Shaving Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠾⢥⡀⠀⠀⠀⠙⠛⣻⡿⣇⠀⠀⠀⠀⠸⠟⠃
⠀⠀⠀⠀⠀⠀⣶⣶⣂⠀⠠⡡⣻⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠞⠲⢆⠇⠀⠶⢄⠀⠘⣷⣤⣬⠆⠀⠀⠆⠀⠀⠑⣆
⠀⠀⠀⠀⠀⣠⡃⠀⢹⡀⠐⢽⣿⣿⡀⠀⠀⣧⡀⠀⠀⠀⠀⠀⣇⣀⡖⡁⠀⠠⠷⠀⠃⠀⣿⡏⡽⠀⠀⢸⠀⠀⠀⡅⣿
⠀⠀⠀⠀⠀⢹⣷⣾⠈⠃⠠⢫⣿⣿⡇⠀⠀⣿⡿⠀⠀⠀⠀⢰⠁⠦⣤⡁⢠⠀⠀⠀⠀⠾⠿⡾⠁⠀⠀⢰⠸⠀⠀⠇⢻
⠀⠀⠀⠀⢸⣼⢿⣿⡀⡀⠀⠋⣿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⠙⠒⠋⠄⠃⠀⠀⠀⣒⠴⠁⠀⠀⠀⠋⠁⠒⠚⢳⠛⠃
⠀⡠⠀⣠⣼⠀⠐⠉⡇⠁⠀⢠⣿⡟⠃⠀⠀⡇⠀⠀⠀⠀⠀⠈⡆⠢⠤⢤⠤⠤⠄⢈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⣿⣿⣿⠿⡀⢴⠞⡃⣇⣀⠘⢿⠇⠀⢀⣀⣧⣀⣀⠀⠀⠀⠀⡇⡈⠐⠃⠀⠀⠔⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⣀⡀⠀⠠⠤⡤⠉⢀⣀⣀⡀⡤⠂⠉⠀⠀⠀⠀⠁⠈⠉⡀⠡⡤⠀⡀⠀⠂⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠁⢀⡼⠁⠀⠀⠀⠀⠀⡰⡧⠒⣩⣴⠯⢊⠴⡋⢰⣀⣀⣀⡈⠑⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠇⣫⠖⠉⠀⠀⠀⠀⡘⡠⢞⠁⠠⠚⢁⠄⢊⠠⡺⠀⠈⠁⠀⣀⠀⠀⠀⠈⠳⣄⠀⠐⣒⣒⣒⣒⣒⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠋⠐⠁⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠠⠟⠁⠀⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠄⡄⠀⡄⢠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠼⠁⠟⠄⡟⢾⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠱⠀⠜


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠐⠦⠆⠀⠤⠠⠤⠤⠭⠅⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⢠⠄⠠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠴⠒⠒⠒⠠⠤
⠀⠀⠀⠀⠀⠳⠤⠤⠄⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠠⠤⣤⠄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣇⣤⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡒⠿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⠙⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣽⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣉⠛⢿⣿⣿⣿⣿⣿⡷⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠐⠉⣿⡇⠘⣿⣿⣿⠋⠉⠀⠀⠀⣀⣀⣀⣀⡀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⢟⠀⢠⣿⣿⣿⠀⠀⠀⣶⡿⠟⠉⠛⢻⣿⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢉⡇⢘⣿⣿⣏⠀⡄⠀⠀⠐⠿⠲⡞⠛⠙⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠙⠛⢻⣿⣿⣿⡀⠀⠀⣸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⣠⣶⣐⡓⢢⣤⡀⠸⡟⢺⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠃⠀⠁⠀⠀⠉⣏⣽⣿⠇⠴⠃⠈⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣾⠃⠀⠀⣟⣀⣉⣁⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠄⠀⣾⣿⣷⠃⣼⣿⣿⣿⣛⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢻⣿⣿⡏⣰⣿⣿⣿⡯⠉⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇⠀⠀⣿⣿⢱⣿⣿⣿⠏⠀⢰⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠃⠀⠀⠀⠈⠛⠛⠉⠁⠀⢀⣀⣹⣄⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⢻⣿⢷⣄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣼⢻⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣧⡙⠳⢮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡁⢤⡐⠀⠀⠀⠀⠀⠀⠀⠠⠄⠿⠿⢻⣿⣿⡈⠻⢦⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⣴⠋⠸⠊⠱⡄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣾⣿⣿⣿⣄⠀⠁⠀⠀⠘⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣿⣿⣿⡉⢿⡿⣯⣳⣄⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠁
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣠⡀⣂⣤⣶⡶⠟⠛⠉⠉⠹⣿⡻⠄⠀⠀⠙⠦⣷⣄⣀⣀⣀⣀⣠⠄⠈⠛⢿⣿⠿⢿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠙⠋⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠷⠿⠇⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣤⣤⣄⣀⣀⣤⣤⡤⣄⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣭⣭⣤⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣫⣥⣬⣄⣈⣩⣙⡛⠻⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠿⠿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠘⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡛⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣦⣤⡶⠶⠶⠦⠀⠈⠙⠛⣿⣿⣾⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢩⣥⣄⣀⢀⣤⠒⠀⠀⢠⣾⠫⠕⢓⣹⣅⠐⠻⠈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠈⠿⠛⠛⠛⠃⠀⠀⠀⣮⠁⣴⣟⢫⣷⣿⣿⡄⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⣀⡺⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⢿⣿⣷⡽⢿⣿⣿⠿⠿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠠⣇⣀⣤⣈⣉⠀⠀⠀⠀⠀⢀⣤⣿⣿⣶⣼⣏⢻⣿⡈⠛⠿⣷⣷⡦⢈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣾⣿⣿⣿⣷⣤⣤⣾⣿⣿⠿⣿⣿⡿⢿⡀⣿⡇⢦⠀⠈⠉⠙⠓⠂⠀⠉⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⠛⠻⣿⣿⣿⣿⣿⣿⠀⠹⣿⣄⣼⣷⣹⣷⠘⢷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠹⣿⠙⣿⣿⣿⡇⢠⣯⣹⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⡇⢿⣿⣿⠘⣿⣿⡿⢿⣿⣯⡙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣧⠴⢿⣿⣇⣿⣿⣷⠀⢿⠻⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣋⡉⣙⠃⠀⠀⠉⢇⠀⠘⣿⣿⣹⣿⣿⡆⠈⠳⣌⠛⠁⢠⡤⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠉⠁⠐⠄⠀⠀⠘⡆⠀⠹⣿⣟⣿⣿⡇⠀⠀⠈⠙⠛⠓⠛⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣧⣄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠋⣸⣿⠋⣿⣆⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣸⠛⠾⢷⣦⣤⡀⠀⠀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣦⣴⣶⣤⣶⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠟⢁⣽⡅⠀⣀⠤⠿⠿⠿⣿⠻⢷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣴⣾⡿⠡⡿⣿⣿⣾⣿⣷⣶⣿⣿⣥⣀⠳⡀⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢜⣻⡍⠉⠀⠀⠀⠉⠀⠉⠉⠉⢙⠫⣝⡶⣿⣔⢭⣿⡿⣟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⣾⠟⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠉⠲⣾⠹⣟⢻⣿⣿⣎⣳⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⠻⣿⣶⣿⣿⣿⣿⣇⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⠾⣋⣉⣻⣷⡆⠀⠀⠀⠈⢦⣭⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⡁⣻⣷⠀⠀⢸⡟⠙⠉⠀⠀⢠⠆⠀⠀⠀⡻⠿⠽⠋⣉⡙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠣⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠦⠀⢠⡄⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠒⠆⠀⢀⡀⠖⠂⢘⣧⠀⠀⢀⠀⠀⠨⣇⢀⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢾⡖⠻⣷⠀⠐⠀⠀⠀⠀⠈⣯⠀⠀⠀⠒⠀⡆⠈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⡈⣽⣆⢰⡤⣀⡄⠀⠀⠀⠀⢻⠂⠀⢀⣤⡀⠀⡦⢰⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⢁⡴⠃⡌⠀⠀⠀⠀⠀⣿⡗⢾⣿⡇⠞⠐⠣⠋⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡄⠀⠀⠘⠀⢋⠀⠀⠀⠀⠀⠈⠋⠘⠛⢢⣷⡈⠛⠓⠄⢥⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⢃⣀⠀⠀⠈⠁⠀⢠⠤⠤⠄⠂⠀⢀⡀⠿⡿⠐⢿⣦⣄⣠⣀⣈⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡟⠀⠀⣸⣿⣶⣶⠷⠘⠛⠇⢀⡄⠀⣠⣦⡿⠀⠎⠀⣀⣀⣀⣈⡀⠀⠀⢀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶⢾⡿⠡⣿⣆⠀⠀⣀⡴⠾⠏⠀⣬⠃⢈⡴⠉⠉⠉⠉⠉⣁⠸⠆⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠎⠀⠀⠀⠀⠁⡼⢡⡖⠛⣿⠛⠿⠋⠀⠀⠀⠐⢁⠂⠈⢁⢠⡿⠂⠳⠿⠟⡀⠀⠀⠀⠀⠀⠘⣇⠂⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣾⡏⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠹⡧⠀⠀⡄⠀⠀⠀⠀⢀⣤⠋⣻⡇⢠⣄⠀⠀⠈⠀⠈⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⣻⣆⡄⠘⡄⠀⢀⣴⠊⠀⣰⢏⣷⢤⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠸⣿⣿⣦⡵⠊⠉⠁⣠⠾⢃⡼⠁⣼⣿⣿⣿⣷⣆⣤⣤⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⠁⡰⠏⢀⡼⡛⠹⠻⣿⣇⠈⠻⠀⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣴⠟⡱⠃⠀⠀⢨⠞⠀⠀⠂⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠊⠁⢠⠇⠀⠸⠇⡀⠀⠈⡄⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠃

⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠃⢠⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⣠⣴⣿⣿⣿⠛⠛⠛⠷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠁⠀⠠⠴⠄⠘⢄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⢠⡄⠈⢢⡀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠈⣁⣀⠀⠱⡀
⠀⠀⠀⠀⠀⢻⣭⣽⣥⣄⣻⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⢣⠀⠁⠄
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠘⣉⡙⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠰⡀⠹⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠖⠊⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⡀⠀⠀⠀⠀⠀⠀⢀⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⢄⠀⠀⠀⠀⢘⣋⡴⠶⢤⣄⠀⠀⠀⠀⠀⠚⠶⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡁⠀⠀⠀⠑⠂⡶⠚⠉⢁⡰⠖⠋⢋⡦⣀⠀⠉⠀⠀⠀⠈⣳⠼⠖⠚⠓⠲⢦
⠀⠀⠀⠀⠀⠐⠉⠀⠇⠀⠀⠀⢀⠈⠀⠀⠈⠁⢀⡤⠖⠁⠀⠀⢦⠀⠀⠀⢠⣎⠀⢀⣀⠠⠤⠐⠈⠢
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡘⠛⠋⠉⣓⡀⠀⠀⢷⠀⠀⠈⡷⠊⠁⠀⠀⣑⡠⠔⠙
⠀⠀⠀⣰⠃⢡⠀⠀⠀⢆⣀⠔⠀⠀⠀⠀⠀⠀⠐⠛⠉⡉⠁⠀⠀⠈⢧⠀⠀⠲⠦⠴⠂⠁⠀⠀⠠⠒⡄
⠀⠀⢀⠃⠀⣘⣦⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⡴⢶⡍⠀⠀⠀⠀⠀⠘⡆⠀⠀⢸⠛⠒⢶⣚⣀⡴⠊⠘
⠀⠀⣸⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠉⠀⠀⢻⡄⠀⠀⠀⠀⠀⢹⡀⠀⢸⠀⠀⠘⣄⠀⢃⠀⢀
⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠇⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢇⠀⠈⡀⠀⠀⠘⡆⠈⡶⠊
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠘⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⡀⠀⣇⠀⠀⠀⠸⡄⢳
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡄⠀⠀⠀⡁⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣇⠀⢹⡀⣀⣠⠴⢷⡈⢧
⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢢⡀⠸⠀⠀⠀⢁⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⠀⠈⣯⠉⠀⠀⠀⠉⠉
⠀⢳⠀⠀⠀⢀⣆⡤⠶⠋⠀⠀⠀⠱⣄⠃⠀⠀⠸⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠓⠒⠛⠁
⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⡆⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡇⠀⠀⠀⠀⡻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠟⠀⣼⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⣼⣿⠋⠁⠀⠉⠻⠟⠀⠀⠘⠛⠛⢛⣛⣛⡛⠛⠁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠀⠈⠁⣠⠶⠲⢤⡀⠀⠀⠀⠀⠀⣼⣉⣠⣤⣽⡆⣼⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⢸⡷⣾⣿⡷⡿⠀⠀⠀⠀⠀⠹⣯⠻⠿⣱⠃⢿⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣷⠀⠀⠀⠈⠳⣌⣛⡵⠃⠀⠀⠀⣆⠀⠀⠈⠛⠛⠁⠀⣼⠁⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⢻⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣇⠀⠀⠀⠀⠀⠀⠀⣀⣠⡖⣀⡀⠀⣿⣳⠶⠤⠀⣀⡟⠀⢼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡄⠀⠸⡄⠀⠀⠀⣠⠖⠾⣥⣤⣛⣩⠭⠿⠥⠤⠶⠋⠉⠉⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⢳⡀⣠⠞⠁⠀⠀⠀⣀⣀⣠⣄⣤⠴⢲⣶⠦⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠈⡿⠃⠀⠀⠀⢀⡟⠻⣷⡶⣶⣖⣛⣭⢯⠶⠃⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠻⣤⣠⣛⣉⣉⡛⣦⠟⠀⠀⠀⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣷⡀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣤⢤⣀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣿⠀⢀⣬⣧⡀⠈⠙⢦⣀⣀⣀⠀⠀⠀⣀⢀⡴⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡇⠀⡟⠀⢀⣙⣦⡀⠀⠉⠉⠈⠓⠶⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠃⢠⡇⢀⡟⢉⣩⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⠃⢸⢠⡏⠀⣀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠁⠀⢿⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣀⣀⣤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠉⠉⠀⠉⠉⠁⠈⠉⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢀⠴⡆⢀⣦⠀⣀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡆⡴⠁⠰⠗⠋⠈⠳⡏⠲⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⢀⡤⠦⡄⢹⡀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠚⣑⠀⠀⠀⠶⠀⠀⠠⠟⠉⢷⢀⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⠘⠀⠀⠀⠀⠀⢀⡀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⡠⠴⠂⢾⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡤⠤⠔⠉⠉⠀⠀⣠⣿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⠠⠤⠚⡏⠁⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡠⣜⢳⡇⠀⠘⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⡰⡇⢸⡀⢷⠀⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⢸⢣⠘⠈⠃⠘⢧⠀⠈⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠸⡄⠈⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠘⢦⠀⠀⢰⠃⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠳⣄⡏⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠈⠳⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⢷⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⣠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣯⣿⡿⠟⢋⡡⠔⣊⣽⣿⡆
⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⡥⠖⣊⣥⣶⣿⠿⠛⠉
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⣋⠥⢖⣫⣥⣶⣿⣿⠟⠉
⠀⠀⠀⠀⠀⢹⣿⣳⣿⣿⣖⣩⣴⣾⣛⣿⣿⣿⣿⣿⡀⡴⠒⠶⣄
⠀⠀⠀⠀⠀⠈⠙⠛⠿⡿⠻⠋⠛⠛⠛⠿⣿⣿⣿⣿⣿⡇⠀⠀⣨⠟⠲⣦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠱⢹⣿⣦⣰⢇⣀⡠⢣⡷⠋⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠐⠂⠁⣿⣿⡿⠃⠀⠀⢠⡿⠃⠒⢪⠾⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡞⠀⡽⠁⠀⠀⢀⡿⠁⠀⠀⡆⠠⠁⡏⠉⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡀⠀⠀⡷⢼⠃⠀⠀⠀⡼⠁⠀⠀⡴⠁⠀⢸⠃⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⢸⡄⠀⠀⢸⠓⠂⠂⢠⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⠈⡇⠀⠀⢸⠀⠀⠀⡿⠒⠒⠀⣿⣿⣦⣄⣠⣤⣤⣄⡀⠈⠙⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡇⠀⠀⡞⠀⠀⠀⡇⠀⠀⢠⣿⣿⣿⣿⣿⣦⡀⠀⠈⠓⢦⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠸⠀⠀⠀⣾⡜⣝⢿⣿⣿⣿⡿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⡀⠀⠀⠀⠀⠛⠘⠈⠀⠈⠛⠋⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⠿⠛⣉⣿⣿⣿⣿⣿⣿⣿⠿⠛⡛⠁⠈⠙⠉⠓⠈⢿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠿⢷⣦⣀⣈⣉⡉⠙⠛⠦⣬⡉⠐⠦⣬⣓⠶⣄⡀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⡤⠀⠉⠳⢦⡈⠙⠷⣌⠻⣷⣾⣶⣄⣤⣄⠀⠙⣿⣿⣿⡝⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠟⠀⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄⣻⠉⣭⣤⢹⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⠛⠻⠶⠶⠆⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⣿⣼⣿⡿⣼⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠟⠁⢶⣶⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⠤⠶⠶⠶⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⣿⣿⣷⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⢁⣤⣶⣶⣿⣿⣿⣿⠉⠀⠀⠀⠀⢀⣤⢶⣒⣛⡛⠓⠶⢤⣌⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣀⣸⠿⣟⣛⣛⣩⣍⠀⠀⠀⠀⠀⣠⠴⠋⣉⣉⣉⣉⣉⠃⠀⠀⠉⠛⠒⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⣰⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠛⢺⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⣴⡿⠟⠋⠉⠁⠈⠉⠙⠻⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢽⠟
⠀⠀⠀⣀⣤⣛⠿⣿⢿⣃⣀⣤⣭⣷⡿⠃⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⠻⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⣤⣉⠻⣦
⠀⠀⣰⣟⣴⠿⣧⠈⠿⣿⣿⣯⣽⡿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣤⣤⣤⡀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠉⠳⡌
⠀⢠⣿⠟⠁⢀⣿⡆⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠷⠄⠚⠛⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⡄
⠀⣸⡟⠀⢀⣬⠼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⠀⠉⠛⠉⢿⣦⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠉⠙⢻⣿⣷⡄
⢰⣿⠁⡄⠈⠁⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠀⠀⠀⠀⡾⠃⠀⠀⠀⣾⠶⢟⡁⠤⢀⡀⠀⠀⠀⣿⣿⡇
⢸⣿⣄⠣⢠⡀⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣠⣥⣤⣤⣄⠈⢦⠀⠀⣿⠟
⠈⣿⡻⣦⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠰⠆⠀⠀⠀⣿⢙⠛⠿⣿⣿⣿⡄⠁⢰⠏
⠀⠘⣷⢨⡟⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⢠⣿⠋⢀⣠⣄⣀⠀⠀⠀⠙⠈⠻⢦⡀⠁⣸⢷⡶
⠀⠀⠈⢷⣄⣀⣀⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⢀⣠⣴⣶⣶⡿⠟⠋⠀⠀⠀⠙⠂⠈⠛⠋⠙⠃⣀⣴⡇⠀⠀⠀⠉⠁⠀⢀⡀
⠀⠀⠀⠀⣿⡿⠿⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢫⡿⠇⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⢀⣿⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢳⣶⣤⣤⣤⣤⣄⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⣼⡿⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⣠⠞⠃⠠⣴⣏⠁⠀⠙⣷⣄⠀⠉⠙⠛⠿⣷⣦⣉⣀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⣸⠃⠀⠀⣀⣀⣄⡀
⠀⠀⢠⣿⠃⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠈⢿⣷⣦⡤⠄⠈⢧⡀⡀⠀⠀⠀⠉⠛⠿⣷⡄⠀⠀⠸⣿⠀⠀⠀⢠⡏⠀⠀⠘⢏⢩⣿⠿⣆
⠀⢀⣾⡏⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠈⢷⡉⠛⠛⠁⠀⠘⣦⡀⠀⠐⠂⠻⡟⢷⣶⣤⣄⣙⣂⡈⠻⠶⠂⢠⡿⢀⡀⣠⡟⠀⠀⠀⢠⣤⣿⣟⠀⠙⣧⡀
⠀⣼⠟⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⣠⠿⣆⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠈⠳⣄⠉⠉⠛⠋⠁⠀⠀⠀⣡⡶⢋⡴⠋⠀⠀⠀⠀⢯⡀⣾⡉⢠⡄⠈⢳
⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⣰⣿⡀⠹⣧⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠈⠁⠀⢀⡤⠶⠶⠤⣞⡋⠐⠋⠀⠀⠀⠀⠀⢀⣤⣿⣛⣇⣈⣠⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡙⠷⠴⠿⣷⣄⠀⠀⠀⠀⠙⢷⡀⠀⢀⡤⠀⠀⢻⡄⠀⠀⠀⠈⠙⠲⠦⣤⠄⠐⢻⣉⣿⡿⠾⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠙⢦⠈⠀⣤⠴⠃⠹⡄⠀⣤⠀⠀⠐⠣⠆⣷⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠹⣷⡀⠀⠀⠸⢦⡘⠇⡀⠀⠀⠀⠀⠙⢎⣿⣧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠈⢿⣄⠀⠀⠈⠀⠈⢿⣆⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠚⠣⠌⢿⣧⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⢹⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠑⠒⠂⠀⠐⡂⠀⠀⣻⠁⠀⠀⠀⠀⢀⠀⣀⠀⠀⠠⢤
⠀⠀⠀⡤⣀⠀⠀⠆⡆⠀⠠⣬⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢻⠀⠉⣉⣁⠀⠀⠤⠤⠤⠄⠒⠺⠆
⠀⢸⣏⡋⣩⣾⠇⢸⡇⢡⠄⢸⠀⠧⠰⢃⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠙⠇⠛⠋⠀⢸⡇⡸⠀⢸⠀⠀⠀⢾⠀⠀⠀⠀⠐⠀⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣽⡀⠀⠀⢠⠞⣫⠾⣾⣧⠀⠀⡄⡇⠀⠅
⠀⠀⠀⠘⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣾⠀⠀⠀⠐⢦⠄⠀⠐⠀⣤⣤⣼⣷⣴⣦⣶⣾⣿⣿⣷⣶⣷⠾⣄⠛⠁⣿⠟⠀⠀⢀⡘⠀⡇
⠀⠀⠀⢠⠀⠀⠀⢸⡗⠃⠀⢸⠀⠀⠀⣿⡀⠀⣀⣄⡀⠀⠀⠛⠰⠿⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡟⣳⠧⣉⣴⣿⣆⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⠀⠀⠀⢸⡇⠁⠀⢸⠀⠀⠀⣿⠰⠈⠉⠉⡳⢦⡦⠀⠀⡀⠀⠀⠖⠀⠒⣰⢠⣿⣿⣿⣿⡷⠿⣊⣿⣿⣿⣿⠇⠀⠘⡇⠀⡄
⠀⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⢸⣀⢠⣤⣽⡀⠀⠀⢰⠃⠀⠀⠀⠈⠃⠀⠋⠀⠲⠀⣾⣰⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡄⠄⢰⠃⠀⡅
⠀⠀⣀⣀⣀⣀⣀⣸⣇⣶⠿⣔⣻⣿⢿⣿⡇⠀⠠⠬⠠⠀⠀⠀⠁⠀⣶⡖⠠⣄⠘⠟⠿⣿⢫⣾⣿⣿⣿⣿⣿⣿⣿⠙⠁⠀⠼⠀⠀⣧
⠀⠀⠀⠈⠉⠉⠙⠛⣢⠞⠋⣹⢳⣶⡿⣿⣇⠠⢶⠐⠐⠀⣆⣿⣾⣿⣿⠟⠂⠀⠀⠠⠀⡁⢨⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⡸⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠆⡟⣾⢈⣇⣿⡟⢰⡾⠓⠤⡀⠈⣽⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⡝⠋⣽⣿⣿⣿⣿⠏⠂⠂⠘⣃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠀⠃⢿⣄⠙⣿⣷⣻⣇⣼⣴⣴⠰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠈⠙⣟⣩⣤⣀⡲⠆⢠⣿⠈⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⠀⢸⡋⢠⣿⣗⡏⣯⢿⣽⣿⣇⣿⣿⠁⠀⡆⠀⠀⠀⠀⠀⠀⠐⠀⠘⡆⣨⣿⡟⠀⠀⠀⣐⣀⡈⠂⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⢸⢡⢾⡿⣧⠁⠰⣾⡿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⢸⣿⣿⠉⠉⠉⢭⠉⣭⠉⠉⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣜⠴⠂⢸⣊⣤⠟⣿⠀⢤⡴⣳⣿⡻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⡶⡿⢿⣶⣤⣀⠀⠀⠀⢨⡦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⢺⠁⣻⣿⣿⣶⣾⣀⡤⣰⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠄⡄⠀⠀⠙⣿⣿⡋⣭⠢⠀⣷⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠁⢀⣰⣿⣿⠯⠟⠛⠋⢩⠞⠉⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠏⣠⢾⣿⣦⡁⠚⠃⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢶⠟⠻⠽⠖⠀⠀⢀⠴⡫⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣾⣽⡿⣿⣿⠇⠀⠀⡿⢨
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠠⣄⡄⣰⣶⣄⣀⣀⠀⠐⠊⠁⠀⢀⣴⣿⣻⢿⡏⢰⣿⣿⡀⣷⣾⣿⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣼⣿⣸⣿⠟⣿⣷⣶⣶⣶⢶⣿⣿⣿⣿⣏⣠⣿⣿⣿⣿⣿⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⢼⡓⢂⣿⣿⣛⣹⣿⣼⢹⣿⣿⠏⣡⣿⣿⣷⣝⠿⢿⣯⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⣺⠞⢃⡯⠀⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣮⣾⠿⠉⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣇⠴⠞⠋⠉⠉⠉⡯⣷⣿⣿⣿⣿⠿⠛⢋⠃⡀⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⣠⡴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡇⠀⡠⢤⠀⢺⢄⡀⢠⠖⠂⠀⡔⢰⠈⠧⣄⠀⣀⢀⠈⠆⠀⡴⢤⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠒⠚⢄⢀⠀⡸⠀⡄⢸⠀⡌⢠⡇⣼⠉⢁⠎⢀⠃⣼⠀⠀⠁⠀⡘⠁⡐⢀⠎⢀⡄⣠⠋⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠶⠐⢀⡘⠘⠐⠓⠊⠀⠈⠉⠉⠁⠀⠀⠉⠁⠀⠀⠉⠀⠑⠒⠊⠉⠀⠀⠓⠛⠄⠚⠶⣁⠼⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡷⠶⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣇⣀⡀⣴⡆
⠀⠀⠀⠀⠀⢀⣾⣿⠋⠁⠀⠀⠀⠈⠙⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡖⠤⢀⣤⣶⣶⣾⣿⠷⠶⢾⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⣠⢾⣿⣋⡁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⡀⠀⠀⠀⠀⠀⠀⡿⠸⠇⠘⣸⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⣿⢸⡟⠛⠛⠃⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⣇⢀⠀⠀⠁⢞⣿⣿⠟
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠙⠎⢧⠀⠀⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠻⣿⣿⣿⣶⣾⣿⠋
⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠜⡄⠀⢻⣿⣿⣿⠿⣿⣶⣄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⡟⠛⠋⠁
⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠀⠈⢿⣿⣷⠀⠀⠉⠻⣷⣄⠀⣼⣿⠃⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠐⡀⠀⠘⢿⣿⣆⠀⠀⠀⣾⣻⣿⣿⠋⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠈⢆⠀⠈⠻⣿⣿⣶⣶⣿⣿⡿⠻⣦⡀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠙⠆⠐⢻⣿⣿⣿⠟⠋⠀⠀⠘⢷⡄⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣀⣀⠀⠹⢿⣧⡀⠀⢀⣀⣤⣾⣿⣿⠟⠻⣿⣄⠀⠀⣠⢖⡈⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⣿⠿⠿⠛⠛⠋⠉⠙⢶⣤⣼⣿⣷⣄⢁⣿⣿⠟⣱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣄⣀⣀⣀⣹⣿⣿⣿⣿⣿⡿⣷⣿⠟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⢿⣿⡿⠿⣿⡜⡇⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠒⢄⠀⡰⠁⢀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⠻⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠜⠰⠣⠃⢆⡜⠜⢱⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣰⣿⢟⣵⡿⢏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⡔⠁⣄⠀⡠⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⣄⠀⠀⠼⣟⣵⣿⠏⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠞⠐⢖⡁⡜⢀⡀⠀⣀⢠⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣟⣿⣿⠏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠋⠀⠀⠰⠓⠁⠎⠸⠱⠁⠏⠸⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂

⠀⠀⠀⠀⢀⠴⠂⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠀⠐⢄
⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠒⠒
⠀⠀⠀⠀⠉⣉⣽⣿⠿⡏⠁⠀⢠⠊⠉⢲⣀⡾⠋⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣽⣿⡿⠏⠁
⠀⠀⠀⠀⠀⡿⠋⣁⣤⡇⠀⠀⢸⠀⠀⣰⠋⢧⡀⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢁⣠⡆
⠀⠀⠀⠀⠀⣶⣾⠿⠛⡇⠀⠀⠀⠠⡀⠂⠀⠀⠁⡐⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠟⣀⣤⣶⡇⠀⠀⠀⠀⠈⢔⢄⠀⠀⠂⠀⠀⠀⠠⡈⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⠇⠀⠆⠀⠐⠀⢠⡤⠴⡆⠀⠀⠀⡟⢁⣤⣶⡇
⠀⠀⠀⠀⠀⣿⠿⠛⢁⡇⠀⠀⠀⠀⠀⠈⡀⢱⠆⠀⠀⠀⠀⠀⠙⢀⣀⣀⡐⠁⠀⠀⠀⠀⠀⡜⠀⡘⠀⠀⠀⠀⢐⠒⠿⠅⠀⠀⠀⣿⡿⠛⢉⣀
⠀⠀⠀⠀⠀⣃⣴⣾⡿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠹⠀⠀⠀⠀⠀⢰⠃⢀⠃⠀⠀⠀⠀⢒⣛⣭⡅⠀⠀⠀⣏⣤⣶⡿⠇
⠀⠀⠀⠀⠀⡿⠋⣁⣤⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢇⠀⠄⠀⠀⠀⠀⠀⠠⣬⣬⡇⠀⠀⠀⡿⠛⢉⣠⣦
⠀⠀⠀⠀⠀⣶⣾⠿⠛⡇⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠜⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠩⠥⢖⡂⠀⠀⠀⣷⣾⡿⠟⠁
⠀⠀⠀⠀⡀⠋⣁⣤⣾⢇⡀⠀⠀⠀⠀⠀⠈⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⠀⠀⠀⠀⠀⠀⠀⣘⢒⣲⠇⠀⠀⣀⠛⢁⣤⣶⠗⡀
⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠠⠠⢀⣀⣀⣀⣀⠉⠉⣉⣙⣉⣁⣈⠭⠂⠀⠀⠀⠀⠀⠂⠉⠭⠅⠀⠀⠠⠠⠤⠤⠤⠤⠂
⠀⠀⠀⠀⠈⠢⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀⠀⢠⠀⠀⢠⠀⣀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣦⡯⠐⠁⠀⠀⠀⠒⢸⣧⡆⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⣁⡟⣤⠀⠀⠀⠀⠀⠀⢠⠙⣅⡔⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠓⠷⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⣀⡱⠃⠀⡀⠀⠀⠀⠀⠀⠀⣀⢘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⡀⠀⠀⠀⠀⠀⠀⡰⠁⢠⣥⣤⡦⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠨⠗⠂⠀⠀⠀⠘⠬⣿⡟⠦⡀⠀⠀⠀⠴⢪⣿⡧⠊⠀⠀⠀⠀⠀⠀⢀⡧⠄⢺⡇⠀⣹⠢⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⣦⢀⠀⠀⣌⣠⡚⣉⠀⠀⠀⠀⠀⢀⣤⠊⠍⢹⣆⡌⠛⠛⠁⠀⣶⠋⠁⢦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡆⠀⡿⠏⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠓⠒⠒⠒⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⣤⡶⠶⢶⣶⣶⣶⠶⠦⣄⣄⣀⣀⣀⠄⠒⠐⠂⣄
⠀⠀⠀⣀⣠⣤⣶⣾⣿⡿⠿⠿⣯⡴⠿⠿⠿⠷⢶⣤⡈⠙⣷⣆⠀⢻⣷⣸⣄⣀⣀⣠⣴⣿⣧⣀⡀
⢠⣴⣿⡿⠟⠛⠉⠁⣀⠀⠰⠚⠁⢴⣄⠐⣦⡀⣠⡙⣿⣆⠘⣿⡆⠿⢿⣿⣿⠠⠤⠧⢼⣿⣿⠻⣿⣷⣤
⢸⣿⡇⠀⠀⠀⠀⠀⠁⠐⢩⠾⣦⢐⣹⣦⡘⣿⣿⣧⡘⣿⠰⣿⠇⣠⣾⣿⣿⠀⣶⣶⢸⣿⡇⠀⠀⣿⣿
⢸⣿⡇⡠⠀⠀⢠⠂⠀⢂⠀⢼⣿⣦⡈⠉⠙⠻⢿⡧⠷⠋⠐⠙⠛⢭⣿⣧⣏⠀⠛⠋⠈⠸⠀⠀⠀⣿⣿
⠸⣿⡇⠁⠀⢀⠃⠱⢰⠃⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⠃⠀⣀⣤⢠⣾⡇⠀⠀⣿⡿
⠀⣿⡇⠀⠀⠙⡄⠀⣾⠀⠀⢸⣿⠃⠶⠶⠒⣲⡀⠀⠀⠔⠶⠶⠶⡌⢿⣿⣼⡇⣿⣿⢸⠏⠁⠀⢀⢿⡇
⠀⣿⢸⠀⠀⢀⡜⡶⢻⠀⠀⢸⡏⠀⡠⢶⡗⠯⠃⠀⠀⠰⢶⡽⢎⠛⠀⣿⠟⠃⠉⠀⠀⣇⡀⠀⢸⢸⡇
⠀⢹⡏⠀⠀⠈⢤⡜⠸⡀⠈⣿⠇⠀⠀⠈⠁⠀⠀⠄⠀⠀⠀⠉⠁⠀⢀⢦⣞⠡⣴⣶⣾⣿⡇⠀⡸⣾⠁
⠀⠸⣧⠀⠀⠀⠸⡇⠀⡇⠀⠱⣇⠀⠀⠀⢀⣀⣤⣤⣤⣴⣀⠀⠀⠀⠀⡟⠁⡾⠿⠿⠛⡇⠀⠀⣧⡿
⠀⠀⢿⣷⠀⠀⠀⡇⠠⠃⠀⠈⢿⣷⣤⢖⣩⣭⣥⢶⣾⣭⣬⣙⠦⣠⣼⡟⠀⠀⠀⠀⣀⣧⡆⢸⣿⠇
⠀⠀⠸⣿⣇⠀⠃⢧⠀⢠⠀⠐⣾⣿⣾⣿⣿⡉⢹⣶⣿⡉⣹⣿⣿⣿⣟⣿⣀⢰⣿⣿⣿⡟⣧⣿⡿
⠀⠀⠀⢿⣿⡄⠈⠃⢦⢸⠀⠀⢼⣧⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡇⣷⠾⠭⡥⢤⣷⣾⣼⠇
⠀⠀⠀⠈⢿⣳⠀⠀⠀⣨⡄⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣼⣿⣤⠤⢧⣼⣿⣿⡟
⠀⠀⠀⠀⠘⣿⣧⠀⠀⢹⠀⡂⠨⡏⠀⠈⣷⠀⡂⢸⠂⢐⠀⡇⢐⡦⣖⠀⡂⢸⡀⠀⢈⢛⡝
⠀⠀⠀⠀⠀⠘⢿⣳⡀⠻⠤⠴⠖⡴⠾⠤⠶⠤⠷⢶⠤⠶⢴⡷⠶⠦⡶⠶⠦⠼⠃⣰⣳⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⡧⠄⠘⡇⠠⠄⢸⠀⠿⢀⡇⠨⡤⠆⠀⠀⢀⣴⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠛⠒⠛⠛⠚⣷⣾⣿⣶⠛⠓⠛⠃⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣄⣀⣀⠀⠛⠋⢁⣀⣀⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣿⡟⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⢉⠛⠿⣿⣾⣿⡁⣠⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢠⣙⠳⢦⣌⡙⠻⢿⣿⣯⣀⣼⣿⠃⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⠀⠀⠙⠳⢦⣌⡛⠶⣤⣉⠛⠿⣿⣧⣤⣿⡟⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⢿⣿⡆⠀⠀⠀⠀⠈⠛⠶⣤⣙⠳⢦⣌⡙⠻⣿⣷⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣌⠛⠶⣤⣉⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣬⡙⠃⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣄⣀⠀⠀⠀⢀⣀⣠⣤⣶⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠿⠟⠛⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠉⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⡀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣄⣀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⢀⣀⠙⠳⢦⣄⠀⠀⠀⢀⣴⣿⠏⠙⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⢀⡾⠉⠻⣦⠀⠉⠁⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠾⣥⣀⡾⠃⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠘⠳⢦⣄⠈⠙⠁⠀⠀⢀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠉⠛⠶⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠘⠳⣦⣀⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⡀⠀⠉⠻⠆⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠒⠲⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡟⠁⠀⠀⠀⠳⠀⠀⡠⠒⠉⠀⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⡐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⡀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠒⣴⣾⣿⡷⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⣀⣀⠤⠒⠁
⠀⠀⠀⠀⢠⡈⠉⠁⠀⠠⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠈⣿⡠⢀⣾⣡⠆
⠀⠀⠀⠀⠈⢧⣀⠀⠀⣀⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠓⠛⠇⠚⣁⣴⠀⠀⣤
⠀⠀⠀⠀⠀⠀⣿⢁⠘⣿⡿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⡼⠁⣸⠆
⠀⠀⠀⠀⠀⠺⠘⢿⡀⠋⢀⣼⣿⡀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠃⠺⣧⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠁⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠀⠀⠀⠰⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣶⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡿⢛⣤⣍⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⣻⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⣼⣿⣿⣿⡄⡤⠀⠀⠀⠀⠀⠀⣻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⠁⣀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⣶⣿⣿⠀⠀⠀⠀⠀⢤⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠻⠟⢻⠄⠀⠀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡸⠛⠚⠋⣡⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣄⡠⠀⣽⣄⣀⣾⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣁⠀⠀⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢍⠉⢉⣹⠟⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣴⣶⣤⣤⣤⣤⠀⢠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠹⣿⣿⣿⣿⡟⠀⣸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣀⣀⣈⣉⡻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⣿⣿⣿⣿⠁⡙⢿⣿⣷⣿⣿⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠏⠷⠀⣿⣿⣀⢑⣻⡛⣮⠙⣿⡏⠀⣠⣬⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⠆⢸⠏⠀⠀⠀⠘⣿⣿⣿⣿⠟⠉⣡⣿⠃⠚⠁⢩⣽⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⠛⠻⣷⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣀⠉⠉⡿⢁⣔⣀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⡇⠀⠀⢀⠔⠉⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡿⠀⠀⡠⠂⠀⠀⢀⣩⣿⣿⣿⣿⣍⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡌⠀⢀⣴⣿⡿⣿⣿⣿⣯⠻⢿⣦⡀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢠⣎⣀⣴⠟⠋⢰⡇⣿⣽⣿⣿⣧⣠⠉⢳⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡏⠙⣿⠁⠐⣣⣾⣿⣦⣿⡹⣿⣿⡿⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢰⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣄⣾⡏⣼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⠋⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⢀⡄⠀⠀⣠⢾⣿⡿⠷⠇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⡇⠀⠀⠀⠸⢻⠇⢀⣴⢶⡯⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣧⣰⡇⠀⠀⢀⣴⡿⢏⣺⣀⣼⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣷⡀⠀⠸⠟⠁⠈⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠈⠉

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



⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠲⣶⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣶⣶⡾
⠀⠀⠘⣿⣀⣤⣤⣴⣶⣿⣿⣿⣷⣶⣦⣤⣄⣸⡏⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡿⠛⠉⠁⠉⠁⠈⠉⠉⠉⠻⠿⣿⣿⡿⠟⠋
⠀⠀⢸⣿⠛⠿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⢀⣀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⢸⣿⣷⡄⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣀⣰⣿⣿⣷⣀⣠⣾⣿⣿⣿⣿
⠀⠀⠈⣿⣿⣿⣿⣦⣤⣠⣿⣇⣠⣴⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⠄⠀⠀⠀⠠⠀⠄⠀⠀⠠⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⠄⠠⠀⠖⠰

⠀⠐⢸⠉⠐⠒⠚⠀⠒⠒⠲⠖⠀⠠⠠⠀⢐⠒⠒⠀⣠⣤⣤⣤⣤⣤⣤⣀
⣠⠤⠤⢬⠇⠡⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠫⡃⢖⢢⠫⡩⡝⢽⡄
⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⡿⡳⢠⢊⠢⠡⣡⢼⣏⣈⣀
⡇⠀⣾⢱⠀⣰⠋⠱⡀⠀⡖⡄⠘⢈⡉⡁⢠⡹⣻⣼⠢⡃⣑⢜⠃⡺⢙⢯⣿⣻
⡇⠀⣟⢀⣭⡁⠀⠀⢈⣈⠁⠓⠐⠋⠀⠈⠾⣟⠏⣥⣴⠟⠉⣯⣀⣷⢦⣯⣷⢿⡇
⡇⠀⠉⠁⠀⠹⣄⡐⠁⢹⣠⠉⢻⣄⠀⢀⡴⣤⡀⣀⠀⠀⠀⠈⠛⠋⢸⣿⣏⣾⡇
⠣⠥⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⢀⢍⣿⢏⡼⣹⣽⢿⣆⠀⢀⢴⣷⣆⢸⢯⣿⣿⡇
⠀⠀⡤⠞⠀⠀⠰⠀⢀⢄⠀⢀⠪⡺⠃⡘⢐⢌⢋⡞⣽⠳⠣⢿⢢⣻⢮⣟⠧⢽⡇
⠀⠀⡇⠀⠀⣀⠈⠀⠁⠀⢠⣃⡼⡑⢅⢌⢪⡂⡕⢸⡡⣑⢫⡡⣹⣧⣏⢨⣷⢾⡇
⠀⠀⠀⠉⠁⠐⠂⠨⠥⠴⣟⣿⣾⢬⣧⣱⣕⢌⠢⠑⢌⡂⡑⢔⣿⢾⣯⣟⠛⠋⠑⠒⠒⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⣾⣶⣮⣾⣧⣣⣼⣷⠠⠐⠒⠒⠒⠂⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣛⣩⠿⣿⣿⣿⣿⡝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠧⢀⢤⣿⡿⠿⠛⠛⠋⠉⠁⠀⠀⠈⠻⣿⣿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀⠀⠈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠉⠉⠙⠛⠿⢷⣦⠀⠀⠀⠀⠾⠟⠛⠉⠉⠉⠉⠉⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠌⠀⡇⠀⠀⠀⠀⠒⢚⣶⠂⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⠒⠀⠀⠀⠀⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⠏⠀⢠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢿⡀⠀⢀⣤⣾⡿⠿⠿⠿⣿⣦⡀⠀⠀⣸⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣧⡴⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠤⠀⠂⠈

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

⠀⠀⠀⣤⣄⣀
⠀⠀⠀⠻⣝⠮⣝⡲⢤⣀
⠀⠀⠀⠀⠈⠳⣄⠙⠲⢬⣙⠦⣄
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠓⠦⣝⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠲⢭⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠉⠓⠿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠐⠂⠀⠈⠙⡦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠈⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡇⠀⠉⠙⠒⠦⠤⠤⠤⠴⠒⠋⢀⡠⠖⠚⠛⠓⠲⣄⠐⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡤⠤⠒⠛⢤⡀⠀⠀⠀⠀⠀⠀⠀⡐⢀⡞⠀⠀⠀⠀⠀⠀⠈⢳⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠈⠛⢷⣤⡀⠀⠡⠘⣇⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠉⠻⣷⢤⡀⠈⠳⢤⣀⣀⣀⠴⠚⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡓⠦⣤⣀⣀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠴⠒⠛⠓⠲⢤⡀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⡼⠁⠀⠀⠀⠀⠀⠀⠱⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⡀⢷⡀⠀⠀⠀⠀⠀⠀⣰⠇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠑⠦⣄⣀⣀⡤⠖⠁⢐⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁


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

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⣀⡀
⠀⠀⠐⠁⠉⢄⠀⠀⢰⠋⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠘⣤⣿⣭⠁⠀⠀⠀⣀⣡
⠀⠀⠘⠦⠠⠓⠀⠀⠺⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠁⠀⠀⡄⠀⠀⠀⠀⢹⣝⠋⠀⠀⠀⠀⠉⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠈⠉⢱⣶⡾⠁⠀⠀⠀⠀⢢⣉⠀⠀⠀⠀⠀⠆⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⡷⠆⠰⢾⡄⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠠⠄⢳⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣥⠒⠒⣬⡞⠀⠀⠀⠀⢠⠭⡄⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣷⡆⠸⢛⠀⠀⠀⠀⠀⣴⣶⢠⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣧⡄⠰⢞⠀⠀⠀⠀⠀⠋⣿⣏⠀⠀⠀⠀⠀⠐⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⢀⣿⣿⣿⡟⢛⡆⠀⠀⠀⠀⣇⡄⢰⠾⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡏⡅⢠⣾⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⡴⠏⠁⢨⠴⢦⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠇⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣞⠻⣿⣿⡏⡀⠀⠀⠀⠁⡇⠀⠀⢸⠈⠀⠀⠀⠀⠸⠆⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠁⠢⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠉⠉⠉⠑⠀⠐⠒⠲⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢤⠴⠒⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡶⠧⠤⢤⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠀⠀⠂⠰⠆⠄⠄⠀⠠⣀⡀⠤
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡀⢻⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⡦⠀⠀⠁⢰⡆⣀
⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠴⠶⠶⠖⢢⡍⠁⢰⠁⠀⠀⠀⠀⠀⠀⢀⡤⣿⠐⣸⢁⣄⠀⠀⠀⢀⠀⡇⠶⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠉⠈⠉⠹⡧⠀⠀⠀⣇⠒⠂⠈⠀⢸⠃⠀⠀⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⡇⠀⠀⠀⠀⡶⠉⠀⢀⠀⠀⠀⠀⠀⠘⢠⠄⠀⠀⣽⣦⣄⣀⠀⣠⠃⠀⠀⠀⠀⠀⠘⠀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⠀⠀⠀⠀⠀⣠⠆⠙⠓⠂⠉⠯⣤⡀⠹⠂⡀⠀⠶⣜⣁⠀⢦⣸⠆⠀⠀⡀
⠀⠀⡠⠋⠀⠀⢀⠾⠊⠋⠈⠣⡄⠀⠀⢸⠀⠀⣇⠀⠀⠀⠤⠰⠁⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠉⢦⡆⠈⠳⡎⠰⠅⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠊⠄⢠⠋⠀⠀⠀⠀⠀⠰⠀⠀⠘⡇⠀⣿⠀⠀⠀⠦⣼⠀⣀⠀⠀⠤⠀⣀⣀⢬⡹⠀⠸⢌⢫⡄⡾⢠⠓⠀⠤⠀⡀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠴
⠀⠀⠆⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡇⠀⢣⠀⠀⠀⠀⠡⠀⢳⣤⠀⠀⠲⢶⠏⡘⠀⠀⠀⠈⢦⡙⣤⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡬⠀⢀⡆
⠀⠀⠀⠀⣦⣼⡄⠀⠀⠀⠀⠀⡸⠃⠀⠀⡇⠀⣺⠀⠀⠀⠀⠀⠙⠢⡄⠐⠂⠂⡠⠖⠀⠀⠀⠀⢀⡌⡁⢬⡳⠖⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠙
⠀⠀⠀⠀⠁⠀⠉⠠⡀⠀⣀⡴⠁⠀⠀⠀⡑⣿⣫⣄⠀⠀⠀⠀⠀⡀⠏⠀⠀⠘⢦⡀⠀⠀⠀⢸⠋⢘⡿⠢⣴⡜⠁⠀⠉⠙⠛⠶⠤⣀⣠⠊⠉⠁⠐
⠀⠀⠀⠀⡈⠁⠈⡀⡏⠉⠁⢠⠀⠀⠀⠀⣧⡀⢩⡄⣰⢄⣴⠋⠉⠃⢀⣄⡰⠀⠀⠉⠉⢃⡀⢈⡟⠉⠡⣠⣼⢓⡤⢄⡀⠀⠀⠀⠀⠀⡁⠀⠉⠀⠐⠆
⣀⡐⠀⠁⣀⠀⣀⠔⠓⠰⠆⠉⠉⠑⠶⡄⣙⠙⠡⢶⣅⠛⢷⣶⡄⠀⠀⠅⠠⠀⠀⠀⠀⠁⠸⠋⠀⠀⡜⠁⠀⠀⠀⠀⠀⠋⠰⠂⡄⠈⠣⣤⠲⠶⠀⠐
⣉⣈⣁⣀⡀⣀⣡⠤⠤⠖⠘⠓⠒⠒⠒⠢⡙⢄⡀⠈⠘⡣⣀⣩⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡷⠊⠀⠻⡑⠉
⠀⠀⠀⠈⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠻⢯⡄⢀⠸⢇⠀⠀⠀⠀⠀⠀⠀⣸⢣⣤⢠⡴⠁⠀⠀⠀⠁⠴⠀⠀⠀⠀⠀⡮⠀⠀⠀⠀⢐⢹⡷
⣀⣀⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠤⠤⠤⠤⢾⠀⠀⠀⠀⠀⠀⠀⡇⢺⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠁⠀⠠⣇⡀⠀⠀⠀⠠⠘⠁
⠀⠀⡌⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡯⠄⠀⠀⠀⣺⠀⠀⠀⠘⠀⠀⠀⡃⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠈⠀⠀⣰⠆
⣀⠀⣐⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡆⢳⠒⠒⠀⠉⢹⠂⠀⠤⠤⠤⠀⠤⠁⠀⠀⠈⠉⠒⠦⣄⡀⠀⠀⠀⠀⠀⠘⠁⠈⢠⠤⠚⠁⠀⠘⠃
⠀⠠⡴⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠤⠀⣀⢀⠀⡈⠠⠉⠉⠈⠁⠂⢉⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢠⣄⠀⣀⠀⠀⠀⠀⠀⠒⠁
⣀⠀⠸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⡇⠀⠀⢀⡀⠀⠀⠈⡏⠀⠠⣀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠈⣡⣈⡐⠇⡀
⠉⠀⠀⠙⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠤⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠢⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠔
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠹⡄⣀⣀⣀⢀⡀⣀⠠⠤⠤⣤⠖⠂⠁⠠⠄⠀⢻⠀⠀⠀⢸⣷⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡄⠂⠁
⠀⠁⠀⠀⠀⠁⠀⠀⢩⠙⢤⣀⡀⠀⣠⠞⠁⡇⠀⢀⣀⠀⠀⠾⠀⠀⠀⢰⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠈⣄⠀⠀⢸⠀⢀⣀⣀⡏⠀⢀⣀⡠⢤⡇⠀⠀⠀⢺⠀⡇⠀⠀⠀⢛⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠞⠻⠀⠈⠁⠉⡟⠛⠈⠁⠀⢀⡇⠀⠀⠀⢸⠀⢰⠀⠀⠀⢸⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⣀⣤⠤⠤⢶⠒⠒⠚⢉⠭⡀⠀⠀⠀⣇⣠⠋⠁⠂⡄⠀⠙⠀⠀⠀⠀⣏⠀⠸⡆⠀⠀⠘⡇⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠠⠀⠁
⠈⠀⠀⠀⠀⠀⠑⠄⠀⣏⠀⠘⠀⠁⠀⠁⠁⠀⠀⢀⠇⠀⣿⢀⣀⣠⣰⡇⠀⠀⢿⡀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⣈⣉⣀⣀⠤⠖⢁⣤⠎⠀⠀⠀⠀⢸⡇⠀⢠⡞⠉⠁⠀⠈⣳⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠁⠀⠸⠶⠒⢤⠒⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣂⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠋⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣛⣛⣟⣻⣻⣿⣿⠛⣿⣟⠛⢛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⣠⠔⠋⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⠀⢀⡤⠊⠁⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡇⢀⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡷⠋⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠖⢉⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠴⠶⠞⠋⢸⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢁⡤⠶⠒⠲⣤⡀⠀⢸⡶⡏⠈⡇
⠀⠀⠀⠀⠀⠀⢠⡟⠀⣰⠋⠀⠀⠀⠀⠀⢿⠀⡟⠀⡗⠚⢣⡀
⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⠀⠀⠀⠀⢀⡟⢸⠃⠀⡟⠀⣈⣉⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠘⣧⠈⠣⡀⠀⢀⣀⡴⠋⣰⠏⠀⢰⢠⡞⠉⠉⠙⢷⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣈⣉⣉⣡⠴⠛⠁⠀⠀⡎⢸⡇⠀⠀⠀⠀⢷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢠⡇⠘⣧⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠙⣦⡀⠀⣀⡾⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠙⠛⠉⢀⣾⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣲⣒⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⢠⣢⣶
⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠭⡍⣹⣿⣿⣿⣿⣿⣿⠿⣫⣖⣻⣿⣯⣀⣠⡤⠾⠃
⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⠀⠀⠀⡠⣖⣵⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠒⠁⣠⢔⣯⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣫⣶⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣩⣴⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣪⣵⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢖⣪⣵⣾⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⣀⠤⣒⣭⣶⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⢠⣾⠵⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠸⣿⣿⡿⠿⠛⠉



⣷⣶⠶⠶⠶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣍⡉⢉⣩⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⡉⠉⢨⢨⠉⡏⣯⡽⠛⠛⣏⣯⣭⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠾⠿⠛⠿⠾⠿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⣠⣾⣿⡋⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣟⠀⣠⣾⣿⠋⠻⣿⣦⡈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠛⢿⣷⣄⠈⠻⣿⣦⡈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠙⢿⣷⣄⠈⠻⣿⣦⡈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠙⢿⣷⣄⠈⠻⣿⣦⡈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠻⣿⣦⡀⠙⢿⣷⣄⠈⠻⣿⣦⡈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠈⠻⣿⣦⡀⠙⢿⣷⣄⠈⠻⣿⣦⡈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠀⠀⠈⠻⣿⣦⡀⠙⢿⣷⣄⠈⠻⣿⣦⡈⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠛⣿⣷⣄⠀⠀⠀⠈⠻⣿⣦⡀⠙⢿⣷⣄⠈⣻⣿⡿⠋⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡁⠀⠀⠀⠀⠀⠛⣿⣷⣄⣀⣴⣾⡿⠻⣿⣦⡀⠙⣿⣿⡿⠋⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠉⠻⣿⣦⡀⠀⠀⠀⠀⢀⣿⣿⠿⠛⠁⠀⠀⠈⠻⣿⣿⣿⠋⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⠀⠀⠀⠀⠈⠻⣿⣷⡄⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠁
⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋
⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠁
⠀⠀⠀⠘⢿⣷⣦⣄⣀⣤⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣮⡼⣛⠛⠿⣶⣄⡀
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠋⠁⠀⠀⠙⢿⣶⣄⡀
⠀⠀⣠⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣤⣀⡀
⠀⠸⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠸⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⣭⣛⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣶⣬⣝⡻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣶⣬⣍⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣷⣶⣬⣭⣙⣛⠻⠿⣿⣿⣿⣿⣟⠉⠙⠛⠛⠛⠛⠛⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣶⣶⣾⣭⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⢨⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡋⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⠿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⣠⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣄
⠀⢀⣾⡿⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⢻⣷⡀
⠀⣼⡿⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣧
⠀⣿⡇⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⢸⣿
⠀⣿⡇⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⢹⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⡟
⠀⠘⣿⡆⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢀⣿⠇
⠀⠀⢻⣇⠀⣿⡇⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⢸⣿⡀⢸⡟
⠀⢀⣼⣿⣾⣿⠁⠀⢰⣾⠿⠟⠛⠉⠉⠉⠙⠛⠻⠿⠀⠀⠀⠀⠀⠀⠺⠟⠛⠋⠉⠉⠉⠛⠻⠿⣷⡦⠀⠀⣿⣧⣿⣧⡀
⣰⡿⠛⢻⣿⡿⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣤⣄⠀⠀⠀⠀⠀⢹⣿⡟⠛⣿
⣿⡇⠀⠈⠿⠇⠀⠀⠀⠀⠀⠛⠋⣿⣿⣿⡏⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢹⣿⣿⣿⠛⠛⠀⠀⠀⠀⠀⠘⠿⠃⠀⢸
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⢸⣿⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣿
⠈⣿⡇⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠹⣿⣄⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣠⣿⠇
⠀⠀⠙⠻⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣠⣤⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠃
⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡿⠿⠿⢿⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡏⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠉⠻⢿⣶⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠛⠁⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⡿⢿⡿⠿⠿⠛⠁⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⣶⠀⢰⣦⡆⠐⡶⠀⢰⠀⠀⣶⠂⢰⣦⣶⠀⣶⠀⢰⣶⠄⠐⠀⢴⡆
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠙⠛⢀⣛⣂⡈⠉⠁⢀⣋⣀⣘⠓⠀⠙⢃⣈⣉⣙⡀⠛⠂⣘⣙⣁⣀⡀⢚⣃⣀⣀⡀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡄⠀⢨⣿⣿⣧⠀⠀⢸⣿⡿⣿⣿⡆⠀⢸⣿⠟⠻⣿⡆⠀⣿⡿⠛⠻⠃⢸⣿⡿⠿⢿⣧
⠀⠀⠀⠀⠀⢸⣿⠁⠈⣿⡇⠀⢸⣿⢿⣿⠀⠀⢸⣿⠀⠀⣿⡷⠀⢸⣿⣦⣶⣿⡃⠀⣿⣷⣶⡆⠀⢸⣿⣇⣠⣼⡿
⠀⠀⠀⠀⠀⢸⣿⣶⣾⣿⡁⠀⣾⣿⢸⣿⡇⠀⢸⣿⣷⣿⡿⠃⠀⢸⣿⠋⠙⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡟⢻⣿⡁
⠀⠀⠀⠀⠀⢸⣿⠿⠿⣿⣇⠀⣿⣿⣾⣿⣇⠀⢸⣿⠋⢿⣿⡄⠀⢸⣿⣦⣶⣿⠏⠀⣿⣿⣿⡿⠇⠸⠿⠇⠀⠿⣷⠄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡿⢸⣿⡟⠛⣿⣿⠀⢸⣿⠆⠈⠿⠟⠀⢛⣋⣉⠉⢀⣴⣶⣶⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⢸⣿⣦⣶⣿⠃⣸⣿⠅⠀⠘⠛⣁⣀⣀⢠⢤⡆⣴⡟⠋⠉⣻⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣿⠿⠟⠁⢀⣀⣀⣠⣤⣤⣭⣭⣭⣭⣥⣭⣥⣬⡥⠶⠞⣋⣁⣈⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠀⣠⡴⣾⠛⢻⣿⡆⢠⡆⠀⠀⢀⣤⡄⢠⣶⠛⣿⠀⣼⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⠋⠀⠹⠦⠾⠟⠀⣿⢷⣷⣴⡏⣸⡿⣾⡇⣸⠟⠀⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⡇⠀⠀⣠⣤⣶⡄⢸⡟⣼⠏⣿⣥⡟⢱⡿⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⣿⣿⣶⣿⡿⠿⣿⠃⣾⠃⠿⠃⠈⢀⣀⣉⣥⣤⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⠿⠛⠁⠀⣠⠏⠀⣋⡤⠴⠞⣛⣉⠉⠁⢀⣽⡿⠁⣺⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⡇⠀⢀⡴⠃⠀⠁⠀⠶⠶⢶⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠸⣿⣷⠞⠋⠀⠀⢀⣠⣤⠶⢞⣛⠉⠉⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣫⣥⣒⡛⠋⣁⣠⣤⣤⣾⣿⣿⣿⣿⣯⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐⠂⠟⢲
⠀⠀⠀⠀⠀⠀⢠⡾⠋⣠⠶⠶⣦⠁⡠⣾⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⣿⣿⣿⣿⣿⣿⠀⣻⣄⠋
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣀⣼⠋⠀⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡼⣧⣀⣠⣀⣽⣿⠂
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢁⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⣸⣿⣿⣿⣿⡟⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣇⢹⠁⠀⠀⠙⠦⠬⣭⣍⣉⣉⣉⣉⣉⣙⣛⣛⣛⠛⠛⠛⡻⢋⣴⣿⣿⡿⠟⠉⡇
⠀⠀⠀⠀⠀⠀⢠⣆⡽⢮⣡⠴⣄⢠⠀⠀⠀⣠⣾⣵⣧⣤⣴⣿⣽⣁⣉⣉⣙⣟⡛⠐⠾⣿⠻⠛⠁⠙⠋⠉⣧
⠀⠀⠀⠀⠀⠀⠸⡏⣽⡿⠉⣿⠟⢸⠀⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣩⣿⣿⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⢸⢾⣀⣿⣐⡇⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢀⡇⠀⢈⡬⣉⣩⢭⡭⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⢸⠀⠀⠹⠤⠻⠟⠘⠇⠨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠾⣉⡛⠿⠿⢿⡿⠃⣸⠤⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠉⠉⠉⢉⠛⠛⠛⠛⠿⠿⠷⣿⡁⠊⠀⣿⡆⠐⠐⣈⣃⡒⠁⠀⠀⠀⠀⠀⠉⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄⡀⠉⠁⠀⠀⣿⡇⠀⠀⠉⢻⣿⣶⣦⣄⢲⡄⠀⠀⢀⣠⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⢶⣶⣿⣯⣤⣤⣤⣭⣭⣭⣭⣥⣴⡶⠶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⡠⠼⣀⠦⠄⠀⠀⢀⠀⣿⡇⠀⠀⢈⡴⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠈⠀⠉⡃⠀⠀⠀⢷⣻⡇⠀⠀⠈⠛⠋
⠁⠄⠋⠠⠄⠀⠀⠈⣄⣿⡇⠀⠀⣈⣴⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⣠⣶⣆⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⣶⡆
⠀⢀⣀⡀⠀⠀⠀⠀⢹⣾⡇⠀⠀⠈⠛⠋⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢠⠞⣩⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⡇
⠀⠛⣹⠟⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⢶⡆⠀⠀⠀⠀⡶⠀⠀⠀⠀⡀⠸⣿⡇⠀⠀⢠⢞⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢚⡴⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢀⣠⠁⣠⡀⠀⠀⢁⠀⣿⡇⠀⠀⠰⢋⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢉⡴⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠂⠀⠟⢄⡈⠀⠀⠈⣄⣿⡇⠀⠀⢴⣯⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⣩⣞⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠹⣾⡇⠀⠀⠀⢈⡀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠙⠛⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠶⢻⣷⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⣶⡆⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⠰⣫⡾⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⡇⠀⠀⡠⠺⣀⠔⠄⠀⠀⢠⠀⣿⡇
⠀⠀⣿⡇⠀⠀⠀⣀⠃⣀⠀⠀⠀⠄⠈⣿⡇⠀⠀⠘⣡⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠀⠉⠢⠄⠀⠀⠀⣦⢿⡇
⠀⢰⣿⡇⠀⠀⢎⢉⡶⡉⠑⠀⠀⢰⠀⣿⡇⠀⠀⠘⣡⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠘⣿⡇
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⢧⢻⡇⠀⠀⠙⠿⠟⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢛⡽⢟⠀⠀⠀⠀⠀⠸⠁
⠀⠈⠛⠃⠀⠀⠀⣤⢿⣦⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⣤⡄⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢈⡴⢛
⠀⠀⢰⣷⠀⠀⠀⠰⢋⣵⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢿⡇⠀⠀⠀⣠⠁⢀⠀⠀⠀⠄⠀⣿⡇⠀⠀⢈⡴⣻
⠐⠀⢸⣿⡆⠀⠀⠰⢋⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢀⠷⡁⠀⠀⠀⠸⡀⣿⡇⠀⠀⣨⣞⣻
⠀⢡⠘⣿⡇⠀⠀⠰⣫⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⢻⣽⡇⠀⠀⠈⠛⠁⠀⠀⠀⠀⠲
⠀⠈⣦⣿⡇⠀⠀⠺⣷⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⢤⢿⣷⠀⠀⠀⠀⠈⢻⠇⠀⠀⠀⣶⡆⠀⠀⠀⠀⠟
⠀⠀⠹⣿⠇⠀⠀⠀⣀⡀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠰⢋⣴⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⡇⠀⠀⠠⠴⡀⠶⢄
⠀⠀⠀⢸⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠰⢋⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠁⠀⠋⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡠⠾⣀⠶⢄⠀⠀⢠⠀⣿⡇⠀⠀⠰⣩⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠁⠐⠉⠢⠀⠀⠀⠀⣆⢿⡇⠀⠀⠺⣷⠞⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠚⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠸⣾⡇⠀⠀⠀⣠⡄⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢘⡵⢿
⠀⠀⠗⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠛⣹⢟⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣿⡇⠀⠀⠀⢀⠃⠀⠀⠀⠈⠄⠈⣿⡇⠀⠀⢊⡴⢻
⠀⠀⠇⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢘⡵⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠔⠉⣦⠋⠰⠀⠀⢰⠀⣿⡇⠀⠀⢈⣴⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣠⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣠⡇⠾⣿⣿⣿⡾⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠉⠁⠦⢀
⠀⠀⠀⠀⠀⠀⢀⡼⠆⠀⠀⠉⠁⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⢀⣀⣼⣽⣷⡀
⠀⠀⠀⠀⠀⠀⢈⣤⣤⣤⣤⣵⣤⣶⣲⣶⣶⢾⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣩⣤⡤⢾⣟⠋⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠸⣇⡡⠈⢿⠀⠀⠘⠟⠽⠃⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⠉⠀⠀⠀⠁⠼⠋⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠹⠦⣀⣸⡆⠀⠀⠀⠐⣲⢷⡅⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⣏⠀⣀⣠⡄⣤⣠⠄⣾⠤⡄⢠⡶⡂⣴⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣦⣞⢻⣿⡅⠀⠀⠀⠀⠀⠀⠀⢰⣿⢉⡿⢸⡇⣾⠁⠛⡿⢰⡟⡼⠣⣿⠈⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠛⢿⣧⣽⣭⢛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⡰⠁⠘⠃⠉⣀⠤⠉⢈⠅⠄⠀⠈
⠀⠀⠀⠀⠀⠀⣺⠁⠀⠀⠀⠨⣿⣿⣿⣿⣿⣟⠡⢀⡀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠸⣧⠀⢰⡿⢀⠀⣴⢲⠀⣶⢰⡄
⠀⠀⢰⠁⣠⣾⣿⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣶⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣽⢏⡏⢸⣟⡼⢸⣇⠞
⠀⠀⢸⣿⣿⣿⣿⡏⠀⠄⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣸⠏⠛⠈⠁⠀⠉⠀⣿⠁
⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⣤⣯⣿⣿⣿⣿⣿⣿⣿⠁⠀⠰⠿⠆⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠿⢿⣿⣿⣿⠇⠀⠀⠀⠈⠀⠹⣿⡿⢿⣿⡿
⣶⣶⣶⣶⠶⣶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⡶⢶⣶⡶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠁⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠉⠀⠀⠀⠀⠉⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠄⠄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⠉⠐⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠂⠤⢀⡀⠀⠀⠠⠊⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠆⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠁⡲⠁⡀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⣈⣁⠠⠔⠊⠀⠉⠰⠁⠀⠂⢠⠏⣓⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⣸⣷⠂⠀⠈⣁⠒⠀⠀⠤⠔⢊⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠠⠁⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠚⠀⠀⡀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠀⠀⠀⠀⡐⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠐⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠰⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠅⠀⠀⠀⠀⣰⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠄⠀⠀⠀⠀⠔⢋⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣀⠀⢈⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⢀⠔⠱⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠔⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⡴⠋⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠛⢂⣠⢄⣁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠋⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣴⣤⣀
⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣷⡆⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣤⣽⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠐⣱⣿⣿⣿⡿⠂⠀⠀⠀⢀⣠⣤⣤⣴⣦⣤⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⢀⣴⣿⢟⣿⣿⣿⣿⣟⠂⠀⠀⠀⣿⣿⣿⣿⣿⣝⡛⢿⣿⣦⠀⢸⣿⣿⡿⠴⢤⡀
⠀⠀⠀⠀⢀⡸⢿⣿⣿⣧⠀⢠⣿⣿⣩⣥⠭⣿⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠆⢰⣿⣿⣠⠚⣨
⠀⠀⠀⠀⢸⠁⣤⢹⣿⣿⡀⣸⢿⣿⣿⣿⣷⣶⣿⣿⣿⣷⠀⠀⢸⣿⣿⡴⢿⣿⣿⠛⣻⡉⠉⢩⣁⣿⣿⠃⠀⡇
⠀⠀⠀⠀⠀⢳⠀⢿⣿⡿⠁⠀⠀⠀⠹⣯⣩⣽⡷⠒⠋⠁⠀⠀⠀⣿⣿⣷⣾⣭⣷⡶⠏⠀⠀⠘⣿⣿⣿⡇⢐⡇
⠀⠀⠀⠀⠀⠘⡆⣾⣿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⢸⠁
⠀⠀⠀⠀⠀⠀⡇⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⣼⣽⣿⡧⠋
⠀⠀⠀⠀⠀⠀⢻⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡟⠉⢦⣀⠀⢀⣼⣟⢘⡿⢿⣷⣄⠀⠀⠀⣠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⡆⠀⠀⠀⠀⣠⡴⠞⠉⠀⠙⠛⠷⢬⣽⣿⡿⠛⠋⠁⠀⠈⠛⣦⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠐⣿⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⣾⢠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢧⡀⢋⢰⢧⣶⡶⠶⠿⣿⣷⣶⣶⣾⣿⠿⠟⠶⣶⣷⡟⣾⢿⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠻⣷⣦⣠⣁⡉⠙⠁⢀⣀⣤⣶⣿⡏⠋⣰⣯⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠉⠘⣿⣿⢿⣿⣿⣿⣿⣿⢋⠏⠀⢰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠘⠧⡀⠈⢉⣩⡿⠇⠈⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠓⠶⣤⣤⣤⣴⣦⡄⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠘⠛⠛⠛⠋⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⢶⠶⠾⠿⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⣠⣶⣿⠿⠟⠛⠉⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⢀⣴⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣾⡿⠟⠁⠀⠀⢀⣠⠖⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣟⡥⠚⠉⠀⠀⢀⣠⣶⠟⠁⠀⠀⠀⣀⣴⡾⠋⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣿⠛⣻⣵⣿⣿⡿⠙⠁⠀⠀⢀⣤⣶⣿⠟⠁⠀⢀⣠⣴⣾⣿⠟⠁⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⣀⣴⣾⣿⣿⠟⠁⣀⣤⣶⡿⢟⣿⡿⠋⠀⣠⣴⣿⣟⣥⠖
⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣠⣴⣾⣿⣿⣿⠟⣡⣴⣾⡿⠛⣩⣾⣿⠋⣀⣴⣿⣿⣿⡿⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣴⣿⣿⣿⣿⡿⠛⣥⣾⣿⡿⢋⣤⣾⣿⢟⣡⣾⣿⣿⠟⠟⠁⠀⠀⠀⠀⣠⠊⠀⢀⣤
⣿⣿⣿⣿⣿⣿⣿⡿⢋⣠⣴⣿⣿⣿⣿⡿⠟⠁⠀⢠⣿⡿⢋⣴⣿⣿⣿⣿⣿⠿⠛⣩⠖⠁⠀⠀⠀⠀⣠⣾⣡⣴⣾⣿⣿
⣿⣿⣿⣿⣿⡿⣫⣴⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠟⠋⣡⣴⠟⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⣁⣴⣾⠿⠁⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⣵⣾⣿⠿⣁⣠⣄⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠚⠛⠲⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠚⠀⠀⠀⠀⠀⠀⣀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠖⠒⠲⢦⡀⠀⠀⠀⣀⡤⠤⣄⡀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢰⡏⢠⣞⣻⢮⠇⠀⢀⡞⣡⣤⡀⠈⢳⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡟⠀⠀⠘⣇⡀⢉⣡⠏⠀⠀⢸⡰⣿⣼⡿⠀⢸⡇⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡿⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠳⣌⣉⣀⡴⠋⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡇⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣀⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣼⡇⠀⠀⣠⡄⠀⠀⠉⠛⠛⠋⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠗⣦⢤⣀⣀⣀⣠⣤⢶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⣧⡛⢦⣠⡁⠀⣇⣠⢜⡿⠁⠀⠀⠀⠀⠀⢠⣤⣤⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠧⣄⠀⠁⢸⣀⡴⠋⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⢌⣙⣛⡋⠁⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⣀⣀⣠⣤⠶⢿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣟⠛⠉⠉⠁⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡟⠻⣦⣄⣀⣀⣠⣴⠟⢙⡷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠈⠳⠶⣤⣌⣉⣉⣉⣠⡴⠛⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⠒⣒⣒⣒⠒⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢀⡄⣴⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣀⡀⠉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣿⣿⣿⣿⣿⡸⣿⣿⣿⡿⣿⣿⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⣾⡿⠿⢿⠛⠧⠈⠻⠿⠷⠘⠋⠘⢋⣵⡿⣟⡅⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⠉⠉⠉⠙⢲⡶⠲⣶⠋⠁⠀⠀⠙⢿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⠀⠀⠀⠀⢀⡇⠀⢷⠀⠀⠀⠀⠀⣸⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡙⢦⣀⣀⡠⠊⠀⠀⠈⠳⠤⠠⠤⠞⣱⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣰⣿⣿⢁⡀⠀⠀⠀⢀⣶⣿⠶⢤⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⢧⠀⠀⠀⠠⣿⣿⣷⣀⣤⣴⣿⣿⣿⣿⣿⣶⣤⣴⣿⣿⣿⠋⠀⠀⠀⢀⣾⡿⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⡀⠀⠀⠘⣷⡀⠀⠀⢾⣿⣿⣿⣿⠛⠛⠋⠀⠉⠉⣽⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⡿⠁⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡓⠶⣦⣼⣷⡀⠀⢸⣿⣿⣿⣿⣷⣤⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⡄⠀⢠⣿⡟⠁⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠉⠙⣟⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⡟⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⠛⠳⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⡇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠁⠀⠈⠋⠻⠿⠻⠿⠿⠿⠟⠛⠿⠋⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⣀⡘⠛⠉⢰⠖⠆⢤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠾⣛⠞⠁⢠⣧⣄⣀⡄⠀⠀⠀⢼⡿⠛⠁⢰⡶⣤⣤⣀⡀
⠀⠀⠀⢀⣞⠋⠀⠀⠉⠀⠀⠀⠀⠉⠙⠃⠶⢦⣤⠟⣀⢀⢠⡌⢱⣿⠏⠘⠀⠿⣷⡶⠠⣄⡀
