Soak Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⣹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣤⣶⣤⡀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣤⣤⡀⠀⣠⣤⣄⠀⢀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠷⠿⠿⠿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⠇⣾⣷⣬⣹⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣛⣋⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣛⣛⣛⣛⣃⡙⠿⠟⠛⣿⣿⣿⡿⠛⠿⠟⣁⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣙⣋⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢰⣾⣦⢹⣿⣿⣿⣿⣿⡏⣴⣿⡆⢻⣿⣿⣿⣿⣿⢡⣾⣶⢸⣿⣿⣿⣿⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠘⣿⣿⡀⣿⣿⣿⣿⣿⡇⣿⣿⡇⢸⣿⣿⣿⣿⡇⣸⣿⡿⢸⣿⣿⣿⠏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⡇⢻⣿⣿⣿⣿⡇⣿⣿⡇⢸⣿⣿⣿⣿⠃⣿⣿⡇⣿⣿⣿⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⢸⣿⣷⢸⣿⣿⣿⣿⡇⣿⣿⡇⢸⣿⣿⣿⣿⢰⣿⣿⢁⣿⣿⣿⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠸⣿⣿⡈⣿⣿⣿⣿⡇⣿⣿⡇⢸⣿⣿⣿⡏⣸⣿⡿⢸⣿⣿⠟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠂⠻⠿⠃⠻⠿⠟⠻⠃⠻⠛⠃⠸⠟⠛⠿⠃⠛⠻⠃⠺⠶⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⢄⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠙⢲⠏⠉⠓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠄⠀⠀⠀⠘⡆⠀⠰⡦⢄⡉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡀⢹⣄⣈⠱⠦⠄⠈⢑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠁⢉⢏⠟⣄⠀⢀⣀⡤⠞⠓⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠎⠀⠈⠋⠉⠀⠀⠀⠀⢀⠠⢻⡖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡰⠋⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠫⠀⠀⠀⠀⠀⣠⠊⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠎⠀⢠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⢠⠞⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠎⠀⢠⠃⠀⠈⠑⡦⠀⡞⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣕⢦⡀⠀⠀⢀⠔⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠎⠀⠀⣇⠀⠀⠀⡼⠀⣼⡃⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢝⣢⡴⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡎⠀⠀⠀⠈⠓⠢⠚⠁⡼⢹⠀⠀⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⣎⡎⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡏⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⢎⡞⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⢀⡞⡎⠀⠀⠀⠀⡀⠀⠀⠀⢠⠃⠀⡜⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⢀⣌⡜⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⠇⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠋⠙⠋⠋⠉⠉⠉⠙⠉⠉⠉⠙⠛⠛⠛⠋⠙⠉⠛⠙⠛⠉⠉⠉⠉⠉⠛⠙⠙⠛⢙⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⣶⣶⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣶⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⣄⣉⠙⠛⠋⣹⠿⢷⣤⣤⣤⣤⠾⠿⡟⠟⠿⠿⠿⠛⠛⢩⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠿⢿⣿⣶⣿⡀⢘⣿⠿⠿⣧⣠⣄⣿⣶⣶⣶⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣶⣤⣄⣉⣉⣉⣭⣵⣶⣶⣮⣽⣭⣁⣉⣉⣉⡉⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢘⣃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡄⣠⠀⠀⠀⠀⢠⣶⠂⠀⠠⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⡀⢧⠀⠀⠀⠀⢠⣶⡆⠀⠔⠂⡀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⢤⠀⠀⠀⢈⠇⡸⢀⡇⠀⠀⠀⠀⠰⠿⢗⣤⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣤⣤⣶⣶⣿⣿⣶⣦⣤⣄⣀⣠⣤⣶⣶⣶⣶⣤⣀⣀⣀⣠⣤⣶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⠉⣿⣿⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣯⣿⣽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡰⢺⢴⡀⢦⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⡁⢧⣎⣡⠋⡗⠀⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂⢣⠁⠈⠘⠣⠄⡸⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡼⣅⠀⡀⢠⣷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠈⠂⠀⠘⠡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠄⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡜⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠤⠄⠊⡐⠀⠀⠀⠰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠰⡀⠀⠀⠀⡁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢠⠀⡘⠀⠀⠀⠀⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⣀⠐⠀⠀⠀⢀⡌⢅⠂⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢡⠔⠈⠃⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡃⠀⠀⠀⠄⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⠁⢀⠀⠄⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠈⠀⠀⠀⡀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠢⢄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⡄⠀⠀⠈⢀⡁⢸⢡
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠰⣹⣿⣷⠡⠀⠀⡇⠀⠀⠀⠀⠀⢀⠄⠀⡀⡁⠀⢀⠔⡏⠀⠄⡐
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠉⠉⠙⠀⠀⢀⡇⠀⠀⠀⠀⠀⠈⠀⣦⣄⠁⡠⠃⠠⠄⡐⡎⢰
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡠⢠⡞⠁⠀⠀⠀⠈⠄⠀⡰⠁⠀⠛⠁⢀⡴⠒⠹⣍⠢
⠀⠀⠀⠀⠀⠈⠐⠂⠤⢠⣀⣀⣴⠗⠋⠐⢂⣾⣿⣙⣾⠃⠀⡇⠀⠤⠤⠴⠻⣄⣦⣧⡌⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠤⢰⣶⡖⢣⣤⣤⡋⠈⠉⠻⣇⢄⣀⣀⣀⣀⣈⣿⡤⢧⣾⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠛⠃⠚⠛⠣⠽⠿⠅⠈⠃⠀⠉⠉⠉⠉⠉⠙⠁⠛⠛⠚⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣼⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢛⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠆⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢠⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣤⣄⠀⠀⠀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣯⣿⣧⣤⡀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠸⣧⠀⢸⡏⣿⣿⣿⣿⣿⡏⢸⡇⢈⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⣇⠀⠀⠙⠁⢸⡇⢹⡷⣿⠃⣿⣹⠸⣇⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⢀⣤⠞⠛⠉⠉⣿⠟⠳⣤⠀⣸⠇⠘⠃⢹⠀⠟⢿⠀⣿⡸⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⡆⢹⠄⠀⠀⠛⢇⠾⠖⠈⠙⠶⢄⠀⠀⠛⢁⢿⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⡀⢱⡎⠀⠀⠀⢀⣿⠂⠀⠀⠀⠀⠈⣳⠀⠀⢸⠀⠇⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠉⠉⠉⠉⠁⠈⠉⠻⣿⠿⡿⢿⡿⠛⠛⠛⠒⣿⠀⠀⢸⡷⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣿⣦⡀⠀⠀⣠⡴⠃⠀⠀⢸⡅⠀⠀⠉⢉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⣀⡴⠟⡙⠳⠶⠛⣉⣀⠀⠀⠀⠀⢸⣅⣴⠶⠛⢛⠛⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡇⠀⠀⠀⣸⠀⠉⢙⣷⠾⠛⠛⠉⠀⢰⣧⠀⠀⠀⢉⠁⠀⠀⠀⠀⠸⠋⢁⣴⠿⣿⠿⠆⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠋⠀⠀⢀⣿⣤⠾⠋⠁⠀⡀⠀⠀⠀⣸⠟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣂⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⣶⡿⢨⠀⠀⠀⠀⠠⣷⣤⣶⡄⢻⠈⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠘⡿⠾⢿⡟⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣶⠟⠋⠀⠙⠟⠀⠀⠀⢀⡄⢿⠁⣸⡇⢼⡆⠀⠀⢸⣄⡿⠀⠀⠀⠀⣶⠀⠀⣧⢄⣸⠃⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣸⢀⣿⠁⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⡘⢧⠀⠀⣿⠀⠉⠀⠀⢸⣿
⠀⠀⠀⠀⢀⣴⠿⠋⣿⠀⠇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⠃⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⠀⠀⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣰⡟⠁⠀⠀⠘⠛⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⢠⡿⠀⠀⠀⢠⡄⠀⠀⠀⣼⠀⠀⠀⠙⠁⠀⠀⣿⠀⠀⠀⢰⡇⣹
⠀⠀⢰⣿⠀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠁⠀⠀⢠⡈⢠⣿⠃⠀⠀⠀⢸⡇⠀⠀⢀⠻⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⣤⣻⣷
⠀⠀⠘⣿⣀⣿⠀⠀⠀⠀⢰⡏⠀⠀⠀⣇⣠⢀⡶⠉⣠⡿⠃⠀⠀⠀⠀⣿⠁⠀⠀⣼⡀⡆⠀⠀⠀⠀⠀⢸⡇⠀⡆⠀⠀⠽⠿⣇
⠀⠀⠀⠹⡿⢇⡀⠀⠀⠀⡌⣣⠀⠀⠀⠛⠛⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⣹⡄⠀⠀⠙⠛⠀⠀⠀⡀⠀⠀⣸⡇⢰⣧⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⢠⣾⠻⠶⣦⣤⣷⣯⣤⣤⣤⣤⣤⡾⠟⢁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⣧⠀⢀⣿⠁⢸⢹⠀⠀⢀⣤⣾⠋
⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⠀⢸⡟⠀⣧⢸⡶⠞⠋⠀⠘⣷
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⡄⣿⠃⠀⠙⠋⠀⡀⢀⣠⣴⠟
⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠘⢿⣷⠶⣶⠶⠶⣾⣃⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠻⠇⣽⡟⠀⠀⠀⠀⠀⣿⠈⠉⠙⠶⣄⢀⡆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⠏⠀⣿⠀⠀⠘⠛⠁⠙⠻⢿⣷⣶⣄⣀⠀⢹⡇⠀⠀⠀⠀⠀⣸⣟⠀⠀⠀⠀⠀⣠⡟⠀⠀⢀⣴⠏⢸⡇
⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⣘⣦⡀⢻⣆⠀⠀⠀⠀⠀⠀⠈⣿⠿⣠⣽⠿⢻⡇⠀⠀⠀⠀⣴⣿⣻⣶⠀⠀⠀⠀⡹⠀⠀⠀⠈⠙⣷⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠛⠚⣿⡆⠀⠀⠀⠀⠀⣾⠈⡀⠈⠻⣾⣤⠇⠀⠀⠀⣼⡟⢉⣻⣿⠃⠀⠀⠀⠻⠟⠀⠀⠀⣴⠟⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢘⣷⠶⠀⠸⣷⡀⠀⠀⠀⠀⢿⡴⠃⠀⠀⠀⠁⠀⠀⢠⣾⣿⣧⢸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⣅⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣹⢿⡟⣻⣿⣋⣀⣀⣀⣀⣀⣤⣀⣀⣠⣼⡇
⠀⠀⠀⠀⢸⡆⠀⠀⠀⣾⢡⠀⣾⠋⠀⠐⣦⣽⠗⢻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣸⡇⠘⣺⡋⢉⣟⠉⠉⠉⠉⠙⣿⠉⠉⠉⢻⡄
⠀⠀⠀⠀⠘⣿⠀⠀⠀⣷⡀⠰⣿⡀⠀⠀⡞⠉⡆⢸⡍⠛⣶⣤⣤⣤⣤⣴⠟⠁⠀⢠⡿⠀⢰⡿⠃⠈⠛⢻⡟⠻⡷⢿⢹⣶⠻⢶⣟⠁
⠀⠀⠀⢠⡆⢿⠀⠀⠀⠉⠁⠀⢽⣟⡛⢻⡇⠀⠑⢾⣦⡼⠛⣯⠀⢀⣿⢷⡄⠀⣀⣾⠃⣠⡟⠁⢐⡀⠀⢸⣯⠀⣷⣾⡘⢿⣄⠀⠉⢷⡀
⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⢠⡿⠋⢻⠃⠀⠀⢸⡇⠀⠀⢻⣤⡾⠃⠀⠹⣿⢿⡏⠰⠋⠀⠀⢸⣇⠀⢘⣿⣴⠟⢿⡇⢈⣿⣷⣤⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢺⡅⠀⣿⠀⠀⠀⢸⣧⣄⣀⣼⠛⣇⣀⣀⣠⣿⣿⠁⠀⠀⠀⠀⡄⢻⠀⠘⣿⠁⠀⠀⠛⠋⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⢸⣷⠀⠙⣿⡇⠀⠀⠀⡾⡏⠉⠉⣿⠀⡟⠉⠉⢙⢹⣿⠀⠀⠀⢀⠀⣧⡘⠀⢠⣾⠀⠀⠀⠀⢀⡸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⡀⣿⠀⠀⠙⢷⣤⡀⢐⡇⠉⠀⣰⠏⣀⣽⠀⠀⢸⣼⣿⠀⠀⠀⢸⡀⠈⠁⠀⠀⢀⣠⣤⡶⠞⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⣧⡿⠀⠀⠀⠸⣇⠘⠛⠷⠀⣰⠏⠀⣿⡟⣀⠀⠀⠈⣿⣀⣀⣀⡸⣧⣤⡶⠶⠟⠛⠉⠀⠀⢸⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠀⢠⣾⠋⠀⠀⢸⡆⢿⡄⠀⠀⠈⠋⠉⢹⡁⣿⠁⠀⠀⠀⠠⢼⡿⠛⢛⣧⡄
⠀⠀⠀⠀⣠⣤⣤⣤⣤⡤⠴⠖⢶⣶⣶⣾⠃⠀⠀⠙⣧⡀⠀⢰⣼⠃⠈⣿⡄⠀⠀⠀⠀⠘⣧⠟⠀⠀⠀⠀⠀⠘⣿⠛⢿⣍
⣤⡾⠛⠛⠉⠀⠀⠀⠀⠀⠰⣿⣋⣈⣈⣿⣛⣋⣶⣶⣼⣿⣆⣀⣠⣴⣾⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⣶⣾⡛⠛⠛⠛⠻⠶⠶⢶⣦⣄⣀⡀
⠙⠻⠶⣶⣤⣤⣤⣤⣤⣤⣤⣀⣈⠁⢉⣉⠉⣉⣀⣤⣼⡇⢸⣿⣟⢉⣉⣿⣟⣛⣛⣻⣶⣶⣦⣤⣤⣤⣤⣤⣤⣷⣦⢀⣨⣿⣆⣀⣤⣤⣤⣤⣤⣤⣬⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⠛⠱⣯⡁⠀⣀⣀⣤⣤⣤⣼⠄⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣅⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⢦⣤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣼⠿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⢻⣿⣿⣿⣿⣿⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⠀⠀⠀⠀⠀⢀⠴⡄
⣿⣿⣿⣿⣿⣿⣧⢿⣿⣶⣿⡿⢼⢭⣿⣿⣷⣾⣿⣏⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠀⢸⣓⡇⠀⠀⠉⠁⠀⡜
⣿⣿⣿⣿⣿⣿⣿⡼⣿⢿⣻⣛⠿⡾⢟⣟⢫⣝⠿⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⣆⢀⡠⠴⢬⣈⣁⠤⢤⡜⢢⣠⠇
⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⠿⢷⣼⡏⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⠉⠀⠀⠀⠈⠀⢀⣀⠙⠉⡼
⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡏⠭⣿⣿⣷⣭⣾⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢴⣲⠀⠀⠘⠚⠁⢀⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣛⣛⣛⣛⣛⣛⣛⣛⣋⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⠤⠤⠤⠤⠼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⣶⡶⠶⠶⢶⣶⠶⠶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠿⢶⡶⠟⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣀⠀⠀⢸⣧⣤⣤⣤⣄⣴⡿⠛⢿⣾⡇⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡘⠛⣠⣤⡿⠛⠉⠉⠉⠙⠛⠀⠀⠀⠿⠟⠿⣶⣄⢳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣏⣼⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⣿⣛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣦⣄⣤⣾⠟⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢀⠀⢠⡆⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣟⠁⠀⢹⡇⠘⣷⣾⣷⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⡏⠀⠈⠉⣉⣿⣿⡶⠿⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢩⡿⠀⣀⠀⢸⡏⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣴⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡇⣠⡟⠁⠀⠉⠀⠀⠙⠷⠶⠶⠶⠛⠋⠁⣀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⢰⡟⢉⣿⣾⠋⢀⣄⠀⠀⠶⠀⠀⠠⡦⠀⠀⢀⡄⠀⠉⠀⠺⠂⠸⣧⣀⣀
⠀⠀⠀⠀⠀⠠⣿⣁⣾⡟⣃⣀⣀⣁⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣁⣀⣀⣤⣤⡶⠾⠛⣩⣽⠇
⠀⠀⠀⠀⠀⠀⠉⣿⣛⣛⡛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣁⣀⣀⣀⣀⣉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⣻⣷⠄⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣿⡏⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣶⣤⣴⡶⠶⠟⠋⣩⣤⣤⣤⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠒⠒⢤⡎⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠚⠉⠁⠀⠀⠀⠀⠀⠸⡆⠀⠲⣤⣈⠉⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠱⣌⣉⣲⣤⠄⠉⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣠⣴⣶⣷⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⣄⠀⠀⣀⣠⠼⠳⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠴⠟⣻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣭⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢉⡷⢀⣾⡀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡄⠿⣤⣀⣤⡿⠁⡼⠀⡇⢀⣤⡦⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠙⠁⡼⠁⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣄⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠾⠤⠤⠭⠿⠿⠿⠿⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⠶⣄⡀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣤⣄⣠⣤⣴⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣋⣭⣥⡀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⡄⠀⠉⠉⢹⡇⠀⠺⣿⠋⣿⠁⠐⢿⡇⢰⡏⠀⠺⣿⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠈⠛⠶⠞⠁⠀⠘⠳⠶⠛⠁⠀⠙⠶⠶⠋⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠲⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⣄⠀⢠⡤⠤⢰⠀⠀⣦⠀⢲⡀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⡇⢸⡇⠀⢸⠀⣰⣻⣄⠈⢧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⡀⢸⡟⠃⢸⠀⠟⠉⢿⠀⡼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠹⡜⠛⠓⠘⠛⠛⡀⠘⢂⡁⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠃⡴⠚⠳⡄⠀⢰⣷⠀⢸⡇⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⢰⡇⠀⠀⣿⠠⡿⠾⡖⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠀⢳⣀⣠⠏⢸⠃⠀⠳⢸⡇⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⢠⡄⢠⡆⣤⠀⠀⣄⠀⢠⡇⡄⣤⠀⢨⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⢸⣿⡸⡇⢸⡄⢸⢻⡄⣼⠀⡇⣿⣇⢸⠀⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣼⢸⡇⢻⡇⠀⢧⡟⠀⣧⡏⠀⡇⣿⠘⣾⠀⡇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠘⠃⠈⠃⠀⠘⠃⠀⠘⠁⠀⠃⠛⠀⠙⠐⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣿⣶⣶⣾⣿⣶⣾⣶⣶⣾⣿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠉⠉⠉⠉⠉⠉⠹⠆



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠊⠉⠁⠂⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠔⠁⠉⢍⠁⠢⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⡆⡖⠀⠶⠀⠂⠰⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢢⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠉⠁⠀⠉⠉⠉⠁⠑⠀⠊⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠚⠀⠀⡳⠤⠒⠂⢸⠀⠉⠈⡏⠉⠀⠳⠆⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⠁⠀⠘⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⡇⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡸⠁⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡌⠀⠀⢇⠀⠀⡇⠀⠀⡅⠀⢠⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⡇⠀⠀⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⠀⠀⠈⠀⠀⡇⠀⠀⠃⠀⠀⡇⠀⢠⠀⠀⠀⢀⠀⠀⠀⠇⠀⢠⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠇⠀⠘⠀⠀⠀⢸⠀⠀⢰⠀⠀⠈⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡀⠀⠀⡇⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⠀⢀⡇⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆⠀⢰⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠄⠀⡇⠀⠈⡆⠀⢠⠀⠀⡸⠀⠀⡆⠀⠀⡇⠀⠀⡘⠀⠀⢠⠃⠀⡌⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣸⠀⠀⠃⠀⠠⠅⡄⠚⠖⠀⠁⠋⠈⠁⡏⠀⠁⠈⠀⠃⠶⠐⢣⡄⢀⡇⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠦⠔⠊⢉⡁⠀⣸⠀⡀⠤⠀⠠⠇⠠⠠⠦⠄⠤⠆⠄⠴⠇⠀⣴⣆⣀⣤⡏⢺⣶⣦⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⢋⣀⣰⣧⣶⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣿⣿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣾⣿⣟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣿⣟⣿⣿⣿⣿⠻⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠼⠚⣻⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄⡀⠿⠀⠀⠀⠀⠀⠀⢰⣤⣼⣿⣿⣿⣿⡟⣿⣿⡧⡈⠃⠀⡂
⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⣿⣿⣿⣿⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡻⣿⢿⣿⡿⣿⣷⢿⣿⣿⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⣻⣿⣿⡿⣏⣿⣾⣿⣿⠿⠶⣀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⡐⠁⠈⠁⠺⠆⠘⣷⠿⠹⢢⣿⣿⣿⠂⠐⠨⠅⠐⠂⠀⠤⣤⣤⣤⡤⠄⠐⣂⠩⠔⠨⠋⣿⡟⢿⢸⣾⠇⢻⠏⠉⠀⠀⠁⡄⠤
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡍⠻⣦⠀⠀⠀⠀⠉⠀⠀⠉⣿⡏⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠋⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠘⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟
⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀
⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠶⣼⣧
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣼⣯⣤⡀
⠀⢀⣾⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣤⣤⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⣼⠃⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠉⢻⣶⣄⡀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⢰⡟⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⣠⡾⠁⠈⠙⠳⣦⣤⣀
⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣀⣤⡾⠋⠀⠀⠀⠀⢠⡿⠉⠉⠛⠷⢶⣦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⣿⠁⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣼⠃⢹⣧⣤⣄⣀⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢿⡀⠀⢀⣠⣴⠿⢧⣤⣀⣀⣀⣤⡾⠁⠀⣿⠁⠀⠈⣭⣉⣙⡛⢷
⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣶⡶⠾⠿⠟⠛⠉⠀⠀⠀⠈⠉⢉⣽⠟⠀⠀⢰⡏⠀⠀⠀⣿⣿⣿⣿⠆
⠀⠙⢷⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠶⠶⠶⠾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣥⣀⣠⣴⡿⠀⠀⠀⠀⠹⣿⠟⣡⡾
⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠷⠶⠶⠶⣶⣶⣿⣋⣁⡀⠀⠈⠉⢁⣿⡁⠀⢀⣀⣀⣤⡴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣶⣶⣾⣋⣛⣛⣛⣉⣉⣁⣀⣀⣀⣀
⣿⣿⣋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢰⣿⠿⠛⢿⣷⣄⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢀⣿⡿⠀⠀⠀⠹⠿⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠿⢿⣶⣦⣴⣾⡿⠁⠸⠿⠀⠀⠀⢀⣀⣴⣾⠿⠁
⠀⠀⠀⠀⠀⣠⣾⠿⠿⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⢀⣀⠀⠈⠉⠉⠁⠀⠀⣠⣴⣶⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⢰⣿⠏⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠘⠛⠀⠐⠿⠃⣀⣴⣿⠟⠋⠁
⠀⠀⠀⠀⢺⣿⠀⢀⣾⡿⠛⠁⢀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠾⠇⢠⣾⡿⠋⠁
⠀⠀⠀⠀⠘⣿⣦⣼⣿⠁⠀⣰⣿⠟⠉⠀⣠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠈⠙⠛⢿⣷⣤⣿⣿⠀⠀⣾⡿⠋⠀⢀⣴⣾⡷⠀⠀⠀⠀⠀⠸⡿⠃⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣷⣶⣿⣇⠀⠀⣾⣿⠁⠀⠀⠀⠀⣀⣀⣀⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣶⣿⣿⣤⣀⣀⣠⣴⣿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⡀⠀⠀⠀⣿⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣾⣷⠀⠀⠀⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢹⡆⠀⠀⠀⣿⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣤⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢀⣴⠟⣩⣍⠻⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠈⠻⣆⠀⠀⠀⠀⠀⠙⠀⣠⡶⠟⠡⣾⡿⢋⣼⠟⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠈⣴⠟⠁⠀⠀⢀⣼⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣿⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣰⠏⠀⠀⢀⣴⠟⠉⠉⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸⡇⠀⢀⣴⠟⠁⠀⠀⢀⣤⠶⠞⠁⠀⢀⣴⠟⠁⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣸⣇⣴⠟⠁⠀⠀⠀⠀⠛⠁⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⣴⠟⠉⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣤⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠈⠻⣦⡀⢀⣠⡶⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠈⠻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢂⠴⡶⢤
⠀⠀⠀⠀⠐⢦⣆⠹⣁⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠔⠁⠐⠁⢈⡅
⠀⠀⠀⠀⢀⣤⠔⢛⠐⣤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠔⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈
⠀⠀⢀⠠⠌⠀⠀⢿⠂⠀⠁⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡈⢈⢞⢾⡁⠀⠀⠀⠀⠀⠀⡔⠲⡀⠀⠀⠀⠀⡠
⠀⠀⠈⠢⠠⠀⠀⠀⠒⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⡈⠢⡄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⡹⣁⣶⠆⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠑⡀⢆⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⢪⠓⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣈⡀⢆⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠄⡈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠃⢰⣴⢴⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂⠽⠀⠀⠀⢀⢅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠴⠂⣠⠁⠲⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠔⣃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣴⠀⠀⠀⠸⠿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠁⠤⢚⡴⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡊⢡⡀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣽⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⢸⠄⠀⣠⡀
⠀⠀⠀⢀⡀⠀⠀⢀⠀⢈⣩⣈⣫⡁⠀⠀⠀⠀⠀⡀⣀⡠⠊⠀⠙⢣⢔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡙⢑⢨⠑⠈⠉
⠀⠀⠀⡄⠘⠀⡌⠘⠀⠋⠀⣿⠀⠉⠁⠀⠀⠀⠀⣿⡇⢀⠠⠃⢀⠀⠀⢨⠉⠉⠥⠀⠀⠀⠀⠀⠀⡇⢸⠠⠞⠉⢰⡈⠙⠶⠄
⠀⠀⠀⠀⢰⠀⠦⡤⠀⠀⠀⠀⢠⠠⠀⠀⠀⠀⠀⣿⡏⠀⠀⡀⠂⠀⣐⢘⠂⣂⡒⠀⠀⠀⠀⠀⠀⡇⢸⢰⠀⠀⠸
