Christmas movie Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⣄⠀⠀⢀⡄⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣯⠀⣿⠸⣍⢸⠀⠀⠀⢸⣤⣦⣼⢡⢿⣆⢸⣧⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠷⠃⠻⠶⠳⠳⠞⠹⠀⠀⠀⠀⠟⠙⠟⠊⠉⠻⠸⠘⠇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠖⣀⣀⠀⠀⢠⡶⢶⡄⣀⣀⡀⡂⣀⢀⢰⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢼⠁⢸⠇⠀⢸⣇⣠⠟⡷⣶⠋⡇⡿⣾⢸⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠉⠀⠀⠈⠉⠁⠀⠁⠈⠁⠉⠁⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡇⡴⠶⣍⡏⠁⠀⢠⡞⢃⡄⢠⢠⠟⠙⡇⣤⠆⡼⠉⢳⣤⠀⠀⢀⣤⡉⣿⣵⠶
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠿⣅⣠⠏⡇⠀⠀⠘⣷⡮⡟⢻⠸⣄⣴⠃⣿⡤⢧⣠⡞⢸⣄⡀⠞⠺⣷⣿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⢀⣀⣀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣆⢸⣷⡧⣶⠙⢷⠀⠀⠸⣆⣦⣸⢃⣾⣦⢸⣰⡟⢻⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠻⠸⠘⠇⠿⠶⠋⠀⠀⠀⠿⠙⠿⠘⠉⠻⠼⠙⠷⠺⠀⠛
⠀⠀⠀⠀⠀⢠⡤⢀⠀⡀⡤⠶⣤⠀⢀⡶⡶⣆⢀⡀⠀⣴⡀⠀⣀⠀⠀⢠⣄⣴⠀⣀⣀⢰⡀⠀⡄⢠⠶⠆⢀⡀⣴
⠀⠀⠀⠀⠀⣿⣀⣿⠴⡇⣷⣞⠁⣆⠳⢄⡇⡿⣾⡇⣸⠽⣧⡘⠤⡀⠀⢸⠿⢿⣼⠁⢨⡟⣧⡼⢰⣾⠟⠃⡳⢄⡿
⠀⠀⠀⠀⠀⠘⠋⠙⠀⠁⠉⠈⠁⠋⠙⠋⠉⠃⠀⠃⠀⠀⠙⠙⠛⠁⠀⠘⠀⠘⠈⠓⠋⠀⠙⠁⠈⠉⠛⠃⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣴⡅⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⢾⣇⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠈⡄⠘⠂⠀⣀⡤⠒⠊⠉⠉⠀⠀⠀⠀⢘⣟⠉⠉⠑⠒⢴⢅⠀⠈⠻⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⢀⡠⠒⠉⢹⣵⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠇⠀⠉⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡀⣀⠔⠉⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⢿⡿⠃⠀⠀⠸⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠽⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢲⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡶⢮⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⡄⠙⠀⠀⠀⡴⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡏⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠅
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣧⣤⣷⣤⢠⡶⢉⣥⠀⢸⣧⠁⢸⣇⠜⠀⣾⡆⢰⣦⠀⣶⡄⢀⣴⠋⣡⠀⢠⠓⣶⠀⢸⡿⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⡏⠉⡏⠁⠻⠷⠊⠻⠦⠜⠻⠦⠔⠻⠶⠊⠘⠿⠃⠻⠟⠙⠷⠚⠿⠖⠻⠷⠊⠀⠻⠦⢺⡿⣅⣀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⢠⡄⢰⡦⢠⡄⣴⠤⡀⢠⢀⡤⣄⢤⣤⠤⢴⣦⠀⣴⡀⢀⣦⠀⣤⢤⡀⠀⠁⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣿⠀⢸⡇⢸⡇⣿⡞⠁⢸⠘⣇⠹⠀⣿⠀⢸⡏⠛⢸⡇⣼⢿⡀⣿⠸⠇⠀⠀⠀⠰⣦⣤⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⠀⣿⠹⣄⣸⠀⢹⡀⠀⣿⠀⢸⡇⠀⢸⡇⡷⢺⡇⠘⡇⠀⠀⠀⣤⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⣀⠁⠤⠀⠈⠐⠘⠇⠈⠀⠛⠀⠀⠘⠀⠻⠃⠀⠟⠀⠸⠃⠀⠘⠙⠃⠘⠇⠳⢏⣤⣤⡀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡛⡁⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣾⣷⣄⠀⠀⣸⣿⢿⣃⡉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠉⠸⣿⡿⠺⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⠏⠉⠙

⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⣤⣄⣀⡀
⠀⠀⠀⣴⣶⠀⣌⣫⣴⣿⣷⣶⣾⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣡⣤⣤⣀⣈⠉⠙⢶⣄⣀
⠀⠀⠀⠀⢀⠀⣨⡻⢿⣭⣭⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⣛⣩⣥⣼⣿⣿⣿⣿⣿⣷⣦⣌⠙⠻⣦⣄⡀
⠀⠀⠀⠀⣾⠘⠿⢃⡌⠁⠀⣈⣀⠸⣦⣤⠀⣠⡴⠖⠊⠀⠉⠉⠉⠋⠉⠉⠀⠈⠛⠛⠿⢿⣿⣿⣷⣦⡈⢻⣿⣷⣦⣤⡀
⠀⠀⠀⢠⡿⣦⣥⣴⣈⢀⠐⠈⣿⢷⣾⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢤⣌⣛⠻⣿⣿⣆⠹⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠛⠳⣎⠻⣿⣿⣿⣎⣀⣀⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡆⠀⠼⣿⣿⣧⡝⣿⣿⣧⠘⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢹⡾⠿⣯⣽⠛⠉⢁⡀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡟⢰⣦⣛⣻⣿⠧⣸⣿⣿⡆⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢨⡟⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠃⠀⢽⠏⡛⣡⡈⠛⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡄⠀⠀⠀⠁⠀⡉⠉⠙⡻⠿⠿⠷⠿⢿⣶⣤⣀⠀⠾⠂⠿⣿⣿⣿⣛⢛⡟⢰⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠙⢀⣤⣄⣀⣈⣙⠶⣦⡀⠀⠀⠀⠀⢀⠈⠛⣿⣷⡀⠀⠀⠉⢩⣭⣾⡿⠁⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⢁⣤⣠⣽⣿⣮⡇⠤⣀⣠⡀⣀⣈⣠⣝⠛⣧⠀⠀⢀⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⠀⠟⠀⢨⣭⡿⣫⡳⠀⡄⠛⠊⠻⢿⣿⣿⡗⣘⡀⠀⣼⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⠃⠀⠀⠈⣉⠁⠀⠿⠍⠀⠉⠒⠨⢷⡾⣿⣃⣿⡅⠁⠀⣽⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⡄⠀⣼⣿⡅⣴⣦⣶⣿⣇⠰⣧⠄⢈⣉⣳⣄⠀⠀⢈⣿⣯⡻⢿⣖⡂⢸⣿⠛⢿⡇⠀⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣠⠌⠀⠰⠴⠂⠀⠡⣼⣦⡀⠀⠀⠀⠙⠷⠛⠚⠛⠛⠉⠈⠳⠶⠿⠟⠋⠁⣾⡙⠻⠂⢿⣄⠀⢀⣠⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠐⠉⢀⡤⠴⠖⠒⠒⠀⠁⠘⠋⠈⠙⠲⠿⢶⣦⣀⣄⣠⣀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡴⠈⣙⣶⣝⢿⣿⠿⣿⣿⣿⣿⣋
⠀⠀⠠⠴⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⢙⣻⣿⣇⣈⠙⠒⠤⠘⠻⠿⠟⠉⠋⠲⣶⣴⣿⠿⠛⠏⠉⠀⠀⠤⢈⢹⣷⣦
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢢⡸⣿⠄⠀⠀⠀⠀⠀⠀⠄⣠⢳⣿⣶⣄
⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠀⠀⡃⠠⡀⠀⠀⠀⣿⣿⣾⠀⠀⠀⠀⠀⠠⣾⣽⣿⣿⣿⡻
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠸⡄⢀⠀⣀⠲⠀⠀⠀⠀⠀⠀⠃⢰⡁⠀⡸⠀⣿⠟⣧⠠⣀⠀⢀⠀⠀⠀⣽⣿⣿⡻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠀⠀⠱⢀⠤⢏⠀⠀⡀⠀⠀⡄⠄⢀⠀⠃⠀⣋⠀⠛⣰⣿⣷⣼⣷⣤⣝⣶⣶⡾⢻⣿⢡⣬⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠆⡎⠀⢸⡇⡀⠁⠀⢠⠇⢠⢟⠀⣰⠀⣹⡄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠀⣿⣷⣙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠾⠀⣐⠑⠉⠀⠀⠀⡼⢠⡌⢻⣼⣿⡇⣸⡇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣧⣦
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢐⡄⠀⠋⢀⠄⢁⡤⠎⠿⢸⣿⣧⢹⣿⣇⡾⠀⣹⣿⣿⢿⣿⣿⣿⣋⠽⢾⡃⣼⣿⠟⢫⣥
⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⢠⡝⠡⠂⠀⠀⠐⠛⠁⢀⢠⡿⣿⣿⣿⣿⣯⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠃⢈⡿⠋⠱⣷⣿⠉
⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠐⠺⠂⡠⠀⠀⠀⠀⠀⢠⣿⠟⣸⣿⣿⣿⣿⠇⣼⣿⣿⣿⣯⣿⣿⠟⠋⠠⡐⠉⠐⣠⣿⣿⣣⡄
⠀⠣⠄⠃⠀⠀⠀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⠄⣠⡿⠋⣰⣿⣿⡟⣙⣧⣿⢿⣿⡻⣿⣚⡟⢥⣶⣥⠶⢴⡆⢘⣿⠟⡵⠏
⠀⠀⠀⠀⠀⡐⡻⠋⢀⣀⡀⣀⣠⣤⡄⠀⣀⣤⣶⠟⠋⣵⣿⣛⣥⣶⡿⣿⣟⡧⣟⣯⠹⠘⢋⠉⠀⠀⣉⣈⣉⣉⣚⠛⠐⠚⠓⠘⠃
⠀⠀⠷⠆⠀⠀⢀⠴⣟⣻⣿⣿⡿⠁⠀⠾⣿⠟⠁⣠⣿⣿⣿⣿⣿⣯⣻⣿⣿⡾⠟⠓⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠀⢢⡄
⠀⠀⠀⠠⠀⠀⠀⠈⠻⢿⣿⣿⣦⠀⠀⣼⣿⠂⣼⠿⢟⣿⣿⣿⣿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠉⠀⣴⡾⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⠀⢀⡀⠰⠠⠀⠛⠀⣈⠀⠀⣤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠈⠀⠃⠈⢈⡁⠀⠤⠤⢠⠀⠈⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⣠⡄⠈⢸⠐⢈⣉⡀⡀⠠⠤⠀⠠⠀⢈⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⠅⠀⠀⠀⠀⠀⠀⢀⡠⠀⠈⠘⠃⣠⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢨⡄⠐⣶⠠⠀⠀⢀⣸⣤⣀⡀⢽⠶⠾⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡮⠃⠀⠀⢠⡀⢰⣿⡙⢿⣿⣿⣿⣀⣼⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿⡇⠈⠛⢄⠼⢻⣿⣿⣿⠻⣜⡄⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠷⣿⠗⣼⣿⣿⣿⣿⣿⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠈⠀⣿⣿⣿⣿⢿⡿⣉⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⠀⠀⢠⣷⡧⢉⡁⠀⠀⣇⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣜⡅⡌⢣⠀⠀⠀⠡⠝⠰⣹⣿⡿⠷⣖⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠉⣼⡆⠀⠀⠀⠀⠐⢿⡟⠀⠀⣁⡀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢴⣤⠆⠈⢽⠁⠀⠀⠀⠓⠈⣩⠉⠀⡀⠈⠀⢸⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⣾⣧⣷⡄⠈⠀⣀⣀⣤⡆⠀⠈⠀⡽⢅⠀⠀⠚⡂
⠀⠀⠀⠀⠀⠀⠀⠋⠉⠒⠾⠃⠀⣿⣿⣏⠋⡀⠸⣿⣿⣾⡿⠂⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣏⡟⠠⠀⠀⠿⢿⣿⠇⠀⠀⠉⠟⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡈⠀⠀⠀⠀⡠⠔⣒⢄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣴⠇⠀⡤⡤⠈⠃⠄⠈⡈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢵⠏⠀⠁⢧⠼⣧⠤⠀⠀⡤⠀⠀⡤⠘⢄⢼
⠀⠀⠀⠀⠀⠀⢀⡤⠦⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠄⢀
⠀⠀⠀⠀⠀⢠⡟⠀⠆⠔⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠈⠒⠂⢧⡴⠁
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢸⢧⡄⢠⠀⠄⠀⠀⢰⠄⠐⠁⠀⢠⠔⣤⢲⠀⠠⢦⠄⠀⡆
⠀⠀⠀⠀⠀⢿⣆⡀⢀⣰⠃⢠⣀⠈⢀⡀⢺⡀⢈⠂⠀⣦⡤⡟⠀⡇⢸⣠⣀⡞⠀⠀⡵
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠀⠈⠁⠀⠀⠁⠈⠁⠈⠁⠀⠉⠀⠀⠀⠀⠈⠁⠉⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⢁⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠉



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

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠹⡉⠓⠲⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠚⢉⠏
⠀⠀⠱⡈⢻⠛⠻⠿⢿⣯⣉⠙⢲⣶⣶⣦⡤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⣴⣶⣶⡖⠋⣉⣿⡿⠿⠗⠒⣟⣀⠎
⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠶⠤⠼⠿⠿⠿⠧⠤⠬⠿⠿⠿⠷⠶⠒⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠈⡟⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡤⣄⢀⣤⡄⢠⣤⡄⢠⣤⡄⠠⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢿⠁
⠀⠀⠀⠀⢷⣤⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣈⣁⣩⣬⡿⠥⠬⠭⠧⢬⣭⣇⣀⣛⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡼
⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢀⣤⠤⣄⠀⣠⠴⡄⠀⠀⠀⢀⡔⡒⠒⠦⢄⣰⠂⠀⠀⢴⢲⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⢸⠛⢻⡇⠀⣰⠏⠸⠥⠞⢠⣟⡼⠁⡀⢀⣄⢀⡏⢁⣤⠀⢠⠏⢉⣓⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⡇
⠀⠀⠀⠀⣟⠀⢸⠀⠀⣿⠀⠀⠀⣀⣼⠋⣹⣀⠙⣟⣁⢸⣃⡼⢷⡄⡟⠀⡾⢩⡏⣟⢰⠏⣽⢁⣸⣇⣠⠀⠀⣇⣀⣳
⠀⠀⠀⢀⣿⣿⡏⠀⠀⠈⠓⠒⠋⠁⠉⠀⠈⠁⠀⠈⠀⠈⠁⠀⠉⠁⠉⠁⠁⠈⠀⠉⠈⠉⠉⠉⠀⠿⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⣀⣀⣠⠤⠤⣤⣶⣶⡖⠒⢲⣶⣶⣖⠛⠒⢶⣶⣶⣶⠒⠶⢤⣤⣤⣀⣀⣀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⣰⠁⣿⣤⣶⣶⡋⣉⣿⣿⠿⠤⠤⠾⠛⠛⠛⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠾⠿⠿⠿⣦⣈⣉⣿⣶⣶⡤⣷⠈⢆
⠀⣰⣁⡤⠴⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠦⢤⣈⣆
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⢀⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⠶
⠰⣿⡏⠀⠀⢀⣀⣤⣀⣤⣄⣀⣠⣦⡀⠘⠟⠀⠀⠀⠀⠀⠀⣴⣿⣷⣶⣶⣶⣶⣶⣶⣦⡄⠀⠀⠀⠻⢿⣷⠀⠀⠀⠀⠀⣠⣿⡿
⠀⣿⣇⠀⠀⢾⣿⣿⣿⣿⣉⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⢿⣿⣿⣿⣿⣿⣿⡇⠀⣀⣀⣤⣤⣁⡀⠀⢰⣷⠸⠿⠇
⠀⠘⠟⠂⢀⣬⣿⡿⣿⣿⣿⣏⠙⠛⢻⣶⣶⣾⣷⣤⠀⠀⢠⡞⠉⠀⣀⣀⠀⠉⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣶⡞⠋⣀⣀⡀⠻⠃
⣀⣀⠀⢀⣾⣿⡿⠃⠀⠈⠻⣿⣦⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⡤⢾⣿⣿⠷⣀⣉⣙⣿⣿⣿⣿⠿⠿⠟⢿⣿⣿⣧⣼⣿⣿⣿⠀⠀⢀⣄
⠿⠋⣶⣿⣿⡟⠀⠀⠀⣀⣴⡿⢿⡿⠟⠛⠛⠿⢿⣿⣿⣿⣶⠖⠓⠼⣿⣿⣾⣿⡿⢿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⡿⠛⢿⣿⣧⡀⠈⠛⢶⠄⠀⡀
⠀⠈⠛⠻⣿⠃⠐⠀⠻⢿⣿⣤⣼⣧⠀⠀⠀⢀⡀⠹⢿⡿⠋⠀⣀⡀⠘⢿⣿⣷⣆⠀⠹⣿⣿⣄⠀⠀⠘⠋⢹⣿⣿⡟⠀⠀⠀⠙⢿⣷⠀⣀⣀⣾⡿
⠀⠀⠀⣠⣤⣤⣄⣠⣶⣾⣿⣿⣿⣿⠀⠀⠀⠸⣿⣆⠀⠀⠀⠈⠻⠿⠆⠀⠙⠻⠋⠀⠀⠋⠉⠁⠀⠀⣀⠀⠸⣿⣿⣿⣦⠀⠀⠀⠀⠙⠿⣏⣠⣀
⠀⠀⠀⠉⠙⠛⠛⠛⢿⣿⣿⣿⣿⣷⡀⠀⠀⣹⣿⣿⡆⠀⠀⠀⢀⣤⠀⠀⠀⣠⣤⡀⠀⠀⣠⣿⣶⣼⠋⠀⠀⠀⣠⡄⠉⣶⡄⠀⠀⠀⠀⢿⣿⠏⠘
⠀⠀⠀⠰⢷⣦⡄⠀⢸⣿⣿⣿⣿⣿⡇⣰⣿⣿⣿⣿⣿⣶⡀⠀⠈⠉⠀⢀⠘⠋⣁⡅⠀⢰⣿⣿⣿⣿⠚⠛⣾⣿⣿⣿⣶⣍⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⣶⣿⣧⣴⣿⣿⣦⣼⣿⠛⠻⣿⣆⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠⠄
⠀⠀⣠⣄⠀⠀⠀⠚⠋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⠟⢻⣿⠟⠛⠋
⠀⠸⠿⠿⠿⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠉⠀⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠿⠿⠿⠷⠾⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⠀⢀⣿⣿⣿⠿⠛⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢁⡝⠛⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠻⣿⣿⡇⠀⠈⢿⣏⠀⠀⠀⢰⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣾⣿⣶⣄⡸⠿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⣤⡿⠋⠀⠀⠀⠀⠙⠓⠀⣤⠀⠘⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠿⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠁⠀⢠⣿⣿⡄⠀⠀⠀⠀⢀⣤⣤⣀⣀⣤⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⠀⠈⠋⢻⣿⣿⣿⡌⠹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣰⣾⣿⣿⣷⣄⠀⠀⢀⣾⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠻⢿⣿⠃⠀⡙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣼⣿⣿⣿⣿⣆⣴⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣀⣁⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠐⣿⠟⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠿⣿⣿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣴⣷⣦⣴⣿⣤⣿⣿⣷⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠉⠁⠀⠀⠀⠉⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⢠⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡌⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⠀⠀⠀⣿⣿⣶⣿⠁⠀⠀⠀⢀⡀⡀⠀⢸⣿⣿⣷⣿⡏⠁
⠀⠀⠀⠀⣠⣶⣦⡀⢻⣿⣟⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣿⡏⠀⠀⠀⠈⠙⠻⠟⠀⠀⠀⠀⣈⡉⢙⣤⣼⢿⣿⣿⠋
⠀⢠⣴⣿⣿⣿⣿⣿⣿⣽⣿⠃⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⣿⣿⡿⣿⣿⣀⣿⣷⣤⡀
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠰⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣦⣴⣿⣿⡆⠉⢹⣿⣿⣩⣿⣿⠆
⠀⠻⢿⣿⠿⠛⠁⣽⣿⣿⣿⣿⣿⣿⣿⠏⠀⠈⢻⡶⠤⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣶⣿⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠐⠂⡀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⠟⠉⠿⠟⠀⠰⠖⣤⣄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠻⣿⣿⣿
⠀⠀⠀⠀⣄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣤⡄⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠻⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⠿⠿⠛⠿⢿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⠀⣠⡀⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡇⠀⡀⠀⣀⣤⣾⡿⠛⣿⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠐⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⣿⡟⠁⠸⣿⠟⠿⣿⣿⣧⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⡀⠀⣷⣄⣼⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⠿⠛⠛⠋⠱

⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀⣀⣀⣀⣄⣹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⡿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠉⠀⢀⣄⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⣀⣀⣀⠀⠀⣤⡖⠀⠀⠀⠠⣾⣿⣽⣷⡀⣀⠀⢠⣄⠈⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⢀⣤⠖
⣿⣿⣿⣿⣾⠟⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣿⠟⠃⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⠟⠃⠀⠀⠠
⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠰⠋⠉⠉⠀⠈⠉⠀⠀⢸⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣄
⣿⣿⣿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠋⠠⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⣶⡿⠷⠀⢠⣾⠿⠏⣉⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣿⣷⣦⡀⠀⠀⠀⠀⠀⢁⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢳⣦⠉⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⡀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣶⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣤⡄⠀⠀⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠉⠙⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠁⢰⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⣠⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣀⠀⠉⠀⠸⠿⠟⠀⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣧⣤⣀⠀⠀⠀⣴⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣟⡛⢛⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⠀⠃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠋⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠈⢟⠀⠀⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢰⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣿⡇⠀⠈⠁⠘⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠉⠻⠿⠋⠙⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⣀⢄⣀⠤⠒⣐⣂⠈⠈⠈⣐⣒⡒⠤⣀⡀⢀⣀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠔⢤⡎⡡⢊⡴⠟⡩⠔⠊⠉⠀⠀⠀⢀⠀⠀⠨⠉⠑⠦⢍⡳⢧⡔⠪⠱⣀⠤⣄⣀
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠐⠚⡃⢴⠂⢨⣵⠏⡤⡊⠀⠠⢴⠒⡒⢄⢀⠞⢶⣶⡶⠒⠒⠲⢶⣿⣦⡙⣶⡙⢁⢴⠄⠞⠶⠀⠀⠀⠀⠀⠀⠀⠠
⢀⠖⢂⡑⡇⠀⢠⠀⢀⠀⢀⢀⡀⠀⠛⢹⠀⡴⢡⣊⠋⠀⠀⠠⢘⠁⣡⠋⠈⣶⡶⢀⢘⠀⠠⠀⠀⡀⠀⠿⣌⢣⡀⢸⠀⢀⡄⠀⠀⠀⢀⠄⠀⠘⠃⠦⢂⡤⡀
⠉⠲⠤⢄⡇⠀⠉⠁⠀⠁⠀⠀⠀⠀⠁⢸⡾⢡⡏⢅⠀⠀⠀⣠⣴⣿⠁⠈⡬⡬⡀⠆⠉⠀⠀⠀⠀⡇⠀⠀⠘⡄⢷⢿⠀⠀⠀⠀⠈⠀⠀⠉⠓⠒⢠⠋⡡⠠⠜
⠀⠀⠀⡔⡧⠄⢠⠀⠤⠄⣄⠀⠀⠀⠀⢾⡇⡎⠑⠒⣶⣶⣿⣿⣿⣿⣤⡀⠀⠀⠈⠖⠃⠀⠀⠐⠂⠃⠀⠀⠀⠸⠘⡽⣤⠤⠀⠄⠠⡤⠤⡄⠤⠤⠴⠃
⠀⠀⠀⠈⠁⢸⣁⣁⠤⠎⠙⠒⢄⠉⠂⢿⠀⠇⠀⠀⣻⣿⣯⣿⡿⡿⠁⠀⢀⡤⡆⠀⠠⠀⠀⠰⠄⠀⠀⠀⠀⠀⡇⣿⠇⠊⢀⠔⠊⠑⢤⣈⣂⣘⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣻⠀⡇⠀⠀⠈⢡⠈⠁⠀⠀⠂⠠⡎⠀⡇⡲⢖⢀⢦⠐⠀⣆⠀⠀⠀⠀⡇⡗⡙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣧⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠰⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⣄⠀⢀⠎⠀⠐⠀⠀⠀⠀⣯⡆⠀⠀⠸⠂⠐⠀⠘⠀⠀⠀⢠⠃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢶⡞⠀⠀⠀⠀⠀⠀⡆⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡑⢤⠀⠀⠄⠀⢈⠅⠈⢿⡆⠀⢠⠀⠀⠀⢠⡠⠊⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣍⡚⠥⣤⣴⣶⣾⣿⡁⠀⣰⣀⡤⠖⣉⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠬⢭⠍⠉⠩⠭⠩⠄⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⢹⡿⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠲⡄⣸⢃⠜⠡⣄⡀⢸⠃⣾⠇⣿⠁⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠎⢀⣤⠄⢠⠇⣿⡠⣦⠀⢹⣇⣿⡴⢻⡴⢧⢼⡿⣻⡷⢋⡿
⠀⠀⠀⠀⠀⠀⡿⠀⠻⠤⠔⠋⠀⢸⣧⣿⣸⠀⠛⠈⠱⡽⠃⠀⠈⠁⠛⢡⡟⣠⠴⣛⣣⡄
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡜⠀⠈⠁⠀⢀⣀⢠⣤⢤⣶⠲⠶⠽⠂⠉⠀⠀⠸⠫⢴⡾⠷
⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠄⠐⣈⠀⠀⢰⣼⣿⢼⣿⠺⣭⣰⣶⣰⣿⣼⡇⠀⠀⠀⢰⣓⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⣷⠆⠀⣾⣿⣾⢿⠟⣿⢻⣿⡛⡏⣿⣧⠀⠠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⡿⡿⠂⡸⠦⠀⣿⡸⣿⢸⢸⡿⡎⣿⡇⣧⢹⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢸⡇⣭⣜⣎⣣⣿⣬⡿⠼⠾⠟⠀⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢉⡍⠳⡌⢠⠀⠈⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡔⢊⠸⡄⠘⡥⢾⣿⡀⣿⠀⡇⡾⡆⣾⢢⢀⣤⠀⣀⠀⢽⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢟⡄⠺⣿⣧⠈⢿⣧⠸⠚⠿⠇⠇⢷⢋⡞⡅⢠⡟⢁⡆⣙⢿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣿⣧⠈⣿⣧⠀⠀⠀⠀⡄⢠⡄⠁⠚⠠⠟⣴⣫⠖⡏⡻⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⢿⣧⠼⠛⠁⠀⠀⠰⠿⢷⡇⠀⠀⠀⠀⠁⠧⣺⠟⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠉⠀⠀⣀⠀⢠⠀⠀⠀⣀⢠⡀⠀⢀⡄⣠⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠀⠀⢨⣏⡼⢸⣷⡻⢾⣽⠀⡧⡟⢸⡇⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡟⠀⠀⠀⢠⠋⡹⠃⠀⠁⠀⠫⠛⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡨⠒⠈⠐⠢⢄⣉⠀⠀⠀⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⢤⠀⠀⠀⠈⠙⠒⠲⠶⠶⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣧⣶⣾⣿⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠙⠻⠿⢿⡿⢿⣿⣷⡆⠀⠀⠀⠠⠤⣤⠤⣄⠀⠐⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠱⣸⢿⣿⠅⠀⠀⠀⠀⠀⠀⠈⠉⢷⣤⣨⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠉⠀⠀⠈⠁⠂⠀⠀⠀⠀⡀⠠⠀⠀⠀⠙⠉⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⣴⣧⠄⠀⠄⠀⠀⠈⣿⣄⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣭⣦⣧⣮⠂⢠⡀⠀⣼⣿⠥⠀⠀⡀⠀⢰⣿⣿⣿⣦⣤⣄⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠉⠁⠈⠀⠙⠳⣾⣵⣸⣿⣿⣶⠒⢊⣀⠀⣿⣿⣿⣿⣿⣿⠇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢤⡀⠀⢸⣶⣿⣿⣿⣿⡇⢻⣏⠀⣿⣿⣿⣿⡿⠋⠀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠉⠀⠀⠀⠀⠀⢈⠀⣿⣿⣿⣿⠃⣿⣿⡀⣿⣿⣿⡟⢤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢀⣤⣤⠀⠀⣘⣱⣿⣿⣿⡟⠰⣿⣿⣇⣿⣿⣿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⢀⠄⢘⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⠀⠀⠀⠀⠘⣿⣿⣿⣿⠏⠀⢰⣿⣿⣿⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣶⡀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠈⠉⢻⣿⣷⡄⠰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣤⣦⡀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⢙⡂⠉⠉⠀⢘⣿⣿⣿⣿⣷⣴⡄⠀⠀⢿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠒⠂⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠹⡌⢉⣳⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡖⢒⣋⠉⡸⠁
⠀⠀⠱⣌⣧⠉⠙⠛⠛⠦⠤⢹⣿⣿⣿⡐⠒⠲⣶⣶⣶⡒⠒⢶⣶⣶⣖⠒⠘⣿⣿⣿⠥⠤⠜⠛⠛⠉⢡⣯⡴⠁
⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠸⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠰⠋⠏⠱⠰⠿⠃⠸⠻⠀⠟⠿⠀⠷⠞⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⡹
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣀⣀⠀⠀⣒⣈⠉⠉⠉⠉⡩⠭⢍⣉⠉⡍⠉⠉⣑⢆⠀⠀⠀⠀⠀⠀⠀⣿⣷⡇
⠀⠀⠀⠀⡟⠿⡇⠀⢠⡞⢱⣀⠇⢰⢏⠎⠀⠀⡀⠈⠖⢃⡀⠀⣹⠓⠢⠤⠤⠞⠀⠀⠀⠀⠀⠀⠀⡿⢿⡇
⠀⠀⠀⢀⡇⠀⡇⠀⣾⠀⠀⠀⣀⡿⢋⣇⠘⣾⠋⢸⢁⡹⣇⢰⠃⢰⠋⡗⡷⢠⠖⡶⢀⣾⢀⡀⠀⣇⠀⢇
⠀⠀⠀⢸⣿⡿⠀⠀⠈⠓⠒⠋⠀⠁⠀⠉⠀⠈⠁⠈⠉⠈⠉⠈⠉⠈⠀⠀⠉⠈⠉⠉⠉⠸⠇⠀⠀⠸⣿⣿
⠀⠀⢀⣿⣿⠃⠀⠀⢀⣀⣠⡤⠤⠤⣶⣶⣶⠒⢲⣶⣿⣏⠉⠙⣶⣶⣶⡒⠒⢦⣤⣤⣤⣀⣀⡀⠀⠀⢻⣿⣇
⠀⢀⠌⠰⢷⣶⣾⣏⣩⠿⠿⠟⠒⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠚⠛⠛⠛⠤⢤⣹⣿⣶⡖⠷⠘⣆
⠀⠮⠔⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠼⠆

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡟⠛⠛⢛⣿⡟⠛⠛⠛⢛⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⠉⢻⣿⣿⣿⣧⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⢠⣾⠟⠀⠀⠀⢀⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣿⣿⡀⣈⠁⠐⢺⣿⣿⣿⣿⣿⣦⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⡇⢠⣿⠏⠀⠀⠀⢠⣾⠏⠀⠀⠄⠀⠀⣀⠔⠀⠈
⣿⣿⣿⣷⣴⣶⣿⣟⢛⠫⢼⣿⣿⣿⡶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠆
⣿⣿⣿⣿⣿⡟⢻⣿⣿⣶⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠐⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠡⠎
⣿⣿⣿⣉⣭⠚⣿⣿⡿⠿⠿⠿⠏⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠘⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⢠⡀
⣿⣿⣿⣿⣿⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠏⠱⠆
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⡠⠄⠘⠁
⣿⣿⣿⡿⠟⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⠐⢤⡀⠀⠀⣾⢠⢠⡄⢆⠈⢧⠀⣿⣂⠀⠀⣤
⣿⡿⣋⣀⣾⣿⠛⢀⡠⡀⠀⠀⠀⠀⢀⡄⢀⣀⠀⡸⢧⣀⠀⠘⣦⠤⣼⠘⠋⠓⠙⠀⠈⠀⠈⣠⣄⣾⣿⡤
⣿⠀⢈⣿⠋⠉⠀⢣⡴⡅⡆⠀⣿⣄⣯⣀⡈⠟⠦⠙⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠙⠋
⡿⠀⢀⣿⠀⢀⠀⠘⣤⠜⠓⠂⠁⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠌
⣿⠀⠈⠉⠁⠻⡓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⡀⠰⣁
⣿⠀⠀⠀⠀⠀⠳⠬⠧⠀⡀⠀⠀⠀⡠⠄⠀⠀⢀⢤⠀⣄⠠⣆⠸⡇⠸⢦⠓⢦⠛⠑⠀⠛⠀⠰⠦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠰⡶⠿⠞⠦⠰⠀⠞⠃⠀⠀⠀⠀⠀⠀⠀⠐
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠄
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠈⢣⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡜⠢⡀⠣⡸⠂⠙⠁⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠷⢤⢐⡇⠀⠀⠀⠀⠀⠀⠘⠈⠳⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⠀⠀⢀⣄⣀⣀⣀⣀⣤⣤⣴⣶⣿⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠲⢦⣄⡀⠀⠀⠀⠀⠐⠈
⣿⣿⣿⣿⣿⣿⣿⠏⢠⠉⠀⠀⠉⠛⠛⠛⠿⠿⢿⣿⣿⣿⣟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠈⠻⣷⣄
⣿⣿⣿⣿⣿⣿⠏⠀⠘⠀⠀⠀⠀⠀⠀⣀⣤⡆⢸⣿⣷⣿⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠈⢿⣦
⣿⣿⣿⣿⣿⠃⠀⠀⠇⣿⣿⣷⣤⣶⣴⣿⣿⡀⣿⣿⣿⣿⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠗⠚⠀⠀⠀⢻⣷⡀
⣿⣿⣿⡿⠁⠀⠀⢰⠀⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣟⣿⣿⠀⠃⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⣿⣿⡟⠀⠀⠀⢀⡌⢸⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⣴⡮⠙⢿⡢⠄⠀⠀⠀⠀⠀⠀⠹⣿
⣿⠏⠀⠀⠀⢀⣾⣧⣾⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠈⠎⠁⠉⠀⠄⠀⠀⠀⠀⢄⠀⠀⠀⠘
⣿⠀⠀⠀⣴⣿⠟⠳⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣯⣽⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⠀⠀⠀⠀⠘⣦
⣿⠀⣤⣼⣿⣥⠀⠗⣹⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣷⢘⠇⠀⠀⠂⠀⠂⠀⠂⠀⠀⠀⠜⠷⠶⠀⠀⠀⠀⠹⣧⡀
⣿⣠⣿⣿⠿⠛⠃⠀⠀⡠⠀⣿⣿⣿⣿⢀⣿⣿⡿⣿⣿⢿⡿⠠⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⣿⡿⠏⠁⠀⠀⣀⠀⠀⠀⡾⠿⢿⣿⡏⢸⣿⣿⣿⣾⣿⣾⣿⠀⠷⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡆⠀⠀⠀⠻⣷
⣿⠀⠀⠂⠠⠐⠁⠀⠀⠀⠙⠻⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠄⠀⠄⠈⠀⠀⠀⠀⠀⢨⣿⠿⠀⠀⠀⠀⠘
⣿⣤⣤⣤⣄⡀⠀⠀⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠈
⣿⣿⣿⣿⣆⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠻⢿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⣴⣿⣷⣿⣦⣤⣴⣤⡀
⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⠋⣰⣾⡿⠛⠛⠛⠿⣿⣿⣿⣗
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣟⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡿⠁⠀⠀⠀⠀⠙⢿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠉⠉⠙⢿⣿⡙⢻⣿⣿⡿⢹⡿⠷⢴⣦⠀⢀⣀⣀⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣷⢾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣀⣿⣧⠀⠈⠀⠀⠈⣇⠀⠀⣀⣀⠀⠈⠉⣿⡿⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠈⢁⣾⣿⣿⣿⣿⢿⣿⡟⠀⢾⣯⠿⢆⠘⠻⠾⠛⠟⠀⠀⠀⠀⠾⣿⡀⠲⠶⠦⠄⠀⣸⠏⠀⣀⣀
⡈⣿⣿⠿⠟⣿⣿⣿⣿⣯⣿⡿⠇⢸⣿⣿⣿⠏⠈⢿⠗⠀⠈⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣄⡀⢀⣠⣴⠃⠀⠀⠹⣿⣯
⣯⢻⣿⣶⡾⠋⣿⣿⣿⣿⣽⣧⣿⣿⣿⣿⠁⠀⠀⠀⠠⣤⣴⣴⣶⡟⠛⠿⡿⢿⠀⠀⠀⠀⠀⠈⠠⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠛
⣿⡇⣿⣿⣧⣴⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⢸⣿⣯⠙⣷⠶⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣄⠻⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡤⣿⣿⣿⣷⡉⠉⠉⣹⣿⣿⠗⠀⣀
⣿⣽⣿⣿⣦⣤⣼⣿⣿⡿⠋⠀⠀⠀⠙⢻⣿⡿⠿⣻⣷⣿⣿⡿⢿⣿⣿⡿⠋⠛⠭⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣻⣾⡿
⠘⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⡀⠀⣯⠿⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢦⢰⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⣿⡉⢰
⣤⣺⣿⣿⣿⣿⣿⣿⣿⣤⢠⡎⣡⠶⠀⠀⢀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡄⣸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⡘⠃⠸
⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⡉⠁⠀⠀⠈⡩⢛⣍⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡟⢿⣦⣿
⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠘⠛⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣬⣉⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣷⣄⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠙⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⣿⣿⠟⠋⠉⠙⠛⠛⠉⠙⢿⣿⣿⣷⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡇⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⣠⠆

⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⣠⡤⣤
⠀⠀⢠⡾⠋⣿⣬⣿⠀⠀⠀⣀⢀⣀⣨⡁⣠⡤⣤⣾⣿⠒⣷⡾⣿⡾⣿⠀⣿⠋⣿⡇⢻⣦⣍⠁
⠀⠀⣾⡇⠀⠛⢩⣿⡾⣿⠀⣿⡟⠟⣻⡇⢻⣷⣌⢹⣿⠀⣿⠁⣿⠃⣿⣀⣿⣀⣿⣧⣾⣌⡿⠀⣦⢠⢤⣰⢃
⠀⠀⢿⣇⠀⠀⢸⣿⠀⣿⣀⣿⠁⠀⢹⣧⣾⣌⡿⠋⠿⠾⠿⠀⠛⠀⠙⠋⠉⠉⠀⠉⠀⠀⠀⢰⢊⡷⣲⢯⣉⣃⣄
⠀⠀⠈⠻⠶⠶⠛⠿⠀⣙⣛⠙⠀⠀⣀⡉⠀⠀⣀⣀⣀⠀⢈⣉⡉⠉⡉⡉⢉⣉⠉⣉⣉⣁⠺⢒⠗⣯⣟⣽⢏⡦⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢈⢣⠀⠎⡅⡇⢠⢎⡥⠒⢌⠱⡈⡆⢃⢠⢁⠇⢘⢸⠀⡃⡖⠚⠀⠈⢃⡝⢬⠜⢣⠅
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠸⢆⠛⡜⡇⡇⢸⠸⡀⠀⢠⠃⡇⢸⡈⠆⡜⠀⠰⢸⠠⡇⡤⠼⠀⠀⡆⠁
⠀⠀⠀⠀⣀⠰⡗⣘⠀⠧⠄⠀⠚⠀⠧⠇⠀⠓⠬⠭⠥⠊⢀⠀⠣⠰⠁⠀⠨⠼⠀⠥⠭⠽
⠀⠀⠶⢼⣇⣽⣯⡍⣧⠶⠆⢰⣶⡄⠀⢰⣶⠄⣤⡀⠀⠀⢿⣷⣤⣾⣟⣷⠀⢀⣶⡆⠀⠀⠀⠀⠆
⠀⠀⢸⡁⢾⡳⡷⣻⢊⣷⠐⢺⣿⣷⡀⢸⣿⢀⣭⡀⢀⣤⣤⣤⠉⣿⣏⣁⡀⢺⣿⠷⠖⠒⠒⡚
⠀⠀⢶⢺⡏⣛⣟⢃⡟⠶⠆⢸⣿⠻⣷⣼⣿⢸⣿⠁⣿⡁⣸⡿⢸⣿⠛⣿⡇⢸⣿⠀⢒⢜⣵⣖⡆⡄
⠀⠀⠀⠀⠉⠰⡧⠉⠀⠀⠀⣿⡇⠀⠙⣿⡿⠈⠛⠂⢙⣋⣿⣇⣸⠟⠀⠿⠃⠙⢛⠀⡵⠾⣿⢿⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⢯⣼⠿⠉⠛⠻⠿⢿⣿⡿⠟⠀⠀⠡⠗⠊⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣿⣿⣿⣿⣿⣿⠟⠛⠛⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⠟⠛⠛⠛⣿⣿⡿⠻⠿⠟⠛⠛⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛
⠻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣽⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⣴⣿⠏
⠀⣈⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⠀⠀⣴⣿⡟⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⣹⣿⣿⣿⣿⠿⣿⣿⣶⣄⠀⠀⣿⣿⣿⠿⠿⠿⠿⠿⢽⡇⠀⢬⠀⣰⣿⠟⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠐⠃⠀⠀⠠⡴⠋
⣾⣿⣿⡏⢛⠫⢤⣿⣿⣿⣿⣷⡦⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶
⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣤⣼⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢉⣿⣿⡿⠛⠉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣆⣴⡆
⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠀⠿⠂
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠙⠃
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⣰⠋⠁⢀⠀⣤⠀⠻⡋⠀⢸⡤
⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠃⠀⠙⠦⡀⠀⠀⢻⡘⣧⣼⡆⢹⡄⠀⠿⠀⠸⠷⠶⠀⠀⣼⣇
⣿⠁⢠⡄⢦⡀⠀⠀⠀⣀⠀⢀⡴⠆⠀⣄⡶⠀⣼⠛⢦⣀⠀⠀⢹⡶⠓⠾⠃⠙⠀⠉⠈⠁⠀⠀⠀⠀⢀⣴⣦⣽⡿⡿⠖
⠁⠀⠸⣦⣼⣄⢰⠀⠀⣿⣦⣸⣅⣀⡀⢿⠛⠆⠙⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠙
⠀⠀⠀⣷⣀⣽⢸⡄⠀⠁⠀⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠹
⢾⣄⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈
⠸⣇⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⢤⡀⠀⣸⣦⡀⠛⢦
⠀⠙⠢⠼⠆⠀⠠⠀⠈⠀⠀⣠⠤⠄⠀⠀⠀⣠⢤⡄⢤⡄⠰⣂⡀⢻⠀⢼⠳⣯⠙⡦⠙⠉⠑⠀⠋⠀⠀⢺⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⡦⠾⠱⠞⢧⠀⣧⠀⠤⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢰⣏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⡴⣄⠀⡇⠀⢧⡀⠱⠆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⣸⠓⢦⡀⢣⣀⡿⠀⠙⠁
⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⠀⠙⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠀⣸⠇
⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠂⣿⢀⡀
⣿⣿⣿⣿⠃⣀⣀⠀⠀⢠⣤⣤⣤⣄⣀⣠⣤⣤⣴⣶⣿⣿⡇⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠈⠙⠷⣦⣄⣀
⣿⣿⡿⠃⢸⠁⠉⠁⠈⠙⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠙⠻⣿⣦
⣿⡟⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⣿⣿⣼⣿⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠈⣿
⠟⠀⠀⢀⠇⣠⣀⣀⡀⠀⠀⠀⢠⣾⣽⡇⢀⣿⣿⣿⣿⣿⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⣀⡀⠀⠀⠈
⠀⠀⠀⣰⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠁
⠀⠀⠀⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⢻⣿⡇⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⢸⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣠⣌⠰⣾⣷⣀⠀⠀⠀⠀⢁
⠀⣠⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡻⠋⠴⠀⠈⠁
⣴⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠸
⠿⠟⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣤⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⢠⣶
⡄⠀⠶⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀⡿⠀⠀⠀⠆⠀⠰⠂⠀⠰⠀⠀⠀⠀⠠⠘⢷⣶⠆
⢿⠦⠀⠀⠉⢉⡉⢻⣿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠐⠈⢡⣾⣿⣿⣿⣿⡇⢸⣿⣿⣿⣄⣿⣿⣋⣿⡇⠀⣿⠀⠀⠀⡄⠀⠰⠀⠈⠁
⠀⢠⠀⠀⠀⠀⣜⣛⡛⠛⠿⠿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣇
⠀⠀⠀⠀⠀⠀⢈⠙⢿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠇
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠈⠉⠁
⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠄⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣼⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠙⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣤⡿⠋⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣽⢷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣺⡽⠖⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⡞⠋⠁⠀⠀⠘⣧⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠈⣧⠀⢀⣤⢶⣸⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⡷⠽⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠿⣿⣦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡃⠀⢀⣿⠛⠿⢿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣾⣏⡐⠀⠀⠈⣹⡟⠻⠾⢿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢯⣿⣖⣤⣿⣁⣀⢀⣀⣼⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⢵⢿⣛⣟⣯⠿⠿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⢶⣲⣹⣭⠧⠓⠚⠊⠉⢻⡀⠀⠀⠀⡅⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠽⣷⠛⠉⠉⠀⠀⢿⠀⠀⠀⠀⢀⣸⡇⡤⢤⣒⣿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢸⡄⠀⡀⣀⢤⠸⡗⣆⣾⣿⠽⠞⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣠⣴⣾⣧⣯⡿⠛⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣟⠋⠙⢻⡿⠯⠿⣷⣶⣲⣦⢤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⢧⣄⣀⣿⡃⠀⠀⠀⣿⠇⠉⠛⠛⠛⣾⠿⢿⣯⣷⣶⣲⡤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠿⠿⣿⣷⣶⣿⣦⣤⣤⣄⣰⣿⠀⠀⠀⠀⣽⡍⠙⠛⢻⡿⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⢿⣿⣵⣶⡿⣧⣤⣄⣸⣧⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢬⢹⣻⣿⣿⡿⠟⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⣤⢴⢲⣖⣖⣺⢹⡭⠧⠷⠛⠚⠛⣏⠉⠀⠀⠀⡇⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⢖⣲⣾⣯⣽⣽⠼⠞⠚⠓⠛⣿⠉⠉⠁⠀⠀⠈⣷⠀⠀⠀⠀⠀⣻⣀⣀⣠⢤⡗⣲⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠉⢹⠉⠀⠀⠀⣧⠀⠀⠀⠀⠀⣿⡀⣀⣀⢀⡠⡤⣿⢠⣲⣖⣞⣯⡭⠷⠗⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣤⡤⡿⡴⣆⣶⣺⣻⣯⡧⠿⠽⠞⠓⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡯⠯⠽⠗⠛⠚⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⢀⣘⣛⣂⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢁⡿⠄⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣹⢸⡟⣇⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡍⠰⠀⠀⠀⠀⠀⠴⣠⠄⡧⣠⠀⠀⠀⠀⣰⢻⡇⣿⢸⡇⢹⡄⠀⠀⠀⠐⠀⠀⠀⠆⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣬⠭⠵⣫⡴⣿⡂⠀⠀⣠⠇⡾⢸⣿⢸⣿⠈⢷⡀⠀⠀⠐⠀⠁⠀⠖⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠥⠞⠑⠃⠀⠀⠉⠈⠁⠉⠉⠈⠉⠀⠈⠁
⠀⠀⠀⢰
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠸
⠀⠀⠀⢸
⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⠃⠐⢩⢔⠀⠀⠀⠀⠀⠚⠥⠀⠀⠀⠀⠀⠀⢠⢋⡂⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⡔⠃⢒⡀⠀⠀⠀⠀⡈⠉⠭⠂⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠐⠠⠠
⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠃⠀⡈⠀⠰⠣⠀⠀⡀⠭⠭⠭⠙⠀⠀⠀⠀⢸⠗⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠄⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠋⠉⠌⠄⡠⠀⠀⢑⣒⣒⢒⣒⣲⠄⠀⠀⠍⠀⠀⠀⠀⡄⠀⠀⡄⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠌⠌⠀⠀⠀⠀⠀⠀⠸⠡⠤⠬⡭⠥⠤⠄⠀⠐⠦⠀⢀⠀⠀⠀⠀⠀⠀⠁⣀⡐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⣰⣲⡄⢄⣛⣡⣴⣿⣶⣶⣾⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣉⣁⣀⣀⡈⠉⠛⠷⣤⣀
⠀⠀⠀⠈⠉⡀⠈⡹⣿⣟⣛⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣛⣉⣹⣿⣿⣿⣿⣿⣶⣦⣄⠙⠻⣶⣄⡀
⠀⠀⠀⠀⢰⠁⢾⡿⠂⠹⠋⠉⠙⠁⣰⣄⣀⠀⠀⣀⣤⡤⠎⠐⠿⠛⠛⠟⠛⠛⠉⠉⠻⠿⢿⣿⣿⣿⣿⣦⣈⠻⣿⣷⣶⣤⣀
⠀⠀⠀⠀⣼⣦⣀⠀⡈⠂⠀⢨⠻⣿⣦⣹⣿⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠻⢿⣿⣿⣶⡈⢿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⣼⣷⣝⢿⣿⣿⣷⣶⢰⠐⠋⠉⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣄⣀⠀⠀⣿⣷⣶⡘⢿⣿⣿⡄⠹⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠹⣦⣽⣿⡿⣿⣷⠷⠖⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⣿⡄⣈⠹⣿⣿⣿⡃⢿⣿⣿⡄⠘⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠛⠉⠙⢿⠏⠀⠀⠰⣦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣟⠀⠻⣷⣶⣾⠟⠡⣾⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢀⡠⠀⠴⠶⣶⣶⣶⣤⣤⣀⣀⣀⡀⠀⠀⠀⢘⣏⢠⣥⢶⣦⣌⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣦⠀⠀⠁⠀⠀⣤⣄⠀⠨⠟⠋⠛⠛⠛⠛⢿⣷⣦⣤⠈⠛⠘⠻⣿⣿⣿⢯⣭⡝⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⣠⣶⣦⡤⠤⢤⣝⡻⣶⡀⠀⠀⠀⠀⠀⢀⡈⣻⣿⣧⠀⠀⠀⠀⣰⣴⣿⡟⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣤⢤⣴⣿⣿⣶⡇⠲⣤⣤⠄⣀⣠⣤⣼⣍⢻⡇⠀⠀⢰⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣤⠀⠟⠃⠀⣬⣽⠟⣽⡷⢀⡔⠛⠋⠻⠿⣿⣷⣿⣃⢱⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣼⣿⠋⠀⠀⠀⢉⡉⠀⠰⠟⠁⠀⠉⠒⠠⢿⡾⢿⣟⣼⣯⠈⠀⠂⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⢀⡄⠀⣼⣿⡅⢤⣧⣶⣾⣿⡀⢼⣥⠔⠉⠉⣶⣄⠀⠀⠀⣿⣿⣝⢿⣷⡦⠀⢻⣿⠛⢿⡇⠀⢹⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢀⣀⡋⠀⢬⣴⠀⠀⣏⣽⣤⡀⠀⠈⠁⠘⠷⠾⠷⠞⠻⠟⠋⠘⠷⣶⣿⡿⠟⠙⢤⡞⠻⠷⠘⣿⡀⠀⠁⢀⣸⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠠⠚⠉⢀⣀⣤⡤⠤⠤⠒⠂⠹⠟⠙⠒⠧⣶⣶⣤⡀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⡤⠘⢶⣬⡻⣶⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣠⡤⠖⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣟⠙⠦⣄⣀⡠⢦⣤⣶⠿⠿⠯⣥⣄⣺⣿⣿⢾⡟⠶⠊⠉⢉⠉⠻⣷⣤⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠙⠋⠙⢿⣷⣦⡄⠀⠀⠀⠈⠁⠀⠀⢠⠈⣿⣿⠋⠁⠀⠁⠀⠀⠀⠀⠀⢲⣿⣿⣅
⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢏⢻⡿⠂⠀⢄⠀⠀⠀⠀⠀⢱⣼⡟⡀⠀⠀⠀⠀⠀⠀⣈⡶⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠘⠁⡆⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡼⠀⡇⠀⠀⡀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠻⠻⣿⣿⣿⣿⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⡄⠀⢿⠀⠴⢠⡎⠁⠀⠀⠀⠀⠀⠀⠁⠀⢷⠀⢰⡇⠰⣿⢸⣷⡈⢢⣄⠐⢤⣀⣀⡒⣿⣿⡿⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢈⠀⠛⡜⠋⣇⠀⢠⠀⠀⠀⡆⠀⣠⡀⠀⠀⢸⡄⠀⢡⣾⣿⣿⣿⣿⣿⣶⣿⣿⡿⡁⢿⣧⢻⣶⢷
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠘⠛⣼⠂⠀⣽⣠⠁⠀⠀⢸⠇⢰⣿⡄⣠⣇⠀⣷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⢸⣿⣷⡜⠻
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁⠀⠸⠇⢠⣍⠁⠀⠀⠀⢀⣞⢠⣆⡘⣷⣿⣾⢀⣿⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣶⡦
⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⢀⣡⠈⠊⠁⡧⠘⢠⡴⠛⠻⢸⣿⣿⣜⣿⣿⣸⠇⠀⣿⣿⣿⣻⣿⣿⣿⣏⡑⠾⣏⢰⣿⠿⠋⣰⣦
⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠀⠀⣬⠃⠜⠀⠀⡠⠊⠋⠁⢀⢠⡿⣿⣿⣿⣿⣿⡅⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣽⠟⠁⠻⣷⡏
⠀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠇⢀⠄⠀⠈⠀⠀⠀⢠⣿⠟⢰⣿⣿⣿⣿⡿⢄⣿⣿⣿⣿⣻⣿⣿⠟⠋⠀⣔⠊⠁⠂⣸⣿⡿⣣⡀
⠀⢧⡰⠇⠀⠀⠀⠀⣀⣀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣀⠄⣠⣿⠇⢠⣿⣿⣿⠓⢯⣶⣿⣿⡿⣿⣿⢺⣿⣯⣴⣯⣴⣤⣦⠀⢻⣿⢟⣼⡏
⠀⠀⠄⠀⠀⢀⠔⡝⠋⠉⠀⠀⠀⣀⣀⡀⠀⠀⣀⣤⣶⠾⢛⣥⣾⠟⣃⣴⣿⣿⡟⣧⡼⢿⣻⣿⠿⠛⡛⠩⠉⠓⠚⠛⠓⠒⠶⠧⠴⠿⠓⠚
⠀⠀⢧⡄⠀⠠⠊⢀⣴⠾⣿⣿⣿⣿⠏⠁⣤⣾⣿⠛⠁⣰⣿⣿⣿⣿⣿⣿⣿⣾⣿⣯⣿⠿⠕⠀⠈⠁⠀⠀⠒⠋⠉⠉⠙⠛⠓⠒⢤⣀
⠀⠀⠀⠁⡀⠀⠀⠉⠸⣷⣿⣿⣿⣅⠀⠀⢰⣿⡅⢂⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠄⠀⠿⠟⠃⢈⣩⣽⣿⣿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣨⣿⠋⣥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣿⣿⣿⣷⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡈⣻⣿⡟⠛⠫⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡾⢿⡏⢙⣧⣀⣴⠛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡀⢀⣿⠟⢿⡟⠛⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⡿⢿⣿⣾⣿⡿⠿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣏⠀⠀⣿⣄⣼⣇⠀⣀⡟⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⢹⣿⡿⣿⡿⠛⠻⣿⣿⠿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣦⣤⣼⣷⣴⣿⣇⠀⣀⣿⣧⣤⣾⣭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⠛⣿⣿⠛⠛⢻⣿⠟⠿⣿⣧⣽⣿⡏⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣴⣿⣿⣄⣀⣸⣿⣤⣤⣿⡟⠉⠹⣿⡿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⠛⢿⣿⡿⢻⣿⣿⣿⣧⣤⣶⠿⠿⢿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣼⣷⣤⣼⣿⣿⣿⣿⣧⣀⣿⣿⣏⡀⠀⣼⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⠋⢻⣿⢉⣻⡏⠀⠈⢻⣿⣿⠋⢹⣿⣿⣾⣿⠟⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣾⣿⣿⣿⣿⣿⣤⣾⣿⣦⣶⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣍⣀⣸⣿⠁⠀⢹⣿⠛⠛⣿⣿⣥⣿⡟⠁⠉⢿⣏⠁⠹⣟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⣿⣿⣿⣿⣦⣴⡿⢻⣶⣾⣿⠋⢹⣿⣧⣄⣠⣿⣿⣿⣿⣿⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⣛⣿⣿⠿⢿⣶⣾⠿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠉⠁⠀⠈⠉⠉⠉⠉⠁⠉⢫⠝⠛⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠀⠀⠀⠄⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⠀⠤⣄⠠⠀⠀⠉⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁

⣿⢿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⣋⣴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣾⣿⣿⣿⡿⠋⣁⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢟⣩⣶⣿⠟⣉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣫⣾⣿⠞⣫⠿⣫⣴⡿⢛⣡⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⣉⣴⣿⣿⡿⢟⣭⣶⡿⠋⣀⣴⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⢉⣴⣾⣿⣿⣿⡿⢞⡿⠟⣩⣴⣿⠿⠋⣀⣴⣶⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠾⢛⣽⣾⠿⢛⣥⢞⣫⣶⣿⠿⢋⣡⣶⡿⠿⠿⠿⡿⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣤⣾⣿⡿⣿⣿⣿⣵⣾⠿⣋⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⣋⣵⣾⠟⣫⣿⣿⣵⡾⠛⠉⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⢉⣴⡾⠟⣩⣴⣿⣿⠿⠛⠁⣀⡴⠞⢋⣡⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⢁⣤⡾⠛⣉⣴⣾⣿⠿⠋⢁⣠⡶⠟⣉⣤⣾⡿⠟⣁⣤⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⢁⣴⣾⣿⠿⠋⣁⣴⠾⢛⣡⡴⢛⣽⢞⣫⣶⣿⣿⣿⣿⢛⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⣤⣾⣿⠿⢋⣡⣶⠟⣋⣥⠾⣋⣵⣾⣿⣿⣿⣿⣿⣿⣿⡟⠋⡛⠛⠃⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣋⣥⡾⠟⣫⡴⠞⣉⣴⣾⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣵⡾⢛⣭⠶⠛⣡⣴⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⣟⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣛⣵⠞⢋⣠⣶⣿⣿⣿⣿⣿⢟⣫⣵⣾⣿⣿⣿⣿⣿⢟⣫⣵⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⢉⣠⣶⣿⣿⣿⣿⣿⢟⣭⣶⣿⣿⣿⣿⣿⡿⢛⣩⣶⣿⣿⣿⡿⣿⣿⣿⣿⠿⣿⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⠟⣋⣵⣾⣿⣿⣿⠿⣛⣭⣾⣿⠿⢛⣥⢞⣭⠶⣿⣅
⠀⠀⠀⠀⠀⠀⠸⠟⠉⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣾⣿⡿⢛⣩⣴⣿⣿⣿⣿⠟⣫⣵⣾⣿⠟⢋⣡⣾⣿⣿⣿⣿⣿⡿⠛⠿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⣯⣷⣿⣿⠿⣋⣥⣶⣿⣿⣿⡿⢟⣫⣶⣿⡿⠟⢋⣤⣾⣿⣿⣿⣿⣿⣿⠿⠋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣫⣴⣾⣿⣿⣿⡿⣛⣽⣾⣿⡿⠛⣉⣴⣾⣿⣿⣿⣿⣿⣿⡿⣛⣥⣾⣿⣿⣿⣯⡀
⠀⠀⠀⢀⣠⡶⣛⣿⣿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⣿⣿⣿⡿⣻⣵⣾⣿⠿⢋⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⢋⣤⠀⣠⡄
⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⡿⠟⢋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣾⣯⣵⡾⣋⣤⡶
⣸⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⠿⠛⣁⣤⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣵⣾⣿⣿⠟⣫⣿⣿⣿⣿⣻⡷⠻⣆⣠
⠉⠛⠉⠛⠁⠉⠁⠀⠀⠘⠋⢩⣽⣿⣿⣿⡿⠟⢋⣡⣴⣿⣿⣿⣿⣿⣿⠿⢛⣩⣾⣿⣿⣿⣿⣿⣿⣿⠻⣿⣛⣠⣾⠟⣉⣴⣾⣿⣿⣿⢛⡿⠛⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟⠉⠀⠘⢋⡽⢿⣿⡿⠿⠛⢋⣡⣶⣿⣿⣿⠟⠉⠉⠋⠙⠛⠉⠃⠺⠏⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⣿⣿⠿⣿⠟⠁⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣡⣞⣥⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢋⡵⢋⣤⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠞⠋⣼⣷⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣻⣿⣿⡿⣿⣿⢻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠄⠐⠒⠒⠒⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠓⠈⠘⠊⠉⠀⠀⠙⠦⠔⢦⠤⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⡼⠃⠀⠀⠀⠀⣀⣀⠠⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣼⠁⠀⠀⢀⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣸⠇⠀⢀⡠⠌⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢻⠿⠖⠛⠓⠉⠉⠁⠀⠰⠟⢢⢠⣄⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣧⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠠⡎⠛⠁⠀⠀⠀⢉⡆⠀⠀⠀⠀⢈⣀⡀⣀⠈⠙⣇⡀⠀⠘
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⡥⠖⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⢁⣀⣀⠈⠒⣿⠈⣧⠻⠂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⣼⠃⢹⠋⠀⡠⢞⣉⠉⠲⣄⠀⠀⠀⠀⢠⡶⢣⣤⡀⠑⣆⢀⢰⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢿⣀⠀⠀⠀⠀⠠⠁⢀⠎⠀⢸⠁⣹⣿⣷⠀⠈⣆⠀⠀⠀⣾⡓⠾⡿⠃⢀⣨⣰⡄⠳⢄
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⢽⡀⠀⢰⡿⢆⠀⠈⢧⡀⠥⢤⣙⣿⡥⢄⡠⠊⠀⠀⠀⠻⡉⠒⠒⠊⠁⠈⠉⠉⠐⠎
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⡴⢇⡏⢹⡦⣉⣠⡾⠁⠄⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠙⣆⠀⠚⠀⠢⠀⠀⠀⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠐⠒⠚⠁⢎⡤⠊⠁⠷⠀⠀⠀⢈⠉⠀⠀⢀⠠⠧⠤⢤⣀⣀⠤⠛⠒⠤⠤⠖⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠐⢄⣀⡴⠋⠀⠀⠠⡆⠀⠀⠀⠀⠐⠄⠐⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⣤⣴⠲⠿⢿⣿⣿⣾⡿⠒⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠃⠁⠀⠉⠳⢤⣀⣈⣻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⣠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⠀⠀⣠⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠓⠚⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⡀⠠⠎⠀⠑⠉⠀⠐⠈⢻⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡌
⢱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠈⢢⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠈⠲⠃⣄⢀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⠀⠀⠀⠀⠈⠺⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢧⡈
⠀⠀⠀⠀⠀⠘⠣⠐⠢⠄⠣⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡟⠦⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣴⡄⠀⣯⠗⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⣘⡆⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠲⢄⣀⠐⠂⢤⣤⡤⠄⠚⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠰⢀⠂⠐⠂⣆⢂⡶⠖⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠈⠉⠈⠛⠡⠌⠀⠀⣄⠀⠀⠀⠍⡲⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠁⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠴⠋⢀⡀⠀⠙⢤⠀⠠⠏⠀⠀⠁⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢀⡔⢄⣀⣤⠵⠶⠚⠛⠛⠛⠻⠷⢶⣦⣥⣐⠀⡀⠀⠀⠀⣀⠀⠬⠦⡀
⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⢀⠦⠠⢞⣭⡶⠛⠉⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠈⠙⢿⣶⣅⡠⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠠⠀⠒⢂⠒⠈⢁⢒⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣮⡢⣾⠆⠐⠤⢀⡵⡀
⠀⠀⠀⠀⠁⠀⠀⠀⢒⡄⣰⣿⣿⡟⣥⣶⡿⠁⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣄⢾⠆⠀⠘⠃⠑⡀
⠀⠀⢀⠂⠀⠀⠀⠀⠌⣼⣿⣿⣿⣷⣿⡅⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠸⡆⠀⢿⣿⣿⣿⣆⠀⣈⡀⠀⠀⠰
⠀⠀⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⣻⣿⣿⣷⣴⣦⢀⣬⡇⠀⢀⡠⠖⠚⠉⠉⠹⠛⠛⢶⣿⣄⣸⣿⣿⣿⣿⣇⢛⡣⠈⠐⠀⠃
⠀⠀⠀⠀⢀⠂⠀⢸⣿⣿⣿⣷⣚⢿⣿⣿⣿⣿⣿⣿⣷⣾⠿⠒⠀⢀⠀⢰⡔⠒⠒⠀⢿⣙⢿⣿⣿⣿⣿⣿⡈⣁⠀⣄⡀⠘
⠀⢸⠀⢀⠀⠀⡗⣼⣿⣿⣿⣿⣿⣈⡿⣿⣿⣿⣿⣯⠈⢹⣤⠶⠛⢻⣧⡀⠑⢐⡀⠀⡖⡂⠉⠙⠹⣿⣿⣿⡇⠀⠀⠄⠙⠶⡆
⠀⠈⠀⠈⠀⢀⠃⣿⣿⣿⣿⣿⡿⡏⢉⠛⣿⣿⡿⣿⣶⣤⣄⠀⠀⢀⡝⢁⣀⡸⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠂⠀⠘⠀⠀⡅
⠀⢐⢀⠀⠀⠎⠀⣿⣿⣿⣧⠈⠄⠁⢀⣴⣿⣿⡇⠀⠙⠛⠁⢀⡒⠋⠙⠁⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠈⣿⡇⠀⠀⠠⡄⠀⡇
⠀⠀⠘⡄⠀⠀⠀⢸⣿⣿⣧⠏⠀⢠⣿⣿⣿⣿⡷⠀⠀⠀⠀⠈⠀⠀⣀⣀⣠⢤⠔⠖⠛⠛⠋⠀⠀⠀⠀⢹⢃⡥⠀⠀⠛⣦
⠀⠀⠀⠰⣄⠀⢀⣠⣿⡟⠀⠀⠀⠀⠀⣀⣹⣿⣿⣆⠀⠀⢀⢰⢾⣾⡟⠛⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⢧⡌
⠀⠀⠘⠀⢈⣴⣿⣿⡿⠃⠀⠀⠉⠀⠀⠀⡈⣿⡠⠙⣿⠀⢄⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣆⡀⠺⠁
⠀⠀⢠⣿⡿⢛⠄⠀⡁⠀⠀⠀⠀⠀⠀⡀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀
⠀⣠⣿⡟⢱⠃⡄⠐⠁⠀⠀⠀⢠⠀⣠⣤⣾⣿⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇
⢠⣿⡉⢀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣿⣿⣿⣿⣟⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⢸⡋⡇⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦
⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣟⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠋⠙⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠻⠟⠿⢿⣿⣷⡀⠰⡧⣠⠄⠀⠀⠠⠀⣶⣤⣾⣿⡿⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠆⠐⣿⠁⢶⣂⣴⣾⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⣿⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⡀⣿⣿⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⡿⣿⣿⣿⣟⡳⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⣯⣿⣿⣿⣿⣋⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣶⣶⣿⣿⣿⣿⣿⠿⠛⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠛⠻⣿⡿⠻⢿⣿⣅⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡿⠿⡀⣴⣿⣿⣿⣿⡟⢿⣿⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣆⣡⣾⣿⣿⣿⣿⣿⣿⡟⠦⠙⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣿⣿⣿⣿⣿⣿⣧⣽⣿⡛⢿⡿⣷⣶⣶⣦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢩⣍⡉⠉⢠⣷⣬⣾⣿⣾⣿⣤⣦⣿⣿⣿⣯⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣽⢿⢷⣈⣤⣽⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣾⣿⣿⢿⠿⡟⢷⣾⢿⣿⣿⣿⢿⣿⣷⣛⠿⢿⣾⣅⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⢟⡀⢀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣜⠛⠉⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡿⡏⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣝⣿⣿⣶⣿⣯⣽⣿⣿⣿⣿⠟⢿⣿⣏⣽⣿⣬⣛⣻⣿⡿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣟⣻⣿⠏⠛⠻⠇⠙⠃
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⠿⠻⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⡟⠫⠈⠁⣿⣿⣿⣿⣶⣿⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⣿⣷⣤⣼⣾⣿⣿⡿⠋⣻⣿⣿⣿⣿⣇⣰⠀⢀⣿⣿⣿⣽⠙⠿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⣿⡻⣿⣿⠿⢿⣿⠏⠹⣿⣿⣿⣏⠛⠻⣿⡷⠉⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠙⠋⠀⠀⠉⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⡀⠀⡀⠀⢀⠀⠀⠀⢠⠀⡄⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠁⠀⠀⠀⠁⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⣱⠎⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢤⣿⣷⡶⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠞⠛⢁⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣶⣾⢹⡅⣠⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⡿⣧⣾⣷⣤⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⡋⣀⡿⠿⠛⢠⣼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢉⣁⣅⣠⠄⢾⠿⠾⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠏⠈⢻⣄⣘⣷⠈⠑⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠠⣤⠟⠁⡈⣠⡴⠋⠉⡾⠆⡷⠂⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢂⣾⣿⠋⡪⣴⣀⢱⡤⠶⢂⣤⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣄⠴⠲⠾⠿⢃⢠⣄⣼⠛⠋⡀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⢠⡶⢠⣀⣈⣉⠀⠴⠶⡅⠀⣀⡤⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠘⡻⠊⣸⡿⠃⠀⠘⠋⠀⠀⡈⠁⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀⣤⡂⠀⠈⠀⢸⡿⢁⠠⣀⠖⠀⠀⠀⠈⠓⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠐⣿⣾⠋⠀⠉⠀⢠⠀⠀⠈⠰⠄⣠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣄⠀⢀⠀⠄⢻⣿⣶⡀⠀⡰⢿⠀⢀⢀⠄⠂⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠀⠈⠁⠀⠀⡰⠃⠀⠀⢀⣧⡀⠀⠀⠦⠈⡀⠰⠊⠀⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠉⠒⠥⠰⠚⠃⠐⠒⠀⠚⠛⠟⢉⣠⠤⠊⡑⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠂⠐⠂⠀⠂⠠⣤⣦⠔⠀⠀⠹⠷⡰⠚⠓⠋⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⣤⣄⣀⠆⠚⢸⣿⠏⠀⠀⠀⣶⡆⠀⢈⡀⠀⠀⠒⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄⣨⡼⠏⠀⠀⡸⢡⠀⣀⣠⠁⠀⠀⠀⠀⠘⢆⠀⢀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠁⣿⣿⣿⡇⠀⢠⡈⠀⢀⠀⠈⠡⠀⠈⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⠝⣿⠁⠐⠈⠁⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣻⠉⠉⠉⠉⠀⠉⠀⠀⠀⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⡄⢀⠀⠀⠀⠘⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⣯⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡤⠴⠖⠹⠒⠁⠙⠉⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢰⠃⠀⠀⣀⠀⠀⠀⠀⠀⡜⠷⠃⠀⠀⠀⠀⠀⠀⠐⠚⣣⠄⠀⠴⢋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣀⠀⠋⢠⠂⠀⠀⡶⠚⠀⡠⣀⡛⡐⠀⠀⠀⠀⠐⣉⢉⠉⢃⡐⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠨⡇⣀⠀⠘⠁⢠⡴⠃⢤⣜⠯⠥⠀⠀⠑⠐⠉⠲⢀⠀⠀⠀⡘⢁⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠻⢿⠀⠀⣄⣸⠀⠀⠀⠲⢀⡀⠀⠀⠀⠀⠀⠀⢀⢹⡀⠉⠀⢼⢌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⡴⣆⣿⠃⠀⠂⠀⣀⡀⠉⢉⡔⠀⠐⠦⠴⠋⠁⠈⠙⠢⠼⠀⡇⠻⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡔⠉⢸⠀⠰⡇⡎⠸⣀⡬⠋⠀⠀⢀⣠⠎⣑⠦⠖⠤⠤⠴⠊⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠆⠀⢰⠁⠰⣇⠀⠀⠀⠙⠦⢤⣄⣤⠖⠉⠁⠀⠀⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠾⣧⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢧⠀⠀⠀⠀⠀⠀⢠⠞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠴⣷⣀⣈⠧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⢧⣀⢀⡄⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡈⠁⠀⠀⠀⠘⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⣷⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠳⠦⠷⢤⣤⣄⣀⡀⣄⠈⡀⠀⠀⠘⢦⣘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠙⣄⠀⢀⡞⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠐⠚⠛⠿⠿⣿⣿⣿⣻⢿⣿⣟⡲⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡠⠆⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣯⡻⣿⣌⡻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⡴⠋⠀⣠⡤⠀⢀⡀⠀⠀⠀⠀⠀⠠⠌⠑⠌⢻⣿⡹⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⣉⣤⣶⣿⣯⡽⠿⠒⠩⢭⣥⣖⣀⣠⠀⠈⢢⡀⠈⢻⡇⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡟⢀⣿⣿⠦⠄⠉⠁⠈⠩⠉⠛⠻⢿⣦⡀⠀⢻⡌⢊⠃⣾⡿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠱⠀⢺⢟⣠⠍⣛⣻⣿⣿⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⣴⣆⣠⣶⣷⣄⢀⠈⣿⡆⢠⡀⠀⢽⡆⠀⢀⡄⠈⠐⠋⠉⠉⣩⣿⠋⠉⠻⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠒⠫⠵⡻⠟⠋⠙⠆⣐⠐⠚⠉⠁⠀⠈⢧⡀⠂⠀⠀⠠⠀⠤⠾⠋⠁⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⢿⢸⣆⣤⠴⠛⠁⠈⠀⠀⠁⠉⠉⠢⣤⣤⡄⡆⠀⢈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠓⠌⣼⣿⠃⢠⠀⡁⣴⣿⣿⣦⡀⢄⡀⠈⠙⠓⠛⠂⠀⢃⡀⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⢠⠀⠀⣤⠻⠀⠁⠙⢿⡿⠋⠈⠀⠙⠒⠒⠂⠀⠀⠀⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠑⠀⠀⢀⠀⠊⣄⠀⠀⠦⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠑⠦⠤⢤⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣨⣦⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢿⢰⣾⣶⣤⡆⣷⣼⡖⠄
⠀⠀⠀⠀⢀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣾⣿⣿⣿⣿⣿⢛⣤⣶
⠀⠀⠀⠀⠘⣿⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣾⣟⠁
⠀⠀⠀⠀⠀⣽⣿⣿⠟⠂⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠢⠛⠛⣽⣿⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠙⢫⠃⠀⢀⣾⠓⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠳⣌⣉⣋⣤⣿⣿⣿⣿⣫⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⡋⢠⡄⢲⡖⢠⣾⣦⣹⣿⣿⣶⣶⣶⣾⣧⣴⣿⣿⣿⣿⣿⣿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠜⠠⢄⣠⠴⠿⠿⠈⠛⠻⠿⠿⠿⠿⠿⠟⠛⠉⠙⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⠀⢈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡋⣳⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠰⡦⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣟⠛⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆⣤⡄⠀⡀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡏⢹⣧⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⡷⠁⠤⠀⠀⠀⢰⣿⠎⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡶⣤⣤⣴⣤⡾⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠀⠀⠀⠀⣼⣋⣻⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⡤⠉⣗⡆⠀⠀⠈⠉⠉⠀⠀⢤⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⡶⢠⡾⠿⣷⣀⡀⠀⠀⠀⠀⢞⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠛⠛⠉⠛⢷⡤⡄⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⢦⣴⣄⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠏⠀⣽⠿⢿⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⡧⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡤⢸⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣖⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⣀⡀⣀⡤⣤⣴⣲⠖⠻⣾⠿⢯⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⠗⣾⣽⣶⣿⠽⠛⠓⠋⠉⢹⣯⣿⡆⠀⠀⠙⠛⠋⠀⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣎⣥⣆⣴⢞⣁⣻⣭⣽⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣀⣈⠳⣤⣶⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⣩⡿⣅⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠉⠉⠛⣽⣶⣖⣦⣠⢤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⢿⠇⠀⠀⠀⠘⣿⣭⡏⠀⢈⣿⣻⡷⠛⠻⣽⣶⣟⠦⢴⣲⠤⣖⠶⠾⡿⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢲⣾⣵⣧⡶⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠃⠀⠘⣯⣭⠇⠀⠀⠀⠀⠙⣦⣴⣾⣷⠚⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡤⣀⣴⣺⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢟⣶⣶⣖⣦⣤⢲⣤⣴⣢⣤⣖⣦⠴⣲⠶⠾⣽⡿⣷⠛⠉⠛⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠗⢲⠿⣿⣪⣿⣾⠇⠉⠀⠨⣿⣙⡧⠀⠀⠀⠀⠀⠻⣶⠿⠀⠀⠀⠀⠐⢿⡾⢿⡖⠽⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣯⣿⢿⣲⣤⣠⢤⡙⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠉⠀⠋⠀⣽⡯⣿⠊⠉⠓⠛⠛⢯⡿⣾⡽⠶⢾⡷⠶⢏⣶⣶⡭⠖⠺⠽⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣊⢥⡶⠃⠀⠀⢀⡴⣤⢈⡟⣻⠀⣠⡄⢠⠿⣸⣯⣿⡁⣠⣄⢰⡄⣴⣯⣬⣧⡀⠀⠀⠙⢶⠬⣳⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠰⠞⠒⢏⡤⠞⠹⠋⠱⠊⠷⠛⠋⠁⠈⡇⠀⠀⠀⢸⠋⠁⠈⠛⠙⠞⠻⠏⠻⠏⠲⢄⡟⠚⠳⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠉⠉⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠉⠉⠉⠉⠉⠉⢛⡳⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠀⣀⠤⠟⠓⠒⡶
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠚⠋⠀⠀⠉⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡃
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢠⣄⣠⣀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠋⢀⠄⠀⠐⠉⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠒⠉⠉⠰⠶⢄⣤⡙⣆
⠀⠀⠀⠀⢠⠞⢷⠀⣀⠴⠃⠈⣉⣉⣉⠶⠲⡶⢦⡤⠚⠃⠀⠀⠀⢀⡤⠶⢛⡉⠁⠀⠀⠀⠀⠀⢠⠤⠠⠦⡀⣿⠋
⠀⠀⠀⢀⡏⠀⠈⢷⠉⢢⣀⣀⡀⢀⡸⠊⠛⢢⣀⠇⠀⠀⣀⠤⣪⠕⠚⠉⠉⠁⠀⠑⢦⠀⠀⢠⠁⠀⠀⠀⠈⣿⡀
⠀⠀⠀⡜⠀⠀⠀⠈⠁⠈⠁⠀⠈⠋⠰⣄⣠⠜⠁⢀⠴⡎⠁⠀⠉⠉⠙⠃⠒⠦⠄⠴⠃⠀⠀⠘⣠⠒⠂⡀⠀⡇⠛⠋⠉⠛⠲⢤⡀⣀
⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡠⠚⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠘⡄⢷⣼⣇⣧⣤⠷⠓⢤⡀⠀⠀⠀⠻⣼⡷⣀⡀
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠊⠀⠀⠀⢀⣇⣀⠀⠀⠖⠀⠀⠐⢒⣼⠗⠒⠲⣤⡽⠚⠛⠛⠻⣇⠀⠀⢀⡇⠀⠀⠀⠀⢹⠃⠹⡍⠙⠲⣄⡀
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⡷⠾⠛⠲⠦⣄⣠⠴⠋⠀⠀⣷⠀⠀⠠⣵⣄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠌⢷⣤⡴⢚⠁⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠈⠳⣄
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠈⠑⠁⠀⠲⠤⣤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡷⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡿⡀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠋
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⢣⡙⠢⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣤⣀⠀⢀⣀⠀⠤⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠃⠰⢇⡨⡍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠉⠲⠤⣀⣀⣀⣀⣀⠠⠔⠚⠒⠛⠒⢯⡑⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢀⣀⣹⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⠏
⠀⢀⡴⠉⠁⠀⠉⠛⠒⣦⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⢀⡾⠋⠀⠀⠀⠀⠀⠀⠈⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠈⠓⠠⣶⣤⣤⠔⠻⠞⠁⠀⠀⠀⠈⠛⠦⢤⣄⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⢀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠛⠉⠲⢄⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠛⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⢰⡇

⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⣰⣣⡄⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠘⣣⠒⠀⠀⠀⠀⢾⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⠀⣴⣶
⠀⠀⠀⢠⡯⠏⠁⡀⠀⠀⠸⢿⡿⠇⠀⠀⠀⠠⠰⠀⠀⠀⠀⠴⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⡂⠀⠀⢀⣾⣿⣿⣧⡀⠀⠀⠀⣷⣿⣆
⠀⠀⠀⠚⠉⠙⠛⠛⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠘⠉⠉⠉⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⢀⣼⣿⣧⠀⠀⠀⢀⠡⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⢾⣿⣿⡂⠀⠀⠀⢀⣽⣿⣄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣴⣿⣿⡄
⠀⠀⠀⣠⣾⣿⣷⣄⠀⠀⠈⠐⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠰⣿⣿⣿⡦⠀⠀⠀⣠⣾⣿⣧⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⣿⣿⣿⡷
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠙⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣧⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⡿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢯⣄⠈⢷⣜⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢫⣿⣾⠁⢳⣄⠻⡷⣝⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣱⣿⣿⡟⢲⡀⠻⣧⡈⢮⡳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢃⣼⣻⣿⡟⣇⠀⠹⣦⡈⢯⡢⣙⢦⡑⢽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⡽⢋⣴⡟⣱⣿⠟⣴⡏⢷⣄⠈⠻⣦⡙⢮⡓⢭⣢⣌⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠖⢋⣴⣿⢋⣼⣿⢋⣾⠳⡇⠀⠹⣷⢄⠙⢿⡳⣽⣦⡈⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠵⠚⣩⠞⣡⣾⠟⣡⣿⠃⢠⡟⢦⡀⠈⠳⡙⢦⣙⢦⣉⠻⢶⣄⡀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⢾⡿⣫⡾⡿⠃⣠⢿⣦⡀⠻⣦⡀⠙⢦⡈⠓⢬⡳⢤⡈⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠿⠛⣱⣟⡴⣫⠞⢁⣼⠟⢸⡏⠻⣦⡈⠻⣷⣄⡙⠦⣀⠈⠑⠻⢷⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢋⡴⠃⣴⣿⠋⣰⢻⠷⣄⠙⢿⣷⣌⠻⣝⠲⣬⡳⢤⡀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢁⡴⠋⣠⣾⠟⢡⣾⠏⣸⠧⣜⢷⣄⡙⠻⣟⢮⣝⣦⣍⠓⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⡴⣋⡴⣚⡿⢁⡴⣫⢏⡼⢻⠀⠈⠳⣝⢿⡦⣌⠙⠮⣙⠻⢿⣶⣤⣉⡙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⣡⠖⢋⡴⢋⡜⣡⠞⠁⢸⠀⠀⠀⠈⠓⢿⣮⣝⠲⠮⣽⣶⣄⡉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣡⠖⢋⡴⢋⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡈⠉⠙⠛⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⠿⠚⠉⢀⡴⢋⡴⠋⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣫⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⣹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡄
⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣟⠦
⠀⠀⠀⠀⣸⣯⣿⣧⠀⠀⠀⠀⠀⠀⢴⣶⣾⣶⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠐⠂⢁⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣿⣟⣆⡀
⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣶⣷⣶⣷⣶⣿⣿⡆⠀⠀⠀⠀⠀⠂⠠⢀⡀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⣿⠛⠄
⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣬⣟⣛⣛⣛⣫⣭⣾⣿⡆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣤⣿⣾⣿⣿⣿⣿⣇⡀