⠀⠀⠀⠀⠙⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠗⢿⡧⢀⣠⠎⡄⣀⢀⣀⡞⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠇⠀⠑⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠛⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠕⠻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠛⠳⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠁⠙⠀⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⢀⣀⣤⣤⣶⡾⠿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣁⣀⣠⣴⣿⡟⠛⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠉⠉⠉⠃⠀⡿⠟⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖⢸⠇⠀⠀⠀⠀⠀⡈⢁⡀⢰⣏⣀⡀⢀⣴⠖⡄⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠾⣤⢠⡾⢺⣿⠐⣿⣾⠃⣸⡏⣸⠇⣼⡯⠊⠠⠋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢰⠏⢸⣇⣸⣯⠀⠀⣿⠄⢿⠄⠋⠀⠻⠆⠈⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢆⠎⠀⠈⠉⠀⠁⠀⠀⠀⠀⣤⠶⠒⠀⣠⡖⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⣰⣿⠆⡀⢠⣾⠓⡄⢰⡇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡆⣿⡏⣼⠃⣼⡿⣠⠄⣿⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⣼⡇⣿⠃⠛⠀⠘⠛⠁⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⡴⢦⣤⣰⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢨⡗⣠⣭⣿⣿⣿⣿⣠⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠞⠙⠋⠁⠴⢾⠿⠿⠟⠉⠀⢈⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢊⠁⠀⠀⠀⠀⠀⠀⠀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣄⣀⣰⣈⣰⣀⣠⣤⠶⣶⣶⣶⣴⣿⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⢻⠛⢿⣿⡟⠛⠛⠛⢿⣆⡿⣿⣿⠉⢷⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⢠⠖⡀⠙⣧⠀⠀⠀⠈⠣⠴⠻⠃⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣀⠘⣿⡄⠀⠀⠀⠀⠀⠰⢶⡾⣶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣧⣀⠀⣀⡀⢀⣴⣿⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣿⣿⡿⢿⡿⣿⣿⣇⢠⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠟⠳⣿⣿⣌⣓⣈⡿⢻⣿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠏⠀⠀⠀⠀⠀⢙⣿⣽⣿⣿⣶⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡀⠄⠂⢁⣴⣟⠀⠀⠀⠀⠀⠀⢉⢿⣶⣿⣿⣿⣻⣿⣿⣥⡈⠒⢤⡀
⠀⠀⠀⠀⠐⢺⠁⠀⢰⣾⣿⣿⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣿⣼⣿⣿⣿⣿⣻⣾⠀⠉⠂⠠⠄
⠀⠀⠀⠀⠀⣾⣰⣾⣾⣿⣿⣿⡇⠆⠀⠀⠀⠀⠀⠐⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⠈⠁⠲⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⣼⣆
⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣆⢀⣀⣀⣀⣤⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⢀⣀⣸⣿⣿⡀⣀
⣿⠿⡿⢿⠿⠿⢟⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠻⠛⢿⣿⣿⡿⠿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠟⠛⠛⠻⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣤⠶⠶⠶⠶⠦⠄⠀⠀⠀⠀⠀⠀⠠⠴⠶⠿⠶⠶⣄⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠟⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠿⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠾⠛⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠛⠷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣏⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣀⣀⣀⠀⠀⠉⠉⠀⠀⣀⣀⣄⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢻⣯⣭⣭⣛⣛⣛⣿⣿⣭⣽⡟⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠈⠻⣄⣀⠀⠉⠀⠀⣀⣠⠞⠁⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠻⠶⠶⠶⠶⠟⠛⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠾⠿⣿⣤⣄⣀⣀⡀⠀⠀⠀⣀⣀⣀⣤⣽⠿⠷⢶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⢶⣤⣀⡀
⠀⠀⠀⠀⠀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣸⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣐⣤⣤⣶⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣭⣍⣭⣭⠴⠶⠀⠒⠂⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠖⢓⣀⠒⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠒⠒⠒⣛⣉⣉⣉⣁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣬⣄⣉⡤⠴⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⠔⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⣷⣤⣤⡤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣹⠉⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠎⣴⠁⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡴⢋⡾⠃⣀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠚⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠐⠀⠂⠲⠆⠀⠄⠠⠀⢀⠀⠀⡠⠀⠂⠠⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠄⠀⠀⠀⣀⠀⠀⠀⠀⠉⠤⠋⠀⢂⠀⠀⠈⡉⢴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⡄⠀⠀⡀⣠⡀⠸⠘⠀⠉⠀⠇⠀⠐⢀⡴⠁⡐⣀⡀⠀⠁⠔⡀⠀⠀⠀⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢂⠐⡄⠃⠀⠀⠀⠀⢀⢠⠀⠈⢄⡀⢴⠋⡠⠀⠀⠉⠐⡠⠂⠀⢀⡐⠀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠈⠄⠠⡘⠃⠀⣀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠅⠀⠈⠐⠆⠀⠁⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠰⡀⠀⢀⠂⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠀⠤⠤⠤⠔⠒⠒⠒⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠃⡤⠊⠀⠀⠀⠀⠄⠀⠀⣄⠀⠠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠄⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠪⣑⠈⢪⠀⠈⢣⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠉⠁⠉⡁⠀⠀⠆⠀⠀⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⢄⠀⠀⠀⠀⠀⢍⠁⠀⠀⣉⠀⢈⡂
⠀⠀⠀⠀⠀⠀⠤⠀⠆⢀⠢⢄⡀⠂⠀⠀⠀⠀⠠⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠩⠉⠀⠉⠐⠠⡀⠀⠨⢤⡀⢀⠀⠀⢰⠒
⠀⠀⠀⠀⠀⠐⠾⠀⣀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣔⠀⠁⢈⡒⠀⠈⠇⡀⠀⡣⢄⡀
⠀⠀⠀⠀⠀⠀⠞⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠁⠄⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠑⠂⠀⠀⠐⠀⠀⠈⠁
⠀⠀⠀⠀⢀⣀⠀⠂⢐⠒⠧⠀⠀⣠⠀⠀⠀⠀⠀⠒⠄⠀⠙⠂⠀⡀⠀⠐⡁⣠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠠⠥⠄⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠁⠛⠁⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⠀⠄⠀⠀⠸⠆⠐⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠄⠀⠀⠀⠀⠀⠈⠀⠈⢐⡤⠂⡤⠊⠀⠀⠀⠀⡆⡰⠂⠀⠀⠀⡉⠄⠀⣰⠃⠀⢀⠂⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢡⠂⠀⠀⠀⠀⠀⢠⠊⠀⠎⠀⢀⢤⡄⠀⠀⠀⠀⠀⢡⠄⠁⠀⠠⠄⠱⡀⠠⠊⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠠⠀⠋⠀⠀⠀⠀⠀⠀⠀⡀⣰⠇⠀⠀⠀⠀⡈⠠⠜⠀⠀⣀⠀⠂⠁⠁⠀⠀⠀⠀⠀⡐⠀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠥⠀⠀⠈⠃⠠⡄⠀⠀⠈⠈⠑⣠⠂⠀⠀⠀⠄⡰⠐⠀⠀⠉⠀⠀⠀⠴⡈⠀⠀⠀⡀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠠⠒⠀⠀⠀⢔⠁⠀⠀⠀⠀⢠⠁⠀⢀⢀⡄⠀⠀⠀⠀⠀⢠⠀⡌⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠖⠀⠀⠀⠀⠀⠀⠞⠀⠈⠁⠀⠃⠃⠁⠈⠈⠀⠀⠀⠀⠀⠈⡄⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠌⠀⢀⠤⠄⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⠒⠒⠒⠒⠒⠒⠒⠾⠉⢀⣉⣉⣉⣉⣹⣉⣉⣍⣩⣄⣀⣀⠒⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡫⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠰⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠹⣿⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠙⠡⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠠⣀⣷⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠑⣽⣷⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⢹⣷⡵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⢿⣿⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⣟⣿⡜⣄
⠀⠀⢀⠔⠒⠶⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⠙⠻⠜⣆
⠀⠀⠸⣶⡀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⢴⣺⡟⣄
⠀⠀⠀⠙⠿⣷⣴⣤⠀⣀⠀⠀⠈⠑⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣉⠀⢀⠀⡑⠂⠀⢀⠉⠐⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣷⣄⠂⠀⠀⢀⡀⢀⣀⠉⠉⠒⠢⠤⠤⣀⡀⠈⢳⠱⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣾⣦⣽⣭⣿⣿⣿⣷⣶⣦⣖⡲⣤⡉⠉⢛⣻⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣾⣿⣿⣿⣿⣿⣶⣦⣿⣯⣀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣜⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⣧⠀⠀⠀⠀⠀⠀⣸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣇⠀⠀⠀⠀⣰⠇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠸⣆⠀⠀⢠⠏⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠹⡄⢠⡏⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢻⡟⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡾⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣼⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣠⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠛⠋⢰⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡾⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⡴⣿⡁⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣾⠁⠀⢀⡴⠋⠀⠈⠻⣆⠀⠀⢙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠛⢦⠟⠁⠀⠀⠀⠀⠈⢳⡴⠋⠁⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⣤⢞⣩⠤⠤⢤⣀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣀⣤⣤⣬⣛⢦⡀
⠀⠀⠀⠀⢠⡞⡵⠋⠀⠀⠀⠀⠈⢷⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⡞⠁⠀⠀⠀⠉⢳⡝⣦
⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⣼⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡆⠀⠀⠀⠀⠀⠀⣿⢸⡆
⠀⠀⠀⠀⠸⡆⢳⣄⠀⠀⠀⠀⢠⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣷⠀⠀⠀⠀⠀⣠⠏⣼
⠀⠀⠀⠀⠀⠹⣦⡙⠳⢤⣤⠴⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣤⣤⡤⠞⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠴⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢠⡖⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇⠈⢶⠀⢠⠶⣍⣹⣿⣿⣿⣿⣦⡀⠈⠀⠠⢀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⢹⣷⢹⣿⠀⠀⠀⢠⠉⠁⠀⠀⢸⠀⢸⡇⠀⢃⣔⠁⠚⠛⠛⠻⠿⠛⠋⠉⠛⢷⡀⠀⠀⠈⠐⠠⢀⡀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⢸⣿⢸⢿⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⢸⠁⠐⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠜
⠀⠀⠸⣿⢸⠸⠀⠂⠀⠀⡉⡁⠀⠀⢸⠀⡟⠀⡀⢿⡇⠀⣀⣠⣤⠀⠀⢠⣤⣤⡄⢸⣿⣿⣾⢸⠀⠈⠁⠒⠀⢀⠀⠀⡀⠀⠀⢤⠄⢠⡄
⠀⠀⠀⣿⢸⢐⠀⠀⠀⠀⠙⠁⠀⠀⢸⠀⠁⠠⠀⣿⡷⠐⢉⣁⡀⢄⠀⡆⣠⣤⡈⠈⢿⡇⠋⢹⠀⠀⠀⠰⠀⠀⠘⣿⠀⠑⡀
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠊⡈⠡⠦⣀⡿⡇⠀⠈⠉⠄⠈⠈⡇⠂⠉⠀⠀⢈⡐⡆⢸⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢃
⠀⠐⠀⢹⡀⢸⠀⠀⠀⠀⠒⠒⠐⠂⠀⡰⢂⠐⣬⠹⢷⣀⠀⠀⡀⢀⡀⠿⠆⣀⠀⠀⡘⠱⠐⠐⠒⠂⠀⠀⠀⠀⠀⠈⠀⠂⠀⢓⠢⢄⢀⡀
⠀⠠⠀⠈⣇⢸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⢇⠾⢦⠴⠐⠏⠀⠎⣀⠠⠄⠤⢀⡌⠀⠀⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⢈⡆
⠀⠀⠀⢠⢹⡸⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⠘⠐⠃⡰⠁⠠⠀⠀⠈⠉⢉⡉⠁⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⢰⢸⢸⡇
⠀⠀⠀⠈⣿⡧⠀⠘⠀⠀⠀⠀⡇⠀⠀⢰⠀⠀⡰⠀⠀⢸⠈⠌⠢⡀⠀⠀⠀⠔⠸⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⢸⡇
⠀⠀⠀⢀⡡⠀⠀⢂⣄⣄⢠⠀⠀⠀⠀⠁⠀⡰⠁⠀⠀⠀⠀⠈⢂⠙⠤⠤⠞⠀⣧⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⢀⠾⠇⢀⣴⣿⣿⣿⣷⣄⠀⠀⢀⠊⠁⠀⠀⠀⣠⣾⡇⢡⠀⠁⢀⠀⠂⡀⢻⣿⣿⣿⣶⣦⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠎⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣷⠀⠂⠤⠀⠀⠀⠈⢺⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠉⠉⠉⠉⠙⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⠀⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠛⠉⠁⢈⣿⣿⣿⣿⣿⣿⢿⣿⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⠶⠒⠂⠀⠈⢹⣿⡿⢟⡻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠜⠙⠛⡷⠒⠀⠀⠀⣾⡿⠱⢇⢁⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⡜⠀⠠⡄⢀⠀⠀⠀⣰⣿⣇⠾⠇⣼⣿⣿⣿⡟⣠⣤⣾⣙⠦⣾⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠛⣿⣷⣧⣼⣧⣴⣾⣿⣿⣿⣷⣾⠟⠿⣿⣿⠀⠈⣿⡿⣿⣷⣮⡻⢶⣶⠌⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠈⠹⡄⠀⠙⣄⣯⣿⣿⡿⣧⣍⣃⠀⠈⠣
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢂⡀⠀⠀⠀⢹⠟⡇⠉⢹⣛⣠⢀⣨⡿⣿⣆⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡉⢀⡀⠀⠉⠀⠻⣄⠀⠙⢵⣿⣟⣴⣬⣯⣷⡀⠀⠀⠙⠲
⠀⠀⠀⠀⠀⣐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠉⠉⠳⣄
⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣂⡤⠔⠊⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⣿⡿⣿⣿⣿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢛⣿⠟⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⡿⠃


⣶⡖⡶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣊⠔⠊⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣊⠵⠊⠁⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢔⣫⠔⠊⠁⠀⠀⠀⢀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⠀⡞⠉⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⢀⣠⡤⠴⢤⣘⠄⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠊⠁⣠⠔⠋⠀⠀⠀⠀⠈⠓⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣆⣍⡓⠾⣅⡀
⠀⠄⠀⠀⠄⠂⢁⣠⠿⣿⣿⣷⣦⣬⣙⠲⠤⣄⣀
⠣⢤⣤⠤⠴⠚⠋⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣦⣬⣉⡓⠲⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣯⣟⣿⢿⣿⣷⣶⣤⣭⣉⣛⠒⠲⠦⠤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣿⣿⣟⣿⢿⣿⣿⣿⣷⣶⣶⣤⣤⣍⣉⣉⣉⠛⠛⠓⠒⠲⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣾⣯⣽⣟⣻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣴⣤⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠷⢶⣶⣯⣭⣭⣭⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⠄⠤⣦⡀⠀⣽⡇⠀⢈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⢀⢠⣶⣆⣰⣥⣈⠷⠒⠂⢩⢆⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⡀⠀⠊⡀⠈⣉⣿⣻⡆⣍⣈⡀⢠⣶⣾⠵
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣾⢷⢰⣿⣿⣻⠑⠿⣿⣶⣿⣠⣭⣦⣆⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠁⠀⠤⢠⠈⣧⠘⣋⢻⣿⣿⣶⣴⣯⣉⣿⠿⠘⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠄⠀⠀⠀⠀⣀⢠⠿⡿⣿⡿⣟⠻⣟⣯⠟⠁⠆⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠀⡴⢤⡒⠋⡿⣿⠿⡷⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠐⠂⠀⠌⠀⠀⠤⠊⠓⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠉⠁⠈⠉⠉⠛⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢛⢾⡇⠁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⡶⢓
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⢿⡇⠀⠉⢀⣙⣻⡶⣦⡀⠀⠀⢀⣤⠶⣟⣋⣉⠉⠀⠀⢸⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⡇⠀⠐⠋⠹⠟⠙⠄⢠⡄⠀⡤⠀⠎⠛⠏⠈⠁⠀⠀⣸⡁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢆⣸⠀⠀⢀⢰⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣀⣬⣽⣶⢲⣭⣤⣤⣀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢰⠛⠋⠉⠉⣀⢊⡉⠉⠉⠛⠻⠆⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢀⡀⠀⠲⠤⢤⣥⡤⠔⠁⠀⢡⣆⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠺⣿⣄⡀⠀⠉⣉⡁⠀⢀⣴⣾⠟⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣾⣿⠀⠀⠙⢿⣿⣷⣾⣿⣿⣶⣿⡿⠋⠀⠀⠀⠙⢻⠒⢤⣄⡀
⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁⠸⣈⠉⠀⠀⠀⠀⠈⠉⠛⠛⠃⠀⠉⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠉⠛⠶
⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠖⠋⠁⠀⡴⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠋⠉⠀⠀⠀⠀⣠⠞⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠄⠀⠀⠀⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣖⡘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣖⡈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢃⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣖⡀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣖⡀⠙⣆⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣖⡀⠈⢧⠀⠀⠀⠀⣠⡿⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⣂⠈⢳⡀⠀⣴⡟⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣔⣂⠀⠳⣾⠏⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡔⣂⠀⠙⣆⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⣂⠀⠘⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠹⣷⣂⣀⣨⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡧⢦⡄⢹⡍⢀⡴⢬⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠈⠿⡭⢿⠏⠹⡜⢯⡿⠃⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢠⠋⠀⠀⠙⡄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣀⣰⠃⠀⠀⠀⠀⠘⣆⣀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⡴⡞⠉⢩⠇⠀⠀⠀⠀⠀⠀⠸⡏⠙⢳⢦⣀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⠞⠁⢀⡇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⡆⠈⠳⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠁⠀⠀⢸⠃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⣇⠀⠀⠈⠳⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⡞⠁⠀⠀⠀⢀⡟⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢹⡀⠀⠀⠀⠈⢳⡈⠳⢤⡀
⠀⠀⠀⠀⠀⢀⡤⢖⣛⣛⡛⠉⠀⢠⠏⠀⠀⠀⠀⠀⡼⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢧⠀⠀⠀⠀⠀⠙⡄⠀⠉⢛⣛⣓⡲⢤⡀
⠀⠀⠀⠀⡴⣫⡾⠋⠉⠉⠙⣧⠠⡏⠀⠀⠀⠀⠀⡼⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⢧⠀⠀⠀⠀⠀⢹⡄⣴⠋⠉⠉⠙⢷⣝⢦
⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⢸⠆⡇⠀⠀⠀⣀⡼⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢷⣀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠹⣯⣧
⠀⠀⠀⣯⣿⠀⠀⠀⠀⠀⢀⡾⣸⡃⢀⡴⣫⣥⣤⣤⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⣤⣤⣬⣝⢦⡀⢘⣇⢷⡀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢿⢻⣄⠀⠀⢀⣠⠞⣹⣅⣹⣫⡾⠋⠀⠀⠀⠹⣆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⠏⠀⠀⠀⠙⢷⣝⢏⣨⣏⠳⣄⡀⠀⠀⣠⡿⡿
⠀⠀⢀⡜⣳⢭⣟⣛⣭⠵⠋⠁⢸⣷⡟⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡀⠀⠀⠀⠀⠀⢻⣿⡇⠈⠑⠮⣽⣛⣛⡭⣞⢣⡀
⢀⡴⢋⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣼⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⢧⡀⠀⠀⠀⠀⢸⣟⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⢦
⠻⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢿⣄⣀⣀⣴⢞⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⢦⣀⣀⣠⡾⣳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣮⣭⣵⣖⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣲⣮⣭⣵⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⣼⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏⣾⣿⠋⠀⠀⠉⠻⠟⠀⠀⠘⠛⠛⣛⣻⣛⡛⠛⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠀⠈⠁⣤⠶⠲⢦⡀⠀⠀⠀⠀⠀⣼⣉⣤⣤⣽⡆⣼⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⢸⡷⣾⣿⡷⡟⠀⠀⠀⠀⠀⠹⣯⠻⠿⣹⠃⢿⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣷⠀⠀⠀⠈⠳⣌⣛⡵⠃⠀⠀⠀⣆⠀⠀⠈⠛⠋⠁⠀⣾⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⢻⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣇⠀⠀⠀⠀⠀⠀⠀⣀⣤⡖⣀⡀⠀⣿⣳⠶⠄⠀⣀⡏⠀⢼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡄⠀⢸⡄⠀⠀⠀⣠⠖⠺⣥⣤⣛⣩⠽⠿⠧⠤⠖⠋⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⢷⠀⣠⠞⠁⠀⠀⠀⣀⣀⣠⣤⡤⠴⣲⣶⢦⡄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠈⡿⠁⠀⠀⠀⢠⡟⠻⣷⣲⣶⣖⣋⡭⢫⠞⠃⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣦⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠻⣤⣠⣛⣉⣉⡛⣦⠟⠀⠀⠀⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣷⡀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡦⢤⣀⠈⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⡿⠀⢀⣬⣧⡀⠈⠙⢦⣀⣀⣀⠀⠀⢀⣀⢀⡶⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡇⠀⡟⠀⣀⣙⣦⡀⠀⠉⠁⠈⠓⠶⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠃⢰⡇⢠⡟⢉⣭⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⠃⢸⢰⡏⠀⣀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠁⠀⢿⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣀⣀⡤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣀⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠂⣠⣾⣿⣾⠃⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣴⣿⣿⣿⠟⠁⠀⠈⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡀⢾⣿⣿⣿⡿⠋⠀⠀⠀⢀⠄⠡⡀⠀⣴⣿⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⣾⣷⡀⠹⣿⠟⠁⠀⠀⠀⠀⢸⠀⠀⣱⣾⡿⠃⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣴⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠘⣤⣾⣿⠟⠣⡀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⡀⠀⠀⠀⠀⣠⣾⣿⣟⠁⠀⠀⠈⠲⣄⡀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⢀⡴⠊⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠘⢿⣶⣦⣄⡻⣿⠋⠘⢦⡀⠀⠀⠀⠀⠙⠓⠶⡄⠀⠀⠙⢦⡀
⠀⠀⣴⠋⠀⠀⠀⠀⣿⣏⣈⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣠⣿⣿⣭⣿⣦⣄⠀⠀⠙⠢⣄⡀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠈⠳⣄⠀⠀⠀⠘⠿⠿⠿⠟⠁⠀⠀⠀⢀⡠⠖⠋⠉⠉⠃⠉⠻⠿⠛⠿⠶⣄⡀⠀⠉⠓⠒⠠⠞⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⠶⢤⣄⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⢠⠞⠉⠀⢈⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣇⠀⠀⠈⢳⡄⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣿⠀⠀⣠⣾⠋⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠹⢷⣄⣀⣠⣿⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡶⠿⠛⠁⠀⡏⢹⢹⠙⠋⡏⡇⣿⢸⢹⠉⡏⡏⠏⢹⠀⠀⠉⢉⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠃⠘⠸⠀⠀⠇⠇⠿⠸⠸⠀⠅⠇⠂⠘⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠞⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⠟⠉⠀⠀⠀⠀⠆⠊⠀⠀⠀⠀⠀⣰⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⠀⢸⡿⠋⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠛⢨⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢏⠀⣤⠠⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢀⡨⠿⣿⣯⣿⣿⣿
⣿⣿⣯⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⢁⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⠀⠥⡿⣟⣿⣿⣿⣿
⣿⣿⣿⣿⣦⠈⠙⠻⠿⠿⣿⣿⣿⣷⣌⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⣀⣛⣻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣾⡆⠀⠀⡙⠻⢟⣻⠿⠟⠋⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⢀⡀⡐⠂⠶⡍⢑⣺⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣭⣦⡀⢿⣷⣿⣵⣶⡦⢀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠣⠈⠓⢛⢬⡟⣹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠻⣷⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠘⠻⣇⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⡱⠂⠀⢘⣶⣆⠎⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢃⡘⠲⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠃⠖⠈⠱⢻⣐⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢳⣀⣻⣦⡄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠤⢵⣾⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣵⣍⣩⠧⡿⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠻⣌⡙⠶⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⢻⣠⠔⣴⡾⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⠛⢣⣸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣟⠻⣿⣿⣻⠛⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣏⡿⣦⢰⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢰⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⡀⡯⢸⣁⠀⣇⡀⡇⠀⣇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠻⠃⠘⠘⠂⠇⠟⠧⠀⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⡄⡄⡄⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⡏⡇⡇⣷⠁



⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⢀⣀⣀⣀⣐⡄⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠠⠤⣀
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠠⠀⡔⠒⠒⠒⠃⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠤⠀⠠⠀⢼⠁
⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⠤⠤⠤⠤⢤⣼
⠀⠀⠀⠀⠀⠀⢇⣀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠐⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠰⣶⣶⠀⢰⣶⠀⠂⠀⠰⣶⣶⣶⣶⡆⢤⡐⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣖⣺⣿⣇⠀⠀⠀⢻⣿⣿⣿⣿⣤⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣻⠿⢛⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣭⣭⣿⣿⣿⣿⣿⣿⠿⠿⠟⢛⣛⣿⣿⣿⣿⡄⢠⣤⣾⣿⣿⣿⣿⣿⣧⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⢾⢿⣿⡿⡻⢿⣝⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡄⠁⢫⣿⣿⣦⣤⣿⣞⣿⣿⣷⡈⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⢡⣿⣿⣿⣷⣶⣟⣻⡾⠛⣻⣿⣿⣾⠿⣿⣧⢹⣿⣇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢻⣷⣿⣿⠋⠉⢻⣿⣿⣯⣴⢟⣫⣼⠿⢿⣧⣿⣿⡆⢻⣿⣿⣿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⣠⣾⣿⡿⠃⠀⠀⣠⣿⣿⣿⣿⣿⣯⡵⠋⣸⣿⣿⣿⣿⣾⣿⡿⢋⡾⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣷⣾⣿⣿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣽⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣄⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡼⣿⣿⣧⣄⣰⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣍⠻⢿⣿⣿⣿⡏⢉⣉⣋⣩⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣫⣶⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡛⠿⣿⣯⢷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣹⣿⠿⣋⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⠈⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣥⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣛⣷⣿⣿⣿⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣩⣶⣿⣿⣿⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡀⢻⣿⠋⠛⠁⠹⣿⣿⣿⣿⡟⢫⡛⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣕⣺⣿⣇⠀⠀⠀⢻⣿⣿⣿⣿⣤⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣿⠿⡛⣛⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣭⣍⣿⣿⣿⣿⣿⣿⡿⠿⠟⢛⣛⣿⣿⣿⣿⡄⢠⣤⣾⣿⣿⣿⣿⣿⣧⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⢶⢿⣿⡿⠻⣿⣝⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡄⠠⢫⣽⣿⣦⣤⣿⣞⣿⣿⣷⡘⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠟⠁⠿⠿⠿⠷⠶⠟⠻⠾⠛⠻⠿⠿⠞⠻⠿⠇⠹⠿⠆⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠀⠠⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠋⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⢠⣤⠄⠀⠴⠖⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⣴⣿⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⢀⣾⣟⡋⢉⢉⡉⢉⠉⡛⣛⠛⢛⡛⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⢀⣾⠏⣾⣧⣿⣸⡏⣟⢿⡇⡿⠁⣹⣼⢿⢿⡽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠞⠁⠀⠠⠄⠠⠀⠀⠀⠤⠠⠀⠠⠄⠄⠤




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣙⣫⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠀⠁⠐⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠁⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠈⠙⠂⠄⣀⡀⠀⠀⠀⠀⠀⡠⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⡁⠢⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠠⠾⠟⣿⣌⡀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡃⢈⠢⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡡
⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⡈⠢⡔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡈⠢⡠⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠠⢄⡐⡠⠀⠠⠄⣈⣀⢀⡱⡌⠆⠙⢦⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠉⠓⠦⢌⣲⡀⡀⠱⡊⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⣈⣁⠀⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⠢⢧⣌⢂⠀⠀⠀⠀⠀⠐⡀⢠⠋⠀⠈⠉⠑⠒⠢⢌⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠩⢘⠀⠀⠀⠀⡉⡑⠂⠀⠄⢂⣀⣙⠀⠁⢀⠀⠀⠀⠀⠀⠈⠆⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠀⠂⢤⣙⡓⠂⣀⠔⠁⠀⠀⠉⠑⠢⠤⣀⣐⠒⠂⣁⣀⡠⠂
⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⢉⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠠⠊⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠝⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠠⠄⡉⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡞⠉⠙⠢⣌⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠘⢄⠀⠀⠈⠙⢆⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢤⡑⠠⠀⠀⠤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠶⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠙⠀⠀⠀⠀⠚⠋⠙⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣶⡄⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠋⠉⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠁⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⣠⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⣁⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⣿⣿⠤⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠻⡋⠉⠉⠉⠙⢛⠿⣿⣦⠀⢀⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠈⠐⠂⠒⠊⠁⠀⠈⣻⣶⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣀⣀⣀⣠⣤⣶⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⣘⡒⠲⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⣭⡭⠉⠙⠓⠒⠦⠬⢭⣍⣂⣒⠂⠤⠄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⡆⢼⡍⠁⠒⠀⠠⠄⣀⡀⠀⢀⣤⡶⢄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠁⠒⠒⠤⣴⣭⣿⣿⡏⠠⠫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⣶⣿⣯⡽⣿⠿⠾⣷⣖⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣮⢿⡿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣤⣤⣴⡶⠿⣷⣿⣿⣿⡽⢻⣯⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠇
⠀⠀⠀⠀⠀⡠⠺⠛⠛⠉⠁⠈⠻⣦⣾⣧⡤⣿⣿⡯⠗⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣷⣶⣤⠔
⠀⠀⠀⠀⠀⣥⣤⣤⣀⣀⣀⡀⠀⠘⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠘⢆⠀⠀⠀⠈⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠙⠂⠈⠀⠀⠀⠀⠀⢀⣄⡐⠤⠄⠒⠒⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡏
⠀⠀⠀⠀⠀⢐⣂⣀⣀⣀⣤⢄⣀⣀⣀⡀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠸⡄⠀⠀⠐⠾⠿⣷⣦⣤⡤⠄⠀⠀⠀⠀⣸⢷⣼⡏
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢄⡀⠀⠀⠀⡰⠛⠉⠉⠁
⠀⠀⠀⠀⠀⣤⣄⠀⠈⠙⣟⣷⠲⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠀⠉⠀⠀⠀⢰⠃⢀⣴
⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢄⠀⣄⡀⢀⣿⣿⣿⠛⠉⠉⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠸⣿⣿⠋⠠⠤⠴⠤⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⡆
⠀⠀⠔⠁⠀⠀⣠⠼⠋⠀⢰⠛⣏⠀⠀⠙⠋⠀⠀⠉⠛⢣⠀⠀⠐⠐⠀⠓⠒⠒⠒⠚⠒⠒⠓⠒⠓⠒⠊
⠀⠀⠀⠀⠄⢘⣅⠀⠠⠤⠼⡀⢻⣄⠀⠀⠀⠀⡄⣠⡀⢈⠀⠀⠀⢀⣀⣀⣄⢀⣀⠀⠀⠀⠀⢀⢀⣀⡀
⠀⠀⠐⠀⠀⢇⠘⣆⠀⠀⠀⠈⣾⣿⣿⣾⡏⠉⣉⣀⢿⡟⠀⠀⠀⠘⠛⠛⠛⠛⠛⠂⠂⠐⠂⠐⠚⠛⠂⠀⣀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⡄⢸⡀⠀⣀⠄⠉⠛⢿⣿⣷⣦⣤⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢊⣂⠀⢈⡉⠉⠀⡗⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣾⠗⠊⠀⠀⠀⠀⠀⠀⠉⠙⠛⢉⠷⢄⠀⠀⠀⠀⠀⠆⠴⠠⠸⠠⠄⠷⠀⠆⠶⠀⠀⠀⠀⠀⣶⡄⠰
