Delectable Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣐⣐⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⢿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⣄⢀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⡷⠾⠿⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿⡿⠿⣿⣿⡿⡿⠿⠿⠟⠛⠉⠀⠀⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣭⣝⠳⢶⣤⣤⣥⣌⡇⠈⢀⠀⠀⣀⣀⣀⣀⠤⠤⠤⠴⠖⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣦⣤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣧⣄⣀⣄⣠⣄⣀⣀⣠⣤⣤⠶⠖⢛⠉⠃⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠉⠛⠹⠛⠉⠙⠋⣉⠉⠙⠋⢉⡉⠉⠹⡋⠉⠛⠉⠰⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣥⣄⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡛⡿⠭⣻⠟⠿⢿⣦⣄⡀⠀⠀⡀⠀⢀⣀⣠⣤⠴⠶⠖⠚⠛⠉⠉⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣦⠄⣀⡀⠉⠉⠉⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣶⣶⡿⠟⠋⠉⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠛⠛⠛⣛⣻⣿⡻⠯⠃⡀⠀⠀⠀⠀⢀⣠⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⡛⢿⣿⣿⣿⣾⣧⣄⣀⡁⠀⡀⢀⠀⡀⣁⣀⣀⢀⣤⣤⣤⣶⡶⠞⠛⣉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣬⡉⠐⠬⣽⣛⡻⠿⢿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⡿⠟⢉⠠⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣦⣄⣈⡉⠀⠁⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠲⠦⠤⠄⠠⠀⠠⠦⠴⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠤⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠣⡀⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣈⡀⠀⠠⢦⡀⠀⠐⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠈⠡⠀⡴⠒⢦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⡄⡟⢀⣤⠀⡇⠀⠀⠀⢀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠄⠻⠷⡇⢸⣷⢆⠇⠀⠀⠀⠀⠁⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢠⠔⠚⠒⠛⠚⠦⠴⠊⠀⠀⠀⠀⠀⠀⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣇⠀⠀⠀⠀⠀⢀⠀⠎⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣠⡀⠈⠳⠶⠤⠶⠖⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⣼⢡⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠣⣄⠀⠀⠀⠀⠀⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠙⠓⠒⠲⡎⠉⠁⢻⢦⡀⠀⠀⢆⣀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⠊⠀⡐⠇⡠⠖⠉⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⣀⠜⢀⠈⠉⠀⠀⠀⣼⠃
⠀⠀⠀⠀⢀⠔⠚⢣⣠⠏⠀⠀⠀⠘⡄⠀⠀⠀⠈⢧⡀⠀⠀⠀⣀⠤⢤⡀⠀⠀⠀⠀⠂⠁⠀⢀⡾⢧⣀⡀
⠀⠀⠀⣠⠆⣤⣀⡞⠀⠀⠀⡘⠀⠀⢱⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠈⠉⠳⢤⡀
⠀⡠⠈⠁⠀⡸⠁⠀⠀⠀⢰⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠓⢶⠊⠀⡻⣤⣤⡤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⢠⠇⠘⢲⠀⠐⢄⠀⠀⠀⡼⠀⠀⠀⠀⠄⡘⠀⢀⡀⣀⡀⡤⠖⠅⢄⠀⡆⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⡆⠀⢠⠊⢦⡠⠎⠀⠀⡾⠁⠀⠀⠄⠀⠰⢁⡠⠂⠀⠀⠀⠀⠸⠤⠼⠀⠀⠀⠈⠢⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⡇⡰⠅⠂⡀⠀⠀⠀⢰⡷⠐⠀⠈⡰⢀⠟⠁⠀⠀⠀⠀⠠⡖⠒⠤⠐⠒⠒⢦⠀⠀⢀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢰⠁⢄⣠⠆⠀⠀⠀⠀⠻⣄⣀⣹⠔⠁⠀⠀⠀⠀⠀⣀⣀⡹⠦⠤⣤⣀⡀⠀⠹⡀⡀⠀⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀⠰⣇⠀⠀⠀⠀⠀⠀⠙⣆⠀⠈⠓⣄⢣⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸
⡇⢌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⠧⠍⠛⠛⠛⠳⢄⠀⠈⠂⠀⠀⠈⡆⢧⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸
⢸⡈⠙⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⠀⠹⢤⣤⣤⣴⣤⡀⠀⠙⠀⠀⠀⠀⠀⠈⠈⡇⢀⠀⢿⠀⠀⠀⠀⠀⢸
⠈⠱⡀⠀⢰⠈⠙⡆⠀⢠⠐⠠⠀⠀⠀⠦⣀⠤⠁⣰⡛⠋⠀⠉⠈⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠑⢾⣇⠀⠀⠀⠀⠸
⠀⠇⠙⢄⠈⠒⠚⠁⠀⠳⣄⡠⠃⠀⠀⠀⡠⠒⠉⠀⠙⠛⠛⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠀⠀⠈⠓⢦⡀
⠀⢸⡀⠀⠉⠒⠢⢤⣀⣀⣠⣄⣀⣠⠴⠋⠀⠀⠀⠀⣀⠤⠖⠉⠀⠈⠙⠒⠶⠤⠤⣤⣤⠴⠞⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⢷⡀⠀⠀⣀⠞⠁⠀⡟⠙⠒⠶⠤⠤⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄
⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠐⠂⣤⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠒⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣔⠖⡈⠀⠀⠀⠀⠀⢠⣸⠃⠀⠀⠀⠀⠀⠀⠰⣶⣤⣤⡠⣪⡾⠀⠀⠀⠉⠐⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠈⠙⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠶⣄⣢⠄⠀⠈⢲⢢⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠗⠀⠀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠀⠢⡀
⠀⠀⠀⠀⡠⢀⠀⠀⢀⣼⠆⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣭⠇⠀⠀⠀⠀⠀⠀⠀⢠⠂⡀⡀⠀⠀⠀⠀⣤⡘⠄
⠀⠀⠀⡰⠁⠆⠀⠻⠋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠏⠀⠀⠀⢰⡾⠀⠈⠄
⠀⠀⣀⠡⣄⠀⠀⠀⢰⢸⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⠀⠀⢰
⠀⠠⡥⠞⠉⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣿⣿⣿⣯⣾⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣴⠀⠆
⠀⠀⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠲⡄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠆⠀⠀⠀⠀⠀⡀⠸
⠀⢰⡀⠀⠀⠀⢰⣵⢳⣦⠿⠿⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⠤⠄⠀⠀⠄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣾⡿⠀⠀⠀⠀⢠⣾⢁⠆
⠀⠀⢆⠰⢶⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢇⣾
⠀⠘⢌⣄⡈⠻⣷⣤⡀⠀⢀⠄⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⠀⣀⢠⠀⠀⠀⠀⢰⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⢛⣥⣾⡟
⠀⠀⠸⠛⢿⣷⡙⡿⣿⠷⢬⣒⠖⠃⠀⠀⣰⣹⠃⣀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⢐⣿⣾⣿⡟⠁⢠⣿⣿⣿⠃
⠀⠀⠀⠆⠀⠉⢻⣮⣭⡀⠀⠨⣤⣤⠄⢠⢌⣥⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣤⡜⣻⠟⠉⠁⠀⣠⣿⣿⣿⠃
⠀⠀⠀⠈⠢⡀⠀⠈⠻⢿⣦⣀⣀⣉⢉⠀⠈⠉⠛⢿⣿⣿⣯⣯⣽⡿⠿⠿⠿⠟⠛⣚⣉⡉⠉⠉⠉⠉⢀⣠⣶⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠈⠙⠛⠟⠿⢿⣿⣷⣄⡀⠉⠉⠋⠉⣁⣤⣶⣶⣶⣷⣿⣿⣿⣿⣶⣾⣾⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠈⠋⠛⢿⣦⣀⣠⡾⠿⠿⠿⠻⠟⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠰⠤⣀⣀⠀⠈⠁⠉⠀⢤⣴⣶⣳⣿⡳⠿⣿⣿⣿⣿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⠋⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⠾⠛⠛⣻⣿⣿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣯⣽⣷⢻⡷⠇⡠⢦⠀⡠⢎⠉⠛⢻⣿⣿⣷⣶⣻⡶⠶⢿⠶⣶⣤⣤⣀
⠀⠀⠀⠀⣠⣾⡿⡟⠋⠋⠁⣢⠄⠈⠴⠒⠊⢀⠘⢂⣶⣾⢯⡻⣿⣮⣷⠁⠀⣀⠀⢠⠞⠴⣋⡽⠷⠦⣄⡀
⠀⠀⢀⣼⡟⠏⠀⢰⠆⠤⠒⢠⠀⠆⠀⠒⠁⣱⡾⠛⣫⢭⡁⣡⠏⠁⠳⠖⠋⠉⢳⣀⣠⣆⡴⠚⠛⣦⡴⢿⡒⢄
⠀⠠⠋⡼⠁⡄⠠⠆⠠⠒⠀⠀⠐⠀⠔⢀⣾⡩⡠⠖⠁⠀⠉⠁⠀⠀⢀⠴⠀⢀⠄⠉⠁⠁⠴⠂⠀⠀⠀⠉⣇⠀⠙⣦
⢠⠧⢀⡄⣈⠀⠀⠀⠐⡀⠀⠃⠀⠄⡴⢿⠟⠀⠠⠆⠠⠃⢠⠖⠀⢀⠀⠀⠀⠡⠀⠐⠁⡤⠀⠀⠀⠃⠀⠰⠙⠋⢳⡀⣷⡄
⡟⠀⠉⠘⠁⠠⠂⢀⡜⠁⠀⠀⣠⣾⠁⠀⠀⢠⠀⠀⣀⠀⠀⠀⠀⠊⠀⠠⠃⠐⠀⠀⠀⡀⠀⠂⠀⡀⠀⢠⠄⠀⢹⠃⡘⠛⣆
⡏⠀⢠⠆⠀⡀⠈⠠⠀⠀⣨⣿⣿⡇⠀⠈⠀⠂⠀⠐⠁⠀⠀⠁⠀⠀⠀⠀⠂⠀⡄⢁⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠷⠋⢹⣰⣿
⢀⠀⠀⠀⠘⠁⡀⠀⠈⣀⣺⣿⣿⠀⡰⠀⣠⠀⡀⠠⠁⠄⠀⠎⠁⢠⢠⡧⣤⡦⣥⣜⣀⡀⢀⠀⠀⠰⠀⠰⠀⠠⠁⢰⣿⣵⣻
⠸⠀⠚⢠⠀⠀⡤⠖⠀⡈⢦⣿⡟⠀⠀⠐⠡⠀⢡⠀⠀⠀⠀⠤⠀⠀⣨⠟⠀⠀⠀⠉⢻⡅⠈⠀⢰⠂⠀⠀⠀⠀⠈⠉⣁⣿⣽
⠀⠐⢦⠀⠀⠀⠠⠚⠀⠁⠈⠛⣧⠀⢀⠖⢈⠀⢸⠄⡦⠀⡀⠀⠈⡼⣿⣦⣀⠀⠀⢀⣸⣿⡆⠰⠒⠀⠰⢀⠀⠀⠀⢸⡿⣟⡁
⠀⠀⠈⢧⣀⠀⠠⠃⢈⠁⠴⠀⡇⠀⢀⡄⠞⠀⠘⡀⠀⠤⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠂⠀⠈⠀⡆⠀⠀⣴⣿⣷
⠀⠀⠀⠀⠙⢶⣀⠀⠘⠀⠴⠂⠈⡄⠈⠀⡾⠀⠘⣁⠀⢂⡴⠀⠀⠀⠀⢸⢹⣿⡿⠿⠿⣿⠃⠀⠀⠐⢠⠄⠀⠰⠀⠸⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠲⢶⣀⠀⠀⠀⠘⡄⠀⠁⡆⠘⢁⡰⢈⡤⠀⠄⠀⠀⠀⠀⡀⠀⢀⡜⠃⡀⠀⠃⠀⠀⡠⠃⠀⠀⣰⠯⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣦⣤⣬⣦⡀⠃⡏⠀⡀⠀⡄⣠⡄⠈⠀⠀⠐⠀⠀⡠⠀⠐⠁⠀⠐⢀⠀⢁⡀⠀⢠⡏⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⢿⠀⠀⠁⢀⠀⠚⠀⠀⠀⠀⠃⠀⠁⠀⠠⠀⠐⠀⠀⠀⣄⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣀⠰⠏⠔⠀⢠⠄⠄⠀⠂⠈⠀⠂⠀⠀⠀⠐⠀⣀⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⡀⠁⠀⠀⠀⠋⠈⠁⠐⠁⠀⢀⣠⠴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⠄⠁⣄⠠⠀⠀⠀⠀⢠⣿⠷⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣶⣶⠆⠒⠛⠛⠛⠉⠀⠀⠀⢒⢶⡆⠍⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡝⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⣀⠉⠀⠀⠠⠦⣷⠀⠀⠀⣾⣿⣷⡀⢳⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡉⢱⡆⢀⠀⣄⠠⣾⣿⠀⠀⠹⡇⢀⠀⠀⠀⠀⠀⠐⣿⣿⡄⢠⡀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⣅⣉⣁⣻⠠⣼⣇⣹⣿⡤⠀⠀⢁⡌⢀⡀⠀⠀⠀⠀⣿⣿⣷⠀⢳⡄⢹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠰⣿⣼⣷⣾⣿⣟⠛⠻⣿⣿⣿⣿⡁⢺⠙⠣⠀⠹⡇⠀⠀⠀⢰⣿⡟⠻⣇⠀⢣⠀⣥⣿⢫
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡉⢹⣧⣿⣷⡀⢰⣿⣿⣉⣧⣬⠀⣀⣦⣇⠀⠀⠀⠀⠈⡟⠀⠀⠈⠃⠀⠀⣿⠙⢾⣇⡄
⠀⠀⠀⠀⠀⠀⣆⣹⣧⣿⣿⡝⠻⣯⣤⣾⣿⣏⡛⠁⠉⣿⣿⣿⡟⠇⠀⠀⠀⢠⠃⠀⣰⣄⠀⠀⠀⢿⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠿⢰⣿⣿⣿⣷⣄⠸⣿⣿⣿⣷⣿⡄⠀⣿⣿⣿⠁⣀⢰⣦⡀⡆⠀⢀⣿⣿⠀⠀⠀⠀⠀⢸⡿⠇
⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣷⠶⡾⠦⠬⣩⣿⣿⣿⣇⣤⣿⣿⣿⣷⣿⣼⣿⣇⣿⠀⢸⣿⣿⡆⠀⡀⢀⡀⠼⢷⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣴⣦⡤⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣧⣿⠀⣼⣿⣿⡇⠀⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠶⣿⣿⣦⠄⣴⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣍⣿⠀⣿⣿⣿⡇⢰⣿⡃⣧⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠛⠻⠛⠻⠿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠏⠉⠋⠉⠉⠉⠁⠉⠉⠉⠉⠉⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⣀⢀⣤⣶⣦⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠿⣸⣿⣿⣿⣿⡆
⠀⢀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⠟⠀⠚⢿⣿⣿⣿⣿⣧⣶⣶⣦⡄
⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆⢠⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⢹⣿⣿⠿⠟⣋⠽⠒⠂⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣄⠉⠉⠑⠿⠟⠃
⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⣛⣋⠁⠀⠻⣿⣿⡿⠿⠋⠀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠉⠛⠃⠀⠐⠂⠐⠒⠂⠀⠈⠁⠀⠂⠈⠐⠀⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⡱⡆⡱⣊⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠀⠀⠀⣀⠀⢀⠀⡀⠀⢀⢠⠀⡀⠀⡄⠀⣤⢀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⠀⠋⠀⠘⠚⠘⠃⠀⠘⠂⠒⠀⠉⠂
⠀⠀⢀⣶⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠦⠤⠶⠶⠛⠛⢥⣤⣤⣔⣀
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣉⣻⢿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢺⣿⣿⣶⠆
⠀⠀⣾⣿⣿⣿⣷⣬⡙⠿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⣴
⠀⠀⠈⠩⣿⣿⣿⣿⣿⣷⣦⡝⠋⠁⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⠉⠉⠻⠿⠿⠿⠃⠀⠀⢈⠿⣶⡄⣀⢀⣀⠀⣰⡿⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣾⣿⣿⣙⣿⣷⣿⣷⡆⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣠⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢀⡬⠙⠛⠿⣿⣿⣿⡿⠿⠛⠁⠀⣀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠒⠀⠐⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠛⠛⠉⠛⠒⠒⠲⠦⠄
⣶⣶⣶⣶⠶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⡶⠶⣶⣶⠶⡶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠉⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠆⠀⠀⠀⠀⣠⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡏⠀⠀⠀⠀⠐⠁⠀⢀⠀⠀⠰⠄⠀⠀⠃⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢄⠀⠀⠐⠂⠂⢒⡴⠖⣂⣀⡀⣤⣬⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠔⠉⠁⠀⠀⠈⠀⠀⣰⢞⣁⡀⠀⠀⢻⠇⣾⣿⡁⠀⠀⠈⣇⠀⠀⠀⠀⠀⠠⡀⠀⠀⢾⣄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣠⠄⢀⡔⠈⠀⢀⠀⠀⠀⠀⣀⣀⣾⢱⣿⣿⣿⣿⣟⣡⣾⣿⣿⣇⠀⣠⠧⠿⠀⠀⠀⠀⠀⠀⠈⣲⣄⠘⣿⣧⣀⠀⡀⠀⠀⠀⠋
⠀⠀⠀⠀⣼⡟⣱⠟⠀⢀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠃⠀⠀⠀⢈⠀⠀⡀⠀⠀⠀⠈⠦⡈⢻⣿⣿⣿⣧⠀⠀⢀⡀
⠀⠀⠀⢸⣿⠛⢫⠀⠀⡏⠀⠀⠀⠀⠀⣠⠝⠛⠛⠿⢿⠿⢿⠿⠿⠛⢋⣽⡻⠿⠿⠷⣶⣿⣶⡄⠀⠀⡀⠀⠀⠀⡈⢓⣹⠀⠙⠿⣿⣿
⠀⠀⠀⢸⣿⣷⣄⢀⠀⠙⠤⢤⠤⠔⠊⠁⠀⠀⠀⡐⠁⠀⠀⠀⢀⠔⠋⠉⠀⠀⢀⡴⠣⠛⢻⣿⣷⡌⠛⢄⠀⠀⣽⣮⢿⡇⠀⠀⠀⠤⠀⠀⡆⠀⠠⡄
⠀⠀⠀⠈⣿⣿⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠂⠁⠀⠀⠀⠀⢠⠁⠀⠀⠀⢨⣿⣿⣿⣆⠀⢀⣤⡐⢾⣿⡇
⠀⠀⠀⢠⠈⣙⠿⣿⣿⣦⣤⣄⠀⡄⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡶⣬⣽⣿⣯⡉⠀⠀⠈⠉⠀⠀⠀⢠⠀⠁
⠀⠀⠀⠀⣅⡈⠛⠮⢙⣿⣿⣿⣿⣧⣤⣄⣤⣀⢀⠀⠀⠀⢀⣀⣤⣠⣄⣠⣤⣤⡶⠶⠟⠛⠙⠉⠴⣿⡿⣷⣄⣿⣿⠉⢿⣄⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠘⠿⣄⠀⠀⠙⠂⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣰⣿⣿⣿⠏⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠉⠁⠁⠀⠈⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠋⠀⠀⠙⠿⡋⠀⠀⠀⠀⠀⠘⠒⠲⠴⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠲⠦⠤⠤⣤⣤⣤⣠⣤⣀⣤⣤⡤⠤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠉⠐⠒⠀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⢀⣀⠀⠀⡀⢀⠠⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⡀⢀⠀⠀⢠⠀⢠⠀⠀⡄⡀⠄⠀⠄⠀⠀⣀⡆⡀⠀⡀
⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⢀⢀⠀⠀⠀⡀⠀⢀⣀⡀⣀⠀⠀⣀⠀⣠⡀⡀⠀⠀⣀⣀⢀⠀⠀⡀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⣀⡀⢀⠀⣀⡀⢀⢀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⣀⠀⢀⣀⠀⡀⠀⣀⢀⠀⢀⣀⠀⡀⠀⡀⢀⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⢀⢀⠀⡀⠀⡀⡀⢀⡀⣀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⣀⢀⢀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠻⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠉⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠠⢜⡫⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⡀⠀⢢⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡐⠙⡢⠀⠀⠐⢒⣫⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠈⠓⠒⠲⠶⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠅⠤⡭⣭⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⢀⣀⣀⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠀⣸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠡⣄⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠚⠠⠶⣿⠿⠿⣷⠦⡀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⡋⢠⡀⠨⣀⠀⠀⠠⠈⠑⡄⠀⠱⡌⢂⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠲⢠⣼⠏⠀⠀⠀⠀⠀⢀⣵⠆⠀⠪⡖⠘⡀⠀⢘⡀⡀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⢈⠔⠁⠐⣿⡿⢀⡐⣆⣀⠀⠀⠙⠿⠃⢠⣾⡿⠞⠀⠀⠸⢀⠇⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⠈⠀⠀⠀⠄⠏⠀⠀⠀⢻⣿⡐⠌⠿⠟⠁⠀⣀⡤⠶⡛⠄⠀⣀⣠⣴⡷⠃⠘⠀⠀⠀⠀⠀⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣀⡡⠐⠀⠀⠒⠒⠤⡁⠀⡄⠀⣾⠿⡟⣧⣡⣤⣤⠶⠾⠋⠉⠀⠀⠈⠛⠛⠟⠛⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠽⣿⣿⣿⡟⠀⢀⡀⠀⠀⠀⠀⠈⢆⠈⠂⠤⠐⣡⡼⣿⠛⠛⠀⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⠠⠀⠈⠹⣧⣤⣶⣿⣿⣿⣿⣷⠄⣿⠿⠆⠀⠀⠀⢀⠸⠀⠀⣠⠎⠀⠛⠛⠀⠀⠀⠙⠁⠀⢙⣆⣀⠠⠤⠴⠋⠳⠤⣀⢂⣍⣟⡆⠀⠀⠀⠀⡇
⠀⠠⠃⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠋⠴⠊⠀⡠⢶⣶⠄⠀⠀⠄⠀⢠⡏⠀⠀⠀⠀⣎⠶⠀⠀⠀⣰⠟⠉⠀⠀⠀⢀⣴⣶⣦⣌⣿⣿⣿⡿⠀⠀⠀⢀⠃
⠀⠈⢦⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠴⠋⠀⠀⠄⡤⠴⠂⠀⠀⠀⢷⡀⢤⡤⠀⠀⣀⠠⣴⣿⢃⡤⠋⠁⠙⠒⠺⢿⣿⣿⣿⣿⣿⠿⡇⠀⠀⠀⠸
⠀⠀⡄⢀⣠⡴⠶⠶⣦⡀⠀⢀⣠⣤⣤⣠⣤⣤⣄⠒⠒⠂⠀⠤⠄⠀⠈⠙⢷⣤⣄⣀⣀⣀⣾⣿⣼⣆⠀⠀⠠⠀⠐⠂⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⡖
⠀⠀⠙⢏⠁⠀⠀⣀⣀⡙⣖⣉⣁⣀⣀⠀⠀⠀⢻⣄⣰⠟⠛⠛⠛⠻⣷⣀⣤⣤⣤⠉⠉⢉⣁⣘⠿⠟⠁⠄⠀⠀⠀⢀⣤⡶⠶⠒⠶⣦⡄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠉⠡⠤⠤⠄⢀⡀⠀⠀⠀⠐⢿⡀⢠⡿⠉⠉⠉⠙⠉⠙⠶⣤⡴⠋⠀⠁⠀⠀⠀⠈⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠂⠉⠁⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⢺⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣿⠉⢠⣿⡍⣙⣿⣯⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⣾⣧⠀⠘⣿⠸⡍⠙⠛⠁⠀⠀⠀⠀⠀⠀⠐⢢⡄⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠝⠉⢿⣷⠀⠀⢿⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠆⠀⠈⢿⣷⠀⠙⣧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠂⠀⠀⢸⣿⣦⡀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡁⠀⠀⠀⠀⠈⠙⠷⡌⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢀⢁⣀⡀⠀⠀⠀⢰⡇⣹⣟⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⣷⣿⡟⢾⣿⢟⠠⠄⠀⠀⡸⣧⡟⡁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠨⣿⣯⣹⠻⠿⡌⠓⠀⢠⡾⠉⠉⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣷⣶⡠⠀⠏⡀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⢀⣀⣤⢤⠶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣻⣿⣿⢶⣿⢷⣦⡤⢶⣤⣄⣠⣤⣦⣥⣥⣴⣢⣤⡤⠞⠿⠋⠉⠉⠐⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠻⠻⡦⣾⡆⠉⠻⠋⠉⠛⠟⡍⠋⢛⡏⠁⠙⡿⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢹⡄⠙⡄⠀⠀⡀⠀⠀⡇⠀⠸⡇⠀⠀⠃⠀⠀⠀⢀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⣷⠀⢱⠀⠀⣇⠀⠀⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠆⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢹⡄⠈⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠸⠀⡀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣧⠀⠀⠀⠀⡅⠀⠀⠀⠀⠁⠀⠂⠀⠁⣠⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣤⣤⣅⡀⣠⣀⣀⡄⢀⣧⣾⡴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠐⠂⢀⠀⠀⠀⠀⠈⠀⠠⣄
⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈⢿⠂
⠀⠀⠀⠀⠀⢠⠛⠛⠛⠛⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠰⢷⣀⡀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢳⢲⡀⠀⠀⠀⠀⢨⣿⣴⣤⠶⠂⠀⠀⠉⠈⠉⢟⢷⠆⠀⠀⠀⠀⠀⠈⠳⡌⠂⣠⣾⠛
⠀⠀⠀⠀⢸⣷⣝⠷⣄⡀⣠⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡽⠟⠅
⠀⠀⠀⠀⠀⣿⣿⣷⣮⣙⠻⠿⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣆⠂⣀⣀⡀⠀⠀⠉⠐⠀⣀⡤
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣔⣀⣉⠐⠛⠿⠾⠶⠦⠶⠶⠶⠟⠛⠋⠉⠁⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⢯⣿⣿⣿⣿⣿⣟⡻⠿⠶⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡁⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣎⠻⣿⣿⣿⡽⠭⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠥⠠⠄⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢿⣿⣛⡛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣆⣠⣴⣤⣠⣤⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣠⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣠⣴⣿⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠘⣦⣤⣤⣄⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠙⢻⠿⢿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠁⠈⠘⠀⠙⣿⣿⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣴⣶⣿⣶⣴⣄⠀⢀⣤⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡀⣿⣿⣿⡿⠿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⢻⣿⣿⠀⠀⠀⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠈⢻⣿⡀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠈⣿⡇⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣈⣿⣿⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡿⢙⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠻⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠁⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠔⠊⠁⠀⠉⠒⢎⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡠⠐⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠉⣢⠠⠤⠤⢀⡀⠀⣂⠠⠤⢀⡀⠀⠀⠰⠄⠒⠚⠀⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠈⠒⡄⠀⠀⠀⠀⠀⠈⢆⠀⡀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⠅⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠢⠄⢀⠈⢄⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⣀⠔⠁⠀⡀⠀⠀⠀⠀⢀⠜⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠂⠉⠢⣄⡀⠀⣀⠤⠊⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠂⠤⠄⠐⠁⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠑⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠒⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡐⠁⠈⢆⡀⠈⡩⠥⡄⠀⠀⠂⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁⠀⠀⠀⠉⠉⠀⠀⠈⠓⠒⠉⠈⠢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡶⠀⠊⠉⠢⠤⠴⡀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠾⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠱⢀⠠⠊⠀⠑⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣀⣀⣀⠠⡎⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⣀⠜⠀⠀⠠⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⡑⠢⡀
⠀⢾⠀⠪⣀⡀⠣⢀⠤⠤⠠⡀⠛⠠⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠑⠒⠢⠄⠤⠁⢀⠇
⠀⠀⠉⠑⢆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⣉⡉⠁⠒⠒⠒⠒⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⠒⠒⠒⠒⠂⠉⠉⣀⡀⠄⠀⠀⠀⠀⢀⠔⠊⠁
⠀⠀⠀⠀⠀⠉⠒⠒⠒⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⣄⢀⡠⠤⠤⠀⢀⠜⠀⠀⠀⠀⠉⠉⠀⠀⢀⣀⡤⠴⠂⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⡉⠉⠉⠁⠒⠤⢀⣀⡀⠀⠀⠀⠈⠉⣲⠀⢀⣈⣁⠀⠄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠐⠒⠒⠒⠒⠒⠠⢄⣀⣀⡀⠀⠈⠉⠁⡠⠔⠖⠐⠒⠂⠒⠀⠀⠒⠒⠒⠒⠒⠒⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣾⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠀⣀⣪⣔⠶⣿⣧⠂
⠀⠀⠀⠀⠀⢰⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣐⣿⣿⣿⣮⣿⣿⣿⣦⣿⣿⣯⡐⠂⠀⠀⠠
⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣶⡤
⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠝⠙⠛⠙⠛⠋⠉⠀⠐
⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢤⣠⣀⣀⣀
⠀⠀⡀⠉⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⢿⣿⡿⣿⣿⣿⣧⣄
⠀⠈⠰⠶⠶⢶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣀⣀⡀
⠀⠀⠀⠀⠀⠉⠉⢉⣹⣟⣋⠙⣿⣍⠿⡿⢧⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⡀⠀⠛⡿⣿⣿⣿⡄⢙⣿⣿⡀⠀⠁⣤⣋⣙⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠅⠉
⠀⠀⠀⠀⠉⠀⠠⠀⠀⠙⠛⠁⠈⠉⠿⣿⢿⣶⣿⣿⠿⠟⠾⢿⣿⡟⠛⠙⠟⠉⣿⠉⠉⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠠⡀⠋⠊⣁⣠⣤⡴⢶⢦⠸⣿⣿⣤⣤⣠⣼⢿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠻⡟⢋⠙⠁⠀⢰⠿⣾⠟⠛⢹⡿⠷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠰⠷⡀⠀⠐⠲⡤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣄⣀⣠⠴⠒⣒⣒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⣀⣤⠤⣤⡴⠋⠛⠋⠉⠁⡴⠋⠀⠈⠉⠉⠉⠙⠓⠦⣌⠳⣄
⠀⠀⠀⠀⣼⠱⠂⠀⢸⠁⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣈⣧
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠙⠯⢽⡲⣄
⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⢠⠪⠀⠀⠀⠀⠀⢀⣠⣤⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁⠀⠀⠀⡠⠚⠁⠀⠀⠈⠻⠝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠂⠀⠀⡠⠴⠖⠒⠶⢾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠠⠤⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠙⠛⠋⠓⠒⠛⠲⢤⣤⣀⣠⡾⠵⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣤⠔⠓⢆⡀⢐⣦⣠⠼⠷⡖⠀⠀⢀⠠⡾⠀⣤⣤⣤⣭⣥⣀⣀⡀⠀⠀⠘⠿⠃
⠶⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⢶⣮⣷⢶⡚⠙⢯⣀⣈⣶⣤⠀⠀⠀⠀⠀⢱⠀⢒⣒⣒⣿⣿⣿⠿⠇
⠀⠈⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⣧⠟⢾⡉⠉⢳⣄⡈⠀⠀⠠⠄⠀⠀⠀⢴⠚⠛⠩⠉
⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠺⠗⠀⠀⠀⠀⠀⣿⠶⠖⠛⠛⠩⡈⠀⠐⡀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣶⢆⣶⠁⢀⣀⠄
⠀⠀⠀⠀⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠰⠷⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⠀⠀⢸⣿⡆⢰⣿⢸⠇⣴⣿⣿⣿⣦⢠⡀
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡭⠋⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⣾⡄⢿⣿⣿⣿⡿⢰⡿
⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣾⢿⣁⠀⠀⠙⠿⢦⣽⡿⣿⡁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠈⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⠀⠀⠀⢸⣧⠀⠀⣀⣠⣤⣤⣤⣤⣤⡶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⣼⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠠⠆⠀⠀⠀⣀⠠⠐⠂⢩⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣥
⠀⠀⠀⠀⠈⢳⠀⣠⡾⠀⠀⠻⢦⡀⠀⡀⠀⠀⠈⠲⢤⣤⠤⠂⠉⠀⠀⠀⠈⣿⠈⢳⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣤⠜⠛⠸⡟⠁
⠀⠀⠢⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠙⢾⣅⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢤⣴⡿⢀⢸⣿⣦⡀⠀⠠⠋⠉⠀⡾⠿⣿⣇⡀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⣨⡶⠒⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⢠⠀⠁⢠⠀⢀⣀⣼⣿⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠈⠳⠀⠀⠀⠀⠀⠠⠄
⠀⠀⢸⡁⢀⡀⠀⠀⠀⢠⡀⠀⠀⢀⣼⣿⡟⠀⠀⠠⠃⠀⠀⠘⠀⠈⠻⢿⣿⡀⠀⠀⢸⡿⠛⠀⠀⠀⠄⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⠺⣦
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⣿⢿⣧⣾⣿⡏⠀⠀⠰⠁⠀⢠⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠰⠀⠀⠘⠉⠀⠀⢀⠠⠾⠗⠀⠈⠷⠀⠀⠹⡆
⠀⠀⢀⡆⢀⣠⠄⠀⠀⠘⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠘⠉⠀⠘⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠈⡳⠚⣻⠶⡖⠠⣾⣤⡀⠀⠀⠈⠁⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⠀⠀⠺⣿⡶⠀⠀⠀⠀⠀⡀⠤⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⣤⡯⠀⠉⠀⢻⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢘⠀⠀⣶⣿⠓⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣾⣶⠀⠤⡤
⠀⠀⠀⠛⠘⠃⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢁⣻⣧⣆⣀⠀⠀⠀⠻⠆⠀⠀⠹⣇⠀⠉⠀⠀⠀⠀⠀⣀⣀⠀⠀⣰
⠀⠀⠀⠀⣀⡀⠀⠀⠀⣸⣷⠸⡇⠀⠀⠷⠔⡇⣤⣽⡿⢿⣯⠳⡄⠀⠀⠀⠀⠤⠠⣿⣷⡄⠀⠀⠀⠀⣠⣄⡈⣀⡀⠉⠀⠀⠀⣀⣀⠀⢀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⣿⡁⠀⠀⠀⠘⠿⠄⠀⠀⠀⠠⠄⠙⡁⠁⠀⠀⠙⣷⢹⡄⠀⠀⠈⠀⠀⢸⣿⣿⡄⠀⠀⣠⣻⣿⢹⣿⣿⠀⠀⠀⠀⠈⠻⠷⠎⢘⠿⠚⠃
⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢀⣲⠔⠀⣴⠂⠁⠀⠀⠀⠀⣿⡞⠁⠀⠀⠀⠀⠀⢸⠿⠟⠀⢀⡀⠘⠿⣿⡆⢿⡿⠀⣀⣀⠀⠀⠀⢀⠰⠃⣴⡶⣶⣶⣆
⠀⠀⠀⠀⠉⠀⠀⢀⣀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⣰⠉⡷⡈⢿⡀⠀⢿⣷⣤⢡⣇⢹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠠⠀⣤⣠⣤⡴⠀⠀⠀⣀⣀⣀⣀⣤⠀⠀⣀⡄⠀⠀⠘⣧⣑⢼⡾⠁⢰⣮⣿⣿⡗⠙⠉⠛⠁
⠀⠀⢀⣀⣤⡴⠖⡁⠀⠀⠀⠻⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢠⡀⠈⠉⠉⠀⠀⠀⠀⢹⡟⡶⠭⠛⣚⣟⣿⡄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠻⠟⠀⠀⢀⡄
⠀⠀⠀⠀⠉⠀⣠⣡⡶⠠⢤⡤⠦⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣿⡆⠀⠀⠀⣐⣦⠀⠈⠐⠫⠔⠃⠉⠉⢿⠓⣀⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⢠⣽⡿⢿⡔⠋⠁⠀⠀⢠⣀⡀⠀⠀⠈⣷⠀⠀⠈⠋⢀⣤⣾⣫⠏⠀⠀⠀⣠⠇⠀⠀⠄⠈⠸⠏⠀⠀⠀⠀⠀⢠⠀⠀⠈⢀⠇⠀⠀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠆⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠘⠓⠀⢠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠌⣀⠀⠀⠀⡀⣀⡼⣯⣄⠀⠴⠀⠀⠀⠀⣤⡄⢤⣤⡵⠶⢻⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠤⠀⠉⠠⠀⠈⠀⠸⣧⠀⠀⠀⠀⠀⠀⡌⠀⠙⠀⠘⠿⢷⡴⠞⣡⣠⠀⠤⡄⠀⠀⠰⢞⡿⢦⠙⠿⣿⣿⠆
⠀⠀⠀⠀⠀⢀⡀⠺⠀⠂⠂⠀⣀⣀⣀⣾⡆⠋⠀⠀⠀⠀⠀⠀⠃⠀⠀⠦⠔⠳⢶⣦⣀⡏⠁⠴⡆⠀⠀⠀⠀⢦⠶⠶⢼⣋⣟⠻⣙⠃⠀⣰
⠀⠀⠀⠀⠂⠈⠙⠧⣦⠀⠐⢢⣀⡎⢍⠀⠈⠃⡰⠀⠀⠀⠀⠀⠨⡀⠀⢠⣤⠓⠂⠉⠀⠀⣠⠶⣴⠇⠀⠀⢠⠘⣴⡆⢯⠘⢣⡋⠼⣧⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠑⠫⠶⠜⣀⠤⠖⠁⠀⠀⠀⠀⠀⡀⢰⣐⣤⠀⣀⣀⣠⣠⡞⠉⠀⠀⠀⠀⠀⠀⠁⠈⠳⠞⠖⠈⠅⠓⠈⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠔⣊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠁⠘⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣀⡔⠚⣄⠀⠆⠀⠀⠀⣀⡈⠇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⢀⣿⠃⠀⢯⣀⠀⠁⠀⠀⠈⢁⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣽⡇⠀⡏⠘⢠⡀⢠⡄⠀⠀⠀⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⣿⡉⡀⢳⠈⠁⠀⠀⠀⢀⠀⢠⡿⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠈⢻⠁⠘⠀⠈⠧⠀⠀⠙⠁⠞⢁⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠷⢄⠀⢀⡀⠀⠀⠀⢀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⣿⡿⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⣼⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠋⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⢁⠋
⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣐⠂⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⢀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⢀⡀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠢⢤⠀⢠⡠⠀⠀⣠⢀⡲⠶⠔⡉⢅⠉⠀⠀⢭⡉⠐⠦⡀⠀⠀⠀⠀⠨⠤⢉⡀⠄⠠
⠀⠀⠀⠀⠀⠠⠔⠀⠐⠀⠼⡋⠉⢹⠈⠑⠐⣀⠤⣶⣧⠴⠄⠊⢠⡦⢆⡷⠀⠽⠀⠀⡔⣩⣉⢀⡀⠴⣾⠇⣐⠀⠀⠀⠀⠤⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢄⠙⠟⠛⣆⣉⡔⣀⡠⣔⣎⠰⠤⠚⠁⠀⠀⠀⡂⠉⣁⠉⣠⣤⣀⠅⠉⣠⠒⠷⠦⠊⡶⠀⠒⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠘⢆⠤⣄⡀⢌⡨⢵⠀⠌⢰⡀⢠⠰⠞⠁⠠⢁⣄⡘⠿⠈⡃⣠⠈⡛⢲⣋⡴⢣⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐⠈⠔⠂⡀⠰⣄⣈⠐⠀⡤⠃⠤⣈⣧⡀⠧⠭⠀⣀⣤⠃⠀⠉⠉⢉⠔⠁⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠀⠀⠢⢄⡄⠀⠐⠃⠀⠎⠙⠛⠛⣛⠉⡑⣲⠐⢀⣣⠀⠀⠀⡀⡄⡀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⠐⢠⡠⠖⠀⠴⠆⣤⣤⣠⣤⠅⡥⢨⠟⠀⠀⠀⡆⣆⡗⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣄⠒⠒⠂⡠⢀⣐⡐⠲⠔⠒⠚⢖⠦⠚⠃⠀⠀⠀⠃⠓⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢀⠀⠀⠀⠀⠀⠈⠒⠀⠄⠀⠀⠀⠀⠀⠐⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠍⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⡀⠀⠀⠀⠀⠈⣀⠤⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠐⢉⠁⠀⠀⠀⠈⡒⠰⢂⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠁⠀⠀⠂⠄⢀⠀⠀⠀⡀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⣤⣾⣸⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠰⠟⠙⠾⣷⣶⣮⣾⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠶⠀⠀⠀⠈⢿⣿⡟⡘⠂⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣋⡭⠀⠀⠀⡀⠀⠀⢠⣠⣀⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣠⠚⣽⡁⣴⣿⠏⠁⠀⠀⣠
⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠹⣿⣭⣽⠁⣠⠔⢩⡇
⠀⠀⠀⠲⣄⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣯⣿⠋⠀⢀⡨⠃
⠀⠀⠀⠀⠘⠷⠟⠛⢷⣤⡄⣢⡀⠀⣀⠀⢀⡜⢁⡔⠀⢀⠀⢰⣶⢬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⢇⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠿⠿⠿⠿⠛⠚⠿⠿⠞⢧⣀⣏⣠⡦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⡉⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⢤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠁⠀⠀⠈⠓⢄⡀⠀⠀⠙⠀⠒⢦⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⠈⠒⢤⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢡⡀⠉⡐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠉⠀⠠⢀⠀⠉⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠈⠸⣧⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢠⡀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣧⠀⡀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠠⠀⠈⣳⠀⢿⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⠀⠀⣷⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠸⡄⠀⠀⣹⡆⢀⡀⠀⠠⠀⠀⣸⠀⠀⡎⠀⢀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⡟⣰⡇⠀⠐⣿⠁⠀⡇⠐⠁⠠⣰⠇⢠⣿⣧⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⠷⠷⠶⣿⣶⣾⣷⢂⣀⣴⣏⣠⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡄⠢⠀⠂⠐⠀⠃⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆⣰⡰⡄⠀⠀⡸⠠⠄⠀⠐⠀⠀⠀⠀⠖⠄⠀⡄⠀⢠⠀⠀⠀⠴⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⢀⠇
⠀⠀⢀⠀⠀⠀⠁⠆⠀⠀⠀⢠⠈⠁⠙⡀⠠⡇⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⢀⡀⠈⠁⠠⠄⡀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠈⢀⠀⠀⠘⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠈⠈⠁⠀⠁⠀⠀⠧⠀⠀⠢⡀⠀⠀⠂
⠀⠀⢀⣀⡤⠀⠈⠀⡠⠄⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠉⠉⠉⠉⠙⠒⠶⣄⠀⠠
⠀⠀⠈⠉⠀⠠⠀⠒⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠠⡄⠀⠀⠀⠀⠀⢠⡇⠀⠐⠀⡀⠁⠀⠀⢰⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠁⢀⡀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠉⠀⢢⡔⠢⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⠀⠀⡀⠐⠀⠢⠄⠀⠖⠚⠃⠁⠀⠀⡈⠂
⠀⠀⠀⠀⠀⠀⠒⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠡⠀⢀⠠⠀⠀⠤⢀⣀⠠⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣿⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⢀⣀⡙⣁⣤⠤⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⢀⠀⠀⡆⠀⠀⠃⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠁⣀⠀⡢⠴⠇⠀⠀⠃⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣤⠀⠦⠀⠀⠀⠀⠀⠂⠉⠀⠀⣀⣠⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⣀⣠⠤⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⡀⠂⠀⠓⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠐⠋⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠛⠀⢀⣦⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠦⠀⠀⠀⣸⣿⣿⢻⣿⣟⣿⣿⣿⡟⣿⣿⣿⣿⡇⠀⢀⢨⠾⠡⠁⠀⠀⣠⣶⣾⣿⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⠃⠀⠸⡉⠁⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⠀⠖
⠀⠀⠀⠀⣀⠀⠻⠃⠀⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠃⠀⠀⠀⠼⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠸⠆⠀⢠⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠟⠀⠈⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠰⡆⠀⠀⡀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡄⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠰⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⠀⣀⡀⠀⠰⠀⠀⠀⠀⡀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠐⠀⠂

