Chilli Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢻⣿⠿⠛⢻⣿⡧⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⣸⣿⠀⠀⣾⣿⠇⢸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⣀⣰⣿⡟⠀⢠⣿⣯⣠⣿⣿⠿⠛⠀⣀⣤⣤⣀⣄⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⠟⠁⠀⠀⠻⣿⣿⡿⠃⢀⣴⠟⣉⣭⣭⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣡⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⠀⠀⠀⣠⣴⢟⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣉⣤⣤⣴⣶⣾⣿⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡼⠿⠚⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣹⣰⣶⡵⢿⣟⡿⠿⢦⡀⠀⠀⠀⠀⠀⢛⢆⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⣹⠛⢟⠁⠀⠀⠙⢳⡀⠙⠟⣆⠀⠀⠀⠀⠀⢈⣇⠀⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡋⠀⠀⠀⢀⡄⠀⠀⢟⠀⡈⢾⣷⡀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡧⠀⠀⠀⠀⠳⡆⠀⠀⠁⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⡀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠎⢷⢀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠙⠶⢄⠈⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡎⢷⢓⣠⠀⠈⠠⡀⠀⠀⠈⠠⡈⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠁⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡊⢝⠤⢀⠀⠀⠀⠀⠀⠀⠀⢂⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣠⠀⠀⠀⠂⠀⠀⠈⠙⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⣀⠀⠈⠀⠀⠀⣙⣧⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣨⣧⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⣀⠈⠀⠘⠛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⢣⠃⠈⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠛⠒⠻⠶⠤⣜⣽⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠺⣄⠀⠀⠠⡙⠃⠁⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠠⣈⠓⠠⡄⣀⣀⠀⢤⠀⠈⠀⠀⠀⠀⠀⣠⣴⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠐⠸⢤⣭⣍⣉⠃⠀⠒⠀⠈⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁
⠀⠀⠀⠀⢀⠤⡄⣠⠀⠀⣀⢺⠠⡄⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣂⣦⣰⣏
⠀⠀⠀⢠⡇⠀⠀⢸⡖⢦⠹⡌⡇⢻⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⣿⠟⠉⠈⢻⣏⣿⣿⡤
⠀⠀⠀⢈⣷⣄⣀⠌⣇⠸⠄⡧⠓⡘⠆⣀⠀⢀⢄⢲⡈⠀⠀⠀⣼⣿⠃⠀⠀⢠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⠿⣶⣦
⠀⠀⠀⠀⠙⢋⣀⠠⣄⠀⠀⢤⠘⢦⢻⠈⢣⢳⡈⢁⠧⠀⢀⣾⡿⠁⠀⣤⣤⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⣽⠷⠇
⠀⠀⠀⠀⠀⠀⠙⡀⠘⢀⠑⠌⣇⡸⢈⣧⠈⠀⠉⠀⣀⣶⣿⡟⠀⢀⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣀⣀⣠⣄⢀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⠦⠐⠸⠀⠀⠉⠀⢀⣴⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⠟⠁⠀⣰⣾⡿⠟⠻⣷⣾⣦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⢀⣼⣿⠟⠀⠀⠀⣸⣷⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⠉⠉⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⣿⣿⡿⠿⣿⣿⠟⠋⠀⠀⠀⠀⠀⣀⣴⣿⠋⠀⢠⣼⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠻⣿⣿⣿⣿⠿⣿⣿⣿⡿⢿⠛⣋⣉⣵⡶⠿⠋⣁⣤⣤⣴⣶⣶⣾⣿⣿⣿⣧⣀⣰⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠙⠛⠓⠚⠛⠛⠛⠛⠛⠋⠉⠉⢀⣠⠶⠟⢛⣩⣤⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣏⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⢛⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠰⠀⣀⠀⡈⡀⠀⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⢀⣠⣼⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠿⠛⠦⠔⣏⡛⠅⡿⢻⠟⠉⠉⠀⠀⠈⠈⣑⣻⣿⣿⣟⠙⠛⣷⣶⣶⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡀⠀⠀⠀⠈⠛⢻⣾⣯⣷⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣿⣿⠿⠟⠋⠉⠙⠛⠿⠿⣿⣿⣿⠿⠿⠿⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠢⢄⠀⠀⠚⠁⠙⠋⠻⣌⠓⠫⠹⠦⠑⡂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠙⡆⣀⠀⠀⣀⣤⣔⠉⠤⠠⡤⠀⢠⠤⢍⡀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠛⠉⠉⠉⠁⠈⠁⠉⠀⠉⠉⠉⠙⠒⠦⢤⣐⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠮⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠆⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣦⣤⣄⣀⣠⣠⣄⣠⣀⡘⠂⠀⠀⠀⠀⢀⠞⠁⣀⡀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⢳⠀⠀⠀⠀⠀⡄⠾⣧⠹⠷⠆⣶⠋
⠀⠀⠀⠀⠀⠁⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢝⠱⠾⠛⠀⠁
⠀⠀⠀⠀⠥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠋⣿⢤⡝⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠈⠁⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠠⠚⡽⠀⠀⠀⠀⣴⡏⣹⠿⠀⣶⡞⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠩⠖⠀⠀⠁⣠⠞⠀⠀⠀⠀⢶⣶⠦⠿⠉⢀⣭⠀⠉⠋⠱⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠚⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠘⢦⣄⣀⣀⠄⠊⠙⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⢆⡀⣀⣀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠿⢿⣤⣀⣀
⠀⠀⠀⠉⠉⠀⠘⠿⣦⣀⣀⣀⣀⣤⣴⠴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⣤⣄⣄⣄⣀⣠⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⡿⠿⠿⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⠋⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣂⣤⣾⣿⡿⠿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠶⠿⠟⠛⠉⠀⢀⣠⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠾⢿⣿⣛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠒⠲⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⣤⠀⠀⢠⠀⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠀⠀⡾⠀⠐⠢⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⡇⠀⠀⠀⡷⠀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣼⠷⠤⠤⠞⠁⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⣦⣤⣤⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠉⠙⠛⠛⠛⠛⠛⠛⠉⠁⣀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣶⣤⣤⣤⣴⣶⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣠⣶⠃⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⢰⣿⡇⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⣿⠀⢠⣿⠀⠀⠈⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣾⡏⠀⠀⢸⣿⣇⢸⣿⡄⠀⠀⠘⣧⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢸⣿⡇⠀⠀⢸⣿⣿⡆⠻⣿⣦⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⣤⣾⡿⣿⡇⠀⢹⣿⣧⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠸⣿⣷⡙⠛⠛⣵⡿⠁⠀⠀⣿⣿⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⢿⣿⢀⣾⡟⠁⠀⠀⢀⣿⣿⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣾⣿⠀⠀⠀⠀⣼⡿⢹⣿⠏⠀⠀⢰⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣦⣴⣏⠀⠀⢀⣿⣿⣿⣤⣀⣤⠞⡉⢀⣿⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠾⣿⣿⣦⣤⣾⣿⢻⣿⣿⣿⣿⠀⡇⢸⣿⡿⣶⣾⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡛⢿⣿⠀⠻⣿⣿⣿⠀⡷⠄⢀⣴⡿⠉⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹⣧⣀⠁⠰⣆⠙⠛⠋⢰⠁⣴⣿⣿⡀⢀⣠⡿⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⣠⢀⣿⣿⣷⣦⡈⠻⣶⣶⣿⠀⠻⣿⠟⣿⡿⠋⠀⠀⠀⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣧⠀⢹⣿⣿⠷⠖⠀⣰⣿⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡙⠛⠋⠁⢸⣿⡏⣴⢿⡟⠁⢀⣴⣾⣿⣿⣀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣄⠀⠘⠋⢸⣿⠘⠀⠀⣿⣿⣿⠟⠛⠛⠋⣹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠘⡆⠀⣿⡀⠀⠀⠈⠉⣡⠖⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣰⣿⠇⢀⣵⠀⣽⠹⣦⣤⣤⣾⣿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡟⠀⣾⣿⢾⣿⡀⢻⣿⠏⣿⡿⢸⣿⣧⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠄⠘⠿⠄⢉⣤⠆⠁⣰⠟⠡⠾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⣷⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣠⠟⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠿⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⣤⣶⡴⠞⠋⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣀⢀⣴⠯⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⡀
⠀⠀⠀⠀⠀⠙⢷⣶⣭⣽⣛⣛⣻⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣟⣛⣛⣯⣥⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁

⠀⠀⠀⠀⠀⣀⣰⣠⣀⢀⣀
⠀⠀⠀⣰⠊⠀⠐⠃⠀⢀⣨⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢹⣿⣿⡿⠿⠟⠛⣿⣿⠀⢠⣤⡄⣀⣀⣀⣀⡻⠿⠿⣿⣿⢀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⣤⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢸⣿⡇⠀⣀⣤⠀⠘⣿⡀⢸⣿⠇⠻⠿⢿⣿⡗⠀⣸⣿⠃⢸⣿⠁⠘⣿⡇⠀⣿⣿⣧⠀⠀⣿⡇⢀⣴⣿⠟⢻⣿⡇
⠀⠀⠀⠈⣿⡇⠀⢻⣿⣶⣾⣿⡇⢸⣿⠀⠀⢀⣿⡟⠀⣰⣿⠃⠀⢸⣿⠀⠀⣿⡇⠀⣿⣿⣿⣆⢰⣿⡇⣼⣿⠃⠀⠀⠉⠁
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠈⢻⡇⣸⣿⠀⠀⣾⡟⠀⢠⣿⠃⠀⠀⢸⣿⠀⠀⣿⡇⠀⣿⡏⢻⣿⣼⣿⢣⣿⡇⢰⣿⣿⣿⡟
⠀⠀⠀⠀⢸⡿⠗⢲⣿⣧⠀⢸⣿⣿⡏⠀⣼⡟⠀⠀⣿⣷⣶⣶⡄⢸⣿⠀⠀⣿⡇⠀⣿⡇⠀⢿⣿⣿⠸⣿⣇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡀⠈⠋⠉⠀⣸⣿⣿⡇⢸⣿⣶⣶⣶⡎⠉⠉⠉⠀⢸⣿⣤⣤⣭⣄⠀⣿⡇⠀⠘⣿⣿⠀⢿⣿⣄⢀⣼⣿⠃
⠀⠀⠀⠀⠀⣿⣷⣦⣤⣶⣶⣿⡿⠃⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠁⠀⠉⠀⠀⠀⠈⠉⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢋⡏⡄⠀⠔⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡰⠁⠀⠐⠀⢠⠇⠀⡡⢢⡄⢀⡴⢊⠭⢍⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡴⢡⠀⡇⡆⡆⡿⠀⠸⣇⠀⠀⠘⠀⣇⣶⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣇⠰⡀⢧⢸⠸⣿⡀⢆⠉⠳⣄⠐⡄⠻⣝⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠁⠈⠢⣳⡈⢆⢣⠻⣿⣌⢧⢀⠘⣷⠸⡄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠪⡅⠀⠀⠉⡡⡄⢱⣌⢿⣾⣷⣧⠹⣿⠘⣾⣦⣿⠇⡇⣈⠀⢸⡆⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣺⢩⣛⢦⣿⣿⣿⣶⣿⢀⣿⡿⣫⠞⠁⡡⢀⣼⠃⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠠⡸⠖⠀⢸⠂⠹⣿⣿⣿⣇⣾⣯⡟⢁⣴⣿⡶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⢀⠀⠀⢤⠀⢿⡄⢆⠹⣯⣿⣿⣿⣿⣷⣿⣴⣆⡀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣋⠍⢹⠒⡟⣰⠺⢿⣶⣷⣿⣿⠟⠛⠋⠉⠉⠁⠱⠉⠃⠙⠒⣶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠿⠃⣼⠀⠀⠁⠐⣠⠼⠛⠁⠁⠀⠠⠀⠀⠀⠰⢆⠀⡀⠀⣼⢡⢺⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠷⠶⠶⣧⠤⢤⣞⣭⣀⡀⠀⠀⠀⢀⡴⣋⠀⢤⠶⢦⠶⢾⣧⣷⡘⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡗⠀⣤⡤⡀⢀⢸⡀⢀⢮⣹⡆⠀⣨⣾⠛⢿⣃⡏⠠⠄⠀⣼⣿⡍⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢉⡇⢀⡛⡇⠀⣀⣸⣧⣐⣊⣿⢷⣿⡁⠠⠀⠘⣿⡷⢦⠤⠼⠋⢙⡃⢁⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠲⠶⢒⣻⠿⢛⠅⠈⢛⢽⠞⠻⢿⣴⣶⠟⠃⠐⠀⠁⠀⠃⠀⠀⠴⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢤⡀⠀⠀⠀⢀⠀⣠⣾⠀⠆⢀⡶⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⠉⠂⣀⡀⢹⠦⣀⠸⠄⢩⠉⠃⢦⣔⡄⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠟⠁⠈⠙⢱⣦⣈⠑⢦⡔⠀⢠⣁⠘⠀⢀⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢻⣦⣀⣀⡀⢀⠈⠻⠃⠸⠃⠀⠸⠻⡄⠘⠈⣠⡁⢠⠇⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠴⠚⠉⢙⠋⢁⠘⣶⡃⠀⠀⠀⠀⠀⠰⠁⠀⠀⢿⢰⡜⠀⠀⠀⡀⣼⣸⡏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣨⡀⠀⠀⠘⠇⠀⡄⡼⣷⠀⠀⠀⠀⡀⢀⡆⠀⠀⠀⠀⠀⢃⢀⣼⡇⢱⣭⠀⡆
⠀⠀⠀⠀⠀⠘⢦⡀⢿⣝⠢⣄⠦⢄⠐⠛⢧⣌⠀⠀⠀⠀⠀⠀⠀⡴⣷⠆⠀⢀⠛⢯⡋⣇⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠙⠦⡙⠷⡄⠁⠀⠀⠀⠀⠈⠃⠐⠀⠀⠀⠀⣈⠻⠱⠆⠀⡾⢷⣄⡉⠋⠁⢠⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣌⠓⣂⡀⢀⠀⠀⠀⠀⠁⠄⠼⠛⢿⡀⠀⠀⠀⡓⠦⣈⠻⠆⠠⠋⢹⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⠳⢝⡷⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⣧⣦⠉⠀⠀⠀⡀⣠⣬⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⣑⣮⠉⠀⣄⠀⠀⠀⠀⠁⠰⢏⠙⠺⢙⠿⣀⠀⠀⣹⡉⢻⡍⠸⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠉⠑⢢⡀⠀⠀⠀⠢⣄⠀⠀⠈⠈⠑⠾⣿⣃⢀⡁⠀⢀⡆⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠌⢢⠀⠄⠀⠈⠓⢤⡀⠀⢂⠄⡂⣹⡯⣷⡀⠉⣤⠤⢄⠀⠠⠐⠙⢒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠙⢦⡑⠄⣀⠀⠙⢶⣧⡈⠀⠈⠀⠈⡷⠈⠙⢆⡉⠉⠉⠉⠒⠭⡥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢵⠂⣠⣈⠲⢾⣶⣤⠁⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⡾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠹⡄⠙⠂⠀⠀⠙⢦⣃⠌⠀⢀⠔⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⠀⠈⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⣀⣤⣄⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⢿⣧⢰⣿⡏⢻⣷⡀⣿⡟⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠘⣿⡆⢻⣿⠈⣿⡇⢹⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⣸⣿⢣⣿⠏⢰⣿⢃⣾⡟⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣿⠃⣿⡏⢰⣿⠏⣾⡟⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠘⣿⡆⢿⣧⠘⣿⣆⢻⣿⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡿⠃⠈⠿⣷⡿⠏⠈⠻⣷⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠋⣉⣠⣤⠴⠶⠶⠶⠒⠲⠶⠶⠶⢤⣤⣈⡉⠛⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋⠁⢀⣠⠶⠛⢉⣡⣤⠶⠶⠖⠒⠒⠒⠒⠲⠶⠶⢤⣌⠙⠳⢦⣀⠀⠉⠛⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠋⠀⠀⢀⡴⠋⠁⣠⡶⠋⠁⠀⠀⢶⣴⠂⠀⠀⠻⣦⡖⠀⠀⠈⢳⣄⠀⠉⠳⣄⠀⠀⠈⠙⠿⣷⣄⡀
⠀⠀⠀⠀⠀⢸⣿⠋⠀⢀⣀⣀⣴⠏⠀⣠⣾⣿⠁⠀⢦⣀⡴⠋⠙⢷⣄⣠⠞⠉⠻⣦⣠⡴⠈⡏⠻⣦⡀⠈⢷⣤⣄⣀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣰⠟⠁⣸⠃⠀⣴⢻⡏⢿⠀⢀⡴⠛⢦⡀⠀⣠⠟⢷⣄⠀⢀⡼⠻⣦⠀⡇⠀⠈⢳⡄⠀⢻⡀⠙⢷⡀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⡟⠀⢰⡇⠘⣇⠘⣧⡀⠀⠀⠀⣹⣾⡁⠀⠀⢙⣷⣏⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠘⡇⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⣇⠀⢸⡇⠀⠻⣄⠈⠛⢦⣀⠀⠁⠀⠙⣦⡴⠋⠀⠙⠃⠀⠀⣼⠀⠀⠀⠀⣿⠀⢀⡇⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⢻⡀⠀⢻⡀⠀⢷⡀⠀⠉⠳⣤⡀⠉⠳⣤⠀⠐⠋⢙⣤⣀⠀⠀⠀⢠⡏⠀⠀⠀⣰⠏⠀⣸⠃⠀⣼⠃⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠙⠳⠾⢷⡀⠈⠳⣄⠀⠀⠀⠙⠷⣄⠘⣇⠀⠠⣏⠀⢸⡇⠀⣠⡟⠀⠀⣀⡼⠋⠀⣰⠿⠶⠚⠁⠀⣸⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠹⣦⡀⠈⠛⢦⣄⣀⠀⢻⡄⠙⢦⣀⠉⠛⢋⣠⣴⠟⣀⣤⠞⠋⠀⣠⡾⠁⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠙⠶⣄⡀⠈⠉⠛⠚⠷⣤⣀⠉⠙⣛⣿⠟⠛⠋⠉⠀⣀⡴⠞⠁⠀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣄⣈⠉⠛⠶⠦⣤⣄⣀⣉⣙⣋⣉⣀⣤⡤⠶⠞⠛⢉⣀⣤⣶⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣷⣶⣦⣤⣤⣬⣭⣉⣉⣥⣤⣤⣤⣶⣶⡿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁


⣿⠿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠈⠐⠠⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢇⡀⠀⠀⠀⠀⠘⠃⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢑⢢⠀⠀⠀⠀⠀⡀⠀⠛⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⣷⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠉⠀⠀⠠⠀⠀⣀⠀⠀⠀⠀⠸⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⣤⢠⠀⢠⣄⠈⠀⡈⠀⠀⡀⠀⠀⠈⠓⠖⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⠈⠳⠀⠄⡀⢀⣍⢠⡾⢃⡘⠻⠃⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡀⠘⠀⠀⠸⠀⠀⠈⠛⠀⠙⠁⢠⠀⢠⡀⢀⠇⡚⠂⢻⠀⠘⢂⠈⠀⢀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠈⠀⠀⡄⠳⠀⠆⢠⣄⡄⠈⢆⢈⠀⣤⠀⠈⡀⠋⣠⠀⠶⠉⠉⠠⠀⢯⠀⡠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⡀⠀⡀⠁⠢⠀⠈⠀⠰⠮⣀⠀⠈⠀⠀⠸⣤⠊⠀⠀⠋⣠⠂⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⢠⡙⠀⣠⠈⠀⢧⠀⠀⠘⡀⠀⢳⡀⢸⡇⠀⢠⡴⠈⠀⠀⢸⠆⢀⣿⡤⠀⠠⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣶⣀⠀⠛⠠⣀⠀⡃⠀⠀⡃⠀⠀⠙⣿⠀⠀⡟⠀⣠⠤⠔⠃⠈⠁⠀⠀⠘⢩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⣠⠀⠀⠀⠀⠠⢄⠉⠙⢦⢀⠇⠀⢦⣸⣏⢀⡴⠃⣾⣠⡤⠄⠀⠐⠀⠀⡀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠉⠀⠰⠖⢀⠀⠀⠈⠉⠉⠙⢻⣄⡄⠀⣿⡿⠋⠀⠀⡞⠁⠀⡀⠤⠤⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠚⠋⠀⠀⣀⡀⠐⠢⠀⢿⣷⣀⣿⠀⠀⢀⡼⠁⠀⠠⠀⣀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⡄⣨⠆⠀⠀⠀⠙⣯⣿⣧⡞⠋⢀⠒⠂⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠢⢤⣤⣤⣄⣀⣿⣿⡏⠤⣠⠼⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣴⣶⣯⣾⣿⣿⣯⣽⣶⣶⣿⣿⣷⣿⡶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣏⣽⣶⣿⣷⣍⡻⣿⣿⣿⣿⣿⣴⣷⣌⣭⣿⣿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣔⣿⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣿⣿⣮⣽⣿⣿⣿⣿⡿⣿⠿⠿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣵⣾⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣿⣷⣌⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠋⠉⠉⠟⠛⠛⠛⠉⠋⠉⠉⠉⠁⠀⠈⠉⠉⠙⠛⠉⠈⠉⠉⠀⠈⠛⠛⠛⠿⠟⠿⣿⢿⠿⠛⠷⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⢿⠛⡿⢻⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠂⠐⠀⠀⠁⠈⠀⠁⠉
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠻⣿⡟⢻⣿⡛⣻⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢀⣄⠀⢠⡄⠀⣴⠄⠠⠦
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⣿⣿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⡀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⡿⢿⠈⡏⣰⣷⡄⢻⣿⠈⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⢀⣈⣿⢰⡏⠉⢻⡆⠀⣿⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣶⣾⠀⣧⡘⢿⢃⣼⣿⣄⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⠉⠉⣿⠘⢧⣄⡼⠃⠀⠿⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠘⣿⣿⣿⡷⠟⢿⢛⡛⠟⢻⢛⣝⣿⢩⢽⣿⣿⣿⣿⠀⠀⠀⠀⠀⢘⠀⠀⠀⢈⣠⡀⣬⢤⣠⣄⡴⠶⠄⣖⣒⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣩⣼⠨⣅⢻⢸⣘⣟⣿⣘⣛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠄⠀⠀⢸⠶⠋⣷⠺⣝⡏⠧⠠⠆⠧⠤⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⣿⣿⠿⣿⡿⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⢀⣄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣠⣿⣧⣼⣿⣶⣾⣿⣶⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠄⠀⠀⠐⠟⠀⠘⠃⠀⠙⠁⠀⠉⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⢀⢀⠀⠄⠠⠠⠀⠀⠀⠐⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⢧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢆⠀⠀⠀⠀⠀⣠⠋⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸⠀⠀⠀⢀⡴⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⢀⡷⠊⠉⠉⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧⡀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠙⢆⠀⢸⡇⠀⠀⠀⠀⠀⡴⠁⠀⢠⡆⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠑⣾⡇⠀⠀⠀⠀⡼⠀⠀⢀⣾⠀⠀⠇⠀⠀⠀⢀⣀⣀⣀⢀⣀⠤⠚⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠈⢇⠀⠀⠀⠠⡇⠀⡰⠋⡏⠀⢸⠀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠸⡆⠀⠀⠀⠳⡼⠁⢸⠁⠀⢸⠟⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠉⠒⢦⡀⠀⠈⣧⠀⠀⢀⠀⢷⠀⠀⠀⠀⡇⠀⠸⡄⢠⠃⠀⠠⡀⡀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠘⢆⠀⢸⡆⠀⢸⡄⢸⠀⠀⠀⠀⡇⠀⠀⢳⣜⠠⡀⢠⡀⢇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⡆⢠⠇⠀⢨⡟⣎⠀⠀⠀⠀⢣⠀⠀⠀⠙⣄⢱⠐⡇⢾⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⡏⠀⠀⢈⡇⠀⠙⠢⡀⠀⣸⠀⠀⠀⠀⠘⡄⡆⣷⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠈⢡⡄⠀⣸⠁⠀⣀⠀⢸⡆⠁⠀⡜⡅⠀⠀⡇⡇⣾⣸⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣇⠀⣿⠀⠀⠘⡇⡰⢻⠀⣰⠃⠀⢸⠇⠀⠀⡇⢸⠀⢀⠇⡏⢁⡤⠒⣉⠟⣂⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄⠹⡄⣯⡜⠀⢠⡷⠁⢸⠀⢟⠀⠀⡟⠀⣀⠀⢃⢸⠃⢸⠀⢡⠋⠀⡞⠁⠈⠀⠀⢀⡠⠟⠓⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠹⡄⠙⢷⣱⡀⢸⣇⢰⠈⣇⢸⢰⡀⢿⠀⠈⢷⡘⣼⢾⣿⡀⡏⠀⠀⡆⢀⣸⡀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⢦⣀⠙⠳⣄⢿⡘⠂⢸⡎⢸⣹⠘⠂⠀⢸⢳⠀⣾⣿⣷⡇⠀⠀⡇⠘⣿⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢧⡈⠉⡳⢌⠙⢿⠀⠀⠁⢸⠀⢀⡠⠴⠋⡸⣤⠛⣣⣿⠀⠀⠀⢸⢰⠿⠇⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠈⠻⣤⡙⣄⢳⠀⠓⢀⣾⣿⢀⡞⠀⠀⠀⡇⡿⡄⠈⠁⠀⣴⡄⢸⢀⣶⣾⣿⣤⣿⣒⢩⠟⣦⣶⡀
⠀⠀⣀⣤⣴⡰⢶⣶⣶⡟⣈⢩⣍⢩⣭⣈⣓⣄⣈⠙⠊⢷⠃⠀⠸⡇⣿⡀⡇⠀⠀⠀⣟⡱⢃⣾⣾⣾⠟⣠⣾⣿⣿⣿⣟⣻⡧⠤⠼⢾⣽⣿⡝
⠀⢼⠁⢸⣿⣻⣦⠿⠿⠓⠐⠋⠉⠁⠀⣩⠝⠋⠉⣒⡒⡾⠀⠀⠀⠳⠘⠏⠓⠂⠀⢘⣭⢶⣾⣷⣤⣶⠿⣯⣤⡤⢤⣠⣍⠁⠀⠀⠀⠀⠈⢻⡏
⠀⠀⠉⠛⠤⠴⠶⠶⢦⣤⣶⣶⡶⠖⠋⢀⣤⣴⣾⡿⠻⠅⢀⣴⠋⠙⠦⣐⠉⠉⠛⠦⠭⠽⣭⣭⢿⣟⣛⣛⡺⠔⣫⠀⣼⣶⡶⣶⣾⡋⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣭⣄⣠⣴⣿⣾⣋⣭⣴⣯⣿⠶⣦⣟⡲⢄⡈⠙⠒⢤⣀⣀⠀⠀⠀⠀⠈⠁⠈⠙⠋⠙⣄⡟⠓⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠞⣹⣯⠶⣿⣿⣿⡾⠟⣯⣴⣾⣿⣿⣊⢿⣿⣿⣛⡿⣦⡀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠍⠉⠉⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠻⣿⠰⠿⠋⠉⠀⠀⠀⠀⠈⠁⠀⠙⠄⠙⢿⣿⣿⣿⣿⣦⣄⣀⠀⠉⢻⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡿⢻⣮⣳⣄⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣯⣄⣠⡼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠄⠀⠀⠘⢌⠢⠀⠀⠰⠀⠠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠊⠀⠘⡠⠀⠀⠀⠠⠑⠀⠀⠀⠐⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠂⠀⠀⠀⠀⠀⣄⠀⠀⡰⠁⠀⢹⠀⠀⠀⣀⠀⠈⠀⠀⠀⠄⠁⠄⠘⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠐⢀⠁⠳⠉⢦⣠⠁⠀⠀⢸⠀⠲⢀⡶⢱⠀⢀⠀⠸⠎⠀⠀⡄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⠀⡉⠀⠀⠹⡀⠀⠀⠸⠀⠀⠰⠁⠀⡇⠈⠴⠃⠐⡆⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢾⠣⣄⣠⠀⠀⠀⠀⠀⠀⠀⢇⠀⠃⠀⠀⡇⠂⠀⠀⢠⠃⠀⠀⣀⢀⡀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠈⠣⡄⠀⠀⢇⠀⠀⢠⢃⣄⣀⣤⡔⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡞⢈⠼⠊⢰⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠙⠀⠀⠀⠀⠀⢄⠀⠠⠀⠀⠀⠀⠀⢀⠔⢠⠀⠀⠱⠁⠀⠰⠁⡾⠁⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣄⣅⣀⠀⠀⠀⠀⠀⠀⠃⠄⠂⠖⠄⠀⡀⠈⠠⠄⠂⠀⠀⠠⠴⠋⠉
⠀⠀⠀⠀⠀⠀⢐⡖⣻⣉⣉⠉⣩⣽⠶⣀⠀⠀⣀⣠⠤⢤⣀⡄⡦⠀⡠⢦⠐⠠⣤⢄⠂⠠⠾⠿⢷⣿⡂⡀⢰
⠀⠀⠀⠀⠀⠀⠹⠷⠼⠷⠶⠚⢧⡇⢠⣾⣶⣛⡻⣄⣵⢠⣼⠃⣓⢲⠀⢘⣧⡂⠚⢺⣇⣀⡀⡶⡞⣮⣭⠴⣾⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠙⠛⠛⠁⠀⠙⠶⠖⠛⠉⠙⠳⠚⠋⠀⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠉⠁⠒⠢⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⠀⠙⢆⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⢄⡀⡰⠁⡴⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠾⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏⢀⡵⣕⠀⠀⠀⠀⠈⢢⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏
⠀⠁⠢⠀⠀⠒⠄⡀⢀⠔⠀⠈⠊⠐⣈⡻⠃⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⠁
⠀⠈⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠌⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡇⠀⠀⠀⠀⢸
⠀⢰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⢂⡔⠛⠃⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⡇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⡀⢀⡠⠃⠀⠀⠀⢀⠤⣚⡐⢛⣤⠤⣀⡀⠀⠈⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠀⠀⢧⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠞⠁⠄⠀⠣⠄⢠⠤⠊⠉⠓⠢⢤⠴⠃⠀⠀⠀⠀⢀⣀⡤⡞⠁⠀⣸⠀⠀⠈⠳⢤⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠒⢉⣉⡀⠔⠀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠀⡸⠀⠀⡱⠦⠴⡄⠀⠀⢰⡆⠠⣥⠀⢀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⣇⣀⣀⣧⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⢸
⠀⠀⠀⠐⠲⢄⣄⣠⢶⠊⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣄⠀⣰⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⡙⠻⣿⣷⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠿⠛⠛⠻⠿⣷⡸⣿⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣤⣶⣶⣶⣶⣬⡛⣿⣣⣿⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣍⡛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⢛⣛⣛⠛⠛⠉⠁⣾⣿⡇⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠇⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣼⡆⠀⠙⠟⠛⠀⠻⣿⡆⡿⠋⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢠⡏⠸⢿⣿⣄⠀⠀⠀⠀⠀⠈⣡⣴⡆⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡶⠒⣴⠟⠐⢺⣿⣿⣶⣶⣶⡶⠿⠀⠀⠀⠐⢶⠄⣤⣄⠀⢸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠄⣴⡆⠔⠒⣨⢞⡽⣻⣿⠟⣯⡁⠠⢀⠀⠀⠀⠀⠀⠁⣀⠀⢸⣿⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠄⠙⣀⠋⢀⠠⣻⣯⣞⡼⣱⣛⠷⣌⠻⡄⠀⠀⠛⠠⡤⠀⢠⣿⣦⠸⠏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀⠀⡿⠛⠉⢠⡏⡿⣾⢱⠁⠙⠳⢌⠃⠀⠀⠀⠀⠀⣶⣄⠀⠻⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣤⣀⠀⢸⣧⡇⡟⠟⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⣷⡀⠀⠀⠛⣵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣾⣷⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢽⣿⣿⣿⣿⣿⣿⣿⢟⣽⣿⣿⠟⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠙⠻⣧⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⠈⣿⣿⣿⣿⣿⣿⢋⣾⡿⠋⣁⡴⠀⢀⠀⠀⠀⠀⠁⠀⠈⢿⣶⣄⠀⠀⠀⠀⠀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣁⡼⠻⣿⣿⣿⣼⣿⠁⠾⠋⣠⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠰⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠿⢐⣀⠼⢻⣿⣿⣿⣷⠀⣼⣿⣿⢛⠄⠀⠀⠀⠀⠀⣠⣶⠏⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠳⣿⡞⠁⣀⠄⢀⠠⣽⡏⠉⡉⠉⠻⡏⢀⡠⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠁⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡠⠸⠃⢁⡼⠲⠦⡄⠙⢁⡞⠋⠙⠦⣤⡾⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⢶⣷⡇⠀⣤⠀⢀⠸⣿⠂⢸⣁⡀⠹⡇⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⡀⢻⣜⡴⠙⠐⠮⣀⣥⣴⠃⠀⠙⠦⣵⡗⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢩⡟⣿⢁⠐⡆⠀⠆⠹⡇⡅⠲⠀⠤⠙⠃⠀⠈⠂⠀⠀⠀⠀⠀⡠⠀⠀⠒⠐⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⡄⠀⠀⠀⠀⠀⣴⠀⠘⠃⠀⠜⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠸⡆⠀⠀⠀⠀⣟⢧⡀⠙⠤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⣌⠇⠀⠀⠀⠀⢳⡨⢘⠦⣄⠀⠑⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⠀⠀⠀⠀⠀⠀⠈⢖⡡⢂⠠⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡡⠹⡄⠀⣠⠀⠀⠀⠀⠀⠳⣅⠪⢀⠤⠛⢆⠀⠀⠀⠀⠀⠀⣠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠑⠪⢒⠩⡇⢰⢳⠀⠀⠀⠀⠀⠀⠈⢲⠁⡊⢔⠨⢣⠀⠀⠀⢀⠔⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡐⠅⣊⠑⣺⠃⠘⢌⠳⣄⢲⣤⣀⠀⠀⠀⢹⡔⠅⣢⡜⡇⠀⠀⡎⡰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠐⡩⠐⠥⣻⠀⠀⠀⠑⠺⠄⢳⠍⢣⡄⠀⠀⠇⢊⡀⢀⢽⠀⠀⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢕⠀⠷⡀⠐⢧⡀⠀⠀⠀⠀⣜⠠⠲⣷⠀⣜⠁⠸⡁⢀⡸⠀⠀⡴⣷
⠀⠀⠀⠀⠀⠀⡶⣄⠀⠀⠀⠀⠀⢹⡡⠀⢇⠈⠢⡀⠙⢆⠀⠀⡼⠑⠀⠀⣷⠞⠁⡠⠁⡆⢀⡇⠀⠀⡏⢸⡆
⠀⠀⠀⠀⠀⠀⠹⡀⠹⠲⣄⠀⠀⠀⢫⠄⠘⣄⠀⠙⢢⡈⢳⣞⠉⡰⢣⠀⠋⢀⠜⠁⠰⠁⡼⠀⠀⣸⡡⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⡘⢆⠀⠀⠀⠳⣄⠈⢆⠀⠀⠱⠀⠁⡜⠀⠈⠄⡴⠃⠀⢠⠃⣼⠁⢀⡴⠏⠀⠡⢸⠀⠀⠀⠀⢀⣻
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣌⡙⠀⠀⠀⠀⠘⣧⠀⢂⠀⠀⢣⡘⠀⠀⢰⠋⠀⠀⢠⠃⣴⣧⠶⠛⠀⣀⠀⠨⣼⠀⠀⠀⡰⠋⣲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡉⠒⢤⡀⠀⠀⢸⡄⠈⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠇⡘⠋⢀⡠⠔⢊⠃⠀⣠⠇⠀⠀⢎⡠⠖⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠕⠈⠹⡄⠀⢸⡃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠀⡠⠊⠁⠀⡰⠁⢀⡰⠃⠀⠀⠘⢁⡤⠞⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠗⡄⢹⣄⣾⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⡜⠀⠠⡞⠁⣠⠤⢀⡴⣃⢢⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠀⡆⠘⡄⠿⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡟⢠⠾⣟⢀⣞⠁⠒⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢠⠁⠀⠶⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⢀⣷⠝⠂⢟⣼⢃⡾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⢸⠀⠀⠘⣦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢇⠈⡇⡼⡇⢻⠟⡎⢠⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠣⣠⠀⠱⡈⢰⠁⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠒⠀⡘⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡑⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠌⢲⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⢚⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⣀⠀⢀⡀⠀⠀⠀⠉⠐⠂⠤⠤⠀⠀⢀⣀⣀⣀⣀⠀⠠⠌⠑⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠒⠈⠄⠂⠡⠐⡈⠔⢊⠡⠐⡀⠄⣀⠤⢐⠀⢀⡠⠄⡂⠄⣂⠀⢂⠠⢀⡀⢤⡒⢄⠂⠐⡂⠤⡰⢄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠈⠀⠈⠁⠈⠀⠈⠀⠀⠁⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡋⠀⠀⠀⢼⣿⡏⠀⠀⠀⢸⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠈⣻⣿⡆⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⠀⣸⣿⡟⠀⠀⠀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠘⢿⣿⡄⠀⠀⠘⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠀⠀⠀⢰⣿⡿⠁⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠉⠀⠀⠀⠀⠀⠉⠛⢿⣿⣦⣄
⠀⠀⠀⢀⣾⣿⠏⠁⠀⢀⣠⣴⣶⣶⡆⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⢀⣿⣿⠃⠀⢀⣴⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⣸⣿⡇⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣄
⠀⠀⣿⣿⠁⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣶⣶⣶⣿⡿⠛⠙⠻⣿⣧
⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⢀⣿⣿
⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠿⠿⠿⠿⠿⠿⣿⣷⣄⠀⠀⠀⣻⣿⡆
⣠⣤⣤⣤⣤⣽⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣶⣤⣴⣿⣿⣤⣤⣤⣤⣤
⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⢹⣿⣇⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⣰⣿
⠀⢻⣿⣦⡀⠙⠻⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⠿⠋⢀⣴⣿⡟
⠀⠀⠙⢿⣿⣦⣄⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⢀⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠏⣻⠇⠀⢀⠃⠀⠀⠀⠀⣰⣿⣿⠟⠛⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡠⠄⠒⠙⠠⠤⠐⠁⠀⠀⠀⠀⠀⢿⣿⣯⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠘⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠐⠒⣸⣿⣯⣐⠒⠂⢰⠦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣤⣀⠔⠊⠁⠀⠀⠀⠀⣸⣿⣿⡿⠁⠀⢀⠆⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⡟⠀⠀⠀⠈⢛⠿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣿⣿⠏⠀⠀⡠⠃⠀⠀⢠⣶⣄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠠⠾⠋⠀⠀⠀⠀⡠⠁⠀⠈⠙⢿⣿⣿⣷⡀⠀⠀⠙⠿⠄⠄⠊⠀⠀⠀⠀⢺⡟⠁⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠉⠐⠢⢄⣴⣿⣿⣷⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢀⡤⠖⢐⣾⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠑⢄⡀⠀⠀⢀⣠⣿⠟⢡⣿⠁⠀⣾⣿⣿⡟⠀⠈⠁⠂⠤⣀⠀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⢰⣿⣿⡿⠁⠑⠢⢤⣤⡾⠆⠀⠈⠉⠉⠉⠉⠀⠀⠘⠛⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣈⠁⠀⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⢦⠀⠀⠀⣿⣿⣿⠁⠀⢠⣶⣶⠀⠀⠙⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⢀⣴⣾⡿⠷⣆⠀⠀⣾⣇⠀⠀⢠⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⡆⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⢠⣿⡟⠁⠀⠀⣰⣶⣤⡹⣿⣦⣠⠃⠀⠀⠀⣿⣿⠁⠀⠀⡆⠻⣿⣀⡠⠀⠀⠀⢘⣿⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⢸⣿⠀⠀⢀⣴⣿⣿⠿⠃⠙⣿⣿⣦⣄⠀⠀⢻⣿⠀⠀⠀⠃⠀⠈⠉⠀⠀⡠⠒⠁⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⠃⠈⠉⠁⣼⣿⠟⠁⠀⢀⠌⠀⠛⢿⣿⣷⣄⠈⢿⣇⠀⠘⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢀⠂⠀⠀⠀⢰⣿⠃⠀⣀⠔⠁⠀⠀⠀⠈⠻⣿⣿⣷⡀⠙⠒⠁⠀⠀⡠⠃⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠐⠁⠀⠀⠀⠀⠈⠃⠄⠊⠀⢀⠤⢤⣄⡀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢹⣿⡄⠀⠀⢹⣿⣿⣧⠀⠀⠀⢁⠀⠀⠀⠀⠀⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠈⠋⠉⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠁⠂⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⣀⣀⣠⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⣠⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⢉⠛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡰⠿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠐⡀⣤⡆⠀⠠⣀⡛⠁⠀⠀⠀⠀⡀⠄⠠⠠⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠈⠛⠤⣄⣀⡀⢀⠀⢀⡉⠁⣤⢠⡄⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⢠⡸⢷⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠴⢠⡀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⡄⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⢤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⣈⣁⠈⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠘⢰⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠷⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⣼⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣿⣿⣿⡿⠋⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠻⣿⡿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡿⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣷⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠻⣿⣧⡀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣼⠀⠀⠀⠀⢻⣿⡇⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣿⡆⠀⠀⠀⢸⣿⠇⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣄⣀⣠⣿⡿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢹⣿⣿⣿⣿⡿⠁⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠀⠈⣿⣿⣿⣿⠃⠀⢀⣾⣿⠏⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⢹⣿⣿⣿⣷⣶⣿⣿⡟⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⢸⣿⣿⡻⣿⣿⣿⣿⡇⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⢀⣼⣿⣿⡇⠸⣿⣿⣿⣿⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠘⠛⠁⢰⣿⣿⣿⣿⣿⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⣿⡄⠀⢰⣿⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⣿⣿⠃⠀⠈⠛⠀⢻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠈⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⣀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡤⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠉⠃⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⡄⡨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣦⡌⠀⠸⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠚⠈⠀⠈⠂⠐⠂⠈⠑⠀⠀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣐⣻⠿⠟⠓⠁⢻⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⠀⠀⡀⠀⠀⣀⣀⠘⠉⢑⠠⠠⠄⣄⣀⣰⡂⠉⠁⠀⠀⠀⠁⠈⠀⠀⢀⡀⠀⣄⣤⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣷⣶⣷⡎⠸⢦⣶⣾⣷⣶⣷⣶⣷⣤⣿⣯⣭⣄⡨⣉⠀⢂⡰⢦⠒⣊⣥⣾⣾⣿⣿⣿⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⣿⣄⡼⠿⠛⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣧⣿⣤⣸⣶⣿⠿⠿⠛⠋⠉⠀⠀⢀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠉⠉⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⡠⠀⠀⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⣀⠄⠀⠖⠒⢀⡉⠙⠤⠤⠥⣱⠀⠀⣀⠈⠐⠒⣒⠂⠀⣀⣊⠈⢧⡈⢢⣽⣷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀⣀⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠚⠀⠀⠀⠂⠀⠀⠀⠀⠈⠄⢀⡬⠩⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠁⢰⠁⢠⡄⣀⡤⠂⠒⠀⠤⢄⠀⠀⠀⠀⠈⠃⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠥⠔⠋⠁⠀⠀⠀⠀⠀⢀⠱⠢⢀⠀⠀⠀⠀⠈⢢⠄⠀⠀⠀⢀⡠⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⡀⠱⠒⠤⠤⠤⠤⠄⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣰⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡏⢀⣿⣿⡀⣸⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠇⣸⠛⠀⡟⠸⣧⠀⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⡎⠁⠀⡔⠁⠀⠀⠀⠀⡿⠘⠃⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠸⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡧⢤⡆⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⢻⣯⢹⣴⠀⠀⠀⠀⠘⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⠋⢣⠛⣿⡆⢿⠀⠀⡀⠀⠀⠸⣿⣇⡠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣶⡌⢈⡝⢋⡄⢀⠀⢀⡌⢳⠻⣆⠀⡇⠀⠀⢀⣍⠣⠙⢶⣿⡀⣠⣏⠷⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣔⣁⣀⡀⠀⠀⢿⡿⠊⣰⣿⣿⠂⠀⣾⠃⡾⠀⢸⡷⣧⡀⣠⣾⡯⣀⠀⠀⡙⢷⡿⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣸⠟⠻⠶⣽⣤⣶⡟⣡⡾⣿⢣⠇⠀⣼⣿⠀⡇⠀⢾⣧⣧⢹⠙⢿⣿⠸⣦⡀⠈⢮⣥⠤⡀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠉⢟⣻⡿⠼⢧⣿⣀⣸⣿⣾⣼⣥⠀⢸⣿⣿⣎⣆⣀⣾⣧⡽⠟⠋⠉⠙⠲⣤⠀⠀⠌
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠎⢸⡟⠀⠁⠁⠀⠀⠀⠀⠀⣾⢸⡗⠁⠀⠉⠉⠁⠀⠀⠹⡇⠀⠖⠊⠁⠀⢿⣦⡔
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢰⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣭⢺⡇⠼⠉⠁⠀⠀⠀⠀⠀⣻⣀⣀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⣗⡤⢾⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢽⠇⠀⠀⠀⠀⠀⠀⢀⠂⠀⢀⠞⠉⡦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠐⠒⠂⠒⠒⠚⠒⠈⠓⠦⠶⠶⠖⢒⣒⠇⡀⢠⠞⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⣤⡷⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣧⣴⣶⣦⡀⠀⠀⠀⢰⣦⠀⠀⠀⡠⠎⠀⠀⠀⠉⠳⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠹⣿⣿⣷⠀⠀⣾⣯⣷⡀⢠⠁⠀⠀⠀⠀⠀⠀⣹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣼⣿⣿⠸⣿⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣴⣶⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡹⣿⣤⠟⠁⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣏⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⣿⣿⣌⠻⣶⣤⣄⡀⠀⠀⠀⢀⣭⣿⣿⣿⡿⠿⠿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢘⣾⣿⣿⣿⣷⣬⡛⢿⣿⣆⢀⣴⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⣋⣿⠿⣿⣿⣿⣿⣿⣧⣙⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⢿⣿⣯⣿⣿⣻⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽
⠀⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠋
⣀⣤⣴⣿⣿⣿⣿⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠈⠹⣿⣶⣶⣤⣬⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈⠙⠿⣿⣿⣿⣧⢵⡀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄
⠀⠀⠀⠀⠀⣰⣿⣿⡿⢯⡻⣦⡀⠀⠀⠀⠀⠀⠀⣽⢦⡀⠀⠀⠀⠀⠀⠀⢀⣀⡴⣺⣿⣿⣿⣶⣶⣮⣄⣀⡆⢹⣴⣼⠗⠁
⠀⠀⠀⠀⢸⣿⣿⣣⣿⣽⣿⣷⣯⣷⣆⣤⣄⣿⣀⣵⠟⠻⡄⠀⠀⠀⠀⠂⠀⣀⡠⠥⠶⣿⠿⣬⠽⢿⣹⣷⠟⢿⣯⣾⣧
⠀⠀⠀⠀⣼⣿⣱⣿⠿⠿⠛⠉⢁⣿⡿⠟⠋⠻⠟⠁⠀⠙⠟⢶⣖⣒⠾⠋⠩⠛⠋⠉⢻⣿⣿⡿⣿⣿⣶⣶⣿⣿⣿⣇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣛⣛⣛⣙⣛⣛⣛⣿⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⡤⠶⠶⠶⠤⠦⠶⠤⠶⠶⠦⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⣛⠩⠅⠒⠛⢉⣀⣤⣒⣦⣂⣠⢀⣤⡴⠶⡶⢖⣠⣤⣀⣀⢀⣉⣛⠻⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⣩⣴⠾⠋⠁⢀⣀⠔⣩⣴⠖⢈⡥⣌⣩⡿⢶⣻⣿⣷⣿⣾⡅⠉⠙⠏⡏⠙⣻⢷⢻⡲⣄⠈⢽⣦⣄
⠀⠀⠀⠀⠀⠀⣰⣟⠴⠛⠛⡇⣠⣴⢔⣺⣔⣺⣏⣵⣲⡿⠚⢥⣞⡫⠽⣧⣻⡤⢶⣝⢷⣤⣀⣀⣄⡀⢀⣴⢘⡣⠛⢿⠓⣌⢻⣕⣆
⠀⠀⠀⠀⠀⢰⣿⠏⠀⢠⣎⣹⠥⡙⢢⠌⢛⣗⣲⣿⣷⢶⡶⢤⡾⢻⣷⣞⣱⣯⣟⢋⣹⡏⠘⣿⢿⣿⣷⡿⠽⣗⢊⡽⢗⢹⠈⠊⣾⡀
⠀⠀⠀⠀⠀⢸⣿⣧⡘⢥⡼⢛⣿⣳⣿⣟⣯⣝⣂⣿⣭⡷⣊⢿⣿⢓⣶⡛⣿⡖⣴⣋⣁⡶⠶⠭⣿⣩⡵⣿⣟⣻⣛⣣⣹⡮⢷⡴⣿⡇
⠀⠀⠀⠀⠀⢸⡇⢻⣿⡲⢿⣻⡿⢿⡶⣤⣉⣂⣉⡥⢾⣛⣶⣥⣿⣿⣶⣬⣿⠽⣖⣿⠭⠄⣀⡤⢺⣗⣾⣭⣠⡾⢿⡿⢿⣭⣯⠞⣽⠁
⠀⠀⠀⠀⠀⠈⣿⠈⠀⠙⡛⣶⢭⣉⡘⠶⣞⣛⠆⣼⠻⣶⣷⡾⡾⠿⠏⣾⠿⣿⣷⡞⣏⣿⣉⣳⠮⣗⣒⣾⣫⠽⣟⣿⡷⡟⡏⢠⡿
⠀⠀⠀⠀⠀⠀⢹⡇⠀⡄⠁⢸⠐⢮⣍⣛⠒⠶⠭⢥⣶⣿⣽⣧⣬⣷⣤⣈⠇⣉⣾⣽⣼⣷⣶⣿⣿⡿⠿⠿⠛⠛⢹⠋⢀⣿⠃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⢻⠀⠀⠀⠐⠂⠬⠭⢑⠒⠂⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠐⠒⠊⠁⠀⠀⠀⠄⡼⠁⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠒⣉⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣈⣉⠑⠒⠒⠒⠒⠒⠛⢉⣉⣠⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⢸⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠙⠁⠀⢇⠀⠀⠀⠘⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢆⠀⠀⠀⠘⣧⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢦⠂⠀⠀⣆⣹⡄⠀⠀⠀⣰⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⡇⠀⠀⢿⣿⠇⠀⠀⢰⣿⡇⠈⠀⡄⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢸⣿⡇⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⣧⡀⠀⠀⠀⠀⠀⡠⠒⢄⡤⣂⢀⡀⠀⠈⠉⠀⣠⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢹⠆⠀⠀⣀⣀⠔⠁⠀⣀⡀⡁⢘⡄⠀⠀⠀⠐⠋⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢼⣿⡏⠀⣾⣿⠟⡿⣽⠟⠁⠴⢦⡀⠀⠀⢃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢀⡧⠛⠂⠈⢉⣷⣤⣿⣿⣿⣿⣿⣿⡅⠀⠀⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡔⠀⠀⡀⣀⢠⣼⣿⣿⣿⣿⣿⡇⠉⢿⣿⠀⠈⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡅⣀⣆⣣⣽⠾⢻⣿⣿⣿⣿⣿⣧⣬⣿⣟⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⢸⣿⣿⣿⣟⠉⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢸⣿⣿⣿⣿⠁⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣃⡀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡻⡟⠢⠚⢿⣿⣿⣿⣿⣿⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣗⣷⠀⣈⣹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡚⠉⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠐⠀⠥⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⢀⡤⠚⢁⠀⢈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⢠⠾⣅⣀⣀⠀⠀⠀⠀⠂⠀⠀⡀⠀⣀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⢻⣀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠒⠧⠤⢬⣄⣀⣆⠀⠀⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⢀⠎⠀⠉⠙⠒⠒⠲⠤⠤⢤⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠲⠤⠤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄
⠀⢸⠀⠀⠀⠀⢀⣠⡤⠤⣄⠀⠀⠀⢀⠉⠉⠙⠓⠒⠲⠶⠤⠤⣄⣀⣀⣀⣀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⢤⣤⠤⠤⠞⢳⡄
⣰⠋⠙⠒⠒⠋⠉⠀⠱⡀⠈⠓⠒⠚⠉⢧⡀⣀⡴⢲⠶⡄⠀⠀⠀⠀⢀⣈⠉⠉⠉⠙⠒⠒⠒⠶⠶⠤⠤⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣜⠁
⢧⡤⠖⡲⠖⢲⠤⢤⣀⣉⣢⣄⣀⠀⠀⠀⢉⣡⠴⠋⠀⠑⢤⣤⠴⠊⠉⢻⡙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⢣
⠀⢠⠞⠲⢤⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⣤⣤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠑⢤⡈⠙⠒⠲⠤⠤⠴⠒⠊⢉⡠⠋⠀⠙⢦⠀⠀⠀⠀⡴⠛⢦⡀⠀⢀
⠀⠘⠦⢤⣸⣄⣀⣀⣀⣤⠤⠤⠖⠒⠚⠛⠻⠤⠤⣤⣀⣘⣟⢦⣀⣀⣠⠤⠖⠒⠛⠓⠢⠤⠤⠤⠤⠤⣶⣉⣀⣀⠀⠀⠈⠳⢄⣠⠴⠁⠀⠀⠉⠙⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠲⠦⠤⠄⣀⣀⣀⡀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠀⠲⢋⡉⠉⠉⠉⠛⠒⠒⠒⠦⢤⣀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠚⠦⠤⠤⠤⣶⣖⣒⣋⣉⠉⠀⠀⠉⠉⠉⠉⠙⠛⠲⢤⡀⠀⠀⠀⠀⢀⡜⠋⠙⡶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠒⠒⠒⠒⠶⠶⠤⠬⠷⠦⣄⣠⠮⠤⠤⠚⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣠⠴⠃
⠀⠀⠀⢀⡔⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠈⠧⣤⣀⣱⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢦⣀⠀⠀⠀⠀⣠⠶⠒⣦
⣰⠒⠒⣦⡀⠀⠀⠉⠃⢀⣤⠶⠛⠉⠉⠁⠀⠀⠉⠉⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⣠⠴⢊⠩⠷⣄⠀⠀⠷⠶⠖⠋
⠙⠶⠚⠉⠁⠀⠀⢀⠖⠋⢀⡠⠤⠒⠒⠒⠒⠒⠂⠤⢄⣀⡀⠈⠉⠓⠲⠶⠦⠴⠶⠒⠛⢉⣁⠠⠐⠊⠁⢰⠊⠁⡠⢖⡩⢷⡀
⠀⠀⠀⠀⠀⠀⣰⠋⡠⠊⠁⢀⡠⠤⠄⠒⠒⠒⠒⠂⠤⠤⣈⠉⠓⠒⠒⠒⠒⠒⠒⡋⢉⣤⠤⠖⠊⠉⢀⣡⣴⡿⢊⣥⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠛⢧⣤⠞⠁⠀⠀⠀⢀⠤⠤⣄⠀⠀⠀⠈⠳⠤⠚⠁⠉⠉⠀⠀⡀⠀⡠⠤⠆⠉⠉⠩⠟⠁⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⡠⠗⠀⠀⠀⢁⣀⣀⢉⣉⣁⣒⣒⣒⣂⠀⢉⣁⡠⠴⠒⠒⠋⠁⢀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠊⠁⠀⣀⠤⢒⣩⠵⠶⠒⠒⠛⠓⠶⠦⢤⣤⣀⣀⣀⣀⣀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣊⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠏⠐⠂⠒⢄⡀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⢢⡀⢀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣀⡀⠂⢀⠀⠈⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⢀⠀⠀⡫⠵⠀⢤⡔⠌⠀⠀⢱⣦⡈⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢼⠡⢠⡶⠋⠀⢀⣀⠀⡀⠀⠁⠀⠀⠀⣸⠃⣝⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢄⠑⠃⠈⠃⠠⠄⣠⡖⢀⡴⣻⠟⠂⠀⣀⣀⣼⣿⠎⣩⠇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠈⠁⡤⠛⠀⢘⣇⣴⣀⣤⣾⣿⣿⣿⠃⠀⡟⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠘⠹⠿⢿⠿⠟⠛⠉⠀⠀⠀⣇⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠀⠀⢀⠰⡀⠀⠀⠀⠀⠀⠀⡏⠀⢢⠈⠉⡇⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⡠⠴⠄⡁⠀⠀⢀⠀⠀⠀⡠⠀⢸⠀⢆⠸⠀⠈⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣂⣤⠖⠁⠀⠀⠀⠁⠀⠀⠶⠀⠊⠀⠈⠢⡁⠈⠂⠀⠀⠸⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣘⠁⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣀⣀⠀⢀⠀⢀⢱⣀⣀⣀⠀⠁⡀⠁
⠀⠀⠀⠀⠀⠀⣠⠃⢃⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢀⡀⠀⠀⠀⢔⠛⣻⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⡴⠁⠀⢸⠀⠀⠀⡐⠸⠀⠀⢀⡝⢶⣄⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢒⣒⠒⠒⠒⠀⠀⠀⠀⠒⡒⠞⠋⠉
⠀⠀⠀⠀⡜⠀⠀⠀⠸⠀⠀⡐⠀⡀⡧⠀⠀⠑⣄⠙⠿⣦⣄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⠃⡉⠛⡋⠉⢩⡍⠀⠡
⠀⠀⠀⠰⠁⠀⠀⠀⠀⡆⠰⠁⠀⡇⠱⣄⠐⢄⣀⠁⠀⠈⠛⢿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⢒⠃⠠⢾⡇⠀⠀⢣
⠀⠀⠀⡧⠀⠀⠀⠀⠀⠱⠃⠀⠸⠁⠀⣨⠗⠸⠁⢀⡀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢦⣀⠠⡄⠀⠀⠀⠄
⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⡠⠖⣿⠐⢉⡤⠛⣧⠀⠀⠙⠳⣦⡀⠀⠉⠒⢄⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠘
⠀⠀⣜⣀⡀⠤⠤⢴⠖⠉⠀⠘⠁⡠⠋⠀⠰⠁⠀⠸⡀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⢤⢚⠀⠀⠀⢀⠆
⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠁⠈⠁⠀⠈
⠀⠀⠀⠀⠈⠒⠦⠠⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡶⠋⢠⣤⣤⣀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠃⠈⠉⠉⠁⠀⠀⠀⠀⣻⣯⣽⣿⠀⠈⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠁⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠛⠃⢀⣀⣠⣤⣤⠶⠶⠶⠛⠛⠛⠀⠰⢶⣤⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣤⣤⡶⠶⠞⠛⠋⠉⠁⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠋⢩⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠉⠙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣯⣽⡿⠀⢀⣠⣤⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⢛⣹⡷⠞⠋⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠶⠾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠟⠋⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠐⣾⣟⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠙⠿⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⢦⣄⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⡀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣀⣠⣤⡾⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠰⣯⣴⣆⣠⡌⠉⣩⡉⠀⣠⠀⠀⣈⠉⢉⡁⠈⢉⠉⠉⣉⠀⠀⣀⠀⢀⣀⠀⢠⡀⢀⣬⠉⣭⣄⣶⣤⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⠉⣿⡾⠛⣧⡼⢻⣇⣾⢿⣄⡾⣿⣠⡿⣧⣸⠿⣆⣰⠿⣆⣼⠿⣦⡿⢻⣾⠟⣷⡏⢹⡏⢨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡀⢹⡆⠀⣿⡁⠀⣿⠁⠈⣿⠁⠘⣿⠁⢹⡟⠀⢹⡟⠀⢹⡏⠀⢻⡇⠀⣿⠀⣼⠇⣼⠇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⡇⢸⡇⠀⢸⡇⠀⣿⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⣼⠃⠀⣿⠀⣿⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣷⠀⣿⠀⢸⡇⠀⢨⡀⠀⣿⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⣿⠀⢰⣿⠀⣿⠀⣿⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡀⢿⡀⠈⣿⠀⢸⡇⠀⣿⡀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⣿⠀⢸⡇⠰⠏⢸⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⣿⠀⢸⡇⢰⡆⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⣧⠘⣧⠀⢹⡆⢸⣇⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⣿⠀⢸⡇⣸⠇⣿⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣥⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⡇⠘⣧⠀⢸⡇⠀⢸⡇⠀⣿⠀⣿⠁⣿⠀⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⡇⢻⡆⠸⣇⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⣿⠀⣿⢸⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⣧⢸⡇⠀⣿⠀⣿⠀⠈⣷⠀⢸⡇⠀⣿⠀⢸⡇⠀⠸⠃⢀⣿⠀⣿⢠⡟⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠈⣷⠀⣿⠀⢿⡆⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡆⢸⡏⠀⣿⢸⡇⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⣿⠀⢹⡇⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⢸⡇⢠⡏⢸⣇⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠾⠷⠾⠷⠶⠿⠶⠾⠷⠶⠿⠶⠾⠷⠶⠿⠷⠾⠷⠾⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⢹⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⡟⡆⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣦⢸⢸⠿⡆⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢰⢿⠇⡇⣴⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣿⠀⡼⠹⠘⢸⠀⣷⠀⠀⠀⠀⣶⡰⣭⡆⠀⠀⠸⠈⡇⣇⠏⢣⠀⣿⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⣡⣾⢸⣧⣙⣡⣧⠸⣤⣿⠀⠀⠀⣰⣿⣷⣜⠃⠀⠀⣷⡆⡇⣼⣌⣋⣼⠉⣧⣜⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣼⡟⣷⢧⡄⡇⢹⣥⣨⢣⣿⣾⠀⢀⢴⣿⡏⣿⢻⡦⡀⠀⣷⣿⡜⣅⣬⠋⢸⢰⡹⡿⢿⣧⢿⡇
⠀⠀⠀⠀⢠⡄⠀⢸⣦⣹⣷⠏⣿⠀⣿⣷⢻⡏⣿⠇⡿⢀⠎⣼⢹⡇⣿⢼⢃⣵⡀⢿⠸⣳⢹⡏⣾⣿⠘⣷⠻⣾⣿⢬⡇⠀⢠⡄
⠀⠀⠀⢠⡿⠀⠀⢸⣟⣹⡏⢆⡏⢸⡏⡏⡌⠑⡎⡀⡇⢸⡟⣿⠘⣇⢹⣧⣸⣿⡇⢸⢀⣿⠸⢡⣿⢻⡆⢣⢁⢹⣏⣾⣇⠀⠀⢿⡄
⠀⠀⢠⠏⣧⠀⣠⠇⡻⣿⣱⡟⣾⢸⡇⢣⢻⣄⠁⡇⢿⡜⣧⡇⣶⢻⡀⣿⣿⣼⣣⡟⣼⠈⣰⡟⡏⢸⡇⣧⢿⡆⣿⣿⠹⣄⠀⡸⢹⡄
⠀⠀⢸⢠⡘⣿⣡⣞⢠⡿⠿⢹⢿⢸⡇⣼⣆⠛⠆⢿⢸⣙⣿⢡⢿⣾⡇⢹⡇⣿⠉⡇⡿⠸⠋⣰⢧⢸⡇⣿⠘⠃⢿⡈⣣⣎⣿⢃⡌⡇
⠀⢠⣼⢼⡇⣹⡶⡽⣾⣷⣇⢸⡿⢸⡇⡟⡿⡜⡟⡌⣼⣿⢃⡿⢸⢸⣇⣼⡇⣏⠿⣏⢃⢿⢱⢿⢸⢸⡆⢿⠀⣼⣾⣇⢯⢶⣏⣼⢱⣧⡄
⢀⡾⢻⡏⣱⠏⣀⡰⣻⢿⡟⣧⢿⣸⠃⠀⠁⡇⡷⣱⢋⣿⣏⣷⡼⢸⡿⣿⢱⢟⣄⠹⣎⢾⢸⠈⢀⠘⡇⠿⣾⢹⡟⣿⣆⣀⠻⣟⢸⡟⣿
⣼⡷⣾⡿⣱⢇⣿⠾⣇⣾⠣⢻⣄⡿⢸⣧⢠⠃⣼⠃⢸⣿⡜⠘⢁⡾⢁⡏⡿⣾⠙⣦⡹⣦⢸⣠⣼⡆⣿⢰⡟⠸⣧⣼⠧⣿⡼⣞⢿⣗⢾
⡏⢻⣿⢣⢻⡘⣯⠘⡿⢣⡶⣸⣿⠇⣦⣿⢸⡾⡏⣶⠸⢀⡖⣸⢹⣧⣾⡗⣧⢣⡆⣼⢣⣿⢿⡇⣿⣰⠸⣿⣇⢶⡘⢯⠃⣽⢣⡟⡜⣿⡞
⢷⡟⢸⡸⠞⠄⣩⢞⣡⡞⣰⠿⡏⢸⣇⢻⠘⠀⡇⠙⠘⠏⡟⢉⠘⠛⡏⢣⠉⠘⠃⠛⠈⣿⠀⠁⡇⢸⡇⢻⠻⣦⠷⣌⠳⡍⢠⠳⢷⡇⣷
⠈⢿⡈⢧⠻⣷⣯⣞⣠⡾⢃⢸⠁⢸⡘⡏⣴⡄⣿⠀⡆⠀⢰⠞⠀⠀⠀⠈⠟⣧⠐⠑⠀⣿⠀⣆⢱⢙⡇⠘⡇⡘⣷⣌⢳⣽⣦⢟⡞⢀⡿
⠀⢬⣳⡌⢳⣿⢋⡞⣿⢣⣿⢸⣸⣇⠻⢠⡏⠀⡿⡄⠀⢸⡄⢶⠀⠀⠀⠳⠀⢹⡄⠎⣸⢿⠀⢹⡌⠟⣼⣇⡇⣿⡘⣟⢿⡙⣷⡞⢠⣞⡤
⠀⢸⣿⠇⢾⡻⣟⠠⣿⢸⣻⠸⣿⠘⡆⢸⢰⢰⡇⢰⣸⡿⡇⣾⢷⡆⢾⡆⢸⠂⣷⢰⡇⠸⠀⣌⡇⢸⠁⣿⢣⣟⡇⣿⢤⣿⢿⡁⣸⣿⡇
⠀⢸⡏⢧⡾⡁⣧⠃⣿⣸⢻⡇⢡⡄⣷⢸⢃⡟⢰⢸⡿⣦⢃⡟⠸⣳⢢⣿⠸⠁⡘⣿⢳⡄⢷⡘⡇⣿⢰⡅⢸⣿⡇⣿⠘⣾⢈⢧⡽⢹⡇
⠀⢸⣷⡜⢱⢃⣿⢷⡟⣿⡟⡴⢛⠃⣿⣴⡟⣠⢿⢸⡇⡿⣸⣇⢠⣏⡎⣿⣴⢸⡇⣿⡞⢳⣌⢻⣦⣿⠸⡛⢆⢻⡇⢿⡾⣿⡸⡎⣡⣾⡇
⠀⢸⡟⣸⠀⣼⣿⢀⡁⣿⢸⢱⡏⢸⢫⢏⣾⡟⡘⣼⠉⣵⡿⡏⣾⢏⡗⢹⡅⣼⠇⡿⢧⠎⣿⣦⡹⡝⡆⢹⡞⡇⣿⢈⡀⡿⣧⠀⣇⢻⡇
⠀⠀⢧⠻⢸⠇⣿⢘⢣⠈⣤⣾⠃⡾⠃⣎⢸⢱⢣⡇⣼⠹⡇⢳⡿⢠⢗⡟⡼⠁⣸⣧⠸⡜⡜⢂⢳⠘⣷⠘⣧⣥⢁⡞⡃⣿⢸⡆⢏⡾
⠀⠀⠈⢷⣼⡆⣿⠸⡾⣇⢿⡘⣸⢃⠸⡹⠘⣿⢸⡿⣏⣆⢳⡀⣷⠺⣬⢡⣽⣼⠋⣿⢿⡇⣿⠸⢾⠀⡘⣇⢇⡟⣸⣣⢇⣿⢸⣧⡾⠁
⠀⠀⠀⠀⠉⠃⠈⢧⡹⣿⣮⢡⡏⣷⡀⣧⡄⢻⠘⠁⣿⢙⣦⠳⣌⠷⠏⢣⡿⣱⡏⣿⠘⢠⡇⢠⡼⡄⣧⢻⡈⡿⣿⢋⡾⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠐⣦⡻⣍⣮⣾⡇⢿⣿⠘⠇⢸⠀⣷⠸⡎⣿⡆⠹⣷⣴⠏⢐⣯⣹⠇⣧⢸⡇⡞⠃⣿⡹⢸⣧⣽⣱⠟
⠀⠀⠀⠀⠀⠀⠀⢫⣳⡹⡏⢿⠀⡟⢸⣠⢰⣾⢰⡏⡄⣿⠘⣷⠀⢻⡏⢺⣘⠃⣿⢰⢹⠸⣇⠇⣄⠃⣷⢸⡇⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣥⢻⣾⡄⡇⣾⡿⡄⢿⡘⣧⡇⣿⠃⡁⣾⢸⣷⠸⡏⢸⣿⢾⣸⢰⡟⢸⣿⢰⣶⢸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣏⣣⢷⠉⡇⢷⢰⣄⠻⠇⣿⢠⣿⡏⣾⠛⣧⢻⢹⣿⠸⠏⣨⡄⣷⢸⢀⡏⣜⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣎⠳⣏⢸⠸⣘⢧⠀⣿⠸⢹⣧⡟⢀⢹⣦⡟⣿⠀⠾⢹⠀⡇⣸⠟⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⡻⠄⢿⡀⡆⣿⡼⣼⢹⡇⡿⢸⣿⡕⡿⢸⣠⡟⠰⢛⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣍⣃⠘⣷⡉⢾⡕⣣⡟⣩⣼⢃⣉⣭⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣾⣷⡟⢞⣵⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⣞⡽⠃⣀⣀⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣹⠀⠀⢠⠀⢀⣭⠶⠛⢉⡵⠟⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡿⠀⢠⣾⢠⡞⠁⠀⢠⡟⣠⠶⢲⡿⠃⢀⡤⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠛⢁⡴⢋⡏⣾⠀⠀⠀⢸⡞⠁⢠⡟⠀⠰⠾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢠⣏⠄⢸⣇⣿⡕⠀⠀⠈⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣇⢸⣿⣶⠀⠻⠏⠀⠀⠀⠀⣠⠞⢡⣶⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠘⢦⣏⠀⠀⠀⢀⡴⡖⠀⢰⠏⣠⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⢠⡄⡾⠰⢁⡀⢸⡟⠁⣸⠁⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⢰⠃⢷⠇⠀⠛⡆⢀⠀⣠⡿⠞⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⢸⠀⠈⠀⠀⠀⠷⡿⠀⡀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⢣⡀⠀⠀⣀⣴⣃⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣤⣤⡴⠶⠞⠋⣠⣶⣶⣄⠙⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⡗⠀⠀⠉⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣈⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣈⣁⣤⣤⠶⠶⠞⠛⠛⠋⠉⠉⠉⠀⠘⠛⢶⣤⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠷⠾⠛⠛⠉⠉⠀⠀⠀⠰⣷⠆⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠙⠿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⣾⣾⣷⠄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠐⢿⡾⣿⣃⣤⠶⠞⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠷⠶⠶⢦⣤⣤⣄⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠿⣷⣿⠗⠀⠀⠀⠀⠀⠀⠀⠰⣶⠄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣟⣉⠈⢛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠶⠖⠀⠶⠶⠞⠛⠛⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠛⡛⢋⣉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡾⢻⣧⡾⣿⣀⣾⣧⢠⣾⡆⣰⣷⠀⣾⡆⢠⣷⡀⢠⣷⡀⣸⣷⣠⣿⣆⣾⢷⣼⠿⣾⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣇⠘⣿⠀⢸⡟⠁⢻⡟⠈⣿⡏⠸⣿⠏⢻⡿⠉⢷⡿⠈⣿⡏⠘⣿⠁⢹⡏⠈⣿⢀⣿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⠀⣿⠀⠘⣧⠀⢸⡇⠀⢻⡄⠀⣿⠀⢸⡇⠀⢸⡇⠀⣼⠀⠀⣿⠀⢸⡇⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢹⡆⢹⡆⠀⣿⠀⢘⡃⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⠀⣿⠀⢸⡇⢸⡇⢸⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣇⢸⡇⠀⢿⡀⢸⡇⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⠀⣿⠀⣸⠇⢼⠇⣿⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣿⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⢠⣿⠀⣿⠀⣤⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⡄⢿⡄⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⢸⣧⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⣿⢰⡏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢨⡄⠀⣿⠀⣿⠀⠘⣧⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⢰⡏⢸⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠘⣧⠀⣿⠀⢻⡄⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⢠⡟⢸⡇⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⡄⣿⠀⢹⡆⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⠛⠀⢸⡇⢸⡇⣸⠇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸⡇⢻⡆⢸⡇⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⢸⡇⣿⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣸⡇⠘⣧⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⢸⡇⣿⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⢷⣀⢀⡶⣠⠄
⠀⠀⠀⠀⠀⢈⡻⠿⠟⢿⣦⣤⣶⠖⠂
⠀⠀⠀⠀⠀⡻⣦⣾⣤⣾⣿⣿⣿⣦⡀
⠀⠀⠀⣀⣾⢂⣴⣷⣿⠿⣿⣿⣿⣿⣷
⠀⠀⢰⣿⣿⠋⠉⠁⠀⠀⠸⢿⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣙⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣛⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⣫⣠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⣵⣆⣽⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣼⣿⠻⣯⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢩⣿⣷⢉⣭⣝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣬⡤⠹⢛⣥⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣤⣤⡙⢻⣿⣿⡟⢛⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡘⣿⡿⣷⣿⣿⠏⣁⡘⠿⢋⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠻⠷⢶⣻⣯⣅⠿⣿⣿⠆⣛⡛⠋⣼⣿⣿⣳⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣡⠘⠿⠏⢿⣟⠺⣦⡰⣶⠽⣿⡟⣠⣤⣤⠻⠿⠟⢹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⢰⡶⠻⣿⣿⠆⢴⡶⣶⣏⢴⡛⢛⣋⣉⢲⣿⣆⣻⣝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡾⣿⡿⢿⣷⣶⣿⣿⣼⣧⢿⣿⡿⢇⣹⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⡐⢿⡇⠀⠈⠙⠻⠿⠿⢿⡈⣭⣥⣬⣝⣿⣿⡿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠈⢉⣿⢿⣿⡟⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠐⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⢿⣧⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⡿⠘⡀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡟⠜⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠰⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠐⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠁⠀⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠻⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡴⠇⠀⠀⠀⠀⢀⠈⣆⣀⣀⣤⠤⠖⠀⢂⡟⠀⠀⠀⠀⠀⠀⡀⢳⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠿⠶⠶⠖⠛⠛⠋⢩⣴⠏⠀⠀⠀⣀⡀⣄⣼⣧⣴⣶⣶⣿⣿⣿⡇⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡷⠶⠖⠛⠛⠻⠶⠿⠞⠛⠛⠉⠉⠁⠀⣤⣸⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⣿⡆⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣿⠀⠀⠀⠀⠀⢀⡏⡇⠰⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⠀⠀⠀⠀⠀⡼⢠⡇⠀⡇⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠁⢹⡄⠀⠀⢀⡼⠁⡄⡀⠀⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⡀⠛⠶⠶⠋⢠⠞⢠⠃⠀⠟⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠸⣿⠀⡇⠀⢇⠀⢰⠃⢠⠏⠀⠀⡇⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⢦⡀⠀⠀⡏⠇⢡⡀⢸⠀⠂⢠⢃⣴⡖⠀⡇⣸⠁⠀⠀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡆⠀⡀⢻⡀⢀⡇⢸⠀⠃⠘⢀⠀⣧⡞⢹⠁⠀⡇⣿⠀⠀⠀⡟⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠃⡇⠀⡃⠈⠷⠼⠃⡘⢀⡴⢻⡟⠙⣿⠁⠸⡆⠀⡇⢻⡄⣀⣼⠃⡇⠀⢹
⠀⠀⠀⠀⠀⠀⢠⠏⢰⠁⠀⢁⠀⠀⠀⢠⢣⡟⢠⡏⠀⠀⢹⠀⠀⢻⡄⠰⠀⠉⠉⣁⠀⣃⠀⢸
⠀⠀⠀⠀⠀⠀⡞⠀⡦⠀⡆⠀⠱⣄⠀⠈⠘⡇⢸⡄⠀⠀⣾⠀⠀⠀⢻⠀⡆⡰⣱⣿⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⡇⠀⣀⠈⢣⠀⠀⢻⡀⠙⠶⠞⠁⠀⠀⠀⡾⢀⠔⣱⠏⡿⠀⡛⣰⠃
⠀⠀⠀⠀⠀⠀⢹⡀⢳⠀⢰⠀⣿⡆⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢼⣁⣤⠞⠁⢰⡇⢠⠃⢿⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠈⢷⡈⢆⠈⢧⣿⠹⣤⣀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡾⢁⠆⠀⠈⢛⠋⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡌⢢⡈⠻⡆⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣡⣤⡄⢀⡠⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⠦⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣩⡴⠛⠐⣩⠞⠁
⠀⠀⠀⠀⢀⣤⣤⣄⣀⠀⠀⠉⠳⢦⣌⡳⣦⣄⣀⣀⣀⣀⣀⣠⣤⠴⠖⠚⣋⣡⡤⠖⠋⣁⣀⣤⡤⠶⢶⡄
⠀⠀⠀⠀⣿⠀⢄⣀⠉⠛⠲⢦⣄⣀⠀⠉⠛⠲⠿⠿⠭⠭⠴⠶⠶⠖⢚⣋⣭⠴⠶⠚⠋⠉⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣿⠀⢀⠀⠉⠒⠀⠀⠤⣈⡙⠲⠦⣤⣀⣀⢀⣀⣤⠶⠚⠋⠉⠀⡀⠀⠀⠐⢀⣀⠤⠒⠂⠀⣀⣼⠁
⠀⠀⠀⠀⠻⢦⣀⠀⠉⠐⠂⠤⠀⠀⠀⢉⣠⡤⠞⠛⣉⠡⠄⠐⠂⠀⠀⠀⠀⠀⠀⠈⠁⣀⣠⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣤⡤⠶⠛⠉⠀⠀⠀⠉⠀⠀⣀⡠⠤⠖⠀⠀⣀⣤⣴⣖⡋⠉
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⢁⡠⠄⠂⢀⡠⠔⠀⠤⠂⠉⣀⣠⡤⠶⠚⠋⠁⠠⢀⡉⠉⠛⠲⠦⣄⣀⡀
⠀⠀⢠⡶⠛⠻⣍⡀⠀⢀⡈⠄⠒⠂⠉⠀⠀⣀⣠⠴⠶⢯⣉⣀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠂⠀⠀⠉⣹⠿⠶⠦⣄
⠀⢠⡟⡼⢬⡑⡌⣷⠀⢀⡀⠤⣒⣮⠽⠖⠋⠁⠀⠀⠀⠀⠀⠉⠉⠛⠲⠦⣤⣀⡀⠀⠀⢀⠀⠀⠀⣼⠡⢊⣑⡀⠈⣧
⠀⡸⢧⣳⣬⢔⣱⣧⡤⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠦⣄⣀⠰⡇⠠⣘⢀⡱⣠⠏
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠐⡐⢠⠀⣀⠀⠈⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⠀⠀⠀⠄⠀⡀⠐⠆⠀⠀⣄⠀⣠⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⠀⠁⢀⠀⠀⠀⠆⠀⠄⠈⠃⠈⠀⡄⠀⠈⡄⠀⢀⡄⠈⠀⠒⠰⠃⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠠⠐⠀⠀⣀⠐⠈⠀⡀⠐⠄⠀⠀⠈⠀⡀⠄⠃⠀⡀⠠⠆⢢⠄
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⡀⠆⠀⠀⠘⠈⠀⠀⠀⠈⡀⠀⣀⠀⢈⠈⡇⠙⢁⠀⠁⠈⡡⡯⡻⢀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠰⠀⠂⠂⠄⢀⢌⠁⠀⠀⡀⣀⣰⢨⣄⠀⣅⠊⠀⠃⠀⣂⣀
⠀⠀⠀⡀⢀⠄⠠⠘⢀⠀⠀⢐⠂⡠⠌⢠⡶⠐⣐⡃⡄⠁⠠⠃⠄⠊⡄⠀⢍⡀⡄⡏⠿⡄⠤⣵⢌⢡⠆
⠀⠀⠀⠀⠀⠃⢠⠑⠈⠀⠰⢤⠀⠄⡀⠀⠁⠠⠈⣅⠀⣬⠠⢏⠌⠀⣻⡀⢘⢿⡍⠁⡌⠃⠃⡀⢠⡃
⠀⠀⠈⠋⠀⢠⠀⠄⠀⢠⠤⡐⠘⢠⡉⣀⣤⠳⠗⠂⣡⠉⠐⡄⣯⠈⡠⢶⢤⠈⣛⠛⡶⠀⢹⠿⠚⢁⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠀⠸⠇⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢻⠀⠀⠀⠀⡇⣸⠃⢀⠀⢠⡈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠁⠀⠀⣰⠃⠀⠀⡞⠀⢈⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠸⠁⠀⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠚⢸⡆⠀⢰⡆⠀⠀⣠⠀⢀⡀⠀⠈⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣞⠀⠀⠀⢻⠀⠀⢸⠆⠈⠁⠀⠀⣇⡔⢰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠋⢠⠀⡄⠀⠀⠀⣀⠀⠀⠀⣼⠀⢉⠀⢸⣧⡇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠇⣼⠃⠀⡠⠀⢀⡀⠆⠀⢀⢉⠀⠀⣧⠁⠀⠈⠀⠏⠏⠀⢰⠀⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣇⠐⡇⠀⠀⠁⠀⢸⣷⠀⠃⣾⠀⠀⠂⠈⠀⠀⠁⠆⠀⠀⠀⢸⢃⡁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠁⣸⡆⣦⢠⠀⠀⢠⠀⠁⠀⠀⠛⠀⢀⢰⠀⡄⠀⣆⠀⠁⠀⠀⠀⠸⠃⢸⠆⠋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⡀⣠⠠⠈⡃⠞⠀⠀⠁⡀⠀⠀⢀⠀⢀⠸⠆⣆⠁⡞⠑⠀⡀⣤⠀⠀⣠⠀⠊⠀⣠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⢳⣉⠀⠁⠑⠰⠓⣠⣷⠇⠀⠰⡌⠂⠰⠀⡄⠉⠀⠇⠐⠾⠁⣸⠀⣶⠀⣶⠄⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠋⡤⠛⠐⣷⡀⠜⠿⣿⡏⣻⣧⡐⠁⢀⠀⢸⡇⠀⠀⠀⠀⢠⣘⡧⣶⠛⣷⣤⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⠘⠄⢀⡀⠀⣄⣀⠈⡄⠙⠉⣿⡄⠉⠀⡠⢁⡯⠀⢠⠀⣸⡟⠁⠇⠞⠟⢁⣀⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡄⠙⣻⡆⢨⢉⣠⣿⣾⣏⢀⣔⢃⠈⣁⠀⡀⠇⠋⠛⠀⠀⠀⠋⠅⢡⣴⡀⢸⡇⢦⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠓⠉⠙⠛⢾⡌⠙⣿⠇⠁⠈⠛⠙⢷⢈⣤⣅⠀⣄⢂⣼⣷⡀⢰⣾⠆⡿⠃⣾⣿⠆⠘⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⢠⣿⡿⠂⠉⠁⠒⢶⡄⠀⣤⣼⡿⠋⠁⠀⠈⠙⢿⣿⡆⠀⠀⠆⠉⠘⠳⣦⣤⡠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣗⡒⣶⠞⣻⣟⡀⢀⠀⣀⣼⣮⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⣸⣆⠀⠀⢀⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠯⢉⣛⣛⡽⣿⣦⢠⠞⣿⣿⠦⡀⢀⡀⣰⡿⠿⣿⣿⣯⣾⣽⣷⣀⣫⣾⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠦⠤⠔⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠈⠁⠀⠀⠤⠀⠁⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠎⠀⠀⠀⠐⠛⠋⠀⠀⠀⠲⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠴⣦⣤⣤⣤⡀⠤⠒⠊⠁⢈⣽⠍⢻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠁⠀⠀⠀⠀⠈⠉⠹⣝⠐⢯⠀⠔⢨⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⣶⠀⠇⠃⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢢⣻⣦⠀⠰⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⢀⣧⠳⣀⠀⠄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠂⠉⠉⠉⠉⠉⠛⠢⣌⠲⢌⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠐⠒⠲⣄⡊⠁⠉⠙⡆⠀⠑⢄⡉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠁⠀⠀⡁⠀⠀⠀⠄⠳⠣⠔⠂⡸⢡⠟⢆⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠂⠀⡰⠂⠙⡦⢤⡐⢄⠠⣦⠃⣰⢓⠛⠀⢸⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢹⠀⠀⡇⠀⠀⠘⠂⣈⠀⣳⣷⣶⣋⠯⢔⣐⠫⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⠀⠀⠑⠶⠖⠒⢶⠞⡭⠗⢻⣍⡑⠶⡃⠈⠀⢰⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⢧⠀⢰⠁⠀⠔⠰⠀⢰⡜⠈⢻⢁⠁⡈⢦⠴⠂⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣇⠀⠀⣣⡀⠳⣀⠀⠀⣠⢎⡄⠖⠈⡄⠀⡅⢠⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢝⢢⣤⣤⠍⠩⠿⠶⠷⠿⠿⠿⣷⣄⣰⠿⣦⣈⠉⠉⠱⢠⣀⣀⡣⠓⠦⠚⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠀⠙⠻⠾⢏⡗⠢⣤⣤⢀⣀⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠉⠁



⡶⡶⢶⢶⢶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⠶⣶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣾⣿⣷⡶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⠿⠟⣉⣩⡭⠤⠴⠶⠒⠖⠶⠦⠬⢭⣉⣛⠻⠿⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡾⢛⣩⠴⠚⠉⠁⣴⠶⠖⠀⢰⣿⣆⢸⡇⠀⠴⣦⣬⣉⠓⠦⣍⡛⠿⣼⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢺⡿⢟⣡⠞⢩⣄⠀⢻⣆⠀⢻⡷⠛⠀⢸⡇⠹⣾⡇⠀⢀⣾⠁⠁⢠⡶⠀⠉⠲⣌⡳⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢋⡴⢋⣤⠄⠈⢻⣶⠛⢻⣦⠘⠷⠚⢃⣈⣃⣀⣙⡃⠀⠸⠏⠀⢠⡿⠁⣠⡶⠶⣤⡙⢦⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⣱⠋⠰⠟⢿⣄⠀⠀⠻⠆⣀⡤⢖⣂⣩⣥⣤⣤⣤⣤⣭⣍⣑⡲⠤⣙⠁⠸⣿⠀⠀⠈⠀⠀⠘⣎⢻⣏⣧⣤⣶⣶⣦
⠀⠀⠀⠀⠀⠀⢀⣾⣿⢋⡜⠡⣦⣄⡀⠀⠙⠷⢀⡴⢊⣥⣾⠿⠛⠉⠉⠈⠀⠀⠈⠉⠉⠛⠻⣷⣬⣑⢦⡙⠷⠀⣀⣠⣤⣴⣾⠿⠟⠛⠉⠉⠙⣿⡆
⠀⠀⠀⠀⠀⢀⣾⣿⢃⡞⢠⣄⠀⠙⢻⡆⢀⠴⣡⣴⠟⠋⠀⠀⣠⣴⣴⠂⠀⢺⡿⠀⠀⠀⠀⠀⣉⣿⣷⣾⣿⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣶
⠀⠀⠀⠀⠀⣾⣿⠇⡞⠀⠀⠙⠻⠶⠞⢡⠎⣴⡟⠁⠀⢴⣴⠀⠀⠛⠉⠁⠀⠀⣀⣤⣴⣶⣾⠿⠟⠛⠉⢁⣀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⢰⣿⡟⢸⢁⣶⠶⢶⡶⠶⢀⢋⣶⡟⠀⠀⠀⠈⠉⢀⣀⣤⣴⣶⡿⠿⠟⠛⠉⠀⠀⣤⡄⠀⢸⣿⣿⣷⡀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⣾⣿⠇⡏⠀⠙⠳⢾⣅⠀⡎⣸⡟⢀⣀⣤⣴⡖⠿⠿⠛⠋⠉⠀⠀⣤⣤⠀⢸⣿⣦⣻⣿⡀⠀⢿⣏⢿⣷⡀⠘⣿⣧⣠⠀⠀⠀⠀⠀⠀⠀⣹⣿
⠀⠀⠀⠀⣟⣿⢰⡇⠀⠀⠀⢀⣉⣴⣷⣏⡿⠿⠛⠋⠉⠀⠀⢠⣶⡾⣷⣆⠀⢹⣿⡆⠀⢿⣿⣿⣿⣧⠀⢺⣿⡿⢿⣷⡀⠻⠟⠛⠁⣀⣠⣤⣶⣶⡿⠿⠛
⠀⠀⠀⠀⣿⣿⣸⣧⣤⣶⡿⠿⠛⠋⠉⣀⣀⣀⡀⠐⣿⣷⠀⠸⣿⣇⢛⣩⡄⠈⣿⣿⡀⠘⣿⡏⢿⣿⡄⠸⠟⠃⠈⣁⣠⣤⣴⣮⠿⠟⢻⡏⣿⣿
⠀⡤⣴⣾⡿⠿⠒⠋⠉⢀⣤⣤⡀⠀⢻⣿⡟⣿⣧⠀⢹⣿⣇⠀⢻⣿⡏⣿⣷⠀⠸⣿⡧⠀⠙⠋⠀⣁⣠⣤⣶⣾⣿⢿⠞⠉⠁⠀⠀⠀⢸⠇⣿⣿
⠈⣿⣏⠀⠀⠀⠀⠀⠰⣿⣏⢻⣿⡄⠈⣿⣿⠿⣷⡄⠀⢿⣿⡄⠈⠿⣿⠟⠛⠃⠀⣀⣠⣴⣶⡾⠿⠟⠛⠉⠁⣴⡏⡸⢰⡗⠀⠀⢀⡀⣸⠈⣛⡟
⠀⠘⣻⡄⠀⠀⠀⠀⠀⢻⣿⡌⣿⣷⠀⠸⣿⣇⢻⣿⠀⠘⠛⠃⠀⣀⣠⣴⣶⣾⠿⠟⠛⠈⠁⢀⠀⠀⠀⠀⣼⡿⣱⠃⠘⢷⣤⣤⡾⢃⡏⣼⣿⠇
⠀⠀⢻⣷⠀⠀⠀⠀⠀⠈⣿⣷⣼⡿⠂⠀⠛⠋⢀⣀⣤⣴⣶⡿⠷⠟⠛⠉⠀⠀⠀⢠⡀⡀⠀⠾⠿⠀⠠⡾⠟⣱⢃⠶⣶⣤⣉⠉⢀⣾⣠⣿⡇
⠀⠀⠀⢬⡆⠀⠀⠀⠀⠀⠈⠉⢉⢁⣤⣴⣶⣿⣏⢿⣿⣉⠀⠀⠀⠀⠀⣼⣖⠀⠐⢺⡿⡅⠀⠀⣀⠶⠶⣣⠞⠱⡿⠃⠈⠉⠻⢇⡜⣱⣿⣿⠃
⠀⠀⠀⠸⣿⣄⣀⣀⣤⣴⣖⡻⠿⠚⠋⠉⠀⠀⣈⠳⢌⡛⠿⣶⣤⣀⡀⠈⠀⠀⢀⣈⣀⣤⣶⡿⢛⡡⠞⢩⣜⠋⠙⠻⣶⡄⢀⡾⣱⣿⡿⠃
⠀⠀⠀⠀⠋⠽⠿⠛⢻⣿⣧⡹⣄⠀⠀⢀⣤⡾⣻⡇⠀⣉⠓⠮⢍⣙⣓⠛⠛⠛⠛⢛⣉⡩⠵⠚⡉⢰⣶⣤⣙⣧⡀⠀⠈⣠⢏⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⠳⣄⠛⠉⢻⡟⠀⣰⡟⠀⣰⡆⣤⣬⣭⡍⢩⡉⠁⣤⠀⢾⡟⣉⡀⠙⣦⡉⠛⠛⣠⠞⣡⣞⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣿⣮⡙⢦⣉⠁⠀⣿⣀⣰⡟⠀⠀⣿⠇⠀⢸⣧⣤⣿⡆⠘⣿⣉⣤⠀⠈⣃⡴⢋⣥⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣥⣶⣬⣙⠲⢬⣉⡉⠀⠀⠀⣿⠀⠀⠸⡿⠀⠸⠇⠀⢘⣉⡥⠐⣋⣥⣾⣿⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠨⢿⣿⣶⣦⣭⣉⣓⡒⠲⠶⠶⠶⠶⠖⢒⣚⣉⣭⣴⣶⣿⡿⠟⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⡿⣿⣿⣿⣧⣴⣿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⡈⠁⠀⠀⠀⠀⠀⠀⡞⠠⡊⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢽⠀⠀⠀⠀⠀⠀⢀⣇⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⡄⠀⠀⠀⠀⠀⢸⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣾⡆⠀⠀⢀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢃⠀⠀⠸⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⡆⠀⠀⠇⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡇⠀⠀⠀⡴⠀⠀⠀⠀⡇⠀⣼⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢤⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠃⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⡤⠀⠄⡀⠠⠀⠀⠀⠘⠀⢸⣆⠄⢼⠀⠀⣀⠄⠂⠲⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢈⠁⡄⠀⣄⠀⠈⡢⣀⡀⠀⢀⠆⠀⠀⠀⢀⠱⡊⠠⠒⢁⠀⠱⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡠
⠀⠀⠀⠀⠀⢠⡕⠀⠀⡇⠀⠘⢄⠀⡈⠪⣻⡦⠉⠀⠀⠀⢰⠀⡄⢇⡀⠈⠁⠔⠀⣱⠔⠂⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢁⠁
⠀⠀⠀⠀⠀⡞⠈⡓⠦⡘⡄⠀⠈⠳⡈⠂⠀⠙⣇⠀⠀⡀⠀⠀⡀⢸⠂⠉⠐⢂⡠⣤⠀⠈⠓⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠐⠬⠒⠤⠉⠒⢤⣀⡈⠢⣄⠀⠸⣆⠀⠡⡀⠀⠁⣼⠋⠩⠭⠅⠉⢀⡑⢦⣠⠔⠉⠀⠀⠈⢢⡀⠀⠀⠀⣀⠄⠀⢀⠜
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘⠀⢨⠃⠀⠱⠾⢋⡳⠢⢼⠶⢞⠡⠔⣒⣉⠩⠠⡨⢉⠓⠨⡀⢄⠀⠠⠖⠀⠧⣀⠤⠊⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢠⠃⠀⡞⠀⠀⠀⠀⠀⠀⢹⠃⠒⠴⠲⠅⢀⣀⢁⡀⡁⣁⡀⠁⠜⢦⡙⠄⣀⡈⢂⡡⠤⠂⠉
⠀⠀⠀⠀⠀⠀⠰⡉⠉⠢⣄⡎⠀⢰⠁⠀⠀⠀⠀⠀⠀⡆⠀⢀⠇⠀⠀⢀⡀⠈⠉⠉⢨⣞⠁⡤⠂⠂⠒⠁⠰⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢣⠈⠳⢬⠁⠀⡧⣀⡀⢀⣀⣀⣀⣸⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⡌⠛⢠⠃⠈⠉⠉⠉⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠫⡑⢾⠀⢸⢦⣀⣀⣀⣀⣀⢀⡏⠀⢠⣧⡀⠀⠀⠀⠀⠀⣀⣠⠃⠀⡜⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⠀⡞⠠⢀⣀⣀⣀⣀⢸⢁⠀⣼⣀⠈⠑⠒⠂⠒⠉⠀⣼⡀⢠⠃⠒⢄⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⡀⠀⠀⠀⠀⠀⡟⡌⢠⣇⠈⠙⠒⠶⠶⠖⠒⢐⢇⠁⡼⢢⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣀⡀⢰⠇⠁⢸⠀⠉⠓⠶⠶⠖⠒⠀⣸⣠⠴⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠒⠚⠶⠶⠶⠶⠖⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⡠⠂⠀⠀⠀⠀⠀⢀⠤⣀⠀⠀⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠊⠀⠀⠀⠀⠀⠀⢰⢡⣄⠸⡇⠀⢇⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡘⢤⡀⠀⠀⠀⠀⠀⡇⢯⡔⢘⡇⠀⠀⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠆⠀⠀⠀⠈⢆⠱⠀⠀⠀⠀⠀⢸⢠⣬⠞⠀⢀⣴⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡡⠂⠀⣤⠀⡠⣣⠆⠘⠃⠀⠀⠀⡎⢼⠁⠀⠀⡇⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⡑⡄⣤⠀⢸⢱⠁⠀⣤⠀⠙⠀⢸⢨⡽⠀⠀⠀⢱⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣀⣀⣀⣀⣀⡀⠠⠤⠤⠤⠤⡇⣼⣇⣀⣀⣀⣈⡋
⠀⠀⠀⠀⡠⠔⣚⡭⡵⢶⠶⢶⠳⡳⣚⡛⡚⡛⣚⠉⠉⢹⢱⡟⣒⣶⢒⠒⠒⠭⢭⡁⠒⢄
⠀⣀⠴⡺⣄⡀⠳⢼⣜⣺⠾⠷⠓⠛⠛⣻⠛⠻⠟⡻⣟⠻⠰⠟⠍⠛⢹⡏⠛⣓⣤⠼⠀⣸⠄
⡞⠁⡾⠛⣟⢝⡲⢴⣤⣨⣩⣉⣉⡉⣉⣉⡉⡉⡉⡉⢉⣉⣉⣉⣉⣉⣉⣉⣭⣤⠦⠾⣚⣽⠀⠀⣰⢒⡆
⠹⣶⡙⢤⡟⠭⡂⠀⠙⠋⠊⠉⠉⠻⢿⣛⣟⣉⢉⣙⡋⠉⠛⠛⠛⠒⣒⣒⣒⣋⢭⣭⣷⡟⣳⢞⢡⡿⠃
⠀⠈⠛⠓⣿⡷⣢⠀⠀⠀⠀⠀⠀⠨⠭⠀⣐⣒⣒⡒⠂⠀⠀⠀⠀⠀⢐⣒⣒⡦⣭⣰⡾⡋⣅⠾⠋
⠀⠀⠀⠀⣗⠬⣒⠀⠀⠀⠀⠀⠀⢈⡻⠥⠶⠲⠄⠒⠂⠀⠀⠀⠀⠀⢀⣒⣾⠋⣬⣠⣴⡟⠁
⠀⠀⠀⠀⢳⡭⡒⠅⠀⠀⠀⠀⠀⠀⠨⠭⠄⣐⣒⣒⠂⠀⠀⠀⠀⠀⢐⣒⣛⣿⣯⣒⢽⡇
⠀⠀⠀⠀⢸⠭⣒⠅⠀⠀⠀⠀⠀⠠⠭⠍⢂⣒⣒⠶⠆⠀⠀⠀⠀⠀⢀⣛⡒⠭⠝⣊⢽⠁
⠀⠀⠀⠀⢸⡪⢐⠅⠀⠀⠀⠀⠀⠨⠭⣒⣒⣒⣒⡒⠀⠀⠀⠀⠀⠀⢐⣒⡀⠭⢕⣊⢽
⠀⠀⠀⠀⢸⡫⢖⠭⠀⠀⠀⠀⠀⠀⡯⠴⢒⣂⣀⣐⠂⠀⠀⠀⠀⠀⢒⣒⣒⠩⠕⣒⣽
⠀⠀⠀⠀⠘⣟⢕⠭⠀⠀⠀⠀⠀⠀⣪⠭⠒⣒⣒⣒⠀⠀⠀⠀⠀⠀⠂⠒⣒⡂⠭⣂⣿⠶⢶⣀⣀⣀
⠀⠀⠀⢠⢾⣟⢵⡭⡀⠀⠀⠀⠀⠀⡨⢭⣐⣒⣂⣀⠂⠀⠀⠀⠀⠀⠭⠅⣐⠢⢭⠺⣗⣒⠒⢒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠸⡝⢭⠷⢯⣒⠀⠀⠀⠀⠀⣻⣭⣉⣉⣂⣈⠀⠀⠀⠀⢀⣀⣵⣶⣶⡿⢷⡿⠳⣏⢷⡀⠀⣀⣈⣉⣉⣉⣉⣉⣁⣐
⠀⠀⠀⠀⠈⠒⠭⣒⠤⣍⡓⠲⠦⠤⣤⣍⣉⣉⣉⠈⠀⠉⠉⣉⣟⣛⣫⡭⠷⠮⠭⣭⡭⢯⡽⠉⠩⠍⠉⠭⠉⠉⠭⠭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠭⣝⣒⡒⠲⠶⠶⠿⣿⣟⣻⣿⣟⣛⣛⡧⠶⠶⠚⢛⣯⠭⠿⠭⠭⠭⠥⠬⠭⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠈⠉⣹⣋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡟⣀⠔⠊⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢒⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢐⣀⣩⣤⠤⠄⣤⣀⣀⠀⠠⡄⠤⢤⣀⡙⢅⠢⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣀⡤⠖⠋⢩⠏⠁⠀⠀⠀⠀⠀⠈⠉⠒⢼⡳⡄⠈⠉⠓⠧⠀⢠⡄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢨⠁⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢸⡄⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⠀⠀⠀⠈⠀⠀⠙⡆⠀⠀⠀⠀⠀⢸⠋⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠙⠦⡤⠖⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣄⡀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠈⠑⠲⠦⠴⠖⠋⡁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠄⣤⣠⡤⡤⠼⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠒⡈⠃⠛⢐⠒⠂⠀⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⡀⠀⠀⠻⣄⡔⠁⠀⠀⠀⡄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⣀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⣇⣀⡴⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡔⠀⢨⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢧⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡉⠂⢀⠠⠒⠈⠁⡴⠃⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠊⢁⡠⠤⠗
