Beans Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡶⣿⠁⠁⠉⠰⠂⡙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢫⡩⠳⣄⠀⠀⠀⠄⠶⣉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠈⠙⠛⠻⣕⢦⠀⠀⠀⠈⠐⢦⡙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠘⢿⡷⣄⠀⠀⠀⠀⠨⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⢤⣤⣀⣀⣀⡀⠀⠈⠻⣆⡀⡀⠀⠀⠀⠳⡬⣳⠦⣄⠀⠈⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣋⠡⠌⠂⠁⠄⠀⠀⠀⠉⠉⠉⢉⠳⣄⠀⠈⠻⣮⣄⢀⠀⠀⠘⢷⡹⣸⡆⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⣰⢟⠁⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡍⢷⡄⠀⠀⠙⠻⢿⣶⣤⣀⣹⣤⣷⣾⠋
⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠠⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣴⡿⣻⠏⣾⠃⠀⠀⣠⠴⠚⠉⠁⠀⠀⠁⠤⣻⣧⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣀⡤⠖⢋⣉⣉⣤⣤⣾⣿⠿⠟⠓⠚⡕⢡⡟⣀⡴⢋⠁⠀⠀⠀⠀⠀⠀⡴⢞⣥⠞⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠈⣷⠒⢛⣩⠴⠖⠻⠛⠛⡋⠉⠁⠀⠀⠀⠀⠀⢠⣟⣼⠋⠀⠁⠀⠀⠀⠀⠀⣠⡞⢅⡾⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠲⠃⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣿⡋⠂⠀⠀⠀⠀⠀⣀⣴⠏⣡⠞⠁⠀⡠⡱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠈⢀⣀⣤⠞⠁⢸⡏⠢⠀⠀⠀⣠⠖⡛⠍⢡⡞⠁⠀⠀⠀⡜⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⠒⠛⠋⠉⠀⠀⠀⢸⣧⠀⠀⣀⡼⠁⢈⡴⠚⠉⠀⠀⠀⠠⢺⣮⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⢛⡡⢒⡴⠋⠀⠀⢀⣐⣔⠕⣡⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣴⣋⢄⢄⠀⠐⣀⣽⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠀⡀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠃⠀⢁⠀⠀⠀⠀⢢⡮⠀⠀⠀⠀⠀⠀⣤⠀⠒⠒⠂
⠀⠀⠰⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⣠⠜⠋⠀⠀⠀⠀⠉⠒⠀⠒⠋⠀⠀⠀⠀⠀⢀⣮⡤⠀⢀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⡴⠂⠈⠉⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⡀⠁⠀⠈⠡⠒⠂
⠀⠀⠀⠀⠀⠐⠚⢁⠄⠀⠀⠀⠀⡲⠗⠂⠀⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡄⠀⠀⠘⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠁⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠈⠻⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⡁⡀⡀⣠⠁⠀⠸⢀⠄⡀⡀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠀⠀⠀⠀⠒⠀⠀⠉⠀⣈⣰⠀⢀⡀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⠀⠀⣠⠞⠀⠛⣦⣄⡀⣀⢤⣶⣄⡀⠀⠐⠦⠄⠀⣤⠼⠋⠀⠘⠄⡤⠤⠓⠀⠁⠀⠤⠀⣤⡀
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠁⡢⠿⠂⠠⠂⠀⢈⣵⠁⠀⠀⠈⠉⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠊⠉⠁⠈⠀⠀⠄⢺⡾⠉⠠⠠⣄⣠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠠⡤⢄⣠⡔⠠⣤⣠⡞⠀⠀⣀⣠⠞⠁⠀⢙⣀⣤⠀⠳⠲⠂⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⢄⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠓⠂⠁⠧⡾⠇⠀⠉⠀⠀⠀⠀⠀⡀⣀⠅⠀⠀⠀⠀⠉⠒⠠⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠳⠀⠀⠐⡀⠀⠀⢰⠆⠀⠀⠀⠐⢀⡀⠘⠀⠀⠈⠁⣀⣁⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⠲⠤⠶⠦⠤⠞⠁⠈⠳⠤⠀⠤⠼⠉⠀⠀⢸⠁⠀⠀⢰⠂⠀⢀⣴⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠘⡀⠉⠉⠁
⠀⠀⠀⠀⡟⠀⠀⢀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠾⡄⠃
⠀⠀⠀⠀⠀⠙⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠹⢇⠀⠀⠀⣀⣼⠎⠉⢈⡝⠃⠀⠀⣀⣶⢒⡖⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠒⡷⠲⢤
⠀⠀⠀⠀⠀⠈⠁⠀⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠛⠛⠋⠋⠛⠛⠓⠘⠛⠀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢢
⠀⠀⠀⠀⠛⠂⠂⠀⣀⡋⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠲⡄⠄⠀⠘⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⣟⠂⠀⠀⠈⣄⡀⢀⡀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠼⠇⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠚⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⡀⠀⢀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⣡⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⢋⣼⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢁⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢣⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⣸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⢸⣿⣿⣿⡿⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠿⠛⢋⣙⣛⡛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⢸⣿⡿⠟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠟⢋⣤⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣋⣭⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠟⣫⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⢉⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣿⣿⠿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠋⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢃⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⢹⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⣠⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢣⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⣠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢛⣋⣉⣉⣉⣉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣉⣉⣉⣉⣙⣛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⣭⣙⠻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⡀⢿⣿⣿⣿⣿⣷⡀
⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠿⢿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣷
⢠⣿⣿⣿⣿⣿⣿⡿⢋⣤⣶⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠿⢋⣵⣶⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿
⣿⣿⣿⠿⠟⢛⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⡿
⢛⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⣾⣿⣿⣿⣿⡿⠟⠛⣉⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣷⠝⠁
⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⣾⣿⣿⣿⠟⢁⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠉⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣫⠉⠃⠀⣴⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⢦⣄⠀⠀⠀⣀⣠⡤⣖⣚⣭⠵⠚⠉⢀⡀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⣿⠱⡇⠀⠀⠀⣿⡄⠀⢈⠈⣷
⠀⠀⠀⠀⢠⠞⢻⡇⠀⠀⠈⢢⠹⣷⣖⢉⡴⠚⢉⣉⣉⣠⣤⠤⠖⠋⢀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣧⡁⠀⣰⡾⠛⠁⠀⠟⣠⣿
⠀⠀⠀⢠⢧⠂⠸⣧⠀⠀⠀⠀⠃⠹⣿⡏⢀⣀⠀⠀⣀⣤⡶⠶⠛⠛⠛⠛⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⡲⠌⠓⠀⠤⢶⣾⣿⠋
⠀⠀⠀⢸⢸⠀⠀⠙⣷⡀⠀⠀⠈⠀⢿⢿⠀⢀⣴⡿⣫⠵⠚⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡄⠀⠀⠀⠀⠀⠙⢿⣿⣶⣶⣶⣾⣿⠟⠁
⠀⠀⠀⢸⠸⠀⠀⠀⠈⢿⡄⠀⠀⠀⢸⠸⣶⣿⠋⠘⠁⠀⠀⠀⠀⣀⣤⣤⣤⣴⣺⠿⠋⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠈⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⢠⣿⡇⠀⠀⠀⠀⣠⣶⠿⠿⠟⠛⠋⠉⠁⢀⣄⡏⠀⠀⣀⠤⢖⣛⣉⣉⣙⣿⣿⣿⣿⣖⣒⡢⢤⡀
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣿⡇⠀⣠⢃⣾⣿⣧⡀⣀⣤⡾⡻⠀⠀⠀⠀⠀⠀⠀⣠⢞⡞⠀⠀⢰⡔⠛⠉⢁⠀⠉⣡⣤⣤⠤⣤⣍⠙⠻⠶⡌⡆
⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣈⣙⣢⣵⡿⠉⢿⣿⠛⠋⠑⠉⠀⠀⠀⠀⠀⠀⠀⣚⡵⠋⠀⠀⠀⠘⣯⣓⠲⠦⠤⠴⣶⣶⣶⠶⠶⠶⣶⣶⣶⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁⠀⠀⠀⠛⢷⣦⣤⣀⣀⣀⣀⣀⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠐⠀⢀⡀⠴⡦⠂⠁⠀⠀⠀⠀⠀⠀⠴⡅⠀⠀⠀⠀⠐⠒⠻⡃⠀⠀⠈⠙⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠁⢀⢶⣟⣦⠰⠃⠀⠀⠀⠀⠀⢀⣤⠀⡄⠱⣄⠀⠀⠀⠀⠀⠀⠘⢃⢶⣤⡄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⣩⣄⢉⡝⠁⠀⠀⠀⠀⠀⢠⣎⣉⢦⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⣿⡦⣄⡀⠈⠁⠠
⠀⠀⠀⠀⠀⣤⠀⠀⢀⠆⢠⢿⣧⣿⡟⠀⠀⠀⠀⠀⠀⠀⢷⣿⠿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⡛⢣⡀⠀⠀⠱⡄
⠀⠀⠀⠀⠘⠃⠀⠐⠇⠀⠚⢀⠉⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣿⠃⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣵⡟⠓⣄⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣬⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢋⣰⡖⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠹⡟⠿⠻⡀⠀⠀⢡
⠀⠀⠀⠀⡄⠀⠀⢠⠇⡇⠙⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢿⡆⠀⠀⠀⢰⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣏⣴⠶⣇⠀⠀⢸
⠀⠀⠀⠀⡏⠀⠀⠈⠀⢉⡴⠶⣾⡇⠀⠀⠀⠀⠀⠀⠀⡼⠹⠿⠟⡰⠁⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣤⣟⣦⡴⠃
⠀⠀⠀⠀⡇⠀⠀⠀⡀⠸⣷⣴⡟⢧⠀⠀⠀⠀⠀⠀⡔⣡⣦⡂⡌⠁⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣀⠞⠊
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠳⢄⣀⣸⠀⠀⠀⠀⢀⣐⣿⢟⣹⠿⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⡰⠋⡉⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⢰⣤⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢆
⠀⠀⠀⠀⠀⠈⡄⠀⡄⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡞⠋⠉⠻⡉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣄⠀⢀⣀⠀⠰⠀⠀⠁⠤⠀⠀⠘⠂
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠦⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡁⠀⠉⠀⠀⠀⠊⠈⠉⠁⠐
⠀⠀⠀⢀⠾⢌⠀⠀⠀⠀⢫⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠑⠀⠀⠤⠃⠀⠨⡁⠀⠀⠁
⠀⠀⠀⣘⠀⠈⢆⠀⠀⠀⠀⠀⠠⢸⠇⠀⠀⠀⠀⠀⡘⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠎
⠀⠀⠀⡟⠁⠒⡈⢻⣉⡙⣻⣄⠠⣈⡀⠀⠀⠀⠀⢀⣄⠀⠘⣆⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠐⠒⢄⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢁⠀⢀⠈⢺⠑⠣⡀⠉⠻⣄⡐⠠⡀⠀⠀⢸⠀⠀⠁⡈⡧⣠⠹⢷⣤⡒⠲⠄⠀⠀⠀⠀⠁⠘⠲⠤⣄
⠀⠀⠀⠀⣅⠂⠆⠈⠀⠀⢰⡖⢠⡙⠎⠀⢬⡀⠀⠀⢂⠀⠀⠈⠇⠈⠲⡈⠈⢻⡤⠍⠢⡁⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⠊⠇⠀⠀⠀⠀⠀⠀⠰⡌⠘⠁⠀⠀⠀⢛⡁⣦⠘⠀⠐⠲⠀⠀⠀⠈⠀⠀⠀⠊⠐⠂
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠟⠢⡀⠀⠀⣠⣶⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣀⡤⠓⢦⣴⣧⡤⢞⣹⣫⠤⣤⣶⣶⣲⣒⣶⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⣦⣼⠉⠹⣦⣾⣄⣈⣶⣿⣿⣴⣦⣤⡶⢿⠯⠙⢫⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣀⣀⣿⠶⠾⣿⡾⠋⠙⣇⣀⣩⣿⣷⣌⣻⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⠟⠿⢿⡀⣤⣼⠶⢶⣿⣁⠉⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠹⣷⣤⣾⡿⠿⡏⠓⣦⡤⠈⠳⠶⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣾⣿⢟⡵⢿⠆⣠⣇⠀⠀⢹⡷⣖⠒⡀⠀⠈⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⣮⢟⢡⣏⣾⡥⡿⣇⠋⢙⢿⢻⣄⡇⠈⠁⢸⠿⣍⠠⣀⠀⠈⢉⣲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⠇⣼⡿⠁⠀⡇⠈⡇⠀⠈⣀⣯⣷⡀⠀⠘⠆⠈⣷⣤⣀⣀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣵⣧⣾⠟⡇⠀⢀⣷⠘⠳⣄⠐⣋⠘⣸⡗⣦⡀⠀⠀⠹⢨⠓⣌⢁⡀⠀⠨⣳⡄
⠀⠀⠀⠀⠀⠀⢀⣸⣿⢿⡁⠀⢳⠀⠋⢿⡦⢸⣏⡞⠁⠀⠙⡷⡀⠹⣄⠀⢀⠀⠆⠈⠷⢄⣀⣤⠀⠹⡄
⠀⠀⠀⠀⠀⡴⡋⢀⡇⠀⢧⡀⠈⢦⠀⠊⢳⣤⢸⣆⣀⢶⠆⡙⣇⠀⠉⠛⠛⠓⢦⡀⠀⠈⠛⢦⡀⠛⢻⡀
⠀⠀⠀⠀⢠⡧⡇⢠⣧⠀⠀⢳⠀⠈⠳⣄⠸⠿⣦⡻⡍⢧⣠⣟⡎⢷⠀⠀⠀⠀⠀⠙⠾⠶⠒⠲⠾⠷⣮⣷⡀
⠀⠀⠀⠀⠈⢧⠀⠘⠙⣆⠀⡊⠳⣄⠀⠈⠳⠦⠯⣧⣽⠀⠈⠉⠙⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠘⡆⠀⢀⣿⣦⠀⠀⠈⣆⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠙⢾⡀⠉⠈⢷⡀⢳⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢧⠘⣷⢸⡏⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⣷⣧⡀⠀⠀⠀⣠⣤⣼⣉⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⠧⣤⣾⠿⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢔⣒⣮⣟⣟⣲⢤⡀
⠀⠀⢠⣞⣯⣉⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡎⠡⠖⠛⠈⠁⠈⠈⠿⢫⣧⣷⡀
⠀⠀⢘⣿⣿⣦⣮⣄⣀⢈⡙⠓⠦⣄⣀⠐⢒⠀⢀⣒⣶⣿⣿⣿⣶⡀⢀⡊⠄⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠋⢙⡄
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⠚⣛⢻⡿⣷⣿⣤⠤⠾⠿⢿⡿⡳⠎⠀⠀⠀⠀⠀⣀⣠⣾⡟⢫⠿⠋⠁⠀⠀⣼⠁
⠀⠀⠀⠘⢿⣿⣿⣿⣿⡏⢤⠂⠉⠀⠀⠘⠁⠐⠋⣝⠿⣿⡿⡟⠸⠁⠀⠀⠀⠀⣀⣾⣿⠿⠋⠈⠁⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠙⠻⢿⡿⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠑⢷⣿⣥⣀⣠⣴⣾⡿⢫⡤⠈⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡙⠿⢿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠙⣿⣿⣛⣫⣡⣶⡞⠄⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠁⠹⠂⠨⠉⠛⠛⠿⣿⣿⣿⣶⣦⣄⠀⠈⣿⣿⣟⣛⣻⣳⣶⣶⣿⣿⣶⣗⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠈⠛⢲⠶⢮⣭⣛⠿⣷⣄⣿⣿⡿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠽⣷⣤⣀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣮⣿⣿⣿⣿⣏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣛⣿⣷⣶⣦⣤⣤⣤⠶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣭⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⡤⠶⢞⣛⡉⠀⠀⠀⠀⠀⠀⠉⠳⣆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠉⠁⠀⠐⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣀⡴⠟⠛⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⢾⣯⡀⠀⠀⠀⠀⠀⠘⣟⠦⣄⣙⡓⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣀⡴⠟⠉⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠈⠛⠳⠿⠿⠿⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠛⠛⣛⡲⠶⠶⠶⢚⡛⠉⠉⠀⣀⣬⠿⣋⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠟⢃⡤⠖⣛⣭⣤⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠉⠙⠲⠶⢤⣤⣤⣤⣤⡤⠴⠖⠋⠁
⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣀⣀⣀⣀⣀⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠶⣤⣀⣀⣀⣀⣀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠿⠿⠛⠛⠛⠛⠿⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣥⠴⢀⣀⣠⣤⡤⣤⣤⣄⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⣀⠴⠋⠉⠁⣀⣀⣀⣀⣀⠀⠙⢦⡀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⢀⠞⠁⣀⡴⢒⣩⣤⠤⠤⣤⣄⡉⠳⣄⠙⣆⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⢠⡏⢀⡴⣡⠞⠉⠀⠀⠀⠀⠀⠀⠉⢦⡘⣧⠘⡄⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⡾⠀⡎⡰⠁⠀⢀⡴⣫⠭⠭⣝⢢⡀⠀⢷⠘⡀⢱⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠰⢴⡇⢸⠁⣧⣠⣀⡾⡜⠁⠀⠀⠈⡇⡇⠀⢸⡄⡇⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢸⡇⢸⠀⡇⠀⠉⢶⡱⣀⢀⣠⢞⡵⠉⠰⣌⢣⣇⣸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⣇⠈⢧⠹⣄⠀⠀⠙⠚⠳⠛⠉⠀⠀⡰⢫⡎⢠⠇⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠹⣆⠈⠳⡌⠳⣄⡀⠀⠀⠀⣀⠴⢋⡴⠋⢠⠎⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡈⠦⣄⠉⠓⠤⠬⢍⡭⢭⠴⠚⠁⣀⡴⠋⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡈⠛⠶⠤⣤⣀⣀⠀⣠⠴⠚⢁⠀⣰⣿⣷⣶⣶⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⠀⢠⡨⠍⠀⣀⣈⣭⣴⣿⣧⣤⣤⣤⡭⣭⣭⣭⣿⣟⡿⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣶⣴⡾⣿⣟⣛⣿⠿⢿⡿⠟⠛⠿⣷⡛⠉⣻⣿⣽⣿⠻⢶⣝⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢁⣾⣿⠛⣉⣿⡟⠉⠉⢿⣴⣾⣷⠟⠛⠻⣿⠛⠻⣯⣀⣀⣻⡶⠶⢿⣷⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠻⢷⣟⠋⠙⣷⠞⠳⢾⣏⠉⣹⡷⠶⠶⣿⡟⠷⣾⠏⠙⣿⣠⣤⠾⢋⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣝⠿⣷⣦⣍⡛⠾⢷⣤⣤⣤⣿⣤⣿⣤⣤⣤⣼⣷⡴⠿⠷⠛⣋⣉⣤⣶⡿⢛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠙⠢⣍⡙⠛⠿⠷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⠿⠿⠛⠛⣉⡡⠴⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠙⠛⠒⠶⠦⢤⣤⣬⣭⣭⣭⣥⣤⠤⠤⠴⠶⠒⠛⠉⢁⣤⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⣿⡇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢠⣌⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣹⣿⠋
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣭⣙⡓⠶⠶⠤⠤⣤⣤⣤⣤⡤⠴⠶⠚⣋⣥⣶⡿⠟⠁
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣶⣶⣦⣤⣦⣴⣶⡶⠿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡆⢻⣥⢡⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣇⣀⡀⠀⠀⠜⠁⡸⠁⠌⣇⠀⠁⠙⠉⠖⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣤⡭⣭⡥⠀⠠⠀⡀⠀⠀⡃⢹⢠⣀⡀⠀⠀⠐⠄⠐⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⡼⠋⠁⡀⠀⠀⢠⢣⡄⠀⡇⡏⠀⡿⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⣾⠟⠁⠀⣠⠞⠀⠀⣾⣿⡾⠃⠸⢠⣷⠀⠁⠘⣄⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢸⡞⠁⠀⢀⠞⠁⠀⠀⣼⣿⠋⠀⠀⠀⠘⣿⡄⠀⠀⠈⠆⠀⠈⠆
⠀⠀⠀⠠⣤⣤⣴⣾⣷⣤⠎⠀⠀⠀⠊⠀⠀⢀⣰⠟⠠⠀⠀⠀⡖⠀⢹⣷⡀⠠⢰⢨⣧⣀⢥
⠀⠀⠀⠀⠀⠘⠛⠛⠓⠛⠀⠀⣴⠆⠀⠀⢀⣼⠃⡀⢰⡁⢀⢰⣉⡁⠰⣿⣿⣬⣈⣌⢸⣮⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣾⠃⠀⠀⠔⠠⠤⣽⠒⠆⠀⢿⣿⠻⣿⣿⣛⠛⠛⠓⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⠀⠀⠀⢀⣼⡇⠀⠀⠰⠁⠀⠰⣏⣉⡇⠀⢸⣧⡀⢳⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡦⠀⢁⣠⣾⣿⠀⠀⢀⠡⠀⠈⡼⠠⠼⠀⠀⢸⠛⢧⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣶⣿⣿⣿⣿⠀⠂⠠⠀⠀⢤⠉⠐⠂⠀⣄⠎⠀⠀⠉⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠛⠋⠀⡄⠩⠤⠀⢀⠖⠤⠍⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣖⠀⣾⡭⠃⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣓⡜⠒⠚⠒⠰⢤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣇⠀⣀⣠⣦⣴⡮⠾⣷⣻⡟⠀⣆⠀⠀⡀⡀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠻⣭⣸⢧⣼⢙⣼⡩⢿⠇⣐⣽⠗⢝⢙⣣⡕⣼⣯⠳⢟⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡤⣼⣧⠠⢸⢺⣷⡯⢭⠝⡯⡟⠙⢷⢼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠓⠤⡀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠙⠶⣄⠑⡜⠾⣨⡸⠁⣤⠹⠊⠉⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⠂⠀⠐⠒⢹⢆⠀⠀⠀⠀⠉⠉⠁⠛⠈⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣺⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⢠⠰⡀⠀⠀⠀⠀⠀⠀⣠⠘⣯⣧⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣴⡶⣾⣷⣿⡝⠽⣟⠳⣶⣖⣄⡀⢦⠤⠄⡀⢠⡠⣤⡄⣀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⠴⠎⣾⣿⡿⠂⣿⡿⡟⢏⡾⠳⢞⣯⠇⡶⣾⠓⠘⣻⡆⢿⣿⠈⣵⠷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠦⠠⠽⠞⢲⠲⣿⡷⣴⠿⣞⠋⢨⣱⠟⠷⢶⡧⢕⣿⠘⠀⠁⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠋⠑⠀⠈⣫⠛⠻⠾⠁⠁⠹⠣⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉

⠀⠀⠀⠀⢀⣠⣶⣾⣿⣭⣽⣿⣷⣦
⠀⠀⠀⢀⣾⣿⣿⣿⢿⣻⣭⣭⠍⣹
⠀⠀⠀⢸⡿⢿⡿⠿⢟⣹⣿⣿⣿⠏
⠀⠀⠀⠀⠙⠶⢶⣾⣿⠿⠿⠟⠁


⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡒⣒⢠⡀
⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⠿⣿⡆
⠀⠀⠀⣾⣿⣿⠟⣋⣭⣭⣴⣶⣿⣿⡇
⠀⠀⠘⣿⣯⣖⣭⣶⣾⣿⣿⣿⡿⠟
⠀⠀⠀⠙⠿⣿⡿⠿⠿⠛⠋⠉⠀⠀⠀⠀⢀⣀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣊⣥⣶⣿⣿⣿⢟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⣿⣿⣿⢟⣿⢋⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣟⣵⢟⣵⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣯⣴⣿⣿⣿⡟⠁⠀⠀⡠⢚⣻⢟⣶⣤⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣏⣼⣿⣿⣿⠟⠉⠀⠀⠀⣼⣴⣿⣿⣼⢹⣿⣇⠀⠀⠀⢰⢻⣿⣿⢸⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠋⠁⠀⠀⠀⠀⢠⣱⣻⣿⢇⣿⣿⣿⡏⠀⠀⠀⣼⣼⣿⣿⢸⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⢏⣾⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⣇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⣿⣻⣿⡿⠋⠀⠀⠀⠀⠀⢿⣿⣿⡀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣟⣛⡭⠿⡟⠛⢲⣶⣶⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣌⠙⢧⡀⠀⠀⠛⠀⢿⣍⠉⢹⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠻⣦⠀⠀⠀⠀⠈⠶⣄⣻⣿⢿⣿⣿⣷⢄⠀⠀⠀⠀⣠⣼⣛⣀⣤⡀
⠀⠀⠀⠀⠀⢠⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣧⣶⣦⣶⣦⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣨⣯⣽⣭⣭⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⠟⠉⠻⣷⡌⠙⠛⠋
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣀⣀⠈⠉
⠀⠀⠀⠀⠀⠙⠿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠟⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠛⠛⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠉⠁



⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⣀⣴⠾⠛⣉⣥⣤⣴⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠁⠀⠀⢀⣠⠾⢋⣥⡶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠿⠛⠛⠛⠛⠛⢷⡶⠶⠛⣡⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡈⠳⠶⠶⠶⠶⠖⢚⣡⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠉⠛⠳⠶⠶⠶⠾⠛⠋⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠋
⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢶⡆
⠀⠀⠸⣧⠒⠂⠒⠒⠐⠒⠐⠒⠂⠒⠒⠐⠒⠀⠒⠂⠒⠂⠐⠒⠂⠲⠖⠐⠶⠂⠶⠰⠆⣼⠃
⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣶⠶⠶⠶⠶⠶⠿⠿⠟⠿⠛⠛⠛⠛⠛⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠄⠁⠠⡀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣼⡟
⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠈⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠘⢧⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⣄⣛⢛
⠀⠀⠈⠘⠛⠛⠉⠀⠀⠀⠘⢦⡄⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠑⢺⡏⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠃
⢂⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠘⣧⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⢆⠀⠀⢀⠔
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠖⠀⠀⠀⠀⠀⠀⠀⠠⡆⠙⠪⢧⣬⡤⠄⠀⠀⣀⠜⠋⠀⠘⠀⡠⠊
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡱⠆⠀⠀⠀⠀⢠⡀⣠⠄⠚⠁⠐⠀⠀⡀⡸⠆⠀⠀⠀⠀⠉⠙⠂⠀⠐⠒⠒⠉⠁⠀⠀⠀⠐⠃⠁
⠀⠀⠀⠀⡠⣿⣤⣴⠀⠀⠀⠐⠉⠁⠀⠀⠀⠤⠊⠁⠈⣡⡀⢀⡶⣶⠌⠛⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠁⣲⠶⣤⡀
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⠵⠊⠁⠄⠂
⠀⠀⢀⣠⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡠⠊⢀⠤⠂⢉⡇
⢠⡄⡀⠙⠒⠒⠒⠚⠛⠉⠛⠒⠒⠒⠛⡀⠈⣧⠀⠀⠀⠀⠀⠀⠄⢀⣾⣿⣿⠏⠑⣶⡀⢠⡄⣤⠁⠀⢀⠌⢀⠔⡡⠔⠁⠠⢖⡀⠀⢀
⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡇⠀⠀⠀⠀⠠⡀⠹⣟⠛⠓⠦⣄⣶⠈⢙⡏⠘⠟⣆⣶⣆⣰⣡⠊⣀⣀⣀⣠⠈⠫⢶⡈
⠀⠀⠑⠦⣦⣦⣄⠀⠀⠀⠀⠀⣴⣤⣤⠤⠟⠀⠀⠀⠀⠀⠀⠙⠻⠞⠓⠒⠊⠉⠉⠑⠒⠛⠉⠀⠛⠀⠟⠉⠁⠙⠛⠉⠉⠡⡥⠶⠖⠂
⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠂⠚⠁⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠠⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣀⣤⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠖⠋⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡌⠗⠚⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠁⠀⠈⢐⠀⠒⢀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘
⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡂
⠀⠀⠹⣦⣄⠀⠀⠀⣀⡴⡮⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠘⠚⠢⡀⠀⠀⠀⠂⠀⣠⠐⠀⠀⠆⠀⠀⢈⣸⡇
⠀⠀⠀⠈⠛⠻⠯⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠟⠷⣄⠀⣀⡀⠐⠂⠒⠐⠁⣤⢐⣾⠟⠒⢪⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠁⠉⠉⠁⠀⠉⠛⠂⡁⠎⠱⠶⠾⠷⠾⠟⠂⠚⠙⠈⠃⠹
⠀⠀⠀⠀⠀⠂⠀⢄⣀⣰⠈⠀⠈⠀⠀⠀⠈⡉⣭⣅⠀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣷⣄⣀⠈⠀⠛⠂⠀⠠⠄⣀⠀⠀⠉⠛⠃⣸⢰⡇⠀⠀⠀⠀⠀⢠⡀⠀⠁⠠⣄⠀⠁⠀⠰⠀⠀⠈⠛⢤⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠙⠻⠗⠳⢦⣄⡀⠠⠄⠀⠀⠠⠤⠤⠐⣣⡟⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⡞⠀⠀⠢⠤⠀⠀⠀⠦⠀⠀⠙⠢⠠⠈⣰⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡉⠀⠀⠠⣤⣤⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⡤⣀⣤⡾⠥⠒⠂⠃⠀⠀⠀⢉⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⣀⣀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠻⣦⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠾⢡⢀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡾⠋⠠⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢉⠙⣦⠀⠀⠄⠀⢀⣾⠋⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠛⠀⢸⡇⠀⠀⢠⡟⠁⠀⠀⠀⠐⢯⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟⠀⢠⡟⠀⠀⠀⠊⣀⡴⠞⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⠤⠢⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠀⠊⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣿⠀⡀⣼⠃⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⠽⣇⠸⢿⠀⢿⡇⠞⠒⠒⠒⠢⠤⠤⣤⣒⠤⠊⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⡤⠖⠚⠉⢁⠀⠀⠘⢦⠻⢇⢻⡔⢀⡀⠒⠀⠠⠄⠀⠀⠁⣠⠞⠻⢷⡦⡀
⠀⠀⠀⠀⠀⠀⢀⠊⠞⠁⢀⠄⠀⠨⠀⠀⠀⠀⢂⡙⢦⣿⣗⠀⡀⠀⠈⠂⠀⠀⢀⡼⠋⣀⠠⠀⠉⢪⢢
⠀⠀⠀⠀⠀⠀⠈⢆⠢⢀⠀⠀⠈⠀⠠⠀⠀⠐⠀⠁⠄⠛⠿⠷⠀⠒⠀⠠⢁⠐⠸⠃⠀⠀⠀⠀⠂⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠈⣟⠮⣵⣀⠀⢀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠂⡀⠈⣀⡯⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠈⠉⠛⠒⠾⠥⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣤⠤⠤⠤⠗⠒⠊⠉⠀⢰⠇
⠀⠀⠀⠀⠀⠀⣠⠶⠖⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠿⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠱⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢡⠞⠁⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣄⣘⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⡾⠃⣀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢍⣉⠛⠓⠶⠤⠤⠤⣤⣤⣤⣤⣤⠤⠴⠶⠶⠟⣛⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⠒⠉⠁

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

⠀⠀⠀⠀⢀⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⡀
⠀⠀⠀⢀⡏⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠙⡆
⠀⠀⠀⠈⢧⣄⣠⠆⠀⢸⠇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣳⣄⣠⣤⣄⣀⣀⣀⡀⣠⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣧⣤⣶⡾⠿⠟⠛⠛⠛⠒⠛⠛⢻⡋⠉⠙⠛⠛⠛⡿⠿⢷⣶⣦⣤⣀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⢦
⠀⠀⠀⠀⣠⢴⣫⠝⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠙⠛⢿⣶⣄⣠⠶⠋⢁⣀⣤⣶⡿
⠀⢀⡴⡿⠟⠉⠀⠀⠈⠣⣀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣾⣤⣤⣀⣀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⢨⠟⠁⢠⣾⠿⠛⠉⠁
⢠⣯⠎⠀⠀⠀⠀⢀⣠⠴⠟⠻⠯⠉⠀⣤⡴⠀⢈⡁⠀⠀⣀⠀⠀⠀⠈⠙⣿⠒⠦⣤⣀⠀⣰⠋⠀⣰⣿⣇
⣾⣿⠀⠀⢀⡤⠞⠉⠀⠀⠐⠲⠆⣠⠤⢤⡄⠀⠫⠭⠇⠘⠾⠃⡴⡆⠀⠐⠯⠗⠀⢠⡼⠛⠁⠀⣰⣻⣿⣿
⣿⡹⡄⢠⠞⢡⢺⡆⠀⠀⠀⠀⢀⢬⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠖⣻⢆⠀⠸⡆⠀⢀⣴⣿⢫⢏⣿
⢻⡻⣿⣯⣀⠈⢉⢤⠸⣹⡆⠀⠈⢦⡗⠀⠀⠀⣴⣶⠆⠀⣴⣻⠇⠀⠈⠉⢁⣀⠀⢀⣽⣶⠟⢛⡷⣫⠟⡿
⠈⣧⣄⠙⠻⢿⣶⣯⣤⣀⣀⣀⡀⠀⠐⠏⡷⠀⠀⠀⠀⠀⠀⠀⠠⣒⣶⡆⠓⠚⠀⣈⣑⣿⣾⠿⠋⠁⣸⠇
⠀⠸⣞⡳⣦⣄⡀⠉⠙⠛⠛⠻⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣒⣛⣺⣿⡭⠽⠟⠛⠉⠁⢀⣤⢾⡏
⠀⠀⠹⣤⠇⣠⣿⢷⠲⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠚⣽⣩⡟
⠀⠀⠀⠹⣗⠿⣯⡾⢡⡞⡭⢹⡆⣩⢿⡻⣟⠛⢒⡶⢶⡖⠒⢲⡶⢶⠒⢛⡿⠿⡍⢡⣿⡏⣷⣹⠞
⠀⠀⠀⠀⠘⠧⣄⠉⠙⠓⠿⠯⣤⣇⣻⣥⠟⢰⡏⣞⣃⡿⠀⡯⣞⣸⠇⠸⡼⣷⣃⣸⢦⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⢷⠦⣄⡀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠚⠛⠛⠛⠋⠉⠉⣉⡥⠖⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠉⠛⠓⠶⠦⠤⣤⣤⣀⣀⣀⣀⣤⣤⡤⠴⠖⠚⠋⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢄⠀⠀⢀⠀⠀⠀⠀⢀⡄⢰⡟⣐⣀⣠⡴⠃⠀⢰⡆⠀⠀⠀⣑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠳⡈⢢⣀⠀⠀⠉⠀⠠⡉⢧⠟⠷⠬⠟⠁⠂⠶⢀⡀⠰⠐⠋⠀⠁⠁⠠⣍⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠈⡈⠛⠦⣌⣠⣸⠭⡆⠛⠈⠉⠑⢶⡶⠾⠟⠽⢓⠈⠐⠈⠀⠀⠀⠀⡠⠖⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⣈⠠⢠⡌⠐⢾⣿⣟⣶⢁⡀⢀⠀⢠⣄⣭⠀⣠⠀⢤⣠⡴⠂⠤⠖⠀⠁⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠤⢱⡄⠛⠓⠀⠤⠀⢀⡤⠜⠋⠁⠈⣿⠟⠙⠆⠀⠳⣜⣦⣀⠀⠠⠂⢀⣈⡀⡿⠎⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⡁⠐⠀⠀⠉⡲⠦⠤⠴⣆⠀⠀⠀⠀⠀⠇⠀⠄⢀⠀⠀⡏⠙⣘⣬⣴⠾⠗⠊⠀⡅⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠀⣉⡡⠬⠙⢛⠰⠖⡆⠀⠀⠀⠈⢢⡀⠀⣀⠄⠂⢀⠖⠀⡄⣔⣤⠜⠫⠿⠋⠁⠈⠀⢀⠵⠀⠀⠶⠞⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠠⠒⡋⠕⠂⢤⠊⠀⠀⠀⠀⠐⠫⡌⠁⣚⢊⢜⣊⣔⠫⠁⠀⡍⠀⢂⠀⡀⠀⡤⠚⠲⣿⠤⠄⠀⢀⠍⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠴⢊⠁⢰⠏⠀⢸⣆⡀⠀⠠⠔⠊⠀⢸⡤⠿⠛⠋⠉⠀⠆⠐⠀⠀⠘⠀⠀⠐⠋⠀⠀⢀⠀⠁⣀⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⣰⢡⣡⠕⠘⠁⠀⠀⢀⠀⡸⠀⡀⢱⠀⠸⠄⠈⣀⠠⠔⠊⠁⠀⠐⠶⠬⢁⠔⠒⠈⠁⠀⢀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢛⠁⠀⠀⠀⢀⠐⠋⠀⠐⠄⣙⠛⡒⡡⠃⡞⠤⠐⠀⠶⠄⠂⠀⠈⠁⠀⠀⠀⣠⠤⠐⠁
⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠘⠶⢀⢀⡄⠁⠀⢀⡜⠀⠑⠂⠐⠂⠀⠀⠐⢄⡀⢀⡀⢀⡀⣀⣀⠴⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣭⣤⣬⣹⣏⣴⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠏⠀⠀⢀⡽⢠⠖⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠤⠖⠒⠒⠲⢤⣀⡤⠖⠋⠉⠳⣤⠴⠚⠉⣁⡤⠋⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⢀⣠⠿⠚⠉⠉⠙⣄⣠⣴⣚⣉⠉
⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⢤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡰⠋⠉⠉⠉⠉⠉⠉⠉⠓⠶⣤⠤⠤⢤⡀
⢰⠋⠁⠀⠀⠀⠀⠀⠀⡴⠋⠀⢠⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⣀⣀⣀⣀⣠⣀⣀⡀⠀⠀⠀⢸⡏⠙⢆⠙⣆
⠸⡷⣄⡀⠀⠀⠀⠀⣰⠁⠀⣠⠎⠀⢳⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠑⣄⢀⡞⡗⠚⠉⣀⡼
⠀⣧⠀⠙⠻⣍⠉⠉⠙⡆⠐⠉⡇⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠚⠁⢠⠟⠛⠉⠁
