Thumbtack Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠛⠛⠛⠓⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠘⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠈⠙⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠶⣤⡀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⣶⠦⣤⣤⣤⣤⠴⠚
⠀⠀⠀⠀⢀⣠⡴⠶⠒⠒⠒⠲⠶⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⣴⢻⡏⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⣸⠃⠘⣧⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⣿⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⢹⡄⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣀
⠀⠻⣄⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠙⢷⡀
⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠠⠤⠤⠄⣠⠤⠾⠁⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠉⠳⣦⣀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠦⠤⣤⣄⣀⣀⣀⣀⣠⣤⠶⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠉⠓⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣠⠞⠉⠉⠛⠳⠶⢦⣤⣤⣀⣀⣀⣀⣀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣴⠋
⠀⠀⠀⠀⠀⠀⣰⠇⢀⡼⠁
⠀⠀⠀⠀⠀⢠⡏⢠⡞⠁
⠀⠀⠀⠀⢀⡟⣠⠏
⠀⠀⠀⠀⡾⣰⠋
⠀⠀⠀⣸⡿⠃
⠀⠀⢰⡿⠁
⠀⢠⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠊⠉⠉⠉⠁⠐⠒⠒⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣤⡀⠀⠀⠈⠓⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢚⣓⠦⢤⣀⠀⠀⠀⠉⠗⣢⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠁⠠⠔⣒⠈⠡⠞⢓⡶⠾⣤⣡⣡⢋⠟⡙⢳⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠀⠒⢀⠉⠀⠐⢚⣉⠥⠟⣛⠶⠶⣥⣎⣐⡔⡽⡹⣹⢛⢖⠶⡶⡶⢶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠈⠡⠄⠒⢂⡁⠤⢚⡡⠞⢉⡿⣻⠶⣷⣥⣯⣞⣴⡵⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⣉⠡⠄⠒⢊⡡⠔⣊⠥⢚⡡⢞⠥⢊⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠤⠤⢐⣂⠭⠔⣒⠩⠔⣊⠥⢒⠥⣾⠃
⠀⠀⣀⡤⠔⠲⡖⠋⠁⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣈⠡⠄⠐⣒⡉⠤⣒⠩⢔⣊⠅⣺⠃
⢠⠞⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠠⠬⠤⠐⣒⢊⠩⠔⣀⡥⠔⢂⠉⣴⣾⠃
⣾⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢤⣄⡀⠀⢚⣒⣀⣩⡭⠤⠒⣋⠅⠀⣺⠍⡵⣿⠃
⢹⣧⡀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠚⡦⢴⣶⣔⣒⣂⠭⠐⢂⡡⢒⡩⣾⠏
⠀⠹⣷⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢰⠘⠃⠉⢃⡏⠉⡟⣿⢾⢿⣽⣾⣯
⠀⠀⠘⢧⣤⠄⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⣰⢁⠇⠆⡘⡘⢸⢸⠀⠇⠇⡌⡼⠘⠛⢿⣧
⠀⠀⠀⠀⠙⠯⣔⡠⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⢠⢃⡎⡆⡸⢡⠃⡘⡎⢰⢠⠀⡄⡇⣿⢠⠘⡿⣧
⠀⠀⠀⠀⠀⠀⠈⠙⠧⣧⣂⠀⠀⠀⠀⠀⠀⠀⠉⢹⠷⣮⣜⣱⢃⡏⣰⢃⠃⡆⡌⠰⢠⠇⡟⣸⠀⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠷⣤⣄⠀⠀⠀⠀⡘⢠⢠⠉⡟⡿⢴⢇⣜⡼⢠⠃⠇⡜⢸⠃⡏⣸⣹⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠩⠿⢾⣴⣰⠁⡎⡌⢰⢠⠅⡾⠠⡏⡏⡛⢺⠲⠧⡿⢼⡗⣯⢫⠱⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠭⠭⠭⠙⢒⣩⠟⠚⠤⢧⡎⣰⠇⡼⢠⢃⠇⡌⢰⠀⡇⢸⡇⣼⢸⢠⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣂⠈⢥⣴⠋⠀⠀⠀⠀⠈⠉⠘⠓⠧⠾⣴⣁⣎⣰⣃⣸⣠⡽⠦⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠰⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣐⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⡾⠁
⠀⠀⠀⠀⠀⠀⠀⡜⢠⡝⠁
⠀⠀⠀⠀⠀⠀⣴⢡⡟
⠀⠀⠀⠀⠀⣰⢧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣂⣠⣤⣤⣤⣤⣤⣀⣐⣒⣂⣀⣀⣀⣀⣒⣒⣂⣀⣀⡀
⠀⠀⠀⠀⠐⠿⠿⠶⠖⠒⠒⠒⠒⠒⠒⠒⡒⠒⠒⠒⠒⠒⠚⠉⠭⠭⠭⠥⠤⠤⠤⠤⠶⠤⠤⠀⠠⠀⠂⢀⣐⣒⣒⠒⠀⠀⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠤⠈⠉⢈⠙⠛⢛⣉⣉⣉⠁⠉⠀⠉⣀⣀⠀⠀⠐⠒⣶⡖⠲⣶⠶⣶⣶⣒⣒⣒⣀⡐⠛⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⣉⣀⣤⣤⠭⠤⠤⠤⣤⣤⠤⡤⣤⣤⠤⠄⢠⠤⠴⡦⠴⠦⠴⠄⠠⠄⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂⠉⢉⣉⣀⣉⣉⣉⣉⣉⣉⡩⠭⠤⠤⠭⠥⠬⠥⠭⠥⠀⢀⢀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠛⠙⠛⠓⠶⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠈⠙⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠶⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠲⠶⠶⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣶⠦⢤⣤⣤⣤⠴⠋
⠀⠀⠀⠀⢀⣠⠴⠶⠒⠒⠒⠲⠶⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⣴⢻⡏⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⡸⠃⠘⣧⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⣧⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⢹⡄⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢦⣄
⠀⠻⣄⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠙⢷⡀
⠀⠀⠘⢦⡀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠲⠦⢤⣤⣤⡴⠞⠁⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⣤⣤⣄⣀⣀⣠⣤⣤⠶⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠉⠛⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣠⠟⠉⠉⠛⠻⠶⠶⣤⣤⣄⣀⣀⣀⣀⣠⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡼⠃
⠀⠀⠀⠀⠀⠀⣰⠃⢀⡾⠁
⠀⠀⠀⠀⠀⢠⠏⢠⡟
⠀⠀⠀⠀⢀⡟⣠⠏
⠀⠀⠀⠀⡞⣴⠋
⠀⠀⠀⣼⡿⠁
⠀⠀⣰⡟⠁
⠀⢀⡟
⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣷⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣦⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠶⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠃⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣤⣄⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠙⠻⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠙⢿⣦⡀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⢶⣦⣤⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠈⢻⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣿⣦⡀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⡾⠋⠉⠻⣦⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⡾⠋⠀⠀⠀⠀⠙⢿⣦⡀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⣴⠟⠁⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡾⠋
⠀⠀⠀⠀⠀⢰⣿⣴⡾⠟
⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠍⠂⠀⠀⠀⠀⠀⢀⠤⠂⠉⠉⠉⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠤⠒⠒⠂⠒⠒⠢⢄
⠀⢠⠄⢀⡂⠬⠐⠀⠉⠀⠀⡀⠈⠠⠀⠀⠀⠀⠠⠁⡠⠰⠎⠉⣉⢦⣠⠌⡄⠀⠀⠀⡀⠀⢀⡀⠠⠄⠐⠀⠉⠑⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠂⠁⠀⠀⠀⠀⢉⣿⣾⣥⡄⠄⠆⠀⠀⠀⠀⠀⡀⢀⠜⠀⠱⠃⠋⡁⢰⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠀⠀⠀⠀⠀⠉⠟⢻⡀⠀⠰⠘⡀⠀⠀⠆⡀⠰⣁⠀⠀⠀⡱⡄⠇⠄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⠀⢤⠚⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠇⠃⠀⠀⢀⡀⠤⠽⠳⠀⠀⢃⢁⠀⠀⢸⣡⢆⠀⠉⠀⠈⠀⢠⠘⠀⠀⠀⠀⠃⢀⡀⠀⠤⠉⠀⠀⠀⠈⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠸⠈⡄⠊⠁⢀⡀⠠⠀⠐⠂⣉⠰⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡇⠑⠒⠉⠀⠀⠠⠔⠂⠉⠁⠀⢀⠤⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⠲⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠞
⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⡔⠒⠈⠉⠁⠀⠀⠀⠀⠀⡨⠂⠀⠀⢠⠖⢉⠇⠀⠀⠀⠀⣰⠖⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⣠⡤⠤⠄⠒⠊⠁⠀⠀⢰⠃⠀⠈⠲⡄⠀⠀⠘⣉⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠭⠀⠛⠘⠓⠈⠉⢡⠀⠀⠀⠀⠟⡄⠀⠀⠀⣇⠀⠀⠐⢓⣙⢒⡷⠀⢺⠀⠀⡔⠤⠄⠀⠀⠒⠒⢐
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠈⠑⢄⠀⠰⡁⠀⠀⠀⢀⣀⣠⠤⠰⠀⠀⠀⠀⠂⣹⢾⣭⣭⡁⠀⠀⠘⣇⠀⠀⠀⡠⢿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠈⡀⠀⠀⠁⠈⠉⡋⠀⠄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠀⠑⣆⡀⢰⢡⠂⠀⠀⣇⣀⡀⠀⠀⠀⢀⣸⡀
⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠘⡀⠀⢀⣀⠀⠀⣠⠗⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠘⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠓⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⡠⠖⠛⢉⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⢡⢀⡤⠶⠂⡡⣀⡴⡊⠁⠀⠐⡀⠀⠀⠀⡠⠶⠀⠐⠃⠄⢀⣰⠀⠀⠀⠀⠀⠀⡠⣀⠤⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⢸⠻⣓⠴⢾⣿⣿⣿⣦⠀⠀⠀⢣⠀⠐⠊⠂⠈⠀⠰⠦⢀⣄⠀⠉⠀⠒⠂⠀⠠⠖⠁⡠⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⢏⠀⠀⠐⣝⣛⡿⠟⢀⡠⠐⠁⠀⠐⠢⠤⣀⠀⠔⠒⠒⠠⡀⠛⠀⠓⠀⠀⠀⣰⡉⡄⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠈⢢⡔⠔⠊⡁⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠀⠀⢀⠧⠀⠐⠐⠒⠒⠉⠁⠈⠁⠀⠀⠀⠀⠀⠈⠂⠤⠤⠤⠤⠤⠤⠤⠤⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⢐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠤⠤⠤⠬⢤⠄⡁
⠀⠀⠀⠀⠀⠀⠀⣇⠤⠔⣀⠔⠐⠀⢲⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⡀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠠⠤⢀⡀⠀⠀⠑⢄⠀⠀⢀⠔⠋⠊
⠀⠀⢠⠀⠀⡠⠁⠉⠢⣀⠋⠀⠀⠀⠈⠀⠀⠀⣠⡄⠼⠏⠁⠀⠚⠁⠉⠉⠉⢆⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠑⠐⠁
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠴⠦⠵⠶
⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⡀⠔⠁⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠇⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠈⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠀⠠⠤⠄⠐⠒⠀⠁⠁⠀⠀⠀⠀⠙⠛⠳⠶⣶⣄⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⡿⠘⠀⠀⢻⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣔⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⣠⡠⠀⠠⢤⠴⠄⠀⠀⢀⣀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢄
⡀⠈⠛⠙⠚⣰⢊⡋⡛⠟⠧⠿⠎⠉⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠁⠀⠤⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⠁⠐⠒⠒⠚⠃⠀⠉⠈⠀⠰⠷⠿⢕⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢄⠀⠀⠀⠸⣉⡀⠂⠄⢠⣢⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠤⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠂⠀⠀⢸⡹⣆⡄⠄⢀⠀⠀⠀⠀⡀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠳⠈⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠘⡩⡺⢠⠀⠀⠀⠀⡀⠀⠐⢀⣁⠀⠀⠸
⠀⠄⠀⠠⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠒⠀⠂⠀⠁⣀⡀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠑⠁⠀⠀⠀⡀⠀⠠⠀⠀⠉⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠤⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠁⠈⠀⡖⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠇⠄⠀⠀⠀⠀⠀⢦⡰⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠂⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⡀⣀⣀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⣰⣀⣠⠤⠤⠠⠯⠿⠯⠿⠗⠀⠀⠀⡇⠀⠉⠉⠉⠛⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢁⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢤⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠒⠀⠄⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡀⡐⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⣀⡀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢻⡟⠉⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣠⣀⠀⠀⣿⢿⡇⠀⠀⠠⠤⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣼⣯⣤⣿⡄⠀⠀⠀⠀⣾⣿⣿⣇⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢳⣶⣦⣀⠀⠀⣀⣤⡀⠀⣰⡿⠛⡙⠛⣷⡄⢀⣴⣾⣿⣿⣿⡋⣴⣿⠋⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠘⣿⣿⣿⡶⣟⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⣅⣶⣿⣿⡇⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⠀⣸⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏⠀⠀⢀⡀⡄⣆⢳⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠀⢀⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⢺⡜⡎⣧⢳⢹⡜⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠧⠼⠘⠈⠁⠀⠀⣶⡄
⠀⠀⠀⠀⢀⣤⣤⣴⡖⠒⣶⣶⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣤⣤⡀⠀⠀⠀⣀⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠷⠤⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢽⣽⣧⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣾⣿⣿⠀⠀⢸⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡷⠿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠚⠋⠁⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄⠀⢱⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⠴⠦⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢳⣀⣀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠛⠙⠯⣭⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠋⠍⠅⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣿⣿⠿⡀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⣴⣿⡿⡟⠻⠆⣽⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣼⡿⠋⡀⠀⢠⣼⡿⠃⢸⣿⣿⣿⣿⡏⠉⢻⡟⠀⢹⣿⡄⠁⠉⠻⠿⡉⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠙⠀⣸⡿⠁⠀⢸⣿⣿⣼⣯⡇⠀⠘⡁⠀⢀⣽⣿⠖⠒⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⢸⣿⠏⢹⣿⠇⠀⢠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠈⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠂

⠀⠀⠀⡋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣧⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠄⠀⠀⠀⠀⠀⠠⠡⠊⠀⠄⠀⠑⠀⠀⠀⠀⠀⢀⠤⠐⡶⢐⡟⠉⠓⣄
⠀⠀⠀⠁⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠈⠢⣌⠥⣝⡭⢯⠓⢺
⠀⠀⠅⠌⣉⠫⠀⠀⠀⠀⠀⠀⠀⠀⠠⡩⢀⠘⣦⠈⡱⠀⠀⠀⠀⠀⠈⢌⠠⠀⠄⠠⢈⠔⠀⠀⠀⠀⠀⠀⠀⠓⢥⠂⠋⠁⠁
⠀⠀⠐⠒⠛⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⣶⠠⠐⣀⠐⢀⠠⢢⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠢⡄⠀⠀⠀⠀⠀⢀⣤⣄⠁⠀⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡃
⠀⡎⢀⡀⠈⡕⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⢔⡛⣆⠀⠀⠀⠀⠀⠀⠀⠿⠝⠠⡚⠙⢲⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠅⣈⠀⠅
⠐⡆⠢⡙⠚⠟⠡⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣠⠺⣠⣀⠇⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶


⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠠⠐⠤⠖⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⠀⠁⠒⠫⡭⠂⠁⠀⠀⠀⠀⠀⠠⢀⠀⠄⠀⢁⣔⡤⠀⠀⠀⠀⠀⡁⠀⡀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⣲⣄⠀⠶⠶⠖⠇
⢀⠀⠀⠀⠀⠀⡏⠀⡀⠀⠀⠀⠀⢐⣂⠀⡄⠀⠰⠍⡀⠀⠀⠀⠀⠀⠀⡈⠀⠓⠀⡀⠈⠀⠀⠀⠀⠀⠀⢀⢀⡁⠀⢀⣀⠀⠂
⠀⠀⠐⠐⠦⠄⠀⠀⠀⠀⠀⠀⠀⠻⠩⠈⠒⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠙⠠⠀⠤⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢀
⠀⣠⠄⣀⣐⠠⡌⠀⢢⠀⠀⠀⠀⠀⠀⡀⠠⠀⠐⠶⣶⠀⠀⠀⠀⠀⢀⠴⠶⠲⡊⠨⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⣠⠔⠀⠀⠁⠀⠀⠀⠀⠀⠈⣿⠈⠁⠃⡀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⠓⠲⠒⢄⠀⠀⠀⠀⠀⠀⠈⠀⠛⠁⠈⠆⡄
⠁⠀⠀⠐⠁⠚⠁⡀⡀⠀⠀⠀⠀⠀⠸⣟⢲⡶⠦⢬⡽⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠠⢲⣀⠀⢀⡐⠋
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠛⠟⠲⠺⠟⠅⠀⠀⠀⠀⠀⠀⠀⣄⣣⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠓⠋⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠒⠂⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠦⠤⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠑⠄⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠉⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠁⠀⠄⠀⠀⠊⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠰⢷⠀⠀⠀⠀⠀⠀⠙⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⡂⠀⠡⠂⠀⠄⠀⠀⠀⠀⠠⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠔⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⠀⠄⠞⠂⡑⢄⠀⠀⠀⠀⠔⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠊⠈⡀⢀⠀⠀⠀⠀⠘⢄⠁⠀⠀⠀⠈⠢⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠄⠰⡄⠀⠄⠈⠕⠀⠀⠀⠀⠀⣬⡤⠴⠓⠀⡐⠀⠀⠠⠑⠀⠀⠠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠢⡑⢄⠂⠀⠀⢈⠢⡀⠢⡑⣌⠢⣀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣦⠀⠊⠁⢈⣣⡚⠀⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠁⠁⢀⠀⠋⠜⠂⠈⠑⠶⠈⢢⡄⠀⡀⠤⣖⡱⠐⠀⠀⢀⠀⠰⡣⠀⠀⠨⠓⠁⠀⢀⢘⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⢌⠢⢃⠂⢄⠈⠀⠀⠒⠲⠦⣤⣤⠖⠋⠁⠰⠂⠁⠘⠛⠛⠃⠁⠀⠀⠁⠘⠁⡤⣌⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢡⣌⠓⡀⠄⠐⣊⣤⠗⠋⡙⠦⢰⢤⠂⡿⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡆⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⠜⢋⡡⠶⡛⠁⠀⠀⣚⣅⢀⡉⡆⠰⡏⠀⠀⠀⠉⠉⠉⠉⠙⢻⠀⢸⠀⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⠡⠂⠀⠀⠀⠢⡙⢯⠀⠈⠁⠀⠀⠀⢠⣘⣇⠀⠀⣠⣤⡀⠀⠀⠀⠌⠀⢸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⣑⠄⠀⠀⠀⠰⡏⠀⠀⣹⣿⠁⠀⢙⣷⣤⣤⣤⣤⡞⢠⠍⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠊⠁⠈⠑⠒⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⡀⠀⠀⠀⢹⠃⠀⣀⠀⠀⣀⣴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢦⡀⠀⢸⠀⠸⣍⡷⠀⣿⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣸⣄⣀⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⢁⣠⡤⣤⣀⠉⠓⢯⡀⠀⠀⠀⠀⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⣦⣀⣀⣠⠎⠀⠀⣰⣫⠞⠉⠓⢮⣳⡀⠀⠘⢦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣋⣳⣀⠈⠛⠁⠀⢠⠇⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⡇⠀⠀⠈⢯⠉⠉⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠛⠛⠛⣿⠀⠀⠀⡞⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⢸⡶⠶⠶⠶⠶⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⢾⣙⡦⠤⠤⠤⠤⠤⢿⠀⠀⠀⡇⠀⠀⠀⢾⠆⠀⠀⠀⡇⠀⠀⢸⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀⣀⣀⣘⣆⠀⠀⡇⠀⠀⠀⠺⠀⠀⠀⠀⡇⠀⢀⡞⠀⢀⣀⡈⠉⠳⡖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠙⢦⡀⠳⠤⠤⠤⠤⠤⠤⠤⠴⠃⣠⠞⠛⠛⠻⠤⠗⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣳⠀⠀⠀⠀⠀⠙⡶⢤⣀⣀⠀⢀⣀⣠⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⡍⠉⢹⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡞⢦⠀⢸⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢠⡇⠀⠙⠋⢠⠾⣆⠀⠀⠈⢳⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠼⠉⠁⠀⠀⠀⠈⠓⠋⠀⠀⠀⠈⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⣰⡖⢥⣃⡂⠄⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⡁⢀⣴⠾⣧⣼⢟⠘⢍⠀⠉⢳⢌⠢⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠊⠀⠈⠀⠉⠑⣊⡟⡄⣡⢀⠀⢀⡼⠀⠑⠌⢂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠂⠡⠀⠀⠀⠀⠘⢄⡌⠙⢂⠀⠒⠁⠀⠀⠀⢈⡆⢃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⡀⠸⠙⠂⠀⠀⠀⠀⠘⡷⢛⡀⡀⢀
⠀⠀⠀⠀⠀⠠⠂⣀⣀⣐⠂⠄⠀⠐⣄⠀⢀⠤⢤⠘⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠚⠇⠁
⠀⠀⠀⢀⠤⠤⠤⠤⠤⠬⠧⠦⢤⡀⠈⠑⠴⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠃⠆⢠
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠀⠀⠀⠀⠙⠉⠁⠁⠢⠄⡀⠀⠀⠀⠈⠀⡙⠠⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢁⢠⡄⠀⠀⠀⢠⠀⡄⠀⡀⠀⠀⠀⠈⠢⡀⠀⠀⠒⠚⠓⠒⠒⠒⠒⠗⠛⠉⠉⢝⡭⢭⣍⢂⠀⠀⠀⠀⡰
⠀⠀⠸⠀⠀⠸⠇⠀⠀⠀⢹⠋⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⡠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⣤⠂⣿⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠄⡀⠀⠀⢃⡈⠀⢀⣘⠂⠄⠀⢀⡀⢸⠄⠄⠊⢂⠈⠉⠁⠰⠀⠀⠀⠀⠠⠂
⡔⠐⡇⠀⠊⠀⠀⠀⠀⠀⠀⠀⠱⠀⠘⠉⢳⠂⠀⠈⠉⠁⠀⠐⠊⠒⠀⠈⠁⠀⠒⡀⠀⠀⠀⠣⠀⠀⡄⠀⢀⡀⠤⠐⡞
⡯⢁⠀⢠⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠍⠁⠀⠒⠀⠀⠒⠀⠉⡀⠀⠀⠀⠀⠀⠛⠀⠀⢀⢔⣑⡀⠃⠀⠀⠀⠀⠀⠐
⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠇⢈⢀⠀⣄⠀⠀⣠⢀⠈⢠⠀⠀⠀⠀⠀⠘⠈⡠⠢⠈⢹⡟⢻⠀⠀⠀⠀⠀⠰
⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠠⠀⠀⠀⠀⠀⠀⠃⡀⣄⣀⣀⣠⠀⠐⣋⢀⣀⣀⣤⠤⠼⠤⠤⠤⠤⠤⠴
⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠉⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠦⠀⠄⠂⠀⠀⠀⠁⠀⠔⠃⠀⠀⠀⠀⠀⡤⠂⠤⡄
⠀⠀⠀⢀⣯⣤⣶⣶⣄⣄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⢠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠆⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠘⠛⠟⠁⠀⠸⡆⠀⢀⡤⢀⣀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⣉⠥⠐⠛⠁⠀⠈⠈⠩⠉⠀⠂⠀⢼⣆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⠁⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠀⣠⠶⠦
⠀⢀⠈⡠⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⠀⠀⠀⠄⠀⠀⡀⢀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠘⠁⠔⠀⠀⠀⠀⠀⣸⠏⠀⠀⢼⣒⠲⠞⢒⣠⣈⡀⣸⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠓⠀⠀⡆⡂
⠀⠀⠀⠰⠖⠀⠒⠀⠀⠞⠁⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠶⠞⠁⠀⠀⠚⠓⠚⠓⠀⠐⠐⠛⠓⠛⠂⠀⠀⠀⠀⠀⠄⠀⢀⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢡⠋⠔⠁⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⣠⢀⡀⢀⣠⣤⣤⣶⣶⡆⢾⠁
⠀⠀⠠⢀⠀⠀⠀⠀⠀⠄⠀⠠⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⠋⡽⠋⠹⠈⠈⢣⡼⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⣀⡀⠂⠀⠀⠤⣤⣤⣤⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠦⠄⠀⠀⠀⡀⠠⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡟⢸
⠀⠀⠀⠀⠀⢀⠴⠶⢾⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠀⠄⠈⠆
⠀⠀⠀⠀⠠⠀⠀⠀⠘⠄⠐⠓⠀⠂⠐⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠤⠤⠖⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡅
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⠀⠠⠤⡄⣤⣤⣤⣠⣀⣤⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠉⠙⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⣰⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣆⠀⠀⢀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣏⣀⣛⣛⠁⣿⠀⠀⠀⠀⢠⣴⣶⣄⠀⠀⠀⠀⣿⠈⣛⣛⣈⣹⠉
⠀⠀⠀⠀⢠⣴⣄⣀⣉⣉⣉⣉⡁⣿⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⣿⢈⣉⣉⣉⣉⣀⣠⣶⡄
⠀⠀⠀⠀⠈⠛⠋⠀⣭⣥⣭⣬⡁⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⢠⣭⣤⣭⣤⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⣶⣤⠄⣿⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⣿⠠⠤⣶⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠙⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠋⠀⠀⠀⠈⠻⠋




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

⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠚⣩⡤⠶⠦⣬⡙⠂⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⢀⡀⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠁⠴⢚⠲⠄⠁⠀⠀⠀⠀⠀⢀⠂⠀⠋⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣇⣀⣠⣿⡇⠀⠀⠀⢀⠀⠀⠤⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠽⠿⠿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠼⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠽⠿⠿⠀⠒⠐⠃⠐⠂⠿⠿⠿


⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣒⣒⠴⠶⣂⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠐⣲⠁⢈⠆⢘⣒⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣍⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢔⣀⡢
⠀⠀⠀⠼⠷⠤⢤⡾⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⣤⡆⠀⠀⠀⠀⠀⢸⣿⣟⣉⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠪⠭⣕
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠿⠛⠛⠃⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⣶⣤⠚⣿⠳⣴⣶⠀⠀⠀⠀⣀⣤⣶⣶⣄⣀⠀⣤⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⣶⠖⣿⣿⡿⢶⡶⠀⠀⠀⠘⠿⠿⢻⠟⠿⠟⠈⠛⠛⢻⡟⠛⠛⠀⠀⠀⠀⠠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠸⠛⣿⣧⣠⣄⣀
⠀⠀⠀⠀⢠⣤⣭⣄⡄⡀⠀⠀⠀⠀⠀⢀⣐⢴⠀⠀⠀⠀⠀⣀⣲⠀⠀⠀⠀⠀⢸⣦⣍⣉⣵⣿⠀⠀⠀⠀⠀⠀⢸⡟⠿⠿⢿⣿
⠀⠀⠀⠄⢤⢤⡤⡤⡌⠄⠀⠀⠀⠀⠀⠀⠀⡭⠀⠀⠀⠀⠀⠁⠭⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⡩⡀⠀⠀⠀⠀⠀⠾⡷⠤⠤⢬⣷⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⣄⡟⠙⢂⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣙⢒⠂⣾⣿⠈⢳⠀⠀⠀⠀⠀⣖⢂⣐⠁⢤⠤⠐⢂⡁⠇⠀⠀⠀⠀⢸⠐⠩⠃⡇
⠀⠀⢀⣴⣆⣿⣿⣿⣿⣴⣦⠀⠀⠀⠀⠀⠀⠲⣒⣆⢇⠀⣿⡟⠀⠀⠀⠀⠀⢄⠀⡶⠔⠒⠀⢀⠁⡦⠆⠀⠀⠀⠀⣸⢖⠤⡖⠇
⠀⠀⠀⠙⠁⠈⠙⠋⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠓⠒⠁⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠈⠒⠚
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢬⠉⠙⠒⠀⠀⠐⡆⠀⠐⠊⠩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣼⠀⢀⣀⣀⠀⠀⢇⣀⣀⣀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠒⠊⠉⢀⠚⠛⠉⠀⠀⠀⡈⠋⠋⠉⠉⠈⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⢄⣀⠀⠀⢠⡉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠨⢆⠈⠀⠀⡸⠐⠒⠒⠒⠒⠒⠲⡀⠄⠀⠀⠄⠀⠉⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢰⡐⠄⠀⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠒⠒⠈⣀⠠⠘⠋⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠔⠈⠀⠉⡠⠂⠀⠀⠀⠀⠀⠠⠐⠀⡡⠔⠋⠙⠉⠻⠲⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠔⠁⠀⡠⠊⠀⠀⠌⠀⠀⠀⠄⠈⢢⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠀⢰⠁⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣠⠀⣤⣤⣶⣆⣀⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⡲⠄⢀⣀⡀⠉⠁⡒⠒⠒⠂⠤⠀⣈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡿⠧⡌⣯⠁⣹⠽⢿⣷⣧⠠⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠁⠀⠀⣕⠂⡤⢠⠄⠀⠤⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠴⠃⣿⣷⣸⡦⠀⣿⠁⠀⠀⠁⠐⠠⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⢀⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⡀⡀⠘⣆⣀⣀⣽⣿⣿⣷⣄⠀⢈⠂⠄⡀⠀⠀⠈⢄⢀⡴⠃⠀⠀⠀⠀⠀⢆⠠⠺⢉⡗⢒⠤⣠⢲
⠀⠀⠀⠀⠀⠀⠀⠈⢠⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠠⠀⠀⠒⠁⠀⠀⠐⡄⠰⢊⡴⢁⡞⠀⣠⢆⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁⠃⠈⠀⠀⠀⢠⣄⠀⢀⡜⠵⣋⣴⢏⣠⠞⠀⠏
⠀⠀⠀⠀⠀⠀⡘⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠄⠀⠀⠀⠀⠀⢈⣕⡶⢭⡋⢥⣠⠯⣣⡀
⠀⠀⠀⠀⠀⠀⠠⠰⠄⠤⠠⣤⠚⠓⢲⡴⠒⠒⢦⡀⣀⣀⣀⣀⣀⣇⠘⠁⠀⠀⠀⠀⠀⠀⠉⠐⠠⣈⡑⠪⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠙⠷⠔⠚⠛⠢⠴⠟⠉⠛⠃⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠮⠀⠀⠀⠀⠑⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣐⡄
⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⠐⠒⠀⠀⠱⠀⢀⠀⠸⣹⣿⢳⡆⠀⣾⣹⣿⡿⢿⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⣤⣦⣤⣦
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣀⣶⠀⠀⠀⢀⣀⣸⣀⣀⠀⠘⠤⠄⢻⠀⠀⠀⠀⠿⠯⠥⠥⠼⠃⠀⠀⢻⡯⣝⠛⠀⠀⠀⠷⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⢠⡶⢶⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠉⣿⣿⣿⣶⡆⠀⠠⣿⢟⣷⡿⠃⠀⠠⠒⠶⠶⠖⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠿⠦⠤⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠙⠟⠁
⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣿⣿⣿⡿⠀⠀⠀⠰⠶⣾⣶⣶⠀⠀⠰⠶⣾⣂⣀⠀⠀⠀⡀⢸⣦⠀⠀⠀⢀⣶⣿⡏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⢺⡿⢷⡦⠀⠀⠀⠰⠴⠿⠿⠿⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⢙⣍⣙⠃⠀⠀⠠⠤⠤⠤⠿⠧⠀⣰⣷⣾⣧⣤⡀⠀⢾⣦⡎⠱⠿⠆
⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠋
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⡥⠙⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⢰⣦⣀⣠⣞⠀⠀⢠⣀⣂⡄⠀⠀⠀⠀⣠⢀⣀⣀⠀⠀⠀⣬⠀⢀⡃⠀⠀⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠈⠁⠈⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠋⠛⠛⠋⠀⠀⠀⠀⠙⠈⠁

⠀⠀⠀⢠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠀⠀⠀⠀⢀⡄⢀⣀⣀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢀⣶⣠⡄⠀⠀⠀⡶⢶⠶⢤⡀
⠀⠀⠤⠀⠀⠙⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠘⠛⠛⠉⠛⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠛⠟⠀⠀⠀⢀⣷⣾⡶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢠⡞⢳⡄⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡀⠀⢿⣴⣇⠀⠀⢹⡏⠀⠀⣸⣦⡿⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡶⠃⠀⠀⠀⠈⣷⠀⢸⡇⠀⣾⠁⠀⠀⠀⠘⢧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣉⡿⠶⠶⣄⠈⠳⣄⠀⢀⣀⣿⠤⠼⠧⠤⣿⣀⠀⠀⢀⡴⠋⢀⡤⠤⢴⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠳⣄⣨⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣏⢀⡴⠋⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣾⢛⣷⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⣾⡛⣷
⠀⠀⠀⠀⠀⠈⠉⠙⢦⣤⣤⣤⣤⣼⠁⠀⠀⠀⠀⢸⡏⠀⠀⢹⡇⠀⠀⠀⠀⠈⣧⣤⣤⣤⣤⡴⠋⠉⠁
⠀⠀⠀⠀⣠⠶⣄⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⡴⠞⠓⠒⠒⠚⠳⢦⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣠⠶⣄
⠀⠀⠀⠀⠙⠶⠋⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⡇⠀⠀⢺⡗⠀⠀⢸⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠙⠶⠋
⠀⠀⠀⠀⠀⢀⣀⣠⠞⠛⠛⠛⠛⢻⡀⠀⠀⠀⣇⡀⠀⠘⠃⠀⢀⣸⠀⠀⠀⢀⡟⠛⠛⠛⠛⠳⣄⣀⡀
⠀⠀⠀⠀⠀⢿⣬⡿⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⢿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠞⠁⣹⠷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋⠙⢦⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣯⣸⠗⠒⠚⠁⣠⠞⠁⠀⠀⠉⣿⠒⢲⡖⠒⣿⠉⠀⠀⠙⢦⡀⠙⠶⠶⣾⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⡄⠀⠀⠀⢀⡿⠀⢸⡇⠀⢿⡀⠀⠀⠀⢠⠾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣾⠻⡏⠀⠀⣸⣇⠀⠀⢹⠟⣷⠀⠈⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠘⢧⡼⠃⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡠⠂⢤⡀⠀⠀⠀⠀⢀⡠⠔⠂⢉⠀⠀⠉⠑⢢⡀⠀⠀⠀⠀⠀⢀⠔⠊⠉⡁⠀⠡⡐⢀⠀⠀⢀⡀⠀⠀⢀⠔⠈⠣⠄⠠⠜⠁⠢⡀
⠀⠀⠣⡀⠀⠈⠐⠂⢀⠍⢠⠀⢁⠚⠥⠈⠀⠁⠈⠁⠀⠕⠀⠀⠀⢀⠃⠀⠀⠀⠁⠀⠀⠀⠑⢆⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⢀⠜⠂⣈⠕⠀⠈⠀⡡⠐⠒⠢⢄⠀⠀⠀⠀⡀⣀⠜⠁⠀⠀⢸⠠⠈⢈⡈⡀⠀⣠⠀⠔⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠘⠒⠫⠤⠄⠄⡤⠊⢀⠀⢀⡠⠚⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⢳⠀⠀⡆⠁⢀⠀⡘⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠮⠤⠴⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣢⡈⢲⡰⠁⠀⢘⠖⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⣀⡀⡠⠔⠢⢄⠀⠀⠀⠀⠁⠈⠀⠀⠈⠀⣀⠀⠀⠀⣀⠀⠀⠈⠒⣄⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠇⢯⡱⢸⠒⠀⠀⠆⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠠⡤⠴⠎⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⢄⡠⠃⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⢀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠛⠁⠀⣠⠄⠒⠒⡤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠆⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠠⠁⣠⡀⡄⠀⢡⣧⠀⠀⠀⢨⠉⠉⡇
⠀⠀⠀⡀⠀⠐⠚⠀⠀⠐⠻⣄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⢰⠃⠀⣙⡷⠃⠀⣿⠇⠀⠀⠀⠈⠳⠞
⠀⠀⢰⠁⠀⠀⠀⠀⠠⡄⢠⠀⠃⠀⠑⠐⠊⠉⠉⠐⠢⠤⠀⠤⠔⠊⠀⠀⢀⠀⠀⠀⠇⠀⠐⠇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠁⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⢸⠂⣠⣀⡀⠀⠀⣼⠇⠀⠀⠀⠈⠁⠀⣀
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠲⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡈⠈⠁⢐⡏⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⡧⠀⠐⠒⠒⠀⢤⢄⠈⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠂⠚⠉⠢⣄⠀⠀⠀⠀⠀⠙⠶⠶⢥⣤⣤⡾⠁⠀⠀⠀⠀⡚⠓
⠀⠀⣇⠀⠀⠈⠀⠂⠈⣾⠘⠃⡅⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⣄⣴⠟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⡀⣠⣖⠠⠀⠐⠀⣷
⠀⠀⠉⡄⠀⠀⠀⠀⠀⠙⠐⡀⡇⠀⠀⠀⠊⠀⠀⠀⠀⢰⠃⠙⠋⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⢠⡄⠀⠀⠀⠀⠄⠀⢀⢸⣿⣿⡇⠀⠀⢸⡅
⠀⠀⠀⠲⠶⢤⠤⠤⡾⠡⡾⠐⠁⠀⠀⠀⠀⠀⠀⣀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠈⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⠛⠶⠶⠶⢴⣄⡺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⣉⠀⢀⡂⠀⠀⠀⠁
⠀⠀⡀⠀⣀⠠⠖⠒⠊⠱⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠁⠀⠠⠔⠒⠂⡄⠀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⣤⣀⣀⣀⡀⢈⠀⠀⠀⠀⠀⠀⠀⢀⢄⠤⠄⠀⠀⠄⠀⠀⠰⠂
⠀⠀⢸⡀⠀⠀⠒⠡⡐⠀⠰⡀⠀⠀⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⡄⠕⠁⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠈⡃⠀⠐⠂⠚⠁⠀⠀⡁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⡈⠀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡀⡀⠀⠀⠀⡧⡀
⠀⠀⠀⢣⠀⣤⠴⠖⠚⠀⠀⡁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣄⢣⣤⠇⠀⠀⠀⠀⠀⡀⠀⠁⠀⠰⡀⠀⠀⠀⠀⠂⠀⠒⠊⠁⠈⠒⢄
⠀⠀⠀⠈⠆⡀⠀⠠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠙⠋⠘⠻⠂⠀⠀⠀⠈⠛⠀⠀⠀⠀⠈⠀⠀⠉⠉⠐⠲⠀⡀⠤⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⣤⣴⣦⣤⡄⠀⠀⢐⠒⠒⠂⣀⡀⠀⠀⠀⢀⣰⣾⣾⡆⠀⠀⠀⠠⢤⠤⠀⠀⠀⠀⠀⣠⣾⣧⣠⠂
⠀⠘⠿⠿⠿⠿⠇⠀⠀⠘⠀⠀⠣⠀⠀⠀⠀⠻⡿⠛⠻⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⠄⣦⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣾
⠀⠀⢸⣿⣤⣄⠀⠀⠀⠀⠸⠿⠿⠿⠄⠀⠀⠀⠀⠚⠁⠟⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢰⣾⣶⣾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣾⣿⣄⠀⠀⠀⠀⠠⢨⡂⠀⠀⠀⠀⢸⡇⣶⣿⣿⠀⠀⠀⠀⠀⣠⡺⠟⠁
⠀⠀⢹⣿⣾⡿⠀⠀⠀⠀⢠⢏⠷⡄⠀⠀⠀⠈⣇⣻⣿⡟⠀⠀⠀⣠⣦⠟⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣤⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣶⡦
⠀⠀⣾⣿⣿⣿⡄⠀⠀⠈⠉⠉⠉⠈⠁⠀⠀⠀⠀⡇⢥⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣒⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡫⠭⠭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠉⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢁⣴⣶⣶⣄⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣀⣀⣀⡀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣋⣹⣟⣁⣀⣀⣀⣀⣻⣟⣘⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠵⠧⠤⠾⠿⠿⠿⠿⠦⠤⠷⠝⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡿⠿⢿⣿⣿⣿⣗⣺⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠀⠀⠀⢀⣠⣀⠀⠀⠀⠬⠅⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡅⠀⢀⣿⣿⣿⣗⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠠⠭⠤⠄⠀⠸⣿⣿⠂⠀⠠⠤⠥⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⡿⠀⢹⣿⣿⣿⣿⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠈⠩⠭⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣤⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠁⠀⠀⠈⠛⠋⠀⠀⠀⠨⠍⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠴⠶⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠠⡐⢈⡠⠖⠊⠉⠉⠉⠉⠑⠲⣄⠀⠀⠀⢑⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⢀⡀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠐⡂⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠴⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠭⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⣚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣒⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠤⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢐⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠍⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠀⠠⣤⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣆⠀⠀⢰⡌⣸⣦⡶⡶⢶⠔⡒⠒⠒⠊⠉⠉⠉⠹⣇⣀⣀⢒⣴⠯⠝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣀⣈⣙⣛⣓⣚⣉⣈⣉⣁⣉⣈⢤⣥⣤⡤⠤⠤⠦⠔⢒⠾⠟⡒⢒⢚⠋⠋⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡁⣍⢨⡉⡀⡄⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡜⡸⢁⠋⡜⡘⠡⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⢠⢁⠂⡜⡠⢡⢁⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⠿⢋⢟⣶⡴⡰⢡⢃⠆⡸⡰⢃⠂⡌⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡘⠣⠡⢣⠺⠃⣿⠣⢃⠎⢠⡱⢁⠎⡘⡰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡵⡥⣱⣱⢷⣿⠃⡌⡠⡱⢁⡞⡰⠠⣡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⣤⣟⣿⣧⡇⡌⣠⣱⢁⡜⣠⢡⣣⢇⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣿⡿⢛⠛⠟⢙⠛⠃⡞⠹⠛⠟⠛⠘⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢈⣾⡧⢤⠶⡶⠦⠄⠾⢱⠶⠆⠄⡼⢷⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠆⡼⡿⣾⢫⣷⣦⣶⣶⣧⡼⣰⢦⢆⡴⣰⢴⣦⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⢀⠂⡌⡘⢸⣽⣣⣯⣮⣦⣿⣷⣿⢇⠃⡎⡘⢠⢃⠆⠌⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⠀⡄⠰⠐⠁⠈⠉⡛⡻⢻⢋⠟⡹⢁⠆⡜⡸⢡⢂⡎⣌⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡀⢠⢀⠀⠄⡀⡸⠱⢡⠏⠎⠴⠁⠂⡜⠰⠁⣃⣚⣘⣈⣨⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠶⠤⠤⠤⠤⠤⠤⠤⠒⠒⠒⠒⠒⠚⠋⠙⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⢀⡤⢀⣄⡠⢀⠔⣠⠄⣠⠖⡠⢄⡤⢀⡤⢀⠄⢀⠴⢀⡤⢀⠄⣠⠄⣠⠄⡠⢀⡤⢀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠖⠡⠖⡵⢪⠖⢡⠞⡵⠊⡤⢊⠴⢫⠔⣩⠖⣡⠞⣡⠔⣡⠖⣡⠞⣡⠊⡠⢊⡴⢋⠔⢋⠔⡡⢊⠔⣡⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠁⠀⠁⠀⠀⠈⠀⠁⠀⠁⠈⠁

