White christmas Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣠⡿⠁⠈⢷⡀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠁⠀⢀⡄⠱⡄⢀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⠟⠀⠀⢸⡇⠈⡇⠀⠀⡗⠀⠀⠀⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣛⠛⠉⠀⠀⠀⢸⡇⠀⣿⠀⢰⠇⡄⠠⡀⠦⡀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⠤⠀⢄⢀⠀⠀⠈⠏⠀⠀⠀⠸⡀⠓⠀⠧⠀⠇⠙⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡼⠁⠀⠀⣸⡇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⢸⠃⠄⠀⡀⠀⠉⠀⠀⢠⠀⠀⠈⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠓⠃⢸⡇⢶⠀⠇⡄⣨⠰⣇⣸⠂⣶⢰⡄⡇⣰⢀⡄⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡴⠙⠃⠾⠈⠸⠇⠙⠂⡹⠻⠎⠛⠘⠸⠇⠉⠉⠧⠀⡽
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠒⠀⠀⣴⠀⠀⠀⠀⠀⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠐⣹⣏⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⣺⡃⠀⠀⣼⣿⠄⠀⠀⠀⠀⠀⠀⠓⠀⢀⣼⣄
⠀⠀⠀⠀⠀⠠⠟⠁⠀⠀⠀⠜⣹⣏⠢⡀⠀⠀⢀⣿⣧⠀⠀⢀⢼⢤⠀⠀⣼⣿⡄⠀⠀⠀⣤⠀⠀⠀⢀⣤⣿⣷⣤⡀
⠀⠀⠀⠀⠀⣠⢤⠤⠀⠀⢀⠴⣻⣟⢧⡀⠀⢀⣾⣿⣿⡆⠀⠠⢺⡣⠄⠈⣹⣿⡍⠀⠀⢰⡿⡆⠀⠀⣀⣴⣿⣿⣤⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠔⣻⣟⢦⡈⠀⣼⣿⣿⣿⣿⡄⠒⢹⡑⠆⠼⢿⣿⠛⠂⠀⣾⣩⣿⠀⢀⣠⣼⣿⣷⠤⠤
⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⢀⠼⠋⡟⢦⠉⣸⣿⣿⣿⣿⣿⣷⡐⢩⠙⠀⢠⣿⣿⣧⡀⣸⣿⡣⣿⣇⠉⢉⣽⣿⣿⡷⠒⠂
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠇⠀⠀⠉⠉⠉⢹⠉⠉⠉⠀⠸⠀⠀⠛⠉⡏⠉⠁⠛⠛⠛⠛⠛⠀⠛⠉⢹⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠱⠎⡁⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣷⣶⠂⠀⠀⠀⠀⠀⠄⠀⡈⠴⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠁⠙⠻⢦⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠰⠀⡀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠶⠀⠀⠰⠄⢀⠈⠀⠀⠂⠀⠈⢀⠀⠰⠀⠀⠰⠆⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⢰⣷⡀⠀⠘⠀⠀⠀⠃⠀⠁⠘⠀⠀⢰⣷⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠈⠁⠄⠀⠀⢾⣿⡇⠀⠀⠐⠄⢀⢁⠀⠔⠀⠀⠀⣿⣿⡆⠀⠈⠀⠉⠀⠀⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠆⠄⠰⠐⠀⠆⠀⠆⠀⠀⠰⠰⠀⠀⠆⠂⠀⠆⠀⠂⠀⠀⠆⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⡀⠀⠀⡀⢄⠠⢤⠉⠆⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠎⢁⠴⢀⠄⣀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠼⢀⡵⢘⡆⣹⠀⡅⠐⠁⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠘⠀⠇⢰⡁⢞⠱⣅⠘⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⠈⠀⠁⠀⠀⢀⠆⠀⠀⢀⠀⠀⡆⢴⣶⠀⡆⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠈⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⠀⠀⢀⠁⠁⢀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠈⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠋⠀⠀⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⢄⠀⠉⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠰⠘⠀⠀⠀⠀⠘⠃⢰⣶⠀⠛⠀⠐⠀⠀⠈⠐⠠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠆⠄⠄⠂⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⢖⠒⠒⠂⢒⠐⢒⡶⠆⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠀⢀⠀⠀⠈⠉⠀⠒⠻⣑⠭⠀⢀⣀⠀⡀⠀⠉⡒⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢁⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⠂⠀⠀⣰⠆⠀⠀⠉⠲⣄⠀⠀⢸⡗⠁⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⠆⢀⠁⠀⣛⣛⣬⣬⣭⣭⣭⣥⣼⣒⡀⠤⢈⡃⠀⠜⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠀⠀⢠⡴⢓⣨⣴⠾⠋⠉⠉⠀⠀⠀⠀⠀⠉⡉⠛⠛⠻⣶⣮⣔⡠⡀⠀⠀⠰⠀⠘⠉⠙⢦
⠀⠀⠀⠀⠀⠀⡠⡁⠀⠀⠀⣠⠞⢃⢜⣥⣾⡟⠉⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣦⣕⠄⡀⢀⡀⠄
⠀⠀⠀⠀⢀⠌⠀⠉⠩⡍⠀⢀⢜⣵⣿⣿⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣔⢾⠟⠀⠱⢆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠘⡤⣢⣾⣿⣿⡿⣤⣾⡿⠃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠹⣿⣿⣿⣧⡹⠿⡀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⡰⣱⣿⣿⣿⣿⣵⣿⡍⠀⠀⠀⠻⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢳⡀⠀⢿⣿⣿⣿⣷⡐⡀⢡
⠀⠀⢐⠀⠀⠀⠀⠀⠠⢱⣿⣿⣿⡿⣿⣿⣿⣷⣦⣶⡀⢀⣹⡆⠀⠀⢀⣤⠶⠚⠛⠉⢹⠿⠷⢶⣮⣇⡀⢸⣿⣿⣿⣿⣿⡘⠻
⠀⠀⡎⠀⠀⠠⡆⠀⢣⣿⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣷⣿⣿⣧⣴⣾⡥⠄⠀⠀⠀⢠⣠⡤⠤⠈⢻⡟⢿⣿⣿⣿⣿⣿⣿⣧⢹
⠀⢠⠀⠀⠀⠉⠀⣤⢸⣿⣿⣿⣿⣷⣗⢹⣿⣿⣿⣿⣿⣿⡟⠛⢿⠃⢀⡤⢤⣷⡀⠙⠇⠀⠀⠀⢈⣉⡉⠻⠿⢿⣿⣿⣿⣿⡈
⠀⠸⠃⠀⣦⠀⢀⠇⣿⣿⣿⣿⣿⣿⣿⡿⣧⣿⣿⣿⣿⣿⣷⡀⠘⠛⠉⠀⠀⠙⣿⡄⠀⠀⡆⠀⠊⠐⠀⠄⠀⠈⠙⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡌⠁⣿⣿⣿⣿⠿⠿⠫⠟⠐⠈⢺⣿⣿⡏⠻⢿⣿⡇⠢⠄⣀⡴⣁⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠸⠀⡀⠀⠘⠁⢠⢻⣿⣿⣿⣢⡐⠀⠀⣠⣶⣿⣿⣿⠃⠀⠀⠈⠀⠀⡬⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⠀⠀⠀⠀⢘⣿⠃
⠀⠀⡆⢇⠀⠀⠀⠀⠸⣿⣿⡟⡼⠁⠀⠰⠿⡿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢤⣤⠖⠛⠛⠉⠙⠁⠀⠀⠀⠀⠀⣿⣰
⠀⠀⢐⠀⢷⡀⠀⢀⣠⣿⣿⠋⠄⠂⠀⠀⠀⣀⢸⣿⣿⣷⡄⠀⠀⢀⢰⣿⣿⣿⡋⠻⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠈⠆⠀⢙⣴⣿⣿⣿⠟⠀⠀⠀⠉⠀⠀⠀⡀⢻⣇⢍⠟⣷⠀⡈⡀⠐⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣴⣾⡿⢟⠁⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⣴⣿⡟⣡⠏⠐⠀⠳⠀⠀⠀⠀⢀⠀⢀⣋⣠⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⣾⡟⠟⠀⡉⠀⠁⠀⠀⠀⠀⠀⠰⠀⣠⣾⣿⣿⣿⣧⣴⣿⠃⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣦⠀⡆⠙⠀⠀⠀⠀⢠⡴⠂⠀⠀⠙⣿⣿⣽⣿⣿⣿⣶⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⢺⠀⠀⠀⠀⠀⠀⠀⠌⠐⠊⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣿⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⡷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣷⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⢏⡟⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⠸⡗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⠺⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣴⣧⠗⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⡁⣔
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡙⠺⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡤⣀⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢺⡿⢿⡿⠞⠿⢦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⢝⠫⠖⣀⢚⣛⡀
⠀⠀⠀⠀⠀⠀⡈⠦⠤⠌⣀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⡇⣉⣤⣾⣶⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⡓⣻⡇⢌⠉⠻⣭⡄
⠀⠀⠀⠀⠀⣀⠛⠲⠀⠤⠁⠐⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣷⡟⣿⣟⡻⣷⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⢉⣵⡖⢹⣄⠻⠆⠖
⠀⠀⠀⠀⢠⠈⠁⠈⠘⠃⠁⠈⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣽⣿⣷⣿⣿⣿⣿⠃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣋⣋⢉⣁⣠⠠⠀⢡⠚⠋
⠀⠀⠀⢀⡀⠙⠓⠶⠒⡄⠙⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⣻⣿⣿⣿⣿⣿⡿⣆⢀⡀⠀⠀⠀⠀⠀⠀⠈⠋⢶⢁⣉⠁⠐⠒⠳⠿⠓⢚⡁
⠀⠀⠀⢨⠇⠠⠄⠰⠶⠰⠆⠀⠀⠞⠀⡀⠀⠀⠀⠀⠀⠀⢤⣤⡤⣭⣿⠋⠵⣿⢩⣯⢹⣟⡟⢋⢠⠀⠀⠀⠀⠀⠀⣫⠖⣩⢿⣷⠞⣽⣄⠑⠎⠞⣁⢀⣀⡀
⠀⠀⠠⠀⠀⢠⢀⠀⠀⠀⠀⠐⠴⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢛⣷⣿⣶⣿⣸⣿⣶⣿⣻⡿⠉⠀⠀⠀⠀⠀⠰⠭⢙⠃⣘⡋⣀⡅⠉⢴⠒⣈⡄⠉
⠀⠀⢤⡀⠀⠀⠈⠁⠒⠀⠐⡀⠀⠄⠄⠂⡠⠀⠀⠀⠀⠘⢦⣢⡼⢟⣭⠟⢩⣿⡎⣿⣷⣿⣿⣤⣆⣤⡖⠀⠀⣈⣰⣠⡆⣀⡘⠉⠹⠗⠂⠀⠜⠣⣿
⠀⠰⢌⠀⠃⠒⠂⠰⠂⠀⠂⠉⠀⡀⠀⠀⠀⠓⠀⠀⠀⣀⣄⣈⣴⣿⣿⢿⣺⣿⣿⣉⣹⣿⠦⣟⡿⣍⠀⠀⠠⣌⣛⣻⡍⢩⣥⡀⣶⡇⢒⡶⣦⡀⣰⢶⣆⠠⠄
⠀⠀⠀⠀⠀⠉⠉⠐⠐⠒⠘⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣹⣿⣏⣤⣿⣿⠻⣿⣷⣲⣦⢼⡿⠒⠋⠀⠀⠀⠉⠋⠀⠃⠺⠞⠟⠃⠂⠀⠈⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠈⠙⠷⠜⠛⠛⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠐⠀⠀⠀⠂
⠀⠀⢀⠨⠄⢠⠀⠀⠀⠆⠀⠀⠀⠄⠉⠀⠀⠀⠐⠀⠐⡂⠀⠢⡀⠀⠁⡀⠂⠀⠠⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠂⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠈⠠⠀⢀⠘⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠕⠀⠠
⠀⠀⠐⠀⠀⠄⠁⠈⠀⠁⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⢐⣤⣰⢄⣤⢐⣂⣤⣤⣄⠤⣀⣠⣄⣀⢀⡄⣀⣠⣴⣤⣦⠖⣰⣶⣦⡤⣂⡴⣶⢦⢀⠂⣤⣬⣤⣆⣀⠤⣀⢀⡀
⠀⠹⠟⡵⢫⣴⡿⠛⠻⠯⠠⠾⠛⠿⡟⡿⢋⣾⠾⠟⠛⠉⠋⠚⠉⠍⠙⠾⠾⢿⠛⠟⠣⠾⠍⠯⠷⠿⡟⠾⣿⠼⠛⠒⠂
⠀⠈⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠀⠀⠀⢠⠀⠀⠀⠃⡀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠘⠀⠐⠀⠀⠀⢠⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠨⠂⠀⠃⠀⠰⠆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⣀
⠀⠀⠉⠁⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠃
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠲⠀⢴⠆⠀⡄⠀⣴⠄⠠⠤⠀⠺⠇⠀⠉⠀⠀⠀⠙⠀⠤⠀⣀⡀⢀⡀⢠⣤⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⢤⣠⠗⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣮⣉⣩⣉⣭⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣤⣶⣤⣾⡿⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣄⢉⣁⡈⣉⣤⢒⡹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢛⡛⢋⣭⡥⠂⠀⠀⠀⠀⠀⠀⠀⠈⢉⡋⠳⠛⠚⣁⣠⣤⠂⠀⠀⠀⣠⠄⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠋⠁⠀⠀⠀⠀⠠⠀⠄⢀⣄
⠀⠀⠀⠀⠘⠋⠙⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢻⠶⠒⠒⣋⠩⠄⡀
⠀⠀⠌⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣤⣀⣈⣀⣠⠴⢟⡁
⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠄⠨⠄⠐⠈⠁
⠀⠘⠂⠀⠀⠀⠀⠀⠀⡹⠁⠀⠀⠀⠀⠀⠀⣀⠉⠀⠙⠁⡈⢁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠈⠙⠢⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⢷⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦
⠀⠀⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⡄
⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠺⠿⡿⢟⣃⡀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣟⠂
⠀⠀⠀⢀⣿⣷⣿⣇⠀⠀⠀⠀⠀⠸⠿⠿⢿⠿⣟⣗⠀⠀⠀⠀⠀⠀⠀⠦⠉⠀⠄⠀⠀⠀⠀⠀⢘⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⣇⠀⠀⠀⠀⡌⠀⠂⠠⠀⠈⠀⠀⠀⠀⠐⣻⣿⣿⣿⡷⠄⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣏⠙
⠀⠀⢰⡿⠿⣿⣿⣿⣿⠀⠀⠀⢰⡾⣭⣽⣽⣷⠾⣿⠿⠇⠀⠀⠠⠀⠀⠀⠀⠀⠐⠡⠀⠀⠀⠠⢿⢿⣿⠿⡷⠀⠀⠀⠀⠠⠶⠿⠿⣾⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠘⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠙⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⠋⡄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣭⠂⠀⠀⡀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⣴⡌⠀⢀⣾⡷⢧⡀
⠀⠀⠀⠀⠀⣺⣁⠀⠀⠀⠀⢀⣋⠉⠀⠀⢠⡀⠀⠀⠜⠉⡀⠀⠀⢠⡒⡄⠙⠁⠀⣹⣿⡶⠉⠛
⠀⠀⠀⠀⣠⡾⠉⠀⠀⠀⠠⠛⠃⠀⣀⠀⣪⢡⡄⣀⠶⠈⠃⢰⠀⣠⡗⣧⡀⠁⠈⡿⢋⣠⠴⠛⡇⠠⡆
⠀⠀⠔⣰⣿⠗⣂⠀⣧⡀⢰⠿⠀⢠⣼⠀⣿⢼⡁⠋⢀⣴⡷⣴⣏⣁⣀⣀⣬⠷⠆⠴⢻⠄⢀⠤⠁⣰⢡⡄
⠀⠰⡐⣿⡉⡖⢀⡈⠀⠕⠆⠀⠻⠆⠉⠰⠁⠈⣧⠀⠘⠈⠰⣯⣤⣠⣄⣤⣤⣾⡷⠈⢼⠏⢩⠀⠀⡍⠀⢭
⠀⠀⠃⣿⣷⣭⠦⠦⢖⠌⠀⠀⢠⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠻⣞⣛⣚⣛⣋⣹⠇⠀⠀⠐⠀⠀⣰⠦⠀⠰⢧⡀
⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⢴⣿⣷⡖⢂⠀⠀⢸⠁⠐⠀⠂⠀⠈⠙⠀⠀⠊⠁⠀⠀⠀⠀⠀⠚⠿⠿⢿⠿⠿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠓⠈⢧⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠄⠀⠘⠘⠀⢰⠁⠘⠀⠘⠉⠃⠀⠀⠀⠘⠋⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣌⠳⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠾⢷⡿⢿⣿⣷⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠋⢁⡾⢿⣿⣾⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣅⢙⢴⣿⡿⠞⣩⠗⣷⡾⠁⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠛⢟⡿⠟⣨⣽⣍⡉⣛⣤⣶⣖⣡⠃⢀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠟⢖⠉⣀⣰⣿⣿⣤⣩⣵⣦⣤⣻⡿⣿⠿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⡴⢂⣠⣼⣷⣿⣿⣾⣿⣿⠄⠼⢯⣹⡻⢿⠷⡾⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠻⣿⡿⠟⢫⡿⢿⡽⠿⠷⣶⣦⣄⣸⡃⡀⢉⣭⣬⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠷⠀⠰⡌⠉⣠⣤⠿⢓⠾⣿⣒⠒⠊⢿⣭⡈⠛⠻⠿⠥⠷⢀⣠⡄
⠀⠀⠀⡰⣰⣶⣤⠀⠻⠴⢿⡦⣰⣿⠺⢋⣠⣶⢽⣦⠈⠻⣿⡀⠀⡜⢻⠿⣶⣼⣶⠟⠋⠉⠁
⠀⠀⠀⢻⠿⣱⠃⣀⣀⠀⢈⣿⠟⢁⣀⣽⣷⣟⣹⣿⣀⠀⠸⢿⣷⣬⣭⣄⠈⠛⠤⣀⠀⢀⣴⣄⣿
⠀⠀⠀⠏⢠⢳⣎⢿⣿⡗⠋⢁⣴⣦⢿⣿⣧⣤⣠⣿⣿⣶⡦⢰⣬⡟⠺⡁⠿⠛⠶⣦⣭⣽⣿⠛⡿⠤
⠀⠀⠀⣇⢺⣀⢿⡜⠟⠛⣘⣟⣩⣬⣿⣿⣿⡟⠉⠀⠀⢹⡇⣿⣿⣇⣞⣥⣦⣄⠀⡏⡞⣻⠇⢠⠛⠁
⠀⢀⣴⣿⠀⢳⣾⣴⣾⠛⢻⣾⣿⡏⠀⠸⣻⣇⡼⠛⠻⡖⠤⠤⣽⣿⣿⠁⠀⢻⣦⠧⢛⣫⣮⢏⣀
⠀⠀⠺⠟⠣⣼⣻⠟⠋⡀⢰⣿⣿⣄⢀⢰⡇⢽⣇⠠⢸⣆⡀⠐⢿⣿⠙⡀⢃⠛⣿⣯⡽⠛⠳⣦⢤⠐
⠀⠀⠀⣰⢲⠙⢟⠻⢿⣥⠏⣩⣏⠈⠑⠚⠁⣸⣱⣤⡴⠷⣠⡖⢾⣿⣷⣦⣅⢖⠁⠙⢂⠧⠄
⠀⠀⠀⡧⢈⢤⣼⣶⣤⢰⢬⣉⠛⠳⢦⣴⣶⣿⣿⣿⣿⣴⠅⣴⠾⠿⠟⠛⠉⠀⠀⠀⣸⣷⠄⢠
⠀⠀⢠⢻⡇⠸⡻⠏⠁⢸⣸⣿⢠⣀⠀⡀⠀⠉⢡⠰⢰⠀⠀⠀⢀⢠⣄⣠⡄⣤⢢⢀⡿⢡⠄⠊
⠀⠀⢸⠋⢳⣰⠑⠦⢤⣀⡈⠉⠃⠸⠀⠇⠯⡇⠸⠀⠸⠽⠼⠸⠏⠞⠒⠛⠋⢉⡀⢼⢀⡞
⠀⠀⠸⣆⡿⠁⠀⠀⠀⠀⠈⠉⠙⠒⠤⣀⢤⣀⣀⣀⣀⣀⡀⠤⠴⠤⠐⠊⠉⠀⠀⠀⠉
⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠗⣌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢦⡘⢷⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣳⣿⣿⣄⠹⣆⠻⣮⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢏⣴⣿⡿⣇⠈⢦⡘⢷⣌⡪⡙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣻⠟⣡⡞⣽⡿⣡⡏⢧⣀⠙⣦⣙⢍⠪⣷⣍⠛⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⣚⡵⡾⢋⣾⢟⣵⠏⣧⠀⠹⣷⢌⠻⣗⢷⣤⡉⠛⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣡⢞⣵⡿⣫⣾⠏⣰⣏⠱⣄⠈⠳⡙⠮⣓⢬⡙⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠷⣫⣯⢞⡽⢃⣴⠋⡟⢷⣌⠳⣦⣈⠢⣈⠑⠻⢶⣤⡉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⣾⠟⣡⠎⣰⡿⢃⡞⡷⣄⠹⣷⣌⡻⣗⠮⣓⢤⡀⠈⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⣡⠞⣡⣾⠏⣴⡟⣰⡗⢌⢷⣄⡙⢿⡾⢷⣮⣙⠻⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣾⠵⣫⠟⣡⢾⢞⡴⢹⡇⠀⠙⢿⣿⡲⢌⣓⢭⣙⠻⠶⣦⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⡤⣚⡥⢚⡕⣡⠊⠀⢸⠃⠀⠀⠀⠈⠛⠷⣬⡉⠛⠻⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠋⢁⡴⣫⠞⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠗⠊⠁⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠐⣷⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠡⢠
⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠑⠈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢈⠃⠀⠀⠉⠉⠀⠀⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠃⠀⠀⠀⠔⠈⠀⠀⠀⠀⠨⠤⠀⠀⠀⠀⡀⠀⠮⠌⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠛⠛⠛
⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢤⠈⣌⡁⣧⡀⣇⣀⠱⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠄⢀⠀⠀⠄⠀⢀⢠⡀⠠⡄⠀⠀⡄⠀⠀⡠⠀⠀⠀⣰⣿⠀⠀⠫⣶⠙
⠀⠀⠔⡃⠀⠀⠀⠀⠀⢀⠀⠐⠤⠀⠇⠳⠘⠒⠄⠂⠐⠅⠀⠳⠀⠇⠘⠐⠃⠐⠐⠒⠀⠀⠊⠟⠫⠀⠀⠀⣿
⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⣀⠀⢠⠀⠠⠄⠐⠂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠿⠇⠀⠀⠀⠀⠈⣠⡄⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⡆⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠂⠀⠀⠀⠀⠐
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠐⠀⠀⠂⠀⠁⠀⠀⠀⠈⠈⠄⠀⠘⠀⠀⠀⠈⠀⠀⠀⠁⠀⠠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣭⣯⣽⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣯⠂⠀⠀⠀⢀⠀⢀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠰⠀⢀
⠀⠀⠀⢐⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⣦⣼⣤⡄⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣔⠀⠤⠀⢀⢁⠋⠀⠌⠊
⠀⠀⢀⠀⠀⠀⢀⠀⢀⠀⠀⡀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠰⠐⢶⠄⣇⠀⠀⠀⠄⢨⡑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⢀⠀⠀⠈⣀⠄⢀⠀⢂
⠀⠀⠀⠁⠀⠀⠁⠀⠀⠐⠀⠀⢆⣄⠁⠀⠂⠀⠀⠀⠀⠀⡀⢈⡁⠀⠀⠐⠁⠀⠀⢁⠠⠀⠀⠀⠀⡗⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⡀⡀⠀⠁⠀⠀⠁⠈⠐⠛⠤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠊⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰
⠀⠀⢀⠀⡀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠱⡄⠀⠀⠸⣉⡇⠀⠀⠀⠀⠱⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠘⠀⠁⠠⠀⠐⠀⠌⠁⠁⠀⠀⡠⡆⡀⠀⠀⠈⠇⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢬⠓⣥⣧⡕⢫⠀⠀⠀⢀⠀⠀⢠⠖⠁⠑⢆⠀⠀⠊⠉⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠺⣧⣤⡟⢐⠛⠀⠀⠉⠀⠀⢇⠀⠀⠀⢨⠀⠀⠀⠀⠀⢀⠼⠀⠀⠀⣀⠀⠀⠠⠇⠀⠀⣀⠀⠈⠠⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠄⡇⠌⠳⠀⠀⡀⠀⠀⠀⠈⠳⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠁⡄⣆⣆⢣⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠠⠤⡄⠭⢰⡏⣛⣗⠬⠀⡦⠄⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡟⡛⡱⠍⠀⠀⠀⢀
⠀⠀⠀⠔⢻⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡾⠁⠀⠀⠀⠀⠐⠂⠀⠀⠘⠃⠈⢠⡅⠁⠈⠳⠀⠀⠜⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⢀⠀⠀⡀⠀⠀⢄⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⣀⠀⠰⠀⢐⠀⠀⠀⠀⠀⠀⡈⢇⣀⠀⠀⠀⠀⠀⣦⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⢀⡨⣩⠕⢭⠭⡀⠀⠀⠈⠁⠀⠀⠈⠓⢂⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠰⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠈⢐⣙⡦⣚⣒⠉⠀⠀⢀⣠⣥⣀⠀⠀⠀⠈⠁⠀⠀⠀⠸⠖⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠉⠀⠰⠀⠈⠀⠀⠀⠈⣿⠜⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡞⠀⠀⢈⡟⠒⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⣇⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠙⠟⣿⣿⣿⣟⢀⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡬⣿⢿⡷⢶⣿⡿⢿⣼⢟⣻⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⠉⠾⣅⢹⣿⣷⣾⣿⣫⣧⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⢀⣤⠆⣸⡯⠶⠋⢿⢾⣿⣿⣷⣿⡿⣿⣿⢶⡿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⣿⣀⣿⠗⠚⡧⣸⣼⣿⣿⣿⡛⣿⠓⢈⠘⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣤⣶⣿⣿⠟⢐⣟⢻⠟⣹⣿⣿⣿⣿⣶⢶⣾⣤⣶⡧⣤⠄
⠀⠀⠀⠀⠀⠀⠼⢹⣿⣿⣿⣿⣿⠟⢁⡴⢢⣥⣴⡇⡋⣹⣿⣿⣿⣿⠿⠚⢭⠉⠉⠛
⠀⣴⣾⣿⣦⡀⡀⠀⠈⣿⣿⣿⣿⠀⢜⢻⢮⣌⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⠛⠁
⠀⢿⣿⣿⣿⡟⢻⣄⣶⢾⣿⣿⣿⣗⣺⢭⣛⠹⠏⢟⣿⣿⣿⣿⡏⢰⠁⣸⠉⣠⣶⣶⣄⢀⣤⣄⣀
⠀⠀⠉⢻⣿⣿⣾⣿⡇⣾⣿⣿⣿⡟⠄⢸⣭⣷⣿⣿⣿⣿⣿⣿⣿⣿⣯⠟⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠘⣿⣿⣿⡟⠀⣿⣿⣿⣿⣧⡴⢻⢛⡿⢾⣿⣿⠀⣾⠀⢸⣿⣿⢟⡂⠛⠛⠛⠛⢉⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣷⣾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡕⠞⠁⡔⠛⠛⣻⣿⡇⠀⣹⣿⣿⣿⡏
⠀⠀⠀⠰⣿⣿⣿⠃⢠⣿⣿⣿⣿⣿⡟⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⡇⠀⢀⠐⠿⠞⢛⠉⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢹⣿⣿⠀⢸⣿⣿⣿⡿⠟⢿⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣦⣼⣿⣧⣴⢿⡆⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⠄⠀⣿⠋⣿⡇⣎⡽⡿⠛⣹⠋⠉⣿⡟⢻⠛⠿⢿⣿⣿⠉⠀⡜⠉⠈⠘⡆⠀⢸⣿⡟⣿⣧
⠀⠀⠀⠀⣿⢹⣿⠀⣴⣿⠀⣼⡟⠃⢀⠀⠀⢻⠠⠀⡧⠄⠈⣎⠀⠀⢻⠉⠀⠀⡇⠀⠀⠀⠃⠀⠸⣿⠇⣿⣿⠆
⠀⠀⠀⠀⠛⢻⣿⣦⠈⠿⠄⠈⠿⠄⠀⠀⠀⠺⠀⠀⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠶⠟⠠⣴⡿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠰⠆⠀⠏⠈⡇⠈⢴⢖⡖⠲⠐⢆⣔⠀⢴⠄
⠀⢀⡠⢄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡎⠉⠈⣀⣀⠀⡄⡴⢀⣰⠖⡉⢧⡁⢀⣀
⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠘⢧⣀⡠⠇⠃⠈⠉⠑⠁⠁⠓⠶⠤⠇⠘⠛⠀⠀⠤⡄⠀⠀⠴⠴⠄⠀⠀⠐⠐
⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡘⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠠⢦⠠⠶⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠁⠀⠀⠀⠀⠀⠈⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⢛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⣀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀⢰⠀⠠⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⠙⠎⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠈⣅⠀⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠜⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⢀⠄⠀⠴⡮⢀⡀⠀⠀⠀⠀⠈⠁
⠠⢴⣆⠀⠀⠀⠐⠀⠀⠀⠀⢰⡄⠉⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠂⢀⡀⠀⠀⠀⡄⠀⠀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⣸⠀⠀⠀⠀⠸⠁⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⡎
⠀⠀⠠⠀⠁⠀⠉⠋⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠐⠀⠁
⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠀⠀⠀⢀⠠⡀⠀⠀⠐⠀⠂
⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣥⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣁⠒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠠⢀⠁⠀⠀⠀⠈⢀⣀⣄⠰⡎⡄
⠀⠀⠀⠀⠀⠀⠀⢔⣤⡄⠀⠀⠀⠀⠀⠀⠀⢇⡨⠂⠀⣼⣠⣿⣸⠃⡇⢻⡁⠸⣆⣽⡖⡈⠒
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⢸⠏⢸⡇⠀⠙⣆⢻⡀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡤⠀⣀⡈⠁⠛⠒⠋⠈⠁⠀⠻⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣠⠂⠀⠀⢠⣟⣀⡾⢡⡾⠋⠀⢀⣶⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠃⠘⠇⠀⢀⣾⠃⣶⣀⠀⠀⠀⠘⠀⠀⢀⡀⠀⠀⠀⠀⠀⠴⢃⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠋⠁⠀⠀⠒⠃⠀⠙⠁⠉⠁⠐⣀⠀⠀⠀⠲⣬⠵⠋⠀⠀⠀⠀⠀⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠲⠦⣤⣀⡀⠁⠀⠀⠀⢀⣀⠀⠀⠀⠛⢷⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡼⠚⡁⠀⠀⠀⠀⠸⡅⠀⢠⠄⠀⠀⠉⠙⠛⠻⠿⠖⠛⠃⠀⠈⠀⠀⠃⠀⠀⢨⣔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⡀⠀⠀⠀⢀⡀⠀⠄⠠⠀⠀⠀⠠⠀⠠⠠⠀⠀⣀⠀⠀⣶⠀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⢻⡇⢠⣄⠀⠈⣿⠀⠀⡄⠀⠀⠑⣤⠀⠂⢰⡆⠀⣽⡇⠀⣿⠀⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⢸⡇⢀⣿⠀⠀⣿⠀⢸⣄⠀⠀⢠⣟⠀⢠⣿⡀⠀⢿⡦⠀⣿⡁⠀⠁⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⢸⣿⡀⠀⢿⡦⠜⠛⠉⠀⠈⠁⣀⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢧⡂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠀⠀⠀⠸⣧⠀⠉⠀⠀⠠⠤⠤⠤⣀⠀⠂⣀⣼⣥⣤⣄⣀⣀⠀⠀⣿⡇⢠⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⢰⡆⠀⠀⢠⣤⠀⠀⠀⠹⠋⠁⢹⠀⠀⠀⠀⢩⣙⣷⠿⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⢸⡇⠀⠀⠘⠉⠀⠐⠂⢀⣤⣄⢸⠀⠀⢦⡶⢶⡿⣦⠀⠶⢶⡄⢠⣤⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠠⢂⠂⠒⠀⠀⠀⢸⣇⣄⡀⢴⡾⠻⢸⡇⠻⣦⡀⢸⠀⠀⢸⠀⢸⠀⢸⢠⡖⢛⡇⠘⢷⣅⠀⣀⡀⠒⠒⠋
⠀⠀⠀⠈⠀⠀⠀⢀⡁⠀⣶⡞⠀⠀⡀⢸⡏⠈⣧⢸⠁⠀⢸⡇⢀⣀⣿⠸⢦⠞⠺⠂⠘⠃⠚⠊⠛⠛⠓⠸⠦⠿⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠄⠙⠇⠀⠘⢷⣤⡤⠀⢸⡇⠀⠿⠼⠇⠀⠘⠁⣈⣉⣠⡤⠤⠒⢶⡂⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡊⠹⢋⠀⠀⠀⠠⠠⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠁⢠⠄⠐⠀⠀⠙⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⣀⡀⢀⣀⡴⠴⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢲⣿⣿⠿⠟⢀⣽⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡀⢠⣿⣿⣿⣿⣻⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠉⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣝⣛⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣦⣶⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠛⠻⠟⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣵⣿⣿⣿⣿⣽⣠⣿⣿⣟⣉⣉⠍⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣿⣿⣿⣿⣿⡇⢿⣿⡿⠛⣿⡿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⢿⣸⡿⣿⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⡇⠈⢿⣇⡏⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⢺⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣿⣿⡿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣽⣿⠛⠛⡙⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣤⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣝⢯⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢦⡞⣏⠰⣶⣶⣮⣴⢞⣤⣤⠰⣦⣀⠐⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣿⣾⣿⠖⢤⢀⣀⡨⣀⣀⣦⣗⢰
⢐⡻⡄⠋⢙⠿⣟⣡⠈⢨⢸⡗⣫⠩⡟⠊⡩⡼⢀⣀⣀⢀⣄⠀⣀⣠⣠⠟⣤⢝⡎⢼⠨⣏⣮⠹⣿⠫⡐⢓⢻⠤⢳⠾⣇
⠈⠇⢑⠦⡋⠏⢹⡼⢎⣽⢨⠶⣨⡷⠚⣶⢑⣯⣘⣿⠹⣏⠋⣿⣿⢿⣟⣿⢍⣜⣄⡼⡔⠿⢆⠓⢿⡧⠙⢠⠙⡏⣸⢳⡟
⠐⠀⠾⡖⣩⠓⢞⠋⠂⠶⢊⢌⡽⠋⢔⣠⠋⡐⣬⡱⢏⢀⣀⠈⠗⠞⠣⠊⢄⣿⡸⢆⡲⢌⣆⠛⠶⢦⢘⠀⢅⠱⢗⠊⢍
⠀⠀⣰⠀⡌⡙⠂⣀⠀⠀⠀⠀⠀⠋⠀⠓⠀⠙⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠀⣆⠁⢲⠌⢺⡷⠄⠈⠀⠁⠙⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠉⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⡆⠀⣀⠴⠶⠥⠤⢤⣄⣀⡀⠀⠀⠀⢸⠀⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠹⡀⠀⠀⠸⡾⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⡞⠀⠀⢠⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠐⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⣀⣀⣠⣤⠴⠛⠀⠀⠀⠐⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⣄⣀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠙⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠲⠤⢤⣄⣀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡴⠋⠀⠀⠿⣯⢽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡿⠄⢠⣠⡄⣈⠉⠲⢄⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢀⡞⠀⣤⡿⢳⣖⠀⢀⡽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⠉⠐⣾⣯⣹⡟⠂⣀⣄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣀⡄⠋⠷⠿⠉⡰⠋⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠒⠒⠦⠤⢄⡀⠛⠉⠠⣾⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣳⡆⠀⠀⡼⢁⣴⣿⡙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⡶⡶⣦⣄⡈⠓⢄⣀⠉⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠁⢀⡞⢁⡾⡏⡿⢇⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⠇⡿⣇⡏⣻⣶⣄⠈⠓⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠔⠋⣠⣿⣳⣿⣿⡈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠇⣊⣠⢠⣷⣷⣏⢃⡟⣿⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣾⢏⣮⡿⢽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠁⠛⢹⡿⡨⣇⡿⡸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⢻⢹⠸⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⢹⡟⣃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢼⣟⠰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠄⠀⠀⠀⢀⣤⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡀⠀⣻⣤⡲⢀⣤⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠀⠀⠠⡼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣿⡀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣌⡆⠀⠀⢠⠟⠞⢃⡾⣹⢧⡹⡏⡜⠛⠁⠀⠀⠀⠀⣀⡤⠒⠋⠘⠁⠠⣶⡀⠀⢸⡞⢻⣀⡀⠀⠀⢠⣷⡄⠀⠀⠀⡩⡀⣥⣾⡀⣠⣿⣄⣾