⠀⠀⠈⠉⠉⠹⠿⠉⠛⠃⠀⠀⠀⠉⠈⠛⠋⠿⠋⠈⠉⠈⠁⠀⠀⠈⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠏⠉⠉⠀⠀⠀⠀⠈⠉⠈⠉⠹⠟⠛⠿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠀⠒⠀⠆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠴⠏⠒⠢⠤⢀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡀⢀⠤⢀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⠀⡁⠱⡄⠀⠀⠀⠀⠀⠀⠀⡎⠲⢉⣉⠬⢋⠀⠉⠐⠲⡐⢄⠀⢰⠀⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠈⢉⣉⢁⠀⣈⠦⠤⣀⠀⠀⠀⠴⠀⠀⠀⠑⠠⣝⠓⠢⢠⡄⠈⢺⠀⠈⠆⢀⡰⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⢀⠌⠁⠀⡀⢌⣱⢠⠊⠀⠑⠢⠄⠀⠀⣠⡅⠀⠒⠀⠄⠓⠒⠀⡶⠚⠈⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⡔⠁⠀⠈⢻⠀⠀⠀⠀⠀⠀⠠⠈⠙⠦⠤⠀⠀⠀⠤⠊⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⡀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠈⠒⠢⠄⠀⠀⠤⠤⠤⠒⢍⡁⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠊⠉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢦⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠇⠀⠀⠀⠑⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⠑⠢⡔⠁⠣⡀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢈⠶⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣖⢉⡁⠀⢆⠀⠀⠈⠢⠄⠠⠲⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢸⠘⠉⠉⠁⢒⠈⡇⠀⠈⠉⠉⠉⠀⠀⠘⠀⠀⠄⡈⠃⣄⠀⠀⠀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⢔⣤⣀⣀⣠⢀⠃⠀⠀⠀⠀⠀⣀⣀⠠⠤⠖⠊⠉⠉⠀⡍⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠲⡄⠀⢨⠀⠀⠀⠉⢉⠁⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡃⢘⠃⠘⢰⠉⢄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣤⡙⠨⠀⡘⠀⠣⡀⠉⠂⠀⠚⣧⢀⣀⠀⠀⠀⠀⣀⣀⠠⠤⠔⠒⠈⠁⠀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡑⢷⣈⣷⠧⣤⠀⠈⢲⠄⠤⠤⡛⢄⠀⠈⠉⠉⠁⠀⠀⠀⣀⡀⠤⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠒⢁⠜⢚⣶⠦⡀⢲⠒⢲⠊⠁⠀⠀⠡⠀⠉⠉⠀⠀⠘⡉⠉⠁⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⡄⠀⢸⠈⡀⢸⠀⠀⠀⠀⠀⡤⣀⠀⠀⠀⣀⠙⠠⠤⠤⠄⠄⠒⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠸⢀⣸⠀⡇⢁⠀⢀⣠⠷⠊⢀⡠⠤⡄⠰⢄⡀⠉⠉⠁⣈⢱⠄⠀⠀⠀⠀⠒⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡌⢖⠉⠀⠀⢠⠃⠀⠀⠈⠂⠀⠀⠉⠉⠁⠀⠀⡄⠀⠀⠀⠀⠀⠄⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣑⣄⠀⢘⠄⡀⠀⠀⢀⠀⠀⢀⣀⠀⠤⢰⢆⠀⠀⠀⠀⠀⠀⠀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠀⠈⠢⠄⣉⣉⣉⣉⣁⠠⠤⠤⠒⠚⡈⠄⣀⣀⣀⡠⠔⠊⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⡠⠄⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠉⠁⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠒⠒⠀⠀⠀⠀⠠⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠋⠉⠉⠉⠉⠒⠂⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢠⠞⠁⠀⠀⢀⠀⠀⠀⣠⠒⠉⠁⠈⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⣁⠄⠀⠀⠀⠀⠀⠂⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠁⠀⠉⠀⠀⣀⠀⠀⠀⠀⠀⣭⡬⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣀⣀⠀⠡⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⣠⠖⠋⠉⠉⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠆⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠉⠂⢾⠁⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⠿⣌⠀⠑⠀⠀⠀⠀⠀⢀⡠⠔⠀⣤⢤⡤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⣤⠈⠃⠀⠀⠀⠄⠀⠀⠀⠀⣠⠾⣁⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⣀⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢸⠹⡇⠘⠂⠀⠀⠀⠀⠀⠤⠴⠒⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠧⣼⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢺⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡄⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠁⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢠⠇⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣠⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⢀⡀⠀⠀⠀⢈⣀⡤⣀⠀⠁⠐⠒⠀⠀⠴⠚⠁⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠀⠈⠳⢤⣀⣀⠀⠀⠀⢀⣀⣤⠖⠋
⣀⣀⣀⢀⡀⡀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⣤⣤⡴⣶⣶⣤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣽⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠉⠉⠉⠉⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⠶⠤⣤⠀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠲⠄⣶⡄⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣭⣯⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠁⠴⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢬⡧⡄⠀⠀⠀⠀⢹⣿⣿⣿⣿⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⠴⢀⣭⣩⠛⢶⣤⠀⠀⠠⣆⣀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠙⣼⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠁⠨⢺⠋⠀⠀⠀⠋⠙⣭⡯⣤⣀⠨⣝⠲⠄⠀⠀⠸⣿⠂⢤⠄⢸⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⣿⢳⣤⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡄⠀⠀⠀⠀⣠⣈⣴⣀⣀⣀⣤⣀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣧⡐⢻⣿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢆⡀⠐⣀⣀⠤⠞⠋⠉⠉⠉⠉⠙⠛⠻⢦⠾⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹⡿⠟⠁⠈⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⣄⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⡄⠀⠈⡆⠁⠐⠀⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠶⣿⣿⣿⣞⡒⣚⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠲⢦⠞⠋⢀⡀⠀⣿⣶⠒⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠠⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠉⠈⠙⠛⠙⠉⠛⠙⡷⣶⣦⣀⣀⣀⠀⢁⣴⡾⠁⠀⢹⠇⡆⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢰⡏⢠⠇⠀⡠⠾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠈⠀⣟⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠈⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀
⠀⠀⠀⠀⠀⠀⠀⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠹⢶⣤⣅⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠈⣦⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⣠⣧⠃⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡾⠋⠀⠀⠀⢀⡰⡇⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠁⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢃⠸⠁⠀⠀⠀⡀⢸⠃⢣⠀⠐⠶⠌⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢰⡤⠾⡁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣴⡞⠀⡞⢀⠀⠙⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣠⠄⣴⢀⠀⢀⣼⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣼⠃⠐⠀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣥⠞⣿⡄⠠⢾⡏⠀⠀⠀⠀⠀⠀⢿⡀⠀⢠⡄⠀⠀⠀⠀⣴⠿⣤⣤⢠⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢠⣽⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢸⠃⠀⠀⠀⡼⠃⠀⢠⡔⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣠⡞⠀⠀⢀⣶⠀⠀⠀⠀⢦⡀⠀⠸⠀⠀⣠⠞⠳⠶⠖⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡼⠋⠀⣹⣇⡴⠖⠁⢿⡄⠀⠀⠀⠀⣀⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠀⣠⡼⢣⡜⠀⠀⢀⣾⡿⢲⣤⣴⡞⢁⣴⠛⠀⢾
⠀⠀⠀⠀⠀⠀⣀⡀⠤⠔⠋⠁⠀⣸⠀⠀⠀⢠⠛⣇⣼⠁⢸⣧⡘⢳⣄⠀⣾⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠇⠀⠀⡠⠋⠀⢻⡏⠀⠈⣿⠀⠀⠉⠀⢸⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠘⠉⠀⠀⠀⠈⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠰⠀⠀⢠⡄⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠠⠋⠁⠀⠀⠀⠀⠞⣿⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠉⣶⠶⠁⠀⠀⢀⠾⣿⠣⡀⠀⠀⠀⢀⣾⡄⠀⠀⠀⣹⡅⠀⠀⢩⣿⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⣹⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡼⣿⢷⡈⠀⠀⢀⣾⣿⣿⡀⠀⢈⠼⢌⠀⠐⢛⣿⡟⠀⠀⠀⣼⣧⠀⠀⠀⠉⣹⣿⣏⣉⡀
⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⢈⡴⣿⣧⡙⠂⠀⣾⣿⣿⣿⣷⠀⡨⢘⠢⡁⢠⣾⣿⣿⡆⠀⢠⡿⢼⡄⠀⠀⢙⣿⣿⣏⣉⣀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢉⡴⢿⢦⠑⠆⣸⣿⣿⣿⣿⣿⣧⠠⠞⠦⠀⠈⣸⣿⣆⠀⠀⣾⡿⢹⣷⠀⠲⠟⣻⣿⣿⣭⣄⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠋⠀⢸⠀⠑⠀⠟⠿⠿⠿⠻⠟⠛⠂⢸⠀⠀⠼⠿⠿⠿⠇⣸⣿⣧⣽⣿⡆⠠⡾⠿⣿⢿⣧⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⠉⠉⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⠙⣆⣠⡤⠖⠚⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⠀⠀⠰⣯⣭⣤⣤⣤⣤⣀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣠⡼⠁⠀⠀⢀⡬⠗⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⢀⡴⠛⠒⠒⠒⠋⠉⠉⠉⠉⠙⠒⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⢿⠹⡏⡶⡆⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠸⣏⠃⠀⡔⢱⣇⠀⢀⡞⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⣤⣸⠞⣇⡴⣆
⠀⠀⠀⠀⠀⠀⢀⣼⣧⣤⡄⣺⣾⣦⠟⣠⠶⡟⠉⠁⣽⠷⢛⡛⣏⠉⠁⠙⠳⢦⣄⠀⠀⠀⠀⠀⠈⠳⣼⠀⠿⠸⠏⢠⢇
⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⠉⠉⢰⣽⠞⢁⣾⣧⡀⠀⢀⣀⣸⣛⣋⣀⡀⠀⢀⡀⣌⢻⢲⣄⠀⠀⢀⣤⣽⠀⢰⡆⠀⢟⣩
⠀⠀⠀⠀⠀⠀⣨⠷⢤⠤⠴⢚⡟⠁⠀⢸⡙⠟⠛⠚⠋⠁⣼⠋⠁⠀⠉⠳⣜⡛⠃⠀⢿⠉⠳⣄⡎⢸⡑⠦⠈⠁⠐⠋⣹
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢀⠎⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⣰⠏⠙⢳⡄⠈⠀⠀⡿⡇⠀⠉⠳⠤⣀⡴⢻⠁
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠈⠲⣄⣀⣀⣀⣀⣬⣿⡖⠚⠁⠀⠀⠀⢿⣀⣀⡰⠁⡟⢦⣀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠦⡤⢬⣷⣀⠀⠀⠀⠀⠀⣼⡿⣵⠀⢹⠀⠈⠉⠓⠒⣞⠁
⠀⠀⠀⠀⢸⡇⠀⢀⡴⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡗⣾⠁⠈⠳⠦⣤⣄⣀⣤⠶⠋⠀⢸⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠈⣧⡴⠋⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⠳⣄⠀⡿
⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠙⣾⠁
⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠸⡆
⠀⢰⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡄⣀⣠⡤⢶⠖⠦⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢠⡞⢧⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡜⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡆
⡏⢡⠈⠳⣄⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡇
⢇⠈⢳⡀⠈⠳⣄⠈⠙⠲⠦⢤⣤⣀⠀⠀⠀⠀⢸⠇⠀⢸⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⣀⡴⢻⡄
⢸⢧⠀⠙⢦⡀⠈⠙⠒⠤⢄⣀⡀⠈⠉⠉⠉⠉⢻⠟⠓⠛⠓⠛⠳⡖⠒⠢⠴⠤⠤⠤⠒⠒⠋⠁⠀⣀⠤⠚⠁⠀⣸⢷
⠈⡎⠳⣄⠀⠙⠲⢤⣀⠀⠀⠀⠉⠉⠒⠢⠤⣤⠼⠧⣄⣀⣀⣀⣠⣃⣀⣀⣀⣀⣀⡠⠤⠴⠖⠒⠉⠁⠀⠀⠀⣠⠏⢸
⠀⠘⢦⡈⠳⣄⡀⠀⠈⠙⠲⠤⣄⣀⣀⣀⣀⡿⠀⠀⣽⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢀⡇
⠀⠀⠀⠙⠦⣄⡙⠳⠦⣄⣀⠀⠀⠈⠁⠈⠉⠀⠀⢀⣜⠀⠀⠓⠶⠤⠤⢤⣤⣤⣤⣤⣤⠤⠤⠴⠶⠚⠋⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢸⠙⠳⢤⣀⡉⠉⠙⠛⠃⠐⠶⠒⠛⠉⠉⠓⠲⢦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢖⡿⠃
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⠉⠙⡳⠶⠤⢤⣤⣀⣀⣀⣀⣀⡀⢀⣀⣀⣉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣥⠤⠖⠋
⢀⡴⠒⠒⠒⠚⢧⡄⣀⡀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢿⠉⠉⠉⠉⠉⠁⢀⡇
⡞⠀⠀⠀⠀⠀⠀⠃⠀⠈⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣸⣀⣀
⠘⢦⣀⡀⠀⠀⣀⣀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠙⠒⣶⠊⠉⠀⠈⠉⠳⣄
⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⣤⠷⣄⣀⣀⣀⣠⠴⠋
⣿⣿⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣠⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣯⣿⣛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⣿⣿⣿⣿⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⡿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⣿⣿⣿⣷⣾⣯⣽⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣿⣷⣽⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣽⣛⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣷⣯⣿⣻⣿⠿⠿⠿⠿⠿⢿⣿⣛⣂⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⢿⣿⣿⠿⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠙⠛⠀⠀⠀⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣀⣿⣏⠀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡈⢿⣿⣿⣶⣤⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣹⣷⣽⣿⣯⣴⢛⢀⡀⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣴⡾⠿⣿⣿⣗⠉⠉⠛⠛⠉⢩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣍⣯⠀⢀⢽⣿⡿⠦⠄⣁⣀⠀⠛⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⣤⡄⠽⢻⣿⡷⣾⠟⠑⢰⣼⣿⣛⣤⡶⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠓⠓⢓⡋⠰⢿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⣠⣄⠐⠏⣿⣿⡅⠀⠙⣿⣍⢁⣁⣈⣀⣤⢶⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠒⢿⡿⠿⢷⣶⣽⡳⢦⠾⠿⢏⣿⣿⣭⣻⠍⠁⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢂⣀⣩⣿⠀⠀⢰⣟⣿⣿⣿⣿⣷⣿⡛⠙⠻⠏⠈⣤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⠛⠿⠿⠲⣿⣭⣿⣿⣿⢿⣝⣿⣟⣶⡴⠾⣻⠿⢷⠶⡦
⠀⠀⠀⠀⠀⠀⠰⢽⣦⣶⣶⣾⣴⣶⡿⠟⢿⣿⢀⣟⣶⣿⣿⣿⠁⠀⠀⡸⣦⠴
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡤⡀⣉⡐⣤⣥⡼⣟⣯⣿⣏⣳⣮⣿⣷⣀⣈⣽⣧⣶⣯⣭⣀
⠀⠀⠀⠀⠙⠗⣶⣦⣄⣉⡉⢁⣭⣍⣛⣷⣿⣿⣿⣬⣋⠻⡟⣿⣯⣹⣿⡷⣶⣤⣬⣍⠙⠓
⠀⠀⠀⠀⠈⠀⠀⣈⠹⠙⢙⣿⢿⣿⠋⠉⠉⢿⣿⣷⣿⣟⣧⣸⣽⣿⠿⢿⣞⣂⣉⣉⡀
⠀⠀⠀⠀⢀⣠⣤⡤⠶⠚⠋⢀⣽⣿⠀⢠⢄⣹⣿⣿⣿⣿⣯⣀⣼⡁⣀⠀⣿⣿⠿⢿⣢⣦⣤⡀
⠀⠀⠀⠈⠉⠀⠀⠀⢀⣠⣶⣿⣍⣿⡿⣛⣿⠻⣿⡿⡏⣿⣏⠿⣼⡿⢶⠾⠻⡷⠄⡀⠈⠉⠉⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠙⠋⠉⠀⣉⣽⠿⠛⠁⣰⠟⠈⣷⡜⢿⣆⠀⠛⢷⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⢹⡧⠀⠈⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠊⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⣈⣏⣹⣋⣏⣫⣙⣏⣙⣉⣋⣟⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣭⣿⣿⣯⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⣱⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣻⢯⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣟⡬⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠌⡀
⠀⠀⠀⠀⠀⠀⢰⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣿⣿⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠲⠿⣿⠂
⠀⠀⠀⠀⠀⢠⠷⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⢻⡟⢚⠓⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡉⢖⣤⣛⡄
⠀⠀⠀⠀⠠⣌⡐⠂⠄⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣰⣶⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠽⣃⢤⠨⣟⣂
⠀⠀⠀⢀⠒⠀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⣿⣯⣿⣷⣾⠇⣀⠀⠀⠀⠀⠀⠀⠀⠲⢶⠺⠇⢘⠃⠙⡴⠖
⠀⠀⠀⡀⠑⠂⠄⠄⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢩⣟⣙⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠓⣎⣑⠋⡉⠐⠼⠗⣚
⠀⠀⠀⠃⠐⠀⠒⠒⠂⠀⠚⠁⠄⠀⠀⠀⠀⠀⢶⡖⣿⣟⡑⣿⣹⣆⣿⣏⣁⠄⠀⠀⠀⠀⢩⣎⡝⣿⠏⡽⡤⠘⢘⠥⡠⠄
⠀⠀⠁⠀⠀⠶⠀⡀⠁⠈⠉⠀⠀⠀⠀⠀⠀⣄⢀⢩⣽⣿⠿⢿⡿⣿⣿⣿⢃⡀⢀⠀⠀⠀⡑⡀⠀⡦⢰⣆⠀⢋⣴⣧
⠀⣌⠃⠄⠤⠀⠄⠀⠀⠂⠀⠁⠀⠡⠀⠀⠀⢀⠙⣫⣾⣯⣼⣿⣗⠛⣿⣟⡿⣟⠃⠀⢀⡺⢿⡓⢒⡂⣠⡄⢤⣤⡀⢩⣤⠀⡀
⠀⠀⠁⠀⠀⠀⠀⠂⠘⠐⠂⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣧⢼⣿⠿⣿⣾⣦⠾⠗⠛⠀⠀⠉⠛⠁⠁⠷⠟⠃⠈⠈⠑⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠓⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠒⠚⠛⠉⠉⠉⠉⠉⠛⠒⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡴⠋⠀⠀⠀⢀⡤⠤⡤⠤⠤⠤⡖⠲⢄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢧⣀⣀⡤⠖⢫⣠⠜⠡⠀⠀⢰⡙⠲⡚⠂⢰⠇⢠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⠰⣋⣀⡀⢐⣇⣰⡿⠇⠀⠿⣷⡦⣇⣀⣈⣄⡨⠇⢸
⠀⠀⠀⢀⡴⠒⠒⠊⠉⠒⡇⠀⣇⠀⠀⠀⠉⠉⠀⢻⡀⠀⠀⣀⠄⠀⠀⠀⠈⠀⠀⢀⠞
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⡇⠀⡸⢧⡀⠀⠀⠀⢀⣠⡼⠋⠉⣓⣤⣄⣀⣀⣠⠤⠖⣹⠀⠀⢀⣐⠄⠠⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠤⠴⠃⠀⢳⠄⠉⠛⠛⠉⠉⠱⣝⢿⡿⢟⠝⠀⠀⠀⠀⠀⠀⡟⠒⠂⠉⣰⡱⠚
⠀⠀⢧⠀⠀⠀⠀⠀⠑⠤⣄⣀⣨⣦⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⣠⠞⠁
⠀⠀⠀⠑⢦⡀⠀⡀⠀⠀⣠⣫⠋⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠠⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⣰⠃⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠃⢀⡴⠀⠀⠀⠈⠳⠤⠤⣄⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣰⠂⢧⡑⠴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡏⠉⠋⠁⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠈⠓⠙⠟⠉⠈⢧⠀⠀⠀⠀⠀⠀⠠⣀⠠⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⠔⠒⣛⣒⡲⣄⠀⣀⣀⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣶⣤⣤⣤⡇⢧⣤⣶⣶⡧⢸⣿⡿⠿⠟⠋⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣉⣉⠉⠉⠉⠙⠮⢭⣤⡤⠖⠁⠀⢀⣀⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣉⡩⠗⠒⠒⠚⠁⠙⠒⠛⠫⢍⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏

⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡍⠠⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⠠⠄⠀⠀⠀⠀⠀⠰⢽⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠠⠫⢝⢔⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠎⠃⠂⠀⠀⠀⢤⣭⠦
⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⢦⣤⠶⠂⠀⠀⠀⠈
⠀⠀⣾⣿⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣤⡶⠞⠂
⠀⢸⣿⣿⣇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⣦⣤⣤⣤⡶⠞
⢀⣿⣿⣿⣿⡄⠀⠐⠐⠀⠀⠐⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠂
⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠈⠀⠀⢀⢁
⠀⠀⠀⠀⠀⠀⠀⣀⣐⡂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣈⣛⣁⠀⠀⠀⠠⢈⠠⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣙⣟⣁⡤⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⢁⢁⠁⠉⠄⠄⠀⠀⠀⠀⠐⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠐⠐⠐⠈⠀⠀⠀⠀⢈⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢨⠀⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠲⣆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠐⣮⡀⠇⢀⣠⣤⡶⢄⡲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⡀⡎⠐⢠⣼⠉⡛⠋⠀⠀⠀⠘⠉⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠁⠁⠁⠀⡈⠛⠛⢁⠀⠉⠉⠛⢶⡄⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠰⣿⣤⡀⠀⣀⣾⣇⣀⣀⣸⠀⠁⠒⠤⠈⠙⣄⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠟⠃⡄⡄⠀⡦⠘⠹⢷⠊⠉⡇⠻⢿⣿⣧⣄⡀⠀⠀⠀⠀⣸⠂⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣷⡀⠀⢣⢼⡅⠀⡀⢰⡀⠀⢻⠀⠀⢸⣿⣿⢹⣿⣶⡞⠋⠁⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⠀⢸⠜⣀⣼⢷⠀⠁⠀⢸⠀⠀⡾⢿⣿⣿⣶⣭⣀⣠⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠘⠧⠈⣢⡼⠁⠜⠉⠀⠀⢆⣀⢤⣤⣤⣴⣾⣿⡿⣯⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠻⡿⣿⢿⢿⠟⠛⠉⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡄⠀⣄⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣀⠀⠀⢀⣀⠀⠀⠠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⠙⢿⣿⢠⠀⠀⠠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠞⠘⠈⠃⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣾⠁⠀⠀⠈⣄⣠⣤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠄⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣹⣿⡏⠀⠀⠀⠀⠀⡿⣏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠙⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⡀⠀⣀⠤⠒⠉⠉⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠚⠉⠁⠀⠀⠀⠠⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣀⣀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡴⠖⠉⠘⠃⠈⠁⠀⠐⢇
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡤⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢀⣔⢀⣤⠋⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡤⠂⠠⡄⠀⠀⠹⡄
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⣾⠑⠉⠀⠀⠀⠀⠀⡠⠀⠀⠀⣣⡤⠔⠒⠠⣄⣈⠳⡋⠚⠃
⠀⠀⠀⢠⠻⡄⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣠⡠⠔⡛⢈⡀⠀⠀⢀⠄⠀⠌⣿⠃
⠀⠀⢠⠃⠀⠹⡄⠀⠀⠀⠀⡆⠀⠀⠀⢀⡠⠔⠉⠀⢁⠎⠀⠀⠈⡀⠀⠅⢀⠀⠀⡻⣄⣀⣀⡀
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠔⠉⡆⠀⠀⠀⠈⠐⢉⢢⠀⡇⢀⢺⣠⣔⢀⣧⣄⡀⠀⠈⠳⣰⡄
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠀⠀⢠⣇⠀⠀⠀⣬⣴⣾⣷⣲⠦⣼⠾⠛⠛⢿⡀⠀⡵⠀⠀⠀⢹⠊⢏⠓⠦⣄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠔⠉⠀⠀⠀⠀⠀⠀⢺⡀⠀⣠⠇⠀⠀⠀⠀⡟⠛⠑⠒⠤⠔⠉⠀⢸⠀⠀⠨⣦
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡞⠀⠀⠀⠀⢀⡤⠒⠀⠀⠙⠯⠄⠀⣀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠈⢧
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠘
⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠢⡑⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢳⠤⠀⢠⠤⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⢀⣄⣠⣤⡴⡊⢁⠹⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢣⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⢀⠀⢨⣁⡄⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠃
⠀⣼⠁⠀⠀⠈⠉⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⡜⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⣇⠀⠀⠀⠀⠀⠀⠀⢸⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠘⢦⠀⠀⠀⢀⣀⡠⠛⠀⠑⢄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠉⠉⠁⠈⠀⠀⠀⠀⠀⠈⠉⠚⢆⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠺⠛⠓⢄⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⠤⢄⣀⣀⣀⠀⣦