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣿⣿⡿⠻⣿⠛⠿⢿⡿⢿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠉⠁⠀⠀⠀⠀⠉⠉⠙⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣡⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠟⠛⠛⠿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢾⡏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣾⢁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣠⣿⠀⣰⡟⡇
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠉⠻⣿⣻⣧⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⣀⣼⡿⣷⣤⣴⣾⣿⣷⣾⣿⣠⠇
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⣾⣿⢿⣦⣶⣤⣶⣾⣿⠟⠛⠛⠛⠛⠉⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢸⣇⣀⣤⣶⣶⣤⡀⠀⠀⢸⣿⣿⣶⣶⣾⡿⠟⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠈⢻⣿⠋⠉⠙⠻⣷⣤⣾⡿⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣄⡀⣀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠒⠒⠒⠒⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⡀⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠈⠉⠓⢦⡀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣸⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⣀⣀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡿⠀⢀⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⠉⠉⠙⠳⢦⡉⠳⡄⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡼⠃⢀⡼⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠙⣦⠙⠆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⣤⡤⠴⠛⠓⠛⠻⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣸⣧⣄⣠⣤⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡴⠶⠦⢭⡀⠀⠀⢀⡿⠦⠤⠴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⠀⠀⢹⣦⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠐⠀⠀⠀⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⡰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢿⠧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⣀⣀⣀⣀⡀⢁⠶⠭⣓⠦⡀⠀⠙⠔⢅⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠖⠀⠠⠖⠛⠉⢉⡔⠣⢌⢙⠣⣍⡒⢌⠳⡄⣄⠀⠀⡀⡀⡢⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌⠀⠴⠐⠁⠀⠀⠀⠀⠀⠑⠰⢯⡈⠑⠠⣁⠡⡄⢉⠀⢣⠙⣌⠳⣄⠀⠘⠄⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠑⠈⠢⠑⡇⠘⡄⠠⡀⠑⢬⣑⠲⣄⠀⠂⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⡀⠀⠘⡀⠡⡘⢤⡘⠂⡀⡀⠂⢤⡑⣌⢣⠁⡀⠈⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⠂⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡁⠑⣄⠳⡜⢆⠱⢌⠢⢤⡂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⡙⢅⠈⢢⠙⠀⡑⠦⡽⠊⠀⢦⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠣⡀⢡⠈⠢⢌⡀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⡐⠀⠌⠀⠀⠀⠈⠃⠠⡄⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢆⡈⠑⢢⠍⠀⠀⠀⠀⠐⠀⠑⡀
⠀⠀⠀⠀⠀⠌⠀⠘⠄⠀⠀⠀⠀⠠⠀⢳⡀⢀⠀⠤⢤⠤⠂⠀⠲⠂⠰⠂⠐⠢⠄⢀⣀⠀⠀⠀⢠⠏⠢⠌⠛⠀⠀⠀⡀⠀⠀⠀⠀⠄⠈⠄
⠀⠀⠀⠀⠌⠀⡐⠀⠘⡀⠀⠀⠀⠀⡡⠤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠰⣏⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠈⢂⠈⠂
⠀⠀⠀⠐⠀⡐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠈⢄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠁⠀⢰⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡉⠐⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠦⣙⣤⠀⠀⠀⠀⠀⠀⠀⢮⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⣦⣫⡶⠋⠀⠀⠀⠀⠈⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⠿⣷⣋⡦⠀⠀⠈⢻⢲⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⡲⠆⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⡾⠀⠑⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡔⠁⠀⢀⠄⣢⡆⠀⠀⠀⠀⠀⠀⠘⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣻⡦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡀⠀⠀⠀⠀⠀⠀⢀⡈⢧⠀⠀⠀⡇
⠀⠀⠀⢀⠊⢠⠃⠰⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣾⡇⠀⠀⠀⢠⣿⡟⠀⢳⠀⠀⡇
⠀⠀⠀⠊⠀⠁⠀⠀⠀⢠⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⠀⠘⠛⢁⠀⠀⡇⠀⡇
⠀⠀⠶⢀⡵⠀⠀⠀⠀⢨⣼⠁⠀⠀⠀⠀⠀⠀⢀⣠⣖⣮⣽⣦⣤⣴⣦⣤⣇⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⢄⡀⠸⡄⡇
⠀⢘⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠛⠷⠟⠀⢻⡇
⠀⢨⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠿⣿⣿⠟⠛⠉⠉⠉⠉⠉⠉⠙⠻⣅⣀⠤⢊⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠁⠀⠀⠀⠰⠀⣠⡇⣸⡃
⠀⣈⡀⠠⠀⠀⠀⣸⡿⢳⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠒⠈⠙⠲⠤⠤⠤⡤⠤⠤⠔⠒⠚⡉⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⣿⡷⠀⢀⡴⠏⢀⣼⣿⢠⠟⡇
⠀⡍⡄⠰⣴⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠐⠀⠀⠉⠀⠀⠀⠀⠂⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⠉⢀⡤⠖⠋⢀⣤⣿⣿⢿⣿⢘⡇
⠀⠳⡙⡄⠈⢿⣷⣤⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⠤⣄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣠⣒⠋⠀⢀⣠⣴⣿⣿⣟⣵⣿⡇⢨⡇
⠀⠀⢹⣿⣷⣦⡙⢿⣿⣷⣤⣘⢮⣭⡹⠳⠂⠀⢀⢾⣷⠀⠀⠀⠀⠀⠀⠸⠶⠾⠿⠋⠀⠀⠀⠀⠀⠿⠏⠀⠀⠀⠚⠀⢀⣴⣦⣴⣿⣿⣿⡅⣴⣿⣿⣿⠇⠰⡇
⠀⠀⠈⡄⠙⠿⣿⣜⠿⠿⠀⠙⢻⢞⡂⢀⡀⠀⠻⠞⣡⣶⡦⠀⠀⠀⠀⠀⠄⠀⣀⠠⠤⠤⠀⠀⠀⠀⠀⣀⣀⣠⣤⢰⣿⣿⣿⣿⡿⠟⠃⣼⣿⣿⣿⠏⠀⢸⡇
⠀⠀⠀⠱⡀⠀⠈⠿⢿⣶⣦⡀⠈⠛⢿⡿⣷⠿⠷⣾⣿⣯⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣯⣤⣴⣾⣿⣿⣿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠛⠿⣿⣦⣤⣤⣤⣤⣀⣀⡀⠈⠛⢿⣿⣿⣿⣿⣿⠿⠟⠻⠛⢛⣿⣋⣽⣷⣶⣬⣟⣛⣉⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠈⠉⠉⠛⠟⠿⢿⣿⣿⣦⣄⡀⠉⠛⠛⢉⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣴⠀⠀⠀⠀⠉⠈⠛⠿⣿⣷⣦⣤⣶⣿⠿⡿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⣤⣤⣀⠀⠀⠑⠛⠛⢿⣯⣶⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠾⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠶⠧⣷⣤⢜⣫⣧
⠒⢬⠂⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢨⠃⠠⠀⠀⠀⠀⠀⠀⠹⣿⣶⣄⣠⠀⠉⠻⠶⠻⠆⠀⠀⠀⠀⡀⠀⡀⢠⡄
⠒⠉⠱⣣⠆⡀⠀⠀⠀⠀⠀⠀⢠⣤⠠⢘⠃⠠⠴⢣⠿⠆⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣿⡟⣴⣤⡀⠀⠀⠀⠀⠀⠶⢀⡉⠉⣱⡂
⢁⠀⠈⠡⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢌⠁⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⡇⠏⢻⡿⢿⡦
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣷⣤⡀⠈⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡠⠒⢩⣤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠟⠺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣄⣀⣀
⠀⢰⠁⠈⡙⢿⣷⡀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⢹⣏⣉⣀⣢
⠀⠈⠀⢇⠀⠹⣿⣧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢤⢀⡄⠀⠀⠈⠛⢯⡙
⠀⠀⠀⠀⠀⡀⠙⠛⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣴⠖⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠙⢥⠖⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠀⢺⣽⠀⠀⠀⠀⠀⠹⠁⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢻⢲⠟⠛
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢡⡶⠠⡄⠀⠀⠀⠀⠀⠀⠀⣋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠀⢸⠁⢀⠆⠀⡐⠂⠲⠀⠀⠀⣿⠁⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⣀⠚⢢⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢷⠀⠀⠉⠁⠀⠀⢣⣠⣀⠀⠀⠀⠋⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⡀⠶⣀⠺⢂⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠀⠀⠀⠀⠳⠄⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠃⠀⠀⠀⠀⠘⠋⡤⠉⠴⢏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣠⡀⠛⠴⠎⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡅⠀⠀⠀⠀⢀⡌⣹⣦⡀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠇⠀⠀⠀⠀⠀⠤⠉⠁⠛⣷⣟
⠀⠀⠀⠀⠀⠀⣠⠾⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡠⠀⡀⠀⢀⠢⢤⡀⠀⠀⡏⠀⢻⢀⡤⣠⠊⠉⠀⣀⠤⠄⠀⢞⢡⠀⠑⢆⠀⠐⠤⡀⠀⠀⡞⢳⡄⠀⢠⠟⠓⠦⣔⠒⠓⠦⣄⣀
⠀⠀⢀⠰⢀⠑⣄⢻⠀⡠⠓⠀⠀⠈⠓⠁⠸⡀⣀⠇⠀⢸⣇⠀⠀⠀⠈⠉⠁⠉⠁⠤⠤⠒⠳⡄⠀⣧⣸⠃⢀⡟⠀⠀⠀⠈⠙⠒⠶⢤⡈⠉⠉⢱⡀
⠀⠀⠀⠂⢢⣽⣿⡻⠋⡠⠔⠒⠉⠉⠓⠦⣀⠀⠀⠀⠀⢸⣧⠤⢄⣀⡀⠀⠀⠀⠈⢉⣀⡈⢽⡃⠀⠼⠁⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⢀⡟⡄
⠀⠀⠀⢊⢀⠔⠁⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⢸⡅⢀⡀⠀⠈⣈⡉⢉⣉⣀⠀⠀⣰⠇⠀⠀⣠⠞⠀⠀⠀⠀⠁⠀⠀⣠⢎⡵⠁⠀⡼⡸
⠀⠀⠀⠀⠁⢀⡎⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⣇⠀⠙⠒⠦⠬⠤⠤⠄⠀⢀⣀⠠⠤⠘⠁⠀⠀⣧⣀⠀⠀⠐⠀⠀⠀⣰⢣⠏⠀⠀⢠⢇⡇
⠀⠀⠀⠀⠀⠸⣇⠠⡀⠀⠀⢀⡞⠁⠀⠀⠙⢦⠀⠀⡿⠀⡴⠾⠶⡆⠀⠀⠀⠀⠀⠠⢤⣄⣀⠀⠀⠀⠘⠫⢽⣲⡦⣄⠀⢠⠇⣾⣀⣀⣠⡼⢸⠁
⠀⠀⢀⣤⣄⠀⠙⢦⡀⠀⠀⠘⣇⠀⡀⢀⡄⣸⠀⢠⠇⢸⠁⠀⣰⠏⠀⢠⠒⠒⢦⡀⠀⠀⠈⣧⣀⣀⣀⣀⣀⡈⠙⠲⣝⣫⠞⠓⠶⠶⠶⠖⠋
⠀⠀⣾⠁⠈⠳⣄⠀⠉⢦⡀⠀⠈⠳⠬⠵⠞⠁⢠⠟⠀⠘⠒⠚⠁⠀⠀⠘⢮⣀⣠⡇⠀⢰⣯⣽⣏⣀⣠⣤⠤⠽⠇⠀⠀⠀⠀⠀⢠⠖⠒⠲⡄
⠀⠀⣿⡄⠀⠀⠈⠑⢤⣀⠑⢄⡀⠀⠀⢀⣠⡴⢋⣠⠴⠖⠒⢛⡓⠒⠲⢦⡀⠀⠀⠁⠀⢸⠀⠀⢹⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠏⠀⠀⠠⢹⡀
⠀⠀⠸⣷⡀⠀⢀⣤⣤⡈⠑⢦⡈⠉⠉⠉⠁⡴⢯⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠲⢄⡀⠀⠸⣄⠤⠤⢷⡤⠤⠤⣤⡇⠀⠀⠀⢠⣾⠀⠀⠀⠀⠃⢦⡀
⠀⠀⠀⡟⣧⠀⢻⠀⠀⡝⣦⠀⢹⡄⠀⠀⡞⠀⠈⣧⠈⠀⠀⠀⠀⠀⠀⡎⠀⠀⢠⢙⡆⠀⣟⠒⠒⠚⣷⠚⠛⢉⡇⠀⠀⠀⠸⡍⢙⠋⠙⠉⠉⣩⠁
⠀⠀⠀⢿⠘⣇⠀⠳⢤⣤⠟⠀⢰⠇⠀⢰⠳⣕⠀⠘⣆⢀⣒⣒⣀⣀⣸⣇⣀⠐⣡⠞⢳⠀⢻⠀⠀⠀⠸⣆⠀⢸⠃⠀⣀⡀⠀⠙⠾⣷⣤⡤⠞⠁
⠀⠀⠀⠈⢧⣈⠳⢤⣀⣀⣠⣴⠟⠀⠀⢿⣐⣨⠷⠞⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⢧⣀⣸⠇⠈⢧⣀⣀⣀⣉⣠⠜⠀⢯⠉⢧⠀⢀⣠⡴⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⠋⣁⣀⣀⡀⠀⢉⣁⡀⠀⠀⠀⠸⡳⣶⣄⡇⠀⠀⠀⠀⠈⠁⢀⣴⣦⣍⡉⠉⠀⠀⠀⠀⠈⠓⠻⣄⣈⡓⠶⠴⠶⠞⠃
⠀⠀⠀⢰⡒⠦⣴⡒⠋⢉⣉⣀⠠⠀⠉⢶⡁⠀⡝⠀⠀⠀⣠⠽⠾⢿⣧⣠⠤⠤⣄⡀⠀⢸⣝⡻⠷⣯⣭⣍⣉⣉⣉⣛⠲⢤⣀⠈⠉⠓⠦⣄
⠀⠀⠀⠘⣧⠀⠱⢭⣒⣤⣀⠀⠠⠄⠀⠀⠱⡀⢨⠆⠀⣼⢡⠊⠈⠀⠈⠁⠀⠀⢠⠹⡄⠈⠻⢿⣷⣶⣶⣯⣭⣭⣽⣛⡻⠶⣌⢳⡄⠀⠀⠘⣆
⠀⠀⠀⢰⡿⡦⢤⡘⣿⣭⢻⢷⣦⣄⡀⠀⠀⠑⣌⠆⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠶⠤⢤⣉⣉⣉⣉⡉⠛⠻⢷⣼⠋⠀⠀⠀⠀⣿
⠀⠀⢀⣀⠓⢬⣒⢵⡀⠛⠉⣟⠀⠛⢿⢗⠦⣄⡈⣧⠀⠸⡆⠀⠀⠀⠏⠘⠆⠀⠀⢰⠁⠀⢸⠭⠶⠶⠶⠶⠶⠦⣭⣍⡓⠤⡉⠀⠀⢰⠋⠉⠉⣹⠂
⠀⠀⠀⠈⣳⡀⠈⠑⠮⣓⢦⣈⠉⠳⡀⢀⡙⠲⠽⠇⠀⠀⠹⣔⠤⡀⠀⠀⠀⠄⣠⠏⠀⠀⠘⠿⢿⣒⣒⣒⣒⣒⠢⠌⡙⣾⠀⠀⢠⡟⡶⠒⢲⣿
⠀⠀⠀⠑⢿⣧⣤⡠⠴⡌⠑⢬⡓⢦⣉⣁⡸⠂⠁⢀⣀⠀⠀⠈⠓⠒⠛⠓⠒⠛⠁⠀⣀⣤⠴⠶⠲⠶⠦⣄⡀⠈⠙⢦⣄⣷⠀⠀⠸⠶⢷⣬⣿⣸
⠀⠀⣠⢾⡿⠋⠛⠒⠋⠀⠀⠀⠈⠓⠖⠋⢀⡠⠞⠉⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⢉⢳⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠈⠁
⠀⠀⠉⢁⡤⠤⠤⠄⢤⣄⣤⠤⠤⡀⠀⠀⡿⠤⠤⠤⠤⢤⡴⢦⣄⣈⣓⠲⢄⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⡰⡠⠃⢀⡴⠞⣻⠋⠉⠉⠉⠓⢦
⠀⠀⡎⢹⠉⣿⠛⠛⠛⠛⠛⠛⠛⡏⠀⠀⣛⣷⠀⠀⣀⠀⠙⠋⠀⠀⠉⠉⢻⠀⠈⢧⡳⢤⡀⠀⠀⠀⠀⠀⢸⠁⢀⡞⠀⢎⠉⣠⠶⢶⣄⠀⠈⢧
⠀⠀⠹⣼⣧⠘⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣯⡀⠀⠸⠽⠂⠀⣠⣤⣄⠀⠀⢳⠀⠀⠀⠙⣆⢷⠀⠀⠀⠀⠀⠘⡇⠸⡇⣠⠊⠘⣇⠀⣸⣽⠀⠀⣸
⠀⠀⠀⢠⡛⣦⣀⡀⠀⠀⠀⣠⠗⡄⠀⠀⣾⠁⠀⠀⠀⠀⠀⠛⠶⠟⠀⠀⢸⠀⠀⠀⠀⢿⢸⡆⠀⠀⠀⠀⠀⢹⠀⢷⡸⢄⣠⣌⢉⣩⣁⡖⣲⠏
⠀⠀⠀⠀⠉⠛⠿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠙⠒⠒⠒⠒⠒⠚⠙⠒⠒⠒⠒⠚⠀⠀⠀⠀⠈⠳⣧⣄⣀⣠⣤⡤⠞⠀⠀⠙⠶⣤⣈⣉⣠⡭⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣄⠀⠀⠀⠸⣁⡲⢄
⠀⠀⢀⠀⠀⠀⠀⠀⠚⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣈⢧⡀⠀⢀⢿⣿⡄⡣⡀
⠀⠀⠀⡀⠲⢄⠀⠀⠠⢹⣌⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣻⡿⣿⣾⣿⣯⡻⣧⣷⢹⡠⡀
⠀⠀⠀⠹⢦⡀⠑⡔⣼⣇⢻⡌⢷⣦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢟⣉⣩⣍⣉⡚⠮⣍⠛⡿⣦⠂
⠀⠀⠀⠀⠀⣛⡚⣉⣛⡻⠗⠙⠈⢻⠻⣿⣦⣲⢄⡀⠀⣀⠤⣄⠀⠀⠀⠀⣴⣫⡼⠂⠳⠦⠭⢿⡾⢷⡇⡈⢻⣦⠚⡋⣦
⠀⠀⢀⢀⣠⠟⠓⠧⢤⣭⣓⣦⣄⠠⢎⠈⠻⣯⠻⣿⡫⣤⣿⡻⡀⣠⡾⠛⠉⠁⠀⠀⢠⡆⠀⠀⢬⡗⣿⣳⠀⢻⣿⣷⡙⡀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⣿⠞⡇⠀⠸⠀⠀⢀⣄⡁⣿⣿⣿⣆⠀⠉⠓⠈⠻⣿⢿⣷⣹⢻⡀⡇⠀⠀⠀⠀⠊⠀⣰⣿⡷⣷⢸⡈⣆⠀⢻⡿⡍⠻⡶⣦⣴⡉⢧⡾⠃
⠀⢾⡇⠸⠀⠀⡤⠀⢰⣿⣾⣷⣿⣸⢸⣯⡄⠀⠀⠀⠀⢸⠀⢻⢹⡜⣷⠃⠀⠀⠀⡄⠀⠀⠉⠩⠂⣏⣾⢿⠟⠗⠀⢣⢸⡆⢹⢘⡟⢧⠛⡄
⠰⣦⣧⠀⠀⠀⠀⠀⠒⠠⠉⠁⢿⡟⢸⠙⠈⠀⠀⠀⠀⣸⠀⣾⠻⡺⣿⠀⣈⣀⢰⡇⠀⠀⠀⠀⠀⣻⠇⢸⡄⠀⠀⠘⡆⡇⢸⠈⣿⣞⡇⢱
⠀⠙⣿⣀⣠⠀⠀⠀⠀⠀⠀⣴⡿⠃⢸⠈⢀⠀⠀⠀⠀⡇⢀⡟⠀⢸⡿⠒⢛⠛⠻⠦⣄⣠⠂⢀⠔⢹⠀⠈⡇⠀⠀⢻⡇⣇⢸⠤⡿⣿⣿⣎⡆
⠀⢸⠤⠤⣬⣅⠒⠤⣶⠀⣴⣽⢡⠀⢈⠀⣸⠀⠀⠀⣸⠁⣼⠇⠀⢸⣿⠙⣿⢹⡏⠑⣦⣉⠲⢧⣼⠃⠀⢰⢃⠀⠀⢸⡇⡟⢸⢀⢳⣷⡇⢻⣧
⠀⡼⢀⣰⣿⠀⠈⣶⠤⣙⠫⣷⡎⠀⣜⣾⢻⣆⠀⢠⠇⢠⡿⠀⠀⣿⡟⠀⠈⠀⠀⡼⠁⠀⠑⣾⠿⢄⣴⣿⡏⠃⠀⢸⢡⡇⡸⢸⡎⣼⣟⣊⢧⡀
⠀⣧⣿⡿⠃⠀⠀⠁⠀⣈⡯⣟⣠⣾⣿⣿⠀⠙⣀⡞⠀⣾⠃⠀⢠⣿⠇⠀⠀⠀⠀⠀⣠⠒⢛⣦⣴⡿⠋⠙⣿⠀⠀⡼⣸⡇⡇⡼⡸⠁⢹⣽⢦⣳⡄
⡼⠀⠉⠀⣠⡀⠀⢠⠶⢿⣴⣿⣿⠋⠈⢻⢀⡴⠋⣠⡾⠃⠀⠀⣾⠃⠀⠀⠀⣀⣀⣸⢙⣤⡞⣿⣿⡿⠀⠀⠻⠀⡰⣳⠏⠈⣷⣳⡷⡄⢸⠀⢻⣿⣿
⢧⠆⠆⢻⡎⢷⢠⣿⣤⡾⣿⣿⣿⣧⣠⠔⠉⣀⣼⠿⠀⠀⢀⡼⣿⡆⢀⠀⠠⣿⡷⡋⣿⣿⣷⡜⡻⣇⣀⢀⣴⠞⣱⠏⢀⣸⣿⣿⡴⡙⣾⠀⠀⢣⡠
⠈⢧⠀⠋⠀⠾⡾⡟⢸⠳⣽⣿⡟⠋⢁⣠⡾⠋⠀⢠⣏⢽⣯⠞⠁⠹⡜⠀⠹⠟⣵⡇⣿⡈⠙⠿⠦⠄⢉⣛⣠⣞⣃⡒⢛⣵⡿⢾⡏⢿⡄⠁
⠀⠘⠿⠦⠤⠴⠓⠃⠈⣷⣤⢉⣻⣙⠿⣿⣷⣶⣶⣿⠿⠟⠃⠀⠀⠀⠳⣴⣄⣈⣴⢵⣿⣿⡦⣄⡤⠀⡖⠚⢋⣉⣥⣶⣛⣽⣦⣤⣷⢫⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⣛⠛⠛⣉⠴⠚⣉⠕⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠳⡶⣇⠀⣿⠛⠛⢉⡰⠇⠁⢬⠴⠚⢁⠞⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣿⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠬⣹⡆⠻⣉⡽⠋⠀⣠⠶⠯⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢻⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⣷⣼⠏⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣻⢖⣛⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡎⠉⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠚⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⡄⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠁⣀⠴⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⡆⠘⠀⡀⡂⠀⠀⣠⡳⣸⠞⠁⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀
⠀⠀⠀⢀⠎⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⠀⠀⠀⠀⠀⣰⢣⡟⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠀⢀⠞⠊⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⡸⢁⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠒⠋⠉⠑⠲⢌⣇
⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⢃⠎⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢢⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡜⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢣⠀⠀⠀⠀⢿⠀⡀⠀⢸⠃⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠄⠄⡀⠒⠈⢯⣆⠀⠀⠀⢸⠘⡡⠀⢸⠀⣠⠀⣰⠃⠀⠀⠀⠀⠀⠈⠠⠠⠄⠀⣀⣀⠀⠀⠀⠉⠣⠀⠀⠘⣆
⢧⠁⠤⣔⡒⡛⠛⣡⣠⠤⠖⠛⠚⠚⠛⠛⠶⠤⣬⣦⣘⣇⣼⠀⠁⢀⠃⠀⠀⠀⠀⠀⢀⣤⠶⠒⠋⡩⠩⠍⠉⠭⠝⠛⠻⣄⡀⠀⠸
⠈⠓⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣸⠇⠂⠀⠀⠀⢠⠖⣩⠤⠒⠚⠉⠉⠉⠐⠓⠒⠢⠤⣀⡀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠄⡀⡄⣴⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⢺⣹⠋⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣾⠇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠴⠖⠒⠒⠲⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠒⠦⣄⠙⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⢘⢧⠀⠀⠈⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⡀⡇⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀⠐⠀⠂⠔⠠⠈⠐⠠⢀⠀⠀⠀⠀⢠⠇⢠⠇⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⣊⣠⠾⠖⠋
⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠛⠛⠶⣦⡘⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⠻⣿⠉⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⠲⠶⠶⣶⣶⠿⠿⠿⠿⠿⠿⠿⣷⣦⣄⠀⠋⠀⠀⣿⣇⣀⣤⣶⠶⠶⠶⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠛⠒⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠈⠙⣿⣆⠀⢠⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⢙⣷⡦⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡉⠁⠀⠀⠤⠤⠤⠶⠒⠛⠛⠉⠉⠉⠉⠉⠛⠳⢦⣄⢘⣿⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣛⣿⣃⣀⣀⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽⡿⠋⣍⣿⣤⣴⡶⠾⠿⠿⠶⣶⣤⣄⡀⠀⠀⠀⠉⠉⢉⣩⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣄⡐⠼⠶⠶⠛⣋⣡⡴⠶⠛⢋⣩⡿⠟⣿⠗⣨⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⣩⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⢟⣋⣥⣶⣶⣿⠟⠁⢀⣾⠃⣰⣿⣿⣯⢠⣤⠤⣂⡤⠄⠀⠀⢀⣠⣤⣤⣤⣤⣤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠟⠛⣻⡿⠛⠁⣠⣶⣿⡏⢰⣿⠉⠻⣿⣦⣝⡛⠛⠛⢉⣀⣀⣠⣤⣤⣭⣝⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⢀⣼⠟⠀⣠⣾⠟⠉⣾⠃⢸⡇⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣾⣏⡀⠀⠀⢠⡾⠁⢀⣼⠟⠁⠀⠀⣿⠀⢸⠁
⠀⠀⠀⠀⣠⣾⡿⠟⠋⡉⠉⢩⡿⠟⠛⠿⠿⣶⡿⠁⣠⡟⠁⠀⠀⠀⠀⣿⠀⢸
⠀⠀⢠⣾⡿⠋⠀⠀⠀⠻⣦⣈⣤⠆⠀⠀⠀⡿⠁⣰⣿⡀⠀⠀⠀⠀⠀⣹⡄⢸⣄⣀⣀⣀⡀
⠀⢠⣿⠏⠀⣀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣸⠃⣰⡏⢿⣷⠀⠀⣠⣶⣿⣿⣧⠈⣿⠿⠿⠿⣿⣿⣷⣤⡀
⠀⢸⣿⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡟⠀⢸⣿⣧⣾⣿⠟⠉⠀⠘⣦⠘⣦⡴⠀⠀⠀⠉⠙⢿⣷⡄
⠀⢸⣿⡀⠸⣿⠀⠀⠀⠀⢀⣀⣤⣤⣴⣾⡇⣼⣥⣤⣾⣿⣿⡿⠁⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⢿⣷⣄⠈⠃⢀⣴⣾⡿⠟⠋⠉⠉⢹⡃⡏⠉⠉⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠈⠻⣿⣦⣴⣿⠟⠉⠀⠀⠀⠈⢷⣼⣅⣧⣤⠞⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠈⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⣸⣿⠃
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋⢀⣴⣿⠏
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢠⣿⡿⠻⢿⣶⣶⣶⣶⣶⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠠⢤⣤⣤⣤⣤⣶⣿⠟⠁⢀⣴⣿⡿⠁⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣤⣬⣍⣉⣉⣉⣤⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⣀⡤⠤⠤⠤⠤⠤⣄⣀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣄⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠄⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢀⠖⠉⡉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡴⠶⠶⠶⠶⠶⠶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⢡⠖⠛⢛⢲⣞⠒⢇⠁⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠛⠋⠉⠉⠓⢦⠤⠤⣤⠶⠤⣤⠤⠖⢉⣨⠽⠛⣿⣆⠀⠀⠇⢴⠊⢱⠛⠉⠁⠀⢓⠩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⢀⣀⣈⣷⢖⣚⡷⢀⡔⠒⠛⠉⠀⢀⣴⡿⠋⠀⠀⠀⠘⠑⠎⡳⠖⢟⡠⠳⡄
⠀⠀⠀⠀⣠⣶⠿⠷⠈⠙⢷⣦⣀⡏⠉⠉⠁⠀⠀⠛⠉⠳⣄⣀⣤⡶⠛⢁⠀⣀⠤⠤⢤⡀⠀⠀⠁⢴⠈⢱⡆⣿⠐
⠀⠀⠀⠀⠛⣿⣄⠀⠀⠁⠀⡀⠉⠛⠷⠶⠶⢤⠶⠶⠶⠛⠋⢉⣀⣤⠖⠃⠰⢑⡄⠆⣤⢹⣄⡤⠦⠉⠙⠈⠡⠃
⠀⠀⠀⠀⠐⠈⠛⢷⣤⡀⣎⠉⠉⠀⠀⠀⠠⠀⠠⣄⢀⣠⣴⠿⠋⠁⠈⠀⠀⣈⠃⣄⠿⣸⠃⣠⠰⢤⡘⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣤⣤⣤⠴⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⢿⢦⡀⠗⣤⡺⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⠀⠀⠀⠠⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣴⣻⡌⡇⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠉⠂⠚⢉⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⢀⣤⣀⠀⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣴⣿⣿⣿⡆⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠒⠤⣀⣀⣀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠛⠿⠿⠛⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⡼⠀⠀⠀⣀⣤⡤⠤⠶⠤⣄⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⢺⡆⠀⠀⠀⠀⠀⢨⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠐⠒⠒⠒⠒⢠⡀⠀⠀⠀⠀⠀⢀⡤⠼⠧⠤⠤⠤⠤⠤⠼⠼⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠃
⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⢀⣀⡠⠤⠔⣊⡅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠐⢬⠭⠉⠉⠭⠭⠁⠤⠐⠂⠉⠀⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠘⡄⢀⣀⣀⡀⠤⠤⠀⠐⠒⠈⢹⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⢀⣀⣀⣀⠀⠻⣃⣟⠘⠃⠀⣠⠒⠢⠤⠭⠍⠉⠐⢲⠋⡇
⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⣤⣦⣄⠀⠀⠀⢸⠀⠀⠀⠀⠓⠐⠒⠀⠀⠀⠐⠒⠒⠒⠒⠒⠐⠀⠐⠃⠀⠀⡇⠀⠀⠢⠠⠐⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⢱⠀⠀⠰⣿⢿⣽⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠁⢀⡀⠀⠠⠤⠤⠄⠚⠃⢰
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⠻⠟⠋⢀⣀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⣔⡉⣑⢏⡠⢶⠀⠀⠀⠀⠘⠁⢠⢰⡴⠄⠶⠦⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠐⢖⠒⠒⠂⠈⠉⠀⠀⠀⡄⠀⠀⠀⡠⠠⠤⢊⠭⠠⠕⠀⣈⠔⠨⠥⠄⠒⠤⠀⠀⠀⠀⢰⣈⡡⡀⠤⠤⠤⠤⠒⠃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⠀⠀⣀⡠⠄⠊⠀⠀⠀⠀⡗⢒⡒⢁⠔⠒⠊⠀⠀⠀⠀⣀⠀⠀⡀⡀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢰⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⠂⠑⢊⣐⢂⡮⠭⠱⢞⢄⡨⠮⡀⢴⠀⠀⠀⠀⠆⠀⠐⠒⠂⠀⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠐⠠⣀⠀⠀⠀⠀⠀⢸⣁⢉⡈⡡⠤⠤⠊⠑⠒⠉⠑⠀⠒⠢⠼⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀⠤⡐⠜⠋⠁
⠀⠀⠀⠀⢀⠴⠈⠠⠤⠤⠄⠤⠤⠤⠤⠄⠤⠤⠤⠬⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠧⠤⠶⡈⢁⡀
⠀⠀⠀⠀⠘⣷⣮⡴⡤⠰⢶⣤⣤⣄⣄⣠⣡⣤⣐⣲⣶⠃⠀⠀⠀⠀⠀⠀⠀⣔⠈⡡⠄⠰⡀⠠⡤⢀⣐⠢⠤⣄⡉⢲⡄
⠀⠀⠀⠀⠀⢏⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⢋⡁⠃⠀⠀⠀⠀⠀⠀⠀⢙⠛⠠⠬⢵⣏⣘⣏⣁⣬⣡⠦⠶⠚⢛⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠀⣀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠀⠠⠦⠒⠂⠀⠐⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⣠⡇⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⢀⠀⡀⠀⠀⠠⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣢⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠐⠀⠁⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄⠒⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠁⠀⠀⢸⠂⠀⢀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠊⠀⠁⠀⠀⠀⠀⠰⠤⠀⠔⠃⠀⠐⢄⠀⣀⡀⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀
⣿⣟⣙⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠙⣼⠀⢀⣼⡞⠀⠀⠀⣀⡤⠤⠶⠶⠦⣤⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⢸⣽⣦⡀⢹⣷⣌⣽⣇⢀⣢⡾⠉⠀⠒⠈⠙⠒⢦⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠶⣌⢻⣿⣧⣠⣿⣿⣿⣿⣿⣿⣷⣶⣞⣭⠄⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠠⢾⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣫⣴⡖⠀⠀⠈⣒⢌⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⣁⠀⠲⣿⣿⡟⠚⠋⠉⠁⠀⠈⠙⠛⢿⣿⣿⡟⠀⠀⠀⢀⣸⣷⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⠤⠀⡸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠻⠀⠀⠒⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠐⣄⠀⠀⢀⣠⡄⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⣦⢠⢟⣳⠿⠛⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠀⠀⠜⠛⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⣄⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢸⠏⠷⣄⠀⣾⡛⠛⠻⠟⠛⠷⣶⣿⣟⢋⣷⠶⠶⢯⡿⢁⠷⢒⢍⣷⡴⠶⣶⡶⣤⣀
⠀⠀⠀⠀⠀⢠⡟⠐⣴⠉⠷⠿⡇⠀⠐⢲⣍⣉⡉⠉⠉⣻⡁⠀⣀⠀⢠⠘⠤⠍⢢⣾⣧⡀⢸⣿⣷⣿⣷⡄
⠀⠀⠀⠀⢠⠟⡀⣒⠠⠶⠂⡈⢤⠀⢝⡩⠙⠓⠚⠿⠛⢛⡀⢈⣁⣤⣪⣿⣲⠶⠻⢯⣹⣿⡿⠿⢻⡴⡀⡿⠹⡆
⠀⠀⢠⡟⢿⣦⡀⠤⠄⠁⠈⣁⠘⣤⠜⠴⠀⠀⠀⠀⠀⠾⠉⠁⠂⡌⠠⠛⣿⡇⣇⣨⡽⢟⣃⠂⣸⢱⡗⠃⢀⣿
⠀⠀⠈⢻⡿⢛⡁⠀⠆⠀⠀⠀⢀⣃⠀⠂⠀⠙⠐⠀⢃⡄⠐⠗⢤⣤⢀⡄⠀⢣⣿⣉⢀⠆⠈⠳⣅⣟⣀⢀⣾⣧⡀
⠀⠀⢰⡏⢠⠈⠘⢆⠀⢒⣀⡴⡤⠽⠂⠀⠀⣄⢀⣠⡴⢷⣤⣄⣀⡀⠀⣀⣀⢺⡿⢆⡤⠀⠀⠀⠀⠠⠀⣨⣟⣾⠁
⠀⠀⣸⠃⣈⠸⠀⡙⠁⠜⡏⠁⡂⡀⢄⡀⠠⣸⣍⡿⠛⠈⠙⠛⠙⠛⠛⠛⣭⣿⣷⣾⢃⡁⠀⠀⠀⠀⢀⣾⢾⣅
⠀⠀⢸⡄⠁⠦⢈⡃⠄⠘⣾⠁⠀⠀⠀⠀⡰⢸⡟⠁⠀⠀⠀⠡⢰⠆⠐⠋⡭⠛⡋⠉⠀⢀⣀⡀⠱⠠⢾⣷⣤⡏
⠀⠀⠈⣷⣤⣥⡈⣅⢀⠞⠂⠑⠁⢀⠀⠀⠲⣼⡄⡀⠀⠀⠀⠀⠂⠁⢠⣎⠀⣀⣤⣶⠿⠛⠉⠀⠀⣿⠆⣿⢯⡄
⠀⠀⢰⣿⣟⢰⣶⠈⠡⡐⠐⡆⢠⠄⣦⠴⠿⣟⠛⠗⠲⠄⣴⠆⣿⣄⠀⠈⡆⠂⢀⡀⠀⠄⠘⠀⠀⢄⣾⣿⣼⣇
⠀⠀⠈⢿⣿⣼⣿⣧⡴⠄⡄⠀⠂⠐⢛⠦⢀⠄⢀⡶⠠⣠⡤⠔⠋⠋⠉⠀⠀⠀⠤⢀⣀⠠⠐⢈⠐⠃⠻⡿⠷⠛
⠀⠀⢠⣻⣿⣷⣧⣀⣷⣄⠢⠐⠆⣸⠂⠀⢀⢈⣿⣇⡔⠹⠲⠂⠀⣠⠄⣀⠴⠆⠀⠮⣄⢀⣶⠿⣤⠀⣱⣷
⠀⠀⠘⣿⠿⠿⠏⠙⠈⣿⡟⠆⢠⣼⣤⠊⠉⠛⢷⠿⠲⢂⡽⠄⠀⠄⠀⠉⠀⠀⡏⢠⣦⠈⣠⡙⢩⠉⣿⣹⡇
⠀⠀⢼⠃⠀⠀⠧⠀⣚⢹⠇⠁⢀⣹⠇⠀⠀⢀⠲⢀⣄⠐⠂⣠⣄⠀⠈⣿⣙⣯⣀⡿⢇⢠⣿⠁⣼⠼⠅⠈⣷
⠀⠀⣸⣧⡠⢀⠬⣀⠠⠤⡧⠂⣾⠅⠙⢦⠯⡏⠀⢻⣿⣦⣴⣿⡿⠀⣰⢿⡟⠓⠀⠰⠠⠤⢥⢀⣧⡀⢐⣠⣿
⠀⢠⡿⠟⣿⠀⠀⠈⠃⠀⢹⠊⡑⢦⢭⣁⠠⣟⣤⡀⠉⠛⠛⠋⢀⡆⠰⠠⠆⠰⠀⣷⣤⠄⢨⣴⡄⠋⠙⣿⠃
⠀⠈⣛⣾⣿⣦⣄⣀⣉⡀⢈⡠⠀⠐⠀⣹⠰⢋⡍⠁⢰⡿⠿⢿⣿⣤⠀⢄⡤⠄⣐⣭⣴⣮⠶⠃⠀⣀⢀⣾⣧
⠀⠘⢧⠄⠚⠁⠈⠟⢁⣴⡿⣩⡾⠒⣃⢹⣶⣋⣴⡿⢼⡧⠦⠀⠰⣦⣾⠟⡿⠿⠿⠛⠛⣡⠀⠿⢂⣽⣿⣿⣟
⠀⠀⠸⣦⣀⡀⠀⣰⣿⣷⠿⡋⠰⠆⠀⢨⣟⠻⠀⠣⣬⡀⠠⠔⡂⠰⣿⡆⠀⠀⠀⠾⠂⠁⢀⡿⢯⠝⢿⣿⣿⡆
⠀⠀⠀⢸⣿⣧⡄⠛⠉⢠⠀⠙⠀⠈⢀⡼⢿⣾⣍⠀⣤⠹⠀⠀⠀⢀⣯⡽⠟⡚⠷⣶⠞⢴⠉⠀⠀⢠⡼⣿⣿⣧
⠀⠀⠀⠀⣿⢟⣡⡌⣃⡀⠀⠈⡠⠈⠁⠀⠈⢳⣹⡄⠘⡐⠁⢠⣿⡿⣽⡧⠄⠀⠠⡇⢈⠘⣶⢛⡊⠍⢓⣛⣩⠿
⠀⠀⠀⠀⣟⠈⠻⣧⠈⣻⡿⠋⠰⠿⣤⣀⣤⠈⣏⠶⣶⡅⠈⢸⣝⣦⡹⠰⠢⠐⠄⣿⡶⣫⢕⠢⡴⢾⣟⣿⣽⠆
⠀⠀⠀⢠⣿⣄⠸⣿⠇⠛⠀⠀⠰⠀⠀⢀⠁⠀⢸⡨⣿⣿⢾⣿⣻⣫⢷⣦⣀⢄⣴⣫⣞⠿⠶⣻⠟⠁⠀⠈⢻⡀
⠀⠀⠀⠸⣭⡿⠗⠒⢯⣄⡀⠀⠀⠀⠀⠈⠀⠀⠈⣇⣿⣿⡐⠗⠦⠚⣂⣿⣿⣷⡟⠁⠈⠇⠀⡡⠳⠘⠢⠀⢸⠇
⠀⠀⠀⠀⠈⠓⢶⣄⡂⠀⠈⠛⠶⠤⣤⣠⣴⠐⠶⠖⠒⢾⡛⣿⣻⣿⣟⣿⣿⣿⣿⣟⣇⣀⣰⢿⣦⣤⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⢤⡴⠶⠛⠋⠉⠳⠶⠤⠴⠶⠚⠁⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡟⠛⠛⠛⣿⠀⢰⡟⠛⢿⠀⢸⡇⠀⡿⠛⢻⡆⠀⣿⠛⠛⠛⢻⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣿⠀⢸⡇⠀⢸⠀⢸⡇⠀⡇⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣿⠀⢸⡇⠀⢸⠀⢸⡇⠀⡇⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣇⣀⣀⣀⣿⠀⢸⡇⠀⢸⠀⢸⡇⠀⡇⠀⢸⡇⠀⣿⣀⣀⣀⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠀⢸⡇⠀⢸⠀⢸⡇⠀⡇⠀⢸⡇⠀⠉⠉⠉⠉⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⣤⣤⣤⣤⠀⢸⡇⠀⢸⠀⢸⡇⠀⡇⠀⢸⡇⠀⣤⣤⣤⣤⣤⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣿⠀⢸⡇⠀⢸⠀⢸⡇⠀⡇⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣿⠀⢸⡇⠀⢸⠀⢸⡇⠀⡇⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣿⠀⢸⣇⣀⣸⠀⢸⡇⠀⣇⣀⣸⡇⠀⣿⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣸⣇⣀⣀⣀⣿⣀⣈⣉⣉⣉⣀⣸⣇⣀⣉⣉⣉⣁⣀⣿⣀⣀⣀⣸⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⢠⢠⠀⢀⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢈⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠐⢄⡀⠀⠀⠀⣀⣀⡉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠇⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠂⠁⠀⠊⠉⠲⡀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠡⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⣈⠉⡆⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⢠⡤⠄⢇⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠆⠠⠂⡐⢦⣍⣠⢟⡀⠀⠀⠀⠐⠐⠠⠀⢀⡔⠁⠀⠐⠀⢑⢠⠔⠀⠀⠀⠀⠑⠄⠀⠀⢁
⠀⠀⠀⠀⠀⠘⠀⠀⢈⡈⠦⡄⠀⣄⠈⠂⡤⠀⠀⠀⠀⠀⣹⡌⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠐⢢⢨⣰⠂⠠⠴⠁⢤⠀⠀⡀⢀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡀⡙⡘⢣⡗⠀⠰⠀⠀⠠⠁⠀⠀⠁⠀⢄⠤⠟⠃⠐⠐⠀⢀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠀⠦⠀⡀⠈⠁⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⣠⠄⠃⠀⠀⠀⠀⠀⠉⠠⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠂⠐⠒⠒⠃⡀⠂⠀⠐⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠐⡀⢠⠖⠀⠀⠀⠈⠀⠈⠑⠖⠀⠠⠛⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠱⡈⠢⠀⠀⠁⠀⠀⠀⠀⠁⠀⡄⠀⠁⠈⠑⠒⠣⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠂⠄⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠠⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠔⠁⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠁⠒⠒⠈⠀⠂⠀⠀⣀⡠⠤⠚⠋⠀⠁⠀⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠁


⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡟⢳⠀⠀⠀⠀⠀⠀⠀⡟⣷⠀⠀⠀⠀⢰⠋⢳⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠷⠾⡀⠀⠀⠀⠀⠀⢰⠓⠻⡆⠀⠀⠀⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣄⣠⠇⠀⠀⠀⠀⠀⢸⢀⡀⡇⠀⠀⠀⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⢹⡏⣇⠀⠀⠀⠀⠀⢸⣌⣱⡇⠀⠀⠀⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣇⡇⢹⣸⡀⠀⠀⠀⠀⢸⢸⠁⡇⠀⠀⠀⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡼⣸⢴⡦⣇⢧⠄⠀⠀⠀⢸⢸⠀⡇⠀⠀⠀⢸⣀⣸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⢣⠇⠀⠀⠸⡸⡄⠀⠀⠀⢸⢸⠀⡇⠀⠀⠀⢸⠀⣹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡞⡼⠀⠀⠀⠀⢧⢣⠀⠀⠀⢼⢸⠀⡇⠀⠀⠀⢸⠉⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠗⠃⠀⠀⠀⠀⠘⢺⠀⠀⢰⣿⢸⠀⣷⠀⠀⠀⢸⠦⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣠⢾⣸⣠⠧⡄⠀⠀⠹⡤⡞⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⣿⠀⠀⠘⢺⣹⣈⡞⠁⠀⠀⠀⣇⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠓⠒⠒⠒⠒⠓⠒⠛⠀⠀⠀⠈⠇⠛⠀⠀⠀⠀⠀⠿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠂





⣏⠉⣹⣛⣛⣛⣛⣻⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣻⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠛⠉⠉⠉⠁⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠤⢄⡀
⠀⠀⠀⠀⢀⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⣽⢋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⢰⣿⣿⣀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠄⠤⠤⠮⠥⠤⠤⠤⠤⠤⠤⣤⣤⡤⠤⡭⠭⠤⠤⣤⠤⠄⠀⠤⠤⠄⣤⠀⠄⣝⠉⠀⠀⢸⠀⠀⠀⠀⣏⠉⠀⠀⠀⣴⣾⣦⠀⢸⣀⡀⠀⡀⠀⢀⣀⣅⡀
⠐⠂⠐⠒⠒⠒⣤⣶⡦⠀⢸⠿⠿⠀⠰⠦⠄⠠⠄⠶⠤⠤⠤⠤⠤⠀⢾⠄⠀⡦⠀⠀⠠⢰⠤⠤⠄⠠⡦⠤⠤⠤⠤⡿⠩⠤⠤⢼⣿⢤⣾⣿⠦⢼⡿⠿⠤⠤
⠀⢲⣶⣶⠒⠐⡟⠛⠓⠐⢸⠒⠀⠂⠐⡇⠒⠂⠒⠒⠀⠀⠐⠒⠂⠀⢺⠂⠒⠿⠶⠶⠶⠾⠷⠶⠶⠶⠗⠀⠀⠀⠀⡷⠐⠂⠀⠚⠻⢾⠇⠀⠀⢸⠆
⠁⣹⠉⠉⠉⠉⣟⠉⠉⠉⢹⡋⢈⣉⣹⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻⠉⠀⠀⠀⠀⠀⣠⣠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣀⣸⡃⠀⠀⢸⡃
⠀⢸⠀⢀⣀⣠⣧⣤⣶⡾⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠐⠠⣀⣀⡐⠿⠋⣀⣀⠤⠘⠁⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠸⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣈⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢴⡶⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠤⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⣼⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣀⡤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⡇⠀⠊⠁⠀⠀⠀⠀⠀⣈⣥⡀
⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢈⣿⣿⡍⠀⠀⠀⠀⠀⢸⣿⣿⡉⠀⠀⢀⣸⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡇⣠⣤⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⡍
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢀⣟⣋⣁⢀⣀⣀⣀⠀⢸⣟⣋⡀⠀⡀⡀⣽⡀⢀⣀⡀⣬⡀⣀⢀⠀⠀⠀⠀⣰⣈⣉⣇⣙⡛⣁⢀⢀⣀⢀⣴⣶⡄⢈⣟⣛⣁⠀⣀
⠄⠀⠠⢸⠄⠀⠀⠀⠀⠀⠀⡿⠿⠇⠀⠀⠀⠀⠀⢸⠿⠿⠁⠀⠀⠀⢼⠄⠀⠀⠠⢼⠀⠄⠀⠀⠠⠀⠤⢼⣿⡤⠤⣴⣾⡦⠀⠀⠀⢸⡟⠉⠀⠀⡿⠿⠇
⠒⢢⣶⣾⠂⠐⠒⠀⠐⠀⠒⡗⠒⠒⠒⠒⠒⠒⠒⢺⠖⠒⠒⠒⠒⠂⢾⠒⠒⢰⣶⡾⠒⠐⠂⠀⠀⠀⠀⠚⠻⠆⠠⡯⠉⠀⠀⠀⠀⢸⠆⠠⠀⠠⡦⠤⠀⠄
⠀⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠐⠀⠀⠀⠉⠁⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣗⣀⣠⣤⣶⣶⣾⠂⠀⠀⠐⠃⡂⠒
⠧⢳⠛⡐⡲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⠰⠽⡯⢇⡰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⣉⣉⣉⡥⠂⠀⠀⠀⠀⣿⣿⡆
⠀⠋⠛⠛⠛⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠐⠓⠚⠞⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡄⣷
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⢀⠀⠀⡾⠀⣿⠀⠀⢹⠀⣀⣀⣤⣤⠤⠤⠴⠶⠒⠒⠒⠲⠶⣿
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⢸⠃⢠⡟⠀⠀⢩⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠀⠈⡇⠀⠀⠀⠀⠀⢠⡏⢀⣿⡗⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡀⢠⣼⠀⢸⡿⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡄⠀⠸⡇⠀⠀⠀⠀⣾⠇⠀⣾⠃⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡿⠀⣼⡏⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡇⠀⢸⡇⠀⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠄⢸⣧⠟⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⢼⠍⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡅⠀⢸⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠇⠀⢸⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⡧⠤⠤⢤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣧
⠀⠀⠀⠀⠘⠀⠈⠙⠓⠲⠤⢀⣀⠀⣀⣀⣀⣤⠀⠺⠃⠀⠈⠉⠒⠚⠟⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣟⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡐⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢦⣀⣀⣀⣠⣄⣀⣀⣀⣀⣈⡈⠁⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡴⠋⢉⠉⠉⠛⢿⣦⠀⠀⢈⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡏⣠⣾⣿⣿⣿⣶⣄⢹⣇⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡁⣿⣿⣿⣿⣿⣿⣿⢌⡏⠀⠘⠀⠘⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢷⡹⣿⣿⣿⣿⣿⡟⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠿⠿⠿⠿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠁⠀⠉⠉⠉⠁⠉⠉⠙⠛⠛⠈⠉⣀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠄⠀⠠⠄⠀⠠⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠧⠤⠤⠤⠤⢤⡤⠤⣤⣤⠤⠤⠤⠤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠶⠒⠒⢒⠚⠛⡉⠉⠉⠁⠀⠘⡋⠓⠒⠦⠤⣄⠠
⠀⠀⠀⠀⠀⢀⣠⡤⠒⢫⡍⠀⠀⡠⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⢈⡀⠀⠘⢁⠦⠂⠀⢀⠢⣄
⠀⠀⠀⢀⡴⣫⠟⠁⡀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⢀⠈⠀⠀⠂⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠉⠉⢪⡑⣆
⠀⠀⢠⢎⠀⡆⠁⠀⠁⠀⠐⠀⠀⠰⠆⠀⠀⠀⠀⠀⠘⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠠⠆⡀⠀⠀⠴⠀⠘⣿⡜⣇
⠀⢠⢯⡟⠃⡀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠂⠠⠀⠐⠀⠀⠀⠒⢀⡄⠀⠈⣹⣇⢿
⠀⣼⢺⡁⡈⠁⠈⠊⢀⠀⠀⠔⠀⠀⠁⠀⠀⢀⠀⠀⣄⠀⠀⠁⠀⠘⠀⠀⠀⡄⠉⠐⠁⠠⠀⢀⠄⢲⣿⣿⡏⣼
⠀⠸⡜⠿⣫⡄⠀⢀⠀⢀⠀⠀⠀⠀⠐⠄⠀⠈⢀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⣄⢀⠐⣠⣒⡤⠾⢛⣋⣥⣾⣅
⠀⣠⣾⣷⣶⢶⣤⣔⠂⠊⠀⠀⠃⠀⠀⠀⢀⠀⠀⡀⠀⡈⠀⠀⠀⠤⠦⠀⠀⣒⣂⣩⣤⣴⣶⣶⣿⣿⣿⣿⡿⡀⢚
⡞⠁⣺⣿⣾⣮⣉⣟⣿⣻⣿⣿⣿⣷⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⠿⠍⠀⠁⢨
⢣⠀⠈⠙⡉⠙⠛⠏⠭⠻⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠶⣷⣿⣟⣻⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⣡⣤⠤⢴⡞⠉
⠀⠙⢲⢏⣹⣄⣀⠀⠀⢀⣠⠶⣲⣖⠬⣩⡥⢤⡉⠉⢁⣰⠟⠛⠛⠋⠋⠉⠉⠉⠿⠷⢄⣤⣴⣚⣩⣵⣶⣾⠋⡆
⠀⠀⠸⡸⣿⣻⢿⡿⢿⢿⣤⠞⠛⠿⢻⠾⠿⣶⣉⠛⣉⣀⣠⡤⣤⣀⣰⣶⣦⣤⣴⠶⠛⠋⠉⠓⠁⡉⠀⠈⡰⠃
⠀⠀⠀⣙⣶⣤⣀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠸⠈⠛⠉⠉⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀⣨⣵⣶⣿⡷⠑⡄
⠀⠀⢰⠀⣿⣿⣿⢿⣷⣶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣠⣤⣤⣤⣴⣶⣿⣿⣿⣿⣯⣿⣾⣿⠿⠁⢀⠇
⠀⠀⠈⣣⣌⣓⠟⣓⡛⠻⢭⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢯⣵⠟⣉⣀⣀⣠⡴⢋
⠀⠀⠀⢏⠈⠽⠷⠤⣭⣽⣦⣈⠉⠀⠈⠙⠻⣿⡿⠟⠛⠋⣉⡭⠭⣭⣉⣉⠉⣉⣡⠴⠿⢋⠭⠵⠾⠟⣛⣋⣡⡞
⠀⠀⠀⣸⠹⠗⠤⠤⢤⣄⡀⢈⡳⠦⠤⡈⣁⣠⣐⠺⠟⠂⠀⠒⠠⢤⣤⣬⣭⣭⣤⣤⣶⢯⣤⣴⠟⠚⠉⠛⢹⠇
⠀⠀⢰⣿⣆⡀⠀⠀⠈⠈⠁⠀⠑⠶⠶⠒⠻⠟⠛⠛⡳⠊⠉⠛⠳⠄⣈⣉⠤⠞⠛⠋⠀⠀⢀⠀⠀⠀⣀⡴⠋
⠀⠀⠘⠛⢻⣷⣤⣄⣀⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣀⣮⠵⠖⠋⠁
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠟⠻⠿⠷⢶⣤⣤⣤⣤⣤⣀⣀⣐⣀⣠⣤⣤⣤⡤⠴⠖⠛⠋⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣿⣿⣿⣾⣷⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣤⠀⢤⡀⠀⣀⡀⠀⠀⠀⠀⠀⠄⢀⡀⠠⢀
⠀⠀⠘⡆⠀⠙⡆⠀⢹⡆⠀⠀⢉⠀⠲⠀⣁⠐⠂
⠀⠀⣶⡀⠚⢦⡉⢤⡀⠇⠀⠀⢈⠀⠤⡄⠈⠀⢤
⠀⠀⠈⠇⠀⠀⢷⠀⠙⡆⠀⠀⠈⠃⠀⡀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠈⠛⠂⠀⠛⠂⠁⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢠⠠⡤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡠⠤⠀⠄
⠀⠀⠀⠄⠠⠀⠠⠀⠄⠁⠀⢨⢳⠇⠀⣀⣀⣐⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠄⠣⢂⡀⠂⡁
⠀⠀⠀⡀⠀⠐⠀⠂⠀⠂⠀⢘⡛⠀⠀⠘⡅⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡄⡤⡄⠠⣋⠰⡆
⠀⠀⠄⠀⠌⠀⠈⠄⠁⠀⠀⠈⠠⠤⠂⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠣⠆⣀⠀⣠⡂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠈⠁
⠀⠀⠄⢠⠤⠀⠤⠄⢠⡄⠀⠀⠀⠄⠤⠠⡀⠀⠤⠀⠀⠀⢠⢠⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡈⢉⢀⠉⡀⡀⠀⠀⠀⠀⠀⠠⠀⠄⠁⠄⠀⠀⠀⢠⡾⢰⠷⠿⡗⠤⠀⠀⠃⠜⠀⢰⣶⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⠀⠀⢸⠃⠐⡇⢠⡘⡀⠀⠀⠓⠀⠾⣧⡤⠿⡂
⠀⠀⡁⠈⠀⠀⠁⣁⠀⠀⠀⠀⠀⠠⠀⠆⠀⠀⢂⠀⠈⠛⡗⠂⢧⠀⡐⡛⠀⠀⠟⣤⣎⣉⣷⣞⠿⠃
⠀⠀⠉⠁⠈⠈⠀⠈⠁⠀⠀⠀⠁⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠈⠉⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠠⢤⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣤⣁⣀⡽⠋⢻⡆⠀⠀⠄⠐⠀⠒⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠉⠝⠻⣦⣆⠚⠀⠀⠀⠀⡄⠋⠀⣄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⠦⡶⣦⠐⠛⢧⡀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠉⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠠⠤⠀⠄⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠒⠄⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣇⠁⠀⣀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡴⠄⠓⠰⢆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⡇⠰⣢⠾⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⠖⣓⡲⡄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣽⣯⠌⣡⡀⠀⠄
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⢒⣺⠿⠟⠒⠒⠲⠶⣤⣤⠤⠶⣒⣺⡿⠿⠟⠛⠛⠛⠓⠲⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠋⣩⠟⠉⢀⡤⠞⠉⠀⠀⢀⣠⠴⠚⠉⣁⡤⠖⠋⠁⠀⠀⠀⢀⣤⠴⠚⠋⢉⣉⣉⣭⣭⣙⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠸⣥⡴⠚⠉⠀⠀⠀⠀⠰⢯⡤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠸⠭⠴⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢀⣧⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠤⣤⡤⢤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⢀⡴⢻⡼⣟⠲⣦⡤⠤⠶⣶⡖⣒⡿⣻⣿⣛⣛⣉⣀⣤⣾⣥⣿⣿⣿⣥⣿⡿⠟⠲⢬⣅⣀⡤⠥⠚⠉⢯⡻⣿⣯⣽⣷⣶⠲⣶⡶⠶⢖⣒⡝⢹⢢
⠀⢸⡀⠀⠙⢮⣉⣁⣀⣠⠴⣟⣛⣓⡮⠕⠋⢈⣉⣁⣤⠤⠤⠬⠭⠭⢭⡽⠶⢦⠤⠤⣤⠤⠤⠤⠤⣤⣤⣄⣉⣓⠻⠽⠿⠽⠟⠋⠙⠒⠶⠶⠖⡫⠎
⠀⢠⣽⣶⠶⠒⣲⣿⡓⠒⠒⠒⠒⠒⢾⣯⠟⢉⡴⠦⣀⣀⣀⣀⠀⣰⠋⣹⢀⣉⣛⣛⠋⠀⠀⠀⢰⠟⡆⠀⠰⡿⠉⠉⠛⠿⠷⠖⠒⢲⡖⠚⡭⢤⡤
⡴⠛⣋⠭⢶⠚⠃⠀⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠁⠀⠉⠉⠀⠀⠉⠉⠛⠓⠚⠋⠈⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⢳⠶⢭⣉⡵⠋
⠓⠋⠁⠀⠈⠢⣄⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣄⣀⣀⣀⣀⠀⠀⠀⣀⣠⠴⠋⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠀⠀⠀⠀⠠⠤⠤⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⢠⡄⣀⠀⡀⣢⠀⠀⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠓⠞⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣄⠀⠀⠈⠉⠉⠹⠟⠁⠀⠠⣤⡠⡀⠈⠐⢤⣄
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠸⡏⠑⢄
⠀⠀⠀⢀⠄⡀⢀⢀⡴⠂⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡵⠀⠀⠀⠀⠀⠀⠀⡔⠀⣀⠀⠀⠀⢀⣄⠢
⠀⠀⢀⠊⠘⠀⠙⠋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠀⠀⠀⢶⠏⠀⠣
⠀⠀⠄⣰⠄⠀⠀⢸⣸⠀⠀⠀⠀⠀⠀⣠⣤⣤⣦⣀⣄⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈⡄
⠀⢘⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠙⠻⠃⢠
⠀⠘⠀⠀⠀⠀⣴⣇⣀⣤⡄⠀⠀⠀⠀⠈⠻⢿⣋⠁⠀⠀⠀⠀⢀⠹⠤⠂⠁⠀⠀⠀⠀⠀⢀⠀⣠⣿⠁⠀⠀⠀⢠⡄⠈
⠀⠈⡀⣠⣄⠀⠋⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⢀⣴⣿⣤⠁
⠀⠳⣣⡀⠻⣷⣄⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⣄⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⢟⣵⣿
⠀⠀⠻⢿⣷⡝⣿⣿⠦⣵⣢⠝⠀⠀⢠⣸⠃⡀⠀⠀⠀⠘⠋⠉⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣶⣶⣿⠿⠁⣰⣿⣿⡏
⠀⠀⠐⡀⠉⠻⣮⣭⣄⠀⠩⣤⣤⣀⢌⣵⣾⠋⣀⣀⣀⣀⣀⡀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣤⣛⡿⠝⢉⣀⣴⣿⣿⠟
⠀⠀⠀⠘⢄⠀⠀⠙⠿⣦⣤⣬⣁⣀⣀⡈⠙⢿⣿⣿⣿⡿⠟⠛⠛⢛⣛⣣⣭⣍⢉⠈⢉⣀⣤⣶⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠉⠙⠛⠛⠿⣿⣦⣄⠀⠈⢁⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⣀⠀⠀⠀⠉⠻⢳⠶⠛⢉⣉⣏⣽⣉⣿⣿⣻⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠠⠤⠤⠶⠬⠿⠿⠿⠿⠯⠛⠛⠛⠋⠉⠉

