Weary Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠈⣿⡛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⣿⣦⣄⣙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣠⠴⠚⠉⠁⠙⣯⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠈⢇⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⢻⠏⠈⣿⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣴⠋⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⠈⢡⣿⡿⠋⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⡴⢶⣶⣤⡀⠀⠐⠋⠉⢀⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠈⠛⠿⠆⠀⠀⠐⠛⠁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠖⠿⣿⣿⡇⠀⠀⠀⠀⠠⢒⣀⣀⡀⠀⠀⢰⣯⣴⠶⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⢰⡈⠉⠀⠀⠀⠀⠀⠀⠛⠛⠁⠉⠀⠀⢨⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣻⡇⠀⠀⠀⠰⡀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠳⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠦⠀⠐⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣠⠄⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠬⠈⠋⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢹⣆⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠓⠶⠾⠃⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡇⠀⠀⢿⡆⠀⠀⠀⠈⠙⠶⣤⣀⣀⠀⢀⣀⣠⡟⢿⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠯⠈⠩⣿⡀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠉⡝⠉⠁⢹⡄⠈⡃⠀⢿⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠄⣁⡀⠀⠘⣇⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢳⡈⢤⠀⠀⠈⢩⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠤⠀⠀⠀⠲⠤⢀⢻⡄⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⢸⡇⠀⣠⣶⣦⣵⠆⠀⡐⠈⢉⢹⡇⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣁⠀⠀⠁⣰⡄⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣈⠳⣦⡀⢸⡇⣰⣿⣿⡿⠿⠞⠀⠀⠀⣀⣽⠇⠀⠀⠹⡄
⠀⠀⠀⠀⠀⣠⡴⠛⠉⠘⠛⢒⣂⠀⠀⠀⠀⠁⠀⠀⢻⠀⠀⠀⠀⢀⣾⣿⣷⣎⢳⣾⣷⣿⠉⠐⠁⠀⠈⢦⠀⠀⢩⡟⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⡴⠛⠋⠉⠉⠉⠙⠛⠲⠮⣥⡀⠀⠑⡀⠀⢸⡆⠀⠀⣠⣿⠿⢍⠉⠻⣧⣿⠁⣿⡀⠀⡆⠀⢳⠀⠈⢑⡾⠁⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⢧⢀⡼⠟⢹⡀⢨⡷⣄⠀⠋⢀⣿⠀⢁⠀⠠⠄⡀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⢠⡈⠙⠁⠐⣿⠗⠐⠽⠛⠶⣴⣾⠧⠀⠨⠁⠐⠀⠀⡼⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠈⠁

⠀⠀⢀⣀⣀⣀⡀⣠⡀⣀⣀⣠⣤⣄⣠⣀⣀⣀⣀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⢀⣿⣏⢈⡙⣻⣿⣿⣿⣿⣿⢩⣏⠙⠋⠉⢙⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⡉⠉⠉⠙⠛⠿⠿⠛⠛⠋⠛⠋⠙⣿⣿⣿⣿⣿⣿⠿⢛⡛⣛⣙⡛⠁
⠀⠀⢸⣿⡿⠈⣿⣿⣿⣿⣿⣿⣿⣿⠛⢁⡀⠀⠀⢈⣀⣤⣤⣤⣄⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⢀⣠⣤⣄⡀⢀⣈⠁⠈⠙⢿⣷⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠘⡻⠂⠷⣿⣿⣾⣿⣿⣿⣯⡅⠀⠀⠈⣠⣾⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠚⠻⠟⠻⢿⣿⣿⣿⣿⣿⣟⡛⠿⣿⣝⠻⢿⣿⣿⡞⠛⢛⡿⠟⢻⡿⢻⡇
⠀⠀⢸⡿⢦⣶⣿⣿⣿⡿⠛⢁⣴⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢠⣾⣿⣿⣤⡉⠙⠛⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⢈⣷⡆⣾⣿⣿⡿⢣⣦⣾⠇⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⠛⠿⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣽⣿⣿⣶⣤⣀⡙⠻⠟⢋⣹⡇
⠀⠀⢸⡿⣡⣼⣿⣿⣡⣾⠋⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠙⠛⠿⣷⣾⠿⣦⣀⣠⣄⡀⢰⣆⡀⠀⠉⠻⠟⠋⠱⠶⠆⠀⠸⠆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠹⢿⡿⠿⠿⠽⠿⠶⠤⠄⠀⠀⣰⣾⡟⠛
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢳⣤⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠉⠙⠋⣿⣿⣿⣿⣿⣯⣭⣭⣿⣿⠇⠐⠆⠀⠀⢠⣶⡶⢦⣤⣅⠀⠀⠒⠒⠂⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⣿⣿⣿⡿⠉⠙⠋⠉⠉⣹⣦⣶⣤⣿⡿⠋⠉⠁⠀⠀⠀⣀⣀⠀⣀⠀⠀⠀⠀⠀⣤⠀⣤⡴⢶⣤⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣼⣿⣿⣿⡇⠀⣀⠀⢀⠶⠿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠙⠃⠀⠀⠄⢡⣄⠉⠀⠈⠙⠳⢶⣾⣷⡆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢹⡟⣱⣿⣿⣿⣿⠀⢠⠍⠄⠛⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣶⣦⣦⣀⣀⡄⠀⠁
⠀⠀⠙⢿⣿⢿⣿⣿⣿⣿⣧⣼⠾⠁⣿⣿⣿⣿⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠹⠿⢿⡷⢺⡏
⠀⠀⠀⣸⡏⢸⣿⣿⣿⠇⢀⣠⣤⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇
⠀⠀⠀⣿⡇⢸⣿⣿⣿⡄⠘⠛⠁⣸⣿⣿⡿⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣄⡙⡃
⠀⠀⢰⣿⠃⢸⣿⣿⣿⡇⠀⠄⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢿⣿⣿⡆
⠀⠀⢸⣿⠀⢈⣿⣿⣿⡇⢀⠀⣸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⣿⡟⡀
⠀⠀⣾⣿⣀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣘⣻⡿⠃
⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⢿⣿⡇⢈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⢾⣿⣿⡇⡀
⠀⠀⣿⣿⣷⡟⢹⣿⣿⣿⠀⣼⣿⣿⣿⠆⠀⠀⠀⠘⠁⢠⣼⣿⣿⣿⣿⡇⠀⠻⠿⠟⠛⠿⡦⠀⠀⠀⠀⠀⣀⠀⠀⣠⣾⡟⢀⣀⣶⡶⠀⠀⠀⣿⣿⢻⣷
⠀⠀⣿⣿⡟⠻⠿⣿⣿⣿⡆⣿⣿⣿⣿⣴⣧⠀⠠⠀⠀⠀⠹⢿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⡙⢿⣿⣿⣿⣿⣿⣧⡄⠀⣿⡏⢾⡟
⠀⠀⣿⡆⠚⣷⣤⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇⠐⠠⠀⡆⠀⢀⣤⣿⣿⣧⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣾⣿⡿⢻⣿⣿⣿⡀⢿⡇
⠀⠀⣿⣇⢸⣿⢁⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡀⠲⠋⢀⣴⡾⠛⢉⣩⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣾⣿⣿⣿⣿⠁⢸⣇
⠀⠀⣿⣿⣸⣿⣇⢹⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡦⠀⢼⣿⣡⠄⢸⣿⣃⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⣿⣶⣶⣶⣤⣀⢀⣤⡿⣿⣿⣿⣿⣿⡁⠀⠀⣿⡀⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣩⠿⠟⠶⢀⣬⣁⢀⣀⣈⣉⣁⣀⣈⣉⣉⣉⣉⣉⣁⣠⣄⠈⠉⠙⠛⠋⠀⠹⣿⠃⢹⣿⣿⣿⣿⣿⠀⠀⣽⣿⣿
⠀⠀⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣾⠿⠴⠦⠤⢄⣀⣀⣀⣀⣀⣁⣄⣈⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢀⣾⡿⠻⢿⣉⣥⣠⣄⢨⣿⣇
⠀⠀⣿⣿⣿⣿⣴⢹⣿⣿⣿⣿⣿⣿⡇⣶⠀⣠⣀⠀⠩⠉⠉⠙⠉⠟⠛⣿⣛⣛⣛⣛⣀⣀⣠⣄⣾⣷⣦⣽⣛⣉⣛⠈⣉⢈⣿⣇⠀⠀⠉⡉⢉⠁⢸⣿⣿
⠀⢰⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣃⣠⣴⣶⣶⣶⣶⣤⠀⣤⣯⣭⣉⣭⣭⣭⣉⣉⣉⣉⣉⣉⣩⣍⣉⣭⣿⣯⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⣿⣾⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⢿⣧⣁⣀⣀⣀⣀⣀⣉⠈⢩⡜⣿⣷⣿⡿⢶⣿⣷⣯⣹⣿⣏⣹⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠹⢿⣿⣉⣴⣿⣿⣿⣤⢸⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣉⡉⢩⠟⣿⣿⣿⣟⠋⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢹⡿⠿⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣦⣿⣦⣤⣤⣤⣿⣤⣬⣥⣤⣭⣤⣤⣤⣤⣴⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⣶⣶⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠉⢁⣈⠙⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⢃⣴⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠸⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠛⠛⠻⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣈⠕⠈⣉⠚⠉⡸⠋⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠤⠀⠀⣤⡄⠀⣠⡤⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠒⠁⢂⠰⢦⡀⠀⢀⠀⠹⣆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠃⠀⢠⠄⠁⠀⢠⢷⡌⠁⠀⠀⠸⣆⠁⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⣀⠖⠚⠉⠁⠈⠉⠙⠒⠶⢿⡆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠠⠀⠒⠂⠒⠰⢬⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠚⢳⣖⠚⠋⠛⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠤⣿⡄⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⣦⣀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⣷⠀⠀⠀⢠⠊⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠤⠋⠉⠁⠒⠘⢯⠙⠛⠶⠲⠿⣷⡀⠤⢀⣀⣟⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠠⠆⠀⠀⠘⠁⠘⠓⠦⣄⠀⠀⠘⠻⢦⣀⡽⠉⠱⣤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡤⠤⢤⣹⣷⡄⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⣄⠀⢠⡏⢹⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢁⡔⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⢸⡀⣾⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⢀⢘⣷⣧⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠁⠀⠀⠀⠀⠀⠐⠈⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⣀⡀⢀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠟⠛⠋⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⢶⡀⢠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⠏⡾⠉⢻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣼⣿⣿⣿⣿⡞⠁⠀⢈⣹⣿⠟⠛⠃⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣿⡿⠿⡇⠀⠀⢀⣾⣯⣴⡄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⣻⡟⢿⡼⡿⢻⡿⠿⠻⠿⠿⣿⣿⣷⣿⠇⠀⢀⣨⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⢀⠔⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡦⠚⠀⠀⠛⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⡥⣴⠀⢰⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢀⠌⠀⠠⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠢⡀⠀⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠂⠀⠀⠑⠢⡀⢠⠔⠒⠀⠄⠀⢨
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢰⠈⢧⠀⠀⠀⠀⠐⡃⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠏⠀⡄⠀⠀⠀⠀⠀⠀⠀⠑⠄⡏⠒⢫⢣⡠⠃⢸⠈⠉⠀⠉⡄⠀⠘⢀⠀⠁⡀⠠
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡆⢣⠀⢰⠀⠀⠀⠀⠁⠀⢀⠀⢱⠎⠀⠰
⠀⠀⠀⠀⠀⠈⠀⠀⠘⠂⠀⠀⠀⠀⠐⢄⠀⠀⢰⠀⠀⠀⢱⡘⠆⠀⢧⣀⣀⣀⠃⠀⠸⠣⡌⠁⡀⠁
⠀⢰⣄⡀⠂⣀⠄⠐⢆⠀⠀⠀⠀⠀⠀⢀⠃⠀⠂⠀⠀⠀⠀⠂⠀⠀⡟⠀⠀⠨⠉⠉⡀⠀⠈⠉⠉⠀⠀⢀⣀⡀
⠀⠆⡀⠈⡂⠸⠤⠤⠈⡷⠶⠶⠶⠖⠂⢁⡒⠒⠒⠒⠒⠂⠈⠁⠀⠠⠃⢀⠒⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢁⡞⠁
⢀⢠⢀⠔⠠⠀⠀⠤⠤⣇⠀⠀⠀⠀⢀⡁⠀⠀⠈⠉⠉⠙⠛⠓⠂⠐⠲⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠞
⠃⠉⠀⠐⠁⠀⠀⠀⠀⢸⠀⡄⠀⠆⠀⠁⠀⠘⠀⠀⠉⠁⠚⠀⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⣀⣀⠀⢀⠀⢢⠀⠄⠀⠈⡅⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠅⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⢄
⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠘⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣴⣢⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⢀⡀⠀⡀⣉⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢰⣿⠇⢠⣀⣨⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠋⠀⠠⡄⠛⠉⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠒⣤⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡗⠀⠀⠀⠀⠀⠀⠀⢠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠏⠀⢷⡀⠀⠀⠀⠀⠖⠈⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠁⠘⢧⡀⠈⠻⣤⡀⠀⠀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠶⡈⢷⠀⠀⠈⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠉⠀⠀⠛⢾⣧⣀⢾⣧⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠃⠀⠀⠀⠀⠀⠉⢿⣳⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣦⠀⠀⠀⠠⠀⠀⠀⠈⣿⠈⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢰⡄⣿⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⣾⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⣿⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⣇⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠃⠀⠀⠀⠀⠀⠀⢸⠂⠀⡀⢸⡿⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣄⠀⠀⠀⠀⠀⠀⠀⢈⣀⡀⠁⠸⣇⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠀⠀⠀⠀⠀⠀⢀⡆⠀⣤⠃⢘⡟⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⣺⡿⡿⢿⠀⠸⢃⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠸⠋⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⢳⣦⣀⣀⣤⣄⣠⡟⢀⠀⠘⠀⠀⣴⣷⣾⣶⣾⣷⢤⣌⣩⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡝⠋⠲⠶⢾⠏⠀⣈⣀⡅⠀⢀⣿⣿⣿⠛⠋⠁⠈⠓⢻⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⠄⢱⣄⢘⡇⠀⠈⠉⠉⠀⠀⣀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀⢟⠀⢿⡿⠿⣶⠾⡿⠟⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣷⡞⡼⠃⢠⠀⠘⠀⠀⠀⢹⣿⣿⣿⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢘⣿⠇⠇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄⠀⠀⠀⠀⠀⠸⣧⣤⣄⣀⣀⣀⡀⠀⣨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠈⢹⣦⡀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡄⢊⠻⡄⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠈⠆⠠⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠸⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⢠⣷⣄⡀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠹⣄⢀⣬⣶⠟⡀
⠀⠀⠀⠀⢀⣇⠻⢿⣿⣿⣷⣶⣿⣷⣶⣤⣤⣀⠀⠀⡀⡄⠚⣀⣴⣿⣿⣿⡏⠊
⠀⠀⠀⠀⠀⠙⠓⠶⢤⣩⣿⡿⠿⣿⣿⣿⣿⠿⣄⣤⣤⣶⣿⣿⣻⣯⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣉⣉⠿⠻⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⠀⢀⣶⣦⡀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⣼⠏⣿⣤⣴⠟⠁⠈⠛⠛⠛⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⢻⡷⠟⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠀⠀⠀⠀⢀⣶⣦⣀⣴⡟⠷⠶⠶⠶⠶⢶⣤⡀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⣠⣤⣤⣼⠇⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢠⡾⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠟⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⠃⠀⠰⠾⠛⠛⠛⠻⣦⠀⠀⠀⠀⣴⠟⠛⠛⠛⠷⠆⠘⢷⣄⡀⠀⣿
⠀⠀⠀⠀⢀⣴⠿⢻⣿⠁⠀⠀⠀⢠⣴⠶⢶⣤⠀⠀⠀⠀⠀⠀⣤⡶⠶⣦⡄⠀⠀⠀⢹⣿⠟⠿⣦⡀
⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠻⣦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠸⣧⣤⡶⠶⣦⣄⣸⡇⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣠⣽⠟⠁⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⠶⠶⠶⠶⠶⠶⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠉⠉⠛⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⠀⠀⠀⠀⠞⠋⠉⠀⠀⠀⢴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡞⠀⠀⠀⠀⢀⠄⠀⠀⠄⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠇⠀⠀⠀⠀⡡⠤⣰⡆⠀⢦⣰⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⠀⠀⠀⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠙⠳⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠶⠶⠶⠖⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⡆⠀⠰⠀⠀⠀⠀⠀⠀⠐⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢱⡄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⢷⡀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⠀⠀⠀⠠⠤⠀⠀⠀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠤⠄
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠀⠤⠤⠠⠤⠠⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⠀⠈⠉⠁⠉⠉⠉⠉⢉⣈⠳⠾⠛⠒⠀⠀⠀⠒⠒⠒⠒⠂
⠀⠀⣰⠏⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠆⠀⠀⢀⣀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠒⠂
⠀⢠⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠶⠂⠈⠉⠃⠀⠰⠶⠶⣦⣴⣶⣶⣶⣤⣤⣤⣤⣤⠤⠤⠤
⠀⡟⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⠀⠒⠒⠤⢠⡤⣴⠂⠀⠙⣦⣀⣀⣀⣀⡀⢀⡀
⢸⡇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢀⣠⡶⠷⣄⠀⠀⠀⠀⠀⠤⠤⠴⢄⣀⢹⡇⠀⠀⠹⠏⠛⠙⠋⠁
⠸⠇⠀⠀⠀⠀⠀⠀⠘⠦⠀⠠⣤⣶⡿⠁⠀⠀⠈⠓⠤⣄⣀⠀⠀⠀⢰⡆⠀⠘⠃⠀⢀⡿⠠⠴⠶⠒⣂⡀
⠀⢀⡄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣀⠀⠀⠀⠀⠀⠀⠉⠀⠚⠓⠚⠃⠀⠀⠀⠀⠈⠁⠐⠀⠀⠀⠉⢹⠆
⢠⡟⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠄⢀⣀⡀⠀⠀⠀⠀⢤⠀⠀⡀⠀⠒⠤⠀⠀⠀⢤⡖⠁
⣸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣳⢄⡈⠃⠤⢀⡀⠀⠠⠂
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⡀
⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠤⠾⠿⡀
⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠤⠶⠒⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⢀⣧⣀
⠀⡿⠶⠶⠄⠀⠀⠀⠀⠄⠀⠀⠀⠰⠀⠘⠷⢢⣄⡀⠀⠀⠀⠀⠀⠀⠛⠛⠲⠦⣤⡀
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠶⠦⣤⣀⣀⣀⣀⣀⣄⣹⡇
⠃⠀⠉⠉⠁⠰⠶⠶⠦⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⠛⠛⠿⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣤⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣷⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠆⠠⢾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠃⠀⠀⠀⠀⠀⢀⠀⠀⠂⢀⠒⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠀⠘⢁⡀⢤⡤⠀⣀⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠒⠒⠋⠁⠀⠀⠀⠀⠀⢸⠉⠙⠛⠃⠀⠈⠟⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡀⠀⠀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣄⣀⣀⣤⣤⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⢷⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠈⣷⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠘⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⠒⠒⠀⠐⠀⠀⠠⠤⠤⢤⣤⠤⢤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠔⠒⠒⠒
⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠲⠶⠤⠤⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣶⡾⠿⠒⠐⠒⠒⠶⠶⠶⠒⠲⠖⠒
⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⣀⡀⠀⠆⠈⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠤
⠀⠀⢰⠋⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣀⡁⠀⢤⠤⠔⠒⢀⡀⠀⣀⡀
⠀⢰⡎⠀⠀⠀⢀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠉⠉⠀⠀⠀⠀⠀⠈⠙⠋⠛⠛⠛⠿⠿⡿⠿⠷⠾⠿⠿⠷⠶⠒⠚⠂
⠀⡎⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠈⠉⠃⠐⠒⠲⢰⣶⠂⠀⠀⢻⣦⣠⣀⣀⣀⣤⡤⠄
⠀⡇⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⣧⡀⠀⠀⠀⠀⠀⠰⠦⠤⣤⢤⣀⡀⢹⣇⠀⠀⠀⡿⠛⠛⠉⠉⠉
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⣀⣠⣶⣿⠋⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠠⣤⠉⠁⠴⠇⠀⠀⢀⣷⢤⡤⠤⠤⠤⠤
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠠⠤⡤⠴⣿⠀⠀⠀⠀⠀⠐⠛⠃⠤⠄⠀⠀⠒⠓⢲⡄