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⠀⠁⠈⠈⠃⠀⢠⠄⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢸⣜⠄
⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠻⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣸⣠⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣼⡇⠀⢀⣻⣁⠀⠀⣿⣴⠆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣳⠞⣋⣍⠉⠛⢶⡟⠛⠛⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣄⣸⠇⣼⠉⠀⠀⠀⠈⣷⣀⣴⣋⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠉⠹⣆⠀⠀⠀⠀⠀⢀⡟⠉⠙⢯⠁⠀⠀⠠⣄⠀⠸⡇⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⠦⣤⣀⣠⡴⠿⣄⣀⣀⠀⠀⠀⠀⠀⠈⠳⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⢿⡇⠀⢀⣽⡀⠀⠀⣿⠻⡍⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠋⢹⠙⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠟⠳⣄⣠⡴⠛⢦⣀⣠⠞⠻⣦⣀⣴⠛⠳⣄⣀⡴⠛⠳⣄⣀⡴⠛⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣬⣥⣤⣤⣤⣬⣤⣤⣤⣤⣭⣤⣤⣤⣬⣭⣤⣤⣤⣬⣭⣤⣤⣤⡿





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣾⣇⣤
⠀⠀⠀⣀⠀⢀⣶⣶⣶⣶⣄⡀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⠟⠀⠀⠀⠘⠿⢶⣄⠀⠀⢀⡿⠀⢸⣿⣿⣿⣿⣠⡀⡽
⠀⢓⣜⡙⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣶⡇⠰⠶⠆⠀⠀⠀⠀⠛⣁⢠⠀⡀⣀⣀⣀⣀⣀⡿⠀⠀⢀⡿⣰⣾⣿⣿⣿⣿⣿⡇⡿
⣾⣿⣉⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣥⠉⠉⢉⠉⠉⠁⠀⠀⢈⡷⢹⣿⣿⣿⣿⣿⣿⠃⡿
⠀⠀⠀⠀⠀⠀⢶⡿⠟⠃⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⡇⠀⠀⠀⠀⠀⠀⠐⠋⠿⠙⠂⠀⢬⣿⣿⠄⠀⠀⠀⠈⠛⠈⠻⣿⣿⣿⣿⠟⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿

⠀⠀⡀⢀⣴⣶⣶⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⢀⡀⠀⠀⠀⠀⠀⣾⢛⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⡀⣤⣤⣤⡀
⠀⢾⠁⠸⣿⣿⣿⡿⠀⣳⠀⠀⠀⠀⠀⠀⠠⢠⡶⠛⠛⢷⣄⠤⠀⠀⠀⠀⣿⢸⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠆⣁⣀⣹⣇⣀⡀
⢐⡏⠀⣀⡙⠿⠟⣁⡀⠨⡇⠀⠀⠀⠀⠀⣠⣿⣤⣀⠀⠀⢹⡦⠄⠀⠀⣤⣿⣼⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠈⣜⣬⡼⠛⡛⠻⣿⣿⣿
⠘⢢⣿⣿⣿⣦⣾⣿⣿⣦⠛⠀⠀⢀⣴⡶⣿⢛⣛⢻⣶⣤⡟⠤⠀⠀⠀⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠀⠀⠀⠀⣿⣿⡇⠐⡿⠂⣽⣿⣿
⠀⣼⣿⣿⣿⣿⣿⣿⡿⣿⡄⠀⠀⠸⣿⣶⣿⣬⣽⣿⣿⡿⠃⠀⠀⠀⠀⠈⢧⣤⣤⣤⣤⣤⣤⣤⡼⠃⠀⠀⠀⠀⣿⣿⣿⣶⣶⣾⣿⣿⡿
⠀⠛⠃⠘⠛⠛⠛⠛⠃⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⡀
⠀⠀⠀⣀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⡴⣚⣃⡿⠃⠀⠿⣃⣓⢦⡀⠀⠀⠀⠀⠀⠀⢀⣆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣠
⠀⢠⣤⣬⣥⣯⣤⣄⠀⠀⠀⠀⠀⠀⡾⠛⠿⢿⡆⠀⣾⠿⠿⢿⡀⠀⠀⣶⠀⠀⢠⣼⣿⣿⡇⠀⠀⣶⠀⠀⠀⣀⠀⠀⠀⠹⣿⢿⠂
⠀⢸⣿⠛⡍⡝⢻⣿⠒⢦⠀⠀⠀⠀⢿⠀⠀⣸⠁⠀⣧⠀⠀⣼⠀⠀⠀⢿⣦⣲⣮⠻⣿⢟⣴⢆⣴⡿⠀⠀⢸⣿⡆⣤⣤⡄⠀⠀⠀⠀⣀
⠀⢸⡇⣽⡀⣸⡅⣿⣠⠞⠀⠀⠀⠀⢸⡆⠀⢻⡀⠀⡿⠀⢠⡏⠀⠀⠀⠀⠙⢿⣿⣿⢴⣿⣿⡿⠋⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿
⠀⢸⣿⣤⣅⣥⣾⣿⠁⠀⠀⠀⠀⠀⠈⣇⠀⣸⠇⢸⣇⠀⣸⠁⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠘⠿⠉⠉⠉⠉⠉⠉⠉⠉⠻
⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⢀⣠⣤⣤⣴⣾⣷⣦⣀⠀⠀⠀⠀⢰⣇⠀⠾⡥⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣿⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣬⣽⡱
⣠⠶⠾⠁⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢋⣤⣤⣥⣀⣘⠁⠀⠀⠀⠀⠀⠀⠀⠠⠟⣿⢿⠉⠀⠀⠀⠀⠀⠀⠀⢰⡄⣿⣿⣿⣿⠄⣦
⠻⣤⣤⣤⣤⣤⣭⣼⠏⠉⠀⠀⠀⠀⠐⣶⣶⣿⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⣶⣥⣶⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣿⣿⠘⣧
⠀⠀⠀⢀⣤⡀⠀⢰⣾⣷⡀⠀⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡀⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣸⣿⣿⣿⠘⣧
⠀⠀⠀⠘⠛⠃⠀⠘⠻⠛⠀⠀⠀⠀⠈⠉⠉⠩⣿⣽⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠻⠄⠻⣿⣿⠏⠘⠧
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠶⠐⠄⠂⠆⠄⠐⠀⠦⠶⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠠⠀⠶⠀⠆⠀⠤⠀⠆


⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⢃⣦
⠀⠀⠀⠀⠀⠀⠀⠰⠂⡰⠮⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢒⣿⣿⠲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠆⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠼⠄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡒⠒⢒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⡀⠁⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢉⡉⢛⡉⢉⡊⢉⡑⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄

⠀⢷⠀⡤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠖⠼⣟⣵⣀⡀⡀⢑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⣀⣠⣤⣤⣴⣶⣾⣿⣿⣿⣞⢧⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢮⡑⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡄⢙⣿⣿⣿⣿⣯⣷⣾⣿⣿⢿⣿⣿⣿⣿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡦⠀⠀⠀⠀⠀⠀⠹⣷⣿⣎⣣⡍⢿⣿⣿⣿⣿⣿⣿⣿⣧⡱⡘⡹⣿⣿⣷⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡻⣿⣿⣿⣿⣿⣿⣿⣷⣑⣀⢹⣿⣿⡏⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢠⣆⠀⠀⢸⠛⠋⠙⢿⠁⢿⣿⣿⣿⣿⣿⣿⡙⠛⠿⣿⣿⣿⣿⣦⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣦⣈⣿⣦⡀⠎⠀⠀⠀⠈⠁⠈⠇⠻⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣭⡉⡇
⠀⠀⠀⣖⣤⣄⠀⣀⣀⢤⣒⣯⣽⣿⣿⣯⣝⣫⣽⡛⠛⠒⣦⡀⠀⠀⠀⠀⠈⠂⠈⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢠⣿⣾⡿⠷⡾⠛⠛⠋⠉⣩⣉⡉⠻⣿⠀⣀⣀⣠⠀⣙⡱⢀⣀⣴⣯⣷⡄⠀⠪⣿⡟⣿⣿⣿⡿⣻⠋
⠀⠀⠀⠀⠙⣩⣍⣲⣴⠻⢿⣤⣴⣶⣿⣶⣿⣾⠆⠀⠁⣀⡔⢫⣴⣾⣭⠻⣿⣿⡿⠀⢰⣽⣿⣿⢟⣫⣾⣶⡿⢠
⠀⠀⠀⠀⠀⠀⠈⡜⣣⡀⠊⡭⣟⣻⣿⣶⠿⠿⣣⣼⣿⣿⣆⠸⣿⣿⣿⣷⣮⣍⣛⣽⣿⣿⣿⠾⣿⣿⣿⣯⣠⣥⡀
⠀⠀⠀⠀⠀⢀⢎⣾⣿⣿⣿⣴⣿⣿⣿⣷⣶⣿⣦⣍⣿⣿⣿⣄⠈⠛⠻⠿⠻⠿⠋⠻⢛⣩⣶⣿⣿⣿⣿⣿⣿⠿⠀⢂
⠀⠀⠀⠀⠀⢘⡬⣺⡿⣿⣿⣿⣟⣻⣿⣛⠿⣨⠻⣿⣿⣿⣿⣿⣿⣷⣶⣤⣴⣖⢲⣾⣿⣿⣿⣿⠿⢛⣵⠿⣿⣿⣠⠇
⠀⠀⠀⠀⠀⡯⣠⢿⣿⣿⣷⣄⠈⠿⣿⣿⠿⢮⢷⣽⣿⣿⣿⣶⣬⣍⣙⣉⣋⣀⣤⡼⠛⣉⣥⣴⣿⣷⣮⡑⠙⢿⣧⡆
⠀⠀⠀⠀⠸⣼⣿⣄⠛⢻⢿⣿⠆⠜⠀⠐⡆⠀⣦⡙⣯⡻⢿⣿⣻⣿⣿⣿⣿⡿⣩⣾⣿⣿⣧⡜⠹⣿⣿⣿⣷⣄⡹⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢟⣩⣶⣶⡿⣿⣶⣬⡙⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⢡⣾⡟⠋⣷⡄⣴⣿⠙⢿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡟⢷⣷⡝⠃⠟⣵⣶⠞⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣆⣴⣶⢖⡀⣔⢶⣶⣄⣾⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣌⡁⣾⠇⢿⣷⢈⣽⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣷⣬⡻⢿⣷⣦⣶⣿⠿⣋⣴⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣶⣶⣶⣒⣒⣒⣒⣲⣶⣶⣒⣒⣒⣒⣺⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠿⠿⠿⠿⠭⠭⠭⠭⠭⠭⠭⠭⠽⠿⠯⠭⠭⠭⠭⠭⠭⠭⠭⠽⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢟⡛⣿⣿⠟⡛⢿⣾⠟⣛⢿⣿⡿⣛⠻⣿⣿⢟⡛⣿⣿⡟⣛⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣾⣿⣮⣥⣾⣿⣮⣭⣾⣿⣦⣉⣴⣿⣷⣍⣵⣿⣿⣮⣋⣼⣿⣦⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁





⣿⠿⠿⠿⠿⡻⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠞⠓⠒⠋⠙⠒⢲⡂⠀⠐⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠠⢇⡄⢠⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡖⠈⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢁⣼⡆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠫⠜⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⡨⡼⣿⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠀⢨⠆⡳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⡒⠒⡫⠊
⠀⠀⠀⠀⠀⠀⠀⣰⠃⡆⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⡲⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠒⠤⣀⢀⠔⠉⡔⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⣶⠀⠀⠀⢠⠏⠀⢀⠀⡘⠕⢠⡎⠀⠀⣀⠅⠋⠀⠚⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢷⡆⠀⢀⡏⠀⠀⢄⠀⠈⠀⠱⠜⠒⢉⡡⠄⢄⠐⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢸⡇⠀⡼⠀⠀⠀⠀⠀⠀⠐⠠⢶⠊⠁⣀⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠉⠉⣆⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⣿⢸⠀⠀⠀⠀⠀⠀⠀⣠⠠⠄⢚⠃⢀⣩⠎⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣉⠛⢆⡀⠀⢀⣠⡴⣋⣁⠀⣴⢉⣹⡀⢸⡀⣏⠉⠉⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⠶⡶⠶⠖⠲⠶⠶⡞⠀⢻⠀⠀⠉⠉⠁⠉⠉⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡾⢰⠓⠒⠒⠒⠒⡇⢃⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠞⣇⠎⠀⠀⠀⠀⠀⢡⡸⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠂⠈⠁

⠀⠀⠀⠀⢰⣿⡲⠶⣄
⠀⠀⠀⠀⠀⢽⣇⠀⠈⢦⣀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡈⠑⢦⡀⠀⠙⢦⣀
⠀⠀⠀⢀⣾⡿⢿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢸⠇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠈⠉⠓⠀⠀⠀⢀⡶⠋⠉⠁⠀⠙⢦
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠘⣦⣸⣷⣶⣣⣄⣀⣀⣘⡇
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢻⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⡟⠛⠿⠿⠻⠿⣯⣍⣹⠇
⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠹⣆⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⣤⡀⠀⠈⢛⡟
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠈⠑⠦⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⣰⠁⠀⣠⡿⠁
⠀⠀⠀⠀⠀⣠⠃⠀⣰⡟
⠀⠀⠀⢀⡴⠁⠀⢰⣿⣷⣤⣀
⠀⢀⣴⣿⣄⣀⣀⣀⡈⠛⠿⣿⣇
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠒⠤⣀⠀⠀⠀⠀⠀⣠⠖⠉⠛⠲⢄⠀⠀⠀⠀⠀⣠⠔⠒⠢⢤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠑⠒⠤⠒⠊⠁⠀⠀⠀⠀⠀⠑⠦⠤⠤⠞⠁⠀⠀⠀⠀⠈⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠤⢄⡀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠦⣄⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠒⠤⠤⠤⠒⠉⠀⠀⠀⠀⠈⠳⢄⣀⣀⠴⠃⠀⠀⠀⠀⠉⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣶⡶⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⡿⣿⠿⡟⣿⡟⠃⠗⠀⠟⢻⡿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠁⠏⠀⠀⠋⠁⠀⠀⠀⠀⠀⠃⢹⣿⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⡧⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⢚⡃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠃⠹⡄⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⠔⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠐⠠⠀⣀⣀⣀⣀⡤⠚⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⢴⠴⢿⣿⣟⠉⢧⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⣿⣽⣤⣤⣤⣤⣤⠤⢴⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣰⠙⡾⢻⣿⣿⠠⢤⣴⣖⣒⣒⣒⣿⣿⡟⠁⣦⡂⠀⠐⠃⠉
⠠⠤⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⣦⣴⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡞⠋⣱⡄⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠
⠀⠀⠀⠀⠀⠀⠠⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣤⡞⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣆⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣈⣩⠛⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢋⣼⡆⠀⠀⠀⠀⠈⠀⠉⠑⠢⢄
⠀⢀⠔⠒⠉⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⠩⣥⠄⠀⠀⠀⠀⠀⠀⢀⡀⠔⠊
⢰⠁⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢒⣤⣿⣋⣉⣉⣉⣉⣩⣭⣭⡭⠭⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡀
⠀⠀⠈⠀⡄⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠘⠓⠚⠉⠭⢤⣦⡤⠤⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡
⠀⠀⠀⠎⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣦⣶⣶⣷⠀⠀⠀⠀⠐⠂⠨⢍⠀⣤⠴⠀⠀⠀⣰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣛⡍⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠘⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠵⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠠⠤⠄⢀⣀⣀⣀⡀⠀⠀⠀⠀⠠⠀⠐⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⢀⡖⣃⢀⣀⣠
⠘⠲⠤⡤⠀⣠⣠⡇⠀⠀⠀⠀⠀⠀⠀⠉⣧⠀⠀⠀⠀⠀⠀⢰⠉⠇⠀⢀⣀⠀⠀⢸⣈⣄⠀⢀⣄⢳⢺⠰⠾⡗⠛⠛⠋
