Chakra Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣦⡜⠟⠹⢧⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣿⣿⡉⠉⠉⠉⠉⠙⣿⣿⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠶⠶⠼⣿⣷⣄⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⠛⠛⠁⣹⣿⣿⣿⣷⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⣿⣧⣠⣶⡦⠘⣿⣟⣼⣧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⡄⢸⣿⣿⣿⣟⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢾⣿⣿⣿⡍⢿⣿⣿⣤⠸⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⡄⣿⣿⢿⣿⣿⣿⣾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠈⢿⣿⠿⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣿⣿⣿⣿⣷⣿⣯⣿⣿⣿⡁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣤⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⠙⢻⣟⣻⡟⠻⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⢪⡂⠤⢀⠇⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣁
⠀⠀⠀⠀⠀⠀⢠⣾⡿⡿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⢷⡹⣮⣗⣯⣾⢱⢇⡶⢡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣾⣿⣿⣿⡆
⠀⠀⠀⠀⢀⣰⣏⠉⠉⠀⠀⡉⠉⢳⡄⠀⠀⠀⠀⠀⠀⢀⡏⠶⣜⣶⣿⢷⡯⠿⣽⡖⣿⣶⣥⡦⡊⠀⠀⠀⠀⢐⢠⣶⣟⣵⠿⠿⠷⠶⠿⠿⣷⡝⣷⣤⠐
⠀⠀⢀⣾⣿⣿⡿⠛⠛⠻⢟⡁⠘⢿⣿⣿⣆⠀⠀⠀⠠⠨⣈⣓⣎⣯⣽⠟⢉⡐⢿⣿⣯⣿⣳⣟⣭⢤⠀⠀⠀⠀⢸⣟⣾⣿⡷⠒⣒⡒⠶⢠⣿⣿⣼⡿
⠀⠀⢾⣿⣿⡏⠠⣟⠛⠒⠛⢻⡇⣸⣿⣿⣿⠆⠀⠀⠀⡰⠴⡽⣻⢾⡏⢻⣿⡁⢨⡤⠀⠰⣏⢾⠦⠆⠀⠀⠀⠀⠆⢸⣿⣿⡀⣿⣿⣿⡷⢸⣿⣿⣿⠀⠆
⠀⠀⠈⢿⣿⣷⡄⠹⣆⠀⢠⡞⣱⣿⣿⣿⠏⠀⠀⠀⠐⢘⢛⣭⢟⣟⣻⣌⡙⢁⣼⣿⣟⣿⡹⣯⣛⠊⠀⠀⠀⠐⢸⣿⣿⣿⣷⣦⣬⣤⣴⣆⣿⣿⣿⣷
⠀⠀⠀⠀⠈⠹⣿⣾⣿⣷⣿⣼⣿⠿⠉⠀⠀⠀⠀⠀⠀⠈⡅⠶⢋⡿⣿⠾⣏⣿⣻⢶⣿⡿⡝⠖⣍⠁⠀⠀⠀⠠⠘⠿⣿⡿⣿⣿⣿⣿⣿⣿⡾⣿⡿⠟⠠
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⡾⣱⢷⡯⣞⢿⡹⡎⢷⡘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⢿⣿⣿⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⢼⠅⠒⠈⡃⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⡉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⢔⡢⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⡐⠱⣻⡤⠭⢩⠤⣛⠆⢂⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣿⣤⠿⠩⠧⣴⣿⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢚⣵⡉⠉⠉⠀⠀⠉⠉⢻⣮⡋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡶⣴⡿⠛⠛⠛⠛⠒⠛⠻⣷⡴⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢏⣿⣿⣿⠖⠂⠀⢀⣀⣐⣾⣿⣷⡹⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⣷⣿⣿⣿⠇⣸⣿⣿⢀⣿⣿⣿⡞⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢵⢺⣿⣿⣿⣷⠄⠉⣉⠙⢿⣿⣿⣿⡇⣮⠢⠀⠀⠀⠀⠀⠀⠠⢱⣿⣿⣿⣻⠏⣴⣿⣿⣿⢸⣿⣻⣿⣿⣿⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡸⣿⣿⣿⣿⠀⢾⣿⣧⣼⣿⣿⣿⢇⡐⠀⠀⠀⠀⠀⠀⠀⠀⠨⡉⣿⣿⣿⣄⠛⠛⠋⣁⡘⣿⣿⣿⣟⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡓⠹⣿⣿⣿⣦⡈⠙⣿⣿⣿⣿⡋⠚⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⢿⣿⣿⣽⣿⣿⣿⣡⣿⣿⣿⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⣊⢙⡿⣿⣿⣿⡿⣿⡋⠑⡋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⣿⣿⣿⡿⢿⣿⢿⣿⣿⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠉⠠⢘⡃⠐⠉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠢⢻⠿⠣⠐⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠄⠠⡀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠙⣦⣇⣀⣀⣸⣔⠉⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠙⢲⠷⠒⠉⠁⠀⠀⠀⠀⠉⠉⠒⢼⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⢨⢟⠒⠒⠒⠒⠒⣲⠶⠶⠶⠶⣒⠒⠒⠒⠒⠒⡻⡍⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⠇⠈⢆⠀⠀⣠⠊⠀⠀⠀⠠⠀⠀⠱⡄⠀⠀⡴⠁⢸⣤⠜
⠀⠀⠀⠀⠀⠀⢀⠔⠉⢹⠀⠀⠈⢣⠀⡇⠀⠀⢉⡭⠿⠉⠀⠀⢸⠀⡜⠁⠀⠀⡏⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠢⢄⣸⠀⠀⠀⠀⢳⡇⠀⠀⠰⡏⠳⠀⠀⠀⣸⠞⠀⠀⠀⠀⣇⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⡆⠀⠀⠀⠀⠹⣦⡀⠀⠈⠑⠀⢀⣴⠏⠀⠀⠀⠀⣸⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣸⣄⠀⠀⠀⠀⠙⡌⠑⠒⠒⠊⢡⠋⠀⠀⠀⠀⣠⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠣⣀⠀⠀⠀⠘⣄⠀⠀⡰⠁⠀⠀⠀⣠⠞⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢼⡗⠦⢄⣀⡈⣆⡰⢁⣀⣠⠴⢺⠧⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣠⠞⡏⠉⠉⣹⠳⣄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠐⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣞⣣⣃⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣉⢫⣱⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢀⣀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡔⠛⠀⠀⠙⢦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠓⠤⠤⠘⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠤⠦⣅⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠔⠚⠉⠠⡏⠀⢀⢘⡆⠉⠓⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠑⠤⠦⠞⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⡀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠂⠀⠀⠀⠰⡟⠀⠀⡞⡆⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸⠀⠀⠀⠀⠻⢤⣤⠜⠁⠀⠀⠀⣿⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢠⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡏⢇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⡏⠀⡇⠀⠀⢀⣴⠚⠑⢶⡀⠀⠀⣸⠁⠘⡆⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⡜⠀⠀⢹⡀⠀⠀⢇⠀⠀⢨⠇⠀⠀⡇⠀⠀⢳⡀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⢀⡜⠀⠀⠀⢸⠇⠀⠀⠈⠉⠋⠁⠀⠀⠀⣇⠀⠀⠀⢳⡀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡠⠋⠀⠀⠀⠀⣸⠀⠀⢀⠤⠎⠱⠄⡄⠀⠀⢸⡀⠀⠀⠀⠳⣄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣠⠞⠀⠀⠀⠀⢀⣰⠃⠀⠀⠀⠟⠀⠀⠙⡅⠀⠀⠀⢷⠀⠀⠀⠀⠈⠳⣄⠀⠘⣆
⠀⠀⠀⠀⠀⡠⠀⠤⠞⠁⢀⡜⠁⣠⠤⠒⠊⠉⠉⠀⠀⠀⠀⠀⠉⠳⠔⠉⠀⠀⠀⠀⠈⠉⠒⠢⠤⣀⠀⠈⠳⡀⠈⠳⠤⠀⢄
⠀⠀⠀⠀⢠⠱⠃⠀⠀⢀⡿⠒⠉⠀⠀⠀⠀⡀⠄⢀⠀⢀⠀⠐⣠⠖⠲⣄⠆⠀⡀⠀⠀⣀⡀⠀⠀⠈⠉⠒⠦⣽⡄⠀⠀⠘⠞⡀
⠀⠀⠀⠀⠈⡱⠀⢀⠠⠁⠀⠀⠀⠀⠀⠉⠀⠐⠀⠀⠀⠀⠀⠸⢧⡀⢀⣸⠀⠀⠀⠀⠀⠀⠀⣀⣠⠆⠀⠀⠀⠀⠙⠄⡆⢀⢋⠁
⠀⠀⠀⠀⠀⠓⡧⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡽⢿⣒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⠒
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠒⠋⠉⠀⠀⠀⠀⠉⠙⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠤⠤⠴⠶⠶⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠕⠃⠁⠛⢕⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠁⠀⠀⠀⠀⠀⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣄⠀⠀⠀⠠⢀⣟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢖⣤⢤⣔⠯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⡎⠄⠤⠄⠤⠌⣮⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡆⠈⠄⠀⠀⠀⠌⠀⡆⢑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡐⡀⠈⢄⢀⠊⠀⡐⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡠⣀⠀⡈⠚⠒⠀⠃⠐⠚⠈⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢉⠌⢁⠉⠲⣄⡀⠀⠀⠀⢀⠖⠋⠉⠈⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⣁⠋⠈⠈⠡⡩⠙⡄⠀⠀⢀⡟⠀⠠⠒⠀⠲⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣠⠛⡀⠀⠀⡐⠣⣀⠎⠀⠀⠘⡆⠈⢧⠀⠀⢀⠞⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⡉⠉⢍⠙⠉⣩⠟⠀⠀⠀⠀⠙⣄⡀⠋⠀⠋⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠫⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠓⣄⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⠤⠤⠤⠀⢠⠀⠀⠀⠀⠠⠠⠴⠊⠰⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡰⢋⠵⠒⠒⠢⣝⢒⠠⠀⠀⠀⡠⠨⠀⠀⠀⠀⠨⠢⠸⠀⠀⢀⢤⢋⣀⣀⣀⣀⣀⢈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢠⠡⠀⠀⠀⠀⡈⠂⠲⠀⠀⢸⠀⠀⠩⡀⠀⡈⠀⠀⢰⠀⠀⠀⡆⠈⢂⠀⠀⠀⢀⠄⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠜⡈⢄⠁⠂⠐⠊⢀⢂⠚⠀⠀⠸⡀⠀⠀⠑⠜⠀⠀⠀⠸⠀⠀⠰⣡⠀⠀⢂⠀⢀⠂⠀⢀⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠦⣅⠂⡐⣢⠵⠒⠊⠁⠀⠀⠀⠄⡀⠀⠀⠀⡠⠀⠈⠀⠀⠀⠈⢓⢀⣀⠱⢁⣀⢔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠛⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠑⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢾⣿⣿⠛⠻⣿⡿⠛⠻⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣿⠃⠀⠉⠁⠀⡄⠀⠀⢠⠀⠉⠉⠀⢸⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡯⠀⠀⠀⠀⠘⡄⠀⠀⣇⠀⠀⡾⠀⠀⣰⠁⠀⠈⠁⢨⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠙⠁⠀⠈⢦⡀⠀⢹⡄⠀⣿⠀⢰⡇⠀⣰⠃⠀⢀⡜⠁⠀⠙⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠂⠀⠳⢄⡀⠀⠻⣆⠀⢿⡄⢸⡇⣼⡇⣰⡏⢀⣴⠋⠀⢀⡠⠆⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢄⣀⠀⠀⠙⢶⣄⠘⣷⣜⣿⣼⣧⣿⣵⡿⣰⡿⢁⣠⠶⠋⠀⠀⢀⡀⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠇⠀⠀⠈⠙⠲⢦⣤⣙⠻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⢟⣁⣤⡴⠖⠋⠁⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠒⠲⠤⢤⣤⣤⣬⣙⣿⣿⣿⣿⠟⣋⡛⢿⣿⣿⣿⣟⣋⣁⣀⣤⣤⡤⠤⠄⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⢀⣀⣀⣠⣭⣭⣿⣿⣿⣯⠸⣿⡿⢨⣿⣿⣿⣯⣭⣉⣁⡀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⠀⠉⠉⠉⠉⠉⣉⣤⣴⢾⣿⣿⣿⣷⣦⣶⣿⣿⣿⡿⣿⣭⣉⠉⠉⠉⠛⠒⠂⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⣀⠤⠖⠛⠉⢁⣴⠿⣻⡿⣿⣿⣿⣿⣿⣿⣿⠻⣦⣌⠉⠛⠶⢤⣀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣀⠀⠀⠀⣠⠾⠋⢁⣼⠟⣰⡟⣽⡏⣿⠹⣷⠙⢷⡀⠙⠳⣄⠀⠀⠀⠁⣀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠊⠁⠀⢠⠞⠁⢠⡟⠀⣿⠁⢿⡀⠹⣇⠀⠻⣄⠀⠈⠑⠄⠀⢾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣤⡄⠀⠴⠃⠀⢀⡞⠀⠀⡟⠀⢸⡇⠀⠹⡄⠀⠈⢦⠀⢀⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣏⣀⣀⡀⠀⠜⠀⠀⢰⠃⠀⠈⡇⠀⠀⠱⠀⢀⣀⡀⣈⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣀⣠⣤⡄⠈⠀⣀⡀⠁⢀⣤⣄⢀⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣷⣶⣾⣿⣿⣶⣾⡿⠿⠛⠋⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⢰⣿⣷⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠂⠀⣰⡿⣦⣀⠀⠉⠁⣀⣤⠿⣧⠀⠀⢺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢀⡴⢋⡶⠶⣭⡛⢶⣟⣭⠴⠳⣜⢷⣄⠀⠀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⣿⢩⣝⣛⣛⣿⢰⡋⠀⠀⣀⣹⣾⣿⣀⣀⡀⠈⣷⡿⣫⣭⣭⣭⢹⡇⠀⢠⣶⣆
⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⣸⢇⡟⠀⠀⣈⣿⠿⠓⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣾⣁⠀⠀⠸⡎⣷⠀⠈⠛⠋
⠀⠀⠀⠀⠀⣤⣀⣤⡴⠶⠻⢾⣧⣴⠞⠉⠀⠀⠀⠀⠀⠀⢰⣿⣷⡄⠀⠀⠀⠀⠀⠉⠳⢦⣠⢏⣾⣛⠶⢦⣤⣤⡄
⠀⠀⠀⠀⠀⢹⡇⣶⠚⠋⠉⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡋⠉⠙⢲⢆⡿
⠀⠠⣿⣷⠀⠸⡇⣿⠀⢀⣜⣁⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣆⠀⢸⢸⡇⠀⢰⣿⡧
⠀⠀⠈⠁⢀⣴⢿⣬⣳⡞⢻⡉⠉⠉⠉⠉⠉⠉⠉⣩⠽⠛⠋⠉⠉⠛⠻⢯⣙⣿⣿⠉⠛⠉⠉⠉⣹⠻⣿⣟⣛⠷⣄⠀⠉
⠀⢠⣤⡴⢟⡵⠋⢉⡟⠀⠀⠹⡄⠀⠀⠀⠀⣠⢋⣠⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡿⠀⠀⠀⠀⡼⠁⠀⢹⡉⠉⠳⣌⠳⣦⣤
⠀⠀⠙⢷⡹⡆⠀⢸⡇⠀⠀⠀⠹⣆⠀⠀⣸⣿⣿⡟⠋⠉⠀⠀⠀⠀⠈⠈⠁⠀⢹⡄⠀⢀⡾⠁⠀⠀⠈⣧⠀⢀⡞⣼⠏
⣴⣦⡀⠈⢷⣹⣦⣿⠀⠀⠀⠀⠀⠘⣆⠀⡇⠘⢿⣧⣤⣶⡶⠶⠤⣤⣄⡀⠀⠀⠀⣇⢀⡞⠀⠀⠀⠀⠀⣿⣶⣚⣼⠋⠀⣠⣶
⠙⠛⠁⢀⡾⣩⠖⢻⠀⠀⠀⠀⠀⠀⠘⢦⡇⣰⣿⡿⠋⠁⠀⠀⠀⠀⠈⢿⣷⡄⠀⣷⠎⠀⠀⠀⠀⠀⠀⣿⠛⢻⡙⣧⠀⠘⠛
⠀⠀⣠⡾⣱⠏⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⣸⠏⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣳⡜⣧⣀
⠀⠈⠛⠳⣮⡳⣤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⠞⣡⠾⠋⠉
⠀⠀⣀⣀⠈⠛⣶⢶⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠻⡝⢿⣦⣤⣤⣤⡤⠴⠚⡽⠃⠀⠀⠀⠀⠀⠀⠀⣰⠟⢯⡻⡟⠁⢀⣀⡀
⠀⠈⠿⠿⠀⠀⣿⢸⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠛⠶⣤⣄⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢸⡇⣧⠀⠸⠿⠏
⠀⠀⠀⠀⠀⣰⣇⡻⠦⣤⣀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢀⣠⣿⢦⣤⠴⢞⣃⣿⡀
⠀⠀⠀⠀⠀⠛⠉⠙⠛⠶⢾⠫⡟⠉⠲⢄⡀⠀⠀⠀⠀⠘⣦⠀⢠⡞⠀⠀⠀⠀⠀⣠⡴⠟⠉⣿⢳⡶⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣤⠀⠸⣇⢳⡀⣀⣀⣽⣷⣶⣤⣤⣀⣈⣷⣏⣀⣤⡤⠴⣾⢿⣏⣀⣀⣰⠏⡾⠀⢠⣴⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⣿⣌⣭⣭⣭⣴⡟⢧⡀⠀⠉⣹⣿⢻⣀⠀⠀⢀⡾⣸⠧⣭⣭⣭⣼⡇⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⢀⡀⠀⠻⣦⡹⠖⣋⣥⠞⠷⣭⣙⠶⢫⡾⠋⠀⣀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠂⠀⠈⣷⠟⠉⠀⣀⡀⠀⠉⢳⡟⠀⠀⢺⣿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠟⠏⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠛⠉⠉⢀⡠⣤⡤⣀⠈⠉⠙⠛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢶⠆⠀⣶⡦⢀⡴⠛⠁⠀⠀⠀⠀⠀⣾⣱⠋⢣⣸⡆⠀⠀⠀⠀⠀⠙⠳⣄⠠⣶⡆⠀⢶⠖⠒⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢠⣾⢣⠂⠀⠈⢣⠻⣆⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⡧⠀⠠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⠿⣸⡤⢴⡋⠿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠰⣂⠉⠀⠈⠉⢲⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣁⠻⣦⡀⣠⡾⢋⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣼⣃⣬⠁⠉⣤⣿⠏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⠋⣟⢃⠤⢤⣙⡏⢰⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠼⣵⠛⢡⠄⢠⠞⢳⡬⡄⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⢞⠈⠋⢰⣦⡠⣯⣀⣠⣧⣀⣷⠈⠫⣙⠄⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡠⠊⠀⣈⡀⠐⠛⠈⠋⠡⠋⠣⠋⠙⠘⠓⠀⣈⡀⠀⠢⡀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠻⣄⠀⡰⠿⢲⠛⠙⣶⣞⢆⡖⢉⠂⠀⠀⠈⠙⣶⣞⣦⡷⠙⢳⠺⢷⡀⢀⡼⠃⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢧⡹⣴⡻⠍⡀⣄⡢⠬⠖⠒⠈⠉⠉⠑⠒⠮⢤⣁⡀⣀⢭⣻⡼⣡⠟⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⣶⡆⠙⠶⣄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡴⠞⠁⣶⡆⠀⢰⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⣦⣀⣀⠈⠀⠀⠀⠀⠀⠈⢀⣀⣠⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠉⣀⡄⠚⠛⠛⠃⣄⡈⢉⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢿⠀⠀⠀⠀⠀⢸⣇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣀⡀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡐⠻⠓⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣴⠾⣦⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡿⠉⢉⣿⣥⣀⣬⣿⡉⠙⢿⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⢩⡿⠓⣿⣅⣠⠿⣄⣨⣿⠛⢿⡉⠉⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢻⡶⠛⡷⠺⡧⣤⢛⣶⢛⡤⢼⠳⣾⠛⢷⡞⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⣼⢧⠾⢷⢚⡦⠉⠀⠀⡁⠈⢺⠓⠞⢷⡾⣧⡀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⣇⣨⠶⣞⡇⠀⠰⢶⡶⣷⣄⠀⢙⡑⠾⣇⢸⡟⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣰⡟⠻⣴⣏⡇⠀⣄⣙⣿⣁⣼⠀⢈⣩⣤⠟⢹⣆⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⢳⡾⣧⣼⣍⣤⠈⠉⠁⠈⢁⢴⣩⣧⡼⢧⡞⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣼⣧⣀⡷⣬⡆⠾⣀⠶⣀⠿⢸⡤⣾⣀⣼⣇⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠙⣧⣀⣿⠒⠻⣦⠟⠓⣿⣀⣼⠋⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣧⠁⠙⣷⠞⠛⠷⣾⠃⢀⣽⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⡾⠋⠻⣶⠶⣦⡀⠀⠀⠀⠀⠙⠟⠛⠛⠷⣶⡾⠋⠛⠻⠃⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⣾⠻⠋⠁⠀⠠⠴⠤⠀⠈⠙⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠲⠶⠊⠁⠀⠀⣀⠀⠀⠈⠑⠶⠖⠶⣦⡀
⠀⠀⠀⣤⢾⣿⠤⠤⠤⢤⣤⠤⢤⣤⠤⠤⠤⢼⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⢰⢏⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢻⡄⠈⢻⣆
⠀⠀⠀⣻⡆⠀⢂⠀⡤⠛⠂⣂⣲⠎⢢⠀⢀⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⢠⠀⠈⢦⠀⠀⠀⠴⠶⣦⠀⠀⠀⣀⠃⢸⠂⠀⢻⣦
⠀⠀⢸⣟⡁⠀⠀⢢⡀⠀⣸⡧⢤⡀⠀⣶⠈⠀⠀⣹⣷⠀⠀⠀⠀⠀⠀⠀⢾⣏⠀⠀⢸⠀⠀⠈⢣⠀⠀⠀⠚⢿⡀⠀⡰⠁⠀⠀⠀⠀⠀⣙⡷⠄
⠀⠀⠀⣹⠇⠀⠀⠀⢳⡀⠙⢆⠀⢀⡼⠃⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠘⠀⠀⠀⠀⢣⡈⠳⠶⠾⠁⠴⠁⠀⠀⢰⠀⠀⣸⠏
⠀⠀⠀⠻⢾⣄⠀⠀⠀⠹⡉⠀⠈⡹⠁⠀⠀⢀⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠙⣄⠀⠀⠀⠳⡀⠀⠀⡜⠀⠀⠀⣠⠇⢀⣼⠏
⠀⠀⠀⠀⠀⣿⣴⣄⡀⠀⠱⣀⡜⠀⠀⣠⣴⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣬⡶⢄⡀⠀⠱⣀⠜⠀⠀⠠⣶⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠿⢶⣟⣷⠾⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠛⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠟⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣴⠶⣦⡀⠀⣀⡀
⠀⠀⠀⠀⠀⣼⠛⢛⡹⠾⠖⢦⠒⠺⢿⡙⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢋⡙⠿⠣⣶⠔⠿⢏⡙⢻⡄
⠀⠀⠀⢀⣐⣧⠐⠁⠀⠀⠀⠈⠀⠀⠀⠈⠑⢬⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⢾⠛⠁⠀⠀⠞⠛⠳⡀⠀⠈⠘⡷⠶⣤⡀
⠀⢠⣶⠛⢩⠿⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⡺⠏⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⢳⠀⠀⢰⡟⠁⠀⠀⠻⡂⠀⠀⡻⢆⣿
⠀⠀⢻⣄⠇⠀⠱⡀⠀⠀⢉⣈⣿⠁⠀⠀⡠⠁⠀⢃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠃⠀⢆⣀⠂⠀⡻⠃⠙⠇⠙⣄⡜⠁⠘⠻⣦
⠀⠀⣠⠿⡇⠀⠀⠱⡀⠀⠸⣏⠁⠀⠀⡜⠁⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣸⠀⠀⠀⠷⠀⢴⠃⠀⣨⡀⠀⡼⣄⠀⠀⢀⣼⠃
⠀⢠⣿⡀⢡⠀⠀⠀⠱⡄⠀⠙⠀⢀⡘⠀⠀⠀⠠⠂⣹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢆⣰⠃⠀⢳⡈⠛⠋⠉⢃⡘⠀⠘⢂⡸⣿⠁
⠀⠀⠈⠻⠶⣦⡀⠀⠀⠘⡄⠀⢀⠞⠀⠀⠀⣰⡷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣫⡀⠀⠈⠻⡄⠀⢀⠟⠁⠀⠀⣹⣡⣿
⠀⠀⠀⠀⠘⣧⠈⠢⢤⡀⠘⣢⡎⢀⣠⠔⠊⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣟⠂⢄⣀⢘⣤⡊⢀⡠⠐⢸⡏⠉
⠀⠀⠀⠀⠀⠙⠛⠛⠛⢹⣄⣉⣁⡼⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠻⣧⣉⣼⠟⠛⠚⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠀⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⣿⠡⠔⠒⠈⣩⠁⠐⠒⠄⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣰⣟⣥⠀⠀⠠⣬⣙⣇⣀⣀
⠀⠀⢀⣴⠖⢶⡟⠓⠒⠀⠈⠙⠛⠒⠒⠂⠛⣷⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⢋⣼⠟⠉⠀⠠⠦⠄⠀⠉⢻⣯⡙⠻⣦⣀
⠀⢀⣾⠃⠰⠁⠻⡄⠀⠘⣛⠛⣛⣿⠋⠀⠹⡇⠐⡀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣰⡟⠧⠀⠀⣀⣀⣀⣀⣀⠀⠀⠟⢻⡆⠈⣿
⢀⡾⠃⠀⠇⠀⠀⠘⡄⣼⠁⠉⠁⠘⡇⠌⠀⡇⠀⠃⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⡇⠀⠀⣠⠦⠩⢼⣏⠀⠀⡄⠀⢸⠀⣿
⠘⣷⡀⠀⡇⠀⠀⠀⠀⢋⠓⠈⠑⢢⡌⠀⠀⡇⠀⡀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠅⠀⠱⣄⠀⢷⣄⣀⣤⡿⠀⡰⠃⢀⠀⢾⡁
⠀⠘⣷⡀⢡⠀⠀⠀⠀⠀⢞⠳⢮⠛⠀⠀⠀⡇⢠⠃⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡀⢣⠀⠈⠓⠠⣌⣉⣤⠵⠊⠁⢀⡜⢰⠀⣿
⠀⠀⠈⠿⣦⣶⣄⠀⠀⠀⠀⣵⠇⠀⠀⠀⣀⣷⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠉⣄⠑⠤⠀⠀⠀⠀⠀⠀⢀⡤⠊⣠⠃⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠑⠤⠄⣀⣀⣀⡀⠠⠔⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣌⣢⣄⡉⠐⠒⠒⠒⠂⢁⣠⣞⣡⣴⠟⠁⠀⠀⠀⠀⠀⠐⠒⣆
⠀⠀⠀⠀⠀⠀⠘⠻⣦⣄⡀⠀⢀⣀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⣧⣁⠒⠐⠒⣃⣼⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁

⠀⠀⠀⢀⣤⣴⣦⣄⡀⠀⠀⠀⠀⠀⠄⠀⠀⡄⢀⠀⠄⠠⠀⠀⠀⡀⡄⡀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣀
⠀⠀⢠⡿⢁⡔⢒⣌⢻⡄⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠂⠀⠀⠀⠀⠀⠙⠀⠅⠀⠠⠀⠄⠀⢞⠌⢤⣤⠁⢲
⠀⠀⠼⣇⢀⣲⡅⠞⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠸⠑⠀⣾
⠀⠀⠀⠙⠿⢯⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⣰⠒⠀⢶⠀⢳⣔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠰⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠘⡚⠁⠼⠑⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡠⠤
⠀⠀⠀⣠⠈⣁⣈⢁⡄⠀⠀⠀⠀⢀⢕⣀⣤⣀⡡⠀⠀⠀⠀⠀⢀⡥⣒⣲⠢⡀⠀⠀⠀⢠⠀⢀⣄⣠⡄⠀⡆
⠀⠀⠀⣠⡐⡎⣇⢂⡔⠀⠀⠀⠀⠰⠀⠰⡖⠃⠀⡇⠀⠀⠀⠀⠚⢰⣀⣸⠀⡗⠀⠀⠀⢠⠃⠰⠄⠀⠓⠐⡇
⠀⠀⠀⠀⠰⠖⠰⢆⠂⠀⠀⠀⠀⠈⠠⠀⠄⠄⠂⠀⠀⠀⠀⠀⠀⠒⠄⠨⠒⠀⠀⠀⠀⠈⠀⢀⣀⣅⣀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠉⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣴⠛⠉⠙⢳⡄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⣤⣤⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡟⠁⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠙⣷⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣇⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢀⡿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠛⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠋⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠺⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⠀⢀⣤⡄⠀⣤⡄⠀⠀⠀⣿⡇⠀⠀⠹⣿⡆⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠟⠉⠉⠙⢿⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣾⡿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⢺⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡗
⠀⠀⠀⢘⣿⣷⣤⣀⣤⣾⣿⠀⠀⠀⢸⣿⡄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷⣤⣀⣤⣾⣿⡃
⠀⠀⣰⣿⠟⠉⠛⠛⠋⠉⢻⣷⡄⠀⢸⣿⣿⡿⠋⠸⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣿⡇⠀⢠⣾⡟⠉⠙⠛⠛⠉⠻⣿⣆
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⣈⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⡗⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣀⡎⠀⣀⡀⢱⣄⠀⠀⠀⠀⠀⠀⠀⡼⠉⠳⣄⣇⣠⠞⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⡠⠃⠀⠀⠈⢀⠜⠙⡄⠀⠀⠀⠀⠀⠀⡖⠒⢒⡾⣿⣗⠒⠒⣛
⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⠀⠀⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠃⠀⠀⠀⢀⠜⠀⢠⠇⠀⠀⠀⠀⠀⠀⠙⣴⠋⠀⣇⠈⠲⣤⠋
⠀⠀⠀⠀⠀⠀⣴⠋⠚⠹⠿⠿⠛⢢⠸⣠⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⠓⠀⢐⠋⠀⣨⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠛⠒⠋
⠀⠀⠀⠀⠀⡰⠃⠀⠑⠢⠄⠤⠒⠋⠀⠁⣃⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠦⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⣰⠁⡀⢀⠀⡀⢀⠀⢀⠀⡀⢀⠈⣗⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⣽⣿⡿⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠘⠀⠋⠈⠃⠙⠀⠃⠐⠁⠂⠁⠂⠑⠉⠂⠀⠀⠀⠀⠀⠀⣤⢠⡇⢠⡟⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠸⣿⣷⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⣆⣸⠟⠋⠉⠙⠛⢦⣴⠂⢀⠀⡀⠀⠀⠀⡄⠀⠀⠈⠛⠛⠛⠉⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢖⡿⠡⠶⠌⠀⠘⣷⡄⢹⡎⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⢸⠃
⠀⠀⠀⠀⠀⠀⡴⢫⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢺⣧⠀⠀⠀⠀⠀⠀⠀⢸⡏⠿⠊⠉⠀⠀⠀⠀⠁⢄⡀⠀⠀⠀⠀⠀⣀⠘⠁
⠀⠀⠀⠀⠀⣼⠁⡾⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠃⠑⢻⡄⠀⠠⣶⠄⠀⣀⡿⢛⠁⠉⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⠀⣿⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⡹⠶⣤⢤⣤⣞⠋⠃⠈⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠘⣆⠙⢷⣄⠃⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠁⠸⠃⠈⠀⠙⠡⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢐⡀⡤⣗⣲⣲⠈⡀⡀
⠀⠀⠀⠀⠀⠀⠘⢧⣄⠙⠳⠦⠤⡤⢤⣴⠞⠁⠀⠀⠀⣀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⡹⠊⣏⣿⢣⠀⡸⣯⡄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠚⠉⠀⠀⠀⠀⠀⢸⠁⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢹⠀⠀⢨⡇⢰⡀⡵⢾⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡜⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠢⠤⣀⠀⠀⠀⠂⠠⠚⡘⣶⠁⡎⢻⣦⠀⣴⡿⡲⠆
⠀⠀⠀⠀⠀⣀⣠⡴⠞⢛⣙⣛⡻⢶⡄⠀⠀⠰⡆⠠⠴⣉⠧⠤⢴⠆⠀⠀⢀⢠⣿⠃⠀⠀⠀⠈⣧⡀⠀⠀⠀⠈⢱⠌⡳⢴⣄⣤⡴⢋⠞⢁
⠀⠀⠀⣤⣯⣭⣥⣶⠋⠩⡟⣧⠉⣳⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⡀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠐⠁⠼⠈⠃⠙⠆⠀⠈⠃
⠀⠀⠀⠈⢀⣄⠀⠈⠙⢲⣿⣿⡞⠁⠀⠀⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠰⠈⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠈⠲⠶⠶⠾⠛⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣬⣧⣤⡤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠙⣧⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠦⠄⢈⡀
⠀⠀⠀⠀⠀⠀⣠⡤⠛⣶⠓⠣⣀⡀⠀⠀⠀⠀⠀⠀⢤⠀⡄⢠⡄⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠘⠁⠂⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⢰⠃⠀⠀⣿⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠈⣶⣷⡞⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⣷⣤⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⠀⡠⠂⡂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠀⠀⠀⠀⠀⠻⠙⠂⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠈⠀⢾⠃⣸⠈⢷⣆⠖⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠭⠤⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣀⣀⣠⣤⣶⠖⠁⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣠⡞⠁⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡼⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣯⠿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⢶⣿⡶⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣉⣉⠤⣉⣉⣛⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣤⣤⣤⣤⠄⠀⠀⢰⣿⡇⠀⠀⠠⢠⣤⣤⣤⠄⠀⠀⠀⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⡏⠀⢸⣿⣿⣿⡟⠛⠛⠛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⣿⣶⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠀⠉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⠀⠀⠀⠀⠀⠀⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠟⠂⠀⠀⠀⠀⠀⠉⠛⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣤⣤⣤⣀⠀⠀⠀⣠⡴⠚⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠉⢳⣄⡼⠃⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣀⡀⣹⠃⠀⠀⠀⠀⣀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣡⣿⠀⠀⠀⠀⣾⣏⣳⠀⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣷⠀⠀⠀⠀⠘⠿⠟⢻⡀⠀⠀⠀⢿⣿⡟⠀⢰⣧⣤⣤⣤⣤
⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢷⣄⠀⠀⠀⠀⣠⠞⢧⡀⠀⠀⠀⠀⠀⣠⠏
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠖⠛⠁⠀⠀⠙⠦⣤⣤⠴⠞⠁⢠⣤⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⡏⠛
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠁⢈⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⢨⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠀⠀⠠⢀⡿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⠤⠄⢤⡀⢀⣼⠃⠀⠀⠀⢰
⠿⠶⠶⠶⠶⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⢩⡿⠁⠀⠀⠀⠀⠀⠀⠀⢈⣷⠞⠁⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠺⠷⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣼⠃

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢐⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⢲
⢰⠀⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠑⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡞
⠈⢆⠀⠀⠈⢦⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⠛⠛⠁
⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬



⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣠⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠤⡧⢤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢳
⠀⠀⡀⢴⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣡⠤⠧⢤⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠒⠚⠦⢊
⠀⠀⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤
⠀⠀⠀⠀⣠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣄⠀⠀⢀
⠀⠀⠀⠀⠉⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⠀⢀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⠀⠀⠣⣀⠜
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠙⠄⠀⠖⠉⠱
⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⢀⠞⠙⢦
⠀⠀⠀⠘⡄⠀⠤⠳⣤⠤⢤⡀
⠀⠀⠀⢠⠫⠿⡄⠀⠙⣆⠒⠉⠲⣄
⠀⠀⢠⣷⠘⠤⠽⣦⠀⠈⢳⣄⠀⠈⠳⣄
⠀⠀⢻⠙⣧⠀⠀⠈⢷⡀⠀⠙⢦⡀⠀⠈⠻⣦⡀
⠀⠀⢸⠀⣆⠘⣆⠀⠈⢳⡄⠀⠀⠙⢦⡀⠀⠈⣙⣦⡀⠀⠀⠀⠀⡴⢋⡙
⠀⠀⣼⢶⡏⠛⠹⣆⠀⠀⠹⣄⠀⠀⣤⠻⣄⠈⠵⠊⢳⣄⣠⠶⠻⠁⠈⠳⡆
⠀⢸⠃⠀⢇⠀⠀⢻⡄⠀⠀⠘⢧⡀⠘⠁⠹⣆⠀⠀⠀⠹⣯⠀⣴⠀⠀⠀⡇
⠀⣼⣄⠀⡸⣆⠀⠀⢻⡄⠠⢶⡨⣷⡀⠀⠀⠘⣧⡀⠀⠀⠸⣷⡇⠀⠸⣿⡇
⠘⠇⠘⣇⡇⠹⣆⠀⠀⠙⣦⠀⠁⠀⢱⡀⠀⠀⠹⣷⡄⠀⠀⢻⣷⠀⠀⠸⣇
⢸⠀⠀⣿⠀⠀⠹⣧⠀⢻⠊⣧⠀⠀⠀⢷⣀⠀⠀⠙⠛⠂⠀⠈⢿⣦⠀⠀⢿⡄
⡆⠀⢀⣿⠀⢧⠀⢹⡄⠀⠀⠘⣦⠀⠀⠀⠙⠗⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠈⢷⡀
⢧⠀⢸⠟⢳⡈⣆⠀⣷⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠘⣧
⠈⢧⠀⠀⠈⢧⠸⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠈⣧⠀⠀⠈⢣⡹⠖⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⣇⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠠⢋⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠉⠉⠉⣙⣿⣿⣿⣶⣦⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠰⠹⣿⣿⣿⣶⣶⣤⣤⣶⠖⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⠟⠋⠹⣷⣄⡀
⠀⠀⠀⠀⠀⣴⠟⢿⣦⣀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⢿⣿⡉⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⢴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⢠⡾⠁⠀⠈⣿⠿⠿⠂⠀⠀⠀⠀⠉⠁⠀⠀⢀⣿⣿⣄⠀⠀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢈⢵⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⣰⠟⠀⠀⠀⣼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⣿⣿⣿⣦⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⢸⡏⠀⠀⠀⠀⣿⣿⣄⣀⠀⠀⢀⣠⣤⣶⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⣿⠃⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠃⠀⠀⠀⠀⠀⢀⣴⣿⠟
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢋⣁⣴⡾⠁⠿⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⢀⣴⣿⠟⠁
⢸⡟⠀⠀⠀⣠⣤⣤⣤⣤⣶⣶⣿⣿⡿⠋⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⢀⣴⣿⡿⠋
⣸⡇⢰⣶⣤⣽⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⣠⣿⣿⠟
⣿⣿⡈⠛⢿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠇⢀⣠⣾⣿⡿⠃
⠘⢿⣿⣦⣄⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣶⣿⡿⠟⠋
⠀⠀⠙⠻⣿⣿⠶⠄⠀⠀⢠⣾⣿⣿⣶⣿⣷⣴⠟⠋⠁⠀⠻⣿⣅⡠⠴⣾⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠈⠻⣶⣶⣶⣶⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣠⡴⢶⣄⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣠⡶⢦⣄⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣼⠋⠀⢠⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣿⣄⠀⠙⣧⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢻⣄⠀⠈⠁⢀⡼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠈⠁⠀⣠⡟⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠛⠛⠛⠲⣤⡀⠀⠀⠀⠀⠀⣀⣠⣾⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢀⣴⠟⠉⠁⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⣿⠃⠀⢀⣤⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣿⡟⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⣿⡀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⣿⡆⠀⠘⠛⠁⠀⠀⠀⠀⠀⣼⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣩⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠈⠻⢶⣤⣄⣀⣤⣴⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡋⠀⠀⠀⠙⠻⠷⠶⠶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⢀⣿⡇⠀⣠⡾⠛⠛⢷⣆⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⡀
⠀⠀⢸⣿⡇⢸⣿⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡀
⠀⠀⢸⣿⠁⠈⢿⣦⣀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣤⣤⣤⣄⣠⣤⣤⣤⡶⠶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠻⣿⣿⣿⠃
⠀⠀⢸⣿⣄⣀⣀⣉⣉⣉⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⠀⠉⠁
⠀⠀⠈⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⢀⣠⣶⠾⠿⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⣼⡿⠁⠀⢠⡾⠋⠀⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⣠⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣰⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⠋⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⣼⣿⠟⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠘⢿⣤⡀⠀⠀⠀⢀⣠⡿⠃⢠⣾⡿⠃⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠾⠛⠉⠀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣠⣶⠿⠛⠛⠛⠛⠷⣦⣄⠀⢀⣼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⡟⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⡿⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⣴⣿⡟⠁
⣴⣿⣿⣿⣿⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⣾⣿⣿⣿⣦
⠛⢿⣿⣿⣿⡟⠁⠀⣠⣴⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢈⣿⠟⠀⠀⣸⡟⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠈⠻⠿⠿⠁
⠀⣠⡿⠃⠀⠀⠀⢹⣇⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟
⢰⣿⣥⡀⠀⠀⠀⠀⠙⠻⠷⠟⠋⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠈⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁
⠀⠀⠀⠈⠙⢿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣄⡀⠀⠀⣠⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣴⣿⡿⢿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣏⣀⣀⣉⣿⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠟⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣠⣀⣠⣄⣤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡶⠶⠂⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠘⡛⣳⠳⢭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣏⢹⡏⣹⢷⡀
⠀⠀⠀⣧⠀⠙⡷⠖⠛⡇⠀⠀⠀⠀⠀⢀⠄⠀⠀⣤⣾⣿⢿⣿⣶⣄⠀⠀⠀⠀⠀⢸⡧⠬⣿⣿⠥⢼⡇
⠀⠀⠀⠈⠛⠚⠐⢶⠶⠃⠀⠀⠀⠀⠀⠀⠀⢀⣼⣵⣄⣩⣹⣿⣿⣿⣷⡄⠀⠀⠀⠀⠳⣎⢸⠈⣢⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⠇⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠈⠉⠉⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣄⠀⡀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢸⣿⣿⡟⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠲⣶⠟⠛⢧⡴⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠿⠷⣤⣶⠄
⠀⠀⠐⠺⣯⣀⢀⣼⠷⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢯⠀⠀⣹⣏
⠀⠀⠀⠀⠛⠙⡟⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢱⢉⡿⢣⢀⠀⠀⠀⠀⠀⠀⠊⠀⠐⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⣿⡧⢊⡾⠀⠀⠀⠀⠀⠈⡄⢠⣴⣦⡔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣻⢗⢾⠟⠁⠀⠀⠀⠀⠀⠀⠠⡸⣷⣾⢇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠉⠛⠿⢿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⡿⠿⠛⠉⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⣿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⢀⡟⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢻⡀
⠀⠀⠀⠀⠀⣼⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⣧
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠸⣇
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⢀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⡀
⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠿⠿⠿⠿⠆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⠿⣶⣶⣶⣶⣶⣶⣶⣶⠿⠟⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣀⣰⣤⡀⠀⣠⣴⣂⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣶⣶⣽⣿⣿⣿⣾⣿⣿⣿⣵⣶⡶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣾⣿⣿⡿⠛⠛⠛⠧⠙⠟⠁⠿⠛⠛⠻⣿⣿⣿⣶⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣤⣽⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⣯⣤⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣿⣟⣉⣤⣤⣤⡄⠀⠀⠀⢲⡄⠀⣠⠆⠀⠀⠀⣠⣤⣤⣬⣙⣿⣧⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣿⢟⠋⢙⣿⣭⠀⠀⠀⠀⠀⢿⣴⡏⠀⠀⠀⠀⢀⣭⣿⠋⢛⡻⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢃⣾⣿⠃⢠⣄⣀⣷⠀⠸⣿⠁⠀⣇⣀⣤⠄⢙⣿⣦⠸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⠏⠀⣾⣿⣿⡆⠲⣚⠿⡇⠀⠀⡟⠀⠀⣿⠿⣲⠆⢸⣿⣿⡇⠈⠻⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢃⣀⠀⠸⣿⣿⣇⠀⠈⢉⡇⠀⠀⠀⠀⠀⢿⠉⠁⠀⣾⣿⡿⠃⠀⣀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠘⢿⣿⣦⢀⠞⠀⠀⠀⠀⠀⠀⠈⢧⢀⣼⣿⡟⠁⠀⠀⢻⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣠⣶⠀⠀⠈⢿⣿⡟⠀⢀⣀⣀⣀⣀⣀⠀⠈⢿⣿⡟⠀⠀⢠⣶⣸⣿⣿⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠰⣄⠸⣿⡇⠀⠀⠉⠻⣿⠟⠁⠀⠀⣸⣿⠁⣴⠂⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣿⣿⣿⠁⡀⠹⣧⣿⣧⡀⠀⠀⠀⡃⠀⠀⠀⢠⣿⣧⣾⠁⡀⢸⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢿⣿⣿⡟⣠⡇⠀⢹⣿⣿⣿⡖⠒⠉⠉⠁⠒⢾⣿⣿⣿⠃⠀⣷⡈⢿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⢀⠀⡿⢿⣿⣧⡄⠀⠀⠀⣠⣾⣿⡟⡿⢀⡀⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣡⣿⡆⢀⠈⢻⣿⣿⣿⣶⣿⣿⣿⠟⢀⠀⣼⣷⣼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠸⣿⡿⢿⣿⣌⣷⣄⡙⢿⣿⣿⣿⠟⢁⣤⡞⣼⣿⡟⣿⣿⠁⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠈⢿⣿⣿⣿⠿⣷⣿⣿⣿⡾⠿⣿⣿⣿⡟⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⢿⣷⣄⠉⠛⢁⣴⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡿⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢰⣶⣶⡆⠀⠀⠀⠀⠀⣶⣧⡄⠀⠀⠀⠀⠀⠷⣤⡆⠀⠀⠀⠀⠀⢉⢈⠇⠀⠀⠀⠀⠀⢾⢿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠈⠻⠿⠁⠀⠀⠀⠀⠀⠒⣯⡄⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢣⡼⠀⠀⠀⠀⠀⠀⢸⠿⠀⠀⠀⠀⠀⢀⣬⢫⣄⠀⠀⠀⠀⠀⣴⣷⡿⠀⠀⠀⠀⠀⣶⣷⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠷⣤⡄⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⠶⠒⠚⠒⠦⣄⠀⠀⠀⠀⣠⡤⠶⠖⠦⣄⡀⠀⠀⠀⠀⣠⠖⠊⠉⠉⠑⠲⣄
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠘⣦⠀⢀⡞⠁⠀⠀⠀⠀⠈⠻⡄⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⡴⠞⠛⠉⠛⠲⣤
⠀⠀⣼⠁⠀⠀⢀⣄⠀⠀⠀⢸⡆⣿⠀⠀⠀⢀⡀⠀⠀⠀⢻⠀⡾⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⣧⢠⡟⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⢻⡄⠀⠀⠈⠋⠀⠀⠀⣼⠁⣿⠀⠀⠀⠘⠟⠀⠀⠀⣾⠀⣇⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⡏⣾⠀⠀⠀⢀⣤⠀⠀⠀⢸⡇
⠀⠀⠈⢳⣄⠀⠀⠀⠀⣀⡼⠃⠀⠘⣦⡀⠀⠀⠀⠀⢀⡼⠃⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢹⡄⠀⠀⠀⠉⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠉⢻⡷⠶⢿⡋⠀⠀⠀⠀⠈⠛⢶⡶⠶⣾⠋⠀⠀⠀⠀⢙⣦⣄⣀⣀⣀⣠⡴⠟⠁⠀⠀⠻⣦⣀⠀⠀⢀⣠⡾⠃
⠀⠀⢀⣤⣤⣿⣥⣤⣼⣇⣀⣀⣀⣀⣀⣠⣿⣀⡀⣻⡆⠀⠀⠀⠀⣾⠃⠈⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⢹⡟⠛⢻⡇
⠀⠀⣿⡏⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⣶⣶⣶⣶⣿⣷⣶⣾⣷⣤⣤⣤
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⣿⡇⠀⠀⣀⡀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⣿⠇⣴⠟⠛⠛⢷⡄⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⣿⠐⣿⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⠀⠀⠀⠀⠀⢠⣾⡟⠁
⠀⢀⣿⠀⠙⠳⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⠤⠤⠤⠤⠤⠤⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠘⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⣿⡿⠁⠀⣠⡾⠋⠉⠀⠀⠀⠈⠛⢶⣄⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⣠⡿⠉⠳⣤⣀⡀⠀⣀⣠⡾⠋⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋
⠀⠀⣠⡿⠁⠀⠀⠀⢠⡾⠋⠉⠉⠛⢦⡀⠀⠀⢠⣾⡿⠁
⠀⣴⠟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧⠀⣰⣿⠟
⣾⣏⡀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢀⣼⢇⣼⣿⠋
⠈⠻⢿⣶⣄⡀⠀⠀⠀⠙⠛⠷⠾⠛⢡⣾⡿⠁
⠀⠀⠀⠉⠻⢿⣶⣄⡀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣄⣀⣾⡿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣻⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔⠲⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⢎⡇
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠛⠀⠒⠒⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢊⠴⠁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠸⠃⠀⠈⡆⡔
⠀⠠⠀⠀⠀⠀⣰⣁⡍⢀⡴⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠓⠒⠒⠒⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠀⠀⠀⠀⠀⡠⠂⠉⠉⡇⠢⡀⠀⠀⠀⠀⠀⡇⡤
⠀⠀⠀⠙⠔⠉⠉⠉⠉⠁⡜⢣⠈⠉⠉⠉⠉⠑⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢘⠀⠀⢀⡴⢆⠀⢸⠀⠀⠀⠀⢀⠈⡀
⠀⠀⢠⠖⠉⠉⢹⠙⠢⡀⠑⠊⠀⢀⠔⠋⠉⠉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⢦⡀⠀⠈⢹⠈⠁⠀⠀⠸⠛⠀⠀⢀⠔⠁⠀⢈
⠀⢀⠁⠀⠀⢀⠾⡀⠀⢡⠀⠀⢀⠃⠀⠀⢀⠼⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠦⣠⡈⢆⡀⠀⣀⠞⢸⡤⠔⠁⠀⢀⠴⠁
⠀⠀⠙⠂⠉⠁⠀⠈⢙⠈⠀⠀⠈⠛⡆⠈⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡠⠒⠛⠂⠘⠓⠢⡀⠀⠀⠀⡤⠖⠛⠃⠘⠋⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡴⣔⢉⢩⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡔⠒⢄⠀⠈⠁⠒⢄
⠀⠀⠰⠢⢄⠀⠀⡰⠛⢤⡣⠄⠃⠈⢣⠀⢀⠀⢀⡆⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠢⡊⠗⢄⠀⠀⠈
⠀⠀⢐⠀⠠⠃⠀⠑⠀⠲⡀⠁⡛⠀⠘⠀⠀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠙⠦⠴⠃⡀⠜
⢀⠄⣈⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠑⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡀⠑⠄⣀⡠⠎⢠⠒⠄
⠣⠠⠆⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢦⠀⠀⠐⠒⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢬⣊⠀⠀⠀⠀⠀⠀⢈⡓⠅⢠⣄⡀
⠀⢠⠄⠤⡀⠀⠀⠠⣁⣀⡠⠚⠤⣀⡇⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢠⠀⠑⢄⠀⠀⠀⡰⠋⠀⠀⢠⠀⠉⢆
⠀⠸⡀⠀⠉⠢⡀⠀⢘⡀⠀⠀⠀⡇⠀⠀⡠⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠴⠋⠢⣀⣈⠀⠀⠀⣇⣀⣀⠤⠊⠢⢀⣈
⠀⠀⠈⠢⡀⠀⠈⠒⠤⠵⠦⠤⠾⠤⠐⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣀⣠⠃⠀⠀⠀⠀⡇⠑⣄⣀⣀⣠⠊
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠗⠉⠀⠀⠀⠀⠉⠲⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣤⣄⣀⣤⣤⣤⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣤⣤⣤⣤⣤⣤⣤⣤⣃⣀⣈⣀⣀⣀⣠⣄⣤⣤⣤⣃
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠛⢻⣿⣿⡟⠛⠿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠁⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠈⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⢀⣴⣿⣿⡿⠋⢸⣿⣿⡇⠙⢿⣿⣿⣦⡀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏
⣴⣦⣄⢇⣰⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣴⣿⣿⡿⠋⠀⠀⢸⣿⣿⡇⠀⠀⠙⢿⣿⣿⣦⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣆⡸⣠⣴
⠙⠻⠿⢿⡿⠛⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡋⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢙⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠿⠟
⢠⣤⣤⣀⢣⢠⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣀⣀⢸⣿⣿⡇⣀⣀⣤⣶⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡄⡜⣀⣤⣤
⠘⠻⠿⠿⠿⢿⠛⠉⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠉⠛⡿⠿⠿⠿⠟
⠀⢠⣤⣤⣄⣈⢧⢰⣿⣿⡿⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢿⣿⣿⡆⡼⣁⣤⣤⣤⡄
⠀⠹⠿⣿⣿⣿⡷⠿⡟⠉⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠉⢻⠿⢾⣿⣿⣿⠿⠏
⠀⠀⠀⠀⣠⣤⣄⣀⡘⢦⡸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢇⡴⢃⣀⣠⣤⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡿⢿⡋⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⢿⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠙⠺⣿⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⢺⣷⣀⠔⠋⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠟⠛⢻⣷⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⣠⣤⣄⠀⣿⣭⡿⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠲⣶⣾⣋⣠⣿⣧⣿⠇⢰⣿⠛⣿⠇⠰⣿⡀⠋⣀⣿⠿⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠉⠉⠉⠻⠷⣦⡶⠿⣷⣾⠿⣧⣤⡿⠛⠛⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣴⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠉⠳⡼⣟⢿⠈⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡚⠁⣠⣄⣀⡤⠬⡟⠳⣄⣹⡈⠩⣿⠗⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠠⠆⠀⠉⣤⡀⠘⠁⣹⠀⠱⣄⡀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢶⣶⡄⠈⠛⣇⠐⠖⠀⢀⣠⢄⠀⢰⠁⢀⡠⠚⠃⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠻⠏⡇⠀⠀⠘⣆⠀⠘⢿⣀⡼⢠⡾⠚⡏⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠶⠿⢧⠀⣀⡴⢾⢓⡶⣶⣲⢶⢋⠿⣄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡋⠁⣀⡤⣼⣿⠿⢳⡿⢯⡠⠜⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⣀⣈⡧⣄⣀⣠⣴⠟⣠⠴⣇⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⠛⠛⠻⡖⠚⠁⠀⢨⠿⠟⠋⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣰⠋⠂⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⠦⢄⣀⠀⠀⠈⠳⢶⠇⠉⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣼⠟⠒⣶⠦⣄⠙⠲⠤⣄⣀⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣠⠞⠁⠀⠈⢢⡀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠋⠀⠀⠀⠀⡴⠟⢦⡀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⢳⠀⠀⠈⠁⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠘⣆⣀⣤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⡽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡄⠀⠀⠀⠀⠀⡞⠻⣽⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣶⣗⠀⠀⠀⠀⠀⢲⣿⠆⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡰⠋⠁⠀⠀⠉⣛⡼⢢⣀⠀⠩⣿⡄⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠐⢷⠙⢿⣯⡷⠶⣦⡁⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢒⣶⠶⠖⠉⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⡾⠿⢻⣢⡇⢸⡟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠸⠀⠀⠀⠀⠀⠀⡇⠀⢼⣴⣋⣽⣟⠮⠇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⢠⣾⢎⣱⠆⢿⣦⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠦⠤⠖⠀⠀⠶⣤⣤⣴⠄⣿⠸⠞⡋⣽⣸⢷⡄⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣼⡶⠟⣩⡥⢨⠃⠀⠀⠈⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡧⢹⡟⢿⣇⢈⠀⠀⠀⠀⢹⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡹⢷⡎⢳⣸⡏⣼⠁⠀⢤⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⡏⢺⣿⢸⠃⣶⢻⣤⡀⢀⣿⠟⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡟⣎⢻⣎⡷⠁⣯⢸⣹⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣷⣿⢀⡿⡁⡄⣿⣤⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣷⢻⡿⢷⣄⢷⠙⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⣸⣿⣼⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢈⣡⣿⠏⣿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣻⠛⠋⣻⠏⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣾⣿⣷⣾⣿⣏⢠⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣾⢻⡿⣰⣿⣏⣸⣿⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠉⠈⠁

⡿⢿⡿⠿⡿⠟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣿⣟⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡠⡀⠤⢀⣐⠀⢀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣋⠑⡟⢉⣛⠧⠤⠍⠰⣼⣿⠆⡔⣻⠁⡫⢄⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠈⡰⢷⣹⢆⡊⠀⠀⠀⠘⠁⠀⠚⡴⢷⡎⢟⠲⣷⢀⡄⠲⠫⠄⢄⡀
⠀⠀⠀⠀⠀⣠⣖⢽⡆⢭⣤⡁⢉⡐⠀⠊⠈⠀⠀⡀⠀⠀⠈⠀⠀⠒⠚⠂⡦⡄⠗⠀⢈⡡⣀
⠀⠀⠀⢀⠀⣽⣟⣏⡱⣇⠏⠁⠀⠀⠀⠀⠀⠸⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⠍⣀
⠀⠀⠀⠎⠀⣴⡭⡩⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⢂⡰⠀⠀⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢠
⢀⡯⠦⠱⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⢀⣀⣍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠋⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⡀⢣⠄⢠
⠀⠀⠂⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠈
⠀⡀⠀⠥⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⡄⠠
⠀⢋⠁⠄⡜⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠰⠁⠀⠀⠳⠂⠀⠠⠘⢃⠀⠄⠀⠀⡄⠀⠀⠀⠠
⠀⠈⢧⣀⠀⠉⠶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⣾⠃⠀⠀⠀⠟⠠
⠀⠀⠀⠂⠀⢡⣡⢀⣠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠋⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠟⡇⠆⣸
⠀⠀⠀⠀⠄⠀⢻⠟⡚⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⢜⢰⡀⠀⠀⠀⠀⠀⠠⠂⠂⠄⠳⢁⣠⡧⠄
⠀⠀⠀⠀⠈⠀⠨⣀⡀⠀⠀⠀⠀⠉⠀⠠⠄⠀⠀⠤⠷⠇⡠⠀⠀⠈⠀⢀⠈⠘⢃⡑⠀⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠐⠍⡀⡀⠘⡂⠂⠁⠀⡴⠆⠀⠀⠂⠀⠐⠀⠈⡀⠹⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢈⣀⠸⠋⢣⠠⠀⡕⠀⠀⠓⠀⣩⠇⠈⠁⠌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠲⣀⣉⢀⣤⢀⠂⠃⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⣠⣤⣤⣤⡸⣷⡀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡟⠁⠀⢹⣷⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⡏⢻⣆⠀⢀⣾⠇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠙⠛⠛⠁⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢁⣴⠶⢶⣦⠀⠀⠀⠀⠀⠀⢹⠉⣭⠀⠀⣶⠄⣴⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⠁⠀⠀⣿⠆⠀⠀⠀⠀⠀⢸⡄⣿⣤⣶⣿⡀⣿⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⣷⣄⣀⣼⠟⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠈⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⡄⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠘⣿⠀⢠⣄⠀⣼⡇⣠⡾⠛⠻⣦⠀⣟⢀⣀⣼⡟⢰⡇⠀⠀⠀⣿⠀⠀⠹⣿⡄⣾
⠀⠀⠀⠀⠀⠀⢻⣇⣿⣿⣷⣿⢡⡟⠀⠀⠀⣿⠃⣿⣿⣟⠉⠀⢸⣿⠀⠀⠀⣿⠀⠀⣠⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡟⠘⣿⡿⠈⢿⣤⣤⣾⠏⠀⣿⠀⠻⣷⠄⢸⣿⣤⣤⡀⣿⣴⡾⠟⠃⠀⡛



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⢠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠀⠀⠀⠘⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠈⢲⡀⡠⠚⠉⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡞⠀⠀⠀⠀⢀⡀⠀⣿⠁⠀⠀⠀⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠙⠛⣇⠀⠀⠀⠀⠈⠃⠀⣿⡀⠀⠀⠀⠛⠀⡼⠶⠶⠶⠶⠶⠶⢶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⣠⠞⠁⠳⣤⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠈⠁⠀⠀⢀⡴⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠶⠶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⣴⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏⠉⠁⠀⠀⠉⠉⠛⠶⣄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⣠⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⢀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡿⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⢀⣀⣤⣶⣾⣿⡿⠟⠋⠁⣿⠀⠀⠀⠀⠀⠀⢻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠓⠶⢤⣄⣀
⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⢸⡆⠀⠀⠀⣀⡴⠋⠁⠉⠻⢷⣤⡀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⢉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⣿⠋⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣴⣟⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⢿⣿⣿⣿⣷⣶⣤⡀
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣴⡶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠘⡷⠶⠶⠟⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⣠⣶⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣦⣤⡀⢾⣧⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣾⣿⣿⣿⣴⠾⠷⠤⣄⣀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣨⣿⣁
⠀⠀⠀⢀⣿⣿⢻⣿⣿⣿⣿⠻⣿⣿⠇⠀⠀⠀⠈⠃⠀⠀⠀⢸⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⠉⠉⠓⠦
⠀⠀⠀⢸⣿⠃⢈⣟⣛⣛⣏⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢻⣿⣿⣿⡇⠻⠿⠃
⠀⠀⠀⠘⠛⠀⣼⣿⣿⠛⠛⠂⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⣿⡇⢀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⣴⠀⠀⣿⣿⣿⣷⣶⣿⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢻⡇⢸⣿⣦⣤⣴⣿⡄⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠈⠀⢠⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠐⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⢛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢷⡈⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣌⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠲⣌⢇⠀⠢⠀⠀⠀⠰⡶⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⡀⢠⠀⠀⣠⣴⠟⠁⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣱⡟⣴⣿⡟⠁⠀⠀⢀⣴⡿⠁⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠁⠀⠀⢠⣾⠏⠀⠀⢈⣁⣀⣤⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠶⣿⣿⠿⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠷⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⡤⠀⠀⠀⠀⠀⠀⠀⡾⠃⣴⣶⡄⠀⠀⠀⠀⠀⠉⠉⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣼⣷⣶⣍⠻⠗⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣸⠛⠛⠿⣿⣦⡀⠀⠀⠔⠤⡀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣰⢿⣦⡀⠀⠀⠉⠻⣦⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢰⠏⠀⠙⢿⣆⡀⠀⠐⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⠉⠙⠂
⠀⠀⠀⠠⠤⣤⣙⢦⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠉⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⡰⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⡼⠁⡀⠼⠁⠀⢀⣼⡟⠀⠀⠘⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡄⠀⠀⠀⠀⠀⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠠⠿⣋⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢄⠀⣀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⣰⡃⠀⠀⠀⠀⠀⠀⠀⠉⠿⠛⠉⠉⠁⠍⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠈⠛⢁⣴⠀⡼⠁⠀⠀⠀⠀⠀⠀⣰⠟⣛⣱⣶⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠼⠇⠀⠀⠀⠈⠠⢄⠀⠉⠻⢿⣶⣿⣯⡼⠁⠀⠀⠀⠀⠀⠀⢠⠟⢋⣫⠝⠻⠟⠀⠀⠀⠀⠀⠿⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣶⣦⣤⣭⣻⡿⠁⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⠁
⠀⠀⠀⠀⠦⠰⠠⠄⠀⠀⠀⠈⣉⠛⠛⣿⠃⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⠃⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢀⣀
⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠀⠉⣹⠃⠀⠀⠀⠀⠀⠀⠀⣼⣅⠀⠀⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠒⠒⠒⠒⠒⠚⠋⠘⠇⠀⠀⠉⠻⢄⡀

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⣴⢿⣿⡟⣦⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⣶⣷⡆⠀⠀⣤⣤⣰⠏⠉⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⢹⡟⠿⢋⣍⠀⠠⠀⠀⠆⠁⠀⠀⠀⠀⠈⠋⠁⣀⣼⣿⣿⣿⠀⠀⣠⣿⣧⣮⣿⢧⠀⠀⠀⠁⠄
⠀⠀⢈⣹⠶⣏⣉⠀⠀⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣻⣟⣿⠿⣿⡆⠀⠻⣿⣿⣿⣿⡾⠃
⠀⠀⠈⠃⠀⠈⠃⠀⢀⠀⠈⠀⠀⠀⠀⢀⣴⠀⠀⠀⢹⢾⡏⠀⣿⡇⠀⠀⠚⠙⠋⠙⠂
⠀⠀⢀⠀⠀⢠⣶⣶⣿⣶⣦⡄⠀⠀⣀⣼⣿⣧⣄⣄⠈⠈⠀⠀⠈⠁⠀⠀⠀⠀⣤⣴⣦⡤⠀⠀⢾⡷
⠀⠀⠘⠃⠀⢸⡿⢼⣿⡧⢿⠇⢀⣾⣿⡿⠟⠻⣿⣿⣷⠀⠀⢀⣤⣤⡀⢶⣷⣤⣿⣿⡿⠃
⠀⠀⢀⡄⠀⠀⠙⠲⠤⠔⠋⠀⠈⠿⣷⣤⠀⢠⣾⡿⠃⠀⠰⢿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢀⣼⣧⣄
⠀⠀⢸⡇⠀⢠⠀⠀⣦⢀⡀⠀⠀⠀⠈⠿⠾⠾⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠀⢠⡄⢀⣿⢿⡿⢿⡄
⠀⠀⠀⢀⣶⣼⣷⣶⣿⣿⠿⠀⣤⣄⡀⠀⠀⣀⣠⣤⣤⣶⢻⠁⠀⢰⣄⣠⢠⡄⢀⠀⠀⢸⣟⠿⠿⢻⡇
⠀⠀⠀⠀⠈⠻⣿⣿⡿⠉⠀⠀⠘⢿⣿⡀⠀⡿⠟⢛⢁⣁⣤⠀⠐⡲⠾⣿⣄⣷⢋⡆⠀⠀⠙⠓⠒⠛⠁
⠀⠀⠾⠇⠀⠀⠈⠉⠛⡆⠀⠀⠀⠀⠈⠁⠀⣶⣾⣿⡽⠟⠋⠐⠒⠛⢳⣿⣿⣷⠞⠁⠠⣤⣠⡄
⠀⠀⠀⣠⣶⣦⠀⠀⠀⣰⣾⣿⣿⣶⠀⠀⢀⣉⣤⡶⡖⡿⡇⠀⠀⠀⢀⡟⠟⣿⠀⠀⠈⠉⠉⠉⢳⣤⡄
⠀⠀⢀⣈⣿⣏⣀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠟⠋⠃⠉⠀⠀⣀⠀⠐⠺⠿⠟⠿⠒⠀⢹⣦⡀⣀⣼⡿⣅
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⡀⠀⠙⠁⠀⠀⢀⣠⡀
⠀⠀⢸⣰⣿⣶⣸⠀⠀⠚⠛⠛⠛⠛⠋⠋⠁⠀⢠⢿⣿⢿⣷⠀⠀⠲⣶⣭⣅⣀⠀⠀⠀⠀⠀⣰⣶⣶⡆
⠀⠀⠸⣇⠀⢸⠿⠀⠀⠀⠀⣠⣾⢿⣿⣆⠀⠀⠘⢿⡿⠾⠿⠁⠀⠀⢠⣿⣿⣥⣶⣾⡅⠀⢀⣷⣿⣷⡇
⠀⠀⠀⢻⠀⢸⠁⠀⠀⣠⣾⣯⣷⣾⢿⡿⠀⠀⠀⠀⠀⠀⠀⠤⣤⢶⣫⣽⣿⣿⣩⣩⣴⡆⢸⣿⣿⣿⠃
⠀⠀⠀⣾⠀⢸⡀⠀⠘⢯⣿⣿⣷⠿⠛⠁⠀⠀⠄⡐⠀⠀⠀⠀⠀⠀⠙⠿⣿⠋⠀⠀⠀⠀⠈⣿⡟⠁
⠀⠀⠀⠁⠀⠈⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠰⢄⠀⠰⡆⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠊⠒⠀⠀⡰⣀⡀⠀⣰⠂⠀⠀⢩⡍⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⡀⠀⠀⠀⠘⡦⢀⡤⠁⠉⣰⣠⠀⠠⠀⠀⣀⡁⠛⢙⠁⢀⡿⠃⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⠠⣤⠈⢠⠄⣀⡑⢀⣽⣻⠄⠀⠈⠆⠍⢤⣦⠈⢀⡜⢧⡘⠀⣴⡁⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡙⣡⡄⠒⠂⢄⡏⠉⡉⠉⠀⠀⣀⠈⠀⢰⣆⡈⡇⠠⢘⡿⢋⣥⠖⡃⠀⡞⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠻⡲⠶⡉⢿⠿⡻⠀⣬⠂⠄⠀⢀⠲⢿⠀⠀⠘⠄⠠⠥⢌⠀⠀⣐⣷⡇⠙⡗⠈⢱⣩⠙⠋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⢠⡶⣇⣤⣴⠄⣈⠀⣤⡀⠈⣢⠐⢠⡤⢀⣤⠾⣀⠠⡄⠒⠀⣸⠀⢀⠀⢪⡾⣡⣶⡦⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⡀⡀⠈⡐⣥⣯⣿⣯⡌⡠⠝⠀⠐⠃⠠⠀⢐⠿⠟⠰⢿⠆⣡⠦⢦⣀⠄⢁⡧⠹⠛⣿⣁⣿⣿⡃⡀⠸⡏⠄
⠀⠀⠀⠀⠀⠀⠩⠁⠻⣈⡂⢩⢻⣏⣇⣄⠀⣌⠈⠗⣶⠾⡄⠰⠠⢂⡩⠆⠀⠉⡾⠛⠐⠑⣶⠁⣦⡐⣿⣿⡋⠁⣀⣪⣇⡀⠃
⠀⠀⠀⠀⠀⠠⣦⡀⠙⣀⢉⣷⠸⠿⣯⠠⠀⡍⠸⡿⠁⠓⠀⢀⠚⢩⣴⢮⠀⣤⣤⠓⠐⢻⠦⠆⡶⠼⣏⠚⣤⠳⢮⠿⠍⢷⡀⠁
⠀⠀⠀⠀⠀⠴⣷⣰⡄⣼⣮⣿⣡⣀⠡⢤⣤⡃⠠⠀⠀⡀⡀⢀⣺⠧⣶⡀⢊⣭⣳⣷⠀⠃⠠⠒⢢⠉⣼⣿⣩⣥⣾⣇⡉⠉⣤⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⢀⣨⠋⣀⢀⢰⣾⣗⡞⠒⢶⠀⢬⠀⠘⠛⣷⣾⠫⢝⣟⣘⠫⣻⡿⠂⡈⠗⠀⠈⠙⣏⣙⣿⣿⣽⣟⣏⡄⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣉⡥⠔⣣⡯⠽⠿⠁⣀⠲⠀⡠⠀⠀⠀⠀⠰⢙⢰⣾⠿⣟⠀⢿⣿⣄⠈⠠⠄⠈⣠⣆⡿⠙⢿⡽⢿⣩⠂⡀⢀
⠀⠀⠀⠀⠀⠀⠐⠌⠤⡽⢉⠀⠞⠇⡔⠩⣆⣠⣴⡀⠒⣶⠁⢶⣀⣬⠔⠁⣿⣿⡿⠉⣉⣇⣝⠣⣤⠓⡮⠈⠱⡷⠟⡾⠿⢄⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠲⠀⠄⡀⢀⣚⣳⣿⣞⠿⢿⣿⡇⣿⢿⠤⣨⡇⠹⠀⢣⡞⢓⣾⣌⠿⣿⡿⠄⢻⣿⣭⠺⣷⣄⣼⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠶⠾⠂⠀⢘⠀⠛⠧⢠⠞⡾⡸⠻⣿⡇⢻⢨⣦⢖⠴⠺⣭⣄⠸⠠⣼⢷⣤⡘⢯⣝⡚⠽⣾⡿⠎⢄
⠀⠀⠀⠀⠀⠀⠀⠙⠫⠀⠀⠀⡈⢀⢵⠞⡠⢺⣭⣴⢀⣼⣦⠄⠛⢷⢛⠓⢀⢰⣫⡀⠐⠂⠀⠨⢧⠈⡙⢋⢆⠀⠀⠰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠊⢤⡄⠁⣈⢡⡾⣿⣿⠿⢶⣡⡄⣠⡴⢹⡇⠭⠁⠀⠖⢁⣶⣮⡇⠈⠂⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠅⠒⠤⣈⠀⣿⣍⢱⠄⠄⢎⡿⠂⠈⠰⢆⡻⠤⢞⣉⣙⠋⠱⠄⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠐⠈⠀⠎⢒⡀⠯⢽⠖⣶⣾⣏⠃⠈⠱⢏⠁⡀⠈⡉⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠑⠃⠈⠺⠆⢹⠍⠡⠀⠀⠉⣅⠀⠈⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⢀⠀⢠⡏⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⢦⠀⠀⠀⠀⣸⣄⠀⢄⠈⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⠇⠀⠀⠀⢻⠘⡳⣄⠈⠀⢜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⠉⠀⠀⠀⠀⠘⢧⢊⠄⠱⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡡⠺⡀⢀⡀⠀⠀⠀⠈⠳⡁⠂⢀⠝⢦⡀⠀⠀⠀⠀⠀⡰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠂⢐⠸⡇⡜⣇⠀⠀⠀⠀⠀⠈⢮⢐⠀⠠⢳⡄⠀⠀⠠⠊⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠑⡨⢐⣼⠁⠑⢌⠣⡘⣦⢄⡀⠀⠀⢳⠁⡨⡄⣷⠀⠀⣁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢂⠅⢊⠐⢹⡀⠀⠀⠉⠓⢸⡆⠱⡄⠀⣘⠈⡄⠀⢸⠀⠀⢡⡆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⡆⠀⢘⢢⡀⠳⣄⠀⠀⠀⡼⠐⠉⡇⣠⠏⠀⠃⠀⣚⠀⠠⡟⢇
⠀⠀⠀⠀⠀⠸⠓⠤⣀⠀⠀⠀⢣⠀⠘⡄⠈⢦⠈⢳⣀⡜⠁⡄⠀⠿⠁⡠⠁⡂⢨⡇⠀⢸⠇⢺⡀
⠀⠀⠀⠀⠀⠀⠣⢈⠈⢳⡄⠀⠀⢳⡀⠙⡄⠀⠑⡄⠛⢀⠎⠘⡀⢠⠎⠀⡰⢁⡜⠀⣠⡯⠂⠙⡇⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠠⢤⣈⡐⠇⠀⠀⠀⠘⣆⠘⠄⠀⠘⣀⠎⠀⢀⠖⠁⠀⡰⢁⣜⣠⠾⠁⠀⠀⢀⡇⠀⠀⢀⠜⢽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠳⢦⣀⠀⠀⢸⡄⠀⠀⠀⢻⠀⠀⠈⠀⠀⢀⠁⠚⠉⣀⠤⢒⠋⢀⣼⠃⠀⠰⢁⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢐⠙⢦⠀⢸⡇⠰⠀⠀⠀⠀⠈⠀⠀⠀⢰⢀⠔⠉⠀⡠⠃⢀⡘⠁⠀⠀⢋⡠⠖⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠣⠈⣇⡾⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢰⠁⢰⠟⢀⣴⠂⣰⠯⢐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⢰⠀⢃⠙⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡏⣰⢛⡏⣸⠅⡈⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠘⡀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢃⠀⡿⢊⠐⣧⡇⠜⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠃⠘⡆⠋⠸⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠈⢢⡅⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠅⠢⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠤⠀⠀⠀⠀⣀⣀⠀⠠⠌⠑⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠐⠈⠔⠈⠄⢂⠡⢀⠠⢐⠠⢐⠤⢂⠠⠐⠠⢐⠠⢐⠠⡰⠜⠢⠄⡂⠅⠢⠒⠄⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣦⠀⠀⠀⠀⠀⠸⣶⣤⢁⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢀⣀⠘⡈⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⣀⣙⣛⠳⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣀⣴⠟⠉⣉⠙⠻⠷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠉⠀⢀⣴⠾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠶⣶⣶⣶⣶⣶⡶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠐⠷⠶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠛⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⣼⡏

⠀⠀⠀⠀⢀⣤⡴⠶⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⢛⡿⠛⠛⠛⠙⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⢦⣄
⠀⠀⢀⡶⠋⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡸⢀⠁⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⣴⠏⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣸⡏⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⡀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⡘⠀⠀⠹⡄
⣿⠄⠈⠁⠠⠀⠀⠀⠀⡼⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠃⠀⠀⢠⠾
⢸⡇⠈⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣁⣀⠀⠀⢀⣈⣡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠃⠀⠀⠀⠰⠃
⠀⠻⣄⠀⠐⠀⠀⡘⠀⠀⢀⡀⡀⡀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠙⡦⢴⡎⠀⢀⠟⣠⠀⡀
⠀⠀⠘⠷⣄⣀⠇⠀⠀⠐⠁⠀⠀⠈⠂⠀⠀⠁⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠈⠉⢻⡣⣤⡿⠉⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠈⠙⠻⠦⣤⣴⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣄⣤⣤⣤⣤⣤⣴⣤⣤⣼⣇⣀⡓⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠈⣦⢰⠀⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡴⠞⠁⠀⠀⠀⠀⠀⠉⠀⠀⣀⠃⠀⠀⠀⠀⠘⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠞⠁⢰⡾⠇⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⠉⣶⠂⠛⠀⠀⠀⠀⢦⠁⠀⠀⠆⠐⠂⠠⡀⠀⠈⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠐⡅⠀⠀⠀⠀⠀⠀⠠⠀⠀⢃⠐⠀⠀⠀⠄⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢣⡤⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠋⠀⣤⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠉⠤⢄⣀⣀⣀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠾⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠘⠃⠀⣠⠞⠙⠀⠄⠀⠄⠐⠀⠈⠀⠀⠄⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣀⡀⠀⠀⠘⠷⠀⠐⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⡀⠀⠈⠀⠈⠀⠀⠀⢀⠈⠁⠉⠀⢀⣀⢀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠄⡀⠤⢀⡀⠀⠀⠀⠈⠁⢀⠉⢢⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠐⠀⠀⠁⣤⠀⠀⡄⠂⠀⢀⡰⠁⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⡀⠐⠀⠀⠐⢂⡈⣙⡋⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠘⢀⠀⠉⢠⠀⠉⢠⠀⠁⠤⠈⢡⠉⠄⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⢸⡀⠀⠂⠀⠀⠂⡀⠈⠃⠀⠐⢇⡀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠈⠊⠁⡄⠈⠠⠄⠈⠠⠀⠀⠀⠈⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠖⠀⠀⠒⠤⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠴⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠠⠀⠀⠀⠒⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠁⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⢀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠂⠀⠀⡸⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣀⣀⣠⡤⠊⡀⠄⠁⠈⠉⠁⠈⠙⢂⠀⠀⠐⠂⡴⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠉⠀⠀⠀⠰⠛⠀⠙⢦⣀⣰⠋⠀⠙⠆⠀⠀⡘⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠄⠀⠀⠀⠀⠀⠐⠲⠞⢀⣹⡗⠒⠁⠀⡇⢠⢁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠸⡄⠀⠀⡜⠉⢻⡄⠀⢀⣾⠋⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠁⠤⢀⡐⣤⡌⣀⠠⠊⣡⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣤⣤⣤⣤⡤⠖⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⡿⡙⣿⢛⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⠘⠋⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣐⡈⢷⣤⣀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠟⠿⠉⠛⠷⣬⡿⣿⣷⣿⣦⡷⣶⣄⣈⣿⡁⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⢤⠚⡟⠋⠙⠻⣷⡌⠉⠙⠋⠀⢫⣼⣿⡿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢠⣤⣽⣶⣤⡶⠊⠉⠁⠀⣀⡀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡷⣄⠀⣸⣿⣷⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡷⠈⠲⢿⡇⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣦⡉⠁⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⢀⣿⣿⣼⣿⣿⡿⠯⢇⠀⠀⠀⠀⠹⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠆⠀⢀⠀⣸⠀⠀⠀⠀⠀⣸⣿⣿⣿⣟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠻⣧⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣀⣧⣀⣀⣸⣿⡿⢷⣿⣶⣶⠾⣷⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠻⣿⣿⠟⠉⣿⠃⠚⠻⠟⠁⠀⢿⡀⣼⡿⠙⣿⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠈⠁⠀⠘⢿⣤⠀⠀⠀⠀⠀⠘⢷⡿⠁⠀⣿⠀⢻⣿⡆⠀⠀⠰⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣿⡇⠀⢻⣧⠀⠀⠀⠀⠀⠀⣦⣄⢀⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠉⠀⠲⣾⣿⣷⠀⠀⠻⣆⠀⡇⠀⠀⠀⠀⠙⢟⣉⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠘⣿⣿⣿⣧⠀⠀⠈⠲⣧⡀⠀⠀⠀⣠⣿⠉⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣸⣿⣿⣿⣇⠀⠀⠀⠀⠈⠓⠄⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠹⣿⣿⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢹⣿⣿⣿⣿⣿⡎⢷⡀⠀⠀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠸⢿⣿⣿⣿⣿⣷⠈⣿⡄⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢹⣿⣿⣿⣿⡆⣿⣷⠟⣼⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢿⣿⣿⣿⣷⣽⡏⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣻⣼⣿⣿⣿⣿⡟⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢃⣼⢿⣿⣿⣿⣿⡟⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⠟⠡⠾⠍⠿⣿⠟⣶⢹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠄⠤⠶⢒⣶⡟⣼⡿⣿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⢤⣴⢁⣿⢟⣼⣿⠃⠙⠲⣇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠱⣾⣏⠚⠛⠋⠀⠀⠀⠀⠳⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⣶⣶⣶⣦⣴⣶⣶⣶⣤⣶⣶⣴⣦⣴⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣶⣤⣶⣶
⣿⣿⣿⡻⠛⠛⡿⠋⢛⠛⠛⡛⠟⣛⡿⢟⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣿⣿⣾⣿⣶⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡦⠀⠀⠀⢀⡴⠒⠲⡄
⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⣿⠀⠀⠀⢻⠀⢠⣾⠂⠀⢠⣿⣧
⠀⠀⠀⣠⠴⠋⣉⣐⣦⣀⢦⡈⠳⣄⠀⠀⠀⠀⠀⣿⡇⠀⠀⣹⣦⣀⣀⣸⣀⣸⣿⠀⠀⢸⠘⣿⡄⠀⠀⠀⠀⢀⣤⣄⡀⠀⣰⣶⣄
⠀⠀⠀⠈⠓⠦⣄⣩⡟⠋⣽⣫⠽⠋⠉⠉⠉⠉⠉⢹⣿⡴⢺⣿⠀⠀⠀⡼⠀⠈⣿⠀⠀⡞⠀⣿⣧⠴⠋⠉⠉⠉⠛⠛⠉⠉⢹⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡤⢺⣿⠀⠘⣿⣄⣀⡼⠁⠀⠀⠙⠓⠚⢁⠴⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠞⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⢀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⢃⣠⠖⠶⠶⠶⠖⠛⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠙⢻⣿⠉⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⣿⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠃⠀⠀⣿⠀⠀⢸⣇⣀⡟
⠀⠄⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣿⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠤⠴⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⢀⣶⣿⡿⠂⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣦⣴⣿⣿⢣⣾⣿⣿⣿⡿⢃⣀⢀⣶⣿⣿⣿⣿⣷⡀⠰⠶⣿⠛⠃⠀⠘⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⡿⢿⣿⣿⢻⣿⣷⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢹⣄⣤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣾⣿⣿⣿⡿⠏⣼⣿⠿⣿⣿⣽⣿⠟⠁⠈⢿⣿⣿⣿⣿⡿⠃⠀⢀⣼⡿⠋⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣸⣿⣿⣿⡿⢣⡀⢹⣷⣾⣟⠗⢶⠟⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⣾⣿⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣛⣽⣿⡏⢀⣾⣯⢿⣿⡫⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣰⣿⣿⣿⣇⣸⣿⣹⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣿⣿⣿⣿⡿⠃⢈⣿⣴⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⡆⢈⣛⢛⣫⣹⣾⣿⣿⣿⣿⣿
⠀⠈⠻⢛⣿⣿⣀⣴⡿⠿⠦⣿⡇⠀⠀⠀⠀⣀⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣉⢻⣿⡿⣿⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣾⣿⣯⡤⠀⠀⠈⠻⢶⣶⣶⠀⣿⡇⠀⠀⠀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⣿⢰⠉⣵⡼⢟⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢹⠟⠛⠃⠀⠀⠀⠀⣠⣾⣿⢠⣿⣧⣐⠚⢈⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣸⣿⡸⣷⣿⣿⣬⣻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠸⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣟⢻⣾⣿⣿⣿⣧⠻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣰⣶⣿⡿⠋⠀⠀⠀⠀⠙⢿⣿⡿⣻⣍⣿⣿⣿⣽⣿⣿⣿⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⡶⣶⣶⢶⡶⠶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⢶⣶⣶⣶⠶⡶⢶⡶⠶⡶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡟⢯⣍⣉⣉⣋⣉⣉⣉⣉⣉⣉⣁⣉⣉⣉⣙⣉⣉⣉⣉⣙⣉⣳⣤⣀
⠀⠀⠀⠀⠀⠀⠈⡃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⠇
⠀⠀⠀⠀⠀⠀⠀⡇⢄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠣⣄⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⡶⠶⣶⠶⢶⠆⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⠠⡢⡵⠋⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⡀⡀⢡⠞⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡙⣨⠋⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⢄⢌⣸⠃⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠢⡢⣵⠁⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠠⡑⣼⠃⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠡⠐⣽⠏⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡀⡞⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠠⣸⠁⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠠⠠⡟⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⢣⡇⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⢺⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣻⠀⠀⣀⣀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⣠⡴⠞⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠷⢏⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠗⠚⠀⢵⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠶⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⠛⠶⠲⣤⣤⣤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣥⣀⠶⣀⣴⠿⠟⠉⠀⣴⠋⠀⢀⡴⣀⣤⣤⣤⣀⣀⠉⠈⠁⣼⣿⡿⣄⡉⢛⣛⣋⣁⣈⡀⣀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⠘⣷⠀⠀⠘⢤⣯⡍⠉⠉⠉⠉⠀⠀⠀⣿⣿⠀⣽⠁⠈⠉⠉⠉⠉⠁⣬⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷⠀⠶⣶⡶⠞⠁⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣋⣿⣋⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣿⣿⣿⣷⡄⠀⣼⣶⣾⡟⢻⡶⠶⣶⡉⢹⡄⠀⠀⠀⠀⣾⠃⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣰⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣿⣿⣯⠙⠿⣤⣀⣀⠀⢀⣙⣃⣀⣸⣿⡇⠸⠇⣸⠋⣿⠛⢿⣦⠄⠀⣴⣿⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢰⠆⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠈⣿⡟⣷⠀⠈⠙⠻⢿⣿⠏⠉⢉⣴⡿⠁⢦⡸⠇⢀⣿⣤⠀⢿⡀⠀⠈⠙⠃⠀⢀⡄⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⡗⡇⠀⠉⠀⠀⠀⠀⠴⠶⢿⣿⠁⣰⣿⠃⠀⣼⣿⣶⣦⣴⣧⣀⣀⣀⣀⣀⣼⠃⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢠⣿⡏⣀⣠⣏⣉⠉⢉⣉⣹⣏⡉⠉⠉⠉⠁⠀⠀⠀⠀⣠⣟⣀⣀⣀⣀⣀⣀⣀⣘⣧
⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⢰⣿⣇⠀⠀⢀⣶⠀⠀⢀⣼⠏⠀⢠⡿⠿⣿⣿⣿⡟⠛⠛⠛⠛⠋⠙⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡟⠛⠛⠛⠛⣋⡹
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡟⢀⣤⡿⠁⣠⣴⠟⠁⢀⣴⣿⠃⠀⠀⠿⠛⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠃
⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣿⣿⣿⣿⣯⣴⡾⠟⠁⣠⡴⠋⣽⠁⠀⢀⠇⢰⡇⠙
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢰⣿⡿⠟⠋⣉⣁⠴⠖⠛⠁⢀⡼⠃⠀⣠⡟⠀⡆⢹⠄
⠀⠀⠀⠀⠀⢠⣿⠛⠃⠀⣸⣿⣤⣴⣿⠋⠁⠀⣀⣠⣴⣟⣁⣤⣾⣏⣤⣿⡼⠃
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢻⡟⠋⠉⠉⠉⢙⡟⠙⠋⠛
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿⣿⣟⣿⣿⡟⣠⡟⠀⠀⠀⠀⢀⣿⠀⠀⠀⣠
⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⣿⣿⣿⣿⣯⣼⡿⠀⠀⠀⠀⢠⣾⠃⠀⠇⠀⣿⡆
⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⣴⢿⡏⢀⣶⠀⢠⣿⡇
⣴⢓⣀⡀⢀⣿⡇⠀⣼⣿⣿⣿⣿⣿⠋⠀⠀⣀⣴⣿⣯⡆⢀⣾⠃⠀⣾⣿⡇
⠙⠿⣿⣿⣾⣿⣧⠸⣿⡅⠙⠛⠉⠀⣀⣤⣾⣿⣿⣿⠟⢀⣾⡟⠀⢠⣿⣿⠁
⠀⠀⠘⢿⣿⣿⣿⠀⠙⢿⣶⠶⣾⣿⣿⣿⣿⣿⣿⠋⣠⣾⣿⠁⠀⣸⡟⠃⢰⠃
⠀⠀⠀⠈⣿⣿⡏⠀⣰⣾⠟⣀⣤⣿⣿⣿⣿⠟⢁⣴⣿⣿⠇⠀⢸⣿⡇⠀⡂⢀
⠀⠀⠀⠀⠙⠿⣷⡤⢟⠁⠘⢿⣿⣿⣿⠟⢁⣴⣿⣿⠿⡏⠀⠀⣿⣿⠇⠀⡇⢸
⠀⠀⠀⠀⠀⢀⠛⢡⣾⣷⣄⠀⠙⢿⣁⣴⣿⡿⢃⠘⣦⠀⠀⣾⣿⣿⠀⢸⣧⠀⣶
⠀⠀⠀⠀⠀⠈⢰⣾⣿⣿⣿⣷⣄⠀⠙⣿⡏⠡⡎⢱⠁⠀⣸⣿⣿⡇⠀⢸⣿⠀⢻
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⡀⠈⠛⣧⣠⠇⠀⣰⣿⣿⣿⡇⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠋⠀⠛⢦⡄⠈⠻⣧⣄⢻⣾⣿⠘⠇⠀⠀⢿⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣶⠟⠀⣰⠃⢙⣦⡀⠈⠻⣿⣡⠙⣤⠀⠀⠀⢸⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣁⣀⡾⠁⢠⣿⠋⡹⣧⡀⠈⢿⣦⡟⠀⠀⠀⣸⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠀⣰⣿⠋⡀⠙⢠⡿⢦⡀⠙⢷⡀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⣀⣾⣿⠁⣠⠉⡄⣿⠁⣾⠛⢦⡀⠙⣷⣀⣿⠀⠀⠀⠈⣅
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⡿⠁⣰⣏⣿⢱⡏⢀⠋⠀⢼⠙⢦⡀⠹⢿⡀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠟⢀⣾⣿⣿⠇⣸⠃⣿⠀⠀⡻⠀⠀⠙⢦⡈⠛⣦⡀⢸⣿⡄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⡿⢠⡟⢀⠀⠀⣼⡇⠀⠀⠀⢸⠻⢦⡈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢻⣿⠟⣁⣴⡟⣿⣿⣿⣿⡇⣸⠃⢾⠀⢀⡿⠀⠀⠀⠀⣿⠀⠀⠙⢦⡘⢻⣷
⠀⠀⠀⠀⠀⠀⢾⣯⣾⣿⠟⣸⣿⣿⢹⡟⠀⣿⠀⢸⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⢨⡿⢦⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣿⣿⠃⣿⠃⣼⡏⢀⡏⠀⢸⠃⠀⠀⠀⢀⠃⠀⠀⠀⢸⡇⠸⣿⣦⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⣼⡟⠀⣿⠀⢸⡇⠀⡎⠀⠀⠀⠀⣸⠀⠀⠀⠀⣿⢱⠀⣿⣏⠙⢠⡙⢧⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⢰⣿⠁⢸⡟⠀⡘⠁⢠⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⢸⠇⣿⢸⡄⠀⠙⢦⡙⠂
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣠⣿⡏⠀⣿⡇⢠⡇⠀⣸⡏⠀⠀⠀⠀⢻⠀⠀⠀⠀⡇⢸⠀⡇⢸⡇⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠁⢸⡿⠀⠈⠇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⡇⢸⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇⠀⣾⡇⠀⣴⠀⠀⣿⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢸⡇⠀⡇⠈⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⡇⠀⣿⠀⠀⣿⠀⠀⠀⠀⢀⣿⠀⠀⠀⣰⠸⡇⠀⠁⠀⠹
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠻⣿⣆⢸⣧⠀⣿⠀⠀⠀⠀⠘⢣⣄⠀⠀⠻⣄⠰⣆⠀⢇⠀⢿⡀
⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠀⢼⣿⣿⣯⣀⣼⡆⠀⠀⠀⠀⢀⣼⣷⣀⣀⣙⣶⣿⣆⣸⣧⣄⣻⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣶⣶⣶⣶⣶⣤⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣬⣥⣬⣭⣭⣤⣬⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠟⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠤⢐⣀⣙⣿⣿⣿⣶⣤⣤⣲⣶⣶⣦⣴⡒⠤⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⡀⠴⠚⣩⣭⣿⣿⣿⣏⠻⣿⣿⣷⣾⣯⣽⣛⣻⢷⣦⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣮⣉⠻⢿⡯⠻⠦⡀⠀⠹⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡽⣫⣾⣿⣿⣿⣿⣟⢻⣿⢿⡟⠻⠣⡀⠙⢦⡀⠀⠀⠀⠈⠻⣿⣿⣿
⣿⣿⣟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠹⠟⢻⣿⡟⠛⠀⠉⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⣮⣻⣿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢦⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⡇⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⡆⡼⠁⠀⠀⠀⠘⣧⢹⣄⡀⠀⡀⣀⣴⣦⠈⡄⠀⠀⠀⠀⠀⠀⠘
⣿⠁⠀⠀⠀⠀⠀⢀⡾⠀⢠⠆⠀⠀⢸⣗⡄⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣴⣿⣿⣿⣿⣇⢳⠀⠀⠀⠈⣆
⡟⠀⠀⠀⠀⡞⢠⣿⠃⣠⠏⠀⠀⠀⣿⣿⢗⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣿⡂
⡇⢰⠀⠀⢠⡇⣾⣏⣴⡏⠀⠀⠀⢠⣿⣿⣾⣿⣿⠧⠤⠤⣤⣀⣀⠀⠀⠀⠀⠀⠙⢷⡙⢿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣷⣆
⠀⠘⠀⠀⠰⣿⣿⠙⠿⠁⡀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⠙⠛⠳⠦⠀⠀⠀⠀⠙⣶⡿⠿⢿⣿⣿⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⢠⣿⠉⠈⠒⢀⡇⠀⠀⢸⣿⣿⠟⣿⣶⣾⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣽⣿⡆⠀⠀⣾⣿⣿
⠀⠀⠀⠀⢸⢿⡀⠊⠁⣾⣧⠀⠀⠈⣿⠋⠀⢹⠛⣟⠙⠛⠛⠀⠉⠀⠀⠀⠀⠀⠀⢰⡿⢿⠏⠹⠻⣿⣿⡇⠀⢠⣿⣿⣿
⠀⠀⠀⠀⡆⠘⣧⠀⠀⣿⣿⡀⢂⠀⢹⠀⠀⠈⢧⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢸⣿⣿⡇⣤⣾⣿⣿⣿
⠀⠀⠀⠀⡇⠀⢿⣷⣄⠉⣿⣧⠀⢧⠨⠆⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡷⣽⣿⡿⣿⣿
⠀⠀⠀⠀⣷⠀⢸⣿⣿⡅⠈⢿⣷⡈⣷⣔⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣻⣿⡇⣿⣿
⠀⠀⠀⢠⣿⠀⠀⣿⣿⣇⠀⠈⢿⣧⡸⣏⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⢻⡟⠁⠸⣿
⠀⠀⠀⢸⡏⠀⠀⢸⣿⣿⡄⠀⠈⢿⣷⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣠⣾⣿⣿⡿⣿⡏⢸⡇⠀⠀⢿
⠀⠀⠀⣾⠁⠀⣷⠀⢻⣿⣿⡄⠀⠘⣿⡟⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢁⠓⢸⠁⠀⠁⠀⠀⠘
⠀⠀⢰⡟⠀⢀⣿⣧⡀⠹⣿⣷⠀⠀⢿⣿⡄⠉⠻⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⣡⡎⠀⠘⡆⠀⠀⠆
⢀⡶⣿⠁⢀⣾⣿⣿⣿⣄⠹⣷⣧⠀⠻⣿⣷⠀⠀⠈⠙⢿⣿⣿⣿⣶⣤⣴⣿⣿⣿⣿⣿⣯⣾⣻⡆⠀⠀⠃⠀⠀⡄
⠞⣽⠃⢀⣼⣿⣿⠟⠉⠹⢧⣘⢻⣧⠀⢻⣿⣇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡏⣸⣧⠀⠀⠀⠀⠀⠁⠀⠐
⣰⠏⢀⣾⡿⠋⠁⠀⠀⠀⠀⠉⠳⢿⣧⡀⢻⣿⣆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣹⡟⢸⡿⠀⣿⣿⣦⠀⠀⠀⠀⠀⠀⢿
⡋⣠⣿⣿⣴⣶⣶⣿⣿⣷⣦⣤⣀⠀⠻⣿⣄⣻⣿⣇⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⣿⣿⡏⢂⠀⠀⠀⠀⢰⡜
⣴⣿⣿⡿⠟⠛⠛⠛⠛⠻⢿⣿⣿⣦⣀⠈⢿⣿⣿⣿⣆⠀⠀⠀⠙⢿⣿⣾⣿⣿⣿⣷⢸⡆⠀⣿⣿⣿⡈⣆⠀⠀⠀⠀⢿
⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⢻⣿⣿⣿⣧⠀⠀⠀⠈⠉⢻⣯⣛⢿⣿⣆⠇⠀⢸⣿⣿⣧⠘⣦⠀⠀⠀⠈
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡙⢿⣿⣿⣦⠀⠀⠀⠀⠈⢿⣿⣦⡈⢻⣿⠀⠈⣿⣿⢿⣧⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⣿⣿⣯⣧⠀⠀⠀⠀⠘⣿⣿⣿⣶⣎⠆⠀⠘⣿⡎⣿⣆⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡳⡩⡝⠶⠺⢻⣿⣿⣿⣿⣷⠀⠀⠸⣧⠘⣿⣯⢿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⡿⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢀⡇⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢸⠁⠀⠀⣾⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⢀⡏⠀⠀⢰⠇⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡜⠀⠀⢀⡞⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡰⠁⠀⠀⡼⠁⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠐⠀⠀⢀⡜⠁⠀⢀⡞⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠠⡘⠆⢠⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⠎⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣤⡙⠆⠰⢫⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡐⠁⢀⣾⣿⡏
⠀⠀⠀⠀⠀⢀⡈⠳⢄⠠⠏⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⣸⢃⣴⠾⠿⢷⣶⣄⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠟⠀⠒⠂⠀⠈⠻⣷⣄⠠⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠈⢿⣦⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠹⣷⡈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢀⡶⢶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠈⠻⣌⣿⠀⠀⠀⠀⠀⠀⠀⢐⣿⠏⢀⡾⠁⠸⠁⠠⠞⠉⣩⡿⠛⠳⢶⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⡞⠀⠀⠀⠀⠀⠐⠋⢁⣠⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠏⠤⠔⠊⠀⠀⠀⠀⣀⡠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⠀⠀⠀⣀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⣴⣶⣦⣴⣶⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⣿⠋⠉⠻⠋⠉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣆⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠿⠿⡿⠿⠟⠀⠀⠀⠙⣿⣿⠏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣀⣀⢀⣀⣰⣿⣿⣿⣦⡀⢀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡀⣿⣿⡇⡀⠀⠀⠀⣠⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⠀⠀⣸⣿⣿⣿⣿⡟⠀⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠐⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⢀⢀⠀⠀⢹⣿⣯⡀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⢠⣧⠈⠉⠀⠀⠀⠈⢻⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⡟⠸⣷⣾⣿⣿⣿⣷⡄⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠠⡀⠀⠀⠠⠂⠒⠀⡄⠀⠁⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠁⢰⠒⠂⠒⠀⠀⢂
⠀⠀⠀⠀⠀⠀⢐⠉⠐⠀⠀⠀⠀⠠⠁⠀⡀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⠀⠀⠠⠄⣄⠀⠀⠀⠤⠐⠀⠠⠀⠠⠠⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠄⠀⠃⠀⡐⠁⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠐⠅⣠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠐⠒⠃
⠀⠀⠀⠀⡀⠀⣀⠁⠄⠀⠀⠉⠀⢀⠄⠀⠀⠰⠀⠀⠘⡀⠀⢀⡀⠤⠒⠇⠀⠀⡄⣀⠈⠀⠀⠀⢡⠀⠀⡰⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠂⠀⠀⠀⠀⠑⠀⠂⠁⠀⡄⠀⠀⠀⠑⠂⠈⠀⠀⠀⠀⡀⠀⠀⠁⠀⠃⠒⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡀⠀⠠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁⠀⠀⠀⢀⠀⡄⠒⢤⢀⠀⠀⠀⠀⠀⢐⣰
⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠃⠁⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⢈⢉⠣⡀⠇
⠀⠐⢄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠒⠐⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠈⠀⠀⠀⠢⢀⠂⠀⠀⠀⠀⠐⡄⠂⡀⠀⡡⢄⠀⠀⠀⠀⠀⡠⠤⠄⡀
⠀⠀⠀⠈⠀⠐⠁⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠑⠰⢉⡈⠄⠐⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠀⢀
⠀⠀⠀⠀⠒⣤⠒⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⡎⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠐⠀⠒⠁
⠀⠀⢄⠈⢢⠦⣔⠁⡰⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠠⠄
⠀⠠⠌⣠⠝⠰⢂⣈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⢀⡠⠣⠀⡀
⠀⠀⠀⠠⠄⠓⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢄⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢀⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⢀⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡀⠀⢢⠀⠀⠀⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠈⠀⠀⠀⠀⠀⢀⠤⠄⡀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⢁⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠄
⠀⠀⠀⠐⢄⠀⠁⠀⠀⠀⠀⠢⠀⠀⠠⠊⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠀⡀⠆⠀⠀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠡⠀⠐⠈⠀⠀⠀⠀⠀⠃⠄⠁
⠀⠀⠀⠀⠐⢠⠉⢒⢀⠆⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⢀⠠⠔⠠⠰⣅⠀⠀⠀⠀⡐⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢄⡀⠄
⠀⠀⣀⡀⠁⢠⠄⠐⠢⢀⡈⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡡⠊⠀⠀⠀⠀⠀⠈⠁⢀⠀⠒⠀⠀⡐⠀⠀⠀⠀⠀⡠⡅⠙⢠⠁⠀⡃⠜⠠
⠀⠈⠄⠈⠖⠀⠀⠀⠀⠈⢃⡀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡅⠀⢀⠂⠉⠁⠢⠀⠀⠈⠀⠀⠀⠜⠀⠀⠀⠀⠀⡀⠴⠐⠦⡀⠆⠬⠁⣀⠈⠰
⠀⢀⠦⠠⢆⠀⠤⠀⠴⠀⡈⠀⠢⠈⠀⠀⠔⠁⠀⠂⠁⠀⢀⠃⠀⠐⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠆⠠⠤⢄⡀⡀⢅⡀⠈⠱⠄
⠀⠈⠰⠁⢀⠱⠤⢤⡄⠊⢋⠐⡆⠀⠀⠀⠈⠢⠀⣀⡀⠀⠊⠡⡀⠀⠀⠈⠁⠀⠀⢀⠴⣀⡀⢠⠀⠀⠀⠀⠀⠀⠈⠚⢀⡠⠎⠀⣧⠀⠑⡈⠄
⠀⠀⠀⠀⠁⠼⡀⣤⠈⠦⠜⠀⠀⠀⠀⠥⠂⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠤⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠐⢂⡠⠂⡐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡿⢿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⣻⣟⣿⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⢸⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣤⡄⠀⠀⣿⡿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⠀⠍⢻⣧⠀⠀⣤⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠙⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⡤⠒⠒⠢⠀⠀⠀⠚⢺⡗⠒⠀⠀⠀⠩⣿⣿⣻⠏⠀⠀⠀⣦⣿⣿⣿⣴⠀⠀⠀⠀⠰⢈⣿⣆
⠀⠀⠈⠳⠤⠤⠔⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠸⣯⣀⣹⠇⠀⠀⠀⠀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠁⠉⠉⠁⠉⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⢀⣲⣼⣷⣆⡀⠀⠀⠀⢎⢰⠂⡇⠀⠀⠀⢰⢠⢸⢩⠉⠀⠀⠀⣴⠡⠯⠌⣦⠀⠀⠀⠀⢠⣄
⠀⠈⡹⢻⡿⡍⠁⠀⠀⠑⠮⣽⡭⠔⠁⠀⠀⠘⢲⢻⢳⢲⠀⠀⠀⢄⣓⣤⣋⡸⠀⠀⣼⣧⣾⢿⣦⣿⡀
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠀⠘⠀⠀⠀⠈⠑⠒⠈⠀⠀⠀⠋⠘⠃⠀⠛⠈⠃
⠀⠒⠐⠒⠒⠐⠒⠀⠀⠘⠒⠒⠒⠒⠂⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⠓⠂⠂⠂⠀⠀⠀⠀⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⢀⡴⠒⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠀⠀⠀⣨⣿⡄
⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡁⠀⠾⠇⠀⠀⠀⢠⠐⢗⣏⡧⠀⠀⠀⢀⣿⣿⣿⡀
⠀⠀⢀⠀⡑⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⡀⠛⠋⠉⠀⠀⠀⢀⠈⢛⢛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢠⠀⢒⣶⣂⣶⣶⣲⢲⡯⣽⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣽⣻⠈⠘⠂⠘⠈⠉⠁⠑⠘⠘⠘⢽⢣⢽⡶⢄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡱⡀⠙⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⡠⡀⠀⠠⡾⠙⣯⢏⣲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣍⡓⠉⠀⠀⠀⢠⣴⣦⠀⠆⠊⠓⠋⠛⠻⠳⣟⣠⡄⡀⡀⠀⠘⠡⣯⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠙⠀⠝⠀⢀⣴⡇⠈⠁⠀⠀⠀⠀⣀⢀⣀⣀⠀⠐⠞⠸⢏⡴⡀⠀⠀⠀⣏⡵⠔
⠀⠀⠀⠀⠀⠀⠀⠰⢧⣛⣳⠂⠀⡐⠿⢲⠄⠀⣀⠰⠞⠂⠀⠉⠈⠉⠙⢳⡦⡤⡀⠀⠈⢫⢆⠀⠀⠀⠖⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠈⠃⠀⣰⣝⡢⠀⠀⠴⡍⠀⠀⠠⠄⠐⠒⠒⢦⣀⠀⠈⢳⣄⠀⠈⠕⠂⠀⠀⠀⣲⣡
⠀⠀⠀⠀⠀⠀⢖⣟⠆⠀⠀⣀⣉⠀⠁⠀⠮⠆⠀⡐⠋⠀⠀⠀⠀⠤⣀⠈⠱⠄⠀⠁⡄⠀⠀⢟⡆⠀⠀⠑⣶⡇
⠀⠀⠀⠀⠀⠀⠋⠉⠇⠀⠀⠙⠳⠀⠀⠲⠄⠀⢠⠉⠀⠌⠀⠀⠀⠀⠀⢀⠀⢰⡀⠀⣱⡀⠀⢸⣷⠄⠀⠨⠭⠷⠧
⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠐⠂⡀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⡤⠄⠀⡴⢴⠄⠀⣠⢦⣒⡀
⠀⠀⠀⠀⠀⠀⢰⢾⡁⠁⠀⢨⣳⠀⠀⠐⡦⠀⠈⢆⠀⠑⠀⡀⠀⠀⠔⠀⢀⠊⠀⢠⡍⠀⠀⡾⠆⠀⠀⣩⢦⡏
⠀⠀⠀⠀⠀⠀⠈⡶⣒⡀⠀⠀⢿⢇⡀⠀⠑⣅⡀⠈⠑⠀⣀⠀⠀⡀⠠⡴⠋⠀⣠⡍⠀⠀⣠⣿⠀⠀⢠⣭⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡿⢇⡄⠀⠈⢓⡧⡀⠀⠈⠳⢂⣀⠀⠀⠈⠉⠉⠀⠠⣤⡶⠛⠀⢀⡤⡌⠁⠀⢀⡴⢯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡳⣃⠄⠀⠈⠙⢟⢤⠀⠀⠀⠁⠋⠃⠖⠖⠛⠷⠁⠁⠀⠀⣤⣕⠭⠀⠀⢀⣄⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢥⢞⡄⠀⠀⠀⠉⠠⣾⢄⢤⣄⢀⣀⡀⣀⡀⡄⢤⢻⣞⡎⠀⠀⠀⢤⡳⡝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢴⣻⠂⢀⠀⠀⠀⠘⠙⠋⠸⡟⠛⠚⠽⠟⠀⠀⠀⣀⢠⡰⣜⡧⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣢⠯⠄⡄⣄⡀⠀⡀⠀⠀⢀⣀⡀⡄⠰⡄⢺⡧⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⡷⠧⠏⢀⣇⣿⣽⣸⡧⡷⢽⠄⠙⠈⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠂⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠐⠀⠀⠐⠀⠠
⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠈⠐⠄⡀⠀⠀⢀⠠⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠂⠀⠀⠰⠁⠀⠀⠀⠀⠈⠐⠠⡀⠀⠀⠈⠐⠀⠁⠀⠀⡀⠔⠈⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠄⠀⠀⠀⠀⠰⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⢀⠠⢄⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠠
⠀⠀⠀⠀⠀⢄⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⢀⠄⠂⠀⠀⠀⠈⠂⠄⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠄⠐⠁



⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣷⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣷⣶⣶⣤⣀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣶⣤⡀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⣾⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠟⠿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⢀⣾⣿⣿⡿
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⡿⠟⠋⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡎⠉⣙⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⢁⠀⠘⠿⠃⠰⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠉⠐⠂⣤⠂⣀⡀
⠀⠀⠀⠀⠀⠀⡰⠀⢲⡿⠟⠀⣤⣶⠂⠀⠀⠀⠀⠀⠀⡀⠉⣣⣄⠘⠿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠶⠶⠶⣄⠈⢳⡄⠙⢶⡄
⠀⠀⠀⠀⠀⢠⠁⣀⠂⠀⢉⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⣆⠸⣿⡿⠠⢀⣠⣇⠉⠛⢶⡄⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⣴⣶⡄⠘⣷⡾⠃⢀⣼⠇
⠀⠀⠀⠀⠀⠸⠄⠰⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣄⠀⣀⠀⢠⡿⠿⣷⣶⣤⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠙⠛⠁⣲⡟⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢹⣿⣿⣿⣦⣤⣄⣈⣙⠛⠓⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠿⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⣤⡀⠀⠀⠠⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠲⣶⣶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠄⠀⣀⣁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠒⢀⠀⠈⠛⠉⠁⢠⡀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠉⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠁⠀⠀⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⢀⡈⠹⡿⠃⢠⣾⣷⠄⠈⠻⠀⠀⠀⠀⠿⠋⠀⢤⣾⣷⣤⠄⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⣾⣶⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢁⠀⠀⠙⠉⠀⣀⡀⠀⠀⠀⠀⢠⣀⠀⠸⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠉⠀⠰⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢷⠄⠀⠼⠟⠁⠀⠀⠀⠀⠀⠙⠻⠦⠀⠠⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣦⣤⣀⡀
⠀⠀⠀⠀⣠⣦⡀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠟⠿⠟⠦⡄⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠐⠀⠀⠀⠀⢦⣀
⠀⠀⠀⠀⢿⣿⠃⠀⠀⠛⠋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠃⣤⣶⣄⠠⣿⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⡋⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⢠⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⠙⠛⠋⠀⠈⠀⠀⠀⠀⠐⣿⡷⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠍⠉⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠤⠴⠿⠛⠀⠀⠀⠀⠀⠀⠈⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⠇⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀⠀⠀⠁⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠀⢠⣾⣷⣶⡄⠀⣼⠉⠀⡄
⠀⠀⠀⠀⠀⠀⣠⠶⠲⣶⢦⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠙⢁⡀⠈⠻⠿⠛⣠⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠶⣤⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⣸⠟⢀⣴⣄⠀⣿⠀⠈⠀⢈⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠒⠚⠛⠄⣀⣈⠛⠃⠀⡀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⣀⠈⠙⢋⣠⣤⣤⠤⠒⠀⢀⡀⠈
⠀⠀⠀⠀⠀⢹⣦⡈⠛⠋⠀⡹⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢈⣿⣿⡷⠀⢿⡁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠳⠶⠿⠋⠀⠀⠀⢀⣬⣄⠀⠀⠀⠀⠀⣀⡀⠐⣤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣆⠈⠉⠉⠁⠀⠶⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠁⣀⣀⡀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠐⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠉⡉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠙⢪⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠊⡐⢀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣤⡀⡴⣋⣶⡀⢿⡟⢠⠀⠀⠀⠀⠀⢸⣏⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⣙⡁⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣛⠾⠇⢰⣷⠎⠀⠀⠀⠀⠀⢿⣏⠀⠀⠀⠀⢀⠀⠠⣢⡜⠛⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣠⣶⣦⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⠄⠀⠀⠠⡌⠀⢰⡁⢰⣶⣶⡈⠙⣷⡄⠀⠀⠀⠀⠀⠀⠸⠁⠀⠙⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⡇⠈⠙⠋⠀⣾⡿⠄⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄⠀⠀⣰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣿⣷⡴⠿⢅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣡⣤⣤⣼⣋⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣃⣸⣯⣹⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⢤⣄⠀⢀⡴⠚⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⠻⡞⠶⠆⠀⠈⣷⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡶⠾⣧⠀⠀⠀⠀⠀⠾⢂⡇⠀⠀⠀⠀⢸⡏⠉⠉⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠤⠤⠖⠋⢳⡀⠀⠀⠀⣼⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠞⠃⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠶⠶⠚⠛⠛⠛⠛⠉⠉⠉⠁⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢰⠒⠛⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢰⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⢾⡉⠉⠛⢦⣄⠀⢸⠰⡏⠉⠉⠙⠛⠛⠛⠷⠶⠶⠶⢦⣤⣾⠁⠀⣠⠞⣤⣄⠀⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠘⢦⡸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢻⣤⠞⣡⠞⠁⠈⠳⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠻⡆⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣡⠞⠁⠀⠀⠀⠀⠈⠙⠲⠦⠤⣤⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠟

⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠆
⠃⠀⠀⠀⠀⠀⠒⠂⠠⠤⠄⠀⠀⢀⠀⠀⠀⡀⠀⠀⡀⢆⠀⠀⢀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠄⠀⠀⠀⠀⠁⠈⠡⢀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠
⠧⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠞⣀⣀⠀⠁⠀⠀⡀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠈⠒⣄⠀⠈⠁
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣰⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠈⠃⠀⠀⣀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠐⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⣀⣀⣀⣀⣀⣀⣀⣀⣑⣤⣤⣀⣀⣀⣠⣤⣤⣤⣤⣤⣔⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠋⠁⠀⠀⣿⣿⠀⠀⠈⠙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣿⣿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⢀⣴⣿⣿⠟⠁⠀⣿⣿⠀⠈⠻⣿⣿⣦⡀⠀⠀⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣀⣴⣿⣿⠟⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠲⣶⣷⣆⣀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⡀⠀⠀⣿⣿⠀⠀⢀⣠⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠠⠐⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⢀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠰⠀⠐⠒⠉⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⣀⣀⣄⣀⣠⣀⠀⠐⠂⠂⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀⠀⠀⢈⠔⠒⠃⠈
⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⣠⡆⣀⣒⢀⠄⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⡀⠠⡐⠀⠀⠀⠈⠄⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠂⠀⢈⢠⠟⠀⠀⠉⠉⠓⠺⡄⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠔⠅⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠼⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠐⠀⡼⠴⠀⠠⢬⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⠀⠉⠀⠀⠈⠉⠁⡤⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠴⠦⠆⠀⠔⠋⠁⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⠠⠄⠀⡆⠠⢸⡀⠀⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⢄⡀⠀⠀⡈⠀⡠⠂⢀⡀⠀⠀⠀⠀⠀⠙⢄⡁⠂⠀⣀⡄⢁⠀⡀⠳⣄⠈⠀⠁⡠⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠢⢄⣀⢥⠚⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠆⠙⠒⠊⣩⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠀⠁⠂⣤⠐⠈⢀⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠒⠓⢤⡀⠀⡠⠓⣥⠤⣇⠸
⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠙⠒⠚⠑⠙⠒⠉⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠁⠂⠤⠤⠅⠀⠈⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⡇⠀⠒⠀⢰⠀⠀⡖⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡠⠂⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠊⠈⠀⠁

⠀⠀⣼⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢛⣿⡿⠁⠁⢸⣿⣿⡄
⠀⠀⢿⣿⣿⣿⣿⡿⠤⠾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢡⣾⡟⠀⠀⠀⠀⣿⣿
⠀⠀⠸⣿⣿⣿⣿⢹⣦⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣵⣿⣟⡄⠀⠀⠀⢸⣿⣿⡀
⠀⠀⢀⣼⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣯⣾⣿⣿⣿⠀⣾⣿⣦⣟⣿⣿⡇
⠀⠐⣿⣿⣿⣿⣿⣿⣿⣦⣘⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⢽⣿⣿⡿⢿⣿⣿⡆
⢀⣄⣽⣿⣿⣿⣿⡿⢳⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⢾⣿⣿⣿⣿⣿⣿⡿⠋⠀⢹⡟⠻⠾⠛⠀⠙⣿⣿⠻⠃
⠀⢹⣿⣿⣿⣿⣿⡇⣾⣿⡟⣼⡟⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⣿⣦⣤⡄
⢠⣿⣿⣿⣿⣿⡿⠧⢸⣿⣷⣿⡇⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⣼⠈⡆⠀⠀⠀⠀⠀⠈⠈⢿⣷⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⢹⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡆⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣰⣦⡀
⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠏
⣿⣿⣿⡗⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠁⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣹⣿⣿⣿⣿⣿⡿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⡇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⡇
⢿⣿⣿⣿⣿⣿⣿⣿⣟⡼⢿⣿⣿⣷⣼⣧⣿⣿⣿⣿⣹⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣤⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⡏⠀⢻⡀
⠀⢹⣿⣿⣿⣿⣿⣿⡿⢇⣼⣿⣿⣿⣿⣿⣿⣯⣿⠛⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢹⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢰⠇⠀⢸⣇
⢀⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠸⠀⠀⠀⣿⡄
⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡆⠀⠀⠀⢙⣧
⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢰⠁⠀⠀⠀⢸⡼
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣾⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡼⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠇⠀⠀⠀⠀⠀⢹⠀⡀
⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣬⣰⡀⠀⠀⠀⠀⠀⢸⡆⢁
⠑⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⣦⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡇⠘⡆
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢊⣽⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣱⠀⠧
⢶⣟⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠸⣿⣿⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠄⠃⠀⠐⢠⠇⠄⠀⢀⠀⠄⠀⠀⢸⣀⡁⠀⠀⢐⢐⠀⠀⠀⢹⠏⠀⠀⠈⠲⠀⠀⠀⠨⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠄⣰⠦⠀⠀⢀⠶⠂⠀⠀⠄⠐⠀⠀⠐⠂⢀⠀⠀⠀⢀⣆⠀⠀⠀⢲⢄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠙⠃⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡉⠀⠀⠀⠧⠀⠀⠀⠀⠉⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢁⢩⡅

⠀⠀⡀⣐⠀⠀⠀⡄⣐⠀⠀⠀⣠⠈⠀⠀⠠⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠂
⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢠⢄⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣀⠀⠀⠀⠀⡤⠄⠀⠀⢀⢦
⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠆⠀⠀⠀⠠⢦⠦⠀⠀⠖⣢⠔

⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠤⡀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠤
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠰⠂⠀⠲⠂⠀⠲⠀⠀⠶⠀⠀⠶⠀⠀⠶⠀⠀⠶⠀⠀⠶⠀⠀⠶⠀⠀⠶⠀⠀⠶⠀⠀⠶⠀⠀⠶⠀⠀⠖⠀⠀⠖⠀⠐⠖⠀⠐⠖⠀⠐⠖⠀⠐⠖⠀⠐
⠀⠀⢠⡀⠀⣠⡀⠀⣠⠀⠀⣠⠀⠀⣤⠀⠀⣤⠀⠀⣄⠀⠀⣤⠀⠀⣄⠀⠀⣄⠀⠀⣄⠀⢀⣄⠀⢀⣄⠀⢀⣄⠀⢀⣄⠀⢀⣄⠀⢀⣄⠀⢀⣄⠀⢀⣄
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁
⠘⠃⠀⠚⠂⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠓⠀⠀⠛⠀⠐⠓⠀⠐⠓⠀⠐⠓⠀⠐
⠀⠀⣤⠀⠀⣤⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⣠⡀⠀⣠⡀⠀⣠⠀⠀⣤

⠘⠃⠀⠘⠃⠀⠛⠂⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠐⠛⠀⠀⠛⠀⠐⠓⠀⠘
⠀⠀⣤⠀⠀⣤⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⠀⠀⣤⠀⠀⣤

⠘⠃⠀⠚⠃⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠐⠛⠀⠀⠛⠀⠐⠓⠀⠘
⠀⠀⣤⠀⠀⣤⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢤⠀⠀⣤⠀⠀⣤

⠘⠃⠀⠙⠁⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠈⠛⠀⠀⠋⠀⠈⠋⠀⠘
⠀⠀⣤⠀⠀⣤⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⠄⠀⢤⠀⠀⢤⠄⠀⣤

⠘⠃⠀⠙⠁⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠋⠀⠈⠋⠀⠈⠋⠀⠈⠋⠀⠈
⠀⠀⣤⠀⠀⣤⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⠄⠀⢠⠄⠀⢤⠄⠀⢠⠄⠀⢤⠄⠀⢤⠄⠀⢤⠄⠀⢤

⠘⠃⠀⠙⠁⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠛⠀⠀⠋⠀⠈⠋⠀⠈⠋⠀⠈⠋⠀⠈
⠀⠀⣤⠀⠀⡤⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⡄⠀⢠⠄⠀⢠⠄⠀⢠⠄⠀⢤⠄⠀⢠⠄⠀⢤⠄⠀⢤⠄⠀⢤⠄⠀⢤⠄
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡀
⠘⠁⠀⠙⠁⠀⠙⠀⠀⠙⠀⠀⠙⠀⠀⠙⠀⠀⠙⠀⠀⠛⠀⠀⠛⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠈⠋⠀⠀⠋⠀⠈⠋⠀⠈
⠀⠀⡦⠀⠀⡦⠀⠠⡆⠀⠠⡄⠀⠠⡆⠀⠠⡄⠀⢠⡄⠀⢠⠄⠀⢠⠄⠀⢰⠄⠀⢰⠄⠀⢰⠄⠀⢰⠄⠀⢴⠄⠀⢴⠄⠀⢴⠄⠀⢴⠄⠀⢴⠄⠀⢴⠄
⢀⠀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⣀⠀⠀⣀⠀⠀⣀⠀⠀⣀⠀⠀⣀⠀⠀⣀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡀
⠘⠁⠀⠙⠁⠀⠙⠀⠀⠙⠀⠀⠙⠀⠀⠙⠀⠀⠙⠀⠀⠉⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠈⠋⠀⠈⠋⠀⠈⠋⠀⠈
⠀⠀⡦⠀⠀⡦⠀⠠⡆⠀⠠⡆⠀⠰⡆⠀⠰⠆⠀⢰⡆⠀⢰⡆⠀⢰⠄⠀⢰⠆⠀⢰⠄⠀⢰⠄⠀⢰⠄⠀⢴⠄⠀⢰⠄⠀⢴⠄⠀⢴⠄⠀⢴⠄⠀⢴
⢀⠀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⣀⠀⠀⣀⠀⠀⣀⠀⠀⣀⠀⠀⣀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡀
⠈⠁⠀⠙⠁⠀⠙⠀⠀⠙⠀⠀⠉⠀⠀⠉⠀⠀⠉⠀⠀⠉⠀⠀⠉⠀⠀⠉⠀⠀⠉⠀⠀⠉⠀⠀⠋⠀⠀⠋⠀⠀⠋⠀⠈⠋⠀⠈⠋⠀⠈⠋⠀⠈⠋⠀⠈
⠀⠀⠶⠀⠀⠦⠀⠰⠆⠀⠠⠆⠀⠰⠆⠀⠰⠆⠀⠰⠆⠀⠰⠆⠀⠰⠆⠀⠰⠆⠀⠰⠆⠀⠰⠄⠀⠰⠄⠀⠴⠄⠀⠴⠄⠀⠴⠄⠀⠴⠄⠀⠴⠄⠀⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡴⠶⠻⠛⠷⢶⠄⠡⡶⠾⠛⠛⠶⣦⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⡆⠈⠋⠀⠀⠁⠀⡀⠂⢀⡠⠀⢀⠀⠀⠀⠀⠙⠁⠰⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⢀⠀⠨⣔⢤⠙⠂⠀⠁⢨⡞⢳⡁⠈⠀⠐⠃⡄⣢⠅⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢇⠀⢀⠀⠈⠀⢨⡏⠑⢶⠀⠑⠀⣺⡰⢆⣗⠀⠊⠀⡶⠋⢹⡅⠀⠁⠀⡀⠀⡹⠧
⠀⠀⠀⠀⠀⠀⢀⣴⡶⢇⠀⣰⣠⠌⠁⠀⠄⠘⣧⡩⠼⠛⣛⣉⣩⣭⣭⣍⣉⣛⠛⠦⣏⣼⠂⠠⠀⠈⠠⣄⣆⡀⡸⠶⣦⡀
⠀⠀⠀⠀⠀⣰⡟⠁⠀⢀⡀⢞⡏⢉⡳⣀⡴⢚⣭⣴⠛⠛⢁⣉⢁⣈⣄⣈⢁⠉⠛⠛⣦⣭⡓⢦⣀⢖⡉⢹⠳⢀⡀⠀⠈⢻⣆
⠀⠀⠀⠀⢰⡿⠀⠀⡀⠀⠁⡀⠛⣪⡵⢋⡶⠛⠁⡤⠔⢋⠬⠁⠔⠂⠀⠐⠪⠁⣘⠂⢄⡘⠛⢶⣙⠦⣕⠋⢀⠈⠀⢀⡀⠀⢿⡆
⠀⠀⠀⠐⡈⠉⡀⡀⡀⣀⣀⠉⣰⢋⣼⠋⢁⠔⠉⠞⠀⢈⡄⠠⠖⠀⠒⠂⠠⣄⠀⠡⠠⠜⠢⡈⠙⣧⡙⣆⠈⣀⣀⢀⢀⢀⠉⢁⠂
⠀⠀⢀⡾⠋⠀⠐⢝⢏⠉⣍⣻⢃⣿⠁⡱⠉⡆⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢀⡀⠠⣜⢄⠈⣿⡘⣏⣩⠉⡹⡫⠂⠀⠙⢷⡀
⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⣰⠃⣿⠀⠴⠠⠂⠁⠞⠀⠀⠀⠀⣀⣀⡀⣀⠀⠶⢀⣤⡀⠈⠁⡀⢀⡈⢦⠀⣿⠘⣆⠀⠀⠀⠀⠀⠀⢘⣷
⠀⠀⢿⡄⠀⠠⠄⠀⠉⠀⡟⢸⡇⠠⢆⠄⠀⠞⠀⠀⠈⢷⣿⡿⠿⣿⠈⠛⠛⠟⠛⠁⠀⠀⢉⠀⠘⠰⡆⢸⡇⢿⠀⠉⠀⠠⠄⠀⢰⡿
⠀⠀⠘⠗⠀⣄⣢⠴⢚⡚⡇⣿⣁⡸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣎⡀⢀⣠⡶⠶⠶⣄⠀⠈⠃⠈⠆⡆⢈⣷⢸⢓⡃⠧⣔⡠⠁⠺⠃
⠀⠀⢠⡦⠀⠉⠙⠶⢪⢵⡇⢿⡁⠸⠰⡀⢠⡀⠀⢀⣠⣄⡀⠀⠀⠈⢻⡅⡅⠀⠀⠀⣹⡀⠀⠃⠀⡀⡎⢀⡿⢸⡮⡅⠶⠫⠉⠀⢴⡄
⠀⠀⣾⠃⠀⠘⠒⠀⡠⠀⣷⢸⡇⠸⠆⣄⠠⢡⠐⠛⠛⢿⣿⣄⡀⣀⣼⠇⣷⣄⣀⣴⡿⠀⡴⢀⠀⠱⠀⣨⡇⣾⠀⠄⠀⠒⠃⠀⠘⣷
⠀⠀⢻⡇⠀⠀⠔⠀⢠⡤⠻⡄⣿⠁⠛⡀⡁⠈⢀⡀⠀⠀⠙⠻⠿⠛⠁⠀⠈⠙⠛⠉⠀⠴⠀⠠⠃⠞⢀⣿⢰⠟⢤⡄⠀⠦⠀⠀⢸⡟
⠀⠀⠈⢿⣤⠀⠠⢹⣏⣀⣋⣽⡌⣿⡀⠑⣍⠀⠀⠉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠰⣀⢊⣀⣿⢡⣯⣙⣀⣱⡍⠄⠀⣤⡿⠁
⠀⠀⠀⠠⢁⣀⠁⠁⠁⠉⠈⡀⠹⣎⢻⣄⡈⠲⠔⠂⢀⡀⡙⠂⠀⠆⠀⠶⠂⠉⡁⠠⡔⣠⠞⢁⣠⠟⣱⠏⢀⠀⠁⠈⠈⠈⢀⡈⠄
⠀⠀⠀⠀⠸⣷⠀⠀⠁⠂⡀⠀⣠⢜⡳⣌⠳⣤⡆⠑⠠⠅⢀⡲⠐⠀⠠⠔⢀⡚⡠⠔⠊⢀⣦⠾⣡⢞⡡⣄⠀⢀⠐⠊⠁⠀⣾⠇
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⠀⢢⣇⣈⡵⠉⠳⢮⣙⠻⣦⣤⣄⣀⡉⢙⡉⢁⣍⣀⣦⣴⠟⣋⡵⠞⠉⢮⣅⣸⡔⠀⠁⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠈⠛⠷⡎⠈⠙⠉⢀⡀⠐⠃⠠⡏⣽⠲⣦⢭⣍⣙⣋⣙⣋⣩⡭⣴⠖⣯⢹⠄⠘⠂⢀⡀⠋⠋⠀⢱⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣏⠀⠈⠀⢀⠀⢈⣇⡠⡾⠀⠤⠀⠼⠰⠆⠧⠀⠄⠀⢻⢄⣸⡁⠀⡐⠀⠁⠀⣹⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⢐⠉⠊⣠⠀⠀⡀⢈⢧⡼⡁⢀⠀⠀⣄⠑⠉⡂⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠇⢀⣄⠀⠀⠀⠀⠁⢀⠈⠅⡀⠈⠀⠀⠀⠁⣠⡀⠸⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠻⠶⢶⣶⠶⠞⠃⠘⠳⠶⣶⡶⠶⠟⠋⠀⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠶⣄⡀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣠⣀⣀⣀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡤⠖⠛⠛⠲⣼⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠻⢿⣷⡖⠋⠉⠙⠳⣄
⠀⠀⣾⠁⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢸⡆
⠀⠀⢿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⠳⢤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠁
⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢠⠄
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢠⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⠀⠀⢸
⠳⣤⣾⠀⠀⢸⠀⠀⠀⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⡇⠀⠀⢸⠀⠀⢸⡦⠴
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠹⣄⠀⢸⡀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣼⣀⣀⡼⠁
⣶⣶⠶⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢏⡻⣄⢀⣤⠀⠀⠀⠀⠀⣦⡀⢠⠞⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠸⣦⣀⠀⠀⠀⠀⠀⣀⣀⡏⣬⠓⣝⠛⢿⠀⠀⠀⠀⠀⣿⠙⢃⠰⢣⠘⢒⡟⠀⠀⠀⣄⠀⢀⡤⣾
⠀⠀⠀⠀⠀⢻⣏⡳⣤⣷⠀⠀⢻⠉⠘⢡⣿⣯⠀⢸⡀⠀⠀⠀⠀⡼⠀⢈⣼⣿⠁⡞⠀⠀⠀⢰⠈⠒⢋⣴⡇
⠀⠀⠀⠀⠀⠘⠻⢷⣄⠀⢇⠀⢀⡇⢀⣾⠃⠘⢷⡀⠻⣆⠀⢠⡞⠁⣠⡿⠁⢹⡆⢣⠀⠀⢀⡆⢀⣴⠟⢹
⠀⠀⠀⠀⠀⠀⠠⠀⢹⡇⠈⠳⣼⠁⣼⡇⠀⠀⠈⣷⡄⠈⠳⠋⠀⢰⣿⠀⠀⠀⣿⠘⣤⠔⠉⠰⣿⡁⠀⠘
⠀⠀⠀⠀⠀⠀⠑⣦⣾⠃⠀⠰⡶⠂⠙⣷⡀⠀⢠⡟⠀⢀⣘⣀⠀⠀⢿⣇⢀⣴⠏⢀⣤⡄⠀⠀⢻⣷⠞
⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⢦⣴⡇⢀⠀⠈⠻⣶⡿⠀⠀⠀⢻⠏⠀⠀⠈⢻⡿⠃⠀⠀⣻⣁⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⢀⣤⣝⡓⠦⣤⣀⡉⠀⠀⠀⠈⠁⠀⠰⢆⠜⢀⠐⠆⠀⠀⠀⠀⠠⠘⠋⣉⣩⠴⢛⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠻⠿⣷⣶⣭⣍⣛⡲⠶⠤⣤⣤⣬⣁⣀⣀⣀⣤⣤⡤⠤⠶⠖⣚⣋⣭⣴⣾⣿⣷⣄
⠀⠀⢀⣴⣿⠟⣧⠞⠛⠋⠠⣌⠉⠛⠻⠿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣶⣶⣶⣾⣿⠿⠿⠛⠛⣩⣥⣀⠻⣿⣦⡀
⠀⢠⣿⡿⢃⡼⠻⠀⠀⢀⣤⣹⣦⣀⠀⢸⣿⡆⠀⠈⠉⠉⠉⠉⣭⠉⠉⣠⡤⠤⣴⣂⠠⠞⠽⠆⡸⠀⠘⢿⣷
⠀⠀⢻⣷⡈⠁⢐⣦⠀⣨⣍⣿⠋⢁⣀⣠⣯⣄⢠⣤⡶⠀⠴⣀⠀⡀⠀⠉⢃⠠⠤⡾⠶⠒⠂⠉⢀⣤⡄⣸⣿
⠀⠀⠈⣿⣷⡀⠻⠉⢻⠁⢸⠃⠀⠀⠀⠀⠉⠉⠘⠛⠡⠤⠦⠬⠤⠤⠐⠈⠀⠀⢀⣀⠀⠀⢘⠗⠀⢀⣼⣿⠏
⠀⠀⠀⠸⣿⣇⢠⣾⡀⡀⠀⠉⠀⠀⠀⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆⠀⢿⡿⣧⠀⣮⠖⢫⣿⡿⠃
⠀⠀⠀⠀⢻⣿⡌⡏⠁⠃⣴⣄⠀⠶⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠠⣀⣿⡼⠁⣰⣿⡟⠁
⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠈⠛⠂⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⢠⣾⡟⢹⣷⠀⠀⠀⢸⡿⠁⣼⣿⠟
⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡀⠀⠀⠀⠀⣰⣿⠏⠀⢺⣿⡇⢀⡴⠋⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣴⣿⠏⠀⣠⠀⣽⡶⠟⠂⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠠⠛⠉⠀⠀⠀⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⢀⡔⢠⡆⠀⢀⣴⡦⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢏⡴⢻⡄⣾⣀⠀⡞⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⣾⣤⠘⠃⠘⠛⣰⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠿⠏⠀⣠⣀⠀⠙⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠛⠻⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⣄⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣬⡾⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣤⡄⠀⠀⣰⠛⢡⠿⣀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠨⠈⠀⠀⠀⡿⢀⠁⠸⠍⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⢠⣦⠀⢾⣤⣴⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣤⣀⣈⠁⠀⠀⠀⠙⠔⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿⣿⣶⣶⣦⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⢿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⢰⠛⠱⡀⠀⠀⠀⠀⠀⠔⠀⢢
⠀⢀⣤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡆⠀⢸⠀⠀⢱⠀⠀⠀⠀⡘⠀⠀⢸
⠀⢸⢁⠀⠠⢌⠳⡄⠀⠀⠀⠀⢀⠤⠒⠉⠉⠈⠉⠑⠠⡀⠸⡴⣚⠧⢀⠼⣴⢤⣸⡀⠀⠀⢀⠃⠀⠀⡌
⠀⠈⢮⡈⠁⠀⣹⣼⣆⠀⠀⠰⠃⠀⡀⠀⠀⠀⠈⠀⠀⠈⢆⠹⡀⢀⠀⣀⠈⠓⠺⡇⠀⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠑⣤⠀⣩⡅⠈⢆⠀⡅⠀⡀⠀⣠⠔⠢⣄⠀⠈⠀⠘⡀⠈⢦⠀⠀⠀⠉⢀⡇⠀⠀⡧⠂⠀⡎
⠀⠀⠀⠀⠈⠻⡻⣠⣤⣼⡄⡇⠀⠁⣰⠃⠀⠀⠈⡆⠀⠄⠀⡇⠀⠀⢏⠀⠀⠀⠨⠇⠀⢸⣣⡠⣰⠁
⠀⠀⠀⠀⠀⠀⠙⡟⠿⠟⢃⢸⠀⠀⠜⢦⡀⠀⡰⠁⠂⠆⢠⠁⠀⠀⢸⣤⣤⣍⣸⠀⠀⣇⣈⣍⡏
⢀⡤⡀⠀⠀⠀⠀⢸⡄⢒⢻⠀⢃⠀⠠⠀⠈⠉⠀⠠⠂⢀⠎⠀⠀⠀⠤⠸⠿⠿⠇⠀⠘⠯⠿⡾⠁
⣏⠀⠈⢂⠀⠀⠀⡸⠁⠀⠉⠀⠀⠳⢄⠀⠀⠀⠀⣀⠔⠁⠀⣀⠴⠚⠫⢲⡵⠃⠀⢀⣤⢬⡉
⠱⡀⠀⠀⡱⣲⢾⡅⠀⢠⠁⠀⠀⠀⠀⠉⠉⠈⢉⠔⠒⠊⠉⠀⠀⠀⣠⠞⠀⠀⢰⠁⢿⡄⢸
⠀⠑⠤⢀⡇⢘⠂⢉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠤⠔⠊⠀⠀⠀⠀⠈⠢⠿⠧⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠁⠀⠀⠀⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣇⠀⢀⣀⠀⠀⠀⠀⢀⡀⠀⣸⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡏⡞⠉⠉⢙⡆⢰⡋⠉⠉⢳⢹⠀⠀⠀⠀⠀⠀⣾⠋⠁⠀⠉⣿⣿⠁⠀⠈⢹⣧
⠀⠀⠀⠀⠀⠀⠀⣧⠙⠶⠖⠋⡞⢳⡙⠲⠶⠋⣼⠀⠀⠀⠀⠀⠀⢿⣷⣤⣴⣾⠏⢹⣷⣦⣤⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠈⠛⠛⠁⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⢈⣿⣿⣶⣶⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⣴⣦⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡷⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠉⠻⠟⣠⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠈⠻⣿⣶⣶⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠛⠻⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠛⠉⠁⠀⠉⠙⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⠟⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠁⠀⠀⠈⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣶⣤⡀⠀⢀⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠿⣿⣿⣿⣿⡿⠿⢿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣇⠀⠀⠸⣷⣦⣀⣀⣀⣀⣴⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣀⡀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⠋⠁

⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢸⡟⠛⠛⢻⡇⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⡇⠀⠀⠀⣠⠴⣖⣷⠦⣀⠀⠀⠀⠀⠀⢤⠀⢀⣀⣀
⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⢻⢫⠝⡟⢹⠀⠀⠀⠀⠀⢿⠀⢀⣿⣽⡗
⠀⠀⠀⣧⣤⣤⣼⡇⠀⠀⠀⠘⠛⢛⣿⣿⡛⠛⠃⠀⠀⠀⠈⠛⢼⡤⠗⠊⠀⠀⠀⠀⠀⠉⠀⠀⠈⠗
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣾⣫⣟⣷⣄⠀⠀⠀⠀⠤⢒⣉⣉⡒⠤⠀⠀⠀⠀⠀⢘⣧⡴⢒⣆⠀⠀⠀⢰⠒⠒⡒⠒⣶⣶⡆
⠀⠀⡧⡧⠴⡧⢾⢾⠂⠀⠀⠀⠀⠡⢒⣒⠌⠁⠀⠀⠀⠀⢰⣫⣤⣿⣿⠏⠀⠀⠀⢸⠹⡇⡇⢦⣬⠍⡇
⠀⠀⠙⢿⣾⣷⣻⠟⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⡀⠀⠀⠀⠸⠤⠤⠥⠥⠶⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠠⢲⢶⣀⣶⡲⡄⠀⠀⢠⠒⠉⠀⠉⠉⠐⡄
⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⢠⢀⠵⠤⠤⢎⠙⠀⠀⠀⢰⣟⠋⣾⣿⡦⢏⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡏⣿⣿⡿⠁⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠀⠀⠀⠈⠹⡶⠾⠿⠷⡾⠀⠀⠀⠀⠉⠒⢦⠔⠀⠉
⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⡀⢀⢄
⠀⡔⠀⣲⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠄⢠⠀⠀⠀⠀⣦⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠑⠠⠿⢁⠔⡠⠤⣄⠀⠀⠀⡄⠀⠉⠉⠀⢸⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠮⠥
⠀⠀⠀⠀⠃⠚⢇⡀⠿⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠀⠀⠀⣼⣛⣿⣟⣟⡿⠀⠀⠀⠀⠀⢰⢲⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠘⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⢀⣤⣶⠿⠟⠛⠛⠛⠛⠛⠻⠿⠷⢶⣶⣤⣤⣀⣀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⣀⣠⣤⣴⣶⠶⠿⠿⠟⠛⠛⠛⠛⠿⠷⣶⣤⡀
⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⡿⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠉⠛⠿⢶⣦⣄⡀⢀⣠⣴⣶⠿⠛⠋⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢀⣨⣿⡿⢿⣿⣅⡀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⠈⠙⠻⢷⣦⣄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣠⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄⣸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢀⣴⡾⠟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠛⢷⣦⡀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⡿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⢿⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠙⢿⣦⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⡿⠋⠻⣷⣄
⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠈⠻⢷⣤⡀⠀⣿⡀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣀⣀⣤⣾⠟⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣤⣾⠟⠁⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣶⣦⣄
⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⢹
⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠙⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣤⣾
⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠉⠛⠿⣶⣄⣀⢀⣠⣴⡾⠛⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⣀⣤⣿⡿⠿⣿⣥⣀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣀⣠⣴⣶⠿⠛⠉⠁⠀⠀⠀⠉⠛⠿⢷⣦⣤⣀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⢿⣦⣀⠀⠀⠀⠀⣴⡿⠛⠿⣦⡀⣀⣀⣤⣤⣶⢿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⡿⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿
⠀⠀⠉⠛⠻⠿⠷⢾⣿⠀⠀⠀⣿⡟⠛⠋⠉⠁⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠷⠶⠶⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⢀⣠⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣄⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠙⢾⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣟⠛⠛⠛⠛⠒⠒⠒⠒⠛⠛⣻⠆⠀⠀⣀⢠⠀⠀⠀⠀⠀⠀⡄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣀⣀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡎⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⢩⠿⡇⠃⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⠔⠁⠀⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡴⠊⠠⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⣠⠔⠁⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⣁⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡤⣠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⠤⠤⠤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠦⠤⢤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣴⡋⣿⡆⠀⣾⡇⠀⣾⡇⠀⢾⡆⠀⠀⣴⠀⠀⣿⡆⠀⣶⣆⠀⢾⣏⠳⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⡴⠋⠀⠀⠀⢀⣀⣠⣤⣤⣤⣼⢿⣿⡷⠾⣿⡿⠿⣿⡿⠿⢟⡗⠒⠒⣻⠖⠶⣿⡷⠶⣾⣿⣤⣾⣿⣶⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⢠⣞⡠⠤⠖⠒⠋⠉⠉⠉⠁⠀⠀⣴⠚⣿⡇⠀⣿⡇⠀⢿⡧⠀⢸⡇⠀⠀⢼⠀⠀⣿⠃⠀⢸⣿⠀⣸⣿⠛⢯⡉⠛⠛⠻⠶⢶⡤⣀⡀⠀⠀⠀⢣⡀
⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣃⣀⣙⣃⠀⠘⠁⠀⠈⠁⠀⠀⠘⠀⠀⠙⠀⠀⠙⠋⠀⠻⣃⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⢼⣿
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⠀⠀⢠⣿
⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⣿⣻⡿⣻⡿⠃⣠⡾⠃
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣖⣭⣿⣿⣟⣻⣽⠟⠉⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠈⠲⢶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣶⣾⣿⣽⣿⣿⣿⣿⡿⠟⠋⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠉⠛⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⢀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣀⡀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⢀⣠⡤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⢦⣀⡀⠀⠀⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠁⠀⣀⣀⣠⠴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠲⠶⠶⠶⠀⠶⠶⠒⠒⠛⠛⠋⠁

⠀⢀⡀⠀⢀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢶⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠀⠀⢀⠔⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⣄⡀⠂⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠑⠉⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠐⠀⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈
⠁⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠄⣀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⣀⡄⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠
⠁⣀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⠀⠀⣀⠀⠀⠂⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠀⠀⠒⠂⠀⠂⠀⠀⠠⠄⡀⠀⠂⠀⠀⠀⠀⠀⠠⠄
⡄⠉⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢁⠀⠀⠀⠀⠀⠀⠀⠈⠃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠄⠤⠠⠤
⢠⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠰⠄⠀⠀⠀⡈⠀⠀⠀⣐⠠⡀⠀⢰⡄⡄⢠⢠⠀⡖⠀⠐⠁⠀⠐⠊⠀⠈⠀⠀⢈⠀⠀⢀⢀⡀
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠃⠂⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠃⠄⡀⠀⡉⠠⠀⠀⠈⠁⠁⣉⠈⠉⠁⠀⠀⠣⡀⠀⠀⢀⠔
⠀⠀⠠⠀⠄⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠈⠢⠔⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣀
⠒⢠⠄⢀⠉⠐⠂⠀⠀⠀⠈⠀⠀⣀⣈⠀⠀⠀⠀⠠⠢⠀⠀⠀⠀⠀⠙⠁⠐⠲⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢠⠂⠀⠀⠀⠀⠐⠆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⢀⣀⣠⠈⠔⢸⠆⠀⠀⠀⠀⠀⠀⡁⠆⢉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⣀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⢄⠀⠂⢀⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉
⡀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡂⠀⣴⣤⣤⣤⣤⣤⣶⠀⠀⠠⠖⠜⡳⡴⢄⠀⠀⠘⠆⠀⠀⠀⠰
⣇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡠⠐⠀⡇⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠁⣀⠀⡄⢠⠀⡀⠀⠀⠀⠢⣀⠘
⠈⠀⢸⠀⠀⠀⠄⠀⠀⠀⠀⢜⠉⣀⠅⠀⠀⠀⠀⠀⠒⠀⠤⠀⠀⠀⠀⠀⢀⡀⠀⠀⡂⠀⠀⠀⢀⠀⡉⠉⠡⠀⠀⠂⣀⡀⢄⣌⡠⠀⠀⠀⠀⠀⢸
⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠁⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠉
⠀⠀⣀⣀⢀⠀⠀⠀⢠⠤⠤⢤⠤⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠠⠀⠄⠂⠄⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⡠⠀⠀⠀⢠⡌⠀⠐
⠐⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠒⣮⠀⠀⠀⠁⠈⠀⠁⠁⠀⠀⢐⠀⠐⠒⠀⠘⠡⠀⠀⠀⠀⠀⠀⠠⠋⠂⠀⠀⢀⡼⠃⠚⠁⠀⠀⠀⠀⡇
⠀⠀⠲⠄⠀⠀⠀⠀⠨⠉⠹⠏⠩⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠄⠀⠀⠀⠆⠘⠃⠀⠐⠄⠠⠘⠢⠂⠂⠀⠐⠔⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠠
⣀⣤⣤⣤⣀⠀⠀⠀⠀⡀⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠿⠀⠀⠀⢿⠀⠆⠀⠀⠠⠀⠀⡀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⢰⣲⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⡀
⠀⡆⣖⢰⢨⠓⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠠⠁⠀⠀⠂⡆⠀⢸⠀⠀⠀⠀⠀⠐⢀⠀⠀⠁
⢄⠃⠓⠘⠨⠉⠁⠀⠀⠀⡀⣀⠀⣀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⡀⠲⠆⠐⠀⠀⠀⠀⢘⠂⠃⡀⠚⠀⡂⠀⠀⡁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⡁⡐⢄
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠁⠘
⠠⠀⡶⠉⢫⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡚⠛⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⢼⣉⠀⠀⢖
⡄⢻⡤⠔⠚⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠴⠊
⠘⢦⣈⣓⣂⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⣜⣂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠠



⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠄
⠀⠀⢸⣄⣬⡀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠒⡂⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⡆
⠀⠀⠸⣿⣇⠇⠀⠀⠀⠀⠀⠀⢧⣀⣈⣉⡁⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠳⡅⠀⠜⢀
⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠘⠛⣿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⠄



⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡍⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡘⢢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⠄⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⢔⠂⢹
⠀⠀⠧⠤⠭⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣒⣒⣒⣲⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⣞⠡⠨⠤⢌⣳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣃⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠒⠚⠉⠉⠀⢁⡏⠀⠀⠀⠉⠉⠙⠒⠤⣄⡀⠀⠀⢀⣠⠤⠒⠛⠉⠉⠉⠁⠈⣯⠉⠉⠉⠓⠒⠤⢄⣀⡀
⠀⠀⠀⢀⣠⠤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⡀
⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢄
⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣀⣠⣤⡤⣼⣤⣄⣀⠀⢀⣀⣠⣼⣤⣤⣄⣀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡦⠚⠉⠁⠀⠀⢀⡇⠀⠀⠈⠉⠉⠀⠀⠀⣇⠀⠀⠈⠉⠓⠦⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡰⠋⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠙⢆⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⡜⠁⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠑⢤⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠔⠋⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠓⠲⢤⡀⠀⠀⠀⠀⣠⠴⠒⠋⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⣄⠀⣠⠞⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠙⠦⣄⡀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣀⡬⠟⢧⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡠⠖⠋⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠉⠒⠲⠤⠤⢷⠤⠔⠒⠋⠁⠀⠀⠀⠈⠙⠒⠦⠤⢴⠧⠤⠤⠖⠂⠉⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣠⡤⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠤⣤⣄⣀⡀⠀⢠⡇
⣤⣤⣶⣴⣤⣤⣤⣤⣤⣤⣤⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⢀⣀⣀⣐⡂⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⡤⠠⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⠆⣠⣾⣿⣿⣿⣿⣿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣟⣀⠛⠀⠤⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢿⣿⠿⢛⣥⣾⣿⣿⣿⣿⡙⠿⡧⠴⠾⠿⠟⠉⠉⠟⠛⠛⠉⠉⠀⠀⠀⠀⠩⠿⣏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⠉⢿⣿⢿⣿⣿⠀⢺⣧⣿⣿⣿⡟⠉⠀⢀⠤⠤⠀⣀⡀⠀⠀⠀⢠⣶⣿⣿⣆⣠⣶⡀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⣠⣴⣶⣦⣄⢻⣄⠈⠻⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⣿⣶⣾⣿⣿⠀⠹⣿⡦⠀⢐⣶⣴⣿⣿⣿⣿⣷⣶⣶⣤⣄⠀⠀⠀⠀⠉⠻⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡆⠀⠸⣿⠿⠛⠁⠀⢴⠿⢃⣠⠘⡛⢹⣿⣿⣻⣿⣿⣺⣿⣿⣿⠃⠀⠀⠀⠋⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⣿⠀⣀⠀⢀⣠⣾⣿⣏⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡏⢸⣿⢦⣿⣿⣿⠂⠈⣘⣀⣼⣿⣿⠿⡿⠿⠸⣿⣿⠇⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣯⠀⠙⢿⣾⣿⡿⠟⠛⡛⢛⠻⣟⠛⠻⠶⣶⣿⣧⣄⡀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡿⠋⢠⡄⠈⣿⡟⡀⢀⢑⡀⠈⠉⠈⠀⠀⢀⡠⠙⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨⣧⠀⢿⣠⣾⠟⠀⡄⠀⠈⠛⠀⢂⠁⠄⠀⠀⢰⠃⢼⣦⣉⠉⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⢿⠂⣲⠛⡃⠀⠠⠉⠠⠀⠀⠠⠌⠀⠀⠀⠀⡀⠀⡈⢁⡜⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣶⣗⣶⠊⠏⠐⠀⠀⠀⠠⡀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠲⠆⠀⠀⠀⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⡆⠀⢾⣶⣆⠀⠀⠀⠁⠀⣀⣀⠀⠀⠀⠀⠐⠀⢀⡀⠀⠐⠆⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠠⣿⡇⠀⢸⣿⡿⠑⢄⠀⠀⠐⠈⠛⠷⢦⣤⣤⠄⠲⣾⣿⢶⡀⠀⠀⠙⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⡇⠀⢸⣿⣧⢰⣄⢁⡀⠆⠀⠀⠀⠀⠀⠄⢠⣤⣼⣿⡘⠳⠀⢠⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣏⡀⠁⣸⣿⣟⣁⣙⣾⣷⡿⠟⢢⣀⣀⠀⣀⣨⡟⠛⣿⡇⡀⢠⠃⣜⢸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡯⣿⢾⢿⣿⣿⣿⣿⣿⣿⠷⠄⠬⠿⠟⢻⣿⣿⣿⣾⣿⣿⡇⢠⡈⣿⢠⠶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣹⠿⣥⣿⣾⣌⣿⣿⠿⣿⣿⣿⣿⣿⡷⠶⠶⣮⣭⣝⣿⣿⣿⣥⣇⠀⢖⡋⣨⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠉⠙⠉⠁⠀⠈⡿⠧⠿⡿⠤⠀⠃⢀⣾⣿⣝⣿⣿⣿⢹⣿⠀⡐⠀⢒⡉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡗⠀⢹⣟⣛⣿⠂⢰⣇⢠⣙⠦⠒⠀⣀⣀⠀⠀⠀⠈⠉⠉⠀⠿⢀⡱⣤⣤⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠈⠉⠉⠉⡠⠋⠙⠃⠉⠉⠀⠀⠉⠉⢹⡆⠀⠀⠈⠀⠀⠀⣩⡇⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣶⣶⡖⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⣺⣶⣤⣠⣤⣤⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⡿⠷⠂⠀⠀⠀⠐⠼⢿⣽⣿⠿⠿⡿⢏⡁⠀⢀⡀⠀⠀⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠃⠀⠡

⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠉⣽⡍⠁⠀⠀⠀⠀⠀⡔⣆⠤⠤⣰⠠
⠀⠀⣠⣝⢃⡈⢭⡄⠀⠀⠀⠈⣿⡇⠀⣴⣿⣿⠃⠀⠀⠀⣿⠶⣟⡷⢦⡀⠀⠀⠀⢈⠚⡛⢷⡾⢛⠓
⠀⠀⠻⣧⣬⢡⣬⡟⠀⠀⠀⠀⢻⣇⠐⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠢⣘⡾⢷⣊⠔
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢀⣴⣶⣾⣿⣶⣶⡄⠀⠀⠀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⢺⣧⠀⣦
⠀⠈⠙⠿⠛⠿⠿⠋⠀⠀⠀⠀⠙⣿⣇⢠⣶⡆⠀⠀⠀⠀⢸⡟⢥⣀⢻⡇⠀⠀⠀⠀⢀⣠⣼⣿⣞⣣⠆
⠀⠀⢂⡘⡘⣀⢃⠃⠀⠀⠀⠀⠀⠙⢺⡏⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣿⣇⠀⠀⠀⢰⣿⠟⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠐⢻⡟⠂⠀⠀⠀⠀⠀⠀⣾⡿⢿⣷⠀⠀⠀⠀⡤⠤⠤⠤⠴⢶⡆⠀⠀⠀⠀⣀⣤⣴⣶⢤
⠀⠀⠀⣀⣼⣧⣀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⡇⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠈⠉⠉⢩
⠀⠀⠀⠹⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⣾⡧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣶⣶⣦⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⢸⢃⣤⠀⠀⠀⠀⠀⠖⠊⠠⠔⠀⠄⠀⠀⠀⠀⣴⣾⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⣀⡟⠁⢀⡤
⠀⠀⠀⠙⣦⡊⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢻⣿⣿⠀⠀⠀⠀⣰⣿⠿⠿⠟⠋
⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠛⠁⠈⠛⠋⠀⠀⠀⠘⠛⠃

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠁⠀⠀⠀⠄⠀⠐⠁⠀⡠⡀⢀⠀⠀⢠⠀⠀⢨⠂⠀⠠⠊⠀⠀⠀⠀⠀⢀⠀⣠⣠⣄⠑⠀⠀⠄⢀⠀⢤⠑
⠀⠀⡀⠠⠐⠂⡀⠀⠀⠀⠀⠁⠀⠀⠸⠀⠀⢀⠐⠂⣀⢀⠀⢀⠀⠀⡀⠀⠀⠀⠸⠀⣦⢰⣴⠀⠀⠀⠀⢠⣀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠊⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠄⠐⠂⠄⠀⠀⢀⠀⠐⠂⠀⠀⠀⠀⠠⠐⠂⠠⡀⠀⠀⡠⠐⠒⠠⠀⠀⠀⠀⠀⠒⠀⢀⠀⠀⠀⠄⠒⠂⢄
⠀⠀⠁⠀⢿⡇⠀⠁⠀⠀⢼⡇⠳⠆⠐⠀⠀⢰⣶⣶⣶⠀⠀⠈⠀⢸⣹⠆⠀⠀⠐⠀⣶⣿⣦⠀⠀⠀⢀⣼⣿⣧⡄⠁
⠀⠀⢀⠸⠿⠿⠆⠀⠀⠀⠈⠛⠿⠃⠐⠀⠠⡘⠛⠛⠛⡀⠀⠀⡀⠈⠋⠀⡠⠀⠐⠀⠙⠿⠋⠀⠁⠀⢌⠻⠿⠟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣰⣾⣿⣿⣷⡄⠀⣠⣾⣿⣿⣷⣄⠀⢠⡾⣿⣿⡷⣆⠀⢀⣶⣿⣿⣿⣦⠀⢀⣴⡿⣿⢿⣦⡀⠀⣴⣿⣿⣻⣶⡀
⠀⠀⣿⣿⣿⣿⣿⡷⠀⣿⣿⣿⣿⣿⣿⠀⢿⡿⣿⣿⡿⣿⠀⢸⣿⣿⣿⣿⣿⠇⢸⣿⠿⣿⠯⣿⡇⠰⣿⡿⠿⠷⣿⡇
⠀⠀⠙⠿⢿⡿⠟⠁⠀⠘⠻⢿⡿⠟⠃⠀⠈⠻⢿⡿⠿⠋⠀⠈⠻⠿⣿⠿⠋⠀⠀⠛⠷⣿⠾⠛⠀⠀⠙⠷⣿⡿⠟⠁
⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⣾⣿⠉⢹⣿⡆⠀⣼⡏⢹⣍⣻⣧⠀⣰⡟⠉⠉⠙⣷⠀⢰⣿⡏⡭⣻⣷⡀⢠⣿⠛⠉⢛⣿⡄⠀⣾⠯⠉⠹⢿⡆
⠀⠀⢿⣏⣁⣈⣻⠏⠀⢻⣷⣌⣉⣹⡟⠀⠹⣧⣄⣠⣤⡿⠀⠸⣿⣷⣠⣿⡿⠁⠘⣿⣤⣒⣤⣿⠃⠈⢿⣄⣀⣰⣼⠇
⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠙⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣶⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇⠛⢛⣿⣇⠀⠀⠀⠀⠀⠀⣸⡟⢉⣤⣤⣽⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣥⣶⡄⢿⣿⣿⣤⠆⠀⠀⠰⣤⣿⡄⢾⣿⡉⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣌⣿⣿⠏⠀⠀⠀⠀⠹⣿⣿⣮⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⣿⣷⣄⡀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⢀⣤⣾⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⡷⣄⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣤⠞⣋⢩⡛⢿⡇
⠀⠀⠀⠀⠀⣠⣾⡿⣿⠒⣴⠤⢽⣿⣷⠀⠀⠀⠀⠀⠄⠀⠀⠀⣦⣾⣾⡿⢿⣿⣿⣦⣀⠀⠀⠀⠀⣿⣿⡇⣶⠉⠈⠾⠎⣷⣄
⠀⠀⠀⢠⣾⣿⣿⣧⣙⣃⡐⠿⣻⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣿⣥⣄⣈⣙⣻⣿⣾⣿⣿⣦⠀⠀⣼⣿⣿⣷⡉⠶⢤⠙⣠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡽⣿⣾⠉⢻⡇⠀⠈⠙⠻⢿⣿⠛⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠹⠿⠿⠿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣷⣤⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣬⣿⣶⣤⣀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⣰⠂⠀⠀⠀⠀⠀⣀⣤⣶⣿⣥⣀⣀⡀
⠀⠀⠀⠐⢶⣿⣿⡿⠿⢙⣉⠛⣿⣿⣿⡟⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠹⠿⠋⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣟⠛⢃⣙⠛⢻⣿⣿⡶⠂
⠀⠀⠀⠀⠀⠻⣿⣟⣂⠿⠿⠂⣽⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠀⢿⣿⠂⢾⣿⠏
⠀⠀⠀⠀⠀⠀⢸⣿⣯⣶⣤⣿⠟⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠻⣷⣆⣠⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠸⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⢿⡿⠛⣿⣿⡄⠀⠀⠀⠀⢠⣿⣿⣽⣻⣿⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⣿⠋⢼⡽⣿⠻⠄⠀⠀⠠⠿⣿⣿⣿⠷⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⠌⢰⢋⣵⡟⠀⠀⠀⠀⠀⠀⢻⣿⣣⡘⢣⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⣷⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠑⢒⣤⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣷⣄⠀⠀⣀⣀⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣵⡆⣄⠀⢈⢻⡇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣿⡏⠂⡝⠻⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠂⣼⡃⠀⢀⣠⠐⢰⣟⣢⡀⠀⢀⡟⠀⠀⣀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢐⡻⣿⠋⠐⣻⠟⠁⢴⠾⡇⠀⠠⣘⡋⡉⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⢻⡞⢻⡇⠄⣰⣿⢂⡦⣾⣰⠃⠀⠀⠈⣉⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣻⣾⣿⣤⠍⠀⠛⠛⠙⠰⣿⣿⣧⠀⠀⠀⠉⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠏⠹⠄⢙⠀⠀⡀⠀⠀⠈⣿⣿⣿⣷⣄⠀⢄⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⣤⣼⣤⠀⠀⠀⠀⣾⣿⣿⡿⣿⣿⢷⣄⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣯⣾⠃⣤⡼⠟⠀⠁⠂⠀⢿⣿⣿⣷⡿⠁⠀⠙⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠋⡻⣆⣽⢛⣿⣡⠊⠉⠀⠐⠀⢢⡘⢿⢳⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡭⠒⠋⠙⠁⠊⣽⣏⡀⠀⠈⠀⣬⣿⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣦⣀⣀⢙⣶⣾⣿⣿⣿⡇⠀⠀⢈⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠉⠈⢽⢿⡟⢻⣿⠿⢿⣜⣿⣿⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠺⡏⠋⠀⣤⡄⠀⠉⠀⠀⠀⢉⠈⢿⡄⠀⠀⢸⣿⡿⠀⠠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡦⠋⢁⣤⣴⣶⣦⠀⢺⡗⠂⣹⠄⠀⣾⠟⠁⠀⢀⣭⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⣾⠿⣥⣿⠟⠀⢲⣾⣿⠉⠀⣿⣇⠀⡀⢤⣁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⠀⢿⠃⠀⠀⠘⡁⠁⢿⡿⢧⠠⣼⣿⣦⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠃⠀⠀⠀⠀⣀⡉⠀⢸⣷⠈⣥⣿⣏⣻⡿⠿⢆⣀⠀⣀⣀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⢸⣾⣟⠁⠈⢹⠀⠈⠉⠉⠁⠀⢻⣿⣿⣷⣿⣿⡿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢦⣶⣿⣿⣿⣧⣄⠸⠀⢈⠉⠙⠉⠈⡉⢹⠋⢉⣽⢟⡹⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣄⡙⢿⠿⣗⣻⣿⣟⠛⠻⡛⣿⣿⢉⠉⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢍⡻⠏⠉⠿⢿⣿⣛⠛⠛⣿⣿⠀⠀⢻⡿⠿⣯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠒⠤⣒⠲⢶⣼⣿⣿⣀⡄⠑⢋⣀⣴⠟⣿⢧⡥⠟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⣀⠂⠄⣀⡁⠈⢽⣵⣿⣿⠆⣴⡿⢿⡟⠰⡷⣮⣄⠠⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠀⠀⠀⠉⠉⠀⢀⠠⣽⣷⣟⡿⢛⣯⣥⣼⣿⣷⠆⠙⠛⠻⠿⠁⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠹⠷⠿⠿⠿⠗⠚⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠇⠆⠐⡱⠄⠆⠐⢠⠀⡴⠀⠁⠀⠈⣽⠯⠀⠠⢊⠦⠰⠠⠺⢁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⢦⣄⣶⣶⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣷⣶⣤⡈⠛⢿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠋⠀⠀⠀⠹⢹⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⠿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣁⢟⡽⠃⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⠈⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣷⣄⡀⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⠘⡟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢐⢃⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣼⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⢁⠜⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣦⣄⣠⣀⣀⣤⣾⣿⣿⠟⠀⠀⠀⠀⠠⢀⣀⣐⡢⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠟⠁



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

⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠈⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡛⠁⠀⠀⠀⠀⠀⠛⣿⣟⠋
⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⡄⠀⠀⢀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣦⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢣⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣧
⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣭⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⠎⣀⣀⣀⠤⠤⠶⣚⡉⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠛⠛⠉⠽⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠆