Forensic science Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠛⠛⠉⠉⠉⠉⠉⠛⠛⠷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠉⠉⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣀⣴⠾⠛⠉⠉⠁⠀⠀⠉⠉⠛⠷⣦⣄⠀⠀⠙⢷⣄⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⢠⡾⠋⠁⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠻⣦⠀⠀⠀⠈⢻⣆⣀⣀⣀⣀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢹⣧⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⣿⡆
⠀⠀⠀⠀⠀⢠⣿⠀⠀⢰⡟⠀⠀⠀⠰⠶⠶⠶⠶⠄⠀⠴⠶⠆⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢿⡄⠀⠸⣧⣀⣀⣀⣀⣀⣀⣿⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠈⣿⠀⠀⢸⡇⠀⠀⠈⣿⠉⠉⠉⠉⣿⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣧⠀⠀⢹⡇⠀⠀⠀⠴⠶⠶⠶⠶⠆⠀⠀⠴⠶⠶⠶⠶⠆⠀⠀⢰⡟⠀⠀⣸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢀⣿⠀⠀⠀⠀⣿⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠀⠀⠀⠀⣠⡿⠁⠀⢀⣾⠃⠀⠀⠀⠀⣿⡇⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⢿⡇⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⠛⠻⠶⣦⣤⣤⣤⣴⠶⠿⠛⠁⠀⠀⣠⡾⠟⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⡿⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⠏⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⠶⠶⠶⠶⠶⠾⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣾⠛⠉⢹⣿⠁⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠟⠋⢻⡇⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠈⣿⡀⠀⠘⣷⣄⣀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠿⣧⣤⣴⣿⠉⠙⣿⠷⢶⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠾⠛⠋⠁⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⠾⠛⠛⠉⠉⠀⠀⠀⠈⠁⠈⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⠛⠋⠉⠀⠀⢀⣀⣠⣤⣴⡶⠾⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣤⣤⡶⠾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣀⣤⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠒⢤⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠛⠒⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢰⡂⢀⠀⡠⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠾⢶⣠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈
⠀⠀⠈⢷⡀⠠⢄⡴⠁⠀⠀⠆⠀⡀⠀⠀⠀⠀⠁⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⢾⣷⢸⣿⠀⠀⠀⢸⠁⠠⠲⠂
⠀⠀⠀⠀⠀⠈⠛⡀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉⠀⠀⠀⠀⢣⡮⠄
⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠔⢤⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠃
⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠐⡼⠀⠀⢀⡄⠁⠀⠀⠀⠀⡔⠀⣀⠀⠀⠀⢤⣤⣾⠄⠀⠀⠀⠸⠐
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠒⠢⢄⣀⠀⢠⡄⠠⡜⣦⠇⠀⠀⠀⠀⠀⣿⡋⠁⠀⠀⠀⣾⣟⡩⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠴⣿⡷⡀⠀⡀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠋⠛⠒⠀⠀⠀⠊⠫⠄⠀⠀⠀⠀⠀⠀⡔⠁
⠀⣸⠉⠀⠀⣲⠚⢿⣿⠀⠒⠘⠁⠀⠀⠡⡀⣀⠄⠀⠀⠀⠀⠀⢀⣠⡤⡄⠀⠀⢟⣷⡴⠀⠀⠀⠀⠀⠀⠐
⠀⠉⠀⠀⠀⠀⠀⠾⠓⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠉⠉⠁⢠⠴⠶⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣄⠀⠘⢄⢤⣤⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⣄⢀⡀⠈⠠⡀⠀⠀⠀⠤⠬⠟⠋⠀⠺⣠⡼
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡄⠤⠄⠀⠤
⠀⠀⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢆⢀⡴⠋⠀⠀⠀⠀⠠⢔⣦⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⡠⢠⠃⠀⠹⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠈⠀⠠⠔⢀⡀⡀
⠀⠀⠀⠀⢀⣊⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠰⡇⣰⠆⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠖⠀⠈
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⢀⣴⣳⣄⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⣰⠟⠉⠛⣦
⠀⠀⠀⣾⣿⡯⣼⡿⠛⠃⠀⠀⠀⠀⠈⡏⠉⠉⡉⡇⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⣿⡿⠀⠀⠀⠀⠀⢿⡀⠀⠀⣼
⠀⠀⣾⣿⠟⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡇⠲⠞⠁⡇⠀⠀⠀⠀⠀⠘⡿⠇⠀⣰⣶⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⢷⣄
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠾⠇⠀⠀⠀⠀⠀⠀⠸⣿⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⢀⣠⣤⣶⣷⢄⡀⠀⠀⠀⠀⠀⠀⠸⣏⢉⡿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣧
⠀⠀⢿⣝⢿⣯⣶⣾⣿⠆⠀⠀⠀⠀⠀⠰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⡻⡿⢃⡀
⠀⠀⠀⠙⢷⡝⢋⠅⠒⠁⠀⠀⠀⠀⢀⣀⣹⣿⣁⣀⠀⠀⠀⠀⠀⠀⢠⣬⣟⣯⣤⠀⠀⠀⠀⠀⠀⢿⣻⣿⣿⣿⣻⠇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀
⠀⠀⠀⣿⢹⣯⣿⢹⡃⠀⠀⠀⠀⠀⢠⣴⣾⣼⣶⣤⠀⠀⠀⠀⠀⠐⣿⣿⡿⣿⣿⡗⠀⠀⠀⠀⠀⠿⡚⠉⡏⠙⠺⠇
⠀⠀⠀⣿⢿⣈⣹⢿⡃⠀⠀⠀⠀⠀⠀⣿⠀⡟⢸⡇⠀⠀⠀⠀⠀⢠⣿⣷⢀⣰⢿⣧⠀⠀⠀⠀⣬⣀⣱⠀⣧⢰⣁⣈⡄
⠀⠀⠀⠿⠼⠯⠿⠼⠇⠀⠀⠀⠀⠀⠠⠿⠤⠯⠼⠧⠀⠀⠀⠀⠀⠀⠻⠿⠦⠶⠶⠀⠀⠀⠀⠀⠈⢭⣥⣴⣿⣤⣭⠍


⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣄⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⢄
⠀⠀⣠⣤⡀⠀⠰⡇⠀⣷⠀⠀⠀⠀⠀⠹⣦⠤⡾⠁⠀⠀⠀⠀⠀⠀⠐⢫⣾⢆⡀⠀⠀⠀⠀⠀⠀⠀⠸⣸⣪⡏⡆
⠀⠸⣅⣀⣿⠀⢀⣿⠏⠁⠀⠀⠀⠀⠀⢀⡬⠷⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠑⢋⣀⡀⠀⠀⠀⠀⠀⠀⣓⣚⢕⡽⡥
⠀⠀⠈⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠾⠒⠒⠺⠆⠀⠀⠀⠀⠀⠙⠁⠀⠰⠿⠿⠿⠆⠀⠀⠀⠀⠀⠘⠿⢷⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡑⠲⠦⣄⣶⣤⡀⣤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠠⣈⠢⡙⢦⡙⢎⠃⠐⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⣀⡶⢲⣤⢄⣴⠒⠀⣅⣤⠑⢈⠢⡁⠠⠑⠈⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⡥⠒⣖⣫⠤⢭⣍⣛⠓⠆⢋⡸⠏⠢⠳⠄⠂⠀⡉
⠀⠀⠀⠀⠀⠀⠀⢀⠐⣣⢴⠉⡀⠁⠄⠀⠐⠀⠈⠁⠀⠂⠀⠐⠈⠁⠆⡃⢀⠀⠀⠀⠄⠁⠀⢀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢈⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠌⠢⠐⠀⠀⠀⠢⠀⠀⠀⠀⠀⠤⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠊⠂⠂⠓⠀⠀⣀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡈⠀⠀⠠⢀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠈⠰⠀⠃⠂⠜⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠌⣠⠄⡀⢀⠀⠠⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡦⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⢀⠆⠈⠀⠀⠆⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠏⣸⠣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⢠⠃⠴⢠⠀⠠⠀⠁⢀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⢩⠄⡀⠀⠀⠀⠀⠀⠀⢀⠈⠰⢀⠜⣀⠂⠐⠂⡴⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠄⠐⠠⠂⠐⠀⠞⡀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠈⠀⠀⠀⠀⠀⠀⠁


⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⡤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣤⣤⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡾⠁⠐⠔⢤⡈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⢉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠧⠓
⠀⠀⠸⡇⠈⠁⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣠⣴⠟⠁⠀⡇⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠉⠀⢀⢀⣀
⠀⠀⠀⠹⣦⣀⠒⢃⣠⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⢿⠁⡠⣀⠀⡇⠀⠀⠀⠀⣀⣤⣿⠟⣉⣤⣴⣶⣿⠿⠛⠙
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠻⣦⡀⠀⠀⠀⠀⠀⠈⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⡤⠆⠃⠙⠀⡇⠀⠀⠀⠀⠈⠈⠇⠗⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠩⣗⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣄⡀
⠀⠀⠀⠀⣤⣳⣄⠀⣿⡟⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡨⡪⠄⠐⠫⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⡋⠉⠁
⠀⠀⢀⡾⠋⠉⠻⣦⢿⡀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡔⡐⡩⡂⢍⢎⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⢹
⠀⠀⠈⢷⣀⢀⣠⠟⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢄⠚⣠⡢⠁⠀⠀⠀⠀⠀⠀⢴⣶⣿⠛⠻⠥⠴⢪⣀
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⠌⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠙⢦⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⣷⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣹⣿⠀⡸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⠿⣷⢱⡿⢿⡇
⠀⠀⠀⠀⢠⢼⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣿⠀⢧⣀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⣃⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣛⣙⣻⣿⡇
⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡭⢭⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⡝⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣺⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣻⣿⣁⢀⣀⠀⠀⠀⠀⠀⠀⠸⠷⢾⢿⢿⡿⡿⡿⢿⠿⠃
⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣌⣉⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⣋⣙⠻⡬⠇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣠⣾⠓⠁⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠿⠿⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋
⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠷⠾⠿⠿⠃

⠀⠀⠀⢀⡾⠿⣆⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⣀⡀
⠀⠀⠀⠐⢿⣿⠇⠀⠀⠀⠀⠀⠀⢺⡿⢿⡿⢻⠟⠛⠉⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⣶⣷⣆⢸⡟⠉⠛⣆
⠀⠀⣠⣤⣄⣃⣤⣄⠀⠀⠀⠀⠀⡞⢠⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢴⣮⣙⣛⣋⣉⣩⣶⡤⠀⠀⠀⣾⠁⠀⢹⡞⢧⣀⣤⠏
⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠳⠶⠊


⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⣀
⠀⠀⢠⠐⢈⠈⠑⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⢀⣀⡀⠀⠀⠀⢀⠾⢿⣿⣿⣾⡿⠷⡀
⠀⠀⠘⡄⡠⢀⡀⠀⠀⠀⠀⠀⠈⠹⠿⠛⠉⠁⢱⠤⢤⡆⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⣿⡷⠀⠀⠀⢧⡀⣸⠀⠉⠸⣄⢀⡜
⠀⠀⠀⠀⠃⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁


⠀⠀⢀⣀⣀⣄⣠⣀⣠⡀⠀⠀⠀⠀⠰⣿⣶⣿⣿⣶⣿⠆⠀⠀⠀⠀⠀⢠⣼⣿⣿⣧⡄⠀⠀⠀⠀⠀⣴⠛⠉⠳⡄
⠀⠀⢻⣿⣿⠛⣭⣝⢳⣾⠀⠀⠀⠀⢠⣿⡿⠋⠙⠻⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢷⣀⠀⣠⡇
⠀⠀⣿⣿⣿⣎⡛⢋⣼⣿⠀⠀⠀⠀⢸⣿⡷⣀⣀⢾⣿⡷⠀⠀⠀⠀⠀⠀⣤⣄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣦⡄
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠻⠿⢿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠖⣤⣦⢤⡀⠀⠀⠀⠀⠀⠀⢰⣶⢶⣶⣶⡆⠀⢀⡀⠀⠀⠀⠀⣠⣶⡄⠀⢠⡿⢻⡄⠀⠀⠀⠀⠀⡾⠭⠭⠭⠭⢷
⠀⠀⢰⠸⣧⡂⣑⡇⠀⠀⠀⠀⠀⠀⢸⣿⣹⣿⣿⣇⠔⡵⠃⠀⠀⠀⠀⢿⣠⣿⠀⠸⡿⣾⠁⠀⠀⠀⠀⠀⡇⢀⣠⠖⠀⢸
⠀⠀⠀⠐⠝⠛⠉⠳⣤⡀⠀⠀⠀⠀⢸⣿⢺⣷⣾⡟⠋⠀⠀⠀⠀⠀⠀⠘⠊⠉⠀⢰⠀⡄⠀⠀⠀⠀⠀⠀⡇⢛⠏⠀⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣬⠇⠘⠛⠁⠀⠀⠀⠀⠀⠀⠣⠬⠣⠤⠄⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣐⣒⣒⣒⣲⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤
⠀⠀⠀⠀⠀⠀⢀⣠⣶⡷⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⡄⠈⡁⠀⠀⠀⠀⠀⠀⢀⠈⡊⠑⠚⠦
⠀⠀⠀⠀⣠⣶⡿⠟⠉⣀⣠⣤⣰⣤⠀⠀⠀⠀⠐⠀⠀⠈⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⣅⣴⡾⠋⠀⠀⠅⠀⠀⠀⠀⠀⠀⠨⣇⢀⠂⠠⠢
⠀⠀⠰⠾⣝⡥⣶⣾⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢉⡁⠸⠳⠢⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠃⣴⠄
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠓⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡀⠀⠀⠀⠀⠀⠀⣠⣂⣍⡒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡧⠙⠀⠀⠀⠀⢀⣤⢿⣤⠀⢱⠀⡇⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣽⠀⢀
⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⣟⣚⣻⠀⡌⠀⡇⠀⠀⠀⠀⠀⢽⣿⡿⢡⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠈⣉⠁⢀⡼⠃
⠀⠀⠰⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠯⣿⡀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢙⣳⠞⠦⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⢰⣶⠶⣶⣶⣶⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢠⡟⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠶⠮
⠀⠀⢸⣿⣛⣿⣛⣿⣇⠴⣫⠞⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣏⣘⡧⠀⢸⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⠆
⠀⠀⢸⣿⣭⣿⣭⣿⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣸⣡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣇⣴⣤⡀
⠀⠀⠸⢿⡶⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡆⠘⠞⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠠⢂⣤⣶⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣤⡀
⠀⠀⠀⣀⣤⣶⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⢆⢸⣧⠐⢅⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣤⡤⡀
⠀⠀⠈⠹⠿⠛⠋⠁⠈⣧⠤⢤⣦⠀⠀⠀⠀⠀⢀⣴⠟⡛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠙⡿⠛⠙⢷⣤⡀⠀⠀⠀⠀⠀⢨⣿⣿⡉⣶⡎⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠘⠿⠿⠷⠮⠶⠿⠇

⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠩⡍⠉⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⠀⢐⣢
⠀⠀⠀⠐⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⠀⠛⠒⠒⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⢀⣀⣀⡰⣄⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢐⡊⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢬⡳⡄⠀⠀⠀⢸⠰⠳⠀⠀⠅⢨⠀⠀⠀⠐⠀⠀⢀⠂⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠒⠚⠒⠂⠀⠚⠂⠀⠀⠀⠀⠀⠐⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⡄⠒⡁⠵⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢉⠀⢀⡖⡀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠈⠰⠤⠃⠈⠀⠀⠀⠀⠀⠆⠩⠈⠅⠏⠸⠡⠂⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠠
⠀⠀⠀⠔⡒⠒⡢⠈⢀⠄⠀⠀⠀⠀⠀⠈⠀⠁⠅⢸⠈⠀⠀⠀⠀⠠⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⡄⠒⠀⠂⠀⠐⠆⠀⠀⠀⠀⠄⢨⢨⠀⢄⠠⠀⡄⠀⠀⠀⢄⠀⠀⠉⢠⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠐⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢰⠀⢠⠀⠠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠰⡀⠀⡦⣅⢀⠀⠀⡀⠀⠀⠀⢠⠂⢠⠄⠀⠎⣀⡭⠀⠀⠀⠀⡈⠁⠈⠄⠁
⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡀⠢⢄⠅⠀⠀⠀⠀⠀⠙⠤⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠰
⠀⠀⠀⠀⠀⠈⠀⢀⠀⡁⠀⠀⠀⠀⠀⠀⠠⠀⡅⠀⠀⠀⠀⠀⢀⠐⢤⠖⠖⠚⠀⠀⠀⠀⠀⠀⠀⠀⡰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠂⠊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢐⠆
⠀⠀⠀⠀⠂⠀⠀⠀⡈⠂⠀⠀⠀⠀⠀⢀⠄⠊⠁⣀⠀⠀⠀⠀⠀⠀⠀⡁⢈⣚⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⡀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠
⠀⠀⠀⣅⣀⣈⣀⣀⣀⣀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡀⢀⠀⠀⠀⠀⠀⠀⠄⡀⠡⠄⠤⢠

⠀⠀⠀⠀⠐⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢛⣛⣛⣛⣛⣛⡛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠆⠀⠰⢦⣀⡀
⢀⠀⣔⠅⠉⠉⠁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢀⣦⠟⣀⣀⣀⡰⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠂⢙⠒⠒⠒⠊⢱⠜
⠀⠘⠀⠂⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⢀⠤⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠂⠀⠀⠀⠀⠀⠀⠀⢑⡃⠈⠉⠀⢸
⠀⠰⡀⠀⠀⠀⢀⢬⣆⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⠄⢨⠀⡇⢘⡇⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡃⠈⠉⠁⢸⠐
⠀⠀⠈⡊⠁⠉⠚⢧⡪⢿⣄⠀⠀⠀⠀⠀⠀⢸⠀⢰⢃⠀⠀⢘⡤⠁⠨⡇⠀⠀⠀⠀⢀⠔⣪⠔⠛⢭⠭⡭⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠈⠉⠀⢸⢈
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣉⠷⠀⠀⠀⠀⠀⢸⣄⡉⠉⠀⠀⠈⠒⢀⣴⠃⠀⠀⠀⠀⠈⠉⠀⢀⠤⠮⠥⠽⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣤⣤⣤⣤⣤⣼⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡠⢴⠂⢉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢀⡔⠚⠣⡤⠔⠮⠤⡇⠀⠀⠀⠀⠀⠀⠀⢠⠐⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠈⣀⣀⣀⣰⠉⡢⠀⠀⠀⠀⠀⠀⠀⡌⠣
⠀⠀⠀⢸⢤⣁⢢⠆⠀⠄⠀⠁⠀⠀⠀⠀⢀⣀⢈⡒⠒⣚⢈⣓⠒⣚⢀⡠⡀⠀⠀⠀⠀⠀⠀⡤⠤⠀⡀⠀⠀⠈⡏⠃⠀⠀⠀⠀⠀⠀⠀⢰⠀⠱⡀
⠀⠀⢀⠀⢁⠙⠉⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢬⠠⡤⠠⡤⠠⡤⠄⡤⠨⡇⠃⠀⠀⠀⠀⠀⡰⠀⣤⠀⠱⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢀⠐⡄
⠀⠌⢉⡉⠉⢉⡢⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠀⡆⠀⡇⠂⠀⠀⠀⠀⢰⠥⠤⠤⠒⠀⢣⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢣
⠀⣀⠠⠇⠀⠠⠄⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⡁⠀⡂⠀⡄⠀⡇⡁⠀⠀⠀⠀⡃⠀⠀⠀⠚⠃⠀⡇⠀⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡔⠆
⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠉⠈⠁⠀⠉⠈⠁⠀⠁⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⣀⠄⢒⣂⣀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣒⢒⡲⠀⢀⡄
⢀⠞⠴⡈⠠⠀⠀⡤⢌⠢⡀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡄⢣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡒⠀⠀⠀⢀⠴⠊⢽⡀⠀⠀⠀⠀⠀⠀⠸⠂⠒⡇⠐⠲
⡌⡈⠰⡈⠆⢠⢠⠑⠀⢀⠰⠀⠀⠀⠀⠀⠙⠤⠈⢉⡗⢹⡆⠤⠄⠄⣒⠠⡀⠀⠀⠀⠀⠀⣏⠀⣠⠂⠈⠁⠙⠢⢤⡺⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⠰
⡇⠇⢰⠄⢀⠤⢅⠀⡸⠜⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠉⣏⠀⠒⣘⠄⢄⠇⠀⠀⠀⠀⠀⠀⠈⠙⣦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢕⠀⠀⡠⠂⠐⠖
⠱⠘⠄⠀⠐⠄⠴⠀⡔⢒⠞⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⡀⠀⣠⣥⣈⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠈
⠀⠀⠠⡀⠠⠤⠤⠤⢖⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡩⣐⠂⠀⠀⠀⠀⠀⠀⠀⢘⣄⣹⡛⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠵⣄
⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈⠈⠁

⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡀⠀⠀⠀⠀⠀⠀⠂
⢘⠀⠀⠀⡀⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⢢⠀⠀⠀⠀⠀⠀⠀⡔⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠒⠒⠒⠒⠒⠒⠒⢒
⢀⠀⠰⡉⡠⠄⠉⠁⡊⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠇⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⢀⠠⠈⠎⢀⣒⠄⠎⠂⠇⡄⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⢠⡡⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡗⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠰⠂⠁⠲⡤⠄⣄⢨
⠀⢸⠀⢀⢀⠸⢸⠀⠃⡀⠃⡇⠀⠀⠀⠀⢠⠎⣀⠠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⢄⣀⠠⠁⠀⠀⣴
⠀⣸⢠⠁⢸⢀⠸⠀⠄⠁⠀⡆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠃⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡀⢀⡴⢨
⣤⣿⣿⣾⣿⣾⣾⣶⣶⣶⣶⣦⣴⣴⣶⣶⣴⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣾⣶⣶⣶⣦⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣈⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢿⣿⣂⠘⠛⠂⠚⠛⠷⠶⣶⣤⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡜⠛⠋⣈⣻⣿⣟⣻⣛⣛⣛⣛⡛⠷⠶⣬⣝⡃⢩⣅⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣴⠿⣛⣩⣭⣭⣭⣭⢙⣻⣯⣭⣟⣻⠶⣶⣭⣛⠶⣮⣁⠶⣭⣻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢘⣟⠉⡴⠟⣋⣭⣴⠶⠶⠶⠮⣭⣭⣬⣭⣍⡛⠷⣬⣙⠻⣦⣙⠳⢦⣝⠳⣬⣀⣤⡀
⠀⠀⠀⣀⠀⠛⢶⡆⠉⡰⠶⠛⣫⣵⡶⠾⢛⣻⣷⣶⣶⣶⣦⣬⣝⡛⢶⣭⡻⢶⣍⡻⢦⣝⠷⣮⣙⠿⣯⡀
⠀⠀⣾⡟⣼⡿⢸⡟⣨⡛⠀⠛⣋⣤⣶⠿⢛⣋⣭⣭⣤⣤⣤⣬⣍⣛⠷⣮⣝⠷⣬⡻⢦⣝⠷⣮⣝⠷⣬⡻⢷⣄
⠀⢸⡟⣰⡿⢡⣿⣰⣿⢁⣤⡘⢛⣩⣴⠾⣛⣩⣥⣤⣤⣤⣤⣭⣭⣝⣛⠶⣿⣷⣬⡻⢶⣭⡻⣮⣝⢷⣬⡛⠆⣙⠿⡄
⠀⣼⡇⢿⡇⣘⢹⣿⢡⣿⠏⣴⠟⣫⡴⢞⣛⣭⣥⣷⡦⠴⠶⠶⣬⡍⣛⠻⢶⣤⣍⡻⢷⣬⡻⣶⣝⠿⣿⡛⢦⣌⠳⣦
⠀⡙⢡⣼⢻⣿⢸⡇⢸⣏⣾⢃⣿⢫⣶⡿⢛⣩⣴⡶⠶⠿⠿⠶⠶⢦⣭⣛⡳⢦⣭⣻⣦⣝⠻⢦⣙⠷⣮⣛⢦⣝⠳⣤⡘⡦
⢸⣿⢸⣇⢸⠇⣿⢧⣿⣿⣿⠸⣿⠈⣉⣴⠿⣋⣵⡶⠿⠟⣛⣛⡿⠶⣦⣍⣛⠳⠮⣽⣛⡻⠷⣦⣍⡓⣬⣙⠷⣽⡷⢤⡙⠇
⠈⣿⠀⣿⢀⣶⢀⣾⠏⣾⢃⣼⠏⣼⡿⢋⣼⡿⢋⣴⠟⠛⣛⣛⡛⠻⠶⣮⣝⣛⠳⠶⢮⣝⡻⢶⣝⡻⢷⣝⣷⣌⠻⢷⡍⠶⣄
⢰⣿⢰⡟⣼⡏⣾⡟⣼⡟⣼⡏⣸⣿⣡⣿⠟⣀⠛⣡⣾⠿⠟⠛⣛⠛⠶⣦⣬⣙⠛⠿⣶⣬⣛⠷⣿⡻⣿⣝⠷⣌⠻⣤⡙⠧⠁⢀⡀⠻
⢸⡇⣾⠇⣿⠂⣿⠇⣿⢰⣿⢱⡿⣱⣿⠉⠺⢋⣾⡟⢁⣴⣿⠿⠛⠛⠷⢶⣬⣝⣛⠷⣶⣬⡙⠿⣮⡻⣷⣝⢷⣄⠀⢬⠱⣦⡘⢦⡀⠄
⣤⠐⠿⢠⣿⢸⡿⣸⡏⣾⡏⣼⡇⣻⡏⢸⣇⠸⣿⢠⣿⠟⣡⣾⠟⠛⣿⣦⣬⣙⡛⠿⣶⣌⠻⢷⣌⡙⣦⡙⢷⡌⢳⣌⠿⣎⠻⣄⡙⣦
⢈⢸⣷⢸⣿⢸⡇⣿⡟⣿⢱⣿⠠⣿⣿⢸⣧⢰⡀⢹⡏⣰⡿⢁⣼⡿⠛⢉⣉⡙⠻⢷⣦⣝⠻⣦⣉⢿⣦⡙⢦⡘⢷⣌⢷⣌⠳⣮⡛⠎⠻⠂
⠈⢘⡏⢸⣿⢸⡇⠈⠁⣿⡞⣿⡈⣿⡙⣇⢻⡄⢿⣇⠁⠻⠇⢸⠏⣴⣿⠿⠛⠛⠻⣶⣍⠻⣷⣌⡛⣷⣝⠻⣦⡙⢧⣿⣧⡙⠳⣌⠒⢀⡀⢀⡀⠶⠄
⠀⢸⣷⠸⣿⢸⡗⠀⠀⡙⢷⡹⣧⢹⣷⣸⡎⣿⡘⣿⡄⢇⠀⣶⠀⠈⠁⠀⠀⠀⠀⢀⣉⠳⠎⠉⠁⠈⠛⣷⣾⠻⣆⣽⢦⡛⢧⡙⢧⣌⡱⣌⠛
⠀⠈⣿⠀⣿⢰⡇⢠⡀⠛⠈⠃⣿⡆⣿⡈⢡⣿⡇⠿⠃⠀⡀⠹⣧⠂⠀⠀⢀⠹⣷⣤⠛⣷⣄⠀⠀⡘⢿⣌⠻⢷⡉⢳⢆⠙⢦⡙⢢⡌⠓⢈⣃
⠀⠀⢠⣦⠻⠀⡃⢻⡇⢿⡀⣦⠘⣗⢸⣧⣸⣯⣿⠀⠀⠀⠰⡄⠙⢧⡀⠀⠘⢷⣌⠻⣷⣞⠻⠷⠎⢿⣆⠻⢷⣈⠥⣄⠻⣟⡀⠙⠦⡀⠂⢠⡍⠓
⠀⠀⠀⠉⢰⡇⢿⠊⣋⠸⣧⢺⡆⢿⣄⢻⣯⢿⣆⠀⠀⢀⡆⢹⣆⠈⣻⡆⠀⠀⡙⢷⣌⠛⠀⠀⠀⠀⣛⢂⡐⠈⢠⡌⠀⠀⠙⠆⣀⠉⠰
⠀⠀⠀⠀⢴⣤⡌⣰⣧⣦⢹⣆⢿⡀⠻⠌⠿⠛⣿⠄⡀⠸⡟⠈⢿⡄⢹⣆⢰⣄⢸⣌⢻⣷⡄⣠⣤⣌⣟⠻⠷⣶⣏⡀⠄⢠⡀⠰⣤⠈⠀⠀⣀⠀⠀⠶
⠀⠀⠀⠀⠀⠈⣷⢿⣏⢿⡆⢿⡌⣷⠀⣆⠸⢷⣹⣆⢿⡆⣏⠀⠀⠀⠈⠋⠄⠸⣦⡝⠳⣬⣙⠻⣶⡏⡙⠃⠀⠀⣈⣙⠶⣠⡙⠳⣬⠁⡀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠹⠊⣿⡘⣿⣌⢿⡘⠇⠻⠀⠀⠝⢿⣄⠻⡜⢦⠀⠀⠀⠀⡘⠔⢈⠛⠀⢠⡉⠋⢵⡌⠫⣀⡶⡤⠍⠛⣶⣌⡻⠰⠦⠀⠻⠀⠀⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣈⣷⢀⡻⣦⡀⡄⠰⡄⠉⣙⠷⠙⢂⡁⣦⡐⢀⡁⠐⠌⠛⠀⠀⠙⠛⢶⣆⣰⠎⠓⢦⣤⠀⣺⠿⢷⠲⣌⠀⠐⠳⠂⢀⡠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢻⡽⡟⣷⡘⢷⠙⠀⠈⠦⠉⢠⡜⢮⡻⢎⡱⠀⠑⢤⣴⠃⠀⠀⠛⢧⣆⣙⠐⠻⢧⣀⣼⠛⡉⠐⠦⣤⣉⠙⠃⠠⠆⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠎⢷⠠⣾⢳⡌⠳⠀⠀⠀⠙⢈⠻⣎⠓⣌⠳⣜⣦⡆⠉⠁⣀⡀⠄⠀⠤⣙⠛⠲⠆⠀⣀⠉⠋⠀⠐⠛⢡⡆⠙⣲⡤⠄⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⠹⠆⠀⣤⠘⢧⣄⠤⠈⠣⡄⢣⣝⠷⡙⢷⠁⢀⡌⠻⣧⣀⠀⠀⠈⣑⠿⠶⢤⣄⣉⡉⠉⣀⡀⠁⠀⠛⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣧⡹⡦⠲⡉⢶⡳⠆⠙⣄⠉⠆⠙⣬⠂⠀⠹⣌⠡⣌⠃⠠⠀⠈⠻⠓⢤⡤⢋⣈⠀⢉⣁⣄⠀⠈⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠀⠉⠘⡇⠀⠀⠀⠃⠀⠀⠈⢳⣄⡄⠈⠓⠈⠛⠦⡄⠀⠈⠀⠘⠀⠨⣅⢀⣠⣁⠀⠀⢄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣰⡆⠇⠀⢆⠻⡄⠠⢦⠀⢂⣘⢳⣄⣀⢀⣘⠳⡖⣀⠀⠑⠶⠶⢤⣄⡉⢄⢋⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡻⣄⠃⡀⠳⡄⠀⠀⠈⢠⡉⠲⣼⡓⠤⡌⢳⡤⢠⣉⠃⠀⠂⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠳⠀⠀⠀⠀⠀⠀⠀⠐⣳⣮⣓⠒⢀⠁⡰⠿⠉⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠈⠃⠀⠀⠋⠠⡼⠂⠠⠌⠀⠀⠀⠠⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⡂⠀⠀⠀⠀⠀⠈⠁⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⠿⠿⠿⠿⠿⢿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⡀
⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⢀⣠⣴⣶⠿⠿⠿⠟⠿⠿⠿⢷⣶⣤⣀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⣠⣿⠏⠀⠀⣠⣶⠿⠛⠉⢀⡀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠻⢷⣦⡀⠀⠙⢿⣧⡀
⠀⠀⣼⡿⠁⠀⢠⣾⠟⠁⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣿⡇⠀⠀⠀⠙⢿⣦⠀⠀⠹⣿⡄
⠀⣼⡿⠁⠀⣰⣿⠃⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢻⣷⡀⠀⠹⣿⡄
⢰⣿⠁⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⢻⣷
⣾⡟⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⣠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⡆
⣿⡇⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠛⠁
⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠻⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣤⡄
⢻⣷⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢠⣿⠃
⠘⣿⡄⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⣿⡇⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣼⡟
⠀⠹⣿⡄⠀⠘⢿⣦⠀⠀⠀⠀⠀⣿⣯⣀⣀⣀⣀⣀⣠⣿⡇⠀⠀⠀⠀⢀⣾⠟⠀⠀⣼⡿⠁
⠀⠀⠹⣿⣄⠀⠈⠻⣷⣄⠀⠀⠀⢿⡿⠛⠛⠛⠛⠛⠛⣿⠇⠀⠀⢀⣴⡿⠋⠀⢀⣼⣿⠁
⠀⠀⠀⠈⢿⣧⡀⠀⠈⠛⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠟⠋⠀⠀⣠⣾⠟⠻⣷⣄
⠀⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠈⠙⠛⠿⠿⠿⠶⠾⠿⠿⠟⠛⠉⠀⠀⣀⣴⣾⠟⠁⠀⠀⠈⠻⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡿⠟⠛⢿⣦⡀⠀⠀⠀⠀⢈⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠙⢿⣦⡀⢀⣴⡿⠋⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡿⠋⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⠤⠤⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡀⠀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡠⠔⠈⠀⠀⡌⠀⠀⡏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⢃⢠⠄⠀⢰⠀⠀⣸⠁⢸
⠀⠀⠀⠀⠀⢀⢀⠀⢤⡀⠀⣸⠀⠀⠈⠁⢸⠀⠀⡎⠀⢀⣿⠀⠘⡇
⠀⠀⠀⣶⠛⡀⠀⠀⢀⡷⢠⠃⠀⠀⠀⠀⠂⠀⢸⠀⠀⣸⣿⠀⠀⣧⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠻⠿⠗⠒⠉⠁⠀⡞⠀⠀⠀⠀⠘⠀⠀⠆⠀⢠⣿⣿⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣷
⢀⠦⠀⠀⠀⠀⠀⠆⠀⢰⠁⠀⠀⠀⠀⠃⠀⠘⠀⠀⣼⣿⣿⡄⠀⠸⡆⠀⢀⣿⣿⣿⠟⠉⠀⠀⢸⣿⣿⣿⣿
⠰⣿⣤⣤⣤⣤⣼⡶⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠛⠛⠛⠲⠖⠃⠀⣼⣿⣿⣏⣠⡄⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣤⣀⣈⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠞⠀⣀⣀⣠⣤⣤⠀⢰⣿⣿⣿⣿⣿⠁⠀⣸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠘⠢⢄⣉⡉⠓⠲⠤⢤⣀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⡏⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠯⣝⠻⣿⠖⠢⡀⠹⢦⠀⠀⠘⠿⠛⠛⠉⠁⠀⣸⣿⣿⣿⣿⣿⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠔⡹⢯⡀⡡⢂⠴⠒⠋⣶⣶⣶⣶⣶⣶⣶⠀⢠⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣦⢉⡴⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⡉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣷⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣻⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⣴⠾⠙⠷⢦⣄⠀⠀⠀⠀⠀⠀⠀⢠⢲⣀⣀⣀⣀⣴⡂⠀⠀⠀⠀⠀⠀⣤⢤⢤⢤⣤⡤⠤⣤⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠰⣽⠗⠉⠀⣨⣳⣿⡆⠀⠀⠀⠀⠀⠀⢸⢙⠒⡒⣒⣒⣪⡽⠀⠀⠀⠀⠀⠀⠛⠉⣉⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⢸⢉⣉⣉⡉⢉⣉⢉⡉⢳
⠀⠀⠀⢰⣏⠹⣉⡏⣈⠁⣨⡧⠀⠀⠀⠀⠀⠀⠀⠫⡧⣄⢠⡤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢣⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠶⢲⣼⣣⢰⢰⠀⢺
⠀⠀⠀⠘⡿⢗⡤⢼⢷⡸⢥⠃⠀⠀⠀⠀⠀⠀⠀⡠⡿⠿⠭⠯⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⠀⠀⠀⠀⠀⠸⣈⣙⣉⣁⣈⣉⣋⣁⠼
⠀⠀⠀⠀⠈⠓⢬⠛⣥⠐⠁⠀⠀⠀⠀⠀⠀⠀⢰⣙⣓⣒⣒⣒⣻⣇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⡓⡄⢐⠀⠀⠀⠀⠀⠁⠀⢀⣸⣀⣀⣋⡀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⠅⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⠐⠀⠂⠂⢺⠅⠀⠀⠀⠀⠀⠀⠲⠤⠤⠤⠤⠬⠤⠴⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡥⠃⣀⢄⠀⠀⠀⠀⠀⠀⢀⠀⡐⣽⠋⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠲⡄
⠀⠀⠀⡎⠀⢡⠄⠔⢤⢂⣠⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣴⠾⠒⠋⠀⠀⠀⠀⠀⡀⠅⢘⣏⢦⡔⢀⢼⡀⠀⠀⠀⠀⠀⠀⢠⡳⠀⣎⡱⢆
⠀⠀⠀⡇⠀⠀⣧⡄⢬⠖⠊⠀⣐⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢘⢠⢈⠳⢿⠓⠣⣙⡆⠀⠀⠀⠀⠀⠸⢦⠴⠁⢣⣀⣃
⠀⠀⠀⠓⠒⠒⠒⠒⠓⠒⠚⠒⠛⠀⠀⠀⢴⡨⢝⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠏⠰⠫⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣏⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠉⠀⠃

⠀⠀⠀⠀⠀⢀⠤⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⣦⣄
⠀⠀⠀⠀⠀⠈⡏⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠭⠭⠭⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢨⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢁⠤⠀⣄⠹⣧
⠀⠀⠀⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠨⢭⢴⡻⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⢿⣛⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠠⢑⠺⠙⢀⣿
⠀⠀⠀⠀⠀⡴⢤⠤⡀⣱⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠭⡇⡣⡄⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⡾⣶⣩⣴⣾⠿⡟
⠀⠀⠀⠀⠜⠀⠛⣧⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡹⣕⢬⣔⡝⢦⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣜⣻⠄⠊
⠀⠀⠀⠈⠦⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⠒⠶⡤⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢠⡀
⠀⠀⠀⡑⠒⠒⣒⡒⠒⡒⠒⠒⡆⡀⠀⠀⠀⠀⡇⢸⡇⢒⠐⠂⡂⠀⠀⠀⠀⠀⠀⠀⢰⠉⡇⠁⡏⢻⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⡏
⠀⠀⠀⠅⠀⠀⡄⠥⢔⡝⠀⠀⡇⡅⠀⠀⠀⠀⡆⢸⡇⢸⢂⠂⡂⠨⠀⠀⠀⠀⠀⢰⠹⠂⠃⠁⠃⡎⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⡧⢄
⠀⠀⠀⢇⡀⡀⠒⢁⠘⣊⣀⢀⡇⡀⠀⠀⠀⠀⡆⢸⡇⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠂⠀⠀⠀⠀⠀⡘⠀⣤⠏⠀⠀⠸⢇
⠀⠀⠀⠃⠉⠈⣒⠀⠉⡖⠀⠉⠘⠁⠀⠀⠀⠀⡇⠸⠇⠀⠤⠄⡤⡬⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⣶⠡⠤⢧⠀⠀⠀⠀⠀⠀⢓⠐⠈⠉⠉⠀⠀⡹
⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠘⠃⠊⠓⠘⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣾⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡆⠀⠀⠈⡻⣿⣁⠀⠀⢰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⡇⠀⠀⠀⠳⠿⠇⠀⠀⠸⣿⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣤⠀⠀⠀⢴⣶⡆⠀⠀⠘⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⢀⣸⣿⣧⣤⣤⣴⣿⣶⣄⣀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⣻⣴⣾⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣶⣤⣻⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⠟⠋⢹⣿⡇⠀⠀⠀⣶⡆⠈⠙⠻⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠃⠀⠀⠈⠛⠃⠀⠀⠀⠿⠇⠀⠀⠀⠀⢹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠻⠹⣿⣿⣧
⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠘⠛⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠈⢫⣿⣭⡃⠀⠀⠀⢸⣿⣿⠃⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⢸⣿⣿⠀⠀⠀⣼⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⠀⠀⢸⣿⣿⡀⢀⣼⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣸⣿⣿⣽⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡆⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣾⢿⣿⣿⣿⣿⣿⠇⢀⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⣒⡶⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠭⢉⡤⢐⣒⣒⣒⡨⣍⣛⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣤⣌⡉⠥⠄⠷⠿⠥⠌⣝⣓⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⡄⠀⠠⢤⡈⢳⣈⣓⡲⢦⡭⣭⣹⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⠠⢠⠀⠄⠀⠠⢀⠀⢠⠀⠲⡦⠭⣭⣒⣒⠂⠩⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠤⠄⠠⠤⡤⠄⠄⠀⠀⠀⠀⠬⣭⣒⡲⠭⣝⡒⢮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠀⠠⠠⠀⠀⠀⠠⠀⠤⠤⠠⠀⢐⣲⡮⢭⣓⠲⢭⣙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠠⠤⠤⠄⠄⠄⠠⠠⠠⠄⠄⠄⢀⣐⠍⢟⡺⢯⣁⡈⢍⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⠠⠰⠠⠀⠀⠆⠠⠠⠀⠀⢀⣆⡩⠭⣓⠠⢕⠲⢍⡒⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠶⠶⠂⠠⠀⠰⠠⠀⠀⠀⠰⠀⣠⣾⣏⣻⠲⢭⣀⠜⡲⠌⣐⢬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⡄⠀⠂⠐⠀⠐⠰⠀⠀⢀⣤⡾⠿⣟⡒⠈⢍⣺⢬⡿⣾⣗⠢⡉⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡷⡦⣄⣀⡀⠀⠀⠀⣀⣾⣿⣿⣀⠀⠀⢶⡒⠤⡂⢑⠬⡺⡟⣟⠦⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⡰⣢⢋⢞⡽⢋⠟⡽⣿⠪⣛⡭⠋⣡⡄⠀⠻⢿⣮⡱⢕⢝⠆⠬⡳⢝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠡⢃⠎⡴⠠⣢⡿⠋⠔⢑⣦⣭⢤⣾⣦⣴⣿⣿⣿⡋⣳⡳⡕⢌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⢢⡄⡹⣳⠚⡿⣱⢡⣎⣻⠁⡉⣳⢗⡺⣿⣿⣿⣿⣿⣿⡙⢎⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡊⣃⢹⢡⠰⢡⢟⢸⢼⣾⠀⡇⢷⣌⡟⢣⠻⣿⣿⣿⣿⣷⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡟⡀⡦⠖⡀⣸⠛⣴⢭⢼⢿⠘⡼⡝⣄⢳⢳⡻⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⡛⠓⠌⠀⠋⠐⣯⣨⡄⡚⠦⣳⡑⡙⣯⢃⡻⣌⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⡂⡅⡟⢗⣯⢧⣿⢓⣀⢅⢃⢿⣜⣮⠳⡝⠧⠃⠙⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⢀⢹⢳⠸⠸⡌⡆⠿⠘⣌⢫⢪⡣⡳⠞⠀⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠿⣣⢱⠑⠜⠇⡄⠫⡶⣑⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣿⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⡀⠀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⠲

⠀⠀⢀⣀⡀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⣀⡀⠀⣀⣀
⠀⠀⣿⣯⣬⣉⣉⣥⣾⠗⠚⠛⠀⠀⠀⠀⠀⠰⠄⠀⠀⢻⡏⢉⠀⢼⣇⣦⠀⣿⡅⢻⠁⣺⠏⢍⠈⣯⣔⠢
⠀⠀⠛⠛⠛⠛⣛⣋⣡⡄⠀⠀⠀⠀⠀⠀⠐⣦⡄⠀⠀⠈⣿⡶⢀⣼⣧⣦⡀⢸⡇⠞⡀⢻⣾⡟⠂⢬⣝⣿⡀
⠀⠀⠀⠀⠀⣻⣷⡿⠿⠷⠶⠒⠐⠒⠒⡐⠀⠉⠀⣤⡆⠀⠛⠉⠀⠈⠁⠈⢩⠀⠉⠋⠁⠀⠉⠋⠀⠈⠛⠋⠁
⠀⠀⠦⠃⠀⠀⠀⠀⡀⢄⡀⡀⠀⠘⠿⠃⠀⠀⠀⠈⠀⠀⠀⠀⡀⡀⠀⠨⠀⠀⠀⠀⣰⣎⠑⠠⢾⡿
⠀⠀⠂⠉⠀⠑⠀⠉⠉⠁⠀⠀⠀⠀⢰⣻⣿⣿⣻⣿⣿⣧⡁⣈⣑⣖⡀⠀⢠⣴⡄⠀⢉⣀⠶⠴⠊⠀⡄⠒⠠⠇
⠀⠀⠀⠀⣼⠄⠀⠀⠀⠉⠉⠁⠉⠉⠈⠛⠛⠛⠛⠿⠿⠷⠦⡯⠝⠋⠀⠀⠺⠞⠀⠀⢰⠯⠇⣠⣤⣄⠈⡅⠀⡀
⠀⠀⣶⡀⣿⠀⣠⡆⠀⠀⠀⠀⢒⣈⡉⠀⠀⠀⠀⠀⠀⢀⣀⣈⣀⡀⢀⠀⣄⠀⠀⠀⠀⢀⣠⣈⠉⠉⠉⠀⢀⡰⠠⠤
⠀⢠⣽⣷⣿⣾⣿⣶⠏⠀⠀⠀⠉⠠⠂⠀⠀⠀⢀⡴⠋⠉⠀⠉⢙⣃⣘⠷⣌⣙⠀⠀⣾⣿⣿⡿⠀⠀⣾⡓⠊
⠀⠀⣙⣿⣿⣿⣿⣭⡄⠀⠀⠀⣤⣾⠻⣼⡄⢀⡎⠀⢠⠤⣤⢤⣿⣿⣻⣷⡍⠋⠀⠀⠀⠋⠁⠀⠀⠀⠀⠑⢄
⠀⠀⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠙⢿⣦⣉⣛⡿⠁⠀⠀⠀⣿⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠺⣧⠀⠀⠀⠀⠀⢠⠂⠒
⠀⠀⠀⠀⡄⠀⢀⣶⣄⣀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣿⣿⠈⠟⠟⠟⢋⣠⡾⠋⠀⠀⠀⠀⢱⠤⠄⣴⣶⢃
⠀⠀⠀⠀⠀⠀⡘⣿⣽⢉⣼⠟⠆⠀⠀⣼⠉⠢⠀⠀⠀⣿⡿⠀⠀⣠⠀⠀⠉⠑⠀⠐⠈⠀⡀⠁⠀⠀⠈⠉⠹⣆
⠀⢠⣬⡀⡀⠈⣛⢻⡿⣿⠁⢀⠀⠀⣨⡍⡆⢠⠦⡀⠀⠿⠃⠀⠀⠀⠄⡀⠠⠀⡀⠀⠀⠀⢳⡀⠀⠀⠀⡀⢀⠜⠉⠉
⠀⠘⠿⠷⠖⠀⢹⣟⠀⠈⠻⡿⠃⠀⢹⣶⡷⢏⡇⠀⠀⠀⠀⠀⠀⠆⠆⠉⠠⠘⠁⠀⠀⠀⢾⡗⠐⠶⢾⣷⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⠿⠿⠿⠛⠛⠘⠓⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠘⠇

⠀⣠⣴⣾⣿⣿⣿⣿⣷⣶⣤⡀
⣾⣿⣿⡖⢠⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⢻⣿⣿⣷⣖⡒⠂⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠛⠳⣦⡀
⠈⢿⣿⣯⣥⣤⡀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⣀⢒⢌⠀⠹⣷
⠀⠈⢿⣿⣿⣿⣷⣂⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⢀⣈⢳⣴⣿
⠀⠀⠈⣿⣿⣿⡭⣁⣤⡄⠠⣾⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠲⣔⣆⢹⣮⣿⣿
⠀⠀⠀⠘⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠿⢿⣷⣤⡀⠀⢄⠨⣧⣽⣿⣿⣿
⠀⠀⠀⠀⣹⣿⠋⠀⠀⢀⣀⣤⣤⣀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠲⢽⢿⣦⢠⣳⣽⣿⣿⣿⠿
⠀⠀⠀⣼⣿⠃⠀⢠⣾⣿⠿⠛⠉⠛⠻⣦⡀⣿⣧⣤⣄⣀⡀⠀⢀⣤⣄⣀⣀⣤⣤⡀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣤⡀⠀⢀⢻⣿⣿⣿⣿⠟⠁
⠀⠀⢸⣿⠇⢠⣤⣿⣿⠄⠀⠈⠉⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣟⣿⣟⣿⣷⡦⣤⣾⣿⣷⠤⠄⠀⠙⣝⢿⣄⠀⠊⣿⡿⠛⠁
⠀⠀⠸⣿⡄⣾⣾⣿⣿⠛⠀⠀⠀⢀⠀⠀⠈⢿⣿⡉⠉⠉⠀⠀⠈⠛⠚⠛⠉⠙⠛⠉⠙⣿⣿⡿⠛⠂⠀⣤⣀⢮⠁⠹⣇⠂⣿⡇
⠀⠀⠀⠙⣿⣿⣿⣿⣧⠔⠈⠀⢀⣤⡀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⢲⣿⣿⠇⠀⠁⢻⣦⡿⠃
⠀⠀⠀⠀⠈⠛⢿⣿⣿⡇⠀⠀⠀⠸⣿⠰⣦⣀⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⣀⢀⠀⣾⣿⠃⠈⠀⠀⡸⣟⠁
⠀⠀⠀⠀⠀⠀⣼⣿⡿⡒⠀⠀⠀⢚⣿⣧⡻⣧⣄⣤⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢍⣉⣷⣾⡟⠉⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⢰⣿⣿⡧⡡⠀⠀⠀⠉⠙⠻⢿⣶⣭⣽⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠻⣄⣈⣵⡟⠀⠀⠀⢀⣤⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣏⠁⢙⣷⣦⠠⣄⠀⠀⠀⢉⠋⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⠀⠛⠁⠀⣀⣠⣴⠿⢛⣿⣿⠂
⠀⠀⠀⠀⠀⢸⣿⣏⠸⣦⡉⢿⣿⣶⣤⣄⣤⣶⣷⣾⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡛⠻⢷⣷⡶⣟⠛⢉⣥⣤⣾⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣶⣄⠙⢿⣦⡈⠉⠻⠿⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣄⣉⣬⣵⣾⠿⠛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣦⣿⣿⣶⣦⣶⣴⣔⣾⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣈⠛⠿⠟⠋⣁⣨⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣿⣷⣍⡛⠛⢿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⡿⠿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⣿⣿⣿⣿⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⣛⢟⣙⠛⣻⠋⣿⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠈⠻⣿⣶⣄⠀⠀⢀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠈⠙⠻⣿⣿⣟⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠙⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣦⣄⠀⠀⠀⠀⠙⠻⢿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠟⠛⠻⣿⣶⣦⣄⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣤⣀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⠿⠿⠿⣿⠏
⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⡿⠿⠿⠶⠶⠾⠛⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⣄⠀⠀⠀⠀⢀⣼⣿⣿⡇
⠀⠀⠀⠀⢸⣿⡏⠻⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣴⣿⠟⢸⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣄⡀⠉⠛⠿⢿⣿⣷⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣽⣿⡿⠋⠁⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠻⢿⣷⣦⣀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠙⠃⠈⠁⠘⠉⠉⠉⠉⠉⠉⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣽⣛⣻⡿⣶⣦⣖⣒⣲⢗⣶⢯⣄⠀⠀⠀⠀⠀⢀⣠⣦⣴⡿⠿⣷⣴⣤⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⢿⣏⠀⠀⠀⠉⠁⠛⠋⠉⢉⡏⣄⠁⠀⠀⠀⠀⡶⡿⣹⠖⠋⠁⠀⠀⠙⠻⣿⡄⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠⠤⠌⣳⣇⡇⠀⠀⠀⠀⠀⠀⠀⡘⣠⡉⠛⠀⢀⣰⠟⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠊⠐⠒⠀⠀⠀⠦⠈⠑⠤⡀⠀⠀⠀⠀⠀⣧⡉⠀⠀⢀⣼⣷⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢠⣬⣦⠀⠀⠀⠀⠙⡿⠦⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⣨⡇
⠀⠀⠀⠀⠀⣠⠊⠀⠀⣠⣤⣶⠦⣭⠭⣿⣋⣲⣶⠤⣀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢰⡅⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣰⣿⠁
⠀⠀⠀⠀⣰⠃⠀⣠⣾⡿⠛⠋⠉⠉⠉⠉⠓⠩⢝⡳⢤⣛⣶⣤⠟⠁⠀⠀⠀⠀⠀⢳⡀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡘⠷⢣⣿⠇
⠀⠀⠀⠀⡏⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⡄⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠙⢦⣀⠀⠀⠀⠀⣀⡠⠖⠋⠀⣨⣷⣮⣿⠋
⠀⠀⠀⠀⣧⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠈⠉⠉⠛⠋⠉⢀⡀⣤⠾⠋⠉⠉
⠀⠀⠀⠀⠸⣦⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⠦⠤⣠⣤⠶⠾⠒⠋
⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠈⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠙⣷⣷⣦⡀⢰⣆⡌⣷⠤⢤⣀⣀⣀⣀⣤⠤⠖⠉⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⠒⣺⠯⠄⣀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣓⡶⠋⠉⠓⠒⠒⠒⠒⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⣿⢿⡿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣿⣟⢛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⡿⣟⣿⣷
⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣷⣀⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣈⣿⣄⠀⢀⣼⣟⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣰⣶⡶⠶⣶⣤⣤⣤⣤⣶⠶⢶⣶⣦⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠛⢷⣄⣀⣼⠟⠻⣧⣀⣀⡾⠛⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣼⣿⢿⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⠿⣿⣿⠿⠟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠿⢶⣿
⠀⠀⠀⠀⠀⢿⣏⠉⠈⢉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣴⣾⣷⣦⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣄⡀
⠀⠀⢀⣴⣿⣿⣿⣷⣶⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠙⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠘⣿⢛⣛⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠛⠻⢶⣶⣶⡶⠟⠛⢻⣿⡿⠀⠀⠀⠀⢸⣿⠉⠙⠛⣿⣟⠛⠉⢹⣿
⠀⠀⠀⠀⠀⠀⣰⣶⣴⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠟⠋⠙⠻⠶⠶⠟⠁⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠛⠛⢿⣿⣿⠃
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⢀⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣄⠀⠀⢠⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣾⣶⣶⣾⣯⣁⣀⡀⠀⣀⣈⣽⣷⣶⣶⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣿⣿⣿⣿⢻⣯⣯⠇⠸⣽⣿⡏⣿⣿⣿⣿⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠤⠤⣸⡠⠤⢄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡉⠉⠀⠀⠀⣼⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⣄⣀⣀⣠⣴⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈
⠀⠀⠀⠀⠀⢤⣤⣤⣤⠄⠀⠀⣹⠀⠀⠠⢤⣤⣤⡤⠄⠀⠀⢰⣶⣶⣶⣶⣶⣿⣷⣤⡤⣤⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠙⠛⣻⣿⣿⣿⣿⢹⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠛⠛⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠛⠛⠛⠛⠛⠛⠛⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠲⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⢿
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠫⣉⠀⠸⣄⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠁⠀⣀⠟
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⡄⢸⠉⠛⠒⠒⠶⠤⠴⠤⠶⠶⠖⠚⠛⠃⣠⡤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⣜⣀⣤⣀⡀⠀⠀⠀⠀⠹⣼⡇⠢⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⢰⣿⠧⠤⠤⠤⠤⠴⢿
⠀⠀⡴⡿⠋⠉⠙⢦⡹⣄⠀⠀⠀⠀⢹⡇⠀⠀⠀⡠⠤⡄⠀⣿⠀⠀⠀⢸⡿⠶⠤⠤⠤⠴⠶⣏⣀
⠀⡼⢰⡇⠀⠀⢀⣈⣧⠘⡆⠀⠀⠀⠀⡇⠀⠀⠀⠷⢮⡤⠀⣿⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⡏⠉⠓⠦⢤⣀
⠀⣧⢸⡇⠀⢾⣉⣈⢿⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠀⢸⠀⠀⠀⣿⠤⣀⣀⠀⠀⢀⣀⡤⣇⠀⠀⠀⠀⠈⣹⠦⡄
⠀⢹⡀⢧⠔⠋⢩⠿⣼⢀⡏⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡿⠀⠀⠉⠉⠉⠉⠀⠀⡷⠀⡀⠀⠀⡞⢁⡀⣿⡀
⠀⠀⠳⣜⡦⣴⣋⣴⣣⠞⠀⠀⠀⠀⠀⢿⠈⠙⠒⠲⠶⠶⠖⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠁⡄⣸⠀⡏⠀⢀⣽⡆
⠀⠀⠀⠈⠉⠉⠉⠉⠳⢤⣀⣀⣀⣀⡤⠾⠦⣄⣀⠀⠀⠀⠀⠀⠀⣀⡤⣿⡀⠀⠀⠀⠀⠀⠀⢀⡏⠛⠲⢧⣿⣤⠽⠳⠾⠋⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠤⠒⠛⠉⢹⡄⠉⠉⠀⠀⠀⠀⠀⣉⣉⣉⡙⠛⠋⠉⠀⠀⠈⠳⣭⣓⣒⣒⣒⡪⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤
⡞⠋⠉⡟⠛⠓⢦⡴⠒⠋⠀⠀⠀⢀⣀⣀⡀⡾⣒⣂⣉⣹⡆⠀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡖⠉⠉⠉⠹⠉⠓⢦⡀
⠉⠒⠲⠦⠤⠤⠼⠁⠀⠀⠀⢠⡎⠁⡸⠁⢉⡷⠦⠭⠭⣾⠖⠫⡁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⣼⡤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣙⣛⣛⣭⠇⠀⠀⡼⠁⠀⠀⣸⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠯⢭⣭⣭⣭⣍⣉⣉⣉⣉⣉⣭⣭⣿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⠙⠳⠦⢤⣤⣄⣀⣀⣀⣀⣉⣁⣀⣠⣤⡤⠾⠛⢳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢻⡶⢶⣶⣦⣤⣤⣭⣭⣭⣥⣤⣤⣴⣶⡶⢿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣼⣧⣸⣇⣸⣇⣸⣇⣸⣇⣸⣇⣸⣇⣼⣧⣾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢾⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⠆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠰⠷⢶⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⠿⣿⡇⠀⠀⠀⠀⢀⣴⠟⡽⠊⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⣶⣦⣤⣄⡀⠀⠀⢸⣿⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⣿⣇⠀⠀⠀⣰⡿⠛⠳⣄⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣸⠒⠲⠍⣿⠀⠀⢸⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠄⠀⣿⡀⠀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣶⣦⣶⠟⠀⠀⠀⠙⠿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡿⠟⠋⠀⠀⠈⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⢿⡟⠛⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠈⠉⢹⡇⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣎⡑⠦⣀⣀⡀⠀⣀⡤⢊⣡⣧⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢀⣠⣾⣿⠿⠛⠛⠛⠛⠒⠚⢿⣿⣿⣿⠓⢋⣉⣩⠉⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠉⠙⠛⠿⣿⣿⣿⣿⣧⠐⠉⠹⠿⠏⠑⠄⢸⣿⣿⣿⠐⠁⠛⠟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠞⠉⣉⣢⠀⠈⢿⣿⣿⣦⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠊⠀⢠⠁⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠙⡍⠛⠛⠛⠛⠛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡅⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⠀⣠⠤⣄⢀⡠⠦⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⣄⠀⠀⠀⠀⠀⠐⠉⠀⠈⠙⠁⠀⠈⠉⠀⠀⠈⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢿⣿⣿⣶⣦⣤⣤⣤⡤⠤⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⣀⣀⣀⡠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠶⠶⠶⠶⠦⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠚⠋⢿⡄⠉⠉⠋⠛⠓⠒⠒⠒⡖⠒⠒⠛⠋⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠈⠳⣄⠈⢳⠶⢤⣤⣄⣀⣀⣀⣀⣃⣀⣥⣤⠴⡞⠁⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⢷⣸⡄⣀⡀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣷⣞⠉⠉⠉⠉⣙⣷
⠀⠀⠀⠀⠀⣀⣴⠷⠛⢿⡓⢦⠀⠀⠀⠀⠈⢿⡇⠀⠉⠉⢙⣛⠛⠛⣷⠀⠀⠀⢀⣿⣈⣉⣉⣉⣉⣉⡼
⠀⠀⠀⠀⣰⢫⠇⠀⠀⠀⠹⡌⢷⡀⠀⠀⠀⠘⡇⠀⠀⠀⣯⡬⢃⠀⣿⠀⠀⠀⢰⠏⠀⠉⠉⠉⠉⠀⠙⡖⠶⢤⣀
⠀⠀⠀⠀⡇⢸⠀⠀⡠⠖⠲⣿⠈⣇⠀⠀⠀⠀⡇⠀⠀⠀⠉⠽⠫⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⣇⠸⡆⢀⡵⠶⣞⢹⠀⡟⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠉⠒⠶⠶⠶⠶⠖⠋⣿⠀⠀⠀⠀⢠⠞⠉⢳
⠀⠀⠀⠀⠸⣆⢿⡁⣠⠟⢉⡟⣸⠃⠀⠀⠀⠀⣿⠲⢤⣀⣀⣀⣀⣀⣼⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡖⠀⢀⡎⢠⠂⠛⢢⡄
⠀⠀⠀⠀⠀⠈⠳⠯⠿⠶⠿⡚⠁⠀⠀⠀⠀⠀⣻⡀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣇⡇⢸⠀⢻⣄⣰⣷⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠬⣟⠲⠶⠶⠖⠛⠉⠉⠓⠦⢤⣤⣤⣤⡤⠤⠖⠋⠹⣶⠤⣀⣀⣀⣀⣀⣴⠏⠀⠈⠙⠛⠓⠋⠀⠉⠁⠀⠈⠙⠲⠦⣄⡀
⠀⠀⠀⣠⣤⣤⣤⣴⣋⠀⢘⣀⡼⠁⠀⠀⠀⠀⠀⠀⣠⣒⣲⠶⢦⡀⠀⠀⠀⠀⠀⠈⠛⠲⠾⠶⠷⠚⠉⠀⠀⢀⣤⠤⠤⢤⡤⣀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢧⣀⣀⠃⠀⠈⢻⠉⠁⠀⠀⠀⡠⠴⢶⠶⢤⣟⠲⠤⠤⢞⢇⣠⠶⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⡇⠀⠉⡧
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠀⠀⠀⠀⢸⠆⣀⣎⣀⡴⡏⠉⠛⢋⠟⠋⠉⢆⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠓⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠦⠶⠚⠁⠀⠀⣯⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⣩⡛⣉⣝⣙⣯⣏⣿⣫⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡼⢧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣽⣶⣾⣷⣾⣷⣶⣯⣗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣁⣛⣛⣛⣛⣛⣛⣛⣛⡻⣿⣿⣷⣎⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣯⣽⣿⣿⣿⣿⣿⣿⣯⢻⣿⣇⡟⣿⣿⣿⣷⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⢡⡿⢿⣟⣝⢿⠿⣦⢹⢸⣿⡗⣿⣎⣿⣿⣿⣇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣌⣻⣿⣷⣵⣿⣿⣋⣾⢸⣿⡇⣿⣿⢸⣿⣿⣏⣽⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡧⣿⢏⣿⣿⣿⡏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡏⣥⢿⣛⣛⡛⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠶⠶⠶⠶⣹⣿⣿⣿⣿⣿⣿⣿⣿⢧⡖⣛⢿⣿⢟⡳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⡯⣭⣭⣭⣭⣭⣭⣭⢻⡏⣿⣷⣝⢷⣵⢟⣵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣘⣛⣛⣑⢻⣿⣷⣶⣶⣶⣶⣿⣇⢿⡟⣵⢟⣙⢷⡝⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡎⠳⢷⣿⣿⣷⠷⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢦⣶⣶⣶⠀⠀⣴⣿⣶⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠘⣿⣿⡿⣰⣦⢻⣿⠿⠃⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⣀⡀⠀⡿⢿⠄⢀⣀⣀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣯⡄⠤⠤⠄⣹⡏⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠓⠾⠷⠞⠋
⠀⠀⠀⠀⠀⡀⠄⠒⠋⠁⠀⠀⠉⠒⠤⡀⠓⢄⠀⠀⠀⠀⡀⠔⠀⠀⠤⠒⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠉⠉⠁⠀⠀⠊
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⢦⣶⣶⣶⠄⢰⣶⡆⠀⠀⠀⠀⢸⣦⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠸⣿⣿⡿⢠⣾⢿⠇⠀⠀⠀⠠⡠⣿⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠠⡀⠀⠠⣤⣤⣁⣀⣀⠀⠀⣿⠿⣂⡈⠀⠀⠀⢀⠅⠀⠀⣑⣂⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠢⠤⠼⡏⠉⢹⡅⢀⣄⠠⠄⢤⠀⠀⠀⠀⣀⣀⣀⣸⡇⠉⠉⠋⠈⢀⠀⠀⢀⡀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠙⠁⠲⠶⠶⠾⠀⠀⠀⠀⢲⠷⠷⠛⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀⠠⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣤⣤⣤⡀⠸⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢤⠉⠁⠈⠻⢿⠃⠀⠈⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢠⣇⠀⠀⢨⠇⠀⠀⠀⢀⣤⠀⢀⣤⣦⡌⠻⣦⡀⢀⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣮⠀⢨⡦⡀⢸⠛⠓⠒⠁⠀⠀⢀⣴⣿⠋⠀⣼⣿⣿⡿⠀⠉⢁⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢰⠋⠉⠙⡇⢣⠇⡇⠀⢣⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⣿⣿⡟⢁⣀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠸⣄⣀⣠⠇⠀⠀⢲⠀⢸⠇⠀⣠⣾⠟⠋⠀⠀⠀⢰⣿⣯⠡⣿⣿⣷⡌⠛⠛⠛⠛⢋⢯⣿⣦⠀⢀⣦
⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⣀⡀⠙⠋⣠⣾⡿⠃⠀⠀⠀⠀⠀⠈⠛⠋⠀⠻⣿⣿⣿⡀⠀⠀⠀⢈⡻⣿⣿⣿⣼⣿
⠀⢀⣿⣿⣧⣤⡶⠟⠉⠁⠀⠀⠀⠰⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣻⣥⢀⢠⣴⡄⣙⢿⣿⣿⣿⣿⣇
⠀⠈⠿⠿⠟⠋⣠⣾⡷⠀⣦⢰⡗⣤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠘⣿⣿⣿⣿⡄⠉⠛⠿⣿⠟
⠀⠀⠀⢀⣤⣾⠟⠉⠀⠀⣿⢸⣇⡟⡾⠀⠀⠀⢀⠠⠒⠀⠂⠈⠁⢀⠂⠀⠀⠀⠀⢀⣀⡸⣿⣿⣿⣧⠀⢀⣠⣤⣤⣴⡄
⠀⠀⠰⡿⠛⠁⠀⠸⣷⠀⣽⣮⣭⣝⠁⠀⠀⠀⢀⠀⠀⠀⣠⢠⢀⣀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⠀⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠻⣿⠀⠀⠀⠀⠀⡄⡀⠀⡄⡀⡀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣈⠹⠿⠟⠃⠀⠀⠀⣤⣄
⠀⠠⣾⣿⣄⣀⣀⣤⠀⢿⣿⣶⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣷⣶⣆⠀⠀⠀⢰⣿⡏
⠀⠀⠈⠉⠀⠀⢈⣉⣀⣀⣉⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠿⣿⣿⣿⣿⣷⣦⡀⣾⡿
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡽⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠰⠿⣿⣿⠿⠋⠛⢦⡭⠞⠙⣿⡿⠛⢹⣿⠇
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⣠⣦⣶⣿⣿⣿⣷⠀⠘⢷⣦⣄⠀⠉⠀⠀⢀⣠⡤⡄⠀⠀⠀⠀⠀⢿⡟
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠏⣰⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠙⢿⣷⣤⡀⠀⠘⢾⣻⠷⠤⢤⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠙⠿⣿⣄⠀⠀⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠃⠀⢠⣴⣾⣿⣷⣦⡀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠉⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠻⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⣿⣿⣿⣿⣿⣿⣿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣟⣿⣟⣛⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠖⠒⠒⠲⠦⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢿⠒⠦⠤⠤⡤⡤⡤⢤⠤⠴⢾
⠀⠀⠀⠀⠀⢀⠔⠋⠁⠙⢦⡘⡶⠤⠤⢤⣀⣄⣤⣤⠤⢴⢋⣀⣀⣀⡀
⠀⠀⠀⠀⢀⣯⣤⣄⠀⠀⠀⠱⣟⠐⠤⠤⠤⠤⣄⠀⠀⢸⡿⠤⠤⠤⢽⡆
⠀⠀⠀⡜⡏⠀⠈⢣⢳⡀⠀⠀⢻⠀⠀⢰⡵⡀⣿⠀⠀⡸⠋⠙⠛⠋⠙⡶⠤⣀⡀
⠀⠀⢸⡀⡇⠠⣊⡹⡇⡇⠀⠀⢸⠀⠀⠀⠊⠁⢸⠀⠀⡧⢀⣀⣀⣀⣀⢼⠀⠀⠉⢑⠦⡀
⠀⠀⠀⢧⢳⢈⡼⢳⢣⠇⠀⠀⢸⠦⣄⣀⣀⣀⣸⠀⠀⡇⠀⠀⠉⠁⠀⢸⢠⠀⢠⢃⠄⠧⡀
⠀⠀⠀⠀⠙⠛⠛⠻⣅⣀⠀⣀⣨⢦⣀⠀⠉⠁⠀⠀⣀⣇⠀⠀⠀⠀⠀⡽⠾⢤⣼⣸⠦⠾⠳⢤⣀
⠀⠀⢀⣀⣀⣠⠒⢋⣹⠈⠉⠁⠀⠀⣈⢭⣭⡙⠉⠉⠀⠙⠿⠷⠶⠶⠟⠃⠀⣀⣀⣀⣀⡀⠀⠀⠈⠉⠒
⠀⠀⠧⢄⣅⣈⣹⠉⠀⠀⡠⠐⡖⠲⣟⣒⣖⣧⣤⠒⣦⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⢀⣆⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣒⣓⡾⠃⠀⡜⠁⢀⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉

⡘⢿⣿⣿⣿⣿⡟⡿⢰⡈⢿⣷⡹⣿⢿⡤⡄⠀⢸⢸⡇⠀⠀⢠⢤⣤⣤⣤⠤⠤⠄⠀⢸⢸⡇⠀⠀⢤⣤⣤⣤⣤⠤⠤⠄⠀⠀⢸⠀⠀⠀⣤⣤⣤⣤
⣧⠸⣿⣿⣿⣿⠃⠃⣿⣷⡀⠹⣷⡜⣎⣗⡂⠀⢸⢸⡇⢰⠀⢐⣒⣒⣒⣒⣒⣒⡂⠀⢸⢸⡇⡄⠀⢒⣒⣒⣒⣒⣒⣒⠂⠀⠀⢸⠀⠀⠀⣒⣒⣒⣒
⠘⣧⠹⣿⡟⣿⠀⠀⣿⣿⣿⡄⡘⣷⣹⣮⠁⠀⢸⢸⡇⠸⠀⠨⠭⠭⠭⠭⠭⠭⠅⠀⢸⢸⡇⡇⠀⠬⠭⠭⠭⠭⠭⠭⠄⠀⠀⢸⠀⠀⠀⠭⠭⠭⠭
⠀⠙⣧⢹⡄⠋⠀⣄⢻⣿⣿⣿⣿⣼⣇⣿⡇⠀⢸⢸⡇⢘⠀⢘⣒⣒⣒⣒⣒⣒⡂⠀⢸⢸⡇⠁⠀⢒⣒⣒⣒⣒⣒⣒⠀⠀⠀⢸⠀⠀⠀⣒⣒⣒⣒
⠀⠀⠸⡎⣇⠀⢀⡿⠀⢩⠉⠀⢀⣀⣀⣉⣁⣀⣸⢸⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡀⠀⠀⣧⡿⣀⠞⠀⠀⡀⠔⠈⠁⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣷⡀⠀⢹⠁⠟⠄⠀⠈⢀⣀⣤⡴⠖⠒⠂⠉⠉⠉⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡴⠁⠀⠉
⡿⣧⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⠋⠁
⠇⣿⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠈⡄⠀⠸⢸⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣯⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢻⠀⠀⡀⠀⠀⠀⡇⠀⠀⠈⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⣇⢀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡈⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠀⠀⠀⠀⠀⢿⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡤⠖⠋⣠⣤⠘⣿⣿⣶⠶⢀⠠
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡥⠙⢛⣡⠈⢩⠀⠂
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢒⠈⠡⠀⠀⢸
⠀⠀⠀⠀⢸⡀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣿⠀⢀⢀⠁⠀⠀⠀⠀⠀⠠⢀⠔⠀⠀⠀⠘⠁⠀⠀⠀⠀⢸
⣀⣆⣀⠀⠀⡇⢀⣴⡗⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠈⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢰⠀⢀⣀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣀
⠭⡯⠤⠀⠀⡷⡫⢊⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠆⠀⠠⠤⠄⠀⠀⢸⠀⠀⠀⠭⠭⠭⠭
⣛⣛⣃⠀⠀⢰⠃⣼⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠸⠀⠀⣘⣚⡀⠀⠀⢸⠀⠀⠀⣛⣓⣛⣛
⠤⠦⠄⠀⡄⣏⡜⡽⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠠⠦⠤⠀⠀⠀⢸⠀⠀⠀⠤⠤⠤⠤
⠉⠋⠁⠀⣷⢋⡜⠁⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠋⠉⠉⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠉
⠀⡆⠀⠀⢉⡞⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⡶⣦⡄⠀⠀⠀⠀⠀⢀⡤⠤⠦⠴⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⢸⣉⣉⣀⣉⣉⡇⠀⠀⠀⠀⠀⢸⣃⣀⣀⣀⣀⣀⢀⣻⠀⠀⠀⠔⠉⠉⠉⠑⠒⢄
⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⢼⡁⠀⠀⠀⠀⢀⡴⠃⠈⠀⠀⠀⠀⠈⠳⢄⠀⠸⡣⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠒⡂⡐⠈⡇⠀⠉⠒⠦⠤⠤⠒⠋
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡇⠆⡬⠀⠙⠒⠐⠀⠀⡀⡀⡇⠀⡴⠒⠒⠛⠛⠻⣿⣿⣿⣶⣆
⠀⠀⠀⢸⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡇⠀⡇⠄⡋⠀⣀⣾⣆⡀⠀⡘⠀⡇⠸⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿
⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠠⠃⠀⠉⢿⠏⠁⠀⠅⠂⡇⠀⠉⠛⠒⠒⠒⠛⠛⠛⠛⠉
⠀⠀⠀⣼⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⡇⠁⠙⠤⠤⡠⠤⡠⠔⠁⠂⡇⠀⡔⠋⠉⠉⠉⠉⠉⠉⠉⠒⡄
⠀⠀⠀⣾⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⡇⠁⠈⠓⠖⢂⡈⠡⡃⠂⠀⡇⠘⡅⠀⠀⠀⠐⠀⠀⠀⠀⢀⡇
⠀⠀⠀⣻⠀⠀⠀⠀⢶⣿⠶⠀⠀⠀⠀⡇⠀⠈⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠁⠀⠈⠉⠛⠓⠒⠒⠒⠒⠋⠉
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣤⠤⠲⠤⠠⠴⠲⡤⠤⠶⠒⠖⠲⠖⠒⢲⠲⠶⠒⠒⡶⠦⡀
⠀⠀⠀⣿⣶⢶⣦⣶⣦⣤⣤⣶⣶⣶⣖⡇⠀⠀⢀⡞⢡⣀⠀⠛⣄⡀⠈⠃⢥⡄⠀⠉⢠⣤⠀⠈⠀⢦⠀⠀⠁⡼
⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣟⡇⠀⢀⠎⠀⠀⡀⡀⠀⠀⡀⠀⠀⠀⡄⢀⠀⠀⣠⠀⠠⠀⠀⣀⠀⡰⠁
⠀⠀⠀⢻⡻⣽⣿⣿⣿⣿⣿⣿⣿⣿⢟⠇⢠⠋⢠⣤⠀⠋⢤⠄⠈⠁⢢⡆⠀⠀⠰⠖⠈⠈⠀⠲⡖⠈⠈⣰⠃
⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠓⠒⠒⠛⠉⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠦⠴⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⠀⠀⠈⠀⢡⠀⠍⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡊⠐⠀⣶⡆⡄⢰⠀⢀⣶⠀⠰⣦⠀⡆⠖⣶⠶⠀⣶⠒⠆⣶⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢻⣷⣿⡌⠀⣜⣻⡆⢸⢻⣷⠇⠀⣿⠇⠀⣿⠶⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠹⠁⠠⠇⠹⠷⠸⠀⠻⠇⠀⠿⠆⠀⠿⠤⠆⠿⠴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⠀⢰⣰⣶⠄⣴⣲⢰⣶⠀⠐⣆⣲⠀⣶⢸⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⣀⠀⣁⢀⠀⣀⢀⢈⠈⣀⡀⠀⣈⣀⣀⠀⣀⣀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⡟⠀⣿⠄⠀⢻⣾⡆⢠⢻⣧⠀⣿⡧⣟⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢛⡃⠙⢀⠋⢈⡀⠈⣁⠁⠘⡀⠙⢀⠛⠃⣙⠀⢋⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⣍⠀⣿⢰⡇⣿⣼⠩⣧⣿⢹⣶⡏⣿⣼⠉⣧⣿⢹⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠻⠞⠀⠛⠈⠓⠛⠘⠳⠃⠙⠜⠉⠣⠟⠘⠲⠋⠛⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⣀⢀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣚⠉⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⣫⠵⠊⠁⠀⠀⠀⠀⢈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢛⠥⢖⡩⠔⢀⠀⠀⢀⣠⢴⡺⠟⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⢀⡤⠖⠋⠀⠀⣁⡴⣚⠽⠊⠁⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⡫⠀⢀⡤⠞⠉⠀⠀⣀⣴⡾⠕⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢀⡠⢞⡩⠖⢋⠤⠚⢉⠤⢊⣠⣴⡾⠛⠁⠀⠀⠀⢀⠀⠀⠀⢠⡞⠋⠉⢳⡄⠈⢧⡀
⠀⠀⠀⡞⠉⠐⠉⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⣇⣀⣠⡼⡷⠀⠀⠹⣦⡀
⠀⠀⠀⠙⢦⡀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⣀⡤⠤⣄⣀⣤⣤⣀⡤⢶⠿⣿⠷⠊⠁⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠙⣦⡴⡪⠟⠁⠀⠀⠀⠀⠀⠀⡞⠢⠤⢤⣌⣿⠀⠀⣾⠀⢸⠀⢸⡏⠙⣢⠤⠤⣄⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣄⣀⣠⡴⣿⣤⠖⢿⣦⣼⣦⡞⢡⡎⠉⠲⣤⡾⡆⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⡼⡏⠣⡈⠙⣦⢰⡇⢿⡴⠋⣷⠀⠘⣬⠿⠿⢾⣋⣩⠴⡧⠶⠶⣄⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢷⠳⣄⡙⢆⡿⠈⣧⣈⣳⣤⠟⠀⣾⠓⠲⢖⣰⢿⠀⢸⡄⠀⠀⣸⡇⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠈⠳⠮⠽⠟⢡⠾⡀⠉⢍⢱⣴⡟⠻⣿⣭⣭⠵⠋⠀⠘⠮⣭⣭⡽⠃⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠈⢧⣉⠓⢶⣿⡘⢿⢦⡈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⣽⡉⠁⠀⢸⡆⠑⠿⠞⠃⠀⢀⠤⠶⢤⡀⠀⠀⠀⠀⢀⣴⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠈⢧⣙⡲⢶⡿⠁⠀⠀⠀⠀⢠⣇⠀⠀⣨⡇⠀⠀⠀⢐⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠉⠉⠀⣠⠴⡶⢤⡀⠈⠻⢿⡭⠟⠁⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢰⡇⠀⡇⢀⣿⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠸⣿⣶⣷⣻⠏⠀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠉⠉⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋

⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠒⠒⠲⠤⣄
⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⣰⠒⢲⡀⢀⣀
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢸⠁⠀⣧⡇⠀⠀⣷⡏⠈⡧
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢳⡀⠀⠀⠀⠀⢸⠀⠀⢹⡇⠀⠀⣿⠀⠀⣇⣠⢤⡀
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢄⠀⠀⢹⡄⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠀⡷
⠀⠀⠀⡏⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⢸⡇⠀⠑⠤⠞⠁⠀⠀⠀⢸⡇⠀⠸⡇⠀⠀⣿⠀⠀⡟⠀⢠⡇
⠀⠀⢸⠃⠀⣼⠀⠀⣿⠀⠀⢸⡆⠀⢸⡇⠀⠀⠀⢀⡴⠒⢤⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⣸
⠀⠀⢾⠀⠀⣿⠀⠀⣿⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠘⣇⠀⠀⠑⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠈⠓⠋⢹⠀⠀⣿⠀⠀⢸⣇⠀⠀⡇⠀⠀⠀⠀⠈⢧⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢸⡀⢸⢿⠀⠀⢸⢻⠀⢀⡇⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠉⠁⠈⠧⠤⠏⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⠤⠤⠤⠤⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⢘⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠉⠀⠉⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣁⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣾⣿⣿⠇⠀⣀⡀⠀⠀⠀⠀⠀⣈⠂
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣸⣿⠟⠃⣴⣿⣿⣿⣷⣤⠀⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⣿⣿⣿⠿⠿⠃⠀⠈⣻⠹⠁
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠍⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⣟⠻⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⣷⣾⢿⠿⠓⡄⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⠗⠲⢤⣄⣄⡤⡔⢲⢢⠈⡈
⠀⠀⠀⠀⠀⠀⠀⢠⠒⠮⣓⣤⡀⠀⠀⠀⢀⣧⠀⠀⠀⠉⠀⠂⠀⢠⢦⢷⢷⡜⠛⠊⣄
⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠉⠛⠀⠀⢣⠈⡟⣆⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠙⠛⠒⠮⢆⡇⠹⡶⣤⣤⠤⢀⣀⠀⠰⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣷⠀⢻⡼⠿⣿⡃⣼⠅⡑⠒⠤⠤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻⣿⠶⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⣂⡌⠀⠀⠌⠀⠂⠀⠀⠑⢦⡉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣏⠀⠀⠸⠀⠆⠀⠀⠀⠲⠁⠈⠂⢜⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠸⡄⢿⠀⠂⠂⠸⠀⠀⢀⢃⠇⠀⠀⢀⣎⢳⢨⡟
⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠀⠈⠑⠢⣘⡎⡀⣤⣼⣴⣤⣤⣈⣈⡀⠁⠀⠉⠀⠀⠀⠉⢦
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠁⠀⠙⢿⣿⣿⣿⣿⡟⠉⣻⣶⡖⢲⡶⠒⠊
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣾⡿⠟⠻⢿⠃
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊
⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⠊
⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡞
⠀⠀⠀⠀⠉⠁⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢀⠜⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡀⠀⠀⠊⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣀⠃⠀⠀⠐⠀⠀⠑⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠐⠀⠀⠀⠂⠀⠀⠙⠒⠢⣄⡀⢀⡀⠀⠀⢀⡠⠐⠶⠶⠖⠶⠶⠶⠂⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠒⠒⠶⠦⣤⣤⡄⣀⠀⢀⠀⠙⣿⣦⡀⠆⠀⠜⠁⠀⠀⠀⠀⠈⠁⠉⡉⠙⢻⡅⠀⠀⣀⢀⣀⣀⣀⣀⣠⣈⢳
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⢘⠃⠘⠉⠋⠉⠉⠉⠉⠉⠁⠈⣷
⠀⠀⠀⠀⠀⣬⡁⠀⠀⠀⡀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⡀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⢸⣤⣄⡀⣀⣀⣀⣀⣤⣤⡤⠘⠣⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢴⡃
⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠖⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⠀⡆⠀⠀⠀⢻⡤⣄⣀⣤⣠⣤⠀⠠⠶⠶⠶⠀⢹
⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠉⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠃⠀⠀⠀⢸⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯
⠀⠀⠀⠀⠀⠀⠱⣏⡆⠀⠈⠥⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⡄⠀⠀⠀⢠⠀⠀⠀⠈⣷⠰⢄⣤⣠⣄⣠⣄⣤⣤⣶⡀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⢀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣣⡀⡀⢠⠀⠀⠀⢢⡀⠀⠀⢠⠀⡀⢀⠘⠇⣀⢠⣀⡘⠆⢀⢠⢤⣰⣴⣤⣿⡁⠈⠀⠚⠛⠛⠓⠛⠳⠖⠛⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠶⣾⠿⠶⢷⣿⣿⣶⣶⣶⣿⡾⢿⣤⠾⠄⠌⠷⠂⠴⠿⠂⠛⠛⠛⠙⠻⣷⣶⣤⣀⣀⣀⣠⣤⣠⡶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⣀⠀⠀⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⢀⡀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠞⠁⠀⠀⠈⡀⠀⠀⢣⡼⠋
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠛⠛⠛⠛⠛⠛⠋⠐⣄⣀⣀⣀⢤⣤⠐⠚⠉⠁⠀⡀⠀⠃⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡼⢠⣴⣤⣤⣤⣤⣄⣤⣠⢀⣠⠟⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠰⠾⠋⠀⠄⠀⠤⠰⠶⠶⠚⠛⠉⠉⠀⠙⠀⢄⡀
⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢡⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠈⠃⠀⠀⢹⠆
⠀⠀⠀⠀⢠⡞⠁⠒⠿⠶⠦⠤⡤⣦⠤⠶⠤⡟⠀⠀⠀⢀⠀⠀⠀⠀⠳⠀⠀⠀⠀⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⡄⠀⠀⠀⢾⡀
⠀⠀⠀⠀⢀⡟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢻⡀⠙⠛⠛⠊⠀⠶⠶⠶⠦⠴⠲⣞⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠀⣀⢾⠁
⠀⠀⠀⠀⡿⠐⠾⠶⠾⠶⠶⠦⠶⠴⠒⠂⠺⡅⠀⠀⠀⡐⠀⠀⠀⠐⠆⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠶⠀⠀⠀⡀⠀⠀⠐⠂⠀⣸⢂⠛
⠀⠀⠀⠀⠳⣤⣤⡀⠀⣀⡀⣀⣠⣀⠀⡄⠀⡟⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⣤⡀⢸⠾⠃
⠀⠀⠀⠀⢰⣏⠉⡉⠉⠉⠉⠉⠉⠉⠁⠀⢀⣹⣦⣃⣾⣤⣆⡰⠄⢉⡤⠄⣠⠐⢁⠄⣀⡀⠰⠀⢀⠀⠘⠁⢀⡀⠼⢀⠔⡄⢦⠝⠃
⠀⠀⠀⠀⠀⠉⠛⠿⠶⣶⣶⠶⣴⠾⠶⠿⠛⠁⠈⠉⠉⠉⠉⠑⠈⠙⠂⠙⠚⠲⠗⠚⠧⠶⠾⠷⠧⠞⠚⠃⠛⠛⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢿⠋⠉⣩⣭⣽⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⡉⠉⢹⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠸⣿⡏⠀⠀⠀⠀⠀⠉⠙⢿⣿⡿⢿⣿⠟⠋⠉⠀⠀⠀⠀⠈⣽⡿⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⣸⣿⠀⠸⣿⣄⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣤⣶⣿⠟⠁⠀⠀⠙⠿⣷⣶⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣴⣄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣠⣄⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⢀⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⠛⠿⠷⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠟⠛⠉⠁⠀⠀⠐⠛⠛⠉⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⢀⣤⡀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⢀⣴⣾⡄
⠀⠀⠀⣼⣿⣿⣶⣤⣀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⣠⣶⣿⣿⣿⡟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⣠⣶
⠀⢸⣷⣄⡀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⢀⣠⣾⣿⣿
⠀⠿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿
⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⣀⡀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠻⣿⣶⣤⣀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⢀⣀⣴⣾
⠀⠈⠛⢿⣿⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣧⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢰⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠤⠐⠷⠼⣀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣀⡀⠀⠈⠑⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠿⠿⢷⣶⣴⣶⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠿⣆⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⢸⣼⠖⠖⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⢸⢟⣷⢄⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣧⣄⣦⣤⣠⢠⣿⣿⠃⠀⣠⡟⠹⠃⠘⠒⠊⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⠠⡆⠀⠀⢠⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣶⣿⣿⢀⣃⣀⣠⡏⢀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⢿⣯⡽⠟⣿⣿⣷⣶⣿⣿⣿⣿⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠛⠛⠛⠻⣿⠿⣿⡄⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⣤⣴⣤⣴⣶⣿⣦⠐⣴⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣽⠯⠛⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣵⢂⠀⠀⠀⠀⡈⠀⠘⢶⣿⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠃⠀⠀⠀⠀⠶⣀⠻⠟⢂⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⠂⠀⠀⡀⢠⡤⠜⠛⢯⡉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⢆⠀⠠⠀⣀⡀⠀⠀⠰⠂⠉⠀⡄⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⣪⢀⠀⢀⠉⠀⠐⠀⠀⠀⢠⠀⣠⡄⠀⠀⡠⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⣁⡨⠀⠀⢀⡁⢀⣤⣄⠀⠀⠀⠙⡃⡀⢠⡙⠧⠘⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠤⠈⠀⠈⡂⠒⢨⠁⠘⣹⣦⡂⠀⠀⠀⠠⠌⢦⠭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠒⠛⠟⢻⣿⣿⣶⣿⣷⣶⣿⣤⡉⢠⣄⠊⢡⡄⠀⠀⠀⠀⠀⠓⠀⠀⣠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣽⠱⣄⠀⠀⠀⠀⠛⠛⢿⣿⣿⣿⣿⡟⢸⡟⠁⠄⠀⠐⠀⠒
⠀⠀⠀⠀⣠⠞⠉⠻⣿⡁⠀⠑⣌⠣⡀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⠛⣥⡠⠶⠀⡀⠀⡀
⠀⠀⣠⠊⠀⠀⠀⠀⠙⢿⣆⠀⠈⠳⣌⠢⡀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣾⣃⡁⠀⠂
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠈⠳⣌⠢⡀⠀⠀⠀⢈⣿⣿⣿⣿⣮⡑
⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⢷⣝⢆⠀⢀⣹⡿⣿⣿⣿⣋⡑
⠀⢸⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠙⢷⣟⠉⠀⢐⣿⣿⣯⠼⠀⠓
⠀⠈⢧⡀⠉⠒⠤⠀⣀⡀⠀⠀⠀⣀⡀⠹⣧⢀⣠⠤⠛⠀⢀⣴⣿⠟⠅
⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉⠁
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠁⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣌⠁⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠃⠘⣧⠈⠛⣛⢀⡀⢐⣒⠂⠀⢤⣤⠀⠀⠤⢶⡴⢦⣄⣙⣼⠃
⠀⠀⠀⠀⠀⢀⠀⠀⢠⣿⢞⠳⠄⠀⡀⠐⡀⠀⠲⠖⠀⡄⠀⠈⠀⠘⠿⠾⢿⠁⠾⠁⣄⣀⠀⠀⠀⠀⢀⠨⠶⣾⡷⠚⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣶⣾⣿⣡⣴⠾⠋⢉⡽⠋⠀⠲⠀⠀⠀⣈⡀⠀⠀⠶⠶⣶⣶⠆⣤⡀⠉⠛⠶⢄⠾⠃⢀⣛⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠈⠁⢀⡀⣠⡤⠶⢈⡁⠁⠀⠀⣀⣠⣶⣄⣈⠙⠛⢰⠀⢶⣄⠀⠀⠠⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣀⢶⠉⢅⠄⢀⠀⠈⠇⠀⠀⠀⠈⠉⠉⠉⠻⠟⢡⣶⠀⣀⠹⡆⠀⠀⠀⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠘⠁⠀⣴⠊⠉⠆⣧⡄⢠⠀⠆⠀⠀⠀⠐⠲⠶⢿⣶⣦⣄⡈⠀⠁⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣒⡀⣴⢫⡴⢿⣿⡼⢃⡜⠀⡄⠀⠀⠀⣈⠑⠲⠦⣄⠈⠙⠁⢶⣶⠀⣶⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⣤⠌⠷⣙⣂⣀⠀⠀⣤⠞⠁⠀⠀⠀⠻⠿⣶⣶⣵⡖⠲⢶⣦⡌⢀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⣨⣉⠙⠻⠂⢁⡀⠀⠀⠀⠀⣤⣤⡀⠉⢙⡻⠟⠂⣀⡀⠛⠿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠻⢧⡀⠰⠖⠀⠧⠄⠂⠀⠀⢠⡝⠛⠿⢷⣬⣀⠀⠈⠙⠿⡷⠀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⡰⠀⠀⠸⠟⠂⠀⠀⠈⠙⠻⠶⠀⠀⠀⠈⠻⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠀⠀⠀⠀⣀⠀⠀⠀⠠⠆⠘⢿⡆⠀⢠⡄⠀⠀⠀⣀⠈⠁⠀⠘⠻⠆⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢀⣀⣀⣤⠀⠀⠀⠈⠀⠀⠈⠀⠤⠆⠀⠀⠀⠈⠻⣦⠃⠀⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠓⠒⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠈⠀⠀⠠⠈⠃⣀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣶⡀⠀⠀⠀⠀⠀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⢾⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠐⢦⡀⠀⠀⣀⠀⣿⣌⢀⣇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠀⠘⢦⡀⠀⢪⠛⢀⠀⣽⡤⠴⠒⢚⣿⣯⣥⠬⠉⠭⢷⣶⣾⠾⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠈⠁⠀⠸⠆⢨⣽⣿⣶⠐⣔⣶⣾⣄⣀⠰⢶⣶⣶⣬⡽⣿⢿⣶⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠷⠀⠀⠀⠀⣠⣶⡶⣤⡄⠨⡿⢃⠈⠷⣌⡻⣿⣛⣤⣼⣟⡿⠿⠅⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⠀⠀⣤⣠⠋⠀⠀⣩⡄⠐⣷⠀⡄⠀⠈⠳⣮⡙⢿⣌⣀⠁⠀⠈⢀⠈⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠄⠈⠻⠀⠀⠀⠁⠴⠆⣿⠐⣻⠀⠀⠀⠀⠙⢶⣌⠛⠶⣍⣄⡾⡚⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠳⡀⠀⠀⠀⠀⠀⣯⣭⠿⠁⠀⠀⠀⠀⠀⠉⠛⢷⣄⣨⡉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠻⣤⣤⣌⡀⠀⡀⠀⠰⣇⣄⠀⢇⣠⡀⠀⠀⠀⠐⠀⠳⣯⣟⠷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠙⡁⠀⠀⣿⠒⢻⣭⠀⡘⡟⠳⡀⢠⠀⠀⠀⠀⠘⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠁⢀⣤⠟⠀⠘⡛⠀⣿⠁⠀⠀⢶⣀⣄⠀⢸⣦⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠚⠃⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠻⠛⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠸⢷⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣖⣒⣦⡤⠭⠭⠭⠭⠭⠥⠤⠰⠲⠖⠀⠒⠒⠒⠒⠒⠛⣏⠉⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠑⢄⡀⠀⠀⠀⠀⠀⢠⠛⠈⠯⡿⢽⡟⠒⠒⠠⠤⠄⠀⠀⠤⠤⠤⠀⠰⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠙⢦⡀⠀⠀⣀⠸⣀⣀⣀⡤⠏⠀⠀⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠙⠶⣄⣯⢕⢄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣿⠶⠖⠢⠤⠤⠤⠤⢼⢹⣿⣵⣴⡄⢀⡤⣎⡙⣁⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠑⢌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠀⠀⠈⣟⡥⠛⠀⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡠⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠢⢐⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠊⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠓⢌⠢⢀⠀⠀⠀⠀⠀⡠⢊⠔⠊⠀⠠⠎
⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠣⣄⠀⠀⠙⠤⡑⢀⠀⠀⡠⠖⠁⠀⠀⡰⠁⠀⠀⡤⠤⠀⠀⣴⣀⡠⣄
⠐⢍⣿⣿⠟⠓⢺⢻⣛⠑⠋⠐⠈⠀⠀⠀⠙⠲⢄⠀⢈⣦⣴⠊⡀⠀⢀⡤⠊⠁⠀⠀⠀⠉⠉⠉⠉⣤⣤⡏⢬⣿⣷⣄
⠀⡼⣿⡧⠚⠛⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠷⠏⠲⠷⢾⡒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡿⢿
⠰⠷⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣦⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡩⣰⡞⢅⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡌⠺⣦⣦⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠁⠀⠀⠀⠀⢠⣏⠰⣶⡶⢦⣀⠀⠀⠀⠈⠂⠘⢟⡛⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⡟⠀⠀⠀⠀⠀⠀⢠⡟⠀⠑⢬⣑⠘⠁⡆⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⢀⣼⡿⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡿⢷⡄
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡀⣿⡀⠀⣠⣾⣿⣷⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⣿⡄
⠀⠀⠀⠰⠛⠛⠙⠓⠛⠟⠀⠀⠀⠀⡠⠊⣾⡿⠟⠾⠟⢻⠯⣧⣼⣟⡿⠂⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣷⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢊⣠⣄⣊⣉⣽⣁⣈⣁⣤⡿⠋⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢀⠀⡠⠒⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⣾⠀⠀⠀⠀⣬⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠠⠁⠀⠀⠀⡖⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠂⡇
⠀⠀⢀⠤⠤⠤⡀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⡜⠀⠀⠀⠠⠁⠀⠀⠀⠐⡠⠒⠀⠒⠄
⠀⠀⡃⠀⠀⠀⠈⢢⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢠⠏⠀⠀⠀⠀⡆
⠀⠀⠐⡄⠀⠀⠀⠀⠑⢄⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢀⠔⠒⠳⡀⠀⠀⠀⡸⠁⠀⡠⠒⠉⠉⠳
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢑⠄⠚⠃⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⡇⠀⠀⡰⠁⡠⠚⠀⠀⠀⠀⠰⣠⠔⠒⠢⡄
⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡼⠀⢀⢞⡤⠊⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⢀⣇⡶⠟⠁⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⡠⠜⠓⢄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⣀⡀⠜⠋⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⡠⠘⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⡔⠁⠀⠁⠐⠀⡀⡀⠀⠐⣂⣴⣃⠀⢠⡖⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣀⣔⡩⠤⠤⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠉⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠉⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠇⠀⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⡜⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠚⠁
⠀⠀⢸⣀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒⠉
⠀⠀⠈⠻⢿⡶⢤⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠒⠉
⠀⠀⠀⠀⠀⠈⠑⠛⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇⢀⣴⢀⣿⣿⣿⣿⣆⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⢹⣿⣿⣿⠄⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢴⣶⣿⡇⠀⠠⠆⠄⠤⠀⡀⣾⣤⣤⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠋⢸⣏⠶⠒⠀⣦⡀⠠⢽⣿⠃⢉⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠈⠛⠁⠀⠀⠀⠀⠀⠙⠛⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⣀⣠⣄⣀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢰⣌⠉⣉⡶⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢠⡈⠉⢉⣠⠤⠖
⠀⠀⠀⠀⠀⢀⣠⣄⢀⡀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠶⢂⡉⠀⠈⠀⡶⠞⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⣀⡀⣀⡀
⠀⢠⡆⠀⠩⠓⠲⠶⠶⢖⣒⠂⠉⠛⢦⣄⣀⣀⡤⢊⣀⠐⠶⠦⠖⠿⠛⠋⠛⠴⠴⠶⠂⣀⠁⢤⣀⢀⣠⣴⡚⠉⠀⣒⡲⠶⠶⢶⠚⠯⠁⢢⡄
⠀⠀⢡⡀⣀⡀⠀⢀⣠⢤⡌⢉⡲⠞⢛⡦⣍⣙⠓⠻⠤⠴⠖⠒⣧⣼⠶⠶⢲⡶⠒⠶⠶⠴⠷⠚⢛⣉⢴⡛⠷⠖⡋⢉⣤⣄⡀⠀⢀⣀⢀⡌
⢀⡴⠆⠉⠉⠙⢶⠉⠀⠀⢈⠁⠀⣠⠋⠰⠉⡙⠳⣄⠀⠀⠶⠞⠉⠉⠛⠛⠉⠉⠳⠦⠀⠀⣠⠞⠟⠉⠦⠙⣆⠀⠈⡉⠀⠀⠉⡷⠋⠉⠉⠱⢤
⢸⡇⠀⠀⠀⠀⢸⠋⠀⠰⠁⣠⠴⣿⡀⣇⠤⠗⢒⠉⠙⠲⣆⡀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢙⡂⠰⠤⣰⠂⣿⠣⣄⠈⠆⠀⠘⠏⠀⠀⠀⠀⢀
⠈⢧⠀⠀⠀⣸⢿⠀⠀⠀⢰⢃⡤⠜⣷⣈⠲⣶⠀⠲⠴⠀⠈⢱⡀⠀⠀⠀⠀⢀⡞⠁⠀⠶⠖⠒⣸⠶⢃⣼⠣⣤⡘⡆⠀⠀⠀⡿⣦⠀⠀⠀⡼
⠀⠀⡆⠀⠸⠃⠀⠳⣄⠀⡎⢾⡵⠟⣀⠭⡷⠢⣍⣉⠻⠉⠛⠀⠷⠤⠴⠦⠤⡾⠀⠙⠁⡎⢉⣉⠥⢾⡍⣀⠻⢮⡷⢺⠀⢠⠜⠁⠘⠇⠀⢠⠁
⠀⠀⠇⠀⠀⠀⠀⠀⠘⠀⣷⣌⠷⣎⡁⠔⠛⢂⡀⠀⠛⠃⠀⢈⣷⠀⠀⠀⠀⣾⡁⠀⠘⠛⠀⢀⡐⠚⠆⢈⣡⡾⢃⣾⠀⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢰⠀⠀⠀⠀⢰⣷⠸⡙⢷⣌⡓⠾⠉⠀⠀⠠⢤⣤⡌⠛⣁⠀⠀⠀⠀⢈⠛⢡⣄⡠⢄⠀⠀⠉⠶⠞⣡⡴⢋⠃⣾⡄⠀⠀⠀⠀⡆
⠀⠀⠀⠸⠀⡇⠀⠀⣼⢹⡆⢱⡊⣉⠿⢲⣄⠉⠙⠛⢂⣁⠰⣦⣥⠀⠀⠀⠀⣼⡴⠆⣈⡙⠛⠛⠉⣠⡴⠿⣍⠑⡮⢡⡏⢷⠀⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣽⣦⠙⢶⡞⠁⠀⠠⠶⠞⢋⣉⣡⡶⢣⡄⠀⠀⢠⡜⢶⣍⣉⠙⠲⠶⠅⠀⠉⢓⡴⠋⣠⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢛⡀⢓⣄⡉⠛⠁⠴⣀⣈⠠⠔⣉⣴⠇⣀⠀⠀⢀⠸⣦⣉⠂⠄⣁⣐⠤⠌⠛⠋⣠⡚⢁⡻⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢻⣄⠙⣁⠀⠙⢡⢤⣠⣩⡅⠖⢫⡾⠳⣿⠇⠀⠀⠸⣿⢎⢷⡝⠂⢀⣉⣀⣤⡄⠛⠠⡈⠋⢀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠉⢶⣌⠒⢮⣀⡔⠋⠁⣀⠴⣋⡞⡲⠃⠀⢀⡀⠀⠘⢞⠳⣝⠦⣀⠈⠙⠢⣀⡱⠖⣡⡴⠋⢧⠄⠀⠀⢰⡄
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢰⡙⠇⢈⠿⠦⣀⠐⠀⣀⠀⠘⣩⠟⡀⢟⣴⠉⠙⢦⡹⢀⠻⣍⠃⠄⣀⠐⢂⣈⠴⠾⡉⠸⢏⡜⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⢳⡤⠑⢤⡔⢉⣉⠛⢁⣠⠞⡵⢺⠏⣼⠧⠶⠶⠤⣦⠰⡗⢬⡳⢄⡈⠛⣉⠙⢢⡤⠚⣡⠞⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⡌⢦⠙⢠⡈⡹⠉⠋⣉⠔⠛⣰⢃⡼⡇⠀⢀⡀⠀⠘⢣⡙⣆⠛⠦⢉⠑⠈⢋⢁⡔⠛⣤⢃⡇⠀⠀⠀⠘⠇
⠀⠀⡀⠀⢰⠀⠀⠀⠀⣾⠟⢄⠳⣅⡸⠛⠛⠋⠀⣠⠞⠁⣼⠃⣾⠋⠉⠉⠉⣱⠈⣷⠈⠳⣄⠀⠙⠛⠛⠇⣈⠞⣡⠞⣷⠀⠀⠀⠀⡆⠀⢀
⠀⠰⡇⠀⣼⠀⠀⠀⠀⡟⡆⢎⡱⢌⠳⠮⠤⠴⠚⠁⠀⣸⠇⠀⣏⢀⣤⣤⣀⣻⠀⠙⣇⠀⠈⠓⠤⠤⠴⠖⠁⢎⡱⢠⣿⠀⠀⠀⠀⣷⠀⠸⠆
⠀⡀⠀⠀⣿⠀⠀⠀⠀⢇⣿⡌⢣⡜⠓⠄⡴⠖⣲⠀⣰⠋⠀⠔⢟⠉⠀⠀⠈⢻⠀⠆⠘⣆⠀⣦⠐⠦⠠⠚⢃⡜⢡⣯⡸⠀⠀⠀⠀⣿⠀⠀⣠
⠀⠃⠀⠀⢻⠀⠀⠀⠀⠈⣼⢉⣤⠳⣤⣦⠴⠊⣡⣾⠃⠀⠀⠀⣤⠀⠀⣀⣀⣸⡀⢀⠀⠈⣷⣈⠑⠦⢴⣤⠞⢠⡉⣯⠁⠀⠀⠀⠀⡟⠀⠀⠉
⣸⠀⠀⠀⠀⠰⡀⠀⠀⠀⠛⢾⡱⣁⠠⡤⠶⠛⠁⠀⠀⠀⠁⠀⠏⠀⠉⠋⠉⣠⠏⠠⠀⠀⠀⠉⠛⡶⠤⡤⣘⢌⣧⠻⠀⠀⠀⢀⠄⠀⠀⠀⠐
⣇⠀⠀⠀⠀⢤⡸⠀⠀⠀⠘⡈⢻⣎⣉⣉⡁⢀⠂⠀⠀⠀⠀⠀⣄⣀⣠⣤⣀⣈⠀⠀⠀⠀⠀⠀⡀⠀⣈⣉⣡⡞⢁⠃⠀⠀⠀⢎⡤
⠸⣧⣤⣦⣄⣸⡇⠀⠀⠀⠀⠹⣦⡙⠦⢞⡡⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠑⢌⡳⠤⢛⣴⠏⠀⠀⠀⠀⢸⣇⣠⣴⣄⣼
⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣷⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣾⣷⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠖⠒⠚⠛⠛⠛⠛⣛⠳⢦⡀⠀⠀⠀⢀⣠⣄⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⢀⡔⡉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⠳⣌⢳⡀⢀⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠘⠁⢀⣠⣠⡀⠀⠀⠀⠀⢴⣶⣿⣭⣿⣧⣥⣾⣿⣿⣿⣿⣿⣿⠀⠐⡒
⠀⠀⠀⠀⢀⠈⣉⡉⠉⠁⠀⡶⠋⠀⢰⣿⠳⠁⠙⠀⣠⠔⣫⣿⣿⣿⣿⡿⠋⢓⣻⣿⣿⣿⣷⣿⠇
⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⢀⡟⠀⢀⢀⣿⠳⣦⠀⠀⠁⣾⣽⣭⣤⣿⣿⡆⣠⣾⡍⢉⣿⣿⠿⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠻⣾⣇⣠⡟⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡎⣇⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠘⠀⠉⠉⠀⢀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⡆⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠠⠄⠀⣰⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⢾⣿⣿⣿⣿⣿⣿⣿⣿⠴⠥⠀⠘⣿⣿⣹⣀⣂⣀⣀
⠀⠀⠀⠀⠀⠐⠂⠐⠚⡳⢤⡀⠀⢀⣀⣀⣴⣾⣽⠀⠙⠿⢿⣿⣿⣿⣿⡉⢉⠋⠛⠛⠛⠛⠟⠍⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠹⠶⢶⢿⠚⠟⠛⢹⡀⠀⠐⠶⣦⠀⠀⠀⠙⣆⠀⡄⠠⠀⠦⠬⠉⠁⠈
⠀⠀⠀⠀⠀⠀⣬⠤⢤⣤⣤⣥⣤⣀⣀⡀⠀⠐⠂⢳⡀⠀⠀⠉⠀⢄⠀⠀⠀⠙⢦⡐⠒⠚⠓⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢨⠉⠈⠉⠉⠉⠉⠉⠀⢷⡀⠀⠀⠀⠸⣄⡀⢀⠀⠀⠙⣇⠀⠀⢀⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠄⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠸⡽⣿⡇⠀⠀⡏⠀⠀⠠⠤⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠤⣄⣤⣤⣦⠤⠀⠐⠒⠀⣳⣤⠀⠀⠀⣷⢏⠀⠀⢀⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠈⠉⠛⠻⠗⠙⠃⣠⠄⠁⠁⠀⢀⠼⡁⣸⠀⠀⢸⡆⠀⠀⠀⣤⣤⣤⣤⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠜⠁⠀⠀⢀⠔⠃⠀⠀⣿⣀⣠⣼⣷⣤⣀⠀⣿⣿⣿⣿⠿⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⡀⣠⠖⠁⠀⠀⣀⣴⡋⠋⢉⣛⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⢺⣿⣶⣄⣀⣴⣧⣼⠍⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⡗⠒⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⠶⠤⠤⠤⠤⡴⣶⣶⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣤⣤⢤⡄⣤⣥⣾⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⣮⣭⣵⣿⣭⣍⣩⠼⣯⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⠉⠀⠐⠃⠉⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠣⠄⠁⠀⣤⣀⠀⠈⠐⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠈⠁⠀⠈⠉⠀⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⠟⠛⢳⡖⢂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡔⠀⠀⠀⣸⢿⢈⡽⠓⠶⣤⡴⠆⢀⣐⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠰⠷⣴⣉⡠⠈⠸⠊⠁⢀⠔⠁⠀⣀⣽⣯⣽⡷⣷⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢳⣤⣉⣁⠓⢤⣰⣶⠃⠀⠀⠚⠿⣿⡿⣉⣃⡈⠑⡀⠀⠀⠀⢀⣠⣴⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠿⠍⠉⣙⠁⠰⣶⣿⢰⠀⠛⠁⣀⠘⢿⣿⣭⣿⣿⡟⠛⣤⡀⠜⡿⣾⣇⠀⠀⢀
⠀⠀⠀⠀⢠⣶⣾⣿⣯⡖⠶⠓⠒⢒⣼⠿⢿⣥⣠⣮⢹⣶⣆⠀⣩⡗⠐⠀⣸⣿⡿⣷⡿⢅⡥⠂⣠⢞⣿⡆
⠀⠀⠀⠀⠈⢲⣭⣭⣭⣭⣭⣭⣥⣤⣴⡖⠁⠀⠉⠻⡞⢿⡙⣧⠍⠉⠐⢀⣈⣽⠛⣡⠛⠿⣧⣽⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠛⡛⡛⠩⠍⠙⠀⠀⠀⠀⠀⠘⠂⢛⢿⣄⡰⢶⣆⣍⡠⠊⠁⣡⢞⣽⢿⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠰⠛⠠⣼⠀⡘⠑⠛⠉⠐⡛⠛⢹⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⢀⡄⡙⠒⠇⣀⣰⡛⠋⠁⣰⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⣀⡠⢾⠿⣋⡄⣶⣾⣿⠧⣼⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡔⠛⠛⢡⡾⠿⠿⠚⣿⣶⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠏⠁⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣬⣍⣈⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⢠⣶⣶⣶⣶⣤⡀⠀⠀⠀⣤⣴⣶⣤⣶⣤⣄⠀⠀⢰⠀⠀⢸⡆⠀⠀⡆⠀⠀⠀⠀⢰⠛⠇⠀⠀⠀⠀⠀⠀⠛⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠘⢻⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⠇⠀⠀⢸⠉⡏⢹⡏⢹⢩⡃⠀⠀⠀⠀⢸⠤⡆⠀⠀⠀⠀⠀⠀⣼⡿⢻⣷⡀⠀⠀⠀⠀⠀⢀⣘⡇
⠀⠀⠚⠛⠋⠉⠙⠁⠀⠀⠀⠈⠛⠛⠉⠉⠛⠀⠀⠀⠘⠤⠧⠤⠄⠼⠼⠀⠀⠀⠀⠀⢸⠀⡅⠀⠀⠀⠀⠀⢸⣿⣯⣬⣿⡇⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⠀⣠⠠⠀⡄⠠⢤⢠⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠊⠀⠀⠀⠀⢸⠀⠲⢿⡷⠆⠀⡃⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⠀⠀⠀⡇⣿⣿⠿⡿⣿⣾
⠀⠀⢠⠔⠀⠀⢤⠀⠀⠀⠀⠀⡸⠗⠑⠾⣄⠀⠀⠀⢸⡀⠺⣉⣉⠆⣀⡇⠀⠀⠀⠀⠶⠤⠴⠀⠀⠀⠀⠀⢸⣿⣤⣿⣿⣿⠀⠀⠀⢧⠿⠿⣶⡷⠿⢿
⠀⠐⠁⠀⠀⠀⠀⠙⠀⠀⠀⠋⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⢿⣿⠿⠀⠀⠀⠀⠀⠠⠭⠤

⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⢠⣤⣤⣤⣄⡀⣀
⠀⢰⣶⣶⠿⠷⠶⣦⠀⠀⠀⡇⠀⠀⣦⠀⠀⠀⠀⠀⠀⣄⡀⣼⡇⢀⣠⠀⠀⠀⠀⠀⠀⠀⣨⡿⠖⠀⠀⠀⠀⢺⡟⢛⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡃
⠀⢸⡶⠟⣶⠲⠞⡏⠀⠀⠀⣧⣀⣈⣋⣀⣤⡄⠀⠀⠀⢘⣿⣿⣷⣟⠁⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⣰⠀⠀⠀⠁⣦⠀⠀⠀⠘⠛⠛⠿⢿⣿⣿⠆
⠀⠈⠉⠛⠁⠀⠈⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙

⠀⠀⣠⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⢰⣿⣿⠏⢿⣿⣷⠀⠀⠀⠀⠀⣤⣤⣄⠀⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠰⠖⠀⢸⠦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣿⣷⣾⣧⠀⠀⠀⠀⠀⠀⣿⣸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⣠⠞⠁
⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠃⠀⠀⠘⠃

⠀⠀⠀⡄⠀⠀⠄⠀⠀⠀⠀⢠⣾⣿⢰⣧⡲⡄⠀⠀⠀⣴⠞⠋⡉⠓⢦⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⢀⣀⡀⢀⡀⠀⠀⠀⠀⢀⠶⡿⣟⡈⣱⣄
⠀⣠⣀⣀⠀⢀⣠⣦⠀⠀⠀⣧⣿⡿⠸⣿⣿⢡⠀⠀⢸⡇⣴⠀⡇⢰⠈⡇⠀⠀⢸⡟⢻⠲⣿⢹⣇⠀⠀⠐⣛⣛⣛⣛⣛⣛⠀⠀⠀⣦⡔⣾⣿⣻⡞⢿⠄
⠀⠹⣿⠟⠀⠘⢿⡿⠂⠀⠀⠳⣋⣾⢸⣦⣻⠃⠀⠀⠈⢷⣌⡒⠐⣊⡼⠁⠀⠀⣿⣷⠾⠙⠿⣾⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁⠀⠀⠘⣿⠊⢛⣯⣦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢓⠆⠶⠃⠀⠀⠀⠀⣠⣴⣶⣶⣄⣄⠀⠀⠀⠀⠄⠤⠄⢠⣤⣤⡀⠀⠀⣶⣶⣶⣴⣤⣤⠀⠀⠀⠀⠼⡧⠤⠤⠤⠦⡄⠀⠀⡏⠀⣠⣤⡄⠄⢸
⠀⠀⠀⠯⢀⡠⠇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠗⠀⠀⢰⣦⣤⣶⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡄⠁⣂⠈⢺⡀⡇⠀⠀⡇⠀⠻⡿⠇⠀⢸
⠀⠀⠀⠈⢡⠉⠀⠀⠀⠀⠀⠘⠋⠉⠙⠏⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠻⠿⠿⠿⠿⠟⠋⠀⠀⠀⣷⣶⣴⣶⣴⣶⡇⠀⠀⠈⠉⢉⣿⡍⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⠻⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠘⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠆⡠⠀⠠⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣷⣿⢇⠰⡀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⢦⡘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡧
⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠻⣿⣷⠆
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣶⣤⡤⠖⠋⠉⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⢀⣤⠀⠀⣀⡰⠟⠋
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢼⡀⠀⠀⢀⠠⠤⢤⡀⠀⠀⠀⠑⠒⠊⠉⠻⡷⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⠰⡟⠒⠒⠉⠀⠀⠀⡃⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠳⣤⠤⠞⠁⠹⡄⠀⠀⠀⢀⣠⠶⠋⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⣀⡤⠖⠂⠹⡀⠀⠀⠘⣄⢄⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡜⠁⠘⠓⠒⣲⠀⠀⠀⠀⠀⣹⠚⠉⠙⠆⠀⠀⡀⢳⡀⠀⠀⠀⢩⠃⡠⠚⠻⠿⠿⠇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢽⡂⠀⠀⠀⢠⠇⢀⣄⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠸⡰⠁
⠀⠀⠀⠀⠀⢿⠿⠿⠟⠳⠶⠤⠤⠤⠿⠁⠀⠀⣠⠞⠀⡼⠀⠙⢦⣀⠀⠀⢀⡟⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠈⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠑⣿⠀⢀⡤⠴⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⠄⠙⢿⡄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣠⠷⠊⠁⠀⢀⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠈⠂⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣽⠀⢀⡜⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡿⠠⠚⠛⠷⣦⣀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠉⠢⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⢀⠈⠙⢷⣏⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠢⡀⠙⣷⡄⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢻⣦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣡⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠹⣷⡀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠠⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠑⢄⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠘⣷⡀⠀⠀⢀⠤⡲⠋⠁⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠂⢌⣆⠘⣿⣶⡨⠕⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠘⣿⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⡀⠀⣀⢤⠞⠋
⠀⠀⠀⠀⠀⠀⡠⠃⠰⡀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⣩⠔⠁
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠣⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠊
⠀⠀⠀⠀⢰⣯⠀⠀⠀⠀⠱⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⢀⣾⡇⠆⠀⠀⠀⠀⠈⠢⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣾⡘⣿⡘⢄⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠙⢿⣦⣀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠸⣧⠘⣷⡈⠢⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⣿⠛⠷⣮⣄⡀⠀⠀⠀⠀⣀⠔⠈
⠀⠀⠀⠀⠹⣷⡈⢿⣆⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠉⠛⣻⠖⠚⠉
⠀⠀⠀⠀⠀⠘⢿⣄⠙⢷⣄⡀⠈⠁⠀⠀⠀⠀⠀⡰⠃⠈⠒⠤⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡙⠻⢶⣤⣄⣀⣠⠤⠞⣁⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⡈⠙⠻⠶⣤⣄⡠⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠭⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠠⠘⠛⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠚⠦⣁⠉⠻⠝⠏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠒⠠⢀⠀⠀⠀⢀⣠⣤⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠠⠍⠃⠀⠁⠀⠀⠉⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠄⢰⡄⠀⠘⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀⠀⠀⠠⠆⠈⢢⠀⠀⠊⠀⠀⠀⠀⡔⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠑⢄⣀⣤⠁⠈⠀⠀⠀⠀⠀⠀⠑⠚⠘⠄⠈⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⡑⠤⠤⠂⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⢀⠈⢆⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣆⢠⢄⡀⠈⡆⠀⠀⡀⠀⠀⠀⠈⠑⠦⡄⠖⠁⡰⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠁⠀⢂⠈⠂⠀⠀⠀⠀⠁⠲⠷⡪⠊⠀⠀⠄⠈⠒⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠦⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⢀⠚⣦⡉⠁⠪⣑⠂⠀⠀⠀⠀⠀⠁⠀⠈⠀⣀⠔⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠃⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡤⠖⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠆⠲⣄⠀⠘⠦⡀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢀⠀⠐⡶⠀⢄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠂⠀⢀⡏⠀⠀⠳⠀⠀⠀⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠠⠁⣀⡀⠜⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⣠⠄⠠⣄⣨⣭⡏⠄⠀⠁⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠷⣚⣤⠴⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠠⣤⣀⡀⠀⢀⠀⢀⣼⢶⡚⠳⠀⢀⣀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠾⢶⣦⣽⠀⠈⠉⠙⠁⠂⠉⠁⢰⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠠⠐⢢⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠰⠀⠀⣸⢳⡖⠀⠀⠉⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⢠⠃⠀⠀⡿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⣀⡤⠚⠁⠉⠙⠛⠁⠀⠀⢀⡎⠀⢠⠴⠃⠀⠉⠓⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢠⡀⠓⠋⠟⠀⠀⠠⣀⠄⠀⠀⣸⠃⠀⠀⠀⠰⠋⠘
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠠⢟⡧⣀⣀⣀⠰⡄⠀⠀⠀⠀⠒⣼⠶⠴⠚⠈⣳⢄⡀⡄⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠨⠃⠀⠀⠀⠀⡿⠇⠀⠀⠱⣈⣹⠋⠈⠙⠄⢰⣶⣄⣀⣅⠀⣀⣠⣴⣧⠀⠀⠈⠁
⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⠀⠀⠐⠤⣄⢶⡾⠛⠛⠀⠀⠀⠈⣟⠉⠉⢹⣛⣛⣯⡅⠀⢢⠁⣠⠄
⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣌⢻⣋⠙⠻⠿⢿⡄⠂⠈⠚⠁⠀⠀⡀
⠀⠀⠀⠤⠤⠂⠀⠀⠀⡀⠐⢢⣄⠀⣀⠀⠀⠀⠹⣀⠀⢠⠞⣿⢽⣯⣷⡄⠀⠀⢉⡄⠀⢠⣄⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠹⠀⠈⠀⠒⠒⢂⠀⠉⠁⠈⠀⠀⠀⡳⠳⠾⠛⠋⠀⠉⠀⠈⠀⠀⠈⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⠀⠀⠀⠀⢀⡴⠀⡠⣇⠀⠀⠁⠀⠀⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⡔⠊⠀⠀⡀
⠀⠀⢀⠠⠂⠀⠀⠀⠹⠿⠄⠀⠀⠀⠀⠀⠀⠜⠀⠀⡁⠁⠀⠀⠀⣠⡴⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⠁⠀⠠⡀
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠔⠉⠀⢠⣀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣠⡷⠾⠛⢹⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢇⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠖⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠈⠁⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢦
⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⡾⢶⠆⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⣦⣼⣷⠀⢀⣀⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⡀⣀⠀⠀⣸⡏⣰⡄⢰
⠀⠀⠀⠀⢸⡇⢀⣻⢭⡿⠀⠈⠋⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⣷⡟⠃⢸
⠀⠀⠀⠀⢸⡟⠛⣿⡟⠁⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣶⣍⠛⠛⡁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣦⣄⡀⠙⠂⠈⢿⡟⠀⢹
⠀⠀⠀⠀⢸⡇⣶⣿⣁⠄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢂⢀⣿⡄⢸
⠀⠀⠀⠀⠈⣧⣿⡇⠯⣸⡟⣿⣻⢯⡟⠷⠿⠿⢿⣿⡿⣿⣿⣇⠀⠀⠀⠠⠄⠀⠗⠒⢄⣀⠀⣸⣿⣿⡿⠟⣻⡛⣟⡿⣿⣿⣿⣿⣿⣧⢼⡾⠙⣇⠨
⠀⠀⠀⠀⠀⢸⣿⢉⡆⣼⣿⣿⢿⣶⣾⠳⠀⠚⠃⣨⣠⣬⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢲⣿⣟⣯⣤⣶⠎⠀⠉⠑⠠⢾⡿⣿⣿⣷⠀⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠈⡿⢸⠃⣿⣸⡷⣨⣿⣅⠀⢠⠈⠰⢉⠻⣿⣿⡇⠀⠀⠀⣀⣄⠀⣤⣄⡀⠀⠈⢿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠈⢳⣾⢿⣿⡄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣰⠁⢼⡁⣿⣯⣷⢹⡿⢿⣆⢰⣧⡂⢤⡁⣻⡿⠇⠀⢠⣾⣿⣿⣿⣿⣿⣧⠀⡀⠸⣿⣿⡟⠩⢀⣤⣔⣀⣀⣠⣼⣇⣹⣿⠇⠀⠀⠸⡇
⠀⠀⠀⠀⢠⡟⢰⡟⠓⠈⠻⣿⣾⣥⣤⣿⣿⣷⠶⠾⠛⠋⠀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡖⢦⡈⠉⠛⠻⠿⠷⣮⣭⣿⣧⣿⣿⠿⠃⢀⡀⠀⠀⢻⡆
⠀⠀⠀⠀⢸⣧⣾⠁⠀⠀⠒⠀⠈⠁⠀⠐⠀⠁⠀⠀⠀⠸⠿⠀⢸⣿⣿⣿⡿⣿⣽⣿⣿⣿⡇⢰⡏⠐⠒⠀⠀⠀⢀⠛⢨⣍⣉⠀⠀⢰⡟⠤⠀⠀⢶⡄
⠀⠀⠀⠀⢸⡇⣧⡀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⣾⣿⣿⣿⡝⣿⣿⠻⣿⣿⣧⡄⠉⠀⠀⠀⠛⣿⠉⠛⠀⠈⠙⢦⣄⠐⣿⡂⠀⠀⣼⡇
⠀⠀⠀⠀⠈⠳⣌⢻⣶⣦⡔⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢼⣟⡉⣹⣿⣿⣿⣿⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠀⠘⣧⣀⣴⡽⠃
⠀⠀⠀⠀⠀⠀⠙⢶⣬⣿⣋⣀⣀⡀⠀⢲⣶⣦⣀⠀⠀⢀⡀⠘⢿⣿⣿⣿⣿⠇⣿⣿⣿⣿⡿⣯⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣾⣷⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣄⠈⠻⣯⡳⣴⠿⡟⠀⠀⠈⠉⠉⢹⡖⠀⠉⠉⠉⠀⠁⠸⠃⠀⣾⡿⠛⠽⢦⣿⣿⣿⣿⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡻⠿⣿⣿⣿⣧⠀⡙⢿⡃⣔⠃⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⢸⣶⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠂⢺⣿⣿⣿⣧⡿⣾⡟⣿⠁⡞⠀⣶⠀⢀⡀⠀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠳⠀⣿⣿⣿⣿⣿⣻⣏⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⣿⣿⣿⢿⣼⣿⣿⣿⣼⣇⡀⣿⣄⢸⣇⣄⢸⣆⣀⣣⣤⣤⣧⣤⣄⣀⣀⣽⠟⠉⣻⡏⣹⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠺⢻⣷⡄⠉⢻⠉⢇⠉⠈⣿⠁⠈⣿⠁⠈⣿⠁⠉⡿⠁⢨⡏⠀⣿⢻⢿⡇⠀⣠⣿⢁⡆⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠰⡆⠉⠻⣦⣙⣀⠻⢧⠀⡏⠀⢠⡏⠀⢸⡇⠀⠀⠁⠀⠀⣧⠀⣿⢸⣞⣃⣴⢻⡿⢸⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢠⡇⠀⠀⠈⠛⠟⡻⣿⢴⣷⣴⣿⣧⣶⣾⣷⣶⣶⣿⣶⣶⣿⣦⣿⠿⡟⠉⠙⠋⠀⢸⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⡇⠀⠀⠀⠀⠀⢁⡀⠀⣿⡄⢿⡍⢿⣥⠹⡏⢻⠁⣿⡏⢹⣿⢻⡼⠿⠀⠀⠀⢠⠈⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⠸⡇⠀⠀⠀⠀⠀⠈⠙⠖⢹⣤⣬⣧⠾⢯⣬⠥⠼⠠⡿⠷⣬⠿⢬⠁⠀⠀⠀⠀⡘⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣀⠀⢀⣤⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⣰⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣶⣶⣶⣶⠶⡤⢄⣀⣀⣀⡀⢀⣠⠴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠀⠈⠀⠁⠀⠉⠀⠉⠀⠁⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣼⡐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠇⢹⣇⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣿⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠿⢤⣶⣾⣿⣿⣶⣶⡖⠲⢴⣶⣿⣿⣶⣦⡤⠇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢻⢸⣿⣿⣿⣿⣿⠿⣃⣘⠘⣿⣿⣿⣿⣿⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠈⠶⠝⠛⠉⠉⢒⣰⣿⣿⡖⠦⠍⠛⠉⠋⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⠀⢀⣀⡄⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⡏⢷⠀⠀⠀⠍⠩⠀⠀⠀⣾⢹⡿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢳⣜⠒⠢⠤⠤⠤⠤⠶⢖⢣⡞⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠈⠱⡄⠀⠀⣸⠀⠀⠛⠉⠑⠒⠀⠢⡒⠋⠙⠚⠁⠀⠇⠀⠀⠀⠔⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠱⡀⠀⠘⠧⣀⠀⠀⠀⠂⠤⠤⠀⠀⠀⢀⣀⠔⠁⠀⠀⠚⠀⠰⢦⡉⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣶⣄⠀⠀⠑⠄⡀⠀⠀⠑⢦⡀⠀⠀⠀⠀⢀⡤⠊⠀⢀⠠⠊⠀⠀⠀⠀⠈⠃⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠠⡀⠀⠀⠐⠄⡀⠀⠈⠉⠉⠉⠉⠁⠀⡠⠔⠁⠀⠀⠀⡠⠄⠒⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠑⠀⡀⠀⠀⡠⠔⠉⠀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣀⠀⠀⠁⠃⠀⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠉⠢⢄⡀⠀⠀⠁⠂⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠈⠀⠀⢀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⠁⠈⠁⠀⠀⠀⠀⢀⠠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⡀⠀⠀⠀⠀⠀⡀⠈⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⡶⠂⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠐⠇⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣄⢀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢃⣀⣤⣤⣤⠤⠶⠶⠶⠶⠶⠶⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣌⠻⣄⠀⠀⣠⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣾⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣒⡶⣽⣧⡹⠀⣰⣿⣿⣿⣿⣿⣿⡧⠀⡀
⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⣀⠀⠀⢀⡇⠀⠀⣤⡾⡶⢦⡀⠀⠀⢀⡠⢽⣿⣿⣷⣿⣿⣿⢾⣻⣿⣿⣿⣿⢿⣿⡇⠀⠀⠤
⠀⠀⠀⠀⠀⠤⠠⠠⠄⠀⠀⠀⢸⠋⠡⠀⠸⣿⣱⠁⠀⠁⢠⠖⣩⣶⣿⠿⢿⣿⣿⠋⠀⣩⣿⣿⣿⣿⣿⣿⠿⠀⠀⢀⡀
⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⣼⠁⠀⣀⢠⣿⠛⢷⡄⠀⠀⢠⣯⣼⣶⣰⣿⣿⣿⠀⣐⣿⣄⢀⣿⣿⠿⠟⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠈⢿⣿⣇⣠⡾⠁⠀⢀⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢹⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠘⠀⠉⠉⠀⠀⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣷⢳⠀⠉⢉⠉⣉⠉
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡼⠉⠀⠘⣿⣿⣏⣇⠀⠄
⠀⠀⠀⠀⠀⠁⠤⠉⠩⠼⢷⣅⠀⠀⠀⠀⠀⠀⣀⣴⣛⡃⠙⢿⣿⣿⣿⣿⣿⣿⡟⠒⠺⠗⠶⠲⠿⠿⠿⠟⡛⠛⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣲⣾⣿⣿⣿⠿⡇⠀⠀⠈⣉⣉⠉⠉⠛⠻⣆⠘⠐⠂⠀⠀⠠⠤⠄⠀⠀⠰⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠁⠀⠠⠁⠉⠛⠒⠂⠀⠁⣀⢱⡀⠀⠀⠈⠽⠀⡀⠀⠀⠈⠳⣬⡓⠀⣀⣀⣬⡀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠺⠿⠿⡾⠷⠶⣶⠤⢤⡤⢬⡄⢳⡀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠙⢦⣀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠳⣤⣄⣤⠀⠀⠀⢧⠀⠀⠀⠦⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢧⢻⣿⠆⠀⠀⣜⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠒⠶⣤⣤⣴⣶⡦⠀⠐⠒⠂⢀⣷⣤⡄⠀⠀⢸⡖⡇⠀⠀⠀⡇⠀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠛⠛⠷⠄⠋⠀⡠⠊⠁⠉⠀⠀⣠⠏⠁⣧⠀⠀⢈⡇⠀⠀⠀⢀⣄⣀⣀⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⣠⠞⠁⠀⠀⡇⠀⣀⣨⣷⣄⣀⠀⢸⣿⣿⣿⣿⣿⣯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠟⠁⠀⠀⠀⣠⠞⠰⠲⠶⠶⢦⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠚⠞⣱⣿⣅⡀⠀⠀⣠⠟⣓⡖⠀⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣶⡞⣿⣟⣚⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠹⣿⣿⣿⣿⣃⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣿⣿⣫⢿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠲⣄⢀⡠⣄⠀⠀⢸⣿⣿⣿⣬⣯⣿⣿⣿⡇⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠂⠀⠸⣀⣀⠔⣏⠀⠈⡇⠀⢸⣿⣿⣿⡿⢻⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣷⠀⠀⠈⡏⠀⢘⡟⠊⠀⠀⢸⣦⣿⣯⣽⣥⣽⣿⣿⡇⠀⢰⣷⣿⢸⣿⣏⢰⣤⡆⠶⢶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠺⠿⠻⠀⠀⠀⠛⠶⠋⠀⠀⠀⠀⠸⠿⢿⡿⡿⢿⣿⣿⡿⠇⠀⠀⠉⠁⠀⠉⠁⠈⠉⠁⠀⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⡷⣤⡀⠀⠀⠀⠀⢠⣾⠿⢿⡆⠀⠀⠀⠀⠀⢰⣷⣾⡆⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠠⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⢷⡄⠀⠀⢸⣷⣶⣶⡇⠀⠀⠀⠀⠛⠿⠿⠿⠿⠛⠀⠀⠀⠀⢰⣾⣿⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣷⡏⠀⠀⠀⣀⣿⣿⣿⣃⠀⠀⠀⠀⠀⣠⡀⢀⣄⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⣀⣾⣿⡛⢛⣿⣷⣀⠀⠀⠀⠈⣻⣿⣿⠃⠀⠀⠀⠀⠀⠿⠿⠿
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⡛⠛⠐⠿⠛⠻⠃⠘⢻⣟⠻⠂⠀⠀⠀⠻⢟⠿⠃
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⡟⣿⣿⡄⠀⠀⠀⠀⠀⢀⠎⠡⡤⠉⢆⠀⠰⢣⣬⡿⣥⣌⠆⠀⠀⠀⢀⣿⣿⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⡿⠛⠻⣿⡇⠀⠀⠀⠀⠀⠈⣅⠀⠃⢀⠍⠀⢀⢸⣿⣤⣾⣟⡀⠀⠀⠀⠘⠻⠿⣷⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⡀⠀⢰⣿⣇⠺⠆⣸⡇⠀⠀⠀⠀⣠⠞⠁⠉⠉⠁⠀⠀⡄⣿⣍⣿⣻⣿⢣⠀⢰⠶⠶⠶⡴⣿⡟⠈⠉⠉⠙⠛⠁
⠀⠀⠀⢸⣿⣿⣿⣿⠏⣅⡙⣿⣿⣿⣿⣯⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠪⢟⢿⡛⠝⠃⠀⢸⠄⠂⠁
⠀⠀⠀⠸⣿⣿⣿⣿⣷⣭⣶⣿⣿⣿⣿⡟⠀⠈⠉⡭⠿⠿⡍⠁⠀⠈⣡⣤⣤⣤⣤⣤⢤⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣶⠟⠓⠒⠒⠒⠒⠚⠷⡆⢿⣿⣿⣿⣿⣿⡇⣿⢰⡞⡀⡐⣲⣶⡒⠒⡒⢶⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⢤⣶⣶⣿⣇⡇⡇⠀⠀⠀⢖⡆⠀⠀⠀⡇⢻⣿⣿⣿⣿⣿⠇⡿⢸⡅⠔⠰⣼⣿⠽⠰⠆⢸⠀⡆⡆⠁⡇⢰⣲
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡞⠻⠿⣿⡏⠇⣇⠀⠀⠀⠚⠃⠀⠀⢀⡇⣸⣿⣿⣿⣿⣿⠀⡇⢸⣷⣶⣶⣾⣿⣶⣶⣶⣿⠀⡇⡇⡟⢻⣾⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠀⢿⣶⣶⣶⣶⣶⣶⣶⣾⡇⠻⠿⠿⠿⠿⠿⠠⠇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠘⣧⡗⢧⡜⢹⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠲⣶⣶⣶⡶⠆⠀⣴⣤⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣾⣗⡆⠀⢀⣤⠶⣾⣗⡒⡄⠀⠀⣠⣶⣿⣿⣿⣿⣿⣦⠴⡶⠀⠀⠾⠿⢿⠿⠿⠆⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⢊⣽⣶⢍⢿⣿⡍⣷⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢼⡛⢻⡆⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠻⠿⠆⠚⠿⠋⠅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣤⣬⣿⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣵⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠙⢦
⠀⠀⠀⠀⢠⡞⠙⢦⣰⢳⡅⠀⠀⣧⠞⠉⠻⡄
⠀⠀⠀⠀⢰⡅⠀⢸⡏⠯⠀⠀⠒⡏⣐⠀⢰⡇
⢀⠶⠒⢦⡈⠇⠀⢸⣥⠦⠤⠤⣤⡇⠿⠀⠘⡇
⡎⣰⠀⢸⠇⠒⠒⠺⡇⠀⠀⠀⣠⣄⣀⣀⣀⡇
⡇⠀⣀⣈⡂⠀⠀⢸⡇⠀⠀⠀⣼⠀⠀⠀⠀⡧
⡟⠁⠀⠀⠁⠀⠀⠸⡇⠀⠀⠀⠈⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡠⢄⡀
⡇⠄⠀⠀⡇⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⡲⡑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠈⢣⡰⢋⠄⠀⢻
⢳⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡇⢠⠋⡘⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡜⡼⠀⠀⣸⠣⠜⠀⠀⣾⠖⠒⢆
⢸⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⡀⠬⠡⠀⠀⠀⣾⡀⠀⠁⠀⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⡾⣈⡀⠀⢤⠇⣀⠀⠀⢸⣏⠆⠀⣸
⠸⣀⠀⠀⢿⠀⠀⠀⠀⠈⢪⡀⠂⡠⠂⠀⠀⠁⠘⢦⡀⣠⠎⠀⢀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠉⠓⣎⠁⠀⠉⠓⠯⠘⠀⢰⡇
⠀⣷⠀⠀⠈⠑⣆⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⣠⠟⠁⢀⠎⠉⠙⢦⠀⠀⠀⡼⠡⠀⠀⠀⠰⠌⠀⠀⠀⡌⠓⠤⣀⣸⢗⠒
⠀⠧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢸⠀⡆⠀⢈⣧⠀⡐⢦⠁⠀⠀⢀⠗⠀⠀⠀⠻⠀⠀⠀⣸⢃⠏
⠀⢸⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢸⢠⠃⠀⠀⠸⣢⡑⠂⠀⠀⠀⡜⠀⠀⠀⠀⠁⠀⠀⣠⠍⠑⠦⣘
⠀⠀⡆⠀⠀⢀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡇⠂⣠⡴⠶⠻⠈⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⡟
⠀⠀⢣⠀⠀⠀⢸⡇⡀⣀⣀⣀⣉⣀⣀⣁⣀⣿⠀⠀⠀⠀⠀⠘⡎⠁⠀⠄⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⢸⡗⠒⠚⠚⠛⠛⠛⠯⠭⠭⠍⠉⠉⢉⣿⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡽
⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣈⡿⠀⠀⠀⠀⠀⠀⡘⠆⠀⠀⠀⠀⠀⠀⣸⠁⠈⠂⢀⢨⠐⠆⠀⠀⠀⡾⠁
⠀⠀⠘⡏⠉⠉⠉⡛⠉⠛⠙⠙⠉⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⢀⠷⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⠀⠁⠀⠀⠀⢤⡾⠁
⠀⠀⢘⠁⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⡼⠛⠒⠦⣄⣀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⣆⠀⠀⠀⠀⣿⢈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣼⣁⡀⠀⠀⠀⠉⠛⠺⢧⣄⡀⠀⠀⠀⠀⠰⡱⠋
⠀⠀⣿⠀⠀⠀⠀⣿⢠⠔⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠛⠿⣤⣒⡤⢠⡀⠀⠀⠉⠛⠶⢤⣀⡞⠁
⠀⢠⢹⠀⠀⠀⠀⣿⠸⢀⡃⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠠⡐⠀⠟⠙⠳⣦⣤⣀⠀⠀⠀⠀⣌⠂
⠀⢸⡇⠀⠀⠀⠀⣿⢸⠘⠇⠀⠀⠀⠀⠀⡄⣰⠏⠀⠀⠀⠀⡰⠡⠗⠀⠈⠀⠀⠉⡙⢛⠶⢦⡤⠎
⠀⡇⠀⠀⠀⠀⠀⣿⠘⠀⠀⠀⠀⠀⠀⢈⣼⡧⠀⠀⠀⠀⡰⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠱⡼
⢰⠘⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⡼⢣⡻⠵⠀⠀⠀⠀⠀⠀⠀⠀⣶⠃
⣆⣀⣀⡀⠀⠀⣀⣿⣀⣀⣀⣀⣀⡴⠁⠀⠀⠀⠀⠀⡼⢁⠳⡙⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠟⠀⠀⠀⠀⠀⢈⡼⠡⠢⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄⠀⠀⠀⠀⢢⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⣤⣀⣀⡼
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣠⣄⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢭⠭⣷⣴⣶⣤⣤⣀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣐⣿⣯⣿⣿⣿⣿⣿⣷⣾⡯⣗⡢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢆⠨⢗⣺⣯⣽⣿⣿⣿⣿⣿⣿⣯⣿⡿⢷⠅
⠀⠀⠀⠀⠀⠀⠀⠀⢔⡅⠨⠀⠙⡺⣷⣾⣿⣿⣿⣿⣿⣿⣆⣽⣿⣿⣿⡗⡻⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢊⠈⣀⢐⣁⣶⣾⣼⣿⣿⣿⣿⣿⣏⣻⣿⣿⣿⣿⣿⠮⡸
⠀⠀⠀⠀⠀⠀⠀⠠⡡⠂⠀⢴⣟⣿⠟⠋⠉⠉⠛⠛⠙⠀⠈⠉⠛⢿⣿⢿⣟⠢⣀
⠀⠀⠀⠀⠀⠀⠘⠖⠄⣤⢠⣏⢝⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⢝⠮⠫
⠀⠀⠀⠀⠀⠀⠀⣠⡳⡁⠩⣻⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⡅⠐
⠀⠀⠀⠀⠀⠠⢃⡹⡷⣱⣇⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢾⢌⢃
⠀⠀⠀⠀⠀⠠⢸⠳⣿⠙⡖⣏⠀⠀⡂⢄⠦⣄⠀⠀⠀⢀⣤⢤⠾⠖⠀⠀⣺⡢⠶⡀
⠀⠀⠀⠀⠀⠠⣰⡛⢷⣤⠄⢯⡁⠀⢹⡟⠟⣿⠆⠀⠀⢾⢣⠇⢆⠂⠀⠘⠉⠑⡄⡁
⠀⠀⠀⠀⠀⢠⣡⢱⣭⣸⡄⡛⠁⠀⠀⠁⠙⠈⠀⢀⡀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⢧⣟⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠚⠙
⠀⠀⠀⠀⠀⠀⠈⡼⠾⣿⢻⣛⢿⣿⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⢰⢱⣿⡳⣍⣻⣷⡟⡰⠄⢀⠀⠀⠀⢀⠀⠀⣼⣎⠛⡠⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠃⣎⢻⣿⢿⢽⣿⣥⠁⢇⢉⢷⣶⠶⡬⣴⢞⣻⣽⡅⠐⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠘⣮⣟⡞⡿⣿⣿⣻⢣⢋⠋⢮⡯⣽⣛⣽⡟⣝⡞⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠜⡜⡿⣝⣿⣏⢯⠮⢣⡗⣽⣯⣿⣿⣽⣿⡮⠨⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⠿⢝⣳⣽⣶⠺⠻⡿⠹⠿⢿⣽⠛⠶⡊⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡄⠉⠉⠠⢴⣯⠁⠈⣅⠐⠂⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠣⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠉⠉⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⡆⠀⠀⠤⠤⣤⣄⡀⠀⠀⠀⠀⠀⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣧⡀⠀⠀⠀⠀⠉⠻⢷⣤⠀⣀⠔⠟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⣾⡃
⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⣧⡀⠀⠀⠀⠀⠀⠀⢙⠶⣄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠊
⣾⠃⠀⠀⠱⡄⠀⢀⣀⡀⠀⠀⠀⠸⡆⠙⠿⡄⠀⠀⢀⡤⠊⠀⠀⠀⠙⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠃⠀⠀⠀⠈
⢷⡀⠂⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⣄⢦⣀⣀⣤⣄⣤⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠂
⠈⢷⡀⢠⡀⠀⣱⡀⠀⠀⠀⠀⠀⠀⡖⣤⠀⠀⠀⢀⣠⣀⠀⠀⢀⡀⠀⠀⠀⢘⡜⠛⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠋⠀⠀⠀⠀⠀⢠
⠀⠈⢷⣴⣿⠉⣿⣿⡄⠀⠀⠀⠀⣸⡿⣾⢀⡴⠢⠎⠝⢭⢗⣶⣴⢾⣥⣶⣵⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠟⢋⣩⡄⠀⠀⠀⠀⣀⣤⠟
⠀⠀⠘⣿⣿⣧⠈⢿⣿⡄⠠⠤⠴⠟⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⠛⠻⠻⣛⣤⠀⠀⠀⠀⠀⠀⠀⣏⢉⣁⣤⣾⣿⣿⣧⣤⡴⠖⠋⠉
⠀⠀⠀⠘⣿⣾⣧⠘⣿⣿⠀⠀⠈⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⢠⡇⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠘⢿⣿⣿⡁⠋⢠⠆⠄⠀⠀⠈⠳⣄⡀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢠⢀⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠑⣼⣿⣷⣄⠀⠀⠐⠒⠒⠒⠛⠋⠁⠀⠀⢀⢴⣿⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢄⡈⠛⢶⣤⣿⡏⠛⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠻⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢷⣤⣤⣀⣀⡀⠠⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢄⠉⠻⣿⣷⣆⠈⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠈⡄⠀⠈⠛⢿⣿⣦⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰
⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠸⠷⢤⣀⣠⢀⡀⣠⠔⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⢧⣽⣿⣷⣴⣼⡴⠋

⠀⠀⠀⠀⠀⠀⠀⡐⠉⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠣
⠀⠀⢀⠒⠀⠢⡀⡀⠀⠀⠘⠀⢀⠔⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠢⠀⢀⠁⠀⠀⠰⢀⠔⠀⠐⠄
⠀⠀⠀⡀⠀⠀⢠⠁⠀⠀⠀⡆⠌⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⢸⠀⠀⠀⠐⡌⠀⠀⠀⡆
⡀⢀⠀⢡⠀⠀⠀⡈⡀⠀⠀⠇⡆⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢘⠀⠀⢀⠃⠀⠀⠀⡐⠀⡀⠄
⡇⠀⠑⡄⢂⠀⠀⠃⠡⠀⠀⢀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠸⠘⠀⠀⡌⠸⠀⠀⠠⢁⠎
⢃⠀⠀⠘⡌⡀⠀⠘⠔⠀⠀⠉⠀⠀⠀⠀⡔⠉⠑⠀⠀⠰⠁⠉⢆⠐⠀⠀⠀⠉⠀⠀⠓⠂⠀⠀⢃⠎⠀⠀⢠
⠈⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⢀⠀⠀⢡⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⢠⠃⠀⠀⠀⠡⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡀⢤⢪⠀⠀⠀⠀⠀⠀⠀⠀⢀⢓⠤⢀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠏⠑⠂⠂⢒⠀⠈⠅⠀⠀⢣⠱⠀⠀⠀⠀⠀⠀⠠⢁⠊⠀⠀⠨⠉⠁⠒⠀⠐⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢡⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠠⣜⠀⠀⠀⠂⠀⠀⠅⠀⠀⡅⠀⡄
⠀⠀⠀⠀⠀⠢⢀⣄⣀⣀⣄⠀⠬⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠰⠤⠀⢠⣀⡀⢠⠄⠄

⣁⣴⣿⠿⠋⢀⣴⣿⡿⠋⢁⣤⣾⡿⠛⢁⣠⣾⡿⠛⠁⣠⣾⡿⠟⠁⣠⣾⣿⠟⠉⣠⣴⣿⠟⠋⣀⣴⣿⠿⠋⢀⣴⣿⠿⠋⢁⣤⣿⡿⠋
⠿⠟⠁⠠⠾⠿⠟⠁⠠⠶⠿⠟⠉⠠⠴⠿⠟⠉⠀⠴⠿⠿⠋⠀⠴⠿⠿⠋⠀⠴⠾⠿⠋⠁⠤⠾⠿⠏⠁⠠⠾⠿⠟⠁⠠⠶⠿⠟⠁⠠⠶
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⣤⢤⡤⣤⠤⡤⢤⡤⣤⢤⡤
⠀⠀⠀⠀⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⡀⠂⢀⠠⡀⠀⠀⠀⣿⠀⡇⣿⠀⡇⢸⠀⡇⢸⡇
⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠆⡑⢈⠐⠀⠀⠀⣿⣠⣇⣿⣠⣿⡞⣷⣧⣸⡇
⠀⠀⠀⠀⠀⢈⣽⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠢⠀⢠⠀⠀⠀⣿⠀⡇⣿⠀⡿⢷⠿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⢀⣿⣤⣇⣿⣠⣧⣼⣄⣧⣼⣇

⠀⢰⣿⣶⣿⣷⣶⣶⣶⣷⣾⣿⣷⡄⠀⠀⠐⢀⠒⠀⠀⠀⢈⠀⡀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠻⠟⠳⢶⣤⡀⠀⠀⢰⣶⣶⣴⣿
⠀⠈⠙⠉⠉⠻⠛⢻⢿⣿⣿⣿⡉⠀⠀⠀⠢⠈⠤⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⣿⣿⠀⠀⠀⢠⣶⣶⣀⣽⣯⣤⣤⣼⣿⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠈⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠙⠉⢹⣿⣿⣿⣿⣿⣿⣦⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣯⣉⣉⣍⠉⠛⠻⠿⠟⠛⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢒⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⣀⠂⣔⠂⢀⠀⠠⠄⠀⠒⠠⠨⢁
⠀⠀⠀⠀⣀⠀⠀⢀⣀⣀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣐⠂⠀⠅⣂⠀⣤⠍⠑⠒⢀⠉
⠀⠀⠈⠉⠙⠛⠿⠿⠿⢿⣾⣿⣿⣿⡶⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠁⠀⠀⠀⠀⠀⠒⠀⢀⡠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡄⠂⠈⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⣰⡀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⣾⣷
⠀⢠⠞⠉⠙⢢⡀⠀⠀⡞⠁⠀⠙⣆⠀⠀⠀⠀⣀⠚⠿⢿⣿⣿⣿⣿⣿⣿⠟⠀⢤⣤⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢰⣦⠀⢹⣿⡇
⠀⠸⡄⠀⠀⣸⠃⠀⠀⢧⣀⣀⣠⠇⠀⠙⠛⠛⠿⠛⢁⣀⡀⠉⠉⠉⠉⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠈⠉⠀⢠⣦
⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋
⣁⣴⣿⠿⠋⢀⣴⣿⠿⠋⢀⣤⣾⡿⠛⢁⣤⣾⡿⠛⢁⣠⣾⡿⠟⠁⣠⣾⣿⠟⠉⣠⣴⣿⠟⠋⣀⣴⣿⠟⠋⣀⣴⣿⠿⠋⢀⣤⣾⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢦⠖⠂⠂⠀⠒⢒⣢⡄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣄⣀⣀⠁⣄⡿⢹⡀⠻⢁⣤⡀⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⠞⠉⠀⠀⠉⠀⡘⠀⡀⠃⡄⠀⠊⡑⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⣠⠚⠁⠀⠀⠀⠀⠀⠀⠉⡈⠀⠆⠀⢀⠰⠤⠆⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡀⠉⠀⠈⠀⠀⠀⠐⠦⣄⠀⠀⠀⠀⣀⣀⣀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⣀⠈⠑⠏⠁⣄⠀⢰⣿⣿⣿⣿⡆⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠆⠀⠀⣴⣶⣦⡙⠇⠀⠀⢀⡀⢀⠸⣿⣿⣿⣿⠃⡃
⠀⠀⠀⠀⠠⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⡀⣀⠻⣿⡿⠃⠀⠀⣀⢆⣴⣾⣦⠀⠈⠉⠁⠀⡷
⠀⠀⠀⠀⢀⡗⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠙⠛⠃⠙⠓⠀⠀⢀⠔⠈⠁⠀⠉⠈⠁⠀⠀⡄⡐⠉
⠀⠀⠀⠀⠈⠓⢄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⡄⡤⢀⣀⣀⣀⡼⢲⠆⠀⠀⠀⠀⠀⣀⠈⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢭⣽⠐⠃⠀⠀⣈⠶⡂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡖⠾⢋⠀⠆⠀⠀⡀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢸⡁⠀⠀⠀⢈⡝⢀⡐⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠢⠀⠀⢀⡠⠤⠠⡤⠒⠂⡹⠀⠈⠧⠀⡀⢲⡀⠀⠀⣷⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣢⠀⠒⠀⢀⣧⡀⠀⠈⠂⠀⠻⡆⠀⣷⠒⠚⣿⠲⡤⠾⠛⠃⠀⠀⠀⠀⠈⣓⠆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣤⡴⢸⡟⢻⡀⠀⠸⡗⠛⠃⠀⢿⠁⠸⢛⠀⠀⢰⣆⠀⠂⠄⠀⠀⣀⡍⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠚⢛⣢⡄⠀⠀⣆⡀⣂⠀⢸⠀⠒⢾⠁⠀⣩⠉⠢⡄⠈⢀⠈⡛⠀⠌⢀⡽⠶⠋⠙⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⡀⠹⠁⠀⠀⡏⠀⣠⠀⠀⡄⠠⣴⠁⠭⠻⣄⠘⠒⠒⢸⠂⠀⡔⠀⠹⠁⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠃⢰⠀⠀⠰⣟⠒⠙⠀⠀⡃⢀⣠⠀⢼⡃⠈⣆⢁⡢⠉⠀⠈⠠⠐⠂⠀⠐⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣀⡀⡶⠀⠀⠀⠙⡀⠘⠀⠀⡇⠐⢃⣠⠎⠑⢠⠤⠋⠁⠠⠞⠀⠀⢀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⢱⡆⠀⠀⠀⠀⠀⠈⡀⠀⡇⠘⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢴⣟⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⠃⠀⠀⠀⠀⠀⠀⠁⠐⣇⠆⠀⠀⡀⠀⠀⠀⡀⠤⡶⠬⣄⣸⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⡉⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⣤⠔⠃⠀⠀⠔⠊⠀⠀⡦⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠁⡜⠋⠁⠀⠰⠂⠀⠀⠀⠀⠀⢃⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠈⠀⠀⠀⠀⠀⠀⢠⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠙⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⠋⠀⠀⠈⠉⠙⠢⣄⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡾⠁
⠀⢀⡤⠶⠶⠶⠦⢤⣄⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁
⢰⡏⠀⢠⡀⠀⢄⠀⠈⠙⠲⣼⡟⠛⠒⠒⠚⠛⠛⠉⠉⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⣿⢇⠀⠀⠙⢢⡀⠱⣄⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⢿⠈⢆⠀⠀⠀⠈⠢⡈⠳⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⣴⠃
⠘⢧⠈⠳⡄⠐⠠⠀⠈⠳⣌⠳⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠁⠀⢀⡼⠃
⠀⠈⠳⣄⠈⠲⢄⣀⠀⠀⠈⠳⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠈⠙⠦⣄⣈⡉⠛⢛⣋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠛⠉⠁⠀⠀⣀⣠⣤⠤⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⢀⣤⠞⠋⣉⠀⠀⢀⡠⠖⠋⣀⠽⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⢀⣴⠋⠀⠀⠀⠀⡠⠖⠉⣀⠴⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⢤⣤⣄⣀⠈⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠔⠁⢀⡤⠚⠁⠠⠂⠀⣠⢺
⠀⠀⠀⠀⣠⡞⠛⠳⢤⣀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢴⣎⠁⢀⡴⠋⠀⢀⣀⡤⠔⠋⣠⡞
⠀⠀⠀⡾⠁⠉⠓⠦⣄⡈⠙⠲⢤⣀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡈⠉⢉⣉⣉⣉⣀⣤⠴⠞⠉
⠀⠀⠀⣷⣄⠀⠀⠀⠀⠉⠓⠦⣄⡈⠙⠲⢤⣀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⢻⡌⠑⠢⢄⣀⠀⠀⠀⠀⠉⠓⠦⣄⡈⠙⣲⠞⢻⡆
⠀⠀⠀⠀⠻⣄⠀⠀⠈⠉⠒⠒⠲⠶⠶⠖⠚⠋⠉⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣓⣶⣶⣦⣶⣶⣖⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢯⣍⣉⣙⡒⠦⣄⡀⣀⡤⠶⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠉⢉⣉⣷⣤⢿⡁⣀⡴⢋⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠰⣏⣁⣠⠼⠖⠋⠁⠀⠀⠙⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠠⢤⣀⠀⠀⢹⣇⠰⡂⠀⠀⠀⠀⠀⠀⣸⢌⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢴⣶⠾⠛⠉⠀⠀⠀⣸⢳⢖⣡⣿⡆⢳⡀⠀⠀⠀⠀⣀⣴⠞⠙⢦
⠀⠀⠀⠀⣠⠶⣟⠉⢀⡴⠋⠀⠀⠀⠀⠀⠀⢠⣃⣈⡥⢿⠺⢿⡄⢉⣠⠴⠖⠛⠉⠀⠀⠀⠈⠳⣄
⠀⠀⢀⣾⠥⢄⠀⠀⠉⠀⠀⠀⢠⠇⠀⠀⢀⠞⠛⠒⣾⣏⣃⠀⡿⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⡠⠋⡤⠤⢿⠹⡈⠉⡼⢁⠞⠀⠙⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠈⢷⣀⡶⠂⠀⠀⢀⡴⠛⠒⠚⠋⠀⠈⠋⠉⠿⠷⠇⣸⠟⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠈⠻⢧⣤⣤⠶⣿⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⢤⣌⣉⣓⡲⠤⠤⣄⣀⡀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⣆⠹⢦⣀⠀⠈⠉⠙⠒⠦⢤⣉⠉⠛⡇⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⢀⠩⠟⣳⠶⣤⣤⣀⣀⣈⣉⣹⢿⡀⠀⠀⠀⠀⠀⡏⣠⣾⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⠀⠀⠃⠀⠁⠀⠀⠙⠉⠁⠈⣼⢷⣄⡀⠀⢀⣰⣷⡏⠈⠀⢸⠇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠈⠉⠉⠉⠁⣿⠹⢶⠶⡏⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣈⠀⠀⠀⠀⢀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠒⠶⠄⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠐⢶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠙⠀⠀⠀⠀⠀⠀⠀⠐⠏⣿⡿⢦
⠀⠀⠀⠀⠀⠆⠨⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡄⡷⣦⠀⠀⠀⠀⠀⠀⢠⡞⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠛⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⣚⠸⢄⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⡇⠀⠀⠀⠀⠀⠀⢠⣤⣀⣤⡄⠀⠀⠀⠀⠀⠰⣇⠀⠀⡀
⠀⠀⠀⠛⠛⠛⠛⠃⠉⠁⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠛⠓⠃⠀⠀⠀⠀⠀⠀⠈⠑⠒⠁


⠀⠀⠀⠀⠠⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢸⠀⠷⠀⠀⠀⠀⠀⠀⠀⠰⠀⢩⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⣜⡟⠸⣷⡄⠀⠀⠀⠀⠀⠀⣴⢿⡃⢸⠘⡿⢷⡄⠀⠀⠀⠀⠀⠶⠆⢼⠰⠶⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠂
⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⡟⢹⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣖⣠⣿⣻⣿⣿⣿⣟⣖⠠⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣓⡒⠒⢒⠒⣶⠶⣶⣤⣬⣤⣤⣾⠿⠿⠾⠻⡅⠀⢀⡀⢠⣤⣤⡤⠤⠤⣤⣤⠤⣼⣷⣶⣤⡄
⠀⠀⠀⠀⠀⠈⢩⣿⣿⣿⣿⣿⣤⣾⣿⣿⣟⣋⣉⣿⣿⣏⣅⠀⠀⠀⠀⣬⡒⠛⠻⣿⡿⣿⣧⣤⣿⣶⣦⣶⢿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⣾⣿⡿⠃⠁⢹⣿⣿⣿⣿⣿⣿⣍⣀⠶⠶⠛⠆⠀⠀⠘⡟⢻⢛⣫⣥⣾⣿⡿⣻⣿⣿⣿⣿⡀⢻
⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⢼⣿⣿⣿⣿⣻⡋⠉⠉⠙⠋⢿⣷⠄⠀⠀⣞⠻⢿⣿⠿⠛⣉⣴⣿⣿⣿⣿⣿⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣴⣿⣿⣝⣿⣿⣿⣷⣶⣌⠁⠀⣉⡁⠀⠀⠀⠈⠰⣈⣤⣶⣿⣿⣿⠟⣸⣿⣿⠏⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣾⣿⣿⣯⣿⡋⠁⢀⣀⣀⣼⠇⠀⠀⠀⢺⢦⣬⡿⠿⠟⢉⣠⣾⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢿⣾⡿⣿⣿⣿⣿⠷⠀⠀⠀⣤⡀⠀⠀⢠⡖⠀⠂⢴⣶⣿⣿⣿⡿⠋⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣸⣿⣿⣿⣟⣋⠀⣀⠤⠂⠀⠛⠁⠀⠀⠀⢈⢑⡂⠀⠙⠛⢋⣁⣴⣿⣤⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣯⡉⠛⠿⣟⣛⠿⠀⠀⠀⠀⣶⠂⠀⠀⠀⠈⣀⠁⠀⠛⠿⢿⣿⣿⠀⢉⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⡿⠿⣯⣽⣉⠀⠀⣠⣴⡾⢀⡴⣠⣆⣀⠁⠐⣌⡻⠶⠤⠀⠀⢉⡠⢴⣧⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣤⣀⣀⣉⢿⡿⠇⠀⠀⡤⠈⣾⣿⣏⣙⡿⣧⠈⠀⠀⠐⠲⣾⣅⡀⠀⠈⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡿⠿⣿⣿⡠⠀⠀⠂⠘⣡⣤⣿⣿⠿⢿⣇⣀⠀⠐⢌⠲⢤⠀⠙⢉⣠⣞⡁⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣦⣤⡴⠾⠓⠒⢂⣴⠾⢋⣹⣿⣿⣿⣺⡟⠛⢷⣄⠀⠑⠐⠒⢾⣧⣄⠙⢛⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣟⠛⠷⠶⡤⡔⠋⣁⣴⠚⠫⣿⣿⣅⣸⣧⣀⠀⠙⠷⣦⡀⢀⠠⣩⡤⢴⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣙⠛⠿⢐⠞⢡⡾⠋⠁⣠⣀⣿⣿⡿⢿⡇⠈⠻⣆⠀⠈⠻⣦⠀⠀⠀⢀⣉⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠈⠀⠀⠀⠈⢹⣿⣿⣷⢶⡟⠙⠀⠈⠧⠀⠀⠉⠈⠑⠄⠩⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⡟⠙⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡉⢈⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆
⠀⠀⠀⠀⠀⡤⡀⠀⠀⠸⠀⢠⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉
⠀⠀⠀⣀⠀⡇⢡⠀⠀⠀⡇⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⠃⠀⠀⡠⢲
⠀⠀⠰⡁⢱⣣⠈⡄⡇⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⢸⠀⢀⠀⠇⠘⡠⢤
⠀⠀⠀⠇⠀⠋⠄⠁⡁⠀⢇⡸⠀⡇⠀⠀⠀⡠⠒⡆⠀⠀⢀⠀⠀⠀⠀⢰⠀⡇⢸⠀⠘⠸⠀⢳⠁⠸
⠀⠀⠀⠸⠀⠐⠃⠀⠁⠀⠀⠁⠀⠁⠀⠀⠜⠀⡰⠁⠀⠀⡇⠑⢄⠀⠀⢸⠀⠰⠌⠀⠲⠂⢰⠆⠀⠇
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⡐⠁⠀⠀⠀⠈⢆⠈⢢⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⢆⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⢠⠆⠀⠂⠀⠉⣠⣀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠖⠀⠠⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠸⠴⠀⠓⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠃⠐⠼⠀⣄⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⡀⠀⣄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣹⣿⡿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣶⣶⣶⣶⣥⠙⠋⡿⣱⠶⣴⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⢻⣿⣿⣷⡃⠬⠐⠅⠩⠞⣻⣿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⣿⣾⡭⠀⢉⣑⣛⣩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⠻⠞⣁⢠⡖⠠⣍⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⠀⠀⢿⡆⣥⣀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠐⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠁⠠⠖⣻⡗⠃⢬⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⠀⠀⠀⠉⠀⠀⠠⣼⡯⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢩⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠦⠂⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠳⡀⠀⢰⣿⣿⣶⡿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡚⢿⣿⣿⠷⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢾⡏⢳⣽⡆⢸⣷⣌⠂
⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⢻⡿⠈⠻⠋⠀⠀⠘⠃⠘⡀
⠀⠄⠠⠌⣽⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣴⡆⠀⠀⠀⠀⢠⣾⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⠆⠀⠀⠀⠀⠈⢿⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⡄⠀⠀⠀⠀⠀⠀⠈⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡘⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠴⠶⠶⠶⠶⠚⠞⠻⠒⠒⠒⠛⠓⠒⠒⠒⠛⠿⠿⠿⠶⠿⠂

⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣡⣴⣿⠟⠉⣠⣾⡿⠛⢁⣤⣾⡿⠋⢁⣴⣿⠟⠋⣠⣴⡿⠟⢉⣠⣾⡿⠛⢁⣤⣿⠿⠋⣀⣴⣿⠟⠋⣠⣼⡿⠟⢁⣠⣾⡿⠋
⠟⠋⠀⠰⠿⠟⠉⠀⠶⠿⠟⠁⠀⠾⠿⠋⠁⠰⠾⠿⠋⠀⠴⠿⠟⠉⠀⠶⠿⠛⠁⠠⠞⠛⠋⠁⠰⠾⠿⠋⠀⠴⠿⠟⠁⠀⠾
⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠐⣶⢲⠖⡶⢶⠖⡶⢲⠶⡖
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠡⢀⠀⡀⠂⠀⠀⠀⣿⢸⠀⡇⢸⣀⣇⣸⠀⡇
⠀⠀⠀⠀⠘⢿⣿⡟⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠔⠀⠐⠌⠀⠀⠀⣿⢺⠖⡗⢺⣿⣹⣿⠲⡗
⠀⠀⠀⠀⠀⠒⠛⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢂⠀⠈⡐⠀⠀⠀⣿⢸⠀⡇⢸⠉⡟⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⢠⣦⣤⣤⣤⣤⣤⣤⣤⣴⣦⠀⠀⠀⡀⠀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠴⣶⢤⣄⣀⠀⠀⠀⣀⣀⣀⣰⡄
⠀⠘⠿⠿⠿⡿⠿⣿⣿⣿⣿⠟⠃⠀⠀⣂⠃⡘⠀⠂⢐⠀⠂⠀⠀⠀⠀⣿⡇⠀⠀⠀⣠⣤⡈⣻⣇⣀⠀⣹⡿⠛⠿⠃
⠀⠀⠀⠀⠀⠐⠀⠐⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠛⠛⠻⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠐⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢿⣏⣉⣉⡙⠛⠻⢿⠿⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠡⠂⣸⠂⠠⠀⣐⠂⠀⢃⡀⠁
⠀⠀⢀⣠⣄⣀⣀⣀⣤⡀⣤⣄⣀⠀⠀⠀⠀⠀⠀⣒⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠂⠤⢁⡄⠒⠧⠈⠖⠀⠈⢁⠄
⠀⠀⠀⠀⠉⠙⠛⠛⠻⣿⣿⣿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠂⠉
⠀⠀⠀⠀⢀⣀⠄⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⡐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢀⣄
⠀⠀⡰⠿⢦⡀⠀⠀⢰⠟⠓⢦⡀⠀⠀⠀⠀⠀⣴⣶⣶⣿⣿⣿⣿⣿⣦⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⡁⠸⣿⡇
⠀⢸⠀⠀⠀⣳⠀⠀⢿⠀⠀⢀⡇⠀⠴⢶⣶⣶⠦⠉⠙⠻⠿⠿⠟⠋⣁⣈⡉⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠻⠃⠀⣛⡃
⠀⠈⠳⠤⠖⠁⠀⠀⠈⠑⠒⠋⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠐⠿
⢂⣠⣶⡶⠂⢀⣴⣶⠶⠂⣀⣴⣶⠖⠀⣠⣶⡶⠖⢀⣠⣶⡶⠂⢀⣴⣶⠖⠂⣠⣴⣶⠖⠀⣠⣶⡶⠒⢀⣴⣶⠶⠂⣀⣴⣶⠖
⣿⣟⣁⣠⣾⣿⣋⣁⣤⣾⣿⣋⣀⣴⣿⣟⣉⣠⣶⣿⣟⣁⣠⣾⣿⣋⣁⣴⣾⣿⣋⣀⣴⣿⣟⣉⣠⣶⣿⣛⣁⣤⣾⣿⣋⣁⣴
⣿⣿⠉⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠶⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠚⠋⠉⠉⠑⠒⠲⠦⠤⠤⣄⣀⣈⠉⠉⣛⣻⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣾⣛⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⠿⢻⠉⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣶⣟⣉⣀⣀⣀⣀⠀⠀⠉⠉⠉⠙⠓⠒⠢⠤⢤⣤⣤⠶⠛⠉⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡟⠉⠁⢹⠁⠀⠀⠀⠈⠉⠉⠉⠉⢙⣻⣷⣶⣶⣶⣦⣤⣶⣟⡉⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣤⣾⣯⣥⣤⣿⣽⣶⣿⣿⠀⠈⠹⡇⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠀⢀⣠⣴⣞⣋⣥⣸⣿⣷⣶⣖⣺⣿⣿⠿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⣠⠶⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡿⠟⠋⠁⣾⠭⠤⣀⣈⣹⣧⣴⠛⠛⠛⠛⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠻⣦⣤⣄⣀⣉⣛⣒⠶⠖⠛⠙⠷⢭⣭⣭⡭⠾⣿⠀⠀⠀⣿⠀⣀⡴⠞⠉⠀⠀⠀⠀⢀⣠⠶⠛⢉⣉⣛⢦⣄
⠀⠀⠀⠀⠀⠀⠸⣇⣠⠶⠛⠁⠻⣿⣿⡻⠛⣿⡟⣿⠛⠲⠶⠶⣤⡴⠞⣿⠀⠀⠀⣿⠞⠋⠀⠀⠀⠀⣠⡴⠞⠋⢁⣤⠶⢿⣿⣿⡇⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢷⣶⠤⣤⣤⣀⣈⣉⠉⠉⠀⠈⠛⠛⠛⢛⣫⠴⠛⣿⠀⠀⢀⡿⠀⠀⢀⣠⠶⠛⠁⢠⡞⢛⣻⡿⠿⠟⢉⣡⣶⣒⣲⡮⠙⣦
⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠉⠉⠉⠉⠛⠛⠓⠲⠶⠚⠋⠛⠒⣲⡿⡴⠶⠛⢁⣠⠶⠋⠁⣠⡴⠶⣾⡿⠿⠛⢁⣠⣾⣯⣝⣿⠟⠋⣡⡶⠋
⠀⠀⠀⢰⠛⠛⠛⠓⠶⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠀⢀⣠⠶⣋⣡⣤⣀⣸⡧⠴⣖⣿⣁⣀⣰⣯⣥⣤⣿⠏⢁⣴⠞⠉
⠀⠀⠀⠸⠧⢤⣤⣀⣀⡀⠀⠀⠈⠉⠛⠓⠶⣤⣀⠀⠀⠀⣠⠞⠉⠀⢀⡴⠛⠁⣾⣋⣀⣀⣭⣿⠉⠉⣹⣟⠉⢉⣹⡏⠉⢉⣡⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⠶⣤⣤⣀⣀⠈⢷⣠⡾⠋⠀⠀⠀⠀⠻⣦⡀⠛⠶⠶⢶⣿⠷⠶⠶⣾⠿⠿⠿⠟⣡⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣶⠖⠶⢦⣌⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⠀⢿⡳⠤⠶⣚⡿⠀⢀⣤⠞⠉
⠀⠂⠀⠀⠀⠀⠀⠀⣾⠧⣄⣀⣏⣇⡤⠔⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠉⠛⠋⠉⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣤⠶⠞⠋⠀⣠⡴⢶⣶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠥⠤⠧⠿⠴⢒⣩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠶⠖⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡉⠝⠐⠁⠀⠰⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡐⠂⠄⡀⢺⣆⣭⣭⠭⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣚⣃⡀⠀⠀⢟⣼⣥⡶⠶⠶⢶⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣉⣉⣉⠁⠀⣤⡬⠴⠶⠛⠚⠶⢶⣤⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣭⣭⣩⡍⠂⣀⣁⣶⠛⡻⠿⠿⠷⣦⣌⣯⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣁⣀⣼⡛⠛⣉⣤⡿⣷⠶⠶⣦⣌⢹⣥⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⢉⣉⣹⡟⠛⢉⣠⡞⣻⢐⣮⠹⣏⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐⡶⣟⠛⠛⢉⣁⣾⡟⢋⣴⠏⣼⠸⣯⣍⣉⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⡹⢿⡆⢹⡟⠛⠛⢋⣁⣿⠛⣥⢾⣏⠛⢦⣌⠻⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⣳⣤⣙⣜⡇⣾⡟⠛⢉⣉⡄⣿⠈⣿⠰⡀⠻⣷⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⣄⣉⣉⣷⣾⡏⢛⠛⠋⢹⣿⡄⢻⡇⢻⣆⠙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⠿⣭⣥⣤⣤⣴⡏⠉⠀⠀⢻⡇⢸⡇⢸⣿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡶⠾⠏⣥⣤⣴⣾⡀⠀⠀⢠⠿⢿⣿⣿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⡟⢛⣩⡤⣌⢉⡀⠀⠀⢿⡦⢻⣿⣷⣼⣷⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣥⣶⣶⣦⣬⣭⣁⡛⠦⠐⠚⢇⠸⣷⣶⣷⣶⡹⡛⢿⣿⡿⢸⡷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⣟⣿⣿⡇⠹⢿⣿⣿⡸⣿⣎⠿⠁⣸⣄⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠘⠊⠑⠺⣧⡽⠟

⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢰⣭⣵⠀⠀⠀⠀⠀⠀⢀⣸⡀⠠⠑⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⢾⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣶⣶⡆
⠀⢀⣀⡙⣁⣀⠀⠀⠀⠀⠀⣿⣋⣿⢸⢀⠀⠀⠀⠀⣠⣜⣛⢛⣛⣛⣤⠀⠀⠀⠀⣴⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢟⣿⣽⣿⡇
⠀⠸⠿⠿⠿⠿⠆⠀⠀⠀⠀⠻⠭⠿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⣤⠀⠀⠀⠀⢾⣿⣿⢿⠿⠟⠓⠀⠀⠀⣰⣯⡿⣻⣿⣷⠀⠀⠀⠀⠀⣄⣈⣁⣂⠀⠀⠀⠀⠀⣠⣤⡀⣶⠷⢦⡀
⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢠⣾⣟⣶⠟⠁⠀⠀⠀⠀⡎⡸⠁⠀⠀⠀⠀⠀⠀⠀⢻⣷⣳⣶⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⢰⡋⠈⣳⠻⣄⣤⠇
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠈⠙⠟⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠀⠀⠸⠿⠿⠶⠿⠟⠀⠀⠀⠈⠓⠒⠁

⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⡾⢟⡓⢯⡆⠀⠀⠀⢀⣀⡤⠊⠀⠀⠀⠀⠀⠀⢓⠀⡀⡠⢄⠃⠀⠀⠀⠀⢀⣠⣤⣶⡖
⠀⢠⣴⣦⣶⣤⡀⠀⠀⠀⠀⠀⣭⣄⣭⡀⠀⠀⠀⠀⢸⣿⣷⣙⣛⣼⡇⠀⠀⠀⠘⠿⠑⣷⣶⣾⠇⠀⠀⠀⠄⠀⠸⠅⢨⡇⠀⠀⠀⠘⠿⠛⠉⠉⠦⠠⡆
⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡄⠀⠀⠀⠀⠀⠀⣄⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡀
⠀⢀⣸⣧⣤⡄⠀⠀⠀⠀⠀⢀⣼⣤⣧⡀⠀⠀⠀⠀⠀⣿⠿⠛⢿⣏⠀⠀⠀⠀⠀⣦⣻⣶⣦⡄⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⢀⠼⢿⣾⡷⡿⠧⡀
⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⠛⣿⡷⠀⠀⠀⠀⠸⣿⣧⣤⣾⣿⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠈⠓⠶⠚⢧⡀⠀⠀⠀⠐⠄⠔⠀⠀⠣⠠⠂
⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⢰⢶⣴⣤⢴⡄⠀⠀⠀⠀⣲⣶⠶⢶⣶⠀⠀⠀⠀⢠⣤⣤⣤⡄⠀⠀⠀⠀⠀⢀⣤⡀⠀⣰⢷⠀⠀⠀⠀⠀⠀⠀⠀⣠⡢⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⠟⣿⣟⢿⡇⠀⠀⠀⠀⢸⣿⣬⣼⣿⠀⠀⠀⠀⢸⣟⣿⣿⣇⡺⠋⠀⠀⠀⠸⢧⡇⠀⠹⠿⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⣀⡴⠞⠁⣀
⠀⢸⣶⣏⢹⣿⡃⠀⠀⠀⠀⣸⣿⣭⣿⣿⠀⠀⠀⠀⢸⣟⣿⣿⡏⠀⠀⠀⠀⠀⠀⢈⢡⠀⣆⡆⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠚⠩⠶⠚⠋⠁
⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⠄⠀⠀⠀⢀⡔⠊⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⠀⠀⠀⠶⠶⠶⠖⠆⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢸⡀⢀⡀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢞⠁⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⢦⡀⠀⠀⠀⢀⣴⠛⠩⠀⠀⠀⠀⠀⠀⠀⡉⠩⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣡⣬⡻⣆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓⢠⡙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡻⣮⡻⠟⠑⠽⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢤⡙⢦⣠⣾⣗⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⣪⢻⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⡝⣆⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠁⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢾⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠇⠀⠀⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣰⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⡾⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠤⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⡀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣿⣇⠸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⠀⢿⣿⠤⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣿⣿⠇⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣿⡀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⡿⢤⣴⣶⣿⣿⣿⣿⣿⣷⣶⡿⠖⠳⣴⣲⣿⣿⣿⣿⣿⣷⣦⣤⢼⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⢿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⡆⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠄⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡅⢿⣿⣿⣿⣿⣿⠿⠛⠙⣱⡆⣶⠀⡘⠻⢿⣿⣿⣿⣿⣿⠇⡀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠹⠶⠯⠍⠉⠀⠒⠒⢡⢦⣿⣿⣿⣇⠙⠶⠤⠈⠉⠉⠉⠐⠊⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⡀⠀⠀⠀⠀⠀⠒⠀⠻⢺⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⣦⣶⣤⢤⡀⠀⠀⠀⠈⢿⡟⢿⠟⠀⠀⠀⠀⣠⣴⣶⣾⣦⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⣿⣏⠈⡧⠀⠀⠀⠀⠐⠆⠐⠀⠀⠀⠀⠰⡏⢀⡿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢻⣆⠻⠶⠤⢤⣤⡤⣄⡠⣤⣤⣤⡤⠞⣃⣾⠁⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢻⣟⠲⢦⣠⢀⡀⢈⠀⣀⣠⢠⠶⢎⣸⠇⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠃⠐⠢⢅⢁⣉⢹⠙⣉⡀⠄⠒⠁⠋⠊⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⡀⠀⠀⠀⠀⠐⠢⠤⣤⡤⠄⠂⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⣀⣀⣀⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉


⠀⠁⠀⡁

⣿⠟⣩⣭⣽⣿⡏⢩⡉⢻⣿⠉⣿⣿⠋⠹⣿⠋⢹⣿⡏⢩⣭⣽⣿⣿⣿⣿⣿⣿⡏⢹⣿⡿⢋⣭⣭⣿⡿⢋⣭⡍⠻⣿⡏⠙⢿⡏⢻⣿⠋⣭⣭
⣿⡀⠿⣿⢿⣿⡇⢠⡀⢿⣿⠀⣿⣿⠀⣄⠁⣤⠘⣿⡇⠠⡦⣾⣿⣿⣿⣿⣿⣿⡇⢸⣿⣇⠸⢿⡿⣿⣇⠸⢿⠟⢠⣿⡇⢰⣄⠁⣻⣿⠷⣦⠈
⠛⠛⠒⠒⠚⠛⠓⠛⠓⠚⠛⠒⠛⠛⠒⠛⠒⠛⠒⠛⠓⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠓⠒⠒⠛⠛⠓⠒⠒⠛⠛⠳⠾⠛⠓⠛⠛⣳⣶⠚
⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⣤⢀⡴⠆⠀⢠⣴⣶⣶⠄⠀⢠⡖⠀⠀⣠⣾⠗⠀⣠⡖⠒⣶⡀⠀⢠⡄⠀⢤⣄⠀⠀⠀⣲⣶⡀⠀⠀⠀⣿⢿⠇
⠐⠛⠛⠛⠉⠉⠉⠉⠀⠀⠸⠋⠈⠀⠀⠀⠸⠏⠀⠀⠀⠀⠋⠀⠠⠊⠈⠁⠀⠀⠒⠣⠧⠟⠊⠀⠘⠤⠃⠘⠴⠃⠀⠾⠿⠿⠷⠄⠀⠲⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢠⡄⡠⠀⠀⠀⢠⡄⠀⠀⠀⡤⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡆
⠠⢿⡏⠀⠀⠀⢠⡿⠶⠀⠀⠀⣿⠃⠀⠀⢀⣾⢶⣄⣄⠀⠀⠠⣶⣿⠒⢂⣤⠀⠈⢻⣿⠇⠀⠀⠀⢀⣀⣰⠄⣠⢽⡏⠀⠀⠰⠚⢶⣤⣾⣭
⠀⢸⡇⠀⠀⢀⣼⢻⡆⠀⠀⠘⡿⢧⠀⠀⢸⣿⣿⣄⠈⠀⠀⠲⠾⢙⡆⠀⠀⠀⠀⢸⣿⠀⠀⠒⣲⠟⠉⠻⠆⠀⠈⠳⡄⠀⠀⠰⡏⠸⠏⠰⠦
⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠠⡔⠒⠒⠀⢀⠀⠀⢀⣀⡾⠃⠀⢠⣶⣼⣶⣶⠀⠀⢰⠋⠈⠙⡄⠀⠀⠉⠍⠁⠀⠀⣴⣷⡄⠀⠀⠚⣿⣿⠷⠀⠀⠀⣶⣿⣿⠀⠀⠀⣠⠄
⠀⠀⠈⠙⠒⠿⠀⠐⣿⡿⠁⠀⠀⢸⣿⣿⣿⣿⠀⠀⠘⢄⣀⡠⠃⠀⠀⡀⠁⣀⠀⠀⣤⣤⡇⠀⠀⠀⢻⡟⠀⠀⠀⠸⢿⡿⠃⠀⠀⢸⣿⣀⣀

⠀⠀⢠⡀
⠀⠀⣷⣷⠀⠀⣀⡀⠀⠀⠀⠐⠄⠤⡄⠀⠀⡠⠢⠀⠀⠒⠂⠀⠠⠄⢶⠆⠦⠀⠀⣶⣶⡄⠀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣩⣍⢂
⢀⡀⣿⡇⠀⣴⣿⣿⣦⠀⢦⣰⡋⢚⣤⡔⠀⠀⢄⠀⠴⠎⠀⠀⠒⠂⢸⠐⠒⠂⠀⣧⣿⡇⠀⢈⣏⣏⣏⡇⠀⠀⢀⣤⣤⠠⠒⢢⠀⢡⠙⡋⡌
⠈⠁⠉⠁⠀⠉⠛⠛⠁⠀⠀⠉⠁⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠉⠙⠁⠀⠛⠋⠙⠛⠛⠀⠛⠛⠟⠟⠀⠙⠋⠀⠀⠈⠉

⠀⠀⠀⢀⣤⠀⠀⠀⠀⡀⠀⠀⣠⢤⠤⢤⡄⠀⢠⣤⣤⣤⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⢀⣴⠟⠁⠀⠰⣰⣿⡷⠀⠀⣻⡆⠀⢸⡇⠀⢸⣿⢛⣿⠀⠀⠲⡞⠛⠳⠆⠀⠀⣤⣤⣤⡉⠁⠀⠀⣿⣿⠂⠀⢀⣴⣰⣷⣶⡆⠀⠀⣬⣋⣄
⠀⠛⠈⠀⠀⠀⠀⠙⠀⠁⠀⠀⠻⠒⠶⠾⠃⠀⠀⠉⠛⠁⠀⠀⠀⠓⠀⠖⠀⠀⠘⠉⠛⠋⠛⠀⠀⠀⠛⠚⠀⠀⠙⠋⠉⠉⠛⠁⠀⠀⠛⠛⠛

⠀⢀⠤⡦⢄⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢠⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠷⠄⡁⠤⡆⠀⢠⣼⣿⣿⣤⠀⠀⠀⡋⢉⠀⠀⠀⣾⣤⣼⠀⠀⢰⠶⢶⠀⠀⢴⣾⣿⡿⠀⠀⣿⣿⠛⣿⡇⠀⠀⢰⣿⡷⠀⠀⠀⢴⡿⣍
⠀⠈⠒⠗⠊⠀⠀⠀⢿⠙⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠸⢶⠾⠀⠀⠈⠿⠉⠁⠀⠀⠹⠟⠛⠿⠃⠀⠴⠋⠀⠀⠘⠀⢠⣤⡄⠈⠓