⠀⠀⠀⠀⠈⠁⠂⠠⠌⠉⠀⠀⠀⠀⠴⠃⠿⢲⡤⠤⠤⣶⠀⠙⣃⡿⠀⠐⢸⡇⠀⠨⣃⣤⣉⠉⢻⡄⠀⡀⢠⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⡆⠚⠉⠀⠀⠀⠢⠐⠀⠀⠀⡉⠺⠀⢸⠧⠀⢠⠎⢣⠀⡆⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠗⠁⠀⠀⠠⢄⢠⢴⣇⠄⠀⠀⠀⠀⠀⠀⢀⠀⢠⡾⠀⠱⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠘⡇⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡀⠈⢁⣀⠆
⠀⠀⠀⠀⠀⠀⠈⡔⢁⠭⡉⢦⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠆⡠⠒⠊⠀⠓⠺⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢠⡀
⠈⠠⢉⣉⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⢔⠀⠀⡃⠀⠀⠀⣀⣀⣀⠀⠀⠤⠀⠀⠠⠴⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠉⠐⠀⠀⠁⠉⠁⡈⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠂⠀⠀⠂⠒⠒⡀⠀⠀⢀⠂⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠉⠙⣦⡖⠋⠉⠳⡀
⠀⠀⠀⢀⡤⢄⠀⣠⠔⠚⠛⠲⡇⠀⣠⠞⠉⠑⢤⡀⠀⢱⠖⠒⠢⢄
⠀⠀⢰⠁⠀⠀⠻⡄⠀⠀⠀⠀⢹⠀⢧⠀⠀⠀⠀⠙⢢⠇⠀⠀⠀⢸⠦⠦⣄
⠀⠀⠈⢧⡠⠤⠤⠼⠦⠤⠤⠤⠮⠤⠤⠵⠦⠤⠤⠤⠼⠤⠤⠤⠤⠼⠤⠤⠼⣄
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⢹⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⢁⣀⣀⣀⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠉⠉⠑⢆⠀⢣
⠀⠀⠀⢸⠀⠀⠀⠀⢀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠈⠛⠟⠁⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⢸⠀⠀⠀⢀⡼⠀⣸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠒⠒⠉⢀⡴⠃
⠀⠀⠀⠈⡆⠀⠀⠰⣉⣉⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣉⣉⡹⠀⠀⠀⡾⠤⠤⠴⠒⠉
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⢀⣤⣶⣾⠿⠿⠿⣶⣶⣤⣀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣰⣿⠛⠁⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢰⣿⠃⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣼⣿⣴⡿⠟⠛⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⣷⣦⡀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⣰⣿⠃⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣰⣿⠃⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠃⠀⠀⢰⣿⠃⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣋⣁⣀⣤⡀⠀⠘⠋⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠛⠻⠿⠛⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣤⣄⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠀⢀⣠⣤⣤⣀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢀⣀⣀⣴⣿⠟⠉⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠘⠛⠛⠋⠐⠿⢠⣦⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⡀⠘⢟⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⣴⠂⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠻⣦⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢹⡇⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⡸⠃⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠐⣒⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⡂⠀⠀⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⣎⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣈⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠁⠀⠶⠤⡀
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠇⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢠⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠠⠒⠁⢀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢤⢤⢄⡂⠁
⠀⠀⠀⢠⠂⠉⠁⣤⣴⣶⣶⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣷⣶⣶⣦⣤⠈⠉⠐⡄
⠀⠀⠀⠈⠁⠂⠠⠌⢉⣛⣛⠛⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠛⣛⡛⡉⠡⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠥⠈⡁⠀⣀⣀⡀⢀⣀⣀⠀⣀⠁⠭⠁⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡴⠶⢶⣄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡏⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⡿⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠲⢦⣄⠀⠀⢸⡇⠀⠈⠁⠀⠈⠁⠀⠀⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄⠀⠙⢧⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⠻⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣄⡀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣟⠀⠀⠀⡿⢠⣤⣤⣤⣤⣤⣤⠀⡇⠀⠀⢘⣻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣇⠸⠿⠿⠿⠿⠿⠿⠂⣿⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⢛⡿⠛⠛⠛⢻⡟⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣠⡴⣿⠀⠀⠀⢠⡞⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠉⠀⢿⠀⢀⣴⡏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣸⡷⠋⠀⠻⣦⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠾⣯⡁⠀⠀⠀⠀⠈⠙⠶⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠶⠿⠿⠷⠶⠶⠶⠶⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⢤⣀⡼⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⢿⡁⠀⢀⡀⠉⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠹⣟⠶⣤⣀⠈⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡏⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠙⣦⣤⣭⠽⠶⠆⠀⠈⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⠀⢀⣤⣴⣾⣿⡗⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⣦⠀⠀⠀⢀⣠⣴⠟⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⣸⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠛⠋⠁⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠘⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠉⠛⣶⠀⢠⣆⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⣿⠁⠀⠀⢠⣿⠏⢠⡏⢹⠀⠀⠀⣀⢀⣦⡴⠟
⠀⠀⠀⠀⠀⠀⠀⣸⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡀⠉⠛⠷⣾⣿⠏⢠⡟⠀⠈⣇⣰⠽⠚⠉⠁
⠀⠀⠀⠀⠀⠀⣰⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣧⠀⠀⠀⠀⠉⢀⡞⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣰⢧⣄⣠⣤⣤⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⢀⡞
⣀⣀⣀⣀⣰⣯⣤⣤⣍⣻⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸
⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡟
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣼⠁
⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠇
⠀⠀⠀⢻⣿⣿⣿⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣀⣀⣤⣴⣶⣾⡟
⠀⠀⠀⠈⣿⣿⣷⣤⣀⣽⣿⣶⣦⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠿⠿⣿⠇
⠀⠀⠀⠀⠀⢻⡄⠉⠛⠿⢿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⡿⠿⢿⡟⠋⠉⣡⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠈⣿⢴⡄⠀⠀⠈⠉⢹⣏⠀⠈⠷⣿⠛⢹⡏⠀⠈⠷⣿⠀⢸⡏⠀⠈⠷⣿⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠻⢦⣤⠾⠋⠀⠀⠻⢦⣤⠾⠋⠀⠀⠻⢦⣤⠾⠋⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣾⠃⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣼⣧⣤⠄⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣹⣇⣠⡤⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⠀⢀⠀⢦⣤⣈⣹⡏⣁⣤⠦⢀⡀⢀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⡀⣿⠀⣿⠀⠀⠀⠉⢻⡟⠉⠀⠀⢸⡇⠸⣇⣈⣷⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⣈⡽⠿⣦⣿⠀⠀⢀⣠⡾⢧⣄⡀⠀⢀⣷⡴⠿⣯⡈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣠⡴⠛⠉⠻⡾⠋⢁⣠⣄⡈⠛⣶⠛⠉⠳⣦⡈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠀⠀⠀⠀⡇⠀⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠙⠷⣄⣀⣴⣷⣄⠈⠛⠋⢁⣠⣿⣄⣀⣴⠞⠋⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠙⣷⣶⠟⣿⠁⠀⠉⠛⢶⡾⠋⠁⠀⠈⡿⠷⣶⣟⢉⣠⡄⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠉⣿⠀⣿⠀⠀⠀⣀⣼⣇⡀⠀⠀⢸⡇⢠⡏⢙⡿⢧⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢙⣀⡈⠀⠶⠛⢉⣹⣏⠙⠻⠶⠈⠁⠈⠁⠘⠃⠀⠀⠀⠀⠐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠶⠶⠛⠛⠉⠀⠀⠀⠞⠛⣹⡏⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⢻⡟⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠞⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠳⠶⠶⣤⣤⣤⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠷⠶⣤⣤⣤⣤⡤⠶⠞⠛⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠆⠀⠠⡄⢊⢹⣿⣷⣶⢖⣓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠓⠛⠓⢚⣿⠻



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠄
⣶⣦⣴⣿⣶⣶⣦⣤⣤⣤⣤⣤⣶⣦⣦⣤⣤⣶⣤⣤⣼⣿⣧⣬⣽⣿⣦⣤⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣛⣻⣿⣿⣟⣻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠋⠛⠉⠉⠉⣉⣉⡉⠉⠉⠉⣉⣉⣉⡉
⢿⣿⣿⡿⠿⠿⠿⠿⣻⣛⣟⣿⣿⡿⠿⡿⣿⣿⠿⠟⠛⣿⣿⠿⣻⣿⡋⠀⠀⠁⠀⠀⠀⠠⠶⠶⠾⠿⠿⠶⠀⢻⣟⣻⣿⡿⣿⣿⢻⡿⣿⡿⢟
⣾⣤⣿⡾⢶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠿⠿⠁⢾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉⠉⢠⣶⣴⣶⣷⣴⣶⣶⣿
⣾⣿⣿⣷⣼⣿⣭⣯⡙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣦⣀⣀⡀⠀⢙⣿⣿⣿⣿⣽⣿⣻
⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠐⠂⠀⠀⢰⣆⢈⣿⣿⣿⠿⠿⢿⣷⣶⣤⣝⠙⣿⡏⠉⣻⡿
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣾⣟⢿⣿⣋⡁⠀⠤⠖⠒⠲⢦⣤⣤⣄⣀⣀⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣶⣦⣶⡀⢭⣿⣿⣿⣿⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠋⢈⣿⢳⣶⡼⣇⠐⢀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠸⣿⣾⣏⠲⣿⣻⠾⣵⡆
⠀⠀⠀⠀⢰⣀⠀⠀⠠⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠟⠀⣧⡉⠈⠠⣓⡌⡠⢄
⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢶⣶⣾⠀⣢⣀⣻⡿⣦⣶⣄⢠⡀⢂
⠀⠀⠀⠀⢸⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⡦⠀⡀⠐⣽⣿⣿⣿⣾⣿⣿⣯⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡦⢠⡤⣿⣶⣶⣿⣿⣷⣄⣿⣶⡿⢯⣝⣷⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⢠⡀⠀⠀⢼⡟⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠼⢿⣿⢻⣟⢻⠳⠀⠿⣿⣿⢄
⠀⠀⠀⠀⢄⢸⣿⠀⠀⠀⢸⠉⠂⠀⢸⣷⣮⣟⠻⣿⣿⣻⣿⣿⣿⣿⣿⣷⡛⢋⣙⣻⠏⠁⡁⠀⠀⠑⡀⠉⠈
⠀⠀⠀⠠⣸⣿⠁⠀⠀⠀⢻⠄⠀⠀⠠⠛⠀⠹⡆⠉⠘⠺⠻⣿⣿⣿⣿⣿⡇⢡⠈⢾⣽⣽⣷⢿⣶⠼⣿⣿⡿
⠀⠀⠀⠰⣹⡟⠇⠀⠀⠀⢸⣄⠀⠀⠀⣄⠀⠈⢡⣤⣤⣤⣄⣸⣿⣿⡿⠿⣉⡛⢾⢻⣿⣿⣷⣿⣿⣶⣬⣁⢄
⠀⠀⠀⠸⢿⣿⡆⠀⠠⠀⠾⣏⠂⠀⠈⠫⠀⠀⠀⠀⠚⠙⠛⠋⠿⣯⣽⣁⣉⣁⣀⣬⣽⣿⣷⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢤⣼⣥⠲⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣤⣄⣉⣛⠛⠛⠛⣻⣟⢻⣿⣾⣿⣿⣿⣿⣿⡤
⠄⠀⠀⡀⣦⣽⣿⡁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⢉⢠⡤⡎⠛⢻⣽⢻⣻⣻⠗
⠀⠀⠈⠙⠳⠿⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣤⣤⣴⣤⣶⣦⣄⣐⠛⠲⠂
⠀⠀⠀⠀⠀⠀⠘⠛⢛⣻⣶⣶⣶⣦⣼⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶
⠀⠄⠀⠀⠀⠀⢂⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣤⣿⣏⣿⣿⣿⣍
⠀⠀⢰⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻
⠀⠀⠀⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⢿⡿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠙⠻⠿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣤⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠛⠛⠁
⠀⢀⣤⣀⢒⣾⣷⣤⣤⣤⣭⣽⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠤
⠀⠀⠐⠀⠉⡹⢟⡛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠛⠻⠧⠅⠈
⠀⠀⠀⠀⠀⠁⠑⠋⠀⠀⠐⠾⣿⣿⣿⣿⣻⠿⢿⣿⣿⣿⣿⣿⣿⡿⣿⡿⢿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠧⡿⢅⡀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣤⣴⣶⣤⣤⣦⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣦⣴⣦⣤⣦⣴
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠠⠴⠛⡻⠛⠉⠙⠻⢿⣿⣷⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⣿⡿⠟⠋⠉⠛⢿⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⣿⣿⣿⣦⣴⣤⣄⣀⡀⠀⣾⣿⣿⣿⣿⣷⠀⠀⣀⣠⣤⣴⣤⣾⣿⣿⡿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣘⢺⡶⢦⣝⣿⣿⣿⣿⣯⣿⣿⣯⣯⣿⡛⠛⣋⡤⢶⣗⢓⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⣒⡒⠒⠿⠯⢿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣽⣿⣶⣿⣿⣿⡿⠭⠿⠓⢒⣒⣒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⢛⡉⠉⠉⠉⠽⠧⠿⠿⠛⠛⠛⠛⠻⠿⠼⠿⠍⠉⠉⢉⡛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⠿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣅⣤⣤⣤⣼⡏⠀⢀⣤⣤⣤⣤⣤⣾⣿⣿⠇⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡏⠉⢹⣿⣿⣿⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠉⠉⣿⣿⣿⡇⠀⢸⣿⣿⣿⣶⣤⣿⣿⣿⡇⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⢸⣿⣿⣿⣀⠀⠛⢿⣿⣇⡀⠉⠛⠿⣿⣤⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⠀⠻⢿⣿⣿⣿⣿⣶⣤⣈⠉⠛⠷⣦⣄⡀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣦⣄⣈⠙⠻⢿⣿⣿⣿⣿⣶⣤⣀⠙⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⡉⠛⠿⢿⣿⣿⣿⣷⣶⣤⣈⠙⠻⢿⣿⣿⣿⠇⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⢷⣦⣄⡈⠙⠿⢿⣿⣿⣿⠇⠀⣸⣿⣿⡟⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⢀⣿⣿⣿⣇⣀⣸⣿⣿⡟⠀⢀⣿⣿⣿⠁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣼⣿⣿⡟⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡟⠀⢠⣿⣿⣿⠋⠉⠉⠉⠉⠉⠀⢠⣿⠋⠉⠁⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡦⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠸⣿⣿⡗⠀⠀⠀⠀⠀⠀⣿⣿⢏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⣀⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣀⠀⣠⣼⣿⣿⣾⣿⣿⣷⣀⣀⡀⠀⣀⣀⠀⠀⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠿⠟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠾⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡇⢾⣷⣶⣿⣿⣿⣿⣿⡿⣫⣭⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣭⣦⣙⠿⢛⣴⣯⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⢛⡛⣿⣿⣿⣿⣿⢟⣛⢿⣿⣿⣿⣿⠟⣛⢻⣿⣿⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⣿⣿⢸⣿⣿⣿⣿⢸⣿⡿⣿⣿⣿⣿⢸⣿⢇⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢸⣿⡎⣿⣿⣿⣿⢸⣿⣯⣿⣿⣿⡇⣿⣿⢸⣿⣿⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡜⣿⣇⢿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢣⣿⡏⣾⣿⡟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⢿⣿⢸⣿⣿⡿⢸⣿⠿⣿⣿⡿⢸⣿⠇⣿⣯⡵⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⡿⢿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠋⠙⠛⠿⢿⣶⣤⡀⠉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣼⣿⠏⠙⢿⣷⡀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠈⢹⣿⣦
⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⠙⢿⣷⣶⡿⠟⠁
⠀⠀⠀⠀⣾⡿⢹⣿⡀⠀⠀⠀⠙⣿⣧⡀⠀⠀⣠⣴⣿⠟⠋
⠀⠀⠀⢀⣿⡇⠘⣿⡇⠀⠀⠀⠀⠈⢻⣷⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⣿⡇⠀⢹⣿⡄⠀⠀⣀⣴⣾⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠉⠙⢻⣷⡄
⠀⠀⠀⠀⢿⣧⠀⠀⢻⣷⣤⣾⡿⠛⠁⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠘⣿⣧⣤⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣶⣾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⣿⣷⣶⣶⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠠⣄⣀⠀⣀⣴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⡈⠉⠉⠉⣀⣤⣤⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠈⣿⡇⠀⠀⠙⠓⠒⠛⠉⣀⣀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⢠⣿⡇⠀⠀⠳⠦⣤⠶⠟⠉⠉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⢦⣄⣀⣠⡶⠛⠛⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠁⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠚⠋⠙⠚⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣩⣵⠄⠀⣀⣀⠤⠼⢦⣈⢧⣚⣻⣤⡷⠧⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣧⡔⠊⠉⠁⠀⠀⠀⠘⣮⠏⠃⣟⠉⠀⠀⠀⠀⠀⠈⠙⢢⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⡀⢣⣀⣀⣀⠤⢤⡄⠀⠿⠃⠯⢿⠀⠀⢠⢤⠤⠤⠤⠤⣼⣇⡤⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠿⣯⠉⡇⢀⢤⠔⢁⠇⠀⠀⠀⣀⡸⡆⠀⠈⣎⠳⣄⠀⢀⣷⠈⣅⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣿⠀⡇⡎⠀⠀⡾⠀⠀⠀⠀⠳⠋⡇⠀⠀⠘⡄⠘⡆⡏⠁⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⠀⡇⡇⠀⢠⠇⠀⠀⠀⠀⣀⡀⡇⠀⠀⠀⢹⡀⢧⢳⠀⠀⠸⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⡏⠀⠧⠃⠀⢸⠀⠀⠀⠀⠀⠓⠃⣃⣀⠀⠀⠀⣧⠈⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⡏⣳⢒⡶⠶⠿⢏⠉⠉⡟⠛⠯⢿⡁⠀⠀⠀⠀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢠⣧⠀⠀⡇⡇⢹⡀⠀⠀⢺⠀⢰⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠈⠛⠀⠀⢧⠃⢠⠃⠀⠀⣞⣀⣸⠀⠀⠀⢹⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠳⠛⠉⠉⠉⠀⠉⠉⠩⣇⣀⣀⠔⠉⠉⠈⠢⠤⣀⣀⣹⠀⠀⠀⠀⠀⠉⠉⠉⠉⣙⡦
⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠒⠒⠦⣀⠀⠀⠀⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢐⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠁⠀⠀⠈⠉⠉⠉⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡄⠀⠀⢸⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⠒⡼⠰⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⣤⡈⠃⡾⠸⠈⣁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡔⠁⠀⣸⠀⠁⠁⠀⠀⠀⠐⢀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡓⠒⠒⠓⠒⠔⠢⠄⠀⠠⠀⡶⠀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠱⢄⣀⣀⡀⠐⠃⠀⠀⠀⠀⣢⠴⣒⡒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡸⠴⠊⢁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠓⠀⠀⠜⢉⠀⠀⠀⠀⠀⠀⡄⠸⢡⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢋⡅⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡀⠀⠀⡆⠀⠟⠀⠀⡰⠀⠀⠀⠀⠸⠀⠀⠀⠘⠀⠠⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⠀⠈⠀⠀⠂⢀⠔⠀⠀⠀⡄⠀⠃⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⡄⢰⣤⣀⣀⠀⠀⡄⠀⠀⠓⡾⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⢸⣄⣧⠀⠀⠈⠹⢯⡶⡃⠀⠀⡼⣧⠀⠀⣈⠀⠘⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⠇⢸⣆⠀⠀⠦⠀⠈⠁⢠⣾⣷⡏⠀⠀⠁⠀⢸⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠠⠟⣦⡀⠀⠀⣀⠔⢡⠏⡘⢓⣒⣒⣒⠂⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⣆⡆⠀⣷⠘⠋⡙⢃⣀⡏⡜⠁⠀⣷⢸⡌⢇⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡘⢁⠃⠀⣧⠀⠘⠁⢀⡷⠀⠀⠀⠇⢸⠁⠑⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠘⡦⣀⡣⢠⢃⡄⠠⢿⠀⢀⠈⢀⣉⡤⢴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠀⡸⠂⡵⠃⢸⠱⡀⠘⠒⢚⡏⠉⠀⢶⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠂⠉⠉⠉⠯⠹⠷⠴⢼⣦⣬⡴⣇⣀⡀⠈⠀⠀⠀⢸⠼⡓⠒⠲⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⡗⠲⠀⠠⠭⠤⠉⠀⠉⠉⠉⠉⠉⠉⠁⠉⠀⠻⣌⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠆⠰⠂⠐⠀⠀⠐⠐⠀⠀⠐⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠻⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡄⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⢀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⡻⣿⣿⣿⣿⣿⣷⣦⡻⣿⣿⣿⣿⡿⠿⠛⠋⢹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⡿⣿⣿⣿⡄⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠛⠿⠟⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⡀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠈⠛⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠈⠛⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⢠⣾⣿⣿⣿⣷⡄⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣶⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⢺⣿⣿⣿⣿⣿⣿⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠁⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠉⠻⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣆⠰⠶⠶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢻⣟⣛⣛⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣄⠹⣶⠶⠶⠦⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢻⡄⢉⣉⣉⣹⣦
⣿⣿⣿⣿⣿⣿⣿⣟⠻⣆⠹⣦⡿⢷⣶⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⡄⢻⣄⠛⢀⡉⢉⣉⣉⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢠⡿⢁⡾⢁⣾⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⠟⣰⡟⢠⡟⢩⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣤⣤⣤⣾⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠛⠛⠀⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠿⢡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣿⣷⣦⣄⣀⣀⠀⣀⠀⢸⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡷⠶⠾⠶⠷⠾⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⣴⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⠏⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⢿⣿⣷⣤⡀⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⣀⣀⠀⠈⠙⠿⣆⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣴⣿⣿⣿⣿⣦⠌⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠟⠁⠀⠀⠀⠈⣻⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⣀⠀⠀⠀⠀⠠⠾⠛⣛⣩⣥⣶⣶⠶⠷⠶⠶⢦⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⣟⣁⣠⡤⠶⠚⠋⠉⣉⣉⣉⣉⣙⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣬⣉⣉⣉⣁⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⠶⠀⠀⠀⡇⠀⠀⡇⠀⠀⡤⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢱⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⣴⠃⠀⡕⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠈⡆⠀⠀⠰⠁⠀⢀⡇⠀⠀⢠⡏⠀⢸⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⡆⠀⠀⢸⠀⠀⣽⠀⠀⢰⠃⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢣⠀⠀⠀⠀⠀⢸⡇⠀⢀⡞⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢡⠀⠀⢸⠀⠀⡿⠀⢀⠇⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⡀⠀⠀⡄⠀⢸⠁⠀⡸⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⡀⠀⣸⠀⠀⡇⠀⡸⠀⠀⡼⠀⣠⠖⢦⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢇⠀⢰⠇⠀⢸⠀⢀⠃⠀⢀⠇⢀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢻⠀⡏⠀⠀⡇⢀⠇⠀⢰⠇⢰⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠸⡄⢸⠀⠀⢸⠀⡼⠀⠀⡿⢀⠎⠀⢰⠇
⠀⠀⠀⠀⢠⠖⠲⣄⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠙⠋⠀⠀⠟⣠⠃⠀⣸⠃⠀⡶⠒⢦⠀⠀⠀⠀⣿⠀⠀⠉⠁⠀⠀⠈⠛⠁⠀⠸⢃⡞⠀⢠⡟
⠀⠀⠀⠀⠸⡀⠀⠘⣆⠀⠀⠀⢃⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠛⠁⠀⢰⠁⠀⠀⣇⠀⠀⢳⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢮
⠀⠀⠀⠀⠀⠳⡀⠀⠈⢦⡀⠀⡸⠀⠀⢀⡞⠀⠹⣆⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠘⣄⠀⠀⠱⣄⠀⢈⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠉⠛⠁⠀⢠⡟⢰⣶⣶⡝⣆⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⠛⠉⠀⠀⠀⠘⠃⣈⡀⠋⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⢠⠏⠐⠛⠾⢿⡓⠘⢆⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠈⢉⡠⠛⠃⡈⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢰⣏⣀⣀⣁⣀⣀⣉⣀⣈⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⠁⠚⠀⠉⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⠀⠀⠀⢸⠀⠀⢈⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⠀⠀⠀⠐⡇⠀⠀⡇⠀⠀⠀⢀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢱⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⣼⠃⠀⡵⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡆⠀⠀⢀⠀⠀⢀⡆⠀⠀⢠⠏⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⡆⠀⠀⢸⠀⠀⣼⠀⠀⣰⠃⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢱⠀⠀⠠⠀⠀⢸⡇⠀⢀⡏⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢱⡀⠀⢸⠀⠀⡟⠀⢀⠇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠸⣄⠀⢠⡆⠀⣸⡅⠀⣼⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⡅⠀⣸⠀⠀⡇⠀⡸⠀⠀⠸⠀⣠⠚⣷⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⡀⢸⡇⠀⣿⠁⢰⡇⠀⣸⠇⢠⡾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢻⡀⡏⠀⠀⡇⢠⠇⠀⢰⠇⡰⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⡧⠀⠸⠇⣾⠃⠀⣿⠀⡿⠀⢀⡿⢰⡟⠀⣸⡇
⠀⠀⠀⠀⢰⠛⠳⣄⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠙⠋⠀⠀⡟⣰⠃⠀⣼⠃⢠⡖⠀⣦⡀⠀⠀⠀⣴⠀⠀⠘⠃⠀⠀⠘⠛⠁⠀⣼⣣⡟⠁⢰⡟
⠀⠀⠀⠀⠸⡀⠀⠘⢆⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢰⠋⠀⠈⢇⠀⠈⢳⡀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠋⠀⠀⡏
⠀⠀⠀⠀⠀⠹⡄⠀⠈⢣⡀⢀⡼⠀⠀⣀⡀⠒⠒⢠⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠈⢆⠀⠀⠱⣄⠀⣠⠃⠀⠀⠀⡔⠁⠀⢀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠉⠉⠁⢀⣺⣿⣷⡀⢠⣿⣿⣧⠀⠀⠀⡎⠀⠀⠀⠀⠀⠈⢳⡀⠀⠈⠉⠁⠀⠀⠰⣦⠀⠀⠰⠏⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢨⠿⠿⠿⢿⡾⠿⠿⠟⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠘⢄⠀⣰⣿⣷⣄⠀⡜⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠑⠛⠻⠟⠛⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠁⠤⠤⠤⠂⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢡⣾⡟⢛⢛⠛⣛⣛⣛⣛⣛⣛⣛⣛⢛⣛⠛⣿⣦⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⣿⡇⢸⣿⡿⠿⠿⠿⠿⢿⣿⠀⣿⡇⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡘⢿⣷⣿⡇⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣬⣿⡇⢸⣿⣄⡀⠀⠀⠠⣼⣿⠀⣿⡇⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠈⠛⠿⣿⣦⣍⡛⢿⣷⣤⣀⠀⠉⠀⢿⡇⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣶⡄⣀⠀⠙⠻⢿⣶⣬⣙⠿⣿⣦⣄⡀⠀⠙⠿⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⣿⡇⢠⣄⠀⠈⠛⠿⣷⣦⣍⠻⣿⣷⣤⣀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⣿⡇⢸⣿⠛⠀⠀⠀⠀⢹⣿⠀⣿⣯⡙⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⣿⡇⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⣿⡟⣿⣷⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⣿⡇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠀⣿⡇⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣌⠻⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⢋⣼⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠀⠁⠀⠈⠁⠁⠀⠀⠂⠈⠀⠈⠈⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣄⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠉⠙⠻⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⡼⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢰⠸⢿⣷⣦⡀⠀⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⢀⠀⠀⠀⢈⠳⡀⠈⡹⠃⠂⠀⠄
⠀⠀⢀⠀⢀⠤⠒⠒⠊⠀⣠⡤⠤⠔⢒⣒⠒⢚⠒⠂⡟⢸⣇⣐⣒⡛⠓⠦⠤⣄⡀⠓⠒⠂⠂⠀⠠⠀⢀
⠐⠉⢠⠎⠁⠀⠀⠄⢹⠋⠀⢰⡞⠉⠀⠰⡀⠀⠐⠊⠀⢀⡲⢄⣈⠉⠝⢲⣮⠀⠙⢶⣄⠀⠀⠀⠈⠲⡄⠈⠐⠠⡀
⠀⠀⠸⡔⠀⠀⠀⠠⠸⡆⠀⠀⠳⢄⢀⠀⠈⠐⠢⠤⠠⣌⡉⠍⠂⠐⢁⠼⢉⠇⠀⡨⠙⠄⠀⢀⠀⠀⡸
⠀⠀⠀⠈⠒⠦⠐⠀⠀⠈⠲⠖⠤⡄⠈⠉⡑⡛⠛⠻⠷⠛⠻⠿⠛⠓⢒⣉⠡⠔⣝⠤⠊⠀⠀⣀⠡⠓⠁
⠀⠀⠀⠂⠄⢀⡀⠀⠉⠑⠒⠒⠤⠤⢄⣈⣉⡑⣒⣛⣿⣟⣓⠖⣂⠒⢒⣉⣉⠡⠤⠶⠒⠃⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠀⠀⠀⠀⠠⠄⠀⠈⠉⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠐⢂⡀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣤⠖⠀⢖⠂⢰⠀⡀⠐⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⣤⣤⣠⣄⣬⣄⣠⣸⡀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣯⣤⣤⣬⣤⣶⣴⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣶⣶⣶⣿⣿⣿⣯⡿⠛⠛⠛⠉⠉⠻⠯⠯⠿⠯⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠛⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣬⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣶⡞⠛⢿⡛⠛⠙⠉⣿⡏⢰⡖⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⡇⣰⠼⠁⠀⠀⠀⣿⡇⢙⡋⠃⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢲⠿⠗⠻⢳⣶⣶⣶⡶⠿⠷⣿⣷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⠀⠹⢿⣦⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣄⢀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣶⣶⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣾⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠿⠿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⣿⣿⠁⢀⣾⣿⡟⠉⠁⠐⠲⠿⢿⣛⡺⠟⠿⠿⠟⠛⠟⠛⠿⠍⠛⠉⠉⠀⠀⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⢸⣿⡟⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠀⣴⣿⣿⢃⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⡇⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠿⠿⠄⢴⣿⡿⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢠⡀
⢀⡴⢂⡀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠘⢇
⢿⡆⠘⠀⠀⣴⢿⣷⣾⡟⠀⠀⠀⠀⠀⠀⢣
⢀⣷⠀⣠⣾⠏⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠸⣛⣛⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣠⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⣀⠀⢰⣦⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣠⣾⠋⠀⠀⢹⡇⠀⣿⣾⣯⣴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠉⠉⠉⠛⢛⣷⣶⣼⣷⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⢻⡿⠀⠀⠀⢀⣾⠛⠉⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢠⡿⠁⠀⠿⠀⠈⠻⣦⣄⣀⣀⣤⣴⡶⠟⠛⠉⠉⠉⠉⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣾⡇⣴⠟⠁⠀⠀⠀⣀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠰⠆⠀⣀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡇⢠⣿⣿⠋⠀⠰⠆⠀⠀⠉⠀⠀⠀⠘⠃⠀⠀⠰⠦⠀⠀⠀⠀⠉⠀⣘⣿⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠸⣯⣤⣿⣟⣁⣤⣤⣤⣤⣤⣴⡶⠶⠶⠶⠶⠶⣶⣦⣤⣤⣤⣤⣤⡶⠾⠛⠋⠉⢠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣋⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⣠⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⣻⣷⠄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣈⣿⠏⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣥⣤⣤⣤⣴⡶⠟⠛⣉⣥⣤⣤⣀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣥⣤⣬⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠉⠉⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠰⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠒⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⣀⠀⡠⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠒⠒⢤⣠⠤⡤⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⢀⠔⡠⢂⠆⢠⠀⡀⠀⡀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡠⢀⠄⡠⢂⠔⡡⢚⠔⢁⠔⡡⢂⠔⢁⠜⢠⠊⡔⠁⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⠠⠂⠔⡡⠊⢐⠑⡠⢀⢐⠕⢈⠔⡪⠂⡥⢊⠔⡡⢊⠔⡡⠊⠔⢁⠔⡡⢊⠔⢁⠔⢁⠔⡡⢊⠔⡀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣀⣀⣁⣈⠀⠁⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠊⠀⠁⠊⠊⠚⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⢠⡀⠀⠀⡤⠀⢀⣠⠀⠀⡀⠀⠀⠉⢉⠏⠉⢩⡭⠧⢉⠜⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣔⠃⠀⠉⣀⠀⠘⠁⠀⠈⠁⢀⡸⢃⡀⠀⡰⠃⠀⠀⠚⡠⠀⠁⠀⠀⠀⠓⠁⠀⠀⠀⢠⠞⠰⠋⣠⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠞⠀⠸⠁⠀⠀⢠⠞⢁⠜⠁⠀⠀⠀⠚⠒⠒⠈⠉⠭⠭⠄⡠⠋⠀⠀⢀⡤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣴⠃⠀⢀⡄⠀⣠⠀⠀⣀⠖⠀⠀⠰⠇⠐⠒⠒⠒⠲⠒⠀⠀⠀⡨⢁⠔⠉⠀⡼⠁⠀⠀⣠⠄⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⠀⠀⠟⠀⠸⠂⡀⠀⠉⠀⠀⠀⠀⠀⡴⠀⢀⠔⠁⠀⠀⠠⢢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡨⠀⠀⢀⡦⠀⠈⠃⠀⢠⠆⠀⢠⠄⠀⠀⡠⠊⠀⠀⠀⡀⠜⠁⡀⠀⠀⠀⣰⠂⠀⠀⠰⠃⢠⡆
⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠈⠁⠶⠀⠈⠀⡾⠀⣠⠀⠀⠀⠀⠋⠀⣠⠊⠀⠀⠀⡠⠊⠀⠀⠞⠁⡴⠀⠀⠁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠓⠀⠀⠀⠈⠁⠀⠁⠀⠐⠟⠀⢀⠔⠁⠀⠀⡠⠊⠀⠀⣀⠀⠀⠀⠀⠀⣠⠀⠉⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡮⠁⠀⠀⣤⠀⡴⠁⠀⢀⠄⠉⠀⠀⠀⠜⠁⠀⠀⠀⠀⠈⠁⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⡌⠐⠊⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡘⠀⠀⢀⠤⢛⡩⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢠⠀⠀⣢⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠒⠉⠀⠀⠠⠞⠀⠀⠀⠀⠀⡴⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢀⠀⠄⡐⠠⠀⣒⠠⠄⡒⠠⢄⠢⠐⡂⠄⠒⡈⠄⢀⠨⠐⢈⡡⠄⢂⠁⠄⢐⠠⠅⠒⠨⠀⠶⠒⠶⠂⠆⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠈⠀⠀⠁

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⣀⠁⠑⡌⠦⠀⣤⣀⠉⠀⢀⡷⣤⡀⠀⢕⣠⣀⣛⣀⣄⡊⠀⠀⠰⣿⡍⢿⡀⠀⠀⠈
⠀⠀⠁⠀⠐⠀⠀⠀⠈⠁⠀⠀⠛⠛⠋⠁⠀⠀⠀⢀⣄⠀⠀⠀⣀⣠⣄⠘⢷⠈⣿⠀⠠⠴⠤⠴⠄
⠀⠀⠀⠀⣠⡟⠛⢦⠀⠈⠉⢉⣳⣦⣶⠀⠀⠀⢀⣸⣇⣈⡂⠈⠁⠶⢿⣦⣀⣀⠈⠀⢀⡀⠀⠀⡠
⠀⠀⠰⣾⣿⣿⠃⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠉⠉⢉⠉⠁⠀⠀⡀⠀⠈⠉⢉⠀⠊⠛⠶⠖⠚⠁⠁
⠀⠀⠀⠀⢈⡄⣠⠄⠀⠀⠀⠰⠤⣄⣠⣄⣤⠤⠆⠀⠀⠉⢟⣿⣾⣿⣿⣗⠏⠁⠀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⢀⣼⡿⢋⣵⠕⠀⠀⠀⠈⠉⠉⠉⠉⠙⢉⣁⡀⠀⡤⠤⢤⣀⣶⣀⡤⠤⡤⠀⠀⢀⣄⠈⠙⠦
⠀⠀⠉⠉⠚⠋⠁⠀⠀⢠⣾⣿⣦⠀⢀⠀⣤⣟⣵⠍⠃⠀⠐⠘⠛⠛⠛⠃⠀⠀⢠⣶⣮⣩⣶⣦
⠀⠀⠑⢦⣀⣀⡤⠊⠀⣼⣁⣦⣹⡄⣀⣀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⣤⢿⡿⣤⠀⠀⠀⠀⠀⠀⠀⠙⣿⢻⠟⠀⠀⠀⠀⠀⣠⣀⡀⠈⠀⢠⣤⠀⢀⣼⣿⣥⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠔⠀⠀⠦⠀⣠⣛⣌⡂⠀⠀⣠⣆⣼⣷⠤⠤⠀⠀⢾⣿⣆⠀⠀⠉⠙⠛⠛
⠀⠀⠶⠀⠀⠛⠀⠀⠁⠀⠋⠀⠀⢀⣉⡉⠉⠀⠠⢄⡉⠉⠉⢀⡠⠄⠀⠀⠀⠉⠛⠿⠶⠶⠶⠦⠤⠄
⠀⠀⠀⠀⢀⣤⣾⠅⠀⠀⠀⢀⣴⣟⣉⠉⠓⠀⠀⢀⣙⣿⣿⡉⠀⠀⠙⠑⠶⣤⣀⡤⣄⣠⡴⠖⠙⠁
⠀⠀⠀⠈⠋⠙⠃⠀⠀⠀⠀⠋⠙⠛⠋⠀⠀⠀⠀⠘⠋⠛⠋⠁⠀⠀⠀⠀⠈⠚⠛⠛⠛⠛⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⣩⡿⢿⡏⠉⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠶⣾⡏⠀⢠⣿⠁⠀⢁⣠⣤⣼⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢻⣷⣄⠈⣿⣦⣠⣿⠟⠉⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠛⠉⠙⢿⣷⣬⣿⣿⣧⣴⣶⡶⢶⣤⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⣤⣀⣀⣤⣿⡿⣿⣼⡿⣿⣧⡀⢀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠉⠙⠛⠛⢉⣴⣿⢿⣷⠈⠻⣿⡆⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣴⣿⠟⠁⢈⣿⠀⠀⢹⡏⠴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣉⠉⢸⣆⣀⣼⠟⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠙⠛⠛⠛⠉




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣤⣭⣽⣵⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⢲⠆⠀⠀⠀⠀⠀⢹⡀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠁⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⣀⣠⣴⣄⡀⠀⠀⠀⠀⢸⡃⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡤⠀⠀⢀⡀⠀⣴⣿⣿⣿⣿⣿⣯⣿⡶⣤⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠰⠚⠁⢠⠀⠈⢷⠀⣿⢻⣿⣿⣿⠏⢸⠃⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡴⠀⠀⠀⢀⡤⠶⠒⢻⡧⠤⡀⠀⣿⠸⠇⡇⢸⢶⠸⣇⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⡇⠀⠈⠻⠷⠖⠐⠓⠦⡈⠐⠛⡽⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⡤⣄⣄⠽⣀⣀⣀⣸⣇⣀⣀⣀⣀⣹⠀⠀⣇⠂⠀⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠟⢹⡝⠋⠉⠉⢉⡝⠀⠀⡛⠀⠀⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠘⣦⣀⠀⠀⠀⠀⢀⣠⠞⠿⣅⠲⠂⠉⠀⠀⠀⣇⣀⡤⠶⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣩⠀⠀⠀⡎⠉⢑⡶⠖⠒⠉⠁⢸⠀⠀⠀⣉⠀⠀⠀⠀⠟⠁⡴⠶⡿⠆⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣸⡷⠞⠋⠀⢠⠀⠀⠀⡾⠅⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣰⣧⣴⣇
