Doofus Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⢠⡤⢤⠀⡖⠦⠀⣴⡀⢰⠆⠀⢀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⠚⣏⠁⣿⡤⠈⡷⣏⢸⣗⣶⣸⠿⣧⣼⣀⢠⣾⣷⢠⠏⠙⠀⣳⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⡷⣷⡀⠹⠄⠘⠒⠃⠉⠈⠁⠉⠁⠀⠀⠀⠉⠉⠉⠀⠙⠘⠶⠃⢠⡏⢀⡞⣠⠞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠎⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠧⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠙⠓⠦⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠛⠛⠛⠲⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣠⠄⠀⠀⠀⠀⠀⠉⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⣀⣀⣀⣀⣀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠃⢠⠞⠁⠀⢀⣶⠀⢀⡄⠀⣀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡴⠚⣉⡥⠤⠄⠀⠀⠈⠛⠺⡅⠀⠀⠀⠀⠀⠀⢸⡴⣿⠶⣶⣶⣟⣇⣀⣸⠀⠀⣿⠀⠀⢹⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢠⠏⠀⠀⣁⣀⣀⠀⠀⠀⣠⠴⠒⠻⣄⠀⠀⠀⠀⠀⠀⣙⣿⣎⡁⠀⠀⠙⠢⣍⠛⣟⠻⢤⣀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⡟⠀⣰⠊⠉⠀⢈⡹⡄⠀⡇⢀⣀⣘⢻⡆⠀⠀⠀⠀⡼⢥⠄⠀⠙⣆⡴⠚⠉⣛⢯⡁⠀⠀⢸⠛⠳⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡇⠀⢻⣀⣠⠴⠾⢧⠇⠀⣝⠁⠀⠀⢻⡀⠀⠀⠀⠀⢳⡖⠲⢤⣰⠻⡄⣀⡈⠉⠀⡇⠀⠀⡼⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣇⠀⠀⠉⠀⠀⠀⠀⠀⠀⣘⡆⠀⠀⠈⡇⠀⠀⠀⣀⣠⠯⡄⡞⠁⠀⠛⠉⢉⣳⡴⠃⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣹⠆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣷⠀⢀⡎⡴⠋⢳⣔⣧⠞⠁⠀⠠⣤⠤⠖⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⠀⢀⢀⣤⣄⣀⣠⢤⡀⡆⣿⠀⠘⡞⡇⠀⠸⣄⣹⡗⠛⢉⡟⠉⠙⣎⢧⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠀⠀⠀⢸⡸⣤⢤⡀⠀⢀⠇⢙⡇⠀⠀⠙⢿⣞⡶⡬⣍⣓⡾⣉⡉⠲⣄⡏⡼⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠈⠛⠓⠚⠉⢀⠞⠀⠀⠀⠀⠀⠹⡟⠙⠓⠻⠽⠽⠯⢗⣫⠔⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣤⣀⣤⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠈⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢲⡄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⡀⢀⡴⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣻⢰⡧⡄⢰⡋⠿⠈⡿⠛⠀⢰⡖⠒⣶⢲⡄⡇⣾⠋⠀⣷⣼⡇⣇⢹⡎⣓⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠓⠂⠦⠽⠆⠰⡇⠀⠀⢸⡛⠃⣿⢯⡀⣇⢻⣭⡄⠷⠙⠷⠙⠋⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⢠⣶⣶⠀⣶⣄⣶⠀⣤⡄⣠⡄⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⢀⠀⠀⠀⠀⠀⠄⠀⠁⣀⢀⣶⠀⢶⡆⢿⡄⠿⣯⡁⠀⠀⠈⣻⣦⢀⣿⢀⣿⠀⣿⣷⣿⡇⠀⠀⠙⠿⠿⢿⣿⣷⠀⠀⠀⠀⠀⢀⣀⠷⠀⠠⠐
⠀⠀⠘⣁⣴⡈⠂⣀⣤⡄⠘⣿⣿⣇⡀⠘⣧⣸⣿⢰⣾⡿⠀⠀⠘⠿⠟⣈⠻⠟⠋⣸⡿⠸⢿⠃⠀⠀⠀⣀⠀⠘⣟⠿⠒⣤⣄⣠⣾⣿⣿⣴⠀⠄⠤
⠀⠀⢁⡍⠠⠄⠸⣿⣿⣿⡀⢸⣿⡝⠿⠀⠙⠋⠁⠀⠀⣴⣆⣰⢷⡀⣼⠋⢳⡄⣼⠋⢻⡄⣼⢷⣸⡇⢺⣿⣯⠐⠉⢾⣶⡜⢻⣿⣿⡟⠉⠉⠃
⠀⠠⠀⢠⣾⣷⣦⠈⣿⡟⠿⠊⠉⠀⣤⠀⣾⣛⣁⠀⠀⢻⡈⠟⢸⡇⢿⣤⣼⠇⢿⣤⣼⠃⣿⠸⣿⠃⠘⢾⣄⠀⠀⠈⠉⢠⣿⡿⠀⠀⠀⠀⠀⢠⡀
⢀⣤⣤⣀⠻⣾⣿⠃⠈⢁⣤⡄⢻⡄⠈⣧⢨⣉⣹⠇⠀⠈⠃⠀⠈⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠉⠀⠀⢀⣽⣷⠦⠀⠀⣸⣿⡇⠀⠐
⠛⢿⣯⣹⣧⠈⠉⣄⡀⢺⡁⠰⣆⠻⠶⠟⠀⠉⡁⢀⣠⠀⠀⠀⠆⣴⠀⣷⠈⢋⠀⡏⠀⠂⣸⠃⣴⢠⠆⢾⣿⠃⢸⡿⠛⠉⠀⠂⣀⡀⠀⡀⠀⣀⠌
⠀⠀⠹⠿⠃⠰⣶⢮⣿⣌⠻⠶⠛⠀⢀⡄⢠⡀⢹⡀⠁⢀⠀⠀⣄⣹⠀⣅⢀⣼⢠⡇⣴⢠⡿⠘⢃⡴⠆⠀⠀⡀⢸⡇⢀⠀⠀⠀⣴⣿⣶⣄⠀⢠⠔
⠀⠀⠀⠀⠻⣄⡙⢦⡌⠙⠁⣀⠀⠀⠈⣥⠀⠓⢸⡇⢠⡿⠀⠈⠋⠀⠀⠁⠈⠁⠰⢠⢁⣄⠀⠈⠋⢰⣧⠀⣡⡄⠴⠟⠋⢀⣠⠀⠘⠿⠿⠋⠀⡀
⠀⢠⡞⠛⢶⣌⠻⠂⠀⣀⠀⠙⠀⢸⡄⣥⠀⠘⠋⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠄⠉⠀⢂⡉⠀⠀⠀⠀⠙⠏⢾⣭⢀⡀⠂⠈⠉⢳⡶⢾⡅⠀⠺
⠀⠈⠙⢶⣤⡟⠀⠀⠀⠁⠀⣶⠠⠾⠃⠀⠀⢠⣶⠿⠛⠻⢿⡿⢿⠿⠿⣶⣌⠁⠀⠑⡄⠀⠀⠀⣤⣀⣤⠀⠀⠉⠉⠁⠀⠀⢿⣏⠙⠚⠛⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠲⠀⢠⠀⠚⠋⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠰⢸⠀⠀⠈⠹⣦⡀⠀⠀⠀⠀⠸⣿⣿⣿⣧⣀⠀⠀⠀⢀⣀⣈⠓⢾⡀⠐⠊⠁
⠀⠀⡀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⢰⡿⠃⢰⣟⠟⣧⠀⠀⡇⠆⢀⣰⣶⣌⢻⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⡀⠀⠄
⠀⠀⠀⠆⠀⠀⠀⠀⡈⠀⠀⠀⠀⢀⡠⠸⡇⠀⠀⠙⠋⠁⠀⢀⡇⠣⡀⠙⠚⠋⢀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠓⢀⡾⠋⠀⠄⡁⠆⠀⠀⠀⢀⠔⢉⣤⣄⡈⢳⡀⠀⡘⠢⣀⠀⠀⠀⠀⠀⣸⠿⣟⠁⠀⠀⠀⠙⠻⢿⣧⠀⠛⢦⣄⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣷⣀⢀⠀⢿⡀⠀⣀⢀⡎⠀⠸⣿⣿⡷⠀⣇⠀⠀⠀⠘⣷⠀⠀⠀⣠⠟⣀⡿⠀⢀⡀⠒⢶⣾⣦⡙
⠀⠀⢀⣠⣄⡀⢀⣀⣀⠀⠀⠀⣼⡏⠀⢣⡈⢇⠠⠁⠀⡇⠀⠀⢹⡏⠀⠀⢻⣀⡼⠀⠀⣹⡇⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠈⠛⢿⣿⣧
⠀⢰⣿⣿⣿⣤⣿⣿⣿⣷⡀⣸⣿⠃⠀⠀⠹⣮⠻⠆⢠⡇⠀⣠⣿⣿⣦⠀⣸⣿⠁⠀⢠⣿⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⠃⠀⠀⠠
⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠀⠀⠈⢻⣿⣬⣷⣾⠟⠉⠏⠉⢻⣿⡟⠀⠀⣾⣿⠀⠀⠀⠀⣴⠀⠀⠀⠀⢀⣲⠄⠀⠀⠀⣿⣆
⠀⠀⠀⣾⢿⡿⠋⠀⣿⣿⡿⠋⠀⡃⠀⠀⠀⠀⢠⡿⠀⠙⠿⢇⡀⠀⠀⣼⡿⠀⠀⠀⣿⡿⢄⠀⢠⡀⢉⣀⠃⢀⠀⠙⠛⠀⠀⠀⠀⣹⣿⣷⡀
⠀⠀⢾⠇⣼⡃⠀⣠⡾⢻⡇⠀⢸⣿⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⡛⠛⠛⠙⣷⠀⠀⠀⢸⡷⠀⣀⠀⠈⠉⢿⣞⣋⠀⠀⠀⠀⠀⣀⡴⠋⠈⠙⠻⠦
⠀⠀⠀⠀⠈⠀⠈⠉⠀⠻⠀⠀⢸⡏⠀⠀⠀⠀⡧⠀⠀⠀⠀⢐⡅⠀⠀⠰⣿⠀⠀⠀⢸⣧⠀⠈⢰⠀⠀⠀⢈⡉⢻⡿⠟⠈⠈⠹⣇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠇⠀⠀⠤⠤⠚⠓⠤⠤⣀⣿⣆⠀⠀⠘⣿⡀⠀⣾⠀⣊⠀⠨⠀⠟⠀⠀⠀⠀⠀⠙⠋⠐
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠸⣿⣇⡀⠀⣠⡞⠀⠀⠀⡀⠀⠀⠂⠐⠊⠁⠀⠈⢧⣄⣠⣿⣷⡄⢻⡀⠀⠀⠃⠀⠀⠀⢰⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⠉⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠙⠿⠿⠛⠁⢘⡇⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣄⣀⣴⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢰⡀⠀⠀⡄⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⢀⢱⠲⠀⠀⠀⢠⡾⠀⢀⠀⠐⠀⠀⢶⣶⡄⠈⠀⠀⠈⢻⣿⣇⣈⣩
⠀⠀⠈⢀⠀⠀⠀⢠⠞⠀⠘⠈⣛⠁⠠⡈⠢⠄⣀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢸⠘⣷⠀⠀⠀⢸⡇⠀⠀⠐⠀⠁⠀⣿⣟⠛⣿⣿⣽⠶⣴⣿⣿⠉
⠀⢀⠀⠀⠄⠀⢀⣸⡶⢢⠤⠄⢀⡀⠀⠀⠀⠀⠈⣧⠀⠀⣬⡇⠀⠀⠀⠀⣠⣿⡰⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⣀⠀⠈⣿⡇⢙⡟⠋⠀⠈⡻⠁
⠀⠈⠛⠁⢾⡾⢿⣷⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣆⠀⢨⣷⣤⠀⠀⣄⣻⡿⠁⠀⠀⠀⢻⡄⠀⠀⠀⠀⣾⠟⣿⣶⣤⣟⣁⣀⣀⡄⠐⠁
⠀⠀⠀⠀⢀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⡯⠿⣄⣼⣿⣷⣤⣼⣿⠧⠀⢀⣤⠴⢈⣧⠀⠀⠀⠀⠀⠙⠉⠛⡁⠀⣉⣬⣿⣦⡀⢀⡀
⠀⠀⢂⣰⣿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⢉⣩⣟⣤⣤⣿⣿⣶⡿⠿⠟⠛⠁⠀⢃⠀⠠⣞⣩⣽⣄⠀⠀⠀⠀⣠⣤⣀⣅⠀⠈⡿⣿⣿⣿⣿⠃⠀⠄
⠀⠀⠀⠉⠒⠄⠤⣤⠠⣴⣤⡀⢠⠲⠄⠐⠭⡏⣸⢅⣀⣥⣨⠏⠀⠠⠀⠀⢈⠃⠨⠀⣠⣄⣉⣤⣄⣈⣐⠂⢄⠘⢿⣿⣿⡖⠃⢼⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠘⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠳⠧⢤⡍⠀⣍⣁⠈⣥⠟⠀⠀⠈⠉⠀⢿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣴⡟⠋⢉⣉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡏⠈⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡲⠤⢤⠞⢫⣭⣷⡄⠀⠀⠀⠐⠊⢿⣿⣅⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⠀⠐⣋⠀⠀⢉⡠⠀⠀⠀⠀⠰⢾⣿⣆
⠀⠀⠀⢸⣆⠀⢰⡆⢀⣴⠶⠶⣤⡀⠀⢸⣿⠀⢀⣴⠶⣿⣦⣀⣿⠀⠀⠀⠀⠀⠚⢛⣛⡃
⠀⠀⠀⢨⡿⢦⣸⡇⢸⡁⠀⠀⠀⣷⠀⠘⣿⣼⣿⣷⣾⣿⣿⣿⣿⡦⢤⣔⠤⠀⠀⣀⡬⠁
⠀⠀⠀⠀⡷⠈⠛⠇⠈⠳⢤⣤⠾⠃⠀⠐⠘⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⣀⣀⣀⠀⠀⣀⣤⣄⡀⠀⠀⣠⣤⣀⡀⠀⣀⣀⡀⢀⡀⠀⣠⡀⢀⣤
⠀⠀⠀⠘⡏⠉⢙⡧⢸⠟⠁⠈⠙⣇⣼⠋⠀⠈⢻⡄⣏⣉⣡⢸⡇⠀⢸⡧⣯⣤⣤⡄
⠀⠀⠀⠠⣧⣴⠞⠁⠘⣦⣀⣀⣠⠏⢹⣄⣀⣀⣼⢇⡟⠉⠁⠘⣷⣀⣼⠇⢀⣀⣤⠇
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠁⠀⠁⠀⡀⠀⠈⠉⠁⠀⠈⠉⢁⡀
⠀⠀⠀⢰⡾⠛⢛⠀⢿⡄⣸⠇⢠⡶⠟⠀⣾⠀⠀⣿⢠⡞⠁⡀⢰⡇⠀⢸⡇⣴⠋⣀
⠀⠀⠀⢸⡿⠛⣋⠀⠀⣹⢷⣄⣿⣀⣠⡤⢿⡄⠀⣿⠈⠉⠉⣻⠞⣧⠀⢸⡇⠉⠉⢙⡷
⠀⠀⠀⠘⠟⠛⠉⠀⠰⠋⠀⠉⠙⠛⠉⠀⠘⠻⠿⠃⠀⠒⠛⠋⠀⠙⠿⠟⠁⠐⠚⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⠐⢄⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠂⠑⣄⢀⠄⠀⠤⠠⡀⠀⠱⠁⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡉⣁⠀⠈⠑⠆⠀⢀⠤⠄⠠⢀⠀⠎⠀⠔⠑⠀⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⢀⠬⠤⠤⣀⠀⢠⢁⣶⣿⣷⠈⠄⠀⠀⣠⣄⡐⠀⢍⠁⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠂⠈⢰⣿⣶⡄⠀⠀⢡⠘⠯⠅⠠⠬⠛⠀⠂⠀⠀⠀⣙⡄⠀⠑⡒⠂⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠈⢿⣋⠉⢉⡩⠃⡔⠂⠀⠀⢱⠀⠐⡖⠒⠒⠻⢿⡇⠀⠉⡏⡔⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡈⠈⠀⠑⡤⠒⠋⠀⣠⠤⠃⠀⠀⠀⠀⠉⡂⠈⠑⠒⠶⢉⡀⠀⣆⠙⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⣈⡉⠓⣲⠤⠀⠇⠀⠀⠀⠀⠀⠀⠐⠁⢀⡀⢄⠠⠄⠜⠀⢪⠉⠙⡄⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠂⡉⠀⠈⢲⢡⣾⣶⠄⡄⠀⠀⠀⠀⠀⠀⢄⢿⣿⡇⡷⠊⠉⢢⡘⠢⣰⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠘⢄⠀⠈⠛⠯⠕⠒⢇⠀⠀⠀⠀⠀⢀⠞⠸⠿⠖⠁⠀⣠⠆⢘⡀⣀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠒⢺⠉⠒⣦⢄⡀⠀⠀⠉⠒⠒⠒⠊⠁⠀⣀⠤⠄⠒⣾⢅⣀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠂⢘⠢⠊⢁⠀⠉⢉⢗⠒⠒⠒⣶⠒⠊⠉⠹⠲⠤⡤⠃⠀⠼⠤⠔⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠁⠄⢬⢉⡍⡅⠈⠢⠤⠔⠉⠦⠀⠤⠧⠀⠀⠣⠤⠒⢂⣁⡒⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠘⠁⠀⠄⠔⠀⢂⠀⠀⠀⠀⠈⠒⠒⠒⠒⠋⠁⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⠇⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣛⡀⠀⠀⠀⠠⠐⠂⠐⠂⠀⠤⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠤⠤⠀⠀⠀⠈⠓⢊⠂⠀⠀⠀⠈⣋⣠⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠐⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠐⠀⠳⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠇⠀⢈⠁⠀⡆⠀⣸⠀⢠⡄⡀⠀⢰⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠁⢰⣳⠅⠀⠀⠈⢲⢨⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠘⠁⠂⢁⠂⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠁⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⠔⠀⡐⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠊⠀⠀⠀⠀⠁⠀⠒⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠈⠄⠀⠀⠀⡀⠓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⠸⠀⠀⠀⢡⠘⣀⢤⣈⣿⣿⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣼⡏⣈⣽⣿⣆⠡⣿⣿⢀⢀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⢏⣀⣸⡿⣗⣀⡻⣀⠹⢀⢿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠿⣏⣀⡸⠍⣁⣅⠀⠈⣉⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⠚⠛⠳⣤⠾⣿⣧⣤⡿⢿⠠⢢⡈⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠙⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣶⡆⠀⢀⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⠿⣿⣿⣶⣤⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢻⡏⠀⠈⢉⣉⡉⣰⣟⣻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣇⣰⡏⢷⣀⡄⠛⠻⢃⠘⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡙⣿⠁⢰⣿⣿⣡⣾⣷⣤⣄⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠈⡀⢹⣶⣿⠿⣿⣧⡬⣳⣿⣯⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⡇⢸⣿⣿⣦⠴⣭⣀⣻⣿⣿⣿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠆⠀⣨⣿⠸⡏⢫⡌⠓⢼⣟⡿⠿⢣⢴⡄⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡘⠻⠛⠟⠛⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢠⣿⢻⠀⡇⠀⠙⢤⣈⠍⢡⣾⣧⡈⣷⡀⠻⠆⡀⠀⠀⠀⠀⠀⠀⡀⠊⢠⣴⡿⣷⡄⢀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣿⠉⣼⢖⡇⣠⠴⠂⠉⠁⠀⠙⠛⠃⢙⣿⣄⠘⢌⠢⡀⢀⡠⠴⠊⣹⣤⣿⣿⣿⣿⡇⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⡀⠀⢠⠆⠀⠤⠼⠿⡿⠛⢿⣿⠟⣀⠀⣤⡀⠘⢿⣿⣶⣼⣿⣿⣧⣈⡦⠛⣉⣤⣾⣿⠿⠧⠼⠻⠟⠿⠥⠤⠼⠿⠿
⠒⠀⠀⠀⠁⠀⠀⠀⡎⠀⠀⠀⣀⣴⣣⣾⣷⡉⢻⣏⣓⣿⡗⣈⠿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠈⠉⠐⠂⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⠟⠁⠀⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣍⡀⠀⢀⠐⢀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠳⣄⠀⠀⢀
⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡀⠀⠙⢧⡔⢀⠀⠀⠀⠀⠀⣾⣿
⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢹⣮⠱⡄⠈⠻⣄⢇⣀⡀⣰⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⡯⠀⠘⣿⣷⡀⠈⠻⢿⣿⣿⣿⣿⠻⢿⣿⣠⡀⣠⣿⡇⠈⠑⣄⠙⢿⣘⣷⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠸⣾⣿⠃⠸⠇⢸⣿⣇⠀⠀⠀⢙⣿⣿⣿⡆⠀⠙⢿⣿⣿⣿⡇⢀⠀⠀⠑⢾⣿⣽⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡄⢀⣿⣿⠀⠀⠀⣸⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣀⣠⣾⣿⣿⡿⠃⠈⢆⡀⠀⠀⠙⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣤⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠙⢦⡀⠀⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠭⣀⣀⡀⠀⠀⠀⠙⢦⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣥⣁⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⢀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⠐⢶⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⣋⡉⠁⠀⠀⠀⠉⠙⠲⢿⣄⠉⠳⢦⡠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣙⡷⠶⠂⠀⠀⠀⠈⠂⠀⠀⠙⢿⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠒⠒⠶⣦⠄⠀⠀⠀⠀⠀⠀⠀⢸⣇⡠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⡢⠤⠄⣀⠀⠀⠀⠀⠉⠀⠀⠀⢤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠞⠁⠀⠈⠓⠤⢄⡀⠀⠀⠀⠀⠀⣄⠠⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⠩⣅⣠⠤⠶⠒⠚⠋⠉⠑⢢⣠⠴⠊⠁⠀⠀⠀⢀⣴⣖⠁⠀⠀⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢤⣀⠀⠀⢺⣀⠀⢀⡖⠉⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢆
⠀⠀⠀⠀⠀⠀⠀⢧⣀⡀⠀⠈⠙⢶⡖⠋⠉⠹⡔⠆⠀⠀⠀⠀⢀⠠⠚⠁⠀⠀⠀⠀⠀⠀⢀⣾⠤⠴⠶⢤⡀⠀⠀⠀⠀⠀⣄⠱⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠛⠂⠀⠀⠁⠀⠀⠀⠙⠒⠛⠓⠲⠋⠁⠀⠀⠠⣄⡀⠀⠀⠀⠀⠉⢠⠖⠛⠟⠂⡿⠀⠀⣠⡆⠀⡏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣳⣶⢤⣄⣀⣀⣀⣀⣀⣠⣤⣤⠶⠶⠒⠒⠉⠉⠁⠀⠙⠀⠀⢀⡴⠛⠲⠤⠤⠶⠛⠛⠒⠚⠁⠧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⠤⠞⠙⢧⣩⠟⠁⠀⠀⠀⠉⠓⠦⢄⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣼⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠽⠤⠤⠴⠓⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠂⠀⠀⠀⢠⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡀⠀⠀⡾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣸⠀⠀⠀⡇⢸⠇
⠀⠀⠀⠀⠀⣠⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠛⣶⣶⠚⠃⢸
