Muscle Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣾⣿⡇⠀⠀⠀⠀⠀⣤⣾⠟⠛⣿⣾⣯⣤⣤⡄⠀⠀⠀⠀⠀⠀⠘⠓⠲⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢼⢻⣲⡄⠀⠀⠀⢸⡇⢹⠀⠼⣿⣿⣩⠽⣿⠇⠀⠀⡠⠂⠐⠊⠀⠴⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣯⢼⡯⣬⠇⠀⠀⠀⠀⣇⢸⣤⣾⡏⠉⠠⠞⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⢸⡀⠊⣿⠦
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢿⣶⠀⠀⠀⠀⠀⠸⡘⢇⣸⢿⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠀⣮⢀⡆⠋⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⣟⠉⠀⠀⠀⠀⠀⠀⢻⡿⢿⣭⣇⡃⠀⠀⠀⠀⠀⢀⠀⠠⠂⣷⣾⡇⢀⠃⣼⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣸⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⣿⢻⡇⠀⠀⠀⠀⠀⠈⢀⣼⣆⠸⣿⠄⢀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣿⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣿⣶⡇⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⢿⡔⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡜⢿⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⣦⣻⠃
⠀⠀⠀⠀⠀⠀⡔⣴⡔⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠛⣇⡫⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⠆⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣿⠀⢀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⢀⣻⠁⠀⠈⣆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠴⠤⢤⣤⠶⠋⠁⠀⠀⠒⣩⣄⠀⠀⠀⠈⠪⡆⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣨⡉⠀⠀⠀⠀⢿⣖⠁⠀⠀⠀⠈⣲⡿⠀⠀⠠⠐⡄⢻⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣡⣦⣉⢀⠐⠤⣀⠈⠻⢿⣶⠂⠈⣽⣿⡀⠀⠠⠀⡝⠄⠹⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣷⣤⣈⣷⡀⠈⠛⢻⣿⢩⣙⣿⡄⢠⡀⡀⠀⠢⡄⠀⣴⢒⣈⠥⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠦⠾⠿⣿⣿⣿⠿⠾⠿⣄⣠⣼⣿⠀⣽⣿⡀⠀⢻⣧⠀⠀⠹⢄⡇⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠻⡄⠀⠀⠀⢻⡟⠀⢠⣿⣿⠀⠐⠀⠈⢷⡄⠐⡌⣧⠀⠠⠏⣰⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠹⡄⠀⠐⣼⣷⣰⣾⠏⠤⠀⠁⠀⠀⠀⠙⢷⣦⡘⠄⣀⠖⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠙⠂⣶⣿⡟⢿⠃⠀⠒⠀⠁⠀⠀⠀⠀⠀⠈⠙⠆⠐⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⡿⠟⠁⠀⢡⣀⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠁⠀⠀⠀⢤⣻⡇⠀⠁⠀⠠⠖⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣻⡿⠃⠀⠀⠀⠤⢊⣥⠠⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⣀⠘⠟⠃⠀⠀⠀⠀⠖⠋⣶⣦⡄⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠠⣿⣿⠀⠀⠀⠐⠀⠀⠐⠋⠀⠀⠀⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡖⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢹⣿⣾⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠛⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠰⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠛⣛⠒⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣀⡁⠀⠐⠚⣂⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠁⠀⠀⠀⠀⠀⠀⢘⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⡶⡄⠀⢀⠖⠂⢸⠖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣦⡈⠉⠁⢀⡴⠁⠘⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⣈⠀⠸⠑⠒⠒⠁⠀⠀⠀⠠⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⣠⠄⠀⠲⣄⠀⠀⠀⠀⢀⣀⠀⠀⢃⣠⣤⠴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠐⠁⠀⠈⠑⠄⢀⠎⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⡜⠀⠀⠐⠒⠤⠀⠀⠀⠙⠦⣤⣤⠀⠀⠀⠀⠀⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⢦⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠈⠢⡀⠰⢻⢡⣄⣀⠀⢀⣀⣀⣀⣀⡤⠖⠋⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠣⡅⠀⠀⠈⠉⠀⠁⠀⠀⡠⠀⠀⠀⠘⢦⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠂⠤⢄⣐⠀⠁⠒⠒⠀⠀⠀⠉⣠⠔⢁⠀⣀⠤⠓⠁⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢄⡀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠈⠀⠔⣣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢍⣉⠓⠢⢄⠀⠀⠁⠀⠐⠒⠒⢤⡀⠊⠁⠀⠀⣀⠀⠀⠀⡠⠀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⡒⠄⠀⠀⢄⠀⠠⠀⠀⢱⠀⠀⠀⠊⠁⠀⣠⡬⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠢⡄⢣⣸⠊⠀⠒⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⡙⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢡⠤⠤⠤⠔⠒⠒⠲⠾⠌⠉⠉⠙⠶⠒⠒⠂⠤⠤⠤⢤⠃
⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⢀⣀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠄⣀⡀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⢦⣀⣠⠤⡶⠛⢄⡀⠀⣀⠄⢠⣀⡀⣀⡠⠿⣦⢄⣀⡠⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⢀⢉⠿⠛⠂⠀⠛⢿⡁⡀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⡦⠄⠀⠀⠀⢴⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢤⠤⠀⠀⠤⡤⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⣬⣄⣠⣀⣠⣤⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢹⣿⣿⣿⣿⠉⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢸⡇⢻⠃⢻⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢻⠀⠈⠀⢸⠀⠘⠀⠀⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄⠀⢠⠎⢣⡀⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠛⠀⡸⠀⠸⡇⠀⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⠀⡧⠀⠀⣇⢀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠰⣇⠀⢀⡿⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⢸⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠟⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢻⣿⣿⣿⣿⣿⣿⠏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠃⠀⢸⣿⣿⣿⡿⣿⣿⣿⣿⡀⠀⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢼⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⣠⣤⡔⠿⠟⣠⣤⣄⠀⠀⠀⠀⢀⣤⡴⣾⠀⡷⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆
⠀⠀⠀⠀⢸⣿⣿⣷⡇⣰⣿⣿⣿⠆⠀⠀⠀⣿⢿⣷⣾⠀⣷⣿⡟⣿⠀⠀⠀⠀⠀⢠⣶⣆⡻⢛⣴⣶⡄⠀⠀⠀⠀⣠⣶⡺⡅⡟⣶⣦⡀
⠀⠀⠀⠀⢸⣿⣿⠛⠃⠛⠻⣿⣿⠀⠀⠀⠀⡿⣿⣯⡿⠀⣿⣿⡿⢿⠀⠀⠀⠀⠀⣻⡁⠀⢠⠁⠀⢸⡇⠀⠀⠀⠀⣿⣿⠿⠇⠿⢿⢿⡇
⠀⠀⠀⠀⢸⡟⢻⢤⡄⣤⣀⡏⣿⡇⠀⠀⢠⣄⠹⡿⡃⠀⢻⢿⠃⣸⠀⠀⠀⠀⠀⣿⡇⣶⣮⣷⣶⢸⡇⠀⠀⠀⠀⠛⢸⣶⠄⣴⣎⠘⠃
⠀⠀⠀⠀⣾⠇⢸⣿⣧⣾⣿⡇⢸⡇⠀⠀⢸⣿⢀⡷⠀⠀⠀⢾⡀⣿⡆⠀⠀⠀⠀⣿⢀⣿⣗⣿⣯⠘⣷⠀⠀⠀⢸⡿⢠⠟⠀⠸⣧⢰⣷
⠀⠀⠀⠀⣛⢰⣿⣿⣿⣿⣿⣿⠈⡇⠀⠀⠸⠇⢨⣦⣄⢀⣤⣮⡁⠹⠃⠀⠀⠀⢰⡏⣼⣿⣿⣿⣿⣦⢻⠀⠀⠀⢸⠇⣽⣦⣀⣤⣾⡄⣿
⠀⠀⠀⠀⠟⠘⣿⣿⡛⢻⣿⢿⠈⠇⠀⠀⢸⠆⢸⣿⣿⣿⣿⣿⡇⠴⡆⠀⠀⠀⢨⠀⣿⣿⡟⢿⣿⣿⢨⡄⠀⠀⣘⢀⣿⣿⣿⣿⣿⡇⢘⡀
⠀⠀⠀⠀⠀⠀⡌⣿⡇⢹⣿⣰⠀⠀⠀⠀⠀⠀⢉⣿⣿⠈⣿⣯⡅⠈⠀⠀⠀⠀⠈⠀⣷⣿⡇⣾⣿⡇⠀⠁⠀⠀⠋⠘⣸⣿⡇⣿⣯⡀⠈⠃
⠀⠀⠀⠀⠀⠀⣿⣿⠃⢸⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠃⢿⣿⡇
⠀⠀⠀⠀⠀⠀⡀⣹⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⢸⣻⡇⠀⢻⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⢸⢻⠇
⠀⠀⠀⠀⠀⠀⣇⡿⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⢸⣦⡇⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠈⡀⠁⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⢀⡰⠈⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠸⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣟⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣴⡇⠈⣷⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⣿⣟⠻⠉⢋⣽⣿⣿⣿⡲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡿⠇⠈⣧⢈⣧⠰⠶⣿⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣷⣤⣽⣿⣾⣏⣠⣸⣿⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣧⣤⣾⣿⣦⣽⣦⣠⣿⢳⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⣿⡿⣿⣿⣸⣿⣿⣿⣿⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⢿⣿⠈⡿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⣿⢻⣿⡥⠐⡧⠠⣿⣿⢻⡇⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡏⢻⣿⣦⠟⠞⢧⣼⣿⠛⡇⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠇⢸⡿⡗⠂⡕⠠⡾⣿⠀⢇⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣸⠇⢸⣿⡇⠀⠀⠀⣿⣿⠀⢷⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⠀⢸⣧⡗⠒⠗⠒⢧⣿⠀⠸⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡟⠀⢼⣿⡀⠀⠀⠀⠹⣿⡄⠘⣿⣩
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣾⡟⢰⣀⣀⡀⠀⣽⡆⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣼⣿⣿⣦⣤⣾⣿⣿⡀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⣀⡈⠉⠀⠀⢀⡿⢿⣶⡿⠿⣷⣾⠿⣇⠀⠀⠈⡁⣀⡀⠀⠀⠀⢀⡘⢙⠀⠀⠀⣿⡏⠀⣻⣿⠁⠈⣿⡇⠀⠀⠘⡊⣀
⠀⠀⠀⠀⠀⠀⠙⠙⠋⠀⠀⢸⣇⣀⢻⣧⢠⣿⢁⢀⣿⠀⠀⠈⠃⠈⠁⠀⠀⠀⢻⣑⠛⠀⠀⢠⡿⣷⣶⣿⢿⣷⣾⡿⣴⠀⠀⠈⢓⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢹⡄⣿⢸⡇⡞⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⢸⣿⣿⣷⣿⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠷⡏⠈⡿⢿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡏⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣄⠇⠀⣇⡜⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠾⢿⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣻⠀⠀⢹⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠾⠃⠀⠙⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠋⠀⠀⠈⠃⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠙⠀⠀⠈⢻⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⣿⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⠀⠀⢠⣾⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿⠀⠀⢸⡇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⠀⠀⢸⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠇⠀⠀⠈⠃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠟⠀⠀⠐⠆⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡇⠀⠀⠀⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠶⠶⠿⣿⠻⡿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⢠⡀⠀⠀⠀⠀⣀⠀⠀⡀⠀⡀⢰⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣼⣷⣶⣴⣶⡞⠛⠀⢀⡇⢸⡇⢘⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⡤⠞⠋⣻⣿⠟⠉⠙⠻⠷⣾⣿⣧⣾⣥⣼⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⠀⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⢀⣴⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡇
⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠖⠛⠉⠀⣿⣿⠃
⠀⠀⣰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿
⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢃⣀⣀⣀⣀
⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⠿⠟⠛⠋⠉⠉⠛⠻⢶⣤⡀
⣿⣿⣿⣿⣦⣀⠀⠀⠀⠉⠉⠁⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣤⣤⣶⡶⠶⠶⠶⢶⣤⣄⣀
⠹⣿⣿⣏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣦⣀
⠀⠈⠻⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣿⣿⠿⢶⣤⡀
⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⠷⠶⠒⠀⠀⠀⠙⠷⣦⣤⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⠋⠁⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣦⣤⣀⣀⡀⠀⠀⠀⢀⣀⣠⣶⣄⡀⠀⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⡇⠀⠀⠀⠀⠹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⢀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣄⠀⠀⠀⠀⠀⠙⠿⣿⣦⣼⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣠⣤⣀⣀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⠟⠛⠃⣒⠆⠀⠀⠰⣶⣶⣶⣶⣶⣶⡦⣤⣤⣤⣶⣶⣶⣶⣦⣬⣉⡙⢛⡻⠷⣿⣿⣿⣿⣆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠠⡒⣩⣥⡶⠚⠂⠒⠦⣬⣕⡂⠀⣀⣈⣻⣿⣄⡀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣿⣷⣌⢶⠤⡉⠛⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠾⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠁⠙⠹⠉⠛⠳⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣄⠀⢀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⣿⣿⡀⢰⣿⣿⠍⢛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠃⠀⣿⠃⠀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡃⠀⠀⣿⣤⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠈⠀⠀⠀⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⣿⠘⣿⣿⣿⠱
⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣦⡀⠻⣿⡇⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⢻⣛⣭⣿⣭⣽⣍⠉⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣬⣿⢷⣿⣿⣦⠳⣶⣶⣶⢶⣆⣠⣶⣿⡿⠋⠁⠀⠖⣦⡀
⠀⣿⢛⣩⣼⣿⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢾⣿⡇⠈⢿⣟⣷⡗⠒⢛⣩⣾⡿⠋⠁⠀⣠⠆⠃⠐⣾⣷
⠀⠻⠛⢻⣿⣦⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⢸⣿⡓⠠⣾⣿⡿⣿⣽⣿⠟⠋⣀⣤⣠⡎⠀⠀⠀⢈⣿⣿
⠀⠀⠀⠀⣹⣷⡆⠀⠀⠘⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⢾⣿⣇⠠⣿⣿⡳⣿⡀⢠⣶⣿⣿⣿⣿⣆⡀⠀⢀⢾⣿⡏
⠀⠀⠶⠶⠿⠟⠃⢤⣶⣶⣶⣿⣿⣆⠀⢤⣀⣀⡠⠄⣠⣤⣤⣦⡀⢠⣿⣿⣿⣿⣿⣮⣿⡟⣿⣷⣾⣏⡛⣩⡿⠋⢙⣿⣦⣽⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⡋⠙⠋⠻⣧⡀⠀⠀⠀⠀⠀⠀⠉⠀⠁⢈⣿⣿⣿⡏⡿⣿⣿⡇⣿⣿⣿⣧⠿⠋⠀⣠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⣿⣿⢻⢱⣿⢿⡇⣽⣿⣴⡏⢀⣠⢾⣫⣿⡿⢻⢹⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⢀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢨⣼⠿⣧⠇⣿⡟⢸⣿⣿⣿⣯⡟⢿⣿⣿⣿⠏⠀⠀⢠⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣾⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣨⡟⢠⡿⡞⣿⡃⠀⠛⣿⣯⣿⣧⢹⡟⢿⡇⠀⢀⣤⣾⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢢⡙⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣀⣧⣿⣿⡜⣿⠁⣷⣴⣿⣿⣷⣼⡿⠀⠰⣧⣼⡿⠟⠋⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠱⡀⢨⡳⡝⣷⡄⠀⠀⠀⠀⣡⣿⣿⡿⠟⠋⣠⣤⠾⣿⣿⣭⣽⡇⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠱⠀⠰⡜⢦⠻⣄⠀⠀⠀⡇⢹⡇⠀⠀⠀⣉⣤⣾⣿⣯⣯⣼⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠁⠀⠈⠂⠀⠘⣧⠀⠀⣅⠸⡃⠀⠀⢶⣿⣿⣿⣍⣧⢼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣋⠀⠀⠀⠀⠀⠀⡀⢻⣧⠀⣮⠀⠃⠀⠰⠚⣿⣿⣿⡇⢿⢈⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠙⡀⠉⠒⢴⢰⡇⣀⣴⠿⡟⡿⢫⣇⣼⠺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢂⠀⠀⠀⠐⡇⣸⣿⡟⠁⠀⠀⣿⢸⣿⢾⡗⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣀⢰⣆⠀⠀⠀⠀⠀⢀⠀⢸⠘⢇⢀⣶⢸⡇⢸⣿⡆⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣿⣶⣿⡀⠀⠀⠀⠀⠈⠀⡆⠠⠈⢪⣿⣾⡇⠸⢿⣷⣸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡼⣿⣾⣿⣿⣉⠁⠀⠀⠀⠀⠀⠀⠘⠦⣰⣄⣹⣿⠆⢴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠿⢿⣿⣿⢿⡿⣄⠀⠀⠀⠀⠸⠀⠀⠀⣻⠟⣻⣿⣯⡉⠉⠙⠿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣰⡾⠋⠀⠀⠉⣿⣧⠀⠀⠀⢠⠀⠀⣰⢋⡶⠻⠛⠮⠙⢷⣤⣄⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠈⢻⣦⣤⠀⢀⠀⢠⣿⣮⡕⠀⠀⠀⠀⠀⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣇⢤⣠⣿⣿⠛⠂⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡕⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣾⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠠⡄⣵⣺⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢱⠀⢀⠀⠀⠀⠀⢀⡜⣿⣿⣯⣿⣿⣿⣿⣦⡀⣀⠀⠠⣄⣶⣽⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⣄⣦⡐⡀⣠⢏⣼⣿⢿⣿⣋⣾⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣿⣧⣾⣿⣵⣛⣷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠾⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠻⢉⢻⡏⢩⡟⢛⣟⠋⣽⡿⢫⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡀⢠⢦⠤⣀⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠
⠀⠀⠀⠠⢂⠠⣦⣝⣣⣀⢌⢳⡙⣟⢏⢮⠲⣤⣤⠶⣔⡲⢶⡒⢶⢰⣦⣤⣄⡀⠀⠀⠀⢨⣽⡁⠀⢀⣀⠀⡀
⠀⠀⠀⠉⢻⣙⣬⠙⡟⡏⣾⠇⣇⣽⢸⢮⠳⡝⢮⣑⣄⣝⢷⡙⣮⢿⠁⢧⣹⡝⢯⣹⡗⣨⣁⠸⠇⠘⠛⠚⢏⣀⡀
⠀⠀⠀⠶⣬⣿⣿⠉⠚⣱⣣⣘⠉⣫⣾⠋⣸⠟⣦⢻⢿⢻⢺⡃⢹⣨⣧⣚⢨⡿⣿⣇⠀⠙⠠⣽⠆⢄⣀⣀⣠⠛⠃
⠀⠀⠀⠀⠙⠀⠜⣽⢭⡑⡔⢽⣁⣻⡘⣦⢵⠋⣤⡻⡈⠜⠐⢁⣼⣭⠮⣁⣍⢘⣙⡟⢆⠀⢀⠀⠑⠛⠇⠨⠟⠃
⠀⠀⠀⠀⠀⠀⣼⠖⢂⠔⣋⣵⣭⠟⠋⠁⠒⢒⡟⢉⡇⠠⡖⠉⠀⢳⠄⠈⠙⠢⣭⣥⡏⠉⡛⠑⡀
⠀⠀⠀⠀⢀⡚⠓⢠⠧⠚⠉⠠⠀⠐⠀⠋⠉⠉⠉⠀⠀⠠⠚⠀⡠⠜⠁⠀⠀⠀⠀⠉⠉⠉⡇⠀⠰⡀
⢀⠀⠀⠀⠉⣀⡸⠁⠀⡀⠐⠀⠀⢰⠁⠀⠀⠀⠀⠤⠴⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣄⣄⠠
⠈⠒⠂⠂⠂⠁⠀⠀⠀⠛⠐⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⡿⠋⠀⠀⠀⠀⠀⢠⡖⢀⣴⡟⣁⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢸⡇⠀⠀⠀⠀⠁⢰⣿⣧⣿⣿⣿⣿⣿⣏⣠⣴⣶⣶⣟⣁⣀⡀⠀⠀⠀⣷⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⢧⡀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡁⢀⣰⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠉⠙⢉⣽⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⠉⢀⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡟⢁⠀⠀⠘⢿⣿⡏⢻⣿⣿⡟⠉⠉⠉⠉⠉⣙⡛⢿⣷⣾⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠠⠋⠀⠀⠀⠈⠻⣧⠀⠙⢿⣧⠀⠀⠀⠀⢀⠀⠙⣆⢻⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠁⠀⠰⣏⡈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠊⣁⡽⠆⠈⠈⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⣸⡏⠀⠀⡰⣿⡿⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣿⢿⢦⠀⠀⢻⣇⢨⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠟⢣⣠⣿⡇⠀⠐⠀⠁⠻⣄⣙⣱⣦⢶⣃⣶⠶⣴⣏⣃⣴⠏⠀⠀⠀⠀⢸⣿⣄⡝⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠉⠍⣿⣠⣤⣉⣡⣤⣬⡯⠍⠉⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⣆⡀⠀⠀⢄⣸⣿⣇⠀⢀⣠⡴⢶⡿⠲⣦⡭⢭⣿⡿⢭⣴⠶⢾⣿⠶⣤⣀⠀⠀⣿⣿⣇⡠⠀⠀⢀⣸⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⠈⢻⣿⡿⣄⢸⣿⣇⢸⡇⢀⡟⠀⠀⢹⠁⠀⢻⠀⠀⣧⢀⣼⣿⠃⣲⢿⣿⡏⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠻⣇⠸⠀⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠃⣾⠟⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡇⠀⣠⡞⠀⠀⣹⡄⠀⠸⣏⢻⣿⡹⢿⣿⣿⣿⣿⣿⡿⠟⢸⡏⢻⡿⠃⠀⣼⣏⡀⠈⢷⣄⠀⢰⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠀⢸⣇⠀⠀⣿⣿⡀⠀⠘⠦⣏⠛⣷⠛⠛⣿⠛⠛⣷⠀⢹⣦⠞⠁⠀⢠⣿⣿⠀⠀⣼⡇⠀⣸⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣶⣿⣿⣶⣾⣿⣿⣷⡄⠀⡀⢈⡙⠛⠦⠴⠿⠦⠤⠟⠚⣉⠀⢀⠜⣠⣿⣿⣿⣷⣶⣿⣿⣶⣿⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⡟⠛⠁⣹⣿⣿⣿⣦⡈⠢⠍⠛⠲⠶⠶⠶⠖⠚⠉⠠⠐⣡⣴⣿⣿⠿⡏⠘⠛⢻⣿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢰⡇⠀⠀⠙⠻⣿⣷⣦⣤⣀⣀⣀⣀⣀⣠⣤⣴⣾⡿⠛⠉⠀⠀⣹⡄⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡄⠀⢈⣿⡿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡇⠀⠀⢀⡴⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⣛⡛⣻⣿⣆⣀⣀⣈⡉⠉⠁⠀⠀⣸⣿⡟⢛⡛⠛⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠿⠿⠶⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠉⠙⠻⣿⣿⣷⣴⣿⣦⡿⠿⠷⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠑⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠣⠄⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠐⠦⢤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠖⠁⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠒⠒⠤⠤⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠒⠛⠋⠉⠉⠉⠉⢳⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠉⠒⠢⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠶⠒⠒⠲⠶⠦⠤⢤⡜⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠚⠉⠉⠒⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠦⠤⠤⠤⠤⠤⠤⠶⠶⠒⠚⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠁⠈⠉⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠀⡎⢀⡌⠀⠄⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠳⠼⣤⣌⢠⣆⡀⢀⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⢿⡇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣿⠀⠀⡴⠀⠀⠀⣿⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠀⣄⠀⠘⠆⠀⠙⠀⠸⠁⠀⡤⠀⣿⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⣤⣈⡙⠻⣷⣦⡈⠑⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠀⠀⠀⠐⡆⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠈⢻⣿⣶⠾⠟⠛⠛⠛⠻⠶⣤⡀⢸⡇⠀⠀⠀⠀⠀⠘⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣇⠀⠀⠀⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣄⡀⠀⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠙⠛⠻⠿⠟⠛⠋⣁⣤⣤⣤⣤⡤⠶⠖⠃⠀⠀⠀⢈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⡿⠛⠷⣦⣤⣄⣀⣀⣀⣀⣀⣤⣤⣶⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢾⠋⢁⣤⢀⣨⢏⠙⠛⠃⠶⢦⡀
⠀⠀⠀⠀⠀⠀⣠⣄⠛⣉⣡⡤⣍⠉⠛⠁⢀⣤⠁⠛⠲⣬⡅
⠀⠀⠀⢀⡠⠟⠈⣉⠀⠍⣁⡤⠄⠀⡀⢀⠀⠄⠀⣠⡶⠽⠗
⠀⠀⠀⠈⢧⣤⣤⠀⢰⣾⢋⡴⠛⠀⠀⢀⣶⣾⣦⠸⣁⠀⣠
⠀⠀⠀⣴⢿⣿⠁⢀⣿⣟⢸⣿⠋⡀⠀⣿⣋⡁⠉⣰⢉⣾⠟⠳⠄
⠀⠀⣸⣿⠀⣿⡀⠈⠻⠹⢿⣿⡰⣧⡀⠈⠁⢙⡄⡉⠌⠁⠀⠀⠆
⠀⢠⣿⡟⣧⢹⣟⠀⠀⢦⡀⠈⠛⢦⡕⠀⠠⢀⡁⠠⠦⠄⠠⠈
⠀⣾⢻⣧⠈⠳⠛⠷⠀⠀⠙⠷⣤⣠⡄⣴⡟⠁
⠀⣿⣄⠙⠳⣦⠀⠀⣠⠇⣠⠀⠀⠀⡼⣻⣦⡀
⠀⣼⠻⣦⣄⠀⣷⣾⡅⢀⡿⠀⠀⡜⠀⠁⠈⣩⡿⠷⠖⣠⡀
⠀⢸⣷⣄⢻⣟⠂⠀⠀⠘⠁⠀⠰⠀⠙⠆⠘⢫⡶⣻⡿⠟⣛⣲⣄
⠀⠸⣿⣏⠂⠻⢦⡀⠀⠀⠀⢰⠀⢀⡴⠂⣰⡿⣸⠏⣠⣾⣿⢋⣽⣷⣤⡀
⠀⣠⡏⢻⣷⡀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⣿⢃⠁⣼⡏⠸⢁⣾⢁⡄⠘⣿⣦⡀
⠀⣿⣤⣦⣬⡉⠛⠳⠄⠘⠀⠀⠀⠀⠀⠀⢻⡈⠀⢻⡇⢴⡈⣿⡟⠁⠙⣀⣿⣿⣦⡀
⠀⣿⣉⣀⣉⣿⣿⠛⠦⠀⠠⡀⠀⠀⠀⠀⠀⠙⠀⠘⠇⠸⣿⢹⡇⢀⠘⠙⢋⡽⠿⠛⢧⠄⣀⡀
⠀⢻⣿⢟⣛⣿⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⢤⡙⠃⠀⠙⠋⠐⠢⠤⠤⠄⠀⠀⠀⢈⠁⠘⢃⣤⡀
⠀⠀⣷⣿⣁⣉⠛⠛⠷⠄⠀⠐⠂⡄⠀⠀⠀⠈⠉⠛⠁⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠈⠓
⠀⠀⢸⡟⣋⣍⡉⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⣄⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠁⢀⠀⠀⠈
⠀⠀⠈⢿⣟⠉⠉⠁⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢱⡾⠟⣾⡿⣿⡄⣿⠉⣿⠀⠀⠓⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀
⠀⠀⠀⠀⢹⡿⣂⡀⠀⠑⠂⠀⢀⡀⠀⠡⣦⡀⠀⠀⠀⠀⠀⠀⠀⠚⢀⠾⠁⢠⣿⣇⢿⣧⠟⢠⣿⠟⣶⣿⡄⠀⠀⠀⠀⢠⡀⠀⠀⠃⡀⢠⡀
⠀⠀⠀⠀⠈⣿⢫⣴⣶⡚⠳⡄⠈⠓⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣿⠋⡼⠋⢠⡾⠋⢠⡿⣻⡆⠈⠑⠢⣄⡠⣥⣦⣤⡸⢧⣼⠃
⠀⠀⠀⠀⠀⠈⢻⣿⠋⣁⡀⠸⣆⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣤⠤⠚⠁⠀⠀⠀⣹⣇⡰⠋⣴⣫⡿⠦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⠁⢀⡹⡄⠀⠀⠑⠤⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⣉⠡⣴⡾⠿⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡄⠘⢦⠀⠀⠀⢹⣿⠷⠖⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠴⢛⢻⣶⣾⠿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⡀⠓⠀⠀⠀⢱⣶⠦⢤⡀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⣤⣴⣶⠾⢛⡛⠛⢀⣀⣀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣦⣀⡄⢀⡀⢳⣶⣤⡉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣏⠹⢡⡾⠋⣠⣾⠿⠛⠛⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⢫⢉⡴⠛⣻⣧⡉⢶⣦⣄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠸⠐⢦⣿⠇⣰⡟⢁⣴⠿⠛⠷⢶⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣴⣿⡀⠀⠀⠉⠀⠈⢻⡄⠀⠀⠀⢠⡀⠀⠀⢀⣀⠀⠀⠀⠀⡿⢠⣿⠀⡿⠁⠀⣴⢶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⡆⠀⠀⠀⡀⢀⣧⠀⠀⠀⠀⠹⠆⠐⠀⢉⠑⢦⣀⠀⠰⣆⠉⠳⣧⡟⢠⣤⣤⣌⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠀⠂⠀⢎⠀⠀⠀⠀⢁⡌⠀⠂⠠⠈⠂⠀⠚⠄⠀⠀⠀⠉⠉⠛⠢⣌⡳⢸⣿⢀⣿⠃⡀⠀⢹⣷⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⣄⣠⣄⣠⠞⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⢇⣾⠃⣾⠇⠀⣀⠙⠿⠇⣸⣿⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢀⠙⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠂⠤⠄⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⣿⣰⠞⣿⠀⢷⣶⣿⡿⠁⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣧⢀⡀⠐⠀⠛⠀⣄⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠼⣶⣇⡀⠈⠇⠈⠳⣦⣾⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠑⠛⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡴⠾⢦⡀⠈⡏⠉⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠀⠘⣿⣿⣿⠋⠉⠀⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⢻⣛⡁⢸⣿⣀⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⠀⠀⢀⣿⣿⣿⣀⠀⣠⣿⣿⣿⠀⠀⢹⣿⣶⠖⠶⣾⣟⣴⠆⠙⠟⠀⠙⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠘⢿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠂⢴⣿⣿⡟⠀⢠⣿⣿⡁⣠⠀⠀⢷⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠛⠛⠛⢻⣥⠶⠂⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠁⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠙⠋⠉⢳⣦⡄⠐⠚⢿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣴⣿⣿⠇⢀⣀⡼⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⡠⠿⣀⣤⠺⠇⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⠄⠀⡆⢠⠀⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣀⣀⣰⠧⣤⣊⣡⣊⣐⣣⠞
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡼⠛⠳⠶⠴⢧⣠⣤⠼⠃
⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠿⠷⢶⣦⣄⠀⠀⢀⣠⡴⠆
⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠘⠁⢿⣤⡴⠶⠶⠲⠶⢦⣤⣀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠈⠙⠻⣾⠟⠉
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠈⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠛⢦⡀
⠀⠈⠷⣤⣤⣤⣤⣤⠤⠤⠄⠈⠙⠛⠒⠶⠾⠛⠛⠻⣦⡀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠉⠙⠛⠶⢶⣤⣤⣀⣀⣀⡀⣀⣀⣀⣤⣬⡿⠷⣦⣬⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠁⢸⣧⠀⠀⠉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠰⣄⠀⠀⣜⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡈⠻⢦⣼⣆⠙⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡐⠦⣽⣷⣄⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣿⠿⠿⠷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡾⠀⣀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠙⡁⠀⠈⠉⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠂⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣤⣄⣀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡄⠈⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⠁⠀⠀⠀⠈⠀⠂⠱⣸⠃⠻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠢⠀⠽⡀⠀⠀⠿⠢⣄⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⠳⣄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⠀⢢⣤⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠈⣦⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠈⠓⠄⠉⠉⢹⣧⡀⠠⠄⠀⠀⠀⢀⣰⠿⢍⣀⡀⠀⢠⣼⡏⢻⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣿⠻⢿⣿⣿⠿⠛⢋⠉⠀⢠⡤⠈⣉⣿⣿⣿⡇⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠎⠀⠀⠀⠁⢀⣾⣧⡀⢶⣽⣄⡀⠈⢀⡀⠀⢀⡄⢀⣨⢿⡿⣿⣇⠀⠀⢂⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢶⡶⠟⠈⠛⢿⣿⣿⠉⠛⣦⠀⡉⠉⠁⠀⠉⠋⢺⠇⠈⠻⣷⣶⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠹⡄⠀⠠⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⡁⠛⠒⠐⠃⠚⡑⢿⠀⠀⠀⢸⣇⠀⠀⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢐⡆⠀⠀⠀⠀⠀⡿⢦⡀⠀⢇⠀⠀⠠⠀⠀⡇⢸⠀⠀⠀⢸⡇⠦⠀⢀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠳⡀⢰⠎⡀⠀⠀⠀⠀⢸⡟⠀⠿⡅⠀⠁⠀⠀⠀⣰⣃⠨⡆⠀⠀⠸⣇⠀⠁⡜⢱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠑⣌⢰⠃⠀⠀⠀⢀⣾⠖⠀⠀⠄⣶⠄⠀⠀⠀⢩⣱⡆⠵⡀⠀⠀⠙⣇⠀⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⠀⠀⠀⠀⠀⢸⣿⠁⠀⠉⣚⠈⣷⡀⠀⠀⠟⠁⡈⢔⣇⠀⠀⠀⠈⢃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⣼⣯⡂⠀⠀⠉⢀⢏⠳⣄⠀⠀⣠⠁⠀⣸⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠤⠢⢷⡘⣠⠞⠋⠀⠀⠻⡇⠀⢀⠠⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠬⢿⠀⠀⠀⠀⠀⠀⠀⢀⡟⠠⠴⠒⠀⠀⠀⣧⠀⠀⣤⣤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠖⠀⠘⠀⠀⠀⠀⠀⠀⠰⠻⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⣷⡟⢳⡶⠴⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢳⡟⢃⣿⠷⠀⣿⣷⡌⠉⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⣿⠁⢸⣿⠀⢸⡿⠉⠃⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣼⣿⠀⣿⣿⠀⢸⡇⠀⠀⢳⣤⣄⠀⠀⠀⠀⢻⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⣿⡀⣿⣿⠀⣸⣧⠀⠀⣾⣿⡉⠉⢠⣄⣤⠞⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡻⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣶⡔⠋⠁⢀⣀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⠏⠉⠙⠛⠛⠛⠛⠛⠛⠁⠈⠻⣆⠀⠀⠙⢿⣦⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠙⢿⣷⡀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢸⣧⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣦⣤⣄⣸⣿⣧⡀⢻⣷⣄⠀⠀⠈⠛⢶⡄⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠁⠈⠉⠉⠛⠻⢿⣿⡿⣿⣷⣿⣿⣷⣤⡀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣬⡉⠙⠛⠻⠿⣷⣤⣄⣀⠀⠀⠀⢀⣀⡀⢘
⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣾⡿⢣⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠊⠙⠿⣷⣤⡀⠀⠀⠈⠉⠛⠿⠷⠀⠀⠉⣴⡿
⠀⣀⣤⠶⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⣿⡿⢋⣴⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⣀⣀⣀⣀⣠⣼⠏
⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠂⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⣿⣿⡏⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⣫⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣼⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠁⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⡡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠃⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠙⠲⠶⠄⠢⣰⠰⠞⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⣀⠀⠀⠀⢄⠀⠀⠠⠄⠉⠀⢦⡄⠀⠀⠙⠀⠀⣄⠀⣤⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⠉⠲⣄⠀⣀⣀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⡴⠂⠁⠀⠀⢄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣠⠅⠀⠒⢠⢀⣀⣀⠘⠛⢧⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢁⠟⠛⢀⣀⣀⠠⠐⠂⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠐⣫⣤⠶⠄⠀⠀⠈⠳⣌⡆⠈⠀⠀⠀⡈⢀⣔⡴⠀⠀⠴⠶⣤⡙⠀⠀⠀⠀⢴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⠜⠁⠀⠀⠀⠠⣀⠈⠳⢬⣧⠀⠀⠀⣀⠠⠚⡉⢀⡠⠄⠀⠀⠀⠙⢆⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⣌⠲⠄⡀⠀⣈⣀⠴⢪⡴⠋⠀⠀⠀⠀⠀⠀⠈⡆⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⡆⠀⠀⠀⠀⠀⠀⠀⠙⢟⠀⠀⠀⠉⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⣾⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣶⣤⡀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢰⡟⢛⣿⣦⠀⠀⣷⠁⠀⢰⠈⢦⣰⡆⠀⢀⣾⣿⠛⣷⡤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠀⠸⠿⠋⣧⣼⡛⠳⠶⠾⢦⡤⠚⣻⣷⡿⠺⠿⠄⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⠤⠄⢸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠂⠤⠤⠤⠤⠆⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⢶⣄⡀⠀⠈⠳⣄⠢⡀⠀⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣷⠖⢠⠀⣴⠛⠁⠀⣀⣤⡞⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠑⠂⠤⣽⣆⣵⣄⠟⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⢤⣈⢿⣶⣅⣼⡥⠴⠒⠉⠁⠉⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⢀⠀⠁⢀⠟⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣈⡙⢣⡀⠁⠀⠀⠀⠀⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢄⠘⢲⠖⠋⠀⣀⣀⠤⠂⠀⠀⠀⠀⠈⠉⠒⢄⣈⠙⠛⠓⣾⠚⢀⠄⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢾⠀⠀⠀⠀⢠⡴⠶⠀⠀⠀⠀⠀⠀⠗⢦⡀⠀⠀⠀⢸⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢧⡀⠀⠛⠃⢈⣀⣀⣀⣀⣀⣀⠀⠛⠃⠀⣠⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡟⠛⠛⠻⠿⠛⠋⠙⠛⡻⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠰⠐⠀⠀⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘

⠀⠀⢀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⣀⡀⡠⠀⠀⠀⢀⠔⠁⢠⠊⠀⠰
⠀⠀⠀⠀⠀⢰⢀⣀⣀⣀⠤⣄⣀⠀⠀⠀⠰⠀⠐⣄⠤⠐⠂⠀⠈⠁⠀⠀⠰⡋⠀⠀⠒⠒⠲⠖⠒⠊
⠀⠀⢀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠋⠀⠀⠄⣤⡀⠀⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠉⠒⠒⠤⠤⠴⠢⡄⠀⠀⠀⠀⠀⢇⠀⣀⣀⣀⠤⠚⡄⠡⡀⠀⠀⠀⠀⠀⠀⠉⠒⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠄⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀
⠀⠀⠀⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣰⣃⣀⡀⣀⠠⠄⠀⠀⠀⠁⠀⠀⠀⣀⡀
⠀⠀⠀⡆⠀⠐⠋⠉⠉⠙⠁⠉⠁⠀⠀⡼⠁⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠙⠉⠉⠃
⠀⠀⠀⠈⠒⠤⣀⡀⠀⣨⠓⡄⠀⠀⠀⠈⠒⠤⢀⣀⣀⡰⡟⠄⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡄⢸⢱⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢤⠀⡀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⣴⣷⡀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠻⢿⡶⠀⣀⣴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂
⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠺⠀⢹⣿⠿⠑⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣱⣄
⠀⠀⠻⣿⣦⣄⣀⣀⡄⠀⠀⠀⠢⣄⣀⠀⡀⠀⠀⣀⣾⢻⡄⠓⠄⠀⠀⠀⠠⡔⠀⠀⠀⠀⠀⡉⠛⠳⠄
⠀⠀⠀⠈⣿⡏⠉⡉⠘⢧⣀⠀⠈⠁⢀⠀⠀⠀⢠⣿⡿⠈⣷⠀⠀⠀⠰⣤⣴⡇⠀⠀⠀⠀⣼⡷
⠀⠀⠀⠀⢹⣿⣴⣽⣤⡀⢿⣀⡀⠀⣼⠀⡔⢀⣾⣿⠇⠀⢻⡷⣄⠀⠀⠈⢻⡇⠀⣠⣴⣾⡿⠁⢀⡄
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣌⣻⣧⣰⣿⣾⣾⣿⣿⠟⠀⠀⠘⣷⡽⣿⣗⣀⣠⠓⠛⠋⣭⠗⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠘⠻⢮⣍⠉⠁⠀⣴⡈⣡⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠛⠉⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠡⠀⠀⠀⠀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⢄⠀⠸⣆⠸⣆⢱⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠰⡶⠀⢞⡙⢀⡏⢠⡇⣰⠯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⣤⡴⠦⣤⣈⣛⣫⣿⡿⠟⠃⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⡴⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⠞⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣀⣤⣴⠶⠶⠶⣦⣤⣀⠀⠀⢀⣠⡴⠶⠛⠉⠉⠉⠉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡿⠟⠉⠐⠂⠙⠓⠒⠒⠉⠛⠶⣟⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣼⠃⡀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⢿⣆⠙⠂⠀⠀⠀⠠⣤⣴⡶⠟⠁⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢶⣦⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠦⠤⠀⠀⠀⠀⠀⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⢯⣍⡉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣭⣭⣭⣭⣭⣭⣴⣶⣾⠿⠿⠛⠛⠋⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡄⠀⠀⠀⠸⠿⠿⠿⠟⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⢀⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠟⠉⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀⡀⠈⠙⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆
⠀⠀⢀⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠁⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠁⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣆
⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣶⠄⠀⣠⣾⣿⣿⣿⠿⠛⠉⠉⠀⠀⠀⠈⠉⠙⠻⢿⣿⣿⣿⣦⡀⠀⢴⣾⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⠏⠀⣴⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡄⠈⢿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣆⠀⠻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠃⢀⣤⣄
⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣦⠀⠙⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡟⠁⢠⣾⣿⣿⣿⣶⣤⡀
⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠈⠻⢿⣿⣿⣿⣶⣤⣤⣀⣀⣀⣠⣤⣴⣾⣿⣿⣿⠿⠃⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠈⢻⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⢀⣄⡀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⣀⣄⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⡤⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⢠⣴⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠁⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⣻⡟⢳⡖⠲⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢱⡟⢃⣿⠗⢀⣿⣶⠈⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣿⠇⢸⣿⠀⢸⡏⠈⢁⠀⠀⠀⠀⠀⠈⣓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⣸⣿⠀⢸⡇⠀⠀⣷⣶⣄⡄⠀⠀⢸⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣿⣿⡀⣼⣧⣠⣴⣿⣧⣀⠀⣲⠶⠋⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣻⣿⡿⠿⣿⣿⣿⣿⣿⣿⡿⠟⢿⡄⠀⠐⢶⣤⡀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⢿⣦⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠙⢿⣦⡀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠻⣄⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢻⣆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⣤⣸⣿⣦⡘⢿⣧⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠉⠁⠀⠀⠀⠈⠉⠛⠿⣿⡿⠿⣿⣿⣿⣷⣄⡀⠀⠙⠆⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⢷⣦⣈⠉⠙⠛⠻⠷⣦⣤⣀⡀⠀⢠⡤⢀
