⠚⠩⢯⡘⢿⠿⠖⠒⣸⣿⣿⡷⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⢛⠁⠀⠛⠢⠰⣇⠀⠀⢀⡼⠹⣿⢟⣾⡀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠴⣾⠏⠀⠀⠈⠳⣤⠏⠀⠀⢵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠿⠙⠁⣴⣿⣿⢿⠿⠇⠙⠁⢀⣀⣤⡄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣅⣀⣀⠐⠿⠂⠀⠀⣤⣌⠲⠾⠿⠟⠁⣰⢺⡆
⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⠀⠠⠤⢠⡶⠊⠉⠉⠁⢀⡠⠀⡠⣶⡌⢣⡀⠸⣿⣷⡿⠛⠻⠦⣀⠠
⢠⣤⣤⣶⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠈⢠⣮⣾⣿⣿⣿⣿⣐⡼⠋⠀⠀⠀⠀⠈⠉⠁
⠈⠿⠛⠛⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠸⣳⠋⣹⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠉⡁⣀⢀⠠⢤⣄
⠀⠀⠀⢀⣦⠀⣠⣤⣤⡶⠛⠉⠙⠓⠶⣶⢦⣷⠀⠰⠞⠳⡄⠙⣻⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣀⠀⠠⣤⣤⠆⠀⠤⠼⠟
⢀⣀⣴⠋⣠⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⢹⡌⡏⠐⠀⠀⠀⠀⣰⠌⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⠀⠘⣤⣴⣤⣴⡄⣀⠷⠆⢀⡄⠀⠀⠀⠀⡀
⠈⠽⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⢰⡆⠀⢀⣸⠟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠘⠿⠯⠁⣴⠿⠛⠇⠀⠘⢡⣾⣌⡙⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠠⠴⠾⠧⠦⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣴⣶⠖⠀⠀⠀⠀⠀⠚⠂⠀⠀⣿⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠒⠛⠓⠲⠒⠂⠀⠀⠀⠀⠀⠀⠀⣀⣰⣼⣿⡿⠂⢰⡾⠟⣀⣶⡽⠋⠀⠀⠀⠀⠑⣆⠀⠐⠆⠀⢻⣿⣄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢾⣿⡿⠿⠛⠉⠀⠀⢋⠀⣸⡟⠁⢀⠀⠀⠀⠀⢰⢠⣿⣿⣷⡆⢻⣶⡿⠛⢧⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠎⠀⢠⡏⠀⠀⣿⣄⡟⢛⣷⢸⣿⣿⣾⣏⠀⠀⠀⠳⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⣀⣴⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣻⠚⠇⠀⠀⢠⣾⣿⠿⠛⠉⢠⠷⡄⠴
⠀⠀⠀⠀⠀⠤⢒⣲⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⢤⣤⣤⣴⣟⢣⠀⣀⢀⣤⠟⠉⠢⢤⣾⣿⣤⣈⣢⣄⠀⢂
⠀⠀⠀⠠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡿⢛⡿⠀⠉⠁⠈⠉⡷⠖⠒⡾⣆⣀⠀⣀⣀⣉⠳⠄⠈⠀⠀⠀⢀⣾⣿⠋⠀⠉⠉⠙⢿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣻⣿⣷⠄⠀⠺⠦⠀⠀⠀⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢠⣴⣤⣶⣶⡲⠄⠀⠾⠿⣥⠀⠀⢀⡀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⠀⠀⠀⣠⠾⠛⠛⠛⠛⠻⢷⣤⣄⣀⠀⠀⠀⢠⣷⡀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣲⣾⣿⠟⠉⢀⣠⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣶⣾⣿⣷⠀⠀⠀⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢉⣦⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡗⠒⠶⣦⣌⠳
⠀⠀⠀⠀⣀⣤⣐⣢⠞⠁⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠉⠉⠁⠀⠀⡰⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡓⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⣄⡀
⠀⠀⠀⠀⣀⣄⡀⠀⡀⣿⡆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿
⠀⠀⠀⠚⠛⠛⠟⠿⠛⠿⠛⠿⠟⠛⠛⠋⠝⠛⠛⠋⠉⠛⠙⠛⠁⠀⠀⠀⠀⠀⠀⠠⠤⣶⠄⢠⢄⠽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⡿⠋⠀⠀⠀⠀⠀⢠⡖⢀⣴⡟⣁⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢸⡇⠀⠀⠀⠀⠁⢰⣿⣧⣿⣿⣿⣿⣿⣏⣠⣴⣶⣶⣟⣁⣀⡀⠀⠀⠀⣷⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⢧⡀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡁⢀⣰⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠉⠙⢉⣽⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⠉⢀⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡟⢁⠀⠀⠘⢿⣿⡏⢻⣿⣿⡟⠉⠉⠉⠉⠉⣙⡛⢿⣷⣾⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠠⠋⠀⠀⠀⠈⠻⣧⠀⠙⢿⣧⠀⠀⠀⠀⢀⠀⠙⣆⢻⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠁⠀⠰⣏⡈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠊⣁⡽⠆⠈⠈⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⣸⡏⠀⠀⡰⣿⡿⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣿⢿⢦⠀⠀⢻⣇⢨⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠟⢣⣠⣿⡇⠀⠐⠀⠁⠻⣄⣙⣱⣦⢶⣃⣶⠶⣴⣏⣃⣴⠏⠀⠀⠀⠀⢸⣿⣄⡝⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠉⠍⣿⣠⣤⣉⣡⣤⣬⡯⠍⠉⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⣆⡀⠀⠀⢄⣸⣿⣇⠀⢀⣠⡴⢶⡿⠲⣦⡭⢭⣿⡿⢭⣴⠶⢾⣿⠶⣤⣀⠀⠀⣿⣿⣇⡠⠀⠀⢀⣸⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⠈⢻⣿⡿⣄⢸⣿⣇⢸⡇⢀⡟⠀⠀⢹⠁⠀⢻⠀⠀⣧⢀⣼⣿⠃⣲⢿⣿⡏⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠻⣇⠸⠀⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠃⣾⠟⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡇⠀⣠⡞⠀⠀⣹⡄⠀⠸⣏⢻⣿⡹⢿⣿⣿⣿⣿⣿⡿⠟⢸⡏⢻⡿⠃⠀⣼⣏⡀⠈⢷⣄⠀⢰⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠀⢸⣇⠀⠀⣿⣿⡀⠀⠘⠦⣏⠛⣷⠛⠛⣿⠛⠛⣷⠀⢹⣦⠞⠁⠀⢠⣿⣿⠀⠀⣼⡇⠀⣸⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣶⣿⣿⣶⣾⣿⣿⣷⡄⠀⡀⢈⡙⠛⠦⠴⠿⠦⠤⠟⠚⣉⠀⢀⠜⣠⣿⣿⣿⣷⣶⣿⣿⣶⣿⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⡟⠛⠁⣹⣿⣿⣿⣦⡈⠢⠍⠛⠲⠶⠶⠶⠖⠚⠉⠠⠐⣡⣴⣿⣿⠿⡏⠘⠛⢻⣿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢰⡇⠀⠀⠙⠻⣿⣷⣦⣤⣀⣀⣀⣀⣀⣠⣤⣴⣾⡿⠛⠉⠀⠀⣹⡄⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡄⠀⢈⣿⡿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡇⠀⠀⢀⡴⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⣛⡛⣻⣿⣆⣀⣀⣈⡉⠉⠁⠀⠀⣸⣿⡟⢛⡛⠛⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠿⠿⠶⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠉⠙⠻⣿⣿⣷⣴⣿⣦⡿⠿⠷⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⡒⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡳⠞⣿⠫⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣿⣿⣷⣶⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⣠⠔⠒⠚⠷⠒⣀⣹⣿⣿⣿⠧⣆⡀⠤⠀⢀⠀⣀⣀⣀⣀⡤⠐⠂⠀⢀⣴⣤⣶⣤⣀⣿⣻⡿⣠⠁
⠀⠀⠀⢀⣾⠃⠀⠀⢀⡀⠀⠉⠙⣿⣏⣴⣧⣤⠀⢤⣤⠚⠉⠁⠀⢀⣼⣿⣷⣤⣾⠿⠛⠉⠉⠉⠉⠉⠛⠛⠁
⠀⠀⠀⢸⣿⠀⠀⣠⠿⣅⠀⠀⢀⡞⣿⣿⣿⢧⠃⣼⣿⣷⠦⣶⡾⠟⠋⠉⠉⠉
⠀⠀⠀⢀⣿⠀⠀⠋⠀⢹⣿⡤⠷⢿⡿⠛⠛⣷⡿⠁
⠀⠀⠀⢸⣿⣅⠀⠀⠠⠚⠉⠀⣴⣿⣤⣀⡰⠟⠁
⠀⠀⠀⠈⣿⣿⣷⡄⠀⠀⠀⣼⠟⠈⣿⢋
⠀⠀⠀⠀⢸⣿⣿⣷⣴⣤⡴⠛⠃⠀⣿⠎
⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⣚⡋⢡⢼⠃
⠀⠀⠀⠀⣜⣿⡿⢷⣶⣾⠉⠭⣿⠏
⠀⠀⠀⠊⠈⠉⠉⠛⠓⠚⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣟⠟⠉⣳⠀⡄⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠛⠓⢤⡿⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠙⢷⣄⣻⡟⠷⠀⢢⠄⠀⠀⢉⠉⠙⠡⠄⠶⣄⠀⠀⠀⢷⣄⡀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⠀⠀⠤⠉⢿⣤⠄⠈⠀⢰⣧⠾⠃⢠⡇⢸⡦⠋⠐⠾⣭⡦⣿⣍⣧⣀⡀
⠀⠀⠀⠀⠀⠀⡷⠶⠾⠧⣤⡀⠀⠠⠈⠁⠀⠤⠀⠄⠒⠀⢸⡟⠀⢀⡾⣩⣿⡽⡦⠾⠿⣷⣶⣚⠣⡀⠁⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠿⡄⡀⠀⢀⣠⡔⠁⠀⠀⠀⢀⡞⠀⠀⣸⣧⣿⢽⡄⡼⣷⢼⣿⣏⠉⠛⣤⣬⣾⣟⢻⡛
⠀⠀⠀⠀⠀⣼⠋⠀⠀⡀⣰⡇⢤⣴⣟⡉⠰⣾⡗⢀⣀⣺⡔⠃⣤⠙⣿⣻⡾⣷⡍⠘⠀⠙⠻⣶⣦⣼⣿⣿⣿⣧⠱⣄⠀⣄
⠀⠀⠀⠀⠀⢁⠂⢠⣶⣹⡏⢷⢠⡉⢻⣿⣿⣿⣿⣶⣯⡙⠓⢊⠁⡀⣿⣿⣧⣿⣷⣶⡦⠀⠀⠈⠻⢤⡈⣿⣿⣿⡟⠻⠂⠻
⠀⠀⠀⠀⢠⡆⡦⢸⠁⣶⣿⡜⣧⢳⣀⡝⠉⢻⣿⢿⣧⣷⠂⠈⢳⡉⣿⣿⣿⢿⣿⣿⠃⠀⠀⠀⠀⠀⢿⣻⡼⠃⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠓⠀⠀⠀⢹⣷⣿⡈⠂⠻⡇⠀⠘⢻⣷⣙⣯⠅⢠⣤⣿⣿⣿⣯⡘⣿⣿⠀⠀⠀⠀⠀⠠⡆⢻⣃⠀⠀⣰⡷⠂⠀⣰⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠃⠀⢀⠀⠀⠀⠀⠹⣿⣥⣶⢰⢿⣿⣿⣿⢻⡆⢿⡟⠀⠀⠀⠀⠀⠀⠃⡼⠁⠀⠰⣿⣄⡀⠀⠹⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⡖⠁⠀⠀⠀⠀⠀⣿⣿⡟⣼⢿⣿⣿⡟⠻⣿⡈⣿⣄⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣿⠇⠀⢰⠋⣧⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⡇⢲⣦⣀⠀⠀⠀⢻⣿⠞⠁⠈⡛⠘⣷⣤⣬⣁⡉⣻⣆⠀⠀⠘⣶⠃⠀⠀⠀⠀⠀⣸⠇⠀⠈⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣀⢣⡀⠙⣿⣷⠀⠀⠘⠁⢲⣄⣾⣷⠠⢤⣿⢿⠛⠿⢿⠿⣿⠀⠀⠁⠀⠀⠀⠀⠀⢀⣿⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠦⢼⡋⠀⠈⠙⠢⣼⣿⣿⠁⠀⠀⣿⣈⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⣼⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⣀⡄⠀⠀⢺⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⣴⡀⠀⠀⠈⣿⠇⠀⠀⠸⡇⠀⠀⠀⣼⠛⣿⣹⠛⠀⠀⣶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⣿⣷⡀⠀⠀⢸⣷⣇⠀⢸⣿⠀⠀⣤⣿⠷⠛⠋⠀⠀⠀⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⠿⡷⠤⢶⣀⠙⢿⣷⠜⣿⣶⣜⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛


⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⠿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣷⣟⣿⣷⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠛⠛⠋⠉⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡿⠿⠁⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠁⠀⠰⣶⣶⠦⢴⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠃⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡠⠞⠻⣻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠁⢀⣾⣄⠀⠀⠀⢀⠰⣶⢾⣶⣿⣿⣶⣦⡀⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢺⣿⣿⣷⠀⣄⡈⡈⠁⣾⣿⣿⣿⣿⣿⣷⣄⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⢀⠀⠒⠰⠒⠔⣼⡿⠋⣤⣦⣿⣿⣦⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⢟⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣠⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣭⡶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢀⣤⣿⣿⣿⢿⣿⣿⣿⣗⠀⠀⠀⠉⠉⠁⠸⣿⠹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣄⢀⣿⣾⣿⣿⣿⣋⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣿⠀⠻⠧⠕⠊⠑⡄⠠
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⢁⣙⣿⣿⣿⣿⡿⠻⠛⠋⠀⠄⠀⠀⠀⣸⣿⣧⠀⠐⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢁⣾⣿⣿⡟⣼⡟⠁⠀⠀⠀⠀⠀⣀⣠⠔⠁⠙⣿⡷⣴⠀⠀⠀⠀⢈⣦⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠀⠀⠀⠀⠀⠀⠀⠙⢁⡾⣿⣿⣿⣇⣿⡇⣀⣤⡶⠚⣿⣿⠿⠋⠀⠀⠀⠈⠑⢿⡆⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢀⠠⠖⠁⠀⠀⠀⠁⢛⡛⢻⣿⣿⣿⠿⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠠⣄⣤⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠮⠀⠀⠀⠀⠀⠀⣠⡟⢁⡿⠉⠉⠀⠀⠠⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣀⠀⠀⠀⠀⠤⡍⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠺⠿⠀⣀⣴⣾⠟⢀⣾⠏⠀⠀⠀⢀⠀⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠐⡳⢿⠛⢠⣾⠟⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣴⣷⣮⣥⣶⣒⣖⣶⣶⣮⣿⣤⡦⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢹⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣥⣴⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⢡⣈⡁⠘⣿⠿⡀⢻⣿⣿⣿⣿⣿⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⢿⡀⡀⣿⣿⣷⡸⡆⢰⣷⣿⣿⣿⣿⣿⡝⣿⡷⠈⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣝⢷⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⢿⣿⡈⠁⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡻⡛⠃⠙⢷⣌⣻⢿⣿⣿⠿⠋⠛⠛⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⢉⣥⠞⠋⠁⠉⠉⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⣷⣄⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⣀⣀⣀⣤⣤⡤⠶⠶⠶⠶⠶⠖⠒⠒⢀⣄⣀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⠙⠛⠷⠶⠶⣶⣿⠷⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢮⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⣠⣤⣤⣶⣶⣾⣷⣶⣶⣶⣶⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣴⠾⠟⠛⠉⠉⠁⠀⢸⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣦⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣶⠟⠋⠉⠀⠀⣿⡇⠀⠀⠀⠉⠙⠻⢷⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡇⢀⣴⠟⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣾⠏⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⡀⠀⠀⠀⠀⢀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣆⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣦⣄⣀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⣻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣿⣿⣯⣷⣦⣄⠀⠈⢿⣆⠀⢀⠀⠀⣀⣀⠤⣀⡠⣀⣠⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⣽⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠛⠟⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠋⠹⠏⠙⠛⠛⠛⠟⠛⠛⠛⠛⠉⠙⠏⠙⠫

⠀⠀⠀⠀⠀⠀⠀⣾⡶⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⣜⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣘⠦⢏⡀⠀⠰⠂⠀⠀⠉⠀⢀⠀⠤⠤⣴⠆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠁⠀⣏⠙⢳⣴⡯⢐⡱⠀⠀⠀⠀⠀⠑⠳⠀⠀⢀⣿⣿⠿⠋⠢
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠶⠚⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⣼⣿⠁⠀⠀⢧⡄
⠀⠀⢺⣿⡀⠀⠀⠀⠀⣠⣄⣢⣜⢿⣛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣶⡶⠂⡐⠀⠀⠀⠀⠀⠀⠀⠙⠝⠻⠟⠀⠀⣼⣿⠇
⠀⠀⠀⣿⣧⠀⠰⠒⠚⠁⠈⠙⠛⠛⠪⣻⣗⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠌
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣽⠆⠀⠀⢹⣆⠀⠀⠀⢀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣂⡦⣄⡠⠿⣏⠀⠀⠀⠈⢿⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⢀⣿⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠠⠜⠃⠀⠀⢀⣀⣠⣀⣀⣠⠠⢐⡈⠀⣿⣤⡼⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣰⣿⣄⣠⣾⡿⠀⠀⠈⠁⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠋⠀⠀⠀⠀⠀⠀⡆⠄⠈⢠⡖⢦⠈⠈⠁
⠀⠀⠀⠀⢠⣾⠓⠛⠻⠿⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠃⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠈⠁⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣸⠛⠚⠙⠁
⠐⠿⠿⣤⠋⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠘⠀⠀⠀⢹⣏⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣄⣠⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⢾⣟⣳⣢⣴⣦⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢠⣤⣶⣆⠀⠀⣨⡿⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠉⠛⠛⠻⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⠋
⠀⠀⠀⠀⠀⠯⠀⠹⣿⡀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡜⠃
⠀⠀⠀⠀⠀⠀⣥⣀⠷⡇⠀⠀⠀⠀⠀⠀⠀⠀⣘⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢡⣦⡌
⠀⠀⠀⠀⠀⠀⢼⡿⠆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⠜⠢⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣿⣷⡶⣿