⠀⠙⣆⠀⠀⠈⠑⠦⣀⠘⠦⣄⠣⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣯⡤⠤⠶⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⢀⡾
⠀⠀⠈⠢⣄⠀⠀⠀⠈⠓⢦⡈⠳⣄⠀⠁⠀⣀⣀⣀⣠⣤⠤⠴⠖⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠊
⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠙⡋⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠶⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⣠⣤⣤⡤⠤⠴⠶⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠄⠀⠒⠂⠀⠀⠀⠀⠐⠒⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠠⠚⠐⢄⢀⠜⠢⢄⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠁⠀⠀⡐⡉⠛⠠⠀⠠⠤⠁⠛⠒⠒⠒⠒⠒⠒⠒⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠠⠤⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠒⠈⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠶
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠠⠀⠠⠀⠀⠠⠄⣀⡀⠀⠀⠀⠀⠀⢀⡠⠐⢉⠔
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠈⠁⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⣑⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣁⣐⢂⡀⢀⣐⣈⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣠⢊⡌⣀
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⢰⠁⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠂⠀⠀⠈⡆⡀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠐


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡴⢊⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠄⠂⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣀⡁⠔⠈⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣿⡿⡆⠀⡿⡇⢸⠶⢶⣶⣶⣤⡤⠃⢀⡔⠉
⠀⠀⠀⠀⢀⣤⣶⣿⠿⠛⠋⢉⡉⠀⠀⠤⠀⠀⠀⠀⡇⡐⡲⢃⠆⠀⠀⠠⢠⠜⠁⡰⠛⠻⢷⣶⣄
⠀⠀⠀⢰⣿⡟⠉⠠⠐⠈⠀⠀⠐⠀⣠⣥⡀⢋⠀⠞⠨⠾⢁⠁⠒⠚⠀⠀⠁⠀⠌⠀⠀⠠⠀⣘⣿⡇
⠀⠀⠀⠘⢿⣿⣦⣦⣀⣘⣛⡁⠈⠤⠄⠐⢀⠲⠖⠈⠠⠤⠀⠀⠛⠡⠀⢶⣂⣀⣠⣥⣴⣾⠿⠟⡟
⠀⠀⠀⠀⠈⢦⠉⠙⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⡜
⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⣠⠖⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⢄⡀
⠀⠀⠀⢠⡞⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠱⡄
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⣿⡿⢻⠿⠻⠿⡿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠤⣄⣤⣀⡤⢤⠀⢀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⣯⡈⠛⣦⣉⣣⠘⣇⣼⢳⠈⠒⣉⡿⠴⢟⣒⣒⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣒⣓⣾⣿⣿⠈⠧⣈⠭⣤⣙⣻⡾⠛⣛⡥⠖⠛⠉⠛⠣⠍⠙⡚⠭⢷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣾⣿⣿⡿⣫⢟⣻⣿⣶⣦⣼⣿⡿⢋⣠⡜⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢶⠱⡙⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣄⣾⠛⣭⣭⣭⣗⣓⠿⣿⣿⣿⣿⠟⣡⠶⢛⣡⣤⠦⠄⠀⠀⠀⠀⣤⣤⡀⠈⠻⣈⠷⢸⡿⣄
⠀⠀⠀⠀⠀⠀⠀⢧⣀⣰⣟⣻⠿⣭⣿⣿⣷⡌⣿⣿⢏⡜⠁⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⡀⠈⠉⢦⠓⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢧⣤⣚⡡⠖⠉⠉⠉⠉⠛⢿⣿⣫⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⢆⠀⣿
⠀⠀⠀⠀⠀⢀⡴⠻⠋⠈⠳⢄⣤⣤⣴⣶⣤⡿⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡀⢣⣿
⠀⠀⠀⠀⠀⠀⣿⠓⠲⠴⢚⣿⠿⣿⣿⣿⡿⣹⣁⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⡄⠉⢁⣼⡟
⠀⠀⠀⠀⠀⢀⣘⣦⣠⣾⢻⣸⡇⡏⢸⣿⣿⠉⠈⠓⠀⠀⠀⠀⠀⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠟⠛⠁
⠀⠀⠀⠀⠀⢯⣿⢁⣿⣿⣿⣿⣧⣿⣿⢳⠇⠀⠀⠀⠀⣀⡀⠀⠀⠈⠍⠁⠀⣀⣠⣤⠶⠞⠛⠋⠉
⠀⠀⠀⠀⠀⡼⢿⣿⢿⣿⢿⡟⢿⣿⢿⡿⠖⠢⠄⠈⠉⠉⠁⢀⣀⣤⡴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⣧⠀⣿⡆⢻⡌⣷⣼⣏⣾⣋⣀⡤⣰⣦⣤⡶⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠹⠤⣿⣽⣾⣷⣿⣫⣿⣿⡷⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⣀⢴⡄⠀⢀⣶⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣧⠼⢿⡛⠋⣽⣿⣿⣿⣿⢛⠛⣲⣶⡴⠟⢾⡇⠀⠈⢛⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣴⣾⠯⠙⠀⣇⠀⠀⠙⠚⠁⢻⢱⣿⣟⡍⠛⢩⢟⠀⠀⣨⠿⠛⣿⣭⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⣿⣲⣴⡟⠿⠏⠀⣺⢶⡞⠋⣠⠖⢯⡲⣀⣸⣿⣿⡷⠶⠶⠿⠾⠶⠾⠷⣤⣤⣴⣴⣿⣿⡽⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⠻⣷⣽⣶⡞⠋⡹⣷⣣⢙⣷⣿⡿⠛⠋⣍⣙⢲⠂⠀⠄⠘⠃⣤⠀⠙⠛⠒⡾⢭⣛⠿⣿⣦⣬⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡄⠀⢿⣿⣿⣝⣦⣐⣿⠿⠛⠙⠿⠃⡀⠀⠶⠾⠍⠀⠀⠆⠀⣤⠀⠀⠛⡇⠠⡍⠛⠾⣝⠲⣝⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣁⣀⡿⣀⡎⡼⣻⣿⡿⢋⠁⠀⠀⠀⢰⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠤⠈⢁⠈⠓⢮⠻⣆
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢸⡆⠀⠊⠉⠚⠛⣇⣷⣿⡿⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⢀⠀⠌⠳⡜⣧⡀
⠀⠀⠀⠀⠀⠀⡾⢿⠿⢀⡿⢹⣄⣤⣤⣖⣴⠟⠛⠻⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠐⣾⣼⣤⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠙⠦⣤⡿⠟⣻⣷⣤⣤⣿⡁⠀⠈⠀⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⢠⠀⠀⠀⡀⡌⠀⠀⠀⠘⠉⠉⢁⣠⡴⠞⠛
⠀⠀⠀⠀⠀⠀⠀⣀⡟⠉⠀⠹⢿⣾⣿⠟⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⡀⠀⠀⣤⠀⣀⣬⡶⠞⠋⠁
⠀⠀⠀⠀⠀⢀⣼⡧⣶⣲⣿⣿⣾⣿⣟⡤⠀⠀⠀⠀⠀⠈⠀⠀⠐⠍⠀⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⣀⠀⠀⠋⣀⣼⠶⠛⠋⠁
⠀⠀⠀⢀⡴⣫⣷⣿⣿⣿⠿⣿⣿⣿⡞⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠂⢀⣠⣤⠶⠛⠉
⠀⠀⢠⣟⣾⣿⡿⠛⣡⡶⣶⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠐⠾⠀⠀⠠⠀⠀⢠⠀⢀⣠⣴⢾⣯⣡⡴⠞⠛⠉
⠀⠀⠈⠿⣯⠿⢤⣠⣘⠿⣿⢯⡇⠘⠟⠀⡤⠀⠀⠀⠀⣄⣀⠀⠐⠃⢁⣠⠴⠚⣉⣤⡿⠟⠋⠁
⠀⠀⣠⣖⡲⡄⢰⠋⣡⠶⣿⢸⢡⡄⠀⡶⠀⠀⠀⠀⣰⡓⡾⡄⠀⠚⣉⣤⠶⠛⠉⠁
⠀⠀⢧⣙⠛⣁⡼⢏⠁⢀⣿⢸⠺⠀⠀⠀⢀⠀⢀⠀⠌⢙⣩⣴⠶⠛⠉
⠀⠀⠀⠈⠉⠉⠀⡼⢰⢯⣿⣼⣆⢀⡀⠀⢈⣌⣨⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⢰⣳⣎⢿⣿⣷⢿⣼⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠻⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡠⠤⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣶⣶⣶⣾⣥⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣴⣾⣿⡿⡗⠛⢉⠉⠻⢿⣿⣿⣭⣟⣿⣿⣿⣷⣖⢯⣶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣷⣾⣿⠿⣟⣿⣿⣿⠐⣹⣿⣿⣷⡖⢈⣿⠛⣯⢻⣿⣿⣿⣿⠿⢿⣿⣯⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠙⢟⣏⢻⣾⣿⣿⣿⣿⣷⡃⢚⢛⣢⣵⣾⠁⣰⢿⡏⢿⣿⣿⣿⣷⣾⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠈⢁⣿⣿⣯⣶⣟⠛⠻⠿⣿⠿⢟⣵⣿⣾⣇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⣧⡀
⠀⠀⠀⢀⣤⣤⣶⣾⠀⠀⠀⠀⢠⣿⣛⣽⠟⣿⣩⢰⣶⣞⣩⣴⡿⣹⣿⡿⢫⣾⣿⠿⠋⠉⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⣿⡟⠉⢩⣷⡆⠀⠀⣠⡿⠏⠉⠀⠠⣻⣿⣿⢿⣿⡿⠋⣼⣳⡟⠁⣿⡿⠃⠀⠀⢠⣿⣿⣴⡿⣹⠿⢿⣿⣦⡀
⠀⠀⠀⢿⣿⣧⡿⣿⣷⣦⣾⣿⣶⡁⠀⠀⠀⣹⣿⣿⣾⣿⣿⡲⣾⣿⣆⣤⠠⡆⠀⠀⢀⣾⣿⣿⣿⣷⣿⡄⡜⠛⢻⣿⣦
⠀⠀⠀⠘⢿⣿⣷⣽⡿⣷⣸⠿⣿⣿⣷⣤⣼⣿⣿⣿⢿⣿⠟⠀⠈⡹⢿⣿⣤⣇⠄⢠⣮⣼⣿⣿⠿⠋⠉⡴⠀⠀⣸⣿⣿
⠀⠀⢠⣶⡟⠻⣿⣿⣧⡹⠿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣏⠙⠧⢀⠀⠐⢋⡽⣿⣿⣿⣿⣿⠟⠛⠉⠀⣰⣾⣤⣶⣾⣿⣿⠏
⠀⠀⠀⠹⣷⣄⠈⢿⣿⣿⣆⠀⢻⣿⡻⠿⠿⡿⠇⢻⡿⣿⣶⣦⣤⣤⣤⡴⣿⠋⠉⣁⣠⣤⣶⣾⣿⣿⣿⡿⠿⠛⠛⠁
⠀⠀⠀⠀⠘⢿⣷⡾⠿⣿⣿⣷⡄⠈⠙⠳⠦⣤⣴⣿⠟⠓⢿⣄⣀⣀⣤⣾⣿⣾⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠈⣳⣤⣶⣶⣾⣿⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢟⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⢒⣲⡾⠿⣭⡵⠋⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠟⠋⠀⢰⠋⢉⡴⠞⠁⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣠⡤⢤⣘⡴⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣠⡾⠋⠀⠀⣠⡿⠂⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠋⠀⢠⠃⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠋⣁⡤⠤⢤⣀⠀⠈⢇⢀⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⢠⠊⠁⠀⠀⠀⠈⢳⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⢀⡼⢋⠔⠊⠉⠉⠓⢄⠀⡇⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⢀⡴⠋⣰⠃⠀⠀⠀⠀⠀⠸⡀⢇⢀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⢸⡀⠀⡇⠀⠀⠀⢀⣠⠤⠞⠛⠋⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠹⣄⢧⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⢀⣠⠴⠒⠚⡓⠲⣄
⠀⠀⠀⠀⠀⠈⠲⣽⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⣠⠞⠉⠀⣤⠶⠛⠋⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠶⠶⠚⠋⠉⠁⠀⣀⡤⠶⠶⠶⠶⠤⠤⠤⣤⡀⠈⠓⠶⠴⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠋⠙⠛⠟⠃⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⡒⡚⢒⠚⣋⢁⣈⡑⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢊⣅⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⢛⣟⣽⡆
⠀⠀⠀⠀⠀⠀⢀⠔⣨⣵⣿⣿⣿⣿⣿⣿⣿⣿⡏⣤⣰⣷⡿⣿⠿⠛⣛⣦
⠀⠀⠀⠀⢀⠔⣡⣾⣿⠏⣹⣿⣿⣿⠿⠋⠕⣠⣶⢿⡿⠞⠋⠁⣠⣾⡿⣿
⠀⠀⠀⢠⠃⣼⣿⣿⣿⣿⡿⠏⣡⣴⣶⠶⣛⣻⡷⠋⢀⣠⣤⣿⡿⠟⢱⡿
⠀⠀⠀⡇⣴⣿⣿⠿⠛⣉⣴⣿⣯⣶⡶⣿⣿⢋⣠⣾⣿⣿⡿⠋⣀⣴⡿⠁
⠀⠀⠀⢳⠿⠿⣏⣴⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠛⢉⣤⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⣠⡶⢶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠘⣷⣶⣿⠷⢻⡿⡿⠛⠙⢿⣿⣿⠿⠟⣁⣤⣶⣿⣿⣿⣟⠓⠀⠀⠀⠀⠀⢠⣾⣯⣻⣷⣤⣈⠙⠛⠛⠛⠿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠽⢧⣤⣴⣟⣷⢶⢶⣾⣿⣯⣷⣶⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⢻⢀⠈⠛⢿⣿⣶⣤⣤⣀⠀⠘⠻⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠻⢾⣶⣷⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⢉⣀⡀⠀⠀⠉⠻⢿⣿⢿⣦⣀⠀⠉⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠙⠿⣶⣶⣄⡀⠀⠈⠛⢿⣿⣿⣶⣆⠀⠨⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣆⡈⠙⠿⣿⣶⣦⣄⠀⠙⢷⣾⣿⣿⣄⡙⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣻⣿⣿⣿⣦⣀⡀⠈⠛⠿⠷⣤⣀⠀⠉⠛⢿⣿⣻⣍⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡻⣿⣿⣿⣿⣿⣷⣴⣆⣄⣡⣈⡀⢉⣀⣈⣧⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢾⣿⣿⣉⡛⣻⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠻⠛⣁⣼⠿⠟⠛⠋⢸⣗⣿⣄⡠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣠⣽⣾⡟⠛⠁⠀⣀⣀⣴⣿⣿⡿⠛⣿⣿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⢪⣿⣿⠏⠀⠀⠀⣸⣿⣿⡿⠛⠁⠀⢸⣿⣿⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⣼⣿⣿⠃⠀⠀⣠⣼⣿⣿⣿⠃⠀⠀⢀⣼⣿⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⢀⣾⣿⣿⠏⠀⠀⣼⣿⢿⣿⠟⠁⠀⠀⢠⣾⣿⡿⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⣿⣿⠋⠀⣠⣶⣷⣿⣿⣟⠋⠀⠀⠀⣼⣿⣿⡟⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠋⢠⣾⣿⣿⡟⠛⠉⠀⠀⢀⣀⣿⣿⠏⣽⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢰⣿⣿⣿⠏⠁⣀⡤⠴⠿⠟⠋⠉⣀⣾⣿⣿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡾⣿⣟⡁⣀⢈⣉⣠⣤⣤⣤⣶⣿⣿⣿⣿⣛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡀⢀⡀⠀⢀⡀⣠⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⡄⠀⠀⠀⣀⣴⣤⣷⣶⣷⣿⣽⣴⣾⣿⣧⣿⣿⣇⠀⣀⣀⣥⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣿⣦⣴⣿⣽⣿⣿⣯⣿⡟⢹⡿⢋⣱⣻⣿⣻⡿⣿⣾⣿⣿⣿⣿⢻⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢱⡎⣍⣿⣟⡛⠋⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⠾⣾⣿⣿⣿⠿⣿⣼⣶⡿⣿⠟⣛⣿⣯⣷⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣷⣶⣾⣿⣿⣿⣿⣿⣿⠿⢻⠏⠀⠀⣿⡟⠁⠻⠀⠘⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢱
⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣶⣾⣯⣿⣿⣿⡟⢿⡛⠛⣆⠀⠀⠀⢀⣿⡎⢉⡱⡄⢀⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⣿⣿⣿⣿⣿⡧⠬⠁⢬⣭⣷⣤⡶⣾⣿⣶⣿⣤⡿⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣯⣿⣿⡿⣿⣯⠀⠠⠃⢀⣠⡭⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣯⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⢫⡿⣱⣿⣿⣷⣤⣖⣛⣁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡿⠋⠁⠈⠉⢁⣤⠟⣿⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⣀⠀⡰⠏⠀⢰⡟⠿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠶⠶⠒⠋⠀⠀⠀⠀⠀⠉⠙⠋⠉⠛⠛⠛⠛⠛⠛⠋⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠼⠇⠀⠀⠀⠤⣀⠀⠠⠄⠂⠁⠀⣁⣀⣈⠐⠠⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠸⢿⣷⣦⡀⠀⠀⠀⠀⠀⣰⡀⠀⢉⠄⠿⠋⡭⢙⣷⠒⠛⣏⢤⣿⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡄⡀⠀⠀⠀⠻⣿⣤⣏⢠⣧⠴⠿⠿⡬⢹⣿⠩⡿⠤⠐⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣾⡏⣹⡷⢦⡜⢿⣮⣷⠀⡙⡿⡟⠛⠛⠛⠙⠛⣯⣵⣫⣾⡯⠭⣐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⣿⣿⣿⣿⢷⣦⣿⡟⢿⣿⣿⣶⡇⠀⠀⠂⢰⡀⢀⣿⢯⣿⠏⠉⠀⢸⡀⢤⡈⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣬⡛⠛⠲⣿⠏⠛⢷⣙⣷⣹⣿⠋⠀⡆⣠⣼⠞⣿⣷⠿⣅⠔⢌⣠⠏⠀⠀⢉
⠀⠀⠀⠠⣄⡐⠂⢀⠀⠀⠈⠙⠿⣿⣦⣝⡷⣤⣤⣽⣿⢛⣿⡟⠚⢻⣿⣩⣶⡿⢛⡄⠀⡸⠛⡅⠀⣸⢰⠙
⠀⠀⠀⠀⠙⠳⠤⣀⠀⠂⢀⣀⡀⠈⠙⠻⢿⣾⣽⣍⠛⠳⢿⠛⢒⡒⠛⠟⡯⢭⣁⠟⠆⠠⡀⠀⡔⡻⠋⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢧⣮⣁⠀⠀⠀⠀⠈⠛⠿⣷⣦⣙⢦⣼⠇⠀⠂⠙⠶⣿⣉⢦⣤⣾⡿⠋⠀⢔⠍
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣾⣝⡊⠀⠀⠀⠀⠉⠛⠋⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠘⠛⠦⠤⢤⣀⣁⡒⠠⠶⠄⠀⠀⠀⠲⢤⣀⠠⣦⣄⡉⠻⢿⣷⣤⣄⣀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠂⠀⠀⣀⡀⠂⠀⠈⠙⠷⢮⡓⢄⡉⠛⠿⠥⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠦⠀⠙⢦⡙⠂

⣿⣿⢛⠿⣛⠛⣟⢛⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡤⠤⢄⡀⢀⡠⢎⠉⠳⣄⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⡴⠁⠈⢳⢤⡤⠃⡠⠴⡌⠭⠀⠈⠂⠀⢈⠭⣓⣂⣚⢳⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠉⠉⢆⠀⠀⢦⠃⠀⠘⡀⣀⣱⠰⡶⠧⠴⢊⡵⠮⢭⢤⣍⣿⢀⠎⠓⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠒⠲⠤⣀⠀⢦⡱⣀⠮⠁⠀⣊⣉⠭⢵⣿⣭⠥⠤⠽⢊⣉⣠⠤⢾⠋⠉⠑⢤⣀⣠⢷⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣇⠀⠀⢲⠤⠞⣓⡩⠭⠭⠝⣒⣦⣀⣹⡀⣁⣠⠤⠔⠚⠛⢛⣛⣛⣛⣛⣒⠲⠦⢄⡀⠀⢃⠄⢀⣇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠤⠤⢩⠖⣡⣶⠋⣠⡂⠤⠔⡺⠉⢉⡥⠖⠉⢁⣠⠴⠒⠚⠉⠉⠁⠀⠀⠀⠉⠉⠙⠒⠮⣕⡺⡀⠁⠈⠉⡧
⠀⠀⠀⠀⢀⡖⠉⢱⠀⠙⠲⢡⢞⣥⣶⠦⢤⣛⡏⢈⡱⠚⠁⣠⠴⣚⠥⠂⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠲⢍⠻⣦⡀⠘⢑
⠀⠀⠀⠀⠀⡇⢀⡴⣒⣤⢣⣣⠟⠋⠘⡄⠀⢜⡵⠋⢀⡴⢋⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⢷⡔⠋
⠀⠀⠀⣠⠚⠉⡰⡪⠃⣟⢿⣧⠀⠀⡠⣇⡴⠋⢀⠔⢩⠞⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢷
⠀⠀⠀⢇⠀⡰⠃⠀⣸⡟⢀⡾⣖⠊⢠⠎⢀⡴⠃⠔⠁⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡆
⠀⠀⠀⣸⠙⠁⢅⠮⠊⡠⢪⡄⠀⡴⠃⣠⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢰⠃⢠⡌⡆⠠⡪⡾⠛⠃⡜⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⢀⡏⠀⢾⡀⠀⡇⡇⠳⢄⡞⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤
⠀⡼⠀⠖⠀⢨⠂⢄⠀⠀⡜⢀⡞⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠋⠉⠉⠁
⢠⠇⡆⠠⣀⡴⠒⢲⠁⡼⢀⡞⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡠⠤⢒⣊⠭⠐⣂⡥⠴⠒⠉⠁
⣸⢰⠉⡀⢀⠀⠐⠊⡼⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢒⣊⠥⠖⠊⠉
⡇⠘⢀⠀⠘⢰⠄⡰⠁⣰⠁⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀
⡇⡇⠈⠓⢚⣜⡼⠁⣰⠃⠀⠀⢀⣠⠤⠖⠋⠉⠀⠀⠀⠀⠀⠉⢻⡏⠉⠀⣸⣧⠀⠀⣴⠟⠋⠙⠁⣴⠟⠉⠛⢷⡄⢼⡏⠉⠋
⢻⡿⣌⣩⠥⠊⢀⣜⣡⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⣯⣼⣇⠀⣿⡀⠀⠀⠐⣿⠀⠀⠀⢸⡇⠈⠙⠻⣦
⠀⠙⠒⠶⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠠⠟⠀⠀⠹⠆⠘⠻⠶⠶⠆⠙⠳⠶⠶⠋⠀⠲⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣀⠀⣐⠀⠀⠀⢀⠀⢠⠂⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢘⣆⣀⣳⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠃⣰⠋⠹⠉⠉⠟⠴⠃⡀⠀⠀⣀⣀⣤⣤⡤⣀⣀⡀
⠀⠀⠀⠀⠀⣤⡄⢀⡀⠀⡤⢼⣄⣿⠀⠀⠀⠀⣀⣤⣴⠿⣿⣿⣭⢽⣶⢛⣿⣉⣿⣉⣿⢲⣄
⠀⠀⠀⠀⡶⣍⠂⢸⡓⡆⠻⠸⠁⠀⠀⣠⡿⠛⠛⠻⡻⠿⠽⣷⣻⣏⣹⣿⡾⣧⣼⣯⣽⠟⠟⠃
⠀⠀⠀⠀⡨⢎⠃⠂⠀⠈⡀⢢⢠⠀⢠⣿⣵⠠⢀⠀⠀⠀⠀⠀⠉⢻⢷⣿⡶⣿⣰⠏⠙⠀⠀⡄
⠀⠀⠀⢀⠭⠒⠿⣲⣾⠿⣿⣾⣿⡿⠟⢛⠛⠿⢿⣶⣤⣀⠀⠀⠀⠀⢀⣻⠃⠁⠁⠀⠀⣰⣚⡇
⠀⠀⠀⠐⠋⠠⢺⠋⠉⠀⢸⣿⣏⠀⠀⠀⠀⠂⠀⠀⠙⠻⠿⣶⣼⣦⣬⣾⣀⢀⣤⡶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⢸⣦⡀⢴⣿⣿⣼⣦⣤⣀⡀⠁⠀⠀⠀⠀⠀⢂⡉⢿⣿⣿⢿⣿⡁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠸⢷⣿⡿⣿⡷⠟⠿⣷⣬⣽⣶⡦⠀⣄⣠⣤⣤⢛⣿⣿⡼⠱⡉⣇⠀⡠⠶⣅⣿⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣉⡛⠿⢿⣬⣦⣄⠈⠛⠛⠿⢿⣿⣿⡟⢛⣭⣘⡙⢶⠶⠞⠃⣸⠒⠲⣿⣧⡔⠉⠘
⠀⠺⠇⣠⡶⠛⠛⠲⢌⡁⠀⠀⠈⠙⠦⣍⣹⣷⠉⢙⣿⠿⠋⠻⢬⣹⠇⠀⠀⠀⠀⢙⢀⠰⠛⢻⡃⠉⠻
⠀⠀⠰⣟⡉⠉⠁⠒⠤⣵⡀⠛⠓⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⢤⣴⠉⠀⠀⠀⠀⠀⠀⢷⣥⣖⣚⣓⣫⡞
⠴⡶⠀⢀⣍⡓⠒⠒⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉
⣀⣀⣀⣀⣉⣀⣀⣀⣀⣁⣀⣀⣀⣀⣾⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣦⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⣋⣩⣵⣶⣦⣤⣾⠏⠁⣉⣭⣭⣵⣶⣿⣿⣿⣿⣭⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢡⣴⣾⡿⠛⢻⣿⠋⢉⣡⣶⠿⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⣀⣴⣿⣧⣌⣡⣤⣾⠟⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢀⣾⡟⠛⠀⠉⣹⡿⠉⢡⣾⠟⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣿⣟⠁⠀⣠⣤⣿⠇⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⢸⡿⠀⣠⣿⠿⠿⢿⣿⠁⠀⣼⡟⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠁⠀⢀⣼⡿⠀⣼⡟⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⢠⣿⠁⣾⡟⠀⠀⣿⡏⢀⣾⡟⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣠⣿⠇⠀⣀⣿⣇⣾⡟⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⣠⡀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⢰⣿⢉⣾⠏⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠹⣿⣄⣀⣼⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣶⣶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠉⠉⠓⢦
⠀⠀⣀⠴⠖⠒⠂⠄⡀⡖⠛⠉⠑⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠒⠭⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢠⠤⡴⠖⠒⠖⠛⠛⠛⠛⠓⠒⠒⠒⠤⠤⣠⡔⠉⠀⠀⠀⠀⢀⣀⠤⠒⠋⠉⠁
⠀⠀⠘⠑⠂⠠⠀⠀⢀⣀⣀⡀⠀⣾⡶⠔⠂⠈⠉⠉⠛⠛⢲⣤⣤⣄⣀⣀⡀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠈⣧⠤⠄⠀⠀⠂⠈⠀⠀⠀⡉⠉⠀⠉⠙⠻⠶⢦⣌⠑⣄
⠀⠰⢤⣤⣤⣶⣤⣤⠴⠿⠛⠒⠛⠛⠛⠃⠁⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡆
⠀⠀⢠⠸⣿⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠣⠤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠸⡦⡙⠷⢤⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢄⢀⠀⠀⠀⠉⠑⢒⡠⠔⣫⢾⠃
⠀⠀⠀⢧⠙⠢⢄⣈⠙⠛⠲⠶⠤⠤⢤⡀⣀⣀⣈⣀⣀⣀⣀⣀⣤⠤⠴⠖⠒⣋⣡⠴⠚⠁⡼
⠀⠀⠀⠘⣆⡀⠀⠀⠉⠁⠐⠂⠀⠠⠄⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠒⠒⠋⠉⠉⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡟
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠓⠲⢤⣀⠀⠀⢀⣀⣀⣠⣤⣤⢤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣠⡇⠀⠀⣠⠬⠛⠋⠉⠁⠀⠀⠀⠀⠈⠉⠓⠒⠮⢭⣓⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣌⠙⠲⣄
⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣙⣦⣀⣀⣀⡀
⣰⠋⠀⠈⠓⣄⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠈⠉⠉⠓⠦⣄
⣇⠀⠀⠀⡴⠋⠙⠲⢤⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠈⠙⠢⠤⡇⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢀⡤⠚⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣠⠴⠚⠉⠀⠀⢸⣄⡠⠤⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢰⠃⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⡄⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⣤⣀⡀⠀⠀⠀⢸
⠀⠳⢄⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣀⠤⠖⠒⠒⠒⠒⠶⠤⣄⣀⣉⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡰
⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠀⠀⠀⠀⢰⣣⠔⠋
⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠖⠛⠺⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠋⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠐⡦⠄⣠⣀⢀⣤⣦⣀⣀⣠⣂⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠚⠻⠿⠿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡐⢴⣤⣤⣭⣉⣛⢛⠩⣁⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣁⣮⣿⣿⣿⣿⣟⣛⣷⣉⡛⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠿⢿⣿⣿⣿⡿⠿⠿⢿⣿⣿⡟⣾⣿⠋⠈⡀⠃⠀⠀⢂⡀⢂⠰⣶⠀⣾⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⠉⠛⠀⢴⣶⢾⡿⡿⠛⠁⠀⠁⠀⠀⢌⠙⢻⣷⣮⣝⡺⡷⠙⢤⣿⡗⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣖⣾⣿⣶⣿⣛⠻⠿⠦⣦⡀⠈⣿⣅⡀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣉⣛⣻⣋⣀⣴⣾⣿⣷⠆⠁⣼⣾⣶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠾⠚⠡⠚⠚⠿⠿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⡆⣾⣿⣿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠛⠛⠋⠉⠉⠀⠀⠀⠈⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠿⣭⡾⣿⠛⠛⣻⠶⢦⣍⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣭⠿⠶⠒⠶⠶⢬⣟⡛⢿⣤⣄⣭⠿⣤⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⠈⠀⣀⣁⣀⣤⣀⣈⡙⠷⣄⡘⢧⣤⡀⠻⣎⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠡⣀⣤⣶⣛⣽⡿⢶⡴⣬⣍⣙⠳⢦⡙⠷⣼⡷⢦⡉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣻⣯⣥⣬⣭⣍⣳⠶⢾⡏⠈⠙⣻⣆⡙⢦⡨⠙⢾⣷⡾⠇
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠁⠀⠀⡀⠀⠈⠉⠓⢦⣝⡋⢿⣉⡈⠹⣦⢻⡞⠋⠁
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠠⡄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠛⢦⣍⣿⣄⡘⠻⣿⡀
⠀⠀⠀⠀⠀⢀⡿⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⣀⠀⠈⠻⢮⣿⣤⣿⠃
⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠐⠀⠀⠃⠀⠐⠂⠀⠠⠀⠀⠀⠀⣀⣤⠴⠟⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠰⠀⠀⠀⠄⠀⠠⠄⠀⠀⢀⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⡄⠀⠀⠀⣀⣤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢻⣄⠀⢀⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁






⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⢀⡀⠤⠀⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠀⠠⢈⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⠠⠖⢋⣁⣀⣤⣤⠶⠖⠀⠀⠀⠀⠐⠂⠠⠴⣶⣦⣤⣤⣤⣤⣀⡀⠈⠑⠢⢈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢄⣪⣵⣶⣿⣿⡿⠿⠉⢁⣠⠴⠶⠖⠒⠀⠰⠦⢤⡀⠀⠝⢬⡿⠿⣿⡿⠿⠿⠿⣶⣦⣅⣠⡙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣰⣿⣿⣿⣿⡿⠻⠉⠀⢐⡊⠉⠍⠈⠈⠭⠁⢒⣀⠤⠀⢀⠀⠀⠀⠃⠂⠁⠶⠒⠒⠒⠲⢿⣿⣿⣿⣆⠙⣆
⠀⠀⠀⠀⠀⢀⣀⣼⠁⣼⣿⠿⠏⠁⠫⠥⠈⠉⠁⠤⠤⠤⠄⠀⠀⠠⠤⠀⣀⠀⠘⡏⠉⠉⠉⠐⠢⡀⣠⣤⠀⠀⠀⢄⠙⢬⡛⣛⣷⠘⣆
⠀⠀⣠⠖⠋⠉⠁⡿⠀⣿⠃⠀⠀⠀⠀⠀⠀⣤⡄⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⢨⡋⠀⠀⠀⠀⢀⣠⣴⡈⠁⢀⠚⠂⠀⠀⠉⠉⣩⣿⡆⣿⡀
⠀⢸⠃⠀⠀⠀⣴⢿⠀⢷⣀⣀⣀⣠⣤⠀⠀⠉⠁⣠⣤⡍⠀⣤⠄⠀⠀⠀⠠⠄⠀⠀⠧⠴⠚⠁⡀⠊⠙⡀⠻⠿⠃⠀⠀⠀⠀⣾⣷⡇⢸⢧
⠀⠈⢧⠀⠀⣼⠋⠈⣧⡘⣿⣯⣿⣯⣍⣀⠀⠀⠘⠛⠡⣬⣤⠰⣿⠏⣠⠀⠀⠰⠛⠀⢀⡀⠀⠀⢱⠀⠠⠴⠀⢀⣄⡠⠔⢣⣾⢿⡿⢠⣾⡀⠱⣄
⠀⠀⠀⢳⡼⠃⠀⠀⠈⢷⣜⠻⣿⣿⣿⣿⣷⣤⠀⠀⠀⠙⡋⠀⠀⢾⣿⣿⣿⡷⠀⠐⠿⠓⠀⠀⠈⠀⠀⠀⠀⠛⣉⣠⣶⣛⡷⠋⣠⡿⢿⠍⣤⠘⣆
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠘⠷⣌⠙⠿⣮⣍⣙⠉⠀⢈⣿⣿⣿⣦⠀⠙⠿⠿⠇⠀⠀⠀⠀⠐⠒⠐⠀⠀⣀⣤⣾⣿⡿⠟⢉⣤⠞⠁⠀⠀⣸⠈⠳⢿⣀
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡈⠙⠛⠲⠿⢭⣽⣿⣿⣴⣦⣤⣀⣤⣤⣶⣾⣿⣷⣶⣾⣿⠿⠶⠚⠉⣁⣤⠾⠋⠁⠀⠀⢀⡴⠋⠀⠀⠀⠉⢷⣄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠦⢤⣄⣀⣀⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣠⣤⡶⠞⠛⠉⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⣻⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠓⠒⠒⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠜⣹⠃
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠶⠖⠒⠶⠶⠶⠖⠒⢛⣡⠾⠋
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣤⠶⠶⠻⣿⣿⣿⠋
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠤⠤⠤⠴⠲⠖⠶⠆⣰⡔⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⡰⠞⠋⠀⠀⠈⠙⠋⠐⠂⠐⠔⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠹⣿⡾⠀⢀⣀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣭⣿⣥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣶⣦⡈⠋⠀⣠⣴⣦⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢙⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣯⡰⡌⣿⡿⣿⣷⣿⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⠿⠾⠿⠿⠛⠛⠋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣧⡜⢀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠘⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣏⠄⢴⣠⣴⣿⣿⣿⣿⣴⣶⡒⣤⣾⣿⣎⣷⢪⣆⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡏⠀⢀⣠⣴⣷⣷⣿⡿⠟⠝⠛⠩⡿⠿⣿⠟⠟⠻⠟⠛⢻⣿⠀⠁⣄⠈⢫⣷⡀
⠀⠀⠀⠀⠀⠀⣀⠀⣠⣤⡉⠁⣠⣴⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠉⠐⠺⠿⣦⣀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⣚⣿⢏⣽⣻⡿⠛⠛⠉⠉⠁⠀⠀⠀⢀⠀⠀⠀⢀⡈⣴⡴⢿⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⠟⢿⣷⣦
⠀⠀⠀⠀⢸⠏⠻⠈⠀⠁⠀⠀⠀⠀⠀⠀⢀⣶⣠⣼⣿⣿⣷⡟⠀⠍⢀⡚⣄⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣬⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠻⠃⢻⣿⠁⠀⠀⢸⣿⣿⢡⣧⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠻⠄⠀⢼⡿⠄⢠⠀⢻⡿⡧⡿⠉⠈





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣭⢻⣛⣿⡟⡟⣟⠿⢻⡻⠏⡟⣿⢻⢸⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠐⡫⠀⡀⠀⠄⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠒⣄⠀⠈⢂⠀⠀⠀⠊⠈⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⢰⠁⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡠⠀⠀⠀⠀⠀⠁⠀⠀⢀⠄⢀⣀⡿⠋⠀⠀⢀⠂⠀⠀⠀⠀⠀⡀⠂⠁⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡄⠀⢀⠄⠁⢠⠜⠁⠀⠀⠠⠀⠁⠀⠀⠀⠀⠠⠀⠀⠈⢀⠔⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠁⠀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠸⣬⠀⢨⣴⠁⡰⠁⠀⠀⠀⠀⠀⢀⠀⠀⡠⠈⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⣠⠥⠔⢺⡀⢛⡆⢘⡟⠘⠉⠉⠉⠉⠉⠓⠒⢶⠼⠊⢀⣴⡉
⠀⠀⠀⠀⠀⠀⠠⣂⠥⠒⠋⠁⡀⠀⠠⠀⠻⣜⠿⣞⣷⠋⣠⠈⠉⡁⠒⠒⠀⠀⢀⡔⠉⠑⠫⣒⢄
⠀⠀⠀⠀⠀⢊⡞⠁⢀⠄⠂⠁⠂⠀⠀⠀⠘⣈⠻⣾⣿⡄⠠⠀⠀⠀⢈⠉⡄⠀⠔⠁⠍⠀⢄⠈⠣⠑⡀
⠀⠀⠀⠀⠈⢆⠢⣀⠁⠀⠀⠀⠠⠄⠀⡀⢂⠈⠀⠘⠿⠿⠦⠐⠂⠠⠌⡤⠀⠶⠀⠀⠀⠄⠀⠆⡠⡰⠃
⠀⠀⠀⠀⠀⠘⣷⡮⣕⡠⠀⠀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠠⠀⠈⠂⠀⠈⣁⠭⣾⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣶⣯⣝⣳⠶⢬⢥⣄⣦⣀⣀⣀⣀⣀⣀⣀⣀⣐⣂⣤⡤⢤⠤⣶⣚⣯⡵⢾⠏
⠀⠀⠀⠀⠀⠀⠠⠘⢿⠆⣉⠛⠛⠛⢶⠶⠾⢾⣥⣭⣬⣽⣼⣵⣾⣷⡾⠶⠿⠻⡟⡛⠉⣇⣸⠟⠀⠀⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠈⠻⣭⡀⠥⢀⡀⡃⠀⠀⠀⢀⡇⠠⣢⣳⠅⠶⠦⣘⠶⠸⣧⣂⣀⡴⠃⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠁⠢⣄⡀⠀⠈⠻⣟⠁⠙⣦⣼⣀⡠⠞⠉⠈⢹⣏⠀⠀⠀⢺⢱⣦⣈⡽⠋⠀⠀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠂⠈⠷⢾⣱⠠⣂⣐⣈⠈⢀⣌⣿⣀⠁⣀⣘⣮⡾⠉⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⣀⡀⠈⠉⠉⠉⠉⠉⠉⠉⠀⢉⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⠒⠒⠚⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠒⠲⠤⠤⠤⠴⠖⠒⠒⠚⠛⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢰⠚⠉⠉⢀⡄⠂⠐⠒⠒⢤⡠⠐⠂⠉⠉⠀⡈⠉⠘⠳⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣒⠉⣀⣀⣠⢚⡁⠤⠶⠁⠀⠀⠠⠤⣤⡎⠀⠠⢒⠉⡁⠉⠉⠛⠳⠄⠀⠁⡙⡦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠄⠊⡀⣄⢀⡠⠈⠁⢀⣠⠠⠄⠀⣀⣀⣀⠀⢈⡟⠁⠀⠉⠃⠀⠀⠀⠀⠐⠺⡦⠜⠀⠀⠛⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠁⠀⡴⠱⠋⠀⠠⠉⠀⣠⢚⠷⠒⠆⡂⣹⠃⠀⠀⠉⠉⣀⡠⠴⠒⠂⠀⠀⠈⠉⠁⠀⠒⠂⠄⣀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⢀⠜⢰⠁⠀⠀⠀⡧⠁⢀⠀⠀⣠⡈⠷⠀⠀⠀⠈⠑⠀⣠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠂⢄⠙⡄
⠀⠀⠀⠀⠀⢀⠎⠀⢺⠀⠀⣰⠃⢸⠀⡈⠀⠘⡠⠲⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾
⠀⠀⠀⠀⢀⡌⠀⡖⠉⡖⠈⠁⠀⠀⢂⡇⠀⠇⡇⠀⠀⠀⠰⠊⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⡼⠀⡔⢀⣼⣇⣀⡀⠀⠀⠈⢁⠀⡇⡃⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢰⠁⠠⠋⠀⠀⠀⠀⠀⠀⣠⠛⠉⠀⠋⢀⠌⠀⠀⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⢀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⢀⡐⠂⠀⠀⠀⢰⠁⠉⠁⠀⣠⠊⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠇⠈⠀⠀⠀⠀⠀⣈⠀⠀⣠⠞⠁⢀⠜⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠸⡀⠐⢄⡀⢀⡔⣃⡴⠚⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡾
⠀⠀⠀⠀⢳⢄⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⠠⠄⠤⠴⠶⠶⠶⠶⠶⠖⠚⠁
⠀⠀⠀⠀⠈⢦⡈⠀⠈⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠤⠒⠒⠒⠒⠒⠚⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠒⠶⠶⠶⠶⠒⠚⠉⠉⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⠿⠛⡿⠿⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠚⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣰⠋⠁⠀⠀⢀⣀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⢧⠀⠀⣼⠁⠀⢀⣴⡾⠟⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠻⣷⣄⠀⠀⠘⡇⠀⢿⣀⢖⡽⠋⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⢻⡳⣄⠀⡿⠀⣼⢫⢾⣤⣀⣠⣤⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠙⣎⢷⠇⣼⢱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠛⣎⢷⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠦⠼⠧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣤⠀⠀⢀⡀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠆⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡀⠀⠒⠀⢀⣀⣤⠀⠀⠀⢀⣄⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡏⠈⠁⠀⠀⢰⡏⠉⠉⠛⠲⢤⣄⣁⠀⠀⠲⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠙⢦⡀⣸⠁⠀⠀⠛⠀⠀⠻⣄⡀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠺⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠉⠛⠦⣤⣤⣤⡤⠤⠶⠾⠿⠶⢦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉⠳⢦⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠢⢤⡔⠲⣤⡤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⠀⣸⣧⠀⠀⠀⢀⡠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡀⣰⠾⣻⠏⠐⡢⠄⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠋⡴⡢⠊⣀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠊⠉⠁⠀⠀⠀⠀⠀⢀⡀⠈⢉⡓⠦⣴⣋⠉⠀⣙⣦⠾⠿⠒⣾⡫⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢱⡶⠦⠝⣷⠟⠉⠀⠀⠀⠀⠈⠉⠁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢠⠃
⠀⠀⠀⢀⡴⣒⠲⣤⣀⣀⡎⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡾⠏⣰⠋
⠀⠀⢠⡏⣼⠋⡔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡠⠞⡁⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠠⠄⣀⠀⣄⡤⠏⣠⣾⡁
⠀⠀⠘⠯⠵⠾⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠖⠋⠀⠈⠀⠀⣤⠴⠶⢤⡀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠙⠀⠘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡁⠀⠀⠀⠀⢀⠄⢠⡇⠀⠀⠀⠀⠀⠀⣠⢤⡾⣻⡯⠴⠚⢉⣩⠷
⠀⠀⠀⠀⠀⡠⠒⠒⠲⡟⠹⢷⣦⡁⠁⢠⠀⠀⠀⠀⠀⠀⠀⠘⢇⠳⢆⠀⣠⡤⠞⣠⡞⠀⠀⠀⠀⣀⣠⣥⡵⢚⣿⣿⣶⣴⣦⡼⠒⠢⢄
⠀⠀⠀⠀⠸⡗⠁⠤⡖⠚⠓⠚⠻⠿⢦⣀⠂⠀⢤⣄⠀⣠⠖⠒⠒⢟⠒⠲⠶⠖⠚⠁⠀⠀⣀⣿⣴⠳⠚⣫⣶⠟⠉⠉⠉⠉⠙⠣⣶⠀⣨⠇
⣀⠤⣔⡒⢈⡻⠦⢤⣾⠭⠛⠃⠐⠒⠖⣚⣻⡶⢤⣬⠵⠿⢤⣀⣠⠾⣦⠀⠀⠢⠴⠶⠟⠂⠚⣈⣥⣴⣾⣉⠉⠉⣥⡴⠶⢤⡤⣄⣿⠛⠙⡶⢦⡀
⠉⠓⠧⣄⣀⠀⠀⠀⠈⠉⠓⠒⠢⠤⠬⡤⠤⠒⢺⡁⠄⠀⢀⣼⡀⠀⢸⣷⠶⠶⠿⡿⠟⢛⣉⣩⠿⣏⣠⠟⠙⠾⠛⠤⠶⠞⠀⠀⠈⠒⠋⠁⣸⠇
⠀⠀⠀⠀⠈⠉⠒⠒⠤⠄⣀⣀⡀⠀⠀⠉⠓⠒⠤⠍⣛⣒⣉⣈⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉⣁⣀⣀⣀⣀⡀⠤⠤⠤⠒⠒⠀⢀⣀⣀⡤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⡄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠡⢏⡐⡠⢃⣜⡀⢄
⠀⠀⢰⣶⠤⢄⣀⠒⠀⠀⡀⠀⠀⠀⠻⢦⣴⡬⠉⠓⠎⠴⠠⠄⠀⠀⠀⠀⠄
⠀⠀⠈⢳⣝⢢⣄⡀⠓⠂⠀⠀⠀⠀⠀⠀⠉⠛⠓⠷⠾⠶⠴⠖⠒⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠙⠻⣮⣝⣒⣤⢤⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⢀⢀⡈⠱⣆⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠲⠍⠘⢃⠥⢹⣿⣭⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠄⠆⠀⠀⠀⠀⠈⠋⠳⠂⢤⠁⠤⠹⠯⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠘⢒⣿⢁⠰⣖⢙⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠠⠶⠒⠻⠿⢟⠚⢂⣶⠾⠀⠀⠀⠀⣄⠐⣒⣖⢛⣛⠛⣓⣀⣀⣀⣀⣀⣉
⠀⠀⢤⣠⣌⣀⡠⠀⠠⠐⣋⠭⠳⡇⠋⣸⠆⠈⠀⠀⠀⠀⢔⢀⡐⣛⠭⠖⣋⠁⠬⠀⠀⠉⠠⠔⣺⢿⠄
⠀⠀⠀⠈⠙⠫⡂⢥⣲⡩⠬⢒⣉⣠⡾⠃⠀⠀⠀⠀⠠⠤⣬⣨⣥⣤⣦⣿⠁⢀⠀⠈⠁⣠⡴⣪⡵⠋
⠀⠀⠀⠀⠄⢤⣍⣨⣤⣠⣦⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣐⣾⣭⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⢀⣀⡠⠖⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠒⠠⠤⣀⡀
⠀⠀⠀⡏⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣨⡦⢄⠀⠀⠈⠓⡄
⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⢹⠓⢚⡫⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣀⠔⠁
⠀⠀⠀⠀⠑⠲⠒⠙⠲⢤⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⢠⠖⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⣀⠤⠎⠉⠙⢢⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⣀⠀⠀⢧
⢸⠁⠀⠀⠀⠀⠘⠒⠲⢦⠤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠈⠣⢄⣀⠀⠀⠀⠀⢸⠀⠀⠘⣆
⠈⠲⡤⠀⠀⢀⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⣠⠞⠁⠀⠀⠀⣠⠃⠀⠀⠀⠘⢆
⠀⠀⠑⠢⠔⠊⠙⢿⡚⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⢀⠔⠁⠉⠁⠒⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠽⢦⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠙⢦⡈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠑⠤⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⠛⢄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⡶⢄⡈⠙⠒⠤⠤⣤⣠⠤⠤⠒⠋⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢀⣠⠴⠊⠀⠀⢈⡆
⠀⠀⠀⠀⢳⣄⡀⠀⠀⠀⠀⠀⠀⠑⠢⢍⡑⠲⠦⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⢤⡤⠶⠒⠚⠉⠁⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠉⠻⢷⣢⠤⣀⣀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠂⠀⠀⠀⠀⠀⠀⠐⠂⠒⠂⠉⠀⠀⠀⠀⠀⢀⣀⠤⣔⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠯⣭⣕⣒⣒⡒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣒⣒⣒⣒⣭⠭⠗⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠫⠭⠭⠭⠭⠍⢉⣉⠩⠭⠭⠭⠭⠽⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⣠⠞⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⢀⣤⠞⢉⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⢀⣠⠴⠞⣋⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣠⡴⢞⣫⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⢀⣤⣶⣿⡷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣞⣫⡥⠶⠶⣦⡀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣚⣭⠶⠛⠉⠀⠀⠀⠀⢸⡇⠀⢠⡾⠁
⠀⠀⠀⠀⠀⣀⡴⣛⡵⠞⠉⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣰⡟⠁
⠀⠀⠀⣠⢞⣭⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⢾⡟⠁
⠀⣠⢞⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢋⣴⠞⣡⠟
⣼⢫⡟⠁⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⣁⡴⠋⣡⡾⠃
⣿⠈⠷⣤⣤⣤⣤⠶⠾⠛⠉⣀⡴⠞⠉⣠⡾⠋
⠘⢷⣤⣄⣀⣠⣤⣤⠶⠞⠛⢉⣀⣴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡖⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⠴⠶⠶⠶⠦⠤⣤⣤⣤⣖⣶⠟⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⡀⣠⣾⣿⣭⣴⣶⡾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠀⢀⣴⣿⣿⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢾⣾⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠶⠆⠀⢠⣤⠀⠀⠚⠰⣠⠏⠈⠙⢷⣄⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⠋⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠘⣇⡆
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡙⠃⠀⠀⢀⠀⠀⠾⠃⠀⡀⠀⠗⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠰⡍⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣀⣀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠿⣮⡲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣺⣿⣿⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣝⡢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⣪⣵⣿⣿⣿⣿⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣯⣥⣐⣒⡀⠠⠄⠀⢀⣀⠀⠠⠤⠤⠤⣖⣚⣋⣭⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠇⠁⠉⠙⠻⡿⣿⣿⡿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣇⠁⠀⠀⠀⠀⠀⠁⠐⠁⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣿⣻⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠿⠭⣯⣽⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣴⣾⣇⢁⣴⣄⣿⣿⣿⣽⣶⣶⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⢺⡄⠀⢠⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣰⡖⠀⠀⢀⣄
⠀⠀⠀⠀⢿⡄⠀⠘⢧⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⢠⠾⡄⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⣠⠟⠁⠀⣀
⠀⠀⢀⠀⠀⠙⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡖⠠⣀⠀⠀⡰⠁⠈⠢⣀⡰⠃⠀⠹⣄⣀⠔⠋⠁⠸⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⡾⠋
⠀⠀⠻⢷⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⠀⠙⠖⠁⠀⠀⠀⠈⠀⠀⠀⣀⣈⣀⢀⠀⠀⠀⣇⡀⠔⠋⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡴
⠀⣀⡀⠀⠉⠀⠀⠀⠀⠀⠀⢸⠉⠐⢒⣤⠼⠤⠄⢆⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⠀⢸⠀⠀⠀⠈⣀⡀⠤⠤⠷⠤⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠈⠉⠀⠀⠀⢀⣀⠤⢦⡀⢸⠀⠊⠁⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⡴⠁⠀⠀⠀⢀⣀⣀⣸⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠙⣭⠃⠀⠀⠀⡤⠤⢄⡀⠀⠘
⠀⣀⠠⠔⠚⠉⠁⠀⠀⠈⣧⠺⠀⡆⠀⠀⠀⣦⠀⠀⠈⣆⠀⠀⢰⠃⠀⠀⠀⡴⠉⠀⢠⠀⠀⢺⠏⠀⠀⠀⡤⠒⠦⡀⠀⠀⠈⣆⠀⢀⣾⠁⠀⠀⠈⢳⠄
⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⢆⠁⠀⠀⠀⠿⠆⠀⠀⠘⡆⠀⣿⠀⠀⠀⠀⢷⠛⠒⠚⢠⠇⢽⠀⠀⠀⠼⠠⣄⠀⢹⠀⠀⠀⢹⡖⢩⠃⠀⠀⠀⢀⠎
⠘⣆⠀⠀⣀⠀⠀⠀⢻⣁⠀⠈⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡟⣆⠀⠀⠀⠀⠉⠉⠙⡏⠀⢻⠀⠀⠀⠇⠀⠈⣆⠜⠀⠀⠀⡾⢠⠃⠀⠀⠀⢠⠃
⠀⠘⠾⢋⡈⡆⠀⠀⠀⢇⢑⠀⠀⡇⠀⠀⠀⢰⡞⢷⡀⠀⠀⠸⣇⠈⠳⣄⠀⠀⠀⠀⠀⢧⠀⠘⡆⠀⠀⠈⠒⠚⠁⠀⠀⠀⡰⢣⡏⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠉⢘⡄⠀⠀⠸⡆⡆⠀⢠⠀⠀⠀⠀⢮⠈⣷⣀⣠⠤⠼⡄⠀⠈⠙⠒⠶⠖⢒⣏⠀⠀⠙⢦⣀⠀⠀⠀⠀⣀⣤⠾⣳⣉⣑⠒⡤⠊⠙⡲
⠀⢠⡔⠒⠉⠁⢱⠀⠀⠀⢻⡘⠄⢸⣀⡠⠤⠚⡃⡁⠱⣄⣀⣀⣼⣷⡶⢶⣮⣧⢒⣓⣾⣦⡷⣤⣀⡀⠉⠉⠉⠋⡋⠝⠁⠾⡏⠀⠀⠙⣄⡴⠊
⠀⠀⠈⠲⣄⠀⠘⣇⢀⡠⠔⠛⢱⡀⠁⣀⡀⢤⣿⡿⠻⠍⢉⣍⠑⠣⡔⢉⡬⣷⣄⣴⣛⣶⣦⠤⡛⠳⢶⡐⠂⠉⠀⠀⢸⠀⠈⠲⡶⠚⠛⠤⣄
⠀⠀⠀⠀⢀⠵⠀⠈⠁⠐⠀⠐⡠⢤⣖⣩⣵⣾⣶⡿⠶⠶⠶⠶⠶⠦⣷⣎⡐⠁⢤⣮⣙⡯⣶⠶⠻⣶⣤⣌⠽⣦⡀⠀⠀⠁⠒⠊⠀⠀⠀⠀⠀⠙⣲⠆
⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⢠⣲⣷⠟⠋⠉⠀⠠⠀⠀⠔⠰⠂⠀⠀⢀⠙⠊⠍⢻⢦⣄⠠⠄⠨⣇⢰⡵⢟⠉⡖⢈⢿⡷⣄⠀⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠈⠐⠒⠤⣀⠀⠀⠀⣠⠟⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠈⡁⠉⠳⣦⡒⢉⢠⣸⣏⣀⠙⡇⠘⣿⢬⡳⣄⠀⠀⠀⠘⢄⡀
⠀⠀⠀⠀⠀⡠⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠁⠪⠻⢦⡈⠛⣤⡙⣛⠟⢮⡉⢑⡙⢮⣧⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢠⠎⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠜⢪⠻⣮⠁⠙⣆⣠⣴⣹⠉⢱⠊⢯⠳⡄⣀⡤⠖⠊⠁
⠀⠀⠀⠃⠠⠤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢌⢳⣏⡾⠀⣉⣹⢛⡿⠻⡏⢣⢻⣅
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠢⠹⣦⠘⣭⠐⠊⡃⣠⢦⡾⡆⢻⣑⡄
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠣⠘⣧⠲⢈⠟⢋⠩⠀⠸⢿⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠱⡘⣷⡄⡀⢨⠀⠀⣀⣾⢸⡇
⠀⠛⠀⠀⠈⠛⠶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢌⡻⣤⣠⣴⣮⡿⢣⣾⠃
⠀⠴⠂⠀⠀⠀⠀⠀⠘⠛⠉⠉⠛⠛⠛⠓⠒⠲⠶⠶⠶⠶⠦⠤⠤⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⣒⣺⣉⣥⡶⠟⠁⠀⠀⠀⠀⠀⠙
⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡠⠔⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢹⠋⠉⠙⠛⠻⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠱⣤⡀
⠀⠘⠁⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠘⠂⠁⠀⠀⠸⡄⠀⠀⣠⠖⠄⠀⠀⠀⣠⠆⡀⠀⠀⠀⡜⠑⢦⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠋
⠀⠀⠠⡾⠋⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡴⠋⠀⠀⠈⢆⢀⠔⠁⠀⠙⢦⡀⢰⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢻⣄
⠀⠀⠀⠀⠀⣰⠟⠁⠀⢠⡾⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠰⡆⠀⢸⡄⠀⠘⣧⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢠⣤⣖⡒⠐⠂⠰⠶⠰⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠄⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠒⡒⠉⠉⠉⠀⠒⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠀⠤⣸⡀⠀⠀⠀⠀⣀⠤⠖⠙⠁⠀⠈⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠨⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠖⠚⠛⢧⡀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠒⠀⠠⢀⡀⠰⠀⠤⠎⢀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡐⠄⡀⠀⠀⠄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣧⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠇⠀⠈⠢⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠳⣄⠀⠀⠀⠀⠈⠑⠦⣤⣀⠀⠀⠀⠀⠰⢁⠀⠠⢀⢀⠁⠁⠁⠄⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣧⠀⢈⠳⣄⡀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠳⡄⠀⠀⠀⠀⠀⢀⠃⠀⢹⣧⠀⠀⡀⡉⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⣀⡀⠀⠀⠀⠂⢂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢸⠀⠀⠀⢯⢳⡀⠀⠄⠁⠀⠈⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⡟⠀⠀⠀⠀⠀⢸⠄⠀⠀⠈⢣⡙⣄⠃⠀⢀⠀⠀⠀⠀⡍⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠙⢎⢻⡪⡄⠢⠀⠀⢀⠇⠀⢁⡩⣑⡶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡙⢮⡊⠀⠀⠉⠀⡐⠁⠀⠀⠀⠀⠙⢍⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢳⡀⠉⠲⢤⣀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠃⠀⢉⠷⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠄⠀⠀⠀⠀⠀⡹⣄⠀⠀⠈⠑⢦⣑⠠⣄⡀⠈⠀⣠⠏⡰⠁⠀⠀⠀⠀⠀⠉⢛⠲⠦⢤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠓⠲⣄⠀⠀⠀⠀⠈⢧⠀⠀⠀⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠓⠦⣅⣉⣉⡀⠀⠘⢄⠀⠀⠀⠱⠘⢄⡾⠈⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠐⠀⢸⠆⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠦⣥⣀⣠⣴⣴⣫⡵⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠊⣠⠟⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠤⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠂⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠀⠀⠀⠀⡀⠠⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⠀⠀⠀⣈⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⣷⣶⣶⣶⣶⡷⢿⣾⣷⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢷⣾⡷⠾⠿⣷⣶⣷⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⡌⢫⢁⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⢷⡴⡾⠁⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⢾⡜⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡁⢳⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⣼⡧⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⣛⣟⣛⣛⣛⢓⣚⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⡞⣿⠻⢗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣡⢦⣄⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠏⢯⣭⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⠿⠛⢉⣇⣴⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⢀⣀⣾⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⣉⣠⣤⣤⠶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⣥⣶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣡⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠉⠉⠻⣿⠉⢠⡟⠁
⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠃
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⡿⠋
⠹⢴⣤⣤⣤⣤⣶⣶⣞⣻⠭⠭⠚⠋⠁

⠀⠀⠀⠀⠀⢀⣤⡶⣶⣦⡀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠂⢀⢀⡀⠲⢤⣄⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⢀⣤⣞⡿⠷⠝⣻⡧⣸⣷⣷⠦⢤⣄⠀⠀⠀⠀⠀⣴⣿⠿⠿⢳⡌⠛⢏⡨⢧⠈⣿⢿⣷⡄⠀⢻⡟⠿⣿⣶⣄⠀⢾⣷⡌⠉⠉⠋
⠀⠀⠛⠿⣿⣶⣦⣞⣵⠟⣹⢻⡿⡝⣿⣖⡶⢙⢦⡀⠀⣾⣷⣿⡿⢿⢨⠴⠘⠦⣤⡼⠁⣿⣻⠟⠀⠀⠈⠻⣿⣮⣻⣿⣆⠀⠘⠟