⠀⠀⠀⣀⣠⣤⣴⣶⡶⠶⠶⠶⢶⣶⣾⡿⢋⣼⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠉⠻⠿⣦⡀⠀⠀⠀⠉⠛⠋⠀⠀⣰⡿
⣠⡴⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠋⣠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣶⣶⣾⠟
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⣻⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⣿⣿⣿⣧⣤⣤⣤⡴⠶⠾⠛⠉⠁⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⡿⠟⣩⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠉⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⢁⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢿⡟⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣤⣤⣀⣀⣀⣠⣤⣤⣴⣶⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠁⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⢀⡀⢀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣄⠀⠈⠻⠟⢁⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⠋⢀⣼⠀⠀⠀⠟⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⢲⣮⡏⠀⣰⠆⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠉⠁⣸⠇
⠀⠀⢉⣽⠦⢂⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⡿⢦⡀⠀⠀⠐⢀⣄
⠀⠀⠛⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢍⣠⡿⠀⠀⠀⠀⠀⠀⠀⢠
⣄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠳⣦⡶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢹⣶⣾⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⠂⠑⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠃⣀⡄
⠈⠉⢹⡷⠀⠀⠀⠀⠀⠀⠀⡸⠀⣴⢇⣠⡄⠀⣀⣴⣷⡄⠀⡄⠂⠀⠀⠀⠀⠀⠸⠟⠀⣰⠛⢡⣤⠀⠀⠀⠀⣦
⠀⠀⠈⢠⡄⠀⠀⠀⠀⠀⢸⡇⠀⢀⣸⡦⠰⣤⣀⠀⠀⠇⠀⠀⡤⠀⠀⠀⠀⠀⠴⠀⠴⠀⢰⠞⢁⡷⠄⣀⣰⣿
⠀⣾⠀⠘⠶⢀⠀⠀⠀⠀⠾⢀⡶⠋⣷⣦⣄⠈⠙⠻⢶⣶⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠁⠐⠋⣤⡾⠉⢀⡏
⠀⠉⢀⠀⠀⠘⢣⡀⢀⠛⢠⡝⠀⠀⢿⣿⡿⠷⠐⢤⠾⠟⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢸⣿⠃⠀⠸⠁
⠀⠀⠿⠃⡀⠀⠈⠳⠀⠀⢈⠁⠀⠰⠟⢻⠗⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡄⠀⠀⠀⠀⢈⠁⠀⠀⣨⡿
⠀⠀⠰⠟⢡⣄⠐⢦⠄⠀⣠⣶⣄⠂⠀⠘⣷⡄⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣤⢰⣷⡄⠈
⠀⠀⠀⠀⠀⠻⣷⠀⠀⠖⢉⠑⠁⠀⠀⠀⠈⣁⠶⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠘⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡶⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠂⠀⠀⠈⣷⠀⠠⡶⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢶⡖⠀⠀⠀⠀⠃⣤⠀⠀⠰⠾⠁⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠐⠀⡀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⠰⢃⡀⠀⠀⠀⠀⠀⣤⡉⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠃⠀⠀⠊⢉⡤⠀⢀⣴⡆⢩⡟⠃⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠈⠳⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⡅⠘⠃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣼⠂⢀⢀⣴⢿⣿⢦⠄⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣈⡁⢠⣟⠻⣿⣸⡿⠶⠦⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠⣶⣿⠃⠸⠃⠀⠈⣿⣧⣤⠀⠀⠈⣧⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣼⠁⠀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⢀⣴⠃⣸⡏⠀⠀⠀⠀⠀⡿⢡⣿⠆⠀⢸⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⢿⠀⠀⠀⠀⠀⠀⡗⠀⢹⡇⠀⠈⣿⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠀⠀⡇⢀⠀⠀⠀⠀⠘⠃⠀⠀⠏⠀⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⠀⠈⠀⠀⠀⠀⢀⣧⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⡘⠀⠀⣿⠀⠀⠀⠀⢀⡀⢸⣿⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠁⠀⠰⣿⡇⠀⠀⠀⢸⡧⣾⡇⠀⢀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠁⣶⡆⠀⠟⠀⠀⠀⠀⠘⡧⠞⠁⠀⠉⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⣤⠀⠀⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣇⠀⠀⠀⠀⠀⠈⠀⠈⠀⡟⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠁⡀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢁⡿⠁⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣤⡼⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠖⠀⣦⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⣤⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣤⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⢶⡄⠀⠠⠈⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢄⡀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⢀⡠⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡲⢄⠀⠀⢀⣀⣬⣥⣀⡀⠀⠀⡠⢖⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢊⡴⢞⣭⠶⠖⠲⠶⢭⡳⢦⡑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣵⠋⠀⠀⠀⠀⠀⠀⠙⣮⢳⡀
⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⠤⠤⡄⢸⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⡇⢠⠤⠤⣀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠉⠛⠁⢸⡸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡇⠈⠛⠉⠉⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡹⣄⡀⠀⠀⠀⠀⢀⣰⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠙⠦⣝⣓⡶⢶⣒⣫⠴⠋⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠔⠋⠀⠀⠀⠉⣉⣉⠉⠀⠀⠀⠙⠢⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠈⠀⠀⠀⠀⠀⠀⠀⢇⡼⠀⠀⠀⠀⠀⠀⠀⠁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠻⣿⣿⣧⡀
⠀⠀⠀⢠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄
⠀⠀⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦
⠀⢠⣿⣿⣿⣿⣿⣷⣤⣶⣶⣦⣴⣿⣿⣿⣿⣿⣿⣦⡀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣤⣶⣶⣦⣸⣿⣿⣿⣿⣿⡆
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣍⣿⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠶⣿⠀⣸⠷⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣿⢿⣝⣃⠈⣴⡿⢿⣿⡏⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠃⣸⡆⢸⡆⠛⢻⣏⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣽⡿⣿⣴⣮⣿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣶⣶⣇⣷⣾⣿⣶⣾⣟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⣿⣿⡹⠟⢳⠛⢿⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⣿⡟⢻⡇⣸⠛⣿⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢸⠏⣿⣿⠀⢹⠂⢺⣿⠋⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢸⠃⣿⣿⠞⠙⠙⢾⣿⡇⢻⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡼⠀⣿⢽⡉⠸⠁⣙⢽⠀⢰⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡾⠀⣿⣿⠀⠀⠀⢸⣿⡇⠘⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠃⠀⣿⡿⡁⠉⠀⠁⢿⡇⠀⢻⣍⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠁⠀⣻⣧⣄⠀⢀⣠⣽⡇⠀⠹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠁⠀⢸⣿⣄⢳⣴⣶⢀⣾⣇⠀⠀⠻⠧⠀⠀⠀⠀⠀⠀⢸⢻⠁⠀⢠⣿⡟⠻⣷⡿⠛⢿⣧⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⡦⣦⡄⠀⠀⣾⠋⢻⣿⠉⣻⡿⠋⢿⡀⠀⠀⡄⠐⠆⠀⠀⠀⣴⢆⡄⠀⠀⢈⣿⣄⣠⣿⣷⣀⣼⡏⠀⠀⠀⣦⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⡄⢻⡇⣿⢁⢠⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⢾⣼⣿⣿⡏⣿⣿⣯⣸⡇⠀⠀⠀⠚⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣿⣸⠃⣿⣾⣄⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡟⣿⣿⡇⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠹⠀⢸⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠹⡇⠀⢸⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠸⠃⠀⠀⠇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠄⠀⠀⠸⣿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡄⠀⠀⣦⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⡄⠀⠀⣠⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⢹⡇⠀⠀⣿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⣷⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠁⠀⠀⠏⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢈⠃⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠶⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⠀⠀⠀⢻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⢻⡟⠚⠓⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣭⠀⣴⣿⣾⣿⣷⣶⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣲⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠼⣁⢈⠿⠿⢿⣿⣿⣿⡿⠋⠀⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠏⠁⠀⠀⠀⠀⠘⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣊⣤⣶⣾⣿⣿⣿⣿⣥⠈⠉⠒⠚⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⣫⣴⣿⣿⠿⣻⡿⠛⠋⠉⠉⠉⠛⠢⠀⠀⠀⠈⠐⣶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢥⣾⣿⡶⠛⠁⠞⠁⠄⣒⠒⠒⠂⣀⣀⢀⠀⠀⠀⠀⠀⠈⠃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⣾⣿⣟⠁⠀⠀⣠⡴⠟⣛⣻⣶⣜⢿⣿⢎⠳⡷⠀⠰⠶⣷⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡀⢸⣿⣿⠃⠀⣠⣿⠋⣰⠟⠉⠉⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠈⣿⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣇⠈⠐⣶⣆⣰⠃⢡⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢷⣿⣿⣿⠋⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠈⠻⠻⠷⣾⡷⠀⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣿⣷⣄⣷⣦⣾⣟⠁⠀⠀⠈⣀⣀⣀⣉⠙⠛⠃⠀⠀⠀⠴⣛⣉⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣮⣿⣿⡇⣙⢿⣷⣪⠂⠀⠀⠙⠻⣟⣿⠏⠳⠄⠀⠀⠀⡴⠻⡟⠉⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⡿⡟⠱⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣅⢻⣄⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⣿⣿⣷⣦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⡠⠞⠙⣿⣿⣿⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠶⠾⡦⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠩⠷⠶⠖⠀⡰⠁
⠀⠀⠀⠀⠀⠀⣸⣿⡄⢀⡴⠞⠀⢰⣯⣾⣿⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠘⣿⣏⡀⣨⣿⠁⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣻⣿⠃⢀⣨⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡿⢷⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⠟⡀⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠛⠋⡀⠀⠀⠁⠘⠢⠤⣀⠀⠈⠂⠀⠈⠻⠮⡳⣌⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠘⠻⠧⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠑⠀⠀⠈⠪⡓⡙⠛⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠡⠀⠀⠀⢀⠀⠀⠀⠀⠀⣒⣶⣦⣀⣄⣤⣴⡴⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⡇⠀⠀⠀⣤⡄⢠⡀⣤⣶⣿⣿⠿⠿⣿⣾⣿⣿⣿⣦⡄⡀⣨⡄⠀⠀⠀⢸⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠸⡇⠀⠸⣄⣿⣿⣾⣿⣿⣛⠛⠛⠃⠀⠀⠈⠉⠉⠉⠛⢿⣷⣿⣷⣠⠄⠀⣸⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠛⠶⠶⠿⠿⠟⠛⠛⠋⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⡿⡿⢶⠖⠛⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⣀⡄⠀⠀⣴⠋⠉⠑⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠒⠛⢷⡄⠀⢀⣀⣤⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⢊⣵⣿⣷⣄⠀⠀⠲⣦⣤⣰⣶⡶⠚⠀⢀⣴⣾⣶⣝⢷⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⣶⣿⠋⠀⣿⣿⡟⠉⠻⣷⣄⣀⣷⣿⣿⣿⣿⣥⣴⡾⠋⠙⣿⣿⡇⠙⣿⣶⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢻⣿⡟⠋⠁⠀⢿⣿⣇⠐⠦⠀⣹⣿⡿⠿⠿⠿⣿⣿⡁⠠⠖⢀⣿⣿⠇⠀⠉⢻⣿⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡟⠁⠀⠀⠀⠀⠉⠙⠳⠶⣾⡿⢗⣛⣥⣬⣽⣻⣿⣿⡶⠶⠟⠋⠁⠀⠀⠀⠀⢻⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣄⠀⠀⢀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⠀⠀⢠⣿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡄⢠⣿⣿⣿⠉⠉⠹⠉⠙⠛⠛⠉⠛⡿⠋⠉⠛⠛⠉⠹⠋⠉⢹⣿⣿⣧⠀⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠟⣿⣿⣸⣟⢿⣿⣆⢠⣾⣧⣼⣷⣦⣤⣤⣧⣤⣤⣶⣿⣤⣿⣦⣀⣼⣿⢻⣿⣼⣿⠛⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡌⢿⣿⡿⠁⢿⣿⣿⣿⡿⠿⣿⢿⣿⣿⣿⣿⡇⠘⣿⣿⠏⣾⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋⣿⣷⡸⣟⢧⡀⠈⡟⠀⣃⠀⠀⡇⠀⠀⡇⠈⣏⠀⢀⢿⡏⣼⣿⠙⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⣷⣌⡓⠮⠍⠛⠚⠛⠓⠛⠛⠓⠛⠛⠚⠋⠭⣖⣫⣾⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢸⣿⡿⣿⣷⣶⠀⠀⠐⠈⠛⠛⠧⠀⠀⠀⢲⣿⣿⢿⣿⡏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠙⢿⣿⡿⣶⣶⣆⡀⢀⣤⣶⣶⢾⣿⡿⠃⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠻⠌⠳⠈⢻⣿⠋⠞⠡⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣤⣄⡀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⠋⣠⡾⠋⠀⣠⡾⠛⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢸⣿⡾⠋⠀⣠⡾⠋⠀⣠⡾⠋⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠘⣿⠀⣠⡾⠋⠀⣠⡾⠋⠀⣠⡾⢻⣇
⠀⠀⠀⠀⠀⢀⣤⣤⣠⣠⡀⠀⢸⡏⠀⠀⠀⢻⣿⠋⠀⣠⡾⠋⠀⣠⡾⠋⠀⣨⣿⠀⠀⣠⣠⣀⣀⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⢸⣧⠀⠀⠀⠀⠻⣦⣾⠋⠀⣠⡾⠋⠀⣠⡾⢻⣿⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⠻⢶⣾⣏⣀⣠⣾⣋⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢋⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⠿⣦⣄⣀⠀⠀⠀⢀⣀⣤⡾⠟⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⣀⠒⠁⠀⢉⠑⠄
⠀⠀⠀⡠⠌⠀⠀⠀⠀⠀⠈⢢⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠶⠶⠤⠤⡀
⠀⢀⠊⠀⠀⠀⠀⠀⠐⡀⠀⠀⢦⠠⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁
⠀⠸⠀⠀⠀⠀⠢⡀⠀⠈⢆⠀⢼⠀⠆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⡇⠀⠀⢀⠀⠀⠈⢢⠀⣰⠠⠊⢁⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⢰⠀⠀⠀⠀⡕⠢⠀⠘⠊⠁⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⡇⠀⠀⠀⠀⠀⠀⡄⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⣀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⣀
⠐⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣀
⠀⠡⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢣⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡼
⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢠⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⣀⠀⠀⠀⠀⣀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⢀⣠⣴⣾⣿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⡿⠿⠶⣤⣀
⠀⠐⠚⠉⠉⠉⣿⠛⠿⠿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⣤⡄⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⡿⠿⠛⠋⠉⢹⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⣿⡿⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣾⣿⣿⣿⣿⣿⣟⠋⠁⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠘⠧⠴⠿⠋⠀⠈⠉⠙⢻⡏⢿⡋⠉⠹⣇⣀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⢻⡏⠉⠉⠀⠀⠀⠀⢸⡏⠉⠉⠙⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠛⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡍⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⣤⣤⠿⠤⠴⠶⣶⠶⠶⠾⠷⠶⠒⠒⠒⣶⠟⠛⠛⠛⠋⠉⠉⣹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢰⣿⠹⣦⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⣀⣀⣀⣀⣀⣿⣄⣀⣀⣀⣀⠀⢰⡿
⠀⠀⠀⠀⣆⠀⠘⣿⠀⠈⢷⣄⣀⣤⠴⠶⠞⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠲⠦⠤
⠀⠀⠀⠀⢻⡄⠀⠙⠳⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠙⠶⢤⣤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⢀⣀⣠⣄⣀⡀⠀⠀⢠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣴⣾⡿⠿⠛⠛⠿⢿⣷⣦⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⡀⠀⠀⠀⠀⢀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠙⠿⣿⣶⣶⣶⣶⣿⠿⠋⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠘⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡼⠋⠐⠚⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠘⢷⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢷⣤⣠⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠷⠒⠒⠢⠴⠂⠉⠉⠙⠛⠳⠄⠀⠀⠀⠸⠖⠊⠉⠉⠐⠦⠖⠒⠒⢾⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣀⣀⠀⠁⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣤⣤⡀⠈⠁⢀⣀⣼⢿⡄⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⣠⣯⣀⠀⠈⠉⠀⣠⣤⠴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠘⣇⢀⡜⠲⠶⣶⣾⡁⠀⣹⣶⣶⠶⠚⢁⢀⡟⠉⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣆⠞⠉⠀⠀⠀⠀⠀⠙⢢⢢⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠸⠶⠀⠀⠀⠀⠐⠾⢆⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⣦⡤⠀⠀⠀⢤⣦⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⣩⡀⠀⡀⠀⢀⣁⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢹⣿⣿⣶⣷⣾⣿⢿⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡀⠀⢸⡿⣿⣿⠿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠃⠀⢸⠃⠘⡏⠀⣿⠀⠀⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠈⠀⠀⡇⠀⠘⠀⠀⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣇⠀⠀⢀⡼⢳⡀⠀⠀⢀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠹⡇⠀⢸⠁⠀⢿⠀⠀⠿⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡿⠀⠀⢸⡀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡤⢀⡿⠀⠀⢸⡇⢠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠼⣧⠀⠀⢸⡿⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢹⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠸⠀⠀⢹⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣤⠁⠀⠀⠀⢣⡄⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠃⠀⠀⠀⠀⠈⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠟⠋⠉⠉⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣃⠠⠤⠠⠤⠀⣤⠀⢈⠠
⠀⠀⠀⠀⢀⡀⢂⣾⣷⣦⣈⡙⢆⠐⡀⠀⢰⠿⢤⡀⠀⠀⢲⡤
⠀⠀⠀⡀⠀⠐⠋⢩⣿⣿⡏⠀⠈⣀⣉⣠⡎⠀⢀⡴⣤⣄⡀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋
⠀⠀⢺⠁⡠⠴⣾⢿⣿⠁⠻⡄⠀⢈⣿⡏⠀⠀⠞⠂⣸⣿⣆⠀⠘⠇⠀⠀⠀⠀⢰⣧⣤⠲⠃⠂⡀⠀⠀⣼⡟⠀⠓⢤⠀⠀⣦⢀⣠⡀⠀⡀
⠀⠀⠘⠢⢄⠈⠁⠰⣿⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢠⡏⠙⣿⣿⡗⠀⠀⠀⠀⢠⣿⠹⠋⠀⣀⡄⠂⠀⠀⠸⣿⡄⢸⡀⠀⠘⠁⠀⠁⠙⠶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠰⡈⡏⡐⠀⠀⠀⢸⠁⠐⠊⠁⠀⠀⠀⠀⠀⣸⠟⣀⠀⢀⠀⠀⠀⠀⢠⣾⠉⠁⠀⠀⠀⠀⠀⠘⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢳⠞⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠃⠀⠈⠄⠀⣠⠀⠸⠀⡖⠀⠀⠀⠀⠀⠀⠀⠤⠂⠘⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠘⢻⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠘⠃⡇⠀⠀⠀⠸⡦⢿⠀⡖⠀⠀⠀⠀⠸⠀⠀⠀⠀⠂⠀⢠⠀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠐⠀⠀⢀⣇⠀⣠⠀⢀⠠⣄⠖⠓⠁⡀⢀⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠐⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠁⠀⠇⡁⠀⠀⠀⢸⣿⣀⠁⠀⠂⠱⠹⠘⠃⢔⣥⠁
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢼⡃⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠁⢺⣟⣋⠃⠀⢀
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢠⣿⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠔⠈⡀⠀⠀⠀⣼⣿⣿⣧⣀⣀⣤⡆⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠺⣇⠀⠀⠀⠀⢀⣾⠻⠿⢷⣀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠃⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠃
⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣘⣉⣉⣶⣐⣊⣉⣉⣀⣉⣻⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣀⣀⣘⣿⣿⣿⣿⣿⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠙⠛⠛⢉⣿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣽⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣯⣠⠴⠛⡾⣶⣶⡄⠀⠀⠀⢀⣴⡾⣿⣻⣿⣿⣿⣿⣿⣿⣾⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⡿⢷⣬⠙⣷⠀⠀⣴⣿⣿⣶⢯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡛⡛⠛⠀⠀⠀⠀⢹⣿⣿⣿⣈⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠂⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣠⣴⡾⠿⠿⠿⠿⢷⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠙⠛⠛⠛⠛⠛⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⣠⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⠙⢿⣿⣿⣿⣿⣿⡿⠟⡅⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠤⠟⠸⡿⠃⠈⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣤⢀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢠⣸⣿⣿⣿⣿⢋⡁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣿⣼⡇⣦⠀⠀⠙⠋⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡟⢿⣿⣿⣿⣿⡇⠘⠇⠲⢤⠀⢾⡀⠀⠀⠀⡘⠀⠀⢸⣿⣿⢿⣿⡴⡖⣢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⡉⠇⣿⢿⣿⣿⣿⡇⠀⠀⠀⠈⠀⠈⠀⠀⠀⠐⠋⠀⠀⣸⠟⢿⡇⢹⠃⠃⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡇⢸⣿⢾⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢈⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠻⢻⠸⡏⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⢀⢸⠀⢣⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠈⢸⠀⠀⣿⢿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠿⢸⠇⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢰⢸⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠐⣿⠂⠀⢰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⢸⠓⡸⢿⣿⡟⣿⣧⠀⠀⠀⠀⢠⡏⠃⠀⠀⠀⠀⠀⢠⣿⠀⠀⣼⡇⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡇⡀⢳⢸⣿⡇⣿⣿⠀⠀⠀⠀⢼⡇⠸⡀⠀⠀⠀⠀⢸⣯⠀⠀⣽⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢈⢈⣿⣷⣾⣿⣇⠀⠀⠠⣿⠃⠀⣷⡀⠀⠀⠀⢸⡇⠀⢠⡉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠸⡏⣼⣿⢿⣿⣿⡀⢀⣼⠋⠀⠀⠘⢿⣆⠀⠀⣾⠇⠀⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠉⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⣀⣀⣀⣤⣴⣶⣶⣿⡿⠋⢻⡇⠀⢀⣤
⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣿⡇⠀⢾⣿
⠉⠛⠿⠿⣿⣿⣿⣿⣿⣷⣽⣟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢸⣟⣀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠻⠿⠿⢿⣿⣏⡁⠀⠈⢻⣷⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⣾⣃⣔⣀⣼⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣄⣀⠀⢀⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⠀⢸⣿⠏⠟⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⢸⣿⡄⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠃⠀⠀⢸⣿⣷⣅⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⠈⢻⣿⣿⣧⠁
⠀⠀⣀⣤⣴⣶⣦⣤⣤⣤⣤⡀⠀⢀⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢷⣄⡘⣿⣿⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⢸⣻⣿⣽⣿⣿⣿⣉⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⢤⡻⠷⣾⣷⣿⡄⢳⣶⣶⡶⣶⣖⢢⣴⣾⣿⠟⠉⠀⢀⡞⢶⡄
⠀⢾⠛⢋⣩⣿⣥⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣿⣿⡇⠈⢻⣿⣿⣾⡷⠶⠞⣉⣽⣿⡿⠛⠁⠀⣠⡖⠁⠀⢺⣿
⠀⠼⠿⣿⣯⣭⡄⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣮⣿⡗⢀⣀⢾⣿⣿⣟⣲⣿⣿⠟⠉⣀⡀⣀⠆⠉⠀⠀⠰⣾⣿
⠀⠀⠀⠈⠙⣿⣧⣤⠀⠀⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⢸⣿⡇⠀⣹⣿⣿⢹⣿⠛⠋⣠⣴⣿⣿⣿⣿⣟⠀⠀⠀⣶⣿⣿
⠀⠀⣤⣤⣼⣿⣯⣁⡀⢀⣀⣀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⣿⣦⣌⢹⣿⣼⣿⣧⣾⣿⣿⢿⣿⣿⠟⢿⣤⣀⢶⣿⣿⣿
⠀⠀⠈⠋⠉⠉⠁⠀⠻⣿⣿⣿⠿⣿⢿⣧⠈⠛⠖⠒⠂⠀⠒⠚⠛⠿⠗⠀⠸⣿⣿⣿⣿⢿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠟⠁⢀⣼⣿⣿⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠄⠀⠈⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣾⢻⣿⣿⠀⣿⣿⣿⣿⠟⠁⠀⣠⢞⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢺⣿⣿⣹⠃⣿⡿⣿⡄⣿⣿⣿⣿⣀⣠⡞⣣⣿⡿⠛⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⢀⡀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⡟⣯⡏⣸⣿⠀⢿⣿⣿⣿⣿⡟⣻⣷⣿⣿⠟⠁⠀⠀⢀⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⢾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⢰⡟⣦⣻⣟⠀⠘⠉⢛⣹⣿⣧⣈⣿⠻⣿⠀⠀⣠⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠶⣍⢻⡿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡁⣾⣿⣿⣼⢿⡏⢰⣄⣴⣿⣿⣧⣼⣿⠃⠀⢿⣤⣼⣿⠟⠛⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⣆⠙⣝⢫⡻⣷⡀⠀⠀⠀⠀⠀⡉⣿⣿⣿⡿⠿⠋⢁⣠⣿⣾⣿⣙⣏⣿⠀⠀⠈⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠈⣆⠈⢦⡹⣎⢻⡄⠀⠀⠀⠀⠋⠿⠟⠉⠀⠀⠛⠋⣉⣶⣿⣛⢿⢹⡇⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠈⠀⠀⠱⡈⠂⠙⣆⠀⠀⠈⠃⢸⡇⠀⠀⢀⢴⣿⣿⣿⠿⣟⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠈⠀⠀⢼⣷⠀⠰⠓⠀⡇⠀⠀⢘⣿⣿⣿⣿⠂⣿⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠃⠀⠀⠀⠀⠀⠀⢠⡈⢿⣧⠸⡇⠀⡇⠀⠀⣋⣹⣿⣿⣿⡆⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠱⠀⠉⢰⡇⢈⡇⣠⣴⠿⢿⣿⡟⢺⣿⣿⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠇⣸⣿⡟⠁⣀⢸⢹⡇⢽⣿⣹⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠰⡆⠀⠀⠀⠀⠀⠀⢠⠀⠁⠈⠇⠀⣿⡇⣾⠃⣸⣿⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⢻⣿⣧⣿⠀⠀⠀⠀⠀⠀⠈⠀⡃⠀⠀⢳⣿⣇⣿⠀⢹⢿⣷⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣼⣧⢾⣿⡿⠛⠇⠀⠀⠀⠀⠀⠀⠀⠹⣄⠰⡄⠙⣿⣯⠀⢨⣿⣿⣗⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣿⣿⡿⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠿⣿⣿⣿⣷⡶⠶⠶⢾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⣀⣿⠟⠋⠳⢱⣿⡄⠀⠀⠀⠀⠈⠀⠀⢀⡜⠁⣼⢿⢝⠻⣦⣄⠀⢙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⡟⠁⠀⠀⠀⠀⠛⢿⣆⠀⠀⠀⠀⠀⠀⣼⡰⠟⠉⠁⠋⠑⠢⢙⢿⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣆⠀⠀⠀⣼⣿⡟⣧⠀⠀⠀⠀⠀⠀⠈⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⢦⣰⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣯⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⢠⣆⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⡿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣤⣠⣮⢷⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⣀⠢⠀⠀⢀⡞⣱⣿⣿⣽⣿⢋⣿⣿⣿⣿⣿⣿⣷⣔⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣿⣿⣧⣡⣶⠏⣴⢫⣿⣿⣻⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋⠉⠋⠉⠉⠉⠙⠛⠉⠉⠉⠙⠋⠉⠋⠙⠛⠛⠉⠉⠉⠛⠉⠙⠋⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢀⣄⣼⣿⣿⣿⣿⡄⠀⠀⠈⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⢸⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣈⢉⣿⠟⠀⠉⠹⣿⣿⠏⠉⠀⠙⣿⡉⣁⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏⣶⣶⡅⠢⠘⠋⠔⢈⣶⣶⠸⢿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣫⠀⠈⠻⣄⣻⡶⠦⠴⠶⣟⣠⠟⠁⠀⠙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡡⠢⢤⣤⣤⣀⠹⣧⣴⠇⢁⣠⣤⣤⡑⣮⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡀⢈⢿⣦⡁⣿⣶⣶⣾⣇⢠⡾⡁⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⡈⣿⣷⣿⣿⣿⣿⣿⣿⠃⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣿⣿⣿⣿⣿⣇⢸⡇⣿⣿⣿⣿⠸⡏⣸⣿⣿⣿⡿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⣿⡿⣿⣿⣄⢧⣘⠻⠟⣏⡰⣡⣿⣿⣿⣿⢣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡌⠛⢿⣧⣄⣯⣭⣅⣴⣿⠛⢣⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⠿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⡟⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣛⠛⠋⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡴
⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃
⠀⠀⠀⠀⠀⠈⢩⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡃
⠀⠀⠀⠀⠀⠸⢿⣿⡄⠀⠀⠀⠰⡄⠀⠐⡄⢘⣿⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡿⣷⡀⠀⠀⢱⡘⣆⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠿⠋⡆
⠀⠀⠀⠀⠀⠀⠠⠃⠩⠿⠀⠜⠂⠟⠈⠛⠛⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡧⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⣈⠿⡃
⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⣀⡀⠀⠀⠀⠀⠠⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢮⣿⣇⠀⠀⠀⠀⢸⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣧⠀⠀⢀
⠀⠀⠀⠀⠠⣶⣶⡌⠀⠀⢠⣿⣿⠀⠀⠀⠀⡞⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠉⠠⠀⢨⣷
⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⢸⣿⠃⠀⠀⠀⠐⡆⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣀⣀⣀⣴⡏⠀⠀⠀⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢷⡀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠺⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⡇⠀⠀⠀⠀⠈⢁⣀⣠
⠀⠀⠀⠀⠀⠚⠛⠛⠛⠓⠂⠀⠀⠈⠛⠛⠛⠛⠛⠒⠒⠒⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⡣⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⠋⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣐⠋⠐⢿⠀⠐⢉⠭⠓⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡐⢞⣻⣿⣠⣀⠈⠀⡄⠊⣶⣄⡀⠠⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣣⣼⣿⠞⢻⠁⠀⢰⠀⣹⣿⣿⡦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣹⠃⠀⡿⠋⠀⠀⣿⠀⣴⠇⣠⣽⣿⡟⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠏⠀⠘⣿⡧⠀⠀⣧⣼⡏⢴⠤⣾⣿⠃⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣴⠏⠀⠀⠀⣧⣇⢀⣴⣿⠁⢀⣤⣴⣿⡟⠀⠀⠀⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣤⣴⠏⠀⠀⠀⢠⣸⠟⠋⢹⡿⠀⣸⠉⣽⡿⠀⠀⠀⠀⢱⡀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⡇⢸⣿⡇⠀⠀⠀⠀⠈⣿⣷⠿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣾⡿⠃⠀⠀⠀⠀⠀⠀⢸⠇⠀⡸⢠⡟⠀⠘⣿⡇⠀⠀⠀⠀⠀⠘⣿⡠⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠊⡱⠋⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⠀⣧⠊⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠈⢷⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠀⠀⠐⣿⣤⣉⣻⡷⠍⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡇
⠀⠀⠀⠀⠐⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠁⠈⢻⣿⣿⡇⠀⣄⣀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⢿⣿⠀⠀⠀⢻⣿⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⢸⡏⠀⠀⠀⠘⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⠀⠀⠀⠀⠸⣿⣷⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣿⠀⠀⠀⠀⠀⢟⢹⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢩⡇⠀⠀⠀⠀⠀⠘⢶⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣾⣇⠀⠀⠀⠀⠀⠀⠸⡆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⣿⡇⠀⠀⠀⠀⠀⠹⡿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⠀⠀⠀⠀⠀⠀⠀⢣⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡏⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠚⠛⠛⠉⠁⠀⠀⠀⠀⠈⠋⠛⠒⠶⠦⠄⠶⠖⠒⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣉⣻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢿⡇⣿⣿⣿⣿⣿⠟⠉⢉⣀⣠⣭⣙⠻⢿⣟⢿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢺⡇⣿⣿⣿⡿⢣⣄⠚⢿⡿⢿⣿⣿⣷⡔⠛⣧⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣾⡇⣿⣿⣿⣷⣼⣿⣦⠈⢿⣾⣿⣿⣿⣧⠀⠚⠛⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⢋⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⡄⠈⠙⠻⣿⣿⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⣀⣤⣿⣯⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⣈⡉⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢽⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⢠⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠉⠈⠁
⠀⣊⣼⣷⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡼⢣⡄⠰⣷⣄⣰
⠀⠀⠘⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠿⣦⣯⣩⣿⣟⡽⠃
⠀⠀⠀⠀⠻⣿⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣹⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣷⣤⣤⣤⣴⣶⣶⣦⣶⣿⣟⣿⣶⡟⣿⠿⣿⣿⣿⡿⠟⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⢿⣾⣿⣿⣿⣿⣿⣿⣘⣿⣿⢦⣿⠟⠁⠴⢶⠿⠛⢳⣾⡟⢛⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠿⠿⠿⠿⠿⠿⡿⠷⢼⣿⡿⡦⠃⢰⡿⢶⣾⣛⣧⣬⣿⣵⣮⣤⢸⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡏⣌⠋⢑⣾⢿⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⣼⣁⣙⣛⣿⣿⣣⣽⣷⣿⣿⠙⢯⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢱⠺⡟⢏⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠈⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⠀⣿⣄⠠⡴⠶⢾⣿⣿⣿⡟⠁⠸⣶⣿⣿⢿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⠐⢿⣿⣖⢒⣲⣯⡿⡿⠋⠀⠀⠀⠈⢻⣿⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣿⠀⠘⢻⣿⣷⣿⡟⠠⠇⠀⠀⠀⠀⠀⠀⢽⣠⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⠀⢺⠋⣿⣿⣿⣧⣜⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠟⠠⣦⣿⡿⢟⠋⠑⠏⣆⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⢚⡛⣼⡇⢀⡰⣿⣟⡴⢦⣴⡏⠐⣿⡆⠀⠀⠀⠀⠀⢹⣿⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣋⣡⣴⣾⣿⣿⠏⣴⠢⠀⠐⢻⠆⢻⣿⡇⢰⢿⣷⠀⠀⠀⠀⠀⢸⠟⢰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡟⠉⢻⣽⡿⠁⠄⠀⠀⠀⠀⠊⠀⠘⣥⠁⠀⢸⣿⠀⠀⠀⠀⠀⠿⠆⠘⠄
⠀⠀⠀⠀⠀⣀⢾⣿⣿⣿⣿⣿⣿⣷⣴⡿⢿⡾⢷⣶⡟⠲⣶⡆⠀⣎⢳⡏⠀⠘⣿⠃⠀⠀⠀⠀⠘⢀⡀⣤⢂
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⣟⡽⢿⣿⣷⠌⠛⠉⠀⠀⢀⣿⢀⣰⠟⠉⠁⠀⣼⠃⠀⠀⠀⠀⠀⠰⠋⠙⠾⢻
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣥⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠸⣿⢸⡇⠀⠀⢸⣦⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠉⡛⣯⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⡴⠀⣸⣯⠇
⠀⠀⠀⠀⠀⢿⠹⣿⣟⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢀⣿⣿
⠀⠀⠀⠀⠀⠸⡆⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⣿⣾⡿⣿⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢳⣼⣿⣶⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⡟⢻⢧⣿⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠛⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⣾⢺⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⣏⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣡⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣠⡿⣜⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⣀⢤⣴⣎⣭⣾⣿⣿⡿⢄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⡇
⣤⣻⣵⣶⠿⠟⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⠁
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠉⡯
⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠲⣌⣸⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠓⠛⣽⣋⠽⠿⠿⣦⣤⠤⠤⠤⠤⢤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠂⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⢰⡆⡀⢀⣰⡗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⣿⠀⢇⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠊⢁⣿⠇⠀⠸⣆⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠒⠒⢰⣶⣞⣉⠀⠀⢘⣭⣷⡆⠀⠀⠉⢻⡌⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣄⠀⠀⠻⢿⣦⣀⣀⣂⣾⣿⡀⠀⠃⠸⡄⢷⣇⠀⠐⠀⠐⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣍⡙⠲⣄⠉⢿⣯⣴⡿⢿⣧⠀⠁⠐⠿⢦⡟⠀⢀⣀⡠⠖⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣾⣿⣮⠀⢘⣿⡟⢻⣿⠇⠘⣾⡆⠀⠻⡄⡾⠃⠚⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣟⢿⡉⠉⠉⠳⣿⠟⢃⣿⣿⠇⠀⠛⢻⣄⠀⣝⣗⠀⢰⠟⣠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢷⡀⠀⢢⣿⣄⣴⡟⢭⠁⠂⠀⠀⠙⢶⣾⡹⣆⣀⠶⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⢲⣾⣿⠿⡟⠁⠸⠀⠀⠀⠀⠀⠀⠈⠛⠦⢴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⡿⠋⠁⠸⣄⣇⠀⡀⠀⠀⠀⡠⠤⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⣰⣿⣏⠀⠀⠀⠔⣻⡶⠀⠄⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⣊⣵⠖⠀⢄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣦⣦⡝⠁⠀⢠⠀⠘⢉⡿⠟⠒⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣰⣿⡿⠃⠀⠀⠠⠀⠀⠁⠀⠘⢠⠎⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⠃⠀⠀⠀⠔⠀⠀⠀⠀⠂⠘⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡾⠿⡛⠀⠀⠀⠀⢈⠀⠀⠀⢀⡆⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠢⡀⠀⠀⢀⠀⠀⠀⠜⣠⠎⠁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡆⠀⠈⠀⠀⠈⢀⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠧⠀⠀⠀⠀⠷⠈⣀

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣷⣶⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠛⠉⠉⠉⠉⠛⠿⢿⣿⣿⣿⣶⣶⣦⣶⣿⣿⣿⡿⠿⠛⠛⠋⠉⠛⠻⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣠⣤⣶⣿⣿⣿⡿⠿⣿⠿⣿⣿⣿⣶⣤⣤⣀⡀⠀⣀⣠⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡿⠿⠿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⢿⣿⣿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠒⠛⠢⣄
⠀⠀⠀⠀⢀⣠⡤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⢀⡞⠉⢀⢰⣶⣠⡀⠀⠀⡰⢅⠀⠀⢀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⣼⠀⠀⠀⠀⠀⢙⣿⣀⣤⣶⣀⠲⡠⠁⣠⣢⠤⠀⠀⠀⠀⠘⠳⢦⣄
⠀⠀⠘⣄⠀⠀⠐⠋⠋⠉⠉⣙⣻⣿⡷⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⠶⣄
⠀⠀⢀⡞⠱⠦⠤⠒⠊⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠝⠳⡄
⠀⡰⡯⠐⠲⣆⠀⠀⠀⠾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢰⡄⠀⢀⡃⠒⠀⠀⠀⣈⣴⡶⠶⠶⠾⢤⡄⣠⡤⣴⢂⠠⠲⠆⢀⠀⠀⠀⠀⠀⠀⠃⠠⡌⢳⡶⠾⠟⠒⠦⠴⢿⡓⣦⣄⣀
⠀⠙⣺⠛⠳⠶⢒⣋⣉⣉⣀⣀⢀⣤⣴⠿⣿⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠘⣷⣆⢸⠀⠀⠀⠬⢷⠁⠍⠏⠉⠓⠲⠦⢤⣤⣤
⠀⢀⣇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⡔⠀⠘⣉⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣋⢸⠀⠀⠀⢀⣸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠻⣧⣸⣠⣄⣀⢀⣐⠀⡉⠠⣩⣴⡼⠟⢫⣤⡙⢿⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡧
⠀⠀⠀⠈⢹⡋⠛⠛⠛⠛⠒⠛⠛⠛⢩⠀⠀⣼⡯⠔⠉⠉⠓⠆⣤⣶⣤⣀⡀⠀⠄⢄⡀⢀⠀⠀⠀⡞⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢷⣦⣀⡀⡀⠀⠀⣄⣀⣷⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠰⠀⠀⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠁⠈⠙⠿⢼⣐⡜⣁⢀⠀⠀⠀⠀⠀⠀⠉⠀⠁⣠⣀⠀⠀⢠⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠚⠿⢧⣤⣥⣶⠾⠶⠛⠛⢶⣙⡀⡏⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⠀⠀⠀⠀⠀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣄⣀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣻⣿⡟⡇⡀⡄⠀⠂⠀⠂⠀⡐⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣱⣹⡡⡐⣐⡀⡀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣵⣿⣿⣿⣟⡙⠐⢡⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⠮⢼⣔⣐⣃⣩⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠈⠉⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠀⡔⠀⠀⡼⠀⠀⠀⠀⠈⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠈⢑⠒⢽⡄⣲⠇⠀⠐⠀⢰⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠾⡁⡐⠀⢚⣔⠪⢀⠀⠀⣀⡀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣞⠖⢾⡪⠔⠋⠉⠉⠉⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠄⠄⠀⠀⠀⠀⠀⠀⠙⠤⡓⠀⠀⠀⠀⠀⠀⠀⠀⢢⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠰⡄⢂⠀⡀⡀⢁⠶⠂⠀⠀⠀⠀⠀⠈⢣⡀⠢⠀⠀⠀⠁⡀⡂⣬⠂
⠀⠀⢀⣀⠀⠤⠄⠐⠂⠈⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠈⠑⠦⣈⣾⣆⢮⣶⢋⡀⠀⠅⡀⠀⠀⠄⠡⠀⢳⠅⠔⣐⣀⡤⠔⠶⡅
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⡀⢝⢛⡿⣦⣏⠀⢀⠅⡒⠦⠀⠀⠈⠀⢸⠒⢋⢁⠠⠀⠀⠐⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡄⠂⢀⢀⠀⠀⠀⠀⠀⡀⠠⠆⠖⠋⢡⢞⣯⡟⣟⠠⠐⠂⠁⠀⠀⠀⠀⡏⠀⠈⠀⠀⠀⠀⠀⢠⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠀⢳⠹⣄⠂⢂⢀⠀⠀⠂⠀⠀⠂⠂⢀⠨⢠⠜⣼⣧⡿⠀⠀⠀⠀⠀⠤⠡⢄⡇⠀⠀⠀⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⢄⢣⠈⠑⢄⡂⠈⠌⠀⠈⠐⠀⠀⣤⠞⢃⢾⠋⠻⣿⡐⠀⠀⠚⠣⠤⠔⠚⣧⣀⣤⣤⡤⣤⣤⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣈⢁⠳⠀⠄⠄⡘⡰⠀⠠⢠⡴⠋⠜⣰⢧⣿⢄⠀⠹⡂⠀⠀⠀⣞⠁⠀⠀⠘⣦⠀⠀⠐⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠠⠀⠀⠈⠀⠀⠀⢣⠂⠐⠂⠄⠤⡐⠏⠣⠃⣠⢞⢌⢃⣟⣇⣠⠔⢻⣅⡀⠑⣄⠀⠀⠀⢀⡜⠀⠁⠀⠀⡀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡀⠀⠄⠀⠅⠀⠠⠚⡿⠞⠀⣾⡿⠽⣿⡟⡈⠙⠿⣿⣮⣿⣿⠿⠋⠀⠀⠀⠀⠀⢟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢱⠀⠂⠀⠀⡀⠀⠠⠀⠁⡀⠀⣀⡄⢄⠆⣸⣑⣺⣶⣀⠀⣀⣤⣀⣴⣴⣴⣴⢾⣾⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⢊⠄⡀⠀⣠⡀⠄⠘⠛⠈⠙⠏⠙⠛⠋⠑⠉⠚⠚⠙⠙⠙⠋⠋⠺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⡁⡀⠀⠠⠁⠘⡄⠄⠀⠀⠀⠈⠐⠜⠿⡞⡞⠾⢼⠿⣷⣞⠹⠿⠂⠉⠋⠈⠑⠃⠀⠀⠀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠔⠠⢀⠂⠜⡄⢦⠀⠆⠈⠂⠄⠀⠀⠰⠁⠀⠀⠃⠘⠦⣤⣠⣄⣄⣤⣠⣠⣦⠶⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠔⡐⡡⢂⣈⣪⣧⠭⠤⠦⣀⠈⠎⡐⠀⠀⠀⠐⡘⡐⡠⡠⣁⣲⣡⠥⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠚⠚⠚⠈⠉⠁⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⡀⢀⡠⢀⡀⢀⡀⢀⠤⠀⠀⡀⢀⠀⡠⠀⠀⣀⠄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠂⢐⠄⡊⠔⡠⢐⠡⢂⡡⢂⠅⡈⠄⡨⠔⡨⢐⠥⢂⠡⢂⠥⢊⢄⠢⢒⠅⡪⠔⡡⢂⠅⡪⠔⠩⠊⠔⠁⠊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠁⠊⠀⠊⠐⠁⠐⠁⠈⠁⠂⠀⠊⠐⠁⠐⠁⠀⠁⠀⠁⠊⠀⠀⠀⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆⠀⢺⣿⣷⡶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⢿⡇⠀⢸⠈⢿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠘⣿⡎⣷⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣹⣷⢿⠀⠀⡄⠀⠸⠋⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠋⢻⣼⡄⢀⠃⠐⠀⠀⢾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠠⠋⠘⠃⠈⠀⠀⠀⠀⠈⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠆⠀⠀⠀⠀⠀⢀⡄⢠⠀⡖⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⢀⡔⠚⠁⠆⣨⠀⣅⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡠⠀⡞⢀⢃⠘⢠⠈⠀⣿⡇⣙⢯⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠐⢀⣼⠁⢸⢹⠀⠀⠀⠀⡡⠆⢸⠇⠉⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⣰⣿⠇⠀⣾⢸⢰⠀⣠⢤⡀⢠⣾⠂⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣴⡿⠂⠀⢀⡿⠀⢾⡆⢸⣼⠇⢸⣿⡇⣀⡄⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⢀⣴⣿⡆⠀⠀⢸⡇⠀⢸⣧⠀⣿⠄⢸⣿⣿⣿⡇⠀⢀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⢿⣿⠟⠀⠀⠀⢸⡇⠀⣾⣿⠀⢹⠀⢸⣿⣿⣿⡟⠀⠈⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠅⠀⠀⣼⣷⡈⡇⠀⣿⣿⡄⠀⡄⢸⣿⣿⠉⠙⢆⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡝⠀⠀⡀⠸⡟⠁⠃⡄⢹⢻⠿⠀⣵⠊⣿⡟⠀⠀⠀⠂⠀⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⠞⠀⠀⠀⠃⣴⠀⠀⠀⡇⠸⡿⠀⠀⠸⠂⢸⡇⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⣇⡼⠋⠀⠀⢀⠇⢠⡇⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⢺⡟⠀⢸⠟⠀⢸⣧⠎⠙⡆⠀⠐⣴⠇⠀⠀⠀⠀⠀⠈⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⢀⠃⠀⣾⠁⠀⠘⠀⠀⣸⡟⠀⠀⢡⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠆⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠎⠀⠀⢿⠀⠀⠀⠀⠀⣿⠃⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢠⠇⠀⠀⠔⠀⠀⡟⠀⠀⠀⠃⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⢠⡏⠀⠀⠰⡆⠀⠀⣴⠀⠀⠀⢄⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⡇⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠘⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⡆⠀⡾⠀⠀⠀⠀⠀⠀⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠀⠀⠀⠀⢠⡀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠏⠀⠀⠀⠀⠈⠃⠛
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊

⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡄⠀⠙⢿⣷⣦⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠙⠻⣾⠟⠻⡿⢿⣷⡈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣄⡀⢸⠗⠂⠀⣜⠛⢇⣀⠘⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢶⠀⣠⢟⣴⣿⠟⠁⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣵⣿⠋⠁⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠉⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠀⠀⠈⢿⠀⢠⡄⠀⠀⠀⠐⣆⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡃⠀⠀⠀⢀⠁⣿⣿⡄⠀⠀⠀⣿⣇⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠘⣷⡜⢿⣿⡀⠀⠀⣿⣿⠀⠀⠙⠉⠉⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⣭⣝⣈⢿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⣿⣿⣿⡜⠇⠀⢰⣿⣿⠐⣿⣿⡇⣶⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠛⠛⠛⠛⢸⣿⣿⣿⡇⣦⠀⠈⠁⠀⡄⣿⣿⣇⣿⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢇⣿⡇⠀⠀⢸⡇⣿⣿⢸⡇⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⢀⣀⠀⠀⢸⣿⣿⣿⢸⡟⠀⠀⢀⣿⡇⣿⣿⣸⡇⠀⠀⠀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⢸⣿⣷⡦⢸⣿⣿⡇⡾⠁⠀⢀⣾⣿⢧⣿⡇⣿⢁⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⠃⣾⣿⣿⡇⠁⠀⣠⣾⣿⡏⣾⣿⣧⣿⠈⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠀⠀⠀⢀⣼⣿⣿⠃⠀⢿⣿⣿⣴⣶⠆⣨⣭⣭⣼⣿⣿⢸⡟⠀⠀⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⡿⠁⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠸⣿⣿⡈⠛⠛⠿⠿⣿⣿⠼⠇⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠻⢿⣷⣶⣤⣝⡋⠀⠀⣀⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡄⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⣀⣤⣦⣤⣤⡀
⢰⣶⣶⣶⣶⣶⣶⣶⣿⣋⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠚⠿⠿⣿⣿⣿⣿⣆
⣾⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠛⠙⠛⠛⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠈⠙⠿⣿⣧⡀
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠙⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠀⠀⠙⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠙⠄
⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⣠⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠀⠙⢿⣿⣿⣿⣷⣦⡀⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠻⣿⣿⣿⡿⠃⢹⣿⣿⣿
⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠛⠁⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣶⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣽⣿⣷⣤⡎⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣏⢹⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣴⣿⣿⣿⣿⠟⠛⣸⡿⠛⠙⠻⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠶⠶⢤⣀⠀⠀⠀⢠⣾⣷⣿⣿⣿⣿⡿⠛⠁⣤⣶⡉⣶⣄⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡡⠀⠀⡰⠋⠀⠀⠀⣈⣭⠓⢒⣾⠏⢉⡿⠛⡋⠁⠀⢠⣾⣿⡇⠉⣦⣀⣱⣀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⡈⠀⠀⣰⡔⠀⠀⠀⠀⠀⠁⠐⠋⠁⠀⠜⠀⡞⠀⠀⠔⢋⡀⢸⣇⣴⣯⡿⣏⠃⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢸⡄⢀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⡟⠀⣿⡟⠛⢦⣏⠈⢀
⠀⠀⠀⠀⠀⠒⠶⣾⣿⣿⣿⣿⠀⣇⢸⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠁⠀⣼⡟⢀⣼⠏⠀⠀⠀⠙⣦⣛⣷⡞
⠀⠀⠀⠀⠀⠀⠞⣿⣿⣿⣿⣿⣧⡈⠀⣿⣧⠈⠻⢦⡐⢤⣄⣠⣾⠃⠀⣧⠀⠀⠀⣼⡿⢡⡟⢁⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠁⠁⠘⣧⢠⠾⠿⢷⣄⠀⠈⠑⠈⠉⣿⠀⣠⡞⠳⣤⡀⠹⠵⠋⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣱⣶⠇⠈⢳⣶⣶⣤⣤⣿⣾⢿⡇⣰⡿⠧⢀⣐⡺⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⣿⠋⠀⠀⡼⡿⢻⠋⠙⢻⠁⠀⠹⡿⠷⠻⢿⣟⠓⢤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣻⡇⠀⠀⠀⠹⣼⢿⡀⠀⠘⣧⡄⠐⡇⠀⠀⠀⠉⠻⣆⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠈⠻⡹⣄⡀⠘⣿⡄⢃⠀⠀⠀⠀⠀⠘⢿⡦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡽⣆⠀⠀⠀⠀⠀⠙⣿⣷⡀⢸⡅⠈⡆⠀⠀⠀⠀⠀⠀⠹⡜⣦
⠀⠀⠀⠀⠀⠀⠉⠁⢀⡀⠀⠀⠀⠀⠀⢷⣿⣇⠀⠀⠀⠀⠀⢹⣿⠛⣸⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⢻⡈⣷
⠀⠀⠀⠀⠈⠙⠓⠀⣈⣛⠛⠛⣓⠒⠒⣾⡟⠛⡆⠀⠀⠀⠀⢠⡾⣏⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠙⠊⠉⠛⠁⠀⠀⠀⠀⢸⣿⠟⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠘⠶⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠙⠛⠛⠓⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉⠁⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣄⢰⡄⡆⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢀⣈⣃⠘⠦⠼⣴⣧⠋⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⠋⠁⠈⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⣠⢋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⠾⠛⠭⠭⠽⠓⠦⣤⣴⡿⠋
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⢀⣠⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣾⠱⡀⠀⠀⠀⠀⢀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠘⢧⡈⠀⠀⠀⠘⠋⠁⠀⠀⠀⠈⠢⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣤⣤⣤⣄⣀⡀⠀⠀⠈⠙⠓⠶⠶⠤⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⣀⠠⠤⣄⣀⣀⣀⣀⣠⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠶⠶⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠚⢛⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⣠⣴⠇⠛⠱⢷⡦⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡄⠀⢀⡭⠷⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢰⠇⠀⠀⠀⢀⣀⣠⡤⠤⣤⣸⠁⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⢸⠀⢠⠔⠋⠉⠉⠁⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡤⠊⠉⢹⣴⠇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠃⠀⠀⠀⣀⣤⠤⠶⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠒⠻⢿⣯⡉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⣷⣦⣄⣀⣀⣠⡤⠖⠋⠀⠀⠀⣼⠲⢤⠒⠋⠁⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠀⠀⠀⠀⠀⠈⠐⠨⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠙⣟⢦⠀⠀⠀⣠⠇⠀⢈⡇⠀⢀⣤⣤⠤⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣳⠴⠚⠁⠀⣤⠊⣠⠊⡡⡊⣽⠃⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⠋⠁⠀⠀⠀⠀⠀⢸⣀⡇⠐⣼⡿⣃⡴⠁
⠀⠀⡴⠒⠶⢶⣶⣤⠼⠄⠀⡀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠉⠉⠁⢰⡏
⠀⠀⡇⠀⠀⠀⢻⣏⡇⢀⡤⠛⠲⣤⣟⢦⡀⠀⠀⠀⠀⠀⣾⠛⠲⠶⠶⠖⠒⠚⠋⠉⠀⠙⠦⣄⣀⣤⢤⣀⣀⣠⡤⠊⠁⠀⠀⣠⣄⣘⣧
⠀⠀⠹⡄⠀⠀⣼⣿⣧⠎⠀⠀⠀⠀⢹⢿⢳⡒⠦⢄⣀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⣀⣲⠤⠖⠉⠀⠉⠁
⠀⠀⠀⠙⢦⣰⠃⡜⠁⠀⠀⠀⣀⣀⣜⡼⠀⣧⠀⠀⠈⠓⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠉⠁⢹⡟⠒⣶⠇⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠚⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣒⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠛⠻⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠉⠛⣳⣦⣤⣄⠀⠀⠀⣀⣤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠇⠀⠀⠐⠿⠿⣿⣿⣿⣿⠿⠟⣰⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣟⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⢿⠉⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣌⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣄⣠⣿⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣬⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠻⣷⣤⡀⠀⠀⠀⠀⠀⠱⢶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣇⠀⠀⠈⢿⣿⣶⣤⣀⡀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢻⡄⠀⠀⣰⠟⠈⠻⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠃⠀⢠⡟⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠛⠶⣤⣀
⠀⠀⠀⢀⣿⣿⠿⠿⠛⠻⠿⠁⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠛⠀⠀⠀⠀⠀⠤⠄⠀⠀⠐⠒⠒⠛⠛⠛⠂⠉⠉
⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠚⠓⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⢻⡄⣠⡤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣇⡀⠀⠀⠀⣸⡏⠀⠀⠀⠈⣷⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⠉⠙⠛⣿⣛⣛⠛⠛⢳⡄⣼⠋⠉⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡾⠀⡿⠀⠀⠀⠀⣿⠒⠒⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠢⣄⡀⠀⢀⣠⣤⠾⠛⠁⢰⡇⠀⠀⠀⢰⡇⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⣹⣿⣿⡟⠁⠀⠀⠀⣼⠃⠀⠀⠀⣾⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠐⣤⠶⠻⠿⣿⣇⠀⠀⠠⣴⣿⠀⠀⠀⢠⡏⠀⠀⠀⠀⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠙⠻⢿⣷⣶⣾⡧⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣐⡲⣶⣿⣇⡒⣤⣤⣿⠿⠦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⠉⠉⠉⠙⠉⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠰⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠶⠒⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠛⠿⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠓⠶⠒⠀⠀⠀⠀⠒⠲⠶⠒⠺⠿⢿⣷⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣤⠤⡤⡴⠛⠉⠄⠈⠑⠒⠋⠉⠀⠀⠈⠉⠳⠆⠈⠩⠍⠛⢿⡦⠤⣤⣤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠁⠉⠀⠀⣀⣀⣤⡤⣤⡶⢀⣴⡶⣶⡆⢰⣶⠆⢶⡶⢤⣤⣤⣤⡄⣀⠀⠀⠉⠉⠙⠻⢷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠈⢻⣧⣬⠁⢸⣧⢸⣿⡀⣾⣷⢸⣿⠓⣿⡇⠈⢰⣿⠘⠋⣿⠇⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣆⠀⠀⠀⠀⠀⠀⢸⠿⠆⠀⠘⠛⠂⠉⠛⠉⠋⠈⠙⠁⠛⠛⠀⠛⠿⠀⠰⠧⠀⠀⠀⠀⠀⠀⢀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠻⣆⠀⠀⠀⢀⣀⡀⠀⠀⢀⣴⡦⠤⠀⠀⣀⠰⠴⠀⢀⠀⠤⠤⢶⡄⠀⠀⠀⣀⡀⠀⠀⠀⢠⡟⠛⠙⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠉⠀⠀⠀⢻⣷⣾⡿⠛⠉⠉⣿⡏⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠀⠀⠀⠀⠉⣿⣿⡏⠉⠛⢻⣿⣶⡿⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠈⠛⣻⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣼⣿⠛⠁⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠠⠃⠀⠀⢤⣖⡀⠀⠛⠋⠉⠉⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠉⠛⠛⠛⠣⠀⣐⡤⠀⠀⠈⢃
⠀⠀⠀⣰⠡⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠄⣳
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣼⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠻


⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⢠⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣴⣶⣤⣶⣤⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣾⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣴⣏⣴⣾⣿⡷⢛⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⠰⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠖⠀⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣧⠶⣶⣦⣽⣿⠛⠛⠟⢿⢿⡿⡿⠻⠛⠛⣿⣯⣴⣶⠖⣸⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡈⠛⠻⣿⡛⡋⠀⠀⠀⠈⠈⠁⠁⠀⠀⠀⢘⢛⣿⠛⠛⢁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠈⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⣿⣟⢻⣿⣤⣀⠀⠀⢨⣄⣠⡅⠀⠀⣀⣤⣷⡄⣹⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡭⠀⣿⣿⠈⣳⣄⣀⢸⣏⣀⣠⣾⠅⣿⣿⠀⠭⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣿⣿⣅⡄⠀⠈⠛⠓⢶⣟⣍⡉⢉⣩⣻⡶⠚⠛⠁⠀⢠⡨⣿⣿⣮⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣰⠀⠀⡴⠶⢦⡙⠛⢿⡿⠟⢋⡴⠖⢦⠀⠀⣆⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠏⢹⣿⣿⣿⣿⡄⠠⠹⣾⢠⣍⣖⣦⣴⣒⣩⡀⣷⠏⠀⢠⣿⣿⣿⣿⡏⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⡟⣼⣿⣿⡇⡀⠀⡿⢻⣿⣿⣿⣿⣿⣿⡟⢻⠀⠀⢸⣿⣿⣿⢻⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠇⣿⣷⡇⠀⢧⣬⣩⣩⡭⠭⣍⣍⣥⡼⠀⢸⣾⣿⠘⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⣿⣧⡀⢀⠀⢀⠀⠀⠀⡀⠀⢀⣼⣿⣿⠏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣏⢻⣿⣴⣼⣇⢰⣷⣧⣾⡟⣹⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣥⢲⠴⠶⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⠾⢻⢾⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠦⠤⠴⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠲⠾⢶⠲⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠶⠾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣦⢰⢢⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠠⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣒⠆⠀⠀⠀⢑⢒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠄⠀⠀⠀⠀⠀⠢⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢈⢡⢸⠖⣓⣶⡈⢁⡀⡀⢀⡤⡞⠻⡦⠲⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠠⠸⡿⢿⠹⢿⠻⠹⠿⠿⠿⠿⡟⢿⠇⠠⢒⡔⠀⡰⠹⠷⡿⠿⠾⠾⢾⡷⠇⠠⠤⠄
⠀⠀⠀⠀⠀⠀⢠⣶⣺⣶⣶⢲⢶⣶⣷⣶⣶⣿⢸⢼⡷⣇⡇⣟⢻⢼⠼⢿⡿⢧⢤⠤⣦⢤⣦⡄⣬⡥⠐⠒⠒⠂
⠀⠀⠀⠀⠒⠒⠒⠶⠀⠂⠶⠆⠲⠒⠶⠶⠶⠠⠶⠖⠶⠶⠦⠆⠶⠶⠲⠶⠒⠈⠰⠶⠖⠰⠶⠠⠶⠖⠶⠆

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⡇⠐⢸⠻⣿⣿⡙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⢻⡇⠀⢸⠄⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠃⠹⣿⣾⣇⠀⠀⠀⠀⠘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢠⣿⡿⣿⠀⠀⡆⠀⠀⢿⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠚⠉⣿⣼⡄⢀⡛⠔⢂⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⡼⠋⠻⠇⠘⠁⠀⠈⠀⠀⢫⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠀⠀⠀⠀⠀⠀⣀⢠⡄⢠⡬⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣏⠀⠀⠀⠀⣠⡴⠞⣅⠀⡃⢘⠃⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣝⠀⡀⠀⡾⠁⣇⠠⢏⠘⠇⢸⣷⠘⢿⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⡳⠉⣸⠇⢸⢾⡄⠸⠃⠀⠸⢻⡄⣷⣝⢿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠔⣠⣾⡏⠀⣾⢸⠀⠀⡄⠀⠠⠋⠁⣿⠉⠀⠀⠩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠁⠈⢠⡿⠟⠀⠀⣿⢸⢰⡇⢰⡖⣧⠀⣾⣿⡇⠀⠀⠀⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠁⠀⣸⣿⠟⠀⠀⢰⡿⠀⢾⣇⠀⣷⡿⠀⣿⣿⣇⣈⣶⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⢀⣴⣿⣷⠀⠀⠀⢸⡏⠀⢸⣿⡄⢹⣷⠀⣹⣿⣿⣿⣿⠀⠀⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⢠⣿⣿⡿⠃⠀⠀⠀⢸⡇⠀⣸⣿⡇⠈⡷⠀⢿⣿⣿⣿⣿⠃⠀⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠌⠀⠈⢈⣋⠀⠀⢀⣶⣆⢸⡇⠀⣿⣿⣧⠀⢁⠀⢸⣿⣿⠏⠙⢦⠀⠀⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢻⠀⠀⠀⠈⣿⡿⠀⠇⡆⢹⣿⣿⡆⠘⣦⠼⣿⣿⠀⠀⠀⢁⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣀⢈⣤⠁⠀⠀⣾⠀⡍⠀⠀⠀⡇⢸⣿⠇⠀⠀⣧⠀⢻⡟⠀⠀⠀⠀⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⣰⠀⣼⡇⠀⠀⠀⡟⠀⡿⠁⠀⠀⠘⠀⣾⠃⠀⠀⠀⠀⡀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠀⢠⣿⡞⠁⣠⣠⠀⢰⡇⢠⣧⡀⠀⠀⠂⢠⡟⠀⠀⠀⠀⠀⠰⠄⠠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡈⠀⢸⡿⠀⠀⣿⠡⠀⢸⣿⠏⠀⢻⠀⠀⠹⣾⠁⠀⠀⠀⠀⠀⠀⢠⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⡀⠁⠀⣸⠃⠀⠀⠉⠀⠀⣿⡏⠀⠀⠸⡆⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠘⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⠏⠀⠀⠠⠇⠀⠀⢿⠀⠀⠀⡀⠀⠀⣿⠇⠀⠀⠈⠁⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⢀⡞⠀⠀⠠⠶⠀⢀⡿⡄⠀⠀⠸⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⡟⠀⠀⠀⣧⠀⠀⠀⣱⠂⠀⠀⢠⡀⠀⠀⡼
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⡇⠀⠀⠀⠘⠇⠀⢠⡟⠀⠀⠀⠈⠧⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠌⠀⠀⠀⠀⣼⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⣧⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠰⠄⣼⠃⠀⠀⠀⠀⠀⠈⢀⡇
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢸⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉
⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢐⣷⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⣿⣿⣷⣵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠸⡟⢿⣿⡏⠀⠈⢁⡄
⣬⣥⣀⠀⠀⠀⠀⠀⠀⠈⣸⣟⡀⠀⠀⠀⠀⠀⠠⢴⣾⣷⣆⠀⠀⠀⠀⠀⠀⠻⣾⡛⠳⣦⣶⠋
⠻⣿⣿⣷⣤⣄⡀⠀⢀⣼⣿⣿⣿⣦⣄⠀⡀⣠⣾⡿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠋⠃
⢼⣶⣿⣿⣟⣿⣿⣤⣿⣿⣿⡇⢻⣿⢿⣷⣿⣿⣷⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠻⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⡇
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠰
⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠈
⠀⠀⠀⠀⠀⠞⣁⣤⣄⣠⣤⠆
⠀⠀⠀⠀⢠⠆⠙⠋⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⢶⡇⠀⢠⣤⠀⠀⣀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⣼⡇⠀⢰⣷⠀⠀⡉
⠀⠀⢀⣶⠇⠀⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⡆⠀⣿⡇⠀⠀⣿⠀⢠⡏
⠀⠀⢸⡇⠀⠀⢠⣶⣾⠀⠀⠀⠀⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠀⢹⣿⠀⠀⣿⠀⢸⡇
⠀⠀⢸⣷⡀⠀⠀⠀⠉⠓⠀⠀⠀⣀⣠⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⢿⡷⣄⣼⡸⣄⣀⠟⠳⠞⠙⠁
⠀⠀⠈⠻⠿⠶⠦⣴⢶⣦⣤⣶⡾⠟⢻⣿⡋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠓⢦⡉⣻⣿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠻⠿⣿⡛⠻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠸⣷⠀⢹⣯⢿⡆⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢺⣧⡀⠀⣿⣂⣼⣿⣿⠃⠀⠀⠀⠈⣿⡉⠁⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⢿⡿⠛⠛⣿⠛⠁⠀⠀⠀⠀⠀⢸⣧⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢘⣷⣶⠾⠋⠀⢀⣠⣤⡀⠀⠀⠀⢿⡆⣸⡇⠀⠀⠀⠀⣰⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠶⠶⠶⠶⠿⢿⡏⠀⠀⠀⢴⣿⡋⢹⣇⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⢠⣿⠁⢉⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠙⢿⣮⣿⡄⠀⠀⠀⢹⣿⠀⠀⠀⠀⣾⣧⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠁⠀⠀⠀⠸⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣀⣤⣶⠾⠛⠛⠋⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢀⣤⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉



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