⠀⠀⠀⠀⠀⠁⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠈⠿⠿⢗⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣶⠆⠀⠀⠀⣴⣿⣿⡿⠋⠹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣶⡦⠀⠿⠇
⠀⠀⣠⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢋⠀⠀⠉⠁
⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷
⠀⠀⣬⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠘⢿⣿⠆⠀⠀⠀⠀⠀⠀⠈⣙⡅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢀⣀⣠⣤⡄⢠⣤⣤⣀⡀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠃⢀⡀⣾⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣷⢀⡀⠘⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⣿⣿⡘⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⢃⣿⣿⣦⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡘⣿⣿⣿⡇⢸⣿⣿⣿⢃⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣮⡛⠿⡇⢸⠿⢛⣵⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢰⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣴⣴⣤⣤⣤⣤⣤⣤⣤⣴⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢋⡛⣉⣉⠙⣉⣉⣉⣉⠙⣉⣉⡉⣋⣉⣉⣉⡙⣉⣉⠉⣉⣉⣉⣉⡙⢉⣉⢋⡉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⢋⣾⣿⣿⣿⣿⣷⡙⠏⣴⣿⣿⣿⣿⣷⡜⢋⣾⣿⣿⣿⣿⣿⡌⢣⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣌⢿⣿⣿⣿⣿⣿⢃⡜⢿⣿⣿⣿⣿⡿⢡⣌⢻⣿⣿⣿⣿⡿⣡⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠻⣿⣿⡿⢣⣾⣿⣎⢻⣿⣿⡟⣱⣿⣿⣦⠻⣿⣿⠟⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡹⠟⣱⣿⣿⣿⣿⣧⠹⠏⣼⣿⣿⣿⣿⣷⡙⢏⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⡘⢿⣿⣿⣿⣿⣿⢃⡘⣿⣿⣿⣿⣿⡿⢣⡜⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢋⣾⣿⣌⢿⣿⣿⡿⢡⣾⣷⡌⢿⣿⣿⡿⣡⣿⣿⣎⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠻⡟⣱⣿⣿⣿⣿⣎⢻⠟⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇⠸⣿⣿⣿⣿⣿⣿⠇⠸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠋⠀⠀⠘⣿⣿⣿⡿⠃⠀⠀⠙⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠈⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡯⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣷⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⠀⠀⠀⣴⣶⣦⣄⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⠀⢳⣄⠀⠈⣉⠉⠁⠉⡇
⠀⠀⠀⠀⠀⠀⣀⣤⠖⠢⢤⡴⠒⠒⠒⢊⣡⡾⢯⣿⣧⠀⠙⠀⠊⠙⠢⢤⣼⣁
⠀⠀⠀⠀⢠⣾⣿⡟⠀⠀⠀⠙⣶⣾⣿⢿⡿⠃⢸⣷⣿⣷⣄⠀⠲⣯⣽⣿⡀⣈⣱⡦⢤⣀
⠀⠀⠀⢀⣿⡇⣿⡇⠀⠀⠀⠀⡿⠤⠤⢀⠀⠢⠈⡏⠙⠿⣿⣦⡙⢲⠶⣿⣿⠉⠀⠉⠂⠈⠙⢦⡀
⠀⠀⠀⢸⣿⡇⣿⡇⢠⡄⠀⣸⣁⠀⠀⠀⠈⠐⢤⡁⢀⡤⢞⣻⣽⣟⢤⣿⣿⣦⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢏⣿⣿⣷⣾⣿⣿⣷⡀⠁⠀⠀⠀⠀⢰⡽⢫⣾⣿⣿⣿⣿⣿⣌⢻⣿⣷⣄⣀⠀⡄⠀⢸⡆
⠀⠀⠀⢀⡾⣿⣿⡟⠉⠉⢻⣿⣿⣶⣄⡀⠀⡹⢋⣴⣿⣷⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣷⣧⣴⣿⣇⣀⣀⡀
⠀⠀⠀⣾⠀⣿⢸⣿⡆⠀⠈⣷⣝⠻⢿⣿⠞⣱⣿⣿⣿⡻⣻⣿⣿⣿⣿⣿⠀⣿⡿⣆⠉⡿⣻⣿⣿⣿⣿⣦⡈⠳⡄
⠀⠀⠀⣿⣤⣿⢸⣿⣧⣀⣤⣿⡻⣿⡷⠋⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⠀⣿⣷⡽⣿⣿⡿⣿⣿⣿⣿⣿⣿⡄⠈⢆
⠀⠀⠀⢿⣿⣿⠀⢻⣿⣿⣿⡿⠋⡁⣠⣦⣻⣿⣿⣿⣿⣿⡿⢉⣽⣿⣛⣋⣠⡿⢋⡟⢿⡟⣾⣿⣿⣿⣿⣿⣿⣿⡄⠈⡆
⠀⠀⠀⢸⣿⠁⠀⠀⠀⠉⠉⣧⣿⣿⣿⣿⣷⡻⢿⡿⢟⣩⣾⡿⠿⠿⠿⠛⠋⠀⢺⠀⠈⢿⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀⢱
⠀⠀⠀⢸⣿⣿⣶⣤⣤⣴⠿⣻⣿⣿⣿⣿⣿⣿⣷⣖⣻⣿⣿⣿⣦⣤⣤⡄⠀⠀⠸⡆⠀⠈⢷⢿⣿⣿⣿⣿⣿⣿⣿⡇⡸
⠀⠀⠀⠈⢻⡿⠿⢟⣋⣵⠾⣻⣽⣿⣿⣿⣿⣿⣟⠛⠋⢻⣿⡛⠛⠉⣿⣠⠆⠀⠀⣧⠀⠀⠀⠳⢭⣛⣻⣯⣭⢭⠖⠊⠁
⠀⠀⠀⠀⠘⣷⣯⣭⣽⠾⠟⠛⣿⣿⢿⣝⡏⢿⣿⣦⣤⣼⣿⣷⠶⣶⣟⣡⠄⠀⢀⡟⠀⠀⠀⠀⠈⠛⠻⠿⠗⠋
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢻⡹⣆⠹⣿⡆⣿⡿⠛⢻⣾⡞⠋⠉⣿⠏⠀⠀⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣌⢳⣼⣇⣿⣷⠀⡸⢟⣁⣀⣤⣿⡤⠖⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣝⡻⢷⣦⣴⣿⣿⣿⣿⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣦⣤⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠹⣯⣩⠟⡀⠀⠀⠀⠀⡀⣀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠰⡀⠀⣅⡰⢀⡠⡷⠐⠚⠛⠋⢉⡀⠙⠟⣦⣄
⠀⠀⠀⢀⢀⣀⣀⣀⠀⠀⠀⣀⠄⠀⢀⡴⠤⠀⠠⣠⠊⠀⢀⣷⡄⠘⢾⣌⢙⣒⡬⠟⠛⠿⠾⠷⠔⠴⠾⣟
⠀⠀⠀⠀⢿⡉⠻⠏⢛⡻⠟⣵⡿⠟⠁⠀⠀⠀⠀⣸⣦⠀⠈⢳⠛⠁⡤⣽⡯⠁⣄⡀⠀⠀⠀⣴⠀⣠⡄⠈⡧
⠀⠀⠀⠀⠻⣭⡛⠛⢉⣤⠞⡩⡜⠀⠀⠀⠀⠀⠐⠃⠉⢣⡀⠀⠂⠀⠙⠠⠝⠋⠀⠀⠐⠤⠘⣏⢣⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⠁⡇⡇⠀⠀⠀⠀⠀⢁⡶⠖⠉⢁⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠢⠰⣶⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠘⠃⠀⠀⠀⠀⢠⣙⣰⠆⠸⡙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢙⠀⠀⠀⠈⢂⣉⡠
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⢸⣦⡀⠀⠀⡠⠊⠑⠷⣌⡄⠀⢣⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠡⠼⠃⠉⠂⠈⠀⠀⠀⠀⠀⠁⠐⠒⠒⠷⢄⣀⢢⡀⠀⠀⠀⠲⣯⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠢⣀⠀⠀⠑⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠐⡧⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠀⠈⠀⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢷⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄⢉⣢⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠈


⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣬⣿⣶⣥⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⣿⣿⣿⠃⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠻⠛⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠋⠀⠈⣧⠀⠀⠀⠠⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⡤⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⡟⠁⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠿⠋⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠁⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠉⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢈⣧⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡄⡤⠤⣤⣤⣤⣤⠴⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⢼⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡍⠻⣿⣷⣿⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⣀⣀⢿⣿⡘⣟⠻⣦⣄⣀⣀⡤⣀⣀⠀⣀⡴⠿⣿⡄⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠾⢏⣼⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣷⣠⣾⣿⣾⣿⡿⣿⣯⣿⠳⠀⠀⠀⠈⠉⠉⠀⠘⠛⠋⠉⠀⠀⠙⣿⣦⡀⠀⠈⠣⣀⠀⠀⠀⠀⣠⡾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠇⣿⣿⣿⠁⠀⠀⠈⠉⠒⠊⠀⠒⠂⠀⠀⠀⠀⠀⠈⠻⣿⣶⠀⣠⡟⠀⠀⠀⣘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡏⣠⣿⣿⣿⡀⠀⠀⠒⠒⠒⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣧⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣴⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⠰⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠆⠘⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⢿⣿⣿⣿⣿⣿⣷⡦⠱⣌⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣿⣿⣿⣿⣿⣿⣿⣃⣨⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⢻⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡷⠈⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⡴⠃⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣴⣟⣼⠃⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣳⡄⠻⣿⣧⡀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⢀⡀⠈⠻⣿⣦⣀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⢷⣤⣀⠉⠁⢀⣿⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠉⣿⣬⣉⠷⣸⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠄⠁⠀⠈⠻⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣶⣦⣤⣄⣂⣀⣀⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣝⢿⣿⢮⠻⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢮⡑⠌⠳⡀⠙⡄⢻⣿⠸⠟⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠯⣍⡙⠻⢮⡂⠀⠀⠀⠀⠀⢻⠀⠀⢿⡟⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⡛⠛⠒⠢⠄⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢻⢫⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⣫⣭⣭⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢣⠞⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⡫⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡄⣤⢄⠀⠀⠀⠀⠁⣠⡾⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡠⣀⣤⠔⠂⠀⠀⠀⠀⠀⢸⡏⡟⠻⠜⠧⠘⠋⠀⠀⠀⠀⠐⣉⣴⠿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⠟⣉⠀⠀⠀⠀⠀⠀⡄⢠⣶⣶⡇⢸⡇⠀⠀⠀⠀⠀⠠⠔⠋⠉⠚
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡾⠋⢀⠀⠀⠀⠀⠀⠑⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣪⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡴⢣⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣛⣛⣯⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣣⡧⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠐⠢⠤⣤⣤⣬⣽⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣼⣷⠀⠀⣧⠀⠀⠀⠀⠀⠨⡳⣦⣬⣙⣲⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣿⣿⣴⡄⣿⣧⠘⣄⠈⢦⡐⢌⡳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣼⣦⡳⣿⣷⣝⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⣄⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡮⠁⣴⠠⢹⡅⠀⠀⢠⣤⣤⣴⣿⣾⣷⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⠂⠒⠨⣛⣇⣀⠀⣿⡿⠿⢛⣥⣾⣿⣿⣿⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡗⢿⡿⠿⠛⠛⣷⣴⣵⣾⠿⠿⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣟⠀⠸⠀⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠀⠉⠑⠛
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⢀⣀⠀⢀⠀⢰⠂⠀⠀⠀⠈⠶⣿⠖⢠⡆⠠⣤⣀⣸⡿⠁
⠀⠀⠀⠀⢀⣾⣿⣿⢿⣆⠰⣦⣿⣷⣮⣻⣿⣿⡀⠀⠀⠀⠀⣿⠖⠚⠀⠑⢮⣿⣿⠇
⠀⠀⠀⢀⣿⣿⡿⠛⠚⠉⠀⢸⣽⣜⠯⡁⣿⡟⠿⣷⣦⣤⣤⣽⣇⠰⠇⠀⣠⣿⠿
⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠉⠻⡓⠀⣿⠇⠀⠀⠈⠉⠉⠉⠛⠛⠛⢼⣷⣟⣶⣤⣀⡀
⢀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣏⠉⠻⣿
⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠐⠂⡀⠀⠀⠴⠶⠾⠟⠁⠀⠀⠀⠀⠘⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢾⠋⢁⣤⢀⣨⢏⠙⠛⠃⠶⢦⡀
⠀⠀⠀⠀⠀⠀⣠⣄⠛⣉⣡⡤⣍⠉⠛⠁⢀⣤⠁⠛⠲⣬⡅
⠀⠀⠀⢀⡠⠟⠈⣉⠀⠍⣁⡤⠄⠀⡀⢀⠀⠄⠀⣠⡶⠽⠗
⠀⠀⠀⠈⢧⣤⣤⠀⢰⣾⢋⡴⠛⠀⠀⢀⣶⣾⣦⠸⣁⠀⣠
⠀⠀⠀⣴⢿⣿⠁⢀⣿⣟⢸⣿⠋⡀⠀⣿⣋⡁⠉⣰⢉⣾⠟⠳⠄
⠀⠀⣸⣿⠀⣿⡀⠈⠻⠹⢿⣿⡰⣧⡀⠈⠁⢙⡄⡉⠌⠁⠀⠀⠆
⠀⢠⣿⡟⣧⢹⣟⠀⠀⢦⡀⠈⠛⢦⡕⠀⠠⢀⡁⠠⠦⠄⠠⠈
⠀⣾⢻⣧⠈⠳⠛⠷⠀⠀⠙⠷⣤⣠⡄⣴⡟⠁
⠀⣿⣄⠙⠳⣦⠀⠀⣠⠇⣠⠀⠀⠀⡼⣻⣦⡀
⠀⣼⠻⣦⣄⠀⣷⣾⡅⢀⡿⠀⠀⡜⠀⠁⠈⣩⡿⠷⠖⣠⡀
⠀⢸⣷⣄⢻⣟⠂⠀⠀⠘⠁⠀⠰⠀⠙⠆⠘⢫⡶⣻⡿⠟⣛⣲⣄
⠀⠸⣿⣏⠂⠻⢦⡀⠀⠀⠀⢰⠀⢀⡴⠂⣰⡿⣸⠏⣠⣾⣿⢋⣽⣷⣤⡀
⠀⣠⡏⢻⣷⡀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⣿⢃⠁⣼⡏⠸⢁⣾⢁⡄⠘⣿⣦⡀
⠀⣿⣤⣦⣬⡉⠛⠳⠄⠘⠀⠀⠀⠀⠀⠀⢻⡈⠀⢻⡇⢴⡈⣿⡟⠁⠙⣀⣿⣿⣦⡀
⠀⣿⣉⣀⣉⣿⣿⠛⠦⠀⠠⡀⠀⠀⠀⠀⠀⠙⠀⠘⠇⠸⣿⢹⡇⢀⠘⠙⢋⡽⠿⠛⢧⠄⣀⡀
⠀⢻⣿⢟⣛⣿⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⢤⡙⠃⠀⠙⠋⠐⠢⠤⠤⠄⠀⠀⠀⢈⠁⠘⢃⣤⡀
⠀⠀⣷⣿⣁⣉⠛⠛⠷⠄⠀⠐⠂⡄⠀⠀⠀⠈⠉⠛⠁⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠈⠓
⠀⠀⢸⡟⣋⣍⡉⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⣄⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠁⢀⠀⠀⠈
⠀⠀⠈⢿⣟⠉⠉⠁⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢱⡾⠟⣾⡿⣿⡄⣿⠉⣿⠀⠀⠓⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀
⠀⠀⠀⠀⢹⡿⣂⡀⠀⠑⠂⠀⢀⡀⠀⠡⣦⡀⠀⠀⠀⠀⠀⠀⠀⠚⢀⠾⠁⢠⣿⣇⢿⣧⠟⢠⣿⠟⣶⣿⡄⠀⠀⠀⠀⢠⡀⠀⠀⠃⡀⢠⡀
⠀⠀⠀⠀⠈⣿⢫⣴⣶⡚⠳⡄⠈⠓⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣿⠋⡼⠋⢠⡾⠋⢠⡿⣻⡆⠈⠑⠢⣄⡠⣥⣦⣤⡸⢧⣼⠃
⠀⠀⠀⠀⠀⠈⢻⣿⠋⣁⡀⠸⣆⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣤⠤⠚⠁⠀⠀⠀⣹⣇⡰⠋⣴⣫⡿⠦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⠁⢀⡹⡄⠀⠀⠑⠤⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⣉⠡⣴⡾⠿⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡄⠘⢦⠀⠀⠀⢹⣿⠷⠖⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠴⢛⢻⣶⣾⠿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⡀⠓⠀⠀⠀⢱⣶⠦⢤⡀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⣤⣴⣶⠾⢛⡛⠛⢀⣀⣀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣦⣀⡄⢀⡀⢳⣶⣤⡉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣏⠹⢡⡾⠋⣠⣾⠿⠛⠛⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⢫⢉⡴⠛⣻⣧⡉⢶⣦⣄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠸⠐⢦⣿⠇⣰⡟⢁⣴⠿⠛⠷⢶⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣴⣿⡀⠀⠀⠉⠀⠈⢻⡄⠀⠀⠀⢠⡀⠀⠀⢀⣀⠀⠀⠀⠀⡿⢠⣿⠀⡿⠁⠀⣴⢶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⡆⠀⠀⠀⡀⢀⣧⠀⠀⠀⠀⠹⠆⠐⠀⢉⠑⢦⣀⠀⠰⣆⠉⠳⣧⡟⢠⣤⣤⣌⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠀⠂⠀⢎⠀⠀⠀⠀⢁⡌⠀⠂⠠⠈⠂⠀⠚⠄⠀⠀⠀⠉⠉⠛⠢⣌⡳⢸⣿⢀⣿⠃⡀⠀⢹⣷⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⣄⣠⣄⣠⠞⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⢇⣾⠃⣾⠇⠀⣀⠙⠿⠇⣸⣿⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢀⠙⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠂⠤⠄⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⣿⣰⠞⣿⠀⢷⣶⣿⡿⠁⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣧⢀⡀⠐⠀⠛⠀⣄⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠼⣶⣇⡀⠈⠇⠈⠳⣦⣾⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠑⠛⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡴⠾⢦⡀⠈⡏⠉⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠀⠘⣿⣿⣿⠋⠉⠀⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⢻⣛⡁⢸⣿⣀⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⠀⠀⢀⣿⣿⣿⣀⠀⣠⣿⣿⣿⠀⠀⢹⣿⣶⠖⠶⣾⣟⣴⠆⠙⠟⠀⠙⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠘⢿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠂⢴⣿⣿⡟⠀⢠⣿⣿⡁⣠⠀⠀⢷⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠛⠛⠛⢻⣥⠶⠂⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠁⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠙⠋⠉⢳⣦⡄⠐⠚⢿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣴⣿⣿⠇⢀⣀⡼⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⡠⠿⣀⣤⠺⠇⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⣀⡀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢸⣿⠀⠉⢁⠀⠛⡹⢾⠍⠀⠀⠀⢀⣀⣀⡠⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢶⣿⡆⠀⢹⣤⠈⠁⠀⢀⣀⣴⣾⡿⠟⠋⠻⡌⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⠂⢀⣿⠁⠀⣠⣴⣿⡿⠟⠁⠀⠀⠀⠀⠘⣿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⠤⠤⡤⠀⠠⣄⣀⡀⠀⠐⠉⣠⡖⢀⣀⡉⠛⠻⠟⠉⠀⠀⣠⠆⢀⣤⣀⡀⠹⣿⣄⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠿⠶⠀⠀⠀⡏⠀⠀⠀⠈⣿⡟⣀⣾⣿⠇⣠⣬⣭⣷⡶⠋⠑⣰⠞⠓⢒⣾⠋⠙⠶⠞⣷⡟⣿
