Tartaruga Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⢀⣤⢘⠳⢿⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠴⠲⣼⣧⠁⢻⣷⡠⣴⣦⡤⣰⡞⣯⠂⠉⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⠀⢘⣿⣷⡄⠀⢰⣿⣧⣾⡿⠿⠛⠿⠋⠀⠪⠔⠰⢀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠇⠼⡿⠀⢸⠻⡏⢀⣴⣴⣿⠟⠻⡁⠀⠀⠐⠒⠀⠮⠧⢴⣋⣙⠱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠗⢀⣧⣤⣐⣾⣿⣿⡟⣛⣻⡷⠀⠀⢵⣷⠀⠀⡠⠠⣃⠀⠀⠘⡏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⢿⣿⣿⣻⣿⠽⠖⠻⠉⠔⠚⠀⢤⣽⢿⡞⡑⢢⣋⠞
⠀⠀⠀⠀⠀⠀⠀⠛⢵⣟⠟⠩⠭⠉⠀⠈⠀⠀⠀⡄⠀⠀⣷⣿⢏⢏⠞⣧⣤⠆⠀⠀⠀⠀⠀⠌⠄
⠀⠀⠀⠀⠀⠀⢀⣌⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡔⣯⣞⠞⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣎⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⡶⡿⣯⠿⢿⣿⣭⡭⣾⣺⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡹⣿⣾⠻⣧⡀⠀⠀⠀⠀⣴⣿⣙⣿⣽⡗⢯⣽⣾⣽⣾⣯⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣨⣭⡸⣷⡓⡀⠀⠀⠸⣿⣯⡇⣿⣻⣿⡾⠟⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣶⣝⢟⣣⣾⣝⠿⢄⠀⢀⡷⢤⡭⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢊⣋⢿⣷⢟⡛⠀⢈⡞⢁⣠⠼⠦⣄⣼⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡁⠉⢀⡤⠚⢁⡱⠺⡇⠀⠀⠀⠉⠛⣶⣇⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣲⣤⠀⡠⠞⠒⣇⣠⠔⠋⠀⠀⢻⣀⠀⠀⢀⡴⠁⠀⠙⢯⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣯⣽⣓⢭⡏⣟⣀⡴⠚⢯⣀⠀⠀⣠⠔⠋⠉⠉⠉⡟⠀⠀⠀⠀⠀⠹⡎⠙⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⣻⢾⣿⣿⡷⢷⠁⢹⠁⠀⠀⠈⢳⠋⠁⠀⠀⠀⠀⢰⡃⠀⠀⠀⣀⣤⠴⠻⣦⠼⡄
⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⡿⣷⡿⠋⠀⢨⡗⢺⡀⠀⠀⠀⡏⠀⠀⠀⢀⡠⠖⠉⠉⠉⢻⠋⠁⠀⠀⠀⢻⠀⢱
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣯⣿⠏⠀⠀⠀⠀⣇⣼⣧⠤⠖⠚⠙⠒⠒⡶⠉⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⢸⡗⢻⡄
⠀⠀⠀⠀⢠⡿⣣⡾⣿⠿⠃⠀⠀⠀⠀⠀⢳⠀⢻⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠤⠊⠉⠉⠙⡶⠒⠚⢻⣇⣸⠇
⠀⠀⠀⠀⣾⣽⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⠙⢧⠀⠀⠀⣠⠞⠉⠙⠒⡞⠁⠀⠀⠀⠀⢀⣇⠀⠀⢸⠉⢹⠤⢤⣀⡀
⠀⠀⠀⠀⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡼⠳⣤⠞⠁⠀⠀⠀⢠⣇⠀⠀⠀⣠⠖⠋⠉⠳⣤⡿⢤⠏⢀⠂⡓⡭⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣸⠷⣄⡀⠀⠀⡼⠉⠉⠛⡏⠀⠀⠀⠀⢠⣟⢀⡞⡈⢸⣦⠗⠜⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣈⣹⠳⠾⣅⣀⣀⣀⣹⣀⣀⣤⠴⠋⢈⠏⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢼⣁⣈⣹⣍⣁⣈⠧⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⣅⢦⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣵⡷⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣾⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣆⡀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢀⣤⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⣿⣿⣿⡋⠉⠉⠁⠀⠀⠈⠙⠻⠟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡓⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⣶⢄⡀⠀⠀⠀⠀⣀⡀⠀⠀⣠⣤⣤⣤⣤⡤⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢉⡿⢯⣠⠿⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣥⢻⣽⡶⣢⠴⠛⠋⢉⡴⠛⠉⠀⠀⠁⣠⠴⠛⠉⠉⠉⠙⣩⣹⣶⣤⣀⠀⠀⣷⣞⢙⣶⡏⣳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠶⠫⠤⠞⠁⠀⠀⠀⠊⠀⠀⠀⠀⢀⡚⠁⠀⠀⠀⠀⣀⠞⠉⠀⠀⠀⠀⣁⣰⣧⣼⢽⡥⡯⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⢤⠄⠐⠀⢀⣠⠶⡂⠀⠀⠀⣀⡀⠈⢀⠀⠀⠀⠀⠜⠋⠀⠀⠀⠀⣠⠞⠁⠀⠉⠙⢾⣓⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠠⢀⡴⠋⠁⠀⠀⡰⠞⠉⠉⠙⠂⠀⣀⡤⠶⠇⠀⠀⠀⠀⡠⠈⠁⠀⠀⣠⠖⢉⣩⢿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠩⠖⢲⠂⠋⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣴⠞⢂⠀⠠⠀⠰⢁⡾⠉⢛⠋
⠀⠀⠀⠀⣀⣠⣤⣠⣶⡀⢀⣣⡀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⠁⠀⠀⡀⠀⠀⡀⣸⠃⣴⣿⣷⠦⠀⠠⡄⠀⡀
⠀⣀⣴⣻⣾⣺⣾⢛⣾⢺⣏⣄⠙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣵⡟⠘⠛⠻⠏⠀⡷⡤⣓⠖⢿⡋⣠
⣼⣣⣞⣻⣾⢛⣾⣛⣼⣻⣢⣟⡿⢧⡀⠑⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠞⡼⠁⠀⠀⠀⠀⠀⠙⠦⠞⠶⠚⠏⠑
⠙⠛⠹⠿⠼⠿⠼⠛⠘⠛⠈⠁⠀⠀⠉⠻⢤⣀⣉⡗⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣻⡡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠦⠤⢾⣍⣲⣶⣒⣉⣉⣁⣈⣿⡵⢺⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣨⣿⢻⣴⢟⣷⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣽⢻⣾⡻⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡽⣏⣽⢿⣹⢿⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⣽⣟⣻⣿⣙⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣟⣷⡿⢳⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣭⡿⣇⠽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣭⣿⠽⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠿⠋⠋⠓⡆⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣻⡭⡻⡄⠀⢠⣶⣾⣿⣿⣿⣿⣦⡀⠀⣠⢾⣻⣭⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣾⡷⣛⡵⣿⣾⣿⡿⣿⣿⣿⣿⣿⣼⣿⣞⣞⣷⡛⣏⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢳⣯⣶⠟⣶⣿⣿⣿⣿⣮⣾⣿⣿⣿⣿⣿⣿⣿⡞⢿⣼⣞⣽⣆
⠀⠀⠀⠀⠀⠀⠀⣸⡛⣿⣣⡿⢁⣼⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡌⢻⣿⢾⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣝⣶⠟⠀⣺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⢻⣧⠀⠹⣯⣲⣷
⠀⠀⠀⠀⠀⠀⠸⣧⡿⠁⠀⠐⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⢿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣽⣽⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣫⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣷⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡳⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣞⣿⣯⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣗⢽⣻⡿⠟⢻⣿⣿⣿⣯⡿⠻⣽⣺⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠙⠛⠟⠋⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠠⠀⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢤⣤⣤⠀⠀⠀⠀⠀⠀⡀⠀⠩⠌⠀⢠⣶⠔⠣⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⣀⣠⠿⣤⣆⣀⣀⣄⢾⣿⣿⠀⠀⣽⢗⠶⠒⠀⠀⠀⠐⠤⠙⠁⠁⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡠⣾⣭⣵⣿⣿⠀⠀⠈⣾⠿⠀⠀⠡⣿⣿⡿⠛⣵⣧⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠾⢻⣤⣾⠏⢈⣿⣟⠋⠋⠀⠀⡀⠁⠀⠀⢀⣼⢟⡰⠁⡀⠟⠋⠀⠀⠀⠀⠀⡰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠀⣲⣿⠿⠉⠀⢸⣯⠋⠀⠀⠀⢠⡇⠀⠀⣰⠟⠁⠈⢉⡌⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣾⠏⠀⠰⠟⡀⠀⠀⠀⠉⡇⠀⠀⠀⢀⣼⣸⣶⠿⣿⣧⠊⠀⠐⠀⠀⠀⠀⠀⠀⡰⠛⠒⠒⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⣿⡱⡴⠃⠀⠁⡠⠃⠀⠀⠁⣤⣴⠾⠟⢫⣝⡂⠀⠈⠁⠀⠀⠀⠀⠀⢀⣀⣠⠘
⠀⠀⠀⠀⠀⠀⠔⡀⠀⣩⠟⣦⣿⣿⣁⣫⣤⣴⠿⠷⠿⠶⠛⠋⣵⣊⠄⠚⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢳⣏⣽⠋⣿⣋⣹⣾⣇⢀⠾⣟⣃⡼⠭⠀⠀⠉⠁⠀⢀⣠⣴⣖⡀⠀⠀⠈⢁⣍⠁
⠀⠀⠀⠀⠀⠀⢀⠰⠏⠀⠀⠀⠉⠉⠉⠉⠩⠠⠅⠊⠉⠉⠉⠀⠀⠀⠒⠺⠋⠈⠀⠈⢩⣙⠃⠀⠀⠀⠨⠍
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡛⡃⢀⠐⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢁⠀⠀⠈⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠐⠀⠀⠈⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⡟⡹⣟⣦⣀⡀⠀⠀⠀⡀⠀⠶⣏⠉⣉⡻⣭⣲⢦⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠻⣿⣟⢿⡯⡙⣦⣔⣠⢀⣀⠀⠀⣶⣷⣦⣬⢻⣿⣿⣷⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢜⣧⣈⠻⣿⣷⣿⣉⡱⠔⢴⣽⣽⣯⣓⣙⣻⣿⣭⣭⣾⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⡳⣥⣌⣛⣿⢿⣶⣤⣿⣤⣁⣿⣹⣬⣿⣿⣽⣷⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣫⣿⢟⣾⣞⣽⣿⢼⣙⡿⣿⢿⣿⣯⣿⣿⣭⠯⠛⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠛⠋⠀⠉⠙⠛⠛⠛⢻⣜⣟⡟⠛⠉⠉⠀⠀⠀⠀⠀⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣡⣶⣿⣿⣷⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡞⠉⣠⣤⣤⣄⠙⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⢠⣾⠟⢛⡛⢿⣷⡀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠸⡿⢇⣚⣙⣉⣛⣛⣉⣛⣃⠸⣿⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣿⡇⢹⣿⢓⣦⡀⠀⠀⠀⢻⣿⣩⣯⡬⢧⠼⣧⣼⣏⣹⠃⠀⠀⠀⣠⣶⢻⣿⡏⢹⣿⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⣿⣧⡘⢛⣡⠶⣟⣦⠀⠀⣀⡇⢀⠀⠀⠀⠀⠐⠀⡀⣇⡀⠀⢠⣞⣹⡦⣜⠻⠁⣼⣿⣤⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣤⣤⣤⡈⢋⡴⣏⣹⠖⢉⠈⠷⢚⣥⠴⠶⠲⠛⠛⠙⠛⠖⠧⢬⣹⡶⠏⠈⠙⠦⡞⢳⣌⠛⣉⣡⣤⣈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⣉⠟⣤⠻⣦⡏⡁⠠⣈⡵⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⢁⠄⠈⣻⣉⡷⡝⢟⣛⢿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⢸⠟⣾⣹⠞⠁⠙⣮⡶⠋⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⢦⡞⠁⠙⢦⡿⣌⢿⠆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⣾⡽⠃⠀⢀⣾⠏⠀⠀⠀⠀⢀⡤⠶⢿⣉⣫⣥⣌⣛⡉⠿⠶⣄⠀⠀⠀⠀⠀⠻⣦⠀⠀⠙⢿⡆⢿⢟⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢠⣿⠋⠀⠀⠀⠀⡴⠿⢁⡕⠋⣩⣤⣀⠀⠈⠉⠓⣦⡹⠷⣄⠀⠀⠀⠀⢻⣧⠀⠀⠀⠉⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⡏⠀⠀⠀⠀⡼⠟⣲⠃⠀⢸⡃⠀⢹⠀⠀⢀⣄⡀⠻⡔⢻⣆⠀⠀⠀⠀⢟⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⡃⠀⠀⠀⢰⣗⣱⠇⠀⠀⠀⠙⢦⣸⢀⡴⠋⠀⢹⠆⢻⠁⣿⠀⠀⠀⠀⢺⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣻⠀⠀⠀⠀⢸⣅⢸⠀⢠⠖⠒⠒⣲⢿⣟⠛⠛⠓⠋⠀⢸⢀⣽⠀⠀⠀⠀⢸⠖⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡆⠀⠀⠀⠘⣏⠘⣇⠹⢦⣠⠞⠁⢸⡏⠳⣄⠀⠀⠀⡼⢀⣿⠀⠀⠀⠀⣸⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠤⢯⠀⠀⠀⠀⠹⡷⠘⢦⡀⠀⠀⠀⠸⣇⣀⡼⠁⣀⠞⠱⡾⠁⠀⠀⠀⢀⡯⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠐⠚⣧⠀⠀⠀⠀⠘⢿⣄⡙⠦⢤⣠⣀⣀⣩⡤⠞⠁⣼⠟⠁⠀⠀⠀⢀⡞⠠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠈⡈⢧⡄⠀⠀⠀⠀⠉⠛⠷⢴⣦⣔⣤⣶⠴⠟⠋⠁⠀⠀⠀⠀⣠⢎⠉⠆⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣏⣹⣄⠀⠊⢙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠓⡀⢀⡴⢻⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠌⢧⣼⠷⣄⠁⠐⠉⠷⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⡚⠃⠈⢀⣤⣏⣹⠟⣱⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣤⣶⣷⠙⠶⣏⣳⢦⣀⠀⠀⠨⠉⠛⠛⠖⠲⠲⠖⢻⠋⡍⠐⠀⠀⣠⡔⢻⣤⠟⢱⣦⣄⣉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢩⣿⠃⣸⣷⢍⣛⠶⣏⣹⡶⢦⣤⣀⣐⣀⣁⣀⣀⣀⣠⡤⢶⡞⢻⣤⠟⣋⢵⣾⡌⣿⡟⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢿⡀⢻⣿⣌⢿⢷⣬⣉⠓⡺⢦⢿⣥⣿⣉⣿⣨⣯⣼⠷⠞⢋⣩⣴⢿⣇⣼⣿⠃⣼⠇⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡘⣷⡀⠙⠿⢿⠄⠉⠳⣾⣿⠀⣶⣶⠲⣶⡦⢰⣶⠆⣿⣷⡴⠋⠁⣸⠿⠟⠁⣰⡏⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⣿⣷⣶⣼⠃⠀⠀⠀⠉⠓⠾⣯⣷⣶⣶⣿⡯⠖⠋⠁⠀⠀⠀⢸⣤⣶⣿⣿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠉⠁⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⢠⠄⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠊⢀⠔⢈⠀⠀⠀⠀⢀⠀⢀⡀⠀⢘⣆⡀⠀⠀⢀⡔⢷⠆⠀⠀⠶⢣⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⣀⠀⠀⠈⡁⠢⣄
⠀⠀⠀⠀⠀⠀⡸⢀⡔⣥⡒⠁⠀⢀⡴⠊⠀⢀⡤⠚⠉⣋⢀⠀⣈⠉⠓⠤⡄⣀⡀⢀⣀⣀⠱⠖⠒⠉⠉⠉⡗⠢⢶⡉⠀⠉⠢⡀⢈⠢⡀⠱⡄
⠀⠀⠀⠀⠀⠀⣇⢾⣿⠏⠀⢀⠔⠁⠀⢀⠔⠃⣠⠶⠛⠉⣈⣀⡛⠀⠃⠰⠦⠤⠤⠤⠤⠄⠐⠀⠛⠀⠁⡘⠳⢦⡀⠙⣦⠀⠀⢧⠀⠳⣌⢆⠘⡀
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠔⠊⠁⠀⠀⢠⠊⠀⠄⠉⡤⢒⠩⠀⠀⠀⠩⡁⠓⢒⣒⡀⣐⠒⠒⢈⠩⠉⠈⠭⢑⠢⣄⡹⢃⠀⢳⠀⠸⡀⠀⢻⣿⣢⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠛⢀⠎⠔⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢈⢣⠀⠶⠀⢃⠀⠙⠦⠠⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⣤⠤⡌⠀⠄⠀⣀⣀⡀⠀⠔⢀⠀⠁⠠⠀⠀⠈⡀⠈⠀⣀⣀⣀⣀⢀⠀⠀⡤⢨⡄⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣿⠀⠂⠊⠁⠀⠐⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠐⠀⠀⠀⠀⠀⠇⢸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣹⡀⣠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠀⠀⠀⠀⠀⢀⢠⡀⣸⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠸⡈⠒⠀⠀⠀⣀⠂⡁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠉⠀⠀⠀⠀⡸⢀⡈⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢴⡄⠱⡘⠀⠀⠀⠀⠀⢰⠀⠈⠐⠒⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⢠⠃⣶⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠐⠀⢱⡀⠀⠀⠀⢀⠀⡀⠀⣀⠀⠀⣀⣀⠀⠀⡄⠐⠀⠐⡄⢠⡃⠐⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠚⣇⠘⢎⢀⠀⠀⠀⡁⡀⠂⠀⠀⠀⠀⠂⡀⠂⠀⠀⠀⣡⠃⣼⠃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠳⣌⠷⣀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⢄⡴⣁⡜⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡁⠀⣈⠳⢅⠠⠀⡉⠀⠉⠀⠀⠀⠁⢀⡴⠊⡀⠈⡲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠢⣄⠺⠆⡉⠒⣦⢢⠤⢤⢠⠶⠒⡁⠲⠆⣀⠞⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⢠⠋⠑⠢⢄⣀⠓⠲⠤⠤⠖⠆⠀⡁⠔⠊⠘⡄⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠁⠊⠀⡞⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠉⠁⠀⠀⠀⠀⠇⠀⠈⢌⢀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣸⡦⠁⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠐⣣⣢⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⠃⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠰⢳⡃⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⠾⠗⠋