⢀⣄⢠⣿⠀⣰⡰⡚⠛⣸⣦⣴⣃⣶⣤⣴⣏⠀⠙⠿⠀⠘⠁⢀⡠⢔⣾⡏⠀⠀⠀⠀⠀⢴⣿⣆⠀⠼⣇⣾⡍⠁⠀⠀⣦⢷⣰⢀⠀⢰⡟⢻⣠⡏⢳⣿⣸⣇⢿
⢸⣿⣫⣭⣷⣿⢉⣿⢉⣻⡿⢩⠇⢸⣧⠁⣾⡆⣸⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⣎⣻⢸⡤⣀⣽⣏⠹⠶⠀⣠⣤⣿⡉⢼⠁⠼⣷⣚⣡⣇⣸⣧⣹⠙⡛
⢸⣿⣽⡉⣿⠟⠁⣈⢈⡉⢹⣿⣠⣾⣧⡖⢁⣿⣏⠈⡝⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣷⢿⣌⣥⠈⠾⡟⠳⠇⠀⢈⣡⣯⡛⠆⣉⠰⣾⡿⢋⣍⠁⢹⡻⡦⣿
⢸⣯⣴⣿⡝⠛⠋⠈⣿⡛⠛⣿⣿⣿⡿⣥⣿⢿⣿⣆⣻⣿⣾⣧⣦⣷⣶⣴⣄⣤⣄⡤⠚⣇⣼⣮⣙⣥⡀⡇⠀⣀⣴⡌⢀⣧⢥⠔⢣⡴⢤⣿⣯⣿⣷⣼⣿⣁⣿
⢸⣿⣿⣿⠧⠾⣀⣼⠞⢷⣶⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣟⣯⣛⣧⣿⣿⠖⣉⣿⠈⣧⡙⢯⢻⣿⣟⡣⠾⣙⣿⠅⠞⡔⠢⠗⢤⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⡄⣠⠏⠈⣄⠀⢳⣀⣻⡇⠿⠛⢿⠈⠟⠻⡿⣿⠟⠙⠛⠙⠿⠿⣿⣿⡿⢛⣡⣻⠆⠁⣹⡚⠻⡿⢓⠦⠓⠆⣹⢇⡠⠃⡤⠧⡘⠾⢿⡿⡿⡻⠻⠻⣿
⠈⠋⠻⢯⣀⡴⠦⠤⠊⠓⠛⢿⣄⣀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⣡⠀⣥⠀⢈⣍⠠⡬⠃⠈⠋⠸⡀⠤⣹⠇⡴⢳⠀⡉⠙⡈⢃⠁⠀⠀⠀⠁
⠀⠀⠦⠴⠁⠀⠦⠀⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣀⡀⢠⠄⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠈⡀⠀⠀⢀⡀⢹⢦⣘⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠄⠄⠀⠀⡁⠤⠘⣁⠠⠐⠂⢢⣁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠀⠰⠤⠐⠞⢀⣀⡀⠀⠀⠠⠉⠱⠒
⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠶⠵⢆⡠⠀⠠⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⡠⠐⠀⠀⠀⠀⠒⢄⠀⠀⠀⠐⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠⠴⠀⠔⢄⡀⠀⡠⢁⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡀⠘⠀⢰⡆⠀⠰⡦⢠⠁⢘⠅⠕⣨⣀⠀⢀⡄⠀⡇
⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⡀⠀⢨⠀⠂⠀⣤⠀⠀⣀⠁⠁⠀⢀⡔⠁⠈⠑⢀
⠀⠀⠀⣠⠀⠀⠪⡜⠒⠒⡄⠀⠀⠀⠀⠈⠆⠀⠀⠀⠤⠈⠃⠀⡄⡠⢚⠀⠀⠀⡀⣶⢓
⠀⠀⠀⠁⠀⠀⠁⠉⣉⣩⢇⡀⠀⠀⠀⠀⠀⡀⠆⠤⠌⢠⠄⠊⢰⠀⢐⠦⢈⠀⠈⠃
⠀⠀⠀⠀⠀⠄⡀⠤⢤⣤⠺⠂⠀⠀⠀⠀⡀⡘⠉⠁⠐⠉⢄⣧⢀⠓⠂⠉⠁⠒⢘⠀⠀⡄⡀⡌
⠀⠀⠠⡏⠈⠁⢀⠀⠮⠘⠈⢀⠀⠀⠀⠀⠳⡃⠠⠬⠉⠐⠀⢙⠃⠀⠨⠀⣍⡍⢩⠑⠉⠉⠁
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⣴⠊⠁⠀⠀⠀⠀⠑⡒⢒⠒⠊⠀⢠⠀⠀⠀⠐⠈⣣⣼
⠀⠀⠀⠀⠀⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⣃⢇⠀⠀⠢⢀⠀⠀⠀⠉⡩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠳⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠿⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠙⠇⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠘⠋⢧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠐⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⢇⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⠤⠞⠉⠉⠉⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠯⢷⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠴⢟⡻⢦⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⣀⣤⠴⠞⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠑⢛⢀⡀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣯⠋⠁⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡷⣻⠓⠀⠀⠀⠀⠉⠓⠲⠶⠶⠒⠻⡄⠀⠀⠀⠉⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢡⣾⣿⠁⣠⡤⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠉⢨⣿⠖⠰⡶⠤⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢀⡀⠀⠀⣤⠀⠀⠀⣤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢄⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠀⠀⠈⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⡀⠀⢁⠆⠀⠀⠁⠆⠀⠀⠀⠁⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠤⠀⠀⠀⠀⠀⠀⠀⠀⢑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠑⡠
⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⠀⠀⠀⡆⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠌⠢⡀⠀⠈⠃⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⡏⠀⠀⡠⠂⠀⠀⠀⠈⢆⠀⢀⡀⡨⠓⠀⠀⢀⣀⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡆⠀⡀⠀⠀⢠⣮⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⣰⠀⠀⠀⠀⠈⠸⡁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠈⠉⠀⠸⠀⠠⠀⠀⡀⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠀⢀⡀⠀⠍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠀⠀⠠⠀⣀⠀⠀⠰⠀⡠⠀⣀⡠⠜⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠑⢄⠀⢱⡁⠉⠀⢀⣀⡀⠄⠁⠀⣠⠂⠀⠀⡀⢠⠀⠀⡈⢁⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠠⠆⠀⠀⠀⡀⠀⢀⠀⢅⠊⠓⠤⣄⣀⣀⡠⠤⢚⠀⠠⠒⠉⠁⣈⢀⠀⠘⢇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠁⠈⠐⠐⠀⠉⢄⡰⠁⠀⠀⠑⠂⠤⠀⠤⠤⠵⠒⣉⠒⠤⠤⠤⠚⢉⠀⠄⠀⠀⠉⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢁⠠⠄⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡣⣀⠐⠁⠎⠐⢀⠂⠈⠀⠀⢠⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠊⡑⠒⠒⠒⠚⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠎⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⠧⢄⡠⠤⠤⢀⣰⡦⢄⣰⡖⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠔⠁⠀⠀⠠⠤⠤⠀⠢⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠀⠀⢠⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣠⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⡄
⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢻⣿⣿⣿⡇⠋⢠⣿⣿⣿⡇⣝⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⡆⠀⠀⠀⢀⣧⣿⠇⠀⠀⠀⣿⢧
⣿⣿⣿⣿⣿⣿⣷⣴⣶⣶⢸⣿⣿⠋⠀⢰⣿⣿⣿⡿⠃⣿⣿⣿⣿⡟⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⡇⠀⠀⣴⣿⣿⠁⠀⠀⠀⣠⣼⠈⠀⠀⢰⣶⡆
⣿⣿⣿⣿⣿⡿⠿⢸⡟⠉⢸⣿⡇⢰⢋⢈⣉⠹⣿⣷⣴⣿⣿⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⣠⣶⣧⠀⢸⡿⣱⢿⠿⣷⡄⠀⠙⠋⠀⠀⢠⢸⡅
⣿⣿⣿⣿⣿⣧⣠⣾⣷⣶⣿⣿⣧⡘⢾⡟⡿⢀⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠈⠉⠁⠀⠘⣿⣝⠸⡄⣸⡇⠀⠀⠀⠀⠀⠈⠻⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠒⣆⣶⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⡙⣟⠉⠀⠀⢀⠀⠀⣀⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣃⣿⣿⣿⡇⠀⣸⣿⣠⣤⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⡄⠀⠻⢿⣧⠟⠀⠀⣸⣿⡿⠀⡿⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣁⣀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠇⠀⠀⠸⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡜⠀⠀⢠⠇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣰⠃⠀⠀⠁⠀⠘⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠁⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠚⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢠⡀⠀⠀⠉⠒⠷⣄⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠆⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣠⠂⠐⠁⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠓⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⣦⡄⡈⣢⡄⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠐⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⣀⣠⠀⠀⠀⠀⠀⠀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠶⠤⠚⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠂⠄⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠠⡭⡄⠐⠈⢡⡀⢠⠀⠥⢈⣱⡄⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣄⠀⣠⣾⣿⣿⣆⠀⠀⠀⠁⠈⠋⠀⠀⠰⠦⣉⣙⣣⣿⣧⠉⠺⣄⠀⢤⣼⣷⡏⠳⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡆⠠⠀⠈⠰⠀⠀⠀⠈⠀⠀⠠⢭⡿⠖⠠⢉⣀⠀⠨⣿⡦⠀⠻⣄⣴⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⡄⠈⠓⢾⣿⡀⣆⠀⢹⣇⠀⠀⡂
⠀⠀⠀⠀⠀⢀⣴⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⠚⠆⠀⠀⠀⢐⠤⠄⢨⠙⠒⠶⢼⣿⣇⠸⣧⠀⣿⡜⢀⡏
⠀⠀⠀⠀⣼⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢔⠒⠦⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡖⢿⣄⢸⣧⣸⣇⡠
⠀⠀⠀⢠⣿⣿⣿⠟⠋⣿⣟⣛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⢳⣤⣤⣠⣠⣶⣾⡟⢻⣿⡿⠀⢹⣿⣿⣿⡏
⠀⠀⠀⠈⣿⠏⠁⠀⠀⠈⠨⢻⣷⡀⠈⡿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣻⠀⠴⢊⣠⣾⠋⠉⠙⠛⠛⠻⣿⣷⡀⠞⣿⣿⡿
⠀⠀⠀⣼⣃⣀⣔⣦⢀⣤⣴⣿⣿⣿⠀⣿⣿⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣼⣩⣷⡀⢐⢀⣲⡄⢠⣄⣿⣿⡧⢠⠿⣟
⠀⠀⠀⠈⠉⠀⠚⢻⣛⣛⠛⠟⠋⠉⠈⠀⠙⠁⠁⠃⠛⣡⣽⣏⣩⣿⣭⣴⣤⣬⠤⠄⠉⠀⠀⠀⠘⣷⣤⣼⣿⡈⢳⣜⡁
⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⠞⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣠⣶⣿⣿⠟⠛⠁⠀⠛⠉
⠀⠀⠀⠀⠀⠴⢶⣾⣧⣤⣤⣤⣤⣤⣄⣤⣤⣤⣄⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣷⣶⣾⣿⣿⠟⠛⣋⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠙⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⣀⣄⠤⢤⡤⠼⠒⢒⡀⠀⡀⢀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠐⠒⠭⠉⠀⠀⠀⢠⡄⠃⠀⠂⠀⠈⠡⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣀⣀⣄⣯⣼⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⢉⡉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡒⣉⠀⡆⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠉⣥⡤⢴⢶⣤⡤⠤⠴⠶⠿⠟⠛⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈⠁⠢⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡡⠅⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠠⠟⠳⠦⠀⠀⢀⡇⠀⠰⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠨⠅⠨⠿⠿⠓⠐⠀⠀⠀⠀⠈⢧⡄⢍⡀⠂⠀⠉⠁⠀⠀⡻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠑⠊⠇⠱⠀⢀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠠⠀⡤⢊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢣⠀⠀⣠⣤⠀⠀⠀⠀⠀⡜⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣸⠀⠀⣿⠗⠀⠀⠂⣀⣀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢅⢀⠀⠈⠀⠀⠀⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⡟⢛⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⣈⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣍⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠤⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡔⢋⠭⡙⢦⠀⢀⡠⢤⡀⠀⠀⠸⣈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⡇⠀⠖⢦⠀⢣
⠀⠀⠀⠀⠀⠈⠀⠘⢄⡁⠜⠀⠸⡀⠰⠃⠀⠀⠀⢀⡉⠑⠓⠀⠉⠉⠁⠒⢠⣤⠒⠋⠉⠉⠉⠉⠛⠲⣄⡀⠈⠒⠒⠁⡠⠃⠀⠀⠀⡀⢀⡀
⠀⠀⣠⠤⢤⡀⠐⠤⠄⣀⡀⢠⠄⠘⠁⠀⠒⠀⢀⠋⢀⡀⡀⠀⠀⠀⠀⠀⡞⠀⢱⠀⠀⠀⠀⢀⠤⣄⠈⢫⠁⠀⠙⠓⠤⢀⡀⡠⠈⢀⣠⠇
⠀⠐⣇⠐⠖⠃⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⡏⠠⠈⡂⠀⠀⠀⠀⢧⣀⣼⠁⠀⠀⠀⡃⠠⡀⠃⢸⢆⠀⠀⢀⠀⠀⠈⢋⠉⠁⠀⠰⣁⡄⢹
⠀⠀⠈⠑⠒⠂⠉⠀⠥⡱⠀⠀⠀⠀⡜⢠⠀⡸⠀⠓⠬⣤⠚⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠦⡤⠖⠋⠈⠢⢐⣂⡕⠀⠀⠈⡁⠒⠤⢤⠬⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⣄⠀⠀⠀⠘⠦⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣇⣀
⠀⠀⠀⠀⠀⠀⠉⢻⣿⣏⠈⠀⠀⠀⠀⠈⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠞⠋⠄⠀⢰
⠀⠀⠀⠀⢀⡧⠀⠺⠀⠈⠃⠀⠀⠀⠀⣠⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠁⠀⠀⠀⠀⢀⣀⣀⣤⠀⠀⠀⠀⢨⣅⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣌⠻⣶⠀⠀⠀⠀⠐⠆⠀⠀⢿⣿⣻⠃⠀⠀⢀⠀⠀⠀⠀⢠⡄⢀⣧⣀⠤⠀⠀⠀⠈⠀⡩⠤⠀⠀⢀⣴⣾⣿⡆⣄⠀⠀⠀⠊⢺⡹
⠀⠀⠀⠀⠙⡿⡖⠀⠀⢀⣠⣾⣷⠄⡀⠀⠉⠁⠀⠀⠀⠛⢤⠀⠀⠶⣔⡃⠞⡃⣡⡴⠤⠄⠀⠀⠒⠁⠀⠀⠀⡾⠛⣧⡜⠛⣾⡇⠀⠀⠙⠀⠁
⠀⠀⠀⠀⡀⠿⠀⠀⣰⣃⢤⣄⣤⣤⡟⡄⠀⠀⠀⠀⠀⣰⡀⠀⠀⢠⣈⡿⢹⠟⢴⣖⢣⡀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣟⣳⣾⣻⠃⠀⠀⢀⣶⣄
⠀⠀⠀⠐⠤⠄⠀⠀⢿⣭⣧⣟⣳⣎⣹⡇⠀⠀⠀⠀⠐⣯⢕⠀⠀⠀⢠⡵⢟⢹⠦⢻⡀⠀⠀⠀⢠⣾⣤⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⣠⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣏⣹⡽⠟⠁⠀⠀⠀⠀⣠⠗⠋⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠖⢶⡿⡿⣓⣤⠀⠀⠀⠀⠀⠀⠀⢠⣮⣭⣭⢭⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⠐⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠛⠿⠶⠧⠵⠗⣣⠀⠀⠀⠀⠀⠀⠸⣿⣻⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠝⣒⢒⢒⣒⣪⡽⠀⠀⠀⠀⠀⠀⠀⠑⢿⢭⡦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠃⢊⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⣁⣁⣡⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠈⢧⠘⣧
⠀⠀⠀⠈⠢⡈⠃⠉⢣
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠣⡀⠀⠀⠀⠀⠀⠀⣀⠐⢀⠀⠄
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠙⢆⠀⠀⠀⠀⠀⡀⠐⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠲⢄⠑⠺⣆⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢌⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠚⠻⢷⣦⡀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠄⢉⣛⣛⣿⣦⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠐⠀⠈⠠⠀⠐⢚⠇⡀⠈⠛⠳⢤⠄⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⡁⠀⠀⠀⠁⠀⠉⠕⣂⠔⢠⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⣀⠠⠀⢀⠀⠀⡠⠈⠐⠾⣿⠆⠀⠀⢀⡇⠀⠀⠀⢀⠀⢠⣾⠋⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠐⠈⡀⠜⠁⠀⠀⠀⡼⠀⡀⣀⢠⡿⠐⣺⣏⣂⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⣤⣾⡀⣴⠃⣰⡟⠵⠂⠩⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣾⠏⣤⣼⡇⠚⣛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣾⠟⠣⢤⡽⣻⣿⣯⣥⣤⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣻⢾⣿⣿⡿⣰⡆⡸⠚⢩⡄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠡⠺⡿⢯⣿⣿⢏⣡⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⡟⣻⠟⣛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⣀⣭⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣸
⡇⣠⠿⠀⡀⣤⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢹⣿⡉⠀⠺⣳⠀⢸⡟⠀⡄⣸⠀⠀⠀⢸⡄⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐
⣾⣥⣌⠉⣠⡼⠂⠸⢁⣘⢻⣯⢀⣼⣴⡯⠤⣾⣠⠀⠀⡀⠁⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡿⢿⡶⠂⠠⠤⠀⢀⣬⡍⠼⣥⠀⣿⣿⠛⠂⢹⣿⣄⣼⢁⡀⢀⠁⡀⠀⣰⠀⡀⠀⢠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⢀⡀⣀⠀⢠
⡷⠨⣙⣟⡳⠒⠀⢸⡟⣀⣘⠟⡯⣿⠛⡘⣋⡀⣽⣿⠟⣛⣦⣿⣼⡇⢰⡿⢻⣧⣶⡾⠇⣴⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣠⣶⣿⣡⡏⣾⣿⣵
⠀⠀⠀⠀⠈⠈⠀⠘⠛⣷⠍⠂⣼⡻⠂⠠⠤⠀⢐⣿⣧⠆⠰⣿⡗⠭⢿⡥⠄⢿⣿⡷⠼⣿⡿⢲⣾⣆⡀⢀⣄⣤⣿⣿⣿⣿⣿⣿⡿⠥⠽⠿⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⢯⡃⢦⣤⠄⠀⠉⢻⣇⠐⠈⢻⡿⢠⢽⣏⣀⣋⣿⡥⠀⢹⣇⣈⣿⡿⣿⣿⠿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠶⠾⠿⣯⣘⡛⣷⣀⠊⢿⡧⢴⢺⣄⣐⠈⣷⡆⢸⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠚⠛⠶⠿⠿⠿⠒⠛⠉⠉⠁





⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣿⣵⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣶⡾⠿⠿⠿⣶⣦⣄⡀
⠀⠀⣾⡟⠁⠀⠀⠀⠀⣴⣿⠛⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⣿⡇⠀⠀⠀⠀⢰⣿⠃⠀⠀⢈⣽⣿⣿⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⡟⠛⠛⠛⢿⣦⡀
⠀⠀⣿⡇⠀⠀⠀⠀⣾⡿⠀⠀⠀⣼⣿⠁⠀⠀⠉⠙⠛⣿⡿⠿⠿⠿⣿⣿⡿⠿⠿⠿⢿⣿⡟⠉⠁⠀⣿⡇⠀⠀⠀⠀⢿⣧
⢀⣾⡟⠁⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⣼⡿⠀⠀⠀⢸⣿⠁⠀⠀⠀⢀⣾⡏
⣾⡟⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣸⣿⠀⠀⠀⠀⢰⣿⠃⠀⠀⢀⣿⡏⠀⠀⠀⢠⣿⠟
⣿⣇⠀⠀⢀⣾⡟⠁⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⢀⣿⡇⠀⠀⠀⢠⣿⠏⠀⠀⠀⣼⡿⠀⠀⠀⠀⢸⣿
⠘⢿⣷⣄⣼⡿⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⣼⡟⠀⠀⠀⠀⣾⡟⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠉⠻⢿⣷⣤⣀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⢀⣾⡟⠁⠀⠀⠀⢰⣿⠃⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣤⣄⣀⡀⠀⢰⣿⠁⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡁⠀⠀⠀⣀⣤⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⣶⣶⣤⣼⣿⣧⣤⣀⣀⣀⣀⣸⣿⣄⣤⣤⣼⣿⣷⣶⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣀⣀⣀⠀⠀⠀⠈⣿⡏⢿⣧⣀⣠⣤⣴⣶⣶⠿⠿⠿⠛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠛⠛⠿⠿⠿⣷⣶⣿⡇⠈⢿⣿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠈⣿⣇⢀⣀⣠⣤⣶⣶⣶⣿⡿⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣤⣤⣄⣼⣿⠀⠀⢸⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠛⢻⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⡀⠀⠀⠀⢸⣿⠀⠀⠀⢿⣿⣤⣶⣾⠿⠿⠛⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠛⠛⠛⠻⠿⠿⣷⣶⣾⣿⠀⠀⠀⢸⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⣼⣿⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⣿⣿⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠿⠿⢿⣶⣶⣶⣤⣤⣀⣿⡇⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢛⣿⡇⠀⠀⣼⣿⢀⣀⣤⣤⣶⣾⠿⠿⠿⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⢸⣿⠀⠀⢠⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⣷⣾⡟⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣾⣿⣶⣶⣾⠿⠿⠿⠿⠿⠛⠛⠛⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⡿⣿⣶⣦⣤⣄⣀⣀⣀⣠⣿⠏⠀⠀⢿⣯⣥⣤⣄⣀⣀⣀⣀⣀⣀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢡⡟⣼⢇⣿⢹⡟⣿⢻⡟⣿⠟⠋⠀⠀⢀⡿⢹⡟⣿⢻⡿⣿⠟⡿⣿⢿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⡿⢱⡟⣼⠇⣾⢡⡟⣼⢇⣿⠀⠀⠀⠀⣾⢣⡿⣸⡏⣾⢃⣿⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠿⢰⡟⢠⡟⣸⠇⣿⢸⡇⠀⠀⠀⠈⠋⠾⢡⡿⢰⡟⢸⡇⣾⢡⡟⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠃⠿⠸⠏⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠘⠃⡿⢸⡇⣿
⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣆⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⡇
⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⣀⡀⡿⠀⢸⠃
⠀⠀⠀⠀⢠⡏⠀⢻⠀⠀⠀⠀⢸⠉⠙⠇⠀⡾
⠀⠀⠀⠀⠀⣧⠀⢸⡄⠀⠀⣀⣸⡀⠀⠀⠀⣷
⠀⠀⠀⡴⠶⠾⡄⠀⢧⠀⠰⡏⠉⠃⠀⠀⠀⢸⡀
⠀⠀⠀⢳⡀⠀⠀⠀⠈⢧⠀⢳⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⢀⣳⡀⠀⠀⠀⠈⠳⣖⠻⠄⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⢀
⠀⠀⠀⣯⠁⠉⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⡘⢷⠛⠋
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠑⢄⠀⢠⠖⠈⠑⠲⡀⠀⢸⣧⠸⣇⣻⣔⠛⠋⠁
⠀⠀⠀⠀⢀⡼⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⠀⠈⠗⠁⠀⠀⠒⠀⠁⠀⣨⠷⢾⣭⣍⠛⠛
⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⠖⠀⣰⠃⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢓⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣤⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⡀⢀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠤⠒⢧⠤⢺⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠿⠷⠟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠸⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⢀⣶⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠘⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣛⣿⣿⣟⣿⣻⣻⣿⣋⣁⣤⣤⣤⣄⣀⣈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢟⢿⣿⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣼⣿⣥⣧⣼⣧⣧⣿⣽⣷⣭⣤⣵⣶⣧⣴⣾⣧⣶⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣬⣤⣤⣭⣄⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠞⠛⠛⠛⠛⠶⣦⣤⡾⠋⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣤⣤⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡶⠶⠶⠶⢶⣤⡀
⠀⠀⠀⢀⣀⣴⠟⠋⠉⠻⠋⠀⠀⠈⢻⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⣴⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⡀
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⡀
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠙⠛⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⠟⠉⠉⠉⠉⠻⡶⠛⠛⠛⠷⣦⣰⡟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠷⣦⡀
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⣽⣿⢭⡇⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢤⡆⠀⠀⠀⠀⠀⠀⠀⠈⠾⠽⠶⠁⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⣺⣗⢺⠂⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣻⣿⣻⡟⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠃⠀⠀⠀⠀⠀⢾⡷⣽⠆⠀⠀⠀⠀⣴⡖⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠹⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠸⣗⡿⣗⡟⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⢲⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⠙⠀⠀⠀⠈⠩⠣⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⣺⢿⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡿⠿⠻
⣿⣿⣿⣿⠿⠟⠛⠉
⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣈⣷⣌⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣸⣿⣿⣿⣿⣧⡀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⡄⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠾⣿⣿⣿⣷⣄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⠈⠻⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣶⣮⣿⣿⠿⠛⠉

⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⣶⡿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇
⣿⣦⡀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣏⣹⣿⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⣿⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣯⣩⣿⠋⠻⣿⣿⣿⡓⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣟⠁⡀⣠⡞⣁⢹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⣨⠟⠉⢋⣹⡯⣀⠨⠉⠃⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠙⣧⠒⠒⠁⢩⠥⠈⢶⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿
⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠘⠢⣤⣶⠒⠷⠠⠴⠈⢂⠀⠀⠀⠀⠀⠀⠀⣸⣧⣿⣿⣿
⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠾⣫⣀⣤⡤⢴⣿⣷⣾⣿⠶⠖⡦⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣩⢥⣀⡦⣍⣽⡟⡟⠛⢦⢶⣿⡏⠛⠛⡇⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢲⣷⡿⣿⣄⠀⣿⠋⠑⠓⠄⢈⢹⢉⣦⣷⣶⣷⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⣿⡿⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣷⢀⠘⢷⣶⡃⠘⢾⣼⣻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⣿⣿⡯
⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⣇⠘⠻⠇⠒⢻⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⢀⣿⣿⣷
⣿⣿⠀⠀⠀⢀⣠⣾⣿⡿⡿⢿⣟⣿⣿⡿⣿⠿⠯⠅⣽⠿⠽⣿⣿⢼⣿⢿⡿⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿
⣧⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⣿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⠏⠘⢧⡘⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⣿⠃⠀⠀⠀⣀⣀⡀⡜⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣀⠀⡤⢠⣤⠄⠰⠾⠂⣐⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣴⣿⣿⠏⠀⠀⠀⣾⣿⣿⡿⠁⠀⠀⠂⠀⠀⡰⠀⠀⢀⣈⣑⣤⡇⠈⠉⠀⠺⠂⣀⠩⠆
⠀⠲⠶⠶⠶⢶⣶⣿⣿⣿⠟⠒⠒⠒⠒⠚⠛⠛⠓⠛⠛⠛⠋⠉⢩⠋⠉⠉⠁⠈⢅⣀⡀⠀⢀⠀⠲⠂⠉⠀⠃
⠀⠀⠀⢀⣴⣿⣿⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣐⣪⡟⢲⠦⠍⠂
⠀⢀⣠⣿⣿⣿⣋⣀⣽⣄⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠖⡾⠒⠒⠋⠉⠉⢩⠻⡁⢀⡤⠖⠂
⠀⢨⣿⣿⠟⠉⠁⣠⣼⣷⣶⣶⣄⠀⠀⠀⠀⠀⠀⡴⠛⠛⣳⠁⠀⠀⠀⠀⠀⣸⣴⠟
⠀⢸⣿⠏⠀⢀⣾⣿⠿⣿⠿⢿⣿⣷⣄⣀⣀⣀⣀⣓⣾⣿⡥⠤⠤⠶⠒⠒⠛⢿⠛⣆
⠀⢺⣿⠶⠶⢾⣿⠛⠛⢻⡛⠛⠙⣿⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⣣⡞
⠀⠈⢿⡀⠀⠘⣿⠀⠀⠸⡇⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠞⢾⠋⠁⢇
⠀⢀⣈⣷⣄⣠⣬⣷⣦⣤⣷⠤⢴⡿⠷⠖⠒⠚⠛⠋⠉⠉⠉⠀⢀⣤⣤⣄⠀⠀⣠⠏
⠀⠈⠉⠉⠉⠛⠦⢤⣤⣤⣿⡶⠋⠁⠀⠀⠀⠀⢰⣾⣿⡷⠀⠀⣸⣿⣿⡿⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣠⣤⣤⠤⠼⠿⠛⠟⠛⠉⠉⠀⠀⠁
⠀⠰⠶⠶⢲⣿⣿⣟⠛⠋⠉⣿⠉⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⣠⡌
⠀⠀⠀⠀⠻⣿⡟⠋⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⣶⡖⢂⣀⣀⣐⣒⡒⠒⠂⠠⠤⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣭⠔⠒⠈⠉⠩⠃⠀⠀⠀⠀⡏⠁⠀⠈⠉⢙⣛⠒⠂⠬⠅⣂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⡃⣆⠀⠀⠀⢠⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⢹⠀⠀⠀⠀⠀⣹⡏⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⡈⡳⠤⣀⣀⡼⠴⢦⠴⠶⠶⠷⠶⢖⡒⠀⠼⠬⢀⣀⣀⣀⣠⠟⢟⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢨⡟⠢⠄⠀⢉⠙⠑⢤⠀⠀⠀⠀⠀⠈⠣⡉⠉⠙⡏⡏⠁⠀⠀⠘⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣞⠀⠀⠀⠀⠘⡄⠀⠀⠱⡀⠀⠀⠀⠀⠀⠱⡀⠀⡇⢁⠀⠀⠤⠊⢳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢼⣍⠀⠀⠀⠀⢆⠃⠀⠀⠀⢧⠀⠀⠐⢒⣒⡈⢇⠀⠁⢨⠀⠀⢀⠠⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⡸⠒⠒⠒⠒⢺⠈⠉⠁⠀⠀⠀⢘⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠪⡒⢄⡀⠀⢸⡗⠒⠂⠀⠀⠐⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠑⠬⡒⢤⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠀⠀⠀⠀⠀⠀⢀⠽⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠢⢕⠢⣀⠀⠀⢀⠎⠀⠿⠿⠿⠿⠋⠓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠉⠒⠅⣲⠎⠀⠀⡇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⠢⡀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢧⠤⠀⠀⠤⠤⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠀⠀⠑⠪⣖⢄⡀⢀⠞⠀⠀⠀⠀⠀⢘⠀⠈⠉⠉⠉⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢪⠊⠝⡲⢤⡀⠀⠀⡁⠞⠁⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢃⢎⠗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠠⠤⢐⡒⠊⠩⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡁⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⣠⠚⢍⢫⠫⡑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⠀⣜⡄⡀⠀⢢⠂⠸⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡁⢀⣼⣿⣧⠀⠘⠀⠀⡇⣻⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣜⣡⣿⠏⢠⠆⠀⣰⢃⡆⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡿⣸⣿⣿⡟⠀⢟⠀⣰⢃⡬⢀⡼⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣯⣿⣿⠁⢀⡟⢐⣣⡾⣰⡿⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡟⢠⠾⣁⣾⢋⣽⠋⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⣿⣿⣿⠃⠙⠄⠸⢵⠟⢁⡶⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢸⣿⣿⣿⠋⢠⡴⠁⠀⣤⡴⢋⣠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣜⡻⠿⠶⠓⠘⢿⣿⠁⠀⢋⣴⠖⠁⢠⣾⠟⢋⡠⠖⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣷⠀⠀⠈⠄⠙⢷⡄⠈⣁⣤⠆⢨⣴⠿⠏⣁⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⡇⠀⣉⣠⣤⣴⠶⠾⡟⣉⣀⠐⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣸⡿⠁⢀⣁⣉⣙⣛⣛⣟⣋⣉⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡄⠈⠀⠀⣄⠲⠦⡤⢴⣭⣉⣛⣟⣁⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠈⢙⣄⠲⣶⣬⣉⣛⣛⡉⡁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⠀⠀⠘⢿⡿⢿⣥⢌⣁⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣀⡀⠀⠻⢶⣽⣓⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⡀⠀⠀⠙⠿⣿⣿⣿⠀⠀⠀⠙⠛⠛⠁⠆
⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⡘⠃⠙⢿⣦⢀⡰⣦⡀⠉⠻⣧⣀⢀⣀⣤⣄⣤⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣰⡖
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠹⢯⣑⣓⣢⣽⢏⣃⠘⣷⣄⣷⣜⢿⣿⣿⣿⣿⡿⠿⠉⠛⠿⣿⣿⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⠉⢳⣆⣈⣑⣺⣿⡙⢿⡆⢹⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣭⡁⠙⢺⣟⣘⣆⠸⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣵⠞⠋⠉⢟⣧⣤⣧⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢂⣀⣀⣙⣿⢿⣌⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣯⢀⣠⣴⠟⠉⠛⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⡎⠀⠀⠀⠀⣀⡤⠀⠘⡇⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠴⠀⣀⡸⠀⠀⣠⠖⠋⡁⠠⠆⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⢀⡶⠄⠀⠚⢀⡀⠘⠛⠃⠀⠈⡀⠴⠀⣡⠤⣤⣄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠌⠇⠀⢾⠁⠀⠾⠀⠀⢀⣄⠀⠠⡄⠀⠁⠀⣀⠧⣊⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⠉⠀⠀⢁⣦⠀⠀⠀⠈⠁⠀⠂⠂⠐⢏⡂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢈⣡⡌⠀⠐⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡤⡶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠀⠀⢠⡄⠀⠀⠀⠀⢰⠛⠛⠳⢶⠄⠀⠀⠀⢻⣄⠛⢾⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠤⠀⣀⠄⠀⠀⠀⠺⠷⠁⢀⣤⣠⠂⠀⠀⠀⠀⠀⠉⠉⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⡾⠀⠂⡞⣵⠢⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⢀⣬⡄⠘⢁⡙⠀⢨⡋⠁
⠀⠀⠀⠀⣀⡀⢀⡀⢠⡞⠁⣄⠁⢤⡈⠉⠀⢀⠈⠁
⠀⠀⠀⢀⡍⠭⠉⠉⠉⣃⠈⠋⠀⠀⠶⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣣⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠰⣏⣪⢳⠀⠀⠀⠀⠀⣠⣤⡶⠶⠆⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⠃⡄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠹⣦⣌⡭⠓⣄⠀⠀⠀⠀⢨⠀⠀⣀⡄⠀⠀⠀⠈⡒⢏⡀⢀⡁⠤⢀⣀⠀⢀⡴⡁⠴⢀⡀⠓⠀⠠⠤
⠀⠀⠀⠀⠀⠉⠀⢠⢖⣓⠦⠔⠀⠀⠀⠀⠋⠀⠀⠀⢴⡾⠀⠈⠋⠁⠀⠀⠀⠀⠉⠛⠱⠎⠠⡀⠛⠛⠀⠈⣄⠛⠀⠈⠁⠛⠦⠤⣔⣚⡀
⠀⠀⠀⣤⠀⠀⠰⠆⠫⠕⣀⡤⠀⠀⠀⢤⡁⠃⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢠⡶⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⢠⡄⢀⡄⠀⣀⡴⠚⢁⠀⠖⠀⠀⠀⠀⠀⢛⠁⠀⢀⡀⠻⠁⠀⠂⠀⡄⠒⠒⢀⡙⠆⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠘⠃⠀⢁⠰⠆⠉⠀⠀⠀⣠⠾⠷⠄⠈⢀⡀⢛⠁⢀⣵⠄⠀⠄⠰⠚
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠞⠁⠀⠀⠖⠂⠉⠁⠀⠲⠟⠁⢰⢩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠁⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣇⠀⠐⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖
⠀⠀⠀⠈⠉⠉⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠉⠉⠀⠠⠿⠿⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠿⠿⠷

⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⡄⠀⠀⠀⠀⣀⡀⢠⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⠀⣀⡀
⠀⠀⠀⠀⢿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠿⠿⠿⣿⣿⠗⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠈⠉⠛⠃⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠛⠋⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠐⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡤⠀⠠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶
⠀⠀⠀⠠⠿⠿⠏⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⣿⣿⣧⠀⠀⣿⣿⡟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠿⠿⠧

⠀⠀⠀⠰⣤⣤⡄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣶⣶⡶⠀⠐⣶⣶⣆⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣤⣤⡶
⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠧⠀⠠⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣦⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⡿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⡧
⠀⠀⠀⠠⣴⣶⣿⣿⣿⡿⠿⠟⠛⠉⠉⠀⠀⠉⠀⠀⠀⠠⣴⣶⣦⣴⣶⣾⣿⣿⣿⣿⣿⠿⠿⠋
⠀⠀⠀⢀⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⡿⠿⠟⠛⠋⠉⠁

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

⠀⠀⠀⠀⠀⠀⡤⠒⠀⠀⠐⠢⣀
⠀⠀⠀⠀⠀⠈⠀⠰⠆⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⢤⡂⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠉⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠳⣤⣷⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⢀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣄⡀⠀⠀⢹⣇⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢷⡀⠀⣠⠒⠂⠀⠐⠒⠚⠻⣍⠛⠳⠾⢿⣀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠈⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠙⠓⠲⠶⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠈⠐⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠾⠇

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠛⠉⠉⠉⠛⠛⠒⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣦⡀
⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠙⣦⡀
⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠁⠀⠀⠀⠠⠬⠷⣄
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠻⢷⣶⣶⣾⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠤⠞⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⣼⡟
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⣋⡁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⢿⣦⡀
⢀⣴⠶⠋⠀⠀⠀⠀⠲⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⠛⢉⡴⠋⠉⠉⠳⣄⡴⠋⠉⠉⠙⢦⠀⣀⣀⣤⣽⣷
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠠⠀⠀⠀⠀⣠⠟⠁⠀⣾⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⣧⠘⠻⠿⢿⣥⡀
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢶⡇⠀⠀⣴⠏⠀⠀⠀⢿⠀⠀⣴⣷⡄⠀⠀⠠⣾⣷⠀⠀⣿⠀⠀⠀⠀⠙⢿⣦
⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠻⢦⣴⣷⣶⠀⠀⠀⠘⣇⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⢰⠇⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠘⢧⣀⠀⠀⠀⣀⡴⠛⠉⠀⣾⣿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠙⢦⠀⠀⠀⣰⢆⠀⠀⠀⣰⠋⠀⠀⠀⠤⠀⠻⢿⣿⣿
⠀⠀⠀⠉⠙⠛⠋⠉⠀⠀⠀⢀⣈⣿⡷⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠙⠒⠛⠁⠈⠛⠒⠋⠁⠀⠀⢀⣴⠃⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⠁⠀⠀⠀⠀⠀⠀⢻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⡀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣤⣀⣀⣀⣀⣀⣀⣤⣶⣿⡿⠃⠀⠀⠀⠀⠀⢤⣿⡿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⡿⠿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢄⣀⡠⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣬⡿⠃
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄
⠀⠀⠀⠀⢿⣿⣛⡉⠉⠁⠀⠀⠀⢻⣍⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣩⣟⠁⠀⠀⠈⠙⠿⣶⣄⡀
⠀⣠⣤⣤⣴⣾⠿⠋⠀⠀⠀⠀⠘⠃⡶⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢶⠜⠓⠀⠀⠀⠀⠀⠀⢉⣿⣿⡆
⠀⠘⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠀⣰⣆⢰⢦⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡤⡆⢠⣆⠀⠈⡆⠀⠀⠀⠀⠀⠀⣠⣤⣼⠿⠁
⠀⠀⠀⠈⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⢣⡴⠃⠈⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠁⠈⢢⡼⠁⠀⠀⠀⠀⠀⠈⢿⣇
⠀⢠⣶⠿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠙⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⣿
⠀⢠⣿⣋⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠋⠉
⠀⠈⠛⠛⠛⠛⠛⠛⠛⢿⣶⠆⢀⣠⣤⣤⣶⠶⠶⠶⣶⣤⣀⣀⣀⡀⠀⢀⠀⠀⢀⣤⣶⣶⣦⣄⠀⠀⢀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣶⡶⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠿⠛⠉⠉⠛⠁⠀⠀⣀⣽⠉⠉⠉⠙⠿⣦⣿⡿⣿⠿⠃⠀⠀⠈⠻⣷⣶⡟⠋⠙⠛⠃⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠈⢹⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠓⠒⠒⠚⠃⠛⠒⠒⠚⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⣹⣿⣭⠬⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠏⠀⣸⣯⡴⡤⠤⠘⠓⢂⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣸⢯⢻⠗⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⡷⠿⠅⠀⠁⠉⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠲⢤⡀⠐⡉⠙⡗⠀⣿⣨⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠚⢛⠋⠀⠀⢰⠀⠈⠉⢹
⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⣀⣠⡾⢻⠶⢦⣶⡿⣷⣛⣓⣀⣠⡠⠉⠯⠁⠠⠤⢀⠀⢀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⣉⡀⠀⠛⢋⠁⠀⠘⡇⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠰⣤⡤⠖⠛⠉⠙⣦⣈⣽⠀⣾⢀⣿⣷⠛⠛⠋⣉⣩⣍⡉⣁⣠⣤⡤⣀⠈⢩⠭⠟⠛⣛⠛⢍⣉⣥⣴⣶⣾⣤⣄⣴⢮⣌⣽⠖⠀⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢈⣉⣁⣀⣈⣛⣻⣿⠿⢋⡙⠿⠁⠐⢀⣬⣁⣚⣷⠾⡿⠓⡺⠛⠛⠻⢷⣦⣶⣿⣍⣏⠉⠉⣻⣿⣏⡏⢱⣸⣆⡂⡆⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⢶⣅⡌⢹⣷⣶⣼⣿⠒⠚⠛⠈⠉⠉⣉⡥⠞⠉⢰⠀⠀⢤⡉⠙⠋⢁⣄⣀⣀⡉⡉⣋⣛⣲⣶⣿⠿⡇⢸⣉⣉⡉⢩⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠈⠉⠋⠉⠉⠉⠀⠀⣻⠀⠀⢀⢀⣴⠞⠉⢠⣲⣶⠾⡄⠀⠀⣻⣔⣷⡔⠒⢒⣶⣤⠦⣌⡉⠉⠙⠛⠑⠀⠀⢻⠀⠀⠸⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠆⠀⠹⣿⣤⣤⣄⣤⣤⡴⠟⠛⠉⠉⢩⠜⢣⡙⣷⣷⢿⣷⣤⡴⠀⢰⣯⣀⣉⣭⣤⠈⠩⣤⣭⢀⣿⠀⢠⡄⢀⡄⠛⠈
⠀⠀⠀⠀⠀⠀⢠⢢⡤⣤⣀⠀⠀⣀⡯⠉⠋⠩⠣⣳⣄⡠⠐⠉⢁⣴⣿⣽⡿⢟⣮⣽⣃⢠⡘⢿⡄⠙⠛⠿⠿⠶⠤⠴⠚⠋⣹⠀⣴⠖⠉⣀⣣⠖
⠀⠀⠀⠀⠀⠀⠀⠑⠂⠉⠉⠓⡂⠘⣂⡃⣠⡴⠶⠆⠠⣔⠋⠀⢴⣾⣿⣿⣵⣦⣨⣍⡩⠟⠁⠈⢅⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣰⠃⢀⡔⢁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠿⠄⠉⠁⠀⠀⠀⠀⠀⠈⠳⢶⣾⣏⣽⣿⣻⣿⣥⡎⠀⠀⠀⠀⠲⣍⠓⠒⠒⠒⠒⠚⠋⢉⡔⠁⣠⠉⡤⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉⢩⣧⡾⠉⠀⠀⠀⠉⠂⢀⡀⠀⠀⠀⠠⠊⢠⠞⣡⠀⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠔⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠵⠋⠉⠀⠀⠠⠤⢤⣀⡀⠀⠀⠈⢻⣄⡼⢁⠜⠑⠠⠇⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⣉⠛⢓⠚⠋⣴⠋⠀⢀⠀⠃⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠈⠋⠱⠿⢶⣦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣶⡄⠀⠟⠀⢘⠀⢠⡇⠀⠀⠈⢐⣵⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣄⠰⢟⡙⢇⣐⣄⢘⡤⠉⡉⠐⡟⢿⡳⣶⠦⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠀⠀⣼⡄⢸⡇⠀⠀⢁⡾⠁
⠀⠀⠀⠀⠀⠀⣠⣿⣤⢾⡁⣰⠂⢙⡾⠳⠮⠧⠐⣗⢀⣧⠊⠭⠉⢧⠬⢶⠂⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢹⡄⢸⢀⠔⠀⡼
⠀⠀⠀⠀⣠⣞⣉⡾⠉⢰⠶⢹⣷⢛⣆⠤⡴⠈⠀⠉⠒⠚⢆⣴⣄⠤⠀⣠⡎⠈⢳⠎⢩⠙⠶⠶⠔⠒⠀⠀⠈⠉⢘⣇⠸⠋⣤⠞
⠀⢀⠤⣾⢏⣁⢀⠓⣇⣽⡰⣏⢀⡜⠛⣳⢧⣍⣀⡴⢤⢠⣟⠁⢸⣶⠛⣩⣭⠶⠼⢶⠀⣿⠆⣀⣤⡄⠚⠉⢳⣦⣾⣿⠀⣰⠋
⣼⣇⠂⢩⠾⣧⣍⡉⠓⠩⣙⠿⢠⣴⣄⢳⣠⣟⠉⣵⠿⢇⣼⣶⡋⢃⣠⢍⠃⡼⢦⡆⣾⣤⣽⢟⣻⡿⠁⣠⠋⣼⣿⣿⣰⠃
⢇⠀⠉⠑⣾⣣⣤⢮⠕⣲⠀⠀⠀⠈⠉⡛⠃⣭⣴⠋⣦⡼⣇⢈⡿⠿⢷⡶⢿⡗⢀⣽⣥⡟⠃⠸⠋⡠⣪⠏⡼⢻⣿⣿⠁
⠈⠛⢶⣄⡀⠉⠻⠿⣶⣦⣈⠑⠢⠕⠢⢄⡉⠂⠤⣙⡛⠻⣾⣏⠙⣦⣾⣁⠀⡽⠋⠘⡦⠴⠇⠀⠜⢰⠏⣾⢧⣾⣿⣿⡄
⠀⠀⠀⠈⠉⠛⠢⢤⣀⠍⠛⠿⢷⣦⣄⡐⠂⠁⣒⠤⣉⠑⠀⢬⣛⠋⠡⣴⣻⡓⢂⣴⠐⠂⢟⠀⢰⠏⣼⠃⢡⡿⠭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⣀⡉⠙⢿⣿⣦⣌⡉⠒⠭⣶⡀⠀⠉⠐⠢⢄⣉⠋⠀⠶⡤⠎⣰⠏⣰⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⠉⠛⠻⢿⣦⣤⣀⠀⠛⠳⠶⠤⣬⣷⠀⠋⠀⣰⠋⣼⠃⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣆⠙⠛⠻⢿⣶⣤⣶⠂⠹⠆⠀⢀⡴⢃⡾⠃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠀⠤⠉⠛⠿⠦⣄⣀⡀⢠⠎⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣀⡀⠈⠙⣧⣠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠲⢦⣄⠀⠀⠀⢀⣠⡤⠶⠶⠒⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠻⣆⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠛⠛⠧⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢀⣀⣀⣀
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠉⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠈⣻⡶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣀⣀⣀⡴⢾⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠿⢫⡟⠛⠛⠻⣤⣞⡃
⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠙⢦⣤⣀⣀⣠⣤⡞⠉⢳⣦⣤⣀⣀⡀⠀⢀⣀⣤⣏⡿⣠⣭⡗⠀⠀⠀⣿⠉⢿⠄⢀⣄
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣯⠹⣤⣤⣤⣽⡜⢯⣩⣟⢻⣍⣧⠙⣴⣧⣴⠟⣦⣤⡾⠡⣿⡭⠤⠟⠼⠧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣶⢂⡶⢶⣲⡶⣠⠶⠶⣦⡴⢶⡶⣀⠰⡖⢶⣔⡶⠒⣆⢲⡒⠒⠓⢶⣘⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⢾⡉⢰⡇⠀⠀⢸⣗⠛⢰⠋⣷⠙⣻⡏⠀⠀⠘⣧⠛⢶⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⢧⣤⣤⣤⠟⡴⠞⠛⠳⢷⡙⣦⣀⣀⡴⣷⣦⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡴⠒⢲⢆⡘⣇⠀⠀⠀⣰⢃⡄⢶⠶⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡞⣷⠙⢲⣶⠚⢡⡏⠹⣎⢧⣍⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⢁⢺⡄⠙⠶⠛⠹⠷⠖⠋⠀⣸⠂⢰⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠟⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠷⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⠉⢩⣭⣭⡉⠈⠉⠙⠛⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢀⡀⠀⠀⠀⠄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣶⡌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠛⣛⣡⣼⡶⢶⣾⡶⠛⠓⠲⣦⣀⣀⠀⠀⠀⢙⣿⡿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⣶⣾⡿⠛⣿⡶⠶⣋⣷⡄⠀⢀⣤⣈⣿⣶⣤⣱⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣶⣶⣤⣄⡀⠙⠋⢼⣿⠇⢀⣾⠋⠙⠏⠛⠻⠇⢠⣿⣯⣉⡁⢹⣿⣿⣁⣀⣀⣴⣾⣿⣿⣿⡶⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⣻⠁⠀⠸⣟⣀⣀⣀⣤⣦⣀⣈⡉⠉⠉⠁⢘⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠚⣿⠟⠛⣿⡿⣿⣿⣿⣿⣿⣿⡿⠃⣤⣄⣤⣼⣿⣛⣯⣁⣸⣏⢛⣱⣦⣤⣤⣿⡽⣿⣿⣿⣿⠏⠉⢉⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠀⢄⠙⠻⢿⣿⣿⣿⣇⠀⠶⠟⢁⣙⡿⢋⣻⣛⠻⠿⢿⣿⣷⣄⠼⢿⣤⣿⣿⣿⢇⣀⡀⠷⠶⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⣶⣦⡀⠀⠙⠿⠋⢀⠀⠈⠁⠀⠚⠻⣿⣿⠿⣿⣿⣿⣿⣽⠛⣿⡭⠟⠻⡟⠋⠉⣅⠁⣴⡆⢈⣿⣿⣿⡛⠉⢠⣄⣀⠈⢻⣿⣿⣿⣿⣯⡀⣀⣴⣄
⠀⠀⠸⠳⣾⡿⣦⣀⠀⢀⡌⠁⠀⠀⢠⣴⠿⠋⢀⣴⣿⣧⣸⠀⢻⣤⠉⡀⠀⠀⣿⡆⢀⠙⣶⠛⢳⣾⣿⣿⣿⣧⡀⠛⠛⠋⣠⣼⡿⠟⠿⡿⢿⣿⡉⢹⣅⣠⡀
⠀⠀⢠⣀⣹⣿⡾⠛⠋⠉⢷⣶⣶⣶⣿⠋⠀⣴⡿⠃⢸⣿⣏⠀⠚⠁⠘⣿⠀⢸⡿⠃⠈⠁⡀⢠⣌⣿⣿⣏⠉⠛⢻⣶⣶⡾⠛⠁⠀⢀⣠⣴⡾⢿⣭⡄⠉⣿⠇
⠀⠀⠀⠉⠉⠀⠓⢀⠀⡀⠀⠀⠀⠉⠛⢻⣾⠋⠀⢀⣸⣿⣯⠄⢠⣀⣤⠙⡀⠘⠻⣄⠀⢠⣟⣾⣿⣿⣿⣄⡰⣆⣸⣿⣏⢁⣤⣤⡾⠛⢹⣿⣧⣀⣉⣹⣶⠁
⠀⠀⠀⠀⠀⠀⢤⡾⢿⣷⠀⢤⡀⣈⡀⢀⡁⠀⣠⣿⣿⢿⣿⣦⠉⣟⠁⡀⢹⡔⠛⣿⡤⢀⣿⠻⣿⡿⠟⠛⠛⠋⠙⢉⣍⠉⢉⣻⠀⣦⡼⠀⢋⣿⣿⢿⣟⡆
⠀⠀⢀⡄⠀⠀⠀⠀⠘⢛⣿⣶⣷⣬⡿⢿⣿⣿⣿⣿⣿⡌⣿⣿⣷⣿⣼⠷⠾⡷⣂⡏⣰⡿⠛⢁⣤⣴⣦⣄⠀⣠⣤⣾⣿⣌⠹⡿⠀⣻⣿⣾⣿⣽⣿⣾⣿⣿⡄
⠀⠀⡄⠀⠀⠀⡀⠀⣾⠛⠁⠀⠀⡴⠶⠾⠟⢀⡸⢿⣿⡤⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⠁⠀⠦⣿⡿⢿⡍⢻⣿⡛⢷⣤⣤⣿⣇⠘⣿⣏⣻⡟⢿⣽⡃
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣴⣤⣤⣤⡔⠂⠀⠀⢰⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠰⠉⢁⣨⠄⠀⣿⣧⠂⡙⣿⣿⣿⣾⣏⢙⢿⣥⡿⠹⡇
⠀⠀⠀⠀⠂⠘⢦⡘⣿⣄⡀⠀⠀⠀⢴⣦⣼⠿⠷⢶⣶⣶⣴⣿⣿⣿⡗⠒⠈⢉⣭⣭⣛⣿⠟⡩⠛⠋⠀⢠⠞⢃⡤⠀⣿⣿⣾⣧⠸⣯⣼⣿⣍⠀⠀⠉⣷⡆
⠀⠀⣀⣀⣀⠔⠛⠇⢨⡿⠛⢿⣷⣶⣾⣿⠁⠀⠀⠀⠁⢀⣼⣿⣿⣿⣷⣠⡿⢿⣿⡾⠛⠟⠂⠀⠀⡀⠀⠀⢠⣼⣀⡼⠟⠻⣧⣈⠂⢉⡉⢿⡇⣴⡀⡀⣿⣇⣀
⠀⢠⣶⣿⣥⣤⣴⣶⣿⠃⠀⠀⠀⢁⣴⣿⡇⠀⠀⠀⠀⣿⡿⠛⢛⣻⣿⣿⣶⣿⢯⣶⣾⣷⣤⣜⢻⣟⣙⣿⠁⢸⣿⣶⣶⡾⠛⢛⣻⡿⠃⠸⠿⢿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣎⣿⣿⣿⡇⠀⠀⠀⢾⣿⠃⣿⣿⠟⣽⣿⠟⢁⣼⣿⣿⣿⣿⣟⣿⣧⡈⣿⡗⠿⠿⠻⠋⣠⣾⠟⣉⣤⣾⣷⣦⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧⣶⠶⣶⣾⣿⣀⣿⣿⠀⣿⣧⡀⢻⡿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣁⣰⣧⡼⠿⣿⣿⣯⣭⣅⣤⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠉⠛⢻⣯⢻⣿⠀⣿⣿⣿⣿⣿⡷⢌⣻⣿⣿⣽⣿⣸⣿⣻⣿⣦⡁⠀⠛⢻⠿⣿⣿⣿⣿⣿⣿⣿⡛⡯⢹⣤⣿⣿⣿⣍⡙⣿⣌⡹⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡏⢰⣿⣿⣿⣿⣿⣧⠘⢻⣿⣿⣿⣿⠛⠻⢟⣫⢬⣷⣤⣤⣴⣄⡈⠉⠙⠿⠉⣷⣿⣿⡿⢋⣿⡇⢻⣿⣿⣷⣦⣉⣷⠃
⠀⠀⠀⠀⠀⣠⣴⣶⠛⠀⠨⢀⣿⣿⣿⣿⡟⠛⢿⣇⣻⣿⣿⣿⣿⣿⣶⣈⠁⠀⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣬⣽⣿⡖⠛⠁⠀⠘⠋⠹⣧⣶⡿⠃
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⣄⡈⢿⣿⣿⣿⣿⡷⠀⣿⡏⠉⢻⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠟⠿⣿⣿⣿⣿⠿⣿⠟⣶⣶⠘⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣶⣾⣿⣿⣿⣿⣿⠶⠋⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⣠⡶⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣏⣽⣟⣻⣽⣿⣾⣿⣿⣿⡿⠗⠀⢁⣤⣿⡿⢿⣿⣶⡶
⠀⠀⠀⠀⠀⣼⠃⠰⠀⠀⠌⠉⢛⡿⠿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣽⣿⡿⣿⣿⣿⣿⠿⠖⠚⠙⠛⠁⠀⠀⠀⠙⢿⣶⡀
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⡟⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣿⡇
⠀⠀⠀⠀⠀⣶⣠⠄⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣼⠃⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⢹⣿⡿⠃
⠀⠀⠀⠀⠀⢻⣇⠀⠀⡀⠀⠀⠀⢢⠙⠄⠀⠀⠀⠀⣿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⡃⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠛⢿⣤⣆⠀⠀⠀⠀⠤⠀⠀⠆⠀⠀⢿⡋⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⣿⣷⣶⣤⣴⣦⣄⢘⣿⣇⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣀⣀⣄⣀⣄⣀⣀⣠⣀⣤⠜⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡛⠻⢿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡄⠀⢰⣿⡿⠿⣿⣿⣿⣿⣿⣿⠟⠋⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠉⠁⢀⠀⠸⢿⣿⡇⠀⠀⣯⣿⠆⡀⠈⠉⠉⠉⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⠀⠀⠀⠐⠛⢁⣀⣴⣾⡟⠀⠀⠀⠈⠛⠿⣦⣤⣤⣤⣤⣴⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣉⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣤⣼⣿⣿⣿⣿⡿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠋⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠢⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠠⠎⠑⢢⡍⠀⠁⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⠞⣄⡐⠆⠀⠁⠄⠠⠙⠀⠢⡀⠀⠠
⠀⠀⡀⠄⠀⠀⠀⠀⢀⠀⠒⠒⠀⠀⢀⠝⠐⠀⠔⠐⠤⠀⠀⠀⢀⡄⠀⠃
⠀⠀⠁⠀⠀⠀⠐⠊⢱⡦⠛⠣⢀⠔⠁⠀⠀⠀⠀⠈⠠⠃⢰⠆⡀⠄⠀⠀⠀⢄
⠀⠀⠀⠄⣂⠀⠀⠀⠀⠀⠴⠚⠁⠀⠀⠀⢀⠀⠀⢀⠖⠀⠈⠀⠁⠨⢁⡂⢠⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⡈⠀⠲⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⢣⠤⠁⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠀⠠⠀⠀⠉⠠⡀
⢐⠂⠀⠈⠀⢀⠀⠀⠀⢀⡄⠀⠀⠀⡃⠀⢂⢀⠠⠀⡈⡹⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠩⠀⢀⠀⠀⠈⠠⠂⠀⠁⠠⠂⢰⠀⡊⠈⠈⠇⠀⠑⠐⠁⢐⣄⠀⠠⡀⠀⢀⢀⠄⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣢⠄⢤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠠⠄⠉⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠃⠼⢀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠠⠊⣠⠀⡀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⢀⠒⢀
⠀⠀⠀⠂⠉⠀⠈⠖⠀⠀⡠⠒⠄⠀⠀⡀⠀⠀⣀⠀⠲⣤⠈⢙⠂⢠⠒⡄⠠⡀⢂⠈⠄⠑⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠁⢀⠊⠃⠀⠘⠂⠀⠉⠀⠈⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠆⠐⠒⠊⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⣿⠶⠜⠤⡶⣿⣷⣴⣷⣀⣀⠠⢀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⢀⠸⠿⢷⠿⢿⠤
⠀⠀⠀⠀⠀⢸⣿⠁⡰⠀⠂⠂⢰⢷⠀⠪⠩⠟⢋⠙⢭⠿⠦⢷⣦⡟⢤⠾⠆⡨⣁⠀⠏⠇⠈
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠐⠤⠌⠀⠘⠀⠀⢈⠁⠠⠀⡁⠀⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢠⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢘⠿⣦⡈⠳⣤⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠘⢿⣮⣿⣶⣤⡉⠓⠶⣤⣀⠀⢰⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⢿⣿⣿⣿⣿⣿⣷⣦⠀⠈⢳⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢿⣿⣿⣿⣿⣿⣿⣠⠴⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠻⢿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣿⣿⣿⡿⠛⠁⠀⠘⠃⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡘⠿⠿⠿⣿⣿⣿⣧⣤⣤⠀⠈⠉⠀⠀⠀⠀⠀⣴⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠐⢾⣿⢿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⣤⣩⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢒⣒⣒⣒⡒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣈⣩⣭⣭⠭⠍⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠌⣩⣤⠶⢚⡩⢔⠀⠀⠀⣾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡍⣴⡾⢋⡴⣫⢫⢢⡷⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣆⠴⢏⣼⢃⡾⢸⡇⢿⡆⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠧⠵⣮⣥⣮⣡⡶⠴⠋⠁
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣼⣶⣶⣶⣶⣷⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢧⡄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣁⣹⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠂⠀⠿⠆⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢈⣽⠛⢻⣿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠂⣴⡄⠀⢀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⣿⡏⠙⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⡍⢁⡀⠉⠁⠀⠀⣴⣶⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⢻⣿⡟⠛⠚⠛⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⡄⠀⢰⣼⣭⣦⣮⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠓⢓⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠈⠀⠠⠀⣶⡶⠀⠀⠀⠠⠀⠀⠀⠀⠐⠖⠓⠶⠀⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣭⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠚⠓⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠑⠛⠙⠚⠓⠓⡁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠀⢠⠖⠀⠉⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠚⠐⠳⠖⠤⠂
⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠌⣠⠖⠈⠈⣧⣽⠋⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡐⠧⠢⠔⠦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡯⠀⠈⠐⠉⠈⠒⠚⠀⠙⢆⢀⡠⢀⢀⠀⢠⡄⠤⠒⢲⣴⣧⣶⡜⢚⣒⠘⢀⠀⡀
⠀⠀⠀⠀⠀⡠⠐⡐⠒⠊⠉⠁⢡⣹⣷⢰⠀⣰⣶⡆⠀⡄⣯⡴⣄⢅⠠⡌⡛⣿⣏⣴⠀⠌⠁⢉⡁⡀⣀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⣌⠄⠈⠉⢅⣐⣋⠏⡁⢹⠀⢀⡇⠀⣠⠤⠼⠂⠂⡁⠀⠈⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⢀⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠋⠉⠁⠉⠐⠄⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠤⠒⠒⠒⠒⢊⡩⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠭⢤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠑⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⣑⣦⡄
⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠒⠒⠊⠉⠉⠉
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠓⠉⠉
⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⠁
⠀⢠⠞⠀⠀⠀⠀⠀⣀⣀⣀⠠⡤⣤⣤⠄⠀⠀⠀⠀⠀⣀⡤⠒⠉⠁
⣠⣋⡤⠔⠒⠊⠉⠁⠀⠀⢀⡠⠒⠉⠀⠀⠀⢀⡠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠐⠒⠒⠒⠒⠒⠂⠤⠤⣀⡀
⠉⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠚⠉
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⠁
⠀⠀⠀⠀⢀⠜⠁⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⣰⠋⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⡴⠃⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⣀⣀⢣
⠀⡰⢁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢠⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉
⠉⠀⠀⠀⢶⠒⠒⠒⠓⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡼
⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⣰⠃
⠀⠀⠀⢠⣤⣤⠤⠒⠉⠀⠀⣀⣠⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠰⠧⠤⣠⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠲⠤⣤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠧⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⠚⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡏⠁⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⢃⣹⡴⠇⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢫⠛⢻⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⣇⣤⣰⢲⣿⢧⡛⣷⠈⠢⣀⠀⠀⠀⠀⢀⡴⠋⡧⡭⣷⣜⡜⣅⠻⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠙⠤⠒⣴⣯⡇⠀⡯⠹⡄⠀⠈⠓⠲⠶⠒⠋⠀⢠⡏⣭⣉⢻⠑⢧⠰⠆⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⢳⣦⣰⢿⣏⡇⣠⡟⢵⡿⣄⠀⠀⠀⠀⠀⣀⣴⣿⢷⢜⢯⢿⡇⡤⢍⣊⡉⣷⣏⣿⣿⣿⣿⣿⣷⡀
⠀⢠⣿⣿⣿⣿⣿⣿⡿⢻⡷⢋⣳⡏⠻⣿⣯⠚⢃⣌⡁⣝⢦⡀⢀⡴⢚⣡⡮⣋⠝⡬⢻⡼⡠⢌⠛⢷⠺⡿⣿⣏⡿⣿⣿⣿⣿⣧
⠀⣾⣿⣿⣿⣿⣿⢏⣤⣾⠴⠜⢸⢣⢿⣾⢻⡈⠂⠛⣧⠠⢧⣿⣾⣒⣛⣛⢦⡥⠴⢴⠿⠤⣌⣻⣀⣷⣜⢁⠇⢿⡇⣿⣿⣿⣿⣿⡆
⢀⣿⣿⣿⣿⡟⣙⢧⠗⡗⡐⠈⠹⡟⢸⣷⠋⡟⢺⢒⢃⠖⠀⢌⡏⢿⢽⡁⣘⠄⣍⣁⣜⠒⡢⡜⣀⠘⣊⣾⡗⣼⡷⢋⣿⣿⣿⣿⡇
⢸⣿⣿⣿⡟⢁⡭⣭⠽⢿⣙⠩⣄⣯⡺⠁⠀⢸⡇⣎⡙⡄⢟⠏⠛⡘⠈⢳⡏⠉⡆⣛⠃⡺⠇⣳⣩⣼⠋⣿⣽⡿⣾⢿⡿⣿⣿⣿⣿
⢸⣿⣿⣿⡗⢋⣴⠿⢄⡼⢳⡇⣎⣼⣷⡿⢏⡗⣱⡚⠧⢻⡯⢠⡕⢱⡆⣄⢿⣲⣬⣤⢵⡵⣞⣹⣷⢏⣤⣿⠟⣰⣿⡏⠀⠘⣿⣿⡟
⠈⣿⣿⠻⣗⠏⡎⡎⣉⠒⢡⢱⣍⢶⢙⣧⠛⣰⣅⠟⣮⡤⠖⢉⠒⢁⣤⢣⢸⠙⠀⡉⣛⡾⢾⣿⣿⣿⣿⠿⣛⡿⢿⡇⠀⠀⠘⣿⡇
⠀⢿⡏⠀⢿⡔⡡⢖⡸⣫⡳⡬⢌⡷⣍⣩⣎⠡⢁⠼⣒⣛⠲⣽⠖⣣⢿⢨⡶⠃⣎⢁⡿⣗⢺⠘⢆⡛⣋⡭⢋⣟⡾⠁⠀⠀⠀⢹⠃
⠀⠘⠁⠀⠈⢻⣄⣉⠣⣉⠴⣇⢫⢶⠁⡙⠺⣷⣤⣤⣅⠠⢇⡺⢿⠷⣯⠌⢀⠱⢚⣇⣣⠊⣉⠴⢉⢖⣲⣀⣠⢾
⠀⠀⠀⠀⠀⠀⠙⢾⣌⡡⢋⣒⣊⢜⣀⡰⡴⡡⣩⠟⣻⡷⣶⣛⣿⣻⠋⡿⡾⢿⣵⣂⣴⣁⣘⣃⣯⡿⠷⣫⡜⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣊⡤⢲⠃⣠⡇⢰⣥⣧⠏⣏⡴⣫⢵⣿⣮⢸⡰⣷⠿⠓⡩⣯⣔⠼⠏⡏⠶⠤⠂⠯⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣋⢡⡼⢃⣺⣿⣥⡿⢯⣴⣧⣿⣋⣻⣽⢊⡑⠠⡒⠂⣈⣀⡟⠊⠀⠤⡄⠒⡌⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠛⠛⠛⠻⣍⣟⠉⣛⣥⣄⣠⢿⠈⢠⢠⡑⢾⠒⢺⢃⠤⢲⣔⠳⣺⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⣛⡿⠋⢴⣷⠓⠆⠥⠙⡌⢦⢢⡾⢣⣘⢸⢋⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣨⣆⣠⡾⣀⡒⠑⢢⣱⠿⣥⢰⠢⡇⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⡡⢜⡃⠈⣰⠦⢤⠀⠰⡀⢢⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⡇⠀⣛⡚⠉⢴⣶⡵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣅⣚⣽⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣷⡾⠀⠺⣶⠀⠀⠀⠀⠀⠀⠀⠀⠤⣈⣠⣡⣄⠉⣿⡟⠻⠛⠀⣠⠃⣨⠀⣉⠀⠘⠛⠛⠄⠐⠀⡆
⢿⣷⣤⣴⠿⠀⠀⠀⢀⣀⣀⠠⠄⢾⡏⠀⠈⣿⠆⠀⠐⠀⠀⠀⠉⠆⠀⠖⠌⠈⠀⠀⠀⠀⠀⠀⢤⠶⠀⠶⡄
⠀⠀⠄⠀⠀⠀⠀⣼⣿⡿⢿⡇⢠⠸⣧⣄⣴⡿⠁⢠⠶⠶⢦⠀⡐⢀⠀⣀⠀⠰⠆⠌⠈⡀⠄⢠⡌⢦⣤⡶⠁
⠀⠀⠀⠀⠀⣀⠀⠹⠂⠀⠂⠇⠀⠀⠉⠉⠀⠀⠀⡄⠶⠰⠦⡆⠁⠈⠁⠉⠀⠀⢲⡿⠋⠄⣷⠀⠀⠈⠉⠁⡀
⠀⣀⣭⣅⣀⠉⠀⠤⠠⠤⢄⠀⠀⠀⠀⠤⠀⠀⣀⠘⢢⡄⢈⡏⠀⠀⠀⠀⠉⠀⡸⠀⣤⡄⠋
⠘⠉⠂⠘⠉⠠⠀⠀⠀⠀⠀⠀⢀⣦⣀⢀⣀⡄⠉⢀⣏⠉⠈⣧⠒⣰⣟⠋⣿⣏⠁⠀⠁⠉⢀⠀⣠⣴⣶⣦⣀
⠰⢠⣤⣤⡀⠀⡄⠀⢀⣈⣀⠀⢀⣷⡍⠍⣩⣿⠁⠀⠁⠀⠀⠀⠀⣿⣏⠀⢩⡿⠀⠀⠀⠀⠀⠀⣏⢩⠉⣀⠉
⠂⠀⠀⠀⠉⠀⠀⠸⠟⠋⠙⢿⠐⠙⠋⠀⠙⠋⠀⠀⠐⠀⠂⠀⠐⠀⠚⠛⠛⠀⠀⣠⠄⠰⣄⣀⠀⣆⡀⠀⠰
⠀⣠⣤⣀⠀⠀⢀⡁⢠⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠄⠶⡷⠀⠀⠀⠀⠀⠀⢐⣯⣤⢤⣼⡏⠀⠛⠉⠉⠉
⣎⠹⠠⠌⣡⠁⠈⠀⠀⠀⠀⠀⠀⠴⠞⠻⠦⠰⠇⣿⣿⠂⢾⣇⠃⣞⣉⣉⣢⠠⠌⠿⠀⠄⠼⠿⠂⠀⠠
⢸⣄⢠⣀⣾⠀⠀⠀⠀⠀⠀⢀⠀⢄⠂⠐⠂⠆⠀⠀⠀⠀⠀⠉⠐⡏⠀⠈⠹⠀⠀⠀⠀⠁⠀⠀⣰⠞⠓⢦⠠
⠈⠙⠀⠘⠁⠀⠀⠀⠀⠀⠀⠜⠃⠘⢦⡤⠞⠀⠀⠀⠀⠐⠀⡀⠐⠀⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠄⠠⢸
⠀⠀⠀⠄⠀⠀⠀⠀⣌⠀⡐⠂⡀⠀⠀⠀⠀⠀⢠⡚⠄⠡⣀⡁⠀⠉⠐⠛⠊⢰⣤⡠⠥⣀⡀⠰⠄⢠⠤⡀⡁
⠀⢀⣴⣦⣄⠀⣠⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⣼⡗⠂⠐⣿⠆⠀⠐⠂⠂⠀⠐⣏⠲⠀⠒⡡⡀⠀⠉⠀⠉⠁
⠀⠾⡿⠿⡿⡇⠀⠀⠀⠀⠄⠀⠒⣠⣿⣦⣄⠦⣿⣿⡦⢶⣿⠀⠀⠀⠀⠀⠀⣤⠈⠿⠿⡞⠉⠀⠀⢀⣀⣀
⡤⠘⢤⣀⡴⠁⠁⠀⠀⠀⠠⠀⠰⣿⠯⠭⢻⡄⠉⠁⠀⠀⠀⠀⢰⢤⠀⣠⢵⠀⠐⠏⠁⠁⠀⠀⢰⣭⡍⠉⣃
⠀⠀⠛⠉⠓⠀⢀⣤⣶⣦⡄⢀⠀⢁⣤⣤⡌⠀⢀⠀⠀⣀⡀⠀⣀⠉⠠⠀⠘⠀⠀⣀⣀⠀⢀⠀⠘⠇⠀⠀⠟
⠁⠀⠀⠀⠀⠀⠘⠒⠀⠀⠻⠀⠈⢉⠉⠉⠉⠁⠀⣼⡿⢿⣿⡆⠉⣄⠈⠠⠀⢀⣚⡓⢚⣓⠘⠃⠀⠀⠈
⢀⣴⠶⢦⡀⠀⣤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠈⠛⠃⠀⠀⠀⠀⠀⠘⠏⠁⠈⠙⠀⠂
⣾⣶⣶⣶⢶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣷⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⢶⡶⣶⣶⣶⣶⣶

⣀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠿⠿⣿⠿⠗⠀⠀⠀⠀⠉⠀⠀⢀⣴⣄⠀⠹⠇⢀⡀⠀⢰⣿⣿⣷
⣿⡀⠀⠀⣿⣿⣿⠄⠀⠆⠀⣴⡁⠀⠀⠀⣾⣿⣷⠄⠦⠀⣾⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⠀⠉
⣿⣷⡀⢴⣿⣿⣿⣧⠀⠀⣼⣿⣷⡀⠀⣠⣿⣿⣿⡄⠀⠰⣿⣿⣿⡷⠀⢀⣄⣀⠺⠿⢿⣿⠿⠟⠂
⠻⠋⠁⣼⡿⣿⣿⣷⠀⢀⣼⣿⣷⡀⠀⣹⣿⣿⣿⡏⠀⠼⠿⣿⠿⠿⠄⣹⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⢈
⠀⠠⠄⠀⠀⠙⢠⡄⠀⢉⣿⣿⣿⡉⠀⠛⠛⣿⠛⠛⠂⣄⢠⠘⠀⠀⠰⣿⣿⣿⠧⠀⠉⣀⠀⠐⢻⣤⡞
⣤⣤⣤⠄⠀⠀⣾⣿⡀⠚⠛⣿⠛⠛⠂⠀⠀⠉⠀⣸⡄⣏⢹⠀⣼⡆⠼⠿⣿⠟⠷⢀⣾⣿⣷⡀⠿⠛⣿
⠿⢿⢿⠀⠀⣼⣿⣿⣧⠀⠀⠉⠀⢦⡼⣆⠀⠃⢠⣿⣧⠈⠉⣸⣿⣿⡀⠀⡉⠀⠀⣸⣿⣿⣿⡇⠀⠀⣿
⠀⠸⢸⠀⣰⣿⣿⣿⣿⣇⠀⠋⠀⣾⡿⠉⣀⠀⠛⢿⠿⠓⢰⣿⣿⣿⣷⠀⠉⠀⠰⢿⡿⣷⣿⡿⠆⠀⠏
⠀⠀⣰⡀⠛⠛⢻⡿⠛⠋⠀⠀⠀⠙⠶⠿⢫⣾⣿⣄⠀⢀⣿⣿⣿⣿⣿⣇⢰⡆⢰⣿⣿⣿⣿⣿⡆
⠀⣰⣿⣷⡀⠀⡀⡀⠀⠀⣴⣶⣶⡀⠀⠀⢿⣿⣿⣿⠆⠀⠀⠉⠻⠉⠁⠀⣾⣷⡀⠀⠀⠟⠀⠀⠀⠀⠈
⠀⣹⣿⣿⡉⢰⡍⣵⠀⢈⣿⣿⣿⡅⠀⣰⣿⣿⣿⣿⢧⠠⢶⣀⡴⠃⠀⠾⢿⡿⠷⠀⢀⡄⠀⣼⣶⣶⡀
⠴⣿⣿⣿⡿⠆⠉⠁⠀⢾⣿⣿⣿⡿⠄⣹⣿⣽⣿⣽⣯⠀⣾⢿⣷⣀⣁⠀⢀⡀⠀⢠⠀⠀⢠⣿⣿⣿⡄
⣰⣿⣿⣿⣷⣄⠀⠂⢀⣾⣿⣿⣿⣷⡄⠛⠛⢻⡟⠛⠛⠃⡄⠀⣿⣿⣿⣿⣿⠀⢠⣿⣧⡀⢻⣿⣿⣿⡟
⠀⠀⢻⠁⠀⠀⠀⠀⠈⠁⠀⢿⠀⠀⣄⠀⢄⠈⠁⢀⠀⢀⣿⡀⡟⡇⠀⢸⢸⠀⣠⣿⣿⡀⠟⠛⣿⠛⠛
⠀⣧⠀⠀⠂⣰⣶⣶⡀⠀⠃⠀⠀⠀⣸⣶⣏⠁⠀⣀⢀⣾⣿⣧⠀⠁⠀⠈⠈⠘⣻⣿⣿⡛⠀⠀⣉⠀⠤
⣼⣿⣆⠀⠀⣽⣿⣿⡏⠀⣶⣶⣶⣶⣿⠋⠙⠁⢠⣿⣎⠙⠟⠉⠁⢠⣾⣦⡀⠘⠛⢻⠛⠛⠂⣴⣿⣷⡀
⠻⡿⠟⠂⢾⣿⣿⣿⡿⠄⡟⡟⠛⢻⣿⠀⠀⠐⢻⣿⣟⠂⠀⠀⢰⣯⣿⣿⣷⡀⠀⡈⠀⠀⢠⣿⣿⣿⡅
⢠⡄⣀⠀⣼⣿⣿⣿⣿⣄⠧⠃⣤⠘⠟⠀⢀⣠⣿⣿⣿⣦⡀⠀⣴⢿⣷⢿⣿⣄⢰⡟⣳⠀⣹⣿⣿⣿⣏
⢾⣽⡅⠈⠁⠀⢹⠁⠀⠀⠀⣸⣿⡆⠀⠤⠀⣰⣿⣿⣿⣆⠀⠸⣿⡿⣿⣿⢿⡿⠀⠓⠃⡄⠉⠉⢿⠉⠉
⣤⣙⡟⠀⠐⠂⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠘⠋⠉⣿⠉⠉⠃⣰⣿⣿⣿⣿⣿⣿⡆⠀⣸⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⡄⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⢧⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⢳⡄⣤⡀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢣⡴⠋⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠶⠿⠼⠇⠟⠹⡄⠀⠠⠐⠓⢦⣄⠀⣠⢋⠔⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡀⠑⠬⠀⠃⣀⠀⣤⠀⠀⠘⣷⢡⠃⠀⠀⠀⠀⢈⡹⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠉⢁⡀⠀⠈⠡⢴⢀⣀⠀⠀⠀⠈⠀⠀⠤⠞⠉⠉⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠑⢦⡀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣙⣆⣀⣀⡤⠔⠉⠀⢁⣤⣥⡽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧

⠋





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀⣀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠙⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⡿⠟⠋⠉⠀⢀⣀⡀⠀⠐⠒⠲⢶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀⣤⣬⣭⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣫⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⣠⡾⠟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡟⠳⠄⠀⠀⢿⣿⣿⣿⣿⠇⠻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠉⠒⡗⠊⠁⠀⠀⠀⣤⣶⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠙
⠀⢰⡏⣰⠿⢷⡄⠀⢠⣾⣿⡇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣾⣿⠆⢠⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⢀⣀⣀⣸⠇
⠀⠈⢣⠁⠈⣀⠏⠀⠀⠉⠉⠀⠀⠀⢠⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠉⠒⠐⠃⠀⠀⠀⢀⣤⣤⣤⡄⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⢀⣈⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡟⠿⠿⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠉⡇⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠸⠟⠠⠂⠀⠸⠛⠻⠀⠀⠀⣠⣶⡆⠀⠀⠀⠀⢰⠇⠀⠀⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡇⠀⠀⠀⠀⣴⣶⣤⣤⣤⡄
⢀⡀⠀⠀⠀⠀⢀⣈⢀⣀⡀⠀⢠⡿⠛⢻⠀⠀⣴⣶⣾⢀⣤⣤⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠛⢿⣧⠀⠀⠀⠠⣾⣿⠇⢿⣿⠟⠀⠀⠀⠀⣿⣭⣭⣭⣽⠁
⢸⣷⣤⡀⠀⠀⠙⠋⠈⠛⠁⠀⣾⠁⠀⣼⠀⠀⠉⠛⠁⠘⠛⠛⠁⠀⢰⣿⣿⣿⠀⠀⠀⠀⣿⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠛⢻
⠈⡇⠉⠳⠀⠀⢀⠀⠀⠀⠀⠀⣯⣠⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⣀⣀⢠⡏⠀⠀⠀⠀⠀⢰⠃⠀⠀⡶⠆⠀⠀⠀⢀⣤⣤⣿⠀⣀⣀⣸
⠀⣧⠀⠀⠀⠀⡸⠀⠀⠀⢀⣤⣿⣿⠏⠀⠀⠀⠀⠀⢰⣶⣶⣶⣷⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡼⠀⠀⠘⠀⡼⠀⠀⠀⠹⢿⣿⠟⠸⣿⣿⡿
⣿⡿⠀⢀⣠⣀⠇⠀⢀⣴⣿⡿⢻⠁⠀⠀⠀⠀⠀⠀⠘⣷⣶⣶⣿⡄⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⢠⣶⣶⠇⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠉⠁⠀⠈⠛⠛⠀⢀⣾⡟⠁⢀⣸⣦⣀⡀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣇⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠈⠙⠉⠀⠀⢰⠟⣠⣤⣄⠀⠀⠀⢘⠛⠛⠀⠀⢀⡄⠀⠈
⠀⢀⣀⠀⠀⠀⠀⢸⡟⠀⢰⡿⠻⡟⠻⣷⡄⠀⠀⣠⣶⣾⡀⣾⣿⣿⠀⠀⠀⠀⢸⠉⠉⡇⠀⠀⠀⢀⣠⣤⡄⠀⠸⡀⠏⠠⢸⠇⠀⣀⣸⢀⣼⠀⠀⣼⠃
⠀⣼⠻⢿⣶⡤⠀⠸⣇⠀⢻⡀⠀⡇⠀⢹⡇⠀⠀⠙⠿⠛⠁⠉⠋⠁⠀⠀⠀⣀⣨⢀⣤⣧⠀⠀⠀⢸⣿⠶⢧⠀⠀⠈⠒⠐⠋⠀⠀⠙⠋⠈⠉⠀⣰⠏
⣸⠻⠿⣷⣾⠃⠀⠀⠘⢦⣀⠑⠀⣧⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠉⠁⠀⠀⠀⣀⡆⣠⣼⡀⠀⠀⣿⢆⠐⠀⠀⠀⠀⠀⠀⢠⡟
⠃⠀⠀⢠⠏⠀⠀⠀⠀⠀⠈⠉⠉⢻⠁⠀⠀⢠⣶⣾⠿⢿⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣄⣾⠁⠀⢠⣾⣶
⢸⣿⣷⡟⠀⠀⠀⠀⢀⣴⣶⡄⠀⢸⠀⠀⠀⠘⣇⠀⠀⠈⣧⠀⠀⠀⠀⣾⠉⣃⠀⠀⠀⠘⣷⣄⠀⠀⢀⣄⣤⣤⣤⣤⡄⠀⠀⠀⠘⢿⣿⣿⣿⠇⠀⠀⠀⠙⠛
⠀⠉⠉⠀⠀⠀⠀⠀⠘⣿⠿⠃⢀⡼⠀⠀⠀⠀⢹⡀⠀⢀⣸⡆⠀⠀⠀⠅⢠⡟⠀⠀⠀⠀⡇⠈⠀⠀⢸⡿⠿⠿⠿⠿⣧⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣤⣤⣤
⣴⣦⣤⣤⣤⣄⡀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⣤⣶⣧⠸⣿⣿⠇⠀⠀⢀⣼⡿⠁⠀⠀⣠⣶⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢿⠶⢿
⣿⠛⠛⠻⠿⢿⡇⠀⠀⠀⢀⣄⠀⠀⠀⠀⠈⠛⠟⠋⠀⠀⠀⠀⠀⢠⣿⠋⢃⣀⠀⠀⠈⠋⠁⠀⠀⠀⠀⣏⠉⠉⠉⠉⢿⠀⠀⠀⠸⡿⣶⣄⠀⠀⣤⣼⢠⣶⡆
⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣿⠁⣴⠟⡛⢿⡆⠀⠀⠀⠀⠀⣀⡀⣿⠀⠀⣀⣀⣸⡄⠀⠀⠀⢳⠀⠈⠃⠀⠉⠉⠀⠉
⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⢿⣿⣦⡀⠀⠀⠀⠀⢹⠳⣆⠀⠀⠹⡄⠛⠄⢃⢀⡇⠀⠀⠀⢰⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⣇
⠀⢀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣿⠀⠉⠻⣧⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⠓⠒⠚⡏⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠙⠛⠛⠋⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⣠⣶⣶⣤
⠰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⣴⣾⡇⠀⠀⠀⠀⠀⣴⣦⠀⢡⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠙⠃⠖⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣤⡄⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡿⣶⣦⣤⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⡴⠋⢹⠀⠀⠀⠀⠀⡿⠿⢷⠆⠀⠀⠀⠀⢠
⢿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⡿⣷⣶⣭⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣴⣶⡿⠀⠀⠀⢸⢁⣴⠏⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠸
⣠⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢀⣤⣄⣸⠃⠀⠀⣹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠀⢀⣤⣶⣿⠟⠁⠀⠀⠀⠰⣿⡿⢰⣶⣾⠀⠀⠀⢠⣤⣆
⠛⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⣴⣿⣶⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠿⠋⣁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⡼⠛⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡏⢠⡾⢻⠻⣷⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⣧⡾⠁⠀⠀⠀⢸⡆⠀⠀⠀⣄⡀⠀⠀⠀⠀⢀⣤⣤⣤⣼⡇⢀⣤⣶⣶⣼⡇⠀⠀⠀⠀⢇⠘⠀⠘⠀⣿⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡿⢿⣦
⣶⣶⣶⣾⣿⡷⢷⣾⣾⡿⣾⣷⣶⣷⣿⣿⣿⣿⣾⣶⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣾⣷⣾⣿⣿⣿⣾⡶⠶⢶⠾⠿⣾⣶⠶⠾⠿⡿⠾⠿⠾⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣖⡶⠖⠒⠒⠒⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⢩⢃⠔⠉⠉⠁⠐⠒⠤⠤⠤⠍⣙⣛⣶⣶⣦⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⢀⠔⠝⠻⣄⡀⠀⣀⣀⣤⠤⠶⢶⣶⡖⠲⢬⡉⠉⠁⠀⠀⠀⠈⠉⢿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡞⠌⠀⠀⡟⠀⠀⠉⠲⣄⡀⠀⠀⠉⠫⡢⡀⠈⢦⣀⡠⠤⢀⠀⠀⢸⢀⠟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢹⢳⠦⡄⠀⠰⠀⠀⠀⠀⢻⠙⢷⣄⠀⠀⠘⣝⡄⢸⠀⠀⠙⠲⠚⠛⢺⢋⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢅⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢷⡄⠀⠘⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣙⣿⣶⣄⣀⣀⣈⣷⣀⣀⣾⡄⠀⠀⠈⢿⡄⠀⢹⣿⠀⠀⠀⠀⠀⣀⣰⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢩⡟⠘⡜⡄⠀⠀⠈⣿⡀⠘⡏⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠐⢰⠀⠀⠀⠀⠃⢰⣁⡰⡐⠀⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠆⠀⠀⢸⣸⣟⠙⠛⠷⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢠⢀⠀⢀⡜⠁⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢠⠀⠀⠀⠀⢠⣸⠔⠈⠀⢀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠙⠀⠀⠀⠀⠆⠄⠀⠀⣠⠞⠁⠀⠀⢠⠃⡄⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠠⠰⣀⡾⠉⢧⠀⠀⣠⠃⡜⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡾⡁⠀⠀⠀⠂⠀⠀⠀⠀⠀⡷⠋⠀⠀⠘⣇⡴⢁⠌⠀⠀⠀⡸⢸⣿
⠀⠀⠀⠀⠀⠀⢠⡟⠇⠠⠀⠀⠀⠈⠀⠡⠀⢀⡾⠁⠀⠀⠀⠀⢻⣤⠃⠀⠀⠀⡰⠁⡘⢻⡀
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⡀⠀⠀⠀⠀⢡⡼⠁⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⡴⠁⡐⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⣸⠃⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⠜⢀⠎⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠁⡠⠁⠀⠀⡰⠁⣷
⠀⠀⠀⠀⠀⢸⡇⠑⠄⠀⠀⠀⠀⠈⢄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⠀⠀⢀⠜⢀⠜⢻
⠀⠀⠀⠀⠀⢸⡇⠑⠀⠢⡀⠀⠈⢄⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡠⠋⡠⠂⠀⢸
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠂⠀⠀⠲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡠⠊⢀⠔⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢻⡃⠀⠀⠈⠢⡐⢄⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠞⠁⠀⢀⡤⣶⡿
⠀⠀⠀⠀⠀⠀⠈⣿⠘⢔⡀⠀⠈⠢⡑⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠤⢒⣪⡵⡎⣿⢷
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠥⠾⢶⣤⣬⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡏⣿⠀⡇⣷⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠲⡟⠻⡏⢸⡉⣟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡧⠿⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢳⡀⣇⠈⡧⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠮⢠⠇⠀⠀⠀⣤⢠⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣥⡎⢸⠋⠀⠀⠀⡇⠿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣤⣤⡀⠠⠶⡶⠏⠐⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⢻⡿⠿⢿⢷⣶⡿⣟⣻⣿⡲⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡿⠉⢁⣶⠓⡿⣛⣫⣽⠟⠾⡛⣿⣿⠴⠶⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣀⡑⢸⣿⣷⣶⣿⣿⣿⣭⣾⣿⡭⢉⣀⣀⣚⣿⣁⣠⠖⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⡶⣻⣷⡿⠿⡿⣫⣿⠃⢰⠟⠵⣿⢻⣿⣿⣿⣿⡿⢱⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠸⣽⡇⠠⠐⠀⠀⠋⠀⠈⢂⡀⣠⠿⢟⣾⣿⣿⣷⢿⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡟⡰⣭⣿⣷⣤⣴⣯⣧⣀⣰⣁⣧⣤⠿⠟⣯⠹⣿⣿⡄⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⣷⠁⠨⠷⢤⣴⣼⢿⣿⣿⠟⠉⠉⠐⡽⠓⠂⣉⠙⢧⡀⢺⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣽⣻⣿⣦⣁⣍⣽⣷⣾⣿⣷⣾⣥⣵⣽⠗⡶⠏⠀⡄⠀⠙⢶⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣫⣽⡛⠻⠟⠋⠉⣻⠟⢹⣿⣿⠘⠆⠙⠟⣦⠄⠈⠀⠁⠀⠀⠀⢱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠋⠈⣤⠔⠀⠀⠉⡡⢴⠷⢊⣙⢂⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠋⠀⠄⠀⣠⣦⠔⢀⠘⠋⠅⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡰⠿⢸⠀⡄⠀⠟⠰⢋⡄⠚⡀⠀⣬⠁⠀⠀⠀⠀⠀⠀⢠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠈⠇⠃⣘⠇⠂⢢⣼⠀⠈⠀⠠⠛⠀⡀⠀⠀⢀⡔⠀⠀⠀⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠁⡂⠄⢌⠁⠉⠀⠀⠀⠘⠏⠀⠀⠒⠀⠀⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠐⠀⠀⣀⣀⠀⠁⠀⠙⣀⡾⠥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠄⠀⠀⠈⠁⠀⡠⠙⠒⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⠕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠓⠋⠉

⠀⣰⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⡀⠠⢠⠀⠀⠀⢀⠀⡤⠋⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢰⡿⠂⢀⣤⠀⡀⠠⢀⢀⡆⠊⢛⣿
⠀⠈⠂⠸⣃⠰⠋⠐⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⢦⣄⠈⠑⡼⣠⠰⠋⠀⢠⡼⠁
⠀⠀⠂⠀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠆⢑⠀⢦⡄⣰⡏
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢌⠛⡋⠤⠋
⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠
⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠆⢨⣀⠔⢉⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡤⡈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠
⠀⠀⠀⢠⡂⠐⠙⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⡴⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠠⡄⠀⠀⠀⠠⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⢠⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠉⠁⠂⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⠟⠋⠩⠐⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⢛⠩⠁⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⡆⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠘⠻⠟⠋
⠀⠀⠀⠀⢀⣾⣿⠟⠀⢀⣸⣦⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟
⠀⠀⠀⠀⢸⣿⡏⠀⢠⣿⠟⡏⠉⠙⢿⣷⡀
⠀⠀⠀⠀⠸⣿⡇⠀⢸⡇⠀⣷⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠙⠀⢸⡀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⣟⠛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣼⡷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⢰⣾⣿⡿⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⢀⣶⣿⣿⠇⠀⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠙⠋⠁

⠀⠀⠀⠀⠀⢀⣠⠤⢤⣤⣀
⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢰⣷⡆
⠀⠀⠀⢸⣇⣴⣶⣦⡀⠀⠀⠈⣿⣿⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠻⣿⣿⡿⠃⠀⠀⠀⢿⣿⡇⠀⢠⣤⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣇⡀⠀⠀⠀⠀⠀⠀⣟⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠛⠁⠀⢸⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠛⢦⣘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⠀⠀⠀⢸⠀⣀⣀⠀⠀⠀⠘⢻⡏⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⠋⢙⣿⠀⠀⠀⢠⣼⣷⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡼⠋⠀⠀⠀⠘⢻⡋⠉⡇⠀⠀⠀⢀⣴⣾⣿⠀⠀⠀⢀⣴⣾⣿⡇⠀⠀⠀⣠⣶⣿⡇
⠀⠀⠀⠀⠀⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠘⠃⠀⠁⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠛⠛⠋

⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⣾⢸⡇⠀⠀⠀⠈⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⢀⣄
⠀⠀⠀⠀⣽⠞⠀⠀⠀⣆⡄⠈⠀⠀⠀⣀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠛⠃⠀⠀⠀⢦
⠀⠀⠀⡞⣹⠾⢷⣆⠀⠴⠃⠀⠀⠀⠘⠛⠁⠀⠀⠺⠿⠃⠀⠾⠿⠃⠀⠾⠟⠀⠀⠀⠀⢠⣤⠄⠀⠰⠿⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⣫
⠀⠀⠈⢇⠙⠏⢨⡟⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡖⣱⡜⣱⣼⣰⠀⠀⠣
⠀⠀⠀⠀⡅⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣀⠀⠘⠃⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠱⠏⠡⠏⠁
⠀⠀⠀⠀⠑⠋⠀⠀⠀⣠⣤⢀⠀⠀⠀⠀⢰⠒⠀⠀⠀⢻⣧⠄⠸⠟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⣀⠀⣀⡄⠀⢀⡀⢠⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠾⠏⠀⠛⠁⠀⠟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠸⠟⠁⠀⠈⠉⢀⠀⠀⠀⢸⣽⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠾⠏⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⣤⡄⣀⣤⠀⠈⠆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤
⠀⠀⠀⠀⣶⠗⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⣤⣠⢀⡤⡄⠀⠀⣝⠳⡀⠀⠀⠀⠘⠋⠀⠛⠋⠀⠀⠐⡀⠀⠀⠀⠀⠀⠿⠿⠿⢿⠀⠀⢠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠾⠟⠀⠀⠀⠀⠋⠛⠛⣸⠒⠃⠀⠀⠈⠻⠄⠀⠀⠀⣠⠤⣤⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⢸
⠀⢀⢯⠇⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⠀⢷⡆⠈⣿⠠⠄⠀⠀⠀⠀⠀⠀⣴⣶⠇⢠⣶⡾⢰⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠰⠿⠋⠀⠀⢀⣤⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⣤⠀⣤⠠⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣤⠀⣀⣄⠈⣭⡆⠸⠟⠀⠈⠀⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⢈⠀⠰⢏⡴⠀⠀⠐⠛⡇⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠛⠋⠀⠛⠁⠀⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠾⠟⠀⠀⠀⠀⢿⠟⠀⠀⠀⠛⠀⠀⠀⠈⠋⡇⠀⠀⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⡧⠔⡅⠀⠀⠀⣿⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡀⠘⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠖⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠌⠉⠚⠿⢗⣶⠀⠀⢰⡏⠸⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠰⠶⠶⠶⡦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡶⠶⠀⠀⠀⠀⣴⡶⠞⢀⣀⣀⠀⠀⠈⠀⠀⠘⠧⠄⠂
⠀⠀⠀⠀⢀⠀⠀⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⢠⣶⠆
⠀⠀⠀⢀⣯⠀⠀⣤⠞⡀⠀⠀⣠⣤⠀⢀⣠⠀⠀⣀⠀⠀⠀⠀⢰⡟⡿⠇⠀⠀⣷⡿⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⣀⡀⠀⠀⠀⢀
⠀⠀⠀⠚⠉⠀⠀⡴⢛⡅⠀⠀⢉⠁⠀⠛⠉⠀⠺⡛⠀⠀⠀⠀⠈⠻⡦⠀⠀⠀⡇⠀⠀⠀⣴⡶⠀⢘⡇⠀⠸⠿⠛⠃⠀⠀⠐⠚⠗⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠉⠒⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠶⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡶⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⡟⠄
⠀⠀⢀⣤⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⢴⠋⠀⠀⠀⠀⠠⠠⠎⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⣼⠀⢴⢎⡠⢤⡀
⢠⣆⣼⡗⣠⠀⠀⠀⠴⠂⠠⠀⠀⠈⢀⡀⠀⠀⠠⠠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⢀⣼⡆⠛⠛⠿⣤⣠⡄⣀
⣿⣿⡟⣷⣿⣠⣠⠔⢐⠁⡄⣀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠐⣆⠀⠀⣯⠃⠀⡜⢛⠉⣻⡁⣀⠹⣀⠠⣿⣀
⣸⡿⢀⢻⣿⣿⣿⢞⠁⠈⡁⠀⠡⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠕⢀⢘⣤⠓⣄⡔⠀⠶⢌⣯⠈⠣⠈⢳⣮⣿
⢨⡿⢛⢰⣿⠟⠃⠌⠁⡀⠀⠀⢂⢘⣿⣷⣿⣶⣼⣤⣄⣀⡀⣠⣤⣤⣷⣤⣼⠴⠅⣘⣋⠀⣀⡥⠋⣁⠀⠈⢿⠀⠀⠠⡀⠈⣿
⢠⡀⠊⢈⣥⠖⠋⠀⠚⠀⠀⣄⡀⠑⢿⣿⣟⣹⣽⣽⣿⡿⠟⠟⠿⠿⣿⣿⠟⠀⡀⣈⡑⡆⢈⡶⢋⡠⢦⠜⢦⡂⠀⠀⠉⠙⢿
⠘⠋⠀⢯⡥⠀⠁⠚⠁⢤⡜⢳⡀⠱⠦⢷⣀⢼⠷⠛⣡⠤⣄⠀⠀⠀⠈⠻⠀⠀⣀⢉⠻⣄⣾⢉⢁⠀⠀⡁⠂⢄⡀⠀⠢⢀⠠
⠀⠀⢠⡎⠀⠂⣏⡀⠔⠀⠈⠤⠷⠀⣀⢀⠀⠈⠰⣭⡼⠏⢿⣽⠶⠾⠟⠁⣨⠈⢁⣁⣉⣻⣿⡷⢛⣧⣀⣿⣷⢄⣻⣇⣧⠈⢿
⠀⠠⠋⠀⣀⡤⠶⣅⠀⠀⠈⢳⡠⠲⡍⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠀⢩⠉⠛⠙⢦⠠⠉⠻⣏⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠠⠀⠊⠁⠐⠄⠀⠠⠳⠄⠸⠿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠁⢠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢪⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠰⠀⢀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⠀⡀⠀⠠⠀⠀⠀⠈⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⢁⠀⠁⠉⠁⢀⠕⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠈⠂⠀⠀⠀⠀⠐⠁⠀⡀⠢⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠊⠀⠀⠀⠂⢀⡄⠐⠠⠃⠀⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢱⢃⢴⠤⠀⠀⠺⠕⢀⠀⠃⠀⠀⣀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢆⠰⠂⠀⠀⠀⠀⣣⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠴⠢⢀⡄⡠⠄⠀⢀⢀⠄⢀⠔⠙⠠⠤⠄⠁⠀⠀⠄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠔⠁⠄⠁⠂⡁⢐⠁⡀⠋⠀⠀⡰⠞⠣⠉⠀⠀⠀⠀⢈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡪⢠⠄⠠⡌⢠⠁⠁⠁⠉⠘⠄⠀⠐⠁⠀⠀⠀⠈⠀⠄⠀⠈⠆⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢈⠆⣉⠐⠀⠤⠀⠀⠀⢀⠄⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠒⠀⠠⠄⠒⠀⠤⡀⠀⢀⠀⢀⢖⡒⠀⢀⡒⠂⠀⠀⢀⠀⢀⡈⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣀⠀⡀⢁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠰⠔⠁⠀⠀⠀⠤⠀⠀⠨⠀⣀⠀⠠⠃⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣉⠐⠠⡀⠀⠀⠀⠀⠀⡀⠀⠠⡈⠁⠁⠐⠂⠃⠀⠀⡀⠀⠀⠀⠀⣅⡄⠀⠤⠘⠡⠀⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠜⠋⡠⠊⠣⡄⡁⡐⢨⡄⣀⠊⠀⠠⠄⠰⠀⠀⠀⠐⠀⠀⠀⢀⠂⢈⠀⠀⠠⢀⡁⠀⢆⠄⢈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡣⡄⠀⠀⠀⠁⠈⠉⠁⠘⡀⠀⠀⡐⠀⢠⠀⢀⠁⠠⠄⠈⠑⠀⠀⠀⢠⡀⠀⠀⠟⠐⠄⠀⠀⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠛⢃⠴⠲⠲⠛⠠⠂⠠⠰⠏⢀⠀⠁⠀⠀⠴⠂⠀⠑⠂⠀⠀⣄⠀⠀⠀⡀⠘⠐⠁⠀⠀⠀⡄⠔
⠀⠀⠀⠀⠀⠢⠁⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠤⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠂⢀⡀⠁⡀⠀⠡⠐⠁⠄⠀⠂⠁⠀⠀⠀⠀⠁⠀⠀⠁⠠⠉⠢⠀⠀⠀⠬