⠀⠀⠀⣀⣄⡀
⠀⢠⣾⣿⣿⣿⣦⠀⠀⢠⣿⣿⣿⣿⣷⡀⠀⢀⣀⣀⣴⣷⣦⣤⡀⠀⢠⣤⣆⣶⣿⡄⠀⠀⠀⣄⡀
⠀⣸⣿⣿⣿⣿⡿⠀⠀⠘⣿⣿⣿⣿⣿⢃⣤⣿⣿⣿⣿⣿⣿⣿⡿⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⡧⠄
⠀⢻⣿⣿⠿⠋⠁⠀⠀⠀⠈⢿⣿⡿⠁⠈⢿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠙⢿⣿⡿⠁⠀⠀⠀⢨⠁
⠀⠀⠈⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣄⢉
⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⠀⠀⢠⣾⣿⣿⣿⣿⣆
⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⣼⣿⣿⣿⣿⣿⡏
⠀⠀⠸⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡶⠀⠀⠀⢿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠁⣀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⡀⢈⣀⣠⣤⣤⣀⠀⠀⢀⣠⣀⣾⣿⣿⣧⣠⣄⡀⠀⣾⣿⣿⣶⡀⠀⣴⠿⣿⣿⡃⠀⠰⣶⣾⣿⣿⠆
⠀⢸⣗⣾⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⠛⠛⠛⠛⠓⠀⢻⣶⣴⣿⠗⠀⠈⠉⠛
⠀⠈⠻⣿⣽⣿⣿⠿⠋⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠁⠀⢠⣤⣤⡦⠀⠀⠈⠿⠿⠇
⠀⠀⢀⣠⣤⣆⣤⡀⠀⠀⠀⠀⢀⣀⢈⡙⠛⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⣤⣤⣄⡀
⠀⠀⣽⣿⣿⣿⣿⣯⡀⠀⠀⣺⣿⣿⣿⣿⡀⠀⠀⣶⣦⣶⣶⣦⠀⢀⣾⡹⣼⣏⢿⣷⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠛⠉⠋⠋⠋⠛⠃⠀⠸⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⡇⠘⣿⣷⣽⠿⣸⣿
⠀⠀⢠⣤⣤⣤⣤⡤⠀⠀⠀⠻⣿⣿⣿⡿⠁⠀⠘⢿⣿⣿⣿⠟⠁⠀⠈⠻⠿⣿⡿⠇
⠀⠀⠀⠙⠿⠿⠟⠀⠀⠀⠀⠀⠙⠛⠋⠀⡀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⢀⡤⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⣿⣿⣧⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣁⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠃⠀⠄⣦⢲⠆⠀⠀⠀⠀⠁⠀⠀⠂⠄⠀⠀⠀⠀⠀⢤⡄⡀⠀⠀⠙⠀⠠⡀⠀⠀⠀⢀⡀⠀⠀⢄⣀⣓⣠⢂⣘⣀⠂⠀⠁
⠀⠀⠀⠀⠀⣀⠀⠘⢧⡀⠀⠰⠄⠀⠀⠀⢥⠁⡔⠊⢀⠄⠀⠀⠀⠘⣣⡔⠀⠀⣀⢄⡟⠂⣄⠀⠈⠁⠀⢹⡋⠀⠀⡈⠉⠁⠀⢙⡇
⠀⠀⠀⠀⢘⠃⠀⠀⠀⠘⠂⢀⣀⣴⡀⢀⠈⠖⢀⠞⠁⢀⡄⠊⠀⠀⠀⠀⠀⠠⣡⠜⢧⢗⡫⠂⡀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⣼⠁⣀
⠀⠀⠀⠀⠀⢠⠁⠀⠈⠀⢀⠌⠈⠈⠁⠀⠀⠐⠦⠀⢀⠎⢠⠞⠆⠀⠀⠀⠠⣔⣄⣶⠨⠍⡠⠖⡧⠀⠒⠀⣀⠀⠀⠀⠀⠀⠀⠇⠀⠁
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠀⠐⢤⡆⠀⢧⠀⠂⠀⠀⠄⠀⡅⢤⣥⠔⠚⢀⡠⠆⠀⠀⠀⠙⠠⠤⠄⠠⠤⠤⠂
⠀⠀⠀⠀⠀⣀⣰⡄⠀⠀⠀⢤⠀⠀⣀⡀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠸⢗⣸⢩⡍⢮⡛⠀⠀⣄⢶⠆⠀⠀⣀⠀⠀⢦⣆⡄⠈
⠀⠀⠀⠀⡶⠚⠁⢃⣀⣀⠀⣀⠀⠀⠘⢡⠾⣼⠀⠀⠀⠀⢀⡀⠀⠀⠀⠦⠀⠀⠈⠉⠛⠃⠉⠀⠀⠷⡉⠀⠀⠀⠀⠁⠀⠀⠀⢊⢱⡆
⠀⠀⠀⠀⠀⠀⡞⠋⢀⡠⠴⠬⢤⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠰⡦⡀
⠀⠀⠀⠀⠀⠀⠻⣶⠃⠁⠀⠀⠀⠈⠆⠀⠀⣥⠀⡆⡄⢐⢰⠀⡄⢀⠇⠀⠂⢰⠀⠰⣀⠂⡦⢀⠀⠀⠀⠀⠀⠘⢼⣁⠄⡓⣧⢂
⠀⠀⠀⠀⠋⠀⠀⠘⠀⠁⠀⠀⠀⢀⠀⠉⠀⠤⠐⠳⠁⠬⠸⠄⠇⠨⠤⠀⠀⠇⠱⠐⠻⠀⠗⠀⠇⠁⠀⢲⡦⡄⢩⢽⣶⠲⠨⠇⡰⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠠⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡀⢀⠀⠀⠉⠹⢾⣤⡜⠿⢭⠘⣧
⠀⠀⠀⠀⣠⠀⠀⠢⡇⡀⣀⠀⣠⠅⠀⠀⠙⣣⡆⣀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠎⠃⠀⠀⣄⠀⠀⠀⠀⠘⠳⠴
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠂⠀⠀⠚⠃⠀⢀⠀⠀⡠⠠⠎⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡠⠀⠄⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣠⣄⣀⣄⣀⡀⠀⠀⢠⡄
⠀⠀⠀⢠⠀⡈⠀⠀⠀⢠⡀⢠⠔⠀⠀⠀⢄⠀⠀⠁⠀⠀⠁⢀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠛⠛⢻⣿⣧
⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠉⠲⠀⠀⠀⠀⠘⠀⣴⡀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠁⠀⢰⠇⠀⠀⠀⠀⠻⡿⢤⠂⠀⢠⣿⣿⠃⢠
⠀⠀⠀⠀⠀⠘⡀⢀⠄⠎⠁⡀⠀⠰⠄⠀⢡⣹⠓⠀⠀⣀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣦⣟⠁⠀⠀⠂⠀⡜⠀⠀⠀⠀⠾⠋⠐⠀⡀
⠀⠀⠀⠀⠀⠲⠈⠈⢀⢀⠉⠈⠀⠙⠀⠀⠈⠁⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠂⣠⡂⣤
⠀⢀⠤⣀⣀⢐⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠀⡀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢈⣻⠔⡻
⠀⣧⣿⠿⠿⢷⣅⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣠⠖⠋⢀⠠
⠀⠻⣏⣀⢀⡸⣫⠁⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⢰⠠⠆⠀⠠⠄⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡚⠍⠀⠀⠀⣀⣀
⠀⠀⠈⠉⠗⠋⠁⠀⠀⠀⠀⠀⠀⣀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀
⠀⠀⠀⠀⠀⢐⠀⣠⡤⠴⠖⠒⠂⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡡⠌
⠀⠀⠀⠀⢀⡿⠊⠁⠀⣄⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠗⠒⠀⠀⠀⠀⣤⣳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣀⠀⠀⠀⠀⠰⣿⡆⠀⠀⠠⠀⠀⠀⠀⣀
⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⢄⠑⠠⡀⢤⣀⣸⣿⡎⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠿⣿⣦⡱⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣄⣌⣙⣿⡇⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣧⠀⠀⢸⣿⡇⡁⡀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠈⠄⠋⠀⠀⠀⠀⠀⠀⢀⠎⠙⢻⡀⢸⣿⡟⣰⠀⠠⠀⠀⠀⠀⢀⡴⠀⠀⠀⠘⢮⡳⢴⣶⠿⠊⠀⡈⠀⠀⠀⠀⠶⠄⠀⠄⠀⢰⣍⣳
⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⣈⠂⡂⢻⣷⣿⠏⡔⠉⠀⠀⠀⠀⠠⠂⠙⠁⠀⠀⠀⠀⠀⠉⠉⠉⠒⠉⠁⠀⠀⠀⠀⠄⣀⠀⠄⠀⢀⣮⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠚⠶⠿⠶⠿⠛⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠒⠘⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⢬⡍
⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠈⠃⠀⡠⢀⣠
⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⠀⠀⠀⠀⠂⠄⠈⠀⠀⣈⠅⠀⢠⣾⣦
⠀⣀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠖⠀⢀⠌⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠙⡬⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒⠦⢄⡤⠖⠚⣛⣉⣉⡉⣉⣉⣉⠉⠙⠛⠒⠲⢤⡀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡴⣋⠤⠒⠀⠀⠀⠀⠀⡞⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡠⠖⠊⠉⠁⠐⠊⠭⢝⡒⠤⣀
⠀⠀⠀⠀⠀⡞⠘⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠪⠙⠢⣄
⠀⠀⠀⣠⠖⡇⢸⠀⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢀⡞⠁⠀⢻⠈⡆⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢀⡎⠀⠀⠀⠸⡄⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⢎⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⡠⠤⢔⣒⣉⣭⠤⢤⣷
⣼⠀⢀⠀⣀⡀⢧⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⢠⢣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢔⣪⠵⠒⠋⠉⠀⠀⠀⠀⠀⠀⠙⣄
⡇⠀⠔⢋⣠⠴⠚⢦⣀⣠⡤⣤⣀⣧⠀⠀⠀⠀⢰⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡭⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠳⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣄⠀⠀⠈⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠹⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢕⣦⣔⡀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⢭⣓⣂⣀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠤⠤⠤⠄⣒⣒⡒⠠⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⠔⣡⠤⠖⠒⠒⠛⠉⠉⠉⠉⠉⠉⠉⠙⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⠓⢤⡀⠀⠀⠀⠀⢀⡴⢃⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⣉⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡠⣠⠞⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠠⣤⣦⣶⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠉⠒⢦⣀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡤⢄⠐⠀⡀⠈⣆⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠖⠀⢀⡆⠀⠐⠒⠂⠰⠦⠂⠀⠀⢸⡶⣌⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢺⣀⠐⣯⠂⠾⠋⠀⡀⠲⣀⣤⠀⠀⠀⠀⢀⣼⢡⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠈⢄⠉⢰⡞⠀⡾⢛⠛⠁⢀⣤⣤⣾⡿⠃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠰⠉⠀⠀⣷⣾⣶⣶⣿⣿⡿⠟⠁⠀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠁⠀⠀⠀⠀⡦⠒⡂⢶⢤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⠊⠢⡀⠀⠀⠀⠀⠀⠡⠀⢓⢈⠈⡆⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⠀⣀⠔⠉⠁⠂⠀⠀⠑⣀⡀⠰⠋⠢⡘⠀⠣⠁⠀⠀⡌⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠖⠘⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠅⠀⠢⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠚⡁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠁⠘⡆⠀⠀⠀⠀⠀⠀⠀⡠⣴⣶⡶⠧⠶⠓⠒⠒⠛⠒⠒⠷⠴⠂⠀⠀⣀
⠀⠀⠀⠀⠀⡠⠃⠀⡇⠀⠀⠀⠜⡄⠀⠀⡲⣤⡀⠀⠺⡄⠀⠈⠀⠀⠀⠀⠁⠒⠻⢧⡤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠤⣤⠶⠞⠃
⠀⠀⠀⠀⡰⠁⠀⠀⢡⠀⠀⡄⠀⢱⠀⠀⠙⣈⠻⣶⣄⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡷⡒⢲⠶⠒⠒⡒⠂⠢
⠀⠀⠀⡰⠁⠀⠀⠀⢸⠀⡐⠀⢰⠸⡄⠀⢄⠈⠓⠀⠙⠻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢱⠈⠢⠌⠀⣰⡇⠀⠀⢁
⠀⠀⠀⡇⠀⠀⠀⠀⠘⠴⠁⠀⠞⠀⠈⡲⢈⠏⢀⠀⠀⠀⠈⢃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢢⡀⠈⡁⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⢶⡖⠒⣋⡵⢲⠀⠀⠙⢶⣄⠀⠈⠓⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠐
⠀⢀⡜⣀⣀⣀⠤⣤⠒⠉⠀⠞⢁⡔⠁⠀⡎⠀⠸⡄⠀⠈⠻⢦⣀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠠⡄⠀⠀⠈⢠⡐⠂⠀⠀⠀⡀
⠀⠀⠀⠈⡁⠀⠀⠉⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠊⠈⠑⠒
⠀⠀⠀⠀⠑⠲⠄⠤⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠶⠶⠶⠛⠛⢻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠶⠶⠞⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣹⠇
⠀⠀⠀⠀⢀⣤⡴⠶⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡞⠁
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠾⡋
⠀⠀⢸⠟⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢀⣴⣿⣴⣿⣁⣤⢶⣆
⠀⠀⣿⠀⠀⠀⠈⠉⠛⠛⠶⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⣠⢾⣿⣡⣾⡿⠋⠀⠰⣿
⠀⠀⠙⠷⠶⢶⣴⣦⣤⣄⣀⠀⠀⠀⠈⠉⠛⠛⠶⠶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⡾⢃⡼⢿⡿⠋⠀⠀⣠⣼⣇
⠀⠀⠀⢀⣀⣀⡻⠶⠾⣿⣉⣹⡷⢦⣤⣤⣤⣀⡀⠀⠀⠀⠉⠉⠛⠳⠶⢦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣾⣯⡴⢋⡵⠋⠀⢀⣴⣾⠋⢀⣿
⠀⠀⢠⣾⣯⣉⣉⣛⣒⠿⢿⣥⣤⣀⣘⣧⡀⠉⠙⢛⡲⠶⣤⣤⣀⣀⠀⠀⠀⠈⠉⠙⠓⠶⢦⣤⣤⡴⠋⢀⣴⣾⣵⢟⡵⠋⠀⣀⣴⣿⠟⢁⣤⠞⢻⡇
⠀⠀⣿⠀⠀⠉⠉⠉⠉⠛⠓⠶⠶⣭⣭⣍⣉⣛⡿⢿⣿⣦⣀⣀⣉⡉⠛⠳⠶⣤⣤⣀⣀⠀⠀⠀⢻⠄⣠⣿⡿⢛⡵⠋⠀⣠⡞⢉⣀⣤⠾⢿⡷⠚⠻⡧
⠀⠀⣿⣟⠳⠶⠶⢦⣄⣀⣀⣀⡀⠀⠀⠀⠉⠉⠛⠛⠳⠾⠯⣭⣍⣙⣦⠀⠀⠀⠀⠉⠉⣿⠷⢦⣼⣿⠿⣻⡴⠋⢀⣤⣾⠏⣿⣋⣭⠟⠛⠛⠁⠀⣰⡗
⠀⠸⣧⠙⠳⠶⣤⣤⣈⣉⣉⣉⣽⡿⠿⠟⠛⣶⡶⣤⣀⣀⣀⠀⠉⠉⠙⢷⣄⣀⣀⣴⣾⣙⣛⣛⣩⡴⠟⠉⠀⢠⣿⡿⠋⣴⠋⠙⣷⠀⠀⠀⣠⡴⠋
⠀⠀⣿⠻⣦⣤⣤⣀⣈⣉⣉⣀⣀⣤⣤⠶⠟⠻⣆⠈⠻⢧⣍⠻⠟⠳⣦⣄⣉⡉⠉⠀⠈⠉⠉⠉⠁⠀⠀⣠⠾⠋⣁⣤⣾⣥⠶⠟⠋⢀⣤⠞⠋
⠀⠀⢻⡆⠀⠀⠀⠈⠙⢿⣍⡉⠁⠀⠀⠀⠀⠀⠈⠳⣤⡀⠉⠛⠶⣦⣤⣀⣉⣙⠛⢛⣛⣶⣤⣤⣀⣠⣾⡷⠶⠛⢁⣿⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠶⠶⠶⠶⠾⠿⢦⣤⣤⣄⣀⣈⠉⠉⠉⠉⢉⣁⣤⡼⠟⠁⠀⠀⢀⣴⠞⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠉⠙⠳⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣍⣉⣙⣛⣛⡛⠛⠉⠁⠀⠀⠀⠀⢀⣾⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠳⣤⣀⣀⣀⣤⡾⠃⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠷⠶⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠁⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠳⠶⠶⠶⢦⣄⣨⣧⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉








⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣉⣙⣋⣛⣉⣛⣉⣀⣉⣙⣿⣋⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣯⣤⣭⣭⣭⣭⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠨⠁⠀⠀⠹⡖⢢⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠁⠀⠀⠀⠀⠀⠀⠀⡠⣷⠚⠀⠀⠀⠀⠀⡖⠉⠁⠀⠀⡟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡴⠶⣄⣤⡤⠴⠴⠶⠖⠁⠀⠀⠀⠀⠀⠀⠙⠒⠋⠓⠚⠉⠉
⠀⠀⠀⠀⠀⣀⡀⢀⡤⣊⣳⣦⣀
⠀⠀⠀⡠⠞⠀⠹⣍⣁⠀⢀⣀⠈⡇
⠀⠀⠚⠃⢀⠀⠒⡛⣷⠄⠤⣦⡀⡇⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⠒⠒⢲
⠀⠀⠀⢈⣿⠀⡧⠇⡿⠁⠀⠓⠂⡇⢀⠊⢳⠀⢀⠀⠀⠀⠀⠤⠤⠤⠤⢄⡿⠀⠤⠁⠭⠀⢽⡀⠀⠀⠄⠀⠀⣄⠀⠀⠀⠀⠀⡠⢶⡄
⠀⠀⠀⠀⠁⠀⠁⠁⣇⠀⢀⢀⡀⡇⢸⠀⢸⡇⡟⠛⠉⠉⠛⢛⣙⣉⡋⠙⣷⡀⠐⡂⡀⢀⣺⣰⣾⣆⠀⣾⣧⣿⢠⠲⢙⣳⡾⠤⠤⠽⣆
⠀⠀⠀⢰⡶⠀⠀⠀⣗⠂⠘⢸⠵⡇⠈⢲⡾⠁⣇⡤⡤⠄⢰⠯⠿⠯⠿⠄⣧⠀⠀⠀⠄⢀⢸⠀⣀⡀⠀⢀⡀⢹⠘⢴⡥⠞⡣⠤⠀⠬⣿
⠀⠀⢀⠈⠛⠀⡀⠀⠟⠀⠀⠈⠀⡃⠀⢠⣇⠀⡇⠃⠃⠃⠐⠴⠶⠶⠿⠀⣿⠀⠀⠁⠁⠈⢹⡄⠛⠃⠀⠙⠙⢸⠉⠻⠏⡏⡃⢀⠀⠂⢹
⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠉⠈⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠀⠁⠉⠉⠁⠈⠁⠀⠈⠁⠀⠒⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠰⢀⠉⠀⠀⡀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣉⠒⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠥⢄⣉⠈⠁⠀⢠⡘⠀⠇
⠀⠀⠂⠀⠐⡀⠁⠠⠁⠀⠀⠁⠀⠀⠀⠀⢉⠈⣉⠂⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠲⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡆
⣠⡐⠃⢀⢁⡄⠀⠀⠀⠀⠀⠁⠤⣀⠀⠠⠀⣠⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀
⠋⠁⡉⠡⠀⢠⣄⣐⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠦⠊⠀⢀⠀⠀⠀⢆
⠀⠤⠁⠐⠃⠀⠀⠁⡠⡙⠄⠀⠀⠀⠀⠀⠀⡀⠐⠀⠒⠀⡤⠀⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠤⣄⠀⢠⡆⡆⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢳⠀⡿⣤⡈⠻⢅⡀⢸⡄⢸⠀⣿⢸⡇⣰⠛⢸⡇⢠⡄⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢳⡒⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⣧⠀⠈⣿⡄⢸⡇⢸⣇⠾⠼⡟⣼⠁⣿⠀⢸⡧⢸⡇⠀⠐⣄⠙⠦⠤⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠸⡆⢄⡿⠃⠈⠁⠀⠁⠀⠀⠀⠻⠀⠉⠀⠘⠇⢸⡇⠀⠀⠀⠁⠢⠀⠊
⠀⠀⠀⠀⠀⠁⠂⠠⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠚⠁⠀⠀⠄
⣄⠀⠀⠠⠤⠀⠀⠀⢠⠀⠀⠲⢘⠉⠀⠀⠈⠉⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠄⠈⠀⠀⠀⠀⠀⢠⠈
⡍⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠀⠀⠀⢀⣀
⠈⠂⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠠⠾⠄⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⡀⠁⠀⠀⢢⠋⣹⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠱⠀⢁⡂
⠔⣂⠸⢟⢠⠄⢀⢹⢰⠄⠉⠴⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠂⢀⠈⣣⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠙⢍⣘⠁⠀⠀⢘⣸⡀⠠⠀⢈⣫⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⢶⠈⣁⠌⠀⠀⠀⠌⠀⠀⠠⠀⠀⠀⠀⠀⠐
⠢⣀⠉⠀⠀⠜⢻⣤⡵⢶⡏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠁⢈⠈⠁⠀⠀⠀⠊⠘⠄⠀⠐⠀⠀⠀⠀⠀⠁⡀
⠀⠈⠪⡰⠄⡀⡊⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠉⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠉⠁⣲⠐⢎⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠃⠀⠀⠱⠄⠀⠀⠀⠀⢠⣾⣿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠻⣿⡿⠏⠀⣀⣀⣤

⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⡄
⠀⠀⠀⠀⠀⠀⢠⠄⠈⢄⣀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠛⠫
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠐⠀⠀⠴⠋⠀⠠⠀⠀⠀⠠⠄⠤⠄⠻⣿⣿⣿⡇⡀⠀⠀⠀⣀⣤⣶
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠑⠀⢄⡀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⠉⠡⠂⠁⠀⠀⢰⣿⣿⣿⠃
⠀⠀⠀⠘⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠉

⠀⠀⠀⢀⣀⣰⣶⣶⣶⣦⣄⠀⠀⠀⠀⠔⠀⠰⡀⠀⠀⠀⠀⢀⣀⣠⣤⣠⣀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⡴⠆⠀⢀⡴⠂⠀⢀⡰⠆⠀⣀
⠀⠀⠐⠛⠿⠿⡿⠟⠛⡋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠈⠠⢈⡯⢻⣷⣧⠀⠀⠀⠈⠉⣷⠀⠀⠀⣀⡀⠀⢁⣀⠀⠀⣀⠀⢀⣈⠀⢀⡈⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⠀⠇⠀⠀⠀⠈⠀⠁⠀⠈⣟⣻⣾⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣷⢺⣾⢟⣷⠒⣿⠻⠶⣖⠛⠛⠷⠲⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⠁⠀⠀⠀⠈⢀⡿⠀⠀⠀⠈⠉⠀⠒⠒⠚⠂⠀⠘⠓⠛⠒⠛⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠁⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⢀⢔⣡⣶⣾⠿⠿⠿⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣻⣳⣴⣾⣿⣷⣿⣿⡷⠀⠈⠒⠒⠉⢹
⠀⠀⠀⠀⣀⠂⠄⢀⠀⠠⢀⣐⠢⠀⠀⠀⠀⠀⠀⢤⣵⠿⠋⠁⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⡿⠿⠛⠛⠀⠈⠁⠀⠀⠀⠀⠀⠨⡄
⠀⠀⠀⠢⠤⣉⠐⠀⠈⣁⠀⢈⠔⠃⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢸⠀⢠⠁⠀⢀⡀
⠀⠀⣶⠤⢄⣀⠈⠑⠂⢈⠐⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠋⠀⠀⠀⠀⢘⠀⠄⠀⢰⠁⣿
⠀⠀⠙⠛⠛⢻⣌⣹⠶⠋⠀⠀⠀⠀⠀⣰⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⣀⢄⣄⠀⠀⠐⠒⠀⠂⠀⠀⠐⠁⠀⠀⢈⡜⠀⢀⠘⠓⠛⣴
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠂⢺⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⢨⡷⠈⠀⠀⢀⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡇⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠶⢶⣶⡶⣶⠶⢚⠂⠀⠀⠀⠰⡀⠁⠀⠀⣾⣿⡏⢀⣮⠀⠀⠀⠀⠀⠀⠞⠀⠀⠘⡁⡛⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⢀⡿⠿⠾⠿⠾⡿⠿⢿⠀⠀⠀⠀⠀⡈⣴⠂⠀⠘⠿⠷⠄⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆⠃⠀⡇
⠀⠀⠀⠀⠀⢤⠦⠄⠀⡄⣠⡀⠀⠀⡠⠾⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠁⠀⠁⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠄⠀⡇⡄⠠⡇
⠀⠀⡀⠀⠀⠀⣰⠏⠀⡔⠉⢧⠙⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢸⡇⠀⠀⡀⣀⣴⣴⣶⡆⠀⢀⣀⠀⠿⠆⠀⠀⢣⡅⢸⡇
⠀⠀⠀⠀⠀⠀⠁⢰⠞⠀⠀⠈⠇⠘⣁⠤⠀⠀⠀⠀⠀⣠⣴⢆⡜⢉⠝⠗⠦⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⡛⠋⠋⠉⠙⠇⠀⠀⠉⠁⠀⠀⠀⠀⠸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣴⣽⢿⣶⣭⣏⣷⣄⣬⣵⣤⠀⠰⠀⠀⣠⠋⠀⠀⠘⡧⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣴⣯⣟⡻⢶⣤⡀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⢀⡴⠊⢁⠶⠀⠀⠀⢲⡈⠀⠰⠀⢀⠀⠉⢻⡀⠀⠸⣧⣤⣤⣉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⣤⡙⡿⣿⠟⢱⡶⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠂⢀⡎⠀⢠⡾⠁⠀⠑⠦⠈⡻⢟⣁⡿
⠀⠀⠀⠀⠠⣤⣤⡀⠀⠈⠙⢿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠃⢀⡾⡀⢰⡿⠁
⠀⠀⣼⠀⠀⢸⡏⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠤⠀⠂⠀⠀⠀⣰⣶⣺⠟⠀⠀⠀⠀⠀⠌⠀⢠⡿⠁⠀⠀⠀⠀⢤⡂
⠀⠀⣹⠀⠀⢈⡁⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡷⠎⠁⠀⠀⠀⠀⠀⠀⠠⢤⡾⠁⠀⠀⠀⠀⡆⠸⢁⡐⡇
⠀⠀⣿⠀⠀⠈⣿⠄⡄⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⢀⡴⣾⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡄⠀⠀⠀⠀⠀⢀⠃⠸⠀⡇⡇⢄⠔
⠀⠀⣿⠀⠀⠀⠡⡆⠁⠀⠀⠀⠀⠀⠒⠀⠒⠐⠒⠀⠒⠒⠀⠐⠀⠀⠀⢐⣿⡷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠈⠀⠀⠀⠀⠘⠀⠂⠚⣛⣽⡌
⠀⠀⢿⠀⠀⠸⣷⠀⠈⠀⠀⠀⠀⠀⠠⣀⣤⠀⣄⣀⣀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢠⣀⢀⠀⡀⢂⠐⡀⣂⠀⠀⠀⠀⠀⠀⠀⢸⣶⠆
⠀⠀⠸⡀⠀⣸⢡⠄⡄⠀⢀⡀⠢⠄⣻⣾⠏⣹⣁⡼⣤⣾⡚⣩⣤⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠘⡟⡌⠁⠉⡀⢆⠰⡐⢂⠀⠀⠀⠀⠀⠀⠈⣿⡶
⠀⠀⠀⠈⠰⢯⠎⠉⠀⠀⠈⠡⠤⠈⠭⠭⠬⠩⠍⠠⠭⠌⠭⠉⠁⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠉⠁⠉⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⠋⠉⢭⠉⠙⠟⠛⠻⣿⣶⣆
⠀⠀⠀⠀⠀⣠⣿⣿⢿⣶⣴⣄⢤⡈⠂⠀⠀⠠⡈⠏⠻⣷⣄
⠀⠀⣀⣀⡾⠇⡜⢟⠾⢿⣿⣿⣷⣿⡄⡄⡆⣦⠉⡆⠀⠹⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠙⣿⣁⡾⡀⠀⣹⣷⣻⣿⣿⣿⣧⣿⡵⣏⠀⢀⢶⡆⡲⢽⣧⣀⣀⡴⠚⠛⠛⠛⠒⠛⠛⠻⢿⣿⣶⣤⣀
⠀⠀⢰⢡⣿⠒⢙⠛⢋⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣸⠇⠀⡀⠿⣏⡁⠀⠀⠛⠂⠀⠀⢀⣀⡀⠀⠈⠻⣿⣿⣦
⠀⢀⣼⣿⣿⣆⠈⠀⠋⡼⣟⣿⣻⡟⣿⣿⣿⣿⣧⣄⣿⣿⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠘⢿⣏⠻⡀⠀⠀⠸⣿⢿⣷⡄
⠀⢸⣿⣿⣿⣿⠀⠦⡄⠳⡿⢡⣿⣿⣸⣿⣿⣿⣿⣿⣿⠏⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⠀⠀⠀⠀⣿⣿⣽⣿⣤⡄
⠀⢰⠟⡴⠿⠟⣅⠠⢷⢻⣇⢸⣿⣿⣷⣿⣿⣿⡟⠉⠁⠀⣠⠇⢸⣿⡆⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⢀⠀⣿⣿⣿⣿⣷⣍⣓⠄
⠀⠀⠀⠀⠀⠀⠈⠒⠖⠚⢁⣾⣿⣏⣿⣟⣿⣿⣿⣄⡀⠘⠏⠀⢘⣿⡇⠀⢀⣀⣤⣾⣿⡿⢿⣿⠀⠀⡼⢰⣿⣿⣝⣿⣿⣯⣭⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢋⢩⣉⣉⣹⣿⣾⣿⣿⣿⠁⢀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠀⢾⠁⣿⣿⣟⣿⣶⡍⣷⣦⣈⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⣟⠉⠉⠉⠀⠉⠙⠛⢿⣦⠁⠐⡿⠿⣿⣽⣿⣿⠟⠋⠁⠉⠉⣿⣿⡄⠀⢠⣿⣿⣿⣝⣿⢽⣿⣿⣿⡿⠿⠷⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⢦⣤⣤⡀⠀⠀⠀⠀⢻⡆⢸⡇⠀⣼⣿⣿⡃⠀⠀⠀⠀⠀⠈⢻⣿⡆⣾⣿⣿⣿⣿⣿⣷⣿⣻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣿⣿⡌⣆⠀⠀⠀⢸⣿⠐⡇⣠⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣧⠁⣿⣿⣿⣿⣿⣿⣟⡻⡿⢿⣿⣿⣶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠗⠋⠀⠀⠀⠀⢹⣷⣿⠟⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⠋⠻⣿⣿⣿⣯⣽⡷⠤⠀⠁⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⡸⣄⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢳⡄⠀⠀⠉⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣾⣿⡇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠆⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⢣⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⠉⢋⣲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⣤⠖⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠰⠆⠀⠀⠰⠶⠂⠀⠀⡀⠀⠀⢠⠖
⠀⠀⢀⠋⠀⢀⣀⡀⠀⠳⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠠⠆⠀⢀⠀⠀⠀⠀⠀⢟⠊⠀⠀⠔⠀⣴⠖⠀⠀⠀⢀⡀⠙
⠀⠀⠇⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠒⠀⠀⠀⢀⠋⠀⣰⣆⡠⠔⠉⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⡆⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⡤⠶⠭⠤⠤⠖⠊⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⣄⠀⠀⢀⣀
⠀⠀⢿⡄⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⣀⡀⠀⠈⠁
⠀⠀⠀⠐⠀⠐⠻⠀⠀⠀⠀⡶⠀⡾⠃⠀⠀⣀⣀⠀⠀⢀⣀⣀⠀⠀⢀⡾⢹⠀⠀⠀⠀⣀⠀⠀⠀⠁⢤⣤⡀⠀⠙⠃⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⢋⣼⠇⠸⠟⠛⠛⠛⠷⣾⢟⡋⣘⢂⣤⡼⣧⠃⠀⠀⠀⠀⠈⠁⠐⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⣿⡏⢠⡶⢲⢰⡟⢹⣸⣿⣡⠏⣿⢸⡗⡡⣈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⢁⣿⡧⣼⢻⣏⡞⠻⠏⢷⣄⡭⠥⢥⡈⠣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣀⡴⠋⠙⠓⢡⣤⣤⣴⡶⠟⠋⠙⠳⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠴⠆⠀⠚⠛
⠀⠀⠀⢠⠄⠀⢀⡄⠀⠀⠉⠛⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢠⠄⠲⠒⠒⠈⠛⠢⠀⠀⠀⠀⠀⠀⣠⣏⣴⠶⠂⠈⠁⠒⠀⠁⠀⠶
⠀⠀⢀⡄⠘⠳⠀⠀⠀⠀⠴⠖⠀⠀⠉⠉⠀⠀⠀⠀⡴⢈⡃⠀⠐⠛⢂⠀⠀⣤⡄⢠⡄⠀⠀⠀⢹⡯⢠⣦⣀⣤⣧⣠⣤⡤⠀⠠⣄
⠀⠀⢸⣷⣄⡀⠶⠂⠺⠷⠀⠀⠾⠳⠂⠈⡜⠀⠀⠼⠁⢨⣅⠀⠀⠀⠈⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⣿⡿⠋⠋⣩⡍⠁⠈⠙⠲⣶⠉
⠀⠀⢿⠿⠿⠿⠿⠿⣦⣿⠶⠶⠦⢶⣖⠛⣀⠀⠀⢎⠀⢀⡀⢀⠀⠀⠀⠈⠷⠐⠁⡄⠀⠀⠀⠀⠰⠦⠁⠴⠤⠀⠀⠀⠲⠶⠀⠈⠧
⠀⠀⠀⣽⣿⣶⣛⣠⣀⣀⣀⣠⣤⣿⠿⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⣸⢀⡄⠀⠀⠀⢀⣄
⠀⠀⢀⣭⣍⡉⠁⢈⢉⡉⣩⠿⠀⢀⣀⣤⣴⠂⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠙⠀⠀⢸⡇⠀⣤⡀⠀⠀⠉⠀⢠⠈⠇
⠀⠀⢈⣜⡋⠩⠿⠉⠉⠹⠏⠿⠉⢉⢉⣉⡥⡄⠀⠀⠀⠀⠄⠀⠰⠚⠀⠂⠈⠀⠁⠀⣤⠀⠀⠀⠀⢻⡀⠈⠁⠐⠂
⠀⠀⠺⡿⣩⣟⠻⢶⣶⠶⠓⠓⠚⠉⣩⡄⢐⡴⠀⠀⠀⠀⠀⡄⠀⠀⠀⠷⠖⠀⠀⠀⠁⢠⡆⠀⠀⠀⠙⠒⠠⢄⣀⡘⠃⠔⠒⠁
⠀⠀⠀⠙⠛⠛⠲⠮⠉⢤⡤⠤⠶⠤⠶⠚⠉⠀⠀⠀⠀⠀⠤⠀⠶⠆⠀⠀⠀⠀⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠄⠀⠀⠈⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠄⠀⠰⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⢠⠄⠁⠀⡄⠈⡝⠢⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⣀⣠⠤⠤⠶⡿
⠀⠀⠀⠀⣠⠞⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠒⠊⠉⠉⠀⡀⠀⢎⠜⠁
⠀⠀⢀⠞⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⠒⠋⠉⠁⠀⠀⠀⠀⠄⠀⣊⣠⠴⠗⠚⠻⣄
⠀⠀⣧⣂⡀⠰⢀⣀⣁⣠⠤⠤⠖⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠚⠋⢁⣀⡴⠚⢻⠉⠁
⠀⠀⣧⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⢶⠶⢶⣾⡯⠏⠙⠉⠉⠉⠉⠉⠉⢀⣠⠴⣪⡟
⠀⠀⠓⡶⢬⠤⣄⣀⣀⣀⣠⣤⣴⣒⠋⠉⠅⠈⠂⣀⠞⠀⠠⠀⠀⠀⠀⢀⣀⡤⠴⠒⣋⡡⠴⠚⠁⢣
⠀⠀⠀⡹⢌⣀⡀⠀⠀⠀⠀⠀⡀⠈⣉⣉⣉⣉⣉⣡⠤⠤⠤⣖⣒⣚⠭⠥⠔⠒⠊⠉⠁⠀⠀⠸⢀⠜
⠀⠀⠘⢧⠤⠤⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠴⠚⠁
⠀⠀⠀⢼⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠖⠊⠉
⠀⠀⠀⠈⠒⠲⠷⠤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠛⠛⠉⢀⣠⣴⣾⣿⠿⣿⣿⣿⠣⠉
⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⣤⣾⠋⠁⠀⠀⠀⢀⣠⣶⣿⣿⣿⡛⢡⠎⠽⣿⠙⠂
⠀⠀⠀⠀⡄⠀⠀⢀⣴⡿⠿⠋⠁⠀⠀⣠⡶⣿⣽⣿⡿⠋⣠⡟⠳⡂⠀⢸
⠀⠀⠀⢰⠃⠀⣴⠏⣿⠀⠀⠀⢀⣴⣿⣯⣿⣿⣿⣿⣣⣼⡿⠁⠀⠀⢀⠄
⠀⠀⠀⢸⢀⣾⣇⣼⠉⠀⣀⣼⣿⣿⣿⣿⣿⢏⣽⣿⣿⡿⠃⠀⠀⢀⠈
⠀⠀⠀⢸⣿⠿⠛⠛⢠⣾⣿⣭⣿⣿⡟⠋⢻⣿⢿⣿⡿⢁⡠⣃⠔
⠀⠀⠀⢸⡇⠀⢀⣼⣿⢁⣴⣿⣛⠿⣷⣤⡿⢛⣯⣿⣧⡼⠚⠁
⠀⠀⠀⢸⠙⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣤⣤
⠀⠀⠀⢸⣧⣟⣿⣿⣿⣿⣿⣿⡟⠉⠙⠛⠿⣿⣿⣿⡿⠋
⠀⠀⠀⢸⣿⣿⠋⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠁
⠀⠀⠀⢰⢿⡬⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢

⠀⠀⠀⠀⠀⣀⠀⠗⡀⠈⠀⠀⠀⠀⠀⠠⣠⣤⣤⢦⡠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡤⡄
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢐⠓⠊⠨⠁⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⠠⠀⠀⠀⠀⢎⠰⠈⡟⠉⡵⡄⠀⠀⠀⠀⠀⠀⠀⠃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⡔⠀⠀⠀⠀⠀⠈⠢⣤⠤⠔⠉⠀⡀⠀⠀⠀⠂⠟⢳⡋⠛⣲⣾⠁⠀⠀⠀⢠⠊⠵⠇⢪⠓⠂⢑
⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠎⠀⠀⠀⠀⠀⠀⠑⠂⠤⣀⡀⠤⠊⠀⠀⠀⠀⠀⠲⠦⠉⠙⠀⠁⠀⠀⠀⠀⠈⠂⠤⠠⠛⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⢀⠀⠀⠀⠀⠀⠀⠀⣁⢓⡀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠈⠉⠀⠐⢄⠀⠀⠀⠀⠈⠤⠀⢀⡦
⠀⠀⠀⠀⠀⠈⡸⠶⡇⠀⠀⠀⠀⠀⠀⠢⠜⠫⠙⣉⣃⠖⠀⠀⠀⠀⠀⠀⠘⢃⡀⣄⣀⠀⠀⠀⠀⠦⠊⠀⠉⡉⢢⣀⡠⠀⠀⠀⠰⠃⠀⠀⠈⣄⡀
⠀⠀⠀⠀⠓⢺⠀⠀⠐⠂⠀⠀⠀⠀⠈⢈⠃⡉⢰⡈⡐⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠂⣸⠉⠀⠀⠀⠀⢠⡀⠀⣠⡟⡠⢤⡱
⠀⠀⠀⠀⠀⠀⠁⠐⢀⡠⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠬⠉⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⠿⠃⠀⠀⠀⠈⠓⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠣⡊⣉⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⢀⡜⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⣿⣿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠐⠶⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⣀
⠀⠀⠀⠐⠁⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⡡⠀⠀⠀⠀⢺⠮⣿⢿⠆⠀⠀⠀⠀⠀⠀⣢⣄⡏⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⢛⣬⣇⣐⣨⣻
⠀⠀⠀⠠⡀⠀⣠⠶⠲⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⢜⢎⣷⡦⠀⠠⠀⠀⠀⠀⢰⣿⣿⣳⣦⣀⠴⠋⠀⠀⠀⡈⣿⢫⠞⢁⠏⡠⡅
⠀⠀⠀⠀⠈⠉⠛⢄⡑⠖⠊⠀⠀⠀⠀⠀⢀⣦⣄⡀⡠⠃⠀⠀⠀⠀⠀⠣⣳⢟⡅⠨⠀⡀⠀⠀⠀⠀⠑⢿⣿⣿⠿⠒⠀⠀⠀⠀⠀⠈⠻⠴⠚⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠜⢂⠍⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⠾⣷⣶⣤⠀⠀⠀⠈⢢⡀⠀⠀⠀⣤
⠀⠀⠀⠀⢆⣤⣶⣶⣆⡘⠂⠀⠀⠀⢠⡄⢠⣀⣀⣠⣿⣀⡀⠀⠀⠀⠀⢘⠉⠉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣬⡻⣿⣇⠀⠀⢀⡀⠃⠐⢓⠚⠁
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⡿⠁⠀⠀⠀⢀⠄⠀⠀⠹⡀⠀⠀⠀⠀⢀⣀⣀⣠⡬⣿⠿⣿⡾⠏⠀⠀⠘⡀⠀⠁⣴⠋⠉⠲
⠀⠀⠀⠻⣿⡿⠻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠠⣠⣀⣠⣀⠇⠀⠀⠀⠀⠈⠈⢷⣼⣀⣼⡷⠃⠀⠀⠀⠀⠀⠀⠒⠚⠻⣄⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⠿⣿⠿⢿⢿⡿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣦⣶⣶⣶⣷⣶⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣚⣽⡶⠾⣷⣶⣬⣍⠓⢤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢚⣉⣀⣀⡀⠀⠈⠉⠻⣿⣦⣝⢦⡉⠲⢤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠚⠉⠉⠀⠀⠉⠙⢿⣶⣄⠈⠙⢿⣧⣳⡄⠀⢳⡀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⣃⠀⠐⠶⣤⣀⡀⠀⠀⠈⠻⣷⣦⡀⠙⣿⣿⠀⠀⢷⡄⠀⢿⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣴⠾⠋⠙⠛⠶⣿⣿⣻⣧⡀⡈⠱⡄⠈⢻⣿⣦⠸⠟⠀⢠⢸⡇⠀⣸⣧⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⠍⠋⠁⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⣷⣿⣿⡿⣿⣶⣌⠀⣈⣀⣿⣼⡾⣿⣷⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡵⠓⠛⣓⠶⢶⣤⣄⠀⢄⣀⠀⠈⣿⣿⣿⡞⢖⣯⣷⣻⣿⣿⣿⣿⣿⡿⠃⣠⣏⣡⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠉⠉⠙⠳⢾⣿⣷⣬⣛⣹⣷⣾⣿⣿⣿⣾⣽⢯⢿⡿⢟⣿⣿⡿⠶⠏⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣠⣲⣽⣿⣿⣾⣿⡻⡟⣿⣿⠙⠖⢛⣸⣿⡿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡣⠖⠛⣉⣉⣙⠺⣷⣿⢻⢿⣿⢟⣟⣀⣿⣷⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣴⣶⡶⣾⣿⢿⣺⣷⣯⣴⡾⠿⠝⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣙⠿⠞⣛⣵⣾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡞⠁⠀⠀⠀⠀⠀⠀⢃⢠⡴⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢂⣁⡗⠢⠤⡀⠄⠒⢢⠤⠒⠧⡀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣠⣎⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠂⢁⡨⠀⢠⠤⠀⠈⠀⠠⠖⢂⠀⢸⡀⠀⡏⠢⠄⠀⠀⢀⢎⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠡⠄⠀⣠⡤⣀⠀⠀⣀⠀⠀⠑⠈⠁⠀⠐⠛⠉⠉⠀⠈⢱⡔⡑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢒⡞⡀⢤⡬⢤⣄⡇⠐⠉⠀⠀⠐⠀⠀⠤⣀⣀⡤⠀⠀⠀⣀⢄⡦⠪⢊⡘⣖⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢾⡓⠐⠒⠒⠒⠀⢘⢣⠀⣰⣓⣱⠀⠀⠀⠀⣄⠀⠀⠠⠾⡠⠥⠤⠖⠐⠁⣈⠆⢀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⠀⠀⠈⠢⠄⣀⡀⠴⠭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠥⠤⠐⠒⠊⠉⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠲⢆⠠⠤⠤⠄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠠⠤⠤⠄⡴⠖⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠐⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣟⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⠯⠦⠷⠓⡾⠼⠵⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣔⣿⣿⣿⣿⣿⣧⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⠉⡠⡸⠽⣶⠽⢂⣀⠓⡥⡀
⠀⠀⠀⠀⠀⠀⢰⠧⣿⣿⣿⣿⣿⣿⣿⣿⡦⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡅⢠⢗⠁⢠⠻⡀⠙⡯⡀⢹⢥
⠀⠀⠀⠀⠀⠀⣬⡿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⠀⣜⡇⠀⣼⠀⣧⠀⡽⣅⢈⡯⡀
⠀⠀⠀⠀⠀⠀⠸⢭⣿⣿⣿⣿⣿⣿⣿⣿⠖⡀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡇⠀⣜⡄⠘⡏⠀⢻⠁⣨⢥⠀⣉⡁
⠀⠀⠀⣤⢶⣶⣷⡞⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡳⠀⢀⣐⡾⢀⣠⣢⠀⡔⡄⢨⡅⢘⠙⠀⣔
⠀⠀⢎⣽⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣜⠳⢃⢠⢢⠛⠞⠀⡬⡌⠀⠀⠈⡠⢄⠁⠫⠇⡤⡈
⢀⣘⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢬⠉⣠⡯⠏⠉⣠⡶⠁⠈⢡⢄⠀⡠⡈⠁⠑⣴⣀⠉⠻
⢀⢩⣝⣻⣼⣟⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣯⠤⠀⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⢀⢕⠁⡠⢚⣩⣤⠶⠒⠈⢉⣄⠑⠁⠦⣤⣌⡛⠢⣀
⢿⣶⣶⣶⣬⣜⣛⣛⣛⣿⣽⣽⣿⣾⢿⣿⣿⣿⣻⣿⣿⡟⠿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢿⠞⢺⢿⡋⠉⢀⡀⡠⣄⢯⠇⠁⠸⡽⠠⢄⢈⠉⠙⠚
⠈⠙⠛⠻⠿⣿⣷⣷⣶⣷⣾⠿⠛⠛⣿⣤⣛⡻⢛⣿⣛⣿⣧⣬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠿⣏⠛⠙⠽⠛⠚⠉⢁⣄⠐⡿⠂⣤⡈⠉⠫⠓⠯⠎
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠳⣳⠴⣲⠜⠽⠊⠚⠀⠀⠈⠓⠑⠽⠢⣖⡤⣖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⠖⡒⣶⣤⣤⣤⠀⠀⠀⠀⠀⠀⡈⡆⠍⠁⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠂⠉⠁⠀⠄⠐⡀⠀⠀⠀⠀⢀⡊⠐⠁⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠄⠀⠸⠀⠀⠀⠠⠈⡉⠰⠃⠰⠆⠘⠂⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠐⠘⠁⢰⠂⠀⠱⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⣀⣀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠞⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠘
⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠂⠀⠀⣁⣀⣀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣯⡀⢋⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡘⢱⡖⠂⢀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⢰⠀⠙⣄⠁⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠰⠙⢘⡠⢇⡠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠈⠅⠘⠀⠀⡇
⠀⠀⠀⠀⠀⠸⠀⠀⠈⠷⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡄⠠⢀⠹⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⠁⠝⠀⠒⠂⢈⡉⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⡀⢒⣾⠀⢤⡀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠠⠐⠂⠌⡐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠸⠀⠀⠀⠀⢀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⡉⠤⠈⡁⠂⠀⠀⠀⠀⠀⠀⠀⠑⢄⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⣀⡀⠄⢀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠐⠂⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠓⠠⡀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢼⡤⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠉⠄⠡⣄⡗⠋⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡐⠀⠛⠛⠈⠀⠐⡽⠦⣀⣀⡀⠀⡀⠈⣃⢀⠊⢉⢧⡀⢁⠨⠧⡀⠀⡀⠈⠀⠀⠀⠀⢀⠀⠀⣾⠛⠂⢤⡠⢴⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠉⣈⠀⠀⠿⠀⠀⠐⠲⠀⢀⠀⠐⢌⠂⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠂⡂⢀⡍⠀⠠⠋⠈⢠⠠⡂⠌⠐⠀⠀⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⢐⠡⠀⠐⠁⠄⠉⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡜⠀⠀⠀⠀⠂⢀⣀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡠⠒⢒⡄⠈⡄⠀⠀⠀⠀⠀⠴⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠓⠒⠉⢀⠄⢰⠀⠀⠀⠀⠀⠠⠀⠀⠅⡀⠀⡀⠀⠀⠀⡀⠀⢊⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢅⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⡑⠀⡀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠁⠀⠀⣀⣀⡀⠄⠀⠈

