Africa Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⣤⣴⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⡀⠀⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿⢋⡀⠹⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣆⠈⠻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⢀⡆⠸⣷⣦⣤⣴⡂⠀⣼⣿⣦⣄⣠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠈⠙⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣿⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠙⠋⢀⡀⠀⠀⠀⢈⣿⣿⣿⠀⠀⠀⠀⠀⢀⣿⣧⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣠⣴⣺⣷⣞⣉⣉⢷⣶⣶⣿⣿⣿⡟⣠⠘⣿⣿⡆⠘⣽⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠿⠛⢻⣿⣿⣿⣞⢿⡿⣹⣿⣿⣿⣿⣿⣿⢱⣿⡇⣿⣿⡇⡄⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⣹⣿⣧⢰⣿⣿⣿⣿⣿⣿⣟⣾⣿⣇⣿⣿⣇⣷⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣧⣼⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣸⣿⠁⣰⣆⠀⣶⣤⣤⡀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣴⣿⣏⣰⣿⣿⣆⣹⣿⣿⣃⣢⣨⣿⡇⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢠⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⣿⢽⣿⣿⣿⣽⣿⣿⣿⡋⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⣿⣿⣿⣿⣷⣾⣿⠇⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠀⢈⠏⠉⠀⠤⠀⠰⠂⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠨⣄⣀⣃⠈⠀⠘⢤⡆⢈⠉⢩⠉⠑⢒⠂⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⠀⢸⡇⡌⠀⣀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⡟⢀⠜⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢻⣧⠏⠀⠀⠀⣀⣠⠤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠⣄⣈⠰⢺⠿⡏⠭⠤⠀⠀⢸⡌⠀⠀⠋⠉⡀⠁⠠⡁⠀⢀⠃⡄⠔⢀⢀⠁⠂
⠀⠀⠐⠙⠀⠀⠀⠂⠈⡀⠹⡀⠀⠁⠔⠀⠀⠀⠀⠀⢸⢇⠀⠀⠀⠀⠑⢄⡀⠘⣄⠀⣶⠁⠀⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⢠⡧⠀⡔⠀⠀⠀⠀⠀⠀⣸⣶⠀⠀⠀⠀⠀⠀⠙⢦⣨⡤⠃⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⠊⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠠⠊⠀⠀⠀⠀⣤⣠⡄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠓⠒⠀⠀⠐⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠃
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠦⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣭⣯⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣤⢒⣬⡴⢬⢿⡇⠀⠀⠀⠀⠐⢄⣸⣶⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠞⢿⡛⠋⠀⢻
⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠐⠀⠀⠀⠀⠀⣽⢸⣟⣷⠋⡹⠿⠛⢿⠀⠀⠀⣿⢠⠝⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⠾⠟⠋⠹⡉⠀⢠⡠⠀⠄⡀⣶⣾⣇
⠀⠀⠀⠀⠀⡜⠈⣿⡄⠀⠀⢻⡄⠀⠀⢠⠹⣻⠏⠁⢾⠧⠀⢴⠸⡆⠀⠀⣧⠉⠷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠦⢁⢀⠀⠀⡄⠀⠈⠀⠀⠀⡁⢀⣠⡴⠛⠻⠋
⠀⠀⠀⣀⠜⠀⢀⣿⣿⣦⡀⠛⢿⣦⣤⡟⠛⢹⣆⠠⠸⡿⢧⠿⣰⠃⠀⢸⣿⣽⡆⠀⠀⠀⠀⠀⠀⠀⠄⣄⠀⠀⡁⠀⠀⠘⠀⠐⢻⣀⠀⣀⣾⠁
⢀⣶⡞⠁⠀⢠⡼⡿⠛⢿⡄⠀⠀⠙⠻⣧⡌⢳⡘⢇⠀⠑⠀⣠⠋⠀⠀⣸⡞⣿⡅⠀⠀⠀⠀⣀⠀⣀⣤⠃⠳⣀⠀⠀⣼⠴⠾⣦⡙⠻⣷⠘⠃
⠀⠿⠃⠀⠀⢸⠀⠀⠀⠈⣷⠠⠀⠀⠀⠈⢷⠀⣿⠸⣧⡴⠋⠁⠀⢀⣼⣿⡚⢲⠁⠀⠀⠰⠿⡿⣿⣿⣇⠀⠄⠈⢢⣀⡟⠀⠀⠀⣬⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣴⣿⠟⢸⠏⠁⠀⠀⠀⢠⣾⣯⡽⣛⡏⠀⠀⠀⠀⠀⠀⣻⡟⠁⠀⠀⠀⠀⣿⡇⠀⠀⢼⡏⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠻⠿⠟⠀⠀⠤⠠⠄⣠⠔⣫⠶⣬⢳⡟⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠈⠻⣦⣤⢼⣧⡄⠀⠀⠀⠀⠛⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⡿⣾⣻⣟⣴⠆⠀⠀⢀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⠶⡝⡆⠇⣟⢰⡔⣿⣿⠀⠀⠀⠙⣶⠃⢶⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣊⠛⣖⢆⠹⠚⠳⢤⡻⠾⡋⢝⡄⠀⣠⣼⢿⡁⢷⢋⢻⢿⣳⣀⣀⡀⠀⠀⢀⣀⣤⣄⡤⠤⠄⠀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⣐⢁⡞⢈⢰⣼⡇⡾⡒⢈⢫⣼⠁⢰⣿⣤⠗⠂⢾⣿⣻⣧⣿⣥⡿⡏⣏⡝⢪⡵⣫⡵⢖⣛⣫⣙⣛⠳⣄⡀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠡⣩⡘⠄⣚⠡⠕⠺⢡⡤⣉⢵⣗⡀⠀⠉⠁⠀⠀⠘⣜⣏⡽⢫⣷⢵⡹⣿⣿⢸⢰⢏⡖⡩⠤⠤⠬⠍⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⣀⠟⣄⢨⢄⡞⠁⠈⠀⢠⠋⠢⠿⣫⢅⣀⠀⠀⠀⠀⠀⠈⠻⠆⢋⣬⡲⣽⡟⢧⢸⠸⠈⢻⣺⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⡀⠃⠁⠚⠄⠊⠁⠀⠀⠀⠯⠒⠀⠀⠀⠈⠢⢁⠀⠀⠀⠀⠀⠀⠀⠊⠁⣱⡷⢇⠸⠿⠃⠐⠁⠉⠛⣍⢡⠄
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠁⠀⠘⢂⠜⠀⠀⡗⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢔⠇⠑⠄⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠃⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠗⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢙⡆⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡄⠀⢀⠔⠁⠈⢨⡥⠀⢀⠀⠀⡀⠲⣴⣾⣿⣿⣤⠀⠀⠀⠀⡀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢉⢀⣠⠏⢀⣣⣴⣷⣶⣶⣿⢷⣠⣼⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⡗⠀⠀⠀⢈⢂
⠀⠀⠀⠀⣷⣤⣴⠆⡁⠈⠙⠟⠃⠀⠂⠀⠀⠀⠀⢀⠀⢤⠁⠀⠈⢱⡎⣩⡏⣤⠀⠉⢿⢿⣿⣿⣽⣾⣿⣿⡿⠁⠀⠀⠀⢠⣼⣷⠀⠀⣤⣄⣷⣅⡀
⢀⡀⠀⠀⠈⣿⡟⠀⠀⠀⠁⠀⠀⢼⡀⡄⠀⠀⠈⠠⠜⢸⣷⠀⢱⡄⢻⡁⠀⠊⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⣀⣾⣿⣿⠀⣼⣿⣿⡇⠈⢷
⢸⡇⠀⡄⠰⣿⠃⠀⠘⣧⣤⡀⠀⡘⣷⣿⡄⠀⠀⢀⣤⣿⣿⣀⣴⡇⠀⠉⠓⠉⠀⠀⠀⡼⠛⠃⣉⣿⣿⣶⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣇⡀
⠀⢻⣄⠙⣶⡃⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣻⡿⢷⡀⠀⠀⠀⠀⢀⡴⠁⣠⣴⠟⠉⠹⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣧⠀⠀⠙⠻⠛⠂
⠀⠀⠙⣿⡏⠀⣀⠤⠚⠻⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⠠⣧⣴⡿⠋⠁⠀⠀⠀⢿⢿⣿⠀⠀⠀⠀⢸⣿⣿⣯⠹⣿⠓⠁
⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⢻⣿⡟⠀⠀⠀⠀⠀⢠⣿⣿⡷⠀⠀⠀⠀⠀⠘⠀⠻⣆⠀⠀⢀⣼⣿⡟⠁⠀⢹⣇⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⠀⢾⣿⠇⠾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⢞⡀⣘⠆⢠⣾⣿⣿⠃⠀⠰⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠄⠀⠒⠙⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⠄⠀⠀⠀⠒⠒⠒⠄⢀⠀⠠⡄⠀⠀⡠⠶⠤⢀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⠀⠀⠒⠀⠉⠉⠁⠰⠆⢉⠂⠌⡀⠙⠢⡈⠢⢌⠑⠀⢀⠀⠩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢿⣿⠗⠒⠶⣶⣶⣿⠿⠛⣷⣶⣤⡀⠁⠢⢀⠁⠒⠀⠄⠀⠀⠈⡀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠈⠀⠀⠙⠋⠀⠀⠻⢟⡋⠹⣷⣤⡀⠀⠘⠂⠀⠒⠂⣁⣠⢤⣶⣾⡄
⠀⠀⠀⠀⠀⠀⠀⡁⢉⡷⢒⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣟⠛⠳⣶⣤⡤⠶⢾⡿⠧⠐⠊⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣁⡙⡄⢖⠒⣝⠢⣐⢂⠠⠋⠱⡍⢀⠒⣠⡄⠀⠈⠁⠘⠛⠒⠋⠁⠀⢀⣀⣀⡀⠤⣴⣄
⠀⠀⠀⠀⠀⠀⠸⠒⠨⠅⣈⢎⣁⠜⢊⣉⠣⣀⠫⠕⢈⠶⠎⠠⡌⢄⢒⡠⠜⡋⠣⢄⠥⠉⡠⢫⠬⢔⡒⢩⠭⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠉⠑⠀⠤⠤⠤⠬⠶⠖⠀⠈⠀⠳⠖⠈⡓⢄⠙⠬⢃⡠⠕⠪⢄⡓⠴⢊⠔⢡⣒⣢⡌⠀⢲⠶⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣒⡐⢠⣤⣀⣀⣀⠀⢤⣴⡆⠻⠿⠋⢁⣠⣤⡁⠙⠒⠁⠤⠤⠁⠀⠘⠒⠓⠐⠀⠂⠒⠒⠉⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠾⠿⠟⠛⠁⠀⠈⠉⠃⠀⠄⠺⠿⠟⠁⢠⣴⣦⣄⣀⣀⡀⢤⡤⠤⢤⣤⣤⢤⣤⣤⠔⠂⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠛⠀⠈⣉⠁⠴⠆⠻⠟⠋⠀⣤⣭⠁⠚⠿⢡⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠑⠒⠛⠿⠿⢇⠀⣼⣇⣼⣿⣿⣋⣀⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡀⠨⡉⠀⠒⠀⠤⠠⠀⢀⣀⣀⠀⠀⠀⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠥⠀⣉⠀⠓⠈⠆⠰⠄⠰⠄⢠⡌⢩⡁⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⠀⠀⠐⠒⠒⠠⠄⢈⡃⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡈⢻⣿⡟⠉⠷⠒⣶⣶⡶⠛⣻⠖⢢⣤⣀⠀⠀⡀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡘⠄⠉⡠⢊⣀⢂⠈⠋⢀⠔⠀⡄⡘⠋⠁⠀⠠⡁⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠊⠂⠔⠁⠈⠀⠑⠊⠤⠊⠃⠘⠀⠀⠀⠀⠐⣝⢄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠩⠭⠁⠀⠠⠤⠉⢑⡢⠆⠀⠀⠀⠀⢀⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠶⢀⡠⠆⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣶⣭⣷⡀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣾⠉⣦⡞⢣⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠰⢏⠀⠀⠀⠀⠀⠀⠀⠈⣹⣾⣿⣿⣿⣿⣿⣿⣿⢿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⢻⣿⣿⣇⠀⠈⠳⢤⡀⠀⠀⠀⠘⢿⣿⣿⣷⠀⠀⠀⢰⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠋⠁⠀⠀⠀⠸⠟⣿⠉⠀⠀⠀⠀⠉⠳⣤⡄⠀⠀⣿⣿⡿⠆⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⣤⣤⣤⣄⣿⢀⣤⠄⠰⠞⠛⠋⠉⠁⠀⣰⡛⢻⡇⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣉⣿⣧⡀⠀⠀⢀⣠⣴⣶⠿⣿⣿⣄⣤⣿⣿⣶⢀⣀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⢡⡞⠛⠉⠉⠀⠀⠀⢿⣿⣿⣿⠿⠀⠀⠀⠙⠛⠲⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡏⠀⠀⠙⠳⣦⣄⡀⠀⠀⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠉⠛⣶⣤⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢧⠀⠀⠀⠀⠀⠀⠰⠋⣸⠿⠿⢿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠰⣿⣟⣿⡿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣮⣻⣾⣕⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣷⡿⣿⠀⠀⠀⠀⠀⠀⣴⣷⣶⣶⣾⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣽⣯⣿⣿⣿⣟⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⡿⡟⡯⡍⣯⡝⣻⡽⢟⡿⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣭⣧⣭⣷⣿⣷⣛⣧⣓⣧⠀⠀⠀⠠⣯⣷⣭⣾⣭⣮⣽⣮⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡟⣿⡟⣷⣼⣿⣿⣿⢿⣿⣿⡅⠀⠀⠐⢷⣫⢾⡻⣕⣵⢟⢫⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡿⣿⣭⣷⣿⣿⣿⣿⣚⣿⣾⠃⠀⠀⢈⣿⣿⣿⣛⣛⣛⣛⣛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣷⣿⣶⣷⣿⣿⡿⣯⣽⣯⣿⡇⠀⠀⣬⣵⣬⣬⣼⣭⣭⣥⣅⣠⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣴⣽⣿⣿⣿⣛⣿⣟⣿⣟⣿⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠈⢹⡏⠉⠉⠁⣤⣾⣧⣽⣭⣭⣭⣿⣽⣭⣽⣭⣭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡿⠉⠉⠙⠛⠛⠛⠛⠛⠛⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠘⠛⠛⠓⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠒⠚⠉⠉⠉⠉⠉⠙⠒⠲⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡖⢶⡶⠙⣿⠏⢻⡿⠋⣿⡶⢢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⠺⢿⡁⠹⣇⠒⠃⠀⠉⡱⢎⠁⠐⠛⢂⣼⢋⢸⣿⠶⣤⡀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⣾⠀⠀⠀⠑⢦⡀⢀⣴⠛⢻⣆⠈⠁⢠⠉⠑⢦⣀⠊⣀⠀⢑⣀⠔⠁⠠⠀⠉⣣⣴⠟⢻⣦⠀⣴⠞⠀⠀⠀⢻
⠀⠀⠀⢻⠀⠀⠀⠀⠈⢳⡛⢛⡖⠁⠈⢁⣰⠶⠖⠚⠈⠉⢠⠭⢏⠈⠉⠘⢓⠦⢶⡊⠉⠀⠑⡾⢋⡿⠁⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡋⡍⢠⠟⠉⠀⡄⠁⠐⠄⡀⠌⠀⠈⠖⢀⠐⠀⠃⠀⠉⠺⣆⠈⢁⣾⠁⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠘⣏⡿⠋⠈⠐⢀⣣⣤⣴⠾⠟⠗⠛⠓⠻⠾⢦⣤⣠⡂⡠⠒⡈⠳⣻⡇⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⢻⡇⠠⣢⡴⠋⢩⠁⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⡍⠙⢦⣄⠀⢠⣿⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠸⣧⣰⡟⠀⠀⠠⡗⠀⠀⠠⡀⠀⢀⠠⠐⡀⢲⠄⠀⠀⠙⣧⣸⡏⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣾⣿⣿⠁⠀⣠⠶⢶⣦⣍⣐⠠⢉⠤⠓⣨⣶⡶⠶⢦⡀⠀⣽⣿⣧⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡄⠈⢀⣠⣤⣄⡉⠛⠉⠐⠁⠙⠋⣁⣤⣤⣀⠁⠀⣿⣟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢼⡇⠈⠻⣭⣽⣟⣻⠄⠀⠀⠀⠀⢼⣻⣿⣍⠿⠋⠀⣿⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢻⡆⠀⠀⠀⠀⠘⠞⠳⠦⠖⠳⠆⠀⠀⠀⠀⠀⣿⠛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⣧⠀⠀⠀⢀⣀⡠⠖⠀⠒⢦⡀⡀⠀⠀⠀⢰⠇⠀⠀⠡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠠⣀⡀⠿⣦⠀⠀⠈⠫⡈⠉⠁⠉⢉⡘⠉⠀⠀⣠⡿⢀⣀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠈⢳⣄⠀⠀⠉⠛⠒⠚⠋⠀⠀⢀⣾⠋⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⢤⣀⡀⠀⠀⣀⣠⠖⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠉⠙⠛⠋⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠴⠖⢻⠛⠉⠀⠁⠒⠀⢠⡀⠀⠖⢠⠀⡄⠄⠀⠀⠀⠈⠙⢻⠟⡶⢦⣤⣤⣀⣀⡀
⠀⠰⣠⡴⠞⣛⣛⡉⠽⡥⡶⢢⡈⢢⡑⢤⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠜⢠⠞⢮⡱⠯⡽⣛⣛⣲⣦⣄
⡴⣿⣋⣃⣭⠭⠤⢶⠂⡜⢾⡓⣵⢦⠐⠦⣙⠲⠤⣄⣀⣀⠀⠀⠀⠀⣀⣀⣤⠴⠞⣉⠔⢃⡼⣣⡛⣔⢣⠘⡖⠲⠦⠭⢭⣉⡻
⢹⣎⠥⠀⠤⠀⢠⡏⡰⢀⠃⣹⢥⡾⢸⡲⢌⣉⠒⠀⠬⠍⢉⠉⢉⡉⠩⠥⠐⠒⣉⡡⢞⣅⢶⡴⢃⠈⡄⢣⠹⡄⠀⠤⠤⠄⣰
⠀⢻⡄⢠⡟⢀⡟⢠⠃⢈⡩⠤⢚⡉⢯⢴⡏⣺⡭⣽⠒⣲⣆⣴⣦⣔⣒⢲⡯⢭⡖⢻⣦⢹⢏⣑⠒⠤⣁⡀⢆⢹⡄⠀⠆⣰⠏
⠀⠠⠙⣦⠁⣼⠁⠞⣋⡥⠖⠊⠉⡇⢰⠈⣈⠝⣺⠟⣬⠧⣼⠧⠿⣤⢿⡬⠿⡞⠻⣅⡈⢐⢨⡏⠉⠒⠦⣌⣙⠂⠻⢂⡼⠃
⠀⠰⠀⠈⠳⣄⠲⠋⠁⠀⢤⠀⠀⣇⢻⠘⣡⠞⡡⠒⡌⢦⠘⣆⣠⠃⡜⡰⠓⢌⠳⣄⠋⢈⢸⡇⠀⣤⣀⠀⠉⢛⡴⠋
⠀⢠⠀⠀⠀⠈⠛⢦⣌⠀⢀⠀⠀⣿⢰⠞⣡⠞⠁⠀⠘⢆⢳⡈⢁⠞⡼⠁⠀⠀⠑⢬⡳⠮⢸⠀⠀⡇⢈⣡⠴⠋
⠀⢰⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣛⠠⠞⠁⣠⠊⢡⠀⠈⢦⠹⢎⡜⠀⠀⠌⢣⣄⠀⠙⢆⣈⡡⠴⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣤⣤⣤⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡟⠉⠋⢹⠀⠀⢸⣾⣧⣀⠀⢀⣤⣄⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣁⣇⣀⣀⣸⣀⣀⣸⣄⣈⣙⣿⣾⣏⣹⣟⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡉⠣⣄⠙⢦⣈⠙⢦⡈⠳⣄⡉⠳⣄⠙⢦⣈⠑⢦⣈⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⠝⠉⡩⠛⠉⡹⠋⢉⡽⠋⢉⠝⠉⣩⠋⠉⡩⠋⢉⡽⠋⣩⠋⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⠛⠛⠛⠛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡛⡛⠛⠋⠛⢛⠙⠛⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠬⠤⠤⠤⠤⠥⠤⠤⠤⠤⠤⠬⠤⠤⠤⠤⠥⠥⠤⠤⠤⠬⠤⠤⠤⠬⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣖⢶⠒⠒⢲⡖⠒⠒⣶⡖⠒⠒⣶⠒⠒⢲⡖⠒⠒⣶⠒⠒⣶⡒⠒⢲⡖⠺⢿⣤⣴⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣌⣧⢠⠇⠹⡄⣰⠃⠹⡄⣰⠁⢧⠀⡞⠘⡄⢸⠙⣆⢠⠃⢣⢀⡞⠸⡄⣰⠹⡄⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠶⠶⠿⠛⠛⢻⣿⣧⣄⣈⣿⣁⣀⣹⣇⣀⣸⣯⣤⣬⣿⣤⣤⣽⣧⣤⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣙⣉⣹⣉⣏⣹⣉⣏⣹⣉⣏⣙⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⣍⣽⣀⣇⣹⣀⣏⣸⣀⣇⣭⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⠿⠿⠿⠿⠿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠤⠤⠤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⡷⢏⡺⣿⡿⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣯⣾⣎⣿⣿⣟⣽⡱⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣟⡿⣛⣿⣿⣿⣿⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⢗⣛⣿⡿⢪⡝⣯⣽⡿⣻⣿⣛⣧⡟⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣿⣮⣿⣿⣿⣿⣷⡿⣻⢫⣭⣿⣟⣵⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣽⣿⣿⣿⡿⢏⡿⣫⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡿⣿⣿⡿⣫⡕⣶⡿⠷⣹⣋⣽⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡣⣭⣽⣿⣿⣿⣻⣿⢟⣭⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⣽⣜⣿⣿⣿⢟⠻⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠞⣟⡻⠟⠋⠀⡄⠈⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡐⠁⠂⠄⣰⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣾⣿⣿⠏⠉⠛⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⢝⢿⣿⣿⣿⣿⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡇⠀⠘⢌⠳⣝⡾⣽⣿⣯⢿⡣⢠⡇⠀⢳⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡇⠀⠀⠈⠳⣦⣍⡓⠒⢚⣋⣴⠟⠀⠀⢐⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⢀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠄⠀⣀⠀⠀⠀⠀⠀⡯⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⡞⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⠀⢀⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠙⣞⠙⠿⣧⣀⡀⠀⠀⠀⠀⢀⡴⠖⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣻⣟⣗⢀⠀⠀⠀⠀⠀⢰⢤⠀⠀⠀⠀⠀⠀⢠⠆⣸⠀⠀⠀⠀⢀⠀⢤⠜⠿⣾⣟⣻⡁⠁⠀⠀⠀⠹⢫⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣠⣿⣿⣿⣿⣾⣟⣷⡀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣀⠜⠚⠃⠀⠀⠠⠼⢿⡧⡄⠀⠀⠀⣸⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠉⢾⣿⣿⣿⣿⣿⠣⡟⢻⡟⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⢺⠁⠙⢃⠀⠀⠸⠇⢠⢸⡗⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⢏⣈⣩⣀⣞⢀⣷⡇⣿⣿⠃⠠⠀⠀⠀⠀⠰⡆⠀⢸⡇⠀⠙⡀⠀⠀⣀⠀⡁⠀⣸⡀⠷⠀⣴⠛⣢⡰
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠀⣰⣻⣴⢯⡮⠬⡌⡇⣸⣏⣿⢷⡿⠐⣴⠂⠀⠀⠀⢀⡦⠈⠄⢻⡟⠠⣗⠀⠀⠙⡌⠁⡔⣸⣥⣷⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣠⣤⣭⣾⣟⣩⠭⢲⣵⣷⡿⢾⣫⠿⠁⣰⡯⣧⠀⠀⠀⡾⣷⡄⠐⣌⣡⠀⠉⣇⠀⠉⠑⠀⠻⡍⢝⣇⡀⠀⢈⣤⡴⠶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⢹⣿⣿⣿⣿⡀⣽⣿⠟⢛⠀⣸⠻⠃⠰⣿⠃⠸⠄⠀⢀⠏⣽⠁⠀⡻⣿⣿⣆⠈⠛⡛⠋⣡⣿⠛⢸⡟⠛⠷⡌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠏⠘⠁⠀⠸⠘⠏⠀⡀⣦⠛⠀⢠⣇⠀⢘⠀⠛⡄⠀⠃⠀⠀⠛⠿⡟⠻⢷⠿⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠃⠀⠀⣭⣤⡿⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⠀⠀⠀⠀⠠⠀⠀⠀⢃⠀⢠⠤⣤⠤⢠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⡀⠀⠀⠀⠈⠁⠒⡈⠆⠰⠒⣬⣿⣁⣀⣶⣦⣤⣶⣴⣴⡄
⠀⠀⠀⠀⡀⠀⣀⣀⣀⣀⡀⢀⡀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠁⠊⢦⠈⠉⠂⢼⠁⠆⠈⠁
⠀⠀⠀⠈⠉⠒⠛⠻⠿⠑⠲⠞⠓⠒⠂⠀⠀⠀⠈⠱⠄⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠐⠀⠄⠀⠐⠠⠀⠀⠄⠀⠀⠀⠀⢠⠀⠀⠇
⠀⠀⠀⠀⠀⠿⠻⡉⠈⠛⠀⠀⢀⠀⠠⠀⠀⠀⢸⠀⠰⣦⠀⠀⠀⢰⡄⠃⠐⣶⠀⠀⠀⠈⡆⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠈⠈⠀⢸⠀⠀⠹⡄⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠠⠁⢀⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⢠⠀⣷⠀⠀⠀⢠⠰⠆⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠀⠀⠙⠀⠀⠀⠈⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠐⢦⣤⣶⣶⣦⣤⣠⣄
⠀⠀⠀⠠⢤⣀⡀⣙⣻⣿⢿⣿⠷⣿⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠉⠛⠛⠛⣟⠦⢬⣴⢩⠿⢯⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢲⢶⡄⣤⣀⡆⠀⠚⠚⠙⠃⠀⠀⠀⡀⠀⢀⣀⡀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⣡⠜⠀⠀⠀⠀⠀⢀⣀⣄⣤⣀⠈⠙⠋⠻⡛⢛⡛⠳⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣸⠁⠀⠀⠀⠀⠀⠀⠈⠉⠈⣍⠉⠁⠀⠀⠀⠘⣏⡀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠿⠶⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⣄⠀⣐⠀⠀⢀⡤⠀⣀⣂⡀⢶⣴⣢⠥⣠⡀⠒⠒⠦⢠⣤⢀⣠⡔⠂⣀⠐⠀⢀⠕
⠀⠀⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣭⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣴⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠘⠿⠿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⢘⣿⡇⠀⠀⠀⠀⠀⠀⢦⡠⠠⠐⢉⠀⠀⠀⠀⠀⠀⣈⠹⣭⠡
⠀⠀⠀⠀⠀⢹⣿⣿⢁⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠃⢰⠂⠀⠋⠈⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠘⠿⠃⠈⠀⠀⠀⠀⠀⠀⠠⠋⠇⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠛⠋

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠐⠟⠛⠛⠻⠷⠰⣿⠆⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣣⡞⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣆
⠀⠀⠺⠿⠟⠷⠶⣿⠿⠶⠦⠀⠀⠀⠀⢿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⠀⠀⠀⠀⠀⠻⣇⡎⣉⣸⠚
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠙⠃⠀⠀⠀⠀⠀⠀⠀⡟⠛⡇⠀⠀⠀⠀⠀⠀⠑⣆⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡘⠳⣤⡙⠆⠀⠀⠀⠀⠀⠀⠀⣿⣹⣶⡄⠀⠀⠀⠀⢠⢾⡷⡄
⠀⠀⠀⠀⠈⣨⡍⠁⠀⠀⠀⠀⠀⠀⢰⣿⢶⠈⠛⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠛⢓
⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⣬⡅⠀⠀⠀⠀⠀⠂⢾⡍⠀⠈⠿⠀⠀⠀⠀⠀⠀⢰⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⢀⠀⣽⠇⢸⠇⠀⠀⠀⠀⣾⣿⣯⣿⣿⡆⠀⠀⠀⠀⠀⠐⢼⠆⠀⠀⠀⠀⠀⠀⠀⠤⠀⡄⠤⡆
⠀⠀⢠⣿⠿⠿⠿⡿⠋⠀⠀⠀⠀⠀⠿⠇⠈⠨⠿⠇⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⡀⣀⠂⣶⠚⢓⡚⡃
⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠀⠀⠀⠀⠉⠓⠉⠉⠉⠑⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠘⠿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠈⠻⢿⣿⣿⣿⡿⣿⣿⣿⡟⠛⠉⢿⣿⣿⣿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠠⣿⣆⠀⠀⠀⠀⠀⠀⣿⡿⣿⠁⠀⠀⠀⠉⠛⠋⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⢀⣤⣶⣤⣶⣾⣿⣿⣿⣿⣿⣄⣴⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠉⠛⠻⠿⢛⠋⢻⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠱⣄⠛⣿⢿⠿⠇⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐⠀⡒⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢀⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠈⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄⣠⣶⣾⣿⣷⣤⣤⣤⣄⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠛⡿⠿⢛⡉⠻⢿⣿⣿⣿⠗⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠁⠙⠻⡟⢡⠎⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⡊⠁⣡⣾⣌⠣⠾⣓⣾⢻⣶⣦⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡗⣬⠑⢯⡡⠇⢟⡢⣓⡈⠑⣾⡝⣦⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⠛⢘⡞⠷⣮⣳⡅⡟⢿⡿⠱⡶⡿⡋⠀⣜⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡮⡿⣏⠸⠖⣴⠙⣖⢓⡜⣻⣶⣦⠞⠷⣻⢯⡦⠏⢢⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣝⣽⢹⣯⠥⢻⡀⢶⣦⣿⣏⣮⣬⠢⣝⣌⡼⡽⠽⢆⠲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡿⠫⣴⣻⠏⡽⠛⢆⣟⠾⢔⢙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡣⣄⢥⢉⠆⢫⣼⣽⢺⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⢲⣸⢄⣂⡭⣿⣦⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠛⢪⣥⣴⡺⣯⣟⢮⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⢒⣾⡐⢮⣺⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⠘⡃⣾⠱⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠦⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⢴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣉⣯⣉⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⠸⣿⣿⡿⣼⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⠛⠻⠟⠷⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⣀⣀⣀⣁⣀⡀⠛⠿⠿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠛⠛⠋⠀⠀⢠⠀⠹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⢀⣤⣴⣾⣷⣶⣶⣶⡆⠀⠀⠙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣤⣤⣈⣡⣤⣤⠀⠈⢻⠀⠀⠀⠀⠘⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠛⠛⢋⣥⣶⣿⠇⠀⠀⠀⠀⠀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣝⠁⠀⠚⠛⠛⣋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⣀⣠⣤⣶⣾⠿⠛⠛⣠⣴⣴⣾⡿⠋⣠⣼⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣽⣿⠿⠟⠟⠋⠉⠀⠀⠀⠀⠀⣿⡟⠛⠉⣠⣾⠟⠛⠉⠀⣡⣦⣀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠠⠴⠾⠿⢿⣿⡿⠟⠛⢉⣠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⢀⣤⣶⠆⠀⣿⣿⡿⠗⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⡟⠁⠀⣠⣿⡿⠀⠀⠀⠠⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣿⣿⠋⠀⢀⣴⣿⠿⠃⠀⣦⠀⠀⠀⣦⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠻⡏⠀⠀⠸⠋⠀⠀⢀⣴⡿⠃⣀⣼⣿⣧⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣯⠀⠟⢀⣿⡀⠀⠀⠀⣀⣴⣿⠃⢠⣾⡿⠟⠋⠁⠀⠀⠀⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⡿⠟⠁⠀⣰⣿⣿⡿⠃⢀⣼⡿⠁⠀⠀⠀⠀⣷⡀⠀⠈⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠶⠀⠘⣿⣿⠋⠀⠀⠘⠉⢀⣴⣿⠀⣠⣾⣿⣧⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⠀⠀⢹⡇⠀⠀⣠⣶⣿⠿⠋⣡⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⣿⣿⠋⡉⢻⣿⣷⠀⠀⠰⢀⣾⣿⠟⠁⠠⡿⠛⢉⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⣿⣷⣀⠛⣸⣿⠟⠠⠀⠀⢻⣿⠃⠀⠀⠀⣠⣾⠟⠁⢀⣴⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣄⠀⠀⠀⠈⠛⠿⠿⠿⠋⠐⠧⠀⣀⠈⢁⣀⣤⣶⣿⡿⠋⠀⢀⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⢹⣷⠀⠀⠶⠀⠀⠀⠀⠀⣤⠰⠚⠁⢴⣻⣿⣿⠿⠋⠀⠀⣠⣿⠟⠁⠀⠀⠀⣠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⠈⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⠻⠁⠀⠀⠀⠘⢋⡁⠀⢀⣠⣴⣾⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⠟⠁⠀⠀⡀⠀⠀⠀⠀⠃⠠⠆⠁⠰⣤⠉⠀⠀⠀⠀⣀⣴⡟⠁⠀⠈⠛⠙⠏⣱⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠛⠀⠀⠘⠇⡌⠀⠀⠀⠤⠾⠛⣿⠟⣤⣄⡀⠀⠀⠈⢠⠟⡉⠳⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⣠⣴⣶⣤⡀⠀⠀⣄⠀⠀⠒⠀⠛⠀⣶⡧⠖⢀⠤⠀⢀⠐⢀⠏⠈⠉⠻⠛⢷⣶⣯⡜⢁⠀⠘⢧⡍⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠛⠻⣿⠟⠠⠀⡀⠓⠀⠀⠶⠀⠈⠀⠈⠉⠲⠢⣄⣤⠃⡰⠀⠀⠀⠀⠀⠘⡿⠷⣉⢤⣾⡆⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠐⠓⢀⠏⠀⠀⠀⠱⠀⠀⠀⠀⠐⢻⡀⠀⠀⠀⢠⣯⡙⠂⢁⠆⠀⠀⠀⠀⢹⡆⠀⢋⣸⠷⠋⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⣶⠟⠀⠘⠀⢀⠐⠀⠘⠁⠀⠇⠀⠃⠀⠀⠀⠷⡌⠁⠀⠈⡐⢃⠀⠀⣸⣇⠀⠠⠘⢉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⡐⠀⠀⢀⡘⢦⠁⠀⠀⠀⣡⣬⣄⠸⠋⠁⠀⡤⠀⠀⠑⣠⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣄⠀⠂⠠⠈⠀⠀⠠⠄⠀⠈⡄⠁⠀⢠⠌⠓⠂⡴⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠉⠻⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠁⠐⠻⠀⠀⠀⠀⠀⠀⡆⠀⠀⠠⠀⠀⠀⠆⠀⠀⠘⠀⠀⠀⠀⠡⠖⠀⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠀⠀⠚⠓⠀⠀⠀⠀⠀⠀⠶⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⡿⠿⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⣠⠄⢠⠞⡴⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣡⠞⣫⡴⠃⠜⢁⡴⢋⡴⠀⠀⡀⠀⣴⠀⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⣱⠟⢉⡶⠀⣠⡞⣱⢏⡴⢀⡾⢛⣼⢡⡶⢋⣠⠞⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⣡⣾⠟⣡⡴⢛⣵⡾⢋⣼⣿⠏⠀⠚⢡⣾⠃⢉⡴⠋⠁⠴⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⣉⣴⡿⠋⣠⡾⠋⠰⢋⣼⡏⡴⠋⣰⣿⠏⣰⡟⢀⣤⡾⠀⢀⣡⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢴⠚⠉⠁⣰⠟⢋⡴⣠⣴⡿⢋⣼⠃⢰⣿⠟⣼⡟⢀⣾⢻⡇⣰⠏⣤⠶⣛⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡆⠈⠻⡏⠻⠋⢠⣶⣯⣾⣤⣼⣧⡼⠿⣤⠾⠇⠾⢡⣟⣼⣿⠶⠧⠞⢔⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⢁⠄⡀⢉⣀⢠⡙⠻⣿⠛⠟⠻⣯⠁⠀⠉⠀⠀⠀⠘⠀⠘⣛⡀⠠⣄⠁⢢⠖⣠⠆⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⠿⠃⠛⣶⡿⠓⢤⡀⠘⠲⢆⡼⣦⡄⠐⠳⢀⡠⠹⠟⣠⡀⠹⠶⣈⣡⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡩⠀⠀⠴⢻⣗⣀⠀⠀⢻⠀⠀⠀⣽⠯⠵⠞⢋⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⣴⠆⢀⠀⢸⣿⢻⡇⠀⠀⢐⡿⠃⣿⠶⢒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⠎⢀⠾⠿⠈⠿⢲⣆⠈⠀⠀⠂⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢠⣿⠿⢯⣤⣤⣌⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣤⠀⠀⠀⢸⡃⠀⠀⢹⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⠁⢀⠴⠀⢀⡟⠉⠦⠀⠀⡾⡸⠁⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⣠⡄⠈⠁⣸⡇⠤⣤⡄⠀⠀⠐⠀⠀⠀⢘⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣭⡄⢋⣄⣽⢿⡄⠀⠀⠊⡅⠀⠀⠀⠀⢚⣔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠘⠛⣡⠟⢹⠦⠐⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⡶⠿⢃⡴⢃⡠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠶⠛⠁⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠄⢴⣧⣤⣤⠄⠀⠈⢓⣒⣶⡖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠐⠀⠀⣵⣟⡈⠀⠀⠀⣀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⠄⠀⠀⠀⡘⠀⣰⣿⠟⠀⠀⠀⠀⡅⠀⢸⣶⣖⠂⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⣹⠁⠀⠙⢻⠗⠁⠀⠀⢰⡄⣤⣿⣿⠂⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⢹⡟⠒⠀⠀⠀⠐⠉⢹⢃⠀⠀⣸⣶⢲⣶⠶⠆⠀⠀⠀⣹⣿⣃⡀⠀⠀⠀⢰⣾⠃
⠀⠀⠀⠀⠀⢀⣬⠇⠀⠀⠀⠀⠀⢀⣼⠴⠒⠟⠿⢿⡞⠃⠀⠀⠀⠀⣤⢿⠉⠛⠒⣶⠶⠶⠾⠛
⠀⠀⠀⠀⠀⠐⣾⡆⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠘⡦⡄⡀⠴⠂⠀⠈⢄⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠻⢧⠄⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢀⠀⠀⠀⠀⢸⣆⠀⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⣯⠈⢂⣦
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣠⣾⣿⢉⡤⡾⡿⠲⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠣⠏⢱⢳⣇⡴⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⡋⠀⠀⠀⠀⠾⠀⠁⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠛⠉⠁⠀⠙⠦⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⠝⠋⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⠵⠊⠁⠀⠀⠀⠀⠀⣠⠔⡫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡽⠚⠁⠀⠀⠀⠀⠀⢀⡤⣊⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⣤⢖⣩⢾⣥⣀⣀⣀⣀⣀⢀⡠⢖⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⢉⠔⠛⠓⣯⠔⠋⠀⠀⠀⠀⣀⣠⣠⠝⢋⠴⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠸⣄⠀⣀⡜⠀⠀⠀⠀⠀⣼⠁⠀⠈⡟⠁⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠘⠦⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢸⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⡇
⠀⠀⠀⠀⢸⡷⣌⠙⠲⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠖⠚⠉⢀⣠⠔⡇
⠀⠀⠀⠀⠀⡇⠈⢑⡦⢄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⡠⠤⡖⠊⠉⠀⢠⡇
⠀⠀⠀⠀⠀⡇⠀⡆⢡⠀⠀⠉⠉⢉⠗⠒⠒⠒⠒⠒⠒⡚⢏⠉⠉⠉⠉⠀⠀⠂⢱⠀⠀⠀⡎⡇
⠀⠀⠀⠀⠀⡇⣸⠀⠘⡆⠀⠀⢀⠎⠀⡇⠀⠀⠀⠀⡰⠁⠘⡄⠀⠀⠀⠀⢸⠀⠀⢣⠀⠀⣇⡇
⠀⠀⠀⠀⠀⣧⠇⠀⠀⢣⠀⠀⡌⠀⠀⢃⠀⠀⠀⡰⠁⠀⠀⠱⡀⠀⠀⢀⠇⠀⠀⠈⡇⢰⢹⡇
⠀⠀⠀⠀⠀⢻⢦⡀⠀⠘⡄⡸⠀⠀⠀⢸⠀⠀⢰⠁⠀⠀⠀⠀⢳⠀⠀⡘⠀⠀⠀⠀⠸⣸⢸⠁
⠀⠀⠀⠀⠀⠘⢦⡙⠢⣄⣁⠁⠀⠀⠀⠀⠇⢠⠃⠀⠀⠀⠀⠀⠈⡆⢠⠃⠀⠀⠀⠀⣀⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣉⠑⠲⠤⢄⣀⣀⡈⠀⠀⠀⠀⠀⠀⠀⢠⣌⣠⠤⠤⠒⣉⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣌⣉⣉⣉⣉⣉⣉⣉⣉⣥⠤⠴⠖⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣿⢰⣿⣿⣦⣿⣿⢰⣿⣿⣾⣿⣿⢹⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣿⣿⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿⣶⣶⡆
⠀⠀⠀⢳⣶⣿⣎⣁⡄⣤⣤⣥⣤⢿⣿⢸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣸⣿⡟⠿⠿⢿⡿⣿⠿⠿⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠻⠿⠿⢿⡿⡿⢿⣿⣿⡛⠿⠛⠋⠉⠉⠀⠐⣾⢾⠿⢶⠶
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠁⣰⣤⣿⠿⠿⣧⣴⣀⠀⡀⠀⠀⠀⡇⣸⡀⢸
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡃⢰⡶⠛⠉⠀⠀⠀⠀⠀⠀⠉⠳⣶⠀⠀⠀⣿⣛⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⡿⠋⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⣴⡿⣷⠛⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡷⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡯⣿⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠫⢀⣀⣁⢜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢜⡇⢀⡙⠆⢀⠀⠁
⠀⠀⠀⠀⣀⠴⢊⠁⠀⠀⠀⠀⠁⢸⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⡏⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠁⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⡄⠉⠛⠶⠶⠶⠶⠶⠾⠋⢁⡄⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠟
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠐⣧⡈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠈⠛⢦⣄⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣌⠙⢦⣤⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠓⢲⣶⠋⣠⣾⣅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠑⠦⣉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣡⠞⠁⠀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠑⠦⣙⠲⣄⡀⠀⠀⠀⢰⠟⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡙⠓⣆⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⠉⠛⠓⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠉⠀⣠⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⠙⠳⢦⣄⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⣤⠶⠟⢩⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡄⠀⢀⠌⠉⢟⠛⠒⠶⠶⠶⣶⠶⢶⡶⠶⠶⠶⠶⠶⢻⠟⠛⢍⠀⠀⢀⡎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢱⡴⠁⠀⠀⠈⢦⠀⠀⢀⡔⠁⠀⠀⠑⢄⠀⠀⠀⡰⠃⠀⠀⠈⠳⣄⡼⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⠊⢣⠀⠀⠀⠀⠈⢧⡠⠋⠀⠀⠀⠀⠀⠀⠱⣄⡜⠁⠀⠀⠀⠀⠀⣸⢧⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⣀⠀⢧⠀⠀⠀⣠⠞⢣⡀⠀⠀⠀⠀⠀⠀⣠⠞⢦⡀⠀⠀⠀⠀⣰⠃⠀⢙⡶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⠳⢦⣧⣴⠊⠀⠀⠀⠱⣄⠀⠀⠀⢀⡴⠁⠀⠀⠙⢆⠀⠀⣰⣁⣤⠶⠋⠀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠉⠙⠳⠶⢦⣤⣬⣦⣀⣠⣎⣀⣀⣀⣤⣤⡤⠷⠞⠛⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢤⣄⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠲⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠉

⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠊⠉⠉⠉⠉⠉⠉⠓⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⣀⣠⣶⢀⡴⣛⣥⠾⠋⠉⠉⢻⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⢻⣾⣿⣏⣡⢼⡟⠉⠳⡄⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡃⠀⠀⠘⠛⠻⢿⣿⡧⣟⠀⢰⠇⠀⠀⢀⡾⠀⠀⠀⠀⠀⢠⡶⠶⠶⢤⡴⠋⠙⠓⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⡟⠀⠀⠀⢀⡴⠛⢉⢿⣰⠋⠀⡏⠀⡴⠞⠋⣤⣤⠀⠀⠀⠀⣸⠃⣴⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡾⠀⠀⠀⠀⢾⡀⣴⠋⠦⡧⠀⠈⠧⠞⠀⠀⠀⠉⠉⠀⠀⠀⢿⣁⠐⡏⢳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⠁⠀⠀⠀⠀⠀⣿⠃⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⢉⣙⣧⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⢀⡴⠒⢦⡀⠀⠀⢠⠃⢠⡏⠀⠀⠀⠀⠀⠀⠀⠠⢶⡞⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢯⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡴⠶⢦
⡞⠀⠀⠀⢹⡄⠀⣼⠀⢸⠃⠀⠀⠀⠀⠀⠀⣠⠴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⠀⠀⣀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢠⡏
⡇⠀⠀⠀⠀⣇⠀⡇⠀⢸⣆⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠎⣁⡈⢿⠷⣮⠓⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸
⡇⠀⠀⠀⠀⢸⠀⡇⠀⠘⣿⠀⡴⠒⢦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠉⠁⠙⢦⣠⠶⠲⠖⣁⠀⢰⡛⣦⡴⢦⠀⣾⣿⠀⡇
⡇⠀⠀⠀⠀⢸⡀⡇⠀⠀⠸⣆⣧⣤⢶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣝⢇⠀⠉⢩⠇⠸⣿⣿⣿⠀⡇
⡇⠀⠀⠀⠀⠈⡇⢳⠀⠀⠀⠈⠳⣼⡆⠙⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡀⣠⠞⠀⠀⣿⢹⡏⢸⠁
⡇⠀⠀⠀⠀⠀⡇⠘⡀⢠⠔⠢⣄⠈⠻⣄⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣇⣠⠖⢦⣄⣼⠁⢸
⡇⠀⠀⠀⠀⠀⢳⠀⢻⡏⠀⠀⠈⢣⠀⣸⠇⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢹⡟⠀⡾
⡇⠀⠀⠀⠀⠀⢸⠀⠸⡇⠀⠀⠀⠈⡆⢧⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⢀⡞⣻⠀⠀⠀⠀⢸⠇⠀⡇
⢳⠀⠀⠀⠀⠀⢸⠀⠀⢷⠀⠀⠀⠀⠸⡌⢧⡀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⣡⠇⠀⠀⠀⠀⡿⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠘⡆⠀⠘⡇⠀⠀⠀⠀⠹⣆⠙⢦⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⡜⣿⠏⠀⠀⠀⠀⣸⠃⠀⢰⠃
⢸⠀⠀⠀⠀⠀⠀⢧⡀⠀⢻⡀⠀⠀⠀⠀⠙⢦⡈⢇⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡴⢋⡼⠃⠀⠀⠀⠀⠀⡟⠀⢀⡼⠀⠀⠀⠀⠀⢀
⠈⡇⠀⠀⠀⠀⠀⠀⠉⠲⣜⣇⠀⠀⠀⠀⠀⠀⠙⢾⣆⢀⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢋⡴⠋⠀⠀⠀⠀⠀⠀⣸⣣⠖⠉⠀⠀⠀⠀⠀⠀⢸
⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠤⣤⣤⣄⣀⣀⣠⣤⣤⠤⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⣉⣁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠⢈⠉⠉⢓⣶⣶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⢾⠷⠐⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡾⠁⠁⠀⠉⠢⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣈⡑⠒⠒⢺⠒⠒⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠉⠀⠀⠀⠀⢠⠀⣾⡹⠀⠤⠄⢻⢈⡆⠀⢸⣀⠠⠤⠧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣿⣧⣤⣤⣾⣼⣧⠀⡚⡂⠀⠘⠋
⠀⠀⠀⠀⠀⠀⣤⢤⡮⢤⡤⣤⣤⣄⠀⠀⠀⠀⠀⣸⠀⠹⡁⠀⠀⠒⣿⣿⠟⠈⣯⡇
⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⡟⣸⡗⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣾⣏⢙⡃⠀⠰⣿⣿⠀⢀⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⣄⣿⡇⠀⢘⠿⠫⠉⠀⠉⠹⢆⠀⠿⠿⢿⣿⣿⡟⣿⠅⠀⠙⡂
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⣿⣿⣧⡾⠉⠀⠀⠀⡀⠀⠀⠀⠈⢲⠀⠀⢉⢿⠀⢹⠀⠀⣴⣧
⠀⠀⠀⠀⠀⠀⢹⣟⠋⢺⠋⠋⠛⡿⠁⠀⠀⠐⢧⡀⠠⡂⠀⠀⠀⢃⠀⢀⣏⠀⣿⢆⡔⣾⠂⠱⡀
⠀⠀⠀⠀⣠⠴⢻⢻⢧⣸⠀⠀⠀⣇⠀⠀⠀⠐⠉⠀⠀⠃⠀⠀⠀⣸⠀⠀⠏⡸⢹⠞⢳⡟⠂⠒⠙⡄
⠀⢀⡤⠚⠣⣠⠎⢸⡸⣿⣄⠀⠰⢅⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢉⡦⠤⣶⣤⣸⣀⡼⣿⠁⠀⠀⠈⢄
⡴⠉⠀⠀⠘⠀⠉⠙⠀⠈⠾⢢⠃⠀⠙⠈⠂⢄⣀⣀⣀⣤⡄⠘⠋⠀⢱⠈⣹⠛⠼⠛⠒⠛⡆⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠒⠀⠤⠴⠒⠒⠋⢀⣀⣀⣠⡄⣾⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡍⠉⠋⠋⠉⠑⠤⢄⣀⠠⡄⠀⠰⠤⠤⢤⡤⠖⠒⠀⣀⠀⠀⠈⢻⣷⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠉⠀⢀⣤⣇⡀⠀⠀⠀⠀⠈⠰⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡴⠛⠁⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⣤⣀⣀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⣿⣄⠀⠀⠈⠸⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠺⠀⢹⠀⠙⠢⢄⠀⢂⡤⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣤⢒⡟⣉⣧⣄
⠀⠀⠀⠀⢀⡠⠚⠁⠃⠀⠀⠙⠁⠀⠀⠀⠉⠺⠇⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⡨⢠⣃⣤⠟⠫⠄⡇⠝⠦⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠑⠳⢦⣀⣀⣀⣸⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⢀⡆⢠⢠⠏⢾⢦⠪⣃⡜⠁⠖⠒⠋⠀⢠⢠⠇⠠⠃⡷⠼⢂⡛⠀⠞⠠⠆⠴⢀⠀⠄⣈⣀⠠⠤⠠⡷⠈
⢈⡀⡉⣜⠦⣼⠾⠀⠁⡀⠰⠐⠐⠬⣄⢏⠆⠶⢤⠛⠁⠀⠇⠈⡆⠀⢷⠘⠛⠛⠃⢠⡀⠲⠄⠠⠤⠄
⠀⡇⠀⡟⢀⣸⠀⡾⢶⠶⠆⠀⣇⠄⢀⠀⠀⣄⠀⡀⠂⢱⠀⢜⠀⠶⠀⠖⠰⠀⠆⣸⠃⠂⠀⠀⣀⠀⡜
⠘⣀⣘⣁⠀⢰⣀⡇⣸⢀⡄⡴⠀⠀⠒⠒⠒⠀⢰⠉⢀⣁⣀⢈⠉⠀⢰⠀⡉⠉⠈⡁⣀⠀⠃⠈⠉⢠⠅
⡄⡂⢀⢀⠀⠆⠨⠤⠅⢈⠁⠖⠛⠠⠂⠆⠰⠐⠂⠀⣉⣈⡠⠀⠀⠹⣙⡆⢀⡤⡄⠀⠠⠀⠠⠖⠒⠀⠰
⠀⡁⢨⢤⡥⠀⠆⠄⠄⢸⠄⠂⠐⠀⠒⠖⠂⠀⠀⠁⢁⠀⠀⠀⠋⠀⢈⠀⡀⢀⠄⠀⠀⠆⠂⠁⠁⠒⠂
⠐⡇⡀⣸⢀⡄⠀⠈⠓⠀⠀⡘⠀⠠⠠⠆⠄⠀⣀⢀⡘⠀⠈⡁⡀⡉⠡⠀⠀⡄⠰⠃⠀⠆⠀⠱⠀⠤⠤
⣀⡀⠀⢀⠀⠠⢤⡘⠀⠂⠈⠁⠰⠒⠀⢀⡀⠈⠀⠠⠤⠤⠀⠀⠀⠠⠶⠂⠰⠁⠀⣀⣀⠀⣀⠀⠠⠠
⠀⠄⠀⡘⠀⠠⠀⠄⠀⠐⠒⠀⠀⠈⠀⣬⠁⠀⠀⠔⠠⠠⢀⣠⡄⠠⠀⠂⢀⡀⠠⠠⠠⠀⡉⠁⠀⢀⠇
⡄⠀⠈⠁⠠⠖⠂⠀⠀⠀⠈⠀⠸⡀⠀⠀⠀⠉⠓⠀⠀⠀⡈⢀⠀⠀⣀⠀⢀⢉⠀⠀⡀⠀⠀⣬⠄⠉⠰
⢠⠀⠀⠀⠄⠰⠀⠀⠊⡁⠀⠀⠀⠉⠀⢀⡀⠁⠘⠀⠀⡄⠁⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⢀⠀⠄⠀⠰
⠀⠀⠀⠀⡤⠀⠀⠈⠀⠀⠐⠲⠀⠀⢀⠁⠄⠀⠀⠀⡤⠁⠀⠀⠀⠁⠈⠀⠀⠀⠘⠀⠁⠀⠀⡤⠄⠀⢀
⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢠⠀⠄
⠀⠀⠀⢀⡀⠀⠑⠀⠈⠄⠀⠀⠀⠠⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠⠄⠀⠀⠠⠀⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠈⠐⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠔⠄
⠀⠀⠀⠀⠈⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠀⠀⠈⠵⠿⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠴⠢⠤⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⣠⠄⠀⠀⠄⢁⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠁⡀⠈⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡁⠀⠀⠁⠀⠀⡀⠀⠀⠙⠔⢫⠂
⠀⠀⠀⠀⠀⠀⢠⠗⠀⠀⠆⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⡸⢀⠁⠀⠐⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠎⠀⢠⣀⠀⢀⡀⠄⠊⠀⠸⠀⠀⠀⠁⠀⠀⠂⠀⠀⢸⢠
⠀⠀⠀⠀⠀⠀⡈⠁⠠⡀⠀⠀⠁⠀⠀⡠⠊⠀⠘⠅⢀⡀⠚⠀⠠⠂⠀⠀⢄⠀⠀⠈⠀⠀⠀⠀⢀⡬⠃
⠀⠀⠀⠀⠈⠃⠄⠀⠀⡀⡁⠰⣀⠀⠉⠀⠀⠀⠀⢀⠁⠚⠀⠀⡀⠀⠀⠀⠀⠁⡦⢀⣀⡀⠠⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠔⠈⠈⠀⠑⠆⠀⠀⠀⠀⠸⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠤⣀⢺⣀⡀⠀⠀⠀⠀⡄⠀⡐⣂⠀⡠⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠄⠀⠀⠀⠠⠀⠀⠀⠀⠈⠈⠀⠨⠉⠀⠀⠀⠀⠀⠀⠠⣄⠀⣀⠌⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠂⠘⠂⠈⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⡠⠀⠀⠘⠀⠰⠀⠀⠀⠀⠀⠘⠀⠂⠀⠂


⣿⣿⣿⣿⣟⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣟⣛⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠐⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠂⠀⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡰⠀⠈⠈⠀⠀⠀⢀⣀⠀⠢⠈⠀⠀⠀⠠⠄⠉⠑⠀⠁⠈⠄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠠⠊⡐⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠂⠐⠳⡄⠀⡀⠘⡀
⠀⠀⠀⠀⠠⠁⠌⠠⠂⠀⠀⠀⠊⠀⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠱⠀⠀⢙⣈⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢹⣃⠀⠸
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠁⠀⠀⡤⠂⠀⠀⠀⠀⠈⠀⠀⢀⣤⠦⣬⠀⠀⠀⢸⡆⠐⡈⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠁⠀⠀⠀⢠⠣⠀⠀⠈⠂⠂⠀⠁⠀⠀⠈⠑⠈⡑⢫⠋⠩⠂⠈⠙⣗⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠁⠈⠈⠙⠉⡊⠈⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠉⢪
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡠⡀⣀⠂⠀⡁⠀⠀⠁⡀⠀⠁⠉⠡⡞
⠀⠀⠀⠀⡂⠀⠀⠀⠈⠀⠀⠂⡀⠂⠀⠀⠐⠀⠀⠀⠀⠎⠁⠉⠈⠉⠉⠠⠀⠀⠈⠑⡦⠂⠀⢠⡁
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠐⢠⠐⠈⠳⠤⠜⠱⢃⢑⣦⡌⢀⠐⠀⠱⠰⠄⠖⠄⠤⡤⠄⠀⠀⠘⣠⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢀⠀⠀⠉⢏⠀⠈⡛⠋⠋⢻⣀⠀⠠⠂⠐⢂⠀⠐⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⡜⠃⠠⠔⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⢀⠀⢀⢀⠄⠀⠀⠠⠄⡀⢡⡄⠀⡀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⡄⠀⠂⠀⠂⠀⡄⠆⡬⢡⢀⣶⠄⠀⠀⠀⠀⠈⠀⠀⣀⠙⠦⡀⠸⠘⠀⠠⡀⠂⡦⡤⢀⠀⢠⡀
⠀⠀⠀⠀⡇⠀⠀⠳⠖⢂⡤⠤⠄⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣀⣁⣀⡀⠠⠤⣤⡰⠶⠾⢊⡇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⡄⠈⡧
⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡄⠀⡧⠀⡠⢀⠄
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡠⢁⠌⡀⠀⠂⠀⡀⠁⠀⠀⠄⠀⠀⠀⡠⠊⠐⡁⠐⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣤⢄⣀⡀⠀⠀⠀⠀⠄⠡⠊⠔⠁⠈⠀⠡⠂⠈⠀⢀⣀⡄⡤⡶⢚⠁⢂⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠲⠒⠐⠒⠒⠒⠃⠙⠩⠀⠉⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢰⣆⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⡿⠯⠽⠿⠛⠿⠿⠿⠿⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⠁⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣬⡙⠛⠿⣿⡷⣶⢶⣦⣤⣤⣤⣤⣤⣤⠴⠶⠚⠛⠉⠁⠀⣡⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣶⣄⣒⣘⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠿⣿⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡏⢹⠛⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⢿⣿⣿⣿⣿⣿⣗⡋⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠘⣿⣿⣿⣿⣟⡺⠀⠀⠀⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢻⡿⣿⡏⠀⠀⠄⠸⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣟⠃⠀⠀⠀⠸⣿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣹⡇⠊⠀⣿⣍⠐⠂⠠⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⡏⠀⠀⠀⠀⢀⣿⠀⠉⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠀⠀⠀⠘⣿⣷⢶⣤⠄⠈⣿⣿⠁⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⣀⡴⠛⡅⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠸⣤⣉⠛⠿⣾⣤⣽⣧⣄⣀⣀⣀⣀⣀⣀⣹⣧⡴⠶⠛⠉⢀⣴⠇⠀⠀⠀⠀⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠘⠷⣧⣀⡑⠈⠉⠙⠩⠍⢉⡉⠉⠉⠉⠀⠀⠀⢀⣤⠶⠛⠁⠀⠀⠀⣠⢄⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠶⠤⠤⠶⠴⠶⠶⠖⠒⠋⠉⠀⠀⠀⠀⠀⣠⣾⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⠦⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠃⢹⣿⣿⣿⣿⣿⣿⣿⣧⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢧⠠⢀⣠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣄⣰⣷⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣷⣦⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⣿⣦⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣠⣴
⣿⣿⣿⣦⡀⠀⢸⡇⠀⢀⣤⣾⣿⣿⣿⣿⣷⣄⠀⠀⢸⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣀⠀⠀⣿⠀⠀⣠⣾⣿⣿
⣿⡿⠋⠁⠙⢦⣸⣇⡴⠋⠈⠙⢿⣿⣿⡿⠋⠈⠑⢄⢸⢀⡴⠋⠈⠻⣿⣿⣿⡿⠋⠈⠳⣄⣿⣠⡞⠃⠉⠻⣿⣿⣿⠟⠁⠙⢷⡄⣿⣠⠞⠁⠙⢿⣿
⡏⠀⠀⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⢀⣽⣏⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⢈⣿⡁⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⢉⣿⡁⠀⠀⠀⠀⠙
⡇⠀⠀⠀⣠⣾⣿⣿⣷⣆⡀⠀⠀⢸⡇⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⢸⠀⠀⠀⢀⣴⣿⣿⣿⣷⣀⠀⠀⠀⣿⠀⠀⠀⣀⣴⣿⣿⣿⣦⡀
⡇⢀⣤⠞⢿⣿⣿⣿⣿⡿⠟⢦⡀⢸⡇⢀⡴⠻⣿⣿⣿⣿⣿⡿⠳⣄⠀⢸⠀⢀⡔⠻⣿⣿⣿⣿⣿⠟⠳⣄⠀⣿⠀⣠⡞⢿⣿⣿⣿⣿⣿⠿⢦⡀
⣷⠟⠁⠀⠀⠙⢿⡿⠋⠀⠀⠀⠙⢾⡶⠋⠀⠀⠈⠻⣿⡿⠋⠀⠀⠈⠳⣾⠖⠁⠀⠀⠈⠻⣿⠟⠁⠀⠀⠈⠳⣿⠞⠃⠀⠀⠙⢿⣿⠟⠁⠀⠀⠙⢶
⣿⣦⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣾⣷⣄⡀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣴⣿⣦⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣴
⣿⣿⣿⣦⡀⠀⠀⡇⠀⠀⣠⣾⣿⣿⣿⣿⣷⣆⡀⠀⢸⡇⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⣿⠀⠀⣠⣾⣿⣿
⣿⡿⠋⠀⠙⢦⣀⣇⡴⠋⠀⠙⢿⣿⣿⡿⠋⠀⠙⢦⢸⣇⡴⠋⠈⠙⢿⣿⣿⡿⠋⠀⠙⢆⣿⣠⡾⠋⠈⠻⣿⣿⣿⠟⠉⠈⠳⣄⣿⣠⠞⠁⠙⢿⣿
⡏⠀⠀⠀⠀⢀⣽⣯⡀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⢀⣽⣯⡀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⢀⣽⣏⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⢈⣿⡁⠀⠀⠀⠀⠙
⡇⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⢸⡇⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⢸⡇⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⢸⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀
⡇⢀⡴⠛⢿⣿⣿⣿⣿⡿⠛⢦⡀⢸⡇⢀⡴⠛⢿⣿⣿⣿⣿⠿⠛⢦⡀⢸⡇⢀⡴⠻⣿⣿⣿⣿⣿⠟⠳⣄⠀⢸⠀⣠⠔⢿⣿⣿⣿⣿⣿⠟⢦⡀
⣷⠋⠀⠀⠀⠉⢿⡿⠋⠀⠀⠀⠙⢾⡷⠋⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠙⢾⡶⠋⠀⠀⠀⠹⢿⠟⠁⠀⠀⠈⠳⣾⠞⠁⠀⠀⠘⢻⣿⠟⠁⠀⠀⠙⢦
⣿⣷⣄⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣿⣷⣄⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣿⣦⣀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⣾
⣿⣿⣿⡷⣄⠀⢸⡇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⠀⠀⣀⣴⣿⣿⣿⣿⣿⣷⣦⡀⠀⢸⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⡇⠀⠀⣠⣾⣿⣿
⣿⠟⠁⠀⠈⠳⣼⣇⠴⠉⠀⠙⢿⣿⣿⠟⠁⠀⠙⢦⣸⣠⠞⠁⠀⠙⢿⣿⣿⠟⠁⠀⠙⢦⣸⣠⠞⠁⠈⠻⣿⣿⣿⠟⠁⠈⠳⣄⣇⣠⠞⠁⠙⢿⣿
⡇⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠀⢀⣽⣧⡀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⣀⣿⣅⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⣨⣿⡁⠀⠀⠀⠀⠙
⡇⠀⠀⢀⣴⣿⣿⣿⣿⣧⡄⠀⠀⢸⡇⠀⠀⣠⣴⣿⣿⣿⣿⣦⡀⠀⠀⢸⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⣿⠀⠀⠀⣠⣾⣿⣿⣿⣶⡀
⡇⣠⠖⠙⠻⣿⣿⣿⣿⠟⠉⠣⣄⢸⠇⣠⠞⠙⢿⣿⣿⣿⣿⠟⠙⢦⡀⢸⠀⣠⠞⠛⢿⣿⣿⣿⣿⠟⠳⣄⠀⣿⠀⣠⠞⠻⣿⣿⣿⣿⣿⠟⠢⣄

⠀⠀⠀⢀⠀⠄⠀⠀⢤
⠀⠀⠈⠀⠢⡀⠀⠀⠒⠀⠀⠩⢂⠐
⠀⠀⠀⠀⠀⠈⠙⠋⣩⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⠒⢲⣾⣿⣿⡟⢻⡆⠠⢤⣶⣶⣾⡏⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣌⠀⠀⢸⣿⣿⠛⡇⠘⢓⣢⠄⠠⠻⣿⡇⡆⣿⣿⠀⠀⠀⠀⢠⣶⣒⣚⠡⠤⢤⣼⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢹⠀⠀⠈⣿⣿⠀⡿⢸⣾⠁⣾⢠⣿⣿⣇⣧⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⢀⣀⣸⡿⣧⠿⠼⢿⡇⣿⣠⣿⠿⢟⣚⣾⡿⢃⣀⣀⣀⣀⣀⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠀⢸⡀⣀⣈⣃⣈⡑⠛⠒⠚⠓⠚⠉⠀⣿⣈⣯⡉⠀⠀⢸⠷⣷⡶⠶⠶⠶⢾⢿⣿⣾⡇
⠀⠀⠀⢸⣿⣿⡟⣿⢉⡏⠩⢭⣭⣿⡇⠀⠀⣠⠤⠒⠒⢚⡷⠞⠛⠛⠒⠲⢤⡀⣿⡇⠀⠀⣇⠀⣾⣿⣿⡇
⠀⠀⠀⢸⣿⣿⡇⣿⢸⣇⣸⠘⠻⣿⣇⣴⣯⣄⠮⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⢿⣻⣻⠟⠛⠋⢉⡧
⠀⠀⠀⢸⣿⣿⡇⢿⢸⣿⣿⣰⠀⠏⣿⣿⣿⣿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡅⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣿⣿⣷⠀⢸⣿⣷⣼⡆⣸⣿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⢀⠴⡷⣄
⠀⠀⠀⠈⣿⣿⣿⣀⣸⣿⣿⣿⣇⣿⣿⣿⣿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡇⠀⢠⠏⠀⡇⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⣰⠯⢤⣀⣇⣠⠼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⣷⠃⠀⠀⠈⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡼⢧⡀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡼⠁⠀⠀⠀⣴⢃⡄⠀⠀⠈⣆
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⠤⣀⣀⡠⠤⠖⠋⠀⠀⠈⠁⠀⠀⠀⠀⡏⢸⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠈⢸⠂⠙⠀⠀⠀⠀⠀⠄⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⢀⠤⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⠐⠀⢴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢁⣤⡖⠂⠉⠉
⠀⠠⠀⠠⠀⠁⠀⠉⠁⠀⠀⢐⡿⠀⠐⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢃⣼⠶⢋⣌⠀⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠊⣠⡾⠋⠀⠀⠆⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⡤⢠⣆⠀⠈⠀⠀⠀⠁⠀⠀⢀⣼⠃⠀⠀⠀⠐⠖⠛⡟⠛⠏⢁⠀⠀⠊⠁⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣟⠀⠔⠀⠀⠀⠀⠀⠀⠈⠂⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡂⣨⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⡈⠥⣀⠀⠀⠀⠀⠀⢈
⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠙⣧⡈⡀⠀⠀⡀⠀⠀⢠⡀⠀⠛⠀⢀⠀⠀⠀⢀⠀⠀⠀⢈⡓⠀⠀⠀⠔⠁
⢀⠀⠀⠀⠀⠐⠂⠘⠀⠉⠀⠄⢀⡷⢄⠘⣦⡀⠨⢽⣦⡀⠱⢄⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉
⢁⠤⠴⠥⢤⣀⡠⠄⠀⠀⠀⠀⣨⠁⠁⢨⠁⠑⠀⠀⠹⡝⠿⡸⣇⠀⡄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⣠⠛⠀⠉⡇⠀⠀⣧⠀⠘⡜⡆⢡
⠀⠀⠀⠀⠀⠀⠐⠃⠈⠀⠀⠀⠀⡴⠂⠀⠀⡰⠂⠀⡀⠊⠃⢀⠂⣴⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢤⠄⢁⡠⠞⠁⢀⠤⡂⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠑⢲⠆⠀⠀⠈⠔⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠘⠋⠉⠀⠈⢀⣀⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⠇⠀⢠⠃⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⢤⣾⢁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠀⠄⣼⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀⠀⠐
⡤⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠚⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠰⠞⠁⠀⠀⢁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⣉⣷⡄⠉⠂
⠀⠀⠀⠀⠺⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣀⡴⢾⣿⠋⠉⠀⠠⠀⢆
⠀⠀⡀⠀⠀⠨⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠒⠉⢁⠪⠕⠁⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⠀⢀⡀⣀⡄
⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⣀⣠⡤⢴⠠⠊⣀⠀⠠⠄⠀⠀⠀⡆⣇⠀⢳⣁⡠⠂⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⡠⢖⣪⡷⠋⢈⡤⠞⠏⠀⠒⠒⠒⠲⠲⠾⠓⠛⠘⠋⠁⠀⠀⠈⢀⣼⣤⠏
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠤⠌⠉⠐⡡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠤⠀⡀⠀⠀⠀⠀⠠⠆⣸⠁⢻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠠⠀⠈⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⢀⠄⠉⠀⠤⠄⠀⠀⠈⠀⢀⣀⣀⣉⠀⠒⠀⠀⡀⢀⡀⠀⢤⠶⠊
⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠁⠀⠄⡠⠀⠁⠠⠠⠆⠊⠀⠀⠀⠀⠀⠁⠀⠀⠁⠒⢄⣀⡀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠂⠐⡀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠉⠀⠘⠠
⠀⠀⠀⣀⣠⣤⣤⡤⡜⠉⠀⠀⠁⠘
⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠰⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⢀⣤⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⣠⡟⠉⠳⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠈⠉⠉⠀⠀⣇
⣴⣶⣦⡿⠀⠀⠀⠀⠀⠉⠑⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣠⣯
⠉⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⢀⣴⣶⣴⣾⣿⣿⣿⣿⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄⠀⠀⠘⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⠟⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⡿⠃⠀⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣈⠄⠀⠀⠀⠀⠘⠛⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠚⠲⠄⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠆⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢀⠤⣀⠴⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⡿⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⠟⠉⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⡄⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡼⠁⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣾⠁⠀⠀⠀⠀⣼⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋⠉⠉⠁⠀⠻⠇⠀⢠⡿⠀⠀⠀⢀⡼⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⣾⡥⢤⡴⠞⠁⢀⡴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿⡀⠀⠈⢷⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠙⠛⠶⣦⣤⣤⣤⣤⣴⣶⠶⠞⠛⠁⠀⠘⣷⡀⣨⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣵⣤⣤⣴⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡴⠋⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢰⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢹⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣠⡀⢸⡀⠀⠀⠀⠸⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⠶⠿⢿⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣥⡼⠏⠉⠛⠀⠀⠀⠀⠀⠈⣿⡉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡆⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⣤⣤⣭⣬⣯⣬⣥⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⠈⡏⠀⠀⢹⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡇⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⢾⡇⠀⠀⢸⡷⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣾⡇⠀⠀⢸⣷⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣇⠀⠀⣼⣿⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣧⣿⢹⢳⣿⡿⣿⢸⡟⣿⢸⡇⣿⡈⣿⣸⡏⣇⢹⠻⡇⠀⣷⠀⠀⠀⠀⠀⠀⣰⠇⠀⣠⢞⡥⢖⣊⡩⠭
⣸⡿⡾⡞⣿⣿⡿⣼⠃⣿⢸⡇⢸⡇⣿⡇⣿⢹⢸⡆⠳⢿⠈⠢⣤⣀⠀⠀⢰⡏⢀⣼⢟⣵⣾⣷⠶⢿⣿
⣯⢣⣧⢇⡿⣸⠇⣿⢠⣿⢸⡇⢸⡇⣿⡇⣿⢸⠸⡅⠀⢸⡆⠀⠈⠙⠦⡄⣸⡇⣼⣿⣿⣿⢕⣪⡭⠵⠶
⠃⣼⢝⡾⢱⡟⣸⠇⢸⡟⢸⡇⣼⠇⣿⡇⣿⢸⢠⡇⠀⣼⡅⠀⠀⠀⠀⢌⢻⣽⣿⣿⣿⡷⢋⠤⣒⣨⣭
⣾⢏⡾⣱⠏⢠⡟⢀⡿⢁⡿⣱⠏⢠⣿⢳⣿⣿⡾⠀⠀⡟⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⣱⡷⠛⠉⠁
⡱⢋⡴⠋⣠⠏⣠⠟⣡⠟⣱⠏⣠⣾⣷⢋⣾⠞⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⡟
⡥⢛⡴⢊⡥⢞⣡⢞⣡⠞⢁⣴⣿⣿⣷⡿⠃⠀⠀⠀⢹⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣚⡵⢚⡭⣖⡯⠖⢋⣠⣴⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⣵⣾⠿⣚⣥⣴⣿⣿⣿⣟⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⣽⣶⣿⣯⣅⣈⠀⠀⠈⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠐⣾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⡀⠀⠈⠈⠉⠙⠛⠷⣦⡀⠀⠀⠀⠈⠑⢝⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⡛⠻⠶⣦⣄⡀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⢹⣧⣠⡾⠋⠂⠒⠀⠀⠀⠀⠀⠀⢀⢀⣠⠃
⠙⠷⢤⣀⡉⠛⠷⣦⣄⠀⠀⠈⠳⣄⠀⠀⠀⣿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠠⠀⠐
⠀⠀⠀⠈⠉⠓⠲⢬⡙⢿⣦⡀⠀⠘⢧⠀⣠⣿⠂⠀⠀⠀⠀⠀⠀⠀⢰⣾⢏⠠⠂
⠛⠛⠛⠲⠶⢤⣀⠐⠙⢷⣝⢷⡄⠀⢸⣿⣻⣥⣄⣀⠀⠀⠀⠀⠀⡄⠀⣿⠃⠀⠀⠀⠀⠀⠀⢀
⣭⣭⣭⣕⣢⢄⡈⠓⢆⠀⢹⡎⣷⣠⡟⠡⠄⠀⠀⠉⠙⠲⣄⠀⠀⢹⣾⣏⠄⠀⣀⠠⡰⠄⠊⠀⠉⠉⠉
⣶⠿⠳⠶⢮⡑⢯⣢⡈⢳⠀⣿⣿⣷⣦⠶⠤⣔⡢⠀⠀⠀⠈⠳⡄⠸⣿⢏⣴⠪⠔⠁⢀⡠⣶⠿⣛⣛⣛
⣉⣉⡛⠛⠦⡍⢲⣹⡱⡀⡇⣹⣿⣿⣭⣽⣛⢶⣭⡳⢅⠢⠀⠀⢹⢸⡿⠃⠁⢀⣠⡾⣛⠭⣒⠭⠒⣒⣲
⡤⢭⣉⠓⠦⡈⢆⢹⣧⢱⣧⡿⠉⠀⠀⠀⠈⠙⢪⣻⣮⠳⡱⡀⢸⣼⠃⢀⣴⡿⡫⡪⠔⠉⣀⣴⡿⢗⡯
⠤⢭⣐⠭⡢⡙⡼⡄⣿⢸⣿⣷⣖⡒⠲⢄⠀⠀⠀⢹⣟⢇⢻⣧⣾⠃⣴⣻⡫⡪⠊⢀⣴⣾⢟⡵⣪⠵⣚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠤⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠐⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠁⠒⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠠⣈⡠⠂⠀⠀⠀⠀⠀⠀⢀⠤⠤⠂⠀⠀⣀⣀⠀⡠⢀⡀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⢀⠠⠃⠀⠀⠀⠀⠀⣀⠤⠔⠈⠐⠁⠀⠀⠀⠀⠊⠀⠈⡀⠀⠅⣂⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠔⢁⠖⠁⠀⢀⠄⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⣈⣗⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠀⠃⠀⠀⢠⠆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠈⣥⣂⠡⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⡈⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡀⠀⠠⠂⠀⠀⠀⠄⠔⢀⡆⠀⠀⠈⢏⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠁⠀⠀⢲⠂⠀⠀⠀⠀⠀⢈⠀⢀⣠⣴⢢⢤⠁⢊⡂⣈⠇⠈⡭⣅⠮⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠜⠢⡈⡀⢀⡐⢁⣀⠀⠀⠀⠀⢉⠐⠉⠩⠉⠁⠒⠈⠉⣳⠭⡂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠈⠀⠁⠩⠍⡉⣁⡂⡠⠀⠀⢀⡀⠀⠀⠂⠁⠀⠀⠀⣈⠻⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣌⡤⣨⣌⠀⣀⡀⠀⠀⠁⡀⠀⠒⠉⠁⢾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠐⢄⠐⢂⠁⠠⠊⠀⠐⠀⠀⠌⠊⠁⠈⠉⠉⠄⢄⠀⡄⠑⢦⠐⠐⠠⡎⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡕⠊⢛⠦⡴⠋⠛⣘⣹⣭⡁⢨⠂⠀⠖⡲⢀⠖⠔⠤⣤⠄⠠⠊⠚⢀⢗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠍⠊⠀⠈⠳⣅⠀⢉⠚⠉⠩⣒⣀⠁⠨⠊⠐⢂⡈⠀⠊⠈⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⢐⠛⠂⠈⠓⠐⠀⠀⠁⠀⠀⠐⠂⡀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠐⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠒⣻⣿⣶⠃⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠌⣣⠀⠀⠀⠀⡋⠀⠀⠀⠐⠛⠛⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠆⠀⠀⠄⣄⣲⢠⢟⡀⢣⣄⠐⠁⠀⠀⠈⢒⠀⠁⠈⠻⣍⠙⢻⠻⠛⢁⠃⡆⣶⢡⡉⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠤⣀⡤⠀⠀⣈⣈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠉⠀⠃⠃⣂⣁⡀⠉⢠⣄⣡⡤⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠚⠛⠋⠉⠉⠁⠀⠨⠭⠭⠍⠉⠉⠉⠐⠒⠒⠂⠈⠉⠉⡁⠀⢠⢁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⢆⢀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⡐⢁⠈⡐⡠⢂⠈⡐⠀⠠⠁⢀⠀⠊⡡⢋⡠⢊⠄⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⠀⠠⠐⠀⠈⠀⡠⢊⠔⡠⠊⠐⠡⢂⠔⡠⠁⠌⠐⠐⢀⣁⡤⠖⡈⠐⡁⢀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠒⠦⠦⠤⣤⣬⣄⠁⣈⡀⢈⣤⢤⣤⠄⡠⠶⠂⠜⠛⠉⠊⠔⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢿⣦⠀⢀⣴⣿⣿⣿⣿⣿⣶⣤⡀⣤⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠈⣿⠀⣸⣿⣿⠉⢿⣿⣿⣿⣿⣿⣾⣿⣿⠁⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠃⠀⢻⡿⣿⣄⠈⣿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢰⡏⠉⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣯⣍⡛⠷⠋⢻⣿⣿⣿⣿⣧⡀⠀⠸⣧⣀⣀⣿⡿⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠌⢹⣿⣉⠙⠓⢺⠛⠻⣿⣿⣿⣿⣷⡀⠀⠈⢿⠛⠋⠐⠶⠶⢀⣠⣾⡆
⠀⠀⠀⢠⡀⡀⣤⣦⡄⣀⣤⡤⡲⣧⣿⣾⣟⡖⣿⣿⣤⠄⣠⣖⠴⠋⢻⣿⣿⣿⣧⣀⣯⣐⣆⣀⣐⣰⣼⣿⣾⣿⣧
⠀⠀⠀⣿⣿⣿⣿⣷⣤⣿⣿⡿⢛⣛⣿⣿⣿⣧⡿⠿⠶⠒⠛⠁⠀⠀⠀⠛⠻⢿⣿⣷⣿⣿⣋⡭⣁⠙⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣿⣿⣷⣽⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣴⣤⣤⣤⣶⣦⣠⣶⣤⣤⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⠒⢶⣴⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⣀⣸⠁⢈⣀⣉⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡝⢁⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠁⠀⠀⠀⠀⢱⡞⠁⠂⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⡾⡯⠀⠻⠛⡳⠆⠉⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠤⠤⢼⡇⠀⠐⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⡴⣿⣅⣀⣀⣠⢤⡶⠞⣍⠉⠉⠛⠻⢲⡧⣭⠤⠀⣀⣐⣒⣒⣤⣤⡄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣸⠀⠀⢠⠀⠀⡇⠀⣿⠀⣀⠄⠀⠨⡇⠀⠀⢨⠀⠀⠀⠀⠉⢹⠇⢠⣼⣃⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠄⠀⢸⠀⢠⡅⠀⢉⣀⣻⣀⣟⢸⠋⠉⠀⣸⠀⠀⠀⠀⡇⢸⡧⠤⠼⡧⠀⠀⢒⣛⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⢸⠀⠀⢠⡆⠀⠙⠿⣷⠿⢿⣿⣂⣀⣀⣤⣤⣀⣠⣤⣼⠁⠀⠀⣇⠀⠀⠁⢘⡂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⣄⠀⢸⣇⡀⠀⠀⣇⢀⣀⣘⣀⣠⡇⠀⠀⠉⠉⠙⠁⠉⠀⠈⡬⠒⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡄⢀⡾⡿⡄⠀⠉⠙⠛⣸⢿⡍⠉⠁⠀⡨⠙⠓⣀⠀⠀⢠⠀⠆⠀⠀⡇⠀⠀⠀⢀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣼⣁⣧⣨⣆⣀⡀⠀⠑⠚⠷⣂⠀⠀⠀⠀⠀⠙⢿⣆⠰⡄⣇⠀⢀⣟⠀⠀⠀⢸⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠋⢉⣿⡷⠾⠟⠯⠭⣤⣀⠈⠁⠀⠀⠀⠀⠀⣠⠆⢻⣇⠢⣿⡀⢈⣉⣀⠀⠀⠸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢰⡿⠉⠀⠀⠀⠀⠀⠀⠉⠙⢦⡑⣄⢀⡤⠀⠀⠀⠈⣿⡄⠬⠀⠈⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⠹⣯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⠎⠀⠀⠀⠀⠀⡿⠃⠀⠀⠴⡆⢀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠍⢐⠃⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢃⠀⣀⠴⢁⡟⠀⠂⣠⣘⣻⠈⠉⠁⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢀⡄⠉⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⡀⠉⠀⠈⠃⢠⠚⠁⡹⣷⠤⠧⠂⢀⡀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡾⠏⠀⢠⠿⠛⠒⠶⠫⠾⠒⠀⠀⠉⠓⠤⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⢍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠃⢀⠴⠁⠀⠸⠆⠀⠀⢠⣦⠀⠀⠀⠀⠀⠈⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠱⢿⣶⠀⠀⠀⡀⡀⢀⣴⠟⠉⢀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⠉⠐⠈⠛⠊⠈⠀⠀⠀⠘⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡀




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠏⠲⠇⠱⣄
⠀⠀⢀⡤⠒⠒⠤⡀⠀⠀⢀⡇⠀⢀⠀⠀⠀⠈⢑⠒⠦⢄⡀⠀⠀⢀⠤⠒⠒⢤⡀
⠀⠀⡎⢰⠋⠉⠲⣌⡦⠊⠁⠲⠀⠈⠀⢠⠀⢀⠈⢠⠀⣀⠈⠓⢴⣡⠞⠉⠙⡆⢱
⠀⠀⢳⡘⢄⠀⡠⠊⠀⠀⠐⠀⢀⠈⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠑⢄⠀⡠⢃⠞⠀⠀⠀⠀⢀⣠⠤⠴⢒⣆⣤⠤⢤
⠀⠀⠀⠙⠢⡝⠁⣠⠤⠤⠤⣄⠀⠄⠀⢁⠀⠐⠀⠀⢀⡠⠤⠤⠤⣀⠈⢫⠔⠉⠀⣠⠴⠒⠉⢁⠀⠈⠀⠀⠀⠄⠀⠛⠒⣤
⠀⠀⠀⠀⡜⢀⠎⣠⣶⣿⣷⣦⡱⡄⠠⠈⠀⠄⠘⢠⢋⣴⣿⣿⣶⡌⢳⠀⢳⠒⠉⠀⠂⠀⠀⠁⠀⠀⡀⠀⠀⣀⠀⠀⠀⡑⠦⡀
⠀⠀⠀⢸⠁⢸⠀⣿⣿⣿⣿⣿⣧⠷⡖⠉⠉⠉⢲⠦⣾⣿⣿⣿⣿⣿⢀⠇⠸⡀⠀⠠⠄⠀⠈⠁⠀⠀⠠⠀⢀⡀⠀⠀⣀⠀⠀⠈⢫⡒⠢⣄
⠀⠀⠀⡇⠀⠈⢣⡙⢿⣿⣿⡟⠁⠀⠑⠒⣤⠖⠊⠀⠈⢻⣿⣿⠿⣃⠞⠀⠀⡇⠠⠄⠀⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⡀⠀⠱⡀
⠀⠀⢸⠀⠀⠀⠀⠈⠙⠒⢺⠀⠀⠀⠓⠖⠛⠲⠚⠀⠀⠀⡗⠒⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠈⠁⠀⢀⠒⠀⠀⠀⠀⢳⣀⣀⠇
⠀⠀⠸⠖⡆⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⢠⠓⠃⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠊⢀⡴⠊⠁⠀⠀⠀⠈⠀⠈⡆
⠀⠀⠀⠀⢸⡀⣀⡄⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⣀⠀⡜⠀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⣧
⠀⠀⠀⠀⠀⠉⠀⠘⠖⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣆⠜⠉⠛⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⡖⠲⣖⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠓⢤⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢹⠒⠤⣠⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⢠⠏⠀⠀⡸⠀⠀⠀⠀⠀⢠⣇⣀⣀⣀⣀⠤⠤⢤⠖⠚⠉⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⠀⠀⠀⠀⢀⡴⠋⠀⠀⡰⠁⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠸⠻⢄⣀⣀⡠⠊⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠑⠒⠒⠒⠉⠀⠀⠀⠰⡁⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣀⣀⣀⣀⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠓⠗⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢠⣾⣿⠁⠀⠀⠀⢰⣷⡄⠀⠀⠐⠦⢄⠴⠖⠀⠀⣀⣶⣷⡆⡀⠀⢀⡀⠘⠿⠀⠀⠠⠚⠁⠈⠁⡆
⠀⠀⠘⠸⠟⠀⠀⠀⡀⣈⢛⡁⡀⠀⠀⠀⢸⠀⠀⠀⠀⠨⠻⡿⠣⠀⠀⠸⠋⠠⣴⡤⠀⠀⠈⠉⢩⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢀⣴⠙⢁⣦⠀⠀⣿⣿⡇⠀⠀⠀⠀⣠⡞⢶⡆⠀⠀⣀⣀⢀⢀⡀⠀⠀⠀⣤⡅⠂⠀⢠⣤⣤
⠀⠘⡩⡄⠘⣏⠃⠀⣻⣿⣻⣿⡇⠀⠈⣿⠀⢀⡟⠀⠰⣏⣿⣸⣮⡇⠀⠈⠈⣿⡃⠌⠀⠀⣾⣿⠟⠻⠆
⠀⠀⠈⠑⠀⠁⠀⠀⠛⠀⠛⠛⠃⠀⠀⠛⠉⠛⠃⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠉⠀⠀⠃
⠀⠀⠀⡐⢂⡀⠀⠀⠀⢀⡗⢠⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⡴⢰⣀⠀⠀⠀⡤⠀⢄⡀⠀⠀⢀
⠀⠀⢀⡁⣉⠀⠀⠀⠀⣾⡉⠛⠀⠀⢀⣴⣧⣀⠡⠀⠀⢹⢐⡍⠔⠂⠀⠘⠌⣮⢚⠃⠀⢠⣾⡂⢐⣿⠂
⠀⠀⠈⠻⠟⠀⠀⠀⠰⠃⠃⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠻⠛⠛⠉
⠀⠀⡀⠀⠀⣀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡀
⠀⠘⠿⣀⣘⠿⠃⠀⢰⣿⠟⠈⠀⠀⠀⣤⣿⣿⡇⠀⠀⣔⡀⠰⣄⠀⠀⣶⠒⢲⡄⠀⠀⠀⣿⢰⡆⣤
⠀⠀⠀⠧⠟⠀⠀⠀⠸⠏⠀⠁⠃⠀⠀⠸⠦⠤⠃⠀⠀⠻⠛⠛⠿⠃⠀⠻⠗⠚⠛⠿⠂⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣄⣤⣠
⠀⠀⢠⢖⡢⡄⠀⠀⠀⢸⡆⢹⠀⠀⠈⠃⠸⠀⠛⠀⠀⠒⣿⠷⠄⠀⠀⠀⢠⡾⠅⠀⠀⠀⠈⣿⢁⣀
⠀⠀⠣⣘⣃⠎⠀⠀⠀⢼⡧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⣶⠂⠀⠀⠀⠙⠀⠀⠀⠀⢰⢻⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡴⢣⡐⢦⠀⠀⣴⣧⢀⣿⡆⠀⠀⢰⣶⡇⠀⠀⠀⡔⢉⣥⠛⡄⠀⠀⣠⣼⣶⡄⠀⢀⣾⡏⢹⣷⡄
⠀⠀⢣⡉⢃⠞⠀⠀⢣⣨⣶⣀⠇⠀⠀⠲⠌⠶⠃⠀⠀⢧⡘⢛⣠⠃⠀⠀⠟⠁⡼⠃⠀⠈⢿⣿⣺⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⡿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⣁⠀⢹⣧⣿⣿⣿⣿⣿⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠼⣿⢷⣿⣸⡟⣿⣧⣻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠰⣿⣾⣿⣿⣧⢹⣿⣿⣿⣿⣿⣏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢹⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢠⣀⢲⣶⣶⡓⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣎⢿⣆⠿⠛⠁⠁⣤⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣏⢻⣿⣿⣷⡄
⠀⠀⢀⡀⠀⠄⠀⠀⠀⠠⠷⠿⠛⠛⠋⠉⠟⠛⠛⢦⠀⠀⠀⠀⠀⢻⣿⣻⢹⣿⣾⣿⣿⣻⢿⣿⣿⣆⣿⣿⣿⠇
⠀⠀⠀⠀⠈⠀⢲⠀⠙⠢⡀⠀⠀⠀⠀⠀⠈⢣⠀⢻⣥⡄⠀⠀⠀⣼⣿⣿⣾⠿⠛⠉⠁⠙⡏⢢⣿⡻⣿⣼⣿⡀
⠀⠀⠀⠀⠀⢠⡊⠀⢀⡀⠑⠆⠀⠀⠀⠀⠀⠐⡗⠀⢸⡇⠀⠀⣨⣵⣿⣿⠋⠀⠀⠀⠀⠀⠃⠀⠙⣿⣷⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠛⠻⠄⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⡇⢠⣿⣿⢟⡿⠉⠀⢠⡤⠴⠛⠀⠀⠀⠀⢻⣿⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣏⡵⠋⠀⢠⣀⠀⠀⠀⠀⢀⣜⣥⠠⣯⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⣷⠘⠋⠀⠀⠀⠀⠙⣿⣶⣤⡀⣼⢿⠁⣼⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠷⢤⣾⣿⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠏⠙⠈⡡⢂⣘⣿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⠀⠀⣀⣠⣶⡛⣿⠿⠛⡛⠋⠀⣀⡴⢡⣾⣿⣿⣿⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⠛⠉⠙⣿⣿⣿⣷⣷⣻⣆⣀⠙⣶⣿⣟⣹⣿⠁⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣈⣿⣿⣿⡙⠛⡋⠁⠛⠿⢾⣿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⠋⠉⠉⢱⣿⣿⣳⣶⣬⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠾⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⣠⣤⣠⡄⠘⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⠀⠠⢄⣀⣀⣨⣿⣿⡏⠉⠉⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⣶⡛⠋⠙⢛⢿⡛⠛⠓⠒⠒⠒⠒⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣾⣿⣶⣻⡶⠀⠀⠀⢠⣤⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢉⣉⣭⡌⣇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠛⠉⠀⣀⣾⠂⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⠈⠓⠀⠀⠀⠀⠲⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃
⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣤⠤⠴⠶⠶⠶⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠶⠶⠶⠤⠤⣤⣀
⠀⠀⠠⣽⡉⠀⠀⣠⢀⣀⠤⢾⣥⡤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣬⡳⠤⣀⡀⣄⠀⠀⢉⣿
⠀⠀⠀⠀⠈⢉⡟⠁⠀⠀⠀⡌⠀⠀⠀⠸⠉⠙⠓⢦⡀⢠⡴⠚⠋⠉⡅⠀⠀⠀⢡⠀⠀⠀⠈⢻⡉⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠤⠄⠀⣀⡇⠀⠀⣾⣆⠃⠘⣸⡇⠀⠀⢰⣀⠀⠠⠬⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡸⡄⠀⡘⠛⠛⠒⢦⣤⣀⠈⠁⠒⣯⡟⠀⠀⢯⡵⠒⠉⠀⣀⣤⡴⠚⠛⠛⢂⠀⢠⢇
⠀⠀⠀⠀⠀⡇⡇⠐⣃⠐⠂⠠⢎⡀⠈⠙⢯⡀⢻⡁⠀⠀⢸⡇⢀⡽⠋⠁⢀⡡⠄⠐⠂⡸⠂⢸⢸
⠀⠀⠀⠀⠀⢹⠁⠀⠀⠑⠂⢴⣆⡈⠑⠢⡜⠀⢸⡃⠀⠀⢸⡇⠀⢢⠔⠊⢁⣰⡦⠐⠈⠀⠀⠈⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠀⣠⠞⠛⠉⠉⠙⠳⣤⡀⠉⠉⠁⠘⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡜⡆⠀⠀⠀⠀⢸⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠸⡇⠀⠀⠀⠀⠰⢣
⠀⠀⠀⢀⡴⢳⠷⣄⠀⢀⡴⢿⠷⣄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⢶⡿⣆⠀⠀⣠⠾⡞⢦⡀
⠀⢀⡴⠋⠀⠘⠀⠈⣷⡞⠁⠘⠃⠈⢳⣄⠀⠙⢦⣄⣀⣀⣀⡤⠞⠁⣠⠞⠁⠘⠃⠈⢳⣾⠁⠀⠃⠀⠙⢦⡀
⠀⠈⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡁⠀⠔⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⡘⠀⠀⠉
⠀⠀⣀⣤⠴⠶⠦⢼⡀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠙⢄⠇
⠀⠀⠈⠳⢄⡀⠀⣸⠳⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠜⠀⠀⠈⡎
⠀⠀⠀⠀⠀⠀⠉⠐⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠤⠄⠒⠒⠀⠉⠀⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠑⠂⠤⠄⣀⣀⣀⡀⠤⠔⠒⠈⠁⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⢠⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢿⣿⣿⣿⠀⠀⠀⢠⠏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠙⣛⠟⠀⠀⡰⠋⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⠋⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠜⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⡀⢧⣄⣠⣤⡴⠛⠁⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠂⠤⠤⠤⠀⠄⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡴⠻⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣀⣠⠞⠓⠤⣉⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡋⠀⢠⠾⠄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠱⠤⠞⢁⡤⢄⣀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡀⠑⠴⠂⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠟⠉⠉⠉⠉⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⡹⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠜
⠀⠰⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡰⠁⡜⠉⢢⡀
⠑⢮⠀⠀⣎⢉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣤⣴⣶⣶⣤⡀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣱⢦⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠙⢿⣿⣿⣻⠗⠚⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢳⣤⠖⠛⢤⡙⢦
⠀⠀⠣⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⠀⠀⠀⠀⠁⠸⣄⣠⣶⣇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⠤⡄⢀⣹⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠁⠁⠀⠀⠀⠀⠀⠒⠀⠀⠈⢻⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢴⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣹⠤⠖⠙⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠹⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⡄⠀⠀⠀⠼⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⢠⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⠂⠄⣠⠁⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⣿⡏⠛⢧⡀⠀⠀⠀⠀⠀⠀⢹⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⢀⡀⣀⠀⢸⣷⣄⠀⠈⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠠⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠷⠄⣀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⣿⠿⠟⠋⠉⠂⠈⠓⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠊⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣠⠀⠀⢰⣾⣦
⠀⠀⠀⠀⠀⠀⠁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⣿⣿⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⣿⣿⡿
⠀⠛⡁⢀⡐⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣯⣹⡏⠃
⠀⠈⠡⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠉⠈⠉⠉⢉⠉⠐⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⣤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣺⡞⢯⣭⣷⣐⡲⠤⣄
⠀⠁⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⣎⢧⡀⡀⠀⠀⠁⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠁⠠⢝⠢⢉⡳⣯⡛⡶⢾⡷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠘⣧⢻⡄⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⡗⠫⠟⢆⣁⡘⠿⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣷⣿⣆⠀⠀⠀⠀⠀⠀⠲⢤⡤⠤⠂⠀⠀⠢⢄⠉⠁⠀⠀⠙⠲⠞⠄⠋
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⡿⢿⣄⡆⠀⠀⠀⠀⠐⠄⠛⠀⡘⠃⠀⢀⠀⠀⠈⠀⢠⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠦⢊⣟⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⢀⣼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠈⠉⠁⠀⢻⣿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠠⠀⢰⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠃
⠀⠀⠀⢀⣾⡿⢿⣿⠁⠀⠀⠀⣿⡏⣹⡿⠀⠀⠀⠀⠈⠁⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢻⡇⠀⠙⢷⣄⠀⠀⣽⡇⢿⡇⠀⠀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠟⠀⠀⠀⢤⡄⠀⢠⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⣄⣄⡀
⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠁⠀⠀⠀⠐⠾⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢉⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣋⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢋⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⣁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢋⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⡴⠋⣡⣴⣿⣅⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⢋⣭⣭⣤⣄⡴⠛⢁⣤⠞⠉⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⣻⡶⠛⠁⣠⣾⣉⡀
⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠀⠀⠀⠀⣴⠋⠀⠀⠀⠘⣷⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⠞⠁⣀⣴⠟⠁⠀⠉⠉⠛⠶⢦⣄⡀
⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠙⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣿
⣿⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⣿
⣿⣆⠈⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠛⠁⠀⠀⢀⣴⣿
⣿⠈⠳⣄⠀⠈⠉⠛⠳⠶⠶⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠋⠉⠀⠀⠀⠀⢀⣠⠴⠋⢸⣿
⢸⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠉⠀⠀⠀⡾⣿
⢸⠀⠀⠀⢀⡏⣇⠉⠛⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠚⠋⠍⢦⠀⠀⠀⠀⠀⠀⢠⡇⡿
⢸⠀⠀⠀⣼⠀⢸⡀⠀⠀⠀⠀⠀⠉⠉⠉⣹⢻⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⢯⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⣆⠀⠀⠀⠀⠀⢸⢰⡇
⢸⡄⠀⢰⠇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣰⠃⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠸⣆⠀⠀⠀⠀⡟⢸⡇
⢸⡇⠀⡟⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢹⡄⠀⠀⠀⡇⢸⡇
⠸⡇⢸⠃⠀⠀⠀⠈⣇⠀⠀⠀⠀⣰⠃⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢳⠀⠀⢸⡇⢸⠇
⠀⣇⡏⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⠏⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⣇⠀⢸⠁⣿
⠀⣿⣦⠀⠀⠀⠀⠀⠘⡇⠀⢀⡞⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿⠀⣿
⠀⢻⡌⠳⢤⡀⠀⠀⠀⢹⠀⡼⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡿⢀⡿
⠀⠈⢳⣄⠀⠙⠲⣄⡀⠘⢀⠇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⣻⠇
⠀⠀⠀⠈⠳⢤⣀⠀⠙⠓⠾⣤⣀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⡏⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠈⠙⠓⠶⠦⣤⣀⣀⣈⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣼⣁⣤⡤⠴⠶⠚⠉⠁⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣄⣀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠒⠒⠓⠒⠒⠒⠚⠛⠛⠛⠉⠉⠁⠀⢀⣀⣠⣤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠖⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢰⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠰⠶⠶⣶⡾⠿⠿⣿⡶⣶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠴⠂⠐⠤⡉
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢇⣄⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⣿⣷⣶⣿⢡⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠸⡟⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠉⠉⢱⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠾⠀⠄⠾⠸⠲

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣳⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠙⠟⡇⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠐⡂⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡷⣄⣿⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⣌⡃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⠀⠀⠀⢀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣶⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⠀⢸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⡀⣠⢠⠄⡀⢀⠄⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡉⢀⠈⠁⣀
⠀⠀⠀⠀⠀⠀⠇⠧⠸⠜⠧⠸⠄⠿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠄⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠓⠀⠛⠛⠀⠒⠛⠚⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠶⡄⣴⠄⣠⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠒⠖⣡⣾⢟⠕⢮⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠶⠦⠴⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠾⢿⠗⢡⢀⢠⠘⢮⠻⠷⠀⠀⠀⠀⠀⢀⡴⠚⢻⣿⡑⠤⡀⢀⢃⣽⡞⠛⢶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣏⠈⠙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡠⠄⠚⠀⠀⡀⠀⠀⠀⠀⢸⡅⠀⠋⢀⡗⠀⠈⠛⢺⡗⠛⠂⠀⣿⠀⠀⠀⠀⠀⣿⣿⠏⢹⣿⡶⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠁⠶⢀⠀⣀⢈⠀⢀⣤⢄⠲⠈⠀⠀⠀⠀⠈⠻⠶⠶⠟⠁⠀⠀⠀⠀⠛⠶⠶⠞⠁⠀⠀⠀⠀⠀⣿⣿⣦⣤⣿⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣚⣉⡉⠚⣿⢿⣛⣿⣭⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣼⣿⣷⠈⡦⢸⠈⢻⣿⣿⡭⢅⡀
⠀⠀⠀⠀⠀⣀⡤⠴⠒⢒⣒⣒⣒⣒⣉⣁⠠⠴⠀⠻⣿⣟⣠⢏⡎⠀⢸⣿⣯⣍⡀
⠀⠀⠀⣴⡿⠿⠭⠽⠗⠒⠒⠚⠛⠛⠛⠓⠒⠦⡀⠀⠀⠀⢀⠞⠀⠀⡾⡝⢝⢮⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⣠⠔⠁⠀⠀⢸⠁⠘⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠉⠀⠀⠀⠀⠠⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠴⠀⠀⡀⡉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠉⠀⠀⡄⢀⡤⠬⠥⣄⡀⠘⠛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡰⠃⢀⠀⠢⠂⠈⠲⡄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠚⠀⠀⠀⠴⠀⠈⢦⠁⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡇⠠⠆⠀⢠⠀⠀⠀⡄⠀⢣⠒⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢧⠀⠀⠁⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡇⠠⣰⠀⢹⠀⠙⠀⠀⠈⠃⠘⡆⠄⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢳⠀⠀⠀⠘⡆⠀⠀⢠⠀⠀⠀⢳⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠘⢆⠀⠀⠀⢱⡀⠀⠈⡆⠀⠀⠸⡄⠀⠀⢰⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣬⣷⣄⣀⢀⣳⡀⠀⣸⢄⡀⣠⠃⠀⢃⠀⢇⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠈⢻⡟⠁⠉⠉⠁⠀⠉⠙⡄⠀⢸⠀⠸⡄⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⡇⠀⢧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠭⢟⣛⣫⣥⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡇⠀⠸⡄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠩⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⣇⠀⠀⡇⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣄⢀⡟⠦⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡿⣗⣿⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣗⢾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣭⠟⠂⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠘⢿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢷⡀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⣀⣠⣤⣤⣤⣤⣦⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⠿⢻⠟⠉⠉⠉⠙⠛⠿⠛⢛⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣳⣾⣷⣿⠟⡡⠚⠁⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠟⠭⠓⠉⠀⠀⠀⠀⠀⠀⠐⠾⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠛⠙⠛⠻⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠀⠀⠀⣀⣠⠶⠾⢿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡫⠄⠀⠀⣀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣧⠲⣄⣀⣠⣴⠞⣁⡤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢋⠁⠀⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣄
⣀⣀⣀⣀⣤⣤⣶⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⢿⣿⣿⣿⣿⠿⣿
⣿⣷⣾⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⣂⣠⣄⡀⠀⠀⠀⠀⠀⠐⠒⠈⠉⠈⠀⠀⠀⠀⠀⠀⠈⢿⡟
⠿⠟⠛⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠵⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟

⠀⠀⠀⠀⢴⢖⣴⣶⠀⠀⠒⠀⠀⠀⢀⣀
⣠⣤⣴⣦⡾⣆⢻⠟⠀⠀⣠⣤⣬⣶⣿⣿⣿⣶⣶⣤⠄
⠉⠙⡿⠿⣦⡷⠄⠀⠀⣤⣽⣿⣿⣿⣿⣿⣿⡯⠙⠁
⠀⠀⠙⠿⠛⠀⠀⠀⢠⣿⣶⣿⡿⢿⡿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠸⣟⣢⠀⠀⠉⣿⣿⠁⠀⠀⠑⢂⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠙⠙⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠒⠒⠂⠀⠀⠀⠀⠀⢀⣀⢀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠀⠉
⠀⠀⠀⠀⠀⢀⡀⢀⣄⣀⢀⡀
⠀⠀⠀⠀⠀⠾⠿⢽⠷⠿⠾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡆⠀⠀⠀⣀⣀⣀⣀⣀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠘⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⢪⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡲⠤⠀⣒⡣⠀⢐
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡻⠇⠀⠈⠭⣓⠀⡩
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣄⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠻⡟⠲⠶⠤⣄⣀⣀⣠⠀⠀⠐⢣⡄⠦
⡐⠊⠢⡀⠀⢡⡌⠛⢬⣶⣶⡄⠀⠀⣄⠀⠐⠣
⠁⠈⣀⣽⣇⠀⠘⠁⠀⠃⠀⠂⠀⠀⠨⠐⠷⠁
⠂⠒⠰⠲⠀⠃⠀⠀⠀⠀⠈⠃⠀⣆⣔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠉⠄⢐⠂⠀⠠⣤⠀⠀⠘⠃⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⣀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠎⠣⠄⠀⠀⠀⠀⠀⢀⣼⠟⠛⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠐⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⢀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠐⠢⡈⢣⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣷⠀⠀⠀⡀
⠀⠉⠃⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣤⠀⣸⣿⣿⡇⢀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣤⣤⡿⠀⠀⢻⣧⡇⢳⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⢿⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠿⢿⣿⠃⠀⠙⣻⣮⣿⡘⣼⠆
⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⡀⡇⠀⢀⡇⣀⢀⣀⣠⡄⠀⠀⢀⢀⣀⣀⢀⣿⡇⠀⣀⣀⣀⠀⠀⣀⠀⢀⣨⢙⠀⣀⠀⢁⡀⠀⢰⣿⣿⣿⢧⣿⠀⣀

⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣶⣾⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠿⣿⣿⣿⡿⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠴⢿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡆⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⣤⣴⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡦⡷⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⡷⠀⠀⠀⣿⡂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⡷⢶⣶⣶⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠛⠛⠿⣯⠀⡇⠀⠀⠀⣿⣿⣿⠿⠟⠛⠛⠿⣧⠀⡇⠀⠀⠀⣽⠋⠁
⠀⢠⣶⣿⣁⣀⣀⣿⣿⣿⣷⣾⣿⣶⣶⣿⣤⣷⣶⣶⣴⣿⣿⣿⣷⣾⣿⣶⣶⣷⣤⡷⠀⠀⢠⠿
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⠿⠟⠛⠉⠈⠁⠉⠉⠛⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠺
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⢟⠝⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠪⣻⣿⣿⣿⠿⠿⠿⠿⢇⡀⢸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⠃⠀⠀⠀⠀⢀⣴⣿⣶⡆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶⣶⣶⣾⣧⡼
⠀⣿⣿⢿⠛⠛⠛⠛⢻⣿⠃⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠰⣿⡉⠉⢹⡟⠉⠉⣿
⠠⣿⣿⣿⣷⣷⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠰⣿⡇⠀⢾⡷⠀⡰⡿⣆
⠀⠙⠿⣿⣿⣿⣿⡿⢻⣿⡄⠀⠀⠀⠀⠀⠘⠠⣟⣿⠠⠇⠀⠀⠀⠀⠀⢠⣿⡧⡀⣸⣧⠞⠁⠿⠈⠳
⠀⠀⠀⠀⠀⠈⠉⡄⠀⢻⣶⡀⠀⠀⠀⠀⠀⠐⠿⠿⠀⠀⠀⠀⠀⠀⢀⣷⡟⢈⣮⠛⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⣞⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⠟⣴⠟⢉⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠀⠈⠻⣷⣢⠄⣀⠀⠀⠀⠀⢀⢠⣰⣽⠟⠁⠚⠋⠀⣰⣦⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠾⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢠⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⡇⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣎⣤⠀⠀⢾⣿⢿⣿⣿⣿⡟⣩⣷⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⣿⡷⠆⠁⠀⣼⣿⣿⠉⡁⢩⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠈⠂⠓⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣄⡀⣿⠙⠛⠀⠀⣾⡿⠉⠂
⠀⠀⠀⠀⠀⠀⠰⠆⠶⠴⠰⠲⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣟⣻⣤⣤⡀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢻⣿⢿⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⢿⠏⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⡔⠂⠄⣀
⢿⡷⠁⠀⠀⠀⠉⣲
⠀⠀⠀⠀⠀⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠒⠒⠛⠋⠉⢿⣠⣄⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢉⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠿⠇⠀⠈
⠀⠀⠀⠀⢀⡶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠛⠓⢦⣄⣧
⠀⠀⠸⢧⡄⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠛⠉⠉⠛⠒⠦⣄⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⣾⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⡏⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠈⡇
⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⣇⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⢸
⠀⠀⠀⠀⢻⣄⣤⣄⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⣾⠁⠄⢀⠈⠁⠀⠀⠀⠀⠀⠐⠒⠋⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠓⠶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⠀⠀⡠⠖⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠿⠟⢿⡆⠀⠀⠀⡼⠃⠰⣯⠛⠄⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⢀⡽⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢾⡯⢴⢶⣤⠶⠋⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠉⠉⠛⠶⠶⠶⠖⠛⠛⠛⠉⠉⠀⠀⠈⢷⠀⠸⡇⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧⡐⢄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢹⡶⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠻⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⣦⡼⠚⢷⡶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠋⠉⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠷⠃⠀⠀⠈⠳⣦⠀⠀⠀⣀⣀⣀⡀⠀⠀⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢀⡀⢀⣈⣹⡅⣀⡀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡖⠒⠒⢻⠗⠒⠒⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡅⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⢀⣀⣀⣀⣤⣄⣠⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣶⣶⣶⣶⣿⣦⣤⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡏⠉⢹⡋⠉⠀⢙⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠉⠲⠤⠼⠦⣄⣸⣬⣭⣿⣯⣿⠟⡋⠉⠋⢉⠠⠤⠀⣂⡀
⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⣩⠏⠁⠀⣀⣄⡠⠔⠁⠀⣠⣾⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⢀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⡟⠉⠉⠀⠀⠀⠀⣰⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠀⠀⠀⠀⠀⠀⠀⢿⠁⡀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⠏⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣿⠏⢿⡇⠀⠀⢻⡟⢻⣇
⣀⣀⣿⣀⣠⣧⣀⣠⣀⣀⣠⣀⣤⣤⣤⣤⣤⣾⣿⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣤⣼⣿⣤⣤⣼⣷⣤⣿⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠈⠉⣉⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠐⠒⠚⠿⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⢰⣿⡅⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠛⠉⠛⢷⡄⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠐⠒⠂⠉⠉⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠻⣄⠀⠀⠀⠶⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⢠⠀⢳⡀⠀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⢨⠿⠦⠤⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣤⣄⣀⣀⣀⣀⣀⢠⣤⠔⠋⠁
⠀⠀⠀⠀⢸⠀⡀⢢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⢸⠶⣽⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠄⣸⠀⠈⠙⣦⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠚⡏⠙⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⡜⠁
⡄⠀⠀⡆⢹⠖⠒⠒⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠀⠀⠀⡇⠀⠀⠀⠈⠉⠲⠦⣤⡀⠀⠶
⢧⡀⠈⠉⢸⠆⠀⠀⠀⠈⠳⢦⠀⠀⠀⠀⠀⠀⣀⠴⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢦⣀⣀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠙⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠀⠀⠀⡘⠃⠀⠀⠈⠉⠛⠛⠢⠤⣄
⠀⠀⠀⠀⠀⣿⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠙⡄⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡇⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⣆⠀⠀⡗⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠻⢧⣄⠐⠒⠈⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠈⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣀⣰⡆⠀⠈⣧⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠳⣤⡄⠄⠒⠠⠄⠘⢷⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠀⠀⠈⣿⠀⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠙⠠⠤⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠛⣦⣤⣤⡔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠀⠈⣤⠁⠀⠀⠀⠀⠀⠀⡍⠀⠀⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠘⠁⠈⠁⠈⠀⠈⠉⠀⠈⠉⠁⠈⠈⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣠⣤⣤⣤⣤⣤⣀⡀⠴⠖⠒⠒⠶⢦⣤⠀⠖⠒⠒⠖⠢⣄⡀⠰⠶⠒⠶⠶⣤⣀⡄⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⠃⠀⠀⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⡜⢁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⣸⡆⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⢀⡆⠀⡄⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠸⡇⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⢀⠀⠀⣦⠀⠀⠀⢠⠀⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⢿⠀⠀⠀⠘⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢰⡀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⢩⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⢸⠁⠀⠈⠀⠀⠀⠀⠇⠀⠀⠸⡇⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⡄⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⢁⠀⠀⠘⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⠇⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⠀⠀⠀⠃⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⡄⠀⠀⠀⠀⡆⠀⠸⠀⠀⠀⠀⠸⠀⠀⡆⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⡇⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⢄⠀⠇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⠀⣀⡿⢧⣀⠀⠀⠀⣀⡿⢆⣀⣀⣀⣀⡠⠘⠛⠲⠶⠶⠶⠖⠉⠛⠑⠶⠆⠉⠁⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠀⠀⠀⠈⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠶⡄⠐⠂⠀⠀⢲⡆⠐⠀⠀⠒⠒⠒⢶⠀⠐⠂⠐⠒⠀⠒⠀⠒⣤⠔⠂⠀⠀⠀⠀⢠⢰⣧
⣤⣤⣀⣤⣤⣤⣄⣠⣤⣤⣄⣤⣸⣷⣤⣷⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣾⣧⣠⣤⣤
⣿⣿⣿⣿⠿⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⣴⣦⣽⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠁⠘⠛⠻⣿⣿⣿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡⡜⣿⣿⣷⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⡦⠼⠗⠉⣀⠉⠓⠌⣙⢣⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⣈⣭⣥⡄⠀⠿⠿⡛⠆⢤⡉⠉⠙⢻⣧⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠐⢊⣿⣿⡄⢠⡀⠈⠳⠀⠀⠀⠺⢿⣿⣿⡇
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⣹⠿⢁⣿⣷⡀⠀⠀⠀⠀⢀⣼⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠀⠀⠀⠚⠁⠀⡸⢹⠟⠰⣧⠀⠓⠈⠀⠀⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⣈⣁⣀⣤⣴⣿⢿⣿⣾⢟⡛⠾⠦⠄
⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣻⣿⣿⠟⣫⣕⣿⡿⣻⣿⣷⣈⣂⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠁⠀⠈⢙⣿⣿⣿⣟⣉⢙⣋⣉⣿⣿⣷⣿⣿⣿⣿⣧⣿⣿⣿⡟⢿⣧⣼⡾⠛⠙⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠵⢚⣩⣭⣿⣿⢿⡿⢟⣿⠿⠛⠙⠛⠉⠃⠈⢡⡤⠈⠙⢧⡀⠄⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢊⡽⠁⡠⠶⠒⠀⠁⠀⠀⠁⠀⢀⡴⢂⣆⡤⣧⣞⡀⠀⠀⠀⠙⠦⣔⠀⠀⠀⠀⢝⠢⠀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠏⢤⠈⠀⠲⡄⠀⠒⢿⣶⣿⣷⣿⣾⣯⣽⣿⠏⠈⡇⠀⠀⠀⠀⠀⠙⢳⣤⡀⠀⠈⢦⠀⠙⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠂⠀⠀⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠤⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠤⠤⢤⡼⣅⡀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠹⡱⡀⠀⠀⢠⣤⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡀⠤⠄⠀⠀⠀⡴⠁⠱⡀⠀⡾⠈⠃⠀⠀⠀⢱⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⡠⠀⣀⠀⢞⠀⠀⠀⠱⣀⠇⠀⠈⠀⠀⠀⠀⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠢⢶⣷⣠⡄⠁⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠤⠬⢿⠀⣀⠈⡀⠉⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢉⡖⠀⠀⠀⠀⠄⠈⡆⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠣⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠⠦⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠐⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠉⣰⠔⠀⠀⠀⠀⠑⡄⠀⠀⠀⠲⠏⠀⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⢎⣠⡤⠤⢠⣤⡄⢀⠐⠀⢀⠀⠀⢀⡜⠹⣇⣉⣀⣁⣀⣀⣈⣂⣤⠴⠦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⢮⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠠⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣜⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡀⠀⠀⡀⠀⠄⠀⢠⠀⠀⠀⠈⢳⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡤⠀⠀⠀⠀⠀⠀⣢⠄⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠀⠉⠲⠀⠐⠚⠀⠀⠈⠀⣄⠀⠀⠀⠙⢄⠀⣿⣀⣀⡀⣤⠤⠖⣋⣵⢊⢿⣶⠀⠀⢀⠄⡑⢉⣄⣰⣈⠱
⣘⠀⠐⣈⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡤⠀⠈⠢⢀⠀⠀⠀⠳⡇⠀⠀⠀⢨⠄⢚⣿⣿⡷⣔⣊⢉⠒⠀⠚⡋⢟⣿⣿⡥⠄
⠙⠐⠂⠪⠛⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠁⠀⠀⠈⠉⠂⠂⠈⠀⠀⠀⠀⠀⠓⠥⣨⡀⠤⠋⠀⠈⠄⠀⠀⠙⢬⣌⣁⣩⠞
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠈⠈⠁⠀⢐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠤⣤⡤⢴⣾⣇⣠⡶⢾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡠⠶⠿⠿⣟⣿⠿⡦⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡎⢼⠀⠀⣄⠀⠀⠸⠀⠰⠀⢠⡀⣀
⠀⢐⠀⠀⠀⠀⢠⠀⠀⠀⢐⠡⠀⠀⠠⡀⠀⠀⠀⠱⠈⢏⠒⠀⠀⢦⠀⠀⠚⠉⢠⡀⡐⠄⠉⣀⣀⣈⠁
⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠸⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼⢷⢶⡶⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠤⡖⠀⠀⠀⠐⠖⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠈⠐⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣟⣛⣛⣛⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠛⠛⠛⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠳⣮⡟⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣿⢉⣤⣤⡀⠀⠀⠀⠀⠀⠀⢘⣻⡆⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠙⠋⣴⢮⣷⠀⠀⠀⠀⠀⢠⣾⣏⣤⣀⣀⣀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣉⠉⢹⡇⠀⠀⠹⡆
⠀⢀⡶⠛⢷⠀⠀⢸⣇⢰⡞⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠉⠉⠻⣎⠛⣷⣠⣄⣿⠀⠀⢰⠟⠳⣆
⠀⣼⠁⠀⢸⡇⠀⣾⠻⣮⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⣤⡈⠀⠈⢹⡆⠀⣿⠀⠀⢹⡆
⠀⡟⠀⠀⢸⡇⠀⢹⡄⠀⠈⢻⡏⠛⠛⢦⡄⠀⠀⠀⠈⠛⠶⠟⣦⠀⠀⠀⣠⡞⠁⠀⠀⣸⠁⠀⣿⠀⠀⠘⡇
⢸⡇⠀⢀⡸⠇⠀⠘⣇⠀⠀⠸⣇⠀⠀⠀⠙⠛⣻⠄⠀⠀⠀⠀⢸⡇⠀⢀⡿⠀⠀⠀⢀⡟⠀⠀⠟⣀⠀⠀⣿
⢸⡇⢠⡟⠙⢷⡀⠀⢹⣆⠀⠀⠙⠳⣆⠀⠀⣀⡿⠀⠀⠀⠀⠀⢸⠀⢠⡟⠁⠀⠀⢀⣾⠁⠀⣰⠟⠙⣧⠀⣿
⣼⠀⠈⣧⠀⠀⠻⣆⠀⠹⣦⠀⠀⠀⠹⣦⣸⠋⠀⠀⠀⠀⠀⠀⠿⠶⠋⠀⠀⠀⢠⡾⠁⢀⡾⠋⠀⢠⡏⠀⢸
⣿⠀⠀⠘⣦⠀⠀⠈⠳⣄⡈⠳⣄⡀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣀⡴⠋⠀⠀⢠⡟⠀⠀⢸
⢿⡀⠀⠀⠈⢷⣄⠀⠀⠈⠻⣦⡈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⣠⡾⠋⠀⠀⢀⣴⠏⠀⠀⠀⣸
⠈⠻⣦⡀⠀⠀⠙⣷⠀⠀⠀⠈⠻⣆⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⢀⡾⠋⠀⠀⠀⢰⡟⠁⠀⠀⣠⡾⠋
⠀⠀⠈⠻⣆⠀⠀⠘⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠙⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠻⠀⠀⣀⣀⣀⣀⣠⣬⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣤⣀⣀⣀⣀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⢸⡛⠛⠋⠉⠉⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠉⠉⠛⠛⢻
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣀⣀⣀⣤⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣄⣀⣀⣀⣀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠠⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠻⠛⠓⠚⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠊⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣶⣾⣶⣶⣶⣶⣶⣦⣤⣤⣶⣶⣦⣤⣤⣴⣶⣶⣴⣶⣿⡷⠾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣷⣶⣿⣷⣶⣦⣿⣿⣤⣤⣤⣴⣿⡷⣤⣾⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⣷⣿⣿⣿⣿⣿⣿⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⣰⣿⡏⠀⠻⣿⠏⣩⠝⠉⢹⣿⠉⠙⢻⣿⣙⣿⣿⣿⣿⣿⣬⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠃⢹⣿⡇⠀⠘⣿⠇⠁⠀⠀⢸⣿⠀⠀⠽⢿⣩⡿⠟⢹⣿⣿⣯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠸⣿⡇⠀⠈⣿⠀⠀⠀⠀⢸⣿⠀⠀⢼⣿⣤⠴⠚⢹⣿⣿⢋⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠄⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⢸⡿⠀⠀⣸⡟⠀⠶⡶⢼⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡤⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⢸⣇⠀⠀⢹⣿⠒⢶⡦⢸⣿⡿⣦⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣧⠀⣿⡀⠀⠀⠿⡇⠀⠀⠀⢹⡿⠀⠀⢸⣿⢰⠞⠀⣼⣿⡇⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢉⣿⠀⠀⢰⡇⠀⠀⠀⢸⡇⠀⠀⢸⡟⢀⣿⣴⣿⣿⣤⣮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠸⣿⡄⠀⢸⣧⠀⠀⠀⣾⡏⠀⢀⣿⣿⣋⣿⣿⣿⣿⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢹⣷⠀⢹⣷⠀⠀⠀⣿⡇⣤⣸⣿⠏⣛⣩⣿⡟⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠸⣿⡄⢨⣿⡅⠀⠀⣿⣿⡂⣾⣿⣰⣿⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣹⣷⣤⣽⣧⣤⣤⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡟⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠻⢶⣿⡏⠹⢿⣿⣿⣿⣿⣏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠤⠀⠉⡿⠟⢿⣿⣿⣟⡵⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⠀⠀⠀⠉⠀⣛⣛⣿⣿⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠳⠛⠶⠶⠶⠦⠾⣿⣿⣿⣿⣿⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡤⠰⠿⣿⣿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⣃⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⠟⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣿⡿⢂⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠺⣟⣿⣏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠻⣶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣧⣶⣿⣿⡿⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠙⠯⣹⣿⣿⣶⡁⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢲⣤⣿⣯⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠈⠋⠐⠲⠛⠋⠹⠟⣻⣧⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠶⢦⣤⣬⡥⠾⠿⠟⠛⠉

⠶⠚⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢌⡑⢌⣿⣿⢷⣝⡡⡢⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡤⠶⢛⣉
⠞⣋⣩⡄⠀⠀⠆⣠⠆⣠⠆⣠⠦⣹⠦⣹⠦⣽⣿⣻⢮⣼⠔⣡⡦⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⣋⣥
⣛⣩⣤⠦⠀⠀⠀⠁⠞⣡⡾⣍⡷⣍⡶⣍⡶⣿⣿⣝⡷⣡⣾⢋⡴⣿⣿⣳⣀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣏⣭⣴
⣏⣥⠶⠞⡀⠀⠀⠠⡘⢮⡽⢯⡵⢯⡵⢏⡵⢿⣿⣯⣿⢟⡵⢫⣾⢿⣿⣿⡿⣿⣽⣹⣱⣱⣻⣹⣻⣿⣿⣿⣿
⣽⣟⣾⣿⣅⡠⣌⢳⣌⣲⡌⣫⣽⣯⣽⣯⣽⣯⣹⣮⣱⠊⣠⠊⣠⢟⣛⣻⠑⠳⠳⢣⢣⢣⢣⢧⢯⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⣿⢿⣷⠿⣫⠞⣡⠞⣡⣟⣻⣿⠁⠀⠀⠀⠀⠀⠉⠋⢾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠞⢁⠜⢁⠜⢁⠔⢁⣟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⢋⡴⢋⡴⢋⣹⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠋⡠⠋⣹⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠢⠢⠘⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠞⣩⣭⣽⣄⣀⣀⣀⣀⣄⣄⣤⣄⣤⣌⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣩⣽⣿⣿⣿⣿⢿⠝⠿⠿⠟⠽⠽⠿⠝⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⢰⡄⢻⣄⢿⣿⣿⡿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠀⣶⠈⣷⡈⣷⡈⢷⡈⢷⣬⣛⢿⣿⣿⡿⣫⠿
⠀⠀⠀⣀⠠⣄⠰⣦⠸⣇⢸⣧⢸⣧⡸⣧⠘⣧⠘⣧⠘⣧⣸⣽⣻⢿⣿⣿⣟⣷⠇⠀⠀⠀⠀⠀⠀⢀⡤
⡄⢻⡄⢻⡄⢻⡆⢻⡆⢹⡆⢻⡇⢹⣇⢹⣆⠻⣶⠹⣇⣽⣾⣝⡻⣷⣾⣿⣯⡞⠀⠀⠀⠀⠀⠀⢰⣿⠶⢿⣏
⢿⡈⢿⡀⢿⡀⢿⡀⢿⡄⢿⡄⢻⡄⢻⡄⢻⣾⣿⣶⣿⣿⣿⣿⡿⣿⣯⣿⡻⡝⣆⢧⢳⡸⡎⣷⢮⠶⢺⣋⣿
⡘⣧⠘⣷⠘⣷⠘⣷⠈⣷⠈⣷⡈⣷⡈⢿⡈⣿⣿⣿⣿⣿⣯⣿⢿⣿⣿⣿⢧⢹⡸⡌⣇⢧⠹⠘⠶⠞⣋⣻⣿
⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⢻⡹⡜⣇⢧⠹⠸⠜⠃⠃⠉⠀⠀⠀⠀⠐⢛⣩⡤⠶
⣭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⡟⠆⠃⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⣥⠶⠞
⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⠢⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⠴⠞⣛
⣶⣚⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣌⣿⣯⣷⣽⣣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣴⣖⣛⣩
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⣄⣐⣲⣶⣶⣶⣶⣶⣶⣖⣂⣀⣤⣄⠀⣀⡀
⠀⠀⢀⠔⢀⠀⡔⠢⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣢⣬⣷⣾⣿⠿⠿⠿⠛⠛⢛⣛⣛⣛⣛⡛⠛⠛⠻⠿⠿⢿⣿⣾⣯⣴⣢⢄⡀
⠀⠀⠈⡗⠜⠀⢻⠐⠀⠀⠀⠀⠀⡠⢒⣵⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⢸⡿⠿⢿⣿⠿⠿⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣮⣕⢄
⠀⠀⠀⡇⠀⠀⣼⢀⠀⠀⠀⠀⡜⣵⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠈⠙⠻⠿⠬⠿⠛⠋⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠉⠻⣧⢱
⠀⠀⠀⡇⠀⠀⣿⢸⠀⠀⠀⢠⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⠦⣤⣀⣤⡴⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢆
⠀⠀⠀⡇⠀⠀⣿⢸⠀⠀⠀⡇⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡶⠶⠛⠋⠉⠈⠉⠛⠳⠶⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⢘
⠀⠀⠀⣇⠀⠀⣿⢸⠀⠀⠀⠸⡉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠋⢉⡏
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⢰⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡆⠀⠀⠀⢀⣴⣿⡄
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⢸⢸⣿⣿⣷⣦⣤⣀⡀⠉⢹⣯⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⢵⡏⠉⠀⣀⣠⣴⣾⣿⣿⡇⡇
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⠘⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣦⣤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃⡇
⠀⠀⠀⣿⢀⠀⣿⢸⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣟⣗⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⠱⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡜
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⠀⠑⢄⡀⠙⠻⢿⣿⣿⣿⣿⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⡿⠟⠋⠁⡠⠊
⠀⠀⠀⣿⠸⠀⣿⢘⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠈⠉⠛⣻⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⠛⠉⠁⣀⡠⠔⠋
⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢭⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡭⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠐⠄⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⠀⠀⡀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⢱⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠠⡆⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢠⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠄⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠄⠀⢸⡧⠀⣠⣄⠑⡄⢃⠈⠀⣀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣄⣾⠟⠋⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⠀⠀⢸⠀⠠⠛⠁⠀⠀⡌⠀⠈⠀⠀⠀⠉⠉⠛⠂⠀⠀⠀⠆⠀⠀⠀⠀⢱⡦⠉⠙⠂⠀⠀⠀⠀⠈⠁⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠠⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⣀⣾⣏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠐
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⠶⠲
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⡌⣀⣀⠐⠦⠦
⠀⠉⠈⠀⠴⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠂⠾⣤⡀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠠⡤
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⣿⣿⣿⣿⣿⡟⣛⢻⡿⣿⠻⡿⢿⣿⢻⡿⣿⡿⢻⣿⡟⢻⣿⢿⡟⣿⢿⡿⣿⠿⡟⣿⡟⣛⡟⣿⣿⢿⠟⣿⢿⠻⢿⡿⢿⡿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⢿⢀⢨⢼⢸⢺⠀⠀⠀⢸⠄⢸⢸⣿⢐⡘⠃⡆⡇⠀⡆⡇⠓⡇⡅⣿⡷⡌⠃⡱⡇⡥⡄⠰⡶⢰⡆⢨⠤⢌⢺⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣶⣿⣶⣾⣶⣶⣾⣿⣾⣷⣾⣶⣾⣿⣾⣷⣾⣷⣷⣿⣷⣿⣷⣶⣷⣿⣷⣶⣷⣿⣷⣾⣷⣷⣾⣿⣿⣷⣾⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣇⣀⣀⣿⣃⣈⣇⣇⣼⣌⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢹⡿⠍⠉⠉⡏⠉⠉⠉⣩⠍⠉⡉⠉⠙⠋⠉⠍⠉⠉⠉⢹⢻⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⣮⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⢞⣃⠀⠀⡇⠀⠀⢰⣷⣖⠲⣧⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠀⠀⠀⠀⠀⠀⠶⠿⠂⠀⠄
⠀⠀⢘⠓⢰⡆⠈⠁⠀⠀⢀⡀⠐⢪⣙⠀⠀⡇⠀⠀⠈⠉⣻⢰⣿⡄⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⣤⣤⣤⠄⣀⢰⣇
⠀⠀⠀⠀⠀⢻⡄⠀⣄⣼⢩⠖⠂⣨⡿⠀⠀⡇⠀⠀⠀⢀⣻⢮⣭⣉⠻⣷⣄⠀⢀⡀⠀⠀⢸⠀⠠⠁⠒⠲⠿⢧⠴⠎⠋⠐⣾⣆⣨⢿⡀
⠀⠀⠀⠀⠙⠈⢻⣄⢻⠇⠈⠀⠚⠋⠄⠀⠀⡇⠀⠀⠀⠸⣷⠬⣹⣿⣷⣜⣿⡆⡞⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⢾⠏⠁⠉
⠀⠀⠀⠀⠂⠀⠘⠇⠀⠀⠀⠀⠐⠛⠀⠀⠀⡇⠀⠀⠀⠀⠙⣿⡿⠾⠛⠋⠉⣀⠀⠀⠀⠀⢸⡤⠶⠒⠋⠀⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⡀⠈⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⣇⠀⠀⠀⣀⡀⠀⠙⢳⣤⣀⡀⠀⡶⣦⠀⠀⢸⠀⢀⠀⣀⠀⣠⢀⡈⢱⣄⠀⠀⠀⠀⠀⣘
⠉⠉⠉⠉⠉⡉⠉⢉⣉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⣉⡍⢩⠉⣉⠉⠉⠉⠉⠉⠉⠉⢹⠉⠉⢉⢉⣉⣿⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠋⢾⣿⣮⡳⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⡀⠂⠘⠉⠰⣤⣀⠀⠀⠀⠀⢸⠀⢀⣯⣭⣍⣛⠀⠀⠁⣄
⠀⠀⣰⣾⡀⠿⢯⣾⣿⣄⣴⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣘⡻⣏⣹⣧⣤⣄⡜⠓⢠⡢⠄⠀⢸⠀⢸⣷⣈⣉⣽⡷⠈⠀⠘⣷⡄
⠀⠀⠧⠸⡀⠄⠀⢀⣅⣘⡿⠀⠀⠀⠀⠀⠀⡇⠀⠖⠺⡟⠀⢈⠹⣿⣷⠋⠁⠀⠐⠂⠀⠀⢸⠀⡾⣯⣩⣿⣿⡧⣠⣶⠿⣿⣿⣦⡀
⠀⣇⡇⠀⠁⠀⠀⠀⢉⡉⣵⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⣷⠈⠀⠘⠃⠀⠀⠀⠀⣰⡇⠀⢸⠀⠀⠉⢿⣿⣿⠁⠏⠃⠀⠀⢿⣿⢧⠀⢲
⢸⣅⠉⠢⠀⠀⠀⠖⠀⠂⠹⡃⣤⣄⡀⠀⠀⡇⠀⠀⠀⢻⡘⠃⠀⠈⠀⠀⠀⠀⠈⣭⣢⠀⢸⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⣸⡏⣠⣤
⠸⠐⠆⠀⠀⠀⠠⠆⠈⠀⠀⢠⡟⣀⡻⣦⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣧⣣⣼⠀⠀⣶⣇⠀⢸⠆⠀⠀⠀⠀⢹⢃⣿⣯⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢉⡆⠀⠀⣎⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡷⠀⠀⢾⢷⢦
⠀⠀⠀⠀⠀⠀⠀⠸⠓⢒⣁⡀⠀⠀⡠⠔⠚⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢛⣁⡀⠀⠀⢠⣤⣶⣇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢇⡀⠉⢦⠀⠘⢶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢺⣿⣿⣧⠀⠈⠻⣿⡟
⠀⠀⠀⠀⢸⠹⡀⠈⢆⠀⢹⡰⠃⠀⠀⢀⡷⠁⢰⢉⡆⠀⠀⠀⠀⠀⣷⠀⠈⢦⡀⢸⣿⠃⠀⠀⠀⡹⠁⠀⣾⠇
⠀⠀⠀⠀⠘⡄⢡⠤⡀⠑⠢⠤⠤⠤⠒⠉⣠⠤⡌⢸⠀⠀⠀⠀⠀⠀⣿⣦⣤⡀⠉⠂⠤⠤⠤⠔⠊⢀⣠⣤⣿
⠀⠀⠀⠀⠀⢣⠀⠱⡌⠓⢄⠀⠀⠀⡴⠊⢡⠊⢀⠎⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣄⠀⠀⠀⢠⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠢⡄⠀⢸⡀⠀⢀⠁⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠀⠀⠀⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⣉⡱⠀⢏⣉⣚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡶⠀⢴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢀
⠀⠀⠀⠀⢰⣶⣶⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠓⠼⣀⡄
⠀⠀⠀⠀⢘⡉⠉⣁⣸⠀⠀⠀⠀⠀⠀⠀⢠⢊⡔⠛⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠸⣿⠇⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠚⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⣠⠿⢄⠀⠀⠀⣀⣠⣔⣀⡀⠀⠀⠀⢠⠒⠩⠿⠔⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⢻⣱⠊⠀⡄⠀⠈⠈⠋⠀⠀⠀⠀⠈⠉⠉⠉⠁⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⡎⠁⣀⡼⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠉⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⠷⡄⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⡎⠀⠀⠀⣄⣀⣀⣀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡠⠃⠀⠀⠈⠢⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢸⠀⠀⡘⠁⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠙⢄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠖⠋⠀⠀⣧⠄⣀⠠⠤⠤⠒⠒⠂⠉⠉⠹⡀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠑⠒⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠲⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠺⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⢿⣤⠤⠤⣤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣤⣖⡺⠭⠷⠶⠟⠉⠻⣏⠉⠉⠉⠉⠍⠛⢛⠚⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠛⠿⢦⣄⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠈⠙⠲⢄⣀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣦⣬⣑⠲⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⢀⣠⣴⣾⠟⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢯⣾⠛⠿⢯⣿⣒⣲⠦⠤⠤⢤⣤⠤⠤⠤⠤⣖⣒⣾⡭⢷⡏⣿⡴⢛⣥⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢘⡏⠙⠳⠶⣤⣌⣉⣉⣉⠙⢻⢸⡛⠋⣉⣉⣉⣤⠧⠿⠛⢻⡃⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠉⠩⠭⢹⢻⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢞⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠖⣦⠀⢐⢂⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣽⣠⠴⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠶⣏⡛⠲⠦⣭⣔⣀⡀⠀⠀⣠⣄⠀⠀⠀⠀⣀⣐⣲⠭⠷⢿⣀⡷⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠥⣭⣩⣿⣿⣇⣸⢟⣛⣋⣉⣉⣤⠤⠴⠒⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠋⠉⠉⠉

⠀⠀⠀⠀⠀⣀⠤⠒⠂⠈⠉⠉⠉⠑⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠉⠉⠉⠉⠒⠦⣀
⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠒⢄⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⡯⠉⢳⡀⣱⣄⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⡌⠀⠀⠀⠀⠀⠀⠀⢰⠉⠹⣶⡄⠀⢸⡏⠀⠸⣿⣿⣿⣧⣶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢹⣾⣿⣇⠀⢹⣿⠛⣿⣿⣿⣿⣷⣤
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢸⣿⣿⣿⣦⣘⣿⡄⢿⣿⢋⣶⣿⠿⣷⣶⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣷⣿⢩⢻⡿⣿⣿⡧⠌⣟⣿⡟⢁⣤⠾⢻⣿⣦⣤
⢡⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣾⣿⡏⣧⢸⠏⣷⡌⢹⢇⣶⡿⢸⡼⣿⣋⣴⣿⣿⠿⣿⣧⣄
⠈⡄⠀⠀⠀⠀⠀⠀⣠⢿⣿⡟⢸⡇⠇⠟⡦⡿⣸⣦⠸⢹⢳⡿⢰⣿⢹⣿⠋⢠⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠘⡄⠀⠀⠀⠀⣼⢃⣵⣾⠘⣼⣷⣶⣮⠁⣵⠛⡾⡇⡞⢸⢃⣿⠃⣸⡏⢹⣿⣿⠿⢛⣹⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠙⡄⠀⢀⡼⠡⣪⢟⢕⠟⢉⠛⢛⣣⣴⡇⢸⡇⡇⠁⣿⢸⡟⣰⡿⠁⣸⡟⢳⣤⣿⣿⠿⠛⢿⣷⣄⠀⠀⠀⢀⠄
⠀⠀⠀⠘⢄⢞⡵⠉⠑⢁⡵⠾⠛⠛⢋⡽⢹⣧⢸⡇⠃⠀⡇⣿⢱⡿⠁⣴⣿⠇⣼⣿⣟⠁⢀⣤⣶⣿⣿⡿⠂⢀⠎
⠀⠀⠀⣠⣿⣫⣀⣠⣞⣥⣶⣶⣶⠿⠋⠀⣾⣿⡄⣷⠀⣸⢃⣿⣿⠄⣼⣟⡞⢸⣿⠇⢘⣿⣿⣿⣿⠟⠋⠀⡰⠃
⠀⠀⣼⣿⣿⣿⣿⣫⠥⠤⣀⡉⠛⣿⡧⠀⣿⡇⢣⣹⠀⣿⣿⣿⠏⣼⠏⠊⣰⣿⢣⠀⣨⣿⣿⠟⠁⠀⠀⠔⠁
⢀⡼⣼⣿⣿⣿⣿⣿⡿⠟⢾⣿⡆⢸⡇⠀⣿⣇⣆⣿⣰⢏⣾⠏⣼⣳⠃⣰⣿⣿⡆⢰⣿⣿⠏⠀⠀⠀⠊
⣾⣿⣿⣻⣿⣿⣿⠿⢶⣶⣤⢸⡇⡼⡰⢠⣿⣿⣿⣿⢃⡾⠃⣼⣿⠃⡰⣿⣿⡟⢀⣿⡟⡏⠀⢀⠔⠁
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⡼⠑⠋⠁⠀⠀⠹⣿⠟⠁⣼⣿⠃⡔⣼⣿⡟⢀⣾⣿⢱⠁⡠⠃
⠀⠙⠿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠘⢄⠀⠀⠀⠀⠀⣿⠀⣸⣿⠃⣘⣾⣿⡟⠁⣼⣿⣏⡯⠊
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠘⣶⡟⢁⣴⣿⠫⠋⢀⣾⣿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠀⠉⠉⠛⠓⠒⠛⢛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠔⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣱⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣯⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣿⣯⣽⣯⣭⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣿⣿⣿⣿⣄⠀⠀⠐⣶⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣀
⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⢿⡿⢿⡿⠛⣯⠡⠘⠿⠿⢿⣟⣿⣿⣿⣿⡿⠿⠏⠙⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⠟⢻⣍⣠⣄⡴⠋⠀⠀⠐⡄⠹⣧⡀⢀⡠⢮⡂⠀⠀⠈⠉⣂⣠⠿⠛⠥⡀⡈⠻⠿⠿⠿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠉⠉⠉⠁⠉⠀⣿⠀⠀⠀⠀⠀⠈⠒⠛⢷⡇⢀⢀⡿⠀⠀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⢶⠶⣶⣄⠀⠀⠘⣿⡈⠻⣷⣠⣾⢁⡤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢬⠿⠷⣶⣄⠙⢿⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛

⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠛⢡⠜
⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣆⣸
⢻⣿⣿⣿⣿⣿⡿⠀⢀⣤⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⠟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠉⠀⣠⠿⠛
⠈⣿⣿⣿⣿⠏⠀⠐⠉⢀⣀⡀⠀⠀⠀⠀⢶⣯⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⢀⠎
⠀⠸⣟⠙⠿⠆⠀⠀⠈⠻⠷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⡴⠁
⠀⠀⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡠⠋
⠀⠀⢨⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡄⠀⢠⠔⠂⠀⠀⠀⠀⢀⠔⠊
⠀⠀⣼⣿⣿⣿⣷⡄⠀⠀⠀⠀⠙⢲⣶⣶⣾⣁⣠⣠⣴⠗⠈⠀⠀⠀⠀⠀⢀⠖⠁
⠀⣴⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡶⢁⠀⠀⢷⠀⠀⠀⠀⠀⠀⡰⠋
⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⡔⢀⣐⣦⣾⠀⠀⠀⠀⢠⠊
⢀⣼⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠻⢻⣿⣿⠄⠀⠀⡰⠂
⠈⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣦
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶
⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⡄
⠀⠀⠈⠉⢻⣿⡀⠀⠀⠀⠀⠰⠀⠀⠀⠠⠀⠀⠀⠀⠈⠻⠿⠀⠃
⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠈⠻⠿⢧⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄⠀⠀⠀⢈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣴⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⡏⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢠⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢸⠇⠀⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣤⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠉⣄⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡁
⠀⠀⠀⠀⠀⢀⣤⣥⣀⣠⣠⡆⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣄⣠⣥⣄
⠀⠀⠀⢀⣴⣟⣋⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡛⢻⣷⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⣛⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠄⡻⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠈⣿⠛⠛⠛⠉⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠉⠛⠛⠛⢻⠇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣡⣤⠐⠒⠒⠒⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⢀⣀⣀⠠⠦⠶⠖⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇

⠀⠀⠀⠀⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⠀⠀⠤⡴⠣⠂⣤⣀
⠀⠀⠉⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠃⠀⠀⠀⠀⠔⠃⠀⢀⠼⠁⠀⠀⠀⠈
⠠⠄⡁⠁⠀⢸⠀⠀⠙⠀⠢⠴⠀⠀⠀⠀⠽⠀⢀⠦⡀⠀⠀⠀⣀⡤⠤⠂⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠂⢰
⠈⠀⠀⠀⠀⠘⠆⠀⠠⠀⠀⠀⠀⠀⠀⠐⢢⡄⠀⠀⠀⠠⠤⢐⡏⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠐⠁⠀⠈⠀⠉⠁⠀⢀
⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠⠎⠀⢀⠄⠀⠀⠀⡀⡠⠴⠒⠀⠉
⢀⠠⠄⠀⠐⠐⠖⠄⠄⠠⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡠⠀⠁⠀⠀⢠⠾⠀⠠⠟⠋⠋⡡
⠀⢀⢀⠀⠀⠰⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠐⠚⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⡤⠊
⠀⠃⠀⠀⠀⣀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⡐⠀⠀⡟⠀⠀⠌⠀⠠
⠀⠀⠀⢠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠐⠁⠀⠀⠀⠀⠀⡖⠁⠀⢀⠃⠀⠀⠀⠠
⠀⠀⢠⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⠀⠀⠀⠀⠃⠉⠀⠀⠂⠀⢀⠀⠀⠀⢀⣀⢳⡀⣠⠂⠠⡄⠠⠀⠤
⠀⠀⣬⣇⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⡸⡆⠀⠂⢓⠀⠐⠂
⠀⠀⠏⠀⠀⠀⠀⠀⠩⠃⢀⣀⠀⠀⠂⠀⠀⠀⠀⠉⢠⠈⠀⠀⠀⠀⠀⢀⠀⠀⠠⣤⠀⠀⠀⠈⠀⠀⠀⠀⢀⣶⡀
⠀⣸⠀⠀⠀⠀⠀⡰⠁⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠉⠱⡀⠀⠀⠀⠀⠈⣧⠔⠚⠉⠢⠀⠀⡄⣀⢀⠀⠑⠌⠀⠁⡀
⠀⠧⠀⠀⢄⠀⠞⠀⠐⠀⠀⠠⠀⠠⠀⠀⢁⠐⠀⠀⠀⠀⢠⡀⠈⣀⠀⠂⠀⠇⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠘⠣⣄
⠀⠀⠀⢠⠀⠞⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠈⠀⡀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⣤⠔⠂⠐⠃⠂⠤⣀⠀⠀⠂⠈⠉
⠀⡆⠀⠀⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠇⢀⠀⠢⡀⠀⠀⡤⠤⠰⠂⠀⡥⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠙⠒⠂⠤⣤⡀
⠀⢣⡀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣇⡀⡜⠂⣀⠀⠀⠀⢀⡀⣆⡀⢤⠠⠀⢀⣀⡀⠀⠀⠠⡁
⠀⠀⣧⢠⡅⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡘⠉⠁⠉⠙⢍⡀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⢹⠂⠆⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣟⠙⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣏⠏⣡⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⠙⠋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡄⠀⠀⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣾⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣸⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⣰⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⣿⣿⣿⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣵⡿⣿⣿⣿⣿⣿⣫⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⠆⢻⣿⣿⣿⣿⣿⣾⣿⣽⣽⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣇⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⢛⣿⣿⣿⣿⣿⣿⣿⣴⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⣿⣿⠻⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⡞⠽⢽⠿⢿⣿⣿⣿⣿⣿⣿⣼⣯⣿⣿⣿⣿⣿⣿⣡⣾⣿⣿⣿⡿⢹⣿⡿⢿⣿⢿⣾⠟⣙⡻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠻⢷⠛⣿⡟⠺⠒⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⣿⣿⣧⣿⣼⡇⠀⢾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠂⢺⣿⣏⢰⣦⣬⣿⣿⠟⣿⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣟⢀⣿⣝⣿⣿⣿⣿⣧⡀⢛⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣤⣿⡿⣿⣿⣿⣷⣾⣾⣷⣜⢿⣿⣿⣿⣿⣿⣿⣿⣯⡽⢽⡟⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡟⣿⣯⠹⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⢁⣿⣹⣧⢹⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣯⢿⣿⣿⣻⠋⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠁⣿⣿⣆⣻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⣸⣷⣿⣼⣴⡿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠠⣠⣿⢻⣿⡏⠸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣿⡿⢿⢹⡟⠳⣿⠸⡟⣿⣿⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⣸⣿⣿⣿⣿⡟⣿⣇⣾⠠⠀⣤⣿⠨⢾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡖⢸⣷⠀⣟⡄⣷⣿⣿⣿⣿⣿⣿⣿⠹⡿⢿⣿⣿⣿⠁⣿⠯⣿⣿⣿⢠⣿⠋⡄⢠⠀⣿⣷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⣯⣧⠙⣻⣿⣿⣿⣿⣿⣿⡇⢿⠸⣿⣿⣿⣧⣿⡌⢿⣿⡇⡜⣿⠉⣇⠸⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠀⠏⢻⣿⣿⣻⣿⣋⣷⠘⡄⢯⠘⣿⢻⣿⠃⡎⢻⡷⠃⠿⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠸⠯⠰⣿⠀⡇⠀⠄⢿⠀⠿⠐⠛⠊

⠀⠀⠀⠀⠀⠀⠀⠈⠲⣦⣌⣿⣿⣷⣶⣦⠤⢤⡄⠀⠀⢀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⠾⣽⣿⣿⣿⣿⣿⣧⣚⣲⢶⣶⣶⠶⡄
⠀⠀⠀⠀⠀⠉⠛⠷⣢⣤⣒⣠⣿⣟⠈⠧⡜⠉⡭⢿⣿⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠹⣋⠛⠶⣦⣾⡀⡿⠿⠃⣿⡿⣿⣯⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢳⣦⣤⣄⢀⡀⠀⣖⠀⠀⠗⠶⠃⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢹⠉⠛⠛⣫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠶⣷⠶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀⠁⠀⣲⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⢶⣶⣶⣤⡀⠀⠀⠁⠐⠢⣄⢀⠖⠈⠘⠛⠋
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡉⠉⠁⠀⠀⠀⠀⠀⠘⣏⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⢀⠀⠀⠀⠀⢆⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠑⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠠⣴⣧⡀⢈⣌⡀⠀⠀⢈⣴⠒⣀⢀⣈⣁⡀⠚⣷⣽⣛⡦⠒⢠⣦⡀⠉⠉⠙⠓⠀⣲⣇⣤⣠⢖⠁⠀⣈⣤⠋⡈⠀⢀⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠋⠀⠀⠈⠉⠀⠉⠉⠉⠙⠋⠀⠉⠙⠛⠋⠀⠈⠙⠋⠈⠀⠉⠀⠀⠁⠘⠈⠁⠘⠀⠚⠙⠛⠈⠁⠁⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢠⡿⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⣠⣴⢶⣤⣤⣤⣄⣀⠀⠀⠀⡾⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⢫⣾⡿⢣⣶⡙⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⣱⣿⣿⢡⣿⣿⣷⠸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⣼⣿⣿⣿⡜⢿⣿⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣠⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⣭⣙⡛⠟⠼⢿⡿⠛⠿⣿⣶⢠⣾⣿⠟⠛⣿⡟⡛⠿⢛⣻⣿⣿⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⢰⣿⣶⡆⢹⣷⣦⡙⠘⣫⣶⣿⡇⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡀⠀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⡘⢿⣿⣿⠰⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣶⣭⣥⢠⣬⣥⣾⣿⣿⣿⣿⣿⣟⡻⠟⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣥⣤⣤⣤⣤⣤⢠⣤⣤⣤⣤⣤⣤⣤⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠏⠹⣿⣿⣿⠟⡹⣿⣿⣿⠟⡙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⢠⣶⣿⣿⣶⣬⣴⣾⣿⣶⣤⣴⣾⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⠸⠿⠟⣛⣛⣛⣉⣉⢸⣿⠟⢿⣿⣿⡿⠻⣿⣿⣿⠟⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣩⣥⣶⣦⠹⣿⣿⣿⣿⣿⣿⢈⣵⣾⣦⣭⣭⣶⣷⣬⣭⣵⣾⣦⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⢸⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣌⢿⣿⣿⣿⢘⣋⣵⣦⣙⣛⣫⣴⣬⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⡙⢿⣿⢸⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠖⠲⠾⠤⠄⣀⡀⠀⠀⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠈⠀⠀⣘⠁⠀⠀⠀⠀⠄⡀⢀⠀⠀⠀⠀⡀⠀⠱⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⢃⣀⠠⠴⠒⠄⠀⡁⠸⠆⠀⠀⠀⢠⡀⠀⡇⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠉⠀⠀⢀⢿⠟⠁⠀⡀⠀⠀⠀⠀⠈⠂⠀⠁⢉⠓⠃⢀⠇⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠴⡋⡨⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡜⠀⠠⠤⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡐⠠⠀⠀⠀⠘⠁⠀⠀⢠⠜⠀⢠⠁⠀⠘⡁⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠛⠒⠓⠀⠤⡀⠀⢀⠠⠀⠔⠒⣁⠉⠀⢄⣑⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⡀⠀⢀⢐⠎⠤⠉⠀⠀⠀⢱⠟⠲⠞⢋⡵⣲⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⡦⠀⠀⠀⠀⠀⠀⢀⣦⡀⠈⠓⠒⠒⠋⠀⠁⣸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠠⠀⠤⠠⢤⣀⣘⠀⠑⠄⡄⠤⠄⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⢴⠀⠁⠀⠀⠐⢢⡠⠶⠦⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⡶⡄⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠄⢣⠈⡦⢀⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣉⣀⣈⣨⠤⠃⢹⠀⠄⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⡤⠤⣄⣀
⠀⠀⠀⠀⠓⠢⠤⣌⣉⠓⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⠬⣉⣓⠲⠤⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠛⠉⠑⠒⠮⢭⣛⡾⠛⢯⡉⠉⠉⠓⠒⠲⠤⢤⣀⠀⣀⣠⠤⠴⠒⠊⣱
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⣀⣀⣠⠤⠤⠴⠖⠒⠒⠒⠛⠒⠚⠳⠦⠤⣤⣤⠤⠴⠒⣋⣉⡥⣤⠖⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⢰⡇⣠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢔⣒⣪⠭⠴⠒⠒⠉⠉⠓⠦⣄⢸⡆
⠀⠀⠀⠀⠀⠀⢠⡟⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢻⡇
⠀⠀⠀⠀⠀⠀⢨⡗⢄⡈⠙⢒⣦⠤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⣴⡒⠋⢁⡠⢺⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠑⢻⣬⢧⣀⣀⣀⠈⠉⠉⠉⡽⢯⠉⠉⠉⠁⢀⣀⣀⡼⣥⡟⠊⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠈⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠁⠀⠀⠀⡇⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠧⡀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⢀⢼⡇
⠀⠀⠀⠀⠀⠀⠈⢧⡈⠓⠢⣼⢼⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡧⣧⠴⠚⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣽⣴⡋⠉⠉⠑⠒⠒⠒⠒⡟⢻⠒⠒⠒⠒⠚⠉⠉⢉⣦⡯⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠲⠶⠿⠿⠶⠖⠒⠒⠒⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠰⠦⢶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣋⣃⠑⠧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣾⣿⣯⡳⠄⠂⠀⠀⠀⠀⠈⠉⠀⠀⠁⠒⠁⣱⣿⣿⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡿⣡⡦⢏⠤⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠉⠸⣿⣧⣶⡆⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡈⠻⠋⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠿⡁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠰⣿⣿⡾⡃⠀⠀⠀⠀⠀⠐⢿⡻⠷⠀⠀⠀⠀⠀⠀⠀⠀⢴⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠿⡟⠁⠀⠀⠀⠀⠀⠈⠓⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢠⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⣢⠀⠀⠀⢠⠸⣿⣾⣧⣾⣟⠯⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠄⢰⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠁⠀⠀⠀⠠⠀⠈⠓⢮⠋⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⡀⠳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⢀⠀⠀⠀⠁⢠⣀⣬⠷⠦⠤⠤⠴⣋⠀⠀⠀⠖⠀⡇⢠⠙⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠂⠠⠄⣀⢀⡀⠀⠤⠚⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠄⠀⠀⠀⠀⠀⣿⠁⠀⠈⣿⠿⠟⢻⣿⣿⡟⠉⠀⢈⣿⣟⠉⠉⠙⠋⢈⣿⣿⣿⣟⠀⠀⠀⠀⠰⠷⠄⠸⢿⠋⠀⠐⠀⣹⣿⣿⣿⣿⡇
⣇⠀⠀⠀⠀⣰⣿⣷⡖⠈⠀⠀⢀⠘⠛⠛⠻⠶⣾⡍⠉⠉⣠⣿⣷⡶⠈⠉⠙⠿⣄⣰⣶⡖⠀⠀⠀⠀⠀⢀⡽⣿⠏⠀⠐⠀⠀⢹⣿⣿⣶
⣷⡀⣀⣴⣶⡄⠀⠀⢱⡤⠀⠺⡟⠀⠀⠀⠀⠀⠈⡫⠈⠻⣉⣻⣿⡇⠀⠀⠀⠀⠹⠋⠀⢁⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠘⠉⠁
⠻⠿⠿⣯⣿⠃⠀⠀⠀⠁⠀⠀⢰⣄⠾⠀⠀⠀⢠⣧⡈⠂⠈⠁⠀⠸⣶⡦⣴⣶⣿⣶⡾⠟⠃⠀⠀⠀⠀⠀⣠⣎⣀⣀⣀⠴⠒⠂⣀
⣤⣀⠀⠀⠀⠀⣰⣶⡿⠈⠀⠀⣘⢡⣿⣷⣶⣶⣿⣿⠇⠀⠀⠀⠀⠀⡿⠦⢿⡏⠉⠙⣀⠀⠀⢀⣿⣶⡶⠞⠉⠉⣿⣿⣿⣦⣤⣴⣿⣿⣶
⣻⡿⠛⠛⠉⢉⣻⣿⡿⠿⠿⠿⠿⠿⠋⣻⣿⣥⣀⣠⣦⣀⣀⣠⣤⡀⠀⠀⣠⣦⣷⠤⠛⠋⠀⠽⢻⡿⠛⠓⢲⣴⠟⠋⠉⠉⠉⠛⠛⢿⡜
⣻⣇⣀⣠⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣟⠛⠉⠛⠛⠛⠛⠉⢉⣶⣿⣿⣏⣁⣀⠀⢀⣀⠐⠛⠃⠀⣴⡿⠿⠂⠀⠀⠀⠀⠀⠀⢀⣷
⣿⣿⠿⠛⠲⣄⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠈⠻⠧⠀⠈⠹⣿⠟⠋⠠⢹⣶⣤⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⣼⠿
⠉⠀⠀⠀⠀⣾⠷⠀⣴⣦⣄⠀⠀⠀⢀⣴⣿⣽⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣀⣠⣆⢠⣿⣷⣦⣴⠿⠛⠦⡀⠀⠀⠀⣠⣆
⣀⣀⣀⣤⠂⠀⢀⣠⣷⣿⣿⡋⠀⠀⠀⠀⠉⠙⢻⡿⠋⠉⢡⠄⣄⣀⣀⠀⢀⠴⠟⠛⠻⠏⠉⠉⠙⠛⠟⠁⠀⠀⢀⡿⠖⠀⠈⠉⠉⠐⠚
⣿⠿⠛⠃⠀⠀⠿⣿⣿⠛⣿⣷⣦⣄⣀⣀⠀⢀⣠⡀⠀⠀⣀⠠⠚⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣿⠉⠀⠁⠀⢹⣿⣿⠋⠁⠀⠉⠋⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠗⠀⠀⠤⣾⣿⠿⠀⠀⠀⠀⣠⣿⠿⠿⢦⣄⣀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣀⣤⡆
⣧⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠁⠀⠀⠀⠈⢋⣠⡤⠀⠶⢾⡏⠀⠀⠀⠀⠈⠹⣿⣿⡅⠀⠉⠙⠛⠋⠀⠈⠉⠙⣄⡀
⣿⣷⡶⠖⠡⠛⡃⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⣇⡀⠀⠀⠀⠈⢢⣄⣀⣀⣀⡋⠟⠛⠓⢀⡀⢀⠀⠀⠀⠀⠀⢠⣽⡿⠦
⡴⠋⠀⠀⠀⣰⣿⣶⣤⣤⣀⣀⣀⣀⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢿⣿⣿⡟⠁⠀⠀⠀⠈⣿⣿⣷⣶⡶⠞⠛⠐⠉⠁⠀⠀⠙⠂⠀⢀⣀⣠
⣀⠀⠀⠀⣰⣿⣿⣶⡤⠀⢿⣿⣿⡏⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⢈⡏⠈⢻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⠇⣌⣴⡶⠀⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠘⢿⣿⠀⠀⣸⡿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢯⡙⢄⡹⣿⠀⠐⠇⢀⠞⠋⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⢿⣦⢻⡀⡁⠀⠀⣀⣴⡏⠀⣠⣿⢋⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⢹⢸⡇⣿⢸⡇⣟⣧⠀⣾⢻⠁⣿⠟⠙⢻⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢨⣰⠃⡿⢸⡇⠿⢸⣇⣯⠟⣰⡌⢸⢰⣿⠀⣴⠏⢉⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣚⡽⣡⡾⠃⣼⣿⣶⣦⣨⢎⡼⢋⡇⣿⢸⡇⢸⣿⠰⠏⠈⣽⣿⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣋⡾⠋⣁⣈⡙⠻⠾⢫⠌⠈⣇⢾⢰⡟⢸⡇⣼⡿⢰⣦⡾⠛⢻⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣩⡶⠟⠋⠉⠙⠷⣦⡄⠘⡆⢻⠀⡿⢰⢸⣀⡿⢁⣾⡏⡀⣠⡿⢻⣿⣗⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣯⣿⡿⠗⠛⠛⠿⣿⣦⡀⠀⢻⡄⣧⢸⡄⢁⠈⢸⡿⢡⣾⡿⠁⣿⡏⠀⣻⣿⣿⡀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣿⡿⣵⠾⠿⣷⣦⣄⠀⠹⣷⡄⠸⡇⢸⠈⣧⡘⠆⣾⢣⣿⡟⠁⣼⣿⡇⡞⠉⢼⣿⠁⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⣥⣦⣄⠀⠉⠛⢷⡄⢈⡿⠀⡇⢸⡆⡎⣧⢸⠏⣾⡟⠀⣼⣿⡿⠁⣷⡤⢿⡇⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣿⣿⣧⠉⠛⠻⠿⣶⣤⠈⠇⣼⠃⣸⠇⣸⠀⣷⢨⠞⣼⣿⠁⣼⣿⡿⠁⣸⣿⣷⣿⡃⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⣾⡇⢹⣿⠳⢒⠲⢤⣄⠸⠀⣼⣣⢴⡟⢠⡟⢠⡏⠀⣰⣿⠃⣸⣿⣿⠃⣸⣿⡇⢸⠿⣇⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⣿⣛⣠⣿⣿⣿⣿⣷⣇⣿⣼⣸⡷⠿⠏⢰⡋⠶⠟⣠⣼⡟⠁⣰⣿⣿⠏⢀⣿⣿⡇⣾⢸⡏⣷⣀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠁⠀⠀⠀⢀⠻⣶⠿⠟⠋⢠⣾⢟⣿⠇⢀⣾⣿⣿⠁⣿⣾⣷⡿⣧⢹⣦⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⣉⣴⡿⠃⢀⣼⣿⣿⡇⢰⣿⣿⠃⣿⠸⡆⢯⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⢰⣾⣿⠟⠀⢀⣾⣿⣿⠟⠀⣼⣿⡇⢰⣿⠀⢿⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠁⠀⢠⣿⣿⡿⠋⠀⣴⣿⣿⠇⣾⣿⠀⢸⡆⣿⣿⡏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢠⣿⣿⠟⠁⢀⣾⣿⣿⡏⢰⣿⣿⠀⢸⣧⢹⣿⡇⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠻⣆⠀⠀⠀⣿⣿⠋⠀⣴⣿⣿⣿⡿⠀⣼⣿⣿⠀⠀⣿⠀⣿⡷⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⢿⣶⣾⡟⠁⣠⣾⣿⠟⠋⠀⢀⣴⣿⣿⣿⠀⠀⣿⡀⣿⡗⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣄⠉⠉⠀⠠⠛⠉⠀⣀⣤⣴⣿⣿⣿⣿⡟⠀⢸⣿⡇⣿⡷⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠉⠉⠛⢦⣤⣤⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣾⣿⠀⣿⣧⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠉⠙⠛⠛⠋⠉⠀⠀⢀⣼⣿⡏⢠⣿⡇⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠀⣾⣿⡧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠟⠁⢀⣼⠿⣿⡇⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⢻⣷⢦⣤⣤⣤⣤⣴⣾⣿⡿⠟⠛⠁⠀⣠⡾⠋⠀⣿⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠹⢦⣽⠿⠛⠋⠉⠉⠉⠀⣀⣀⣀⡾⠋⠀⢀⣴⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡋⣽⡿⠿⠏⠤⠿⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠉⠀⠐⠀⠀⠀⠀⠀⠈⠻⡦
⠀⠀⠀⠀⠀⠀⢠⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⣼⡖⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⣶⣤⣀⡀⠀⠀⠀⢀⣀⠠⠀⠀⠀⡇⠀⠀⠾⠛⠃⠀⠀⠰⣦
⠀⠀⠀⠀⠄⡀⠋⠀⠀⢋⠉⢹⣿⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠇⠀⡀⢀⡀⠰⠆⣀⡀⠘⠷
⠀⠀⠀⠀⠂⠀⠀⢄⠀⠘⢿⣿⡏⠑⠀⠐⠁⠀⠀⠀⠀⠀⠐⠂⠐⠀⠙⠁⠀⠀⠉⠁⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠙⠧⢀⡀⠀⠀⠀⠀⠀⡠⠂⣴⡿⢁⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⠂⠤⠀⣀⣀⣀⣤⣄⠀⠀⠀⢿⠀⣾⡿⠁⠐⢀⠀⠀⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⢲⠠⢀⡄⠀⠀⣂⢡⣀⠠⠄⠀⠑⠀⠀⠀⠓⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠉⠁⠀⠉⠀⠐⢄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠂⠀⢀⠀⡀⠐⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⢄⠀⠀⡐⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠈⠃⠊⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄⠀⠐⠤⢀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠀⠀⠀⠀⠳⣶⣶⡄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠈⠁⠀⠈⠀⠈⠉⠑⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢊⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠘⠁⠼⢀⡀⠀⠀⠀⠀⠀⠃⠒⡆⠀⠀⠀⠀⢸⣿⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠀⠀⠀⠀⠀⠀⠀⠧⢤⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡁⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠉⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠇⠀⠀⠀⠀⠀⢀⠵⠂⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠷⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢠⡀⡀⣄⣤⣠⡄⠀⠀⠀⠀⠀⢀⣄⣠⣠⢀⠀⡠⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠂⠀⠀⠈⠐⠠⣀⠀⠀⢀⡈⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠐⢆⠿⠿⠛⠿⠛⢷⠇⠀⠀⠀⠘⡾⠙⠿⠋⡿⠻⡱⠂⠀⡠⠔⠁⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣙⠮⡅⠀⠒⠀⡦⠄⠉⣈⠉⣁⠉⠠⣤⠂⠒⡄⢈⠵⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠨⣖⠀⠀⢑⣼⣶⡐⢄⠀⠉⢀⣴⣶⠂⡱⡌⠀⠀⢲⠅⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠐⠚⢃⣶⠠⡎⢿⣿⡇⡠⠀⣀⡘⣿⣿⢃⠆⠰⠄⢲⡘⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠀⢇⠀⣩⡀⠆⠉⣠⠁⠙⠁⣤⠀⡸⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠂⠀⠀⠀⠐⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠲⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡘⠀⠀⠀⠀⠓⠀⠂⠐⠂⠀⡖⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠁⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⢃⠀⡀⠀⠀⠀⠀⠀⠠⢆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⣢⣀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣆⡌⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣛⣲⣤⣄⣀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡆⠁⠑⣤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⢘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⢀⣚⡳⠿⠿⣿⡷⢶⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢅⠀⣷⡆⠀⠀⠋⠉⡆⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡋⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⢀⣌⠀⠃⣤⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣈⡔⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠉⠻⣾⠀⢠⠿⠷⢶⣶⡶⠛⢻⣧⡘⠁⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⡄⣷⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⣠⣿⣟⢿⣿⡶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠹⣧⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⢸⠃
⠀⠀⠠⣼⣀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠀⠙⠿⣶⣭⣾⣿⢹⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⡷⡜⣿⡛⣫⣿⣦⣄⡀⠀⠀⢠⡬⠀⠀⠤⠀⠀⠀⠀⠰⠌⠀⠀⠀⠤⠌
⠀⠀⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣴⣽⡗⣹⣛⣃⣻⣟⠿⠶⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠛⢷⣾⣭⡸⣿⣿⡸⡿⠷⠼⡿⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠁⠀⠛⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣦⡭⡾⣿⢿⢇⣶⣩⣽⢛⡶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠐⢦⡀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣿⡏⣀⠾⣟⡩⣷⡟⣿⡝⣫⣽⢟⡲⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⣶⢿⡍⢿⡧⢛⡛⣷⡷⣬⣛⣼⢫⣿⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣙⡻⠎⢵⣾⡩⣿⣗⣬⢁⣭⡜⣫⣾⢯⣵⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠒⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢸⣷⢍⣰⣦⢜⣫⣼⣿⡷⢩⣽⢘⣵⡇⢸⡄
⠀⣤⣤⣤⣶⣤⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠾⣟⠛⠛⠛⠉⢫⣶⡷⣭⣹⡵⠇⠸⡇
⠀⢛⣃⠘⢰⣼⣏⣉⣟⣟⣿⣽⣶⣶⣤⣦⣦⣤⢴⡿⠶⠛⠂⣄⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠑⠶⢹⡄⠀⠀⠀⠀⠻⣾⣿⣥⣽⢃⠀⣇
⠀⠀⠍⠀⡈⠹⡟⢙⠿⣻⡟⣎⢽⡟⢻⡴⢁⣦⣯⠞⠋⠉⠛⢻⢷⡀⠀⠀⠀⠀⠀⣸⣹⠇⠀⠘⣥⣇⠀⠀⠀⠀⠀⠘⣿⣷⣆⣘⠃⣸⣦
⢀⠀⢀⣨⠟⠓⢦⣀⠀⠉⡹⠮⠃⢿⢯⠀⠀⠁⠫⠶⠛⠛⠛⣿⠘⡇⠀⠀⠀⠀⠀⠉⠘⠀⠀⠀⠹⠟⢂⠀⠀⠀⠀⠀⠘⠋⢯⠹⡄⠈⢻⡇
⠘⠶⠎⠀⠀⠀⠀⠈⠳⡈⠀⢷⠀⠀⠀⠀⠀⣠⡆⢙⢫⡍⣿⡏⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣬⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢡⣾⠁⠀⠀⠀⠀⠀⠈⣭⠀⠀⠈⠻⡄⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢸⡟⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠈⠃⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⢠⡔⠀⠀⠀⠀⢠⡤⠃⠀⠀⠀⠀⠀⠀⣤⠅⠀⠴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆⢠⣶⠇⠀⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠉⠁

⠉⢉⣽⣿⣿⣿⣿⠿⠟⠛⠉⠉⠛⢯⡉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣽⣻⡟⣽⢫⡟⣽⣋⡽⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠋⠉⠁
⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠉⢻⡻⣿⣿⣿⣷⡙⣮⠗⠉
⣿⣿⡟⣉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢱⠀⠉⠻⣿⡟⠁
⣿⡿⠋⠉⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡸⠀⠀⢠⡟
⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣀⣠⣤⠴⠷⠤⠤⢄⣤⡀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠴⠋⠁⡞⠁⠀⠀⢀⡴⠋⠁⠉⣑⣦⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⢶⣶⣶⣶⣶⣶⣦⠤⢤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⡴⣏⢀⣀⣈⣳⣄⣀⠀⢸⡇⠀⣰⡾⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⢀⡴⣻⣿⣿⣷⣦⣤⣾⣿⣿⡿⠛⠉⠁
⢀⠔⠈⠉⠉⠲⣄⣸⣶⡿⢻⠏⣩⣿⠿⠛⠛⠛⣳⣦⣿⣠⠴⢲⢻⢹⣇⠀⠀⠀⠀⠀⡾⢶⣿⣿⡋⠁⠀⠈⠉⠻⣉
⠃⠀⠀⠀⠀⠀⠸⣿⣿⠁⣿⢀⣿⠃⢀⣴⠞⠋⢉⣩⢻⡁⠸⣌⢯⣧⣿⣆⠀⠀⠀⠀⠹⣄⡈⠌⡯⡑⢄⠀⣠⣤⣬⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⠀⣿⡬⣧⢠⣿⡇⠀⣴⠟⡇⠀⠳⢄⣈⣓⡟⠁⠙⣷⡀⠀⠀⠀⠙⣿⡿⠾⠃⠀⢻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀
⡀⠀⠀⠀⠀⣀⣀⣸⣿⣆⣙⣷⣿⣾⣿⣧⣼⠃⠀⠙⢤⣀⠀⢈⡃⢹⡄⠀⠈⠻⣦⡀⠀⠀⠈⣷⡄⠀⠀⣰⣿⡿⠛⠛⠛⠛⠿⣦⣴⣾⣿
⣿⡄⠀⢀⡎⠀⠀⠈⠙⠿⣟⣋⣁⣠⡴⠋⠈⢧⠀⠀⠀⠈⢉⠋⠙⠻⣷⠀⠀⠀⠉⠳⣄⠀⢀⣿⡿⢶⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⡁
⣿⣿⣄⢸⠀⠀⠀⠀⠀⠠⠚⠉⠁⠐⡄⠀⠀⠀⠙⠢⠤⣤⠞⠛⠻⢶⣼⡆⠀⠀⠀⠀⢙⣷⣿⣿⡠⣄⠈⠩⣾⠛⠷⣦⣀⠀⣠⣶⣾⣿⣿
⣿⣿⣿⣿⡄⠀⠀⠀⢠⠁⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣴⡿⠛⣛⣿⣷⡿⠷⣒⣶⣶⡾⢿⣟⠁⠘⣿⣮⣵⣾⠃⠀⠀⠀⠙⠷⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⡀⠀⠸⡆⠀⠀⠀⠀⠀⠙⠦⣄⣴⠞⠁⣴⢿⠫⠈⠁⠀⣼⡛⠋⠁⠀⠘⣿⣄⣠⣬⣽⣿⣧⣄⠀⠀⠀⠀⠀⠙⢿⣿⡿⠛
⣿⠻⢿⣿⣿⣿⣿⣷⣤⣹⣦⣀⣀⣀⣠⣴⣾⣟⡁⠀⠀⠙⢮⣶⣄⡀⢄⡻⣷⣤⡄⣀⣀⣈⣛⣻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠈⢿⡄
⢻⣧⡠⢈⣙⣻⣿⣿⣿⣿⣿⡿⠿⠟⠛⠁⢙⣿⣿⣿⠿⠟⠛⠉⠉⠉⠋⢹⠉⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣷⠞⢻⣆⠀⠀⠀⠸⣧
⠘⣿⣉⠓⠀⣁⣼⣧⢠⢿⣟⠀⠀⠀⣠⢾⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⠾⢻⣆⠀⠀⠀⢿⣦
⠀⢿⡟⠷⠤⠉⠀⣿⣼⠘⢿⣦⣤⣾⣏⣿⣿⠁⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣾⣆⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⠟⢻⡄⠀⠀⢸⣿
⠀⢸⡇⠀⠀⠠⠴⠛⣿⣆⠈⢉⣩⡿⢾⣿⣿⠀⠀⠀⢀⣴⣿⣿⠟⠛⠛⠛⠛⠿⡄⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⢾⣧⠀⠀⣸⣿
⠀⣸⠷⣤⣀⠀⠀⠀⣿⣷⣒⣺⡿⡗⠺⣿⣿⡄⠀⠀⢸⣿⠏⢀⣀⣤⣤⣄⣀⢀⠇⠀⠤⠀⠀⣰⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠖⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠂⢁⡀⠀⠀⠤⠈⠀⠛⠿⠿⠶⣶⠶⢤⡂⠀⠂⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⣠⣀⠀⢦⣶⠀⣸⣤⣶⣶⣶⣾⠷⣿⣿⣭⡿⣿⡆⢇⠐⣶⢧⣢⣤⣄⠒⢠⠤⣤⠠⠄⠀⠀⠩⢆
⠀⠀⠀⠀⠀⡐⠁⠢⣤⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣻⣉⣉⣀⣈⡉⠓⢦⣾⣧⣾⣿⣿⣿⡿⣶⣾⣧
⠀⠀⠀⠀⣠⣡⣾⣶⣿⡿⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⢠⣿⣿⣿⣭⣿⣶⣿⣿⣿⣿⣾⣿⣿⣧⣼⣿⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡛⠿⢿⣿⣾⣶⣶⣮⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣶⣈⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣥⣼⣯⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⢉⣭⣭⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢿⣿⣿⣿⣿⣧
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠈⠿⠟⢿⣿⣿⣿⣿⠟⠇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠛⠛⠻⢿⠿⠻⠛⠛⠛⢻⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠠⠶⣿⣿⣧⠼⠄⠀⠻⣿⣿⣿⣶⣶⣴⣶⣶⠼⠾⠟⠿⠿⣿⣿⣷⣤⣀⠀⠀⠀⢀⠀⠈⠉⠉⢋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⣽⣿⣿⠟⢿⣷⣶⠀⠀⠀⠀⠀⣤⣿⣿⠟⠉⠛⠿⠀⡄⣿⢿⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⣡⠀⠈⠻⣿⣆⠀⠀⠀⠀⠉⠉⠀⠀⠐⠦⣄⡀⣷⣾⡿⣳⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣴⣿⣤⣀⣀⣌⣿⡶⣢⣤⣤⣴⣦⣤⣤⣴⣤⣼⣿⣿⣿⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣟⡿⠛⠋⠀⠀⢢⠈⢻⣧⣿⣽⣻⡏⠀⠀⡀⠉⠉⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠹⢤⣬⣿⣿⣿⣯⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣴⠋
⠰⣟⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⡾⠁
⠀⠈⠳⣄⡈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠟⠁
⠀⠀⠀⠈⠛⢦⣄⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣳⣦⣙⠷⣄⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⡾⣡⣞⣁⡀
⠀⠀⠀⢀⣀⣤⠤⠴⠒⣚⣛⣻⣯⣭⠭⠭⠿⠷⠿⠷⣿⠉⠙⠳⣶⠲⠶⠶⢶⡾⠛⢻⡿⠿⠭⢭⣭⣽⣛⣛⣒⠲⠤⢤⣀⣀
⣠⡶⠛⠉⣡⠶⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⣷⠀⢰⠏⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠶⢦⡈⠙⠳⣦
⣿⣄⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⢻⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠿⠀⢀⣼
⣿⠈⠛⠶⢦⣭⣝⣛⣲⠶⠦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠶⢶⣚⣛⣭⣥⠶⠞⠋⢸
⣿⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠲⠶⠿⠿⠿⠯⠭⢭⣭⣭⣭⣭⣭⣭⣭⣭⠭⠭⠿⠿⠿⠶⠶⠒⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢸
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿
⣿⠀⢿⡙⠛⠲⠶⢤⣤⣤⣀⣀⣀⣀⢠⡟⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢷⡀⢀⣀⣀⣀⣠⣤⣤⠴⠶⠚⠛⢉⡿⢸
⡟⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡟⠙⠛⠛⢿⡛⠛⠛⠒⠒⠒⠒⠒⠚⠛⠛⣿⠛⠛⠛⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸
⣷⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸
⣿⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸
⣿⠶⣄⣀⠀⠹⡄⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣸⠃⢀⣀⡴⢾
⣿⠀⠀⠉⠙⠛⢿⡶⠤⣤⣾⣅⣀⣀⣀⣀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣧⣤⣤⠴⣶⠟⠛⠉⠁⠀⢸
⣿⠀⠀⠀⠀⠀⠀⢷⣀⡾⠁⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⢻⡟⠛⢻⡟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠹⣆⠀⣰⠏⠀⠀⠀⠀⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡏⠀⠀⠀⠀⠀⠀⣸
⠈⠛⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠶⠶⠦⠤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠶⠶⠖⠚⠛⠋⠉⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⡶⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣴⠋⠀⠀⣷⠀⠈⠹⣄⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠳⣤⣼⠁⠀⠀⠀⣿⠀⠀⠀⠘⣦⡤⠞⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⢸⠋⠛⠒⠶⠶⣿⠴⠶⠒⠋⠹⡆⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠒⠒⠒⠺⡟⠒⠒⠒⠒⠚⣿⠒⠒⠒⠒⠒⢻⠓⠒⠒⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⡏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⣸⣷⠖⠛⠉⠉⣍⠉⠉⠛⠲⣾⣅⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣏⠁⠸⣆⠀⠀⠀⣿⠀⠀⠀⣴⠃⠈⣻⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⣷⣄⡀⣿⢀⣠⣾⣥⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⠢⠀⢀⠀⢀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⠒⠒⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢠⠴⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣱⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⠤⠤⠤⠤⠤⠤⠔⠀⠒⠀⠀⠀⠈


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

⠀⠀⣶⡄⣠⣾⠇⠀⠀⠀⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⣠⣤⠀⠀⠀⡀⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣤⣦⠀⣤⠀⠀⠀⣠⣴⣆⣠⣤⣄
⠀⠀⠹⣿⣿⣿⣶⠆⠀⠀⠘⠋⠁⠀⠀⢶⣄⣀⠢⡔⠶⢬⣃⣠⣴⣾⣿⡿⠇⠀⠀⠳⣄⠀⠻⣄⠘⣣⠄⠀⠀⣠⢠⡌⠉⡁⣸⠃⠀⢀⣾⣿⣿⣿⣟⠋⠁
⠀⠀⠀⣿⠋⠉⢁⠚⠀⠰⠆⠀⢀⣴⣶⣦⠉⠉⠛⠛⠛⠂⠹⢿⣿⣿⠟⠁⠀⢆⢢⠀⠀⣍⠐⠛⣦⡿⠀⠀⢰⢣⠟⠀⣸⠗⠁⣠⠀⠁⠉⠉⣩⠉⠁
⠀⠀⠀⠿⠀⣼⣿⠟⠀⠀⠀⠀⠈⠙⠛⠛⠀⠠⡿⢀⣰⣤⣄⠀⠣⠀⠃⠈⣳⡈⠛⠦⣤⣀⡉⠳⣼⡇⢻⠀⡷⠁⡀⣰⠃⣀⠴⠃⢠⠄⠀⠀⡏⣠⠞⢀⡼⠃
⠀⠀⠀⣠⡄⠙⠋⢀⣴⡄⢠⡶⢛⡛⠓⢀⣀⠀⠀⠈⠿⡿⠻⠆⠰⠤⠤⠤⠟⠋⠉⢉⣉⡉⠙⠳⣦⣽⣆⢳⡇⢈⣵⠷⠾⠥⠶⠚⣉⣀⣤⣾⡿⠿⠒⠉
⠀⠀⠀⠘⠃⢠⡄⠘⢛⡇⠈⠻⢮⡁⣶⠈⠋⠀⠀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⣠⡄⣼⣿⠁⠀⣦⠈⣿⣿⡈⡇⡘⠁⣀⣤⡶⢿⣟⣋⢉⣁⣀⣤⠀⢰⣸⣇
⠀⠀⠀⠀⠺⠀⠁⠀⡾⠁⢾⡗⢀⠳⠄⠈⣡⣶⣿⠿⠛⠛⢛⠻⣿⣶⡀⠀⢸⣿⡀⠘⢋⣀⠀⢸⣧⢸⣿⣿⣷⡷⠛⠋⠁⠀⠀⠀⠉⠉⠉⠉⡐⣿⣿⣿⣿⠇
⠀⠀⠾⠇⠀⠀⢀⣄⠀⠴⣾⣷⠿⠁⠀⢰⣿⢟⠁⣀⠂⠀⣠⠆⢬⣿⣇⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⡟⠀⠐⣶⣄⡀⠀⠀⠀⠀⠀⠀⠻⠿⠛⢋⣥⣶
⠀⠀⠀⠰⣦⢠⡿⢁⡀⠀⠀⢀⣴⣶⠀⢸⣿⣮⠀⡉⠁⠠⣀⠀⣿⣽⡿⠀⢠⡆⠈⣿⣿⣿⣿⡇⠀⠀⢿⣿⡇⠀⠀⠈⠻⣿⣶⣤⣶⣿⣿⣦⡀⣠⣾⣿⣿⠏
⠀⠀⢴⡄⠘⡿⠛⣉⠀⠀⢠⣿⣿⣿⠀⢸⣿⣿⣦⣀⣀⣤⣴⣾⣿⣿⠃⠰⣿⣿⠀⠙⢿⣿⡿⠁⠀⠀⣀⠀⢰⣶⣄⠀⠀⠀⠙⢿⣯⡇⠸⣽⣿⣿⣿⠿⠋
⠀⠀⣘⢿⣸⡵⣛⣥⠀⠀⣼⣿⣿⡇⠀⠈⠻⠿⣿⣿⣿⠿⠿⠟⠋⠁⠰⠚⠿⠿⠇⠀⠈⠛⠃⠀⠀⠈⣿⣷⣾⣿⣿⡆⠀⠀⠀⠀⠛⠶⠾⠛⠉⠁
⠀⠀⢻⣦⡿⣻⣽⠶⢀⣀⣿⣿⣿⣿⣶⡦⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣀⣀⣘⠻⠿⢿⣿⣿⣿⣿⣀⣠⣴⣾⣿⠷⣄⠀⠀⢀⣴⣿⣆
⠀⠀⠀⣿⠋⠁⠀⢰⣿⡿⠿⢿⣿⣿⡏⠀⢀⣴⣿⡏⠠⣤⣿⣿⢿⡿⣿⣿⡿⠿⡿⠿⢿⡿⠿⢿⣿⣿⡷⣶⣤⡈⠉⢻⣿⣿⣿⣿⣿⠀⠀⠀⣰⣿⣿⣿⠙⠇
⠀⠀⠀⢁⣤⣾⠃⠀⠉⠀⠀⠀⠹⣿⣧⠀⣼⣿⣿⣿⡶⠌⣯⢒⣀⣇⢀⣿⢐⠰⡇⢰⣆⡇⣶⢈⡇⠖⡀⢿⣿⣿⣦⡀⢹⠉⠛⠉⠁⠀⠀⣰⣿⣿⣿⠏⢠⣆
⠀⠀⠀⢸⣿⣿⠆⠀⠀⠀⠀⠀⣤⣈⠻⠇⢹⣿⣿⣿⣦⣴⣿⣿⠟⠹⡿⢻⡿⢿⣛⡿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⡄⠰⠄⠀⣠⡄⣿⡿⢟⠁⡆⢸⣿
⠀⠀⣀⣿⣿⣿⠇⠀⡀⠀⠠⣶⠀⠻⡧⠀⠈⠻⣿⣿⣿⣿⣿⣿⣠⣄⣇⣾⣧⣜⣣⣄⣻⣌⣠⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣄⠀⣤⠀⠻⠈⠁⠀⢸⠀⢁⠀⠇
⠀⠀⢿⣿⣿⣿⡆⣠⡟⢻⡇⠘⣶⣾⡷⢀⣴⡦⣬⠙⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⢉⠀⠀⠐⠚⢆⣿⠋⣰⡀⠀⠀⢸⡀⠘⠶
⠀⠀⠈⢿⣿⠏⠀⠛⠳⠛⢀⣀⠈⠉⠀⢼⡋⣼⣆⡀⠀⠀⠀⣀⣤⣶⣾⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣄⡄⠦⢸⣌⣳⡆⠀⢸⣿⢚⣉⣉⣤⡶⣊⠛⠂
⠀⠀⠀⠀⠀⣠⡴⢶⡀⣤⡛⠉⠑⣶⠀⠀⢿⣾⠽⠟⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠒⠷⡼⣿⣿⢀⡉⠉⠿⣷⣄⣸⣿⣾⠿⢟⡉⠉⠃⠔⣠⣖⠂
⠀⠀⠀⠀⠀⠹⣷⠸⡧⠈⠹⣷⡼⠟⠀⠀⠀⠀⠀⢠⠆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣼⣶⡆⠰⣄⣈⢻⣾⣅⣀⣀⣹⣿⣿⣿⣿⡷⣶⣶⣶⣾⠿⠿⢿⣓
⠀⠀⠀⣀⡀⠀⠀⢀⣤⠀⠀⣶⣄⠀⣧⠀⠀⠀⣰⠏⣠⠀⠀⠉⠙⠛⠛⠿⣿⣿⣿⣇⠀⠉⣿⠀⠀⠋⠀⠀⣈⠿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⢳⠤⣄⠈
⠀⠀⢸⣿⣿⣶⣿⣿⣿⡄⠀⠸⣿⣧⠸⣆⠀⣰⠏⣰⣿⠃⠀⢀⣤⣴⣶⣶⣿⣿⣿⣿⣷⡀⠈⠀⠀⠉⠉⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣦
⠀⠀⢀⣿⣿⣿⣿⣬⢻⡷⡄⠀⠙⢿⣤⣿⣾⣿⣾⡿⠋⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⡄⠀⢠⣾⣿⣷⡄⠀⠀⣸⣿⣿⣿⣿⣿⡄⢸⣿⣮⡴⠀⠈
⠀⠀⢸⣿⣿⣿⣿⠿⣾⠷⠃⠀⠀⠀⢻⣹⣿⣯⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠁⠘⣿⣄⣈⠉⢿⣿⡟⠀⠀⣿⣿⣿⣿⣿⣿⡇⠺⠻⣿⠁⢠⠀⡠
⠀⠀⠸⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⢀⡼⠹⣿⣿⣿⣿⣿⡟⣀⠀⠀⠀⠀⠈⠉⠉⠀⠘⠿⠉⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣾⣤⠇⣠
⠀⠀⠀⠉⠉⢀⣴⣤⣴⣮⠤⠴⠶⠖⠀⠸⡟⢹⡇⠀⠿⢡⣿⠈⣽⠛⠛⢉⣦⢻⣦⠀⠀⠀⢾⣿⣶⣾⣶⣶⣾⣷⠀⣿⣿⣿⣿⣿⣿⣿⠘⣄⢀⡀⣿⣿⣴⡇
⠀⠀⠀⢀⣠⣿⣿⣿⣿⣶⡿⣿⣗⣒⠢⠄⠙⠉⠀⠀⢀⣿⠏⢸⡿⠀⠀⠀⣿⡇⢻⣧⡀⠀⢠⣿⣭⣿⣟⣴⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⠀⠹⣮⡧⣿⣿⡟
⠀⠀⠐⠛⠛⡟⠻⡟⠛⠛⠋⠛⠾⢍⡛⠳⠦⠄⠀⠀⠘⠿⠃⠸⠟⠀⠀⠀⠸⣿⠆⠉⠁⠀⠀⠻⢯⣧⣼⡿⠋⠀⠀⠻⣿⣿⣿⣿⡿⠋⠀⠀⠻⠟⠘⠟

⣶⣶⣶⣶⣶⡾⢶⣶⣶⣶⣾⣷⣶⣶⣿⣾⣿⣷⣾⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⡿⢿⡿⠿⠿⠷⢶⣶⠶⠶⢶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣤⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣼⣷⢄⡀⠤⠐⠂⠉⠉⠀⠀⠀⠀⠀⠉⠀⠐⠒⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⠶⣾⣿⢡⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠘⠋⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⣤⠀⠒⡄⠠⡖
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢤⣀⢸⡆⣠⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋⠁⠀⠉⠂⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠠⠒⠀⠂⠀⠀⠀⠀⠀⠐⡳⠦⡀⠀⠘⠄⠀⠀⣠⠞⢳⡄⠀⢀⣉⣀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠠⣾⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠁⠶⠚⣠⠈⠁⠀⠀⣇⠠⠊⠀⠀⠀⠀⢶
⠀⣷⣶⠀⠀⠀⠀⡆⠀⠀⠀⢇⡠⠔⠠⢤⠜⠀⠀⡒⣠⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠸⠇⠀⠀⠀⣡⠞⠁⠀⢀⡴⢣⠀⠀⠀⠀⠀⠀⢔⡃
⣤⠆⢀⠀⠀⠀⠀⠃⠀⠀⠀⠸⡇⠀⠀⠛⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠠⠊⠀⠁⢴⠄
⠀⠀⠟⠀⠀⠀⠀⠸⠀⠀⠀⠀⠘⣄⢳⡀⠀⠀⠀⢄⠀⠿⡒⠤⢀⡀⠀⠀⠀⠀⢀⠀⠐⠧⠀⠆⠀⠀⡰⠁⠀⠀⠠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠹⣿⣄⠀⠀⠈⠲⡄⢀⠀⠀⠀⠀⣁⣾⣟⣻⣿⣆⢀⠊⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠂⠬⢤⣀⡀⣸⣻⡶⢟⣻⣿⡁⠀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠷⠶⣤⢀⣀⣀⡀⢼⢹⡿⠋⠉⢉⡉⢃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠛⣄⠀⠀⠀⣼⠃⠀⠀⠀⠀⡀⢻⠁⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⢠⡌⠁
⠀⠀⠀⠀⠀⠀⠢⠒⠤⠊⠀⠀⠀⠀⠀⠀⡠⠂⠰⠀⠀⢠⠀⠈⢠⣿⡴⠋⠀⠀⠀⠀⢀⣈⠷⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⠀⠀⠀⠤⠀⠊⠁⠀⠀⣀⠀⠙⠄⠒⢻⣧⠀⡀⣠⠄⠤⢴⣷⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠄⠀⠀⠀⢸⣿⣶⣛⡟⠀⠀⠀⠻⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣡⣄⠈⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⢻⣿⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠱⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠢⡉⠢⡀⠀⠀⠀⠀⠀⠀⠁⠚⠳⠶⣤⣤⣀⣀⠀⢀⣤⣇⠦⠑⠁⠀⠀⠉⠀⠒⠶⠒⠶⠀⠀⠈⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣨⠄⢑⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠈⠉⡱⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠎⠀⠇⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⢠⡄⠘⡀⠀⠀⠀⢀⡐⠁⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⡘⢀⠮⠤
⠀⠀⠀⠀⠀⠀⠐⠘⠀⠀⠀⠀⠐⢀⠈⠒⠂⡐⠮⡑⣠⡦⠐⠂⠀⠀⠀⠀⠀⠀⠀⡠⣔⢂⠁⠀⠓⢀⠔⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⡼⠋⢸⠀⠀⠀⠀⠠⠔⠂⣤⡴⠓⠁⠀⠀⣸⠒⠛⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠀⠀⠐⠂⠤⠭⡌⠤⠅⠒⠒⠊⠁⠀⠀⠀⣠⠚⢋⡄⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡆⠀⠀⣠⠾⡈⢀⠈⢌⡰⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠒⠐⢇⠖⢲⠉⠢⡐⢠⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⡀⠀⢻⠀⠗⠐⢶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡡⠒⠀⠺⠀⠐⣌⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡠⠤⠇⠸⡇⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠰⢺⡀⠈⠀⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⢁⠤⢷⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠂⡠⠿⢶⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⡘⠀⣇⠐⡛⠦⡤⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠁⠈⠣⠴⠁⠔⠁⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠙⣉⡙⣏⢀⣿⢉⣹⣙⢉⣿⠟⣭⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣿⣿⣿⡦⣄⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⡿⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⡀⣹⣿⣿⠷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣧⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣱⣿⣿⣿⡏⢻⢯⡙⢗⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⣬⣿⣷⣅⠘⢃⣹⣿⡏⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⠀⣄⠀⠀⣴⣿⣾⣿⣿⣿⣶⢟⣹⣿⣿⣷⡆⠠⣾⣷⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡉⢹⡟⢻⡿⢿⢿⣿⣿⣿⣿⣿⠛⠛⠛⢛⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣿⡟⣁⣼⣷⣿⣟⣿⣿⣿⣿⣿⡅⢬⠅⣼⣿⣿⠟⣭⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣀⡼⠉⠻⣴⣾⣿⣿⠿⡟⢿⣿⣿⡿⠹⣶⡆⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡿⣿⠰⢣⠀⠀⣿⣷⡿⣿⣶⣯⠀⠘⢿⣷⣹⣿⣻⡶⣿⣿⣿⣿⣿⡿⢻⡇
⠀⠀⠀⠀⠀⠾⠍⠁⠀⠀⠀⠀⠉⠈⠀⠀⠀⣄⢨⡿⣷⠛⠟⠚⢿⠀⠈⢸⣿⣻⣿⣷⣬⡟⢹⡿⠋⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⣤⣰⣿⣿⣿⣇⠀⠀⢀⡎⠀⠀⠀⢸⣿⣿⣿⠟⠗⠓⠿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣍⣿⡉⣿⣆⠀⣭⠁⠀⠀⠀⢸⣿⣿⡟⠀⢰⠷⡄⠛⠙
⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⢰⠈⣿⣿⣿⠿⣋⣉⣸⣛⠃⠀⠛⠀⠀⠀⠀⣿⡏⣼⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡄⠻⢿⣠⣾⣆⣿⠻⡇⣠⣾⠿⣿⣋⠁⠶⠒⠀⣠⣶⣷⠛⠛⠠⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣀⡀⠀⠀⠒⠶⢿⣭⣿⣧⣤⡷⠎⣿⣾⠿⢿⣿⣿⣶⣿⣿⣏⡷⣀⠚⠢⡄⠀⠀⠀⠸⠏
⠀⠀⠘⢿⣿⣤⡀⠀⠼⠿⠿⢾⣗⠦⠁⠀⣉⠉⠐⠦⢤⣀⣽⣟⣿⣎⣷⡄⠀⠀⣁
⠀⠀⠀⠀⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣴⣆⠀⠋⠁⠁⠁⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠶⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣇⡀⠀⠀⠀⣠⣴⠦⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃⠀⠁⢉⡺⡉⠓⠀⢀⡀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠆⠀⠈⠛⠿⠿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠰⠟⠋

⠀⠀⠀⠀⠀⠀⢠⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣀⡤⠴⠒⠋⢉⣀⣤⠤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠙⠢⣄⠀⠀⠀⠀⠀⢸⢡⠞⠁⠀⠙⢶⣋⣉⣤⠤⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣌⠑⢦⡀⣀⣀⠀⢸⡀⠀⠀⢀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⠏⠁⠈⠙⣆⠙⠒⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠒⢻⡀⠀⠀⢀⡾⠉⠉⠉⠉⠉⠛⠒⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠙⠶⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡈⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠋⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠲⢤⣀⡈⠉⠙⠓⠚⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠴⠶⠒⠚⠋⠉⢁⣀⡴⠞⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢧⠀⠈⠉⠛⠒⣶⡤⢤⣤⣤⣀⣀⣀⣀⣀⣠⣄⣤⠤⠶⢶⠾⣏⡉⠁⠀⢠⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧⠀⠀⣠⠞⠁⠹⣆⠀⠀⠀⠀⣰⠏⠙⢦⡀⠀⠀⢠⠏⠀⠀⠙⢦⣀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣧⠞⠁⠀⠀⠀⠈⢧⡀⢀⡾⠁⠀⠀⠀⠙⢷⣴⠏⠀⠀⠀⠀⢀⡿⣦⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠴⠋⠙⣇⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⣴⠛⢦⡀⠀⠀⠀⡜⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⡀⠀⠀⠸⡆⠀⠀⢀⡴⠋⠀⠹⣄⠀⠀⠀⢠⡞⠁⠀⠈⢻⣄⠀⡼⠁⠀⣀⠜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡈⠓⠲⠤⣿⣤⡞⠉⠀⠀⠀⠀⠸⣆⢀⡴⠋⠀⠀⠀⠀⢀⣹⣾⠥⠖⠋⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⡀⠉⠉⠙⠓⠒⠒⠒⠛⠛⠒⠒⠒⠛⠉⠉⠉⢀⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠶⠶⠶⠶⠶⠤⠤⠶⠒⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⣠⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠛⠷⠀⠟⠋⠀⠀⠀⠀⠀⠀⠀⠠⡿⣱⡿⠃⠀⠈⢷⡝⠷⠀⠀⠀⠀⠀⣼⣿⣿⡇⠈⢻⣿⣆⠀⠀⠀⠀⠀⢸⠀⣘⠀⣁⠀⡃⢈
⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠄⢤⣤⣤⠀⡐⠀⠀⠀⠀⠀⢸⡟⠙⢿⠁⠀⣸⣿⣿⠀⠀⠀⠀⠀⠠⠠⠤⠠⠤⠠⠤⠠⠄
⠀⠀⠘⠿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠘⠦⣈⣙⣻⠔⠁⠀⠀⠀⠀⠀⠀⢻⣦⣬⢧⡾⢋⣼⠏⠀⠀⠀⠀⠀⢈⠬⠭⠬⠏⢈⡉⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋


⠀⠀⠀⣄⡞⣉⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠒⢦⡄
⠀⠀⠠⠠⠄⢤⠀⠤⢰⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠙⢆⠂⠁⠔⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⠿⠋⠀⠀⠀⠀⠀⠀⠀⣰⠃⣀⠀⣠⣶⠶⠹⡆
⠀⠀⢀⣔⡆⣰⣠⣶⣆⣀⡀⠀⠀⠀⠀⠀⢡⣄⣀⠀⡈⣧⠴⡞⡃⠀⠀⠀⠀⠀⠈⠻⠷⠀⢠⡶⠆⠀⠀⠀⠀⠀⠀⢻⡀⠉⠁⠉⠁⠀⢠⡇
⠀⠀⠀⠙⠻⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠉⢿⡜⠛⣻⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣿⣶⣶⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀
⠀⠀⢠⣾⠏⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⢀⣀⡴⠉⠉⠉⠉⢳⣀⡀
⠀⠀⢸⠫⠠⣤⣤⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⢬⣿⣻⣖⢄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣽⣛⣿⣿⣻⣿⣛⣻⠁
⠀⠀⠰⣦⣤⣽⣿⣷⣷⣦⣄⠀⠀⠀⠀⠀⠀⡆⠞⠁⠆⠰⠠⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠹⠿⠋⠉⠏⠉⠻⠟
⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⢿⠀⠀⠀⠀⠀⠀⠐⢿⣀⣃⡜⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠛⠂


⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⡀⢈⣤⠶⣦⣌⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠾⢿⣿⣭⡇
⠀⠀⠀⠁⢾⡁⣾⠍⣹⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⣤⣾⣇⡸⡿⢠⡿⠟
⠀⠀⠀⠁⢈⠻⠷⠾⢋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⠖⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣂⣈⣉⣉⣽⡷⢶⣤⠀⢶⡶⠀⢰
⠀⠀⢶⣶⣶⣶⡆⢰⣶⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠒⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢉⡛⠛⠟⢋⡠⢾⠃⠰⠈⠁⡠⡊
⠀⢰⣦⣬⣛⠻⢷⣸⣿⡀⢀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⢧⣿⣿⢶⣤⡷⢟⣣⣤⣤⠾⠃⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠈⣿⡇⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⢉⣁⣠⣿⣿⡿⠉⣀⣤⢠⠜⣧
⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣶⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡟⠃⢀⣘⣯⣿⠆⢀⣠⡄⣀⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⠧⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠄⠛⠋⠾⡴⢬⠅⠀⠘⠓
⠀⠠⠤⡤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⡀⠉⠛⢤⡀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿
⠀⢠⣿⡓⢄⠈⢯⡔⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣾⡿⣿⡿⣷⡿⣿⠿⣿⡿⣿
⠀⢠⠄⡀⠈⢤⡂⠐⠱⡊⣷⣦⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⡿⣿⡿⣿⡿⣿⡿
⠀⢸⡄⠀⡨⢀⣎⡂⠀⠑⢮⣙⣋⣉⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⣿⣿⣿⣟⣿⣟⣿⣟⣿⣟⣿⣟⣿
⠀⢀⣀⠀⠀⣈⠉⠈⣀⠈⠀⢀⣀⣉⡁⠀⠀⠀⠀⢀⢀⡀⡀⡀⡀⠀⠀⢀⡀⠀⠀⣈⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋
⠀⢸⣿⣦⡀⠘⢷⡀⠙⢆⣄⠨⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⣻⣋⢿⣠⣼⢿⡝⠤⣙⣻⡣⡥⡏
⠀⢸⣿⡌⠳⣦⡘⢷⠀⠀⠻⣷⡌⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⣥⣷⣁⣨⣿⣽⠣⣟⣷⣟⠘⣦
⠀⢸⣿⣿⣆⠙⣷⡌⢷⣄⠀⣸⣿⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣿⡿⣶⣿⣿⣿⣷⣦⣻⣟⣰⣡⡇
⠀⢸⣿⣿⣷⠀⠀⠻⣿⣿⣷⡈⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣿⣿⣿⣿⡟⣾⢏⣿⣧⣝⢣⡔⣃
⠀⠈⠉⠉⠉⠁⠀⠀⠙⠋⠙⠉⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠚⠃⠀⠀⠀⠀⠛⠓⠛⠛⠛⠚⠛⠛⠚⠛⠒⠒⠋
⠀⠘⠢⠘⠿⣿⣜⢿⣷⡄⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣶⠐⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠈⠀⠙⠛⠣⡙⢷⡀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⠟⠋
⠀⢰⣤⡀⠀⠰⡀⠀⠀⠀⠙⡄⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢄⠀⠀⠙⢷⣄⢾⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠉⠛⡀⠀⢳⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⣀⣿⣿⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠉⠛⠁
⠀⠀⠀⠀⠈⠀⠀⠘⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠿⠷⠿⠿⠿⠿⠿⠂⠀⠀⠀⠀⠶⠿⠿⠿⠇
⠀⣶⣶⣶⣶⣶⡰⣶⣶⢶⣶⣶⣶⣶⣶⠄⠀⠀⠀⠀⡶⠂⠀⡄⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣻⣿⣿⠂⠀⠀⠀⢀⣷⣄⠀⠻⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⠛⡶⢲⢦⣆⡀⠒⣠⠁⠀⠃
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣼⣿⣷⡄⢀⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣝⣆⠉⠚⢁⠿⢩⣄⢀⡀⠄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⢸⣿⣏⢻⣿⣿⣿⣧⣀⣀⣤⣾⣿⡇⠀⠀⠀⠀⠀⡀⠈⠣⠝⣶⢟⣻⡟⠀⣀⠐⠂
⠀⠿⢿⡿⢿⡿⢿⡿⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⠀⠙⢿⣿⠷⠙⢻⡿⠿⠛⠁⠀⠀⠀⠀⠀⠉⠃⣴⠓⠚⠆⠀⠳⠈⠣⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠀⠁
⠀⣰⠆⢠⡶⣶⡶⢂⣤⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⢀⠀⠠⠀⠂⠄⠀⠀⣠⣤⠀⠀⠀⠠⡄⠀⠀⠀⠀⣠⠶⣰⢆⡴⠦⣴⢶⠶⣀⠶⡀
⠀⣿⠀⣼⠀⣿⣿⢋⡾⠟⠳⢶⣍⣏⡻⠇⠀⠀⠀⠉⠔⠣⠀⡔⡂⠀⢀⢌⡳⠆⠀⣱⣾⡇⠀⠀⠀⠀⣩⠿⣽⢮⡵⢿⣽⢿⠿⣩⠿⡅
⠀⣿⣆⣿⡀⢿⣿⣾⠃⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⢠⠐⣈⠂⠐⠠⠀⡀⣾⡗⣀⠂⠀⢨⠀⠀⠀⠀⠀⣭⠿⣽⢯⡝⢛⣽⣿⠟⣤⠻⡅
⠀⢿⣿⣿⣷⣄⡻⣙⠷⣤⣤⣾⢃⣿⣿⡇⠀⠀⠀⢦⣷⣴⣷⣤⣷⣦⣿⣿⣿⣎⠡⡈⠠⠀⠀⠀⠀⠀⣤⠛⣼⢫⡜⢋⣿⣿⠻⣤⠻⡄
⠀⢷⡙⠿⣿⣿⣿⣿⣿⣿⡿⠟⣡⣿⣿⡄⠀⠀⠀⢌⣵⡫⣭⣻⣿⣿⣿⣿⣿⣿⣶⢬⣶⡂⠀⠀⠀⠀⣤⠛⣼⢣⡜⢫⣽⣿⠛⣤⢟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠃⠘⠁⠉⠉⠋⠀⠙

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠒⠂⠀⡝⠉⡀⢀⠂⠀⠃⢀⣀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⣘⡀⡀⠀⠣⡟⢨⠀⢠⠈⠁⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡇⡜⠀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡟⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣄⣀⡀⠀⠀⢸⣷⠏⠀⣀⣠⡦⡴⠒⡰⠰⠦⡤⡀
⠀⠀⢀⣀⡀⠀⠤⠲⡤⠶⡦⠄⢉⠛⠛⠛⠒⠂⠀⠸⣏⠁⠀⠉⠈⢅⢈⠙⢆⠀⠰⢀⠇⠁⡄⠌
⠀⠀⠈⠀⠀⠀⠀⢁⡀⠀⢱⠄⠀⡀⠁⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠈⠒⢆⠀⢆⠀⡏⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⢸⢃⡐⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⣴⡊⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠀⢀⣀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠎⠀⠀⠀⠀⠐⠒⠚
⠀⠀⠠⠆⠀⠀⠀⠀⠀⠠⣿⠄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠢⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