⠀⢠⡟⠀⠈⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⢰⡏⠀⠀⠀⠀⠀⠈⠛⠳⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂⠠⠤⠤⠤⠄⠀⠀⠀⠘⢧⡀⠀⢦⡀⠀⠙⠓⠀⠀⠀⠐⡿⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠶⢦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⡀⢈⡵⢦⣄⠉⠒⠂⣠⣀⠀⠀⠊
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢀
⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⣀⣠⣾⡇
⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠤⢿⣧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠐⠀⠰⠿⠶⣤⣤⣀
⢀⡸⠛⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠹⠋⡶⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣦⣀
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⢦⣤⣄⣠⣤⣤⣤⣤⣤⣈⡿
⠿⠋⢉⣉⠁⠀⠀⡦⢴⣦⣤⣤⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠉⠀⠉⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠛⠉⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⢀⡴⢫⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠃⠀⠀⠀⠀⢠⠞⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠃⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⠖⠒⣶⣶⠶⡶⣦⣤⣀⣀⠀⢸⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠙⠛⠃⠀⠀⠀⠉⠙⠛⢾⠀⠀⠀⠀⠀⢸⡿⠤⠤⠶⠶⢶⣶⣶⣦⣤⣤⣤⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠈⠻⠿⠂⠀⠀⠀⠉⠙⠛⢷⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠓⠶⣦⣤⣄⣀⣀⣀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⡙⢀⠄⠀⠀⠀⠀⢸⡷⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣴⠟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠛⠛⠷⠶⠶⠶⣶⣤⣤⣤⠇
⠀⠀⠀⠀⠀⠐⠶⢤⣤⣻⣆⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠶⠖⠒⠂⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣤⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢀⣀⣤⣤⣴⡶⠶⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡯⠥⠤⠤⠤⠭⠍⣛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡶⠆⢠⡄⠀⠓⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⣿⠀⢠⡇⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⠹⡇⢸⡇⠀⡶⠛⠳⣦⠈⣁⣈⣿⠉⣿⠋⠉⠋⠹⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⠀⢿⣾⠀⠸⣿⠛⠛⠛⣾⠉⠈⡏⠀⣿⠀⠀⠀⠀⠘⢷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠘⠋⠀⠀⠙⠶⠞⠃⠈⠛⠛⠻⠛⠛⠛⠀⢀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠿⠿⠿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠟⠛⠋⠉⠙⠿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⢿⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠆⠀⠀⠀⠀⠀⢀⡾⠁⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢠⣿⣇⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢸⣿⠁⠀⠀⠀⢹⣿⣿⣏⠛⠉⠛⠻⠿⢷⣶⣤⣄⣘⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠶⠶⠛⠛⠛⠛⠻⣿⣿⣿⣿⡷⠶⣶⣶⣤⣤⣄⣀⠀⠀⢸
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡦⠀⠀⠀⠀⠈⠉⠛⠻⠿⣷⣾
⠀⠀⠀⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⣿⣯⠀⠉⠛⠻⠷⢶⣶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠀⣠⠀⠀⠀⠀⠀⠀⠀⣿⡷⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠛⠛⠿⠶⣶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⢿⠃
⠘⠿⢶⣶⣦⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⡿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣇
⠀⠀⠀⠀⠉⠙⠿⣿⣷⣦⣄⡀⣀⣀⣤⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⡶⠾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣤⣶⣾⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠿⠿⠿⠿⠿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠴⠶⠶⠀⠐⠲⠶⠶⠤⣉⣙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡶⠶⠀⢰⣦⠀⠈⠛⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⣼⣿⡀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⣿⢿⣧⠀⢸⣿⠀⠀⢀⣀⣀⣀⠀⠀⢰⠟⠛⢿⣶⠾⣿⣷⣴⠾⢷⣾⢿⣿⠛⠀⠛⢻⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⣿⠀⣿⡀⣾⡇⠀⢠⣿⠉⠉⠙⢿⡄⢀⣠⣤⣴⣿⠀⢸⣿⠁⠀⠈⠁⠀⠻⣧⡀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⡏⠀⢹⣧⣿⠃⠀⢸⣿⠿⠿⠷⠿⢷⣾⠋⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⢿⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠇⠀⠈⢿⣿⠀⠀⠈⢻⣦⣤⣤⣶⠈⠻⣦⣤⣼⣿⣤⣾⣿⣶⠆⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢴⣄⠀⢀⣰⡟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠛⠙⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣠⣤⣤⣤⣤⣄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣛⠛⢻⣿⣝⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣋⣤⣴⣾⣿⣿⣿⡿⠿⣿⢿⡿⢿⣿⣿⣷⣾⠿⠿⠿⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⠿⠛⠉⠁⠀⠀⠈⢧⡀⠸⣧⣿⠿⠋⠁⠀⣀⣀⣀⣀⡀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣴⠾⠛⠋⠉⠉⠉⠙⠻⢦⡀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⣋⣹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⠿⢿⣿⠿⢛⣩⡽⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⣿⠳⣄⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠚⠋⣠⠶⢋⣡⣴⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⡈⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⡟⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⣿⣿⡿⠛⠉⠀⠉⠛⠷⠶⢶⣤⣴⣶⠶⠿⠛⠉⠀⠀⠙⠿⣷⡄⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣥⢷⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠟⠁⠀⠀⠀⠀⠐⢆⠀⣄⠀⠀⠀⣰⣤⣀⡀⠀⠀⠀⠀⠀⠈⠙⠀⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣠⠴⠚⠋⠉⠈⢧⠘⢶⠀⢠⡏⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡿⣿⡿⠾⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢀⡠⠛⢁⣠⣤⣤⣤⣤⣼⠀⠈⢧⣸⣥⢤⣶⣶⣶⣾⡶⣶⠒⠒⠤⠀⢹⣷⣍⢻⡄
⠀⠀⠀⠀⠀⣴⣾⣿⢋⡴⢖⣂⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡩⠴⠾⠥⠤⠽⠿⠿⢳⡟⡠⠀⠀⠻⣏⡛⠛⠛⠉⠀⠀⠀⠈⠑⠂⠀⢸⡇⠹⢊⡇
⠀⠀⠀⠀⠈⠉⠟⣿⢸⠓⣿⠛⠃⠸⠇⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠡⠊⠀⠀⢀⣀⣀⣀⡠⠔⣫⡴⠃⠀⠀⠀⠙⢿⣥⣖⣒⣒⣀⣀⡤⠴⠀⠀⢸⡇⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢿⡘⡀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⠴⠒⠋⣡⠖⠒⠶⢄⠀⠀⠀⠉⠛⣿⡆⠀⠀⠀⠀⠀⠀⠘⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣤⣤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⢀⣀⣠⣤⣶⠟⠋⢹⡄⠀⠀⠀⠀⣆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠛⠷⢿⢿⣿⢿⣿⣿⣿⣦⡀⢿⡄⠀⠀⠀⢿⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠃⣄⣰⣶⣾⠿⠸⠋⣀⣈⡉⠉⠻⢷⣾⣿⡀⠀⠀⢸⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣇⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣰⠏⠼⠟⢉⣠⡴⠖⠂⠀⠉⠉⠉⠛⠶⣤⡉⢹⡇⠀⠀⠸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢹⡄⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢠⣏⣠⡴⠞⠋⠀⠀⢀⣀⣤⡴⠶⠂⠀⠀⠀⠙⠻⡇⠀⠀⠀⡟⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⢿⡄⠀⠉⠳⠄⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⡾⠛⠁⠀⠀⠀⠀⠀⠉⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⣼⡏
⠀⠀⠀⠀⠀⠀⣴⣿⡟⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⣸⡟⠙⢷⣦⣀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠸⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⢀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣿⣶⣶⣶⠿⠿
⠀⠀⠀⠀⢀⣿⠃⠀⠀⠈⠙⠷⣦⣄⡀⠀⠀⠉⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⣀⣤⣂⣤⡶⠟⠋⠀⣸⡏
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠾⠿⠟⠛⠉⠀⠀⢀⣴⠟
⠀⠀⠀⠀⢸⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⠶⠶⠶⠶⠾⠿⠿⠿⠛⠶⠶⠶⣾⠿⡿⣿⠁⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⣼⡟⠈⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢻⣸⣿⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⣾⠀⢰⠇
⠀⠀⠀⢐⡟⠃⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣬⣯⣽⣿⠗⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⡏

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣤⢢⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⢀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢈⠓⣿⠟⣾⢴⠂⢀⣰⣦⠀⠀⠠⠀⢠⡟⠈⠀⠀⠀⢑⣶⡤⣄⡀⠀⠀⠀⠀⠀⢠⡄⡂⠂⠀⠄⣠
⠀⠀⠀⠀⠀⠐⠀⣀⢻⣯⠽⠋⠲⢄⠀⢸⠟⠁⠀⠀⠀⠀⢸⡇⠀⠂⠀⠀⢨⣿⡟⢡⠹⠆⠃⠀⠀⢀⠙⢈⣳⣴⣞⡴⢻⡆
⠀⠀⠀⠀⠀⣧⢠⣾⣿⣱⣾⣳⡆⣀⠀⠀⠀⠀⣀⣤⡄⢰⢶⡄⠀⠀⠀⠀⢸⣿⣷⠿⠶⠀⢗⠀⠎⣩⢿⣿⠋⢛⣿⣆⢎
⠀⠀⠀⠀⠀⠑⠋⢿⣿⣃⢿⣾⡏⠩⠂⠈⢧⡀⠀⢀⠈⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⢉⣈⢐⠘⢳⡞⢻⢾⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠒⠓⠻⣯⣤⣉⢡⠀⡀⠀⠀⠉⡙⠟⣾⡟⠀⠈⠂⠀⠀⣀⠀⠁⠹⣿⣧⣼⢯⣿⠛⢾⣾⣵⣿⣿⠛⠂
⠀⠠⣺⡄⠀⠀⠀⠀⠀⠈⠛⢿⣿⡏⠻⢶⣄⠼⠒⠀⠹⡀⠀⠀⠀⠀⠰⠿⠀⠀⠀⠈⠉⠓⠰⠟⡁⣤⣸⡟⠿⠟
⠀⠀⠏⠟⢂⡀⡀⢀⠁⢒⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⣆⡀⢀⣀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡀
⠀⠀⠀⠚⠈⠻⣷⡄⠀⠀⠀⠈⠃⠀⠀⠀⠀⢰⡖⠀⠀⠀⣀⡄⠁⠀⢹⣿⣷⣄⡀⠀⠀⠀⠀⠋⠀⠲⡂⠀⠀⠀⠀⠀⠀⡾⡯⠐⠀⠠
⠀⠀⠀⢂⠴⢤⣿⣎⢉⣴⣄⠀⢀⣀⣀⣀⣴⣬⣄⣀⣼⡏⠀⠀⠀⠀⢀⣸⣿⣿⣿⣾⣤⡄⠀⠀⡀⢀⠀⢀⣤⡴⢶⣼⡁⠀⢀⠀⣀
⠀⠀⠀⠐⠁⠚⠻⣿⣿⣟⣋⡀⠊⠙⢿⣿⣿⡿⣿⣽⣿⠯⠠⣆⡀⠀⠚⢻⣿⣛⣿⠿⡽⢿⡾⣾⣉⡽⢋⣄⣹⣷⠟⠘⠃⠆⢳⠂
⠀⠀⠀⠀⠀⠈⠀⠈⢿⣿⠉⠋⠀⠈⣈⡹⠟⢺⣯⣽⣏⣀⣼⣷⣾⣾⡰⠿⣿⡿⣯⡴⣿⣾⠀⠸⢿⣦⣤⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠐⠐⠘⢋⠝⠐⠆⠁⠉⢿⣿⣿⠟⠉⡛⠻⣿⣿⠁⠻⠃⢻⣇⠀⠀⣠⠟⠁⠀⠆⠀⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣼⣭⣷⠀⣿⣷⠾⢘⣛⠀⢸⣿⡄⣸⣿⣿⣿⠀⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠛⠋⠀⠛⠑⠀⠸⠟⠀⠘⠛⠀⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢦⢀⡤⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⡄⣾⡗⢐⠉⡭⡰⣳⢔⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢐⡖⠒⠛⠀⠈⣖⢶⣲⢶⢤⡟⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⢠⡎⠀⠀⠀⢀⠄⠛⠀⢠⡤⠂⠻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⢣⡀⢀⡄⡎⠀⠀⠀⢀⣡⣤⣴⣘⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡆⠀⠙⣄⠙⠧⡠⠄⠖⠿⠻⠿⠟⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⣀⠈⠗⠦⣄⣀⣸⣅⣀⣀⡠⠎⠁⠑⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠓⠲⠼⢿⣿⣭⣻⣍⣀⣀⠠⣞⣵⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠏⠁⠈⠻⢦⠘⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡟⣛⣋⣸⡀⠀⠀⢮⠳⡌⠓⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡛⣿⣝⡋⠀⢤⢄⣸⡀⢗⣀⡇⠀⣀⣠⣤⡃⣒⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣩⣯⣏⣙⣓⣆⣪⠳⣧⣬⠉⡉⠈⡿⠟⠋⠑⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠁⠀⠉⠉⣉⡠⠔⠚⢣⠹⣿⣀⡧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⠔⠒⠉⣄⡤⠔⢒⠉⢇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠔⢂⠩⣀⡨⢥⣶⡊⠀⠀⣸⢿⠀⠘⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣊⣁⠤⠤⣒⠮⢭⡔⣲⣿⣭⡽⢖⠲⣷⠝⢛⣖⣢⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣉⣩⣥⣤⡶⣾⠶⠷⢷⡿⣥⡤⠤⠿⠆⠖⠓⠛⠛⠛⠛⠛⠉⠀⠉⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣶⠿⠿⠛⠛⠛⠛⠿⠿⠿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⠉⠉⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣟⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⠶⢶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⣴⡾⠟⠉⠁⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠹⣿⡄⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡇⠀⠀⠀⠀⠀⢠⣿⠃⠀⠘⢿⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⢰⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣾⠟⠁⠀⢀⣤⣶⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⣶⣤⡀⠀⠀⠀⢻⡇⠀⣼⡟⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⣼⡟⠀⠀⠀⣾⠏⣁⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣈⠹⣷⠀⠀⠀⢸⣿⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣿⡇⠀⠀⠀⠀⢰⣿⣿⣈⢹⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣍⣙⣿⡆⠀⠀⠀⠀⢸⣷⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠈⣿⡇⠀⠀⠀⠀⢸⣧⡙⠛⢻⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⡌⠛⠛⣿⡇⠀⠀⠀⠀⢸⣿⠉⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣀⢀⣿⡇⠀⠀⠀⠀⠀⠛⠿⠾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⠿⠟⠀⠀⠀⠀⠀⢀⣿⣇⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠸⣷⡀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⠿⠷⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣶⣶⣤⣤⣤⣤⣤⣶⣶⠾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⡴⠶⢶⣴⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣷⣶⣾⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠀⠀⣉⣉⣉⣽⣿⡟⠛⠁⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠠⠔⣊⣉⣩⣭⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⠶⠛⠉⣉⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣋⣠⠴⠒⠋⣉⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣋⣁⣤⣴⡾⢿⣟⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⣋⣡⣴⣿⣿⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣵⣶⣿⣿⣿⣛⣉⣡⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⣶⡶⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡙⢏⣿⣿⡛⠖⠛⠋⠀⠀⠀⠀⠀⢀⡶⢛⣡⣾⣿⣿⠃⠐⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡘⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠛⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠙⠋⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⣾⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣠⣶⣴⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⢺⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣼⠿⡯⠋⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣤⡀⠀⠀⢠⣾⢛⡻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠛⢏⠙⠌⠋⠻⣯⡀⠀⠀⠀⠈⠙⢷⣤⣀⠀⣀⠀⡀⢤⣀⠀⠀⠹⣿⣦⡀⠹⣷⠀⠀⣹⡯⠀⠀⢹⠶⣄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢿⡣⠕⠨⠢⠁⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⠪⡦⡆⣽⣿⣿⣿⣿⣿⣷⡀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢀⣾⣿⢿⡳⢝⠢⡈⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠨⣟⢿⣶⣶⣿⡟⠛⠉⠉⠈⣿⣿⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⡿⣙⠦⡉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⡀⠀⠀⠀⠀⡠⢋⣴⣿⠿⣿⡇⠀⠀⠀⠀⣿⣿⡷⠛⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⣸⡟⠑⢮⡃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣄⠀⠀⣰⣿⣿⠗⣋⣽⡇⠀⠀⠀⠀⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⣭⣟⠿⣿⣾⣿⣿⣿⣯⣵⣾⠃⠀⠀⠀⠀⠸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠉⠀⠈⠛⣿⣿⡿⠟⢻⡇⠀⠀⠀⢠⣶⣾⣿⣿⡀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢋⣠⠖⠛⣧⠀⠀⢰⣿⣿⣿⠿⠿⠿⠶⣶⣤⣄⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠋⠀⠀⠀⠹⡄⢰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁
⣿⠿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣶⠶⣦⣄⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣶⣥⣀⠰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⡀⢻⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⢟⡿⠉⢿⡀⢹⡇⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⡿⢋⡞⠀⠀⢸⣧⠀⠇⠸⣿⣿⣿⣿⣿⣿⣿⣶⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣹⡏⢰⡞⠀⣰⡇⠈⠿⠀⠂⢀⣿⡿⣿⡏⢾⣿⣦⣦⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡬⠙⢻⣿⣿⣿⣯⣿⣿⣿⣾⣿⣿⠃⢸⠀⢰⣿⠁⠀⠀⠀⢀⠞⣿⣿⣟⡹⣿⡄⢀⡙⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠳⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⠿⠀⠈⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣾⠟⠉⠛⠟⠙⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢉⢐⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢰⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠓⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣦⡊⠀⠀⠀⢹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣼⣿⣿⣿⡧⠀⠀⠀⠀⢻⣿⣿⣷⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⠙⠛⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠘⢿⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣹⣿⣿⣿⠃⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣓⣦⣼⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠛⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⣤⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠅⣰⠀⠀⣤⠀⠀⢳⣀⠀⡆⠈⠁⡀⢶⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠶⡘⢃⣤⡘⠊⠷⢿⣽⢶⡗⡈⢠⣼⡇⠉⣿⡆⢆⡀⣴⠆⠀⢸⡀⠀⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⠤⢻⠇⠀⢀⠰⣿⡁⠐⣴⣇⣈⡈⢸⡯⣠⣿⣠⣾⣿⡵⣾⢷⠿⢃⡆⣈⡅⠰⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⠉⡀⠀⠈⠁⡼⣄⠀⢾⡄⢰⣾⣽⣿⣧⣾⣷⢙⡿⢿⣿⣿⣿⣷⡛⡷⣾⠁⢻⡅⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢰⠛⢙⣻⣯⣿⣸⣷⣎⣿⣿⣿⣿⣿⢿⣿⣿⣾⢿⣿⣾⣿⣿⣴⢄⣴⡮⣱⣦⢀⡔⠀⠈⠁⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢀⡀⢀⣿⣿⢾⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⠿⢿⣾⣿⣷⣾⣿⣿⣿⣿⣷⣏⠈⠁⠀⣀⠘⠆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⡟⢻⠀⡉⢽⣀⣽⢛⣼⣿⣿⣿⠿⣿⠇⠀⠠⠆⠀⠀⠀⠙⢿⣻⣿⣿⣿⣿⣿⣿⡿⣿⣿⠀⠾⠋⠲⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⠾⣣⣴⡞⢿⣩⣿⢻⣿⣿⣏⣶⡀⠀⠀⠀⠀⠀⠀⣀⣴⡾⢿⣿⣿⣿⣿⣿⣿⣿⣼⣯⡘⣷⢻⡿⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢹⡍⠳⠛⣏⢰⡟⣻⣷⣿⣿⣾⣿⣁⣀⣀⣀⠀⠀⢠⣾⣿⣤⣶⣊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢰⣆⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢢⠓⢶⠲⣷⣸⣽⣿⣿⣿⣿⣿⣿⢿⣯⣿⣿⣿⡆⠀⡿⠿⢿⡿⡿⠟⢿⣿⣿⣿⣿⣿⣿⣭⣿⡿⣠⣤⣀⣔⡂⠰
⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠐⣤⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣥⣿⡿⠋⠀⠀⠀⠀⠀⠁⠀⠀⠾⣿⣿⣿⣿⣿⣽⣟⣹⠏⢿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡦⣄⣶⣎⣿⣿⣷⣿⣿⣿⣿⠆⠉⠛⠉⠀⠀⠀⣤⣴⣶⡶⠆⠀⠀⠀⠀⢻⣿⣿⣿⢿⣿⡟⣥⢸⡿⠁⠈⠢⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⡏⠩⠞⠙⠶⢿⣿⣷⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠈⣉⣹⣤⣤⣤⡔⠁⠞⢻⣿⣿⣿⣿⣿⣿⡟⠟⠓⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠘⠂⣠⡼⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣧⣂⡄⢠⠀⠀⢀⣾⣟⣛⣩⣤⡶⢀⠂⣤⣼⣿⣿⣿⣿⣿⣿⢉⡆⠊⠀⠀⠀⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠠⢄⠀⢡⣼⡶⣽⢲⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⠀⠴⠛⠿⠟⠛⠲⡾⠇⣸⣿⣿⣿⣿⣧⠙⠷⠼⣧⡾⠃⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢿⡷⣟⣹⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡄⠀⠀⠀⠀⠀⠲⣾⣤⣾⣿⣿⣿⣿⣹⣿⣷⣠⡄⠈⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢞⡉⢛⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⣶⡴⠿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣛⣉⣡⠾⣴⠦⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠒⠈⣡⠿⠏⠉⠙⠉⠙⣛⣛⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⠛⠻⠆⠀⠄⠈⠈⠉⢻⣿⣿⡉⠛⠿⠋⣁⠘⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠉⠉⣙⣿⣿⣿⢿⣿⣿⣇⣀⠘⡷⠄⠀⠀⠀⠀⠀⠀⣹⣿⣇⠙⠀⣤⠀⠉⢠⠆⠀⠀⠘⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢛⣿⣿⣿⢎⠛⠛⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⣸⠧⡎⠁⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣭⣿⣷⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠉⠉⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢡⠏⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣠⣶⠟⠉⢐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣵⣿⣿⣧⣀⠆⠠⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠂⡠⢂⢏⡠⠈⣀⡠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠋⠛⢤⢾⣷⠀⠬⠄⢲⡛⠒⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⠿⠀⠓⠤⠊⠀⠉⠋⠀⠀⠀⠀⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠉⡀⠀⠄⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠆⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡅⠀⠀⠠⠀⠐⠤⡀⠀⠀⠀⠠⠀⠀⠀⢠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠀⠀⢀⠀⠀⠈⠑⠢⠄⣀⣀⣀⣀⣤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⢠⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢁⠀⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠑⠢⡀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠈⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⡤⣀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠰⢀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠷⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⣀⡄⢀⡞⣧⠀⠀⠀⠀⠀⠀⠀⣤⠊⠥⠈⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⡶⡊⠑⠃⡀⠘⡀⠀⠙⠉⠀⡀⢊⠈⠑⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠧⣤⢦⡀⡀⠀⢰⠠⠂⠀⢰⠀⠈⠫⠬⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⡼⠀⠈⠙⢼⢃⣠⢼⡖⢶⣲⣺⠿⠲⡆⠐⢢⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠃⠀⠚⠿⠿⠿⣯⢴⣯⣾⣿⣿⣿⣿⣷⣿⣷⣿⣿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎

⠀⠀⠀⠀⠀⢠⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⡤⣀⢀⣠⡄
⠀⠀⠀⠀⠀⢠⣿⣟⢻⣿⣿⣯⣹⣿⣿⣿⣾⣯⡿⣖⡿⣿⢿⣿⣏⣿⡷⣯⣯⣹⣿⣻⣿⢿⣿⡷⣟⠻⣿⣷⣦⣤⡄
⠀⠀⠀⠀⠀⢘⡛⢯⣽⣿⣿⣯⢣⢇⣦⣳⣸⣯⣿⣯⣟⣿⡮⣿⣻⣯⣿⡮⣻⠛⢫⣿⣓⣱⢇⡨⢙⢯⣞⢿⣾⠃
⠀⠀⠀⠀⠀⢘⡹⡟⣉⣟⣿⣿⣏⣸⣿⣿⡿⣜⠋⣿⣿⣾⡷⢹⣿⣿⢟⣿⢃⣼⣛⢿⣿⡉⣿⢿⢻⣷⣿⡿⣾⣷
⠀⠀⠀⠀⠀⠸⣏⣻⣿⢉⣿⢃⣹⣽⡿⠙⢃⡠⠚⣹⣿⣿⣥⣈⣯⣵⣯⣧⢟⢶⡣⢴⠊⣴⢟⣿⣾⡾⣷⣿⢖⣭⠁
⠀⠀⠀⠀⠀⢸⣿⣿⢯⣿⣯⣿⡷⣃⣤⣾⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣚⣿⣯⡷⢿⣱⡾⠅
⠀⠀⠀⠀⠀⠀⠙⡘⠐⣷⠻⣿⣴⣿⣿⣛⢛⡙⣿⣿⣿⣿⣿⡟⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠻⣯⡝⣫⣿⡿⣏
⠀⠀⠀⠀⠀⠀⡀⠅⠀⣿⠀⠇⣿⣿⣿⠈⠉⠛⣿⣿⣿⣿⡟⠉⠋⢹⡎⣿⣿⣿⣷⣦⣿⣿⣿⣿⣾⡋⠉⠁
⠀⠀⠀⠀⠀⠸⡄⣠⠀⠀⠄⢘⣿⣿⠿⠦⢐⠐⠿⢿⣿⣿⣷⣄⡀⠘⠿⠟⣿⡿⠛⣿⡿⣽⠿⢟⡽⠗
⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢀⠘⠉⠀⠀⠀⠈⠂⠀⠀⣿⣿⡉⠀⠘⠃⠀⡀⠈⠁⠰⣿⢃⣾⠿⠛
⠀⠀⠀⠀⠀⠸⠉⠑⠂⠠⠸⣷⢤⣀⣂⠀⠀⠀⠀⡀⢽⣿⡷⠀⠀⠀⠀⡟⠀⠠⣷⣽⡸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣿⠏⠛⠙⠛⠉⠁⣸⡿⠁⠀⠀⠀⠀⠁⠀⠸⣿⣿⠧⠉⣀⣄⠀⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⣸⡄⠀⢰⣥⣞⡀⡠⠤⣀⠠⢄⣤⡒⠉⠙⠿⠰⡿⠁⠐⡼⣀⠛⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠘⠿⠿⠿⠃⠀⠛⠄⠈⠿⢿⣷⣄⠀⠀⣇⠀⠀⠀⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠋⠑⡀⠸⢿⣳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡆⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⣷⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣾⣷⣆⠀⡃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢠⠠⢻⡿⣿⣿⣁⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⢛⣾⣿⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠙⠻⣯⡟⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⢀⣴⣾⣿⢭⣓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣭⣽⣿⣿⣷⣶⣶⣿⣯⣭⣍⣙⠺⣷⣝⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠿⠛⠛⠉⠉⠀⢠⣾⠋⠀⠀⠀⠙⢿⡄⠉⠉⠛⢷⣮⡻⢟⡺⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⠿⢸⡇⠀⠀⠀⠀⠙⢿⣆⠈⢮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠈⢻⣆⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣭⣛⠛⠛⠋⠀⢀⣦⡀⠀⠀⠀⠀⠈⣿⡘⡏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠂⢀⡞⠀⢿⠀⠀⠀⠀⠀⢻⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠿⠀⠀⠀⠀⢀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠶⣶⣶⣶⡶⠶⠶⠾⢿⣿⠛⠛⠉⠉⠁⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⡤⠶⠛⣋⡉⠉⠛⠳⣦⡀⠀⠀⠀⠸⣷⣶⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⠞⠉⠙⢦⠀⠀⢸⣷⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢹⡄⠈⠉⣽⢻⠙⠛⢛⡿⡟⠛⠛⢻⡟⠉⠉⠉⠉⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠈⢷⣀⡼⠉⣆⣧⣠⣼⡴⢳⡀⢠⡞⠁⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠈⠛⠁⠘⢧⡀⣠⠴⠃⠈⢷⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⢇⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⢆⡽⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⡀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡟⠛⠛⡛⠛⠛⢻⣿⠷⢦⡤⠾⠿⠾⠾⠿⠿⠿⢿⣿⠿⣿⡿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⣠⣤⠶⠒⠚⠛⠒⠒⠒⠶⢤⣄⡀⠀⠀⠀⠸⣧⣀⣰⣿⡿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⡇⠀⢴⣷⣶
⠀⠀⠀⠀⠘⠿⢧⣀⣀⠀⠀⠀⢀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠈⣉⣉⣉⣀⣀⣸⣿⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣬⣿⡀⠈⠛⠛⠛⠛⠁
⠀⠀⣀⣤⠤⠤⠤⠤⠄⣉⡩⠉⠁⠈⠉⠁⠐⠒⣲⡿⠂⣴⡋⠉⠁⠀⠉⠉⣩⡙⢶⡀⠀⢠⡞⠉⠉⠀⠀⢀⣀⡈⢳⡄
⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁⠀⠀⠈⠻⠶⠶⠶⠞⠛⠙⠷⠾⠃⠀⠈⠛⠶⠶⠷⠛⠋⠉⠷⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⢾⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⣀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢠⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠧⣈⣳⡀⢠⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⠿⠻⣿⣿⣿⣧
⠀⠀⠀⠀⠐⢏⡩⠽⠃⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢛⠻⢿⣿⡏⠀⠀⠀⠀⣶⠖⡤⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⣿⡿⠁⠶⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢠⣿⠃⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡴⠁⢘⢷⡀⠈⢦⣉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡸⠃⠀⠀⠀⠀⠀⣤⡼⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⡴⠾⣿⣿⠏⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠚⢷⣄⠀⠀⠀⠀⠀⠉⠀⠒⠙⠀⠀⠀⠀⠀⢠⡾⠳⣶⣦⣀
⠀⠀⠀⣀⡴⠖⠲⠞⠉⠀⢀⣀⣀⡾⠉⠛⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡿⠠⣄⡼⠃⢙⢧⡀
⠀⠀⢸⡋⠀⠀⠀⠀⠠⡶⠋⠁⠉⣹⠖⠂⠀⠙⠿⢭⣿⣷⣶⠖⠒⠻⣏⠁⢀⡿⠀⠈⠓⢶⡁⣰⠇
⠀⠀⠀⢿⣀⡀⠀⠀⢀⡟⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⡜⠷⠋⠀⠀⣴⠦⠴⠛⠁
⠀⠀⠀⠀⠙⢧⡴⠷⠋⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⠶⠞⠁
⠀⠀⠀⠀⢀⣤⢤⡀⠀⠀⢸⡦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⣠⠞⡱⠋⢹⣆⡀⠘⣇⠀⠈⠉⠙⠛⠶⠦⣤⣄⣀⠀⢀⡟
⠀⣠⠞⡡⢊⡤⣴⠏⠈⠙⢦⣾⡳⣤⣀⠀⠀⠀⠀⠀⢠⠟⠻⣿
⣼⣣⠎⢱⣭⡵⠛⢦⡀⠀⠀⠈⠙⢲⡽⠛⠓⠶⠦⢤⡏⠀⠀⢈⡇
⠈⠛⠛⠋⠁⠀⠀⠀⠙⠶⣤⣤⠴⠋⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢧⣄⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠠⣀⠸⢭⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣅⠢⢍⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣍⡾
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠣⠀⠄⠀⡀⠈⠀⠀⠀⠐⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⡁⠐⠉⠂⠀⠀⠀⠄⠀⠀⠀⢀⣴⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣆⠀⠠⠀⠀⠀⡈⠀⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣾⡘⡟⡷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⣿⣿⣿⣿⣿⣿⣿⢹⣧⣳⢃⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⢣⢿⣿⣿⣿⣿⡿⢃⣾⢧⣏⢸⣿⡿⣿⣿⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⣸⡇⢸⣾⣿⣿⣿⢏⣶⣫⣿⣿⣿⢸⣿⣽⣿⣿⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢰⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣷⣿⣿⣿⣿⢹⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣼⣧⣸⣿⣿⣿⣏⡿⣿⣿⣿⣿⣿⣿⢹⣿⢿⣿⣿⣼⣿⣈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣧⣻⡌⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⡻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⡟⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣷⣾⣿⣿⣿⣟⣿⣿⣯⢻⣿⣿⣿⠟⢳⣿⣿⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣿⡿⣿⣿⣿⣿⣿⡟⣿⣿⡿⠁⣼⣿⣿⣿⣿⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠛⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣥⣾⣟⣿⣿⣿⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠻⣿⢿⣿⡿⢛⠯⠋⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⣿⣷⡾⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠁⠀⠈⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢚⡧⠴⣶⠿⠗⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠏⢠⡴⠇⠒⠋⡝⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⡜⢁⣀⠴⠊⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠯⣅⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡟⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⠤⠤⠞⠀⡷⢻⢀⣀⣀⡀⠤⠄⢲⠖⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⡸⡶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⣢⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡀
⠀⠀⠀⠀⣀⡤⠄⠠⢤⡞⠀⠀⠀⠀⠀⡠⠞⠉⠁⠙⠲⢆⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣢⡀
⠀⠀⡠⠚⠁⠀⠀⠀⡘⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⢀⡴⠃⢀⡴⠊⠀⠀⠈⢆⡰⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢀⠎⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⠀⡴⢊⡤⠖⠁⠀⠀⠀⠀⠀⠀⠙⠎⢇⠀⡀⠀⢠⠀⠀⢀
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⣀⣀⣾⠁⢀⡴⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢸⠀⠈⡆⠀⢸
⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠎⠀⠀⠀⠀⣰⠋⠉⢀⠏⠴⠋⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⡞⢢⠀⠸⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠃⠀⠀⠀⠀⡼⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⠀⠀⡇⢸
⡆⠀⠀⠀⠀⠀⠀⠀⠘⡶⠁⠀⠀⠀⠀⡜⢁⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠘⠀⠀⢰⡜
⢣⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢀⠞⠀⢸⠁⢠⠇⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⢣⠄⠈⠀⠀⢸⠃
⠈⣆⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⣼⠖⠂⢸⣠⠋⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠏
⠀⠈⢦⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⣏⠀⠀⠸⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⡴⠀⠀⠀⠘⡄
⠀⠀⠀⠑⢄⠀⠸⡀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⢦⠄⠀⠀⠀⠀⠀⠀⣀⠤⠔⠐⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠳⣄⠙⢄⠀⠀⠀⢀⡤⠊⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣀⣀⣀⣀⡀⠤⠵⠄⠀⠀⠀⠀⠀⣠⡎
⠀⠀⠀⠀⠀⠀⠘⡆⠈⡦⡀⢠⠏⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠳⠃
⠀⠀⠀⠀⠀⠀⠀⣧⠜⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠤⢄⣠⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⠤⢖⡈⠉⠓⠤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠙⢦⡀⠉⠻⣉⠉⠁⣉⠦⣄⠀⠉⠂⠀⠀⠀⡏⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠙⣄⠀⠈⢫⡉⠀⠀⠈⠳⡀⠀⠀⠀⠀⠦⠎⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠈⢧⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⣆⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡠⠄⠤⠤⠴⠆⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠂⠉⠉⠉⠉⠉⠉⠐⠒⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠎⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠉⠒⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⡘⠀⠀⠀⠀⠀⢀⠔⠉⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣰⣶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⣇
⡇⠀⠀⠀⠀⡰⠃⠀⢀⠊⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠱⠈⠙⠢⣀⠀⠀⠀⠀⠉
⡇⠀⠀⠔⠊⠀⠀⠀⠈⠄⠈⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠈⠀⡰⠀⠀⠀⠈⠑⢢⠀⠀⢳⠄⠄⠤⠤⠤
⢓⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⠁⠀⡿⠀⠀⠀⠈
⠘⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠐⠀⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⡇
⠀⢇⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡼
⠀⠘⡄⠀⠀⠀⠱⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠎⠀⠀⠀⢠⠃
⠀⠀⠱⡄⠀⠀⠀⠙⢄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⡰⠃⠀⠀⠀⢠⠃
⠀⠀⠀⠹⡄⠀⠀⠀⠈⠳⣄⠀⠡⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠌⠀⡠⠞⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠙⡉⠐⠒⣲⠀⢠⡖⠒⠂⢉⠉⠀⢀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⣹⠻⠯⡍⠉⠀⢀⠤⠚⠀⠈⠗⢤⡀⠈⠉⡩⠭⠛⣍⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⡘⢢⡓⠂⠴⠃⣄⠀⢊⠃⠀⣆⠙⠤⠐⢳⡐⢣⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠑⠒⠊⠀⠇⠀⠀⢸⠀⠑⠒⠈⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠒⠒⠒⠚⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⢀⡞⠀⠀⠀⠀⠀⠀⢣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⠤⠤⠤⠤⠤⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣴⣇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠄⠛⠉⠉⠀⠀⠀⠀⠘⠛⣿⣦⡌⠙⠶⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⣁⠙⣿⣔⡐⠚⠛⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⣀⣀⠀⠤⠤⠤⠶⢋⡡⡢⢝⣿⣤⣀⣈⡁⠚⣦⡀⠀⠀⠀⠀⠀⣠⠞
⠀⠜⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢀⡴⠁⠀⡀⠐⠂⠉⣀⣀⠀⠁⣒⣒⣂⣭⣉⣷⣾⣿⣭⣤⣤⡄⣀⠈⢳⡄⠀⠀⣠⠔⠁⠠
⠈⠈⠌⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠟⠀⠀⠀⠀⠀⢈⡁⢠⣤⣴⣶⣿⠿⠟⠛⠉⠉⠙⠻⢿⣿⣧⠄⠀⠁⠀⠹⣤⠊⠉
⠀⠀⡐⠀⠀⢉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢰⠏⠀⠀⠀⡀⠔⠃⠈⣡⡴⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⢱⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠀⠀⠀⠀⣠⠇⢀⠃⠀⠀⢀⠂⡠⢔⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⢇
⠀⠠⠠⠀⠄⠀⠄⠀⠀⠡⠠⠀⠌⡼⠉⠉⢣⣠⡏⠀⠀⢀⠀⣰⡗⣩⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⢂⠸⡧
⠀⢡⢡⠉⡌⠈⠌⠀⠀⠁⠠⠉⡜⡵⡤⠠⢼⣿⠁⠀⠀⡌⢠⣟⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⢢⠈⡄⣿⠀⠀⠈
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡐⠀⠀⣾⣭⣄⡀⢰⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢻⣿⣆⢃⣰⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡿⣅⢀⡸⡀⣿⣿⠃⣤⠶⠶⣦⣤⣄⡀⠀⠀⠀⠀⠀⣠⠴⠾⠛⠛⠉⠛⠒⠀⠀⠀⠀⢸⣿⣿⡎⢹⣾⡇
⠀⠌⠌⠀⠡⠡⠡⠀⠀⠌⠌⠤⠡⠡⠀⢸⣷⠀⠀⠀⢱⣿⣿⡀⠀⣠⣴⣦⣤⡢⠀⠀⠀⠀⠀⠀⠀⣠⡶⡶⠿⣷⣦⡀⠀⠀⠀⠀⠸⠹⣿⣷⠈⣿⡉
⠀⡈⡈⠀⠀⠀⠁⠀⠀⠀⠀⠀⢡⠀⠀⢠⣼⡆⠀⠀⠸⢹⣿⡇⠘⠛⠯⡭⠀⠉⠶⠀⠀⠀⠀⠀⠘⠙⠲⠂⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⡀⣿⠟
⠀⠐⠐⠀⠂⠀⠂⠀⠀⠀⠀⠀⢂⠂⠀⠀⣸⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣾⣿⣧⣸⡇
⠠⠡⠡⠈⠄⠄⠌⠀⠀⠠⠠⠀⠌⡄⠄⣀⣸⣇⡿⢀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠀⠀⠀⠹⣿⣷⣶
⠀⢁⢃⠈⠀⠀⠀⠀⠀⠀⠂⣀⣼⣾⣿⣿⣿⣿⡇⠰⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⢰⠀⠀⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠉⠻⣿⣿⣷⣦⡀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠐⠂⠀⠀⢸⠀⠀⣿⣿
⠀⠄⠄⠠⠀⠠⠠⠀⢀⣿⣿⠁⠀⠀⠀⠀⠹⣿⣿⣿⣇⠀⣿⣿⣿⣧⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠃⠀⡀⠀⠀⠸⠀⠀⠘⠀⢰⣿⡇
⠀⠈⠈⠀⠁⠀⠁⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⢹⣿⣿⣿⣧⡀⠀⠀⠀⠈⠛⠿⠷⠾⠋⠁⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡆⠀⡀⠀⣸⣿⡇⣾
⠀⡀⡐⣀⠀⢀⢂⠀⣿⣿⡇⠀⠀⣄⠘⣆⠀⠀⢻⣿⣿⣇⡈⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠸⢀⣥⣾⣿⣿⣰⡿
⠀⡸⡀⠀⠀⠀⠀⠰⣿⣿⡇⠀⠀⠸⣦⠘⣧⠀⠀⢿⣿⣿⡠⠀⠀⠈⡹⢿⣿⣿⣷⡀⠀⣀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡗⠂⠸⣿⣿⣿⣿⡇
⠀⠽⠡⠀⠄⠄⠀⢸⣿⣿⡇⠀⠀⠀⢹⣧⡹⣷⡀⠸⣿⣿⣧⢂⠀⠀⠈⠙⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⡟⣿⡟⡀⠘⠀⢿⣿⣿⣿⣷
⠀⢱⢣⠉⠈⠈⡌⢸⣿⣿⡇⠀⣤⠀⠀⢻⣷⣿⣷⠀⣿⣿⣿⣟⣦⠀⠀⠀⠸⡞⣹⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⢠⣾⣇⣀⠀⠂⢻⣧⠇⠀⠁⠈⠹⣿⣿⣿
⠀⢃⢂⠀⠀⠐⡀⢸⣿⣿⡇⠀⣿⣿⣄⠈⢿⣿⣿⣧⢸⣿⣿⣿⣿⣧⢆⠐⡀⢻⣣⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢿⣿⣼⣰⠃⠸⣧⠈⠀⠀⠀⠀⠹⣿⣿
⠀⢚⠆⠀⠀⠀⠀⢸⣿⣿⡇⣀⢹⣿⣿⣷⣜⣿⣿⣿⣿⣿⠙⣿⣿⣿⣾⣦⢳⡈⣿⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣼⠀⣿⣆⠐⠀⡄⠀⠀⢿⣿
⠀⠜⠄⠀⠠⠠⠠⢸⣿⣿⡇⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡟⣿⣿⣿⣿⡎⣷⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡞⣼⣿⣆⢠⣷⠀⡆⢸⣿
⠀⣼⡌⠀⠀⠠⠀⢸⣿⣿⡇⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡆⢻⣿⢿⣿⣷⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣿⣶⣧⣼⣿
⠐⡸⡐⠂⠂⠂⠂⢺⣿⣿⣧⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣼⠃⢠⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡁⠀⠀⠀⠀⢶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠿⠛⠋⠀⠀⠻⣆⠀⢀⣴⠋⠉⠙⠻⢿⣶⣤⣀
⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⣷⡿⠁⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⢦⣤⣀⣀
⠀⠠⠤⠶⠛⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁
⠀⠀⠀⠀⠀⣿⣠⡤⠤⠶⠒⢶⠖⠻⡟⢻⡿⠉⠉⠉⠉⠉⣿⠉⠙⢿⠉⣿⣻⡷⠒⠒⠒⠲⠤⠤⢼⡇
⠀⠀⠀⠀⠀⣿⡀⠀⢀⠀⠀⠈⠑⠚⠛⠋⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠸⣷⠊⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡀⠀⣸⣧⡄⠋⠀⠀⣀⠀⠀⠀⠀⠤⣄⡀⣰⡟
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠈⣇⣴⠏⠈⠳⣤⡀⠐⣯⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣼⡀⠀⠀⢀⣀⣠⠴⠛⠁⠀⠀⠀⠀⠉⠛⠓⠲⠶⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⢤⣤⣤⣤⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⣯⣓⠾⣽⣳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠶⠞⠋⠉⠛⠷⢿⣝⣺⣽⣗⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⣿⣓⢯⣝⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣭⣗⠪⣽⣲⠦⣄⣀⡴⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢮⣟⡲⢭⣠⣴⡶⢦⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢿⣿⣗⣾⠇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⠙⠛⢉⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣉⣉⠁⠐⠒⠈⠉⠉⠁⠀⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣒⣈⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⠪⢭⣯⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡶⠤⠤⠤⠬⣥⠤⠔⠒⡶⠚⣁⣴⡠⠴⢠⠀⠀⠀⠀⠀⠈⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣹⠇⠀⠀⢀⡏⠀⠀⠀⠈⠉⠁⠀⠀⢀⠈⣆⠀⠀⠀⠀⠲⣌⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠸⢄⡀⣠⠊⠀⠈⠀⠀⠀⠀⠀⠀⠒⠁⠀⠘⢆⠀⠀⠀⣀⡏⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠉⠑⠦⠔⠀⠀⠀⠀⠀⠀⠀⡤⠴⠉⢹⠳⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⠒⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠊⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢⢄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠤⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⢍⣙⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠞⠉⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠰⡪⠗⡋⠐⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⠿⡧⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠠⠼⠄⠐⠉⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠧⢄⣀⣀⣀⡠⠔⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⣉⣤⣴⣀⣀⣀⣀⣀⣀⣀⣐⡂⣄⠈⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠄⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠃⡰⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠡⡘⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡿⠿⠶⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣠⡀⠔⢮⢴⠠⡰⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢽⣹⠞⡁⢸⡘⣿⠚⡵⢟⣿⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣟⣓⣇⠈⠸⣦⠀⢰⡄⢤⠈⠀⣍⠉⠭⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡀⠄⠀⡉⠖⠁⠀⠘⠈⠈⢁⠘⣆⠁⠸⣀⣋⠤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢈⠄⡐⡘⠂⠉⠀⣘⠈⠀⠃⠈⠀⠉⠀⠀⠀⡐⡂⠀⣈⢇⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢏⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠌⠠⢀⠆⡥⠑⠀⠁⢲⣽⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠘⣫⠀⡀⠀⠀⠁⠸⠀⠊⠈⠈⠈⣑⢠⡔⣆⠘⣼⢻⣶⡂⠗⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠷⠌⠉⢀⡠⣤⠠⣴⠞⣰⢿⢶⡾⠲⠎⠀⢩⠟⠝⣧⡑⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⣞⢂⡇⠧⢪⣀⣾⣾⣼⣪⣷⠟⣦⠀⢩⠆⠲⣆⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⠆⡳⣯⡯⣞⢿⣻⠽⡖⠟⢏⠱⢧⣘⣧⣼⡛⠓⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⠀⠀⠐⠳⠛⣯⢎⠰⢇⢃⡴⡴⣚⣷⣯⠌⡁⠀⠀⠀⠀⠑⠪⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢯⡍⢁⣏⣈⢡⣞⣿⡵⣶⣮⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣭⢭⣽⠙⣝⠌⡶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⢏⢴⡥⡔⢦⣴⣚⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣚⣻⡯⠄⠈⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⣉⣁⣤⠴⠖⢋⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣉⣁⣤⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠋⣉⣤⣤⣤⣀⡐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⢸⡇⠀⠉⠢⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⡆⢃⠀⠀⠀⠹⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡇⠀⠀⣠⠞⠀⠈⢆⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢸⠀⣠⠖⢡⠀⠀⠀⣄⠳⢄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠐⠹⠖⠀⠰⠮⠁⠀⠙⢺⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢹⡄⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣼⠋⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⢸⢧⠀⠀⣰⣿⣿⣿⡆⠀⠀⣰⠋⡇⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⡆⠈⣧⣀⣿⣿⣿⣿⣿⣀⠼⡁⠀⢳⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡜⠀⠀⡇⣹⡛⠿⢧⡈⠇⠩⡄⢸⠀⠀⠳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢃⠑⡄⢠⠁⠀⡞⡇⢘⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠢⢤⣁⠈⡇⢺⠉⠛⡜⠦⠔⠁⡇⣸⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡇⠸⠀⠀⢹⠀⠀⢰⠀⡎⠀⠀⣻⠈⠉⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠈⣆⣀⡎⢰⡧⠖⠋⠉⡆⠀⠀⠀⠀⠈⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠸⠀⠀⢠⡇⠀⣇⣸⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣇⠨⠴⢞⠭⠄⠾⡥⢊⠔⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠈⡗⠒⠉⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⢧⠸⡀⠀⠀⠀⠱⡈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⣄⠀⠀⠀⠱⡈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠉⠛⢻⣦⠀⣠⢤⡿⠤⣙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠛⠉⠀⠳⠤⠵⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋



⣿⠿⠿⠿⡿⡟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠟⠛⠛⠿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠈⠉⠉⠉⠉⠀⠒⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠠⠄⠐⠀⠐⠀⠒⠀⠀⠠⢄⡀⡠⠀⠒⠒⠀⠤⣀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⣀⣀⠀⠤⠤⠤⠀⢀⣀⡈⢆⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣒⣛⣉⡉⠀⠒⠒⠒⠀⠤⣀⡀⠀⠉⠐⠒⠤⢄⡀⠀⠀⠇⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⢄⡀⠀⠈⠑⠢⢄⡀⠀⠀⠈⠐⠤⣘⡀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢺⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⠀⠈⠑⠦⡀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠸⠸⡎⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⠈⠑⠢⣀⠀⠀⠀⠉⠢⢀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡄⠀⠙⣆⠀⠀⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣑⠢⠄⡀⠀⠀⠑⠦⡈⠆
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢀⠁⠀⠀⠈⠳⣄⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠱⢦⣤⣤⢘⡘⡄
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⡰⠉⠛⢦⣄⡀⠀⠀⠀⠈⠐⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⠉⡙⠘⡄
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⠜⠀⠀⠀⠀⠉⠊⠕⡒⠠⢄⣀⠀⠀⠀⠈⠁⠒⠀⠀⠠⠤⠤⠤⢠⣤⡤⠤⠐⠒⣛⣱⡚⠤⠚⢁⠇⠘⢄
⠀⠀⠀⡠⠊⠀⠀⠀⣀⠔⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠬⢉⣀⠒⠒⠒⠒⠐⠒⠒⠂⢈⣁⡠⠤⠴⠖⢉⣉⡀⠠⠤⠐⠃⢤⠀⠀⠉⠀⠈⠁
⢀⡴⠊⠀⠀⣀⠄⠊⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠒⠀⠤⠤⠀⠀⣠⠤⡄⠒⠁
⠈⠉⠐⢶⡭⢤⠤⠔⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠸⣄⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⡸⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢆⠉⠐⠒⠈⠁
⠀⠀⠰⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢂
⠀⠀⢡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜
⠀⠀⠀⠑⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠉⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠠⠤⠀⣀⣁⣒⣀⣀⣒⣉⠀⠠⠤⠐⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠚⠛⠛⠉⠉⠁⠀⠈⠉⠉⠉⠛⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⣤⣄⡀⠀⠀⣠⡴⠚⠉⠉⠲⠀⠀⠀⠀⣀⠀⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⡏⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠀⡉⠿⠿⠟⠛⠿⡲⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⣠⣤⣭⣀⠀⠀⠀⠀⠀⠀⠘⣏⢿⡈⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⠻⢿⣿⣷⣦⡀⠀⡀⠀⠀⠙⡎⢳⡘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⢿⡍⢻⣿⡄⣿⡖⠆⠀⠃⠸⠁⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠸⠃⠈⣋⡇⠸⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣿⡇⢆⠀⢀⠀⣠⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠚⠉⠨⠀⠈⢻⣄
⠀⠀⠀⠀⢀⡼⠋⢀⣶⣿⣿⣿⡆⢹⣤⣿⣿⡄⠀⠀⠀⢠⡾⠻⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⡞⠁⠀⢈⣿⣿⣿⢿⣷⡌⠉⠛⠻⠃⠀⠀⠀⠋⠀⠊⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠈⢿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠘⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⣾⠀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣧⡀⠙⠿⣷⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣉⣠⠄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡗⠀⠀⠀⠈⠙⠏⠑⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠑⠓⠒⠦⠤⣤⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠘⢷⣆⠐⠀⠀⢀⢿⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈⠀⡷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠠⠀⠀⢰⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠐⣿⡿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⣿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠈⢷⣼⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣧⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣷⣴⣿⣶⣶⣤⣤⣴⣾⣿⠋⢻⣿⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠿⢿⣿⣿⡀⣹⣿⣿⣿⣿⣷⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣾⣿⣿⣿⢿⣿⣿⣿⠿⠿⢿⠛⢿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠛⠋⠀⠀⠁⠈⠀⠀⠀⠀⠙⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⢀⡀⠀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⢀⠦⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠋⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡄⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡴⡶⢤⠞⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠐⣄⣀⣼⡕⠁⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠸⠂⢀⠄⠁⠀⠀⠉⠓⢄⡀⢰⠲⠤⣀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠉⠁⠈⠀⠀⠉⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢆⠘⢄⣆⠀⣦⠸⣆⡳⠔⠈⠠⡀⠀⠐⠐⠺⠤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠘⠉⢄⡀⠣⠆⠉⡤⠀⠀⠨⡄⠀⠁⠀⡄⠙⠁⠄⠀⠀⠀⠀⠀⡐⠈⠑⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡁⠀⠀⠈⠄⠐⠙⠀⢠⡀⠈⠀⠠⠰⠀⠘⡀⠀⠀⠀⠀⣰⠄⣄⠀⠀⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠙⢀⡈⢆⡀⠀⠈⠒⠄⠀⠃⠀⡀⠰⠀⡇⠀⠀⡇⠀⠀⠀⡎⠁⢄⠀⡗⢄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢄⣠⠋⢀⠀⠙⠀⠀⠀⠀⠀⢀⡀⠑⡀⠈⠁⠀⠀⠃⠀⠀⠀⠉⠒⡄⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⡌⠁⠁⢠⡃⠀⠀⠀⠀⠀⠀⡸⡉⢱⠀⠀⠀⢘⠀⠀⠀⠀⠀⢠⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠁⠁⠈⠢⡀⠀⠀⣀⣚⢅⠈⠲⣄⠀⠀⢁⣬⣠⡄⠀⠀⡈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⠋⣁⣁⠀⠀⠑⣀⠎⡙⢤⠃⠁⠻⠿⣃⡘⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠉⢠⠿⣦⣶⣦⣠⣄⣉⣮⣈⣦⣼⡄⠀⠈⠉⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀⢸⡉⠀⠈⠁⠙⠋⠹⠟⠛⠿⠏⠛⠉⠐⣂⠀⡠⠒⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⠟⠓⠐⠀⠸⠈⠂⠂⠔⠖⠚⠚⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣀⡀⠀⠸⡀⠀⠀⠀⠀⣠⠤⠤⠀⠀⠀⠀⠀⡤⠤⠤⠄⠰⠦⢄⡀⠀⠀⠀⠀⢈⡵⠋⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⣨⣓⠺⣧⡀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⣤⠞⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⣤⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣥⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⢀⡀⣀⣀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠨⠥⠚⠛⠛⠉⠙⠓⠀⠀⠋⠉⠀⠙⠀⠉⠛⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠄⠀⢠⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⢄⠀⠀⢀⡈⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠒⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⢀⣀⣀⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⢨⠀⡠⠄⡸⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠈⠁⠁⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠿⠿⠿⠿⢶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣰⡟⠉⠀⢙⣷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣴⠾⠃⣀⣤⣤⣀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠈⢻⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣧⣀⠀⠀⠀⠀⠀⠘⡇⠀⢀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⣋⣭⣿⣿⣿⠉⠉⠛⠿⢷⣶⣶⣶⡾⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⢁⣠⣾⠿⠋⢹⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⣠⣾⠟⠁⠀⠀⣼⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⢀⣾⡿⠁⠀⠀⠀⠀⣿⡇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⢠⣿⠏⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢿⡟⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀
⠀⠀⠀⣰⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣤⣶⣿⣿⣿⣿⣶⣤⣄⢀⣀⣠⣤⣤⣤⣶⣾⣿⣿⣿⣯⣭⣭⣥⣦⣴⣶⡶⠶⠒⠒
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⣛⣿⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣯⣥⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⠿⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠠⠻⠟⠛⠛⠛⢋⣽⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣤⣤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⠿⡿⠟⠛⠛⡟⠻⠿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⣛⣩⣡⣴⣶⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠰⣋⠀⠙⢦⡀⢰⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡔⠁⢀⣹
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠸⣿⣿⣿⣿⠏⠀⠀⠀⠀⣼⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢳⡀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⣞⢉⡩⠟⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⣉⠥⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠀⠘⣇⠀⠀⠘⠫⣉⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⢀⡞⠁⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣠⠖⠲⣄⠀⢿⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⢸⣀⣀⣀⣤⣤⡴⢾⠀⠀⠀⠀⠀⠀⠷⠶⢤⣼⠀⢸⡗⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠘⣏⠉⠁⠀⠈⣠⠏⠀⠀⠀⠀⠀⠀⠉⠲⠖⠃⠀⢸⣇⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⢿⣿⣧⠀⠀⠈⠓⠶⠶⠚⠁⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠖⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠞⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢀⣤⣶⠤⣄⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢠⡾⣿⣿⣷⣾⠃⠀⠀⠀⠀⣠⠟⠀⠀⠀⣠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠺⠤⠬⣿⣿⡏⠀⠀⠀⣠⡞⠁⠀⠀⢀⠴⣅⣠⠟⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡼⠋⠀⠀⠀⠀⡇⠀⣀⣽⢀⡼⠁⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡽⠛⠻⣏⣉⣹⣧⡀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⢹⡄⠈⠁⢈⠷⠎⠁
⠀⠀⠀⠀⠴⢂⣐⡶⠶⠶⢴⠃⢸⡇⠀⢠⡏⠉⣿⢫⡟⠛⠶⠶⠶⠶⠛⢿⢹⣦⠀⠀⠀⠀⢀⡴⠋⣠⠞⠃
⠀⠀⠀⢠⠇⠰⣏⠤⠀⠀⠀⣀⣈⠉⠙⢻⡇⣸⡿⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠷⣶⣴⣶⡯⠴⠋⠁
⠀⠀⠀⠊⠀⢰⡇⣠⠆⣠⠞⠑⢈⢳⡴⢾⣶⣻⡥⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠘⠀⠀⢀⡟⠁⢰⠃⠀⠀⠀⠀⢈⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⢀⠃⠀⠀⠈⠧⠴⠋⠀⠀⠀⠀⠀⣜⣿⢧⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠈⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⣁⠀⠀⠉⠉⠙⠛⠳⠶⠦⣤⣼⠁
⠀⠀⠀⣠⡞⣩⠖⣤⣀⠀⠀⠀⠀⣼⣹⣟⠳⣄⠀⡀⠀⠀⠀⠠⣄⣀⣴⠃
⠀⢀⡼⢋⣴⠁⢀⣿⠏⠹⣆⠀⢰⣭⠿⣄⡴⠟⠛⠷⣦⣄⣀⢀⣠⣼⣿⣅⣀
⣠⠟⡡⠋⢀⣧⠞⠁⠀⠀⠙⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⢶⡍⠛⣦
⠛⠻⠷⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣐⣾⣥⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣁⣀⣉⠙⠛⠃⠀⢀⡝⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⣧⣬⣤⣍⣉⣁⣛⠒⠖⠦⢽⣆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣡⣾⣏⢉⣿⣿⡇⠀⠀⠀⣰⠋⠀⠀⠀⠃⠀⠀⢠⣀⣋⡀⠿⢸⢟⡙⠛⢳⣿⣿⡟⢛⣉⣿⡟⢉⡿⣿⡿⣷⣿⣿⠷⠛⢠⣜⠻⡄⢹⣷⡿
⣿⠾⢿⣭⢠⠟⠀⠀⠄⣠⡇⣀⣠⢀⡄⢀⠀⣤⠀⣩⣥⣤⣄⣤⣤⣤⣷⢨⣷⡾⢏⣿⡟⢻⡇⠭⣵⣿⣱⢏⠙⠣⣴⡷⣾⡟⠉⢲⣿⣩⣿
⣤⣤⠾⡇⠀⠀⠀⠀⡼⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⢠⣥⠹⠄⢻⣄⠰⠀⠀⣨⠉⠸⢥⡤⢨⠉⣭⣥⡼⠟⢀⠠⣿⡷⣶⣋⠿⣾⡽
⣿⣿⠞⠁⠀⡠⠀⣀⠁⣄⣀⠀⣀⣀⡀⠀⢀⠀⢀⢠⣄⢡⣀⠤⢠⠀⣤⠼⠧⣠⣄⣠⣿⣶⡀⣤⠀⢸⣷⡏⢰⡾⣦⣤⣌⣅⣹⣻⣤⠙⣧
⣉⡉⠀⠀⡐⢀⡾⠁⠀⠀⠀⡘⠁⣼⡃⠀⠀⠀⠈⠀⠐⠀⠉⠀⠀⠀⠧⠀⠀⢠⠴⠿⠛⠈⣃⠀⢀⣬⡉⣼⡛⡿⣅⠁⢨⣿⣇⢹⣅⠑⣷
⠋⣁⡁⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣠⣀⣀⣠⣆⠰⣾⡂⢁⣼⢿⣤⢤⢉⣉⠛⣦⣕⣠⣝⢁⣼⣭⣿⣼
⠛⠛⠻⠟⠛⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣉⠙⣋⠉⢉⡉⠉⠋⠑⠉⢹⣿⠛⠛⢿⣿⢿⣿⡿⠋⢻⡦⢯⡙⠻⣿⣿⣹⣧⡙
⣀⣤⣦⣤⣀⣶⣀⣤⣴⣆⣀⣰⣆⣠⣤⣴⣰⠶⠶⠂⢉⣥⣤⣿⣤⣤⣍⣻⠿⠷⣶⣯⣍⡰⣦⢬⣍⣛⠿⣜⣿⣄⣤⠀⠿⠷⢦⡬⠿⠿⣿
⡿⠽⠿⣍⣉⣩⣏⣉⣁⠩⠿⣯⣤⣬⡟⠋⣉⣴⣶⡿⠿⠛⠛⠛⠉⠙⠛⠿⣶⣤⣉⠿⠻⣿⢛⡷⠾⣿⡶⣾⠤⢾⡟⠳⢄⢶⡀⠓⠲⢦⡝
⠄⠠⠴⠿⠄⠤⠼⠿⠥⠤⠤⢿⡿⠁⢀⣴⡿⠋⢁⣤⣾⣿⣿⣿⣿⣿⣿⣶⣦⡙⠻⣶⡄⠉⢹⣧⣶⣯⡟⠿⣶⡾⠷⢦⡤⣼⠿⡶⢤⣄⠻
⣦⣶⣤⣤⣴⣶⣤⣤⣴⣦⣴⠞⠀⢠⣾⠏⠀⣰⣿⣿⣿⡿⢿⠻⣿⠿⢿⣿⣿⣿⣦⡈⢻⣿⡀⠈⢻⣯⢶⣶⣾⣷⠤⢦⣴⡿⠶⢶⣤⡻⠶
⢶⡶⠾⠷⣶⡤⠷⢶⣶⠉⡟⠀⠀⣾⠏⠀⣼⣿⣿⡟⠋⠀⠀⠀⣿⠀⠀⡙⢿⣟⢿⣷⡀⠙⣿⡄⠈⠹⣷⣿⣿⣿⣿⣷⡿⠦⠠⢤⣿⠿⢶
⣾⣅⣀⣠⣿⣄⣀⣼⠃⠴⠃⠀⣾⡏⠀⣸⣿⣿⡿⠁⠈⠃⠀⠀⣷⢼⠁⠟⠈⢿⣾⣿⣷⡄⠘⣿⡄⠀⠻⣶⣤⣥⣈⢿⠦⠀⣀⣄⠻⠆⠈
⠁⠀⠀⣸⠁⠀⠀⡉⠀⠃⠀⢠⡿⠀⢠⣿⣿⢿⣇⣀⣀⣀⡀⠀⡟⢺⣀⣀⣠⣨⣿⣿⣿⣷⡀⠸⣷⡀⠀⠘⣧⣘⣧⣀⣀⠀⠨⣷⣤⣀⣤
⠐⠒⠒⠛⠖⠒⢲⠃⡀⠀⠀⣿⠃⠀⣾⣿⡿⠈⢨⡿⠀⢹⠀⠀⣿⠈⣿⡇⢸⣿⣏⣿⣿⣿⣧⠀⠹⣷⡆⠒⠻⣧⣨⣿⣿⡿⢦⣀⣀⣬⣴
⠠⠀⠀⢀⡀⠀⠈⠀⠗⠀⣰⡇⠀⢸⣿⣿⡟⠀⢰⡗⠀⢹⠀⠀⡿⠀⠘⡇⠈⣿⣆⢻⣿⣿⣿⣇⣠⡿⣷⡀⣸⣷⣾⣿⡅⢠⣌⣽⣁⡀⢀
⠀⠀⠀⠈⠁⠀⠁⡀⠀⠀⣿⠁⠀⣾⣿⣿⠃⠀⠀⣏⣀⣠⡀⢀⣷⣹⣴⣧⣤⣿⢀⣹⡏⣿⣿⣿⡟⢦⣽⡅⠈⠐⢾⣯⡄⢉⣿⣥⣼⣷⣀
⠀⠀⢀⡀⠀⢀⡀⠀⠀⢸⡏⠀⢠⣿⣿⢿⠀⠀⣴⡏⣹⠉⣩⣿⡿⠿⠿⣿⠏⣽⣟⠁⣿⣹⣿⣿⣷⠀⠙⣿⡗⠚⠻⣯⣸⣿⣿⣿⣿⣯⣿
⠀⠀⠀⠁⠀⠘⡃⠀⠀⣾⠁⠀⣸⣿⡏⠀⠆⡀⢸⡏⢻⣾⡏⢹⠀⠀⠀⠘⠀⠀⢻⡀⢯⡝⠛⣿⣿⣧⠀⠹⣧⠀⢀⣸⣿⣿⣿⣿⣷⣿⣿
⠀⠀⣄⠀⠀⣮⠀⠀⠈⡟⠀⢠⣿⣿⡇⠀⣄⠁⢸⣇⢹⡄⠀⠈⠀⠀⠀⠀⠀⠜⠀⠑⣺⣷⡄⣿⣿⣿⡆⠀⢻⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡧⠀⠀⢸⡇⠀⢸⡏⢿⠁⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢸⡇⠃⢿⣿⣿⣿⡀⠈⣷⠀⠋⠻⣿⣿⣿⣿⣿⣻
⠀⢶⠀⠀⣾⠀⠀⠀⢼⠀⠀⣸⡇⡘⠀⠀⠀⠀⢀⡏⣿⠀⠐⠀⠀⠀⠀⠀⠀⠘⠆⡀⢸⡇⠂⠂⣿⣿⣿⡇⠀⠸⣇⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠠⢀⡟⠀⠀⠠⣮⠀⠀⣿⢠⣇⠀⠀⡆⠀⢸⡇⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⡉⠸⣧⢀⠀⣽⠉⣿⣿⡀⠀⢻⡄⠀⠘⣿⣏⣽⣏⢻
⠀⠖⠀⠀⠁⠔⠂⠀⠁⠀⢀⣿⠾⠉⠃⢀⡇⠀⢰⣇⣸⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⣶⣾⣬⣆⢿⡇⢻⣿⡇⠀⠀⣿⡀⠙⠻⣿⣿⣿⣿
⠀⣀⣀⣤⣤⣴⣆⠀⢀⠀⢸⣿⠀⡇⠀⠘⡇⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣼⣷⠘⣿⣿⠀⠀⠘⣷⡀⠰⣿⡿⠋⢻
⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠺⠏⠘⡇⠀⠀⡁⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣹⣿⣿⣿⣿⣿⣆⢹⣿⣇⠀⠀⢿⣄⡀⢸⠁⠀⠸
⠀⠛⠻⠟⣿⣿⣛⣿⡦⠀⣾⠇⠀⡇⠀⠀⣥⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣸⣿⣿⡄⠀⠈⣿⠀⣶⣴⣾⣶
⠀⠀⠀⠈⠈⠋⢩⡯⠀⠈⣿⡆⠀⣇⠀⠀⢩⣸⣿⣿⣿⣿⠀⠀⠘⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⡇⠀⢀⣼⣿⣿⣿⣿⣿
⢀⡁⠀⠄⠀⠛⢠⣤⣤⣴⣿⣷⣤⣿⣤⣤⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣷⡠⠾⣿⣿⡿⣿⢿⠛
⣮⣿⣶⣤⡀⠀⠋⠉⠻⣏⠻⠁⢁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣠⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⢿⣿⣿⣿⠛⠉⢹⣿⡆⠈⠀⣤⣴⣶⣶
⣿⡿⣿⣿⣯⡀⣀⢤⡶⣿⠄⣀⣾⣄⣀⠀⣤⣠⣿⣿⣿⣿⣿⣿⡀⠀⠀⠰⡀⠰⣄⣴⢾⣷⣿⣿⣾⣿⣿⣧⣄⣀⣤⠿⢧⣴⣾⣿⣿⣿⣿
⢿⡿⡿⢟⢻⣉⣉⣀⣀⡀⠀⢉⣉⡉⣋⣙⣋⣹⣿⣿⣿⣿⣿⣿⣏⣀⠀⠀⢰⣦⣤⣴⣿⣿⣿⣿⠟⠛⠋⠉⢽⣯⣿⣀⣀⠿⢿⣿⣿⣿⣿
⠀⠀⠥⠀⣾⡟⠛⠙⣿⠛⠟⠛⠛⠛⠙⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣡⣼⣿⣿⣿⣿⣿⣿⣅⡠⠀⠀⠀⠺⠋⠈⠛⣿⠃⠚⠉⣀⠉⠁
⠷⡄⠀⢀⢋⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣖⣾⡁⠀⣾⠀⠛⡀⠀⠀⠉⠄⠒
⣆⠀⠑⢀⠘⣿⣄⣀⣀⣀⣀⣼⣦⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣟⢉⣛⣄⣻⣄⣈⣿⣿⣴⣤⣤⣤
⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠏⠀⢿⣿⢿⣿⣿⡞⣻⡿⠛⠿⣿⠿⣾⣿⣏⠉⠉⠉⠉⣉⣉⣈⣁⣀
⡿⠛⣿⠟⠿⢻⡶⢶⣶⣶⡶⢴⣶⡿⠿⠻⣿⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣽⡇⠀⠀⢻⡆⠁⠀⠙⣿⡛⠛⠛⢿⠉⠉⠉⠉
⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⠀⠈⠁⠀⠈⠉⠉⠀⠀⠘⠛⣿⠛⠋⠉⠈⡇⠈⠁⠘⠳⡀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢀⠁⠀⠀⠀⠘⣿⣤⠟⠉
⣧⣀⣀⣀⣀⣀⣤⣤⣿⣤⣤⣤⣴⣦⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠓⠒⠞⠳⢶⣿⣁⣴⣄⣀⠀⠀⠀⠀⢠⣶⢀
⡙⠛⠛⠛⠛⠋⠙⠋⠉⠒⠀⠙⢊⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠈⢏⠉⠉⢉⡹⠿⠿⠉⣹⡛⠻⠿
⠛⠀⠀⠀⠀⠠⣀⡤⠤⠐⠶⣾⣿⣅⡀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⣿⣥⠔⠂⠀⠀⠀⠀⠀⢀
⢠⠟⡀⠒⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⢷⡄⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣱⡄⣠⣴⠖⠒⠛⠉⠀⠀⠒⠀⣈⡉
⠂⠘⠃⠀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠂⠰⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⡙⣋⣀⠀⠀⠆⠀⠀⠀⠀⠀⠘⠛
⡭⠉⠀⠀⠈⠙⠚⢢⡀⢀⡀⠁⠀⠀⠀⠀⠒⠀⠒⣀⠀⠹⣦⣤⣀⣀⣀⣀⣀⠴⠊⠉⠀⠈⠉⠉⠉⣰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⣄⠀⡀
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠁⠉⠙⠓⢦⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢤⡦⠟⡶⠄⠙⠒⠤⣆⡀⣰⠀⠀⠀⣠
⠸⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⣳⣌⠀⠉⠘⡀⠀⠀⠉⠉⠹⠔⡦⢊⡏
⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⠖⢦⠀⠀⠀⠀⠀⠀⠀⠴⠶⢄⠈⠁⠀⢀⣄⣤⣾⣄⣄⣀⠀⠀⡰⡇
⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡴⠃⡜⠁⢠⠎⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⡏⡷⠤⠚⠉⠀⠀⠈⠉⣻⣦⡄⠸⢤⡄
⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠊⠉⠀⠀⠣⠔⠁⠀⡌⠀⡜⢀⠊⡇⠠⡲⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣉⡁⠀⠧⣀
⠀⠀⠀⠈⠳⠤⣄⣀⣀⣀⣠⡶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠈⠊⠀⠀⠀⡰⠋⠀⠀⠤⣀⠀⠀⠀⠀⢀⣿⡟⠛⠂⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡞⠰⢦⣀⡈⡙⠀⠀⢀⣸⠋⠁⢰⡚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠸⢮⣁⡬⠃⠀⡴⣋⣉⣠⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣩⡀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣡⣦⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠈⠀⠠⡀⠀⠀⡠⠤⠴⠃⠁⠈⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠐⢄⠀⠀⠐⡀⠀⡀⠘⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣄⠀⠀⠉⠉⠓⠃⠀⣧⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠉⠉⠓⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⣀⠴⠊⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠉⠁⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠘⢾⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⡉⠁⠀⠀⡼⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠹⢄⣀⣠⠾⡁⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⢠⠜⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣐⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣭⣥⣤⣤⣾⣄⣀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⢀⣀⣀⣀⠀⢀⣀⡤⠀⠀⢀⡀⠠⠶⣀⠈⠡⠄⠠⠤⢔⣀⣀⠀⠀⠀⠀⠀⠀⠒⢋⣠⠶⠋⠀⠀⠀⠀⡀
⢁⡝⠁⠀⠀⠀⠈⠲⠀⠀⠀⠀⡤⠊⠀⠀⠀⠀⠀⢠⣶⡉⣀⣀⡀⠀⠐⢐⣨⠖⠉⠀⠀⠀⠀⠀⠀⠀⢷⠀⣀
⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣞⠀⢀⠠⢐⡤⠖⠋⠀⠀⡀⠀⠀⠀⣶⠀⣀⠤⠚⠉
⢤⡀⣀⠀⠀⠀⠀⠀⠐⡖⠒⠁⠚⠁⠀⠠⠀⠀⠀⣤⣬⡠⡟⠘⠉⠀⢣⡀⠀⠀⢃⠀⣀⠤⠚⠉
⠒⠒⠒⠒⠶⠶⡒⠒⠂⠀⢈⡁⠀⠠⢮⣤⢤⣤⢤⡬⠉⡁⠀⢸⡆⠀⠈⣷⣀⠤⠚⠉
⠀⢠⡏⠀⠀⢰⠇⠀⠀⠀⢸⠁⠀⠀⠀⡟⠀⠀⢸⡇⠀⢻⡀⠈⣷⡤⠒⠉⠁
⡀⣸⡀⢀⠀⢸⡀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠸⡇⠀⠸⣗⠯⠁⠈⠀⠀⣀
⠒⠒⠒⠒⠒⠂⠒⠒⠒⠒⠐⠒⠀⠀⠀⠈⠀⠀⠀⡽⠒⠁⠀⠀⠀⠂⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠂
⠶⠲⠶⠴⢆⠲⠀⠀⠤⠤⠤⠤⠄⡀⠀⡄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠄⣿⠀⠀⠀⢀⣠⠴⠞⠋⠁⠀⢀⣠
⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣿⠀⠰⠋⠁⠀⣀⣠⣴⡶⠿⠛⠉
⠄⠀⠘⢤⠀⠀⠑⢄⠘⠀⠀⠀⢀⣁⡀⡇⠀⠀⠀⠄⠀⠀⢠⣷⠀⠀⠀⣾⣿⠀⠀⠶⠶⡟⠛⠋⠁⣀
⡀⠀⡁⠀⠀⠀⠀⠀⠉⠂⠐⢆⠈⠉⠁⡇⠀⠀⠀⠂⠀⠀⠸⣿⣿⠀⢀⠿⣿⠀⠀⠀⠀⠃⠀⠀⠀⢀⢈⣁⠐
⠷⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠘⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡙⠛⠩⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠙⢆⡇⠀⠀⠀⠀⠀⠁⠀⣀⣿⠀⠀⠀⢀⡆⠀⠀⢸⠁⠙⠦⠈
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠁⠀⠠⠬⠴⠖⠛⠋⠁⠀⠀⠀⠀⠸⠇⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⡀⣶⡆⠀⠀⠀⠀⠀⠀⠤⣠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⡧⠄⠀⠀⠀⠀⠀⠀⢀⠴⠶⣾
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⡀⠀⠀⠄⠀⠀⢀⢈⡆⠠⡀⠀⣦⠀⢤⣀⠀⠠⠁⢀⣀
⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢀⠀⡔⠉⠁⠻⠏⣿⡀⣹⡇⢀⠆⠘⡟⠀⣠⢳⠀⠈
⠏⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⢈⣦⠀⠴⠀⠚⠑⡿⠀⣀⠈⠻⡄⠰⠀⠀⠀⢚⠱⠉
⡄⠃⠈⢧⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠁⢀⣀⠰⣄⠀⠀⢀⠀⢁⠀⡈⢀⡄⠀⢀⠎⠀⢀⠘⢰⡦⠆
⠁⢀⣀⠤⠗⠒⠉⠉⠉⠉⠉⠓⠤⠀⠀⠀⠀⠀⠞⠏⠀⢀⣸⣿⡆⢀⣠⠈⠀⠈⠀⠙⠖⠋⠀⡠⢻⡀⠈⠳⣆
⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⠇⣀⣀⠀⣸⠃⠀⠚⣿⡰⡦⠀⠠⠤⠀⢀⠴⠶⢤⠇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⠄⡿⠀⠼⣿⢀⣽⠂⠉⢰⣿⡠⢁⣄⡄⢀⡄⠿⠀⠼⡇⠀⢀⡷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⢻⣄⠀⠀⠉⠁⠀⠀⡾⠈⢑⡛⣉⠁⡶⣀⠀⠀⢀⣤⣼⣤⣤⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣐⡿⢥⠄⠀⠒⠺⣿⣵⡄⠷⠶⠭⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠘⣿⣭⡇
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠙⠻⠦⠠⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠐⢀⠀⠈⠳⢄⣀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⠟⣉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠄⠓⢀⡴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⣀⣽⣿⣿⣭⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠌⠛⠻⠟⠿⣷⣶⣍⡢⣀

⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣰⠀⠀⠀⠀⠀⡤⠖⠃⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄
⠀⠀⠀⡼⠛⠛⠛⠛⠙⢦⠀⠀⣀⣠⣿⡿⢿⣆⣿⠀⠀⢀⢤⣤⣷⣤⣄⠀⠀⠀⢀⣽⣿⣿⣿⣾⣿
⠀⠀⠀⡇⠀⠔⡀⠢⠀⡸⠀⠀⣿⠀⠀⡾⢻⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⢲⠀⠀⠀⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠘⠦⣀⣉⡠⠔⠁⠀⠀⣏⠀⠸⠃⣠⠚⢂⠀⠀⠈⠻⣿⣿⠿⠋⠁⠀⠀⠀⠈⠉⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⢦⡀⠀⠀⠀⠈⣧⣿⣧⡀
⠀⠀⢠⠀⠀⣰⣷⡀⠀⠀⠀⠀⡯⠐⡖⢀⢠⡄⢹⡀⠀⠀⢰⢿⣿⡏⣧⠀⠀⠀⠀⢰⠉⠉⠉⣿⡂
⠀⠀⢸⣠⣼⣹⣿⣽⣦⣄⡄⠘⢧⣈⠁⠚⠈⠁⡴⠁⠀⠀⠀⢸⡏⣷⠀⠀⠀⠀⡄⣈⠩⢉⡉⢉⢀⡄
⠀⠀⠸⠉⠉⠹⠩⠏⠉⠉⠇⠀⠜⠋⠙⠋⠉⠉⠀⠀⠀⠀⠀⠸⠇⠿⠀⠀⠀⠀⠁⠉⠀⠀⠉⠈⠉⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠈⠓⠑⠀⠀⠀⠀⠀⠀⢀⡾⣿⣷⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣉⣋⣻⣷
⠀⠀⠀⣿⣿⢻⢻⣹⡟⢦⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢰⠀⠠⠏⣰⠪⠇⠀⠀⠀⠐⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠘⢧⣽⣿⡿⠉⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣶⣶⣶⣶⣶⣶⣶⣷⠀⠀⠻⠿⣿⣿⡏
⠀⠀⠀⠐⠒⠚⠓⠒⠂⠀⠀⠀⠀⠀⠈⠋⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢙⠛⠉
⠀⠀⠀⠀⠀⣠⣦⠀⠀⣤⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⢀⣿⣗⡀
⠀⠀⠀⢀⣨⣿⡿⣷⣄⣿⠀⠀⠀⠀⢾⡿⣿⠆⠀⠀⢀⣼⣿⣿⠿⠋⠻⢷⡄⠀⠀⢠⡟⣿⣿⢧
⠀⠀⠀⣿⠀⢀⡾⣻⣿⣿⠀⠀⠀⣠⣤⣿⣷⣤⠀⠀⢸⣏⠀⠤⢀⠠⠄⢸⣿⠀⠀⠈⢠⡟⣿⠈
⠀⠀⠀⣇⠀⠼⠁⡠⠓⣄⠀⠀⢰⡟⣿⣿⣿⣿⠇⠀⠸⠿⠦⣄⣙⣀⡤⠿⠿⠀⠀⠀⠸⡇⣿

⠀⠀⠀⠀⠀⠀⠀⠐⠂⡐⠖⠒⠈⠉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠋⠀⠀⠀⠒⠤⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢰⠀⠀⢰⣶⠀⠙⢃⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠠⠭⠀⠀⠀⠀⠠⠄⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠈⠑⠤⡝⠓⠦⠤⢀⡀⠤⠔⠚⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢫⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢆⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠘⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠈⢢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠄⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠠⠤⣀⠀⠀⢸⢀⠔⠡⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠑⠚⠁⠂⠁⠀⠤⣄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠈⢢⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢡
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠔⠀⠀⠀⠀⠀⠀⠁⢇
⠀⠀⢦⣴⠂⠠⠀⠀⠀⢄⠂⡀⠴⠘⠦⠰⠂⠄⠂⠠⠁⠠⠀⠠⠀⠀⢔⠀⠀⡠⠄⠀⠠⠘⡄⠈⣀⡀
⣄⢀⠀⢄⣀⣀⠀⠃⡀⠀⢀⠈⠀⠀⡀⢀⠀⠀⠐⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠤⠗⠀⢀
⠘⠚⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠻⣿⣷⣄⡀⢀⣀⣠⣤⢿⣿⣿⡦⣤⣄⣀⡀⢀⣠⣾⣿⠟⠋⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢠⣀⣈⣙⠻⢿⣿⣇⠀⠀⢸⣿⣿⠁⠀⠀⣼⣿⡿⠟⣋⣁⣀⡤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣈⣉⡉⠀⠀⠈⠿⠀⠀⠀⣿⡟⠀⠀⠀⠟⠁⠀⠀⢉⣉⣁⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠛⠛⠉⠀⠀⠀⣀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⠀⠀⠀⠙⠛⠛⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣀⡀⠀⠀⣀⣤⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⣤⣀⠀⠀⢀⣀⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠃⠀⠉⢹⣃⣀⣵⡄⠀⠀⠀⠀⠀⠀⢰⣯⣀⣹⡍⠁⠀⠘⠛⠛⠛⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣠⠇⠀⢀⣀⣀⡀⠀⠹⣶⣆⡼⠁⠀⠀⠀⠀⠀⠀⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⠿⣿⣿⣷⡶⠶⠖⠀⠀⠀⠀⠉⣡⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⠀⠀⠐⠲⠶⢶⣿⣿⡿⠿⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣀⡤⠖⠒⠛⠒⠦⢤⡀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣀⡤⠖⠒⠛⠒⠲⢤⣀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡄⢠⠞⠁⠀⠀⠀⠀⠀⠀⡀⠉⠻⣶⣤⣄⣀⣀⣀⣀⣤⣴⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⣤⣤⣾⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠛⠀⢠⠀⠀⠀⠀⠀⠀⠀⣷⠀⠉⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠂⠀⠉⠛⢻⡟⠛⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⢳⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⢠⠏⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣙⢦⡀⠀⠀⠀⠈⠓⢤⡤⣤⡤⠤⠞⠁⠈⠳⠤⢤⣤⢤⡴⠚⠁⠀⠀⠀⢀⡴⣋⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣀⣀⣀⣀⡤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⣄⠈⠉⠉⠀⣠⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠛⠁⠀⠀⠀⠀⠉⠛⠿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠲⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⡲⢤⣳⡄⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠈⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⣉⡭⠤⠠⢀⡀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⡴⡉⠀⠀⠀⠀⠀⢱⠀⠂⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠈⢷⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡩⢤⣀⠀⡟⠀⠈⢦⡀⠀⠀⠀⠂⠀⣀⠘⠷⣄⠈⢷⡄⠀⠀⠀⠀⠀⠀⠘⠏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢸⠀⠀⠸⡀⣇⠀⠀⠀⠣⡀⠀⠈⠁⠀⠀⠉⢦⠈⠁⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡄⠀⠣⠀⠀⠁⠈⠑⠒⠶⠎⠑⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣜⠀⠀⠙⢲⠖⢤⡀⠀⠀⠀⠀⠘⠀⠀⠈⡄⠀⢀⡴⠋⠀⠀⠀⠀⣀⣀⣈⣧⠀⠀⠀⠀⢀⡏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⣄⠘⡏⢱⠀⡞⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⢘⡻⠇⠀⠀⠀⠁⠀⠀⠀⢸⠇⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠐⠄⢬⣸⠁⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⢀⢆⠈⠉⠠⠀⠀⠘⠀⠘⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⡉⠂⠀⠀⠒⡊⠄⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⢀⡀⠀⠀⠀⠀⠀⢀⠔⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢖⣢⡀⠈⡄⠀⠀⠀⣠⠎⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⢰⠇⠉⠙⠓⠒⠶⠦⠤⠤⢤⡤⠤⠴⠖⢺⠶⢤⣄⣀⣥⠔⠙⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⣄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⠀⠂⠐⠠⠤⠤⢻⡒⣶⠶⢶⡶⠻⡧⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠓⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⡃⢻⣄⣀⣰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⡀⣯⣠⢨⢭⡉⠛⠓⠒⠶⠦⠤⠤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠃⠒⠂⠄⠤⠤⠤⠤⠴⠶⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣆⠤⣤⣄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠻⡿⠛⠉⠀⠈⠀⢄⠀⠀⠘⠛⠛⠛⢲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠛⠀⠀⠀⢦⠀⠀⠀⢿⠁⠀⠀⠀⣀⠀⣀⡉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⢀⡀⠀⠀⠈⠀⠀⡀⠀⢀⡀⠀⠀⠋⠈⠛⠃⠈⠁⠱⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⡹⠋⠀⠀⠈⠁⠞⠢⡦⠀⡀⠁⢤⠀⠁⡀⠀⢠⡀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⣄⠀⣅⢠⡀⠰⠆⣩⠀⠀⠁⠀⠀⠘⠃⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠹⠂⠠⠄⠀⠈⠁⠰⠆⠀⠲⢾⣼⡥⣆⠭⠄⠀⠟⠀⠀⡀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⡀⠀⠀⠀⠀⢠⠄⢀⡀⢋⠐⢊⠙⠛⣭⡶⢂⠌⠁⡀⠄⠉⠀⠀⠘⠁⠀⠚⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠰⠠⠌⠼⠽⠠⠓⠀⠁⠀⠄⠚⠂⢀⠀⠈⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠐⠋⠁⠀⠒⠀⠐⠂⠀⠆⠐⠂⠰⠀⠖⡀⠃⠀⠏⠀⠀⠀⠈⠉⠂⠀⢣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠰⠄⠁⢀⠒⠀⣅⠘⡇⠈⠙⠁⠀⠀⠀⠀⢶⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⠀⠀⠐⠖⠀⠀⠁⠀⠀⠀⢀⡄⠈⠀⠰⡀⠀⠑⡄⠠⠖⠢⡄⠂⠀⢜⣥⠂⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣄⠀⠀⠀⠀⠀⣀⠰⠆⠈⠁⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⣴⡟⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣿⣤⡀⠀⣀⠉⠀⠀⠀⠀⢰⡆⠀⠀⠀⡛⠀⠀⠀⠐⣤⠞⠛⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⡦⠀⠀⠀⡄⠀⠀⠉⢻⠾⢿⡠⠶⠄⠀⠀⠀⠀⠀⢀⡀⠀⣀⣤⡶⠉⠀⠸⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⣿⢿⡿⠷⢾⡟⠋⠉⠀⢀⡠⠄⠘⠆⠀⠀⠀⠀⠀⠿⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⡀⢀⣠⠠⠄⠐⠀⠀⠀⠂⠀⠀⠀⢀⣤⠀⠲⠀⠀⠐⠀⠀⠀⢛⠵⠦⠁⡀⠀⠀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠁⢸⠰⣤⣄⠀⠀⠀⠃⠀⠀⠀⠰⡇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠐⢮⣑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⣶⡟⢻⠀⠀⠁⠀⠀⠠⠀⣠⡔⠀⠺⠁⢀⡀⠀⠀⠓⠀⠀⠰⠆⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢶⡠⠀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢱⣆⠀⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠉⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿⡟⠛⠸⢯⣷⡆⠀⢀⡜⠀⠀⢐⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠉⠀⠀⠀⠐⠀⠟⠁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠘⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⢰⡄⠀⣠⠄⡄⠀⢀⠀⠀⠸⣷⠀⠀⠰⠄⠀⠛⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠀⠀⠘⠂⠀⠀⠀⢣⠀⠸⠁⠰⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠺⠁⠠⠄⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡠⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠀⠀⡀⠐⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠤⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠬⣉⠐⠁⠀⠂⠐⠠⢂⠄⡀⡀⢀⠀⠀⠀⠊⠊⠄⠀⢁⠁⠈⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠒⠀⠀⠐⠢⠁⠬⠀⠡⠒⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠠⠄⠄⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠁⠀⣈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢱⠁⠈⢢⢀⠀⢱⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⠈⠀⠀⠐⡇⢁⠈⠀⠀⠀⠁⢠⣀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢘⡠⠀⠀⠀⠀⠀⠁⠙⢢⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⢴⡀⠀⠆⠀⠀⠀⢘⡠⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠣⡄⡎⠀⠀⠀⡬⡀⠄⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⢱⠘⠀⠀⠑⠀⠀⠀⡜⠩⢐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢤⠀⠰⠧⠱⠘⠆⠀⠀⡠⠾⠶⠭⢖⣞⡮⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣟⢀⣤⠀⠌⠠⠶⠚⠄⠀⠀⠀⠀⠈⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡂⠑⢖⣲⠀⠀⠄⡂⠐⠠⢂⠔⡠⢂⠖⠔⠕⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢐⠁⠀⠘⠀⠀⠤⠂⠰⡶⠤⠬⣄⣡⡀⢤⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠅⡪⠀⠀⠀⠀⡨⢐⠀⠇⢨⠀⡀⠀⠐⠄⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠄⠨⠂⠀⡄⠂⠀⠀⠐⠀⠘⡈⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⡀⣀⡀⢀⣜⡽⠀⠀⠁⠠⠀⠃⠃⠀⠀⠠⠪⠂⠀⠐⡆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠲⣁⡀⡀⢀⣠⡤⠚⠝⠒⠒⠂⠚⠉⠛⠑⠋⠀⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠠⠀⠀⡏⠐⠁⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢋⠀⠀⠀⢀⣾⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⠁⣀⣿⣄⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⣠⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡐⣴⣦⢐⣒⣶⣯⣤⣶⣾⣿⣿⣦⣽⣷⣧⣠⣧⡀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢼⣿⣿⣷⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡘⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢡⣤⡄⢚⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⣯⣿⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣜⣿⡿⣀⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠐⠶⣞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠠⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⢻⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣀⣠⣿⣆⠀⠈⢻⣿⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣏⢻⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣌⠙⠛⠳⠄⠀⢿⣿⠀⠈⠻⠟⠟⢿⣿⣟⢻⣿⣄⠁⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠻⢿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣧⡀⠀⡀⠀⠈⣿⠛⠀⠀⠀⠐⠿⣿⣿⣀⠈⠙⠿⠋⠀⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⡏⠻⣿⠶⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⣤⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠸⣿⡇⠻⠿⠇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⠷⣶⡆⠀⠀⠈⢿⣿⣿⣦⣄⠀⠀⠲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠠⣿⣿⣦⣿⣿⣿⣿⣷⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⠿⠟⠿⠁⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⢠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⣼⣷⡀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⢀⣾⣧⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣿⠈⠛⠟⠘⠟⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠻⠃⠻⠛⠁⣿⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣛⠉⠀⠀⢠⣤⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⡄⠀⠀⠉⣛⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⡶⠀⠀⠙⠿⠛⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠛⠿⠋⠀⠀⢶⣿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠷⠀⠀⢀⣀⣀⣀⠀⠀⢀⣤⡶⠟⠛⠛⠻⢶⣤⡀⠀⠀⣀⣀⣀⡀⠀⠀⠾⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡿⠋⠉⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠉⠛⢿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠁⢸⣧⠀⣠⣴⣦⣄⣠⣴⣦⣄⠀⣼⠇⠈⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⢹⣿⠋⠀⠀⣽⡏⠀⠀⠙⣿⡏⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣿⣧⡀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⣠⡿⠃⠙⢿⣄⠀⠀⠀⠉⢛⡂⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⠏⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠙⣷⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠁⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠈⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠢⠄⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠐⠒⠉
⠀⠀⠀⠀⠀⠀⡀⠀⣀⢀⠀⠀⠈⠄⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠂
⠀⣈⣀⠀⠀⠀⣀⠀⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⡸⠆⠀⠀⠀⣈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠁
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠁⠠⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢴⡇
⣷⣤⡀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠀⠘⠦⣄⡀⠀⠘⡆⠄
⣟⠻⢿⣶⢛⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠷⣼⣧⠰⠀⠀⠀⠀⣤⣀⣀⣀⡀
⠛⠻⢧⣽⣿⣷⠶⢿⣶⣤⣤⡀⠀⠀⠀⠀⠈⠲⣿⣿⠗⢦⣄⡀⠀⠈⠉⢻⠿⠟
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠿⣿⡷⣶⣤⡀⠀⠀⠀⠘⠏⠲⠀⠉⠀⠀⠀⠀⠈⠀⢯⠀⠀⡀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣽⣿⣿⠶⡄⣢⣀⣦⣠⣀⡔⠘⠀⠀⠁⠀⢸⡀⠀⢩⣤⣦⠀⢰⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣿⣿⣻⣿⣿⣿⢆⢆⠀⣤⣡⡶⠀⠀⠀⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣯⣰⣿⣿⡄⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡉⡛⣿⣿⣿⣁⡀⣧⡄⠘⠻⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣻⣿⣿⣿⣿⡿⠁⠀⠀⣾⣿⢦⠀⠀⠐⣾⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⣿⣿⣿⣿⣿⣇⠀⠀⣼⣿⣿⡗⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⣀⠀⠀⢠⣀⣠⣾⣿⣿⡏⢿⣿⣷⣿⣿⠀⣼⣿⣿⠿⢀
⠀⠀⠀⠀⠀⢀⣄⣠⣾⣟⡯⠤⢩⣶⣶⣿⣿⣿⣿⣿⣿⣷⢸⣿⣮⣿⠿⢠⣿⣿⣷⠆⠄
⠀⠀⠀⠀⠀⠐⣿⣿⣅⡠⠟⠒⠛⠛⠉⠙⠿⣿⣿⣿⣿⣿⣿⡿⢿⡇⠛⢸⣿⣿⣟⠁⠀⠀⡄
⠀⠀⠀⠀⢀⣠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣛⣿⣿⣿⣿⣶⣿⣶⡂⣸⣿⣿⣿⠀⠀⠀⢘
⠀⠀⢀⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⠿⠛⠃⣿⡇⠛⢻⣿⢿⡓⣧⠀⠈⢷⠤⠠⡀
⣤⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠉⠀⢀⣄⣠⡶⣟⡀⠀⠀⠻⣿⠇⡟⠡⡀⠘⠓⢦⣄
⣯⣁⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠈⠻⠿⠿⣿⠀⠀⢠⣶⡶⠂⠀⠀⣿⣄⠀⠀⠈
⢇⡘⠇⠀⠐⢿⣷⡆⢀⣀⣦⣤⣶⣤⣤⣴⣿⡿⢛⣇⠂⠀⠀⠀⠀⠈⠁⠛⠉⠀⠀⠀⠀⠀⢠⠀⠉⠑⠾
⠋⠁⣀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣭⣿⠟⠁⢸⣟⠃⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣤⠤
⠓⠆⠉⠹⢧⡄⠰⣿⣿⡿⣿⡟⣻⠿⠛⠁⢀⡠⣿⣿⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠙⢤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠋⠻⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠈⠻⠛⠃⠈⠛⠟⠁⣽⣿⣿⣿⡇⠀⣠⣶⣤⠀⠨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠁⠀⢀⣀⣀⡀⠀⠈⠻⠿⢿⣿⡆⠀⠻⣿⡿⠀⠠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⣴⣾⣿⣷⣆⣠⡀⣀⣰⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣷⡄⠀⠀⠉⠉⠀⠀⢠⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣶⣶⡄⢠⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⡞⠉⣧⣤⠤⠤⠤⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣇⡀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⢸⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⡀⠀⠀⣸⡀⠀⣸⡀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢳⢤⠗⠒⢚⡡⠝⠒⠒⠓⠞⠓⠚⠓⠛⠿⣇⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣻⠒⣦⠀⠀⠸⠀⠀⠛⠁⠀⠘⠃⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠓⠚⠳⡤⠜⠀⠀⠠⡂⢀⠀⢀⡀⠄⠑⠄⠀⠀⠀⢀⡤⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⡀⠀⠈⠉⠀⠀⢀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⠔⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠺⠤⠼⠶⠃⠀⠀⠀⢀⡠⠞⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣴⠃⠀⠀⠀⡀⡸⠧⣀⡀⠠⠤⠀⣀⣀⣀⣤⠤⠤⠖⠚⠉⢰⡦⣤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠋⠘⠧⣤⡤⠞⡿⠁⠀⠀⠉⠙⠓⠒⠒⡧⣤⣤⡤⠶⠶⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣠⠴⠒⠒⡆⡖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢱⡀⠀⠀⣧⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢣⠀⠀⢸⡄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢣⠀⠀⢇⠀⠧⢤⡀⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⡠⠄⠒⠳⣄⣉⣹⣏⣉⣙⣻⢩⡏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣾⣥⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⠁⣸⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠄⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠼⠛⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⢿⣿⣧⠀⠀⠀⠀⢀⡀⢴⡆⠰⠀⢂⡀⠀⢠⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⣷⣿⣘⣿⢆⠠⢀⠀⠁⠈⠁⢚⣐⠃⡀⣀⠀⠀⠄⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⢁⣥⣿⡗⣤⣃⢦⠘⠳⣦⠠⠖⡀⠰⠊⠁⠀⠀⠀⠀⣠⡀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣸⣿⣿⣧⡞⣼⣫⠏⠳⠠⠾⠨⠁⠚⠰⠗⠀⠊⠀⠄⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢀⣠⣼⣿⣾⣿⡛⣿⣽⡘⠧⠥⢁⣶⡘⠁⢈⡀⢀⢉⡄⠈⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⣻⣿⣼⣿⣿⣿⡿⡏⣿⣿⣫⡿⢊⢓⣈⠁⠄⠙⠡⠠⠀⠄⠀⠐⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣿⣧⣶⣬⣉⠻⠛⠉⠀⢀⣹⣏⣻⢾⡤⠜⠀⣈⢃⡚⠂⠠⠄⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠛⠻⢿⣿⣿⣶⣦⣄⡆⠺⣿⣯⣻⣓⡒⡐⠏⠁⣅⠘⢒⠀⡀⠈⠁⠀⠐⠃
⠀⠀⠀⠀⠀⠳⡄⠀⠀⡀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣦⣴⣴⣶⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁⢀⠈⠲⠀⠀⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⢷⠶⠛⠻⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠿⠿⠿⢿⠻⢿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⣤⣤⣦⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⢿⠁⠀⠀⠀⠀⠀⠘⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣃⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠃⠐⣿⣿⣿⣗⣂⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣾⢿⣿⣿⣯⣍⠙⣿⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣟⠲⢿⣿⣿⣿⢿⡀⢿⡈⣟⠿⠍⠀⢀⠏⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠉⠛⢻⣿⣿⣿⣦⡦⣤⣹⣷⣶⠶⠂⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡶⣾⣿⡛⢃⣐⣞⣛⠀⠀⠙⢿⣦⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣵⠟⡟⠛⢛⣿⣿⠟⠛⠳⠞⠟⠀⠀⣸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣤⣠⣤⣤⣀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⠿⢿⣿⠿⠇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠻⡟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠾⠿⠷⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠦⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣆
⠀⠀⠀⣸⡟⠀⣀⡀⠀⠀⢀⣴⠟⠁⣿⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⢀⣼⣏⣴⣾⠏⠀⠀⢀⡾⠃⠀⠀⠸⣇⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣿⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠰⠿⠿⠋⢹⡟⠀⠀⢀⡿⠁⠀⠀⠀⠀⠹⣦⠀⠀⠘⣧⠀⠀⠀⠀⠀⣿⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⣼⡇⣀⠀⣾⠃⠀⠀⠀⣤⠾⠷⠙⢷⣄⠀⠘⢷⣄⠐⠛⢿⣷⡀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⣿⡿⢿⣷⡏⠀⠀⠀⠈⠋⠀⠀⠀⠀⠉⠻⠶⣦⣽⣿⣶⣾⡟⠁⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠻⠇⢸⣿⡇⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⢻⣷⠟⢻⣆⣰⡟
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠈⠻⣦⣀⠀⠀⠀⠀⠀⢤⠤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠙⠛⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⢿⡁⠀⠀⠀⣀⣀⣤⣶⠶⠶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⢹⡟⠳⠶⠶⠾⠛⠋⠁⠀⠀⠘⠷⣶⡾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠘⠷⣦⣤⣤⣤⣤⣤⡶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠉⠻⣶⡄⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⣤⣤⣤⣤⣤⣤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠁⠀⠈⠉⠉⣹⡟⠉⠉
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠁⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣼⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢀⣼⠏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡿⢷⣤⣤⣾⡃⠀⠀⢀⣿⠛⠷⠶⠶⠶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠉⠉⠻⣷⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣻⣶⣶⣶⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠒⠢⠄⡐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⢀⠄⡤⠂⠚⠁⠀⢀⠀⠁⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡀⡠⢀⢄⠈⠀⠀⠋⡄⠀⠀⠈⢷⡀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡿⠀⠀⠀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠙⣦⡀⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠈⠉⠉⠀⠀⠀⠹⢧⡀⠀⠀⠈⠛⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠉⠳⢄⠀⠀⢄⠀⠀⠀⠀⡿⠀⠀⠀⠀⠘⡇⠘⠄
⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⢣⠀⠈⣇⡒⠶⠞⠁⠀⠀⠀⠀⠀⢷⡰⣦⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⣄⡐⠶⢄⡀⠀⠀⣄⣸⠃⢀⠤⠤⢾⡀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠻⣝⠛⠾⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡧⢀⡵⠀⠀⡏⠓⠢⡅⠀⣠⠚⣷⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣾⠃⡀⢠⣬⣷⣄⡇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢊⠥⠚⠉⣠⡞⠐⠠⣀⠀⠰⢿⠁⠀⠹⡜⠄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢿⡀⠈⠛⢷⡇⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠺⠷⣶⣶⣿⣄⣀⣀⠀⢱⣧⠙⣧⠀⠀⠈⠦⠠⠄⠀⠀⠀⠀⠀⣸⡃⣀⣀⣙⣧⠀⠀⣾⣷⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠉⠉⢙⣻⢷⣿⠀⣿⣧⠀⠀⠀⢨⡖⢀⠂⠤⣤⣤⡿⠋⣿⠉⠉⠛⠂⣼⠃⢻⡄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⣾⣋⣶⣿⣯⣄⣼⠘⢧⣄⠀⡾⠀⠘⣿⣿⣶⠟⠀⢀⡇⠀⠀⠀⣼⠁⠀⠈⣷⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠈⣽⠏⠙⣻⡟⠛⠛⣿⠿⢧⡀⠀⢸⣿⣿⡀⠀⢸⡇⠀⠀⣰⡏⠀⠀⠀⠈⠂⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⣸⡏⠀⣰⡟⠀⠀⢈⣙⢶⡿⣶⣄⠉⣉⣉⡷⠀⠸⢇⣆⢰⡟⠀⠀⢀⣀⣶⡶⠶⣆⠤⡀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⠟⠀⠀⠀⢸⡿⠻⡉⠛⠛⠉⠉⠉⡇⠤⠤⣠⣷⡶⠟⠛⠛⠉⠉⠀⠀⠀⠈⢳⡌⠳⣄⠀⠹⣆
⠀⠀⠀⣠⠋⠙⠛⢿⠀⠀⠀⠀⠀⠀⢀⡠⢁⣟⣴⠿⠶⠶⠶⠶⢾⠁⡴⠀⢠⠀⠀⠠⠤⠗⠂⠀⠈⠳⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠎⠿⠧⠀⢻⡄
⠀⠀⣴⠃⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠘⡄⠓⠀⠀⡃⠀⠠⠤⠄⠂⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠀⠀⢫⡓⠒⠒⠀⠀⠀⠀⠀⣰⡿⠀⢀⣀⣀⣠⣤⣤⠶⠖⠒⠚⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠓⠓⠲⠖⠒⠒⠊⠉⠙⠛⠛⠛⠋⠉⠉



⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠶⣶⣒⠲⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠲⠶⠶⠶⠦⠤⣄⣀⣀⡀⠀⠤⣄⣀⡀⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⠁⠀⠀⢀⣀⠀⠠⠄⠽⠏⠉⠉⠙⠛⠛⠛⠓⠀⠀⠀⠀⠜⠀⠀⠀⠈⠉⠑⠚⠂⠈⠉⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠰⡊⠀⠀⠀⠀⠀⠀⠛⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⡘⢧⠀⠀⠀⠀⣭⠷⠀⠀⠀⠀⣀⠤⠴⠖⠒⠒⠒⠒⠒⠤⢇⢀⣠⠴⠒⠚⠋⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢧⠈⠳⣄⠘⡷⣌⢷⡀⠀⠀⡞⠁⠀⠴⠄⠀⠀⠀⠀⠀⢀⣸⠏⡄⠀⠀⠀⣠⣾⣁⣠⣤⡤⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠈⣯⣷⠀⠀⠙⠒⠶⠶⠤⠤⠤⠄⠒⠚⠉⠙⠲⠶⠶⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠲⢶⣖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣁⠤⣤⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠚⠉⠉⠁⠈⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⠒⠒⠚⢛⣻⠿⠓⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⣠⡤⠴⠚⠋⠁⠀⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⣯⡽⠛⠉⠀⢀⣠⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠬⠤⢴⡖⠠⡤⢤⣤⣀⣀⣈⣊⠥⠤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠲⣀⠀⠇⠀⣰⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠙⠳⢤⣸⠀⡤⢽⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⢢⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠲⢤⡀⢀⡍⢹⡇⠀⡉⠑⠒⠒⠒⠒⠂⢸⠓⠃⠀⠀⢀⡀⠀⢰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡀⠀⢙⠛⠀⠘⢦⠀⠘⣦⣄⡀⠀⠀⠀⢺⢤⣄⠚⢳⡏⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡁⠀⢸⡄⠀⠀⢸⠀⠀⠸⡄⠉⠛⠒⠤⢼⡿⠘⠦⡼⠁⠀⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠀⠈⡟⠀⠀⢳⠀⠀⠘⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢠⠋⠈⠉⠁⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠀⢰⠇⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣠⣧⣴⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⣀⡀⣀⣠⡼⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠠⣾⣿⣿⣿⣿⣿⠿⠛⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣎⣡⣌⣠⠥⠀⠀⣼⡀⠀⢀⣀⣤⣴⠇⠀⠀⠙⢿⠿⠋⠁⠀⠀⢀⣠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟⠓⠒⠲⠶⠶⠶⠖⡾⠉⢻⠋⠉⠁⢠⡏⠀⠀⠀⠀⠀⠱⣄⠀⣠⡔⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣀⣇⣀⣾⣀⡀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠈⠛⡹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠧⣄⠀⠀⠀⠤⠬⠭⠕⣾⠉⢁⣀⣁⡬⠷⣠⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⠈⠉⢓⡶⠶⠒⠒⠋⠙⠣⠍⢡⡀⠀⢐⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠒⠀⠐⠀⠀⠆⠂⠂⠀⠀⠂⠀⠀⠐⠐⠐⠂⠀⠀⠀⠀⠘⠀⠃⠁⠐⠀⠃⠘

⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠘⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠷⠙⢿⣿⣿⣾⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⢛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡸⣿⣿⣿⣿⣇⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣤⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⢿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣉⣻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⡟⢰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡏⠘⢿⣿⣿⡟⣋⣉⣥⣼⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣙⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⠿⠿⠿⢿⣿⣿⣿⡿⠿⣛⣫⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣋⣥⣴⣶⣦⠀⢠⣤⣤⠄⠀⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣾⡿⠟⠉
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣷⣿⣿⠿⢟⣉⡄⠀⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⡀⠀⠺⠿⢶⣶⣤⣤⣤⣤⡶⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠛⣿⣿⣿⣿⣿⠟⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡧⣶⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢶⠞⠋⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⠀⠉⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⣀⣀⣀⡀⢹⣿
