Hongo Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠿⠛⠛⠛⢛⡛⠛⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣋⠡⠠⠶⣶⠂⢰⣷⣦⣀⣠⣬⣾⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣿⡿⠟⢟⠋⡏⠉⠉⠉⠋⠛⠿⢿⣏⢀⣰⠜⢶⣤⣄⠈⠉⠉⢹⠆
⠀⠀⠀⠀⠀⠀⣰⣿⡷⠛⢉⠀⣤⣤⣌⢿⣄⣤⣄⣠⣤⣤⣄⣼⣿⣿⣿⣀⣘⠛⠛⠒⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣿⠋⠀⠠⠀⢼⣚⣿⡇⠀⣿⣿⣿⢫⣤⣮⣿⡷⠿⠟⠛⠛⠻⠿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⣙⣦⣄⣀⠈⠉⠿⢿⠀⢀⣿⠿⢢⣿⠟⠋⠀⠀⢰⣶⠀⢰⡀⠀⠀⠀⢉⡇
⠀⢀⣤⣶⠾⠾⠛⠛⠛⠛⠻⠿⢦⣀⠀⠈⠁⠀⠈⠣⠀⢀⠀⡀⢀⣿⠀⠈⣿⡿⠶⠶⠊⠁
⠀⢿⠉⠀⠤⠴⣷⠒⢠⡤⠀⠀⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⢀⣠⣤⣴⣦⣤⣄⣀
⠀⠈⠙⠲⠶⠶⠟⠀⠈⠶⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⣩⣭⠋⣭⣍⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⣤⠀⣀⠈⠙⠛⠛⠛⠀⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⣀⢈⣰⣶⣥⣼⣿⢷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣹⣿⣿⣿⣷⣿⣿⣿⣿⡿⠟⠀⢀⣴⡥⡴⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡧⠈⠁⣤⣴⣾⣾⣿⣾⣷⣄⢈⡠⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣜⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣾⣽⡹⠁
⠀⠀⠀⠀⠀⠀⠀⣀⢶⣾⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠹⢈⡻⠟⠿⠿⠿⠟⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢒⡲⠶⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣶⢿⣯⣷⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⡀⠄⡾⢣⡤⠚⠁⠀⠀⣀⣀⠀⠀⠱⡀
⠀⠀⠀⢀⣤⣶⣞⣻⡟⡽⣿⣿⣿⣵⣿⣿⣿⣿⣿⣦⣶⣭⢻⣿⠀⠀⠀⠀⢀⡴⠚⠉⠰⠤⢰⠿⠒⢀⠔⠉⡀⠀⠀⠀⡀⠈⠉⠙⣦⠀⠁
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⢀⠔⠁⠄⠀⠁⣀⠀⠀⠀⠀⠐⠀⢁⡄
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣬⣏⠀⠚⠂⠀⢀⠀⠈
⠀⠀⠈⢿⣿⣟⣿⣿⣿⣿⣿⣯⣿⣿⣷⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⡲⠄⠀⠀⠀⠈⠉⢆⠀⠈⠁⠀⠂⠀⠄⠈
⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⢿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠤⠄⠠⠈⣄⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣷⣄⣀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠁⠙⢻⢦⣴⡆⢤⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣭⣯⣿⣻⡿⣿⣷⣿⣿⣿⣿⣼⠶⠚⣭⣦⣤⠀⠀⠀⠀⠀⠀⠀⠰⢷⢛⠘⠁⡆⢤⡀⠈⠁⠈⠀⠠⢩⣽⣿⠛⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣾⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠡⠀⠉⢑⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣯⣯⣹⣟⣿⣯⠿⠃⠀⠀⠀⠀⠀⠀⢤⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠘⠙⠒⠲⠒⡒⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠛⢻⣿⠛⠟⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠉⠀⠀⡀⠂⠀⠒⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⢀⠀⠀⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢦⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡄⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⡁⡀⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡄⢠⠀⠀⠀⢠⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣿⡿⣿⣿⣿⣿⣿⣷⡷⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⢧⣹⠀⢴⡀⢸⠀⠀⠈⠢⣑⣤⡤⢄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠙⠠⠟⢿⣿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠐⢌⣛⣮⣷⣿⣾⡄⢀⠟⠋⠁⢐⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⣠⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣽⣿⣿⣿⣿⣿⣧⣀⣸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠟⠋⠉⣉⣑⠂⢍⣢⠑⡜⠀⠡⣡⢞⠩⡙⠙⢛⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣗⢂⡉⠅⠀⠀⡨⢿⣿⠟⡛⠙⠛⣿⣕⣠⣵⣶⣤⡴⠤⣤⣉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠻⠶⠶⠷⣿⢴⠙⠈⠐⢻⠿⠛⠓⠚⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⢀⣐⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠋⠛⠋⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⢠⠇⠀⠂⢀⠀⠀⢸⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢻⢁⡏⢀⠂⢠⡿⠀⠀⣘⡇⡰⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣈⡟⠀⣿⢀⡏⡇⡀⣸⠛⡿⣁⢴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡘⡿⣷⣸⡿⣼⢰⣴⣷⡷⣿⠟⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠷⠘⠟⠃⠃⠸⠋⠛⠀⠉⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⣠⠤⠖⠲⡏⠉⠙⣏⠙⠛⠒⠦⣄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣿⠀⣼⠍⡐⡐⠠⠠⡇⠀⠀⡿⠀⢀⠠⡑⣜⡇⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠳⡜⢦⡈⠺⢴⣁⣇⣦⡗⠀⠀⣷⣸⠬⠖⣛⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠙⢦⡉⠓⠲⠶⠦⠶⡇⠀⠀⣿⠒⠛⢉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⣀⣤⣤⡤⢤⣤⣄⣀⠀⠈⠳⡀⠀⠀⢸⠃⠀⠀⡏⢀⠖⠉⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣞⠍⡀⠀⢼⠀⠀⣇⠀⠈⠝⢳⡀⣿⠀⠀⡞⠀⠀⢸⠟⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⣛⠶⠴⢼⠀⠀⣯⠤⠮⠶⣋⡴⠃⠀⣸⠃⠀⠀⡜⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⠈⠉⠉⢻⡄⠀⢿⠒⠛⠉⠁⠀⠀⢠⠇⠀⠀⢰⠻⣍⣈⢀⣸⠀⠈⡏⠉⠑⣲⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠙⢦⡀⠈⡇⠀⠸⡆⠀⠀⠀⠀⢠⡏⠀⠀⢀⡟⠓⠦⣬⣉⡏⠀⢈⣛⣋⣉⣥⠴⠋
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⣀⣀⣀⣀⠀⠙⣆⢻⡀⠀⢻⡀⠀⠀⠀⡾⠀⠀⠀⡾⠀⠀⠀⠀⢰⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠘⣇⢰⣞⣋⣹⠁⢸⣩⣭⠿⢂⡜⠈⣇⠀⠈⢧⠀⠀⢸⠃⠀⠀⢰⠃⠀⠀⠀⢠⡏⠀⢀⣮⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠈⠓⠲⠭⢽⡀⠸⡶⠒⠚⠉⠀⠀⠸⡆⠀⠈⢧⠀⡏⠀⠀⠀⡟⠀⠀⠀⢀⡞⣠⠞⠉⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢳⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢿⠁⠀⠀⢰⡇⠀⠀⣠⠏⢸⠁⣤⠤⢤⠤⢤⣀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢳⡀⠀⠀⠀⠀⠀⢳⠀⠀⠈⢆⠀⠀⢸⠃⢀⡞⠁⠀⠀⣱⠮⣝⡞⠀⣶⠤⠟⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠙⣆⠀⠀⠀⠀⠈⣇⠀⠀⠘⣆⠀⢸⡴⠋⠀⠀⢀⡞⠁⢀⡼⠁⣰⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠈⠳⣄⠀⠀⠀⢸⡄⠀⠀⢹⡄⢸⡅⠀⠀⣰⠋⠀⣠⠎⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⢧⡀⠀⠀⣧⠀⠀⠀⣇⠈⡇⠀⡼⠁⢠⠞⠁⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠳⡄⠀⢸⠀⠀⠀⢸⠀⠹⡞⠁⣴⠋⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢹⡄⢸⠀⠀⠀⢸⠀⠀⣷⣸⠃⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⢠⡇⢻⡀⠀⢠⣇⣀⣠⠏⢿⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠙⠛⠉⠀⠁⠀⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣻⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⣿⣷⣾⣿⣯⣿⣿⣿⡿⣿⣟⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣷⣾⢿⣯⣿⣛⣿⣿⣿⣟⣿⣿⡻⣿⣻⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⢿⣷⡿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣽⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⢉⠍⠉⠉⠙⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⠀⠀⢁⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠋⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠤⠀⢀⣀⣬⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠈⠉⠁⢀⣀⠀⠀⠀⢰⡆⠀⠀⡧⠶⠖⠛⠋⠉⢁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠦⠶⠶⠶⠶⠖⠛⢻⡇⠀⠀⣿⠀⣀⣤⣶⣞⣿⣿⣿⣛⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣾⣿⣷⣿⣿⣷⣿⣿⣿⣿⣯⣿⣆
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣆
⠀⠀⠀⠀⣠⣾⣿⣯⣿⣿⣽⣿⣷⣦⣄⠀⠀⠀⠀⠀⣼⡇⠀⠀⣿⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣼⢿⡷⣿⢿⣿⣿⣿⣻⣷⣿⣿⣷⣄⠀⠀⠀⣿⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿
⠀⢀⣾⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣿⠀⠀⠀⣿⠶⢤⡴⠁⠀⡔⠀⡀⠠⠀⠀⠠⡙⢿
⢀⣾⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠿⣷⢸⣿⠀⠀⠀⣿⢀⡾⠁⠀⣼⠣⢤⣅⣀⣀⡛⣀⣀⣼
⢸⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠂⣸⣿⡇⠀⠀⢀⣷⠟⠀⠀⣰⠇⠀⠀⠀⠀⠉⠉⠉⠉
⢸⣿⠋⡁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣠⡴⠞⢁⣿⠃⠀⠀⢸⡏⠀⠀⣰⠏
⠈⠳⣤⣀⣀⣐⣀⣀⣀⡆⠸⡗⠚⠋⠉⠀⠀⠀⣸⡟⠀⠀⠀⢸⡇⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⢸⡇⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⣿⣸⠀⣸⣿⠃⠀⢀⡴⡄⠀⣿⡁⠀⢰⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡀⣿⠀⠀⢸⡟⢀⣿⡟⠀⢠⠏⢸⠀⠀⣿⠹⡄⠈⡇⣾⠀⣠⠔
⠀⠀⠀⠀⠀⠠⡀⠀⢳⣿⡀⠀⢸⡇⣸⣿⠃⠀⣾⠀⣾⠀⠀⣿⠀⢹⡄⢿⣿⡞⠁⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠈⢳⡄⣿⡽⡆⢸⡇⣿⣿⠀⢠⣿⠀⣿⠀⢸⣿⡆⠀⣧⣸⣿⠁⣴⠋
⠀⣀⣀⣀⣀⣀⣠⣤⣿⣾⣿⣿⣿⣿⣿⣿⣦⣾⣿⣶⣿⣶⣾⣿⣷⣴⣿⣿⣿⣿⣿⣤⣤⣄⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⣴⣾⣹⣿⣭⣿⣏⣹⣷⣫⡷⣦⠔⢤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣷⣿⣿⡁⣸⣏⣻⣷⠟⢻⣿⣟⢹⡿⣿⠟⢻⣭⡷⢤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢛⣿⠟⣯⡄⣿⡿⠛⣿⣿⣋⣿⣷⢾⣯⣽⢯⣥⣿⢶⡟⠽⡗⣼⣿⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⡤⢭⣷⠿⣿⡾⠿⡾⣿⢿⣿⣟⢻⣧⣤⣿⠛⣿⡷⠲⣿⣧⣤⣿⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣹⡟⣻⣷⣾⣯⣀⣼⣷⣼⣃⣸⣿⣤⣭⣞⣿⡩⡝⢻⣿⠷⣴⣿⡷⣿⣟⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⣿⣏⠛⣿⠟⣿⣽⣿⣇⣠⣯⣩⠟⣿⢀⣿⣯⣿⣶⡾⠷⣤⣿⣻⡦⣭⡿⢾⣉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⠿⣿⣿⢿⣟⠱⢷⣀⣿⠟⠻⣶⡶⢿⣏⣹⣟⠑⢟⣰⣿⣉⣿⣶⣿⣄⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣾⣧⣀⣸⣿⡛⣿⣿⠿⣧⣿⣦⣴⣿⡷⠾⠷⠿⠟⠛⢻⢿⡛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⣿⣹⣏⣙⣣⣿⡿⠷⠛⢻⡏⠀⠸⠀⡀⠀⠀⠀⠀⠀⡆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣿⣿⠿⠛⠉⠀⠀⠀⠀⢈⡇⠄⠀⢀⡄⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡰⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇⢸⡇⠀⡆⠀⠀⠀⠀⠀⠀⢠⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠱⠄⢣⡀⠑⢄⡀⠀⠀⠀⠰⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣤⣙⣦⣤⣄⣉⣤⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠉⠉⠁⠈⠀⠀⠈⠉⠲⣌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⢀⠤⠒⠚⠓⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠂⡱⡀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⢠⡟⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⠀⠀⠀⠀⠀⠀⠋⠱⡄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⣸⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠺⠀⠀⠀⠀⠀⡀⢄⠑⣤⠞⠋
⠀⠀⠀⠀⠀⢹⡀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⠀⠀⣄⠀⠀⡗⠼⠉⢿⠁
⠀⠀⠀⠀⠀⠈⢷⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠞⠓⠚⠀⠑⠚⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠾⣄⣀⢀⠀⠀⠀⢀⣀⣀⣠⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠉⠉⠉⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡶⢤⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⢃⡿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠗⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠺⠦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⣦⣤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢤⢤⣤⡤⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡶⠻⣿⣭⣉⣭⣤⢶⠇
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡶⢶⣶⣤⣤⣴⢖⢚⢍⣀⣠⣤⢤⠬⣧⣶⣶⣤⡴⣧⣤⠶⣀⠀⢀⠠⠅⠀⣤⠠⣐⣦⠤⠭⠁
⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣿⣿⣯⣿⣍⠀⠀⠀⠀⠀⢀⣽⣧⣿⣥⢀⣹⡤⢬⡖⣛⢛⣛⡷⡿⠿⠋
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣽⣿⣿⡿⠞⠛⠁
⠀⠀⠀⠈⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠎⠷⠤⣄⣀⣈⣙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢈⣇⠀⣀⠠⣼⣿⣿⣿⣇⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣽⣿⣿⣿⣿⣿⣯⣽⣽⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠟⠛⢿⡛⢻⡛⣋⡏⡹⢋⡽⣉⣈⠔⠊⣁⡬⠟⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⠛⠤⢤⣈⣐⣢⣽⣶⣷⣾⣿⣿⣿⣿⣽⣶⣟⠯⠥⠔⠒⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠉⠉⣉⣰⣿⣿⣿⣿⣿⣿⡷⣾⣧⣿⣿⣿⠿⡟⠒⠒⠒⢒⡶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣮⣩⠴⠞⡫⢟⠟⠋⣿⣿⣷⣯⣍⣿⡏⢆⠁⢠⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣥⣴⣴⣽⣿⣿⣿⣻⣿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⢿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⣠⣴⣷⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄⣀⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢰⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⢿⡷
⠀⠀⠀⠈⠛⠋⢹⣿⣿⡿⠛⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇

⠀⠀⠀⠀⠀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣄⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣈⣉⠛⣿⠿⢿⣿⣿⡿⠇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⢠⣾⣿⣿⠋⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠐⠿⠛⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠛⠛⠋⢻⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⡄⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣤⣄⡀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠺⣿⣿⣿⣿⠿⠿⠛
⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠈⠛⠻⠿⢿⣿⡟⠛⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⢠⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠉⠉⣻⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢳⡄⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠓⠢⠋⠀⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠤⢤⠀⠀⠠⣏⣤⣝⡆⠀⢸⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢧⠀⠙⠒⠛⢀⣀⣤⣠⣄⣀⠀⠀⠀⠓⣤⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡋⣠⠊⠀⠠⢉⣩⢽⣖⠲⡶⢲⣾⢭⣛⠡⢀⠀⣰⠟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⡡⣊⣕⣚⠻⠪⣀⣧⣳⣵⣡⣇⡠⠲⢛⣒⣝⡪⡙⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⣾⣗⠀⠀⢌⣱⣾⣿⠉⠉⢹⣿⣶⣉⠄⠀⢰⣿⣦⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⢶⣿⣛⣫⣟⡩⠀⠀⠘⣙⣯⣛⣻⣷⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡒⢢⣤⣀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⠈⠘⠈⠓⡶⢤⡀⠌⢣⣶⠅⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢢⠀⣤⡽⡖⡖⡺⣛⠗⡾⠀⢱⢿⠛⠙⣟⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠯⢚⡴⠋⢱⡄⠸⡾⠟⠓⠉⠀⠀⠃⠈⠀⠀⢿⢼
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⠴⠚⠋⣇⡆⠘⣆⠀⠀⠀⣠⣠⠀⡄⣠⣨⠈⠀⠀⠀⠀⠠⠠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠐⠈⠢⠀⠀⢫⠛⠙⠙⢿⠉⠀⠀⠀⡠⠔⠛⠋⢢⣑⢽⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣡⣮⠖⣾⠀⣤⠸⡠⣀⣜⣰⠁⠀⢠⡀⠳⡺⠖⣒⡦⠄⣨⠝⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⠀⡈⢲⣧⠀⢱⡶⠖⠲⠐⠲⢺⢻⣴⣿⣽⠋⡿⣽⡛⣯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⣦⣬⡿⡄⣴⠘⠀⡘⡴⡇⡼⣼⡗⣷⡢⣰⡇⠈⠉⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠹⢿⣌⣷⣽⣻⣰⣴⡀⣻⠿⢵⣹⣿⡟⣷⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⢤⠤⣤⣤⣼⠿⠛⠁⠀⠙⠻⣧⣄⣈⣀⣙⠛⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⢀⣸⡿⠀⠙⢿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢃⣴⣿⣿⠇⠀⢻⣧⡈⠛⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣏⢠⠘⣇⣼⣿⣿⣷⣼⣿⣿⣿⣿⠶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⢻⣿⣿⣿⣾⣿⣾⣿⣿⣤⣀⠈⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⣻⣿⣿⣿⣿⡆⠀⠛⣿⣿⣿⣿⣿⣍⠻⣿⣿⣷⡄⠹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⠿⠛⠁⠐⠉⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣷⣼⣋⡉⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠖⠀⠀⠀⠘⣿⣿⣄⣠⣾⡿⠟⠿⣿⣿⣿⡉⠙⢷⣾⡟⠷⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠙⣿⣿⣿⠛⠁⠀⠀⠀⠀⠈⠉⠈⠃⠀⠙⠻⣷⡌⢷⣄⠰⠄
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⠁⠴⣚⣿⣿⣿⣿⡿⠛⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠋⠀⠘⠿⠟⠟⠋⠁⠂⠀⢆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⢠⡀⢣⡜⠃⠀⠀⠀⠀⠀⠀⠛⢄
⠀⠀⣠⢀⣤⠀⠀⣿⣿⣿⣿⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⠁⢠⣀⠃⠀⢹⣿⣧⡀
⠀⢈⡅⠉⠹⣦⣠⣿⣿⣿⠟⠋⠀⠀⢀⡄⠀⠀⠀⠐⠀⠀⠀⣆⢀⠋⡀⠀⠀⠀⠁⢸⡇⢸⡄⠸⠀⢀⣾⠀⢀⣿⣥⣾⣿⣿⣿⠀⠀⢰⠃⠀⠀⣠⠀⠀⠀⢃⠠
⠀⢸⠿⡗⠀⠉⠉⠉⠙⣃⠀⡄⠀⠀⠈⠀⠀⠀⠀⢀⠀⡀⠃⠀⠋⣠⣧⠀⢠⣤⡄⣼⣣⡼⣷⡀⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⡾⡟⠀⠋⢀⡤⠞⠁⠀⠀⡀⠀⢀
⠀⢇⠀⢁⠘⠀⣀⠉⣰⡅⠀⠀⠀⠀⠀⠀⠀⠠⢂⠌⠶⣠⠞⣦⡀⢻⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣛⣛⣂⣀⡀⠉⢉⡠⠖⠀⣀⡀⠀⠀⣸
⠀⠀⠀⠈⢠⡀⠀⠀⠈⠣⡀⠀⡀⠀⠀⠀⣀⢀⣈⢦⣴⣿⣿⣾⣿⣷⣽⣿⣠⣿⣿⣿⠿⢿⢟⣿⣿⣻⣿⡙⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣭⣶⣾⠛⠉⠁⠀⡤⢋
⠀⠡⠀⠀⠀⠛⣤⣄⣀⣀⡉⢾⣷⣶⣶⣤⣾⣿⣿⣿⢿⣿⣿⠿⣿⣾⠙⣛⠉⠟⢹⠇⠀⠐⢈⣿⣿⣿⡶⣽⡷⢻⣷⣏⠻⣿⠿⠿⣿⣿⣿⣤⡀⠀⠀⠀⠘⠀⠌
⠀⠀⠈⠁⠀⠢⠬⠿⣿⡿⣿⡶⠿⠿⠿⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⣿⠀⢹⠀⠇⠘⠀⠀⠀⢸⣿⠟⡻⢷⡤⠙⠁⠌⠛⣣⠤⠀⠀⠉⠉⠉⠤⡄⠀⠙⣦⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⠀⠀⠀⠀⠀⠀⢸⡟⢀⣀⣀⡀⠀⠀⠀⠠⣤⣔⣉⣀⣀⣤⣤⣤⡶⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⠟⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠁⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄⢀⡀⢰⡷⢠⣄⡀⢱⡴⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠁⠸⠀⠈⠁⡇⠈⠁⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⡇⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡟⣶⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠑⠄⠀⠀⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡄⠀⠀⠀⠀⠀⠀⣀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⣷⣄⠙⠂⠀⠀⠠⠀⡀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⡘⠋⠀⠀⠀⠀⠀⠈⡀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠈⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠶⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⡀⠀⣀⡀⢀⠀⣠⣤⠤⠤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠤⠤⣤⣀⡀⠀⠈⠉⠈⠁⠀⠀⠀⠈⠀⠁⠀⠐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠋⠉⠋⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣛⣩⠍⠁⠀⠉⠁⠀⠀⠉⠙⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢟⣩⠴⣒⣒⣒⣐⣒⡛⠿⣟⠻⢿⣿⣿⣿⣷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣴⢏⠀⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢦⡉⠻⣦⣝⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣦
⠀⠀⠀⠀⠀⣼⠇⠈⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⠀⠈⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣽⣧
⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⡟⣿⣿⢿⣯⠻⡷⡙⠻⣿⣯⢳⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⣿⡇
⠀⠀⠀⠀⠀⢿⣾⣦⣹⣿⣿⣿⠻⡌⠘⠀⠉⢣⡀⠈⠀⠁⠹⣧⠳⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⢄⣀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⣏⠀⠀⠘⠄⢤⠣⠁⠀⡠⣇⢱⣽⣧⡑⠀⠀⠀⠀⠀⠀⠀⠀⠚⠳⢬⣛⠷⢮⣥⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣌⠀⠀⢠⡀⠈⠳⣄⡀⠉⠈⢳⣿⣿⣷⡀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠈⠙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⣿⣇⢠⡀⠉⠣⡷⣌⠙⢶⣶⣬⣝⣿⣿⣿⣦⣄⠙⢿⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡙⢿⣿⣷⣾⣷⣴⣤⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣭⣁⣒⣒⣂⣀⣀⣠⣤⣤⣤⣶⣤⣤⣢⣏⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣍⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠿⠛⠛⢿⣿⣿⣿⡛⠓⠂⢀⣉⣙⣿⣿⣿⣿⣼⣷⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠬⠙⠛⠿⠿⣿⣿⣿⣿⣡⠀⠉⠉⠉⠁⠀⠀⠀⠀⡦⣿⣿⣿⣿⣿⠁⠀⠀⠀⠤⢬⣭⣿⣿⡟⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢀⣤⡆⠀⢸⡇⠏⠆⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣿⣿⣟⡛⠒⢈⣉⣛⣛⣻⣿⣿⠟⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣰⠿⣿⡀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠷⣶⣶⣶⣾⣿⠿⠿⠛⢋⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠇⠀⠀⠀⡿⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡎⠋⢳⡄⠒⢀⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣼⠃⠀⡏⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠦⠑⢂⣄⠹⣦⣻⠟⢹⣇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⣆⢰⡇⠆⠰⡇⠀⣿⠆⠶⠀⠀⠀⠀⠀⠘⠍⢀⠱⠤⠀⣯⠀⢰⡌⢿⡄⠉⢻⣾⠆⢐⣯⣤⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢻⡇⢸⣧⡈⣿⣄⠀⠀⢳⡄⣿⠐⠀⠀⠀⡀⠀⠀⠀⠀⠙⠀⢴⡀⢈⡓⠚⠀⠃⢻⡄⠈⣿⡶⠛⠛⢁⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣼⣿⠿⣿⣿⣿⣤⡀⠙⣿⠈⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠅⣠⠀⠀⣀⠀⠀⣿⣾⣿⣶⣄⣀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⣤⣙⣿⡀⠀⠙⠻⢿⣷⡈⢿⡄⠀⡄⡀⡸⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⣠⡶⠛⢉⣽⠿⢋⣽⣥⣤⣤⣾⣭⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⣟⢿⣆⠀⠀⠀⠙⢿⣦⣷⡀⣿⣌⠘⡼⡄⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⢉⠠⣾⢟⣿⡿⠛⠁⢀⣠⣴⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠁⠈⠳⣾⡄⢳⣄⠙⢟⢷⡗⢈⡑⢦⡙⢆⠀⠀⣠⠶⢛⢭⣡⣴⡾⠃⣼⢃⡾⠛⢃⡴⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣠⣙⡳⢼⣷⣿⣦⡻⣄⠻⣄⢻⣦⢠⡞⠁⢀⣼⠟⠋⠁⢀⣾⣃⠞⣡⣤⣾⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠙⠛⠉⠁⠈⠉⣙⠛⠛⠷⣞⢧⣹⣿⣥⡾⠻⠋⣤⡵⠞⠋⠈⠀⠐⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡀⠥⠐⠒
⠀⠀⠀⠀⠅⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⠿⠀⠀⠠⠠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠄
⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡂⢠⣤⣴⠀⠀⡈⠀⠈
⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠠⢤⠅⠙⠒⠁⠀⠀⠀⠀⠖
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠠
⠀⠀⠁⠀⠐⠐⠐⠀⠐⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⡄⠀⠼⠂⣀⣄⡄⠤⠆
⠀⠀⠄⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⡝⡂⠀⠀⠀⣠⣽⡍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢀⡀⠉⠀⠀⡜⣁⣀⢉⠠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⡉⡱⠂⠐⠞⣿⠔⡀⠀⠀⠉⠁⠀⠀⠀⠀⢀⠐⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⢤⠤⣷⣿⣠⡋⠙⠛⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⠡⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣔⣤⡄⣀⣀⡀⣉⠆⡐⠲⠀⠰⠄⠀⠀⠀⠀⠀⠀⠧⠌⠀⣄⠀⠗
⠀⠀⠀⢀⣀⣀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠄⠽⠁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⢀⡌⠔
⠀⡐⢶⡷⠅⣭⡂⠀⠀⠀⠆⠀⠀⠀⠀⠐⡂⠀⠠⢀⣠⣤⠄⠀⠀⠀⠀⠀⢀⡐⣈⡲⡀⢚⠲⠈⠀⠀⠀⢰
⢠⣽⣾⡭⣱⡶⢟⢂⡉⠤⠔⢈⠆⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠠⢧⠱⠚⠤⣥⠀⠑⠢⡁⢠⡆⠞
⢿⡿⣿⣿⣿⣿⣽⣖⣿⣺⣿⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠇⢃⠐⡑⠀⠀⡓⠚⠋⠉
⠀⠹⢿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠆⠴⠟⠁⠀⢀
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢌⣀⣡⣤⣶⣋⣀⣤⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣷⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠘⠁⠀⠀⠀⠀⠀⢦⠀⠇⢚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢰⡀
⠀⠀⠀⠀⠈⠻⠆⠰⣤⣠⢸⣼⣶⠀⠀⠀⠀⠘⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠨⡀⠃⡴⠀⠐⠗⠀⠀⠀⡶⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣉⢣⡎⠀⠀⠀⠀⠀⣠⣌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣀⠀⠀⠀⠀⠉⠀⠾⠀⢀
⠀⠀⠀⠀⠀⢠⡀⠀⠈⠙⠄⠀⢠⡆⢠⣿⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⣱⠟⠀⠘⠋
⠀⠀⠀⠀⠀⠘⣿⡆⠘⠃⠀⢀⠉⠑⠈⠡⣀⠀⠆⠒⠀⡀⠀⢰⣌⡥⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢸⡆⢀⡆⠀⠀⠮⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⡄⢦⢀⠆⠘⠋⢰⡶⣷⡀⣧⠔⢀⠏⠀⠀⠀⠐⡆⠐⠀⣠⠀⠀⠀⠶⠀⠄⡀⠀⠀⠀⠀⠀⠀⠁⠈⠋⠀⠈⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠇⠀⡜⡈⢧⡀⠐⠚⡎⣿⡷⣾⠊⠀⣀⠀⢀⠘⠁⠀⠘⠀⠘⠀⠀⠀⢀⣠⣤⡄⠐⠃⠀⠀⠀⢄⢀⡔⢠⡆⠀⠀⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⡄⡆⠁⣤⡄⠻⡆⠈⣄⢀⢠⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⢂⡀⠀⡻⡗⠀⠀⠀⠠⠿⢹⠸⢠⡿⠃⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠑⣄⠀⢿⡿⠀⠀⠀⠹⠎⢸⠻⠒⠒⠂⢀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠃⠋⠀⣀⡁⡾⠃⠀⣠⣄⠈⡠⢈⣤⡤⠆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠐⢧⡈⣄⢸⠆⠀⠀⣈⠛⢄⠀⣶⡄⠀⠀⢯⠀⠀⠀⠀⠀⠀⢢⣶⢠⣾⡞⢻⠟⠀⠀⠀⢀⣿⠏⠀⡥⢋⡡⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠄⡀⠻⡀⠐⠁⡀⠈⠢⣤⠁⠸⢇⡀⠀⠀⠑⢀⣀⠠⠄⠘⠶⢿⡟⣀⠻⣠⡶⣏⠀⠀⡀⠛⢠⠈⣀⡧⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⢤⣞⠲⠀⢀⠀⠀⣀⠀⠀⢈⠰⠀⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⣿⠁⠋⡱⠋⠀⢤⠁⢠⡟⠈⢁⠠⣘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠉⠙⠃⠀⠁⠀⠈⠓⡀⠀⠁⢡⡀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠐⠈⡨⠈⠁⠀⡈⠃⢀⠀⢸⣧⡀⢈⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠢⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡤⠐⠁⠀⠞⠁⢀⠂⠀⠘⠛⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠠⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢄⠀⠐⠀⠀⠶⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⣄⠈⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣣⡀⠉⠀⡀⠀⠀⠂⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢃⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠑⢦⣀⠀⠓⢤⠀⠤⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⣠⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠒⠊⠑⠤⢤⣥⡀⢀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⢁⡔⠓⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠤⠤⠤⠀⠀⠁⢃⡠⢞⡁⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠘⠛⠉⠁⠠⡤⠐⠉