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⡀⡄⠤⠴⠒⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢊⣡⢄
⢁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⡵⠊
⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠤⠤⠤⠤⠶⠶⠒⢻⡉
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣢⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠉⠙⠒⠐⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣠⣤⠤⠼⢰⣤⣤⡤⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⣸⠀⢸⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠁⠀⠀⠘⠢⠚⠡⠀⠀⡘⠀⠙⠋⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⣀⠀⠀⣀⣀⣀⣤⠴⠖⠒⠃⠀⠑⠦⣤⣀⡀⠀⠀⠀⠈⢢

⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢲⣤⡀⠀⠀⠀⠀⠀⠀⠀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⣱⣿⢻⣧⠀⠀⠀⠀⠀⠀⢀⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿
⠀⠀⠀⠀⢸⣿⠘⣿⡀⠀⠀⠀⠀⠀⠐⢠⠘⣏⢻⢿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢁⣾⡏⠀⠀⠀⠀⠀⠰⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣦
⠀⠀⠀⢰⣼⠀⣤⠛⡇⠀⠀⠀⠀⠀⢰⢿⣿⢃⡀⢸⣶⠂⠀⠀⠀⠀⠀⢸⡇⠋⢀⢸⣯⣔⡇⠀⠀⠀⠀⣾⠉⡀⢸⡇⠀⠀⠀⠠⢠⣠⡾⣿⢠⡌⠹⣷⡀
⠀⠀⠀⢈⣏⠃⡝⣆⣷⠀⠀⠀⠀⠸⣿⠆⢁⡞⣇⠀⢛⡟⠀⠀⠀⠀⠀⠘⢿⣄⢼⣦⣿⡟⠀⠀⠀⠀⠀⣾⠄⣧⣈⡇⠀⠀⢀⣴⠛⢛⣄⣁⣼⣧⡆⡹⠁
⠀⠀⠀⠀⠉⠛⠓⠋⠁⠀⠀⠀⠀⠀⠈⠑⠂⠉⠉⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⠃⠀⠀⠀⠀⠀⠈⠑⠛⠉⠀⠀⠀⠘⠛⠀⠛⠈⠉⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠰⢃⣴⡎⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠴⠀⠠⠀⠀⠀⠀⣶⣄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠄⢀
⠀⠀⠀⠀⢾⡟⢿⣷⡈⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠀⣠⣴⣶⡿⠀⠀⠀⠀⠀⠀⢿⣿⠿⣷⣆⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣴⠁⠀⢀
⠀⠀⠀⠀⣸⡇⠀⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣆⡿⠋⣜⣥⣠⡌⠀⠀⠀⠀⠀⠻⣆⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢃⡴⣿⠠⣶⡿⢻⣿⢀⡴⣯
⠀⠀⠀⠀⠿⣟⢸⡄⡿⠀⠀⠀⠀⢀⣠⣰⣶⠏⣸⠟⢡⠜⠙⠻⢛⡵⠆⠀⠀⠀⠰⡶⣿⢠⡄⢸⣦⠀⠀⠀⣠⣴⣤⡴⠋⣸⡿⢚⡉⣠⢿⡿⠋⣴⠏⡀
⠀⠀⠀⠀⠀⠈⢿⠟⠁⠀⠀⠀⠠⣿⢋⣛⣡⢦⣠⠶⠻⠴⣦⣤⣀⡶⣧⡀⠀⠀⠀⢹⡦⣾⡁⡸⠃⠀⠀⠞⡋⡉⣩⣞⣀⡀⢴⣋⣈⣥⣠⡄⠚⠟⣻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉
⠀⠀⠀⢀⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⣟⣻⣦⢀⠀⠀⠀⠀⠀⣀⠂⢀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠃⠘⣿⡏⢻⣆⠀⠀⠀⢸⣿⣷⣄⡠⢀⣁⣀⣒⣠⣤⣌⡁⠀⠀⠀⠀⠀⠀⣿⡿⣦⣤⣇⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣧⠈⡀
⠀⠀⠀⣠⣦⡿⡁⠘⠉⡇⠀⠀⠀⠻⢿⣟⠛⠟⠛⠻⠿⣿⣿⠛⣿⣤⡀⠀⠀⠀⠀⠹⣧⠈⠙⠻⡿⣦⣷⣦⠀⣂⠀⠀⠀⠀⠀⠀⠀⣿⠻⣷⣿⣦⣤
⠀⠀⢼⡏⠋⣼⣧⠄⠸⣿⠀⠀⠀⠀⠐⠉⢻⣷⣶⡄⢲⣤⣈⡆⠙⢿⠇⠀⠀⠀⢀⢳⣾⢷⡄⠰⣤⣤⣍⣉⠻⡟⢷⠀⠀⠀⠀⢀⣗⠾⣆⢉⡛⠈⠻⠿⣆
⠀⠀⠈⠻⠦⠿⠸⠧⠖⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠧⠼⠷⠽⠷⠦⠟⠀⠀⠀⠰⠾⠿⠋⠁⠠⠶⠾⠧⠉⠉⠳⠤⠌⠦⠀⠀⠀⠸⠿⠆⠠⠾⠟⠓⠛⠲⠼⠇

⢰⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⢰⢶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶
⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢈⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣀⢠⣀⣄⢠⠀⢠⣠⣤⣴⡆⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠝⠋⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠐⠂⠀⠀⠀⠀⣀⡀⡀⣀⢁⡁⢈⣉⡉⠘⠈⠉⠙⠛⠛⢃⡀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣈⣠⡀⠀⢿⢧⣢⠸⠻⠃⢠⠼⠧⠶⡇⣿⡾⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠛⠛⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣏⢫⣾⣿⡻⢟⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⢸⣿⣿⣷⣮⣟⣠⢿⣌⣍⡿⠻⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⢀⠀⢤⣤⣤⣤⡀⣄⢠⣄⢰⠀⣠⣄⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣯⣭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠐⠒⠀⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⡀⠀⣀⡀⡀⣀⡀⣀⣀⣀⣀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠁⠈⠉⠁⠁⠉⠁⠉⠉⠉⠉⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡤⠄⡄⠠⠤⠠⠄⠴⢴⠶⣶⠦⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠶⠶⠶⠶⠶⠾⠷⠾

⣤⣤⣤⣤⣤⡤⠄⡠⠤⣤⡠⣤⡤⠄⠤⡄⠠⡤⡤⢤⣤⠤⠀⢠
⣿⣿⣿⣿⣿⡧⠄⡇⠀⣿⠀⠀⠀⠀⡄⣿⡄⠀⣿⠜⣿⡄⣩⢸
⠉⠉⠉⠉⠉⠉⠁⠉⠈⠉⠉⠉⠁⠁⠁⠀⠀⠃⠀⠘⠋⠀⠉⠙⠁⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢩⡈⠾⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣆⡀⠠⠁⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠔⢀⠤⠞⠀⡛⠢⠜⠛⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠉⠻⠷⡨⠦⠄⠀⠀⣈⡀⠀⠀⠀⠀⡀⠄⠒⠈⡴⠃⠀⠀⠰⠆⠓⡀⠀⠰⡄⠀⠀⠈⠀⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠀⠀⠣⠘⠄⠀⠀⠀⠀⠟⠀⠀⠀⠤⣀⠀⠈⠛⠀⠀⢀⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⢣
⠀⠀⠀⠀⠀⠂⠀⠀⢠⠀⠠⠀⡀⠀⢀⠀⠀⠈⠆⠀⠀⢀⠛⣼⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠢⠀⠀⠀⠀⠀⠀⢠⠀⢁
⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠈⠀⠀⠲⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠸⡆⠀⢠⣧⣄⠀⢀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⣀⣡⣤⡤⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠄⠀⣀⡤⠂⡠⠐⠁⠀⠀⠀⠀⠀⠘⡄⠀⠃⠀⠈⢀⣀⡤⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠋⠉⠐⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠇⠀⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠤⠠⢍⡒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⡈⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⣈⠁⠀⢧⠀⠀⠀⠀⠀⠀⠀⠃⠀⡁⠈⢈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⣄⡀⡄⠀⢠⣤⠀⠀⠀⢀⠊⠀⠊⠁⠀⠀⠐⠈⢳⡄⠀⠀⠀⣀⠀⠀⢸⠇⠀⠀⡁⠒⠒⠀⠀⠀⠐⠢⠍⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠰⡀⠇⢆⡀⠐⠐⢻⡟⢃⣤⡞⠃⠀⠀⠀⡟⢀⡀⢸⠀⠀⠀⠀⠀⠘⣟⠄⠀⠀⡀⠸⠆⠀⠀⠰⢺⠁⠀⠀⣠⡖⠀⠀⣰⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣙⠓⠂⣈⠓⠄⢠⣁⠀⠀⠀⠀⢯⡀⠀⠘⠀⠐⠃⢀⠊⠀⢻⠈⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡥⠀⠁⢀⡱⠎⠀⠀⠀⠀⠀⠉⢙⣢⣦⡤⠴⠦⠀⠐⠁⠀⠀⠀⠀⠈⠢⣀⠰⠆⠤⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠢⡀⠀⠀⠀⠠⣄⢀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠤⠄⠀⠀⠈⠁⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⡀⢹⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠓⠒⠦⣤⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⠱⢄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣆⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢀⠈⣀⠴⢻⡠⡀⠀⠀⠀⠀⠀⠀⠀⠔⣉⠽⠓⠀⢀⣩⢶⣤⡗⣦⠀⠀⠀⠀⠀⠀⢀⡶⠹⠀⢹⣌⢏⡓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣟⢋⢸⣦⣟⢀⡅⠀⠀⠀⠀⠀⢠⣶⣞⠁⠀⡠⠊⡸⠋⠁⠀⠀⣼⡆⠀⠀⠀⢀⠴⠋⢁⠁⠀⠸⠈⠢⠀⠂⠹⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠹⡉⠉⠉⢁⡀⠀⠢⡄⠀⠀⢸⡿⢁⣤⣬⠄⠊⠁⠀⠀⠀⠘⢹⠋⣧⠀⠀⢀⠘⢀⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⡔⣰⠚⢀⠆⠀⠀⢱⠀⠀⠀⢏⢦⠀⠜⡆⠀⡾⣻⡜⠟⠁⠀⠀⠀⢀⡤⠂⠔⠃⡸⡇⠀⠀⠂⠃⠸⠀⢸⠀⠀⡀
⠀⠀⠀⠀⠀⡇⢀⠔⠋⠀⠠⡴⠋⠀⠀⠀⠀⠈⠀⢰⡇⠀⠁⡉⣈⠃⠀⣀⣀⣤⣬⣤⠄⣠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢇⠀⠀⣴⣰⠠
⠀⠀⠀⠀⠀⢁⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⡀⠃⠀⠀⠁⠉⠑⠀⠉⠉⠉⡀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⢀⡀⠖⠀⠀⠈⠈⠀⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠳⠉⠓⠂⠀⣀⠰⠖⠒⣀⠶⠐⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠤⠴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠤⢄⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠁⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠠⠤⡀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⢸⠁⢀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⠓⠄⡀⠀⢧⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⢀⡈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⠟⠃⠀⠰⠟⠛⠛⠦⡈⠐⣄⠃⠀⠀⠂⠠⠆⠀⠀⠀⠀⠀⠞⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠽⠀⠀⠻⠥⠀⠀⠀⠀⢹⡆⠀⠀⠀⡨⢄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⢀⠁⠄⠀⢀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⠀⢠⠀⣀⣀⣤⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢊⠄⢁⠀⠀⠸⠀⠀⠐⠿⢿⣛⣛⢻⡷⠖⠀⠀⠀⢸⣷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡀⠀⠀⣠⠜⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⡠⠚⠀⢨⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⢦⣤⡤⠔⠊⠀⠀⡠⠋⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠐⣾⡇⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠈⠙⠲⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⢊⠁⡘⠃⠀⠉⠿⠦⣤⣀⣀⢐⣐⠧⠀⢀⡴⠊⣴⠁⠀⠀⢉⢑⠖⠦⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠠⢋⡜⠁⠀⠀⣠⠀⠀⠀⠀⠀⠉⣉⠉⠉⠆⠁⣸⠏⠀⡆⢰⠃⠏⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠐⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⣦⡿⠀⠈⢀⡟⣀⣔⢀⡀⠀⢰⡀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣸⣿⢣⡇⢠⢾⠇⣿⠟⠀⠀⢠⣤⠑⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠰⠙⣧⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢡⠏⠞⢸⣠⠇⡜⠸⢻⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠀⢸⠁⠀⠀⢸⡟⠀⠃⠀⢿⠀⣠⠄⢠⣤⠀⠈⠢⠀⠠⠂⠈⣁
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠐⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣸⠃⠀⠀⠀⠈⠣⡄⢰⡿⠋⡁⠀⣤⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢰⢸⠇⡀⠳⡄⠘⠧⡀⠀⠀⡈
⠀⢀⡆⠀⠀⠀⠀⠀⠀⢰⠀⠘⡷⠸⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⢀⡄⠀⡇⠀⠀⠘⠀⣰⠀⠀⠀⠀⢠⠂⠀⠁⠷⠄⢁⡀⡀
⠀⡇⠠⠀⠀⠀⠀⢀⠔⠠⠁⡤⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡇⠀⠀⠀⢐⡅
⠀⢱⠐⠀⠀⠀⡰⠃⠀⡴⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢼⡇⠀⠀⠀⠘
⠀⠀⢣⣄⠀⠈⠀⣠⡌⠔⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠁
⠀⠀⠀⠐⠢⠔⠊⠁⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠆⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⢀⠀⡀⣀⡀⠀⣀⠀⢀
⠀⠀⢀⡗⠀⠀⠀⠀⢀⣴⣿⣾⡆⠀⣼⣿⣷⠀⠀⠀⢀⡀⡀⠀⠀⠐⣾⣿⣷⣿⣿⣿⣾⣿⣿⡿⢀⣉⡇⣨⡽⢃⡀
⠀⢠⣾⣿⣴⡄⠀⠀⣸⣿⣿⣿⣿⡀⠁⠹⠏⠀⡐⡜⣨⡵⣿⣷⣤⡠⠙⠛⠛⡛⠛⣛⣛⣛⣋⠀⡸⣿⣜⡿⢳⡞⣖
⢀⣿⡟⠉⢿⣷⠀⠀⠻⣿⠿⢿⣿⠇⠀⠀⣠⣾⣷⣿⣿⣿⣿⣯⣿⣿⣦⣄⢼⣿⣙⢺⣟⡋⠛⠁⠘⠿⠻⣷⣾⠟⠋
⠀⠛⠄⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣼⣿⣿⠋⠀⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⢰⣠⢼⡰⢠⢠⣀⠀⠀⠁⠊⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⠀⠢⠀⠀⠰⡄⣠
⠀⠀⠐⡄⠀⢰⣺⣽⣿⣿⣾⣽⣼⣿⣿⠀⠀⠀⠀⠀⣠⣦⣤⣄⠀⠀⠀⠀⠐⠀⠈⠼⠷⠁⣨⢷⣿⢕⠗⠀⢀⣘⡆
⠀⡄⣾⢁⠀⠈⠙⠛⡉⠉⠉⠙⠛⠛⠋⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⠀⠀⣠⢞⢢⡇⠀⠀⢀⣈⣿⣿⣏⣀⠰⣞⣮⣷
⠈⢷⣙⣾⣦⠀⠀⣴⣻⣶⡄⠀⢀⡺⣿⣎⠠⣄⠈⠻⣿⣿⣿⣿⡿⢠⣶⣿⡏⣿⣷⡇⠀⠉⠉⠉⠉⠉⠉⠀⣤⣿⠇
⢀⣾⣼⣿⡏⠀⣼⡏⠹⢿⣿⠀⢢⣹⣯⣟⣵⡝⡄⠠⣿⡿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣗⠀⠀⢀⣠⡄⠀⠀⠈⠻⣧⣄
⠀⠛⠿⠟⠃⠀⠈⠓⠀⠘⠃⠀⠘⠛⠛⠛⠛⠛⠋⠀⠘⠇⠸⠋⠀⠰⠿⠿⠿⠿⠿⠟⠀⠶⠿⠿⠿⠷⠶⠀⠶⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡒⠓⠛⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢸⡴⠒⢲⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡜⢀⠎⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡴⢹⠇⢀⡞⠀⢠⠇⣰⠉⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡜⠀⡘⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡼⢱⡇⡀⠺⠤⠤⠋⠀⠓⠚⣁⡤⠤⠤⢴⢀⡞⠋⠉⠉⡉⢰⠁⡸⠁⡌⢉⠇⡰⠋⠃⠒⠲⡄⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠰⡇⠀⠻⣞⡆⠀⠀⡰⠉⡲⠀⠧⣤⠀⢀⠎⢀⡽⠋⡠⠞⢡⠃⢰⠃⡸⠀⡜⢰⠁⡰⢺⠃⣸⢁⠴⠋⣀⣀⠉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠙⠦⣤⡃⠀⠀⢰⠃⢠⠃⢠⠞⢁⠔⠁⡰⠋⠀⠾⢰⠀⡞⢠⠇⢠⠃⡸⢀⠃⣰⢃⡏⢠⠇⡏⠀⣞⡡⠦⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠹⡄⢠⠏⢠⠏⡰⠁⠀⢉⡹⠐⠧⠀⠒⠒⠋⠀⠉⠉⣀⣀⣉⣀⣀⣉⠁⠈⠙⠊⠀⠣⣤⣀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣹⠀⠀⡇⢼⣀⠞⠀⠗⠒⠋⠉⠀⠀⠀⠀⠀⠀⢠⣎⣉⣉⣭⣄⣤⣤⣤⣍⣉⣉⣙⠓⠒⠤⠤⠃⣰⠁
⠀⠀⠀⠀⠀⣀⡤⢤⡀⠀⠀⢠⠟⢤⣸⠁⠀⣴⣶⣶⣶⣦⠀⣶⣶⣶⣶⣶⢰⣶⣶⣿⣶⠄⠀⢀⣶⣶⣦⠀⠀⢠⣾⣿⠙⣲⣶⡖⣡⣶⣶⣶⣦⡄
⠀⠀⠀⠀⢰⠃⠀⡼⠁⠀⣠⠏⠀⣸⡧⡄⢰⣿⣿⠋⣿⣿⠀⠛⣻⣿⡟⠋⢸⣿⣿⠛⠛⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣰⣿⡿⠀⣿⣿⡟⢻⣿⡇
⠀⠀⠀⠀⣟⠀⠈⠧⠤⠚⠁⢀⡴⠃⡷⠇⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⢸⣿⣿⣿⡿⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠘⠲⢤⣀⣀⡠⠔⠋⠀⡞⠀⠀⠘⢿⣿⣦⣄⡀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⢻⣿⣇⠀⢸⣿⣿⣿⣇⠀⠀⠻⣿⣷⣄
⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣤⣽⣿⡿⣿⣤⣤⡿⡿⣧⣤⣼⠿⠿⠿⠿⣦⣴⡿⣿⣼⣿⣿⣤⣼⣿⢿⣿⣿⣤⣤⣤⣬⣿⣿⣿⣦
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⡇⠉⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⣿⡏⠁⠘⠛⠈⠉⣿⣿⠀⢸⣇⠉⢻⣿⣿⣿⣿⡯⠛⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢱⣿⣿⣿⠛⢻⣿⡇⠀⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⣿⠃⠀⣴⣶⡄⠀⢻⣿⠀⢸⣿⡀⠘⣿⡟⠛⣿⡇⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣮⢻⣿⣿⡀⠈⠉⠀⢀⣿⣿⡇⠀⣿⣿⣿⠀⠈⠉⠉⣿⠀⢰⣿⣿⣿⠀⢸⣿⠀⢸⣿⣇⠀⢻⡇⠀⠉⠁⠀⣼
⣿⣿⣿⡿⠿⠿⠛⡟⢻⣿⠿⠿⠿⠟⠘⢛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠟⣿⣿⣿⣿⣿
⣿⣿⣿⢁⠒⠗⢨⣥⣾⣶⣿⣿⡿⣇⣙⡋⢭⣿⣿⣿⣿⣶⣶⢦⣼⣕⡂⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⡿⠣⢿⡟⣸⣿⣿⣷⡿⢿⡏⠜⣽⣿⣿⣿⣿
⣻⣿⡧⣄⠱⠄⣹⢿⡏⣛⠛⣂⣭⣶⣶⢜⠻⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⡧⡪⣳⣿⣿⣿⡿⢛⣉⡉⢀⣶⡏⠸⠉⡿⢱⠏⠼⣋⣴⣌⣵⣿⣿⣿⣿
⣼⡟⢘⡕⣪⣧⡼⢤⡑⢀⣈⣩⣭⣭⣭⣼⣿⣿⢿⣛⣙⠀⣽⣿⣿⠛⠛⠺⡴⠁⢻⣿⣿⣿⣿⡿⢡⣿⣿⣀⣾⣤⣶⣶⣿⣿⢿⣿⣿⣿⣿⣿⢿⣿
⣿⣷⡘⣻⠞⣱⢱⢇⡻⣿⣿⡟⣿⠏⠛⢛⣋⣙⣉⣢⣩⣤⣾⠿⠄⠄⠐⠊⠀⢀⣿⣿⣿⣿⡿⣷⣾⣿⢟⣭⠛⣿⣿⣿⣿⠿⡿⣫⡭⢋⡼⢁⣾⣿
⣿⣿⣿⣦⣙⠾⠈⠺⢇⣉⣉⠉⠉⠉⠉⠉⠉⠘⠛⢛⣻⣿⠷⠁⠀⠀⢀⣠⣴⣿⣿⣿⣿⠣⣶⣾⣿⣇⣼⡏⣰⡏⡼⠱⣋⣸⣤⣭⣾⠶⠂⣘⣻⣿
⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⢁⣾⠍⣽⣿⢟⡋⣰⣿⣷⣿⣿⣿⣿⣿⣟⣰⣫⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣵⣾⣿⡧⢞⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡿⢻⣿⣿⣿⣿⢶⣾⡿⠍⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠺⣭⣙⣧⠘⠟⢟⠿⣙⡇⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣣⣾⣿⣞⣶⣿⣿⣦⣤⡈⠙⠛⠀⢠⣾⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⠈⢹⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣻⠿⡿⠍⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⡀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡀⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⣿⣿⣿⣿⠾⣿⣿⣿⣄⠀⠈⠙⠻⠿⣿⣿⣿⡏⢿⣧⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠉⠀⠀⢹⣿⣿⣷⣶⣶⣶⡄⣠⣶⣾⡇⢾⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠋⠈⢡⣾⠟⣿⣿⡿⣿⣿⣿⣟⣾⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣷⡟⢛⣿⣿⣷⣿⣿⡿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⢸⠏⠛⢻⣿⡟⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⢤⣼⡤⡰⠛⠁⠀⠀⠁⠠⣀⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠰⠁⢸⣿⣿⢶⣶⣿⣷⠁⣄⣀⡀⠀⠀⠀⠀⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⣿⣿⣿⣲⣬⣭⣾⣿⣶⣿⣿⡷⣄⠀⠀⠀⠘⣏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⣿⣧⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠅⠀⣼⣿⣿⣿⣿⣿⣿⡿⡽⠋⠁⠰⠏⡌⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠋⡙⢛⣿⠋⠉⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⣿⣷⠎⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣭⣙⠃⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⡏⣀⣠⡤⢀⣈⣀⠀⠁⠀⢀⣀⡀⠬⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣋⡉⠉⠀⠈⠉⢙⡖⠋⠀⠀⣬⡿⣷⣿⣿⣆⠘⡄⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢤⣵⣦⡀⠀⠀⠀⠀⠀⢸⠿⠿⢿⣿⣿⣷⣶⣤⣀⠉⠉⢈⣧⣄⣿⣿⣿⣿⣿⣿⣆⢰⣄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣀⠀⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠙⠻⣧⣙⠂⠀⠀⠀⢀⣰⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠀⠚⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⠂⠀⠀⠀⠀⠀⠀⠀⡜⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠇⠀⠀⠀⢸⠀⠀⢠⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠛⠀⠀⡴⠀⠀⠀⠰⠋⠀⠀⠀⠠⠀⢀⡞⠀⠀⠀⠀⠐⠐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⢡⢧⠂⡆⠀⠀⠀⣴⣄⠀⠀⠀⢁⠀⣀⡆⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠃⠐⠀⣴⠀⠙⡀⠀⡀⠈⣶⡈⠙⠀⢸⠅⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⣰⠃⠀⠀⡅⠸⠁⠀⠉⠃⠀⠀⠈⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠖⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠀⠐⡈⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠁⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡆⠳⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡸⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣙⢾⡿⠀⠀⠀⠀⣴⠞⣻⢧⡀⠀⠀⡀⠀⣿⠀⠀⠀⢀⡂⢀⡾⠀⠀⠀⠀⢸⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠁⠀⠀⢀⡴⠃⢀⣾⡿⠁⠠⠜⢻⡎⢹⣿⢄⠀⣸⠿⠽⠟⠶⣴⣦⣀⣀⣀⣂⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣧⣇⢀⡾⠋⠁⠀⡾⠉⠀⠌⠀⡠⣾⠂⢠⣷⠈⠛⢟⡀⠀⠀⠀⠀⠀⠀⠀⠩⠋⢿⠌⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠋⠀⠀⠀⣼⠃⠀⠀⠀⠀⡱⣿⠀⢸⣿⡆⠀⠹⣣⡀⠀⢠⡤⠀⠠⢰⠡⠚⢈⣵⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢛⡟⠁⠀⠀⣰⠇⠀⠀⠀⠀⢠⡇⣿⠀⠀⣏⣷⡀⠀⢻⢿⣿⠶⢶⣾⣤⣽⣥⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠁⠬⡀⠀⢠⣾⡏⠀⠀⠀⠀⠠⠋⢠⢛⠀⠀⡏⢹⣧⠀⠈⡀⠸⣄⣈⡛⠚⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣆⠀⣈⢷⣾⡿⠀⠀⠀⠀⢠⠃⠁⠈⢸⠀⠀⡧⠋⠐⡄⠀⢱⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢻⠿⢳⠶⠖⠂⣰⣄⡀⠀⠀⢸⠀⣰⡇⠀⠀⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠚⠋⠀⠓⠀⠀⠘⠋⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⣼⡟⠁⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁⠀⠀⢀⣼⠀⠀⠀⠀⢀⣼⣿⣿⡟⠀⠀⠀⠀⣼⣿⠁⣸⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠁⠀⠀⣠⣾⠇⠀⠀⠀⣠⣿⣿⣿⡟⠀⠀⢀⠃⢰⣿⣿⢀⡇⢸⠏
