Protein Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠑⡀⠀⠄⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠂⠀⠐⢀⠌⡀⠤⠄⠀⠴⠊⠀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⣀⡠⠂⢐⠁⢄⠀⠀⢀⡖⠁⠀⠀⠈⠑⠤
⠀⠀⠀⠀⠀⠀⠰⢱⣶⣿⠆⡀⠀⠈⣷⣦⣤⠈⠠⣀⣈⠉⠀⠀⠀⠀⠘⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⢀⣥⣤⡄⢈⡹⢻⠀⠀⣠⣿⡆⠀⠀⡄⠔⠁⢸⣿⣓⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⣿⣦⣁⠀⠘⣿⣏⣀⠀⠀⢿⣷⣶⡌⠿⠿⣿⡛⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠿⠿⠀⠀⠈⢉⣿⡧⠀⠀⠀⠁⠀⠀⣼⣿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣖⠏⠀⠀⠀⢻⣿⣥⣤⠰⣿⣶⡄⠀⠘⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠉⠉⠃⠀⠀⠀⠉⢩⢻⡷⠀⠉⠁⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠶⢷⡦⡀⠀⠸⢤⡄⠹⣿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⠀⠲⢤⠤⠄⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠈⠁⠀⠀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣩⠓⠃⠀⠀⢐⢀⢀⡀⠀⠀⠺⣿⣄⠙⠻⣿⣿⠇⠀⠀⠀⠸⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡆⠀⠀⠀⠿⠿⠿⡆⠀⠀⢙⠋⠀⠀⠘⢩⡿⢿⣷⡄⠀⢀⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⠵⠃⠀⠀⠀⠀⡐⠉⠐⢿⣧⡀⠀⢠⣴⣄⡀⠀⢾⣿⣁⠀⠘⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢊⠩⠀⠀⠀⠉⠛⠃⠀⠀⠛⠛⠀⠀⠈⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡽⠀⠀⠀⢀⠀⠸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠠⡈⡉⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠠⡀⠀⠅⠀⢀⠀⢀⡀⡀⠄⠂⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⢘⠛⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣭⣭⡑⠂⠉⠣⢀⡀
⠀⠀⠀⠀⠀⠀⡐⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠲⣷⣿⣿⡛⠷⣶⢄⣀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠢⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠝⢛⠶⠿⠶⢷⣹⣿⡷⣆⡈⠠⠄⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠈⠘⠛⠷⡏⣰⠞⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠻⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠛⠉⠛⢿⡀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠀⠴⠾⣳⠄⠀⠈⠑⠐⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣎⣀⣠⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠁⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠈⢙⠌⠁⠴⢀⠈⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠄⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⣠⡿⠁⢸⠁⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠒⠃⠀⣚⠒⣺⢻⣶⣶⡿⢶⣶⣾⠇
⠀⠀⠀⠀⠀⠀⢀⠤⠀⢀⠒⠚⠉⠐⠒⠒⠦⢄⠤⠕⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠁⠀⠉⠉⠀⠀⠉⠁
⠀⠀⣀⡤⠶⠏⠀⢓⠍⡭⠤⠀⠉⡤⠬⠉⠒⣄⠌⣐⡠⠀⣀⠀⠀⢀⠄⠀⢒⢢⡶⠃⠀⠀⢀⠀⢀⢰⣿⣶⣔⣶
⣠⡾⡿⠝⠧⠀⠀⠘⡆⢀⢈⣓⣎⡀⠀⠀⠀⠀⠀⠀⠘⠙⠒⠒⠂⠀⠀⠈⠥⣏⡀⠀⠀⢰⣽⠿⣷⣾⣫⣿⣿⣏⠀⡰⣶
⠑⢌⠈⠃⠀⠀⠀⢀⡘⠋⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⠈⠁⠀⠨⠽⠶⠀⠈⠋⢼⣵⣿⡿⣿⡾⢿⡘⠾⢃
⠀⠀⠀⠒⠤⣐⠒⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⣭⣽⠿⣛⡿⠻⠏⢰⣵
⠀⠀⠀⠀⠀⠀⠀⠁⠒⠷⣶⣄⡐⠂⠐⠒⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠏⠀⠺⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣞⡋⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣷⠀⠀⠀⣿⢿⡇
⠀⠀⠀⠀⠀⠀⣤⣔⡶⡷⣷⣿⡽⣽⣿⣿⣳⣦⠭⠭⠥⠀⠀⠀⣧⡌⠃
⠀⠀⠀⠀⠀⠀⠈⠉⢋⣛⣾⣿⣿⠟⠚⠚⠙⣺⣻⣿⡇⠀⠀⠀⣿⣿⠀⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠐⠻⠟⠿⢿⠿⣦⣀⡠⣾⣯⣻⡛⠃⠀⠠⡆⠁⠈⣆⠀⠀⠀⢁⠀⣀
⠀⠀⠀⠀⢄⠀⠈⠀⠲⡟⠀⠀⠀⠀⠀⢠⢭⡿⠯⠑⠉⠉⠀⠛⠆⠐⠛⠒⠀⠒⠲⢶⡾⠟⠁
⠀⠀⢀⠀⠻⠾⢧⣄⣀⣀⠀⠀⠀⠠⠀⢠⠀⠑⢄⣠⣤⣅⠀⠀⠀⠀⠀⠀⠒⢤⠀⠀⠀⠀⢀⣶⣤⡄
⠀⠀⠈⠙⠓⠶⢶⣶⣶⣶⣦⡀⠀⠰⠚⠞⠒⠚⠋⢩⠽⢟⣤⡤⠀⢠⣤⣤⣤⣾⡇⠀⣶⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢧⣤⣤⣿⣿⣿⣟⣉⣀⠀⠈⠛⠙⢂⡘⠓⣌⣘⣼⣃⣤⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠓⠚⠿⠶⣌⡀⣡⣾⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⡄⢤⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤
⠀⠀⠀⠀⠁⠀⠀⠀⠁⠠⢴⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡄⢠⠀⡔⠀⠀⢀⣾⢹⣆⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠐⠚⠉⠩⣝⡲⠄⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠰⠧⣀⣀⠀⠈⠀⣸⣷⣶⠼⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⢹⡦⠤⠀⠤⠾⢿⡆⠀⠀⠀⠉⠀⠀⠂⢨⡖⠏⢹
⠀⠀⠰⠲⢤⣀⠈⠉⠓⠠⠁⠉⣤⠃⠀⠀⠀⠀⣆⠀⢀⢠⢀⣀⡀⠈⢻⠀⠀⠀⠀⢸⣿⡦⠀⠀⠒⣾⡇⠀⠀⠀⠀⣭⣉⣛⢸⠀⠀⣸
⠀⠀⡃⢠⠀⠀⠰⠖⠀⢸⠒⠁⠒⢀⠀⠀⠀⠀⣩⠀⠀⠀⠆⠀⠴⠀⢼⠀⠀⠀⠀⠘⠧⠁⠀⠈⠰⣿⠂⠀⠀⠀⡆⠀⠠⡝⢾⣿⠻⣿
⠀⠀⣀⣈⡀⠀⠀⠀⠀⠚⠤⣶⠀⠘⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠐⣾⠀⠀⠀⠀⠰⣷⣶⢿⣷⣾⣿⡇⠀⠀⠀⡷⠄⠶⠉⣷⣿⣧⣼
⠀⠀⣷⡜⠀⠤⠀⣴⢀⢠⣞⡿⠥⢰⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⢸⣿⡛⠋⠙⢻⣿⠀⠀⠀⠀⡃⢀⡀⣠⣿⣏⢉⡿
⠀⠀⠃⠀⠠⢴⠄⠠⠈⠰⠾⣵⡞⠋⠀⠀⠀⠀⠿⣷⣤⣀⣀⣤⣴⣿⡿⠛⠀⠀⠀⠙⠷⣄⣀⠀⠠⠟⠃⠀⠀⠀⠈⠁⠋⠝⠿⠛⠁
⠀⠀⠊⠁⠦⠀⠀⢂⣀⣨⠀⠛⠀⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢘⡟⠋⢙⣏
⠀⠀⠀⠀⢰⠀⠀⠀⠂⠀⠤⣄⣀⡀⠀⠀⡀⠒⠒⠀⠄⢀⠀⠀⠀⠀⢣⣄⣀⣀⢀⡀⣀⠀⠀⣀⣀⣠⢴⡇⠀⠀⠀⢠⣔⠚⠉⠉⠉⠋⠂⣀
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢊⠀⠀⠀⠀⡹⣿⣿⣬⣀⣉⣉⣉⣀⣀⣰⣿⡷⠇⠀⠀⠀⢈⣉⠁⠀⠀⠂⢈⣿⣱
⠀⠀⠀⢸⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⢮⣟⠃⠀⠀⠀⠀⠀⠀⠁⠠⣼⣿⡼⠀⠀⠀⠘⠻⠶⠦⠤⠤⢿⣿⡟
⠀⠀⢀⡏⠺⡇⠀⠀⠰⣏⠀⡶⡆⢠⢰⡵⢐⠆⠀⢰⠀⠀⠂⠀⠀⠀⢸⣿⢯⡁⡀⠐⠒⢒⡀⠀⠀⣹⣿⠀⠀⠀⠀⢰⣀⠀⠀⠀⠀⣰⣿⣧
⠀⠀⡼⠀⠀⡇⠀⠀⠐⠤⢀⡀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢸⢰⣿⠀⣌⣁⣈⣈⡉⣀⠀⢺⣿⣿⡆⠀⠀⠀⠐⠛⠿⠖⡖⠺⠿⣿⡗
⠀⠀⡿⠀⠀⡗⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣙⣿⢤⠀⠁⠀⠀⠉⠀⠀⣙⠿⣿⡉⠀⠀⠀⢸⣀⢇⠀⠇⠀⢰⣿⣷
⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠔⠷⠀⠀⡎⠀⠀⠀⠀⠈⠙⠢⠤⣀⣀⢀⢀⣀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠙⠛⠓⠒⠒⠻⣿⡷
⠀⠀⡗⠀⠀⡇⠀⠀⠀⣴⠶⠀⠀⠀⠠⠅⠀⡀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⢠⣿⣧
⠀⠀⢱⠀⢰⠃⠠⡀⠠⠂⠀⠀⠠⠔⠀⢤⣔⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠈⠉⠉⠉⠉⠻⣽⣷
⠀⠀⠸⡀⡽⠀⠀⠈⠉⠀⠉⢶⣅⣆⡀⠉⠀⡗⠘⠁⠀⠀⡄⠀⠀⣇⡄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢻⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⣉⣀⠠⠊⢀⡀⠁⠀⢰⠄⠀⠠⢹⡇⠀⠈⠈⠈⠁⢙⠁⣌⣦⠄⠀⠀⣨⢁⡆⠀⢸⠀⠀⠀⠀⠀⠐⣿⡷⠂
⠀⠀⠀⠿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁⠀⢠⣴⢺⠀⠀⢀⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⡇⠀⢸⢀⠀⠀⠀⢠⣽⣿⡟
⠀⠀⠀⠩⠥⠀⠀⠀⠀⠀⣀⣀⣀⣀⡂⣀⣈⣀⣀⣀⣂⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠤⣄⣀⣀⣠⣾⠿⠇

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

⠀⠀⣀⠀⣠⣤⣄⡀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⡤
⠀⠀⠛⣿⣿⣿⣿⣷⠀⠀⠀⢺⣼⣿⣿⣧⠀⠀⠀⠀⠀⢠⣾⣿⣶⠋⠀⠀⠀⠀⠀⣤⡦⢤⣤⣤⠄⠀⠀⠀⢾⣛⢿⡿⡿⠊
⠀⠀⠀⠈⠉⠈⠉⠁⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋

⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⣀⣴⣿⡿⢦⠀⠀⠀⠀⠀⣄⢠⣄⢠⡄⠀⠀⠀⠀⠀⠄⣀⣢⣄⠀⠀⠀⠀⢀⣤⣶⣄⣀⡄
⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⢸⡏⢀⡀⢰⣿⠀⠀⠀⠀⠺⡿⢿⡷⢿⠗⠀⠀⠀⠀⢷⣦⣿⠛⠛⠃⠀⠀⠀⠘⠿⣿⠟⠛⠃
⠀⠀⠲⠶⠛⠀⠀⠀⠀⠀⠀⠘⠓⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠻⣷⡿⠃⠀⠀⠀⠀⠐⣿⠀⠸⢿⣶⠀⠀⠀⠀⢰⣶⠛⠛⣻⠃⠀⠀⠀⠀⣰⣾⣷⣝⢦⠀⠀⠀⠀⠀⠀⣿⣿⠆
⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⠀⠻⣶⣤⣾⠟⠀⠀⠀⠀⠈⠻⠿⣠⣿⠇⠀⠀⠀⠀⠻⠿⣿⡿⠟⠁⠀⠀⠀⠀⢠⠮⡟⡆
⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁

⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⢰⡏⠁⠉⣷⠀⠀⠀⠀⠀⣼⠙⣿⠙⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⢀⢿⣿⡇
⠀⠀⠸⣇⣸⣀⡿⠀⠀⠀⠀⠀⠹⣄⣁⣰⠇⠀⠀⠀⠀⢠⣶⣶⣿⠟⠀⠀⠀⠀⠀⠀⣿⣿⣾⠃⠀⠀⠀⠀⠀⠸⣸⣟⡿
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡐⠠⠀⠠⠐⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠴⠤⠄⣄⡀⠀⠀⠀⠸⣷⣤⢄⣒⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠡⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠈⠙⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡐⢦⣤⣤⢤⣄⠀⠀⠀⠀⣀⣠⣶⣦⡀
⠀⠀⠀⠀⢾⣽⠆⠀⠀⠀⠀⠀⠀⠘⠷⡆⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠑⠈⠉⠁⢈⠛⠇⠀⠀⡸⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠒⠁⠀⠀⠒⠀⠀⠀⠐⠙⠛⠁
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⡁⠀⠡⠈⠀⢀⠀⠀⠀⠀⠨⠉⠉⡀⠀⠀⠀⠀⠀⡊⠁⠐⠄⠀⠀⠀⠀⢀⣰⣶⣖⠒⠲⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⣾⣶⣶⢶⡆⠀⠀⠀⠀⠀⠘⠀⠂
⠀⠀⠀⠀⠀⠀⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡀⣤⣤⢤⡀⠀⠀⠀⡂⠘⠄⠀⠀⠀⠀⠀⠀⣀⣤⡤⣄⠀⠀⠀⠀⠀⠠⠠⢤⡀⠀⠀⠀⠀⢀⡐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠌⠡⠰
⠀⠀⠀⠀⠠⠀⠠⠀⡀⠂⠀⠀⠀⠈⠀⢈⠀⠀⠀⠀⠀⠨⠀⠠⠀⠀⠀⠂⠀⠀⠀⠅⠀⠀⠹⠷⠀⠀⠀⢸⣿⣟⣛⣿⣿⠀⠀⠀⢶⢴⢰⣶⣶⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⣐⠰⠶⠶⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠂⣤⣲⡄⠀⠀⠀⢠⢢⢀⡼⠁⠀⠀⢠⣴⣶⣿⣷⣶⣧⣄⣀⠔
⠀⠀⠀⠀⠑⠛⠻⠟⠛⠊⠀⠀⠀⠘⢤⣀⣠⠜⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠘⠚⠁⠀⠀⢀⣤⣶⡶⠀⠂⠀⠀⠉⠁⢈⡉⠀⢀⡄⠀⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢿⣦⡀
⣤⣄⡀⠀⠀⠀⣠⠎⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁
⠷⣼⡳⠀⢀⣼⡟⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣉⠀⠀⠀⠀⠀⣠
⢢⣻⡼⢿⠈⠉⠁⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⢀⡀
⣶⣿⣦⡄⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣇⠀⠀⠀⠀⢸⡇
⣿⣿⣁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢀⠀⠀⠈
⠈⠿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠃⢞⠃⠠⣤⣄
⡀⠀⢰⠁⠸⢹⡷⠤⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⣷⠀⢠⣤⢠⢻⡇⢠⣿⣿
⡗⠀⣿⣤⡆⠘⢁⣿⣄⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡬⣄⣁⡀⢿⡀⠈⣉⡉⠙⠧⡶⠋
⣇⠀⢿⠋⢻⡶⣿⣿⡏⠀⣴⡍⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⣫⣤⣿⢿⡆⠀⢻⡄⣿⣿⣿⠟
⢿⠗⠸⣿⢉⣴⢺⣿⣷⡀⢻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⣹⡿⠿⣿⢇⣭⡀⠜⢃⣸⣿⠋⠀⣀⣤⡀
⢸⡇⠀⢹⡀⣿⣤⣟⣿⣧⡀⠻⣄⣀⠁⠄⢀⠀⢀⣄⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢟⣿⡟⠁⠀⠀⠰⣌⠿⣇⠀⢈⡽⠁⠀⠈⢫⣿⣧
⠀⠹⣏⠀⠛⢼⣿⣿⣏⣿⣷⡤⠙⣿⡀⡠⢈⣿⡘⠣⠀⢨⡏⣼⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣾⣿⣟⣵⣿⣿⠀⠀⠀⠀⠀⠙⣻⣯⢷⣿⠃⠀⠀⠀⣸⣿⡿
⣦⡁⢿⣕⣆⡀⠙⠿⠋⠁⢰⣆⠒⠂⡠⠀⢼⣿⣿⣿⡷⠋⣠⣿⢣⣯⣜⡡⠍⠒⠂⠀⢠⣿⢟⣽⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢾⣿⣧⣿⠃⠀⠀⢀⣾⣿⡇
⡉⣭⣈⢉⣉⣁⠀⠀⠤⣀⡀⠉⡦⠈⠠⣾⠀⠤⠋⡩⠴⢿⣋⣽⣿⡿⠤⠔⣖⡱⠶⣢⡎⠱⠋⠈⢈⣉⣋⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⣇⡀⠈⢲⣾⣿⡿⠁
⠀⢈⣽⠁⣍⠋⠐⠢⠤⠈⠛⠀⢠⠂⠀⣤⠀⢰⡶⣠⡰⣿⣿⣿⣿⡻⣷⣾⢿⣲⣶⣿⠠⣤⢀⣀⣀⣿⣿⣿⣦⠂⣿⣦⣀⣰⣿⣿⣿⠏⠀⠀⠀⠈⠁⠀⢀
⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣷⣶⣾⣶⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⣶⣶⣶⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠿⠿⠛⠛⠛⠿⠿⢶⣶⣤⣀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣶⡶⠶⠶⠶⠶⢶⣦
⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⢀⣼⠟⠁⠀⠀⠀⠀⢀⣤⡶⠟⠛⠉⠉⠉⠈⠉⠉⠛⢷⣄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⣾⠏⠀⠀⠀⠀⢀⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠏⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠙⠻⣆⡀
⣸⡏⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⣿⠃⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⢹⡇⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠐⠂⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠈⢿⣄⠀⠀⠀⠸⣆⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀
⠀⠀⠙⢷⣦⣄⠀⠈⢷⡶⠶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠒⠒⠒⠲⠶⠶⠖⠒⠴⠶⠀⠀⠙⠻⠿⢶
⠀⠀⠀⠀⠈⠙⠻⢷⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⢉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⢳⡀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡞⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡴⠦⠀⣤⣠⡶⠚⢶⠶⠶⢦⣤⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⣄⠀⠀⠀⠀⡸⠃⠘⠷⣤⡀⠀⣠⣄⠀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⣾⠛⠳⠶⢦⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢸⡇⠀⠀⠀⠈⠻⠾⠋⠙⠛⠛⠷⠶⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠉⣿⠀⣸⠛⠻⣦⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠘⢷⠤⠶⠤⠴⠒⠚⠛⠀⢿⡆⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⡀⠙⠶⠾⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠉⠉⠉⢙⣷⠆⠀⠀⠀⢰⡏⠙⠲⢦⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠇⢀⣴⣟⣁⡀⠀⠀⠀⢸⣧⣤⣀⣤⣿⠇⠰⣟⠉⠳⣆⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⣟⣁⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢈⣩⡝⢛⠛⣛⣛⡋⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠉⢿⠿⣿⡟⠉⠻⡆⠀⠀⠀⠀⢠⡞⠉⠉⠙⠛⠲⣤⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠛⠳⠶⣦⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⠛⣿⣿⣟⠁⠀⠀⠀⢀⡿⠑⠾⠟⠀⠻⠿⢿⡅⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠶⠟⠀⢹⡇⠀⠀⠀⠸⠷⠴⠦⠤⢀⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣶⢶⡄⢸⡇⠀⠀⠀⠀⢀⣀⣤⠶⠛⠋⠉⣸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠾⠃⢼⣁⣸⡇⠀⠀⢸⡟⠛⠉⠀⠀⣠⣴⣟⡋⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡟⠙⢻⡇⠈⠉⠉⠁⠀⠀⠸⣧⣀⣀⠀⠒⠈⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⠷⠦⢤⣤⣄⠀⠀⠀⠀⠉⠉⠉⠐⠲⠶⠤⣤⣧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢈⡇⢀⣠⣀⣀⡀⢀⣿⠀⠀⠀⢠⡶⠛⣷⢀⣤⣄⣴⡶⠤⣄⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣇⢸⣇⠀⠀⠉⠉⠉⠀⠀⠀⣿⠀⠰⣿⣿⠁⣾⣿⣧⠀⢹⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠉⣩⡿⠛⠛⢦⡀⠀⠀⠀⠀⠹⠶⠶⣤⣁⣀⡀⠉⠁⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢰⠄⢸⡇⠀⠀⠀⠀⠀⢀⣄⡈⠉⠉⠉⠛⠛⠒⠿⠃⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣀⣠⣾⠃⠀⠀⠀⠀⣠⡾⠉⠙⠒⠠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠶⠿⢯⣿⠻⢦⡀⠀⠀⠀⠛⠷⢶⣤⠀⠀⠀⠉⠙⢶⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⠁⣠⡾⠋⠀⠀⠀⠀⣴⠟⠁⠀⣾⠷⣦⡀⢰⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣇⠘⠀⠀⠘⠛⣷⠄⠀⠀⠀⠀⠙⠷⣤⣼⠏⠀⠈⠛⠟⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣻⠷⣶⣦⠴⠴⠿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡿⠁⠀⠀⠉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⢿⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠻⡦⠠⢤⣤⣤⣤⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠿⠿⢶⣶⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⡠⠄⠒⠀⠀⢀⠂⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠈⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢡⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠠⠐⠀⣠⠤⠤⢄⠠⠀⠀⠀⠈⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠀⠀⠀⠀⠀⠀⠀⡀⠂⠈⠀⡰⠂⠀⠀⠀⠀⡈⠀⠀⠠⢉⠀⠀⠀⠀⡁⠈⡀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠂⠠⠁⠀⠀⠀⠀⠀⠀⠀⠘⠠⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠈⠀⡠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⡰⠠⠤⠤⠤⠀⡀⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⠀⢀⠄⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⢀⠀⠠⠄⠐⠒⠈⠉⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠐⠂⠈⠉⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠔⠁⠀⠀⢀⠀⠠⠂⠀⠀⠀⠊⠉⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡀⠀⡂⠀⠀⠀⠀⠄⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠠⡱⠂⠠⠀⠠⠄⠒⠀⠀⠀⠀⠒⠀⠀⡀⢠⠠⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⠂⠀⠀⠀⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠁⠀⠄⠀⠂⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠠
⠀⠀⠀⠀⠀⠠⣀⠠⠴⡋⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠢⡀⠀⠀⡀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠐⢄⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠈⢂⠀⠐⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠄⠀⣀⣀⠀⠘⠀⠀⠉⠉⠉⠈⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⢀⠀⠀⠀⢀⠠⠤⠀⢀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⢀⢐⠀⣀⠀⠀⠀⠀⠀⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠶⠶⠶⠶⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣉⣉⣉⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿⠿⢿⣿⣿⣿⢹⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣞⠛⠛⠛⣛⡛⠛⠛⠛⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄⣴⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⠀⠀⠀⠻⠋⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠻⣧⣼⠛⠻⣿⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⠀⣤⣤⠞⠣⣤⣄⠀⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡟⠿⠦⠶⠿⠿⠦⠶⠿⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⠀⠛⣀⡙⢋⣀⣀⣙⣋⣀⡷⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠚⠉⠉⠉⠉⠉⠉⠀⠉⠍⠢⣄⣀
⠀⢀⣀⡤⠤⠴⢞⣡⠔⠂⠀⣄⠀⡀⠈⠡⣠⠆⠀⡀⢀⡠⣀⠈⡇
⠀⣏⠀⠀⣠⡔⠋⠑⣤⠀⠀⢈⠿⣅⠠⠀⠀⠀⠠⠈⠁⣀⣠⢴⠇
⠀⠘⡶⠦⢤⣌⣳⣉⡀⢈⣂⣀⣀⡠⠴⠒⠛⣋⣉⣉⣉⡡⠴⠋⢀⣀⣤⣔⣪⣿⣯⣵⣖⡒⣒⢤⡀
⠀⠀⠈⠷⣤⣀⣀⣈⢉⣉⠉⣉⣁⡤⠴⠊⠉⠁⠀⠀⠀⠀⣀⡴⢿⣿⣿⣿⣿⣿⣿⡿⣛⣁⠛⣿⣎⢢
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⠤⣒⣯⣵⣶⢶⢟⣳⣦⣼⣷⣿⣾⣿⣿⡕⣿⣿⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠟⣷⠫⣝⣾⠿⢯⣭⡭⠿⠭⢿⣿⣿⣿⣟⣸⡿⢥⡾
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⣞⢠⣿⡇⠀⣿⣿⡇⠀⠀⠀⢹⣿⡦⢀⣀⡠⠟⠁
⠀⢀⡴⣫⣶⣦⢰⣍⠒⠤⢤⣄⣀⣀⣀⣀⣈⣉⠁⠈⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠛⠋⠉⠉
⠀⣾⣴⣿⣿⣿⠸⣿⣿⣿⠿⢃⣶⣶⣶⣶⣤⣍⣙⠢⣄
⠀⡇⣿⣿⣿⣿⣆⠠⣭⣀⠒⠛⠛⠛⠛⢛⣉⣭⣭⣶⢘⡆
⠀⣇⠹⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣉⣥⣾⡇
⠀⢹⣦⣝⡛⠟⠃⣸⣿⣿⣿⣿⣿⠟⣫⠴⠋⠁⣠⠴⠛⠀⣀⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢿⡄⠉⠛⠒⠬⢭⣭⣭⣭⣶⣿⣿⣤⠔⠋⠀⢀⡔⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠊⠁⠀⠈⠓⠤⡀
⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⣿⣿⠿⠛⠁⠀⠀⢀⠊⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾
⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠓⠤⣀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠔⠒⠉⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠈⠉⠉⠉⠉⠉⠀⣀⣠⡤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠐⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠲⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠈⢹⠉⠉⣻⠛⠛⠛⡟⠛⠛⢿⠛⠛⢻⠛⠛⢻⠛⠛⢹⡏⠉⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⣼⣄⣀⣽⣀⣀⣠⣇⣀⣀⣼⣀⣀⣸⣀⣀⣸⣄⣀⣼⣄⣤⣧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⠤⢤⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣁⡠⠿⣆
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⠷⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⣶⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⠀⠀⠀⠀⣴⠟⠛⠃⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠀⣼⠋⠉⠉⠉⡏⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠟⠀⠀⢿⣤⣤⣤⣤⣿⣤⣤⣤⠾⠁
⠀⠀⠀⠀⢀⣤⠶⠶⢶⣤⣤⣭⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡴⠶⢶⣤⡀
⠀⠀⠀⠀⣟⠀⠀⠀⡼⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠛⠁⠈⢹⣧⠀⢀⡼⠛⢶⣄
⠀⠀⠀⠀⠙⢦⣄⣀⣳⡀⠀⠀⢀⣰⠟⢀⣀⠀⠀⣿⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠙⢦⣠⡾⠋⠳⣮⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢛⣻⣷⠞⠋⠉⢻⡄⠹⣦⡀⢠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢿⡀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠻⢦⣄⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠈⢧⠀⠀⣠⠟⠀⠈⠻⣧⡀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⣼⠷⠛⠁⠀⠀⠀⠀⠈⠻⢦⣀⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⢦⡀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠄⡀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⢠⣿⣬⣿⣦⣧⠀⠀⠀⠀⠋⠀⠈⠐⠓⠒⠒⠉⠉⠁⠀⠀⢱⣄⡀
⠀⢲⡻⣷⣿⠀⠀⠀⢀⣀⣠⣿⠀⠀⠀⢸⣾⣇⣿⣼⣅⣿⡾⠀⠀⠴⢄⣀⣀⡀⠀⢀⣀⣀⣠⣴⡆⠀⣾⠇⠁
⠀⢻⣭⣻⣢⡦⢰⡲⣯⡳⣽⡏⠀⠀⣰⢾⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠈⠙⢿⣇⢸⡿⠿⠿⣿⣦⠸⡟
⠀⠀⠙⠻⣿⣷⣶⣿⣮⣽⠟⠀⠀⠀⠈⢢⡻⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢺⠹⣸⠀⠀⠀⢌⡏⢠⠇
⠀⠀⠀⠀⠈⠻⠿⠿⡿⠃⠀⠀⠀⠀⠀⠀⣼⣩⡭⡯⠍⣹⡄⠀⠀⠀⠀⠀⠈⠈⠢⠋⠀⠀⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠁⠐⡘⠀⢤⠀⠀⠀⠀⢰⠋⠛⠤⡷⠞⠋⢳⠀⠀⠀⢀⢾⢀⠀⠀⠀⠀⢀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠀⠀⠀⣧⠀⠀⠸⡧⠀⢠⡿⡟⣿⡄⠀⢠⡴⣻⣭⣵⡽⢿⣷⡆
⠀⠀⠀⠀⣀⡄⡰⠁⢀⢂⣤⣀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠘⠷⣷⠷⠃⠀⣵⡞⣿⣿⡟⣿⣯⠻⣿
⠀⠐⣿⣶⣌⣿⡇⠀⠀⣹⣏⣠⣶⡷⠀⠀⢀⠀⠀⠀⢆⡀⠀⠀⢀⢀⢀⠀⠀⠀⠀⢠⣁⣾⣿⣧⣿⣻⢄⡿
⠀⠀⠈⠉⠋⢻⣿⠀⢠⣿⠏⠙⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠋⠾⣼⣿⣷⣠⡀⠀⢸⡿⣽⣿⣿⣿⠏⣼⠁
⠀⠀⠀⠀⠀⠘⣿⠃⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣛⡀⠀⠀⠤⢤⣴⡏⠀⠀⢠⡷⣱⡿⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠱⣥⣥⠇⠀⠀⠀⠀⠀⠀⠀⠀⠬⢭⣭⠄⠀⠀⠀⠀⢸⡇⠀⠀⠸⣟⣫⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⣶⣶⣶⡶⠶⣶⣶⣶⣶⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡿⠿⢰⣇⠸⠿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣇⣛⢻⡿⢛⣃⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣘⣃⣸⣿⣿⣿⡇⠀⠀⣿⡴⠖⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣡⣴⣾⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⡴⠶⠶⠶⠶⣿⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠳⠶⠶⠦⠴⢿⣿⣿⣿⡿⠟