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠒⠠⡀⠀⠀⠀⠀⢀⡤⠒⠠⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⢀⣀⠄⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⡀⠠⠀⠈⢀⠀⠈⠆⠀⠠⠀⡞⠠⠀⠈⠀⠀⢉⡉⣠⣅⢠⡄⠀⠀⡔⢎⠀⢸⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠅⣾⠀⠀⠀⠀⠀⠀⠍⢠⡌⢡⡄⠀⠀⠀⠻⣬⠑⡯⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠅⠀⢠⠀⠀⠡⡈⠀⠘⠀⠄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢁⠉⠀⠀⠀⠈⠀⠀⠀⠒⠊⠀⠀⠀⠀⢀⠐⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠚⡆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⢀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠐⠀⠀⠀⠁⢸⣏⠀⠀⠀⡀⢀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠆⠐⠤⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⢰⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠀⠐⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⢀⡁⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠈⠃⠀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠠⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠱⣀⢀⣀⣤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⠂⠉⣩⠉⠑⠠⡀⠀⠀⠀⠰⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⢄⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⣐⣀⠔⠁⠀⠀⠠⢄⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠔⠉⡑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣃⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠑⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠃⠂⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠊⡀⠀⣀⠈⠳⣄⠀⠰⠌⡤⠂⡐⡄⠀⠀⠛⠀⢠⡷⢾⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠉⠁⠢⠀⡇⠈⠋⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠇⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠁
⠀⠀⠀⢠⡀⠢⡈⠀⠀⠰⠂⠀⠀⠀⠀⠐⢁⡄⠀⠑⡈⠀⡠⠁⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠈⠒⠴⣦⣿⣔⡾⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡷⠈⠀⡌⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠤⠤⡀⠀⠀⢀⡀⠀⠀⠀⠀⡰⠁⢷⡀⠌⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⢀⣠⣦⠵⠒⠀⢢⡀⢠⣄⣩⡽⠁⢳⡿⢿⣿⣾⠇⡄⣁
⠀⠀⠀⠀⢀⠠⠀⠨⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⢣⢬⣬⣽⠃⣾⡟⠀⠀⣼⣿⣾⣴⣈⣩⣶⣤⣀⠀⡀
⠀⠀⡤⣊⠐⣖⡂⠙⠂⠀⠀⡖⠨⠆⠀⠀⠀⠀⠀⠈⢯⣽⢇⣶⢻⣿⠶⠙⠫⠟⢿⡟⠙⠻⣿⠟⠻⠇
⠀⠀⢧⠉⡘⡎⢉⠀⠀⠀⠀⡀⠀⠂⠔⠈⠀⠐⠀⠀⠘⡧⠿⢿⡞⠁⣀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⡶
⠀⠀⠈⢆⠁⣽⡌⠀⠀⢀⠠⡀⠀⠄⠓⠐⠂⠔⠁⠈⠀⠹⠀⢰⡆⠀⠀⠁⠀⢤⢄⠀⣿⣶⣄⡀⡼⢁⡀
⠀⠀⠀⠈⠆⢟⣷⠠⠀⠀⠀⣈⠁⠀⠀⠀⠀⠄⢉⡀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⡀⠺⣿⣿⣿⣿⠿⣿⣆⡀
⠀⠀⠀⠀⠘⣤⢿⣧⠀⠢⠄⠤⠀⠀⠀⢀⠀⠀⠠⠄⠁⠂⠈⢦⣀⠀⠂⠄⢠⣤⣾⡅⣀⡞⣽⠿⢿⣷⣿⣿⠷
⠀⠀⠀⠀⠀⠘⡜⣿⣆⠀⠀⢐⢃⡀⠀⠀⠀⡀⠓⢄⠠⠀⠀⠘⣟⡳⣦⣤⣸⣟⣹⣿⠏⣼⠳⣶⣾⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠘⡌⣿⡄⠀⠀⠂⠀⠉⠈⡁⠀⠀⠄⠀⠀⠀⢀⡹⣠⠏⠛⠻⠻⢻⡿⣼⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡘⣷⡀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣠⠀⠀⢈⣀⠡⠈⠉⠉⢈⡁⣹⣏⡍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣧⠀⠀⠀⠀⠤⠰⠀⠀⠈⠀⠀⠰⢀⣩⣄⣠⠄⣤⡈⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢧⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠛⡯⠙⡛⠹⢗⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⠢⠀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠈⠨⠋⠔⠉

⠀⠀⠀⠀⠀⢀⣤⠖⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⣤
⠀⠀⠀⠀⢠⣿⡇⠐⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⢀⢠⡀⡄⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠙⠶⠞⠃
⠀⠀⠀⠀⣾⣿⠓⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣷
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀⠀⠀⠀⠀⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣸⢀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠻⢷⡶⢤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⠁⠀⠀⠈⠀⠉⠒⠌⠀⠀⠉⡓⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠘⠀⠀⠀⠀⠀⢀⣠⣶⠿⢛⡷⢾⣆⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠃⠀⠀⠀⢀⣀⠤⠒⠻⠁⠈⠀⠀⠀⠜⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢀⣠⣴⠦⠘⠫⠁⠀⠀⡖⠀⠀⠀⠀⠀⣀⣤⡶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢄⣿⠀⠀⠀⣴⣾⣿⡿⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⣀⡀⢀⣀⣤⣤⣤⣴⣾⡎⠙⢳⣤⣾⣿⣟⡃⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⣉⠈⣀⣶⣿⣿⣿⡇⠀⢸⣿⣟⣩⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠒⠟⠺⡇⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠈⠘⢿⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡇⠀⠀⠀⠁⠀⠓⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣾⠀⣀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⣠⡿⠉⠁⠻⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣤⣄⣦⣌⡗⠤⡀⠈⠄⠀⠀⠸⠀⣼⠏⠀⠀⠀⠀⠈⢅⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠽⣿⠁⠹⣶⣿⠷⢾⠞⠀⠀⣿⣿⡔⠊⠈⡁⠐⠐⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠈⠀⠀⠀⠀⠹⠜⡶⣶⡿⠉⢹⣶⣞⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠃⢠⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⡀⠀⠰⠀⠀⠀⠀⠀⠀⢰⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠙⢦⡀⠆⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⡇⠀⠀⠀⠀⣦⠞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⢹⡀⠀⠀⢸⣧⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠳⠦⠴⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀⠀⢀

⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠰⠓⠄⠀⠄⠀⠀⠀⠀⠀⢘⣍⡁⣧⠀⠀⠀⠠⡄⠀⠀⠰⠿⠿⠶⠶⠦⠴⠶⠒⠲⠶⠶⠶⠒⠒⠒⠂⠐
⠀⢂⡙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠁⠂⠐⠂⠁⠀⠠⠄⣇⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣾⣁⣯⣦⡄⢆⠀⠈⢰⣆⢨⡄
⠀⠀⠀⢠⡀⠀⠀⠀⠀⠠⠁⠀⠀⠀⣈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠛⠋⣠⠏⠀⠀⠘⠟⠸
⠀⠀⠀⠠⣿⠀⠀⢠⣄⡴⠀⠀⣤⣁⠉⠀⣀⠐⢆⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢏⣻⣿⡷⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⣽⠀⠀⠀⠋⠁⠀⠀⠈⠁⠀⠀⠈⠂⠈⠉⢀⠆⠸⡟⡀
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠇⠀⢨⣇⣀⣀⠀⠀⠀⠀⠀⠀⢈⣇⠀⠀⡀⠁⠄
⠀⠀⠀⣼⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠉⢷⣀⣠⣀⣠⣀⣈⣛⣉⡉⠉⠀⠀⠀⠀⠘⠏⠀⠀⣀⣠⢤
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠂⢀⠃⠀⠀⢀⣴⣿⣿⣿⡭⣌⣏⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠐
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠾⠃⠀⠀⠈⠟⠙⡛⠛⣛⠬⠊⠉⢙⡻⠉⢉⡀⠀⠀⠀⠀⠀⠰⣗⡄⠄⠄⠐⢈⣹⠁
⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠔⠋⢠⣯⣶⡂⠀⠘⠃⠀⠈⠁⠀⡀⠀⠀⠄⠀⠘⠇⠒⠀⠁⠈
⠀⢀⣤⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢰⣦⣞⢭⣿⣿⡏⠀⣾⠀⠀⠀⢰⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠈⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣿⡏⠀⠀⠉⠀⠀⢹⡆⠀⡃⠋
⠀⢀⣾⣿⣦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠇⠾⠀⠖⠀⢰⡀
⠀⢸⣧⣿⣿⡆⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠈⠣⠂⠀⠱⠚⠀⠘⠃
⠀⢸⣿⣾⣿⠇⠀⠀⣀⣿⠇⠀⠀⠀⠀⠀⠀⠈⠁⠓⠀⠃⠂⠓⠀⠉⠉⠠⠶⠀⠀⠰⣶⣟⣛⢓⣙⣋⣓⣶⠆
⠀⢸⣿⣿⣿⠀⠀⣼⡟⠁⣠⣶⣶⣦⣄⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠘⠃⠀⠀⣼⣿⡿⠛⣿⣿⣿⣿⣿⣧
⠀⠀⣿⣿⡏⠀⠀⠹⢷⣾⣿⣿⣿⣿⣿⣧⣷⣶⡾⠀⠀⢟⡆⠀⠀⣰⢫⡽⠀⠀⢸⣿⣿⠀⠠⠘⣹⣿⣿⣿⣿⡇
⠀⠀⠘⠿⠁⠀⠀⠀⢠⣿⢿⡿⠃⢹⣿⣿⡿⠛⠁⠀⠀⠈⢋⡀⣄⠁⠛⠁⠀⠀⠘⣿⣿⣄⠀⢣⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠹⣄⢣⡀⠘⡇⢹⡀⠀⠀⠀⠀⢰⣄⠘⣿⠀⠀⠀⠀⠀⠀⠀⢛⡋⡔⢀⠀⢨⣜⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉

⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠿⠿⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣯⣽⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⣿⣷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⡾⠿⠿⠶⡿⠷⣶⡿⠷⢾⠿⠿⠿⢿

⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣾⣿⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣿⣧⠐⢠⠀⠀⠀⠊⠉⠉⠀⡀
⠀⠀⠹⣿⣿⣿⡷⠀⠀⠄⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⢶⡾⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣿⣿⠂⠀⢀⡔⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠙⠦⠀⠀⠀⠼⣿⡆⣠⣿⢉⢓⣀⠀⠀⠀⠀⠀⢠⣴⡦⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠒⠋⠀⠀⠀⠀⣸⣧⢋⢇⣼⣿⠇⠀⡀⠠⠤⠶⠛⠉⠀⠀⠀⠀⠁⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡿⣿⣿⣿⣾⡏⠁⠀⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⢻⡿⣿⣿⠃⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⣇⣹⣾⣿⣿⡀⠀⠀⠉⣀⠀⠀⠀⡀⠀⠀⠀⢀⡴⢒⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⢘⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⣶⣄⣀⣠⣀⡀⠀⠀⠀⠑⠾⠋
⠀⠀⠀⡇⠂⡒⢤⣄⣀⣠⡀⠀⠀⠈⠙⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁
⠀⠀⠀⢿⣤⣮⣿⣿⣿⣿⡷⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣠⠂
⠀⠀⠀⢸⡉⠉⢉⠻⠋⠁⠀⢀⣤⢤⣖⣋⠉⠀⠀⠀⠀⣀⢐⠀⢠⣾⡿⠄⢠⣾⣏⣀⣀⢀⢲⢩⣵⡗
⠀⠀⠘⢾⣷⣄⠘⢿⣷⣤⣾⣿⠿⠿⣿⠋⠀⠀⠀⣀⡐⠉⠸⢀⣄⣿⣿⣗⡟⠘⣿⠁⣹⡿⠛⢷⣆
⠀⠀⠀⣺⣷⣿⣿⡺⢿⣿⣿⠋⢾⣿⣿⣦⣴⡂⠀⣿⣛⠛⠂⠀⣿⣟⣏⡙⣿⣽⣿⣿⣴⣧⣤⢶⡌
⠀⠀⢀⣡⡼⣿⣿⣿⣾⣿⡿⢿⣾⣿⣟⣿⠋⠀⠀⠻⢿⣷⣌⣾⣿⣿⣿⣿⣋⣙⣿⡮⣤⣯⡶⠻⠁
⠀⠀⠀⠀⠀⠀⠈⠛⣿⠷⠟⠛⠓⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢶⠶⠶⡤⢤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠟⢣⢀⣠⣤⣄⢀⣀⣊⡉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢌⣰⣾⣿⡿⣿⣿⣿⣛⣳⣄⣠⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠈⢯⣻⣿⡇⠀⣿⣿⣟⡟⠛⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⡛⣹⡁⢰⣿⢿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣤⣶⡶⠟⣻⡶⣄⢀⡀⢠⣔⣮⣷⣿⣿⣷⣇
⠀⠀⠀⠀⠀⢀⣴⣿⣷⣷⣾⣿⣿⣿⣿⡿⠿⠋⣁⠈⠠⢀⣤⣿⣛⣻⣿⣻⣿⡆⣈⠁
⠀⠀⠀⠀⠀⠊⠁⠚⣻⣿⣿⣿⣿⣟⠋⠉⠁⠠⠀⠀⢠⡾⣿⣟⣿⣿⣦⣉⠙⢛⢿⣿⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢟⣿⡇⣿⠿⠀⠀⠀⠀⣠⡀⠈⠑⣻⣿⣿⢻⣿⣿⠋⠀⢸⣿⣟⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⠀⠀⠀⣤⣴⣾⣿⣿⣄⠈⠉⠉⡟⢸⠋⠀⢀⣴⣿⣿⣛⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠨⡑⠿⣿⣿⠿⠟⠁⣶⡆⠐⠁⠰⢾⣿⣿⠟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠘⠏⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⣾⡿⠀⠀⡰⠀⠀⠀⠘⠻⢿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡆⢀⡄⣰⢀⣿⣿⢁⣼⠁⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆⢐⣷⣘⣿⡿⢸⣿⣯⣼⣏⡠⠀⠀⠀⠀⢳⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣳⣾⡿⣾⣿⣿⠀⢀⣠⣤⣶⣾⣿⣿⣦⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⢶⣿⡿⢿⠟⢿⣿⣿⣿⣿⣿⣿⡷⢞⡿⠁⠉⠛⡇⠘⣏⠙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡀⠶⠋⠉⡧⣼⠀⠰⢇⠈⢿⠈⣿⠙⠀⢺⠃⠀⡼⠶⠁⠈⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠙⠀⠀⠀⠤⠃⠁⠰⠀⠎⠀⠸⢠⡋⢃⠀⡄⠀⡴⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠁⠈⢆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⢒⠺⠓⠢⠒⠘⠏⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⣀⡀⠀⠀⡰⣴⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⡤⠤⠀⠤⡴⠀⠀⢠⠂⠁⠀⢀⠔⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡛⠚⠒⠐⠢⠐⠚⠁⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠠⠇⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠀⠀⠷⠀⠀⠀⢀⣠⠖⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠂⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢳⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠠⡴⠿⠁⠙⠉⠁⠀⠀⠀⠩⠆⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠙⠉⠃
⠀⠀⠀⠀⠀⠀⢐⠀⢓⠀⠀⠀⠀⠀⠘⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠜⡁⡐⠲⠂⠀⠀⠀⠈⠇⢀⣄⠀⠀⠀⠀⠀⠀⢰⡶⣶
⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⡀⣈⠙⠀⠀⢐⣖⣶⡞⠆⠀⠀⠀⠀⢤⠚⣁⡚⣠⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠌⣄⡀⠀⠈⠉⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠯⣣⠅⠀⠀⠀⠀⠀⠀⠀⣩⠃
⠀⠀⠀⠀⠀⠈⠛⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠄⢺⠠⠄⠀⠀⠀⠁⢀⣦⢤⣤⡤⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣠⠰⠿⠒⠛⠋⡉⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢤⡘⡛⡀⠜⠁
⠀⠀⠀⠀⠤⠴⠦⣐⡀⠁⣠⠦⠤⢀⣀⡸⠂⠘⠀⠙⠤⠖⠀⠀⠀⠛⠋⠙⠁⠀⠀⠀⢀⡀⠀⣴⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⡀⠀⠀⠆⣄⠈⠀⣠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⡀⠀⠀⠀⠀⠙⣃⠅⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠑⡐⠋⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠄⠀⠀⠀⠀⣤⢘⠃⠀⡀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⠀⠀⠀⠀⠠⣶⢈⠀⣠⢫⠄⠀⠀⠀⠀⠒⠤⢄⣀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⡌⠰⠀⠀⠀⠀⠀⠲⢿⡆⣀⡌⠀⠀⠀⠀⠀⠀⠐⠘⠛⡀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⣁⢴⡂⠀⠀⠀⠀⠀⢭⣿⠋⠁⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⡀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠄⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣘⠉⠀⣚⠀⠛⣄⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠰⡆⢁⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠜⠋⠁⣠⣄⣧⠤⠀⢄⠁⠈⣉⢐⣂⡞⠰⣴⣾⣷⣶⣶⡀⡉⠂
⠀⠀⠀⠀⠀⠀⡀⠀⡂⠁⣀⡴⠶⠛⠉⠁⠀⠀⠀⠀⢇⢘⣉⣻⡟⢠⣶⠏⠀⠉⣷⣧⣞⣨⢛⢂⣤⣀⡀
⠀⠀⢀⠤⠂⣥⠤⠐⡎⠉⠀⢀⠤⢰⡀⠀⠀⠀⠀⠀⠈⣞⢻⡟⢀⡼⣿⣦⡄⠾⢿⡿⣽⡿⠻⣿⣿⣿⣿⡦
⠀⠀⢇⠛⠂⣇⠺⠂⠀⠀⠀⠈⠁⠀⠀⡠⠐⠀⢀⠀⠀⠘⡟⣥⣿⣥⠟⠁⠀⠀⠁⠀⠑⠁⠀⠈⠛⠀⠀⢡
⠀⠀⠘⡆⠳⢸⡄⠆⠀⠀⠀⠐⠀⢀⠀⣅⢀⡀⣁⠆⠠⠄⢻⡀⠈⡍⠀⠙⠳⠀⠀⠀⠀⠐⣤⣀⠀⠀⢠⠯⠀⢀
⠀⠀⠀⠘⡄⣻⣷⡀⠀⠀⠂⠉⢠⠊⠁⠀⠀⠀⠑⠤⠀⠀⠄⢆⠀⠱⠀⢀⠀⠀⠀⠙⠂⠀⣿⣿⣷⣦⣯⣴⣷⣄
⠀⠀⠀⠀⠸⡜⢺⣧⠁⢀⠀⠀⡂⢀⠀⠀⠀⠀⠁⠐⠂⠈⠀⠈⣄⠀⠀⢈⠀⠀⠀⠀⣠⠂⠈⣻⢿⣿⣿⣏⣻⣿⣶⡠
⠀⠀⠀⠀⠀⠹⡝⣿⣗⢀⠉⠈⢣⠀⠀⠀⠠⠀⠀⣈⠁⠀⠀⠀⡘⣷⣄⡀⠈⠀⢶⣿⣿⣴⡴⢣⣟⣉⣹⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠱⡘⣿⡄⠀⡀⢂⡰⠂⡄⢤⠄⠂⠀⠱⠈⠀⠀⠀⢹⡦⠹⣿⣶⣿⣦⣾⡿⢡⣿⣨⣹⣿⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠱⡸⣷⡀⠀⠁⠄⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⡆⢣⠘⡄⢄⠉⠩⠿⢧⣿⡻⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⢴⠌⠀⠀⠀⠀⢴⢀⣞⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⣇⠀⠀⠀⠀⠐⠘⠀⠈⠀⠀⠀⢐⣃⣤⣶⣤⡖⠀⣶⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⢝⠀⣩⠁⠛⡻⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠢⠄⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⠍⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠚⠛⠛⠋⠙⠛⠛⠶⠒⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠈⠱⡄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠒⠉⠀⠀⠀⠀⡴⠋⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⡤⠔⠒⠒⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⢤⡀⠘⣧
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠹⡀⠘⣧
⠀⠀⣰⠃⣠⠖⠉⠉⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢰⠁⠀⠸⣆
⠀⢠⡏⢰⠁⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢄⠤⠖⠁⠀⠀⠀⢹⡀
⠀⣼⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⢠⡇⠀⠈⢦⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⢀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠃⠀⠀⠀⠉⠒⠂⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡞
⠘⣧⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠀⠀⢳⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⡇⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠈⠳⣤⣀⡀⠀⠀⣀⣈⣳⣤⡤⠤⠴⠶⠚⠛⠛⠛⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠛⠓⠒⠒⠶⠦⠶⢮⣤⣀⣠⣴⡋⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠉⠛⠻⠿⠭⢤⣤⣤⣤⡤⠤⠤⠤⠤⠤⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠤⢤⣤⣤⡤⣤⣤⣤⣤⢤⠭⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠦⠿⠶⠤⣤⣤⣄⣀⣤⣤⠤⠞⠛⠒⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣴⠇⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣻⣏⣾⣧⣀⠉⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢆⠛⡟⢯⡛⡛⢏⠂⠀⠀⠀⠀⠀⠀⠀⠘⠙⣾⣷⡋⣿⠀⠀⠁⠀⠠⡰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⣶⣨⣆⡓⠄⠸⠀⠀⢠⠀⠀⠀⠀⢀⢰⣶⣏⣿⡷⣶⡤⣄⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⣼⣿⣿⡿⢩⡇⠀⠀⠀⠀⠘⠆⠀⠀⠀⢸⣆⢻⣿⡿⠁⣾⣧⡹⢧⡀⠀⠠⠀⠠⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⡇⢙⡿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⢿⣿⣿⣶⣿⠙⣿⣿⡷⠀⠀⠀⠀⠈⢹⢷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢨⣿⣿⣿⣷⣤⣴⡼⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢲⣶⣤⣿⣿⣏⠀⠈⢻⣷⡀⠀⠀⠀⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡀⠈⠀⣿⣿⣿⣿⡟⣅⣾⣿⡇⠀⠀⢀⣀⣄⠀⠈⡘⡆⠀⡸⠞⢿⣿⣽⣿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠚⠻⣷
⠀⠀⠀⠀⢠⣀⡄⠀⠠⠀⠀⠉⠀⠀⠠⠟⠉⠀⠡⢂⣽⣿⣿⡇⠀⠀⢸⣉⠉⠄⠀⠸⡀⢀⠃⡘⠚⠉⢩⢿⡟⣷⠀⠀⠘⢿⠄⠀⠀⠀⣠⡍
⠀⠀⠀⠀⠈⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣿⣿⠉⣿⣷⠀⠀⠀⠇⠀⣶⡆⠀⣅⢨⠀⣵⠈⢰⢘⡭⡇⡟⠀⠀⡉⠈⢻⠂⠀⠀⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠞⢩⡿⠿⡿⡟⠀⠹⣷⣄⠀⠀⡀⠀⠀⢠⡀⠹⡇⠀⠹⠼⣒⡉⣈⠇⡇⠀⠀⣉⣹⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣴⡀⠱⣄⠱⣾⣦⠀⠘⠻⣴⣄⣁⠈⠀⣾⠳⠀⣿⡄⡀⢸⠖⢹⠉⢸⠁⠀⠀⢹⣿⢿⠒⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⠈⠳⠿⣿⡇⠐⠂⠘⢿⣿⣿⠒⠻⡿⣶⢿⣿⣴⣾⠀⠘⠴⠇⠀⢸⡀⠉⠇⢈⠀⠉⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢻⣷⣿⣿⠷⣄⡀⠈⣿⡀⠀⠀⠀⣷⡄⡀⠀⢱⡜⣾⠛⠛⠃⠀⠀⠀⢸⣦⡼⠏⢠⠀⢈⡀⢠⣷⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢘⠁⢹⡟⠀⠈⠙⣦⢨⣷⠀⠀⠀⢹⡻⣷⡄⠘⣿⣏⢰⡀⠀⠀⠀⠀⠈⢿⣿⣦⣶⡤⠾⠀⠈⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠊⠀⡞⠀⠀⠀⠀⣘⡃⠏⠀⠀⠀⠀⣿⣻⣷⠀⠉⢉⡜⣿⡌⠁⠀⠈⠀⠈⣿⣿⠁⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣡⡤⠖⠚⠉⠀⠀⠀⠀⠀⢸⠃⣷⠈⠐⠛⠚⠋⠻⢁⣀⡴⢿⡿⠃⣯⣦⣬⠉⣷⣄⠸⣽⣿⣇⣀⡀⣿⣧⣶⣷⡆
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠶⠿⠿⠛⠻⠿⠏⠀⣿⣿⣿⣷⣧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠙⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀

⣠⣤⣤⣤⣤⣀⣀⣄⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣄⣠⣤⣤⣀⣀⣄⣀⣀⣀⣀⣤⣤⣤⣀⣠⣤⣤⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣠⣀⣀⣀
⣿⣿⣏⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⢠⡀⠑⠈⠁⠈⠀⣀⣀⠀⠀⠀⠀⠀⠦⠰⠢⠠⠀⠀⠀⠀⠀⠠⠈⠁⠢⠄⠀⡀
⠀⠠⠀⠈⠀⠘⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠄⠀⠀⠂⠀⠐⠤⠀⠄
⠀⠀⢀⠀⠀⠀⠠⣀⡠⠀⠀⠀⡰⠀⠀⠀⠲⠇⠀⠀⠾⠀⠀⠀⠂⠀⠀⠱⠔⠀⠀⠘⠂⠀⠀⢨⠄⠀⢤⠇⠀⢀⣨⠁
⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠀⠀⠈⠓⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⡀⠀⠢⠌⠀⠂⢤⠀⡼⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⢀⡀⢠⠠⠀⠶⡒⠇
⠀⠀⠐⠁⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠀⠀⢢⣄⠀⠦⠀⠀⠀⡈⠀⠀⢁⠀⠀⠀⢠⠇
⠀⠀⠀⠈⠀⠀⠀⠑⠦⠤⠤⠞⠀⠀⠀⠀⠈⠃⠀⠠⠴⠀⠀⠀⠀⠀⠠⣸⢿⠀⠀⠀⠀⠀⠐⠄⠠⠜⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠠⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠃
⠀⠀⠀⠈⠐⠂⠀⠀⡀⠀⠀⠀⠐⠀⠀⠈⠃⠀⠀⠄⠄⣤⠄⠄⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠂⠂⢸⡇⠒⡄⠆
⠀⠀⠀⠀⠀⡀⡀⠀⠁⠰⠀⠂⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠜⢹⠓⠚⠩⣤⡆⠀⠀⠀⡀⠀⣺⣁⠔⢀⠀⠄
⠀⠀⠀⠈⠀⠙⠃⠉⠀⠀⠀⠀⠄⠀⠠⡴⠃⠀⠀⠀⠀⠙⠊⠀⠀⠘⠂⠀⠘⠶⠚⠐⠚⠀⠀⠀⠉⢀⣤⣽⠇⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⢽⡯⠤⠤
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠘⠀⡀⠀⠀⠈⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠔⣁⠀⡠⠊⠀⠀⠈⠀⠀⠇⡀⡱⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠜⡅⠀⡠⠋⠁⣒⠤⠶⠒⠀⠀⠈⡄⠠⠁⢀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠊⠀⡠⠊⠀⠀⠀⠀⢀⡔⠀⠀⢀⢀⡄⣀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⣆⠈⠀⠀⠀⢠⡄⢀⠨⠃⠀⡠⠠⡈⠀⣁⠀⢀⠐⠸⡤⢠⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠘⢷⡟⣿⢀⡤⠈⠒⠀⠰⠋⠁⠀⠀⠆⠀⠇⠀⢃⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢆⡍⠉⣩⡻⡵⢀⠈⠁⠀⠀⢀⡀⠀⠀⠘⠀⠘⡤⠚
⠀⠀⠀⠀⠀⠓⢀⠘⠤⢀⡁⠀⠐⠁⢀⠄⠉⠀⠀⢏⣱⣄⠀⠀⠀⠁⠀⠀⢀⡠⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠪⠔⠀⢀⡤⠀⢅⠈⡔⠀⠀⠀⠙⠮⣢⡀⠀⠀⠀⡀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠑⢊⠀⢠⠟⠁⠒⠀⠀⠀⠀⠘⠍⠆
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠒⠠⠤⠀⠀⢀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠧⠤⠀⣀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⡀⡐⣾⢖⡀⡀⠀⠠⢼⣀⣀⡠⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠻⠂⠀⠀⠀⠀⠀⠳⠋⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⣋⣽⣿⣧⠴⣶⣤⣀
⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⣰⣿⣆⣀⣠⣤⣁⣤⣠⣶⣄⣀⣄⣉⣈⠁⢀⣀⠐⠂⠈⠛⣦⣄
⠀⠀⠀⠀⣰⣏⡴⡛⠛⠉⠀⠈⠪⣻⠁⠙⡭⡹⢹⣹⢹⡙⡏⣉⢋⢋⠟⡯⢴⣦⣀⡉⡤⠈⢣⡀
⠀⠀⠀⠀⡟⠁⠀⠀⠄⢀⠂⠄⡐⠈⢶⣄⠌⣱⡾⣏⣧⣕⣼⣏⣡⣪⢊⡴⠓⠉⡹⠶⣗⣄⠀⢱⡀
⠀⠀⠀⠀⢷⡀⡀⠀⠀⠠⠬⣒⣬⣑⣦⣜⡷⠿⠷⢿⣿⣿⣿⣿⣥⣿⣥⢤⣴⣮⣴⣵⣮⠿⣌⣋⠡
⠀⠀⠀⠀⠀⠻⣤⠒⣒⠺⠷⢶⣬⣽⣻⣿⡇⠀⠀⢰⡟⢿⣿⣶⣦⣤⣄⣀⡤⠍⠉⢉⡬⠟⢻⣥⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠒⢬⣥⠶⢒⣛⣻⣿⠀⠀⠀⢸⠀⢸⣧⣙⠡⡉⠙⡻⠽⢍⠛⡻⠶⠄⣼⡛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢾⣿⠀⠀⠀⠀⡆⢻⣿⣿⡀⠀⠀⠈⠂⡀⣑⣌⣢⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⡇⣼⠻⢽⣧⠴⠖⠚⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠠⣶⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⣠⠂⣤⡀⡻⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣰⣵⣿⣾⣿⣿⣷⣮⣬⣢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣀⡀⣤⣀⣼⠟⠋⠁⠸⠸⡟⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠇⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢈⣗⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⡏⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⣀⣻⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠐⠂
⠈⠀⢀⣄⣀
⢸⣏⣑⡀⠌⢳⡀
⢨⢹⣿⣿⣿⣾⣟⣤⡤⡄⢀
⠈⠳⢹⣿⡿⠿⣿⣿⣉⡀⢼⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⢀⠟⡟⠳⠚⠉⠟⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠰⠁⠀⠘⣆⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢨⣗⠀⡀⠠⠈
⢀⠀⠀⠀⠈⢧⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠊⠀⣀⡹⠚⠁⠀⠀⠀⢀⡀
⠘⠀⠀⠀⠀⠈⢣⡀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⣀⢶⣀⣾⣿⣦⣆⣀⣀⣹⠃⣀⠀⠀⠀⠀⠈⡁
⢀⠀⠀⠀⠀⠀⠀⣷⣾⣇⡀⠀⠀⣀⡛⠄⠊⠀⠀⠀⢭⣾⠿⢿⣭⣹⣿⣽⣿⣙⡛⠛⡷⠀⠀⠀⢰⡇
⢨⣄⣠⣀⣤⣤⣤⣤⣤⡾⣵⣶⣮⡁⠀⠀⠀⠀⠀⠀⢦⢻⣾⣿⣿⠿⢷⣲⠿⠿⣿⣿⡄⠠⠀⠀⣸⠃
⢸⣟⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣟⣷⠀⠀⢺⠀⠀⠲⢊⣾⣿⣿⡟⢰⠋⣽⣿⠀⢻⣿⣷⠀⣶⠟⠁
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⡋⣁⣀⣀⣾⣰⡀⠀⣾⣿⣿⣿⡇⠘⠿⠿⠛⠀⢸⣿⣿⣇⢰⣦⣤⣤⣴
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡎⣿⣿⣿⡿⣛⣿⣟⢰⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⢸⡿⣿⣿⠀⣿⣿⣿⣿
⢸⡿⣽⣿⣿⣿⣿⣿⢋⣾⣧⢿⢹⡋⣯⣞⣿⣿⡇⠸⠿⢿⣿⣿⡇⠀⠰⣆⠀⠀⢸⡿⠋⡀⢨⣿⣿⡇⢸
⢸⣿⣿⣭⢻⡿⣿⡋⣿⠟⣿⠘⡾⠇⣿⣿⣿⣿⣿⢈⣶⣦⡈⠛⡇⠀⢀⡟⠀⠀⠀⢀⣼⣡⡾⣿⣿⣷⠘
⢸⡿⢻⣿⣿⣿⣿⡿⣶⡛⢙⡇⢷⣤⣿⣿⣿⣿⣿⣿⠿⠷⠛⠂⠢⠀⡉⣿⡦⠀⠈⠘⠫⠻⠀⣿⣿⣿⣦
⢸⡶⠟⢛⣿⣇⣿⣿⠉⢳⢾⣯⣼⣿⣿⣿⠾⣿⣿⣿⣶⣾⢷⣦⢀⣨⣤⣿⣶⣷⣶⠂⠲⣿⠀⣹⣿⡿⣾
⢈⠱⡿⢙⣛⣿⣿⣟⡾⢟⢻⣥⣿⣿⣥⡌⠀⣿⣿⡿⠻⣿⣢⡟⣿⣿⣭⡿⠹⣾⣩⣶⣿⠏⠐⠛⠛⢳⣿
⢸⡿⣿⣿⣿⣿⣿⡻⡿⠈⣍⣳⣿⡀⡟⠀⡄⢿⣌⡁⣈⣫⣭⣸⡿⣿⣯⣀⢾⣬⣹⣿⣅⣴⣶⢾⡇⣻⣿
⢸⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣥⢹⣨⡟⢿⣿⣤⣟⠛⠹⣯⣿⡌⠻⠛⢻⡟⣙⣿⣿⣷⠄⡼⣝⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠘⣏⠿⣾⣿⣍⢿⣧⣾⣬⣿⣿⣿⠀⣶⡌⣿⡛⠛⢿⣦⢡⣿⣿⣿
⠨⡄⢻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡄⢻⠿⣓⣭⣧⢀⡿⣍⡐⢫⡅⢻⣷⣾⣿⢿⠀⢀⣿⡏⢹⣿⣿⣿
⢸⡿⠞⣾⣿⣉⣼⡿⠉⠁⢈⣻⣿⣛⣿⠸⣿⣿⣿⣿⣿⣶⣬⡛⠀⠑⡆⢹⣿⣯⢯⣶⢿⡿⣜⣿⠙⣿⣿
⠠⢱⣾⣿⣿⣿⠻⣇⣤⢿⣿⠟⢳⣿⣿⡇⢿⣿⢻⣿⣿⠇⠙⣿⡈⡏⢰⣨⣮⢱⣶⣿⢿⣧⡿⣿⣤⠉⣿
⢘⣸⣿⣿⣿⡿⢿⣿⣤⣀⢘⡰⢿⣿⣛⣯⠸⣿⣛⣽⣿⠁⣲⣿⣷⣻⡦⢤⣴⣾⡿⣁⣌⣻⠃⣿⣗⠘⣻
⢸⣿⣿⡟⢋⠐⢈⢹⣿⣿⣿⣿⣾⠿⣿⣿⣤⣿⣿⣿⡏⠸⣿⣿⣿⣿⣷⣿⣿⣿⣗⢠⣿⣿⣿⣿⣿⡄⠻
⠀⣸⣿⡾⣦⣴⣯⣿⣿⡟⢽⣯⡟⣰⡿⣿⠋⠙⢿⣿⣙⡛⠿⣿⢿⣿⣍⣻⣿⡿⢿⢿⣿⣿⣿⣿⣾⣝⣢
⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣤⣌⡛⢿⠋⠀⣴⣿⣿⣿⣿⣿⡁⢀⡝⢿⣿⣷⣦⡌⢉⣽⣿⣿⣿⣿⣿⣽
⢸⣿⣿⣿⣿⣩⡶⢟⣛⣿⣿⣿⠿⠟⠛⢤⣶⣟⣿⣿⣿⠟⢿⡛⢫⣄⣶⣿⣿⣿⠇⣸⣼⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⡿⠛⣿⣿⣷⣿⠉⠀⠀⡀⢠⣾⣿⡿⢸⣿⡏⠐⠀⢀⠶⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠰⠿⠛⣩⣿⣄⣠⣷⣼⡿⠋⣸⡼⣧⠏⡀⣶⣶⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣧⣴⣦⣶⣿⣿⣿⣿⣿⣏⣠⣾⣿⣷⣦⣶⣀⣿⢿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠶⢖⠛⠛⠛⠒⠶⠦⣼⣍⡉⠉⠉⠉⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠙⠳⢶⣄⡑⠄⠀⠀⠀⠀⠀⠉⠛⣧⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⡁⡠⠀⠀⠀⠈⠹⡗⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠤⠄⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⠛⠛⠛⠛⠓⠲⢦⣧⠀⠀⠀⢈⡉⣀⠤⠀⠚⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⡛⠒⢦⠀⠀⠀⢀⡤⠖⠿⠃⠀⡾⠛⠙⠙⠛⠛⠛⠳⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠉⠃⠀⠀⣼⣷⠞⠛⠋⠉⠛⢛⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠉⠛⠷⣤⣸⣆⠀⢹⡛⣶⡄⠀⠀⡤⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣴⣛⡫⠭⠤⠭⠶⢽⣷⣴⣇⣀⣀⣀⡀⠀⠀⠀⠀⠒⠠⠤⠠⠬⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠒⠉⠀⠀⠀⠀⠀⢀⢷⡿⠛⠉⠉⠉⠉⠙⠳⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠘⠟⠉⠈⠁⠀⠀⠀⠀⠀⢰⢡⡿⠤⡀⠀⠀⠀⠀⢀⢀⠤⠽⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⢘⡆⠀⠀⠀⡼⠧⣄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠤⠤⡈⠀⠀⠀⠰⠁⢀⣬⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠀⠠⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠶⠶⠶⠶⠶⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠙⢦⠀⠈⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⡊⠉⢓⡦⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠒⠋⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⡥⠶⠶⠤⣄⠀⠉⠛⠉⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⢡⠊⠁⠀⠀⠀⠀⢈⡇⠀⠀⢀⡔⠋⠁⠀⠀⠈⠉⠉⢳⠀⠀⠀⢀⡖⠉⠀⠀⠈⠉⠳⣄⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠸⣄⣀⣠⣤⠤⠖⠋⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠴⠶⠒⠛⠉⠁⠀⠀⠀⠀⠀⠢⣄⣀⣀⣠⠴⠚⠁⠀⠀⠀⣀⠙⣦
⠀⠀⠀⠀⢰⠇⠀⠀⢀⣠⠖⠚⠋⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⠙⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠹⡜⡆
⠀⠀⠀⠀⡿⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠘⠦⠤⠇⢿
⠀⠀⠀⠀⣧⠀⠀⠀⠳⢄⠀⠀⢀⣀⡤⠖⠛⣉⣀⡤⠴⢶⡖⠒⢲⠒⢲⠒⡖⠚⡟⢛⠖⠒⢲⡶⠦⣤⣀⠀⠸⡁⠈⠙⢦⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⣴⡋⠉⠑⢤⡀⠀⠹⡄⢸⡆⢸⠀⡇⡼⢠⠎⢀⡴⢋⡴⠊⣠⠜⠛⢦⣌⠓⠒⠋⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⢧⡀⢸⠉⠙⡆⠀⣠⠛⠥⢤⣈⣉⠒⠤⣌⡓⢤⡘⢆⠇⠘⣀⣠⣣⠋⣰⡫⢞⡡⢖⣉⡤⠖⠋⢁⣈⣷⡀⠀⢠⠒⣹⣴⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⠁⢸⡃⠀⠒⠒⠒⠬⢭⣷⣦⣭⣝⣿⠶⠛⠋⠉⠉⠉⠛⠛⠶⣟⣋⣭⣤⣶⡾⠭⠭⠥⠐⣷⠀⠧⢴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠈⢯⡉⢉⣉⣭⣭⠭⠭⢽⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠵⠒⠒⠒⠒⢒⣒⣰⣏⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠽⣶⣤⣤⣒⣚⣉⣡⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠶⠶⠶⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡷⠀⣸⢒⢂⠀⢖⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⠀⡄⠀⢀⣀⣀⢁⡼⢢⣹⣍⠰⠸⣷⣹⣼⣿⡀⠀⢠⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣿⣾⢾⡁⠈⢰⠿⢿⢻⣿⣧⣴⡿⡿⣏⡀⠿⣭⣟⣻⣒⡈⣿⠣⣶⣟⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠹⣟⣏⣳⢀⠘⠂⡀⣴⣯⣧⡂⢸⣟⡂⢛⣿⠻⣯⣿⣿⣯⣾⣾⡀⠸⠟⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣼⣿⣍⣻⣤⢄⠞⡋⢻⡹⣿⡷⡽⡿⠟⠸⡷⡙⣯⡽⢶⠍⢘⠻⠯⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⠊⢉⢋⡽⣾⡾⠫⠁⡾⣇⡡⣰⣕⣃⣾⢄⢣⣶⢷⣼⢷⡠⣾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣼⢷⣦⣼⡌⢼⣯⣤⢸⣧⣙⠀⢼⠉⠷⣯⠘⡿⠎⠉⠛⠁⠿⠝⣧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡛⠙⡩⠰⠒⠀⣡⠞⠓⠈⠷⣶⣶⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⢆⠀⠉⠀⠀⠀⠀⠛⠉⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣜⣾⡷⢠⡀⣴⣰⣠⣀⣸⣤⣼⣠⣼⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⣲⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣛⣿⡏⢹⣿⣿⣯⣜⡍⠗⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⣿⣿⣿⢿⡿⢩⣿⣽⣿⣿⣯⣿⣟⣤⣿⣿⣵⣿⣯⣿⣿⣷⣯⣥⣎⡽⡈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣾⣿⣿⣿⣧⣾⣷⣾⣿⡿⣟⣿⣿⣿⣿⣿⣼⣿⣭⣿⣯⢿⡿⢿⣿⣿⣷⣿⣾⣿⣷⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⣿⣿⣿⣿⣟⣾⣿⣿⣟⡙⡿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⡿⢣⡝⢿⣯⣿⣟⣿⣟⣽⠯
⠀⠀⠀⠀⠀⠠⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣻⣽⢟⣿⣿⢟⡻⣟⣻⡽⡓⣿⣛⣛⣿⣿⣽⣿⡿⣗⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣯⣾⣿⣿⣿⣿⢿⣿⣿⣿⡟⣷⣿⡟⣻⣿⣷⣿⣷⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼⠏⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠹⠟⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣟⣾⣿⣿⣽⡾⣿⢿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡖⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⢿⡿⣿⣷⢟⣿⣿⣮⣿⣿⣿⣿⢿⡟⣞⡿⣿⣿⣿⣿⣼⣿⣿⣯⠍⠳⠝⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠿⠻⠟⠁⠈⠉⠻⠿⢿⣿⣼⠛⠓⣻⣿⣿⣿⣿⡿⠿⠏⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠉⠋⠉⠉

⠀⠀⠀⠀⢀⠀⣇⠀⢀⡷⠀⢀⡤⠀⠀⠀⢠⡀⠀⡄
⠀⠀⢧⣀⣿⣄⣿⣶⣿⣿⣿⢛⠁⣀⡀⠀⢀⣿⣴⣇⣠⠖
⠀⠀⠀⠉⠛⠿⠿⣿⣿⣧⣹⣿⣠⣿⣧⣶⣿⣿⣿⣿⣧⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣛⣯⣁⡀
⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠻⣿⣿⡿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢿⡿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢻⣿⣿⡿⠋⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠈⢿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⢹⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠸⣿⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿⡇⠀⢀⣼⣏⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣧⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⢸⣿⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣟⣿⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠄⠀⢠⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢴⠶⣚⠛⠂⢤⣶⠖⣿⡵⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣀⠀⣀⡤⠄⠒⠂⠀⠐⠁⠀⠀⠉⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠝⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠑⢦⣄⢀⡀
⠀⠀⠀⠀⠃⠀⢀⣰⠆⢐⡢⠠⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠉⠁⠀⠁
⠀⠀⠀⢀⡠⠔⠋⠱⠚⣉⣀⣞⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⣀⡤⠄
⠀⠀⠀⠉⠀⠀⢀⡀⠀⢉⢀⢈⠄⡀⣀⠀⢔⠀⣴⠃⢀⣀⣀⣀⣐⣦⡀⢠⣤⣄⣀⣈⠣⡤⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠛⠁⢠⣇⣀⣶⣿⣷⣿⣠⣾⣦⡟⠰⣿⣿⣿⣿⣿⣿⣿⣏⠙⢿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⣿⣿⠟⠁⠈⢿⣿⣿⠃⠀⣿⣿⣿⣿⡿⠟⠋⢈⣾⣿⣭⣿⢻⡁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠈⠻⣭⡀⠀⠈⠛⠛⠉⠀⠀⠀⣾⣿⣿⠀⠀⠀⢳⠀⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⢿⣿⣶⢰⠶⣤⣶⣾⡤⠀⠀⠉⣁⣀⠀⠲⠄⠉⠉⠉⢰⣇⣀⡍⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠟⠀⠀⣼⠿⣿⣿⢿⣶⣿⣿⠄⠙⠒⠂⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⣀⣶⣤⡄⢠⣿⣿⣿⡟⣾⣿⡆⢻⣿⡞⣿⣿⠄⠀⠀⣾⡇⠨⠨⠠⠤⢹⡿⠛⠿⠆⠐⠦⠤⠤⡄⠶⠒⠈⠁⠁
⠀⠀⠀⠀⠹⡏⠀⠈⠉⠁⠀⠈⢻⣿⡜⣿⣷⡄⢹⣿⡈⠻⡀⡀⠀⠁⠐⠀⠀⢀⠀⣾⢁⣿⣤⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣿⣿⣷⣿⣼⠷⠀⢢⡁⠀⠀⠀⠀⠀⠀⠀⢧⠸⣿⣏⣼⣤⣴⠂⣀⣤⣶⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣾⣿⡻⣿⣿⣿⡿⠋⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⣼⣶⡿⡟⠛⠛⠁⠾⠟⠿⠿⠋
⠀⠀⠀⠀⠀⠀⢰⡶⣾⣿⣷⣦⣍⠁⠉⠉⢋⡿⢶⣦⢄⣤⣤⣤⡾⣷⣾⠾⠿⠻⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠌⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣷⠟⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣇⠀⠀⠀⢠
⠀⠀⠀⢀⡀⠀⣼⣯⠅⠀⠀⠀⠀⠀⢠⣷⡄⠀⠀⢰⣿⣿⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣆⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣧
⠀⠀⠀⢺⣷⢀⢚⣷⣆⢠⣾⠿⠂⠀⠈⣿⣿⠀⠀⢀⣿⣿⡆⠀⣴⣿⡇⠀⠀⠀⠀⠁⠚⠇⡠⢀⠀⠃⠀⠀⠀⠀⣤⣠⡄⠀⠰⣶⣿⣧⡦
⠀⠀⠀⠨⣿⣧⠀⣬⣿⣿⠏⠀⠀⠀⠀⣿⣿⣷⠀⢿⣿⣿⠇⠈⢿⣿⣷⠀⠀⠀⠀⠠⡀⠸⣀⠌⠀⠀⠆⠀⠀⠰⣿⣿⡶⠆⢶⣾⣿⣧⡄⢠⣰⠆
⠀⠀⠀⠀⠀⣟⣷⣟⠃⠀⠀⠀⠀⠀⠀⢘⣿⣿⡄⣸⣿⣿⣆⢠⣿⣿⡃⠀⠀⠀⠀⠀⣸⠃⡝⠀⢲⡀⠀⠀⠀⠘⣻⣿⣟⠃⣴⣾⣿⠏⠠⣾⠿⠍
⠀⠀⠀⠀⠀⠺⣷⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠻⣿⣿⡿⠘⣿⠿⠃⠀⠀⠀⠀⠀⠘⡀⢃⡄⠞⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠈⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠘⠻⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠟⡀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠰⡄⠀⢸⡄⠀⢰⠀⠀⠀⠀⠀⢦⣞⠀⠠⣥⠄⠀⠠⣗⡀⠀⠀⠀⠀⠀⠀⣼⡟⣠⡇⠀⠀⠀⠀⠀⠀⠈⣙⢋⡁⠀⠰⢦⣨⡅
⠀⠀⠀⠸⡀⠀⢹⡀⣼⠁⠀⣾⠀⠀⠀⠀⢨⡿⠇⠀⣽⡿⠀⠀⢈⣛⡃⠀⠀⠀⠀⠀⢸⣿⢰⡿⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣥⣛⣤⢤⣴⠄⡤
⠀⠀⠀⢀⠳⡄⣼⣧⣿⢀⡾⢃⡞⠀⠀⠀⠐⡿⢿⡀⢲⡯⢇⠨⢿⡓⠀⠀⠀⠀⠀⢰⡌⣿⡆⣿⠀⡖⠀⠀⠀⠀⠀⠰⣈⣁⣶⠖⡶⠦⢾⣗⡛
⠀⠀⠀⠀⣧⢹⣿⣿⣿⢏⡴⠋⠀⠀⠀⠀⠀⠨⠭⠄⣰⠯⠭⠈⣷⣿⠀⠀⠀⠀⢀⡾⣱⣿⢃⡿⢸⡿⠀⠀⠀⠀⢈⣝⣿⣤⣿⣟⠴⠦⢶⡿⠃
⠀⠀⠀⠀⢸⡎⣿⣿⡇⣸⠁⠀⠀⠀⠀⠀⠀⠸⠿⢗⡮⣭⠁⣰⢿⡛⠀⠀⠀⠀⠘⣇⣿⡇⣾⢱⡿⠁⠀⠀⠀⠀⠀⠒⣊⢶⡶⢾⣛⡳⠶⣟⠛
⠀⠀⠀⠀⠀⠻⣾⣿⣧⠟⠀⠀⠀⠀⠀⠀⠀⢩⡽⣫⣭⣗⣐⣻⡟⠀⠀⠀⠀⠀⠀⠹⣿⣇⣿⣸⠇⠀⠀⠀⠀⠀⠀⠀⠱⢚⡿⠚⠋⠙⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⢫⣧⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⣠⡇⠐⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⣴⣾⠋⠀⠰⠷⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣋⣭⣵⣾⣯⡐⠷⠀⠀⡀⠀⣿⣿⣿⣿⣦⡀⠀⠘⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠛⠉⣿⣦⣄⠈⠁⠀⠀⠉⠛⠿⣿⣿⣶⣶⢟⣿⠇⠀⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠅⠈⠀⠃⡸⠋⡣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣏⣀⣠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣫⣿⠃⢀⠀⠐⠜⠂⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣽⣿⡿⢛⡛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠠⠀⠤⠜⠀⢀⡠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢹⡿⣷⣌⢻⣿⣿⣿⠿⠿⢿⣦⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⡀⠀⠀⠐⠀⠀⠠⠄⠅⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣾⣿⡟⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠵⡀⢄⣴⣶⣿⣿⠥⢀⠀⠄⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⣿⡿⣏⣿⡇⠀⠀⣀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣴⢬⣿⣿⣝⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⢰⣿⣿⠀⠻⣾⣿⣦⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠋⠈⢿⣝⡻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⣿⣟⢇⠀⠀⠉⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠉⠛⣿⣷⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠿⠟⢻⣮⡓⠦⡤⣶⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⣴⡿⠟⠿⣶⡀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣾⣿⠃⣴⢿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠹⠦⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⢠⣾⡿⠙⣷⠀⣀⣤⣤⡴⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⣴⣿⠟⠀⢀⣿⣿⣿⠟⠁⢀⣤⣾⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⡲⠶⢶⣶⣶⠿⠟⠋⠀⢀⣠⣾⡿⠋⠀⣠⣾⣿⠟⠁⣠⣶⣿⣿⡿⠛⠁⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠳⣝⡶⣤⣤⣤⣤⣤⣴⣶⡿⠟⠋⣠⣴⣿⠿⠋⢁⣴⣾⣿⣿⠟⠋⠀⢀⣴⠶⠶⢤⡈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣛⠶⣯⣭⣍⣉⣩⣥⣤⡶⠿⠛⠉⢁⣤⣾⣿⡿⠟⠉⠀⠀⠀⠀⣾⡇⢰⣆⣼⡇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣍⣉⣉⣁⣀⣤⣤⣶⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣉⣉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣽⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢸⣧⣿⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⢁⠀⠀⢀
⠀⠀⠀⠀⣦⣀⣰⣦⠀⠧⡄⢇⣼⣿⣿⣿⣛⣀⣀⣀⣀⣂⣀⣀⣀⣐⣒⣒⣒⣒⣒⣒⣒⣒⣐⡒⢐⣳⣿⣿⣾⣷⠠⠶⢴⡆
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⢻⣧⣦
⠀⠀⠀⠀⠀⣶⣿⣿⡇⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢸⣿⡏⢰
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸
⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⣰⡟⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠿⣿⣁⣀⠀⣀⠀⠀⣴⡄⠀⠀⠀⠀⢸⣿⣧⠘⢠⡄
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⡿⠃⢀⣿⡶⠛⠉⠀⠀⢀⣀⣀⡀⡰⠃⠀⠀⠀⠀⠰⢾⣿⡏⢺⣿⣧⡐⡇⡆⠀⠀⠀⠀⢸⣿⣿⠀⢸⡇
⠀⠀⠀⠀⣤⢸⣿⣿⡇⠀⢀⡴⠃⣰⠿⣿⣤⠖⠂⠠⡀⠉⣡⠟⣿⣧⣀⣀⠀⠀⠀⠀⢰⣿⣷⣿⡟⡛⣿⣿⣇⠀⠀⢸⡆⢸⣿⣿⠀⢸⡇
⠀⠀⠀⠀⣿⢸⣿⣿⡇⠀⣾⠇⠀⠃⡴⢿⣧⣤⣄⠀⢁⡞⢁⣴⡿⣿⣿⣧⣦⣀⣀⣀⠀⢨⣽⣿⣷⣿⣿⣿⣿⡧⣶⢸⠇⢸⣿⣿⠀⢸⡇
⠀⠀⠀⠀⣿⢸⣿⣿⡇⢠⣿⣶⣷⠎⢀⣿⣿⣿⣿⣿⢏⡀⣻⣿⣾⠛⠛⡛⣿⣿⡯⠍⠁⠘⠍⣻⣿⣿⣿⣿⢿⣇⣿⣷⠄⢸⣿⡿⠀⠸⠇
⠀⠀⠀⠀⣿⢸⣿⣿⡇⢀⣼⣿⠃⠀⢸⠿⠿⣿⣿⠿⣿⣿⠿⠿⠿⠿⠭⠿⣿⣿⣿⣶⣤⡀⠀⠿⣿⣿⣿⣿⣿⣿⣯⣥⠄⢸⣿⡇
⠀⠀⠀⠀⣿⢸⣿⣿⡇⠘⣿⡁⢠⢀⣠⣷⣶⣶⣿⣶⢋⣹⣶⣦⠀⠀⠀⠀⠀⢈⡿⠛⠛⠛⠷⣦⡽⣿⣿⣿⠿⠻⣿⠏⠀⢸⣿⢇
⠀⠀⠀⠀⣿⢸⣿⣿⡇⢀⣤⡀⠤⢿⡁⣻⣿⣿⣏⣤⣿⣶⣮⡉⠛⢶⣄⠀⠳⢿⡷⠀⠶⠀⠀⠀⠛⠻⢿⣿⣾⣿⠛⡃⠀⢸⡿⣼
⠀⠀⠀⠀⠿⢸⣿⣿⡇⢸⣿⣅⣀⣾⣟⣙⣿⣿⣿⡷⠛⣭⡍⢹⠟⠁⠈⠧⠀⣾⣷⣦⡄⣀⠀⠀⠀⠀⠀⠉⠻⠋⠀⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠠⣿⣏⣻⡿⣿⡿⠛⠉⠁⠀⠀⠉⠀⢸⣆⠀⠀⠀⣾⣏⣽⣹⡉⠅⠀⠐⣶⣄⣴⣴⣾⣶⡇⠰⠀⢸⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢠⣿⡤⠹⣿⣵⣷⠄⠀⣠⣦⣤⣀⣤⣾⣿⠀⢳⡀⢻⣿⣿⣽⣷⣾⣱⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠈⣷⢀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠄
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠈⣻⣿⣦⣤⣿⣿⣿⠿⣿⣿⣿⣿⡟⠣⢨⡿⢦⡘⠉⣶⢠⣿⠿⣿⢿⣿⣿⣿⣿⣿⢻⣿⠙⡏⠀⢸⣿⡃
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣧⠙⢁⡴⢏⡽⠟⠋⣀⣉⡛⠻⣾⣿⣾⣡⡿⣻⡿⢋⣌⡿⠹⣷⠸⣿⡄⣿⣾⣗⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣏⠿⢷⣶⣄⣰⣎⣩⠿⣿⣄⣹⣿⣽⣿⢿⣫⣶⣯⣥⣾⣷⣽⢶⣽⢿⣧⢿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢘⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⣿⣿⣷⣦⣤⣙⣿⣦⣬⣷⣾⣿⣻⣿⣿⣿⣿⣿⣧⣽⡦⠟⡛⠛⠀⢸⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⠿⣦⣿⡏⢺⣿⡿⣻⣿⣿⣿⣿⣿⣏⡹⠿⣿⣿⣷⣾⣿⣿⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⡛⠷⣌⣿⡌⢻⣿⣿⣿⢷⣾⣧⡐⣼⣿⡏⢸⣿⣥⣽⣿⣿⡦⢿⣿⡏⠷⣶⣾⣿⣿⣿⣿⣿⠀⢸⣿⡏⢠
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢈⣽⢦⣾⣿⡛⢿⢿⣶⠿⣷⣿⡻⣟⢿⣿⣷⣿⣻⣿⢗⣩⣿⣷⣼⢿⡟⢂⡿⣷⢋⣿⣿⣿⣿⠀⢸⣿⡆⢸
⠀⠀⠀⠀⣤⢻⣿⣿⡇⢸⣿⣭⣝⢿⣿⣶⡛⢻⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⣿⣿⣿⣍⡛⠃⣿⣧⣿⡿⢋⡩⢠⣿⠀⢸⣿⡆⠘
⠀⠀⠀⠀⣿⢸⣿⣿⡇⠘⠿⣿⣿⣾⣿⣿⣿⣿⡟⣿⣿⣿⣿⠿⣿⣿⡿⣻⢿⣿⠿⢿⣿⣽⣿⣾⣿⣋⣴⣾⣿⣶⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⣿⢸⣿⣿⡇⢸⣷⣿⣿⣿⡿⣿⣿⣿⣿⣭⣿⣿⣿⣷⣿⣿⣿⣴⣿⣿⡔⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⣿⢸⣿⣿⡇⠸⢿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⢿⣿⣿⣽⣿⣬⣿⣷⣾⣿⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣸⣿⣿⡇⠈⠙⠛⠛⠘⠛⠋⠘⠛⠛⠋⠉⠛⠛⠛⠛⠉⠛⠈⠁⠛⠛⠛⠛⠃⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢸⣿⡗
⠀⠀⠀⠀⣏⠈⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧
⠀⠀⠀⠀⣿⠀⠾⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠑⠒⠒⠐⠘⠐⠈⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠀⠉⠁⠊⠁⠀⠐⠐⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⠒⠂⠓⠒⠠⣇⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡚⣅⣦⣴⣶⣶⣶⣶⣴⣥⣉⠾⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣼⣷⠿⠟⢉⠋⣦⣤⣤⡀⠉⠙⠿⢷⣽⣧⢂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡤⠀⠀⠀⠀⠀⠀⣸⠋⠁⠀⠀⠠⠄⣿⣿⣿⠇⠀⠀⠀⠀⠈⠛⣗⠨⠀⠁⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡞⠓⡀⠀⠀⡀⠔⠂⠈⠑⠠⣀⠐⠄⠑⠤⠏⠀⠀⠀⡠⠀⡀⠔⠈⠀⠐⠢⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢶⠈⠀⠀⠀⡄⠀⠀⠀⠙⠘⠷⠦⠠⠤⠴⠺⡂⠉⠀⠀⠀⠀⠀⠔⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠑⡀⠀⠰⣱⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢀⠊⠀⠀⠀⣀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠈⢆⠀⠃⠀⠀⢀⠠⠇⠒⠀⠈⣷⡆⠐⠠⠆⡀⠀⠠⠁⠀⠀⠀⣳⣛⣞⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢠⣌⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠠⣾⡀⠀⠀⠀⠀⣰⡟⠀⠀⣠⡴⠟⠐⠣⡀⠀⠀⠀⠁⠿⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠁⠢⣀⠀⠀⠀⠀⠀⣰⠉⠀⠈⠻⣦⡀⠀⠰⣿⠀⠀⢰⡏⠀⠀⠀⢀⣨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢀⣀
⠀⠈⠉⠉⠉⠉⠣⢉⠉⡍⠉⠉⠉⠉⠉⠉⠋⢍⡉⠉⢹⡇⠀⠀⠀⠘⣿⡄⠀⣿⠀⠀⣼⠇⠀⢀⣴⠏⠀⠀⠩⠉⠉⠉⠉⠉⠉⠉⠍⠋⢋⠉⡁⠙⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⡀⠀⠀⠀⠀⢸⠀⠀⠸⠀⠈⠻⢦⣤⣄⡀⠸⢧⣤⣿⣤⣴⡏⣠⡴⠟⠁⠀⠀⠀⠀⢂⡀⠀⠒⠀⠀⠀⠀⢀⠴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠑⠄⡀⠈⢣⠋⠀⡇⠀⠀⠀⠀⠀⠈⠙⠳⡾⠋⠀⠀⠙⢿⡏⠀⠀⠀⠀⠀⠀⠰⢸⠀⠀⠀⠀⢀⠤⠊⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠒⠀⠈⠒⢌⠀⠀⣦⣄⣤⣴⠶⠿⠷⠶⠶⣧⠀⠀⠀⠀⣼⣷⣤⣄⣀⠀⠀⠀⣠⣤⠀⢀⡠⠂⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⠜⢄⡃⠀⠀⠀⠀⠀⠀⠀⠉⠂⣅⠉⠉⠀⠀⠀⣀⣤⣤⠾⢷⣦⣴⡾⠻⣦⣄⠀⠉⠛⢿⣟⡉⣨⠔⠉⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢢⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣠⡾⠋⠁⠀⢀⡾⠁⠘⣷⠀⠀⠹⣷⡀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠸⠀⠀⠀⢀⠀⠀⢀⠠⠄⠀⠰⢌⣶⠏⠀⠀⠀⠀⣾⠁⠀⠀⢹⡆⠀⠀⠘⢷⣄⡠⡢⠄⠀⠀⢀⢇⠀⠀⠀⠀⠀⠀⠇⠘⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⣀⡤⠯⠆⣉⢠⡄⠀⠀⡀⠀⠉⢑⠄⠀⠀⢰⡟⠀⠀⠀⢸⡇⠀⠀⢀⠠⡙⠛⠢⢤⠀⠀⠀⠇⠀⠀⡀⠀⠀⡘⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠔⠊⡓⠁⠀⠀⠀⠈⠲⠀⠀⠰⠀⠀⠀⠀⠱⢦⣴⣏⠀⠀⠀⠀⠈⢷⣀⣠⣕⠈⠀⠀⠀⠀⠆⠀⠀⠀⢀⠊⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠲⢼⠀⠀⠀⠀⠀⠀⠀⡇⢶⠃⠀⠀⠀⠐⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠘⡀⠀⠈⢃⠀⠀⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⡀⠀⠀⠀⢀⠴⠀⠎⠀⠀⠀⠀⠠⠑⠉⠻⠿⢿⣿⣿⣿⠿⡿⠿⠛⠍⠊⠄⠀⡀⠀⠀⠡⠀⠀⠈⢄⠀⠘⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠑⣖⠚⢌⠙⢓⠀⠀⠀⠀⢀⡠⠒⠄⣀⠀⠀⠀⢀⣀⣀⣀⠤⡔⠪⠓⢄⠀⠐⢄⠀⠀⠂⠀⡠⠊⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠦⡀⠠⡁⠔⠁⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠉⡁⠀⠀⠀⠀⠉⠢⢌⠢⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠜⠛⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠄⠛⠡⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⢀⡸⡀⠈⠑⠂⠤⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠄⠐⢈⣹⡄⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠁⠀⠀⠀⠀⠐⠈⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠈⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⡀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣾⣶⣶⣾⡻⢻⣿⣷⣖⠀⢠⣄⡀⡀⠀⠀⠀⣤⣀⣤⠀⠛⣿⣸⣿⠸⠇⢸⠉⠁⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⢶⡄⠀⠀⠀⠀⣷⡄⢢⡐⣒⣺⡿⣿⣷⢸⣷⣿⣿⣿⡥⣼⣿⣿⣿⣿⣶⡘⣾⣿⣿⣿⣀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⡉⢻⣿⣿⣿⠄⠻⣷⣰⣿⣿⣧⣼⣿⣿⣾⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⣰⣆⣠⡄
⠀⠀⠀⠀⠀⠀⣠⣶⣤⣄⣠⣿⣿⣷⣿⣿⣿⣿⣦⣔⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣻⣿⣿⣿⠀⣿⡿⠛⠉⠁
⠀⠰⢾⣾⣷⠈⢥⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧⣿⣿⡷⠘⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠏⣽⣴⣷⣿⣿⣿⣏⠀⣤⣿⣇
⠀⠀⠦⢻⣿⣧⡿⠷⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣆⣹⣿⣏⠀⠀⠀⢹⢿⣿⣿⣿⠿⣿⣿⣿⣿⣦⣿⣿⣿⡟⠋⣿⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢻⣧⣦⡉⠛⣻⣽⣿⣿⣿⣿⣿⣴⣤⠀⠀⠀⣿⣿⢛⣿⣿⣿⡿⣿⠛⠹⠋⠉⠉⠉⠉⠛⠉⠉⠀⠀⠀⠀⢀⣀⣤⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢥⣘⣿⣿⣿⣾⠠⣾⣿⣟⣿⣿⢽⣿⣿⣿⣶⣤⣾⡟⣸⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣸⡦⠙⠻⠻⡟
⠀⠀⠀⠀⠀⠀⠀⠦⣾⣿⣿⠋⢻⣯⣿⡼⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣧⣶⣄⣠⣶⣶⣴⣦⣶⣦⣼⣷⣶⣎⣷⣿⡶⠏⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠖⠘⠛⢿⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⣿⣿⡟⠛⠻⣹⣿⣿⣯⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠾⢻⡏⠀⠈⢉⠁⣿⠻⣷⣿⡿⣿⣿⣿⣿⣿⡿⠿⠿⡿⢿⣿⣿⣿⢇⣀⣀⣿⡝⣿⣿⣿⣿⣿⣿⡿⣿⠄
⢀⣶⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡜⢻⣷⣟⢿⣿⣿⣏⣀⣀⣠⡷⠚⠛⠛⢟⣿⢿⣿⣿⡟⠛⣯⠋⠉⠉⠟⠀⠉⠀⠀⠀⢀⡀
⠀⣿⣿⣿⣧⣶⣄⣠⣴⣆⣤⣄⡀⢀⣠⡀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠙⢳⣿⣿⣿⠟⠉⠉⠁⠀⠀⠀⠀⠈⠙⠀⠉⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣆⣀⣿⣦⣤⣤⣀⡅⣀⣠⢀⣿⣿⣇⣄⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣦⣀⠀⣠⡀⣀⣴⣶⣠⡄⠀⠀⣤⡄
⠀⣿⣿⣿⣿⢿⠯⣽⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣯⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠠⠿⠷⠿⠷⠦⠦⠽⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠼⠧⢿⣾⠿⠿⠷⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⡤⠄⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠐
⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠁⠀⠀⠀⠀⢰⠀⠀⠠⣠⡶⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠉⠉⠀⠀⠠⠤⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠉⠉⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠠⠠⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠇⢨⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢎⣁⡀⢢⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⣂⡤⠀⠀⠢⠈⠇⠆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠃⠀⠈⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⡔⡀⠀⡠⡀⠀⠀⠀⠽⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠁⠀⠀⠉⡀⠘⠁⠱⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⠀⡨⠀⠀⠠⠀⢀⡀⠀⠀⠄⠂⠀⠀⠀⢸⠀⢐⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠀⠒⠒⠊⠉⠀⠀⠀⠂⣀⣀⣀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠂⢠
⠀⠀⠀⠀⠤⠦⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠢⠔⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠨⣳⠄

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠠⠀⠀⠀⠀⡠⠐⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⣀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⠊⠀⠀⠀⠒⠲⣄⠀⠀⠀⠀⠀⠀⡀⠀⠓⠠⠀⠨⠄⠂
⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⡠⠒⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡀⠤⠀⠀⠈⠐⠀⠀⠈⠉⠉⠀⣀⡀
⠀⠀⠀⠀⠂⠀⠁⠉⠉⠀⡀⠠⠔⣈⡀⠀⡀⠀⠠⠄⢂⢠⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⠤⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⢾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠨⠤⢄⣤⣬⣥⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠠⢔⣽⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠄⣀⠀⠉⠐⠒⠀⠄⠠⠤⠀⠀⠀⠀⠤⠄⠐⠷⢤⣤⣀⣀⣀⣀⣐⣐⣨⡧⢀⠠⣊⠔⠒⠉⠉⠉⠙⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⠒⠠⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢁⠀⡀⠀⡴⠃⢀⠠⠀⠀⠄⠠⢌⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⢧⡔⡥⠚⠉⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⠀⠀⠀⠀⠠⠀⠐⡀⠔⡯⠥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣟⠒⠤⠤⣤⠭⢤⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌⡀⠉⠉⠙⠛⠛⠛⠛⠛⢑⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠶⠖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠄⠠⠈⠠⠀⠀⠠⠂⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣦⡀⣀⣀⡀
⣷⣦⣄⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣛⢿⣿⡻⣿⣮⣻⡛⠿⠟⠃
⣿⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿⣿⣿⣟⣿⣭⠝⡻⣿⣷⡶⡆⠈⠻⢿⣿⣷⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣆
⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠈⠉⠅
⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠛⠉⠀⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣶⣄⠀⠀⣮⣿⣷⣿⡛⢽⠿⣟⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣻⣿⣿⣦⡲⢶⣝⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠉⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣶⡆⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡀⠀⠀⠠⠘⣦⠀⣅⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⡈⠉⣀⣖⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠛⢩⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢴⢾⣤⣖⢓⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠂⡼⣾⣿⣿⣿⣿⣄⡀⠠⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠘⠁⡀⠀⣢⣴⣿⡟⠩⢉⡿⠋⡝⠁⠀⠓⠒⠂⠀⠀⠀⠀⡠⢄⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠆⡱⠁⣠⣾⡧⠃⢙⠇⠂⢸⠗⢸⠓⡀⣦⠀⠀⠀⢀⡆⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⣱⠋⠀⠀⣠⡴⠀⠀⠀⠀⠇⠀⠉⠉⠈⠉⠁⣛⡆⠀⠀⠹⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢹⣷⣦⢼⠗⡀⡀⠀⠀⠀⠀⠀⠙⠣⠔⠂⢊⡥⣠⡖⣠⡞⠛⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠋⠀⠀⠇⠃⠀⠈⠈⠀⠀⢀⣠⣀⠀⢲⢰⠉⠁⣿⠟⠶⠇⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⠁⣈⣈⠃⠷⢆⡙⠦⣀⡤⠉
⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⢰⡦⢿⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⣴⠟⠩⣿⠂⠰⣰⡏⡂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⠏⠀⠀⠀⠀⠀⠠⣸⣀⠀⠠⣿⣬⣾⡇⡤⢠⣤⣠⡔⠤⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠈⡧⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⣀⣻⡶⣾⠉⡏⢨⣿⠖⠁⠤⠀⣾⣿⣁⡀
⠀⠀⠀⠀⠠⢿⣦⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡹⠃⠈⡩⡀⡇⡜⠏⠀⣴⣖⢄⠘⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠛⢇⣤⣤⣼⣟⠛⠉⠀⠀⠀⠀⠀⠐⠒⠚⡇⠉⠒⢒⡆⣷⡌⠀⠀⠀⣤⣎⡇⠀⠋
⠀⠀⠀⠀⠐⡀⠎⠀⠉⠉⢿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠤⢈⡁⢻⣿⢠⡀⡁⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠸⣿⠒⠒⠀⠀⠀⠀⠀⠰⠦⠀⠁⣴⠒⠆⠀⢸⣇⡄⠀⠁⠀⣿⡏⠀⡀
⠀⠀⠀⠀⠀⠀⡄⢿⣶⠀⣠⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣭⠟⠀⢸⣿⡇⢸⡄⠄⠉⠁⣰⠇
⠀⠀⠀⠀⠀⠀⠹⢌⠿⣰⣿⢿⣄⠀⠄⠢⢀⣑⠃⢀⣤⠄⠛⠣⡄⠀⣄⣠⡿⠻⢌⠿⡤⢆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠈⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡳⠀⠀⠇⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠂⠀⠀⠁⠐⢀⠉⠃⠀⠉⣉⡀⠀⠀⠀⠐⠈⠠⢶⡷⠆⠀⠀⠉⠁⠉⠉⠏⠉⠛⠳⣶⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠦⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡎⡀
⠀⠀⠀⠁⠀⢀⡤⠒⠩⠀⠀⠀⠒⠀⠀⡀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⡅
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⡄⠀⠀⠀⠈⢮⣐⢄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠘⠁⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡆⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢠⡤⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠖⢻⠇⠀⠀⠀⠐⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠐⠋⡁
⠀⠀⠀⠀⠤⣾⠁⠀⠀⠀⢀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠘⣓
⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⣘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠠⠋⠀⠀⠁⠀⠀⠀⠈⢹⡄⠀⠀⠀⠀⣉
⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⢸⣹⡄⠀⠀⠀⠀⠀⠀⠀⢐⠀⠈⠀⠀⠀⠀⠠⡀⠠⠀⠀⠀⠂⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⢉⠁
⠀⠀⠀⢠⠼⠀⠀⠀⠀⢀⣸⡌⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠀⠀⠀⠐⣀⣀⡀⠀⠀⢀⡞⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠄⠰⡇⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠆⣤⣶⠟⠋⠉⠉⡙⠛⣦⡞⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⢈⣭⣶⣶⣦⣤⣄⠁⠀⠀⠀⠀⢀⡇⠀⠀⣀⢀⣰⠏⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢨⣄⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢧⠀⠠⠴⠀⠐⠏⠋⣈⡠⠟⠋⠀⠀⠀⠀⣼⠀⠀⢐⣭⠞⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠷⣼⣶⡄⠀⣀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡇⣠⣶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣾⡇⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢻⡿⣽⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⡟⢇⢏⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣅⣅⠙⢃⠰⡄⡄⠀⠀⡀⢠⣾⡏⠀⣁⣼⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠀⣒⣀⠡⢘⣄⣠⠁⢃⣙⡀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠋⠁⠈⠛⠲⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋⠡⠀⠰⠶⠀⢠⠜⠛⠛⠫⠹⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⣠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠀⠀⠈⠉⠛⠛⠿⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣤⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⢀⣤⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣤⣄⣤⡀⣀
⠀⠀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣎⠀⠀⠸⣿⣿⣿⣿⣿
⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⣿⣿⣿⣿⡄⠀⡀⢹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣷⣦
⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣻⣿⣿⣿⣧⠀⠀⠀⣿⣿⡏⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⠿⢿⣿⠄⠀⠀⢉⣉⣀⣸⣿⣿⠂⠀⠀⠀⠀⠀⡰⠾⢷⣞⣻⣿⡿⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⡀⢀⡠⣶⠿⢼⣿⣿⡟⠛⠁⠀⠀⠀⣀⠾⣷⣧⣀⠘⠛⠻⣿⣶⠾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⢀⠻⢿⣿⡇⠀⠀⠀⠀⠸⠿⡷⢿⡀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣷⠀⠀⠀⠀⠶⣷⠆⢀⡁⠀⠀⢷⣄⠀⣴⣦⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢛⣿⣿⡇⠀⠀⠀⠀⠀⠘⠉⠁⢀⡄⠀⠙⠀⠻⠏⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠷⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠁⠁⠸⣽⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠐⠿⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡿⣿⣿⡀⠀⠀⠤⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠗⣿⣿⣧⠀⠂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⢹⣿⣿⡄⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣿⣇⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡼⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢒⡰⢾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⢘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⣁⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢳⣼⠿⣻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣆⠙⡶⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡃⢖⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢾⡿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⡸⠂⠂⠀⢺⡿⠂⠀⠀⡀⠄⢒⠞⠳⠒⠤⢀⠀⠀⠀⢿⣿⠀⠒⠐⢿⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢈⠠⠐⠈⠁⠀⢀⠎⠀⠀⠱⡀⠀⢈⣗⠂⠤⡉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡱⠀⠖⢨⡀⡀⠀⠀⠀⢠⠂⢀⣄⠀⠀⠑⡄⠘⡞⠀⢀⢀⡄⠲⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⢃⡉⠀⠀⠀⠀⢓⠐⢤⠅⢠⣼⣿⣦⠠⠄⠨⡄⠃⠘⠈⠀⠀⠀⢉⡀⠀⠰⡇⡱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡠⠀⠀⠀⠀⠀⠀⠄⠂⠀⢸⡿⠙⣿⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡀⠀⠀⠀⠀⠀⢸⡁⠀⢸⡇⡀⢹⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡸⠃
⠀⠀⠀⠀⠀⠀⡆⠀⠀⡈⠀⠘⢄⠀⠀⠀⠀⣈⣃⡀⠈⠱⠷⠯⠄⠀⢀⣀⣇⠀⠀⠀⠀⠀⢨⡄⠃
⠀⠀⠀⠀⠀⡤⠛⡄⢀⠁⠀⠀⠀⠱⡄⠠⠊⢁⣤⣤⣤⣶⣶⣶⣤⣤⣄⡠⠉⢢⠀⠀⠀⢤⡼⢧⡼⡆
⠀⠀⠀⠀⠀⠀⡆⠀⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⣆⡄⠀⠠⠾⢷⡞⠳⠴
⠀⠀⠀⠀⠀⠀⠁⡘⠀⠀⠉⠢⢄⡀⠀⠲⣤⣤⡀⢠⣶⣶⣶⣦⡀⠀⣾⣷⠀⠄⠁⠀⠀⠀⠘⠃⠀⠀⢃⣀⠠
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⢟⡤⢠⣿⣿⣇⣸⣿⣿⣿⣿⣿⣾⣿⣿⠀⢰⡀⠀⠀⠀⠀⡠⠄⠂⠉⡀
⠀⠀⡖⠒⠒⠒⠞⠒⠒⠒⡖⠀⠀⠟⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢸⣤⠄⠀⠀⠀⢰⠒⠂⠒⠳⠒⠒⠒⢲
⠀⠀⠈⢄⠀⠐⠀⠀⠀⠞⠀⠀⠀⠀⠀⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠨⠀⠀⠀⠀⠀⠣⠀⠀⠀⠂⠀⠠⠁
⠀⠀⠀⠈⢛⠛⠓⢓⠛⠓⠛⠒⠚⠓⠚⠓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⠚⠓⠓⠚⣷⠛⠓⠛⡚⢛⡛⡛⠛⠒⠚
⠀⠀⠀⠀⠀⢳⡀⠂⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠳⠀⠀⣠⠛⣄⠀⠀⣰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⡀⠀⠈⣴⠉⠀⠰⢿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠆⠂⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠒⠄⡟⠀⠀⠠⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⡀⠀⠀⢀⠖⢁⡜⠛⠻⠿⣿⣿⣿⣿⣿⣿⠿⡿⠛⢡⡈⠒⡀⠀⠈⠐⢄⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⣇⡤⠀⣙⠠⣎⠀⢀⠃⠀⠀⡇⠀⠀⢸⠀⠀⠘⡀⠀⢱⠄⣋⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⡾⠦⠀⠈⠓⢤⣉⡘⠒⠠⠤⠤⠤⠤⠤⠤⠀⠐⢃⣉⡤⠚⠱⣄⢰⡆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⠟⠁⠀⠀⠀⢸⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢇⠀⠀⠀⠀⢀⠎⠉⠙⠻⠿⣿⣿⣿⣿⠿⠿⠋⠁⠡⡀⠀⠀⣤⠁⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢱⡟⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠒⠀⢀⠠⠄⠙⠂⠄⣀⡠⠄⠀⠀⠠⢄⣀⠠⠀⠻⠠⢄⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠋⠀⠉⠠⢀⢀⠀⠀⠀⠀⠣⡀⠁⠉⠁⡜⠀⠀⠀⠀⠃⡀⠄⠙⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠈⠂⠆⣄⡀⠱⡀⠀⠜⢀⢠⠰⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⣥⡤⠤⠤⣼⣷⠄⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⣴⣿⠤⠤⢤⣴⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⠀⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣟⣙⣹⣿⣥⣉⣿⣏⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣭⣽⣿⣯⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢄⠰⣆⢀⠀⣴⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⡷⣿⣼⣷⣿⠛⣶⢫⣷⡇⣠⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡅⠀⠈⠃⠈⠀⠻⢸⣿⣿⣿⣄⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣤⣤⣤⡄⠈⢿⣿⣿⡿⢿⣿⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⠻⣦⣌⠛⠛⢑⡭⠚⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠹⣷⣼⣿⣅⣾⣿⣧⣤⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣤⣀⣤⣄⣴⡿⠿⣷⣾⠟⠈⠛⣿⢯⣍⡉⠁
⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣦⣤⣤⣤⣶⣶⣾⣿⡿⠛⠋⠉⠉⠙⠻⢿⠟⠉⠉⠉⠀⠀⠟⠁⠀⠀⠈⠲⠿⠿⣿⣿⣶⣄
⠀⠀⠀⣠⣾⣿⣿⠿⠛⠉⢽⣿⣿⣿⠿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠈⣉⠁⢉⣱
⠀⣤⣾⣿⣿⣿⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣦⣤⣼⣶⠦⣬⣷⠶⠾
⣰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣶⣿⣿⣿⢿⣿⣿⣿⡿⠛⠁
⠻⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣆⣀⠀⣸⣿⣿⠋⠀⠀⢸⣿
⠀⠈⢿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠐⣾⣷⡀⠀⠠⡀⠀⣤⣀⣿⣿⣿⣿⣿⣶⣿⣿⡇⠀⠀⠀⠈⠋
⠀⠀⢈⣿⣿⣿⣿⣿⣿⣦⣶⣦⣤⣿⣿⣿⣄⣀⣻⣦⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⢸⣿⣿⣿⣿⠛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠈⣿⣿⣿⠃⠀⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣽⡇
⠀⠀⠀⣿⣿⡏⠀⠀⣿⣿⣿⠎⠙⠿⠿⠿⠟⠛⠛⠛⠙⣿⣿⣯⣿⡟
⠀⠀⣸⣿⡿⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠁
⠀⢰⣿⣿⠃⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⢸⣿⡏⠀⠀⠀⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⣿⠁⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡀⠀⣠⣤⣤⣄⡀⠙⠏⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢀⣸⣧⡀⠙⠛⠛⠛⠁⠈⠋⠀⣀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠈⢻⡟⠉⠀⠀⠀⠰⠶⠀⠀⠀⠙⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠇⠀⠃⠀⠈⠁⢀⣤⣶⣷⣦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⡀⠀⠀⠉⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⡀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⠋⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠉⠁⠉⠀⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣤⠀⠈⠋⠀⢰⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⣿⣿⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⡿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡀⣀⣤⣤⣄⣤⠤⠀⠀⠀⣀⠄⠘⠛⠉⠹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠈⠛⠟⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢂⣂⡄
⠀⡀⠀⠀⠀⠀⠀⠀⢄⢶⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆⣸⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣾⣿⣿⠃
⠀⢠⣆⡀⠀⠀⠐⢌⣿⣯⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣴⠈⠸⣆⠐⣄⡟⢐⣥⣿⣠⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡖⠀⠙⣿⡟
⠀⠈⣿⣿⢆⠀⢬⡻⣿⣿⠯⠭⠀⠀⠀⠀⠀⠀⠀⠀⢤⡘⢆⢹⡆⠀⠻⣤⡿⠛⣫⡥⠄⠀⠈⠁⠀⠀⠀⠀⠀⠈⣿⡟⠀⢠⡄⣿⠀⠀⢀⣤⣤
⠀⠀⠽⣿⣯⡼⣲⣿⣷⣿⣟⠂⠀⠠⠀⠀⠀⠀⠀⢲⣌⣳⣌⣛⢛⣿⣾⣯⠦⠞⣅⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠀⠀⠉⣇⣿⠀⠀⢨⡿⠋
⠀⠀⠈⢿⣿⣿⣷⣿⣿⣿⠷⡀⠐⠶⠄⠂⠀⠀⠀⠀⠀⡽⣿⠛⢻⡿⢹⠓⣞⢻⡟⠻⣖⠲⠀⠀⠀⠀⠀⣺⣶⣿⣹⣆⣶⣤⣿⣹⢲⣴⣾⣴
⠀⠀⠀⠀⠹⣿⣟⣿⣿⠛⠠⠅⣼⣿⣦⠀⠀⠀⠀⠀⠁⢰⠁⠠⠞⢳⡈⠡⠋⠇⠩⠀⠈⠓⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣷⠤
⠀⠐⣫⠠⢙⣏⠽⠛⠋⣿⣤⣶⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠛⠿⠛⠋⠉⠁⠀⠀⠀⠀⢸⡈⠑⠈
⠀⠀⠿⠁⠻⠿⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣤⡀⠀⠀⠀⠀⠀⡀
⠀⢀⡀⠀⠐⠋⣀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣦⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣇⣸⣀⣼⢁⣤
⠀⣦⣉⠀⣤⣄⢉⣤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣯⣿⣿⣮⠄⠀⠠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣿⠿⣿⣿⣯⣝⠃⠈
⠀⠀⠋⣉⠹⠿⣟⣻⣞⠰⠄⠃⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⣭⣿⣵⡯⠀⣀⣰⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⣁⠚⣇⢻⣿⣿⣇
⠀⢠⠾⠺⠻⡎⢛⣧⠍⠰⢾⣶⡀⠀⠀⠀⠀⠀⠀⢰⣀⣼⣦⣴⣿⣿⣿⣯⣷⣿⣿⣯⣅⣤⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠙⢿⣿⣷⣤⠠⡄⡤⢀
⠀⠀⠀⣴⣿⣧⣪⣿⣄⣴⣶⣶⠆⠀⠀⠀⠀⢀⢤⣴⣿⣿⣿⣷⣿⣿⣿⣻⣻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣿⢇
⠀⠀⠞⢋⣾⣿⣿⠹⡏⠛⡏⠛⠀⠀⠀⠀⠀⠃⠒⠁⡯⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠿⠻⣿⣿⡆
⠀⠀⢀⣾⣟⣻⣷⡶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⠿⠆
⠀⠀⠀⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣯⣤⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀
⠀⠈⣿⣿⣴⡄⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣿⣏⣽⣷⣿⣿⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣄⠀⠀⠀⢀⣴⣿⣡⣤⣤⣿⡃
⠀⠀⠘⡿⠿⣽⣿⡵⣂⠐⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢠⡿⠞⠛⠉⣥⣶⣦⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣾⣿⣦⣿⣿⣿⣿⣿⣯⣤
⠀⠀⠀⣚⣿⣿⣿⢟⡥⠀⣀⣠⣶⣾⡿⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⣿⡗⠀⠀⠀⣠⣠⡄⠀⠀⠀⠀⠀⠀⣘⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡷
⠀⠀⢀⠀⢶⣿⣟⣿⣾⣵⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣷⠀⠚⠛⣻⡿⢷⡆⠀⠀⠀⢀⣘⣟⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠐⢶⣤⣤⣥⣾⣟⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣦⢀⣼⡿⠁⠀⠉⠀⠀⠀⠈⢰⠋⠨⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⡧⡄
⠀⠀⠙⠻⢿⣿⣷⣿⠟⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠛⣿⣿⠟⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠋⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠔⠋

⠀⠀⠀⣀⠤⣤⡀⠀⠀⠀⣀⣀⠀⠀⢀⡤⠠⣄⠀⠀⠀⢀⠠⡄⠀⣀⡀⠀⢠⡤⣤⡀⠀⠀⠀⠀⣤⡀
⠀⢀⣼⣇⣀⣀⣿⡄⠀⡜⠈⣿⣿⣧⠈⠗⢺⡏⠀⠀⢀⠈⢉⠁⢴⣥⣷⠠⣿⣀⣸⠏⠀⣤⡀⠘⠿⠇
⠀⣼⣥⣽⣿⡿⠛⢋⢼⣻⣷⣭⠋⠁⠀⠑⠚⣡⠔⠈⠙⢷⣄⠀⠀⢸⠀⢤⡈⠉⠧⠀⢀⣁⡿⣆
⠀⠁⠟⡟⢹⢧⠀⠈⠛⠾⠃⢹⢒⣄⢀⢄⠘⢿⣤⣄⣀⡸⠿⢶⡀⠀⠒⠾⠋⠀⡋⢀⣲⡄⠠⠿⠆⢠⣄
⠀⠀⠸⡇⠉⡇⠀⠀⠈⠀⠀⢘⢿⠋⠛⢺⠆⠀⠘⠋⠟⠋⠁⠈⠁⠀⠀⢸⠀⠐⠃⠈⠉⠀⠀⠀⢴⣛⣿
⠀⠀⠀⡇⠀⡇⠀⠀⠀⢳⣦⡜⡌⠀⠀⢽⠀⠀⢄⡾⢀⠀⠀⢤⣄⠀⠰⡜⠀⠀⠀⠀⠀⠀⢰⡆⠈⠙⠋
⠀⠀⡀⠷⣶⠇⡄⣠⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠞⠙⠩⣷⣦⣼⠶⢓⣤⣔⠒⠒⢦⡀⠀⠀⢀⡆⢀⢨⡄
⠀⡞⢿⡀⠀⠀⠈⢉⡀⠀⠀⡀⠠⡰⣤⠉⠉⠀⢀⣀⠀⠀⢰⠀⠀⠻⣟⠛⠻⢶⣶⣿⣄⠚⠊⢀⣈⣧
⢀⡇⢸⣧⠀⠐⡲⠈⠁⠀⡸⡧⡑⣽⠻⡎⠀⢠⣿⣻⣷⡀⠘⡄⠀⠀⠀⠙⣺⠛⠛⠉⠁⣴⣶⡄⠉
⠘⠷⣾⠟⠃⠀⠀⠀⣤⠀⠈⢺⣿⢿⠨⠆⠀⢾⣾⣽⡿⠋⣤⣊⠉⣷⣦⠠⣽⠀⠀⠈⠻⢿⡶⠷
⠀⠀⠾⠀⠀⠀⠀⠀⠀⣴⠀⢴⡄⠘⡄⠀⠀⠀⠀⠈⡇⠀⠛⠻⣻⠟⠋⠀⣿⠀⢶⡷⠀⠈
⠀⠀⡄⠀⠀⠀⠈⠉⠀⠻⠿⠗⠃⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠁⠀⣆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢿⡟⠉⠉⠉⠉⢹⡏⠉⠙⠛⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⣴⣿⡏⠁⠀⠀⣿⠀⠀⠀⢀⣈⠛⠶⠶⠞⠃⢀⣽⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠈⠳⣦⣤⡾⠋⠀⢠⡾⠋⠉⠻⣦⠀⠀⠀⢾⡁⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢶⡄⠀⠀⠀⠀⠀⣀⣀⣸⣇⣀⣀⣀⣿⡀⠀⠀⠈⠳⢦⣴⠾⠋⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣀⣠⣿⣶⣶⠿⢻⡟⠛⢻⠋⠉⢻⠋⠉⠛⡟⠛⠻⡷⠶⢦⣤⣀⡀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠉⢷⡀⠘⠆⠈⠃⠀⠈⠀⠀⠘⠀⠀⠘⠁⠀⠸⠃⠀⡼⠋⢉⣿⠷⣮⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⠄⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠋⡁⢀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣴⣏⣀⠀⣴⠀⠀⣿⡇⠀⠀⠀⠘⣿⠀⠀⢄⢀⣰⣤⣤⣿⡿⠟⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣿⠁⠀⠀⠀⡄⣿⡿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⣧⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⡟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢀⢀⠇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡜⡘⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣸⠀⠇⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⢠⠇⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠉⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣤⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⡀⠋⠶⠴⠴⣄⡄⠀⠀⠧⢀⠀⡀⠑⢖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠒⠂⢠⢀⣤⠀⠀⣰⠀⠘⣧⡄⠐⠛⠁⠛⠄⠀⣅⠂⠬⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠠⡔⡖⢻⢀⠠⠒⣄⢼⠝⠏⠋⠀⠀⠶⠋⠑⠒⠚⠷⠂⢀⣀⠀⠂⠰⣌⣆⠀⡀⠈⢹⣄⡄
⠀⠀⠀⠀⠀⠀⢠⠹⣦⣼⣧⣶⡿⠟⠉⠀⢓⡳⠂⠡⣄⠀⠀⠀⠀⠘⣧⠀⠀⢉⡧⢀⠀⢹⡦⣀⠀⠈⢿⡇⠀⡀
⠀⠀⠀⡀⢸⠀⠎⢀⣸⣿⢳⣏⣤⠤⠚⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠲⢇⣆⣷⢀⠁⠘⡧⢸⣦⡀⠎⢷⡴⢻⡆
⠀⠀⠀⣧⠀⡆⠀⡸⢋⡿⣾⣉⣆⢠⢠⡠⣴⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢿⠳⠷⣧⡄⣣⡑⡄⢼⣷⣾⠁
⠀⠀⢀⠸⡔⡉⢶⡟⣹⣱⡟⠓⠁⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠂⠳⣄⣏⠘⠻⢷⣿⣀⣿⡃⣨
⠀⢳⠈⡦⢻⣦⡟⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢸⣿⡝⣿⡷⡴⠂
⠀⠈⢷⡄⠈⢿⣾⢃⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠳⣿⡇⢻⣇⢽⠄
⣄⠀⣸⢿⣆⣼⣟⣱⣿⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢻⡷⣼⣿⣷
⠘⣧⡽⣎⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⣸⣿⣼⣿⠏
⠀⠘⣧⣿⢸⣿⣿⢀⣄⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⡏
⠀⢀⢹⣾⢾⣿⣿⣨⠋⠡⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣀⠀⡓⢾⣿⣿⢳
⠐⢾⡤⣿⣤⣿⣿⡄⠀⢸⣿⣫⡀⠀⢠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⡻⡟⣾⣿⣿⢸⠇
⠀⠸⣿⢏⢿⣿⣿⣧⠀⢒⢸⣟⡆⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡇⢈⣿⣿⡇⢸⠀⠠
⠀⠀⢹⡥⡂⢻⣿⣿⣧⠀⢖⣿⢩⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣿⠥⣼⣿⣿⣇⡏⠀⠏
⠀⠀⠀⠘⢷⡈⣿⣿⣿⣷⡨⣯⣾⣿⢘⣠⣀⡆⠀⠀⠀⠀⠀⠀⠀⢶⣷⣆⡀⠀⠈⢀⣤⡸⣿⣏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⣿⣛⠛⣿⣻⣿⣶⣄⢀⠀⢀⠀⠀⠘⢝⢯⣇⢆⣇⡞⣿⠩⢿⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠛⠻⣿⣷⣾⠼⠿⠿⠿⠯⣄⣠⣤⡄⢠⠴⠓⢉⣩⣍⠙⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠦⠤⣼⣿⣥⠤⠄⣨⡤⠐⠀⠀⢈⠗⣅⠀⣲⣊⣹⣥⣾⣿⣿⣿⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡏⠉⢻⣿⣟⣩⣹⣷⣶⣊⣠⠶⢻⣯⠟⠉⠀⠀⢤⣤⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠿⠏⠀⠊⠁⠐⢤⠞⠂⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠈⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⢤⠄⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣏⠉⠹⠉⠀⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠘⠂⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣷⣶⣦⢰⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡏⣝⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠙⢻⣿⣶⣶⣿⠏⠋⠉⠀⣹⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠒⠒⠀⠈⠉⠉⠀⠀⠀⠴⠦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠋⠉⠀⠄⠀⡄⠀⠀⠀⠈⡀⠉⡀⠀⠈⠙⠻⠿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣀⠀⠀⣠⣷⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣸⣿⠟⢣⡴⣿⣷⣾⣿⣾⠛⢷⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⣿⣿⣧⡤⢘⣟⢻⣍⣻⣽⠃⡞⠈⠀⣨⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⠛⢻⣿⣿⣷⣾⣋⣾⢯⣿⣿⣿⣷⣤⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⢀⡤⠴⠋⠀⠀⠀⠀⠀⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⡟⠀⠀⣼⣿⠿⢿⣇⣹⠻⠛⠻⠟⠫⠌⢻⣿⡟⠳⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠐⠀⠘⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣆⠀⠀⣼⣿⠲⣾⣿⠋⠉⠃⠰⣶⠀⣤⡸⢻⣆⠦⢼⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠏⣿⠄⠀⢼⣽⣿⣿⣿⣄⠀⠀⠀⣥⣴⠟⠀⠈⣿⣷⠈⣀⠤⠄⠒⠺⣷⡤⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡶⢦⣴⣿⣿⣿⣿⣶⢠⡀⠲⡏⠀⠀⢀⣨⡧⢻⡇⢀⣘⠗⣤⢦⣄⡛⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣛⣁⠈⠡⠄⠘⠫⣿⣿⣿⣝⢧⣭⣿⡆⠀⣰⣶⣿⢗⡽⢛⣫⣴⣺⣡⡾⠛⠀⠀⠘⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⣿⣧⡀⡀⠑⠢⠌⠙⠛⡿⠾⠿⣿⣷⢿⡛⠭⢂⣡⠴⢛⣹⣿⠋⠁⠀⠀⠀⠀⠀⢀⣠⠠⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣶⣿⡿⠿⢿⣾⣶⣦⣶⣌⠀⠀⠀⠰⠞⣉⣴⠾⣿⣿⣿⣿⠿⠃⠀⠀⠀⡀⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⣾⣿⣿⠿⣧⡀⠀⠀⢠⣛⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢂⣾⠟⢉⣤⣔⣻⣇⣴⣦⣿⡿⠛⣇⣹⡄⢀⣠⢴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣯⣄⣿⣾⠿⢿⡟⠛⠋⠛⠉⠄⢛⣿⣡⣟⣛⣁⣾⢿⣿⣦⣀⠀⠤⠀⠴⠶⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣽⣾⣿⡏⠶⢄⣘⣿⠗⠀⠈⠁⠀⢨⠸⠟⣻⠿⢿⣯⣼⣿⣿⣿⣷⠀⠟⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠃⠈⠛⣿⠋⠀⠊⢀⣸⣿⡇⠀⠀⠀⠀⣸⡄⠀⠁⠀⠾⣿⣿⣴⣿⣿⣿⡔⠈⠁⢲⣄⣴⡟⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡎⠀⠀⢸⣿⡻⢿⣶⣆⣴⣾⣿⣷⣄⠐⢀⣠⡿⣿⡿⢿⣿⠁⠀⠀⠀⠈⢻⣿⣴⣾⣿⡀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⢸⣿⠁⠀⠀⢸⡿⠁⠀⠘⠛⠿⠟⠈⠙⣧⠀⠉⠉⠳⢶⡾⠋⠀⢴⠀⠀⠀⢀⣼⣿⠿⢿⡉⠙⠳⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠰⠿⠐⠃⣄⠀⠀⢿⠁⠀⠀⠁⠴⠖⠋⠁⣿⠀⠈⠛⠿⢷⣄⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢶⠇⠀⣠⠂⣀⠀⢀⣴⡿⠀⠀⠀⠀⠀⠛⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣇⣾⠏⢹⣿⡆⠀⠀⠀⠀⠀⠀⢠⣀⣆⣀⣠⣤⣶⣶⣶⠿⠟⠁⢠⣍⣶⣿⣿⡷⢀⡀⠀⠀⡀⠦⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⣺⣿⠏⠀⠀⢻⣿⡴⡶⣶⣀⠀⠀⠀⠛⠛⠋⠀⢰⣿⢿⡏⣾⣇⣤⣿⣿⣿⣿⠟⣢⣮⡁⠀⠀⣱⠖⣠⡮
⠀⠀⠀⠀⠀⠰⠄⡤⢀⠚⣿⠀⣿⡿⠀⠀⠒⠛⠉⢶⡄⠀⠀⠀⠀⠀⣷⡄⢄⠀⢸⡇⣜⢳⣿⣿⠿⢿⣛⣋⣵⠞⣻⣿⣷⠀⠀⠀⠈⠉
⠐⢀⠀⠀⠀⠀⠀⠀⣸⠶⠁⣾⣿⡇⢀⣄⡀⢀⡀⠛⠿⡶⠀⠀⠀⠀⠈⠗⠀⠀⠸⣿⣿⣷⣿⣙⠳⡟⢫⣵⣴⣾⣿⠿⣿⡆
⠀⠀⠀⠀⠤⠀⠀⠀⣾⠀⣸⣿⠏⠀⠚⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠹⣦⠄⠈⢳⡿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠿⡿⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⣿⡏⠀⠀⠀⠀⠀⠀⢤⣴⣶⣶⠶⠓⠀⠀⠀⣽⢓⠆⠀⠀⣿⣷⣻⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⣀⣠⡤⠤⣾⣿⡟⠀⠀⠀⣠⣶⣿⣿⣇⣀⠐⠘⠃⠘⠻⣿⡟⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡇⠾⣿⣷⠟⠋⠉⠀⠀⠀⠀⠀⠀⢀⠿⣿⡿⢿⣍⡀⢉⣿⣦⠀⡄⠀⠀⠉⠀⠀⣉⠀⠰⠀⠀⣼⡇
⠀⠀⠀⠀⠂⠁⠀⠀⢠⠁⠀⢟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⢷⣦⣀⡿⣷⡄⠈⢿⠆⠁⠂⠘⠥⠀⠈⢫⣀⡀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⠩⢶⣿⣇⢠⡴⠆⣤⠶⠟⠀⠀⠀⠈⠿⠁⠀⠉⣸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠷⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣌⡻⡎⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⣁⣤⣾⣿⡿⠿⠿⢲⣦⣤⣤⣴⣶⣿⣿⣷⣶⣿⡿⣿⣿⣷⣦⣤⡿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⣿⣿⣿⣿⣿⠇⠿⠿⠿⠿⠋⠀⣼⡿⠀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢶⡌⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠿⣋⣤⠀⢷⠀⢴⣼⣿⣿⣿⠟⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠀⣠⠀⠐⠺⡿⠀⠐⠂⠀⠀⢀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⣀⣴⣄⡀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠤⠀⠀⠀⢀⣀⣠⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢷⡶⠤⣶⡖⠂⠀⠐⠐⠲⠾⣿⠿⠛⠛⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠐⣻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠷⠶⢤⣠⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠏⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠙⢳⣄⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⡤⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣁⣤⠶⠓⠀⠑⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⢶⣞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⣥⡀⠀⢠⣶⣵⠞⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠤⠤⠶⠶⠶⣶⣿⠇⠀⢻⣶⣿⣟⠩⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣠⠜⠛⠻⠟⠇⠀⢸⡿⠵⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠀⠀⠀⠂⠉⢇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢀⡶⠿⠳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠞⠁⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⠀⢠⡞⠁⠀⠀⠀⠡⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢿⢲⡏⠳⣈⣰⣤⣠⡄⠀⠀⣰⣟⣀⠤⠀⠀⠠⠠⣶⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠟⠉⠀⠈⠁⠀⠀⢟⡫⣀⣠⡴⠖⣤⣀⠼⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⠃⠀⠸⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡇⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢱⠀⠀⠀⢸⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⠧⠄⠤⠼⠁

⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠈⢋⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⢤⣤⡄⣀⢠⠤⡄⠁⠀⠀⠹⠎⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⠃
⠀⠀⠀⠀⠀⠐⠋⠃⣀⠊⠘⠛⠀⠀⣆⠀⠀⢠⡤⣶⢠⣤⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡈⠀⠀⠀⠀⠀⠀⠀⠑⠐⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠙⡠⠊⠀⠀⢀⠇⠀⠐⠤⢀⠠⠈⡐⠀⠈⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠁⠀⠲⠌⠀⠋⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⣈⡁⢐⠀⠉⠉⠐⠠⡀⠀⠲⠀⢰⡤⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠄⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠋⠀⠀⠀⠀⠀⠐⣶⠀⢈⣤⠿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢂⠀⠀⠈⠁⠀⠄⠠⠀⠀⠀⠂⡀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠀⠦⠀⠀⠀⠀⠀⠀⡀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⠂⠤⠼⠶⠤⠖⠒⠢⠄⠀⠅⠀⠀⠀⠀⡄⠘⠀⣂
⠀⠀⠀⠀⠀⠊⢀⠈⠁⠂⠰⠋⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⠈⣡⠀⢀⣀⠀⠀⠀⠀⠁⠀⠄⣀⠐⡄⠀⠨⠇⠀⢐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢲⠌⣭⠀⠀⠡⡀⠀⠀⢉⢀⣤⠀⢈⣁⠠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⡀⠙⡆⢀⣿⡆⢰⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡣⠭⠀⠀⠀⠈⢁⠀⠉⠁⢀⡴⠁⢁⠀⠰⡞⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠐⢄⠀⠛⠁⢀⡀⠈⡘⠃⠀⠈⠉⠉⠀⠐⢸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⠀⠁⠀⠀⠈⠀⠀⠐⠰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⡻⢘⣿⠂⣀⠄⠀⠀⠀⠙⢢⡈⢢⠀⠘⠂⢛⡴⠈⠸⠷⡆⠀⠀⠀⠀⠀⢶⡂⠚⠁⢰⠀⠚⠘⠈⠁⠈⠀⠠⡀⠀⠀⢀⠲⠆
⠀⠀⠀⠀⠀⠀⡭⠀⡁⠀⡉⠲⡄⠀⢀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠉⢠⠴⠀⠀⠠⠠⣦⠄⠀⠀⠈⠁⠀⠀⢃⣀⠀⠀⠰⣍⠀⠀⢢⠀⠠⡄⠀⠁
⠀⠀⠀⠀⢴⡈⠅⠀⠀⢠⠀⠱⣤⡀⠨⠓⠄⢀⠠⡤⢄⠀⣄⠒⠀⡁⢀⢠⠀⢀⣀⣈⣁⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠓⡜⠢⠀⠀⣐⡀
⠀⠀⠀⠀⠈⠉⠢⣀⠉⠉⠀⠀⡈⠀⣀⠀⢀⠈⠐⠋⢸⠀⠈⠠⠆⢀⠸⣏⠁⠘⠏⠁⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣄⣀⢒⣂⠐⠈⠙⠂⢠⣤⠍⠑⢻⣆⢈⣦⠀⠀⡞⠰⠚⣀⠤⢧⠀⠉⠁⠫⠄⠀⣉⣉⡄
⠀⠀⣤⡌⠁⠄⠀⠠⠗⠀⠋⠙⠉⠙⠆⠀⠀⠀⠀⠀⠙⠀⠘⢶⠒⠠⠙⠂⠀⠀⢆⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⠀⢀⡀
⠀⠀⠉⠈⠀⢀⡀⠁⡐⢤⠠⣀⡀⢹⠾⠇⠀⠀⠀⠀⠀⢀⠤⢄⡛⠒⢶⣾⠃⠀⠀⠉⡀⠲⠖⠤⠀⠀⠀⠀⠀⠿⣁⠀⠁⠂⠈⡟⠋
⠀⠀⠀⡤⠈⠀⠐⠂⠘⠉⠀⢀⡀⠀⠴⢒⢤⠀⠚⠞⠡⠂⣈⠓⡉⠉⠻⠏⠀⢲⠤⠬⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈
⠀⠀⠀⣀⣀⢹⠟⠀⠀⠀⠀⡿⠗⠀⠸⠁⢸⠖⣆⠀⠀⠀⠀⠑⠦⠉⠐⡠⠈⢯⠀⢀⣀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡜⠁⠀⢀⡄⠰
⠀⠀⠰⡷⠀⢀⠀⠀⠰⢙⡮⠀⢢⠄⠀⠈⠁⠑⠊⠀⠀⠀⠠⣤⠆⠀⡄⠀⡀⠀⠑⡌⠉⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠁⣀⣰⡒
⠀⠀⠀⠀⠀⡆⡔⠀⡀⠀⠀⠀⠘⠳⠄⠀⠀⠀⣀⠀⠀⣴⣂⣀⠀⠈⠀⠀⠙⠒⠂⠩⣀⠙⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡔⠋⠀⣀⠤⠤⠄⣠
⠀⠀⠀⠀⠐⢣⡇⢈⠀⢀⡀⠀⠈⠀⠈⠒⠋⠉⠡⢹⡯⡄⢧⠀⠁⢢⠤⠰⡦⠀⡀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣀⡤⠀⢉⡀⠤⣀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⣎⣀⠁⡖⠀⠀⡆⢀⢄⡀⠀⠀⠀⠀⡔⠌⠃⠀⠀⠀⠀⠂⠀⣬⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡡⠖⠉⠁⠀⠀⠀⠀⢀⡐⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠄⠀⠁⣆⠀⠑⠀⠀⢠⠀⠁⠀⣠⣄⠐⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠃⠴⠺⠊⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣷⣕⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢍⣿⣿⣿⣿⣿⣶⣄⠀⢀⣝⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢿⣿⣿⣿⣿⡏⣕⢸⣷⣴⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢯⢹⣿⣿⣿⣿⡧⠦⢹⣿⣿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣮⣧⣇⢰⣿⠬⢾⣿⣿⢻⣿⡇⠿⢼⣿⣿⡏⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣿⢩⢹⣿⡓⣭⣿⡇⣟⢸⣿⣿⣿⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢨⢸⡯⡹⢫⣽⡇⣭⢼⣿⣿⣿⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠤⢼⡇⠀⠀⢿⡇⠲⢸⣿⣿⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣹⣹⡇⠀⠀⠰⡇⠲⠞⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡧⢶⢾⡅⠀⠀⠀⢁⢰⡆⣿⣿⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢇⣶⣸⠁⠀⠀⠀⠘⢀⣀⣿⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣄⢘⠀⠀⠀⠀⠠⣩⣭⣹⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢍⠈⣿⠀⠀⠀⠀⠀⠤⣧⣼⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⢶⡶⠋⠀⠀⠀⠀⠀⠰⣶⢸⣿⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡗⠾⠿⢻⠀⠀⠀⠀⠀⠴⡶⢼⣿⡕⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠃⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠀⢀⣠⣤⣤⡄⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠉⠉⠻⠋⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢹⣄⠀⠀⠀⢀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣶⡀⠀⢀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠉⠉⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⡯⣽⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⢻⣿⣿⣿⣿⠿⢹⣿⣿⣿⣿⣿⠟⠁⢀⠆⢀⡄⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⣿⣿⡇⠀⢸⣿⣿⣿⡟⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡟⣴⣏⣠⠏⣠⠞⣱⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⡇⠀⠈⢿⡿⠟⠁⠀⠘⠁⠈⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠏⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⡿⠿⠟⠃

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢚⠽⠟⠛⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣖⣾⣻⣷⣿⣿⣿⠿⢿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣠⠤⠤⠀⣠⠄⠒⢟⡴⠊⠁⠀⠀⠤⡤⣀⠀⠘⡄
⠀⠀⢀⣴⣾⣿⣭⣿⣱⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣮⣻⡇⠀⠀⠀⢠⠞⠉⠀⠘⠃⠼⠓⢁⠔⠉⠠⠀⠀⠀⠀⠀⠀⠄⡳
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠔⠁⠐⠄⠀⢠⠄⢀⡀⠀⠈⠀⢠⠂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣍⣻⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠘⡃⠀⢀⠀⠀⠀⠀⡠⢤⠀⠘⡿⠂⢈⠉⠀⠠
⠀⠀⠀⠙⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠊⠁⠀⠀⠀⠀⠈⠂⠀⠀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⣉⣸⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢀⠡⠐⠒⠲⣦⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⣿⣿⢿⣿⣿⣇⣶⣿⣷⣤⣤⣠⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠄⡤⢠⣀⠀⠑⠋⠁⠈⠙⢷⣻⣿⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣾⣯⣽⡿⣿⣿⣿⣿⣿⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⡀⠛⠂⣄⡀⠀⠀⠉⠉⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣿⣿⣟⣻⢿⣿⣻⠟⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⠤⢰⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠟⢿⣿⠻⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⠂⠀⠀⠰⠀⠰⠒⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⠀⠀⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⣆⡀⠀⠀⠀⠀⠀⠁⠀⢠⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠀⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣿⣿⣿⣿⣿⣿⣿⡁⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⠀⡦⠀⡀⠀⠂⠀⢀⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⢿⢿⣿⣿⣿⣿⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠳⣷⡀⢧⣠⡆⠀⠀⢀⣥⠷⠖⠒⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠈⠲⠿⠿⠛⠿⣷⠤⠋⠀⠀⠠⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⡿⢟⡿⠛⡇⡀⢰⣤⡀⠆⣴⡷⣾⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠉⠀⠀⠈⠑⠀⠀⡀⠀⠉⠃⠸⠀⠀⠉⡰⡄⠸⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⡧⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠐⠀⠀⠀⠠⡄⠀⠀⢻⡄⠀⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢸⣿⣧⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢻⡄⢃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⠀⢈⠈⠉⡿⠎⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠈⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⡇⣸⡀⣀
⠀⠀⠀⠀⠀⠀⣤⢠⠐⡏⠀⡁⠀⣷⠀⠀⠀⠀⠀⢠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⢧⡟
⠀⠀⠀⠀⠀⠀⢹⣟⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⢆⡀⠀⠀⠀⠀⠠⢠⠀⢸⣧⣄
⠀⠀⠀⠀⠀⣴⣿⠏⠃⡀⠈⡃⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠰⠆⠀⠀⣀⣴⣷⣶⣿⡖⠒⠒⠀⣤⠀⡇⠀⢘⣿⣿⡀
⠀⠀⠀⠀⠀⣿⢿⠃⢄⠁⠀⠃⠀⠀⠀⠀⢠⣶⡤⠤⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣾⣟⢿⢯⣟⠛⠃⠀⠈⠀⡠⠁⠀⢸⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣶⠀⢘⠀⠀⠀⠀⠀⠀⢀⣾⠐⠋⣀⡴⢾⣿⣯⣙⢧⣸⣼⣷⣿⣿⣿⠕⢛⡩⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⡄⠘⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡄⢀⠀⠀⠀⠀⢀⣴⣿⣷⣶⣛⣋⣭⣼⣿⣿⣿⣿⣿⣿⡿⠟⠛⢲⣾⣿⣿⣿⣿⣾⠖⠁⠀⠀⠀⠀⠀⠇⢷⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⢠⠀⠀⠀⢠⣿⣿⣿⣿⠏⠚⠉⠉⠑⠒⢩⡾⣫⣤⣤⣤⣀⣼⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣷⠈⠁⠀⠀⠘⠻⢿⣿⣿⣿⣶⡀⠈⣀⣾⣿⣿⣿⣿⣿⠁⠙⠸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠉⠉⠙⠁⠀⠀⠀⠀⢹⡔⠀⠀⠀⠀⠀⣌⠀⡂⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠤⣄⠀⠂⠀⠀⠀⠀⠀⠘⠀⠀⢁⡀⠀⠀⢠⠘⡇⠀⠀⠀⢀⡠⠎⠀⠉⠀⠀⠀⠀⠅⣠⣸⡿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⢈⡓⢀⡀⠑⠄⠐⡄⣠⠀⠀⠀⠀⠀⠠⡆⠀⣇⠀⠐⣴⣿⣇⠀⠀⠀⠀⠀⠀⠱⠞⢉⡿⠋⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡅⠀⣾⠆⣀⠠⢡⣷⣆⠀⠀⠀⠀⠀⠇⠋⢻⣦⡀⠹⣿⣿⣧⣀⣀⠀⠀⠀⠀⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡞⢶⣂⣘⣧⣼⣿⣷⣶⢿⣿⡦⣄⠆⠀⠈⣿⠀⠀⠈⡊⣹⠿⣿⡻⠒⠒⣢⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢉⣽⣟⣿⣴⡿⢿⣧⠆⠉⣙⣿⣄⡀⠘⣁⡴⠈⠀⣰⡏⠠⣿⣥⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⡰⠈⠁⣷⣶⣿⣿⣿⣞⣧⡄⣠⣶⣿⣧⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠻⠿⠿⠟⠛⠛⠓⠛⠉

⢦⣼⠀⡌⣿⣿⡆⢻⡰⣴⣃⡶⠋⠹⢲⡼⠀⢠⡸⠀⠀⢸⡄⢴⠥⣰⣷⡞⢻⡈⢸⣿⣿⣧⢸⣿⣀⢼⢿
⣛⣿⣇⡿⣿⣿⣷⠋⣷⠋⢸⠀⣥⠟⡼⠓⠂⢩⡤⣰⠀⠈⢷⡘⠮⢼⡞⣷⠀⢻⣾⣿⣿⣿⣾⣿⣸⣽⣿
⠘⢧⣸⣧⣿⣿⣿⣼⠉⣧⡯⣂⡴⠚⠁⠀⢐⠘⠆⣬⡳⢦⡈⢷⡤⣉⣿⡜⣄⡄⠹⣿⣿⣿⣿⣿⢉⣿⣿
⡦⣹⡟⠈⣿⣿⣿⣇⣾⣿⡞⠉⠀⢬⠌⣠⠀⠤⠄⠰⠭⠵⢿⣿⣷⣌⣐⠿⣎⠻⣄⣿⣿⣿⣿⣿⣼⣿⡷
⣷⣼⣷⣞⣿⣿⣿⠞⢡⡿⢀⣤⣉⣤⣞⠚⠯⠀⠀⠉⠊⠀⠀⠀⠉⠻⢾⣝⠾⣿⡟⣿⣿⣿⣿⣿⣿⣯⣏
⠸⣏⣿⣿⣿⣿⣿⢸⡿⣷⣩⣠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣿⣿⣿⣿⣿⣿⣦⣾
⠀⢹⣏⣸⣿⣿⣿⣿⣋⡽⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢹
⢀⡾⠙⣿⣿⣿⣿⣏⠿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⢸
⣼⠁⢰⣿⣿⣿⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⡇⢸
⣏⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⣿⣿⣿⣿⣧⢠
⣿⡟⢰⣿⣿⣿⣿⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠤⢻⣿⣿⣿⣿⣿⣿
⣿⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⡏
⣿⣇⢸⣿⣿⣿⣿⣧⡀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡇⠉⣿⣿⣿⣿⣿⣿⡇
⠘⣿⣾⣿⣿⣿⣿⣿⡘⣌⠂⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡇⣴⣿⣿⣿⣿⣿⣿⣇
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⡿⣝⡞⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⢸⡷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢹⣸⣿⣿⣿⣿⢿⣿⣿⡏⣿⣿
⢋⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡕⠀⠀⡀⢀⣀⣀⣠⣤⣤⡤⣶⣷⣿⣿⡟⢻⣿⣿⣆⣿⣿⣧⢻⣿
⣾⠃⣸⣾⢻⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣦⣴⡷⠛⢋⣭⢿⣉⣴⢿⣿⣿⣿⡿⣠⣾⣧⢻⣇⠟⣿⣞⣷⣻
⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣷⡟⢣⣾⣱⣷⣟⣿⣿⣿⢇⣿⣿⣿⣾⣿⣷⣻⣿⡟⣿
⣷⠀⣿⡟⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣷⣿⣿⣿⣿⡏⣽⣿⣿⠸⣿⣿⣿⢣⣿⢹⣿⣿⣿⢸

⠀⠀⠀⠀⠀⡀⡄⠠⣐⡀⠄⠀⡀⢠⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⣄⠦⣘⠻⢭⠻⣤⠢⣀
⠀⠀⠀⢠⣲⠅⡲⠇⣄⠰⠬⠁⡂⢦⣙⢰⡈⣆⠀⠀⠀⠀⢀⡰⢮⡹⠦⠰⢮⢳⡌⣁⡬⢝⣢⡀
⠀⠀⢨⠑⡌⣞⢬⢡⠼⡘⢇⠠⣴⠂⠌⠤⣁⢂⡡⡄⠀⠠⠴⣞⡳⢌⣃⠻⢖⡘⣏⠻⣊⠯⣑⠄
⠀⠀⠸⢳⢩⢼⠰⣌⡳⠜⢄⠭⢬⠁⡓⠍⡚⣀⣐⠀⠀⠘⣛⢴⢆⢖⡱⠘⡳⡝⠊⡧⢝⣒⡖
⠀⠀⠈⠩⡎⣙⡓⡘⡄⢋⢳⡺⡉⢤⠇⡓⠰⡆⠀⠀⠀⠀⠈⠩⢟⣯⣙⢧⠱⡜⣿⣹⠶⠛
⠀⠀⠀⠀⠀⠉⢢⠑⠐⠈⢇⢘⠹⠤⣕⡬⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠑⢀⢂⠺⠫⠸⠂⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⢢⡡⠌⣑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣈⢭⢤⡔⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡦⡅⢖⡥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣓⢯⢩⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣲⠽⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⢤⣀⡤⢶⠆⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⠈⠀⢀⡞⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⢾⣅⣀⠀⠀⠀⠰⣽⣄⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠐⣿⣿⠖⠀⠀⠀⠀⠈⣧⣔⡞⠛⠛⠛⠃⠀⠀⠀⡾⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⢀⣀⣴⠇⠀⠀⠓⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠐⢿⣉⠁⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠞⢿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⠉⠁⠀⠘⠓⠦⣤⡀⠀⠀⠀⠀⠀⠀⣇⣠⣠⠴⠧⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⡿⠛⠁⠀⠀⠀⠈⢉⣠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠒⢦⣼⣿⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⡼⠁⢀⡴⠖⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠠⢾⣿⡷⠄⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢠⠞⢀⡴⠋⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⡴⠋⢠⠞⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠛⠛⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⢀⠀⣠⢞⣡⠞⠛⠂⠀⣠⠏⠀⠀⣀⡤⠖⠚⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⣳⡀⠀⠀⠀⠀⢀⡴⣯⠀⠀⠀⠀⠙⠷⠋⠀⠀⠀⣠⠞⢁⡤⠖⠋⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠁⠹⣄⠀⢀⣠⠏⠀⠈⠳⣄⠀⠀⠀⠀⠰⣄⡴⣋⡵⠚⠉⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⠻⡉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠋⠁⠀⠀⣀⡤⠖⠋⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣯⠵⠖⠛⠉⠉⠉⠉⠉⣳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⢀⠀⠀⣀⣠⡤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢠⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢳⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣸⣎⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣖⣫⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⢠⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣦⣈⣉⠛⠲⢤⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠃
⡿⠿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿

⠀⠀⠀⠀⠀⣰⣦⠀⣁⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⢠
⠴⢶⣄⡿⠇⢸⣿⢸⣿⠟⠃⣿⠇⣀⠀⣠⠔⠉⠁⡄⠚⠉⠉⡤⠤⠾⠛⠛⠀⢀⣀⣀⣠⠀⠀⠠⠄⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠻⣆
⠰⣿⣏⡀⠀⠘⣿⣬⡿⠀⠰⣿⣿⢋⠙⠉⠁⣀⠀⠀⠀⠠⠄⠀⣀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣤⣀⣠⣤⣿⡄⠀⠈⠛⢆⠀⠀⠠⠻⣄
⠀⠀⠉⣙⣷⣶⣼⢸⣿⣤⣤⣿⡿⢿⣄⢀⠀⠁⠀⠀⣀⣤⣾⣿⣿⡟⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⠀⢄⡀⢀⠀⠀⢦⡀⠙⠷⣦
⠀⣶⣾⣿⠟⠀⠀⠀⢈⣁⣀⡙⠃⠀⠹⢇⡀⣀⣴⣞⣛⠉⢿⣉⡀⠀⢐⣀⣰⠶⠚⠛⠻⠻⠿⣿⣿⣿⣿⣷⣤⣼⠄⠀⠉⠛⠳⣦⣀⠁⠀⠀⠈
⠀⠈⠼⠁⠀⠀⠀⠈⠉⣀⠀⠀⠀⣠⣖⠉⢷⡅⠀⠀⠀⠀⠈⢠⣶⣾⣿⡿⠃⠀⠀⠀⠀⠀⠲⢾⡌⠙⠿⣿⣿⣶⠢⠀⠈⣦⠀⠀⠉⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⢤⣾⠟⠁⠀⠀⢻⣤⠀⣠⣶⣿⢿⡿⠿⠿⣤⣀⡀⠀⠀⢀⠀⠀⢸⣿⣦⡄⠈⠻⠻⠄⠀⣄⠹⣧
⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⣠⡿⣅⣀⡀⠀⣠⣞⣿⠺⣿⣿⣿⡿⠃⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⠉⢸⡏⠀⠠⢦⣄⢠⣼⣆⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠑⢤⣰⠏⠀⠉⠉⢉⣾⣿⣿⣿⡿⣻⣿⠏⠀⠛⣦⡴⠀⠀⠀⠐⢀⠀⠀⠀⠀⣰⣿⡇⠀⣳⡀⠈⠛⣿⣿⣀⣿⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠓⢦⣄⣀⣾⣿⡻⣿⠏⠀⠛⠻⣆⠀⠀⠀⠙⣶⣿⣿⣿⡆⠀⠀⠀⣠⣿⡿⠇⠸⣿⡧⢀⡖⠸⣿⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣽⣿⣿⣿⡿⣄⠀⠀⠀⠈⢷⣀⣰⣾⣿⣿⣿⣿⣷⠆⠠⣿⣿⠁⠀⠀⠀⠉⠀⠘⣿⡄⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠈⠳⠄⠀⢠⣯⣿⣿⣿⣿⣿⣿⣿⢇⣸⣆⣿⠍⡈⠀⠲⠦⣀⠀⠀⢹⣧⠐⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠐⣿⣿⣿⣯⣻⣿⣿⣿⣿⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠁⢸⣿⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⡀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⣿⣿⣿⣿⠛⠧⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣟⠀⠀⠀⠀⠀⠀⣾⡟⠀⣿⣿⣿⡏⢀⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⣿⣿⣽⠗⠁⠀⠈⢦⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢘⡇⠀⠀⠀⠀⠀⢸⣿⠇⢀⣿⣿⣿⠃⢸⠁
⠀⠀⠀⠀⠀⠀⠐⢹⣇⠀⠀⠄⣿⣿⡿⢤⠀⠀⠀⢸⠟⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠁⢀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⡿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⣿⡿⠀⣴⠉⠀⢀⡞⢀⡾⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⠃⣠⢹⣿⣿⡇⣰
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣆⢀⡟⠀⣴⣿⠀⠀⠘⢰⡾⡔⡥⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡏⡰⠃⢸⣿⡟⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡁⠀⠹⣿⡇⠀⠀⠀⠾⠸⠃⠀⠀⠀⠉⠙⠿⠛⠛⠛⠙⠳⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⠀⣦⡀⠸⠇⢰⡇⠀⡀⢀⠀⠈⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣾⠃⠀⠐⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣥⣾⣿⣿⠃⢀⡘⠁⣤⣝⡞⣇⠀⠀⣀⣶⣾⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣿⣿⣿⠃⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠋⠘⣿⣏⣰⡿⣱⣿⣮⣿⣜⢆⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣠⡾⢋⣴⣿⣿⡿⠁⠀⠀⣠⣿⠟
⠀⠉⠀⠀⠀⢶⡀⠀⠀⣿⠀⠀⠀⠙⠋⣹⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡾⠟⢉⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⠀⢶⣷⠀⠰⠏⠘⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡿⠿⠟⠛⢉⣠⣾⣿⣿⠟⠉⠀⠀⣠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠸⡇⠀⠀⢰⣦⡀⠀⠀⠀⣠⣀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣴⣶⣶⣿⣿⡿⠟⠋⠀⠀⣀⣴⣾⣿⠿⠋⢺⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠺⣾⡏⢀⣴⠇⢀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠛⠛⠛⠉⠁⠀⢀⣠⣴⣿⣿⡿⠟⠁⠀⠀⠸⠿⠶⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⣛⡛⠛⣻⠿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⡀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠋⠉⠛⠻⠷⢶⣄⡀
⠀⠀⠀⠀⢀⣠⡞⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⣶⣦⣶⣤⡄
⢀⡾⠟⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣧⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⣿⣿⣿⡆⠘⠈⣿⣿⠏⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠿⢿⡷⠀⠀⢉⣉⣐⣿⣿⠂⠀⠀⠀⠀⡴⢶⣾⣿⣿⠏⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠛⠂⠀⠀⢄⣠⠾⠟⣻⣿⣟⠋⠁⠀⠀⣠⡺⣿⠦⠌⠋⠻⠷⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠙⣿⣷⠀⠀⠀⠐⣙⡛⠻⠄⠀⣀⠀⠐⣜⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢾⣿⡇⠀⠀⠉⠙⠡⠶⠂⠀⠙⠷⠸⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⣶⣤⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⡇⠀⠀⠀⠀⠠⢄⣄⠀⠉⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣹⣿⣧⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠙⠛⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢽⣿⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣿⣧⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢨⣿⣿⡄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⣼⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣺⢙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠟⢽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣃⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣾⢛⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡼⣣⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⣵⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠀⠀⢿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⠃⢤⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⠉⠙⠛⠋⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠣⠄⣒⠢⠤⢒⠓⠴⢆⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢰⣦⠁⡀⠁⠀⠈⠉⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⠂⣀⠀⠲⣮⣧⡀⣿⣦⣾⣗⡂⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢠⣶⣿⣀⣼⣥⠖⠀⣀⠐⠲⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠂⠀⠀⢁⣤⣤⡨⠍⢿⡫⣿⢧⣍⡉⠛⠛⠛⠻⠛⠛⢛⢛⣩⠟⠿⣿⠍⣾⠉⣅⣤⣤⡁⠀⠀⠐⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠛⣩⣽⠏⠀⢀⡤⢭⠉⠁⠈⢀⢘⡇⠈⠡⣮⡙⠻⢿⣿⣿⣿⣿⡿⠟⠁⣴⠌⠁⢘⡗⡀⠁⠈⠭⡫⢤⡀⠀⠹⣯⣍⠛
⠀⠀⠀⠀⠀⠀⠀⣼⠟⡽⠋⢀⡡⠖⢀⣤⣀⡘⣦⡿⣿⣦⢱⣿⡅⠀⠲⡌⣿⣿⢡⠖⠀⢠⣿⡖⣴⣿⢿⣴⢃⣠⣤⡀⠲⢌⡀⠙⢯⠻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⡀⣠⢀⣠⣪⡤⠀⢉⣫⣽⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⣿⡃⠀⣾⣿⣿⣿⣿⣯⣽⣉⠁⢤⣵⣀⡀⣤⢀⡘⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⣸⠁⢀⣮⣿⠿⠛⢁⡤⠈⠉⠁⠉⠙⣿⣆⡤⣝⣿⣿⣫⢤⣰⣿⠋⠉⠈⠉⠁⢤⡈⠛⠿⡿⢷⡄⠈⣇⠙⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⢀⢾⡟⠁⠀⠀⡎⠀⢠⣶⣿⣶⡀⠘⣿⡇⠘⣿⣿⠃⢸⣿⠃⢀⣾⣿⣶⡄⠀⢱⠀⠀⠈⢻⡷⡀⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢀⡞⠛⠠⠀⢴⣿⠀⠸⣿⣿⣿⠇⢠⣿⣿⠀⢸⡇⠀⣿⣿⡄⠘⢿⣿⣿⠇⠀⣿⠇⠀⡄⡏⢳⡀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣧⡏⣶⠘⢰⡹⣷⣄⣀⢀⡀⢠⣾⣿⣿⠇⢸⡇⠸⣿⣿⣷⡄⠀⡀⣀⣠⣼⢏⡆⡇⣷⠹⣾⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢿⢺⡛⣷⣦⣻⢿⡟⠏⠉⢠⣾⣿⠏⠁⢁⣾⡷⣌⠈⠹⣿⣿⡄⠉⠩⢻⣿⣟⣤⣼⢟⡗⡻⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⢬⢧⣿⡻⣟⠳⣙⠂⠀⣾⣿⡟⠀⡾⢩⣈⣁⡍⢷⠀⠻⣿⣷⠀⠐⣫⠞⣻⢟⣿⡼⡥⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡈⢣⢣⡙⡛⢦⡝⠃⢸⣿⣿⠂⠀⢧⠈⢹⡏⠁⡼⠀⠐⣿⣿⡇⠜⢫⡴⢛⢋⡴⡼⢁⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⡀⣹⠁⠘⠟⠓⠈⢼⠁⣿⠀⢀⡎⠀⠸⠇⠀⢱⡀⠀⣿⠈⡲⠁⠺⠻⠃⠈⣏⢀⡟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣏⢳⣆⠀⠀⠲⠌⠀⠐⡀⠘⢳⡀⢀⠀⢀⡾⠃⢀⠆⠀⠁⠞⠀⠀⣰⡞⢹⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⡌⢺⡷⣄⡀⣦⡐⠀⠈⢻⡀⢀⡞⠁⠀⢀⣴⢀⣠⢾⡗⢡⠏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠙⢏⢺⠻⣆⡀⠀⢧⡾⠀⢀⣰⠟⡗⠹⠋⠊⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠹⠻⡦⢠⡄⢴⠟⠏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣄⣠⣐⡿⢿⡷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣶⡂⠁
⠀⠀⠀⠰⠉⣽⣿⣿⣶⡿⠉⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⠰⡦⠀⠀⢰⢾⡍⢤⣶⡆
⠀⠀⠀⠀⢠⠋⠈⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠠⣀⠀⠈⠛⢀⠈⠁
⠀⠀⠀⢠⠂⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠁⠢
⠀⢠⡔⢁⣠⠎⠀⠀⠉⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠐⢦
⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⢸⣿⣷⡄⠀⠠⠀⠀⣐⠁⠠⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡀⠀⠿⣿⠇⠀⠀⢀
⠈⠻⠿⡇⠄⠀⠀⠀⢿⣿⣷⡶⠀⠀⠐⠤⡀⠀⠀⠘⠁⠀⠀⢀⠀⣀⠀⣤⠀⠀⠀⠠⣿⣷⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡀⠀⠓⠀⡤⠒⠀⠁⠀⠀⠀⠐⠆⠀⠀⠈⠁⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠠⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠲⠀⠀⠀⠈⢄⢀⠀⡄⠘⠀⠁⠀⢀⠀⠀⣀⣴⣿⡀
⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⢰⡀⢏⡀⢀⡀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠄⠂⠀⠀⣐⠈⠃
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣆⣛⣩⠀⢠⡁⠀⢀⣀⣀⠀⠀⠀⣀⠂⠸⣿⠆⣐⠀⠺⠃⡀⠀⠀⠀⢠⠃
⠻⠟⠋⠒⠄⠀⠀⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠰⣌⣉⣠⠄⠀⠀⠋⢡⡅⠈⠦⠀⠘⠛⣤⡀⠀⠀⠀⠀⠁⠈⠃⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠄⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠠⣌⣉⣠⠆⠀⠀⠀⠈⢇⠀⠄⠀⠀⠀⠉⠀⠉⢺⠑⡀⣊⠀⠀⠀⠀⠠
⠀⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢐⠈⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣠⣤⣀⣀⠀⠀⠀⠸⠇⠄⠀⠴⠂⠄⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀
⠀⢸⣿⣿⣿⡷⠂⠀⢀⣀⣀⣤⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⣽⢿⡄⠀⠉⠙⠶⣄⠐⠶⠦⠀⠀⠁⠠⠀⠀⠀⠀⣘⠀⠀⠀⠀⠈
⠀⠀⠀⠉⠛⠁⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⣠⠟⠁⠀⠀⢰⡏⠈⢷⡀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡉⠀⠀⠀⠀⢰⣧⣤⣤⣤⣴⣾⠦⠤⠼⣷⡶⠶⠶⠶⣶⣷
⠀⠀⠂⠀⠀⠂⠄⠀⣰⣾⠋⠀⠀⠂⠀⠀⡟⠙⠷⣦⣀⣰⡇⠀⠀⠀⠈⣷⣀⣤⠞⠁⢸⠀⠀⢁⣄⠀⠀⠀⠐⠀⠀⢀⠀⠀⢄⣠⣦
⠥⠐⠁⠀⡂⠀⠀⠠⣿⣿⠀⠀⠀⠀⠃⠀⣇⠀⠀⠀⠉⣿⢶⣄⡀⠀⣠⡾⣿⡁⠀⠀⣼⠀⠈⠸⠁⠀⠀⠀⠀⠀⢀⣾⣷⡀⠈⠙⠻
⣀⠀⠀⠀⡀⠀⠀⠀⠻⣿⣆⡐⢁⠔⠀⠀⢹⡄⠀⠀⢰⠇⠀⢈⣿⠿⣯⣄⠘⣧⠀⣰⠃⠀⠀⠄⣲⠀⠀⠀⠀⠀⠛⠛⠛⠓
⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠹⣄⢀⡿⢀⡴⠛⠁⠀⠀⠙⠻⣿⡿⠃⠀⠀⠀⠷⣌⠀⠀⠀⠀⠂⠐⠀⠂⠀⠐
⠀⠀⠀⠈⠀⠀⠀⠀⠒⡡⠀⠀⢠⠀⠀⢀⠀⠀⠈⠙⠿⣯⣀⣀⣀⣀⣤⠴⠚⠉⡀⠀⡁⠀⢠⠁⢸⠀⠂⠀⢀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⢇⡤⠀⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡐⠀⠄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠒⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⢬⠁⡀⠀⠀⠀⠂⠀⠀⠀⠀⢲⣶⣶⣄⠀⠀⠀⠀⠠⠀⠀⠀⠀⡁⠀⠀⠀⠀⠃
⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⡀⠞⠁⠀⣀⣄⡀⠀⠐⠀⠀⠀⠙⠻⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠋
⠈⠦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠔⢠⣿⣿⡿⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡤⠄⠀⠄⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠔
⢀⡀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠁⠀⢀⡔⠀⠀⠀⠀⠀⠠⠖⠒⠛⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⠂⡸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⡝⠻⢧⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⡏⢠⡎⠹⡆⠀⠀⠋⠒⢤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠡⡘⢷⣠⡇⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⠸⣿⡀⠀⠈⡢⣝⠛⠧⢀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠷⠾⠮⣅⣉⡒⣤⣀⠁⠓⠺⢅⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⠞⠟⠓⣲⣦⣤⣄⡀⠈⠙⠺⣟⡲⣦⡂⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⢀⣽⡿⠛⠉⠈⠹⣆⠀⠀⠀⠹⡄⠙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⣸⣟⠂⠀⠀⠀⢠⡿⠀⠀⠀⠀⡇⠈⠈⢷⣁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡋⠉⠙⢻⠒⢤⣾⡟⠁⠀⠀⣠⠞⠀⢀⠀⢸⣿⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⢄⡈⢿⡻⠶⠃⠀⠃⢈⢿⠿⢷⣿⣅⡀⠀⠈⢀⣾⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⢦⢹⡿⢽⣶⣄⣀⠀⠈⠁⠀⠀⠀⠈⠉⠓⠶⢮⣈⡴⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠙⣿⠁⠄⠈⠹⣿⣭⡒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣦⡀⠀⠀⠻⣷⡙⢦⡉⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⠀⠀⠀⠀⠀⢈⡄⠀⠈⡆⠉⢳⠢⠄⡘⢷⡼⠟⠋⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢤⣝⢦⠈⠳⢤⣀⠀⠀⠀⢇⡀⠀⢱⣀⣨⠦⠖⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡀⠀⠀⠈⠓⢄⣀⠼⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⢀⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠄⠀⠀⠀⠀⠀⠀⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⡗⣊⣉⣦⢠⠤⠤⠤⠀⠛⠚⠉⠙⠛⠓⠒⠠⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣭⣭⡵⠤⠤⠤⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠐⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢻⣿⣿⣿⡉⢻⣿⠙⠦⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣋⠁⠀⠘⣿⡿⣿⣿⢿⡟⠀⠀⠴⠓⠽⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠂⠠⡄⠀⠀⠀⠋⠤⠖⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠡⡑⠀⢸⡀⠐⡄⢀⠀⠄⡆⠰⡰⢀⠁⡠⠁⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠀⢈⠂⡐⢌⢜⢄⢳⣀⣿⣼⣴⣥⣇⣅⣁⠆⡔⢡⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡑⢜⢄⣩⡾⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣅⣞⢔⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡈⢱⡟⠁⢀⣖⡽⠿⣿⣿⣿⣿⡿⣿⣿⡉⠙⢻⣅⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⠐⣰⠏⢀⣞⣿⣷⣷⢞⣙⣭⣭⣟⠻⣿⣿⣿⣦⠀⠹⣆⠡⠤⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠬⠄⠀⠀⢠⡟⢀⣿⣿⣯⡋⢽⣧⣿⣿⣿⣿⣿⡎⣿⣾⣿⣧⠀⣿⣆⠉⠡⠄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⢼⣧⣼⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡆⣿⣿⠤⠄⠀⠉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢼⣿⣿⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣿⡿⠀⠀⠠⠄
⠀⠀⠀⢀⠀⠄⠒⠊⠉⡀⠀⡺⣿⣿⣿⣿⣿⣿⣿⣬⣻⣛⣛⣿⣿⣿⣿⠿⠛⢿⣿⣿⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠀⠐⠙⣿⣿⣿⣿⣿⣋⡛⠻⠟⠿⠟⠛⠋⠀⢀⣠⣾⣿⠯⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⣽⡟⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠿⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣒⠠⠭⢒⣀⠀⣀⣀⣀⣐⡀⠠⠄⠒⡺⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠐⠒⠒⠒⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⢰⠆⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠃⠀⠀⠀⠀⠀⢤⢰⠀⠀⠀⠀⠀⠀⠀⠂⡘
⠀⢀⡎⠠⠀⢸⡁⠀⠀⠀⠀⠀⣄⠀⠺⢢⠚⠃⢀⡄⠀⠀⠀⠲⠀⣀⣀⠐⠃⠀⠀⠀⠠⠴⠀⣀⠄⠀⠀⠤⠀⠒⠂⠀⠀⠀⠈⡟⢻⠀⢐
⠀⠀⢶⣶⠟⡇⡏⠀⠀⠀⠀⠀⠘⡗⣬⣿⡹⡄⡯⠁⠀⢀⣐⠀⠃⢴⣄⠘⡁⢂⡀⠀⣰⣦⠀⣤⠟⠀⠚⡴⠂⠈⠀⠀⠀⠑⢘⣓⣘⡣⠊
⠀⠀⠨⣐⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠗⠉⠀⠀⠀⠀⠀⠠⠧⣙⣁⠴⠑⠂⠀⠈⠘⠁⠀⠀⠉⣀⡘⠁⠀⠀⠀⢠⠀⠀⠀⠏⠹⠄⠀⠀⢀⡀
⠀⠀⠀⢿⡵⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠘⠖⠀⠀⠀⡀⠀⠀⠀⠀⢭⡥⠀⠀⠀⠀⣡⢣⠀⠀⢀⣀⠀⢠⠌⠙⠆
⠀⠀⠀⣘⢠⠀⠀⠀⢀⡷⢯⡀⠀⠀⠀⢀⢀⠀⠀⣀⠤⠄⡄⡀⠀⠸⠗⠀⠀⡀⠠⠀⡁⠠⠀⠀⠀⢰⡆⠀⢀⠀⡘⠔⢥⡏⠀⠨⠍⠀⢰⢚⠀⠈⠀⠀⡀
⠀⠀⠀⢸⡖⠀⠠⣁⠁⣷⣯⡇⠀⠤⠀⠘⢉⠀⠮⢰⡶⠳⣶⠹⡄⠀⡂⠀⠀⠀⠛⠀⡅⠘⠃⠀⠀⠈⣷⠒⠋⠀⠀⣄⣢⠀⠀⢀⣆⠀⠀⠙⠶⣀⡈⠟⠁
⠀⠀⠀⢰⢺⠀⢄⡅⢰⢾⠿⡇⠤⠄⠀⠸⠚⠀⣇⢹⣦⣤⣟⢁⠃⠈⠋⠀⠠⠀⣀⠈⠂⢀⣀⠈⠀⠀⢸⠀⠀⠀⠀⠰⠖⠀⠀⠘⡞⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢬⡷⣾⠔⠡⠀⠀⢰⣠⠀⠀⠃⠠⡅⠄⠋⠀⠀⠀⠀⠀⠐⠁⠀⡀⠸⠁⠂⠀⠀⡟⠀⠀⠀⠀⠈⡏⠀⠀⢀⢦⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣴⠄⠀⢐⣸⠀⠐⠀⠀⢠⠀⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠰⠇⠀⠀⠀⡏⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢐⠀⠀⠀⠘⠿⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠁⠀⠀⠀⠘⠦⠆⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⡄⡄⠀⠀⠀⡁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⡆⠀⠀⢠⠉⠄⠀⠀⠀⠘
⠀⠀⠀⢸⢉⠀⠀⠀⠖⠐⢠⠁⠀⠀⠀⠀⠇⠁⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠷⡆⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢦⡆⠀⠀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢐⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⡞⠰⠀⢠⢠⠄⢄⠄⠀⠀⠀⠀⠀⠀⢠⢠⠄⠀⢀⠄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠄⠰⠒⠚⠀⠀⠀⠀⠀⠞⠀⠐⠀⠰⠠⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁⠉⠎⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠉⠛⠶⠀⡦⠀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠈⢩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⢀⡘⣏⡼⢏⠀⣁⠀⠀⣚⠒⠀⠂⠴⣤⡄⠀⢀⡀⢀⣼⣦⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢴⠰⠆⠶⠀⠀⡴⠄⠠⢀⣈⠀⢻⣷⣾⣦⣸⠀⠀⠀⠀⠉⠂⠄⠀⣰⣽⣶⠀⣴⠿⠺⠟⠋⠀⠀⠀⠀⠿⡄⠀⢸⠆
⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⢦⠀⡀⢠⣯⠀⠀⣹⣼⣿⠉⠠⠁⠀⠀⠀⠀⠀⠘⢋⣈⠷⢟⡧⢴⣦⠜⠓⢶⡎
⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠠⠜⠶⠆⠘⣿⡏⠀⠀⡜⢋⡟⠁⠈⠘⣁⣀⠀⠀⠀⠀⢴⡿⣿⣆⠀⠀⢠⡶⠋⠀⠀⠙⠘⠃
⠀⠀⠀⠀⠀⢀⣀⠖⠀⠐⢂⣸⣤⣽⣧⣼⡇⣄⢸⣷⣿⣿⣿⣿⡘⡁⠘⠀⠀⠀⠀⢹⣶⠿⡿⠀⡴⠋
⠀⠀⠀⠀⠀⠀⢻⣄⡠⡀⠤⣭⣿⣿⣿⢿⡷⣼⠟⠉⣏⣿⣿⡿⠘⡆⠀⠀⠀⠀⠠⢶⡧⠏⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢮⡀⣠⡀⢀⢀⣼⠁⠈⠈⢺⠄⣀⠀⠐⢢⡴⠛⠁⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣶⣦⡄⠙⣦⠘⠀⢻⣿⠀⠀⠰⠊⠁⠈⠀⠀⣻⠟⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢃⡀⢀⡀⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢉⠿⣿⡿⡦⠘⢦⠀⠀⢸⠀⣷⢀⠘⠻⢯⡄⢠⡏⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⣥⣾⠉⣀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢏⣟⠛⠁⠁⠆⠈⣧⠀⠘⣧⣀⠘⠁⠀⠈⠟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⠀⠈⢿⣿⣦⡜⠀⢀⡹⣏⡀
⠀⠀⠀⠀⠀⠘⣁⣤⡀⠀⠀⠀⠙⢦⠀⠀⢈⡄⠘⣇⠀⢸⡿⠀⠀⡀⠀⢀⡈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠠⢀⣿⡹⡿⣧⣾⠟⠉⠠⠈
⠀⠀⠀⠀⣶⣿⣿⣿⣵⠆⢀⠀⠀⠈⢇⠀⠀⠀⠀⠘⡄⠀⠀⣴⣾⣧⠀⠈⣣⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⢌⡀⢀⡶⢤⡀⠄⠀⣾⢿⣷⣾⣿⠧⠄
⠀⠀⠀⠀⣿⣬⡿⠋⠹⠷⠈⣧⣤⠀⠀⠀⢤⠀⠠⠀⢯⡀⢠⣿⣿⡏⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣄⠀⠹⠆⠠⠘⠛⠁⠌
⠀⠀⠀⠛⢻⣾⣹⣽⠇⢀⣼⣿⣷⡆⠀⠀⠀⠀⠀⠀⣎⠀⢸⣿⣿⡃⡀⠀⠘⠁⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠘⢷⡄⢠⡀⠀⣠⠳⠘⠲⠀⠈
⠀⠀⠠⣶⡿⣌⡋⠁⠰⢀⡈⠁⠙⠋⠀⠀⠀⠀⠘⢧⡿⡦⢈⣯⠻⠧⠄⠐⠀⠀⠀⠀⠀⠠⣬⠃⠠⠄⠀⠀⠀⠀⠀⠈⣿⠈⣷⠀⣡⠄
⠀⠀⢰⣮⠀⠈⠁⠂⠀⠀⠈⠣⠴⢄⠀⠀⠀⠃⡄⠀⠙⣧⠈⠡⡶⠄⠀⠀⠀⠠⡴⠀⠀⠠⠀⠀⠀⠶⠈⠀⠀⠀⠀⠀⢸⠀⠏⠀⣷⠀⠀⠰⠂
⠀⠀⠘⢿⣾⣷⡀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣤⣤⡤⠂⡀⠀⡉⠀⡀⠃⢀⣀⢀⡔⢆⡀⠀⡀⠀⠀⢀⠁⢠⠒⠂⠀⠀⠀⣾⡄⠀⢰⠏⠘⠲⠁⠀⠀⠀⡀
⠀⠀⠀⠘⠻⢦⣁⠀⠀⢺⣿⡶⠻⠃⠀⠀⠻⢋⠀⠀⠁⣷⣽⣠⠇⠀⢾⡟⢿⣿⣾⢿⣠⣁⣠⠖⠈⢳⠀⠀⠸⡄⠀⠀⣿⡀⠀⣸⡇⠒⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠉⠛⣠⣞⠁⢀⡀⠀⠀⣀⠀⠹⣶⣷⣄⠠⣿⣿⣄⢀⡀⠉⠂⠙⢿⣧⠛⣿⣄⠀⣰⣾⡧⣼⠟⠃⠀⢠⣿⣿⠀⠉⣠⣤⠞⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠋⠑⠿⠿⢦⣭⣄⣉⡀⠐⢿⣿⣿⣄⠀⠀⠀⠌⢿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⢠⡟⠘⢁⡴⠊⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠄⠀⠐⠀⠀⠀⠀⠀⡈⠙⠛⠿⣷⣦⠈⣙⠻⣧⡀⠀⠀⣈⣿⣿⣟⡁⡀⣠⡄⣀⣀⣤⣿⠀⢰⡛
⠀⠀⠀⠀⠀⠀⠀⠐⣦⠉⠛⡙⠆⣹⡆⠀⣠⣴⠃⣤⠀⠀⠈⠛⢿⣌⣳⠌⢻⣦⣀⠈⣿⣙⠻⢧⣭⣭⡽⠿⠛⣾⣿⠀⠀⡄⠀⢀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡐⠶⣉⠀⢻⣶⣾⣾⡇⡀⣨⢇⠐⢈⡷⠀⠀⠂⠀⠻⣿⡆⢠⣽⣿⣷⡄⠹⢶⣴⣀⣀⣤⢶⣿⣿⡟⠀⠀⠀⠀⠉⠁⣾⣿⣿⡏⠓⢁⡀
⠀⠀⠀⠀⠀⣹⠀⠀⠉⠀⠈⠁⠹⣿⡽⣿⡉⢈⢀⠈⠀⠈⠡⠀⢪⣤⣬⣇⠘⣋⢿⣿⢳⣶⣄⠉⢻⡟⠙⠁⢀⣹⠇⠀⢀⣤⣴⠟⢺⣯⠳⡇⠀⠞⠎
⠀⠀⠀⠀⠀⢠⡂⣠⣄⠀⠀⠀⠀⠈⢻⣟⣿⣦⢧⡐⣂⠀⢼⢇⣸⣿⣿⣇⡘⣧⡘⣿⡀⢿⣿⢷⡌⠁⠀⢀⣴⠏⠀⡼⠟⠉⠀⠀⠀⠙⠃⠉⠁
⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣧⣄⠀⠀⠀⠙⠂⠙⣟⣿⣿⢶⣶⣾⢿⣏⢀⠸⣷⡼⠁⣼⡇⠼⠃⠀⢹⡄⠠⠊⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠛⡋⠄⠀⠀⠀⠸⣦⡌⢐⣧⣽⣿⣿⡇⠹⣿⣶⣿⣿⠀⢿⡏⠀⠀⠀⠘⣧⠠⢀⣾⣿⠀⠀⣰⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠰⣧⣤⣴⣦⣄⠀⠈⠻⣿⣛⣿⣷⡾⠶⣦⡽⣯⣿⣿⣀⣿⣿⠀⢹⠀⠀⠈⡀⢀⣹⣿⢀⡴⠃⠉⢻⡁⠐⠓⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠛⠘⠻⣄⡙⠳⣦⣤⣽⣿⣿⡿⠑⠂⠈⠻⣎⡹⠋⠉⠑⢿⣆⠀⢣⠀⠀⢡⠀⢻⡿⣯⡀⠀⠀⠂⠀⠀⡀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢄⠀⡈⠉⠉⠋⠁⠀⡄⠀⠀⠹⣧⡘⢄⣠⢀⠀⢁⠈⠁⠀⠈⠀⣛⣶⡜⣷⣏⠀⠀⠐⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣻⣿⣷⣾⣿⣣⡄⠁⠀⠀⠀⢻⣇⢠⣴⣿⣿⠉⠀⢣⡀⠀⠀⣹⡯⠞⠃⠘⡄⣀⢀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠋⠀⠳⠉⠉⠀⠀⠀⠀⠀⠸⣿⢺⣏⣿⡛⠇⠀⠈⢁⠄⠀⠀⠀⠀⠀⢸⠧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣯⡀⠽⠲⠀⠀⠀⠄⡀⠀⠀⠠

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⢤⣄⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠛⠾⣄⡀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢍⣿⣦⣄⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢰⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⢹⣿⣏⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠈⠻⣶⠀⠀⠀⠀⠀⠀⢠⠞⠯⣉⠹⠏⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢳⡄⠀⠀⠀⠀⠘⡄⢠⠈⣉⠙⢒⢎⠁⡇
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠛⣉⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⡇⠧⠀⠁⡆⡸⠈⠄⡇
⠀⠀⠀⠀⢀⠔⠁⠀⠀⣤⠜⡁⠀⠀⢈⠴⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⠀⠗⠀⡃⠇⠀⠀⠁
⠀⠀⠀⡠⠊⠠⠜⢣⡄⠀⠉⠀⠀⡰⠁⠀⠀⠀⠑⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⣟⠀⠀⣀
⠀⠀⢰⢅⠀⠀⠓⠚⠂⠀⠀⠀⡌⠠⡖⠃⠀⠐⠈⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⡆⠸⠀⠀⠀⠉⠀⠀⣿
⠀⠀⠈⡀⠑⠠⡀⠀⠀⠰⠴⢆⠣⠘⠈⠀⠈⠀⠀⠰⡤⢇⠀⠁⢠⡀⠀⠀⠀⣧⠁
⠀⠀⠀⠑⠀⡀⠀⢁⡂⢘⠓⠎⠁⠀⠀⠀⠴⡀⠀⠘⠓⠎⠁⠀⣘⡾⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠈⠑⠔⠂⢀⡀⠢⢀⠀⠀⠀⠢⠉⠚⠀⠀⠀⡠⠊⠁⠐⣄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠉⢴⠁⠠⡀⠀⠀⠀⠀⡠⠊⠀⢀⡤⠞⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⣠⠐⢠⡤⠚⠁⢀⡴⠋⠀⠀⢰⡡⠤⠄⠤⠤⠤⠤⠤⢤⣀⠠⠬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⣇⡠⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⢠⣶⣿⣿⣿⣿⣿⠟⠋⠉⠙⠛⠛⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠄⠀⠀⣼⡏⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⡀⠀⠀⠀⠀⠈⠙⠛⠦⢀⡀⣀⣠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⢠⡀⣤⠀⠀⠹⣿⣿⣿⢿⣿⣿⣿⢿⣋⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⣿⠀⢄⠠⠀⠉⢱⣿⣿⣯⣴⣯⣥⣤⣤⣤⣭⣐⠲⢤⣀⣀⣀⣠⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢠⣿⠂⠀⠀⠀⠀⣼⣿⣿⠋⣿⣿⣿⣿⣿⣿⠀⠙⣿⢾⣿⣭⣿⡛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢼⡇⠀⠀⠀⠀⠐⠋⠹⣿⠠⠽⣿⡿⠿⠟⠋⠀⠀⠀⠿⠋⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠉⠹⣿⣷⣶⠖⠀⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⣰⣦⣴⣶⣶⡶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠈⠀⠀⠀⠀⠀⡬⠟⠋⠉⠉⠻⣿⣮⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣶⣾⡿⠋⠀⠀⠹⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⠿⠟⠋⡿⣯⢻⣿⣿⣯⣤⣤⣄⡀⠀⠘⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠟⠈⡈⡙⠉⠛⠻⢿⢿⠷⠄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⣷⣿⡉⢉⣉⣩⣶⣤⣄⣀⠀⣄⣀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠟⠉⠉⠙⠝⠻⠉⣛⡿⠿⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠚⢀⣠⡼⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⢶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⢻⣿⣤⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠠⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠤⠄⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⢄⣀⡈⠢⡙⠦⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⠄⢙⠂⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡈⠀⠄⢈⠘⠐⠣⠰⡀⢠⡀⠀⠀⠀⡄⠀⠀⠠⢀⢀⠀⠀⠀⠀⡐⣆⢀⢢⣎⣾⣇
⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠈⠁⢰⣦⣤⣤⣉⢀⡁⢀⡁⣀⣀⡠⣀⣀⣀⣁⣈⠀⠉⢉⣈⣀⣉⠠⠃
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⣤⠠⢠⠠⠆⠖⠠⠂⡴⢀⢄⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢸⠉⠉⠘⠛⠷⢽⡍⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⠗⡌⢠⣠⡦⡴⣤⣠⣦⡤⢂⠡⢢⢐⠀⠀⠀⠀⠀⢇⠀⠀⢸⡗⠦⠠⡀⠀⠘⠐⢟⡇
⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠬⢥⡿⠟⠀⠀⠀⠀⠸⣿⡷⠖⠁⣀⡐⣀⠀⠀⠀⠀⢸⡄⠀⢸⠀⠀⠀⠙⠖⠳⠞⣾⡟
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣰⡃⢽⡇⠀⠀⠀⠀⠀⠀⠻⣷⣆⠀⠀⠀⢰⡦⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⡼⠀⠀⠀⠀⠀⠀⠀⣰⣦⠥⢿⣇⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠈⠉⠑⠻⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢱⠀⠀⠀⠀⠀⠀⠀⠹⢶⣮⢻⣿⠀⠀⠠⣄⣀⡀⠀⠀⣹⣦⡍⡉⣐⡛⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢟⢔⢿⣤⠀⠀⠀⠀⠈⢢⣾⣿⠝⠻⠦⠀⠀⠀⠀⠀⡼⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠌⡾⡆⡿⠀⠀⠀⠀⠀⠀⠀⢺⠂⠁⠀⠀⠀⢀⡠⠊⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣆⡀⠤⠔⠂⠉⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⣀⠀⠀⢀⣀⡌⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⢀⠆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠲⣤⡀⠆⢣⣿⠀⠀⠀⠀⠀⠀⣾⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⠞⠋⠁⠀⠀⠈⠀⠐⠨⣽⣼⣼⣿⠎⠰⠂⠀⢠⠀⢻⣿⣼⡅⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⣣⠞⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡀⠸⢠⢀⠘⣀⣿⡁⠟⡄⡘⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢀⠟⢸⡇⡿⠏⡷⠀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⢂⠀⠠⢣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣹⡀⡆⠙⡇⡇⢱⠂⠄
⠀⠀⠀⠀⠀⠀⠈⠀⡀⣿⢸⢸⡀⠀⠀⢰⠀⢀⠀⠀⢠⠀⢳⠂⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⠙⣇⠘⡃⣷⠘⠖
⠀⠀⠀⠀⠀⠀⣃⠐⠳⡜⣰⢸⠀⡇⠀⢻⠀⢨⠄⣶⡞⡀⣿⠀⠠⠀⠀⠃⠁⠂⡀⠀⠀⠀⣰⢃⡗⢃⠸⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠉⠈⠀⠓⡿⠞⡔⣇⢻⢸⣧⠉⣆⣿⠰⣿⡛⠃⠾⠀⠂⠀⡀⢠⣼⣤⡼⠛⠁⠀⢳⠁⠀⠀⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠁⠀⠇⠘⠀⡀⠃⠆⢬⠈⣸⠀⣿⢼⣇⣄⠼⠀⡀⡔⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⣴⣠⣀⢀⣀⡀⢀⣀⡀⣠⣄⣦⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣻⠃⡠⠖⠊⢻⠛⡙⠲⣼⣧⠖⢋⠛⣟⠑⠳⢄⠘⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡀⢁⣎⣺⣿⣿⣿⣈⣿⣿⢁⣾⣿⣿⣗⣲⡈⢀⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡇⡟⠟⡐⢠⣤⡘⣿⢻⡟⣿⢃⣴⡄⢂⠹⢻⢸⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣯⣷⣌⠉⣰⡿⢸⡇⢿⣦⢉⣡⣾⣼⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣿⣟⣟⢰⡿⠠⢮⡵⠄⢿⡆⣹⣻⣿⡇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣹⡘⠃⠊⠇⢀⠈⠃⡀⠸⠑⠘⢃⣏⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠰⢤⢤⡈⠃⡸⠁⣤⡤⠆⠊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦⣴⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠋

⠃⢄⢉⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡉⣠
⠃⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠇
⠀⠀⠀⠈⢻⣦⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣠⣴⡿⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⢀⠀⠀⠀⣴⡃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠙⠁⠀⠀⠘⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏⠀⠀⠀⢙⣶⡆⠀⠀⠀⠐⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢀⣀⣤⡀⠁⠐⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠈⠉⠛⢁⡀⠀⠙⠒⠄⢈⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣬⠿⠃⠒⠀⠃⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⠓⡶⢿⡏⠀⠀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠁⠀⢀⣀⢀⡀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣶⣿⡟⢡⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⣿⣿⠿⠿⠀⣟⠉⠀⡸⠃⠀⣠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠁⠀⣴⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠠⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣺⠔⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⣠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠀⠀⠀⢠⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⡀
⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠳⠄⢀⢀⠸⢘⠌⠀⠁⠇⠀⠀⠄⠸⠈⠁⠇⠀⠀⠀⠀⠹⣿⣿⣦
⠄⠀⠀⢀⣼⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠻⣧⡀
⡀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆
⡄⠊⣈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠋⠀⠂⠀⢀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠤⠄⠀⠈⠀⠒⠁⠐⠀⠀⡌⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎⠈⢁⡨⠬⠔⢲⡖⢲⢒⢖⢲⠂⠄⠬⣀⡉⠡⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠴⡘⢪⠑⠠⣀⣠⣼⡗⢷⣦⣾⣿⣦⣄⡤⠀⠀⠁⠛⠐⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠁⠐⠖⠀⢀⡠⠼⠿⢿⡇⠀⠈⣿⣿⣿⣟⡛⠛⠀⢀⡠⠖⠚⡓⠲⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢤⣀⢀⣀⣀⠀⠀⠒⠀⠇⠀⠀⢸⠟⠘⠈⠊⣀⡴⠛⠀⠀⠀⠁⠀⠀⠂⠝⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠚⠺⢤⢍⣉⠉⠍⢉⠉⠐⠁⡆⠀⠀⣼⠁⠉⠉⠁⡼⠗⠠⢂⣀⣠⣤⣬⣤⣤⣤⣬⣭⡑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⡉⠱⠀⠀⠀⠄⢀⠝⠉⠒⢧⡀⠀⠢⠄⠀⠀⡟⣧⠀⢀⣺⠴⠒⠛⢭⢿⡏⠉⢻⣿⣿⢷⣥⣌⡿⠇
⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⣀⡀⠀⠈⠂⠈⠀⠁⠀⠀⢄⣁⠽⠳⡞⠀⠀⢸⠯⣏⢣⡘⠧⠮⢤⡦⠖⠒⣃⠀⢠⠉⠉⠡⡄
⠀⠀⠀⠀⠰⠉⠆⠀⠀⢀⡀⠈⠀⠀⠠⣀⡸⠀⠀⠀⠘⠀⠀⠠⣵⠀⠀⠸⣀⣨⡽⡥⠴⢲⣿⠁⣀⡵⠃⢀⣞⠓⠀⠄⢾⡀
⠀⠀⠀⠀⡏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡒⠂⠐⡼⠂⠂⢡⠻⠁⢁⢄⣀⣸⣟⣼⠯⠤⣄⡼⢹⡏⢄⣠⠞⠁
⠀⠀⠀⠀⠰⠉⠀⣀⡤⠶⠶⣶⣶⡟⠛⠛⠛⠙⣽⡉⢈⠠⠄⠊⠁⠀⠀⡬⠠⠼⣿⣿⠋⡽⠋⠁⢀⠼⠁⠀⠔⠂⠉
⠀⠀⠀⠀⠀⠈⠃⠀⠠⠤⠤⠤⠀⠆⠀⠀⠀⠀⠀⠠⠀⠀⡆⠂⠀⠀⢠⠃⠲⠈⠀⣯⠜⠀⢀⡴⠋⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⢀⣑⠀⠠⡐⠀⠀⣘⠖⡁⢹⡴⠋⠀⢠⠎⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠁⠀⠀⠀⠀⢠⣨⡇⠀⠀⢿⣐⣠⠖⠁⠀⣠⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢅⠳⡄⠀⠀⠀⠀⠀⢻⣧⡆⢀⠠⠿⡇⠀⢀⣴⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠨⣑⠻⡄⠀⠀⠀⢀⠀⠉⠙⣍⢁⢐⣁⣰⣾⠃⣜⡅⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣢⣄⣱⠟⡖⠒⠦⠁⠀⠒⢹⡄⡀⡿⠿⠋⠙⠉⢳⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠉⢲⠇⠀⠓⠰⣖⣁⠀⢀⡸⠗⠻⣁⢤⡴⠒⠋⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢏⠀⠀⠀⠀⣃⣨⡽⠯⠥⠤⠤⠤⠼⠷⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠉⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠠⢤⣤⣦⣔⣰⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣀⣀⠘⢷⣦⣤⣄⣠⣤⣅⡀⠀⣀⣀⣭⣵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠙⣿⣿⣿⣿⣿⣆⠈⠉⠻⠿⠿⣿⣿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡠⠀⢀⣠⣤⣤⣤⣤⢠⣀⡞⠁⢰⣿⠿⢿⡿⠛⣻⣷⡀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣟⢻⣿⣿⡿⠹⡇⠀⠞⢁⡴⠞⢻⣿⠿⡁⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠟⠻⠿⢻⣈⠋⠁⠀⠀⠀⠀⣴⣿⣿⣿⡾⠏⠀⠉⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⡏⠂⠀⠀⢠⠀⠀⠁⠁⠈⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⣢⠀⢡⠀⠀⠀⣾⠀⠀⠄⠀⠀⠀⠀⠀⠊⢡⠄
⠀⠀⠄⠀⢀⠀⡀⠀⠈⡐⢿⣿⣿⣦⠆⢀⠀⣿⣿⡄⢸⡇⣀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⢰⣦⣼⣧⣄⣠⣿⣿⣧⡛⠻⣿⣿⣿⣿⠿⣧⡿⠂⠁
⠀⠀⠀⠀⠀⠀⢈⣧⡀⠸⣿⣯⡟⣻⣿⣿⣶⣿⡟⠔⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⣯⣄⠀⠀⠃⣽⠞⢏⢀⠈⠍
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠠⠆⠚⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⢾⠀⠀⣷⣄⡘⣄⡀⠀⠳⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣤⣤⠄⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⢠⣶⠟⠋⠀⢁⣀⠉⠤⠈⠉⠃⠰⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡸⠁⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢸⣅⣀⠈⠀⠂⠀⠐⠀⠀⠀⢐⡤⣀⣽⡷⠀⠀⠀⠀⠀⠈⠉⢘⡺⢿⠮⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠒⠉⠀⠘⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣽⣳⡾⣏⣽⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡜⠄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣿⣿⣿⣽⣿⣿⣿⣿⣿⠟⠀⠐⠈⠐⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠋⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡟⠀⠀⠀⠀⠀⠀⢀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡿⠿⠟⠛⠀⠀⢠⣾⢿⣿⣷⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⣦⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠒⠿⢿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⣟⠁⠈⠙⠛⣿⣿⢿⣷⣤⠀⠀⠀⠀⠀⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣶⣿⣿⠷⣤⣤⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠋⠉⣥⣾⡿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠁⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⢠⣾⣦⣶⣶⢒⣷⣿⡿⠛⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢋⡙⠛⠋⠉⠐⠚⠶⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡟⠊⠈⠁⠀⠀⠀⠀⠀⠀⠀⣬⡜⢣
⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡿⢠
⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡀⢀⠀⢠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠐⠙⠄⣾⣦⠀⠀⠀⠀⠀⠀⡰⠲⡃⣬⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠁⠀⠀⠀⠀⠀⠈⢡⠓⠠⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⠇⢫⣠⣄⣀⡠⡎⠀⣀⠀⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠁⠀⠀⠀⠀⠀⠀⣈⠀⠴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⣏⠀⠀⣷⣄⣁⣠⣿⣆⢸⡄⠀⢿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠁⠀⣸⣿⣿⣿⣿⣿⡟⣸⠁⢀⣾⡄
⠀⠀⠀⠀⠀⠨⢁⠤⠛⣠⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣠⣀⡀⠀⣿⣿⣿⣿⣿⡿⠁⣡⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⢀⡏⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣾⣿⢿⡿⢻⣿⣧⣼⣿⣿⠏⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠉⡿⠉⢻⡟⠁⠀⠃⠈⠉⠿⠁⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡋⢙⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣹⣿⣿⣿⣿⣯⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣽⣿⣷⣄⠀⠀⠀⠘⠛⠛⠛⠛⢿⠟⠛⢻⡟⠉⠉⠉⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣿⣿⣿⣿⢿⣿⣦⠀⠀⠀⠀⣀⣀⡀⣼⠀⠀⢸⡇⢀⣴⣿⡿⠟⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣷⣾⣿⣿⠀⢀⣴⣿⠟⣿⣿⣿⡀⠀⢸⣧⣾⣿⣿⣿⣦⣿⣿⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⡍⠙⡏⠉⠉⠀⣴⣿⣿⣷⣦⣾⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢦⣄⠀⠈⡇⠀⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣿⣷⣼⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣾⣿⣦⠀⣷⠀⢻⠀⣼⣿⣿⣿⣿⣯⡸⣿⣿⣿⣿⣿⡇⡿⠛⠛⣻⠛⠛⠛⡟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣙⣿⣟⣡⣿⣿⣷⡰⡇⠘⠀⠘⠛⠛⠛⠻⡿⠿⠿⣿⠿⠿⠟⢡⡇⠀⠀⡏⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⠉⠉⠀⣃⣄⣀⣾⠂⠀⠀⠀⠀⡇⠀⠀⡟⠀⠀⠠⢸⠇⠀⢸⠱⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠁⢣⠀⠙⡆⠀⠀⠀⢰⡇⠀⢠⡇⠀⢀⡀⣾⠀⢀⡏⠁⢀⣴⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⣴⣶⣦⡘⡆⠀⢸⡀⠀⠀⢸⠀⠀⠘⣡⡾⣿⣿⡆⠀⡼⠀⢠⣾⣟⣿⣅⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣿⣿⣷⠀⠈⡇⠀⠀⡞⠀⢰⣾⣿⣿⡿⣿⣿⡌⠁⠀⠘⣻⠛⠿⠿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⣿⠿⡟⠋⡇⠀⢸⣶⣶⡇⠀⢸⡟⣻⠛⢻⠿⢿⠇⠀⠀⢠⠇⡌⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡀⢹⡄⢹⢀⣿⣧⣿⣿⠆⢸⠁⠇⠀⣾⡴⠃⠀⠀⢠⠏⣸⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣄⢧⠘⢷⡘⡆⠀⡏⢸⠁⠀⣿⡜⠀⣴⠋⠀⠀⡠⣠⢳⣌⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣼⣟⠈⢆⠀⢳⣇⠀⡇⢸⠀⢠⡿⠁⢰⡏⠀⣠⢞⡴⠇⡁⠈⢳⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣆⠈⢦⠘⣆⠀⠙⣄⡷⢸⠀⡼⠁⢀⡞⠀⡪⣐⢩⠴⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣄⠀⠀⠈⠣⡈⠧⠈⢧⠀⠈⠃⠘⠋⠀⠀⠚⠋⠉⠈⠀⠀⣠⠔⠋⠀⠀⢠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠆⠀⢸⣿⣷⡄⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠰⠀⠈⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⡀⠀⣼⣿⣿⣿⣿⣷⠀⢀⠀⠸⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠛⠛⠛⠉⠛⠉⠛⢲⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠉⢃⣠⣤⣶⣾⣴⣇⣠⣠⣿⢄⡴⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠣⢌⣁⢾⣿⣿⣿⣿⣿⣿⣿⣿⢛⣛⣧⡄⢲⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠐⠤⡚⣿⣿⣿⡏⠁⠈⣿⣿⣿⢿⣿⣿⣿⠿⠇⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠠⠤⠦⠘⣟⣿⣻⡇⠀⢸⣿⣿⣿⡲⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠡⣽⣇⠀⣿⣿⡻⣿⠷⡄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣴⣶⡶⠿⠻⣿⣶⡄⢀⠀⠠⢀⣠⢞⣭⣿⣿⣿⣿⣷⣾⣂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣏⣷⣴⣶⢿⣿⣿⣿⣷⣶⣿⣾⠷⠛⠀⠛⠁⠈⡉⣩⣿⣿⠿⡿⠿⢿⣿⣧⣄⠀⠶
⠀⠀⠀⠀⠀⠀⠀⣴⠿⢛⣻⣿⣿⣥⣾⣿⣿⣿⣿⣟⣿⡍⣡⠄⠰⠀⠀⠉⢁⣿⣿⣧⣾⣶⠿⠛⢿⣿⣿⣧⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠘⢁⠀⠈⢹⣿⣿⣿⣿⡿⣿⣿⣌⠁⠀⠀⠀⠐⠀⠀⠀⣴⡿⣿⣿⣮⣿⣿⣿⣿⣥⡄⠈⠩⠛⣿⣿⣻⡖⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⢿⣿⣿⡇⣿⡿⢫⠄⠀⠀⠀⠀⣠⣀⠰⠌⠵⢿⣿⣿⣿⠿⣿⣿⣿⡟⠁⠐⢀⣿⣿⣟⣿⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⡁⠉⠀⠀⠀⠀⢀⣠⣼⣿⣿⣧⠀⠶⠊⠛⠛⣿⠂⣿⠟⠫⠁⠀⣀⣾⣿⣿⡟⢋⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠰⠟⣿⣿⣿⣿⣿⣿⣷⡄⢀⣀⡀⢉⣸⠃⢀⣀⣤⣾⣿⣿⡟⠻⠷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠐⣎⠛⡟⣿⣿⡏⣿⠃⠀⢸⣿⠁⠈⠀⠀⠘⠻⡿⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠸⡟⠉⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣥⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣴⡧⠀⠀⠀⠀⢀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠀⢠⣾⣧⠁⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠂⠀⠀⣼⢻⡏⡀⠀⢀⠖⠀⠀⠀⠀⠈⠻⢿⡟⢿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⡀⠀⣄⢠⣿⣮⡶⠁⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠀⠀⣿⡁⢼⣧⣾⣏⢸⣿⣿⠇⣼⣿⠃⢀⠀⠀⠀⠀⠀⣄⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧⢀⣿⣷⢸⣿⢟⠉⣸⣿⡿⣲⣿⣿⣤⠎⠀⠀⠀⠀⠀⢉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣷⡘⢷⣿⣿⢳⣿⣿⣿⡗⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣷⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⡾⣿⡟⠉⠙⡿⣿⠁⢹⣿⠙⢷⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠛⠛⠋⢉⣱⠿⠻⣏⠉⡏⠀⠈⣿⠋⠻⣿⠛⢿⣿⠿⡟⠁⣰⡏⠀⠀⣘⢀⡿⠀⡸⠿⡀⠀⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠖⠛⠁⠀⠀⡟⢻⢇⠀⠐⠙⣤⠀⠹⡆⠘⣿⡀⠁⡀⠻⠀⠀⢰⠟⡏⠃⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡞⠉⠙⡄⠀⠀⡀⠶⠑⠀⠀⠷⠀⠖⠁⠀⢸⠃⣶⠁⢙⡀⢠⡆⠀⢰⠇⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⡆⠈⢡⠀⠸⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠃⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠂⠀⣴⠆⠀⣀⡀⠀⡬⠀⠀⠃⠀⠘⡅⠀⠘⡇⠀⠁⠀⡈⠃⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠁⠀⠀⡁⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⠀⠀⠰⠦
⠀⠀⠀⠀⢤⡶⠀⣄⣙⣷⠀⣼⠄⠃⠤⠀⠻⣄⠉⠢⡇⠀⣀⠰⡟⠃⠀⠆⠀⠀⠀⠀⣴⢲⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠓⠃⠀⠉⠁
⠀⠀⣠⣀⠀⠀⠠⢀⣴⠀⠒⠀⠰⠂⠲⣴⠆⠀⠀⠄⠀⠁⢈⠃⠦⠅
⠀⠀⡀⠁⣀⣁⣀⠀⢀⡀⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠛⠺⡝⠱⠾⣦⢜⠀⣈⠶⣅⠀⠀⢀⢀⡀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠂⠴⠄⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⠐⠉⠁⠀⠚⠛⠛⢛⠛⠓⠒⠂⠀⠀⠀⠀⢀⠀⠀⢡⢀⠀⠀⠘⡏⠀⠀⠀⠰⠟
⠀⠀⣠⣇⠐⠀⠀⠀⢠⡀⡀⠀⠀⠶⢶⠀⢈⣨⣍⢰⠀⠀⠀⢄⠀⠄⠀⠈⠂⢃⣈⠀⠀⠀⠀⢧⠀⠀⠀⡄⠀⠀⠤
⠀⠀⡈⠋⠀⠙⠁⠀⠾⠋⠁⠀⠀⠀⠀⠀⣤⠀⣀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠
⠀⠈⢍⡲⠻⡆⢠⣐⣶⢇⢀⣈⠁⢀⠀⣄⢻⣇⣿⣿⠇⠀⠀⣶⡀⢸⡇⠀⠀⣦⠀⡀⠀⡀⢠⠀⣠⢈⠀⡀
⠀⠑⢚⡗⠛⠆⡈⠁⠉⢠⠠⡀⠀⣾⡇⠘⣧⣿⣟⡟⠀⠀⢐⡿⣇⢠⡥⢠⢂⣿⠀⡆⢀⡇⠈⠃⡀⢸⣿⣧⡀⠀⠰
⠀⢠⡀⡅⠁⠀⠩⡐⢸⡜⠃⠑⠸⣿⡇⠀⡎⠙⡟⠀⠀⠠⣶⣿⡵⡀⠀⠈⢸⣿⠀⡇⢸⠇⠀⠀⠀⠂⠽⠞⠁⠀⢈
⠀⢸⠀⢣⠀⡤⠢⠀⠀⠁⠀⢂⠸⣿⡇⠀⢤⡀⡄⢀⡀⢘⡺⢷⣿⣖⡀⠑⢸⡿⠀⠃⢸⢀⡆⠀⠠⠀⠀⢂⠀⡆
⠀⠘⢧⠘⠀⢀⠃⠀⠀⠀⠘⠌⠈⠿⠅⢀⣈⣿⠁⠈⠁⢨⢭⣿⣿⠭⠄⡀⣼⡇⠀⠂⠀⢸⠀⠀⠀⠀⠀⢼⠅
⠀⠈⣴⡇⠀⠉⠴⠷⠆⢰⠶⡦⠀⠠⠒⠃⡾⠃⠸⠿⠇⠰⣴⣤⡄⠒⠀⠀⠉⠀⠀⠀⠠⠿⠆⠀⣀⠀⠀⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣔⣀⠠⠤⣀⡀⣀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⡿⣴⣾⣿⣿⣿⣿⣏⣩⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢿⣻⠿⢿⣾⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣉⠀⢘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠟⢶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠏⢹⣿⣿⣿⣿⣷⣈⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣸⣿⠿⢿⣿⣿⣿⣆⣠⣶⣿⣿⣿⣿⢟⠋⢉⠻⣿⠶⠉⠀⢀⡨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣴⣿⣿⣿⣿⣿⣿⣏⠁⠘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣴⣴⣾⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣰⣿⣿⣿⣿⡟⢿⠿⢟⠥⢀⣠⠀⣖⣤⣤⣶⣦⡀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⢿⣿⣿⣿⣿⣷⣴⡞⠛⠂⣉⣵⣾⣿⣿⣿⣿⣿⣿⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⡟
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⡀
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖
⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⠛⠛⠛⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛⣛⣉⣉⣉⣉⣉⣉⣙⣿⣻⣿⣋⣉⣉⣁⣀⣈⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣹⣛⣉⣛⣟⣽⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⣤⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠘⡄⢠⢃⡨⠛⠧⣽⡤⠃⠠⠿⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣧⡟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⠯⠤⠤⠔⠊⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠰⢿⠀⠀⠀⣄⠫⠑⠀⡀⠀⠀⠀⠠⠀⠀⠈⠇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠐⠐⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠉⠉
⠀⠀⠄⠀⠀⠀⠀⠾⠆⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠄⢀⣴⣿⠟⠁
⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠁⠤⢘⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠘⠀⡒⠈⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⡀⠀⠀⠀⣾⡇⠀⠀⢠⡶⣆⠀⠀⠀⠛⠛⠛
⠀⠐⠂⠀⠀⠀⠀⠳⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠈⠀⠀⠀⢸⣽⣿⡇⠀⢀⢀
⠀⠀⠀⠀⠀⣀⡀⠳⠆⠀⠀⠀⠈⠀⠁⠀⠀⠀⡀⠀⠀⣠⣶⡧⠀⠀⠀⠀⠀⠻⠓⠀⠙⠛
⠠⠀⠀⠀⠀⢙⡃⠀⠀⠀⠐⠂⠀⢠⣤⡄⠀⠻⡟⠀⠀⠘⠁⠀⠀⡀⠀⢀⡴⡶⢢⠀⢀⣦⡀⠀⠀⠀⠀⢴⡷
⠀⠐⠢⠀⢰⣏⣷⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣤⣤⡄⠀⣾⠿⠀⠘⢷⣏⡿⠁⠈⠰
⠀⢈⠡⠀⠹⣾⡿⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠑⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠾⠮⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠛⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⢰⠀⠀⠆⠀⠀⠈⠀⢀⡤⠒⡻⣏⠑⠢⣄⠀⠀⠀⠀⢒⠀⠗⠀⣀
⣐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣏⣤⣤⡧⠼⣦⠤⢬⣇⠀⠀⠠⠈⠀⠀⠀⠐⠀⠀⢀⠠
⠀⠅⠀⠂⠀⢲⡶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠉⠲⣾⣀⠀⣹⡖⠉⢸⠀⠀⠙⠀⠀⠀⠁⣰⣰⡀⠘

⠛⢈⡉⠉⡈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⢀⡉⢁⡀
⡆⢈⣻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣟⠁
⠀⠈⠉⠙⠋⣠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣄⠛⠋⠉⠁
⠆⠀⠀⠀⠀⠙⢿⣷⣤⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠠⠀⠁⣠⣠⣾⡿⠋
⡄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃⠀⠁
⡀⠀⠀⠀⠀⠀⠂⠈⠻⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⡀⠀⢰⣿⣿⣿⣿⡆⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠟⠁⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠀⠘⢿⣿⣿⡿⠃⠀⣾⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⣿⣿⣦⡀⠀⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⣤⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢿⣦⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⣴⣶⡶⣦⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⣀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⣴⣦⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠈⠁⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠴⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄⠀⢸⣷⠀⣠⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡿⢿⣿⣿⣿⣿⣿⣿⡿⢟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⢹⡿⠁⠈⠛⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⣤⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣤⡄⠀⠀⠀⡆
⠄⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠄⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦
⠄⠀⠀⠀⠸⣷⣦⣀⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣠⣴⣾⠃
⠆⠀⠀⢀⣀⣬⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣥⣀⡀
⠄⠀⠀⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⣀⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⣀⣀⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛
⠂⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣉⣽⣿⣿⣿⡆⠀⠀⠀⢀⡀⠀⠀⠀⢰⣿⣿⣿⣯⣙⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠉⠛⠿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠸⠟⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⠿⠋⠉⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣇⣿⡄⠀⠀⠀⠀⠀⠀⢰⣿⢼⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⢻⣿⣄⠀⠀⠀⠀⣰⣿⠏⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣾⣿⣿⣦⣤⣤⣾⣿⣿⣷⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣤⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣄⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠃⣠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡴⠐⢆⠀⡄⢤⡀⠀⠰⠠⡆⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⠐⠀⠂
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠻⠰⠆⠀⠙⠀⠛⠀⠻⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠃⠀⠀⠀⠀⣴⡿⠛⠁⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠻⣿⣦
⡀⠠⣤⣴⣦⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠁⢶⣦⣤⠄
⠃⢠⡾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠐⠂⠤⠶⢦⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠐⠒⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⠀⠠⠄⠀⠀⠀⠀⠛⢿⣿⣿⣯⣽⣿⣦⣄⠀⠀⠀⠀⠀⠀⢀⡴⣀⠀⢀⣀⣀⣄⣄⣠⣤⣦⣾⣾⣾⣿⣳
⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣴⣿⣿⣿⣯⡙⠻⢿⣿⣷⣄⠀⠀⠀⠀⠾⠛⣩⢉⣿⣿⡿⣻⡄⢹⣿⡋⠉⠙⣾⣦⡬
⠀⠠⠖⠀⠀⢀⠀⣀⣀⠀⠀⣀⣀⣤⣤⣦⠶⠾⢶⣿⣿⣿⣿⣷⣿⡿⣿⠿⢿⣆⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⢁⣁⣀⣙⡛⠀⠁
⣴⣦⠶⠤⠴⠋⣉⠉⣥⢘⣻⣯⣵⡶⠂⠚⠷⣷⣶⣶⣮⣭⣤⣄⡒⠂⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢹⠋
⢻⣦⡴⣶⢶⣦⣬⣼⣿⣿⣿⣿⠿⣿⡀⠀⡄⢸⣿⣿⠟⠃⣀⠀⠩⣙⠋⠠⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠉⠛⠻⠶⠶⠞⠉⠙⠛⠋⠉⡉⢛⡃⠀⠙⣼⣿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠇⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠶⠶⣮⣩⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄
⠀⠀⠀⠀⣀⢤⢤⣤⢄⡀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡁⠀⠀⠀⠀⠀⢀⣠⣀⢈⡤⣈⣭⡛⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠻⢝⣦⣄⡀⠋⢙⠂⠀⠀⠀⠀⠀⠐⠁⣿⡇⠀⠀⠀⠀⠀⠈⠻⣟⠉⣹⠃⠈⠃⠰⠛⠁⠀⠂⠀⠀⠀⠀⠀⠀⢰⣾⡀
⠀⠀⠀⠲⣮⣽⣭⣛⢁⣠⡟⠀⠀⠀⠀⠀⢀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡄⠀⠀⠿⠂⠄⠀⠀⠀⣾⣆⠀⣠⣼⣿⡀
⠀⠀⠀⠀⠀⠙⠁⠉⠈⠁⠀⠀⠀⠀⢄⣀⢜⣹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⡀⠀⠀⠢⠕⠆⠀⠀⠀⠈⠉⠀⠀⠛⠹⢷
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠂⢟⡰⠂⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣀⡄⣸⠃
⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⠴⠁
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠀⠀⠠⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠳⠰⣦⠄⠉⠁⠀⠀⢉⣴⣤⣿⣿⠟⠃⢼⠆
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⡻⠛⠀⢠⣴⣶⠄
⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⢿⣼⣿⣶⣄⠀⠀⠀⠀⢧⣖⠀⡈⠲⢦⠀⠀⠀⠀⠀⠀⠀⢾⠟⠀⣂⣴⢒⠆⠰⣭⣿⡆
⠀⠀⣠⣿⣿⣿⣹⣿⣿⣷⣶⣿⣦⣿⣿⣧⠀⠀⠀⡼⠋⢣⡾⠛⢢⡀⠀⠀⠀⠀⠀⠀⡶⢀⢘⣻⣿⠿⠇⠉⠉⠁
⠀⣼⣿⣿⠿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣉⣿⣇⠀⠀⣤⣀⣿⣄⠈⢢⡇⠀⠀⠀⠀⠀⡇⢰⡿⠛⠛⠀⠀⠀⢀⠀⠀⠀⠀⢀⡄
⢸⣿⠿⣿⣿⣯⣿⣽⣿⣿⣯⡿⣿⣿⣹⣿⣽⠆⠀⠈⠻⣿⣿⣶⢿⣷⠀⠀⠀⠀⠀⡑⣼⠇⢦⣀⢲⡆⠠⠄⠀⠀⠀⠀⢾⡇⠀⠀⢀⡄
⢸⡁⠀⡙⠁⡛⠙⣿⣿⣿⣿⡿⠛⠿⢿⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢹⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣿⠷⠟⠁
⠀⠈⠃⠄⠈⠀⣠⣿⠀⠀⣿⡇⠀⠀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣰⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡿⣄⣹
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⡇⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⡀⢘⡃⠐⠺⣟⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣥⡾⡟
⠀⠀⠀⠀⠀⠀⢸⣷⡾⢷⣾⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠀⣿⠋⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢆⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠤⠤⠶⠶⣦⠤⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠋⢀⡀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣒⣀⣤⣤⣶⣤⣄⣤⠲⢾⣿⣿⠛⠛⣃⣽
⠀⠀⠀⠀⠀⠸⠂⠈⠸⠏⣿⠀⠀⠀⠀⠀⢠⡿⢿⡿⣞⣰⣟⣓⣖⣶⣀⠀⣄⣀⠀⠀⠀⠈⠙⠛⠛⠿⠿⢿⣞⣏⣥⣬⣿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇⣽⠀⠀⠀⠀⠀⠈⠙⠿⣿⣾⣿⣽⣯⣽⡽⣞⣻⢟⣾⠻⠶⢄⡀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠈⢷
⠀⠀⠀⠀⣤⣦⢰⡀⠻⠈⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⢻⣷⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠓⠂⠈⣑⣶⠟⠿⠟⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢉⢛⣻⣿⣿⣯⣽⣿⣿⠇⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡘⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠰⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣠
⠀⠀⠀⢸⠀⠀⠀⠰⠃⢉⢐⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠘⠠⣶⣆⣠⣍⡭⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣄⢠⣶⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣥⣶⡶⠏⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣿⣭⣴⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⢀⡷⣄⠀⠠⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⣄⣄⢠⡜⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠘⣿⣃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠏⠀⠀⠰⠀⠀⠀⠛⠿⣥⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀⠢⠀⢀⡄⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠈⢡⣤⠔⢀⣁⠄⠐⠀⠀⢘⣷⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⡴⠗⠀⠈⠉⠗⢦⣤⠛⠿⣶⣏⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⢀⠐⢀⠀⠀⣸⠃⠀⠀⠛⡟⠓⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢢⠀⢬⡆⠀⣿⠀⣴⡀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣀⡸⠻⣠⠀⢰⡿⢷⠄⣰⡾⣶⠾⡋⢃⣧⡀⢾⠛⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⣶⠶⠤⢦⡴⠃⡀⠈⠁⠶⢸⣶⡀⠌⠿⢿⣌⢀⣻⣿⢳⣾⣄⣷⣔⡀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢺⡆⠢⢰⣿⣀⣠⠋⠀⠰⠋⠀⠛⢿⠀⣹⡞⢻⡿⣿⣿⣅⣿⣿⡿⢿⡙⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠟⠀⣼⣾⡁⣹⡿⣠⠀⢀⣤⣤⡀⣀⢠⣿⣷⠾⡿⢿⣿⠻⣽⡗⡼⠋⠳⣠⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⢀⣶⡿⢻⡏⠀⠀⢠⡄⡾⠗⠀⢻⣿⢿⣿⡷⡒⠀⠸⠿⠞⡛⢈⠓⢛⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢰⣿⡿⠀⣞⢡⣦⣀⠘⠁⠉⢤⣠⣻⣟⣏⣿⡦⠀⠀⣤⠀⠸⣛⠾⠃⣻⡆⠸⡀⠀⠈⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢷⣾⣟⠁⣤⣌⠁⠀⡘⢿⠶⡀⢀⡉⢀⣿⣿⡿⢧⡄⠙⠋⠀⠑⢹⠀⠀⠘⠀⠀⢼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠓⠛⣸⣣⣴⣿⠿⡜⢸⣇⠉⠀⢰⣾⣧⠸⣿⣿⣧⢘⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠁⣿⡿⠃⠻⣸⡟⠀⡖⣃⢨⣿⠾⣿⢛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢀⠀⣺⣧⣾⣼⡀⢈⡤⢌⣶⡞⠁⢤⠀⠟⢻⠃⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⠼⣿⠛⠁⡼⢳⣿⡄⢀⡿⠃⠁⢈⣽⣦⡼⡀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠋⠀⠀⠀⠦⠀⠲⡟⢠⣄⠈⢋⢦⠐⡮⠏⣿⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠀⠹⡿⠇⠄⠀⠀⠁⠀⡿⠀⠈⠙⠀⢠⣳⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠈⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁





⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠉⢉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⣀⠀⢀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⡀⠈⡀⠀⠁⢠⢤⣶⣶⣦⢰⢠⡄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃⠀⠀⠈⠁⠃⠀⠀⠀⠋⠉⠉⠉⠈⠉⠛⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠘⠛⠀⠟⠀⡴⠀⢠⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠈⠁⣄⠀⣀⡀⣠⡄⠘⠦⠀⠿⠀⠸⠀⠀⣶⠀⢠⣤⠀⠀⡉⠁⠰⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢶⠆⠈⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠃⠀⡶⠀⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⠀⠦⠀⠆⢠⠄⠠⠀⢤⠀⠆⠀⡄⠠⡄⠠⠄⠀⠀⠀⠀⠘⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇
⠀⠀⠀⠀⠀⠄⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⡇
⠀⠀⠀⠀⠀⠈⢴⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣆⡇
⠀⠀⠀⠀⠀⠀⠤⠤⠬⠤⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠤⠤⠤⠬⠤

⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⡇⢂⣀⣀⣘⣃⣃⠸⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⢻⣿⢉⣿⣏⣽⣿⣉⣿⡏⢛⣿⢟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢿⣧⣨⣿⣶⣿⣿⡿⢿⣿⢿⣿⠟⣿⡿⣿⣿⢿⣿⣾⣿⣵⣿⣃⣹⡿⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡿⢿⣏⢹⣿⣀⣿⣧⣼⣿⣶⣿⣿⣿⣷⣿⣿⣤⣿⣧⣈⣿⡿⢻⣿⢿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣉⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⢩⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣼⣿⣾⣷⣿⣷⣿⣦⣿⣾⣿⣼⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
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