⣶⣶⣶⣶⣶⡷⢿⣾⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⡶⣷⠶⢷⡾⢷⣶⣶⠾⠶⢶⠶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠁⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⡤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡜⠁⠀⠀⠀⠀⠙⠄⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠰⡁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠰⡀⢣⡀⠀⠀⠀⠀⣠⠂⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠑⢤⡉⠓⠲⠒⠊⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⢀⠀⣀⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠹⠒⠂⠈⠉⠉⠉⠛⠋⠉⠁⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠐⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠒⠒⠀⠀⠀⠠⡦⠤⠤⢤⣤⠾⠴⠒⠂⠉⠉⠉⠁⠀⣘⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠚⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⢀⡤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠾⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠦⠀⢀⠀⢀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠆
⠀⠀⠀⢰⡇⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⠂⠄
⠀⠀⠀⡆⣧⠈⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠇⢀
⠀⠀⢸⠁⠹⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢸⠀⠈
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠒⠃⠀⠀⠀⠀⠀⠀⣀⠤⠒⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⢀⠇
⠀⠀⡇⠀⠀⠀⠀⠀⢀⠤⢶⡦⠀⠀⠀⠠⠤⠤⠀⠒⠒⠈⠉⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠄⠀⡼⠀⢀
⠀⠀⠃⠀⠀⠈⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡼⠁⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢛⢦⡀⠀⠀⠀⠀⠀⠀⢀⡀⡏⠀⠀⢸⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⢇⡞⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠢⡑⢌⠓⠲⠤⠤⠴⠒⢉⡞⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠦⠤⠤⠖⢊⡴⣣⠋⢀⠔
⠀⡀⠀⠀⠀⠀⠀⠈⠳⡀⠈⠢⡑⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠲⠤⠤⠤⠒⣩⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠘⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⡀⠀⠀⢀⠴⠋⠁

⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠹⠏⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢨⢎⡱⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠳⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠯⣍⢀⡀⢀⣐⠞⣴⣿⣧⣼⣿⣦⠳⣂⡀⢀⡀⣩⠽⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠤⠒⠿⣩⠘⠙⢿⡼⠱⣌⢿⡿⣡⠎⢧⡿⠋⠃⣍⠿⠒⠤⡀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣠⠖⠁⠀⢠⠷⣚⡿⠯⠀⠀⡘⡼⢧⢃⠀⠀⠭⢿⣓⠮⡄⠀⠈⠲⣄⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⢀⡇⣿⠷⣦⣤⣴⣲⠃⠇⠺⠘⣖⣦⣤⣴⠾⣿⢸⡀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⢾⡳⠞⠋⠉⢉⣉⣀⣰⠀⠀⣆⣀⣉⡉⠉⠙⠳⢞⡷⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢱⣤⡴⠖⠒⠦⢝⠿⠀⠀⠿⡫⠔⠒⠲⢦⣤⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢜⡳⠊⠚⢛⣻⣶⡄⠀⠀⢠⣶⣟⡛⠓⠑⠜⡣⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡇⠀⠴⠛⠉⣯⢻⠀⠀⡟⣽⠉⠛⠦⠄⢸⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡲⣤⣶⡶⠏⢠⢾⡵⡄⠻⢶⣶⣤⢖⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠶⠶⢴⠛⠋⢸⡇⠙⠛⡦⠶⢶⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣙⠿⠸⣦⣤⣾⣷⣤⣴⠇⠿⣋⣴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠏⠀⠀⠀⠈⠙⢻⡟⠋⠁⠀⠀⠀⠹⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠀⢸⠀⠀⠀⢷⡾⠀⠀⠀⡇⠀⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣿⠀⠀⠀⠘⠃⠀⠀⠀⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⠀⠀⠀⠀⠀⠀⠰⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣾⠿⣿⣯⣟⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠇⢼⠟⣿⠟⡋⢿⣿⣿⡿⢻⣿⣿⣿⡿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠖⡋⠉⠀⠀⠀⠉⠚⠫⠾⠇⠒⣬⠉⠬⠑⠩⣀⣽⡟⣿⡟⢵⢷
⠀⠀⠀⠀⠀⠀⠀⡈⠁⣶⠀⠆⠀⠀⠀⠀⠀⠀⠘⠳⣶⠘⠿⢊⣃⣤⠙⠣⢉⡡⠋⠁
⠀⠀⠀⣠⡤⢖⠋⠳⣀⡀⠀⢀⡀⣄⡄⠠⠤⠤⠠⣀⣈⢻⡷⣈⣻⠰⢷⢡⣤⢈⡁⣰⠂⠄
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠒⠒⠒⠒⠋⠈⠉⠛⠿⢇⣼⣦⣄⣶⣶⠾⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣆⢻⣿⣿⡿⠘⢿⣿⣿⠷⣦⣄⡀⣀⠤⠚⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇⢉⣵⣿⣿⣎⢩⣶⡿⠶⠛⠉⠁⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠦⣄⡀⠀⠀⠀⠀⠀⡠⠚⢿⣿⣿⣿⣿⢡⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠈⠉⠓⠦⣤⣀⣴⣿⠷⠿⣮⣿⣏⣥⣬⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⡴⢽⠃⠄⢀⠀⠀⠈⠙⠛⠁⠀⢀⣀⣤⠤⣤⠤⠤⣴⠞⢋⣁⡀⢉⣉⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠉⢠⣦⠀⠾⠐⠀⠀⠀⠀⠀⠐⠉⠉⠀⠚⠛⠃⠾⠿⣧⣿⠿⢧⡿⠿⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⠁⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒⠉⡁⠄⠐⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠐⠆⠀⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⣿⡷⣯⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣶⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠘⠃⠶⡮⢑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣴⢟⣷⣠⣆⣴⣶⣤⡀⠀⠀⠀⠀⠀⢴⢡⣇⣾⡿⡀⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠿⠹⠛⢞⣫⣥⢰⣾⣿⡆⣷⣶⣮⡜⠛⠵⣧⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡜⠿⢟⣈⣛⢛⣥⣬⣟⠋⠰⣿⣷⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡝⠿⣣⣾⣿⣿⣿⣼⣿⣿⡿⣴⣿⣮⡉⣵⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣻⠿⠃⢾⣿⣿⣿⣿⡇⣿⣿⡿⣱⣿⣿⣿⢇⡙⠿⣟⣄⣤⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠟⣼⣿⣷⣮⡝⢻⣯⣥⣝⠻⢱⣿⣿⢟⣵⣿⣿⡎⣿⣿⢼⣟⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢆⣾⣿⣿⣿⣿⣷⢻⣿⣿⣿⣣⣶⣮⡁⣿⣿⣿⣿⡇⣴⣶⡆⠈⢫⣾⣣⣦⡇⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢆⠿⣿⣿⣿⣿⣿⡿⠺⣿⣿⣣⣿⣿⣿⡿⢘⡿⢿⣿⡇⣻⣿⡃⠀⠀⠙⠿⠿⠇⢬⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣼⣿⣷⣶⣭⢽⣷⣿⣿⣾⢁⣙⡿⢟⣭⣾⣿⣿⣿⣾⠅⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠰⣶⡄⣀⠀⠀⠀⢛⡑⣿⣿⣿⣿⣯⢿⣿⣿⢯⣿⣿⣿⡜⣿⣿⣿⣿⣿⡟⣾⣷⡎
⠀⠀⠀⠀⠀⠀⢶⣤⣰⣿⣿⣾⣇⠿⠃⠿⢿⣛⢩⣶⣶⣮⠉⡻⢿⣿⠿⢟⣙⢿⣿⣿⣿⢡⣝⡛⠁
⠀⠀⠀⠀⠀⠀⣀⣈⢸⣿⣿⣻⣽⡼⣷⠸⣿⣿⣷⡎⠻⠏⣾⣿⣷⣸⣿⣿⣿⣿⣮⣝⢋⣙⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠙⠏⠿⠟⠃⠀⠰⣷⣝⢫⣵⣾⡏⣴⡄⣯⣭⣅⢿⣿⣿⡿⣫⣥⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢡⣶⣝⠛⢼⣿⣧⢿⣿⣿⠎⠛⣽⡦⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣛⠰⢿⣶⢤⣶⣶⠰⠿⣃⣭⢴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⢸⣯⣿⣾⣻⢗⡁⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠵⢟⣁⠙⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣄⢀⣄⣀⡀⠀⢀⢀⣀⡀⢀⡀⢀⡀⡀⠀⣄⣀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠔⠊⠀⠀⠈⠑⠄⠀⠀⠀⠁⠀⠻⠞⠠⠓⠀⠛⠞⠛⠃⠈⠞⠘⠈⠔⠀⠿⠛⠰⠋⠔⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠢⣄⡀⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⡤⠼⠤⣄⣠⠂⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠈⠳⣤⢔⣡⢤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⢓⣲⣦⡖⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠁⠀⠀⠀⠀⠀⠶⠆⠀⠸⡏⠀⠀⠀⠀⠈⠓⢆⠀⠀⠀⠀⣰⠋⣀⡤⠖⠋⢉⣠⠞⠋⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠐⣯⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⡿⢋⣁⡤⠴⢚⣉⣤⣤⣴⣶⣄⣠⡂⢀⡇
⠀⠀⠀⢀⣠⠞⠉⠉⠉⠉⠉⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⡼⠓⠦⠤⠶⠶⠚⠁⠀⠀⢀⡠⠖⠋⠉⢀⣴⡾⠛⠋⠉⠉⠉⠉⠍⠛⠻⢾⣄⡀
⠀⢀⡜⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣀⣀⣀⡴⠚⠉⠐⠤⣀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣠⣶⠟⠁⣀⠤⠤⠀⠀⠀⠀⠀⠉⠀⠀⠉⣻⣦
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢸⣿⠃⠀⠀⠁⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢙⣷⡀
⠀⠀⠑⢤⣠⣤⠀⠀⢀⣀⠀⢀⡴⠒⠋⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢾⡇⠀⠀⢠⡞⣁⡀⠉⢷⡄⠀⠀⠀⠀⠀⢀⡾⠋⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣾⣿⠀⠀⠀⣷⠀⠀⠀⠀⠀⣾⣿⡇⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠋⠉⠉⠉⠛⠛⠻⢶⣤⣀⠈⣷⠀⠀⣿⣿⣿⣦⡄⠀⡿⠀⠀⠀⠀⠀⣿⣿⣷⡆⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣿⡇⠀⠘⣿⡿⠟⢀⡼⠃⠀⠀⠀⠀⡀⠘⣿⣟⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⣿⡀⠀⠈⠙⠛⢋⡴⠀⠀⠀⠐⠀⠀⢀⣤⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠈⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠁⠀⠹⣶⠶⠶⣶⡾⠋⠁
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⡄⠀⠀⢀⠔⠉⠀⠈⠐⠄⠀⠀⠀⠀⠀⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⣾
⠀⠀⢀⠄⠤⠊⠀⠛⢓⠀⠀⣾⠛⠄⠀⠀⠄⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢆⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⡥⢀⠀⠀⡂⠀⣨⠆⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⠃⢀⠠⠐⠂⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠺⡄⠀⠐⠂⠀⣸⠄⠀⣾⡇⠀⡠⠊⠀⠀⠀⠀⠑⠀⠠⢀⡀⣀⠀⠄⠊⠠⠁⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠢⠒⠤⠞⠉⣧⠀⢀⡿⠃⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⡴⠖⠛⠛⠒⠶⣤⡀⠀⠀⠀⣠⡶⠟
⠀⠀⠀⠀⠀⠀⣠⣼⢶⡿⠟⠋⠉⠉⠉⠙⠶⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠻⣶⠾⠛⠙⣧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⡀⠀⣆⠘⢻⡟⢁⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⣠⡟⢰⣦⠷⢻⠀⠀⠀⠃
⠀⠀⠠⣽⠟⠋⠑⠦⠻⣧⣀⡴⠀⠀⠀⠀⢀⣠⡴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⠙⣦⠀⠀⢀⠀⠀⢠⡀⠀⣠⡟⠉⠉⠉⠉⠁⠀⠈⠑⠒⠓⠢⠤⣀⡀
⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠉⠳⠶⠶⠚⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣸⠦⠴⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣀⠔⠢⣄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⢠⠃⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⣙⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠻⠤⢄⠀⠤⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⠀⣀⠀⠤⠘
⠀⢀⣀
⠀⠁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⢤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⣀⡀
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠈⠳⢄⠀⠀⠀⢀⡜⠁⠀⠀⢀⣀⡉⢳⡄
⠀⠀⠀⣰⠋⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠑⢄⡤⠊⠐⠄⠀⠀⠀⠉⢠⡼⠁
⠀⠀⢀⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠼⠛⠋⠉
⠀⠀⣞⠁⠀⠈⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⣀⡴⠊⠁
⠀⠀⠈⣷⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠁⠀⠢⠈⡇
⠀⢀⡼⠇⠀⠀⠀⠀⠂⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⣀⡀⠀⠀⠀⢀⡇
⢰⠋⠀⠀⢀⣀⢀⣶⠛⠁⠀⠈⠉⠛⠉⠙⠲⠶⠦⠬⠤⠤⠴⠖⠚⠛⠉⣷⠀⠀⠀⣼⠃
⠘⢦⣤⣀⣸⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡤⠶⠶⠃
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⣶⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣼⣫⣽⣙⣦⣄⡰⢻⢻⢷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡗⡭⠴⢭⠁⢀⣺⣻⡿⠉⠙⢟⡾⢻⣷⢰⣏⢷⡯⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⡂⣾⣧⠛⠉⣷⣿⠟⠋⢳⣤⠲⢻⣷⣄⣙⣮⡃⡫⣝⣥⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣭⣞⢹⠟⠁⠀⣠⣽⡿⠀⣀⣸⣟⠅⠀⡫⣿⠏⠛⣿⡗⠈⠳⣱⢆
⠀⠀⠀⠀⠀⠀⠀⢘⠾⠏⠀⠀⠀⠀⣶⣾⡿⠟⠋⠉⠉⣴⣿⠻⣿⣆⣀⣹⡷⣿⠀⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⠀⠀⠀⠀⠙⠻⣿⣥⠀⢀⣀⣛⠏⠀⠠⣿⠛⠟⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠟⣷⣾⡟⠉⠙⢻⣾⠿⢷⣄⣠⡿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⣿⣶⡼⠖⠘⠣⣀⣘⠛⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠷⣮⣞⣉⣭⠽⢶⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠟⠋⣽⡿⣧⣤⣼⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠲⣄⠀⠀⢀⣀⣠⣼⡤⠤⣤⡤⠤⣾⣤⣀⣀⠀⠀⢀⡴⠚⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢘⡷⠚⠉⠁⢩⣀⣀⣀⣸⣀⣀⣀⣸⠁⠉⠙⠶⣟⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣀⣀⡴⠋⠑⣤⠔⠛⠙⡏⠁⠀⠀⠀⠀⠙⠏⠙⠒⢤⠞⠉⢳⣄⣀⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢀⡼⠃⠀⠀⠀⢀⣠⠴⠞⠋⠉⡽⠁⠀⡰⠁⠀⠀⠀⠰⡀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢱⡀⠀⢹⡍⠉⠛⠶⢤⣀⠀⠀⠀⠀⠻⡄
⣼⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⢰⡷⠤⣴⡧⣀⠀⠀⠀⠀⢳⣄⣀⣀⣴⠃⠀⠀⠀⢀⡠⣷⡤⠶⣷⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢹
⣿⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣼⠀⠀⠇⠀⠀⠀⠈⠓⣶⠋⠀⠀⠀⠈⢳⡖⠋⠀⠀⠀⠀⡅⠀⢸⡄⠀⠀⠀⠀⠀⠈⠳⣄⠀⢸
⠘⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⣼⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢰⠤⠼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣼⠶⠤⠤⠤⠤⠤⢬⡷⠀⠀⠀⠰⣧⠤⠤⠤⠤⠤⠴⢾⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠈⣇⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⢈⣆⣀⣀⣀⣀⣎⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠠⢿⣤⠤⠒⠂⠈⠉⢉⡏⠀⠀⠀⠘⡏⠉⠉⠀⠒⠦⢤⣼⠧⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⡇⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡏⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠈⠙⢆⠀⠀⠀⠀⠀⡸⠗⠒⠒⠿⡀⠀⠀⠀⠀⢀⠎⠀⢰⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⠜⠓⠤⣤⣴⡎⠀⠀⠀⠀⠀⠈⣦⣤⡤⠖⠛⣄⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠳⢤⣀⡈⠉⠙⡶⠤⢤⠤⠴⠞⠉⠉⣄⣠⠴⠋⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢰⡏⠙⠓⠾⣇⣀⣸⣀⣠⡿⠖⠛⠉⣧⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢹⡀⠉⠁⣸⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣀⣠⠟⠀⠀⠀⠀⠘⣇⠀⢠⡟⠀⠀⠀⠀⠘⢦⣀⡀⠀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣳⣶⣚⣉⣀⣀⣀⣀⣀⣀⣀⣙⣦⣞⣁⣀⣀⣀⣀⣀⣀⣈⣙⣲⣶⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠔⣦⡀⠀⣀⠀⠀⣠⡿⡉⠍⣳⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣮⡀⠉⢀⣿⡗⣠⣶⣿⣿⣿⠟⡘⡃⢛⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣻⣷⠀⣿⣿⠃⢀⣈⣿⡿⢿⡋⠉⠀⠤⠄⣠⣤⣠⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⢀⣏⡁⢤⣿⣾⣶⠿⢿⣯⠄⡀⡷⡄⠁⠀⣄⠳⡀⠈⠻⡗⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⢾⣿⣿⣻⣿⠽⠿⢛⠩⠝⠀⣰⣻⣿⠖⠘⣐⡾⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠚⢯⣟⠿⠋⠝⠉⠀⠁⠀⠀⠠⠀⠀⣳⣾⠟⡻⢺⣬⡴⠁⠀⠀⠀⢀⠢
⠀⠀⠀⠀⠀⠀⣬⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢔⡵⡱⠛⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡯⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠠⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⣀⣀⣀⠀⢨⠇⠂⠀⠀⠀⡇
⠀⠀⠀⣾⠿⢛⡥⢤⣬⡉⠉⠓⢤⡀⣠⠧⠒⠲⡏⠉⠉⠉⠉⠲⣄⣀⡀
⠀⠀⢠⠋⠻⢨⡣⠿⠿⠀⠀⠀⠀⠙⣄⣠⠴⠒⠙⢦⡀⠀⠀⡰⠋⠈⠉⠓⠦⣀
⠀⠀⠈⢦⣀⢤⣉⣀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⡠⠊⠉⠒⢺⠀⠀⠀⠀⠀⠀⠘⢗⠢⣄
⠀⠀⠀⠀⠑⠤⣀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⡃⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠈⣆⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⡆⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⣀⠴⢧⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡒⠀⠀⠀⠀⢀⠀⢀⣐⢮⣣⡀⠀⠙⢄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠹⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣖⠀⠀⡜⠁⠀⠀⠈⠑⣇⣇⠀⠀⠀⡵⣄⠀⠀⠀⢣⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢣⠀⠀⠀⠀⠀⠀⠀⠗⣤⠎⠀⠈⠳⣄⣀⣸⡀⠀⠀⠀⠈⡆⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡀⠀⠀⠀⠀⠀⠀⠩⢶⠤⣀⣀⠏⠀⠀⠑⠦⣀⡀⠀⠸⡀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠑⣆⠉⠑⠲⠤⣄⣠⡃⠈⠙⡶⠓⢦⠴⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⣳⡤⢄⠠⠤⠬⠭⠵⠦⢼⣤⣈⣢⡙⡽⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠉⠁⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠈⠱⡄⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢄⡀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠊

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣉⠵⡂⠚⠉⡩⠓⠂⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣗⡛⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⠟⠡⣴⠞⣉⡀⠀⠀⢀⣀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠯⣏⢡⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠨⠛⠁⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠜⢪⠣⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠮⢤⡖⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠐⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⡃⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠰⢾⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⡴⠀⠘⡻⡶⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠘⢧⣀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⢰⣴⣃⣿⣻⡻⡆⣈⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣥⡶⠁⠉⠃⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⣈⣴⡏⡼⣛⢹⣇⠜⢯⡧⠀⡀
⠀⠀⠀⠀⠀⢤⣶⣿⢿⡛⡹⠿⡟⠥⠀⣄⡈⠁⠀⠀⠄⠀⠔⢠⡆⠀⠀⠰⠀⠀⠀⠀⠀⢀⣀⠒⣴⡟⠀⠈⠩⠑⠚⠒⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⢠⣀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⡴⠖⠀⠒⡯⠠⠻⠀⠀⣀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠠⠽⣒⣒⣂⣤⣷⠋⠁⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡞⢠⠖⡂⢀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⡟⣹⠃⢸⡶⣿⣯⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡾⣿⠙⣧⣲⣿⢦⣻⣼⠿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣯⣻⣿⣷⣜⣷⣵⣮⡿⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠛⠋⠋⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢟⡻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⠉⠉⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣥⣾⣷⣬⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣄⠀⠀⣠⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢞⡷⣭⠨⢎⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡱⠅⣭⣟⡱⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣰⠖⣮⡃⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢘⣕⠺⢆⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣬⢛⢯⡟⠉⠉⣼⣿⣿⣿⣿⣯⡿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣧⠉⠁⢻⡽⡛⣤
⠀⠀⠀⠀⠀⢠⣜⠛⡞⠟⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠻⢷⠛⣣⡄
⠀⠀⠀⠀⢠⠟⠙⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠙⠋⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢾⣿⣿⣿⣿⣿⣿⣿⣷⣻⣟⣾⣿⣿⣿⣿⣿⣿⣿⡷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢎⡻⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⢿⡱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢎⣛⣿⣿⣿⣿⣿⣿⣻⡱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠹⠃⣿⡿⢿⣿⠘⠏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣘⢗⡤⠈⠁⠈⠁⢮⠺⣓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡉⡆⠂⠀⠀⠀⠀⠐⢐⣉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠂⠵⠀⠀⠀⠀⠀⠀⠮⠐⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿

⠀⠀⠀⠀⠀⠀⢠⠔⣛⢋⡉⢛⣿⣦⣄
⠀⠀⠀⠀⠀⣀⠋⠸⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠙⣿⣶⣍⠛⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠑⠻⣿⡿⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⡒⢀⠷⡿⢋⣻⠿⢿⣯⡿⠷⡉⠐⢀⣀⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡽⣿⣿⣤⠜⠛⠳⠇⢺⣿⣿⣿⣿⣿⣷⣯⣕⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣾⣿⣿⣿⠁⠐⢀⡀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣦⡌⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⣛⣋⠙⠿⣿⣿⣿⡄⢸⠇⠀⠀⣂⣽⣿⣿⣿⣧⣽⣿⣿⣿⣦⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⢀⡾⠃⢰⣿⣿⣿⣿⣶⣿⣿⣟⣿⣷⠤⣀⣀⣘⣡⣿⣿⣿⣽⣯⣿⣛⣿⣿⣿⣶⡄⠈⠓⣄
⠀⠀⠀⠀⠀⠀⢺⣀⣀⣿⣽⣿⣿⡿⠟⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣷⡿⠿⣿⣿⣷⣄⠈⣳⡀
⠀⠀⠀⠀⠀⠀⢸⣏⣛⣿⣿⣿⠇⠀⠀⠀⠀⠹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣼⣿⣻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⢤⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣷⣿⣾⠀⠀⠉⢿⣻⣿⣿⣿⣿⣿⣿⣦⣭⡿⡿⣟⣩⣿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠁⠀⠀⠂⠀⠀⠀⠀⠘⢯⡻⣾⣦⡀⠀⠻⣭⣛⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⣦⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⡈⣤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⢮⡻⢶⣶⣟⠀⠈⠙⢿⣭⣽⠷⠿⣥⣾⠛⠷⣛⠯⣿⡿⢿⣷⣮⣙⢻⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢭⣻⠿⣶⣠⣄⣹⣿⣥⣴⣿⣿⣿⣾⣿⣿⣾⣿⣿⣧⣽⣿⣿⣿⣶⡛⢻⣷⣶⣦⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⢿⣿⣿⡽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡿⢿⣿⣿⣄⠉⠉⠛⠯⠽⣟⣷⣽⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣀⣹⣿⠈⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠉⠀⣌⡛⠿⣿⣿⣿⣏⡉⠙⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠟⠁⣀⡤⠒⠊⠁⠀⠀⠙⢿⣯⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠇


⣀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠂⠐⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⢲⣶⡤⠀⠀⠀⠀⠀⠠⠀⠀⠔⠠⢀⣶⠖⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡖⢀⣠⡤⠛⢲⢆⣠⠤⢄⢿⣿⡏⠀⣠⡟⡕⠊⠁⠀⠀⢀⠒⠊⠁⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣤⠲⣶⣼⣿⣿⠀⠀⠀⡾⠛⠀⠀⢰⣝⠿⠛⢹⣿⡆⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠘⢋⣶⣿⠋⠀⣿⣏⠁⠉⠀⠀⡄⠀⠀⠀⣠⡾⢣⡎⢀⠀⠉⠁⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⡿⠁⢨⡿⠟⠃⠀⠸⡿⠈⠀⠀⠀⢰⢇⠀⣠⣾⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣼⣟⠀⣠⠋⢤⠀⠀⠀⠀⡇⠀⠀⢀⣠⣾⣾⠿⠛⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣟⢹⣷⣳⢫⡄⠀⣠⣃⣀⣀⣀⣴⠾⣛⠉⢱⡮⠅⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠂
⠀⠀⠀⠀⠀⠀⠁⠆⠀⣾⢙⡷⣿⣷⣴⣶⢞⣏⠋⢛⣍⣩⢀⡾⠇⢂⣁⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢥
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠛⠲⠧⠼⠧⠶⠿⠻⢀⠻⠭⠷⠒⠉⢀⣀⣀⣀⠠⠐⢲⠿⢧⡄⠀⠀⡀⠴⠶
⠀⠀⠀⠀⠂⠤⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢼⣦⡀⠀⡈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢂⠀⠀⠤⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢊⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁

⠀⠀⠀⡀
⠀⠀⣉⡟⡀⢀⣤⡼⣿⢷⣦⣄⡀
⠀⠀⢻⣷⣜⣽⣭⣰⡏⢻⡿⢯⡿⢦⡀⠀⣀⣤⣶⣶⣫⡝⢛⠛⠛⢁⡿⢃⣐⣶⣶⣤⣤⣀⣀
⠀⡀⣨⠈⠛⠛⠎⠉⢤⡟⣽⠇⣤⡸⡛⠿⡙⣿⣿⣿⣿⣧⡶⠶⣶⣀⡀⢩⣽⣿⠛⠻⣿⣫⣿⣟⣶⡦⣤⡀
⠀⠈⠻⣿⣷⣦⣄⣠⣆⣤⠠⡀⣿⣽⠃⠀⠁⢨⡛⠛⠀⠻⠂⠀⠉⣿⣿⣿⣯⢔⣠⠬⠞⢻⣿⣿⣷⣶⠶⠮⠷⡂⢤⡀
⠀⠀⠀⠀⠉⠛⢿⣽⣼⣾⣿⣿⣿⣏⠀⠀⠀⢸⡃⢄⣀⣄⡀⠀⠀⠈⢙⣿⣿⣿⣥⣄⢀⠿⢿⣿⣷⢿⣿⡗⠀⢵⢿⣿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⣿⣿⣿⣿⡠⠀⠀⢀⠈⠀⠀⠉⠉⠙⠒⠢⠬⣏⠀⠈⠉⠻⣧⣀⠀⠉⠀⡄⠙⢣⡾⠀⠀⠈⣤⠹⡿⠿⠶⣄
⠀⠀⠀⠀⠀⣠⣽⣶⣶⣿⣿⣿⡿⣿⣥⣂⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢸⣿⣿⣶⣦⣄⣼⣜⣷⣤⡄⠁⠀⠁⠈⡀⢈⠋⠓
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⠁⠘⣿⣿⡟⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠓⣠⣸⠿⠛⠉⠻⣿⣿⣿⣿⣿⣿⣤⣔⣀⡀⣾⡾⣀⢠⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠟⠀⠀⠀⠈⠃⠀⠀⠀⢀⠎⢽⣿⡿⣋⣧⣖⡀⡀⠀⠀⠈⠉⠲⣦⣤⣤⣤⣍⣙⣋⠉⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠏⠀⠈⠙⠿⢻⢿⣿⣿⡿⣷⣦⣄⠀⠀⠟⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣯⣭⣍⣉⣛⣻⣯⣿⣯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢤⠀⠶⣾⢿⣿⣿⣿⣿⣿⣿⣿⣞⠙⠳⠼⠿⠛⠙⠿⠛⠛⠋⠉⠉⠀⠀⠈⠫⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠛⢀⠔⠛⣻⢿⣿⣻⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢞⣩⣙⡻⣧⢿⣿⣛⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠭⣙⣿⢿⣼⣿⣽⣽⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢿⣿⣿⣛⣽⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣟⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠈⠀⡶⠆⠠⡈⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠈⠁⣎⠰⡷⢽⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠚⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠃⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣴⡀⡀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⡾⠯⡤⠀⠤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠤⠀⢀⠀⠀⠀⠀⡄⠀⡀⠠⠀⢁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠽⣏⡹⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⢃⠃⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣻⣯⡿⣏⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢶⣴⢬⡱⡨⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⢷⣥⢷⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣭⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⠛⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣶⣦⣤⣤⣤⣄⣀⠀⠀⠀⠀⢀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣼⣿⣿⣷⡀⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣤⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⣷
⠀⠀⠀⠀⣾⣿⣿⣿⣷⠿⠛⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠉⠉⠊⠉⠉⠁
⠀⠀⠀⠈⠻⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠿⠿⠿⠿⠿⠟⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢷⢧⡞⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣻⣿⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⣛⣍⣝⣛⡕⠁
⠀⠀⠀⠀⠀⠀⢀⣠⢖⣽⣿⣝⢻⣦⡦⢄⠀⠀⠀⢘⠃⢃⡘⠘⡃⠀⠀⠀⡠⢶⣴⡟⣫⣿⣯⡲⣄⡀
⠀⠀⠀⠀⢠⣾⢿⣿⠸⣿⡓⣪⢁⣿⠳⢦⠀⢀⢄⠕⠛⠃⠘⠛⠪⡠⡀⠀⡴⠾⣿⡈⣗⢚⣿⠇⣿⡿⣷⡀
⠀⠀⢀⡴⣾⢹⢺⢸⣷⣬⡏⠁⡿⠋⢛⣁⣂⣴⣶⣄⣶⣷⣿⣶⣠⣶⣦⣐⣈⡻⠙⢷⡈⢩⣥⣾⡇⡗⡏⣷⣦⡀
⠀⠀⣮⣷⣟⢮⣋⠜⣼⣿⠟⠋⠁⠆⣴⠾⠿⣬⣿⣿⡹⠚⡳⠋⣿⣿⣧⠿⠷⣦⠰⠈⠙⠻⣿⣧⠣⣙⡵⣻⣾⣵
⠀⢸⣿⣷⣯⣵⣶⠾⠟⠁⠀⡐⣩⢟⣩⢟⣲⣞⢈⡷⣸⡿⢶⣶⢸⠇⣷⣖⡻⣍⣻⣍⢂⠀⠈⠻⠷⣶⣮⣽⣾⣿⡇
⠀⣿⣮⣶⣾⣿⢶⠊⠀⠀⠘⣥⣶⣾⡞⣿⠛⢷⡽⢜⡿⣧⣴⠿⠊⢿⡾⠛⣿⢺⣷⣶⣬⠂⠀⠀⠑⣶⣿⣷⣶⣵⣿
⠐⡿⠿⣻⣟⣿⠃⠀⠀⠀⢃⣼⣋⡭⣿⡴⣮⣼⢟⣾⢳⠭⣩⡜⣵⡻⣯⣵⢦⣾⢬⣝⣧⡜⠀⠀⠀⠘⣿⣻⣿⠿⣿
⠀⣿⣩⡿⡹⠁⠀⠀⠀⠐⢠⢿⣁⢼⣼⢧⠖⣒⣽⢓⣧⣰⣶⣙⣮⢯⣒⠲⣼⣧⡧⣈⠿⡄⠂⠀⠀⠀⠈⢏⢿⣍⣿
⠀⣯⢛⡿⠁⠀⠀⠀⠀⠘⢨⢰⢛⣷⣾⡫⢿⡿⣤⣮⢻⣍⣨⣿⢵⢫⣽⡿⢿⣷⣿⡛⣦⡄⠃⠀⠀⠀⠀⠈⢿⡛⣽
⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠣⡜⡟⣧⡻⠉⢻⡤⣬⣿⢬⣭⣎⡥⣼⣧⢦⡟⠉⢚⣼⢛⣿⠜⠀⠀⠀⠀⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⣹⡇⠽⢺⣵⣿⢚⣷⢱⣶⣥⣄⢺⡗⣾⣯⡖⢿⠈⣿⠄⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠴⢼⣯⠝⠋⢿⣿⡹⣾⣇⣹⣯⠸⢩⡿⠛⠯⣽⡧⠧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⡐⢻⣟⠻⢂⣻⣮⠦⡻⢋⠿⣾⣟⡰⠟⣻⡿⢆⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣌⢩⡿⣏⣹⢃⣶⣽⣯⣶⡘⣏⣹⢿⡍⣡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣎⡲⠘⠚⣍⣻⣗⡜⢣⣺⢟⣩⠓⠃⢖⣱⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⢋⡯⣽⣮⡁⠤⠛⠺⢿⡿⠗⠛⠤⢈⣵⣯⢽⡉⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢯⣶⡿⣿⣍⡿⠁⠀⠈⠑⠆⠰⠊⠁⠀⠈⢿⣩⣿⣿⣶⡿⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡘⣿⡿⣯⣽⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⣯⣽⢿⣿⢃⣇⡇
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⢿⣿⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⡿⣫⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⢀⣀⣤⣩⣭⣭⣍⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣯⣭⣿⣿⣿⣿⡿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢛⣽⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡜⣿⣿⣿⣿⣿⣿⣣⣶⣿⣾⣿⣿⣿⣿⣿⣿⣆⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠙⣿⣟⣿⣿⣿⣿⣿⣿⣯⣿⣷⣿⣿⣿⣿⣿⡘⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠀⠀⠈⠛⢾⣿⣟⣟⣿⢿⡟⢿⣿⣟⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⠀⠀⠉⠙⠙⠻⠟⠿⠟⠛⢋⣭⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠲⢭⣷⢤⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣟⣴⣢⣌⣻⣡⣀⣀⣀⣴⣿⡇⢀⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⡻⡟⡏⣻⣟⣿⣿⣿⢦⣉⢻⣵⣿⡇
⠀⠀⠀⣀⣴⣾⣟⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡛⠊⢐⣀⠾⠣⠪⡡⡀⣉⣽⣶⣮⡙⣿⣧⣻⡿⠁
⠀⠀⣴⣋⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣍⠓⠶⢭⣌⠌⡡⢵⣪⣤⣭⡻⣿⣿⣿⡜⣿⣿⣧⣀
⠀⣰⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣆⣬⣉⠶⢿⣿⣿⣿⣿⣶⡽⡿⠃⣽⣿⢉⣿⢀⣀⣀⡀
⠀⠛⣿⣿⣿⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣤⣶⣭⣉⡙⠛⠻⠽⡿⢟⡽⢋⠏⠙⢛⣷⣦⡙⢿⣿⣿⠇⣰⡳⣾⡧⣿⢫⣩⠝⠁⠽⣦
⠀⣰⣿⣿⣷⡿⣿⣿⣿⣿⣄⠀⠀⠀⢀⣿⣿⣿⣿⣾⣛⣿⣭⠖⡰⢋⣀⣥⡶⢿⣿⢏⠈⠽⣮⡛⠁⠠⣧⣝⣾⣷⣿⣿⣿⠖⠀⠀⡈⢣
⠀⠿⣇⢉⣵⣾⢻⡟⣿⣿⢿⣧⡀⠀⣸⠛⠛⠛⠛⠛⠻⠟⠃⠸⢷⣟⡻⠁⠀⣸⡿⠂⠀⣀⡴⣯⡶⣦⠹⣿⣿⢿⣿⣿⣿⠦⠂⣈⠴⠋
⠀⠰⣿⣿⣿⣿⡏⣿⡟⣿⢷⣏⣿⠿⡟⠻⡿⠿⠋⠟⣠⡟⠛⢻⠶⣌⠙⢶⣠⡿⣃⡴⠞⠉⣾⣻⡷⢽⡇⢻⢯⣿⣿⡿⠷⠓⠉
⠀⠀⢻⠃⣡⣿⣿⣟⣿⣿⣸⣽⡟⣟⢉⠛⢈⠟⣸⣰⣉⣷⣴⣮⡔⢉⡹⢦⣌⠛⠉⡀⣐⠕⣼⣿⣧⡏⡵⣢⣾⣿⠁
⠀⠀⠀⠢⣿⡿⢛⣿⣿⣿⠿⣿⣧⡅⠤⢿⣭⠖⢁⠻⢥⣿⣿⣿⣿⣯⠶⣂⣿⢀⠸⣻⢧⣜⣩⡙⣿⢟⣴⢿⣝⣧
⠀⠀⠀⠀⠙⢝⣿⣿⢿⣻⣿⣿⣶⣲⣶⡶⣠⠾⢙⣩⠳⠬⠽⣿⣟⢿⢏⡟⣡⡿⣇⢻⣯⢗⣿⢟⣴⣟⣑⣿⡵⠏
⠀⠀⠀⠀⠀⠀⠉⠻⠙⣿⣯⣻⣿⣏⣿⣴⣷⣮⣭⣉⠀⡆⢛⣉⠿⢒⣿⣿⡿⣻⢿⠘⣟⣭⣾⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⣻⣯⣻⣷⣿⣞⡛⢿⣿⣿⣯⢁⠁⢠⣼⣿⡿⠟⣋⡵⢾⣯⣾⠿⣛⡽⠁
⠀⠀⠀⠀⠀⠀⠀⣀⢴⣛⢷⣼⣿⣍⣙⣩⢛⣿⣶⡞⣽⣿⣯⣤⣗⣭⣷⡞⠋⠭⣢⣾⣟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⣤⣉⣧⣾⣿⣿⣿⢯⡛⢀⣬⣿⢻⣸⠟⠹⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣟⢣⡀
⠀⠀⠀⠀⠀⠀⡶⠦⣿⣿⣿⣯⣿⣵⣿⣷⣿⡿⠚⠁⠀⠀⣥⣼⣿⣿⣻⣿⣯⣿⣿⣿⣿⣽⡝⡦⡀
⠀⠀⠀⠀⠀⠀⢁⠀⠈⠙⠹⠛⠛⣻⣿⠛⠁⠀⠀⠀⠀⠀⢛⣉⣿⣿⣿⣝⣿⣿⣿⣿⣾⡿⢿⣿⡟⣓⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⣯⣿⣿⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣷⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⢿⠻⣿⣿⡦⣿⠿⣷⣿⡿⢿⣿⣯⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⣷⣿⣿⡟⠿⣿⣿⣯⣭⢻⠿⢷⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠟⢿⣷⣿⣿⣛⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠻⠿⢿⣿⣦⣾⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⠄⡊⠶⠆⢹⣿⣿⣟⡓⢶⣦⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠖⠱⠀⠀⠀⡐⠏⢁⠘⣋⡁⢿⣿⣷⣤⣭⡉⣁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⣄⠴⠂⠈⠀⠀⠀⢀⠅⠀⠈⣀⡉⡁⠘⠚⢶⡾⣤⡿⣦⢁⡀⢽⠃⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣤⠖⠈⠁⠀⠀⠀⠀⠀⠀⡈⠀⠓⠀⠙⠋⠇⣶⣦⡀⢤⠁⠁⠈⢿⣧⣴⡆⠄⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣡⢤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠸⣷⢠⠙⣿⣿⣆⠁⠀⠀⣸⣿⣿⣆⠁⠲⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⢉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⠱⢈⠙⠛⡀⠀⠸⢃⣈⡛⢟⣧⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠁⣂⡀⠀⠀⢀⡀⣀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠃⠀⠁⠁⠀⠀⠀⠈⠛⠁⠘⡡⠀⠿⠹⠓⢀
⠀⠀⠀⠀⠂⠈⠡⠐⠀⠀⠐⣤⡛⣟⣧⡯⠐⠀⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠀⠀⠀⠐⣦⣿⣼⣿⡿⡏⠅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣄⠈⣿⣼⣿⣿⣏⠰⣐⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠟⠃⠈⠑⢶⣶⣿⣿⡽⠿⠷⠚⠛⠓⠂⠒⠒⠒⠋⠉⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢄⣀⡐⣰⡄⠀⠀⡀⠀⠀⠀⢰⣿⠸⢬⣛⣳⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⠀⠚⠉⢛⣾⡧⠀⣹⣽⣦⣺⣿⣶⠆⢾⠏⢷⠈⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣅⠀⢨⡿⣾⡦⠀⠀⢾⣿⢷⣿⡿⠿⠏⠛⠛⠁⠈⠑⠉⠐⢀⣈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠠⣿⠟⠀⢸⡻⡟⠀⣠⣦⣾⡿⠟⠻⡣⠀⢀⠀⠓⠊⢰⠞⠧⢶⣋⣉⠉⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠧⠂⣰⣇⣀⢔⣮⣿⣿⡿⠿⢟⣯⡴⠐⡀⣵⢳⡀⠀⠀⡤⡙⢂⠀⠀⠘⣿⠓⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣡⣵⣶⣷⣿⣿⡿⢿⣗⣫⡽⠿⡛⣉⠬⠓⠀⣰⣚⣿⣷⠓⠼⡰⠢⡽⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠿⣛⣿⣾⠶⠛⠻⠛⠉⠛⠀⠀⠂⠀⠀⠀⠰⢦⣾⡿⠿⠣⣖⣐⣉⡙⠀⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣙⠗⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠻⢿⠸⡵⣫⡝⣷⠋⠀⠀⠀⠀⠀⠀⢈⠎
⠀⠀⠀⠀⠀⠀⠀⢸⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⢾⡱⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣟⡷⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡶⠂⡀⠈⠑⠨⠭⡯⠋⠀⠀⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢊⣕⠙⢤⣄⣀⣀⣐⣫⠄⠁⠤⣄⡀⠈⢒⢿⢦⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡬⣀⠋⠀⠀⠀⠀⠉⢁⢈⠅⠀⠀⠄⠀⠉⢢⣰⠎⡛⠙⠉⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠗⠠⠤⠤⡄⡀⣀⠵⠂⠀⠐⣠⠀⠀⠀⠂⡊⡈⠛⠀⡈⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⡀⠀⠐⣋⠀⠈⢁⠀⢀⠔⣱⡧⠀⡄⣨⠞⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣣⠐⢰⠂⠀⢴⡄⠀⠀⢸⠅⠀⠀⠉⠤⠊⡁⠜⡈⠙⢇⠁⣧⢰⣟⣨⣌⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡧⢹⣴⢾⡀⢀⣀⣀⡀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠾⠁⠀⠘⣀⣡⠙⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠘⠭⣥⣤⣵⣭⣷⡭⢉⣽⣭⠑⠛⣁⠳⢄⠒⠒⠋⠀⢅⠀⠙⡤⡀⣽⠁
⠀⠀⠀⠀⠀⢀⡠⠚⡟⣆⠀⣾⡟⡷⡯⠍⠩⡛⠿⣿⣿⣿⣤⣤⣤⣀⣯⣃⡀⣨⠞⠁
⠀⠀⠀⠀⢠⢺⢒⡎⣶⣮⡠⢎⢿⡀⠀⠀⠀⠀⣃⣴⠋⡄⠀⠸⠟⣛⣉⠟⠉⠁
⠀⠀⠀⠀⠻⣝⢤⢒⣹⣾⡁⠈⢉⠀⠀⡉⠐⢠⣟⣹⣷⠁⠀⠂⡐⠩⣅⢤⣀⡀
⠀⠀⠀⠀⠀⢿⡚⣟⡿⠡⢁⠀⠀⢄⣄⣤⠖⠀⠬⠁⠉⠾⣷⣛⢦⣰⡍⠳⡇⢽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠘⠊⠰⠽⢄⣫⡨⠒⣼⡃
⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠁
⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣿⣿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣻⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⠟⢛⣽⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⣶⡿⢋⣀⣀⣤⣴⡲⣶⢿⠶⡶⢶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠁⣀⣿⣿⣶⢛⣿⢿⣿⣿⣟⣛⣾⣷⣤⣽⣤⣯⣿⣿⣿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣷⣿⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⢶⣞⣋⢻⣿⡖⢿⣿⣷⣄⠀⠀⠀⣀⣀⣀⣀⡀
⣠⠤⣾⣾⢳⣦⣷⠀⠀⠀⠀⣫⣿⢠⣿⣛⣿⣿⡅⢠⣯⣭⣿⣿⡏⠉⣿⠟⢷⡿⠿⢛⣶⣿⣿⣿⣮⣿⣿⣧⣨⡽⠷⠛⠋
⠀⢐⢉⣿⣚⣿⠃⠀⠀⠀⣴⣿⣿⣿⢿⣯⣖⣿⣿⣿⡟⢧⣴⣾⣿⣿⡿⢶⣞⣿⣷⣮⣠⣽⣿⣿⣿⡯⣀⡿⣿⡄
⠀⠀⠈⠙⠛⠿⠁⠈⠠⠀⣽⣿⣿⣇⣀⡻⢿⣿⣿⣿⣿⣶⣿⣵⠿⢿⣿⣿⢿⣿⣿⣿⡿⣏⡙⣻⡖⢢⣿⣿⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣳⣤⣋⠉⣻⣿⣟⠈⢹⣿⣿⣿⣿⣼⠉⣨⡟⠙⣯⣴⢿⢷⣿⣿⢿⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣵⣿⢻⣤⣔⣿⣿⣿⣿⣿⣛⣿⠏⣿⣷⣾⣿⣿⣿⣿⣧⣯⣬⣿⡾⢀⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⣀⠙⢿⣿⣷⣿⣾⢅⣼⡿⠁⣿⢻⣿⢇⣴⢸⣿⡿⠏⣿⣹⣿⣿⣿⠟⢿⣴⣿⣿⣯⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡜⢿⣿⡄⠙⠻⣿⣿⣿⣿⣧⣾⣿⣼⣾⣾⣷⣾⣿⣷⣶⣷⣿⣿⡿⠟⠀⠙⠛⠿⢿⣿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣿⣿⡄⠀⠀⠉⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣚⣋⣻⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⣠⠚⠛⠚⣿⡛⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠉⠀⢀⣠⣴⣥⣼⠶⠷⠐⠂⠀⠒⠚⠠⣤⣤⡀⠈⢁⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠅⣠⣤⣤⣴⠟⠛⡉⠁⣀⠀⠀⠀⠈⠁⠑⢄⠠⠈⢻⣝⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⢒⣭⠶⠛⠁⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠠⠠⡀⠠⠄⠐⢆⠀⠀⠀⠈⢷⣶⣶⠤⡀
⠀⠀⠀⠀⠀⠀⢀⣼⠉⠀⢀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⢻⣷⡆⠙⣆
⠀⠀⠀⠀⠀⢠⢊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠈⠂⢘⢿⡅⠀⠚
⠀⠀⠀⠀⠀⡈⠁⠉⠀⠀⠀⠀⢀⣀⠄⠂⠀⠀⠀⠀⠰⠒⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠒⠈⠘⢸⡁
⠀⠀⠀⠀⢨⠤⠠⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠑⠀⠀⠈⠉⢼⡛⣇⠀⢀
⠀⠀⠀⠀⢸⡆⢀⠁⠀⠀⠀⠀⠀⠀⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⡀⠄⢠⠾⣳⣇
⠀⠀⠀⠀⠈⡏⠈⢀⠀⠀⠀⠐⠀⠀⠀⠈⠩⠀⠃⠀⠀⠀⢀⠀⠀⠐⠂⠀⠈⠈⠆⢈⣧⣮⠐⡌⢸⡀
⠀⠀⠀⠀⠀⠸⡵⠊⠁⠁⠀⠀⠀⠀⠀⢀⡀⡀⠀⠠⠐⠂⠀⠑⠀⠠⠀⠀⢌⠀⣸⠇⢠⢠⠒⠁⡼⠀⢸
⠀⠀⠀⠀⢀⡠⣶⠀⡈⣄⡀⠀⠀⠀⣀⠀⢓⡸⠐⠐⠄⠀⠑⠠⠐⠆⠐⢫⡴⠮⠉⢄⡡⠃⠀⠀⢿⢠⠏
⠀⠀⠀⡠⠫⢈⢿⣿⠉⠀⢉⠂⠰⣆⢈⣈⣤⣤⣑⣂⣔⣀⣈⡥⠜⠛⢙⢫⠄⢂⠔⠁⠀⠀⠀⠀⠘⠁
⠀⢠⠊⠑⣠⣆⠯⠋⣄⡠⠋⠀⣙⢸⣣⣤⣍⣀⣯⠀⡈⢋⣁⣿⣿⡻⠔⠊⠁
⢰⣣⠤⠒⠁⠀⠀⠀⠀⠈⠚⠉⠁⠉⠉⢉⣟⠿⠿⡿⢿⡏
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⠀⠷⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⡠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠀⡎
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣞⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢨⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⠀⣀⡀⢀⢀⣠⡟⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⢸⣄⡀⠀⣀⡤⣲⡞⠋⠀⠉⠙⠲⣿⠞⠉⠁⠙⠢⢀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⢀⣤⣴⣦⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⣴⡳⠋⠉⠉⠛⠻⣿⠋⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠑⢦⣄⡀⣨⠇⠠⠰⢤⠔⠁⠀⣠⣴⣾⣿⣿⣇⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠒⠒⠒⠛⠀⡀⢸⣿⠟⢿⣿⣿⣿⣿⣿⣿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⢄⣾⣿⠿⠻⠿⠿⠿⣿⣿⣾⣿⣄⣀⣿⣿⡿⣟⣻⣿⠟
⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣿⣎⣂⣀⣢⣭⣷⣿⣿⣯⣀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⣿⡿⠾⣛⠝
⠀⠀⠀⠀⠀⠀⢸⢠⡁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⣿⣄⠠⠤⠤⢖⣡⣾⣿⣿⣿⡿⠿⠛⠛⢿⡏⠀⠉⠛⠶⣤⣠⣺⣿⡯⠤⠤⠤⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠈⣻⣇⣠⣀⣸⣿⣄⣀⠤⠤⢐⣨⣴⣿⣿⣿⣿⣿⣿⣿⠛⠉⢹⣿⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢀⣿⣿⣶⣆⣰⡄
⠀⠀⠀⠀⠀⢠⠀⣿⠈⣿⠛⠋⣿⠿⠿⣿⣿⡿⠟⠛⣿⠏⠉⠉⠉⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡄⣀⡴⣡⣾⣿⣿⣿⣿⣿⣿⡟⠄
⠀⠀⠀⠀⠀⢸⠀⡿⠀⡿⠀⢸⡏⠀⠀⢸⡏⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⣧⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⢀⣴⣾⣿⣇⢀⡇⠀⣸⠃⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡇⠀⠀⠀⠘⡇⣤⢖⣫⣴⣿⣿⡿⠟⠛⡛⢿⣿⣿⣿⣿⣿⢯⠹
⠀⠀⠀⣿⣿⣿⣿⡿⠛⠇⢠⣿⢀⠀⢰⡇⠀⠀⠀⢸⣇⣀⣀⠀⣀⣸⣇⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⡶⠿⠋⠁⠈⣿⣿⣿⣿⣿⢯⣀
⠀⠀⠀⠉⠛⠛⠛⠀⠀⢀⣾⣿⣿⣿⣿⣤⣿⣷⣶⣿⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣧⠱⠆
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣹⡟⠻
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇⣼⣀
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠹⠏⢻⡍⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠄⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢠⢼⡑⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⢠⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣰⠈⠱⠀⠀⠀⠐⠐⠀⢀⠀⢸⠀⠐⢎⢥⢈⠀⠀⠀⠀⠁⠉⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠢⠀⡒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⡁⠂⠀⢰⠎⠁⠤
⠀⠀⠀⠀⠀⢠⠁⢀⠀⡆⠀⠀⢀⠀⠀⠀⢀⠀⣤⠄⠀⠁⠀⠀⠖⠠⠀⠀⠀⢰⡀⣀⠄⣀⠒⠬⢠⢀⠾⣈⠀⠀⢀⡀⠀⠀⢀⡤⣀
⠀⠀⠀⠀⠀⡀⠈⠀⡅⡗⢀⣆⡀⢠⡴⠊⠀⡿⠛⠀⠀⠀⣴⡽⠯⠀⠀⢀⠀⠘⠁⣥⡀⠘⠃⠂⠈⠀⠈⢠⣤⣴⢩⣂⣀⣤⡿⢏⣿
⠀⠀⠀⠀⠀⡘⢀⣀⢠⢟⠞⣡⣾⣿⣟⣷⢿⠿⣿⣦⠠⢞⢁⡘⠁⢠⠀⠀⢀⠀⣀⣄⢾⣆⡚⢠⣶⣬⣷⣶⣬⣿⣿⣿⣿⣷⡾⠿⠏
⠀⠀⠀⠀⠘⢹⣱⣫⠇⢀⣾⣿⡿⣷⣿⣾⣋⣴⣾⣟⣿⣤⠀⣤⣢⣤⡄⣀⢈⣸⡎⠈⣷⣞⣿⠛⢧⠈⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠈⠉⠉⡀⠀⢠⣿⣿⣿⣹⣿⣿⣿⣿⣿⢿⣷⣿⣷⣾⣾⣷⣿⣷⣿⣴⣿⣀⣠⣿⠋⠿⢷⣶⣿⣿⡟⢁⣀
⠀⠀⠀⠀⠉⠁⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⢿⣿⣷⣦⢀⣿⣿⠄⡐⣦⡉⠁⠈⠀⢠⡰⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⢿⣿⣿⣿⣿⣿⣥⡈⣧⠀⠀⠀⠂⢑⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠵⠟⠋⠉⠙⢿⣿⣿⣿⣩⢆⠀⠐⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣤⣇
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡾⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠁
⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⣯⣭⡟⣿⣿⡿⠿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡦⠶⢶⣤⣤⣤⣼⣟⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣠⣴⣾⣾⣿⣭⣽⣔⢻⣿⣿⣿⣜⢿⣿⣧⣵⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⠾⣿⣛⣏⣾⣿⣿⣿⣿⣷⡝⣿⣿⣿⣷⣝⢿⣿⣳⢏⣴⣶⣶⣶
⠀⠀⠀⠀⢀⡤⢲⣦⢶⡴⡷⣳⣷⣞⠍⠽⠉⣩⣉⣉⡙⠛⢾⠿⣭⣭⣭⣯⠶⠟⠁⠙⠻⠟⠛⠁
⠀⠀⠀⠸⣗⣾⡽⠿⢨⣧⢿⡻⠉⠈⠀⢰⣼⣫⡥⠗⣋⣈⠉
⠀⠀⠀⠈⠹⣮⠾⠯⠾⢃⠣⠀⠀⠀⠀⠲⣧⢺⣳⣇⣩⡝⣤
⠀⠀⠀⠐⠒⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠸⠇⣷⣆⣿⣾⣶⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠛⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⣤⡄⢀⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣩⢾⢛⢓⠚⢠⣽⣬⣛⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣤⣄⠀⣿⣵⣦⣤⢴⣄⠉⣻⣿⡷⣿⡉⢰⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⡚⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣸⠁⠉⠣⡙⠟⠿⣿⡿⢟⣱⡿⢯⢁⢛⣃⣿⡿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣷⢉⡛⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠂⢩⣭⠅⠀⣠⣈⡛⢢⡐⢶⠹⠷⢩⡷⢤⡘⢏⣨⣿⠨⠽⢿⣷⡄⠀⠀⠀⠀⣠⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠉⠲⠴⢄⣀⡊⡳⢍⢇⠪⠱⠐⠠⣀⣹⣿⣿⣿⣦⡀⢠⣴⣿⣿⣆⢻⡻⣿⣿⣿⣿⣿⣿⢸⣟⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣤⣶⢧⣾⣿⣽⣿⣿⣿⣿⣿⡎⣿⣿⡿⠿⠈⠻⢿⡿⠿⠿⣿⣿⠇⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣐⣠⣽⣿⣷⣞⣻⣏⣫⣤⣿⣿⣿⣿⠇⠸⢣⣶⣶⣿⣶⣶⣶⡆⣶⣶⣶⣤⡙⢿⣿⣿⣬⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣺⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣦⠉⠛⠿⣟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡤⠝⣽⣛⣭⡽⢶⣿⣿⣿⣿⣿⡿⢏⣵⠟⣿⡿⣿⡘⣆⠻⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⠏⣼⣿⣶⣮⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣿⣿⡟⠛⣯⣤⣤⣿⣿⣿⣽⣷⣏⣡⣿⢟⢼⣿⣿⢟⣵⢌⠐⡘⢿⣿⣿⣿⠇⢸⣿⣿⣿⣿⡟⣸⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣏⢀⣀⡻⣏⠙⢳⣾⣉⡙⣻⣿⣿⠿⠛⠛⠹⣏⢸⡟⣿⡟⠘⣁⡤⣱⡴⢐⠨⣻⠋⠠⣿⣿⣿⣿⣿⢡⣿⣯⣘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠹⠻⣿⣿⡿⠿⣟⣽⣿⢻⣿⣿⠃⠀⠀⠀⠀⢹⣟⣹⡻⢨⣴⣿⣿⣴⣟⣦⣅⣚⣳⢦⡌⣙⠻⢿⡇⣾⣿⣾⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢻⣦⣚⣼⣿⣧⣾⣿⣿⣷⣿⣿⠏⠀⠀⠀⠀⠀⠘⣟⣷⣥⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣆⡄⠉⠛⠻⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣟⣋⣹⣿⣿⣿⣿⣟⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣉⠙⠻⢿⡿⠿⢛⠉⠉⠛⢻⣯⣻⣿⣿⣿⣣⣌⣀⠀⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣆⣨⣿⣿⣭⣭⡿⢿⣿⣷⣶⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣂⠚⠻⠲⠝⠀⣤⣠⣈⢹⣷⣿⣿⣿⣿⣝⡛⡿⣾⢶⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠛⣿⣆⣰⣿⣡⣾⣬⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠡⠽⢞⣭⣯⣄⢀⠠⣋⢾⣷⣿⣿⣿⣿⠿⣿⠿⠷⢟⠉⢽
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣰⡟⠛⠛⠿⢿⣿⡟⣧⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠚⣛⣆⡶⢶⣶⠄⠠⠌⣸⣿⣿⣿⡿⢿⣾⣿⣒⣈⣾⣻⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣹⠦⣄⣀⣈⣹⡿⣮⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢟⣛⣋⣩⣥⡝⣙⠋⡀⢦⣶⣿⣿⣯⡉⠀⠂⠀⠩⠈⠁⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣧⣀⣀⣠⣴⠞⠉⠉⠛⢿⠞⠉⠉⠂⢀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣤⣦⣤⣶
⠀⠀⠀⠀⠀⠀⠠⠁⣀⡼⠋⠁⠈⠉⠻⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠙⢦⣤⣞⣀⣈⡖⠁⢠⣴⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢢⠚⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⢀⢸⣿⣧⣤⣤⣴⣶⣼⣏⠈⢻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣸⣖⣀⣀⣼⣵⣿⣯⡀⠀⠀⠈⣿⣿⠻⣿⣿⡿⠞⠛⠁
⠀⠀⠀⠀⢀⣠⡄⠀⠀⡇⠀⠀⠀⠀⠀⣿⣤⣐⣒⣪⣾⡿⢿⡿⠛⠛⠛⣏⠀⠉⠓⠦⣦⡿⠗⠒⠒⠂
⠀⠀⠀⠀⠀⢸⠳⣶⠶⣿⣶⣬⣬⣴⡾⣿⡿⠻⠿⣿⠋⠀⢸⡇⠀⠀⠀⢹⡀⢀⡀⣸⣿⣿⣷⣿⣤
⠀⠀⠀⠀⠄⣼⢀⡇⢀⡏⠀⢨⡟⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣧⠀⠀⠀⠀⣇⣯⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣴⣿⣯⣸⠃⢸⠇⠀⣸⠁⠀⠀⡇⠀⠀⠀⢿⠀⠀⠀⢸⠰⣺⣵⣿⣿⢛⣛⡙⢿⣿⣿⣿⡏⠉
⠀⠀⠻⣿⣿⡟⠈⢁⣾⣤⣤⣏⣤⣀⣠⣗⣶⣦⣤⣾⣼⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⢸⣿⣿⣿⣟⠦
⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⡿⠋⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡛⡄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣹⠟⡃
⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠷⠧⠄
⠀⠀⠀⠀⠙⢻⡟⣿⠻




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡀⠀⡉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠆⠀⠀⠀⠰⢟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣠⣆⣄⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢉⡻⢤⠤⠤⠤⢿⣒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠈⠉⢒⡶⠛⣴⠋⠀⠀⠓⠂⠀⠀⢈⡟⢦⢹⣖⢖⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠕⠀⠀⡄⢼⢀⡾⠉⠓⠶⠤⣤⣦⣶⣒⠋⢀⠞⢿⠙⣆⡉⠀⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠳⠤⠞⠋⢩⠋⣿⠋⠛⢢⣴⡏⠁⠀⠀⠈⢳⣼⠀⠈⣇⡸⠉⢧⣑⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡧⢆⡀⠀⡗⡇⠀⣶⡷⠀⢸⣿⠀⣠⡏⢱⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣟⠠⠀⢀⡏⣗⠀⠙⠀⠀⣼⣯⠀⢩⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡀⢀⡼⠁⠸⢤⣄⣀⠤⠏⠹⣄⡼⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡖⢳⡋⢀⠔⠚⠙⠉⠉⠛⠛⢲⡸⠁⡼⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠊⢳⡀⠙⣯⣀⡀⢠⢆⡒⣀⡤⢾⣡⠞⠁⠠⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠂⠀⢀⠟⠒⠳⢌⣉⣉⣹⣉⠥⠴⠚⠉⠲⡀⠀⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠄⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠠⢁⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁


⣏⣈⣯⣿⣿⣯⣿⣽⣽⣿⣏⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡻⣿⣿⣿⣿⡙⣿⣶⣶⣤⣄⣀
⣼⣷⣝⠟⣽⣶⡽⢧⡟⢿⡆⡀⠀⠀⠀⢀⣠⣴⣾⣿⣿⢆⣙⡻⣿⣿⣮⢻⣿⠿⣃⡹⣿⣿⣿⣎⢿⣿⣶⣤⡀
⢿⣿⣿⣦⡻⢟⣥⠿⢷⠎⢿⣣⣤⣤⣄⠿⠿⣿⡿⠟⣵⣿⣿⣿⣶⣝⡻⠗⣰⣾⣿⣿⣮⡻⠿⣣⣦⡻⣿⣿⡻⣷⣤⡀
⢳⣍⠛⠛⠳⠮⡍⠺⠊⠩⠇⠭⣿⣿⣿⢸⣿⣿⣿⣷⣌⢻⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣮⡙⣯⣜⡻⠿⣶⣄
⠀⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠻⠹⣿⣦⣍⣙⠻⣿⣿⠷⠙⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣧⢿⣿⣿⣶⣝⢿⢟⣶⣦⣄
⠀⠀⠀⠀⠉⠙⠲⠦⠤⣤⣤⣀⠀⠀⣹⣿⣿⣿⣷⣄⢱⣾⣿⣮⡻⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡘⣿⠿⣛⣡⢾⣿⡿⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣷⣤⣿⣿⣿⣿⣿⣿⣆⠹⣿⣿⣿⣾⣍⡸⢿⣿⣿⣿⣿⣿⡇⠸⠿⢟⣛⣋⢱⣶⣿⣿⡿⣣⣶⣾⣿⣷⣶⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢻⣝⣏⣿⢿⣿⣯⡗⠆⠈⠛⢿⣿⣿⣿⣷⢰⣶⣶⣶⣶⣆⢻⣿⣿⣿⣿⠗⣈⣭⠵⠺⣿⣿⣿⣿⣯⠿⣿⢎⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣿⣾⣿⡖⣟⣿⡿⢮⣔⡀⠀⠀⠈⠙⠛⠋⠘⠿⠿⠿⠟⣃⣀⣩⣭⣥⣶⣾⡉⠀⠀⠀⠙⢿⣿⣿⡟⣻⣴⡟⣻⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⠾⣹⡿⢴⠯⣷⡇⣀⣀⢀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⠿⠟⠻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠉⠻⠿⢩⣷⣶⡜⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡺⣿⢿⣼⣓⣺⠿⢎⡛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⢾⣻⠿⣭⡹⣶⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣛⣿⢛⣷⡾⠟⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣯⠿⣻⢧⣾⠷⢉⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠛⣿⠷⡞⣛⣭⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠺⣶⢛⣌⣻⣅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣴⣴⠄⣀⣀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⣠⡻⠫⠙⠋⢠⣷⣬⣛⢓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠀⡴⠶⡄⢿⣾⣶⣖⣾⣦⢩⣿⣿⢺⡧⢸⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣶⣾⣿⣙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⠴⡅⠀⠈⠨⣉⠛⢛⢛⣥⣿⣭⠦⢠⣦⠾⣿⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⢇⣬⣍⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠒⠲⢦⣙⠌⠃⠘⣥⠮⣭⢂⢽⡒⠻⠈⢯⣽⡿⣆⠀⠀⠀⣰⣶⡜⢿⣿⣿⣿⣿⣿⣿⡟⣵⣿⣿⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠚⠙⠈⠂⣉⣥⣈⣶⣾⣿⣷⣿⣷⣄⢾⣿⣿⣿⡸⣯⣿⣿⣿⣿⣿⣷⢹⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣾⣿⣿⠿⡿⠟⢛⣿⣿⣿⣿⠘⡿⣋⣉⣤⣌⣭⣭⣉⣉⣙⣋⠘⢿⣿⣿⡝⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣳⣶⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣏⠀⣶⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣆⠛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣌⠇⣿⢜⣛⣯⣿⣿⣿⣿⣿⣿⡿⣭⣿⣿⣿⡏⢆⠻⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⢡⣶⣤⣙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣷⡶⢻⣏⣁⣰⣿⣿⣿⣿⡟⣣⣾⣟⣼⣿⣹⢟⣎⡓⠜⢿⣿⣿⣿⠇⢸⣿⣿⣿⣿⢇⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠀⢈⢻⡏⠛⢦⣿⢙⠻⣿⣿⡿⠿⠟⠻⣟⢲⣿⣿⠇⠛⣠⢑⣬⢄⠽⡿⠏⢠⣿⣿⣿⣿⡟⣼⣿⢝⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠙⠟⣿⣿⡿⢟⣯⣿⡿⣿⣿⡟⠀⠀⠀⠀⢻⣟⣿⡟⣨⣾⣿⣯⣾⣷⣅⣘⣳⢦⡍⡙⠻⣿⢡⣿⣷⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣄⢸⣧⣼⣽⣿⣧⡿⣿⣿⢶⣿⡟⠀⠀⠀⠀⠀⠘⣟⡿⣬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡌⠉⡛⠷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣿⣉⣽⣿⣿⣿⡿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠙⠻⣿⡛⢫⠉⠀⠙⢿⣯⡿⣿⣿⣿⣤⣤⣀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⣼⣴⣿⡿⠦⣶⡟⣻⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡜⣙⣛⠊⠀⠶⣤⢀⣹⣿⣟⣿⣿⣿⣍⣛⢻⣿⣯
⠀⠀⠀⠀⠀⠀⠀⣷⠉⢹⣧⣼⣿⣾⣿⣦⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢒⣒⡛⠵⠾⠶⠤⠜⢧⢻⣿⣿⣿⣿⡟⣻⡟⠛⢻⣅⣻
⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⡉⠉⠙⢻⣿⣯⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣭⣭⣽⡮⢻⣿⣃⠀⡂⢼⣿⣿⡿⠟⠛⠿⠷⠦⠿⢼⣻
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⣯⣒⣶⠶⠾⢛⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣥⠤⠶⢒⣛⠠⣁⣂⣹⣿⣿⣿⣷⣤⡀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣖⢲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣳⣦⡄⠀⠀⢀⣀⣤⠀⣀⣤⡤⣤⠤⠤⢤⣤⣤⣄⣀⡀⠀⠀⠀⠈⣽⠻⣴⢛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣏⡷⣯⠾⣮⠔⠋⠉⢀⠜⠉⠀⠀⠀⡤⠚⠉⠀⠀⠀⡡⠼⠿⠶⠄⠀⣸⢾⣽⢿⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠉⣙⠁⠀⢀⡀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⡴⠖⠿⢾⣹⣾⣏⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⢁⠀⣠⠖⠉⠈⢀⡤⠖⠒⢆⠀⠀⢀⣠⠀⠀⠀⠀⠀⠰⠊⠀⠀⣀⠤⠚⣶⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠬⡄⠞⠁⠀⠀⠠⠋⠀⠀⠀⠀⠀⠞⠉⠀⠑⠀⢀⡴⠃⠀⢠⠀⡜⣠⠞⠿⠚⠁
⠀⠀⠀⠀⣀⣠⡤⣤⣆⢀⣣⡀⠠⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⡀⠀⢀⢀⡟⣰⣾⣷⠄⠀⣠⠀⡀
⠀⢀⣴⣻⢿⣻⢏⣳⣟⣯⣤⣙⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣾⠐⠛⠛⠃⢠⡦⢞⡖⢻⣁⡠
⠠⣯⢾⣽⢾⣹⢾⡵⢯⡷⡿⠛⠦⣄⡵⠦⢄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠵⣫⠃⠀⠀⠀⠀⠀⠓⠚⠙⠋⠋⠙
⠀⠈⠈⠉⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠙⠢⠤⣏⡀⠑⣤⢀⣀⣀⠠⠄⠀⣀⡤⣺⡷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠛⠒⠒⠒⠒⣺⣿⣛⣶⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⣷⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣹⣾⣻⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣿⣻⢿⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣿⣻⣿⣋⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣶⡿⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⠷⠛⠉

⠀⠀⠀⠀⠀⢀⣀⢀⣀⡀⠤⠒⠂⠈⠉⠑⠖⠈⠉⠉⠑⠢⢄
⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⣄⠀⠑⣄
⠀⠀⢰⢃⡤⠖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠂⠀⠀⠪⡱⡀⠈⢆
⠀⢀⠃⣨⠖⣉⠒⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣾⣿⣿⡆⠐⣥⠀⠸⡄
⠀⡘⢸⠃⣾⣿⡿⠊⠉⠉⠉⠉⠁⠂⠀⠀⠸⠘⣿⣿⣿⡏⢀⣿⠀⠀⠘⡄
⢠⠃⢸⡀⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠌⠉⠭⠂⠁⠀⠀⠀⠀⠸
⡆⠀⠈⠁⠂⠀⠂⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡸
⠰⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⠀⡰⠁
⠀⠙⢆⡀⠑⠠⢀⣀⠀⠀⠀⣀⡀⠤⠀⢂⠞⠃⠁⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠉⠢⢄⡀⠀⠀⠈⠛⠦⠤⠄⠂⢁⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⣤⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠐⠀⠀⠀⠀⠀⠀⠀⢧⢀⡠⠴⠒⠛⠉⠉⠉⠉⠉⠫⡉⠘⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢱⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠒⠀⠤⠀⡠⠛⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡃⠀⢀⣀⡀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢰⡏⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⣀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢪⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⡞⠀⠀⠐⠢⢀⠀⡠⠄⢀⡀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠁⠀⠊⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡼⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡈⠀⠑⠄⣀⠀⢀⣀⡀⠄⠒⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠈⠙⠲⣤⠐⠘⠢⣀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠅⠈⠁⠉⠲⢤⠋⠙⢦⡀⠀⠉⠂⢄⠤⠒⠁⠀⠀⠈⠉⠐⠀⠌⠀⠀⠀⠀⠀⠀⠀⡛⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⠈⠑⠆⠠⠤⠤⠐⠊⠡⠂⠀⠀⠀⠀⠀⠀⠙⣄⠀⠙⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣀⣴⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢣⡀⠰⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⡸⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠖⠖⠂⠉⠁⢀⣁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠱⣄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡀⠈⠛⠤⠤⢤⢠⣤⠤⠖⠚⠉⠀⣀⣤⠜⠒⠛⠉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠉⠒⠤⣀⣀⣀⣀⣄⣀⠤⠔⣒⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⢑⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠁⠀⣀⣠⡤⠖⠛⠣⢤⣀
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠶⠶⠖⠒⠛⠚⠉⡿⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠰⣀⣀⡀⠀⠀⠀⠀⠀⢀⡤⠊⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⣀⡀⠀⠀⣠⠞⠁⢧⠀⠀⠀⠀⠀⠀⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠘⢄⣀⣢⡤⠖⠋⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠛⠋⠀⠀⠀⠘⢆⠀⢀⣤⣈⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡎⠀⠀⣀⡉⠀⠁⠀⠀⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠁⠀⠀⠐⠍⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠰⣀⠀⠠⠀⢀⠀⠀⠀⠀⠁⠘⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢢⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⡀⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠐⠁⠀⠈⠐⡘⡀⠀⠈⡄⠀⠀⠈⢆⠀⠀⠀⠀⢾⡗⡃⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠡⠱⠀⠀⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠆⠀⠀⠐⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⢡⠀⢃⠀⠀⠀⠀⠀⢀⠈⠘⠉
⠀⠀⠀⠀⠀⠀⡄⠁⠠⡀⠈⠢⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⢡⠀⠀⣠⠷⠚⠁⠀⠀⠀⠌⠉⠁⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⡈⠀⠀⠀⡐⠀⠀⠰⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⢀
⠀⠀⡖⠀⠐⢇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠰⠀⡀⠀⠀⠀⠀⢀⠀⢢
⠀⠀⠈⠒⠄⢀⣿⡶⣦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢺⡃⠀⠑⢆⠒⠂⠭⠈⢀⠃
⠀⠀⠀⠀⠀⠀⠠⠀⠈⢻⡉⠁⠒⠀⠀⠠⠤⠠⠀⠐⠐⠀⠁⠀⡠⠃⠀⠐⠒⡧⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠱⡄⠀⣄⠀⠀⠐⢀⣀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⢇⠀⠀⠫⠄⠠⠎⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠊⠂⠁
⠀⠀⠀⠀⠀⠁⠂⠠⠤⠌⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⡄⣂⢉⠁