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢻⡜⣿⣖⠉⠳⡱⡄⠀⠙⠻⠿⣿⣧⡍⠢⣁⣈⣡⣾⡿⠋⠀⠀⠀⠀⠀⠙⣿⣿⣹⣿⠀⣀⣀⠀⠀⠀⢸⠃
⠀⠀⠘⢿⣷⠀⢹⣿⡆⠀⣿⢸⣷⠸⣷⠸⣿⣶⡤⠹⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⣠⣤⣤⣤⣄⣀⠀⠈⠛⠛⠁⠀⠻⣽⣿⣦⣠⡏
⠀⠀⠀⠀⠀⣶⡆⠉⠁⠀⢿⢸⣿⣇⢿⣧⣿⣇⠚⢤⠇⣷⠀⢠⣶⠀⠀⠀⠛⠳⠶⢞⠍⠭⡭⠭⢙⡛⠿⣿⣷⣤⡀⠀⠀⠀⠀⠴⠛⠉⢹⠁
⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠸⣇⢿⡻⣿⣿⣿⣿⠃⠶⠂⣿⠀⠿⠃⠀⠀⣠⣿⣄⠀⣯⢀⠊⠄⠠⠀⡀⠋⢰⢝⡻⣿⣆⠀⠀⠀⠀⣠⢖⡻⣆
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⣜⣿⣿⠋⠉⠉⠉⠛⠀⠀⠀⢀⣾⣿⢻⣿⣷⡈⠪⡳⣄⠀⠉⣷⠴⣀⠈⡐⠙⣿⠀⠀⢀⡾⣵⣿⣧⢻⡄
⠀⠹⣆⠀⠀⠀⣀⣀⣤⣤⡤⠤⢤⣄⣈⠉⠛⠀⠰⣶⠶⠀⠀⠀⢀⣿⢹⡏⣸⣽⣿⣇⠀⠘⠺⡿⠶⢦⣀⣇⣯⣴⣶⡼⠇⣠⢟⣾⣷⡟⢾⣿⡧
⠀⠀⢻⡟⠛⢻⣿⡟⣩⢲⢒⡖⡶⡦⣌⣙⠶⣄⠀⠀⠀⠀⠀⠀⢸⡇⣼⠇⣿⡇⣿⣿⠀⠀⠀⠀⠈⠒⠳⠿⠒⠛⠉⠀⢠⣟⣾⣝⣧⣽⣾⣿⠇
⠀⠀⠘⠇⠀⠀⢻⡄⣿⠻⢺⠟⢟⢷⣵⣽⣳⣌⠳⣄⠀⠀⠀⠀⣿⡇⣿⢀⣿⡇⣿⡿⠀⠀⣴⢲⣦⠀⣿⣶⣤⣄⠀⠀⢸⣿⣻⠿⣙⣼⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⣾⡿⡾⣾⣬⣌⣙⣳⣷⡙⣦⠀⠀⠀⢻⡇⣿⢸⣿⣸⣿⠇⠀⢸⣷⡘⣿⠀⢻⣯⣿⠻⣧⠀⣿⣗⣿⢿⡻⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣽⣳⣷⠿⠷⠿⠿⠾⠿⢈⣧⠀⠀⠈⢿⣿⣸⣿⣿⡏⠀⠀⠈⢿⣧⣙⣧⠀⠛⠿⠷⠿⠀⢻⣙⣚⡯⠞⢁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠛⠛⠉⠁⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠋⠙⠻⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⢀⣠⣤⣤⣄⡀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⢰⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠴⢡⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢫⢃⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡵⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⡤⠤⠀⠂⠈⠛⠛⠉⠉⠉⠉⠉⠹⠃⡜⠓⠛⠛⠈⠉⠉⠉⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⢠⣞⠑⠈⠀⠀⠀⠀⠀⣀⡀⠤⠄⠀⠀⠀⠠⠁⡸⠠⠄⠤⠤⠤⠄⠀⠀⠀⠀⠀⢸⣡
⠀⠀⠀⠀⠀⠻⣻⣀⢀⡠⠒⠂⠀⠤⠀⠠⠤⠀⠀⠒⠂⠈⠈⢁⣀⠀⠒⠂⠀⠐⠒⠂⠙⠶⠒⠉⡇
⠀⠀⠀⠀⠀⠀⢩⠉⠛⠋⠀⠀⠀⠤⠤⠠⠄⠀⠀⠀⠀⠀⠠⠌⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⢀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢳⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠋⠉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⡀⣀⠀⠀⣀⡀⠀⠀⣀⣀⡀⢠⣶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠛⠓⠒⠒⠒⠒⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⣀⣀⣀⡀⣀⣀⣠⣄⣲⠟⠛⠛⢿⡟⠉⢈⣹⠆⠀⠀⠀⢀⣤⣤⣴⠖⡄
⠀⠀⠀⢸⠿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢾⣿⡋⠉⠉⢹⡟⠉⢀⢻⣧⠴⠶⠶⠛⠉⠈⠉⠀⠀⠀⠀⠀⠈⠓⠶⠖⠂
⠀⠀⠀⣯⠀⢘⡇⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠈⠛⠛⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⢠⠀⢢⠀⠀⠀⠀⠀⠀⡄⠀⠀⠦⡀
⠀⠀⢀⡯⠀⢸⡇⠀⠀⣤⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠋⣠⠎⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢘⣆⡀
⠀⠀⠈⡧⠀⠠⡇⠀⢀⡟⠈⢷⠀⠀⢠⡞⢋⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⣼⠛⢷
⠀⠀⢸⡇⠀⠀⡁⠀⠀⣳⠤⢸⡄⠀⠘⡇⢸⠅⠀⠀⠀⠀⠀⠀⡄⠠⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢹⠀⠀⠀⣏⢀⣠⠃
⠀⠀⢘⡿⠀⠶⡇⠀⠀⢁⡀⢀⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⢤⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣆⠀⠀⢻⡄⡌⠀⠀⠀⡏⠀⠉⠃
⠀⠀⢸⡇⠀⢀⠀⠀⠀⢸⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢿⡄⠀⠆⠀⢠⠀⠀⠀⢻⠂⢲⡄
⠀⠀⠘⡧⠀⢺⠀⠀⠀⠈⣿⠂⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⢐⡇⢻⠀⠀⠀⢸⣀⣼⠃
⠀⠀⠘⡇⠀⠘⡀⠀⠀⠀⢹⣄⢸⡇⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⣤⣤⡤⠤⠠⣄⡀⠀⠀⠀⠀⠰⡇⠈⠀⠀⠀⣾⠀⢘⡇
⠀⠀⠀⠐⠂⠀⢃⠀⠀⠀⠈⣗⠀⢷⠀⠀⠀⠉⠒⢿⣿⣉⣉⠉⢹⡟⠋⣉⡛⣿⡀⠤⠤⠴⣟⣃⣤⠶⠞⠃⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣽⠂⣿
⠀⠀⠀⠀⠑⣤⢲⠀⠀⠀⠀⢹⠁⠹⡄⠀⣀⣀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠓⠒⠂⠀⠒⠈⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡇⡰⠀⠀⠀⣿⢀⣾
⠀⠀⠀⠀⠀⠈⢾⠃⠀⠀⠀⠘⡀⠀⡅⠀⠹⣏⠉⠙⠶⣦⣄⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⢀⡄⠀⠀⠸⠞⠃⠀⠀⠀⣿⣠⡏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠡⡀⢸⠀⠀⠙⢿⣷⣄⣼⡅⣀⡈⠙⡿⠒⠒⠲⢶⣦⣤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣾
⠀⠀⠀⠀⡤⢤⡤⠀⠀⠀⠀⠀⠀⠙⣮⡆⠀⠀⠀⠈⠛⠿⠿⠿⣿⣾⣷⣤⣦⡤⢴⣧⣀⣀⣈⠙⣷⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣴⣋⠞⠁
⠀⠀⠀⠀⠓⠤⠤⠚⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠛⠛⠛⠛⠛⠉⠁⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣋⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣯⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡏⢢⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠢⢀⠈⠻⣾⡔⢹⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⣤⣇⢱⠢⢸⣻⡏⢭⣑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠩⠯⢻⣷⡶⣸⠀⠑⡢⡶⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠯⠐⠉⢺⠀⢰⣰⡷⣛⢒⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⡙⠛⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡤⠤⢄⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⣤⣤⣤⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⣀⣤⠖⡚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠉⠑⠂⠀⠠⠤⠤⣀⣴⡖⠊⠋⠉⢀⠤⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⠤⠼⠃⢀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡀⠀⠀⠤⠄⠀⠠⠤⠠⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠐
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⢣⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⣰⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⣰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢰⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢣⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠘⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⣸⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⣾⣿⣿⣿⡏⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠿⠛⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⢿⣿⡿⠋⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⢋⣡⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣫⣭⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠟⣫⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢁⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⢰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢋⣼⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡌⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⣱⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⢀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠻⢿⡿⣸⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣫⣭⣴⣶⣶⣶⣤⣉⠛⠿⠿⠿⠿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣝⡛⠿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣌⠙⢿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡈⢿⣿⣿⣿⣿⣄
⠀⣠⣾⣿⣿⣿⣿⣿⠿⠟⠛⣛⣛⣛⣇⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⣿
⣰⣿⣿⣿⣿⣿⠟⣡⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣧⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣌⡛⢿⣿⣿
⣿⠿⣛⣉⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣎⢻⡟
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣸⣿⣿⣿⡿⠛⣉⣤⣶⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠟⠃
⠀⠈⠙⠻⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⠋⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣾⣿⣿⣿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣴⢀⡤⣀⠀⠀⡀⠀⢠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠲⢶⣦⣤⠞⠁⠀⠀⠀⠀⡀⠀⠄⠀⣡⠔⠉⠁⠀⠀⠈⢀⢠⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣊⠅⠄⡀⢀⣿⡇⠀⠀⢀⣤⡖⡁⢀⠀⠄⢸⡍⠀⠀⠀⠀⢀⡠⣣⠎⠀⠀⠀⣀⢀⡀⣀
⠀⠀⠀⠀⠀⢠⢃⠊⠀⠀⠈⢳⢸⡇⠀⢀⢐⠑⠋⠀⠀⠀⣶⡄⠚⠻⢋⣤⠶⠞⣩⠞⠁⠀⣠⠖⠉⠀⠀⠈⠂
⠀⠀⠀⠀⡰⢡⡏⠀⠀⠀⠀⣿⡟⠀⢠⣆⠁⠀⠀⠀⡀⢀⣸⠃⣀⣀⡿⠶⠟⠋⠁⠀⠀⢸⡇⡀⠀⠀⠀⢠⠄
⠀⠀⢀⢜⠁⠘⡂⢀⣄⡦⢻⠟⠀⠀⠃⢺⢦⣄⣈⡶⣖⣫⡥⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠙⣫⠶⠞⠋⠁
⠀⢀⠀⡀⣤⣀⣀⢍⣹⡾⠋⠀⠀⠀⣪⣠⣬⣭⠧⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠐⢢
⠀⣈⡜⠁⠀⠘⠛⠊⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠊⠁⠀⠈⠉⠁⢘⣷⣄⣀⣀⡀⠀⠠⠂⠀⠀⠀⠀⢚
⢰⣿⠁⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⢀⡠⠀⠘⠁⠂⠀⠀⠀⠀⠀⢀⣠⠾⠋⠉⠉⠉⠁⢰⡆⢠⢀⢄⣐⡠⠖
⢘⠹⣢⢀⢀⡰⢿⡿⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠐⠊⠀⠒⢰⡤⠄⠋⠀⠀⠀⠀⠀⠀⠀⠈⠫⠿⠖⠛⠁
⠸⡆⠀⡈⠉⣆⣼⠃⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⣣⠖⠈⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄
⡌⠐⠁⠀⠋⣯⣿⡄⠀⠠⠋⠀⠀⠀⣀⠀⢀⠼⠛⠁⠀⠀⠀⣠⡶⠛⠁⠀⠀⢉⢭⢻⡶⠒⠛⠉⠀⡴⠋
⠇⠁⠀⠀⢀⣼⣸⡇⢰⠀⠀⠀⠀⠀⡵⠋⠀⠀⠀⢀⠠⠄⠬⠈⡀⠀⠀⠀⣠⣸⣾⣋⡤⠒⠐⠂⢉⠈⢆
⠀⠣⢀⢀⣜⣿⢽⡇⣼⠄⣀⣀⠴⠊⠀⠀⠀⢴⠾⠛⠉⠈⠁⢠⣝⣯⣿⣿⣿⠿⠯⠁⠒⠀⠁⠣⠉⢄⢈⠣
⠀⣤⠀⠻⢙⠉⡿⠀⠗⠚⠉⠀⠀⠀⢀⣤⡂⣆⠀⠀⠀⠀⣤⡾⣻⠿⠚⠉⢀⠀⠀⢀⠀⠀⢀⡄⠀⠈⡊⠐⡆
⠀⠈⠻⣷⣈⣼⠃⠀⠀⠀⠀⠀⣠⠚⠁⠀⠰⣙⠓⡴⢾⣿⣾⠟⠁⠀⠀⠀⠈⠆⠄⠀⠀⡠⠊⠁⠆⠀⠀⢺⡇
⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⢠⣞⠉⠀⠀⠀⠀⣾⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⢰⡀⠀⡰⠗⡃⠀⠀⠀⠀⠄⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣠⢰⣰⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⡜⢁⠀⡠⢔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠙⣻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢾⠁⢈⡐⠑⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣏⣊⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⣄⢀⡀⣀⠀⣀⣀⢀⡀⣠⡄⢀⣀⢠⢀⡀⡄⣀⢀
⠀⠸⠻⠸⠟⠾⠯⠿⠺⠻⠟⠿⠇⢸⡿⠸⢻⡇⠗⠿⠹
⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⡀⠀⣀⣰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠋⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢠⣶⠀⠀⠀⠀⠀⢼⣧⠒
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⣤⣠⣦⣶⣷⣴⣿⡿⠋⠀⠀⠀⣤⠶⠛⠡⣨⣖⣀
⠀⠀⠀⠀⠀⠀⣿⢁⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⢰⣿⡟⡋⠻⣿⡿⢋⠀⣀⣀⣠⠾⠁⡀⢀⣤⣾⢋⡹
⠀⠀⠀⠀⡀⢲⡷⣦⡏⠀⢶⣦⠀⢀⣀⣀⡤⠾⣿⠋⠀⠀⠀⢀⣠⣴⣿⣷⡀⢠⣿⣧⠀⠀⣾⣿⡏⢰⣿⣷⣟⠋⠃⠊
⠀⠀⠀⡁⠀⠀⣶⣿⣷⣤⣚⣡⣶⣿⣿⣿⠄⠀⣸⣧⣤⣤⣼⣿⡿⢿⢿⣦⠉⢩⡋⡙⠒⠚⠙⠷⡑⠚⢉⡣⠜⠅⠂
⠀⠀⢠⢷⡾⠟⠛⠛⠻⢻⠛⢻⣟⡛⡻⠿⡷⠒⠻⣏⢙⢈⡿⣌⠑⢽⣈⠫⠃⢀⣤⣶⠦⠠⣼⡧⠱⠀⢄
⠀⠀⠀⠀⠳⠀⢀⣴⠀⢨⣄⡾⠛⠛⠡⢄⣀⠰⠶⠛⠛⠉⠉⠓⠺⠿⠸⠄⠋⣉⡡⠤⠒⠈⠁
⠀⠀⠀⠀⠈⠓⠚⠉⠒⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠒⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢔⠀⠠⠒⠄⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠘⡆⣀⣴⡐⡒⠀⠤⡍⣐⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⣿⣿⣿⣿⣷⡀⠸⣷⣿⣿⣽⣷⣇⠠⠀⣠⣀⠁⡈⠀⢠⣀⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡉⢿⣿⣙⣿⡿⣿⣿⣧⣄⢻⣿⣿⠋⠹⠿⣦⡏⠽⠯⠜⣿⡧⠸⣿⠛⢛⠓⠢
⠀⠀⠀⠀⠀⡾⢿⡿⡿⢿⣿⣶⣍⠻⣷⣦⣿⣿⣿⣿⠆⡁⢐⣦⠏⠀⡆⠀⠠⠄⠀⢠⣴⣦⣉⢳⣼⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠁⡈⠁⣌⠉⠙⢻⣿⣿⣦⣍⡙⠻⠿⠟⢰⣇⠢⡉⢤⣤⣀⠲⣦⡙⣿⣦⠝⣻⣿⣿⡿⠃⠀⡰⠁
⠀⠀⠀⠀⠀⠤⠙⣦⡀⠁⠠⢀⡀⠙⠻⢿⣿⣶⣦⣤⡀⠙⠓⠉⠂⠙⣿⣋⣙⣫⣶⣿⣾⣿⠟⠋⠀⡠⠊
⠀⠀⠨⠁⡤⠚⠛⠛⠛⠷⣦⣤⣀⣉⡀⠒⠹⠿⠿⠿⢿⣦⣄⡀⣿⡿⠿⠿⠿⠛⠛⣋⣁⣠⣤⠶⠿⢶⣦⣤
⠀⠀⠀⠈⠉⠿⠮⠷⠿⠾⠛⠻⠛⠿⢿⡿⠿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⡿⠿⠛⠻⣿⣛⣋⣥⣦⣾⣟⠿⠃
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢄⢠⡄⢠⣤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠁⠀⣶⡾⢠⠀⠀⠀⠶⣆⣱⠿⠈⢛⢱⡄⡠⠀⠾⠋⡋⢁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⠄⠀⣠⡙⠋⠀⢀⢉⠑⠋⢁⠬⠂⠂⠄⠀⠀⠉⠙⠲⢨⣿⡁⣾⡘⠛⡀⣾⡥⠀⠀⡐⢠⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣭⡟⢿⠈⣿⣧⠢⣬⡡⠀⠀⠃⠤⠀⠀⠀⠀⠋⠙⣉⠁⣀⡨⡏⢠⣤⡄⢘⠀⠈⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠫⢬⡄⣥⡴⢇⠨⣤⣶⣄⠻⠄⠛⠆⢀⠀⠀⠀⠈⠑⠅⢁⠇⠟⣡⢉⢀⢄⣀⣀⣀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠡⣀⠀⣘⠀⡟⠁⣎⢥⣤⣤⡼⢟⡆⠀⠀⣄⠀⠀⠙⠣⣄⡘⣒⡀⠯⣨⣿⣧⣚⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⣍⠏⢻⣅⣇⠻⢁⠋⠉⠓⠈⠡⣀⠀⢹⣇⠀⠀⠀⠉⠻⡏⡿⠋⠁⡉⣌⢃⢭⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡟⣯⠉⣦⣔⢻⣶⡠⡔⢂⡿⡔⠘⣿⣆⠀⠀⠀⠀⠙⣿⣄⠀⢀⡍⢁⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢭⣘⣏⠂⠀⠠⠠⣿⠫⣾⣷⡀⠹⣿⣔⠀⠀⠀⠀⠀⠹⠛⢿⣴⡿⠷⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠠⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⢿⠿⢷⣄⣐⡠⠄⣤⠐⠻⣷⠄⣿⣿⣶⡄⠀⠀⠀⠈⠢⣼⣯⠀⢦⣤⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡞⠉⢿⣷⡶⣷⠾⣿⣿⡀⢸⣿⣿⣿⣁⠀⢀⠀⠀⠙⠙⣾⣷⡶⠆⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣤⣤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣼⣯⡁⢠⣄⣈⠻⣇⠀⠀⠉⠛⠿⢿⣶⣦⣤⣀⣳⡄⣿⣵⣔⣗⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠏⠛⠻⠿⣶⣤⣄⡀⠀⠀⠀⠈⠨⢽⣧⣶⣯⣅⠀⢻⡄⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠮⣏⠉⠈
⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣦⣤⣽⣄⢹⣿⣻⡀⢸⡛
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣶⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣝⡿⠛⠇⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢤⣀⣤⠴⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠈⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢷⣦⣀⠁⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠓⠦⠀⣀⠀⠀⠀⠀⠀⢘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠁⠀⠀⠀⠀⠀⠐⣠⡟⠛⠲⣄⠀⠀⠀⠹⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣧⣄⠙⢀⣢⣀⡀⠀⠀⠬⣀⠙⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢻⢛⠯⣻⣿⢧⣾⣿⡿⣿⣿⣶⣤⣄⠙⠓⠶⣌⡑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠋⠀⡞⠈⢰⣿⠀⣾⠿⣿⠟⠁⠁⠙⠟⢯⡧⠀⠀⠀⠉⠲⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢈⠀⠀⡿⠀⢏⠀⠀⠀⠀⠀⢂⠁⠀⠀⠀⠀⠀⠈⠳⡠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⣀⣀⣀⣠⠏⠈⠀⠀⠃⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠘⠋⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⡈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⣿⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⢀⢱⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠈⢀⠣⠐⡄⠠⠀⠀⠀⠀⠀⠀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠃⠀⠀⡈⠀⠀⠂⠀⡄⠠⠋⡧⠘⣆⠐⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢠⠀⠀⠀⠁⠀⠀⠀⠸⠀⠠⣉⡃⠐⣜⢧⡀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡠⣿⠀⠀⣼⠀⠀⣀⣰⠗⠀⠐⣉⡁⠀⠇⡻⢿⣄⠠⡀⡀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⢠⠇⢠⣶⣟⠙⡃⠀⢈⣥⡄⢰⠀⡇⠈⡟⠷⣬⣮⡳⣶⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⣿⣶⡿⡇⠹⣧⠹⡀⠀⢴⡄⠘⠀⠀⢠⠁⠀⠀⠉⠙⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠋⠀⠘⢆⠙⣿⣤⠀⢸⠁⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⠿⣧⣼⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠙⠿⣸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣸⣾⣧⣴
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢇⣽⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣦⣤⡀⠀⢿⣿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⢸⠀⠰⠟⠛⠛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⢀⣤⣤⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣼⣿⡿⠃⠀⠀⠀⢠⣿⣿⣿⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⠙⢻⠋⠀⠀⠀⠀⠀⢸⣿⠟⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠙⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⢠⣿⡄⠀⠀⠀⢀⣤⣄⡄⠀⠀⠀⠀⢀⣤⣴⠂⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠘⠛⠁⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣶⡶⣶⢎⡠⢌⣳⣍⠛⠻⣭⣉⣿⡶⠂⠀⣠⠴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⢤⣬⣭⣿⣶⣬⣭⣍⣙⣻⠧⠤⠶⠶⣮⠿⠟⣛⠩⠉⠈⠛⢷⡇⢸⡷⠳⣄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢠⣤⣀⡀⢀⣀⣾⠵⢚⣩⠵⠚⣹⡶⢞⣛⣭⣤⣤⣤⡴⠚⣩⣴⡞⢉⣉⡛⢿⣍⣁⣨⡇⣾⠞⣩⣟⣷⣶⣶⣾⣿⣃⣀⣀⣠⣤
⠀⠀⠀⠀⠧⢤⣭⣉⣙⣳⣾⠿⠶⠯⠽⠥⠾⢭⣁⣊⡥⠚⢁⣴⠿⠽⢿⡶⠋⢹⡙⢿⣿⠟⢰⢳⣧⣾⣿⣿⡿⠿⠿⣒⣒⡒⢂⡴⠋
⠀⠀⠀⠀⠀⣰⠟⠉⠁⠀⠀⠈⠉⠉⠉⠁⠒⠂⠒⠬⠭⣝⠻⠿⣿⣶⠋⠀⠀⠀⢯⠀⢈⡇⡾⢸⢷⣿⠁⣶⠶⠚⣩⠽⢻⣷⡾⠤⠤⠤⠤⣤⣀
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢍⡛⠦⣄⡀⠈⣧⢸⣧⢇⣼⣞⡁⢸⣧⣖⣋⣤⣤⣾⣷⣿⣞⢶⠲⠦⠴⠋
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⣙⠷⣿⣻⠃⠀⠀⠉⠉⢙⣛⠶⣮⣿⡟⠛⠛⠛⠛⠛⠓⠒⠒⠶⡄
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⠛⢶⡒⠒⠒⠉⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⡶⠶⠇
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡙⢦⣀⢄⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⡄
⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣇
⠀⠘⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿
⠀⠀⠀⠈⠉⠙⠛⠲⠶⢦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠶⠶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡹⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⠶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠶⢶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠈⢌⠻⣿⣿⣿⣿⢟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠷⠶⣦⣤⣀⣀⠂⠈⠛⠟⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠑⢤⡖⠰⡤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⣨⡧⠀⠀⠠⠔⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⢐⠴⢋⡽⢃⡬⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠒⠒⠀⠀⠀⠒⠒⠢⠤⣀⣠⠴⠒⠺⣍⣤⣷⣥⡴⠋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠣⣤⣍⠳⡶⠞⠋⠁⠀⠈⠙⠊⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣨⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⠀⠚⢸⠁
⠀⠀⠀⠀⢀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢠⠃
⠀⠀⢠⠾⡧⢈⠓⠒⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⠁⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠐⡾⢋⡴⠃
⠀⠀⢿⣘⣡⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣡⠔⠋⠀⠀⢀⣀⣄⡀⠀⠀⠹⡄⠀⠀⠈⠐⠂⠳⠋⢠⣾⣷
⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣴⠋⠁⠀⠈⠀⢠⠀⡟⠀⠀⠀⠀⠀⢀⣀⡼⣿⣇⠤⠒⣒⡆
⠀⠀⠀⠀⣠⠐⠒⢴⡋⠿⣦⡃⠁⠠⠀⠀⠀⠀⠀⠀⠻⡘⠆⠀⣤⠤⢃⡴⠃⠀⠀⢀⣀⣤⡾⢞⣿⣿⣤⣦⡼⠗⠠⣄
⠀⠀⠀⠀⣇⠁⠴⢒⡛⡚⠛⠻⠦⣄⡐⠲⢤⡀⣠⠚⠁⠻⡙⠒⠒⠚⠉⢀⠀⢠⢷⡞⠓⣉⣾⣟⠁⠈⠉⠉⠑⣶⣀⣼
⢴⣲⠌⠁⠝⠓⠲⢿⣊⣉⠀⠉⠩⠭⠿⣓⡶⠚⠛⠲⡤⠚⢳⣀⣀⣐⣋⣉⣥⣤⢴⡶⡿⢧⣄⣨⣶⠚⢓⡖⠲⢿⣉⡹⠛⣆
⠀⠈⠑⠒⠤⢀⣀⠀⠀⠈⠉⠉⠙⠭⣉⡉⠢⣥⡤⠜⠧⢤⠴⠿⠭⠭⠽⠭⠥⠶⠚⠙⠒⠋⠀⠉⢉⣉⡉⠀⠀⠀⠉⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠢⠤⠤⠬⠭⠭⣖⣒⣶⣶⣶⣖⣀⣐⣐⣒⣒⣒⣒⣀⣉⣉⣉⡭⠥⠤⠴⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣈⠂⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣬⠅⠈⠉⠉⠉⠚⢶⢄⡀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣿⡀⢀⣤⣴⣦⡶⣽⠛⣿⣽⣿⠁⢹⡆⠀⡀⣀⣄⡠⡄⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢽⡄⡲⣧⣯⢙⢼⣃⢯⠯⣐⡭⡵⠳⡎⢭⢱⡔⣒⣿⡟⡺⣱⠄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣠⣼⡯⡄⠄⡼⢛⢿⢯⢅⢽⠷⢻⠣⠛⣷⢽⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢼⡋⠀⠙⠯⣤⡐⡈⡍⢎⢌⣰⢎⢡⡐⠿⠙⠚⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠤⠄⠀⠤⢾⣄⠀⠀⠀⠀⠉⠒⠒⠖⠳⠕⠊⠈⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡌⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠑⠂⠠⠀⠀⠤⠀⠒⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡜⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠊⠘⠆⠀⠀⠀⠁⠀⣀⣠⢾⣬⢗⣾⡻⣆⡀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⡝⣹⣿⡿⡿⣮⣞⣯⡝⢟⣿⣤⢭⣖⡷⢸⡃⡆⢒⣶⠲⢲⣀⡖⢤⣒⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣴⡯⠕⠣⢼⡿⣿⠕⠐⣿⠏⡏⣝⡼⢉⡋⣷⡜⣂⠟⠿⠋⡬⣉⡇⡛⢿⠇⢁⣟⠛⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠣⠄⠊⠻⠗⣻⡻⡿⡟⣾⠾⢳⡝⡠⠥⣾⢓⠟⡻⡼⢼⣿⠝⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠀⠘⠋⢓⡀⠀⣩⠙⠰⠿⠏⠈⠀⠻⠢⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠉

⠀⠀⠀⠀⠀⣄⠀⢠⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄⠀⢀⡀
⠀⠀⠀⠲⡀⠘⠆⠀⠁⠀⠀⠈⠀⠀⠀⠀⢠⢄⠀⠀⡠⢢⠀⠀⢀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠⠞⠀⣠
⠀⠠⣄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀⠁⠀⠑⠂⠀⠀⠳⠀⠁⠀⢱⠀⡠⠔⡆⠀⠀⠀⠀⠀⠀⠀⠚⠁⣀
⠀⣀⠈⠁⠀⠀⠀⠀⢰⠒⢠⡤⠤⠤⡀⠀⠀⠀⢀⡠⠒⠀⠀⠀⡆⠀⠈⣁⠠⠤⠧⢄⡀⣀⠀⠀⠀⠀⠀⠈⢁
⠀⠀⠀⡀⠤⠔⠒⣆⡀⠈⠀⠀⠀⠀⠐⡀⠀⢀⠎⠀⠀⣠⠄⢠⠃⣠⠞⠁⠀⣀⠀⠀⠙⢇⠀⠀⡴⠐⠠⢀⢈
⢔⠊⠁⠀⠀⠀⠀⢸⠈⠀⠀⠀⢹⠄⠀⠱⡀⢸⠀⠀⠀⡿⠀⠎⡰⡟⠀⠀⡼⢁⠙⡆⠀⠈⡤⡺⠁⠀⠀⡘
⠈⡄⣀⣄⠀⠀⢫⡀⠘⢷⠀⠀⠀⣀⠀⠀⢣⡜⢆⠀⠀⠈⠀⢹⠁⢣⠀⠀⠀⢈⡦⠁⠀⢰⣳⠃⠀⠀⠜
⠀⠈⠑⠒⠆⠀⠈⡖⡀⢸⠀⠀⠘⡏⢧⠀⢀⣳⠈⠓⠤⣄⣀⣬⠆⠘⢄⠀⠀⠀⠀⢀⣠⡯⠧⢀⡠⠮⡀
⠀⢤⠒⠂⠙⡀⠀⠱⡣⠈⣄⠠⠔⢙⠘⣏⣠⣼⣷⣶⣶⡤⣤⣾⣴⢤⣀⠉⠒⠒⠺⠋⣹⡋⠉⢒⡠⠚
⠀⠀⠀⠢⡀⠣⠔⡚⢉⠆⢀⠤⣰⣻⣟⣃⣵⣉⣱⡉⢺⠷⡶⣽⣿⣲⣝⡛⠬⡁⠀⠘⠅⣈⠛⠉⠑⠢⢄⡀
⠀⢀⡠⠊⠀⠀⠀⢀⣤⡴⠟⠛⠋⠉⠉⠩⠉⠉⠙⠻⠳⢮⣁⠊⠉⣏⣸⡟⢻⠙⣶⣄⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠉⠒⠠⠄⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠳⣖⠩⣤⡧⣌⣃⣸⠷⣷⣄⠀⠀⠣⣄
⠀⠀⠀⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠳⣤⠓⢬⠋⡱⡄⢊⢻⣧⠀⢀⡨⠗
⠀⠀⠚⠠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠳⣺⠉⣉⢟⡿⠺⢣⢿⡁
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡹⣎⡓⢬⣥⠖⢺⡏⣯⠂
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⣦⠀⠐⠂⢀⡿⣿
⠀⠁⠀⠈⠓⠦⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣳⣤⡶⢟⣱⠇⠀⠀⠀⣀
⠀⠉⡀⠀⠀⠀⠀⠀⠀⢨⠀⠉⡉⠍⠉⠉⠉⠉⠛⠛⠓⠒⠒⠒⠒⠒⠒⢲⠶⠶⠶⣶⠚⠛⠋⠁⠀⠀⠠⣀⠈
⠀⠘⠁⣠⠄⠀⠀⠀⠀⠀⠂⠉⠀⢸⠀⢀⠴⠀⠀⠀⡠⢀⠀⠀⠰⠓⢄⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠃
⠀⠀⠚⠁⣠⠖⠀⢠⠀⠀⠀⠀⠀⠈⠖⠁⠀⠀⠢⠈⠀⠀⠑⢤⠃⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠘⣆⠀⠳⠄
⠀⠀⠀⠀⠉⠀⠐⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠓⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠾⣀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢼⠖⣞⢁⣦⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢻⡏⠛⢋⠏⢁⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⢠⣴⡆⠀⠀⠀⠀⢀⡜⢷⠷⡌⣤⣞⠯⢺⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡊⠀⣄⣀⡄⠀⢀⠙⣯⡘⢳⣠⡄⢹⣧⠜⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⢠⠊⠈⠀⠀⠀⠀⠀⠄⠉⢧⢸⠤⠋⢋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⠀⠀⠀⠀⠀⠀⢌⣹⢢⠀⡦⠀⣄⠀⠀⠀⢠⣈⡳⢑⠀⡰⢅⣰⣨⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠠⠦⠀⠀⠀⠀⠀⠞⠁⠙⠒⠶⠚⠃⠀⠀⠀⠰⠏⠹⠓⠲⡰⠛⠫⠍⠁
⠒⠀⠉⠁⠀⠒⠒⡄⠀⠀⠀⠐⠀⠁⠙⠂⠒⠒⡀⠀⠀⠀⠐⠁⠁⠙⠓⠒⠒⠂⠀⠀⠀⠐⠘⠋⠚⠛⣛⣶⣆⠀⠀⠀⢐⠻⠿⠿⠿⣿⣶⣦
⠠⠤⠤⠄⠄⠤⠰⠁⠀⠀⠀⠀⠤⠄⠤⠀⠤⠠⠁⠀⠀⠀⠀⠤⠄⠤⠄⠤⢰⠃⠀⠀⠀⠀⠤⠤⠤⠀⠡⠀⠃⠀⠀⠀⠀⠠⠤⠤⠄⠡⠔⠎
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣀⣄⣀⣴⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⡀⠀⣀⠀⠀⠀⢀⠀⡀⢀⣰⡀⠀⢀⠀⠀⠀⢀⠀⡀⢀⣸⡄⠀⢀⠀⠀⠀⢀⡀⡀⢀⣠⡆⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⡤⣞⣡⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⠖⠊⠉⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊⠀⠀⠙⠲⠤⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢀⠄⠐⠂⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠐⠀⠀⠂⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠐⠂⠉⠀⠀⠀⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠠⣤⠤⣤⣤⣤⣰⣀⣀⣂⣀⣀⣀⣀⡀⢀⠀⠀⡉⠀⠀⠀⣀⡀⠀⣀⡀⣀⣀⣀⣁⣀⣀⣀⣐⣠⣄⣀⣠⣥⣤⠤⣤⣤⠤
⠀⠀⠀⠀⠈⠉⠒⠦⠤⣀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣽⣭⣭⣯⣬⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣯⣭⣽⣿⣽⣿⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣶⡚⠁⠀⠈⠀⠊⣡⠖⠉⢁⣰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽⠻⢭⣞⣗⠢⠤⢒⡁⠀⠀⠀⠀⠀⠘⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠇⢆⢳⠀⠙⣁⠢⣬⠐⠦⡤⠤⠀⠀⠨⠉⢉⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢌⣎⠐⠀⠮⠀⠀⠸⡆⠄⠢⣄⠀⠀⠀⠠⠤⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢇⠀⠀⠀⠀⠙⠀⠀⢌⠪⡙⠁⠁⠐⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⠼⣄⠠⠀⠀⠀⠀⠀⠐⣔⠌⠲⣦⡤⠀⠈⣧
⠀⠀⠀⠀⣠⣶⠒⠒⠤⠤⣀⠀⣠⠔⠈⢉⣿⡿⠙⣎⠣⣆⢀⠠⠀⠄⢢⠈⠀⠀⠈⠉⠙⠲⢬⣆
⠀⠀⠀⢼⠈⠛⠿⢷⣶⣦⣤⡟⠁⣴⣾⡿⠛⠁⢰⣿⠀⠈⠢⠧⠷⢌⣢⡳⣄⣲⡜⠒⠗⠒⠁⠀⠁
⠀⠀⠀⠸⣷⣤⣀⡀⠈⢹⣁⣤⣾⡿⠋⠀⢀⣴⣿⡟
⠀⠀⠀⠀⠈⠛⠿⠿⢿⢿⣟⢉⣉⣀⣤⣶⣿⡿⠋⠀⠠⠴⠒⠙⠻⠿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠛⠛⠋⠁⠀⢀⡠⠀⠂⠀⠀⠀⢀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⣀⣀⣀⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠘⠷⣦⡀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⣿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠒⠀⠉⠀⠂⣿⠃⠒⠛⠀⠀⠒⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢛⣀⠀⢀⠒⢂⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⢀⠐⠀⠄⠠⠦⢤⡄⠀⠀⠐⠀⠀⠀⠀⠀⢀⡦⠀⠴⠦⠀⠠⠟⠀⠢⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁⠀⠀⠀⠁⠈⢁⡀⠀⠐⠛⣁⡀⠀⠀⠈⠙⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢀⠀⠉⠁⠀⠀⠀⠀⠀⠠⠸⠆⠀⠨⠀⠀⠊⠀⠀⠀⠀⡀⠐⠊⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠒⠒⠂⠀⠤⠤⠤⠤⠄⠒⠒⠒⠈⠀⠈⠁⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠶⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⡀⡀⢐⡶⢤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠍⠐⡁⣀⣀⡥⣀⣬⣌⢻⣟⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⠴⢿⡿⠿⠿⠷⠿⠟⠿⣿⠿⡟⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠈⢮⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣧⣠⡄⣔⣤⢰⣤⡀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠛⠊⠉⠛⠛⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣿⠃⣀⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⢹⣾⠋⠃⠀⠀⠀⠀⢢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣯⢀⡤⠋⠉⠀⠀⠀⠀⠀⢠⣿⠀⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠻⡏⠀⠀⣀⠀⠀⠀⠀⢠⠏⣽⠀⢻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠲⠤⣤⡤⠾⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢹⣷⡊⠁⠀⠀⢀⣼⠀⠀⠀⢀⡿⢠⣿⠀⠸⠙⣧
⠀⠀⠀⠀⠀⢀⠀⢀⡿⠻⢦⣄⣄⢈⣷⡤⠒⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢟⢀⡴⠈⠊⠀⠀⢠⣴⢢⣾⠀⠀⠀⣼⠃⢴⡧⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠘⢷⣿⣠⣴⠞⠻⣿⡋⠁⢀⣠⠿⣄⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢼⠻⠉⠀⠀⠀⣠⡾⠛⣿⠟⠀⠀⢰⡟⠀⣿⡏⠀⡇⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠁⣀⡽⣤⡀⠉⠛⢿⠙⢻⡄⠀⡇⠀⠀⠀⠀⠔⠛⢳⣦⡖⠀⠀⠀⠀⣠⠴⠃⠀⣾⠏⠀⠀⢀⡿⠁⠀⣿⠁⢀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⣀⢀⡿⠀⠈⠿⠶⠖⠟⠛⡆⠙⠒⠁⠀⠀⠀⡆⠀⣐⡿⠋⠀⠀⠀⢀⡴⠁⠀⠀⣼⠏⠀⠀⠀⣼⠇⠀⣸⡟⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⠦⣄⣀⡤⠜⠁⠀⠀⠀⠀⠀⢰⣿⡾⠋⠀⠀⠀⠀⡠⠛⠀⠀⣠⡾⠃⠀⠀⠀⢰⡟⠀⢠⡿⠀⠀⡜⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠀⠀⠀⠀⢀⠔⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣿⠃⠀⢸⡇⠀⠐⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⠀⠀⠀⠀⡰⠃⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⢸⡇⠀⢠⡿⠁⠀⡠⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⣀⡀⢙⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣿⠁⠀⠀⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠦⠾⠖⠛⠛⠁⠀⠋⠈⡆⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣴⡏⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣼⠃⠀⠀⠀⠀⠀⡰⠃⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⣾⡟⠀⠀⠀⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠀⠀⠀⢠⠃⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣠⡟⠀⠀⠀⢸⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠒⢦⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⠏⠀⠀⠀⠀⢀⠆⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣴⡄⠀⢳⠀⠀⠀⠀⠀⢀⡎⠁⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢠⣿⠇⠀⠀⠀⠀⠀⣜⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠎⠀⠀⠀⠀⠀⢸⣷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠻⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢹⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣍⡉⠉⠓⠦⣄⣿⠀⢸⡇⠀⢸⡄⠀⠀⢀⣠⡤⠴⠶⠖⢒⡚⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠈⣿⠀⢸⡇⠀⢸⠃⣠⠞⠋⠁⠀⣀⡤⠞⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡹⣆⢹⡄⠸⡇⢀⣏⡞⠁⢀⡤⠖⠋⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⠷⣄⠀⢷⢠⣷⢿⢏⣠⠞⠉⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠾⠿⢿⣿⣷⣾⣿⢁⣾⣏⣤⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢒⣺⣷⣶⣯⣿⣻⠟⣁⣀⣤⣶⣚⣛⠉⠉⠉⠓⢦⡀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠡⠖⠛⢉⡴⢞⡉⠀⣸⣽⠟⠛⠿⣭⣉⠉⠉⠉⠛⠒⠶⠀⢹⡄⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⣶⣶⠾⠿⠚⠉⣙⠋⢻⣿⠉⠙⠛⠒⣯⣭⣿⣛⠒⠲⠶⢤⣄⣳⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣠⠟⣉⢠⡖⠀⠀⠘⣉⠁⢸⢸⡀⠚⠳⠀⣤⠀⠀⠉⢳⡄⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡾⠹⡆⠀⠀⠀⠀⠀⣄⡟⣠⡞⠁⠀⠃⢀⠀⣴⠒⠀⠁⠐⠺⡎⠳⠆⠀⠀⠈⠀⠘⠳⠀⠻⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⠠⡶⠁⠀⠀⠈⢻⡇⠀⡶⠀⠀⠉⢠⠄⠀⠀⠖⠀⠤⢀⣀⠀⠒⠂⠀⠙⠂⠀⣤⡹⠀⡀⠈⠹⣧⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣸⠁⠀⢹⡄⠈⠀⠀⣀⣀⠀⢳⠀⠀⠀⠆⢠⠈⠀⠛⠀⢠⠄⠐⣯⡀⠀⢰⡄⠀⠀⠶⠀⠘⠃⠈⡇⠀⠀⢸⡶⠔⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢻⡄⢰⠋⠀⠀⢳⠘⡇⠀⠀⠀⣈⣀⣀⣐⣁⠈⠀⣄⠁⠀⠀⢀⣡⠴⠶⠶⢤⣄⡀⠀⡆⢀⡴⠋⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠻⣼⡀⠀⠀⢸⢦⡧⠤⠶⠛⠉⠉⠉⠉⢉⡙⢦⣀⣠⣤⣤⣯⠤⠤⢤⡀⠀⠀⠉⠳⣤⡾⠁⠀⣀⡀⠀⠰⣼⢿⡇
⠀⠀⠀⠀⠀⠠⣧⡖⠀⠀⠀⠀⠻⣷⣤⣖⣡⠏⠀⠀⠀⠀⠀⠀⣴⠋⠉⣩⢿⠋⠀⠀⠀⠉⠓⢦⣤⠏⠀⠀⠀⠀⠸⡇⠀⡞⠁⠈⡆⣠⠏⣼
⠀⠀⠀⠀⠀⠀⠙⠻⢤⣄⡀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠠⣄⣠⡬⠟⠋⠁⠘⠷⣶⡉⠉⣱⠀⠀⠙⢶⡀⠀⠀⠀⠀⢻⡀⢱⠀⣰⠟⠁⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⡓⢦⣀⡈⢷⠆⠀⣀⣠⠴⠚⠋⠁⠀⠀⢠⠒⠒⣠⣤⠽⢿⡁⠀⠀⠀⠀⠙⢦⡀⠴⠢⡌⢧⣀⣷⠃⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣌⡙⠛⠓⠛⠉⠀⠀⠀⠠⣄⠀⠀⠘⣲⡞⠉⠀⣠⣄⠙⢧⡄⠀⠤⣤⠈⠙⠒⠶⢶⡤⣬⣀⣀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣻⣧⠀⠀⢀⣀⣤⠴⠖⢻⠟⠻⣧⣉⣠⠼⠀⠀⠉⢳⡶⠶⠶⠶⠶⠖⠋⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣁⣀⡤⠖⠻⣄⣀⣀⡽⠶⠦⣤⡴⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⣼⠃⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡏⡏⠛⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣟⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠇⠀⠀⠀⠛⠁⠀⠀⠀⢀⣤⣶⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠴⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠸⢤⠀⠖⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣩⣶⣿⣿⣾⣷⡟⢿⣿⣿⣿⡀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⢰⣡⠷⠐⠋⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣬⣿⡟⣿⣿⣿⠟⣍⠁⠈⠻⣿⣿⣾⠿⢿⢘⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⣃⡜⠈⠁⠀⠀⠀⠀⣴⣻⡋⢭⡍⠛⠁⠈⠽⣻⢿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢰⡄⠀⢸⣿⡿⠀⠀⠀⢹⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⣿⣟⠇⣠⣐⡀⠓⣀⣀⠉⠀⠉⠻⣿⣿⣿⣿⠋⠙⢷⠀⠐⡇⠈⢻⣷⡆⢠⡀⠀⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⣦⣤⣤⣼⣿⣯⣾⣿⣟⢁⡠⠶⠀⠀⠑⢄⠀⠈⠻⣿⣿⢱⣄⠀⠣⡀⠀⠀⠘⠿⠁⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⡿⠀⠴⣾⣶⠋⢻⢿⣿⣿⣿⣯⣥⡤⠀⠀⠀⠀⠄⣉⠀⠛⢿⣿⣿⠀⠒⠡⠀⠙⠀⢰⠄⠀⠊⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠗⠈⠀⠀⠳⠄⣀⠠⠤⣜⠙⠻⠿⣦⣟⣀⠀⠘⠂⠘⠀⠀⢀⠻⣏⠀⠀⠀⠀⠀⡀⢸⡀⠀⣄⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠚⢛⡿⣿⣧⡈⠁⢉⠁⠀⠀⠀⠀⠀⠁⠀⠑⠠⠙⠿⣦⡀⠂⠄⢀⠀⠀⠀⣿⡟⠀⡀⠀⢠⠃⣸⣟⣾⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣿⣿⣛⣽⡷⠉⠉⠒⠙⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠘⠂⡀⠈⠙⢒⡀⣿⣿⡀⠁⠀⡄⢰⡼⠛⠁
⠀⠀⠀⠀⠀⠀⣤⡾⢿⣿⣿⠨⠿⣿⣗⡀⠀⠀⠀⠈⠹⢿⣿⣷⣆⠀⠀⠀⠀⠐⠀⠉⠀⡀⠂⠀⠀⠀⠂⠐⠀⠀⠀⠘⣿⡟⣠⣴⡷⠋
⠀⠀⠀⠀⣠⣾⢟⠟⢛⣿⠁⠀⢀⠸⣿⣗⠀⢒⡄⠀⢉⠀⠘⠻⣿⣷⣦⣄⠀⠐⠀⠐⠂⠀⠀⠐⠀⠀⠀⠀⡀⠳⣶⣤⣿⡿⠛⠁
⠀⠀⢠⡟⠋⠉⡈⠀⠀⣿⡄⠰⠃⠐⠘⢿⣿⣿⣷⣤⣄⣭⡀⠀⠈⠛⢿⣿⣿⣶⣤⣤⣤⣤⣶⣤⣤⣶⣾⣿⣿⣿⡟⠋⠁
⠀⠀⢻⣦⡴⠊⠀⠀⣥⣿⣿⣤⣤⣠⣤⣠⣽⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣝⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋
⠀⠀⠺⢿⣿⣿⣷⣦⣤⣌⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣩⡛⣉⣝⣛⣿⣫⣽⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⠶⠈⠛⠛⠛⠋⠱⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⣤⡄⠀⣴⡆
⠀⠀⠀⣰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿⣇⠀⣿⣷⠀⣿⣿⡀
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⢹⣿⠀⣻⣿⡄⡏⢹⡇
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⣿⡆⢿⢿⡇⠀⠸⣿
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆⢸⢸⡇⠀⠈⣧⠀⠀⣿⡆
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡇⣼⢸⣧⠀⡸⣿⢀⡄⢹⡇
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣇⣿⢸⣿⠀⡇⣿⢸⡇⠘⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⡏⠘⣿⠀⡇⣿⣾⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⡇⡀⣿⡖⡇⢹⣿⠃⠀⢿
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⡇⡇⢻⡇⡇⢸⣿⢰⠀⢸
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⡇⢸⣿⡇⠘⣿⢸⠀⢸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⣷⡇⠸⣿⡆⠀⠿⢻⠀⢸
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⢠⠾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣰⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣰⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢰⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢀⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣀⣀⣠⠞⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠉⠉⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣾⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⢤⣀⣀⣀⠀⠀⠀⣠⠔⢲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠃⠀⠀⣀⣀⣀⡀⠀⠉⠙⠛⠋⠁⠀⢀⡼⠓⠲⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡥⠴⠒⠛⠉⠉⠉⠉⡭⠉⠉⠉⠓⠒⠦⠤⣤⣾⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⠴⠋⠱⠖⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⡽⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢰⣠⠟⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⡾⠁⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠠⢻⡀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢹⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠁⠘⡇
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡼⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡇⠀⠀⠐⠧⠴⠒⠒⠒⠺⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⢸⡀⠑⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⣷⣤⣀⣀⣀⣠⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣈⡳⢤⣄⣀⣀⣀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣠⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⢢⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸⠀⢀⣀⡼⠃⠀⠀⠀⠀⠀⢸⣀⡀⢘⡆
⠀⠀⠀⠀⠀⠈⠓⠒⠁⠀⡏⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⠀⠀⠀⢀⡤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣶⠶⠿⠟⠛⠛⠛⠛⣿⠛⠛⠛⠛⢿⠿⣶⣦⣤⣀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡄
⠀⠀⠀⠀⠀⣠⣴⡿⠟⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢾⠀⠀⠀⠉⠙⠻⣶⣤⠴⠋⣁⣤⣶⡿⠃
⠀⠀⠀⣠⣾⠛⠁⠀⠀⠈⢳⣄⡤⠤⠴⠶⠒⢶⠞⠓⠒⠲⠶⠤⢼⣄⣀⡀⠀⠀⠀⡼⠁⣠⡾⠛⠉
⠀⠀⢰⣿⠁⠀⠀⡠⠖⠋⠉⣤⡍⢀⣈⡛⠁⢴⢢⡀⣶⡆⢀⡀⠀⣽⡎⠉⢓⡦⠞⠁⣰⣿⣿
⠀⠀⢸⣿⡀⣠⠊⠴⡄⠀⠀⠀⢀⣙⠐⠋⠀⠀⠈⠀⠀⠀⠛⠁⡠⢶⡄⠀⢯⠀⠀⣰⡿⣻⣽
⠀⠀⠸⡟⢿⣧⣀⠉⢥⠪⡷⠀⠈⢬⠇⡀⠀⣴⡶⠀⢠⣾⠆⠀⠉⢉⣀⠀⣨⢷⡟⣻⣾⠟⣹
⠀⠀⠀⢷⣤⡉⠛⠿⣷⣦⣤⣤⣄⣀⣘⣝⣀⣀⠀⠀⠀⠀⠐⣽⣶⣙⣋⣤⣬⣽⠿⠛⠁⣰⠇
⠀⠀⠀⠈⢿⢻⠓⣦⣄⣀⡈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⢀⣠⣴⢯⡏
⠀⠀⠀⠀⠈⢷⢼⣿⡹⢠⢟⡻⡗⢲⣶⣶⠤⠤⣤⣤⠤⢤⣤⡤⠤⣴⣶⡒⣻⣟⣦⢯⠟
⠀⠀⠀⠀⠀⠈⠳⣌⠙⠚⠾⠾⣅⣗⣯⡼⢀⡏⣗⣸⠂⡮⣖⡽⠀⣷⣦⣃⣿⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡦⣤⣀⡀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⣉⡥⠔⢊⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡈⠉⠛⠓⠒⠒⠶⠶⠶⠖⠒⠒⠋⠉⠁⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⠶⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠶⠁⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⡐⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠔⠀⠀⠐⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠑⢆⠀⡆⠀⠠⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⠄⠀⠈⡄⢃⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⣀⣧⣨⡄⣤⡄⢀⣄⡀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⢔⠒⠺⢋⠉⢃⠈⠃⡈⠀⠐⠰⢸⡏⠀⠻⠀⡀⠑⠀⠀⠓⠚⠻⣍⡙⢟⠭⣖⣲⡦⢤⣤⡴⠛⠁⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⢀⣤⣖⠒⠉⠅⠀⠀⠉⠀⠀⠀⠠⠀⢀⠀⠠⢀⡥⢀⠀⡞⡀⠄⡀⠀⡀⠡⠈⠒⢄⡠⢄⢀⠉⠓⢦⣀⣁⣿⢿⠧⣝⡷⣦
⠀⠀⠀⣴⠏⠁⠒⠂⠀⠀⠐⢤⡠⢠⡂⠄⠈⠒⠀⠛⠒⠓⠀⢀⡝⠉⠉⠀⢀⠀⠈⢩⡉⠉⠀⠁⠐⡛⠚⠗⣹⣾⣟⣳⣄⠰⢬⣃⡾
⠀⠀⠀⠙⣟⣢⣤⣤⣽⠔⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠰⠂⠀⠀⠀⠆⠀⡀⠀⠐⠀⠀⢀⣴⡾⠛⠄⠀⣈⣻⡷⣾⠋
⠀⠀⠀⠀⠈⢿⣛⣛⠛⠛⠶⠶⣤⣤⣤⣀⣀⢀⣚⠀⡀⠐⠀⠂⠀⠀⣀⣈⣁⣀⣀⣀⣀⣀⢠⡤⠾⠿⠿⠶⠚⠛⠋⠉⠁⡶⡏
⠀⠀⠀⠀⠀⢸⠃⠊⠁⠒⠒⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⣇
⠀⠀⠀⠀⠀⢸⠁⣰⢶⠶⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡤⢢⠀⠀⠘⢢⠔⢡⣏
⠀⠀⠀⠀⠀⢸⢶⠈⡷⠊⢀⡀⠐⢄⣠⠜⠀⠀⠀⠰⡐⠖⡐⠀⠀⠀⣄⠶⣰⠀⠀⠀⠦⣉⠜⠀⣀⡉⠒⠁⠀⠞⣱⡄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠲⠯⢌⠆⠀⠀⠀⢀⢞⡱⡄⠈⠋⠀⢠⠖⢄⠈⠛⠁⢀⢔⡒⡄⠀⠀⠠⡱⠖⠂⠀⠀⠉⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠘⢦⣠⣤⣤⡄⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠑⠊⠈⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⣭⣀⣒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠭⣭⣛⣓⣒⡦⠤⠤⠤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣴⠶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠐⠒⠒⠒⠒⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⢒⠔⢀⡤⠀⣉⠀⢀⠈⢉⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣀⠀⡀⢆⡴⢁⠔⠁⣀⣤⠵⠖⢂⣈⣉⡋⣛⣁⣒⣀⡀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⢁⡠⠐⣘⣃⡔⢡⢎⡴⡡⢊⢐⡩⢴⣺⠿⠟⠻⠽⠉⠉⠉⠉⠩⠽⠗⣒⠯⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⢠⣬⡖⠉⠈⡌⡰⠡⢎⣴⢔⣫⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠑⠩⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠋⠤⣶⠞⠲⠂⣌⢢⡜⡐⣽⡶⣫⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠑⢂⠀⣀⣤⣀⠀⠀⡼⢡⡟⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡏⡀⠔⡊⢈⠁⠀⠝⢈⠳⡾⢫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠡⠀⡕⠚⠁⢆⠀⢚⠶⢁⠞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠓⡆⢀⠁⣘⢁⡤⠚⡠⢃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⢰⡀⢨⠔⠿⠉⠀⡰⣡⠏⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢳⢈⡔⢋⠠⠚⢃⡜⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢖⣒⡫⠭⠴⠒⣚⣃⣭⣤⠴⠶⠖⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢠⠛⠂⠀⢚⠔⠁⠔⢣⢎⡼⠁⠀⠀⠀⠀⠀⠀⢀⠠⠤⢒⣋⠥⠔⢒⣉⣩⡤⠶⠒⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⡸⠂⠀⢠⣀⡠⠚⣠⢋⡞⠁⠀⠀⠀⠀⠀⠀⠈⠀⣐⣈⣥⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠐⠂⢩⠋⡔⣡⠏⠀⠀⡀⠤⢀⣀⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠘⢶⢤⣉⡳⢎⠜⠁⣀⣠⡭⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠶⠷⠚⠋⠉




⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⢀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣡⠴⢾⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄⣀⡤⠴⠚⠉⠀⠀⡇⣀⡿⠀⣛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠉⠓⢤⣀⣀⣠⠼⠛⠉⢳⠎⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⣻⠀⠀⠀⣠⠔⠚⠉⠁⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢀⡴⠒⠚⠋⠉⠉⠀⣠⠃⠀⠀⢰⠁⠊⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠔⠣⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠈⣷⢦⣀
⠀⠀⠀⠀⠀⢀⣴⡟⠈⡟⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢿⡀⠹⣧⡀
⠀⠀⠀⠀⠀⢸⡀⠙⢧⣽⠒⠚⠦⢤⣀⣀⣀⠀⠀⠀⣺⠖⠒⢢⡀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡷⠟⠁⢹
⠀⠀⠀⠀⠀⠀⠙⠦⣄⡈⠉⠓⠒⠢⠴⣄⡩⣍⣉⡽⢧⣀⣀⣴⢋⡉⣉⣉⣩⣉⡭⣽⠖⠒⠋⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣉⣁⣤⠤⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢉⢦⡁⠒⠄⠀⠀⠀⠀⢀⠄⡽⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠥⡀⢲⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⢊⣰⠏⣰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⢠⡄⠀⠞⠀⠀⣼⠥⢈⣵⠁⠀⠀⠀⠀⠀⢀⠴⠋⠀⡀⣀⡠⢀⡀⠠⠚⠁
⠀⠀⠀⠀⠀⠓⠶⠄⠈⠁⢀⡸⠏⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣇⣄⠀⠀⠄⠉⠋⠉⠉⠓⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⢀⡄⣠⠀⣉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⡁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⣜⣆⡵⠖⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢠⡤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠈⢐⠀⢄⡀⡈⠃
⠀⠀⠀⠀⠘⠉⠁⠀⢀⠀⠘⠂⠀⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⠄⠠⣾⡆⠀⠀⠀⠀⠀⠔⠈⠁⠀⡀⠀⡰⠁⠀⠀⠀⠀⠈⠦⢀⣀⡜⠀⢠⡜
⠀⠀⠀⠀⠀⡀⠀⠳⢆⡄⣠⢀⢠⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠰⠀⠀⠀⠀⠀⣰⢀⡴⠛⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠛⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠃⠈⠀⡇⠈⠙⠀⠀⠣⠀⠀⠀⣠⣀⠲⣶⣠⡆⠀⠀⢸⡄⠀⡀⠀⠀⡀⡿⠏⠀⢀⣽⠃
⠀⠀⡀⢂⣀⡠⠂⠀⠀⠀⠀⡇⠀⠐⠀⠀⠀⠀⠀⣀⢤⣌⡉⠉⠘⠃⠀⠀⢘⠓⢀⠉⠐⠆⠉⠁⠂⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⡇⡆⢸⠀⠀⠀⠀⠰⢀⠚⠉⡁⠀⠀⠁⠀⠀⢃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡄⠀⠀⠀⠀⣶⠿⠶⠛⠁⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠁⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡀⠀⢀⠠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠤⢀⡀⠀⠀⠀⠀⠀⠸⠒⠂⠰⠦⠶⠴⢂⡜⠀⠀⠀⢀⠠⠒⠉⡏⠀⠀⠀⠒⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢢⡀⠀⠀⠀⢾⠰⣿⡂⠒⠒⠒⠉⠀⡀⠔⠈⠁⠀⡀⢀⡀⠠⠂⡞⠀⠀⠀⢀⡔⠉⡷
⠀⠀⠀⠀⡀⠀⢀⢰⣀⠀⠦⡀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⠀⠹⠃⠀⠀⠀⠀⢀⠀⣀⢦⣙⠦⠛⠀⣀⠀⠈⠀⠀⢀⡆⠁⢹⠊⠁
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠛⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣲⣯⢛⣽⠆⠁⡀⠀⠊⠁⠀⠀⠀⠊⢀⡼⠀⠉
⠀⠀⠀⠀⠀⠀⣠⣠⠆⠀⠚⠉⠀⠀⠀⠁⢀⡀⠀⢀⠀⢤⢀⠤⡒⠀⠀⠀⠀⠀⠀⠀⠘⣿⡽⠶⠮⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠠⡄⣤⣶⣄⣶⣧⣦⡀⠤
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢢⣠⠄⠀⠀⠀⠀⠀⠀⠓⠢⠄⣀⣀⠀⠀⠀⢄⠙⢕⣾⠖⣟⣿⠎⡾⣿⣿⣼⣻⢵⣄
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠯⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣈⡉⠉⠙⠛⠽⠽⠷⠲⠛⢋⡵
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠂⡀⠀⠐⠀⠀⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣐⡋⣉⢉⣤⣴⣂⣧⢃⢒⡌⢱⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⣿⣿⡿⠻⡿⢻⡿⢿⡿⢿⡟⢿⣿⣿⡇⠀⠀⠮⣋⡶⣮⣹⣺⠶⡟⠾⣾⣸⣷⣻⣾⡷⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠸⣿⣿⣶⣷⣾⣷⣾⣷⣾⣷⣾⣿⡟⠀⠀⠀⠀⠈⠉⠑⠋⠑⠛⠓⠛⠛⠚⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠟⠁⠀⢀⡀
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡱⡯⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⣋⣁⣤⣀⣴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣾⡭⠿⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⣽⠟⢹⠃
⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⢠⡟
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⠼⠖⠋⠀⠀⠀⣠⠟
⠀⠀⣇⠠⢠⣤⠤⠤⠴⠒⠛⠉⠁⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠙⠷⣦⣄⣀⣀⣀⡀⠀⠀⢀⣀⣀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