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣄⠀⣶⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⠁⣐⡂⠀⠀⠐⢒⢖⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢧⣈⢣⠒⡟⠘⣇⡖⡄⠀⠀⠀⠀⠀⠀⡜⠠⣀⠔⢂⠀⢈⣦⠴⠓⡄⠀⠳⣄⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠌⢱⣠⡄⡂⢀⡸⢹⣠⡆⠀⠀⠀⠀⢰⡧⠀⠙⣦⡴⠚⡍⠀⢱⣤⠞⠣⡀⢈⠦⡄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠾⠞⣹⡻⡁⣄⠙⢉⠘⡿⡄⠀⠀⠀⠸⢈⠀⠈⠉⢶⣀⠝⢿⡁⢈⣧⡔⠉⠣⡀⡨⠶⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣧⣿⣿⣿⣯⠾⡍⢐⡀⢸⣦⠀⠀⠈⠃⣱⣀⠀⠋⠱⣄⡠⠓⣏⠀⢙⣦⠐⠉⠂⣠⢎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠿⠛⠋⠃⡀⠀⢻⠚⠓⠊⠘⠂⠀⠈⢿⢁⡃⢄⣀⣉⠃⠀⠬⠓⠉⢀⣠⠴⠋⠀⠈
⠀⠀⠀⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⠁⠀⠀⠀⣶⢉⡓⣚⣹⠃⠀⠀⢐⠀⠀⠀⠁⠀⠀⢈⡉⢋⠉⠉⠉⠉⡉
⠀⠀⠀⠸⠉⠉⠉⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣴⣦⡃⠀⠀⠘⠀⢱⣦⡦⠘⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀
⠀⠀⠀⠀⣤⣤⣤⣴⠷⠤⣤⣀⣠⣤⣤⡄⢀⣀⠀⠀⠘⠝⠙⠀⠀⠀⠀⣪⡙⣷⡴⢐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣡⠶⠺⣕⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣼⣿⣟⣦⢄⡒⣛⣸⠑⢦⣄⡐⢝⣀⡠⣄⡀⡈⠓⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢾⣧⠈⡀⠈⠹⣿⠘⣇⡀
⠀⠀⠀⠀⠀⠀⢀⡙⠏⠞⠃⠢⠀⡁⠰⡵⣸⡏⠀⣷⢟⠛⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⢋⠴⣊⣬⢿⡮⡅⠀⠀⠉⠀⢇⠄
⠀⠀⠀⠀⠀⠀⠠⠙⠲⠽⣟⡳⢀⣺⣡⣎⠉⠀⠀⢡⡼⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠷⣾⢛⣼⡿⠉⠈⠱⡄⠀⠀⠀⠾⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⢻⡿⠺⢖⣤⡀⣠⣬⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢁⣴⠾⣓⡯⠒⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠙⠮⣭⢱⣎⡀⠄⠚⠁⠀⠀⠀⠀⠀⣀⣤⠔⠋⣉⠠⢚⣩⠔⠛⢉⡴⠀⠀⠌⠀⠀⠀⠈⠀⠀⢀⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⢀⣀⣤⣶⡷⠷⠒⢒⣩⡴⠞⠉⠀⠀⠀⢸⣿⣄⠀⠀⢶⣀⠂⠈⠐⠀⠁⠀⠎⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⣧⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⢀⣾⣿⠓⠀⠀⠀⢹⣇⠀⠀⠀⣨⢀⣴⠇
⠀⠀⠀⠀⠀⣠⠖⠀⠂⠀⠀⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⢀⣀⣀⣰⣿⣯⣻⣿⡶⠦⣦⡄⠀⠄⠠⣀⣺⣟⣾⡿⢷⡄⠀⣀⣼⣟⢃⣠⠜⣡⠞⡟
⠀⠀⠀⠀⠀⠛⠃⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠿⠿⠭⠿⣾⠿⠿⣭⣉⣉⣙⣙⠿⢿⣦⣭⣠⣡⡤⠶⣊⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠿⠿⢶⣭⠴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠪⢒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠐⠒⠶⣶⣄⡀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠉⠻⢷⡂⠤⠒⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⢼⡇
⢀⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢸⣦⡀⠀⠀⠀⠀⢀⡠⠛⠲⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⡈
⡟⠁⠀⠀⠘⡄⠀⠠⠄⠀⠀⠀⠀⣻⠀⠈⠳⡀⠀⢠⠔⠁⠀⠀⠀⠀⠙⢄⢀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠁
⢹⡄⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢳⣤⣶⣷⣦⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁
⠀⢹⡄⢰⣄⢤⣼⡄⠀⠀⠀⠀⠀⣸⣽⡄⠀⣀⢀⠆⡶⣄⡀⢐⡤⠀⣀⣀⣨⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠃⠀⠀⠀⠀⠀⣰⠄
⠀⠀⢻⣿⣷⡈⢿⣿⡆⠀⠀⠀⣠⡿⠻⠃⠚⠉⠉⠈⠐⠋⠻⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠟⢋⣡⣤⣶⠄⠀⢀⣠⡴⠞⠋
⠀⠀⠀⢿⣿⣷⡈⢻⣿⡄⠈⠈⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠉⠣⠛⡄⠀⠀⠀⠀⠀⠘⣧⣴⣾⣿⣿⣿⡿⠷⠛⠉
⠀⠀⠀⠈⢿⣿⣷⣼⠛⡇⢀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⢸⡉⠀⠀⠀⠠⡙⣶⣤⡀⠀⠀⠀⣀⣀⡀⣠⡴⠔⠋⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⢢⣼⠿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣿⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣠⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡱⠂⠈⠛⢿⣿⣤⣌⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢆⠀⠙⠻⣿⣧⣄⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⢀⢂⢀
⠀⠀⠀⠀⢺⢆⡀⠀⠀⠀⠀⣀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠢⣍⣿⣷⣏⣍⣇⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠵⠚⠉⠓⠢⣄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠔⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠹⣧⡀
⠀⠀⠀⠀⢀⡴⠋⡠⠒⠀⠐⠈⠍⣻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⣷
⠀⠀⠀⠀⣼⠁⡎⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡶⠦⣤⣤⣀⣠⣤⣤⣖⡯⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿
⠀⠀⠀⢠⡿⠀⢧⠀⠀⠀⠀⢀⡴⠖⢒⢶⡄⠙⢢⡀⢀⡤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿
⠀⠀⠀⢸⣇⠡⠘⢆⠀⠀⠀⡞⠀⠊⠉⢳⣿⠀⠀⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⣿⡀⢁⠘⢆⠀⠸⡇⡆⠀⢀⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣼
⠀⠀⠀⠀⠸⣧⠀⠣⡈⢣⡀⠳⣝⣒⣋⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡜⣿
⠀⠀⠀⠀⠀⠙⣧⠀⠈⠢⡱⣄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠎⢰⡟
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠈⠪⣳⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠚⠁⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠈⠓⢌⡉⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢚⡡⠔⠉⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⠑⠒⠨⠵⣲⡶⠶⠶⠶⠶⠖⠖⠒⠒⠚⣉⡥⠔⠚⠁⠀⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠉⠁⠁⠀⠀⠀⠀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⠟⠛⠛⠛⠛⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠀⠈⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⣀⡤⠖⢒⣶⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠘⠁⢀⠤⠚⣛⡛⢟⣟⣻⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⡄⠀⠀⠀⠁⡰⠟⢛⣿⣮⣭⣧⣤⣤⣝⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⢱⣤⣆⡴⢊⠔⢺⣿⣏⡙⣿⣿⣿⣿⣿⣿⢦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠛⢿⡴⠁⠀⠀⢻⠀⠀⢹⣿⣿⣿⣿⣿⡇⡟⢶
⠀⠀⠀⠀⢀⣠⡤⠶⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠺⠤⠈⠷⡦⠾⠆⠀⣠⣿⣿⣿⣿⣟⡏⢱⣾
⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⠄⠀⠄⠀⠘⠻⣿⣿⣿⣿⡟⠀⣾⡟
⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣉⢿⣿⣀⠀⢀⠀⠀⡀⠀⠈⠙⣿⣿⣷⡦⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠆⠀⠀⠀⠀⠀⠀⠻⢆⣿⡿⠋⠉⠁⠉⠁⢀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢈⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠈⠳⣦⣌⡉⠙⠛⠛⠛⠛⠛⢉⡉⠤⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⡀⠀⠀⠀⠐⡆⠀⠀⠀⢹⡤⡤⠐⠁⠀⠀⠀⣆⡀⠀⠀⠀⢸⠤
⠀⠀⠈⠙⠒⠷⠠⠀⠠⠈⠄⠄⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⠀⢀⣧⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣦⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠐⢶⠶⠆⠀⠠⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠻⢦⣤⣀⣉⣉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣁⣬⠟⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⠶⣶⣤⣬⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣶⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⢸⡇⢸⡏⢹⡏⢹⡏⢹⡏⢹⡏⠉⡏⠉⡇⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣄⣸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⠀⡇⢀⣇⣠⣧⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠙⠛⠛⠻⠿⠷⠾⠷⠾⠷⠾⠿⠿⠛⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡾⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⣦⣤⣤⣀⣀⣀⣀⣤⣤⣤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣤⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣉⣉⣉⣛⣛⡋⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠛⠿⢿⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⠿⠟⠛⢻⡇⠀⠀⠀⠀⢀⣴⣟⡱⠋⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠛⠻⡿⠷⠶⣦⡀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⢰⡿⠃⠈⠹⣦⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⣠⡇⠉⠛⢸⡇⠀⠀⢻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡇⠀⠸⣧⣀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣷⣾⠟⠁⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢟⣛⡛⣟⠛⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠉⣀⣿⣀⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠟⠁

⠀⡴⠟⠻⣦⣤⣤⡀
⣼⠀⠀⠀⠁⠀⠈⢻⣄⣀
⠻⣤⣤⣄⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠈⠻⠦⢦⣤⡀⠀⠀⠀⠀⠈⢙⡷⠶⠾⠿⠷⠶⠶⠶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡸⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠶⠶⠾⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⣉⣤⠤⠶⠶⠶⠒⠒⠲⠦⢄⣉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣤⢾⡄⡀⠀⠀⠀⢀⡴⠋⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠙⢦⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⢿⡀⠀⠉⠀⠀⠀⣠⢏⡴⠉⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠈⣧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠟⠁⠀⠀⠸⡇⠀⠀⠀⠀⣴⢣⠏⠀⠀⠀⣠⡤⠶⠶⢤⣀⠀⠀⠀⡴⠊⠉⠀⠀⠈⠉⠙⠢⡀⠀⠀⢸⡆⢿⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣼⢣⠆⠀⠀⣴⠟⠁⠀⠀⠀⣠⠈⢳⠀⢸⡁⠀⣀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡇⢸⡀⠐⢯⣄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⡆⠀⢠⢇⠎⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣠⠴⠃⠈⠧⣄⠈⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡏⢸⠁⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣿⠃⠀⠀⢸⣇⠀⡟⣸⠀⠀⡼⠁⠀⠀⠀⠁⠀⢠⠎⢁⡤⠶⢤⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠾⠃⠀⠀⠀⠀⣿⢰⠇⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⡏⠀⢸⠁⠀⠀⡿⠀⢹⠀⠀⠀⠀⠀⠀⢀⡟⠀⡼⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⢰⡇⠀⢸⡀⠀⠀⠀⠂⠀⠀⢧⠀⠘⠦⠤⠞⠁⢠⡆⠀⠀⠀⠀⠀⢀⡞⠀⡼⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣠⠋⢀⡼⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⣧⠀⠀⠘⢦⣀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠸⡅⠀⠀⠀⣠⠞⠁⣠⠎⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠸⡄⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⠓⠛⠋⠁⢀⡴⢁⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⣉⣙⠒⠒⠶⠶⠶⠒⣒⣊⣩⣥⠴⠟⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣄⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣼⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠠⠆⠀⠀⠀⣀⠠⠐⠂⢩⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣥
⠀⠀⠀⠀⠈⢳⠀⣠⡾⠀⠀⠻⢦⡀⠀⡀⠀⠀⠈⠲⢤⣤⠤⠂⠉⠀⠀⠀⠈⣿⠈⢳⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣤⠜⠛⠸⡟⠁
⠀⠀⠢⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠙⢾⣅⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢤⣴⡿⢀⢸⣿⣦⡀⠀⠠⠋⠉⠀⡾⠿⣿⣇⡀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⣨⡶⠒⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⢠⠀⠁⢠⠀⢀⣀⣼⣿⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠈⠳⠀⠀⠀⠀⠀⠠⠄
⠀⠀⢸⡁⢀⡀⠀⠀⠀⢠⡀⠀⠀⢀⣼⣿⡟⠀⠀⠠⠃⠀⠀⠘⠀⠈⠻⢿⣿⡀⠀⠀⢸⡿⠛⠀⠀⠀⠄⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⠺⣦
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⣿⢿⣧⣾⣿⡏⠀⠀⠰⠁⠀⢠⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠰⠀⠀⠘⠉⠀⠀⢀⠠⠾⠗⠀⠈⠷⠀⠀⠹⡆
⠀⠀⢀⡆⢀⣠⠄⠀⠀⠘⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠘⠉⠀⠘⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠈⡳⠚⣻⠶⡖⠠⣾⣤⡀⠀⠀⠈⠁⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⠀⠀⠺⣿⡶⠀⠀⠀⠀⠀⡀⠤⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⣤⡯⠀⠉⠀⢻⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢘⠀⠀⣶⣿⠓⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣾⣶⠀⠤⡤
⠀⠀⠀⠛⠘⠃⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢁⣻⣧⣆⣀⠀⠀⠀⠻⠆⠀⠀⠹⣇⠀⠉⠀⠀⠀⠀⠀⣀⣀⠀⠀⣰
⠀⠀⠀⠀⣀⡀⠀⠀⠀⣸⣷⠸⡇⠀⠀⠷⠔⡇⣤⣽⡿⢿⣯⠳⡄⠀⠀⠀⠀⠤⠠⣿⣷⡄⠀⠀⠀⠀⣠⣄⡈⣀⡀⠉⠀⠀⠀⣀⣀⠀⢀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⣿⡁⠀⠀⠀⠘⠿⠄⠀⠀⠀⠠⠄⠙⡁⠁⠀⠀⠙⣷⢹⡄⠀⠀⠈⠀⠀⢸⣿⣿⡄⠀⠀⣠⣻⣿⢹⣿⣿⠀⠀⠀⠀⠈⠻⠷⠎⢘⠿⠚⠃
⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢀⣲⠔⠀⣴⠂⠁⠀⠀⠀⠀⣿⡞⠁⠀⠀⠀⠀⠀⢸⠿⠟⠀⢀⡀⠘⠿⣿⡆⢿⡿⠀⣀⣀⠀⠀⠀⢀⠰⠃⣴⡶⣶⣶⣆
⠀⠀⠀⠀⠉⠀⠀⢀⣀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⣰⠉⡷⡈⢿⡀⠀⢿⣷⣤⢡⣇⢹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠠⠀⣤⣠⣤⡴⠀⠀⠀⣀⣀⣀⣀⣤⠀⠀⣀⡄⠀⠀⠘⣧⣑⢼⡾⠁⢰⣮⣿⣿⡗⠙⠉⠛⠁
⠀⠀⢀⣀⣤⡴⠖⡁⠀⠀⠀⠻⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢠⡀⠈⠉⠉⠀⠀⠀⠀⢹⡟⡶⠭⠛⣚⣟⣿⡄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠻⠟⠀⠀⢀⡄
⠀⠀⠀⠀⠉⠀⣠⣡⡶⠠⢤⡤⠦⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣿⡆⠀⠀⠀⣐⣦⠀⠈⠐⠫⠔⠃⠉⠉⢿⠓⣀⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⢠⣽⡿⢿⡔⠋⠁⠀⠀⢠⣀⡀⠀⠀⠈⣷⠀⠀⠈⠋⢀⣤⣾⣫⠏⠀⠀⠀⣠⠇⠀⠀⠄⠈⠸⠏⠀⠀⠀⠀⠀⢠⠀⠀⠈⢀⠇⠀⠀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣫⣽⣶⣶⠒⡠⣔⣫⠽⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣭⣾⡿⠛⠛⠛⢁⣦⡉⣓⠾⣭⣿⣓⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣥⠏⠁⡰⣜⢮⡿⣷⣽⣾⣧⠈⠻⢿⣓⣶⠾⢤⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣉⡁⣴⣿⣿⣮⣳⣝⡮⣷⣿⣿⡬⣧⣄⠺⣭⣽⣛⡂⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⣮⣳⡜⢿⡯⣟⢮⣗⢿⣚⠩⣿⣣⢿⣾⣦⠙⢳⢯⣟⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠻⣿⣧⠈⢿⣮⣟⡮⣝⡾⢈⣓⠾⣽⣲⠏⠀⠈⣓⢏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠻⠃⠄⣶⣝⡲⣝⡛⣭⣐⠮⣝⡆⣬⣛⠂⣐⢎⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠃⣼⠀⢿⣿⣝⡲⣽⡷⢮⣙⠾⣱⢰⣝⡷⠌⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣾⣿⠀⣷⣽⡻⣽⣻⢏⡳⣭⣳⣇⡻⣮⠃⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣿⡿⢠⡹⣿⣿⡳⢍⡳⣝⢮⡻⣯⡻⠀⣤⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⣿⢃⣺⣷⣝⢟⢿⡳⣝⢮⣳⡽⠎⢁⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡾⠏⡰⣝⢮⣿⡇⠠⠅⠈⠀⠁⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⣶⠁⣰⣿⣾⣯⠗⠀⠀⠀⠀⣀⣴⢸⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠟⡃⣾⣯⣿⡿⠃⠊⠀⣠⣴⡇⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣎⡙⠙⠛⢋⣀⣠⡆⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣇⣷⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠳⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢠⠞⣧⠀⠀⢀⣀⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⣠⠋⠀⠘⠚⠉⠉⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠙⠲⢤⠞⠉⠉⠙⠣⡀⠀⠉⠳⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⣧⠀⠀⠀⠀⡰⣋⣞⠉⣱⣄⡀⢸⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠙⣄⠀⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⢠⡾⣽⠣⣼⠃⢌⣇⡜⡸⠀⢀⡤⠲⡈⠳⣄⠀⠀⠀⠀⠘⠧⠤⠤⢤⣤⣀⣀⣀
⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠒⠲⢧⡦⢴⠛⣬⣶⠁⡰⠛⢢⠰⢇⣠⠟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⢾⣠⣿⠏⠻⣅⣀⠼⣀⡠⠃⢱⠛⡆⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⣨⠆⠀⠀⢠⡇⠀⠀⠀⠀⠠⠁⠀⣼⠟⠁⠀⠀⠘⠷⣤⢼⠒⠒⡇⢀⠇⠀⠀⠀⠀⠀⠐⠛⠲⠤⣄⣀
⠀⢀⡠⠖⠋⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠳⠿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠽⠗
⠶⣉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉
⠀⠀⠉⠙⠒⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡛⢏⣙⡦⢵⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠉⢫⠈⣇⣸⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡏⣧⣞⡁⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⣀⣀⣈⣶⡄
⠀⠀⠀⣴⣁⣤⠤⠤⠶⠒⣶⠂⠀⠀⠀⡠⢿⡀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠉⠙⠲⢤⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⠔⠋⠀⠀⠙⠓⠒⠚⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠑⠢⢌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡠⠞⠁⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡽⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⢀⡴⠋⠸⡆

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⢤⣤⣤⣤⣀
⠀⠀⠀⣴⣾⠟⠋⠉⠉⠉⠉⠉⠙⠛⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣄
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣥⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢦⣤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠘⣿⣆⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⣴⣾⠟
⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⢺⣷⡶⠾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣶⣿⡷⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣷⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠶⠦⠤⢤⣤⣤⣀⣀⣀⣀⣤⣶⠿⠛⠛⠛⠛⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠛⠛⠓⠒⠲⠦⢤⣀
⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠿⠟⠁

