Odoriferous Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⠄⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠒⠋⣍⠀⠀⠒⠂⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢊⠡⠐⠀⢋⠔⠁⠨⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠒⠈⠀⠀⡐⠡⠁⠀⠀⠀⠈⠢⡑⢄⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢣⢆⠜⣐⢠⠀⠀⠀⠀⠀⢶⡈⢆⡛⠊⡃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣀⡡⠶⡰⠂⡀⠀⠀⡠⠈⢻⣇⡠⠓⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠠⠀⠀⣳⢄⣀⣃⣠⡴⠒⣁⡠⠘⠉⡠⠉⡍⢀⠡⡀⠀⠻⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠄⢀⣀⠄⠈⢀⠊⠋⠹⠑⠤⢀⠀⠀⡠⠞⠀⣰⠁⠘⠀⢱⠀⠀⠀⠑⢌⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⠔⠀⠉⠀⠀⢠⠃⢰⠀⠀⠑⢄⠀⠉⠛⢄⡠⠊⠀⠀⡆⠀⠀⡆⠀⠀⠀⠀⠁⠀⠑⢄
⠀⠀⠀⠀⠀⢀⠤⠊⢁⡰⢊⠂⠀⠀⠀⠀⡎⠀⡆⠀⠀⠀⠠⣢⡀⡠⠊⠀⡆⠀⢰⠀⠀⠀⠹⡄⠀⠀⠀⠘⡀⠀⠀⠣
⠀⠀⠀⠀⢠⢃⠔⠈⠀⠀⡈⠀⠀⠀⠀⡐⠀⠀⡇⢀⠀⠀⠀⢻⡨⠁⠀⢠⡆⠀⠘⠀⠀⠀⠀⠙⣄⠀⠀⠀⠱⠄⡀⠀⠁
⠀⠀⠀⠀⠞⠀⠀⠀⠀⢠⠁⣀⠤⠒⢹⠀⠀⠀⠋⠢⡐⢀⠀⠈⠇⠀⠀⡞⡆⠀⢠⠐⡀⠀⠀⠀⠂⠈⠒⠤⡀⠑⣌⠢⣴⡀
⠀⠀⠀⠀⠘⣦⠀⠀⢀⢣⠊⠀⠀⠀⠈⡀⠀⠀⠀⠀⣈⠑⠳⡀⠀⠀⠰⠖⡸⡀⠘⡀⢱⡀⠀⠀⢀⠀⠀⠀⠈⢢⡈⠑⣤⣇⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠸⡆⠀⡞⠁⠀⠀⠀⠀⠀⢁⠀⢀⠀⠀⠘⢇⡠⠰⡐⡀⠀⢋⠀⠐⡀⢡⡀⠁⠀⠀⠘⠀⠀⠀⢸⣧⡉⢳⡞⣽⣛⣒⣶⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⢀⣀⣀⠀⠀⠀⢄⠐⣀⡙⡄⠀⠈⠉⠙⢵⠀⠸⡀⢀⠸⠄⣳⡀⠙⡀⢸⠀⠀⠀⠀⠙⠙⠿⠞⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠉⠁⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⠈⢣⠀⢓⠀⠀⠀⠉⠻⢦⡙⡈⢄⠀⢙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠲⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡈⠀⠀⠀⠀⠀⠀⠹⠜⢦⣍⠡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠴⠗⠒⠒⢲⣄⣤⣤⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢰⠙⠂⠀⢀⣀⠤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣤⠏⠸⡄⠀⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠸⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠷⠲⠥⡀⠀⠁⠀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⡔⠅⠀⠐⠀⠘⢄⠀⠀⡠⠛⠉⠙⢋⠿⠛⠉⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢄⠂
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⡠⠊⠁⠀⠀⠀⢘⡤⠄⠀⠠⣇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢰⣁⣠⣄⣤⣤⣤⣤⣤⣤⣤⣱
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠀⢀⠴⠂⠉⠉⠀⠈⠛⠋⠉⠀⠀⢀⣠⣿⣿⣿⢃⠠⠔⠊⣩⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⢀⠠⠐⠶⠾⠿⠿⠛⠋⠁⠀⢀⡠⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠒⠐⠒⠂⠒⠐⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠈⠙⢦⣀⡤⠖⢒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠀⠈⠈⣧⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠄⠀⠀⠈⠉⠀⠀⡠⠤⡴⠛⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡷
⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠤⠤⠤⠤⢿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠐⠓⠢⣄
⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⢤⡀⠀⠀⠀⠀⠀⠀⠐⢤⣀⠴⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠆⢸
⠀⠀⠀⡇⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⣤⠤⠤⠖⠋
⠀⠀⠀⡇⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⠤⠖⠋⢧⠀⠀⠀⠀⠀⣄⠀⠈⠀⣠⠇
⠀⠀⠀⣧⠀⠀⠀⠀⠐⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⢒⣊⣀⣙⣒⣛⣩⡆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠑⡖⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⡇
⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣇⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠤⠤⠤⠤⠴⠶⠶⠖⠒⠒⠒⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄⠀⠀⢀⠀⡄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⣆⢁⡇⠀⠀⠀⠀⠀⢸⣾⠃⠀⣄⢁⢿⠇⠀⠠⠀⠀⢀⠀⠀⡇
⢰⣿⠃⠀⠀⠐⢉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡀⠀⠄⣠⣤⣤⡚⠃⠀⠀⢸⣿⣹⣷⠰⠀⠀⠀⠈⡄⠀⡷⠀⠄
⢼⣿⠀⠀⠀⠀⠉⠀⠄⠀⢀⠀⠀⠀⠠⢄⠐⢄⠀⠀⠨⢸⡧⣀⠘⢿⣶⡿⠃⠚⠠⣀⣈⠉⠟⠋⠀⠐⠀⠀⠀⠃⠀⠁
⢸⣿⡄⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⢀⡀⠀⢦⠀⢳⣄⠘⣾⡇⢠⠈⠀⢁⢷⡦⠴⢀⣸⡗⠁⠰⠀⠀⠀⠀⠱⠀⢺⠱⡟⡀⠠⣴⢾⣶⣦⠈
⠈⣿⣿⡄⠀⠀⠀⠀⢀⣀⡀⠀⠤⡴⡀⢰⣜⠆⠀⠙⠀⢿⠇⡌⠀⠀⠀⢰⣷⠖⠋⠉⠀⠀⠀⠀⢀⠀⠀⢦⡀⣸⣄⡄⣚⡋⠼⢯⣽⣿⡗⢄⠀⢀⡀
⠀⠘⣿⣿⣦⣀⠀⠀⠀⠀⢄⠀⠀⣀⣉⣠⣤⣴⣤⣴⣤⣼⣄⣻⡃⡸⠃⣘⡣⢀⢍⣅⠀⠈⡡⠀⠢⠄⠀⣈⠙⠵⣿⠀⠉⠄⠀⣜⣩⣿⡋⠱⣶⢋⡀
⠀⠀⠈⠻⣿⣿⣷⣦⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣔⠢⢍⡀⠀⠀⠀⠀⢀⡀⠈⢈⠀⢀⡽⠄⡠⠐⠒⣫⣾⣿⣿⣿⣯⡿⡀
⢐⣒⣀⣦⣈⡻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠞⣀⡵⠾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⣒⡍⡀⠀⢀⣾⡥⠷⣶⣛⣉⣫⣥⣿⠛⠛⠩⠹⠿⠦
⢀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⢸⣿⣿⣯⣲⠀⡴⠋⠀⠀⠙⠿⣟⡿⣿⣿⣿⣦⣴⠀⠀⠠⣮⣤⣬⡿⢷⡐⣈⡿⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠
⢺⢠⡺⠿⠿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡥⠀⢰⠁⠀⠀⠀⠀⠀⠙⡃⣾⣿⢿⣿⣿⣷⣄⢔⠾⠿⠋⠀⢀⣈⠙⠁⠀⠀⠀⣀⣀⣠⣴⣿⡿⠁
⠀⠀⠻⣤⣨⣟⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⠀⠀⢠⣃⡀⠀⠀⠀⠈⠈⠝⠿⣿⣿⣿⣷⡀⠀⣁⠀⠂⠈⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠁
⠈⠀⠀⠹⣆⠀⠀⢠⠟⡿⣿⢿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣧⣄⣨⣿⡁⠀⠀⠀⠀⠰⡶⠖⠚⢻⣿⣿⣿⣄⡀⠀⠀⣀⣼⣿⣿⣿⣿⣋⣿⣿⡿⠁
⠀⢀⡓⡂⢷⡀⠀⠸⡀⠀⠹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠟⢀⣤⠄⠊⠡⠀⠀⠀⠀⠜⣻⡟⢻⠟⠟⠉⡟⠘⣱⣺⣽⣿⣿⣿⣿⡟⠀⠀⠀⡀
⠀⠀⠈⠁⣸⣷⠀⡿⠁⠀⢤⣀⡛⠃⠈⣿⣿⣿⣿⣿⣿⠞⣿⣿⣿⠛⢁⡆⠀⠀⠀⠀⠀⠀⢀⡍⠁⠘⠀⠂⠀⠁⠰⢬⣛⣿⣿⣿⣿⣿⠃⠀⢠⡴⠖
⢰⠶⣦⣄⣙⣻⠸⣯⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⣨⡿⡆⢻⡇⡧⢀⣼⣷⣶⣄⢀⠀⠀⠀⠀⠧⡀⠀⠀⠀⠀⠀⡀⠈⠙⣿⣷⣿⣿⡟⣰⣬⣾⣷⣶
⢨⣰⠏⠉⢳⣿⣞⣿⣿⢿⣧⣻⣿⣿⣟⢿⣿⣿⣿⣿⠟⢉⠈⣿⣷⣿⢹⣿⣿⣿⣧⠘⠃⠀⠀⠀⠋⠀⠄⠀⠣⣀⡀⠀⢘⣳⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠛⠛⠂⠈⢛⣿⣿⢯⠀⣸⡿⢿⣿⣹⣿⣾⣿⣿⣿⣶⣾⣾⣷⠸⣿⡟⣸⣿⣿⣿⣿⠀⠀⠀⠐⠀⠀⣀⠘⠀⠀⠈⠀⢀⢬⣛⣿⣿⣿⣿⣿⣿⣿⢿⡿
⢰⣶⢄⣖⣩⣭⡉⢠⠲⣿⣿⣾⣿⣿⣿⣿⣿⣿⢽⣿⣿⣿⣿⣧⣿⢿⣯⣿⣿⣿⡿⠟⢛⣾⣿⣿⣿⣼⣶⣶⣷⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣘⡛⠟
⢸⢱⡎⣹⠳⣫⠟⣮⣁⠈⡿⣿⣿⣿⣿⢿⣿⣿⢿⣿⡿⣷⡿⣿⣿⣾⠇⣼⣿⣿⣶⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⢟⣉⢉⡉⠉
⠀⠀⠀⠛⠶⣫⣽⣷⢏⣶⡈⢤⡎⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣮⣻⣯⣼⣿⣿⣿⣿⣿⣿⣧⣈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠽⢿⣿⠿⠁⠀⠀⠀⠉
⠀⠀⢰⡄⠘⠿⠿⣷⣿⣿⣾⣽⣷⡿⢿⣟⣽⣿⣿⡿⣼⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣶⣌⠛⠟⠻⢹⣿⡿⣞⣫⡿⠖⠻
⠀⠀⠀⠀⠐⠀⠂⠠⠨⠁⠢⠘⠁⠰⡞⢩⣿⢿⡿⢾⣿⣿⡿⣿⣿⣆⢻⣿⡿⢿⠷⠿⢷⢄⣂⣴⡚⠛⠶⠿⠄⠌⠛⠚⠃⠐⠋⠒⠁⠀⠂⠀⠚

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠻⠟⢻⣟⠙⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣴⡦⠀⠈⠛⠃⠐⠿⣷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡄⠀⠀⢀⠖⢁⡀⠰⠂⠀⠀⠀⠰⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡆⠀⠀⠀⣀⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⣿⡿⠀⠀⢸⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢁⣤⣉⠉⢶⠋⣱⡄⠀⠀⠉⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⣙⠟⠉⠉⠙⠳⡄⠀⢉⣁⡶
⠀⠀⠀⢀⡄⠀⢀⡀⠉⠀⢰⣿⠗⠒⠒⢠⡄⠁⠀⠀⢁⣄⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⣧⠀⠘⢧⣤⣤⣠⣿⣦⣄⠀⣸⣧⣤⠜⠓⠈⠉⠿⠀⠛⢠⣤⣤⣤⠤⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡿⠹⣿⢿⡟⠡⡔⠓⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠤⠀⠀⠠⣴⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⢶⣦⠀⠀⠊⠀⠀⠈⠙⠷⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠰⢆⡀⠀⠐⠲⣌⠉⠹⠶⠶⠦⣤⣤⣠⣤⠴⠶⠶⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡇⠸⣆⠘⢧⣄⠐⠂⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠛⣦⠈⣿⠄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡁⢀⡀⠀⠙⠋⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠘⣿⡌⠋⠉⡳⠄⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⡄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠉⢉⡁⠈⣻⠶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⡇
⠀⠀⠀⢠⡆⠀⠺⣅⣀⣉⣴⣦⣼⡏⠀⣿⡄⢰⡆⠀⠀⠀⠀⠀⠀⠀⢆⡘⠃⠸⡇
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⡀⠙⡿⢳⣶⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠚⠀⠀⠉⣿⣷⠈⢣⠀⠀⠀⠀⢀⠏⠀⡌⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⡁⠀⣸⡇⠀⠀⢀⠟⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠃⠀⢸⡇⠀⠀⠿⠀⠀⡀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⡀⢾⠀⠀⠠⣿⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠇⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢸⡆⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠸⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣈⣉⣉⣉⣉⣱⡿⠟⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⣀⣀⣀⣀⣀⣶⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠛⠉⠙⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣟⣀⣀⣀⣀⣀⣀⣀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣸⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢿⠟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣂⣹⣿⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣟⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡾⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⢰⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⢿⡄⠀⠀⠠⠄⡠⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣀⠀⠀⢀⣰⣦⣞⣻⠉⠀⠤⠀⠀⠀⠀⠀⠀⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣤⣿⣿⡃⠀⣈⣻⣿⣿⣿⣛⠻⢦⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢶⣾⠛⠿⢿⡏⢨⡏⢻⣿⣷⣀⣻⣛⣠⣤⣿⣩⣷⣤⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣿⡏⢹⣧⠘⣿⠀⠀⣾⣧⣼⡇⠀⠀⠛⣿⣿⣫⣤⢠⡿⠿⠿⠿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⢰⡞⠁⢀⣀
⠀⠀⠀⠀⠀⠀⣠⣾⡅⢠⡿⠃⠸⣧⠀⢿⠀⠀⠿⠃⠸⠃⠀⠀⠚⠉⣿⠛⠧⣤⠤⣴⣂⡤⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⣿⣁⡠⠬⢿
⠀⠀⠀⠀⢀⣾⣿⣿⠋⠘⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⡿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠓⠀⣀
⠀⠀⠤⣤⠎⢹⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣷⣤⠈⠛⠻⠿⠆⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠀⠀⠈⠉⣿⣿⣿⡟⠠⡤⡤⠙
⠀⠀⠐⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠂⠀⠀⠀⠀⠀⠀⠀⠛⣿⢿⣿⣆⠀⠀⠀⠀⠀⣠⣴⣿⠿⣿⣯⣽⡶⠖⣀⣴⡟⠭⠙⠋
⠂⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡎⠻⣿⠄⠀⠀⠀⠀⠻⢏⡠⠄⠀⠻⠿⠒⠚⠋⠉
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠰⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰
⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⠞⠀⠀⢈⣭⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡸⠇⠀⠀⠀⠀⠀⠀⠀⡀⠖⢀⠀⠀⡀⠀⠙⡙⢛⣦⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣤⣶⣠⠟⠀⠀⠀⠀⠀⠀⣀⣀⣀⡁⠀⣈⣀⢀⡁⠀⢸⣿⡟⢻⣧⠀⠀⠀⠀⠀⠠⠜⣻⣿⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢙⡿⠁⠀⠀⠀⠀⠀⠀⢸⡿⢏⠛⣧⡶⣘⣷⡈⠻⠇⠘⣿⡿⠦⠛⣧⠀⠀⠀⠀⠀⡜⢋⣠⣶⣶⣾⢷⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠋⠀⠀⠀⠀⠀⠀⠀⢰⡟⣉⣼⠛⣹⣧⠙⣿⡇⠺⣶⠀⣿⠿⠂⠀⠘⣧⠀⠀⢀⡾⣡⣾⣿⣿⣿⣤⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⣾⣿⣙⣻⣿⠰⣷⠌⠑⢿⠞⠀⠀⠀⠀⠀⠀⠀⣴⡾⢿⣷⣾⣷⣿⣿⣿⣿⣷⣿⣿⡇⠈⠓⠂⢤⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⢀⠉⠋⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠓⢞⣻⢷⣿⡿⠃⠾⠁⠘⠋⠀⠀⠀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⢉⡉⠉⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠘⣇⢸⣿⡁⠀⠀⠀⠀⣀⣀⣼⣄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢤⠈⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡟⠁⠀⢀⣤⣾⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⢠⣾⣷⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⣿⢉⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⣿⣴⣿⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣴⣟⣳⡶⢺⣿⡿⢿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⡟⠙⠿⠁⠀⢁⡾⠃⠈⢷⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⠿⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡟⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣾⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⢸⣿⣿⣿⡿⢇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢛⡀
⠀⠀⢸⣿⣿⣿⠰⣿⣿⣧⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡟⢀⣉⠉⠢⡔⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃⣾⡁
⠀⠀⠸⠿⠿⠿⠀⠘⠿⠏⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠇⠀⠀⠘⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠂⠙⠛⠖⠂⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉



⠀⠀⠀⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶
⠀⠀⠀⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠟⣂⡀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠧
⠀⠀⠀⣿⣿⣿⡇⢠⣾⣄⣀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢸⣏⠘⡀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⢠⣦⣄⢄
⠀⠀⠀⢿⣿⣿⠇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠀⠉⠁⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠈⠛⠁⠀⠁⠊⠁
⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁




⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⣀⡀
⠀⠀⠀⣿⣿⣿⡇⡀⣿⢰⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢠⣶⣶⣦⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢾⡁⠈⡰⣖⡀
⠀⠀⠀⠻⠿⠿⠃⠉⠱⠆⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠇⠀⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠇⠈⠙⠋⠀⠳⠖
⠀⠀⠀⠠⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⠆⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣹⡃⢸⣇⠀⠀⣤⠀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠟⣸⡧⠀⠀⠻⣟⡿⠄⠀⠀⠀⠀⠀⣿⣿⡟⢛⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢾⣦⡔⠀⠀⠀⠀⠀⠋⣀⣀⡀⠰⣉⠃⡀⠀⠀⠀⠀⠀⠛⠋⠙⣿⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⢿⠀⣠⡀⠀⠀⠘⣷⣟⣿⢇⠀⢻⠟⠁⣄⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣬⡷⣬⠁⣀⠀⡓⢼⣧⠀⡀⢤⠀⣹⢲⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠻⡹⠈⠃⠘⣭⣾⡿⣤⡌⠻⣧⣀⠀⢺⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⡿⠀⠛⢯⡐⠅⠀⠉⢾⡟⢚⣤⡗⠉⠀⠚⠏⠀⠀⣶⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣿⣘⣧⡽⠃⠸⡇⠛⢦⡀⣋⢸⡇⢸⠁⡷⢂⠀⠰⠾⠃⠀⣌⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢻⠾⡷⠀⠀⣥⡞⡀⢻⣷⣿⡇⢀⣴⠟⠻⢦⡑⠀⣇⢈⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣰⣆⡿⠊⣻⡍⠋⠀⠹⣇⠀⣿⡀⢦⠀⠀⠚⠻⣾⢡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢻⢀⡍⠶⠉⠘⢦⡗⠀⢻⣾⠃⢏⣰⠆⣑⠂⣠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠻⠄⠲⡤⢞⣆⣹⣶⣾⣯⠀⢴⢳⡌⡉⢠⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⢦⣰⠾⠟⠩⢟⠹⣷⠀⠸⣾⡗⢡⠟⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣤⣄⠘⠢⣀⠀⠀⢠⣿⣇⠀⠘⣧⣯⠠⣾⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡞⠉⠀⠀⠈⠓⢦⣀⢸⣿⡄⠀⣸⠄⠈⠛⠟⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢦⡀⢳⣤⡴⣄⠙⢿⡍⢷⠀⣯⡄⠀⢀⣄⣰⠟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⢠⠔⢺⠷⡾⢷⣦⡼⣧⡠⣿⢿⣿⡏⠃⠀⣤⡜⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠻⢾⣁⣼⢛⡿⢻⣾⣿⡇⠀⠠⢻⣟⠀⠀⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣤⣀⠙⢦⡈⠀⠈⠀⠙⣷⠀⠀⠀⠻⠁⠘⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠈⠙⢦⡀⠀⢀⡘⣷⡄⠸⢆⠴⢀⣿⡶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⣌⠷⣻⣿⢷⡞⠛⣤⠈⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⣽⢻⡟⢲⠗⠲⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠠⣏⠟⣵⡄⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⣿⠓⠛⠗⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢰⡟⠀⠀⠀⠛⣯⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠷⣬⡞⢰⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⠶⢦⣴⣤⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⡉⠛⠷⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠈⠀⠉


⠀⠀⠀⠀⠀⠀⠘⠀⢰⡀⢸⠀⡇⣾⠈⠃⢸⣸⠄⠁⠀⠀⡁⠀⠈⠃⠘⠁⡄⠁⢰⡇⠈⠀⡆⠀⠨
⠀⠀⠀⠀⠀⠀⢠⡀⠸⠄⠺⠀⠟⠏⠰⡄⠸⠻⠀⠀⢀⡀⡁⠀⠰⡄⢀⠄⠀⡀⠈⠁⢆⡀⠃⡀⠀⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⢿⣿⣿⠿⡿⢿⡻⢿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⠿⠿⠿⠟⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿
⣿⣶⣶⣶⣶⣶⣶⣾⣷⣶⣾⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠙⠲⠤⠤⠴⠒⠚⠋⠁⠀⠘⡄⠀⠀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣃⡴⡟⠛⠳⣤⣖⠛⢻⡗⢶⡤⢤⡈⢆⢾⣟⡉⠁⠈⠉⠉⠹⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⢠⠉⠻⣿⠟⢷⣌⢳⣀⠻⣬⣤⣽⣿⣿⠷⠖⣲⣦⣦⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⣤⠖⠁⠀⠀⠙⣤⡄⠈⢿⣻⣻⡻⣍⣤⣨⠕⠚⠛⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠐⢉⠴⠋⠀⠀⠀⠀⠙⢍⡁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⡄⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢄⡀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⣎⡁⠀⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⣿⣥⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡴⠶⠶⠟⠛⠛⠛⠛⠛⠛⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠉⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠚⠛⠛⠻⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⣴⡟⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢧⣄⣠⡾⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣼⣟⣵⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢠⡿⢻⣅⣀⣀⣰⠏⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⣤⡀⠀⢀⣤⣴⣶⣦⣄⠀⠀⠀⣠⣶⡄⠀⠀⠀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⢸⡇⠀⠈⠉⠁⠀⠀⡄⠀⠀⠀⠀⠀⢺⣿⣄⣀⠙⠃⠀⣿⡏⠀⠀⠈⣿⡇⠀⢠⣿⠙⣷⡀⠀⠈⣿⡀⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣾⡇⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⣀⡉⠙⠛⢿⣦⠈⣿⡇⠀⠀⠀⣿⡏⢀⣿⣧⣤⣽⣷⠀⠀⣿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⡾⠏⠀⠙⢿⣦⣤⣾⠟⠁⣼⡟⠉⠉⠉⢿⡧⠠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⢻⡇⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⢸⡿⠚⠶⢦⣄⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⢠⣿⠂⠀⠀⠀⢹⡆⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠤⠤⠤⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠶⣦⣿⠇
⠀⠸⣎⠁⠀⠀⢀⣼⣥⣤⣀⠀⠉⠛⠻⠷⠶⠶⠶⠶⠒⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠈⢿
⠀⠀⠉⠛⢻⠟⠉⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣝⠀⠀⠀⣠⠟
⠀⠀⠀⢀⡏⣼⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠤⠤⠤⠤⢤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣿⠟⠋⠁
⠀⠀⠀⠘⣷⣝⠃⠀⠀⠀⢀⣰⣷⣶⣴⣶⣶⠶⠶⠿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠷⠶⣶⣶⣶⣶⣶⡶⠾⠟⠋⠁
⠀⠀⠀⠀⠈⠛⠿⠶⠶⠶⠛⠁
⠀⣀⣀⡀
⠀⡃⠀⡁⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠀⢰⣿⡿⢿⣿⣆⠀⠀⢀⣴⣿⣿⣿⣿⣦⠀⠀⣿⣿⣿⣿⣷⣄⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⢸⣏⣿⣿⣹⣹⠃
⠀⠀⠀⠀⠀⣠⣿⠻⣿⡄⠀⠀⠀⢸⣿⣧⣾⡿⠏⠀⠀⣾⣿⡟⠉⠉⠙⠁⠀⠀⣿⣿⡏⠙⢿⣿⡇⠀⢸⣿⣿⣀⣀⡀⠀⠀⢸⣿⣿⣁⣀
⠀⠀⠀⠀⣰⣿⣽⢿⣽⣿⡄⠀⠀⢸⣿⡿⢿⣿⣦⡀⠀⣿⣿⡄⠀⠀⣀⡀⠀⠀⣿⣿⡇⢀⣼⣿⡇⠀⢸⣿⣿⠿⠿⠃⠀⠀⢸⣿⣿⠷⠾
⠀⠀⠀⣰⣿⣿⠏⠈⢿⣿⣿⡄⠀⢸⣿⣧⣴⣿⡿⠃⠀⠹⣿⣿⣿⣿⡿⠟⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⢸⣿⣿⣷⣶⣶⡆⠀⢸⣿⣿
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠁⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉
⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⣶⣶⠀⠀⠀⣶⣶⡆⠀⣶⣶⣶⣶⣶⣶⡆⠀⢰⢶⡶⡖⣶⢲⣶⠀⠀⣶⣶⠀⠀⣴⣶⠄⠀⣴⢶⣶
⠀⠀⢠⣿⣿⡿⠛⠻⣿⠿⠀⠀⣿⣿⡀⠀⢀⣿⣿⡇⠀⠛⠛⣿⣿⡟⠛⠃⠀⠘⠛⠛⣿⣿⠛⠛⠀⠀⣿⣿⣤⣾⣿⡟⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⢰⣦⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣍⠀⠀⠀⣿⣻⣿
⠀⠀⠘⣿⣿⣷⣤⣴⣿⡿⠀⠀⣿⣿⠇⠁⢹⣿⣿⡇⠀⣤⣤⣿⣿⣷⣤⡄⠀⣠⣶⣾⣿⣿⠀⠀⠀⠀⣿⣿⠀⠻⣿⣧⠀⠀⣽⣿⣽⣶⣶⣶⡆
⠀⠀⠀⠈⠛⠻⠿⠛⠋⠀⠀⠐⠛⠛⠀⠀⠘⠛⠛⠃⠀⠛⠻⠿⠿⠻⠿⠃⠀⠈⠛⠿⠛⠁⠀⠀⠀⠀⠟⠛⠀⠀⠿⠿⠀⠀⠻⠿⠿⠷⠿⠻⠃
⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢸⣿⣿⣧⡀⣠⣿⣿⣿⠀⢸⣿⣿⣷⡀⢸⣿⣿⠀⠀⣴⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣽⣟⣄⠀⠀⣴⢿⣿⣿⡿⣷⡀⠀⢸⣿⡿⢿⣿⣆
⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⠀⢸⣿⣿⣿⣷⣸⣿⣿⠀⢰⣿⣿⠉⠈⢹⣿⣧⠀⢸⣿⣿⣤⡿⣽⠀⢸⣟⣿⠉⠉⢻⣟⡇⠀⢸⣿⣷⣼⣿⡟
⠀⠀⢸⣿⣿⠙⢿⠟⢹⣿⣿⠀⢸⣿⣿⠹⣿⣿⣿⣿⠀⠘⣿⣿⣄⣀⣼⣿⡇⠀⢸⣿⣿⠯⠟⠋⠀⠸⣟⣷⣄⣾⣿⣿⡇⠀⢸⣿⣿⣿⣿⡄
⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⠀⠘⢿⣿⣿⠀⠀⠻⣿⣿⣿⣿⡿⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠻⣯⣿⣿⣿⣿⠄⠀⢸⣿⡏⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⠀⢠⡶⣶⢶⣶⢶⡶⡆⠀⢰⣶⣶⠀⠀⢰⣶⣶⠀⠰⣶⣶⡆⠀⣴⣶⣶⠀⠀⢶⣶⣦⠀⠀⠀⠀⠀⣴⡶⡶
⠀⠀⠀⠀⠀⠰⣿⣿⡏⠙⠋⠀⠈⠛⠻⣿⣿⡞⠛⠃⠀⢸⣿⣿⠀⠀⢸⣿⡏⠀⠀⢻⣿⣷⠀⣿⣿⡇⠀⠀⠈⣿⣟⡇⠀⠀⠀⢰⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⡀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠸⣿⣿⡄⢀⣿⣿⡇⠀⠀⠘⣿⣿⣼⣿⣿⠀⠀⠀⠀⠸⣟⣿⣤⣶⣤⣿⣻⠇
⠀⠀⠀⠀⢠⣶⣤⣤⣿⣿⡿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⢹⡷⣿⠿⣿⢿⡏
⠀⠀⠀⠀⠈⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠙⠻⠛⠋⠀⠀⠀⠀⠀⠈⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⣠⡀⠀⠀⣤⢤⡄⠀⣤⠤⡄⠀⢠⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣼⣿⡿⠀⠀⠹⣟⣷⣰⣿⣿⠃⠀⠐⠟⠟⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⠀⠹⣾⣿⡶⠃⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣄⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢀⣼⣿⣿⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠁⠈⢻⠟⠁⠀⠀⠀⢿⣿⠇⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡚⠛⠛⠛⠛⠛⠛⡛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢾⣿⣿⣯⣭⣭⣽⠆⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡖⣒⣒⣚⣛⣳⡆⠀⠀⠘⣷⠶⠖⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠖⣻⡿⠟⠋⢿⣨⡭⠽⠿⠿⠿⢿⡄⠀⠀⢀⡿⣤⣤⠶⠶⢶⡶⢶⣶⣶⡶⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⢀⡴⠞⣫⡿⠋⠀⢀⣤⠞⠉⠀⠀⣠⣾⢉⣻⣿⣿⣿⣿⣿⡅⠀⢀⣼⡷⠋⠀⢀⣴⠏⠀⠀⢠⡼⠋⠛⢶⣄
⠀⠀⠀⠀⠀⣿⣤⠞⠉⠀⣠⡴⠋⠁⠀⣀⡴⠛⢁⡿⠘⠷⠶⠶⠶⠖⠚⣋⣴⡾⠋⠀⢀⡴⠛⠁⠀⢀⡴⠋⠀⢀⡴⠋⢹⡇
⠀⠀⢀⣀⣀⣿⠳⢦⣤⣞⠉⠀⠀⣠⠞⠋⠀⣀⡼⠟⠛⠛⠛⠛⠛⠛⣩⡿⠋⠀⣀⡴⠋⠀⠀⢀⣴⠋⠀⢀⣴⣋⣤⠶⢻⣇⣀⣀⡀
⣴⠛⣉⣉⣉⣉⡀⠀⠈⠉⠛⠓⠿⠷⢤⣤⣾⣯⣀⣀⣀⣀⣀⣀⣠⣞⣉⣀⣠⣾⣋⣀⣠⣤⣤⣿⠵⠶⠖⠛⠋⠉⠀⠀⣈⣉⣉⣉⡙⠻⣆
⢿⠀⢿⣭⡭⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⢭⣭⡿⢀⡿
⠈⠳⣤⣤⡴⠶⠖⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠒⠲⠶⢦⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠙⠓⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠞⠋⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠛⢻⡟⠉⠉⠉⠛⠻⣿⣛⣛⣛⣿⠟⠋⠉⠉⠛⣿⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠁⠀⠀⠀⠀⠀⠀⣿⠉⢹⠇⠀⠀⠀⠀⠀⢹⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡆⠀⠀⠀⠀⠀⠈⣧⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢰⡏⠉⠛⠳⠦⣤⣄⣀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣀⣠⣤⠶⠞⠛⠉⣷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣾⠛⠲⠦⣤⣀⡀⠈⠉⠛⠳⠶⣿⣄⣸⡷⠞⠛⠉⠁⣀⣠⣤⠶⠞⢻⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡇⠀⠀⠀⠀⠈⠉⠛⠲⠦⣤⣀⣀⣸⢿⣧⣤⠶⠞⠛⠉⠁⠀⠀⠀⠈⣧⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣴⣞⣿⣷⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⢸⣷⡏⠀⢙⡿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣾⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣷⣶⣻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣠⠟
⣶⣶⣶⠶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⢀⠀⠀⠈⡀⠠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⠀⠀⠐⡎⢈⣀⠀⠠⠊⡐⠁⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠆⡄⢢⡚⡼⢻⠖⣆⣰⠀⠠⠀⠀⠈⡄⠀⠀⠄⠁⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠊⠀⠑⠀⠺⣉⠟⠈⢠⠔⠂⡈⠁⠀⠉⠉⠀⠐⠰⣖⣰⡒⡂⡠⠀⠁⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠠⠐⠈⠀⠀⠮⠍⡁⠶⢠⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠙⠫⡛⡇⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠰⠏⠘⠡⠀⠊⢀⡅⠀⠂⠀⠀⠀⠀⠀⠀⣈⢳⡴⡇⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⡹⡀⠉⡄⠀⣠⡔⠊⣙⡆⢀⣀⠀⠐⠀⠀⠈⠉⡈⢇⡵⢁⣴⢀⠀⠀⠀⠢⠈
⠀⠀⠀⢀⡀⠀⠀⠀⠢⣼⣇⠁⢤⠇⣤⠽⢹⡌⠒⢞⡵⣋⠄⠀⠀⠀⠀⡀⢉⡘⡀⠛⢹⣦⣞⣐⠀⠂⠀⠀⠀⠀⡀
⠀⢀⢀⡀⢈⠂⢠⠿⡂⠀⠋⣠⠀⢀⡼⠀⠂⡞⠀⣠⣴⣬⠄⠀⠀⠀⠀⠁⣨⠋⢉⣩⣄⢸⡭⣧⣤⠀⣠⠄⣀⠀⠀⡀⠐
⠀⠈⣫⠂⠋⠔⢙⣺⠇⠀⠐⠀⢉⠏⠀⡤⠤⠞⢁⢴⢦⡨⢀⢄⠐⢀⣠⣛⣁⡀⢹⢉⣉⣬⣽⣿⢿⣿⡟⢿⣿⠾⡛⠿⡿
⠀⠉⢤⡌⢁⡉⡍⢀⣠⠞⠈⢀⡤⠚⠒⢄⠀⠀⢿⠷⢦⡵⣂⣈⠫⠎⠡⡿⡄⠤⠄⠘⠤⠠⡤⠤⣤⣩⡿⢶⡶⠛⠿⠷⠶
⠀⠀⠈⠉⠈⠀⠈⠀⠀⠀⣠⠌⠀⠘⡻⠚⠂⠀⠀⠲⠀⢉⠁⠀⡠⠀⠐⢂⠉⢲⣤⡄⡐⣀⡭⠊⠁⡂⠊⣁⠴⠒⠒⠄
⠀⠀⠠⠤⠄⠂⠄⠒⠒⠉⡤⣄⠞⣃⡳⠏⠑⠢⠤⠆⢰⠈⠁⢁⣀⡀⠣⠦⠐⡈⠐⠅⠐⣋⠔⡶⠀⠈⣉⣁⣀⣀
⠀⠀⠐⠒⠒⠚⠉⠩⠁⠚⠀⠁⠠⢌⣀⠄⠖⠧⠠⡶⠈⠓⠚⠁⢀⠠⢉⡭⠄⣉⢹⡒⢋⠁⢀⠉⢛⡨⠠⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠉⠉⠀⠐⠃⠀⠊⠀⠀⠁⠁⠀⠈⠉⠉