⠀⠀⠀⠀⠀⠁⢰⠚⠲⣄⠀⠀⠀⠀⠀⠀⢀⠈⠀⣀⣀⣀⣀⣰⠛⡏⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⢈⣦⠀⢤⠤⠄⠒⠋⢁⡶⢻⠟⠙⢻⣇⣸⠷⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢧⡤⠞⡇⠀⠀⠀⠤⣀⣀⡊⣰⠃⠀⠀⠀⠀⠀⡠⠀⢹⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢻⡀⠀⠀⠒⠀⣉⣿⡙⠓⠶⠀⠀⠀⠒⠁⢠⡞⢈⡏⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣷⣠⡴⠖⠋⠀⠈⠳⢤⣀⠀⠀⠀⣀⡴⠋⣠⠞⣹⠳⠤⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⡾⠶⠖⠋⣹⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠲⠀⠀⠐⠰⠐⠐⠀⠀⠂⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠂⠘⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠙⠉⠛⠛⠙⠋⠛⠀⠛⠙⠙⠛⠀⠘⠘⠋⠚⠛⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⣤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠤⢤⠀⠀⠐⠂⠤⠄⠀⡀⠠⠤⠤⠔⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠂⠀⠈⠻⠀⠀⠀⠀⡄⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⡰⢊⠗⠤⢤⠁⣵⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠤⢾⣧⡥⢠⠀⠀⠀⠀⠀⢖⢾⠀⢁⡆⠠⡟⠀⠀⠀⠀⠀⠎⢧⡄⠀⠀⠠⢃
⠀⠀⠀⠀⠀⠉⠁⠉⠉⠁⠊⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⢨⠀⠀⠀⠀⠀⠠⡄⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⠮⡁⡍
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⢐⡠⠀⠀⣤⠉⠃⠙⠀⠀⠈⠀⠁⠀⠀⢰⢀⢼⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣇
⠀⠀⠀⠀⠀⠀⢐⣄⠀⠀⠀⠀⠀⢀⠠⠀⠱⢄⡀⠀⠀⠀⠀⠐⠠⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠁⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⢀⠀⡁⠀⡸⡦⠄⠀⠀⠀⠀⠀⠃⠀⣰⣸⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠀⠐⠁⠀⣀⡀⢨⡃⠀⠀⠢⠄⡀⠀⠀⣀⢀⣼⡺⠁⢀⢸⠗⠀⠀⠀⣀⡄⡲⠖⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠅⠀⠀⠀⠲⠦⠬⡵⣶⢟⠂⠀⠄⠀⠀⠑⠒⠚⠫⢼⣅⠀⠴⠁⠀⠀⠀⠈⠉⠉⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⢰⡇
⠀⠀⠀⠀⠀⠀⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡘⡀⠀⠀⠀⠀⠀⠀⠀⣻⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣴⠾⠂⠀⡅⢇⠀⠀⠀⠀⠀⠀⠀⠈⢻⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠈⠩⢉⡩⠋⡙⠒⠲⢶⠊
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢰⠟⠀⠅⣄⠀⣣⠘⠖⠒⠒⠃⠀⠀⠀⠐⢸⠇⠐⠌⠐⠦⠄⠀⠀⠀⠀⠀⠀⠴⠁⠠⠠⠬⠠⠔⠌⠠⠀⠀⠀⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠠⠀⠀⠄⠠⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠆⠠⠄⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢿⡆⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⡾⠀⠀⠀⠀⠀⡀⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⠀⠠⠤⠀⠀⢀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠁⠀⡀⠀⠀⢀⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⢀⣞⠄⠘⠋⠢⣄⠀⡇⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⡡⠄⡰⠀⠈⣦⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⢐⡀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠐⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⠒⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡶⠊⢀⠌⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣫⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣊⡷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢈⣁⣀⣀⣀⠈⠦⢶⣤⡸⠟⢃⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠾⢧⣬⡼⠿⢷⡀⢾⡿⢇⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢿⣤⣼⠟⢿⣤⡝⣧⠔⢣⡼⡏⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣤⡼⠛⢧⣤⡜⠧⣡⣤⡄⠀⢤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣤⣦⣉⣉⣙⣶⣛⣿⣷⣤⣞⢻⡁⠳⡤⠤⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢀⠀⠀⢈⠂⣂⡀⠀⣀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⡔⠒⠀⠁⡄⠖⢲⠀⡲⡆⢰⠂⡗⢠⠖⡆⠚⠀⠀⡔⠂⢠⠒⡄⠐⠲⠲⠀⠀⢠⣚⡀⢠⣫⠀⡁⠇⢸⠦⡅⡶⢨⠀⡡⠃
⠀⠀⠀⠈⠉⠀⠀⠀⠋⠉⠀⠉⠁⠈⠀⠁⠀⠉⠁⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠰⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠛⠛⠉⠉⠉⠉⠉⠙⠛⠾⣿⣄⠉⠓⢦⣀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠙⠛⠓⠲⢶⣦⣤⠀⠀⠀⠀⠙⠂⠀⠀⠉⠳⣯⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣭⣭⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠘⡷⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠊⠉⠉⠉⢛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣶⡤⠂⠐⠋⠉⠉⠁⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣀⠀⠀⢧⣀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠳⣶⠚⠋⠉⠹⣖⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡥⠴⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠟⠛⠒⠀⠀⠈⠀⠀⠀⠀⠙⠒⠒⠒⠲⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣤⠶⠿⠷⠀⡿⠀⠀⢀⣴⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠶⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣠⠶⠦⣄⣀⣠⡴⠞⠷⠦⠴⠞⣹⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢻⣌⣩⠟⢿⡛⢛⣿⠛⠉⠉⠉⠑⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠚⠛⠵⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⢀⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣄⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣹⠀⠀⠀⣸⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠠⠷⠤⢀⠤⠟⠀⡟
⠀⠀⠀⠀⠀⠠⠉⢠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡿⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠙⣆⠀⠀⠀⠀⠀⣠⡴⠀⢀⣴⣶⠂⢰⡶⠎⢠⣦⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣿⠂⠈⠛⠃⠀⠀⠀⠀⠂⣠⠁⠀⠀⠙⠂⠈⠉⠐⢦⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡞⠉⢿⠀⠀⠀⠐⠒⠲⣦⣄⣼⣁⠀⠀⠀⠀⠀⢀⡔⠀⢠⡿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⣆⠀⠀⠈⠉⣀⣤⠿⣇⠉⠙⠓⠀⠀⠀⠉⠀⢠⡞⠀⣼⣨⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣦⡴⠞⠋⠁⠀⠀⠙⠷⣤⣀⠀⠀⢀⣠⠔⠋⣠⡞⢁⡟⠳⠞⠋⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡿⠶⠖⠛⣉⡷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⣀⣴⣿⣠⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠿⣿⡿⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡙⢁⡈⠁⣤⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⠀⠀⠀⢠⠔⠁⠀⢹⣿⠟⠫⠈⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠄⠀⠀⠀⢘⣿⣤⠖⢛⡹⠟⡀⠁⢀
⠀⠀⠀⠀⠀⠀⠈⠁⢆⡄⢀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠁⠀⠺⠿⣦⡀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢦⣄⡀⠰⣤⣴⣿⣥⡭⠉⠀⠀⢹⣞⣷⡄⠀⠀⡑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡿⣻⣿⣿⣿⠏⠀⢀⣴⣿⣟⣽⡏⢢⠄⠀⡀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠸⣿⣿⣄⣀⣀⣸⣿⣿⣿⣿⡇⠀⢀⡠⠃⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠰⠗⠀⠀⠒⣾⣿⣿⣷⣦⣿⣿⣟⣟⠉⣴⠖⠉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⢀⠔⠊⢁⠔⠋⠀⡸⠁⠸⡀⠙⢿⣴⣿⣃⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡁⣀⣴⣏⣀⣀⢀⠃⠀⠀⠁⠀⠀⢀⠁⠈⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣤⡖⢀⣤⣴⣄⣰⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠿⡿⠛⠁⠀⠀⠀⠛⢻⡿⡁⠀⠀⠀⠀⠐⠠⠄⣀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣄⠀⠀⠀⠀⢀⣀⣿⠅⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⣿⠀⠀⠀⠀⠀⢻⣿⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿⣷⣶⣾⣿⣿⣀⣀⣤⣶⣤⣄⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⠶⠦⢤⣙⢷⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠳⠶⠤⠤⣤⣀⣀⠀⠈⠙⠳⠀⠈⠓⢼⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣭⣁⣀⣀⡀⠀⠀⠀⠀⠀⠙⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⣭⣀⣀⠀⠀⠀⠀⠰⠛⠉⠉⢉⣙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠋⠀⠸⠯⣄⣀⠀⠀⠀⠀⢀⣈⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⢲⣖⣤⠤⠴⠒⠒⠒⠒⣦⣩⠴⠋⠀⠀⠀⠀⣠⡖⠋⠁⠀⠀⠀⠀⠈⠳⣄⠁
⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀⠀⠸⣀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⢀⡤⠟⠁⠀⠀⠀⠀⠀⠀⠠⣄⡈⣆
⠀⠀⠀⠀⠀⠘⢧⣄⡀⠀⠉⠳⡖⠋⠉⢹⡚⠂⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⢀⡾⠴⠖⠲⢦⡀⠀⠀⠀⠀⣤⣱⠈⠉
⠀⠀⠀⠀⠀⠀⢴⣞⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⣀⣀⣉⣲⢦⡀⠀⠀⠀⢀⠠⠚⠛⠛⣠⡇⠀⣠⢾⠀⡿⠙⠁
⠀⠀⠀⠀⠀⠀⠸⠬⢿⡓⢶⣤⡤⠤⣤⠴⠖⠲⠾⣋⡉⠉⠁⠀⠀⠀⠁⠀⣠⠞⠉⠉⠓⠒⠋⠁⠉⠉⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠑⠖⠁⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣶⣇⣀⣰⠯⣱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠂⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡀⠀⠀⡏⢸
⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢨⠃⠀⠀⡇⢸
⠀⠀⠀⠀⡰⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠘⣿⣿⠛⠁⡾
⠀⠀⠀⠀⠀⡾⠉⠑⣄⠀⠀⠀⠀⠀⣰⣃⣀⣤⣤⣤⣴⠋⡏⠀⠀⡇
⠀⠀⠀⠀⠀⢹⡀⢀⣼⠒⠒⠒⠊⠉⠁⡼⢫⠟⠉⠈⠻⠼⠟⠻⢥⡇
⠀⠀⠀⠀⠀⠀⢻⡋⢸⡀⠀⠀⣐⠒⠶⣧⢯⣀⡀⠀⠀⣀⠔⠀⡼⢲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢄⢣⡀⢀⣠⠴⠒⠻⣄⠀⠉⠀⠀⠀⢀⡼⢁⡼⣾⣇⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠈⠙⠲⠤⣴⣚⡥⠴⢯⡴⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠖⠋

⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠸⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⢠⠀⠀⢀⠔⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠀⣠⡟⠀⠀⠋⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣵⣟⣋⣤⡀⡄⣼⡏⠁⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⣿⣿⣿⠛⠁⠸⡟⠃⠀⠀⠀⢻⡿⠲⢋⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡛⠻⡷⠄⠀⠉⠐⠀⠠⠐⠉⣀⠔⠁⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡜⢋⠔⣊⠀⠘⠂⠈⠈⠈⠀⣤⣴⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠏⠩⠹⣿⣇⠾⢓⠂⠀⠀⠀⠀⠀⠈⣿⡿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠀⢼⣿⣧⣄⣰⡦⠀⠀⠀⣠⣰⣟⠀⠀⠙⠀⢀⠀⠒⡄
⠀⠀⠀⠀⠀⠀⢠⢁⣀⡀⠀⠀⠀⠀⡿⡿⠛⠛⢯⣲⣼⡃⡀⠾⠁⠀⠀⢃⠴⠋⠐⠲⡄
⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠈⠀⡠⠏⠐⠄⠀⠀⠀⠀⢙⡿⠖⠂⠿⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢼⠃⠀⠀⠀⣸⠓⠒⠐⢻⡂⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡿⣝⡲⢖⣭⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣄⡤⡤⠦⠼⣩⣿⠟⢶⣌⡘⠄⠀⠀⠀⢰⠓⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢛⠀⠀⠀⣠⣿⣻⠀⢺⣿⣿⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⢻⡅⠀⠀⡿⣿⣶⣶⣶⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠈⠀⠀⠘⠃⢤⣀⣠⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠉⢁⡀⠀⠀⠀⠀⠈⡟⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⡰⠀⠀⠀⠀⠀⠀⡤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠖⠉⠠⡟⠃⠀⠀⠀⠀⠀⠠⣥⠖⡇⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢀⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠈⠙⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⡀⠀⠀⠀⠀⣠⠖⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠶⠦⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣠⣶⣤⡈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⠀⠀⠀⠀⢀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢡⠇⢀⡀⠀⠀⠀⢀⡀⠘⡇⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣸⣯⣭⣭⣄⠀⠘⢮⡉⠀⣿⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡸⠋⠀⠀⠀⠙⠳⣦⣄⣀⣼⢿⡆⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠘⠉⠙⠲⣤⣀⠀⠀⠉⠉⢁⣼⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠙⠻⠶⣶⣾⡿⠟⠁⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⠰⠄⠀⠀⢀⣀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠹⣿⣷⡄⣠⠾⣿⣿⡇⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣌⠻⢿⡇⢀⣿⡿⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠷⣤⣙⠻⣟⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢊⣽⢷⣌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⣿⠁⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣄⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠠⠅⠰⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⠃⠠⠄⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⡇⠀⠀⠀⠰⠂⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠐⢂⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡸⠁⠤⠄⠙⠋⠀⠀⠀⢢⡈⠁⡸⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢣⡈⠁⡸⠀⠀⡿⢏⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠉⠁⠀⠀⠘⣧⡼⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡄⢤⡀⣭⠱⣄⠀⣀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠊⣭⠀⡤
⠑⢂⡈⠻⠆⠈⢶⠉⣨⠃⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠘⡅⠈⡶⠃⡰⠟⢁⡀
⠷⢄⣉⠒⠈⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠬⠽⠿⠿⠿⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠁⠒⣋⡠
⠀⠀⠉⠙⠢⠄⣀⣀⡸⠛⠦⢀⡀⠀⠀⠀⠀⠙⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡀⠴⠚⢇⣀⣀⠠⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢆⡀⠀⠈⠃⠦⣀⠀⢠⡤⣤⣈⣙⣛⠲⠶⣦⡤⠤⠤⢤⣤⠶⠖⣛⣉⣁⣤⢤⡄⠀⣀⠴⠚⠉⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠉⣻⠀⠀⠀⠀⠀⠉⠋⠛⠀⣀⣀⠀⠛⠉⠉⠀⠀⠀⠀⠀⣟⠉⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣤⣤⣤⣤⣤⣤⣤⣤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⡴⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢳⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢸⢿⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣾⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠈⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡇⠀⣷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣣⢀⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢹⢸⡇⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣾⢸⣇⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣹⠋⠀⠀⣍⣹⣼⡇⡅⠀⠈⢙⣏⠻⣦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣶⣶⣶⣾⣭⣷⣶⣶⣶⣶⣶⣶⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠶⠒⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣋⣡⣤⠤⠤⠴⠒⠒⠒⠦⠄
⣀⣤⣴⣶⣾⣿⣿⣿⣗⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢟⡻⠟⠉
⠋⠉⠉⠈⠀⠀⠀⠉⠉⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣡⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠛⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠁⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠨⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⡀⠀⢠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⣰⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠔⠛⠳⣶⣿⣿⣷⠀⠀⠀⠀⠀⣤⢄⡐⢶⣤⣾⣿⣿⣇⣼⡇⠀⠀⠀⠀⣠⢺⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⢻⡟⠻⣿⡿⢿⡿⢿⡿⠛⠛⠛⣻⣿⣿⠆⠻⢿⣿⣦⡀⠀⡜⢱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⢸⣿⢆⠀⣀⣾⠀⠀⢣⣀⣴⡾⠛⠻⣿⣇⠀⠀⠉⣿⣿⡈⠁⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⣠⣿⠃⠀⣹⠟⠉⣳⠶⠉⠨⣿⡁⠀⡀⠹⣿⣷⣤⣴⣿⣿⠉⠙⢻⣦⡀⠀⠀⠀⠀⠀⠀⡠⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠋⠀⢠⠇⠀⣼⡷⠀⠉⠘⣿⡅⠀⠘⠀⠈⠛⣯⣍⣽⢧⠀⣀⣼⡏⢳⡀⠀⠀⠐⢤⣜⣣⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢯⡿⣤⣄⡆⠀⢸⣧⣿⠀⠀⠀⢸⣿⣤⠁⠀⢠⠞⠋⢻⣻⣧⣴⣿⠟⠁⠈⠀⠀⠀⠀⣐⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⢛⠋⠀⠘⠛⣷⠀⠈⡏⣹⡷⠚⠷⣟⠟⢻⡀⡰⠂⠀⠀⠸⣿⣿⠋⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠻⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⠀⠀⡠⠸⣿⡟⠿⢻⣿⣆⠀⠀⢰⠦⣾⣷⡖⠀⠀⠀⠀⣼⠿⣧⣀⣀⣀⣠⣤⠟⣿⣿⣿⡄⠀⣿⡇
⠀⠀⠀⠀⠀⢠⣿⡟⠻⣆⠀⠀⠀⢀⡄⠰⣿⣀⠀⢉⣿⣯⡤⠀⣠⡟⣠⠟⠱⢀⣤⡤⢾⡟⠀⠄⠉⠋⠩⡝⣣⣾⣿⣿⣿⠁⠀⣾⣿
⠀⠀⠀⠀⢀⣿⠏⠀⠀⠈⢻⣦⣴⡟⠀⠀⠹⡧⣼⣿⡟⢿⣿⡿⠋⠁⠘⣦⠶⠫⠁⠀⢸⡇⠘⠀⣴⣷⣷⣾⠟⠉⣰⣿⣿⠄⢀⣿⣿⠂
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⣿⡏⠀⠀⠟⣧⡀⢀⣼⠇⠀⠀⠀⠀⢸⡇⠀⠘⢋⣿⠋⠑⠖⠚⣹⡿⠙⠀⣸⣿⡟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⠀⠀⣿⠀⡀⠀⠀⢸⣿⢌⡁⢈⡤⠀⣀⡴⠻⣧⠀⣼⠋⠀⠀⠀⣠⡾⠋⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⢸⣷⣤⠶⠛⠉⠉⢻⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡇⠀⠀⡞⠻⣧⡿⠀⠀⢿⣌⠁⠀⠀⢰⡿⠋⠀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⢻⠀⠀⠀⠀⢻⣅⠀⠀⣧⡴⠪⡟⠲⡄⠈⣿⠀⠀⢠⡟⠀⣠⡾⠛⠉⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⢀⣿⠿⢿⣆⠀⠀⢸⠀⠀⠀⠀⠀⠙⠛⣬⣿⣆⣀⣾⡀⢹⡄⠘⣆⠀⢸⠇⣼⣿⣶⣶⣶⣦⣤⣞⣛⣻⣿⣽⣿
⠀⠀⣼⣿⡿⠋⢻⡆⠀⢀⣤⠞⠁⠀⠀⠹⣧⡀⠀⡆⠀⠀⠀⠀⢀⣴⠟⠁⣿⠛⠉⢉⣿⡇⠀⠙⡄⠈⠀⣿⣿⡿⣶⣤⣾⡉⠙⠛⠛⠛⠛⠉
⢀⡼⢿⣿⠁⠀⡘⢧⣶⣿⠃⠀⠀⠀⠀⠀⠈⢿⣄⣿⠀⠀⠀⠀⣼⠛⠳⣄⡿⠀⠀⢸⣿⡇⠀⠀⠘⣆⣀⠘⣿⣀⡼⠋⠙⢯⣷⣄⠀⠀⠀⢀
⣼⠁⠀⡇⠀⠀⢀⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢧⠀⠀⢰⣿⠀⠀⠀⠉⠳⣤⣿⣿⣿⡀⠀⠀⢿⣿⢀⣿⣯⣴⣶⣶⣦⠉⠛⠿⠿⠿⠋
⡇⠀⠀⠃⠀⠀⢿⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⠀⠈⣿⡀⠀⠀⠀⢀⣿⠟⢙⢻⣷⣤⣄⣤⡿⠿⠿⡿⣟⠀⠈⣿
⠀⠀⠀⢸⠀⠀⠸⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⢿⡇⠀⠀⢀⣾⠏⠀⠀⠑⠿⠷⢿⣧⡄⠀⠀⢈⣿⠀⠀⠏
⠀⠀⠀⠀⠁⠀⠀⣽⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⣇⢀⣠⣿⡿⠀⠀⠀⠀⠀⣠⣾⣿⣁⠀⣀⣼⡇
⠀⠀⠀⠀⠘⣤⡾⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢻⡇⠀⠀⠀⠀⡼⠏⢀⣼⢏⣸⣿⠋
⠀⢀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡇⠻⣆⣀⣀⣤⣼⠷⠞⠋⠑⣦⡿⠁
⠺⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⣶⠾⠛⠛⠛⠋⠉⠀⣷⠄⡀⢹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠿⠋⠑⠀⠀⠀⠀⠀⠀⠀⢠⠤⠐⠋⠀⠙⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⠿⡋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣼⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠉⠁⠒⠢⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⠀⠙⢆⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⢄⡀⡰⠁⡴⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠾⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏⢀⡵⣕⠀⠀⠀⠀⠈⢢⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏
⠀⠁⠢⠀⠀⠒⠄⡀⢀⠔⠀⠈⠊⠐⣈⡻⠃⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⠁
⠀⠈⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠌⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡇⠀⠀⠀⠀⢸
⠀⢰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⢂⡔⠛⠃⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⡇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⡀⢀⡠⠃⠀⠀⠀⢀⠤⣚⡐⢛⣤⠤⣀⡀⠀⠈⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠀⠀⢧⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠞⠁⠄⠀⠣⠄⢠⠤⠊⠉⠓⠢⢤⠴⠃⠀⠀⠀⠀⢀⣀⡤⡞⠁⠀⣸⠀⠀⠈⠳⢤⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠒⢉⣉⡀⠔⠀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠀⡸⠀⠀⡱⠦⠴⡄⠀⠀⢰⡆⠠⣥⠀⢀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⣇⣀⣀⣧⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⢸
⠀⠀⠀⠐⠲⢄⣄⣠⢶⠊⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣄⠀⣰⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠒⠒⠋⠉⠉⠉⠉⠉⠉⠓⠒⠒⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠁⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡆⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡤⠤⠖⠋⠁⠀⠀⠹⣄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡞⠀⣀⡀⠀⠀⠀⢀⣀⠈⢧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡤⡼⠁⢛⣛⠛⢦⠀⠀⠘⣛⠛⢺⡠⢤⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⢻⠀⠀⠁⠀⢸⠀⠀⠀⠈⠀⢸⠁⣸⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠘⠛⣆⠀⠀⠀⠙⠒⠀⠀⠀⠀⢸⠗⠃⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⣠⠼⢆⠀⠀⠀⠛⠀⠀⠀⣠⠯⣀⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⢷⡚⠉⠀⠀⠀⠙⠦⣤⣤⣤⠴⠚⠁⠀⠀⠉⢓⡶⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠉⠓⠦⢤⣀⣀⡤⣤⣤⡤⣤⣀⣀⡠⠴⠚⠉⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢳⡘⣞⢷⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢹⣣⡑⣌⢢⠳⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⡄⠀⠀⡞⠀⠀⠹⣝⠎⠀⠀⠀⠘⡆⠀⠀⢀⡄⠀⠀⠀⠀⠀⢧
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡼⠀⡠⢼⣁⠀⠀⣠⠼⠓⢤⡀⠀⠀⣹⠤⡀⠀⢧⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢠⡷⠊⠀⠀⠈⠙⡿⠀⠀⠀⠀⠸⣟⠉⠀⠀⠈⠲⣜⡆⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⢣⡀⠀⠀⣀⠴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⢤⣀⠀⠀⣠⠇⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⣧⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⣤⠤⠤⣼⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⠴⠧⣤⠤⠧⢤⡿⣧⡤⠼⠤⠤⢤⡿⠤⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡏⠀⠀⢸⡇⢸⠀⠀⠀⠀⠈⣇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⢸⠁⠀⠀⡼⠁⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠸⡄⠀⠀⣇⠀⠀⠀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢾⣀⢀⡼⠁⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠙⣆⣀⣹⠤⠒⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣒⣒⣠⣼⣧⣼⣤⣤⣤⣀⣒⣒⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⠛⠙⠛⠻⠛⠛⠛⠿⠿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⠤⠀⠀⡤⠔⠠⠤⣤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⣀⣀⡠⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠶⠶⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⠠⠴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⠤⠠⢄⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠁⠀⡶⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠤⣤⡔⠋⠉⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠹⣿⣿⣿
⣿⣿⣿⣿⡇⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢹⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⠟⠛⠛⠉⠉⠛⠛⠿⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠐⠒⠒⠀⠀⠤⠴⠶⠄⠀⠐⠒⠛⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠛⠒⠒⠂⠀⠀⣀⣀⣀⣀⣀⣠⣀⠀⠀⠐⠒⠚⣏⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠣⣄⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⢀⡨⢿⣿⣿⣿⣿
⣿⣿⡿⠋⠁⠀⠀⠈⠑⢦⣀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣠⠶⠋⠀⠀⠙⠻⣿⣿
⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠜⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻
⣿⣶⡾⠀⢀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⣿⣿⣄⣼⠟⠀⣠⡆⠀⣠⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣀⡀⠀⢠⡀⠈⢷⡄⠈⣿
⣿⣿⣿⣿⣄⣰⣿⣀⣼⣿⣿⣿⣶⣶⡦⠤⠤⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠤⠤⠤⠀⢸⣏⣁⣴⣿⣿⣄⠈⣷⡀⢈⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠲⠂⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⣸⣀⣄⣠⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠙⢿⡴⣤⣤⠴⠦⠤⠤⣬⡉⠀⣿⠃⠈⢫⣤⣤⣤⣤⣤⣤⣬⠙⠛⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠐⠛⠁⠉⠁⠐⠒⠛⠛⠛⠋⢀⡹⢆⠘⠛⠉⠁⠀⠀⠀⠀⠀⠃⠒⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⠂⣀⠀⠀⠀⠀⠀⣀⡀⠒⢉⣿⣄⠑⠂⣀⠀⠀⠀⠀⠀⠀⣀⠐⢢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠙⡦⠄
⡞⠁⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠁⠀⠀⠀⠀⢀⠊⡠⠖⠚⠉⠉⣉⠂⠀⡥⢤⡤⠤⠤⡄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡭
⡂⣷⠤⠀⠀⣤⠘⡥⠀⠂⣥⠖⠁⡠⢾⣤⠂⢂⠄⢀⠈⣄⣶⢰⡟⢸⠁⠀⠄⠀⠀⠀⣠⢠⡀⠀⠂⠀⠀⠂⡈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰
⠛⠀⠀⠀⢀⣿⢢⠑⢀⡴⠋⣠⠊⠠⣄⠈⠦⢓⠀⣀⣿⠸⣿⢸⡇⣼⢾⣅⣀⠠⠊⠀⠀⠀⠑⠀⠀⢈⡉⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⣼⠁⣼⣷⠎⠠⡾⢏⠂⠀⡍⠀⣞⠶⠎⠀⠈⢀⠛⠘⢫⣹⣿⣲⢗⠾⠿⣿⡦⠀⠘⠀⣇⠈⠀⠀⠀⡘⢠⠀⠀⠀⠂⠘⠙⠎⠑⣦
⢀⣦⡀⢰⣿⢸⣿⡇⢠⠁⢱⡦⢉⠂⡍⠉⠁⡀⠀⠀⠀⡀⢥⠰⠙⠓⠘⠗⢩⣿⠿⠿⡇⢤⡠⡄⢸⣕⡋⢤⣡⠄⠈⡌⣀⠀⠀⠲⢀⠀⢸⣟⣷
⡉⠀⠀⢸⣿⡜⣿⠀⡆⠀⣄⣹⡾⠍⠀⠀⢀⣴⠝⠈⡐⢏⡌⠄⠸⠇⣴⢳⣤⠯⡄⣄⠚⡆⠳⣀⠋⡳⣴⠆⠑⠓⠀⠀⡭⠐⠂⠀⡈⠀⢀⠹⣾⡇
⠧⠄⠀⠈⢿⣷⡘⡆⠘⢄⠘⠁⠀⠀⢀⣴⢩⣱⣦⠀⠀⢄⠁⢄⠀⠀⡙⢮⣏⢑⠻⡃⢚⢡⡆⠻⠆⢹⣝⣿⣷⠟⡀⠠⠀⠀⠀⠀⠀⣠⣾⣷⣿⢻⡀
⡀⠀⠀⠀⠀⠙⢿⣬⣦⣀⠉⠁⢀⠀⢮⣶⣿⣿⣿⣦⠐⡈⢲⣦⡑⡀⡘⠉⠻⡬⠷⣀⣿⠇⣿⣿⣣⢘⡮⢈⡟⢎⢀⠀⠀⠀⢀⠀⡐⢿⣏⠂⣽⣿⠇
⠷⠀⠀⠀⠀⠀⣸⣿⡏⣭⠀⠀⡪⠶⢀⡉⠁⠋⠟⠟⢡⢨⠮⣧⣷⠋⠀⣾⠛⡆⣿⢮⣵⠀⢿⣌⣉⠀⢐⠟⢘⢄⡼⢀⣾⣿⡏⠀⠌⣠⣿⣤⣿⠃⣆
⠀⠀⠀⠀⠀⣰⣿⠛⠋⢓⡆⠀⠀⡷⢆⠉⢉⠛⡔⠒⠴⠟⠋⢠⠃⠀⣾⣫⣶⡧⣝⢶⣎⣄⢽⣟⣿⠀⢘⠓⣰⠃⠀⠘⠋⠉⠀⣡⡆⠈⡷⣸⡏⠀⠿
⠀⠀⠀⠀⢰⣿⣿⣆⠘⠿⢷⠀⠀⣏⣀⡈⠙⡳⠼⢷⣶⣯⠀⠎⠀⠀⣿⣽⣞⣎⠁⣐⢹⡄⣍⠁⢿⡆⢩⡜⢪⡀⢘⠆⠀⠀⣰⠿⠇⣌⢳⣟⠁
⠀⠀⣠⠀⠀⣿⣿⣿⡆⠀⢸⣇⢠⣬⣩⣵⣒⡀⠙⠛⠍⣿⡿⠀⠀⠘⢤⣿⣿⡇⢣⡉⠸⠇⢹⡗⢿⡷⠈⠖⠉⣠⣇⠴⢄⣴⣯⠴⢦⠤⠘⠈⠁
⣁⣴⠏⠀⠀⠹⢿⣿⣿⡄⠈⠋⣿⣇⣱⣥⡛⣥⠀⡗⣀⣩⣦⣼⣷⡦⢌⡻⣿⣵⢾⠉⠛⠶⢤⣤⣤⣤⡴⢶⠛⠁⠉⠳⣤⣈⣁⣀⣤⡔⠁
⣿⣃⣠⡾⢁⣼⠃⠈⣻⣷⢀⠀⠉⠁⠀⠀⣿⠦⣍⠛⠻⠯⠀⠚⠋⢰⣟⣛⡻⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠒⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀
⣎⡿⣿⢛⣾⣃⣃⢠⡟⣽⣧⣁⠩⠔⢚⣭⣶⡿⠛⢿⣶⣤⣴⡶⠗⠁⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⣿⣿⣿⣿⠃⢉⢱⡿⠳⠿⣿⣷⣾⣿⠟⠋⣁⠠⠐⠁
⣿⣿⣿⣷⣵⢾⠬⣧⣀⣀⣹⣿⡭⠴⠚⢋⡡⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣠⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣽⣿⣯⠛⣻⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠉⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡀⠠⠶⠍⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁⢴⣤⣄⠀⢙⣇⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡆⠀⠀⠀⠀⠀⣨⣟⡍⢀⠇⣠⢆⣴⣶⣖⣠⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠈⠉⠉⠁⠀⠀⢰⡏⢿⠀⠀⠀⢠⠀⠀⢉⡀⣼⣴⣱⣷⣾⣿⣿⣿⣿⣿⣿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠽⣦⣤⣴⡯⣤⠴⠋⣸⣿⠟⢉⣥⡀⠀⠉⠉⣿⣶⣾⣿⣯⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⠋⠀⠀⠀⣼⠏⠁⣠⣤⣤⠀⠀⠀⠀⠀⣽⣿⣿⣟⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠘⠳⣄⡀⠀⠀⠀⠀⡋⢀⢿⣿⠇⠀⡆⠀⠀⠠⡄⠙⠛⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠐⣠⡿⠁⠀⢠⡟⠙⢢⣠⠂⠸⠂⣿⡀⠀⢀⡰⠀⠀⣿⣷⢰⡀⠀⡷⣶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡶⠞⠛⠁⠀⢀⣴⠟⠁⠀⠀⢹⡀⡀⠀⠈⠙⠛⠁⠀⠀⠘⣍⠁⢨⡇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⢻⣷⣦⣤⡴⠞⠛⠁⠀⠀⠀⠀⣸⣿⠄⠀⠀⠀⠀⢷⣄⠀⠀⡙⠻⠃⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢸⡇⠀⠀⠀⢻⣷⡈⠛⠒⠀⠀⠀⢀⣤⣾⣿⠁⣠⣶⡶⠠⡄⢈⠛⠿⠛⠁⠀⠀⠀⠀⢄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠘⣿⡀⠀⠀⠀⠹⣮⡁⠀⠀⠀⠀⠻⢿⣿⣷⠐⢿⣽⣳⣶⣀⡜⠑⠦⠀⠀⠀⢀⣒⡐⠂⠀⠸⣦⣌⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡽⡆⠀⠀⠘⣷⡀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠈⣹⣷⣤⠄⠀⠀⠉⠻⣟⣷⡇⢠⡴⠋⢁⣿⠶⠠⠠⠧⣹⡧⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣹⣄⠀⠀⠈⠻⣄⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠘⠻⣿⣿⣄⡀⠀⠀⠀⣉⣉⡠⠊⠀⢀⣾⠟⢶⣶⣶⣦⡀⠀⣰⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡛⠿⣳⣼⣦⣄⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠉⠛⠷⣦⣄⣀⠉⠉⠁⠙⢿⠟⠉⠁⣀⣠⣴⡿⠃⣎⣀⣀⣀⠀⠀⣰⣿⡇⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠣⠜⠂⠉⠙⢳⣬⣟⣲⢤⣀⣀⠉⠛⠳⢶⣤⣀⣀⠀⠀⠈⠉⠛⠛⢶⠶⠖⠚⠛⠛⠛⠛⢥⣄⠀⢈⡟⠛⠉⠁⣰⣯⣽⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⠔⢋⡠⢀⠠⠄⡒⠬⠳⠙⡫⢝⡳⣿⢽⣏⣿⣒⣶⣤⢭⣙⣛⣻⣶⣶⣶⣮⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢏⣹⡿⣿⡿⠞⠋⠀⠀⠀⣠⠞
⠀⠀⠀⠀⢀⣴⡻⠝⡂⢥⡐⠬⢓⠨⢐⢈⡁⣄⢢⣥⣬⣦⣍⣭⣿⣿⣿⣿⣿⣿⠿⣍⠷⢉⣣⣤⡾⠿⢓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠦⠤⠤⠒⠋⠁
⠀⠀⠀⣰⡿⢅⡲⠍⢂⠡⣐⣨⡴⢞⠯⢒⠉⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠿⡿⠶⣳⣶⣾⣉⠫⠕⢢⡑⢮⡳⠬⢛⣄
⠀⠀⡴⠏⠐⢡⠀⢌⡠⠕⠋⠡⢚⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣟⡋⢀⣡⣴⠶⣛⢃⣍⠦⣔⠶⠟⠛⣛⣦⡀
⠀⠀⣷⠐⢈⡀⠈⣠⠴⣲⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣦⡷⠛⠋⠈⡑⠬⡑⢊⣉⣤⡽⢿⡀
⠀⢀⣿⢂⣡⡲⣝⣶⣿⠻⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠺⣷⣼⣷⠿⢟⡻⢍⡳⢸⡇
⠀⠸⣯⢿⣿⠿⢛⢩⠲⣵⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣽⣧⣙⣦⣽⣶⢿⣿⠃
⠀⢸⡿⣻⡷⠞⣺⣿⢾⠮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⡹⢌⣣⣴⣟⣿⠋
⠀⣷⠞⣿⣤⠞⢽⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣭⣷⢾⣻⣿⣿⣼⡇
⠘⣷⢺⣯⢔⡪⢼⡟⣧⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⠿⣿⢟⣿⠁
⠀⢹⣯⢲⠟⡩⣴⠟⠺⢑⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⡻⢝⢢⣽⣟⣧⡿
⠀⢨⣿⡾⢓⠭⢂⣵⠞⢪⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢜⢣⣝⣾⣿⡿⢛⣸⡇
⢀⣼⣷⣻⣷⣾⠿⢶⣶⣿⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⡿⣟⣣⣼⣿⣿
⠸⡀⢉⣿⣿⣧⣤⣤⣸⣿⠞⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠛⠻⣧⣿⡾⣿⣵⣾⣿⡄
⠀⠈⣿⣾⣟⣛⡛⠿⣟⠻⢢⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠙⢿⡟⣉⣀⣼⣁
⠀⣼⣿⣿⣭⣭⣿⣷⡚⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣦⣤⣤⡶⣻⡶⣋⣉⣽⡇⢀
⢰⡿⠟⠉⠀⠀⠀⠀⠈⠙⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣯⣄⣡⣿⣷⣏⣠⡴⠶⣧⣄
⡏⠑⠶⢤⣤⡀⢀⠀⠀⣀⡸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠞⠛⠉⠑⢦⡀
⠘⠢⠤⠤⢀⣀⣀⣀⠀⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⣶⣶⣤⣤⣬⣍⣙⠛⠻⠿⣡⣀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣤⢭⡭⠽⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠰⡇⢸⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣤⣠⣶⣼⣧⣤⣄⣄
⠀⠀⠀⠀⠀⠀⣠⣴⠿⠟⠻⠿⢿⣆⡀⠀⠀⠀⠀⠀⠀⣼⡟⡁⠁⠀⠈⠉⠉⣙⠻⣷⣯⡀⠀⠀⠀⠀⠀⠀⣤⠀⣄⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢸⣿⠙⠋⠀⠀⠀⠀⠀⠙⠓⠘⢻⣇⠀⠀⠀⠀⠀⠀⠻⣿⣷⡏⠀⠀⠀⠀⠠⣷⣿⣾⠃
⠀⠀⠀⠀⢸⣧⣤⣤⣤⣀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡇⠰⠆⢠⣾⠀⠀⠀⣴⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢼⣿⡆⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠷⣦⣤⣠⣿⠀⠀⠀⣶⣾⡇⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠈⣟⠁⠀⠀⠀⠀⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣤⣤⣬⣿⣿⣆⠀⠀⠻⣿⡇⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⢐⡿⣷⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡄⠀⢹⡧⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢘⣣⣿⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⠃⠀⠈⣿⠀⠀⣤⣶⣶⣤⣄⠀⠀⢠⡿⠋⠁⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⠀⢨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣦⢰⣾⣷⣆⣶⣄⠀⠀⠀⠀⠀⠀⠙⢧⣀⠻⠟⠛⠿⠿⠃⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣦⣿⡾⠟⠛⠛⠛⠙⠛⢿⣷⡄⠀⠀⠀⠈⠙⠛⢶⣶⡶⢶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣦⣀⠀⠀⣰⣿⡅
⠀⠀⠀⠀⠀⠀⣸⣟⣡⣄⠀⠀⠀⢀⣀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡁⠀⣀⣀⠀⠀⣠⣴⣶⠿⢿⡟⢋⣤⠚⠻⣧⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡟⢁⡈⢡⡄⠀⠀⠉⠃⠀⢸⣿⣄⠀⠀⠀⣾⣷⣴⡟⠀⠘⣷⣾⣿⣿⣧⣾⠿⠟⣁⣰⠶⠀⠿⢿⣤⣀⠀⣬⣽⣿⠁
⠀⠀⠀⠀⠀⢀⣿⠀⠈⠁⣸⠇⠀⠛⠁⠀⠀⠸⣿⢿⣧⠀⣸⣿⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⡶⠛⠋⠁⠀⠀⠀⠀⠀⣉⠓⠦⢭⣧⢀⡀
⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⢿⣶⡦⠀⠀⠀⠀⠀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣟⢻⠄⣷⠀⠀⢰⢞⣉⠃⠀⠀⠹⣿⣻⠆
⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⢀⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⢻⢟⣿⣿⣿⣿⣿⣿⢹⣿⢻⣏⠀⠀⠉⠀⢠⣿⠃⠀⠀⠈⠉⠀⠀⠀⠀⣹⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠛⠃⠀⢀⣴⣿⣿⣿⣿⠇⢎⡎⣽⣿⣿⣿⢿⣿⡇⣿⡄⢿⣆⠀⠀⠀⣸⣷⣾⡟⠀⠀⠀⠀⠀⠀⣠⣿⡿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣿⣦⣤⣀⣤⣿⠟⠛⠻⢿⣿⣆⠘⣰⣿⣿⣿⣿⣿⢿⠇⠈⠀⣿⡏⡟⢹⢃⡟⠉⢸⠟⢻⣟⣛⣿⣷⠞⢻⡟⠁⠀⠈⣳⣄
⠀⠀⠀⠀⠀⠀⣰⡟⠉⠈⠉⢿⡏⠁⢨⡿⠀⠀⠀⠀⠻⣿⣤⠟⣡⣿⣿⣿⣿⡄⠀⠀⢠⣿⣇⣀⡉⢸⡇⠀⠉⠀⠀⢨⣿⠋⠀⢠⡟⠁⠀⠀⠀⠟⠹⣆
⠀⠀⠀⠀⠀⢰⡟⠀⡄⠀⠀⠘⢿⣶⠟⠁⠀⣤⢀⠀⠀⠙⣧⣰⣿⣿⣿⣿⣿⣿⣤⣤⣴⣿⣿⣟⠿⣿⣿⣿⣿⠖⢷⣾⡟⠀⣰⡿⠇⠀⠀⠀⢀⡄⠀⢻
⠀⠀⠀⠀⠀⣿⣧⣠⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣷⡶⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⡏⠻⣧⠈⢿⣿⢀⣿⣇⣀⡀⠀⠀⢿⡇⠀⠈
⠀⠀⠀⠀⣴⠏⢠⣿⠀⠀⠀⠀⠀⠀⠠⢀⠰⠆⠸⣿⣷⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⣿⠿⣇⠀⣹⣧⠈⢻⣿⣿⣿⡿⢿⡆⣴⣾⠃
⠀⠀⠀⣰⡟⢀⡾⢻⠀⠀⢠⠀⠀⠀⡄⠘⡄⢀⠀⣿⣿⣇⠀⢿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⠁⠀⠚⠁⣠⣿⣶⣿⣿⣧⠀⢻⣿⠏⠀⢸⣧⣾⣿⣾⣷⣶
⠀⠀⢰⡏⢀⣾⠁⣼⠀⠀⠚⠀⠰⠚⠇⠀⠛⠘⢂⣸⣿⣿⠀⠘⣿⡟⣽⣿⣿⣿⣽⣿⣿⡇⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡇⠀⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠁⣾⠃⢠⣿⣿⠿⣿⡿⠿⣿⣿⣷⣶⣾⣿⠻⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣾⣿⣿⣿⣿⣟⣛⣿⣿⣿⣷⡀⢀⣴⣿⣿⢿⣿⣿⣿⣿⣿
⠀⢰⣿⢀⡇⠀⢸⣿⣫⣾⣿⣿⢶⣿⣿⣿⡝⣿⣿⠀⣿⣿⣿⣆⠈⣿⣿⣿⡿⣣⡿⣫⣿⣿⣿⣿⣿⠻⣿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣯⣿⠷⣿⣿⣿⣿⣿
⠀⢸⡇⢸⡁⠀⣾⣿⣿⣿⣿⣿⡺⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⡆⠘⣿⣿⣿⣿⡾⠻⣟⣿⣿⣿⣿⠿⠛⠻⣦⢹⣿⢟⠉⢻⣷⣦⡝⢿⣿⣿⣿⣿⣿⡿
⢠⡟⠀⢈⣻⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⠋⡁⠀⠘⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⠀⠀⣰⣿⡆⣿⣿⣇⠀⠸⣿⣿⡤⣿⣿⣿⣿⠏
⢠⣴⣦⣾⠏⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣷⣿⡇⣀⢴⣼⣿⣿⣷⢟⣿⣿⣿⣿⡏⠀⣴⣿⣿⠿⣿⣿⣿⣆⠀⢻⣿⣷⣿⠿⠛⠁
⠋⡿⠻⡿⠀⠀⣿⣿⣿⣿⣿⣿⡿⠘⣿⣿⣿⣿⣿⣷⢸⣿⣷⢿⣷⣿⣼⣿⣿⣿⢲⣿⣿⣿⣿⣿⡇⢠⡏⠀⠀⠀⠀⠉⠙⢿⡆⠸⣿⡋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣿⣿⣿⣿⣿⢃⣩⡾⣿⣿⣿⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈⢿⡀⢿⡇
⠀⠀⠀⠀⠀⢼⣿⣏⣭⣿⣿⣿⣿⠐⣿⣿⣿⣿⣿⣿⣿⣿⣟⡳⣿⣿⣿⣿⣿⡿⣋⣴⣿⣿⣿⡿⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⡇
⠀⠀⠀⠀⠀⠈⠉⢹⣿⡆⢻⡏⠀⠀⠀⢹⡧⠀⢻⡏⢸⣿⣿⣿⢿⣿⣿⣿⣋⡾⢋⣿⣿⣿⣿⠃⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⠀⠀⠀⢸⣧⠀⣻⡇⢸⣿⢟⠅⣸⣿⣿⣿⣿⡴⢛⣵⣿⣿⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⡏⠀⠀⠀⢸⣿⠀⣿⣇⣸⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⢟⣛⡙⣧⣼⣿⡆⠀⠀⠀⠀⠀⠀⣀⣀⣹⣿⣧⣼⠟⠳⣦
⠀⠀⠀⠀⠀⢀⣀⣸⣿⣷⣾⣷⣤⣴⣶⣿⣿⣶⣿⡟⠛⠋⠁⠀⢹⣿⣿⡿⠿⢛⣿⡎⢿⣷⣿⣿⣿⠀⠀⠀⠀⠀⢀⣾⡿⢿⣿⣿⣄⣩⣤⣾⣿
⠀⠀⠀⣠⡾⠛⠛⣻⡟⢹⣿⡿⠋⠀⢉⣙⠃⢹⣿⣇⢀⣀⣀⢀⣾⣿⣿⣄⠛⠻⠿⠷⠾⠛⠛⣿⣿⠀⠀⠀⠀⠀⢸⣿⣒⠚⠛⠟⠛⠛⣛⣉⣿
⣀⣀⣀⣙⣷⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣉⣉⣀⣉⣀⣀⣈⣉⣉⣁⣀⣀⣀
⣿⣿⣿⢿⠻⢿⡿⠻⡿⢿⡿⢿⠻⢿⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣶⣼⣷⣴⣧⣦⣾⣤⣼⣿⣤⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠙⠳⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢻⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⡀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⡞⠁⠀⠀⠀⠀⠀⠀⢠⣿⠻⣦⣀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⣠⣤⠉⠛⠛⠿⠿⠿⠿⠷⠶⠶⠶⢶⣿⣟⡋⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠉⢀⣤⣤⣄⠀⠀⠀⠀⣸⣶⣦⣤⡈⢿⣍⠛⢷⡄⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⣠⣴⣿⡿⠁⠀⣠⡾⠋⠉⢛⣛⠳⣄⠀⣼⠋⠀⠀⠈⠙⢾⣿⣦⡀⠳⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⢠⣤⠶⠟⣫⣾⡟⠀⠀⣼⡏⠀⠀⠀⣿⣿⡷⢸⣦⣿⡀⣾⣿⡆⠀⠈⣿⣿⢳⣄⠀⠀⠀⠀⠈⠻⣦
⠠⠤⠤⠤⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠀⠀⢀⠹⣇⠀⠀⠀⠀⢉⣠⡟⠁⠻⣷⣌⡉⠀⠀⣰⡟⢹⡆⠙⣦⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠛⠀⠀⠈⠻⠾⣿⣿⡿⠿⠛⡁⠀⠀⠀⠈⠻⣿⢿⣿⣿⡾⠛⣷⣤⡜⣧⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⠀⢀⣠⣿⠂⠀⠀⠀⠀⠹⢩⡿⠘⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⣦⡀⠀⠀⠀⠀⢸⣇⢸⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢶⣾⠃⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣈⠻⣦⣄⠀⠀⠀⠻⣤⠙⠻⣿⣦⡶⠀⠀⣠⣶⠾⣿⠟⠻⣿⠟⠛⠻⣿⠛⠛⣿⠿⢷⣦⡀⠙⣧⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠷⠶⠶⠶⠶⠿⣿⠋⢀⣤⡾⣿⠁⠀⢿⡀⠀⣹⡆⠀⢰⣿⠀⢀⣿⣀⣀⣾⣿⠀⣿⡇⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⢤⡀⢰⡏⠀⣿⣿⣧⣼⣷⣴⡿⣿⡟⢿⠟⢷⣾⣿⣾⠛⠿⣇⠉⠻⣿⢷⣿⣧⣤⣤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠻⣾⣧⣀⣈⣽⠟⢁⡾⠃⠀⣿⠁⣾⠆⢸⣿⠀⠙⣷⡀⠙⣷⠀⢻⡆⢻⡏⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⢻⡟⠀⣾⠃⠀⢰⡇⠀⠛⠀⠈⣿⢧⡀⠈⠓⠀⠸⠇⠘⠿⠘⣿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡤⣄⢻⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⢻⣆⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣸⡇⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⣻⣶⠆⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣯⠀⠀⠀⠀⠀⠸⣿⣤⠶⠶⠶⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⣤⣤⣼⣿⣡⡾⠛⠁⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠙⢷⡄⠀⠀⠀⠀⠈⠻⣤⣤⣄⡀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠞⠋⠉⠀⠀⠀⣹⡿⠋⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠈⠻⣭⡙⢷⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣀⣤⡶⠾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠈⠿⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⣾⠏⠁⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠙⢿⣷⣶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣠⣤⠶⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠒⠒⠲⢼
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡇⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢿⣿⣿⣷⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠤⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠞⠚⠀⠋⠛⠋⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠳⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠔⠀⠀⠀⠀⠀⠀⠀⢣⠀⡀⢹⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠇⠀⠀⢠⣿⠆⢸⠋⠉⠊⠀⠀⠀⠀⠈⠙⠲⣄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠀⢸⡿⢀⠜⠀⠀⠀⠀⣀⠀⣀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣠⣥⣤⣤⡼⢿⣷⣦⡤⠤⠔⠊⢉⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⠟⠁⡾⠵⠜⠓⢀⢘⠐⢢⡟⡆⣀⣀⠴⡅⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠒⡞⠺⠶⠷⠯⠴⠿⠶⠖⠛⠉⡎⠉⠀⠀⠀⠙⣄⠀⣀⠤⠒⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠈⣑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠕⠀⠀⡠⠊⠈⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠄⣈⠷⣀⣀⠀⠀⠀⠀⣀⣴⡊⠥⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠉⠛⠻⠿⣿⣿⡿⠿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠠⣏⠁⠀⠀⠀⠀⠀⠀⠂⢄⡀⠈⠀⢄⠀⠉⠠⢀⢀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠃⠀⢀⣴⣿⠟⠁⠀⠙⣄⠀⠀⠄⡀⠀⠀⠀⠀⠉⠓⠄⡀⠀⠀⠀⡀⠉⠻⣷⡄⠀⠈⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⠏⠀⠀⠈⠀⠈⢢⣀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⣂⡬⢷⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⣼⠏⠀⠀⠀⠀⠀⠔⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡴⠖⢀⣠⠄⣛⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⢀⡀⠉⠀⠔⠈⠀⠄⠋⠀⠀⠀⠠⠀⠀⠀⠐⠤⠀⠀⠀⠀⠄⠤⠀⠀⠛⠁⠛⣛⠄⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠈⣿⡶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⣿⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⢿⣿⣿⣿
⣿⠿⠿⠋⠀⢀⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠛⠛⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆⠀⠈⠻⢿⣿
⣧⣀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠁⠀⠀⠀⠀⠀⠀⠒⠀⠙⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠂⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡖⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣧⣾⣷⣤⠄⠀⠠⠀⠀⢀⣀⣀⣀⡀⠀⠠⠄⠀⠀⣦⣴⣿⣄⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀⠀⠀⢀⡀⣀⢀⣀⣀⣄⣀⣀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠍⠛⠛⠛⠻⠇⡿⠠⠟⠒⠒⠒⠩⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⣀⣤⣿⣤⣀⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢈⡁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢁⠀⣄⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠐⠤⠀⠉⠻⢕⣀⣀⠀⠀⠂⢠⠆
⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣴⠞⣡⠆⠀⣁⠠⢖⠀⠠⠄⡛⢿⣷⣇⠀⠦⠀⠀⢀⠞⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣧⠋⢤⢖⡠⢄⠦⠠⣤⡶⠦⡉⠚⢻⣬⣳⣧⠔⠒⠌⢀⠀⠀⣠⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⢀⡾⢃⣠⠖⠀⠉⠛⠂⠁⠀⠀⠈⠀⡩⢴⣿⣮⣿⣦⣐⣀⡘⡀⠀⢀⡀⢀⡠⠰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⢸⢋⠀⠁⠀⢈⣤⣶⣶⣶⣃⠀⠀⡘⡴⠃⠙⣿⣷⣝⣿⠖⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠈⢦⠀⠀⢀⢴⡿⣊⣤⣡⣾⣿⡟⢹⠉⠉⠉⠛⠀⠴⠁⢄⠀⡞⣿⣜⣧⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⣸⡟⢛⣋⣈⢸⠟⡉⠀⠀⡀⠀⠀⢀⠂⢰⠆⠀⣠⢹⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠈⠁⠀⣠⣿⠛⠟⠛⠙⠉⠀⠃⠀⠀⠀⠀⠠⠁⠀⠋⠰⣰⢧⣿⣿⣳⣤⡄⡀⠀⠀⠀⠀⡊⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣡⣿⢿⢳⠀⠀⠀⠉⠀⠠⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠠⠀⠀⠀⠁⠀⠢⢀⠀⠀⣤⣷⡆⠀⠀⠀⠀⠀⣠⢾⠟⠀⣈⡸⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡿⣦⣧⠀⠀⠀⠀⢾⡟⢠⡇⣿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡀⠠⠄⠀⢀⠀⠀⠀⠙⢿⡟⣧⠀⠀⠀⠘⠆⣾⡏⢸⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⣎⠆⠀⠀⠄⢲⣽⡿⠘⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⣠⣜⠞⠀⠀⠀⠈⢿⠇⠀⠀⠁⢀⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠘⢿⠝⠻⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠙⢄⡃⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠚⠶⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠠⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣖⡀⣀⣄⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢀⡀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣼⡿⢀⡇⠀⣼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣟⣿⡇⠈⠀⠀⣿⣿⣿⣻⣿⡆
⠀⠀⠀⠀⣸⡄⠀⠸⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⢈⣿⣧⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⡟⢢⢀⠀⢀⣠⣤⣠⣀⠀⡤⠘⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠄⢠⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⢀⣸⣷⣾⣸⣦⣾⣿⣿⣿⣿⡄⡀⠀⠀⠀⢠⣄⠀⠀⠐⠂⠤⢀⢸⣿⣿⡿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠰⣚⣛⣙⣿⣻⣿⠿⢿⣥⣌⣉⠹⣿⣟⢲⣀⣀⠠⠙⠿⣦⡄⠀⠀⠀⠸⣿⣿⣧⣣⡿⠿⠛⠛⠛⠛⠿⣿⣿⣧
⠀⣿⠉⡭⠩⣿⣿⣴⣾⠿⢿⣿⣿⣿⣿⣿⠟⠻⢶⡗⢦⣼⣿⡆⠀⠀⠀⢹⣿⣿⣿⠀⣠⣤⠆⠲⣴⣤⣄⣘⣻
⢨⣿⠀⠃⠀⠈⠙⠋⢁⡀⠀⣸⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠀⠀⢸⣿⣿⣿⣾⠋⠀⠀⢰⣿⣿⢿⡿⠿
⢸⣿⠀⠒⠀⢀⡒⠀⠒⠀⠀⢻⠒⠒⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣇⠀⢀⠄⣾⣿⣿⣾⣿⣿
⢸⣿⠀⠀⠀⠘⠂⠀⠀⠀⠀⣾⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢸⣿⣿⣿⢻⣄⡌⠀⣿⣿⣿⣿⣿⣿
⡿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡂⠂⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⠀⠀⢸⣿⣿⣿⣿⣿⣳⣠⣿⣿⣿⣿⣿⣿
⠘⡿⠀⠁⠀⢠⠄⠀⠀⠀⠀⢼⠅⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠀⠀⠺⣿⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿
⢸⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⢿⡃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠸⣿⣿⣿⣷⣶⣶⣶⣿⣩⣭⣽⣿⣿
⢸⣿⠀⢀⣀⣁⣀⣀⣀⣀⣠⣿⣍⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⣿⣿
⠸⣿⣀⣻⣿⣿⣿⣿⣿⣿⣟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣯⣿⣿⣟
⠀⠉⠛⠛⠻⠿⠿⠿⠿⣿⠯⢤⣤⣼⣿⣿⣿⣿⣿⣛⣯⣭⣿⣿⣿⣿⣶⣿⣿⡿⢿⣿⣿⣿⡿⢿⠿⣯⣿⡉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣬⣭⣭⣿⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠛⠛⠛⠛⠲⢦⣄⡀
⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠔⠄⠀⠀⠈⠻⣦⡀
⠀⢀⡾⠁⠀⢴⡄⠀⠀⠀⠀⠀⠀⢄⠌⣀⡀⠀⠀⠈⢷⡀
⢀⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⣸⣧
⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⢆⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣇⣀⠠⠀⢠⠂⠀⢂⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠶⢾⡇
⣿⠁⠀⠀⠀⠈⠤⠤⠊⠀⠀⠰⠡⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⣿⡤⠤⠀⠀⠀⠀⠀⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⢆⠄⠀⣿⣄
⢻⡄⠀⠀⠀⠀⠀⠀⣠⠊⠉⠈⢢⡀⠀⠀⢅⡁⠀⠀⢠⡟⠈
⠸⣇⠀⠀⠀⠀⠀⢠⠃⠂⠉⠀⠀⢃⠀⠀⠀⠀⠀⠀⣾⢽
⠀⢿⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡸⠀⢰⣒⡖⠀⢸⠇⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡤⠤⡀
⠀⠸⡇⠘⡆⠀⠀⡸⡆⠀⠀⠠⣶⠇⠀⠈⠈⠁⢀⡿⣤⠜⠀⠀⠀⠢⠆⢠⠏⢱⠀⣴⣣⠤⠖⠁
⠀⠀⣿⠤⣷⠒⠀⠳⣜⣲⣤⡶⠋⢀⣀⣀⣀⣀⣸⣷⠏⠀⠀⠀⣀⡠⠔⠋⢀⣾⠤⠖⠒⠛⠓⡆⠀⠂⡀⠀⠀⠀⢀⠠
⠀⠀⣿⠒⡿⠉⢄⠀⠈⠉⢉⣴⣟⣛⣉⡛⣍⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⡀⢀⣈⡥⠴⠾⠷⠒⠒⠒⣒⠆⠀⠃⠈
⠀⢀⡟⠀⠁⠁⠘⠀⠀⠀⣼⡿⠟⠒⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠛⠫⡄⠐⠯⣩⣏⣙
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢻⣷⣀⣀⣀⡀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠂⠒⠒⠶⠶⢶⣋⣀⠀⠀⠀⠈⠁
⢰⡏⠀⠀⠀⢀⠤⡀⠀⠀⢸⣧⣤⣀⣐⣁⣀⣀⣀⣁⣀⣀⡀⠀⠀⣄⠀⠈⠉⠀⠂⠒⠠⣄⠀⣀⣀⣀⣀⣈⣹⠆⠠⣰
⣼⠀⠀⠀⠀⡁⠀⢨⠀⠠⢸⣿⣤⣴⡶⠿⠛⠛⠛⠿⢿⣯⠉⠉⠛⠓⠤⣄⡀⠐⠲⡤⠤⠼⠂⣀⣀⣀
⣿⠀⠀⠀⠀⠈⠁⠁⠀⠂⠂⠙⠛⠉⠀⠀⠀⠀⠀⠒⠒⢼⡇⠀⠀⠀⠀⠀⠉⠓⠤⠏⠀⠀⠀⠉⠢⢄⣹
⣿⠀⠀⠀⠀⠀⠀⣁⡱⠀⠀⠀⠀⠀⠀⢀⠀⠀⠪⠅⠀⠀⣷⠀⠈⠁⠄
⢹⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡁⠀⠀⠀⠀⣿
⠀⢻⣄⣨⣦⡤⠤⠤⠀⠀⠀⢄⡠⡄⠀⠈⠀⠁⠀⠀⠀⣸⠇
⠀⠀⠙⢶⣤⠷⠶⣒⠀⠀⠀⠈⠷⠉⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠉⠛⠶⠶⣤⣤⣤⣀⣀⣤⣤⣤⣤⠶⠞⠋⠁
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⠀⢰⠀⠀⠀⠀⠈⠳⣄⠀⡀⠀⠀⠈⠲⡄⠀⠀⠀⣀⠤⡁⠈⣾⠀⠈⣦
⢂⠀⠈⠘⡄⠀⠀⠀⠀⠈⡆⠈⠐⠀⢀⠄⠉⠀⠀⠀⢡⠀⠈⠠⡤⠄⠐⠛⢄
⠘⢄⡀⠀⠙⢢⠀⠀⠘⣅⠈⠂⠀⠀⠀⠀⠀⡠⢀⡀⠸⢄⡤⠂⠀⠰⡄⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢧⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠔⢋⠜⠁⢀⢣⠀⠀⠀⢹⡖⠋⠀⠄⢀⢠⡤⠀⢠⠀⠀⢀⡴⠖⠒⠚⠉⠉⠉⠓⠒⢻⠇
⠀⠀⠀⠀⠈⢧⠀⢀⠔⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁⠈⠀⠀⠀⠀⠁⡟⠀⡆⠒⠂⡴⢋⠀⢠⡎⠤⠄⠐⠃⣁⡴⠋
⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⡗⠀⣴⠤⣤⠶⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠔⠉⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠉⠑⢠⡬⠴⠚⠉⢲⡀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⠉⠈⠀⡐⠁⠘⠁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡴⠃⠀⢦⣀⣀⣴⠈⠇⡅⠀⠉⡆
⠈⠂⠐⠲⢀⠀⡟⠀⡀⠄⠠⠤⠚⠉⠀⡤⠾⠶⠴⠀⠂⠀⠀⠘⠲⢤⡀⢀⠀⠀⠀⠈⠛⠋⢱⠀⢀⣠⠏⠀⠀⠈⠓⠒⠶⠶⠗
⠀⠑⠢⠤⢈⡿⡗⠀⠄⠀⢀⠀⠀⠈⠣⢄⠀⠀⠀⠀⠀⣀⡀⠀⠀⡠⠚⠘⠀⢡⠀⠀⠀⠀⠀⠈⠉
⠤⠀⠀⠂⡘⢡⣿⠀⠒⠂⢤⣠⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠉⠻⡀
⠀⢀⡰⠊⢠⡞⠈⢧⡀⡄⢤⡑⠄⠑⠢⡀⢳⡀⠀⠀⠀⠀⠀⠀⠙⢆⠈⠉⠉⠃⢢⡤⠀⠀⠀⠀⠀⢂⠄⠤⠤⠤⠔⠒⠢⠤⠠⠤⡀
⠀⠊⢀⡴⢻⠀⠀⠀⠉⠀⠀⠙⠦⣄⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⡴⠋⠀⠃⠁⠀⠀⠀⠉⡏⠦⡀⠒⠶⢂⠀⠀⠀⠀⠘⢦
⠀⠒⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢘⠈⢢⡀⠀⠈⠳⣄⠀⠀⠘⣇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠀⠀⠀⠀⢄⠀⠀⠀⢀⠇⠀⠸⡀⠀⢻⡳⣦⡀⠉⠙⠆⡀⠈
⢀⡄⡴⠲⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⠁⠀⠀⢰⠀⠢⠈⡟⣵⡙⠲⣄⠀⢹
⠚⢀⠀⢀⡾⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡠⠊⠀⠀⠀⠀⠀⢠⠐⠠⢄⡀⠀⢀⡂⠀⠀⡌⠡⡀⢰⠁⠘⣷⠀⠀⠉⠚⠒
⠀⣀⠐⠇⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠁⣽⠃⡄⢠⠣⠀⠃⡺⠀⠀⢸⡃⢀
⠚⡏⣠⢄⡠⢤⠜⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⣃⠀⠁⠀⡖⢻⡍⠈⠳⠠⡀
⡼⢍⠀⠀⡇⠀⠁⢀⡆⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠃⠀⠀⡎⠀⠀⠛⠀⣠⠔⢧
⠀⠀⠑⠀⣉⡠⠴⠊⢁⡆⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⠀⠀⠋⠀⠀⠀⠀⡽⠀⠐⢀⠛⣄⠀⡀
⢄⣠⡖⠉⠈⠀⠀⠀⢫⠱⡦⠀⠀⠙⢆⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⣼⣀⢤⡀⢀⡨⣦⠀⠀⣸⠀⠀⢹
⠀⠘⡁⢀⡀⠀⢸⠚⡅⠀⡓⣄⠀⠀⠈⠤⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢣⡞⠀⠈⠉⣶⠉⠉⠉⠁
⠀⠀⠈⠉⢀⠉⠁⠀⠘⢄⡁⠈⣧⡈⠀⠀⢣⠀⠀⠀⠀⠀⣠⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢱⡄⠀⠀⠸⡄⠀⣠⣴⠁
⠀⠀⠀⠀⣼⠑⠀⠀⠀⠀⢿⠄⠀⠑⢄⠀⢿⠀⠀⠀⠀⢀⡇⠀⠀⠀⢘⠀⢀⠔⠁⣀⡤⠚⠁⡾⠀⠈⠂⢀⠃⠀⠀⠁⠈⢣⡄
⠀⠀⠀⡐⠙⡂⠀⠀⠀⠀⢸⡄⠀⠑⠀⠡⠟⠀⠀⠀⠀⢸⠀⠀⠀⢀⠏⡴⠁⠀⡜⠉⠀⠄⢸⡇⠀⠀⠀⡼⠀⠈⢣⡀⠠⠚
⠁⡀⠊⠀⠀⣿⡄⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⠔⣨⠞⠀⠀⡰⢧⠀⠀⠈⡄⠉⠐⣦⠰⢣⡀⠀⠀⣳
⠀⠸⡀⠀⢀⡝⡉⢄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠵⡎⠁⠀⠀⡴⠁⠘⢆⠀⠀⠙⠄⠀⢸⡄⠀⠉⠁⠀⠉
⠀⠀⠀⠱⠈⠀⠰⠀⠈⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⣠⠞⠁⠀⠀⣿⠀⠀⠠⡆⠀⢸⡆
⠀⠀⠀⠀⠀⢠⡀⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢹⡀⠀⠀⠘⠄⠀⡷
⠀⠀⠀⠀⠀⠈⢇⠘⡀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠙⡛⡄⠀⠀⠀⠀⠀⠀⠀⠜⠟⠁⠀⠀⠀⠀⠄⢂⠀⠀⠀⠀⠀⠀⠀⡐⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⡄⠂⢄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠤⠤⠤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠴⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⢠⣄⠌⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠁⠤⡀⠄⠀⢀⡁⠄⠀⠀⠀⠀⠀⡀⠇⠆⠈⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢀⠊⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠢⠐
⠀⠀⠀⠀⠀⠀⡸⠀⢠⠂⠀⠀⠀⠀⠀⠐⠂⠀⠘⠀⠂⠀⠚⠀⠀⠀⠑⠂⠄⠀⠀⠀⠤⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣐⣀⣁⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠰⣦⠀⣧⠀⡼⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡿⠀⢈⣤⣤⣄⠀⠰⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉⠀⢰⣏⣤⡴⢿⣷⠀⢠⡞⠳⡆
⠀⠀⠀⠀⠀⢠⡴⠿⠿⢦⡄⠀⠀⠀⠈⠉⠳⣤⣤⠟⠀⠈⠓⠾⠃
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢀⡾⢻⡆⠀⠀⠀⡴⠛⠛⢦⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢀⡾⢡⡟⠀⠀⠀⠈⢧⡴⠶⣆⠻⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⡾⢡⡟⠀⠀⠀⠀⣠⡞⠁⣠⠏⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢸⣇⣀⣀⣸⣿⣡⣟⣀⣀⣀⣀⡴⢋⣤⣞⠻⣦⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⡏⠉⠉⢹⡯⠭⠭⠭⠽⠿⠯⠾⠿⢭⡽⠗⠈⢷⡤⠶⠛⢹⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⡿⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⣸⠃⠀⠀⣰⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡗⠒⢲⡶⠚⠛⠉⠀⠀⠀⠀⢠⡏⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠛⠛⠛⠒⠶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠾⠷⠶⠶⠾⠷⠆⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⣀⣀⣠⣤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠙⠛⠛⠛⠛⠛⣿⠿⢶⣶⣾⣿⠾⣿⡟⣻⠿⣻⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡼⠟⠋⣿⠈⣿⠻⣧⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⡀⢀⣀⣿⣤⣿⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠳⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⣧⣠⣄⣤⣀⣰⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠞⠋⠁⠀⢸⣿⣧⣽⣧⣿⣽⡇⠈⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠀⠀⢀⣴⠞⠋⠉⣀⣀⣀⡀⠉⠙⠷⣦⡀⠈⠙⢿⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣰⠟⠁⣠⡾⠛⠉⠉⠉⠙⠻⢦⡀⠈⢻⣆⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢰⡲⠀⠀⠀⠀⠀⠰⣲⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠿⠛⠛⢻⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡟⠛⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡦⢀⡾⢹⣿⣿⣿⣿⡻⠶⣦⣀⣠⡶⠾⣻⣿⣿⣿⢿⣏⢷⡀⢴⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⡟⠹⠿⡿⠷⠛⣡⣿⠃⠈⠻⣿⣿⣿⣾⣿⣿⣾⣿⣿⡿⠟⠁⠘⣿⣌⠛⠿⢿⠿⠏⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣸⡿⢁⣀⣤⣴⡶⠿⠛⠁⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠈⠛⠿⢶⣦⣤⣀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊
⡶⡶⢶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⢾⣶⣶⡶⠶⠶

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

⠀⠀⠀⠀⠀⠀⣸⠞⢼⣿⣮⡓⢀⠀⠀⢰⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⡞⣿⠻⡏⠙⢦⣉⣲⣿⣍⢻⣿⣿⣗⣹⣿⣿⣿⣿⣿⡟⠉⠙⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣇⠀⣀⢻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡄⡀
⠀⠤⡤⢤⣄⠀⡀⠀⠀⠈⠁⣘⣽⣿⣟⣯⣿⣿⣿⣿⡿⢿⣷⣿⣯⣹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠃⠤⣶⡿⣿⣿⣿⣿⣧⢽⣿⣿⣯⣿⣿⣿⣟⣿⣿⡀
⠀⠀⠀⠈⢳⣠⢠⣤⡈⠀⣤⡀⠀⢀⡐⢀⣈⡽⠛⠛⣿⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠈⠔⠁⠀⢊⠽⠿⢦⣴⣏⣙⣿⣧⡇⠴⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠈⠀⠀⠀⡠⠉⠀⠠⡀⢼⡷⠿⣩⣑⣠⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⣠⠤⠇⢠⡄⠀⠟⠋⠠⣠⣿⣿⣧⣿⣟⣿⣿⣿⣿⣿⣿⣿⣧⣿⣯⣿⠆
⠀⠀⠀⠀⠀⠘⠁⢸⣴⡥⠊⢁⣠⣤⣄⣤⣤⡵⣿⣿⣿⣿⢻⣿⡿⣿⡿⣿⣿⣿⣿⡏
⠀⠀⠀⠐⠀⠀⠴⡾⠋⠤⠌⠠⣶⣿⣭⡷⣬⢻⣿⠿⣿⣿⣿⣿⣲⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠣⡤⠀⠀⢶⣶⡾⡉⠐⠀⣌⣈⣰⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⣈⣉⣙⣩⡷⠶⢭⡋⢰⠐⣻⣿⡿⠻⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠳⡄⠀⣠⡿⠚⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣴⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣭⣛⡛⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⣿⠉⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⠷⢣⡖⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⠏⠉⠈⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⡍⠿⣶⣅⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠈⢢⡀⠁⠈⠉⠻⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⢢⡀⠉⢲⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⠃⠀⠀⠈⣇⠙⢦⡀⠰⠧⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢱⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠰⠁⣿⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡇⢯⠸⡀
⠀⠀⠀⠀⠀⢠⠃⢸⣿⠀⠀⠀⠀⠀⣸⣠⣀⣤⠖⡀⠈⡇⡇
⠀⠀⠀⠀⠀⡜⢠⣿⣿⠀⠀⠀⠀⢠⣭⣽⡧⠔⣹⠇⣠⡇⢇
⠀⠀⠀⠀⢀⡇⠘⢻⣿⠃⠀⠀⠀⠀⡿⠛⣞⣿⣵⡿⠁⢿⣾
⠀⠀⠀⠀⠘⡇⣦⢸⣿⡀⠀⠀⠀⢠⣇⠀⠀⣼⡯⠃⠀⢸⢻
⠀⠀⠀⠀⠀⣷⣿⡄⣿⠗⣉⡑⠛⢦⣹⠀⢠⣿⣷⡀⢀⣼⠆
⠀⠀⠀⠀⠀⡘⢻⣿⣿⣾⣯⠍⠙⢦⡻⠀⢸⡈⠓⠞⠒
⠀⠀⠀⠀⢀⡇⠀⠹⣿⣿⢹⡆⠀⠈⠁⠀⠈⡇
⠀⠀⠀⠀⠈⢧⣳⣤⣹⣿⣆⣇⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠄⠀⣶⣾⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⢻⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢂⠈⠻⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡇⠁⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣦⠆⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡻⢿⣿⣿⣧⣤⡙⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡳⣍⠀⠉⠙⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠁⠒⠠⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⢀⡤⠒⠉⠛⠀⠈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠛⠶⠛⠋
⠀⡔⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠀⠀⢀⡀
⠀⢇⣠⠀⢿⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠁
⠀⠈⠙⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⢸⠉⠒⡂⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⠶⠤⠴⠤⡯⠔⠒⠀⠐⠒⠄
⠀⠀⠀⢸⠀⠀⢃⡀⣀⣀⣰⠖⠛⠋⠉⠉⠁
⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠘⠧⣀⣀⣀⡀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠔⠊⠁⠀⠀⠀⠀⠀⠈⠉⠒⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⠄⠠⠠⠄⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠘⠒⠒⠀⠐⠀⠚⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡇⠀⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢄⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣡⠈⠛⠐⠤⡀⠀⢀⡤⠒⠘⠇⠀⠀⡀⢤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠑⠋⠀⠀⠀⢣⡔⠉⠀⠀⠂⣀⠔⠁⠀⠀⠈⠚⢢⡀⠀⠀⠑⢄
⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠋⠀⠀⠀⠔⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠘⣆
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢁
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠘
⠀⠀⢠⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠃
⠀⠀⡜⠀⡴
⠆⠈⠀⠀⠐⠔⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘
⣿⣿⡿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠠⣤⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠉⠈⠳⣄⠀⠀⠀⣠⠞⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⣷⣠⡾⠃⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣄⣀
⠀⠀⠀⢀⣠⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣶⠶⠤
⠀⠀⠋⠉⠉⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⣿⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡷⠖⠒⠛⠉⠉⠹⣏⠉⣹⣍⣿⠟⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣤⣼⣭⠿⠁⠀⠉⠉⠉⠉⠒⠒⣿
⠀⠀⠀⠀⠀⠸⣿⠀⢀⡠⠄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢹⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣿⣷⠀⠁⠀⠀⢠⠀⠀⠀⠀⠐⠲⣄⡀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⢽⣠⡾⠁⠘⠷⣄⡀⠀⣮⠁⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣬⠇⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠈⠙⠛⠧⠤⠤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠙⠻⣿⣿⣓⠾⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣈⣷⣮⣝⣷⢯⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠉⠛⠻⢧⣍⣷⠮⣝⣓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣯⣙⢲⣭⣛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣮⣿⡒⢬⣝⡲⢤⣄⣀⣤⠶⠶⠶⢤⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢧⣽⣗⠮⣭⣀⣤⣶⠦⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⢿⣿⣷⢤⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⠿⠿⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠉⠲⠶⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣄⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⠖⠈⠑⣤⠀⠀⠀⠀⠈⠓⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⡀⠀⠀⢀⡘⠈⠙⢦⣀⠀⠀⠀⠀⠈⠃⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣩⡷⣦⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⡾⠋⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠈⡇
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠂⠀⠀⠀⣠⡟⠀⠀⣠⡶⠿⠄⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⣀⠀⠰⠿⢶⡄⠀⢀⣠⣾⠋⠀⠀⣿
⡄⠒⢤⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⠶⠷⠶⠋⠁⢸⡇⠀⠀⣿⠀⢀⠐⠒⡀
⠁⠰⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢸⠇⠀⠀⣿⠀⠘⠃⠐⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠶⠶⣦⣸⡟⠀⠀⠀⢰⠏⢀⡀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣀⠀⠿⠄⠀⠀⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⠟⠁⣤⠀⠀⠛⠁⠀⠀⠀⠘⣄⠈⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⢀⡶⠀⠀⢸⣿⡄⠀⠀⡟
⡏⢀⠜⠀⠀⠀⢸⡄⢸⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⢿⣀⡿⠀⢸⡇⠀⠘⢃⠈⡁
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠄⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⠇⢠⡝⠃
⠀⠀⠀⢀⣀⣰⠳⢆⠀⣠⢄⠙⢶⡠⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⡾⠁⠘⣾⠁⠀⠀⡠⣄⠀⣠⠞⣦⣀
⠀⠀⠀⠻⣍⡙⠳⠈⠛⠁⢸⠁⠈⠻⣆⠀⠀⠙⠻⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠋⠀⢂⡿⠁⠀⠀⠈⠣⠈⠟⠁⠾⢛⣉
⠀⠀⠀⠰⣯⣁⡀⠀⠀⠀⠘⠧⣤⣄⠉⢳⣄⠀⠀⠀⠈⠉⠙⠓⠒⠶⠒⢶⡤⠀⠠⡤⠶⠶⠶⠞⠋⠉⠀⠀⠰⣹⠞⠁⠀⣀⣤⡸⠇⠀⠀⠀⠀⣉⣽
⠀⠀⠀⠀⠀⠀⠈⠛⠃⠚⠲⠦⣄⡀⠙⠋⠻⠷⠌⣡⣄⣀⣠⣶⡶⠶⠶⠾⣄⠀⢀⡴⢶⠶⠶⢶⣆⣀⣀⣀⣴⠿⠔⠛⠋⢀⣀⠴⠖⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢤⡄⣀⡀⠀⠁⣹⡇⠀⠀⠀⠂⠀⠤⠌⠀⠀⠀⠀⢸⡏⠈⠀⢀⣀⣠⡤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠻⢷⣴⡆⠀⠀⠀⠀⠀⠀⠀⢸⣦⣼⡿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠛⠛⠛⠛⠛⠛⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⡀⠀⡀⠀⣀⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠋⠻⠟⠻⠟⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢰⣿⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠹⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣾⠀⣧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⣿⠀⣿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣿⠀⢸⡃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠾⣟⠛⣿⠀⣸⡛⣿⠿⠶⣦⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣷⣶⣶⣦⣤⣟⣀⣻⣤⣤⣶⣶⣾⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣁⣀⣀⣀⡀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢚⣉⠤⡤⠤⣄⣈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢸⣷⠀⠀⠀⠈⡦⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⡞⠀⣎⣀⡜⠁⠀⠀⠈⢣⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠛⠒⠒⠒⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢆⠈⣿⠉⡉⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠦⠆⠀⢣⠀⠀⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠈⠀⠀⠤⠶⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠐⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠐⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣴⠲⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⢰⠲⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠢⠄⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⣛⡛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣛⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣤⣉⣵⣯⣿⣿⣧⠌⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⢤⡘⢩⡷⠠⣤⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⣻⡾⣿⢿⣿⡟⠺⠿⠆⠀⡇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢺⠀⢸⠀⢰⠋⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣹⠞⠛⠛⠑⠀⠀⡀⡀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⡄⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⣀⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⣀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠈⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠉⠁⠑⢦⠈⠹⡕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠈⣿⠄⠀⣼⡏⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠐⠲⢤⠤⠲⢄⠴⠚⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣤⠠⠤⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡠⠤⠀⠒⠀⠀⠀⠁⠒⠒⠒⠂⠀⠀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡏⠒⠶⠤⠤⠤⠤⠤⠄⠠⠀⠀⠒⡈⣇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠉⠓⠲⠶⠶⠶⠶⠂⠁⠀⡼⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⢓⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⠤⠒⠉⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣨⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢁⣄⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡼⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡿⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⣠⠁⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢐⡄⠀⠀⠀⠀⢱⠈⠳⢄⡀⠀⠀⠀⠀⠀⣀⡀⠈⠁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢫⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠤⠀⢠⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠘⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠠⠇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠰⠷⠂⠀⠘⠀⠀⠀⠀⠀⠘⡆⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢀⠃⠀⠀⢀⢤⡀⢤⡖⠚⢀⠉⡆⠉
⠀⠀⠀⠀⠀⡎⠐⠐⠐⢀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⡸⠀⠀⠀⠀⠀⢧⡀⠀⡘⠀⠀⠀⢠⡀⠊⠄⣟⠡⠖⣂⢀⠀⠐⠢
⠀⠀⠀⢠⠴⠀⠀⠀⠀⠀⠑⠢⢤⣀⡈⠀⠃⢠⡄⠐⢀⡀⠤⠤⣼⡜⠀⠀⠀⠀⠀⠀⠀⠙⡀⣇⠄⠐⠒⠢⠉⠂⠁⠀⠀⠠⡅⣂⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠈⠀⠀⠁⠀⣈⠁⠒⠐⠒⠒⢀⣤⣈⣀⣈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠚⠛⠋⠁⠀⢠⣴⣄⣍⣥⣤⠄⠦⠝⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣔⣒⣀⠒⠶⠶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣺⣽⠟⢉⣩⣤⣤⣤⣤⣤⣬⣿⣒⠂⠀⠀⠀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠯⠛⠁⠢⢴⡿⠟⠙⣓⡚⢓⠛⡛⠳⠮⢤⣀⣄⡐⠻⡢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⡫⠂⣰⡆⢴⣆⣤⣤⣈⠛⠿⢛⠉⠁⢀⣅⢀⡀⠀⠈⠙⠻⣦⣌⠐⠕⢄
⠀⠀⠀⠀⠀⠀⣠⣿⢿⠏⠠⣺⠟⣡⠀⠸⠟⠉⣤⢶⣳⣬⡛⢿⠿⣶⣦⣬⣉⠒⢦⡀⠈⠻⣷⣄⠲⡡⢀
⠀⠀⠀⠀⠀⣰⣳⠋⠁⢀⡼⢁⠞⡱⠃⡨⢤⣼⡾⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣙⢾⡀⡈⢿⣷⣔⡹⣷⡀
⠀⠀⠀⠀⢰⢳⡏⢀⢠⡿⢠⢃⠠⠀⠐⣁⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⣄⢩⣍⡈⠛⠻⠎⢻⢳⡀
⠀⠀⠀⠀⣿⢿⠇⠆⡟⠁⠋⢁⡄⡸⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⣆⢻⡟⣦⣶⠀⠈⣏⣧
⠀⠀⠀⠀⣿⡞⢠⠁⠀⠁⠠⣿⢣⠃⣿⡇⣀⡤⡄⠀⠀⠀⠀⡠⣄⣀⣀⣃⡀⠄⠀⠀⢸⡘⡈⠗⢇⡀⣴⢻⢸⣿⡀
⠀⠀⠀⣶⣿⡇⡀⢸⣠⡆⣄⣈⠘⢢⠟⠋⠁⢀⠇⠀⠀⠀⡜⠀⢿⣿⣶⠏⡘⠀⠀⠀⠈⠓⠐⣿⢹⠆⣿⠸⢸⣿⡇
⠀⠀⠀⣿⣿⡇⣷⠈⢿⡇⡇⢻⠞⠁⠀⠀⢀⠎⠀⠀⠀⠀⣇⠀⠀⠀⣠⠜⡤⠊⠙⠀⡘⢠⠰⣿⠈⡄⡿⢀⢹⣿⡇
⠀⠀⠀⢻⢿⠀⢸⣇⠸⣟⢰⣸⡄⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⣜⣠⣄⡸⡰⢃⠃⣦⠁⠘⠄⠁⡈⣸⣻⠃
⠀⠀⠀⠸⣾⡇⠈⢏⡆⢿⣖⠗⣷⡀⠀⠀⠈⠷⢄⠀⠀⠶⠮⣥⣦⠀⠀⠈⠉⣩⡞⠁⢃⣼⣿⠃⠈⠀⢠⢡⠇⡿
⠀⠀⠀⠀⢿⣿⣆⠘⣺⡜⢿⢆⠻⡷⠀⠀⠀⠀⠀⠉⠒⠦⠤⢤⡤⠤⠤⢶⠻⠗⢀⡀⠾⠙⠁⠞⠀⠐⣿⠏⡼⠃
⠀⠀⠀⠀⠈⢻⣎⢆⠱⡹⣎⠻⣧⣿⠀⠀⠀⠀⠀⠀⢀⡠⠤⢶⡲⢤⣴⣋⣤⠴⣪⡵⡣⢦⠄⠀⠀⠠⢊⡾⠁
⠀⠀⠀⠀⠀⠀⠛⢯⣧⡙⢿⢧⡈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⣤⣤⠴⠾⣋⠤⠖⢁⠔⡹⢋⡴⡫⠂
⠀⠀⠀⠀⠀⠀⠀⠪⡻⣟⠎⠳⣝⡻⡆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠷⠾⠋⣀⠤⣚⡵⢊⠕⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⣤⡀⠙⢳⠀⠀⠀⠀⢸⠍⢉⣳⣄⣀⣴⣛⡤⢄⣠⣶⠟⢉⣪⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡻⣶⣧⠀⠀⠀⠈⣗⣒⡲⠞⠒⠒⢒⣉⣉⣥⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⡀⠀⠀⢸⣽⡿⠿⠟⠛⠛⠛⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣵⣄⡀⠀⢀⣥⣤⡄

⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⢿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢿⣯⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⡿⠟⠋⠀⠀⠀⠀⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣥⡀⠀⠀⠀⠀⢀⣬⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠌⠁⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠄⠀⠀⠠⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⢰⡆⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠒⠒⠒⠒⠒⠶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠖⣀⣤⣴⣶⣖⣀⣤⢦⡀⠀⠀⠀⠀⠀⢰⡾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠚⠁⣼⠷⠋⠁⠈⠉⠉⠉⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⢾⡀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢠⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⣿⡈⠿⢁⡤⢤⡀⠀⠀⠀⠀⢀⡴⠒⠲⣄⠀⠀⠀⠀⣿⠞⢻⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⡇⢠⡏⠀⠀⢳⠀⠀⠀⠀⢸⣁⣀⠀⠈⢧⠀⠀⠀⠀⠀⣼⡤⢤⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠸⣯⣷⠀⢸⠀⠀⠀⠀⢸⣷⣼⡆⠀⢸⠀⠀⠀⠀⠀⢩⣤⡀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠛⠋⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣸⣠⠇⣼⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣁⡼⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢤⣖⠒⠒⠓⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⣼⢉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢸⣤⠤⠤⣤⣤⣈⣷⠀⠀⠀⠀⠀⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⣦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⢮⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⡾⠀⠀⠀⠉⠉⠉⠉⠀⠀⣰⠃⠀⠈⡿⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢉⡟⢹⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⡞⠁⢀⣠⣤⢽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡞⠀⢸⡄⠀⠸⣄⠀⠀⠀⠀⣀⡴⠋⠀⢀⡰⢋⡴⠚⠉⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⠁⠀⠀⠹⣄⡀⠈⠙⠓⠚⠋⠁⠀⣀⡴⠋⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣼⣀⣀⣀⠀⠀⠉⠓⠲⠶⠶⠖⠚⠋⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠒⠒⠲⠶⠶⠤⠤⢤⣽⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⠒⠛

⠀⠀⠀⠀⠀⠀⠘⢍⡉⣑⣶⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡏⠉⠁⢀⠀⠀⠀⠀⠄⡀⢨⠀⠉⣹⠟⠓⠈⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠤⠤⠤⠤⠤⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠐⠁⠀⠀⠀⠀⠐⢹⡛⠟⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠊⠢⡀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢸⠋⣷⣦⣀⣀⣀⣀⡇⡜⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠊⠀⠀⠀⠙⣄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠊⣉⣸⠿⢧⠄⠀⠀⠀⠈⠉⠙⠒⠢⢬⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢸⠀⢀⡴⠊⠓⢤⡀⠀⣲⠶⢀⣴⠅⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⣄⢃⣀⣬⣄⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡠⠚⠁⣀⡤⠔⠒⠒⠉⡿⢴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣸⣿⣿⡤⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡋⣀⣴⠊⠁⠀⠀⣀⡀⢰⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠬⠭⠭⣬⣭⠤⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣉⣎⣀⣼⠀⠉⠛⠯⢅⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⠤⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣦⠀⡠⠞⡉⠠⠀⠂⠀⠀⠀⠉⠓⣄⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣧⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠁⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢂⡈⢃⠀⠀⠀⣠⠔⠋⢳⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠐⢇⣛⡩⢖⣿⠆⠀⣰⠃⠀⡀⣾⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠀⢰⠃⠀⠀⡼⠧⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢿⠇⢠⠿⣀⡠⠎⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⢰⠋⢺⡉⠉⡼⠀⡎⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡐⠧⠀⠀⢸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠆⣠⡴⣅⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⡈⠑⠚⠉⣉⣉⣒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⡖⠤⣈⡑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⣏⣤⣄
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⢸⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⡆⠀⠀⠀⠾⠻⡿⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡟⠋⠉⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠈⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠸⠀⠐⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⠠⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠋⠀⠀⠢⣦⡀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⠀⠀⠉⠢⡈⠲⣌⣦⡖⠠⢥⡒⠷⠠⠀⠀⠈⣶⠏⠀⠀⠀⠀⢁⠀⠁⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠡⠀⠀⠀⠀⣻⣋⠐⠓⢠⣐⠁⠈⠆⠀⠂⠀⡀⠻⠀⠀⠀⠀⠀⠘⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⢸⡁⠑⢌⡀⣈⠬⠘⠂⢬⠀⢀⡼⠀⠐⣇⠀⠀⠀⢸⡔⡀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠀⠌⢀⠄⡞⠀⠒⠀⠠⠤⠂⠀⠀⠊⠉⢀⠛⠀⠀⣤⠆⢀⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠤⠈⡔⢀⡀⠛⠀⠀⠀⠀⠀⠀⢀⠒⠂⠐⠁⠀⠡⡐⠈⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢷⠊⠉⠁⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠈⠐⠒⠒⣲⠚⣯⡀⠁⠱⠒⠁⠁⠺⠀⠂⠀⠀⠠⠠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡔⠓⠒⠭⠧⡐⠀⠀⢃⣘⡀⢬⠈⠀⠀⠀⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢀⣀⢪⠁⠀⠔⠢⠤⠅⠀⠈⠙⡛⠁⢘⣀⠤⡐⠐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠒⠚⠦⠤⠄⠓⠶⠤⣲⡄⢠⠄⠀⠄⠠⡤⣀⡉⠀⣀⣡⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠄⣰⡃⠠⠀⠉⠐⠺⡁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄⡀⠀⠀⠈⠸⡁⠈⠈⠀⠤⠆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠈⢄⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⡀⠀⡠⡀⠀⠤⠀⠤⠐⠁⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡁⠀⠛⠀⠀⠀⠰⠄⠀⡀⠴⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠀⠠⡀⠀⠀⡀⢄⠀⢰⢏⣠⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⠀⠀⠀⢀⠄⠀⢈⠱⠖⢓⠷⢖⣣⡀⡄⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⢐⡤⠐⠁⡠⠊⡡⠒⠉⠇⠀⠈⠄⠀⠈⢢⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⣈⠉⠐⠺⡀⡌⠀⠀⠘⠓⠆⠀⠞⠀⠀⠀⢂⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠉⠐⡦⡈⠉⠀⠂⡄⠀⠰⠖⠬⠀⠀⠀⠈⡄⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢁⠈⢢⠀⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢄⠀⠀⠀⠑⢌⠓⠓⠏⣐⠂⠠⠄⢀⣀⡀⠠⠜⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠠⡵⠂⠄⡀⠈⠁⠐⠒⠀⠀⠐⠒⠒⠀⠈⠉⢉⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⠀⠀⠀⠀⠉⠁⠀⠐⠒⠒⠂⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡐⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⢅⠢⡀⠀⢆⠑⢄⢹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡑⠢⣄⠳⡀⠑⢄⢣⠈⢢⡆⠱⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡢⢄⠀⠈⢢⠀⠑⠬⣆⠀⠳⣇⠀⢻⠀⢣⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠈⠢⡉⠒⢄⡱⡄⠀⠈⠳⠀⠈⠂⠈⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠒⢄⡲⣬⣢⣄⠉⠺⢆⠀⠀⠀⢀⣀⠀⢤⠂⢀⣴⠆⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡌⠊⠻⢍⡓⠄⠀⠀⣴⠉⠀⠀⠀⠠⠃⣈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠆⣐⠒⠒⠷⢄⠀⠁⠀⠀⠀⠁⠀⠀⠀⢀⠗⠀⠀⠈⠀⣀⠔⠒⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⣛⣁⡀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢠⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠂⠠⠀⠈⠉⠂⠀⠀⠀⠀⠀⠈⡀⠀⡆⠀⠀⠀⠀⠀⠘⢣⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠶⠄⣀⠃⠀⣠⠔⠒⠒⠢⢄⡇⠀⡀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠁⠠⠀⢠⡄⠀⠁⠀⠙⠢⠤⠤⠒⠊⠙⠢⠤⢤⠔⠊⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⠋⠁⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠄⣀⡀⠁⠺⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣴⣿⣟⣓⣒⣂⠒⠋⠑⠢⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡎⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡜⠉⣠⣤⡈⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣄⡈⢟⡄⢇⣀⠀⢀⣀⠀⢠⡀⠀⢀⡀⠀⢀⠔⠉⣀⠀⠀⠀⡜⢀⡊⠁⢀⣈⠢⠄⠠⠜⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠑⠻⠆⠀⠀⠈⣉⠁⢤⡈⢑⠀⠉⠁⠸⡀⠁⡰⠎⠀⢂⠀⠀⣏⠉⣉⠝⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠂⠜⠁⠀⢀⣯⠋⠀⠀⠀⠀⠈⠉⠁⠘⠢⣈⣦⡀⠈⣉⡤⠖⡡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢨⠆⡀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⢙⡛⢄⠀⡰⢁⠞⠐⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠜⠀⠀⠈⣂⠉⠁⢓⡀⠤⢐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠑⠒⠢⢤⢤⠄⣆⠀⠀⠀⠀⣀⡧⢠⠀⠀⠉⢉⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠢⠄⠑⠒⢶⠀⠀⠀⠀⠠⠦⣼⣘⠀⠈⠈⢑⠀⠘⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣖⠁⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣢⣔⠈⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⠀⠉⠉⡠⡶⠖⢶⠘⢻⠉⠀⣈⢽⡈⠸⢒⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠐⠊⢥⣍⣬⢷⣒⣒⡒⠈⠅⠰⠖⠛⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣳⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠠⡀⠀⠀⢀⣀⠀⣀⣀⠀⢠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠄⠈⠁⠐⠲⠌⠑⠒⠊⠉⠀⠁⠉⠀⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣉⠛⠛⠛⠛⠻⢷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠈⠉⠉⠉⠉⠉⠛⢶⣤⡀⠀⠀⠙⢿⡉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣄⠀⠀⠀⠻⣆⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⢻⡄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣤⠒⢶⣿⣿⠁⠀⠀⠀⢈⣿⣿⠛⠶⠶⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠆⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠻⠶⢤⠤⠶⠀⠀⠀⠀⠘⠓⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠛⠛⠛⢷⣦⣀⣠⠞⠛⢷⣦⣀⣰⠞⣿⠉⠁⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⣦⣄⣀⣀⣀⣀⣀⣀⣀⣴⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⢦⣀
⠀⠀⠀⣠⡄⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠚⠋⠀⡀⠈⠙⠳⣄⠀⠀⠀⠀⠰⠦⠠⠤⠴⠆⠠⠠⣤⡀
⠀⠀⠀⠿⢀⣀⠠⠤⢤⣄⡀⠉⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠒⢂⣀⢠⡤⢤⡤⠤⢀⣀⡀⢀⣈⠃
⠀⠰⠶⢶⣍⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠹⠀⠀⠀⠀⠀⢉⡽⠋⢹⣆
⠀⣦⠀⠈⣿⠛⢶⣤⣤⡄⠀⣀⠀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⡀⢀⡀⣀⠠⠴⢤⡄⠉⠹⠇⠀⢸⡇
⠀⠸⢧⣤⣿⡄⠀⠈⠁⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠋⠁⠀⠀⠐⢢⠀⠆⢸⣧⣤⠞⠁
⠀⠀⠀⠈⣿⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢁⠠⣤
⠀⠀⠀⠀⢸⡆⠀⠉⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⢀⡾⠁⠰⣿
⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠈⠛⠷⣶⣄⣠⣄⣀⣀⣀⣀⣠⣤⣤⢤⣤⡴⠞⠋⠀⠀⢀⠘⣿⠀⣠⡟
⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠋⠀⠀⠀⠀⠀⢸⣾⠃⠀⣿
⠀⠀⠀⠀⣿⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣿⠀⠀⣿
⠀⠀⠀⠀⣿⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠋⠀⢠⠟
⠿⠛⠶⣶⣌⡀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢰⡇⠀⠀⠀⣀⣠⠶⠖
⠸⣧⠀⠀⢉⡿⠓⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣿⠀⠀⢸⣤⡴⠶⠊⠉⠁⠀⢠
⠀⢹⡆⠀⢀⡧⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠹⠖⠛⠣⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠸⣷⠀⠀⣁⠤⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⡁⢠⣤⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢿⣇⣾⠿⠤⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃⠴⠴⠿⣦⠀⠀⠀⠀⢠⡏
⠀⠀⠘⣿⣇⠀⠀⠀⢰⣤⡐⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣤⠟⠀⠀⠀⠀⣾⠁
⠀⠀⠀⣿⣿⡄⠀⠀⠀⠈⠛⠷⣶⣟⣛⣁⣤⣤⣤⡤⠤⠤⠀⠀⣀⣀⣀⡠⠤⠄⠁⠀⠀⢄⡹⠂⠀⠀⠀⣠⠋
⠀⠀⠀⠸⣿⢻⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠒⠒⠒⠚⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⢀⡿
⠀⠀⠀⠀⣿⡈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⢸⣇⠀⠉⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠈⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢻⠆⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢰⡯
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⠀⠀⠀⣿
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣹⣄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢰⣋⣀⠠⠴⠤⠴⠶⠶⠴⣦⡀
⣤⡖⠋⠉⠉⠀⠀⢈⣩⡿⠟⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠗⠀⠻⢿⣥⣤⣤⣤⣤⣤⠤⠶⠊⠁
⣾⣿⣿⣿⣿⣿⣿⣷⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣶⣶⣦⣤⣤⣶⣶⣶⣶
⣿⣿⣿⣿⠿⠋⠹⠟⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣁⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣄⠀⢲⢀⣠⡶⢶⣿⠣⣽⣆⣮⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢰⠿⣄⡰⢣⢞⣽⣿⣿⣴⣿⣿⣡⡴⣣⠶⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢔⠫⣹⣮⡿⣷⣿⣿⣿⣤⣷⣿⣿⣿⣿⢟⣱⣦⡴⢤⣴⢟⣾⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣨⣿⣿⣻⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⢷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢨⡙⢮⣻⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⡵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢱⣼⡼⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣯⣶⡟⢁⡔⣠⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠋⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣭⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⠶⣾⣿⣿⣿⣿⡿⠻⢡⠛⠙⠟⢿⣿⣿⣿⣽⠿⠟⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⡴⠛⢻⣿⡽⣯⠁⠀⠀⠜⡀⠈⠈⣿⡟⣿⢿⣻⣦⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢒⡀⢹⣟⡇⠉⠀⠀⠀⢠⡀⠀⠀⠘⠃⡛⠒⠈⠙⠓⠵⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠭⠄⠱⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢤⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠂⠄⠀⡴⠂⠉⠀⠀⢣⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⡆⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠊⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢄⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠈⠓⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠊⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⠒⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠀⠉⠁⠀⠲⢄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠴⠀⠀⠡⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠈⣀⣀⡀⠼⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠔⢪⡁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⢄⠁⠀⠀⠀⠀⠀⠀⠘⠁⠂⢢⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠈⣤⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡀
⠀⡐⠂⠀⠂⢴⠂⠉⠀⠀⠀⠘⠿⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⡇
⠀⡇⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠙⠒⢳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣰⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠻⣁⡠⠊⠉⠳⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⡐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣧⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⡜⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡴⠋⣠⠀⠀⠀⡜⢀⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⢋⠔⠊⠀⠀⠀⡘⡠⠎⠀⠀⠀⡴⠁
⠀⠀⠀⢀⢀⣀⣀⣀⠠⠄⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⢄⡀⠀⠀⠀⠀⠘⡅⠀⠀⠙⠢⠴⣅⠴⡴⠋⠀⠀⠀⠀⢸⠁
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠚⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⢆⡠⠀⠀⡀⣠⠏
⠀⠀⠸⢅⣀⣤⠤⣤⠤⠤⠠⠤⠔⠒⠒⠚⠛⠉⢉⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⡀⠀⢀⡀⣀⡤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠶⠖⠒⠒⠒⠛⠛⠛⠉⠉⠛⠛⠛⠓⠒⠒⠶⠤⢄⡀⢠⠞⠙⠲⣄
⠀⠀⠀⠀⠀⠀⢠⠏⠁⠉⢷⣠⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣀⠤⠒⡶⠀⢀⣀⡤⠶⠒⠛⠉⠉⠉⠉⣑⣲⠦⠈⠉⠀⣀⣀⡸⠇
⠀⠀⠀⠀⠀⠀⠸⠤⣀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡴⠋⠁⠀⣼⠤⠚⠉⠀⠀⠀⠀⠀⢀⣤⣖⡋⠭⠄⣀⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⣰⡀⠀⠀⠀⠀⢠⠋⠰⢃⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢀⠏⡘⠁⡇⠀⠀⡀⠀⡆⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠄⠀⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⡈⠰⠿⠿⢧⠀⠀⡁⠠⠁⠀⠀⠀⠀⠀⡠⠴⠒⠋⠉⠉⠉⠉⠁⠊⠁⠀⠀⠢⡀⠀⠁⠄⠀⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣼⠀⠀⣘⡄⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡠⠔⠒⠒⢮⡓⠀⠰⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠋⠙⢻⡆⡞⢁⢤⠀⠀⠀⠀⠠⡓⠀⠀⠀⠀⢠⠞⠁⠀⢀⠎⠀⠀⠀⠀⠀⠹⡄⠀⠱⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡀⣿⣶⣾⣿⣿⡇⠙⠁⡏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⠂⠀⠢⡎⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠇⣅⣠⣤⣤⣼⡇⠀⠘⠷⠒⠒⣚⢳⡀⢀⡇⠀⠀⠀⠀⠀⠀⣤⠚⡖⢤⣤⣀⡀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⡇⠀⢠⠖⠉⣉⣄⣈⡉⠉⠀⠀⠀⠀⠅⠀⠀⠈⣰⡇⠉⠁⠀⠈⢉⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⣦⣤⣶⣶⣶⡇⠀⣿⠀⠈⠙⠒⠚⠉⢀⠀⠀⠀⠀⠀⠀⠀⡾⠏⠙⠿⠶⠶⠚⣋⣀⡀⠀⠀⣸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡄⡏⠉⠉⠉⢉⡇⠀⠘⠳⠦⠤⠶⠒⠛⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠉⠀⠀⣹⠀⢀⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡁⣿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠉⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢳⣡⣤⣤⣤⣶⡇⠀⠀⠀⠀⠀⠀⠀⠘⢀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂⠦⠄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠸⣸⡏⢉⣉⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⠲⢄⡀⣀⣀⣤⡄⠀⠀⠒⠈⠁⠈⠒⠁⠀⠀⠀⠀⠐⠿⣿⣷⡖⠢⡄
⠀⠀⠀⠀⠀⠀⢣⢻⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠳⣄⠔⠀⢻⡠⠊⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠋⢳⣶⣿⠁⠀⠀⠀⠀⢀⡴⢢⠞⡿⢲⡆⣀⠀⠀⠘⠛⠀⡄⢣⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⣀⡤⠂⠉⠉⠈⠉⠉⢭⠉⠉⠓⠘⠢⠓⠊⠀⠁⠉⠉⠉⠉⠙⠛⠛⢆⣸⠀⡏⢳⣄⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⡀⠔⠋⠁⠀⠰⣖⠒⠀⡔⠁⢸⣤⠄⠤⠐⣦⡤⠤⠤⠤⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢆
⠀⠐⠉⠀⠀⠀⠰⠚⠉⠉⠣⠀⡀⠀⠰⠊⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠂⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠑⠒⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡉⡆⠀⠀⠀⠀⠀⠚⠃⢂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢴⡥⠤⢀⣀⠉⠀⠒⠒⠚⠋⡠⠒⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⡁⠒⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠉⠁⠉⢐⠉⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⣤⣀⣈⠤⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠟⠛⢿⡟⠋⠁⠀⠀⠀⠀⠈⠸⣟⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⣀⣀⣀⣌⣥⣤⣿⠆⣠⣾⣿⠴⠤⠿⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠈⠻⣦⣤⣶⣴⣤⣤⣶⣶⣶⣤⡄⠈⠒⣦⣬⣷⣶⠶⠶⣤⠤⣤⣤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠈⢦⣾⣿⠁⠈⠉⢧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⢸⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⣿⣟⡄⠀⠀⠈⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣾⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⡀⠀⠀⠀⠀⡟⡔⠀⠀⠀⠀⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢞⣡⠀⠀⢠⡀⡷⢄⣠⠄⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡅⠠⣤⠀⣿⠁⡾⠴⠀⠰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠓⠀⠛⠛⠓⠊⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡤⠖⠛⠓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢠⠖⠛⠒⠚⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠳⣄⠹⣤⣀⣀⣀⡴⠚⣁⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⡄⠀⠸⡆⠀⠉⠉⠁⢦⡀⣈⠳⡄⠀⠀⠀⢸⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢳⠀⠀⢹⡀⠀⠀⠀⠈⣇⠸⡆⠹⡆⠀⠀⢸⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⡼⢧⠀⠀⠷⠤⠤⠤⠤⠤⠤⢤⡀⢹⠀⠀⢸⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣀⠀⢀⡴⠋⠀⠘⢦⣀⣀⣀⡀⢀⣀⣀⣀⣠⡇⣀⣀⡀⣸⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠈⢳⡋⠀⠀⠀⠀⠀⠉⠉⠉⠙⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣉⣉⣉⣀⠀⠀⠀⢀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠼⠆⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⣿
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠤⠤⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠨⠀⠸⠰⠁⢘⠀⠀⠘⠀⠈⠀⠈⠸⠂⠂⠀⠒⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣆⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⣠⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠻⣿⣿⡄⢰⡣⠔⠃
⠀⠀⠀⠀⢠⡿⢻⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⢠⣄⠘⢿⣧⠀⢀⡤⠤⣄
⠀⠀⠀⠀⣾⣧⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⡄⠈⠓⠲⠊
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣆⠘⠫⢳⡄
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠐⣾⠁⠀⠀⠀⡤⠦⣄⣀
⢎⣉⡭⠖⠈⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢹⣦⡧⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠸⣦⡴⠶⣼⠁⢠⠏⠉⣿⡀
⠀⠀⢀⠠⡄⠈⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⣿⢧⠀⠙⡖⢧⣠⠜⠉⣹
⠀⠰⣅⡼⠁⠀⠀⢹⡿⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣄⠀⠀⠀⣿⠘⣧⠀⠀⠀⢁⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢾⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄⠀⢠⣷⠞⠁⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣩⠕⠋⠉⠀⣠⣟⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢿⡷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣤⡞⢁⠈⢳⡴⠋
⠀⠀⠀⠀⠀⠀⣞⠉⠛⠻⡄⠈⢷⡈⠙⢲⣦⣤⣤⣤⣤⣤⣴⣖⣿⠽⠛⠀⠀⠙⢯⡀
⠀⠀⠀⠀⠀⣰⠛⠑⠒⢲⠓⠀⠈⠉⠉⠉⠁⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠘⣶⠒⠒⠛⠀⣀⣤⠤⠤⠤⠤⠾⠷⢲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠙⠶⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢶⡟⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⣀⣤⠴⠚⠛⠉⠁⠀⠀⣇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣅⠀⠀⠀⠀⣀⣠⡴⠖⢻⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⢤⣄⣀⣈⣹⣟⣛⣉⣿⠀⠀⠀⠀⣧⣤⣤⡿⡠⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡄⠀⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣇⢀⠀⠉⠉⠭⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠇⣦⣻⠶⠤⢤⣤⠴⠞⠀⠀⠀⠀⠀⠳⠖⠒⠒⠒⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠙⢷⠎
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣼⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠒⠶⣒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡟⠈⠀⢽⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣾⣿⠉⠉⠈⣹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡝⣿⣿⠐⠚⣳⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣿⡟⠀⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠄⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡀⣸⣿⡇⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡯⠁⠀⢀⡠⠐⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣼⣋⣉⣿⣿⢱⣶⣿⣟⣿⠿⠟⠿⢿⠛⢿⣿⣼⣃⠉⠒⠉⠀⠀⢀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣤⣻⣿⣿⡋⠀⡀⢀⠀⠀⠄⠜⠋⠁⠀⠀⣀⣤⡾⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⠓⠴⠐⠴⠆⠲⣚⠉⠀⠪⠥⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠛⣈⣉⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠋⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠭⠇⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠗⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠆⠀⠾⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠲⢾⠂⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠻⣿⣿⣿⣯⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢀⣿⣿⣿⣿⣿⠿⠛⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣫⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿
⣿⣿⠀⠀⠀⠀⠀⠳⠢⠄⠤⣀⠠⠤⠤⠤⠴⠾⠿⠿⠟⠁⠀⠛⠋⠙⠃⠖⠋⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠠⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⣠⡶⣿⡿⢿⣿⣿⣶⣦⡀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠹⣷⡘⣿⡌⠻⣯⡻⣿⡻⣷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣿⣷⡘⣿⡄⠹⣷⡘⢿⣎⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠸⣷⠀⠹⣷⠈⢻⣆⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢻⡇⠀⢻⣇⠀⢿⣆⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⠀⠈⣿⡀⠈⣿⡄⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠿⠆⠀⠻⠇⠀⠹⠇⠀⠸⠗

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠁⠈⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⢤⣶⣶⣶⣶⣄⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠴⢒⠒⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⡄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠈⠙⠳⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠋⠁⠹⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠐⠀⡀⠀⠑⠠⠀⠈⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠑⣄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠐⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⢀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⢒⠹⣿⣿⣿⣿
⣿⣿⣿⠋⠀⠀⡠⠀⠀⢀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⡀⠭⠚⠁⣉⠃⠘⠿⠿⠿
⣿⠟⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⡄⠈⠀⠀⠀⢀
⣿⣿⡗⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣠⣴⣿
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠛⠛⠻⠿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠐⠂⠀⠀⠀⠀⢀⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢠⠔⠈⠈⠃⢀⣸⡄
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠁⠀⠠⠐⠛⠉⠉
⠀⡎⠉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠠⢄
⠀⠳⡶⠀⠩⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠸
⠀⠀⠈⡗⠤⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⡁⠤⠤⢀⡀
⠀⠀⠀⡇⠀⠸⡀⠀⠀⢀⣤⠤⠶⠶⠚⠛⠛⠉⠉⠉⠉⠙⠉
⠀⠀⠀⢁⠀⠀⠓⠂⠈⢹
⠀⠀⠀⠸⣀⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠹⡁⠀⠀⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⡋⣄⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠑⠢⢄⡀⠀⡠⠔⠒⡆⠀⢀⡠⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠙⡆⠀⠀⠀⠈⠋⠀⠀⠀⠱⠞⠁⠀⠈⠠⠔⠀⠀⠀⠀⠙⠢⡀⠀⠀⠱⡄
⠀⠀⠀⠀⡰⠃⠀⠀⡀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠈⠢⡀
⠀⠀⠀⡘⠀⠀⠀⠈⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⢱
⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⡄
⠀⠀⢸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢁
⠀⠀⡆⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡄⠐⠁⠀⠥⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