⠒⠒⢲⢲⢲⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠄⠉⠢⣄⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢠⣶⣾⣏⠀⢠⣿⣷⣆
⠴⠤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣅⠀⠀⢄⡠⠀⢀⡀⠠⡤⠠⡄⠘⠿⠟⠀⠒⠄⠈⠉⠿⠓
⠻⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠆⠀⠈⠀⠀⡄⠠⡀⠀⠘
⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣀⡀⣀⡀⠀⣇⣀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠘
⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡈⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⣀⠀⠀⣀⡀⠀⠈⠁⠉⠉⠉⠉⠉⣉⠉
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠏⠈⠻⠏
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣤⣄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠟⡃⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠵⠦⡄
⡀⠀⠀⣀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⡀⠀⠀⠀⠑⠢⢄
⣥⠐⠊⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠚⠃⠀⠀⠀⠠⠀⠀⠱⠄⢀
⢹⠀⠀⠔⣠⠼⠒⠁⠒⠒⠒⠤⠤⠤⢄⣀⣀⣀⣼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⣔⠠⠂⣿⣉⣛⣒⠒⠒⠦⠤⠤⢤⣀⣀⣀⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣾⠀⠀⠉⡆⢧⠈⠉⠉⠉⠉⠀⠒⢖⠀⠤⠍⣩⣉⣙⡓⠒⠒⠲⠤⠄⠁⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠐⠀⠠⠄
⠀⠀⠀⠀⢷⠸⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠀⠠⠀⡀⠀⡀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠘⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⣟⡉⠻⣿⠛⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⢯⡁⠀⠀⠀⠀⠈⠑⠀⠈⢷⠈⡖⡽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠙⣆⠀⡀⠀⠀⠀⠀⠀⠈⠃⠸⡆⣟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⣞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣴⣶⢶⣼⡼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡶⢾⣿⡷⠀⢷⣷⢂⠀⠀⢀⣷⠀⠀⠀⠀⠀⢸⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡼⠃⠈⠛⠃⠀⠀⢹⡄⠆⠀⠸⣿⠀⠀⠀⠀⠀⣸⢺⣿⡇
⠀⠀⠀⠀⠀⣯⣀⣠⣦⡄⠀⠀⢸⣿⢁⣠⡄⠀⠀⠀⠀⠀⢻⡼⠀⣸⣿⣰⠀⠀⠀⠀⠉⢴⣿⡇
⠀⠀⠀⠀⠀⠀⠉⢉⠈⠙⣆⡀⣾⣿⡛⣯⣁⢀⠀⠀⠀⠀⠘⣷⡆⣿⢻⡏⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠘⢭⢷⡾⠻⢿⡛⠿⠋⠛⠃⣿⣿⣿⣽⣿⡟⠁⠀⠀⠀⣸⣿⣿⡏⣿⣿⡄⠀⠀⠀⢠⣿⣿⣿
⠀⢠⡀⣀⣀⠀⣸⣤⡄⢤⣦⣶⠀⠹⣿⣿⣼⣇⠀⢀⣠⣴⣿⣿⣿⡇⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠘⠛⡛⠛⠀⡈⠉⠁⠈⠁⠀⠀⠀⠙⣿⣷⠙⡛⣻⣿⡗⣿⣯⣻⣿⣿⣿⣖⠀⠀⠀⢸⣿⣿⣿⣿⣄⠀⠀⣤⢢⠄
⠀⠀⠀⢧⣶⢶⠿⠄⠻⡿⠁⠀⠀⡂⣼⡿⢿⣟⣿⣿⣹⣏⣿⣷⢹⣿⣿⣿⣿⡄⠀⠀⢀⣿⣽⡆⢿⣟⣷⣶⣴⣫⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⠁⣸⡿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣯⢿⠀⠀⢠⣿⣿⣷⡘⣬⣿⣷⡟⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢠⡿⣱⣿⣿⣿⣿⣿⢹⣿⣽⣿⣿⣿⣿⣧⡄⡜⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⣃⣿⣳⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣷⣹⣿⡵⣿⣿⣿⣿⣷⣻⡻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⣾⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣝⣿⣿⣿⣿⣷⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣾⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡎⣿⡗⣿⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡇⣿⢯⡏⡿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣏⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⢠⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡄⢿⣇⠀⠘⡇⢸⠄⠸⡇⢸⠃⠀⣸⡷⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠸⣷⡘⠿⠘⠉⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠑⠃⠿⢠⢾⠇⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⡿⠻⣦⡀⠀⠀⠀⠀⠀⠉⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢀⢀⡇⣧⣿⣧⠀⠀⠀⠀⠀⠀⣄⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⣸⢁⡎⡼⢠⡏⣿⣿⣷⠀⠀⠀⠀⠀⣽⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣷⣇⡞⣸⢁⡼⣾⣿⣿⣿⠀⠀⠀⠀⠀⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣽⣿⡿⠿⣿⡀⠀⠀⠀⠀⣻⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠈⠙⠟⠋⠀⠀⣽⣷⠀⠀⠀⠀⠈⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢐⠋⠉⣁⣀⡀⠀⢶⠂⠀⣀⡈⠉⠙⠓⢤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢸⣯⠖⠢⡀⠀⠀⣴⣿⣿⣿⣿⣷⠈⣰⣿⣿⣿⣿⣷⡄⣻⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⡰⠻⡀⢀⠇⠀⠀⣿⣿⣿⣿⡿⠋⣸⡝⣿⣿⣿⣿⣿⡷⢀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠈⠁⠀⠀⠀⣈⡛⠿⢋⠀⣴⡿⣷⡌⠙⠻⢛⣛⣡⠖⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣥⣤⣬⣤⣛⣧⡛⣧⣽⣤⣤⣬⣥⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠁⢀⣨⣿⣿⡻⡿⠛⣿⠛⢿⠿⣿⠿⠛⠁⢀⣀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣾⣇⣷⣶⣿⣿⣿⣿⣝⣷⣶⣿⣶⣾⣯⣥⣶⣶⣿⣿⣿⣟⣶⣴⣦⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠛⠛⠙⠛⠛⠛⠛⠋⠀⠉⠛⠋⠉⠙⠛⠉⠋⠻⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠰⠸⡾⠿⠧⠸⠹⠤⠇⠏⣰⢦⢤⣀⡀⠀⠰⣭⡭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠺⠀⠀⢘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠯⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⣲⣿⣷⣾⡟⣻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⠷⡦⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠵⠺⠚⠩⠽⠿⠿⠟⠋⢙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠐⠈⠃⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠐⡉⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⠀⣀⣠⣤⣶⡾⠿⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠩⣠⣴⣦⣤⣦⣼⣦⣶⣖⢿⣿⣷⢾⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣡⣤⡤⠾⣿⡋⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣿⠈⣦⠙⣿⠀⠀⠀⠻⣊⣿⠇⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠑⢠⠟⠁⢀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣮⠀⡥⢸⡆⠀⠀⠀⠀⠠⢦⣶⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⡽⠁⠀⠀⢀⠀⢀⠀⣀⢀⣟⡐⠀⠀⣠⡤⡀⣤⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⣿⣷⣤⣀⣀⣀⣶⠾⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠞⢳⡆⣼⠋⣿⠃⡿⣾⡇⢸⣿⣱⠇⣼⡯⠚⠨⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢠⣿⣿⣟⢻⡻⢿⣧⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠐⡟⠀⣿⠴⡿⠈⠀⢿⠄⠿⠗⠋⠀⠻⠄⠀⠀⠘⠓⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠁⠀⠈⠉⠻⣿⣷⣾⣮⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠜⠀⠀⠀⠀⢠⡶⠒⠂⢠⡶⢩
⠀⠀⠀⢀⠀⠀⢀⣽⡇⠀⠀⠀⠀⠘⢿⣿⣿⣿⢻⣿⣿⡁⠠⢤⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⣷⡀⢠⣿⣗⣁⢀⣾⠋⡆⣸⡇⢱⠆
⠀⠀⠐⡏⠀⣐⣿⣿⣇⠀⠀⠀⠀⠀⢻⣿⣿⣿⢾⣿⣿⣿⣿⣾⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣿⢸⡿⣸⡇⢼⣟⣴⠇⣿⡡⠏
⠀⠀⠀⣷⣿⣿⣿⣿⣿⠑⠠⡀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⠱⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣰⡟⠸⠇⠙⠁⠀⠉⠁⢸⡟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠉⢁⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠙⠙⠛⠛⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⣴⠂⢀⢀⣀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⠀⣿⣿⠿⠿⢿⠠⡀⢳⠀⢀⠂⣸⣿⡇⠀⣄
⠀⠀⠀⠀⠀⠂⠀⠀⠐⢀⣿⣿⠂⣠⣤⣴⣶⣿⠠⣿⡄⠀⣿⠀⣽⣿⡇⠀⣷⠀⣶⠀⠄⢀⡄
⠀⠀⠀⠀⠀⣶⠀⠀⠀⣼⠏⣿⠁⢉⠛⣉⢡⣿⣸⣿⣿⣄⣿⣆⠘⠿⠁⣸⠇⠀⣿⡆⢀⣼⡆
⠀⠀⠀⠀⠀⢿⣿⠀⢸⡿⢰⣿⠁⣾⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄⠲⠀⣾⠀⢨⣿⡏⠘⢿⡂
⠀⠀⠀⠀⠀⢾⠉⠀⣾⡇⠸⡏⠀⣉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡄⠁⢠⣤⠘⠃⠀⠈⠁⠈
⠀⠀⠀⠀⠄⢀⡀⠀⣿⣷⠀⠀⠀⢹⢩⡍⣭⣻⣿⡟⢩⢉⣭⣿⣿⣿⣟⠀⣻⠘⣧⡀⠐⡇
⠀⠀⠀⠀⠀⠘⠁⣠⣿⣿⠇⠀⠐⣼⣦⣤⣤⣾⣿⣿⣤⣤⣤⣿⣿⣿⠋⠀⣹⡄⠀⠁
⠀⠀⠀⠀⡁⠐⣾⡟⠋⠀⠀⠈⠋⠀⠉⠛⠿⠿⡿⡿⠿⠟⠛⣿⣿⣿⡆⢺⠿⠁⠀⣰⣦
⠀⠀⠀⠈⠀⠀⡀⢀⣾⣿⠀⠠⣶⡾⢦⣿⣿⣿⣶⣷⣧⣤⣿⣿⣿⣿⡇⠈⢀⣴⣧⣿⣿⣧⡀
⠀⠀⠀⢀⠀⠠⣿⣿⠟⢿⣷⣄⣤⣾⣿⣿⢿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣷⣾⡿⠐⠛⠉⠉
⠀⠀⠀⠈⠄⠀⠀⢁⢀⣼⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠘⣦⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢀⣠⠄
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠄
⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣛⠂⠐
⠀⠀⠀⠀⢂⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠿⠿⣿⠟⠿⠟⠿⠇⠚⠛⠓⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠉⠉⠙⠁⠀⠈⠁⠀⡈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣠⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠛⠛⠛⠛⠳⠶⢾⣿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣠⣄⠀⠀⠀⠀⠀⠀⣡⡶⠿⢷⣦⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣴⣤⡀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠹⣿⠀⠀⠀⠀⣿⠃⠀⠀⠀⣿⡇⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⡄⣸⣿⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⣿⠀⠀⠀⠀⣿⡀⠀⠀⣸⣿⠃⣠⣾⣿⣿⣽⣻⣿⣿⣿⣶⣜⠁⠀⣠⣆⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣸⡿⣀⣀⣀⡀⠹⣷⣤⣼⣿⣷⣾⣿⣿⣿⣿⣿⣻⢿⢿⣿⣿⣿⣿⣦⡛⠉⠓⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⡿⣿⣭⣡⣤⣤⣭⣤⣬⣿⣿⣿⡿⢿⡋⠁⠀⠀⢸⡟⠻⣶⣼⣿⣿⣿⣿⣿⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⠋⠁⠀⠈⣇⠀⢀⣀⣼⣧⣤⣿⣿⣿⣿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⣾⣿⣿⣿⡏⠉⠀⠀⣸⣅⣠⣤⣴⣿⣿⣿⡿⠟⠛⠻⣿⣿⣿⣿⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣤⣴⣿⣿⣿⣿⣿⣾⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠏⠉⠁⠈⣿⠿⠿⠿⣿⠟⠛⠉⠉⢿⣁⣴⣿⡟⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⠻⣶⣶⣤⣄⣿⡀⠀⠀⢿⡀⠀⠀⢀⣼⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⠁⠀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣶⣷⣾⢿⣿⣿⣿⣷⣾⣷⣶⣾⣿⠿⣋⣴⣿⣿⣿⣿⣿⣿⣿⣧⣿⠶⣫⣿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣫⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣻⢿⢿⣿⣿⣿⣿⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣭⣽⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣭⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢯⣿⣿⣿⡍⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⠂⢠⡄⣀⠂⠀⠴⡂⠀⢰⠀⡴⠀⠀⢶⠀⠀⢶⡐⢂⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠇⠀⠎⠀⠟⠀⡜⠃⢷⠀⠀⠿⠀⠀⠀⢸⡄⠀⠸⠄⠹⡆⠘⠦⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⢀⣀⡀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢸⡁⠈⠁⠀⠀⠀⠀⠀⠀⢸⣷⡀⢻⠀⠀⡏⠀⢹⡄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⢸⡇⢣⣼⠀⠀⡇⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠄⠀⠀⠻⠦⠾⠁⠀⠀⠀⠀⠀⠀⠘⠃⠈⠿⠀⠀⠻⠦⠞⠁⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⡀⢸⠀⡀⠀⠀⠀⠈⠀⠃⠀⣤⠔⠀⢸⠀⠀⢸⠀⢀⡄⢸⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠈⠀⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠈⠁⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣴⣶⣶⣄⠀⠀⠀⠀⠀⣠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣄⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⡀
⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠀⠀⢠⣘⠛⠉⢉⡠⠷⠄⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⠉⠙⠋⠉⠉⠈⠁⠉⠀⡀⠄⠐⠉⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠑⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣰⢃⡟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢃⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⡇⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣾⠁⠀⠀⠀⠳⣮⡱⡦⠔⠂⠀⠄⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢰⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢡⠀⠀⠀⠀⠤⠄⠀⠀⢀⣀⢤⣤⢀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠖⠋⠁⢀⡀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡌⣄⠀⠀⠀⠀⠀⣸⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣤⡀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⢰⣄⡉⠛⠀⠀⠀⠀⠙⢀⠀⠀⠀⠀⠀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⢸⠛⠻⢶⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⣠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⣿⣿⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠸⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠉⣿⠀⠀⠀⠀⠀⠀⠀⢳⣤⡤⠀⠀⠀⢸⡿⣧⠀⠀⠀⣼⠀⠉⠉⠀⠀⠀⡀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠄⠀⠀⢸⡤⠠⠀⠀⠀⠘⠛⠋⠀⠀⠀⠘⠷⠂⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⣶⣤⣤⣴⣤⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣾⣿⣥⣤⣤⣄⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡿⠟⠋⠈⠻⢿⣷⠆⢀⣴⣿⣿⣿⠛⠻⠿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠁⢀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠀⠀⠀⠀⢠⣿⡿⠋⠻⣿⣦⠀⠀⠀⠘⣿⣏⠉⣇⠀⠀⠓⠀⠀⢺⠀⡶⠄⠠⢶⠀⣿⠂
⠀⠀⢀⣴⣄⠈⠀⠀⠐⢄⠀⠀⠀⠀⠘⢧⣄⣇⠀⠀⠒⠛⠋⠀⠀⠀⠈⠛⠳⠂⠀⠀⠹⠏⢰⣿⣤⣀⣤⣤⠀⢸⣦⣀⣄⣀⣀⣴⣯
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣇⠀⢼⣿⣿⣩⣝⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⠀⠀⠈⢧⠀⠀⠀⣠⣾⠿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠘⠻⢿⠃⠀⠈⢻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠄⠀⠸⠀⠀⠀⠈⠧⠈⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣾⣿⠷⡦⣹⡏⡠⡾⣿⣷⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡾⡛⠥⣣⢇⢻⡟⢿⡘⠤⢍⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠱⡑⠬⠭⣓⣵⣞⢺⣾⣿⡛⠛⢋⢜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⠑⡟⡹⠟⡿⠿⠻⣯⢻⠊⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣄⠱⣧⣢⣑⣚⣴⢝⠎⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡹⡑⠬⣛⡩⢍⣓⠭⠊⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢠⠀⠀⠀⠀⠀⠀⢸⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⣀⢀⣀⠀⢸⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⢠⠛⠊⠉⠚⡇⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⠀⡆⠀⠀⢀⠇⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢇⠀⠘⠦⡠⠊⠀⢸⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡄⠀⠀⠀⠀⢀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠒⠤⠤⠖⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠻⣿⣿⣟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠟⢋⣿⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⢻⣻⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡾⠋⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⢀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠉⠉⠉⠉⠙⠛⠿⢷⡆⠀⠀⠀⠀⠾⠟⠛⠉⠁⠀⠀⠈⠁⠀⠀⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⠀⠀⠀⠀⠐⠒⣲⣶⠂⠀⠀⠀⠀⠀⠀⠀⠐⢖⣶⠖⠂⠀⠀⠀⠀⠀⡆⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠀⡀⠻⣿⠃⡀⠠⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠴⠀⠀⠀⣠⡶⢶⣦⣼⡏⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠳⣦⡀
⠀⠀⠀⢠⡾⠋⠉⠙⢿⣶⣤⣀⣀⣀⣀⣀⣤⣴⣶⣾⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⢻⣿⣷⣄⠀⠈⡇⠀⠀⠀⠀⠀⠀⡴⠛⠻⣿⣷⡀⠘⣧
⠀⠀⠀⠘⢷⣄⣀⣠⣾⣿⣿⣿⣿⣿⠿⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢻⣿⡈⠻⣿⣿⡀⠀⠀⠀⠀⠘⣇⠀⠀⣿⣿⣧⠀⣿
⠀⠀⠀⠀⣠⡶⠿⠿⠛⠉⠉⢹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⢷⣄⢹⣿⣧⠀⠀⠀⠀⠀⠈⠲⠶⣿⡹⣿⣤⡟
⠀⠀⠀⠸⣧⣀⣀⣀⣀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⠀⠉⠛⠶⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠙⢷⣜⣿⣄