⠀⢾⡟⠛⠛⠛⠛⠛⠛⠛⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⣦⡀
⠀⠈⠻⢶⣤⠀⠀⠀⢠⣶⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⠻⣦⣄
⠀⠀⠀⢠⡏⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠙⢷⣄⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⣴⣦⣄⣹⣿⠂
⠀⠀⠀⢸⠁⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣾⠏⠈⠉⠉
⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠙⠛⠁⠀⠀⠀⠀⠀⣠⡿⠃
⠀⢀⠔⠃⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⢸⡏
⠄⠀⠀⠤⠤⢤⡤⣤⡤⠤⠤⠤⠬⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣷⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣴⡟⠁⠈⠛⢦⡀⣸⠇
⠀⠀⠀⠀⠀⢸⠇⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡾⠋⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⠁⢸⣇
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢚⣛⣉⣩⣽⡆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠾⣿⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⣤⡤⠤⠶⠶⠟⠛⠛⠛⠉⠉⠉⣷⠀⠀⠀⢠⣶⠾⠛⠛⠉⠁⠀⠀⠀⠈⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣠⡾⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⠀⢴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠃⠀⠀⣀⡠⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢾⠆⠀⢀⠾⠿⠀⣀⠤⠴⢣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢩⡏⢤⢀⡅⠀⡠⠋⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡠⠔⠢⣄⢸⡆⠀⣠⠥⢍⡡⠖⠊⠉⠉⠱⡿⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢇⠈⠻⠎⡿⠁⡜⠀⠀⠀⠇⠀⠀⠀⠀⡰⠃⠀⠀⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⢙⡦⠀⠀⠄⢡⣄⣸⡎⡴⢀⣀⠔⠊⠀⠀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠆⠀⠉⠓⠢⢤⡀⠀⠀⠀⠀⠘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠙⠢⣄⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⢀⡠⡖⣚⣀⡓⢒⠦⢤⡀⠀⠘⢆⠀⡀⠀⠀⠀⠀⠈⠳⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠉⠴⣄⠀⠉⠉⠁⠀⠀⠀⠙⠂⠀⠈⠁⠀⠈⢱⠤⣀⠀⠀⠀⠀⢀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠸⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠉⠛⠛⠋⠉⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠳⢄⣀⡀⠜⠐⠤⢉⠢⢄⠀⠀⢀⡀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⠌⡙⠫⢑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠈⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠠⠁⠊⠦⣤⠊⢣⣀⠀⠀⠀⠀⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢃⠀⠘⢀⠂⠈⠉⠓⠲⠆⠤⣄⣀⣀⣀⡀⣰⣧⠀⠀⠀⠀⠀⠀⠀⢸⣩⠇⠀⢰⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢀⡏⠀⠀⠀⠀⠀⠀⠂⠓⠼⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠄⣠⡀⠀⠀⠀⠠⢜⠿⠛⠉⠆⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡟⠐⠒⠚⡾⡶⢒⡆⠰⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠦⡀⠀⠀⠈⠉⢉⠏⡸⠀⠀⠀⠜⡔⣁⠀⠉⠐⠺⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠑⠰⢾⣯⣡⣀⠴⠋⠈⠑⠤⣴⠁⠀⠉⠑⣢⠀⠀⠈⠉⠓⠢⢤⣀⡀⠀⠀⠀⠀⢸⠀⠀⢸⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠈⠂⠍⠁⠒⠒⠂⠀⠘⠃⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠲⠢⢄⣘⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠈⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠤⠄⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⠀⠤⠀⠐⠒⠂⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄⢀⠊⠀⠐⠦⡀⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣈⠉⠙⠓⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠤⠀⠀⠛⡀⢠⠇⠀⠀⠂⠂⠼⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠁⠀⠈⠉⠙⠓⠲⠶⢤⣄⣀⡉⠉⠛⠒⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠤⠤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡠⠁⠀⠐⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠱⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢸⡤⣤⢀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⡈⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠋⠔⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠈⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠐⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠠⠀⡀⠀⠀⠓⠐⠁⠀⠀⠀⠀⠀⠒⡌⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣴⠟⢻⣧⢀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠙⠷⠾⣿⡾⠃⢠⡞⢻⡗⣴⠟⠷⣰⠟⣿⠇⣺⠟⢃⡾⢠⡞⢻⡆⣸⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠟⠁⠀⠘⠛⠛⠓⠙⠓⠒⠛⠛⠛⠒⠛⠓⠘⠓⠚⠛⠛⠑⠋⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠄⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡂⠀⠀⣀⠊⠀⠈⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠀⠀⠇⠓⠠⠄⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠦⠀⠀⠰⠀⠀⠀⠒⠀⢤⡄
⠀⠀⠀⠀⠀⣀⣉⡂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠟⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠈⠇
⠀⠀⠀⠘⠿⠟⠛⠃⠐⠿⡷⠆⠀⠀⠁⠀⠀⡀⠀⠀⠀⢀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⣴⣶⣄⡀⠀⠀⢀⣠⣶⣦⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⢀
⠀⠀⠈⠛⠿⠋⠀⠀⠈⠿⠟⠁⠠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⣿⠟⠃⣠⣈
⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠲⢠⡶⠀⡰⠠⡀⠀⠀⠠⠼⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡀⢸⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⠐⠚⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠃⠀⠠⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠂⠈⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠔⠒⠒⠒⠒⠦⣄⠕
⠀⠀⠀⠀⠀⣀⣀⢠⠴⠶⣒⡶⠶⣬⣱⡿⠖⢂⣰⣿⣦⠀⢀⣀⣤⣤⣿⣤⣤⡴⠦⣀⣒⠶⠦⠤⠤⠴⠤⠄⢄
⠀⠀⠀⠀⠀⡁⠀⢀⡠⠎⠉⠀⠘⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣼⠋⠀⢂⠀⠀⢀⠁⠆⡁⢸
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠐⠀⠐⠃⠀⠀⠈⢄⠉⠁⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⢛⠀⠀⠀⠑⢤⠚⠁⢰⡇⢸
⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠈⠉⠉⡇⠀⠀⠠⠀⠀⠠⠉⠀⠄⠀⠀⠀⠀⡈⠁⠀⣿⠀⠀⠀⠀⠀⡑⠄⠜⢣⠸
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⡀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠑⠤⠊⠀⠀⠀⠀⠠⢰⡌⠁⢰⢸⠀⡇
⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇⠆⢸⠘⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⡀⠀⠀⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⠙⠂⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⢀⠀⠀⠵⡠⠀⡉⠏⠀⡇⢸
⠀⠀⠀⠀⠀⠧⠀⠀⠀⠠⡀⠀⠀⡇⠀⠀⠀⠀⠒⠂⠀⠀⠀⠐⠠⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠁⠛⠀⠀⠁⢸
⠀⠀⠀⠀⠀⡄⢈⠢⠐⠈⠁⠀⠀⢡⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⢂⢀⠀⠀⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠢⡇⢸
⠀⠀⠀⠀⠀⠧⠬⠤⣰⠤⠤⠄⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠀⠀⠀⡇⠀⢸⠂⠀⠀⠀⠀⠀⠀⢀⣧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠓⠒⠒⠤⠤⢭⡤⠎⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢒⣂⢩⣭⣭⡍⣁⣐⡲⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢒⣒⠋⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣄
⠀⠀⠀⠀⠀⠀⡰⣿⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢯⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⢀⠔⠒⠉⠋⠛⠲⢦⡀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠑⢄⡀⢀⡎⠀⠀⠀⠀⠐⠀⠀⣇⠀⢀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣤⣤⣤⣄⣀⣀⣀⣀⣤⣤⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠙⠛⠛⠛⠛⠛⠛⢻⡛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠉⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠡⢸⡆
⠀⠀⠀⠀⠀⠀⢠⣃⣀⣀⣀⣀⣀⣀⣁⣈⣉⣙⣂⣘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⠈⣧⣀⡀
⠀⠀⣀⡤⢖⣊⣭⠥⠤⠤⠤⠤⠤⠤⠶⠶⠾⣭⣽⣿⣛⡶⢦⣄⠀⠀⠀⠀⠀⡤⠊⠁⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⣂⢹⡍⠙⠛⠶⣄⡀
⢀⣾⠵⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢾⣷⡄⠀⠀⡜⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⢐⣂⢉⠥⢘⡇⠀⠀⠀⠈⠙⣦
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣄⣀⣀⣠⠤⠴⠒⠉⠀⠀⠀⠄⡀⢀⢸
⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣠⣾
⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⠓⠻⠷⣲⠦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⣞⣿⠽⠋
⠀⠀⠈⠙⠲⠤⢤⣄⣀⣀⡀⠀⠀⠀⣟⠀⠀⠀⠀⣀⣀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⢻⠻⠿⠿⠿⡿⠿⠿⠿⠿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠉⠁⠀⠈⠉⠛⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠚⠛⢻⡆
⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣀⣀⣠⣤⠴⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡴⠀⠀⠘⣇
⠀⠀⠀⠀⢰⡇⠀⠀⢀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠋⠑⠀⠀⠀⢻⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠀⠠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠛⠛⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠴⠾⠇
⠀⠀⠀⣾⣀⣤⡴⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠦⣤⣀⣀⣀⣤⣤⡴⠶⠶⠚⠛⠛⠉⠉⠉
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠤⣀⢀⡴⠚⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠐⠀⠀⠀⠀⡿⠀⡴⠶⡄⠸⡖⠒⠒⠒⠒⠒⠒⠛⢋⠽⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠚⠉⠀⠀⠀⠀⢀⣤⡤⠤⠖⣇⣰⠷⠒⡇⠀⡗⠤⣤⣄⠀⢀⣀⣬⣤⠤⠚
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠐⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⡐⠢⠤⣼⠁⢠⣎⣙⡺⠿⡆⠘⡞⠋⠉⠀⢰⠇
⠀⠀⠀⠀⠀⣴⠋⣡⡤⠭⣍⣙⡛⠒⠒⠒⠢⠤⠤⠤⢤⣀⣀⠀⠀⠀⠀⢀⣉⠭⠟⢻⠀⢸⡼⠴⠚⣉⣿⣤⢧⡀⠀⠀⢸⣄⣀⣄
⠀⠀⠀⠀⢀⡇⣼⣽⠁⠐⠒⠂⠩⠉⠛⠒⠒⠒⠶⠤⢤⣄⣈⣉⠉⠒⢉⣁⡤⠴⢺⠉⠉⣿⠉⠉⠉⠉⠀⠀⠀⢷⡤⣔⡪⠗⠁⡏
⠀⠀⠀⠀⠀⣇⢹⣿⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠭⣟⠋⠉⠀⠀⠀⢸⡇⠀⣻⠀⠀⣾⣿⡟⡆⠀⠘⣏⡥⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠻⣄⣿⢄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢐⡡⢰⡇⢀⠀⠀⠀⠠⠘⡇⠀⢹⠀⠀⠻⣿⣹⠃⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠙⢿⣶⠬⠭⢭⣗⣒⡤⠤⣄⣀⠒⠀⠤⠀⢀⣭⣇⡀⢀⠀⢀⡤⢚⡇⠀⢸⠀⠀⠀⣿⣿⣆⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠦⢤⣉⣉⡛⠒⠿⠿⢶⣦⣤⣈⣳⡠⠔⣪⠵⠚⣩⡇⠀⣸⠀⠀⢠⣿⡿⠾⠆⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⢤⣀⠉⠓⠲⡖⠋⢀⡠⠊⠁⠙⢤⣉⠓⢤⡀⠀⢀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣧⠔⠉⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⢐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⢤⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣀⣠⣄⣀⣀⡀⠈⠙⣦⠡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⡀⠀⢂⣠⣠⣤⣐⣲⣤⣤⣤⣤⣀⣀⣘⣷⣿⢉⣀⠀⠈⠩⠙⣳⣶⠋⠄⢂⣠⣤⠒⣠⣐⡠⠠⠀⠄⣒⣄⣂⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢐⠪⣢⠶⠶⠶⠿⠛⣫⣩⠟⠛⣲⣶⡿⠟⠉⠉⠭⠿⠿⠍⠩⠭⠭⠿⠿⠿⠿⣿⡿⠟⠓⣲⡮⠽⠟⢛⣛⣛⣛⠛⠫⢉⢳⡄
⠀⠀⠀⠀⠀⠀⢈⠰⣿⡀⡀⢀⡠⠞⠁⠀⠀⢸⡏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡏⠀⠀⢎⠀⠀⠀⡀⠈⡆⣯⠀⡗⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠁⣿⠀⡴⠁⠀⠀⠀⠇⠀⠘⢣⠀⠀⠀⢈⡄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡇⠀⢀⠀⠙⢆⡒⠋⠀⢷⡇⢰⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠰⡅⢻⠜⠁⠀⠀⠀⠀⠛⠛⠛⡟⠀⠀⠀⢀⠈⠁⠁⡰⠖⠀⢀⠀⠀⠀⠀⠀⠟⠉⠉⢺⡇⠀⠀⠀⠀⠀⠙⢄⣀⡜⣇⠀⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠠⢺⠀⡀⠂⠁⠂⠘⡀⠀⡠⠇⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠘⢄⠀⡠⠛⠑⠀⠀⠀⠀⠠⡐⠦⠀⠇⢻⠀⣿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠁⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠐⡝⠀⠀⠀⢸⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⢠⡀⣿⠀⠀⠁⣤⠆⠀⣼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠒⠚⠈⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣿⠀⠀⠈⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⣿⠠⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠄⠠⠤⠄⡀⢨⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢨⡀⠙⠊⠀⠂⠄⡀⡇⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⠁⠀⠀⠂⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠅⣁⣀⠌⠙⢺⠃⠃⡆⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣄⠀⠀⠀⠠⡀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠇⠀⠀⠓⠠⠀⠈⠁⠒⢿⠆⢀⠀⡇⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣯⠀⠣⡀⠀⣠⠶⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠠⠀⠀⠀⡀⠀⠀⡀⠀⢡⠀⠀⠀⠀⢀⢸⠀⡇⠈⡇⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡀⣿⢀⣰⠈⢦⡀⠀⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣠⡄⢣⡘⠆⠄⠈⡇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠻⠶⠶⠶⣶⡿⠶⠶⣤⢤⣼⠷⠤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⠀⠀⡠⠂⢐⡦⠀⣠⣿⡶⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠻⠁⠦⠄⠐⠒⠂⠉⠀⠘⠆⠠⠁⠀⠈⠉⠉⠉⠩⠉⠙⠛⠛⠛⠛⠙⠉⠝⠓⠲⠶⠟⠛⠓⠿⠤⣤⣤⣤⡼⠋⠂⠈⠗⠀⡇
⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠀⠀⠠⠬⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠛⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠛⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣀⠀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⢀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⢿⣿⡿⢁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⣛⣴⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡴⠲⢄⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠎⠀⡤⠶⢄⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢄⣀⡼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡄⠘⣄⠀⣸⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠉⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣠⠎⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠁⠈⠶⠁⠈⠉⠉⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⠉⢿⣧⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⢿⣧⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟⠀⠈⢿⣷⡀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣀⡀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠏⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣦⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⣀⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠛⠛⠿⠿⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠀⠀⠀⠙⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣤⣀⡀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣽⣿⢿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣵⣿⠟⠁⠀⠈⠛⢿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣵⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣵⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣵⣿⠟⠁
⠀⠀⠀⠀⠀⢀⣴⡿⢛⣴⣿⠟⠁
⠀⠀⠀⢀⣴⡿⢋⣴⣿⠟⠁
⠀⠀⣴⣿⢋⣴⣿⠟⠁
⢀⣼⣿⣷⣿⠟⠁
⣾⡿⠟⠉

⠀⠀⢀⡤⣶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠄
⠀⠀⣼⡇⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⠛⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠙⠃⣦⠁
⢠⣤⣬⣥⣤⣤⣀⣘⣟⣀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⣻⣷⣿⣺⣿⣿⠛⢿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⣿⣿⣿
⣽⣿⣿⣾⣿⣿⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⡆⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢰⣿⣿⣿
⢺⣿⣿⣿⣿⣿⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢠⣶⠷⠶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⣿⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠉⠻⣷⡀
⠀⠀⡛⣠⣴⣶⣖⣮⣟⠀⠀⠀⠀⠀⠀⠀⠸⢟⣠⡤⣤⣄⡻⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂⣤⣤⣤⣘⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⢠⣿⣱⣿⠟⢉⣍⡙⢿⣧⠀⠀⠀⠀⠀⠀⣴⣿⠟⣙⡙⠻⣿⡆⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⣋⠛⢿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣭⣶⣿⣿⣷⣮⡁
⣿⣧⢾⣿⠀⣿⣿⡗⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⠆⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠁⢾⣿⣷⠈⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣏⢉⣿⣿⣿⡄
⠸⣿⡜⣿⣷⣤⣍⣴⣿⠏⠀⠀⠀⠀⠀⠀⠻⣿⣦⣉⣁⣴⣿⠏⠀⠀⠀⠀⠀⠀⠈⢿⣷⣌⣉⣡⣾⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣼⣿⣿⣿⡇
⠀⠈⠙⠊⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⣠⣦⠤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⣾⡏⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠉⠙⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠈⠙⣿⡄
⠀⣀⣿⣇⣀⣀⣀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⢸⡇⠠⠤⠤⢸⣧⡀⠀⠀⠀⠀⠀⢀⣈⣛⣀⣀⣀⣀⣘⣃⡀
⣾⣤⣽⣥⣤⣤⣤⣽⣥⣼⡆⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣿⡷⠖⠀⠀⠀⠀⠐⢿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇⠀⠀⠀⠀⠠⣿⣿⣿⠁⢈⣿⣿⣇⡀⠀⠀⠀⠀⠀⢀⣺⣿⣿⢛⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣹⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⡃⢸⣿⣿⡏⠅⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⠛⡛⠛⡇⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣾⣿⣿⣿⡀
⠛⠛⠻⠿⠿⠿⠿⠿⠛⠛⠃⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⢠⣴⠿⠿⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠿⠿⢷⣦⡀
⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠘⣿
⢀⣀⣿⣇⠀⠀⡀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠠⠿⠃⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⣿⠆
⠀⣿⣿⣷⣶⢶⣶⣿⣿⣇⠀⠀⠀⠀⠀⠀⣟⡛⣛⢛⡛⣛⠛⡻⡇⠀⠀⠀⠀⠀⢸⣻⢛⡛⣛⠛⡟⢛⢻⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶
⢀⣿⣿⣿⣧⢠⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠠⡇⣿⣿⡇⠤⡇⠊⡧⠌⣿⣿⢸
⠸⣧⣭⣥⣧⣤⣥⣯⣿⡏⠀⠀⠀⠀⠀⠀⣿⣏⣹⣋⣟⣹⣋⣟⡇⠀⠀⠀⠀⠀⢸⣿⣋⣟⣹⣋⣟⣹⣻⠀⠀⠀⠀⠁⢿⣿⣷⣾⣷⣾⣷⣾⣿⣿⠈

