Judicature Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣤⡄⠀⠤⠤⠯⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣄⠀⠤⠤⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⠃⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠔⢚⠁⢈⠓⠛⠩⡤⡤⠈⠉⠁⠀⠀⠀⠀⠀⢀⠔⢚⠁⠀⠁⠘⠩⠤⣄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠆⠘⢦⡙⢤⣤⣴⣇⡿⠉⠁⠀⠀⠀⠀⠀⠀⠇⠀⠈⢦⡙⢤⣤⡴⣁⡿⠉⠁
⠀⠀⠀⠀⠀⠀⡌⢼⣄⠀⢀⠻⢦⣙⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⡜⢸⣄⠀⢀⡙⢦⣈⠉⠈⠁
⠀⠀⠀⠀⠀⢠⢁⣿⣿⣶⣌⡛⢦⣝⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣶⣿⣷⣄⡙⠦⣍⠃
⠀⠀⠀⠀⠀⠈⢹⠋⣿⣿⡿⢿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⢹⠋⣿⣽⡿⢿⣿⣾⣽⡆
⠀⠀⠀⠀⠀⠤⠈⢰⣿⡇⣰⡿⢸⢯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠘⢰⣿⡣⣠⣿⢸⢃⣿⡇
⠀⠀⠀⠀⠀⢠⣴⣸⣿⣿⣿⠇⢸⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣸⣿⣿⣿⠇⢸⡟⣿⡇
⠀⠀⠀⠀⠈⠉⢻⢸⣿⡿⡟⠀⠸⠀⡏⡇⠀⠀⠀⠀⠀⠀⠀⠉⢻⢸⣿⡿⡟⠀⠸⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⢃⣥⣴⠃⠀⡗⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⢁⣡⡴⠃⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣼⡟⠈⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣼⡿⠀⠀⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⣤⣴⣾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⠃⣀⣤⣾⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣾⡏⢰⣿⣿⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣏⢠⡿⣿⠉⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⡇⡿⡆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡾⣿⡇⡿⠀⢸⢻
⠀⠀⠀⠀⠀⠀⠀⣽⣼⣀⣯⣽⣷⣓⣾⣽⠄⠀⠀⠀⠀⠀⠀⠀⠀⢼⣸⣀⣣⣡⣷⣒⣼⣽⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⣿⣦⣄⠀⠀⠀⠀⠀⢶⣾⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠈⠻⣿⣷⣶⣤⣄⣀⡈⣻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡤⠶⠿⠿⠶⠦⣄⣠⠉⠛⠿⡿⣻⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡏⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡆⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣽⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢳⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣫⣿⣿⣿⣿⣿⣿⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢹⣿⣿⣿⣿⡸⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣿⣿⣿⡟⠁⢿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⠁⠀⢨⣿⣿⡀⠀⠈⠙⠻⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠋⠁⢀⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠉⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢲⣄⠙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠈⢀⠰⡐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠋⠀⠁⢀⠀⠭⠠⠅⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠀⠘⢦⡀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⢹⣦⣀⡀⣀⣀⣤⡸⡂⣠⡼⢿⡁⢩⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣠⠞⢿⠱⠁⠈⠀⠁⢈⣼⣤⠜⢁⡴⠀⢤⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠠⣄⠀⠀⠀⢀⣀⠸⠟⢁⡴⢋⡴⡶⠲⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⣠⠟⢁⡞⠀⠃⠠⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⡰⠋⣠⠜⡇⢠⠀⢰⠀⠈⡇
⠀⠀⠀⠉⠙⠋⠉⠉⠹⠋⠉⠉⠛⠙⠀⠀⠈⢠⠈⠃⠀⠀⡰⢁⠴⠉⡆⣿⠀⠀⠰⡤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠂⠀⡄⢠⡏⡀⠀⠀⡧⡞
⠀⠀⢀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢴⠊⢑⠀⠀⠀⣸⠁⢈⠀⠀⡰⠃
⠀⠀⠈⠙⠀⠠⠚⠁⠀⠀⠉⠀⠀⠐⠉⠀⠀⠰⠊⠀⢸⠀⢸⠀⠀⠀⡟⠀⠀⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠀⠀⢠⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⠇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⡀⠀⣠⠔⠂⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣴⣄⣤⣥⣆⣬⣭⣀⣭⣬⣩⣡⣬⢤⡀⣤⡶⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡴⣫⠭⡉⠻⢏⣿⣜⠃⢻⣏⣰⡂⠘⣹⣿⡽⠛⡻⠭⣛⣆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢇⡻⠖⣡⠀⣿⢋⣉⣉⣻⢿⡻⠿⡿⣏⠸⠅⠐⣇⠷⢋⡜⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠉⠉⢁⣼⣥⣦⠴⢤⠶⠄⠴⠦⠴⠦⣤⣼⣦⠈⠉⠉⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠆⠀⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣟⣙⣩⣿⣧⣭⣹⣏⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣭⣭⣿⣯⣭⣭⣭⣯⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⣛⠃⠘⣛⣛⣛⠃⠘⣛⣛⣛⠃⠘⣛⣛⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠀⠀⠿⠿⠿⠀⠀⠿⠿⠿⠀⠀⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣛⣛⣛⣃⣘⣛⣛⣛⣃⣘⣛⣛⣛⣃⣘⣛⣛⣛⣃⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠀⢠⠒⡄⡆⢰⢠⠂⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠈⠒⠁⠑⠚⠈⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠖⠛⠉⠀⠀⠀⠉⠙⠓⠶⢾⣿⡷⠶⠚⠋⠁⠀⠀⠀⠉⠛⠲⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠿⣿⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⠀⣾⣶⣷⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⣛⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⠀⠀⠀⠀⠀⢀⠕⠀⡇⠀⠀⣿⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣶⣿⠀⠀⠀⠀⡀⠄⢠⠀⡇⠀⠀⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⢀⡀⠀⢿⢒⡏⠀⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠘⠀⡇⠀⢘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣴⣤⣤⣤⡦⠀⠀⣼⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⡇⠀⠀⢤⣤⣤⣴⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿⡏⠉⠛⠓⠶⠚⠛⠋⢻⣿⣿⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠠⣤⡤⠀⠠⣤⠄⠀⣭⠽⢦⡄⢠⡤⠤⣤⠤⣤⠈⢭⡭⠀⢀⣤⠴⢦⡄⠀⢤⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⣿⠀⠀⠿⣮⣁⠡⣬⣵⡇⣿⢰⣬⣾⢸⡇⠀⣾⠃⠀⠈⠁⠀⢸⣇⣠⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⣇⡀⢀⡿⠀⢠⣆⠲⢹⡗⣹⣿⢇⣿⠸⣿⡷⢸⡇⠀⢻⣧⣀⣀⡀⠀⣸⣇⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢈⣉⣉⣀⣀⣀⣉⣉⣭⣾⣿⣷⣬⣭⣥⣾⣧⣍⣉⣀⣀⣈⣉⣉⡀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⣥⣥⣵⣾⡼⢽⣦⣿⣼⣯⣥⠽⣧⣧⣧⣽⣽⣼⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠀⠀⠠⠀⣿⠀⠀⠐⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣬⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠘⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠻⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣦⣴⠾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡿⠿⢿⡋⣹⣿⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⠛⠉⠁⠀⠀⠀⢀⣿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠀⠀⠀⠀⠀⢀⠀⠀⣼⡟⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⣿⠀⢰⣿⠃⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠈⣶⠀⠀⣼⣿⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣦⣤⣤⡄⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠹⡄⢰⣿⡿⢰⣿⠋⠉⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢰⣿⣦⣤⣤⣤⣴⣿⣿⡟⣡⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⢋⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣤⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⠋⢠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣧⡀⢀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⠃⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠉⠉⠙⢻⣿⣿⣿⣷⣶⡤⠤⣤⣤⣤⣤⣴⣶⣿⣿⠛⠿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠈⠉⠉⠙⢻⣿⣿⣿⡄⠀⠈⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣠⣾⣿⣿⣿⠃⠀⠀⠀⠀⣴⣶⣶⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣉⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢡⣿⣿⠀⠉⠻⣿⣿⣿⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢠⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⣿⣿⠃⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠀⢸⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⠻⣿⠉⣿⣿⣿⣷⠘⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠁⠀⢻⣿⢹⣿⡆⠹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢈⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠸⣿⠀⣿⣧⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣿⠀⢸⣿⡀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⣿⠀⣸⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠛⠀⢸⣿⡇⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡦⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⣿⣇⠀⠀⣼⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣤⣾⡿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⣀⣠⣤⣶⣆⠀⢤⣾⣿⣿⠿⢋⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠸⣿⣿⣿⠟⠋⠀⠀⣠⣠⣤⣶⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢹⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠘⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢸⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⣿⣿⠀⠀⠀⢰⣆⠀⠀⢸⣿⣇⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⢠⣿⡏⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠘⣿⣿⡆⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⣸⣿⠃⠀⠀⢰⣿⣿⠀⠀⠀⣿⣿⣷⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⡿⠀⠀⠀⢀⣿⡏⠀⠀⠀⢸⣿⡇⠀⠀⠀⢿⣿⣿⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⠇⠀⠀⠀⣼⣿⠃⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢸⣿⣿⠇⠀⠀⠀⣾⡿⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⣿⣿⣿⠀⠀⠀⢠⣿⠇⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢠⣿⣿⣿⠀⠀⠀⣾⣿⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⢰⣿⡇⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⢠⣿⣿⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡏⠀⣰⣿⣿⡟⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⢀⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣰⣿⣿⣿⡇⠀⣰⣿⣿⣿⣿⣿⡀⠀⠀⣸⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠁
⠹⣿⣿⣿⣿⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠿⣿⣿⣿⣿⡿⠿⠛⠛⠻⠿⠟⠋⠛⠛⠿⠟
⣤⣤⣽⣿⣿⣿⣥⣤⣤⣤⣤⣬⣽⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠿⠿⠿⢿⡿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⣿⣧⣤⣧⣤⣤⣥⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⡶⠿⠿⠿⠿⠿⠿⠿⢷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠋⠉⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠈⠉⠛⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠈⠻⣷⣄
⠀⠀⠀⠀⠀⣠⣿⠏⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⢠⣶⣦⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠈⢿⣧⡀
⠀⠀⠀⠀⣼⡿⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠹⣿⡄
⠀⠀⢀⣾⡟⠀⣰⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣆
⠀⠀⣼⡟⠀⣴⣿⣿⣿⣿⣿⠓⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠒⢻⣿⣿⣿⣿⣷⡀⠘⣿⡄
⠀⣼⡿⠀⣰⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⣶⣶⣶⠂⠀⠀⠐⣶⣶⣶⠀⠀⠀⢰⣶⣶⡖⠀⠀⠀⢲⣶⣶⡆⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣷⠀⠸⣿⡄
⢠⣿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠁⠀⠀⢹⣿⣿⣿⠀⠀⠀⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢻⣧
⢸⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿
⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿
⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿
⢿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿
⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠈⣿⡆⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⡏
⠀⠹⣿⡀⠸⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⡟⠀⣸⡿⠁
⠀⠀⢹⣧⠀⠹⣿⣿⣿⣿⠿⠭⠅⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠽⢿⣿⣿⣿⡿⠁⣰⣿⠃
⠀⠀⠀⢻⣷⡀⠙⣿⣿⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣹⣿⡟⠁⣰⣿⠃
⠀⠀⠀⠀⠹⣷⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⡀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⣀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣄⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣀⣀⣤⣴⣾⠿⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠛⠛⢿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠠⠴⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠦⠄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢻⣟⣛⣛⡟⠀⠀⠀⠘⣿⣟⣻⣿⠂⠀⠀⠘⣿⣟⣻⣿⠃⠀⠀⠀⣻⣿⣻⣿⠃⠀⠀⠀⢛⣛⣛⣻⡟
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢘⣛⣛⣛⣟⣀⣀⣀⣘⣟⣛⣛⣟⣀⣀⣀⣐⣟⣛⣛⣻⣀⣀⣀⣀⣛⣛⣛⣛⣃⣀⣀⣀⣻⣟⣛⣛⡇
⠀⠀⠀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀
⠀⢠⡤⠯⡿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⡿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠯⢿⠿⠿⠿⠿⠿⠿⠿⠿⢽⠤⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠴⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠼⠃
⠀⠀⠀⠀⠀⠀⡿⠛⠛⢿⠀⠀⠙⠛⠛⠛⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⡧⠀⠈⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⢹⡦⠰⡏⢀⣀⣀⠀⠀⠨⣈⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄⢼⠁⢀⣀⡀⠀⠀⢀⣀⣀
⠀⠀⠀⢀⡤⠶⡛⠄⠀⡙⠒⠻⠥⢤⢄⠄⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⢚⠃⠀⠈⠑⠚⠫⠤⡤⢠⠘⠈⠉
⠀⠀⠀⡏⢧⠀⢳⡀⠲⡁⠀⢀⣠⢏⢺⡖⠒⠃⠀⠀⠀⠀⠀⠀⠀⢰⠁⡀⠈⢆⠐⢎⠀⠀⣀⡴⠁⣿⠖⠒
⠀⠀⡰⢸⡌⠄⠀⠙⣦⣈⠛⡿⠿⠿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣠⠀⠀⠈⠳⣄⡉⢛⠿⠇⠧⠋
⠀⠀⡇⣚⣷⣌⢀⢷⣌⡻⢷⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⣧⡀⠐⢦⣈⠻⣦⣍
⠀⣸⢰⣾⣾⣻⣧⣅⣩⡻⢦⣍⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⣷⣿⣿⣦⣀⢝⠷⣬⡈
⠀⠹⢹⠋⣿⣿⣿⣿⢿⣿⣷⣯⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡿⠉⣿⣫⣾⡿⢿⣿⣶⣝⣷⡀
⠀⡀⡞⢀⣿⡿⡫⢁⣾⠏⣿⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡇⢰⣿⠟⠁⣰⣿⢹⡏⣻⣿
⠀⠃⡇⢸⣿⡿⣡⣿⡿⠀⣧⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⡀⣿⣯⢎⣴⣿⠇⢸⣱⣿⣿
⣀⣵⡿⠾⣿⣿⣿⣿⠃⠀⡿⢹⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠷⢿⣿⣿⣿⡟⠀⢸⡏⢹⣿⡄
⠀⠀⢷⢀⣿⣿⣟⡏⠀⢸⠃⠀⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⣿⣿⢹⠃⠀⠸⠀⢸⡟⡇
⠀⠀⠘⣾⣿⡿⢹⣁⣤⡎⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣿⠏⠌⣀⣴⠃⠀⠘⡇⡇
⠀⠀⠀⢹⣿⠁⣿⡿⠁⠃⠀⠀⢹⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡏⢀⣾⡏⠀⠀⠀⠀⠇⣇
⠀⠀⠀⢸⣯⣾⣿⡇⠀⠀⢀⣠⣾⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⠀⠀⠀⣀⣴⣧⣿
⠀⠀⠀⢸⡿⣿⡿⠀⣶⣾⣿⡟⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢻⣿⡇⢠⣶⣾⣿⠏⣿⣿
⠀⠀⠀⢸⣷⣿⡇⣸⣿⣿⣇⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⢿⠀⣾⡿⣿⠀⠀⣿⡿
⠀⠀⠀⢸⣿⣼⣿⣿⡇⣿⢣⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣼⣿⣿⢰⡿⠀⠀⡏⡇
⠀⠀⠀⢸⢛⡟⠸⡟⡇⣿⣸⣇⣿⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢹⠁⢻⠹⢸⣇⣴⣀⡇⠟
⣤⣤⣤⣽⣿⣷⣶⣷⣿⣿⣿⣾⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣶⣶⣷⣾⣿⣷⣼⣿⣭⣥⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠑⡖
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠠⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠒⠒⠒⠒⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠃⢀⣀⡤⠴
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠠⠤⠒⠉⠀⠀⠀⡀⠀⠀⠀⣾⠀⠀⢀⡤⠴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠰⡁⠀⠀⢠⡇⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⡐⠈⠁⠢⠅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢡⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠐⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣥⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠴⠤⢤⡀⠀⢧⡀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠱⡌⢆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⠀⣿⢸⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠉⣀⠴⠃⠘⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠰⡄⠀⠈⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠘⢆⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣀⠐⠀⠀⠀⠀⡆⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣄⠀⠈⢧⠀⠀⣠⠞⠁⠈⠉⠒⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⡐⠀⠈⠉⠛⠛⠛⠋⠀⠀⠀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⡿⠿⡟⠛⠿⡿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣛⣛⣛⣿⣿⣛⣿⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡔⠋⣻⠁⠀⠀⠀⠁⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡆⠀⠀⠀⣠⣼⣿⣶⣶⣆⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⢹⣿⣿⣿⡛⣿⡌⢿⡄⠙⣦⡀⠀⠀⠀⠀⠀⠸⣇⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⠀⠀⠀⠀⠈⠀⠙⢷⡏⢹⣿⣤⡙⢦⣈⠳⡄⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠂⠄⠀⠀⠀⠀⠀⠀⠻⣦⣿⣿⡟⢦⣝⣷⣌⠤⡀⠀⡀⠁⠀⢀⣤⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢿⣿⠻⣦⡈⠻⣿⡄⠁⢀⡄⠀⢰⠉⢠⣄⣠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠈⢿⡄⠈⠻⡄⢠⠉⠙⠁⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠘⣿⠀⠀⠀⠀⠓⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⢀⣀⣴⣾⠋⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡆⢸⡇⠀⠀⠀⢀⠀⠰⠀⠄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⢿⣿⣿⣶⣿⣷⣶⣶⣦⣤⣄⣤⣿⣿⣿⣿⠁⡎⢳⡀⠀⢀⣤⡶⠒⡄⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠈⠀⠀⠀⠀⠈⠻⣿⡿⠿⠿⠟⠛⠛⠛⠉⠉⠁⠈⣿⣿⣦⠙⢦⠁⠀⣿⣉⣄⣠⣅⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⡯⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢸⣿⡈⠻⣦⡀⠀⠘⠋⠁⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⢰⣿⣿⣷⣤⣧⡤⣿⣤⣿⣷⡀⠀⠙⣶⡀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣾⠏⢀⠀⢠⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠀⡈⠁⣴⠃⠈⠁⠘⠃⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⣼⣇⣠⣼⣿⣿⠛⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⣿⣦⣅⣠⣾⡿⠛⠶⠶⠿⠶⠶⠶⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠁⠀⠻⡄⠐⢶⣦⡀⠀⣠⣿⠀⠀⠀⢸⣿⠛⠟⠉⠉⠀⠀⠀⠀⠀⠀⢂⡄
⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢳⡀⠀⠉⠻⣿⣿⡏⠀⠀⠀⠀⠉⠙⠿⠶⠶⠶⠶⠶⠶⠶⠖⠈⠁
⠀⠀⠀⠀⠀⠀⠳⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠙⠢⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣦⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣀⠀⠀⠾⠉⠀⢀⡤⠚⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠸⢧⡤⠄⢰⣶⠛⠛⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⠀⠸⡇⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠸⡆⠀⠀⠀⠀⢻⣿⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠿⡀⠀⠀⠀⢸⣿⣧⠈⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢣⡀⠀⠀⣾⡟⣿⣇⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠘⢣⠀⠀⣿⡇⠈⠻⣧⡀⢻⣇⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠊⣦⠀⣿⡇⠀⠀⠈⠳⣄⠻⣦⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢡⣶⡻⣷⡀⠀⠀⠀⠈⠳⡄⠁⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠼⢃⠘⠻⣷⣤⣀⡀⠀⠸⠆⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠟⣀⠀⠀⠉⠛⠿⣿⣦⡄⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢋⡄⢀⡀⠂⣤⡈⠙⣿⣆⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢱⡄⠘⠳⢤⡙⠲⢈⢿⣆⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⠀⠀⣤⡁⠀⠀⠈⠻⣦⠀⠀⠀⠙⠷⢶⠶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠐⡀⠀⠻⣦⡀⠀⠀⠈⠁⠈⠳⣄⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⢀⣈⡛⠦⠀⠀⠀⠀⠀⠘⢻⣆⡀⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⢷⣤⣀⠀⠓⠀⠠⣿⡉⠙⠛⠒⠚⠃⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠀⠘⠳⣤⡀⠀⠀⠀⣀⣤⣶⣷⣦⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠈⠛⠻⠿⠿⠛⠋⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⣤⣿⣿⠟⠀⠈⢹⣶⣌⠉⠀⠈⢳⣾⡃⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣟⠀⠀⠀⡟⠙⠓⠀⠀⠈⠁⠈⠃⠀⠀⠛⠛⠧⠀⠀⠈⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠀⠉⠙⡿⠛⠓⠀⠀⠀⣀⣠⣤⣤⣤⣶⣦⣤⣤⣤⣄⣀⠀⠀⠀⠚⠛⣿⠉⠁⢠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣄⠀⠀⠁⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⢀⣤⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠈⠉⠹⠏⠀⠀⣠⣶⣿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠉⠉⠉⠙⠛⠻⢿⣿⣩⣿⣿⣷⣆⡀⠀⠀⢻⠋⠉
⠀⠀⠀⢤⣴⣯⣀⡀⠀⠀⣠⣾⣿⣿⣿⡿⠛⠉⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠻⣿⡿⢿⣿⣦⡀⠀⠀⣀⣠⣷⣤⠄
⠀⠀⠀⣸⠿⣿⠉⠀⢀⣴⣿⣿⣿⡿⠋⠀⠀⢀⣤⠶⠛⠉⠉⠁⢀⠀⠉⠉⠙⠳⠦⣄⠀⠀⠈⠓⣿⣿⣿⣿⣄⠀⠈⢹⡿⠿⡄
⠀⠀⢀⠀⠀⠈⠀⢀⣾⣿⣿⣿⠏⠀⠀⢀⡴⠋⠀⢠⣦⠀⠀⠘⣿⡟⠀⠀⢠⣶⠄⠈⠻⣦⡀⠀⠈⢻⣿⣿⣿⣆⠀⠈⠀⠀⢀⡀
⠠⣴⣿⣧⠶⠀⠀⣾⣿⣿⣿⠃⠀⠀⣰⣏⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣈⣷⡄⠀⠀⢻⣿⣿⣿⡆⠀⠐⢶⣾⣷⡤
⠀⠘⠛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠙⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠙⠛⠛
⠀⠀⠀⣿⣿⠀⠀⢰⣶⠂⠀⠀⠀⣶⠀⢀⣴⠒⠒⠒⠀⠒⠐⠂⣀⡖⠒⠒⠀⢰⣶⡆⠀⢀⣤⡶⠂⠀⠒⠶⠀⠀⣶⣶⠒⠒⠒
⠀⠀⠀⣾⣿⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠛⠿⣶⣦⡄⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠠⠄
⠀⠀⠀⣾⣿⠀⠀⠘⠿⢦⣀⣀⠴⠟⠀⠸⢦⣄⣀⠿⠋⠀⠀⠨⠿⠇⠀⠀⠀⠸⠿⠇⠀⠘⠿⠦⣄⣀⡀⠴⠃⠀⠿⠿⠀⠀⠤⠂
⠀⠀⠰⠟⠁
⢘⣻⣷⣾⡛⠛⠛⣿⣿⣿⣿⠛⠛⠛⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣖⣿⣿⣿⣿⡟⠛⠛⢛⣿⣿⣿⡿⠛⠛⣻⣿⣶⣋
⠀⠉⢿⠋⠉⠀⠀⠹⣿⣿⣿⣧⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢀⣾⣿⣿⣿⠃⠀⠈⠉⢻⡇⠉
⠀⠀⠀⣀⡀⣠⠀⠀⠹⣿⣿⣿⣷⡀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣠⣿⣿⣿⡿⠃⠀⢠⣄⣀⡀
⠀⠀⠀⣨⣿⣿⡶⠄⠀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⡿⠿⠟⠛⠉⠀⠀⢀⣠⣾⣿⣿⣿⠟⠁⠀⠤⣾⣿⣿⡀
⠀⠀⠀⠉⠈⠿⠀⠀⢀⡀⠀⠙⢿⣿⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡿⠋⠀⣀⠀⠀⠈⠏⠈⠁
⠀⠀⠀⠀⠀⠀⠐⢶⣾⣧⣤⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣴⣶⣾⣿⣿⣿⢻⣿⡿⠟⠁⠀⠠⣤⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠾⠛⢿⠀⠀⠀⡄⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣏⣽⣿⣿⣿⠿⠟⠛⠁⠀⠀⣄⠀⠀⢨⡿⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⡶⠃⠀⠀⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⣀⠀⠀⠲⣾⣿⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠛⠀⠀⢠⣿⣶⠏⠀⠀⢲⣤⣴⠂⠀⠈⢳⣾⣯⣄⠀⠐⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠓⠀⠠⠺⢟⡉⠲⠀⠐⠛⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠡⠒⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣭⡗⠄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⡟⠁
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣤⣤⡀
⢸⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠂⠘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠠⠙⣎⠐⠒⠠⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣟⣯⢻⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⣿⡏⠀⠀⣀⣀⠠⠔⠯⠂⠀⠀⢀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⡿⣿⣿⣯⣾⣿⣿⣿⣿⡆
⢸⣿⣿⣿⣿⣿⣇⠀⠀⢀⣠⣄⠴⢆⣀⡀⠀⣄⠀⠙⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢟⣿⣛⡿⠿⣿⣟⢿⣿⣆
⢸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⡎⠀⠀⠀⣿⣧⡀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣺⣿⣿⣿⠀⠹⣿⡆
⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⠀⢆⡴⠀⠀⣿⣿⣷⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣭⣿⣾⣿⣿⣿⠀⠀⢹⣿⡄
⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡟⠀⠀⠀⣿⣏⣻⣤⣦⡟⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⠀⠶⠚⠿⠃⠠
⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢻⠀⠀⠀⣿⣿⣿⣿⣿⡟⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣿⠸⣿⣿⣿⠀⠚
⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⠀⠀⠀⣯⣿⡇⣿⣿⣧⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠙⣿⡇⣿⢀⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⡆⢀⠀⠀⠀⠀⠀⠀⢸⣥⣌⣻⣬⣏⣙⣉⣡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⣿⠃⣿⢸⡿⠘⠁⠒⠀⠀⠀⠀⠀⠴⠖
⢸⣿⣿⣿⣿⣿⣿⣿⡇⠘⠃⢰⡀⣽⣀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣿⠀⡿⠀⠥⢼⡄
⢸⣿⣿⣿⣿⣿⣿⣿⡇⢀⠆⣯⠰⡻⣯⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⢺⣿⣄⣾⣿⠇
⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠂⣉⡆⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠏⣻⣿⣿⣿⠇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⢥⡿⢿⣿⣯⡄
⢸⣿⣿⣿⣿⣿⣿⡿⠏⠀⣀⡀⠀⠀⣀⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠯⢿⣿⡿⢻⣿⣦⡀
⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠉⠉⠁⠈⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣶⣿⣷⣿⣾⣾⣿⣇
⢸⣿⣿⣿⣿⣿⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡁⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡿⠿⢿⣿⢿⠿⠿⠿⢿⢷⣿
⢸⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣒⣀⣒⣒⣒⣒⣐⣂⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠭⠽⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠀⠀⠀⠄⠄⠀⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠆⠀⠀⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠶⣾⠃⢿⡶⠶⣾⠇⠸⡷⠶⢶⡿⠘⣷⠶⠶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⢸⡇⠀⢸⠀⠀⡇⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⢸⡇⠀⢸⠀⠀⡇⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⢸⡇⠀⢸⠀⠀⡇⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⢸⡇⠀⢸⠀⠀⡇⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣤⣿⡀⣸⣧⣤⣼⡄⢠⣧⣤⣼⣇⢀⣿⣤⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠤⠤⠾⠧⠿⠤⠤⠼⠷⠾⠧⠤⠤⠿⠼⠷⠤⠤⠿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣴⠓⡆⡆⢰⢰⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈⠊⠁⠙⠈⠈⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⠉⠀⠀⠙⠛⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠁⠀⣶⠀⠀⣶⣄⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣄⠀⠙⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⡀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠛⠻⢿⠀⠀⠿⠀⢀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⣄⣀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣄⣀⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢐⡆⠰⢆⢲⣖⠊⠴⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠐⠀⢨⡏⠂⠀⠐⠀⢹⣨⡷⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⡀⠐⢀⠀⢁⠀⡈⠀⢀⡱⠓⠁⠁⠪⢆
⠀⠀⠀⠀⠀⡠⠔⠒⠒⠂⠄⣘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡐⣥⣼⣷⢨⢬⡣⠸⣇⠀⠂⠀⢀⡕⢣
⠀⠀⠀⢠⠊⡠⠂⠉⡀⠀⠀⠈⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢏⠌⠸⠺⡼⠶⠚⢸⣦⡇⡠⡪⠊⠉⠈
⠀⠀⠀⡆⠐⠀⠀⢏⣈⠂⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠟⠛⠚⠛⠚⠢⣮⢏⢴⠎
⠀⠀⠀⢃⠘⡀⠀⠢⡤⠃⢀⠄⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠁⡠⠊⠉⠉⠉⠢⡈⢻⠁
⠀⠀⠀⠈⠢⡈⠒⠄⠀⠔⢉⢠⠁⠉⠉⠉⡩⠭⠭⠭⠭⡉⠉⢉⢟⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠑⢄
⠀⠀⢰⠉⠅⠭⠩⠩⠉⠉⠉⠉⠉⢢⠤⠤⠭⠬⠭⠬⠭⠥⠤⡎⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠊⠛⠻⠛⠛⠋⠉⠉⠉⠉⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢆
⠀⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠒⠒⠂⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠂⠛⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣽⣉⣉⣋⡦⢰⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠋⠑⠛⠀⢸⣀⡠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢀⡠⠄⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⡇⠀⠠⠔⠁⠀⠰⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣐⣀⣀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠥⠚⢧⠀⢀⠗⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠂⠈⠓⡴⠡⠤⠀⠓⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠠⠀⡀⠀⡀⡀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⣄⢋⣰⠈⢢⠠⠤⠤⠤⠼⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢀⠨⠈⠂⠀⠀⡀⡀⠀⠈⠁⠆⡈⢸⣀⡀⠀⠀⠀⠀⡎⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡘⠂⠀⠀⠀⠀⠈⠧⡟⢱⠒⡦⢄⡡⠂⠀⠀⠉⠉⠁⠐⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣧⡠⠊⠀⠀⢀
⠀⠀⠀⠀⠀⢀⢤⠖⡫⠉⠉⡐⠠⢍⡁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⡠⠊
⠀⠀⠀⡠⠚⡱⠃⠀⠀⠀⡠⠃⢠⢄⡌⢝⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠇⠀⢀⣠⠴⡁⠑⡄⢀⡠⠖⠉
⠀⠀⡜⠀⠘⠂⠤⢀⣠⠌⠀⣐⠣⠆⣠⠋⠲⡑⢨⠃⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⢇⠀⠘⣄⠜⠉
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠉⠒⠠⢉⡒⠦⢌⠔⢀⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡆⠀⠀⠀⠑⢄⡇⠀⠀⠃
⠀⠙⠒⠠⠤⠤⠤⠀⠀⠀⣀⣀⡀⠈⠑⠢⣤⠎⠀⠀⠰⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠢⠄⣀⢤⣀⠀⡀⠀⢀⡁⠤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠤⠤⠤⠄⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡤⠤⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠤⠴⠶⠖⠒⠛⠛⠉⠉⠉⠉⠀⠀⠈⢧⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠉⢹⡏⠁⠀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⠉⠓⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣠⠟⠀⠀⠀⠀⠀⣰⠏⠀⢀⣼⣷⣤⠀⠈⢳⡄⠀⠀⣠⣤⡀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁⠀⢀⣀⠀⠀⠀⡏⠀⠀⢿⣯⣤⣀⠀⠀⠀⣷⠀⠐⣇⣀⡽⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡇⠀⠠⡏⠉⣳⠀⠀⣧⠀⠀⠀⠉⠙⣿⡷⠀⠀⣿⠀⠀⠈⠉⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠀⠀⣇⠀⠀⠙⢛⡥⠖⠒⢻⣆⠀⠘⠛⢿⡟⠁⠀⣰⠃⠀⠀⠀⠀⠀⡴⠋⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢿⠤⣤⠞⠉⠀⠀⠀⣸⠋⠛⠦⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⢀⣸⣇⣀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⣀⣠⡼⠞⠁⠀⠀⢀⣠⠞⢁⣀⣀⣀⣤⡤⠤⠴⠶⠖⠒⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⢀⡴⠛⠋⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡴⢦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⣩⣴⡏
⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠳⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠘⢧⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡞⡷⣤⣀⣀⣀⡴⠒⠒⠒⠲⠦⠤⠤⠤⢤⣤⣾⢳⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⢠⢏⡞⠀⠀⠀⣨⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡳⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⢫⠎⠀⠀⠀⡴⠃⠀⠀⡰⠖⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢳⡹⡄⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠿⣄⡀⠀⡰⣣⠏⠀⠀⠀⠘⠦⠤⠖⠋⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠀⠳⡝⣄⠀⢀⣠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⢀⡴⠙⠛⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⣠⠖⠲⣄⡤⢤⡀⠀⠀⠀⠀⠀⣄⡀⠀⠙⠲⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠁⠀⣠⠏⠀⢀⣽⣄⡀⠀⢀⡀⠀⠙⠢⣄⣀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣶⠁⠀⣠⠞⠀⠀⣷⣀⠀⠙⠢⣄⣀⣸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢺⡇⠀⢠⠞⠁⠈⡧⣄⠀⣸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⢯⡀⣠⣞⣁⡸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⠀⢸⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⢠⡜⢛⠛⠀⠀⠀⠀⠀⢸⡗⣿⢺⡇⠀⠀⠀⠀⠀⠀⢀⡞⠉⢻
⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣟⠀⠀⠀⠀⠀⠀⢀⣿⡧⠄⠀⠀⠀⠀⠀⠀⢘⡋⣿⣙⠃⠀⠀⠀⠀⠀⢀⣿⡗⠓⢺
⠀⠀⠀⣨⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⠀⠀⠀⠀⠀⠈⠱⣿⣿⡇⠀⠀⠀⠀⠀⣨⣿⣷⡶⠚
⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠅⠀⠀⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠄⠀⠀⠀⠀⠀⠙⠻⠟

⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠿⠿⣿⣯⣽⠿⠿⠃⠀⠀⠿⠿⣿⣿⣽⠿⠿⠃⠀⠀⠿⣿⣿⣿⣿⣿⠿⠃⠀⠘⠿⠟⡍⣉⡩⠻⠿⠃⠀⠀⠿⠟⠍⣉⡩⠻⠿⠃
⠀⠀⠀⠀⣿⣿⣿⠀⡶⠀⠀⠀⠀⠀⣿⣿⣿⠀⣦⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣏⠉⢱⠀⡶⠀⠀⠀⠀⠀⣊⠉⢱⢀⣆
⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢄
⠀⠀⠀⠀⢸⣿⡇⢜⢦⠀⠀⠀⢀⠞⠛⢷⣦⣄⠀⠀⠀⠀⠀⢸⣶⣶⣶⡇⠀⠀⠀⠀⠀⢸⠛⠛⠛⡏⠀⠀⠀⠀⠀⠀⣤⣴⣦⣾⣧
⠀⠀⠀⠰⢁⣀⠀⣇⠀⠀⠀⠀⡜⠠⠸⡋⠉⠈⡄⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠘⢤⣀⡠⠃⠀⠀⠀⠰⢶⣤⣠⢣⠀⠀⠸⡀⠀⠀⠀⠘⢻⣿⡿⠃⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⣶⣿⣿⣿⣯⣤⣤⡀
⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠷⠶⠛⠃⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠸⠶⠿⠷⠇⠀⠀⠀⠀⠿⠛⠛⠛⠻⠿⠿

⠀⠀⠀⠀⣤⣿⣤⣀⣤⡀⠀⠀⠀⠀⣤⣿⣦⡀⠀⠀⠀⠀⠀⠀⣤⣿⣤⡀⠀⠀⠀⠀⠀⠀⣤⣿⣤⣀⡀⠀⠀⠀⠀⠀⣤⣿⣤⣠⡀
⠀⠀⠀⠀⣇⣀⣸⠉⠄⢣⠀⠀⠀⠀⣇⣀⣸⣿⣿⡇⠀⠀⠀⠀⣇⣀⣨⣿⣷⠄⠀⠀⠀⠀⣇⣀⡰⠛⡑⢄⠀⠀⠀⠀⣇⣀⣹⣿⣷⡀
⠀⠀⠀⠀⣿⣿⣿⢗⢂⣀⡆⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠍⠉⠀⠀⠀⠀⠀⣿⣿⣿⠆⣤⡄⠀⠀⠀⠀⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠻⠿⠟⠈⠋⠉⠀⠀⠀⠀⠻⠿⠟⠈⠉⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠈⠁⠀⠀⠀⠀⠻⠿⠟

⠀⠀⠀⠀⠀⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⣀⡀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⣀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡫⠂
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠲⣤⠤⠤⠤⠤⠐⠒⠚⠉
⠀⠀⠈⠑⠶⠶⠤⠤⠶⠶⠤⣤⡎⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡳⠦⠤⠰⢄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠤⠤⠴⠀⠄⠈⠆⢀⠀⢄⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣀⡀⠐⠁⡀⠀⠀⢀⢀⠌⢀⣤⡤⣤⠀⠀⠀⠐⠀⠀⠀⠀⢀⢄⣠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢄⡠⠞⠴⠚⠓⠈⠁⠀⠙⠥⠤⠤⠤⠤⠒⠓⠚⠉⠉⠉

⣶⡶⢶⡶⣶⣶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣄⠀⣀⣀⣀⣀⡀⢀⣀⣠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡏⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠙⢷⣤⣤⣖⠊⠁⠀⠀⠈⠛⣔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⠋⠀⠀⢄⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⡤⠒⠊⠉⠙⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠑⠄⠀⡀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠈⠢⣀⡠⠒⠁⠀⣡⣀⠙⢿⣿⣷⣄⠀⢀⣴⣦⡈⠳⣄⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⡠⠖⠁⠀⣣⣀⠙⢿⣿⣷⣌⠻⡿⠋⠀⠈⢳⠊
⠀⠀⠀⠀⠀⠀⠀⢆⠀⡠⠊⠀⠀⡠⠊⠀⠀⡧⢄⠙⢿⣿⣷⣄⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⡠⠊⠀⢀⡤⠋⠀⢀⠇⠀⠙⢿⣿⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⣀⡴⠋⠀⢀⠔⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠔⠉⠢⣀⢀⡴⠋

⠀⠀⠀⠀⠄⡀⠀⡀⠄⠀⠀⠀⠀⠀⡀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣀⣤⠖⠒⡄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡋⢰⠀⡆⡀⢹⠀⠀⠀⠀⠀⣠⣾⢿⡁⠀⠀⠀⠀⢀⣌⠀⠉⠓⠐⠁
⠀⠀⠀⠙⠋⢀⣇⠈⠛⠁⠀⠀⠀⠃⢸⠀⡇⡇⢸⠀⠀⠀⠀⢀⣈⣁⡀⠙⢦⡀⠀⠀⡞⠙⡆
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠁⠀⠀⠑⠒⠁

⠀⠀⠀⠀⡠⠠⠄⠤⡀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⣀⣀⣀⣀⡄
⠀⠀⠀⠀⡃⢠⣦⠀⡁⠀⠀⠀⠀⢿⣿⡿⠿⣿⣿⠀⠀⠀⠀⠀⠰⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠱⢄⣀⠔⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠶⠶⠶⠷⡶⠀⠀⠀⠀⠒⠓⠚⠓⠇


⠀⠀⠀⠀⢰⠒⠒⢶⠀⠀⠀⠀⠀⠀⠰⠆⠐⠶⠀⠀⠀⠀⠀⠀⠠⣶⠄⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀
⠀⠀⠀⠀⢸⣒⣶⢾⠀⠀⠀⠀⠀⣙⡃⢘⣫⢪⣛⡀⠀⠀⠀⠀⢰⠄⣦⠀⠀⠀⠀⠀⠀⠀⡬⢡⡄
⠀⠀⠀⠀⠘⠛⠓⠊⠀⠀⠀⠀⠀⠛⠃⠛⠛⠘⠛⠃⠀⠀⠀⠀⠘⠛⠛⠀⠂⠀⠀⠀⠀⠀⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⣻⢹⣻⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⠀⠀⠀⠘⡿⠤⠤⢧⠇⠀⠀⠀⠀⠀⠸⠿⠾⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣤⣤⠀⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠶⠶⠶⠆⠠⠶⠶⠄⠀⠀⠀⠀⠀⢀⡴⠖⠛⡛⠳⢦⣄⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⣈⣁⡄⠀⠙⣧⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠤⠤⠤⠴⠤⠦⠤⠤⠴⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⠦⠤⣤⠀⠀⢹⠆⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡄⠀⣤⠀⢠⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⢷⠹⣄⠀⠐⠒⡒⠋⠀⢀⡿⠀⠀⠀⢸⡇⠛
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣌⠳⣝⠶⣤⣈⣁⣤⠴⢋⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣇⣙⣛⣛⣋⣉⣉⣉⣉⣉⣙⣙⣛⣋⣸⡇⠀⠀⣤⣤⡄⠀⠀⠈⠳⣬⣛⡲⠶⠶⠶⠚⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⡴⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⢷⡘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣡⠟⣀⣀⣈⠉⠉⣁⣀⡈⠉⠉⣁⣀⡉⠉⠉⣀⣀⡉⠉⢉⣀⣀⡉⠉⢉⣀⣀⠉⠉⢉⣀⣀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⢁⣀⣈⣉⣉⣁⣀⣈⠉⠉⠁⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣏⣉⣉⣉⣉⣉⣉⣙⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠖⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠐⠛⠛





⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢻⣇⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢀⡾⠛⠷⣦⣤⣄⣠⡾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠾⠛⠻⣦⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢠⡿⠁⠀⠀⠀⢀⣼⠟⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣠⡿⠁⠀⠀⠀⢀⣾⠃⠀⠀⣠⡿⠋⠉⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠸⣧⣤⡴⠟⠋⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢸⣏⡀⠀⠀⣰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⢘⣷⡀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠉⠻⢶⣼⠏⠙⣷⣀⣤⣤⣄⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠈⠙⢷⣤⣀⣴⠟⠉⢿⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⢹⡿⠋⠈⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⡀⠀⠀⠈⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣴⠟⠁⠀⣠⡾⠛⠻⣦⠀⠀⠀⠀⠘⠷⣦⣀⠀⠀⠈⠛⢷⣦⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣏⠀⢀⣴⠏⠀⠀⣠⣿⣤⡀⢀⣀⠀⠀⠈⠙⠻⢦⣤⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣇⠀⢀⣼⠏⠀⠈⣿⠀⠉⠛⠶⣦⣄⣀⣠⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣧⡀⢀⣾⢿⣤⣄⣀⣀⣴⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢒⣊⣉⣍⣍⣭⣭⣙⣒⡰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡛⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢨⣀⣸⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠽⠋⠋⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⡈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠒⠐⠀⡀⢀⠀⠀⠂⢀⠔⠂⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠈⠀⠀⠠⠁⠀⢹⣿⣟⣽⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠰⠁⠀⠀⠔⠁⠄⠠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢼⢃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠂⠀⠘⡀⠀⢰⣽⠘⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢐⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄⠀⠊⠀⠀⠀⠀⠀⠀⠠⠀⠸⠀⠀⢸⣿⢈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠄⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⡷⠊
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⡇⢪⡀
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣟⠃⡏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⡶⠶⠲⠲⠶⠖⠖⠖⢿⠄⠀⢿⡶⡲⠆⠆⠠⠄⠦⠄⠄⠦⠦⠤⠤⢤⡴⢿⠭⠀⡇⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⠈⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⡀⣸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⢁⢸⡋⠀⣗⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣆⠀⡇⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣀⠄⠀⠠⢸⡵⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢸⠖⠀⡇⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠨⠈⡧⠀⡇⡔⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠟⠙⠛⠛⠃⠀⠀⠸⡧⠀⣿⢸⣿⡄⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢍⡀⠀⠀⡁⠠⣿⢸⣿⠅⣿⣿⣿⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡯⠀⣿⣃⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠈⠉⠁⠀⠒⠂⠠⠤⠤⠄⣀⡀⣀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⣻⡦⢯⠿⠙⣀
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡁⠐⠒⠠⠆⠀⠤⠀⡤⡤⠄⠀⠄⠈⠅⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






⡿⢿⡿⠿⣿⡿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣟⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠂⠀⠀⠀⢒⠤⢹⡶⡖⠒⠒⠒⠀⠂⠀⠀⠐⠒⢒⡊⢳
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⡦⡸⢹⡄⣀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠇⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄⠀⠉⠁⠀⠘⣡⣿⡞⠃⠀⠀⠀⠀⠀⠈⠀⠈⢉⡀⠀⠀⠇⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⠀⠀⠀⠀⣽⡏⢡⡀⠀⠀⠀⠀⠀⡤⠀⠀⠻⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣀⠀⡈⠀⠀⠻⢣⣸⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣯⠘⠋⠁⠀⠀⠀⢀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⢘⡫⠀⠀⠀⠀⠐⠂⠎⠸⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣯⠀⡀⠈⢉⣭⣿⣶⠆⡖⠰⣦⠴⠆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⣿⣯⣿⣟⢚⠸⡱⠿⢿⣻⣖⠀⠀⠀⠀⡀⠀⠀⠐⠛⠆⣳⣾⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣯⣭⣭⣟⣻⣿⠏⠁⣀⡀⠙⠛⠳⠶⠶⣤⣤⣄⣀⣀⠚⠛⠉⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠤⠤⣈⣀⣐⠶⠯⢥⡀⠉⢟⣛⠿⢿⣶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠙⠛⠛⠙⠒⠚⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠔⠒⠊⠉⠀⢀⣀⡨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠒⠒⠉⠁⠀⣀⣀⠤⠔⠒⠂⠉⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢀⣀⡠⠤⠒⠒⠊⠉⠀⠀⠀⠀⠀⢀⠤⠤⠒⠒⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡒⠉⢡⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠰⠠⠀⠆⠀⠀⠀⠰⠀⠀⠀⠀⠰⠰⠰⠰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠠⠤⠤⠒⠶⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⠤⠤⠤⠤⠤⠤⣄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠀⠀⠀⠀⠒⠒⠒⠛⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠚⠒⠒⠒⠀⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡇
⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⡔⠐⠒⠀⠒⠒⠒⠒⡄⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⡱⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠰⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⣇⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⡿⢋⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⡵⣾⣿⣿⢀⡸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣯⢻⠁⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⣿⡟⠛⠛⠓⠒⢂⡠⢤⢀⣤⣾⣿⣿⠻⢿⠏⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⣀⠤⠚⢶⣤⠎⠐⠈⠻⠿⣿⣿⠋⠓⣀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠙⠃⠀⡠⠿⡟⠀⠀⠀⠀⠱⡨⠀⠀⢀⣿⣿⣶⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠀⠀⣧⢳⠀⢀⠀⠠⣇⣀⠀⢸⠉⠛⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢑⣧⠈⣄⠱⠋⠉⠀⡞
⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠶⠶⠶⠾⡊⡍⠀⡘⣷⡄⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣾⣄⣘⣬⡷⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⣿⡋⠈⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⣿⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣻⢹⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢽⢸⣇⢸⡆⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⢸⣿⠸⡇⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⢸⣟⡇⠃⠀⢸⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⡇⡇⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⢸⡇⣇⠀⠀⢸⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⢸⡇⣿⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢸⡇⣿⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⣿⡆⠀⠸⣧⠸⡰⡀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⠀⣾⡇⢽⣧⠠⢽⣿⡀⠓⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠝⠓⠒⠽⡛⠼⡏⠓⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⡤⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡉⠈⠣⡀⠀⠁⠉⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠾⢷⡋⠀⠈⠲⢶⣀⠠⢂⠘⡇
⠀⠀⠀⠀⣰⠋⠀⠀⣀⣤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢁⡄⠀⠀⠙⢆⠀⠀⠸⣇⢈⠤⠀⠸⡀
⠀⠀⠀⢰⠃⠀⢠⠞⠉⠀⠀⠀⠐⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⢠⠎⠡⠂⠀⠉⠳⡄⠀⠀⠈⢳⡄⠀⠹⣆⠀⠀⠘⢷
⠀⠀⢀⠏⠀⢀⡏⠀⠀⠀⠀⢠⠂⠀⠀⢈⡆⠀⠤⠔⠒⠁⠸⠛⠣⡀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠹⡖⠂⡀⠀⠀⠀⢸
⠀⢠⠎⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡄⠀⠀⠀⠀⢠⡀⠀⠀⠁⣀⠘⠀⠀⠀⢸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⡀⠤⠶⠦⣀⠀⣂⣳⣄⡀⠀⠀⠀⢻⠀⣠⠒⠡⠀⠀⠀⠀⢸
⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡠⠞⢻⣆⠀⠀⠘⠣⡹⣀⠔⠃⠀⠀⠀⢸
⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡠⠊⠁⠀⠀⠋⠳⢄⣤⢴⢍⡀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠙⢦⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⠑⠂⠤⠄⠠⣆
⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⡠⠖⢙⣧⡀
⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠩⠀⢀⠎⠀⣿
⠀⣠⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣅⣀⠠⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈
⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢴⡶⢫⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣭⡍⢩⡍⡉⠉⡿⠉⡍⢩⣷⣿⣿⣿⠿⡟⢸⣿⠟⣥⣯⣭⣭⣿⣿⣿⣿⣿⣿⣶⠸⣧⠻⠴⠛⠚⢻⢀⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⠾⠛⠓⠈⠁⢀⠈⠉⠁⠈⠉⣉⠝⣡⡖⢁⡇⡟⣵⠋⠀⢋⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠓⠀⢸⢸⢿⡵⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⢀⣤⠠⢄⠀⠀⠀⠀⡜⣠⣾⣿⢃⡼⢱⡿⣿⠀⠀⢸⠀⠀⠀⠀⣄⣠⣤⣠⢤⠀⠈⠀⠀⠀⠀⡜⡜⣼⣏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠁⠠⠄⠀⠀⠸⣌⣉⠴⠀⠀⠀⢸⣵⣿⣟⣡⢃⡆⣼⣼⣿⡄⠀⡼⠀⠀⠀⣸⢻⣿⠿⢯⠏⠀⠀⠀⠀⠀⡰⢱⠡⣽⣾⣏⡝
⠀⠀⠀⠀⠀⠀⡠⢣⠎⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡻⣿⡿⢁⢾⡇⣿⣾⡿⠁⡴⠁⠀⠀⣆⠀⢀⠄⣀⡀⣤⣤⣤⣴⡖⣰⢣⢣⡜⣽⣿⡿⠁
⠀⠀⠀⠀⠀⡔⣱⣿⣶⣶⣶⡖⠰⡆⣴⣠⣞⣪⠉⡥⣤⣤⣤⣿⡟⢠⢋⣾⡇⣿⣧⣶⡧⢤⣤⢤⣤⢀⣤⣤⠄⠤⢤⡥⣭⠵⠒⠺⢃⣯⢻⣰⣿⡿⠁
⠀⠀⠀⠀⢾⢤⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⣤⣤⣭⣉⣉⣉⣹⣡⣏⣤⣼⣄⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣭⣭⣭⡴⠶⠒⣒⣛⡿⣱⣷⣿⡿⠁
⠀⠀⠀⠀⢘⡿⣿⡿⢗⣒⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠤⢤⣬⣿⣿⣋⠈⠀⡐⠒⠲⠶⠶⠶⢴⣶⣿⣽⣿⠃
⠀⠀⠀⠀⠸⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣽⢿⣯⠿⠿⠿⠿⠿⠿⠦⠶⠶⠿⢿⣾⣿⣿⣿⣿⣿⢿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣄⣀⣄⣀⢀⣀⣀⣀⣠⡀⣼⠀⡓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢸⠀⡇⠀⡉⢉⠉⢉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣀⠀⠁⠀⠀⠀⢸⠀⠱⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠈⠑⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠳⠚⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⢤⡀⠀⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡷⠀⠄⠘⠦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⢒⡒⠒⠒⠒⠒⢺⠀⠀⠀⠀⠀⠀⠸⣆⠀⠘⠒⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⢒⠒⠒⣒⠒⣐⢒⠒⡒⡒⠒⡒⠒⢲⠀⠀⠀⡠⠶⠉⠋⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠋⠀⠀⢠⠟⣡⠌⠻⡄⠀⠀⠙⡆⢸⠀⠀⣜⠃⠀⢰⠛⠄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⢸⡀⢈⡦⢀⡇⠀⠀⠀⡇⢸⠀⠀⢏⡀⠀⢀⣙⠆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠳⠤⠤⠤⠷⠦⠴⠾⠤⠤⠤⠖⠁⢸⠀⠀⠈⠣⣤⠀⡁⢀⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣤⣤⣤⣤⣀⣠⣤⣤⣤⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣷⣶⡣
⠀⠀⠀⠀⢀⣀⣮⣤⣦⣤⣤⣤⣤⣀⣀⠈⡒⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠙⠛⠾⣿⣿⡿⠛⠋
⠀⠀⢀⠀⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢶⡌⢙⡄⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠛⠾⣶⣦⣽⣷⣷⣶⣾⣾⣿⣿⣿⠷⠋⠀⠀⠀⠀⣠⣾⣿⣻⠿⠿⣫⣦⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠋⠉⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠿⠋

⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⣄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠳⣄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⣧⣤⣤⣤⣤⣬⣷⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⣿⠟⢻⡟⠛⣿⡲⠶⣤⣀⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠰⠖⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠆⠀⣠⣶⣋⡁⠀⣠⠞⠁⠀⢸⡇⠀⠈⠳⣄⠀⢈⣙⣶⣄⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠉⠉⣿⠟⠓⠲⠶⢾⡷⠶⠖⠒⠛⣿⠉⠉⠀⠈⠻⣦
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢀⣾⣁⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣈⣣⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠉⠙⠓⠶⢤⣤⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣤⣤⠶⠞⠛⠉⠀⢻⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣩⡽⠟⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢠⡾⢋⣥⣤⣍⡙⢦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⢀⣠⡴⠞⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⡿⢠⡏⣠⠞⠉⢻⡌⣷⠀⠀⢸⣄⣀⣀⣀⣀⣀⠀⠀⢀⡀⠘⠛⠋⠁⠀⠀⠀⠈⠙⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢿⡘⣿⣡⡞⣁⡾⢁⡟⠀⠀⢸⡁⠀⠀⠀⠀⠉⢳⡶⠋⢹⡶⢶⣄⠀⠀⠀⠀⠀⢤⡀⠙⢿⡟⠁⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⣤⣉⣉⣩⡴⠟⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠘⢷⡾⠋⣠⡼⠋⣻⡄⠀⠰⣤⡀⠙⣶⠟⠀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡄⠛⠾⣏⣠⡾⠋⣻⠶⣄⣈⣿⠞⠃⠀⣸⠇⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢻⡄⠀⠈⠉⠳⠞⢻⡴⠟⠉⠁⠀⠀⢠⠏⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣤⡤⠿⣶⠒⠚⠛⢻⡟⠛⠛⠲⣶⠷⢤⣤⣀⣠⡾⠋
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣀⠈⠷⣄⠀⢸⡇⠀⣠⠞⠁⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠿⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⢤⣤⣤⠄⠀⠀⠀⠀⠉⠛⠓⠿⠷⠾⠷⠾⠿⠚⠛⠉


⡖⠒⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠒
⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⢠⠤⢄⠀⡀⢤⠤⠀⣠⠀⢠⠤⡄⢠⠀⠀⡠⠄⠀⢀⠠⠄⠠⡤⠄⡤⢄⡀⡠⠤⡀⢠⠀⡄⢠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⢸⠀⢀⠂⡇⢸⡃⢰⠄⠆⢸⠒⢣⢸⠀⠀⡗⠀⠀⠈⠑⠄⠀⡇⠀⡇⢺⠰⡇⠀⣸⢸⠻⡀⢸
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣷⣶⣶⣶⣶⣶⣾⣷⣶⣾⣶⣶⣾⣶⣶⣶⣷⣶⣶⣷⣶⣷⣶⣶⣾⣶⣶⣾⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢡⣤⣴⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⠉⠀⠀⠈⠁⠀⠁⠀⠀⠐⡤⠀⢸⣿⣿⣫⡭⣿⣿⣷⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⣧⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⢸⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⠄⠹⣄⠀⢠⡄⠀⠀⠢⡖⠀⠀⢸⣿⣿⢸⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⢸⡇⠀⣶⣷⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣷⢰⡶⠀⠒⠀⢀⣶⡤⠤⠤⠀⣿
⠀⠀⠀⠀⢸⣇⣀⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠛⣿⣿⣿⠯⠭⠉⣿
⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠛⠉⠛⠤⠤⠠⣿
⠀⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢰⡿⠛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⢸⡟⢉⣉⡉⣉⠉⡉⠉⠉⢉⣩⠿⣷
⠀⠀⠀⠀⢸⡇⢠⣿⡦⠀⠀⡁⡀⠀⡀⡀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠐⠒⠀⠒⠒⢂⡤⠖⣉⣤⣾⣿
⠀⠀⠀⠀⢸⡇⢀⣀⣀⠀⣀⣀⣀⡀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⡴⠚⣡⣴⣾⠛⠛⣿⣿
⠀⠀⠀⠀⢸⣇⠐⠀⠂⠐⠀⠀⠂⠂⠘⠟⠀⣿⠀⠀⠀⠀⠀⠀⢸⣧⣴⣋⣡⣶⣿⣿⣿⣿⣤⣤⣿⣿
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠂⢠⠒
⠀⠀⢤⣤⣄⠀⠀⠃⠀⠈⠀⠀⡴⠆⠀⠀⠀⢀⣠⣶⣦⡀⢀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠛⠛⡿⠿⠿⠧⠤⠤⠤⠤⠴⣦⡉⠍⠉⣁⣄⣀⣀⣀⣘⣿⣿⣿⣤⣤⣤⡄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠰⣶⣶⡇⠐⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣅⡀⢰⠆⠰⠀⠀⠀⠀⠀⠀⣤⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡟⣁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⡴⠊⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⢀
⠀⠀⠀⠸⣏⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠠⠿⢀⣀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⠛⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⢸⡃⠈⡷⠴⡏⠀⢹⠀⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⠀⠙⠛⠁⠀⠉⠛⠉⢀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⡀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠈⡟⠒⠶⠶⠶⠚⡏⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣇⠀⠀⠀⠀⢀⡇⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣻⠇⠀⢻⠉⣻⠿⡏⢹⠃⠀⢿⡳⠤⣄⣀⡤⠖⠚⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢴⡃⠀⠀⠈⣇⢳⣶⢋⡏⠀⠀⠀⣹⠄⣰⠋⠳⡄⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢦⣀⠀⢹⡞⠘⣿⠀⠀⣠⠞⠁⢠⡇⣾⠁⢿⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢷⢰⢇⡴⠋⠁⠀⠀⠘⡇⠉⠀⢸⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠇⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣤⠴⠞⠙⡶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠛⠛⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⢠⡞⠛⠉⠉⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣷⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⣿⣿⣿⠀⠀⠀⠀⢀⡀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⣿⠀⠀⢐⣁⣸⠆⠀⡄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⠀⠀⢸⠄⢼⠤⠤⠭⠭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿⠀⠀⠀⠑⠚⠒⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿⠀⠀⠀⠆⠰⠀⠆⠰⠀⠀⠢⠘⠄⠐⠐⠂⠀⠐⠆⠶⠐⠀⠆⠀⠰⠀⠶⠐⠔⠒⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠁





⡿⢿⢿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣶⣶⣶⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⢸⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿
⠀⠀⠀⢰⣶⣦⣬⣍⣉⣛⠛⠛⠃⣿⣿⡟⠛⠛⣛⣉⣉⣥⣤⣶⡆⠀⠀⠀⢸⣿⣿⣿⣿⠉⠙⠛⠶⠶⠦⣤⣤⣼⠀⠀⢸⣤⣤⡤⠶⠶⠞⠛⠋⠉⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣛⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢸⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣤⣴⣶⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣶⣦⣤⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⢠⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣦⡀⠀⠀⠀⠸⣿⣿⣿⠃⠀⠀⠀⢠⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣦⡀
⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⣰⣿⣿⣿⣄⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⣿⣿⣿⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠘⣦⠀⠀⠀⠀⠀⢀⣠⣤⣤⣠⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡸⠀⠀⠀⣀⣰⣿⣿⠿⠻⠿⣿⣿⣶⣶⣶⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣶⣄⣀⣤⣤⣤⣼⠁⢸⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠉⠉⠉⠁⠀⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣴⣿⣿⠟⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠛⠁⠘⠛⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠘⠂⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠁⠀⠈⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠸⠇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢿⡷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣂⠙⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⢘⡃⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠘⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡇⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠇⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠇⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡄⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⡅⠰⣨⠇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢐⡂⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠠⠄⠙⣆⠀⠀⠀⠀⠀⠀⣼⡤⠴⠒⠒⠛⠛⠉⠉⠉⠙⠛⠛⠛⠒⠶⠤⢤⣇
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⣆⠀⡼⠀⠀⠀⠀⠀⠈⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀⣐⣓⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡾⢁⣉⠈⠈⢳⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⣶⣶⣦⣤⣤⣤⣴⣶⣶⣶⣾⡿⠛⠁
⠀⢀⣼⡤⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⠤⣥⡀⠀⠀⠀⠀⡸⠏⢀⡈⠀⠀⠼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠀⢯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠁⠀⠀⡴⠋⠁⡀⠀⠀⠀⠀⢐⠈⠙⢦
⠀⠀⠀⠙⠿⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠿⠛⠁⣀⡴⠻⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡈⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⢸⠁⠈⠀⠠⠀⢈⣀⣀⡀⠀⢀⣀⡀⠠⠄⠐⠁⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⣥⠴⠶⠶⠦⣬⡛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⣿⣷⣞⣉⣀⣀⣀⣀⣀⣀⣙⣶⣾⣿⣇⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⡴⠋⠳⡄⠀⠀⠀⢸⠀⡇⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣤⣧⣼⣤⣤⣤⣤⣇⣠⣀⣧⣤⣤⣤⣼⣤⣧⣤⣤⣴⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠁⠀⢹⣻⣟⡏⠁⠈⠹⣏⣉⣉⡿⠁⠀⠙⣟⣟⣻⠉⠀⢹⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡟⢻⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠛⣿⠀⠀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠸⡇⢸⡇⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢿⠀⣿⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠘⢦⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⠞⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣵⣿⣿⣿⣿⢿⣾⣙⣻⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣽⣿⣿⣿⢿⣵⣿⣿⣾⣿⣿⣿⣭⣭⣝⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⠀⠀⠀⠈⠿⠋⠈⠋⠁⠉⠉⠛⠛⠋⠉⠉⠉⠛⠛⠻⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣏⠀⠀⣀⣠⣤⣶⠿⠛⣉⣠⣤⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣶⡿⠟⢋⣁⣤⣴⡶⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣶⡿⠿⠛⠋⠁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⣏⣿⣿⣿⣿⣿⣏⣀⣤⣤⣶⡶⠶⠂
⠀⠀⠀⠀⠨⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⢻⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⣀⣀⢀⠀⡀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠰⡄⠇⡰⢄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡿⠁⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠊⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠠⡀⡆⢠⠄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠀⠘⡶⠁⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠇⡠⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⡀⡇⢠⡀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠈⢰⠂⠐⠂⠁⠬⠐⠀⠁⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠂⠀⠱⣶⠃⠈⠂⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠂⠀⠀⡨⠠⠞⠀⠆⠇⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠦⠀⠅⠀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠈⡌⡔⠀⡤⠤⠤⢤⣀⠀⠀⠈⠀⢳⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣃⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⠭⠭⣙⣒⠀⠸⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⠭⢝⣒⣂⠀⠰⠿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣊⣩⣍⣙⣓⣒⠦⠥⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣛⣛⣒⣒⣒⠭⠭⣓⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣒⣒⣒⣒⣒⡦⠭⢭⣒⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠉⠙⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣁⣉⣀⣉⣉⣉⣉⣓⣒⣤⣼⣤⣤⣐⣀⣀⣀⣀⣀⣀⣀⣈⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠿⠿⠿⠿⠿⠿⠛⠛⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠿⠇⠀⠀⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣦⡀⢰⡆⢀⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠧⠼⠆⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⣾⢣⣶⢢⣶⢰⣚⡇⢀⣾⡆⠀⣿⢀⣶⡆⣶⣦⠆⣆⡆⣶⠆⡶⠂⠹⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠉⠀⠉⠀⠁⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠈⠀⠛⠃⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⢨⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠃⠘⠿⠿⠿⠿⠿


⣟⠉⣹⣛⣛⣛⣛⣻⣛⣛⣛⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣻⣿⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠶⠾⠛⠋⠉⠉⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠛⠛⠉⠉⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣀⡀⢠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠛⠋⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⡀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠉⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢸⡇⢠⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄⣤⢤⡀⣤⠤⢠⡤⣄⢠⡄⣤⡤⠀⢀⡤⡄⢠⣄⢀⡤⣤⢠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡄⣿⢺⡁⣿⣃⢸⣧⡿⢸⡇⢸⡇⠀⠸⣄⣄⡿⢿⡬⡟⢿⢸⣧⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠁⠀⠀⠈⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡴⠒⠋⠙⠛⠒⠲⢤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣔⡴⠶⣶⠴⢦⣠⣴⣲⠶⠤⡤⣿⣦⡤⠤⠤⢤⣤⣤⣴⣧⣤⣤⡴⣆⣀⣀⢶⣶⡤⠤⢤⣤⣤⡤⠤⣄
⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⣠⠞⠛⠁⠠⠾⠋⠉⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠉⠁⢠⡾⠋⠡⡄⠀⠉⠁⠐⠢⡀⡆⢸⡆
⠀⠀⠀⠀⠀⢸⠑⢊⠔⠋⠁⠀⣇⠀⣀⣧⠀⠀⠀⠠⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⠀⢸⡇⠀⠀⠙⢦⡀⣈⡦⠀⣧⡇⢸⡇
⠀⠀⠀⠀⠀⢸⣀⠏⠀⠀⠀⠀⢧⠤⠤⣤⠀⠀⠁⠀⠓⢄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢤⡇⠀⠀⠀⠀⠙⢦⡀⠀⡿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢀⠠⠀⠀⠀⣿⠀⠀⠀⠑⢀⠈⠐⠉⠀⠀⠂⠂⠀⠀⠀⠀⡃⠀⠀⡼⠇⠀⠀⠀⠀⢀⢠⣙⠐⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡃⠀⠀⠑⠔⠉⠀⠀⠀⠀⠀⠐⣠⢦⠀⠀⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠠⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠰⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⢀⡀⡀⠀⠸⠀⠀⠀⠀⡀⠀⠀⠀⠄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘⠀⢫⡀⠀⢀⠀⠀⠅⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⢀⠀⡈⠃⠀⠀⠀⢰⠀⠈⠁⠂⠀⠤⠤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⡳⠠⠀⠀⠤⣤⠆⠀⡆⢀⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢡⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡆⠀⠀⡆⡀⠈⠑⠪⡥⢤⡌⠀⠀⠃⢸⠃
⠀⠀⠀⠀⠀⢠⠋⢢⠀⠀⠀⣡⡠⠀⠀⢸⠀⠀⠐⠀⠀⠀⠉⡀⠀⠀⠀⠈⠉⠀⠀⠀⡁⠁⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠀⠼⡄
⠀⠀⠀⠀⠀⢸⡀⢀⠑⣄⠉⠀⢸⠀⠀⢸⠀⠀⠀⢀⠀⠀⢀⠄⠀⠀⢀⠀⠂⡘⡄⡀⠑⡇⠀⠀⡆⠀⡘⠀⠀⠀⠀⠂⠈⢶⡄⢸⡇
⠀⠀⠀⠀⠀⠸⣅⣨⣤⣈⣷⠤⢬⣀⡀⣸⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠁⠁⠀⠀⡇⠀⠀⡗⠈⠀⠀⢀⠀⠁⡀⠀⢟⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠧⠤⠀⠷⠦⠠⠴⣁⣀⣐⣁⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠰⣤⣭⣧⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠵⠀⢄⠀⠀⠀⠀⠃⡟⠀⠀⣠⠂⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⢣⡀⠀⠔⠧⣙⡄⠀⠉⠏⠀⠀⠀⠀⠈⠳⠀⢀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠈⡝⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⡿⢷
⠀⠀⠀⠀⠀⠀⢸⠇⣻⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡟⠸⡇
⠀⠀⠀⠀⠀⢀⡟⠀⣿⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣗⠈⣿
⠀⠀⠀⠀⠀⣸⡇⠀⣿⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⣯⠀⢸⡇
⠀⠀⠀⠀⠀⡾⠀⠀⢾⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡧⠀⠀⢷
⠀⠀⠀⠀⢸⠇⠀⠀⢿⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠐⡷⠀⠀⠸⡇
⠀⠀⠀⢀⡟⠀⠀⠀⣻⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣏⠀⠀⠀⢿
⠀⠀⠀⣸⠃⠀⠀⠀⣿⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸⡆⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣯⠀⠀⠀⢸⡅
⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠈⢷⣶⣿⣿⠁⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⣷
⠀⠘⠻⣿⠟⠛⠛⠛⢛⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⢸⡛⣿⡇⠀⠀⠀⠀⠀⠘⢿⠷⠞⠛⠛⢛⣻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣿⣿⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠊⠁⠈⠁⠀⠁⠈⠉⠉⠉⠉⠻⣿⣆⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⣉⣭⣿⡿⠿⠟⠛⠛⠿⠿⠷⣶⣶⣤⣤⣄⣀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⣰⡿⠻⠿⠷⠶⠾⠿⠿⠛⠛⠉⠁⠀⠀⣀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⢿⡇
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⣇⣠⣴⡾⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢠⣿⠛⠉⠁⠀⠀⠀⠉⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⢀⣀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⢠⣿⠃⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⢻⣇
⠀⣾⣧⣀⠀⢀⣴⡿⠛⠛⢿⣦⣀⣾⠟⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣀⣀⣠⣼⡿
⠀⠀⠉⠛⣿⣾⠋⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⣸⣷⣶⣤⡀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠉⠛⢿⣦⣄⠀⠀⠀⠀⠀⢹⣿⠛⠛⠉⠁
⠀⠀⠀⢰⣿⠁⠀⠀⠀⢀⣾⠟⠀⠀⠀⢀⣼⠟⠁⠀⠈⣿⣄⣀⠀⠀⠉⠛⠿⣶⣄⡀⠀⠀⠀⠈⠙⠿⣶⣤⣀⣠⣾⠏
⠀⠀⠀⠘⢿⣆⣀⣀⣴⡿⠃⠀⠀⠀⣠⡿⠋⠀⠀⠀⣰⡿⠛⠛⢿⣆⠀⠀⠀⠈⠙⠻⢷⣦⣀⠀⠀⠀⣸⡿⠙⠋⠁
⠀⠀⠀⠀⠀⠙⠛⠛⣿⡅⠀⠀⢀⣼⡟⠁⠀⠀⢀⣾⠟⠀⠀⠀⢨⣿⠿⣶⣤⣀⠀⠀⠀⠉⠻⣿⠶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⠿⣿⡄⠀⠀⣰⣿⠃⠀⠀⠀⣰⣿⠃⠀⠀⠉⠛⢿⣶⣤⣤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⢿⣄⣀⣠⣾⠟⠛⠿⣶⣤⣤⣤⣴⠿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠉⠉⠛⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇⠀⢀⣤⠶⠶⡄
⠀⠀⠀⠀⠀⠖⠒⠚⠛⠛⠛⠓⢸⡋⢠⠶⡆⣸⠁⠀⠀⣆⣲⡄
⠀⠀⠀⠀⢸⠇⠀⡶⠛⠛⠛⠃⢈⡁⣘⠷⠶⠋⠀⠀⢠⡟⢫⡀
⠀⠀⠀⠀⢸⡇⠀⠇⠀⠀⠀⠀⣟⠸⣧⣴⠟⠈⢳⠶⠟⣥⣤⠏
⠀⠀⠀⠀⢸⠁⠀⠀⠀⣠⡶⠚⠛⠁⠀⠱⢦⣤⣿⠀⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠲⣤
⠀⠀⠀⠀⢸⠀⠀⠀⡼⠃⠀⠀⢠⡄⠠⣄⡀⠀⠀⢀⡤⢰⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣦⣀⠀⠀⠈⣧
⠀⠀⠀⠀⣼⠀⠀⢸⠃⠀⠀⠀⡼⠁⢰⠁⢹⠛⣿⠉⠃⢸⡤⢤⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠉⣧⠀⠀⢻⣆
⠀⠀⠀⠀⢿⠀⠀⠘⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠁⠀⠠⣍⠀⠀⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣷⣄⣠⡀⣿⠀⠀⠀⠙⣧
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⢿⣤⣤⠤⠖⣢⠴⠶⠦⣴⣿⠿⢾⣧⡶⠀⠴⠶⠶⠒⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⡏⠾⠋⠀⠀⣀⣠⡿
⠀⠀⠀⠀⠈⠁⠀⢰⡀⠀⠀⠀⠀⠀⢀⡾⠱⢶⣦⣤⡤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣸⠇⠀⡹⠙⠋⠛⠧⣀
⠤⠤⠄⠒⠒⠚⠉⠉⠙⠷⠶⢦⣤⣄⣼⠁⠴⣻⠞⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠳⣄⠀⠀⠀⠀⠀⠸⠀⢠⠋⠓⢚⠞⠁⠀⠀⢀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠀⠀⠐⠷⠤⠤⢱⠀⠀⠀⠀⠀⠀⠘⠦⣤⣤⣼⠘⣆⠀⠀⠀⠃⢀⠏⠀⣰⠏⠀⠀⠀⠀⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠃⠈⠀⠀⠀⠀⣀⣀⣀⡀⠀⠘⢧⠀⠈⢧⣄⣀⡜⢸⠀⣸⠏⠀⢀⡄⠀⠀⢻⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⠤⠶⠶⠖⠚⠛⢉⣉⣉⣉⣤⡤⠀⠈⢧⠀⠀⠉⠉⣇⢸⣷⡏⠔⠚⠿⣤⠀⠀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠒⠒⢛⣉⣉⣉⣠⣤⣤⡤⠶⠶⠖⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠘⣧⣀⠀⠀⣿⠀⣿⠃⢠⡤⢦⣾⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⢹⡆⣿⠀⠞⠀⠘⠱⠲⢦⣄⣀⡀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡼⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⣁⡀⠀⠀⠀⠀⠀⠀⠀⣀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠉⠙⠛⠲⠶⠶⠟⠛⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠛⢲⡶⢶⠖⠛⠉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠞⠛⠀⠀⠀⠸⣇⠀⠀⢸⡇⢸⡄⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠾⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡇⠀⢹⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⠃⠀⠸⣇⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⠀⠀⣿⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡏⠀⠀⠀⢿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠃⠀⠒⠒⠒⠂⢬⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠔⣠⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⡠⠒⠒⠂⠉⠁⠀⠀⣠⣾⣿⡿⠋⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠐⠤⣀⣴⣁⣀⣀⣀⣀⠀⣠⣾⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠒⣩⣾⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠉⠀⠀⠈⠺⣿⡿⠋⡠⠎
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⡀⠀⠑⢄⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠊
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣦⡀⠀⠑⢄⠀⠀⡠⠊
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠑⢎
⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠐⠋





⣏⠉⣹⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣟⣛⣿⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⢤⢤⠤⠤⣄⢀⡤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠚⠚⠚⠒⠒⣿⢸⣛⣛⣛⣛⣛⣛⣛⣛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣄⠀⠀⣿⢸⢐⠶⡶⡶⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠘⠧⣌⠠⠄⣹⠘⠒⠖⠒⠶⢒⣲⣒⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠶⠟⠀⠀⣿⠀⠀⠀⢀⡼⠋⣭⡙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣠⣀⣄⣿⠀⠀⠀⠸⣇⠀⣽⡆⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠿⠧⠤⠤⠟⠀⠀⠀⠀⠙⠳⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡿⠿⣿⣿⣿⡿⠿⢿⣛⡇⠀⣰⠞⣛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣏⢺⡇⡷⠀⠀⣿⡇⢸⡇⠀⣻⡅⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣙⣶⣶⣾⣿⣿⣷⣶⣾⣉⡇⠀⠻⣦⣭⣥⠾⠁





⠀⠀⠀⠀⠀⠀⠀⢰⣖⡆⣴⡄⢦⡴⢰⣄⣶⢰⣒⢰⣆⣴⢲⡖⠀⣦⣠⡆⣶⡒⢲⡖⢰⣀⣆⡔⢲⣰⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠘⠉⠙⠘⠁⠘⠉⠘⠘⠒⠘⠈⠋⠈⠁⠀⠃⠁⠃⠛⠒⠈⠃⠘⠀⠃⠑⠚⠘⠓⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡼⠛⠛⢧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣓⣚⣛⣛⣛⣛⣉⣉⣉⣉⣁⣸⡅⠀⠀⢨⣇⣈⣉⣉⣉⣉⣛⣛⣛⣛⣓⣚⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠖⢲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢳⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢻⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠸⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛
⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠈⠛⠲⠤⠤⡤⠤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⢤⠤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⣄⣸⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⡿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣼⣿⠇


⠀⠀⡀⠀⠀⡀⠀⠀⢀⠀⢀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠂
⠀⠀⠀⠀⠐⢀⢀⡄⡀⢀⠀⢠⣴
⠀⠀⠀⠀⠠⠞⣡⠖⣁⣸⠀⠈⠙⢀⣀
⠀⠀⠀⠀⠴⢞⣡⣿⡏⠤⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠈⠹⠶⢻⠏⠐⠢⠀⢀⣀⡠⠆
⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣦⡈⠀⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡀⣹⣿⣿⣿⣯⣿⣿⣿⣿⣦⡀⠉⠻⣿⣿⣶⣦⣴⣷⣶⣶⣿⣿⣿⣿⣿⣿⣧⠁⢀⣤⣤⠰⢄
⠀⠀⠀⣻⣿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣦⡀⠀⣉⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣍⣿⡭⢍⢅
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠉⠚⠛
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠁
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣰⣿⣿⣿⣿⣿⣿⣷⣾⣮⣝⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣳⣾⣿⡟⡻⠛⠋⠉⡁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⢿⣿⣇⡟⠀⣀⣐⢬⠟
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣷⣽⡏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣿⣿⢿⣿⣿⣿⠀⠀⠀⠀⠒⠲⠀⠀⠀⠉
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠧⠭⣽⣼⠿⠿⣭⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣆⣴⣷⣳⣶⢲⣾⣉⣿⠀⠀⠀⠀⠀⠀⣠⠞
⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⡇⠙⣶⡶⠖⡀⠖⢋⡤⠲⠙
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠉⠁⠋⠋⠛⡁⢻⣿⠇⠀⣨⡴⢛⣡⣶⡟⡧
⠀⣼⣿⣿⣿⣿⣿⣿⠿⠶⡷⢤⣶⣀⢐⠀⠀⠠⡴⠚⢁⠸⣏⡴⠚⣡⣴⣿⠟⠛⠁⠷
⢰⡿⣿⠿⠟⠛⠛⢿⣧⣄⠀⣄⢰⣠⢉⡴⣷⣐⠔⢲⡽⠎⣩⣴⣿⣿⢿⡋⢤⡅
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠛⠓⠉⠁⠞⢉⡴⢚⠋⠉⡅⠙⠏⡇

⠀⠀⠀⠀⠀⡰⠫⠍⡒⠤⣀
⠀⠀⠀⠀⡰⠑⠀⢠⣀⠁⠀⠉⠒⠤⣀
⠀⠀⠀⡰⠁⠀⢠⡏⠻⠿⣶⣤⣀⠀⠀⠉⠒⠤⣀
⠀⠀⡰⠑⠀⢠⣿⡃⠀⠀⠈⠉⢿⣓⡄⠀⠀⠁⠂⠍⡒⠤⣀
⠀⠠⠀⠀⢠⡿⠷⠹⠦⣄⡀⠀⣼⡟⠃⠀⠀⠀⢀⣤⣀⠀⠀⠁⠀⠄
⠰⠀⠀⢀⡻⣏⠀⠀⠀⠈⠐⠂⠚⠃⠀⠀⣀⣄⢻⢿⣿⡇⠀⠀⠀⠁⠂⠅⡢⢄⡀
⠡⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠾⡿⠛⠁⢸⣿⡇⠀⠀⠀⠀⠀⣄⣀⠁⠀⠕⡢⢄⡀
⠀⠀⠁⠒⠤⡀⠀⠀⠀⠀⠀⢀⣴⣶⡍⠿⢳⣦⣀⣨⣿⡇⠀⠀⠀⠀⣼⡿⠏⠀⠀⣤⣀⠁⠊⠕⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⢻⣿⡇⠀⠀⠀⢸⣾⠏⠀⠀⠰⣿⡿⢿⣶⣤⡀⠁⠊⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⣀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠤⣄⠉⠀⠀⠐⣿⡟⠀⠀⠈⠙⣿⣷⡀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⣄⡟⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⠂⠀⠀⠈⠀⠀⢀⠂⣌⡀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠳⢗⣦⣠⣶⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠈⠉⠉⠀⠀⠀⠐⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣿⣿⡿⠿⠿⠛⠛⠛⠛⠻⠿⠿⣿⣿⢷⣶
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣶⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣾
⠀⠀⠀⠀⠀⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⣉⣉⣉⡙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠸⣿⣿⣿⣿⡇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣡⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠛⠛⠛⠛⠃⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⡞⠉⠀⢠⣀⣀⠀⠀⠠⠀⠀⠀⢀⠀⠤⠐⠉⠀⠀⠀⠀⠉⠐⠂⠠⠤⠤⠄⠲⡎⡇⠀⠀⠀⠐
⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡸⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠒⠋⠻⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⡆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠑⢠⠇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⠞⠀⠀⠀⠀⠀⠙⢄⡀⠄⠀⠀⠀⠀⠀⠀⡂⢣
⠀⢄⡀⡀⠄⠠⠀⠀⢸⠀⢀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠞⠃⠀⠀⠀⠀⠀⢄⠀⠀⠈⠳⣖⠀⠉⠀⠈⠀⢣⠸⡆
⠀⠀⠈⠉⠑⠳⠦⠤⢬⠷⠬⣌⡧⣄⢀⡀⠀⢀⣀⣀⡀⠀⠀⠀⢠⡀⠀⠱⣄⠀⠀⠑⢤⣀⠀⡀⠛⢦⡠⠖⠈⠉⠓⠓⠤⠤⠤⠬⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣜⠉⢩⠀⠀⠹⠆⠨⣄⠀⠙⣤⡀⠀⠳⢤⡀⠀⠈⢻⢤⣈⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢⡈⢧⡀⠸⣄⠀⠈⢦⡀⠀⠱⣦⣔⣂⡛⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠛⠉⠉⠉⠉⠛⠛⠁

⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⢈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠛⢄⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⢴⡁⠀⠀⠀⠈⢺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠑⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠤⠄⠠⠄⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠰⠁⡠⠐⠢⡀⠙⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢃⠀⢠⡀⣀⠄⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠢⢀⣈⣁⡠⠊⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠐⠠⣤⠀⠀⠤⠔⠚⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣙⠳⣤⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⠤⠄⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣭⠆⠀⠈⢩⠋⠝⣋⡛⢍⢫⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢁⢠⠸⡘⢅⠸⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⢀⠔⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡜⢠⣤⡄⠊⠀⡈⢆⠢⢍⣒⣒⡡⠊⠳⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢄⠁
⠀⠀⠀⠀⠀⠀⡜⠠⠁⠀⠀⢀⠀⠘⡌⢆⠀⠀⠉⠢⡀⠀⠈⠢⡀⠀⢀⠊⠀⠀⠀⢠⠊⠉⠑⡄⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡼⠠⠁⠀⠀⠀⠑⠃⠀⠘⡌⢆⠀⠀⠀⠈⠦⡀⠀⠈⢦⡌⠀⠀⠀⠀⠐⢄⣀⡠⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡰⢡⠃⠀⢠⡆⠀⠀⠀⠀⠀⠘⡜⡆⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣦⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⠀⣀⡀⠀⢤⣄⣀⡠⠤⠿⠐⠒⠺⠄
⠀⠀⡰⢁⠂⠀⠀⠀⠀⠀⠀⡔⠉⡆⠀⠀⠀⠱⡘⣄⠀⣀⣀⡀⠤⢤⣗⠒⠦⠭⠙⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢰⢃⠂⠀⢠⢢⠀⠀⠀⠀⠈⠛⠁⠀⡠⠠⡀⠱⡘⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢠⢃⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠁⠀⠡⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣣⣤⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠋⢹⡄⠀⣀⡤⠖⠋⠉⣉⠛⠷⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢀⣀⣀⠀⢷⣉⠀⣀⠴⠉⠊⠍⠀⠁⠀⠀⢀⣠⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣰⠞⠉⠀⢘⣧⠤⠾⠋⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠇⠀⢹⡠⠴⠚⠉⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⣸⠤⠴⠒⠋⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⣉⡤⠒⠺⢉⡍⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣰⣶⣏⣁⣀⡀⠼⠂⠡⠐⠊⠁⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠉⣟⠙⠛⠓⠶⢤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣇⠀⠀⠀⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠲⢿⣄⣀⠀⠀⣼⠀⣀⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠒⠲⢦⣄⢈⡟⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⠀⠀⡴⠚⠉⠉⠙⣄⣸⠩⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⣼⠃⢠⠀⠀⠀⢹⢸⠆⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡆⠀⡏⠀⡞⠀⠀⠀⠘⡎⢀⡄⣷⠀⠀⠀⠀⣀⠀⠀⠀⠀⡴⣄⠀⠀⠣⢄⣠⠼⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠇⠀⠛⠦⠤⢤⡄⠠⢼⡀⣿⠀⢀⣧⡆⣸⣼⡆⠀⠀⠓⢼⠦⣀⡔⠉⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢴⣴⣒⣒⣒⣚⣒⣓⣒⣒⣒⣒⣒⣒⣚⣓⣒⣒⣛⣻⣾⣿⣿⣶⣾⣿⣿⡆⠀⠀⠘⣆⣾⠁⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣤⡤⠶⠶⠦⢤⣼⠇⠀⣸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠈⠉⠉⠉⠉⠹⢱⡏⠀⠒⠲⡖⠚⠋⠁⠀⣠⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢿⣄⠹⣍⠉⠒⣒⣈⣉⣁⣇⣀⢀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣶⠻⣦⡙⠋⠉⠉⢀⣀⣀⡇⠉⠁⠀⠀⠀⢳
⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⢿⡖⠛⠋⠉⠉⢹⣶⣤⣤⣤⠴⣾⠋
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠿⠀⠀⠈⠒⠒⠒⠒⠒⠂⠀⠀⢸⠀⠀⠈⣿⠀⠀⠀⠀⡾⠁⠀⣰⠋⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⠀⠀⣿⡆⠀⠀⣸⠃⠀⣴⢻⡀⠀⣿
⠀⠀⠀⠀⢀⣿⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⢹⠀⠀⠀⣿⡇⠀⠀⣿⠀⢰⡏⢸⡇⠀⢸⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⡀⠀⠀⣿⡇⠀⠀⣿⠀⢸⡇⠀⢻⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⠀⢻⠀⠀⣧⠀⠘⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⢸⡄⠀⣿⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⢸⠀⠀⠀⠘⡇⠀⡏⣀⣀⣼⠄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⢸⠀⢀⣠⠴⠷⠺⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⣉⡭⠷⡄⠀⠀⠀⠀⢠⡏⠗⡦⣄⣀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣉⡡⠴⢚⡉⠀⠀⠀⢹⠀⠀⠀⠀⡼⠛⠲⢤⣈⠈⠟⢲⢤⣀⡀⠈⠙⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣉⡤⠖⢋⡥⠖⣚⣨⠵⠚⠛⣆⢸⠀⠀⠀⢠⠇⢀⠀⠀⠈⠙⠓⠦⣌⡁⠿⢳⠦⣄⣀⠀⠉⢣
⠀⠀⠀⠀⠀⠀⠀⡞⢀⡜⠁⢀⣠⠤⠒⠋⠁⠀⠀⠀⠀⣿⢸⠀⠀⠀⢾⡀⠈⠁⠲⠄⣀⠀⠀⠀⠉⠛⠲⢤⣁⠛⠹⠖⢾
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠉⠓⠦⢄⣀⠈⠁⠲⠄⣀⠀⠀⠀⠈⠉⠓⠦⣼
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⡇⠀⠀⣠⠖⢋⣉⣉⠙⠲⠄⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣨⠀⡀⢠⠞⣡⠚⣉⣤⣍⠙⠂⠀⢺⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⡏⡼⠁⡴⠁⡠⠄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡋
⠀⠀⠀⠀⠀⠀⠀⣧⣽⠘⡇⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⣚⣩⠷⠒⣿⡉⣿
⠀⠀⠀⠀⠀⠀⠀⡿⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⣀⣠⠤⠖⠚⣉⣡⠤⠖⠋⠁⠀⢀⣀⣻⡧⠿⢤
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⢀⣠⠴⢺⡇⢸⢸⠀⠀⣴⣋⣁⣠⣤⣖⣋⣭⡤⠤⠴⠖⠚⠋⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡆⡇⠀⢀⣠⢴⢚⣭⠰⣼⠀⡇⢸⢸⠀⠀⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⢚⣹
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⣿⣾⢉⣿⣯⡇⢹⢸⡇⢸⢸⠀⠀⡗⠶⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⢹
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⣿⣿⠉⠸⠇⢃⣨⠼⠃⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⢺
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⡏⠋⣀⠴⠚⠉⠀⠀⠀⣼⢸⠀⠀⡷⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⢾
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⠛⠉⠁⠀⠀⠀⠀⣠⠞⠁⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃⡇⠀⠀⠀⢀⣠⠤⠖⠋⠀⢀⡴⠋⠀⠀⣟⠒⠒⠚⠉⠉⠀⠀⠀⠀⣀⣀⣠⠤⠴⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠙⠒⠚⠋⠉⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠙⠶⠶⠒⠒⠒⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠾⣇⡀⣀⣀⡤⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠿⢿⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⡿⠿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⣿⣿⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠎⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠱
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠉⠛⠛⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠛⠛⠉⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⡤⠤⠶⠶⠶⠒⠒⠒⠒⢛⣛⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣖⣒⡛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⢋⣭⡅⠀⣷⡤⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡥⠤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⢸⡥⢋⡴
⠀⠀⠀⠀⠀⠀⠀⡼⣇⠉⠛⠿⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠈⣷⠏⡰⠂
⣤⠤⠤⠤⠶⠖⢺⡷⠄⠀⠀⢠⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠞⣡⠄
⢻⠀⠀⠀⠠⢯⣿⣙⠀⠀⢠⣟⣛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢁⡴⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⢸⡇⠀⠀⠐⣾⢧⡈⠀⣰⣏⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠋⣠⠀⠀⠀⠀⠀⠀⢀⡞⣡⣍⠋⢽⠋⡛⣦⡀
⠈⡇⠀⠀⣉⣟⠦⠀⣴⠧⠶⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡤⠤⠆⠀⠀⠀⢀⣀⣀⣀⣀⡄⠀⠀⠀⣧⡞⢡⠄⠀⠀⠀⠀⣰⢋⡞⢹⡃⠀⡸⢛⡿⣿⡇
⠀⡇⠀⠠⢼⡏⠁⡼⠷⠖⠀⢛⣋⣋⣉⣩⣭⣥⠤⠤⠶⠶⠒⠒⠛⠛⠉⣉⣉⣁⣀⣄⠀⠀⠀⠀⢻⡴⢋⡄⠀⠀⢀⣼⡵⠋⣴⠏⠳⣞⣱⢋⡴⣹⠃
⠀⣧⠀⢐⣺⠀⣼⡛⠉⠀⠀⢉⣉⣁⣀⣤⣤⠤⠤⠴⠶⠖⠒⠚⠛⠉⠉⢉⡉⠀⠀⠀⠀⠀⠀⠀⢸⣥⠏⡤⠀⣠⢿⡟⣡⠞⠁⣤⠳⢤⣩⠟⠛⠁
⠀⢿⠀⢨⣽⣼⡋⠁⠀⠀⠀⠉⣉⣉⣁⣀⣤⣤⠤⠶⠒⠒⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⡞⣡⠞⠁⣼⢾⠁⢀⡰⢬⣑⣲⠋
⠀⢸⡀⠐⣿⣯⠁⠀⠀⠀⠀⠈⠉⠉⠉⢁⣀⣀⣠⣤⡤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⣛⢲⡾⠉⠀⡐⢦⣙⣲⠟⠁
⠀⢸⡇⢈⣿⠆⠀⠀⠀⠀⠀⠐⠛⠛⠋⠉⠉⠁⠀⢀⣀⣀⣀⣀⣤⡤⠤⠤⠴⠶⠖⠒⠀⠀⠀⠀⠀⢸⣷⡵⠋⠀⠀⠦⣙⢢⡼⠃
⠀⠈⡇⠀⢾⡆⠀⠀⠀⠀⠀⠰⠶⠶⠒⠚⠛⠛⢋⣉⣉⣡⣤⣤⠤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢤⡙⠲⣤⠏
⠀⠀⡇⠀⣩⡇⠀⠀⠀⠀⠀⠀⠶⠒⠒⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢠⡙⠦⣬⡟⠁
⠀⠀⣷⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⡰⢜⡲⣭⡶⠋
⠀⠀⢻⠀⢈⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠹⢤⣙⣲⣿⢋⡞⠁
⠀⠀⢸⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⡀⠀⡀⠀⠀⠀⠀⠶⠤⠤⠶⠚⠀⠀⢠⡏⡀⢬⣳⠞⠁⢹⠏⡴⠂
⠀⠀⠸⡇⠀⢚⡇⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡾⢁⡼⢁⣾⢁⡖⠀⣴⠀⠀⡀⢠⠦⣤⣯⢗⡿⣋⡵⠀⠒⠸⢟⡱⠋
⠀⠀⠀⣧⠀⣩⣧⠀⠀⠀⠀⠀⠀⣰⠏⠀⣼⣡⠞⠧⠋⠳⠛⠧⠊⠙⠒⠋⠉⠉⣀⣀⣀⣈⣠⣥⣽⡤⠚⣁⣟⡟⣡⠆⣰⠂⣠⠀⣀
⠀⠀⠀⢻⠀⠐⣻⠀⠀⠀⠀⠀⣈⣘⣤⣞⣤⣥⣤⠤⠴⠶⠶⠶⠒⠚⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⣿⡼⢁⡞⢁⠞⢁⡼⠁⡴⢃⡤⢀⠀⡀
⠀⠀⠀⢸⡆⠠⢽⡆⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠏⣰⠋⣠⠟⣠⠞⣠⠏⡴⢋⡜⢡⡞⣠⠆⡀
⠀⠀⠀⠀⣇⠀⢚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⡴⠁⠀⠁⠀⠁⠈⠁⠞⠀⠋⠀⠏⣰⠃⡼⢡
⠀⠀⠀⠀⢻⠀⢩⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡟⣠⠊
⠀⠀⠀⠀⢸⡆⠐⣺⡄⢧⠻⠰⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡤⠤⠴⢶⣒⣒⣚⣛⣿⣋⡟⢉⡟⢁⡴
⠀⠀⠀⠀⠀⣇⠀⠺⣧⣤⣤⣤⡤⠤⠤⢴⣶⣶⣒⣒⣺⣿⣭⣭⠭⠤⠶⠖⠒⠚⠋⠉⠉⠉⢁⣀⣀⣽⠋⣠⠏⠀⠋
⠀⠀⠀⠀⠀⣿⠀⠐⠒⠒⠒⠚⠛⠋⠉⠉⠉⣉⣉⣀⣀⣀⣠⣤⡤⢤⠶⢶⡖⢒⡾⢛⡿⢋⡿⠉⡽⠁⡴⢡⠖
⠀⠀⠀⠀⠀⠸⠶⣶⢶⡖⣲⠞⣻⠛⣻⠋⣹⠋⣹⠋⣠⠞⣱⠋⣰⠋⢰⠏⠠⠏⠀⠞⠀⠛⠀⠈⠁⠈⠁⠉

⡴⠊⠉⠉⠙⠓⠒⠂⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⢤⡀
⡇⠀⠀⠀⠤⢄⠄⠒⠄⡠⠄⡄⠤⢄⡠⠤⢤⠀⠀⢄⠀⠀⠤⡀⣀⣀⠀⢀⡀⡀⢀⠀⠀⢱
⡇⠀⠀⡇⠀⢏⠀⡁⠆⡁⠸⡅⠀⠀⡭⠀⢇⠁⠀⠊⠀⠀⠰⣁⢐⠂⢄⢈⣼⡀⠆⡁⠀⢸
⡇⠀⠀⠈⠐⠊⠑⠋⠊⠙⠒⠂⠒⠈⠑⠒⠂⠀⠊⠀⠀⠈⠐⠉⠀⠈⠉⠉⠈⠁⠉⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⣶⠀⠉⠟⠃⡆⠀⠀⠀⠄⡄⠀⠠⡄⢠⡀⡀⠀⡄⠀⡀⠀⢄⢀⡀⢀⠀⠀⠀⠀⢸
⡇⠀⠀⣧⠀⠀⡀⠀⡇⠀⠀⠀⠁⠁⠀⠈⠁⠀⠁⠈⠀⠀⠈⠓⠀⠃⠘⠃⠉⠃⠀⠀⠀⠸⣄⡀
⡇⠀⠀⠋⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡄⢀⠒⠀⠠⠀⢄⠄⢄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄⡀⠀⠉⠒⠤⣄⡀
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⣅⣈⠁⣀⡁⠅⢸⠸⠈⣄⠑⠁⡔⣔⣢⣒⡄⠀⠈⠚⠧⣨⠒⠂⠀⠀⠀⠈⠑
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠓⠀⠐⠓⠀⠚⠀⠂⠀⠈⠀⠉⠉⠁⠉⠁⡀⠀⠀⠀⠀⠉⠚⠌⠘⠀⠀⢀
⠀⠓⠦⠤⠤⠤⠤⠤⠤⠤⠤⡄⠠⠤⢤⠤⠤⠤⠤⢠⠀⡤⠤⡤⠀⠤⠤⠤⠤⠄⠠⠀⠊⢐⡐⠠⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⠈⠉⠙⠃⠢⡘⡐⠐⢂⡄⠀⠀⠔⠢⡀⠀⠀⠀⠀⠀⠀⠓⠐⡟⢠⠄⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢤⣀⡀⠈⠉⠁⠀⠀⡄⠘⢀⠴⢈⡵⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣄⡀⠈⠉⠀⢄⣛⠀⡀⢠⢀⠂⠠⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⡀⠀⠙⠒⠂⢀⠁⠤⠈⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⠈⠉⠀⠀⢠⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠻⢷⣶⣤⣴⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣤⣦⣠⣤⣤⣤⣬⣭⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠑⠂⠀⠀⣸
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠒⠒⠲⠶⣤⣤⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⢻⣿⣿⣿⣿⡟⠩⠐⠆⢻⣿⣿⠇⠀⠀⠀⠐⠛⠻⢷⣤⣄⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⣆⣴⠀⠸⣿⣿⠀⠀⠀⠀⠀⠲⠆⠀⢸⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⢸⣿⣿⣿⣿⡆⠸⠇⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠙⠻⢿⣿⣿⣿⡿⠠⡄⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠆⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡟⠲⠄⠀⠀⠀⠀⠀⠀⣈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣄⠀⠉⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠓

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣫⣴⠾⠛⠛⠻⣿⣿⣤⠤⠴⠶⠤⢃⠈⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⠟⠁⠀⠀⠀⠀⠈⣏⢹⠦⣀⢀⠀⠠⠀⢙⣢⠀⠰⠊⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠋⠀⠀⠀⠀⠀⠀⠠⢈⣁⡠⢤⠳⠒⠃⠉⠻⣿⠠⠰⠰⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠤⠈⠁⠀⠀⠀⠀⣀⠁⠀⢐⣀⣠⣱⡶⢻⠛⢹⠸⠃⠀⠀⠀⠀⢀⠴⣿⠀⠀⢀⠴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⢀⠀⠄⢐⣢⣄⠀⢱⣶⢿⡛⠋⠉⣈⠡⠀⠀⠘⠀⠀⠀⣤⠀⠀⠈⢦⣻⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⢀⠔⢈⣥⣶⣿⣿⡿⡿⠐⠸⣏⠐⠁⠄⢈⣴⡶⠈⠀⠀⠀⠀⠀⠂⠑⠄⠀⠀⢹⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢽⠀⠀⠀⠎⢠⣿⣿⡿⣯⣿⣧⣿⠠⠀⡏⢀⠀⣰⣾⡿⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠸⢸⣿⣿⣟⡿⠋⢿⣷⠶⠚⡇⠀⠀⠟⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠸⠀⣿⡿⡿⠇⠀⠈⢿⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⠖⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⣇⣟⡠⠒⠦⢼⠆⠰⠃⠀⠀⣟⣃⡒⠤⢀⠀⠀⠀⠉⡅⠀⠀⠀⡀⠀⢘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⣿⣿⡜⢱⡁⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠑⠃⠀⠀⠓⠀⠢⢄⠘⠆⢸⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⣿⢻⡆⠮⡁⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⢸⠀⠀⠀⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⢻⣨⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⢈⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⡁⢸⡇⡀⡀⠀⠀⠠⠀⠀⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⣴⣿⣿⣟⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⢸⣿⢁⡇⠀⠀⣰⡆⠀⢺⠉⠢⡄⠠⠐⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠟⠏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⡀⠀⠀⢸⠈⣿⣿⣿⡶⠀⡿⡇⠀⢸⡄⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⡿⠛⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣶⣤⣀⠸⠅⣿⣻⣿⣇⡆⡇⣿⠀⢸⣷⣀⣠⣴⣶⣿⣿⣯⡛⠲⢭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⠛⠿⢿⣦⣿⣿⣿⣩⣛⣧⣽⣤⣾⣿⣿⢿⣿⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠍⠓⠦⢶⠝⠻⠿⣿⣿⣿⣿⣿⣿⡿⢿⡻⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⢁⠀⠀⠀⠉⠋⠛⠙⠢⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⢀⣄⣀⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣷⣶⣶⣤⣤⣄⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⢀⣾⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠋⢀⣾⡿⠁⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⢀⣾⡿⠁⠀⢰⣿⣿⣿⣿⠋⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⢀⣾⡿⠁⠀⢠⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⢀⣾⡿⠁⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣆⠈⢻⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠃⠀⢺⣿⣅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠉⠻⣿⣿⣿⠟⠀⠀⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠀⠉⠻⣿⣦⣀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⣿⣿⣿⣿⣤⣤⡀⠀⣀⡀⠀⠈⠻⢿⣦⣄⠀⠀⠙⢿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣀⡙⢿⣶⣄⠀⠀⠙⠻⣷⣦⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣿⣦⣄⣀⣼⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠸⣿⣿⠟⢿⣶⣴⡿⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⠉⠀⠀⢀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠋⠉⠀⠀⠀⣀⣠⡴⠶⠛⠋⠉⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠞⠛⠉⠁⠀⠀⣀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⣀⣠⣼⣇
⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉⠁⠀⠀⣀⣠⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠋⠉⠀⠀⢻⡄
⠀⠀⠀⣴⠟⠋⠉⠀⠀⢀⣀⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⢀⣀⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⢻⣄⣤⡴⠖⠛⠉⠁⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠈⣯⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠸⣧⡤⠶⠚⠋⠉⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡴⠚⢛⣷⠞⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⡟⠀⠀⣾⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣿⠀⠀⠀⢳⣄⠀⠹⣆⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⣀⣿⠀⠀⠀⠀⠉⠛⠛⠋⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠋



⠀⡁⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠋⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡁⠀⠀⠀⠀⠀⠀⢀⣴⣴⣶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⢦⣤⠀⢰⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⣿⣷⣜⣧⣾⣵⠀⢠⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣤⣶⣾⣿⣿⢿⠟⠛⠷⣤⣿⣶⣦⣦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢉⣾⢻⢏⡇⠞⠈⡠⡀⠀⠆⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠸⣿⠛⠘⠀⣴⠀⠀⡀⣤⣾⣤⣤⣤⣤⡔
⠀⠀⠀⠀⠀⢀⣤⣲⣖⣂⡖⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⠈⠀⠀⢿⡷⠖⠋⠉⠅
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⢠⣶⡀⢀⡀⢰⣿⣀⣤⣤⣤⡾⠁
⠀⠀⢸⣿⡿⢿⣿⣿⣿⡇⡇⠈⣄⣀⡀⠀⠀⠀⠀⣠⣼⠏⠀⠀⢀⣴⣿⣿⢿⠿⠟⡩⣥⠀⠀⠀⠀⠘⡄
⠀⠀⢘⠛⠀⠈⠈⢠⢸⡇⡇⠀⢻⣀⣀⡀⢠⣴⣿⣿⠋⠀⠀⠀⠸⣿⣅⣿⢰⣄⡀⣠⣿⣤⣤⣤⣴⣶⡇
⠀⠀⣼⣀⣧⣧⣼⣾⣿⣇⢳⠀⠸⣿⣛⣿⣭⣽⣿⠇⠀⠀⠀⠀⠀⠙⢿⣿⣼⣟⠙⠉⠉⠉⠉⠉⠉⠉⢱
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠸⡀⠀⢻⣿⠿⠿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡇⢧⠀⠘⡇⠀⠀⢸⡃⠀⠀⠀⠀⠀⢠⣶⡄⣿⠀⣿⢘⣿⡄⠀⠀⠀⠀⢤
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡸⣄⠀⠙⠀⠀⢺⣵⣄⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣾⠿⠁⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣷⣻⠀⠀⠀⠀⣄⣿⣿⣿⣶⡄⠀⠀⠀⠀⠻⠉⠿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠆
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣦⣤⣤⣤⣄⡀⣀⣀⠀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁


⣾⣷⠿⠿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣷⣶⣦⣤⣄⣀
⠀⠀⠀⢀⡀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⢀⣀
⠀⠀⠀⠿⣿⣿⣾⣿⡿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠉⠉⠛⠛⠿⢿⣿⣿⣾⣿⡿⠃
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠀⠀⢀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡃
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢁
⠀⢶⣤⣤⣤⣤⣦⣤⣤⣤⣦⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢤⣧⣤⣤⣤⣦⣤⣤⣤⣴⠄
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦

⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠶⠒⠒⠒⣛⣛⣛⣉⣩⣭⣭⣭⠭⠥⠄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⣚⣛⣉⣉⣉⣉⣀⣠⣤⡤⠤⠤⠶⠶⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠭⠭⠭⠭⠥⠴⠶⠶⠒⠒⠒⢛⣛⣛⡁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠲⠒⠒⠒⠚⠛⠛⣛⣉⣉⣉⣉⣭⣭⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣛⣛⣋⣉⣉⣉⣩⣭⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⣉⣉⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣛⣛⠶⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣾⣿⣿⠿⡜⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠻⣿⣻⡿⢣⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠷⠖⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⣿⡿⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢟⣯⣷⣿⣿⣿⣿⣿⣮⡻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠛⠛⠛⠛⠛⠻⠻⠿⠿⠿⠿⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡿⢟⣣⣴⣾⣿⣿⣿⣿⣶⣶⣤⣀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⣫⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠙⢿⣿⣿⣿⣷⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣷⠦⣀⠀⠀⠀⠀⠀⢀⡀⢴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠁⠀⠀⠉⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠿⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⠿⣟⣿⡇⠀⠀⠀⠀⠀⠠⢤⡤⣤⠤⠀⠂⠀⠀⠀⠀⣿⣛⡿⢿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⣟⣫⣭⣷⣾⣿⠟⠁⠈⠢⡀⠀⠀⠀⡴⠋⠀⠈⠳⡀⠀⠀⠀⡠⠊⠀⠙⢿⣿⣶⣯⣽⣛⡿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⢟⣫⣶⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠑⣦⠀⠰⠀⠀⠀⠀⠀⠰⠀⢠⡖⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣯⣻⢿⣦⡀
⠀⠀⠀⠀⢠⣾⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠅⠂⠀⠀⠀⠀⠀⠀⢊⠍⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣝⣿⣆
⠀⠀⠀⢠⣿⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣇
⠀⠀⠀⣾⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⡆
⠀⠀⣸⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠡⠀⠠⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⡀
⠀⢠⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿⣧
⠀⣾⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿
⢸⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻
⠿⣧⣼⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣴
⣶⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣈⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⣭⣭⠿⠀⠀⠀⢠⠈⠡⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠠⢜⠀⠀⠶⠖⣿⠍⠡⠤⠤⢵⠀⢚⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣉⣿⣉⠘⣛⣛⣽⡇⠀⢒⣚⡿⠀⠥⠵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢤⣷⠀⢈⣁⣀⢽⡇⠈⣉⣭⣿⡀⢒⣲⡧⠀⠀⡀
⠀⠀⠀⠀⢀⠀⡀⡤⢄⡐⢒⡲⠒⠚⠓⠒⠛⠂⠀⠂⠤⢾⡃⠀⠶⣺⣯⠄⣉⡥⣿⢉⣽⡇
⠀⠀⠀⠀⠘⠙⠓⠻⢶⠀⠈⠀⠀⠀⠀⣀⣠⠴⠖⠒⠈⢉⠉⠈⢻⣽⡻⡤⠤⣶⣷⢴⣿⣧⡆⢀⣤⣤
⠀⠀⠀⠀⢀⡀⠀⢈⢹⠀⠀⠀⠀⠀⠀⡿⠿⠷⢤⣬⣦⡶⣰⣭⣿⣶⣒⣖⣒⣲⣾⣽⠿⠿⣯⣽⣿⣿⠷⠂
⠀⠀⠀⠀⠈⠙⠊⠛⠻⢦⣀⠀⠀⠀⠀⣏⠂⠀⠀⠀⠀⡏⠁⠉⠈⠉⠩⣏⠉⠉⠾⣁⣐⣐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠤⠹⠷⢶⣦⣄⣀⣇⣠⣤⣶⣾⠽⠏⠂⠐⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦
⠀⠀⠀⠀⢠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡄
⠀⠀⠀⣠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣄
⠀⠀⢰⣿⣿⡟⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⢻⣿⣿⡆
⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧
⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢹⣿⣿
⢸⣿⣿⡇⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⡇⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿
⠈⣿⣿⣇⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣸⣿⣿
⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠟⠛⠉⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠙⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠘⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠖⠶⠲⠖⠲⠖⠶⠲⠖⠶⠒⠶⠲⠖⠶⠲⠖⠲⠖⠶⠲⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠤⠥⠤⠤⠬⠤⠥⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠲⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⣢⢴⠀⣦⡆⠐⢰⠀⡦⠐⢐⢀⡄⢴⠰⣴⠀⠂⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⣧⣀⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡗⡀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⢀⡤⠴⠶⠦⠤⢤⣖⠢⣤⣀⡀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠉⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣾⣶⣾⣾⣷⡀⠀⠀⠀⠀⠀⣠⠴⠶⡿⠷⠦⣄⡀
⠀⠀⠐⠦⠀⠀⠀⠈⢳⣀⣈⣉⢀⠞⠀⠀⠀⣀⠴⠚⠁⠀⠈⠉⠉⠉⠉⠀⠙⠢⣄⠀⠀⠀⠈⢦⣀⣀⣀⣀⡜⠁⠀⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⣀⠼⢿⠿⢿⠿⣄⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠿⢿⡿⠿⠿⢷⡀
⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⢀⡤⠶⠧⢤⠀⠀⠀⠀⠀⠀⠈⢷⠁⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⡴⠁⠀⠀⠀⣠⠤⣄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣠⠦⡄⠀⠀⠀⠈⢢⡀
⠀⠀⢸⠁⠀⠀⠀⠀⢧⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢆⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⣅⣀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⣇⢀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⡏
⠀⠀⠘⡄⠀⠀⠀⠀⠈⠛⠉⠀⠀⠘⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠉⠛⠁⠀⠀⠀⢀⡜⠁
⠀⠀⠩⢞⡦⢄⣀⣀⣀⣀⣀⣀⣀⣠⣼⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣴⡯⣀⡀⠀⠀⠀⠀⢀⣠⡔⣋⣀⣀
⠀⠀⠀⠀⠀⠞⠁⠼⠋⡽⢋⠜⢋⠔⢁⠔⢁⠞⣩⢟⡿⢛⡿⠋⣹⠟⣹⠋⡹⢋⡽⠁⠎⠰⠋⠞⠙⢫⠟⡽⠁⠞⠙⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠀⠈⠀⠀⠁⠀⠁⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠠⠐⡢⠒⠠⠀⠀⠀⡀⣀⢀⠀⠀⡀⠀⠀⠀⢀⣀⢀⠀⠈⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠄⠵⠁⠔⠀⠀⠀⠉⠊⠘⠀⠁⠁⠀⠀⠉⠋⠉⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠰⠀⠆⠆⠠⠰⠀⠂⠆⠰⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠵⠞⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⡷⣿⡷⣿⣿⣿⣹⡭⣯⣿⣻⣽⣿⣿⣋⢋⣉⣿⣿⣿⡇⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠉⠉⠉⣿⣿⣿⣿⣿⡿⣿⣿⢿⢿⡿⣿⣿⢿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣷⣿⣿⣾⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢛⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⡿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠁⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠻⠿⣤⡼⠿⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠙⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⠀⣀⡀⣀⣀
⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣴⣿⣷⣤⣤⣤⣤⣤
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