⠀⠀⠀⠀⠈⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠈⠱⡄⠀⠀⠀⠉⠛⠿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠉⠙⡄⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣦⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⢀⣀⠀⠀⠀⠀⠘⠿⣦⣀⣤⡷⠀⠀⠀⣰⠞⠉⠉⠛⢿⡛⠻⠿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⣼⠟⠙⢿⣶⣄⠀⠀⠀⠈⠻⣿⡇⠀⠀⠀⠹⣄⠀⠀⠀⠀⣽⣦⣀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⡟⠀⠀⢻⣆⠀⠀⠙⠻⢷⣄⠀⠀⠀⢹⣿⠀⠀⠀⠀⠈⠙⠓⠚⠉⠉⠉⠉⠙⠛⠶⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠳⠦⣄⣀⠀⣹⣷⣦⣄⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⡠⠞⠛⢿⣿⠟⠛⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣿⣿⣿⡇
⠀⢰⠁⠀⠀⠈⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡙⠻⢶⣄⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⢀⣠⣿⡇⠀⠀⠀⠀⠀⠀⣷⡄⠀⠹⣿⣿⡇
⠀⠸⡄⠀⠀⣠⢻⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠈⠙⠂⠀⠀⣰⣿⡟⠀⢠⡾⠉⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠙⣿⡇
⠀⠀⠙⠓⠚⠁⠀⠙⠃⠊⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠘⣇⠀⠀⣠⢡⣦⣄⣀⡀⠀⢀⣿⣿⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢠⣾⣿⠏⠀⠀⠀⠀⢹⣷⡞⠁⢀⠟⠁⠈⠙⡆⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⢠⣾⠋⠀⢀⣾⣄⣀⣀⡼⠃⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⣄⣠⣾⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⠀⠀⠀⠀⢸⡇⣠⡴⠟⠋⠉⠉⠁⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⢠⡴⠋⠁⠀⠀⣠⡿⠃⣬⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁⠀⠀⠀⠀⠀⢀⡿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⢙⡇
⠀⠀⠀⡴⠋⠀⠀⠀⠀⣰⠟⠀⢀⡿⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣼⣷⡟⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⡿⣿
⠀⡀⢸⠃⠀⠀⠀⠀⣼⡏⠀⢀⣾⠃⠀⠀⢸⡄⠀⠀⢀⡀⢀⣀⣤⠞⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣼⡟⠀⢸⡆
⠀⠀⢺⠀⠀⠀⢀⣼⠟⠀⢠⣾⠃⠀⠀⠀⢸⡇⠀⢰⠋⠀⠀⢹⠇⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠈⣿⠟⢁⣿⣧⠀⠀⡇
⠀⠂⠘⠷⣤⣶⠿⠋⠀⢀⣿⠃⠀⠀⠀⠀⡼⠀⠀⠘⢦⣄⣠⠞⣠⠒⠛⢿⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣼⣿⡿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢰⣿⡿⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣾⠏⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣀⣀⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠸⣶⣦⣤⣀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⢹⣿⣿⣿⠋⠛⠿⢶⣶⡤⠤⢀⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⣴⣶⠀⠀⠀⠀⠀⣀⡀⠉⢅⣀⣀⣹⣿⣿⣷⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⢰⠇
⠀⠀⠀⠀⠸⠻⣟⡁⠀⢀⠀⠀⢺⡇⠉⠁⠀⠀⢸⣿⣿⡿⠀⢰⣾⣿⠿⢦⣄⡀⠀⢰⣼⠀⡇⠀⠀⠌
⠀⠀⢠⣦⣰⣴⣿⣷⠠⠀⠈⠉⠨⡇⠀⠀⠈⠉⢸⣿⣿⣷⣿⣿⡿⠿⣷⣦⣌⣻⣶⣼⣿⣿⣅⠄⡘⢠⡀
⠀⠀⢸⣿⣿⣿⣹⣿⠂⠀⠀⣤⣤⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠄⠀⠀⣰⠟⢻⡟⠈⢹⣿⣿⣴⣥⣿⠇
⠀⠀⠸⣿⣿⣏⢻⣹⣀⢸⣿⣿⣇⡀⠀⠀⠀⠂⢸⠿⣿⣿⡏⣷⡀⠀⣼⡿⢾⣿⠃⢨⣟⠈⣿⣿⣿⣿⡇
⠀⠀⠀⢿⣿⣿⣻⢻⣿⣿⣿⣟⠛⢀⠉⠉⡈⠁⣻⣿⡁⣹⣧⣿⣶⣿⣿⣿⣿⠁⠀⡮⠄⠐⠁⠉⠉⠙⠃
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣶⣾⣦⠘⠀⠀⠀⠉⣿⡟⣷⣟⣛⣿⣿⣿⣏⡉⣍⠀⡸⡇
⠀⠀⢸⣿⣿⣿⣿⢃⣼⣿⣿⡇⢈⢠⡄⣶⣷⣄⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⡏⣼⢳⡇⠀⠄
⠀⠀⠸⣿⣿⣿⣿⡾⠛⠛⢿⣥⠙⠙⠻⠿⢿⣿⣯⣿⣶⣶⣃⢸⠉⠁⠀⠀⠈⠛⠾⣿⣄⠠⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣶⠿⠟⠁⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣠⣾⣿⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡴⡿⠓⢀⠀⠀⠩⣿⢿⣦⣀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⢘⡇⠤⣴⡧⠐⠀⠀⠁⠋⡽⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃
⠀⠀⢀⣠⠔⢲⠦⢾⣷⡀⢺⣷⣀⠀⠀⠀⠂⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⢀⡀⢠⣰⣷⡿⠋
⠀⠀⣎⣼⣿⡿⢷⡈⣿⣷⣄⡘⢿⢿⣦⣤⠞⠈⠳⣄⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⡿⠟⠉
⠀⠀⢻⡿⠋⠀⠀⠹⢤⡛⠽⠋⠲⣿⣿⣋⡀⠀⠁⠈⠳⣄⠀⠀⣼⣁⡀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠋⠁
⠀⠀⠀⠳⣄⠀⠀⠀⣰⣿⡇⠀⠀⠈⠻⣯⣄⡀⠀⠀⠀⢈⣳⠾⠥⠄⠉⠓⠄⠀⡀⠀⣵⣾⠟⠁
⠀⠀⠀⠀⠈⠑⠶⠾⠿⠋⠀⠀⠀⠀⠀⠘⢿⣶⣤⡄⠀⠙⡿⠛⠁⠀⣄⢲⣶⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣢⠞⠁⢀⣶⣿⣿⣾⣟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⠟⠁⢀⡴⣏⡵⠛⠋⠁⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣂⡀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠠⣀⣘⣿⡝⢿⣯⣤⡀⠀⠀⠴⠣⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣿⣿⣧⠀⠙⢿⣿⣿⠀⠀⠐⠀⠘⠀⠈⠛⢦⣀
⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣫⠞⠁⠀⠀⠀⠻⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠙⠲⡀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣤⡤⠄⠀⠀⠀⠀⠀⠈⠳⣄
⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣋⠀⠀⠀⠀⠀⢠⠂⠈⢧
⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢭⣯⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣒⣀⣀⠀⠁⠀⠀⣸
⠀⠀⠀⠉⠳⣤⣀⣀⢀⣀⣀⣤⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⣤⠤⣂⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⠷⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠉⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠹⣆⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣸⠃⣠⡴⢦⣄⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣰⠇⢸⠃⠀⠀⠘⡇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠘⢧⣤⣤⠞⢁⡞⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣰⠏⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠉⠛⠦⣄⡀⠀⠀⢀⡞⠀⠀⠀⠀⣨⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠉⠓⠦⣾⡀⠀⣀⡴⠟⠁⠀⠘⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠞⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠦⣤⡀
⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⢀⣴⠞⠉⠀⠀⠀⣀⡴⠖⠛⠛⠛⠳⣦⡀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠛⠲⢦⣄⠀⠀⠀⠈⠻⣄
⠀⣰⠏⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠉⢷⡄⠀⠀⠀⠘⣧
⣸⠃⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⣼⠃⠙⣦⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠸
⣿⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢠⡏⠀⠀⠈⢧⡀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣾⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⢀⡿
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⣠⡤⠞⠁⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠉⠛⠲⠶⠶⠚⠋⠀⠀⠀⠀⢠
⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣤⡴⠖⠋
⣿⠿⠿⠿⠿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣟⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣟⣻⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠒⠒⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣀⣀⣠⣤⣄⣀⣠⣀⣂⡀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠙⣿⠉⠀⠀⠀⠀⠀⠈⠉⠉⠓⠢⣤⣀⣀⡀⠀⠀⢀⣀⣈⡀⠠⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠁⠀⠀⣿⢀⣀⣀⡠⣤⣤⣤⡀⠀⠀⠀⢈⡿⠛⠻⣿⠋⠉⠉⣉⠉⠉⠙⠓⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠀⠈⠁⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠉⠉⠈⠁⠀⠀⣼⠃⠀⠐⣇⠠⠤⣴⣶⣤⣁⠀⠀⢐⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠁⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢱⠀⠀⠀⠈⠉⠈⠁⠀⠋⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⠀⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠉⠛⠛⠉⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠒⠤⠀⠀⠀⡠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⢀⣀⣈⡓⠒⠀⠐⠒⠂⠀⠀⢀⠀⠀⠆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⣈⣉⠉⠉⠉⢉⣉⠩⠁⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣷⣿⣿⣿⣷⣄⢀⣀⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⡀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿
⠀⠀⠀⣸⣿⠟⠉⠙⠻⠿⠟⠛⠛⠿⣿⡷⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠹⣿⣿⣿⡄⠀⠀⢸⣿⡿⠿⠋⠋⠁⠈⠁⠀⠀⠙⢿⣿⡆
⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠠⣶⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁
⠀⠀⠀⠘⡀⠐⠛⠛⠛⠀⠘⠛⠛⠓⠈⠇⠀⠀⠀⠼⢿⣿⣿⣿⡿⠛⠒⠆⠀⠐⠒⠒⠂⠀⢸⡿⠀⠀⠀⠀⠘⠀⠚⠓⠶⠄⠀⠰⠖⠚⠢
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⣷⠀⠀⠀⣀⣀⣀⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⢸⣇⠀⠀⣀⣤⣤⣤⡀⠀⠀⣿
⠀⠀⠀⠀⣽⣿⡒⠻⠛⠉⠙⠻⢓⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⢻⡔⠦⠶⠿⠿⠿⠶⠤⠆⣾⠂⠀⠀⠀⠀⠀⠸⣿⡆⠾⠋⠉⠉⠉⠛⠆⣰⣿
⠀⠀⠀⠀⢹⣿⣿⣷⣾⣿⣿⣶⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣤⣶⣤⣄⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣴⣶⣶⣶⣤⣾⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⠿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⡿⠻⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⠿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⡀
⠀⠀⠀⠀⢦⣈⠻⣿⣿⣷⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣲⣶⣿⣿⣷⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⣶⣿⣷⣷⣷⣦⡄
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠠⣾⣿⣿⡿⣿⡿⢿⡿⡿⣿⡿⢛⢿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠂⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠠⢿⣿⠋⠀⠁⠀⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢰⣿⠇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣰⣿⡿⠉⠉⠙⠛⠛⠛⠛⠉⠁⢻⣿⣷
⠀⠀⠀⠈⣟⠀⠶⠲⣤⠄⠀⢠⡤⠖⠶⠀⡿⠀⠀⠀⢸⡿⠀⠀⢙⣿⣿⢿⠻⣿⣻⣿⣿⠈⠉⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡁
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠃⠀⠊⠉⠉⠉⠀⠈⠉⠉⠁⠑⠀⡀⠀⠀⠀⠀⠹⠀⠠⠚⠛⠳⠶⠀⠰⠶⠛⠛⠢⠀⡿⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⣷⣤⠀⣀⣤⣤⣤⡀⠠⣤⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⣿⣿⣟⠛⠛⠉⠛⠛⣫⣿⡿⠀⠀⠀⠀⠀⠀⣿⡶⢶⣾⠿⠛⠻⠿⣶⠶⢶⣿⠁⠀⠀⠀⠀⠀⠘⣇⢀⡴⠾⠛⠁⠙⠻⢶⣄⢠⡏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣾⣿⣷⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀⣀⣠⣄⣀⣀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣏⠀⠀⠀⠀⠀⠀⠀⢘⡿
⠀⠀⠀⠀⠀⠀⠉⠻⢻⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣶⣆⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⢠⠁⡒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢀⠔⠩⣂⢌⣕⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢌⠀⠻⣿⣷⣤⣒⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢈⡒⠤⣀⠉⠻⣿⣯⣶⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢀⠈⢻⣧⡈⠙⠒⢤⡉⠁⢡⠜⣗⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠛⣿⣷⠀⠀⠀⠈⠉⠁⠀⠉⠃⢀⣙⣶⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⠠⠝⠛⣹⣿⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠢⡐⢌⣿⣧⡀⠠⠄⠒⣀⣡⣶⣾⣶⣿⣿⠿⠿⣷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡬⣵⠶⠔⠖⢒⠉⠩⠅⠀⢠⣶⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠛⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⣀⣀⡤⢀⣠⣴⣬⣦⣴⡶⠿⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠷⠿⠿⠟⠛⠛⠛⠉⠉⠉⢵⣄⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⠋⠀⠀⢀⡤⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⣾⡏⡰⢃⡀⠀⠀⢠⣶⣿⠛⣷⢠⣿⡏⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣰⠏⡀⠀⠀⣠⣤⣦⣼⣿⠀⢰⣿⡇⣼⣿⠁⢸⣿⣿⠛⢁⣿⠀⢠⣿⡟⢁⡴⠃⢘⡿⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡾⠛⢻⣿⡀⣴⣿⠏⠀⣿⣿⠀⢸⠿⠿⣿⡏⠀⢸⣿⠿⣄⣼⡟⠀⣾⣿⠗⠉⠀⡠⠋⠀⢸⣿⠇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⢀⣼⡏⠁⣿⣿⠀⢸⣿⡿⢠⠏⠀⢸⣿⠇⡠⢿⣿⢀⣼⠟⠀⠈⢿⣿⣀⡤⠊⠀⠀⠀⠘⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⡰⢋⡿⠀⠀⢿⣿⡴⠺⣿⠗⠁⠀⠀⠘⠿⠛⠁⠘⠛⠋⠀⠀⠀⠀⠀⠈⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢁⡾⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠓⠂⠀⢀⣴⡟⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣧⠀⠀⠀⠀⢠⣿⣿⢁⡴⠃⠀⠀⣀⣤⣄⡀⠀⠀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⢀⣾⣿⡟⢋⣀⠀⠀⣼⣿⠏⠀⣷⠀⢰⣿⡇⠀⣸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠖⢩⣿⣿⠋⣿⡿⠁⢠⣿⣿⡆⢠⡟⣠⣿⣿⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⡆⢸⣿⡟⢰⣿⠇⡰⢻⣿⣇⠳⣼⠋⢡⣿⣏⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣸⣿⡇⣾⣿⠇⠸⡿⠏⠀⠀⠛⠛⠛⠁⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⣿⠟⠀⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢻⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠭⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣩⠄⢣⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠘⡀⠀⠄⠘⣿⣷⡀
⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠈⢁⠘⣿
⣼⣿⣻⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠈⡌⢆⠐
⣿⣇⢭⠿⣌⠿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠁⠀⠀⠈⡄⠀⠘⡀
⢹⣿⣿⣶⠿⠖⠂⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠘⠄⠀⠐⡄
⠀⢻⡿⣴⠀⠀⠀⢄⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠈⢎⠀⠘⡄
⠀⠀⠹⣮⣆⠳⠀⠀⡀⠀⢀⠀⢝⢦⡀⠀⠀⠀⠀⠀⠀⠀⢣⡀⡄⢀⠀⠤⡈⢆⠂⠹⡄
⠀⠀⠀⠈⠻⣧⡐⠒⡀⠈⢈⠂⢨⠓⠙⠦⣄⠀⠀⠀⠀⠀⠀⢳⡈⠄⠀⠀⠀⠈⠂⢦⣽⣄
⠀⠀⠀⠀⠀⠉⠿⣦⠁⠈⠀⠀⠀⡄⠴⠀⠨⣷⣄⠀⠀⠀⠀⠀⢳⡀⠀⠠⠀⠀⠂⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠁⠄⠠⠚⠀⠈⠙⢕⠤⡀⠀⠀⠀⠳⡀⠀⠁⠀⠀⠀⡀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠐⠶⠴⠤⠌⠡⠀⣀⠈⣲⢄⠀⠀⠙⣄⠀⢰⣶⣾⣿⡄⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⡑⠪⠝⠓⠀⠨⡓⢌⡑⢅⡠⢀⠈⠑⠚⠛⠛⠉⠑⢾⣷⣯⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠤⡀⠈⠙⠂⠀⠈⠀⠁⠂⣀⠀⠀⠀⠀⠀⠹⣿⡹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⣄⡀⠀⠄⠒⠦⠀⠀⠀⠀⠁⠢⣀⠀⠀⠀⠘⣷⡘⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⠐⠀⠠⡐⢂⡁⠀⠀⠈⠂⠄⡀⠘⣧⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢤⣄⠛⠂⢠⡅⢷⠀⠒⢤⠀⠚⢧⣹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⠀⢐⡊⠀⠿⡀⠑⠈⠉⠻⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣑⡂⠁⠘⠦⠋⢫⡧⡙⠗⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠦⣤⣌⣁⡁⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣲⣶
⣿⣿⠛⡿⢛⣛⠛⣟⢛⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀⠀⣠
⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠉⠻⣷⣄⣀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢸⢹⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠙⠿⢠⣾⣿⣉⡀⠀⠀⠀⠀⠉⠛⢿⣦⠀⠀⠀⠀⢸⠈⠈⠘⢿⣿⡟⠁
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠾⣿⡏⠉⣥⣤⣄⡀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠸⣿⣿⣶⣾⡟
⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠁⢳⠀⢻⣿⣿⠿⣷⣤⡀⠀⠀⣿⠄⠀⠀⠀⢠⣿⠉⠁
⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠐⡀⠀⢻⣿⡄⠀⠉⣻⣦⣼⡿⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠈⢄⠀⠻⣿⣦⣤⣾⣿⡿⣧⡀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠱⠆⢺⣿⣿⡟⠋⠀⠈⢳⣄⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣄⣀⣙⣿⣄⣀⣤⣴⡾⠿⣋⡙⣷⣤⡐⢮⣶⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠙⢿⣯⣄⣀⠀⠈⢻⣿⣿⣿⣿⣿⣾⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠟⠛⢿⡟⠛⢷⠃⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠈⠁⠀⠀⢠⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢠⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