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⠶⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡄
⠀⠀⠀⢰⠒⣚⣛⡓⠒⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠋⠿⠀⠀⠀⠀⠀⢹⡟⠀⢠⡐⣆⠀⠀⠀⠀⠀⠀⠀⣸⣽⣽⣽⡇
⠀⠀⢰⠿⣀⠻⣿⠏⣀⠿⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠙⠛⠷⠆⠀⠀⠠⡾⡳⢼⣽⣽⣽⡧⢷⡆
⠀⠀⢸⠀⠈⠓⠶⠊⠁⠀⡃⠀⠀⠀⠀⣿⣧⣿⣿⣿⣿⣤⡄⠀⠀⠀⣼⣿⣿⣿⣀⣀⣀⣀⣀⡀⠀⠀⣠⣵⣼⣿⣼⣿⣧⣧⡀
⠀⠀⠈⠓⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⢀⣠⢀⣀
⠀⠀⠀⠀⣼⣿⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢪⠙⠩⠂⠀⠀⠀⠀⠀⠀⠀⠀⣦⠉⢵⠂
⠀⠀⠀⠰⠏⣿⣿⡿⠻⠀⠀⠀⠀⠀⠀⠀⠀⡘⠑⠑⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠴⠙⢶⣄⠀⠀⠀⢀⣴⡶⠗⢆⣴⡾⠗⢶⣄
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⢸⢹⠀⠀⠀⣿⡏⠀⠀⣿⠏⡆⠀⠀⡟
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⡠⢫⠞⠀⠀⠀⠻⣇⣀⣠⠻⣏⣁⣠⢞⡼
⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⠁⠈⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⡴⢊⣉⡓⠶⢋⡉⠓⢄⠀⠀⠀⠀⠀⢺⠃⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡄
⠀⠸⡅⢿⣿⣿⢠⣿⣿⡇⢸⠀⠀⠀⠀⠀⣿⣇⣸⣿⠀⠀⠀⠀⠀⠀⠀⣐⠻⠁⠇⣀⣀⢀⠀⠀⠀⢀⣤⡄⠀⠻⠿⠃⠀⣀⡀
⠀⠀⠑⢾⣿⡏⠘⢻⣿⡷⠋⠀⠀⠀⠀⠀⣿⡟⢻⣿⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⢳⣿⣿⡿⠒⠀⠀⠘⢻⠉⠀⠀⠀⠀⠐⣉⠟
⠀⠀⠀⠀⠙⢿⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⠉⠉⠀⠀⠀⠀⢠⣼⣧⣤⣤⣤⣤⣤⣷⣤
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣄
⠀⠀⢠⣠⢠⠞⠉⠀⠈⠑⢆⢠⡆⡀⠀⠀⠀⢠⣟⣧⢠⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⢃
⠀⢿⠊⡏⡎⠀⠠⣶⡆⠀⠈⡞⡇⡇⠀⠀⠀⠘⠈⠈⢼⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⣉⡕⠀⠀⠀⠀⠀⠿⠿⠿⠿⠂
⠀⢸⠀⡇⠱⡄⠀⠀⠀⢀⡼⠁⡇⡇⠀⠀⠀⢸⠿⢿⠈⠛⠁⠀⠀⠀⠀⢀⠘⡿⡿⡿⠋⡁⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡆
⠀⠘⠀⠃⠀⠈⠑⠒⠒⠉⠀⠀⠃⠃⠀⠀⠀⠐⠒⠒⠀⠒⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠐⠉⠉⠋⠉⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠦⠀⠹⣿⣿⣿⣿⡿⠋⠀⠶⠶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠈⠻⡿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣦⣀⣴⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⡿⠿⠿⣿⣿⣿⣶⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⣀⠀⠀⢸⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠙⢿⣿⣿⣿⣦⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠙⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⡀⠀⠀⠀⣠⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠁⢒⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⠀⢠⣎⣀⠀⠀⠀⠀⢀⠔⠉⠀⢛⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠀⢝⠂⠀⠀⠈⠙⢲⡀⢠⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡉⠉⠋⠀⠀⠀⠀⠀⠀⢱⣇⣀⡀⠀⠀⠀⡇⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⡀⡰⣖⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⠀⣸⠋⠑⢄⠀⡇⢀⠀⡾⠀⠀⠀⠀⠀⠀⠠⣀⡳⠜⠀⡏⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠈⢀⡀⠀⠀⠀⣰⠃⠀⠀⠈⠷⠋⡠⠈⠴⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣴⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⢤⠠⠒⣓⡄⢠⢀⠤⢡⠒⠢⢴⠃⠀⡠⠀⢀⡠⠊⡱⠷⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠈⠖⢥⠀⠀⢠⠃⠀⠀⠀⠑⡏⣀⠴⢋⠔⠙⠦⠤⠊⢀⡀⠀⠀⠀⢀⠔⢲⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢽⡖⠨⡷⠔⠂⠄⠀⠀⡾⠀⠀⠀⢀⣠⢼⡥⢚⡁⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⡠⠎⢳⣀⣠⠗⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠃⠠⡀⠔⠀⠐⢦⣀⣇⠀⡠⠞⠉⠀⢀⡟⡉⢁⡀⠀⠀⠒⠀⠀⠀⠢⡀
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡿⠀⠄⠉⠋⠀⠑⠀⡇⠀⣏⠀⠀⠀⠀⣼⠁⡀⠄⠩⡒⠀⠒⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠒
⠀⢀⡗⡁⠀⠀⠀⣀⣂⣀⣃⣤⣼⢥⣿⣿⣱⠁⣠⣨⣁⡖⡀⣉⣞⣀⣑⣤⡠⠾⠷⠏⠘⠚⣢⠈⠂⠀⢄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣳⡜⢻⡄
⠀⡌⠁⠁⢠⠀⣀⡤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠙⡖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⠉⠂
⠀⡃⠀⠘⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠇
⠀⠿⠲⢶⠃⠀⠀⣠⠔⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏
⡴⠛⠈⠚⠲⡀⢸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣠⣇⣀
⡇⠀⠀⠀⠀⡇⠀⠀⢠⠁⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡆⡴⠁⠀⠀⠈⡷⠀⠀⠀⠀⠀⢈⠀⠀⠃
⠘⠠⠄⣀⠴⣧⠃⠀⠘⢀⡀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⠀⡠⠀⢁⢧⠀⠀⠀⡰⠃⠰⠄⠉⢹⡂⠘⠆⠁⢀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠙⡤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠈⡏⠍⠉⣉⢱⠀⡔⠀⠀⠠⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠈⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡇⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⡀⠀⢰⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⢉⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣟⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⣴⣈⣿⡿⠃⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣴⣶⣾⣿⣿⣤⡀⠀⢀⣿⣶⣾⣿⡿⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡈⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡿⠃⠺⢿⣿⣿⣿⣿⡀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⠻⠿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣤⡄⣨⣿⠻⢿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣤⣄⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠉⠉⠀⠀⠀⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠙⢦⡄⣀⠈⠑⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠂⠄⠁⠋⠔⣲⠶⣤⣍⣙⢒⠢⠤⣤⣀⣀⡀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⡀⠂⠉⢐⠩⢜⠻⢻⢾⢥⣴⣪⣫⡯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠐⡀⠠⠐⠈⡂⠁⢊⡪⠊⠁
⠀⢀⡀⠠⠄⠂⠉⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⡁⠠⠑⢀⣀⡼⠁
⡐⠁⠀⠀⡄⠀⠀⠀⠀⠀⠐⠠⢄⠀⠀⠀⣀⠀⠠⠌⢀⠠⠒⡀⣡⡾⠁
⢷⡀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠈⠐⡦⣤⣤⣈⣄⠐⠈⡂⠁⡽⠁
⠈⠳⡀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⢠⢀⠁⡀⠉⡉⣟⠻⣿⣿⣇
⠀⠀⠙⠧⣀⠀⠀⠀⠀⠁⠢⢄⡀⢀⢂⠢⠰⠀⢲⢠⠀⠀⣀⡄⢻⣇
⠀⠀⠀⠀⠈⠛⠧⣄⡀⠀⠀⠀⠈⠛⠶⣥⣡⡠⠈⡀⠄⠀⢸⠡⠸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠦⣤⡐⠘⡄⠄⠉⠉⢛⠺⠦⠼⡆⡦⡟⠻⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠉⢉⡽⠙⠚⠮⡜⣐⡄⡄⠆⠆⠀⡇⣇⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠏⠀⠀⠀⠀⠀⠀⠈⠉⠈⠘⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡸⠃
⠀⠀⠀⠀⠀⠀⡀⢀⡼
⠀⠀⠀⠀⠀⠰⢀⡞
⠀⠀⠀⠀⢀⢃⡞
⠀⠀⠀⠀⣎⠞
⠀⠀⠀⢼⡯⠤⠄⠀⠀⠀⠀⠄⠀⠤⠠⠄⠀⠒⠀⠀⣬⢭⡍⠙⠁⠘⠉⠋⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣀⣀⠌⠉⠩⠍⠉⠀⠀⠠⠄⠀⠄⠻⠒⠓⠒⠶⠶⠄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⢓⣈⣛⣃⣒⣛⣛⠉⠀⠁⡀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠘⢳⠊⠀⠀⠀⠀⣴⠛⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⡧⠀⠀⠀⠀⠀⢫⠂⠀⢨⠀⠀⠀⢸⠀⠀⢰⡆⠀⠈⡟⠁⠀⢱⠁⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⠀⠀⢸⠀⠀⢐⠀⠀⠀⣇⠀⢰⠁⠀⠀⠀⠳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⣁
⠀⠀⠀⠀⠀⠀⡴⠲⣄⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣤⣼⣿⣤⣤⣤⣼⣷⣤⡄⠀⠀⠀⠀⢸⣏⣉⣉⣛⣀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠓⠚⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⡄⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡤⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⠁⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠴⠋⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⠀⠀⢸⡗⠒⠒⠪⠂
⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⠔⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠶⢶⡶⠶⣶⠶⠶⢶⡶⢦⠶⢶⡶⠶⣶⠶⢶⡶⠦⠼⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠇⠀⢸⠀⠀⢸⠀⢸⠀⢸⡄⠀⣷⠀⠘⢄⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣹⠦⠤⠤⠴⠃⠀⠀⠿⠂⢰⠁⠀⢸⠀⠈⠁⠀⣿⠀⠀⠀⡇⠀⠀⠀⠸⣅⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⢠⠞⢦⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⡿⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠁⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⡀⠀⠀⠀⠈⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⢀⣤⣶⣿⣿⡿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠻⢿⣿⣿⣶⣤⡀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠀⠀⠀⢀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣦⡀⠀⠀⠀⠙⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠈⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⢠⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡄⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿
⢀⣀⣀⣀⣀⣿⣿⣧⣀⣀⣀⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣀⣀⣀⣼⣿⣿⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⢀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠙⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠙⠿⣿⣿⣶⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡄⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣤⠄⣀⣀⣀⣀⡀⣀⠀⣠⠄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠠⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠛⠀⠳⣤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⢸⠀⣿⢦⡾⣇⠀⣧⢀
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡆⢹⠀⠀⢻⡆⣿⠉
⠀⠀⠀⠀⠀⠐⠛⠗⠀⠀⠉⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠃⠘⠃⠀⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠕⠈⠉⠉⠓⢌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡤⠤⡤⠤⡮⠀⠀⠀⠠⠀⠬⣮⣇⣤⡤⢄⡠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⠔⠁⢀⠇⡸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⣆⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⢮⠥⡽⠀⠀⠀⠀⠀⠀⠀⠀⠮⡅⡽⠂⠈⠢⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡾⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⠀⠀⠀⢿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⣄⢾⣼⡷⠀⠀⠀⠀⠀⠀⠀⠰⣞⣐⡳⠀⠠⠔⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠈⡎⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⠀⡰⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠛⠛⠒⠒⠒⠂⠀⠀⠀⠐⠓⠓⠒⠓⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⣿⠙⢱⡎⢹⡆⠀⠀⠀⠀⠐⠒⡶⠒⠂⢶⢲⠀⠀⢀⣶⡀⢰⡀⢰⡇⢺⠉⠀⣴
⠀⠀⠀⠀⠀⠀⢿⠀⠸⣇⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡿⣄⠀⢼⠗⢷⠀⢳⣼⠁⢸⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠃⠈⠀⠛⠀⠈⠀⠀⠋⠀⠘⠋⠁⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠉⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣀⣠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠒⠂⢼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢁⡾⠃⡠⡶⠄⠘⣿⣟⣠⡀⠂⠀⠠⣄⠀⠘⠫⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⡤⣨⠟⠀⢤⡓⢿⠉⠉⠁⣴⣜⢁⡟⣠⣦⠄⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⣆⡠⠀⠀⠀⠙⠦⠙⠀⡀⣴⠗⣤⠖⠋⠾⠛⢁⣴⡿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣅⠶⣣⠘⢛⣀⣀⠀⠻⣿⣆⠸⡇⠖⢋⣠⡜⠛⢥⡶⢦⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣄⠈⠳⡤⢈⣁⣠⣤⣤⠀⠀⣀⠠⣕⣄⣛⣉⣀⠤⠖⣪⠷⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠛⠁⠶⢶⡄⠀⠀⢩⣡⡘⢦⣉⣥⣄⠰⠋⠁⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⣣⡀⠙⠛⠁⡤⠶⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣠⣤⣀⣠⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣽⣽⣽⣭⣭⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣯⣭⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢀⣠⣤⣤⡀
⢀⣴⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⠀⢠⣿⠋⠉⠙⣿⡆
⣾⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠘⣿⣦⣀⣠⣿⠃
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣶⣶⣶⣶⣾⣿⠁⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠉⠉⠉⠻⣷⣤⣶⡿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣤⣶⣶⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡏⠀⠈⢻⣷
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣷⣤⣤⣾⠏
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⢿⣷⣄⣀⣀
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⣿⡆
⠙⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⣿⣇⠀⢀⣼⡟
⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣈⣻⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠳⣆⢤⣠⠤⢔⢂⣰⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠠⠀⡀⢀⠠⣄⣠⣤⣿⡚⡟⢁⡰⡎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡡⠒⠉⢈⠹⠎⠉⠀⠀⠙⢾⣥⣹⡀
⠀⠀⠀⠀⡀⠠⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢢⣴⠒⠒⠿⠋⠑⠄⠂⠙⣿⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠤⠄⢢⡀⠇⠀⡰⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠉⡆
⠀⠀⠀⡀⢠⠔⠚⠒⠒⠒⠒⠻⠟⠀⠘⠷⠯⠤⢒⠷⢾⣶⣦⠢⣑⡀⠀⠴⠀⠐⠄⠀⠀⠈⣴⡠
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠄⠀⢈⠏⠀⠀⠀⠀⠀⠐⣄⠀⡔⠜⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢸⡄⠀⠀⠐⠒⠖⠙⠐⡥⠤⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢸⠄⠀⠀⠀⠀⠀⠀⢆⠸⡐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠸⠀⡀⢀⢀⠀⢀⠠⢸⢐⠌
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⣷⠶⠤⠤⠬⠠⠄⣊⡜⠋
⠀⠀⠀⠿⠦⢤⠤⣄⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⢀⢒⠂⢄⣔⣵⣷⣭⠨⠤⠴⠚⠉⠀⣀⣠⡠⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠙⠃⠒⠒⠻⠚⠛⠋⠋⠁⠀⠠⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠈⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠱⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⣤⠴⣤⡁⠂⢄⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⡤⢾⣦⡰⡟⠻⡉⠉⠙⢦⢌⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡕⠁⠈⠁⠈⠛⠒⣻⢏⠎⣂⢰⠀⠀⣼⠀⠑⡈⢂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠒⡄⠀⠀⠀⠀⠎⢀⠇⠯⡀⠘⠔⠁⠀⠀⠀⠈⡄⢂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡔⠀⠀⠀⠀⠀⠉⠀⠖⠺⠀⠀⠀⠀⠀⠸⣮⠿⡀⡀⢠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⢀⠀⠀⢄⠀⠀⢀⣀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢐⠵⠾⡇⠁
⠀⠀⠀⠀⡀⠨⠰⠉⠉⠉⣷⠨⠀⡀⠈⠓⢄⣇⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡄⡀⢀
⠀⠀⠀⠎⠇⠀⠀⠀⠀⠀⠀⠀⠀⡉⡄⠀⠀⠀⠙⠶⠒⠂⠀⢀⠀⠀⠀⠀⠀⠑⠂⢲⢁⠁⠈
⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⡄⠵⠤⠤⠤⠤⠤⡧⠾⠴⠓⠲⢟⣚⡛⢍⠉⠙⠛⠛⢋⠆
⠀⠀⢰⠀⠘⠠⡷⠀⠀⠀⠰⡧⠈⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡏⠈⢻⡆⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠘⣄⡀⠀⠀⠀⠀⠀⠄⠁⠰⢢⢀⠀⠀⠀⠀⠰⠀⠀⡠⢄⠙⠒⠚⠁⠃⠀⠀⠀⠀⡔
⠀⠀⡄⠀⡠⠐⠒⠒⠒⠒⠒⠒⠠⠀⠀⡷⢤⡁⡐⠂⠀⠒⠀⠒⣘⡜⡠⠀⠡⠄⠀⠬⠅⠀⠀⠈⢆⠀⠀⠠⠀⠀⠀⠀⠀⡠⠂
⣆⠀⠀⠀⠃⠤⠤⠤⠤⠤⠤⠤⠄⡆⠀⠂⢀⠧⠀⢀⡀⠀⠀⡀⠠⠖⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⢀⠦⠀⡖⠐⠀⠂⠈⠀⠠
⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠰⠠⠀⣠⠑⠶⣷⣴⡀⠀⠀⠀⠀⠀⡔
⠣⣘⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⡠⠂⠐⠒⠓⠒⠒⠓⠒⠀⢀⠀⠀⠀⠀⠀⢀⡤⠊⠈⠁⠐⠃⠀⡀
⣀⣈⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣌⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀⣎⣶⣶⣶⣦⣬⣤⣦⣤⣖⣒⣒⣚⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⣋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠛⠛⠛⠛⠛⠛⠋⠉⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⡄⠀⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⠀⠀⠀⠀⣴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢹⠉⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⡏⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡄⣀⠂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠉⠁⠈⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠶⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣰⡦⣶⣤⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠸⠷⠿⢿⣯⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡌⢳⢛⡅⠀⠀⠀⠀⡄⠀⣴⢳⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⢰⣤⣦⣤⢠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡞⢣⡿⠋⠉⢙⡟⠀⠀⠀⠀⠁⠀⠹⠟⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠹⠿⠿⠇⠿⣼⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣡⣤⣤⣼⣅⠀⠀⠈⡠⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⣶⣴⣆⣶⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠀⠀⣀⡀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠘⠟⠿⠿⠿⠦⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠶⠒⠀⡀⠀⠰⣷⠘⠟⠉⠉⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣴⡆⣶⣰⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢁⣃⣀⠁⠀⠁⠁⠘⣛⣻⣛⣂⣘⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠸⠿⠧⡿⢽⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠳⠖⠐⠖⣿⠩⠏⢉⡿⠛⠛⠛⢻⣷⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠁⠀⠀⠀⣿⣀⣤⣈⠃⣾⣶⣷⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿⡿⠀⠀⠀⠀⣿⠻⡿⠿⠀⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣴⣾⣷⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠛⠛⠛⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠁⠈⠓⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣴⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⡟⠃⠈⢙⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠉⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠇⠀⠀⠈⠘⢃⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠛⢻⡟⠛⠻⠛⢿⠚⢇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣼⠟⠛⠛⠛⠛⠋⢿⡌⠷⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⢃⡺⠃⠀⠀⠀⠀⠀⠀⠀⢹⡆⢳⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⠄⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣦⣶⣾⠃⡏⠀⠇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣭⠀⠒⠀⠀⣀⠀⢀⣀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⣧⣶⣴⡆⢻⡆⣽⠀⢸⣿⠀⢸⣿⠀⠘⠻⠀⠙⣿⠀⠀⣴⡌⣿⢠⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠉⠀⠉⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠿⠸⠿⠇⢾⣷⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⣄⢠⡄⣠⡄⠠⣦⠀⣴⡆⠀⣴⣦⠀⢰⣶⡀⢠⣤⠀⣠⣄⠀⣤⢀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡿⢼⡇⣿⡇⠀⣿⠀⢸⡇⠀⣿⣿⠀⢸⣿⡇⠈⢿⡆⣿⣿⠀⠐⠻⠟⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⢀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠉⠉⠀⠀⠀⠈⠉⠛⠻⣿⣿⣟⢿⣿⠙⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣍⢎⡳⡅⢬⡛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡙⠀⠀⠁⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⢟⣿⢿⣿⡿⠛⠋⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠘⡄⢳⡘⡆⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣟⣾⢣⠟⣰⡿⠋⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⡇⢠⠈⣧⠸⠌⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⣾⣿⡎⠸⢇⠇⣸⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⡇⢸⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡟⠃⡿⢠⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣷⢠⠀⣦⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⣿⡟⠇⠀⠹⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠇⢸⠀⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⠀⣼⠀⡿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢻⣧⠳⡄⢄⢦⡹⣧⣀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⢰⣿⡏⣸⠇⡌⢸⡇⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣾⣷⣝⣿⣿⣷⣶⣶⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⡏⠸⠃⣾⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⡿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⣸⠁⣤⢀⡇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢰⡏⢸⠃⡀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢁⡿⢠⡆⣸⢁⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⡼⢡⡿⢡⡏⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⣼⢁⣿⢡⣟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢸⡇⣼⢣⡿⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢡⡿⠘⢋⡾⣱⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢣⡟⣱⠇⣼⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠛⣱⡟⣰⢣⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢡⡾⣰⡟⣰⢏⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢡⡿⢡⡿⣰⠏⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢁⡾⢁⣾⣱⣟⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⡾⢡⡿⢹⣟⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⣰⡿⢡⡿⢡⣿⢁⢉⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⣴⡟⣱⡿⢡⡿⢁⢣⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⣴⡟⣸⡿⣡⡿⣹⣯⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣏⣴⣿⣿⣯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⠀⢠⣍⣁⣈⣉⣭⣉⢁⣀⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠶⠆⠰⠶⠶⠶⠶⠶⠶⠆⠀⠶⠶⠶⠶⠶⠶⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡒⠒⣚⣛⣛⣛⣛⣒⣒⣛⣛⡒⠚⠒⠛⢛⣛⣚⠛⣓⣚⢛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣉⣉⣉⣈⣉⣉⣉⣉⡉⠉⣉⣉⣉⣉⣉⣉⣁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠶⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠉⠉⠙⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢁⣤⣾⣿⣿⣿⣶⣤⣤⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣴⣿⣿⠟⠉⠀⠀⠀⠙⠻⣿⣷⡀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠸⣿⣧⣶⣶⣿⣿⣿⠿⠿⠿⠿⠿⠷⠶⠿⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣀⣽⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣮⡙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢴⣗⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⣠⠄⠀⠀⠀⠀⠀⠀⡀⡀⠀⣀⡤⠶⠒⠒⠶⠶⣤⣤⡽⠟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣄⡛⠂⢀⣀⣠⡴⠖⠋⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠈⠉⠉⠉⡩⢁⣀⡠⠀⠀⠀⣾⡡⠀⠄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⣠⡾⠞⠋⠁⠀⠀⠀⠀⠶⠶⠚⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡁⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠅⡟⢹⠀⠀⢠⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⢷⡄⠀⠀⠀⣠⠞⠁⠀⣰⣶⣿⣶⣶⣖⣛⣋⣉⣤⠤⣠⣧⠸⠀⠀⢸⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠈⠛⢦⣤⠞⠁⠀⠀⠀⢿⣶⣿⣿⠟⠋⠉⠉⠀⠀⠀⢻⣿⣦⣄⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⢠⠆⠀⠀⢈⡿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⡀⣾⣋⡁⠀⠀⠀⠀⠀⣿⣗⠀⠀⠀⠈⠛⠛⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣮⡙⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣽⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣤⣴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠞⠻⠿⠛⠛⠛⠛⠛⠛⠉⠛⠉⠁

⠀⠀⠀⠀⢀⣠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠐⠒⠒⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠉⢹
⠀⠀⠀⣠⠋⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⠤⠴⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠄
⠀⠀⢠⡇⠀⣰⠒⡆⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢆
⠀⠀⠈⠇⠀⠈⠉⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⢧
⠀⠀⠀⠀⢢⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠒⠒⠒⠒⠐⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣀⣀⣀⣈⡧
⠀⠀⠀⠀⠀⣇⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡥⠤⠤⣤⣤⣤⠤⠤⠤⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠹⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⣆⠉⢁⡠
⠀⠀⡌⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⠴⠒⢄
⠀⢀⡇⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠙⡀⠀⣨⠋⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⣯⠀⠀⣰⠇
⡆⠉⠳⠤⠴⠞⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠋⠉⢻⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣧⣈⠹⠛⣻
⠐⠂⠤⣤⣤⠄⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠈⠀⠁⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⢀⣰⠃
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠈⢃
⠀⠀⠀⢿⠉⠉⠉⠉⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⠉⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⣿⠁
⠀⠀⠀⠘⡆⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠖⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸⡇
⠀⠀⠀⠀⡡⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣠⠞⠁⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢄⢀⣾⢣⣄
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠷⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠉⠁⠀⠨⢷
⠀⠳⠆⠀⠀⣶⠶⣶⠀⠀⠰⠾⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⢀⣀⣀⣀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠤⣤⣤⣤⡤⠴⠊
⠀⠀⠀⠀⠀⠘⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⡠⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠂
⠀⡤⢊⠄⠀⠀⠀⡀⠠⠀⠀⠀⡡
⠀⠄⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⠀⡆⠄
⠀⠠⠀⠀⠀⠀⠀⠈⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⣿⣿⣿⣿⣿⡇⡄⠀⢀⣶⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣿⣿⣿⣷⠁⠀⠀⢿⣿⣿⣿⣿⣿⡾
⠀⠀⠀⠙⠿⠿⠿⠛⠁⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡾⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠟⠃⠢⡤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠛⠛⠛⠛⠛⠠⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⣠⡴⠟⠛⠛⠛⠳⢦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⣠⡤⣤⠾⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣠⡾⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠶⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⡀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣀⠀⠀⠀⠀⠀⣸⡇
⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣶⣦⣄⣰⡿
⠀⣧⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣘⠋
⠀⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡿
⠀⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣰⠁
⠀⢸⡆⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡿
⠀⢸⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼
⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⣿
⠀⠀⢁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢠⡇
⠀⠀⢸⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇
⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⢀⡎
⠀⠀⢸⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇
⠀⠀⢸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡜⠁
⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⢰⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣼⠁
⠀⠀⠀⢸⡄⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡿
⠀⠀⠀⢸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣴⠁
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⡟
⠀⠀⠀⠸⡇⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣼
⠀⠀⠀⠀⣠⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡿
⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣼
⠀⠀⠀⠀⢿⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠛
⠀⠀⠀⠀⢸⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇
⠀⠀⠀⠀⠀⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢀⡜⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⡜⠁
⠀⠀⠀⠀⠀⠹⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣧⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⡏
⠀⠀⠀⠀⠀⠀⠻⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣶
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⠀⠀⠀⢀⣀⣀⡠⠤⣀⣀⣀⣀⠀⠀⠀⠀⠀⡸⠁⢀⠟
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣤⡤⠟⠉⠉⠀⢀⣀⠀⠉⠉⠉⠻⢷⣄⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠋⢀⡴⠋⠀⢀⣤⠛⠋⠀⠉⠉⠛⠓⣦⡄⠀⠙⢷⣤⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢸⡏⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡈⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠏⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢄⣀⠈⠓⠤⠤⠤⠤⢤⠼⠋⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⢄⣀⣀⣀⣀⣀⠶⠟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠋⠈⠛⠛⠋⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣠⣀⣄⣠⣤⣤⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⢠⢂⡐⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢈⣌⡠⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠎⡀⢀⢔⠜⠀⠀⠀⠀⠀⡰⠃⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠃⠠⣀⡡⠊⠀⠀⠀⠀⠀⠠⠀⠡⠊⠠⠂⠀⠀⠀⠀⠈⠢⡡⠠⠀⠀⠀⠀⠀⠀⠀⠢⠤⠘⠀⠤⠂


⠀⠀⠀⠀⠀⢀⡴⠶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢄⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠧⠀⠆⠠⠅⠀⠀⠀⠀⠀⠰⠌⠉⠉⠉⠁⠀⠀⠀⠀⠈⠂⡈⠐⢡⠂⠀⠀⠀⠀⠀⠀⠃⠤⠤⠃
⠀⠀⠀⠀⢐⡄⢀⡄⠀⡈⠀⠀⠀⠀⠀⢠⣡⣕⣢⡖⠀⠀⠀⠀⠀⠠⢔⠁⠒⡈⠄⠀⠀⠀⠀⠀⠀⣢⠤⢤⡄
⠀⠀⠀⠀⠈⠛⠋⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠇⣶⢲⡄⠃⠀⠀⠀⠀⠀⣾⡆⠀⠀⣿⠆⠀⠀⠀⠀⠀⢗⢄⠠⢪⢆⠀⠀⠀⠀⠀⠀⠀⠛⠘⢆
⠀⠀⠀⠀⠀⡇⠙⠚⠁⠀⠀⠀⠀⠀⠀⠈⠑⢲⠐⠙⠀⠀⠀⠀⠀⠀⢀⢕⠕⠈⠁⠀⠀⠀⠀⠀⢖⠁⡲⠀⠛⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠋⠈⠈⠋⠀⠀⠀⠀⠀⠀⠓⠀⠀⠒⠁⠀⠀⠀⠀⠀⠐⠁⠃⠒⠂⠁


⠀⠀⠀⠀⠀⠀⠒⠘⠢⠀⠀⠀⠀⠀⠀⠒⠀⣀⣠⡀⠀⠀⠀⠀⠀⠠⠽⢄⠀⡸⣀⠀⠀⠀⠀⠀⠠⣄⠓⠚⡦⠄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⠸⠀⢸⠀⠀⠀⠀⠀⠀⠚⡐⠆⠘
⠀⠀⠀⠀⠀⠠⡁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠲⠃⠐⠳⠇⠀⠀⠀⠀⠀⣤⠀⠈⢰⠊⠀⠀⠀⠀⠀⠤⠚⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⣄⡀⠀⢀⡀⠀⢻⠁⠀⠀⠀⣼⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢸⡇⠰⣏⡽⠀⢸⠀⠀⣠⠞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣇⣀⣼⣀⡀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣹⠆⠀⠀⠀⠀⣸⠗⠋⠁⣀⣀⣀⡀⠉⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠞⠀⠀⢠⡾⠋⠉⠉⠻⣦⠀⠀⠈⢦⣀⣀⣠⡖⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠉⢩⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢧⠀⠀⠙⠋⢀⣰⣋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣹⠶⠶⠶⠶⠶⣾⠀⠀⠀⢰⣿⣷⣶⣶⣶⣶⣿⣷⡄⠀⠀⢸⡟⠛⠛⠛⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽⠀⠀⠀⢸⣿⣿⣿⡁⣹⣿⣿⣿⡇⠀⠀⢸⠧⠤⠤⠤⠤⠤⣞⣹⠄
⠀⠀⠀⠀⠀⠀⠀⣴⠓⡶⠋⠉⣀⣀⠀⠘⣆⠀⠀⢸⣿⣿⣿⣇⣿⣿⣿⣿⡆⠀⢀⣞⣀⣀⣀⣀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠧⠼⠛⠛⠛⢦⡀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠁⣠⠞⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣤⣀⣀⠀⢀⣀⣠⠴⡞⠁⠀⠀⠀⠀⢰⡋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠉⠉⣽⠀⠀⡗⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢸⠀⢠⠞⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢲⠋⠀⠀⢀⡾⢦⠈⠛⠁⠀⡇⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠉⠹⠤⠏