⠀⣤⡤⢤⠀⠀⢠⢤⠄⡀⠀⠤⣤⠠⠀⢠⠠⠤⠠⠄⠠⠄⠠⠄⡄⠀⠄⣤⡤⠀⢠⠤⠀⠀⡀⢀⢴⣤⣀
⠀⢒⡄⢢⠑⢴⣾⡅⠠⠁⢒⠀⡅⢰⠂⠜⠇⠸⡄⠗⠺⢠⠇⠸⠃⠐⡆⢨⠀⡒⠈⠄⢨⣦⠀⠪⡈⠩⣛
⠀⣀⣼⠐⡟⠀⣈⠊⣄⠻⠈⣃⠇⠂⢠⠀⢦⢠⠘⢿⢻⠣⡔⡤⠀⣄⠐⠸⣘⠃⠟⣠⠱⣉⠢⣌⠀⣁⡒
⠀⠉⠿⠆⢱⡄⠈⠃⠎⡁⠛⠛⢡⡓⠀⠀⠀⠈⠋⢁⡈⠉⠁⠀⠀⠀⢞⡬⢛⠛⢈⠰⠈⠁⢀⡎⠼⠟⠋
⠀⠋⣤⠹⡄⢆⣆⠈⠋⠑⠘⠀⠄⠀⠀⠀⠀⠀⠀⠶⠴⠀⠀⠀⠀⠠⡀⠠⠀⠃⢚⠙⠁⢰⡠⢠⠏⣤⢘
⠀⡛⢨⣶⢇⣛⣇⠀⠀⠀⠀⠋⠚⠁⠀⠀⢀⡀⣄⡐⢈⣀⢀⡀⠀⠀⠈⠵⠑⠀⠀⠀⠀⣸⣋⡸⣦⡔⢟
⠀⢍⣭⢱⠅⠀⠀⡄⠀⠀⢀⠀⢤⠅⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⡤⠀⠀⠀⠀⠠⠀⠈⠠⡎⣭⡩
⠀⠉⠉⠙⠠⠤⠞⣣⢀⣤⠈⠀⠀⢀⠂⠀⠄⣤⢀⣀⣀⡀⣤⠠⠀⠀⠀⠀⠀⠁⢤⡀⢌⠷⠠⠄⠫⠉⠛
⠀⠀⠀⠀⠀⠀⠀⡴⠀⠁⠀⡀⢠⢄⣤⡶⡿⠁⠙⠿⠛⠉⠘⢿⣖⣴⡠⡄⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠄
⠀⠁⠤⠀⠄⠐⠂⠀⠀⠠⠀⣤⡳⠁⢀⣍⣡⢒⣬⢢⡔⣭⡒⡀⣩⡁⠈⣞⣤⠀⠄⠀⠀⠐⠂⠀⠀⢤⠈
⠀⣑⠾⡓⠄⠟⠀⠀⠀⢖⠝⠛⢋⣈⠱⡗⠈⠀⠡⠀⠀⠈⠀⠡⢺⠎⣃⠙⠒⠫⡲⠠⠀⠀⠺⠤⣐⠷⣘
⠀⠀⠀⠀⠗⠀⠀⠀⢠⡿⡆⢴⢵⠦⠁⠄⠁⠀⠀⠀⠀⠀⢀⠘⠠⠈⠴⡮⡆⢰⢿⡄⠀⠀⠀⠶⠈⠀⠈
⠀⠈⠀⠘⠀⠀⠀⣅⠿⠋⡠⢚⠂⠁⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠈⠐⡃⡄⠙⠻⣬⠀⠀⠀⠆
⠀⡤⢄⠻⠀⢀⠈⢰⣻⡀⠢⣑⠁⠂⠀⠄⠀⠀⣐⡿⢿⣦⠀⠀⠠⠀⠐⠈⣊⠔⢀⣾⡆⠀⠀⠠⠏⡠⢤
⠀⠃⠈⣲⠀⠀⠀⠸⡿⠁⠔⡥⡀⠄⠀⠂⠀⠀⠻⣷⣾⠝⠀⠀⠐⠀⠨⠀⢭⠢⠈⢿⠇⡀⠀⠐⣖⠁⠚
⠀⢀⠀⠰⠀⠀⠀⡋⣶⣄⠘⢬⠅⡀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠠⡥⢃⣀⣶⢛⠀⠀⠀⠆
⠀⠀⠀⠀⡦⠀⠀⠀⠘⣷⠇⠺⡺⠖⡀⠀⡄⠁⠀⠂⠐⠀⠈⢠⠐⢀⠲⢗⠇⠸⣾⡃⠀⠀⠀⡶⠀⠄
⠀⡉⢶⡄⠒⣤⠀⠀⠀⠮⣢⢤⣍⠹⡰⡧⢂⢁⡀⠀⠀⢀⡀⡀⢼⢆⡅⣡⡤⣔⠵⠀⠀⠀⣴⠐⢭⡾⢉
⠀⡀⠒⠀⠀⠠⠄⠀⠀⠐⠀⠛⡵⡀⢈⣉⡙⠬⢋⠜⠣⡝⠥⢋⣉⣁⢀⢦⠛⠀⠂⠀⠀⠠⠄⠀⠀⠚⢀
⠀⠐⠀⠀⠀⠀⠀⠓⠀⠀⠀⠁⠘⡊⠛⠷⣷⠄⣠⣴⣦⣄⠠⣾⠾⠛⠙⠃⠀⠀⢀⠐⠟⠀⠀⠀⠀⠀⠂
⠀⣠⣀⣠⠐⠒⢦⡙⠈⠛⢀⠀⠀⠀⠀⠀⠂⠛⠉⠉⠉⠁⠛⠐⠀⠄⠀⠀⢀⡀⠚⠁⢪⡴⠐⠀⡄⣀⣤
⠀⣊⣛⡘⠂⡁⠀⠂⠀⠀⠀⠀⠛⡀⢠⡀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣄⢐⠛⠀⠀⠀⠀⠘⠀⢀⠐⢇⣛⣑
⠀⣵⠛⠿⡎⢭⠏⠂⠀⠀⠀⠄⢶⡀⠀⠀⠈⠀⠉⠃⠌⠉⠈⠁⠀⠀⣁⡦⢠⠀⠀⠀⠀⢩⣭⢱⠿⡡⣬
⠀⣔⡛⣰⠃⡊⠏⢀⣄⡠⢤⠀⠂⠈⠀⠀⠀⠀⠀⠖⠶⢀⠀⠀⠀⠀⠁⠠⠀⡴⢄⣠⡀⠻⠱⠘⣆⠛⣨
⠀⣠⣴⡆⡸⠃⢀⡀⢂⡁⣤⣤⠚⣥⠀⠀⠀⢀⢄⠈⢁⡀⡀⠀⠀⠀⣬⠓⣬⣥⢈⡱⢀⡀⠘⢇⠰⣦⣀
⠀⠉⠛⠠⢧⠄⢉⢄⠋⣴⢀⡍⡆⠌⠘⠀⠞⠘⢢⣤⣾⡐⠃⠳⠀⠋⠠⢰⢩⡄⣦⠙⢠⣉⠤⣴⠄⠟⠋
⠀⠤⠁⠌⠀⠺⢿⡃⠐⡀⠬⢀⣁⠸⠄⢠⡆⢰⠃⣦⢴⠘⡆⢰⡔⠠⠆⣘⡀⠡⢀⠂⢸⡿⠗⢀⠣⠈⡤
⠀⠛⠓⠒⠀⠀⠘⠛⠒⠁⠀⠚⠛⠐⠀⠐⠐⠒⠐⠀⠐⠂⠒⠂⠃⠀⠀⠛⠃⠀⠈⠐⠒⠃⠀⠈⠒⠚⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣦⣄⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⠟⠛⠋⠹⠟⢻⣽⣿⣿⣏⠉⠙⢿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⠀⠀⡀⠀⠀⠠⠀⢸⣿⣿⣿⣷⣶⣾⣿⣷⣭⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⣻⣿⣶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢏⡐⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⠀⠁⣗⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡛⢛⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣹⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠟⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠛⠶⣶⠒⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠈⢳⡖⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈⣷⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⠀⠀⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡿⠟⠛⠛⠛⠛⠿⢿⣶⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⢸⣇⣀⣴⠗⠒⠻⠧⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠘⣷⣀⣠⣴⠟⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣴⣾⠿⠛⠛⠛⠿⢶⣄⠀⣠⡟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠋⠀⠀⠀⠀⠀⠀⠀⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⣤⣴⣶⣦⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣟⠉⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠴⠞


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠁⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠃⢀⣀⡀⡀⠀⠀⠀⠀⠈⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⠀⠉⠑⠦⡀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⢀⣀⡀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢰⠂⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠈⠁⠲⣄⠀⠀⠈⣄⠀⠀⠈⠁⠸⠄⠀⠀⠀⠀⠀⡈⠀⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣠⣤⠀⠀⣠⣤⣤⣤⡉⠢⢀⠘⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⣼⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠍⢠⡄⠀⢈⣤⠀⠀⠉⠂⠀⢷⡄⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠁⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠃⠀⠀⠉⠂⠀⠀⠀⠀⢸⣧⠀⠀⠀⠸⠐⠢⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠁⠈⠄⠁⠤⠀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢄⠀⠀⠀⢀⣠⣤⣴⣆⣀⡀⠀⠀⠀⠀⢸⣤⠀⣈⠆⠀⠀⠈⠂⠀⠀⠀⠆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⠂⠠⠄⠀⡀⠂⠀⠈⠙⠻⠯⣭⠍⠿⠛⠋⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢀⠴⠋⠀⢰⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠦⠤⠶⠒⠁⠀⠀⡰⠋⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⢲⣿⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠉⠓⠦⠤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠒⠁⢠⠋⠀⠈⠸⣖⣤⣤⣤⠠⠄⡐⡅⠀⢀⣠⠂⣠⡊⠀⠀⠈⡉⣲⠶⢄⠀⢰⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠞⡵⠃⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⢉⠀⠐⡎⠃⢰⡟⠀⣦⠀⡾⢰⠃⠀⠀⠑⢬⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠔⡀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠀⢰⠀⣤⣼⠁⠀⠁⢸⠃⢀⡔⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⠁⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⣿⡏⡜⠀⡠⡿⢸⣿⡧⠊⠀⣀⣈⢢⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⡟⢹⢹⡇⣼⢡⠇⡾⡏⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠶⠀⢸⠚⠀⠀⢸⣿⠃⠸⠈⢱⡇⠀⠁⡀⠀⡂⠀⠰⡀⠀⠀⡰⠀⠦
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠄⢰⡏⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡏⠀⠆⠀⠀⢧⡀⠋⣀⣾⡗⠄⠀⢈⠉⠉⠀⢀⢰⠂
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠙⡀⣿⢣⠐⣄⠀⠻⣆⠀⠀⠈⠈
⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡏⠀⡀⠀⠀⠀⠀⠀⠇⠹⡈⡄⠈⢗⠀⠙⠂⠀⠀⡐
⠀⠠⠇⠀⠀⠀⠀⠀⠀⢀⢮⠀⢈⡇⠘⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⡇⠀⠀⠈⠀⢰⠁⠀⠀⠀⠀⡞⠀⠀⠀⠛⠒⠠⠀⠀⠀⠐⠈
⠀⡧⠘⠀⠀⠀⠀⢀⡔⠁⠰⠰⠊⡅⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡏⠀⠀⠀⠀⣽
⠀⠸⡀⠃⠀⠀⡠⠋⠀⠠⠃⡦⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠃
⠀⠀⠑⣦⠤⠐⠁⢀⡴⠕⠊⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠈⠒⠶⠒⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⣀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠔
⠀⠀⠀⠀⢰⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢸⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆
⠀⠀⣠⠔⠃⣉⠸⠉⢧⣦⠀⢲⡀⠀⠀⠀⣠⡄⠰⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡷⣄
⠀⠀⠁⠘⠛⠉⠁⠀⠙⠻⠤⠤⣽⣤⣤⣶⡛⠫⢤⣄⡀⠀⠀⠀⣀⣀⠀⠠⠤⠴⠤⣤⢠⣤⠿⢴⣆⠃⠐⠀⢀
⠀⠀⠀⠀⠀⣙⠓⣻⡶⢦⡤⠤⣈⣧⠀⠀⠈⠑⠀⠀⠀⠀⠀⠐⠚⠃⠀⢀⡠⠤⠄⠒⠈⠀⠀⠁⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠠⣶⣦⡤⠘⠦⣉⠓⠤⢄⣠⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⠓⠒⣂⣲⡤⠤⢀⡀⠈⠁
⠀⠀⠀⠀⠀⠘⢿⢀⣦⣠⠘⣩⡿⠟⠚⡻⢤⠀⠀⠀⠀⣠⡊⠁⠈⡉⠙⠛⠉⠉⠉⠀⠉⠉⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠺⠥⠀⠄⠀⠀⠐⠂⠀⠑⠄⢠⣊⠁⠀⠀⠀⠙⢤⠀⠁⠀⣇⡤⠄⠞⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠂⠀⣠⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢰⠇⠀⡈⢀⡀⣀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⣾⠘⣇⠀⠀⢠⡏⠀⣸⠇⣸⠇⠇⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣷⣤⡀⠀⠀⠀⠀⢻⠀⠈⠀⠁⢸⠀⠀⡟⠀⡟⠀⠀⠸⠁⠀⠀⠀⠀⠈⠁⠙⢲⣿⣷
⠀⠀⠀⠀⣤⣾⣿⣿⡄⠀⢻⣯⣏⠢⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠤⠶⠶⠿⠿⠿
⠀⠀⠀⠌⠁⢸⣿⣿⠃⠀⠀⠹⣿⣷⡇⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣶⣤⠴⠶⠶⠶⠆
⠀⠀⠀⡀⢀⣿⣿⠟⠀⢀⣴⣶⣾⣿⣿⣠⠀⠀⣴⣀⢀⣀⣨⣿⣦⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠙⠿⠛⠁⠀⢀⢸⠛⢿⣿⣿⡝⢿⣧⠀⠻⢿⣿⣿⣿⣿⠟⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣀⡈⠈⡄⠈⣿⣿⣧⠈⠳⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⢼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠠⠞⣿⣿⣿⠂⠘⠷⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁
⠀⠀⠀⠐⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠤⠤⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢰⣴⣿⣿⡟⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⣿⠿⢳⣶⣶⣶⣶⣶⣶⣾⣷⡀⠈⠁⠄
⠀⠀⠀⠀⠙⠛⠉⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡟⠀⠀⠀⠀⣰⠟⠀⣠⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⣼⠀⠀⠀⠀⡞⠋⠀⠐⠋⠁⠀⠁⠀⠀⢀⢰⡛⣛⡿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⡏⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡖⡐⢠⡏⠐⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠻⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣇⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⢟⣷⠟⣼⢁⣿⠀⠀⠀⠀⢸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢇⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢿⢻⠁⠀⠀⠀⠀⠀⣠⣿⣭⣿⢋⠀⠀⢸⡇⠀⣠⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡼⠀⠀⠀⠀⠀⣴⣣⡼⠟⣳⠋⠀⠀⢸⣇⣴⣿⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠟⠋⠀⢀⠄⠀⠀⠀⢸⡏⣿⡇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⢀⡀⣰⡿⠀⠀⠀⠀⠀⡿⡿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⣴⣿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠁⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣤⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠐⠀⠒⠒⠒⠂⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⣀⠀⠀⣀⡀⠀⠀⢀⣀⢀⣀⡀
⠀⢠⣿⣿⣿⠆⢸⣿⡆⠈⠋⣀⡀⢀⣸⡇⠀⠛⠛⠛⢛⣻⣿⣧⡀⢀⠀⢶⣴⣿⣿⡿⠿⠃⠀⠛⠁⠸⠿⠟⠀⣼⣝⣿⣿⡟⣷⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠐⠄
⠀⠘⠿⠿⠋⠀⠈⠉⠁⠀⠈⠁⠽⠏⠋⡡⠀⠀⠀⠀⢈⢿⣿⠍⠀⡸⠁⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⡀⠀⣤⠘⠿⣿⣼⡽⠟⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠴⠒⠖⣋⣻⣾⣗⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠓⠉⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣧⣀⣿⣄⠀⠀⠀⠀⠀⠀⣤⣶⣤⣶⣶⣶⣾⣶⣄⡀
⠀⢀⡆⠀⢠⣟⢙⠿⣤⣿⡆⠻⢀⠀⠰⢶⣦⣿⣄⢀⣾⣿⣿⣿⣿⣶⡄⢀⠀⢠⡔⠊⠒⠄⠀⠀⡀⠰⢽⡼⣿⣹⣦⠀⠀⠀⠀⣦⣋⣻⣿⠏⠛⠚⣉⣉⣩⣧
⠀⢸⣿⡀⠀⣋⠀⠀⠀⠈⠃⢉⣹⠀⠀⠛⢶⣯⠭⣭⣭⣭⣭⣽⡷⠋⠀⠸⡄⠸⣇⣀⣀⡞⠀⢀⠃⠀⠞⠛⠛⠋⢡⣤⡟⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⢧⣙⣦⣌⣀⣀⣀⣠⣴⣿⡟⠀⠻⠇⠀⠉⠉⠙⠛⠛⠉⣩⢤⡤⠄⢠⠁⡀⠈⠉⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⢈⣿⠇⢠⣼⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠙⠻⠭⠉⢀⣩⠥⠴⠋⠠⣶⠤⠄⠀⠀⠀⠀⠒⠒⠚⠷⣶⡶⠗⠘⠦⠤⣄⣤⡴⠴⠶⠖⠀⠀⠀⠦⠤⠤⠼⠿⠀⠀⠐⠿⠻⠿⣿⠟⠛⠛⠋⠉⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣤⢠⡄⢀⡄⢠⡀⢀⡄⢀⡄⢠⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠁⠈⠉⠈⠉⠈⠀⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⠨⠬⠬⠼⠼⠭⠧⠥⠥⠥⠽⠇

⠀⠀⣠⣤⣶⣶⣶⣶⣤⣤⡀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⣶⣦⡀⠀⢀⣀⣀⠀⠀⡠⠢⣀⢀⣀⣠⣤⣀⡀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⣤⢀⣤⣄
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⡇⣠⠒⠒⠤⣀⠀⣀⠀⠰⠀⠙⠻⠟⠠⣿⣿⡿⠀⢀⠈⠠⢿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣍⣐⣶⣶⣷⣶⣶⡇⠀⢤⣙⣺⣻⢿⡀
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⣠⠤⢄⠀⠀⢡⠄⠀⠻⣃⣤⣤⣤⡤⣄⡀⠛⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣹⣿⣿⣿⠀⠀⠈⠈⠉⠒⢻
⠀⠀⠸⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢧⣀⡀⠁⠀⢀⠀⣠⣾⣁⣿⣿⣿⣟⠉⠹⣔⡀⠸⢿⡿⣿⣿⣿⣿⣿⣿⡏⠀⠐⠐⠚⠛⠟⠟⠛⠀⢠⣤⠀⠀⠀⠁
⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠟⠛⡠⠄⣸⢿⣃⠀⠑⢝⠿⢿⣽⣿⡿⠋⠀⠀⠀⠉⠗⠀⣠⡶⠶⠂⠀⢀⣀⠀⠸⠟
⠀⠀⠀⢑⠀⠀⠀⠀⠀⡀⢀⣄⡀⠀⠀⢀⡀⠀⠀⠄⠀⠈⣿⣿⣿⣷⣶⣶⡴⠟⣱⣾⡟⠀⠀⢀⣀⠀⢀⣠⠴⠒⠀⠀⠀⢀⣔⣛⡋⠉⣳⣦⡑⣬⣷⣄⣤
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣟⣷⠀⢿⣿⠗⠀⠰⣶⡀⠘⣯⣿⡯⢭⣤⡶⣾⡽⠟⠁⣴⣾⠿⠿⡟⢉⠀⠀⢴⡶⠀⠐⢉⢳⣄⣿⣦⡸⣿⣿⣸⣿⣿⣿⣿⠃
⠀⠀⠀⠈⠛⠒⠀⠀⠈⠀⠀⠙⠛⠁⠐⠀⠁⠈⠀⠀⠀⠀⠀⠀⠉⠁⠓⠘⠛⠉⠁⠀⠀⠛⠓⠒⠉⠀⠀⠀⠀⠀⠈⠀⠈⠘⠛⠛⠻⠟⠛⠛⠛⠛⠋⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣤⣝⣥⣮⣭⣿⣥⣿⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⢦⡔⢀⠎⠀⣤⢀⠄⠀⡱⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⠀⠀⠁⠊⠀⢀⠁⠈⠀⢀⠅⡀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⢤⣡⠎⠀⢀⠀⠄⠀⢀⢎⠜⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⡄⠀⢀⡖⠀⠀⠀⡠⠂⠀⢀⠔⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡃⢐⠎⠀⢀⠊⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠐⠀⠀⠀⠀⠁⠀⠠⠔⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⡀⠀⠀⠀⡁⠀⢀⢂⠅⢄⠈⠒⠀⢠⠄⢡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⡀⢖⠁⠄⠀⠀⠁⠀⢨⡌⠓⠃⠀⠀⠀⠈⠡⠂⠀⠀⠇⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠙⠸⠒⢁⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣀⡄⠤⢉⢀⡔⠘⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠈⡀⠀⠀⣰⠆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⡀⢉⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⢀⠄⠉⠁⠻⣚⠠⠧⠄⣄⠀⠘⢆⠀⠀⠀⠀⣀⠀⠁⠈
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⡠⠇⠀⠀⠀⡁⠀⠀⢄⡀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠴⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡇
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠍
⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠎
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⢄⡀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣿⣿⣻⣿⣿⣟⣻