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣴⣶⣿⡿⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢲⠧⠄⠀⠀⠀⢈⠘⠻⣿⠋⢠⠀⠀⠀⠀⠀⠠⡾⠃⠀⢠⣾⠃⠀⠀⠀⠀⠈⠻⣿⠄
⠀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠠⣿⣃⠀⠀⠀⠀⠀⠀⠀⢮⠳⠒⠲⠤⣄⡀⠉⡖⠼⠂⠀⠀⠀⠀⠀⣤⡀⠁⢀⣴⠟⠁
⠀⢘⣿⣿⣿⣿⣿⣿⣿⡟⣽⠏⠀⠀⠀⠀⢿⣿⣷⣤⣤⣤⣤⠄⠀⢸⢀⡀⠀⠀⠀⠀⠈⢁⠘⠂⣀⣀⠀⠀⠀⠸⣿⣶⣿⡇
⠀⠀⠈⠒⠛⠋⠁⠐⠋⠉⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⠻⠿⠅⡀⠀⢸⠘⢿⣶⣄⡀⠀⠀⣿⠀⠸⡇⠀⠀⠀⣀⠔⢹⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢀⠀⠀⠈⢸⣷⠚⠽⣿⣿⣶⣿⣀⠀⡧⣠⠏⢀⣠⣀⣼⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⡄⠀⠀⠉⠙⠿⠿⠿⣇⣿⠂⠀⠙⣿⣏⣀⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡀⠈⠋⠉⠉⠉⠉⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⢀⣀⣉⢀⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣧⣸⠆⠀⠈⠻⢿⣤⣤⣤⣤⡘⠷⠀⠀⠀⠀⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣤⣀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏

⢿⣿⣷⣦⣀⠉⠪⡳⣤⡈⠳⣕⢄⠙⢿⣧⡀⠳⡱⡱⠱⢱⡱⢻⡀⠀⠇⡇⠘⡇⠀⢸⠀⠸⠈⢀⡟⢌⣎⠎⢎⢎⠞⢀⣼⡽⡿⡫⣪⠞⢁⣴⢞⠕⠋⣀⣴⣾⡿
⣄⡀⠉⠙⠿⣗⣤⡀⠙⠝⢦⡈⠢⡑⢀⠙⣟⣄⠑⠄⠀⠁⠳⠈⢇⠀⢠⢻⠀⡇⠀⠘⠀⡆⠀⣸⠁⠜⠈⠀⠠⠊⡠⣻⠫⠊⠈⠜⢁⡴⠫⠃⢁⣤⣺⠿⠋⣁⣴
⢈⠙⠳⢤⣀⠈⠙⠪⣖⠀⠀⠙⠲⣄⠀⠀⠈⠻⡢⡈⠀⠀⠀⠁⠘⡀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠃⠘⠀⠀⠀⢁⢔⠝⠁⠀⠀⣠⠴⠋⠀⠠⣲⠕⠋⠁⣀⣬⣾⡿
⢄⡀⠀⠀⠈⠛⠦⣄⡀⠉⠂⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠐⠈⠀⡠⠔⠪⠛⠋⠁⢀
⣦⣌⣉⠂⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣈⣥
⠀⠀⠉⠙⠓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠉⣁⣠
⠷⠶⢶⣤⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⣚⣋⡭⠥
⣉⣀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠈⠁⠀⠀⢀⣀
⠓⠶⠶⠭⠥⠄⠂⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠁
⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤
⠶⠶⢶⣒⡒⠬⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⣀
⠉⠉⠀⠀⠀⢀⡀⠀⣀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉
⣶⣖⣾⣿⡯⠷⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶
⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢄⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠋⠁⠀⠀⢀⡤⠖⠋⠀⡀⠄⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⠢⢄⠀⠀⠂⢄⡀⢀
⢁⣠⣴⡚⠉⢀⣠⢖⠏⠂⠀⣠⠖⠋⠀⠀⢀⣴⠕⠀⠀⠀⠀⡄⢠⠁⠀⢸⠀⠀⠀⠀⡃⠀⠈⡄⢁⠀⠀⠀⠀⠨⣦⡀⠀⠈⠻⣶⣄⠀⠀⠹⡲⣄⡀⠈⢓⠯⣷
⡿⠟⢉⣠⣶⡯⠛⢁⢠⣪⠞⢁⠔⡡⠀⣰⡻⠫⡢⠂⠀⠀⡼⢀⡏⠀⠐⣼⡄⡇⠀⢠⣇⠂⠀⢹⡀⢣⠀⠀⠀⢔⠝⣽⣕⢄⠈⢮⡫⣷⣕⡄⡈⠛⢽⣶⣤⡉⠻
⣤⣾⡿⠟⠉⣠⢔⡵⠛⢁⡴⡻⠊⣠⣾⡟⡕⡵⡱⠀⠀⡼⡱⣼⠁⡀⡆⡇⣿⡇⠀⢸⢸⢰⢀⠈⣧⢎⢧⣆⢆⠈⢮⢪⢻⣷⣷⣄⠙⢮⡻⣟⢮⡢⣄⠉⠻⢿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢛⣿⢏⣿⡏⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠋⠀⠘⢁⣾⡟⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠈⠛⠠⠏⣾⢻⣿⣇
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠸⣧⣴⣶⣿⡶⠀⠀⡀⠀⠉⢈⡾⣿⡄
⠀⠀⢀⣴⠞⠋⠉⠉⢻⣆⠀⢨⣿⣿⡋⠀⣠⠎⠀⠀⢸⡏⠀⠸⡟⣦⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⢠⡟⠁⠀⠀⠀⢰⡀⣿⠀⠼⢽⡛⢿⣿⠿⠀⠀⠀⢈⣷⠀⠀⠃⠈⣿⠿⠛⠋⠉⠁⠉⠛⠋⠉⠉⠓⢦⡀
⠀⢸⠁⢀⣠⣤⣠⣾⡇⣿⠀⠀⠀⠻⢾⡷⠆⠀⣠⣶⣿⣿⣷⢤⣀⡀⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠙⣆
⠀⢸⡀⣿⣿⡿⠿⣿⣿⠃⠀⠀⠀⠀⠈⠓⠶⢞⣻⠻⣿⣿⣿⡄⠉⠋⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⢸⡆
⠀⠘⣇⢻⡟⠀⠀⠈⡇⠀⣀⣤⣶⠶⠶⢿⡿⠟⠁⠀⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣈⢿⣷⣄⡀⠀⢾⣧⡀
⠀⠀⠸⡌⢻⡀⣀⣼⠃⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣿⠛⠛⠋⠸⣷⣿⠟⠋⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣷⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⢸⡄⠀⠀⣀⣿⡏⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣴⡿⠁⢸⣿⣧⡀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⣿⠿⣿⣿⣿⣇⠀⠀⠀⢰⡿⢻⣧⡀⠀⠀⠀⠀⢀⡼⠿⣿⡛⠶⠶⣶⡿⠛⠉⠀⢀⣾⠛⣿⡟⣷⡶⠶⠞⠛⠛⢻⡆
⠀⠀⠀⠀⣿⠀⠀⠀⢹⣿⣆⠀⣠⣿⠃⠀⠙⢿⣶⣤⣤⠖⠁⠀⠀⠈⠙⠀⠀⠀⠉⠳⣦⡴⠾⠟⠛⣿⡉⠁⠀⠀⠀⣀⣴⣿⣿
⠀⠀⠀⠀⢸⡄⠀⠀⢸⣿⣿⣾⣿⠏⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠘⣷⡀⠀⠒⠋⠉⠁⣰⡿
⠀⠀⠀⠀⠀⠻⣆⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⣿⡄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⠀⠀⠀⢠⣿⣷⣶⣶⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡙⠿⢷⣶⣶⣶⠿⠿⠛⢛⣉⣡⣿⣶⣶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣤⣤⣴⣶⣶⣿⣿⣿⡿⠟⠋⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣧⣬⣙⣿⣿⣟⣛⡛⠉⠁⠀⢀⣀⣠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠙⠲⠶⠄⠢⣰⠰⠞⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⣀⠀⠀⠀⢄⠀⠀⠠⠄⠉⠀⢦⡄⠀⠀⠙⠀⠀⣄⠀⣤⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⠉⠲⣄⠀⣀⣀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⡴⠂⠁⠀⠀⢄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣠⠅⠀⠒⢠⢀⣀⣀⠘⠛⢧⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢁⠟⠛⢀⣀⣀⠠⠐⠂⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠐⣫⣤⠶⠄⠀⠀⠈⠳⣌⡆⠈⠀⠀⠀⡈⢀⣔⡴⠀⠀⠴⠶⣤⡙⠀⠀⠀⠀⢴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⠜⠁⠀⠀⠀⠠⣀⠈⠳⢬⣧⠀⠀⠀⣀⠠⠚⡉⢀⡠⠄⠀⠀⠀⠙⢆⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⣌⠲⠄⡀⠀⣈⣀⠴⢪⡴⠋⠀⠀⠀⠀⠀⠀⠈⡆⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⡆⠀⠀⠀⠀⠀⠀⠀⠙⢟⠀⠀⠀⠉⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⣾⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣶⣤⡀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢰⡟⢛⣿⣦⠀⠀⣷⠁⠀⢰⠈⢦⣰⡆⠀⢀⣾⣿⠛⣷⡤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠀⠸⠿⠋⣧⣼⡛⠳⠶⠾⢦⡤⠚⣻⣷⡿⠺⠿⠄⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⠤⠄⢸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠂⠤⠤⠤⠤⠆⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⢶⣄⡀⠀⠈⠳⣄⠢⡀⠀⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣷⠖⢠⠀⣴⠛⠁⠀⣀⣤⡞⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠑⠂⠤⣽⣆⣵⣄⠟⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⢤⣈⢿⣶⣅⣼⡥⠴⠒⠉⠁⠉⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⢀⠀⠁⢀⠟⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣈⡙⢣⡀⠁⠀⠀⠀⠀⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢄⠘⢲⠖⠋⠀⣀⣀⠤⠂⠀⠀⠀⠀⠈⠉⠒⢄⣈⠙⠛⠓⣾⠚⢀⠄⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢾⠀⠀⠀⠀⢠⡴⠶⠀⠀⠀⠀⠀⠀⠗⢦⡀⠀⠀⠀⢸⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢧⡀⠀⠛⠃⢈⣀⣀⣀⣀⣀⣀⠀⠛⠃⠀⣠⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡟⠛⠛⠻⠿⠛⠋⠙⠛⡻⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠰⠐⠀⠀⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠒⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡆⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠈⢯⡛⠉⠻⣤⣴⠞⠉⢀⢀⣀⡁⢔⣺⣿⠿⣯⣄⡀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⢄⠘⠛⣿⣤⠞⣽⡿⣇⣠
⠀⠀⠀⠀⠠⢶⣶⡿⠟⠉⠉⠀⠀⠀⠀⣶⣟⣤⣌⠁⠀⠀⢺⡇⣸⣿⣯⠻⠛⠁⠀⠀⠈⠁⠲⠾⠿⠋⣾⠀⠸⣋⣠⣿⠜⣦⣄⠀⠀⣰⠋⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠰⢿⣪⣿⡿⠀⠀⣴⣿⣿⣿⣝⣏⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣋⡴⠞⠉⠙⠻⠿⢟⣛⡶⢼⠇
⠀⠀⠀⠀⠀⠀⠀⣤⠴⠀⠀⠀⢀⡤⢿⣗⠟⠓⠒⢦⣤⠁⠉⠉⠉⠙⣯⠓⠦⣄⠀⠀⠀⠀⠀⠸⠈⠁⠀⠀⠀⠀⠀⠀⢀⡞⠁⠐⢄
⠀⠀⠀⠀⠀⣠⢎⠐⠋⠁⠐⠚⡡⠊⠉⠀⠀⠀⠀⠀⠈⠂⠀⠀⠉⠂⣜⣆⠀⠀⠈⠃⠀⠻⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠈⠉⢡⠁⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⠂⢀⣀⢀⣀⣀⢈⡀
⡀⠀⠀⠤⢤⣤⣤⣤⠤⠬⠭⠭⠭⠯⠽⠽⠿⠿⠿⠿⠧⠛⠓⠉⠉⠁⠀⠁⠁⠀⠈⠀⠉⠉⠀⠈⠙⠋⠍⠸⠿⠭⠿⠉⠉⠉⠙⠁⠉⠉⠁⠉⠁⠈


⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⣿⣟⣛⠃⠀⢸⡿⢿⣦⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣭⣄⠀⢸⣿⣛⣀⠀⣾⡟⣿⡏⠀⣰⡿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣦⡀⠀⣼⣼⡿⠀⠘⣛⣉⣉⣀⣉⣛⠛⠁⢰⡿⢷⣾⠃⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡿⢿⠆⢀⣤⣶⠾⠛⠛⠋⠉⠉⠙⠛⡛⢷⣶⣄⡘⠃⢠⣾⠟⠁
⠀⠀⠀⠀⣠⡶⣶⣄⣀⠙⠓⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠰⣄⡿⣞⠿⠹⡿⣶⣌⠻⡷⠀⢀⣠⣴⠆
⠀⠀⠀⠀⠻⢷⣾⣩⡿⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⢘⣾⣋⠉⠀⠀⠀⠀⠈⠻⣧⡀⢾⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠛⢡⣾⣿⡴⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⡓⠂⠀⠀⠀⠀⠀⠀⠀⠘⣷⡌⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠲⢷⢋⠀⠀⠀⠀⠀⠀⠠⣼⣯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⣤⣀⡀⠀⢸⣿⠀⢈⣻⣏⡄⠀⠀⠀⠀⠀⠲⣾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣀⣤
⠀⠀⠒⢿⡿⣇⠀⢸⡇⠀⠀⢭⣯⡤⠀⠀⠀⠀⠀⠐⢾⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⢿⡿⠓
⠀⠀⠀⠈⠀⠀⠀⢸⣿⠀⠀⠢⣿⠖⠀⠀⠀⠀⠀⠈⣙⣯⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⢀⡑⡿⠂⠀⠀⠀⠀⠀⠀⠈⣽⣶⢡⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⣀⠘⣿⣤⣹⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣼⢠⠀⠀⠀⠀⠀⣠⡿⠁⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣿⣷⡈⢻⣿⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠿⢾⣧⣦⣲⣼⡟⠁⣾⣏⣴⠆⣴⡄
⠀⠀⠀⠀⠰⠾⠛⠁⢙⣷⣤⡙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⣿⠿⠋⢠⣾⣦⡙⠻⢿⠏
⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⡝⠿⣠⣌⡙⠿⢶⣦⣤⣤⣤⣤⣤⣤⣴⡶⠿⢋⣁⠰⣿⣶⣿⣟⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠙⢷⡎⢀⣾⡏⣹⡷⣰⣤⠀⣉⣉⠉⠉⢉⣉⢠⣤⢰⣿⠻⣷⡈⢿⣮⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⣿⡟⢁⣿⣿⣴⣿⡟⠀⠀⢸⣿⣧⣿⡄⢻⣿⠿⡷⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠟⢸⠏⣿⠇⠀⠀⠀⣿⡜⠿⠇⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢰⣮⡥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢖⡎⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⣳⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡮⠭⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠭⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣭⣭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠩⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠔⣛⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠬⠽⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⣎⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠭⣟⡉

⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣭⣿⣷⣵⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⠦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⠻⢿⣿⣿⠿⠟⠋⠀⠀⠀⠿⣿⠏⡐⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⣀⠀⠀⠀⢀⣤⣴⠖⠀⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠙⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠙⠲⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠈⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣙⣄⢐⣀⣂⡁⠀⠀⠀⡴⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠀⠉⠙⢧⠉⠉⠁⠀⢀⡜⠀⠀⣠⠞⠀⠀⠀⢀⣴⠶⠟⠛⠲⠾⣿⣿⣄
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⣶⠾⢃⣤⡶⠋⠁⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡀
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠛⠋⠒⠋⠉⠚⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⡟⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡄
⢰⣏⠀⠀⠀⠀⠀⠈⢻⣦⣤⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣆
⣾⣿⣷⣤⣤⣤⡄⢸⣹⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⣿⠻⢿⣿⣿⣿⢇⡼⠁⣿⣿⣿⣿⡄⠀⠀⠀⠀⠻⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢹⣀⠀⠀⢸⡀⠀⠀⠀⠈⣆
⣿⡛⠢⠤⠤⠴⠋⠹⣠⣿⣿⣿⠋⢻⣦⣀⠀⣀⣠⣾⣿⣿⣿⣄⡀⠀⠀⠀⢀⣠⣿⠏⠀⠀⠀⠀⡜⠉⠉⠛⠛⠿⢯⣭⡉⠛⢾⣆
⠜⠁⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⠟⠁⠀⠀⢀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⢸⡄⠀⠀⠀⠀⠀⠀⢸⡟⠁⣠⠋⠹⣿⡿⠋⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢹⡅⠀⠀⢠⠀⠙⠛⠛⠻⠛⠿⢿⣿⣿⣿⣦⣤⣾⣿⣦⡀
⠀⣷⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠻⡀⠀⠘⣧⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⢦⣀⣀⡠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠄⠸⡷⠶⠤⠀⠀⢦⡄⠈⠙⠋⠀⠀⠈⠻⣿⣿⣿⠈⠛⠢⠤⢤⣀⣀⡀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡦⠀⠀⠇⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⡀⠀⣀⣈⠻⡿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠷⠤⠤⠤⢤⠤⠤⠒⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣧⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢿⣇⠻⠀⠀⠀⢀⣷⠀⣠⣤⡤⠤⠄⠒⠒⢲⡶⠒⠒⠶⣶⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⡄⢀⡈⡿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢀⠙⡄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢀⣾⠃⠋⠀⠀⠀⠀⠀⢀⣠⡶⠃⠀⢠⡾⠁⠀⠀⠀⠀⠸⠀⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣡⣾⣿⣾⠀⠀⠀⠀⢀⣴⣿⠏⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢀⣴⣿⠟⠁⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣀⣴⣿⡿⠃⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠃
⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠿⠿⠿⢿⢿⣿⣿⠿⠿⡿⠿⣿⣿
⣿⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠰⡄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠉⠓⠢⣄⡵⠶⡉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡤⠴⠒⠒⠲⣤⣔⠋⠙⢦⡈⠑⠠⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣐⡠⠐⠒⠐⠒⠤⣧⡖⠁⠀⠀⠀⢀⡴⠋⠈⢧⡀⠀⠙⢆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠲⢄⠀⣠⠏⠀⠀⠀⠈⣇⠀⠀⠈⢣⠀⠀⡄
⠀⠀⠀⠀⠀⡰⠱⣿⣿⡿⠟⠉⠁⠀⠄⡀⠀⣀⣀⣀⠀⠀⠰⡃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢧⠀⠇
⠀⠀⠀⠀⠘⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠡⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡜
⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡷⠄⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸⠁
⠀⠀⠀⢸⠀⠀⠀⣠⣦⣄⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⠟⠁⠀⠀⡇⠀⠀⠀⣸⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠧⠄⠀⠠⠧⠄⠒⠉
⠀⠀⠀⠀⠠⡀⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠑⠄⠈⠿⠿⠟⠛⠀⠀⠀⠀⠀⣀⣤⣶⣶⣄⠌
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠤⠤⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⢄⠀⠀⠀⠀⡔⠀⢐⡢⠀⠀⠀⠀⠀⣐⠈⡁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⣀⡑⡀⠀⠀⡅⡎⠉⠲⢤
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡆⠀⠀⠀⠆⠀⠀⠸⠀⠀⠀⠀⠀⢟⠉⠀⢸⠀⠀⠀⠀⣐⠁⠀⢱⡄⠀⠀⠀⠀⠉⠀⠀⡗⠀⠀⠀⠑⢇⡀⢼⠛
⠀⠀⠀⠀⠀⠀⢠⠇⠀⣠⠁⠀⠀⢀⢱⡄⠠⡇⠀⠀⠀⠀⠄⠘⣆⢀⡦⠀⠀⠀⠀⢫⠀⠀⢈⠃⠀⠠⠊⢉⣹⠂⠘⠗⠠⠄⡤⠐⠈⠀⠄⠁⠈⠐⠄
⠀⠀⠀⠌⠀⠀⠀⠉⠉⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⢀⠄⠄⠧⠄⠺⠠⠤⣔⠓⠉⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠
⠀⠰⠀⢀⡀⠀⠀⠀⠀⠀⢀⡸⢆⠀⠀⠀⠀⠀⠀⠸⠀⣅⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠀⠀⠀⠀⠸⠢⠄⠀⠀⠀⠀⠀⢰⠀⠀⡀
⠀⢸⠂⠚⠃⠀⠀⠀⠀⠀⠜⠒⠫⠀⠀⠀⠀⠀⠀⢂⡁⠄⢨⡄⠀⠀⠀⠀⠖⠈⠖⡀⢠⡄⠀⠀⠈⠠⢸⠀⣀⣀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⡀⢰⠀⠀⠀⠀⠀⢀⠀⠀⠠⡄⠀⠀⢄⣤⠠⠾⠅⠀⢠⡇⣤⡀⢀⠀⠀⠀⡇⠀⠀⣞⠀⠀⡏⠀⢸⡌⣀⠀⢡⡀⠀⡜⠀⡤⢀⠄⠀⠠⡀⢸⠦⢀⠃
⠀⠀⠃⢸⡀⠀⠀⡠⠐⠉⢀⠀⡰⠀⠀⢀⠊⠀⠀⠀⢑⠀⡚⠀⠀⠀⡠⠀⠓⠀⢷⣀⣰⢅⡥⠠⢇⠀⢸⠇⠀⠀⣾⠀⠀⢰⢀⡀⢸⡄⠂⠑⡎⠙⠏⠘
⠀⠀⢀⠀⡅⣜⠁⠀⠀⠀⠀⠀⠀⠀⢀⠆⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠛⠳⠠⠤⣷⠞⢖⠸⠂⠀⠈⣇⠀⣠⠃⠺⢄⢀
⠀⠀⠈⠋⠁⠄⠱⠐⠀⠀⠀⠀⠀⠀⡌⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠈⢀⡦⠉⢆⣀⠀⠀⠈⠒⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠰⠀⣔⠀⣀⡀⠀⣀⠀⡇⢰⣄⠆⢀⠀⠀⠀⠀⢀⡜⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⣧⡄⠀⠀⠀⠀⠀⠔⠀⠲
⠀⠀⠀⣅⠀⡐⠀⢃⠀⠀⠀⠀⡀⠀⠀⠆⠏⠀⠀⡈⠀⠐⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢠⡄⠀⠀⠀⠀⠀⢸⣀⢀⡇⠃⠀⠀⠀⠀⠀⢸⠐⢚
⠀⠀⠀⠰⢠⡤⠚⢋⢀⢼⠉⠣⣄⠠⠀⢸⠄⠀⠀⣰⠒⡒⢠⠁⡀⡳⠀⢂⠀⠀⡀⠀⠒⠻⠑⠆⠰⠇⠀⠀⠀⠀⢀⠜⠀⡠⠅⠀⠀⢀⡀⠄⠂⠁⡀⠼
⠀⠀⠀⠀⠀⠅⠠⠼⠁⠀⡀⠸⠈⢂⠀⣀⠀⢀⠈⠀⠀⠁⠟⠁⠈⠇⠀⠄⠀⠀⠀⠄⠀⡄⠂⢸⠀⠒⠒⠒⢸⠁⢀⠂⠁⠀⢸⡗⣎⠀⡡⠄⠺⠁⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⢣⠀⠀⠁⢰⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠌⠒⢒⠠⠀⡇⠀⢀⡌⠠⠐⠇⠀⢸⠀⣡⠀⠀⠀⠦⠾⣀⡜⠀⠈⡇⠉⠉⠄⠐⠦⠔⠂⠐⠁
⠀⠀⠀⠀⢀⠣⠀⢸⡀⡘⠀⢸⠀⠀⠀⡌⠢⠀⠰⠀⠀⣶⠀⢀⠌⠀⠀⠇⠀⠁⠁⠄⠀⠀⠀⠄⡽⠉⠐⠀⢀⡂⠀⡄⢈⠀⠀⡇⠀⠒⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠔⠨⠀⠄⠘⠇⠇⠀⡘⠲⠀⠠⠀⠀⠑⣀⣳⡄⣏⢀⡂⠀⢰⠒⣃⠀⠠⡤⢳⠀⠇⢰⠂⢳⡆⠀⠠⣜⠘⠀⠅⢾⠀⡀⠴⠀⠀⢰⠁⢰⡆⠀⡐
⠀⠠⠊⠀⠐⠀⠘⠀⡳⠈⠆⠀⠺⡂⠑⠀⡀⠠⠁⠠⠞⢇⠀⢂⠀⠀⠁⠁⠀⢀⠒⠀⠠⠆⢎⠐⠀⢇⠀⠀⡟⡡⠀⠐⠲⠠⡏⠸⠀⠀⠰⠀⢸⠇⣴
⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠃⠀⠂⠀⠁⠂⠁⠀⠀⠀⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣹⠷⠾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣼⠷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⢿⣿⣯⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢻⣬⠿⣏⢈⣛⡻⠻⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣃⣟⠋⢹⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠉⠛⠣⠻⣿⣏⣄⣠⢾⣄⣸⠿⢽⡦⠀⠀⣀⡀⢈⣱⠾⣍⠰⠗⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠹⣄⢀⢸⣄⠀⠀⠀⠀⠀⠐⠀⢈⣙⣛⠾⠶⢤⣄⣬⣇⡀⠻⡬⢵⡆⣄⠀⡀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⡀⠉⠀⠻⣇⣰⣀⣀⣠⣿⠤⣯⣽⣟⣹⡄⠀⠈⠉⠙⢿⣴⣿⣧⣁⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠉⠉⠉⣥⣰⠐⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣧⣤⣿⣯⡙⠀⣤⡇⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⣮⠻⣶⣤⣤⣶⠶⠛⠁⠀⠻⢷⣇⡐⣦⣤⣀⠀⠀⠀⠀⠉⠙⠿⢁⠀⠈⠙⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠒⢻⠿⠿⣷⣶⠶⠆⠀⠀⠀⠀⠈⠹⣾⣿⣿⣶⠀⠀⠀⠀⠀⠀⠨⢳⣼⣿⣶⣶⣗⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡃⠉⠁⠠⠾⢿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⡥⠤⠄⠉⠛⡿⠧⢦⠤
⠀⠀⠀⠀⠀⠀⠀⢐⣈⡁⠀⠀⠀⠀⠀⠀⠀⢽⡁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣤⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⣼⠁⠉⠀⣀⠴⠋⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠉⠀⣀⠞⠁
⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⡀⠀⠀⠀⠁⢠
⠀⠀⠀⠀⠈⠀⠀⠀⠃⣠⢾⡇⠀⠀⠁⢆⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠈⣷⡦⠀⢀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡲⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⢎⣴⣾⠟⠿⢦⣈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠙⠋⠀⠢⠀⣈⠚⠁⠈⡙⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠒⡂⠖⠲⠁⠀⢿⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡄⠀⠠⠀⠠⢀⠀⠂⠀⣼⠃⠹⠁⢲⡏⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡰⠒⠒⠀⠄⡃⠀⢢⠐⣅⠀⣧⡸⣅⡀⣠⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠍⢀⠄⠀⠀⡴⢹⠋⠀⢈⠉⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡕⠀⠀⠀⠱⠂⠁⠈⠀⠄⠤⢠⠋⠶⡌⠀⢀⢴⠆⠒⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡈⡀⠰⠀⠀⠀⠨⠀⡐⠆⠰⢀⠀⡺⠀⠀⠀⠄⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠜⠀⠀⢀⠀⠀⠀⠀⠀⣷⣤⣽⡀⠀⠀⢀⠀⠀⢨⣤⡤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡄⡃⢦⡸⠄⠀⣂⣀⠀⢉⠈⡻⡇⠀⠀⠀⠀⠀⠐⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢁⠀⠀⠁⢧⠀⠀⠀⠀⢸⡴⣿⢵⡖⣒⡲⠒⢪⣇⠼⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠂⠀⠔⣆⠀⠀⢀⠈⠳⠀⠀⠀⠈⠀⠀⠀⢈⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⣂⠀⠀⠈⣄⠀⠀⠀⠀⠐⡆⠀⠀⣄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⣍⠂⣈⢆⠈⢀⠀⠀⢂⠀⢀⡉⠆⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠠⠁⠑⠀⠐⢤⡎⠀⠸⡇⡘⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡄⠀⠀⢱⠇⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢸⠆⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡥⠀⠹⠇⠨⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡤⠴⠆⠰⠤⢤⡤⠆

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⡉⣐⢰
⠀⠀⠀⠀⠀⢰⠋⠀⠉⠙⡇⠀⢀⠤⠤⠤⡆⠀⡞⢀⠀⢀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢤⠂⡀⠀⡉⢷⠁
⠀⠀⠀⠀⠀⠎⠀⠁⠈⠁⠁⠀⠊⠄⠀⠄⠓⠀⠀⠀⠀⠀⠔⠀⠂⠐⠀⠂⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠤⠤⡊⡀⠂⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣴
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⡀⠠⡒⠀⡢⠄⢀⡀⠈⠀⠀⠀⠢⡉⡉⠣⠀⠀⠀⠀⠀⠈⠂⠀⠈⠁
⠀⠀⠀⣄⠀⠀⠀⠀⠀⡖⠈⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣦⡀⠀⠀⢠⣄⣄⢀⣠
⠀⠀⠀⠈⠓⠀⠀⡰⠎⠈⣳⡤⠀⠀⠀⠀⠙⠀⡤⡇⠈⢣⠀⠀⠀⢄⢀⠒⢀⣐⡚⠋⢛⠓⠀⣺⡇⠙⣟⠉⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⡁⠃⠀⠀⡔⠒⠃⠀⠁⠸⠀⣸⠀⠀⢘⢉⣏⠁⠀⠈⠁⠀⠈⠉⠠⠟⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠊⠀⠀⠀⠀⠀⠈⠀⠀⠁⠉

⠀⠀⠀⠀⠀⠀⣤⡴⠆⢀⠀⠀⠀⢠⣴⡶⢂⡀⠀⠀⣀⣤⠖⣠⣤⡴⢀⡀⠀⠀⠀⠀⠀⣠⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤
⠀⠀⣀⣠⣤⣤⣽⣷⣾⡟⣷⣤⣤⣼⣿⣷⣿⣿⣤⣤⣹⣿⣴⣿⣿⣶⡿⢿⣄⣀⣀⣀⣀⣹⣽⣶⣿⢿⡄⠀⠀⣀⣤⣤⣿⣿⣶⣿⣆
⠐⠛⠉⠙⣿⠿⠿⢿⣿⡿⠛⣿⠿⠿⣿⣿⢋⣽⠿⠿⠿⣿⣿⣿⣿⣿⣇⠉⠉⣩⢿⡿⠿⠿⣿⣿⣇⡀⠐⠛⠛⠻⣿⡿⣿⣿⣟
⠀⠀⠀⠀⠙⣆⠀⠀⠃⠁⠀⠈⠦⠤⠼⠬⠞⠁⠀⠀⠠⠇⠘⠧⠄⠈⠳⠤⠊⠁⠏⠀⠀⠀⠀⠘⠄⠑⠀⠀⠀⠀⠹⣦⠀⠘⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠟⢙⣿⣿⣿⣿⡿⠛⠋⣹⣿⣿⣿⠿⠟⢙⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⣿⣿
⣿⣿⣿⡿⠿⠷⠄⠘⠉⠀⠈⠟⠛⠃⠀⠈⠁⠈⠻⠯⠀⠀⠋⠀⠀⠈⠀⠈⠻⣿⣿⠿⡿⠄⠀⠋⠉⠈⢻⣿⣿⡿⠿⠯⠀⠀⠊⠉⢻
⣝⣁⣠⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⢰⣷⣎⣉⣁⡀⠀⠀⠀⠀⢀⣶⣤
⣿⣿⣿⣿⣄⠻⣿⢓⣠⢸⣿⣤⠚⠿⠆⠂⠁⣰⣿⡿⠿⢰⣄⠐⢖⠂⠀⠿⣣⣔⢡⣾⣿⣷⣶⣆⢲⡌⢿⣿⣿⣿⡀⠚⢖⠆⡠⣸⣿
⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣦⣼⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣦⣾⣿⣷⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠡⢴⣿⣴⣤⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢴⡆⠹⠆⠋⠉⠈⠘⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢀⣤⣾⣤⡂⠉⠀⠀⠀⠀⠰⣾⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡀⣼⡿⠿⠛⠻⢿⠟⠿⠿⣦⡀⠀⢨⡹⣯⠱⣿⡇⠀⠀⠀⢀⣠⣴⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⢀⣤⠿⣇⣸⠂⠀⠀⡠⠃⠀⠀⠀⠀⠉⣳⣮⣠⠚⠋⠉⠋⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠙⠃⢭⣷⠱⣀⣄⡴⡁⠀⠀⠀⠆⠀⠀⠘⣿⣿⣿⣙⡿⢈⠁⠈⠛⢏⠀⠈⠉⠉⠉⠙⠉⠙⠛⠉
⠀⠠⠬⡥⢦⣼⡋⣹⣴⠟⠋⢸⣿⠃⢰⣿⠂⠒⠀⠀⠧⣽⣿⣥⡽⠋⢁⠀⠀⠀⠡
⠀⠀⠐⠀⠀⠈⠉⠁⠀⠀⠀⠀⠹⡤⢈⠛⠢⡀⠀⣀⡴⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣧⣄⠈⠙⣿⣄⣀⣀⣀⡤⠔⠒⢠⡀⠀⠀⠀⡔⠂⠄⠛⢿⠲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⠀⠈⠀⠈⠀⠈⠒⠀⠊⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⣄⣤⡾⣻⠀⣶⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣾⣷⣿⣿⣾⡿⢿⣿⣿⣷⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢛⢻⣿⠿⠻⠏⠀⠀⠘⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠾⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢄⡀⠀⠀⠀⠀⠰⢿⡶⣤⢤⣶⠏⠀⠀⣀⣄⣀⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠎⠀⢸⣇⠀⠀⠀⠀⠀⠀⡠⠀⠘⡿⠁⠀⣾⣿⣿⣿⣿⣦⢌⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠈⢿⡄⠀⠀⠀⡠⠎⠑⠲⣶⣇⢀⢀⣭⣭⣛⠿⣿⣿⣿⣷⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠘⣦⡈⠻⢤⡀⠀⠀⠀⢬⠭⢹⣿⣷⠎⠙⢿⣿⣿⣮⡻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡠⠒⠊⠁⠀⠀⠀⠀⠀⠠⣄⣙⣻⣆⠀⣿⣷⣶⣤⣄⣠⣿⣷⣿⣾⣷⣤⠀⠙⢿⣷⣼⣿⣷⡆
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠉⠋⠳⡆⣼⡟⣷⠀⣿⣿⣿⣿⣿⣹⣧⣿⣿⣷⡏⢼⣿⣿
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡈⢳⣻⡀⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠃⡀⢀⣹⣿⣻⣿⣿⡿⠛⣿⣿⣏⣿⣿⠟
⠀⠀⠀⠠⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠈⠋⣿⣮⣟⣿⣿⣿⡋⠀⣠⠀⠉⠉⠉⢙
⠀⠀⠀⠠⣜⢿⣷⡀⠀⠀⠀⠀⠀⢀⣴⣿⣇⣀⡀⠀⠀⠀⠀⠠⣜⠿⣿⣿⢏⣾⣿⢣⡔⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢳⠀⡿⣿⡦⠀⠀⠀⠀⠁⠊⠀⢻⣿⣿⣶⣦⣤⡴⠓⠿⣿⣶⣯⣾⡿⠃⠸⣇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⣇⣷⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠋⠁⠀⠀⠀⠀⠉⢙⣟⠁⠀⠀⠙⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠘⣿⣷⡹⣧⡀⠀⠀⠀⠀⢀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣦⡀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣜⣷⣤⡲⢤⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢹⠏⠛⠻⠿⢿⣤⣄⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣮⣥⠀⠀⠀⠀⢀⣀⣤⠄⢀⡾⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠏⠳⣍⠛⠁⠀⢴⣶⣿⡿⠛⣁⡴⠋⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠰⢤⣀⠀⠛⠁⣠⠞⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠉⠛⠒⠚⠁⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⣀
⠀⠀⠀⢀⡴⠊⢁⠔⠊⠙⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠒⠒⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⡠⠺⡅⠀⠀⠀⠀⡿⢢⡀
⠀⠀⠀⣾⣣⡴⢁⡠⠤⠒⠚⠛⣿⣆⠀⠀⠀⠀⡠⣞⢹⣀⣠⣤⣤⢤⢤⢤⣤⣽⠹⡦⡀⠀⠀⠀⢸⡷⢀⡇⠀⠀⠀⠀⡇⠀⣷
⠀⠀⠐⣿⣉⢖⣉⡤⠤⠤⢄⣀⣿⣿⠀⠀⠀⢸⣿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣿⡿⠀⠀⠀⢸⢛⡾⠁⠀⠀⠀⠀⢹⣄⣿
⠀⠀⠀⢿⣧⣼⠣⡀⠀⠀⣴⣮⣩⠇⠀⠀⠀⠈⠻⣷⣿⣠⣤⣂⣤⣀⣀⢀⠤⣤⣧⠟⠁⠀⠀⠀⠈⢿⣷⣄⡀⢀⢀⣀⣤⣿⡟
⠀⠀⢠⣤⣿⣿⣿⣽⣾⣾⣿⣷⣧⠄⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠄⠀⠀⠀⠀⠀⣠⣽⣿⣿⣮⣿⣿⣿⣯⡄
⠀⠀⠀⠈⠛⠿⠿⢿⠿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⣠⡴⠚⢻⠿⢗⡲⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⡞⠍⠋⠻⣿⠿⠛⠢⡀
⠀⠀⢀⡞⠁⡇⠀⢸⠤⢄⠈⠓⢵⡄⠀⠀⠀⠀⠀⢠⠏⠀⠰⠟⠀⠾⠃⠀⠀⠘⣧⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢸⠀⠀⠀⢽⡄
⠀⠀⢸⣷⠀⢱⢀⡸⠤⡀⠉⠢⣀⣷⠀⠀⠀⠀⠀⢸⣂⠀⠀⣠⡆⠀⠀⠀⠀⠀⣺⡇⠀⠀⠀⠀⣿⡟⠁⠀⢠⣶⣿⡦⣀⠤⣿⡧
⠀⠀⠸⣟⡧⠚⠉⢀⡠⠊⢑⣴⣯⡟⠀⠀⠀⠀⠀⠸⣿⣄⠀⠈⠀⠀⠀⠀⠀⣤⣿⠃⠀⠀⠀⠀⢻⣧⣀⠀⢈⡿⣯⣰⠀⠠⣴⠇
⠀⠀⠀⠹⣿⣯⣖⡉⣀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⣀⣠⣤⣴⣴⡿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣷⢯⣂⡠⡀⣱⣶⠏
⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣷⣾⡿⠟⠃⠁⠀⠀⠀⠀⠀⠰⠿⠿⣿⣿⣿⣷⣾⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⣶⠾⠿⢷⠿⠾⢾⠾⢶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⡾⢶⣶⣷⡶⣾⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣊⣤⣶⣾⣿⣿⣿⣿⣥⠈⠉⠒⠚⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⣫⣴⣿⣿⠿⣻⡿⠛⠋⠉⠉⠉⠛⠢⠀⠀⠀⠈⠐⣶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢥⣾⣿⡶⠛⠁⠞⠁⠄⣒⠒⠒⠂⣀⣀⢀⠀⠀⠀⠀⠀⠈⠃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⣾⣿⣟⠁⠀⠀⣠⡴⠟⣛⣻⣶⣜⢿⣿⢎⠳⡷⠀⠰⠶⣷⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡀⢸⣿⣿⠃⠀⣠⣿⠋⣰⠟⠉⠉⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠈⣿⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣇⠈⠐⣶⣆⣰⠃⢡⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢷⣿⣿⣿⠋⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠈⠻⠻⠷⣾⡷⠀⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣿⣷⣄⣷⣦⣾⣟⠁⠀⠀⠈⣀⣀⣀⣉⠙⠛⠃⠀⠀⠀⠴⣛⣉⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣮⣿⣿⡇⣙⢿⣷⣪⠂⠀⠀⠙⠻⣟⣿⠏⠳⠄⠀⠀⠀⡴⠻⡟⠉⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⡿⡟⠱⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣅⢻⣄⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⣿⣿⣷⣦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⡠⠞⠙⣿⣿⣿⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠶⠾⡦⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠩⠷⠶⠖⠀⡰⠁
⠀⠀⠀⠀⠀⠀⣸⣿⡄⢀⡴⠞⠀⢰⣯⣾⣿⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠘⣿⣏⡀⣨⣿⠁⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣻⣿⠃⢀⣨⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡿⢷⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⠟⡀⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠛⠋⡀⠀⠀⠁⠘⠢⠤⣀⠀⠈⠂⠀⠈⠻⠮⡳⣌⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠘⠻⠧⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠑⠀⠀⠈⠪⡓⡙⠛⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⠀⠀⢠⣿⣿⠄⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⢀⣶⣦
⠀⠀⠀⠀⣠⡟⠀⠀⢀⣉⠉⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⣴⡟⠀⢀⣴⠏⠉⠃⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⡀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣿⣿⣿⡀
⠀⢠⣾⣿⡄⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⡎⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠘⠛⡻⡻⠟⠿⠟⠙⢶⣄
⠀⣼⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣴⣷⣼⣟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣿⠇⠀⠀⠀⢠⣾⣷⣼⠏⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠙⢷⣄
⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⡆
⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣤⣤⣄
⢰⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⡆⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀⠈⠉⠁
⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧
⠘⢿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣏⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧
⠀⢸⣿⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠘⢿⣆⣀⠀⠀⠀⠀⢠⣿⠏⢸⣿⠁⠀⠀⠀⠀⠀⢀⣸⣿⣿⠻⣿⡟
⠀⢸⣿⠀⢿⡇⠀⠀⢀⣾⣷⡄⠀⢠⡿⠁⠀⠀⢸⣿⣿⣶⣶⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⣀⣴⡿⠟⠋⠀⠀⠙⣿⣆
⠀⠀⣿⡄⣸⣷⡀⠀⠀⢻⡏⠀⢺⣿⠃⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠸⣿⠀⢸⡿⠀⠀⣿⣿⠟⠁⠀⠀⣾⣿⣷⡀⠈⢻⣧⣀
⠀⠀⠻⡷⠉⠛⠓⠀⠀⢸⡇⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⣼⡇⠀⠀⣿⠁⠀⢀⣤⠞⠛⠛⢻⣄⠀⠀⣿⣿⣿⡶⠆
⠀⠀⠀⠀⠀⠀⣾⣷⣄⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⢿⡇⠀⠀⠀⠀⠰⣿⡃⠀⣼⣿⣦⢻⡆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣽⣿⣡⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⣼⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠈⢿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠀⢨⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⠈⠻⠿⠛⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣠⣄⣠⣶⣿⡟⠋⠁⠀⠀⠀⠀⣬⡻⣿⣿⣿⡉⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣾⠿⠟⠋⢸⣿⠃⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⣿⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠁⠀⠀⢀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢈⣿⡀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⢠⣼⠟⠛⠀⢀⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠡⠀⠀⠀⢀⠀⠀⠀⠀⠀⣒⣶⣦⣀⣄⣤⣴⡴⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⡇⠀⠀⠀⣤⡄⢠⡀⣤⣶⣿⣿⠿⠿⣿⣾⣿⣿⣿⣦⡄⡀⣨⡄⠀⠀⠀⢸⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠸⡇⠀⠸⣄⣿⣿⣾⣿⣿⣛⠛⠛⠃⠀⠀⠈⠉⠉⠉⠛⢿⣷⣿⣷⣠⠄⠀⣸⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠛⠶⠶⠿⠿⠟⠛⠛⠋⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⡿⡿⢶⠖⠛⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⣀⡄⠀⠀⣴⠋⠉⠑⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠒⠛⢷⡄⠀⢀⣀⣤⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⢊⣵⣿⣷⣄⠀⠀⠲⣦⣤⣰⣶⡶⠚⠀⢀⣴⣾⣶⣝⢷⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⣶⣿⠋⠀⣿⣿⡟⠉⠻⣷⣄⣀⣷⣿⣿⣿⣿⣥⣴⡾⠋⠙⣿⣿⡇⠙⣿⣶⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢻⣿⡟⠋⠁⠀⢿⣿⣇⠐⠦⠀⣹⣿⡿⠿⠿⠿⣿⣿⡁⠠⠖⢀⣿⣿⠇⠀⠉⢻⣿⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡟⠁⠀⠀⠀⠀⠉⠙⠳⠶⣾⡿⢗⣛⣥⣬⣽⣻⣿⣿⡶⠶⠟⠋⠁⠀⠀⠀⠀⢻⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣄⠀⠀⢀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⠀⠀⢠⣿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡄⢠⣿⣿⣿⠉⠉⠹⠉⠙⠛⠛⠉⠛⡿⠋⠉⠛⠛⠉⠹⠋⠉⢹⣿⣿⣧⠀⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠟⣿⣿⣸⣟⢿⣿⣆⢠⣾⣧⣼⣷⣦⣤⣤⣧⣤⣤⣶⣿⣤⣿⣦⣀⣼⣿⢻⣿⣼⣿⠛⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡌⢿⣿⡿⠁⢿⣿⣿⣿⡿⠿⣿⢿⣿⣿⣿⣿⡇⠘⣿⣿⠏⣾⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋⣿⣷⡸⣟⢧⡀⠈⡟⠀⣃⠀⠀⡇⠀⠀⡇⠈⣏⠀⢀⢿⡏⣼⣿⠙⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⣷⣌⡓⠮⠍⠛⠚⠛⠓⠛⠛⠓⠛⠛⠚⠋⠭⣖⣫⣾⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢸⣿⡿⣿⣷⣶⠀⠀⠐⠈⠛⠛⠧⠀⠀⠀⢲⣿⣿⢿⣿⡏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠙⢿⣿⡿⣶⣶⣆⡀⢀⣤⣶⣶⢾⣿⡿⠃⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠻⠌⠳⠈⢻⣿⠋⠞⠡⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⣾⣿⣿⣾⣦⠀⠀⢀⣀⡸⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢛⣯⣏⠙⢿⣿⡗⣶⢒⣿⣿⣧⣾⣿⣿⡣⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⡶⠚⢻⣧⣬⣿⠠⠼⠋⠀⠀⣉⣉⡻⢦⣿⡿⢿⡝⢿⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣿⡃⠀⢉⣡⣴⢾⣧⣀⠼⢧⣾⡏⣸⣿⣟⠂⢨⡎⢳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣯⠁⠀⢀⡽⠁⠀⠀⠈⠙⠛⠶⠝⠘⠿⣋⡌⢛⣻⡄⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣈⡻⣯⣳⣶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⠿⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢸⣿⣻⣧⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣼⠛⣛⣂⢼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣼⢃⣿⠷⣼⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⡡⢖⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣹⢘⣟⣻⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄⣴⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣽⣿⠻⣿⠿⢿⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⡇⠰⡾⠿⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡿⠀⠀⠀⠀⠉⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠀⠓⣷⣿⣿⣤⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣪⣡⡾⠙⠃⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠉⠀⠉⠙⠛⠷⠟⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠘⠁⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠄⠀⠀⠀⠀⠀⢠⣠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣻⡇
⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⠀⠀⠰⢂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠾⠿⠇

⣿⣟⣻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⡿⠟⠛⠩⠛⣿⡘⡋⡈⠛⢻⣏⣉⡘⢹⠘⡿⠬⠛⣻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠄⣤⣄⣠⠤⠄⠘⡟⠋⠋⠛⠛⡏⠉⠉⠛⢻⣃⡀⢀⡿⠤⡞⣻⣿⡟⠢⠉⠛⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠀⠙⠀⡄⣈⣧⠀⠀⠀⠀⣀⠤⠄⠄⠐⠇⠀⠀⠀⠀⠀⡉⠛⠁⠀⠀⢻⣤⣀⠀⠀⠁⢀⣼⠟⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⡁⠀⠀⠀⠀⠈⠁⡀⠔⠒⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⡆⠀⠀⠈⠉⠒⣵⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⠛⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⣠⠞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⡉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠑⠒⠀⠀⠀⠀⢠⠎⠀⠀⠀⠠⠊⠀⠀⠀⠀⠤⠒⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⡿⠌⠊⠙⡧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠖⠓⢤⣉⠉⠀⠲⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣟⣿⢻⡅⣿⡿⠀⠀⢸⠁⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⠉⠙⠛⠂⠀⡀⡀⠂⠀⠀⠀⠀⠀⠀⠻⠋⠁
⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣶⣿⣧⣀⡀⣸⠀⠀⠀⠠⢀⡀⠸⠓⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⡯⠉⡉⠉⢙⠛⠒⠢⠤⠄⢀⡉⠁⠐⠠⠄⣀⠉⠓⠀⢀⠀⡠⠒⠊⠀⠀⠐⠁⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠄⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠢⠍⠀⠀⠀⠀⢀⠄⠐⠉⠀⠐⠒⠂⠀⠒⠒⠒⠂⠀⠂⠈⠈⠉⢁
⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⠛⣉⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠒⠒⠦⠤⣄⣀⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⠛⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠄⣁⠒⠒⠚⠉⢀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍
⣿⣿⣿⣿⣿⣯⣿⣿⢬⣦⣀⣠⣩⣦⠀⠀⠀⣀⣀⡤⠔⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠃⠀⠒⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣏⢋⠉⠈⠘⣷⠛⠉⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⠀⢀
⣿⣿⣿⣿⣿⣿⣽⣟⣹⣿⣽⢷⣀⣶⡿⠟⠀⢀⡠⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠘⠲⠶⠲⠶
⣿⣿⣷⣴⣿⣿⣿⣿⡧⡿⢿⣧⠉⠁⢀⣴⠚⠉⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⢠⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐
⢿⣿⣿⣿⣿⣿⣿⣿⣳⡿⣛⠟⢰⣿⠟⠋⠁⠢⠀⣠⡶⠉⠀⠀⠀⠀⡘⠀⢣⠧⠄⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠐⡀
⣬⣿⣿⣿⣿⣿⣿⣿⣯⡿⣅⣶⢟⠛⠷⣢⡀⠀⠀⠉⠠⣀⠀⠀⢀⣼⣀⣀⡆⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⡤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢫⠖⠀⣠⡿⠇⢄⠀⠀⠀⡙⠲⣾⠟⠀⠀⠀⠉⠈⠃⠀⠀⠀⠀⡏⠉⠉⠀⠀⠀⠀⠀⠤⠒⣏⠀⠀⠈⣆⣀⣠⡴⠞
⣿⣿⣿⣿⣿⣿⣿⣿⢛⣔⣿⣵⡳⣬⣾⠟⠁⠀⠀⠉⠲⠞⠀⣰⡏⠀⠀⠀⠀⠀⠀⢤⢄⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⢂⠀⠀⠈⢧⣤⡾⢿⣿⠡⠄⡠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠐⠉⠐⣿⣤⣀⣀⠀⠀⠀⠀⢸⠀⠀⠀⢉⡶⠒⠒⠒⠈⠉⣄⣀⣶⡴⠶⡋⣼⠀⠀⣠⣷⣎⣹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣸⣿⣿⣖⣛⠿⡶⢰⡤⣾⣤⣤⣄⣠⡷⠀⠀⠀⠀⠀⢿⣄⣀⢤⣍⣷⣼⣦⡶⢟⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⣿⣮⣶⣷⣠⣦⣬⣭⠀⠀⠀⠀⢀⠀⢼⣿⣽⣾⣿⣿⣿⣿⣿⣈⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣇⣴⣦⡞⢹⣷⣶⠛⢷⣴⣷⣶⠞⣿⠛⠻⢿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣥⠾⠛⣿⣧⠾⠙⢿⣤⡼⠏⢻⡿⠀⢿⡶⠻⣿⡤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠄⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢴⡿⠟⠀⠀⠀⠀⠒⢄⢸⣇⣾⡯⠴⠒⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⣒⡲⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠙⢟⠙⢦⡂⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⠩⠄⢢⠠⠈⠉⢷⣆⣄⠀⠀⠁⣾⣿⣿⣶⠀⠀⠀⠸⣷⡄⠈⠑⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠀⠀⠀⡈⣀⣤⣴⣾⣾⠉⣿⡄⠀⠀⠛⠻⠿⠃⠀⠀⣸⣷⡘⢿⣶⠞⠁⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠁⠀⠀⢀⣼⡿⠋⣀⠄⠠⠈⠐⣹⣷⣀⡢⠀⠀⡀⠀⢀⣤⣯⣿⣷⠀⢉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⣿⣿⣠⠎⠀⠀⠀⣰⣷⡿⠋⠉⣵⣿⡏⡀⢰⠟⠁⠀⠈⣿⣿⣧⣀⣄⠀⠀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⡁⠀⠀⢠⣾⢻⣿⣟⠂⣸⣿⣿⣿⣧⠂⠀⠀⠀⣠⣿⠛⢿⣿⣧⣤⣴⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣶⣾⣾⣯⣿⡿⠾⠿⣿⣿⣯⡹⣿⣽⣀⣠⣴⣿⠃⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢀⣴⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀⢠⡄⠀⠀⠀⠀⠀⠀⢠⣾⠏⡜⣦⠛⢽⣉⣷⣤⠀⠄
⠀⠀⣠⣴⡾⠿⣶⣶⣡⠞⢭⡧⠀⠃⠀⠀⠀⠀⠀⣴⢯⣵⡄⣽⠘⢷⣄⣘⡁⠀⢡⣐
⠀⣴⣿⣿⣿⣇⡀⢘⣻⢷⣶⣦⣠⣦⡀⠀⢀⣀⠀⠙⠊⠁⠘⣿⡆⠀⠙⡿⣏⡙⢿⢾⣦⡒⠀⠀⠀⡀
⣰⢿⣿⠯⠟⠻⣿⣶⣬⣛⠿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣆⣤⣄⣹⣿⣄⠀⠁⠙⠿⣼⣾⣿⢃⣷⠀⠀⠘⠿⣷⣦
⠙⠛⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⣉⣿⣿⣿⣿⣿⣿⡟⠉⠋⢹⣿⣿⡀⠀⠀⢂⠀⣠⣾⢿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⡟⣷⣿⠿⠛⢿⣿⣿⣿⠃⠀⠀⠈⢿⣿⣷⣤⠀⠛⠻⣏⣉⡲⣿⢀
⠀⠀⠀⠀⠀⢀⠘⣡⠞⠛⠙⠋⠉⠉⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠻⠜⠃
⠀⠀⠀⠀⡀⢈⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣧⠀⠀⠀⠀⢸⡵⠧⡘⠇⠀⠰⡆
⠀⠀⢀⡴⠙⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠘⢧⣤⡄⠀⠀⠁⠈⠳⡀⠀⠛⠼
⠀⠀⠀⠀⣰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⢀⣠⣴⣾⣿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⡿⠿⠶⣤⣀
⠀⠐⠚⠉⠉⠉⣿⠛⠿⠿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⣤⡄⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⡿⠿⠛⠋⠉⢹⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⣿⡿⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣾⣿⣿⣿⣿⣿⣟⠋⠁⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠘⠧⠴⠿⠋⠀⠈⠉⠙⢻⡏⢿⡋⠉⠹⣇⣀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⢻⡏⠉⠉⠀⠀⠀⠀⢸⡏⠉⠉⠙⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠛⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡍⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⣤⣤⠿⠤⠴⠶⣶⠶⠶⠾⠷⠶⠒⠒⠒⣶⠟⠛⠛⠛⠋⠉⠉⣹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢰⣿⠹⣦⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⣀⣀⣀⣀⣀⣿⣄⣀⣀⣀⣀⠀⢰⡿
⠀⠀⠀⠀⣆⠀⠘⣿⠀⠈⢷⣄⣀⣤⠴⠶⠞⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠲⠦⠤
⠀⠀⠀⠀⢻⡄⠀⠙⠳⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠙⠶⢤⣤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⣹⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⣰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣿⣿⣿⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣆⠽⠟⠋⠛⠀⣨⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⣿⣿⣿⣿⠗⠶⠂⠀⠀⠀⠀⠈⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⠈⢻⡟⠁⠀⠀⠀⠀⠀⢲⣄⠀⢀⣿⣿⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢲⣾⣿⡿⠃⠀⠸⣦⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣹⣿⣿⣿⣦⡤⠀⠻⢷⣶⣾⡿⣿⣿⣷⡈⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣟⢻⣿⣤⣦⡀⠀⠀⠀⠀⠨⣺⣿⢿⣷⣤⡈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢼⣧⣤⣤⣀⠀⠀⠀⠀⢠⣿⣿⠟⠉⠁⠠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠏⣾⡟⠁⠈⢉⣠⣴⣶⣶⣾⣿⢿⣦⣀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⣸⣇⣀⢠⠏⠁⠀⠀⠀⠀⠘⣶⡬⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⢾⣄⠀⠀⢀⣠⣤⢞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠛⠛⠓⠹⣿⣾⣿⣫⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣀⠀⠀⣠⡦⢰⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶
⠰⠐⠇⠙⢙⣿⣻⢏⠈⡄⠀⢠⠀⡀⣀⠈⠈⠀⠀⠀⠀⠀⢀⣀⠀⠐⠆⠀⡀⠀⣠⠀⢀⠀⠃⠀⠀⠹⠿
⢠⣴⣿⣶⣿⡏⠀⢧⠶⣧⠀⢘⡟⢱⣯⠅⣿⡳⠀⠀⠀⠀⠈⠙⠁⢀⡦⠞⠛⠉⠉⠙⠻⢤⣴⣚⣶⡄
⠘⠉⣉⠙⠻⠷⠒⠆⠀⠸⠀⠈⠁⠘⢳⡘⠮⠇⠀⢀⠀⠀⠀⠀⠐⡞⢀⠀⠀⠀⢀⠃⡀⢀⢈⡿⢁⠙⠆
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠴⣇⠀⠤⣦⣷⣨⣤⣷⣧⣾⡵⡏
⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠘⢦⣐⣚⣭⠭⡙⠛⣫⠭⠕⠃
⠀⠀⠈⠕⠂⠶⠠⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠤⡀⠀⠀⠀⠸⠀⠁⠀⠡⠀⠀⠀⢰
⠐⠽⠆⠀⢂⠁⠀⠀⠀⠈⠨⢉⠀⠻⠀⠀⠠⠁⠈⠀⣀⣤⡤⠀⣄⡠⠇
⠀⠀⠁⣰⠜⡿⢬⡀⠀⠀⠀⣁⠈⠃⠀⠀⠁⢰⠾⠛⠋⠉⢀⡶⠳⠶⠾⠶⠄
⠀⠰⣀⠻⣗⡀⠀⠀⠑⠀⣠⣄⢀⠀⠀⠀⠁⠀⡀⠠⢰⠺⠁⠉⠂⠀⠀⠀⠀⠀⠀⣀⢀⣀⣤⣤⣤⣄⢆
⠀⠀⠁⠠⠄⠑⠀⠤⠒⢚⠙⢋⢐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣵⡿⣿⡆⡿⣿⣿⣯
⢠⣶⣶⣶⣶⡖⠊⠁⠛⠃⢓⠃⠈⠐⠆⠀⠀⠀⢀⡤⣤⡂⠀⠀⠀⠀⠀⣶⠀⠀⠈⣉⠛⣍⠳⠷⣯⡽⢻
⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣶⣴⠤⡶⠀⣿⢾⣦⠡⣿⢰⡧⡄⠀⠈⣶⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⡆⠀⠀⠀⠀⡠⡈⣄⣙⣋⠤⡛⢛⣀⠒⠂⣒⠘⠛⢐⣻⣝⣳⠇⢀⣴⣿⣿
⠸⠒⠀⠀⢀⣠⠝⠻⣿⣿⣿⠕⢾⣶⣶⣶⣷⣎⡺⠊⠙⠲⠮⢭⣤⣉⠉⠉⣉⣉⣁⣭⣭⡵⢾⣿⢿⣿⡿
⠀⠀⠀⠐⠐⠀⢲⡀⠀⠀⢀⡖⠀⠀⠀⠀⠈⠈⠈⠈⠀⠀⠈⠉⣩⣉⣧⣮⣭⣿⡅⡉⢩⠉⠉⠉⠉⠈⠘
⠀⢠⢿⣤⣏⣠⡀⢁⣄⢀⣀⡀⡀⣀⡴⣦⣸⢳⡟⠟⠦⣤⢠⣠⠄⢻⣿⣿⣿⣿⣧⣿⣇⣤⡖⢀⡀
⢠⠟⠈⠿⠻⠼⠃⠿⠿⠿⠼⠛⠯⢋⣛⡏⡟⠚⠀⠀⠀⠀⠀⢀⣠⣿⣿⢿⡛⣛⢿⣋⣯⣹⣙⣫⡀
⢀⠢⠤⠂⣀⣄⣀⣄⣀⢤⣴⣶⣽⡿⠔⠓⠀⠀⠀⠀⠐⠀⠀⠚⠛⠛⣿⣶⣿⣿⢿⡿⢿⡿⢯⠙⠃
⠾⠿⠧⠀⠠⡥⠿⠇⠀⠀⠉⠻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠴⠽⠿⠿⠿⠿⠼⠇⠸⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠁⠀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⣿⣿⡟⠉⣼⣿⠿⠛⠛⠛⠿⠿⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢤⣀⡒⠻⣿⣷⣄⠈⠛⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠶⢂⠞⢻⣀⡤⠀⠙⢯⣽⣶⢤⡟⠀⠀⠀⠀⢒⡫⠽⠟⠂⠀⠀⢠⡄
⠀⠀⠺⣷⣀⡴⣫⠴⢋⠁⠀⠀⠀⠀⠻⣿⣷⣤⡀⠀⡆⠀⠀⠀⠀⠀⣀⣠⣤⡾⠁
⣾⠁⠀⢿⡟⠋⣁⣤⠿⣧⠀⠀⠀⠀⠀⠈⠛⠿⠿⠗⠘⣶⣶⣶⡿⠿⠛⠛⠉⠀⠀⠀⠒⠒⠒⠚⠛⠛⠓⠢⣄
⠻⣦⡴⠈⠳⠆⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡀⠀⠐⢶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢿⣿⣷⣄⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⠿⠛⢉⣀⣴⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠸⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠿⠛⠻⣿⣿⣿⣿⣿⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⣠⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠿⣦⡀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣤⠴⠚⠉⠀⣼⡄⠀⠀⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡁⠀⣀⠀⠁⠀⠀⠀⠀⣼⠟⠛⠛⠉⠀⠀⠀⠀⢰⣿⣷⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠿⠛⠋⠉⠉⠀⡴⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⢀⣼⡇⠀⠔⠁⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠿⠛⠁⢀⣤⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢀⣀⣸⣿⡿⠃⠀⢀⡴⠂⠀⣀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠿⢿⠟⠁⠀⠀⢀⡞⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠟⠋⠉⠉⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣃⠠⠤⠠⠤⠀⣤⠀⢈⠠
⠀⠀⠀⠀⢀⡀⢂⣾⣷⣦⣈⡙⢆⠐⡀⠀⢰⠿⢤⡀⠀⠀⢲⡤
⠀⠀⠀⡀⠀⠐⠋⢩⣿⣿⡏⠀⠈⣀⣉⣠⡎⠀⢀⡴⣤⣄⡀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋
⠀⠀⢺⠁⡠⠴⣾⢿⣿⠁⠻⡄⠀⢈⣿⡏⠀⠀⠞⠂⣸⣿⣆⠀⠘⠇⠀⠀⠀⠀⢰⣧⣤⠲⠃⠂⡀⠀⠀⣼⡟⠀⠓⢤⠀⠀⣦⢀⣠⡀⠀⡀
⠀⠀⠘⠢⢄⠈⠁⠰⣿⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢠⡏⠙⣿⣿⡗⠀⠀⠀⠀⢠⣿⠹⠋⠀⣀⡄⠂⠀⠀⠸⣿⡄⢸⡀⠀⠘⠁⠀⠁⠙⠶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠰⡈⡏⡐⠀⠀⠀⢸⠁⠐⠊⠁⠀⠀⠀⠀⠀⣸⠟⣀⠀⢀⠀⠀⠀⠀⢠⣾⠉⠁⠀⠀⠀⠀⠀⠘⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢳⠞⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠃⠀⠈⠄⠀⣠⠀⠸⠀⡖⠀⠀⠀⠀⠀⠀⠀⠤⠂⠘⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠘⢻⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠘⠃⡇⠀⠀⠀⠸⡦⢿⠀⡖⠀⠀⠀⠀⠸⠀⠀⠀⠀⠂⠀⢠⠀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠐⠀⠀⢀⣇⠀⣠⠀⢀⠠⣄⠖⠓⠁⡀⢀⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠐⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠁⠀⠇⡁⠀⠀⠀⢸⣿⣀⠁⠀⠂⠱⠹⠘⠃⢔⣥⠁
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢼⡃⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠁⢺⣟⣋⠃⠀⢀
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢠⣿⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠔⠈⡀⠀⠀⠀⣼⣿⣿⣧⣀⣀⣤⡆⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠺⣇⠀⠀⠀⠀⢀⣾⠻⠿⢷⣀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠃⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠃
⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣘⣉⣉⣶⣐⣊⣉⣉⣀⣉⣻⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣀⣀⣘⣿⣿⣿⣿⣿⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣤⡤⠖⠚⠛⠛⠽⠿⠛⠈⠉⠉⠻⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠻⢦⣶⣶⣶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣸⠟⠉⠀⣄⣤⣶⣶⣶⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⡈⠉⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡯⠋⠀⠀⢀⡄⢿⣿⣿⠿⠿⣫⡾⡇⠀⠈⢀⣤⡾⣿⣾⣿⣿⣷⡄⠀⠀⠈⠣⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⠀⠀⠀⠀⠈⠟⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠹⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠿⢾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠻⠟⣳⢶⣦
⠀⠀⠀⠀⠀⠀⢀⡾⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣛⣭⡾⠇⠀⠀⠀⠀⠀⠐⣾⢹⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣇⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢰⢻⣧⠀⠀⠠⠀⠀⣴⡀⠀⠁⠀⠀⠀⠞⣿⣿⣿⣶⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⡘⣿⡷
⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⡾⣿⠂⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⠀⠀⠀⠀⣾⣿⠾⠃
⠀⠀⠀⠀⠀⠘⢿⡦⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠀⠀⠀⠀⠐⣻⠇
⠀⠀⠀⠀⠀⠀⠀⢹⣆⡀⠀⠀⠀⢿⣿⣿⣿⣄⡀⠈⠛⠋⣭⣙⡫⣿⠋⠀⠀⠀⠀⢐⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠰⠀⠙⠻⡟⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⢠⢶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡗⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣴⣂⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣾⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⡆⣿⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⣈⠻⣦⠀⠁⠀⠀⠀⠀⠀⠸⠿⠯⠍⠀⠁⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣶⢶⣶⣤⣤⣟⣻⣧⣠⣼⣏⣡⣀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠿⣯⡽⠿⠁⠀⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠁
⢀⣀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⢀⡀
⠀⠈⠙⠲⢤⣀⠀⠀⠀⠠⣄⡉⠲⢄⠙⠢⣄
⢄⡀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠉⠲⢤⡉⠂⠈⠑⢄
⠤⣄⣁⢦⣤⣀⠀⠀⠈⠙⠳⠦⣄⠀⠈⠓⠄⢀⡀
⠀⠀⠀⠉⠓⠛⠙⠒⠦⢄⡀⢀⣈⠙⠻⠿⣶⣤⣉⠲⢄⡀⢀⡀
⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢄⣀⡈⠉⠒⠢⢄⡉⠛⠷⢦⣈⠐⠉⠢⢄
⠤⠬⢍⣉⡙⠛⠓⠶⠦⠤⣤⣀⡉⠉⠒⠢⠤⠀⠀⠐⢤⣌⡑⠢⠑⠢⢁
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀
⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⣒
⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠒⠂⠐⠒⠒⠒⠩⠭⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⠀⠄
⠤⠤⠤⠶⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⢊⣉⡉⠤⠀⠈⠉⠀⢀⠤⠔⠂
⠔⠒⣒⣀⡀⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⠽⠟⠒⠉
⠉⠉⠁⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠁
⠀⣀⡤⠴⠒⣊⡩⠥⠒⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠔⡀
⠀⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⣀⡤⠒⠉⢀⡠⠖⣩⠔⠋
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠐⠊⠁
⠀⠀⢀⡠
⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠇⡟⢳⢼⠋⢸⡏⠁⡟⢻⠀⢸⠂⢸⡇⣿⣸⢸⡋⠱⣏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⡆⡟⠋⢸⡉⢸⡏⠁⣇⣸⠀⢸⡀⢸⡇⡏⣿⢸⡋⢠⣝⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠈⠉⠈⠉⠁⠉⠉⠀⠈⠉⠉⠁⠁⠈⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⣿⣿⠟⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠘⠛⠁⠀⠀⠀⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠦⠀⠀⠀⠀⣀⡀⠈⠛⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢦⡄⠰⠿⠿⠇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠘⡇⠀⡀⠀⠀⠼⠛⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠖⢣⡖⠂⠀⠀⠀⠀⢀⡟⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣪⣭⡍⠉⠀⠀⣀⠜⠀⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⣤⣤⣠⣾⣿⣄⣀⣀⣴⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠶⢄⡀
⠀⠀⠀⠀⠀⢀⣤⡄⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⣴⣲⣄⡀⡄⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣼⣿⣧⣹⣿⣿⣿⣿⢿⡇⠉⠉⠉⠀⢀⠀⠉⠻⢿⣿⣷⣮⡅⢄⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢏⡾⠁⠀⠀⠀⣶⣾⣷⣤⢀⡀⠀⠛⠿⣿⣦⠡⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⡭⠊⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣶⢄⣷⣿⣇⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⢘⣿⡟⣿⣿⡿⣾⡇⠀⠀⠀⠀⠀⢸⣿⣽⡿⣿⣿⣿⣿⢏⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⢠⣶⣯⢀⣻⣿⣷⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣶⣿⣿⣿⣿⣿⡀⠀⢀⡀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠸⣿⡿⢰⣿⣿⣿⢿⣿⣄⠀⠀⠀⠀⠀⠹⣿⣿⡿⣿⣿⣿⣼⣿⣿⣿⡿⢇⣴⣀⠘⣦⣀⠀⠀⠀⢷
⠀⠀⠀⠈⠈⠁⠈⠋⠉⠁⠞⣟⢿⣷⣤⣤⡄⠐⠘⠀⠑⢄⡈⠻⣿⣿⣷⡰⣶⣶⣿⣿⡿⠃⠀⠙⠳⢦⡀⢸
⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣫⠞⢻⡟⠿⠁⠀⠀⠰⠀⠀⠀⠉⠈⠻⢿⣿⡟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠁⢸
⠀⢸⠀⣾⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠳⣐⠆⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠘⢷⣥⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠹⣦⣁⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠙⠷⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣶⣤⣀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⡇⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡗⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⣻⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣛⡛⢟⣿⣿⣿⣿⣶⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⠿⣿⣿⠻⠿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠉⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⣀⣀⣀⣤⣴⣶⣶⣿⡿⠋⢻⡇⠀⢀⣤
⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣿⡇⠀⢾⣿
⠉⠛⠿⠿⣿⣿⣿⣿⣿⣷⣽⣟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢸⣟⣀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠻⠿⠿⢿⣿⣏⡁⠀⠈⢻⣷⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⣾⣃⣔⣀⣼⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⡄⠀⠀⠀⠀⠀⣀⣤⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⣩⣿⣿⣷⣄⣀⣀⡀⠀⠀⠀⠀⠀⣠⣿⣿⣯⡁⠀⠀⠀⠀⠀⢿⣿⡻⢿⡇
⠀⠀⠀⠀⠀⠰⠶⣶⣾⣿⣿⣿⣷⣶⡶⠆⠀⠀⠀⢀⣘⣿⣿⣿⣿⣿⣿⣿⣭⣭⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃⠀⠀⠀⠀⣀⡼⢻⡇⣾⠙⠳
⠀⠀⠀⣴⢻⣿⢿⡁⠻⣿⣿⣿⣿⣿⣳⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠄⠀⠀⠀⡘⢹⢉⣿⠏⠀⠀⠀⠀⣠⢿⣧⡿⠟⢻⠉⠢
⠀⠀⠀⠙⠲⠶⠛⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠙⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠋⠀⠀⠀⠀⠁⠆⠈⠏⠀⠀⠀⠀⠀⠀⠐⠛⠻⡄⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡐⠈
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⡀⠀⠀⠀⠀⠀⠠⣦⢹⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣄⠀⠀⠀⠀⠀⠀⢠⣮
⠀⠀⠀⠀⠀⠀⣞⣿⣿⡏⣷⠀⠀⠀⠀⢠⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⣠⡟⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠘⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠉⣿⣿⡟⠉⠀⠀⠀⠀⠀⠈⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⠀⠀⠀⠀⠀⠈⢩⣿⣿⡇⠀⠀⠀⠀⠀⠀⢘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡟⠿⢦⣀⠀⠀⠀⠀⠀⣿⠁⠘⢷⣄⠀⠀⠀⠀⠀⣸⢿⣿⠁⠀⠀⠀⠀⠀⠀⣾⠟⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⣠⡇⠀⠀⠀⠿⠀⠀⠀⠀⠀⡟⠸⠃⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⣄⣄⣀⣀⣀⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢠⣶⣷⣦⠀⠀⠀⠘⠛⠛⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⢳⡀⠀⢠⡄⠀⠀⠉⠓⠲⣾⣿⣄⡤
⠀⠀⠀⠀⠀⠀⠀⢟⣼⣿⣿⡇⠀⠀⠀⠀⠠⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⡁⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣻⡿⠉⠻⣿⡀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣁⣴⠄⠀⠀⠀⠀⣾⡿⠉⠛⠀⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⣿⢹⡇⠀⠀⠀⠀⠀⠀⢼⣧⣿⠟⠁⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⠁
⠀⠀⠀⠀⢀⡀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡯⠀⡇⠀⠀⠀⠀⢀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣮⠷⠀⠀⠀⠀⠀⠀⠀⠿⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⢈⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢸⣿⣿⣿⠀⠀⠰⣶⣶⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣧⡴⠀⠀⠀⠀⠀⠸⣄⣴⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⢸⣿⣿⡅⠀⠀⠀⣶⡦⠀⠀⠀⠈⢳⣿⠋⠁⠀⠀⠀⠀⠀⠀⠸⣿⡃
⠀⠀⠀⠀⠀⠀⢠⡾⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣷⠀⠀⠀⠀⠀⣿⠋⠻⣤⣀⠀⠉⠁⠀⡀⠀⠀⣼⡿⢷⣄⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⡰⠋⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢹⡆⠀⠀⠀⣼⠃⠀⠈⠉⠉⠀⠀⠀⠋⠉⠙⠛⠋⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠓⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠄⠀⠀⠀⠀⠋⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡂⠕⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⢤⣶⢰⣿⠆⢀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠶⠋⠁⢤⣾⣿⢼⣿⣧⣤⡖⠀⠀⢤⠀⠀⠀⠀⣠⡆⣀⣧⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⡀⠀⣤⣠⣴⣫⣿⣿⣏⢻⡍⣠⣀⣤⣤⠈⠀⠀⠀⠘⢿⣿⣿⣿⣷⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀⢻⣷⡀⣼⣼⣿⣿⣿⣿⣿⣆⠀⠙⣉⣹⠁⠀⠀⣠⣾⣦⡀⠉⠀⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠷⠎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣘⣿⣿⡆⢀⣠⣿⡿⠛⢡⡎⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠿⠿⣿⠃⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢶⣾⣿⡟⠁⣰⣸⣿⣠⡾⠀⠀⠃⠀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠰⡄⢀⠔⠀⠂⠊⠉⠀⠀⠀⠉⢻⣿⣿⠟⣉⡘⣷⣽⣿⡟⠀⢰⣟⡘⣿⡛⠃⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠂⠄⠀⢀⠀⠀⠸⣄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣽⣿⠏⠀⠀⡴⠺⣷⡼⣿⡿⡄⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠘⠀⠉⢄⡀⣴⣿⣿⣦⣾⣿⡿⢋⣩⣵⣿⡿⠉⠀⠀⠀⢀⢁⣿⣿⣹⡷⠄⢀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⡀⠚⣿⣿⣿⣿⡋⡩⠁⣠⡿⣻⠿⡻⠁⠀⠀⠈⢠⡟⠼⢿⣽⢷⡑⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⣿⣿⡟⢡⠞⠁⠀⠀⡰⠀⠀⠀⠀⠀⣾⠁⣸⣿⣾⣿⣿⣋⡗⢶⣦⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠛⠋⠀⣠⡎⠀⠀⠀⠀⠀⡸⠀⣸⣿⣿⢏⣼⣏⡐⠁⠀⠀⠀⠀⢇⠀⠀⠀⠀⢰⡏⠀⣿⣿⣿⣿⡏⠉⠑⠠⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⣿⠇⠀⠀⠀⠀⢠⣧⣴⣿⣯⢱⣿⠋⠁⠙⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣾⡇⢸⣿⣿⣿⣿⠃⠀⠀⠴⠿⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠙⠀⢀⢀⠀⣾⣿⣿⠀⠀⠈⡼⠃⠀⠀⠀⠀⢀⠀⠀⢀⠈⡀⠀⠀⣸⣿⠃⣾⣿⣿⣿⣿