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⣠⣀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡴⢒⠒⠢⡑⣇⠀⠀⠀⡄⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠁⡀⠀⢸⠀⠘⠁⢠⠄⠐⢂⣜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠈⠳⠤⠤⠤⠀⠋⠀⡁⣠⡆⠀⠀⡿⠞⠋⣀⣀⢠⠴⢺⡇⠀⠀⠀⣀⠰⠄
⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⢀⠈⠀⠀⣴⡾⠓⠂⠁⠀⠀⣦⠉⣿⠀⠀⢡⡇⠀⣠⡇⠀⠀⡇⠀⠁⠀⢁⠀⠀⢰⣉⣠⠖⢢⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡠⠖⣇⠀⠀⠀⠀⠀⡒⠚⡄⢸⠇⠀⣿⣠⣤⠾⠟⠛⠉⠁⠀⠀⣇⠀⠀⠀⢸⠃⠀⢈⠉⠀⠀⠀⠁⠀⠀⠀⣀⡆
⠄⠀⠀⠀⠀⠲⣆⢹⠀⢀⠸⠦⠒⠋⠀⠀⠒⠀⢐⣁⣀⣅⡾⠞⠚⠋⠄⠀⠀⠀⠀⠀⢤⡄⠂⢹⣀⣤⡴⠟⠓⢀⠀⠀⠀⠀⠀⠀⠀⠰⣈⡴⠗
⠀⠀⠀⢠⠤⣒⣺⣿⠄⢈⠀⠀⠀⠀⢀⣀⠤⠴⠈⠉⠻⠭⠽⠓⠒⠀⠀⠀⠀⠀⠀⢠⢴⠀⢽⣟⠉⠀⡀⣤⢤⡀⠀⠀⠀⠠⠀⠀⠀⠛⢅⡀
⠀⠀⠀⠘⠛⠁⠀⠈⠁⠈⠀⣀⠠⢴⣧⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡏⣼⠰⠿⠷⠠⢴⠿⢶⣯⣁⠠⢐⡠⠔⢀⣠⡴⠠⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠒⢊⡩⠤⠒⠊⠉⠁⠀⠀⣴⡊⡏⠀⠀⣤⣴⠾⠄⠀⠀⠀⠠⠷⢛⠒⠒⠂⠐⠐⠴⣶⠿⠟⣀⠼⠯⣾⣿⣿⣿⡀⣿⣿⣧
⠀⠀⠀⠈⠀⠀⠀⡇⠰⠉⠀⠀⠀⠀⠀⠀⢀⣀⠄⣿⣆⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⢀⠈⠉⠂⠄⠈⠂⠁⠠⠀⠁⠀⠀⢴⣻⡟⠋⠁⠈⠙⠉
⠀⠀⠀⠀⢀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡶⠛⠋⠓⠦⠟⠁⠀⢃⡀⠐⠢⢁⣀⠄⠒⠊⠸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢖⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡠⠄⠀⢁⡇⠄⠀⢰⠇⠀⣀⡴⠟⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠸⢀⣀⡤⠶⠃⠉⠀⠚⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣠⠴⣄⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡾⠁⠀⣸⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢀⡞⠁⠀⢀⡏⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣀⠀⠀⠀⠀⣠⠟⠀⠀⠀⡼⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢳⡶⠖⠛⠁⠀⠀⠀⣸⠃⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡾⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣄⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢫⡾⠃
⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⣰⠃⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠋
⠀⠀⠀⠀⠀⣰⠃⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠟⠁
⠀⠀⠀⠀⢠⠏⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁
⠀⠀⠀⢀⡟⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁
⠀⠀⠀⣾⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁
⠀⠀⣸⠇⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣠⠞⠁
⠀⢠⠏⢀⣏⣀⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠞⠁
⠀⡿⠀⣼⣋⣡⣤⣤⣄⡉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠟⠁
⢸⠇⢠⡟⠉⠁⠀⠀⠈⠙⠳⣄⠈⢻⡄⠀⠀⠀⣠⠞⢁⡴⠋⠁
⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣿⠀⣠⠞⢁⡴⠋
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠟⢁⡴⠋
⢿⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⣿⡴⠋
⠘⢧⡈⠳⣤⣄⣀⣀⣠⣤⠞⠁⣠⠟⠋
⠀⠈⠛⠶⢤⣈⣉⣉⣁⣠⡴⠟⠁
⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣈⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠋⠉⠉⠉⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠋⠉⠉⠉⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣷⣤⣤⣤⣤⣤⣤⣼⣿⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⡿⠿⠿⠿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⢁⣠⣤⣤⣤⣤⣄⣈⠙⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠛⠓⠶⢤⡀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⣠⣾⠿⠛⠉⠉⠉⠉⠛⠿⣷⣦⡀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢰⣿⠋⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣿⡏⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣴⠶⠿⠿⠶⣦⣄⠀⣸⠃
⠀⠀⢠⣶⣿⣷⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣾⠋⣠⣴⠶⠶⣦⣄⢙⣷⡇
⠀⠀⣾⡇⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡄⢸⣿⠀⠀⠀⠀⠀⠀⣼⡇⢰⡟⠐⠶⠤⠶⢿⡏⠹⣷
⠀⠀⣿⡇⠰⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠆⢸⣿⠀⠀⠀⠀⠀⠀⢻⡇⢸⣧⠀⠀⠀⠀⣸⡇⢰⡿
⠀⠀⣿⡇⢘⣛⣛⣉⣁⠀⠀⠀⢀⡶⠛⠛⢶⡀⠀⠀⠀⠀⢐⣚⣛⡃⢸⣿⠀⠀⠀⠀⠀⠀⠈⢿⣄⠙⠿⠶⠶⠾⠋⣠⡿⠃
⠀⠀⣿⡇⢨⡭⠥⠤⠄⠀⠀⠀⢸⣇⠀⠀⣸⡇⠀⠀⠀⠀⣭⣭⣭⡅⢸⣿⠀⠀⠀⠀⠀⠀⢀⣾⠏⣰⡶⣶⣴⡶⠿⠋
⠀⠀⣿⡇⠰⠒⠒⠒⠒⠀⠀⠀⠀⣿⡅⢸⡏⠀⠀⠀⠀⠐⠲⠶⠶⠆⢸⣿⠀⠀⠀⠀⠀⢠⣾⠋⣴⠟
⠀⠀⣿⡇⢈⣉⣉⣉⣉⡁⠀⠀⠀⢺⣇⣸⡟⠀⠀⠀⠀⣈⣉⣛⣛⡃⢸⣿⠀⠀⠀⠀⢠⣿⠃⣼⠏
⠀⠀⣿⡇⠰⠤⠤⠤⠤⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠠⠤⠤⠬⢭⡅⢸⣿⠀⠀⠀⣰⡿⢁⣾⠋
⠀⠀⣿⡇⢘⣛⣒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠲⠆⢸⣿⠀⠀⣴⡟⠁⠺⣧⣄
⠀⠀⣿⡇⢨⣭⣭⣭⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⣉⡁⢸⣿⠀⣼⡟⢠⣄⡀⠀⢉⣿⠆
⠀⠀⢻⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡿⠀⠙⠿⠛⠙⠻⢶⣾⠃
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣀⡀
⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣄⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣠⣿⣿⣿⣿⣷⣶⣤⣀
⠀⣠⣴⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣦⣄
⣴⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿
⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿
⠙⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿
⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⢹
⠀⠀⠀⢀⡴⠉⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⣤⣤⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀
⠀⠀⠀⡾⠀⠀⡤⢤⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇
⠀⠀⠀⣇⠀⠀⠳⠜⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠘⢦
⠀⠀⠀⠈⢆⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⢀⣤⠄⠤⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢃⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠖⠒⠶⠐⠛
⠀⠀⠀⠀⠀⠘⡄⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠉⠉⠺⡖⢳⠀⠒⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆
⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⢪⡁⠀⣉⠂
⠀⠀⢨⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣤⡴⠂⠠⡀
⠀⠀⡌⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⡀⠀⢀⡟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⣿⠀⠀⢀⡗
⡔⠘⠳⣤⣤⠤⠟⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠍⠑⠺⡇⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⣅⠈⢳⠖⢻⡇
⠱⢤⣀⣀⣀⣀⢠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠈⠀⠘⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠉⠉⠀⢀⡾
⠀⠀⠀⠘⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣤⣤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠶⠚⠋
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⣀
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣨
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢹⡅
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠘⣧
⠀⠀⠀⠀⣸⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢘⡇
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⣄⠀⣸⠃⣤⡀
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠷⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⠉⠉⠀⠌⣻
⠀⠾⠤⠤⠤⢤⣤⢤⡤⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠚⠋⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⣀⣀⣀⣀⣀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⢣⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣯⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⢴⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡨⡀⢀⢁⠀⣠
⠀⠀⠀⠀⠀⣴⣯⣿⣟⣻⣷⣯⣷⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠁⣡⠞⣋⣑⠷⡁⠈
⠀⠀⠀⠀⠀⢸⣿⣿⠈⠅⢹⣿⡟⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢧⢾⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣌⠻⢟⣴
⠀⠀⠀⠀⠀⠀⠀⠙⠯⡽⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠊⠙⠉⠑⠁⠾


⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⡇⠀⠀⠀⡔⠲⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⢠⣤⣤⣤⣤⣤⣤⡄⢸
⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⠇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠠⠤⠤⠤⠤⠄⠀⢸⠀⠀⠀⠀⡇⠀⠈⠉⠀⠈⠉⠁⠀⢸
⠀⠀⠀⠀⠷⠤⠤⠤⣤⣤⡦⠤⠤⠼⠀⠀⠀⠀⠷⠤⠤⢤⣤⣤⡦⠤⠤⠾⠀⠀⠀⠀⠷⠤⠤⢤⣤⣴⡤⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⣶⣶⣶⣶⢲⡆
⠀⠀⠀⠀⢰⣶⣶⣶⣆⣀⣀⣀⣀⡀⠀⠀⠀⠀⢰⣾⣗⣶⣶⣶⣶⣺⣇⡀⠀⠀⠀⠀⢰⣶⣶⣶⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠉⠁⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⠉⠉⣿⣿⣿⡇⠀⠀⠀⠀⢸⣷⣤⣥⣴⣦⣬⣤⣾⡇
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣉⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢀⡐⠈⠏⠀⠀⠀⠀⠀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠋⡰⠁⠀⠀⠀⢀⠘⢝⣂⢴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⣶⠂⠍⠀⣵⣀⠀⠀⠀⠈⠿⠁⠀⠂⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠚⢀⣼⣿⣿⣿⣦⣄⣠⠂⠣⡈⡴⠑⢄⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⣠⣿⣿⣿⣿⣿⣿⣿⠀⡀⢄⠔⠥⠂⠰⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠉⠑⠢⡀⣴⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡿⢠⠀⠀⠂⠀⠀⠹⠤⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠺⠟⣿⡀⠘⡟⡿⠁⢀⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣐⠀⠈⠶⢤⠢⡀⠀⠆⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠀⢸⠇⠤⡉⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠉⠓⢁⣀⣘⡆⠀⠘⢀⠜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙⠀⠀⠉⠚⠠⠟⠀⠀⠟⠛⠉⠘⣿⣿⣿⣿⣿⡷⠀⠀⠀⠎⠉⢹⣿⡖⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⢂⠀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉⢁⠀⠀⠆⠀⢠⣤⠀⠀⡻⢃⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠄⠁⠀⠀⠀⠈⠀⠀⢢⠀⡇⠀⣠⠀⠀⢀⠠⡀⠈⡄⠀⢦⠀⠀⠐⠶⠔⠁⠸⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⢊⠠⠀⠀⠂⢠⠀⠀⠀⠀⠀⢈⢢⠁⠞⠁⠀⢀⠌⠀⢳⡀⠀⠀⠀⠇⠀⡌⢀⠐⠀⢀⡄⠀⢠⣃⣠
⠀⠀⣀⣤⣤⣤⠃⠀⢠⣼⣀⠇⠀⠀⠀⢀⠀⣮⡜⠀⠀⠀⠀⡌⠀⠀⠀⠱⡄⢀⡀⣀⠐⠀⣰⠀⠀⢀⢙⣈⢉⣴⠟
⠀⠘⠻⠿⠿⣧⣤⣶⣿⣿⣿⠀⠒⣻⣿⠁⢀⣿⡇⠀⠀⠀⣼⣿⣷⣦⡀⠀⣈⠃⡈⠀⡤⠊⠑⢄⣀⣤⣾⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⢠⣿⣷⠀⢾⣿⡇⠀⠀⢰⣿⣿⣿⣟⡁⠀⠀⠀⠀⠂⠋⡆⢠⠞⠛⢿⣿⣿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠻⢷⣾⠛⠇⠀⢀⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠘⠴⠃⠘⠀⣠⠂⠈⡁⠀⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠢⣄⠒⣾⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠐⠁⢀⠀⠈⠲⢤⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠥⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⡁⠰⠃⡡⠄⠀⠀⠀⠀⠑⣄⡈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣱⠀⠤⠄⠁⠀⠀⠀⠀⠀⠏⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠢⣆⠐⠢⠀⢺⠦⠄⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣌⢌⡂⠄⠊⢈⠀⢑⡢⠀⠀⠀⡄⠑⠂⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠩⠂⠀⠁⠀⠙⠀⢌⠃⠄⠑⠀⢀⠡⠘⠙⠀⠀⠀⠈⠀⠀⠐⡄⠀⠀⠠⠖⠂
⠀⠀⠀⠀⠀⠀⠒⠆⠑⠃⠒⠀⠏⠐⠀⠐⢶⠁⠀⡀⠙⢦⠐⠐⡄⠀⠣⠀⠀⠂⠁⠠⠀⠀⠀⠀⣴⠆⠀⠈⠀⢠⣴⡄⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⢂⠂⠁⠀⠀⠀⡅⠀⢀⠀⠁⠀⡉⠀⠁⢅⠠⠄⠂⠋⡀⠀⠄⠀⠐⢀⣀⠤⠾⠽⠂⠢⠐⠀⠈⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⢀⠀⡀⠀⠑⠀⠀⠈⠑⠀⠀⠀⠐⢄⠀⠱⢱⣒⡠⡤⠛⠃⠀⠀⠠⢖⠒⠀⣄⣖⣷⣦⡖⣀⠘⠒
⠀⠀⠀⠀⠁⢠⠀⠀⠈⠀⠀⠰⡠⣀⣀⠀⠂⢥⣤⠄⠤⣌⣤⡄⣀⡼⠖⠒⠚⢁⠀⠈⠛⠁⢀⡀⣒⠋⡈⠛⠙⠛⠛⠓⠾⠃
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠂⠴⠦⠶⠠⠤⠾⠂⠈⠁⠄⠐⠂⠀⠉⠀⠀⠁⠈⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠒⠒⠒⠒⠒⠢⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⠳⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢦⠄⠀⠀⠀⠀⠊⢀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⡀⠀⠀⢂⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠑⡄⠀⠀⢖⠋⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣠⠴⠚⢄⠀⠈⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠐⢁⡤⠒⠉⠀⠀⠀⠀⠁⡀⣜⠖⠢⡀⠸⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢐⠁⢺⠗⠐⠒⠀⠑⠀⡇⢸⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢡⠈⠀⠀⠀⠀⠀⢅⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠄⠒⠒⠒⠒⠒⠒⠒⠒⠶⠖⠒⠒⠒⠒⠂⢄⠀⠀⠀⠀⣇⢀⡀
⠀⠀⠀⠀⠀⠀⢠⢤⣄⣀⣀⠀⢠⠈⠀⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠂⠀⢦⠄⠀⢈⣂
⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠉⠉⠉⠙⢲⠒⠚⠒⠒⡲⠾⠒⠶⡖⠦⠤⣄⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠸⠀⣤⣤⣼⠀⠐
⠀⠀⠀⠀⠀⠀⢘⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠰⠟⡁⢷⡌⠆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡶⢘⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣸⠱⠔⠒⣀⠀⠀⢸⠀⠀⡀⣄⠰⣏⣀⡼⠁⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠔
⠀⠀⠀⠀⠀⡔⠛⠒⠒⠒⠒⠚⠒⢢⣽⠁⠀⠀⠘⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠉
⠀⠀⠀⠀⠀⡇⠀⣀⣔⡖⢄⡀⠀⠀⢸⡀⠀⠀⠀⠐⡀⠀⢀⠁⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⡇⢰⠉⡅⡇⢩⠈⠀⠸⣬⢀⠀⠀⠀⠐⠃⢀⠟⠀⠀⠀⠀⠀⠈⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢘⠁⠁
⠀⠀⠀⠀⠀⡇⠈⠖⣆⡆⡲⠔⠀⢸⣨⣌⢁⠴⠄⠙⣻⠷⣦⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠂
⠀⠀⠀⠀⠀⡇⠠⠤⠬⠭⠤⠤⠄⢨⣇⠈⠀⠀⠀⠀⠋⠀⢈⠀⠄⠀⠀⠀⠀⠒⠒⠒⠲⠖⠶⠶⣶⠒⠒⠒⠒⠓⠊
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⠰⠚⠋⠛⠋⠛⠛⠛⠛⠀⢩⡁⠀⠠⠤⠤⠄⠀⢈⣶⣶⡍
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠤⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⣒⣐⣂⣶⣀⣐⠴⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⣠⠴⠒⠛⠉⠙⠛⠒⠦⣄
⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⡼⠁⠀⠀⠀⠀⢀⣠⠤⢤⣄⠀⠀⠀⠀⠈⠛⢦⡀
⣼⠁⠀⠀⠀⠀⡴⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⡏⠀⠀⠀⠀⣼⠁⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄
⢷⠀⠀⠀⠀⢳⡀⠀⢀⣀⡀⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢇
⠈⢳⡀⠀⠀⠀⠉⠉⠉⠀⠙⢦⡀⠈⠙⠦⣄⣠⡤⠖⠚⠉⠈⠲⣄⠑⢦⡀
⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡙⠢⣄
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡈⠑⢄
⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⡼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡼⠧⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣌⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠛⠦⣜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠲⣄⠀⠀⠀⠀⠀⠀⠀⡰⢋⠔⠋⠉⠉⠉⠒⢤⡙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣌⠓⢄⡀⠀⠀⠀⡸⢁⠏⠀⠀⠀⠀⠀⠀⠀⠹⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡉⠲⣄⠀⢧⠸⡄⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡙⠺⣧⡱⣄⠀⠀⠀⠀⠀⣠⠎⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣉⠛⠳⠤⠤⠔⣊⠵⠊
⣴⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣽⣿⣷⣶⣿⣴⣶⣶⣶
⣿⣿⣧⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠋⠁⠀⠉⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⠛⠻⢿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣤⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⠀⠀⣸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⢨⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡖⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢈⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠐⠁⠈⢠⠀⠀⠀⠀⠓⠈⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠀⠠⠈⠀⠀⠀⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡌⠃⠀⠀⠀⠀⠀⡮⡄⠆⠀⠀⠀⢠⠄⠂⣠⠀⠀⠀⢀⡄⠀⠂⡄⠀⠀⠀⠐⢣⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⠙⢁⠆⠀⠀⠐⠆⠠⠀⠆⠀⠀⠀⠀⠊⠀⠂

⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⢀⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠃⠘⠀⠌⠀⠀⠀⢈⡠⠔⢙⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠁⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣀⣀⠀⠀⠠⠀⠀⠀⠂⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⣀⡄⠂⠀⠀⠐⠀⠀⠀⡊⠀⠀⢆⡄⠀⠀⡇⠀⠢⠤
⠀⠀⠀⠀⠀⠄⠈⠀⠁⡠⡀⠀⠋⠡⠠⠂⠀⡀⠂⠈⠈⠁⠐⢀⣀⡝⠥⠄⣠⣇⠀⠀⠠⠆
⠀⠀⠀⠀⠀⢀⠁⠶⠩⠀⣰⠈⠀⠐⢈⡁⠀⠄⠀⠈⠀⠀⠀⡉⣛⠚⢂⠀⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠠⠀⠈⢁⡈⠁⠨⠐⠂⠈⠍⠉⠭⠅⠩⠤⠀⡽⠁⠄⡠⡎⡅⠺⠏⢛⠓⠀⠔⠀⡀⠠⣠⡀
⠀⠀⠀⢠⠀⠀⠄⠈⢁⣠⠤⠐⢀⠑⠀⠁⠀⣠⡀⠀⠀⠀⠀⡄⢰⠅⢙⣋⠊⠉⠉⠫⡀⠈⠀⠉⠀⠉⠋
⠀⠀⢁⠀⢦⡀⠂⠡⠠⠧⠤⠄⠀⠁⠀⢁⡄⠉⠁⠀⠀⠀⠀⠁⢈⠀⠅⠿⠀⣀⠀⠱⡀⠀⢁
⠀⠀⠀⠐⠀⠈⢀⢰⠛⣻⣧⣃⢀⣚⢁⢈⠁⠐⠲⡉⠀⠈⠙⢓⣂⡖⢀⠀⠐⣩⠀⡲⠚⠊⠀⣀⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠠⣀⠀⢀⠀⠀⢨⠀⠜⠨⠄⢣⢠⠁⠈⠈⠁⢆⠀⠠⢰⠅⠀⠘⣤⣧⠃⢡
⠀⠀⠀⡀⠆⠀⠀⠆⠈⠂⠐⠠⡴⠢⠀⢀⡄⠂⠐⠘⠆⢄⠐⠨⠀⠀⠡⣤⠦⠴⠆⠋⠉⣡⣼⣻⡀
⠀⠀⡜⠀⠑⡀⠀⠑⢀⢀⠀⢸⠂⠀⠀⠈⠉⠱⡠⢀⢀⣼⡆⣁⣤⣶⣦⠹⡄⠐⠠⠄⢪⣄⠀⠀⠅
⠀⠀⠇⠀⠀⡏⢰⠌⠆⡚⣎⠠⠀⠀⠀⠀⠀⢀⡀⠐⠈⠊⠑⠘⣷⣙⡼⠀⠹⣄⠒⠀⢀⠉⠵⠖
⠀⠀⠀⠀⠀⠀⠘⡆⠆⢱⡶⢴⠀⣠⡄⣀⠈⠀⡀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠉⢆⢠⣀⣬⣱⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⢁⡄⢠⣯⠀⡇⠀⠂⠈⠰⠤⢀⠀⠀⠁⢀⣀⣠⡤⠴⠀⠉⠂⠁⠃⠃
⠀⠀⠀⠀⠄⠠⡞⠉⠁⠀⠻⡀⣸⠇⠂⠀⠀⡰⢀⠀⠀⠈⠀⠐⠀⠭⠁⠀⠀⢀⡀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠈⠀⠎⢘⡈⠉⠠⡤⠂⠀⠀⠄⠀⠀⠀⠀⠓⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠠⠤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⠚

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠂⠈⠉⠉⠉⠉⠑⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⢀⠀⡠⢠⢠⡤⡄⡄⡀⢀⠀⠀⠈⠙⠲⡀
⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⣤⢢⣳⣡⡴⠾⠐⠒⠛⠶⠷⢴⣐⠀⠀⠀⠀⢪⢧⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⡔⢕⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠐⡯⢷⡀
⠀⠀⠀⠀⢸⠀⠀⠀⣈⣢⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠸⠴⣧
⠀⠀⠀⠀⣧⠀⠀⢰⡆⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠡⢹⡀
⠀⠀⠀⠀⡇⠀⠀⠠⠬⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⢬⡇
⠀⠀⠀⠀⡇⠀⠀⢽⣌⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠭⠅⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⡇
⠀⠀⠀⠀⣧⠀⠀⡄⠀⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢤⠔⡇
⠀⠀⠀⠀⡟⠀⠀⠏⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⠠⡇
⠀⠀⠀⠀⠇⠀⢀⡅⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠠⡇
⠀⠀⠀⢀⡂⠀⢀⣿⡦⣿⣀⣀⣀⡀⢀⣀⢀⡀⣀⣀⣀⣀⣀⡀⠀⠀⢸⠀⠨⣤⣉⡇
⠀⠀⡞⠣⠽⠶⠶⠶⢶⣿⠟⣥⡬⣴⣶⣤⣴⡄⣀⣀⣀⣀⡁⢀⠉⣉⣿⣀⣾⣵⣿⣷⣶⣄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢉⠩⢝⠊⢿⣯⣽⣯⣟⣿
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⣒⠭⢒⠭⣐⡾⢟⣾⣿⣿⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠨⢖⠩⠢⠌⢚⣯⣒⣯⣘⣿⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⣔⢬⣐⠨⠐⠠⢐⡯⣿⡯⣿⣽⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⢵⡦⣈⠫⢄⢄⡀⠩⣿⡭⣿⢽⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠁⠑⢲⡪⣗⠨⡚⠵⠨⠒⠌⡚⢽⡏⢽⡿⣽⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣠⣖⣾⣿⣿⣿⣿⣑⠿⣶⣅⠀⠀⠁⠀⠐⣀⣙⡂⣅⣇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⣿⣿⣿⡟⣿⣿⣿⣿⣟⣿⣿⠢⠀⠨⠀⢹⣾⣵⣯⣽⣧
⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣪⣿⣿⠩⡒⢌⡒⢤⣿⢥⡿⣿⣿
⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⣤⢹⣿⣿⣿⣟⢻⣿⣿⡟⢅⡌⡊⠑⠨⡐⢯⣿⠯⣿⣿
⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢼⣿⣿⣿⡿⢽⣿⣗⠨⣐⠌⡈⢐⡁⢀⣿⡯⣽⣻⣿
⠀⢸⠀⠀⠐⠤⠁⠀⠀⠀⠀⠀⠀⠀⣾⣺⢿⣿⣿⣏⡲⣿⡧⢁⠪⣑⠠⣑⠬⣂⣿⣿⣗⡿⣿
⠀⢸⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⡀⢽⠸⡛⢽⠻⢗⠈⣿⣏⠀⢉⡒⢍⡂⢅⠢⢑⣺⣓⡿⣿
⠀⣼⣄⠀⠐⡈⠀⠀⠀⠀⠀⠀⢀⠬⡙⢷⣾⣧⣾⣲⣗⣿⣵⠩⡒⠬⡒⠬⡒⠩⣲⣿⢺⣯⣷⣗⣖⣒⣒⣒⣒⣂⣀⣄⡀
⠀⠀⠘⠛⠲⠶⢭⣤⣤⣤⣀⣀⣐⣵⣮⣕⣢⣍⣉⣙⣍⣨⣑⣅⡂⣕⣢⣕⣢⣥⣺⣭⣿⣽⠿⢿⠮⠤⠬⠿⠭⣍⢙⡛⠑

⠀⢰⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠈⠳⢦⡄⠀⠀⢠⡶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⡆⠀⠀⢰⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡆⠀⠀⣾⡿⠛
⠀⠀⠀⢸⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢿⡇
⠀⠀⠀⣿⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠸⣿
⠀⢀⡼⠃⠀⠀⠀⠀⠹⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠻⣷⣄
⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠻⠶⠲⠶⠲⡶⣶⠶⠶⠖⠶⠓⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠻⣿⣿⠟⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠻⣿⡟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢧⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢋⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⣠⣿⠟⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠟⠙⠻⠟
⠀⠀⠀⠀⠀⣤⣶⣶⣶⣾⡿⠋⠁⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣄⠉⠀⠀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣨⣿⣷⣄⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠈⠻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠈⠻⣿⣿⡇
⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠀⠀⠀⠀⠈⠛
⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋
⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠊⠉⠉⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⢤⡀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⠲⣦⢤⣤⣀⣀⠀⢀⡟⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⡴⠃⠀⠀⠀⠉⠉⠋⠀⠰⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠳⠦⣤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⠇
⠀⠀⠀⠀⠀⠀⡸⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡞
⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡾
⠀⠀⠀⠀⠀⢸⠁⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢸⠀⠀⢸⢀⡤⠶⠒⠒⠒⠶⣄⠀⠀⠀⠀⠀⢠⠞⠀⠀⡼⠁
⠀⠀⠀⠀⠀⣾⠀⠀⠚⠋⠀⣀⡠⢄⣀⠀⠈⠳⡄⠀⠀⣰⠏⠀⢀⡞⠁
⠀⠀⠀⠀⠀⢿⠀⠀⠀⣠⠞⠁⠀⠀⠈⠳⡄⠀⢱⢀⡼⠁⠀⢠⠞
⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣷⠀⠘⠋⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠻⡄⠀⠀⠀⠀⣀⡟⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠳⠶⠖⠛⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⣀⠴⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢂⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠁⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⢶⣦⣤⣄⡀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠘⣧⣀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠈⡙⠁⠀⢠⠀⣈⢙⠛⠿⣦⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⡀⠼⠞⠒⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣚⡁⠀⠀⠠⠌⠐⢄⠈⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⣀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣼⠀⠨⠀⠀⠉⠃⢤⡀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋⠂⠀⠀⢆⡄⠀⢀⣈⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⣅⡼⠃⢠⠐⠃⠀⠀⢀⣿⡖⠁⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡁⣀⠗⡐⠀⠀⠀⠀⠘⠙⢢⢀⣄⠈⡄⠶⡀⠀⢀⠚⠁⠀⡀⠐⠀⠀⢀⣀⣤⡠⠛⢂⡀⠀⠚⠁⠀⠰⡖⠠⠦⠀⠀⣠⠜⢋⡿⠂
⠀⠀⠀⠀⠀⠀⢀⠆⠭⠀⠈⠀⠀⠠⠔⠀⠀⢁⣤⠾⠯⢤⡨⠀⠉⣵⡆⠀⠀⠘⠿⠂⣶⠠⠛⠁⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠿⠃⠰⠀⡈⣥⠞⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠠⠎⠀⠀⠀⣰⢫⠁⠀⠀⠀⢹⠀⠀⠀⠈⢂⡀⠀⠐⠀⣀⠀⠀⠀⠀⠐⢀⠤⠀⡄⡀⡤⠀⠈⠠⢉⡀⠆⣀⠘⠠⠀⢠⡀⢠⠼⣍
⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢀⠀⠠⣲⠶⡯⠀⠠⡄⣦⣄⣚⠤⠀⠀⠀⠀⠁⢠⠀⠀⠀⢡⠀⠀⢀⣀⢈⠱⠚⠁⠀⠀⢀⠀⠀⠰⠃⠀⠉⢎⣀⡊⣉⠁⠁
⠀⠀⢀⠄⠒⠒⣀⠉⠋⠀⠀⠔⠈⠁⣀⠄⡤⠄⠀⠈⠋⠤⢧⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⢠⠇⠸⠡⠀⠒⣃⠀⠃⠉⠀⠪⢀⠈⡀⠀⠀⢴⠶⠁⠀⠀⠆
⠀⣀⠛⠛⠋⠱⠚⠄⠀⠀⡠⠞⢹⠶⠋⠀⢀⣶⡆⠀⣴⣶⠇⠕⠀⠀⠀⠀⠀⠀⠀⠘⠌⠃⠈⠁⠉⠩⡑⠀⢀⠀⠀⠀⠈⠐⠀⠀⠀⠃
⢀⠀⣐⡀⠀⠀⠀⢰⡔⡀⠑⠠⢀⠦⡀⠀⠈⠉⠄⠀⢁⠘⠉⠐⠂⠀⠴⠤⣤⠦⠴⠂⠀⠀⠀⠀⠀⠀⠈⠂⢠⡑⢄⡀⠀⡹⠄⠀⠁⡠⠀⠀⠀⠀⢀⠌
⡔⠀⠀⡥⣀⣌⠉⠆⠀⠀⠀⠀⠀⠉⠈⠐⠠⡄⠀⠆⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣷⣶⣞⠒⣶⣯⣽⣖⣉⣭⡵⡆
⢐⠀⠐⠀⠀⠐⠂⠁⠀⠀⠀⠂⠀⢠⠀⠘⢟⠲⡦⢤⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⠤⠀⠀⠀⠀⠉⠉⠁⠀⠻⣿⣯⢉⣫⣶⠶⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠁⣠⠀⠈⠀⠐⠷⠔⠲⠤⣄⡀⠀⠀⡀⠀⠀⠀⠀⡠⠄⠉⠀⠀⣀⣠⣰⡞⠛⢂⠀⠀⠀⠀⠀⠀⠀⠿⡅⠂⠰⠚⠁
⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⣬⠡⠙⠙⢿⢙⠀⡀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣤⡬⡷⡀⠠⡊⠀⠈⠿⣤⡤⠉⠀⣀⢠⡄⠀⠡⡀⠀⡀⠀⠀⠠⠈
⠀⠀⠀⠈⠀⡀⠀⠀⠀⠓⠬⣅⣀⣉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢉⡚⠛⢋⡙⠠⡌⢣⠀⠀⠀⠀⠀⢀⡠⠅⢲⡤⠀⣄⠠⠄⠑
⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠙⠧⠘⠃⡀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢉⡀⠀⠁⢤⡄⢳⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⣄⣤⣄⣌⢠⠀⠩⠄⢰⠅⠈⢹⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣠⣤⠶⢦⣄⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⣼⠋⠀⠀⠀⠈⢻⡄⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⢧⣄⣀⣀⣤⠟⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠉⠉⠉⠁⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⠙⣧⡀⣠⡟⢡⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣋⠉⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠈⢉⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠋⠁




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

⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⡀⠐
⠀⠀⠀⠀⡌⠀⠀⢀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠠⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⠛⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢀⡀⣠⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡃⠁⠀⠈⢆⠰⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠌⠂⠀⠀⢆⡈⠀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠃⠁⠄⢄⢂⠄⠘⠠⠬⣃⡑⣀⣀⠿⢉⣤⢤⠤⢤⠀⢸⡟⠝⠛⠛⡏⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠦⣞⡦⠔⣂⠴⠚⠩⡦⠀⠀⣀⣤⣀⠀⠀⡾⠉⠈⢄⣠⡄⠀⢢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠸⠀⢠⠔⠂⠀⠐⠉⢄⠀⠀⠀⠀⠀⠃⣧⣀⣼⠀⣰⣿⡤⠖⢊⠁⠐⣰⡧⠀⡀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠈⠀⠀⠀⠘⣿⠠⠓⣀⡥⠖⠋⣿⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠲⣦⣅⣀⡀⢰⡇⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠒⠀⠀⠀⠀⠈⠀⠁⠰⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠤

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