⢀⠀⠀⠀⢀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠚⠂
⠐⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠨⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠂⠀⠐⠀⠀⠀⠀⠀⠤⠤⠄
⠀⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠄
⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠁⠀⠀⠀⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀
⠀⣄⣤⠛⠁⢠⡴⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐
⢄⠀⠀⣁⠔⠀⠀⠀⠀⠀⠠⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠘⠢
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠅⠀⠄⠈⠈
⠀⠀⠀⠀⠅⠀⠸⠒⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠂⠀⠈⠀⠀⠀⠁⠀⠀⡀⠀⢀⣼⠉⡪⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢇⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⡀⠀⠀⢄
⠀⠀⠉⠀⠀⠄⠀⠠⠀⠀⠁⠀⡰⠋⠑⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠘⠀⢠⡀⠊⠀⠀⠀⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⢶⣳⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄⠠⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢣⡞⠉⣩⣆⠈⠛⣿⣄⠀⠀⠀⠀⠀⠀⠓⢄⠀⣑⡈⣳⣤⣶⡛⠋⠁⠈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⣨⠇⠀⠎⠛⠁⠀⠀⠀⠛⡆⢀⠀⡀⠀⠉⠻⣅⢠⣶⠟⣉⠙⡛⠛⠉⠈⢁⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣶⣄⣀⠀⠀⠉⠲⣄⠀⠀⢲⣄⣼⠦⠶⠃⠄⠀⣈⠀⠀⡀⠈⢉⡟⠉⠛⠉⠙⠋⠀⠐⠋
⠀⠀⠀⠰⠂⠀⠀⠀⠆⠀⢾⣿⢁⠹⣜⢦⠀⠀⠙⠶⣽⡮⠹⣿⣍⣷⠶⠶⠟⠿⠿⢷⣬⡤⠀⠁
⠀⠀⠀⠀⣨⠥⠀⠀⡤⠀⠈⢿⣟⣷⡜⢿⡇⠀⢀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠐
⠀⠀⠐⡤⠿⠔⠀⠀⢅⠀⠀⠸⣿⣿⣿⣛⣣⡄⠀⣀⠀⠀⠀⠀⠀⠀⠠⠄⡦⢀⡀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠡⣀⠀⠄⠀⠀⠀⢄⡀⠈⠻⠿⣿⠟⠁⠀⠁⠀⣀⠤⠀⠀⠀⣠⠠⢲⠇⢰⡆⢀⠀⠀⠀⠀⠀⠈⢶⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⢠⠊⠀⠀⠀⠄⢀⠘⣠⣷⣤⣼⣿⣿⠀⠀⠀⠀⠘⠛⠠
⠀⠀⠀⠀⣀⠀⣈⡄⢦⣤⠀⢄⡀⠀⠀⠀⠀⣰⡤⠀⠀⠀⢀⣠⣾⣿⡿⣿⣿⣿⣿⣿⣷⡦
⠀⢀⠀⣀⡿⠟⣹⢧⣭⣶⣛⣳⠛⡆⠀⠀⠀⠋⢒⠘⠃⡀⣿⣿⣷⣿⡿⣟⠙⠙⠏⠙⠢⣅⣾⠁⠀⠀⠀⢀⠔
⠀⠘⣿⢿⠳⢞⢧⢺⣿⣟⠻⡿⣷⢤⣀⠀⠀⠐⠚⠒⠶⠚⠛⠛⣁⣈⠉⠻⡗⢲⠖⣢⣴⣯⣿⣦⠒⠆⠀⠁⠐⠶
⠀⠀⢰⣾⡖⣸⢋⡧⠌⠹⠇⢰⡜⣧⣌⣧⣄⠀⠀⠀⠀⠀⠀⢀⡿⠉⠀⠀⠀⠘⠓⠴⠼⠿⠟⠁⠀⠀⠀⠂
⠀⠀⠈⢛⣿⣽⡿⢶⠐⣆⡀⢈⡷⣼⠟⣿⣟⣆⠀⠀⢀⣄⣴⠰⣾⡆⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡤
⠀⠀⠀⠨⠙⢛⣛⣛⠁⣬⣿⣟⣿⣯⣯⣿⣿⠷⠂⠀⠀⠙⢿⠂⢩⠃⠈⠉⠉⠀⢧⣤⣡⣶⣿⣿⣧⣄⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⡀⠀⠀⠈⠁⢲⣍⢡⠀⡀⠀⠀⠿⢻⣧⣽⣷⢾⣤⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣾⡓⢸⠂⠀⠀⠀⠼⡦⠀⠁⠀⢀⡄⠈⣿⣿⣿⣸⣯⡐⠀⠀⠀⢠
⠀⠀⣀⠀⢀⡤⠴⠶⠚⠓⣒⠈⢉⠀⠀⠀⠀⠀⠸⣿⣷⠲⡤⣤⡤⠀⠁⠲⠂⠀⠀⠐⢃⡟⣿⣯⣿⣷⣷⡖⠀⡐⠀⡀
⠀⢠⣏⢳⣛⡅⠶⠾⠿⣷⡎⢠⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠤⠂⢉⡭⣿⣇⠀⠀⠀⠩⡰⠛⣿⣿⣿⣿⣷⣵⣤⡟
⠀⠀⣿⣿⢽⣷⣿⣾⣿⡟⠑⠁⠀⠀⡠⠞⠀⠀⠂⠘⠂⠀⡰⣯⣿⠿⠷⠛⠁⢀⠀⠀⠀⠠⠾⣛⠛⠿⢿⣛⠷⠚⠁
⠀⠀⠘⣿⣵⣿⣿⣿⣿⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠈⠐⠀⠒⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢷⠀⠀⠀⠀⠾⢥⣶⣴⠟⢿⣦⡀⠀⢀⠀⠀⠀⠀⢀⡤⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡒⢉⣡⢤⡾⠀⠀⣴⣤⣀⠂⢤⡀⠀⢾⠿⣿⠟⠻⣷⣄⡀
⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⠀⠀⠀⣀⠤⢰⣏⢱⠍⠐⠋⠀⠀⢶⡶⢶⣿⢿⣦⣈⠿⠛⠳⢤⡀⠀⠀⠙⠳⢦⣄
⠀⠀⠀⠀⢤⣉⠂⠠⠊⠀⣠⠶⣯⣵⡿⠟⠋⠀⣀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠻⢿⣳⣦⡀⠙⠧⠀⠈⠀⠤⡉⠻⣿⣶⡆
⠀⠀⠀⠀⠠⢌⠲⢠⣰⡾⣱⣾⡽⠋⠀⠀⠀⢀⠀⠈⢰⣶⣾⣿⡷⠤⡀⠀⠀⠀⠀⠈⠙⠛⠦⠘⢦⠀⠀⣀⣀⡁⠠⠙
⠀⠀⠀⠀⠀⠀⠈⠒⠿⠾⠛⠁⠀⠀⠒⠂⢀⣷⣮⣓⣞⣟⣛⣻⣥⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⡆⠈⠻⠲⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣯⣒⠚⠛⠛⠛⠻⣿⣿⣿⣿⠟⠎⢀⠀⠀⠀⠀⡄⡠⠀⠀⠈⠓⠠⠄⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡟⣾⣿⣷⣷⡶⣞⣾⣶⣾⠗⡶⠁⠀⠀⢀⣻⣴⡌⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⠛⠛⠿⠇⠒⠋

⠀⠀⠀⢀⣀⢀⣀⢀⡀⠀⠀⠀⠀⠠⣼⣶⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣰⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⣴⣿⣿⣿⣽⣿⣿⣷⡀⠀⠀⠀⢻⣿⣧⠀⣿⡟⠀⠀⠀⠀⢰⣾⣿⣿⣿⣷⣶⠀⠀⠀⠀⣰⣶⣿⣿⣿⣶⡄
⠀⠀⠙⠿⠿⣿⣿⠿⠿⠛⠀⠀⠀⠀⠘⠿⡿⠀⠛⠃⠀⠀⠀⠀⢸⣟⠩⠍⠍⢛⡇⠀⠀⠀⠀⠿⠉⠉⠛⠉⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠈⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣼⣧⣴⠆⠀⠀⠀⠀⣠⣴⣶⡤⠖⠋⠉⠀⠀⠀⠀⣰⣿⣿⣿⠄
⠀⠀⢀⣤⠤⣦⣒⣦⣀⡀⠀⠀⠀⠀⢀⣾⣿⣿⣯⡀⠀⠀⠀⠀⠸⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠙⠓⠶⠷⠶⠿⠟⠋⠀⠀⠀⠀⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟


⠀⠀⠀⠀⢆⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣆⡦⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⢀⡤⢦⣤⠶⢤⡀
⠀⠀⣾⣷⣾⣿⣷⣦⣯⣄⠀⠀⠀⢠⣤⣤⣤⣤⣿⡇⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⣿
⠀⠀⠘⠉⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⢹⣦⣾⣛⣲⡶⠏
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⣤⣶⣶⣿⣿⣿⣿⡿⠀⠀⠀⠀⢞⢐⠩⠖⡘⢀⠀⠀⠀⠀⢠⣿⣿⠿⢿⣿⣦⠀⠀⠀⠀⠀⢱⣶⣄⠀⠀⢁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠋⡈⠂⢦⠁⠊⠀⠀⠀⠀⢸⠿⠇⡀⡠⠿⢿⠀⠀⠀⠀⡄⠀⠙⢿⣷⡔⠁
⠀⠀⠸⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠙⠈⠃⠞⠀⠀⠀⠀⠀⠈⠢⢀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣾⣿⡇⠀⠀⠀⠀⠀⣤⣾⠟⠛⣿⣾⣯⣤⣤⡄⠀⠀⠀⠀⠀⠀⠘⠓⠲⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢼⢻⣲⡄⠀⠀⠀⢸⡇⢹⠀⠼⣿⣿⣩⠽⣿⠇⠀⠀⡠⠂⠐⠊⠀⠴⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣯⢼⡯⣬⠇⠀⠀⠀⠀⣇⢸⣤⣾⡏⠉⠠⠞⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⢸⡀⠊⣿⠦
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢿⣶⠀⠀⠀⠀⠀⠸⡘⢇⣸⢿⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠀⣮⢀⡆⠋⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⣟⠉⠀⠀⠀⠀⠀⠀⢻⡿⢿⣭⣇⡃⠀⠀⠀⠀⠀⢀⠀⠠⠂⣷⣾⡇⢀⠃⣼⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣸⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⣿⢻⡇⠀⠀⠀⠀⠀⠈⢀⣼⣆⠸⣿⠄⢀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣿⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣿⣶⡇⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⢿⡔⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡜⢿⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⣦⣻⠃
⠀⠀⠀⠀⠀⠀⡔⣴⡔⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠛⣇⡫⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⣭⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢸⣿⠂⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡁⠀⠀⠸⣿⠀⠀⣧⡈⠳⣄
⠀⠀⠀⠐⡄⢀⡤⠖⠉⠀⠈⠳⣄⡀⠀⠀⠀⠀⢀⣠⢞⡥⣸⡷⠀⠀⠀⢿⠀⣄⣿⣧⡄⠈⣳⡄
⠀⠀⠀⠀⠘⠾⡀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣤⡴⠋⢠⢸⡇⣿⡇⠀⠀⣠⠀⢰⣯⣿⡟⢃⢞⠝⠁
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⠀⡑⠈⢿⣧⠹⡄⢸⢸⡇⣿⡇⠀⠀⣿⠀⢸⠛⣃⢔⠕⠁
⠀⠀⠀⠀⠀⠀⣈⡆⠀⢐⠆⠰⡁⠘⣆⠈⡿⠛⠛⣆⢼⡇⢹⠀⠀⠀⡟⢀⣠⡾⠗⠁⠀⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣦⡀⢳⡀⢹⣄⣸⠖⠃⠀⠀⠈⢆⠁⢸⠀⢀⣠⡶⢟⣋⠴⢒⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠙⠲⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢧⣴⡾⠛⠉⠀⢘⠇⠀⠏⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡎⠀⢀⣸⠀⠀⢠⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⠤⢤⣤⣄⣀⣠⡤⠶⠒⠚⢛⣒⣚⠛⢧⠀⢧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣥⣤⣍⡉⠉⣤⣤⣄⠲⣶⣶⣶⣶⣶⣿⢃⣾⣿⡿⣋⣴⣿⢀⡟⣷⠇
⠀⠀⠀⠀⠀⠀⣼⠁⠾⣿⣶⣯⣽⣿⣦⡘⢿⣿⣷⣮⣿⣿⣿⡿⠃⣚⣻⣻⣿⣿⣿⣿⡿⢁⣿
⠀⠀⠀⠀⠀⠶⣿⠗⠲⠶⠤⣭⣛⠿⣿⣿⣌⡛⢋⣭⣭⣭⣍⠐⢋⣭⣭⣭⣭⡭⠀⢀⣴⡿⣿
⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠆⡖⢩⣛⠶⣌⡙⠻⠶⣦⣭⣭⣭⡴⠿⠟⣋⣉⣥⠴⠞⡻⢟⣼⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⡲⠶⣾⣿⣑⠈⠉⠛⠒⠶⠶⠶⠖⠚⠋⠉⠉⠀⠰⠖⣋⡵⢛⡁⠦⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠁⢀⠍⠙⠶⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠛⠋⠇⠀⠢⠀⢁⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠙⠛⠉⠉⣉⠁⠁⠀⢀⠠⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠉⠁⠀⠉⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠛⠛⠶⣦⡘⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⠻⣿⠉⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⠲⠶⠶⣶⣶⠿⠿⠿⠿⠿⠿⠿⣷⣦⣄⠀⠋⠀⠀⣿⣇⣀⣤⣶⠶⠶⠶⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠛⠒⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠈⠙⣿⣆⠀⢠⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⢙⣷⡦⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡉⠁⠀⠀⠤⠤⠤⠶⠒⠛⠛⠉⠉⠉⠉⠉⠛⠳⢦⣄⢘⣿⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣛⣿⣃⣀⣀⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽⡿⠋⣍⣿⣤⣴⡶⠾⠿⠿⠶⣶⣤⣄⡀⠀⠀⠀⠉⠉⢉⣩⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣄⡐⠼⠶⠶⠛⣋⣡⡴⠶⠛⢋⣩⡿⠟⣿⠗⣨⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⣩⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⢟⣋⣥⣶⣶⣿⠟⠁⢀⣾⠃⣰⣿⣿⣯⢠⣤⠤⣂⡤⠄⠀⠀⢀⣠⣤⣤⣤⣤⣤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠟⠛⣻⡿⠛⠁⣠⣶⣿⡏⢰⣿⠉⠻⣿⣦⣝⡛⠛⠛⢉⣀⣀⣠⣤⣤⣭⣝⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⢀⣼⠟⠀⣠⣾⠟⠉⣾⠃⢸⡇⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣾⣏⡀⠀⠀⢠⡾⠁⢀⣼⠟⠁⠀⠀⣿⠀⢸⠁
⠀⠀⠀⠀⣠⣾⡿⠟⠋⡉⠉⢩⡿⠟⠛⠿⠿⣶⡿⠁⣠⡟⠁⠀⠀⠀⠀⣿⠀⢸
⠀⠀⢠⣾⡿⠋⠀⠀⠀⠻⣦⣈⣤⠆⠀⠀⠀⡿⠁⣰⣿⡀⠀⠀⠀⠀⠀⣹⡄⢸⣄⣀⣀⣀⡀
⠀⢠⣿⠏⠀⣀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣸⠃⣰⡏⢿⣷⠀⠀⣠⣶⣿⣿⣧⠈⣿⠿⠿⠿⣿⣿⣷⣤⡀
⠀⢸⣿⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡟⠀⢸⣿⣧⣾⣿⠟⠉⠀⠘⣦⠘⣦⡴⠀⠀⠀⠉⠙⢿⣷⡄
⠀⢸⣿⡀⠸⣿⠀⠀⠀⠀⢀⣀⣤⣤⣴⣾⡇⣼⣥⣤⣾⣿⣿⡿⠁⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⢿⣷⣄⠈⠃⢀⣴⣾⡿⠟⠋⠉⠉⢹⡃⡏⠉⠉⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠈⠻⣿⣦⣴⣿⠟⠉⠀⠀⠀⠈⢷⣼⣅⣧⣤⠞⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠈⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⣸⣿⠃
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋⢀⣴⣿⠏
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢠⣿⡿⠻⢿⣶⣶⣶⣶⣶⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠠⢤⣤⣤⣤⣤⣶⣿⠟⠁⢀⣴⣿⡿⠁⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣤⣬⣍⣉⣉⣉⣤⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠒⠉⠁⠀⠀⠢⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⢓⣄⣀⣀⣀⡀⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠲⠟⠀⠀⢀⡀⡴⠉⢀⣠⣄⡉⠀⠀⠀⠀⢢⠄⠀⡁⠁⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠚⠓⠒⠛⠩⣉⠁⣀⣐⢠⣌⣴⠄⢹⣦⠀⣐⡤⠤⠤⠤⠼⠷⠷⠂⠀⠀⠀⠀⠤⠄⠀⡀
⠀⠀⠀⠀⢀⡴⠛⡋⠛⢓⣤⡀⢀⠀⣩⠛⠋⠙⢻⠈⠉⠑⠤⢀⡌⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⢠⠊⠄⢀⠄⠚⠓⢆⠉⠉⠀⠈⠦⠀⠀⠀⠀⠀⠀⢰⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠀⠀⠀⢄
⠀⠀⣰⠃⠀⢀⡎⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢓⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡴⣦
⠀⠀⠃⠀⠠⠾⠒⠒⢉⡉⠀⠈⡏⠉⠑⠢⢀⣀⠀⠀⠀⢠⡟⣇⡀⠀⠀⠖⠈⠁⠀⠀⢠⠀⡀⠀⢉⠑⠢⠄⠀⠀⣠⠀⡀⣀⣀⡐
⠀⣀⠤⡰⠆⢀⣀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⠃⠑⠠⣴⡯⠉⠉⠹⠃⠀⣀⠄⠐⠀⠰⠒⢋⡀⠆⠀⠀⠀⠀⠙⣏⡝⠋⣁⢤⡈⠁⠠
⣾⠀⠀⠀⠀⠀⠀⢀⡄⢀⣼⠀⠀⠀⣴⣄⣀⣀⢵⣄⡈⢣⣀⠀⠀⡤⠊⠁⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⢸⡑⠐⠀⠈⠀⠀⠀⠁
⠉⠁⠀⠀⠀⠀⠀⢈⣶⠁⠀⠈⠠⡕⠉⠹⠍⠁⠁⠈⠙⡾⣉⣼⡏⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠀⠦⠠⠄⠐
⠈⠒⠀⢰⠖⠂⠒⠛⠚⠦⠄⣀⠀⠀⢀⠤⠇⠀⠀⢀⣀⡀⠀⠀⠀⠐⠠⢴⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡠⠔⠒⡤⠤⠀⣀
⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠈⣛⠀⡀⠔⠀⢀⡴⠋⠉⠀⠁⠢⡀⠀⠀⠀⠁⠠⠀⣀⠀⠀⠀⠠⠀⠀⠈⠀⠄⠀⢉⠭⠤⠤⠄⠐⠒⠺⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠖⠒⠒⠫⡀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⠀⠀⡼⠇⠤⡀
⠀⠀⡀⠄⠀⠀⠀⠀⢀⠀⠈⢁⠞⠁⠀⠀⠀⠀⠈⣦⠄⠀⠈⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⠀⡄⢤⠀⠀⠉⠂⠀⠀⠀⠀⠀⢀⠔
⢀⠎⠀⠀⠀⢀⠀⠤⠴⠃⠀⠈⣇⠀⠀⠀⠀⠀⢈⠟⠶⠶⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠒⠒⠈
⠸⠀⠀⠀⠋⠀⢀⠠⠀⠀⠠⠀⠨⣑⠂⠠⠤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠑⠐⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⠀⠤⠤⠤⠐⠁
⠀⠀⠀⠀⠈⠀⠀⢀⡤⠖⠂⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠉⠻⣿⣦⣄⣠⣤⣤⣴⣶⣶⣶⣾⡿⠿⠿⠿⠟⠛⠻⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⠻⣿⣯⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣽⣿⣷⣄⡀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣟⡛⠛⠋⠉⠉⠉⠙⠻⣿⣶⣤⣀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠀⠈⠙⠻⣿⣷⣤⣀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⠁
⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣿⠃
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⣶⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣿⣿⡿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣦⣄⣀⣠⣾⠟⣿⣿
⠀⠀⠀⠀⠘⣿⣧⡈⠙⠻⣿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣿⣿⠟⠋⢠⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠖⠒⠒⠒⠒⠒⠛⠋⠙⠛⡒⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠀⠀⠉⢀⠉⠓⠲⣄
⢀⡤⠴⠖⠚⠛⠉⣥⡴⠞⠉⠀⠀⠳⣄⢀⡔⠀⠀⠢⡴⠋⠀⠢⡀⢀⠴⠃⢄⠀⣼
⢸⠀⠀⠀⣡⣴⠛⠁⠈⢢⣤⠀⠀⠀⡨⠻⣄⠠⠖⠁⠀⠀⠀⠤⠈⠁⢀⣀⡤⢾⡇
⠈⣷⢤⣤⣀⡈⠙⢦⡚⠁⠈⠑⠄⠀⠀⠀⢈⣡⡤⠖⠒⠒⠒⠒⠚⠛⠉⣁⡴⠏⠀⠀⢀⣀⣠⠔⣫⣽⣭⣓⣒⠤⠤⠤⢄⡀
⠀⠈⢳⣄⣈⠉⠉⠛⠓⠶⠖⠲⠖⠒⠚⠛⠉⢀⣠⠴⠖⠒⠚⠛⠋⠉⠉⠁⠀⢀⡴⣻⣿⣷⣶⣿⣿⣿⣿⣿⣿⡿⠆⣿⣶⣌⠳⣄
⠀⠀⠀⠙⠛⠷⠴⠤⣴⣤⣤⣤⣀⡴⠴⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⣘⣻⣿⣛⠿⠿⣛⣛⣿⣛⣡⣶⣶⡆⣹⣿⣷⣜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⣫⣽⡶⢿⡿⠿⣟⣩⣿⣿⣜⣿⣿⣿⣿⣿⣿⣿⡷⢹⣿⣿⠏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠻⠿⡗⠮⣍⣲⣟⣿⣯⣭⣭⠭⠽⣯⣼⣟⣛⣛⣛⣿⡿⣃⣿⣋⣥⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⡄⠀⡿⠀⣾⣿⠁⠀⢸⣿⣿⡄⠀⠀⠀⠀⢻⣿⣧⡉⠉⠉⠁⢠⡿⠃
⠀⠀⢀⡴⢒⣛⠛⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠶⠚⠳⠶⣿⣿⣥⣤⣼⣿⣿⣧⣀⣀⣀⣀⣼⣿⣿⠧⠶⠒⠚⠉
⠀⣴⢋⣾⣿⣿⣇⢸⣧⣄⣉⠓⠲⢶⠶⠶⠶⠶⠶⠶⠦⣤⣀⡀
⢸⢃⣾⣿⣿⣿⣇⠸⣿⣿⣿⣿⠿⠋⣰⣿⣿⣿⣿⣶⣶⣤⣉⣙⠶⣄
⣿⢸⣿⣿⣿⣿⣿⡄⠈⣉⣉⠁⠴⠾⠿⠿⠿⠿⠟⠛⠋⣉⣉⣩⣤⠘⣆
⣯⠘⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠟⠋⣠⣿
⢹⣆⠹⣿⣿⣿⣿⡟⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣡⠤⠶⠞⠻⣿⡏
⠀⣿⣷⣬⣉⡛⠋⠀⢸⣿⣿⣿⣿⣿⣿⡿⢛⣡⡖⠋⠁⠀⣠⡴⠞⠋⠀⣀⡤⠔⠒⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠸⣿⡀⠈⠙⠓⠲⠬⣭⣉⣉⣉⣭⣴⣾⣿⣿⣧⣠⠖⠋⠁⠀⢀⡴⠊⠁⠀⠀⠀⠀⠐⠀⠀⠉⠉⠉⠉⠉⠛⠉⠁⠀⠀⠉⠓⢄⡀
⠀⠀⠹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢠⠞⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⣀⣤⠴⠟⠛⠉⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠖⠒⠋⠉⢀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠈⠉⠓⠒⠚⠛⠉⠉⠀⠀⢀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⡀⠀⠀⣠⡤⠴⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⣄⣠⠞⠁⠀⠀⠉⢷⣀⣀⣴⡋⠉⠉⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⡼⠁⠀⠀⠀⢀⡴⠞⠉⠁⠈⠙⢦⡀⠀⠀⠙⢦
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣠⡴⠚⠛⠳⣦⣼⠷⠒⠶⠦⣤⡞⠁⠀⠀⠀⠀⠀⠈⢧⠀⣠⡤⠼⢧⡀
⠀⠀⣴⠋⣩⣯⣍⣻⢦⣴⠋⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠙⣆⠀⢀⣀⣀⡀
⠀⠈⣏⠓⣧⠀⢀⣉⣻⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠴⠚⠛⠛⠛⠛⢦⡀⠀⠀⢹⡶⣯⡽⢥⣽⡆
⠀⠀⠈⠳⣍⣻⡟⠛⢻⣦⣀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢸⣿⣷⣤⢾⣴⠃
⠀⠀⠀⠀⠈⠹⡗⠾⣭⣀⣉⠽⠟⠒⠶⢾⣧⣤⣤⣤⣤⣤⣴⣯⣤⣤⣤⣤⣤⣤⡤⢴⡶⠶⠚⡿⠛⢻⠇⢀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⢻⡚⠶⠤⢉⡛⠶⠶⢾⣥⣄⣠⣔⣋⣁⣀⣴⣊⣁⣈⡴⠊⠁⣀⣔⣁⣀⣠⣮⣤⡤⠾⠚⣿
⠀⠀⠀⠀⠀⠀⠈⣧⠐⠶⢬⠍⠛⠛⠀⠲⠤⢭⡍⢉⣭⣍⣉⠉⢩⣉⣉⣙⠉⠉⣭⣭⣭⠤⠀⠰⠶⠚⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠠⣤⣀⣈⠙⠛⠂⠀⠲⠶⠤⠄⠀⠴⠦⠤⠄⠀⠠⠤⠤⠄⠀⠠⠶⠶⠚⢋⣁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⢦⠈⢉⣀⡐⠒⠶⠶⠀⠐⠲⠶⠶⠀⠶⠶⠶⠶⠀⠀⠶⠶⠶⠆⠀⠋⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠲⠤⣬⠉⣛⠁⠘⠓⠒⠒⠀⠐⠒⠒⠒⠀⠐⠒⠒⠒⠀⢐⡒⠚⢋⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣉⣛⣓⠀⠀⠛⠒⠒⠂⠘⠓⠒⠒⠂⠀⣉⣉⣭⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁


⣿⣿⢻⢿⢛⠿⡿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⠆⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣿⠀⢀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⢀⣻⠁⠀⠈⣆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠴⠤⢤⣤⠶⠋⠁⠀⠀⠒⣩⣄⠀⠀⠀⠈⠪⡆⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣨⡉⠀⠀⠀⠀⢿⣖⠁⠀⠀⠀⠈⣲⡿⠀⠀⠠⠐⡄⢻⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣡⣦⣉⢀⠐⠤⣀⠈⠻⢿⣶⠂⠈⣽⣿⡀⠀⠠⠀⡝⠄⠹⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣷⣤⣈⣷⡀⠈⠛⢻⣿⢩⣙⣿⡄⢠⡀⡀⠀⠢⡄⠀⣴⢒⣈⠥⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠦⠾⠿⣿⣿⣿⠿⠾⠿⣄⣠⣼⣿⠀⣽⣿⡀⠀⢻⣧⠀⠀⠹⢄⡇⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠻⡄⠀⠀⠀⢻⡟⠀⢠⣿⣿⠀⠐⠀⠈⢷⡄⠐⡌⣧⠀⠠⠏⣰⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠹⡄⠀⠐⣼⣷⣰⣾⠏⠤⠀⠁⠀⠀⠀⠙⢷⣦⡘⠄⣀⠖⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠙⠂⣶⣿⡟⢿⠃⠀⠒⠀⠁⠀⠀⠀⠀⠀⠈⠙⠆⠐⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⡿⠟⠁⠀⢡⣀⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠁⠀⠀⠀⢤⣻⡇⠀⠁⠀⠠⠖⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣻⡿⠃⠀⠀⠀⠤⢊⣥⠠⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⣀⠘⠟⠃⠀⠀⠀⠀⠖⠋⣶⣦⡄⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠠⣿⣿⠀⠀⠀⠐⠀⠀⠐⠋⠀⠀⠀⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡖⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢹⣿⣾⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠛⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠰⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢢⠅⠆
⠀⠀⠀⠀⠀⠀⠑⢦⣀⠠⠀⢀⠀⠀⠀⠀⠀⢐⠰⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠠⠍⠀⠀⠀⠨⠀⠀⠀⠀⢘⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣴⠒⠂⠉⠁⠀⠀⠀⢘⠠⠔⠂⢉⠅
⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⡂⠤⡞⠊⡛⠚⠺⠄⡀⡀⠀⠀⠀⠀⠀⠐⡇⠀⠐⢺⠁⠀⣀⡀⠤⠒⠊⠍⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠆⠒⢥⣄⣖⡃⠀⡀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠒⡀⠞⠋⠉⠂⠀⠀⣀⡤⢂⠻⠯⠿⢳⠖⣈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠶⡄⠤⠀⠀⠒⠀⣉⣅⠀⠂⡗⠀⣘⠀⠀⠀⠀⠀⠀⠀⠘⠤⠐⠂⠈⠁⠀⠰⠧⠘⣂⠥⣐⠖⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠠⠈⠁⠀⠈⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⡍⢻⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⡈⠇⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡆⢤⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡠⠸⡜⣆⠃⠘⡆⢤⣆⡀⠀⠀⠀⠀⠀⠀⠀⢈⣁⡸⠋⠄⠀⠙⠆⠀⢀⠄⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠍⢂⠌⠀⠁⠔⡹⣄⠀⠃⠐⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠉⣇⠀⠀⠉⠀⡌⠀⡤⠐⠀⢁⣀⣾⠾⢶⣤
⠀⠀⠀⠀⠀⠀⠢⣒⣨⡛⣇⠀⠀⠀⠠⡈⠓⡆⣡⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡈⠅⠂⠀⠍⢰⣶⣾⠇⣱⠀⣾⡁⠀⣠⣿⠃
⠀⠀⠀⠀⠀⡄⠄⠒⡰⠌⠛⠥⠉⡉⠀⡘⠒⠏⡀⢁⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠅⠀⠀⠘⢅⣨⡉⠙⢆⠘⠹⠋⠛⠉
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠈⠩⡲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠃⠈⠂⠝⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠱⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠹⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠁⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⢀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠄⠐⢲⠒⠉⢹⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡭⣽⣿⣆⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣀⠀⣠⡄⠤⢼⠒⠂⠛⠀⠀⠀⣀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣿⡿⣻⣛⡃⣿⣽⣥⣿⣿⣿⡿⢟⡅
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠇⠀⢸⣀⣀⣤⠤⠀⠂⢀⠴⠊⠁⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⣿⡿⠿⠿⠇⣿⣿⣿⣿⢟⣭⣾⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⡴⠃⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⢟⣵⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠸⡀⠑⠦⢄⣀⣀⣀⣀⡤⠔⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣻⠿⣿⣿⣿⣿⠿⣛⣵⣿⠟⠁
⠀⡀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⡤⠤⠄⠀⢤⣤⡤⠀⠤⢤⡶⠚⠉⣁⣴⣶⣿⠟⠛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⣠⣶⣯⣴⠶⠞⣿⠟⠁⠀⠀⣰⣿⣡⣴⡾⠿⣽⡿⠁⠊⠀⣀⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⠶⣶⠟⠛⠉⠁⠀⣠⡾⠁⠀⣀⣤⣶⠿⠟⠋⠁⠀⣰⠋⢀⣠⣶⣿⡿⠋⢀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡾⠃⠀⠀⠀⣠⣾⣿⡴⠾⠛⢻⡟⢁⡤⠀⠈⣠⣾⣧⡶⠿⠛⣡⡟⠁⠈⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡋⣀⣠⠤⣶⠾⠋⡁⠀⠀⢠⣼⠏⢐⣁⣴⣶⠟⠛⠛⠁⠀⠀⣼⣿⣥⡶⠶⠶⣿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⣠⠞⠁⠀⠀⠀⣠⣞⣩⣴⠞⠉⣼⠟⠁⠈⠀⢀⣨⣤⠞⠉⠡⠀⠀⢀⡾⠃⣀⣷⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⠻⣄⠀⠀⣀⣠⠞⠛⢩⠄⠂⣠⡜⢁⡠⠔⠒⠋⢉⡼⠃⠠⠄⠂⠈⣠⠎⢀⣠⣾⣿⠀⠈⠲⠦⠞⠀⠴⠺⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣪⡟⠁⠀⠀⠀⣠⣾⣿⣾⣿⣗⣶⣶⣶⣿⣤⣤⣄⣐⣦⣾⣧⣶⣿⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⣿⣿⣧⣠⣤⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢫⣑⣟⣹⠟⣿⣷⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣟⣿⣿⣿⣿⡿⠿⠛⠋⠉⠙⠛⠛⠛⠛⠿⠿⣿⣧⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⣿⠆⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣸⣻⡿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣄⣐⣚⣍⠴⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⣧⡄⠀⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠚⠋⠉⠉⠀⠈⠉⠁⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢠⠞⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁





⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠥⠤⠤⣀⡀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠏⠀⠀⠀⠀⠀⠉⠓⣄⠀⢀⡼⠁
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡞
⢀⡴⢏⠉⠋⠉⠙⠓⠦⣄⠀⣷⣀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠒⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⣾⡇⠀⠑⢦⡀⠢⡀⠀⠈⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⣯⠱⡀⠀⠀⠙⢦⡈⠢⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠹⡄⠱⣄⠀⠄⣀⠙⠢⡘⢦⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠐⠀⢀⡾⠁
⠀⠙⣆⠈⠣⣄⠀⠀⠀⠈⠢⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠈⠛⢦⣀⠉⠒⠦⠤⠤⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠴⠶⠚⠉⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⢀⡠⠞⠋⣻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⢀⣴⠋⠁⠀⠈⠁⡠⠖⠉⣠⠔⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠈⠛⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⣼⠃⠀⠀⣠⠔⠁⢀⡴⠊⠡⠀⠀⢀⢼
⠀⠀⠀⠀⣠⡞⠻⢭⣀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⣴⠋⠀⣀⠔⠁⠀⢀⣀⠤⠞⢁⡾
⠀⠀⢀⡾⠉⠙⠲⢤⣈⠉⠒⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡉⠉⠉⠋⠉⣉⣉⣀⡤⠶⠋
⠀⠀⢸⣧⣀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠒⠤⣄⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢿⡈⠓⠦⢄⡀⠀⠀⠀⠈⠙⠲⢤⣀⣉⣳⠖⢻⡆
⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⠑⠒⠒⠒⠒⠚⠋⠉⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣶⣶⣶⣶⣾⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠒⠊⠉⡁⠀⣀⡀⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⡉⠀⠀⠀⢙⣷⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢄⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠖⠃
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠁⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣀⡽⠖⠊⠉⠉⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠢⣄
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠘⢦⣀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡿⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣠⣤⣴⣶⣾⡿⠟⠁⠀⠀⠀⠀⠸⡏⠠⠀⠀⠀⠀⠤⢤⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⣀⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣤⣀⣤⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⡀⢀⣰⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⠴⠯⡽⠚⠳⠋⠹⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⢧⢀⣼⡇⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠛⡟⠘⢿⠛⠓⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠁⠀⠀⣥⣾⡿⠁⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣠⣬⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢶⣶⣤⣀⣀⣀⣤⣤⣴⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠀⠠⠦⠒⠂⠀⠐⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⣠⡇⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⢀⠀⡀⠀⠀⠠⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣢⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠐⠀⠁⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄⠒⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠁⠀⠀⢸⠂⠀⢀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠊⠀⠁⠀⠀⠀⠀⠰⠤⠀⠔⠃⠀⠐⢄⠀⣀⡀⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀
⣿⣟⣙⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠂⡆⢰⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣘⣀⣁⣃⣘⣘⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸⠉⢉⣉⣉⣉⣈⡀⠀⠀⠀⡴⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢨⡇⠀⢀⠎⠈⠢⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠘⣀⠤⡄⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠤⠤⠤⠤⠤⠤⡇⢠⠊⠀⠀⢱⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠣⢀⣀⣀⣀⠠⠔⠊