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

⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⠉⠡⠭⠈

⠀⠀⠀⠀⠤
⠀⣠⣴⡶⠳⣤⣤⡀
⣴⠟⢿⣷⣿⣿⠟⡿⣷⢤⠀⢀⣠⣤⣀⠐⠒⠆⠤⣀⣤⣄⣀⡀
⡆⢴⣿⣟⠉⠁⠚⠹⠳⢾⠙⠉⢿⣿⢿⡖⠺⣦⣤⡼⠟⠙⣿⣿⣿⣖⣦⣀
⡇⠀⠉⠛⠿⣿⣯⣶⣾⣟⠀⠀⠺⠀⣠⡀⠀⠙⢻⣿⣿⣉⠀⠿⣿⣿⣯⠉⠤⣷⣤⣄
⠀⠀⠀⠀⢠⣀⣈⣿⣿⣿⠆⢀⠄⠀⠀⠀⠁⠀⠸⢅⠈⠙⢷⣤⡁⠀⠘⣾⠀⠀⠦⠛⠛⢦⡀
⠀⠀⠀⠀⢻⣿⣿⣿⡟⠘⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡸⠿⠻⢿⣿⣿⣿⣄⣀⡀⣤⣆⢀⣀⣀
⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠈⠀⢀⣀⠘⢿⣯⣷⣦⣤⣀⠀⠀⢲⣶⣦⣬⣭⣉⣙⡛⢻⠿⠿⣿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣽⣿⣿⣿⣷⣶⣅⠟⠻⠿⠛⠿⠿⠛⠛⠛⠁⠲⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⣶⣿⣿⣷⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⣄⣤⣀⣀⠀⠄⠀⠀⠐⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣲⣿⣿⣶⣿⠿⠗⠛⠛⠛⡛⠄⠀⠀⠀⠐⠶⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠛⠋⠉⠉⠉⠡⠀⠄⠂⠁⠂⠀⠠⠁⠀⠀⠀⣀⠲⡖⣭⠨⠰⣿⢯⠓
⠀⠀⠀⠀⠀⠠⠐⠡⢀⡨⠒⡈⠀⠠⠀⠀⠀⠀⠢⡀⠄⢀⠀⠀⠀⠀⢢⣌⠋⠀⠠⠘⠣⡈⠀⠀⣀
⠀⠀⠀⠀⠐⠁⠐⠠⣌⠑⢄⠈⠂⠀⠑⠀⠀⠀⠀⠀⠢⢀⢑⠸⠄⠓⠈⠀⠠⠀⠀⠂⠐⠀⠴⠂
⠀⡄⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠄⠊⠑⠠⢄
⠀⠀⠁⠐⠒⠀⣠⠊⠀⠂⠤⠤⠤⠤⠤⠀⠀⠀⠀⢀⣀⠀⠀⢀⡁⠀⠊⠁⠀⠄⢀⠀⠈⠵⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠑⠆⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣸⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣶⣦⣦⣽⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⣸⣿⣿⣿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣷⣌⢿⣿⣷⠀⣠⣿⣠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⡏⠻⢿⣿⣿⣿⣿⡿⣇⣼⣿⣿⣿⠟⢻⣿⡆
⠀⠀⠀⠀⠀⠀⣠⣞⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢿⣿⣿⣿⡟⣮⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⡿⠛⠛⠻⡟⢁⣾⣄⣿⣿⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠘⢿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⣶⣿⣿⣧⢸⣿⣿⣿⣿⡇
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣠⣿⣿⣿⣿⡛⢿⡟⡛⣿⣯⣭⣿⣿⣿⣿⣿⢻⢻⡏⣥⣹⣷
⠀⠀⣼⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣯⣥⣤⣿⣿⣿⣿⣿⣿⣶⣿⢿⣿⠏⢌⣿⣾⣧⣿⣿⣿⢻⣷⣦
⠀⢸⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠛⢛⣿⣿⣿⣿⣿⠻⠛⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣚⡻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⣴⣿⣿⣟⣹⣷⣾⣿⣿⣷⡾⠿⣿⣾⣿⣿⣿⣿⠟⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣍⣉⡛⣿⣿⡿⢻⣿⣶⣿⣿⣧⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⣿⣿⡟⠁

⠀⣤⣤⣤⣤⣤⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣟⢿⣿⣿⣿⣻⣿⣶⣦⣤⣀⡀
⣾⣷⣝⢻⣷⣮⢳⡟⢻⣂⡀⠀⠀⢀⣤⣶⣿⣿⡟⣨⣝⡻⣿⣷⡻⠿⣟⣥⡻⣿⣿⣧⠻⣿⣷⣦⣀
⡻⢿⣿⣾⡛⢷⢛⡻⡧⡫⣷⣶⣶⡘⣻⣿⣿⡉⢾⣿⣿⣿⣷⣝⢋⣾⣿⣿⣿⣮⠛⣵⣷⣽⠿⣿⡻⣷⣦⡀
⠙⢦⣄⠉⠉⠓⠈⠁⠐⠁⢻⣿⣿⣇⠿⣿⣿⣿⣮⡻⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡆⣿⣿⣿⣷⣰⣶⣮⣟⢿⣷⣄⣀
⠀⠀⠉⠻⠦⣄⣀⣀⡀⠀⠀⠀⠹⣿⣷⣶⣌⡛⢛⣥⣜⢿⣿⣿⢸⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣇⣿⣿⣿⠷⣹⣽⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠈⠉⠙⢛⣲⣄⣰⣿⣿⣿⣿⣷⡙⢿⣿⣷⣽⡿⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⠿⠿⣨⣭⣶⣿⠏⣫⣵⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣟⣿⣿⣿⣿⣟⠯⠈⠻⢿⣿⣿⣷⡎⣭⣭⣭⣭⣯⢰⣶⣾⣿⣿⠍⢛⣻⠵⣾⣿⣿⣿⣿⣿⡾⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⣿⣯⣾⣿⡿⣕⡂⠀⠀⠉⠛⠛⠃⠸⠿⠿⠿⠟⣈⣉⣩⣭⣴⡞⠉⠀⠀⠹⣿⣿⡿⢗⣮⣾⢿⣙
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡽⣯⣿⢷⡏⣷⡇⣀⣈⣀⣀⣀⣀⣠⣤⣤⣶⣶⣿⣿⠿⠟⠻⣿⣿⣿⡄⠀⠀⠀⠀⠉⠿⠏⣿⣶⣎⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢿⣟⣯⣻⣼⠿⣊⠙⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣞⣿⢟⣷⡽⢿⣻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡯⣿⣯⡿⢫⣹⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠈⣽⣽⣟⣭⡼⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠙⢽⣵⡖⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠿⢿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⣹⣇⠀⠀⠀⠀⠀⠀⠀⠱⣈⣷⡦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠿⣄⣀⣠⠂⠉⠉⠉⠙⠉⢲⣶⠒⠈⠉⠁⠀⠈⠀⢌⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠉⠉⠿⢦⣀⠀⠀⠀⠔⢰⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⡀⡓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⡤⠄⠀⠀⠀⠀⠀⡠⠞⠋⠙⠢⣤⡴⠟⠚⠓⠒⠒⢶⡖⠅⠀⠘⠟⣧⠀⠘⡄
⠀⠀⠀⠀⠀⣀⠀⠠⠄⠀⠚⠀⠛⠶⣶⡖⠛⠁⠀⠀⢀⢀⡠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠈⠄⢠⡌⠁⠅⠀⠀⠀⠀⠈⠳⣀⠀⠷⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢗⠀⠀⠀⠀⠀⢰⡀⠀⣇
⠀⠀⠀⠦⣤⢤⣄⣉⣿⠈⠀⠄⡀⠀⡀⠀⠙⢶⡆⠀⠈⠓⣤⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⠄⣨⣿⠻⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠧⠾⠗⠛⡻⠿⣧⣤⡝⢦⣀⣠⡈⠉⠉⠉⠓⣦⡼⢿⣛⠀⠀⢀⠠⢦⣴⠿⠷⠒⢾⢞⠋⢀⣀⣴⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠜⠋⠀⠀⠉⠿⣷⠶⠤⣤⣤⣳⡀⠀⠀⣸⡇⢀⠀⣸⣧⣤⣤⣤⣴⡿⠖⠋⠉⠙⠳⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠎⢰⠇⠀⠀⠀⠀⠀⠴⠿⠒⠀⠀⠀⠘⠋⠉⠉⠉⠉⠉⠉⠉⠀⢀⡤⢐⠛⠓⢠⣐⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠓⠀⠂⠁⠀⠀⠈⠝⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠾⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠢⡀



⣶⣶⣶⣶⣶⢶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣿⣵⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡠⠒⠊⠉⠉⠑⠢⡀⠀⠀⢀⠄⠒⠀⠀⠐⠢⣄⠀⠀⠀⢀⡠⠒⠒⠒⠲⢄⠀⠀⠀⠀⡴⠊⠉⠉⠙⠲⢄
⡜⠀⣀⠀⠀⠀⡄⠀⢹⠀⠰⠁⠀⠀⠀⢀⠀⠀⠈⢂⠀⢠⠊⢀⠀⠀⠀⣀⡀⢳⡀⠀⡜⠀⠀⠀⠀⠀⠀⠈⢳
⡇⢀⠉⠀⠀⠀⠀⡄⢸⠀⡇⠀⡀⠘⠃⠙⠀⢤⠀⢸⠀⠈⠀⠉⠁⣀⣀⣀⣀⠀⡇⠀⡇⠰⡀⣤⠀⠶⢈⡇⠈
⠱⡀⠓⠤⠤⠤⠊⢀⠜⠀⠰⡈⠃⠂⠔⠢⠒⠊⠀⠌⠀⠸⡄⠻⣽⡯⣇⣳⠟⢀⠇⠀⠱⡀⠑⠤⣤⠤⠚⢀⡜
⠀⠈⠒⠤⠤⠤⠒⠉⠀⠀⠀⠈⠂⠄⠀⠀⠠⠔⠊⠀⠀⠀⠈⠲⠤⠌⠁⠤⠔⠁⠀⠀⠀⠈⠒⠤⠤⠤⠒⠉
⠀⠀⢀⣀⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢄⣀
⢀⠔⠁⠀⠀⠀⠀⠙⢆⠀⠀⣠⠊⠁⠀⠀⠀⠉⢢⡀⠀⠀⡴⠊⠀⠀⠀⠈⠑⢄⠀⠀⢀⠔⠉⠀⠀⠀⠈⠳⡄
⡎⠀⠀⠰⠀⠒⠀⠀⡈⡆⢰⠃⢀⡄⠀⠀⠶⠀⠀⢳⠀⡘⠀⠀⠐⠀⠀⢀⡀⠈⡆⠀⡎⠀⣀⠀⠀⠀⠀⠀⢸
⡇⢆⠀⠀⠀⠀⠀⢀⢇⠇⢸⠀⣤⠤⢤⣤⣒⡶⠀⡼⠀⢃⠀⢀⡤⡖⠒⠓⠣⠀⡇⠀⡇⠀⠛⣠⠤⡀⠛⠀⣸
⠘⢮⡒⠤⠀⠠⢔⡪⠊⠀⠀⠣⡈⠛⠥⠤⠚⢁⡴⠁⠀⠈⠢⡀⠈⠁⠀⠀⢀⠔⠁⠀⠘⢄⡀⠀⠀⠀⣀⠔⠁
⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉⠁⠀⠀⠀⠀⠀⠈⠑⠒⠒⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⢠⠔⠊⠋⠑⠒⢤⡀⠀⠀⠀⢀⠔⠚⠉⠑⠢⣄⠀⠀⠀⣠⠔⠒⠉⠙⠲⢄⠀⠀⠀⠀⣠⠔⠚⠉⠙⠒⠦⡀
⡰⠁⢀⡀⠀⢀⠀⠀⠙⡄⠀⡰⠁⠀⠀⡀⠀⠀⠈⢆⠀⢰⠁⢀⠀⠀⠀⠀⠀⢳⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠈⠁⠀⠉⠃⠀⠀⡇⠀⡇⠀⠊⠂⠀⠈⠑⠀⢸⠀⢇⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⡇⠀⠀⣀⣀⣁⡠⠤
⠱⡀⠈⠉⠉⠉⠛⠁⡠⠃⠀⢣⠀⠀⠀⠿⠀⠀⢀⠏⠀⠘⣄⠀⢀⠤⠠⠄⢀⡜⠀⠀⢣⡀⠘⣍⠇⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⡗⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣖⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⣿⣾⡉⣻⣿⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⣞⣱⣾⢣⣀⡤⢀⣴⢶⠟⠗⣠⣤⠞⠛⠉⠙⠋⠙⣩⡬⠟⠛⠛⠿⣷⠒⠦⢤⣀⠀⠀⠀⠘⣧⣠⠷⣇⣼⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣞⣳⠾⠛⣳⡾⠋⠁⠂⠈⣴⠋⠀⠀⠀⢂⢀⣴⠟⠉⠀⠀⡠⠀⠀⣠⠾⠛⠛⠛⠛⠂⠄⣀⡟⠻⡖⡞⢺⡆⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⢉⢉⠭⠋⠄⡈⠀⣀⠊⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⢀⠄⠔⡰⡾⠁⠀⠀⠀⡠⢂⣤⠶⠾⣷⣿⠉⣻⣸⡋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠒⢲⠂⢉⠄⣈⣴⠞⠻⢣⠐⢀⣀⣤⣴⡌⠀⠢⠀⠀⠄⢂⠜⠋⠀⠀⠠⠊⠌⣴⠟⠁⡠⢂⠄⣈⣙⣧⣨⢯⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡂⠐⣠⡾⠋⠀⡠⠡⢀⡿⠋⠉⠀⠀⠻⠀⠐⣠⡴⠟⢻⡀⠀⠐⡠⢀⣡⠊⠁⠀⠈⠀⡵⠊⢁⣐⡼⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡵⠒⢾⠈⢋⠀⠀⠐⠀⢠⡵⠁⠀⠀⠀⠀⠀⠀⢝⠁⠀⠀⠈⠪⢀⢁⣴⠟⢂⠄⠀⠤⠈⡜⢀⡾⠋⠸⠋⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡄⠀⠸⣂⢀⠢⠄⠀⠐⡘⠀⠀⠀⠀⠀⠀⠀⢢⠂⠀⠀⠀⠀⠐⡠⡙⠁⠀⠈⢎⠀⠄⠀⡇⢸⠇⣠⣾⣿⣦⡄⠀⢀⣀
⠀⠀⣀⡴⣮⣯⢿⣽⠽⣭⡯⣶⢋⠈⠳⣕⠐⠀⠊⠀⠀⠀⠀⠀⠄⠀⡔⠅⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠀⠒⡲⠚⣦⣯⢰⡿⣿⣿⠃⠀⣆⣀⠿⣤⣏⡷⠆⢀
⢠⣾⢛⣷⣟⣳⣿⣙⣶⣋⣗⣿⣻⣷⣦⡎⠙⠳⣄⠠⠀⠂⠄⠌⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢤⢆⡼⣽⠃⠀⠀⠀⠀⠀⠘⢳⡉⣳⣅⣀⣗⠶⢯
⣿⣼⠻⢶⡟⢧⡼⠻⣶⠟⣧⣗⠻⠞⠋⠙⠶⣤⡔⠛⠲⢤⣁⣈⡐⢀⠂⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⣈⡴⢊⡴⠃⠀⠈⠐⠀⠀⠀⠀⠈⠉⠉⡎⠁⠈⠄⠈
⠀⠉⠈⠉⠉⠋⠉⠙⠉⠈⠉⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣏⠀⠈⠳⣄⣀⡀⠀⣀⣈⣄⣁⣀⣀⣀⡥⢾⣿⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠿⠷⣬⣽⣿⣭⣤⣤⣄⣀⣬⣿⣞⠯⣭⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢻⣶⢿⣷⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣯⣿⢿⣾⣿⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢛⣿⡏⣷⣾⢻⣷⣟⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⣾⣯⣿⣽⢿⣧⣯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⡿⣷⣟⣽⣿⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣿⣿⢿⡿⣯⣼⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢶⣿⣛⣟⣾⣿⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢿⣭⡿⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢣⠄⢀⠀⢀⠠⠋⠀⠀⠀⠀⢀⡡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣄⣀⡔⠁⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠈⡤⠀⠀⠀⠀⠀⡹⠄⠀⠀⣀⠀⡀⠙⠀⠀⣀⠀⢠⠀⠘⠢⡀
⠀⠀⠠⠀⢰⣿⣿⠇⠀⠀⠉⠉⢳⣦⣀⡀⢠⣤⣀⠲⠦⢄⠀⢀⠎⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠈⡇⠀⠀⠀⢈⠂⡀
⠀⠀⠀⠠⠤⣤⣄⣀⠀⠀⣠⣴⣾⢏⡟⠉⢙⠟⢿⣦⡀⠠⠛⡏⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢱⠬⢦
⠀⠀⠀⠀⠀⠈⠓⠲⠾⠷⠿⠿⢿⡯⣀⣀⣀⣀⣿⣿⣷⡄⠀⢣⠀⠀⢀⡀⠀⢀⠴⠃⠀⡀⠀⠀⠀⠈⠁⠀⡀⠀⢀⠔⠉⠓⠂⢠⠃⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣄⠀⠑⢄⠀⠙⠚⠁⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⣀⣤⠖⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠀⣈⢻⣿⣷⣤⣈⠉⠒⢦⠠⠤⣤⠤⢤⡤⠥⠤⠖⠒⠋⠑⠒⠒⠒⠚⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣻⡏⠀⠀⠰⣿⢿⣿⣿⡿⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⢠⣀⣠⣤⣴⣶⣤⣤⢤⣬⡿⠶⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢷⠀⠀⠀⠠⠁⠈⠉⠈⠀⠀⠀⠀⠛⠛⠛⠙⠛⠋⠉⠉⠉⠹⣿⣿⣿⣿⣇⠀⡇⠀⠀⡻⢻⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠁⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣽⣿⠀⠈⠳⣌⣠⣀⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢽⣿⣿⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣿⣿⣻⣯⣿⠄⠀⠀⣸⡿⠿⠛⠑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⣯⡿⠟⠻⠦⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠠⣼⡧⣀⣰⣌⣦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣼⢢⣖⢰⡴⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⢀⡤⠤⠒⠒⠠⢤⡀⠀⠀⠠⣶⠀⠀⠀⢀⡤⠔⠒⠒⠤⢤⡀⠀⠀⣶⣶⠀⠀⠀⢀⡤⠔⠒⠒⠢⢤⡀
⠀⠀⢀⡔⠉⢀⣀⠀⠀⣀⡀⠈⠢⡀⠀⠀⠀⢄⡔⠉⡤⢤⠀⠀⡤⢤⠉⢪⡀⠈⠁⠀⢄⠔⠁⣀⡀⠀⠀⢀⡀⠉⢢⡀
⠀⢠⠏⠀⠀⣿⣷⠀⠀⣾⣿⠀⠀⠹⡄⠀⢠⠏⠀⠘⠁⠘⠀⠀⠃⠈⠃⠀⠹⡄⠀⢠⠏⠀⠀⣧⡿⠀⠀⣧⡽⠀⠀⠹⡄
⠀⣾⠀⠀⠀⠈⡁⠀⠀⢈⠁⠀⠀⠀⣿⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⣷
⠀⠘⣆⠀⠀⠚⠹⠞⠳⠞⠓⠀⠀⢠⠇⠀⠹⣄⠀⠀⠈⠳⠤⠤⠞⠁⠀⠀⣰⠏⠀⠹⡄⠀⠀⠈⠙⠉⠉⠋⠁⠀⠀⣰⠃
⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⢀⣠⠔⠁⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⢀⣠⠔⠁⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⢀⣠⠔⠁
⠀⠀⢀⣤⡀⠉⠙⠛⠛⠛⠉⠀⠀⣴⡄⠀⣀⡀⠀⠀⠉⠛⠛⠋⠉⠉⠀⠀⢀⡀⠀⠀⢀⣀⠀⠉⠛⠛⠛⠋⠉⠀⣀
⠀⠀⠈⠛⠀⣀⡠⠤⠤⢄⣀⠀⠀⠈⠀⠘⢿⠿⠀⠀⣀⡠⠤⠤⢄⣀⠀⠀⠺⠿⠂⠀⠘⠋⠀⣀⡠⠤⠤⢄⣀⠀⠉
⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⠀⡤⠊⠁⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⢀⡴⠊⢁⡀⠀⠀⢀⡈⠑⢤
⠀⢀⡞⠀⠀⢦⣴⠂⠐⢦⡴⠂⠀⢳⡁⠀⢀⡞⠀⠀⠤⠤⠄⠠⠤⠤⠀⠀⢳⡀⠀⢈⡞⠀⢰⣫⣿⠀⠀⣿⣹⡆⠀⢳⡀
⠀⣼⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⣷⠀⢸⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡇⠀⣾⠀⠀⠈⠉⠉⣀⣀⠉⠉⠁⠀⠀⣧
⠀⠸⡄⠀⠀⠙⠶⠤⠤⠶⠋⠀⠀⢀⡏⠀⢸⡄⠀⠀⠀⡴⠋⠙⢆⠀⠀⠀⢠⡇⠀⠹⡀⠀⠀⠀⠸⣅⣹⠇⠀⠀⠀⢠⠏
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⢀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⢀⡀⠉⠛⠒⠒⠒⠒⠚⠉⠀⠀⠀⠀⣠⡀⠀⠉⠛⠒⠒⠒⠒⠚⠉⠀⠀⠀⠿⠇⠀⠀⠉⠓⠒⠒⠒⠒⠛⢉⣀
⠀⠀⠘⠃⠀⢀⣀⠤⠤⣀⡀⠀⠀⠿⠂⠀⠙⠃⠀⠀⢀⣀⡠⢤⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⠤⢤⣀⡀⠙⠛
⠀⠀⡠⣠⠖⠉⠀⠀⠀⠀⠉⠳⢄⢄⠀⠀⠀⠠⣢⠖⠉⠀⠀⠀⠀⠉⠳⣄⢄⠘⠋⠀⡠⡢⠞⠉⠀⠀⠀⠀⠉⠲⣔⢄
⠀⢀⡜⠁⠀⣄⣀⠀⠀⣀⣠⠀⠈⢳⡀⠀⢀⡞⠁⠐⢦⣀⠀⠀⣀⡴⠂⠈⢳⡀⠀⢀⡞⠁⠀⠙⣶⠄⢠⣖⠃⠀⠈⢣⡀
⠀⣸⠁⠀⠀⠘⠋⠀⠀⠈⠋⠀⠀⠀⣷⠀⢸⠁⠀⠀⠀⠛⠀⠀⠛⠀⠀⠀⠀⡇⠀⣾⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠈⣧
⠀⢹⡀⠀⠀⠐⢦⣀⣀⣠⠆⠀⠀⢀⡟⠀⢸⡀⠀⠀⠴⠻⡴⢦⠟⠦⠀⠀⢀⡇⠀⢹⡀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⢀⡏
⠀⠀⠱⣄⠀⠀⠀⠈⠁⠀⠀⠀⣠⠎⢀⣤⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢀⣄⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠙⠲⠶⠦⠤⠶⠖⠋⠀⠀⠈⠋⠀⠀⠀⠙⠲⠶⠦⠤⠶⠖⠋⠀⠀⠈⠁⠀⠀⠀⠙⠲⠶⠤⠤⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣒⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣾⢶⢧⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣙⡴⡸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⢿⡣⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⠯⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⣞⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢔⢒⣒⢒⣶⣄⣦⢤⣄⠠⣷⣫⠿⢞⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠙⢮⢳⠃⠀⣀⠀⢹⣿⣿⡏⠀⢈⣿⣯⣿⢿⣄⠀⠀⠀⠀⣀⣤⡶⣶⡲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⣀⠀⠀⣠⣶⡼⣷⣿⣧⡤⣀⢼⣿⣦⣀⣼⣦⡿⢥⣀⣨⣡⣾⠟⠙⣿⣮⣵⡟⠒⣶⣾⢻⣻⣾⣛⣿⣴⡿⣦
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⢧⣽⣿⡝⠉⡱⡹⣅⠀⠞⠻⡿⣿⢾⡿⡿⣻⣿⣿⣯⣾⣏⣇⠀⠉⠉⠀⠹⣛⢄⠀⠙⣷⣻⠫⢽⡼⣁⣒⡭⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢀⣴⡿⣟⣿⠟⠀⡹⣯⣅⣒⢞⡵⣿⣭⣊⢾⣿⣿⡿⣷⡟⣿⣾⣟⣿⡋⠁⢀⣻⣿⣧⠴⠛⠛⠛⠏⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣿⣽⣿⣛⣲⡿⣗⣿⣿⣿⣿⣷⡝⣻⣿⢟⡻⢶⣣⣽⣾⣿⣭⣛⣿⣦⣤⡞⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⣿⣼⡫⣫⣿⡷⣮⡿⣲⣿⣽⣿⡟⢯⢛⣿⣿⣿⣧⣟⠋⢿⡏⣩⣿⡿⣯⣽⡟⣓⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⠉⠻⡻⣇⣾⡏⢹⠇⡿⡽⠋⢁⠀⠙⠏⣆⡎⢀⠿⠁⠈⢜⠐⣄⠁⠀⣹⢿⣿⣟⣾⣿⣿⣿⣵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣻⣄⣴⣹⣷⣤⡞⣰⡀⠀⡿⡇⢀⣈⠻⣶⣌⣇⣬⣷⣾⣿⣿⡻⠋⠲⣿⣿⣽⣷⣛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢾⡿⣯⣿⢻⣸⣿⠀⣹⣿⣿⣧⣿⣿⣁⣠⣿⣿⢫⣿⣶⣟⡻⠋⠀⠀⠀⠈⢿⣽⢶⣭⣟⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⡾⣟⢟⣿⣻⠿⣿⣷⣾⣿⣩⣟⣛⢻⣿⣻⣻⣭⣦⣿⣿⠿⠙⠃⠁⠀⠀⠀⠀⠀⠀⠀⢹⣾⣷⡛⢾⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣻⣯⠿⡿⠂⠀⠈⠙⠛⠿⠽⠿⠿⠻⠿⠿⠿⠿⠛⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣶⡿⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣾⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡶⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⠤⠤⢀⡀⠀⢀⡤⠒⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨
⠀⠀⢀⠤⠊⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠈⢀⢀
⠀⠀⠈⢆⠀⠀⡠⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠂⠀⠀⠈⠑⠒⠄
⠀⠀⠀⠈⠉⠡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠄⠂⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⡅⠀⠀⠈⠑⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠤⢐⠈⠀⠀⠆⠀⠀⠀⣀⠀⠀⠀⠀⠈⠁⡀⠀⠁
⠀⠀⡸⠀⠀⠀⠸⡀⠀⠀⠉⠒⠒⠒⠓⠢⢤⡴⢒⣉⠉⠁⢀⡴⠁⠀⠀⠀⠀⠀⠒⠉⠀⠉⠁
⠀⠀⡇⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠃⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠉⠉⠉⠁⠀⠀⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣤⡀⠈⠃⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠛⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠘⠃⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠠⠠⡀
⠀⠀⠀⠁⢄⠀⠙⠀⡀⢂⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠔⠁⢀⠂⡀⠜⠁⢀⠄⠁
⠊⠀⠁⠂⠤⢑⠠⡀⠈⠂⢕⢄⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠠⠂⠀⠀⠔⠁⢀⢔⠅⠊⠀⡠⢐⠡⠄⠂⠁⠈
⠀⠀⢀⠀⠀⡀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠑⠄⠀⢠⡀⠀⣀⣀⠀⠠⠐⠀⡠⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡀⠠⢐⠀⠁
⠣⠀⠀⣀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠐⠂⠀⠠⢀⠀⠀⠀⠀⠀⢀⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⢄⠀⠀⠀⠀⠀⢀⠠⠀⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣬⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣬⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⣤⠤⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄⠀⠀⠀⣴⣶⡄⠀⠀⢀⠀⠀⠀⠈⠿⠈⡐⢙⣿⠓⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⠀⠀⠈⠀⣠⣴⣶⠀⠀⠙⠀⠀⠀⠀⡉⠕⠁⠀⠀⠀⠡⢡⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⡴⠂⢀⣼⡿⠀⠀⠀⣸⣿⡿⠛⠄⠀⠀⣠⣶⠀⣼⠋⠀⢀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠆⣠⣾⡏⠀⢀⣾⣿⠷⡄⠀⠀⠻⠿⠀⠀⢀⠀⠺⣿⡋⢀⠴⠃⠀⠃⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⢀⡤⢂⣴⠏⠴⡟⢿⠇⠠⠛⠿⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⡷⠂⠜⠋⠁⠀⠀⠀⠀⠠⠴⠿⠃
⠀⠀⠀⢀⡀⢀⡘⢻⢀⡀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣶⣿⠗⢉⠅⠀⢀⠀⠀⠁⠀⣴⠞
⠀⠀⢀⡮⠟⠀⠀⢻⡾⢧⣦⢄⣴⣶⠆⣠⣶⣶⡆⢰⣿⡿⠿⠘⠋⠀⢂⣤⣤⡤⠌⠃⠀⠐⡲⠄
⠈⠿⡋⠀⠀⣴⣾⠾⠟⠛⠁⠚⠛⠛⠐⠛⠉⠉⠀⠀⠀⠀⠀⠠⠤⠤⠚⠛⠋⠓⠀⠀⠌⠉⣿⠅
⠿⢤⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⢀⠸⠌⣱⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠡⠭⠃⢀⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠰⢐⣦⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠏⠤⢊⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠶⡰⣳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⣔⡝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⠍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢿⣻⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣻⠿⢟⣷⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⠟⣠⣴⡿⠋⠀⢀⣀⣤⢤⣤⣴⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⣸⣿⣦⡼⢿⣿⣿⣾⣷⡿⠾⣥⣿⣈⠻⣏⢹⢻⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⣾⣿⣿⣿⣶⣺⣶⣶⣻⣿⣷⣶⣿⣿⣿⣿⣏⣉⣿⠿⣿⣿⣙⣿⣷⣄⡀
⠀⠀⣀⣀⣤⣀⢠⡀⠀⠀⠀⠀⡴⢾⠛⣿⣿⣿⣿⡿⠛⢻⡿⠿⣿⣿⡿⠟⢿⣿⣟⣫⣿⣷⣦⣹⣿⣦⣽⡻⣿⣿⣦⢀⢤⣴⣲⣶⠶⠚⠋
⢚⠛⠻⢿⣁⣿⣾⠇⠀⠀⠀⢀⣾⣿⣼⣿⡴⢟⣻⣿⣤⣾⣿⡟⢛⣻⣧⣀⣾⡧⣽⣭⣻⣁⡜⠛⣿⣿⣿⣿⣿⠛⣿⣶⡚⠋⠉
⠀⠐⠠⢾⣶⣽⣇⠀⡄⠀⠀⣿⣿⣿⡿⠛⣿⣿⣾⣿⣿⣿⣍⢳⣿⡿⣿⣿⣯⣹⣿⣿⣿⣿⣶⠶⣿⣿⣿⣿⠛⣶⣿⡻⣷
⠀⠀⠀⠀⠉⠙⠃⠀⠀⠐⠱⣿⣿⣿⣷⣦⣼⠛⢿⣟⣿⣿⣿⣿⢿⣾⣛⣻⣿⣿⠛⣿⠿⣿⣿⣿⣧⠤⣽⡏⣹⡟⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣹⣶⣮⠀⠨⣽⣿⣇⡀⣹⣟⣿⣿⣿⣶⠇⣀⣾⡃⢘⣴⡾⣻⢻⣿⣾⡛⣭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⣿⣿⣷⣿⢏⡷⢶⡿⣿⣿⡿⣿⣿⣯⣿⠏⢼⣿⣾⣿⣿⣿⣿⣿⣶⣧⣴⣿⡿⢁⣸⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡆⢀⣀⡘⢿⣿⣿⣾⣿⡯⣠⣾⡟⢡⣿⢻⣿⢏⣴⢰⣿⣿⠿⠹⠯⣼⣿⣿⣿⠟⢻⣾⣿⣿⣿⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠘⢿⣿⡄⠉⠛⢿⣿⣿⣿⣿⣦⣾⣿⣾⣾⣿⣿⣼⣿⣷⡴⣶⣷⣶⣿⡿⠆⠀⠀⠛⠿⢿⣿⣿⣿⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣻⣿⣂⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠛⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⣿⣿⣿⣻⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⠔⠉⠑⢎⠉⠉⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⢻⠀⢠⣄⠈⡇⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠸⡀⠿⠟⢀⡇⠀⠀⠀⠙⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⠒⠒⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠋⠉⠉⠀⠈⠉⠉⠙⣳⢦⣄⡀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⡠⠚⢽⠦⣤⣄⣀⣀⣤⣤⣤⣖⣫⠛⠁⠀⠙⠳⣄⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠙⢄⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⡜⠁⠀⣼⡞⠉⠉⠙⣶⡏⠉⠁⠀⢻⣧⠀⠀⢀⣠⡾⠳⣄⠀⠀⠀⢷⠀⠀⣠⠖⠲⠶⠿⠷⠛⠋⠁
⠀⠀⠀⠀⠀⣼⠀⠀⢰⡿⠀⠀⠀⠀⢿⡇⠀⠀⠀⢈⡟⠒⣒⢹⠋⠀⠀⠘⣧⠖⠛⢺⡀⠀⣿
⠀⠀⠀⠀⢰⠛⠻⢶⡋⠳⢤⣤⣤⣤⣞⣙⠒⣲⠿⠛⠋⠉⠈⢳⢧⠀⠀⡰⠋⠀⠀⡼⠁⠀⡿
⠀⠀⠀⠀⣸⠀⠀⠀⢹⡼⠉⠉⠉⠉⠉⠈⢳⡇⠀⠀⠀⠀⠀⠀⢫⡷⠊⠀⠀⣠⣞⣉⠙⣶⠃
⠀⠀⢀⠤⡏⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢀⣀⡤⠞⠉⠀⣀⡴⠚⢭⣀⣸⠞⠁
⠀⣀⣿⣀⠈⠓⠒⠤⠿⢧⣀⣀⣀⣠⡤⠤⠤⠟⠒⠋⠉⠀⣀⣠⠴⢾⠁⠀⠀⠀⠉⠳⣄
⠈⠓⠤⣬⣝⣶⡦⠤⣤⣀⣀⣀⣀⣀⣀⣀⣤⡤⢴⠖⠚⠛⡍⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣸⣅⡀⣀⣅⣀⣀⣀⣬⢤⣤⣤⠞⠓⠒⠛⠙⣧⡀⠀⢀⠤⢊⣴⠟
⠀⠀⠀⡼⠃⡀⠀⠀⠀⢠⡟⠈⠉⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣬⣤⡶⠛⠁
⠀⠀⠀⠙⠶⣬⣁⣒⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⠗⠶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠉⠋⠚⢞⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠐⢚⣛⣛⣳⣧⣀⡐⠌⠀⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣦⣤⠾⠚⣋⣛⡳⣿⣿⣟⣹⣯⣤⣝⣧⡀⡀⢀⣿⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣭⣴⣶⠻⠿⠿⠍⢉⡉⢸⣿⣍⠉⣹⡇⡵⢾⠟⣋⣁⡸⠃⣾⢗⡚⠠⢿⠉⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⡍⠀⠀⠀⡅⢷⣶⠀⠀⣼⢇⠠⢙⣿⣿⣿⠇⢸⡇⣼⣓⠈⠀⣢⠄⠎⢡⣴⡶⠸⣇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡽⠋⢡⣄⣻⣇⡀⣠⡞⣿⣇⠉⠓⣊⢁⣹⣿⣷⡍⠅⣤⣷⢙⣻⡏⠃⠀⠀⠀⠀⠀⣘⠩⢄⡜⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣤⣾⣯⣿⡇⠈⢿⣸⣴⣿⡇⠀⠀⣿⠋⢻⣿⣿⡇⣶⣿⡟⣤⣾⡇⠀⠀⠀⠀⣀⣉⠨⠔⠉
⠀⠀⠀⠀⠀⠀⡀⣻⣿⣿⠁⢠⣿⠀⠃⠀⢻⣾⣿⡿⣇⠀⣤⣿⠀⣷⣿⣿⣷⠛⢉⣴⠟⡉⠀⠀⠒⠂⠈⠁
⠀⠀⠀⠀⠀⣰⡇⠻⠛⠻⢶⣼⣿⡀⠀⠀⠈⣿⣿⡇⢹⣶⣿⢻⣿⣿⣿⡿⣋⣤⢛⡁
⠀⠀⠀⠀⠀⠈⢹⡏⣲⣦⣸⣿⠉⣻⣶⣤⣴⢸⣿⣿⣿⣿⣿⡿⢿⡿⣭⣆⣾⣿⡂⢅
⠀⠀⠀⠀⠀⠀⢀⡸⠷⡾⢝⠉⠻⣾⣿⣿⣿⠟⣿⠿⠛⢛⢉⣶⢾⣶⠿⢋⡜⢻⣯⠆⢢⡀
⠀⠀⠀⠀⠀⡀⡟⠂⠀⢙⠸⠿⢾⣮⣭⣥⣀⣥⣴⣤⣿⢵⠿⠛⠈⠁⠠⠛⢤⡼⠃⠀⠼⣷
⠀⠀⠀⣰⠞⠱⢈⠉⠇⣫⠄⠀⠀⠀⠀⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠰⠠⣼⡎
⠀⠀⠀⣼⣧⢀⣤⡰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢩⠬⡐⠱⣾⠃
⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠇⠁⠄⢂⣕⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⢾⡟⠧⢈⠘⣬⢁⣤⡸⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⡃⢰⠆⣰⡁⢨⣤⠺⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⢿⣤⣿⠧⠿⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⠌⠀⠉⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣣⡀⣴⠞⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠠⢰⠀⠠⢀⠀⠀⡀⠀⠀⠀⠠⡋⠀⠀⠀⠀⠀⠠⠄⣀⣐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠠⢤⠶⠄⢘⠀⠀⠠⡀⠀⠀⠠⠂⡠⣤⡀⠉⠙⠛⠉⢠⣆⡀⡀⠀⣀⣄⠠⠤
⠀⠀⠀⠀⠀⠀⣀⡄⠀⣰⠀⠈⠀⠀⠀⠄⢨⠄⠀⠀⠄⢄⡀⠀⠚⠁⣀⡽⢴⡤⢶⡾⣫⠀⠛⠀⠀⠀⣽⣿⣿⣖
⠀⠀⠀⠀⠀⠀⠙⢀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢰⣠⠴⣾⣏⣴⠀⣿⠐⠁⠣⡐⠈⠀⣀⣶⡯⠛⢿⠟
⠀⠀⠀⠀⠀⣀⠞⢽⠁⢀⡇⡄⠀⠀⠀⡇⠀⠀⢘⡠⣴⡾⢉⣁⣮⠄⠀⠈⢳⠉⠘⠖⠠⡂⠌⢽⡟⠙⠁⡀⠙⠂⣄
⠀⠀⠀⡠⣞⣛⠀⡈⠳⣼⣃⠁⠀⠀⢀⡁⣀⡒⠉⠀⢹⡇⣾⠋⠀⠀⠀⠀⢸⡄⣀⣊⣥⡴⢂⠀⣹⣴⠀⠀⠀⢂⡀⡀
⠀⠀⠘⠚⠬⠵⠟⠴⠖⠁⡈⠲⢬⣕⡢⣁⠀⠀⠀⠛⢠⣹⠇⠀⠀⠀⠀⠐⣹⣇⡉⡤⠀⠀⠀⠖⠋⠈⢧⡄⢠⠐⠖⠂⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠀⠀⠉⠙⠛⠛⠂⠙⠀⠀⢠⢆⡀⠔⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⣙⡔⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣃⠠⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠡⠀⠞⢢⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣻⠀⠊⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠁⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣩⣿⡓⠒⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠕⣷⣂⠂⢀⠄⡄⠀⠀⠀⠀⠰⠁⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣦⢘⢕⡷⣂⠀⠀⢳⡀⠀⠀⠀⡰
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢱⡆⠸⣝⢯⢠⠀⠀⠙⠲⠖⠚⠁⠀⢀⣀⣠⣄⢶⡦⢀
⠀⠀⠀⣄⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠰⣿⡶⠐⢘⠐⠚⡧⠀⠀⠀⠀⠀⢀⡰⢎⠡⡼⢷⢼⡟⣯⡰⢆
⠀⠀⠀⠰⡀⠀⠀⣠⠇⠀⠀⠀⣀⣴⡶⢖⡽⢿⣿⣶⡿⣰⣏⣹⣶⡶⣄⢔⣏⣭⣜⠂⢰⠩⣖⡾⡛⡙⠻
⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⣾⣯⣽⣣⣝⠂⠿⡤⣡⣾⡏⢀⢔⢫⠛⣼⣷⠇⠛⠁⠀⢰⢴⠊⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢾⣷⡴⡿⠊⡩⣀⣷⠄⠉⠛⠛⠾⣆⡀⢀⡜⡈⢜⣳⡄⠀⠁
⠀⠐⠀⠀⠀⠀⢀⢾⣟⡬⠫⣏⠸⡐⣲⢟⣾⠁⠀⠀⢤⠈⢳⠐⠙⠋⣞⠙⣦⡿⠡⣄
⠀⠀⠀⠀⠀⠀⣸⣯⣫⠖⣉⠉⢳⢆⡤⠾⠯⢵⣦⢄⣀⡠⠎⢀⡴⠛⢿⣷⣻⣷⣷⠽⠭⣓⣤
⠀⠀⠀⠀⠀⡜⣥⣹⠃⢀⣸⠇⣰⠋⠀⠀⣀⠀⠈⢳⡠⣠⣴⠟⡠⡀⠀⡏⣩⠓⣻⣘⢳⣜⡵⣾⠤
⠀⠀⠀⣠⣤⢟⣽⢉⣻⠿⣥⠀⣿⣄⠀⡉⠉⠀⠀⠀⣧⣬⡿⣄⠩⡧⢸⣧⣯⣻⣳⣶⣴⣤⢈⠱⠋⠅
⠀⢀⣚⣝⣇⣫⠋⣞⠐⠠⣷⠚⠉⠋⠓⠻⠦⠤⠄⣠⠋⠀⠈⢆⡡⡤⢟⣿⢸⡏⠉⠉⠘⠂⠻⣿⡈⠤⠱
⠀⣾⡽⢻⣾⣹⢻⢸⣴⣸⡇⢀⡞⠋⠉⠙⢲⣤⣼⣧⠀⢆⠁⠆⡇⣠⣏⣿⠛⠁⠀⠀⠀⠀⠀⠙⠿⣦
⠈⠉⣤⡬⠻⢉⡻⡎⠦⣿⣇⠘⠆⠉⠂⠀⢸⣿⣤⡍⣧⣀⣀⣼⣻⢟⢻⠏⠀⣠⠴⠶⠤⡀⠀⠀⠀⠷⣦
⠄⣀⣾⠂⠀⠰⣟⣽⢷⡙⠻⠦⣄⣀⣀⡤⣿⡛⢛⠇⠿⢟⣫⣾⡿⠎⠁⠀⡾⠃⠀⠀⠀⠈⠀⠀⠀⠀⠋
⢠⣼⡆⠀⠀⠀⢈⢱⣛⠾⣻⢷⠶⣾⣤⡴⣾⢯⣿⣻⣼⠧⠿⠋⠀⠀⠀⠀⢿
⠸⡝⠀⠀⠀⠀⠆⠄⠀⠀⠉⠉⠛⣋⣻⣷⠟⡌⣹⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⠋⣷⣫⢹⠟
⠀⠀⠀⠀⠀⠀⠀⣀⢤⡞⠾⠛⠉⡁⣼⠷⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⡈⠀⠀⠁⠀⠀⠀⠀⠰⠒⢉⠁⠀⠀⠀⠀⠀⠠⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠓⠶⣀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⢀⠴⠂⠀⠀⠀⢣⣀⣀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⣤⡀⢀⡴⠋⠀⠀⠑⡀⠀⢀⠀⠓⠀⠀⠀⠀⠀⠀⡠⠓⢄⣀⠴⠊⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⣄⡀⠀⠊⠀⠀⢸⡀⠀⠀⠀⠀⠢⠔⠊⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠤⠒⠁⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⣠⠶⠚⢉⠉
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡀⠒⠉⠀⡄⡠
⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⢄⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⢄⠀⠈⠁⠂⠀⠠⠤⠤⠤⢨⠤⠤⠴⠚⡹⠀⠀⠀⠀⢡
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠀⠀⠈⢦⡀⠀⠀⠀⠤⠄⠠⠄⠂⠀⠐⠒⢸⠁⠀⠀⠄⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢳⠀⠀⠒⠒⠒⠒⠂⠂⠒⠒⠒⢸⠀⠀⠀⠀⠀⠃⡇
⠀⠀⢀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠀⡇
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⡁⢘⠀⢂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠸
⣤⣤⣤⣤⣤⣤⣠⣠⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣸⣯⣝⣭⣭⣝⣽⣭⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⣾⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠹⣿⣿⣿
⠀⠀⠀⠀⠤⣶⣯⣟⣿⠿⠿⠿⠿⠿⠿⢿⣛⣭⣾⣿⡿⠛⠛⠛⠁⠀⠀⠀⢸⣿⣿⣿⣿⣟⡋⠁⠢⠤⣀⣀⣀⣀⣀⣀⣠⠤⠔⠋⠀⢀⣤⣦⣤⣼⣿⣿⣿
⠀⠀⠀⠀⠐⢿⡿⢿⣿⠿⠛⠛⠛⠛⠛⠻⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣀⠀⠀⠀⢀⣴⣤⣤⣤⣤⣤⡀⠀⣀⡀⣨⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡿⠿⢿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠠⠀⠠⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⢥⣤⣥⡀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠤⠀⠤
⠀⠀⠐⠒⠚⠀⠀⠀⠀⠀⠠⠊⠂⠀⠀⢐⠒⠚⠀⠀⠀⠜⠋⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃

⠀⡔⠀⣴⢀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠉⠉⠀⠉⠉⠀⠀⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛
⠀⠀⠠⠀⠠⠀⠀⣁⠀⡀⠀⡀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⣀⡀⠀⠀⢿⠀⠀⠀⠀⠸⠇⠀⠀⣀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈⠀⠈⢀⠀⠀⣠⠉⣉⠑⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠤⢀
⠀⠀⠐⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡂⠀⠀⠀⠻⠃⠻⠃⠀⠀⠐⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⣒⠀⠀⠀⣐⠀⢀⡂⠀⠀⢄
⠀⠀⠠⠀⠠⠀⠀⠀⠀⡄⠠⡄⠀⠀⠺⣷⠖⢶⡿⠀⠀⠀⠄⠀⠄⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡄⠀⡄
⠀⠀⠀⠒⠐⠀⠀⠀⠀⠁⠆⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠲⠶⠂
⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠉⡀⠀⠐⡐⠆⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠄⠀⠀⠐⠛⠻⠟⠓⠀⠀⠐⡄⠀⡠⠠
⠀⠘⢗⡂⣚⠇⠀⠀⠀⠀⠀⠂⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠠⣬⣴⣤⣥⠀⠀⢆⠴⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⣀⣠⣤⣶⠶⠛⠛⠛⠛⠳⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠿⢻⣿⡟⠁⠀⠀⢠⡀⢨⣝⣶⡽⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢟⡽⠋⠁⠀⠀⢻⡇⠀⠀⠀⠀⠑⠀⠉⠉⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠛⠙⠁⠀⢀⣀⠀⠠⣌⣀⣀⣀⣤⣴⣆⠤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡏⣤⣤⠖⠛⠋⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⣻⣽⣿⣿⣿⠇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠾⠛⠉⠁⢀⣴⣿⣿⣿⡟⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡟⢡⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣿⣿⣿⣿⣿⣯⣴⠟⠁⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⢿⡿⠿⠟⠛⠉⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠿⠋⠀⠀⠈⠉⠉⠉⠉⠉⠉⢁⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠾⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣤⣴⣶⣶⠶⠒⠺⢿⣿⣿⣶⣶⡶⠆⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⢟⣫⣬⣝⠿⣛⣭⣶⣿⣿⣿⣶⡌⣭⣷⣶⣶⣶⡝⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⣠⣬⣝⢋⣵⣾⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⣎⢻⣿⣫⣿⣦⣤⣴⣾⡿⠋⠉⢀⠀⣨⣝⣷⡄
⠀⠀⠀⢀⣀⣠⣶⣿⡿⢫⣾⣿⣿⣏⣾⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡿⣽⣿⣿⣟⣿⠟⠉⠈⢿⡀⠀⠀⠘⠂⠙⠛⠁⢹
⠀⠀⠰⢿⣿⣿⣟⢫⣴⣿⣿⣿⠟⣼⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⢟⣭⢞⡿⠛⠉⠙⢁⣀⣀⣀⣠⣥⣀⣠⣤⠶⠄⠐
⠀⠀⠀⠈⠉⠻⠿⣷⣭⣿⣿⠣⣾⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣟⣯⣷⠟⣵⣿⡇⠰⣶⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣶⢠⠦⠭⠭⣿⣿⣧⣿⣟⣛⣛⣛⣛⣿⣿⣿⣾⡿⠿⠛⠉⣺⣿⣿⡇⢀⡇
⠀⠀⠀⢀⣠⣤⣤⣴⣾⣿⣿⡷⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⣠⣾⣿⣿⡿⢁⡾⠁
⠀⠀⠺⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⡵⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣶⣶⣾⣿⣿⡿⠿⠿⠟⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣠⣾⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢛⢛⠻⣿⣿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⣾⡿⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣶⣤⣼⣿⣿⣿⣴⣿⣿⣿⣿⣿⢿⣿⣟⢻⣿⣿⡏⠀⢸⡏⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⣷⣿⣿⣶⣷⣿⣧⠀⣿⣿⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣝⣿⣶⣿⣿⣇⠀⣽⣿⣿⣿⣿⣿⢿⣿⣿⡿⢋⡄⢻⣿⡄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠘⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢧⣠⣤⣼⡂⣾⣿⣾⣿⣧
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠈⣿⣿⣇⢈⣿⣿⣿⣿⢿⣿⠿⢿⣿⡟⢿⣿⣿⣿⣿⣷⣿⣿⢟⣹⣿
⠀⠀⣰⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣾⣿⡿⠛⣉⣷⣿⣷⣶⣿⣯⣽⣿⣿⣿⠟⢣⣦⣿⣸⣿⣿⣶⣦⣄
⠀⢰⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣁⣶⣿⣿⣿⣿⣿⣿⡵⢿⣿
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣎⣩⣿⣿⡟⢷⣦⣼⣿⣿⣬⣿⣿⣯⣾⣿⣿⣿⡿⢷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣛⢙⠟⣿⣿⡾⣿⣷⣴⣾⣿⣭⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣽⣿⣿⣿⣟⣿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢩⡉⠉⠉⣩⣍⡩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⠄⠉⠀⠀⡀⡢⠊
⠀⠀⠀⠀⠀⠀⠀⠐⡡⠀⠀⠀⠀⠀⡔⠊
⠀⠀⠀⠀⢀⠠⣋⠄⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠘⢇⠀⠀⠈⡢⠊
⠀⠠⠀⠀⠁⠀⣀⢈⡠⠊
⣇⠀⠀⠀⠀⠨⡦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠒⢰⣲⣶⠒⠂⠀⠀⠀⠀⣀⡀
⡇⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⣘⡀⠀⢀⣀⣤⣿⣿⡀⠀⠀⠀⠀⣀⡀⢹⡗
⡇⠐⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣿⢿⡿⡿⣿⡿⢿⡟⠛⠛⠋⠛⡛⣶⣶⣶⢿⣿⡿⣿⣿⣄⡀⠀⢀
⡧⠂⠀⠀⠀⠀⠀⠀⠀⢠⣇⣸⠿⣿⣿⣿⣿⠌⠃⠀⠀⣸⣇⠨⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢈⠻⣿⣿⣶⣿⣷⠀⠀⠀⠀⢀⠤⠀⢴⡤⣴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠁⠀⢨⣿⣿⡿⠀⢀⢠⠴⣿⡟⣀⠀⠀⠀⠀⠀⣄⣿⣿⣿⠀⡄⢠⢀⣄⣿⣿⣿⣿⠋⠉⠨⣄⡀⢬⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⡴⣼⣿⣀⠀⢠⣿⣿⣿⣧⣄⣠⣤⣴⣿⡯⣿⣶⠆⢖⢶⣿⣿⣻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⠿⠗⠢⣶⣿⣿⣾⣻⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠠⣿⣷⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡀⠀⣠⣿⣿⣿⡿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇⣵⣾⣿⡟⠙⠛⣄
⠀⠀⠀⠀⠀⢴⠈⡻⠀⠚⠙⢿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣾⣿⣿⣿⣷⠠⢠⣿⡄
⠀⠀⠀⠀⠀⠘⠋⠉⢿⣷⣦⣼⣿⡀⠀⢸⣿⠉⢹⣿⣿⣏⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⣿⠁⠠⠆⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠙⣿⣿⣿⣶⣶⣿⣶⣶⣥⣿⣿⣤⣭⣏⣿⣿⣿⣏⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣀⣶⣾⣿⣧⣴⣦⣀
⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⣠⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⡿⠿⠿⠿⠿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠈⠙⡏⠛⠛⠿⣿⣿⣿⣿⡄
⠀⢀⠀⠀⠆⠉⠛⠿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