⠀⣀⠀⠀⢲⠄
⢠⠭⠔⠓⡎

⠀⠀⠀⠀⠀⢠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢦⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢩⠞⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⡧⠘⠀⠀⠈⡇⠀⠀⠀⠀⢀⠀⣠⡄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣄⣄⣀⡴⠃⠀⠀⠀⢀⠚⠁⠲⠧⠤⠚⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠀⠀⠰⠆⠀⡀⠀⠀⠀⠀⡀⢠⣲⢀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠐⠐⠓⠈⠶⠻⠀⠀⡴⠋⠀⠈⠣⡠⠶⡄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣠⠇
⢀⡀⠀⠰⢂⡳⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠃⠀⣸⠁⠀⠀⠀⠀⢈⡷⠁⠀⠀⠀⠈⢧⡀⠀⠀⣀⡼⠋
⡜⠒⠒⠛⠁⠐⡆⠀⠐⡄⠀⠀⠀⠀⣰⠋⠀⠀⢩⡀⠀⠀⢀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠉⠛⠊⠁
⠠⡄⠀⠀⠀⠀⣿⠀⠀⠙⠳⠄⠒⠈⠁⠀⠀⢀⠀⠙⠒⠚⠉
⠀⠳⣄⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢰⡷⢏⣄⣞⣒⡄
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⣀⣰⡀⠠⠯⢶
⠀⠀⠀⠀⠀⢀⣦⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠺⣼⣤⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⠳⢤⣠⣠⠜⠁
⠀⠀⠀⠀⠀⠀⣄⡘⣧⠞⠀⠀⠀⠀⠀⢀⣤⠖⣖⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠉⡿⢭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡜⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⣠⣎⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⡠⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠕⠆
⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠒⠀⠀⠚⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠐⠀⠀⠣⣄⠀⠀⠐⠠⣾⣡⣀⡤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠤⠈⠀⢳⣦⡵⡈⠉⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠷
⠀⠀⠀⠀⠐⠀⠃⣀⠗⠁⢤⡍⡉⠋⠁⢀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⢰⣄⠐⠠⠄
⠀⠀⠀⠀⠀⣈⠋⠉⠢⣨⠂⠁⠁⠤⠐⠈⠑⠐⠀⠀⠀⠀⠀⠃⢀⡀⢀⣀⡤⠚⠙⠚⠀⢀⠃
⠀⠀⠀⠀⠀⠋⠀⠀⠈⠁⠇⠀⠀⢀⣴⠀⠐⠒⠀⠀⠀⠀⠀⠀⠈⠉⠙⠇⠀⠀⢀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠤⢤⡀⠘⡤⠠
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠁⠀⠀⠈⣿⣗⣼⠏⠉
⠀⠀⠀⠀⠀⠀⢀⢈⣄⠨⠇⠠⢏⡀⠕⠀⠀⠀⠀⠀⢠⠂⠀⠀⡀⠀⠖⢤⢐⡯⠁
⠀⠀⠀⠀⠀⠐⠏⠀⢙⠉⢢⡌⠈⢱⡄⠀⠀⠀⠀⠀⠈⠁⠀⠄⣠⠚⠁⠉
⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠴⠛⠦⠀⠀⠀⠀⠐⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠀⠐⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐
⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣠⠴⠐⢁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⡠⠀⠤⠂⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠂⠐⠒
⠀⠄⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⡠⠀⠀⠀⠀⠀⠆
⠀⠠⠀⠀⠀⠀⠀⠀⠨⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠐⠀⠀⠀⠀⠀⠀⠂⠐⠊⠂
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⠀⠀⠀⠀⠴⠏⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡃⠁⠀⠀⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠘⠄⠰⠅
⠀⠀⠀⠀⠀⠀⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠐⡀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡤⠢⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄
⠀⠘⠀⠁⠁⠑⠀⠀⠀⠀⢀⠒⠀⠃⠴⠁⠒⢲⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠐⠀⡀⠐⠂⢄
⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠐⠌⠀⠚⠈⠈⠀⠸⠀⠀⠀⠀⠁⠃⠀⠈⠀⠀⠀⠀⠀⠐⠀⠐⠂⠐⠀⠒⢈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠐⠠⠤⠥⠤⠔
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠤⣤⠤⠖⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⢀⠔⡠⢊⠕⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⡜⢈⡞⡵⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢍⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢸⠁⢠⠀⠀⠀⠀⢸⠃⡞⡜⠀⠀⠀⠀⠀⠀⢀⡤⠖⣛⣉⣍⣓⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡀⡄⣷⡀⠀⠀⠀⠀⣼⢸⡹⠀⠀⠀⠀⠀⢀⡴⣫⠔⣫⠥⠒⠂⠈⠑⣞
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢣⢳⡸⣷⡡⠀⠀⠀⣿⣾⡃⠀⠀⢀⢄⡴⣫⠞⡵⠋⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢀⢿⢧⠀⠀⠣⡱⡽⣷⡄⠀⠀⢸⣿⠁⠀⠀⢠⢎⡼⠣⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢠⠃⠈⠳⣷⣦⣀⣀⣘⣿⣿⣦⣔⠈⢿⠀⠀⠠⢣⠞⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠸⡆
⠀⠀⠀⠀⠀⢸⣀⣀⡴⠚⠉⠉⠁⠀⠀⠀⠈⠙⠛⠛⢆⡀⢠⠏⠀⠀⠀⠀⠀⢐⣒⡭⠭⠶⠶⠦⢭⣍⡁⣇
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣶⡬⢶⣺⠭⡶⠾⠿⠿⠿⢿⠒⢦⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢀⣀⣀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠛⠯⣟⡦⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣹⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⠶⠋⠁




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠖⠒⠛⠋⠉⠛⠟⠏⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠛⠉⠁⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠄⢠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠈⠀⢀⠀⠐⠀⠠⠀⠀⠀⣀⣀⣀⣀⠀⠀⠁⣰⠃⠀⢉⣳
⠀⠀⠀⠀⠀⠀⢠⠖⢿⣯⣰⣀⣦⠤⠤⠤⠴⠖⠒⠒⠚⠉⠉⠉⠁⠀⠀⢀⣀⣰⠃⠀⠐⢿⣅
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢀⣀⣀⣠⡤⠤⠤⠴⠖⠒⠒⠒⠛⠋⠉⠉⠉⠉⠉⠋⢀⣤⣤⣤⡿
⠀⠀⠀⠀⢀⡴⠞⠉⠋⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡖⠲⠦⠀⠒⠒⢾⢦⢷⠀⢹⡆
⠀⠀⠀⠀⠸⠷⣾⠇⢀⣴⡚⢧⠀⡠⠊⠙⡄⣠⠞⠱⠶⠉⠓⠊⣃⣐⣤⣖⡠⠗⣦⠠⠋⠀⡼
⠀⠀⠀⠀⠀⠀⠿⠞⢁⣷⠤⠬⣭⣴⣂⣀⣈⣓⣒⣈⣭⠭⠭⠟⠓⢋⣉⣡⠤⠚⣷⡜⠀⡼⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠓⠶⠤⠤⠤⠤⢤⣤⣴⠶⠒⠒⣒⣿⣯⣉⣉⣀⣀⣀⣠⢾⠄⣼⠃
⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⣀⣤⡶⣛⢁⠙⡛⣛⣛⣉⣁⣀⣀⣩⣭⣽⣫⠴⠋⣰⠃
⠀⠀⠀⠀⠀⠘⠳⢶⣶⠖⡛⠿⠯⠤⠭⠭⠭⠭⠭⠭⠭⠭⠴⠶⠖⠚⠋⠉⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⢼⣇⢄⠀⠐⠀⠐⠐⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⠶⠶⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⡾⠋⠉⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣼⠛⠭⠀⡀⠀⣀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣾⣤⡀⣠⠞⠃⠀⠉⣉⣉⣉⣀⣤⣿⣦⠀⠀⠀⠀⠠⠀⠀⠙⠿⣎⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⣠⠀⠀⢠⣿⠀⠀⠀⠙⠷⠌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣿⣦⣤⣾⣿⣿⣿⣟⣻⣟⠂⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢀⣾⣿⣷⠤⢀⠆⢀⠀⠸
⠀⠀⠀⠀⠀⠀⢠⠴⣶⣿⣤⣤⣾⠛⣻⠿⠿⠧⠞⠛⠈⣥⡤⠾⠷⠶⠿⠿⠿⠿⠿⠿⣿⣿⣷⣶⣾⣿⣿⣿⡶⠛⣶⡟⠀⡀
⠀⠀⠀⠀⠀⠸⡄⢀⣾⣿⣿⣿⣿⡄⠀⠀⣀⣀⣴⣶⣾⣿⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⠛⣿⠟⣷⣸⣷⣿⣾⣿⡄
⠀⠀⠀⠀⠀⠀⢻⡟⢫⣷⢩⣿⣿⡿⣦⣸⢿⣍⣭⣽⣿⣿⣿⣏⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠗⠒⠒⠿⣿⣿⣿⣤⣠⣶⣄
⠀⠀⠀⠀⠀⡄⣠⣽⣧⣼⣿⣯⡥⣿⣾⣿⣧⣿⣿⣿⣿⠟⠛⠟⢷⣶⣤⣶⣶⠂⠀⠀⢀⠀⠀⠲⠀⠠⠙⢀⣸⣶⣶⣿⣿⣿⡛⠛⠛⠛
⠀⠀⠀⠀⠘⢿⡿⠿⠛⠿⠿⠿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣧⣸⣿⣿⣿⣶⣶⣿⣷⣄⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠉⠉⠉⠙⠻⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠉

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

⣿⣷⣄⢀⣀⣤⣤⣀⠀⢀⡄⢀⣤⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠉⠘⠀⢀⡼⢀⣀
⣿⣿⣿⣿⡀⠉⠷⣏⠀⢹⡷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠩⡖⠉⣨
⠻⣿⠿⣿⣿⠠⣤⣽⡗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠀⢟⠁
⣶⡾⠀⠈⠙⢯⡿⡛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⢀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⡦⠠⠄⠀⠀⠀⠙⣤
⢳⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⢰⠇⠀⠀⠀⠀⠀⠀⠘⢧
⠚⣨⠛⠦⠤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⢾⡁⣰⡞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⢹
⡾⢿⣛⣿⣾⡃⠀⠀⠀⠀⠀⠀⠋⠛⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠊⠑⢷⣷⣦⣈
⠀⠀⠀⠀⠉⠱⡀⠀⡠⠀⠀⠀⠔⠚⠁⢟⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢤⡖⠀⠀⠀⠘⣯
⣤⡀⠀⠀⣼⣮⣶⠎⠀⠀⠀⠘⠀⠀⠀⠘⢻⠇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢿
⠛⢷⣴⣲⣿⣿⣿⠀⠀⠀⠊⠐⡄⠀⣠⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣶
⠀⠀⣿⣿⣿⣿⣏⠀⠈⠃⢀⣜⡁⢾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⣿⣿
⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⢠⠄⠙⠛⠛⠀⠀⠀⠀⠀⢀⣤⣴⣾⣯⠀⠀⢤⣄⣉⣠⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⣽⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣦⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠟⢁⠀⠀⠀⠓⠠⣬⡧⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⢰⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣤⡀⠀⠀⠀⣠⡀⠰⡇⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠐⠁⠀⠂⠀⠄⣰⢧⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠙⢹⣧⣠⣶⠃⢸⣇⣠⡅⢀⡀⣸⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠰⠀⠀⠀⠀⠀⣠⡞⠁⢀⡀⠀⠀⣀⣸⠶⠀⠀⠀⠀⠀⢉⡿⠿⠷⠼⢿⠃⠀⠐⣺⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⡟⠋⠀⠀⢀⠞⠁⠀⠀⣀⣀⣾⣿⣿⡶⠞⠛⠆⠈⠛⡉⢀⡀⠀⣀⠀⢀⡿⠗⠀⠀⠀⠤⠒⠂⠀⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉⠋⠀⠀⠀⠀⠀⣀⠘⡻⢿⡇⡺⠛⠕⠢⠀⠀⣤⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿
⠀⠀⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿
⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡇⠀⠀⢈⡄⠀⠀⠀⠀⠀⢹⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⣿⡀⠀⠸⠑⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⠀⠀⠀⠉⠛⠄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣀⡻⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠌⢥⡕⣷⢾⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣆⠀⠀⠀⠀⠀⠀⣀⡚⢦⣙⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡆⠀⢆⠀⠀⣴⣬⣁⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⣿⣷⡄⠹⣿⣿⣿⣿⣿⡧⢰⣝⣇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣶⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣿⡇⢸⣿⣿⡇⢀⣧⠆⣿⡀⠢⡇⢰⡆⢻⡇⢿⡁⢴⠀⡏⢠⡄⢰⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣀⣀⣾⣧⣀⣼⣿⣿⣇⣀⣛⣋⣠⣿⣄⣃⣽⣙⣃⣆⣸⣇⣸⣇⣸⣁⣘⣀⡇⠀⠐⢾⣿⣿⠟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⣤⣤⡈⠻⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⠗⣠⣿⣿⣀
⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠀⠟⣋⡛⣿⠛⡟⢻⡄⢙⠛⣿⠸⠟⡉⢛⠉⣹⠛⣙⣻⣿⣷⣾⣿⣿⣿⣿
⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠋⣠⡀⢿⠇⢨⠀⡯⠸⠆⢻⠇⣸⠀⡘⠿⢿⠀⢿⠦⡄⢹⣿⣿⣿⣿⣿⣿⣿
⠀⣹⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣶⣾⣿⣿⣶⣾⣶⣷⣾⣷⣶⣿⣶⣿⣶⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⣿⡿⠿⣿⣿⠿⢿⡿⣿⡿⣻⢿⣛⣿⡿⠿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⠀⠈⠡⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⠀⠈⠁⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⣴⠥⠀⠰
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠠⢄⣒⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠂

⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⣆⠄⢀⠀⠀⠀⢰⠀⣰⠀⠀⠀⢰
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠏⠾⠈⠳⠁⠼⠸⠇⠘⠠⠿⠑⠼⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⣀⡀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠱⡀⠈⠁⣀⣼⢀⡀⠜⠀⠀⠀⠀⠀⠀⠇⢀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠀⠀⠀⠀⠀⠀⠈⠉⡋⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣀⠄⠐⠀⢠⠀⠐⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠠
⢀⡀⠀⢰⢁⡈⢀⡶⢾⣹⡾⡿⠂⠀⠀⠄⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢐⣒⠀⠂
⠈⠻⠀⠈⠁⠠⢭⣙⣤⣄⣘⣭⡼⠶⠞⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠐⠪⢙⠒⠀⠤⠤⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠉⠂⠒⠒⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠒⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢠⠁⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⡷⠦⠤⢾⠧⢴⠚⠻⠛⠶⠤⠤⠤⢤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠠⠀⡸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠳⢤⡤⠃⠀⡎⠠⠀⡤⠄⠰⣄⣠⡀⠀⠙⣆⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠀⡀⣤⠔⢀⣀⢴⠾⠀⠙⠀⠓⡘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡷⡇⠘⠛⠋⡀⢀⠀⢀⠀⠀⣅⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⡼⠂⠀⠘⢛⠘⢻⡀⢋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⢀⡇⠳⡄⠀⠈⠈⠁⠉⠉⠉⠁⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠇⠙⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡀⠀⠀⡀⠀⠀⡇⠘⢄⡀⠀⠀⡠⠻⣀⣀⡠⠏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢄⠀⠀⠑⠢⠄⣀⠀⠀⠀⠀⣾⠈⠭⠍⠩⠍⠁⠀⡇⠀⣀⢀⢉⣉⣀⡀⠀⣀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠠⡀⠀⠀⠀⠉⠁⠐⠾⠿⢀⣟⡀⠘⠓⠀⣠⠀⠀⢩⡄⣬⠩⠉⠁⠁⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⣀⣀⡑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠛⠧⠰⠴⠶⠶⠶⣦
⠀⢀⡴⠛⠛⠙⠋⠁⠉⠙⢶⣄⠀⠀⣿⠶⠛⠋⠉⠉⠉⠙⠲⢬⡒⠦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡌⠀⠀⣀⣠⡀⢸
⢰⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠷⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠉⠉⠉⠙⠂⠚⠉⠉⢀⠀⢻⠋⠀⠀⠈⢋⣉⡡⠼
⡏⠀⠀⠀⠀⣰⠋⠉⠙⢳⣶⡿⠘⠁⠀⣠⠖⠓⠉⠙⢦⡀⠀⢀⣀⡿⠷⠤⠴⠆⠀⠲⠒⠴⠶⠦⠤⢤⣴⡟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⡀
⠃⠀⠀⠀⠀⠇⠀⠰⠊⣸⡏⠀⠀⠀⠀⠻⣀⡀⣀⡤⠞⠋⠉⠀⠀⢀⣀⣀⠤⠤⠤⠤⠤⠄⣀⣀⡀⠀⠀⠉⠙⠒⠤⣀⠀⠀⠀⢀⡴⣋⡿⠷⢦⡀
⠰⡄⠀⠀⠀⠈⠓⠲⠟⠛⢧⠀⠀⠀⠀⠀⡼⠛⠁⠀⣀⡤⠖⠊⠉⢈⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠉⠒⠦⢄⡀⠀⠀⠑⠢⣴⣩⡟⣹⠤⠖⣶⠟⢧⣀
⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⢀⡷⢦⣀⡰⠋⠀⠀⡠⠊⡥⢤⣀⡔⠊⠉⠙⣿⠯⢤⣞⣿⡾⣻⣏⢉⡵⣶⢦⣀⠀⠉⠒⣄⠀⠀⠈⠻⣿⡧⣶⠋⢉⣽⠛⢯⣧
⠀⠀⠀⠀⠉⠑⠒⠒⠒⠚⠉⠀⠀⡟⠀⠀⢀⠞⠀⢀⡴⠾⠿⡆⠀⠀⣴⠃⠀⢠⠋⠉⠉⢉⠏⠀⠀⠘⡟⠋⠑⢦⠀⠀⢳⡀⠀⠀⠹⣿⣛⣻⣋⣉⣩⠟⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡎⠀⣰⢻⠀⠀⠀⣸⣏⣉⣽⣇⡴⡞⠀⢀⡰⠛⢳⡤⡤⣾⠥⠤⢄⡠⢳⡄⠀⢷⠀⠀⠀⢋⣸⣏⣋⡽⠉⠙⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇⢠⣏⠙⠤⡤⠴⠛⠌⢻⡶⣏⠀⠈⠉⣁⣹⣀⣸⣄⠑⣷⠀⠀⠘⡷⠋⡿⣄⢸⠀⠀⠀⢸⣇⣀⣝⠹⢓⡶⠧⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢯⠹⡏⠹⢶⣿⣻⣦⠀⣀⣽⠛⢋⣩⣿⣏⣿⢯⢤⠀⣑⡛⡶⡖⣾⣁⣼⣾⣇⡎⠀⠀⠀⣾⣻⣏⡉⣹⢿⢦⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣀⣀⣈⣿⣷⡧⠴⢟⡛⣛⣉⠉⣉⡀⠘⠻⠾⠀⠈⠁⣼⣼⣭⡟⠓⢻⣁⠀⠀⣸⠟⠀⠀⠀⣼⠉⠑⣴⡙⠙⢳⣾⣁⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⢉⣩⢭⣭⣵⠶⠒⠛⢿⣍⠁⢽⠁⠀⠈⠛⡿⠟⠉⠒⠒⠚⡄⠀⠀⣷⠤⠿⣯⡷⠛⠁⠀⠀⣠⣞⠛⢳⢶⠋⢉⠟⢧⣀⡝⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠒⢄⡀⠀⠈⠑⠂⠕⣶⣿⠞⢣⣜⠄⠒⠒⢲⣧⣄⡴⠥⠔⠊⠁⠀⢀⣠⣴⠿⡁⢈⠷⣅⢀⡽⠧⢤⣀⠟⠦⣤⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠄⠀⣀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠀⠀⢀⣀⠠⠤⠒⢊⠉⠉⠑⣤⡜⢧⣀⣼⢋⠛⢣⡴⠛⣆⡠⠊⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠒⠒⠀⠉⠉⣁⡀⠀⠀⠀⢮⣋⠆⠀⠈⠣⣤⣏⡽⣻⡔⠛⢧⡠⠚⢷⡤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣈⡼⢟⠻⢳⠀⣡⠦⡄⠀⠀⠀⠀⠟⠉⣀⡡⠋⠉⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣩⠟⠓⠋⠉⠁⠀⠛⠴⠀⠀⠀⠀⠀⠱⣴⠿⠧⣄⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠋⢉⣭⣍⢩⣭⣭⣭⣝⠓⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠁⣀⣐⣿⢷⣿⣿⣿⣟⣿⣽⣿⡧⣿⣿⣶⢆⣉⡓⠦⣄⡀
⠀⠀⠀⠀⠀⢀⣾⠁⣼⣿⣟⣿⣿⣷⣾⣿⡿⣿⣼⡿⣷⢷⣿⣿⣞⣯⣻⣷⣦⠍⠲⣄
⠀⠀⠀⠀⠀⣼⣿⡆⠸⣿⣿⢿⣿⣿⣷⣾⣿⣾⣿⡋⠀⠀⢭⣻⣋⣽⣿⡷⣿⣿⡷⣈⠱⣄
⠀⠀⠀⠀⢠⣿⣿⣿⣄⠀⢶⣿⣿⡷⣿⣿⣿⣿⣿⣾⣷⣞⣿⣯⡿⠾⢷⣾⣭⣽⣚⣲⠷⠘⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⠙⠿⣿⣿⢟⣽⣾⣿⣿⣿⢿⠿⢁⣼⣮⣄⠱⠀⢉⠝⢿⣷⠀⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣘⣽⣿⣿⣿⣿⣍⠀⣥⣽⣿⣿⠆⠀⠀⠂⠀⠉⢾⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⢿⡟⠁⠀⠡⠄⠀⢀⣠⣷⣶⣾⣿⠿⠷⢶⣦⣄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡵⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⠶⡒⠀⣉⣻⣷⣾⡀
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣷⣭⠲⠆⣘⡃⠂⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣿⡿⣽⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⢏⣌⣭⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⢶⡝⠷⣿⣿⣿⣿⣿⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⡦⠀⢠⣴⣶⠒⡄⠙⣿⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡦⠀⠀⠀⠈⢠⠀⠀⠀⡏⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣄⡀⠀⡀⣠⢾⠈⠃⠀⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡭⣙⠙⠻⢿⣿⣿⣿⣽⣿⡴⠃⠀⢀⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠉⠻⣿⣿⢟⣣⣞⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⢟⢿⠿⢿⣿⣿⢿⣿⣾⣤⣀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡋⠀⡁⠐⠀⠄⢀⢀⡉⠁⠙⣿⣿⣧⣄⣤⣤⣠⣀⠙⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠠⠈⡀⢈⠠⠀⢀⣤⣤⡘⢿⣿⣿⣿⡿⢥⡉⣃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣰⣆⣤⣴⣾⣿⣏⠠⣶⢽⣿⣿⣿⣿⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣦⣍⡳⣶⡆⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢛⣉⣩⣿⣮⡙⢶⡆⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣉⠀⠈⠙⠻⠿⠿⣿⣄⠑⢦⠀⠀⠀⢻⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣛⣿⣿⣿⣧⡀⠀⠀⠀⠀⠈⢿⣆⠘⠏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢿⣿⣿⣿⣿⣽⡻⡦⠀⠀⠀⠀⢌⢿⡆⢣⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢚⣋⣀⣰⣿⣿⣿⡿⢿⣿⣦⡀⠀⠀⠀⠈⣮⣷⢨⡀⠀⠀⢈
⠀⠀⠀⠀⠀⣠⣭⣷⣿⣿⣍⡉⠀⠉⠀⠀⠈⠙⢿⣦⠀⠀⠀⢻⣾⣼⣿⠂⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠋⣹⣿⣿⠟⠋⠉⠳⢄⠀⠀⠀⠀⠀⠹⣧⠀⠀⢸⣿⢹⡿⠀⠀⠀⠀⠀⠀⣼⣡⡴⣄⡱
⠀⠀⠀⠀⣴⣿⣿⣟⡛⠧⡀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢻⡆⠀⢸⣟⣿⠣⠄⠀⢸⣀⣦⣿⣿⣿⡿⠟⠁
⠀⠀⠀⢖⣿⣟⠛⠫⢁⠀⠀⠀⠀⠀⠀⠈⢣⡀⢀⣀⣿⣿⣴⣿⣿⣿⣧⣄⣰⣾⣿⣿⣿⠛⠉
⠀⠀⠠⣿⣿⡿⢷⣄⠀⠑⡄⢀⢰⣄⡀⢀⣼⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠁⠀⡀⠄
⠀⠀⠀⢿⣿⣄⣈⣿⣧⣴⣿⣿⣾⣿⡵⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠀⠀⠀⢀⡀⠀⠈
⠀⠀⠀⠀⠻⠿⠿⠟⠛⠛⠋⠩⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⡙⢦⠯⠟⡠⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⡼⠧⠤⠄⠂⣆⡀
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢰⣄⠀⠘⠒⠣⠴⣧⡴⠷⣦⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⡀⢈⣀⣀⣀⠀⠀⠀⠛⠁
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡀⠘⡉⠩⠗⡆⠀⠀⡸⠛⠚⠣⡴⠀⠀⠀⠀⠰⢯⣬⡷⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠸⣤⣈⡁⠀⠀⠋⡿⢃⡤⠁⠀⠀⠀⠀⠆⢹⣻⡆⠦⠀⠀⠀⠀⠀⠀⢱⣤⡄⠀⠀⠀⢀⠀⠰
⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠉⠓⠒⠄⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⣤⣛⠀⠀⠀⠀⠀⠀⠛⠂
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠋⠙⠛⠛⠉⠀⠀⠀⠀⠀⣠⣔⣲⡇⠀⠀⢀⡂⠀⠀⠀⠀⠖⠁⠁
⡷⠉⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⢶⣰⣤⡀⠀⠀⠸⢿⡽⠻⠄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⢦⡙⠾⢷⡷⡽⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠭⣄⣾⠯⠐⠀⡀⢷⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠠
⡄⡨⠤⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⠶⣄⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠰⠗⠄⢀⣤⡺⡄⠀⠀⠈⣧⠀⠀⠀⢠⠀⡀
⢦⣤⣠⠆⠀⠀⠀⠀⠀⠀⠀⣟⠳⣶⢶⣆⠀⠀⠀⢰⣶⣾⢿⠿⠂⠀⠀⠀⠀⠀⠰⠂⠀⠱⡤⠀⠀⣴⡇⠀⠀⠀⠀⣆⠙⢏
⠀⣀⣤⠤⠒⠋⠀⠀⠀⠀⠀⠙⠳⠿⠚⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⡀⣤⡴⠟⠁⠀⠀⠀⠀⠘⠀⠴⠖
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⡡⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠐⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⡡⠎⡱⢎⡱⢆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠌⠡⢞⣹⡃⣱⠎⣡⡶⡀⠀⠀⢠⠀⠀⠀⠀⣠⢖⣶⡶⢶⣄
⠤⢤⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⠱⢎⡹⠟⣹⣾⣷⣿⡇⠀⢠⣾⠀⠀⠀⠠⠿⣾⣻⣿⣿⣽⠇⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣽⣦⣌⣩⣾⡿⣯⣿⡤⠄⠀⠈⢸⠀⠀⠀⢠⣲⣈⠉⠙⠻⣬⠀⠰
⠀⠀⠾⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢫⠎⣹⠿⠋⠀⠀⠀⢀⣦⣀⠀⠀⠀⠈⠙⠿⠛⠙⠇⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠴⠼⣣⠿⠗⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠰⣿⣭⣯⡥⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣤⠐⢌
⠁⠉⠈⠋⠀⠀⠀⠀⠀⠀⠀⢫⠐⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⣿⣄⠀⠀⠀⡄⠐⠒⢣⡌⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⢁⣀⠀⠀⠐⢄⠀⠀⠀⠀⠐⢿⣿⣿⡻⠃⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠾⠿⣯⠋⣉⡙⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⠀⠈⢢⠱⠀⠄⠀⠀⠰⠶