⠀⠀⠀⠀⢠⠆⠀⢀⠠⠤⢤⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠠⢀⠤⢲
⠀⢀⠀⠀⠀⠀⠀⢘⠉⠉⠉⡁⠀⠀⠀⠀⠀⢠⠕⠠⢀⡀⠀⠀⠀⠀⢠⠖⠉⠉⣐⣂⠤⠚⠚⠉⣰⠔⡇
⡠⠫⣻⠫⣀⠼⢟⣟⣀⣀⠀⠅⠀⠀⠀⠀⠀⠸⠀⠄⠀⠇⠀⠀⠀⠀⡄⠿⣴⡪⡽⢁⡠⢂⣲⡮⢔⡾⢃
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠧⠄⠀⠀⠀⠀⠀⠈⠲⢤⣉⣙⠚⠛⠕⢊⣉⣡⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⢀⣤⣄⡠⠊⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠰⢂
⠀⠀⠈⣷⠋⠀⢀⠀⠋⢂⠀⠀⠀⠀⢀⣀⠀⣦⣴⣋⣉⠖⢫⠀⢀⡀⠀⠀⠀⠀⢨⠛⠓⠚⠓⡀
⠀⠀⠐⡯⠒⠒⢒⠚⢓⢿⠇⠀⠀⠘⣄⣦⣏⣏⠁⠀⡘⡾⢞⢳⣿⡌⠀⠀⠀⠀⠸⣶⣦⣤⣾⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⣽⢧⢭⣋⠽⠶⢺⣭⣠⠵⢊⡇⠀⠀⠀⠀⠀⡆⠀⠀⢠
⠀⠀⠀⠱⣄⡀⢀⠀⣀⡜⠀⠀⠀⠈⢫⡑⠒⠦⠵⡀⢀⠾⠗⢚⡽⠁⠀⠀⠀⠀⠀⠇⠀⠀⠆
⠀⠀⠀⠀⠘⠛⠋⠛⠛⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠿⠿⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⡆
⠀⠀⣠⢴⠒⠒⢲⣤⠆⣰⡶⣢⠀⠀⠀⠀⠘⠣⣄⣀
⠀⡘⠁⡐⣰⡾⠋⡉⣞⠋⢞⠉⠀⠀⠀⠀⠀⡖⠋⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠢⢄
⢰⣷⣚⢿⣽⠀⠮⣀⠴⢀⣼⡆⠀⠀⠀⠀⠰⠛⠒⠒⠒⠚⢢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡁
⠘⣟⠻⠿⢶⣿⣶⡶⠿⠟⢻⠇⠀⠀⠀⠀⢸⠸⣄⠀⣀⠆⡀⠀⠀⠀⡤⠎⢠⡀⠁⠀⠀⠀⠀⢁⠀⠘⠄
⠀⠘⢿⣒⠤⠄⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠀⠑⢲⠛⡗⠊⠁⠀⠀⠀⠀⢇⠻⠿⠟⠀⠀⠀⠀⠰⠀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⣀⠠⠔⠒⠉⢹⣻⠆⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⣀⠄⠒⠉⢀⣠⠴⠂⠉⠀⠀⠀⠀⠀⡞⠁⠓⢄⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⣀⠄⠒⠉⠀⣀⠤⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢆⠀⠑⢄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⣀⠔⠊⠁⡀⠤⠖⠀⠀⢀⢀⢤⠠⠄⠀⠀⠀⢀⣤⡴⣶⢶⡀⠀⠱⡄⠈⢦⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⡀⠀⠈⠀⡠⠔⠊⢀⠀⢄⠠⢐⠢⠉⠀⠀⠀⠀⠀⠘⣯⠁⡆⣨⣿⠇⠀⠀⠘⢄⠈⠇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⠄⠊⣀⠤⠂⠁⠀⠀⢀⠘⠢⠈⠁⠀⠀⠀⠀⢠⣤⡀⠀⠀⠘⢷⣀⣟⠁⠀⠀⠀⠀⡈⠀⢸⠀⠀⠆
⠀⠀⠀⠀⠀⢀⡴⠋⠀⡠⡞⡁⠀⠉⠀⠀⠀⠀⠀⠜⠂⠁⠀⠀⠀⠀⠀⠀⠀⢸⡷⣼⣷⣄⠀⠀⠻⣿⡆⠀⠀⠀⠀⠓⠐⠃⠀⠀⢸
⠀⠀⠀⢀⣴⠏⠀⠀⠀⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣋⣙⢓⢄⠀⢸⠛⣰⣿⢾⣭⣶⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⢠⡾⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⡄⠀⢾⠄⡇⠈⠳⣬⠆⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡑⡆⠀⠀⣸
⠀⢠⣏⠀⢹⠀⠀⠀⡏⡍⢳⠀⠀⠀⠀⠀⣼⠁⠀⢈⣁⣀⡘⣧⡘⠤⣀⡿⣦⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡴⠁⢀⣾⡟
⠀⡞⠀⠀⠀⡆⠀⠀⡇⠈⠘⣆⠀⠀⠀⠀⠻⠾⠉⠓⠒⢀⣻⠈⠛⠶⠾⠟⠁⠀⠀⠀⠀⠀⣀⢠⠲⠒⠂⡠⠔⢊⠔⠁⢀⣴⣯⣿⡇
⢸⡇⠀⠀⠀⠸⡀⠀⠹⡀⠀⠘⡄⠀⠀⠀⠀⠀⢀⣠⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⣰⢅⠩⠑⠙⢀⠤⠊⢁⠔⠊⢀⣤⣾⣿⣿⣿⡿
⠈⡇⠀⠀⠀⠀⠱⡀⠀⢣⠀⠀⠘⣄⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⢀⣠⠤⠋⠀⣁⡠⠌⢁⠄⠀⢀⣤⠾⣽⣹⢿⣾⣿⡟⠁
⠀⢧⠀⠀⠀⠀⠀⠹⡄⠀⠳⡄⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠄⠐⠉⢀⠤⠊⢁⣤⢾⣿⠁⠜⣵⣱⣾⡿⠋
⠀⠈⣆⠀⠀⠀⠀⠀⠙⢆⠀⠈⠢⣀⠀⠭⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠊⠁⣀⠔⠊⢁⣠⢾⣿⣛⠛⣋⣿⢿⢷⡿⠋
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠳⡀⠀⠈⠙⠂⠁⠀⠀⠀⢀⠀⡄⠘⠁⢀⡠⠐⠀⣀⣤⢶⣿⠏⣴⢧⠀⣸⣿⣿⠿⠋
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠒⠈⢁⣠⢶⢛⡿⣏⡨⣃⡜⣸⣣⣳⠿⠋⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⠀⠀⣀⣀⣠⣤⣶⣾⣿⣗⣢⢾⣿⣽⣉⣻⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠐⠠⠀⣼⢫⡫⠋⢯⣽⣿⣿⣿⣻⣿⣿⣯⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⡀⢀⣾⣱⡰⢫⣿⣾⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠪⢶⣄⣜⣳⣆⣼⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠤⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠐⠤⣄⡀⢀⣀⣠⡤⠤⠷⠶⠶⠒⠲⠒⠳⠶⠤⣤⣀⣀⣀⡤⠊⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⣀⠤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⢤⡀⢀⣀⠄⠀⠀⠈⠳⣄
⠀⠀⠀⢀⡴⠋⠀⢀⣀⣀⣀⣀⣠⠊⠁⠀⠀⢀⣤⠶⠶⠀⣠⡴⠆⣄⠀⠀⣠⣤⡀⠀⢠⡴⢦⡀⠀⠀⠀⠙⢯⡀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠉⢹⠁⠀⠀⠀⠀⠘⠳⢤⡄⠀⣏⠀⠀⣿⠀⣼⠷⠾⢷⠀⢸⡗⠛⠁⠀⠀⠀⠀⠀⢷⡤⡤⠤⠄⠀⠀⠀⣷
⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⡬⢧⡀⠀⠀⠀⠀⠶⠞⠁⠀⠈⠛⠛⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⣟⢧⡀⠀⠀⠉⠁⠀⠀⠀⠉⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠚⠛⠦⡀⠀⠀⠀⠀⠀⣠⠞⠁⡇
⠀⠀⠀⣷⠀⠙⢦⡀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠉⠙⠓⢶⠲⠶⠶⠶⠶⠶⡶⠶⠒⠒⠛⠉⠉⠱⡀⠀⠀⠀⠀⠀⠈⠁⠀⣀⡤⠞⠁⠀⢀⡇
⠀⠀⠀⣿⠀⠀⠀⠙⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠋⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠒⠛⠋⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⣛⢟⣉⠛⣛⢛⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠉⠂⠀⡛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠒⢤⡀⠘⣜⣷⣦⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⡿⢶⣄⠀⠀⠀⠉⢦⡈⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⣀⣼⣿⡻⢇⢤⡀⠀⠀⠀⠹⣄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣯⣛⠇⠫⠻⠏⠁⠀⠹⢿⣶⣄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⡀⠀⠀⠠⡤⣤⣸⡟⠋⠉⠛⢯⠈⠳⣄⡄⡀⣼⣿⣿⡆⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠈⣡⣶⣺⡟⢶⣜⠦⣴⣿⡝⢯⣾⣺⣿⠇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⣴⡾⠟⣿⣿⠷⢋⣙⣷⠉⠿⠻⡾⠉⠉⠀⠀⢲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⢀⣨⡍⠀⠜⠁⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⡀⠀⣤⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡳⣄⠀⠈⢛⡞⠁⠀⠀⠀⠀⢠⠾⠋⡀⠄⣠⣾⣿⣬⣾⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⢠⡛⢦⣄⡀⠀⠏⠀⠀⠉⠀⣘⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⡀⢷⣄⠈⠙⠛⠓⠒⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠖⠢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀
⠀⠀⠀⢀⡴⠒⢋⠉⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠐⠀⠀⠠⢹⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⠄
⠀⠀⠀⡎⠤⠠⠂⠲⠅⠀⣧⠀⠀⠀⠀⠐⣿⠽⠯⠿⠽⢿⡇⠀⠀⠀⠀⠀⠐⣏⠴⣠⣄⣰⡿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸
⠀⠀⠀⢿⠀⠘⠢⢴⡁⢀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠒⢸⠀⠀⠀⠀⠀⠀⠀⠸⡆⠡⠈⣼⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣰⠁
⠀⠀⠀⠈⠓⠧⣤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠾⠃
⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢠⠖⡛⠉⠉⢉⡉⡙⢦⠀⠀⠀⠀⠀⠀⢀⣖⢒⣆⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠚⠓⠲⣄⠀⠀⠀⠀⠀⠀⣴⣞⣿⣿⣿⡶⡄
⠀⠀⠘⣎⡔⠜⠓⢼⠵⢈⠟⠀⠀⠀⠀⠀⢰⡉⠤⡠⠭⣿⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣘⡇⠀⠀⠀⠀⠀⢿⣯⢿⢹⢻⣽⠇
⠀⠀⠀⠘⣄⣸⡃⣭⣰⠏⠀⠀⠀⠀⠀⠀⢸⠄⣀⡼⡀⣼⠂⠀⠀⠀⠀⠀⣷⣦⣤⣼⣧⣤⣴⡇⠀⠀⠀⠀⠀⢻⠛⠛⠛⠛⡗
⠀⠀⠀⠀⠘⣏⡵⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡤⠜⠁⠀⠀⠀⠀⠀⠈⠲⣀⣓⠉⢀⢛⡸⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠿⠂⠀⠀⠀⠀⢠⠋⣿⢁⣾⡇⢳⠀⠀⠀⠀⠀⡶⢷⣾⡶⣄
⠀⠀⢠⢴⠞⠛⠛⠛⠻⢦⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⡜⠀⡟⠻⣴⡇⠘⡇⠀⠀⠀⠀⡏⢦⢻⣿⣿⣆
⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⢴⡖⠚⣛⣛⡒⢲⡆⠀⠀⠀⠀⠀⣧⠀⣫⡴⢛⡆⠀⡇⠀⠀⠀⠀⡇⠀⠱⡻⣿⣿⣄
⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⢸⣤⣀⣀⡀⣀⣸⠃⠀⠀⠀⠀⠀⢣⠀⢿⠙⢺⡇⢠⠃⠀⠀⠀⠀⡇⠀⠀⠙⢝⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡏⠛⠛⠻⠟⢻⠀⠀⠀⠀⠀⠀⠘⣄⣸⣞⢋⣧⡼⠀⠀⠀⠀⠀⠧⠤⣤⣄⣈⣮⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⡏⠉⢳⠏⠉⢷⠀⠀⠀⠀⠀⠀⠰⠀⠀⣀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢱
⠀⠀⠀⢠⣇⡀⣀⢆⣀⣾⠀⠀⠀⠀⢀⡤⠚⠉⠁⠆⠉⠑⢦⡀⠀⠀⠀⠀⣀⢔⠚⠿⠟⢷⢦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⡾⡛⠁⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⢸⡀⠀⣸
⠀⠀⠀⢸⡀⠀⠀⠀⠀⡞⠀⠀⠀⠸⣽⡀⠀⠀⠀⠀⠀⠀⢀⣞⡶⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠀⠸⡀⠀⡸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⠻⣖⡒⠒⣲⠾⠛⠁⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠓⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠻⢶⣶⣦⠈⠙⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠁⠤⠚⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⢀⠔⠉⠐⠈⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢁⡴⠈⠛⠿⠿⠂⠀⠀⣰⠃⠺⠿⠟⠋⠲⣄⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠉⠀⠀⠀⠀⠀⢰⡇⢰⣿⠀⠀⣤⠀⠀⠀⠈⠙⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⡾⠀⠃⣸⣿⡆⠀⡏⣠⡆⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠸⢴⣿⡇⢠⠀⣿⣿⣿⢀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄
⠀⠀⢀⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣦⢸⣿⣇⣿⣿⣿⣿⣿⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠘⠛⠋⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⡀⠀⠀⠀⠈⠙⠛
⠀⠀⠀⣀⣀⣀⣠⣤⣤⣿⠫⡭⣻⢯⡹⡏⠲⢎⡇⣒⢸⡇⣲⡟⣴⣮⡇⣿⢹⠫⣭⣿⣧⣤⣤⣀⣀⣀
⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣐⣓⣼⣴⣦⣷⣿⣮⣧⣭⣵⣧⣭⣿⣬⣵⣧⣝⣼⣔⣛⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣷⣤⡀
⣀⣀⣀⣚⣋⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣛⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⣠⠞⠁⢀⡤⠖⠒⠛⠓⡲⠦⣄⠀⠙⢦⡀
⠀⠀⠀⡼⠁⢀⡞⠁⠀⠀⣠⠖⠺⣧⠀⠈⢳⡀⠀⢷
⠀⠀⢸⠁⠀⡞⠀⠀⠀⠀⠹⣄⠀⠈⠁⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢻⠀⠀⣇⠀⠀⢀⠀⠀⠈⡇⠀⠀⠀⢠⠇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⣉⣩⠤⠤⠤⠤⠤⠤⣭⣉⡛⠲⢤
⠀⠀⠘⣆⠀⠙⣆⡀⠈⣳⠒⠛⠁⠀⢀⡴⠋⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢰⡋⠥⢴⣂⣀⡴⠚⠉⠙⡶⠚⢙⡆
⠀⠀⠀⠹⣦⡀⠀⠙⠲⠦⠤⠤⠶⠚⠉⢀⣠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⡉⠓⠲⠮⠥⠤⠤⠤⠴⠖⠚⠉⣀⣴
⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣀⣤⣤⣴⣶⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⡿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⡿⣷⣿⣭⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⣋⣉⣉⣉⣉⣉⣉⣙⠓⠶⢿⣽⣿⣷⣾⣿⣽⣿⣿⣿⣿⣿⣽⣿⣾⢿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢠⠞⣉⣩⠄⠀⣠⠖⠒⠲⣭⠽⠳⡄⠈⣿⣿⣾⣿⣯⣽⣿⣿⣿⣭⣿⣷⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡈⠓⠦⠬⣽⣉⣁⣀⣀⣬⠥⠴⠚⢁⣠⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⢿⣿⢿⣾⣿⣯⣿⣿⣿⣿⣭⣿⣿⣾⢿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣷⣯⣿⣟⣻⣿⣿⣿⣿⣿⣻⣿⣽⣿⢟⣿⣿⠷⠛⠉⠉⠉⠉⠛⠻⢿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⣻⣷⣿⣽⣿⣟⣿⣿⣿⣿⣛⣿⣯⣿⣾⣿⡟⠁⠀⣠⠴⠒⠒⠒⠦⣄⠀⠙⢿⣿⣫
⠀⣠⠴⠚⣋⣉⣭⣥⠤⠤⣭⣍⣉⡙⠒⠿⣿⣿⣿⣽⣿⣻⣿⣿⣿⣿⣛⣿⣿⣿⣾⡟⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠹⣯
⡞⠁⢴⠋⣥⣴⣀⣀⡴⠚⠛⢳⡶⠏⢳⡄⠈⣷⣯⣿⣟⣿⣿⣿⣿⣿⣿⣻⣿⣽⣿⡇⠀⣾⢀⣠⡖⠀⠀⢀⡤⢤⡀⠈⣇⠀⢹
⣿⣤⡈⠙⠲⠮⠭⠭⠤⠤⠤⠤⠶⠒⠋⣀⣴⣿⣿⣿⣿⣛⣿⣿⣿⣿⣟⣿⣯⣿⣿⡇⠀⢻⠈⠙⢧⣀⣠⠏⠀⢀⡷⠦⢹⠀⢸
⣯⣻⣿⣿⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⣿⡿⣿⣿⣾⣯⣿⣿⣛⣻⣛⣛⣿⣿⣭⣿⣿⣿⡄⠈⢧⡀⠀⠉⠁⠀⠀⠘⠁⢀⡟⠀⢸
⣿⣻⣿⣿⣽⣿⣿⣿⣛⣻⣿⣿⣿⣽⣷⡿⣫⡿⠙⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠋⠙⢿⣿⣄⠀⠙⢦⣀⡀⠀⠀⣀⣠⠞⠀⢀⡞
⠙⢿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⠈⠉⠉⠉⠉⠀⢀⡴⠋
⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣶⣶⠶⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢶⣏⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣎⠡⠞⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠄⠁⠀⠀⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠠⢣⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠛⠋⢧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⣶⣆⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣼⣀⣀⣀⣸⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⣿⠏⠼⠿⠿⠿⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⣧⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠳⣿⣿⣿⣿⡇⡻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣿⠈⢷⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣴⣦⣀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣀⣀⣀⡁⢤⠀⡄⢠⠀⠀⠒⢠⣄⢠⠀⠂⠀⠀⠀⠀⣿⠀⢹⡆⠘⢧⡀⠀⠠⠛⠶⠀⠀⠀⠂⠀⠀⠀⣠⠶⣥⣛⠶⣮⠃⠀⠀⠀⠀⠀⠀⢀⣰⢶⣄
⠀⠀⠉⠙⠉⠁⠘⣄⠏⠸⠀⠀⠰⠶⠶⠶⠶⠛⠀⠀⠀⠀⠻⠇⠈⠁⠀⣰⠇⠀⠀⠤⠰⠶⠔⢠⠀⠀⠀⠰⢫⣾⢱⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⣾⡗⣬⠟
⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠦⠶⠞⠉⠀⠀⠀⠀⠀⠀⢄⠶⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠔⠢⡀
⠀⠀⠀⠸⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⣄⠾⠂⢠⠀⠀⠀⠷⣿⣷⣿⠿⠃⠀⠀⠀⠴⠀⠉⠒⢄⢀⡀⠀⠀⠀⢰⣶⣶⣶⣾⣧
⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⢠⠖⠐⣮⠙⠃⠀⠀⠀⠀⢷⡆⠉⠀⡠⠋⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⣀⣾⠃⣀⡀⠙⠷⠀⠀⠀⠈⠉⣙⠿⡇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢰⠈⠁⠈⢁⡀⠐⠀⠀⠀⢀⡀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠶⠀⠰⠏⠁⣀⣉⣁⡀⠀⠀⠀⠀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠉⠛⠉⠉⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠰⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠋⠉⠁⠂⠀⠀⠀⢠⡀⠀⢛⠿⠀⣀⠀⠀⠀⠀⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⣠⣄⢀⡀⠀⠀⢴⠁⠐⠀⠀⠀⠀⢸⡇⠀⠀⠙⠂⠀⠐⠀⠈⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⠀⢉⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠉⠁
⠀⠀⠀⠀⠀⢨⣯⡹⡭⠀⠀⠀⠁⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠟⡷
⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢐⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣦⠀⠀⠀⠀⠀⢀⡀⠲⡀⠚⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠛⠃⠀⡘⠛⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠀⠀⢁⡇⠀⠀⠀⠄⠘⠃⢀⡵⠚⠃⠀⠤⠀⠀⠈⠂⠀⠃⠀⠀⠀⡟⠻⠿⠿⠛⢋⢳⡆
⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⡄⠀⠀⣤⠀⠀⠀⠀⠀⢤⡄⠀⠠⢦⣤⠇⠀⠐⠲⠖⠲⠟⠁⠀⠀⠀⠀⠲⠐⠆⠀⢠⡤⠄⠀⠀⠀⠈⠢⠄⠀⠴⠋⠉
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⣮⠀⠀⠀⠀⠀⠈⢀⠀⠐⠀⠀⠀⠀⠀⢀⠄⠴⣿⡟⠀⠀⢀⠋⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠒⢠⡆⢠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠠⠀⢀⡤⠈⠀⠀⠀⠈⠀⣄⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⣄
⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠⠀⠀⠠⢾⣥⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠊⠀⠀⠀⠀⠀⠀⠉⠉⡆⠀⠀⠉⠀⠀⠀⠈⠀⠂⠰⣆⣶⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠀⠀⠘⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠊
⠀⠀⠀⠀⠸⠷⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢟⡻⢿⣧⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠶⠶⠀⠀⠰⠾⠦⠀⠀⠀⢰⠆⢀⡛⡄⠀⡀⠀⠀⠀⣠⣶⣶⣴⡆⠐⠓
⠀⠘⢶⣶⣄⣤⣴⣶⡞⠀⠀⠈⠛⢧⣼⣿⡶⠟⠀⠀⠀⡀⠈⠁⠈⠀⡀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠂⠀⠰⠃⣠⢻⡿⠓⢾⠉⢓⡄⠘⠿⠶⣤⡞
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠞⠃⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⢾⣤⠴⠛⠓⠋

⠀⠀⠀⢀⣤⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⢿⠆
⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠤⣴⡄⠀⠀⠀⠀⠘⠿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢷⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠖⢠⡀⠀⠀⠀⠀⠀⠂⠀⡈⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠄⢀⡤⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢲⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠓⠇⠀⠀⠄⠀⠀⠀⠙⠷⣶⣶⠶⠟⠁
⠀⠀⠀⠀⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠆⠈⠄⠀⠑⠄⠀⠀⠀⠀⠀⠀⣀⠀⠠⠔⠀⠀⠀⣤⣙⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣧⡄⠀⢀
⠀⠀⠀⠴⠆⠀⣤⣾⠃⠀⠀⠤⠤⠤⠌⠉⢫⠀⠀⠀⠀⣤⣹⣿⣧⣄⠉⠀⠀⠀⠀⠀⢠⡄⠐⢳⡄⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢠⡤⠲⡦⠴⠶⠀⠀⠂
⠀⠀⠀⠤⠄⠻⣄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⠋⣻⡑⢫⣧⠀⠀⠀⠀⠤⠸⠃⠶⠊⠀⠀⠀⠀⠿⠟⠀⣀⢠⣦⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠐⠒⠂⠐⠛⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠉⠀⠀⠀⠈⠤⢀⣀⣀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠉⠉⠉
⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡦⠄⠀⠀⠀⣀⠐⠂⠀⢠⡤⠀⠀⠀⠀⠠⠶⠶⠦⠀⢀⠀⠀⠀⠀⠛⠏⠀⠀⣤⢶⠒⣄
⠀⠀⠀⣤⣴⣶⡆⠀⠀⠀⠀⠰⠀⠀⠀⠹⠀⠀⠀⠀⣴⠿⠛⠉⠀⠀⠀⠘⠀⠀⠁⠀⣄⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡌⠄⠀⠀⠀⠁⠂⠂⠀⠐⠫⠜⠋
⠀⠀⠀⢉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠟⠃⠀⠀⠀⢈⣿⡷⠲⠶⠿⠓⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠐
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⡿⢿⣿⡿⠛⢿⡿⣿⡿⣿⠿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣥⣤⣼⣷⣠⣾⣠⣴⣷⣬⣴⣿⣥⣾⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⣯⣓⢦⡀
⠀⠀⠀⢀⡴⢋⠡⠈⠄⡀⠍⠙⣦⠀⠀⠀⠀⠀⣰⣒⣚⣛⣛⣶⡀⠀⠀⠀⠀⠀⢸⡇⢸⡀⠀⠀⠀⠀⠀⣠⢞⣩⠴⣿⣼⣚⣍⣳⣽
⠀⠀⠀⢸⡧⠬⠤⠤⠤⠤⠤⠤⣼⠆⠀⠀⠀⠘⣧⡤⠤⠤⠤⣤⠇⠀⠀⠀⠀⠀⣞⡉⠉⣧⠀⠀⠀⠀⣼⢡⢏⠤⡅⡏⠿⢟⣫⡵⣆
⠀⠀⠀⢸⡗⠛⢒⣒⣛⣒⠒⠓⢺⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡟⠀⠀⠀⠀⠀⠸⡇⢹⡇⣸⠀⠀⠀⠀⣇⢸⠘⢒⠋⠷⢚⣭⡀⡇⣿
⠀⠀⠀⠀⢿⣙⣯⡉⠉⢉⣽⣋⡟⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡟⠉⠀⣿⠀⠀⠀⠀⠹⡜⢧⡻⠗⣡⣌⣋⡼⣡⠏
⠀⠀⠀⠀⠀⠉⠉⠙⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣸⠃⠀⠀⠀⠀⠀⢠⣇⣀⣀⣻⠀⠀⠀⠀⠀⠙⠦⣍⣛⣓⣛⣩⠖⠋
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⣠⣞⢻⡏⣿⢓⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢿⡳⣄
⠀⠀⠀⢸⠷⠾⠮⠷⠧⠾⠶⣿⠀⠀⠀⠀⠀⠀⣤⣤⣯⣯⣤⣄⠀⠀⠀⠀⠀⣠⠖⠋⡀⢀⠐⡈⠓⢦⡀⠀⠀⠀⠀⣰⢋⠀⡁⢈⠙⣿
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⢛⡧⠀⠀⠀⠙⣶⠓⡶⠚⠲⡖⢳⡞⠁⠀⠀⠀⠀⣏⠠⠀⠄⣠⡼⠁
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⣀⣿⣀⣀⣀⣀⣟⣋⣀⠀⠀⠀⠀⢸⡄⢹⠀⢸⠃⣼⠀⠀⠀⢀⣤⠞⣩⠷⠖⠛⠁
⠀⠀⠀⠀⠸⢤⡤⠤⠤⢤⠼⠀⠀⠀⠀⠈⠧⣄⣀⣀⣀⣀⣀⣀⣤⠟⠀⠀⠀⠀⠀⠳⣼⣧⡾⡴⠃⠀⠀⠀⠘⢦⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣉⣳⣆
⠀⠀⠀⠀⠀⢀⣀⡤⠞⣧⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣄⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⡄⠀⠀⠀⣸⣥⣤⣽⣧⣬⣧
⠀⠀⠀⠀⣰⠯⠤⠤⠤⠬⢷⡀⠀⠀⠀⣰⠋⠀⠀⠏⠀⠀⠟⠁⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣠⡏⠙⠓⢿⠀⠀⠀⠹⡞⠛⢚⣷⢶⡏
⠀⠀⠀⠀⡷⠶⣤⠴⢤⡴⢶⡇⠀⠀⠀⣟⣶⣒⣲⣒⣲⣒⣒⣖⣒⣶⣻⠀⠀⠀⠀⠀⢀⡞⠉⠙⠦⣄⣸⠁⠀⠀⠀⢻⣚⣩⡴⡾
⠀⠀⠀⠀⢹⡀⢸⠀⢸⠀⣸⠁⠀⠀⠀⣿⠦⠭⡽⠯⢥⡬⠵⠮⡭⠴⣻⠀⠀⢀⣠⢾⡉⠙⢦⡀⠀⢈⡗⠀⠀⠀⠀⠈⣏⣡⢴⠃
⠀⠀⠀⠀⠈⣧⣸⣄⣼⣠⡇⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠒⠚⠛⠛⠉⠁⠀⠀⠘⠲⢤⣷⣀⣠⠿⠖⠋⠀⠀⠀⠀⠀⠀⠹⣄⡟
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⢶⣤⡀⠀⠀⠀⠀⢀⣤⡀⠀⢀⣤⡀⠀⠀⠀⠀⢠⡤⠤⣤⢀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡿⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠛⣧⡾⡶⣴⢧⡀⠀⠀⣴⠋⣄⠙⣶⠋⣄⠙⣆⠀⠀⠀⠈⡏⠉⡇⠸⡷⢾⡇⠀⠀⠀⠀⠀⠀⣠⢞⡼⣳⠋⠈⢹⡄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⣿⠶⢿⣹⠋⠀⠀⠰⡇⢀⡼⠀⢸⢀⡞⠀⢸⠀⠀⠀⢀⡇⠸⣧⡞⠁⠈⢳⡆⠀⠀⢀⡴⢞⡵⣫⠟⠁⠀⠀⣸⠃
⠀⠀⠀⣴⠋⠀⠀⢀⡴⠃⣠⠞⠁⠀⠀⠀⠀⢳⡈⠃⣠⢾⡈⠇⣠⠟⠀⠀⠀⡏⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢿⢚⡿⠞⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠘⢦⣀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠙⠶⠃⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠈⠻⣄⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠲⠚⠓⠒⠶⠚⠃⠀⠀⠀⠀⠈⠉⠉

⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣻⣛⣿⣿⡿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡾⠀⠀⠈⢳⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠒⠒⠲⢤⡀⠀⢳⣷⣀⡀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠉⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠙⠛⠷⠒
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡴⠂⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠁⠀⠀⠀⠈⠓⠦⠴⠄⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⠆⠀⠠⠭⠷⣄⠀⠸⣷⣤⣀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡤⠒⠚⠉⠉⠉⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⡀⢿⡆⠀⠀⠀⠀⡌⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢠⠀⠈⠙⡆⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠤⠤⠄⠒⠉⠀⢦⡀⠀⠙⢦⠙⠶⠦⠔⠊⠀⣄⠈⠙⠀⠖
⠀⠀⠀⠀⠀⠘⠷⠤⠊⠀⢀⡎⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠁⠀⠀⠈⢧⠀⠀⠠⠤⠤⢌⡁⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⣾⣰⡇⠀⠀⠽⠃⠀⠀⢀⣠⡄⠀⠀⣸⢿⡄⢀⣶⡶⣶⣦⡀⠀⢸⡆⠀⠀⠀⠀⠀⠘⠄⠙⠂⠁⣤
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢻⡄⠀⠙⠛⠛⠟⠛⠷⢶⣄⢰⡿⠛⠻⢶⣤⡏⠀⣿⡀⣿⣠⡤⢻⡇⠀⠀⡇⠀⠀⠀⠠⠐⠲⡄⠀⢀⣤
⠀⠀⠠⣇⣈⡇⡄⠀⠀⠀⠈⡇⠀⠀⠀⣠⠺⣶⠀⠀⢹⣿⡁⠀⠀⢸⣿⠗⠶⢾⡇⣿⠟⠛⠉⢠⠒⢺⠗⠶⣤⠀⠀⠀⢠⠜⢤⡉⠁
⠀⠀⠀⠈⠸⠟⠹⠆⠶⠀⠀⢸⡀⠀⠀⢿⣦⡀⠀⣀⡼⠛⠷⠶⠔⠋⣹⡥⠀⢸⡧⢼⠤⠀⢀⠸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣜⠦⠄⠀⠀⠀⠀⠀⠀⠀⡤⢛⣆⣀
⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠰⠙⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⡁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠉⡇
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠉⠉⠛⠻⠿⢷⣶⣾⣿⣟⣟⣓⣚⣋⣩⡭⠽⠖⢿⠈⠓⠂⠀⡔⡆⢠⠀⠀⠀⠀⠀⠘⠀⠀⠐⠁
⠀⠀⠀⠀⠳⡀⠂⠀⠀⠀⠘⢦⣂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠳⠖⠐⠀⠙⠦⠼⣆⡠⠤
⠀⠀⠀⠀⠀⠈⠓⠒⠀⠂⠀⠀⠈⠙⠓⠒⠂⠀⠀⠀⠈⠛⠒⠒⠛⠓⠒⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠈⠁




⣿⣿⠿⠿⡿⠻⠿⢿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢒⣁⡤⣠⣦⣤⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠛⠛⢻⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⣤⠀⡏
⠀⠀⠀⠀⠀⠀⢠⡲⣿⣿⡆⠀⠀⠀⣠⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⣀⢰⣿⡄⠀⠀⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠟⠁⢀⣀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠀⠀⠀⡀⠒⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣇⠂⠀⠀⠀⠁⡴
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⡻⣿⣿⣷⣄⡀⠶⣾⡁
⠀⠀⠀⠀⠀⢀⠖⠛⢢⡀⠙⢮⡛⠿⠿⢿⣿⣿⡻⡢⡀
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠹⡄⠀⠱⡄⠀⢰⣿⣿⣿⣇⢡⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢹⡀⠀⠙⣄⠀⢹⣿⡿⣿⣿⣇⠀⠀⠀⠀⢾
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⢧⠀⠀⠈⠉⠈⢿⣷⠘⠿⠸⡄⠀⠀⡰⠛
⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢱⠀⡸⠁⠀⠘⢿⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣰⠁⠀⣤⠀⢺⡏⠁
⠀⠀⠀⡇⠀⠀⠀⢀⠀⢾⣣⡟⠀⠀⠀⢀⣀⣠⠀⠁⠀⠈⢁⡠⠂⠀⠙⠛⠛⠀⢀
⠀⠀⠀⢡⠀⠀⠀⡡⢖⢛⡷⠒⠘⠉⠉⠁⠀⢰⠀⠀⠀⠀⢸⣽⣇⠤
⠀⠀⠀⠘⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠖⠒⠉⢸⠐⠄⣾⣛⠛⠛⠛⣛⠛⠃⡄
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠏⠀⢀⠀⠀⢸⡄⡆⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠈⠲⠆⠶⢶⠒⢚⠋⠍⠅⠤⢨⡧⠷⠶⠶⠾⠶⠶⠞⠣⠶⠀⠀⠀⠀⠀⠠⠀⠼⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡆
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡀⣰⠋⢳⡄
⠀⠀⠀⠀⢀⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⣩⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣠⠤⣄
⠀⠀⣤⠐⠿⡞⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡞⠛⣡⢼⡛⠷⢯⣷⡀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠒⠚⠉⠀⠀⢸⡆
⢀⡞⣹⠷⠒⠿⣄⠀⠀⠈⠓⠦⢤⣄⣀⡄⠀⠀⠀⣸⣴⣣⢾⠷⠶⢿⢻⡞⠁⢧⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⡴⠛⠛⠳⣄⠀⠳⣄
⠉⠉⢻⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⣀⠿⣽⠀⠀⠀⢻⠙⠛⠻⠶⠶⢾⠏⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⡼⠀⢸⠀⠀⠀⠀⣹⠀⠀⠈⢳⡀
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠉⠛⠛⢦⣴⡆⠀⠀⠀⠈⢦⠀⠈⠁⣰⠋⠈⠀⠀⠀⢻⡀⠀⠀⠀⣰⠋⠀⠀⠈⠳⢤⣤⠴⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣤⣤⠤⢿⣸⠇⠀⠀⠀⠀⠈⢧⡀⣰⠃⠀⠀⠘⠀⠠⠈⣇⠀⠀⠀⢿⡀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⢀⠀⠀⠀⢸⡆⠀⠀⠀⠙⠓⠚⠉⠀⠀⠀⠀⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠤⠀⠈⠀⠀⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠙⢦⣀⣀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢰⢡⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⣾⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⣠⠤⡎⢉⡆
⠀⠀⠀⠀⠀⠀⠀⡟⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⣀⠀⠀⢀⡞⣠⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠙⣦⠟⠳⣦⡟⣏⠹⣞⣦
⠀⢠⠶⠶⠶⠶⠚⠓⠓⠒⠒⠒⢲⡄⠀⠀⠀⠀⠀⣞⣛⣛⣛⣛⡇⠀⠈⢇⢹⢀⣤⣄⣠⡿⠋⠀⠀⠀⣏⣛⠛⢦⣞⢹⢠⡿⣆⠘⢧
⠀⠉⢻⠉⠉⠉⠉⠉⠉⠉⠉⢉⡏⠁⠀⠀⠀⠀⡴⠛⠉⠉⠉⠉⠑⢦⠀⠞⢀⢸⣄⠈⢹⡇⠀⠀⠀⠀⢀⣈⡟⣆⠈⢻⣸⠃⣼⣄⣨⣷
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⣞⠉⠙⢿⡳⠞⠁⠀⠀⠀⠀⠀⢻⠉⠉⠙⠛⠛⠉⠉⠁⢠⡇
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡼⠶⢿⡶⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣤⣀⣷⣠⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⡏⠐⠈⠙⣏⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠒⠒⠒⠶⠟
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠻⠶⠶⣶⡿⢦⣧⡴⠋
⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠓⠲⠶⢤⣄⡀
⠀⠀⠀⣰⠿⢞⠳⠮⣍⡹⣯⣽⣇⣤⣀⠀⠀⠀⠀⠀⠁⠀⢀⣤⠶⠒⠒⣖⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⣀⡀⣠⠀⠈⠉⠳⢤⡀
⠀⢀⡞⠁⠀⠈⠀⡄⠀⠙⢦⡸⢭⣳⣈⠱⡄⠀⠀⠀⠀⣴⢋⡀⠳⠀⠶⠀⠡⠆⢈⠳⡄⠀⠀⠀⠀⠀⢸⡷⣤⡈⠉⠓⠦⣤⡄⠀⠀⠙⢦
⠀⠸⡆⠠⠀⠀⠀⠀⠀⠂⠀⠹⡄⢿⣽⣇⢹⠀⠀⠀⠀⢷⣌⣁⣀⣀⣀⣖⣀⣈⣋⣁⣹⠀⠀⠀⠀⠀⢸⡷⢈⡇⢀⣩⡤⣤⣙⠶⣞⠀⣈
⠀⠀⠘⠦⣀⠈⠁⠀⢀⠀⠈⠃⠹⡄⣹⢾⣼⠀⠀⠀⠀⣸⣭⣭⠭⣍⣉⣩⠽⢯⣍⣩⠿⡄⠀⠀⠀⠀⢸⠟⠋⢠⣯⡴⡖⣟⣮⠷⠞⠋⠉
⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠠⠄⠀⢷⡷⢺⠃⠀⠀⠀⠀⠹⣦⠤⠶⠶⠿⢶⣶⠶⠶⠖⢺⠁⠀⠀⠀⠀⢸⠶⣄⢘⣦⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣄⣸⡴⠃⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣀⣈⣹⣦⡴⠋⠀⠀⠀⠀⠀⢾⠴⠛⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡤⣤
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠲⡟⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡀⣤⣋⡷⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣤⠤⠞⣻⡥⣤⡷⣄⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⠦⣤⠴⠚⠉⠉⡇
⠀⠀⢀⡼⢣⡤⣼⣆⠀⠀⠙⠢⣄⣀⠀⠀⣀⠀⠀⠀⢸⣡⢇⣴⣏⣭⣷⢯⡽⠛⡆⠀⠀⠀⠀⠀⠸⣅⠀⠀⢀⡤⠶⠦⣄⠀⢳⣀
⠀⠐⠛⠲⡏⠀⠀⠉⠳⣄⡀⠀⠀⠈⠉⣻⣹⠀⠀⠀⢾⠻⠿⢾⣁⣄⣽⡞⠁⠀⢻⠀⠀⠀⠀⠀⠀⠈⡷⠀⡏⠀⠀⠀⢸⡆⠀⠈⢳⡀
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⠙⠛⠻⠿⣏⣁⡍⠀⠀⠀⠘⣆⠀⠐⠂⢠⠏⠐⠀⠀⠈⣇⠀⠀⠀⠀⣠⠞⠁⠀⠳⣄⣀⣀⡼⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠙⠦⢤⣀⣀⣀⣀⣀⣼⣁⡇⠀⠀⠀⠀⠘⢦⠀⢠⠏⠀⠀⠰⠀⠀⠸⡄⠀⠀⠀⡇⠀⠀⠀⠀⣀⣉⣁⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⢳⠇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠙⠲⠶⠒⠋⠉⠁⠉⠉⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡀⠀⠘⠀⠀⢂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠹⣄⡀⠀⣄⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡽⠛⠉⠁⠀⠀⠀⠀⠀⠀⠘⠧⣽⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⢠⡤⣼⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⡀⠀⠀⢠⡞⣠⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⢳⡼⠛⢦⡼⢾⠙⢶⣳
⠀⠀⠀⣠⠶⠶⠶⠖⠛⠛⠒⠒⠒⠒⣦⠀⠀⠀⠀⠀⣾⣛⣚⣛⣛⡇⠀⠘⣇⢳⢀⣤⣄⣴⡿⠋⠀⠀⢸⣛⡛⠳⣴⡋⡇⣾⢧⠈⢻⡀
⠀⠀⠀⠙⣟⠉⠉⠉⠉⠉⠉⠉⠉⡿⠉⠀⠀⠀⠀⣠⠟⠉⠉⠉⠉⠓⢦⠀⠟⣀⣻⡀⠈⢿⠄⠀⠀⠀⠈⣠⣸⣳⡀⠙⣧⡏⣸⣇⣠⣷⠆
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡞⠉⠙⢿⠳⠛⠀⠀⠀⠀⠀⠘⣏⠉⠉⠛⠋⠉⠉⠁⠀⡟
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡾⠶⢿⡶⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⡤⣤⣷⣠⣁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣹⠀⠀⠰⣇⠐⠀⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠚⠛⠛⠓⠛
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⢛⣿⡿⠲⠵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⡤⣤⠤⣀⣠⣀⣠⣀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⣠⠟⠛⠉⠛⠲⢤⡿⠶⣿⣶⠦⣄⠀⠀⠀⠀⠀⠀⣠⠶⡛⠉⠉⠯⠛⠲⣤⡀⠀⠀⠀⠀⠀⠀⢸⠾⠧⣤⣠⠆⠀⠀⠉⠳⣄
⠀⠀⠀⣼⠁⠀⠀⠀⠰⠀⠀⠙⢾⠓⣼⡧⣌⢧⠀⠀⠀⠀⡾⢡⠆⠙⠀⠛⠀⠘⢁⣤⠹⡄⠀⠀⠀⠀⠀⣼⠹⢦⠀⢈⡙⠲⢦⡀⢀⡀⠳⣄
⠀⠀⠀⠹⣄⠂⠀⣤⠀⠀⠈⠀⣌⢳⡘⢾⣻⣸⠄⠀⠀⠀⢹⡦⠤⠤⢤⣼⣭⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⣿⣃⡾⢀⡴⡒⣒⡶⣝⢻⣄⡤⠿
⠀⠀⠀⠀⠈⠳⢤⣀⠀⠠⠆⠀⠀⠀⢧⣸⣹⣾⠀⠀⠀⠀⢼⡳⠒⠛⣲⣶⣖⣛⣓⣶⣞⣹⠆⠀⠀⠀⠀⣟⠡⠀⣟⢾⣹⣽⠽⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠃⠀⢸⡏⡽⠃⠀⠀⠀⠀⠈⢿⡛⠉⠉⠉⠛⠯⣍⠉⢉⡟⠀⠀⠀⠀⠀⡏⣙⡦⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠚⠛⠋⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⢹⢁⣿⡈⡏⢸⠅⢫⡀⢼⢙⣾⠭⠀⣷⣼⣀⣿⡈⡏⢹⡏⡯⢸⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠈⠀⠁⠁⠈⠁⠉⠀⠈⠉⠈⠀⠀⠁⠀⠉⠀⠁⠁⠈⠀⠉⠉⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂⠨⠽⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡇⢀⣀⣀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠃⠀⠀⠈⡷⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠒⠓⠲⠦⣄⠀⠀⢹⣹⣄⣀⡀⠔⠁⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠙⠿⢷⡶⠶⠖⠂⠁
⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢹⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⣄
⠀⠀⠀⠀⠀⠺⣤⡀⠏⠀⠀⠀⠀⠻⣄⣀⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⡤⠒⠒⠲⢤⡀⠀⠀⣾⡅⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣭⠤⠤⠤⣤⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢀⡏⠀⠀⠉⠉⠉⢹⠀⠀⠘⠿⠿⠀⠈
⠀⠀⠀⠀⣀⠤⣄⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⢀⡴⠷⠖⠒⠲⢤⡈⢿⣄⠀⠀⠀⢀⡜⠀⠀⠤⡄
⠀⠀⠀⣴⠀⠀⢈⠇⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠒⠋⢁⠀⠈⠱⡄⠀⠀⠘⣦⠉⠛⠛⠛⠁⠀⠰⣀⠀⠉⠘⠁
⠀⠀⠀⠘⠓⠒⠈⠀⠀⡾⠀⠀⢀⠀⣠⡤⢴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠦⠄⠀⠁⠀⠀⠀⠘⣇⠀⠀⠠⠔⠰⠦⢄⡀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⡇⣾⡁⠀⠀⡨⠏⠀⠀⠐⢀⣠⡄⠀⠀⢠⡿⣷⡀⢠⣶⠶⢶⣶⣄⠀⠀⢹⠋⠀⠀⠀⠀⠀⠀⢹⠀⠓⠂⠁⣤
⠀⢀⠀⠀⠀⠀⠀⠒⠺⡇⠀⠈⠛⠿⠿⠶⠶⠶⢶⣤⡀⢴⠷⠚⠻⣶⣄⣾⠁⠸⣷⠀⣿⣀⣤⣹⣿⠀⠀⢸⡆⠀⠀⠀⢠⣄⠶⢦⡀⠀⢀⣄
⣔⠉⢻⠀⡔⠀⠀⠀⠀⢿⠀⠀⠀⠀⣠⢶⣦⠀⠀⠙⣿⣿⠀⠀⠀⢈⣿⣿⠤⢤⣿⡀⣿⡿⠾⠚⠁⡠⠲⢾⣡⣄⣀⠀⠀⠀⠀⣀⠧⣀⠛⠊
⠉⠻⣟⠆⣧⣀⣀⡀⠀⠸⡄⠀⠀⠰⣿⡀⠋⠀⠀⢀⡿⢿⣦⣀⣠⠞⢹⣇⠀⠀⢹⡇⣿⣀⠀⠀⠐⣧⡄⠁⠀⠀⠉⠁⠀⠀⠀⠁⠀⠈⢳
⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠙⠻⠶⠶⠖⠋⠀⠀⠉⠉⠁⠀⠊⠉⠁⠀⠛⠉⠉⠉⠀⢀⡎⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀
⠀⠀⠸⢷⠄⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠚⠲⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠗⠲
⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⠛⠷⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡴⠶⢶⠾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⣧⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⣯⣿⣭⣭⣭⠭⠶⠖⠛⠋⢹⡀⠈⠉⠁⢰⣽⠇⢰⠁⠀⠀⠀⠀⠀⠌
⠀⠘⢧⡀⠂⠀⠀⠀⠀⡳⢮⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠂⠈⠙⠲⠼⢧⣤⠤⠄⠀⠀⡀
⣤⣤⣤⣭⣷⣶⣶⣦⣥⣤⣤⣬⣭⣷⣶⣶⣶⣶⣬⣥⣬⣷⣶⣶⣿⣷⣶⣶⣬⣷⣶⣦⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣷⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⣿⠁⣹⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⢀⣤⡶⠶⢶⣄⡀⠀⠀⠀⠀⠈⠛⠛⠿⠋⠀⢀⣤⣤⣄⠀⠀⢠⡟⠉⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠈⢿⡄⠀⢀⣠⣄⡀⠀⠀⠀⢰⡟⠁⠀⠙⣷⠀⠘⢷⣄⣀⣴⣿⣄⢀⣀⡀⠀⠀⠀⢀⡶⢶⡄⠀⠀⠀⠀⢺⣅⣀⡿
⣤⣀⠀⠀⢿⡅⠀⠀⠀⠀⠀⢸⡇⠀⣾⡉⢙⣷⣤⣄⣀⠘⢷⣄⣀⣴⠟⠀⠀⢠⡿⠋⠀⠀⠉⠛⠉⠻⣦⠀⠀⠈⠻⠞⠃⠀⠀⠀⠀⠀⠉⠉
⠀⠉⠻⣆⠘⢷⣄⠀⠀⢀⣠⡾⠁⠀⣼⠟⠋⠁⠀⠈⠙⢷⣄⠉⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣴⠟
⠀⠀⠀⢹⡆⠀⠉⠛⠛⠛⠉⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠈⠻⣦⣄⣠⣴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣠⡴⠶⢶⣤⡀
⠀⠀⠀⣼⠃⣠⡴⣄⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡶⠶⢦⣄⠀⠉⠉⢀⣤⠶⠾⠛⠷⢶⣤⡀⠀⠀⠀⠀⢿⣼⣿⠋⠀⠀⠀⠈⢿⡄
⣠⣤⠾⠃⠀⠻⠶⠟⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣾⠃⠀⠀⠀⠘⣷⠀⣰⠟⠁⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣼⠇
⠉⠀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠶⠞⠋⠀⠀⠀⣿⡀⠀⠀⠀⢀⣿⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣠⠾⠛⠶⣤⡶⠞⠃
⣴⠟⠋⠉⠀⠀⠀⠀⠉⠙⠻⣦⣤⡶⠟⠛⠶⣦⣀⣠⣴⠶⠶⠛⠛⠻⠿⢦⣤⣶⠟⠁⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢰⡏⠀⠀⠀⢸⡇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠻⣦⡀⠀⠀⠀⠀⠀⢀⣴⠏⢀⣤⣤⣄⠻⣦⣤⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠈⠻⠶⣤⣤⣴⠶⠛⠁⠀⣿⡀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⢠⡶⠷⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀⢀⣴⠶⠶⣤⠀⠀⠈⢳⡾⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣼⣇⠀⠀⣹⣧⡴⠶⠟⠷⢤⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡙⠛⠛⣿⠏⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣀⡀⠹⣦⡀⠀⠀⠀⠀⣰⠟⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣬⡛⠷⠶⠶⠛⠉⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⠶⠶⠶⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⢟⣏⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠉⠁⠀⠀⠀⣀⡠⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⢠⣤⣴⢒⣊⣉⣀⡠⠄⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣮⢥⣶⣷⣿⣽⣿⣿⣿⣽⣿⣿⣯⣷⣶⣯⢴⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣯⣽⣛⣺⠿⢿⣷⡿⠿⠿⣿⡿⠿⠿⣚⣋⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⣿⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣿⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣀⣿⣀⣀⣀⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠯⣍⣉⣿⣉⡭⠿⠛⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠉⣿⠁⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣤⣶⣾⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠻⠿⢿⣿⠟⠉⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣷⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠈⠉⠉⠛⠻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⡀⠀⠀⠀⢀⠀⠀⢀⣠⣶⣿⠿⠋⠉⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣀⡀⠀⠙⠳⣦⡙⣿⣦
⠀⠀⠀⠀⣀⣤⣤⣄⣀⠙⠻⢿⣿⣿⣿⢿⣷⣶⣿⠟⠋⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠈⢷⡘⣿⡇
⠀⠀⣠⣾⠿⠛⠛⠛⢿⣷⡀⠀⠀⠀⢠⣾⡿⠛⠁⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⢦⡀⠀⠀⢿⡀⠀⢸⡇⣿⣿
⠀⠀⣿⡏⠀⠀⠀⠀⠀⣿⣿⠀⢀⣴⣿⠏⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠘⣧⠞⠁⠀⠀⢸⡇⠀⣸⠇⣿⡏
⠀⠀⢿⣷⡀⠀⠀⠀⣠⣿⡟⢠⣾⡿⠁⠀⢀⡾⠃⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⣿⢙⣷⣄⡘⠷⠀⠀⠀⣾⠁⠰⠟⣸⣿⠃
⠀⠀⠈⠻⢿⣷⣶⣾⡿⠋⢠⣿⡟⢀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣾⠉⠀⠈⢻⡄⣿⠋⠀⠈⠁⠀⠀⠀⣼⠃⠀⠀⣴⣿⠃⣠⣾⡿⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢠⡟⠀⣸⠃⠀⠀⣴⡋⣁⡀⢻⣄⣀⣀⡼⠁⠈⠀⠀⠀⠀⠀⣠⡾⠁⠀⢀⣾⡿⠃⣼⣿⠋⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⠀⠀⢻⡀⠀⠀⠈⠉⠉⢹⡆⠈⠉⠉⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣴⣿⣿⣅⠀⢿⣿⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⡄⠀⠘⢷⡀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⣀⣴⣿⠟⠉⠙⢿⣷⡌⢿⣷⣤⣤⣤⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡌⠻⣄⠀⠀⠙⠶⣤⣄⣀⣀⣀⣀⣤⣤⠶⠞⠋⠁⠀⣀⣴⣾⡿⠋⠁⠀⠀⠀⠀⢻⣿⠀⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣍⠛⠆⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢀⣀⣤⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⣴⣶⣾⣷⣤⣀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠛⠿⢿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡶⠒⠒⠒⠒⠒⢲⡄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⡇⠠⠴⠆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠒⠒⢒⣒⡒⠒⠒⠒⠒⠒⠚⠃⣠⠗⠒⠒⠒⠒⠒⠚⢧⡀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣠⡽⣛⡛⢦⡀⠀⠀⣞⢓⣒⣒⣒⠒⣒⣒⡒⣒⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠿⠞⠹⡟⠁⠉⢻⠇⠀⠀⠉⠉⠉⠉⠉⣩⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⡶⢶⣏⣀⠀⠀⠀⠀⠀⢯⠀⢸⠆⢸⡄⠀⠀⠒⠒⠲⡖⠒⢲⡖⠒⢶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠉⠉⠀⠈⣷⠀⠀⠀⠀⢘⣀⣤⣄⣘⠁⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⡴⠀⠀⡶⢳⢄⣿⡀⠀⣀⣰⠯⠥⠤⠤⠭⢷⣀⡀⠀⠀⠀⡇⠀⣸⣇⠀⢸
⠀⠀⠀⠀⠀⠀⢸⣅⠀⠘⠳⢦⣠⣟⡏⢠⠈⠓⠶⠯⣿⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⡞⢹⣆⡏⢹⢰⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⡶⠴⣇⣹⠇⢺⠶⠶⠶⠶⢾⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠙⠚⠙⠓⠚⠈⠓⠋
⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠳⠶⠶⠶⠶⠞⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⢸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⣶⠶⠶⣶⣶⡶⠶⢶⡶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⣿⠸⠭⠇⠸⠭⠭⠿⠀⠯⠟⢸⠀⠘⠚⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢛⣛⣛⣛⣛⣛⣛⣛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣈⣉⣉⣉⣉⣉⣉⣉⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠁⠀⠀⠐⢀⠠⣄⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠈⠁⠈⠃⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⡆⢀⠀⠲⠒⢲⡶⠶⠶⣶⠶⠖⠠⢀⣀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠓⠀⠈⠀⡀⠈⠀⠀⠀⠈⠀⠀⠀⠤⠀⠀⠰⠀⠈⢩⣟⠻⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠞⠃⠉⡀⠀⠀⠊⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠅⠀⠐⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠤⠀⠀⠀⠙⠁⣽⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢤⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣘⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠐⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣟⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣷⣤⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢛⠿⢶⣦⣤⣀⣀⣀⣀⣄⣀⣀⣠⣤⣀⡀⠀⢀⣀⣀⣀⣀⣀⣤⣀⣀⣤⣤⣤⣤⣶⣿⣿⣿⠿⣟⣛⡭⣉⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣷⣦⣈⠉⢉⣛⣛⠛⢛⣻⣟⠛⠟⠻⢿⢿⣿⣯⣿⣯⣍⣭⣭⠽⠈⢦⡉⠁⢀⣰⡟⢉⠾⠋⢗⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠰⠚⣉⡙⠿⠿⠍⣉⣿⣿⣿⡟⢻⣿⣿⣦⣤⣤⣴⠿⣷⠾⡯⠟⠛⠿⢿⡴⠋⠳⢍⣻⡿⠻⠷⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣭⡥⣄⡀⠀⠋⠙⠿⠻⠿⠛⢻⠿⢻⡿⡯⠋⠀⠋⠉⠀⠐⠂⠒⠉⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠧⣿⣿⣷⣶⣏⡋⠉⠀⠀⠒⠓⠀⠊⢀⣠⣤⣴⣶⣶⠶⣶⣶⣶⣾⣿⣿⣿⣿⡿⢛⣽⣿⣇⣼⢧⣼⡏⣨⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣌⢙⣧⣄⡙⢟⣷⣄⠀⠀⢀⣠⣾⣿⣯⣤⣶⣿⣭⣼⡿⢃⣾⠏⢀⣾⠿⣿⢹⣟⠉⣸⣿⣏⣾⡯⣴⡏⢀⡀⠘
⠀⠀⠀⠀⠀⠀⠂⠈⠉⡃⠛⢮⠛⠿⢙⣛⢻⣿⣧⣴⣿⣿⣛⠉⢻⢿⣭⣼⣿⣋⣾⡿⠿⠿⠿⢛⣴⣟⣨⡿⣿⣿⠿⡿⡗⠓⢄⣁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠶⡥⢄⡠⢀⣀⠀⠀⣈⡉⣉⣉⣤⣤⣞⡀⠒⡢⠈⠟⠀⢐⠊⣀⣀⣉⣡⣽⣄⣤⣤⣏⣀⣿⡿⠶⠾⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠙⠛⠻⠙⠿⠟⠋⠛⠻⠗⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠁⠉⠹⡟⠀⡀⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣄⠀⠀⠀⣰⣀⣼⣦⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠤⠤⠤⠤⣤⣤⣤⣄⣤⢀⣀⣠⣤⣄⣴⣶⣴⣤⠤⠴⠾⠿⠿⠒⠶⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⠀⢰⣿⠀⠀⣴⣦⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⣾⣿⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⢿⣿⣿⣿⠀⢸⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡀⠀⢠⣿⣿⣿⣀⣸⣿⣄⣀⣿⣿⣀⣸⣿⣇⣀⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠿⠿⣿⣿⠿⢿⣿⡿⠿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠀⢸⣿⠀⠀⣿⡿⠀⠘⠿⠃⠀⠘⠋⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⣄⠀⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠏⢀⡤⠴⠦⢄⡀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠤⡤⠖⠋⠀⠀⠀⠀⢱⣙⢚⣠⣄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡎⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠓⠒⠢⣄⠀⠸⣁⡹
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠉⠲⠤⠖⠀⡤⠒⠛⠒⢦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠸⠦⢄
⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⣀⣤⡴⠒⢺⡀⠀⠀⠀⠀⠀⠀⠉⢳⡴⠚⠉⠉⠒⢀⠀⠀⣷⣀
⠀⠀⠀⠀⢀⣠⠇⠀⢠⠎⠀⠀⠣⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⠂⠀⢀⡬⢷⡀
⠀⠀⠀⠀⡇⠀⠙⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣹⡄⠀⠀⠀⠀⣀⠀⠀⣀⡀⢀⡟
⠀⠀⠀⠀⠈⠛⠉⠓⠚⠙⠲⠤⠴⠚⠲⠤⠤⠞⠉⠙⠒⠒⠋⠁⠈⠓⠒⠒⠊⠁⠉⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡯⢿⡀⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⢀⣀
⠀⠀⠀⠀⠀⡀⣴⡻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠼
⠀⠀⠀⠀⢯⠟⣴⢦⠀⢀⡤⢄⠀⠀⠀⢠⣦⠀⠀⢀⡠⠴⠒⠒⢶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠳⠊⠀⠀⠀⠀⣁⠴⠚⠉⠀⢀⡀⠀⠈⢧⠀⠀⠉⠳⡀
⠀⠀⠀⠾⠇⠀⠰⣋⠇⠀⠀⠀⢀⡠⠖⠋⠀⠀⢀⡀⠔⠉⠀⢱⠀⠈⡆⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⡠⠐⢤⡁⠈⠳⡄⢰⠉⠀⠀⠸⡄⠀⠀⠀⣧
⠀⠀⠀⢰⡋⢳⠘⠷⠀⡏⠀⢀⠎⠩⣶⡁⠞⢨⡧⠟⠒⠉⠉⠀⠀⠀⡠⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠉⠁⠀⠀⠀⢳⠀⠀⢑⠆⢀⠗⠒⠁⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠤⢼⡆⠀⠰⠔⠊⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⣀⠔⣋⡀
⠀⠀⠀⠀⣠⠊⠉⠁⠀⠀⢱⡀⠀⠀⣀⣀⠤⠒⠉⡀⣀⠴⠖⠒⠐⣎⠁⠀⠙⢿⠆
⠀⠀⠀⡞⠉⠀⠀⠀⠀⠐⠉⠉⠓⠁⠀⠉⠒⠊⠉⢆⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⡹⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠒⠒⠒⠒⣒⣦⠤⠤⠤⢄⣀⡀
⠀⢀⣸⠤⡄⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡶⠤⠤⠤⣄⣉⣒⠆⠀⠀⠀⠀⠹⡀
⢮⣁⣀⠀⠉⠈⠈⠀⠉⠉⠈⠉⠉⣀⡬⢭⠉⠁⠒⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡩
⠀⠀⠀⡹⠂⠀⠀⠀⠀⠀⠀⠀⠘⠤⠦⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠂⠉⠁
⣀⣀⣈⣓⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣒⣒⣒⣒⣒⣚⣉⣉⣉⣉⣉⣣⣤⣴⣒⣊⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⣾⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠹⣿⣦⡀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡄⠀⢻⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠋⠀⢀⣼⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⢴⣶⣶⣶⣶⣶⠶⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠉⠙⠛⠛⠻⠿⠛⠛⠛⠉⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⠏⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⣿⣟⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⠛⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠘⠛⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⠟⠏⠛⠋⠙⠈⠛⠙⠻⡟⠻⡲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠏⠛⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡹⣦
⠀⠀⠀⠀⠀⠀⣼⠏⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⣴⠅⠀⠀⠀⠀⣠⣴⡾⡟⠟⠻⠹⠿⠟⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡄
⠀⠀⠀⠀⢰⡯⠀⠀⠀⣰⡾⡛⠏⠁⠀⠀⠀⠀⠀⠀⠘⠱⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⣿⡐⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣇
⠀⠀⠀⠀⣿⠀⠀⢸⣿⠉⢀⣠⣶⠾⠿⠟⡿⣶⣤⠀⠀⠀⠀⠀⠀⠸⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⣿⣄⠀⢸⣿⢠⡾⢋⣠⣤⣄⡀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠸⣷⠀⢸⡏⣿⣷⣿⣿⣯⢙⣿⡅⠀⠀⢸⣷⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠹⣦⠀⢿⣿⣿⣿⣿⣿⡏⣿⡇⠀⠠⢸⡟⠀⠀⠀⠀⠠⢵⣿⠁⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠻⣿⣿⣿⡿⠾⠋⢀⣀⣶⡟⠁⠀⠀⠀⢀⣲⣿⠿⣦⣤⣤⣤⠤⠶⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣙⠛⠿⠶⠿⠿⠛⠉⠀⠀⠀⢀⣰⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣴⣤⣀⣠⣠⣤⣠⣵⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣼⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣿⢿⡿⠿⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠂⠀⢀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣶⠳⡖⢋⠕⢊⣾⠟⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⢀⣿⠃⠀⠉⠉⢠⣿⠇⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢧⠀⠀⠀⣾⣿⣀⣀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⣧⣀⣼⡇⢀⣿⣿⣿⣿⣿⣿⠀⠀⠸⣄⠀⣀⣠⣤⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣟⢶⡀⠀⠀⠀⡼⠁⠀⣰⠋⠉⠁⢸⡇⢐⣿⣿⣿⣿⣿⣿⠀⠀⠀⣽⢫⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢠⠿⠀⢹⡆⠀⡰⠁⢀⡜⠁⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣇⢀⣠⠗⠉⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣦⡿⣄⢀⣀⡼⢀⠔⠁⠀⡎⠀⠀⠀⠀⠀⠈⡇⢸⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⢷⡿⡆⣤⠀⢁⠀⡀⢰⠃⠀⠀⠀⠀⠀⣰⠃⣬⣿⡿⠟⠋⠉⠀⠀⣀⣤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢹⡿⣔⣿⡻⠛⢡⡀⠀⠀⠀⠀⡴⢁⣾⣿⣿⡄⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⠀⠈⠁⠙⠉⠳⠤⠏⢹⣄⣀⢀⣜⡡⢟⣫⣥⣾⣤⣤⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠉⠀⢀⣺⣷⣤⣦⡀⠀⢀⣿⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠸⣿⣏⠀⠉⣷⢀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠓⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣙⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠈⢻⣿⣆⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠙⣿⣷⡀⣾⡿⠋⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠉⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⢠⡾⠋⠉⠛⢷⡄⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠈⢿⣷⡄⠀⠀⠘⢷⣤⣤⣤⠾⠁⠀⢀⣀⣀⣀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠉⠀⠀⠀⣴⠟⠉⠉⠙⢷⡄⠀⠀⠘⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣧⡀⠀⠀⠀⢻⣿⣤⣀⠀⠀⠀⠀⠀⠘⠷⣦⣤⡶⠟⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⢿⣷⡄⣀⣤⣾⣿⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠋⠁⠀⢨⣿⣿⡿⠟⠋⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠻⣦⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⣶⠛⠶⣦⣄⠀⠀⠀⢻⣿⣆⠀⠀⠀⢀⣠⣾⣿⣿⣦⠀⠀⠀⠀⠀⠛⢶⣤⣴⠾⠃⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⢻⣆⠀⠀⠙⠳⣦⡀⠀⠹⣿⣦⣤⣾⣿⠿⠛⠁⠙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀⠙⢷⡀⠀⠀⠈⢻⣄⠀⢿⣿⡟⠋⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣠⡶⠶⠶⣦⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠙⢷⣄⡀⠀⢹⡆⠘⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢰⡏⠀⠀⠀⠈⣿⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠈⠛⠳⠾⠃⠀⠘⢿⣿⣦⣄⡀⠀⠀⢸⣿⡇⠀⠀⠘⣧⡀⠀⠀⣠⡟⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣾⣿⠇⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⡠⠀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⢀⠠⢐⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⣀⣢⢠⣈⣀⣥⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠀⡿⠂⠀⠀⡐⢁⢲⡀⠀⠸⠀⠀⠀⠀⣠⠤⠶⠶⠿⠿⢶⡄
⠀⠘⠈⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠾⠀⠀⠀⠀⠀⠛⢾⡡⠀⢁⡆⠀⠀⠀⢹⠀⠀⠀⠄
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠁⠠⠀⠀⠀⠀⣶⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢠⠀⠀⠀⢨⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡄⠀⠀⠐⠀⠀⠀⠐⠒⠘


⠀⠀⠀⠀⢤⠉⠀⠀⠀⠀⠀⢀⣠⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢘⠆⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⠰⣆⠀⡀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⡄⠄⠀⠀⠀⠀⠸⠁
⠀⠀⠠⠂⠀⠀⠂⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠲⢎⠣⣄⠀⠀⠈⠀⡃⠈⠠⠈⠀⠠⠀⠑⠠⠀⠀⠀⢀⠀⠐
⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⢙⡧⠄⠀⠀⠀⠀⠐⠨⠤⠒⠀⠡⠀⠀⠀⠀⡁⢈⠀⠠⠋
⠀⠀⠈⠀⠈⠁⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣠⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠠⠊⠀⣀⣤⣤⣀⡈⠀⠀⠀⠀⣿⢿⡄⢸⣟⠖⠁⠀⠀⠀⠀⢀⡨⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄
⠐⠀⠂⠀⡰⢛⡅⠈⠀⠁⠀⠀⠰⠀⠀⢨⠉⠃⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡜⢋⣀⣠⡠
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠨⡀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠠⡬⠀⠀⠀⠀⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠚⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢀⠀⠀⠀⠀⠀⢀⣠⠄⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⢀⠂⠈⠂⠀⠀⠀⢠⡏⠃⠀⠀⠀⠙⡆⠀⠀⠀⡀⠔⠒⠂⠀⡀⢀⠠⠀⠀⢀⠔⠀⠀⠀⠑⠄
⠀⣀⠀⠀⠐⠚⠛⠛⢲⠇⠀⠀⢀⠈⠀⠂⠂⠀⠁⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⢈⠀⠀⠒⡄⠶⠀⠒⢄⡜⢂
⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢆⠤⠁⠔⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠈⠘

⠀⣀⠀⠀⠀⠀⡀
⢀⠀⠀⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⣰⣤⣤
⠀⠀⡀⠙⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⠃
⠀⠀⡇⠀⢣⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⢀⡜⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠⠦⡀
⠀⠀⡇⠀⢸⠀⠀⡇⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠶⠒⠃
⠀⠀⢀⠀⠸⠀⠀⠠⠀⠘⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠄⠄⠤⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠴⠛⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠁⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠠
⠀⠀⠀⢤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠛⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⡀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠴⠀⠒⡁⠀⠀⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠐⠒
⠀⡀⠀⠀⠀⠈⠤⠠⠴⠒⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡀⠉⠁⠀⠀⠀⣤
⠀⢁⠀⠀⠀⠀⠀⢀⡀⠄⠠⠀⠀⢉⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠔⠈⢀⡂⠀⠀⠀⠀⡅⠀⢀⡀⢀⡤⢠⠠⠀⠁
⠀⠈⠀⠬⠀⠈⠀⠃⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠘⡀⠀⠁⠀⠀⠀⠇⠀⢀⣤⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠛⠁⠀⠀⠄⠀⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠀⠀⢛⠉⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⢀⡀⠀⣁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢨⢀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠁⠈⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠄⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠲⠠⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠘⠆⠘⠃⠂⢂⠄⠀⠀⠀⠀⠀⠤⠄⢙⣐⣒⣉⠠⠤⡆⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⡀⠀⠀⠀⠀⣀⠤⠀⠀⠠⠤⢀
⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠸⡀⢆⢀⣄⠲⠀⢂⠀⡇⠀⠀⠀⠀⠰⢠⠂⠈⠲⠂⠀⠀⠀⠀⠀⢀⠈⠀⣀⡀⠁⡁⣀⣀⢱
⠀⠀⠀⠀⠛⠄⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⢀⠜
⠀⠀⠀⠀⠁⠀⠈⠠⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠒⠒⠒⠒⠒⠁⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⣉⣤⣤⣤⣤⣤⣉⡙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣠⣶⣿⣿⠋⣡⣤⣌⠙⢿⣿⣷⣄⠙⢷⣄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣼⣿⣿⣿⡇⢸⣿⣿⣿⡧⢈⣿⣿⣿⣷⡈⢿⣆⠉⠛⠛⠻⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣼⣿⣿⣿⣿⣷⡈⠻⠿⠟⢁⣼⣿⣿⣿⣿⣷⠈⣿⡀⢿⣿⣷⣦⣌⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⠿⠿⠿⠷⠶⠾⠿⠿⠿⢿⣿⣿⣿⡄⢹⡇⢸⣿⣿⣿⣿⣷⡄⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⠁⣾⠇⣸⣿⣿⣿⣿⣿⣿⡆⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠸⣿⣿⣿⠿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⠏⢰⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠘⢿⣿⠀⣤⣤⣴⣷⣦⣌⡉⠛⢿⡿⠋⣠⡿⢁⣼⣿⣿⣿⣿⣿⠇⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⠀⠛⠛⠛⠛⠛⠛⠛⠃⣼⣷⡾⠋⣠⣾⣿⣿⣿⣿⣿⡟⢀⣿⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣠⣾⣿⣿⣿⣿⣿⠿⠋⣠⣿⡟⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢤⣄⠘⢿⣿⣿⣿⣿⣉⣩⣤⣴⣿⠿⢋⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣌⡉⠛⠛⠛⠛⠛⣉⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⣸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣉⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⡿⢿⠙⠶⠖⠀⠀⠈⢛⡄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣌⣀⠠⠀⠀⠀⠀⠈⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⣄⠤⠀⠠⠀⠀⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣠⠆⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⢷⣶⠾⠛⣉⡤⠞⣡⣿⡿⠛⠷⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢡⣾⡟⠁⠀⠿⢤⡼⠥⢤⣴⣿⡿⠁⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠀⠀⠀⠀⣾⣿⠁⠄⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣯⡀⠀⠀⠀⠀⢀⣼⣿⡏⠀⠀⠀⠀⠀⠈⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠈⢷⡀⠀⠀⠀⣸⡿⠛⢿⣶⣤⣤⣤⣤⣤⣼⣿⣿⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⣷⡀⠀⣠⣿⡇⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⣷⡀⠀⠀⢀⣀⣀⣀⣀⣿
⢀⠀⠀⠀⠀⠀⠀⣾⣆⢠⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⠞⠿⠷⠛⠉⣿⠃⠀⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣷⣴⠞⠋⠉⠉⠉⠙⣿
⣼⡄⠀⠀⠀⠀⠀⢻⡜⠛⣧⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣻⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢰⡿⠹⡆⠀⠀⠀⠀⠀⢿
⣿⣧⠀⠀⠀⠀⢠⡜⡇⠀⠸⣷⡇⠀⠀⠀⡼⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⡧⠚⠁⠀⠀⠀⠀⠀⣸
⣿⣿⣆⠀⢰⡄⠸⠀⠀⠀⠀⣸⠇⠀⢀⡼⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡿
⢹⡟⢿⣆⣺⢷⡜⢦⣀⢀⣠⠟⢀⡴⠋⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠈⣧⣈⣿⣯⣠⣿⡆⠈⢉⠁⠀⢸⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⠛⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠁
⠀⠘⣿⣇⠙⢿⠙⢿⣀⣿⡀⢀⠘⡴⢀⣴⢆⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢀⣶⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡏
⠀⠀⠈⣿⡄⠀⠀⠘⣿⡿⣇⢸⣷⡙⡟⠼⠋⢁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⢠⣾⣿⣿⣿⣡⡄⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡀
⠀⣧⠀⢹⣷⡀⠀⠀⢻⡇⠹⣾⡏⢿⡆⠀⠀⣼⠿⣆⠀⠀⠀⠀⠀⢀⡾⠁⣤⣿⡿⠿⠛⢉⡝⠀⡔⣠⢀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⢳⣾⣿⡇⠀⠀⠀⠀⡀⠈⠀⠀⠙⠳⠛⠋⠀⢹⣧⣀⣀⡀⢠⣞⣡⠶⠋⣁⣤⣶⣿⣿⣤⣾⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢿⡀⠈⠋⠋⠀⠀⠀⢦⣿⣦⡀⢠⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠘⣧⠀⠀⠀⠀⠀⣷⣾⣿⡯⠻⢿⣿⣦⠀⠀⠀⢸⣿⣿⡿⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠘⣧⡀⠀⠀⠀⢻⣿⣿⡁⠀⠀⠈⣿⣿⣀⣤⢿⣛⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⢷⣤⣤⣀⠹⠿⠿⠶⣖⣒⣭⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠈⢻⡉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠙⠷⣤⡀⠛⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣭⣷⣶⣾⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠁⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⣉⣉⣉⣉⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣴⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣝⡛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⣻⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⣧⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⠞⠁⢹⠄
⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⠁⠀⣠⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⣼⠋⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡼⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⣼⠃⠀⢀⡿⠖⠲⣦⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢀⡾⠉⠙⠛⢓⡶⠲⠶⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢰⠇⠀⠰⠋⠀⣠⠴⠛⢦⣄⠀⣀⣴⣻⣦⡄⠀⠀⠀⠀⠀⠠⠞⠋⠀⣀⡼⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣧⡤⣤⣤⣤⣤⡤⠬⢥⣄⠘⠭⠟⣿⠋⢻⣁⡽⠿⠄⠀⠀⠀⠀⠀⠀⠰⠛⢉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣿⠚⠛⢧⣄⠀⠀⠀⢀⡴⠚⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢀⣹⣤⠞⠋⠉⠂⠀⢠⡟
⠀⠀⠀⠀⣰⠋⠀⠀⢴⣶⣤⣾⣿⠆⠀⠀⠈⢷⣦⡀⠀⣆⠀⠀⠈⠀⠉⠃⠀⠀⠀⠀⣼⠁
⠀⠀⠀⢠⡏⠀⠀⠀⢀⣽⣿⣿⣷⡄⠀⠀⠀⠀⠻⣷⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⢸⡇⠀⠀⠀⠈⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⢻⡆⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⡟
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠸⠃⠀⠀⠈⢉⠉⠉⠉⠉⣷
⠀⠀⠀⠸⣷⣀⣠⡴⠞⠛⠛⠻⣦⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⣄⡀⠈⠙⠛⠛⠋⣿
⠀⠀⠀⠀⠘⠿⠯⠤⢤⣤⣄⠀⢸⡆⠀⠀⠀⠤⠟⠁⠀⠀⠀⠀⣈⡿⢦⣄⣛⠛⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡼⠃⠀⠀⠀⠀⢀⣀⣠⡤⠶⢾⡉⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠒⠚⢻⡏⢿⡀⠀⢦⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣹⣦⠴⠿⠛⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⣠⠖⠋⠀⠀⣀⣄⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⢿⣿⣿⣿⣶⡄
⠀⠀⡼⠁⢠⡖⢲⡀⢧⣬⠇⡴⠶⡄⢹⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠛⣿⣧⣀⣼⡿⠛⢿⣿⡄
⠀⢸⠃⠀⠈⠓⠚⠀⠀⠀⠀⠓⠖⠃⢸⠃⠀⠀⠀⣀⣴⠂⠀⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣦⣾⣿⠇⠀⠀⠀⠀⣠⠆
⠀⢸⠀⢰⡏⢹⡄⠀⠀⠀⠀⣀⣀⣠⠞⠀⢀⡤⢞⡟⠁⠀⠀⣿⣿⡏⠉⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⣀⣴⡿⠁
⠀⠘⣇⠀⠉⠉⠀⠀⡴⠋⠉⠉⠁⠀⢀⡴⠋⣰⠋⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠋⠉⠉⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠈⠳⣄⡀⠀⠀⣧⠀⠀⠀⠀⢾⡉⣳⠞⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠰⢾⣧⠟⠁
⠀⠀⠀⠀⠀⠉⠙⠒⠛⠀⡴⠋⠉⢳⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠀⢠⣾⣿⣷⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣵⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣼⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⢸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡟⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠦⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⡱⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠘⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣟⣻⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠝⡿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⠻⣿⢡⣶⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡈⢿⣿⣮⣿⡻⢿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣽⣿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣢⣙⢿⣿⣿⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠟⣼⣿⢃⡹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣝⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣮⣷⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡉⢻⠽⣟⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠹⣟⣿⣿⡃⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⡑⠢⠭⣑⡂⢥⣡⣞⡩⠿⠛⠛⠛⠛⠛⠛⠛⠉⡳⠀⣻⣿⣿⠦⠀⢸⣿

⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⡄⢀⠐⠁⠔⢸⡄⠀⠀⠀⠀⠀⠀⠁⠀⣵
⠀⠀⠀⢀⡠⠈⠋⠆⢰⠆⣝⠀⠀⠀⠐⠐⠶⠚⠙⠿⠿⣖⡂⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⢀⡼⠎⠀⠀⡀
⠀⠀⠈⠐⠂⠀⡧⠐⠒⡠⠡⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠚⣿⡷⠀⣤⢠⠌⣍⢀⣀⣈⣉⡀⢀⡀⠀⠀⠈⠙⠒⠒
⠀⠀⠀⠤⡀⢶⠸⠟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣴⣿⡇⠀⠈⠻⠴⣧⣰⣄⣤⣠⡷⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠃⠀⢰⡄⠀⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣷⣄⣤⣾⣿⣿⡿⠀⠀⢀⡀⠡⠔⠤⠀⠀⠀⡇⠀⠀⠀⠐⠒⠀⠡⡀
⠀⠀⠀⠀⠀⡠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠙⠁⠀⠀⠀⢀⣀⠀⢐⠇⠀⢠⠃⠀⣄⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠸⠀⠀⠹⡇⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠁⣀⣴⠋⠀⠀⠘⢄⣀⢀⣠⣠⣶⡞
⠀⠀⠀⠀⢀⠀⠀⢠⡇⠀⠀⠀⠀⢘⣁⠀⠀⠀⠀⠻⣷⣶⣾⣿⢃⡖⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠨⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠖⠋⠀⠀⠀⠔⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⡀⡇⠀⠀⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣸⡀⠀⠀⠀⠀⠀⠀⢂⣿
⠀⠀⠀⠀⠀⠑⠀⡾⠀⠀⠐⢦⣀⣧⣀⣴⡷⠃⠠⢂⣱⣯⢧⠷⠾⣷⣄⠀⢐⠀⣏⠀⠀⠈⠻⣤⣀⣀⣀⣰⣦⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠃⠁⠈⠉⠆⠀⠀⠈⢻⡆⠀⠀⢹⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠡⠀⠀⠀⢴⡇⠀⠈⠒⢤⣄⣠⣼⣶⠿⠋⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣤⣀⣠⡶⡀⠀⠀⠇
⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⣾⣎⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⣄⠀⠀⠏⠀⠀⠀⠀⠠⠟⠃⠙⠁⡧⠠⠄
⠀⠀⠀⠀⠰⡿⢿⣿⣿⣿⣿⣟⠓⠀⠈⠳⣦⣄⣀⣀⣀⣀⣀⣴⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠶⠒⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⣶⣿⣝⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣖⠚⠉⢡⠶⢦⡸⣧⡴⠋⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣶⣟⠉⢠⡟⠉⢰⡄⢸⡗⠋⠓⠈⣃⣤⠴⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠀⠸⣧⣤⣤⡘⠳⠦⠞⢁⣀⡥⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⡇
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠶⢞⣣⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⡇
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡏⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢸⡇⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⢸⡇⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠓⠲⠦⢤⣤⣀⣀⣀⣠⣤⣤⠤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁
⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⢤⣤⣄⣀⣀⣀⣠⣤⠤⠶⠚⠋⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣏⣏⣼⣩⣽⣥⣿⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⠿⢿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠙
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⡿⠿⡿⡿⢻⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠙⢀⠀⠀⢀⡈⠙⢿⣿⣿⡇
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⡄⠀⠰⢀⠤⠀⠈⢹⣿⡇
⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡖⠀⠀⠃⠌⠁⢈⠁⠀⠀⣾⣿⣿⡀
⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⡀⠀⠀⠀⢀⣤
⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣠⣾⣿⣿⣿⣿⣯⣶⣤⣴⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠟⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠠⢖⠀⠀⠀⠀⠀⠀⠀⢐⡶⣶⣶⣶⣶⣶⣶⣶⡶⠒⠀⠀⠐⠲⠶⣶⢶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⡿⢿⣿⣿⣯⠀⠀⠀⠀⠀⢠⣴⣾⡿⢻⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣄⠀⠀⠀⠀⠀⠸⣿⣿⡤⠤⣿⣾⣿⣿⣷⣄⡀⢀⣀⣤⣾⣥⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⢳⡄⠀⠀⠀⢰⣿⣽⢷⣶⣿⡏⣮⣹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡁⠀⠀⠀⢸⣿⣿⡿⠿⣿⣷⣮⣹⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠈⠓⠚⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⡀⢀⡾⠗⠓⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡖⠒⢈⣥⣀⠀⠀⢡⠋⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠰⣦⣠⣬⣅⠀⣸⠿⠿⢁⠀⠀⠰⠒⠂⠀⠒⠂⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣟⢿⠻⢷⡄⠀⢰⠀⠀⢀⣀⠀⠀⠇⠀⠀⠀⢀⢀⠈⠀⠈⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠿⣿⣿⣶⣭⠀⠈⣦⡀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢙⣿⣿⡟⢃⣴⣿⣿⡆⠀⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠰⠛⠛⠋⠁⠀⠀⠀⠀⠙⠛⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⠀⣀⠀⣀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⠺⠛⠻⠼⠻⠚⠥⠸⠻⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠘⡤⠔⠀⠉⠉⠉⠉⠉⠁⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⣀⢀⣀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠏⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣠⡟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢸⠀⠀⠀⠀⠠⣄⡀⠀⠀⠤⠄⠒⣩⠏⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢻⠀⠀⠀⠀⠀⢀⠶⢆⡀⣀⠰⠚⠁⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⣷⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠖⠓⠀⠤⠤⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡛⢌⡦⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠻⣦⠈⠡⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⢶⠶⠶⠶⠆⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠘⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠔⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢋⡆⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡺⠏⠦⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠰⠀⠳⣄⠀⠀⠀⣀⢠⠔⠒⡝⠁⠈⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⠈⠷⠐⠊⠁⠀⢀⡠⠟⠲⠦⣤⣀⡀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠈⠱⣤⡜⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⡠⠛⠻⠉⠉⠉⠁⠀⠀⠈⠒⢄⡀⠀⢀⠞⠀⠈⠙⠃⠤⠤⠌⠿⠿⣧⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠙⠓⠤⣀⣀⣀⣀⣀⣀⣀⡀⠀⡨⠛⢷⣄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢈⣳⣤⡚⢠⡄⠀⠀⠀⠀⠀⣤⣤⣤⡀⣀
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠛⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠒⣒⣖⡚⠛⣻⣿⡛⢻⣿⣖⠒⠒⠖⠲⠚⠷⠲⣶⣶⡶⠶⣶⣶⡿⢵⣶⣤⢶⣤
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠘⠋⠉⠃⠈⢉⠉⠃⢹⣏⣹⠃⠀⠀⠀⠀⠀⠰⣿⣽⣿⢺⣿⣹⡿⢺⣯⣽⣿⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡦⠤⠴⣦⠤⠤⠤⣤⡤⣥⣤⣤⣤⠤⠤⣤⣤⣤⣬⢭⣥⣤⣭⣭⣥⣬⣭⣉⣠⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⢠⣀⣀⣠⣘⣀⣀⣀⣀⣀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣻⣛⣛⣛⣻⣻⣯⢸⡇
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠐⠒⠚⢦⠔⡄⠤⠀⠀⠀⠤⠄⠤⠤⠤⠤⠤⠤⠤⢤⣤⣄⠀⠀⠈⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠰⣆⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠍⢹⠀⠀⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⢘⣄⠀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠭⠥⠤⠬⠤⠤⠤⠤⠤⠤⠤⢴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠓⢤⣤⣇⠀⠱⢦⣄⠀⠠⣄⡀⢀⣀⣀⡀⢀⣀⣀⣀⣀⣀⢀⣀⠀⠀⢀⣀⣀⣀⣀⡀⣀⣀⣀⢀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠛⠓⠒⠚⠛⠛⠛⠒⠛⠓⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣄⠒⣶⢤⠤⡴⢶⢦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⠞⠫⠚⢡⠮⢂⠕⣡⢄⡡⢊⠔⠩⢫⡞⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡵⠏⠀⡀⢐⠍⡠⢀⡠⢀⠔⡠⢀⠄⡀⠐⠀⠐⢀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣺⠅⡢⢑⠬⢂⠔⡨⢒⠥⡪⡔⠝⡪⠕⠊⠄⢐⠅⡨⠚⡠⠂⠀⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠕⠠⠀⢡⡠⠀⡁⠢⣁⠼⠋⠔⡠⠐⡡⠊⣰⠀⢂⠔⢁⠔⡡⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠴⢊⢴⠎⢊⠔⡣⢲⡁⠡⠄⠀⢊⠔⠞⠡⠊⡡⠔⡡⢊⠄⢊⠔⠂⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢕⡡⢊⠔⠠⡊⡜⠆⢐⠠⢊⠄⡢⡄⢴⡈⠔⡡⢐⠡⢊⡠⠒⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡁⢈⠁⡠⠊⡔⠱⡑⣤⢈⠔⣡⠔⡹⢆⠑⢎⡐⠁⠔⡁⠒⠁⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡖⠡⠊⢔⡡⣊⡠⢁⢚⡡⢊⠔⠈⢔⡢⢻⡀⢩⢔⡈⡂⠀⠀⠀⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⣊⠐⠁⡨⠈⠁⢞⡡⢂⠁⡊⠔⠙⢊⡶⣡⠊⠑⡩⠀⠀⠀⠀⠄⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣎⢂⠐⡡⢘⠈⡷⠁⢊⠀⠡⢀⠍⢰⡉⠀⡠⠠⠂⠔⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣦⡩⣪⠃⢞⠥⡊⢐⠅⣊⢄⠆⠀⠸⠅⠀⠁⡀⠂⣀⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠉⠙⠿⣤⡀⠹⣊⡐⠁⠊⠁⠁⠁⠀⠀⠀⣀⠠⠀⠀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⡉⠲⠷⠿⢦⣴⡾⢛⠀⢈⠻⠛⠋⡀⠛⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣐⣂⣭⠟⣁⣤⣤⢶⡲⠟⠿⣧⣒⢚⠓⣶⢾⠜⠓⢚⣱⣟⡖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡗⣼⣏⡀⠠⠤⠈⢽⣌⠋⠉⠒⠶⠤⠉⠛⠋⢡⠄⠒⣠⣿⢫⡿⣽⣧⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣣⡿⢯⣞⢟⣦⣁⣒⢲⣛⣛⣺⣷⣦⣀⣀⣀⣀⣀⣴⣿⣟⡟⣿⢳⣛⣽⠿⢿⢿⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⣻⠿⢞⡭⣘⣶⠋⠿⠿⢻⣿⠏⣛⣩⣿⣛⢟⣯⡽⢿⠿⠟⠘⣃⣋⡩⢥⡲⠍⠩⠍⠱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⡠⠈⠉⠓⠚⠛⠋⠽⡭⠦⠤⠤⢼⠭⡮⢭⠉⠟⠟⠟⠘⠲⠂⠈⠁⡄⣞⣛⣋⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠂⠐⠀⠀⡔⠰⢁⠌⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠁⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡀⠀⠆⡀⠠⠀⠄⠐⠰⢀⠃⠄⠐⠠⢀⠁⠂⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡦⣀⡁⠁⠰⣱⠃⠞⠾⠲⠷⢷⣆⣤⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡟⠁⠀⠉⠐⠗⠀⠲⠒⢢⣄⢸⣼⣧⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⣠⣦⡄⣄⣀⣀⣼⣽⣄⣨⣅⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⣼⢋⠉⠀⠉⣖⠽⠬⡌⠁⢭⡤⠭⡘⣟⡙⠫⢨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢰⡟⢀⠖⠉⢀⠠⠐⠠⠚⠀⠀⠀⠀⡇⢧⢘⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀⠟⡁⢫⠐⠠⠢⢄⠠⠐⠀⠀⠀⠀⠀⢀⠘⣇⢀⣴⣧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡵⢄⡉⠀⠡⠒⠂⠀⠠⠤⠤⠤⠀⠐⠒⠻⢄⡟⠋⣙⣿⡟⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢦⢤⡜⢰⠁⠀⠀⠉⣴⣤⣤⠀⠀⢀⣀⣀⣀⣀⣠⢤⡏⣹⣨⣿⠆⠛⠛⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡔⣥⣶⢀⣇⢤⣔⠾⠝⠃⠉⠀⠀⠠⠍⠩⠭⠀⠀⠀⠀⣣⢸⡷⣟⣉⣁⠉⠁⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⡣⣾⣧⣬⣬⣭⣥⠖⣆⣒⣒⣄⣤⣥⡬⠭⢭⣭⣭⣁⡀⣽⢐⠞⠁⠠⠛⢆⠐⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣷⣝⣻⣟⢵⡦⠦⠄⢀⣀⣀⣁⠀⠰⠛⠳⠗⠒⠲⠶⠒⣯⣼⢀⣴⡆⣿⠸⠆⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠛⠛⠛⠛⢻⠀⢠⡛⠏⡶⠓⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠳⠞⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠀⠂⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠠⠉⠚⢠
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠈⠠⠀⠀⠀⠀⣀⠀⣠⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⠒⠒⠒⠤⣄
⠀⠀⠀⠈⠀⠀⢸⡋⡛⣻⠀⠱⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁⠀⠀⣀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⣿⡉⠉⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠠⠀⠀⠀⣀⠉⠉⢻⡀⠐⢄⠈⡆⠀⠀⠀⠀⠀⠀⢸⡀⢿⠁⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠀⠀⠸⠛⠛⣏⡃⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢸⠁⠀⠠⠄⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠
⠑⠀⠀⠀⢀⡠⠜⠂⣴⡿⠇⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⢷⡈⠢⣀⠠⡄⠀⠑⠂⠀⠀⠀⢀⠎
⡄⠀⠀⢀⠈⢳⣄⡸⣿⡇⠰⣀⠀⠀⠈⠀⠘⣆⠀⠀⠀⠀⠀⠙⠦⣄⠀⠈⠢⡀⠀⠀⢀⠔⠁
⠀⠀⣠⡌⠛⠦⠋⠀⠙⢥⠖⠋⢦⡀⠀⠀⠀⠈⠓⠠⠤⢀⡀⠀⠀⠈⠉⠐⠒⠊⠲⡈
⡀⠂⠻⠿⠶⠃⠀⠀⠀⠀⠳⠚⣩⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⢀⣀⡂⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠙⠳⠦⣈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⠒⠳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿
⣿⣿⣿⣁⣈⣀⣀⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠤⣀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⣤⣤⣀⣀⣀⠁⠰⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠚⠉⠉⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠁⠒⠖⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠒⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠁⠠⠄⠒⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣙⣦⣀⣀⠀⠀⠀⠀⠀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⠒⠂⠀⠶⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠂⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠳⠁⠀⠀⢀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⠁⠀⠀⠀⢤⡶⠏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣼⣤⡀⠀⠀⠱⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠂⠠⠤⠭⠀⡄⠀⠁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⣠⢞⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡇⢰⡀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠠⣄⣀⠀⠀⠂⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡷⣀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⢀⠘⢑⣂⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⠁⠈⠣⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠆⡇⠀⠀⠀⠀⠐⠄⠀⢰⠀⡇⠀⢀⠀⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣀⡯⡰⠁⠀⠀⡠⠊
⠀⠀⠀⣀⣤⠤⢶⣶⣶⣖⣾⣓⣂⡀⣛⡂⣲⣶⠀⠦⠤⢄⡀⠀⠀⠀⠀⠀⠀⢰⠥⠟⠀⢀⡠⠊
⠀⢰⡏⢡⣶⣶⣶⣯⠛⠋⠛⣁⣈⣭⡍⠉⣩⣿⣶⣦⣤⠀⢀⣦⠀⠀⠀⠀⠀⢸⡆⠐⣶⡉
⠀⠘⠉⠛⠻⠯⢭⣥⣸⣛⣛⣛⣉⣉⣀⣀⣈⣍⢭⢭⣄⣲⣭⣰⠀⠀⠀⠀⠀⠈⠘⠛⠛⣇
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢧⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⡟
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣹⣿⣿⣿⢯⣿⠇⠀⠀⠀⠀⠀⠀⡀⠀⢰⡇
⠀⠀⠀⠀⠑⢤⣤⣀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⠟⠁⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠺⠿⣿⣷⣶⣿⣿⣿⣿⣿⠿⠊⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⡏⠀⠀⠸⣆⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣖⣚⠋⠀⠂⢤⢿⡄⠀⠀⠀⣄⣀⡄⠀⠀⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠛⢝⢻⣿⣿⣦⣶⣶⣶⠟⢷⡀⠀⠀⠀⢿⢷⣶⣤⣀⡀⠀⠀⢀⠀⢿
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠉⠙⠛⡟⠛⠉⠁⠀⠘⣧⠀⠀⠀⢸⠀⠉⠙⢿⠿⠿⠿⠏⠀⢸
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⡄⠀⠀⠘⡆⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠊⠁⠯⠽⠭⠇⠤⠀⠿⠿⠼⠵⠸⠥⠁⠸⠿⠫⠼⠯⠼⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠁⠀⠀⠀⠀⠈⠶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⠂⠀⠀⠀⠢⣤⣀⣠⣴⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠙⠢⢄⣀⡀⠀⠀⢀⣀⣠⠟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠅⠠⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢁⡄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡊⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠒⠂⠀⣀⠄⠀⠀⣀⡄⣀⡤⠞⣹⠁⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠣⠀⠈⠉⠁⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⣉⠙⠛⠛⠉⠁⠉⠁⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢅⣀⠤⣤⢤⣤⢤⡤⠼⢑⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⡄⠀⠜⣀⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠼⡇⠂⠀⠒⠒⠒⠒⠂⠀⠐⡁⠀⠠⠀⠀⠀⠀⠀⠀⡆⠀⠃⠀⠀⠀⠀⠀⠰⠀⠀⠀⠣⠜⠉⡤⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡖⠂⠉⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠢⣀⡠⠆⠀⠀⠀⠀⠀⠀⡠⢶⠁
⠀⠀⡀⠀⠠⠀⠀⠀⠀⠇⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢⠀⠀⢀⣀⠀⠀⠀⠀⠑⠢⢄⣀⣀⣀⣀⣀⣤⣤⡲⠯⠖⠋
⡀⢀⡍⠀⠐⠒⠒⠂⠀⢸⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⢀⠀⣤⢯⡑⢢⣀⠀⢀⡔⠙⣏⡙⠉⢹⠏⠀⠀⠙⣄⠀⠀⡀
⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡦⢀⢀⡀⠀⠀⡀⠀⣀⠠⠤⠌⠀⠠⠀⠘⢯⣻⠄⠈⠒⠚⠳⠆⠀⠀⠉⢹⠀⠀⠀⠀⡇
⡇⠠⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠓⠠⠀⢀⠀⢀⡀⠀⢀⠤⢄⠀⠀⢀⡤⠴⠖⠊⠷⠀⠚⠀⠘⠒⠲⠤⢄⣸⠒⠤⣀⡴⠁
⡇⠀⠈⠦⠤⢤⠤⠤⠴⠏⣄⠀⠀⢩⠀⠀⠀⠁⠀⡯⠉⠐⠅⢤⣀⠀⠀⠘⡄⠀⠈⠁⠠⠠⠠⠄⠀⠀⣀⡤⠞⠋⠀⢠⣧⡴⢲
⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣟⠀⠀⠀⠐⡬⠤⣈⡤⠼⡆⠁⡀⠀⠀⠀⣺⠌⠉⠀⠀⢀⣠⠔⠉⢹⣤⠶⠥⠤⠤⠤⠤⠤
⡇⠁⠈⠁⠀⠉⠁⠉⠉⠉⠉⠁⠈⠈⠉⠁⠁⠁⠁⠉⠐⡀⠀⡌⠁⠉⠉⠁⠉⠁⠐⠠⠄⢀⣀⠳⣊⣤⠤⠚⠉⠀⠀⠀⠸⡎⡅⠉⠁⠉⠉⠈⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡖
⡇⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠉⢹⠃⠀⠒⠂⠒⠁⢧
⡇⠀⠀⠀⠀⠀⠈⠉⠒⠂⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡄
⠧⠤⠤⠤⠤⡌⠀⠀⠀⠀⢨⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡔⠉⠑⠲⠤⢤⡼⠀⠀⠀⠀⠀⠀⠀⢣⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢣⡀⠀⠀⠀⢀⠷⠤⣄⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠐⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⣄⠀⠀⠀⠉⠉⠐⣒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠂⠉⠁
⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⡎⠀⠀⠂⠉⠢⠀⠀⠦⢉⠀⣁⠠⠈⢤⠦⠅⠀⠀⠀⡀⠀⠀⠐⠀⢓⠠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠁⠀⠀⠀⠀⠀⠀⠂⠂⠀⢁⡨⠀⢁⠀⡠⠀⣞⢑⠰⢂⠀⠀⢀⡂⢡⢌⡀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣄⣀⠀⠐⠃⢤⠀⣊⢦⣭⣠⠒⠻⡁⢠⠂⡀⠠⠺⡈⠁⠠⢁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⠢⣄⠀⠈⠉⠁⠈⠙⣦⠓⠀⠀⠤⠠⣭⠘⠤⠤⡁⠈⡂⣁⡄⠀⡠⠰⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠦⠀⢠⢀⠈⢙⠈⠟⢺⠀⠈⡉⣉⠀⠦⠀⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠐⠚⢀⠀⠀⣀⠔⠈⠣⠀⠀⠀⠀⡁⢠⢥⠰⠱⠀⠁⠈⣤⠈⠀⣄⠀⠐⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⡀⠠⢀⠀⠀⠀⠀⠀⠀⡀⢀⡐⠗⠀⢀⡄⢀⠄⠂⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠸⢄⡀⡜⠁⠀⠀⠀⢢⠔⠚⠓⠢⡀⠀⠈⠁⠀⠀⢀⣀⡊⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠑⠄⠀⠀⠀⠀⠀⠀⠀⢀⠽⡉⠀⠈⠑⠂⠁⠀⠈⠀⠔⠉⠁⢀
⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠠⠀⡀⠀⠀⠀⢆⠢⡀⠀⡀⢀⠆⠀⠀⡀⠀⠀⠸⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣠⠾
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠔⠋⠢⢀⡀⠷⣄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣤⠴⠊⠀⠀⠙⠑⠉
⠀⠈⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁



⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠢⠀⢀⠠⢀⠀⠀⠀⠀⠀⡀⠀⠆⠐⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣢⠀⠀⠀⠀⠀⠀⠐⠤⣶⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠐⠀⠨⠴⠔⠆




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⣀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⢉⣀⣀⣀⣀⡉⠙⠛⠛⢿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠻⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣾⣶⣶⣶⣤⣤⣤⣄⡀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠉⠉⠙⠛⠛⠿⠿⢿⣿⣷⣦⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠛⠉⡀⠀⣀⣀⡉⠉⠛⠛⠻⢿⣿⣦⣀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣶⣤⡀⠙⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣤⠀⠈⠻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣶⣤⣤⡀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⠋⠀⠀⠀⠉⠙⠛⠛⠻⣿⣷⡄⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⣤⠙⢿⣷⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣸⡿⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⣴⣿⠃⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⣸⡿⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⣸⣿⠁⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⣿⡿⠀⠀⠀⣸⣿⡿
⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⣿⣿⠇
⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⣿⡿
⠀⠀⢀⣿⣿⠁⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣿⣿⠇
⠀⠀⣼⣿⡟⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢰⣿⡇⠀⠀⠀⢰⣿⡏
⠀⠀⣿⣿⠇⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⣸⡿⠁⠀⠀⢀⣿⣿⠇
⠀⣸⣿⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢸⣿⡏
⠀⣿⣿⠃⠀⠀⢀⣤⣿⣿⣤⣀⣀⠀⢸⣿⠀⠀⠀⢠⣿⣿⠃
⣰⣿⣿⣠⣴⡿⠿⠛⠉⠉⠉⠹⠿⢿⣿⣇⠀⠀⠀⣸⣿⡏
⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡀⢀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⣾⣿⡟
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋
⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁
⠀⢸⣿⣷⣤⣶⣶⡄⠀⣰⣿⡿⠃
⠀⠘⣿⣿⠟⠀⠙⣿⣿⣿⡟⠁
⠀⠀⣿⣿⠀⠀⢠⣿⣿⠋
⠀⠀⣿⣿⣀⣴⣿⡿⠃
⠀⠀⢹⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣠⠴⠒⠒⠒⠦⣄⢀⣀⡼⠁⠀⠀⠀⠀⠀⠙⡦⠖⠒⠲⢤⡀
⠀⠀⣠⠖⠋⠉⠀⠀⠉⠓⢶⠋⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⢀⣠⠶⠛⠛⠛⠲⢦⡀⠀⠀⠀⠹⡤⠖⠛⠉⠛⠲⢤⡀
⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⢤⣄⡀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠹⣦⣤⠶⠶⠦⣄⡀⠀⠀⠀⠀⠱
⡴⠟⠛⠋⠛⠛⠲⣦⣀⡴⠋⠁⠀⠀⠀⠀⢀⠈⠙⠞⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⣦⣀⣤⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣭⠛⠂⠀⠀⠀⠛⢻⡶⢶⡄⠀⠀⢀⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣸⠷⠾⣿⠶⠶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⢴⠆⠀⠺⠋⠀⢰⡿⠶⠶⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠻⠆⠀⠀⠀⠀⠘⣷⠶⠶⢾⡏⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢸⡇⠀⠘⠛⣶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣤⣤⣽⡇⠀⠀⠀⠀⠀⣼⣧⣤⣤⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣿⣤⣤⣤⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣤⣾⡇⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣶⠄⢸⡇⠀⢠⣾⣧⣤⣤⣤⣤⣤⣤⣴⣶⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⢶⡆⢸⡇⠀⢸⡇⠀⢀⣤⡶⠶⢦⣤⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⠀⢸⡇⠀⠸⢧⣤⣠⣤⣤⣤⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠀⢀⣉⣀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣼⡇⢸⡇⠀⢸⣧⣠⣤⣤⣌⣭⣍⣭⣤⣤⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣭⣭⣤⣼⠇⠀⠘⢿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡽⠃

⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠉⠉⠉⡍⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠉⠉⠉⠉⠻⣍⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⡅⠀⠀⠀⠀⢀⣀⣈⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡑⠉⣷⠏⠢⣄⡠⠚⠉⠀⠰⡀⠙⠑⡶⠒⠢⢄⠀⠀⢸⡴⠚⠉⠉⠁⠒⠲⣤⢄⡀
⠀⠀⠀⠀⠀⠀⢠⡞⡎⠀⠀⠿⡇⠀⢸⣀⠀⠀⠀⠀⠱⡀⠀⢧⠀⠀⠀⠑⢄⡎⠀⠀⠀⠀⢀⣀⣠⠇⠀⠈⢱⢤⣀
⠀⠀⠀⠀⠀⢠⠏⡐⢈⠆⠀⠸⣥⡀⢸⠙⠷⠦⡴⠒⠺⢝⢦⡈⠣⣀⠀⢀⡠⠿⣦⠤⠒⠉⢀⣴⡏⡖⢒⣨⡥⡤⠿⠓⢄
⠀⠀⠀⠀⠀⢸⠛⠋⠁⠀⠀⢀⡗⣟⣿⢤⣀⣒⣦⣤⣀⣈⣧⠈⡖⠊⠹⡥⠔⠈⠀⢀⠠⠖⠋⢀⡇⡗⠁⡼⡹⢁⠃⢀⠔⠙⢦
⠀⠀⠀⠀⠀⢨⣤⠆⣀⣀⠤⡌⡯⠌⠉⠛⠚⢷⠿⠶⠍⠊⠁⠀⠈⠓⠚⠓⠒⠒⠊⠓⠤⢶⣚⠁⠇⡇⠞⡐⠀⠌⢜⣁⣀⠤⠊
⠀⠀⠀⠀⠀⠀⢿⠰⡇⠀⢀⠃⠇⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⡇⠐⠀⡸⠉⠁⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⡎⢸⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢫⣻⣴⣅⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢤⡇⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠳⣌⠙⠶⣄⡀⠀⣸
⠀⠀⠀⠀⠀⣠⣖⠵⠋⠀⠀⠇⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠈⠙⠶⣾⡀
⠀⢀⡤⠴⠞⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣄⣽⠉⠛⠒⠲⠦⠤⢤
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣙⣃⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠈⠩⣶⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⣠⠶⢦⡀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡡⣄⢠⣀⡀⢜⠁⠀⠀⠀⢸⢱⠊⠠⢧⠞⡁⢤⡀⡇
⠀⠀⠀⠀⠠⢤⢄⣠⢀⠀⣠⢢⣀⣠⠲⣄⣠⡿⣀⣑⠐⠂⠋⢷⠃⠀⠀⠀⢸⡐⠀⠀⠀⡔⠁⢸⠀⡇
⠀⠀⠀⠀⠀⠐⢸⣻⢠⠉⠁⢀⠉⢁⡀⠠⡏⠂⠎⣇⠀⠀⠀⢸⡆⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⡟⢰⠃
⠀⠀⠀⠀⠈⠐⠊⢹⢸⠀⠀⢸⠀⠘⠁⠀⠉⣤⠒⠃⠀⠀⠀⣸⠁⠀⠀⠀⠀⠹⣌⣀⡴⠃⢸⠃⠌
⠀⠀⠀⠀⠀⠀⠀⣾⠘⠀⠀⠀⠀⠘⠀⠀⠠⣋⠀⠀⠀⢀⠞⠁⠀⣴⣦⠄⠀⠀⠀⠉⠀⢀⠎⣰
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡇⠀⠀⠀⠂⠖⠊⢀⡼⠀⠀⠀⠀⠀⠀⡼⢰⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢾⣟⠁⠒⠒⠉⠀⠀⠀⣀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⠱⡗
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠈⢳⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠋⢀⠀⠀⠀⠀⠀⠀⢰⠁⠐⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⠁⡄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢸⠀⠀⠀⠀⡆⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠓⠒⠂⠀⠀⠀⠀⠀⠀⠄⠀⠈⠑⣄⠀⠀⡇⢸
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠠⣴⠀⠀⠀⠀⠘⠀⠀⡇⢀
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣴⣤⣄⣧⣸⣀
⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⣠⣤⠴⠶⠚⠛⠋⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠐⡂⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢂⡴⠏
⠀⠀⠀⡇⠀⠀⢀⡤⠞⠋⠁⠀⣀⣠⠤⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⢘⡀⠀⠘⠧⣄⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠸⡏⠓⠒⢄
⠀⠀⡰⠋⠀⢀⠄⠀⣟⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⣶⠇⠀⠀⠀⠁⠐⠤⡀
⠀⠰⠇⠀⠀⠉⢀⡞⠁⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⣤⣠⣯⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠡⡀⠀⠀⠈⢣⡄⢺⣍⣁⣀⣀⣠⣤⠤⠴⠶⠒⠛⠋⠉⠉⠉⠉⠉⠀⠒⠙⠥⠥⠜⠁⠀⠀⠀⠀⢀⡀⠔
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣭⣄⣀⣀⣀⣀⣀⣒⣂⣒⣒⣒⣀⣀⣀⣒⣈⣉⣉⣉⣉⣈⣉⣁⣀⣀⣀⣉⣉⣉⣀⣀
⣿⣿⣿⣿⣯⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣭⣯⣿⣯⣽⣿⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣤⣀⡤⠤⢤⡀⠀⠀⠀⠙⠆⣰⠿⠃⠀⠀⠀⢀⣶⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢱⠀⠀⢀⡀⠀⠀⢠⡞⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠞⠀⠀⢸⣧⠀⢰⡿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⣦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⢧⡉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⡴⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣇⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⠃⠀⠀⠀⠚⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠈⠲⠤⠞⠋⠛⠿⠿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠈⢧⣄⣀⣠⡞⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⢀⣤⠀⠀⠀⢸⡏⠉⠉⠉⠉⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⡂⠀⠀⢸⣧⣤⣤⣤⣄⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣴⣿⣿⣿⣄⡀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⠇⠀⠀⢸⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⡿⣿⣿⣿⣷⡀⠀⠀⠛⠻⠛⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣇⣽⣿⣿⣿⡇⠀⣀⣀⣀⢀⣀⣀⠀⠨⣿⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡿⠁⠀⢨⡟⣱⣿⡇⠀⢿⣿⣿⣿⣿⣿⠁⠈⣿⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣷⠀⠀⠀⠁⠙⢿⡇⠀⠀⠛⠻⠿⠛⠁⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣯⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢠⣾⡇⠀⢀⠀⠀⠤⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣷⣾⣿⣿⡇⠀⢈⠀⢁⠀⠆⠀⣠⣴⣿⣤⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠄⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⢠⣄⣀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠙⠻⠟⠛⠁⠀⠀⠀⠙⠛⠟⠋⠁

⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢄⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠠⡏⠀⠀⠀⠀⠈⠉⠑⢆⠀⠋⢰⠋⡠⠄⠀⠓⣄⠀⠀⠀⢘⠁⣀⠥⠀⠀⣌⢮⠑⠁⢀⠀⠀⠀⠊⢄⡊⠀⠠⠀⠁⠄⢱⠀⢛⣌⠂⠀⢀⡠⠤⣤⣄⡁
⠀⠀⠀⠀⠀⠀⡀⣀⠠⢤⣋⠃⠀⠘⡄⠈⠂⡄⡀⠈⠆⠀⢠⠁⠀⣁⠀⡸⠀⠀⠀⠐⡂⠀⠐⠀⠠⠀⠀⠀⠆⠀⠀⠐⢀⠆⠀⣀⠑⠃⠰⡯⡤⠄⡹⢺⠁
⠀⠀⠀⠌⡄⠡⠀⠀⠔⡠⠌⡳⠀⠀⠘⢆⡀⠠⣈⠢⣘⠀⠀⠙⡝⡣⢬⠁⠀⠀⠀⠠⠅⠀⠀⠸⠀⠀⠠⠁⠀⠀⠑⡠⠋⠀⠀⠉⠓⢄⠀⡄⠀⠀⠀⢸
⠀⢀⡀⠰⣉⣉⡀⠤⠂⠂⠉⣀⣀⢀⣀⡀⠈⠒⠈⠔⠀⢠⢶⠀⢛⠽⣳⠀⠀⠀⠡⡡⠊⠘⢄⢀⠰⢀⠅⡀⠀⢂⠜⠁⠈⠦⢇⣀⣀⡘⠀⠃⠠⠤⠔⠈
⠀⠈⠇⠀⠀⣤⠀⠀⡠⠔⠋⠃⢠⠃⠌⠠⠉⢲⡀⠀⠀⣿⠙⠀⠈⣞⡜⠀⠠⠬⡶⠁⠀⠀⣸⡀⢀⠐⢄⡩⠔⢁⠄⡀⠀⣀⠀⣀⠌⠀⠄⢠⡀⠀⠀⢒⠄
⠀⠀⢀⣄⡀⠁⠀⡈⢁⣀⣎⡀⠦⠬⠐⡦⣮⠵⡎⠀⣀⣤⣄⢀⠀⢻⡃⠀⠀⠀⠙⡂⠐⡦⠛⠬⠂⠀⠀⠀⠀⠌⡔⢀⣀⠀⠉⠉⠀⠐⠂⠁⠀⠆
⠀⠀⢿⠀⢹⡄⠘⠰⠒⠠⠐⠚⠒⠒⣉⣀⡤⠞⠀⢘⠇⠀⠀⢉⡗⠀⢠⠄⢠⠀⠀⡀⠼⠀⠀⠀⠀⢀⣤⣄⣤⣎⠀⠈⠋⠀⠊⠑⠀⠀⠈⠀⣀⡀⣀⠂
⠀⠀⢸⢀⠤⠑⡄⠀⠉⠓⠃⠉⠉⠉⠉⠀⠀⠀⠐⡼⠀⠀⠀⢸⡇⠀⢠⠰⠀⠀⠎⠁⠃⠀⠀⠠⣿⣿⣝⣿⣵⣿⣷⠀⢠⠄⠀⠀⡆⠀⠤⣼⡁⢠⠍
⠀⠀⢱⡈⠒⢀⡇⠀⠀⠀⢳⠚⠒⠀⠤⠄⠒⢶⠂⠈⠃⠒⠶⠃⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⡀⠀⡼⡅⠀⠀⠀⠀⢃⠀⢄⣀⡠⠃⠀⠀⠉⠉
⠀⠀⠀⠙⠙⠋⢀⠴⡀⡀⢄⠉⠀⠐⠒⠒⠊⠁⠀⢀⢠⡠⢂⡀⠀⠀⣀⠁⠀⠀⠒⠒⡶⢦⠀⠉⠉⠀⠀⢘⠻⠀⠉⠁⠀⢀⠇⠀⠀⠀⡀⠀⢣⠈⠂⢢
⠀⣰⠂⢐⠀⢠⠃⠀⠉⠁⢈⠀⠀⠀⠠⠄⠤⡄⡖⣆⡿⠿⢉⡕⢀⡔⢱⠊⠉⠌⠉⠉⠐⠁⠀⠈⠒⠀⠤⠀⠀⠐⠀⢠⠌⠁⠀⠁⠀⠞⠁⠀⢀⠗⠲⣎⠄
⠀⠈⠉⠀⢀⠆⠀⠀⠀⠀⠄⢠⢤⠀⢲⠒⠀⠛⠁⣀⠠⠔⢋⡔⠹⠃⠈⣰⣹⠟⠉⠀⠒⠀⠒⡄⠀⠶⠀⠀⣠⣀⡀⣠⡄⠀⣀⠀⢰⠁⠂⣰⠁
⠀⠀⠀⢀⠎⠀⠀⠐⠀⠈⠀⠀⠁⠀⠈⠁⠀⠈⢉⡀⠀⣠⣈⡥⠴⠖⠚⠋⣡⣄⢠⠀⠀⠀⢀⠇⠀⠀⣤⢒⠇⠨⠄⠤⠀⠅⠘⠠⡀⠙⢁⠼⠂⡀⣀
⠀⢀⠀⠰⢈⠋⠣⣀⠄⠀⠀⢀⡂⠀⠤⠄⠀⠀⠙⠁⠀⠁⠀⣤⠀⠀⠀⢞⣭⠍⢸⡀⣀⠔⠁⣀⠀⠀⡀⠀⠠⡈⠉⠉⠁⠉⠉⠁⡁⠀⠀⠀⡘⠀⠀⢁
⠀⠘⢤⡀⠀⠈⠁⣁⡀⠀⠀⠰⠺⣾⠪⠀⠁⢠⣶⠶⠒⠒⠠⢟⡦⣄⡀⠘⠁⡠⠀⠉⠀⠀⠈⠊⠀⠀⡆⠁⠀⠀⠀⠉⠀⠁⠀⢹⠁⠀⠀⠀⠇⠀⢀⡘
⠀⠀⡀⠄⠲⢈⠜⠁⢀⠀⠀⠀⠀⠑⠄⠀⠠⠋⠀⠀⠀⠀⠂⡈⠉⠉⠀⠐⠏⠄⢀⠀⠄⠠⠔⢈⣆⠈⠑⠂⠠⠤⠤⠤⠤⠤⠔⢚⠳⠀⠀⠰⠈⠀⠁⡇
⠀⢨⠁⠙⠀⡁⠆⠀⡠⢀⡠⠄⠤⢄⠀⠀⠁⠀⠀⠀⠀⢈⡄⠀⠔⠈⠉⢂⠀⠈⠁⠐⠀⠄⠂⠉⠀⠀⢀⡄⡄⠀⠀⠀⠀⠀⠈⠀⢀⣎⡆⠐⠤⢤⠰
⠀⠀⢅⢐⡂⠠⠔⠁⠠⡃⣰⠉⢱⠀⢧⠀⠛⠀⠀⠐⠚⠛⠁⠜⠀⠀⠐⠸⡄⠐⡄⠠⠜⠀⢂⠉⠆⠀⠏⠐⡀⠄⠀⠀⣀⠐⠠⠀⠀⠓⠇⠀⡇⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢱⠈⠈⢁⢀⡠⠃⣾⠑⠒⠒⠄⠠⠊⠈⠀⠀⠀⣸⠃⢀⠀⠀⠀⠀⠘⠁⠀⠀⣀⠼⠒⠀⠀⠒⠺⡀⠀⠀⣈⡆⠀⠀⠀⠀⡀
⠀⠀⡁⢨⠽⢹⣀⢓⡀⠈⠢⠚⠁⠀⠀⢰⠁⣀⡤⠊⠀⡇⡀⠰⠿⠀⠃⢸⠀⠈⠄⠉⡁⠀⠀⢀⠴⠊⠁⠀⠀⢀⠔⠂⠀⠉⡹⠊⠀⠀⠀⢾⣟⣀⠤⠂
⠀⠈⠁⠀⠠⡀⠸⢘⡀⠀⠰⣉⣈⡳⠀⠀⠁⠘⠘⡄⠀⠡⡐⠀⠀⢈⣠⠎⠀⠀⠀⠈⠁⢀⠄⢃⠀⠀⠀⠤⢀⠀⠀⠀⢀⡼⠡⡮⠀⠀⠒⠊⠉
⠀⠀⠀⠓⠀⠃⣉⠉⠁⠀⠀⠁⠀⢠⠄⠀⡄⠀⠤⠂⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠨⠦⠀⣑⠬⠤⣉⣒⣒⠚⣥⢊⡛⠋⠥⠐⠊⠁⡄⠄⠀⠄⠀⠀⢀⡀
⠀⠀⠀⢀⠀⠁⢈⠀⠀⠀⡉⡕⢀⠈⠒⠬⠀⢀⣤⡀⢰⡋⠉⠉⣷⠀⠉⢱⠀⠀⠃⠀⠀⠀⡀⠀⠀⢀⡠⣤⠀⡠⢦⡀⣀⠤⡀⢀⠉⠀⠀⠀⠀⠀⠈⠖
⠀⢠⠀⡀⠢⠀⠠⠐⠀⠀⠱⢡⠞⠁⠀⠀⠀⠱⡤⠜⠀⠇⠀⢸⠇⠀⠀⡆⠀⠈⠢⠀⣠⡔⠁⠀⠀⠈⠠⠋⠈⠄⠎⠀⠅⠠⠁⠀⣴⣤⢂⡘⠃⠁⢆⣀⡄
⠀⠀⠉⣀⣤⠐⠤⡀⠀⠀⢀⢰⠀⠀⡠⠒⠒⡄⠀⠀⠀⠉⠉⠉⠉⠀⠈⠀⠰⠆⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢇⠠⠤⠀⠀⠀⠀⣠⠃
⠀⠀⣼⢳⣽⣐⡟⠉⠀⠼⠄⢈⠔⠉⠀⠀⠂⡄⠀⢀⠔⢁⡀⠤⠵⠆⠀⠄⠠⠄⠂⡚⠀⠀⠀⡀⠔⠈⠈⠉⡶⠻⡿⠥⠀⠀⠀⠀⠈⠒⠂⠤⠤⠤⠔⠁
⠀⠂⢻⡶⠒⠉⠀⠀⡄⠀⠀⠸⢀⠀⣀⡠⠊⣤⠀⡇⡰⠋⠀⢠⢤⠀⠑⠒⠒⠒⠊⠀⠀⠐⠀⣀⣀⠤⠤⠔⠁⠀⠈⠢⣀⠀⡀⠀⠠⠄⠁⠀⠀⣀⣀⢺⠃
⠀⠀⠀⠈⠐⠐⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠀⠀⠀⠀⠀⠉⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡿⠟⠛⠉⠁⠀⠈⠙⠳⠦⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⠿⠏⠁⠈⢻⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠛⠛⠿⠋⠉⠻⣿⣿⣿⡿⠟⠻⣿⣿⠿⠛⠁⠀⠀⠀⠘⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣋⣀⣠⣤⡼⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⢿⠿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠙⣆⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⢣⡀⠀⢀⣴⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣦⡀⠀⠀⠈⢀⣿⡁⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠙⣿⣿⣦⠀⠀⢸⣿⣿⣷⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢻⣿⣷⣄⠘⠿⣿⣿⣿⣿⣿⣦⣤⣤⠞⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⠻⣿⣷⣄⡀⠈⠙⠛⠿⠿⠏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⣀⠀⠀⠀⠈⠉⠓⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣶⣦⣤⣤⠄⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠓⠶⢾⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠙⠛⠛⠷⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⢿⢻⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣟⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠳⢦⡀⠀⠀⠀⠀⡠⠔⠚⠉⡷⠀⠀⢀⡴⠊⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣹⠚⣇⠀⠀⠀⠀⡳⣲⣶⡾⠁⠀⠀⠺⡽⣾⣜⡇
⢤⡄⢠⡆⠸⡇⠀⠀⠀⠀⠀⠀⢳⢸⡆⠀⢀⠎⣾⠋⠉⠁⠀⠀⠀⡼⣰⠇
⠀⢻⣼⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡌⣴⠃⠀⠀⠀⠀⢀⡜⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶
⠀⣰⢿⡀⠀⣿⠀⠀⠀⠀⠀⠀⡜⢰⡇⡸⢀⡏⠀⠀⠀⠀⣠⠎⣠⠟⠒⠒⠣⠤⠾⣄⣴⣂⣠⣠⣆⣒⣦⠽⠖⠉⠀⠀⣠
⠀⣿⢨⠧⣀⡏⣧⠀⠀⠀⠀⡼⢡⡾⡰⠁⢼⠁⢀⠤⢂⠝⠁⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⢸⠹⣿⠀⢹⣇⢹⡇⠀⢀⡞⢠⣾⡵⢁⢌⢃⠔⢁⠔⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡛⠁
⢸⡇⢻⡇⠀⢿⡄⢿⢀⡎⢠⡾⠋⡠⣪⣾⡗⠊⠁⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠁
⠀⣧⠈⠛⠤⣤⠁⠈⠈⠀⠊⠀⠈⠘⠋⢉⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⢉⡇⢀⣬⠟⡹⠁
⠀⠈⠀⠀⢰⣷⠆⠀⠀⠀⠀⢼⣹⣯⠞⠉⠁⠉⠒⠦⢤⣄⣀⣀⣀⣀⣀⡤⠤⢖⣊⡭⠿⠽⠋⢠⠞⣫⠎
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣤⣤⣄⣀⣈⣀⣤⡤⠾⠟⠋⠁⠀⠀⠲⠖⠋⠛⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠴⠖⠛⠋⠙⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠋⠉⠉⠀⠀⠀⠀⣀⠠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣁⣠⡤⣄⣀⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⡖⠚⠻⣛⠉⠙⠋⠉⠉⠓⠒⠲⠤⠤⣤⣀⣀⠠⠤⠒⣿
⠀⠀⠀⠀⠀⠀⢀⡠⢾⡉⠀⠀⠉⠢⣄⡀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠳⣄⠀⠀⠀⠀⠉⠓⢦⡀⠙⠒⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠙⠒⠢⠄⠀⠀⠀⠶⣄⡀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠳⢤⣄⡀⠀⠉⠉⠉⣹⠃
⠀⠀⠀⠀⢀⠔⠉⠻⣄⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⠀⠀⠀⠀⠀⠉⠑⠒⠒⡾⠃
⠀⠀⠀⠀⢘⠀⠀⠀⠈⠱⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⣀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢰⠓⢆⠀⠀⠀⠀⠉⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣶⠞⠁
⠀⠀⠀⠀⡴⢦⣈⠑⢢⡀⠀⢀⠀⠀⠉⢉⣻⢧⣄⢀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⡤⠤⣬⠷⠤⣉⣭⠽⠓⠋⠉⠉⠁⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠆⠀⠀⢀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣴⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣳⡄⠀⣠⣿⣿⠟⢡⡀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣀⣀⢠⣾⢟⣿⣇⣐⣻⣿⣛⣿⣿⣋⣈⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠏⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣮⣭⣽⣿⣯⣤⣤⣤⣤⣽⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⢿⣿⡿⣿⣿⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⣿⡇⣿⣿⢸⣿⠃⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢸⣿⠀⣿⡇⢸⡏⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⡀⣿⡇⣿⡇⢸⣿⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⣿⡇⣿⡇⣿⡇⣸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣷⣿⣷⣿⣷⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⣳⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠿⠿⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠴⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⣀⣒⢉⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⣠⠀⣶⣰⡇⣀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⢘⣿⡿⠟⠛⠉⠀⢀⣾⣇⣸⣯⣀⣸⣿⣀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢩⣥⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗⣾⣉⡽⢁⡴⠁⡴⠋⣠⠎⢀⡜⠁⡸⢋⣙⣖⣾⡆
⠀⠀⠀⠀⠀⠀⢸⣯⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⣯⣝⣛⣓⠚⠷⠾⠷⠶⠟⢒⣛⣛⣯⡏⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠻⠿⠿⠿⠿⠿⠛⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠈⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢰⡆⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠉⠉⠉⠉⠉⠉⠉⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢷⡶⢶⡶⢲⠟⢻⣿⣲⠶⣶⠶⣿⣇⡀
⠀⠀⠀⠀⠀⠀⣴⡿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣞⣴⣋⣴⢃⡴⢉⣿⢁⣜⣡⣞⣩⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣷⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠉⠉⠉⠉⠉⠙⣿⠋⠉⠉⠉⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣬⢰⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠤⡀⠀⠀⠀⡀⣴⢠⠌⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠂⠛⠀⠁⠉⠙⠲⣄⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠨⢄⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢅⡀⠇⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠱⠌⠒⠒⠀⠀⠠⠤⠀⠀⠀⠀⠀⡀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⠀⠑⣄⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⡐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠸⠀⠀⠈⢂⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⢌⣀⣀⣀⠠⠄⠪
⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢂⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠙⠢⣄⡀⠀⠀⠀⢀⠀⠔⢃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⢄⠀⢀⡀⠀⠀⣀⠀⠒⠒⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠲⠊⠀⠠⠤⠄⠀⠀⢀⡀⢀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠱⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠇⠰⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢣⡀⠖⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠆⠀⠀⡀⠘⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣬⣶⣮⣥⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣭
⣿⣿⣏⣉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠛⠛⠉⣤⣄⠀⠛⠋⠉⠙⠛⠻⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⢀⣤⡤⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠈⠻⠀⠀⣀⠉⣛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠠⣶⡶⠁⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠙⠃⠉⠃⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣽⣾⡗⠀⣴⡆⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠉⠉⠀⠀⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⡄⢠⣾⠗⠀⠘⠟⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣽⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡛⠁⠰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⡿⠦⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⠿⠟⠛⠀⠀⢰
⠀⠀⠀⠀⠀⢠⣋⠟⠃⠀⠀⠀⠀⣀⣤⣤⣀⣠⣤⣠⣶⣤⣶⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠉⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠻⠋⠁⠀⠀⠀⠀⠀⠉⢀⡜
⠀⠀⠀⠀⠸⡏⠹⣇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠏⠙⠋⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢿⡄⠈⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠻⠟⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠈⢷⣀⡀⠂⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣄⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣶⣟⣷⣾⣿⣶⣦⡄⣄⡀⣤⣤⣦⣤⣵⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢠⣿⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠠⣿⡀⠸⣿⣆⠸⣿⣆⠀⠀⠀⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⢀⡄⢠⣿⠁⢠⡄⠀⠀⠀⠙⢿⣦⠙⣿⡆⠈⢿⣧⠀⠀⣠⠄⣸⡏⠀⣠
⠀⠀⠀⠀⠀⠀⢺⣇⠘⣿⣆⠻⣷⡀⠀⠀⠀⠈⣿⠀⢸⣧⠀⠘⣿⠀⠀⣿⣄⠸⣷⡀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠙⣷⠈⢿⡄⠙⣿⠀⠀⠀⣸⡟⠀⣸⡇⠀⢰⡟⠀⠀⠈⢻⣦⠘⣷⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣸⠇⠀⣿⠀⠀⠀⠋⠀⢠⡟⠀⠀⠋⠀⠀⠀⠀⢸⡏⠀⣿⠀⢨⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⡟⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣸⠇⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠴⠶⠶⠶⠶⠲⠒⠒⠒⠲⠲⠶⠶⠶⠦⢤⣤⣄⣀⡀
⠀⠀⠀⢀⣠⡴⠖⠛⣋⣩⡥⠄⣠⣶⠶⠶⡶⠛⢛⣿⠟⢉⡱⠿⡟⢛⣿⠟⠃⣲⡶⠦⣭⣭⡛⠲⢦⣄
⠀⣠⠞⢋⠤⠒⢉⣵⠿⣋⣤⣾⠉⠻⠦⣠⣣⣾⣋⣤⡞⠛⠷⣀⡾⢋⣀⣴⡞⠛⠷⣀⠞⢁⣼⠿⢆⠈⠻
⣼⠁⢰⠁⣠⡶⢻⠛⠛⣷⠄⠘⣁⣴⣾⠟⠛⢿⡄⠘⢋⣠⠾⠟⠚⢿⡄⠘⢋⡠⠒⢁⣴⠟⠁⢀⠜⠀⣠
⣿⣄⡀⠙⠣⢤⡞⣢⣤⠤⢒⣩⠟⠋⢀⣦⡄⠠⣴⡾⠛⢁⣴⠦⠀⣬⡴⠞⠁⣀⣴⡿⠡⠒⣊⣡⣴⣾⣿
⢹⣿⣿⣷⣶⣤⣄⣉⣉⠙⠛⠃⠒⠲⠛⠓⠒⠞⠛⠲⠾⠛⠒⠒⠚⠋⢉⣉⣈⣭⣤⣶⣾⣿⣿⣿⣿⣿⣿
⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢿⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠘⣇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠈⣷⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⣧
⠀⠀⠀⠀⠀⠀⢰⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢹⡆
⠀⠀⠀⠀⠀⠀⣾⠀⣾⠀⠀⠀⣀⣀⡤⠴⠒⠛⠉⣿⠉⣿⠙⠓⠦⣄⣀⠀⠀⠀⠘⣧⠀⢿⡀
⠀⠀⠀⠀⠀⢰⠇⢸⣷⣒⣚⣉⣁⣀⡀⠀⠀⠀⠀⣿⣀⣿⠀⠀⠀⠀⠈⠉⠓⠶⢤⣽⡆⠘⣇
⠀⠀⠀⠀⠀⣾⠀⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⠀⢹⡄
⠀⠀⠀⠀⢸⠇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣷
⠀⠀⠀⠀⡿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠸⣇
⠀⠀⠀⢸⠇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢻⡄
⠀⠀⠀⡿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠊⠉⠁⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠶⣄⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋⣥⠶⠲⣆⠈⠛⢦⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⢴⠳⣤⡙⢷⣴⠋⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢋⣁⠀⢸⠛⢷⡀⠸⡟⠉⠙⠆⠉⣀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⠹⡆
⠀⠀⠀⣠⠋⣇⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⢿⣀⣤⣜⢧⣄⡽⠀⠉⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⣿
⠀⠀⢠⡇⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣤⠼⠃⣀⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⢀⡴⣿
⠀⠀⢸⣇⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⡤⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠀⠀⠀⠀⣠⠴⠋⣰⠏
⠀⠀⠸⡏⢧⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⢀⣠⠴⠋⢁⣠⠞⠁
⠀⠀⠀⢳⡀⠹⣦⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠀⢀⣠⡴⠞⠋⢀⣠⠖⠋
⠀⠀⠀⠈⢳⣄⠀⠙⢦⡀⠀⠀⠀⠀⠙⠒⠤⢤⣤⣤⣤⣤⠴⠶⠛⠉⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠙⢦⣀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⢀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠙⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠉⠛⠲⠶⠶⠒⠚⠋⠉⠀⠀⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠋⠉⠁

⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⣠⣤⣄⢾⡷⣰⣶⣦⡄⢠⣤⡀⣾⣿⡇⠀⢀⣤⣤⢠⣾⣿⣿⣿⣷⣄⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣮⡻⣿⡇⣾⣿⣿⣿⣿⠘⣛⣁⢬⢉⡀⡄⣿⢟⡋⣿⣿⣿⣿⣿⣿⣿⢚⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣵⣿⣷⡝⣿⣿⡿⢱⣿⣿⣿⣷⡌⣵⣶⣰⣿⣿⣜⢿⣿⣿⣿⡿⣫⡜⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣿⣯⡍⢿⣿⡿⢧⡶⢘⣦⣽⣛⢿⣿⣿⢧⣿⣿⡼⣿⣿⣿⢗⣬⣭⣵⣾⣿⣿⡎⣻⢟⣵⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⢰⣶⣾⡟⣴⣿⣿⣿⣿⣷⠉⣑⣊⡛⠟⣵⣶⣶⣶⣾⡽⣿⣮⡛⣫⣵⣶⣶⣬⡻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠿⠿⢋⣨⣭⣯⣥⡻⣿⣿⣿⡿⣱⣿⣿⣿⣿⣷⡽⣿⣿⣿⣿⡿⣙⣛⣙⠿⣿⣿⣿⣿⣷⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⣾⣿⣿⣿⣿⣿⡜⠿⠿⠇⣿⣿⣿⣿⣿⣿⡇⠿⠿⢿⣟⣾⣿⣿⣿⣿⣎⣿⣿⣿⡿⣠⣭⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⣿⣿⣿⣿⣿⣿⠧⠋⢉⣵⣾⣷⠮⠻⠿⣡⣾⣿⣿⣷⡼⣿⣿⣿⣿⣿⡿⠸⠟⣫⣥⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣟⣻⣬⢻⠿⠿⢟⣫⣶⡄⢸⣿⢏⣶⣿⣿⣿⣮⣻⣿⣿⣿⣿⡝⠻⠿⠿⣛⣵⣄⣠⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⡇⣤⣛⢾⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⣁⣀⣴⣿⣼⠿⣿⡏⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣎⠻⠿⣿⠿⣛⣾⣿⣿⣌⠿⣿⣿⣿⢟⣤⣽⡉⠁⣾⣿⣿⣯⣽⣷⣤⣾⣷⣄⣀⣴⣿⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠛⢻⣿⡿⣶⣶⣾⣟⣿⣿⣿⠾⣿⣿⣿⣏⣹⣿⣿⡟⠙⢿⣿⣿⡿⣱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡹⢿⣿⣧⣀⣼⣿⣿⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡟⣿⣷⣴⣿⣿⢟⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣭⣟⣛⣛⣻⣭⣭⣽⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣯⣭⣟⣛⣛⣻⣭⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⠤⠔⠢⢤⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡤⣤⢤⠀⡠⢠⣄⢀⣄⢀⣀⠀⢠⣤⡀⣠⠄⡤
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠁⠉⠉⠉⠉⠈⠁⠁⠈⠉⠀⠉⠉⠁
⠀⠀⢀⡴⠊⠁⠀⠀⢀⡤⣄⠀⠀⠀⠀⡼
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠙⠒⠒⠊⢳
⠀⠀⠈⠓⠒⢷⠒⠂⠸⠤⠖⠉⠛⣄⢀⡮⣀⡀
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⡨⠋⠀⠀⠉⠓⢤⡀
⠀⠀⠀⢀⠎⠈⠙⠒⠒⠒⠒⠋⠁⢀⠀⠀⠀⠀⠀⠀⠘⢉⡆
⠀⠀⠀⡜⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠑⠄⠀⠐⠠⠀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⢳⠤⣀⣈⡠⢄⣀⠈⣀⣤⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠈⡄⢀⠀⠀⠀⠀⠀⢧⠀⠤⠀⠀⠈⠉⠁⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢐⠀⠀⠀⠀⡀⠠⠤⣌⠢⡀
⠀⠀⠀⡇⠠⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠉⠉⠉⢨⠀⡠⡖⠁⠀⠀⠀⠀⠀⠈⡖⢄
⠀⠀⠀⡇⢠⠀⠀⡆⠀⠀⡇⠀⠀⢀⡀⢀⡠⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠳⡌⠉⠭⠭⠭⠭⠍⠋⠥⠎
⠀⠀⢴⠃⠠⠀⠀⠀⠀⢀⣇⣀⣀⠀⠈⣉⣠⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡀⠀⠀⠉⠳⣄⣠⣄⡀⢸⠉⠙⢏⠙⠛⢡⡟⢦⡀⠀⣀⠄⢢⢧⠄⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⣤⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢘⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠘⡏⠉⠀⠰⠚⠂⠀⠀⢴⠋⠂⠄⡀⠀⠀⠐⠂⠤⠔⠒⣁⠀⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠶⣼⣄⣀⠤⢞⡥⠒⡂⠉⠁⠙⠁⠀⠊⠓⠲⠦⣅⠀⠀⠁⠀⠀⠀⠈⠀⠈⠗⠀⣤⠀⠀⣸⡀⠀⠀⠀⠀⠀⢸⠗
⠇⠀⠀⠀⠀⠀⠈⠂⡤⠏⠡⡀⠈⠑⠨⠑⣒⠀⠀⠀⠀⠒⠒⠂⢁⠅⠀⠀⢠⣀⣀⣀⣠⠀⠰⡀⠀⠇⠀⡀⠩⣷⣦⣤⣤⣤⣶⡧
⡆⠀⠀⠀⠀⠀⠀⡔⢸⠁⠀⠘⠢⠄⡀⠀⠀⠰⢁⠀⠀⠀⢀⡠⠂
⠁⠀⠀⠀⠀⠀⠀⣗⠃⣀⠀⠀⠀⠀⠈⠐⠒⠚⡳⣶⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡬⠤⠭⢭⡭⠄⡾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠙⢄⠀⢀⡾⠄⡀⢥
⠁⠀⠀⠀⠀⢀⠞⠁⠉⢦⠤⠉⣽⠂⠠⠐⠚⡦⡀
⠁⠀⠀⠀⠀⠘⡀⠀⠀⠘⢄⣀⠼⡄⠀⠉⠈⠀⠈⠆
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡳⠄⣀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠚⠁
⣶⡶⢶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⢶⣶⣶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠽⣇⠉⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⠽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠳⠋⢴⣢⠀⢠⠖⡄⠀⠀⠀⢶⠄⠀⣀⡤⠖⠚⠙⢻⡓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠈⠛⠁⠀⠀⢀⡠⠖⠋⠁⠀⣀⠀⡀⠀⢣⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠥⠃⠀⠀⣀⡤⠚⠉⠀⠀⡠⠤⣐⠀⠀⣸⠀⠀⢇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⡴⠊⠁⣀⣄⢠⢊⡐⣆⢀⣈⠧⠸⠀⠀⠀⠘⠆⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠘⠧⠼⠉⠉⠀⡇⠀⠸⣀⠘⢺⣈⡡⠟⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡜⢁⠜⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠞⠉⠙⠆⠀⠈⠀⠀⠀⢀⡠⠔⠈⠀⣀⣀⣀⡠⠞⠁⢶⢦
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠠⠼⠦⡤⠐⠒⢎⡡⠠⢆⠐⠁⠀⠀⠀⠱⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢹⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣳⣀⣀⣀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⡀⢀⣀⠐⠪⢍⠉⠉⠉⠉⠓⡄
⠀⠀⠀⢠⡒⠉⠉⠑⠀⠒⠉⠒⠒⠑⠂⠂⢐⣒⡒⠒⠤⠔⠒⠒⠊⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⢒⡄
⠀⠀⠀⠀⠉⠉⢳⠄⠀⠀⠀⠀⠀⠀⠀⠸⡩⣀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠴⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠳⠤⠤⣤⣄⣀⣀⣀⣠⣤⡤⠤⠤⠤⠤⠴⠶⠒⠒⠒⠋⠉⠙⢦⡤⠤⠔⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣾⣿⣷⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⣀⣯⢺⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠟⣋⣭⣤⣶⣶⣶⣶⣶⣤⣬⣉⡛⠿⣿⣿⣿⣿⣶⣾⠛⠁⠱⡽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢛⣡⣶⠿⠛⢋⣩⡾⠛⠁⢀⡀⠺⠟⠛⢿⣷⣦⡙⢿⣿⣿⣿⣧⡀⠀⠘⢾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢋⣴⡿⠛⠁⣠⣶⠟⠉⠀⣠⣴⠟⣰⠀⣠⣴⠟⠋⠙⣿⣦⡙⢿⣿⣿⣿⣄⢀⣤⡿⢿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⢡⣾⠏⣀⣴⠟⠋⠀⣀⣴⠾⠋⠀⢰⡇⡾⠋⠁⢠⠇⡾⠋⠙⣿⣆⠻⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⢡⣿⣷⠞⣋⣁⠀⠤⡾⠛⠁⢀⣤⡶⢸⣇⠀⡤⢂⡞⠀⢀⣠⡶⠟⢿⣆⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⣾⡟⠁⣸⠧⣭⣙⠓⢶⡄⣶⠟⠉⠀⣈⢻⡷⡾⠟⣡⣴⠟⠋⠀⣠⣼⣿⡄⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣡⢠⣟⠒⠦⢬⣍⣿⣿⡄⢀⣴⠿⠋⣾⢁⣴⠾⠋⠀⢀⣴⡾⠋⠁⣿⡇⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⠋⠺⣭⣉⡓⠲⣾⣿⣿⠓⠶⢤⣤⣼⣇⠛⠁⠀⣤⡾⠛⠁⢀⣠⡾⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡈⣿⣦⡾⠒⠈⢉⣛⡋⠉⠀⢀⣠⡶⢨⡿⠛⡻⢷⣶⣶⣦⣤⣤⡛⠉⠀⣿⡇⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣇⢹⣿⠀⣠⣶⠟⠉⠀⣠⣴⠟⠋⢠⣿⠇⠟⠋⠀⣈⣍⠛⠿⣿⢟⣴⣾⡿⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⢻⣿⡋⠀⣀⣴⠾⠋⠀⢀⢀⣾⣿⠀⢀⣤⡾⠋⠁⢀⣤⡶⠛⣱⡿⢁⣿⣿⣿⡟
⠀⠀⠀⠀⡠⢄⣴⠿⠻⣿⣿⣿⣦⠹⣿⣾⠛⠁⢀⣤⡾⢃⣾⣿⡏⡼⠟⠁⢀⣠⡶⠟⠉⣠⣾⠟⣡⣿⣿⣿⡿⠁
⠀⠀⠀⠘⢿⣮⡣⡀⠀⠙⣿⣿⣿⣷⣌⠻⢿⣶⣟⠉⠀⣠⡛⠛⠁⠀⣠⣴⠟⠋⣀⣤⣾⠟⣡⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠈⠻⣿⣝⣄⠀⠈⣻⣿⣿⣿⣿⣦⣍⡛⠿⣿⣿⣤⣤⣴⣾⣯⣴⣶⠿⠟⣋⣤⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡾⠛⠉⠛⢿⣿⣿⣿⣿⣷⣶⣤⣭⣭⣭⣭⣭⣤⣴⣶⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠛⠉⠉⢢⡀⠀⠉⠙⠻⣿⡛⠋⠙⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠈⠳⣌⡀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠷⣄⠀⠀⠀⠆⠀⠀⢀⡠⠿⠶⣉⣉⡭⠿⠓⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣼⠁⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠊⠻⡄⠀⡗⠀⢠⠎⠀⠀⠀⣸⣿⡂⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⢰⠋⠉⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣶⣷⣶⠋⠀⠀⠀⢀⡿⡏⡁⠀⠀⢼⠿⡇
⠀⠀⠀⠀⠀⠀⣾⡀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡼⠃⠛⢻⣿⠀⠀⠀⠀⠈⣿⣿⣄⣀⡴⠋⢀⣽⣆
⠀⠀⠀⠀⠀⢠⠏⠘⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⠀⢀⣴⣿⡟⠀⠀⠀⠀⠀⣿⣿⠻⣿⠁⠀⢹⢃⣼
⠀⠀⠀⠀⠀⢸⣤⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⠄⣀⠀⠀⢸⣿⣿⣇⣾⡿⢿⡇⡀⠀⠀⠀⢸⣿⡯⠀⣿⣀⡤⠵⠖⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠹⣆⣀⣀⣀⣠⣤⣤⡾⠛⢦⣠⣴⠞⠙⣾⡏⢻⣼⡿⠟⠋⠀⠀⠘⠳⠤⠤⢤⢤⣈⣿⠿⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠓⢶⠟⠁⠀⠀⢿⡃⠀⢻⣆
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠦⠬⠿⠴⠖⠋
⠀⠁