⠀⠀⠀⠀⠀⠀⠀⣦⢀⣠⡴⠛⢧⡄⠀⠀⠀⢸⡷⢻⡇⣧⠀⠀⢠⠹⠄⠀⠀⢠⡄⠀⣟⠉⡇⠀⢹⡄
⠀⠀⠀⠀⠀⠀⣠⡟⡟⠁⠀⠀⠀⠀⠀⠀⠘⡃⡇⣾⠁⠉⠀⠀⠈⠟⠀⠀⠀⠸⡇⠀⣏⠘⠃⠀⢸⡇
⠀⠀⠀⢀⡴⠟⢹⡀⠃⠀⠀⠀⢠⠀⠀⠀⠀⠙⢠⡟⠀⠀⠀⠀⠀⠀⡀⠀⠸⣦⡇⠀⡇⠀⠀⠀⣸⡇
⠀⠀⣰⠟⠀⠀⠈⠉⠀⡀⠀⠀⢸⠆⠀⠀⠠⢀⡾⠀⠀⠀⣆⠀⠀⢀⡇⠀⠀⠈⠀⠀⡇⠀⠀⠀⠃⣧
⠀⠀⣿⡀⣦⠀⠀⠀⢠⡇⠀⠀⡇⠀⢠⠸⢂⡾⠁⠀⠀⢀⡟⠀⠀⡆⢁⠀⠀⠀⠀⢰⡇⠀⠀⠘⢻⣿⠃
⠀⠀⠘⣿⣏⠀⠀⠀⡘⠀⠀⠀⠿⠗⠉⣠⡞⠁⠀⠀⠀⢈⡃⠀⠀⠳⠎⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠈⢣
⠀⠀⠀⢠⡿⠳⠶⣤⣿⣧⣤⣤⣤⠴⠞⢩⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢠⡆⠀⣾⠀⡌⡇⢀⣠⣴⡏
⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡿⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⢠⡟⠀⣧⠟⠋⠁⠀⣹⠆
⠀⠀⠀⠀⢰⠇⠀⠀⠀⢼⣧⣤⣤⣄⣀⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠄⣾⠃⠀⠀⠀⣠⠰⠾⢯⡀
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⡿⠁⣿⠀⠘⠿⠛⠻⢶⣶⣤⣀⠀⢸⡄⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⡿⠀⠀⣠⠟⢺⠄
⠀⠀⠀⠀⠻⠃⠀⠀⠀⢀⣿⣄⢻⡄⠀⠀⠀⠀⠀⣽⠯⣬⠟⠊⡇⠀⠀⠀⣼⣿⣷⠀⠀⠀⢘⠃⠀⠀⠙⣳⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡉⠈⣿⡀⠀⠀⠀⢰⣀⡀⠀⠙⠶⠃⠀⢀⣾⡏⣬⡿⠃⠀⠀⠘⠋⠀⠀⢸⡏⠾⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡞⠉⠀⢸⣷⡀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣰⢿⣿⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⢰⡄⠀⠀⣸⠃⢠⡟⠁⠰⣼⡷⢻⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁⣾⠙⣿⠛⣻⠛⠛⠛⠛⣿⠛⠛⢻⡇
⠀⠀⠀⡘⣧⠀⠀⣿⡄⢸⣄⣀⣠⡏⢳⢸⡏⢻⣶⠴⠦⣴⠞⠁⠀⣼⠃⢸⡟⠁⠉⢹⡏⢻⢿⢻⡿⠲⢿⡁
⠀⠀⠀⣧⡝⠀⠀⠀⠀⠀⢻⡭⣽⠁⠀⢻⡿⠋⢹⠀⣠⠟⢷⣤⣴⡏⣰⠋⠀⣼⠀⢸⣇⢸⣿⡌⢻⣄⡀⣹⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣟⠀⡞⠀⠀⢸⡅⠀⢈⡿⠇⠀⠀⢿⣿⠀⠁⠀⠀⢸⡆⢠⡟⠋⠈⠷⠶⣿⣏⠉
⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠉⢶⠇⠀⠀⡸⡏⠉⢻⠁⡟⠉⢹⠛⡏⠀⠀⠀⠈⣤⠃⢠⡷⠀⠀⠀⢀⢻⡿
⠀⠀⠀⠀⠀⢰⡇⠀⠈⡇⠀⠘⢷⡤⣀⡇⡀⢀⡎⢀⡷⠀⠘⢶⡇⠀⠀⢰⡀⠀⠀⢀⣀⣤⠶⢚⣿⣸⠆
⠀⠀⠀⠀⠀⢸⡄⠀⠻⠃⠀⠀⠘⣦⡈⠛⣠⠟⠀⢻⠇⡆⠀⠀⠷⠤⢤⡾⡷⠚⠛⠉⠁⣠⣤⣼⡇⠉
⠀⠀⠀⠀⣀⣈⣁⣀⣀⣤⣤⣤⡴⠋⠁⠸⣏⠀⠠⣸⠆⢻⡄⠀⠀⠀⠀⣇⡇⠀⠀⠀⠀⣹⡦⢶⡛⠁
⣤⠖⠛⠋⠉⠉⠉⠀⢠⣞⣋⣙⣿⣒⣦⣤⣼⣧⣀⣩⣤⣾⡋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⣧⣼⡛⠛⠓⠲⠶⠤⠤⣄⣀
⠙⠛⠶⠶⠶⠶⠶⠶⢤⣤⡄⢠⡤⠤⠤⠤⠾⠃⡿⠿⣉⣼⣿⣻⣿⣿⠶⠶⠶⠶⠶⠤⠴⠷⠆⠤⢿⣤⠤⠤⠤⣤⣤⣤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣏⣈⠓⠂⠐⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡄⠀⠀⠀⠀⠀⢀⠁⣠⡔⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⢿⡀⠀⠀⠀⣀⣴⠾⠙⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣴⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠈⢳⣤⣴⠾⠛⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⣿⣶⣿⣿⠟⣫⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠋
⠀⠀⠈⠻⢆⣿⡄⠀⠀⠀⠀⣆⠀⠀⠀⠀⣿⣿⠉⠙⠻⣿⡿⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠉⠙⠙⠲⠦⠴⠁⡆⠀⠀⠀⣿⡏⠀⠀⢠⡿⠀⠀⠙⢷⣤⣀⢀⣠⣤⣶⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠈⢻⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿⡟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⢰⡿⠁⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡇⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡉⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠗⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠄⠀⠀⠀⣶⠟⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⠟⠁⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠋⠀⠀⠀⢀⣀⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣤⣤⣤⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢀⣀⡀⣀⠤⠴⠞⠋⢉⣧⣄⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢳⣄⠘⠄
⠀⠀⠀⠀⠀⠀⠐⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠤⠶⢶⣖⣒⣚⣒⣲⣦⠼⠧⣀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⢢⣻⡤⠴⠒⠒⢒⣒⣛⣛⣛⣯⣭⡭⠭⠭⠶⠶⠒⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠳⢦⣄⠙⠢⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣹⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠊

⣀⠀⠀⠠⡄⠀⠀⠀⠀⠀⡀⠰⡆⠀⠀⠀⣤⠀⢰⠄⠀⢀⠀⢰⡄⠀⠀⢀⠀⠀⠀⠀⠦⠀⠀⠀⢀⠀⠀⠀⣀
⠼⢀⡄⠈⠃⠀⠘⡇⠀⠀⣧⠀⠀⠀⡄⠀⠛⠀⠀⠀⠀⠿⠀⠈⠁⢀⠀⢽⠀⠀⠀⠀⠚⠀⢀⡀⠸⠆⠀⠀⣽
⠀⠘⠗⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠨⠆⠀⠀⠀⢶⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⢠⡄
⢀⠀⠀⠀⠀⠛⠀⠀⠀⡧⠀⠀⢀⠀⠀⣠⠀⠈⠆⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡀⠀⣆⠀⢘⡄⠀⠘⠖
⢸⠀⠀⢼⡄⠀⢀⠀⠀⢀⠀⠀⢨⡆⠀⠸⠀⠀⠀⠀⢸⡇⠀⠋⠀⠀⢸⡅⠀⠀⠀⢀⠀⠀⠸⣥⠀⠁⠀⠈⠁⠀⠀⣆
⠀⠀⠀⠈⠀⠀⢻⠆⠀⠘⠇⠀⠀⣀⣀⡀⠤⢀⠲⠒⠺⢧⠤⠄⣀⣀⣀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⢀⡄⠀⢠⠀⠉
⠀⠀⣷⠀⢾⠀⠀⠀⣀⠔⢂⡩⡾⢝⢉⡒⢕⣊⠅⠀⠀⢨⡆⢅⠂⢀⠈⠉⠛⠳⢖⡒⠠⠤⣀⡀⠀⠀⠸⣾⠀⢪⠄
⢀⠀⠙⠀⠀⣠⡐⠙⠀⡠⢺⠭⡢⢋⠥⡂⠅⠂⠀⠀⠀⠀⠘⡀⢇⠂⠈⢀⠀⠀⠀⠈⠳⣕⠆⠈⢉⠢⢄⡀⠀⠀⣤
⠸⠆⢀⡔⡫⠔⠁⠀⡰⠁⢑⠥⡪⢔⠨⢒⠁⠀⠀⠀⠀⠀⠀⠃⢑⠄⠀⠀⠢⠀⠀⠀⠀⠈⢧⡢⢀⠠⠀⠈⠢⡀⠙⠁
⠀⡠⢧⣀⡄⠀⠀⢰⣁⢔⢕⣪⡲⠑⠍⡀⠄⠀⠀⠀⠀⠀⠀⢸⠴⠊⡀⠠⠀⠁⠀⠀⠀⠀⠂⢳⣕⡠⢁⡀⠀⢜⢆
⣼⠖⠓⠓⠒⠧⢀⡨⢔⢅⣕⠵⠮⠖⠒⠖⠤⠦⣤⣠⣀⠀⠀⢐⠨⡐⠠⣘⣪⣥⣀⣂⠤⠀⣀⠄⢳⣊⣔⠤⠙⠒⠺⡆
⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠓⠦⣘⡵⠟⠋⠁⠀⠀⠀⠀⠀⠉⠈⠉⠺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⡇⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣇⠀⢀⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣉⣉⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠐⠒⠄⠢⠀⠄⡪⠐⡡⢂⠅⡪⢀⠔⡠⡂⢔⠔⡠⢂⢔⠔⡠⡢⢀⠀⡢⢀⢄⠀⡀⡀⡀⢀⠄⡀⡀⢀⢀⠀⠀⡀

⠼⣅⠃⠐⢄
⢫⡀⠀⠤⠎⠦⠀⢢
⠀⠙⠦⣄⣀⡀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠠
⠀⠀⠀⠀⠉⠹⣆⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⣇⡤⠒⠀⠀⠀⠀⢀⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠈⠓⢄⡀⠀⣹⣦⣤⣀⢀⣤⣶⠟⢈⡄⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠶⣄⠀⠀⠀⠀⠑⢦⡘⣿⡉⠉⠉⠉⠀⠀⠘⣷⡀⢀⡴⠃⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠘⢷⣄⠀⠀⠀⠀⠉⠸⣿⣦⠐⠀⠀⠀⢀⣿⣷⡈⠷⣄⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠳⢄⡀⠀⠀⠀⣿⣿⣦⣀⣀⣠⣾⣿⣿⣿⣇⠈⠓⢄⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⢯⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⢦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠋⢻⡀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠈⢧⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⢸⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⣄⠀⠀⣤⡄⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⢎⡰⠀⠘⣧⡉⠦⠠⢑⠄⠐⢅⣸⠀⡊⢁⢀⡰⠇⣀⢀⠀⠠⢻⠈⠀⠀⡀⠀⠂⠈⢠⡈⠀⠉⣁⡁
⠀⠀⠁⣀⠀⢈⠈⢹⠋⠉⠁⣀⣀⡀⠀⢀⡙⠉⠙⡄⠀⠀⠀⡠⡦⠞⢳⠀⠀⠀⢁⡀⠸⡇⠈⠆⣆⡤⠸
⠀⢠⠞⡙⡑⢠⡀⠘⠂⠀⠰⢿⣯⣿⡆⠑⢂⠤⠠⠕⠤⠀⠀⠚⠙⠛⠉⠊⠁⠀⠈⢁⡀⠡⡀⡟⠀⡸⠁
⠀⠘⢶⡰⢀⠁⡜⣢⠀⣤⠀⠻⠚⠋⠀⠀⣀⡄⠀⠀⡠⠀⠈⠘⠀⠀⠠⠊⠈⠢⡀⠨⢯⡆⠀⠢⠔⣡⠄
⠀⠠⢢⠈⠪⣔⡼⠃⠀⢠⣠⡀⠀⠂⢠⠀⢠⣥⣨⣩⡝⢤⠀⢀⡁⠀⠡⡀⠀⢀⠃⠀⣠⠒⠂⠒⠐⣌⠁
⠀⢠⣥⡄⠀⠈⡀⠂⠀⣸⢿⣾⡀⣠⢌⠉⠉⠉⠉⢉⣁⡉⠀⠠⣴⠀⡀⠈⠈⢁⡄⢸⠂⡀⠀⠀⢀⠠
⠀⢸⣭⣧⢤⠾⢕⠀⢠⣿⣿⣿⡇⢀⠔⢱⡄⠘⠁⣠⠤⠶⠤⢄⠸⡇⠀⠠⠶⢌⠁⡀⠀⠈⢈⠁⠁⣠⡀
⠀⠈⠉⠁⠉⠀⠉⠀⠀⠿⣿⠿⠁⡀⠀⠮⡃⠀⣶⣁⠔⠑⡂⠊⠁⡽⠀⠀⣠⣀⠄⠀⠀⠀⣿⠇⠀⠙⠁
⠀⢸⡇⠀⡐⠁⠀⢈⣢⣄⠀⡦⠀⡁⠀⠀⡏⠀⣀⢀⠀⢳⠇⠀⢀⠳⢀⡈⢻⡻⡀⢀⡀⠘⡯⠂⠀⠛⠃
⠀⢸⡇⢰⠶⡎⡵⢄⣁⡜⠀⣀⠀⠉⢹⠛⠃⠀⠼⠯⡀⠀⠒⠁⠑⡁⣨⠁⠀⠪⠁⠈⢯⣀⡇⢀⡟⠐⡆
⠀⠐⠇⠀⠓⠔⠁⣀⠀⡤⢎⡉⡧⢈⣩⠀⠀⢀⡆⠀⣠⡀⠉⠋⠠⠽⣿⣏⠁⠘⢛⡀⠀⠑⠘⠉⡄⠀⠠
⠀⠀⠄⣤⣢⢀⠈⠯⠗⠠⡞⣼⠁⠘⠯⠉⣠⡯⡷⠤⠷⠟⠁⠠⠀⠀⠃⠈⠞⢁⠰⣻⢦⠀⠉⠀⢀⠀⡆
⠀⠀⡒⣧⣽⢫⢀⣤⢠⡶⠕⢼⡄⠀⡤⠉⢀⣀⣀⣀⣀⣤⡾⠄⠰⠀⠰⠸⡆⠀⠠⠀⣂⠁⠧⣉⣀⡤⠁
⠀⠀⠀⢐⡊⠀⠀⣁⡀⠙⠚⠁⣴⠄⠁⠀⠛⠙⠺⣥⢰⠊⠀⠀⡀⠀⠀⡀⠀⠐⠃⠐⠁⠠⢄⣤⡤⣤⣤
⠀⠠⠊⡏⢙⠒⠄⢓⡩⣠⠠⣶⢿⠁⠀⠀⠀⠐⠀⠈⠋⢀⡀⠀⠐⠀⠀⠁⢀⡁⡀⣝⠀⣀⠀⠀⡀⡈⠁
⠀⠐⠦⠇⠸⠤⠂⢀⣿⡎⠁⠌⣳⣦⠐⡁⠀⠀⠈⠀⢸⢽⣪⡅⢰⠀⠈⢆⠀⡠⠑⠂⠬⠊⠀⢀⠀⡀⠁
⠀⠐⠴⠒⠤⠢⠔⠂⠩⠓⠂⠚⠛⠀⠀⠀⠰⠰⠀⠀⠀⠉⠊⠀⠸⠤⠂⠘⠂⠀⠀⠀⠆⠀⠠⠈⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠨⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠱⠂⠐⠒⠒⣄⠠⢄⢀⡀⠧⠀⢀⢀⢀⠀⠀⠄⠀⠔⠉⠀⠀⠀⠀⠑⡀
⠀⠀⠀⣀⠀⣐⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢤⢀⢀⠀⠁⢀⢀⡠⠆⠦⣠⠆⠂⢦⠎⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠌⠀⡀⠀⠀⡀⡀⡠⠀⢊⡄⠴⠐⠚⣊⠝⠛⡛⠛⠛⠡⣙⡛⠠⠤⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠂⠤⠤⠤⠀⠀⡲⠖⠈⠁⠀⠀⡠⠊⠀⠀⠨⠐⡄⠀⠀⠐⠈⡓⢤⡀⠈⢑⠒⠤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠠
⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠠⡀⠉⠢⡑⠁⡠⠂⠉⡒⢄⠀⠀⠀⠀⠀⠈⠁⢢
⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠐⠀⠈⢢⠂⠐⠨⡂⠴⠵⢦⠀⡀⠀⠀⠀⠈⠐⠄
⠀⠀⢠⠀⠀⠀⣀⠀⠄⡐⢁⡀⠀⠠⠤⠄⠀⠀⢠⡬⢀⣰⡠⠐⠂⠉⠉⠈⠉⠐⠂⠵⡔⠉⢀⠀⠄⠀⡐⡐⠀⠀⢂⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠋⠁⠀⢠⠂⠁⠀⠄⠀⢨⣧⠀⡀⠀⠀⠀⠂⠂⠀⠄⠀⠀⠠⠡⠡⠀⠈⠌⠔⠠⠠⠁⠂⠂⠢⠐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠘⣿⠂⠐⠐⣀⠀⣀⠀⢀⡄⠀⠀⡂⠒⠀⠒⠐⠒⡐⠒⠒⡒⠒⢒⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⠂⢀⠀⣿⠀⠀⢰⠃⣠⡁⠀⡤⠀⠀⣀⠅⠀⡐⠃⠀⡬⠃⠀⡬⣰⠂⠍⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠉⢠⡉⠀⠏⠀⠀⠴⠀⠉⠀⠄⢠⠂⠀⢁⠀⠀⠩⠄⢀⡐⠀⢀⡆⠀⠀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⢀⡀⠀⠀⣨⠀⠘⠃⠀⠀⠰⠁⠀⠀⠀⡎⢀⠀⠛⠀⠈⣁⠀⠈⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⡏⡇⢠⠄⢠⡂⠀⢀⠄⢀⠄⢀⡌⠀⡀⠀⠰⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣡⡜⡀⠀⠀⠀⠗⠀⠈⠀⠈⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢔⠄⡂⢐⠀⠠⡀⠀⢐⠀⡂⢀⠄⡀⠀⠔⠠⠀⠀⢔⠀⠀⠔⠠⠀⠀⠄⠀⠀⢀⠄⡠⠀⢀⢀⠀⡠⢀⠤⣀⢄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠈⠈⠀⠀⠈⠐⠁⠀⠊⠐⠁⠁⠈⠈⠀⠀⠀⠁⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⣶⣶⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⠀⢸⡇⠀⢠⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠻⣦⣸⣇⣴⠟⠁⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠷⠶⠶⢾⣿⣿⡷⠶⠶⠾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⣴⠟⢹⡏⠻⣦⣀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠁⠀⣸⣇⠀⠈⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠛⠛⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡙⠿⣿⠿⢋⣉⡙⠻⢿⡿⠛⣉⣉⠛⢿⡿⠟⢋⣉⡙⠿⣿⠿⢋⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣻⣿⣟⠛⢛⣿⣿⣟⠛⢛⣿⣿⡛⠛⣻⣿⣿⡛⠛⣻⣿⣟⠛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣼⣿⣿⣤⣤⣼⣿⣿⣤⣤⣿⣿⣧⣤⣤⣿⣿⣧⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠭⠐⡂⠭⠄⣂⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢡⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⡭⠐⡀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⣩⣥⡖⡞⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⡋⠩⠐⣁⠅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄⣒⣨⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣶⣺⣯⢿⣖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⠗⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣗⣿⡿⢿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣤⣾⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⣿⣷⣿⡿⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠼⠇⠀⠀⠀⠀⠀⢀⠀⣀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡄⠀⡠⠔⠠⠘⠛⠀⠀⣰⠭⠏⡀⢀⢞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠉⠓⠄⠀⠀⠉⠀⠀⠑⠤⠗⠁⡠⠖⣋⠀⢀⠤⣈⢹⣗⡌⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠤⣠⡗⠁⠀⢀⡔⠒⣢⣀⡠⠔⠫⠒⠚⡾⠁⣟⡴⡿⠀⠉⠰⠉⠱⡄⠀⣎⣱⠄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠁⠀⡔⠁⣨⡳⠛⡇⠀⠀⠱⢤⡾⠃⣴⡌⠁⠀⡠⠴⠃⡤⠤⡘⢆⠌⢥⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢃⢌⣀⡳⠾⠃⡠⠒⣠⣤⡄⠀⢠⢄⠀⡜⠀⡠⠜⠣⣴⠃⠈⢦⣰⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠐⢤⣠⠔⠊⠁⠀⠳⠞⠉⡕⠀⡠⠗⠒⢿⠉⢀⡀⠈⠛⠀⡇⢨⡀⠀⡄⡐⠀⠀⠀⠻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡔⠚⢇⠀⡄⡰⠰⡘⠸⣰⠀⠀⣄⢀⣴⡟⠀⢧⠮⠀⠀⠀⠘⡄⠉⠀⠋⠀⠀⠀⢠⡤⠱⣄⡄⠀⠀⣠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠙⠛⠒⠋⢰⢡⢁⢢⢁⠗⢣⠀⠀⠈⠉⠉⢠⠖⣽⠂⠀⠀⢆⡆⢈⢀⣄⠀⠀⣖⣢⠀⢱⠠⡟⠀⣠⠼⣄⢀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢃⠃⠃⠇⠎⡀⡜⠓⠋⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⡠⠖⠶⡁⠘⠶⠀⠀⠉⠁⠂⢠⠀⢱⠀⢇⡠⢀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠁⠀⠀⢢⠄⠈⡘⡜⡘⢠⢱⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠓⢄⣸⢂⠀⠀⠀⢀⡀⡃⡀⠈⠐⣸⠀⠀⠉⠠⣎⡶
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠀⡇⠀⠡⢂⠱⢀⢃⠇⡌⡜⡠⠀⠀⠀⠀⠀⣀⡀⠀⢠⣇⠀⢔⠌⠁⠿⠀⢀⢀⠰⠟⠁⠀⢶⠀⡼⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡄⡆⠙⢢⡀⠱⢂⠘⡘⢠⢡⢁⠃⠀⠀⠀⠀⠮⡀⡼⠀⡇⠈⠉⠉⠀⢀⠀⠀⠐⠈⠀⠀⠀⣀⠙⠤⠁⡋⢈⡇
⠀⠀⠀⠀⠀⠀⠀⡤⠠⡄⠘⣷⠀⠀⠱⡈⠡⡂⠀⡎⡈⡌⠀⢠⢊⡹⣄⠴⡌⢀⡴⠁⠀⠀⢠⠰⠘⣀⠤⡀⡎⠉⢧⡀⡇⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠘⠤⠨⠇⠀⠘⡗⠀⠀⠳⡀⠀⠀⠀⠁⠁⠇⡇⠈⠁⠀⠀⠉⠉⠀⠀⠀⠀⣨⠔⠋⠀⢰⠁⢰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣄⠀⢇⠆⠠⡀⠘⣄⢀⠴⢦⡈⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⡇⠀⣦⠂
⠀⠀⠀⠀⠀⠘⠶⠃⠀⠀⣀⡀⠘⠊⠣⡁⡀⠈⠁⠀⠀⠓⠔⠃⠀⡀⠄⢂⡠⠖⣆⠰⡇⠀⠀⠀⠀⠀⠱⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢌⠂⠀⠀⠙⢌⠠⠀⠀⠀⠀⢀⠄⢂⡤⠚⠁⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠑⠢⡄⠀⢀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢸
⠀⠀⠀⡠⢄⣠⣄⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⡴⠒⣠⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⡀
⠀⠀⠮⠠⠄⢴⣃⣃⣘⢤⠀⢉⣒⣒⣂⣨⣭⡑⢤⣤⠔⢒⡈⡉⠑⠢⣤⠤⠌⠁⡭⠁⠤⠤⠔⠋⠀⢝⠦⠤⠤⠤⡄⠀⣄⠈⠂⢐⣚⠂⠀⠀⡠⠠⡖⠒⣄
⠀⠀⠀⠀⣆⢁⣀⣀⡀⠤⠴⡖⠒⠒⠲⢶⣒⣒⢒⡐⠂⣘⣒⣂⣨⡼⠧⢀⣀⡀⣀⣀⣀⣀⡉⠀⠉⠉⢀⢄⠈⠵⣎⣵⣯⢍⡡⢤⣤⣌⣓⡌⣁⡚⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠄⠒⠒⣲⡦⢤⣀⢱⠗⠫⢀⣀⣬⠿⢍⣁⡞⢛⣦⠔⠚⠛⠛⢄⡘⠃⢘⠀⠙⠯⡛⢃⡎⠀⠈⠉⠀⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⢤⣶⣮⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢂⢀⠀⡀⠀⠀⠢⡀⠀⠒⠠⠀⠀⠀⠀⠈⠑⠀⠐⠀⠀⠈⡁
⠀⠀⠀⠀⠂⠈⠀⠤⠀⢀⠀⠀⠀⠠⣀⣂⠀⠁⢀⡆⠀⠀⡇⠀⠀⠀⠄⠀⠀⢠⠠⡠⠤⢄⡀⠀⠄⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠁⡀⡄⠀⠉⠙⠉⠛⠁⠀⠀⠐⡆⠀⠀⠀⢡⠅⠀⠀⠀⠀⠀⠓⠒⢄⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⢀⣘⠁⠀⠉⠑⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⠀⠀⣀⢀⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐⠐⠒⠀⠐⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠒⠒⠐⠂⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠒⠀⠀⡄⠀⠀⡀⠀⠀⠄⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠆⠀⠀⠀⠂⠀⠣
⠀⠀⠀⠀⠀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠈⠦⠀⠐⠂⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠁⠀⠆⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⢈⡀⠀⠀⠀⠀⡀⠀⠀⢈⠀⠀⡀⠀⠦⠅⠀⠈⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⡀⠀⣀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⠀⠀⠘
⠀⠀⠀⢀⡁⠀⢀⠀⠀⠒⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠐⠀⠈⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⢐⡀⠀⠀⠀⣀⠀⠀⡀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠋⠀⠘⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡅⠀⠀⠐⠀⠀⠊⠀⠀⠂⠀⠀⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣠⣄⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣻⡿⣿⣿⣿⣿⣿⠷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢄⡀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⢀⣈⣳⡀
⠀⠀⢀⣀⠀⠀⢰⠃⠀⡠⠔⡺⠍⠉⠈⢯⣳⠀⢀⠀⠀⠠⠾⠃⠀⢀⣤⡄
⠀⠀⠈⠁⠉⠀⡇⡠⢊⡀⠀⠁⡞⠷⠀⠈⣿⠰⠟⣁⡠⠴⠒⠒⢦⠈⠁⠀⠀⢀⣤
⠀⠀⠀⣠⣀⠀⢹⡕⢁⢤⡄⠀⠀⢠⡶⠀⣿⠔⠊⠁⠀⢀⣤⠀⠀⢣⡀⠀⠀⠛⠁
⠀⠀⠀⠈⠉⠁⠀⠻⡄⠀⠘⠂⣀⠀⠀⢀⠇⠀⠀⠀⡠⠊⢀⣱⢤⡤⡧⣀⣐⣦⡀⠐⠊⠉⠉⠉⠑⠂
⠀⠀⠀⢀⡤⣄⣀⣀⠼⣦⣤⣀⣙⣅⡴⠛⠓⠚⠣⠴⢽⠦⣜⣡⣀⡫⢧⣄⣁⣼⡁⠒⠓⠒⠒⠊⠑
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠁⠀⠀⣄⢄⡠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠉⠀⠀⠴⠤⠤⠤⠤⠤⠤⠦⠄
⠀⠀⠀⠀⠀⠑⠒⠉⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣀⡤⠤⠤⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠉⠒⠒⠓⠤⠤⠤⠄

⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡘⠁
⠀⠠⠖⠀⠀⠡⠀⠀⠈⠂⠄⠠⡀⠀⠀⠆⠀⠀⢀⠀⠄⠙⠀⠀⠋⠀⡁⢴⠀⠈
⠀⠀⠈⡀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠳⠈⢧⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢂⠀⠀⠀⠀⠄⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠉⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠚⠀⠀⢷⠀⠀⠈⠀⠀⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠢⢀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡠⠂⠀⠀⠀⠀⠀⠉⠁⠢⢄⠀⠀⠀⢀⠠⠐⠠
⠀⠀⠀⠀⠈⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⢀⡠⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡖⠁⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢞⠀⠀⠀⡐⠀⠐⢀⣀⣀⣀⣀⣀⠀⠀⣀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠄⢀⠀⠀⠀⠀⢀⠎⠀⢀⡀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠡⡀⢿⣟⠿⡻⣿⣿⣿⣿⡟⢻⣿⣛⢟⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡼⠀⠈⠀⠀⠀⠈⢀⡀⢀⠀⣠⠀⣀⠀⠀⠀⠀⠓⢄⠀⢡⠀⠹⣷⣵⣯⣿⣿⠏⠁⠀⠙⣿⣮⣵⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠂⠀⠀⠀⠂⠀⢠⣿⣷⣶⣦⣤⣤⣄⣀⣉⠙⠐⠋⠈⡤⠀⠒⠉⠶⠚⠀⠀⠀⢀⠆⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠊⣿⣧⡜⢋⡸⢿⣿⣿⣿⣿⣿⣿⢶⣶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠷⠀⠀⠀⠀⠠⠄⡀⠀⠀⣄⠀⡀
⠀⠀⠀⠀⠀⠀⢁⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣻⠿⣷⡊⣳⡞⢿⣿⣿⣿⣿⡟⠀⢰⣿⣿⣉⣿⠛⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⡄⠀⠁⠉⢣
⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠹⣄⡛⠛⠿⠿⠿⣿⣿⠛⢣⠌⡂⠉⣽⣿⣿⡴⢟⡨⢿⣿⣿⣿⣿⠿⠀⠀⠀⠀⡀
⠉⠉⠁⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡄⠙⠲⢠⢄⣀⠀⢀⡠⠀⠀⠀⠈⡄⠻⠿⠷⣾⣧⣼⣿⡿⢏⠝⠀⠂⡀⠀⠀⠃⠀⠀⠘⠀⣘⡆
⠀⠀⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡉⠉⠁⠀⠀⠀⠀⠻⠀⣠⠀⠀⡀⣈⣹⠃⠈⠁⡀⠀⠈⠀⡀⠐⡀⠀⠀⣻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢡⠑⢄⠀⠀⠈⢄⠀⠀⠀⠄⢁⠀⠀⠋⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡀⠤⠔⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⢁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠄⡀⠀⠈⠂⡀⠀⠂⠀⠀⠀⠀⠚⡉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠈⠓⠈⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠳⡄⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⣴⠀⢀⠀⠀⠈⠀⠀⠀⠀⢀
⠈⠀⢄⠀⠀⠠⢂⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡇⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠘⢀⣀⣀⣀⣀⣀⣀⣀⠋⡀⠀⠡⠀⠀⠀⠀⠀⠀⠁
⢤⡤⣤⣴⣦⣥⣬⣶⠶⢶⣶⡶⢶⡶⣤⣤⣤⣤⣧⣄⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡻⡹⣿⣿⣿⡿⠉⣿⣟⢫⢻⣿⣿⡿⠃⠀⡄⠁
⠛⠶⣯⣽⣿⣿⣿⡿⠃⠀⠙⠛⠿⣶⣏⣽⣻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠞⠾⠿⠋⠀⠀⠈⠻⠷⠷⠿⠟⠠⡀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡠⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⢤⠤⠐⠀⠈⠀⠀⠀⠑⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠐⠒⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⣀⣀⣀⣤⣤⣶⡾⠿⠿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠉⠉⠙⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠛⠁
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢀⣤⡀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠙⠻⢷⣤⣤⣶⠟⠋⠁⠀⢠⣄⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢹⣯⠀⠀⠀⠀⠀⢸⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⣸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⢿⣦⣄⣀⣤⣾⣿⣦⣄⣠⣴⡾⠟⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠈⢹⣿⠋⠀⠀⠈⢿⡟⠁⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⢀⣾⣇⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣤⣶⠿⠛⠙⠻⢷⣾⠟⠋⠙⠻⣶⣤⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣸⣗⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⣴⡾⠟⠻⢷⣦⣄⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⣠⣄⡀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣤⡆⠀⢹⡏⠀⢰⣤⠄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡈⠉⠙⢦⣸⡇⣠⠞⠉⢉⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⣿⠀⢒⡿⠒⠒⠒⣿⣿⡓⠒⠒⢿⡒⠀⣿⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢹⡄⠀⢀⣀⣤⠞⢹⡇⠙⢦⣀⣀⠀⢠⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠳⣄⠀⠛⠇⢀⣼⣧⡀⠸⠛⠂⣠⠞⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠠⣔⢂⡄⠀⠀⠀⠀⠈⠳⢤⣀⡀⠈⠁⢀⣀⡤⠞⠁⠀⠀⠀⠀⢠⡐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠛⠛⣛⣛⣛⣛⣛⣛⠛⠛⠛⣛⣛⣛⣛⣛⣛
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢩⣭⣭⣭⢭⣭⣭⣭⣤⣤⣤⠌⠉⢩⣭⣭⣭⣤⣤⣤⣭⣭⣭⣭⠉⠁
⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⣠⠞⠳⣄⠀⣠⠞⠳⣄⠀⣠⠞⠳⣄⠀⢀⡾⠛⢦⠀⢀⡴⠛⢷⡀⠀⢠⠞⠳⡄
⠀⠀⠀⠀⠀⠐⠋⠀⠀⠈⠛⠁⠀⠀⠈⠛⠁⠀⠀⠈⠓⠋⠀⠀⠈⠓⠚⠁⠀⠀⠙⠒⠋⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠉⠈⠀⠁⠁⠈⠈⠈⠁⠁⠈⠈⠀⠁⠁⠈⠈⠀⠁⠁⠈⠈⠀⠁⠁⠈⠈⠀⠁⠁
⠀⠀⠀⠀⠀⣀⣈⣈⣈⣁⣁⣉⣈⣈⣁⣁⣈⣈⣀⣁⣁⣈⣈⣀⣁⣁⣈⣈⣀⣁⣁⣈⣈⣀⣁⣁⣀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣿⡫⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⣀⠤⠒⠐⠒⠒⠢⠤⣄⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⢀⣴⡾⠋⣨⣿⣿⣿⣦
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠘⡆⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢶⣿⠋⣠⣾⡿⠋⣿⣿⡿⣷
⠀⠀⡀⢀⠀⢀⢄⢠⡄⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠇⠀⠀⠀⣿⢿⣿⣿⡿⠻⣿⣿⠻⣿⢿⣿⡏⢻⣿⣿⣿⡇⢀⣿⣿⣇⠾
⠀⠀⠀⢘⢳⢸⠀⠈⣅⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⢘⣿⠉⡇⠀⠉⢽⠀⣿⡖⢿⣥⡾⢻⣿⣿⣵⣿⣿⣻⡏
⠀⡄⡀⠀⢸⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⠀⠀⣿⡇⢸⣿⠁⣾⠃⣿⠿⣿⢻⣏
⠀⡇⡇⠀⠘⠤⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⢿⠀⠀⠀⠿⠀⠀⠀⠀⢿⡇⢸⣿⠀⣿⣠⣿⣠⣿⣿⣿⣷⡄
⠀⠷⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣼⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⣦⢀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⡿⣿⣿⣿⡁⠐⣿⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣫⡄⣿⣿⣿⡇⠀⣿⠂⠉⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠐⠠⡄⠀⠀⠀⢧⠀⠱⠀⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣿⡟⠛⢿⣿⠀⣿⡄⠀⢘⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣄⡟⠂⠀⢸⠀⠀⠀⠈⢦⠀⠄⠘⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢿⠐⠃⠀⠈⣿⣇⢻⣿⣄⠰⣿⡇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠻⠇⠀⠀⠀⠳⡀⠀⢀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠘⠀⠀⠀⠀⠘⣿⣎⢻⣿⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠙⠢⡀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⢴⣒⣓⣂⣀⢀⣀⣐⡬⠂⠀⠀⠀⠪⠤⢤⣤⣤⣤⡄⠒⠃⠀⠀⠀⠀⠀⠠⡤⠼⢿⣿⣿⣿⣿⠓⠀⠀⠀⠀⠰⠟⠿⠿⠿⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠾⠛⠛⠛⣛⣛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣡⣴⠾⠛⢛⣛⣻⠿⠀⠀⠈⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢡⡾⢋⣤⡶⠛⠛⠉⠛⠛⠷⣤⡀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⡟⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⣡⣤⣤⡴⠶⠶⠶⢦⣤⣄⡀⣿⣶⠀⠹⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠙⠛⠁⠀⠀⠀⠀⠀⠀⠉⠛⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢻⡄⢻⣆⢰⣶⡶⠶⠶⠶⢦⣤⡀⣿⣷⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡌⢻⣄⠹⢮⣍⡁⠀⠀⠀⣰⣾⡿⣦⡙⢷⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⠷⣤⣉⠛⠛⠿⠛⠛⣻⣥⡘⣷⠈⣷⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣿⣦⣌⣉⠛⠛⠛⣻⡛⠛⠛⠁⣿⠄⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡏⢿⡋⠈⠙⠛⠛⠛⠿⠟⠀⠀⢀⡿⢀⣿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣜⣧⡈⢷⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⢁⣾⠃⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡈⠻⣦⡙⠻⠶⣤⣤⣤⠶⠟⢋⣴⠟⢁⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣌⠙⠳⠶⣦⣤⣴⠶⠞⢋⣡⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⣶⠶⠶⠾⠛⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