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃⠀⢀⣼⣿⡏⠀⠀⣠⣾⣿⣿⣿⡟⠀⠀⠀⣿⠀⢸⣿⡟⠸⡇⠋
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⠀⣰⣿⣿⡏⠀⣠⣾⣿⣿⣿⣿⠋⢠⠂⠀⢰⡿⠀⢸⡿⢁⡆⠁
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⡇⣸⢿⣿⡟⢀⣾⣿⣿⣿⣿⠟⠁⡰⢃⠆⢠⣿⠇⠀⡞⢁⡾⢱⠆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣿⠟⠁⢸⢻⡿⢰⡏⣸⣿⣰⣿⡿⢋⣾⡿⠋⠄⡼⣡⠟⣴⡿⠃⠀⠀⣰⣟⣷⠏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⣿⠏⠀⠀⣼⠈⡇⣿⠀⣿⣿⣿⡟⢀⣾⣿⢃⡏⣼⣷⣯⣾⡿⠁⢀⣴⣾⣿⠟⠁⠀⠀⣀
⠀⠀⠀⣶⠀⣼⢿⡀⠀⠀⢸⣿⠀⢠⠀⡟⠀⠃⠟⢸⣿⣿⠛⠀⣾⣿⠇⣾⢡⣿⣿⣿⡟⠀⣰⣿⣿⠟⠁⠀⣠⣴⣿⠇
⠀⠀⠀⣿⠀⠸⣟⣧⠀⠀⢸⡇⠀⣾⣴⡇⠀⠀⠀⢸⣿⠇⠀⢸⣿⡿⢸⣿⢸⣿⣿⣿⡇⣼⣿⣿⠃⢀⣴⣾⣿⡿⠃⢀⠆⢀⠄
⠀⠀⠀⣿⡄⠠⢹⣿⡆⠀⢸⠃⢠⣿⣿⠃⠀⠀⠀⢸⡏⠀⠀⣿⡿⡇⣿⣿⢸⣿⣿⣿⣷⣿⡿⠁⢀⠼⡫⠓⢉⣠⡶⣋⠔⠋
⠀⠀⠀⣿⣧⠀⣧⣻⣷⠀⢸⠀⣼⣿⡿⠀⠀⠀⠀⢸⠃⠀⢠⡿⢰⢱⣿⣿⠸⣿⣿⢹⣿⣿⡇⢀⡾⠋⣠⣶⣿⡿⠛⠁
⠀⠀⠀⢻⣿⣧⢸⣿⣿⠀⡀⢠⣿⣿⠃⠠⠀⠀⠀⣎⠀⠀⢸⢇⡟⣼⣿⣿⢸⣿⣿⠸⣿⣿⡇⢸⠇⣰⣿⣿⡩
⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⢃⣾⣿⠃⠀⢰⠂⠀⠀⠋⠀⢸⣿⡼⢡⣿⣿⢇⣾⣿⡟⣸⣿⣿⡇⠀⢰⣿⣿⠋
⠀⠀⠀⠀⠀⢻⣻⣿⡟⠀⣾⡿⠃⡀⠀⣸⠃⠀⠀⠀⢠⣿⣿⢃⠟⣸⢏⣾⣿⣟⣵⣿⣿⡟⠀⣴⣿⠟⠁⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠈⢿⣿⠃⣼⡿⠁⣼⠃⣰⣿⠀⠀⠀⠀⣾⣿⢃⢂⢠⣫⡾⣻⣿⣿⡿⠟⠋⡠⠞⠋⠀⠀⠀⠀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡟⣸⡟⠀⣼⠃⣰⣿⡏⠀⠀⢀⣼⡿⠁⢂⣾⣰⣟⣽⣿⠿⠋⠀⢀⣠⠄⠀⢀⣠⣴⠾⠟⢉⣴⠋
⠀⠀⠀⠀⠀⠀⢀⢱⡟⠀⣸⢃⣼⣿⡿⠁⠀⢀⣾⠟⠁⠀⣼⡇⣏⣿⡟⠁⠀⣠⡾⠋⠀⣠⡶⠟⠉⠀⣴⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡟⠀⢰⣿⣿⣿⡟⠁⠀⢀⣾⠋⠀⠀⣼⣿⣇⢻⡟⠀⢠⣾⡿⠀⢀⠾⠋⠀⢀⣴⣾⢿⣿⡏⢀⡿
⠀⠀⠀⠀⠀⠀⢸⠁⠀⣾⣿⣿⠋⠀⠀⠀⡾⠁⠀⠀⣼⣿⡟⢻⢸⡇⣰⣿⣿⠃⠀⠁⠀⠀⣰⣿⣿⣳⣿⡟⠀⡜⣯
⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⣿⠃⠀⠀⠀⠀⠁⠀⢀⡾⢋⣿⡧⣀⣼⣠⣿⣿⠏⠀⠀⠀⢀⣼⣿⡿⣷⣿⠟⠀⡰⠁⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣰⠏⣴⣿⣿⡇⣿⣿⣿⣿⠋⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠀⡰⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡰⢡⣾⣿⣿⣿⣼⣿⠿⠋⠁⠀⠀⠀⣰⣿⣻⣿⡿⠋⠀⢀⡔⠁⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⡄⠀⠀⢀⣴⢟⣽⣿⡿⠛⠉⠀⠀⠀⠀⠀⣠⣾⣿⡷⠛⠁⠀⣠⣴⣿⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⣼⠁⠀⢀⠞⣱⡿⠋⠁⠀⠀⣀⣤⣔⣀⡴⠞⠛⠉⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠁⣿⣳⠏⠀⠀⢊⡞⠁⠀⢀⣤⡾⠿⠋⠁⠉⣁⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡘⣼⣿⠏⠀⠀⠀⢋⠀⢀⡴⠟⢁⡀⠀⠀⠔⢊⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⢀⡴⢃⠤⢊⣠⠞⠁⠀⠀⣠⣾⡿⠿⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⣼⡿⠋⠀⣠⣴⣾⠟⠀⢀⣴⡿⠁⠀⠀⣠⠾⠛⠁
⠀⠀⠀⠀⣼⠋⢀⣴⣿⡿⠟⠁⣠⡴⠿⠋⠀⠀⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠐⠀⠴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠚⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣄⢠⡀⣠⣄⢀⣤⡀⣠⣄⢠⡀⠀⣠⡀⣤⣤⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⣿⣼⢿⡇⣿⣾⠈⠃⡇⣿⢸⡇⠀⡿⣇⢸⡇⢸⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠇⡿⢸⠟⣧⡿⢺⣴⠇⣧⡿⢸⣥⣴⠗⢿⢸⡇⢸⣤⡄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠦⠤⠤⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠄⠈⠀⡽⢇⡠⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡂⠀⠀⠀⠈⠉⠉⠒⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠂⠀⠤⠀⠀⠀⠀⠄⠀⠀⡰⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⡞⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠒⢩⠔⠒⠞⡰⠁⢠⣟⠁⠀⠠⠤⠤⣄⢀⣰⣠⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡴⠁⢠⠇⠀⠉⠁⠒⠐⠊⠀⠠⢙⠥⠚⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⡖⠀⠀⠓⠂⠤⠼⠁⠀⡿⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⡀⠀⠀⠀⠁⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡶⠒⠦⢤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣶⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⢠⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠠⠤⠤⣄⣀⣀⣀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢠⠶⣄⡤⢤
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠉⠛⢶⣼⡐⢁⣤⡞
⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⣊⣉⡂⠀⡀⠀⠙⣷⠲⠖⠃
⠀⠀⠀⠀⠀⣾⡧⢤⢤⣉⠘⠿⠃⠁⠀⠀⠀⢹⡆⠀⠀⣀⣠⡶⢦⡾⢳⡶⢲⣦⢶⣄⡀⠀⠀⠀⢀⣀⣠⣤⣤⡆
⠀⠀⢀⡴⠚⠉⠀⠂⠀⠙⣧⠀⠀⠀⠀⠀⠀⣘⡇⠀⠀⣯⡉⠳⠀⠳⠀⠃⢸⠃⢰⠟⢹⣇⣰⣞⣋⠭⣄⠓⣼⠃
⠀⠀⠿⢶⡶⠾⠿⠂⠀⢀⣾⠀⠀⣀⣠⡴⠞⠫⣿⣤⠶⠾⠿⠦⠀⠀⠀⠀⠀⠀⠈⠀⠋⢩⡿⣤⣤⡀⣈⢱⡏⠀⣠⡴⠖⠚⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠙⠶⠦⠤⠶⠛⠛⣾⡟⠩⠃⠀⡀⠋⠀⠠⣤⣶⣴⣦⡄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⢻⣧⡿⠛⢠⡾⠁⢀⣤⣴⣶⣦⣄⠀⠙⢦⡀⢠⡶⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢌⡂⠤⠓⠻⣾⣿⣶⠀⠀⣠⠀⠀⡰⠏⠁⠀⠀⣴⣏⣀⣀⢀⣾⠀⣰⡿⠋⣥⣤⡉⢿⣷⠀⠈⣟⣩⣦⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⢩⣿⣿⠀⢠⣿⡶⠊⠀⠀⠀⢀⣴⣦⣬⣭⣉⣩⣉⠀⣉⣧⡀⣈⣭⡥⢠⣶⠂⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠁⡂⠀⠀⠀⡀⠶⠛⠻⠀⡼⠋⠀⠀⠀⠀⢻⡌⢿⣿⣿⣿⣿⣿⣿⣦⡈⠛⠿⠿⠛⢁⣼⡟⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠁⠈⢷⡀⠀⠀⠀⠀⠀⢀⠜⠁⡀⠀⠀⠀⠀⠀⠘⣤⠛⠛⠋⠙⠛⢛⣿⣿⣶⣤⣴⣶⡿⠋⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢀⠈⠛⠛⠂⠒⢀⡠⠛⢾⣿⠁⠀⠀⠀⠀⠀⠀⡈⣣⡄⠀⠀⣠⣎⠁⠀⠉⠉⠉⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⡈⠀⣈⡉⡠⠄⠀⢹⣿⡄⠀⠀⠀⠀⠐⠻⠥⣤⠟⠩⠝⢯⡤⠽⠗⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢰⠀⠉⠀⠀⠊⠡⢷⡷⣄⠀⠀⠀⠀⢀⠶⣉⠀⠀⢀⢈⡣⢄⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡆⠀⠀⠀⠀⡜⠀⠻⢮⣳⢦⡄⡴⣫⣾⣿⣿⣶⣾⡿⣿⣨⣵⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣼⠶⢶⣶⣤⣽⣧⣿⣼⣿⣯⣿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠀⠀⠈⠛⠓⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⡀⠀⢀⣠⡶⠤⣦⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠅⣠⢴⣿⠟⠁⢀⡾⠁⠀⠀⠈⢧⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢡⡞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⠀⠀⠀⠘⠀⡰⠇⠀⠀⠃⠀⡠⠀⠀⠀⠀⢓⠘⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠈⣄⠀⠀⠁⠀⣠⡀⠀⠀⠃⠀⢀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠈⠀⠂⣦⠉⠃⠀⠀⠠⣤⠉⢀⢰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⢰⠋⠀⢠⠀⠇⠀⠉⠀⢀⠈⠀⠠⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡤⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⠦⠀⠀⠀⢁⡄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣔⡫⢿⠇⠀⠀⢀⠖⢻⣧⠀⠀⣀⠀⡇⠀⠀⠀⣀⣠⡆⠀⠀⠀⠛⠆
⠀⠀⠀⠀⠀⠀⠀⣐⣀⣀⣀⠀⢀⣠⠞⠀⣼⠞⢀⠐⢻⡇⢹⡿⣄⣼⠓⠛⠙⠲⠷⠤⠤⣴⣴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⢿⡤⠞⠀⠀⡼⠁⠀⠀⠠⣼⡇⢸⣧⠀⢩⡄⠀⠀⠀⠀⠀⢰⢳⠸⣥⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠅⠀⠀⣰⠃⠀⠀⠀⣸⢽⠃⠸⢻⡄⠀⢿⣶⣧⣯⣤⣄⣧⡠⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠥⢁⠀⢀⣼⡏⠀⠀⠀⠀⠃⡼⠁⠀⠏⣷⠀⠘⡈⢇⣀⡛⠻⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⣹⣾⡟⠀⠀⠀⠰⠁⠈⠀⠀⢀⡞⠀⠆⠀⣁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠙⠙⢃⣴⠢⣀⠀⠀⣄⣼⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⡆
⠀⠀⠀⠀⠀⠀⠀⣰⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣾⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠒⠛⠒⠶⠶⣤⣀
⣠⠴⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠶⢤⣤⣶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳
⣿⠦⣤⣀⣀⣀⣠⣤⣤⡤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣄⣀⣀⠀⢀⣺⣿⡦⢾⡇⣿
⢿⠀⣀⡈⠉⠉⠛⠛⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠛⠛⠋⠉⠁⠀⠀⣾⠶⠿⠶⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⠞
⠸⡇⠈⠉⠙⠓⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⠛⠋⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠙⠦⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⠛⠛⠛⠓⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠛⢛⡛⣛⡛⠛⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⢛⣩⣭⣯⣿⠿⣿⣭⡉⠉⠉⠛⠓⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⡭⠥⢬⡀⠁⣸⠀⠐⠂⠀⠀⠒⡂⢹⡖⠀⠀⢠⠖⠒⠻⢿⣟⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⣡⠞⠉⠀⠀⠀⣿⠀⢸⠀⠀⠀⠂⢀⠀⠀⢸⡇⣆⠀⣿⠀⠀⠀⠀⠉⠳⣌⠳⣄
⠀⠀⠀⠀⠀⣠⠏⣠⠞⠁⠀⠀⠀⠀⢀⡿⠀⡿⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⠀⣛⠀⠀⠀⠀⠀⠀⠈⠳⡌⢷⡀
⠀⠀⠀⠀⣰⠃⣴⠷⠶⠶⠶⠶⠶⠶⣏⠀⠀⡿⠒⠒⠒⠒⠒⠶⠟⠳⢾⠀⠐⣿⠀⢀⠀⠐⠄⠀⠀⠀⠙⣆⢳⡄
⠀⠀⠀⢰⠇⣸⠃⠀⠀⠀⠀⢀⠀⠀⠈⢷⠀⡏⠀⠀⠀⡀⠀⠁⠨⠅⠈⡇⠀⣿⠀⠁⠀⠀⠀⠄⣀⠠⠡⢹⡄⢷
⠀⠀⠀⡾⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠉⠙⠳⢼⡇⠸⡆
⠀⠀⢀⡇⢸⣇⣀⣀⣀⣀⣀⣀⣀⣒⣂⣼⠀⣇⣐⣠⣤⣤⣤⣀⣀⣀⣠⠇⠀⢹⣄⣀⣀⣀⣀⣀⣀⣀⣊⡼⢻⡄⢳⡀
⠀⠰⢯⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⡭⠭⠥⠤⠬⠭⠭⠭⠭⣭⣭⡭⠭⠥⣤⣤⣤⠭⢭⣭⢭⡭⡭⠭⢭⠍⠠⠤⠥⣤⠷⢀
⣎⣲⠼⡀⠀⠛⠛⠛⠛⠛⠛⠲⠶⣄⠀⠀⣶⡇⠀⠐⡟⠛⠋⠉⠛⠻⢦⠘⠋⠉⢹⠋⠉⠉⢉⠛⠀⠀⠀⠀⢀⣆⢸⠲⣉
⠈⠁⠀⣇⠘⣇⠀⠀⠀⠀⠀⢸⠀⣽⠀⢸⡟⡇⠀⠀⡇⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣸⣇⡏
⠀⠀⠀⠘⡄⢻⡀⠀⠀⠀⠄⠀⠀⣿⠀⣼⠁⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢹⠀⠀⠀⠀⢠⡿⣼⠁
⠀⠀⠀⠀⢻⡀⢳⣀⣈⣀⡀⠀⣀⡟⢠⡏⠀⣻⠀⢀⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⠸⡇⠀⠠⠸⡄⠀⠀⢠⡿⣱⠇
⠀⠀⠀⠀⠀⢳⡄⢻⡉⠉⠉⠉⠉⠀⡼⠛⠋⢻⡀⢸⡷⢶⣖⠶⡶⠒⠋⢀⠀⠀⠀⡇⠀⠀⠀⠘⠶⢶⡞⣱⠏
⠀⠀⠀⠀⠀⠀⠻⣄⠱⣄⠈⠀⠀⢰⠇⠀⠀⠸⡇⢸⡇⠁⠙⢷⡁⠐⠂⠡⠀⡀⠀⡇⠀⠀⠀⢀⡴⢋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠷⣜⢦⠀⠀⡟⠀⠀⠀⠀⣷⢸⡇⠀⠃⠀⠹⡄⠀⠀⠀⠀⠀⡇⠀⣤⠔⢋⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣘⠃⠀⠀⠀⠀⠹⢸⡇⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠁⣠⣴⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠗⠲⢦⡤⠤⠌⣁⣀⣀⣠⣠⠤⠤⣤⠶⠒⠺⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣅⣀⣀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⣴⣿⣦⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡻⡏⣿⢹⠟⡶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠻⡌⠃⠉⠀⠀⠈⠑⠋⠟⡹⢶⣄⡀
⠀⠀⠀⠀⠀⠀⣄⣠⣴⣾⢟⠪⠀⠀⣖⡆⣶⠂⣴⣲⠀⣽⡿⠷⣯⣟⣗⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⣰⠞⣻⠟⠉⠀⠀⠀⣀⡄⠸⢠⡟⢧⠒⡀⢐⣿⠀⠀⢤⣌⣉⠛⠾⣯⣯⣿⣻⣻⣿⠟⠟⣿⣿⣿⣍⣀⣀⡈⠳⡄
⠀⠀⠀⠐⣮⣿⢋⣀⡀⠀⠀⠈⣿⠆⡍⢸⠁⢸⡇⠥⠐⣿⡆⠀⠀⠹⣿⡿⢶⣤⣤⠉⠻⣿⠁⢀⣀⣀⣀⣈⠙⠇⠀⢹⡄⡇
⠀⠀⢀⣤⡝⠁⠙⠋⠁⠀⠀⢠⡏⢁⡒⣾⡡⠀⠙⣎⠥⠠⢿⡄⠀⠀⠘⢿⣾⠟⠁⠀⠀⣿⠂⣿⡏⠉⠙⣿⡇⠀⠀⣸⢰⠃
⠀⠀⡾⠿⠁⠀⠀⠀⠀⠀⠀⢸⣿⢠⠥⡏⠀⠀⠀⣿⢘⠂⣿⡇⠀⠀⠀⣨⣅⠀⠀⠀⠀⣿⡂⣿⡅⠀⠀⣿⠇⠀⠀⣯⣾⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡿⠐⣂⢷⡀⡀⡀⡿⣬⠁⢸⡇⠀⣠⣾⣋⠹⣷⡀⠀⢀⡿⠀⢻⣇⠀⠀⣿⠀⠀⠚⣀⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠘⣧⢇⠉⢞⣳⣔⢸⢓⠆⠘⣿⣧⣈⣉⣉⡙⠛⠿⠟⠀⣠⣿⡆⠈⢿⣆⠀⣿⠀⢰⡟⣏⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣿⣿⣦⠈⠱⣙⠟⣬⠁⠉⠉⠉⠉⠉⠛⠻⣿⣶⡶⠟⠋⠁⠀⠀⠀⠙⠷⣿⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠃⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣆⠀⠀⣸⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣇⠀⠀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡟⠀⠀⡿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⡆⠀⠀⣅⣠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣥⣔⠁⠀⠰⣦⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠀⢠⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⠀⠀⠠⠠⠤⠤⠄⠤⠤⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠂⠈⠌⡁⠀⠰⠀⠀⠐⠋⠉⠈⠀⠀⡤⣀⣄⡀⢔⡂⠚⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠠⠀⠋⠀⠀⠀⢄⠀⠘⠛⠀⢀⠀⣐⡷⠘⠋⠀⠀⠄⣈⣄⡈⠉⠁⠀⡀⠀⠀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠀⠲⠀⠀⠀⠀⠀⠈⠀⠀⢀⣤⠀⡀⠌⣈⢡⢶⠅⠘⠟⠛⠁⠘⢶⠂⠑⠉⠀⠖
⠀⠀⠀⠀⣴⡄⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢰⠈⢋⣁⣀⣰⣚⡉⠛⠁⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠠⠀⠁⠀⠁⠀⠐⠾⢃⡁⠒⠃⠀⢄⠀⠀⠀⠀⠀⠊⠁⠀⠈⠁⠀⠀⠀⢉⠀⢀⣀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠁⠠⠁⠐⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠈⠁⠀⠀⠈⠀⠠⢄⠄⠀⠐⠁⢈
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢅⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡏⠉⠉⠛⠛⠲⠶⠶⠶⣶⣤⣤⣤⣤⣤⠄⠀⠰⠀⠦⠀⠐⠶⢶⣶⣶⠒⠒⠲⣶⣶⣶⢶⣶⣶⣶⣶⣴⣤⣦⡴⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠤⠤⡁⣀⠀⣈⣩⣨⡶⠿⠛⠛⠉⠏⠀⠀⠠⠀⠀⠀⢚⠁⢘⣄⣄⣀⢠⣌⡇⢀⣀⢠⣿⣍⡉⠉⠉⠉⠈⠉⠉⠉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠉⢛⠙⠀⠀⠁⠐⣤⣤⣄⠀⠆⠐⠙⢄⣩⡤⢯⡟⢢⠉⣉⣉⣉⣿⣿⣓⠒⠒⠤⠤⠤⠤⠄⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⣴⡧⡖⠀⠂⠃⠀⡀⠀⠰⠚⢿⣿⣿⡷⠊⠀⡀⠀⣤⣬⣏⣆⣩⣈⣁⡀⠈⠉⠉⠉⠁⠉⠀⠉⠁⠉⠉⠁
⠀⠀⠀⠀⠈⠀⢁⠉⠀⠀⠠⣴⣿⣈⠁⠐⠊⠥⢷⢳⡞⢛⣻⣿⣿⣿⣧⣾⣇⣀⣼⣆⣭⣏⣛⣛⣻⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⡀⠀⠈⠁⠀⠠⢟⣳⣌⣹⡿⠏⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠀⠉⠉⠉⠀⠀⣀⠵⢿⠫⠹⣟⣿⢿⠶⠒⠂⠀⠐⢉⠃
⠀⠐⠿⠏⠰⡀⠀⠢⣈⡇⢫⣫⠀⠀⠁⢀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢺⡄⠀⠃⠀⣞⠿⡉⠀⠀⠀⠀⢀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠻⢣⠍⠏⠐⠀⢀⣰⣶⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠏⠡⡶⠅⠐⠀⠄⠱⢂⡀⡀⠀⠐⠀⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⢁⣀⡅⠀⣯⢹⠻⡟⢦⡀⠀⠀⠀⠀⡠⠎⠁⠀⠉⣒⣀⣀⡀⠲⡆⠀⢨⠶⠤⠟⢀⣘⣚⡳⠆
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈⠛⠿⠷⠜⠁⠀⠀⠆⠑⢄⠀⠀⠀⢴⠄⠑⠋⠀⠀⠀⠨⣉⠀⠀⠀⠀⠀⠀⠀⠀⢉⠙⠂⠀⠀⠠⠁
⠀⠀⠀⠀⠄⠀⠀⣰⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⢔⣤⣶⣶⣶⣶⠲⠴⠖⠒⢲⢶⣦⣤⣠⡄⠤⠐⠠⡆⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠐⠈
⠀⠀⠀⠀⠀⠈⠒⠼⢤⣀⡀⣀⣀⡠⣄⠀⠀⣀⠉⢚⣛⣓⠛⠒⠒⠒⠂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠀⣰⢀⣀⡀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠒⠒⠛⠁⠁⠒⠒⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡋⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢣⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⠖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⣾⠓⠒⠓⠂⠂⣴⠲⠤⣤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⢉⡅⠀⠀⠈⠀⠀⠾⠀⠀⠉⠀⠀⠀⠈⠁⠛⠡⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠂⠀⠀⣀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠉⣀⣀⣀⣀⣀⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠀⠐⠃⣀⣀⣀⣠⣠⡴⠒⠖⠲⢶⢿⡛⡟⢛⠛⣯⡉⠛⠛⠛⠛⠁⢉⣉⠉⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⡶⣛⠹⠿⠟⠷⠀⠤⠀⣏⣀⣀⠀⢀⠃⠘⠻⠄⢡⡜⠀⠃⡚⢺⠏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠐⠒⠳⠒⠐⠒⠒⠚⡆⠉⠈⢉⠀⡤⢄⣀⡄⠀⠈⠉⠉⠉⠉⡏⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠲⠆⣴⣤⣤⣄⣠⣤⠷⠔⠓⠋⠉⠀⠀⠉⠉⠓⠛⠒⠶⠶⠖⠓⠛⣈⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠚⠹⡍⠀⠀⠀⠈⠉⢙⣳⣋⣭⣉⡲⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠋⠁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⢻⡄⠀⠀⣀⣾⡿⠗⠉⠉⠉⠉⠲⣤⣢⡀⠀⠀⠀⠀⣠⠿⠁⠀⠊⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⣄⠃⠀⢹⣄⣰⣷⠉⠀⠐⠢⠄⠀⠄⠀⠀⠉⢦⡠⠀⢀⡼⠁⢀⡀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠙⠳⠦⡀⠈⠁⠀⠣⠀⠂⠀⠀⠀⠄⠀⠄⠀⠘⢒⣠⢟⠁⠤⠄⠀⠀⣈⠀⣶⣗⣀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⢷⣦⣄⢉⠁⠉⠃⠦⠤⠤⢤⣀⣀⣤⣀⣠⣀⠠⠤⡤⢦⢤⡔⣆⡴⠻⠛⠛⢁⡹⠬⢣⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⣄⠈⠁⠀⣍⠸⠿⠷⠶⣤⣤⣀⣀⣀⠀⠁⠈⠉⠉⠀⠀⠈⠁⣀⢀⣤⠴⠶⠲⢟⡁⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣁⠀⠠⠖⠁⢀⡦⠀⣀⢁⠀⠉⢉⣉⡉⠙⠃⡳⠋⠉⡉⠙⠛⠉⣉⡠⠤⠀⠀⣀⢴⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠏⠁⠀⢄⠀⠀⠂⣠⠉⢀⡠⠂⠏⠀⠀⢤⡁⠀⠀⠉⠀⢦⠄⠀⠐⠒⣀⠦⠤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣔⠂⠘⠶⠤⣦⣤⡀⢚⡄⢀⠀⡴⠒⠂⢰⢀⡀⢦⣀⣤⠲⠴⠶⠄⠈⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠈⠛⠉⠉⠿⡆⠀⠀⢀⡀⡼⠁⠁⠈⠀⠀⠀⠀⠀⡀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⠀⠀⠀⠀⠀⠈⠓⢮⡴⠛⠁⠀⠀⠀⠀⠀⢀⡀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠲⠦⠀⠤⠤⠀⠄⠀⠒⠚⠉⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⢿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢴⡄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠉⠀⡠⡀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠱⢂⠘⢦⠖⢀⠲⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⡄⠀⢾⠳⣾⠺⣯⠿⠖⡋⢢⣵⣥⠊⠛⠾⢯⡿⢺⡶⢺⠆⠀⡔⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡢⡐⡾⡢⢄⠈⠁⠙⠒⢶⣄⡈⠰⡋⠘⠈⠳⠈⣀⣴⠖⠚⠁⠉⢀⠬⡪⡖⣠⣚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠉⠳⠔⠆⢰⠃⠹⠶⡾⣭⣽⣾⣞⣯⣟⣾⡟⣭⣝⡶⠾⠁⢲⠠⠖⠴⠋⠁⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⢲⠖⠊⡀⣺⡋⣶⢫⠿⣿⣬⣻⣻⡿⢫⠻⣿⣻⣫⣼⡿⢯⢳⡎⣚⡂⡈⠒⢶⠚⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢠⡤⡄⠘⠄⢟⠙⣏⣽⣿⣿⣾⣿⠷⠗⠟⠓⠚⠒⠛⠳⠯⢿⣿⣾⣿⣿⣍⡿⣙⠇⠼⠀⡤⣤
⠀⢀⣤⡀⠀⣰⢿⡟⢷⡄⢳⣄⢰⣯⡿⣿⠟⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⣯⣷⠀⣴⠃⣴⠟⣿⢷⡀⠀⣠⣄
⠀⠸⣟⢁⡀⠔⢺⠀⠚⢳⠄⣨⢷⣻⠋⠀⠀⠴⠚⣹⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣳⢯⡀⢴⠛⠂⢸⠀⠄⢀⢙⡿
⠀⠀⢁⡀⢀⢸⠢⢅⡁⠔⠒⣿⣿⠁⠀⠀⠀⠀⢀⡏⡾⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡗⠒⠄⣀⠥⢺⢀⠀⣈⠁
⠀⠸⢿⡅⠐⢗⠵⢪⡐⠢⠸⣇⢾⠀⠀⠀⠀⠀⢸⢱⢃⠎⣤⢀⣦⡆⡀⠊⠀⡦⢲⠠⠰⡆⠀⠀⢸⣆⡿⠐⠒⣨⠲⢵⠗⠂⣽⠿
⠀⠀⣀⢀⡀⠸⢎⠥⠌⠑⠂⣯⢿⡄⠀⠀⠀⠀⠘⢠⣾⠰⠋⠉⠉⠀⠃⠀⠀⡄⡟⠀⠀⠁⠀⠀⣼⢯⡗⢒⠉⠤⢝⠾⠀⢀⢀⡀
⠀⢸⣯⡄⠀⣁⢼⡀⢶⡞⠁⣹⢗⣿⣦⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠐⠁⠀⠀⠀⣠⣾⣖⢿⡁⠙⣶⠆⣸⢄⡁⢀⣬⣿
⠀⠀⠉⠀⠀⠘⠿⢷⡟⡁⠾⠁⢸⣯⣷⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣷⣯⣿⠀⠹⠆⡙⣷⠿⠟⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⢰⠁⠯⡴⣯⡽⣿⣿⠻⣿⡟⡗⣶⡖⢲⠒⣶⡒⡏⣿⡿⢻⣿⡿⣥⡷⠬⠇⢱⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠚⠒⠠⠀⠲⢦⠻⠮⠽⣿⣩⣾⣾⣿⣮⣾⣿⣾⣮⢹⡿⠭⠾⢣⠰⠀⠠⠀⠚⠲⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⡈⡤⡞⡲⢂⠘⠆⠴⠛⢳⣝⡵⠯⡳⠯⠷⡫⠿⣝⡵⠛⠳⠄⠞⢀⠲⡚⡦⡌⣀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠊⠐⢫⡢⠃⢠⠄⣴⠒⡛⠁⠠⡘⢦⣰⢠⠞⡠⠀⠙⡓⢲⡄⢤⠀⠳⣺⠓⠈⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠃⠀⠻⠖⠻⠾⢯⣻⠷⢦⠎⠝⠍⢦⠶⠼⣫⠿⠾⠓⠾⠃⠀⠑⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂⠐⡋⣳⠀⠚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠀⢈⠀⢠⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢄⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠈⠓⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠊⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⠒⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠀⠉⠁⠀⠲⢄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠴⠀⠀⠡⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠈⣀⣀⡀⠼⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠔⢪⡁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⢄⠁⠀⠀⠀⠀⠀⠀⠘⠁⠂⢢⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠈⣤⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡀
⠀⡐⠂⠀⠂⢴⠂⠉⠀⠀⠀⠘⠿⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⡇
⠀⡇⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠙⠒⢳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣰⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠻⣁⡠⠊⠉⠳⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⡐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣧⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⡜⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡴⠋⣠⠀⠀⠀⡜⢀⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⢋⠔⠊⠀⠀⠀⡘⡠⠎⠀⠀⠀⡴⠁
⠀⠀⠀⢀⢀⣀⣀⣀⠠⠄⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⢄⡀⠀⠀⠀⠀⠘⡅⠀⠀⠙⠢⠴⣅⠴⡴⠋⠀⠀⠀⠀⢸⠁
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠚⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⢆⡠⠀⠀⡀⣠⠏
⠀⠀⠸⢅⣀⣤⠤⣤⠤⠤⠠⠤⠔⠒⠒⠚⠛⠉⢉⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⡀⠀⢀⡀⣀⡤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠳⠞⠙⢿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⠛⡅⠀⠘⣅⣸⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠹⣦⣬⠲⣶⡵⠉⠙⠯⢟⣛⡒⠒⠶⠶⢤⣀⠀⠸⡏⠉⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠃⠐⠀⠑⠀⠀⠀⠀⠀⠀⠁⠀⠲⠄⠉⠳⣄⢳⠀⠘⡆⠸⡄⣀⣠⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⡩⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠘⣯⢧⠠⢱⣠⢋⢡⠴⠖⠲⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡏⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠳⣾⣿⣮⡅⠀⠴⠚⠋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡞⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠾⢋⣵⢚⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠼⠶⣞⡩⣁⢀⡽⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⣉⡀⠀⠀⢠⠃⠀⢓⣠⢇⣾⢻
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠋⣀⢔⠑⠖⠇⠀⡔⠀⢸⠆⠀⡙⢈⣼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠛⠁⢀⢼⠀⠤⣀⡀⢼⡉⠁⡶⠓⣆⠽⣘⢊⡹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠳⢄⣘⢢⡄⣀⣬⣀⠄⠨⠂⢳⡔⢻⡤⢇⡠⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣏⡳⠀⢸⠁⠀⠈⠗⠒⠾⠒⢦⡀⠜⣁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⣴⣭⣤⣶⣾⠿⠿⠗⠙⠳⣤⡀⠀⠂⠃⠢⠒⠃⢀⣊⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠛⠉⠉

⠀⠀⠀⠀⠀⢀⣠⣤⡤⣤⣄⣀⣤⢤⡤⢶⣤⣀⣀⣀⣤⠤⠤⡶⢤⣤⣄⣀⣠⣤⣴⠶⠦⣤⣀
⠀⠀⠀⢀⡴⠋⠀⠀⠳⠤⣤⣤⠤⠞⠀⠘⢆⡀⠀⣀⣀⡤⠞⠁⠘⢦⣀⡀⣀⣠⠜⠀⡤⡀⠈⠳⣄
⠀⠀⠀⣼⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠀⠀⠀⠈⠉⠉⢀⣀⣀⣀⣀⣀⣀⣹
⠀⠀⣴⠛⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⢲⠲⠒⠒⠒⢒⡒⠒⠛⣉⣭⣍⡉⢉⣭⣍⣉⣉⣉⣀⣠⣤⣉⡗
⠀⠀⠈⣛⣿⣙⣲⣒⣉⣶⣖⣟⣶⣖⣉⣉⣓⣶⣫⣭⢯⣉⣭⡤⠤⠤⠽⠧⠤⠤⠤⠤⠤⠧⠤⠖⡟⣶
⠀⠀⠀⣏⠓⠒⠊⠓⠤⠴⠒⠲⠒⠊⠓⠒⠒⠛⠒⠒⠛⠉⠉⠈⠙⠋⠙⠋⠉⠙⠋⠉⠑⠒⠊⠉⢀⡏
⠀⠀⠀⠙⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞
⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠋⢀⣿⣿⣿⣎⢿⣿⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢃⣴⣿⣿⢻⣿⣿⣼⣿⡇⢀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣻⣿⣾⣿⣿⣣⣿⣿⣿⣿⣿⠃⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣿⠇⣿⣿⣿⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⡟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢸⣿⣼⣿⣿⣿⣿⡏⠋⢰⠸⣿⣿⣿⣿⣿⠇⣾⡟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣼⣿⣿⣿⡿⢿⣿⡇⣄⠸⣷⡙⢿⠟⣿⣿⣿⠟⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣎⣿⣿⣿⣿⣧⠈⡛⠳⠙⠀⠘⡇⠀⢰⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣦⠳⣄⠈⠀⠀⠁⠰⠟⡛⢉⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠇⠀⡀⠀⠀⠀⢀⠀⠀⢀⠈⠁⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣾⣇⢀⣾⣆⣸⣷⣼⣷⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢫⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣷⡾⣿⣿⣷⣽⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⣴⣿⣿⣟⣿⣏⣿⣿⣭⣿⣿⣿⣿⣿⣿⣷⣟⣽⣿⣿⣷⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⡿⠿⠿⠛⠛⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⡄⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⣀⣀⢨⡤⣀⣀⣖⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⡀⠀⠀⠀⠳⠇⠀⠈⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠘⠋⠋⠈⢀⠲⠘⠚⠁⠈⠙⠃⠀⢠⠈⠀⣤⣤⠀⠓⠦⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠤⣤⠄⠤⡤⠤⠰⠦⠄⠀⢀⢀⣤⠀⠙⢀⢷⠀⠰⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠙⢱⠀⠀⠈⠀⠄⠀⢀⠀⠀⠸⠀⠀⠀⠀⢉⢠⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠯⠉⠉⠉⠉⠋⠉⡋⠁⠀⢀⠀⠲⠟⠀⠸⠆⠰⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣼⣤⣀⣀⡀⣀⣀⣩⡀⠀⠘⠓⠒⠂⠚⠁⠀⠀⠀⠄⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠁⡀⢀⠀⠀⠀⠀⠀⢀⣠⡀⠠⡜⠋⠀⠠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠴⠶⠤⠤⠤⠮⠧⠤⢒⠄⠀⢈⠉⠀⣀⡉⠈⡧⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠞⠀⡀⡀⠀⠠⠖⠀⠀⠀⡐⠬⠀⠈⠉⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠐⠂⠉⠁⠀⠀⠐⠒⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢋⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⢰⠃⢠⠂⠖⢀⠞⡥⠒⢍⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢸⠘⣸⣿⠀⠘⢦⡀⠈⠀⣼⢷⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠱⡘⣄⢧⢣⢿⣦⢳⡀⠙⣦⢣⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⠀⢂⡄⢢⡻⣦⣷⣆⢻⣧⢧⣦⣿⠆⡇⡀⠈⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡺⢽⡛⢽⣿⣿⣶⡟⣸⣿⣏⠞⢁⠁⣠⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠓⠀⣽⠈⢿⣿⣿⣷⣿⣿⢡⡾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠤⣤⣀⣱⣀⣿⣜⣌⣿⣿⣿⣿⡿⠿⠿⠇⢄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣾⢀⡇⠁⠊⠈⢛⡿⠛⠛⠁⡀⠀⠀⢀⠀⠀⠀⢠⠟⢝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⣤⣞⡁⣀⡴⣂⡁⠀⠀⠀⢀⣴⠀⢀⣤⣄⣬⣦⣻⡸⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⣤⠄⠀⢸⠁⠉⣝⡇⠀⣰⠾⠿⣇⡌⠠⠀⢠⣿⡏⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠈⠻⢀⣠⣼⡧⣬⣽⡷⣿⡄⠐⠀⣽⠿⠲⡒⠛⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠽⠋⢩⡀⠀⠝⠋⠉⠻⠟⠟⠉⠈⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⣠⣿⠀⠀⠀⠀⣸⣧⠀⠀⠀⢀⣰⣷⡄⠀⠀⢀⢰⣧⠀⠀⠀⠀⣾⣧⠀⠀⠀⢀⣼⣧⡄⠀⠀⠀⣾⣄
⠀⠀⢾⠿⡹⣿⠀⠀⢰⣿⣿⣿⠀⠀⢾⠟⠙⣿⠀⠀⢸⣾⣯⡤⠀⠀⢰⣿⣿⡆⠀⠀⣽⡟⠻⣷⠀⠀⣿⠏⢻⣶
⠀⠀⠈⠘⠃⠉⠀⠀⠈⠛⠛⠁⠀⠀⠈⠃⠐⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠐⠃⠉⠀⠀⠈⠀⠀⠁
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣄
⠀⠀⣰⣾⣿⣧⠀⠀⠀⣼⣿⠀⠀⠀⢀⣿⣷⡀⠀⠀⢀⣿⢷⣄⠀⠀⢠⣾⢿⡆⠀⠀⣠⣿⣿⡀⠀⠀⢠⣶⢿⡄
⠀⠀⠻⢿⡽⠟⠀⠀⠰⣿⣿⠟⠀⠀⠸⡏⠘⠟⠀⠀⠸⣧⣆⠟⠀⠀⢸⡇⠸⡇⠀⠀⢻⣿⣿⡟⠀⠀⢸⣣⡌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣰⣷⡄⠀⠀⠀⢰⣦⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⢰⣶⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣾⣆
⠀⠀⢴⠟⡈⣷⠀⠀⢨⡟⢹⣶⠀⠀⢰⡿⠹⣧⠀⠀⢀⣿⢿⣀⠀⠀⣰⣿⣿⡇⠀⠀⣤⣿⢿⣧⠀⠀⣤⡟⢹⣧
⠀⠀⠘⠚⠃⠋⠀⠀⠘⠃⠀⠋⠀⠀⠘⠃⠀⠃⠀⠀⠈⠃⠈⠃⠀⠀⠘⠻⠿⠋⠀⠀⠙⠋⠈⠋⠀⠀⠙⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠨⣿⣿⣿⣿⣷⣶⡤⢠⣤⢀⣀⣤⣄⠀⢤⣤⣤⣤⣤⠄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⢚⠦⣀⡞⠝⢦⠀⠀⠀⠀⠉⠛⠋⠀⠉⠃⠘⠿⠦⠿⢿⣿⣷⣦⣄⣀⣾⣿⠀⠌⠀⠈⣰⢄
⠀⠀⠀⠀⠀⠀⠀⣇⠴⢆⠋⣆⣛⡌⠀⠀⠀⢀⠀⢠⢄⣀⡤⣤⣀⣀⡀⣀⡀⠀⡀⠀⠀⠉⠉⠁⠀⢥⢀⢰⡟⠭⣧
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡟⠀⢸⡿⠀⠀⣴⠤⣃⣭⢭⣴⠆⠀⠀⢌⠲⣷⡭⣉⢽⣛⣒⠺⠦⠴⠤⠤⢀⠉⠓⠊⠩⠛⡃
⠀⠀⠀⠀⠀⠀⢀⡀⠘⠇⣀⡀⣃⣴⣄⠐⠁⠉⡿⣿⡏⠀⠀⠀⠀⠐⠹⢎⣵⣷⡑⡽⠟⠅⠀⠁⠉⠁⠠⠠⠤⠤⠀⠐
⠀⠀⠀⠀⠀⠠⣾⣿⡶⠺⣿⠏⠀⣠⣶⣤⠐⡀⠈⠀⠄⡀⢀⣤⣤⣅⠢⡢⠻⠟⡩⠲⢐⣀⣀⡀⢀⠀⢀⣿⣯⣗⡻⡿
⠀⢠⣦⣤⣀⣀⣈⣋⣀⣀⣤⡤⠖⢋⡿⠟⠀⠂⠀⠀⢈⣴⣿⣯⣠⣾⡇⠅⠀⠀⠀⢠⣿⣄⣠⢟⠀⠀⠀⣏⣯⣹⡰⠇
⠀⠀⠻⠏⢿⠟⡻⠟⣩⡙⢡⣆⣴⠏⡾⠋⠀⠀⠀⣰⣿⠟⡕⣽⣿⠏⠀⠀⠀⠀⠂⣿⣿⢾⣷⣿⠐⠀⠂⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⢿⣶⠿⠿⠛⠋⠁⠀⡰⠀⠀⠂⣼⣿⣇⣹⣾⣿⠡⠀⠀⠀⠀⢸⢸⣿⣿⠾⣿⣏⠀⢀⡀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢻⣳⣀⣄⣿⣟⢶⣶⣫⠇⠐⠀⠀⠄⢀⣀⠈⢿⣷⠖⣫⣏⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣄⢿⣾⣿⣓⣾⡿⠊⢮⣿⡿⠷⠛⠀⠀⠀⢀⣟⣯⣿⣿⡿⡭⢿⡿⢟⣥⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣝⠳⠏⠁⠀⠀⠀⣶⡶⡿⠻⠆⠀⢀⣓⠻⢿⣟⡻⠁⠙⣥⡾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠿⠓⢶⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣠⡀⣄⢠⡄⣠⣄⠰⢤⣄⢠⡄⣀⣤⣠⠆⣀⠀⡀
⠀⠀⠀⠀⢀⡀⣀⣘⣇⣼⡿⣯⣾⠋⢿⣾⠟⣷⡾⠋⠙⢷⠿⠛⠷⠿⠛⡿⠾⠛⣿⣤⣭⡦
⠀⠀⠀⣀⡈⢃⣼⠿⢿⡿⣤⣤⣤⣤⣠⣤⣤⣴⣶⣤⠄⢰⡲⠤⣶⣶⣢⣶⢖⣶⠿⠀⣾⢁⣀
⠀⠀⠀⠀⠺⠿⣷⣜⢛⣛⡛⢿⣝⠓⢿⣛⣛⠙⣛⣮⣁⠛⣫⣽⣭⣍⣉⣬⣭⣉⢠⠾⡛⠓⠈
⠀⠀⠀⠐⠂⠐⠛⡿⠉⠹⠟⣫⠛⠛⢫⠍⠙⠛⠉⠰⠙⠛⠉⢰⠂⠉⠉⠐⠮⠉⠛⠀⠉⠃
⠀⠀⠀⠀⠦⠲⠖⠤⠶⠶⠂⠓⠂⠘⠒⠒⠓⠒⠂⠀⠐⠒⠂⠂⠚⠀⠐⠓⠀⠀⠓⠀⠑⠋
⠀⠀⠀⠀⠔⠞⠚⠂⠘⠚⠒⠂⠚⠒⠛⠒⠒⠁⠘⠃⠘⠚⠚⠛⠓⣚⣀⠉⠓⠚⠓⠚⠐⠓
⠀⠀⠀⠀⢀⡴⠛⢉⢉⠑⠖⠋⠉⣧⣤⣤⠔⠒⠒⠖⢚⠳⣄⣤⣼⠋⢭⣙⠶⠒⠶⢤
⠀⠀⠀⠀⢿⡈⢂⣛⣭⡆⢟⣁⣤⡤⠜⣇⢰⠶⠶⠿⠈⢀⣉⠉⣦⠒⠄⠈⢙⠓⠶⢆⣟⡛⠲⢄
⠀⠀⠀⠠⣏⠘⢉⣉⠰⠞⠛⢉⣡⢶⠎⠰⠶⠖⠂⠘⠓⠙⣫⢡⡏⠘⡃⠤⣬⣉⡂⢶⣬⣙⡳⢚⠇
⠀⠀⠀⢰⣏⢘⣋⡡⠞⠉⠉⠉⠀⠈⠓⢲⠀⢰⠷⢷⠿⠧⣈⡰⢷⡄⠛⠆⣤⣈⠉⠐⠶⣉⣁⠏
⠀⠀⠀⠀⠉⠉⠁⠀⢻⣎⣉⡉⠉⡉⢉⣉⠉⠛⠙⠒⠒⠚⠉⠉⠉⠉⠛⠒⠒⢥⣘⣠⠾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠙⠙⠀⠁⠈⠩⠭⠥⠌⠩⠭⠭⠄⠑⠘⠙⠊⠁⠀⠀⣇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⡤⢀⠒⢁⡈⠲⣂⢤⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠂⡈⡀⠞⢡⠔⣱⠀⣠⡄⢀⣄⠀⣎⠂⡍⠳⢀⢁⠐⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⢆⠐⣔⣲⢂⡄⡀⠳⠈⣗⡘⢏⢉⠉⡙⢃⣺⠁⠞⢀⢠⡐⣖⣣⢊⡰⣤⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⡉⠇⡊⢉⠋⣎⢲⡦⢆⢼⡉⠓⢀⣋⡩⢍⣙⡈⠛⢋⡇⡰⢴⡖⡰⠙⡉⢑⠘⣉⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠌⠈⢱⠧⡂⣾⣀⣾⠂⣸⢓⣺⣎⣩⣖⣺⣍⣹⣗⡚⣇⠐⣷⣀⡗⢒⢼⡎⢁⠡⣰⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠦⣬⣻⠂⢂⣠⡙⠳⣛⠚⡵⠆⣂⡉⠤⢛⣊⠅⡨⣑⡻⠤⢉⣑⠠⢮⠓⣚⡞⢋⣄⣐⠐⣟⢥⡴
⠀⠀⠀⠀⠀⠀⢀⢼⠳⡝⠂⡘⠀⢡⠼⡁⡩⠀⣧⡾⠭⠈⠃⣉⢀⠘⠓⡀⣙⠘⠁⠭⢷⣼⠀⢍⢈⠣⡬⠀⢃⠰⢫⠞⡧⡀
⠀⠀⠀⠀⠀⠀⡜⠿⠻⢮⠡⠄⢠⠠⠚⠩⢏⠴⣷⠍⣹⢐⣛⣆⣀⡖⢲⣀⣰⡋⡀⣅⡹⣾⠶⡹⠍⢓⠆⡄⠤⠌⡵⠏⠿⢣⢀
⠀⠀⠀⠀⢸⠦⢣⠓⠒⡖⠓⣞⣂⠁⠁⢉⠯⣀⠉⠉⢸⣌⢠⢿⡛⢃⠘⢻⡽⡄⣡⡋⠉⠉⢀⠽⡉⠈⠈⣐⣳⠚⢲⠚⠚⡜⠴⡇
⠀⠀⠀⠈⠄⣶⣜⣄⠀⠮⣬⢷⣿⠳⢀⣮⠶⢭⡄⣰⠌⣶⣆⡲⡖⣀⣀⢲⠖⣜⣶⠠⣆⠀⣭⠶⣽⡀⠞⣿⡼⣤⠵⠀⣠⣣⣶⠠⠁
⠀⠀⠀⠤⢂⣡⡠⢔⠂⠁⡂⢨⣤⣾⡿⠻⢼⣄⡸⠥⡿⠏⠛⠫⡵⠇⠸⢮⠜⠛⠹⢿⢬⢇⣠⠣⠗⣿⣷⣤⠅⢀⠈⠐⡢⢤⣌⡒⠤
⠀⠀⠈⠲⠮⡙⠑⠠⢅⡀⠁⢘⠛⢷⣷⣲⢲⠋⢱⡒⣳⣆⣤⣔⡲⡆⢰⢞⣢⣄⣰⣾⢒⡎⠙⡕⡞⣾⡿⠛⡃⠈⢀⡨⠕⠚⢊⠵⠒
⠀⠀⠀⢀⠂⠿⠫⠋⠀⢖⢓⡾⣿⡴⠈⣟⠶⢞⠂⠹⠂⠿⠏⠥⢦⠉⠉⢼⠮⢹⠿⡐⠏⠐⣳⠶⣻⠁⢦⣿⢳⠛⡶⠀⠙⠝⠿⠐⡀
⠀⠀⠀⠀⢸⠖⡜⠤⡤⠷⠴⢯⢉⡀⡀⣈⣖⠅⣀⣀⢸⢃⡘⢞⣧⡄⣠⣨⣷⢃⡑⢇⣀⣀⠈⣲⣁⣀⢀⠩⡽⣤⠴⠤⢴⢣⠲⡇
⠀⠀⠀⠀⠀⠀⢣⣶⣴⢚⡐⠐⠘⠐⢤⣰⣎⠶⡿⢆⡙⠈⣌⠏⠉⠧⠼⠉⠹⣍⠁⣟⠰⢿⠶⡱⣖⣴⢶⣃⢶⢶⣷⣾⣶⡾⠂
⠀⠀⠀⠀⠀⠀⠈⢺⡴⣣⠆⢡⠀⡘⠲⣁⣙⢀⡟⢳⣒⢀⡄⣉⠀⢄⡤⠁⣉⢠⡀⣷⠲⠶⠐⣖⠺⠟⠛⠛⠟⢸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠖⢳⣽⠀⠈⠉⣡⡼⣯⢥⡳⠆⠍⣃⠒⣬⢭⣊⣁⡍⣥⠒⣈⠩⠘⢟⡽⣽⢧⣈⠋⠉⠁⢻⡛⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⢂⢀⡸⡗⠁⢾⠉⢿⡄⢹⡬⢽⢏⣙⡯⠽⣋⣹⡯⢥⡏⢠⡿⠉⡷⠨⢲⢇⡀⡐⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢁⡄⢅⣈⣄⠆⠬⠗⠎⢸⣡⣤⡈⡍⣑⢊⢩⢁⣤⣌⡇⠱⠺⠇⡹⣠⣁⡨⢴⣉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠛⠎⠠⢫⠽⠈⠃⠁⡔⢀⡯⢡⣎⣈⣁⣡⡈⢼⡀⣦⠈⠈⠡⠯⠝⢄⠱⠛⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠄⢁⠁⢦⣘⠢⣹⠀⠹⠃⠘⠋⠀⢏⠄⣃⡴⠈⡈⠠⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠓⠌⠦⡈⢡⠴⠁⠚⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣋⡥⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢲⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠛⠁⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠾⣇⢀⣀⣤⣤⡿⠓⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⢀⡶⠛⠛⠳⢦⡀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠛⢶
⢸⠁⠀⠀⠀⠀⣙⣃⠀⢀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠹⡀
⢸⠀⠀⢀⡴⠋⠁⠀⠉⠃⠀⠀⠀⠱⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢣⡤⠒⠠⣀⠀⠀⠀⠀⡠⠞⠙⡆⠀⠀⢀
⠈⣇⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠆⠀⢠⠞⠀⠀⠀⠇⠀⣠⡟
⣠⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠞⢠⣧
⢇⠀⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠈⢻⠆⠈⠉⠐⠒⠤⠤⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠖⠊⠉⠀⢴⠞
⠀⢸⡆⢠⠒⠂⠤⠄⣀⣀⣿⠀⠀⣼⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⣷⠀⢰⣏⡠⠤⠔⠲⡄⢸
⠀⠈⡇⠘⡀⠀⠀⠀⠀⠀⠉⠲⠖⠋⠀⠒⠲⠶⠖⠒⠖⠒⠒⠒⠖⠒⠒⠛⠛⠉⣿⠀⣼⠁⠀⠀⠀⢰⠁⢸
⠀⠀⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⣸⠀⡏
⠀⠀⠸⡆⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡏⢰⠃
⠀⠀⠀⢻⡀⢷⠀⠀⣠⠔⠛⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠹⠆⠀⠀⠀⣼⢁⡟
⠀⠀⠀⠈⢷⡈⢧⠀⠘⣄⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢀⢰⠇⠀⠀⣰⢃⡾⠁
⠀⠀⠀⠀⠈⢷⡈⢳⡀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣰⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠻⣆⠙⣄⠀⠀⠀⠀⠀⠀⠀⢠⠉⠉⠉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣄⠀⠀⠀⠀⠀⠈⠳⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣌⡑⠢⢤⣀⡀⠀⠀⠀⠀⢀⣀⡤⢖⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣷⠀⠀⠉⠉⠉⠉⠉⠁⠠⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣍⣒⣠⡤⠦⢤⣤⠤⠤⢤⡤⠤⠤⠤⠶⢶⣒⣫⡽⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠀⠀⡰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢄⠀⠀⡆⠀⠀⠀⠀⠀⢀⡄⠀⢰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢠⠀⠀⠀⠀⣠⡤⢠⠀⠃⠀⣘⠀⢀⡏⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⠦⠂⢠⠊⡜⠁⠈⠀⢡⡀⠘⠦⠚⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⢧⣀⠄⠀⠀⡇⠀⢀⢻⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸⣆⠀⠀⠇⠀⠀⠀⠀⢰⡥⠔⠁⡸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠸⡌⠂⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴⢃⠀⠈⠓⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣫⠟⣠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢀⡈⠑⠊⠿⣶⣲⠤⠶⠦⠖⠶⠭⠐⠋⠁⠀⠒⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠐⠂⠀⠀⠩⣒⡦⢤⣀⡈⢉⠥⠔⠒⠈⠁⠀⡀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠦⣄⣀⠀⢀⠀⠀⠐⠊⠉⠀⠀⠀⠠⠂⢀⡀⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠚⣁⠄⠀⢀⡀⠠⠀⠀⡀⠀⠒⠊⠉⣁⠒⠄⠤⢀⡀
⠀⠀⠀⠀⠀⠀⡰⢚⡑⢂⠀⠀⠀⠀⣀⡠⠤⠒⠊⠁⠤⢀⣐⡦⠤⢀⠀⠀⠀⢀⠞⠩⣁⢀
⠀⠀⠀⠀⠀⠀⠳⣎⣁⣾⠠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⣄⡀⢸⠀⠄⣰⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡆⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⢻⣿⣿⣇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⡀⠀⠀⠀⠀⠈⢻⡟⠀⠀⠻⣿⣿⡄⠀⠀⠀⠀⠈⢻⣿⡟⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣄⡀⠀⠀⢀⣿⡀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⡿⠃⠈⣿⣿⣿⣿⡿⠋⣀⡴⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣦⣤⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⣿⣿⡿⠋⣠⡾⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⣿⣿⣿⣿⣿⣿⠈⠻⣿⣿⡄⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⣰⠟⠉⠐⠚⣁⡤⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⡆⠀⠀⠉⠀⣴⣄⠸⣿⣿⣆⠀⡀⠀⣠⣾⣷⡶⣶⣶⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⠻⣿⣿⣿⣷⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣾⣷⠈⠉⠉⠀⣰⣿⣿⣿⠏⢠⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⢿⣿⣧⠀⠀⢻⣦⣿⣿⣿⣿⣿⠟⢁⠀⠀⣠⣾⣿⡿⠟⢁⣴⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣶⣄⡙⠻⣷⣄⡘⢿⣿⣿⣿⣿⣿⡿⢏⣤⣾⡿⠟⢉⣠⣴⣟⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣿⣿⣿⣿⣿⣶⣄⡉⠛⠚⠻⠿⠿⠿⠟⠛⢛⣉⣥⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣄⠻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠰⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⢀⠀⣠⣤⣄⠠⠀⠀⠀⠀⠀⠀⠀⡼⠟⢂⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠠⠞⠾⠿⢟⣛⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡟⠀⠀⠀⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⠀⠀⠀⠀⢀⣴
⠀⠀⡆⠆⠂⠐⠂⠚⠛⡛⠉⡃⠻⠁⠀⠀⠀⣠⣆⣿⣶⣾⣿⣷⣿⡇⠀⠀⣿⣿⠀⠆⢐⣶⣿⣿⣧⡀⠈⠻⣧⣌⣣⣌⣿⣿⡇⠀⢴⣿⣷⣿⣇⣴⣿⡿
⠀⠀⠀⡰⠞⠀⠐⠂⠚⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⡈⣻⣿⣿⣿⣿⠇⠀⠀⣿⣿⠻⡞⢿⣿⣿⣿⣿⣿⣦⡀⠀⣀⣀⡀⠙⣿⡇⢰⣿⠿⡻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⣿⣿⡤⢼⣿⣿⣿⣿⣏⠛⠛⠻⣦⡙⠊⠀⣼⣿⡇⢸⣿⣿⣶⣿⣯⣶⣶⣶
⣿⣿⡿⠛⣿⣿⡛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⢙⣫⣽⡟⢻⣿⣿⣿⣿⣿⣿⠘⣿⣮⢉⣼⣿⡇⡈⢻⡿⠋⠀⠀⠀⠀⠐
⡿⠋⠀⣲⣿⣿⢧⣄⠈⠙⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⡁⡆⠀⢠⣿⣿⣿⡄⣼⣿⣌⡏⢻⣿⣦⣿⣿⢿⣿⣿⡧⡂⠀⣅⠀⠀⠀⠀⣼⡶⠇
⢀⡤⢿⡏⢹⣿⣿⣿⣷⡄⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⣇⢸⠀⡼⠏⢻⡏⣄⣼⣿⣿⣷⣀⠙⠛⠛⢁⡼⣿⣿⣗⠀⢠⡿⣷⣤⣴⣾⠟⠁
⠰⠳⠎⠁⠸⣿⠟⢿⠟⠛⢺⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣧⣿⣼⣬⠃⠰⠖⠻⠛⣿⣿⣿⣿⣿⣉⣿⣶⣾⣿⠀⢹⣿⣗⠀⣾⡇⢸⡙⢩⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠀⠈⣿⡧⢸⣿⣇⣀⣉⣉⣀⣀⣀
⣿⣿⣿⣿⠿⣿⢿⣿⣿⢿⡿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣀⣤⡀⢸⣧⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣯⣤⣥⣧⣧⣤⣥⣥⣧⣴⣦⡘⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣇⣀⡽⠿⠇⠉⠉⣩⠟⠁
⣿⣿⣿⡏⠉⠏⠫⠛⢫⡏⢻⣟⡟⠝⠛⡝⠻⠉⣟⣯⢋⢋⡍⡙⢋⢹⠋⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣻⣯⣙⣀⣹⣿⣿⠏⣿⡄⠀⠠⠾⠷⠶⠶⢶⣦
⣿⣿⣿⡿⠿⠿⣿⣿⢿⠿⡿⣿⢿⠿⣿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣧⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⣿⣧⣤⣴⣬⣼⣤⣧⣷⣧⣥⣧⣧⣬⣴⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⡀⣿⡏⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣷⡟⠁⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⡇⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠃⠀⢀⣾⠟⠁⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣌⠀⠀⣠⣿⡏⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣠⡾⢻⡿⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣠⣾⡿⣡⣿⣧⠆⠘⠁⢸⣿⠀⡀⠀⠷⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢻⣟⣵⣿⣿⣇⠀⠀⣠⡾⣿⣿⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣿⣿⣿⣿⣿⣦⡾⣽⣿⣹⣟⡽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢯⣾⡟⣱⣿⣿⣿⣿⡟⣱⣿⣿⣧⠻⣯⠜⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⣿⢣⣿⡟⣼⢿⣯⣿⣿⣽⢣⠟⣿⣿⠹⣦⠙⢧⣀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⡀⡆⢸⣿⠃⣿⡿⢰⡟⣿⡿⣿⣿⣏⠸⡆⢸⣿⣿⣿⡄⡈⢻⣧⡈⢀⣳⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣼⣿⡇⣿⢱⣿⣿⣿⣿⡿⠇⠁⣾⡈⣿⣿⣧⢰⡀⢻⣿⣆⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⣿⣿⣿⠀⢿⣰⡇⡿⠁⣘⡇⢀⣼⣿⠇⢸⢻⣿⢸⡇⠀⣿⣿⠆⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢿⡄⢿⢻⣿⣿⣀⢸⣿⡇⠀⠀⣧⣷⠘⢿⡏⠀⣿⢹⡏⣸⣇⡀⢸⣿⡇⠀⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⣷⣿⣿⣿⣌⣿⣧⣀⠀⠻⣿⠆⠈⣿⣰⢃⡾⢡⣿⣿⡁⢸⣿⡇⠀⣿⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠻⣿⡿⠿⣧⣼⡻⣷⡄⠉⠀⣰⣯⣿⡾⢡⡞⢹⣿⠁⣸⣿⡇⠀⣿⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣙⠓⠪⡿⠶⣿⣷⣿⣮⡻⣦⣬⣿⡿⠋⠐⠋⠀⣼⠟⣰⣿⡿⠁⢠⡏⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢽⣿⣿⣿⣷⣬⣉⠉⠁⠀⠀⠴⠊⣅⣼⣿⠋⠀⢠⠟⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⡏⢿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⡿⠟⢀⡴⢋⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣽⣿⣿⣿⣿⣿⣿⠿⢛⣋⣤⠶⠋⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣷⠶⣿⠶⠶⠊

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡟⠿⡿⠛⡿⠿⣿⠿⢿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣷⣾⣶⣷⣶⣿⣶⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠛⣯⣏⡂⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣩⠤⣤⢤⢦⣤⣬⣬⣈⣁⣈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠥⠦⣤⣴⡃⣈⠘⣋⢈⢉⣙⣋⣛⡿⠗⠢⢤⡤⢤⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠍⠛⢛⠛⠻⠯⠭⠤⠤⢄⣤⠞⠋⣠⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡚⠉⠀⢂⠉⠀⠀⠀⠀⠉⠙⠁⠀⢀⠀⠙⡆⣼⢛⣿⡈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠁⠨⢣⣿⣼⢹⣧⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠕⡹⣿⢧⡀⠻⣾⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠎⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣦⣡⣹⡌⠻⣄⡹⣎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠀⢀⣀⣉⠓⠦⣈⠳⢾⣚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡠⠒⠒⠒⢤⡌⠅⠀⠈⢳⠀⠈⠓⣄⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⠀⠸⠀⠀⠀⠀⠄⠱⠜⡠⠒⢀⠏⣠⠓⠌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⠀⠀⡴⠂⠳⡤⣠⡀⢄⠀⣧⠐⡀⡾⠞⠃⠀⢹⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡆⠀⢸⠁⠀⠀⠴⠠⠍⢂⠲⣿⣠⡞⣳⣗⠂⠵⢇⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠀⠘⢤⣤⡤⢠⣤⡽⢒⣿⢼⡟⠻⢥⡴⠍⡑⠀⢥⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⡇⠀⢷⡀⠀⢰⠂⢀⡠⢈⠐⢠⣠⠁⢿⡻⠐⠈⢢⡄⣄⠌⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣣⠀⠈⢱⡀⠸⡄⠈⠀⠀⠁⡨⢋⣠⠀⢇⠁⡰⠀⢉⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠊⠄⣀⠃⠄⠨⠤⠤⠘⣧⡀⠈⣻⣄⠈⠒⠤⠤⢞⠁⠃⠘⠤⢸⣀⣘⡦⠃⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠲⠶⠶⠆⠨⠭⠭⠭⠵⠖⠺⠛⠶⢾⣭⣿⡶⣄⡀⠀⠙⠒⠒⠚⠁⠈⠁⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠁⠬⠷⢶⡴⠴⠶⠶⠚⠉




⡟⠉⢹⡟⠻⠿⠿⠟⠿⡛⠿⠻⠿⠿⠟⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⢰⡇⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣾⣿⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡜⣡⣾⣿⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣴⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⣄⢠⢣⣿⣿⣿⣿⠟⠋
⠀⣸⣦⡄⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⡤⢊⣡⣴⣶⣿⣿⣿⣿⣶⣝⣿⣿⣿⠟⠁
⠀⢿⣿⣿⣿⣿⣷⣦⣭⣑⣒⣲⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢎⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⣤⣥⣵⣿⣿⣿⣿⣿⣿⣿⡇⠀⡰⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠉⠛⠙⠛⠉⠉⠀⣹⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢔⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⣿⣿⠏⠀⣿⣿⣿⣝⣿⣿⡟⣿⣿⣬⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣇⠀⠐⣿⣿⡿⢿⣿⣿⣿⣝⠏⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡞⣿⣿⣿⣿⣿⣷⢠⡀
⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣯⠝⠋⡀⠀⠹⣎⣛⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⢸⠁⣽⣿⣿⣿⠹⡟⢰⡇
⠀⠀⠀⠀⠀⠀⠸⠛⣿⡿⠇⠀⠐⠀⠀⠀⠹⠿⠟⠻⣿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠃⡟⠀⢱⣿⠻⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄⠀⠀⡌⢠⣿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⡸⠁⠀⠀⢳⡀⠙⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠻⠿⠿⠃⢀⠜⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⡟⠁⠀⠀⠀⠀⠙⢦⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠲⡟⠲⡞⢇⢀⣼⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡄⠀⡘⣦⡙⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⡿⣉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⡼⣿⣿⣿⣿⣿⣧⢻⣿⡿⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢤⣴⣾⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣎⠭⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠐⠀⠘⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠺⣻⣿⠿⢿⣛⣻⣯⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠾⣄⣀⡰⠤⠜
⠀⠀⢪⡀⠀⠀⠀⣀⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⠛⠉⢩⢤⡍⠘⠩⠞⠁
⠀⠀⠀⠈⠉⠓⠚⠁⠰⡀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠼⠤⠂
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⣀⠴⣲⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢼⠀⠈⢆⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠢⣈⣦⣤⣜⠝⠻⣖⡠⢤⡤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠓⠄⠙⠄⠀⠈⡻⢭⣅⠂⠐⢤⡈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡳⠃⠀⠀⠀⠀⠀⠀⠈⠢⠘⢷⣗⣄⢧⡈⢀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⡀⠹⣯⣎⢿⣯⣧⢷
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡠⡹⣇⣜⣿⣿⣾⡇⠀⠀⢀⡴⡒⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡵⠣⢏⢹⣻⣿⡿⣧⠴⣊⡏⣥⡼⣌⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢱⣀⡼⠞⣫⣥⣤⢷⡶⢾⣶⣚⣒⣾⣼⣡
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⣨⣼⠟⢃⠴⠯⢳⢈⣉⣘⣋⣹⡿⣽⣿⣯⣿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢀⠀⠀⢀⠀⠈⢰⢟⡋⠁⣴⡻⠚⠳⣼⠿⢯⣤⣼⣭⣭⣤⣤⢭⡼⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠁⠈⠀⠈⠂⠈⢦⡉⠢⣿⡷⠜⢿⠧⠶⣾⢫⣬⡿⣭⠿⢻⡾⢃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⡀⢠⡀⠀⢰⣿⣿⣾⣥⡝⠻⠿⣾⣶⣾⣇⢙⣯⣉⡾⠋⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠛⠛⠋⠉⠁⠀⠈⠓⠤⣀⣀⣀⣉⣁⡠⠤⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡟⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠑⠦⣄⠈⢷⡈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠻⣎⠻⣦⣀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠸⣆⠀⠀⠀⠀⢹⡆⠈⠙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠙⣦⠀⠹⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⠁⠀⠘⣇⠀⠘⢷⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢻⡀⢀⡾⠃⠀⠀⠀⠘⢧⣀⣸⡇⠀⢻⣆⠀⠀⠀⠀⠀⠀⣇⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡼⠃⣾⡁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠙⢦⡀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⠀⠀⠈⠻⢦⣄⣀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠻⢦⣄⡀⠀⠀⠹⣦⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣆⠀⠀⠀⠀⠈⠉⢿⠓⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡟⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣰⠃⠀⠀⠀⠀⣠⠞⣿⠇⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⡆⠀⢀⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣠⠞⠁⠀⢀⣤⠶⠛⠁⣠⡟⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⡇⢰⡏⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠐⣇⠀⣤⠞⠋⠀⠀⠀⠰⠋⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡇⢸⡇⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠈⣻⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⡀⠀⡼⠃⠀⢀⡼⠁⢸⡇⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢰⡄⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⡇⠊⠀⠀⠀⣾⠤⠒⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⡄⠀⠰⠾⣡⣶⣟⣯⣤⣦⠀⠀⠈⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣦⡀⠀⠀⠀⠀⢀⡀⠛⢿⣷⣦⣤⣠⣴⣿⣷⣿⣿⠟⠃⠀⠄⠀⠂⠈⢁⣤⣶⡆⠀⠀⠀⠀⠀⣠⡟⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠶⠾⠿⣶⣤⡀⠰⠄⠠⣤⠄⢈⣩⣥⣾⠿⠿⠛⢉⣤⣄⡀⠰⣶⡿⠿⣿⣛⡉⣁⡀⠀⠀⢀⣤⡾⠛⠛⠛⠛⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠒⠛⠛⠋⠉⠉⢛⣿⣶⣦⣤⣤⣿⡿⠛⠿⣶⣶⣦⣬⣽⣿⣿⣿⣶⣄⠈⠙⠛⣛⣩⣷⡶⢾⣿⣿⣿⣿⡿⠛⠛⠛⠛⠉⣿
⠀⠀⠀⠀⠀⠀⠀⢧⣤⣤⣤⠶⠶⠒⠛⠛⠋⠉⠀⠀⠀⠀⠀⣀⡤⣶⣿⣿⣿⡿⠿⢿⣿⣿⣛⣛⣻⣛⡻⠿⢿⣶⡶⠶⠦⠬⣭⡉⠛⠛⠓⠶⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠋⢁⣚⡯⠟⠛⠉⣀⣤⣄⠀⠀⠀⠉⠁⠉⠉⠙⠒⠲⠤⣀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⣻⣅⣤⡤⠶⠚⠋⠉⠁⠀⠰⠞⠛⣁⣀⣘⠛⠉⠙⠛⠒⠒⢦⣤⣤⣤⡶⠒⠾⢭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋⠉
⣀⡀⣀⣀⡀⣀⣀⣀⢀⣀⢀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣦⡙⢿⣷⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡁⠀⠀⠀⠀⠀⢹⣿⣿⣷⡌⣿⡆⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢀⣦⠀⠀⠀⠀⠀⣿⣿⣿⡇⠸⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠙⢶⣦⡀⠀⠀⠀⠀⠸⠀⣾⣿⡀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠋⢻⡀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣀⣤⣴⣿⣿⣿⠁⠀⠘⣷⣦⡀⠀⠀⠹⣿⣷⡄⠹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣆⠀⠀⢿⡇⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⢠⣿⣿⠃⢀⡀⠀⠀⠀⠹⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠀⠀⠈⠇⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣿⣿⡏⠀⠈⣿⣦⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡄⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣠⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣷⣄⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣖⣲⣶⣶⣶⣶⡄⢀⣶⣶⣾⣯⣴⣶⣶⣶⣶⣶⣶⣶⣶⠆⡼⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⡿⢟⣵⣿⣿⣿⡿⢿⣏⣭⣭⣿⠟⢻⡿⠿⠛⠛⠀⠈⠁⠚⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡝⢻⣿⢿⣿⡇⠰⣿⣿⣿⣿⠋⠁⢀⣠⠄⢀⣀⠀⠀⠀⣠⣶⣶⣄⣠⣄⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢣⣴⢶⢶⣄⢷⣄⠙⠿⠃⠉⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠸⣿⣿⣿⠟⠀⣻⡷⠀⣸⣷⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⢀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢹⠏⠁⠀⠀⣛⣥⡶⢰⠂⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⢠⣶⣤⣾⣿⡿⠙⠘⢀⣼⣿⣿⣿⡟⣿⣿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣯⠈⠻⣯⣿⣿⠿⠛⠛⠻⣶⠳⠶⢾⣿⣧⣀⠀⠀⠀⠀⠘⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡼⠋⣠⠀⣹⡟⠁⢂⣀⡀⠈⠉⠀⠀⠠⠉⠛⠿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⡀⢻⣾⠟⠁⢆⠀⠈⠁⠁⠀⠀⠀⠀⠁⠸⠗⡄⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣙⣡⡮⠨⠀⠈⢀⠀⠀⠀⠁⠀⠀⠀⠐⠀⣀⡐⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⣦⣤⠀⠀⠑⠂⢀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠠⢸⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣼⡇⠀⢿⣿⠓⢄⠀⠀⠊⠻⠷⣤⣤⡤⠠⣾⣷⣆⠀⠀⠙⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⡇⠀⢸⣿⡔⣆⢀⢀⣀⠀⠀⠀⠀⠀⣤⣼⣿⡘⠂⢠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣤⣠⣼⣿⣥⣬⣿⣿⡛⠱⣤⣤⣠⣶⣿⢩⣿⣧⠄⠁⣬⠸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡟⣿⣽⢻⣿⣿⣿⣿⣿⣥⣌⣉⣁⡿⠿⣿⣿⣿⢻⡄⣢⠽⠺⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠙⠛⠿⠒⠛⠁⣿⡟⣿⣟⡉⠀⢈⣿⣿⣿⣿⣿⢻⡇⠀⠂⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⣖⣲⡆⢀⣇⢀⡳⠄⠀⢀⣀⠈⠉⠉⠉⠋⠸⠇⡰⣤⣨⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠉⠉⢡⠊⠉⠁⠉⠀⠀⠉⠉⣻⠀⠀⠈⠀⠀⢈⡟⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣥⣤⣤⣄⡀⠀⠀⠀⢠⡀⢽⣷⣶⣶⣶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠉⠙⠛⠏⠉⠏⠱⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣰⢾⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢰⠟⢷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡟⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠸⠁⠈⠀⠀⠀⠀⣀⡀⢀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠁⠀⠀⠀⢀⠀⣠⢤⣀⡴
⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠻⢦⡀⠀⡠⠖⣷⠞⠙⠿⠛⠋⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⡀⠀⠒⠉⠟⠁⠈⠉
⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠀⠀⣀⣀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣠⡀
⠀⠀⠀⣰⢿⠋⠉⣷⣠⡴⠞⠋⠉⠁⠀⣹⠀⠉⠻⠷⠄⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⠉
⠀⠀⠘⢷⡈⢷⣾⠟⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢷⡞⢁⣠⠶⠛⠓⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡿⢿⣿⠿⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠖⠒⠲⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⠯⠥⠤⢤⣀⡀⠀⠀⢀⣀⣤⣤⣈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠋⡼⠁⠀⠀⡀⠀⠀⠉⠙⠉⠉⣀⣤⠀⠈⢦⠀⠀⠙⢦⣀⠤⠤⣤⢄⣀⡀
⠀⠀⠀⠀⠀⣴⣃⣀⣰⠃⠀⠀⠻⠟⠀⠀⠢⠔⠀⠀⠈⠉⠀⠀⠀⠳⣤⣔⠷⠐⠒⣈⠁⣠⡖⢉⣦⢄
⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠠⠤⠤⠀⣀⣀⣀⣀⣤⣄⣀⠀⠀⢺⠀⡔⠉⣥⠘⣧⠀⣬⠀⣩⠱⠮⢡⡌⢂
⠀⠀⠀⣼⠇⠀⠀⠀⣀⣀⠀⠀⡠⢊⡥⠞⠋⠉⠉⠓⠺⣵⡄⠈⣰⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⢠⠏⡇⠀⠀⠀⠀⠀⠀⡼⢡⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢸⠀⢳⠀⠀⠀⠰⣆⢰⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢸⠀⢸⠂⠀⣀⡤⠉⠈⣇⣇⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠘⡆⢸⠀⠀⠁⠀⠠⣤⡜⢿⣆⠀⠀⠀⠀⠀⠀⠀⢉⡼⠃⢡⣀⠔⢄⡀⠀⠀⠀⠀⢀⢀⠀⠀⠀⡄⠒⠈
⠀⠀⠀⢳⢸⠀⠀⠀⠀⢠⡀⠀⠀⠙⠳⠦⣤⣀⣠⠤⠖⠋⢀⣘⠀⠀⠀⢀⣀⡈⢇⡀⡠⠃⣀⡉⠂⠈⠀⡇
⠀⠀⠀⠈⢯⢧⡀⠀⠀⠀⠙⠀⠀⣷⠀⠀⡀⠀⠀⢦⡀⠈⢹⠻⡆⠀⠀⠘⠻⠃⠀⠴⠂⠀⠛⠃⠀⡆⢰
⠀⠀⠀⠀⠀⠳⡝⠢⢤⣀⡀⠀⠀⠁⠐⠛⠁⠰⣄⠀⠁⠀⡜⠀⢳⠀⠃⠀⠀⡀⠀⡄⠀⢀⠀⠀⢰⠀⡂
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠉⢢⡀⠀⠀⠀⠀⠀⡸⠋⠙⠾⠥⡖⠺⠋⠉⢢⠀⠁⠀⠃⠀⠈⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡙⢦⣀⣀⣀⡤⢧⠀⠀⠀⢀⡇⠀⠀⠀⣸⣀⠀⠀⠀⠀⢀⣀⡀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠳⠶⠶⠶⠟⠒⠒⠋⠉⠓⠒⠋



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

⠀⠀⠀⠀⠺⣟⡛⠛⢲⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠙⢾⡓⠿⠉⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣤⢤⣤⢬⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠦⠉⠃⠉⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢈⣛⣶⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⡏⠉⠛⠿⣍⡉⠉⠛⠛⢩⣭⡭⠿⣿⣿⠳⢶⡶⠶⠶⠒⣶⣤⠴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⡀⠀⠀⢸⡇⠀⠀⣤⣈⡙⢄⠀⠀⠀⠬⣽⣿⡗⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠜⠉⠀⠀⠀⠀⢷⠀⠀⠞⠀⠀⠀⠀⠀⠀⠈⣿⡋⠉⠀⠀⠀⠘⠁⢀⣠⡤⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢻⣻⣿⡹⠀⠀⠀⠀⠀⢸⣿⣿⣳⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠉⠁⠀⡀⠀⢀⠀⠠⠉⠉⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠘⠀⣀⣉⣶⣏⣀⠐⠀⠂⢀⣠⠴⣿⠋
⠀⠀⠀⢀⣾⠇⠀⠀⠀⣸⣿⣦⡀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠉⠓⢦⣄⣀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠘⢧⡀
⠀⢀⡤⠞⠁⢀⡀⠀⢰⡟⠀⠈⠻⣦⡀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⣴⢋⡤⠖⢠⡏⠀⢀⡿⠀⠀⠀⠀⢸⣿⡋⣀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠈⠙⠷⠤⠴⠛⠛⠛⠁⠀⠀⣀⡴⠋⠁⠘⢛⣷⡄⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⢀⣰⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⣢⡀⠀⣨⣿⠿⠒⠛⠋⠛⠓⠶⠶⠶⠶⠚⠛⠋⠉⠛⠻⣦⡀⠀⢀⡐⢿⣿⠑⠀⠈⠛⢦⣄⡀
⠀⢀⡤⠶⠖⠚⠛⠉⠁⠀⠀⠀⢀⣾⠿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⡟⠃⠀⠀⠀⣶⣄⣈⠙⠳⠶⣤⣀⡀
⠀⠿⠧⣤⠖⠀⠀⣠⡆⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣼⡋⠀⠁⢀⠀⢠⣄⡀⠉⠙⢶
⠀⠀⠀⠻⠤⠴⠚⠁⠹⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡅⠈⣳⣄⢈⡿⠶⠖⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣯⣥⣬⣽⣧⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