Saucer Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⢀⡴⠆⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢰⣿⠁⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠸⣿⡀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠹⣿⣆⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠘⣿⡆⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣿⠃⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠼⠥⠤⠤⠤⠾⠧⠤⠤⠤⠴⠯⠤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢋⣩⣥⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⡶⢶⣤⣤⣭⣉⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⠾⠿⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠴⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡠⠤⣄⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡰⠃⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣣⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣀⣀⣀⡀
⠀⠀⠀⠀⡤⠖⠚⠉⢉⣥⣤⣶⣶⣾⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣶⣶⣤⣬⡉⠉⠓⠲⢤
⠀⠀⠀⠀⠑⠦⣄⣀⡘⠛⠿⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⠿⠿⠛⢁⣀⣠⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⢶⡤⠬⠭⠭⢭⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⡭⠭⠭⠥⢴⡶⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢒⠂⢄⠴⠂⠠⠈⠭⠀⠀⠐⡂⣠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⢗⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠝⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⣿⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠈⢒⠶⠀⠤⠤⠄⠀⠀⠤⠀⣀⠀⠤⠤⠤⣴⡤⠼⠛⠉⡠⢐⣒⣒⡲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠈⠀⠈⠊⠈⠈⠀⠉⠁⠀⠀⠀⠀⢠⡶⠋⠀⠈⠹⡆⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⡿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠤⢠⠀⠀⢀⠉⡑⠂⠁⠀⠀⢀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡔⠁
⠀⠀⠀⠀⠀⢀⣀⣤⣴⡖⣶⣞⠭⠀⠈⠉⠐⠒⠨⠶⣶⣽⣤⣶⡿⢧⠀⢀⣶⣣⠟⠀⠀⠀⢀⣶⠤⠀⣀⠄
⠀⠀⠀⡠⢞⠟⢃⠔⠘⠉⠊⠁⠀⠀⠀⠀⠀⠀⣴⠄⣹⡿⣤⣽⣿⠿⠃⢿⠛⠀⠀⠀⠀⠀⠔⣠⠴⣾⡷⣿⣵⣦⣄
⠀⢠⡎⠔⠁⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⣹⣿⡿⣏⠙⢄⠰⣖⢀⣀⠀⠀⡴⠋⠐⡾⢿⠳⠈⠉⠛⡾⣿⢦
⠀⢸⡐⠀⠀⠀⠦⣼⡷⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠋⠀⠈⠀⠀⠀⠉⠈⠁⠀⣼⣿⣿⢿⣿⡆⢀⣤⣴⠀⠀⢩⡿⡇
⠀⠀⠑⢵⣄⠀⠀⠘⠛⢿⠻⠿⠂⠀⠀⠀⠀⠀⣀⣀⣀⣠⡠⠤⢤⡤⠠⠄⣀⡀⠠⠷⢻⠿⣿⠿⣱⠾⠛⠛⠁⠀⢠⣖⡼⠁
⠀⠀⠀⠀⠙⠓⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠻⠻⠿⠿⢿⣿⣿⣿⣿⣿⢃⡄⠀⠀⠀⠀⠈⠆⠀⠀⡀⡤⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠤⠤⣤⣤⣌⣒⣒⣠⣤⣤⣤⣤⣭⣵⣶⣫⡤⠤⠤⠶⠖⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢤⡀⠀⠄⠀⠀⣀⣀⣀⣀⣀⣄⠀⠀⣤⣠⣤⡤⠄⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⠒⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⢠⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣈⠓⠻⠷⢶⣦⣥⣀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣡⣰⣴⠶⠟⠒⠉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣤⣄⣀⣀⡉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⣀⣀⣀⣤⣴⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢋⣸⣿⡻⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⡏⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⡴⢋⡠⠔⠊⠉⣿⣿⣷⣈⣩⣤⣶⣦⣝⣿⣿⣿⣿⡿⣩⣟⢿⣿⡿⢙⣛⡻⠿⠇⢿⣿⣿⣿⣿
⠀⠀⠀⠀⡸⢠⠋⠀⠀⠀⢀⣸⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣅⣼⣿⣿⣿⣷⣾⣿⣿⢿⡇
⠀⠀⠀⠀⡇⠆⣀⣤⣶⣿⣿⣿⣷⡀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁⠀⣼⣿⣷⣶⣤⡀
⠀⠀⠀⢀⣥⢸⣿⣿⣿⡿⠋⠁⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠙⠻⣿⣿⣿⣷⣤⡀
⢀⠀⣴⣿⣿⣆⢻⡿⠁⠀⢀⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣄⠀⠈⠻⣿⣿⣿⣿⣆
⡄⣼⣿⣿⣿⣿⣧⣑⠤⣀⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡄⠀⢹⣿⣿⣿⣿⣇
⡇⣿⣿⣿⣿⣿⣿⠈⠑⠢⢭⣙⣛⢛⡿⠿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⡿
⠱⡹⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⠄⠊⠀⠈⠙⠛⠿⠿⠿⠋⠀⣠⣿⣿⣿⣿⣿⠃
⠀⠑⢝⢿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠑⠿⣻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⢀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⠀⠀⠀⠤⢂⣆⠃⠀⢰⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠀⠠⠀⠀⢈⣙⣋⡙⠂⠀⠁⠀⢀⠈⡘⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⡀⠀⠀⠀⢀⠀⠒⢀⠀⢀⡈⣰⠄⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⢀⡈⠐⠛⠀⠀⠀⠀⠙⠂⢀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⡈⠀⠀⠁⡐⠉⠀⠁⡀⠂⠉⠂
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠆⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡀⡑⡄
⠀⠀⠀⠀⠀⠀⠙⠒⠠⢀⠀⠀⠀⠀⠰⣄⣀⣀⣀⠀⢀⣀⣀⡼⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⡀⠐⠀⠠⠬⠥⣭⡭⠭⡥⠤⠴⠖⠶⠶⠶⠎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠂⠒⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠈⠁⠀⠀⠀⠈⠉⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⢱⡎⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⡀⠄⠒⠊⠉⠀⠀⠀⠒⠄⢏⡀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⣶⠀⢠⠊⠀⠄⠀⢠⠀⠀⢀⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢸⡤⠑⠇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠸⡄⠀⣀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⣌⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠔⡛⠀⠀⢀
⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠠⠺⡁⠈⠉⠉⠉⠁⠒⠒⠈⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠸⠃⠀⠀⢸
⠀⠀⠀⠀⢘⠓⢄⠀⠀⠀⠀⠀⠀⠈⡒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠀⣤⠤⠤⠔⣂⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠢⣀⠉⠂⠄⡀⠀⠀⠀⠈⠁⠂⠬⣁⠒⠤⠤⣆⣀⠄⠂⠀⠀⠀⡄⠀⠀⢠⠬⠕⠲⠤⣤⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠈⠁⠒⠠⠬⢁⡐⠂⠤⠤⢄⡐⣀⡪⠀⠀⠀⠀⠀⠉⠛⠶⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠍⡒⠤⢀⠀⠈⠉⠒⠀⠤⣀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠠⠬⠍⢙⣒⠒⠒⠒⠒⠒⠒⡻⠋⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡈⠁⠂⠠⢀⡀⠀⠀⠉⠐⠂⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⠀⠀⠉⠑⠒⠠⠤⢀⡀⠀⠀⠀⠉⠁⠒⠀⠀⠠⠤⠤⢀⣀⣀⠀⠀⠀⢀⣀⠠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢀⡀⠀⠀⠀⠀⠉⠀⠒⠀⠀⠠⠤⠀⢀⣀⣀⠀⠀⢀⣀⣀⡀⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⠙⠛⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⠴⠚⠋⠉⠉⠉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡴⠖⠚⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⣿⠟⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⠤⠤⠤⠤⠤⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⣴⡶⠶⣤⡀⠀⠀⠀⠉⠛⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠶⠞⠛⠉⠀⠀⠀⣿⠇⠀⠀⠒⠲⢶⣤⣀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⢹⣷⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢈⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⡴⠋⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⢀⡴⠋⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣴⠞⠛⠛⠳⢶⣤⡴⠛⠁
⠀⣰⡟⠁⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠻⠤⠤⠠⠖⠛⢿⣷
⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⢤⣤⡤⠴⠶⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⡶⠾⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠰⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠉⠛⠛⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣻⣇⣽⣵⣶⣶⣦⣤⣤⣤⣧⡴⠂⣠⣴⣶⠦⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠟⠉⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡠⣄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠷⢷⡶⠶⠾⢿⣶⣤⣀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⡿⠉⠠⠄⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠳⠬⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⢶⡶⠿⠷⠿⠶⠀⠀⠀⠀⠰⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⢀⣀⠀⠀⢘⣷⣦⣴⣦⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣋⠤⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠿⠟⠛⠛⠻⠿⠿⠷⠶⠶⢶⣶⠶⠾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣶⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡄⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⢀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⣸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⣀⣤⣶⣿⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⣰⣿⣿⣿⣿⢿⣿⣷⣶⣄⡀
⠀⢀⣴⣿⣿⠟⠋⠀⠀⠙⢿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣴⣿⣿⣿⡿⠁⠀⠀⠉⠻⢿⣿⣷⣦⡀
⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣷⣦⣤⣤⣤⣤⣴⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿
⠀⠀⠉⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⡿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠐⠂⠀⠀⠤⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⠠⠤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠈⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠵⠤⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠠⢀⠈⢀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠂⠉⠆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⢀⠠⠂⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠌⠵⠄⠀⡤⢐⠠⣥⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠛⠛⠉⢉⣘⣇⣼⡀⣇⡀⣸⣽⣷⣷⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣠⣤⡟⢢⣀⣰⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢻⣟⣻⡵⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣾⠃⠈⢷⠸⣿⣿⢝⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⢧⡀⠸⣆⠹⡆⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣍⡂⠹⣆⠙⢆⢀⡻⢿⣧⡀⠀⠀⠀⠀⠀⢀⣾⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⣷⣬⣉⣡⣼⣿⣷⣀⠀⠀⠀⣠⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣴⡶⠖⠚⣫⣾⣿⣿⣭⣝⣛⣛⣛⣺⣽⣏⣿⣐⣀⡀⠀⢀
⠀⠀⠀⢀⣀⠀⠀⢠⣾⣿⣿⣿⣏⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠾⠿⠛⢋⣹⣿⣿⠿⠃⣡
⠀⠀⠓⠾⠿⣿⣿⣶⡟⠿⠟⠿⠿⢷⣶⣬⣭⣉⣙⣛⡛⠛⠒⠒⠂⠀⠀⠀⣈⣉⣡⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⢬⣁⡀⠀⡀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡛⠶⠾⠾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⡟⣛⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠯⡤⠙⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣺⠏⠉⠀⠀⢀⡀⠀⠀⠉⠹⣗⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣭⣿⡧⣓⠃⠀⠀⠟⠛⠀⠀⠘⣛⠼⣿⣬⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡯⠿⠿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠟⠿⢽⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⢹⢁⣀⣠⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣄⣀⡈⡏⠙⠓⠦⣤⣀
⠀⠀⠀⠀⠀⣠⠶⠋⢁⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡈⠙⠶⣄
⠀⠀⠀⠀⠀⣇⣴⡾⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢷⣦⢸
⠀⠀⠀⠀⠀⠙⠻⢼⣑⣫⠞⡿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣶⣶⡶⠶⠶⠶⠶⢶⣶⣶⣶⣤⣤⣴⣿⣿⣿⣿⣿⣿⢯⠳⣝⣊⡧⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⣽⠏⣿⣿⣿⣿⣯⡉⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢉⣹⣿⣿⣿⣿⠹⣯⠞⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⡛⣽⣿⣿⣿⡿⠿⠛⠛⠻⠿⠛⠛⠿⢿⣿⣿⣿⣯⢛⡾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠯⣽⣭⣉⣀⣀⣀⣀⣀⣀⣀⣈⣉⣭⣯⠽⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⢤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠛⠛⠳⠿⠭⠭⠭⢽⣭⣤⡤⠤⠤⠤⠤⠤⠖⠚⢛⡆⢀⡠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⠖⠒⠒⢄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠷⠁⠀⠀⠀⢸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣳⣶⣿⠀⠀⠀⠀⠀⣸⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣽⢷⣿⡇⠀⠀⠀⠀⡴⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠂⡀⠈⠀⠈⠀⠀⠀⠀⠰⠀⢀⠻⢿⣛⣻⠁⠀⢀⡠⢊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢠⢟⣒⣭⠵⠞⠉
⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡾⠋⠀⠀⠀⠀⠀⠀⠀⣈⣴
⠀⠀⠈⠙⠒⠶⢶⣤⣤⣤⣤⣌⣧⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣄⣀⣀⣼⣤⣤⣤⣤⣶⣶⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢿⣿⣿⣿⡯⢉⣉⣉⣉⣙⣛⣛⣿⣿⣭⣭⣽⡿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠒⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⢀⣠⣤⣴⣶⣶⢖⣀⣀⣀⣀⣀⠀⠒⠒⠢⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠒⠒⢤⡀⣯⣿⣿⠟⠶⠶⠚⠿⠿⠭⠍⠭⠭⠿⠷⠦⠭⠶⠾⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⣧⣸⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡔⢄⠀⠘⠿⡼⣿⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⠀⠀⢣⢻⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⡚⠌⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⢻⣶⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠱⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⢀⣀⣙⣓⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠉⡦
⠀⠀⠀⡤⠦⠤⠤⠠⠀⠀⠀⠉⠙⠒⠦⢤⣹⡿⣿⣮⣿⣄⡀⠀⠀⠀⠀⠀⢀⣠⣕⣀⣀⣀⣠⣤⠤⠖⠚⠉
⠀⠀⠀⠉⠓⠲⠦⠤⠤⣀⣀⠀⠀⠀⢀⠀⠀⠀⠺⠿⠿⠛⠛⢟⣿⣄⣀⣨⣤⡴⠶⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣦⠉⠉⠉⠀⠀⢄⠀⠀⢣⣉⣲⡆⠑⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠙⠛⠛⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣡⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⠀⠀⠀⢈⣗⠁⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠺⡋⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣈⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⢿⣿⣿⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣳⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣦⣤⣀⣄⣀⣄⣠⣀⣄⣤⣴⣴⣾⣿⣿⣿⣿⣿⡟⠯⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣯⣿⠛⠟⠻⠟⠿⠻⠟⣿⣿⣿⣿⣿⣿⣿⣯⣟⠂
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠁⠀⠛⣻⣿⣿⣷⣶⣶⣶⣶⣾⣿⢿⠛⠉⠛⠫⢻⣿⣿⡆
⠀⠀⠀⠈⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠻⡿⠦⢄⣀⣀⣀⣀⣽⣿⡿⠃⠀⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠒⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣟⡻⠉⠉⠉⠉⠉⠙⠻⡶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣛⣒⣟⣛⣄⣀⣀⣀⣀⣀⣠⣚⣛⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⣻⣉⣹⣧⣤⣽⣤⣤⣧⣤⣤⣥⣤⣬⣇⣉⣏⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡾⡿⠟⣟⡉⠀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠘⠉⣛⣿⣿⣿⣓⠲⢤⣄⡀
⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⢻⣛⣛⣛⣟⣻⣛⣫⣿⣫⣫⣏⣉⣝⣉⣉⣉⣙⣙⣙⣛⣻⣟⣿⣻⣟⣿⣶⣾⣿⠳⠶⣤⣄⡀
⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣻⣙⣿⣿⣿⣿⣿⣫⣙⣿⣿⣿⣿⣿⠩⣹⣿⣿⣿⣿⣩⣉⣿⣿⣿⣿⣿⣍⣹⣿⣿⣿⡧⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣭⣻⣦⣅⠀⠹⣤⠍⠉⠉⡭⠉⢩⠍⠉⠉⡭⢍⣽⠉⠍⡭⠉⠍⣡⡿⢓⣶⣿⣿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡟⣳⣎⣢⡀⠑⡀⠀⠘⠀⠀⢰⠁⠀⠉⠀⡰⠁⣀⣟⢻⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣖⣿⣹⠃⠀⠙⢿⣿⣿⣿⣶⣷⣶⣾⣶⣾⣷⣿⣿⠿⠉⠉⠫⣽⢶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠓⠿⢦⣤⣤⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣰⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠶⠚⠛⣿⣿⣿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⣿⠃⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⢠⡿



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⡛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣴⣿⠿⠟⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⣀⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣠⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁
⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉






⡶⢶⢶⠶⡶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠤⢀⢐⣒⣒⣠⣤⣠⡀⠀⢀⠀⠠⡀⠒⠀⡒⠒⡂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣴⣶⠖⠃⠀⠀⠈⠚⠉⠈⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣙⣛⣯⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⡄⣀⣀⣤⠵⠂⢀⠠⠤⠄⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣭⡋⠛⠛⠻⠷⠶⠷⢶⣶⢴⡴⠶⠶⠒⠶⠶⣿⣟⠟⠛⠉⣉⡅⢰⣎⢡⢋⣤⠶⠂⢠⠄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠠⡏⠀⠐⠢⠤⣠⣴⢟⡾⣯⡖⠋⠉⠉⠉⠳⠌⢷⣉⣷⢭⠀⢄⠈⠋⢘⠀⣏⠃⠀⠀⠑⠤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠍⠉⠃⠀⡠⣤⢀⣌⢹⣠⣤⡀⢈⡹⣉⣅⣘⠃⢃⡈⡛⠄⠬⠀⠀⠀⠀⠐⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠬⠉⡐⡟⠸⣼⡖⠃⢀⣐⢊⣄⡙⠠⣭⠀⠀⠠⣿⠇⠙⡇⡟⠀⠠⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠈⢱⠀⠩⡇⢠⡎⣸⢧⡮⣻⢆⣺⣦⡀⠀⡷⠀⠀⡇⠻⠀⢈⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⢣⡝⣯⡥⠟⣠⡏⠉⣁⣄⣠⡀⠀⡇⢸⠀⠸⠀⠀⠀⠀⠀⠠⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⢘⣀⠀⣇⣀⡀⠁⠳⠊⠘⢙⣶⣦⡼⢟⠛⣿⡀⣧⣼⠀⢈⣀⠀⠀⢀⠄⠁⠁⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠐⢀⢸⡟⠳⢿⣿⢻⣦⠠⣄⣐⣿⣿⣿⡖⡸⣠⣿⣟⣹⣄⠀⢨⡀⠈⠈⠀⠀⢀⠔⠁
⠀⠀⠀⠀⢀⠀⠄⣀⣀⡐⢢⣥⡀⠀⠀⣝⢿⡋⣴⣆⠈⢻⣸⢪⠁⠛⢿⡟⠾⡟⢧⣯⢤⠄⣴⣍⠁⠀⢸⣟⣭⣥⠴⢖⣃⠤⣀⡀
⠀⢄⣬⣵⢾⠻⢏⢩⢻⡿⣝⡾⢁⠀⠠⠽⣻⡗⠻⣽⣣⣼⢻⣻⣄⠘⣿⣷⣶⣿⢈⣙⣢⣴⡏⠛⠀⢀⠌⢖⣋⡶⠶⠶⡍⣟⡶⠯⢭⡗⢄
⢄⣿⢻⣥⠗⠀⠀⢺⠉⡰⠤⠀⠈⢳⣀⠸⢶⣿⣷⣼⣈⢁⡼⣻⣯⠈⠉⢁⣹⣜⢿⣿⢿⠉⠁⠀⣠⠎⠀⠉⠀⠀⠀⠈⠵⣇⢚⠻⡑⣿⢌
⠀⣻⠼⣌⣀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠳⢄⠈⣿⣾⣿⣿⣿⡏⠙⠃⣐⣻⣇⣯⣽⣷⣃⡠⡴⠊⡠⣀⢤⠄⠀⠀⠈⠁⠐⠉⡀⠠⠚⡥⠎
⠀⠈⠙⠒⠶⢦⣤⣖⠄⣀⣤⠄⠒⠐⠀⠀⠀⡉⣶⣜⣟⠁⠙⠃⠀⠀⠹⣯⠿⠿⠿⠿⡛⠑⠁⠄⠀⠂⠥⠆⣘⣛⣤⠤⠄⣂⠥⠚⠁
⠀⠀⠀⠀⠈⠂⠈⢉⠙⠓⠒⠛⠛⠛⠓⠓⠒⠲⠶⠖⠲⣶⠭⣬⣬⡥⠤⠴⠶⠖⠶⠶⣾⣿⣿⣿⣿⣿⡿⠟⠋⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⢤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠤⢀⡶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠤⠤⠭⠭⠉⠉⠉⠀⠀⠀⠉⠉⠩⠥⠤⠤⠶⠒⠒⠛⠉

⠀⠀⠀⠀⠀⠀⣠⠤⠤⠶⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠶⣶⣶⣶⠤⣄⡀
⠀⠀⠀⠀⠀⡮⣐⣤⣤⣶⣶⣶⣶⣶⣿⣷⣾⣿⣾⣷⣦⣦⣤⢤⣀⣀⡥⢿⣆
⠀⠀⠀⠀⠀⡇⡃⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠶⠒⠚⠛⠉⠉⠀⢀⢸⣿⣠⠴⢖⣒⣒⠶⣦⣄
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⣒⡤⠬⣅⡀⢸⡿⣡⡔⠋⠉⠉⠙⣆⠹⣧
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⣓⡀⢸⣇⣸⠇⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣖⡂⠀⣼⠏⠁⠀⠀⠀⠀⢀⡜⢠⡏
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣀⠒⢠⡿⠀⠀⠀⣀⣠⠴⠋⣰⠟
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⡒⠀⣾⣧⢶⣚⣩⣥⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⠿⢫⡿⠛⠋⠉
⠀⠀⠀⠀⢀⣀⣤⡤⣧⠀⠀⠀⠀⣁⡀⠀⠀⠀⠀⠀⠀⠀⣘⡂⣼⠿⠶⠯⣤⣤⣄⣀⡀
⢀⠴⠒⠋⠉⠀⣀⡀⠸⣇⠀⠀⠀⢚⣿⢤⣤⠁⠀⣤⣈⣹⣾⣤⣿⣤⣤⣿⣿⡛⠛⠛⠛⠳⣦⠀⠠⠄
⢃⡀⠀⠀⠀⠊⠋⠁⠀⠻⣤⣀⣐⡒⠛⠒⠒⠒⣚⣛⣋⣭⣵⡾⠛⠛⠁⠀⠈⠉⠁⠀⠀⣠⡿
⠀⠉⠓⠦⣤⣄⣀⣀⣀⡀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⣀⣀⣀⣤⣤⣀⣤⣤⣴⠶⠛⠉⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠉⠙⠛⠻⠭⢭⣭⣉⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣉⣩⡭⠵⠞⠛⠉⠀⠀⠀⠒⠀⠀⠀⠒⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣀⣠⣄⣠⣄⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⠾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠾⠛⣻⡏⠉⠁⠀⢀⣠⣴⡶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣿⠀⠀⠀⣴⡿⠋⠁⣠⣾⠟⠀⣀⠀⠀⣠⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⡀⠀⠀⢸⡟⠀⠀⣾⣿⣤⡶⠿⠋⠁⣠⣾⠏⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣿⡟⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⢀⣤⣾⡿⠋⠀⠜⢁⣤⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⠀⠀⢠⣿⠀⢀⠀⠻⢷⣶⠾⠟⠛⠁⠀⠀⠀⢠⣿⠋⠀⠀⠙⢿⡄⠀⠄⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠈⡋⠀⠈⠀⠀⢀⣀⠀⠀⠀⢀⣀⠀⠀⠘⣿⡀⠀⠀⠀⢸⣷⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠻⣦⣰⡿⠀⣀⠀⠁⢀⡀⠀⠀⠈⠋⠀⠀⠐⣯⡙⡇⠀⠀⠘⠻⣦⣤⣤⣼⣿⡿⠛⢁⣴⠆
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣿⣿⠃⢠⡿⠀⠀⣼⡇⠀⠀⣼⠃⣠⡾⣷⠈⢙⡟⢶⡄⠀⣴⠏⠉⠉⣹⠇⠀⠚⠋⠁
⠀⠀⠀⠀⣰⡟⠀⠀⠀⢀⡼⣿⡟⠀⣾⠇⠀⣸⣿⠀⠀⣼⡟⢠⣿⡼⠃⢀⡞⢠⡿⠁⣸⡟⠀⠀⢠⡟⠀⠀⠉
⠀⠀⠀⢠⡿⠀⠀⠀⢠⡞⢱⣿⠁⢸⡿⠀⣰⣿⡇⠀⡼⣿⠃⣾⡃⠀⣠⠞⠀⣿⠁⢠⣿⠁⠀⢠⡾⠁
⠀⠀⠀⢸⡇⠀⢀⡴⠋⠀⣿⡟⠀⢸⡇⣴⠃⣿⣧⠞⠸⣿⡼⠙⠷⠚⠁⠀⠀⠧⠔⠉⣿⣀⣠⠟⠀⠀⠄
⠀⠀⠀⠘⠷⠖⠋⠀⠀⢸⣿⡇⠀⠘⠟⠁⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠉⠁⠴⠟
⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣿⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡏⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠐⠃⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⡉⠳⢄⠀⠀⠀⠀⣰⠃⠀⡠⠋⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠉⠤⠀⠱⡀⠀⢰⠃⠀⡐⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠐⠄⠁⠀⠸⡄⠜⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠶⠶⠤⠤⣄⣀⠀⠈⠙⠺⣆⠀⢀⡿⠛⠉⠁⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠈⡆⢸⠁⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠀⠑⠦
⠀⠀⠀⠀⢠⣟⣉⣉⠉⠉⠉⠉⠙⠚⠓⠦⣦⡀⠀⠈⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⣀⣀⠤⠄
⠀⠀⠀⠀⠛⠁⠀⠀⠉⠲⢤⣀⡀⠀⠀⠀⠀⠈⠙⢦⡀⠀⢸⡇⠀⠀⢀⣠⣤⣤⣄⣠⣤⣤⡦⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠒⠒⠒⠻⢷⡘⡇⠀⣰⠋⠀⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣷⠀⡿⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⡟⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⣡⣴⣦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣼⣿⣿⣰⣿⡏⠁⠀⠀⠄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠑⢤⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⣀⣽⠂⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⡏⠒⠈⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠠⠐⠀⠀⠂⠀⣼⠀⠀⠁⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡗⠀⢃⠀⠀⠆⠀⠀⠀⠀⢸⠀⠀⠘⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣿⣿⣿⣿⣿⣿⣧⠈⠋⢰⡄⠀⠀⠀⠀⠀⠈⡆⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣉⠀⣿⣿⣿⣿⣿⣿⣿⠈⣂⣌⣁⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠌⠀⠀⠃⠲⠄⣠⣧⣿⣤⣿⣿⣿⣿⣿⣿⣿⣤⣴⣿⣿⠀⠀⠀⠀⠀⡄⣨⣥⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠄⠀⢠⣍⠉⡛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠻⠓⠒⠒⢒⣂⣉⣉⠁⢤⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠐⡈⠷⠹⡟⠛⠀⠰⠀⡀⢠⠀⠀⠄⠀⢪⣧⣟⡀⠀⡀⢀⠌⠙⠿⠋⢰⠿⠇⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠁⣀⣠⠍⠁⠀⠀⠰⣿⣿⣿⠄⠀⠦⠘⠻⠛⠁⠈⠁⠀⠐⠀⠀⠀⠈⠛⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠐⠈⠀⠖⠀⠀⠀⠄⠀⠀⠾⢠⠀⠀⠀⠀⠁⠀⠀⠀⣦⣠⡆⠀⢉⣀⣤⣤⢀⡀⠀⠲⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⢀⠀⠀⠀⢀⣴⣦⡀⠀⠀⡀⢀⠀⠀⠠⠀⠀⠉⠀⡀⠛⠙⠣⠀⢈⡛⠛⠛⠈⠇⠀⠀⠐⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠆⢻⣉⠁⠀⠀⠇⠀⠀⢿⢶⣷⡄⠀⠀⠈⠖⠀⠀⠀⠀⠁⠈⠀⠠⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⣠⣀⡀⡤⠄⣼⣄⠈⠉⠑⠚⠀⠀⠀⠸⢀⠁⠀⠀⠀⠀⡆⠀⠐⠀⠀⠀⣰⣦⠀⠀⠀⢢⡅
⠀⠀⠀⠀⠀⠀⠀⢀⣘⣈⣁⣤⣤⣤⣤⣤⣤⡄⠀⠀⠠⠀⠀⠈⠀⠀⠈⠀⠈⠀⠐⠿⢿⠟⠀⠀⠀⢘⣻⣧⣤⣤⣤⣬⣤⣄⣀⣀⣀
⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣷⣶⣄⡀⢀⣤⣂⣀⠀⠚⠓⠂⠐⠒⠒⢛⠂⠀⣀⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡷⠾⠛⠙⠿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⠐⢄⠁⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡤⠤⠴⠶⠶⠖⠒⠒⠒⠒⠚⠛⠛⠒⠒⠒⠒⠒⠒⠒⠖⡤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⣠⣶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢩⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⣀⣰⣀⢀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⠶⠶⠶⠶⠴⠦⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠛⡛⠛⣛⡋⢻⣯⣿⡀⢀⡴⠋⠠⠤⢌⣳⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⢍⡁⠘⠾⠽⠘⠧⠶⠒⠚⠞⠓⢒⣂⣉⣉⡉⢉⣉⠉⠁⠀⠀⠀⠁⠈⠉⠉⠈⡇⡾⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣾⣻⣿⠿⣽⣿⣨⣵⣄⠀⠀⠠⠄⠀⣦⡀⢠⣿⣿⣿⣧⣼⣿⣯⠙⣷⣷⣦⣤⡀⢀⠠⠀⢄⠀⣿⡇⠀⠀⠀⠀⠀⢸⢿⡁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣶⣿⣼⢿⣆⢹⣿⣿⣿⡽⡚⣿⣖⢿⣿⣿⣿⣿⣷⣤⡈⠁⠀⢻⡇⠀⠀⠀⠀⠀⢎⣿⠄
⠀⠀⠀⢰⣧⣿⣿⣿⣿⣿⡛⢛⠨⢿⡟⣝⠻⣿⣣⣤⡈⠉⢻⣆⠈⠈⢿⣿⣿⢿⣿⠾⣿⡛⣿⣿⣿⢿⢿⣿⣿⣦⣶⢸⡇⠀⠀⠀⢀⡜⣼⡏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡗⠐⠀⠀⢸⠀⠀⠀⠻⣿⣿⣿⣆⠀⢱⠀⢠⣾⣿⣿⣧⡉⠁⠀⠀⢁⡄⠀⠈⠠⠟⡿⢿⣿⣯⠻⣦⣀⣒⣿⣾⠋
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡷⠄⠀⠀⠀⣷⡀⠀⠒⢺⣿⣿⣿⣆⣸⠀⣻⣿⣿⣟⣿⣿⠀⠀⠃⡘⠇⠀⠀⠀⠀⠘⠀⢿⣿⠀⠀⢈⣽⠏⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⠀⠀⢠⠀⢸⣟⡀⠀⠈⠁⠻⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⡧⠀⠀⢂⣴⣂⠀⠀⢰⣆⠀⠀⠉⠀⣠⡿⠂
⠀⠀⠀⠸⣿⣿⣿⣿⣣⡿⠷⢴⣼⣿⣻⣟⠿⣗⠀⠐⣠⣮⡿⣿⣿⣿⣿⣿⣿⡿⠋⠐⠀⠀⠀⠫⡜⢹⡽⠋⠀⠀⠀⣰⣼⠟
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣇⡁⠀⠉⠛⠛⠿⠀⠀⠉⠉⠓⠁⠀⠈⠉⠯⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠈⢻⠚⢿⣿⣎⣷⣖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⢣⣿⣿⣿⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣿⣖⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡻⣿⣿⣯⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢝⣿⣿⣿⣭⣶⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣿⣿⣿⣟⡫⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣲⣴⣿⣿⣿⣿⢿⣯⣷⣶⣤⣤⠄⠀⠀⠀⠀⣀⣀⠀⡀⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠻⠻⠛⠿⠿⠿⠿⠿⣯⣿⡍⠉⠉⡉⠉⠉⠁

⠀⢀⣤⣴⣶⣶⣶⣶⣾⣿⣷⣶⣶⣤⡀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣤⡀
⠀⣿⣿⠟⠛⠛⣉⠉⣉⣍⣉⣉⣻⣿⡇⠀⠀⠀⣴⣾⣿⣿⡟⢻⣿⣷
⠀⠻⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡿⠃⠀⢀⣾⣿⠟⢡⣦⠁⢸⣿⣿⡀⠀⣤⣾⣿⣿⣷⣤⣤⣄
⠀⠀⠈⠉⢹⣿⣿⠀⣾⣿⣿⣿⣿⣷⣄⠀⣾⣿⠏⢀⣿⡟⠀⢸⡿⢿⣿⣿⣿⠟⡁⢹⡿⢛⠿⣿⣷⣤⣀
⠀⠀⠀⠀⢸⣯⣿⠀⣿⣿⠃⡄⡿⢿⣿⣿⣿⡟⠀⣾⡿⣰⠀⢸⡇⢸⠟⠻⠏⣼⠁⢸⡇⢈⡇⢿⡶⣿⣽⡆
⠀⠀⠀⠀⢸⡷⡿⠀⣿⣿⢠⣶⢣⣿⣿⣿⣿⡇⢠⣿⢣⣿⠀⣿⡇⠀⣆⠀⣼⣿⠀⡟⠀⣶⠅⣤⣾⣿⠞
⠀⠀⠀⠀⢸⡟⣷⠀⣿⣿⣦⣵⣿⣿⣿⣿⣿⡇⢸⢇⣾⣿⠀⢿⣇⣰⣿⣴⣿⣿⣆⣰⣧⣭⣴⣿⡿⠁
⠀⠀⠀⠀⢈⣿⣹⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣧⣈⣻⣿⣿⠿⠟⠛⠛⠋⣉⣉⣉⣉⣙⣛⣓⠶⢤⣄⡀
⠀⠀⠀⠀⠹⣿⣯⣛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⣀⣠⣤⣶⣶⠿⠿⠟⠛⠛⠛⠛⠛⠻⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠙⠛⠿⣷⣶⣦⣤⣤⣤⣤⣀⣀⣤⣤⣤⣴⣶⡾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠉⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠒⠚⠉⠙⠩⠭⠉⡉⠉⠁⠤⠈⠀⠀⠈⠉⠉⠩⠍⠉⠉⠙⠒⠒⠒⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣿⣿⠥⠤⠒⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢵⣿⣿⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣙⠛⠲⠦⠤⠤⠤⣤⣄⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣥⠤⠤⠴⠶⠞⠛⣉⣴⣶⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠛⠶⠤⠤⠶⠤⠄⠈⣉⣉⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⢈⣉⠉⠀⠤⠴⠤⠤⠴⠞⠋⢩⣿⣿⡁⠤⠀⠀⠉⢧
⢀⡄⡂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡴⢲⡆⠀⠀⣠⡄⠀⠀⠀⣉⠉⠉⠉⢉⡉⢉⡁⠀⠠⠀⠀⠀⠀⠀⠀⣴⢶⠀⠀⣿⣼⢻⢧⠀⠘⠓⠀⡼
⠘⢦⣥⠄⠲⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠷⢬⣠⡀⠀⠙⠂⢠⠟⠛⢿⣷⡄⠀⣨⠟⠄⠀⠀⢶⣦⣀⣠⣿⣿⢶⡔⠁⠀⡜⠙⠶⠿⠋⢠⠃⠀⣰⠃
⠀⠀⠀⠉⠐⣄⠈⠀⠡⡀⠀⠀⠀⢀⡀⠀⢸⡀⠀⠁⠈⣠⡴⢦⡀⠘⠄⠇⠀⣿⠁⠊⣤⣶⡄⠓⠀⢩⢿⣯⡉⠉⠃⠀⠄⠀⣼⠀⢀⣀⠠⠚⢁⣤⠖⠁
⠀⠀⠀⠀⠀⠈⠛⠻⢦⣌⠂⣠⢂⠥⠂⠁⡼⣷⡴⠀⠰⣏⣰⠀⣹⣤⡆⢀⡼⠑⠒⢠⣧⣁⣶⡤⣾⣁⣄⠛⠁⣼⡦⣄⠀⣸⡵⠟⣉⣡⢴⠺⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡳⣎⡙⠀⠯⠤⠆⡈⢳⠀⠀⠈⠁⠴⠛⣻⣧⢸⠆⣘⣯⡙⡆⢻⣻⠑⡋⣽⣷⠆⢰⣿⣤⠉⣰⠋⣰⣿⡉⢉⣭⡻⢮⡢⢄
⠀⠀⠀⠀⠀⠀⠀⡰⢡⢎⠔⠉⠻⣦⡑⢤⡒⠛⠏⣷⡈⢠⣄⣀⠀⠨⣿⣿⠀⢉⠛⠻⠿⣾⣿⡿⠋⣭⠦⡄⢠⣿⠁⢠⢧⣀⣉⡾⠃⠻⣄⣀⣠⡈⢦⠳
⠀⠀⠀⠀⠀⠀⠀⠆⡇⠀⣠⣶⡄⠈⢿⣆⠙⢇⠰⠼⣷⣄⠁⠋⢀⣠⡁⠈⠓⢬⣀⢀⡀⠈⣯⣀⠀⡓⢂⣀⠼⠁⢠⠏⢦⡉⠁⢀⣠⣶⡄⢹⣿⡅⠀⡇⡇
⠀⠀⠀⠀⠀⠀⢀⡃⢧⣳⣅⠈⣠⠄⣰⢻⣧⡨⣳⣦⡈⢿⣢⠀⠀⢨⠤⣦⠴⠢⣬⡍⠉⣴⠂⠹⠺⠑⠋⠁⢀⣰⠋⣾⠛⠻⣿⢻⡍⣰⣧⠼⢋⣴⣿⠇⡣
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣮⣿⠧⠤⢯⠄⢿⣿⠈⠛⠛⠓⠛⠿⠶⠦⣄⣉⣶⣂⡻⠀⠀⢾⡅⠶⠀⠤⣣⣾⣟⣁⠄⠉⠀⠀⣿⡘⠿⢃⡞⠰⣫⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⢷⣄⡰⠆⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠳⢦⣤⣤⣭⣤⣶⣶⡾⠿⠿⠛⠃⠀⣠⡤⣤⣌⣉⠋⣉⣴⠿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⡛⠳⣌⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠓⠂⠀⠀⠀⠀⠀⠐⢯⡠⠀⣀⠴⠚⣋⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢮⣝⣿⣷⡶⢤⣤⣀⣀⣀⠀⠀⠀⠀⣀⣠⡧⢀⣀⡀⠀⠀⢀⣀⣠⠤⠄⣒⣭⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠮⠭⢭⣭⣭⣿⣿⣿⣿⣞⣓⣲⣦⠭⠭⠴⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⢰⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠓⢻⡆⢸⣧⣄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣤⡴⠟⣡⡞⠁⠙⣧
⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠉⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠈⢻⣟⠶⣤⣄⣀⡀⠀⠉⠳⢦⣤⣀⣀⣀⣀⣤⣤⡴⠞⠋⠀⠀⠀⣀⣠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣌⡉⠙⠛⠓⠶⠶⠦⣬⣭⣭⣤⣤⣤⣤⡤⠶⠶⠚⣛⣭⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠖⠛⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠳⢤⡀⠀⠀⢸⣃⢀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⣦⣶⡄⠀⠘⢿⠢⣧⡄⠀⠸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⢱⡿⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠀⠀⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⣶⡦⣬⠐⠄
⠀⠀⠀⠀⠀⠀⠀⣤⣰⡀⠀⠀⠀⠀⠀⡰⣿⣿⣿⣿⣟⠉⠁⠀⠀⣈⣉⣁⣬⣤⣴⣤⣤⡈⠙⢛⣛⣿⣿⢇⡤
⠀⠀⠀⠀⠀⠀⠘⢿⣆⠹⣆⠀⠀⠀⠀⠑⠉⡙⠛⠛⠫⠾⠿⢄⣤⣈⣉⢙⣿⣽⣄⡙⠿⠿⠖⠛⠛⣋⣤⣾⣷⡾⠧⠯⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⢧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠶⢦⣶⣶⣿⣿⣿⢿⡿⠓⠒⢄⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⣿⣿⣿⣿⢁⡾⠀⠀⠀⠨⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠋⡄⢿⡃⠀⠀⣠⠇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⠋⣼⣿⠿⠏⢀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⡍⢰⣷⣉⣠⣬⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⣦⡈⢤⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⣉⣑⠈⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡴⠞⠉⠙⠻⢦⡑⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⡒⢀⣴⣿⣿⣿⣿⣿⣩⠙⠲⢄⠀⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡾⠁⠀⠀⠀⠀⡴⡀⠈⠺⣿⣿⡆⠤⢄⣀⣀⣀⠀⠀⣐⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⠰⠋⠀⠀⢡⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠘⠐⠀⠘⠅⠀⠀⠀⠰⣎⣰⣋⡠⣀⠈⠻⣏⠢⠤⠍⠛⡉⠻⠿⠟⠛⠛⠿⠟⠛⠛⠋⠈⠁⠀⠀⠀⢀⠠⠃⣴⣟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠓⠆⠀⠀⠙⠲⠖⠃⠀⠀⠸⢆⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣔⡻⠇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀⠀⡀⠀⠀⠀⠀⠀⢄⣶⣦⣄⣀⣀⢹⠟⠒⠂⠀⠀⠀⠀⠁⠀⠀⡠⠤⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠛⠛⠛⠟⡻⢿⠿⣯⠾⠿⠟⠃⠀⠘⠛⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠈⢉⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠃⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⠀⠀⠀⠀⠀⢿⡀⠀⠔⠁⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢎⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⡇⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠤⠀⠒⠒⠒⠒⠂⠐⠒⢎⠀⡞⠓⠒⠒⠂⠠⠤⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⣀⣠⣤⣴⣶⣶⣶⣿⣿⣿⣿⡿⣿⠿⠦⠘⠻⣶⣶⣶⣶⣤⣤⣄⣀⡀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢸⠀⢴⣿⡿⠛⠛⠉⠉⠉⠀⠀⠀⠀⢀⡀⠙⠶⠶⠦⠴⠀⢈⣉⣭⠿⠛⠛⠿⢿⣷⣦⡀⠈⡆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠀⠘⠿⢷⣾⣿⣷⣶⣶⣶⣤⣝⠦⠤⠤⢬⣽⣒⣒⣒⣻⣙⢧⣀⣀⣀⣀⣀⣠⣼⣿⠇⣸⣉⣠⣤⣀⡈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣯⣭⣓⣒⣛⣻⣟⣛⣩⣭⣽⣷⠿⠿⠿⠛⠛⠉⠁⠀⠀⡟⠁⠀⠀⠀⠉⢢⡀⠹⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠴⠊⠀⠐⢄⠀⠃⠀⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⠀⠀⠀⠀⠀⠀⠐⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠇⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⠄⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⡔⠁⡘⢠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⡠⠊⢀⠜⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠒⠙⣄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠳⠄⠤⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠈⠁⠀⠀⠒⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠿⣤⠀⠠⠴⠚⠉⠒⠢⣀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠐⠛⠿⠿⢿⣲⠤⣄⣈⠑⠒⠠⠤⢀⣉⡁⠀⣀⣀⠲⣄⡀⠈⠅⢀⣀⠀⠀⠀⢐⣀⡩⠄⢒⣠⠆⠀⠈⠉⣉⡃⠤⠄⠒⠉⢀⣀⣤⣾⣿⣶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠬⡉⣓⠒⠦⠤⣤⣄⣉⣉⣉⣁⡈⠛⠛⠒⠒⠲⠶⠤⠴⠶⠒⠒⠚⠛⠓⣂⣉⣉⣡⣤⡤⠴⢶⣲⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⠀⠤⠭⠍⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠥⠭⠥⠀⠐⠒⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⠤⡤⣤⢠⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⡜⡟⢯⣋⣀⠀⠈⠉⠉⠓⡲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠆⠓⡀⠈⠁⠀⠀⠀⠀⠀⣿⣽⢶⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠙⠒⢠⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⢆⠀⠈⠁⢀⣀⣀⢀⠠⢠⣝⣦⣾⢢⠁⢀⣤⢶⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿⣷⣤⣼⣥⣤⠤⠀⣀⣠⢿⣿⣿⣯⡍⢁⡁⠀⠀⢀⣤⠇⣆⡀⡾⣿⣿⠏⣿⣩⣋⣼⡿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡗⢉⣿⡇⠘⢿⣓⣲⠞⠾⡟⣸⠿⣿⣾⠿⣍⣁⣀⣉⠛⡖⠰⡟⠤⠜⠑⡇⣰⣿⠛⠋⠉⠀⣸⣻⡟
⠀⠀⠀⣀⡤⡤⠲⠲⣖⠻⡉⠊⣧⣸⠇⠁⠀⠈⠻⣷⣶⡿⢂⣁⣀⡀⠀⠀⣏⢣⣤⣉⡛⢃⠏⠸⣶⠘⠃⣣⣽⡟⠂⡀⣤⡴⢿⡿⠁
⠀⢠⣾⣿⣿⡷⡀⠀⣌⠣⣌⢠⠺⣯⣆⠀⠀⠀⠀⠈⠓⠯⠍⠉⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠉⠉⠉⡹⣸⠔⣊⣍⣭⣾⠟⢨⠙⠒⢄⡀
⠀⢿⣿⣿⣿⣿⣛⠿⠿⠯⠈⠳⠄⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠿⠷⠄⠀⠀⠀⠀⠀⢀⣰⣿⣷⠿⠿⣿⣻⡟⣩⠄⣂⣴⣖⠭⡆
⠀⠈⢿⡻⢿⣿⣯⣿⡧⣌⢄⣀⣀⢀⡈⠻⣿⣵⣄⣰⡄⢠⡄⠀⢀⢐⡀⠀⢀⠀⠀⢀⣠⠀⣴⢀⣭⡿⠛⠋⠁⠀⠛⠛⣫⣾⣁⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠉⠳⣮⣭⣙⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣾⣖⣤⣼⣧⣾⣆⣢⣷⣶⣯⣾⡷⠛⠁⠀⠀⠀⠀⣀⣈⢉⡥⠬⣴⢚⣫⠵⠋
⠀⠀⠀⠀⠀⠀⠉⠻⢿⢿⣶⣶⣮⣭⣭⣯⣛⣛⣛⣛⣿⠿⠭⠭⠭⠭⠉⠉⠩⠉⠩⣁⣒⣚⣒⣒⣤⣘⣩⠭⠭⢒⣒⢶⣻⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠯⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣭⣭⣿⢿⣛⠓⢊⣂⣤⣤⣅⣸⣤⠥⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣫⣬⣥⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠗⣾⣿⣿⣿⣿⣿⢿⢹⣿⣿⣭⣌⡛⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣮⣍⠙⠻⣿⣿⢸⣾⣿⣟⠉⠉⠙⠷⠖⠄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡻⣿⣿⣿⣶⣮⣿⡿⣻⣿⣿⡆⣀⣀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢟⡿⢿⣿⣿⣿⣿⡇⣹⣾⣿⣿⣯⣛⣀⣄⡑⢆⣠⡀
⣀⣾⣷⣦⡄⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡾⣷⣿⣿⣿⣿⣿⢻⣿⣿⣯⣽⡆
⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠉⠉⠉⠹⣿⣿⣯⣟⣷⡖⣎⣿⠿⣿⣿⣿⣿⣷⣿⣿⣿⡃
⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠏⣿⣿⣿⣿⣿⣿⣿⠛⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⡿⣺⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡍
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠻⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⡗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣤⣿⣿⣿⣿⣿⠀⠉⠛⠿⠿⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠋⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⠉⠓⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡞⠋⠉⠉⣁⣠⣤⣴⡶⣶⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣠⣶⣿⣿⣿⣿⣟⠻⠗⠾⠇⠿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡯⠭⠽⠿⣟⡛⠛⠛⠛⠛⠛⠟⣛⡻⠿⠯⠭⢿⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⡛⢝⢶⣄⢸⣿⣯⣴⣶⣿⣿⣋⢀⣀⣤⣴⣤⣴⣤⣄⣀⠈⣻⣿⣷⣶⣦⣽⣿⢀⣠⢞⡯⣋⡦
⠀⠀⠀⠀⠀⠀⠈⠛⢦⣕⠯⡙⡿⣋⣋⡛⢿⣿⠇⠟⠋⠉⠉⠉⠉⠩⠉⠙⠷⢹⣿⡿⣛⣙⡻⣿⣋⠕⣡⡾⠋
⠀⠀⠀⠀⠀⣀⣀⣀⣾⣿⣷⣼⢙⡍⠯⣍⢹⣿⠀⡇⠀⠀⠀⢘⠀⢰⠀⢸⢸⢸⣿⢉⡈⠯⢍⢹⣵⣿⣿⣆⣀⣀⣀
⠀⠀⠀⢰⠁⠀⠀⠀⠀⣰⣾⣿⣷⣤⢤⣼⣛⣺⢰⡗⠷⠿⠿⠿⠟⠻⠛⠻⢿⢸⣗⣛⣤⣤⣼⣿⣿⣶⡄⠀⠀⠀⠀⠹⡄
⠀⠀⠀⣯⣶⣴⣴⣦⣶⣿⣿⣿⣿⣭⣼⣻⣿⣿⢸⡇⠀⠀⠀⢀⠀⠐⠀⠀⢸⢸⣿⢿⣻⣤⢭⣿⣿⣿⣿⣶⣴⣦⣶⣴⣧
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣾⣏⣾⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣧⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣭⣛⠿⠟⢻⢨⡷⢶⠶⠶⢾⠶⠶⠶⠶⢾⢸⡟⠻⠿⣻⣽⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡟⠀⢸⢸⡇⢀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡿⣿⣿⣿⣿⣿⡿⢻⡟⠀⠀⠀⣷⣾⣾⣇⠁⠨⠀⠀⠰⠀⠀⠀⠀⢈⣼⣾⣶⡧⠀⠀⠀⣿⡽⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢹⡉⠉⠉⠉⠉⢻⣿⠃⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠸⣿⠋⠈⠉⠉⠉⢡⠇
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣀⣀⣭⣿⠆⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣤⡀⠀⣏⠀⢹⠁⠀⣿⠀⠀⡇⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⠻⢦⣽⡶⠿⠶⠶⠿⠶⠶⣧⣤⣞⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⢀⣤⣤⣼⠇⢠⣤⣤⡶⠶⢶⡶⠶⣟⠛⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠳⠶⠛⠳⠟⠁⢠⡟⠀⠀⠀⢸⡇⠀⠀⡇⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢀⡼⠃⠀⠀⣧⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡛⠷⠶⣤⡾⣯⠀⠀⠀⠀⢿⡀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣠⡾⣳⣤⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⣤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠒⢒⣒⣛⣛⣛⣛⣒⣒⡒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠭⠖⠚⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠦⢭⣓⠦⣄
⠀⠀⠀⠀⠀⠀⣰⡿⠞⠁⠀⢀⣀⡤⠤⠴⠖⠒⠒⠚⠛⠛⠛⠛⠛⠒⠲⠦⢤⣀⡀⠈⠑⢮⠳⡄
⠀⠀⠀⠀⠀⣼⢹⠁⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠓⠦⡀⠀⠀⠀⠈⠙⠲⣄⣸⢀⡇
⠀⠀⠀⠀⠀⣿⢎⠳⣅⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠉⠁⢠⠖⠳⡄⢹⠀⠀⠀⠀⠀⣠⠞⣡⢟⣧⡤⠶⠒⠒⠒⠲⠤⣀
⠀⠀⠀⠀⠀⢹⠀⠑⠮⣝⠲⢤⣄⣀⡀⢧⠀⠀⠸⡍⠉⠉⠀⠀⡟⣼⣀⣤⠤⠖⣫⠴⠋⣰⠋⠁⢀⣠⠤⠤⠤⣄⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠉⠓⠦⢬⣉⣉⣛⠒⠒⠛⠒⠒⠒⢛⣉⣉⡥⠴⠒⠋⠁⠀⠀⠘⢶⠖⠋⠀⠀⠀⠀⠈⢧⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠛⠒⠒⠖⠒⠋⠁⣀⠴⠋
⠀⠀⠀⠀⠀⢀⡤⠞⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣷⡦⠤⠤⠤⠖⠒⠉
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋⠀⠀⠙⢦⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⢌⡙⠲⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⣩⠏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠬⣭⣉⣉⣉⣉⣉⣉⣡⡤⠴⠒⠋⠁⠀⠀⠀⢀⣠⣿⠃
⠀⠀⠀⠀⠈⠻⣗⡦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣖⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠭⢭⣒⣒⠦⠤⠤⣤⣤⣀⣀⣀⣀⣤⡤⠤⢶⣒⡯⠽⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣖⣒⣒⣛⣛⣉⣉⣉⣉⣉⣛⣛⣛⣛⣋⣉⣉⣛⣛⡒⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⢀⡿⢷⡛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠶⣽⢷⡀
⠀⠀⠀⠀⠀⠀⢱⠦⢍⣓⠒⠶⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠚⣡⡞⠓⠢⣄
⠀⠀⠀⠀⠀⠀⡾⠀⢀⡈⠉⠓⠲⠤⠤⣤⣉⣉⣉⠛⠓⠚⠒⠒⠒⠚⠛⠋⢉⣉⣀⣤⠤⠖⠚⢹⡇⢰⡆⠈⠱⣄
⠀⠀⠀⠀⠀⠀⡇⢠⣿⣿⡷⠀⠀⣶⠄⠀⠀⠈⣉⡉⠉⠛⠛⠛⠛⠛⢉⣩⣍⠉⠀⠀⠀⣶⡆⢸⠉⠙⣆⠀⠘⠛⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠆⠀⠀⣾⣿⣿⡇⠀⠄⠀⠉⠁⢸⠀⠀⢸⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⠀⠀⠁⠀⣠⣶⣄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠘⠻⠟⠁⢀⡀⠀⢠⠀⡏⠀⢀⡞⠘⠁⢰⠇
⠀⠀⠀⠀⠀⠀⣧⠀⠸⠿⠃⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠙⠛⠀⠀⣠⣦⡀⠀⢰⣿⡿⠀⠀⢰⠷⠚⠉⢀⡀⢀⠞
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⢀⣀⣀⠀⢀⡀⠀⢿⣿⠇⠀⠀⠉⠁⡀⠀⣼⠘⢁⣀⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⣶⣶⡄⠀⠀⠀⠀⢾⣿⣿⠆⠀⠁⠀⠀⠀⠀⣀⠀⠀⠸⠿⣰⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠿⣿⠏⠀⠀⠀⠀⠈⠉⠉⠀⢀⣤⣄⠀⠀⠘⠟⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⢀⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡝⠓⠲⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠖⠛⢩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠶⠦⠤⢤⣤⣤⣤⣤⣤⠤⠴⠖⠋

⢀⠀⠄⢀⡤⢤⣤⣀⠀⢀⡀⠀⠤⠶⣶⠶⠖⠀⠄⠀⠀⠐⠶⠀⠀⢠⣆⡤⠄⣠⡤⢤⡤⢤⣤⣤⠶⠦⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣂
⠘⠀⠀⠀⠀⠀⠐⠀⠁⠐⠄⠀⠀⠀⠱⣶⣶⣆⣀⠊⠀⠀⠀⠻⣆⣛⠻⠶⠺⠛⠟⣒⠲⠮⠯⣽⡛⢻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⢿⣿⣿⠒⢀⣀⠀⠀⠀⠀⠈⠋⠉⠉⠙⠓⠒⠾⢿⣭⣽⣿⣿⣿⡁⠈⠉⣁⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠙⠤⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⡿⣿⣿⣿⣿⡿⡿⠷⠿⣿⣿⣿⡟⠻⣿⠧⣼⣹⣽⣻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣿⣯⣍⣽⡿⠛⠋⡄⠶⣴⣿⣷⣿⣿⣿⣿⢿⡿⢿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠒⠉⠛⠙⠛⠝⠛⠉⠀⠁⠀⣾⣛⡶⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⡶⢾⣿⣶⣦⢀⣀⣤⣤⣶⢤⣀⣀⣴⣿⣷⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣍⠻⠿⣿⣿⣿⣿⣿⣿⣿⣾⣟⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠀⠈⠹⣿⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣶⢰⣦⠀⠀⠈⠛⠿⠾
⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋⠀⠀⠀⣉⣿⡷⠦⠤⢄⡀⠀⣀⣀⣀⣠⣤⣤⣤⣤⢤⠤⢴⠲⠖⠛⢑⣦⡤⢤⣀⣀⢀⡀⢀⣠⣴⣿⣿⣿⣿⡟⠛⠋
⠀⠀⠀⠀⠀⡔⣫⣰⡞⠁⠀⠀⠀⠙⠉⠀⠀⣰⣾⡇⠀⠘⠀⣠⣦⣤⣷⣶⣄⡠⣤⣶⣶⣄⡀⢈⣿⣤⣠⣤⣤⣤⣴⣶⣾⣏⡼⣿⣿⣿⣟⣄
⠀⠀⠀⠀⠈⣄⣿⣿⢸⣾⣷⣤⣀⣀⣀⣴⣿⣿⣿⣧⣾⣷⣶⣽⣿⣟⠻⣿⢻⣿⣾⣮⣿⣿⣇⣘⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣽⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠠
⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣭⣀⣀⣠⡤⠔⠛⠋⠙⣿⡿⠛⠋
⠀⠀⠀⠀⡀⠀⢸⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣾⣿⠿⣽⣿⠟⣡⠄⠀⠘⠟⠻⢦⡀
⠀⠀⠀⠀⢿⡀⢸⣿⣿⣈⣿⣿⣿⣿⣿⣿⣿⣷⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠰⣿⣿⣷⡔⠪⣴⣿⢀⠀⠀⠑⠀⢢⠁⣠⡄⡀
⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿⠈⢉⡩⠋⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⡿⠋⢠⣴⣷⣶⡘⣿⣿⣿⣿⣿⣿⣦⣦⣤⣶⣷⣶⣬⣿⡋⠉
⠀⠀⠀⠀⣸⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀⠐⠉⠀⠀⠀⠀⣀⣤⠄⠀⠀⣿⣿⣷⢀⡸⣿⣿⣿⢓⣼⣿⣿⣧⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠈⠉⠙⠁
⠀⠀⠀⣀⠩⣏⣿⣿⣯⣿⣿⠻⠟⠛⠛⡴⠁⠀⠈⠀⠀⠀⣼⡿⠁⡀⢀⠀⠙⣿⣿⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⢟⣿⡮⣿⣶⣶⡦⠦⠤⠴⢶⣶⣶⣶⠄⠶
⠀⠀⠀⠀⠀⠉⣿⣻⣿⣿⡟⠒⠒⠛⠙⢀⠀⡀⢉⠵⢻⣿⡟⢓⣠⣤⣤⣶⣦⣿⣿⠼⢻⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⠛⠛⠿⠿⢷⣶⣶⣿⣾⣿⣿⣿⡷
⠀⠀⠀⣀⠀⠀⠘⠃⣿⣿⣿⠀⠀⠀⠐⠊⠀⢬⣌⣠⡾⢿⢟⡫⠴⣋⣀⣈⢫⣿⣿⡠⢼⣿⣿⣿⣿⣿⣿⣟⣹⣋⣽⡿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛
⠀⠐⡄⠘⢏⣴⠀⣈⣿⣿⣿⣆⠀⠀⠀⠀⠀⢠⣿⠁⣤⡿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣯⣛⣉⢛⡛⠋⠙⠙⠛⠿⠿⠿⠿⠿⣿⣷⣦⣶⣾⣿⣿⣶⣦⣦⣤
⠐⠀⠙⢷⣾⢻⣶⡿⠛⠟⠻⠿⣛⠿⢿⡿⢷⣿⣷⣾⣿⠄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠚⠀⡀⣹⣏⣿⣰⣷⣴⣾⣿⣿⣾⣿⣿⣦⣿⣿⣿⣧⣤⣴⣿⣿⣿⣇⣾⣿⡏⠉⠛⠛⠿⠿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⡿⢿⣿⡿⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣶⢾⡷⢷⣶⣶⣾⣷⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣶⣶⣷⣷⣷⣾⣿⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⡇⣾⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣾⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣠⡶⢛⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⣳⠀⣳
⠀⠀⠀⠀⠀⠈⣧⡸⣆⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⣧⢻⡆⠀⠀⠀⠀⠀⠀⠀⠸⣆⡻⣤⣹⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣃⣾⣁⢺⣏
⠀⠀⠀⠀⠀⠀⠈⢷⢹⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⢸⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⢙⡇⢈⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠚⠜⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣇⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⢀⡆⠀⣸⠇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⣰⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡘⢦⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣸⢀⡇
⠀⠀⠀⠀⠀⠸⣇⢰⡏⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣩⡼⢃⡼⠃
⠀⠀⠀⠀⠀⠀⢻⡜⢇⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢩⠵⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠇⠼⠁⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣾⢠⣼⣧⢦⡄⠀⠀⠀⠀⠀⠀⣀⠀⢸⡃⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⡴⠶⣮⠹⣿⠋⠀⡿⠀⠀⠀⠀⠀⢀⣤⣶⣤⠀⡾⢉⣠⣍⣿
⠀⠀⠀⠀⡴⠶⢦⡉⠀⣾⢁⢀⡇⠀⠀⠀⠀⠀⢸⠉⠻⡾⣇⢉⡷⠶⣄⠀⠀⠀⠀⠀⠀⠓⠄⢸⡇⡟⠀⠚⠁⠀⠀⠀⠀⠀⢺⡁⢉⡿⡗⡇⢸⡀⢘⡿
⠀⠀⠀⠀⠧⠀⠈⣷⢀⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⡿⢸⠘⣧⠦⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣸⠃⢱⠀⠉⠉
⠀⠀⠀⠀⠀⢀⡴⠋⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠘⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠂⠐⠊⠊⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡖⠀⠀⠀⠀⠀⠂⢀⠉⠁⠀⠃⠀⠠⠂⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣤⣬⣉⣴⣀⡀⠀⠤⠿⠟⠓⠰⢂⣠⣀⣠⣔⣿⡫⠇⢀⣠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⠉⠉⠈⠉⠉⠉⠙⠉⠙⠛⠛⠛⢿⣿⣿⣿⡏⢽⠀⣚⠁⣀⠤⠤⣈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡬⢭⣭⣿⡷⠇⢐⠄⣧⠊⠁⠀⠀⠘⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠬⢵⣒⣺⠍⢀⠁⡇⠀⠀⠀⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠥⠬⠽⣖⣒⠁⠐⢠⠇⠀⠀⢀⡴⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⣗⣒⠠⠀⢸⡆⠂⠉⢁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠨⠭⢭⣷⣒⠀⠁⣼⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠨⣈⣭⣿⡷⢂⢊⡜⠁
⠀⠀⠀⠀⣀⣀⡠⣤⠤⠀⠒⠀⠚⣦⣀⠀⠀⠀⠠⠀⣦⠡⣶⣿⣿⢟⣁⣴⣿⣶⣶⠶⠦⠤⣤⣄⣀⡀
⠀⠀⡞⡉⢁⠀⠀⡄⠀⠀⠈⠉⡀⠈⠻⢉⣀⡀⠀⠀⠚⡒⣿⣿⣍⣩⡽⠋⣉⠉⠉⠉⠛⠒⣢⠀⢈⠙⣟⡆
⠀⠀⠙⠻⢿⣓⡲⠤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣈⣉⣉⣉⣉⣉⣭⣉⣁⣀⣴⣶⣶⣦⡤⠤⠤⠖⣚⣻⠽⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠶⠤⠤⢤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣥⣤⣤⡤⠤⠤⠶⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⡶⠛⢉⡵⠖⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡏⡀⢾⣥⣤⣬⣷⣀⣿⡀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠓⣵⡤⣀⣴⣾⣟⡛⠋⢟⢷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⢀⡠⠋⠀⠀⠀⠩⣿⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⣄⡘⢉⣽⡁⣇⢀⣒⡰⠀⣴⡟⢀⣼⡋⡀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠓⣿⠀⣸⢿⣆⠀⠀⠀⣿⡀⢼⡟⠿⣬⡛⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣼⢧⡀⠀⢤⠀⣣⡘⠀⠠⣘⣿⣟⠲⣄
⠀⠀⠀⠀⠀⣰⠚⠛⠒⠒⢒⡛⠉⣇⠈⠀⢠⣾⣿⣿⣿⣶⠀⠹⡌⠙⠌⠂
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠲⢽⡄⠀⠺⠿⠛⠛⣿⣿⠈⣼⢃⣠⣾
⠀⠀⠀⠀⠀⢷⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠸⠃⢺⣿⡃
⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣤⠤⠀⣨⠙⡎⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⠀⠀⠀⡆⠀⣠⣴⣿⣿⣿⣿⣿⣿⢷⣄⠈⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣄⣳⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠉⠂⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣠⠴⠒⠒⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣷⠀⣸⠃⠀⡶⠦⠀⠀⠀⠀⠀⠠⠤⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣻⣿⣿⣿⣿⣿⣿⣷⣻⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⣼⣇⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠻⠬⠤⣄⠸⠀⡤⣿⠂⠀⠀⠀⠀⠀⢀⡠⠔⠀⠀⠐⠛⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⣠⣄⠀⠀⢀⣀⡤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⢴⡶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⣿⣿⡿⢀⣶⠟⠁⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣮⣭⣭⣭⣍⣙⣛⣩⣭⣭⡭⠇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠐⠋⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠤⠤⠶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡇⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣠⣦⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠃⠀⠀⠀⠀⠈⠙⠛⣶⣖⣪⠥⠿⠛⠉⠙⠓⠒⠚⠉⠁⠀⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣦⣤⣀⣀⣀⠀⠀⠀⠀⣀⣀⣠⣤⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢿⣷⣦⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⢹⣷⡀⢰⣿⡄⠀⢸⣿⡆⢀⣀⣤⣤⣤⡌⢻⣿⣿⣿⡿⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⣸⣿⡇⣼⣿⡁⠀⢸⣿⡇⠈⠉⣿⣿⠀⠀⠀⠈⠻⠿⠁⢸⣿⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠙⠛⠀⣿⣿⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⢀⣴⡾⠛⣶⡄⣼⣿⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⢹⣿⡄⠀⣿⣿⡇⠀⢠⣿⣿⠀⣾⣿⣀⣠⣿⠇⣿⣿⣀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⣴⡦⠸⣿⣇⣼⠿⣿⡇⠀⢸⣿⣿⠀⣿⣿⠉⠉⠁⠀⣿⣿⠉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⡶⠟⠁⠀⠈⢉⣅⠀⠈⢠⣄⢸⣿⡏⠀⠘⢿⣆⣀⡾⠃⢹⣿⠀⠀⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠠⠶⣶⣶⠖⠂⣾⣿⠀⠀⣿⣿⠀⠉⣤⣶⣤⡀⠀⣰⣦⣠⠴⣦⣄⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⢠⣿⣿⠀⠀⣾⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠸⡿⠀⢹⣷⠀⣿⣿⠃⠀⣿⣿⠀⣼⣿⡆⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠚⠛⣿⣿⢀⣴⡶⢦⣿⣿⠀⣿⣿⠀⢠⣿⣿⠀⣿⣿⣿⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠀⠀⣿⣿⠀⠀⠻⠃⠀⠀⠻⠟⠸⣿⡇⢀⣿⣿⠀⢿⣿⠀⢸⣿⡟⠀⢹⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⣴⠶⣬⡁⢀⣾⣷⠀⠀⣾⣿⡄⠀⠈⠉⠉⠿⠇⢀⣶⡄⠀⣠⡶⠶⣦⡈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⣿⣷⢸⣿⣟⠀⠀⣿⣿⡇⢰⣿⠋⠛⣷⡄⠈⣛⠁⢠⣿⡇⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠿⠟⢸⣿⣿⠀⢠⣿⣿⡇⣸⣿⠀⠀⢹⣿⢰⣿⣇⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢿⣿⣤⡾⠻⣿⠇⣾⣿⣀⣤⣿⠏⢸⣿⣷⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⢀⣾⡆⣴⣦⡀⠀⢀⣤⡄⢻⣿⠉⠉⠀⠀⢸⣿⡟⠸⣿⡇⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠛⠋⠀⠸⣿⣿⣶⣿⣿⠃⠸⠿⠀⠀⠀⠀⠈⠛⠃⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠃
⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⢠⣜⣍⣬⣭⣿⣆
⣿⣿⣿⣿⠟⡉⢿⡞⢛⡛⠞⠛⣋⣷⡶⢶⡞⣙⢛⠛⢻⠋⣳⡶⢶⣾⢋⠛⠛⠛⠞⣛⣷⡶⢶⣶⠛⢋⠛⠞⣛⣶⠶⣶⡞⡙⠚⣙⠲⢋⣳⡶⢶⡞⢉⡟⡙⠒⢛
⣿⣿⣇⣬⣤⣾⣿⣟⣨⣧⣰⣆⣁⣽⣷⣾⣇⣉⣀⣠⣙⣠⣿⣷⣼⣿⣰⣾⣠⣄⣄⣁⣿⣷⣾⣟⣠⣠⣶⣌⣁⣿⣶⣿⣅⣶⣇⣋⣴⣰⣿⣷⣾⣇⣐⣃⣴⣟⣀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣽⣽⣿⣿⣿⣿⡟⢿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⣿⣿⡟⠻⡟⠻⣿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣤⣽⡿⠿⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠁⠟⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⣽⣿⣿⣷⡀⠀⢀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣥⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⡇⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢈⣿⡿⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠉⠁⠈⠛⠳⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢠⣴⠀⠀⠶⠿⠃⠀⠀⠀⠀⠉⠉⠉⠰⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠤⠀⠀⠀⠀⢿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠛⠋⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⡀⠀⠸⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⠈⠀⣮⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠰⣿⣿⣤⢰⡄⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠋⠰⠤⡈⠉⠉⠀⠀⣙⣿⣿⣿⣿⣿⣿
⣿⣿⣦⠀⠀⠀⠀⠿⣿⣿⣿⠁⠀⠙⠛⠉⠁⠁⠸⠛⠏⠁⠈⠟⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢮⣍⠃⢉⣿⣿⣿⣿
⣿⣿⣿⣿⡆⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠈⣿⣿⣿
⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠑⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⡀⠸⠁⠀⣸⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣾⣆⠀⠀⢄⣠⠀⠀⠀⠀⣀⣀⣠⣴⠟⢀⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣠⣴⣿⣿⣿⣿⣿⣶⣤⣦⠀⠀⠀⠶⣌⠉⠉⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣇⢠⣶⣿⣠⣾⡷⣤⣀⣀⠀⠀⠀⡀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠒⢶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣶⡿⠋⠀⠀⠠⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⢀⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⠄⠈⠛⢻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠉⠛⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠙⣿⣼⡟⠋⠛⠻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣥⣼⣦⢸⣿⣿⣿⣿⣿⣿⠿⠋⣉⡉⢁⠈⠙⢷⣶⡾⠛⠉⠀⠀⠀⢀⣄⡉⠻⠿⠋⠀⠀⠀⠀⠀⣠⡿⣤⡻
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠘⠃⠀⠙⠛⠿⠿⠋⠀⠀⠈⠉⠉⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠿⠨⠟⠀⠀⠀⠀⠀⠀⠀⠈⠳⡾⠟⠘
⣿⣿⣿⣿⣿⣿⣿⣵⡄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠈⠀⠀⠖⣀⠀⠀⠀⠀⠀⠈⠁⠤⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠰⠾⢿⡷⠀⠀⠀⠀⠀⠐⠷⣾
⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠤⠤⣤⠀⠀⠀⠈
⣿⣿⣿⣿⣀⠀⠀⠀⣴⠻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣧⣄⠀⠈⠉⠉⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⢸⡏⠹⠗⠐⠀⠀⣀⣤⣶⠞⠃⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠠⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⡀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⣤⣶⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣤⡀⠁⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣶⣿⣿⣿⣿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀
⣿⣿⣿⣿⣿⣿⡿⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠶⠶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⢟⠿⠛⢁⡀⠀⣀⠐⠒⠒⠬⢍⡛⠽⡲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⢶⣟⠗⠡⣂⠅⠂⠀⠈⣺⢁⠀⠀⠀⠀⠪⡑⢮⡻⣷⣙⣢⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⡡⠊⢠⣿⢋⢀⠞⠀⣠⠆⠁⠀⠀⠃⠀⠀⠀⢀⠀⠈⠢⠙⡌⣿⣄⣠⡤⠘⡆
⠀⠀⠀⠀⠀⠀⣰⢋⠞⠀⢀⣏⠇⠎⠊⠀⠀⠐⠂⠀⠀⠀⣀⠀⠘⠀⠀⠐⢀⠀⠀⠀⠸⣹⡆⢀⣴⢣
⠀⠀⠀⠀⠀⢰⠃⠃⠀⠀⢸⢸⠀⠀⠀⠄⠀⠀⠀⢦⡀⢀⠁⠐⠁⠋⡈⠰⠸⠀⠀⠀⠀⡏⣧⡊⠉⠉⠁⠀⠀⠠⣄
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢾⡌⡇⢇⢂⢘⡔⢄⠀⠀⠀⠁⠻⠇⣠⠆⡠⡡⠃⠀⠀⢠⢠⢳⣿⡿⠿⡒⢦⣤⣀⡀⡨
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣧⠹⡈⠀⠀⠙⢷⣵⠶⣤⣄⡐⠐⣡⣯⠎⠂⠀⠀⣠⢃⠎⣾⠿⠇⠀⠀⠀⢵
⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠙⣷⡙⣄⠀⠤⡀⠈⠉⠒⠬⠻⢴⠟⠁⠀⠀⣠⠞⡡⢋⣾⠏⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⣿⣮⣓⢦⣈⠑⠢⠤⠄⠈⠘⠀⠀⢀⣠⠴⣋⡴⣿⠏⠀⠀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⠸⢿⣯⡓⠮⢭⣓⣒⣒⣒⣒⣒⣩⡭⠖⢛⣥⠞⢁⢄⠀⠀⢠⠂⠀⠀⣾
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢂⢠⠙⢝⠷⣦⣤⣀⣉⣉⣁⣬⣤⡴⠞⠋⠀⠀⡠⠂⠀⢠⢂⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠑⡑⢄⠑⠤⡉⠉⢋⠉⠉⠀⠀⠀⠀⠀⠀⡨⢀⢀⠔⡡⠃⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠠⠀⠀⠂⠀⠄⠤⠄⠀⠀⣀⠥⢊⠔⡁⠈⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠐⣂⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢾⡢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⢔⣪⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠯⣿⣶⡦⢤⣤⣄⣀⣀⣀⣤⠤⠤⣖⣺⡭⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠒⠛⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠘⠧⣄⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣶⡖⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡀⠀⠀⠀⣠⠞⠀⠀⠈⠓⣦⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠏⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⠀⠀⠓⣄⠀⠀⠀⡖⠋⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡄⠀⠀⠀⠤⠞⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⣀⣀⠀⠀⠀⠀⠀⠋⠁⣀⣀⣀⠀⠀⠀⡼⠀⠀⡼⠁
⠀⠀⠀⠀⣠⣤⠤⠤⠶⠶⡶⠋⠁⠈⠙⣖⠒⠒⠒⢲⠎⠁⠀⠈⠳⡖⢲⠇⠀⣸⠷⠶⠤⢤⣤⣄
⠀⠀⠀⠀⡏⠉⠉⠉⠉⠙⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⢸
⠀⠀⠀⠀⡇⠀⡴⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦⡀⢸
⠀⠀⠀⠀⣇⠘⣇⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⠇⡟
⠀⠀⠀⠀⢹⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠈⠋⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣸⠃
⠀⠀⠀⠀⠀⢳⡀⠀⡤⠶⣄⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡤⠶⣄⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠳⣄⠳⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠣⡤⠞⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠠⣄⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⠀⠀⠀⠀⡔⢲⡄⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣀⡤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣏⣉⣉⣉⣉⣉⣉⣉⣉⣹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣀⠈⠉⠁⡐⠠⡀
⠀⠀⠀⠀⢀⢀⡀⡀⠠⠀⠀⠀⠀⠀⡠⠴⠁⠉⠀⠀⡀⠀⠉⠃⠱⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡌⠀⠀⢡⠀⢹⡆⠀⡇⠠⡀⢠⠊⠀⡇⠀⠀⣤⠀⠀⣤⠀⠀⡎⠀⠈⠑⣄⠀⠀⠀⠀⠎⠑⣀⠀⢠⠈⠢⡈
⠀⠀⠰⡀⠈⠓⠀⠁⠄⠃⡐⠁⡇⠀⠀⠃⣠⠀⠀⠀⠀⡄⠀⣄⢇⠀⠀⠀⠈⡆⠰⡉⢱⠀⠰⠃⢀⡊⠠⡀⠇
⠀⠀⠀⠈⠑⡀⠀⠀⠀⣠⠁⠀⡇⠀⠀⢰⠀⢀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠸⠀⢡⠀⠁⠀⠀⠉⠀⠠⢃⠃
⠀⠀⠀⠀⠀⠈⠁⠆⣀⡄⠁⠀⡇⠀⠀⢸⢲⣄⠀⠀⠂⠀⢀⣴⠀⢡⠀⠀⠀⠀⡇⠘⠢⠀⡀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠢⠀⣧⠀⠀⢸⠘⣿⣿⠿⠿⠿⢿⡏⠀⢸⠀⠀⢀⠀⣇⠀⢷⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⡞⠀⠀⢸⠀⠘⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣉⣡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡘⠀⠀⢠⠊⠈⠀⠀⠀⠀⢸⠀⠀⠁⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇⠀⠀⠠⡀⠃⠀⠀⠀⠀⡘⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⢀⠅⠀⠀⢀⠇⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⠀⢂⠀⢂⠀⠀⠀⡸⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠄⢁⠈⠀⠀⢀⠃⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣸⠀⠀⠀⠄⠰⠀⠀⠀⢘⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠠⡨⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠊⠀⠁⠐⠀⠀⠚⠢⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣸⠠⠀⠀⠀⠀⠀⠀⠄⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢢⢀⠖⠠⡀⠀⠀⢀⠄⢣⠀⡔⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡀⢤⠆⠀⠀⢧⢄⡀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠐⠎⠑⠈⠁⠀⠀⠀⠀⠉⠂⠡⠔⠀⣀⣄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⣼⣿⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣸⣿⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠤⠤⠴⣿⠥⠤⣾⢯⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡩⢭⣶⣶⡶⠒⠊⠉⠉⠙⢏⠉⠻⠓⠒⠒⠒⠨⠭⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣧⣜⣻⠿⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢨⣍⠙⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⠉⠁⢀⣿⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⡴⠶⠶⣌⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⡟⠀⠀⠀⠘⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣸⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠖⣫⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⢾⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⣱⠯⣅⣀⣀
⠀⠀⠀⣀⡤⣖⣺⠭⠗⠚⠉⠉⠀⠀⢸⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢧⠀⠀⠐⠚⠭⢭⣒⡢⢤⣀
⠀⢠⡞⣵⠟⠉⠀⠀⢀⣀⠤⠤⣒⣒⠚⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠦⣬⠷⠤⢤⣀⠀⠀⠈⠙⠳⣮⢳⡄
⠀⠸⡎⣿⠀⠀⢠⠖⣩⣶⡾⠋⠉⠀⠀⣿⡸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠈⠙⠲⣦⣍⠲⡄⠀⠀⠘⣰⠃
⠀⠀⠙⢮⡳⣄⠸⣄⠻⣿⣦⣀⠀⠀⠀⠈⠷⣬⣓⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠟⠀⠀⠀⠀⠀⠀⠈⠿⣣⠇⠀⣠⠞⠁
⠀⠀⠀⠀⠉⠒⠭⣂⡉⠒⠪⠭⢟⣒⣲⣤⠤⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⣀⣀⡀⠤⠴⠒⣉⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⠤⣄⣉⣉⣉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠚⢋⣉⣉⣉⡥⠤⠤⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⠀⠁
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢡⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠉⠛⠶⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠀⠀⣠⠶⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢀⠏
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⢀⣤⡈⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣧⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢤⣀⡀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠉⠀⠀⠀⠽⠁⠀⠀⠀⠀⠀⠀⠀⢴⣤⣠⣴⣚⣋⣁⣀⣀⣀⣀⣈⣉⣉⣓⣲⣤⣠⣴⡶
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⣽⡟
⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣿⠛⠀⡀⠀⠀⠀⠀⠀⠛⠿⣤⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣞⣁⣀⣀⡀
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠙⠻⠯⣭⣉⣉⣉⣩⣭⠽⠿⠟⣛⣩⣍⣀⠤⢌⡾⠛⠉
⠀⠀⠀⠀⠈⠛⠻⠿⠿⠟⠻⣇⠀⠀⠀⠀⠈⠙⣿⡟⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣽⠏⠁⠀⠀⠛⠋⠉⣭⣥⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡤⠶⠶⠶⠶⢾⣷⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣾⠃⠀⠀⣠⣤⠶⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣤⣤⣀⠀⢀⣤⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⣿⠁⣸⡏⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡉⠉⠙⠛⠻⠿⣿⣧⣄⡀⠀⢀⣾⠇⢠⡿⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠉⠛⢿⣷⣾⠋⢀⣿⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣾⠃⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠀⠉⠻⣷⣄⡀⠀⠀⠀⢠⣾⠏⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠁⠀⠀⢠⣾⠏⠙⠿⣶⣄⣠⣿⠏⠀⠀⠀⢀⣾⣿⢤⣤⣤⣠⣤⣄⣤⣤⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⣰⡿⠁⠀⠀⠀⠉⠉⣧⠀⠀⠀⠀⢀⣸⣿⣍⣷⣿⣟⣻⣿⣿⣟⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠘⣿⣿⡛⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣁⣈⣉⡁⡉⠉⣰⣯⣥⣽⡻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠿⣷⣶⣤⣄⡈⢿⣷⡀⠈⠙⠻⢷⣦⣄⣀⠀⠀⠀⢀⣼⣿⠋⠀⠈⠁⠉⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠈⠙⠻⢿⣾⣿⣷⠀⠀⠀⠀⠈⠙⠻⣿⣶⣤⣾⠟⠁
⠀⠀⠂⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⠁⠁
⠀⠀⢀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣀⣤⡤⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⣿⣏
⠀⠀⠀⠀⠀⠀⠸⣿⡇⣀⣤⣶⣿⣋⣥⠶⢻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⣿⠉⠁⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⡄⠀⠉⠆⠀⠀⢸⡃⠀⣈⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢀⡴⠃⠀⠀⠈⠓⠚⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢠⡎⠀⠀⠀⠀⠀⠐⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠘⠻⣶⣄⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠠⡄⠀⠀⠀⠐⢀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠁⢀⠔⠒⠀⠀⠈⠀⠀⠀⢡⠈⠒⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠈⢴⣄⢀⠔⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠀⠈⠀⢀⡠⢀⠔⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣥⠐⠈⠠⠉⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆⠀⠀⠀⠀⠘⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠀⠀⠀⠄⠂⡸⠁⣰⡟⣶⣶⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣂⣄⢠⡖⠁⠀⣀⣤⣤⣶⣶⣶⣾⣿⣄⣿⣷⣶⣾⣿⣿⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠎⠙⣧⢈⡱⣦⠞⠛⠛⠻⠿⠿⠿⠯⢭⠽⠿⠿⠿⠿⠛⠋⠙⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡀⠀⠛⠋⢤⡙⣿⣿⣿⡶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡐⠠⠀⠀⠀⠈⠛⢿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠐⢂⠀⢬⠻⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣰⣦⣴⣦⣴⣦⣄⠙⠻⠦⣤⣀⣀⣀⠀⠀⣰⠞⠃⠀⠒⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⠉⠿⡷⢶⣮⣿⡷⠿⠋⠉⢱⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⣀⣴⣧⣄⠙⣿⣿⣿⣿⣿⣿⣦⣤⣭⣠⣄⣀⣀⣀⠤⠖⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⢶⣿⣿⣿⣿⣿⣿⣽⣯⣟⣻⣿⣿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠤⣤⣐⠢⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣶⣶⣶⣶⣶⡶⠾⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣭⣛⠛⠿⠿⣿⣿⠿⠿⠿⠿⠿⠟⠋⠩⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣦⣶⣶⣆⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠖⠛⠛⠉⠉⠁⠉⠉⠛⠓⠒⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⣠⠶⠿⠉⠀⠋⣼⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⠳⣤⡀⠀⠀⠀⠀⠀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡖⣠⣤⠙⣿⣟⡛⠀⢋⣽⣿⣿⣧⠄⡀⣭⣿⡄⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣼⣿⣿⣿⣿⣾⣷⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡙⢸⣿⣿⣿⣿⡿⠛⡿⣿⣿⣿⣛⠋⠸⣻⣿⣿⠀⠀⠀⠀⣷⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣟⡊⠉⣿⣷⠶⠀⣀⢤⣾⣿⣿⣿⣤⣲⣾⣿⣿⣯⣤⣴⣾⣿⣿⡇⠀⠀⠶⢿⠶⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣿⣫⣾⣷⣽⣿⣿⣷⣿⣿⣿⣿⡿⠛⠋⠉⠉⠛⠿⣿⣿⡻⠃⢻⣿⡇⠀⠀⠀⢸⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⣿⡿⢻⣿⣿⡿⣿⣿⣿⡟⠁⣤⡤⠒⠋⠀⣤⠀⠈⠙⢿⣶⣼⣿⠀⠀⠀⠀⢸⡆⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢺⣷⡆⡨⣽⣏⠀⠸⠿⠋⠀⠀⠀⠳⠐⠂⣰⡇⠀⠀⣄⠀⠻⣿⣿⡆⠀⠀⠀⢸⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⡾⠷⠀⣰⣿⡏⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠺⠩⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠀⢂⣼⣿⠟⠁⠀⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⢠⣾⣷⣦⣿⠟⠀⢠⡄⠀⠀⠀⠀⢀⣄⡰⣄⣴⠋⠀⢀⣴⣿⣿⠿⠁⠀⠀⠀⠀⣴⠋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠋⠀⠀⠈⣿⡀⠀⠀⠀⢼⣿⣷⠋⠀⣀⣴⣿⣿⠿⠁⠀⠀⠀⠀⣠⠞⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠋⠉⣸⠃⠀⠀⠀⠀⠘⢿⣦⡀⠀⢀⣙⣥⣴⣾⣿⡿⠟⠁⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠀⠀⢀⡘⠃⠠⣄⠀⡀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⣀⡠⠴⠚⠁⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣀⡆⣼⡟⠁⠀⢀⡟⢶⣁⠆⢠⠂⠈⣿⣿⣿⣿⣿⣿⠒⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⡆⣷⣿⣿⣿⠀⠀⠀⠘⢀⡇⡼⠳⣏⠀⠀⠘⣿⣧⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢀⢸⣷⣿⡿⠋⠘⣇⣀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠛⣋⡀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⣶⢸⣾⣿⣿⠁⠀⠀⠹⠉⠓⠦⡄⣀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢌⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢀⣾⠂⠀⠀⠀⠀⠀⣿⣸⣿⣿⠇⡄⠀⣆⡀⣦⠀⠀⠀⠉⠙⠲⠶⠦⠤⠤⠤⠿⣿⣷⡘⣆⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡠⠲⣌⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢣⠀⠈⢂⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠑⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠸⡆⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠼⢷⡀⠀⢹⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⢙⣦⡀⠹⣤⠤⠀⠒⠒⠀⠒⠒⠒⠂⢀⠀⠀⠀⠀⠀⠐⠂⠒⠒⠒⠠⠤⣀⣠⣾⣠⠴⠖⠒⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠳⠶⢯⣥⣄⣹⣆⣈⣳⣀⣀⣀⣀⣀⣀⣀⣴⣋⣉⣓⣦⣶⣶⣦⡤⠤⠤⠶⠖⠛⠉⠁⣿⣁⡤⠴⠶⣄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢠⡏⠁⠀⠀⠀⡿⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣠⠞⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡿⠀⣴⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢰⠇⣼⠁⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⡿⠛⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗⣚⣉⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠚⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣀⣀⠀⠉⠉⠉⠉⠙⠛⠲⠦⣄
⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⢀⣀⣀⣤⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠉⠉⠉⠉⠉⠒⠒⠠⠀⠀⠈⣷
⠀⠀⠀⣴⠟⠁⢀⡠⠔⠒⠉⠉⠀⠀⠀⠀⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠲⠶⠶⠶⠦⢾⠶⠶⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⢛⣽⠋
⠀⠀⠀⠹⣗⠲⠤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡀⠀⠀⣀⣀⣀⣠⣤⣤⠴⠶⠖⠚⠛⠋⠉⠉⣀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣿⠉⠉⠙⠛⢯⡉⠁⠀⢀⣀⣀⣠⣤⠤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠲⠶⠶⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣏⠀⠀⠀⠀⠀⠙⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⢢⢸⡄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⢸⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢰⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡤⠖⠒⠒⠒⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠉⠁
⠀⠀⢀⣀⣀⣀⣠⣤⠀⠀⠤⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⢸⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣀⠀⠀⠀⣀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⠀⠀⡀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡇⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢀⠀⡀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⠠⠀⠔⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⡤⠚

⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠂⠀⠀⠉⠉⠉⠉⣉⡉⠉⠉⠉⠉⠉⠑⠐⠒⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢑⠈⠡⠀⠒⠒⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠒⠒⠀⠄⠁⡒⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠉⠉⠉⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠢⡀⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⡀⠀⢸⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣹⠀⠀⣏⣉⢵⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠹⡑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡡⠔⡉⠄⢧⡀⠹⣄⠠⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⣀⠀⢀⢻⠦⡈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⡜⢁⠎⠀⠈⠀⠀⠀⠑⢤⡈⠑⠂⠵⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡡⠖⠉⠁⠀⠀⠀⠸⠗⢌⠢⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠒⠠⠄⢠⣀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠎⠀⠀⠀⠀⠀⢀⡴⠃⠐⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠙⠖⠉⠐⠒⠢⠤⠤⠤⠄⠀⠐⠂⢈⡠⠴⢒⡲⠷⠶⢤⣤⠤⠖⠋⠀⠀⠀⢀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠠⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠢⢤⣤⠴⠒⠂⣉⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢑⠢⠌⡑⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⠈⠁⢀⠠⠒⠋⠁⠀⠀⠀⠀⣀⣀⣠⣄⣤⠤⠤⠒⡁⠤⣲⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡁⠀⠀⠠⠍⠉⠐⠒⠒⠂⠤⠤⠤⠤⠤⠠⠀⠤⠤⠤⠶⠶⠶⠛⠛⠛⠉⠉⠡⠄⠐⠀⡉⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄⠒⠒⠒⠁⠉⠉⠉⠉⠈⠁⠈⠉⠈⠉⠉⠉⠙⠒⠒⠂⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠪⣅⢠⣤⣤⣶⣦⣤⣀⣀⢠⠔⢲⣠⣶⠲⣀⢤⣤⡤⣀⣤⣤⣄⠀⠀⠀⠈⣕⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡇⠻⡯⣿⠿⠟⠿⣿⣷⣷⡿⠀⣰⠶⢐⣧⣶⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⢠⠀⣻⠝⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠑⠢⠤⢟⡀⠀⠀⠀⣹⡏⠡⠞⢱⢶⣼⣇⢻⣵⣷⣿⣿⣟⣿⣿⡷⠾⠧⠒⣫⡄⢀⣥⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠲⠤⠬⣉⣁⣛⣃⠒⠒⠒⠒⠙⠛⠛⠛⣋⣛⣭⣉⡩⠤⠔⣶⠆⠉⠀⣠⠏⠀⠀⠀⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣶⣶⣶⣬⣉⡉⣀⡁⠈⠉⢁⠉⠉⢉⣉⡉⣀⣤⣶⣾⡇⠀⣴⢹⣏⣀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⣀⡤⠴⣶⣛⣭⣔⣷⣼⢿⠛⠹⣿⣷⣿⣟⡅⣳⣻⣜⣏⢝⣾⣿⣿⣿⣿⣶⡴⢿⣧⣤⣿⠀⠀⠀⡈⠉⠁⣠⠟⢀⣀
⠀⠀⠀⢀⡤⠊⠁⣶⣷⣿⣷⣽⣽⡿⣿⡯⠀⠀⢠⣼⣿⡼⣬⣧⣝⡾⠿⣾⣿⣿⣿⣿⣟⣽⣾⠟⡱⣿⡟⠀⠀⣶⣤⠶⣾⣽⣾⣷⣬⠳⢄
⠀⠀⣰⠋⠀⢀⠀⠈⠿⡟⠿⢻⣿⠛⠹⣿⡳⣄⠘⠁⠀⠀⠀⠉⠘⢛⣿⣿⠷⣾⣽⣿⣿⡟⠀⠶⣵⡯⠇⢀⣰⠟⢙⣿⣿⣿⣿⣿⣿⣷⠈⣧
⠀⠀⢿⡈⠀⠀⠙⠛⢶⣾⣶⣮⠛⣿⣗⣻⠛⠛⢀⣀⠈⣇⣠⠅⠀⠈⢡⣇⣸⢿⣿⣿⣟⣵⣿⣯⡾⡰⣾⣯⣶⣾⡋⢉⣿⣛⣿⣿⠿⢟⣴⠃
⠀⠀⠀⠛⢦⣴⣶⣷⣿⢿⣿⣉⠋⢿⣇⣀⣀⠀⠀⠈⠙⠿⣿⡷⠶⠾⠟⠻⣿⣿⣋⣼⣿⣷⣿⣺⠟⠁⠀⣙⣿⣹⣤⣾⣿⣷⢿⣷⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⣬⣟⡿⠶⠿⢯⣷⣯⣼⡻⣇⡒⠈⠓⠶⠦⠴⠐⠶⠏⣿⣿⣿⠟⣃⣉⡽⣾⢿⣿⣛⣋⣯⡭⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠒⡒⠒⠓⠒⠦⠭⠶⠤⠤⠤⠤⠤⠬⠽⠬⠽⠖⠒⠖⣒⣻⣯⡽⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⢒⣒⡶⠶⠶⠶⠶⣶⣶⣶⠾⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡖⠂⠀⠀⠀⠒⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢾⣉⣙⠛⠒⠒⠒⠒⠒⠛⠋⣉⣳⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠓⠒⠦⠭⠭⣭⣭⣭⣭⣭⡭⠭⠥⠖⠚⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣖⣦⠤⠤⣤⣄⣀⣀⣀⡠⠤⣴⣶⣾⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢻⣿⣿⢿⡯⡆⠉⠀⠀⠉⠙⠹⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⠛⠛⠶⠥⠠⠤⠤⠤⠤⠴⠧⠶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣦⢦⢻⢳⢀⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⣦⣤⣤⡄⠑⠃⠈⠭⣤⠤⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⣻⠁⠀⠀⠀⠰⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⣿⣿⣿⡇⠀⠀⠀⠰⡇⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣘⣿⣿⡿⡇⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠙⠓⠀⠀⠀⠀⠀⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣶⠂⠒⠒⠒⡶⠊⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠒⠒⠒⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡤⣭⣍⡑⡄⢠⠔⣲⣤⣬⣭⣥⠭⠄⠶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⡟⢸⣶⣿⣿⣿⣿⣿⣄⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⡇⠈⣿⣛⣛⢛⡩⡭⢭⢥⣤⠤⠘⠇⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣖⡶⣶⣶⣄⡀⠀⠀⠐⠀⠈⠉⠁⠈⠉⠋⠁⠀⠀⠄⠐⠀⠉⢠⡴⠀⢀⠀⠀⠀⢤⣿⣼⡼
⠀⠀⠀⠀⠀⠀⣠⠟⠛⠻⣿⣎⢿⣿⣿⣄⠀⢚⡀⠀⠀⠀⠀⠀⠀⢰⡁⠰⠀⠀⠀⠀⠀⠀⣰⡖⠾⣯⡉⢠⣏⣾⡛⠂⠀⠀⣤⡀⣀⣀
⠀⠀⠀⠀⠀⢠⠇⠀⣠⢤⡈⢿⣇⢿⣿⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠄⢤⣶⣿⠂⢰⠛⠾⠚⠡⠀⣶⣶⣶⣿⣿⢿⣧⣄
⠀⠀⠀⠀⠀⠘⠀⣸⢁⣾⣿⡜⣿⡜⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣃⠴⠛⠛⠙⠓⢶⣝⢿⣷⣄⠀⠀⠇⢸⠐⣿⣿⣿⣿⡟⠀⠹⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠸⠑⠈⣇⢻⡇⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⣤⠴⣤⣄⡀⠈⢿⣽⣿⣷⡄⠀⠸⠀⣿⣿⣿⣿⣷⠀⠘⣮⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣇⢠⡀⠀⣽⢸⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⡞⣱⣾⣿⣏⢷⣄⠀⢻⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⠀⠰⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠘⠀⠸⡈⢷⣴⡇⣼⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⢰⣿⠫⢤⡉⢣⣹⡆⠈⣿⣿⣿⣧⢀⡄⢻⣿⣿⣿⣿⡀⠀⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢇⠀⠙⠦⠏⢀⣿⢡⣿⣿⣿⠀⠀⠰⠆⠀⠀⠀⠀⣿⠀⣿⢸⣇⠀⠀⡁⠈⣿⣷⠀⢹⣿⣿⣿⡌⠁⠸⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⣠⣾⢇⣾⣿⣿⣃⣀⣄⡀⠦⡄⢀⠀⠀⢹⡀⢹⡌⢿⣦⡀⢑⣼⣿⡏⠀⣼⣿⣿⣿⡷⢈⣧⢻⣿⣿⣿⣧⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠵⣿⣿⠿⢿⣿⣿⣿⣶⣯⣷⡐⣶⣤⣤⣷⡀⠹⣌⠻⢿⣿⣿⡿⠁⢠⣿⣿⣿⠇⠀⠈⣻⡦⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠶⠶⢶⣾⣷⣄⠈⠛⠲⠒⠊⠁⢠⣾⣿⣿⣿⣷⣦⣤⣤⡄⠒⠈⠙⣿⣿⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⣤⣾⣿⡿⠿⠿⠿⠿⠿⣿⠿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣽⣿⠗⠓⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⣙⣛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣟⣟⣿⠿⣿⢿⣿⠿⢿⡿⠿⣿⢿⢿⣿⠿⡿⡿⣿⢿⣿⣟⣿⣏⣽⣧⣶⢶⡿⣿⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠿⣷⣧⣿⣿⣮⣿⣷⣷⣵⣯⣿⣯⣜⣧⣵⣽⣾⣷⣿⣾⣿⣿⣿⣷⡿⠾⠿⠿⣾⣫⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣲⣿⠥⣾⡿⣿⣯⣛⣙⡻⡿⠟⣻⣿⣿⣿⡋⢍⣫⢓⣢⣿⡟⢻⣿⠁⠀⠀⠀⠀⠈⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⣒⣽⣿⣡⡼⣿⡗⣓⣯⣒⣫⣿⣏⣌⣿⣿⠽⣖⡡⣒⣿⣷⣮⡏⠀⠀⠀⠀⠀⢀⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣲⣿⡿⣿⣷⢹⣿⡤⢭⣒⣾⣿⣿⡿⣟⣿⣎⢕⣊⠔⣿⣇⣿⠁⠀⠀⠀⠀⣠⣾⣍⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣷⣿⢻⣇⣿⣖⠭⣒⣿⣿⡿⢇⣿⣿⣷⢒⡴⣪⣿⣿⡏⣀⣀⣤⣶⡟⣿⡼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⠈⣿⣸⣿⠭⢽⣿⣿⡏⠉⢻⠋⢹⡧⠬⢴⣿⣿⣻⣿⣯⣿⣼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠿⣟⣻⣯⡿⠭⢿⣿⣿⣿⣿⣿⣻⣯⣿⡿⠋⣀⣠⢴⡦⠀⠻⣭⢽⡿⠿⠿⠿⠿⣭⣽⢿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡯⣹⠋⠉⠀⠀⠀⠀⠈⢿⣿⡿⢿⠀⠈⢛⠉⣳⠦⢴⡇⢡⣿⠤⢶⣿⣿⠁⠀⠀⠀⠀⠀⠉⣿⣺⡽
⠀⠀⠀⠀⠀⠀⠀⢻⣍⣷⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⢀⣳⢾⣳⣴⣇⣀⠐⠋⠙⠁⣠⣼⣿⠝⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⡁⠙⢶⡟⢮⠽⡟⢸⣿⣷⣴⣷⣒⡿⠟⢸⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣄⡀⠀⠀⠈⠋⠹⣝⣳⣦⣼⣿⣿⣿⣿⠾⢋⠟⠛⠁⠀⣀⣠⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⡶⠤⣤⣬⣄⣈⣓⣋⣙⣒⣡⣭⠥⠴⣖⣺⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠿⠶⠶⠶⠖⠚⠛⠋⠉


⠀⠖⠐⠆
⠀⠓⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣀⡄⢄⣀⢐⠀⡀
⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣄⣲⣶⠛⠓⣻⠀⠀⠀⠀⠀⠠⠂⡔⠂⡜⡲⡠⣼⣴⣸⣧⣾⡷⣧⢢⡎⢠⡀
⠀⠀⠀⠀⠀⢿⠛⢦⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡔⣻⣭⡿⣃⣾⠏⠀⠀⠀⠀⠈⠀⢱⣧⣼⣦⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡉⠻⠿⢿⣯⣽⣿⣏⠁⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠛⠶⠶⠿⠳⡟⠃⢹⢻⣧⡀⠀⠀⠀⠀⢠⠴⡻⠛⠀⠸⣿⣿⢿⣿⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢯⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⣴⣎⠀⠀⠀⠀⠀⠀⠙⢶⡿⠀⠹⣿⣦⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⡟⠀⣿⠐⠙⢻⡟⣻⡟⠉⠉⢹⡿⢿⣦⡭⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣮⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢳⡀⠀⠀⠀⠀⠀⠠⠀⢸⠗⠾⡟⢀⣰⡟⠀⣿⠁⠀⢀⡟⠀⠀⠗
⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠻⠉⠋⣝⠳⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠹⣹⡅⠈⠱⣼⢺⠀⣤⣿⠀⣠⠏
⠀⠀⠀⠀⠀⢴⣛⣿⣿⡃⠀⠀⠀⠀⢻⣿⣿⣷⣦⠄⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠉⢳⡀⠀⠺⠀⠀⣿⢏⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣶⣄⠀⠘⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠳⣄⠃⣂⠀⢉⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢹⠁⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢪⣾⣷⠎⠙⠓⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⢿⣿⢿⣧⣶⣶⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡸⣿⣷⣴⣷⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣀⣀⣤⣶⣶⣶⠿⠿⠿⠛⠋⠉⠉⠙⢾⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⠛⣿⡿⢿⣿⠛⠛⠛⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠉⠁⠀⣀⡀⠀⠈⢿⣷⠦⠠⣼⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠚⠛⠋⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡄⠀⠀⠀⢀⡰⠞⠁⠀⠠⠶⠀⠀⠀⠀⠀⠰⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠃⠀⠀⢀⡘⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⣠⣶⡦⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⡇⠀⠀⢀⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⣀⡘⠻⠓⢦⣴⠛⣤⡀⠀⠀⠀⠛⠋⠀⠀⠀⠘⠿⠟⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⣾⣿⡴⢿⣿⣷⣄⠈⠉⠠⠿⠿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠀⣠⣾⠉⠀⠀⠀⠀⠉⡇⠀⠀⢠⣤⠼⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⣦⠀⠀⢀⡄⠀⠀⠀⢰⣿⡿⠛⠛⠛⠀⣤⡄⠁⠀⠀⠀⠀⣠⡄⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⣿⡀⠀⠸⣧⣀⣀⣀⢀⣽⣷⣄⡀⠀⠀⣸⣇⣀⣀⠶⠶⠀⠉⢻⣶⠀⠶⠦⣀⣀⣀⣀⣀⡀⠀⠴⠶⠶⠆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⣷⠀⠀⠀⠀⠉⠉⠀⠀⠈⠛⠛⠛⠟⠛⢿⣿⢿⣶⣶⣿⣿⡟⠉⢠⣤⠀⠀⠄⠁⠀⢠⠀⠉⠉⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠈⠿⠋⠻⠋⠀⠛⠃⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣷⡶⠀⠀⠀⠀⢶⣷⠀⠙⢿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠈⣶⣶⡄⠀⠀⠤⠛⠀⠀⠀⠘⠻⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⡄⠀⣠⡀⠀⠀⠀⢀⣀⣈⠻⠿⢧⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⠀⡄⠉⢱⡆⠀⠀⠀⠈⠻⠆⠀⠀⠉⠉⠉⠉⢻⣿⠗⠺⠟⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡇⠀⣿⣄⠘⢳⣦⡄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠙⠿⣆⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠉⠷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡎⢿⣦⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣿⣿⡄⠀⢀⣈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠉⠁⡀⠀⢙⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⢸⡇
⠀⠀⠀⠀⠀⣤⠀⠀⢼⡏⠀⠉⠉⢡⣤⣀⣿⡿⠀⢸⣿⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠈⠿⠀⢸⡇
⠀⠀⠀⠀⠈⠿⠀⠀⠀⠛⠟⠛⠛⠛⢻⣿⣿⣧⡀⢸⣿⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠙⣶⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣨⠏⠉⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡘⠁⢀⡿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠜⠃⠀⠀⠈⠁⣿⣿⠆⠉⣤⠀⠀⠀⠀⠀⢀⠟⠁⢀⡌⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⣿⡀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠸⡆⠀⠀⣿⡇⠀⠀⢀⣾⠀⠀⠀⠸⠏⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⠀⠀⠀⠀⢻⡇⠀⠀⣈⠋⠀⠀⢠⡆⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠸⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠘⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠘⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠋⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠉⠉⠙⠛⠛⠉⠉⢠⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⡶⠃⠉⠉⠉⠉⠉⠁⣶⡆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣷⣾⣿⣿⣿⣥⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠋⠁⠈⠛⠋⠋⠉⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣔⡀⠀⠀⠀⠀⠀⠀⠀⠠⠼⢆⠀⠀⠀⠀⠀⡂⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠆⠠⠤⠅⠀⠀⠀⣶⣺⣇⠀⠀⠚⢚⠄⠀⠀⠀⠀⢉⢀⣰⢉⠏⠀⠀⠀⠀⠀⠀⡤⢡⡤⠆
⠀⠀⠀⠀⡄⠲⣚⠃⠀⠀⠀⠀⠸⠍⠇⢀⠰⡶⢰⡋⠀⠀⠀⠀⣟⣬⠁⢸⠁⢠⡖⠀⠀⠀⠔⢂⠀⠰⢭⠡
⠀⠀⠀⠐⠋⠁⠹⣆⠀⠀⠀⠀⠘⠓⣀⠩⠔⣢⠦⠀⠀⠀⠀⠀⠙⣡⣀⣿⡰⠋⠀⠀⠀⠀⠀⡀⢳⠜⠉⣁⡲
⠀⠀⠀⠴⠆⠢⠀⠇⠀⠀⠀⠀⠀⠀⠐⠓⠠⢈⡶⠂⠀⠀⠀⠀⠙⡛⢸⣟⠁⠀⠀⠀⠀⠀⠀⠁⠹⢛⣋⡼⠁
⠀⠀⠀⠀⠀⣀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣀⢂⠈⠀⠀⠀⠀⠀⠀⢠⣼⢟⣡⠀⠀⠀⠀⠀⠀⠀⠀⢘⣭⡈⠁
⠀⠀⣀⡄⠚⠛⠃⠉⠂⠀⠀⠀⠀⠀⠀⠙⢉⡛⠀⠀⠀⠀⠀⠀⠀⠛⠼⠁⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣘⣇⣀⣠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣒⣒⡻⠭⠭⠭⠿⠻⣶⠒⢺⠧⠽⡭⠭⠭⣍⣛⣒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣴⠋⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡜⠀⢀⡇⠀⠀⠀⠀⠉⠙⠓⠶⣭⣓⢤⡀
⠀⠀⠀⠀⠀⢠⡟⣇⠀⠀⠀⠀⣀⣀⣀⣠⣤⠤⠤⡴⠥⢼⡧⠤⡾⠤⢤⣤⣤⣀⣀⣀⠀⠀⠀⠉⠳⣣
⠀⠀⠀⠀⠀⠸⡟⢦⣑⡲⠤⢭⣍⣁⣀⣀⣀⠀⠀⠃⠀⠈⠁⠀⠃⠀⠀⠀⣀⣀⣀⣈⣉⣤⣤⣒⠾⡿
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠉⠓⠒⠒⠪⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢻⠷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢼⡏⣉⣉⣉⠓⢦⡄
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠙⠓⠶⠦⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠖⠚⠉⠀⣸⠛⠉⠉⠉⠳⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣰⠇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⡴⠋⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣤⡴⠞⠋⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠉⠉⢀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠗⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠮⠭⣭⣍⣉⣛⣛⣋⣩⣭⠭⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⣀⡤⠖⣋⣭⠥⠶⠚⠋⠉⠉⠉⠉⠓⠢⣄⠉⣦⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⣠⣴⣋⡥⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢸⠡⠤⢭⣝⡲⣤
⠀⠀⠀⠀⠀⠀⣴⣾⡆⢀⣴⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠏⣀⣀⣀⠀⠙⣦⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢛⡟⢁⡴⠋⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠙⣧⠀⠸⠈⡇
⠀⠀⠀⠀⢀⣤⣶⣿⡄⠀⣾⠀⣾⣀⣠⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡠⠀⠀⠀⠀⢻⠀⠀⢸⠀⠀⣸⠃
⠀⠀⠀⠀⣽⣿⠟⠉⠀⠀⠹⣄⠙⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣖⠋⠉⠉⣉⡽⠋⠀⠀⠀⠀⠀⢺⠀⣠⡟⢀⡴⠃
⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠘⣷⢄⡈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣧⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⢸⡞⢋⣴⠋
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠚⠛⠛⠛⠛⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⢙⡗⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁⠀⣠⡴⠋⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⢁⡠⢀⡠⠔⠉⠙⠛⠓⠒⠚⠛⠋⠉⠀⠀⣀⣠⠶⠛⠁⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⡰⠁⠀⠸⣄⣀⣀⣀⣀⣀⣀⣠⣤⡴⠶⠚⠋⠉⠀⠀⠀⠀⠀⢀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⡀⠐⠤⠄⣀⣉⣉⣉⣉⠉⠁⢀⠀⠀⣀⡠⠤⠀⠀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣉⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣡⣤⡤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⡷⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣪⢔⣠⣴⣿⣿⣶⣼⣿⣿⣿⠏⠀⠀⠀⠀⠠⠀⠆⠀⠀⠀⠷⠂
⠀⠀⠀⠀⠀⠀⣴⣤⣉⣉⠙⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐
⠀⠀⠀⠀⠀⠐⣿⢸⢸⢹⡘⡘⡇⡇⠱⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠊⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠆⠃⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠖⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⢰⡀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠘⠈⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣾⣿⣿⡷⠄


⠀⠀⠀⠀⠀⠀⢀⢄⢠⣖⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⢷⣿⢠⣿⡄⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡄⢀⠀⠀⠀⠀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⠈⠻⠃⠿⡧⢘⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢵⣄⠐⠛⠛⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢤⡀⠀⠀⠈⠙⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠈⠀⢀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣳⠾⠦⢤⣴⣤⣀⣠⣤⣤⣤⣤⣀⣤⣤⣤⣶⡶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠠⠐⠀⠘⢁⠉⠉⠉⠩⡿⢿⠷⣴⣤⣴⣉⡛⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡐⠀⣀⠀⠀⠀⠣⠤⠶⢶⠀⣠⣀⡀⠉⢴⠉⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡣⠄⣀⠀⠀⠰⠀⢒⠂⡀⠈⠹⠶⢤⣴⠿⢋⢤⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠶⡬⠁⠀⢀⠀⠒⠓⢂⣛⠈⢭⣅⢀⣿⣶⣿⠉⠉⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠘⣹⣇⣀⣠⠟
⠀⠀⠀⠀⠀⠀⣠⣤⡶⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣭⣅⡀
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠠⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣛⡎⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢽⢤⡽⢷⣤⣀⣄⠀⠀⠀⠀⠀⠸⠀⢀⣤⣝⢻⡿⣾⣷⣀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣚⠁⡛⠛⠿⢶⢶⣶⣶⣶⡾⢿⣿⣳⣛⡺⠟⠉⠁⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠊⠀⠾⠃⠀⠀⠀⠀⠀⠀⠈⠉⠋⢉⣁⠀⣴⠶⠹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠛⠛⠂⠀⠦⠷⠖⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢢⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡔⢆⡀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠘⣥⡟⣵⣧⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠈⠉⡅⣧⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡿⡍⣷⢸⠹⣻⣧⢲⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢃⡹⣄⣧⡿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡄⠀⡇⡴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠁⠀⡏⢰⣹⣾⣾⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠰⠀⡇⢃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠉⠓⠦⠏⣌⣏⡿⢁⠜⠁
⠀⠀⠀⠀⠀⢀⡀⠠⢾⣤⢅⢣⠶⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⠦⣆⣗⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢤⣀⣀⣀⣀⠬⠠⡎
⠀⠀⠀⡰⡪⠃⡁⢍⣿⡭⠭⠭⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠐⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣓⣙⣛⣹⣼⠦⠷⠟⢟
⠀⠀⠀⠙⠘⢋⢟⣵⢿⡉⠉⠹⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢧⣭⣭⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠊⠀⠀⠀⠀⠀⢆
⠀⠀⠀⢠⣺⢵⠫⠃⢃⣱⠸⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⡰⠺⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣫⣓⣤⠂⠀⠀⠀⠀⢀⡠⢔⣩⡏
⠀⠀⠀⠀⢀⢎⠃⡀⡾⡅⣿⢤⠸⡄⠀⠀⠀⠀⠀⠀⢀⠔⢂⡍⠉⠛⣹⢋⣌⣼⠃⡌⣿⡄⠛⣒⢤⠀⠀⠀⠀⢀⠷⠊⠙⠛⢶⣤⣤⣒⡩⠖⠋⠀⠃
⠀⠀⠀⠀⡌⠄⠰⠀⣗⠙⢦⡀⣾⣗⠡⠀⠀⠀⠀⠀⣯⠗⠐⠒⠂⣹⣏⠟⠇⣟⣧⡷⠻⣿⡆⠦⢐⣳⡀⠀⠀⢹⣀⢠⣤⣶⠽⠛⠓⠫⢴⡂⠤⢄⣄
⠀⠀⠀⠀⢧⢃⠃⠹⢾⡀⡱⢱⢧⣿⢧⢡⠀⠀⠀⢀⡡⢤⠐⣶⠿⠄⡇⠂⡼⣿⠛⢅⢡⣇⠚⢢⡀⠈⠁⠀⠀⠻⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠞
⠀⠀⠀⠀⠀⢎⠀⡈⠧⣛⡄⠸⡌⣩⡘⠃⠁⠀⢠⣏⡯⠬⡴⠱⠀⣴⢧⢶⠠⡿⡌⠨⢣⢙⠦⣉⢷⡦⣀⠀⠀⢾⡂⣤⠄⣀⡀⠀⠀⠀⠀⡀⣀⣀⣬
⠀⠀⠀⠀⠰⠄⠀⠁⠀⢛⢇⣾⣉⢿⣇⠆⡈⡄⠀⠁⡠⠞⣥⡔⡛⣭⡌⠛⢸⣇⢽⣦⣀⡻⢣⣏⣾⠓⠂⢧⠀⢸⣀⣉⣉⣉⣘⣛⢻⣷⣿⣾⠷⠿⡟
⠀⡁⠀⢰⡁⢱⠀⣴⠀⢹⣼⠸⡎⣾⣻⠎⠁⠄⠀⢰⠃⡈⢩⠋⡿⢹⢊⣲⢞⢻⣯⣻⠝⠁⡟⣃⠉⠐⢜⠲⠀⠻⠖⠒⠒⠒⠒⠋⠉
⠀⠀⠀⠋⢹⡝⢣⠃⠑⣾⠸⠐⢇⠋⠹⣁⠴⡇⠀⠀⣇⡱⠃⢀⡷⡀⢳⠃⠂⢸⡼⣷⢢⣾⣃⠱⠑⠄⠈⠁⠀⠐⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠘⠺⡼⠶⠻⡶⢾⡇⠓⣴⢸⣷⢺⢧⣀⡐⠀⠀⠉⠀⠀⡏⠄⡇⢘⣨⢀⠟⣄⣧⡋⡱⠾⢣⠈⠺⠀⠀⠀⡜⢸⠸⠉⢲⠒⠲⡄⠤⢠⠶⡎⢣⢡⢱
⠀⠀⠀⠀⠀⠀⠀⡅⣄⣷⣾⣀⣘⣧⣬⡞⠘⢥⠀⠀⠀⠀⠀⢳⢓⣿⡜⣄⡏⢰⠇⢻⣷⡇⡦⠀⠶⠂⠀⠀⢠⡀⡏⢢⣧⠎⢱⠀⣧⣶⠊⠀⣇⠀⣿⠴⠂
⠀⠀⠀⠀⠀⠀⠈⠈⠁⡼⢹⡎⡍⡅⠉⠣⠺⠎⠀⠀⠀⠀⠀⠀⠻⠝⠃⠈⠀⠘⣏⣠⡟⠹⠃⠀⠀⠀⠀⠀⠀⠶⠗⢇⠈⠇⣸⣀⣿⠞⣄⡾⢜⣧⡟
⠀⠄⠀⠀⠀⠀⠀⠀⠐⠳⣮⣧⢧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⠙⠾⠺⣎⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡼⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⠀⠀⠀⠀⣆⠀⠇⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⡏⠳⣈⠐⠠⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡈⠀⠀⠀⠀⠹⡐⠄⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠒⡇⢀⡄⠀⠀⠀⠙⢤⠀⠀⠙⢆⠀⠀⠀⠀⣠⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⠀⠀⡏⢸⢣⡀⣀⠀⠀⠀⠱⣄⠀⠈⣇⠀⠀⠎⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠄⠸⡇⠀⠑⠹⠌⣏⢦⡀⠀⢹⢀⠈⢸⠀⠘⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠲⡀⠳⣄⠀⠀⢀⡞⠐⡷⣠⠋⠸⠀⢈⠀⢠⣿
⠀⠀⠀⠀⠸⠗⢄⡀⠀⠀⢳⠀⠡⠈⠢⡈⢢⣀⡞⢠⠀⡿⠁⡐⢀⠀⡞⠀⢸⠁⣇
⠀⠀⠀⠀⠀⠑⢄⠉⢦⠀⠀⠱⣄⠡⠀⠈⢄⠛⡰⠁⠃⡠⠊⠀⠀⡼⠁⣠⠏⠊⢹⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠐⠲⡤⣜⠀⠀⠀⠘⡆⠱⠀⠈⣤⠀⠀⠞⠀⠀⠌⣴⠵⠚⢁⡀⠀⣼⠀⠀⡠⢫⡞
⠀⠀⠀⠀⠀⠀⠀⠈⢂⢉⢲⡀⠀⢸⠀⡄⠀⠙⠀⠀⠀⠀⢈⠀⡡⠔⠈⡑⠀⣰⠃⠀⠬⠐⣉⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢆⠹⡀⡟⢠⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⡎⢀⡞⠁⡀⢀⡴⢉⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠆⠛⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⣾⣴⣾⢣⣮⡐⣾
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠘⣀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠐⠀⡿⢛⠛⢾⠻⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠣⢉⠙⡍⠋⠁⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠒⠀⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠕⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠄⠀⠀⢀⣀⡀⠀⠤⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠐⠈⠀⠄⠂⠀⠠⠀⠄⠀⠀⠠⠀⠌⠠⠊⠑⠀⠀⠐⠂⠒⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡒⠂⠈⠉⠉⠉⢻⠋⠙⠀⠉⠀⠐⠒⠢⠤⢀⣀⠀⢠⢶⡆⢲⡆⠒⣦⠠⢤⠄⢤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣢⣵⣶⣿⢋⣀⣠⣤⣤⣤⣤⣼⣦⣤⣠⣄⣄⣀⣀⣀⡀⠀⠀⠈⠙⠺⢿⡛⢻⡛⠛⣟⠚⢷⠶⢶⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠈⣻⢿⡋⠻⡟⠛⢷⠺⢷⠾⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣟⣿⣿⣿⣿⣿⡿⣷⡬⠛⢷⣄⣸⣮⣿⣟⠻⡛⠛⢷⠶⢷⠾⢆
⠀⣏⠉⢐⠂⠲⠿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣿⣿⣿⣵⣿⣿⠿⠛⠑⢋⡠⢀⠀⣿⣿⣿⢿⣿⣞⠿⠟⠛⢷⠶⢷⡾⢦
⢹⡈⢆⠀⣀⠀⡀⠀⠘⢟⡛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣷⣾⣯⡾⣟⠟⡍⠀⣿⣿⣴⣿⡶⠾⢷⡶⢷⣤⢷⡀
⠀⢡⡈⠆⠉⡀⠁⠀⠀⠈⠻⣷⡊⠙⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⢫⣵⡾⢃⠎⢀⣴⣿⣿⣿⣿⣿⣿⡾⣧⣤⣽⣄⣸⣄
⠀⠀⠱⡀⢦⠤⢤⡀⢀⣀⣀⣈⣿⣦⣀⠀⠀⠈⠀⠚⠛⠩⠭⠽⠽⠿⠻⠟⠙⠛⠛⠉⠁⣡⣫⡾⣋⠔⢁⢔⠉⠁⠀⠻⣿⣿⣿⣿⢿⣹⣧⣀⣉⣄⡈⣆
⠀⠀⠀⠙⠠⠤⢤⢤⣤⣤⣤⣬⣿⠿⠉⠓⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠛⠉⠀⠀⠁⢸⣄⠀⠀⣠⣿⡿⠟⣣⣿⣏⠈⣫⡉⠉⢋⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠝⡻⣦⣀⠀⠀⠀⠀⠀⠐⠊⠭⠄⣀⣒⣒⡀⠀⠒⣂⣴⣶⣿⣿⣿⣿⣿⠔⣒⡶⠖⢖⡿⠂⠤⡤⢴⡖⢛⣯⣉⣻⡍⠉⢛⡉⠙⠻⡉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠩⠟⣢⠄⠀⠀⠀⠀⠀⠀⢀⣠⣭⣿⣿⠿⠿⠛⠛⠉⠉⠉⠠⠒⢋⣁⠤⠊⠁⠀⠀⠀⠰⡞⣿⣍⣹⣯⣉⣿⣍⣉⣻⡛⠛⢻⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣶⡿⢿⣿⣿⣟⣛⣉⣀⡀⠀⣀⣀⣀⡤⠤⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣐⣄⡠⠐⠯⠭⠧⠿⠾⠷⠿⠶⠮⣍⣥⣴⢶⣒⠤⢄⣀⡀⠀⠀⢀⡴⢁⣒⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⢟⣙⡐⠛⠾⠋⠁⠀⠀⣀⣐⣂⢶⣿⡿⢀⡀⠀⠀⠙⠚⠛⠛⠛⠋⢿⣷⣦⡜⣼⠃⠀⠀⠪⠑⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣿⣿⣄⡁⠁⠀⢀⢀⢰⣶⣤⣻⣦⡴⢊⠸⢁⣾⡌⠓⠢⠀⠀⠀⠀⢀⠶⣀⢽⣿⡻⢸⣆⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣄⢟⢹⣿⠿⠀⠀⠀⠞⡟⣻⢿⡿⠉⠿⠈⣷⡭⠱⠄⠀⠀⠀⠀⠀⠲⣯⠾⠉⢟⡟⣤⣰⣟⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⣀⡆⠰⠾⡰⢀⠀⠀⠉⠁⠀⠰⠄⠻⣆⠀⠙⠀⠀⠀⢀⣀⣀⠀⢀⣠⣤⠖⢙⣑⢺⠋⣍⠀⠀⠐⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡆⠀⠘⠻⠿⢷⣶⢤⣤⣤⣤⣶⣤⣤⣤⣤⣭⣤⣤⣤⢤⡯⠿⠾⠒⠛⠋⠉⠀⠀⣈⢈⡉⠆⠈⢡⡔⠋⠁
⠀⠀⠀⠀⢀⣀⠀⠀⠀⡤⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⡠⢊⡀⠴⠶⠋⠀⠈⣦⡤
⠀⠀⠀⢀⣿⠃⠀⢠⠃⠼⠞⢯⡶⡀⠀⠀⠀⣱⡂⠀⠘⢃⡖⣀⣠⡤⣖⣦⣀⠀⠀⠀⠾⣿⡟⠀⠀⠀⠀⠀⠡⠋⡦⡚⣷⠐⠀⠀⠘⠛⢿⡆
⠀⠀⠀⣾⡏⠀⠀⠠⣌⣴⡳⣷⢊⠀⠂⠀⠰⠟⠿⠦⢘⢋⣸⣑⣊⡵⣹⣥⠏⣱⠒⣤⠦⠚⠶⠀⠀⠀⢀⡶⠃⠒⠙⢊⠁⠀⣄⢄⣶⣠⡟⡇
⠀⠀⠀⠹⣿⣿⠦⡀⠈⠻⠙⠀⠋⣩⡄⠠⡀⠀⠀⠔⠉⠉⠀⣿⡇⠂⠁⠉⢁⡵⠿⡇⠀⠀⠀⠀⠀⠀⢲⣟⣃⠄⠀⠀⠀⠀⣉⣬⣾⡽⠛⠁
⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠠⠊⠀⡄⢠⠄⠀⠀⢐⣀⡬⠕⠊⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠶⣦⡰⣀⣀⣀⡀⠀⠀⠀⢏⠒⠠⠄⣤⣄⣤⣤⣴⣲⣰⣦⠤⠖⢊⣝⣀⣀⣼⣤⠵⠒⡋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡀⠉⠉⠓⠒⠒⠛⠒⠒⠶⠤⠶⠖⠒⠒⠒⠒⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠶⢶⡶⢶⣶⣤⣦⣶⡦⢶⠦⠔⢒⣲⠿⠗⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⣿⠈⠉⠉⢉⡍⠀⠀⠸⡄⢒⠿⠚⢛⠀⠀⠀⢀⣤⠤⠀⣈⠉⠥⠭⠭⠍⠉⠉⢉⡍⠉⠉⠉⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⡋⠉⠉⠙⣏⠉⠋⠉⠉⠉⠛⠛⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢠⡴⢾⠀⣄⣀⣤⠴⠓⠉⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⢿⠷⠀⠀⠈⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠧⠀⠀⠄⠀⠀⠀⠁⠄⡀⢀⣦⠤⢶⠀⠛⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣷⠆⢀⠄⢠⡀⠴⠛⠋⠉⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⡀⠀⢀⣀⣀⣠⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠼⢁⣴⡄⠲⠄⠀⠀⡇
⠀⠀⣿⡴⠿⠛⠋⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠠⠠⠗⠀⠉⠁⠀⠀⢸⠋⠀⠀⠄⠈⠀⠈⠍⠻⠒⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧⠟⠂⠀⠤⠟⠃⠀⠀⠀⠀⡇
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣽⢆⠁⠾⠁⠠⣤⣶⡶⢿⠋⠐⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇⠘⠻⠈⠷⠘⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⢠⠀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠆⠀⠀⠈⠀⠐⠤⣆⣀⠂⠀⠒⢀⠘⠀⣀⢀⠀⠀⠰⠄⠈⢁⠀⠀⢠⠀⠀⠀⡈⢄⠊⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⡇⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠀⠉⢰⠀⠀⠀⠀⠀⠈⠠⠀⠂⢀⠀⠀⡸⡧⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠠⠨⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠚⠐⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢈⠀⣀⢀⡦⠀⠀⠀⠀⠒⡀⠖⠀⠙⠁⠀⠀⠀⠀⡠⠐⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣀⢠⢀⣠⢶⠒⠊⠧⠄⠀⠀⠀⡀⢀⠠⠄⠐⡡⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣼⣿⣯⡍⠁⠁⠉⠒⠐⠂⡶⠀⡀⠀⠀⠑⢄⠀⠀⠀⠀⠈⠀⠀⠊⠀⠀⠀⠀⠐⠀⠐⠏⠀⠀⠂⠀⠈⠁⠊⠁⠀⠀⠀⠀⠤⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢹⣿⣿⣶⣤⠀⠀⠀⠀⠀⢆⠀⢀⣄⠀⠀⠀⠁⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⡀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠂⢁⣰⣿⣀⡀⠀⠀⠀⠀⢀⠰⣇⠀⡾⠁⠀⠀⡅
⠀⢸⣷⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣆⠀⠀⢸⣿⡷⠈⣿⡇⠀⠀⠀⡆
⠀⢸⡿⡏⣿⡿⢂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢾⡆⠀⠛⠉⣿⣿⢬⣝⣿⣧⣀⣻⣧⣤⣿⠇⠀⠀⠀⡇
⠀⢸⣿⣷⣿⣷⣞⣿⣦⣠⣤⡄⠀⠀⣀⡀⢨⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣸⣷⣄⣽⣓⣼⣿⣿⣿⣯⣻⣾⣿⠋⣤⣤⣤⣀⡇
⠀⢸⡿⣿⣿⣿⣿⣿⠿⠿⠟⠾⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠹⣻⣿⣿⣮⣽⣿⣿⣟⣿⣿⣿⣿⠿⠷⢽⣷⣾⣧⣿⣿⣿⡇
⠀⢸⡇⢸⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⢄⣀⠀⢀⡀⠀⠀⠀⣀⡀⠰⣛⣿⢦⣶⣿⣿⣿⡏⠙⠲⠾⠿⠟⠉⠙⠻⠿⠿⠟⠋⣿⣿⣿⣿⣭⡇
⠀⢸⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⣤⣸⣦⣼⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⡾⠇⣼⣿⣿⣿⣿⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⢿⣿⣿⣿⡇
⠀⠘⠓⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠚⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠻⠛⠓⠚⠛⠛⠛⠛⠛⠓⠒⠒⠒⠲⠂⠦⠶⠶⠶⠶⠶⠢⠰⠤⠤⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡛⠷⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒⠶⢤⠀⢠⡀⠙⣿⣧⡄⠀⡀⠀⢠⠊⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣼⣇⠀⣠⡀⠀⠀⠐⠋⠀⠀⠉⠁⠀⢀⡀⣀⡀⣤⣥⣄⣏⠺⠞⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡀⠀⠙⠄⢿⡇⠀⠀⢠⡴⢤⣤⣴⣶⣶⣿⣿⠁⠀⠀⠀⡿⠛⠷⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⣿⣿⠛⠂⠘⢄⢀⠀⠈⠃⠀⠈⣿⣻⡿⢿⣿⣧⡀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠃⣠⡄⠀⠀⠉⢸⣄⠄⠁⠀⠸⢿⡻⣾⡏⠉⠉⠂⢸⣧⣼⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣿⠿⠘⠃⣱⢾⡿⠁⠀⢹⣄⣀⡀⠐⣻⠀⠀⠀⠀⢠⡶⠾⣿⣿⢠⠦⠀⣀⣀⣐⡀
⠀⠀⠀⠀⣠⣴⣶⡬⠚⠛⠋⠉⠉⠸⣿⣧⠀⠘⠅⠀⠀⠴⠻⠏⠀⠀⠀⠿⣯⣀⣀⣀⣤⠆⠀⢀⡛⠙⠀⢰⣾⣿⣿⠿⠿⣷⣶⣤⡀
⠀⠀⣴⣶⣼⠋⠀⠀⠀⠀⠐⠢⢴⣎⣿⣿⣷⣄⠐⠷⡀⠀⠀⠀⠀⠀⠀⠈⠁⠋⣙⣿⡦⠀⢀⠈⣠⣤⣤⣼⣿⣿⣿⡇⠀⠀⠀⠙⢳⣤⣤
⠀⠀⢻⣿⡁⠀⠀⠠⢄⣸⠗⢀⣀⠋⠉⠙⢿⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⣀⣠⣴⣾⣿⠿⣿⣏⣀⣘⣿⢁⠀⣰⠂⠀⠀⡟⠻⠆
⠀⠀⠀⠉⠳⠦⣄⠀⢸⠟⠲⠥⠀⠀⠀⠀⠀⡟⣿⣿⣿⠌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⣿⠳⠀⠀⠉⠉⠘⣡⠾⠯⢃⡵⢘⣾⠖⠛⠁
⠀⠀⠀⠀⠀⣠⣤⣉⠙⠲⠤⢐⣂⡀⠀⠀⠠⡀⠙⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠀⠀⠀⠀⠀⠀⢁⠴⠤⠅⠘⡉
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣶⣶⣦⣤⣤⣄⣀⣈⣓⡒⠒⠀⠤⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠐⣒⣂⣀⣀⣠⣭⣤⣴⡶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⢛⣋⣭⣿⣿⠿⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⢿⡿⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⠶⠶⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⢀⢀⣀⠀⡀⠀⡀⢀⡀⡀⡀⡀⢠⣀⡀⢀⢀⣠⠀⠀⣠⠀⠀⠀⣤⣀
⠀⠐⠠⠀⠄⠀⠀⠀⠀⢀⣨⣭⣍⠈⠉⠈⠀⢀⣐⣶⠀⠀⠀⠀⡆⠀⡐⠀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠠⠐⠂⠲⠒⣐⣰⠖⠒⢦⣄
⠀⠀⠀⠀⡀⠀⠀⣠⣴⣿⣿⠿⣿⢶⣶⣦⣴⣿⡀⢀⠀⠀⠀⠨⡇⠀⠂⠀⣰⠂⣤⣶⣼⣿⣿⡿⠃⠈⠈⣨⣶⠊⢩⡿⠶⣴⡎⠁
⠀⠀⠊⣾⡑⢀⢰⡎⠛⣿⣯⣶⡇⢼⣟⣹⣿⡟⣿⣿⣖⠠⠠⠄⡇⠀⢀⣾⣅⡀⢏⠙⠿⣯⣾⣷⡂⠀⠀⢹⣷⣦⠛⠁⠐⣷⣆⣞
⠀⢠⢸⣿⠟⠋⠀⠐⠚⣽⣿⡿⡿⢸⣿⣿⣿⣀⣼⣿⣷⢜⠀⢸⡇⠀⣿⣿⡟⠙⣤⠀⠙⣾⣿⠟⠇⠀⣤⣿⣿⠏⠀⠇⢺⣿⡿⣿
⠀⢀⢸⣿⡄⠀⠀⠤⡼⣿⡇⡁⠃⢸⣿⣿⡇⠈⠿⣳⣄⡀⠀⠀⡇⠀⠉⠁⠀⠸⣷⣀⢠⡸⣷⠾⡆⠠⢿⠿⢁⣿⡆⢀⣼⣾⣿⣿
⠀⢀⣾⣿⡇⠀⠀⠴⢿⠿⢉⣿⡭⢸⣿⣿⣧⠀⠀⠿⠛⠁⠀⠀⠅⠀⠶⠄⠒⠂⠀⢺⠬⠱⠟⠊⠁⠀⠀⢠⣨⠉⢿⣜⠛⠻⠧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⠃⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⠀⠀⠀⠀⣤⣤⡄⣤⣤⣤⣤⡤⠤⠤⠠⠄⠀⢀⠀⣤⣤⡄⢠⠀⠀⠀⠠⣤⡄⠠
⠀⢀⠁⠀⠈⢀⠀⠀⠄⠠⣦⢰⠆⢸⢿⣟⣏⣥⣤⣀⡀⡀⠈⢻⡇⠈⠙⠿⠁⠂⠀⣤⠈⠀⠀⠤⠄⡿⣧⣷⣷⣶⠆⠀⠘⠛
⠀⠀⠀⣠⣴⣏⣠⠄⠂⠐⠻⠀⢀⠘⠀⠘⢫⠿⠒⢻⠀⢨⡀⡀⠀⠀⢀⢴⡀⡄⢀⡷⣷⣾⣦⡟⠀⣇⣭⣤⣼⣧⡄⠴⠂⢀⣼⣦
⠀⠀⢿⣿⣿⠟⠃⢀⡀⠐⠀⠀⢹⠐⠀⡄⠀⠐⢶⠺⣮⣿⣜⠃⠀⠀⠸⣿⡇⠁⠘⣧⣾⠇⠉⠓⠁⣿⡏⣽⣿⣇⠀⠀⢀⡨⣿⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠁⣷⣠⠠⡠⠀⠀⢞⡷⢄⣸⡃⣿⣭⣯⢴⡆⠀⠀⣿⡟⠀⠀⢹⣿⠀⠀⡀⠀⣿⣷⣼⡝⡇⠀⠀⠊⣤⣿⢿
⠀⠀⠀⠴⠇⠀⠀⠀⠾⠟⠻⠶⣦⠀⢠⡙⠿⣾⣿⣶⣷⣾⣿⠻⠅⠀⠀⣿⡇⠀⠀⠼⠷⣌⣘⣥⠀⠁⣠⡼⣿⠂⠀⠀⠐⣻⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠟⠉⠙⠽⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠒⠒⠒⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⢉⢽⢍⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠘⠪⠞⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⠒⠦⠤⠤⠖⠋⠈⣆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⡈⠙⠲⢤⣀⢀⣠⣤⣴⣶⣶⣿⣿⣷⣶⣖⡢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠈⠉⠓⠦⣄⠈⢙⡆⣼⣿⣿⣿⣝⣿⣿⣯⣗⣿⣿⢯⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠈⠓⠲⠤⠤⠤⠤⠖⠋⠀⠈⠙⠲⢄⡀⠀⠀⢛⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡿⣽⣿⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣁⠤⠤⠤⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠤⠤⢤⡙⢆⠈⢿⣿⣿⢟⡻⣿⣿⡪⣿⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡜⠙⠓⠒⠒⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⠒⠒⠛⠙⣎⢦⠀⠻⣿⣷⣯⡓⢽⣯⣻⣷⣟⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠘⡜⡆⠀⠈⠛⢿⣯⣿⣟⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢱⢹⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡸⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠸⡆⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⡄⠀⢠⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡆⠀⢠⠇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⡀⠀⡴⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠳⢄⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠛⠁⠐⠃⣀⡥⠊⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠓⠒⠛⠉⠉⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⠤⠴⠖⠒⠋⠉⠉⠉⠑⠒⠶⠤⠴⠒⠋

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⡀⠂⠀⠀⠀⠉⠀⣤⠤⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠉⠉⣇⠀⠈⠁⠀⡅⠰⠴⢂⠀⡀⠈⠦⣀⣵⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠠⠀⠀⠀⠀⢰⠀⣀⡀⠀⠀⠀⢻⠟
⠠⣀⠀⠀⠀⠀⠰⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠙⠓⠀⠀⠈⠀⢀⠇⠠⠇⠋⠀⠀⣀⡀⠀⠀⠀⠀⠰⢮⠋⠽⠓⠀⠀⠀⠀⠀⠀⢀⣤⣼⠿
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣠⠶⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠈⠘⢏⣹⠂⠀⠀⠀⢡
⠀⠚⠁⠀⢙⡀⠀⠀⠈⠲⡄⠀⠈⠀⠀⠀⠀⠀⠐⠚⢁⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠁⡀⢀⠀⠁⠀⠀⠀⠀⠘
⠀⠆⠀⠀⠀⠑⠂⠀⠠⠄⠈⠒⠀⠀⠀⠀⠀⠀⠘⢏⣩⡙⠅⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠁⠀⠀⠀⠐⠀⣀⠀⠀⠀⠈⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢀⠀⠀⠈⠛⠉⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⡋⠁⠀⠀⠀⠘⣶⠯⠃⠂⠀⠰⠂⠼⢻⡾
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⠳⠀⠀⠈⢀⣠⠀⠀⡗⠀⠀⡀⠀⠀⠀⠠⡟⠁⠀⠀⠀⠃⠀⠈⠀⠷⢦⣤⠾⠧⠔⠋⠀⠀⣀⡄
⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⢟⣦⡀⢀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠈⠀⠀⢀⠀⠀⢁⠀⣢⡠⡆⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠠⠌⠀⢀⠀⠈⣸⣷⠄⠈⢦⣄⡀⠀⠀⠀⠀⠀⢰⣶⢀⡀⠉⠀⣆⡰⠀⠀⣰⠟⣁⡶⢀⣴
⠀⠊⠀⠀⠀⠈⠁⠀⠀⢉⣽⡷⣴⣁⠀⠈⠙⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣯⣄⠀⠀⠀⠀⠀⠠⠄⠀⣴⠊⠉⣹⣆⠀⠉⢀⡤⠚⢁⡴⠋⢠⡞⠁
⠀⠀⢄⠀⠉⠀⠀⠀⠀⠁⠹⠙⠆⠈⢠⠤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠈⠀⣀⠠⣿⣥⣀⠀⠀⠂⠀⠀⠀⠀⢹⠆⢀⡡⠀⢀⡴⠟⠀⢠⡞⢁⣴⠏
⠈⣀⠀⠁⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢌⣉⠿⠐⠧⠀⠀⠀⠉⠃⠀⠸⡻⣧⣀⡀⠡⣀⡀⠀⠀⠀⠘⠓⢁⡴⠋⠀⠀⡴⠋⣠⠞
⠰⠀⠀⠠⡀⠐⢀⡀⢢⡀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠄⠀⢄⣀⠀⠀⠿⢿⣥⡀⠀⠀⠀⠀⠀⠠⠔⠉⠀⠀⣠⠞⣡⡾⠃⠀⠀⠈⢲
⠀⠀⠁⠀⠈⠲⡌⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠜⠁⢐⡡⠞⠁⢀⡆⢀⠈⣂⢀⣄
⠀⠀⠀⠀⠀⠰⣇⠓⢄⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠠⢀⠀⠀⠀⠀⠀⠠⢆⣿⠀⠠⠘⠃⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢠⡀⣰⣖⡛⠉⢋
⠀⠀⠐⠀⠀⠀⠀⠁⠀⠉⠂⡀⠀⣀⢴⡇⠀⠀⠀⡘⠿⢧⡄⠈⠀⠀⠀⠀⠰⣋⣳⠏⠀⠐⠆⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢊⠙⠃⠀⠅⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⢈⣦⣾⡟⠁⢀⣄⡀⠀⠐⠂⠀⠀⠄⠐⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠉⠉⠋⠀⠀⠘⠿⠛⠀⠀⠀⠀⢢⠂⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⣤⣴⡤
⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⣯⠀⣏⡅⢩⡅⠘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠓⠀⠃⠁⠑⠂⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢠⣉⠀⠀⠈⠀⡅⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠒⠈⠂⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠡⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣧⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⣀⣠⡤⠶⠒⠋⣉⣁⢄⠀⠙⣆⠀⠆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠚⠉⣁⠠⠄⠒⠊⠉⠀⠀⠀⢣⠀⠈⢧⡀⠀⠑⠒⠠⡢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠊⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢳⡀⠀⠀⠀⠈⡞
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠀⢀⡜⠁⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⡆⠀⠀⠻⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠢⡀⠘⠀⢠⠎⢀⣀⠀⢸⠀⠀⠀⡠⠔⠒⠈⠉⢙⣦⠠⠐⠉⣀⠴⠃⠀⠀⢀⢷⡀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⡐⢁⠄⢠⠃⢀⠈⢉⢧⡾⠤⢄⣸⣷⠤⠤⠔⣒⡽⣋⠤⠖⠋⠀⠀⠀⣠⠔⠁⠈⡇⠀⠀⠘⠮⣂
⠀⠀⠀⠀⠀⡈⠀⠁⢠⢃⢞⣫⠊⢉⣹⡿⠤⠤⠤⠤⠝⠛⠛⠋⠉⠀⠀⠀⢀⡠⠔⠋⠀⠀⣠⠔⠃
⠀⠀⠀⠀⠀⠁⠀⢀⠆⠸⠬⢿⡀⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠊⠁⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⢀⡠⠴⠚⠤⠤⢄⣀⣉⣉⣉⣁⣤⣤⡤⠤⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡏⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⡰⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⡠⠃⠘⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡡⠃
⠑⠬⡓⠤⢄⣀⡀⠀⠒⠒⠒⠋⠉⠁⢩⣡⡴⠀⢀⣀⣠⡤⠴⠒⢋⣩⠤⠒⠉
⠀⠀⠀⠉⠐⠂⠭⠭⢉⣙⣛⣓⣒⣚⣛⣉⣉⡉⠭⠤⠤⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠄⠐⠂⠐⠄⠀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡠⠀⠐⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠤⠄⠤⠄⡀
⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢃⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠠⠀⠠⡀⠡
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⢈⠆
⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠡⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠉⠢⡀⡠⠂⠁⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⡈⠀⠁⠐⠒⠂⠁⡠
⠀⠱⠀⠀⠀⢠⡀⠤⠤⢀⡀⠀⠀⠔⠒⠛⠛⢇⠀⣣⠀⠀⠀⠀⠀⠐⠀⣤⡄⠀⠀⠀⠀⠸⢀⣤⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⡐⠀⠀⢀⡀⠤⠐⠊
⠀⠸⠀⠀⢰⠋⠀⠀⠀⠀⢈⣲⠁⠀⠀⠀⢀⣤⡝⠀⠣⡀⠀⠀⠀⢀⠀⠻⠿⠁⠀⠀⠀⢠⠘⠿⠿⠀⠀⠀⠀⠀⠆⠀⠀⠜⡠⠐⠘⠣⢄
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠰⢿⢿⠀⠀⠀⠀⠘⠛⢁⠀⠀⠘⠆⡀⠀⠀⠢⡀⠀⠀⠀⠀⡠⠊⠢⡀⠀⠀⠀⠀⢀⠔⠀⡠⠊⠁⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠈⠀⠈⢄⠀⠀⠀⠀⢀⠌⡢⢀⠀⠀⢀⠀⢂⠀⠀⠀⢀⠈⠂⠄⡀⠀⠁⠁⠈⠁⠀⠉⠁⠀⠁⠐⠀⠈⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡄⠀⠀⠁⠐⠒⠈⠁⠈⠁⠀⠀⠉⠀⠀⠐⠀⠀⠐⠠⠀⠀⠀⠀⠉⠀⠐⠒⠂⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢠⠤⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁
⠀⠀⠀⠐⢄⣀⠤⠤⠂⠒⠒⠉⠉⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⡿⠿⣛⣿⣿⣆⠀⢀⠀⠀⠀⠀⠀⣀⣀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡌⠀⢀⣌⠙⣿⣿⣿⣷⣦⢤⣁⣀⠲⢤⠙⡿⠷⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⡆⠙⠓⡀⠀⠀⢹⣿⣿⠿⣧⣀⣀⠒⠉⠛⠃⠑⢂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣤⡄⠭⣤⣾⣿⠿⣿⠿⠋⠀⠀⠀⢀⠴⠐⠂⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠈⠀⠀⠀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡆⠈⠋⠡⢄⠑⢄⠀⢀⣠⣄⣤⠁⠀⣀⣀⣀⣀⣠⣤⣶⣿⣿⠀⢲⢯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠈⠣⠇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢤⠀⠈⠉⢩⣁⣭⣿⣿⣿⣿⣿⢀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢶⣶⣭⣻⣶⢯⣿⣿⣿⢯⣿⣇⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠓⢉⣭⠟⣩⠟⠠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠓⠦⣤⡄⠀⠀⠀⠀⣀⡠⢴⢶⣿⠿⣄⣀⡀⣈⣱⣵⣾⣯⣴⡟⠁
⠀⠀⠀⠀⠀⠀⣰⡤⣀⣀⣀⣀⣀⢀⣌⠈⠁⠒⠋⠋⠉⣁⣀⣤⠷⢄⣿⣿⠿⣟⣛⣻⠶⠟⠋
⠀⠀⠀⠲⠶⠊⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡜⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠱⡄⠀⠀⠀⣶⡀⠁⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡷⠀⠀⠀⢇⠙⠤⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡅⠀⠀⠀⠀⠘⢦⠀⠈⠢⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠈⢹⢀⣖⠀⠀⠀⠈⠳⡄⠀⠈⢢⠀⠀⠀⡠⢊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡾⠈⢌⠢⠰⣤⡀⠀⠈⣆⠀⡀⣇⠀⢈⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨⢀⡀⢡⠀⠀⠈⠃⣱⢙⡆⠀⡜⢠⠀⢸⠀⢀⡖
⠀⠀⠀⠀⢠⣀⠀⠀⠀⠸⡀⠠⠑⢄⠑⢄⠀⣠⠃⠀⣧⠞⢀⢈⠀⡅⠀⣸⢱
⠀⠀⠀⠀⠈⠈⠑⢦⠀⠀⠳⡀⠣⠀⠑⡌⢷⢁⠜⠄⠉⠐⠁⡀⣰⠁⢀⠏⠨⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⣁⡀⠇⠀⠀⠙⣄⠡⠀⠈⠄⠄⠀⡠⠊⠀⠐⣠⣃⠴⠋⠀⠀⡇⠀⠀⡠⢺
⠀⠀⠀⠀⠀⠀⠈⠫⠑⠦⡀⠀⢸⡀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠋⠁⠀⡚⠁⡸⠃⠀⡊⠴⠋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡓⢤⠘⡄⢸⠁⡀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠔⢀⡼⠁⡀⢀⡔⢪⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠡⠹⠏⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡎⣠⡾⢡⠋⠠⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠄⠀⠆⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⢠⡟⡃⣇⡏⠴⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠣⠈⠙⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠂⠘⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠄⡀⠀⠈⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠐⠉⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⢀⠄⡄⠀⠀⠀⠀⡾⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡀⠀⠀⠀⠀⠈⣼⠀⠀⠀⠁⡘⠀⠀⠀⠀⢀⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣄⡀⠀⠀⠀⠀⠰⣴⣻⠟⢦⠀⠀⠸⣶⡿⡆⠀⢰⣿⣇⠀⢀⠐⠊⡡⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢉⣭⣷⡄⠀⠀⠀⠙⠀⠎⠀⠀⠀⠘⠇⠆⠀⢠⠃⠌⠀⢸⠸⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢁⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⢂⠴⠃⠀⣀⠠⠀⠂⠀⠠
⠀⠀⠀⢠⣄⠀⢀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⣀⡤⠖⠛⠉⠉⠁⠀⠈⠉⠉⠙⠲⠦⣄⠀⠀⠀⢀⠎⠀⡴⠒⠂⠈
⠀⠀⠀⢰⡏⠀⣞⠁⠀⠀⠀⠀⠀⢸⠀⠢⠴⠶⠦⡄⠀⠀⠀⠀⠀⢀⡴⢊⣥⡶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢾⡝⢦⡀⠀⠠⠞⠁⠀⠀⠀⣀⣀
⢰⡄⠀⣰⠏⢀⡼⠃⢅⠀⠀⠀⠀⢸⡀⠀⣲⣏⠁⠘⣄⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⢦⠀⠀⢠⠞⣿⢇⣀⠴
⢻⡄⠀⣹⠂⢈⣷⠠⣏⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠲⠄⠀⠀⡼⠁⠀⢀⠞⠉⠙⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠓⡄⠀⠀⠀⠈⢳⡀⠀⠈⠉⠉
⢻⡄⠀⢻⡄⢘⡧⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣼⣾⣷⠄⡇⠀⢸⠀⠀⠀⠀⠀⠾⠿⠷⣀⣹⠀⠀⠀⠀⠈⢧⠀⠠⠤⣤⣀
⠸⡅⠐⢿⡀⢸⡧⣾⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⡇⠀⠠⠂⠀⠀⠀⠀⢠⠞⠉⠀⠐⠀⢀⠀⠀⠀⠉⠑⠣⣄⣀⠀⠀⠀⢸⡄⠀⠨⠟⣋⠁
⣾⡁⠀⠸⠀⠈⠀⡞⠁⠀⠀⠀⠤⡀⠀⢉⠷⠖⠋⠁⠀⢸⠀⠀⠀⠀⠀⣠⠞⠀⠈⠓⠒⠛⠛⠀⠞⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⢸⡏⠀⣴⠋⢳⠀⢀⣠⡄⠀⠀⠀⠙⢦⣲⣷⡯⠂⠀⠀⢸⡀⠀⠀⣤⠞⠁⡀⠀⠤⠤⠤⠤⠤⠤⠤⣄⣙⠳⠶⠶⢶⠀⠀⠀⠀⠀⢸⡇⠀⠠⢶⢄⣀
⠀⠙⠂⠙⣻⠿⠤⠮⣅⡀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠹⠀⠙⢦⣀⠐⠒⠀⠐⠂⠀⠀⢀⣠⣽⠟⠂⡎⠀⠀⠀⠀⠀⣼⠀⠐⠲⣤⠀⠁
⠀⠀⢠⡞⠁⠀⡀⠀⠀⠙⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠈⠉⢉⣉⣉⣉⣉⠉⠉⠁⠀⠀⠘⠀⠀⠀⠀⠀⣰⠃⠀⠤⣄
⠀⠀⣿⡀⠀⠀⢿⡄⠀⠀⢱⡀⠀⣈⠉⠙⠳⠦⠤⠄⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠈⠂⢄⣀⣀
⠀⠀⠸⣧⠀⠠⠘⣷⠁⠀⠀⢧⠀⠀⠈⢦⡠⣤⣶⢁⠄⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠰⠮⠵⠦⠄
⠀⠀⠀⢹⡆⠀⢀⡸⣷⠂⠀⠘⡆⠀⠀⠀⠉⠛⠓⠉⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠙⠦⢤⣄⣀
⠀⠀⠀⠀⢿⡀⠀⠀⢻⡆⢀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠶⠶⠖⠂⠈⠉⠀⠀⠀⠀⠀⠀⠠⣔⣿⣃⡀⠈⢡
⠀⠀⠀⠀⠘⣷⠀⠀⠚⢿⡀⠀⠀⢧⠀⠀⠀⠀⠀⢀⣾⣄⢠⠈⠀⠀⠀⠀⡠⠂⢀⠀⠀⢀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⢀⣤⣄⠀⠈⠙⡀⠰⣄
⠀⠀⠀⠀⠀⢹⣇⠀⠀⢘⣧⠤⠀⠘⡆⠀⠀⠀⠀⡜⠉⠀⢻⡄⠀⢠⠂⠀⡠⠐⠁⢀⠔⠁⣠⠃⠀⢀⠀⠀⡄⠀⢰⡀⠀⠀⠻⠶⠈⢳⠀⠀⠑⠂⠂
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢹⣆⠀⠀⢹⡀⠀⡠⢊⣀⠠⠤⠚⠀⠀⣇⠿⠱⡇⠀⢠⢋⣄⣠⡇⠀⢀⢧⣤⣄⢧⠀⠀⠉⠋⠉⠉⠳⡄⢸
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢿⡎⠀⠀⢧⠀⠀⠁⠀⠀⠀⠀⠀⢀⡝⠀⠀⠹⠀⠸⡜⠞⠟⡇⠀⠀⠻⡉⠈⢨⠁⠀⠀⠀⠀⠀⠀⠇⠈⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠈⣷⡀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠐⠁⠀⠀⠸⠄⢠⠆⠀⠀⠐⢁⠖⠁⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⣀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠠⣿⣿⡆⠀⠹⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⢄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠉⣉⣤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣒⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠹⡜⣆⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢹⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⠇⣾⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠎⣸⠇⠀⢸⢹⡌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢠⡏⠀⢠⠇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣾⠁⠀⠎⢀⡟⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⢿⡀⠀⠤⣾⠁⠸⠁⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⣛⣉⣉⣠⡤⠈⠷⣄⠠⠿⣄⠠⣤⣤⣤⣈⣉⠉⠓⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣋⡉⠀⠀⠀⠀⠀⠈⠁⠀⠉⠑⠀⠀⠀⠀⠀⢉⣓⣴⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠈⠉⠛⠛⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠛⠉⠉⢀⣈⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣉⣤⣬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⡏⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢋⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡴⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣣⠞⣉⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡶⠶⠿⠟⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢇
⠀⠀⢀⣤⣶⠿⠛⠋⠉⠉⠀⠀⠀⠀⢀⣀⣀⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀
⠀⢰⡿⠉⠀⠀⠀⢀⣠⣤⣶⠶⠿⠟⠛⠋⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⡀⠀⠀⠀⠀⠀⠈⢣
⠀⠸⣧⡀⠀⠀⢰⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⠀⠀⠀⡼
⠀⠀⠙⢷⣤⡀⠈⠻⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠉⠛⠿⣶⣤⣍⣙⡛⠛⠿⠿⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠴⠶⠖⠚⠋⠉⠁⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠶⠶⠶⣦⣤⣤⣤⣤⣤⣌⣉⣉⣉⣉⣉⣉⣉⣀⣀⣠⣤⣤⣤⣤⠤⠤⠶⠖⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠰⡄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⢤⢈⠛⠓⣠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠼⢿⠁⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⡐⣯⣼⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣀⡀
⠀⠀⠀⠛⠿⠆⢸⡏⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⣰⡀⠉⣹⣶⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣶⡦⡧⠀⠀⠀⠀⠀⠶⣆⠈⠙⠁⡀
⠀⠀⠐⠷⠔⣛⡿⣅⠘⠁⠀⠀⠀⠀⠀⠸⣟⣙⣀⢥⡴⠟⠋⠀⠀⠀⠀⠀⣰⣤⣾⢿⡟⣿⣷⣐⣴⠀⠀⠀⢀⣀⠀⢾⣻⣀⠈⠋
⠀⠀⠐⣶⣋⡍⣴⡏⣁⣤⡀⠀⠀⠀⠐⠄⡿⢿⣧⣖⣯⣰⣿⡆⠀⠀⠀⠨⢿⣯⡿⣭⣯⡟⡳⠏⠁⠀⠀⠀⠈⠁⠺⣻⡏⠟⠷
⠀⠀⠰⣯⣻⣿⡿⢹⣿⡿⠀⠀⠀⠀⠐⢼⣿⡟⣿⣿⠶⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠇⠁⠀⠁⠀⠀⠀⠀⠲⣛⣿⣿⣷⡀
⠀⠀⠀⠀⠐⠾⡿⠊⠉⠁⠀⠀⠀⠀⠀⠘⣿⣷⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⣯⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣰⡆⣀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣰⣣⣾⡴⣀⡀⠀⠀⠀⠀⢠⣤⣄⣀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢨⣿⣹⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢠⡰⠆⢸⣿⣿⣿⠟⠛⠁⠀⠀⣠⣷⣬⠾⠛⣿⠚⣠⣦⣴⡄⠀⠀⠀⢶⣤⣿⡷⠞
⠀⠀⠀⠀⠀⣾⠷⣿⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⢀⣿⣄⣴⣾⡿⠉⠀⣴⣌⡟⠃⡇⠔⠁⠀⠀⠉⠛⠃⠀⠀⠐⢿⢯⣿⣽⡿⠐
⠀⠀⠀⠀⠀⠃⡞⣙⡾⢧⣤⣤⠭⠉⠀⠀⣤⣤⢀⠀⠘⡧⣀⣉⣩⠅⠀⠀⠀⠙⠷⢶⢠⡏⡰⠆⠀⣠⠆⠀⠀⠀⠀⢻⣿⡘⢦⣙⡁
⠀⠀⣠⡀⢠⣟⣩⡅⠰⣿⡟⠋⠀⠀⠀⠰⣤⡷⣛⣿⣾⣿⣿⠿⠛⠂⠀⠀⣀⣾⠛⣸⠏⠀⡅⣴⣶⡆⠀⠀⠀⠀⠤⢬⣥⡿⣿⣿⡛⠉⠁
⠀⠀⢿⠿⣿⣿⣣⣿⡀⡿⢿⣵⣆⠀⠀⠀⠀⣾⡗⣼⡗⡠⡀⢓⣤⠆⠀⠈⢙⣷⣆⡟⠇⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠈⣭⣿⣿⢧⡀
⠀⠀⠀⠘⢿⣭⣟⠻⠷⠤⡀⠀⠀⠁⠀⠀⠀⠛⠿⠿⠋⣷⣾⠏⠁⠀⠀⠀⠀⠁⠘⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⢻⠭⠓⠈
⠀⠀⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣄⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⠀⠀⣠⠤⠀⠀⠀⠀⠴⠛⡿⠿⠟⢿⣿⣷⣆⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠰⣷⢠⣴⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠳⣧⣮⣿⣷⣶⣖⠚⠀⠀⢀⣿⣿⣿⠙⣿⢫⣝⠏⠉⣥⣄
⠀⠀⠀⠀⣀⣸⡿⠿⠟⢶⣻⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣀⣘⣿⡓⠈⠃⣠⣠⣀⡀⠀⠛⠉⠉⢠⣶⢧⠰⡟⠂⢀⠈⠹⡇
⠀⠀⠀⠀⢩⣷⡿⣾⡼⣿⠯⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠨⣽⣿⣿⢷⡝⠧⠉⠉⠀⠀⠀⠀⠀⠀⣸⣯⠆⡞⠉⠓⠈⢡⣶⣄
⠀⠀⢀⣶⣮⣛⡸⣿⣿⡏⢯⣿⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠦⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⢻⣷⠇⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠚⠉⠁⠸⠇⢷⢾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛

⡿⢿⠿⠿⠿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⣁⠤⠶⠛⢛⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⣠⣴⣞⡓⣦⣬⣉⡽⠟⠛⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣫⠕⣪⠴⣇⣀⡒⠐⠦⣨⡼⠋⠀⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⢞⣫⠵⢚⣭⡏⠨⢥⣉⠢⠌⢩⣦⠞⣡⠀⠀⠀⠀⠀⠀⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⣚⣫⠭⠖⣻⠏⢁⡙⠛⠯⣙⠓⢲⠌⣣⡴⠛⣡⠞⠁⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠛⣋⡉⠥⣒⣚⡿⠟⠒⠾⢽⡋⠓⠦⣍⠛⠖⠦⣤⡴⠞⣉⡴⠋⠁⠀⠀⠀⠀⠀⡀⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⣁⣀⣠⡤⣾⡉⠙⣛⠷⣞⠓⠦⣝⠻⣶⠮⢥⣔⣈⡵⠾⢋⣩⠴⠛⠁⠀⠀⠀⠀⠀⠀⡀⣤⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣾⣧⣴⡞⠁⡈⠙⠚⠎⠙⢢⠈⢓⣂⣤⣁⣬⡤⠼⠗⢛⣉⡥⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢰⡟⠁⠀⠻⡄⠱⣄⣱⠶⠖⠛⠛⠉⠉⠉⠀⠀⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⣿⣿⠟⠁
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣿⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢸⣰⣸⣾⣿⠿⠋
⠀⠀⠀⠀⠀⣷⠺⠀⠀⠀⣯⣼⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⣠⣰⣼⣼⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠘⠷⣤⣴⣾⣿⣿⢯⣆⡆⡜⡄⡆⡄⡆⡆⣴⡦⣸⣦⣮⣧⣶⣷⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣷⣿⣿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣇⠀⢸⡇⠀⠀⡾⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿⠀⢸⡇⠀⢰⡇⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢻⠀⢸⡇⠀⣼⠁⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡆⢸⡇⠀⡟⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡇⢸⡇⢸⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡇⢸⡇⣾⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⢸⢇⡏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣼⣼⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠟⠛⠳⣄⠀⣠⣴⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣄⠀⠀⠘⣾⠋⠀⠀⠀⠀⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠹⣧⠈⣇⢀⠈⢁⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠟⠀⠞⠁⠀⢀⡗⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣤⣤⣤⣄⣙⣳⣦⡀⣀⣠⡶⠛⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⣴⠆⢀⣀⠉⠉⠙⠛⠛⢿⣅⠤⠒⠉⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡇⠀⣿⢰⣿⣩⠗⠀⣤⠦⡄⢰⣻⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⡀⠀⢠⡀⠀⢀⣿⠃⠻⠤⠂⠈⠷⣤⣼⢤⡅⠀⣠⣶⢶⠚⠁
⠀⠀⠀⠀⠀⠀⣼⠏⢁⣤⡶⠙⣦⢸⡿⠀⠀⣀⠀⢀⣭⢠⣾⢛⡦⠘⠸⠟⠐⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⢿⣆⣾⣿⠀⣀⣼⡟⢱⣿⠀⣿⠃⢸⣿⠸⣿⣊⣰⠃⢻⣂⡾⠙⣓⠋
⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⠋⠉⣼⠁⠀⠻⠊⠙⠖⠋⠛⠀⠀⠁⠀⠀⠀⠀⠀⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣳⡾⠷⠶⠶⠶⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣤⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⡆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠄⠀⢠⡀⠀⢠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠘⣦⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢆⢈⠀⠀⡄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣿⢳⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢈⡁⠘⣴⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠳⣧⢾⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢸⡇⣾⠏⣈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠷⠿⠾⢟⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢱⣼⣿⣏⣼⢃⡀⢀⠴
⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠈⣿⣿⣀⣠⡀⠀⠀⠠⠶⣷⣴⣿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠰⣟⡅⢈⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠿⠿⠧⢀⡹⢫⢈⣀⣬⡻⣀⣈⣭⣿⣿⣿⢿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠙⢻⣧⠁⡠⠚⣆⠀⠀⠀⠀⠀⠀⢠⣤⣒⣲⣮⣽⣶⣶⣿⣿⣿⣿⢻⡭⠙⠉⠀⣴⠋⢸⣿⡿⡟⢻⡓⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠐⢾⠋⠀⠀⠈⠀⢎⠂⠀⠀⠀⠀⠸⣿⡿⢿⣿⣿⡿⣿⣿⡿⣿⡆⠀⠀⢠⠞⢡⠄⢸⡿⣷⠀⠀⠹⣶⡆⢀⠀⣤⡾⠋⠐
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⡆⠹⠃⠀⢿⡿⡇⣿⢿⡁⠀⣆⣰⠀⠀⠀⣾⢀⡄⠚⢻⡇⢰⠀⠙⠁⣼⡷⠈⡛⠶⠗⠂
⠀⠀⠀⠀⠀⠘⣄⠃⠀⠀⠀⠀⠀⠈⠄⠀⠀⢡⠀⠀⢸⠀⠁⠀⠙⠈⢣⠀⢹⠹⡇⠈⣿⡟⣾⠃⠀⢰⡏⣼⡇⡼⠀⢋⠀⠀⢨⣿⣟
⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠷⠀⡿⢹⣧⡀⠀⢞⣼⠟⡰⠁⠀⣼⣿⡶⠺⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⠶⠶⠤⠲⣖⣚⡫⠜⢵⡀⠤⠀⠀⠀⣁⣼⢾⣯⣔⠀⠀⡘⢁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠚⠛⠉⠉⠙⠓⠲⣬⣻⣆⠤⠖⠋⠉⠀⠀⠀⠀⠉⠈⣥⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠉⠉⠐⠲⢤⡀⠀⣀⣠⣴⠾⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⢈⣿⣷⣦⣤⡴⠚⠉⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⠿⠉⠀⢀⡠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⣴⠿⠛⢌⣿⣿⣿⣶⣤⣠⡀⢤⣴⣾⠟⠛⠉⣿⣿⠖⣲⠋⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠿⢟⡿⢿⣿⣿⠅⠈⢂⡀⠐⠂⠉⠀⢀⢷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠼⠔⠋⠁⠀⠀⠀⠀⠀⠑⠛

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⠶⠶⠶⠶⣶⡶⠶⢶⡶⢶⡶⢶