⠀⠀⠀⢠⣤⣄⡀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣇⠸⢿⣿⣿⣿⢶⣤⡃⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠁⠀⣰⣿⡏⣴⣿⣿⣿⣿⣿⡆
⠀⢀⣠⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⡈⣿⣿⣦⣌⢻⣏⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⣦⣤⣈⣉⣉⣩⣩⣿⣿⣿⡇
⠀⢏⣩⣿⣯⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣷⡄⠀⣉⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣧
⠀⠈⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢠⡀⠀⠀⢠⠘⣿⣿⣿⣿⣿⣿⣄⣿⣿⣸⠀⢸⡆⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⡿⠋⣸⣿⣿⣿⣿
⠀⠀⠀⠴⣿⣿⣿⣿⠿⠿⠿⠿⠛⠁⠀⠀⠘⣧⠀⠀⠘⣷⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⠀⠸⣿⣦⣀⣄⠀⠁⠀⢀⣿⣿⣿⠏⢀⣾⠿⢿⣿⣿⡇
⠀⠀⠀⢬⠽⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⠇⠀⠙⢿⣿⣷⣿⣿⠀⠀⠻⠿⢻⡏⢸⡆⣄⣼⣿⣿⣿⠀⠞⠁⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⣿⣿⠀⠀⠀⠀⠉⠛⢿⣿⢠⣶⠀⠀⠋⢠⣿⣿⣿⣿⡿⣀⠀⠀⢀⣴⣿⣿⣿⠿⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣔⣾⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⡄⣿⠃⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣶⡀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢻⣷⡆⠀⠀⠀⠰⡄⢻⣿⡿⢿⣿⠿⠛⢿⣿⣿⣃⠀⣄⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠹⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⣽⣶⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠙⠇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⡄⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⢀⠀⠀⠀⠀⠘⢿⣿⡼⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⣶⠋⠀⠀⣠⡄⠀⠠⠻⣧⣷⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠀⠐⠾⣿⣿⣠⢀⣄⠉⣉⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⢠⠈⠉⠙⢾⡿⠿⠿⢿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡁⠠⠂⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢀⣄⣼⣿⣿⣿⣿⡄⠀⠀⠈⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⢸⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣈⢉⣿⠟⠀⠉⠹⣿⣿⠏⠉⠀⠙⣿⡉⣁⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏⣶⣶⡅⠢⠘⠋⠔⢈⣶⣶⠸⢿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣫⠀⠈⠻⣄⣻⡶⠦⠴⠶⣟⣠⠟⠁⠀⠙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡡⠢⢤⣤⣤⣀⠹⣧⣴⠇⢁⣠⣤⣤⡑⣮⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡀⢈⢿⣦⡁⣿⣶⣶⣾⣇⢠⡾⡁⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⡈⣿⣷⣿⣿⣿⣿⣿⣿⠃⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣿⣿⣿⣿⣿⣇⢸⡇⣿⣿⣿⣿⠸⡏⣸⣿⣿⣿⡿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⣿⡿⣿⣿⣄⢧⣘⠻⠟⣏⡰⣡⣿⣿⣿⣿⢣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡌⠛⢿⣧⣄⣯⣭⣅⣴⣿⠛⢣⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⠿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠟⠛⠋⠉⠉⠉⠛⣿⡟⠛⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣯⣥⣤⣤⣤⣤⣤⣤⣀⣀⢠⣿⡏⠀⠀⠀⠈⢙⣿⣷⣤⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠉⠉⠉⠁⠀⠈⠉⠉⠙⢛⣿⣿⣿⣦⣀⠀⣠⣶⡿⠋⠀⠙⢿⣦⡀
⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠿⠋⠁⠈⠙⢻⣿⣿⣋⠀⠀⠀⠀⠀⣹⣿⣆
⠀⠀⠀⣼⡿⠁⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⠋⠉⠀⠀⠀⠀⢀⣴⣿⠋⠙⢿⣷⣄⠀⣴⣿⠟⠿⣿⣧
⠀⠀⣸⡿⠁⠀⢀⣴⣾⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠉⢿⣿⡿⠁⠀⠀⠈⢿⣧
⠀⢠⣿⠃⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠘⣿⡆
⠀⣸⣟⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⢿⣧⠀⠀⠀⢹⣧
⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢿⣧⠀⠀⢸⣿
⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠈⣿⣧⠀⢸⣿
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠸⣿⡄⣸⡟
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⣿⣇⣿⠃
⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠈⠻⣷⣄⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⣡⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣼⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠃⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⡇⠀⣉⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⡀⢀⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢠⣽⠍⢻⠟⢁⠀⠀⢀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠁⠀⠀⠀⠀⢀⣾⣿⣿⣇⢸⣇⠘⠂⠀⠀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣤⠀⠀⢀⣼⣿⣿⣿⣿⡿⢿⣦⡀⠰⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠘⢷⣦⡜⠿⣿⠉⠛⠉⠙⠀⠙⠻⣘⠁⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠘⠋⠀⠀⠀⠓⠒⠀⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⢀⣶⠈⠋⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣱⣿⣿⡿⣿⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣴⣿⡟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡾⣸⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⢿⣿⣿⠟⢹⣿⣿⣿⠉⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⡠⠁⠀⠈⠉⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠜⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠋

⠀⠈⠒⠄⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠈⠂⢄⠀⠀⠈⠠⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠈⠂⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⠄⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠒⠒⠈⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠀⠠⠤⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⡀⠤⠐⠊⡡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠔⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⣠⠞⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⡠⠖⠉⠀⡠⠂⠀⠀⢀⠞⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢀⡤⠚⠁⠀⢀⠔⠋⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⣿⣷⣶⣾⣾⡿⣾⣶⣶⣾⣿⣾⣾⣶⣶⣶⣶⣿⣶⣾⣷⣶⣾⣶⣶⣿⣶⣷⣾⣷⣿⣶⣶⣶⣾⣿⣶⣾⣷⣶⣶⣶⣾⣷⡷⣷⡾⢾⣾⢷⣶⣷⠾⠶⡿⠶⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣟⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣶⣦⣴⣶⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣠⣉⣠⣿⣿⠿⠷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⣿⣿⣿⣿⡿⠹⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣾⡿⣿⣿⣿⣿⣿⡟⢿⣆
⠀⠀⠀⠀⠀⠀⢰⣿⠣⢿⠿⠿⣿⣿⡇⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣦⣾⡟⠒⠛⠋⠀⠘⣿⣿⣿⡿⠇⣾⡇
⠀⠀⠀⠀⠀⢠⣿⠏⠀⣀⣠⣠⣀⣹⡇⣼⠏⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⠿⠇⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣧⡿
⠀⠀⠀⠀⠀⣸⡇⠀⣿⣿⣿⣿⣿⣿⠇⠏⡦⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⢿⣿⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⢀⣸⣿⢿⠗
⠀⠀⠀⠀⠸⠁⠀⢸⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠁⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣽⣿⢿⣿⣷⡄
⠀⠀⠀⢀⣤⠀⡠⢿⣿⣯⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⢸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠙⠿⣿⣦
⠀⠀⠀⠘⢿⣿⣿⡖⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠻⠿⣿⣿⣷⣦⣀⣰⡀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⣿⣿⣵⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠹⣷⡟
⠀⠀⠀⠀⠀⣟⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢠⣴⣿⠟⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⣼⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⣄⣸⡒⠃⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⣻⡽⣿⠁⠀⠀⣠⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠱⣿⣄⡻⠦⣀⡴⣻⣿⣿⡿⠿⢿⣿⡄
⠀⠀⠀⣀⣠⣴⣶⣦⣄⠀⠀⠀⠀⡠⠴⢶⣶⣄⣀⣀⣀⣀⣀⣠⢌⠋⢁⣤⣶⣷⣿⣿⣿⠇
⠀⠀⠘⢿⣿⡿⠿⠿⢿⣷⣤⣤⡾⠃⠀⠈⠛⠿⡿⣩⢙⣵⡿⠛⣤⣴⡿⡿⠁⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⣦⣶⣾⣉⡆⠘⠿⠗⢺⣿⣿⣿⣿⣷⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣦⣿⡿⣿⣿⣿⡗⣦⣤⣼⣿⡿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠃⣿⡿⠋⠃⠽⢿⣿⡿⣽⣦⣨⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠋⢀⣾⡟⠁⠀⠀⠀⠀⠀⣾⡿⠁⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠉⠙⠲⢤⣀⠀⠀⣸⠏⠀⠀⠀⣈⣿⠇
⠀⠀⠀⠀⠉⠁⠲⠀⠄⠀⠀⠀⠀⠀⠘⣷⠀⢀⣀⣀⣉⣶⠞⠉⠘⠻⠶⠋⠁
⠀⠀⠀⠀⠀⠀⢂⢢⡄⢠⠀⠀⠑⠀⢈⠀⠸⢭⠉⠉⠉⠀⠀⠀⢀⣰⣄⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠙⠙⠻⠿⠛⠛⠻⠿⠟⠛⠒⠛⠛⠘⠒⠂⠋⠑⠛⠻⠉⠋⠛⠙⠃⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⡟⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣛⠛⠋⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡴
⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃
⠀⠀⠀⠀⠀⠈⢩⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡃
⠀⠀⠀⠀⠀⠸⢿⣿⡄⠀⠀⠀⠰⡄⠀⠐⡄⢘⣿⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡿⣷⡀⠀⠀⢱⡘⣆⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠿⠋⡆
⠀⠀⠀⠀⠀⠀⠠⠃⠩⠿⠀⠜⠂⠟⠈⠛⠛⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡧⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⣈⠿⡃
⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⣀⡀⠀⠀⠀⠀⠠⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢮⣿⣇⠀⠀⠀⠀⢸⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣧⠀⠀⢀
⠀⠀⠀⠀⠠⣶⣶⡌⠀⠀⢠⣿⣿⠀⠀⠀⠀⡞⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠉⠠⠀⢨⣷
⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⢸⣿⠃⠀⠀⠀⠐⡆⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣀⣀⣀⣴⡏⠀⠀⠀⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢷⡀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠺⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⡇⠀⠀⠀⠀⠈⢁⣀⣠
⠀⠀⠀⠀⠀⠚⠛⠛⠛⠓⠂⠀⠀⠈⠛⠛⠛⠛⠛⠒⠒⠒⠂

⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⢶⣆⠀⠀⠀⣀⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢘⠁⠀⠈⠏⠀⠀⠀⠉⠁⠉⠁⠀⣸⣷⣤⣷⣾⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠘⡛⣿⣿⠀⠰⣴⣷⣶⣾⡇⣠⠠⣶⣿⣿⡿⠟⠛⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠊⣩⣿⣄⣰⣛⠋⠀⣹⢧⣭⣇⠉⠉⠉⠀⠀⣠⡀⠘⠻⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠁⠋⠈⢉⣿⡿⠿⠃⢸⠃⠈⠃⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⡦⠖⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠿⠿⠲⠶⣶⣶⣾⡿⠃⣤⣠⡿⠋⠀⠀⣀⠀⠀⡼⠀⠀⠀⢀⠐⢤⣀⠀⣈⣷⣄⡀⠀⣀⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠁⢀⠤⢺⡿⢁⣠⣿⠟⣡⣴⣾⣿⣿⣏⣴⠇⠀⠀⠀⣾⣠⣾⡿⠋⠉⠉⠛⢇⠘⠛⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠠⣦⠀⠀⢠⠀⠀⠀⣠⠟⣀⣴⠟⠁⣼⣿⣿⡿⠃⠸⣿⡏⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠰⡿⠀⢰⣧⣖⡀⢀⢡⣴⣟⠁⠀⠸⢻⡇⠈⠁⠀⠀⢿⣧⠀⠀⢀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠧⠀⠀⠀⣽⡄⠋⠉⠻⣄⡀⠙⠿⠛⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠙⠷⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢋⡿⢷
⠠⣄⣀⣀⠀⢀⣠⣴⣿⣿⣧⠈⠆⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠙⡿⠚⠁⡿
⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡀⠀⠀⢸⣿⣷⣄⡀⠀⠀⢹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠸⠿⠟⠛⡿⠟⠁⠀⠀⠀⠀⠹⣦⡄⢸⡏⠻⣿⣿⣦⣄⡈⢿⣿⣧⡔⠀⣀⠀⡀⠀⠀⠐⠢⣿⣾⣁⣠⠆⠀⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠌⡇⠀⠈⠻⣿⣿⣿⣿⡟⢿⣷⠾⡁⠀⠙⢶⡲⠤⠶⢿⣿⣿⠁⠀⠀⠲⣤⣌⣙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢇⠀⠀⠀⠘⣿⣿⣿⠀⠘⠿⢦⣀⠀⠀⢄⠙⢄⠀⠀⠙⠿⠿⠖⠚⠋⠉⠉⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣠⠘⣦⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠈⠉⠓⠾⣷⣌⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣄⡜⢷⣄⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣍⠙⠮⠻⣿⣿⠀⠀⢠⣤⣴⣦⣀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡆⠀⠙⢿⣇⠀⠀⠙⠿⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣎⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣦⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣻⣛⣟⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠒⠛⠢⣄
⠀⠀⠀⠀⢀⣠⡤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⢀⡞⠉⢀⢰⣶⣠⡀⠀⠀⡰⢅⠀⠀⢀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⣼⠀⠀⠀⠀⠀⢙⣿⣀⣤⣶⣀⠲⡠⠁⣠⣢⠤⠀⠀⠀⠀⠘⠳⢦⣄
⠀⠀⠘⣄⠀⠀⠐⠋⠋⠉⠉⣙⣻⣿⡷⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⠶⣄
⠀⠀⢀⡞⠱⠦⠤⠒⠊⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠝⠳⡄
⠀⡰⡯⠐⠲⣆⠀⠀⠀⠾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢰⡄⠀⢀⡃⠒⠀⠀⠀⣈⣴⡶⠶⠶⠾⢤⡄⣠⡤⣴⢂⠠⠲⠆⢀⠀⠀⠀⠀⠀⠀⠃⠠⡌⢳⡶⠾⠟⠒⠦⠴⢿⡓⣦⣄⣀
⠀⠙⣺⠛⠳⠶⢒⣋⣉⣉⣀⣀⢀⣤⣴⠿⣿⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠘⣷⣆⢸⠀⠀⠀⠬⢷⠁⠍⠏⠉⠓⠲⠦⢤⣤⣤
⠀⢀⣇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⡔⠀⠘⣉⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣋⢸⠀⠀⠀⢀⣸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠻⣧⣸⣠⣄⣀⢀⣐⠀⡉⠠⣩⣴⡼⠟⢫⣤⡙⢿⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡧
⠀⠀⠀⠈⢹⡋⠛⠛⠛⠛⠒⠛⠛⠛⢩⠀⠀⣼⡯⠔⠉⠉⠓⠆⣤⣶⣤⣀⡀⠀⠄⢄⡀⢀⠀⠀⠀⡞⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢷⣦⣀⡀⡀⠀⠀⣄⣀⣷⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠰⠀⠀⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠁⠈⠙⠿⢼⣐⡜⣁⢀⠀⠀⠀⠀⠀⠀⠉⠀⠁⣠⣀⠀⠀⢠⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠚⠿⢧⣤⣥⣶⠾⠶⠛⠛⢶⣙⡀⡏⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⠀⠀⠀⠀⠀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣄⣀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣻⣿⡟⡇⡀⡄⠀⠂⠀⠂⠀⡐⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣱⣹⡡⡐⣐⡀⡀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣵⣿⣿⣿⣟⡙⠐⢡⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⠮⢼⣔⣐⣃⣩⡄

⠀⠀⠀⢀⠀⠀⠀⠄⢧⠀⡄⠀⠀⡀⠀⠀⠀⠀⢰⠰⠀⠀⠀⠀⠀⢆⠀⡄⠀⠀⠀⣀⢀⡀⠀⠄⢠⠀⢀⠄⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⢘⡆⢑⠀⠀⠀⠀⠀⠀⠀⠸⠸⡆⡆⠀⡄⡄⠐⠀⠃⠀⠀⢰⢃⡞⠀⠂⠀⠈⡠⠀⣠⠊⠁⢀⡴
⠀⠀⠰⡀⠀⠀⠀⠁⠀⢣⡈⡆⠀⠀⣰⠀⠐⠀⠀⡇⡇⣷⢸⡇⠃⠀⠀⣠⠀⢀⠏⡼⠀⠀⠀⢀⡜⢡⠞⠁⢀⡴⠀⠀⠄
⠀⠀⠀⠙⢄⢀⠡⡀⠰⡜⣧⠸⠀⡀⢿⡀⠀⡆⠀⢇⣷⠈⢸⡇⢰⠀⢰⡏⠀⡜⣰⠁⠀⠀⠀⠊⠐⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠳⡙⠄⠈⠘⣧⢁⢡⢸⡇⢀⠃⢸⢸⢸⣀⣼⡇⣸⠀⣾⠁⣸⢱⡇⠀⣠⠈⠀⠀⠀⠀⠀⣀⣠⠊⠀⠀⡠
⢀⠀⠀⠀⠀⠰⡄⠙⢔⣄⠀⠘⢧⠀⡜⣷⠀⠸⢸⡎⣘⡇⢹⣷⡇⣇⡇⢆⢃⣿⠁⣤⠏⡰⠁⢀⠄⢀⠼⠋⠁⠀⠀⠊⡄
⠀⠃⠀⢠⠀⠀⠙⣆⠸⣾⡣⣜⠘⢄⠃⣿⣧⠀⣿⡇⣏⡇⢸⣿⡇⢹⡸⠈⡾⢃⣾⢃⡼⠁⣠⠞⠐⣡⡾⠃⠀⠀⠀⠀⡀
⠸⠆⠀⠀⠣⠀⠀⠘⢆⠀⢦⠸⡷⣌⠀⢻⣿⡆⠸⡇⣿⠃⠘⡇⠁⣿⣷⡼⢡⠟⣡⡿⢁⣴⣿⣶⣿⠋⠀⠀⣠⠆⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣎⠎⡆⢽⣮⡀⠘⣿⣷⠀⠣⢿⡄⠘⡟⠀⣿⣿⡅⡫⣺⡿⡡⣿⡿⢻⡿⡡⢊⡴⢞⠑⢉⡥⡪⠞
⠀⠀⠀⠠⡀⢠⡄⠀⠀⠸⣿⡌⠈⣿⡇⠀⠘⣿⢀⠀⠸⡇⠀⠃⠀⣿⡟⠁⡼⠛⢃⣾⠟⣴⣿⢋⣀⣬⢞⠏⠐⠋⠀⢀⡀
⠀⠀⠀⠀⠈⢷⣌⠀⡄⠀⢻⣿⡄⠘⣷⠀⢀⠸⢸⡀⠀⠃⠀⠀⠀⠋⠀⠘⠁⡠⢈⠁⣺⡟⣱⣿⠟⣡⠞⢉⣤⠴⠚⣋⠄
⢈⡓⠂⠀⠀⠀⠙⠷⣌⡀⠀⢻⣷⡀⠘⠄⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⢃⡼⢋⣼⡿⣫⣾⣫⠔⣋⣀⠴⠊⣁⠤
⠀⠉⠉⠓⠂⠄⢀⣳⣬⡻⣆⠀⠙⣗⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠾⢋⡼⢛⣩⣶⡟⠉⠀⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠉⠉⠛⠛⣛⣂⠈⠀⠈⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⣾⣟⣋⣀
⢰⡶⠶⠶⠶⠶⠶⠤⠤⠜⠛⢁⣀⣠⠶⢋⣡⣶⠆⢀⡀⢀⡿⢂⠀⣤⡄⣀⠒⢦⡘⠚⢛⣋⣉⣉⣿⣿⣿⠿⣿⣿⣿⣿⣿
⣾⣷⡶⠶⠿⠿⢀⡀⣐⣛⣛⣓⣠⣴⣾⣿⡽⢂⣴⡟⣰⣞⣥⣣⣿⡹⢥⡌⠷⣔⢌⡻⣷⣍⠻⣯⣙⠊⠉⠉⠉⠛⣛⣻⣿
⣿⠧⣤⣄⣠⣤⣤⣤⡈⠉⠉⢉⣉⡀⡛⢉⣴⣿⡋⣼⣭⠉⠉⠛⣿⣿⣌⢻⣦⠀⠁⠉⠊⠉⠀⠈⠉⠁⠐⠻⠶⣦⣄⣙⣿
⣿⣶⣦⣬⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣭⣴⣶⣤⣤⣤⣬⣭⣭⣬⣭⣥⣶⣾⣿⣦⣾⠿⠿⠷⠶⠶⠶⠶⠶⠾⠿⠟
⣿⣿⢟⠿⢛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣯⣉⣂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣟⠿⠿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⡜⣿⡟⣉⠉⢀⣀⣀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣏⣣⠉⠉⢯⠀⠉⠉⠻⠟⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣰⡿⢻⣿⠀⠀⠈⠀⠀⡠⣦⣁⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡾⠋⠀⠀⠈⠉⠻⣶⡽⣿⣧⡀⠀⠀⢒⡰⢴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣰⣿⣷⣆⠀⠀⠀⠀⠀⠀⠑⡜⢿⣿⣦⣀⣀⣩⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡖⣸⣼⣿⣿⣿⣿⣾⣷⣂⣤⣴⣿⠀⠇⠈⢛⣻⣿⣭⣭⣀⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡡⠊⢀⣄⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠊⠘⢿⠿⢿⣶⣶⣭⡢
⠀⠀⠀⠀⠀⠀⡰⠛⣿⣿⣿⣯⡻⣿⣿⣿⣿⡟⠁⠈⠁⠁⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢰⡎⣦⣶⣶⣦⣤⣶⣿⣮⣠⡇
⠀⠀⠀⠀⢠⡞⠁⣀⣸⣿⣿⠟⣠⣽⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠸⣧⢿⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⣰⠟⠒⠉⠀⢈⣏⠁⣼⣿⣿⣳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢸⣇⠀⠀⠙⠿⡛⣿⣿⣿⣿⣿⢿⣿
⠀⠀⣸⠋⠀⠀⠀⠀⣾⠛⣹⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣼⣿⣿⡏⢱⣿⠁
⠀⢠⡇⠀⠀⢠⣴⠿⢁⣈⣹⣾⣿⣿⣿⣿⣿⣷⣤⣤⣶⡶⠊⣿⡀⠀⠀⠀⠀⠀⣀⣤⣾⡿⣿⣿⡆⠀⠀⠀⠀⠘⢧⠀⠀⢠⣿⣿⣿⠇⣼⣿
⠀⠸⣿⣶⣾⠟⠁⠀⢒⣾⠻⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠙⢿⣿⣶⣶⣶⣿⣿⡿⠟⢡⡟⣿⣿⡆⠀⠀⠀⠀⠈⣇⠀⣾⣿⣟⠏⠀⢿⡟
⠀⠀⠻⣿⣿⣾⣯⣵⠟⠁⠀⢷⣼⣿⣿⡿⠬⣉⣄⡀⣀⣠⡀⠀⠀⠈⠉⠩⠽⢟⣫⠔⢠⠟⠀⠸⣿⣿⣤⣀⠀⠀⣠⣿⡾⠁⠉⠛⠳⠆⡏
⠀⠀⠀⠙⠻⠿⠟⠁⠀⠀⠀⠘⣿⣻⣭⠀⠀⠋⠁⠀⠈⠉⠍⠉⠉⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⡅⠀⠀⢀⣀⢀⣬⠀⠀⠀⠀⠀⠐⢚⣵⠋⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣬⡏⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⠀⠀⡀⠌⠉⠉⠉⠙⠓⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣃⠀⠀⢠⣠⡄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣀⠀⠐⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠽⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣿⣆⡀⠀⠀⠉⠙⢛⠛⠃⠀⠀⢀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢿⣿⢿⣿⢶⣶⣶⡤⣶⣶⣶⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣼⡘⢛⡘⠻⠸⠟⢿⡇⡿⠻⠇⠻⣛⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣿⣿⣿⣷⣤⣶⣴⣴⣶⣶⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿⣿⣿⣿⣿⣿⡿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⠐⡶⣦⣤⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣾⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣾⣿⣤⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣴
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⡿⠛⠻⡿⢿⠟⡛⠿⠟⠿⡿⢿⠿⠃⠻⠛⠛⢻⣿⣿⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠂⠤⢁⠀⠸⢃⡀⠻⠇⣠⠀⢼⡇⠟⡁⠠⠆⣩⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣷⣿⣿⡆⣺⣿⣿⣷⣾⣿⣿⣿⠷⠟⠛⠒⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⠁⡔⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣀⠀⠀⠀⠈⠋⠀⠈⠁⠀⠀⠺⡞⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠟⠉⠉⠛⠉⠻⢿⡟⠛⠁⡠⠐⠀⠀⠀⠙⢢⣤⡀⠀⢰⣾⣷⡆⠀⠀⠱⣄⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣠⠴⠶⠿⠸⣧⠀⠀⠙⠛⠁⠀⣼⣇⠙⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡔⠃⠀⠀⣠⡟⡠⠀⠀⠀⣐⡿⠒⠀⣤⠀⢀⠘⠉⠻⣧⣄⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣏⠀⠀⢠⢺⣿⠀⢰⣾⣿⡄⠀⠀⠀⢰⡟⠻⠷⣤⣴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠷⠷⠟⠃⠙⠛⠛⠊⠿⣧⣤⡶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡄⠀⠙⢿⣷⣦⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠙⠻⣾⠟⠻⡿⢿⣷⡈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣄⡀⢸⠗⠂⠀⣜⠛⢇⣀⠘⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢶⠀⣠⢟⣴⣿⠟⠁⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣵⣿⠋⠁⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠉⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠀⠀⠈⢿⠀⢠⡄⠀⠀⠀⠐⣆⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡃⠀⠀⠀⢀⠁⣿⣿⡄⠀⠀⠀⣿⣇⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠘⣷⡜⢿⣿⡀⠀⠀⣿⣿⠀⠀⠙⠉⠉⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⣭⣝⣈⢿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⣿⣿⣿⡜⠇⠀⢰⣿⣿⠐⣿⣿⡇⣶⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠛⠛⠛⠛⢸⣿⣿⣿⡇⣦⠀⠈⠁⠀⡄⣿⣿⣇⣿⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢇⣿⡇⠀⠀⢸⡇⣿⣿⢸⡇⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⢀⣀⠀⠀⢸⣿⣿⣿⢸⡟⠀⠀⢀⣿⡇⣿⣿⣸⡇⠀⠀⠀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⢸⣿⣷⡦⢸⣿⣿⡇⡾⠁⠀⢀⣾⣿⢧⣿⡇⣿⢁⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⠃⣾⣿⣿⡇⠁⠀⣠⣾⣿⡏⣾⣿⣧⣿⠈⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠀⠀⠀⢀⣼⣿⣿⠃⠀⢿⣿⣿⣴⣶⠆⣨⣭⣭⣼⣿⣿⢸⡟⠀⠀⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⡿⠁⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠸⣿⣿⡈⠛⠛⠿⠿⣿⣿⠼⠇⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠻⢿⣷⣶⣤⣝⡋⠀⠀⣀⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡄⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⣀⣤⣦⣤⣤⡀
⢰⣶⣶⣶⣶⣶⣶⣶⣿⣋⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠚⠿⠿⣿⣿⣿⣿⣆
⣾⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠛⠙⠛⠛⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠈⠙⠿⣿⣧⡀
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠙⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠀⠀⠙⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠙⠄
⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⣠⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠀⠙⢿⣿⣿⣿⣷⣦⡀⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠻⣿⣿⣿⡿⠃⢹⣿⣿⣿
⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠛⠁⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣶⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠋⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢠⣿⣿⣿⣿⣿⣿⡏⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢉⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⢻⣆
⠀⠀⢀⣠⣴⣶⠿⠋⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠙⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠸⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠉⠛⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠙⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣤⣀⣤⣴⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣦⣀⣀
⠀⠀⠀⠀⣿⣿⣿⠟⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⣿⣿
⠀⠀⠀⠰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣹⣿
⠀⠀⠀⠀⠙⠁⠀⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣤⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⠛⠿⠿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣠⣤⡀⢠⣿⠛⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⠋⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠙⠛⠋⠁⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠉⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠉⠁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⣹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⣿⡅
⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⢸⣿⣿⡟⠁
⠀⠀⠀⣿⣿⠀⢀⣤⣀⡀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠟⣰⣾⣿⣿⡿⠃
⠀⠀⠀⠙⠋⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣯⠊⢻⣿⣿⣿⠸⠀⡇⢹⢸⣿⣿⣿⠟⠀⢰⢿⣿⣿⠟⣴⡿⣫⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡙⢿⣷⡀⠹⣟⣿⠀⠁⠁⢸⢸⣿⣿⡏⠀⢠⠏⣾⡿⢣⡾⢋⣾⢟⣫⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣆⠙⢿⣄⠙⡞⣇⠀⠀⠀⢸⣿⣿⠀⠀⠈⢠⠋⡴⢋⣵⢟⣵⢟⢽⡯
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⣙⢷⣄⠙⢦⡈⠘⠄⠀⠀⠘⢻⠇⠀⠀⠀⠀⠊⡠⢋⠴⠋⠀⠐⠝⣋
⠀⠀⠀⠀⠀⠀⠀⢨⣁⠉⠛⠿⠿⣇⡈⠳⢄⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣬⠷
⠀⠀⠀⠀⠀⠀⠀⠠⣬⣍⣒⠀⠀⠀⠈⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⣁⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠨⡥⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⠠⣦⠀⠀⠀⠀⠀⠐⠒⠭⠽⢷⣞⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠶⠶⠶⠤⠀⠀⠀⠀⠈⠟⠋⠙⣈⠀⣡⠀⠀⠀⠀⠀⠀⠨⠭⠽⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠯⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⠸⠟⠟⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠭⣙⡓⠶⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡙⢦⣈⢷⣦⣄⡀⠉⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠐⣋⢀⣠⡶⣀⠔⡀⢀⡴⠁⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠹⣄⠙⢷⣽⣿⣿⣷⣦⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⢟⣭⢞⣵⣞⣵⠟⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠘⣎⣆⠘⢷⣄⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⡷⣛⣵⣿⣿⡿⠃⣴⠃⡼⠀⠀⡿⠀⣾⢠⠀⡄⠀⢹⣞⣧⠈⢻⣷⣌⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡟⢁⣼⣟⣼⢁⡾⢸⡇⣿⣿⢸⠀⣷⢀⢸⣿⣿⣷⡀⡹⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠈⠉⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠀⡔⠀⠀⡼⠀⠀⠀⠀⠈⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠈⢑⠒⢽⡄⣲⠇⠀⠐⠀⢰⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠾⡁⡐⠀⢚⣔⠪⢀⠀⠀⣀⡀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣞⠖⢾⡪⠔⠋⠉⠉⠉⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠄⠄⠀⠀⠀⠀⠀⠀⠙⠤⡓⠀⠀⠀⠀⠀⠀⠀⠀⢢⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠰⡄⢂⠀⡀⡀⢁⠶⠂⠀⠀⠀⠀⠀⠈⢣⡀⠢⠀⠀⠀⠁⡀⡂⣬⠂
⠀⠀⢀⣀⠀⠤⠄⠐⠂⠈⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠈⠑⠦⣈⣾⣆⢮⣶⢋⡀⠀⠅⡀⠀⠀⠄⠡⠀⢳⠅⠔⣐⣀⡤⠔⠶⡅
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⡀⢝⢛⡿⣦⣏⠀⢀⠅⡒⠦⠀⠀⠈⠀⢸⠒⢋⢁⠠⠀⠀⠐⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡄⠂⢀⢀⠀⠀⠀⠀⠀⡀⠠⠆⠖⠋⢡⢞⣯⡟⣟⠠⠐⠂⠁⠀⠀⠀⠀⡏⠀⠈⠀⠀⠀⠀⠀⢠⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠀⢳⠹⣄⠂⢂⢀⠀⠀⠂⠀⠀⠂⠂⢀⠨⢠⠜⣼⣧⡿⠀⠀⠀⠀⠀⠤⠡⢄⡇⠀⠀⠀⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⢄⢣⠈⠑⢄⡂⠈⠌⠀⠈⠐⠀⠀⣤⠞⢃⢾⠋⠻⣿⡐⠀⠀⠚⠣⠤⠔⠚⣧⣀⣤⣤⡤⣤⣤⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣈⢁⠳⠀⠄⠄⡘⡰⠀⠠⢠⡴⠋⠜⣰⢧⣿⢄⠀⠹⡂⠀⠀⠀⣞⠁⠀⠀⠘⣦⠀⠀⠐⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠠⠀⠀⠈⠀⠀⠀⢣⠂⠐⠂⠄⠤⡐⠏⠣⠃⣠⢞⢌⢃⣟⣇⣠⠔⢻⣅⡀⠑⣄⠀⠀⠀⢀⡜⠀⠁⠀⠀⡀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡀⠀⠄⠀⠅⠀⠠⠚⡿⠞⠀⣾⡿⠽⣿⡟⡈⠙⠿⣿⣮⣿⣿⠿⠋⠀⠀⠀⠀⠀⢟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢱⠀⠂⠀⠀⡀⠀⠠⠀⠁⡀⠀⣀⡄⢄⠆⣸⣑⣺⣶⣀⠀⣀⣤⣀⣴⣴⣴⣴⢾⣾⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⢊⠄⡀⠀⣠⡀⠄⠘⠛⠈⠙⠏⠙⠛⠋⠑⠉⠚⠚⠙⠙⠙⠋⠋⠺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⡁⡀⠀⠠⠁⠘⡄⠄⠀⠀⠀⠈⠐⠜⠿⡞⡞⠾⢼⠿⣷⣞⠹⠿⠂⠉⠋⠈⠑⠃⠀⠀⠀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠔⠠⢀⠂⠜⡄⢦⠀⠆⠈⠂⠄⠀⠀⠰⠁⠀⠀⠃⠘⠦⣤⣠⣄⣄⣤⣠⣠⣦⠶⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠔⡐⡡⢂⣈⣪⣧⠭⠤⠦⣀⠈⠎⡐⠀⠀⠀⠐⡘⡐⡠⡠⣁⣲⣡⠥⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠚⠚⠚⠈⠉⠁⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⡀⢀⡠⢀⡀⢀⡀⢀⠤⠀⠀⡀⢀⠀⡠⠀⠀⣀⠄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠂⢐⠄⡊⠔⡠⢐⠡⢂⡡⢂⠅⡈⠄⡨⠔⡨⢐⠥⢂⠡⢂⠥⢊⢄⠢⢒⠅⡪⠔⡡⢂⠅⡪⠔⠩⠊⠔⠁⠊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠁⠊⠀⠊⠐⠁⠐⠁⠈⠁⠂⠀⠊⠐⠁⠐⠁⠀⠁⠀⠁⠊⠀⠀⠀⠁⠈⠀⠈