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

⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦
⠀⣀⡀⠀⣠⣴⠎⠿⣿⡿⠃⢶⣄⡀⢀⣀⠀⢀⣤⡦⠹⢿⣿⠟⠰⣦⣀⠀⣀⡀⠀⣀⣤⡶⠂
⣼⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠉⢰⣿⣿⣿⡌⠁⠀⠀⠀⠀⠀⠀⠈⠁⣾⣿⣿⣧⠉⠁
⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣨⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠘⠛⠛⠁⠶⣦⡀⣠⣤⣤⡀⣤⡶⠎⠛⠛⠋⠰⢶⣄⢀⣤⣤⡄⢀⣴⠾⠛⠀⠀⠀⠀⣴⣦⣴⣦⣴⣤⣦⣤⣤⡆
⠀⠀⠀⠀⠀⠈⠰⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠁⢾⣿⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⢀⡘⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⢰⠾⠛⠁⠀⠉⠁⠈⠛⠷⢠⣾⣿⣶⡀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠁⠀⠀⢩⣭⣭⣭⣭⣭⣥⣭⣭⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣙⠛⠿⢿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⠀⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣶⣾⣿⣿⣿⣶⣶⣦⣤⣍⣙⠛⠛⠻⠿⠟⠛⠛⢛⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠿⣿⣿⣿⣄⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠠⣶⡄⣿⣿⣿⣿⣿⣿⣿⠟⢛⠛⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⣥⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠉⠛⠿⠿⢿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⠡⢐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢂⠡⣐⠈⠀⠔⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢂⡡⢔⡪⢶⡿⠃⡠⠂
⠀⠀⠀⠀⢀⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢂⡡⢔⡪⠕⠈⠁⡰⠋⡠⠊
⠀⠀⢀⠔⠁⠀⠀⠀⢰⠀⠀⠀⠀⡀⠄⠂⡁⢄⡢⠕⠊⠁⡀⠄⡒⠋⠁⠉⠑⢄
⠀⢠⠁⠀⠀⠀⠀⠀⡆⢁⡠⠒⡁⠄⡂⠅⠂⠁⣀⠤⠒⢁⠠⠎⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⣀⡀
⠀⠸⠀⠀⠀⠀⠀⠀⡰⠋⡠⠊⠂⠁⣀⠤⡒⠋⠀⠀⠚⢅⡀⠀⠁⠀⠀⠀⠀⠀⠀⡱⠊⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠊⡠⠊⠀⠠⠐⠉⢀⠄⠀⠀⠀⠀⠀⠀⠡⡀⠂⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠄⠀⢀⠈⡠⠊⢀⠀⠒⠀⠐⠎⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠔⠊⠁⠐⠢⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠂⢄
⠀⠀⣇⠔⡡⠊⠀⡠⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢄⠀⠂⠀⠀⠀⠀⠀⡠⠃⠀⡠⡾
⠀⡔⡡⢊⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⣤⠵⠒⠐⠢⢴⠂⠐⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠐⠂⢉⢤⡢⠵⠈
⠀⢸⡋⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢀⠠⡐⠌⠂⠁
⠀⠈⠢⢕⡢⢀⠀⠢⠤⣎⣀⣀⡀⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⢛⠒⠲⠤⠀⠒⠈⡀⢄⠢⠕⠋⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢹⠲⠭⠠⠥⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⠠⡒⠕⠊⠁⠰⠀⢀⠤⠤⠤⠾⣀
⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠑⠠⣄⠀⠀⠢⢄⣀⡀⢀⣀⠠⠔⡩⢔⠪⠆⠁⠀⠀⠀⢀⠜⠈⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠯⢔⠤⣀⠀⠀⢀⠤⡢⠕⠈⠀⢸⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⡀⠈⠐⠭⠍⠂⠁⠀⠀⠀⠀⡏⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠐⠤⢀⣀⣀⣀⠠⠳⠄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣀⣀⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢄⠀⠀⠄⠀⠀⢀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⣀⣀⡀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠁⠂⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣹⣿⣿⣱⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⡟⢿⣿⣿⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢻⣿⣿⣿⣿⣤⡇⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣏⣹⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣉⣿⣿⣻⠟⠁⠀⠀⠀⠀⠀⠙⣯⣯⣽⣭⣯⡝⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⣟⡿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⣿⣤⣼⣿⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣯⣟⣿⢿⣾⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣿⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣟⡿⣿⢷⣷⡇⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣶⣿⣿⣾⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡷⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠻⣿⢉⣿⣿⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠁

⠀⠀⠀⣀⣄⡀
⠀⢠⣾⣿⣿⣿⣦⠀⠀⢠⣿⣿⣿⣿⣷⡀⠀⢀⣀⣀⣴⣷⣦⣤⡀⠀⢠⣤⣆⣶⣿⡄⠀⠀⠀⣄⡀
⠀⣸⣿⣿⣿⣿⡿⠀⠀⠘⣿⣿⣿⣿⣿⢃⣤⣿⣿⣿⣿⣿⣿⣿⡿⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⡧⠄
⠀⢻⣿⣿⠿⠋⠁⠀⠀⠀⠈⢿⣿⡿⠁⠈⢿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠙⢿⣿⡿⠁⠀⠀⠀⢨⠁
⠀⠀⠈⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣄⢉
⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⠀⠀⢠⣾⣿⣿⣿⣿⣆
⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⣼⣿⣿⣿⣿⣿⡏
⠀⠀⠸⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡶⠀⠀⠀⢿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠁⣀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⡀⢈⣀⣠⣤⣤⣀⠀⠀⢀⣠⣀⣾⣿⣿⣧⣠⣄⡀⠀⣾⣿⣿⣶⡀⠀⣴⠿⣿⣿⡃⠀⠰⣶⣾⣿⣿⠆
⠀⢸⣗⣾⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⠛⠛⠛⠛⠓⠀⢻⣶⣴⣿⠗⠀⠈⠉⠛
⠀⠈⠻⣿⣽⣿⣿⠿⠋⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠁⠀⢠⣤⣤⡦⠀⠀⠈⠿⠿⠇
⠀⠀⢀⣠⣤⣆⣤⡀⠀⠀⠀⠀⢀⣀⢈⡙⠛⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⣤⣤⣄⡀
⠀⠀⣽⣿⣿⣿⣿⣯⡀⠀⠀⣺⣿⣿⣿⣿⡀⠀⠀⣶⣦⣶⣶⣦⠀⢀⣾⡹⣼⣏⢿⣷⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠛⠉⠋⠋⠋⠛⠃⠀⠸⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⡇⠘⣿⣷⣽⠿⣸⣿
⠀⠀⢠⣤⣤⣤⣤⡤⠀⠀⠀⠻⣿⣿⣿⡿⠁⠀⠘⢿⣿⣿⣿⠟⠁⠀⠈⠻⠿⣿⡿⠇
⠀⠀⠀⠙⠿⠿⠟⠀⠀⠀⠀⠀⠙⠛⠋⠀⡀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⢀⡤⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⣿⣿⣧⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣁⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⠉⠉⠉⠉⠐⠤⡀
⠀⠀⠀⠀⠀⡠⠊⠁⡠⠄⠒⠈⠁⠀⠈⠁⠒⢄⠈⠳⡀
⠀⠀⠀⠀⡜⢀⠔⠁⠀⡀⠤⠀⠒⠂⠠⢄⠀⠀⠑⡄⠘⡄
⠀⠀⠀⡜⠀⠏⠀⢠⠊⢀⠔⠈⠉⠉⠢⡀⠱⡄⠀⠘⡀⠰
⠀⠀⠀⡇⠀⠀⠀⡎⠀⠎⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⢇⠀⢁⠀⠱⡀⠰⡄⠀⠀⠀⡠⠃⢠⠃⠀⠀⢀⠀⠃
⠀⠀⠀⠈⢆⠀⠀⢄⡑⣄⠈⢦⠒⠉⣀⠔⠁⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠑⠢⠄⣀⠘⡄⠀⡗⠉⠀⠀⠀⠀⠀⠀⠘⡀⠈⠄⠀⠀⢠⠤⠤⠤⣤⠀⠀⠀⠀⠀⡤⠤⠤⢤⡀⠀⠀⠀⠀⢠⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠒⠲⠻⠀⠀⠀⠘⡓⠐⠒⠐⠚⠃⠀⠀⠀⡟⠒⠐⠒⠚⠃⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⢀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⢤⣤⣤⣀⣰⠤⠀⠀⠀⠤⣧⣤⣄⣠⡧⠄⠀⠀⠤⢼⣤⣄⣀⡮⠄
⠀⠀⠊⠀⡄⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⡄⠀⢀
⠀⠀⠀⠀⠇⠀⠀⠀⠲⡉⠠⠆⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠐⠺⡄
⠀⠀⠤⠠⠄⠀⠀⠀⠀⠸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠠⡇⠀⠀⠀⠀⠀⠸⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⡇⠀⠀⢈⠁⠀⠤⢘⣋⠒⠂⣙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢣⠀⠈⠀⠀⠀⠀⠀⠀⠢⡀⠠
⠘⠉⡀⠂⡜⠇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠴⠃⡀⠂⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡎⠀⠠⠄⠀⠀⠐⠀⠊⠢
⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠂⠀⠤⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠂⠐⡄⠀⠀⠀⠀⡀⠀⠀⠐⡠⠀⠀⠀⠀⢀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠐⢧⡀⠠⠀⠀⠀⠤⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠈⡀⠀⠀⠀⠀⢰⠄⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠰⠄⠬⠄⠀⠩⠀⠤⠬⠥⠤⠤⠤⠤⠬⠤⠤⢤⡥⠤⠤⠤⠤⠤⡥⠤⠤⣬⠤⠤⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠀⠀⠇⠀⠀⠀⠀⠀⣿⡄⠀⣾⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠀⠀⠆⠀⠀⠀⠀⠀⠿⠃⠀⠸⠀⠀⠸⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⢸⠀⠀⠀⡅⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢸⠂⠀⠠⠃⠀⠀⠀⠀⠀⡇⠀⠂⡇⠀⠀⠀⠀⠀⢰⠂⠀⠆
⠀⠀⠀⠀⠀⠀⢀⡤⢒⣪⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣸⠆⠀⠀⠀⠀⠀⣥⠀⣀⠇⠀⠀⠀⠀⠀⢸⡀⠀⣄
⠀⠀⠀⠀⠀⣰⠞⡼⠋⠀⠙⠀⠐⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⡀⠀⣀⣿⣼⡁⠀⡠⠈⠉⠉⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄

⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢠⣞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⡄
⠀⠀⠀⣠⣤⢤⢠⡟⠑⠀⠀⠀⠀⠀⠀⠀⢰⢎⠿⣈⣡⡤⠀⠀⠀⠀⠀⠀⠀⠀⣠⢵⡦⠜⠀⠀⠀⠀⠀⠀⠀⣖⣁⠉⠉⠁⡀
⠀⠀⠀⢀⡞⢸⡦⠟⠄⠀⠀⠀⠀⠀⠀⢸⣎⠿⣤⣨⠿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡔⠊⠀⠀⡁
⠀⠀⠀⠸⣄⠉⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣻⢤⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠙⠦⠔⠁⠇
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⢠⡴⠲⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠓⢢⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⡒⣀⣁
⠀⠀⠘⢍⠓⠢⠴⢞⣝⠀⠀⠀⠀⠀⠀⠀⢼⡛⠷⢒⣿⠄⠀⠀⠀⠀⠀⠀⡔⠁⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠈
⠀⠀⢶⡋⠛⠉⠿⠛⡁⠀⠀⠀⠀⠀⠀⠀⢘⡄⠂⠀⣹⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⡴⡚⠉
⠀⠀⢠⠒⠺⠉⠋⠙⠖⠀⠀⠀⠀⠀⠀⠜⡥⠂⡄⠁⠢⠄⡃⠀⠀⠀⠀⢀⡎⠸⡏⠀⣼⠃⠀⠀⠀⠀⠀⠀⣼⠘⠁⡤⢄
⠀⠀⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡙⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⡝⢶⠉⠉⢱⠖⡇⠀⠀⠀⠀⠀⠸⡦⡀⢱⡊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠙⠺⠥⠜⠓⠁


⠀⠀⠀⢀⣠⣤⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣸⢙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣄
⠀⠀⣠⠟⠙⠀⠇⠐⠉⠢⠀⠀⠀⠀⠀⢸⡶⠦⢶⣶⡶⠟⠀⠀⠀⠀⠀⠀⠸⡽⣾⣿⠘⠁⠀⠀⠀⠀⠀⠀⢰⡿⣯⠉⣼⢷⡏
⠀⠀⢿⣄⡠⢄⣀⠄⠀⠠⠀⠀⠀⠀⠀⢢⠠⠤⠤⠤⠭⢥⠃⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⣿⡇⡄⠰⢡⣀⠇
⠀⠀⠀⠙⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⣉⠉⢉⢉⡁⠀⠀⠀⠀⠀⠀⠘⢿⣦⣼⣦⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣤⣴⣤⣦⣴⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣶⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⢿⣿⣿⣦⡀⠀⠀⣈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⢈⢿⣿⣿⣷⣄⠀⠀⠀⠂⠈⠑⢄
⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⢀⠻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠈⢢⡈⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡒⠲⡄
⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆⠀⠐⠐⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠁⠀⣹
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠐⠘⢦⣀⠀⠀⠀⠀⢀⣠⠏⠀⢰⣏
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⢀⡤⣄⠀⠀⠀⢀⣿⣿⠀⡀⠀⠀⡠⠀⠀⠀⡀⠈⠈⠉⠓⠒⠚⡿⠀⠀⠀⠘⠈⠳⡄
⠀⠀⠀⠀⠈⣿⣿⡄⠀⢠⡏⡞⢢⢳⠀⠀⣸⣿⡏⢀⠇⠀⠀⣇⠀⣠⠀⣇⡤⠂⠀⠀⠀⣀⣽⠤⠤⣄⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠘⣿⣷⡀⠘⣇⢧⡼⡸⠀⣠⣿⡟⠀⡼⡠⠊⠀⣿⠞⣁⡴⠿⠒⠚⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠉⠉⣁⣴⣿⠟⢀⣠⠏⢁⡠⢚⣧⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⣟⠥⠒⢁⣮⣔⡫⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠉⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠉⠁⣲⠐⢎⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠃⠀⠀⠱⠄⠀⠀⠀⠀⢠⣾⣿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠻⣿⡿⠏⠀⣀⣀⣤

⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⡄
⠀⠀⠀⠀⠀⠀⢠⠄⠈⢄⣀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠛⠫
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠐⠀⠀⠴⠋⠀⠠⠀⠀⠀⠠⠄⠤⠄⠻⣿⣿⣿⡇⡀⠀⠀⠀⣀⣤⣶
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠑⠀⢄⡀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⠉⠡⠂⠁⠀⠀⢰⣿⣿⣿⠃
⠀⠀⠀⠘⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣟⣻⡿⣿⠿⠿⠿⣿⡿⢿⣟⣻⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠶⢮⣭⣭⣭⣭⣭⣭⣭⣭⡵⠶⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡶⠛⠛⢶⡞⠉⠀⠈⠉⢉⣭⣭⣿⣿⡛⠉⠉⠉⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣼⣷⡀⠀⠀⠹⢦⣤⣤⣤⣴⣿⣿⣿⣿⣇⣀⣀⣤⣿⡿⣿⠟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠻⢶⣄⣠⣶⣽⣶⡉⠁⢻⣿⣿⣿⡟⠉⠉⠀⣨⣽⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣀⠀⠀⠀⠀⣩⣛⣿⣿⣇⣠⣴⠿⠿⢿⣻⣿⣶⣤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣙⠻⠶⣦⣤⣿⣿⣿⣿⡟⠉⠀⠀⠀⢻⣿⣿⣿⠇⠀⠉⣉⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣌⣉⡙⠛⠋⣄⣀⣠⣤⠶⠟⢛⣉⣀⣤⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠛⣛⠒⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣀⡁⠀⠐⠚⣂⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠁⠀⠀⠀⠀⠀⠀⢘⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⡶⡄⠀⢀⠖⠂⢸⠖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣦⡈⠉⠁⢀⡴⠁⠘⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⣈⠀⠸⠑⠒⠒⠁⠀⠀⠀⠠⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⣠⠄⠀⠲⣄⠀⠀⠀⠀⢀⣀⠀⠀⢃⣠⣤⠴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠐⠁⠀⠈⠑⠄⢀⠎⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⡜⠀⠀⠐⠒⠤⠀⠀⠀⠙⠦⣤⣤⠀⠀⠀⠀⠀⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⢦⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠈⠢⡀⠰⢻⢡⣄⣀⠀⢀⣀⣀⣀⣀⡤⠖⠋⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠣⡅⠀⠀⠈⠉⠀⠁⠀⠀⡠⠀⠀⠀⠘⢦⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠂⠤⢄⣐⠀⠁⠒⠒⠀⠀⠀⠉⣠⠔⢁⠀⣀⠤⠓⠁⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢄⡀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠈⠀⠔⣣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢍⣉⠓⠢⢄⠀⠀⠁⠀⠐⠒⠒⢤⡀⠊⠁⠀⠀⣀⠀⠀⠀⡠⠀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⡒⠄⠀⠀⢄⠀⠠⠀⠀⢱⠀⠀⠀⠊⠁⠀⣠⡬⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠢⡄⢣⣸⠊⠀⠒⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⣠⡆⠀⢀⠆⠀⢀⠄⠀⡀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠒⠒⠀⠁⣹
⠀⠨⢰⣤⢠⠀⠀⠒⠐⡭⠛⠀⠀⠘⠃⠒⠈⠀⠀⠠⠀⠀⠀⢠⡾⠛⠛⠚⠀⠀⠊⠀⠀⠲⠆⠀⠀⠀⠊⠉⣀⡁⠼
⠀⠀⠙⠁⠘⠀⠐⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠴⡮⠁⠁⠀⠀⠀⢀⡀⠀⢀⢠⠔⠆⠀⠀⠀⠀⠀⠤⠄⢒
⠀⡤⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣷⠸⠁⠀⠀⠰⠀⠐⠀⠀⠐⣂⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⢤⣤⣀⣀⠀⠀⠀⠀⠔⠸⠂⢸⠀⣆⠀⠀⠙⠳⠤⠗⠀⠀⠀⠀⢀⡠⠀⡀
⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠉⠈⠻⠛⠀⠀⡒⠂⡃⠘⠂⢸⣴⣿⠀⠀⠀⠀⠀⠀⢠⢋⠔⠡⠈⡠⠊⡠⠂
⠀⠀⢰⣄⠀⠀⣀⣀⣀⣀⡄⠀⠁⠠⡀⠀⠀⠀⠉⠉⠉⣋⠈⣣⣞⡧⠁⠀⠀⠒⠷⣶⡄⠈⠟⠶⣣⢆⣀⠞⢠⠊⢠
⠀⠀⠈⢿⣯⠁⣯⣹⠁⠣⠃⠀⠀⠀⠈⠁⠀⠀⠈⠻⢷⣾⣮⠿⠈⠀⣤⡄⠀⠀⠐⠁⠙⠃⠀⠀⠈⠈⠛⠛⢷⡶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡇⣰⠀⠀⠀⠀⠀⠀⡔⡀⠀⠀⠀⠀⡀⠞⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⡶⢃⠉⠀⠀⠀⠀⢄⡠⠊⠁⠀⠈⠀⣠⠈⠀⠀⠀⠀⠀⠀⠈⣞⡡⠄⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠀⠰⠁⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⣀⡤⠒⠁⠀⠀⠑⢄⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡟⢰⢃⠀⠀⢀⠄⠐⠚⠠⢄⡀⠀⠀⠀⠀⠀⠈⣷⠞⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤⡄⣿⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⠐⢤⡀⢀⣀⡴⣧⡀⠀⢀⡠⠔⠂⠉⠑⠢⣁⠀⠀⡔⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡦⠁⢿⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠹⡤⠾⠒⣚⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣬⣷⣦⡀⠘⡇⣸⢐⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠹⠿⣭⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢛⣛⣿⣧⣭⣁⠹⣼⠀⠅⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⣉⢥⣾⣩⡤⢀⣀⡸⡎⠇⠀⠀⠀⠀⠀⢀⠁⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠸⠀⠀⠤⢼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠩⣐⡭⣥⣀⠹⣄⢀⠀⠀⠀⠀⢠⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠀⠀⢀⢠⠃⡀⠠⠤⠌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣩⣵⢟⡉⠁⠀⢼⣦⣘⠥⢄⢀⠸⡇⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⠎⠀⠀⠒⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⣁⣠⠄⠒⠒⠋⠛⠳⠮⣽⣷⣿⡌⢯⣀⠄⠀⠀⠀⢀⣴⣿⠇⠠⣈⡴⠧⠄⠀⠛⠶⣾⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠠⠈⣉⠁⠰⢖⡶⣋⡀⠀⠀⠀⠀⠀⠹⢦⡙⠋⠛⠟⠋⣠⠾⠿⠶⠚⠉⢀⡀⠀⢦⣴⣶⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠗⠊⠀⠀⣰⡿⠋⡕⠀⣀⣶⠿⢰⣀⠀⠀⠉⠓⠒⠒⠋⢁⡀⠀⢀⠠⡙⠞⠿⣷⡦⣝⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠉⠀⠀⠐⠃⡏⠈⠀⢺⡥⢀⡆⢻⠈⢷⡈⢷⡈⢳⡕⢦⣙⢮⣿⡄⠈⠉⠀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠈⢀⠂⠈⠆⣳⡏⢷⣾⣿⠈⠉⠁⠻⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠐⢸⡇⠀⠈⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠤⠤⠤⠤⢤⢤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⣀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠉⠉⠉⠉⠈⠹⢿⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠤⠤⠤⠤⠶⠾⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠏⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⡌⠐⢲⡶⠲⠂⠒⠒⢒⣾⢻⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⡔⠺⣾⡟⢀⣦⣀⡸⠟⡀⣸⠀⠀⠀⠀⠀⠀⠀⣥⣿⡆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⢈⣿⡋⢁⣼⡁⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠓⠲⣶⠶⣾⠟⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠃⠀⠀⢤⣧⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢙⡆⠀⠀⣞⠉⣷⠀⠀⣚⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠈⠋⠁⣀⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢸⡇⣼⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⢀⠀⡤⠠⡄⠀⠀⣠⠒⢠⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⠉⠉⠉⠉⠉⠉⢸⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠼⠃⠀⠀⠢⠴⠃⠀⠀⠙⠶⠞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣋⣸⡇⠀⠀⢨⣀⣵⠀⠀⠺⣤⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠐⠀⠀⢠⣤⣤⣤⣤⣤⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡄⠀⠀⣔⠀⣆⠀⠂⢠⡁⣰⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢹⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠙⠛⠃⠀⠀⠈⠛⠋⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣈⣀⣀⣼⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁

⣷⠛⠛⣶⢶⡶⣶⠶⠶⣾⠶⢶⠶⠶⣾⠷⠶⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⢶⡶⡶⣾⢿⣿⣿⠾⠿⠷⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣿⡿⣿⣿⣏⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⢿⣿⣧⠈⢿⣿⣆⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣿⣄⠹⣟⣷⡜⢿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣮⣿⣁⠳⣿⣷⡝⢿⣿⣌⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⠻⣿⣯⠙⣿⣿⣿⣿⠛⠛⠿⠿⠿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠙⣿⣦⡹⣿⣷⡙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣷⡜⣿⣷⡈⢿⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣤⡤⢄⣀⣀⣀⣾⣿⣿⡌⢿⣿⡈⢻⣷⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⣿⢿⣿⣼⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢻⣷⡝⢿⣷⡝⢻⣿⣿⠉⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⢻⣿⣎⠹⣿⣾⣿⠋
⠀⢀⣤⣦⣤⣀⣀⠀⠀⠀⢀⣼⣿⣿⠹⣿⣆⢹⡿⣶⣿⣿⠏
⠀⣾⣿⣿⣿⣿⣿⣿⣟⣛⢿⣿⣿⣿⣤⣿⣿⣿⣿⣿⠏⠁
⠀⣾⣿⣿⣿⣿⡻⢿⣿⣟⣿⣿⠿⠿⠿⠿⠛⠋⠁
⢀⣿⣿⣿⡏⢿⣿⣄⣹⣿⣿⠃
⣿⣿⡏⢻⣿⣄⣿⣿⣿⠟⠁
⣿⣿⣷⣿⣿⣿⠿⠛⠁
⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⣀⠀⠗⡀⠈⠀⠀⠀⠀⠀⠠⣠⣤⣤⢦⡠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡤⡄
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢐⠓⠊⠨⠁⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⠠⠀⠀⠀⠀⢎⠰⠈⡟⠉⡵⡄⠀⠀⠀⠀⠀⠀⠀⠃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⡔⠀⠀⠀⠀⠀⠈⠢⣤⠤⠔⠉⠀⡀⠀⠀⠀⠂⠟⢳⡋⠛⣲⣾⠁⠀⠀⠀⢠⠊⠵⠇⢪⠓⠂⢑
⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠎⠀⠀⠀⠀⠀⠀⠑⠂⠤⣀⡀⠤⠊⠀⠀⠀⠀⠀⠲⠦⠉⠙⠀⠁⠀⠀⠀⠀⠈⠂⠤⠠⠛⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⢀⠀⠀⠀⠀⠀⠀⠀⣁⢓⡀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠈⠉⠀⠐⢄⠀⠀⠀⠀⠈⠤⠀⢀⡦
⠀⠀⠀⠀⠀⠈⡸⠶⡇⠀⠀⠀⠀⠀⠀⠢⠜⠫⠙⣉⣃⠖⠀⠀⠀⠀⠀⠀⠘⢃⡀⣄⣀⠀⠀⠀⠀⠦⠊⠀⠉⡉⢢⣀⡠⠀⠀⠀⠰⠃⠀⠀⠈⣄⡀
⠀⠀⠀⠀⠓⢺⠀⠀⠐⠂⠀⠀⠀⠀⠈⢈⠃⡉⢰⡈⡐⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠂⣸⠉⠀⠀⠀⠀⢠⡀⠀⣠⡟⡠⢤⡱
⠀⠀⠀⠀⠀⠀⠁⠐⢀⡠⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠬⠉⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⠿⠃⠀⠀⠀⠈⠓⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠣⡊⣉⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⢀⡜⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⣿⣿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠐⠶⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⣀
⠀⠀⠀⠐⠁⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⡡⠀⠀⠀⠀⢺⠮⣿⢿⠆⠀⠀⠀⠀⠀⠀⣢⣄⡏⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⢛⣬⣇⣐⣨⣻
⠀⠀⠀⠠⡀⠀⣠⠶⠲⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⢜⢎⣷⡦⠀⠠⠀⠀⠀⠀⢰⣿⣿⣳⣦⣀⠴⠋⠀⠀⠀⡈⣿⢫⠞⢁⠏⡠⡅
⠀⠀⠀⠀⠈⠉⠛⢄⡑⠖⠊⠀⠀⠀⠀⠀⢀⣦⣄⡀⡠⠃⠀⠀⠀⠀⠀⠣⣳⢟⡅⠨⠀⡀⠀⠀⠀⠀⠑⢿⣿⣿⠿⠒⠀⠀⠀⠀⠀⠈⠻⠴⠚⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠜⢂⠍⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⠾⣷⣶⣤⠀⠀⠀⠈⢢⡀⠀⠀⠀⣤
⠀⠀⠀⠀⢆⣤⣶⣶⣆⡘⠂⠀⠀⠀⢠⡄⢠⣀⣀⣠⣿⣀⡀⠀⠀⠀⠀⢘⠉⠉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣬⡻⣿⣇⠀⠀⢀⡀⠃⠐⢓⠚⠁
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⡿⠁⠀⠀⠀⢀⠄⠀⠀⠹⡀⠀⠀⠀⠀⢀⣀⣀⣠⡬⣿⠿⣿⡾⠏⠀⠀⠘⡀⠀⠁⣴⠋⠉⠲
⠀⠀⠀⠻⣿⡿⠻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠠⣠⣀⣠⣀⠇⠀⠀⠀⠀⠈⠈⢷⣼⣀⣼⡷⠃⠀⠀⠀⠀⠀⠀⠒⠚⠻⣄⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⠿⣿⠿⢿⢿⡿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣦⣶⣶⣶⣷⣶⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠉⠤⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⠠⠀⠀⠀⠀⠠⠀⠂⠀⡀⢀⡀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠔⣀⣤⡶⠘⠉⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⠖⡒⣶⣤⣤⣤⠀⠀⠀⠀⠀⠀⡈⡆⠍⠁⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠂⠉⠁⠀⠄⠐⡀⠀⠀⠀⠀⢀⡊⠐⠁⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠄⠀⠸⠀⠀⠀⠠⠈⡉⠰⠃⠰⠆⠘⠂⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠐⠘⠁⢰⠂⠀⠱⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⣀⣀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠞⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠘
⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠂⠀⠀⣁⣀⣀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣯⡀⢋⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡘⢱⡖⠂⢀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⢰⠀⠙⣄⠁⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠰⠙⢘⡠⢇⡠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠈⠅⠘⠀⠀⡇
⠀⠀⠀⠀⠀⠸⠀⠀⠈⠷⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡄⠠⢀⠹⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⠁⠝⠀⠒⠂⢈⡉⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⡀⢒⣾⠀⢤⡀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠠⠐⠂⠌⡐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠸⠀⠀⠀⠀⢀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⡉⠤⠈⡁⠂⠀⠀⠀⠀⠀⠀⠀⠑⢄⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⣀⡀⠄⢀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠐⠂⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠓⠠⡀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢼⡤⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠉⠄⠡⣄⡗⠋⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡐⠀⠛⠛⠈⠀⠐⡽⠦⣀⣀⡀⠀⡀⠈⣃⢀⠊⢉⢧⡀⢁⠨⠧⡀⠀⡀⠈⠀⠀⠀⠀⢀⠀⠀⣾⠛⠂⢤⡠⢴⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠉⣈⠀⠀⠿⠀⠀⠐⠲⠀⢀⠀⠐⢌⠂⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠂⡂⢀⡍⠀⠠⠋⠈⢠⠠⡂⠌⠐⠀⠀⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⢐⠡⠀⠐⠁⠄⠉⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡜⠀⠀⠀⠀⠂⢀⣀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡠⠒⢒⡄⠈⡄⠀⠀⠀⠀⠀⠴⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠓⠒⠉⢀⠄⢰⠀⠀⠀⠀⠀⠠⠀⠀⠅⡀⠀⡀⠀⠀⠀⡀⠀⢊⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢅⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⡑⠀⡀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠁⠀⠀⣀⣀⡀⠄⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⢀⡴⠛⠉⠉⠙⢦⡀⠀⢀⣤⠖⠛⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⣠⠿⠚⠛⠓⢦⡀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣤⣀⠀⣼⣃⣀⣀⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⣧
⠀⠀⠀⣠⡤⢶⡶⢤⣄⠀⢀⡴⠋⠁⠀⠀⢙⡷⠋⠉⠉⠉⠙⠳⢿⣄⠀⠀⠀⠀⠀⠀⠀⢻⣤⡶⠛⠉⠙⠳⣆
⠀⠀⣾⣇⣰⠟⠛⠶⣿⣳⡟⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣀⣀⣸⣏⣀⡀⠀⠀⠀⠸⣇⢀⣤⢶⡶⢦⡀
⠀⠀⢻⣌⣹⣆⣀⣠⣬⣿⠃⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡷⠞⠋⠉⠁⠀⠀⠈⠙⢦⡀⠀⠀⣿⣻⡷⠟⢳⡖⣿
⠀⠀⠀⠙⠷⣌⣿⠉⢉⣻⣦⣄⣀⡀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀⣿⣿⢿⣶⣻⡴⠋
⠀⠀⠀⠀⠀⠈⢻⡛⠿⢤⣤⣨⡽⠛⠛⠛⢲⣿⠶⢤⣤⢤⣤⡤⠴⠯⣤⣤⣤⣤⣤⣤⠤⠶⢶⠾⠻⠛⣻⠋⠉⢹⠀⣠⡟⠉
⠀⠀⠀⠀⠀⠀⠈⣧⠛⠶⠤⢉⡛⠳⠶⠾⣥⣤⣀⣤⣖⣉⣀⣀⣤⣚⣁⣀⣠⠖⠉⠀⣠⣔⣁⣀⣠⣴⣥⣦⠴⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠰⠶⣬⡉⠛⠛⠂⠰⠦⣬⣭⡍⢉⣉⣉⣉⡉⢉⣉⣉⣛⣛⠋⠉⣉⣉⣭⣥⠄⠀⠶⠶⠚⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⣄⡀⠀⠛⠒⠖⠀⠀⠤⣤⣤⡄⠀⢠⣤⣤⣍⠉⠉⣭⣤⣤⡄⠀⠀⣤⡤⠶⠞⠛⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣌⡉⠛⠁⠀⠶⢤⣤⣤⠀⠠⢤⣤⣤⡄⢠⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⠀⠰⠚⠛⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⣁⣈⡛⠓⠶⠶⠀⠴⢤⣤⣤⠀⠠⢤⣤⣤⠄⠀⢤⣤⣤⣤⠀⠠⣤⡤⠶⢂⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣉⡁⠐⠶⢤⡄⠀⠠⢤⣤⣤⡄⠀⣤⣤⣤⣤⠀⠀⠶⠶⠶⢂⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠦⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣤⠈⠄
⠀⠀⠀⢔⠐⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡱⠀⠀⢠⣿⣿⠿⠛⠉⣇⠸⡀
⠀⠀⠀⣀⣽⣿⠦⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠱⠀⠀⠘⣿⣇⠀⠀⠀⡇⠀⠈⡄
⠀⠀⢠⣿⣿⡁⠀⠈⠢⡬⠒⢄⠀⠀⠀⠀⠀⠀⡄⢣⠀⠀⠈⢻⣷⡖⠈⢳⠐⢄⡁
⠀⠀⠀⠙⢿⣥⣀⣀⠠⠧⡄⠀⠵⢄⠀⠀⠀⠀⠡⠘⠁⠀⠀⠀⡜⠁⠈⠁⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠔⠁⠀⠀⠀⠈⠢⡀⠀⠀⠙⠢⠄⠐⠘⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠔⠁⠐⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠐⡀⢀⠤⠒⠐⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠀⠀⠀⠀⠀⠘⡳⠀⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠃⠂⠈⠉⠢⡀⠀⠀⢀⡠⠄⠂⠁⠀⢡⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠎⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⡘⠀⠀⠀⢀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡌⠀⠀⡠⠊⠉⢄⠀⠀⠃⠀⢀⠁⠀⡠⠂⠁⠈⠢⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠰⠢⠄⠀⠀⡘⠀⢀⠎⠀⠀⠀⠀⠡⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠈⠢⡀⠆
⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⡰⢀⠔⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠐⠦


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⡙⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢡⠤⠤⠤⠔⠒⠒⠲⠾⠌⠉⠉⠙⠶⠒⠒⠂⠤⠤⠤⢤⠃
⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⢀⣀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠄⣀⡀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⢦⣀⣠⠤⡶⠛⢄⡀⠀⣀⠄⢠⣀⡀⣀⡠⠿⣦⢄⣀⡠⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⢀⢉⠿⠛⠂⠀⠛⢿⡁⡀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⡦⠄⠀⠀⠀⢴⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢤⠤⠀⠀⠤⡤⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⣬⣄⣠⣀⣠⣤⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢹⣿⣿⣿⣿⠉⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢸⡇⢻⠃⢻⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢻⠀⠈⠀⢸⠀⠘⠀⠀⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄⠀⢠⠎⢣⡀⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠛⠀⡸⠀⠸⡇⠀⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⠀⡧⠀⠀⣇⢀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠰⣇⠀⢀⡿⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⢸⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠟⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢰⡇⠀⠀⠀⠀⠉⢳⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⡐⡇⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠻⣤⡀⠈⠳⣶⣤⣈⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡥⠀⠀⢀
⠀⠀⠀⠈⠙⠷⣾⣿⣿⣿⣷⣦⣉⣻⣟⠂⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣼⣷⣔⡁⠀⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣟⣋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⣿⠡⣓⣤⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢜⠁⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠉⠁⢀⣤⠏⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢷⠀⢀⠀⠀⢀⠀⣀⣴⣿
⣿⣶⠦⠄⠀⠂⠀⠈⠀⠀⣀⡠⠴⠚⠋⠁⠀⠀⠀⠀⢠⣄⣀⣀⣀⣤⣆⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⣻⣿⣿⡿⠃
⠙⠻⠶⣤⣤⠤⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠄
⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠒⠐⠚⡆⠀⠀⠀⠀⠀⠀⢠⣾⣿⠞⠓⠂⠐
⠀⠀⠀⠉⢁⢤⠁⠀⠀⠀⠀⠀⠀⠘⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣟⡽⠁
⠀⠀⣾⡤⠴⠐⢦⠀⠀⠀⠀⠀⠀⠀⢻⡧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⡀⢸⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⢳⡴⠀⠚⠇⠰⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠸⡇⢹⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⢹⣿⢀⣄⣄⣰⡶⠀⠀⠀⠀⠀⠀⢹⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⢸⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠻⣏⡭⠵⢿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⣭⣿⡇⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⡠⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⠦⠴⠞⠁

⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣠⣤⣶⣶⣾⡿⠟⠛⠿⠿⠟⠻⡿⡿⣿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⣠⠊⠉⠟⢋⣿⠿⡿⠿⠿⠻⢻⣿⠉⠀⠀⠀⠀⠀⠀⣈⠥⠀⠒⠛⠉⠁⠀⡀⢀
⠀⠀⢸⣷⣿⣿⣳⣿⠋⡞⠁⣠⠆⣴⠎⢠⠆⣠⣂⠀⠤⠒⠈⠀⢀⣀⠠⠀⠀⠀⠀⠀⠀⣈⠁
⠀⠀⠀⠉⠛⢿⣿⣿⣶⣄⣼⣇⣪⣼⣶⣿⣿⣿⠟⠁⢀⡀⢰⢋⣄⠄⠀⠀⠀⠀⠀⠀⡰⠁⠀⣃⣆⡃⢂
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠉⠀⠼⣏⠁⡄⠀⠀⠀⠀⠀⠀⠰⢃⣄⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⠀⠠⠀⡀⣰⠆⠁⠀⠑⠀⡴⠆⢉⡀⣤⣤⣀⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠁⠀⢿⣀⠀⣨⡴⢷⡀⠁⢀⡤⠀⠀⠛⡛⠷⠾⣿⣿⣉⣀⣀⣠⣄⡀⠤⢶⣤⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣫⡍⢙⢻⡄⠀⠷⢡⣷⡟⢩⠧⢴⡖⢈⢁⠠⠔⠀⠂⢉⣫⣙⣷⣾⣿⣟⡋⣴⣴⠦⠓⢒⠼⠀⢠⡟⠿⢶⡄⠀⢦⡴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡀⠀⠀⠙⣿⣷⣄⡈⠺⠿⠿⣿⠻⣜⠖⣬⡾⣿⣛⡯⣝⠻⣤⣅⠀⠒⠂⠂⣁⣠⣶⣏⡄⡈⠎⢀⣰⣿⡿⡾⠂⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠂⠀⠙⠫⢿⣓⣀⣁⢀⣀⣴⡙⠯⠾⠾⠾⠛⠋⠉⠉⣀⣠⣴⣶⣿⠏⠹⠚⠋⢁⣠⣴⡿⠧⠒⠋⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣄⣀⣀⣉⣁⣀⣀⣤⢤⣴⢶⠾⠿⠿⠻⠛⠛⣉⣁⣠⣤⣤⣶⣶⠿⠟⠛⣉⣡⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠘⠶⠤⠴⠶⠖⠒⠚⠛⠋⠙⣯⣉⣉⣡⣬⡤⠴⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢻⣿⣿⣿⣿⣿⣿⠏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠃⠀⢸⣿⣿⣿⡿⣿⣿⣿⣿⡀⠀⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤⣶⡶⣿⠛⠉⠉⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠈⠁⠀⢰⠃⢠⠊⠁⠈⠑⠦⡈⠑⢄⡀
⢠⠖⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⠈⡆⠀⠙⢆
⡏⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢇⠂⠀⠀⠀⢀⠞⢀⠞⠓⣄⠑⡄
⢻⣄⠀⠀⠾⠷⠶⠦⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠙⠲⠤⣤⠏⣠⠋⠀⠀⠈⢦⠘⣆
⢀⡟⠃⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⠉⠒⠢⠤⢍⣉⡐⠒⠒⠚⠀⠘⡄
⣾⠀⠀⠀⣼⠛⠛⠻⠶⠶⠶⠶⠶⠶⠤⠤⣤⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⡀⢳
⠘⠷⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⢄⠀⠀⠀⠀⠀⠀⠀⠱⡌⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡦⢄⣀⠀⠀⠀⠀⠙⣄⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠀⠀⠀⠀⠀⠈⢣⡘⢦⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠙⢦⡙⠦⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢾⣦⣤⣀⣀⠙⢦⡈⠓⠦⣄⡜⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⢿⣷⣤⣀⣠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢼⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⣠⣤⡔⠿⠟⣠⣤⣄⠀⠀⠀⠀⢀⣤⡴⣾⠀⡷⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆
⠀⠀⠀⠀⢸⣿⣿⣷⡇⣰⣿⣿⣿⠆⠀⠀⠀⣿⢿⣷⣾⠀⣷⣿⡟⣿⠀⠀⠀⠀⠀⢠⣶⣆⡻⢛⣴⣶⡄⠀⠀⠀⠀⣠⣶⡺⡅⡟⣶⣦⡀
⠀⠀⠀⠀⢸⣿⣿⠛⠃⠛⠻⣿⣿⠀⠀⠀⠀⡿⣿⣯⡿⠀⣿⣿⡿⢿⠀⠀⠀⠀⠀⣻⡁⠀⢠⠁⠀⢸⡇⠀⠀⠀⠀⣿⣿⠿⠇⠿⢿⢿⡇
⠀⠀⠀⠀⢸⡟⢻⢤⡄⣤⣀⡏⣿⡇⠀⠀⢠⣄⠹⡿⡃⠀⢻⢿⠃⣸⠀⠀⠀⠀⠀⣿⡇⣶⣮⣷⣶⢸⡇⠀⠀⠀⠀⠛⢸⣶⠄⣴⣎⠘⠃
⠀⠀⠀⠀⣾⠇⢸⣿⣧⣾⣿⡇⢸⡇⠀⠀⢸⣿⢀⡷⠀⠀⠀⢾⡀⣿⡆⠀⠀⠀⠀⣿⢀⣿⣗⣿⣯⠘⣷⠀⠀⠀⢸⡿⢠⠟⠀⠸⣧⢰⣷
⠀⠀⠀⠀⣛⢰⣿⣿⣿⣿⣿⣿⠈⡇⠀⠀⠸⠇⢨⣦⣄⢀⣤⣮⡁⠹⠃⠀⠀⠀⢰⡏⣼⣿⣿⣿⣿⣦⢻⠀⠀⠀⢸⠇⣽⣦⣀⣤⣾⡄⣿
⠀⠀⠀⠀⠟⠘⣿⣿⡛⢻⣿⢿⠈⠇⠀⠀⢸⠆⢸⣿⣿⣿⣿⣿⡇⠴⡆⠀⠀⠀⢨⠀⣿⣿⡟⢿⣿⣿⢨⡄⠀⠀⣘⢀⣿⣿⣿⣿⣿⡇⢘⡀
⠀⠀⠀⠀⠀⠀⡌⣿⡇⢹⣿⣰⠀⠀⠀⠀⠀⠀⢉⣿⣿⠈⣿⣯⡅⠈⠀⠀⠀⠀⠈⠀⣷⣿⡇⣾⣿⡇⠀⠁⠀⠀⠋⠘⣸⣿⡇⣿⣯⡀⠈⠃
⠀⠀⠀⠀⠀⠀⣿⣿⠃⢸⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠃⢿⣿⡇
⠀⠀⠀⠀⠀⠀⡀⣹⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⢸⣻⡇⠀⢻⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⢸⢻⠇
⠀⠀⠀⠀⠀⠀⣇⡿⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⢸⣦⡇⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠈⡀⠁⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⢀⡰⠈⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠸⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡗⢺⣆⠧⠔⠀⠀⠀⢀⡜⡏⣿⠋⢀⡴⢾⣟⣭⣿⣿⣿⣧⠀⠀⠀⢹⣧⣄⠀⠉⢳⡄⠀⠀⡇⠀⡃⠀⣇⠀⡃
⠟⠊⠁⠀⠀⠀⠀⠀⠈⢷⣿⣯⣤⢻⠽⣴⣻⣮⣿⣿⣿⠏⠀⠀⠀⠀⠉⠙⠻⢿⣿⡇⠀⠀⢧⣄⢰⡀⠱⡀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠊⠓⠒⠚⠻⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠁
⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⣀⣤⣄⣀⠀⠀⠀⠀⣴⣶⣾⣭⣴⣟⠙⢫⡿⠀⠀⠈⠂
⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡌⠃⠀⣴⣿⣥⠙⢦⣭⣏⡘⣆⠀⠀⠉⠻⠏⠿⠯⠽⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⣠⣾⣧⣿⣷⠀⠀⠀⠀⠀⢰⣿⣟⢿⣧⡄⢸⣿⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⡀⠀⠀⢀
⢠⡞⢹⣿⡯⢞⠛⠀⠀⠁⠀⠀⢸⣏⠛⠊⠉⠸⠿⠁⠀⠀⠀⢀⠠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⣀⡿
⣇⢨⡴⠋⠂⠀⠀⠠⠁⠀⠀⢠⣟⢁⡀⣠⡼⠃⠀⠄⠀⣠⣬⣶⣤⡾⢳⡖⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⢮⣾⠃⠀⠀⣾⡜⢻⡿⠉⠢⠀⠀⠀⢰⡟⠀⠿⠋⣼⣷⣶⣏⠽⣷⠄⠀⠀⠀⠀⢠⠏
⣠⣤⠖⣦⠠⣾⠇⡏⠀⠀⠐⢯⢕⡈⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠘⢻⣿⡏⠷⠴⣯⠘⢦⣀⠀⣰⠏⢀⣀⣤⣤
⢏⠁⠀⣀⣠⣡⡟⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢢⡀⠀⠉⢯⣀⣘⣃⠛⢻⣯⣄⣘⠿⣷⡒⣨⣿⠋⢠⣾⡟⠛⠻
⡄⢀⢈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠈⠛⠯⣽⣏⣚⣷⣶⣿⡿⠟⠁⢠⣿⡅⣳⠆⢂
⣶⡿⣫⣿⠏⠀⠀⢆⠀⠘⡄⠀⢢⠀⢃⠘⡄⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⣿⣱⡀⣠⣼
⣸⣿⠟⠁⠀⠀⠀⢠⣡⠀⣇⡀⣸⡆⢨⡄⣱⠀⠀⠀⠰⡈⠿⣷⠶⣦⣄⣀⣀⣀⣀⣤⣤⡀⠀⠀⢿⣿⣆⣩⡌
⠋⠁⠀⣤⡀⠀⣶⡛⢿⣶⡉⡙⡋⠙⣋⣻⣿⠿⠇⠀⠀⠈⢷⣍⠓⢮⣭⣭⣭⠷⣶⣶⣈⣿⣦⠀⠈⣙⣛⣩⣵
⠀⠀⣴⣹⡿⠀⣻⣧⠀⠛⠿⠷⠲⠾⠛⠿⠃⠀⠀⠀⠀⠀⠀⠙⠳⣔⣏⠉⠈⠒⠈⠛⠻⣿⢿⣶⡄⡀⠈⠉⠉
⡼⠋⢉⣽⠁⣰⠋⠋⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢇⠀⠀⠀⠀⠛⠀⣰⡏⣿⡄
⣤⡶⣼⡇⠀⣿⡒⠠⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡀⠀⠈⢳⡀⠀⢀⠐⣊⣼⣶⡿⠁⠀⠀⣴
⣸⢿⣿⠁⠀⠘⢿⣄⣉⣈⠻⢀⣄⣠⣠⡭⡴⠾⠋⠀⠀⢠⣶⡜⠺⢿⣷⡦⠀⠹⣗⣧⣌⣤⠾⠟⠀⠀⠰⠘⣿
⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⢉⡀⠀⠀⠀⠀⠀⢀⣇⠁⠀⠈⠁⠁⣝⣧⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙
⠄⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⣖⣴⣾⡿⢻⣿⣿⣷⡶⣄⠈⠋⠙⢹⣄⡀⠔⢦⡾⠇⠀⠀⠀⡠⠂⡠⠀⡀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠉⠁⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡠⠚⠁⠀⡏⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢂⠘⡶⠉⠀⠀⠘⠃⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠑⣤⡟⠲⠀⠀⠀⠀⣀⣀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⠠⠤⠀⢤⣤⠁⠀⠀⠀⠀⡞⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠋⠁⠀⠀⠀⠘⠀⠹⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢩⠁⠒⠛⢛⠲⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⣀⡄⠐⠒⡇⢀⡴⠃⠀⠀⠀⠣⡀⠀⡸⠀⠲⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠏⠉⠡⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢃⠘⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠘⡄⠀⠑⢄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠣⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⡇⠀⠀⠀⠀⠀⠀⡰⠀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⠀⢰⠀⠀⠀⢀⡰⠊⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡄⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠰⡀⠀⠀⠀⡆⣠⠔⠉⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢢⠀⡘⠑⠂⠤⠄⢀⣀⣀⠀⠀⣠⠆⠀⠀⠱⠀⠀⠀⠉⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠱⣄⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠤⣬⡷⢀⣀⣀⣀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠋⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠈⠙⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠒⠒⢲⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢇⠀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠑⢄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠎
⠀⠀⠀⠀⢠⠏⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠾⠶⠻⡆
⠀⠀⣠⠒⣇⣀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠈⢦
⢠⡖⠉⠈⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢨⠴⠂⣀⠀⠀⠑⢤⡀
⠣⣝⠦⡀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠵⠶⣤⣭⠆
⠀⠀⠑⢌⣦⡀⠈⢆
⠀⠀⠀⠀⠉⠻⢦⣌⡧
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⣠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦
⣿⣿⣀⣸⣿⣧⣀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣸⣿⠇
⠈⠻⠿⠟⠛⠻⢿⣿⣶⣤⣴⣿⡿⠛⠉⠉⢿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠏⠀⠀⠀⠀⠈⠻⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣿⣿⠛⢻⣿⡆⠀⠀⠀⠀⢀⣿⡿⠀⠀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣀⣀⣠⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣿⣿⣶⣾⣿⣇⠀⠀⠀⠀⢸⣿⠇⠀⢰⣿⡟⠙⣿
⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠿⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠉⠉⠙⢿⣷⣤⣴⣶⣿⣿⣄⣠⣾⣿⣿⣾⣿
⠀⠀⣾⡿⠿⢿⣷⡀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠀⠀⠀⢀⣿⡿⠋⠉⠉⠙⢿⣿⡋⠀⠀⠁
⠀⠀⢿⣷⣤⣾⣿⡁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠉⠛⠙⢿⣿⣄⠀⠀⢀⣰⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⢿⣷⣄⣀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣿⡿⠿⠿⣿⣶⣄⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⣹⣿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠈⢹⣿⣿⠟⣁⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⣀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣰⣿⡿⠃⣼⣿⠟⢿⣿⣤⣤⣴⣿⠟⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠛⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣸⣿⣧⡀⠀⣠⣾⡿⠋⠀⠀⢻⣿⣦⣾⣿⠛⠛⠻⣿⣦⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⡿
⢀⣤⣶⣦⣤⣶⣿⠿⠛⠿⣿⣾⡿⠋⠀⠀⠀⣴⣿⠟⠉⠉⠁⠀⠀⠀⠈⠉⠉⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⣾⣿⠉⢻⣿⡏⠁⠀⢀⣴⣿⠟⠁⣠⣶⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣶⣦⡄⠀⠀⢀⣴⣶⣶⣄⠈⠻⣿⣧⡀
⠹⢿⣷⣿⠟⠁⢀⣴⣿⠟⠁⠀⢰⣿⡏⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⠉⣻⣿⣶⣶⣿⣿⠉⢹⣿⡆⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⠀⠀⠻⢿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠋⠀⠀⠙⢿⣿⣿⣿⡄⠀⠀⠀⠈⢻⣿⣦⡀
⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⢀⠀⠀⠀⠀⠀⠀⢀⡀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⡀⠀⠀⠙⢿⣷⣄
⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⣿⣷⣀⣀⣠⣾⡿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣷⡄⠀⠀⠙⢿⣷⡄
⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣼⣿⠿⠿⢿⣿⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣠⣿⡟⠀⠀⠀⠈⢻⣿
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠙⠛⠛⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠛⠛⠋⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣤⣄⡀⠀⠀⢀⣠⣤⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠛⢿⣷⣶⣶⣾⣿⠛⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣿⠟⠉⠉⠻⣿⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠻⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠏
⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⡠⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠕⠆
⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠒⠀⠀⠚⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠐⠀⠀⠣⣄⠀⠀⠐⠠⣾⣡⣀⡤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠤⠈⠀⢳⣦⡵⡈⠉⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠷
⠀⠀⠀⠀⠐⠀⠃⣀⠗⠁⢤⡍⡉⠋⠁⢀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⢰⣄⠐⠠⠄
⠀⠀⠀⠀⠀⣈⠋⠉⠢⣨⠂⠁⠁⠤⠐⠈⠑⠐⠀⠀⠀⠀⠀⠃⢀⡀⢀⣀⡤⠚⠙⠚⠀⢀⠃
⠀⠀⠀⠀⠀⠋⠀⠀⠈⠁⠇⠀⠀⢀⣴⠀⠐⠒⠀⠀⠀⠀⠀⠀⠈⠉⠙⠇⠀⠀⢀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠤⢤⡀⠘⡤⠠
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠁⠀⠀⠈⣿⣗⣼⠏⠉
⠀⠀⠀⠀⠀⠀⢀⢈⣄⠨⠇⠠⢏⡀⠕⠀⠀⠀⠀⠀⢠⠂⠀⠀⡀⠀⠖⢤⢐⡯⠁
⠀⠀⠀⠀⠀⠐⠏⠀⢙⠉⢢⡌⠈⢱⡄⠀⠀⠀⠀⠀⠈⠁⠀⠄⣠⠚⠁⠉
⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠴⠛⠦⠀⠀⠀⠀⠐⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