⠀⠀⠀⠀⠀⠀⣀⣴⣛⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠶⠶⠶⠶⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠉⠉⠓⣆
⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠠⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠉⠲⠤⢒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡈⠀⣠⠊⢁⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢑⠶⠤⣄⣀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⡇⢠⠇⠀⢸⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠉⢙⠓⠦⢤⡖⣿⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣇⡼⠀⠀⠸⡆⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠠⠁⠀⠀⣀⠴⠚⠉⠑⠢⣄⠀⠸⣝⠒⠺⠷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⢳⡀⠀⣀⠀⠀⠀⠑⣄⠀⡸⠓
⠀⠀⠀⠀⠇⠀⠔⠋⠁⠀⠀⠀⠀⠀⠈⢣⠞⠈⠳⣄⠀⠀⠀⠀⠀⠉⣽⡃⠀⠀⠀⠀⠀⢹⡀⠀⠑⠦⠏⠣⣀⠀⠀⠀⠉⠀⣠
⠀⠀⠀⠘⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠺⣿⠀⠀⢀⣬⣳⠄⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠈⠁⠐⠂⠂⠈
⠀⠀⠀⡇⠀⠆⠈⠀⠙⠓⠦⠤⣤⣤⠤⢿⠀⠀⠈⠛⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡄⠄⠊⠀⠀⠀⠀⠀⠷⠐⠒⠒⠒⢞⠉
⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠀⡜
⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⣟⠲⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⠒⠒⠲⠶⠶⠒⠒⠚⠋⠉⢁⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣁⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⢶⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠦⠤⣤⠀⣤⣤⣤⣤⠤⠤⠴⠒⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡧⢄⣀⠀⢠⠀⠀⠀⠀⠀⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠤⠽⠿⠭⠝⠛⠓⠦⠭⢉⣛⣿⣃⡠⠕

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⢉⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⢀⣤⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⡌⠹⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⢤⣤⠄⡆⠠⢤⠤⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⣠⣿⣿⣿⣿⣧⣿⣿⡆
⠀⠀⠀⣼⣿⣷⣿⣿⣿⣇⠀⠀⠀⠃⢀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⣀⠈⠉⠉⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣯⣈⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⠋⢀⣀⠹⣋⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⡏
⠀⠀⠹⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠉⠙⠻⣿⣿⣷⣾⣿⣔⣢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟
⠀⠀⠀⠙⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⢀⡶⢛⣻⡟⠶⣷⢦⡀
⠀⠀⠀⣿⢀⣾⣿⣿⣧⣼⣷⢿⣦⡀
⠀⠀⠀⣿⢻⣿⣩⡿⣿⡿⣿⣾⡏⣷
⠀⠀⢰⡟⢿⡿⢷⣾⣿⣥⣿⣿⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠈⣷⠘⢿⣛⣒⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣀⡴⠞⠛⠉⠉⠉⢉⠉⠉⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠈⠳⠶⠼⠿⣿⠋⠉⠉⠀⠀⠀⢀⡴⠟⠉⠀⣀⣤⣶⡿⠿⠛⣁⣬⢓⣴⡗⣀⠈⡁⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⢀⣴⠏⠀⣀⣶⣾⢟⠩⣀⠒⣢⠬⢐⣴⣿⢟⣐⠆⣼⣯⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠄⣶⡿⡋⢊⠀⠵⠔⣂⣥⣾⢟⡫⡀⠇⢆⣾⡟⡹⣿⣧⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⢪⣾⠏⡐⣎⣠⣴⣶⠿⢟⡋⠁⢈⠢⢈⣴⠟⢹⣦⡒⢊⢻⣧⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⢢⡿⢃⣵⡾⢟⣋⠁⠒⠔⣀⣁⣠⣴⠾⠛⠁⠀⠀⠙⢿⣮⠀⢻⣿⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠲⠋⣡⣿⢋⡰⣌⣥⡶⠿⠛⠛⠉⠉⠀⠀⠀⢀⡀⠀⠀⠀⠹⣷⡠⢻⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠋⣰⡿⠅⣣⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⠀⠀⠹⣷⡘⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⢰⡿⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣿⣱⠟⠀⠀⡦⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢠⡆⠀⠀⠘⣿⣿⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⢸⡏⠀⠀⠀⢿⣀⣠⡿⠀⠀⢀⣴⠆⠀⠀⠀⠹⣄⣀⣼⠃⠀⠀⠀⣿⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣼⡇⠀⠀⠀⠀⠉⠉⢀⡀⠀⠿⢧⡀⠀⠀⠀⣀⡈⠉⠁⠀⠀⠀⠀⣿⣀⣴⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⢀⣀⡀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⣿⣿⣶⣌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠐⡇⠀⠀⢠⡾⠋⠉⠙⢳⡄⠀⠀⢸⠇⠀⠀⠀⣰⣟⠰⣀⠽⢓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡼⣟⡻⠿⠾⣆⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠞⠋⠀⠈⠙⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⠟⠛⠛⣿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⣛⠛⠛⠁⠀⠀⠉⠙⢋⡟⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⡿⠀⠔⠁⠙⠷⣦⡀⢀⣀⣴⡾⠋⠁⠒⡀⢹⡆⠀⠀⠀⢀⣤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⠷⠆⠀⠉⢷⡀⠀⢸⡇⢘⣀⠀⠀⠀⣿⠉⠉⠉⣿⡇⠀⠀⠈⣕⠀⣷⠀⠀⣰⠏⠀⠀⠲⢾⣟⡀
⠀⠀⠀⠀⠀⣠⠞⠋⠁⣀⣤⣤⠶⠿⣧⠀⢸⡇⣸⡟⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠈⣿⡀⢻⠀⢠⣿⠷⢦⣤⣄⡀⠈⠛⢦⡀
⠀⠀⠀⠀⢸⣇⣤⠶⠛⠉⠀⠸⣆⠱⠹⣧⢸⡇⣿⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⢸⢀⣾⢓⠀⣼⠁⠈⠙⠛⠶⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢃⢻⣿⡇⢻⣷⠀⠀⠀⣿⡆⠀⠀⢸⡇⠀⠀⠀⣿⡇⢸⣾⢇⠂⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⡆⢻⣇⢸⣿⠀⠀⠀⣿⠁⠀⠀⣼⡇⠀⠀⠀⣿⢃⣾⠃⡌⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⠈⢿⢾⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⢠⣸⣿⣿⠏⡳⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠉⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠍⠩⠉⠉⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠫⣝⠛⢻⣟⣻⣿⠽⠷⢿⣛⠿⢯⠿⠿⠽⠟⠛⠛⠛⠛⠛⠛⠋⠉⠋⠉⠁⠈⢿⣄⠀⠀⠀⠀⠀⠀⠨⢍⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠁⠀⠘⣳⢊⠁⠌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢬⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿
⠀⠀⠀⠀⠀⢤⣦⣤⣤⣠⣝⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣾⣿⣿⣿⣷⣧⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠉⢛⣻⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠐⠛⠁
⠀⠀⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢿⡎⠉⠻⣿⣿⣿⣿⣆
⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣄⢙⣶⠋⠈⠉⠛⠿⣿⣦⡀
⠈⠉⢉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢉⣿⠏⠀⠀⣀⣀⣀⠀⠀⠹⠇
⣀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢿⣿⣿⢰⡟⠁⢠⣶⣿⣿⣿⣿⣿⡷
⢠⠿⢻⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡑⠢⡀⠀⠘⣿⣿⠀⠁⠀⣼⣿⠟⠉⠉⠙⠁
⠁⢀⡾⢋⣿⣿⣿⠃⣹⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣸⠂⢠⡸⣿⣷⣡⣠⣿⡿
⠀⠈⠀⢸⣿⣿⡟⢰⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣾⣿⣷⣿⣿⡇
⠀⠀⠀⡯⠉⡿⠁⡼⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢐⠆
⠀⠀⠀⠁⠀⠀⠘⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣶⡄⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⡿⠀⣸⣿⣿⣿⣿⠟⣿⠋⢛⠊⣿⠁⣿⠿⠻⠀⠛⣿⢻⡷⣿⣷⠊⠈⢹⠇⠈⠉⠀⠀⠀⠐⠋⠹⡧
⠀⠀⠀⠀⠀⠁⢀⣽⡟⢰⡿⠋⠔⠁⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⣷⣾⣏⠀⠀⢻⠀⠀⠀⢰⡖⠀⠀⠀⣰⡷
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠗⠀⢻⡄⠄⣼⢹⣷⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣶⢀⣴⣿⡆⠁⠈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⣿⡇⠀⣬⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣏⣼⣿⠀⣴⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠙⠳⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣵⣤⣷⣾⣿⣷⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣼⣿⣿⣿⣿⣿⣿⣾⣿⣿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢟⠿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣀⡤⠀⠀⠀⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⣀⣄⡀
⠀⢠⣾⡿⠁⣠⠴⠛⠡⠐⢾⡿⠆⣠⣤⢄⣠⡾⠖⠒⠀⠈⣻⣿⠷⣍⡠⡀
⣠⡿⠋⣰⠋⠁⠀⠀⠀⠀⠈⣠⣮⡽⢅⣫⡥⣰⢣⢆⡴⡿⢋⣵⠖⣊⣽⡄⡀
⡟⠁⣼⠁⠀⠀⠀⠀⠀⠀⣾⡵⠋⣾⡏⠃⡴⢡⢫⠟⡼⢣⢀⢎⡾⠟⠁⣳⡝
⠀⢸⡇⠀⡄⠀⠀⠀⠀⢠⠋⠠⣫⢾⠛⣳⠇⠟⢈⡽⢣⣯⡀⠈⠀⠀⠀⡽⡿⣠
⢀⣿⠃⢠⣷⣄⠀⠀⠰⢃⢄⣴⠓⢱⣷⢋⣜⢄⠼⡁⣼⣿⣿⠀⣼⡀⢠⣾⡿⠹
⣸⠃⠀⣼⡿⣡⣷⠤⣠⠽⠃⠖⠐⡀⣴⡏⠎⠀⡾⠁⣿⡹⠋⠀⣿⡇⡈⠱⠖⠀⠁
⡏⢠⣾⡟⣰⡟⢁⣴⡟⢠⠌⣖⢸⢳⡇⡼⣒⡄⡼⠰⠈⡀⢾⡤⡴⢮⠁
⢣⣿⣿⠐⠋⢀⡾⠋⠀⠀⠀⠻⠌⠊⣼⡕⡉⠆⠔⠀⡶⣋⠟⠉
⠈⢿⣿⣇⠀⣿⢁⠂⠀⠀⠀⠈⠃⠀⠃⢸⡠⡄⠇⢺⣇⠍⡌
⣦⠀⠙⢿⣷⣾⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠷⠀⣬⠷⠃⠀⠀⠈
⢿⣷⣄⠀⠈⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣎⠿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡟⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⢀⡀⠤⠤⢠⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⢸⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⡠⠀⢄⠸⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⣰⠃⠀⠀⠀⠀⠰⡅⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠉⠀⠁⠸⠀⠀⠀⢠⠄⠀⠢⡀⠀⠀⢧⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⡶⠒⢲⠶⡶⠒⠶⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⡄⠀⠠⠁⠀⠀⠸⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠟⠛⠛⠛⠒⠛⠚⠟⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠗⠶⠶⠶⠴⠴⣿⡩⢽⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣽⠀⠀⠀⠠⠤⠄⠀⢰⠀⠀⠀⢈⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣀⠀⠀⣯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣏⢼⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠒⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⣀⠀⢀⡀⣀⣀⠠⠔⠊⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⡏⠈⠈⠉⡉⡉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢣⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣩⣩⡩⡍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣓⣓⣂⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡼⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⡁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣰⡻⢤⣀⣀⣀⢀⣀⣀⣀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⢹⡇
⠀⠀⢀⣀⣀⣀⣤⣤⣼⣧⣿⣯⣼⣯⣥⣴⣮⣽⣯⣽⣫⣋⣝⣩⣉⣙⣽⣫⣪⣗⣡⣠⣟⣕⣽⣟⢙⢁⡽⡭⣫⠿⡿⣩⠞⠕⣽⠁
⠀⠀⠸⣆⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠂
⠀⠀⠀⠈⠉⠉⠉⠉⠙⠚⠛⠛⠓⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠚⠚⠓⠚⠛⠛⠒⠛⠛⠓⠛⠛⠓⠚⠒⠒⠒⠲⠶⠶⠴⠶⠴⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⠤⠽⠓⠠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⢀⡀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠢⠁⠀⠀⠁⠀⢀⣀⡀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠒⠐⠁⠀⠀⠀⢀⠎⠀⠈⠀⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⣀⡠⠤⡖⠁⠀⠀⠀⠀⠀⠀⠀⢀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢁⠀⠀⠀⠌⠀⠀⠀⠀⠁⠀⢣⠣⡀⠀⠀⠀⠀⡘⠀⢀⠆
⠀⠀⠀⠀⣀⠘⢆⠀⠰⠀⠀⠁⠒⢠⠐⠂⠓⢄⡈⠀⠀⠔⠁⠀⠀⠀⠘⡀⠱⡀⠀⠀⠰⠁⠀⡌
⠀⠀⠀⠄⠉⠁⠈⠉⣠⣴⣴⣿⠖⠘⠌⡈⠁⠒⠭⡈⠀⠀⠀⠀⠀⠀⠀⠃⢀⢸⡄⢠⠃⠀⡘
⠀⠀⠀⠀⠀⢏⡐⠤⣈⠁⣭⣤⢄⠓⢛⡀⠅⡒⠄⡀⠌⣐⠂⠦⢀⡀⠉⢩⠀⠀⠀⠎⠀⢠⠁
⠀⠀⠀⠀⠂⠁⠈⠐⠤⣉⠊⠅⡀⠤⣌⡙⣷⠾⣫⠴⠀⢐⡠⠄⡀⠈⠉⠒⠤⢄⡘⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡁⠂⠍⡃⠦⣔⣁⠉⠁⠂⠀⠀⠍⠂⠀⠀⠀⠈⠉⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⠑⠢⠄⠁⠠⠐⡈⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⠆⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰

⠀⢀⣤⣀⠀⠀⣠⡀⠀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀
⢰⣿⣿⣿⡆⢸⣿⣿⠀⣿⣿⣿⣿⣄⣿⣿⣿⢿⣿⡿⠿⢿⣿⡿⣿⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⢸⣿⠿⠿⢿⣿⠿⠿
⣿⣿⠀⠻⠟⣼⣿⣿⡄⣿⣿⢀⣿⣿⣿⣇⣀⠸⣿⣇⡀⢸⣿⡇⣹⣿⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⢸⣿⣀⠀⢸⣿⣀
⣿⣷⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡿⠁⣿⡿⠟⠀⣿⡿⠃⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⢸⣿⠛⠀⢸⣿⠛
⢿⣿⣄⣶⣾⣿⡟⣿⣿⣿⣿⣿⣿⡀⣿⣇⣀⣨⣿⣧⣄⣸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⡀⣿⡇⢸⣿⠀⠀⢸⣿⣀⣀
⠘⢿⣿⡿⣿⣿⠃⢻⣿⣿⣿⠹⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡀⠀⠀⠀⡆⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⡇⣸⣿⠀⠀⣾⣿⣿⣿
⠀⠀⠉⠀⠉⠉⠀⠉⠉⠩⠍⠤⠩⠭⠩⠉⠉⠍⠭⠩⠉⠉⠭⠁⠌⠭⠅⠀⠡⢠⠥⠠⠄⢤⡀⠤⠠⠄⠀⠀⠀⠠⠄⠨⠭⠉⠉⠩⠭⠭⠁⠭⠩⠤⠠⠭⠭⠍⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠉⠉⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⠈⠀⠀⠀⠀⡤⠀⠠⠔⠒⠐⠙⠄
⠀⠀⠀⠠⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠊⠀⣈⢵⣂⠢⠀⠐⠈
⠀⠀⠀⠄⢈⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠇⠀⢠⣈⢡⠇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠁⠀⠈⠐⠃⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠢⠤⡀⠀⠀⠀⠐⠂⠑⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠤⠐⡀⠀⠠⠐⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⣀⡀⠀⠀⠀⣀⣀⠀⡈
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠠⠠⠄⡐⠁⡢⠥⠈⡅⢰⢠⡄⢐⠁⠊⠁⠉⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠂⠁⠘⠒⠀⠅⠃⠀⠀⠀⠀⢼⠐⠁⠀⠀⠀⠀⠀⠀⠀⠂⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠡⠀⢘⠚⠀⠠⢀⠔⠉⠉⠑⠄⠀⠸⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢈⠄⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠉⠉⠚⠒⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠪⢀⣠⠀⣀⣀⣀⡀⠀⠠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠢⢄⣀⠆⠈⠀⢠⢾⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠂⠄⠀⠀⠀⠈⠲⢇⣀⣄⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠶⣄⠀⠀⢹⡿⡇⠀⡗⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⡒⠈⠀⠀⠀⠀⠀⠐⠓⠙⠂⢀⣣⡰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⡀⠀⠀⣢⠠⢄⡀⠀⠀⠠⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠠⢠⣀⠀⢀⣤⢠⡄⠆⢄⡀⡨⠃⢠⡰⠌⠖⠆⢀⠀⢀⡃⠉
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⡀⠀⠈⠐⠀⡠⢊⣭⡭⣒⠏⣤⣤⢓⣬⡆⣷⡂⣦⠄⡠⣸⠤⠼⡂⠈⠈⢊⠀⢀⡀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⣀⣦⣯⡷⡱⣧⠾⣭⣓⡻⠎⣓⣛⡵⣧⢞⡕⣹⢫⡞⣅⢢⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠙⠛⠛⠙⠓⠚⠛⠲⠻⠻⠙⠛⠚⠛⠳⠿⣯⣶⠿⠿⠟⠓⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠠⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⢂⠀⠀⠀⠀⠂⢀⠁⠀⢄⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠄⠀⠐
⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⡆⠀⠈⠁⠀⠀⢀⠎⠀⠀⠀⡀⠊⠁
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⡅⠦⠠⡀⢠⠄⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⠒⠶⠒⢊⡁⠴⡀⠀⠀⠙⠂⠦⠈⠀⠀⠀⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠀⣤⠄⠐⠠⠊⠀⠸⠴⠦⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠂⠐⠂⠁⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡿⠋⠀⣿⣿⡿⠟⠁⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⡈⠛⠋⠀⠀⠀⠈⢉⡀⠀⠀⠙⠛⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠛⠋⢀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠀⠀⢚⡫⠥⢤⡀⠀⠀⠀⠀⠀⠀⡠⠖⠈⠹⣍⠉⠀⠀⠀⠈⢿⣿⡿⠿⠿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣬⣷⠀⣞⣠⠖⠋⠉⠆⠀⠀⠀⠀⠜⠉⠒⠲⢤⣘⡇⠀⠀⠀⠀⠀⠉⢠⣴⣶⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⣯⣹⡇⡏⠀⣔⣐⣦⠀⠀⠀⠀⠐⠀⡰⢒⣦⠀⠈⡇⠀⠀⠀⠀⠀⠀⢿⣍⡹⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⢻⡇⠜⠀⠌⠿⠋⡖⠉⠚⠀⠀⠠⡝⣛⣋⢠⠔⠀⠀⠀⠀⠀⢀⣀⣘⣋⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣰⠟⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣠⣤⣈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠶⢤⣤⣀⣀⣀⣀⣁⣈⣠⣤⣴⠞⠋⠁⠀⢀⠋⠀⠀⠙⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣠⣤⣭⣭⣍⡉⠙⣛⣶⣿⣥⣤⣄⡀⢀⡞⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⣴⡟⢉⣠⣾⣿⣿⣻⣾⡋⢁⣠⣾⣿⣿⣿⠟⠀⠀⠀⠀⢀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⣾⣿⣿⣿⠿⠾⣷⠌⠙⢻⡿⠿⠿⠿⠛⠉⠁⠀⠀⠀⠀⢀⣼⠃⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⡀⠀⢀⣼⠟⣿⣿⣿⣟⠓⠆⠀⠀⠀⢸⠁⡆⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠈⠉⠙⢧⣹⡏⢸⣿⣿⣿⣿⣭⡍⠀⠀⠀⢸⣄⣧⣤⣤⣤⣶⣶⣾⣿⣿⣿⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠄⠀⠀⠀⠀⢸⡟⠁⠐⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⣾⣿⡦⢠⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠛⠛⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣾⣿⣯⣟⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⢹⣿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⣿⣛⣛⣛⣻⣿⣿⡿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⠒⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣠⠖⠉⣠⠔⠁⣼⡡⠖⠚⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠘⠋⠀⢀⡞⠁⢠⠞⠉⣀⡤⠤⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⣠⣯⠴⠖⠛⠚⠊⠁⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⡠⠠⢄⡀⠀⠀⢀⣀⣼⠁⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠣⠤⠤⠼⠆⠀⣴⠟⠙⣦⠟⠁⠀⢀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠾⠤⠤⠴⠃
⠀⠀⠀⠀⠀⢠⠤⠤⣤⠀⠀⢰⡇⠀⡜⠁⠀⠀⠀⠸⠟⣋⡉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⢻⡀⠀⠀⢠⡤⠤⢤
⠀⠀⠀⠀⠀⠘⠶⠚⠁⠀⠀⢺⠏⣸⡇⠀⠀⠀⢠⠖⠋⠉⠙⠳⣄⠀⠀⠀⠀⠀⠀⢀⣀⣉⠀⠈⣇⠀⠀⠀⠙⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⣇⠀⠀⢠⡏⠀⠀⠀⠀⣀⣹⡆⠀⠀⠆⠀⠀⣿⣁⣈⣳⠀⢿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠖⢻⡀⠀⠈⣷⠖⠚⠋⠛⠋⣼⠀⠀⠀⣦⠀⠀⠣⡌⡩⠋⠀⢸⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣆⠀⢧⠀⠀⠘⠦⣤⣀⡤⠞⠁⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣁⣠⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⡴⠋⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⡼⣿⣿⣷⣼⠆⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠰⠦⣌⣻⣿⡟⠀⠀⠀⢀⡴⠋⠀⠀⢀⣸⡁⢸⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣤⣄⡀⠀⢸⡆⠀⠀⠀⠀⠀⠈⠙⠁⠀⢀⡴⠋⠀⠀⠀⠀⡏⠀⢈⡇⢀⠎⢙⡆
⠀⠀⠀⢀⣠⠤⠤⠴⠊⣉⠴⠚⠉⠉⠉⣿⣉⠉⢻⡄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢳⠀⠙⠉⠙⡶⠊
⠀⠀⡾⠋⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠈⠉⣻⠇⢛⡗⠶⠦⠤⠴⠚⣏⠙⣦⠀⠀⠀⠀⢀⡰⠋⣠⠖⠀⠁
⠀⠀⠳⡞⠀⠀⠀⡀⢀⠇⠀⠀⠀⠀⠀⠀⢠⡟⠁⠘⠀⠀⠀⠀⠀⢨⢻⡷⠛⢶⣦⣤⣶⡯⠴⠋⠁
⠀⠀⠀⠙⢾⣁⣸⠋⠉⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡀⠀⣸⠇⢰⣦⢀⠘⡇
⠀⠀⠀⠀⠀⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣀⡈⠃⠐⠾⠃⠐⠀⡇
⠀⠀⠀⠀⠀⠀⣰⠟⡹⣧⠀⠀⠀⠀⢰⡿⠶⣤⣤⣉⣉⠛⠛⠚⠚⠃⢠⡇
⠀⠀⠀⠀⢠⡾⢣⠞⢀⡼⡆⠀⠀⠀⣸⣯⠀⠀⠀⠈⠉⠙⠛⠳⠶⠦⣼⠃
⠀⠀⠀⣴⠟⡴⢁⣶⣿⡔⢻⡄⠀⣾⠉⠙⢦⣀⡀⠀⠀⠀⠰⢤⣤⣴⡋
⠀⣠⡾⣡⢮⠀⢠⡿⠁⠹⡄⠙⣛⡿⠳⡤⠞⠉⠛⠷⢦⣤⣀⣤⣶⣿⣧⣤⣀
⣴⢏⡔⢁⣠⡷⠋⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⢳⡦⠉⡷
⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠯⣴⣿⣶⡶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠢⠤⢤⡬⣉⣀⣀⣰⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠺⠶⠶⠦⣤⣭⣭⣉⣗⣻⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⣠⡄⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⡛⠶⣶⠀⢻⡟⠀⣶⠶⢛⣁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠖⠂⠀⡈⠀⠘⠃⠀⢁⠀⠐⠲⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠶⠆⠐⣟⢣⠀⠀⠀⠀⡜⣳⠂⠰⠶⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣤⣤⠄⠀⠘⠓⠀⠐⠂⠀⡛⠂⠀⠠⣤⣤⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⠔⠈⠀⠁⠂⢤⣀⣀⣀⣀⡤⠐⠈⠀⠀⠢⡀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠢⡀⠀⠀⢀⡀⣀⠜⠢⣀⢀⡀⠀⠀⠀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⢄⣀⠀⠀⠀⠀⠀⠀⣀⡠⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠈⢀⣑⡀⠀⠀⢀⣊⡀⢁⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢁⠀⢸⣿⣿⡷⢾⣿⣿⡇⠀⡈⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣤⡌⠀⠘⠛⠋⠀⠀⠙⠛⠃⠀⢡⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙



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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣶⣶⣾⣶⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠛⣿⣿⣿⡿⠟⢻⣿⠛⠿⠻⡿⢿⡿⠿⣿⣿⣿⣿⠟⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⢿⡷⠿⡿⠋⠘⠛⠉⠛⠃⠀⠉⠉⠆⠀⠈⠀⠈⠇⠀⢹⡟⠋⠈⠇⠹⠊⠁⠀⠀⠀⠈⠿⢿⣿⡿⠃⢹⣿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠻⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠁⢸⡿⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠈⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠂⠀⠀⠀⠀⠀⠈⠇⠀⠀⠈⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⢹⡶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣱
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣸⣿⣿⡿⠀⠀⠀⠀⠀⠀⢯
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⢀⣠⣤⣶⡶⠶⠿⠿⠿⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢲⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢰⣶⣦⣴⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⢨⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢠⣟⣳⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣶⣿⣿⣿⠏⠉⢠⠆⣩⡀⠀⠀⣠⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡜⡟⢹⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢳⣴⢀⣾⣿⣿⣿⣿⠀⠀⠾⠖⠛⠉⠠⣴⣯⠀⠀⠀⠀⠛⠛⣨⡇⠀⠀⢸⣿⣿⡟⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢠⢱⣤⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⠸⣿⣿⣿⣿⡟⠀⠀⣀⠀⠀⠀⢰⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣿⣿⣿⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣬⣜⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⠀⢸⣿⣿⣿⣿⡇⠀⣰⡿⠁⠀⠀⢀⣽⠂⠺⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣾⣿⣿⣿⡇⠀⣾⣷⣤⣄⠀⠀⠀⠈⠙⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⢧⣿⣿⣿⣿⣿⣧⣀⣿⠁⠀⣀⡖⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠱⡦⠾⠃⢸⣿⣿⣿⣿⡿⠀⠀⠁⠉⠛⠛⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠏⢁⣤⠋⢀⡼⠋⣀⡴⠾⠶⠶⢦⡀⠀⠀⣿⡖⠀⠐⡁⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢠⠟⠁⡠⠌⠁⣹⡟⠛⠛⠒⠂⠀⠉⠐⣆⠈⠙⢆⠀⢻⡄⠀⠻⣿⣿⣿⣿⠀⠀⠠⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠠⠁⠀⣾⠗⠂⣨⣿⡁⠀⠀⠀⠀⠀⠀⠀⢻⠷⠀⢈⡆⠀⠀⠀⠀⢸⣿⣽⠏⠀⠀⡀⠀⠀⠀⠀⢠⠀⢘⠃⠀⣠⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⡇⠀⣴⣻⠿⢆⡀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠸⡧⠀⠀⠀⣣⣿⣿⣿⣧⠀⢠⡅⠀⠂⡆⠀⢈⡇⣍⠀⣰⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⠠⠁⠀⢿⣿⣶⡌⢻⣷⡖⠒⣒⣟⣤⣿⣿⡇⠀⠁⠀⠀⠀⣿⣿⣿⣿⣿⡇⠐⣇⢰⣿⡀⣠⣾⡗⣷⣀⣹⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠘⣿⣿⣧⠀⢻⠁⠘⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣽⢿⣤⢛⣽⣿⣿⣿⣿⣟⢿⣽⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⣏⣿⡀⠀⢰⠃⣸⣿⣿⣿⠃⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡟⣷⣿⣿⣿⣿⣿⣿⡟⣫⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢻⣿⣿⠏⠀⠀⠀⢸⣦⣾⣿⣿⣿⣿⣿⣧⣼⣥⣿⣭⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⢠⣿⡿⠀⠀⠀⠀⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡻⡿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣶⣿⣿⠃⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢻⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣴⣾⣶⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠹⠻⢿⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⣒⠀⢰⣶⣶⣄⠄
⠀⠀⠀⠀⠀⠀⠠⠄⠈⠈⠄⠀⠶⠿⣳⠖⠛⡩⣀⢄⣤⣶⢟⣉⣷⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣵⣶⢋⡔⡵⣱⢫⡏⠐⡟⣡⡶⢻⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣠⣾⣯⣄⡟⢰⠵⢇⡟⡼⠐⠱⢋⠀⠰⣻⣧⡀
⠀⠀⠀⠀⢂⠀⠀⠀⡠⢂⢄⣼⠱⡵⡼⢣⠁⢠⠏⢸⣿⣿⠐⣴⠀⢘⣽⣿⡇
⠀⠀⠀⠀⠀⠑⢤⠀⠁⡦⢸⠁⠀⢉⣵⣻⠃⢠⠇⣻⢿⠿⠀⣿⡆⢹⣯⡥⣴
⠀⠀⠀⠀⠀⠀⠀⢡⠀⣀⢠⡰⡾⣼⢁⡦⠂⣩⢀⠙⠀⣤⣀⣭⣗⠒⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⣇⠳⢣⣏⠖⡧⠡⠂⣤⡷⡿⠋⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠒⠀⠘⠀⡇⡄⡔⢴⡇⡋⡐
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⠷⠁⣸⢧⠆⠃⠀⠐⠂
⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⡴⠚⠋⠀⠀⠀⠀⠛⠠⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠈⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠤⠖⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⡀⢀⡀⠀⠀⠀⠀⢀⠀⠐⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⡿⠿⠿⠿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠉⠉⠁⠀⠀⠀⠀⠀⣀⡙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣶⣤⣤⣶⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡏⠁⠈⠉⠙⠿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡉⢻⣿⣿⠏⠀⠀⠀⢀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣷⡄⢹⡿⠀⠀⠀⠀⠈⣀⡀⠉⠛⠋⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠈⠁⠀⠀⠀⢠⠟⠻⠟⠳⣦⣦⡄⠀⠀⣰⡿⠉⠉⠉⠛⠂⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⣿⠻⠿⠟⢳⡄⠀⣨⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠂⢀⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣇⣀⠀⠀⠀⣠⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠓⠤⠞⠋⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⡴⠶⠶⠶⠶⢦⣄⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣤⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⣨⠛⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠈⠑⠶⠶⠶⠶⢤⣶⡿⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡾⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⣀⡠⠴⠞⠋⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⠁⠀⢻⡀⠀⠀⠀⠀⢉⣷⣶⣶⣿⡇⠉⠁⠀⠀⠀⠀⢀⡿⠉⢻⣄⡀
⠀⠀⠀⢀⣤⠖⠀⠁⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⡞⠁⠀⠀⠈⠁⠉⠻⠶⣤⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢠⡟⠛⢿⣿⣿⣿⠟⢿⡄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠃⠀⣸⣿⣿⣿⠀⠀⠠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠝⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢟⠻⡏⠸⣿⣿⡿⣿⣿⠟⠻⠿⡿⠟⢋⣿⣿⣿⠿⠛⠁
⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠀⠀⠠⠐⢩⠥⠀⠈⣀⠀⠜⠛⠋⠈⠁⢀⣀⣤⣤⣄⣚⣉⡉⠁
⠀⠀⠐⠒⠯⣭⣉⣈⣈⡁⠀⣀⣀⣀⠀⠀⠈⠁⠘⠛⠛⠛⠃⠀⠀⠈⠃⠀⠀⠀⠀⠀⠒⠒⠠⠶⠞⠛⢿⣿⢦⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⣻⠆⠀⠀⠀⠀⠀⢈⣶⡀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢸⣌⠁⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣄⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⠀⠀⣄⣠⣀⡀⠐⠶⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣤⣤⣐⣚⣿⣿⣿⣿⡇⠀⣋⣠⠼⠛⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡆⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣾⠂⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣿⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣰⣿⣿⡄⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠟⠿⠿⠛⠛⠛⠛⠋⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣠⣴⣦⣭⣥⣶⣶⣬⣭⣶⣦⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡽⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢷⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣯⡀⠀⠀⠈⠻⣿⠿⠟⠛⠛⠛⠿⣿⠟⠁⠀⠀⢀⣵⡀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⡀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢀⣴⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⢿⣿⣿⠟⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠁⠀⠀⢀⣴⣿⣶⣤⠤⠤⣤⣶⣿⣦⡀⠀⠀⠈⣻⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⢳⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡠⣞⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠙⠻⠟⠛⠿⠿⠿⠿⢛⠿⠟⠋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⠿⠟⠛⢋⡋⣙⣛⣟⢛⠿⠿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⠟⡻⠋⣉⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣩⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡃⠀⠀⠀⠀⠀⠀⢀⡀⢄⡀
⠀⠀⠀⠀⠀⠀⢜⠁⠀⠉⠳⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⣠⠔⠋⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠃⠀⣾⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⢀⣴⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⡿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⡀⠀⢿⡄⠀⠀⠀⢀⣤⠴⠶⠦⣄
⠀⠀⠀⠀⠀⠺⣄⣠⠶⠋⠁⠀⠀⠀⠀⢸⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⣠⡶⠞⠛⠳⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠘⣧⠀⠀⠀⠀⠙⠲⢤⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⣦⡀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⣀⣠⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⣥⣤⣄⣈⣷⠀⢸⣷⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⢻⡶⠶⠶⠛⠋⠛⠿⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢉⡽⠃⠀⠸⣧⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡆⠀⠀⠈⢷⣄⠀⠀⠀⠀⣀⡼⠋⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠙⠋⠀⠀⠀⠀⣿⣤⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⣿⣿⣿⠄⠀⠀⠀⠉⠛⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠐⠀⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣐⠒⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠾⠛⠋⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⣍⣻⡆⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢠⡟⠙⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⡾⠛⢻⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠘⠛⠶⠶⢿⣿⣿⠃⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⠶⣧⣀⣼⠋⠉⢻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⢈⣿⠀⢰⠏⠈⣷
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡶⠟⢋⣡⡴⠾⠛⠛⠛⢿⡿⠛⠗⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⢻⠛⠾⣯⣤⡼⠃
⠀⠀⠀⢀⡴⠞⠛⠉⠉⠉⠁⢀⡴⠛⠁⠀⠀⠀⠀⠀⠐⠷⣶⣤⣤⣼⢷⣤⣀⠀⠀⠀⠀⠀⣀⣤⠾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⣀⣤⠄⠞⠁
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠄⠀⠈⢛⠿⠿⠿⠿⠟⠋⠱⣦⣼⣿⣦⠀⠀⠄⠠⠖⣀⣴⠞⣁⣴⡾⠋
⠀⠀⠈⢳⣾⠃⠀⠀⠀⠀⠠⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣷⠿⣿⣷⣶⣶⣶⣿⣿⣥⠾⠟⠁
⠀⠀⠀⠀⠻⣦⣴⠃⠀⣾⠗⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⡧⠄⠀⠈⠉⠉⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠻⢦⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣻⣷⡀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡿⢁⡾⠁⢘⣷⠀⠀⠀⠀⠀⠀⣿⠗⠀⠈⠉⠙⠛⠛⠷⠶⣦⣤⣄⣀⣀⠀⣸⡇
⠀⠀⠀⠀⠀⣠⣾⠏⣰⠏⠀⣤⡾⢹⣇⠀⠀⠀⣀⣴⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢻⡟
⠀⠀⠀⢀⣴⡟⢁⡼⠁⣸⣿⣿⣷⠋⠹⣆⠀⢸⣿⡅⠀⠙⢷⣤⣀⡀⠀⠀⠀⠀⠀⠲⠶⣤⣴⣟
⠀⠀⣠⡾⠋⣴⡏⠀⠀⣹⡿⠃⠘⣧⡀⠻⠿⢻⣿⠿⣦⣠⠾⠋⠙⠻⢶⣦⣄⡀⠀⢀⣠⣴⣿⣿⣄⡀
⢀⣴⡟⣡⠞⠁⢱⣆⣼⠟⠀⠀⠀⠘⣷⣤⡶⠛⠁⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠛⠷⢿⣍⡙⠻⢶⣄
⣾⣯⣴⣏⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⣼⠏
⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣍⣩⣿⣵⣤⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠈⠉⠻⠿⠿⠃⠈⠉⢉⡿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣍⣙⣛⠒⠲⠖⠲⠤⣤⣤⣤⣯⣀⣀⣀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠶⣶⣶⣦⣼⣤⣭⣉⣙⣉⣻⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⣚⡛⠙⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡡⢆⠊⣉⠷⠶⣟⢊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡈⠉⠛⠶⢄⠈⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣀⠠⠂⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠒⠦⢤⣀⠀⠀⠀⢠⠖⠁⠐⠦⣠⠖⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⣠⠄⠀⠀⠀⠀⠈⠙⠲⢴⢻⠀⠀⠀⠀⠈⠀⠀⠰⠊⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠴⢾⣞⡁⢙⠤⠤⠤⣀⣀⠀⢀⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⠀⢔⢥⠀⠀⣡⣠⣌⠚⢤⢀⡂⠀⠀⠀⠉⢰⠁⠀⠀⠀⠀⠀⠀⠀⡘⢢⠀⠀⠀⠘⢆
⠀⠀⢴⡞⠋⠀⠀⣌⠈⠙⣦⠀⠷⡕⢄⠀⠇⠀⠀⢀⡤⠈⠓⠄⣠⣴⡏⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠙⢦⠀⠀⠈⠢⡀
⠀⠀⠀⣿⢶⣄⠀⠈⠁⠂⠁⠙⢄⠈⢢⣑⣄⡠⢚⡡⠃⢀⡤⠞⢋⡜⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠑⢄⠀⠀⠈⢆
⠀⠀⠀⢸⠀⠈⠳⢤⡀⠠⢠⠀⠈⠢⣀⠭⠐⠉⣁⡴⠞⠛⠋⠉⢉⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡱⠜⢆
⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⡈⠃⠀⠀⢀⣠⠶⠛⠁⠀⠀⠀⣠⠖⠁⠀⠀⠈⠙⠂⠤⠄⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡦⡀⠑⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⣦⣤⠞⢻⠁⠀⠀⢈⣦⡴⠊⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣈⠒⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢀⡏⠀⠀⠀⢀⢀⡠⣤⠶⠭⠤⡤⣤⠴⢾⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠭⠹⣆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⠀⠸⡇⠀⠀⠈⠘⡅⠀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⠀⠀⢰⠻⡀⠀⠀⠀⠸⡀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢠⣴⡞⢻⠛⡟⠉⡟⢙⡟⣦
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸⠀⠹⡄⠀⠀⠀⢣⣿⠀⠀⠀⠀⣿⠀⠈⣧⠀⠀⠀⠀⠀⢿⣿⡿⡟⠻⡞⢳⡞⣿⡏
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣿⠀⢀⣼⠧⠤⣤⠞⠀⠀⠹⡄⠀⠀⠈⣇⠀⠀⠀⠀⠻⠀⠀⣿⠀⠀⠀⠀⠀⠸⡟⠓⠷⠶⠷⠚⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⣿⠛⠉⠀⣀⣀⣸⠀⠀⠀⠀⠹⡄⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠹⣄⣀⠤⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣰⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣴⣾⣿⣿⣿⣷⣦⣴⡀⠀⠀⠀⠀⠠⠤⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⢁⣈⡆
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⠏⠀⠀⠀⢹⢷
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⣠⡄⠀⠈⢸
⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢹⣿⡆⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠛⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠉⠁⠉⠉⠙⠛⠛⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣀
⣿⠿⠿⢿⠿⠿⠿⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⡿⢿⠛⣿⣿⡟⠿⠿⠿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢄⡞⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⣸⡋⠡⠀⠀⠀⠀⠀⠀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠛⠋⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠰⠶⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠁⠀⠑⠄
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠜⠋⠉⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠀⠀⡿⠁⡘⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⡀⠀⠐⠒⠀⠂
⠀⠀⠀⠀⠀⠈⠓⠦⠄⠄⠒⠚⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⣄⠀⠐⡀⠀⢲⢀⡀⠀⠀⣦⣠⣀⣰⢄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠈⢢⢼⣇⣠⣇⣤⡴⢠⠂⠀⡄⣝⠉⡋⢨⣸⠄⠀⠀⠀⡀⡤⢤⣠
⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠢⣡⡰⠋⠁⠀⠀⠈⢣⠊⠀⠀⠀⠌⠑⠃⠊⠀⠀⠀⠀⠰⠆⠀⠀⠈⠻⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢇⠀⠀⠁⠀⡄⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠻⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠲⠚⠄⠀⠀⠀⠀⠀⠀⠉⠐⠑⢂⠀⠀⠀⠀⠀⠐⠚⠒⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂

⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠒⣁
⠀⠀⠈⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣶⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣠⡤⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢏⠿⣿⡄
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⣀⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⠰⡐⡢⣡⣤⣶
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣠⣴⣶⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠉⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⡿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢺⡯⣿⠷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⡀⢉⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⠀⣼⣿⣿⣿⣿⣿⣿⠉⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⠿⠇⠀⠀⠐⠋⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢀⡞⡴⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⡀
⠀⠀⠈⠄⢃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠘⠿⠟⠛⠉⠁⠀⠀⠀⠀⢀⣠⣴⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡁