Saving Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⣀⡀⡤⢚⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠠⢀⠀⠙⠻⠽⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠦⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢎⣲⣤⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⣥⡤⠤⣤⣤⣄⣀⣀⣠⡤⣄⣠⡴⠚⠋⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠥⠶⠒⠒⠒⠲⠦⠀⡼⠋⠁⠀⣿⡃⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣿⠙⢦⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⣀⠀⠀⣿⠀⠀⠙⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠇⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⣴⠛⢻⡆⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⣇
⠀⠀⠀⠀⠀⢤⡀⠘⢷⣾⣇⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠃⠀⢻⣄⡴⢶⡄
⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠡⠴⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠈⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⠄⢀⣤⢴⣾⣄⠀⢀⠢⠄⠠⢀⠀⡄⠠⠄⠠⢄⠀⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣬⣛⡿⠄⣴⡶⣿⠇⡏⠘⠁⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⡘⠄⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣾⣧⡤⢴⣛⣭⣾⠿⣦⣄⣀⡤⣀⣀⣼⣿⠷⣄⠀⠀⠀⠸⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣙⡾⠤⠾⣒⣿⣿⢳⣬⣓⠿⠤⠶⣚⣿⡏⠑⠮⢵⣷⣶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠭⠥⠔⠛⠁⠀⠙⠚⠧⠭⠥⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⠙⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⠀⠰⣶⣦⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣤⣤⡀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡋⠉⠛⠋⠀⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣆⣀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠛⠁⠀⢀⣀⠀⢀⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⣿⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣾⡿⠿⠿⠋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⢀⣴⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⢰⣿⠁⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠈⢿⣧⡀
⠀⠀⣀⠀⠀⣼⡟⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠹⣿⣶⣤
⣴⡿⠿⣿⣆⣿⣧⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⣿
⢻⣧⣤⣾⣿⢿⣿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢸⣿
⠀⠈⠉⢹⣿⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣿⡏
⠀⠀⠀⠈⠁⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⣀⣀⣼⣿⠿⠿⠿⠷⠿⠿⠿⠿⣿⣧⣀⣀⣠⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢦⡀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠑⣄⠀⠙⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢈⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⢠⠎⠀⠀⠀⠀⠀⠀⠀⣼⢃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠏⠀⠀⠀⠀⡠⢀⣠⢞⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣴⠋⠀⠀⣀⢤⡚⠋⢭⠀⣻
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠒⡞⠀⠁⢠⠞⣁⠞⠀⢀⡞⢠⠃
⠀⠀⠀⠀⠀⠀⢠⣏⣤⠀⠀⣀⣤⠤⠖⠒⡎⠀⠀⡞⠑⣆⣠⠛⠹⢁⠧⠤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣯⣀⡴⠊⠁⠀⠀⠀⠛⠷⠤⣄⡉⠀⣉⡯⠴⠶⠷⠦⣄⡤⢺
⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠁⢾
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠻⣤⠀⠀⠀⠀⣠⡄⠀⠰⢦⠘⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⣤⡙⠀⠀⠀⢰⡇⠀⠀⠀⠈⠀⠉⠀⠠⣤⠴⢧⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢠⡜⣿⢻⠀⠀⠀⠸⣇⠀⠀⠀⣠⠄⠀⠀⢰⠧⡄⣄⡇
⠀⠀⠀⠀⠀⠸⣧⠀⠀⢳⡀⠀⠹⣿⠉⠀⠀⠀⠀⠻⣆⡀⠀⠁⠙⠲⠄⠈⡻⠋⠁
⠀⠀⠀⠀⠀⠀⢿⡆⠀⡾⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⣤⠤⠖⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠙⠳⢶⣤⣤⣴⡀⠀⠀⠀⢠⠚⢻⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⢸⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢘⠈⠀⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⣰⡂⠠⠭⣿⠀⠀⠀⠀⠀⢤⠤⡠⠤⢄⠀⢠⣤⣤⣤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣓⣴⠬⢈⣽⣿⠀⠀⠀⢀⠃⠀⠈⠀⠉⠌⡆⠀⠀⠀⠈⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⡇⣷⣇⣐⣲⣺⠀⠐⠂⠁⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢹⢐⣶⡗⡟⠓⢠⣬⣷⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣔⣻⣗⢠⣽⣿⣟⠛⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⠉
⠀⠀⠀⠀⠀⢀⡤⠀⢀⠈⢱⠨⢭⣽⣗⠚⣿⣷⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣒⣠
⠀⠀⠀⠀⠀⠘⡂⠽⣿⠀⢠⡽⠛⠋⡳⠤⠭⠿⠛⠝⡄⠀⠀⠀⠀⠀⠀⠠⠂⢀⢀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠅⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠣⠯⠟⠃⠒⢗⣻⣟⡿⠒⠂⢀⠀⠲⢾⡀⠀⠐⠀⢠⠤⣤⣤⣉⣅⣀⣌⣉⣩⣽⣿⣿⣿⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢒⢂⣤⠀⠀⠃⠤⠶⠾⠂⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠈⠀⠀⠀⠀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⢀⠀⢀⣀⣀⡀⢀⠄⠀⠒⠒⠀⠀⠒⠖⠠⠀⠀⠀⠻⣰⣴⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⡀⠀⠀⠠⢘⣉⠓⠁⣀⠀⠤⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠶⠖⠒⠒⠂⠁⠀⠀⠀⠀⠐⡞⢩⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣖⢲⣄⣠⣤⣂⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡹⠁⠀⠀⠀⠀⠀⠩⣕⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⣤⢒⣀⠔⡰⢇⠘⡄⠰⡄⠀⠀⠀⠀⠙⢮⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠁⠈⣉⠩⠔⠊⠀⠈⢆⡘⢦⣱⡄⠀⠀⠀⠀⠈⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣰⡔⠉⣀⡀⠀⠀⠀⠀⠀⠉⠒⠘⡏⠀⠀⠀⠀⠀⠥⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡿⠀⣀⡄⢀⠀⠀⠀⠀⠀⠀⠐⠢⠱⣄⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡇⠀⣭⣶⣦⠑⡀⠀⠀⡠⣒⡒⢤⡀⢨⠃⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠢⠀⠀⢁⠈⢿⣿⣿⠗⡀⠀⢠⣿⣿⣿⡄⡏⠸⠀⢸⠀⡎⠀⢘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢄⣄⣆⡗⢤⠈⢆⠀⠀⠀⠀⠂⠀⣀⠙⠿⠟⠊⠀⠀⣡⢍⠺⠀⠀⢸⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⡈⣦⣬⣄⣈⡣⣀⢀⠀⠈⠈⠁⠀⠀⠀⠀⢀⡈⣋⠠⠜⠀⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠉⠀⠀⠊⣡⡤⠠⢤⣤⡄⠀⠂⠁⡀⠀⢠⣶⠞⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠋⣄⡀⠶⠀⠀⠀⠘⢫⣅⡤⠀⠀⠈⠠⡀⠀⠱⣤⠏⠀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆⣹⠆⠀⠀⠀⠀⢼⢤⡇⠀⡤⠁⠀⠘⡶⠈⠁⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠁⡀⠰⠈⠐⠆⣿⢴⡇⢀⡀⠀⠀⠀⣇⣱⠎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠰⢣⣔⣀⡠⠀⡀⠀⠀⠈⠙⠛⠲⠶⠟⠈⡔⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠝⡥⢈⣛⣃⣀⣀⣤⠀⡴⢄⣖⠼⢄⣼⣿⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡁⡈⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⢢⣻⡛⡉⢫⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣥⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠠⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⢂⡩⠑
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠰⢠⡤⠉⠭⠀⠀⠆⢲⠖⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠃⠀⠀⠀⢋⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢲⡀⠀⣀⠀⠀⠀⢰⡲⠄⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⠀⠀⢀⣩⠀⠀⢀⣾⠇⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠶⠴⠚⠉⠀⠀⠀⠐⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⠒⠉⣥⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠖⠒⠈⣉⡠⠤⠴⠒⠊⢻⠀⢹⣷⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⠊⣉⣀⣤⠴⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⢶⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠖⠒⢉⣁⡠⠤⡴⠖⠋⠉⠀⠀⠀⠉⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡄
⠀⠀⠀⠀⠀⣀⠠⠤⠖⠒⠊⣉⡠⠤⠴⠒⠂⠉⠁⠀⡰⠊⠀⠀⣀⣼⣿⣶⣦⡄⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⣇
⠀⠀⠀⠀⠀⢣⠀⠰⣆⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⢰⣿⠋⢿⣏⠙⠁⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠘⡄⠀⠚⢀⡜⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠸⣿⣦⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢇
⠀⠀⠀⠀⠀⠀⢱⠐⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⢿⣷⣦⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⢸⡤⡄
⠀⠀⠀⠀⠀⠀⠘⡇⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠈⣿⡇⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢰⠋⢠⠄⠀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⢱⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⣤⣄⣀⣻⣧⣴⣿⠃⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠵⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠙⠛⠿⠿⣿⡏⠁⠀⢀⠔⠁⠀⣀⡠⠤⠴⠒⠂⢉⣁⠤⠤⠒⡖⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⠀⠀⠀⣀⣀⠤⠶⠓⠒⢉⣉⣠⠤⠔⠒⠊⠉⠁⠀⠀⡠⠤⠇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠔⠒⠦⡀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠛⠋⣉⣀⣤⣤⡶⠒⠉⠁⣠⠃⠀⠀⠀⠀⠀⠀⠀⡜⠀⢴⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⢧⠀⢸⡀⠤⠴⠒⠒⢉⣁⢠⡤⠖⠒⠊⠛⠛⢻⣿⠉⠀⠀⢀⠔⠁⠀⠀⠀⠀⣀⣀⠠⠤⠵⠀⠀⠋⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⠀⠚⣀⡠⠤⠴⠒⠒⠉⠁⠀⠀⠉⠲⠤⢀⣀⣀⣀⣀⣠⡴⠾⠵⠒⠒⠒⠈⣉⣁⡠⠤⠤⠖⠒⠒⠂⢹⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠋⣏⠀⢹⡄⠀⠀⣀⣀⡠⠤⠤⠶⠒⠒⠊⢉⣉⣁⣠⣤⠴⠖⠒⠒⢈⠟⠉⠀⠀⠀⠀⠀⠀⠀⣠⠤⠼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠸⠃⠈⠋⣉⣉⡠⠤⠤⠶⠒⠒⠶⣉⠉⠁⠀⠙⠛⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢴⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠲⠖⠚⠛⠉⠀⠐⠂⠒⠀⠒⠐⠒⠒⠂⠛⠒⠒⠂⠒⠒⠚⠃⠐⠐⠐⠂⠒⠒⠒⠒⠒⠒⠂⠃⠀⠸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠓⠚⠚⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠓⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣡⣶⠾⢷⣦⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠰⣿⡇⢀⣼⠏⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⣿⣿⡄⠀⠀⠀⢸⠇⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⢩⣤⡄⠀⢀⡜⠀⠸⣄⣀⡽⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣀⣈⣉⣤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠘⢤⣤⠝⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⡆⠀⢀⣴⡾⠟⠛⠛⠛⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠀⠘⢢⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠰⠖⠂⠀⠀⠈⣿⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠐⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠃⡘⠀⢀⠤⠀⠐⠀⣀⣠⣀⡀⠀⠴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠂⠈⠿⠶⠤⢤⣀⠆⠰⠙⠆⠀⠀⠀⢀⠔⣛⣻⠉⢃⡀⠱⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⢄⣀⠈⠁⡆⠀⠁⠀⠀⠀⠀⢶⡿⡃⠂⠀⠀⢁⠀⠌
⠀⠀⠀⠀⢀⠀⠁⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⢉⠔⣲⡄⠉⠂⢤⡀⠀⠙⠲⣤⠀⠀⠀⠀⠀⠀⠀⠁⠤⣀⣠⠖⡣⠴⠂
⠀⠀⠈⠉⠀⠀⠀⢀⣀⡀⢀⣀⠀⠀⠈⡇⡴⠋⠀⠀⠀⢀⠔⢳⡮⠵⢡⣦⣤⣿⣂⠓⠤⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠄⠤⠤⠤⠤⠄⡜⠀⠀⠀⡟⠀⠀⠀⠀⡰⠃⡀⠀⠣⠀⣤⢵⢢⡞⠉⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠎⣡⣑⣒⡃⢔⠘⠷⠞⠛⠀⠀⢀⠐⠂⠀⠂⠀⠀⠀⣠⣶⣾⣶⣿⡇⢹⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⢏⢌⠀⣀⠠⠄⠀⠋⠀⢀⠠⠐⠈⢄⠠⠄⡀⠀⠀⠉⠁⠛⠿⠛⠋⠁⠀⠀⠻⣿⣿⡄
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠴⠡⠎⠐⠈⠀⠀⢀⠀⠰⠊⡤⠀⠀⠘⠤⠤⠂⠙⠀⠤⠄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠈⠁

⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄
⠀⠀⠀⢸⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡇
⠀⠀⠀⢸⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⣐⠂⡍⢱
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⣀⡠⠤⢐⣒⡨⠥⠐⠒⠈⠉⠈⠆⣈⡄⠆
⠀⠀⠀⢸⠀⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠀⣇⠶⠒⢉⣉⠉⠁⠒⢄⠀⠀⠀⠀⠀⠀⠸⢸⡄
⠀⠀⠀⢸⠀⣶⣶⣶⠀⢰⣶⣶⣦⠀⣶⣶⣶⡆⢠⣶⣶⣶⠀⡏⠀⣴⠟⣿⠛⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢃⠇
⠀⠀⠀⢸⠀⣿⣿⣿⠀⢸⣿⣿⡿⠀⣿⣿⣿⡇⠸⣿⣿⣿⠀⡇⠀⠙⢷⣿⣦⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠸⡸⣀
⠀⠀⠀⢸⠀⣶⣶⣶⠀⢰⣶⣶⣦⠀⣴⣶⣶⡄⢠⣶⣶⣶⠀⡇⠀⠀⠀⠘⣿⠙⣿⠀⠀⡄⠀⠀⠀⠀⠐⠩⠀⢻
⠀⠀⠀⢸⠀⣿⣿⣿⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇⢸⣿⣿⠿⠤⠧⠄⡀⠻⣶⡿⠿⠯⠤⣔⠀⢀⡀⠤⠄⢚⡀⣥⢼
⠀⠀⠀⢸⠀⣤⣤⣤⠀⢠⣤⣤⣤⠀⣤⣤⣤⡄⡴⢉⠔⠒⢀⠀⠒⢌⡙⢥⠒⠂⡀⠐⠢⡉⠲⡂⠈⠉⠀⡄⡚⢸
⠀⠀⠀⢸⠀⣿⣿⣿⠀⢸⣿⣿⣿⠀⣿⣿⣿⡏⢰⠁⠀⢴⢻⠛⠀⠀⠱⡈⣦⡞⡟⠃⠀⠈⢆⢱⣀⡀⠬⠄⠑⢈⡄
⠀⠀⠀⢸⠀⣤⣤⣤⠀⢠⣤⣤⣄⠀⣤⣤⣤⡅⡆⠀⠀⠈⢹⠦⡄⠀⠀⡇⡀⠉⡷⢤⠀⠀⢸⢈⠗⠂⠉⠉⢉⢹
⠀⠀⠀⢸⠀⣿⣿⣿⠀⢸⣿⣿⣿⠀⣿⣿⣿⣧⠸⡀⠀⠢⣼⡤⠏⠀⡰⢁⠗⢤⣯⠼⠁⢀⠎⡸⠀⠀⢠⠋⠄⢸
⠀⠀⠀⠘⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣱⣌⠒⠤⠤⠤⠔⢊⡤⡛⠢⠤⠤⠤⠒⣡⠼⠭⠭⠭⠭⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠈⠁⠀⠈⠉⠒⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠠⢀⣿⣶⡈⠂⢄⡀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠹⣿⣧⠀⢸⡏⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠁⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣦⣄⣀⣀⣤⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠚⠛⠛⠛⠛⠛⠛⠻⠶⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⡶⠞⠛⣛⠛⠷⢦⣄⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣠⡞⠁⠀⢀⡤⣿⣤⡀⠀⠙⢷⡄⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡟⠀⠀⠀⢿⡀⣿⠀⠁⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⣿⢦⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⣧⠀⠀⢀⡀⠀⣿⠀⣹⠆⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠙⠲⣿⠞⠋⠀⣠⡾⠃⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢙⡷⠶⠤⠭⠴⠶⠾⠋⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢧⣀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠠⢂⡀⠀⠀⠀⠀⠙⠀⠀⠉⠛⠛⠂⠀⠐⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠃⠒⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⣴⣶⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⠝⠭⢍⣤⣵
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡐⠒⠰⢖⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠚⠦⢉⣙⣏⣿⠀⠀⠀⠀⠀⣔⠝⠛⡓⠊⢍⠛⡄⠀⠻⠿⠿⠿⠏⠒⠀⠠⢄⡀⠀⠀⠀⠀⠠⠄
⠀⠀⠤⠀⠀⠀⠀⠀⠀⢹⠽⣾⡗⠠⠴⠮⢾⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⢕⣗⣯⡁⠒⢲⢻⡂⠀⠒⠈⠀⠀⠈⠉⠄⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢘⡚⢤⣼⣔⡶⠶⠤⣩⣭⣷⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠸⡬⣽⣻⣍⣑⣲⣲⣟⣶⣿⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣰⡧⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠓⠶⠾⡗⠨⠽⠵⣿⡚⠠⣤⡵⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠠⠂⠈⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢴⢝⣃⣁⡄⠀⢮⣐⣓⣟⣿⠤⢩⣍⣭⣿⣘⡆⠉⠀⠀⠀⠀⠀⠀⠀⡀⡠⢈⠔⡁⠊⠄⡁⠠⢂⢈⠈⠌⠀⠀⠀⡟⠩⡤⢴⡀
⠀⠀⠀⠀⢸⡒⠿⢷⡇⠀⢠⣺⠛⠉⢊⡑⠤⠶⠶⠞⠥⠟⣄⠀⠀⠀⠀⠀⠄⠠⣊⠔⡡⢊⠔⡠⠊⢀⠄⡁⡀⢪⠀⠀⠀⠀⠫⢬⢽⡿⠆
⠀⠀⠀⠀⠘⠦⠭⡽⠿⠡⠸⡂⢼⡽⢿⣷⠶⠄⠠⡀⢀⢧⣼⠀⠀⠠⠀⠀⢀⣤⣀⣈⣀⣁⣠⢤⢤⡀⣉⢚⡡⠞⠀⢔⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⡈⠉⠉⠹⣗⠀⣤⣤⣧⠀⠁⠈⠭⠒⠶⠒⠲⠚⠀⠓⠅⠂⠔⠂⠂⠁⠊⠉⠉⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠻⠭⣧⡽⠿⠄⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⢤⣀⡀⠀⠀⠀⠀⠀⣆⡀⠉⡐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢀⠤⠤⠴⠖⠐⠂⣩⣐⠍⡡⠄⠀⠀⠉⡎⣁⡐⠀⠀⠀⠑⠢⠽⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⣄⣀⠀⠀⠐⢄⣁⠈⠡⢀⣀⠢⠦⠗⠚⠉⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠲⢯⠴⠶⠺⠒⠒⠋⠁⠀⠀⠀⠀⠀⠘⡷⠋⣁⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠅⣿⣯⡓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣶⠶⠶⠿⠿⠿⠿⠶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠛⠉⠉⠉⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⠉⠉⠙⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠁⠀⠀⠀⠀⠀⢠⣿⣋⣉⣉⣉⣉⣭⣭⣭⣭⣭⣭⣽⡇⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠒⠦⢍⣉⡛⠳⠶⠦⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⡤⠶⠶⠛⣛⣉⡥⠔⠊⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠚⠋⠉⠁⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡠⠞⠳⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠙⣄⢀⡨⠏⠙⠛⠳⠶⠶⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⡴⠋⠙⢦⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⢣⣀⡴⠋⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠛⢳⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠰⣦⠖⣠⣤⠀⣤⢀⣀⢠⣤⡀⢸⠆⠀⠀⣿⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣸⣇⡀⠀⠛⠀⠀⠃⠀⠛⠛⠇⠸⠟⠁⢻⡿⠋⢸⠁⠀⠀⣿⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢴⡁⠀⣱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠑⠊⠙⠛⠶⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠈⢳⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠙⠒⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣌⡙⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣡⣾⠟
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣌⡙⠻⠷⣶⣤⣬⣍⣙⣓⠒⠒⠒⠲⠶⠒⣒⣒⣚⣩⣥⣴⡾⠿⠋⠁⠀⣠⣤⣤⢤⣤⢤⣤⡀
⠀⠀⠀⠀⢰⡿⣉⣁⣳⣚⣓⣀⣩⣿⡆⠀⠀⠈⠉⣙⣛⣻⣿⣿⣛⡛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⢺⣯⣤⣴⣾⣁⡤⣼⣿⠂
⠀⠀⠀⠀⠘⠛⠿⠶⠷⠶⠷⠾⠛⢋⡶⠛⠛⢛⣿⠟⠛⠛⢯⡽⠏⠙⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣽⣿⣤⡤⣴⣾⣿⣿⣽⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠶⠶⠶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢾⡟⠉⢹⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣈⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣤⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⠙⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⡿⠀⢿⣄⡀
⠀⠀⠀⠀⠀⢀⣤⡶⢛⣡⡄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣐⠢⠉⠛⢷⣤⡀
⠀⠀⠀⣠⡾⠛⠡⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣋⠙⠻⢦⣄⠀⠈⢻⣆
⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠙⢷⣄⠀⠙⣧
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠶⠶⢶⣦⣤⣤⣾⣡⣤⣄⣈⡙⠳⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣹⣧⠀⢸⣇
⢸⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⡴⠶⠤⣤⣤⣄⣉⣉⠉⠉⣽⠛⠶⣤⣈⠉⠛⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢁⠀⠀⣿
⣾⠃⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢃⡼⠁⣀⣤⣤⣄⡀⠀⢹⠇⢸⡏⠀⠀⣠⠟⢁⣴⠟⠛⠛⠻⠿⠛⠛⣿⠂⠀⠀⠀⠀⣾⡛⣿⠀⢻
⣿⠀⡏⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣼⠃⡼⢡⠟⡁⠀⣀⡈⠹⣆⡿⢠⡿⠁⢀⡾⠁⣠⡾⠋⠛⠛⠓⣶⠀⠀⣼⠏⠀⠀⠀⠀⠀⡈⠛⠋⠀⢸
⣿⠀⠀⠀⠀⠀⣰⡟⠻⣦⣄⣀⣀⣼⣇⣾⣁⣿⣸⣧⣾⣡⣿⣴⣿⣥⣿⣥⣴⣯⣤⣾⣯⣤⣤⣤⣤⣴⣯⣄⣼⣏⣀⣀⣀⣀⣴⠾⠻⣦⡀⠀⢸
⣿⠀⠀⠀⠀⠀⠙⢷⣄⣼⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢹⣟⢁⣠⣶⠟⠁⠀⢸
⣿⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⣿⠀⠀⠀⢸
⣿⠀⠀⠀⢀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⣿⠀⠀⠀⣾⠋⠙⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢙⣿⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⣿⠀⠀⢰⡟⠀⡼⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⣿⠀⠀⢸⡇⢀⡇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢸
⣿⠀⠀⣿⠃⢸⡃⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⣿⠀⠀⣿⣀⠈⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣾
⣿⠀⠀⠉⠙⢛⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⣿
⣿⠀⠀⢀⣴⠟⢁⣼⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣌⠻⣦⣀⣿
⣿⡆⣴⣟⡁⣴⡟⠁⢀⣽⣧⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡟⠉⠻⣿⣷⢈⣿⡇
⢸⡇⠀⠉⠛⠻⢿⣶⣿⣉⡉⠛⠾⠽⠏⠉⣿⠋⠉⠉⠉⢹⡟⣛⣋⣭⡽⠿⠋⢉⣿⠋⠉⣉⣉⣉⣉⣩⣍⠉⠛⢿⣍⠙⢿⣻⣶⣴⠟⣡⡿⢹⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣶⣤⣤⣾⠃⣀⣀⣀⣀⣾⡋⣉⣥⣤⣶⣾⣿⣿⡯⢰⣟⣿⣽⣯⣭⠿⢿⣇⣤⣤⣽⣷⣄⠉⠛⢁⣼⠏⠀⢸⡇
⢸⡇⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⣶⠮⠙⣿⣿⡿⢟⠛⠛⢿⡿⠿⠛⢻⣿⣿⣿⡿⢶⡶⢿⣤⡿⣶⠿⣶⣟⣋⣽⣆⠀⣹⣿⣷⣴⠟⠁⠀⠀⣿⠃
⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣆⣻⣿⠞⣛⠿⣿⡀⠸⣇⠀⢸⡿⠶⢶⡿⢿⡿⡿⢻⡟⢻⣿⣿⢻⣏⣿⡟⠛⠉⢹⣿⣿⣫⡿⠁⠀⠀⠀⠀⠀⣿
⠈⣿⠀⠀⠀⠀⠀⠀⠘⠷⣭⣿⡿⣧⣙⠳⠎⢻⣄⣛⣠⡿⠛⠛⠛⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⡏⠉⣉⣻⣶⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⣆⠀⠀⠀⢀⡀⠀⠀⣀⣉⡛⢾⣿⣿⣿⡿⠋⠉⠉⠀⠀⠀⠀⣿⣽⣯⣽⣭⣿⣽⠿⠶⠿⢿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⡏
⠀⠀⠙⢷⣄⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠷⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠶⠿⠿⠿⠛⠛⠛⠋⠉⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠙⠻⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡫⠭⢭⡒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⠋⡴⠋⣣⠈⢦⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⣇⣼⠂⣹⠀⡸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡛⢯⣙⡥⢞⡵⠃⢀⡠⠔⢒⣒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠚⠉⠀⢠⢏⡴⠋⣩⢮⡉⠳⣌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡄⠘⣇⠰⡟⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⣵⣔⣏⣀⡟⢀⡼⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⡻⠦⠥⠖⣋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⣛⠛⠓⠦⣍⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠴⠚⢉⣉⠙⠓⢦⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣰⠋⠀⢠⠋⠉⠳⡄⠀⠙⡆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇⡦⠀⠘⣄⠀⢿⡁⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢣⢰⡀⢸⡉⠀⢀⠇⠀⢀⡏⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠳⣝⡂⠙⠒⠋⢀⡠⠎⢠⣞⣀⣀⠀⠀⠀⠀⠀⠀⣀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠶⢄⣉⠉⠉⠉⣉⣠⠖⠉⠀⠀⠈⠉⠉⠓⢲⡴⠋⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⣀⡬⠽⠛⠛⠉⠉⠉⠉⠉⠓⠒⠀⠀⣰⠋⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢸⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⡄⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⢳⣀⣠⠖⠉⠑⢦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠈⠁⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠴⠋⠀⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⡴⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢸⠋⠛⠛⠛⠛⠛⠛⢻⡉⠁⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣠⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢐⠀⠀⣀⡀⠀⠀⠀⠠⡀⠀⠀⣀⣀⣀⡀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠗⠚⠉⠉⠀⠈⠑⠢⠤⢀⡀⠉⠒⠼⣿⠊⠉⠲⣄⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢶⢺⡀⠀⣠⡹⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠋⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⢶⢶⣶⣶⣶⣶⣶⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠎⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣉⣠⠍⠔⣒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠀⠀⠀⠀⣷⡉⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡠⠄⠀⠤⡀⠀⠈⣶⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠜⠀⠀⠀⠀⠈⠂⢰⠇⠀⢀⣄⠦⣤⡀⣾⠋⠁⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢂⠀⠀⠀⠀⠀⢄⡟⠀⠀⡶⡁⠀⢘⣾⣿⣦⣤⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠀⠀⠑⠠⠤⠐⠊⣼⠁⢰⠓⠙⠫⣶⣟⠁⠀⣹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢐⠇⠀⠀⠀⠀⣀⣰⠟⠠⡯⠛⠱⢦⡝⠛⢷⣮⠛⠳⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈⠵⢂⡠⠻⢑⡌⢀⢸⣇⡀⣠⣿⡄⣀⣠⣿⢀⡀⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁⠈⠑⠙⠛⠋⠀⠙⠛⠃⠀⠛⠛⠐⠁

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

⣿⣿⣿⣿⣿⣿⣿⣿⣶⠒⠒⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡧⠤⠤⠤⠤⠷⠶⢤⣀⠀⠀⠀⠀⠀⠈⢢⡤⣶⠖⢒⣒⣲⣄⡀⠀⠀⠑⢤⡀
⣿⣿⣿⣿⣿⠿⠿⠿⠛⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠉⢧⣴⠛⠉⠉⠙⢿⡶⢄⠀⠀⠙⢄
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⡀⠙⡆⠀⠀⠀⢈⡷⢸⠛⡦⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⠻⣧⣀⣀⣠⠾⢃⡼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣦⣄⣉⣉⣁⣴⣎⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣒⣒⣒⣒⣒⣒⣂⣀⣀⣀⣀⣀⣀⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠴⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⢡⡶⠛⠛⠻⢦⡈⢣⠀⠀⠀⣀⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠈⣿⠈⡇⣠⠊⣡⡶⠶⠶⣦⡉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢻⣄⠀⠀⠀⣠⡟⢠⠁⡁⣸⠋⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢦⡉⠛⠛⠛⣉⡴⠋⠀⡇⢹⣄⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠙⢄⠙⠷⠶⠶⠟⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⠴⠒⠲⠦⣄⠀⠀⠀⣀⠽⠖⠒⠲⠮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠋⣴⠾⠛⠛⢶⣄⠱⣄⡜⢁⡴⠞⠛⠳⢦⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢹⡆⢹⠀⣿⠀⠀⠀⠀⠈⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⠸⣧⡀⠀⠀⢀⣾⠃⡾⡀⢻⣄⠀⠀⠀⣠⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠢⣈⠛⠓⠚⢋⣡⠞⠀⠙⢤⡙⠛⠖⠛⢉⡴⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣿⣷⣶⣶
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠱⡸⠀⢀⠄⠀⠀⢤⠀⠈⢀⠎⠈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢉⠄⠒⠉⠒⢢⠹⡐⠁⠀⠀⠀⠀⢸⠀⠏⠔⠀⠁⠁⠢⠌⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⡌⠀⠀⠀⠀⠀⢣⠹⠀⠀⠀⠀⠀⢸⠀⠎⠀⠀⠀⠀⠀⢰⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠱⠀⠀⢀⠠⠤⢄⢃⠱⣊⣁⠈⡲⢸⠀⠄⠒⠠⡀⠀⢀⢂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠡⠀⢁⠤⢂⢑⢟⢆⢣⠠⠦⢜⢽⠀⡁⣓⠒⡌⢀⢂⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠱⢞⡒⠊⣁⠌⠈⣂⣳⢒⣈⣸⠀⡄⢈⣢⢽⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢣⠨⠙⠉⢉⡀⠤⠀⠀⠠⠄⠈⡱⠋⠀⢸⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠠⣀⠔⠢⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠠⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡏⠉⠉⠉⠉⡅⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢈⣿⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀
⠀⠀⢀⣿⡹⣯⡀⠀⠀⠀⠀⠀⣤⣤⣼⣷⣿⣂⠀⠀⠀⠀⠀⣿⣴⣂⣀⡦⡄⠀⠀⠀⣴⣿⣿⠿⢻⣿⠶⠀⠀⠀⠀⢰⣿⣿⣿⢿⡀
⠀⠀⠘⠿⠶⠿⠃⠀⠀⠀⠀⠀⠛⠛⠷⠶⠋⠁⠀⠀⠀⠀⠤⠿⠿⠽⠿⠧⠤⠀⠀⠀⠀⠙⠿⠾⠿⠋⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠁

⠀⠀⢠⣶⣶⣶⡀⠀⠀⠀⠀⠀⢀⣠⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡄⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖
⠀⠀⢸⣿⣿⣷⢲⡄⠀⠀⠀⠀⠾⢀⣞⠹⣿⠓⠀⠀⠀⠰⣟⣿⠀⠀⢸⣏⡷⠀⠀⠀⠀⣶⣒⣚⣛⣓⡂⠀⠀⠀⠀⠀⣰⣟⢽⣦
⠀⠀⠈⠙⠛⠙⠚⠁⠀⠀⠀⠀⠦⠌⠏⠠⠟⠁⠀⠀⠀⠀⠈⠁⠳⠶⠂⠉⠀⠀⠀⠀⠀⠀⠻⠿⠿⠯⠇⠀⠀⠀⠀⠀⠙⠿⠾⠋

⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠀⠠⡀
⠀⠀⢾⣿⣾⣿⡧⠀⠀⠀⠀⠀⢰⣿⣿⣤⣤⠀⠀⠀⠀⠀⢀⡴⣤⡐⢀⠇⠀⠀⠀⠀⠀⣿⣿⣽⣭⣿⡛⠋⠀⠀⠀⢰⠀⣺⠀⢨
⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⠘⠯⠿⠃⠁⠀⠀⠀⠀⠘⠛⠛⠒⠚⠓⠛⠁⠀⠀⠀⠀⠀⠂⠤⠄⠙⠁

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⢀⣄⢤
⠀⠈⠳⠤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⢤⡄⠀⠀⠐⡤⡀⢀⠀⠉⠁⠀⠀⠀⠀⠁⡄⠀⠀⠆⢀⠀⣯⡼⠏
⠀⠀⠀⠀⠀⠀⠀⢐⣦⣄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⡏⠀⠀⠶⠛⠀⠀⢀⡴⢮⣿⣤⠀⠈
⠀⠠⠚⡠⠟⠀⠖⡾⠋⠁⠉⠑⠠⣀⠀⠀⠀⢒⠢⢈⣻⠋⠀⠀⠀⠀⡀⠄⠊⠁⠀⠀⠀⠙⢏⡺⣄⠸⣯⡀
⠀⠑⠚⠁⠀⢀⠋⠀⠀⠀⣿⣶⣄⠀⠑⠄⠀⠉⠀⠞⠩⣠⠄⠀⠀⠄⢀⣴⡶⢶⠆⠀⠀⠀⠈⠙⢅⠀⠈⠁
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⣿⣿⣤⡀⠀⠀⠐⡀⠀⠀⠚⠁⠀⠀⢀⣲⠋⣞⣲⣶⠂⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀⠹⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠀⠈⢾⣓⣒⣲⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠛⠙⣿⡞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠳⠶⠆⠀⠀⠀⠑⠤⢀⡀⢀⣀⠀⠀⠀⠠⠖⠉
⠀⠀⠀⠀⠀⠀⠈⢁⠈⠒⠂⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠃
⠀⠀⢰⡇⡶⠄⠀⠙⠧⠀⠀⠀⠀⠀⢠⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠰⣄⠀⢀⣤⢄⣀⣀⡀
⠀⠀⠀⠙⢤⣤⡀⠀⠀⠀⠀⠀⠁⣤⠾⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⢹⡄⠀⠻⢆⠰⠖⠚⠁
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢠⠞⠁⢠⡀⠀⠀⠠⠖⣸⠆⡇⠀⠀⢸⠀⠀⠀⠈⠓⠦⣄⡉⠉⠀⠀⢷⡂
⠀⠀⠀⠻⠖⠀⠀⣠⠞⠀⠴⠃⠀⠀⢸⠀⢠⠀⠙⠋⠀⠀⡇⠀⠀⠸⡄⠀⠀⣦⠀⠀⠀⠉⠲⣄⡀⡀
⠀⠀⠀⠀⠀⢀⡴⢃⡀⠀⠀⣷⠀⠀⢸⠀⢘⠀⢠⣤⠀⠀⡇⠀⠀⢨⡅⠀⠀⡧⠀⠀⢀⠀⠄⠈⠉⢫
⠀⠀⠀⠀⡰⠋⠀⠀⡇⠀⠀⣿⠀⠀⣸⠀⢸⠀⠈⠳⠂⠀⡇⢠⠀⢘⡃⡀⠀⡏⠀⠀⠀⡅⠐⠀⠀⠀⢧
⠀⠀⠠⠎⠀⢲⠀⣀⣇⣀⣤⣿⣠⣤⣿⣄⣼⠀⠀⠀⠀⠠⠧⠼⠶⠾⠦⠷⠴⠷⠤⠦⢤⣇⣀⠀⠀⠀⠈
⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⡀⠀⠀⠀⠀⠴⣄⣀⠀⠀⠀⠀⡀⠀⡰⣤⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢉⣇⣠⣤⡄⢀⣀⢰⠞⠀⠀⠺⢃⡇⠀⢠⠫⠁⠀⠀⣠⠤⠷
⠀⠀⠀⠀⢀⡱⠖⠁⣀⡟⣹⣖⣢⡾⣿⠃⢨⡿⠉⠻⠀⠀⠀⠛⠃⠀⠈⠈⠉⠀⡴⠕⠀⣀⠐⣄⠀⠀⢀
⠀⢁⢀⣑⣺⠁⠀⣿⣛⠐⠂⣿⠁⠀⠋⣡⡴⠆⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠆⠸⠍⠀⣿⠀⠀⠠⣄
⠀⠘⣠⠄⢸⢇⣀⢿⢟⢀⢀⡹⠖⢋⠄⠉⠁⠈⠁⠀⣠⡨⠤⠀⠠⣀⣀⣈⠀⠀⠀⠈⠂⠶⠚⠁⢀⢔⠀⢀⠠
⠀⠈⠀⠀⠰⠺⠿⠞⠟⣄⣀⣉⣼⣀⠀⠄⠂⠀⠠⡎⣸⠻⢌⣪⡝⢤⠛⠳⠄⢀⡀⠂⠀⠀⠐⠊⠝⠄⣶⠀⠁
⠀⠀⠈⠓⢢⣬⣓⠆⢀⣠⠬⠋⠋⠁⠀⠀⠀⠁⠀⢙⡉⣹⢞⡿⠧⠾⠒⠁⠀⠉⠁⠀⠀⠀⠈⡁⠘⠁⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠡⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⠛⠛⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠁⠀⠀⠀⠉⠻⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠰⠶⢶⣤⣀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠙⠳⣄⠀⣸⠏⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠀⠹⣧⣿⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠉⠈⠀⠉⠉⠈⠉⠉⠈⠁⠁⠉⠀⠈⠈⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠋⠁⠀⠀⠈⠉⠻⠇⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⢠⣤⣤⣼⣿⣿⣥⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠈⠛⢻⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⣀⣸⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣶⣶⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣐⣳⣧⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⢉⡱⠀⠀⠀⠀⡀⠀⠦⠄⠠⢒⣼⠒⢐⢚⣺⠀⣻⡃
⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠈⢉⠥⠤⠄⠒⣒⢲⡂⢎⢫⣻⡦⠭⢯⠟⡖⠋⠉⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⡐⠀⡄⠀⠂⠀⣀⠀⠀⠀⠀⠓⠂⠋⠉⠉⠉⠁⠹⠀⠚⠚⠁⠀⠀⠖⠀⠀⠐⠨⢍⣀⠀⠀⢄
⠀⠀⠀⠤⢶⡤⠀⠀⠀⣠⠐⣲⣾⡞⠨⢵⣶⠅⠐⣖⣈⢥⡀⠶⢁⠁⠀⠀⢀⠀⠀⠀⢈⡐⢁⡢⣀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣈⣤⠴⠧
⠀⠀⠑⠤⡐⠀⠐⠀⢄⠙⠒⢋⣜⣿⣀⢘⣻⠤⢀⣰⣏⡿⣥⡀⠀⠁⢂⡐⠀⠌⣃⢠⠐⠀⠦⢄⠠⠑⠤⣏⣀⣈⣟⣶⡞⠉
⠀⠀⠀⠀⠈⠁⠠⡀⠀⢢⠄⠀⣠⠤⢉⣉⠩⢅⠀⠒⠾⣟⠉⠤⠴⠂⢀⡌⠀⢠⠈⠃⠄⢀⣀⡀⣰⡄⠴⠄⠵⢤⠶⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⢡⣒⡶⠃⠀⠐⠁⠄⢬⡼⠿⣾⠷⣀⠀⠀⢉⡁⠀⡀⢀⣀⡀⣀⣠⣠⣜⣠⠒⠡⢶⡄⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡛⠉⠁⠀⠈⣿⣦⠟⠋⠛⠛⠋⠉⠀⠉⠁⠈⠀⠁⠀⠀⠌⠠⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣢⣀⡰⠆⠴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠴⠖⠚⠛⠉⠉⠁

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣄
⠀⠀⠷⣅⣡⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠀⠀⠠⣆⡀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢄⡀⠀⠀⡾⢋⣷⠍
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢄⡀⠀⠀⡄⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⡀⢡⠆⣴⠇⠀⢀⣠⡄⠀⠀⠀⠀⣈⣦⣤⡀⠉⠀⠆⠀⠙⠋⠁
⠀⠀⠀⣀⠀⣀⠀⠐⠀⣠⣿⡿⢐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢺⣶⠟⠀⠀⠛⠉⠀⠀⢀⡠⠜⠛⠻⠿⣿⣁⠠⣀⠀⢀⡄⠄
⠀⢀⠊⢡⠄⠟⠀⣘⢛⠋⠁⠀⠀⠀⠈⠑⢄⡀⠀⠀⠈⢢⣙⠆⢀⣝⠁⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠈⠫⣘⡛⡆⠈⢷⣼
⠀⠘⠒⠋⠀⠀⠀⠈⠁⠀⠀⠀⢘⢿⣦⣄⠀⠈⠲⡀⠀⠉⠀⢑⠞⢩⣱⡄⠀⠀⡔⠁⣠⡴⣶⣶⡶⠀⠀⠀⠀⠁⠉⠻⡀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣞⣿⣿⣉⠀⠀⠀⠀⢤⡀⠀⠀⠀⠶⠋⠀⠀⠄⣤⡴⢷⠾⡤⢤⡤⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⢓⡄⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠐⠓⣇⣼⣁⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⣸⣷⢿⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢣⡀⠀⠈⠻⠽⠥⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⠏⠀⠀⠀⠀⡘⠀⢀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠑⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠘⠗⠶⡦⠀⠀⠀⠙⠢⢄⠀⠀⠀⢀⡀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠲⠶⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢴⣾⠻⠆
⠀⠀⠀⢠⡤⢀⡀⠀⠀⣶⡆⠀⠀⠀⠀⠀⡀⠀⣸⠇⠀⡆⠀⠀⠀⠀⠀⠃⠀⠘⠓⠂⠀⠀⠀⣀
⠀⠀⠀⠸⡤⠊⠁⠄⠀⠀⠙⠁⠀⠀⠰⡎⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠈⢷⠀⠀⢸⣏⠑⠶⢦⡒⣤⠄
⠀⠀⠀⠀⠈⠠⠵⢦⠀⠀⠀⠀⠀⠀⠀⢀⡶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠂⢸⡇⠀⠈⠛⢦⣊⡋
⠀⠀⠀⠀⡀⠀⠀⣸⠃⢀⠀⠀⠀⠀⣠⠟⠀⠀⣤⠀⢰⠀⢀⡴⠆⣼⠆⣄⠀⠀⠀⢸⠀⠀⠀⠠⠀⠙⠳⢤⣄⡈⠉⠁⠀⠈⢿⡆
⠀⠀⠀⠀⢯⡴⠀⠁⠀⢀⡶⠂⢀⡼⠃⠀⠀⠀⢸⠀⢸⠀⠘⠞⠋⠁⠀⣿⠀⠀⠀⢸⠄⠀⠀⠀⣄⠀⠀⠀⠈⠙⠢⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⡄⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠻⠀⠀⠀⠁⠄⠀⠈⠻⠶⡶
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠒⣦⠀⠀⠀⡿⠀⠀⢀⡛⠀⢸⠀⠐⣯⠓⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⠀⢰⠄⠁⠆⠀⠀⠘⣆
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣿⠀⠀⢠⡇⠀⠀⢸⡇⠀⢸⠀⠀⠀⠉⠀⠀⣿⠀⡦⠀⢨⡄⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠂⠀⠀⠛⡄
⠀⠀⠀⠰⠋⠀⢐⡆⢀⣀⣧⣀⣴⣼⣯⣠⣤⣼⣧⣤⣼⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠾⠷⠾⠷⠶⠶⠦⠴⢦⣼⣦⡀
⠀⠀⠀⠐⠀⠈⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⡠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠚⢻⡇⠀⣀⣀⠀⠀⠄⠉⣨⡿⠀⠀⢀⣶⠀⡷⠀⠈⢠⠖⠆⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠤⡀⢀⡤⠀⢠⡤⢀⢀⡀⢀⣸⣿⠀⠸⣷⡷⣿⣾⠃⠀⠚⢀⣿⡄⠀⠀⢀⣡⠔⠁⠀⣰⠋⠊⣀⡀⠀⠑⠛⠉⠉
⠀⣀⠀⡀⠀⢰⣦⠛⠁⣠⣄⡫⠉⠙⣮⡵⠿⢻⣧⡀⢸⠟⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠜⠁⠁⢀⣤⡀⢳⡄⠀⠀⠀⠄
⠀⠨⡈⣅⣙⣹⡇⠀⠈⣿⣋⡉⠉⢙⡿⠀⢀⠚⢀⣽⡗⠒⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠸⠀⠻⠅⠁⣸⡄⠀⠀⠀⣦⠄
⠀⠀⠃⣀⡀⢸⣗⣀⣀⡿⠶⠂⡀⠀⢯⠔⠋⣀⠈⠙⠁⠊⠉⠀⠀⡀⣈⣄⡀⠠⢀⠀⠀⠀⡉⠀⠀⠀⠀⠑⠠⠤⠾⠊⠀⢀⡜⠁⠁⠀⡄
⠀⠀⠒⠑⠀⠀⣤⣽⣇⣻⣮⠄⠉⠀⠂⢀⠊⠁⠀⣀⠠⠄⢀⣤⡾⣹⣯⡒⠠⡞⢟⣉⠹⣾⣄⠀⠀⠀⢠⡒⠄⠀⢀⡤⠒⢷⡈⣤⠀⡙
⠀⠀⡐⢠⣄⠈⠠⣌⣉⠈⠉⠄⠰⠶⣚⠯⠗⠊⠉⠀⢀⢀⠉⠙⣕⠾⢬⣉⣿⣿⠂⢸⣇⣤⠉⠁⡐⠶⠀⠀⠀⠀⢠⠤⠈⢴⠇⠘⠃
⠀⠀⠀⠀⠈⠘⠳⠮⢅⡀⢠⠴⠛⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣴⣭⣽⣛⣛⣋⣉⠀⣀⠀⠀⠀⠀⠀⠀⠀⢘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠒⠐⠒
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣥⣬⣧⣴⣤⣵⣬⣼⣯⣵⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣦⠀⢀⠀⠀⠀⠀⠀⠀⢴⠆⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⣀
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢺⡷⣾⣿⣷⢾⡇⠀⠀⠀⠀⠀⣿⣇⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡤⣾⡖
⢀⣴⡆⢸⠀⣶⣤⠀⠀⠀⠀⠀⠘⠇⢻⣿⡟⠸⠃⠀⠀⠀⠀⠀⣿⡏⠉⠋⢹⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿⡇
⣾⣿⣿⢼⣧⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⣿⡇
⠛⠛⠛⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⡹⠃⠀⣤
⠀⣤⣴⣯⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠆⠀⠀⠀⠀⠀⠀⠀⡤⠒⠊⠀⢤⣄⣠⣿⡆
⠐⢷⣿⠟⠳⣤⣀⠀⠀⠀⠀⠀⠸⣿⣿⠳⢤⣀⡀⠀⠀⠀⠀⢠⣾⣿⣤⣤⣀⣘⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⢹⣿⡇
⠀⣼⠛⣧⠸⣿⠿⠇⠀⠀⠀⠀⢸⡿⢿⡄⣿⡿⠷⠀⠀⠀⠀⢘⣿⣷⢾⡏⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠸⠏⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄⠿⢠⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇


⠀⢀⣠⡤⣤⣀
⣰⠛⡿⠤⠼⢿⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⢀⣀⣠⣆⣰⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⡷⢺⡗⠒⠒⢾⠒⡇⠀⠀⠀⠐⣶⣾⡏⢯⢹⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣤⣤⣴⣿⣿⣿⣿⣶⡆
⠹⣶⣷⠒⢒⣿⡴⠁⠀⠀⠀⠀⢿⠿⠷⣖⣫⡭⠿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐⠿⠿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁

⠀⢀⡠⠄⢤⣀⠀⠀⠀⠀⠴⠖⠒⠀⠀⠀⠀⢀⣴⡾⣆⠀⠀⠀⠀⢠⢲⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⣧
⠀⢻⣿⣤⣤⣿⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢺⣿⣿⣧⣽⡶⠀⠀⠐⢡⠏⠀⠀⠀⠀⠀⠲⣿⣿⡿⡀⠀⠀⢰⡄⠀⡀⢰⣆
⠀⠘⠿⠿⠿⠕⠀⠀⠀⠀⠈⠁⠀⡇⠀⠀⠀⠻⠿⠛⠁⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⢼⣶⣶⣾⡇⠀⠀⠉⠁⠀⠀⢩⡍
⠀⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⣏⠀⠀⠂⠀⠅⡆⠀⢠⣶⣶⣶⣶⠄⠀⢰⣿⣿⣿⣿⡇⠁⠀⠀⠀⣱⣀⠀⠀⠀⠀⣠⣼⣿⣿⣿⠄⠀⠀⠰⡥⠤⢽
⠀⣇⠀⠀⠀⠀⣃⡇⠀⢰⢘⢘⣋⡃⡄⠀⢸⣿⣿⣤⣼⡟⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⣹⣿⣿⣿⣿⠂
⠀⠈⠁⠀⠀⠀⠉⠀⠀⠘⠤⠤⠤⠠⠂⠀⠈⠻⠿⠿⠿⠍⠃⠀⠀⠀⠈⠛⠉
⠀⠀⠤⠀⠀⠀⠀⠀⢠⠤⠤⢤⡀⠀⠀⣤⠤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⢠⡤⠤⠤⡄⠀⠀⢀⣀⣀⣀
⠀⠀⠨⣧⠀⠀⠀⠀⠀⠐⠲⡆⡇⠀⠀⡇⣤⠦⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡽⡀⠀⠀⢸⡧⠐⠀⡇⠀⠀⠸⠤⠶⢿
⠀⠀⠈⠿⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠔⢃⠀⠀⢸⡀⠀⠀⡇⠀⠀⢸⠩⠭⢹
⠀⠀⠓⠒⠒⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⠀⠀⠈⠉⠀⠉⠁⠀⠀⠈⠉⠉⠉
⠀⡴⠒⢦⣤⣤⡀⠀⣀⣀⣀⣀⣀⠀⠀⠀⣴⣀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣤⣄
⠀⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣧⣤⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠉⠁⠀⠀⠸⡿⣹⣿
⠀⠛⠛⠛⠛⠛⠃⠀⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠀⠐⠒⠶⠖⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⣠⡶⠒⠶⠀⠀⠀⢠⡖⢲⡄⠀⠀⢀⡤⠲⢤⠀⠀⠀⢀⣀⣀⢀⠀⠀⢀⣤⣄⡀⣄⠀⠀⠀⣴⠛⡆⠀⠀⠀⠀⣠⣄⡀
⠨⣿⡿⠏⠀⠀⠀⠀⠺⡷⠂⠀⠀⠀⢾⢦⣈⢻⡇⠀⠀⢰⡎⣀⣰⡆⠀⢸⣿⣿⣅⣙⠀⠀⠀⣿⣛⣿⠀⠀⠀⠐⣿⣿⠇
⠀⠈⠛⠛⠛⠀⠀⠀⠘⠛⠲⠂⠀⠀⠈⠳⠴⠚⠁⠀⠀⠃⠀⠀⠐⠁⠀⠘⠛⠛⠛⠛⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀⠘⢿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣶⣶⡦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠒⠉⡶⠀⠠⣄⠈⠐⠽⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⠊⠀⠀⢀⠁⣐⠀⣇⠱⡦⠀⠈⠜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠀⠀⠀⢸⠺⣿⠇⣇⠼⡄⠀⠀⠈⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢦⡄⠀⠀⠀⠀⡇⠀⠀⠀⡤⠚⠂⠉⠀⠀⠀⠘⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⣴⠶⠞⠛⠉⠀⠀⠀⠀⠙⣛⢦⡀⠀⢇⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢤⠀⢀⢹⠃
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠠⠼⠟⣂⡿⠀⠘⢎⢄⠀⠀⠇⠀⠀⢀⡀⠀⠐⠈⠠⣢⠏
⠀⠀⠀⠀⠀⣠⠴⣲⠿⣷⣶⣶⣶⣶⣿⣿⡟⠉⠀⠀⠀⠈⠳⢕⡤⣘⠈⠀⠀⠀⣀⠠⣢⠕⠁
⠀⠀⠀⠀⠊⣠⠞⢁⡾⠁⠀⠀⠀⠀⠨⠭⠽⣦⠀⠀⠀⠀⠀⠀⠉⠓⠒⠿⠾⠓⠚⠉
⠀⠀⠀⠀⠘⠁⢠⡞⠁⠀⢀⡠⠢⠴⡈⢹⣭⣽⣧
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡆⢴⢰⡆⠤⠄⠤⠤⠼⣇
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠈⢰⢠⡄⡝⠄⠩⠭⠭⢽⡄
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠐⡒⠬⠸⠇⠆⡰⠴⠧⠉⣋⣽⣿⣿⠛⠐⠒⠒⠲⠖
⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠈⠒⠤⢆⠮⢥⣀⣀⡀⢸⣿⣿⡇
⠀⠀⠀⠀⠸⣇⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠙⠓⠀⠤⠀⢤⢀
⠀⠀⠀⠀⠀⠈⠙⠓⠾⣦⡀⠀⠀⠀⠀⠈⠉⠒⠛⣿⣿⣤⠄⣀⢀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠒⠛⠀⠀⠀⠀⠿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠛⠻⠿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠿⢿⣿⡿⢿⣿⢿⣿⠿⡿⢿⡿⣿⡿⣿⡿⠿⢿⣿⠿⠿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣙⠻⣿⠁⠆⢿⡆⠟⣸⡇⢸⡇⣌⠃⣿⢰⡿⠻⣧⣈⡛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣭⣾⣧⣾⣿⣼⣿⣤⣿⣧⣾⣧⣿⣦⣿⣦⣥⣾⣷⣬⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⣿⣿⠟⡛⢿⡿⢛⠻⣿⠟⡻⣿⡟⣿⡟⣿⠛⣿⠟⣟⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⣒⠘⣏⢸⣿⣿⡀⣿⣿⣇⢸⡿⢸⡇⣿⡇⣿⢠⡌⠀⣿⡇⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣾⣿⣷⣿⣶⣶⣾⣷⣶⣾⣿⣶⣶⣿⣿⣶⣾⣿⣾⣿⣶⣿⣷⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣴⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⡀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⢀⡀⠠⡆⠀⠀⠀⠈⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⣿⣸⣧⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠡
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡙⣿⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣆⢻⣷⡘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠹⡷⠈⠉⢀⢐⠂⠀⠀⠀⠀⢀⠀⠺⠇⠀⠀⠂⠀⠀⠀⠈⠉⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⢈⠛⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠒
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡔⠀⠀⠀⠀⠆⠀⠀⠀⠺⢠⣠⠔⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣐⠂⠁⠐⠚⡁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⠂⠂⠐⠒⠁⠀⠀⢀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠂⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⢀⣠⣴⣶⡿⠟⠋⠁⠀⠀⠀⠀⠒⠠⢀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠳⠜⠜⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈
⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠲⠒⠂⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠄⠀⠠⠶⠀⠀⠀⠀⠀⡀⠀⣠
⡿⠋⢁⡄⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⢠⠜⠀⠀⠀⠐⣠⣴⣿⣿
⠁⠀⠺⡤⠀⠀⢸⣿⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⡄⠀⠀⠤⠄⠀⣼⣿⡿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣶⣤⣤⣴⣾⡿⠋⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠠⠄⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⠟⠁⠈⠀⠁⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⡐⠠⠈⠀⠀⠀⠀⠀⠸⣿⣿⣿
⣿⡿⢋⠀⠀⠀⠑⠀⡀⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⠀⠀⢀⠄⠀⠀⠀⠀⠀⢿⣿⣿
⠁⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠈⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠲⠄⠀⠀⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢄⣀
⡠⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢨⠖⠐⠀⠁⢀⠤⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲
⠀⠀⠀⠁⠀⠀⠀⡀⠀⣄⠀⠈⠀⠀⠀⢀⠞⠋⠀⠀⠀⠀⠀⠀⢓⡆⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡖⠀⠘⡅⠀⠀⠀⠀⠀⢉⠂⠐⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠋⠀⠉⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠒⠢⡁
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⢇⠀⠁⠿⢤⡀⣀⠠⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠫⠀⠀⡀⣀⡴⡃⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⣠⡾⣿⢻⡿⢷⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠛
⣇⠀⠀⠐⠀⠀⠀⠀⣀⣤⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣀⣿⢸⣷⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘
⠀⠄⠀⠀⠀⠀⢠⣿⣯⢿⣿⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣾⣿⣶⣄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡆⢀⢶⠀⠰⢄⠸⠂⢸⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡗⠜⠈⡧⢸⠈⠻⠀⢸⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠠⡀⠀⠠⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⡇⠐⠀⠀⠀⠀⠀⢰⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢀⠐⠀⡀⠀⠂⢸⠀⠀⠀⢘⠐⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠐⠀⠂⠀⡁⠐⡀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢸⠀⠀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⣀⠀⠀⠀⢰⢀⠀⡆⠀⠀⠀⠀⡅⢀⢰⠀⠀⠀⢈⡀⠀⠀⡇⢀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⠀⠀⠤⠀⠀⠀⠀⡁⠀⠀⠀⡀⠀⠀⢰⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⡅⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢘
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠙⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⠶⠶⠶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠶⠶⠶⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢀⣀⡀⠀⠀⢀⡀⠀⣀⠀⠀⣀⢀⡀⢀⡀⠀⣀⠀⢀⣀⣀⠀⠀⣀⣀⠀⠀⠘⢷⣄
⠀⠀⠀⢀⠸⠋⠀⠀⠀⠀⢿⣭⣁⠀⢠⡟⣷⠀⢹⡆⣸⠇⢸⡇⢸⡿⣄⣿⠀⣾⠉⣙⡁⠸⣯⣍⡀⠀⠀⠀⠙⢷⡄
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢦⣤⡿⠀⡾⠛⠛⡧⠀⢿⡟⠀⢸⡇⢸⡇⠙⣿⠀⠻⣬⡿⠃⠴⣤⣼⠇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠘⣧⠀⠀⠀⠀⠀⣰⢿⡄⠀⣼⠛⠛⠂⣴⠛⠛⠂⣰⠟⠛⣦⢸⡇⠀⢸⡆⣿⣧⡀⣿⠘⢻⡟⠁⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠈⠳⣄⠀⠀⢠⡿⠾⢿⠀⢿⣄⣠⡄⢻⣄⣠⡄⢹⣄⣠⡟⠘⣧⣀⣼⠃⣿⠈⢷⣿⠀⢸⡇⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⢳⡄⠈⠀⠀⠈⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠈⠉⠁⠀⠈⠀⠀⠁⠀⠀⠁⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠛⠛⠛⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠂⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠚⠋⠁

⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣾⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⢉⣉⣉⣉⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉
⣿⠀⠀⠤⠼⠥⠼⠧⠿⠥⠿⠧⠿⠤⠿⠤⠤⢰⠿⠋⣉⣀⠀⢀⡉⠛⢷⡦⠤⠤⠤⠤⠤⠤⠤⠤⠿⠀⠿⠈⠟⠸⠯⠸⠏⠹⠤⡄
⣿⠀⠀⠀⠒⠂⠂⠀⠐⠂⠂⠐⠒⠀⠀⠀⣤⡏⣴⣿⠁⠀⠀⠀⠈⢳⡀⠙⠆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⣄⠠⢀⠀⡀⠀⡇
⣿⠀⠀⠀⠃⠀⠁⠀⠀⣠⣀⠀⠀⠀⠀⣿⡆⣿⣿⣿⠀⢀⣀⣀⢀⣸⣷⠀⣾⣇⣀⠀⠀⠀⠄⠀⠤⠀⠄⠄⠠⠠⠀⠠⠄⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⢀⡝⣇⢹⠀⣤⠁⢸⠛⣷⣿⣿⣿⠀⠘⠻⠇⠸⡿⣿⡇⠃⡘⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢄⣠⢀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠨⣿⣏⠋⠀⣿⠀⢾⣼⣿⣿⣿⠋⠀⠀⠀⠂⠲⣷⣿⣿⡄⢇⡇⠀⠀⠀⠀⠀⢈⠀⠸⠀⠀⠀⢸⠇⣿⠀⠀⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⢰⡎⡄⢸⣿⣿⣿⣿⣧⠰⣄⠈⠀⠼⣿⣿⣿⡇⢸⠃⠀⠀⡀⡀⠈⠀⡀⡀⡀⠀⡀⡘⠃⡾⠀⠀⡇
⢿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⠄⡇⠀⢿⣿⣿⣿⡏⠑⢦⣶⣶⣶⣿⣿⣿⢧⠟⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⡀⢀⠁⠀⠛⠀⠀⠇
⢸⠶⠶⠦⠐⣠⠶⠆⠶⠖⡶⠤⠛⠳⠖⠆⠀⢤⠰⣷⠰⠆⠿⣿⡿⠟⢿⡶⠶⠀⠂⠞⠶⠁⠼⠳⠶⠶⠒⠤⠷⠈⠾⠷⡬⠶⠳⡧⠖
⢾⡆⠀⠐⠶⠱⣮⠲⢊⣶⡑⠆⠐⠠⣄⢀⡀⠒⢷⣾⣶⣤⡾⠾⠂⣶⠟⢃⡒⣒⡒⣶⣸⢲⢷⢒⣚⠀⣖⣖⣒⣳⡀⢸⡛⣲⡚⠁
⢾⠀⠈⠉⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣨⣭⣭⣭⣭⣥⣬⣵⢸⣷⣭⣭⣭⣭⣭⣭⣭⣭⣽⢸⣭⣭⣤⣭⣭⣭⣭⣵⢸⣭⣯⣥⣤⣤
⠘⣤⠊⠢⡈⠀⠀⠀⣽⠉⢹⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢈⡇⢿⠅⠀⠀⢹⢻⠋⢹⠀⠀⢸⠀⠐⠂⠀⠀⠀⠀⢨⢸⠀⣀
⠀⠻⡄⠀⢣⠀⠀⠀⠈⠀⣼⠀⠀⠀⢸⠻⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠘⠂⠀⠀⡊⢰⠀⠀⠀⢸⢸⠀⠉
⠘⣀⠄⢰⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆
⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠛⠂⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣠⠃
⠀⠯⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠄⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠉⠙⠓⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢠⣄⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⡤⠤⠤⠴⠶⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⡴⣯⣿⣿⣆⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣾⣭⣴⣿⣾⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣠⠾⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⢻⡋⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⠀⠀⠀⣀⡀⣀⠀⢀⣀⣴⣞⡭⠶⠒⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠙⢳⡟⠛⠛⠻⢿⣍⡙⣿⠉⠙⢯⡽⠿⠛⠟⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣤⡴⠟⠁⠀⠀⠀⠀⠈⠙⠋⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣀⠤⠤⢴⢒⡀⣀⠀⠐⠂⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⠋⠁⠀⠀⠘⠛⠉⠙⠛⠛⠋⢰⡶⠶⣄⠉⠑⠲⠄⠀⢂⡬⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡂⠀⠀⠁⠀⠀⠀⠀⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠃⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣀⠀⠰⣶⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⣸⡇⠰⠆⠆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡠⡁⠀⠀⠀⠀⠀⠻⣿⣶⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢰⣤⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⢾⡇
⠀⠀⠀⠀⠀⢀⡀⣠⣶⣟⠀⡈⠹⢡⣾⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏⢸⡇
⠀⠀⠀⠀⠀⣵⣶⣮⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⡇⠀⠠⠼⠃⢀⣶⣚⠈⣉⣻⡛⠋⢀⣸⠁
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⠟⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⡿⠿⠿⢟⠿⡿⠶⢿⣿⣿⣿⠿⠏⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠄⣟⣉⣛⣛⣋⣛⣙⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⠒⠛⢐⣤⣀⠀⠀⠀⢠⣴⣾⣿⣿⣶⡤⠀⠀⠀⠀⢀⣤⠤⢤⠠⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢻⣾⡿⣷⣿⠃
⠀⠀⠀⢾⣹⡇⠀⣿⣧⣿⠆⠀⠀⠀⢹⠀⢸⠀⢸⠀⠀⠀⠀⢰⣿⣷⣬⣿⣠⡾⠀⠀⠀⠀⢰⠾⣧⡿⢷⠀⠀⠀⠀⠀⢸⣯⡿⣯⣿
⠀⠀⠀⠀⠙⠄⣄⠬⠋⠉⠀⠀⠀⢀⣾⣄⣾⣄⣼⣄⠀⠀⠀⠘⠟⠛⠻⠻⠋⠀⠀⠀⠀⠘⠛⠒⠛⠓⠚⠛⠂⠀⠀⠀⠺⣿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣴⠹⡍⢩⠏⢦⡀⠀⠀⠀⢾⢉⢹⡄⢤⡄⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠈⣃⣜⢉⢟⡖⢤⠀⠀⠀⠀⢠⣤⠯⢧⣀
⠀⠀⢺⡇⣻⡇⣿⣿⠀⢸⡧⠀⠀⠀⢨⣧⣯⣵⣼⡄⠀⠀⠀⠑⣺⣿⣿⠀⣿⡒⠁⠀⠀⢺⠋⠀⢹⠺⠤⡎⠀⠀⠀⢶⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠉⠻⣰⣣⣞⣆⠞⠀⠀⠀⠀⢸⣿⣿⣿⣽⡇⠀⠀⠀⠀⠛⠛⠛⠀⠙⠃⠀⠀⠀⠈⠒⠐⠻⣄⡛⣤⠀⠀⠀⠀⠙⣿⠿⢿⡿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⣤⣤⣤⣶⣶⣄⠀⠀⠀⠀⠀⣸⣷⡆⠀⠀⠀⠀⠀⠀⢠⡿⠛⠿⣦⠀⠀⠀⠀⢠⠻⡈⢹⠐⢒⡆⠀⠀⠀⠀⣴⣶⣶⣶⣶⣤
⠀⠀⠀⣯⣽⣿⢺⢺⠾⡷⡇⠀⠀⠀⠂⠙⠛⡉⠛⠶⠄⠀⠀⠀⢘⠗⠀⠐⢟⠀⠀⠀⠀⠙⠒⠃⢸⠀⣎⣘⡄⠀⠀⠀⠘⢿⣿⣿⡿⠁
⠀⠀⠀⠛⠛⠛⠛⠿⠿⠋⠀⠀⠀⣠⣶⣶⡄⠂⠀⠀⠀⠀⠀⠠⣿⣿⣶⣾⣿⡧⠀⠀⠀⠀⠀⣀⣸⣀⡈⠁⠀⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠻⠛⠻⠂⠀⠀⠀⠀⡇⢴⡦⠀⣿⣻⣿⡇⠀⠀⢠⢝⢿⣿⡿⣆⡀⠀⠀⠀⢀⣀⣴⣒⣲⣆⣀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣷
⠀⠀⠀⠀⠀⠘⠂⠊⠀⠀⠀⠀⠀⣥⣤⣤⡄⠛⠛⠛⡁⠀⠀⠇⠀⠑⠀⠛⢩⡁⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣦⢸⡇⢿⣼⡿
⠀⠀⠀⠠⣾⣿⣾⣼⣿⡆⠀⠀⠀⣿⣯⣸⡇⠐⠟⠀⡇⠀⠀⠈⠢⢤⡄⢴⣿⣿⠂⠀⠀⠀⠿⠿⠷⠿⠿⠇⠀⠀⠀⠀⠉⠈⠁⢿⣾⠟
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠐⡘⠘⠈⠙⠁⢊⠃⠋⠈⠅⠀⠀⠀⢈⠑⢁⠺⠁⢘⠄

⠀⠀⠀⠀⠀⣠⣿⣟⠀⠀⠀⠀⠀⠀⠀⢠⣶⢲⡄⠀⠀⠀⠀⠀⣴⡖⣦⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣿⡆
⠀⠀⠀⠀⠸⣿⡷⠸⡆⠀⠀⠀⠀⠀⣠⣼⢻⡟⣧⣄⠀⠀⠀⠀⠻⢧⢿⣥⠛⢠⡄⠀⠀⠀⠸⣿⣿⣷⣿⣿⠗
⠀⠀⠀⠀⠀⢿⣷⣹⡿⠀⠀⠀⠀⢀⣇⣿⢸⡇⣿⣿⡇⠀⠀⠀⠀⠀⣠⡝⠷⠁⠀⠀⠀⠀⠀⢻⠟⠛⠛⡟
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⡠⣾⣿⣿⡦⡀⠀⠀⠀⠀⢠⣦⠀⠀⣴⡄⠀⠀⠀⠀⠒⢲⣤⣠⠤⣶⠄⠀⠀⠀⠀⣀⡀⠀⢀⣀
⠀⠀⠀⡜⢇⢻⣿⣿⢃⡞⡄⠀⠀⠀⠈⣩⣴⣶⣍⠁⠀⠀⠀⠀⠶⠚⠉⠀⣾⡇⠀⠀⠀⠀⠐⠛⣗⣶⣺⡛⠃
⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣸⣹⣆⡀⠀⠀⠀⠀⠐⠓⣛⠚⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⣠⣤⣄⡀⢠⡀⠀⠀⠀⠀⢹⣿⣿⣿⣽⣿⠀⠀⠀⠀⠘⢿⣿⠃⢀⣧⡀⠀⠀⠀⢠⣾⣿⣿⣾⢇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⡛⢛⡛⢛⡛⠀⠀⠀⠀⠀⣨⣅⠀⢸⣿⠀⠀⠀⠀⢸⣷⣾⣿⣼⣯⠃
⠀⠀⠀⠸⣿⣿⠿⢿⣿⡿⠀⠀⠀⠀⢰⣧⢼⡧⢼⡧⠀⠀⠀⠀⢸⣿⣿⡇⠀⣭⠀⠀⠀⠀⠀⢻⠿⠯⠼⠋

⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢾⡧⣿⠆⠀⠀⠀⠀⠀⣠⢴⣿⢿⣦⣄⠀⠀⠀⠀⠀⢿⣟⣾⡏⠀⠀⠀⠀⠀⠀⠉⣤⠶⣦⡈
⠀⠀⠀⠀⠀⣬⠟⢯⣄⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⠀⠀⠀⠀⢤⣄⡉⣡⣴⣤⡄⠀⠀⠀⠈⠀⢿⣧⡿⠃⠁
⠀⠀⠀⠀⠸⠿⠞⠿⠿⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠉⠻⠿⠟⠛⠁⠀⠀⠀⠀⠁⠐⠻⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡀⣿⠘⣧⠀⣿⢸⡟⠛⠃⣾⠛⣷⠚⢿⡟⢳⣷⠀⢠⣿⠀⣿⠛⠃⣾⡆⢸⡇⠻⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣇⣿⠀⣿⢰⡇⢸⣧⣀⠀⢿⣄⡉⠀⢸⡇⢸⣿⣇⣾⣿⠀⣿⣀⡀⣿⣿⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢹⣿⠀⢸⣾⠃⢸⡏⠉⠀⡀⠉⣿⠀⢸⡇⢸⡇⢻⡇⣿⠀⣿⠉⠁⣿⠸⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⡿⠈⢿⠀⠘⡿⠀⠸⣷⣶⠆⠿⣶⠟⠀⢸⠇⠸⡇⠈⠀⣿⠀⣿⣶⡆⢿⠀⢻⡇⠀⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⣀⣠⣤⣤⡴⣶⣶⣶⣶⣶⣾⣿⣿⣷⠶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣄⣙⣿⡟⠛⠛⢿⣯⠉⡁⠘⣿⣟⣋⣉⣩⣭⣭⣭⣉⣉⣙⠓⠾⣭⠛⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣯⣭⣿⠟⠙⣿⠀⠀⠀⢻⣧⠈⠂⠘⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠓⠁⠀⠀⠀⠉⠻⢿⣦⡀⠀⣀⣀⣀⣀⡖
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠈⠀⠀⠀⠈⠛⠀⠀⠀⠀⢠⠞⣻⣿⡿⣿⣦⠀⢠⠀⢲⠆⠀⠀⠀⠀⠈⠂⢝⢿⣎⢴⡞⡏⢳⠒
⠀⠀⠀⠀⠀⠀⢀⡟⢙⡇⠰⡷⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⡋⡇⢸⣿⣰⠏⢠⠟⠀⠀⠀⠀⠀⠀⠈⠂⠀⠹⣮⠓⠠⠊
⠀⠀⠀⣀⣀⣀⣸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣍⣛⣥⣾⣿⠏⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⢠⡾⠛⠉⠛⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⠏⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⢰⣂⢠⢳⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⣆⠉⠘⠟⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠈⠻⣷⣦⣶⣿⣯⣄⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣰⣿⣡⠆⢐⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠈⢻⣟⠿⣶⣤⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣱⣿⢻⠁⣾⢿⣿⢹⣿⠆⠀⠀⠀⠀⠀⠀⠀⢤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣱⣿⠃⢨⣄⠘⠶⠟⣸⣿⠀⠀⠀⠀⠀⠀⠠⣢⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠠⢀⠀⠄⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠉⠛⠛⠛⠛⠁⠀⢀⠄⡠⢂⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠳⣦⣌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⣡⣤⡶⠟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠉⣛⣿⠷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣵⣶⣶⣶⡿⢿⣿⠫⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠖⠲⠶⠾⣼⡿⠀⠀⠀⠀⠉⣿⠉⠉⠉⢩⣿⣿⠿⣯⢤⣄⣸⡟⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⣿⣷⠖⠶⠤⣤⣤⣯⠀⠀⠀⠀⠉⠛⠛⠒⠒⢺⣿⣷⠦⣤⣤⣀⣸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠷⠾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⣶⣶⣼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣷⣦⣍⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⡙⢛⣠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣟⢻⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⣿⣿⣿⣿⣦⣄⠲⣿⣿⣿⣿⣿⣿⣿⣿⡶⢀⣴⣾⣿⣿⣿⠿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣬⣙⠻⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⠿⢋⣥⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⣭⣥⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣭⣽⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣴⣾⣿⣿⣷⣦⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡁⣿⣟⠰⠄⣿⣿⡁⠆⣹⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⢿⠿⠃⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⣿⣷⣾⣿⣿⣿⣾⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠿⣿⣿⣿⣿⠿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣋⣥⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣤⠉⠉⠉⠉⠉⠉⢥⣴⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣾⡄⣾⡆⣴⠛⣦⢰⣷⣰⡆⣿⣛⠸⣦⡞⠀⢰⣟⠂⢰⢷⡄⣷⢠⡇⣶⢰⣷⣰⡆⡶⢛⡃⢴⣛
⣀⣀⣀⣀⣀⣀⣀⣇⣻⣹⣇⣻⣤⣟⣸⣏⣻⣇⣿⣤⣀⣻⣀⣀⣠⣼⣇⣾⣛⣧⣘⣿⣁⣿⣸⣃⣻⣇⣳⣬⣟⣠⣽⣃⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢩⣭⣭⣭⣭⣭⣭⡉⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠿⣿⣿⣿⣿⠟⣋⣴⣠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢃⣼⣿⣿⣿⣿⣿⣿⣇⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⣡⣶⣷⣬⣙⢛⣡⣾⣿⣿⡏⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣴⣾⣿⣿⡇⢸⣿⣿⡟⢻⣿⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡇⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡇⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡇⢸⣿⣿⡇⢸⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣬⣤⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣞⠻⣿⣿⣿⣿⣿⡿⢛⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣷⣦⢙⠛⢛⠁⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⡇⢸⣿⣴⣦⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠟⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⣿⣿⣿⣿⢰⣿⣿⣿⣿⠘⣿⣿⣿⡏⢹⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⣾⣿⠟⠋⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇⣼⣿⣿⣿⣿⡆⣿⣿⣿⡇⢸⣿⣿⣿⠀⣀⣛⣛⣛⣛⣛⣛⣛⣛⡛⢰⣿⣿⣿⠀⣿⣿⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⣧⡙⢿⣿⣿⡟⣡⣿⣿⣿⡇⢸⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣦⣥⣥⢹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣌⠻⢋⣴⣿⣿⣿⣿⡇⢸⣿⣿⣿⠀⠉⣩⣭⣭⣭⣭⣭⣭⣭⣭⡈⢿⣿⣿⡙⠛⠛⣸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠿⣿⣧⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠛⠳⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⠴⠶⠖⠚⠋⠉⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⠉⠉⠉⠛⠛⠛⠛⠛⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⣀⢀⡀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀⣤⣀⣤⣤⣤⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⢀⣼⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⣠⡴⠋⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣠⡴⠛⠁⠀⣼⠇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡾⠉⠀⠀⠀⢀⡏⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣼⠁⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠁⠀⠀⠀⠀⠀⠀⡿⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣿⣀⡤⢤⠶⠶⠶⠶⠦⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⣤⠖⢿⠶⢤⡀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠈⣏⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⣾⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠘⢷⣀⣾⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⣀⣀⣀⣀⣀⣀⡤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠘⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠲⠶⠦⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣠⠶⠶⠶⠞⠛⠛⠁⠉⠁
⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⣰⣦⣰⠰⣆⣰⠆⣶⣒⠠⣞⣓⠐⢲⠖⢰⣶⣰⣶⢰⣖⡒⢰⣦⣰⡆⢲⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠻⠈⠿⠀⠹⠏⠀⠿⠶⠐⠮⠿⠃⠸⠃⠸⠏⠉⠿⠸⠷⠦⠸⠊⠻⠃⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠇⠀⠀⠀⠀⣸⣷⠀⠀⣰⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠧⣤⡼⠁⠈⢳⣾⠁⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡎⠀⠀⠙⠁⣤⣤⠀⠀⢰⢲⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⡏⡇⡶⣦⣿⣿⢀⣀⢸⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠁⣏⡇⣇⡇⣇⡷⣿⣿⢸⣹⢸⣸⢸⣸⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠁⠀⠀⠀⢈⣀⣈⣀⣈⣁⡈⠁⢈⣁⣀⣁⣀⣉⠀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠬⠭⠍⠭⠩⠭⠅⠀⠬⠭⠭⠭⠭⠭⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠭⠩⠭⠭⠭⠭⠅⠀⠭⠥⠭⠬⠍⠉⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠿⡅⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⠶⠶⠶⡄⢹⡟⠛⠛⠛⠛⠛⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣭⡥⠞⣀⣀⣀⡀⠀⠀⠀⠀⢸⡇⠀⠍⠍⠍⠅⠀⡇⢸⡧⡟⡛⠛⠛⢳⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⢉⡓⢦⢸⠀⣼⣡⠔⢢⠏⣄⠀⠀⠀⢸⡇⠀⣭⠍⠉⠁⠀⡇⠈⢧⠳⣔⣐⣢⢞⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣙⣾⣸⡷⠛⠳⠖⠋⠀⠈⠳⡄⠀⠀⠓⠒⠒⠲⢴⠒⠒⠃⠀⠀⢙⣦⠉⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠳⢤⣀⠀⠀⠀⠀⠀⠙⢦⡠⡖⡒⠒⢒⡶⠖⠲⢶⡖⠀⠉⠀⠄⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⡾⣧⣀⠘⠲⡞⠛⠛⠛⠃⠀⢙⡇⢠⡄⣾⠀⠾⣝⠀⢻⠀⣠⠖⠛⠦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡯⢭⡇⠈⣡⣄⡇⡀⠀⠀⠀⠀⢠⡇⠀⠀⠙⣄⡐⢋⣠⡟⠐⠷⠶⠶⠶⠾⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠳⠴⠶⠶⠆⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢨⣭⣭⣭⣭⣭⣭⡇

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣄⢠⡀
⠀⠈⠳⢬⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⣤⡠⠀⠀⠀⠠⣀⠀⢀⡀⠙⠃⠀⠀⠀⠀⠉⠀⠀⠀⢰⠀⡀⠀⣏⣹⠞
⠀⠀⠀⠀⠀⠀⠀⡀⠢⣠⣄⣰⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⣰⣃⡿⠁⢀⣤⠖⠀⠀⠀⢀⣜⣶⣦⡀⠀⠃⠀⠉
⠀⠀⡤⠖⢴⠄⢸⣤⣾⠟⠋⠛⠣⣀⡀⠀⠀⠀⢀⢠⡀⠀⢼⠟⠁⠀⠀⠀⠀⠀⠤⠒⠉⠉⠋⠛⣧⣂⣦⠀⢰⡐⡄
⠀⢰⣠⠎⠁⠀⠸⠔⠁⠀⠀⣤⣄⡀⠙⠢⡀⠀⠠⠗⠂⣴⠿⢀⠀⠀⠀⡐⠉⢀⣀⣀⣀⠀⠀⠀⠀⠲⣧⠃⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢠⣾⣾⠟⠀⠀⠈⠂⡀⠀⠈⢃⣨⠟⠁⠀⠜⣀⣼⣭⣋⣙⣁⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠘⠿⢿⣷⣶⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠸⢽⣰⡗⠛⡛⠁⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣤⣌⣿⣿⡇⠀⠀⠀⠄⠀⠀⠀⠀⠀⣄⠀⠈⠻⢭⡭⠼⠃⠀⠀⠀⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠉⠛⢻⠿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠈⠀⢤⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠁⠀⠀⠛⠲⠆⠀⠀⠀⠑⠢⠀⢀⣀⣤⠀⠠⡀⠠⠔⠊⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠛
⠀⠀⠀⡾⢰⠦⡀⠀⠻⣥⠀⠀⠀⡀⠄⠀⡜⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⢉⠀⠀⠀⢠⡄⠀⠀⣀⣀⡀⢀⣀
⠀⠀⠀⠙⢍⣠⣁⠀⠀⠀⠀⠀⠀⠁⢀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⠀⠀⢷⡄⠉⡽⠀⠚
⠀⠀⠀⠀⠀⠉⢩⡆⠀⠀⠀⠠⢂⣰⠏⢀⡀⠀⠄⠀⢠⣤⣶⡄⡆⠀⠀⠀⣿⠀⠀⠄⠘⠧⣄⡀⠛⠶⠃⠀⢠⣤
⠀⠀⠀⢰⣂⡤⠉⠀⢈⣤⢈⢁⡾⠉⡇⠀⡿⠀⡀⠀⢿⡿⠋⠀⡇⠀⡁⠀⣯⠀⠀⢠⡐⠦⡀⠙⠿⢦⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠉⣄⠀⠀⠀⣇⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢷⠀⠀⠐⠀⠀⠙⠶⢶⣤
⠀⠀⠀⠀⠀⢠⠞⠁⠰⡆⠀⠀⣿⠀⠀⢀⡃⠀⡇⠀⠾⡛⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢉⠀⠀⠐⣄⠀⠀⠀⠀⢣⡄
⠀⠀⠀⢀⡴⠃⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⡇⠀⡇⠀⠀⠉⠀⠀⡇⠀⡆⠀⡇⠀⡄⠀⠀⠀⡀⠀⣧⠀⠀⠀⠀⠀⢣
⠀⠀⠀⡋⠀⣨⣇⣠⣴⠧⠴⠶⠿⠤⠷⠾⠷⠤⠇⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠚⠛⠛⠛⠒⠓⠶⠷⠤⠄⠀⢀⣀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⢤⡀⠀⠀⠀⠀⠒⡦⢤⡀⠀⡠⠀⠠⡀⠀⠔⢶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⢹⣄⣤⣆⡄⠀⣀⠀⡞⠁⠀⠀⠟⢠⠇⠈⢠⠫⠁⠀⠀⣀⡤⠼⠇
⠀⠀⠀⠀⠄⣈⠿⠞⠉⣀⠟⠙⣑⣆⣴⢿⡏⠀⣽⣟⠙⠿⠀⠀⠀⠚⠛⠀⠀⠀⠚⡁⠀⡰⠕⠀⠀⠀⢤⠀⠀⠀⢀
⠀⢓⢐⣒⢲⡟⠀⠐⡿⠟⠓⠈⣿⠏⠁⠼⢃⣢⠥⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠊⠀⡄⢰⣏⠁⣸⡇⠀⠀⢨⡀
⠀⠸⠃⠈⢉⣿⠀⢀⣿⡷⠀⠀⢽⣢⠎⣀⠘⠛⠒⠚⠁⠀⢀⢘⣁⠀⠀⠀⣀⢀⣘⠉⠀⠀⠐⠤⣤⠴⠟⠀⢀⡌⠘⠉⢀
⠀⠀⠊⠃⠀⢭⣽⣧⣷⣞⠈⢀⠳⢀⡞⠉⢁⡠⠤⠀⣀⡴⠹⢿⠄⢠⡶⣏⡩⣶⢇⠀⠀⠀⡄⠂⠀⢀⡤⢒⣏⠨⡄
⠀⠀⠐⣤⣈⠀⢬⣍⠈⠍⠊⢵⢖⠿⠗⠊⠁⠀⡄⡈⠙⢷⠶⢮⣹⣷⡟⢈⣧⡤⠈⠀⠰⠺⠀⠀⠀⠰⠤⠈⢾⠄⠋
⠀⠀⠀⠈⠐⠛⠳⠦⠤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠾⣴⣿⣭⠭⠥⠄⠠⠀⠄⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠄⠁⠄⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠠
⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠂⠀⠂⠀⠀⠐⠀⠀⠀⠂⠀⢀⠐⠀⠀⡀⠂⠀⢀⠐

⠀⠄⠀⠀⠠⠈⠀⠀⠀⠁⠀⠠⠀⠀⠀⢀⡥⠖⠢⢌
⡀⠀⡀⠀⠀⢀⠀⡀⠀⡀⠀⠀⠀⠀⢀⠏⡀⢾⡓⢀⢳⠀⠀⢀⣀
⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⣀⠴⠜⢧⡀⠺⠟⢀⡞⠀⡠⠃⠀⠉⠑⠢⠤⣀
⠄⠁⠄⠀⠀⠠⠀⠀⠀⢴⠛⠉⠠⠀⠀⠀⠙⠓⠚⠉⢧⡔⠁⠀⢀⠔⢒⠒⢤⠀⠉⠑⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⢳⠈⠀⠀⠀⣁⣄⡤⠞⡿⢭⠭⣕⠲⡎⠀⣻⡅⠀⡇⠀⡜
⠂⡀⠀⠀⢀⠐⠀⠀⡀⠂⡀⢣⣴⡖⠛⡉⢀⡤⢬⠧⣬⡭⠽⠤⢼⠤⣉⣁⠴⣧⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠉⠓⠶⢾⣥⣤⡴⠽⠧⡤⣤⣼⠦⠔⠚⠉⠀⠘⡄
⠁⠄⠀⠀⠠⠈⠀⠀⠄⢡⠏⠠⠈⠀⠀⠄⠀⠈⠩⠛⢢⡔⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⡄
⡀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⢲⢤⣄⡀⠀⠀⠀⠀⣼⠻⠉⢆⠀⠀⠀⠀⢀⣀⠤⡖⠊⠁
⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⡀⠀⠉⠓⠶⢦⡜⠁⠸⠀⠈⠣⠤⠖⠊⠉⠀⠀⡇
⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠩⠓⠦⢤⣄⡀⠀⠠⢘⠀⠀⠀⠀⣀⠤⠔⠊⠉
⠀⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⢈⠀⠀⠀⠁⠈⠉⠛⠲⢬⠤⠔⠊⠉
⠀⡀⠂⠀⢀⠀⠀⠀⡀⠂⠐⠮⠿⠭⠿⠍⠋⠉⢉⠹⠭⠭⠭⠭⠭⠥⠤⠤⠤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀
⠀⠄⠁⠀⠠⠀⠀⠀⠄⠁⠈⠡⠈⠛⠛⠟⠻⠟⠻⠟⠿⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠂⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⢀⠐⠀⠂⡀⠀⠐⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⠉⠁⠈⠉⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⣰⣆⢀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⣰⡟⢻⡟⠛⠃⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠷⣾⣷⣦⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⢀⣀⣼⣇⣼⡇⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠈⠉⢻⡏⠉⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⡀⠀⠀⣀⣠⣴⡾⠋⢀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠶⠶⠶⠶⣶⣤⣀⠀⠀⠈⠙⠛⢻⡟⠛⠉⠁⣠⣾⠟⠋⠉⠉⠉⠉⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⢸⡇⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢸⡇⠀⢸⣟⣠⡄⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠰⢶⣼⣷⠀⠀⢸⣇⣤⣾⣿⣯⣁⣀⣀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣦⣤⣶⡶⠿⠛⠷⣦⣼⡏⠁⠀⠀⠉⠉⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⡀
⠀⠀⠀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⣿⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣯
⠀⠀⠀⣿⠀⠀⠀⣿⡁⠀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠻⢶⣤⣤⣤⣤⣤⣴⡿⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠈⣿⣄⣤⡶⠶⢶⣦⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣠⣤⣿⠟⠋⠀⠀⠀⠈⣿⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠛⠻⠶⣶⣦⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣶⠾⠛⠁
⠀⠀⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣄⣀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠛⠛⠁⠘⠛⠛⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣈⣉⣉⣉⡉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠋⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢹⣿⣿⣿⣿⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡈⢉⠉⠉⠉⣀⣾⣿⣿⣄⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣦⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣦⣀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣷⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣶⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⣴⣶⣤⣉⠻⢿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣤⣶⣿⣿⣿⣿⣿⣿⣷⣦⣌⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⠟⢋⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣉⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⠿⠛⢉⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠉⠻⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣄
⠀⣿⣿⡏⢠⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⠀⣿⣶⣶⣶⣶⣶⣶⣦⣶⣾⣿⣷⣤
⠀⣿⣿⣷⣤⣥⣄⢠⣤⣤⣤⣤⣤⡄⣠⡄⢠⣤⣤⣤⣤⣤⡈⣤⡄⢠⣤⣤⣤⣤⣤⠀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⣿⣿⣿⣿⣿⣿⡈⢉⣉⣿⣿⡏⢁⣼⣧⡈⢉⣉⣿⣿⡏⣀⣿⣇⠈⣉⣹⣿⣿⠋⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⡁⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⡄⣿⣿⣿⣿⠀⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⡿⠋
⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠋
⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⠅⣿⣿⡿⠿⠀⠿⠿⠿⠿⠿⠿⠟
⠀⣿⣿⣿⣿⣿⣿⢠⣾⣿⣿⣶⣶⡄⣹⡇⢰⣾⣿⣿⣶⣶⡄⣿⡇⢰⣿⡟⢠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⣿⣿⡟⢉⣭⣥⣤⣭⣭⣭⣭⣭⣤⣬⣥⣬⣭⣭⣭⣭⣭⣤⣬⣤⣬⣭⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣧⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁⢸⣿⣿⠟⢉⣤⣤⣌⡉⣡⣤⣤⣍⠻⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣼⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣿⣿⣥⣤⣤⣤⣼⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡈⠿⣿⣿⣿⠿⢿⣿⣿⡿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣦⣤⣤⣤⣶⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⢀⣄⠀⢠⡀⢀⡀⣀⠀⣠⠀⠀⢀⣄⠀⠀⣠⣤⠀⣠⣤⡀⣠⣤⡀⢀⡄⠀⣄⢀⣀⠀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⣿⠀⣿⣹⡄⢸⠿⣼⡇⣿⣾⡇⠀⠀⣾⣹⡆⢾⠁⠀⢸⡇⠀⢸⡏⠀⣿⢸⡇⠀⣿⢸⡿⣦⡇⠀⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣯⣼⣯⣬⣷⣼⣤⣽⣧⣿⣤⣿⣤⣼⣯⣬⣿⣬⣿⣶⣬⣿⣶⣬⣿⣾⣯⣤⣿⣾⣧⣼⣧⣼⣧⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠈⡀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠅⠀⠀⣠⠂⠀⠀⠠⠎⠀⠈⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⡠⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠄⠀⠀⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢄⡤⠀⡀⠈⠀⠀⠀⠀⠐⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠒⠀⠀⡀
⠠⡀⠀⠀⠀⠐⠈⠁⠀⠄⢠⠀⠀⠀⡄⢰⣉⣃⣘⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⡄⣶⣹⣁⣛
⠀⠀⠀⢀⠤⢠⡄⠀⠀⠀⠈⠃⠀⠀⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡃⠀⢠⠤⢠⠄⠀⠀⠀⠈⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠂
⠉⠀⠀⠌⠀⠼⠀⢠⡄⠀⠂⠀⠀⢀⠀⠀⠀⡀⠂⠀⠀⠀⠰⠃⠉⠁⠀⠀⠌⠠⠼⠀⢠⠄⠐⠂⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⡀⠀⠀⠀⠀⠄⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⣀⠀⢀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢢⡀⠀⢠⠄
⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠂⠰⠀⠊⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⡀⠄⠀⠀⠀⠀⢀⡐⠒⠢⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠠⢂⠐⠀⠂⠀⠀⠀⠀⠀⢀⠐⠒⠂⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠄⢂
⠀⠠⠠⠞⠀⠀⠀⠈⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠉⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠈
⡀⠀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟
⠀⠀⠀⠀⠉⠁⠐⠂⠀⢀⡀⠀⠀⠀⡀⢠⢰⡄⢲⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠔⠀⠀⢀⡀⠀⠀⠀⠀⣠⢰⠄⡦
⠈⢢⠀⠀⡀⠀⡀⠀⠀⠀⠘⠆⠀⠀⠃⠜⠃⠒⠒⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠚⠄⠀⠀⠃⠟⠐⠒⠂
⠒⠊⠀⠠⠉⢩⠁⠠⠔⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⢀⡂⠒⠒⠀⠀⠠⠉⢨⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⣠
⠀⠀⠀⠉⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⢂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠁⠀⠀⠰⠐⠈⣄⠃⠀⠘⠁⠀⡀⢀⠀⡄⠀⠀⠀⠀⠁⢁⠀⠀⠈⠁⠀⠀⠔⠀⠀⡌⠁⠀⠈⠀⠄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡞⢧⠀⠀⣠⢲⠀⡠⠒⠲⠤⣴⠆⢀⠀⠀⠦⠴⠆⠀⢸⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠙⠃⠁⠈⠓⢍⠁⠘⠂⠙⠛⠃⣰⠃⡼⠉⠀⢄⡠⠀⡤⣀⠎⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡒⣹⠃⠀⠀⢀⣀⠀⠀⠀⠀⡁⣀⣀⣻⡄⠉⠑⢾⡃⠦⢤⡔⢀⡼⠁⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⡀⠀⠀⠉⠀⠀⠀⣼⠁⠉⢉⠙⠇⠀⠀⢠⡙⠂⠻⠀⠀⣀⣀⣀⣠⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠙⠃⡶⠒⣢⢄⠀⠈⠙⢏⡁⢬⡀⠀⢰⠞⠳⡀⢷⠶⠾⡝⠒⠁⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠗⠙⠷⣤⣳⡲⠌⣡⣤⡶⠦⠤⠇⠀⠷⢦⣽⡀⣠⠛⠉⠀⠀⠳⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠉⠙⠊⠉⠀⠀⡄⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠀⢧⣠⡶⠀⠐⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢀⡀⠀⣀⠀⠙⠲⠶⠶⠷⢀⠈⠓⠒⠚⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣔⠷⣧⠻⢶⡜⠲⣴⠦⣤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠠⠴⠒⠀⠀⠉⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣾⠏⣹⡟⢹⡿⠉⣿⠒⠚⠉⠙⠳⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⡐⠀⠀⣀⡆⠀⠀⠀⠆⠠⢇⡄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠠⡿⢀⡾⢁⣾⠃⠀⢀⣴⣤⠀⠀⢿⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠀⠀⠀⠀⠀⢀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣃⣸⣃⣾⣁⣾⡁⠠⠤⢈⠉⠛⠁⠀⢨⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⠠⡄⠀⠀⠀⠈⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢫⣟⣿⣋⣿⢋⡴⠒⠲⢦⡘⠄⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⣿⣿⣿⣿⡟⢿⣧⣴⣦⢳⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠈⠌⢃⢘⣠⢏⠙⡂⠀⢀⠀⠀⠀⢠⣿⣿⣽⣿⣿⡿⠀⢿⣷⣄⠀⠸⣻⠏⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⢸⠉⠈⠀⠀⢄⣀⠀⠀⠐⠀⠶⣿⡿⠾⣿⡟⠀⠀⡀⠛⢫⡉⠉⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⡀⠀⢃⠠⣆⠀⠀⠘⠉⠉⠀⠈⠦⠆⠈⠁⠀⣠⡉⣩⡈⠀⠀⠈⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⢸⠀⠇⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢂⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⢠⠀⠀⠀⢠⠓⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠄⠈⠀⠀⠀⠀⠀⠳⠀⠀⠀⠈⡀⠁⠀⠀⠆⠒⠦⢠⠐⠰⠰⠒⠒⡆⠖⠒⠲⠰⠒⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⡆⠀⡀⠀⠀⠀⠀⠈⠢⢀⠀⠡⠀⣆⠀⠂⠀⠀⠂⡆⠀⠀⠀⠐⠀⠀⠰⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⠁⢘⡁⠀⣀⣀⠤⠀⠈⠀⠲⠂⠃⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠁⠁⠀⠀⠈⢀⠀⠀⢈⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢈⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠀⠀⠈⡀⠀⠈⠀⠀⠀⡄⣀⢀⠀⢅⠀⣀⠨⠀⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠄⠀⠀⠠⠐⠀⠀⠰⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠰⠀⠀⠀⠰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡂⡂⢐⠂⢑⢂⠐⡊⠐⠀⠐⠒⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢈⠀⠀⠀⠘⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⡬⠀⠀⠐⢠⡀⠀⠀⡀⢅⠀⠀⠨⢀⠀⠀⢰⠀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠊⠀⠀⠀⠀⠂⠂⠀⠐⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡴⠖⣉⠹⠷⣦⡀
⠀⢀⢳⠃⠀⡀⣤⠀⣦⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠦⢉⡉⠉⠙⠳⠦⣄
⠀⢸⢸⠀⠘⠧⠙⠆⠀⢀⠹⣇⠀⠀⠀⠀⠀⢠⢾⡞⠁⠀⠂⠙⠀⢠⡀⠀⠈⢳⡄
⠀⠀⢞⡄⠀⠙⣃⠸⣇⢸⡇⣿⠀⠀⠀⠀⠀⢿⠏⢰⠀⢠⡿⠘⡟⠘⠁⠀⠀⠀⢿⡄
⠀⠀⠈⢾⣦⠀⣍⣧⠀⣾⢡⡟⠀⠀⠀⠀⢸⣼⠀⠀⠀⠄⣉⠀⠙⠀⢤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠉⠒⠦⠤⣨⠵⠋⠀⠀⠀⠀⠀⢸⠹⠁⠀⠀⠀⠉⠃⠀⡷⡀⣷⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢠⣄⣀⡀⢉⡴⠏⢠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠈⠉⠁⠾⢓⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠛⠋⠁
⠀⠀⣠⣴⣶⣄
⠀⢰⣿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠤⠶⣶⣖⣶⡶⣶⣶⠦⣤⣄⣀
⠀⠀⢿⣷⣴⡿⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣏⡑⢦⣈⠉⠛⠶⣤⣀
⠙⠶⠾⠿⢮⣝⣲⣤⣞⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣾⣌⠳⣄⠀⠀⠉⠻⢦⣀
⠀⠀⠀⠀⠀⣹⡿⢋⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠷⠬⠧⠀⠀⠀⠀⠛⢷⣄
⠀⠀⠀⢠⣾⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⣷⣄⣀⣤⣴⠶⢶⡆
⠀⠀⢠⡿⡡⢋⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣭⡙⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⢉⣤⣶⢸⡇
⠀⢀⡿⠟⡱⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⢾⣿⣷⣄⠙⠿⡳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⣻⠞⢡⣴⣯⡓⠈⢀⣷
⠀⣼⣏⡜⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⣽⡀⠀⠨⣛⣽⣿⣷⣦⡈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠺⠿⣿⢿⡆⠀⣸⠇
⠀⣿⡛⡔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢹⣇⠀⠚⡡⠋⠙⠉⠈⠁⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⢠⣏
⢰⣿⡵⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡴⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠶⣤⠀⠀⠋⠙⣦
⠘⣿⣇⠇⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠞⢉⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣠⠤⢀⣀⣠⣤⣤⣤⣀⡀⠀⠁⠀⠀⠀⠘
⠀⣿⣿⢰⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⢏⡴⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣿⠾⠛⡋⡉⠉⠀⠀⣀⣉⡉⠳⣄⠀⠀⠀⠠
⠀⢸⣿⣿⢸⠀⠃⢠⠀⠀⠀⠀⠀⠀⠀⠜⣿⢨⠞⡡⠀⠀⠀⠀⠀⠀⠀⢀⣸⡏⣼⠃⢠⣿⣿⣟⡆⠀⣼⣿⣷⣾⠀⣻⠀⠀⠀⠐
⠀⠀⢻⣿⢾⣸⢀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⢩⡞⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡹⣇⠘⠿⣿⠿⠃⠀⠘⠛⠛⠋⣠⡿⠀⠀⠀⢈
⠀⠀⠀⢻⣷⣻⢸⡁⡀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠾⣡⠂⡀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣜⡳⣦⣤⣀⣀⣀⣀⣤⣴⠾⠋⠀⠀⠀⠐⢡
⠀⠀⠀⠀⠹⣿⢸⡄⡀⠀⠀⠀⠀⢀⡀⠀⠀⢻⣄⢫⡜⡠⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣉⣉⣉⠉⢉⣉⣉⣁⣤⠶⠋⠀⠀⠤⣠⡟
⠀⠀⠀⠀⠀⠈⢿⡷⠀⠀⠀⠀⠀⠈⣇⡀⠀⠀⠙⢷⣌⡡⠎⡴⢀⠀⡀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠁⠀⠀⠔⣀⡼⠋
⠀⠀⠀⠀⠀⠀⣸⣇⡆⡀⠀⠀⠀⢸⣯⠴⢂⠀⣀⢀⢌⠿⢶⣤⣉⠊⠤⠢⣠⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⡾⠋
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⣾⡶⣚⡥⣂⠙⣿⢋⡜⡠⠈⠙⠛⠛⠶⠶⣦⣥⣤⣭⣤⣤⣤⣴⣶⣶⠾⠟⠋⣹⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣇⣼⢠⠀⠀⠹⣿⣿⣿⣿⣆⢸⣿⡞⡔⡁⠀⠀⣴⣋⣮⣖⡴⣫⠞⣡⣮⣴⣿⣿⠇⠔⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡟⣰⠀⡀⠹⣆⠉⠛⠛⠷⣿⣿⡔⡡⠀⠀⣿⣿⣿⡿⠾⠷⠿⠟⠛⠛⢻⣿⡿⡀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣛⣴⠁⡀⠘⣧⠀⠀⠀⠈⣿⣼⡠⠂⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣗⡄⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣼⣃⣇⣾⠂⠀⠀⠀⠈⢿⣷⣵⣥⣼⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡾⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣭⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡇⠀⠀⣿⠏⣻⡋⢻⣽⣟⢻⣿⡿⢹⣟⣿⡛⣿⢽⣋⣙⣟⣩⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⡞⠡⡼⣹⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⡛⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⣽⢿⡍⣹⣿⣷⣸⣆
⠀⠀⠀⠀⠀⡠⢚⠙⢿⣿⡇⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⡻⣿⡄⠀⠀⠀⠀⠀⠀⢸⣇⠉⠈⢿⣪⠟⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠃⣚⡠⣼⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡲⢍⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣔⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠠⠤⠞⢡⣤⠀⠀⠀⠀⠀⠀⢰⡶⠒⡲⣒⣒⡲⡒⠲⣶⠀⠀⠀⠀⠀⢀⣾⠃⢸⣿⡦⠄⠛⠉
⠀⠀⠀⠀⠀⢠⣅⠐⣿⡇⣶⡆⢸⣿⠀⠀⠀⠀⠀⠀⢸⠀⠶⡹⣯⢜⡷⡱⠆⢸⠀⠀⠀⠀⠀⠘⠛⠃⠉⠈⡆
⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠸⠷⠤⣱⡯⠭⠷⠥⠴⠿⠀⠀⠀⠀⠀⠀⠀⡔⢪⠖⣩⡒⣴⢢
⠀⠀⠀⠀⠀⠈⠩⠤⠉⠡⠉⠡⠌⠩⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡺⣄⡞⣠⣟⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣄⡤⠭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠽⣋⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣶⣄⠁⣠⠟⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⢶⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠦⡝⢻⣿⣿⠃⠀⠀⠀⠀⣠⡞⠁⡘⠷⡧⣯⠞⠁⠻⣧⡄⠀⠀⠀⠀⣿⣿⣿⣿⡀⣫⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣿⣿⠃⠀⠀⠀⠀⠀⠈⠳⣄⣁⣠⠞⡅⣿⣦⣴⡿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠱⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠋

⠀⠀⠀⠀⣶⣶⣶⣶⣶⡆⠲⠤⠤⠄⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠀⡀⠀⠤⢀⢤
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢶⢦⡈⠉⠢⠈
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢀⠀⡀⠠⢄⡀⠀⠀⠀⢢⡀⠀⢠⣄⠀⠉⠧⡀⠠⠆⠅
⠀⠀⠀⠀⡿⠿⠿⠛⠛⠁⠁⠀⠀⠀⠀⠈⠑⠢⣄⡀⠐⢠⡄⠙⣛⣢⣄⠈⡧⠥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢨⣙⠙⣒⣋⣓⠿⠋⠄⠠⠪⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠩⠘⡓⠶⢚⠁⢀⡘⠹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠥⣢⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠯⡈⠑⢄⢀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢸⡚⠇⠀⠀⠖⣒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⢈⣤⣤⣤⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡘⠁⡠⠄⠤⡀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠫⡄⢸⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠈⠳⠬⠴⠋⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠤⠴⠤⠤⠤⠤⠤⠔⠁



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⡶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⣻⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⡤⢄⠀⠀⠀⠉⠛⠻⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠚⠹⣆⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣟⣂⡙⡖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢒⠒⠤⣈⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢂⡡⠐⣺⡏⣿⡗⠠⢁⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠁⠴⠊⠤⠤⠤⠜⠷⠟⠤⠤⠤⠬⠰⠄⠁⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⠥⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠭⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠭⠭⠏⠯⠭⠽⣉⣉⣉⡹⡭⢭⠏⢫⠭⠽⠉
⠀⠀⠀⠀⠀⠀⢠⠤⡤⠤⣀⠤⠤⢀⠀⠀⠀⠀⠀⢸⡅⠀⠀⡨⠄⢸⠀⢰
⠀⠀⠀⠀⠀⡔⠉⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⡇⢸⠀⢸
⠀⠀⠀⢀⠜⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠁⡇⢸⠀⢸
⠀⠀⠀⠸⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⢘⡇⠀⠀⢀⡃⢸⠀⢸⠀⣠⣤⠤⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⡤⠸⠍⠈⠇⠏⠨⠹⠇⠀⠀⠾⠅⠩⠧⠯⢸⣷⣿⣶⣿⠮⠐⣐⠪⡢
⠀⠀⠀⠀⠀⠀⠀⠧⠀⠝⠉⣏⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⢰⡀⡿⠀⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠓⠥⠤⠮⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠕⠀⠀⠁⠪⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⢠⡥⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⣀⣀⡐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠐⠁⠀⠀⠀⠑⠌⠄⡀⡀⠀⠀⠀⡀⠄⠀⠀⠀⠠⠦⠤⠶⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠁⠀⠀⠀⠀⠀⠘⢏⠘⠁⡃⠀⢀⠀⡁⠀⠉⠀⠀⠐⠒⠊⠉⢉⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠓⠃⠀⠀⢣⡁⠀⠀⠀⣀⣀⣀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⣉⣁⠀⢄⠀⡄⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠤⣖⣤⡶⠄⡀⠀⢀⠈⠉⠉⠉⠉⠉⠈⢁⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠔⠁⠀⠀⠨⢆⠈⢀⠰⣠⠀⠀⠀⠀⡃⠠⠀⢀⣀⣀⣀⡠⠔⡇⠀⠸⠦⢤⠤⢠⣤⣶⣾⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⢿⣀⠀⠀⠀⠀⢀⣀⣤⠇⠀⢀⡦⠀⠀⠀⠁⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠈⠓⠒⠲⠅⣀⠀⣇⡀⠉⠉⠛⠛⠛⠙⣸⠀⠀⠀⠁⠀⠀⠀⠤⠰⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⢠⠀⠉⠉⠂⠐⠂⠀⡇⠀⠙⠀⠀⠄⠤⠤⢶⠞⢿⠀⠀⠰⠇⠀⠀⠀⣀⣀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠶⠶⠦⠀⡀⠀⠾⠁⠀⠠⣤⣤⣤⣦⡇⠀⢠⡓⠀⠀⠀⠀⠀⢀⡼⠀⠀⢐⠀⠀⠀⠈⠈⠁⠁⢤
⠀⠀⠀⠀⠀⠀⣆⠈⠀⢐⠀⠒⠒⠉⢀⠀⢠⡀⠀⠈⠉⠉⠉⠁⣧⠀⠀⢀⠁⠀⠚⠓⠛⠩⢰⠄⡀⠈⠀⠀⠀⠤⠤⠤⠶⠟⡇
⠀⠀⠀⠀⠀⠀⠑⠠⠄⠠⢤⢤⣤⡴⠊⠁⠘⠢⠤⣀⣤⢤⡤⠶⠁⠔⠈⠀⠅⣀⡤⣤⣤⠔⠋⠀⠀⠲⣌⣀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⣡⣶⠿⢷⣦⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠰⣿⡇⢀⣼⠋⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⣿⣿⡄⠀⠀⠀⢸⠇⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⢩⣤⡄⠀⠀⡜⠀⠺⣄⣀⡽⠀⢀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣀⣈⣉⣤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠘⢤⣤⠽⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⡆⠀⢀⣴⡾⠟⠛⠛⠛⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠀⠀⢢⠀⠀⢀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠰⠶⠀⠀⠀⠈⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠐⠂⠉⠑⠦⣄⡀⠀⠀⠀⠀⠀⠃⡐⠀⠀⠠⠀⠒⠀⣀⣠⣀⡀⠀⠴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂⠀⠈⠃⠈⠿⠷⠤⢄⣀⠀⠐⠙⠂⠀⠀⠀⢀⠔⣛⣻⠙⢃⡀⠱⠀⢡
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠤⠄⣀⠈⠑⠆⠀⡄⠀⠀⠀⠀⢶⡿⡃⠂⠀⠀⠁⠀⠌
⠀⠀⠀⠀⠀⠐⢁⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⢀⠔⣲⡄⠉⠂⢤⡀⠀⠙⠲⣔⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⣠⠖⡧⠴⠂
⠀⠀⠈⠉⠁⠀⠀⠀⣀⡀⠀⣀⠀⠀⠈⡇⡴⠋⠀⠀⠀⢀⠔⢳⡮⠵⢡⣦⣤⣿⣂⠓⠤⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈
⠀⠁⠀⠀⠀⠄⠤⠤⠤⠤⠄⡜⠀⠀⠀⡟⠀⠀⠀⠀⡰⠋⡀⠀⠣⠀⣤⣵⣦⡞⢉⡠⠔⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⠎⢠⣑⣒⡣⠔⡘⠷⠞⠛⠀⠀⢀⠰⠂⠀⠀⠀⠀⠀⢠⣶⣾⣶⣿⡇⢹⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣯⠌⠀⣀⠠⠀⠀⠉⠀⢀⠠⠀⠈⢄⠠⠄⡀⠀⠀⠉⠁⠛⠿⠛⠋⠁⠀⠀⠻⣿⣿⡆
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠴⠡⠌⠐⠈⠀⠀⠀⡀⠰⠊⡤⠀⠀⠘⠦⠤⠂⠹⠀⠤⠄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠄⠀⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡔⣤⣐⢦⣄⡰⣥⣄⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠠⠤⠄⠐⠊⠀⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣣⠶⣿⡦⠾⣷⠶⣿⣤⡽⠞⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠁⠀⠆⠀⠀⠀⣁⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⢠⡿⠀⣼⠃⣰⡟⠀⠀⢀⣀⠀⠈⢙⣟⣂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⠀⠀⠀⠂⠀⠀⠀⠉⠀⠈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⣼⠇⣼⠇⣰⡟⠀⠀⠀⢟⣿⡇⠀⠀⢳⣍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠆⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣿⢴⣿⣴⡿⣶⡿⠐⢀⣀⠓⠀⡀⠀⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠡⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣿⣵⣿⣶⣿⣴⣎⡍⠉⠙⢦⠈⠄⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠀⣤⠤⡤⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣽⢿⠿⣿⡇⠻⠿⠻⣿⣸⡇⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢀⠄⢠⣠⣨⡜⠂⠑⠀⠀⠀⠀⠀⠀⢠⣻⣟⡯⣿⣿⣿⠏⠀⢻⣿⣦⣀⣘⣩⠟⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠈⡄⠈⠀⠇⢀⠐⠀⡄⠀⣀⠀⠀⠀⠒⢿⡿⠿⠿⣿⠃⠀⠀⣈⠉⠃⡄⠄⠀⣸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠊⠀⠀⠀⠀⡇⢠⠈⠀⢠⡆⠀⠀⠘⢉⠍⠉⠈⠱⠴⠅⠉⠉⢈⣭⠍⣭⡍⠉⠈⠉⠉⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠙⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⢠⢃⠀⠠⠀⡀⠀⠀⠀⠀⢠⣎⢊⣄⠈⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⡄⠆⠀⠀⠨⠀⠆⠐⢠⡄⠀⢠⢠⠀⠀⡀⠀⠀⠀⡄⣄⠀⢠⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠰⡄⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⡄⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠂⠡⡂⠀⠀⠀⢘⠀⠀⢂⠀⣰⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⡇⠀⠀⠀⢈⡀⠀⠀⠀⡆⠀⠀⠀⠤⠀⠀⠤⠀⠄⠀⠀⠀⠠⠠⠀⠤⠀⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠁⠡⠀⠀⠀⠁⠀⠨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠀⠒⠒⠂⠒⠀⠂⠒⠐⠂⠐⠒⠒⠒⠀⠓⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠁⣉⢀⡉⠀⠈⡀⢈⠈⠉⢈⠀⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⠀⠀⠀⠀⢀⣆⣀⡀⠔⢠⣀⠀⠀⣀⠀⡀⠀⢀⡀⢀⢀⠀⣀⠀⡇⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠔⠀⠀⠀⠀⠘⠂⠀⠤⠤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠


⣾⣿⣿⣶⣶⡷⢿⣾⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⢷⡿⠿⡿⢶⣶⣿⡿⠶⠶⢶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⢆⠀⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⡷⠈⠀⠈⠓⠙⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⣠⠶⠶⢆⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡾⠉⡲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡏⢃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⣤⠀⣠⡀⠀⠠⡄⠄
⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠱⠀⠙⡆⡇⠑⡆⢸⠗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠂⠀⠐⠂⠀⠐⠂⠀⠀⢠⣄⡀⠀⠀⠀⠀⠄⡀⠠⠄⠄⠀⡀⢀⡠⠄⠀⠠⠀⠀⣤⡀⢀⠀⠀⠆⠀⠒⠂⠀⠐⡂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⠀⠐⠀⠀⢸⠁⠀⢸⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠃
⠀⠀⠀⠀⠠⠤⠴⠀⠀⠀⠤⠤⠀⡀⠀⠀⠀⠠⠤⠤⠇⠀⠆⠠⠤⠄⠀⠀⠘⠠⠤⠐⢀⠀⠀⢀⠀⠤⠄⠀⠀⠀⠦⠤⠄
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⢠⠀⢠⠀⠀⢠⠈
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠁⠀⠀⠀⡇⠁⡿⠀⠀⠀⢨⠁⢸⠄⠀⢨⠈⠀⠀⠈⠈⠀⠠⠀⠀⠀⡄
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⡀⡁⠀⠀⡀⢀⣀⣀⡇⠀⣟⠀⠀⠀⢨⠀⢸⣀⣀⣐⢀⠀⠀⠀⠈⣀⣀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣴⣶⣦⡄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣿⣿⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⢹⣿⣿⢱⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⣬⣭⣭⣬⣭⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⢠⣶⣶⣶⣶⣶⣾⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠈⣿⡆⢀⣤⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣠⣾⡿⠻⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢀⣤⣭⣤⣤⣤⣤⣭⣥⡍⣿⣿⣿⠿⠃⡿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣔⠿⢾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢡⣷⢸⢏⣾⣿⣿⡇⣿⣿⡏⣾⡇⣿⣮⣿⣿⣿⣿⣿⡜⣿⣿⣙⠿⣪⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣭⢎⡎⣶⡎⣿⡇⣿⣿⣿⣦⡅⣛⢿⣿⣿⣿⣿⣿⠿⠎⠛⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣿⣷⣭⣵⡿⢇⣿⣿⠹⣿⡇⣿⡇⣿⣿⣿⣿⣿⠸⠿⠿⣿⣿⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣿⣿⣿⣷⣮⡅⣭⣶⣿⢟⣛⣛⣛⣘⣛⣛⣛⢿⣟⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣛⣛⣿⣿⣟⣛⣛⢸⣟⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡤⠴⠒⠒⠒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⡏⠙⠉⠀⠀⠀⠀⣶⢲⠏⢹⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠈⠉⢉⣙⣿⠢⣄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠉⠓⠒⠒⡻⠛⠉⠉⠉⠉⠑⢮⣦⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⠤⠤⣄⡀⠀⢱⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡞⢁⡴⡶⠄⠹⡄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⣭⣯⠇⢠⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠀⠀⠈⠳⠤⠥⠴⠋⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼

⠀⠀⠀⠀⠀⠀⠀⢀⡆⡒⠒⠒⣢⢀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠞⢳⡀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠩⠽⡃⠀⠀⠀⠀⠀⠀⠰⠲⠖⠒⠒⠲⠖⢦
⠀⠀⠀⠀⠀⠀⢨⠉⠉⢉⢌⠉⠙⢸⠀⠀⠀⠀⠀⠎⠀⠀⡴⠄⠀⠱⡄⠀⠀⠀⠀⠀⠘⡩⢃⣤⡘⢍⠃⠀⠀⠀⠀⠀⡛⠀⠒⢒⡐⠒⠀⢻
⠀⠀⠀⠀⠀⠀⠰⠤⠄⢘⢢⠀⠠⠼⠀⠀⠀⠀⠀⡇⠀⠀⢭⠆⠀⢠⠆⠀⠀⠀⠀⠀⢸⠁⢈⣳⡄⠘⡀⠀⠀⠀⠀⠀⡇⠀⠀⠛⢌⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠰⣄⢀⢀⣁⣀⣀⡴⠀⠀⠀⠀⠀⠈⠲⣤⠴⢤⡔⠋⠀⠀⠀⠀⠀⠀⠘⠦⠄⡉⠠⠜⠁⠀⠀⠀⠀⠀⢧⢀⣀⢈⢁⠀⠀⡼


⠀⠀⠀⠀⠀⠀⢠⠖⠀⠒⠒⠒⠒⠦⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢒⠢⠄⠀⠀⠀⠀⠀⠀⠀⢠⠒⣶⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⢸⠀⣜⠂⠀⡒⠀⢸⠀⠀⠀⠀⠀⡏⡰⠦⡀⠠⠤⠀⠁⠀⠀⠀⠀⢀⢎⠞⢉⡤⠉⢢⠰⡀⠀⠀⠀⠀⠀⢸⠀⠉⡉⠉⠉⠇
⠀⠀⠀⠀⠀⠀⢸⠀⢠⠕⠀⠤⠄⠘⠀⠀⠀⠀⠀⡇⢈⠊⡀⢀⠉⡁⠀⠀⠀⠀⠀⠸⡸⡀⠈⣑⠄⢠⢂⠇⠀⠀⠀⠀⠀⢸⠀⠂⠐⠀⡰⠦
⠀⠀⠀⠀⠀⠀⢸⣀⣄⣀⢄⣀⣄⣀⠀⠀⠀⠀⠀⠣⠄⠀⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⠁⢙⡢⠤⢔⡫⠊⠀⠀⠀⠀⠀⣴⢺⠀⠉⠉⠁⢈⡳
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠈⢰⠛⠛⡛⠛⢓⢸⠀⠀⠀⠀⠀⠀⠀⠠⣎⠔⠙⢦⡀⠀⠀⠀⠀⠀⣴⠒⠋⣀⠙⠒⡆⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣏⢉⡧⢸⢸⠀⠀⠀⠀⠀⠀⠀⣠⢞⠢⣔⡡⠃⠀⠀⠀⠀⠀⢹⠀⠘⠦⡀⠀⡇⠀⠀⠀⠀⠀⠘⡖⢲⠲⠖⠶⢲⠛
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢉⣋⣁⣸⢸⠀⠀⠀⠀⠀⣠⢞⡵⠋⣀⣈⣀⡀⠀⠀⠀⠀⠀⠈⢧⡈⠛⠀⠜⠀⠀⠀⠀⠀⠀⢀⣃⣘⣀⣀⣁⣘⡀
⠀⠀⠀⠀⠀⠀⠘⠒⠂⠐⠒⠒⠒⠚⠀⠀⠀⠀⠀⠓⠋⠀⠀⠓⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠒⠒⠒⠒⠃


⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢾⡉⠢⡀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠱⠆⠀⠀⠀⠀⠀⠀⡠⡄⠠⡛⠀⠠⢄⠀⠀⠀⠀⠀⠾⠩⠭⠭⠽⡆
⠀⠀⠀⠀⠀⠀⢘⢧⡈⠢⣀⣹⡖⢻⠀⠀⠀⠀⠀⡎⠀⠀⢎⡃⠀⠀⠃⠀⠀⠀⠀⢀⠇⣥⡀⠼⢂⣄⠸⡀⠀⠀⠀⠀⡇⠀⠀⢀⡤⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠠⡑⢝⣊⣽⢀⣯⠽⠀⠀⠀⠀⠀⠣⠀⠀⠢⠇⠀⢀⠇⠀⠀⠀⠀⠘⡄⢇⢱⡀⠌⡜⢠⠃⠀⠀⠀⠀⠇⠀⠀⡎⠀⠯⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠤⠤⠔⠋⠀⠀⠀⠀⠀⠀⠘⠄⠀⠑⠀⠰⠃⠀⠀⠀⠀⠀⠻⠭⠡⠘⠢⠭⠄⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⠤⠔⠂⠒⠀⠀⠀⠠⠆⠒⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣉⠕⠊⠁⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠁⠀⠀⠀⠠⠊⠡⡀⢀⠀⠀⠀⠀⣠⠂⣀⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⠀⠀⡴⢋⠕⠀⣦⣉⡆⠀⠀⠀⠀⠀⠀⢂⢤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠼⠅⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠑⠤⠂⢧⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢰⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡆⠰⣍⠓⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣀⡇⢓⡼⢠⠇⠀⠀⠀⠀⢀⡗⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠦⠴⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢬⡀⣠⠔⠁⠀⠀⠀⠀⢀⡞⠒⠶⠖⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⣥⣀⣀⣀⣀⡠⠦⠤⠤⠀⠀⡀⢹⣷⣶⣶⡆⠀⠀⠀⢠⣀⡀⣀⣀⣠⣤⠴⠞⢓⠀⣀⡗⠘⠀⠀⠂⠈⡍⠀⠀⠇
⠀⠀⠀⠀⠰⠤⣄⡀⢤⣄⣀⣀⣀⣀⡤⢤⠀⠀⠐⠒⠚⣄⠀⠀⢀⡍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⣉⣁⣡⠤⠗⠤⠔⠂
⠀⠀⠀⠀⠉⠙⠒⠓⠛⠀⣀⣁⣀⣀⣠⢼⢀⡀⠀⢀⢀⡌⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⠤⠜⠴⠤⠤⠤⠖⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⣄⡀⠀⠀⣠⢾⡿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⢀⡼⠁⠀⠀⠀⣈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢏⣾⣷⣦⣄⢠⣾⠟⣛⠟⣷⡙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⢯⣾⣿⣿⣿⣿⣮⣛⠀⢩⣞⣸⡇⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣯⣝⡁⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣕⢮⡙⠻⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠙⣿⣿⣿⣿⣿⣷⡝⠳⣄⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠁⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣝⠦⣄⠀⠀⠀⠀⣠⢞⣵⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⡳⣶⣶⡞⣵⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⢸⣿⣿⣿⣿⣿⠈⠉⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠘⠿⠿⠿⠿⠟⠀⠀⠻⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡷⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠄
⠀⠀⠀⠀⠀⠀⠀⡄⢤⡐⣮⢱⣆⡲⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣘⠹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠰⡠⠀⠀⠈⠱⢾⠿⠉⠯⡉⠁⠀⠸⡄⠀⠀⠀⠄⠀⠀⠋⠀⢉⡇⠈⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⣀⠘⣷⠀⠀⠀⠠⠆⠁⠀⢀⠀⠆⠀⠀⠀⠀⣼⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠄⠟⠁
⠀⠀⠀⠀⠄⠐⠀⢿⡀⢂⠉⠀⠀⠐⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⣠⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡍⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠐⠂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢡⠏
⠀⠀⡸⠀⠀⢹⠀⡁⢱⠀⣿⠁⢸⣿⡆⠀⠀⠀⠀⢀⢊⣾⣷⢢⣤⡤⠔⠀⠄⠄⠔⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠈⠀⡠⠋
⠀⠀⢻⡀⠀⠘⠀⠐⠋⠀⠷⠖⠘⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢵⣢⡀⢀⠜⠁
⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣠⠋
⠀⠀⠀⠀⠈⠙⠛⠶⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣠⢄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠅⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠂⡐⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠠⡀⠀⣖⠀⠹⠆⠀⠀⠀⠀⠀⠁⠀⠀⠈⣏⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠁⢠⡖⠆⢱⣀⡉⣡⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢿⣶⣦
⠀⠀⠀⠀⠀⠈⠣⣌⠁⣀⠞⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⣦⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠈⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠀⠤⠤⠤⠀⠤⡄⠀⠀⠀⠀⢀⠜⡀⠠⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠸⣧⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠳⣄⠀⢀⡠⠒⠁⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠁⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠂⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢐⣈⢀⠇⣛⣾⠇
⠀⠀⠀⠀⠯⣄⠀⠀⠀⠀⠠⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠷⠒⠒⠛⠉
⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⣦⠀⢀⡄
⠀⠀⡀⢀⣀⡠⠎⠀⠀⠈⠉⠉⠙⠙⠉⠉⠀⠀⠀⠀⡀⠀⢀⢰⠀⠀⣮⣤⣦⣴⣶⠀⠀⠀⠀⠙⢀⠀⡀⣻⠇⠉
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠚⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣁⠒⠉⠐⠴⠴⠒⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠤⠀⠀⠀⠀⠀⠀⠀⠛⠓⠦⠆⠀⠀⠐⠀⡁⡀⠖⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠆⠀⠀⠀⠀⣠⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⢀
⠀⠀⠀⠀⠀⠠⣤⡦⠶⠟⠉⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠓⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠸⠀⠘⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⡶⠶⣾⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣤⣭⣭⣉⡛⠲⠯⣝⡻⢿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣔⡯⣓⢮⣟⢿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣶⣾⠿⣛⣤⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⡪⡷⢝⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣵⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⡻⣿⣿⣿⡇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣛⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣾⢛⣭⣭⣭⠧⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣇⣿⣿⢋⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣻⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⢸⣿⢣⣿⣿⣿⣿⣿⣿⣿⡟⣛⢿⣿⣿⣿⢸⡇⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢿⡟⣿⢸⣿⣿⣿⣿⣿⣿⣿⡟⣿⡎⣿⣿⣿⣷⣵⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣃⣿⣿⠿⣫⣽⣶⠶⠿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣿⡿⢋⣵⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢨⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣿⡟⣵⣿⣿⣿⡏⣮⢿⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⢏⣦⣿⡇⢋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⣎⢛⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⣝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢯⢿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠉⠉⢭⣭⣶⣿⡶⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠐⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠶⠶⠶⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣀⣀⣀⣈⣆⢲⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣤⣤⣤⣤⡄⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⢻⣿⡏⠉⠉⠂⢻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⡜⠃⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡹⣅⡄⠀⠀⢢⢻⡃⠀⠀⠀⠀⠀⣤⢤⡀⠀⠀⠀⠀⣀⡈⡘⠛⠿⣶⣶⠄⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠜⣧⠄⠀⠀⢋⣯⡁⠀⠀⠁⢸⣱⣶⣟⡄⠀⠀⢀⣥⡰⠁⠀⠰⢿⢋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠂⠀⠈⣿⣇⡀⠀⠐⠀⠳⢤⠟⠀⠀⠀⢠⡄⠀⠀⠠⢶⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠶⡢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡀⠀⢩⣿⡀⠀⣀⣤⣤⢤⣀⠀⢀⡼⠀⠀⠐⢦⠇⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⠉⠛⠛⠣⠾⠿⣿⣿⣦⡈⢳⣸⡇⠀⠀⢲⡟⠀⠀⠑⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡇⠀⠀⠀⠠⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣽⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠁⠀⠐⠦⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⡇
⠀⠀⡀⢀⡀⢀⢀⡀⢀⠀⡀⢀⡀⠈⣧⣀⠀⠀⠀⠀⡀⢈⡉⡉⢉⠉⠓⢒⡒⠒⢣⣀⡀⡗⠉⠹⣯⣈⣁⢀⢀⣁⡀⠀⠀⢀⢀⠀⠀⢀⣀
⠀⠀⠉⠀⠁⠈⠁⠀⠀⠀⠁⠈⠁⠀⠈⡟⠃⠀⠀⠀⠁⡀⠀⠉⠀⠀⠀⡄⠁⠉⢨⡍⠁⠁⠀⠀⣹⢯⣈⣀⡨⠁⠀⠀⠀⠉⠈⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠲⢶⠀⠀⢸⠀⠀⠀⠀⠀⢄⣇⡀⢸⣤⣴⡟⠀⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⡦⡞⠢⠞⠙⠒⠲⠶⠛⠛⠋⢃⡸⠀⠀⢹⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⡀⠀⠀⠀⠙⠁⠀⠀⠀⠀⡆⠀⡄⠀⠀⠀⠀⠀⢷⣼⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠦⢤⣀⠀⠀⠀⠀⠀⢁⡆⢀⠀⣠⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣄⢀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠛⠛⠛⡆⡏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⢻⣤⡄⣀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠇⣇⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⡜⣼⣿⡟⣷⠶⠤⣧⣤⣼⣤⠀⠀⢾⠀⢿⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣰⣿⣏⣇⣿⠀⠀⠀⠀⠀⠀⡀⠐⢸⠀⢸⣄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣿⣇⣉⣸⢹⣀⣀⣀⡀⠀⠀⠀⠀⠘⣤⠾⣼⡄⢎⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢈⡉⠉⠉⣍⢸⣉⣉⣉⣉⠛⠛⠛⠛⠛⠛⠛⢛⣋⣈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣭⠤⠶⠀⠀⠙⡋⠉⠛⠛⠳⠶⠶⣦⣤⣤⣍⣹⣿⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠀⠤⠤⠤⠬⠄⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠉⠀⠀⠀⠀⠀⠙⠻⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠉⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢛⣿⡟⠛⠛⠛⠛⣿⡟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣾⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠸⣿⣷⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⣋⣉⡙⠛⠿⣭⡻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⠛⠉⠀⠈⠙⠲⣄⡀⠙⠲⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡈⠻⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠙⣆⠹⡄
⠀⠀⠀⣀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠙⠲⣄⠀⠀⠸⡆⣧
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⠳⡄⢀⣿⣯⣀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⠤⠖⠋⠉⣹⡏⠙⣿⠛⢦⣄
⠀⠀⠀⣿⠀⠀⠀⣠⡴⢶⣄⠀⠀⠀⠀⠀⣀⡤⠦⣄⣀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⣿⠀⠀⠉⠓⢦⣄
⠀⠀⠀⢸⣤⠴⠋⠁⠀⠀⠈⠛⠶⠤⠤⠚⠉⠀⠀⠀⠈⠉⠙⠛⠛⠉⠹⣤⣀⣀⣠⡾⠃⣠⣄⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠉⠁⠀⣰⣿⠃⠀⠀⠀⠀⡾⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⣼⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⣼⠃⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣼⠃⢀⣤⣶⡶⣄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠘⣧⡀⠀⠀⠀⣼⠃⠀⢸⡷⣿⡷⣿⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿⢦⠀⠘⢧⡀⠀⣼⠃⠀⠀⠈⠻⠿⠛⠁⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣷⣸⣷⡀⠈⢷⣼⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣿⣎⡧⠀⠈⣿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡈⢷⡇⢹⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⢳⣼⠁⠀⠀⡇⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⢀⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢸⡇⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⢸⠇⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣄⣀⣀⣀⣤⣧⣀⣀⣀⣼⣇⣀⣀⣀⣸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣏⠀⣸⣏⣹⣙⣙⣋⣟⣍⣟⣙⣻⣏⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣯⣽⣿⣟⣋⣻⣿

⠀⠀⠀⠀⠀⢀⠖⢦⡀⠀⠀⠀⢀⡴⠂⠉⠛⢖⠤⡀
⠀⠀⠀⠀⠀⢸⣠⠈⣼⣆⠀⢀⠊⠔⠚⡀⠀⠀⢣⠈⢆⣀⣀⣀⣀⣀⠠⠤⠤⠤⠄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⠉⠀⢹⣿⠦⡎⠀⠀⠀⠀⠀⠀⠀⢧⠏⣃⠀⣠⣤⣒⣲⣶⣶⣒⣖⣂⣀⠠⢀⠉⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⣆⠔⠊⠀⠀⠧⣀⠀⠀⠀⠀⠀⠀⢸⣱⢹⡄⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃⠀⠀⠀⠀⠀⠁⠢⢄
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠙⢄⢲⠤⣀⠀⢸⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡲⢶⠀⠱⣄
⠀⠀⠀⠀⣾⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠙⣷⡣⡸⡙⡚⠫⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠊⢍⡛⢶⡅⠀⠉⢷⡀
⠀⠀⢀⠜⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠑⠘⠐⠉⠪⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⣉⠢⡈⠢⡱⠀⠈⢐⣷
⡴⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⢢⠁⡀⠀⠱⠀⠀⢐⠈⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⠑⡌⢢⠱⡈⠄⠂⠀⠀⠐⢴⡿⠻⡕⢆
⣇⣀⡄⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡈⠢⡱⡈⡆⢃⠰⡘⠀⠀⠀⠨⣮⠃⡟⠿⡸⢟
⡿⠻⡇⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠌⠜⠄⢱⢱⠘⡈⠆⠃⠀⠀⠀⠑⡟⠀⠈⠉⠁⠈
⢩⠀⡇⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⢡⢰⠸⠘⡜⡄⠃⡆⣃⠀⠀⠀⠀⢢⢱⠇
⠘⣆⡇⣼⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢸⠘⠘⠈⠀⠇⠁⠡⢸⠃⠀⠀⠀⠀⠠⡱⡟
⠀⠈⠁⠈⢣⡀⠒⠤⠤⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠸⠸⢸⠸⠀⠀⢠⠀⡀⡀⠀⠀⠀⠀⠀⠀⢠⢠⡿⠁
⠀⠀⠀⠀⠀⠙⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠢⠃⠂⠆⠂⠈⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⡼⠁
⠀⠀⠀⠀⠀⠀⡗⣿⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⡀⡠⢀⠀⠄⡠⠀⠀⠀⠀⠀⠀⣀⡀⢼⠉
⠀⠀⠀⠀⠀⠀⡇⣻⢽⠃⠀⠀⠀⠀⠀⠀⣐⠀⢾⠓⠛⠒⠥⠮⠴⠥⣮⡮⠴⢵⢷⢾⡾⣿⠁⠀⠀⠀⠀⠀⡶⠬⢹⡀
⠀⠀⠀⠀⠀⠐⠷⢶⣿⡀⠀⠀⠀⠀⠀⠨⠥⠬⢻⣿⣿⣻⣿⡿⢽⢿⢿⣟⣿⣻⣅⣫⣺⣿⠀⠀⠀⠀⠀⠀⠉⠭⣼⠷⠖⠾⠟⠒⣒⣂⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠖⡠⣤⣤⣬⣤⣽⡛⡛⢛⣛⠃⠀⠉⠈⠉⠉⠀⠁⠈⠉⠉⠙⠲⠶⠶⠶⠶⠶⠇⠌⠁⠈⠉⠁

⠀⠀⠀⠀⡤⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣂⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢂⡄
⠀⠀⠀⠀⢑⣚⣤⣀⣠⡏⡝⡧⠀⠀⠀⠀⠘⠐⢠⡀⠐⢈⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡀
⠀⠀⠀⠀⣼⠋⣶⠙⣧⠙⠊⠁⠀⠀⠀⠀⠐⢀⠈⢣⡠⣦⡝⡁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣁⠸⠁
⠀⠀⠀⠀⠸⠷⣶⠞⠇⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠹⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐
⠀⠀⠀⢠⠀⠀⠀⠀⡀⠄⢠⠀⡄⠀⠀⠀⠀⠀⠄⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠

⠀⠀⠀⠀⠐⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠤⠤⠤⠤
⠀⠀⠀⡀⠀⠀⢠⣦⠀⠀⢠⣠⡀⠀⠀⠀⠀⠀⠍⠀⢈⢁⠤⡀⠀⠀⠀⠀⠀⠘⢻⣽⣄⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢎⠀⠀⠀⠀⠀⠀⢼⠈⠡⠤⠤⠤⠀⡁
⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⢤⠠⣀⡄⣀⡠⠀⣠⠀⠀⠀⠀⢤⢀⠄⠄⢀⣀⡄⡄⣀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠤⠠⠄
⠀⠀⠀⠀⠀⠰⠤⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⢠⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡘⠈⠘⠐⠀⠀⠀⠀⠀⠀⠀⣄⢰⢀⠄⠀⡁
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠹⠀⠴⠀⠀⠀⠀⠀⠀⠀⠴⠔⠡⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠦⠠
⠀⠀⠀⠀⠀⢀⡀⡀⠀⡀⡀⠀⠀⠀⠀⠀⠀⡀⣀⢀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠠⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢⠑⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠆⠠⠭⡆
⠀⠀⠀⠀⠀⠀⠀⢅⢐⣛⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠄⢀⣏⢐⣶⠓⢀⣀⡀⠀⠀⠀⠀⢄⠀⠀⠀⢀⡐⢀⠀⠀⠀⠀⠀⡴⠂⠀⢄⡆⠀⡀
⠀⠀⠀⠀⠈⠅⢺⣟⠘⣇⠐⣒⣺⡇⠀⠓⣄⠀⠀⠀⠡⠂⠍⠀⠀⠀⠈⠀⠀⠀⡇⠀⠼⠭⣷⢰⡿
⠀⠀⠀⠀⠀⠁⠚⠛⠈⠿⢀⣒⣶⠏⠀⢠⡀⠀⠀⠐⢲⣦⠐⠂⠀⠀⢀⠀⢠⠄⣃⣀⣩⣭⣯⣨⡽⠁⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⠀⠴⢬⠉⠁⠉⠉⠉⠒⠒⠒⠒⠀⠀⠘⠒⠒⠒⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠠⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⢨⡭⠶⠒⠛⠛⠋⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣭⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠍⢱⠀⠀⠀⠀⠀⠀⠀⠀⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠈⠁⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡉⠉⣠⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠋⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠈⠉⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡸⠓⠒⠛⡋
⠀⠀⠀⠀⠈⠀⠐⠒⢻⡀⢀⣀⣀⣀⡍⡶⣤⠀⠀⠀⣄⣐⡤⠥⣶⡞⠁⢠⡇⠁⠀⠀⠃⣀⢤⣤⣤⠆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⣉⠁⠘⠦⠿⣀⣀⠸⠄⠀⠁⠈⡙⠃⠤⠊⠁⠀⠘⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠛⠉⠀⠀⠀⠀⠀⠸⠀⠀⣤⣤⣴⠇

⣿⡏⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⠀⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄
⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠛⠉⠀⠈⠙⠳⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀
⠀⢀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⡀
⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⠀⠻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⣠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⡀⠀⠀⢘⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⡄⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣻⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠚⠛⠛⠛⠛⠳⠾⠶⠶⠿⠷⠶⠦⣤⣀⡀⠀⠀⣰⠶⠶⠶⠶⢶
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠋⠉⠙⢷⣄⠀⠀⠀⠀⠀⠀⠘⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣧⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⣛⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠉⠛⠛⠛⠶⠶⠶⠶⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣿⠀⠻⠀⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⣦⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠶⠞⠛⠋⠉⠉⠁⠀⡀⠀⠀⠻⠶⠶⠶⠶⠾
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⢸⡇
⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⡀
⠉⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛


⡏⠉⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡏
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⡖⠐⢢⠰⠀⢲⠂⢀⠶⠀⢰⢐⠆⣶⠀⢰⡆⠂⠀⢰⡂⠀⢰⡆⢐⡆⡶⢠⠂⠒⡄⢀⡰⠂⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⣇⣀⡜⢸⡀⢸⠀⡞⠒⢇⢸⣁⡗⣻⣀⣸⣇⣀⠀⢀⣈⠆⢸⡇⢈⡏⢣⠘⣄⣠⠇⡈⠑⡄⣏⣀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠋⠉⠙⠛⠒⠦⣄
⠀⠀⠀⠀⢠⠖⠋⠀⣠⠤⠖⠒⠒⢒⠦⣄⡀⠈⠳⣄
⠀⠀⠀⣰⠋⠀⣰⠋⠁⠀⠀⡤⠶⢾⡀⠀⠙⣆⠀⠈⣇
⠀⠀⢰⠇⠀⣸⠁⠀⠀⠀⠘⣧⠀⠀⠙⠀⠀⢸⡆⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢸⠃⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢛⣉⣉⣉⣉⣉⣉⣉⣙⠓⠲⢤⣀
⠀⠀⠸⡆⠀⠸⣆⠀⠈⢳⡤⠴⠞⠀⠀⠀⣰⠋⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡴⢋⣉⡵⠀⢀⣠⠖⠒⢲⣭⠽⠳⡄⠈
⠀⠀⠀⢻⣆⠀⠈⠳⢤⣌⣀⣀⣀⣤⠴⠋⠁⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⠲⠤⣽⣛⣋⣁⣀⣀⣨⡥⠴⠚⠁⣠
⠀⠀⠀⠀⠙⣿⣦⣄⣀⡀⠀⠀⢀⣀⣠⣴⣾⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣤⣤⣴⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣯⣟⣻⣿⠿⢿⣿⣿⠿⢿⣿⣛⣯⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠶⠖⠲⠶⠶⠤⢤⣀⣸⡿⣷⣯⣟⣻⣿⡿⠿⠿⠿⠿⣿⣿⣻⣿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣠⠴⢖⡒⠛⠛⣛⣻⣖⡲⠦⣤⡉⠛⣷⣯⣿⣻⣿⠿⢿⣿⣿⠿⢿⣿⣛⣯⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠻⣔⡒⠻⢤⠤⠞⠁⠀⠀⠟⢉⣠⠟⠀⢸⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣭⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣈⡉⠙⠛⠒⠒⠒⠒⠛⠋⢉⣁⣤⣶⣿⣿⣻⡿⢿⣿⣿⣿⣿⣿⣿⠿⣿⣻⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣝⣿⢿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⡿⢿⣫⣾⣿⣻⣿⣿⡿⠿⠿⣿⣿⡿⢿⣿⣻⣽⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣽⣻⢿⣷⣿⣿⣯⣭⣭⣽⣿⣿⣶⣿⢿⣫⣿⡿⠛⠉⠀⢀⣀⣀⡀⠈⠙⠻⣿⣻⣽⣾
⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⣤⣄⣀⢸⣷⣝⣿⢿⣷⣿⣿⣭⣿⣿⣭⣿⣿⣶⣿⢿⣿⡟⠀⢀⡴⠚⠉⠉⠉⠉⠛⠦⣀⠈⠻⣯⣿
⢀⡴⠋⣉⡤⠴⠶⠒⠒⢒⣒⣶⠶⠤⣌⡙⠻⣿⣿⣿⣷⣿⣿⣿⣭⣭⣿⣿⣿⣷⣿⣿⣿⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠘⣿
⣏⠀⠸⣁⠶⠾⣥⣤⡴⠋⠁⠉⠿⠛⢁⡽⠀⢈⣿⣷⣿⣯⣽⣿⣿⣿⣿⣯⣽⣿⣾⣿⣇⠀⢸⡇⣤⣾⠁⠀⠀⡴⠚⢶⡄⠈⣇⠀⢹
⣿⣶⣄⣈⠙⠓⠒⠲⠶⠶⠶⠒⠒⠋⢉⣀⣴⣾⣿⣷⣿⣯⣽⣿⣿⣿⣿⣯⣽⣿⣾⣿⣿⠀⠘⣇⠀⠙⢦⣤⠞⠁⠀⣰⠟⠃⣿⠀⢸
⣿⣝⡿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⣟⣽⣿⣷⣿⣿⣭⣿⣿⣿⣿⣯⣿⣿⣾⣿⣿⣧⠀⠘⣆⠀⠀⠀⠀⠀⠀⠉⠀⢠⠏⠀⢸
⣿⣝⡿⣷⣿⣯⣽⣿⣿⣿⣿⣿⣭⣿⣶⡿⣟⣽⡇⠉⠙⠛⠛⠛⠿⠿⠛⠛⠛⠋⠉⠀⠻⣿⣷⣄⠈⠓⢦⣄⣀⣀⣀⣠⠴⠋⠀⢠⠏
⠘⠿⣿⣷⣾⣯⣭⣿⣿⣿⣿⣯⣽⣿⣶⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠉⠉⠉⠀⠀⣠⠴⠋
⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣷⣶⡶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠶⠤⠤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⣩⣭⣭⣿⣿⣶⣶⣶⣿⣿⡿⠿⠿⠿⢿⣷⣶⣶⣶⠯⢭⣟⡲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⣿⣓⣙⠛⠳⢿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣛⣛⣿⣥⣿⣟⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣈⡙⠛⠿⠿⠷⢶⣶⣶⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⠿⠿⠿⠟⠛⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⠓⠒⠲⠦⠤⠤⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⣤⣴⣶⡶⠚⠛⠛⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡃⠀⠀⢀⠀⡀⣀⣀⣸⡉⢛⣏⡙⢻⡟⠛⣹⣟⡉⣿⣏⣉⣬⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠇⣠⠀⡏⠥⠂⠀⠀⠀⢨⢉⢉⢉⠉⠉⢻⠙⠋⠉⠁⠀⠀⢹⠛⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠲⠃⡃⡷⠀⠀⠄⠀⠀⡈⠈⠈⠈⣀⠀⠀⠀⣀⢀⠀⠀⠀⢸⢀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠃⡇⠁⠀⠀⠈⠀⠄⠀⠑⠄⡁⢀⢷⠀⡁⢸⠀⠘⠢⠀⢸⣽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⢸⠃⡇⠀⠓⠂⠀⢰⠀⣆⡆⡆⢀⡠⠀⡀⠀⢀⣆⠀⠀⠀⠀⣽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡗⠸⣄⠇⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠑⠐⠁⠀⠊⠙⠀⠀⠀⠀⣿⢠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠂⢠⠒⡀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡠⠀⠀⠀⠀⡆⠀⢠⠸⠀⢸⠀⠀⠀⠑⡄⢀⡀⠀⠀⠀⣿⢸
⠀⡀⡀⠀⣀⢀⠀⣀⠀⠀⢀⡀⢀⠀⣅⣯⣀⡄⠁⢡⡄⢀⡁⣁⢀⣨⣍⠀⣨⡀⡀⢀⡀⣥⣬⢸⡀⠁⢀⣍⣈⢈⣀⠀⣀⡀⠀⢀⢀⡀⠀⢀⠀⢀⢀
⠀⠁⠉⠀⠉⠀⠉⠈⠀⠀⠈⠉⠈⠁⣟⡋⢉⠁⡀⠈⠀⠈⠃⠙⣈⠋⠙⡀⡙⠈⠁⠈⠀⠋⠏⢘⢁⡀⠀⣋⢙⢘⠁⠈⠁⠁⠀⠀⠈⠁⠉⠀⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⡗⡶⠀⠄⡄⡠⢀⠀⡉⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠁⠐⠁⠂⠃⠺⠀⠇⢘⠰⡇⣷⠀⠀⠀⠘⣄⠇⠀⣻⢨⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⣔⡆⠰⠦⢤⣀⡀⠀⠀⠀⠐⣶⠀⣠⡔⢠⣄⢠⣶⠁⠾⠆⢸⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⡼⡇⢱⠓⠀⠀⠄⡈⠉⠀⠀⠀⠉⠀⠉⠁⠈⠁⠁⠀⢀⡀⠀⢸⢰⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⣥⠿⣿⣦⣤⣀⡀⠀⠀⠀⠀⠐⠀⠂⠀⠐⠂⠀⠀⠀⣁⣠⣴⠞⠀⡁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠻⢦⣽⣀⢿⡆⢩⠉⠛⠻⠛⠶⢶⣶⢶⣶⢶⠶⡶⢶⡟⣟⠉⡥⢚⢀⡠⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣍⣻⢸⢷⣦⣄⣄⣀⠀⠇⠀⣷⠀⡁⠆⡀⠃⢙⣡⣨⡴⠶⠛⣉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⡉⠸⠦⠄⣉⡙⠛⠛⠻⠿⠿⠾⠿⠾⠟⠛⠛⠉⠁⢀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡃⢸⠀⠀⠀⠈⠉⠁⠒⠒⠒⠂⠒⠛⢦⣀⡠⠖⢄⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠈⠩⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⢉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠂
⠀⠀⠀⠀⢰⣶⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢤⡀⣠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⠀⢸⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⡇
⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠈⢠⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣷⣶⣤⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣯⣿⣿⣿⣿⣽⣿⣷⣿⡿⠃
⠀⠀⠀⠀⡀⠒⡒⢀⡀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⠿⢿⣿⡿⠛⠻⠛⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠰⠖⠞⠦⠢⠔⠀⠶⠂⠷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠺⠘⠃⠂⠃⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣍⠠⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠒⠂⠒⠒⠒⠠⢤⣀⠀⠀⠀⢀⣀⡠⠤⠤⢐⠒⣒⣒⣒⣒⣒⣒⣒⡒⡒⠠⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⢀⣀⠤⠤⠴⠖⠶⠤⠤⣄⡀⠀⠙⠲⢏⡩⠔⠒⠋⠉⠉⠁⠀⠀⠀⣀⡀⠀⠀⠀⠉⠉⠉⠓⠒⠬⣍⡓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠑⢆⠀⠀⠀⠀⢀⣤⣶⠒⠛⠛⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠉⢦
⠀⠀⠀⠀⠀⢀⡔⠁⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠳⡄⠀⠀⠀⠉⠛⠛⠲⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⠀⡌⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠹⡄⠀⠀⠦⠤⢤⣤⡤⠴⠿⠛⠀⠀⠀⠀⠀⠀⣀⡤⠊⢀⡤⢺⡇
⠀⠀⠀⠀⣼⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢱⡤⣄⣀⣀⣀⣉⣁⣀⣀⣀⣤⠤⠤⠖⣒⣉⡥⠶⠚⣡⡴⢫⡇
⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⢉⣉⣀⡤⠴⠒⢋⣡⢴⢿⣇
⠀⠀⠀⢰⡇⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⠛⠛⠛⠛⠓⠚⠛⠛⠛⠛⠉⣉⣉⠠⠤⢒⡺⠽⠚⣡⢞⢽⡄
⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡯⠭⠭⢭⣭⣭⣭⣭⠭⠭⠭⠥⠔⠒⠂⣉⣡⡤⠖⢋⡥⣫⢺
⠀⠀⠀⠀⢳⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⠇⠀⠸⠒⠲⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠋⢉⣉⣠⣴⠾⠛⢉⣴⠋
⠀⠀⠀⠀⠈⢆⠀⠙⡄⠀⠀⠀⠀⠀⠀⠿⣷⣶⣶⣾⡿⠋⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠛⠋⣉⣁⠤⢴⡞⠁⢈⡇
⠀⠀⠀⠀⠀⠈⢦⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⣠⣟⣛⣛⣒⣒⣒⣒⣒⣀⣠⣤⡶⠶⠶⠞⠛⢉⣁⣤⡶⠟⢀⡠⢚⠇
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⢀⣼⠥⠤⠤⣄⣉⣁⣀⣀⣁⣀⣤⣤⣤⣤⡶⠶⠟⢛⣉⡡⠤⠒⢉⡴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀⠀⠉⠓⠒⠤⠤⠤⠤⠤⠐⠒⠋⠁⢀⣠⠶⠥⣈⣉⠙⠓⠒⠒⠒⠀⠭⠭⠭⠔⠒⠒⠒⠒⠛⢉⣉⣠⣤⡶⠞⣉⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢤⣄⣀⣀⣀⣀⣤⠤⠔⠺⢭⣈⡑⠒⠒⠤⠭⢭⣍⣒⣒⣒⣒⣒⣒⣒⣲⣶⠬⠭⠽⠛⠋⣉⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠢⠤⠤⠤⠤⠤⠀⠠⠤⠤⠤⠤⠤⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⢠⣾⠆⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠐⢿⠇⠀⠀⠀⠀⠈⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⠖⢶⠖⠒⡶⠲⡶⠲⠲⠖⢶⠖⠒⢶⠲⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠁⢸⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⢸⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⢸⠀⠀⣿⢀⡧⠖⠋⠉⠉⠉⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⡇⢸⠀⠀⣱⠋⠀⣠⢤⣤⡀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⠸⠇⢸⠀⢀⠇⠀⡸⢵⠚⣎⣳⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠚⠓⠚⠒⠒⠛⠒⠓⠒⠚⠓⠚⠒⢺⡀⠀⠙⢯⢏⡵⢻⣾⢿⣷⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⢧⠀⠀⠀⠀⠀⠘⢾⣾⠿⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠚⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⣀⢀⣀⢀⠀⡀⢀⡀⢀⠀⠀⣀⠀⢀⣀⠀⢀⠀⡀⢀⢀⠀⠀⡀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⢸⠽⠳⡇⢣⡎⢿⣂⢘⣲⠀⡇⢸⢷⠏⢿⡂⠸⢲⡇⢽⠀⢸⣚⣧⠯⡆⠟⢾⢸⠺⡀⡇⡟⣾⢺⣰⠄

⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠘⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⡏⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢅⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠑⠀⠄⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠰⢰⡀⠐⠂⠈⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡀⠀⠀⠀⠀⠀⠂⠀⠉⠂
⠀⠀⠀⠸⠂⠀⡀⢀⠀⢀⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠑⢆
⠀⠀⠀⠀⠠⠊⠡⠌⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠄⠛⠶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠈⠑⠤⠤⠊⠁⣄
⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠶⢂⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⡂⠀⠀⢐⠒⠓⢢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠠⠤⠤⠄⢤⠄
⠀⠀⢰⠀⡠⠤⠤⠀⠀⢠⠁⠉⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠂⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣀⡁⣈
⠀⠀⠀⠄⠁⠀⠠⠤⠤⠼⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⢀⡀⠀⠀⠀⢸⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣇⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⢐⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠂⠠⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⠤⢆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⠢⡀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⡀⠄⡠⢀⣴⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⣦⡀⠀⠀⠀⣀⡀⠼⣦⣐⡒⠙⠁⠁⠈⠉⠈⠙⠲⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡂⠰⠟⠛⠛⠉⠉⠁⠔⠁⡡⢈⢙⣌⣀⡄⠢⠄⠐⠀⠠⣋⠭⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢅⠀⢧⣜⠀⠀⠀⠄⠀⠂⠤⢀⣙⣂⡀⠅⢁⣐⠬⢖⣚⣥⡶⠞⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⢛⠢⣻⣤⠚⣓⣠⡤⢴⠶⣖⡒⢒⠾⠟⢛⠒⢉⢋⣩⣶⣶⣿⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠘⡢⢌⡙⠌⠿⠞⠛⢉⣩⡥⣒⡴⣖⡷⣶⢿⣻⣛⡽⠿⣫⣳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣆⠈⠀⢉⠢⠽⣛⣛⡻⠽⣛⡯⢟⣫⣟⡿⠟⣛⣡⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢁⣻⣀⢖⣒⣂⠭⠶⣾⣟⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠚⠧⠤⠸⠟⠻⠛⠋⠋⠍⠀⠠⠀⠀⠀⡠⢀⠄⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠐⠀⠀⠀⠀⠐⠀⠀⠂⠀⠁⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢸⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⣛⣛⣛⣛⡛⢿⡿⢛⣋⣼⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⡁⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⢀⣴⡿⠋⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣴⡿⠋⠀⠀⣿⡟⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⢰⣿⠿⠿⠿⠿⣿⡇⠀⠀⠀⠈⠋⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⣀⣀⣀⣀⣀⣀⡀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⣿⡟⠛⠛⠛⢻⣷⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⢀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⢸⡷⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⣿⡇⠸⣿⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⢰⣿⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⢸⡿⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⢸⣇⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣤⣴⣿⡇⠈⣿⣦⣤⣤⣤⣿⡇⠀⣿⣧⣤⣤⣤⣾⡿⠀⣿⣷⣤⣤⣤⣾⣿



⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⢀⡤⣄⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⡠⠞⠉⠀⠈⠙⢦⣀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠈⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⣠⠴⠋⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠳⢤⡀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⣀⠴⠋⣁⣀⣀⡤⠒⠋⠉⠉⠉⠀⠈⠉⠉⠉⠓⠦⣄⡀⣀⡉⠲⣄⡀⠉⠳⢄⡀
⠀⠀⠀⢀⡴⠚⠁⢀⡴⠚⠁⣴⠉⠉⠉⠁⢸⣋⣙⡆⢰⢧⠀⣷⡀⣶⣧⡴⠃⠈⠉⠉⠉⢱⡄⠙⠢⣄⠀⠙⠲⣄
⢀⡤⠞⠁⠀⠀⠀⠉⠉⠉⠉⣿⠀⠀⠀⠀⢸⣍⣩⠇⡟⠚⡆⡇⠙⡏⡯⠙⢦⠀⠀⠀⠀⢸⡏⠉⠉⠉⠁⠀⠀⠈⠙⠦⣀
⠛⠒⡖⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⡖⠚
⠀⠀⠧⠤⡤⠤⠤⠤⢤⠤⠤⠤⠤⣤⠤⠤⠤⠤⡤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⢤⡤⠤⠤⠤⣤⠤⠤⠤⠤⡤⠤⠇
⠀⠀⠀⠀⢧⣤⣤⡤⡼⠀⠀⠀⠀⠻⡤⣤⣤⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡤⣤⣼⠃⠀⠀⠀⠸⡤⣤⣤⣤⠇
⠀⠀⠀⠀⢸⢸⠀⡇⡇⠀⠀⠀⠀⠀⡇⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⡇⠀⠀⠀⠀⠀⡇⡇⣿⢸
⠀⠀⠀⠀⢸⢸⠀⡇⡇⠀⠀⠀⠀⠀⡇⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⡇⠀⠀⠀⠀⠀⡇⡇⣿⢸
⠀⠀⠀⠀⢸⢸⠀⡇⡇⠀⠀⠀⠀⠀⡇⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⡇⠀⠀⠀⠀⠀⡇⡇⣿⢸
⠀⠀⠀⠀⢸⢸⠀⡇⡇⠀⠀⠀⠀⠀⡇⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⡇⠀⠀⠀⠀⠀⡇⡇⣿⢸
⠀⠀⠀⠀⢸⢸⠀⡇⡇⠀⠀⠀⠀⠀⡇⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⡇⠀⠀⠀⠀⠀⡇⡇⣿⢸
⠀⠀⠀⠀⣸⣸⣀⣇⣇⠀⠀⠀⠀⢀⣇⣇⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣇⣇⣇⠀⠀⠀⠀⢀⣇⣇⣿⣸⡀
⠀⠀⢀⣀⣏⣉⣉⣉⣹⣀⣀⣀⣀⣿⣉⣉⣉⣉⣇⣀⣀⣀⣀⣀⣀⣀⣀⣏⣉⣉⣉⣹⣇⣀⣀⣀⣸⣉⣉⣉⣉⣇⣀
⠀⢀⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣇
⢀⣏⣀⣈⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣈⣨⣇
⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠓⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⢨⡅⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⢈⠒
⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣠⣈⣀⢠
⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣜⣧⣄⠀⠀⠀⣤⣤⣤⣴⣤⠀⠀⡠⠀⠤
⠀⠀⠀⠀⠐⣤⠀⢀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣍⠀⠀⠀⣿⡿⢿⣿⣿⣷⣄⣨⠆⠀⠀⠀⠀⠀⠀⠀⡗⠀⣈⣁
⠀⠀⠀⠀⠸⣀⠀⠺⡀⠇⠀⢠⠀⠀⠀⠀⣴⣿⣿⣿⡯⠻⣦⠛⠀⠀⠀⠉⢠⣾⡉⠋⣿⣿⡷⠀⠀⠀⠀⠰⣤⠆⠀⡇⠀⠈⠉
⠀⠀⠀⠀⠘⠈⠉⠀⠉⠀⠁⠉⠀⠀⠀⠀⠈⣿⡿⣿⠃⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃

⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠆
⠀⠀⠀⠀⠀⠘⣿⠋⣭⠛⣩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠆⠀⣙⣛⠁⠰⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⡀⢰⣦
⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣾⠟⢭⣷⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⢸⣿⢸⣿
⠀⠀⠀⠀⠀⣠⣭⣤⣿⣤⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⣿⣦⣘⣿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⠿⠸⠿⠸⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣙⠳⣄
⠀⠀⠀⠀⠀⢨⣿⡿⠻⢿⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⣳⠀⣸
⠀⠀⠀⠀⠀⠸⣿⣿⣴⣿⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠚⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣾⣿⣿⣿⣿⢾⣿
⠀⠀⠀⠀⠀⠈⠙⠋⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⠛⠉⠀⠀⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢸⡆⠀⠀⠀⡀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⡇⠀⠀⢸⣭⢉⢡⡀⡄⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢈⣙⣛⡛⠛⢛⣛⣛⠛⠛⣛⣛⣛⠛⠛⣛⣛⡁⠀⢸⡇⠀⠀⠈⣿⣿⣼⣿⣿⡟
⠀⠀⠀⠀⢀⣠⣴⣶⠿⠿⣶⣶⣤⡀⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⢸⠀⡇⠀⠀⢱⢸⠁⠀⠀⡇⢸⠀⠀⠈⠆⢸⡇⡠⠚⠉⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⣰⡿⠛⠁⢀⣶⣦⣀⡈⠙⢿⣦⡀⠀⠀⠀⡇⠀⠣⠤⠤⠼⠀⠧⠤⠤⠞⠘⠤⠤⠤⠃⠸⠤⠤⠴⠃⢘⡟⠁⠀⠀⠀⢠⣦⣀⡀⠀⠀⠀⠳⡀
⠀⠀⣸⡿⠁⠀⣰⠟⠉⠉⠉⠉⠀⠀⢻⣿⠉⠙⠉⢳⠀⡔⠀⠀⢲⠀⡖⠊⠒⢦⢠⠂⠀⠐⡆⢰⠂⠀⠓⡄⢸⡇⠀⠀⠀⣾⠉⠉⠉⠉⠀⠀⠀⠀⢣
⠀⠀⣿⡇⠀⠀⠙⢷⣤⣤⣤⡀⠀⠀⢸⣿⣄⣀⣀⣼⡀⢇⣀⣀⣸⠀⣇⣀⣀⡸⠸⣀⣀⣀⡇⢸⠀⠀⠀⡇⢸⡇⠀⠀⠀⠘⠳⠶⠶⣦⠀⠀⠀⠀⠈⡄
⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣸⡿⠀⠀⠀⠀⢹⡠⠤⠤⢤⠀⡤⠤⠤⣄⢀⠤⠤⠤⡄⢸⠀⠀⠀⡇⢸⡇⠀⠀⠀⣠⣤⣤⣤⡾⠀⠀⠀⠀⠀⡇
⠀⠀⠈⢻⣧⡀⠘⠛⣿⣿⠛⠁⢀⣴⡿⠓⠒⠒⠲⡖⠋⡇⠀⠀⢸⠀⡇⠀⠀⢲⢸⠀⠀⠀⡇⢸⠀⠀⠀⡅⢸⡇⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠙⠻⣷⣦⣤⣤⣴⣾⡿⠋⠀⠀⠀⠀⢀⡷⣄⠉⠊⠁⠉⠀⠉⠈⠀⠁⠀⠁⠀⠈⠁⠈⠁⠀⠈⣀⠜⠑⠦⣄⣀⡀⠀⠀⠀⠀⣀⣠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⣀⠀⠀⠀⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠲
⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠀⠀⢥⡀⠀⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⠦⠀⡁⠀⠠⢶⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠑⠀⡀
⠀⠄⠂⠀⠀⠀⢓⡔⢒⢒⠒⠞⡁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠐⡀⠀⠀⠀⠘⠰⠐⠀⠄⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡉⠁⠐⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡈⠁⠐⠄⠀⠀⠀⢀⠀⢀⡴⠲⠄⠀⠀⠀⠲⡂⠀⠀⠀⢀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠄⣈⡁⠄⠀⠀⠀⠀⠀⠐⠈⠀⠀⢀⡀
⠀⠀⠀⠀⢀⠃⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁
⠀⠀⠀⢀⠦⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠉⠉⠀⢀⡠⡀
⠀⠀⠰⢀⠀⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠉⠉⠂⠀⠀⠀⠨⠐⠀⠀⠉⠘⠏⠂
⠀⠢⡀⠈⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠉⠐⠆⡠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⢠⠀⡠⠐
⢀⠀⢰⠀⠀⠀⠐⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠃⡠⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠤⠎⠯⠉⠀⠀⠀⢼⠜⠀⠀⠀⠀⠦⠀⠀⠀⢠⢱⡀⠀⠀⠈⠁⠄⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠠⠁⢀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡸⠑⢒⡒⡂⢢⠃⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠂⠦⠴⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠤⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⢁⠀⣀⠐⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡈⠠⡠⠀⠀⠀⠀⠊⠉⢈⡀⠀⠀⠀⠀⠀⠈⠌⣀⣈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠗⠁⠀⠀⠀⠀⠀⢀⠀⢄⠘⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢻⡆⠀⠀⠀⢠⠄⠰⠄⢀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣼⠇⢀⠈⣡⡶⠞⠛⠷⢦⣄⠖⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡀⣼⠋⣠⣶⣶⣶⡟⠹⣆⠓⠀⠀⠀⣿⣀⣤⣤⠶⠶⠛⢻⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⣿⠀⢿⣿⣿⣿⣇⣠⣿⣼⣶⣾⠿⣿⠿⠷⠶⠚⠛⠛⠛⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣄⠀⢠⠙⣧⣾⠿⣿⢹⡿⠿⠶⠶⠶⠶⠾⠿⠿⡷⠶⠶⠾⠿⠶⠶⠾⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠰⠆⢹⡆⣿⢸⡇⠀⠀⣶⠶⠶⢶⡶⠛⠙⠻⣶⠶⠶⢶⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠞⠁⠀⠀⠀⠈⣷⣻⣾⡇⢸⡟⠁⠀⠀⣼⠃⣴⡷⠀⢹⡆⠀⠀⠙⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠶⠶⠶⠶⠦⢼⣿⣿⡇⢸⡇⠈⠃⠀⣿⠀⣩⣿⠇⢸⡇⠀⠋⠀⣼⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠶⠶⠶⠶⠶⣿⣿⡇⠈⠙⣧⣤⣤⣼⣧⡀⠁⢠⣾⣥⣤⣤⡿⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣀⣀⣀⣀⣀⣀⣈⣛⣟⣋⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡤⡄⡀⠀⠀⠀⢀⠀⠀⠀⠻⠃⠀⠈⠁
⠀⠀⠶⠶⠶⠲⠲⠘⠲⠖⠆⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠀⠭⣭⡭⣭⠭⠅⢠⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠇⠀⢘⣟⠁⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠠⡜⠓⠛⠓⠈⢳⣶⣾⣿⢷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⣀⣐⣈⣛⣋⠁
⠀⠀⠀⠀⠀⠀⠀⢐⣂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠽⢀
⠀⠀⠀⠀⠀⠀⣤⣮⡕⠃⠀⠀⣨⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠗
⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠻⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠶⠒⠛⠛⠛⠛⠛⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠤⣀⠀⠀⠉⠓⠶⣈⡐⡄
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠶⠀⠀⠀⠀⠀⠀⠈⠳⡇
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣹⡄
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⠀⠀⠀⢿⣾⣀
⢀⣀⡴⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠉⢉⠝⠳⡄
⠘⢾⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⣿
⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠻⠙⢿
⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣀⣀⣀⣠⡴⠏
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠉⠉⠉⣿⠋⠉⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⡟⠉⠛⢻⡟⠉⠉⢹⡇⠀⠀⠀⠀⠀⠈⡏⠉⠁⣹
⠀⠀⠀⠀⠀⢷⣌⠐⠀⠠⠄⠤⢼⣗⣐⣀⣾⠁⠀⠀⠘⣷⠀⠄⢀⡀⠠⢴⣇⣐⣠⠟
⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠟⠀⠉⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠟⠁