Black and white ox jaw Emojis & Symbols

⠀⠰⢶⣐⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⣂⡶⠆
⠀⠀⠀⠈⠙⠲⣄⠉⠙⠲⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⠉⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠉⠙⠛⠖⢶⣶⠖⠛⠛⠲⢤⡀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠲⣶⡶⠲⠛⠋⠉⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⡀⠀⠀⠀⣿⣇⡤⠤⠤⣄⣀⣈⡁⠀⠀⢈⣁⣀⣠⠤⠤⢤⣸⣿⠀⠀⠀⣀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠳⣿⠏⡀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠉⢀⠀⠀⠀⣀⠹⣿⠞⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⣠⣿⠶⣤⠰⠘⣦⠘⡄⠀⣴⣦⠀⢠⠃⣴⠃⠆⣤⠶⣿⣄⣀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⣥⣤⣤⣍⡙⡇⢀⠈⢷⠐⠘⣧⠙⠀⡌⢡⠀⠋⣼⠃⠂⡾⠁⡀⢸⢋⣩⣤⣤⣬⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⢫⠞⡽⠛⣿⣿⣿⠇⣿⣮⣳⡌⠀⠂⢹⡄⠀⢴⡦⠀⠰⡏⠐⠈⢡⣞⣵⣿⠸⣿⣿⣿⠛⢯⠳⡝⣄
⠀⠀⠀⠀⠀⠀⢠⠃⠶⣄⡀⠀⠙⢛⣵⠖⣿⣋⣿⢻⠀⠂⣼⠀⠀⠂⠐⠀⠀⣧⠐⠀⡟⣿⣙⣿⠒⣮⠛⠋⠀⢀⣠⠷⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢳⣶⣤⣄⣠⣤⡾⠁⠀⠹⣯⣁⠈⠈⡀⠀⠀⠀⠺⠗⠀⠀⠀⢈⠁⠁⣈⣽⠏⠀⠈⢷⣤⣄⣠⣤⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠛⠹⠃⠀⠀⠀⠀⠈⡇⠀⢀⣻⡀⠀⠀⢂⡐⠀⠀⢀⣟⡀⠀⣸⠁⠀⠀⠀⠀⠘⠏⠛⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠇⠀⠀⠻⠟⠀⠀⠸⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⢈⡴⣢⠀⣡⣌⠀⣔⢦⡁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⡞⢈⣀⣀⠻⠟⣀⣀⡁⢳⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠿⢦⠉⠻⠟⠉⡴⠿⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣰⣓⣛⣀⡀⢀⣀⣛⣚⣆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⡿⠿⠿⠿⠿⢿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠷⠾⠟⠛




⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⠋⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠒⠛⠛⠓⠒⠒⢾⠀⣇⡀⠀⠀⠀⠀⠀⠀⢀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢄⡑⠲⠝⠒⠒⠢⡔⠂⡡⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⢀⣑⡼⠉⠀⠀⠀⠀⠈⠙⣊⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢖⣲⠂⠀⠐⢒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠺⠛⠀⠀⠀⠐⠊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡰⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⣣⢠⣖⡂⠀⠉⡲⡇
⠀⠀⠃⠁⠀⠀⠀⢰⠘⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠈⠿⢵⢙⣙⣩⠵⠃
⠄⠈⡀⠀⠁⠀⠀⡘⡇⡇⠀⠀⠀⠀⡼⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⠁
⠀⠀⢐⠀⡰⠀⢀⠇⠁⡇⠀⠀⢀⣼⠃⠀⢈⠗⠦⣄⣀⠀⣀⡠⠴⠒⠂⢳⠀⠀⢸⡀⣀⣀⣤⠚⠁
⠀⠀⠈⠒⠡⡒⣊⠌⠀⢸⠀⠏⠉⠀⠣⡀⠇⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠸⡄⠀⡼⠉⢁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣈⡄⠀⠀⢰⠅⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢘⠀⠀⡸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⣰⡼⠀⠀⠣⠤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢅⠨⡄⢰⠉⠉⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⣰⡼⠂⠉⠉⠁
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⡀
⠀⠀⢀⡞⢡⠊⢀⣀⣀⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣀⣀⡀⠑⡌⢳⡀
⠀⠀⠸⣄⠘⣾⡉⠉⠀⠀⠀⠀⠀⠉⠉⠻⣶⡟⠉⠉⠉⡽⠋⠙⢯⠉⠉⠙⢻⣶⠟⠉⠉⠀⠀⠀⠀⠀⠉⢉⣷⠃⣠⠇
⠀⠀⠀⠈⠻⠿⠿⠧⠤⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠈⢧⠀⠀⡼⠁⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠤⠼⠿⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡖⠚⠉⠙⢒⠲⣤⡿⠤⣀⡀⠀⠘⡆⢰⠃⠀⢀⣀⠤⢿⣤⠖⡒⠋⠉⠓⢲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠉⣿⡏⡙⠢⡙⠖⠋⢁⡈⠙⠲⢋⠔⢋⢹⣿⠋⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⠓⠛⢳⢿⡶⣦⢀⡠⠊⠑⢄⡀⣴⢶⡿⡞⠛⠚⠛⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⣼⠀⠀⠀⠀⣧⡏⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⣸⠀⠀⠀⠀⣇⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⠯⠀⠀⠀⠀⢽⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣭⣛⡏⠀⠀⢹⣛⣭⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠀⠀⠀⠀⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠋⠁





⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢒⡀⠀⠊⠁⢀⠀⠁⠀⠀⢐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⣀⣠⠃⢦⣀⡀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⢀⡤⠐⢖⠒⠙⠻⡤⢹⠋⢻⡡⡾⢻⠑⣒⡖⠀⣄⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⢂⣴⣂⠸⠉⠃⠀⠁⠀⠀⠑⠀⠀⠀⠐⠁⠀⠀⠁⠀⠋⢹⢀⣲⣄⠢⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⡁⠞⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠆⣱⡄⢘⣇⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢼⣿⣄⠏⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⢇⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣧⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣣⡿⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⡄⡇⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡅⡄⠀⠸⡀
⠀⠀⠀⠀⠀⠀⡸⠃⠀⡴⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠰⡄⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠠⡄⡀
⠀⠀⠀⠀⠰⠀⠈⡀⢀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡇⠀⠐⠁⠀⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡀⠀⠑⠀⠀⡇
⠀⠀⠀⠈⠢⣤⣀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣠⡤⠊
⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠘⠦⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠞⠀⢀⡴⠛⠉
⠀⠀⠀⢡⠀⠀⠀⠀⠈⠀⠑⢦⡀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣠⠆⠀⠈⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠱⡌⠷⢂⡄⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⣄⢰⠏⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⠋⢣⣷⠂⣠⡴⠲⡀⡄⣱⡀⡠⡁⡀⡰⠶⣠⡆⢲⣦⠋⠷⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠑⠒⠛⠓⠒⠋⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⣀⣀⡁⣀⡤⠶⠒⠲⠦⣄⣀⣀⣀⡀⠀⢠⠀⠀⠀⠀⠁⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⢤⣤⣤⣀⣀⠬⠔⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠚⠒⠤⠄⣀⣠⣤⣤⠤⠶⠊⠁

⠀⠀⠀⠀⡀⠄⢀⡰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⢀⡤⠊⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠀⠀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠈⠡⡔⠢⡀
⢠⠋⠀⠀⠀⠀⠉⠑⠲⠆⠤⢲⠞⠁⠀⠀⠀⠀⠀⠀⢰⣥⠾⠛⠛⠓⠲⠤⣄⣀⣀⣀⡌⠀⠈⢆
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣫⠞⠃⠀⠰⣶⣶⡚⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠘⡆
⢷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡡⠤⣄⣠⡎⢤⣤⠿⣆⠀⠱⡈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠑⢤⣀⣀⠀⠀⢀⣠⣾⢸⠖⢿⣯⣿⠀⠘⠿⣿⡟⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠈⢹⡛⠉⠉⢠⣿⠘⢤⣀⡽⠋⠙⠒⠚⠉⠰⠺⡀⢧⠀⠉⠛⠷⣦⣤⣴⡤⢴⠞⠋⠁
⠀⠀⠀⠀⠀⠙⠒⠚⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢳⣄⠀⠀⠀⣱⠄⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠁⣿⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣸⠀⣿⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⣰⠋⡴⠊⢉⡀⠀⠀⠈⠓⠒⠒⠋⠁⣀⠀⠀⠙⢦⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄
⠀⠀⠀⠀⢠⠁⠀⠙⠦⡬⢛⣷⡀⠀⠀⠀⠀⣰⡞⠿⡁⠀⠀⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⡏⠀⠀⠀⠠⡃⡎⠙⠃⠀⢀⠀⠀⠙⠉⣦⢸⡄⢠⠏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⡇⠀⠀⠀⢰⢧⠘⢦⣀⠀⠘⡇⠀⢀⡤⠏⡸⠁⠸⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⡇⠀⠀⠀⠈⠳⠷⠦⣌⣉⣛⢛⣉⣉⡤⣾⣥⣤⢼⣁⡰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⡟⠓⠶⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠤⣶⢶⡟⠉⠀⠀⠀⣸⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⠀⠀⠀⠀⠀⠀⠈⡏⠉⣿⠉⠉⠉⡏⢸⠀⠀⠀⣿⠈⡇⠀⠀⠀⡞⠁⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠾⣆⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⡠⠛⢻⠀⠀⠀⡿⢤⣷⡀⠀⠀⣇⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⠀⠀⣿⠀⠀⠀⠀⠀⡼⠙⠛⢅⠀⠰⠥⠆⠬⠇⠀⣸⠁⠀⠈⢳⡀⠀⠘⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢯⣤⣠⡟⠀⠀⠀⠀⢰⡇⠀⠐⣄⢳⡀⠀⠀⠀⠀⠀⠛⠾⠷⠶⠖⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣳⣶⣶⣾⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢬⣉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠤⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣀⠀⠀⢀⣀⡀⠀⠀⠀⢀⠀⠀⢀⠬⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠘⠰⠶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠖⠋⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣀⠀⠈⠳⣄⠀⠀⠀⠀⢨⠄⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⠉⠉⠈⠓⠙⠿⡶⠆⠠⣯⡄⠀⠀⠀⠀⠀⠶⣄⠀⠀⠙⢻⣯⠵⠚⠛⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⢰⠀⣨⠑⠒⢤⠀⣠⠖⠒⠲⣌⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡤⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⡤⠴⡻⠠⡃⠀⣾⣬⣷⡇⢰⣯⣤⢸⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠻⢦⡤⠟⠁⠱⢤⣭⡤⠟⠂⠀⠀⠀⢻⣦⣀⡀⠀⠀⠀⠀⣀⣤⠞⠁⠀⡠⠖⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣉⡙⠛⠛⠛⠛⠉⠀⠀⠀⠸⢠⠖⠳⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⠉⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠉⠉⠛⠲⢦⣄⡀⠀⠀⠀⠀⢸⠀⠀⡇⡇
⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⢀⣸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠰⠸⠀⢀⡃
⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠆⠒⠀⠀⠀⠀⠈⠓⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡜⢠⠁⠘⢦⢦⣀⣀⡀
⠀⠀⢀⠞⠁⠀⠀⠀⠀⣾⠁⠀⢠⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⡄⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠈⠑
⠀⢠⡎⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠋⣰⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠸⡀⠀⢀⠀⠀⣀
⢀⡿⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠉⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠶⠛⠛⠛⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢱⠀⠈⠢⣀⠹
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠁⠀⠀⢿⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠲⠶⠶⡶⠶⢶⠶⣦⣤⣤⣤⣤⡤⠶⢾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠆⢰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠘⣦⣤⣤⣤⣤⡴⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢱⡀⢇⡄⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠉⠛⠛⠉⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢳⠈⣡⠋
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⡤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⢱⣹
⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣠⣧⠖
⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⡷⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡞⠛⠉⠁⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢳⡀⠀⠈⠉⠉⠛⠟⠻⠟⠛⠛⠛⠉⠉⣹⠁⠀⠀⣸⠃⠀⠀⠰⠶⠄⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣇⠀⠈⣧⡀⠀⠀⠀⠀⠌⠀⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠢⣀⣈⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢤⣀⡀⠀⠀⠀⠉⢂⡐⡀⠀⠀⠀⠈⡄⠀⠀⠀⣿⠀⠀⢰⡇⠀⠀⠀⠈⠑⠶⠤⠤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠙⠔⠂⠐⠐⢸⠍⠑⠂⠀⠈⠉⠀⠠⣖⣀⡀⠀⡀⢤⣈⣑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⡴⠊⠀⠘⠶⠶⠟⠐⠈⠛⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⣠⠃⠀⠀⠀⠀⣠⠟⠀⠀⢀⡴⠋⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠸⢧⣤⣤⣤⢤⡟⠀⠀⠀⣠⠾⠋⠀⠀⠀⣾⣥⣄⣠⠟⠁⠀⠀⠀⠀⢀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⣤⠴⠶⠚⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠛⠉⠉⠛⠛⠓⠒⠶⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢔⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⣡⠟⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠺⣏⣃⣀⣀⣀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠛⠉⢀⡴⠞⠁⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⣀⣀⣤⠤⠶⠖⠛⠉⠉⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠶⠒⠒⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⢀⣀⣤⠤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠖⢶⡛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠙⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⢰⡇⠀⠀⠀⠀⢀⠴⠚⠛⠲⠦⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⣼⣤⣤⣤⠤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠓⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣟⠳⣄⡀⠀⣴⠦⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣭⣤⡶⠂⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣗⠀⢀⡉⠓⠶⠤⠤⠬⠟⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠉⠙⠒⣤⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⠀⠀⠙⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠄⠀⠀⠀⠀⠀⠀⠸⠤⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⢤⣤⣄⡀⠀⠀⠀⠀⠀⣸
⠘⡇⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⡟
⠀⢻⡄⣿⠀⠈⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠦⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠓⠢⠀⠘⣆⠀⠀⠀⠀⣼⠁
⠀⠀⢣⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⣰⠃
⠀⠀⠀⢳⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣰⠃
⠀⠀⠀⠀⠻⣌⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡷⢀⡼⠁
⠀⠀⠀⠀⠀⠈⢧⡀⠉⠓⠦⣤⣀⣀⠀⠀⠈⠁⠀⣀⣠⡤⠤⠤⠀⠀⢬⣉⠉⠉⠉⠉⠉⠙⠓⠶⣤⣀⣠⠞⠋⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⢀⡞⠉⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⢀⣀⣠⣄⣀⣀⣴⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠉⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣓⣲⣶⣦⣤⣤⣤⣤⣶⣶⣒⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⡀
⠀⠀⠀⠀⠀⣀⡴⢋⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣎⠳⣄⡀
⠀⠀⠀⣠⠞⢉⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠀⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣌⠙⢦⡀
⠀⢀⡞⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡤⠒⣀⣀⣠⡤⠄⠀⠀⠤⢤⣀⣀⡐⠢⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠙⣆
⠀⡟⠀⢠⣿⡇⠠⠤⠤⠤⠴⠶⣒⣒⣒⣻⣽⣿⣿⣿⠛⠁⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠙⢻⣿⣿⣯⣽⣛⣒⣒⡲⠶⠤⠤⠤⠤⠀⣿⣧⠀⠸⡇
⠀⢧⠀⢸⣿⣿⣷⣶⣤⣭⣭⣭⣭⣿⣾⣿⣿⣿⡿⣧⡄⠀⢿⡄⠀⠐⢦⣴⣴⣤⣤⠖⠀⠀⣼⠇⠀⣤⡻⣿⣿⣿⣿⣾⣯⣭⣭⣭⣥⣴⣶⣿⣿⣿⠀⢠⠇
⠀⠈⠣⣄⠙⠿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣭⣭⣶⡏⣀⡦⠀⠙⢷⣤⡈⢿⡇⣿⠏⣠⣶⠟⠁⠠⣆⡈⣷⣬⣭⣝⣛⣛⣛⣛⣻⣿⣿⣿⣿⡿⠟⢁⡤⠋
⠀⠀⠀⠈⠹⢶⣤⣬⣭⣉⣉⡉⠉⠛⠛⠿⢿⣿⠛⢸⠃⠀⠀⠀⢷⡙⣷⣄⠁⢁⣴⡟⣱⠋⠀⠀⠀⢻⠘⠻⣿⠿⠛⠛⠋⠉⣉⣉⣩⣭⣤⣴⠾⠉
⠀⠀⠀⠀⠀⠀⢀⠀⢄⣈⠉⠙⠛⠷⢶⣴⡏⠞⢀⡀⠀⠐⢦⣀⣾⡟⠸⠟⠀⠘⠿⠘⣿⣆⣠⠖⠀⠀⣀⠘⠎⣷⣴⠶⠟⠛⠉⢉⣀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡉⠉⠛⠲⣄⠀⣿⣷⠀⢰⠩⣷⣦⡀⠛⢋⡤⠀⣶⡟⣿⡆⠀⣌⠛⠃⣠⣴⡯⢱⠀⢰⣿⡇⢀⡴⠚⠋⠉⣡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠤⠀⢿⣿⡆⠣⣕⠚⢉⡟⠀⠏⠀⠀⢸⡇⣿⠀⠀⠈⠃⠘⣏⠙⢒⡵⠃⣾⣿⠇⠠⠴⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⡇⡟⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢠⠟⣀⠀⠀⠀⣀⡀⠀⠀⢀⡘⢧⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⢱⠀⡖⠋⠉⠉⠙⠻⠿⠿⠛⠉⠉⠉⠓⠄⢰⠁⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠘⢠⡆⡐⢦⡀⠀⠀⠀⠀⠀⠀⣠⠖⡀⣦⠘⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣧⠹⢾⠿⠀⠀⠀⠀⠀⠸⣿⠾⢡⡏⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⡈⢛⣒⣒⣂⣤⡴⣤⣄⣒⣒⣒⠋⡀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢄⠙⡟⢿⣿⣿⣿⡇⣿⣿⣿⣿⠟⡟⢁⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⡆⢹⠀⠈⣙⣿⣦⣾⣟⡉⠁⢸⠁⡆⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⡆⢸⣿⡻⠟⠿⣛⣿⠀⡎⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣿⣶⣽⣿⣄⣼⣿⣵⣾⡇⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⠿⢿⡿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⡒⠦⡞⠹⠗⣲⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠋⠁⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣄⡀
⠀⠀⣠⡴⠞⠩⠋⠀⠀⠀⠀⢀⣀⡤⠶⠾⠿⠛⠛⠒⠒⠲⠤⣄⡤⡄⠀⠀⠀⠁⠉⠙⠛⠒⠦⣄
⡴⠋⠁⢠⠤⣤⡤⠤⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠒⠒⠒⠒⠒⠒⠂⠀⠀⠈⣧
⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠙⠓⠲⠶⣤⣤⠤⢤⣤⣀⠀⠀⠀⠀⠠⡀⠀⠀⣠⠄⠀⠀⠀⠀⣀⣠⡤⠀⠤⠖⠒⠚⠉⠁
⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠹⢗⠀⠀⠀⠀⣼⣦⣴⣇⠀⠀⠀⠀⠘⢯⡀⠀⠀⠀⠀⣶⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⡆⠘⠅⣑⠸⠀⠀⢹⠀⠀⠀⠠⠝⣓⣒⣊⡥⠞⠁⠀⠀⠀⠀⠀⠀⠀⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠘⠦⠴⠛⠳⠤⠴⠋⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣯⡴
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠉⠑⣆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡗⠿⠀⠀⢧
⠀⢸⠀⠀⠀⠀⠀⠀⡠⠂⠀⢀⣴⡟⠋⠙⠂⠀⠠⠖⡒⣶⣦⡀⠈⢳⡀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈
⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠴⠈⠀⠂⠺⠙⠳⠦⠴⠋⠀⠘⠁⠐⠦⠀⡇⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⢸⠇⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⢀⡴⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿⣷
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⠀⠉⠓⠲⠶⠶⠒⠋⠁⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠏⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣄⣤⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢠⠞⣰⠇
⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠊⠁⢠⠏
⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⣞
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠶⠃⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠖⠛⠃⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡼⢁⣄⠀⠀⠠⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⢧⠀⠉⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡟⠀⠀⠀⠀⠀⣼⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⡀⠀⣶⣄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣶⣦⣶⣤⡽⠄⠀⠀⠀⠀⠀⠧⣤⠼⡞⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠋⠉⠉⠙⠛⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⣀⡠⠊⠁⠀⠀⠈⠉⢙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠉⠛⠷⢦⣞⠉⠀⢨⡇⠀⠀⠀⠀⠀⠀⠁⠊⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠐⠑⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣿⢦⠀⠀⠀⠀⠀⢹⡆
⣠⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⢠⣿⡖⠀⡽⠛⠻⣦⣀⡀⠀⢠⡀⣠⠀⠀⠀⢸⣦⡤⠋⠘⡇⠀⠀⠀⠀⠀⣷
⢧⠈⠀⠀⠀⠀⠀⣄⣰⣧⣄⣀⣀⣀⣀⣤⣾⢿⣏⢤⣼⢾⠓⠋⣿⣿⣍⣯⠉⠉⠙⠛⠛⠻⢦⡀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⡿
⠈⠓⠦⢤⣤⠴⠞⠋⠁⠀⠀⠈⠉⠉⠉⢯⠀⢹⠀⠸⠿⣿⢆⡀⠈⢛⣿⡧⠶⠶⢤⡀⠀⠀⠀⠉⠛⠒⠛⠁⠀⠀⠀⠀⠀⣼⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠦⠼⠶⠤⠖⠃⠀⠉⠉⠉⠉⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠩⠉⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢹⠋⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢣⡀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⠁⠀⠀⢰⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⢦⠀⢀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⠀⠀⠀⢀⣀⣠⠤⠤⠒⣚⡉⠗⠀⠀⠀⠘⢆⡀⠀⠠⢎⡁⠀⠀⠀⠀⠀⠐⠿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣉⣉⡤⠤⠒⠒⣻⡭⠤⠤⠀⣄⣀⣀⣤⣤⠟⠛⡓⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣇⠀⢸⢰⠁⠀⢠⠔⢺⣤⠖⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⣹⢠⣋⡙⡆⠀⠀⠋⠛⠘⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠒⠒⠒⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠀⠐⢒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠄⠀⠀⠀⠀⠀⠀⡈⠑⡌⠀⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⠴⡀⠀⠀⠀⠈⠀⠖⠀⠊
⠀⠀⠀⠀⠀⠀⡌⢀⣀⢆⡴⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠃⠀⠨⠀⡀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠁⠋⢺⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠐⠃⠀⠄⢀⠔⠊⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠲⣀⠤⠆
⠀⠀⠀⠀⢀⠂⠐⠠⡐⠤⠀⠀⢄⠀⣠⡞⠔⠀⠀⠀⠀⠀⠀⣄⡀⠠
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠀⠘⡘⠙⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⡀⠀⠀⠀⠀⠀⠀⢠⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠜⠀⠀⠀⠀⠀⠀⠘⠀⠘⠁
⠀⠀⠀⠀⠠⠀⠒⣒⡒⠒⠒⠠⠤⢄⣀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠈⠁⠒⠢⢬⣑⠒⠤⢀
⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠐⠠⠌⡐
⠀⠀⢸⡀⣸⢸⡎⠀⣨⢽⢸⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂
⠀⠀⠀⠉⠁⠉⠈⠀⠉⠉⠈⠀⠈⠁


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⣠⣤⣤⣴⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⣀⣀⡀⢀⣠⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣭⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⣿⣿⢯⣼⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣆⠀⠘⠿⡿⢻⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠻⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣫⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣿⣷⣿⠘⠛⣛⣭⣥⣽⣿⣿⡿⠛⠁⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⡿⠿⠿⠘⣿⣿⣿⠉⠁⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠀⠉⣛⣛⡃⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⣾⠟⠋⠸⣿⡏⠀⠀⠀⠀⠈⠻⣿⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⠁⠀⠉
⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠈⢳⣤⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⢀⣠⣤⣴⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣤⢀⣤⠦⠄⠀⠈⠻⢶⣤⣴⣶⣷⣶⣦⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠏⠀⠀⠀⠀⢴⣚⣶⡿⢿⣿⡿⣿⣶⣒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣿⡇⠀⠀⠀⠀⠀⠀⢾⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣟⣿⣯⠿⠙⠋⠻⣿⣿⡿⢿⣿⢱⣟⣿⠛⢿⣅⠀⠀⠀⠀⠀⠀⠈⢻⣟⣿⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⠁⠀⠙⣿⣄⠀⠀⣴⣦⣤⣽⡧⠈⢻⣄⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠹⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⢿⠷⠀⠀⠀⠀⠸⠿⠂⠀⠀⠈⠁⠀⠀⠀⠈⠿⠷⠄
⠀⠀⠀⠀⣧⣀⣠⣴⣶⣶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡴⢶⣦
⠀⠀⠀⠀⠈⠉⣿⣯⣿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⢸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢾⣽⡿⢻⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⢈⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣾⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣮⡙⣿⣿⣿⠇⠿⠿⢛⣛⣻⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⠉⣿⠏⠉⠛⠻⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣾⠟⢁⣿⠟⠁⠀⠀⠀⠀⢹⣿⠇⠈⠛⢿⡆⠀⠀⠀⠀⠀⠀⠘⣯⡀⠀⠈⠉⠀⠈⣿⣿⠃⢰⣿⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⣠⣟⠀⢀⣼⠋⠀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢻⠿⠀⠀⠀⠀⠀⣿⡋⠀⠉⠉⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣄⠀⠀⠀⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⣹⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⢰⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣡⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠉⠈⣩⣿⣿⠛⠻⠿⠿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⢠⡞⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠘⢧⡦⢄⣀⣀⡀⠴⣶⣶⠲⡶⠚⠙⠓⠶⣢⡟⠁⠒⣄⠁⠒⠤⢤⣄⡀⠀⠀⠀⠀⠀⠠⠤⠖⠂⠀⠀⠐⢒⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣈⣷⠃⠀⠀⠀⠀⠀⠳⣈⣤⡖⠋⠱⡄⠀⠀⢀⡀⢤⣄⡀⠀⠀⠀⠠⠀⢤⠄⠀⠀⠀⠀⠀⠀⠈⠙⠂⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠂⢛⣹⣿⠀⠀⠀⡀⠀⠀⢰⣹⣧⠛⠁⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⢀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⢋⢴⠀⠁⠘⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠏⠀⠀⣠⡟⠀⠀⢰⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⢀⣀⠰⢠⢆⣿⠀⠀⡀⠀⠀⠈⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣠⠀⢀⠞⠁⠀⠀⠀⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⠀⠞⠉⡼⠃⡅⢠⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠂⣠⠆⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⡅⡼⢀⢜⣇⡞⠁⡄⠀⠋⠁⠀⠀⠀⠀⠀⠀⠠⢴⡿⡀⠀⠀⢐⣲⣦⡀⠀⢐⣿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠁⢃⣾⠿⠀⠀⣠⡄⢀⣼⡀⠀⠀⣀⣠⡤⣤⢤⡄⠄⠀⣀⣿⣽⡋⢹⡆⢸⡇⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠊⢡⡄⡁⢈⠉⢠⣋⣥⠀⠀⠀⠀⠀⠐⠦⠤⢀⠀⠈⠀⠈⠋⠠⣨⢇⣧⣴⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⠅⣯⠇⠀⠛⣭⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡸⡰⠃⢸⠋⢫⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢠⠂⠀⢹⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⡅⠀⠀⠀⠀⠸⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠃⠀⠀⡸⠇⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⡼⠁⠀⠀⠀⠀⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠠⣤⠀⠀⠴⠉⠶⠢⠤⠄⠀⠀⠀⠀⢒⣒⡷⠼⠟⠖⣒⣀⡀⠀⣠⠟⢤⠃
⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠉⠛⠛⠛⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠂⠀⠙⠛⠋
⠁⠀⠁⢀
⠀⠀⡄⠀⠑⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣶⡤⠄⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⡀⠶⠀⠀⠀⠀⠀⠉⠀⠀⣴⣤⣤⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⢰⣿⣶⣦⠀⠀⠀⠀⠀⠀⠈⠉⢉⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣬⣿⣯⣤⡀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⡀⠀⠀⠀⢀⢸⣿⣿⡿⡿⠃
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⡀⠀⠀⠘⡏⠙⠉
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠁
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠐⢦⠀⠀⣤⠄⠀⢀⣤⣀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⣀⣀⡀⢀⣷⡀⠀⠀⠈⠋
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠟⠛⠟⠃⠀⠆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠈⠉
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⢿⡟⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠃⡿⢿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢷⣷⣿⠉⠀⠀⠀⠀⠀⣠⣤⣀⣀⠀⠀⢀⣶⣤⡀⠀⠀⢀⣠⡾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇⠀⢸⣿⡿⣿⣦⣿⣿⣿⡇⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠁⠈⠉⠉⠉⠉⢹⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠹⠿⠛⢛⣿⡆⠘⣿⣿⣿⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠂⠀⠀⠀⠀⠀⠀⠈⠿⡇⠀⣿⣿⡟⣸⣿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣥⡀⢿⣹⡇⢹⡏⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠸⣿⠀⠘⠷⣾⣿⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⢺⡇⠀⠀⠉⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠰⠤⠤⠇⠀⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣭⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣯⣶⡞⠁⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠀⢶⡄⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢶⡀⠀⠛⠟⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣟⢿⡿⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢮⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⢧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⣥⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⢀⣿⣿⢿⣿⣿⠟⠛⠛⠙⢿⣿⡏⢀⡼⠁⠀⠀⢸⣿⡿⢿⣿⡿⠛⠛⠋⠻⣿⣿⠀⢠⡞⠀⠀⣿⣿⡿⣿⣿⠟⠛⠛⠻⣿⣿⠿⣿⡟⢹⣿⡟⠁⠈⠉
⠀⠀⡾⠋⠁⠈⠻⣯⠀⠀⠀⠀⠀⠹⣧⠀⠁⠀⠀⢠⠟⠉⠀⠈⠻⡇⠀⠀⠀⠀⠈⠻⣇⠈⠀⠀⢰⠟⠉⠀⠈⠻⣆⠀⠀⠀⠈⠻⡆⠈⢻⡈⠉
⠀⠸⠧⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠸⠧⠀⠀⠀⠸⠆⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠻⠆⠀⠀⠿⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠹⠆⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣠⣴⡿⠁⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⣿⣿⣿⣷⡼⣭⣿⣿⣏⣉⠉⠁⠀⠀⠀⢻⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣾⠆⠀⠀⣠⡞⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣷⣹⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣿⠏⠀⢀⣾⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⢻⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣆⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣾⣏⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠹⡇⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠋⠉⠻⠟⠿⠿⡿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⠒⠁⠉⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢡⠀⠀⠀⡀⠀⠆⠀⠀⠀⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠁⢀⠂⠀⠀⠃
⠀⠀⠀⠀⠀⢠⣀⡤⣀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠇
⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠞⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠊⠠⠀⠇⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡎
⢧⠀⠀⠀⡀⠊⠩⢿⡤⠱⠤⣀⣂⠂⠀⠀⠠⡀⠀⡇⠀⠁⢄⠀⠁⠀⠀⠓⠀⠀⠀⠀⡀⠀⠀⠀⠊⠀⠀⢀⠀⠃⠄⠀⠀⠀⢀⠌⢰⠇
⠈⠻⠯⣿⠐⠉⠀⠀⠀⠀⠦⠈⠁⠀⠀⠀⠀⢡⠀⠃⠀⠀⠀⠁⣿⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⢀⠀⠀⠀⠘⢤⡾
⠀⠀⠀⠸⠀⠀⢠⠿⠀⠀⠀⠈⡆⠀⠡⠀⢡⠀⠆⢲⠀⠀⠀⠀⠁⡈⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠈⢣
⠀⠀⠀⠘⠆⠀⠀⠌⠀⠀⢀⣠⠇⠴⠄⠁⣊⠴⠛⠻⡀⠀⠀⠊⠁⢻⢀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡄⠀⠀⢠
⠀⠀⠀⠀⠀⣀⡀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡄⠀⠀⠀⠏⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢁⠀⠀⡏
⠀⠀⠀⠀⢾⡿⠈⠷⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢳⠁⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⠀⠀⠀⠀⠀⠈⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇⠀⠀⠀⠀⠀⠀⣠⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡃⠀⣀⡾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣎⡛⢿⡿⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⠶⠶⠛⠻⠮⠝⠛⠱⣮⠻⣄
⠀⠀⣠⣴⠶⠶⠛⠛⠻⢶⣤⣄⣀⣀⣀⣀⣀⣤⣤⣴⡶⠛⠉⠉⠀⠠⣄⠀⠀⠀⠰⡄⠀⠀⡀⠀⠀⠈⣷
⢀⣾⢫⣶⠂⠀⠀⠀⠀⡀⠈⠙⠻⠍⠉⠉⠉⠁⠀⠉⠻⢆⠀⠀⠀⠀⢹⡆⠀⠀⠀⠙⠶⠤⢷⣦⣤⣤⡵
⣸⣇⣿⡇⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⢠⡾⠋
⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣿⠈⣷⠀⠀⣠⡿⠁
⣿⣾⠈⣿⡀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⣿⠇⠀⣽⢡⡾⠋
⣿⣿⠀⠸⣧⠀⠀⠀⣠⣾⣿⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣼⡿⠊⠀⣡⡿⠁
⡟⢻⡄⢠⡿⠀⣠⠞⣿⠟⠁⠀⠈⠻⣿⣛⣿⠿⠿⠿⠟⠛⣿⠀⢰⡿⢷⣶⠟⠋
⣷⣼⠃⢸⣇⢸⣇⢸⡏⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣿⣀⣾⠃⣸⠇
⠈⠟⠀⠈⣿⢸⣿⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⡇⣿
⠀⠀⠀⠀⣿⢸⣏⣧⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⢿⡄
⠀⠀⠀⠀⠻⣧⣽⣿⣧⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣬⣿⣮⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣿⣿⣷⣦⣤⣄⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉⠉⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣫⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠛⠛⠋⢹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠀⠀⢸⣿⣿⠹⣿⣿⠁⠀⠀⠀⠀⠉⠻⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢃⣀⣀⣠⣤⣤⣤⣴⣿⣿⣶⣿⣿⣶⣦⣤⣤⣤⣤⣀⣀⡙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠶⠶⣄⢰⡆⠀⢰⡄⣶⠀⠀⢰⡄⠀⠀⠀⣴⠀⠀⢠⡴⠶⣦⡀⣠⡶⠶⡤⢀⣴⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣧⢸⡇⠀⣸⠇⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢿⡀⠀⣸⡇⣿⡀⠶⣶⢸⣇⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠙⠛⠋⠀⠛⠛⠋⠘⠛⠛⠀⠀⠙⠛⠛⠀⠙⠛⠋⠀⠈⠛⠛⠉⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄⠀⣀⠄⠠⠀⠀⠀⠀⡄⠀⣠⡠⢄⡄⠠⠠⠀⢠⡀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢒⣒⣛⣋⣙⠛⡒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣊⠽⢘⣫⠥⠔⠒⠒⠚⠸⢭⣙⠲⢬⡑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠵⣪⠔⣊⠥⠒⠋⠉⠉⠉⠉⠁⠰⠤⠙⠢⡉⢫⡖⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣔⣡⡎⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⣌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⣲⢸⣇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⠀⠀⣀⡵⡘⡆⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡼⡱⢁⠀⢟⡈⠣⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣸⡟⢳⠮⡁⠀⠳⠼⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⡼⡸⢡⠃⠀⠘⡿⣄⠈⠲⢤⣤⣭⠭⣍⣀⡡⠤⢚⡽⣸⢻⠀⠘⠆⠰⡟⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠷⢱⢃⠏⠀⠀⠀⠘⢮⠀⠀⠀⠀⠀⢀⠉⠀⣉⣉⠥⣖⣵⠷⣦⠀⢸⠀⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⡞⡇⡞⠸⠀⠀⡄⡄⣠⠊⠀⠘⠉⠀⠀⠘⠆⠤⠒⠒⠯⠟⡽⣠⢧⠀⠈⢸⢱⢣⠃
⠀⠀⠀⠀⠀⠀⢠⢣⢇⡇⡆⠀⠀⠃⡃⣇⡀⠀⠀⢀⣴⣒⢦⠀⠀⠀⠀⡴⠯⣞⠁⠘⡆⠀⡼⠼⠼
⠀⠀⠀⠀⠀⠀⢸⢸⢸⠀⠁⠀⠀⠀⢁⣜⠃⠀⠰⠯⣗⣚⠟⠀⠀⠀⠀⠃⠀⢸⠀⠀⡇⢰⢣⢲⠂
⠀⠀⠀⠀⠀⠀⢸⡸⡼⠃⠀⠀⠀⢠⣮⠏⣠⠎⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠀⢸⠀⠠⣦⢧⢏⠏
⠀⠀⠀⠀⠀⠀⠀⣇⢾⢳⠀⠀⡴⠯⠥⡤⢄⠰⠀⠀⠀⠀⣠⣫⠂⠀⠀⢠⠀⢘⠀⣠⢣⢏⠎
⠀⠀⠀⠀⠀⠀⠀⠸⡜⣆⢣⠀⢹⡀⠮⢟⣘⣦⠀⠤⢆⡨⠟⢙⢦⡀⠀⡞⠀⢊⣀⠓⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢆⠳⡄⠙⠻⠽⠯⠋⠉⠉⠑⠺⠽⠊⠀⠱⣞⢀⡴⣫⢞⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠳⣌⠢⣄⠲⠄⢀⣀⣀⣀⣀⡠⠀⢀⡠⢞⡩⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡑⠦⣍⣓⡆⡤⠭⠤⠖⣒⣫⡅⢶⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠬⢤⣭⣭⡭⠭⠵⠒⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠜⢀⡃⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠚⠛⠓⠒⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠹⡇
⠀⠀⠀⠀⡀⠀⢠⠒⠊⠁⠀⠐⠐⢢⣨⠗⢒⠈⠁⠀⠀⠀⠈⠉⠉⠒⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⠁
⠀⠀⠀⢢⡉⠂⢉⡑⠀⠀⠀⠀⠀⠀⢉⣦⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠀⠀⠀⠀⠀⠀⢸⢠⠞⠁
⠀⠀⢀⣀⠬⠭⠥⠂⠀⠀⣀⡀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢧⠀⠀⠀⠀⢀⡌⡼
⠀⠀⢋⣀⠀⠀⣠⢖⠀⠀⠉⠉⢀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⣌⠳⠄⠠⠤⢊⠜
⠀⠀⠀⠈⠉⠉⠁⠈⠢⢤⣤⣄⣀⣣⠀⡐⠁⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⠓⠒⠖⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠙⠢⢤⡶⠋⢐⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣄⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢱⠀⠀⠀⣀⠠⡴⠒⠚⠉⢰⠘⣆⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠒⠦⣤⣄⡀⢰⠀⠀⠀⢠⢋⣠⠴⠊⠁⠀⠘⡄⠀⢀⡼⠀⠈⠓⠦⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣠⣎⡀⠉⠙⡇⠀⠀⡏⠀⠀⠀⠀⠀⠀⢰⠃⠀⡞⠀⠀⠀⠀⠀⠀⠑⢆⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠙⣇⣈⠷⢰⠇⠀⢀⡇⠀⠀⠀⠀⠀⡠⠃⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⢀⠎⠀⡰⠋⠀⠀⠀⠀⡠⠊⠙⠲⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢀⣱⡄⠀⠀⢰⣿⣿⠞⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠳⣤⢦⠇⠀⠀⠀⠀⠀⠑⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠓⢲⡞⠀⠀⠀⠀⠸⣟⣟⣼⢠⣭⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠛⠋⠀⠀⠀⠀⠀⠰⢯⢴⣷⣶⠦⠆

⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠳⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⣼⠟⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⣶⠀⠀⠀⠀⠙⠤⠞⠁⠀⠀⢀⣤⠤⣄⠀⠀⠺⠋⠉⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⢤⡴⠂⠀⠀⣠⠶⢦⡀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣅⠀⠀⠀⠀⢀⡀⢠⡄⠀⠀⢸⣇⠀⢈⣧⠀⠀⠀⠀⠀⢠⠛⠙⢣
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠁⠀⠻⠁⠀⠀⠀⠈⢿⣿⡁⠀⠀⠀⠈⠲⠿⠋⠀⢀⡀⠀⠀⠘⢇⡀⣸
⠀⠀⠀⠉⠻⡷⢤⡀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁⠛⠀⠀⠀⠀⠀⠀⠀⢦⣼⠃⠀⠀⠀⠀⠉⠁
⠲⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⡛⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠛⠓⠀⠀⠀⠀⠀⠀⡞
⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣏⠀⠀⠀⠈⠻⠶⠞⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠛⠉⠀⣠⣤⣄⠀⠀⠀⠀⣀⣸⠀⠀⠀⠀⠀⠀⠈⢻⠛⠓⠂⠀⠀⠀⠀⠀⠀⠠⢄⡟⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠹⡆
⠀⠀⢸⠉⠀⠀⣷⠀⠀⠀⠀⡟⠙⠒⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠓⠂⠀⠀⠀⠀⠀⠈⠳⣤⠾⠃⢀⣀⠀⣰
⠀⠀⠀⠙⠶⠾⠁⠀⠀⠀⠀⠁⠀⠀⢠⡞⠳⡆⠀⠀⠀⠀⠀⢠⠞⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢳⡀⠀⠀⠀⠀⠀⠈⠉⠳⣿⣄⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣠⠏⠀⠀⠀⠀⠀⠘⣦⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣨⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⠁⢀
⠀⢱⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢠⣸⡷⠀⠀⠀⠀⠀⠀⡏
⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣷⡄⠀⠀⠀⠀⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠙⣿⠶⣄⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣷⠀⠀⠀⠀⠀⠀⠖⠲⡄⠀⠉⣿⡷⣀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡾⠋⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠼⠏⠀⠀⠀⠀⠀⠸⣄⣠⠏⠀⠀⠟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠙⡄
⠀⠀⠀⠀⢷⣀⣠⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⡼⠇
⡗⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣴⠂⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀
⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠴⣾⣄⣀⠀⠀⠀⠀⠀⠀⠈⣻⣯⣄⠀⠀⠀⠀⠠⠾⠿⣿⠉⠁⠀⠀⢸⠉⢙⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⠳⣆⠀⠀⢸⡅⠉⠉⠀⠀⠀⠀⠀⠰⠏⠀⠙⠃⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠘⠶⠟⠁
⠀⠀⠀⠲⣦⣀⡾⠀⠀⠈⢧⣀⣠⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣸⣿⣧⡀⠀⠀⠀⠉⠁⠀⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⣇⠀⠀⠀⠀⠐⠒⢾⣤⣀⠀⠀⠀⠀⠀⠙⢷⣴⠏
⠀⠀⠀⠜⠟⠀⠉⠉⠀⠀⠀⠀⠀⠀⠹⣦⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣴⠟⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢀⡿⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢤⣄⠀⠀⠀⢦⡀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠲⠀⠀⠋⠁⠀⠈⠁
⠀⠀⠀⢠⠞⠛⠒⡄⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢹⣿⣅⠀⠀⠀⠀⠀⠀⠀⠰⣦⣰⡆⠀⠀⠀⠘⣄⣀⣀⠇
⣀⠀⠀⠈⢧⣀⣀⠟⠀⠀⢰⣄⣴⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⣼⠃⠀⠑⠀⠀⠀⠀⠀⠀⠀⢘⣿⣄⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⡀
⢏⡀⠀⠀⠀⠈⠁⠀⠀⠀⢠⣿⣿⡀⠀⠀⠀⡏⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠏⠻⠀⠀⠀⠀⣤⣄⣠⡦⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⣀⠀⠀⠀⣠⣶⠿⠿⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠘⠟⢿⣿⠀⠀⣹⡿⠋⠀⠀⣼⡟⢁⠀⠀⠀⠀⠙⣿⣄
⠀⠀⣴⣶⣶⠀⠀⠀⠐⣿⣿⠗⠀⠀⣠⣾⠟⠛⠛⠻⣷⡄⠀⠀⠀⠀⠀⢻⣷⣴⣿⠁⠀⠀⢰⣿⠁⣿⡀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠈⢻⣧⠀⠀⣰⡿⠃⠀⠀⢰⣿⢣⣦⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠸⣿⡀⠻⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢻⣧⣼⡟⠁⠀⠀⠀⣾⡏⢸⡟⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⣾⡿⣿⣄⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⣸⡿⠀⠘⢿⣦⠀⠀⠀⠹⣷⣦⣀⣀⣀⣴⣿⠏
⠀⠀⠀⠀⣠⣿⠋⢻⣷⡀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢰⣿⠁⠀⠀⠈⢻⣷⡀⠀⠀⠀⠉⠛⠛⠛⠋
⠀⠀⠀⣴⡿⠁⠀⠀⢻⣷⠀⠀⠀⠀⠀⠙⠿⣶⣶⠿⠛⠁⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣹⣿⣦⣤⡄
⢀⣀⣾⣿⠁⠀⠀⠀⠾⠿⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠟⠀⠀⠀⠈⠉⠉⠉⠉
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⢠⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣧
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠈⢿⠋⠀⠀⠀⠀⠀⠀⠀⢀⢁⠀⠩⠄⢖⠄⠀⠈⠃⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡀⠀⠈⠻⠏
⠀⠀⠀⠀⠀⠉⠲⠴⣽⣒⣠⣤⣰⣾⡂⠰⠢⠀⠂⠀⠀⠀⠀⢀⢰⠖⢀⣺⣶⣤⣄⣈⣻⡽⠔⠊
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡉⠉⢹⡿⡁⢀⠘⠀⢢⠀⠀⠀⠠⠅⢺⣷⠄⡝⠋⠉⠉⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠻⠩⠉⠉⠉⠉⠛⠦⢹⣾⣄⠀⠀⠈⠀⢀⠀⠀⠀⠀⢀⣨⣤⠡⠞⠉⠉⠉⠉⠉⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⠳⠆⡄⠀⠀⠀⢀⣯⠶⠟⣿⡿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠈⠀⠀⣿⠀⠀⠀⢨⠁⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣽⠀⠈⠀⢸⡀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠊⠁⠀⠁⠉⠁⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠛⢆⢹⢡⡞⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠌⠛⠻⡤⢤


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣶⣶⣶⣶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣆⡀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣄⣄⣀⡀⠀⠀⣀⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣷⡄⠈⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⣷⠀⠀⠀⠉⠉⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢀⣤⣵⣶⣤⣭⡷⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣭⣭⣤⣍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⠹⢦⣀⣀⣀⡤⠊
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠁⠀⠀⡻⢿⣿⣿⣿⣿⣿⣿⣿⡿⣹⣿⡿⣿⣝⣿⠿⠿⠿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣬⣿⣿⣿⣿⣿⣿⣿⣾⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡏⣿⣿⣿⠋⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣿⣿⡏⣦⠹⣿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⠟⠀⠘⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣹⣯⣧⡿⣿⣿⣿⣷⢻⣿⣿⣿⣿⡟⣾⣿⣿⣿⠟⣡⡎⠀⠀⠀⠻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⡿⣿⣿⣿⡇⠙⠾⣿⣿⣿⣦⡙⢿⣿⣿⣿⠛⠋⠉⠀⢀⣿⣧⠀⠀⠀⠀⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣏⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⠀⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⢀⣾⣿⣿⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠆⠀⠀⠀⠉⠉⠁⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⠟



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠄⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠂⠀⠀⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠁⠉⠻⣿⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⢸⣿⡏⠀⠀⠀⠀⣿⣿⠀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⠀⠀⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣼⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣦⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣯⡁⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⢿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⢹⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡇⠀⢸⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢸⣿⣿⣤⣀⣠⣴⣿⣿⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢣⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢃⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣾⡿⢣⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢠⣾⣿⣵⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠀⠀⠲⢶⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⣦⣤⠄⠀⠈⠻⠿⠿⠿⠷⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⢀⣾⡇⠀⠈⢿⣿⣿⣿⣯⣀⣤⣆⡤⠀⠀⠀⠀⡀⠀⠀⠠⢄⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠠⣾⣿⡏⠀⠀⠀⢻⣿⣿⣿⣿⣿⠟⠀⠀⠀⣴⡿⠀⣀⠀⣀⣀⠙⣿⣆⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⣿
⠀⠀⠀⣿⣿⡟⠀⠀⠀⣄⠙⢿⡿⠋⠁⠀⠀⢠⣾⣿⡇⠸⣿⣿⣿⣿⣧⣹⡷⠀⠀⠀⠹⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⣿⡟
⠀⠀⠀⢸⣿⣿⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡓⠤⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠙⠃⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⣠⡿⣱⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣇⣾⠋⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⢀⣴⣿⣷⣿⣿⡟
⠀⠀⠀⠘⣿⣿⣿⣿⣢⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢻⠃⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠘⠁⢸⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢋⣽⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣄⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⣄⣠⣄⣀⣀⡀⠀⣀⣀⣤⣾⣿⣿⣿⣦⡄⠀⠀⢀⣤⣤⣤⡶⠶⠾⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣄⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣶⣤⣤⣄⣠⣤⣤⣶⣶⣾⣷⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠛⠻⠿⠿⠛⠋
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠁
⡿⡈⠛⠧⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠁
⣷⣏⣤⡀⠙⢢⡄⠀⠀⠀⢸⡿⠟⠁
⣿⣿⣾⣷⡄⢀⣘⢦⡀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⢄⡪⢻⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠂⠀⢀⡠⠀⠀⠐⢦⡶⠄⡀⠀⢰⡄⠀⣤⣾⣶⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣦⣾⣏⡀⠀⠀⠀⠈⠳⣤⡁⢻⡼⡏⠀⢮⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣶⠀⣠⡦⠌⠻⣶⣅⣯⣴⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⡀⠀⡰⠒⢶⣤⠀⣿⣿⣿⣿⣿⣿⣿⣿⠉⢺⣿⣯⡃⠀⠘⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠈⠾⢿⣿⡾⣿⣶⣾⣿⣿⣿⣛⣿⣿⡇⠀⢸⣿⡛⠉⠀⠀⠀⡿⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠈⠈⠁⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡘⣿⡛⠁⠀⢄⣼⠀⠀⠈⢟⠻⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠉⢿⣷⠀⣸⡏⠀⠀⠀⠀⢄⠈⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣑⢻⣿⣠⣿⢀⠀⢠⢀⡀⠀⠀⢀⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣀⣈⡉⠁⣹⣿⣿⡌⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣉⣁⣸⣭⣭⢁⠀⠀⡀⡆⠀⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠉⠻⣿⡿⣿⣿⣿⣿⣿⡟⠁⠀⠇⣽⠇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠸⡇⠁⠀⠈⠛⠉⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⡽⠀⠀⠀⠀⠀⠀⠀⢴⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇
⠀⠀⠰⢖⡒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠉⠉⠉⠀⠀⠀⠀⠀⢹⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⢠⡃
⠀⠀⠀⠀⢹⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⡸
⠀⣠⠔⠚⠉⠀⠀⢧⡀⠀⣀⠤⠤⠤⠖⠊⠉⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⢀⡀⠤⢍⠓⢤⡀⠀⠀⢀⣠⢏⡀⠀⠀⡴⠃
⢀⡇⡀⠀⠀⠀⣠⠋⡹⠋⢀⡔⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠁⠀⠀⠀⠉⠊⠉⠉⠉⠉⠀⠀⠀⠉⢲⠁
⡼⢰⠧⠄⠶⠊⠁⢀⠃⡔⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡎
⡇⢾⠀⠀⠀⠀⠀⢈⠀⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠔⠁⠀⠀⡄⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠈⠑⠤⢤⣀⡀
⠱⡈⢦⣀⠀⠀⣀⡼⢰⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⣁⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⡤⠤⠭⠓⢤⡀
⠀⠈⠒⠬⢍⣉⠥⠖⢹⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠈⠉⠞⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠳⣄⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊⠀⠐⡀⠀⠘⢦⡁⠂⠀⠀⢠⠀⠀⠀⣴⢻⠁⠀⠈⠉⠛⠒⠒⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⡦⠢⡀⠀⢠⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣠⡾⢷⣄⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡜⠉⡀⠘⠀⠈⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⠞⠁⠀⢸⡝⢷⣄⡀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⠔⡠⠀⠠⢠⠄⠀⡀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢰⠃⠀⠀⠀⢸⠁⠐⡇⠉⠓⠶⠤⣿⠀⠀⠀⢀⣰⠏⠀⠀⠀⠐⣩⣼⠁⠀⠀⢺⡀⡌⠈⠁⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⡄⠀⠀⠀⡜⠀⠀⣷⠀⠀⠀⠀⣿⠀⠀⠠⡞⠁⠒⠒⠒⠚⠋⠁⠈⢦⠀⠀⠰⡇⠈⠒⠒⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡇⠀⠀⠀⢣⡀⠀⠹⢄⠀⠀⠀⡏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠑⢄⠀⠀⠀⢳⠀⠀⠘⡄⠀⠀⢳⡀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠘⣆⠀⠀⠘⢄⣀⣀⡽⠀⠀⢠⠃⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣠⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⢀⣀⡸⠆
⣿⣿⡿⠾⠷⠿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣾⣿⣿⣿
⣿⣿⣧⣤⣤⣦⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀
⣿⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣷⡀
⣿⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⠻⣦⣀
⣿⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⠶⠶⣶⣤⣄⡀⠀⠘⢷⡀⠈⠛⢷⣦⣄⡀
⢿⡀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⣄⠈⠻⣦⠀⠀⠈⠙⠻⢷⣦⣤⡀
⢸⣧⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠈⠻⣦⡀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⢿⣆⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣰⡟⢀⡄⠀⠀⠀⠀⢀⣠⡴⠖⠋⠀⠀⠀⠀⠀⠀⠚⣿⣷⣄⣈⣹⣷⣤⣀⠀⠀⠀⠈⠻⣷
⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⡿⠀⢸⣧⡀⣀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠟⠛⠉⠁⠀⠀⠉⠙⠢⠀⠀⠀⣿⡇
⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠻⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣄⣀⣀⠀⠀⢀⣠⠤⠤⣤⠞⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠂⠀⠀⢿⠀⠀⠀⠤⠔⢒⣒⠒⠲⠾⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⣿⡟⠁⠀⠘⢁⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡄⠀⢠⣾⣿⣶⡿⠶⠦⠀⠘⣇⠀⠀⠈⠳⣤⡈⠉⠛⠶⣌⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠐⣈⡆⠀⠀⣿⠀⠀⠀⠀⠀⣰⠇⣰⣿⠏⠈⣿⡇⠀⠀⠀⠀⡿⠀⣄⠀⢲⣿⡁⠀⠀⠀⠈⠳⣌⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢀⡠⠔⠀⠹⣿⣦⣀⣿⣧⡀⣀⣤⣾⣿⡾⢿⡵⠀⣰⣿⣡⠄⠀⠀⠀⠁⠀⠸⣦⡀⠉⠛⠂⠀⠀⠀⠀⠘⣆⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢀⣴⣿⣶⣤⠶⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⡅⠀⠤⢶⣟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣄⠀⠀⠀⠀⢀⣀⣽⡆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠉⠁⠀⠀⢸⡟⠀⢿⣿⠏⠀⠀⠀⠈⢿⡀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠻⢿⣷⣶⣶⣶⡶⠾⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣼⡏⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢾⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢿⣦⡀⠀⠃⢀⣠⡴⠶⣦⡀⠀⢠⣴⣦⣤⣀⠀⠀⠀⣠⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⠹⣷⣶⣶⠟⢁⣤⣤⣿⠇⢀⡟⠁⢀⠙⣿⡆⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⠀⢿⡿⠁⠀⢾⣿⡿⠋⠀⢸⡇⠀⠀⣷⡾⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠈⠁⠀⠀⠈⠉⠀⠀⢀⡾⠀⣸⣶⠟⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⣤⣤⣤⣤⣤⣤⣶⣯⠴⠚⠋⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡀⠈⠉⣉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣾⣷⣦⣤⣤⣤⣴⠶⠞⠋⠁

⣤⣶⣶⣶⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣶⣶⣶⣶⣶⣤⡙⠃⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⣶⠄⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣾⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣯⣥⣶⣶⣦⣟⣛⢱⣿⣿⣿⣿⣿⣿⣿⣷⢙⣛⣧⣶⣶⣦⣬⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣝⢿⣿⣿⠟⣋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣈⡛⢿⣿⣿⢟⣽⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⢮⣙⣿⣿⣿⣿⣿⣟⣩⢺⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⡃⣿⣿⣿⣿⡸⣿⡟⣿⣿⣿⣿⣿⡿⣸⣿⣿⡿⠇⠙⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡼⠃⠀⣿⣿⣿⣷⣜⣿⣿⣿⣿⣿⣟⣴⣿⣿⣿⡇⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣼⣿⣿⡟⣼⣿⣿⡟⡾⠁⠀⣸⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⢿⣿⣟⣿⣿⣷⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣿⣿⣿⢡⣿⣿⣿⣸⠃⠀⢰⣿⣿⣿⡿⣿⣿⣮⡔⠿⢟⣮⣾⣿⡿⣿⣿⣿⣧⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⡏⣿⡿⢸⣿⣿⡇⡟⠀⠀⠈⣿⣿⣿⣷⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⡇⣿⡇⢸⣿⣿⡇⡇⠀⠀⠀⠘⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⡄⣽⡇⢸⣿⣿⡇⣧⠀⠀⠀⠀⠐⣮⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣩⡖⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠘⡇⢸⣿⠈⣿⣿⣷⠘⡀⠀⠀⠀⠀⢸⣿⣷⡹⣿⣿⣿⣿⣿⡿⣱⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢃⠘⣿⡆⠸⣿⣿⣇⠱⠀⠀⠀⠀⠀⣿⣿⣷⡙⢿⣿⣿⠟⣸⣿⣿⣇⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣄⠘⣿⣿⣧⡐⠀⠀⠀⠀⠸⣿⣿⡇⣶⣤⡀⠀⣿⣿⡿⣼⡇⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠈⠻⡿⣷⣄⠀⠀⠀⢠⣿⣿⡇⢿⣿⠁⠀⣿⣿⣷⣿⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡀⠉⠂⠙⢧⡄⠀⠸⣿⣿⡷⢸⣿⠀⠰⣿⣿⡿⠛⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠢⠀⠀⠀⠈⠑⠂⣹⣿⡁⠸⣿⠀⠀⣹⣿⣵⣷⢠⡾⢛⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡁⠀⠀⠀⠀⢸⣿⠁⢘⣛⠃⠀⢻⣿⢹⣿⡅⠞⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣼⣿⡀⠈⠋⠀⠀⣸⣿⣌⢭⣵⠿⠛⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠃⠀⠀⠀⠁⠻⢿⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡿⠁⠀⠀⠀⠀⠹⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢉⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠱⣃⡲⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣶⣦
⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⢀⣿⡿⢿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠅⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⡀⢠⠤⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀
⣶⣾⣿⣆⠀⠀⠉⠀⠀⠀⢠⡄⠸⠀⠁⢿⣴⠒⠀⠀⣠⣶⣦⡈⣹⣦⣴⣾⣦⣾⣿⣿⡾⠁
⣿⣿⣿⣿⣷⣄⣀⣀⣠⣴⣿⣿⣠⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠆
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣆⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠘⠃⠀⣼⣄⢰⣠
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠉⢻⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⢿⡿⠛⠁
⣠⣤⣤⣾⣷⣶⣄⠀⢀⣀⡘⠃⠀⠀⠀⠈⢱⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⡛⠻⡇⠀⠈⢿⣿⣌⣷⡠⢮⡀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇
⢿⢿⡿⣿⡿⢿⡿⠀⠰⣿⡿⠿⠿⠶⠤⠤⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠻⠿⠿⠿⠿⠿⠿⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⢀⣀⣤⣤⣤⣤⡀⠀⠀⣠⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⠙⠃⠀⠀⠈⠻⠿⣿⣿⠆⠀⢠⣭⢩⣿⣿⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣿⡟⠁⠀⠐⢿⣿⠃⠄⠂⠀⠀⠀⠀⠀⠀⠈⠟⣤⣩⣍⣬⣛⡭⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⣿⣿⣿⣧⣄⡀⠰⣿⢃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣽⣯⣿⣽⣿⣿⣸⡿⠿⠛⠛⠻⢿⣿⢿⢻⡇⠖⢶⡘⣧
⠟⢙⣿⣿⣿⣿⣿⡇⠈⠀⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡹⢉⣿⣿⣿⢹⡆⢠⣾⣿⣆⠠⠟⠘⠐⠃⡀⡀⠁⠋
⣿⣿⠿⢿⡟⠿⠳⢷⠆⠀⠀⡹⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⢸⣷⠘⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⠲⢶⠰⠒⠲⠶⠶⠀⠰⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠻⣿⢻⣿⡆⠀⠀⠀⠀⢀⣠⡤⠤⡶⠶⠸⡇
⣿⣟⢺⠗⠂⡒⠛⢚⣀⠸⡀⠀⡡⠄⠀⠀⠐⠁⠀⠓⠀⠀⠁⠀⠀⠐⠀⢹⣿⣷⠀⠀⠀⠀⠀⣬⡶⠶⠶⣋⠜⡇
⣿⣗⣛⣛⣛⣙⣚⣈⡛⢀⡨⠉⠃⡀⠀⠀⠀⠖⢀⠁⢀⣶⣰⡄⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠠⣶⢸⡗⢿⢇⡁
⣿⣿⣭⣿⣭⣭⡏⠉⡠⠋⠀⠀⣌⣃⠀⠀⠀⠠⠀⠀⠀⠙⠋⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⠈
⣿⣿⣿⡿⠋⠥⢔⣒⣭⣵⡆⢿⣿⠏⠀⠰⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆
⣿⣵⣾⠁⣷⣿⣿⣿⣿⣿⣿⣦⡙⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠄⠀⡀⠐⡚⣉⡍⠥⠆
⣿⣿⣿⡅⠈⢿⣿⣿⣿⡿⠟⠿⠿⢆⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠛⠉⠈⢡⡄⣦⢰⡇
⣿⣟⠋⠀⣄⠀⣹⣿⣅⣴⠶⠲⢤⡀⠈⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⢀⠀⠀⠀⠊⠉⠉⠀⠆
⠛⠛⠁⠀⢠⣤⣌⡉⠛⠀⠀⠀⠀⠈⠁⠐⠒⠛⠂⠀⠀⠀⠀⠂⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⠤⠔⠒⠒⠒⠲⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⢻⣿⣿⠉⠓⠀⠀⠀⣀⣀⣤⢤⣤⣤⣤⣈⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠓⠦⠼⠿⠟⠂⠀⠀⠀⠀⠀⠦⢤⣸⣿⣿⣀⡈⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣠⣤⠤⣶⣴⠶⢶⣶⡒⠦⠤⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⡏⠀⡴⠃⠈⠛⠁⠀⠙⠲⠎⠉⠓⠚⠁⡟⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡴⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠁⠀⠀⠀⠀⠀⣰⣶⣤⣠⣴⣶⡄⠀⠙⣦⠏⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⠟⠓⠒⡶⠞⠛⢻⡟⠛⠋⠉⡿⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠀⠀⠘⣇⡀⠀⠈⠁⠀⣀⡼⠁⣀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣇⠀⠀⠀⠉⠙⠛⠒⠛⠁⠀⠀⢻⠙⢦⡀⠀⢠⠎⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣙⠞⠙⢦⣀⣠⠦⣄⠀⢀⡀⠀⢸⠀⠀⣳⡖⠃⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣍⣀⠀⠈⠓⢋⣹⡤⠼⠖⠋⠁⠀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠯⠭⠥⠤⠤⠤⠶⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⣶⣽⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠠⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣶⣆⠀⠀⣿⠿⠃⢩⠿⠃⣁⣶⠿⠋⠀⣠⠄⠀⠀⠀⣾⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠁⠈⠉⠀⠀⠀⠀⠁⠒⠄⠀⠀⠘⠩⠉⠀⢀⣔⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣤⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⢠⣼⣿⣿⣿⣻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢟⣟⡉⡋⠙⠳⠤⢈⢄⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⣶⣿⣿⣛⣿⣿⣻⣿⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⡜⠿⠿⠋⠀⠀⠠⠦⠃⠀⠁⠂⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠾⣿⣿⣿⣿⣿⣷⣶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢏⡞⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⡄⠀⠀⠀⠀⣀⣴⣿⣿⣿⢿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣾⣿⡈⠿⣿⣯⣿⣿⣿⡏⡧
⣶⠾⠁⠀⠀⠀⠰⣿⣿⣿⢿⠿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠂⠀⠚⠬⠍⠃⠂⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠙⠁
⠀⠲⡄⠀⠀⠀⠀⠀⠈⠀⠀⠄⣦⣮⢷⡀
⠀⠀⠀⠐⠠⠀⠀⣠⣴⣠⣠⣿⣿⡿⠋⠉⠉⠉⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣶⣷⣾⣧⣀⣠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡉⢹⣿⣤⢠⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠟⠋⠉⠉⠀⠀⠀⠀⠂⠠
⣿⣿⠛⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢼⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣄⠀⠀⢠⠋⠀⠈⠧⠤⠒⠂⣤⡤⠀⢀⡜⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠉⠒⠉⠀⠀⢀⠀⠀⠀⣀⣉⣒⣦⠉⠀⠀⠀⡌
⠀⠀⢠⣄⡀⠀⠀⠀⠀⢈⣦⣀⠀⠀⠀⠀⠀⠀⠀⠑⠿⠒⠚⠒⠀⠀⠀⠀⣠⣞⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠘⢆⠉⠉⠒⠒⠛⠉⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⠈⠉⠡⠒⠈⡉⢁⠎
⠀⠀⠀⠀⠙⠢⠤⢀⡠⠤⠊⡇⠀⠀⢈⡤⢄⠀⠀⠀⠀⠀⢀⣤⠌⡂⠀⠀⡟⠢⣀⣀⣀⡠⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣿⣿⣾⡇⠀⠀⠀⠀⢺⣿⣾⡿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠙⠉⠀⠀⠐⠂⠀⠀⠉⠉⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣶⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⣀⠉⠂⠤⠤⠤⠔⠂⠁⣀⠴⠊⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⠉⠒⠒⠒⠚⠉⠉⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⡄⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢨⠋⠒⡾⡀⠀⠀⢠⠋⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣼⣤⣼⠀⣇⣀⣀⣼⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⣩⠵⠤⠚⠀⠯⣉⣩⠗⠴⠚

⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⣠⡿⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢿⣄
⠀⠀⠀⢠⣿⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡄
⠀⠀⠀⢸⡇⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢸⡇
⠀⠀⠀⠸⣷⠀⠀⠘⠻⠷⠶⠶⢶⣦⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣴⡶⠶⠶⠾⠟⠃⠀⠀⣾⠇
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠉⠉⠉⠛⠛⠛⠻⠿⠶⠶⢶⣶⣶⡶⠶⠶⠿⠟⠛⠛⠛⠉⠉⠉⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠛⠿⠶⣦⣤⣀⣀⠀⠀⠀⢸⡏⠀⠀⠀⢷⡶⠶⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⢶⡿⠀⠀⠀⢹⡇⠀⠀⠀⢀⣀⣤⣴⠶⠿⠛
⠀⠀⠀⠀⠀⠀⠲⣶⣤⣌⣉⠛⠻⠷⢶⣼⣧⣀⡀⠀⠈⣿⡀⠀⠀⠈⢻⣆⠀⠀⠀⠀⣰⡟⠁⠀⠀⢀⣿⠁⠀⢀⣀⣼⣧⡶⠾⠟⠛⣉⣡⣤⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠙⢿⡝⠿⣿⣷⣶⣤⣄⣉⡉⠛⣿⠂⠸⣧⠀⠀⠀⠀⢻⡆⠀⠀⢰⡟⠀⠀⠀⠀⣼⠇⠀⣿⠛⢉⣉⣠⣤⣶⣾⣿⠿⢫⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠙⠻⢶⣭⡉⠛⢻⣿⠀⠀⢻⣇⠀⠀⠀⠘⣿⠀⠀⣿⠃⠀⠀⠀⣸⡟⠀⠀⣿⡟⠛⢉⣭⡶⠟⠋⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠉⠛⠷⣾⣏⠀⠀⠀⢿⡆⠀⠀⠀⣿⡆⢠⣿⠀⠀⠀⢠⡿⠀⠀⠀⣹⣷⠾⠛⠉⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠻⠿⣿⣿⣿⣶⣄⡘⠻⣶⣄⡀⣻⡇⢸⣿⢀⣠⣴⠟⢃⣠⣶⣿⣿⣿⠟⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠙⢿⡻⣿⣷⣤⣉⠛⠿⠇⠸⠿⠛⣉⣤⣾⣿⢟⡿⠋⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠙⠿⣿⣿⡟⢷⣦⠀⠀⣴⡾⢻⣿⣿⠿⠋⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠙⢷⡄⠀⠀⠀⠀⠹⣿⣧⠀⠄⠀⠀⠠⠀⣼⣿⠏⠀⠀⠀⠀⢠⡾⠋⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠈⢿⣆⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⣰⡿⠁⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢿⡆⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢰⡿⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⡀⠘⣿⠀⠀⣀⣿⠇⠀⠀⠀⠀⠀⠀⠸⣿⣀⠀⠀⣿⠃⢀⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⣻⡇⣰⡟⠛⠻⠷⠶⣦⣴⠶⠾⠟⠛⢻⣆⢸⣟⣰⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡟⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠻⣷⡄⠀⠀⠀⠀⢠⣾⠟⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠈⣁⣀⣀⣀⣀⣈⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣴⠟⠛⠉⠉⠉⠉⠛⠻⣦⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⣤⣤⣴⣤⣤⣤⣀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⣸⡿⡶⠀⠀⠀⠀⠹⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⢭⡓⢦⡀
⠀⠀⠀⠀⠀⢰⡿⠋⠩⠉⠻⣷⠀⠀⠀⣾⠙⡛⢷⡟⠋⠙⣿⠀⠁⠀⠀⢀⡤⠤⠤⠐⢶⢤⡀⠀⢀⡄⠀⠀⣠⠖⡿⠀⠀⠀⠀⠘⡀⣷
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⡿⠀⠀⠀⠻⣄⠁⠀⠀⠀⢰⡏⠀⠰⠤⠀⡎⢱⡀⠀⠀⢸⠀⠙⢄⠉⠀⠀⠀⢷⡀⠰⠄⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⢨⡷⢦⣄⣰⣿⠀⠀⠀⠀⣸⠠⠤⢧⠖⠒⡟⠚⠋⢩⠃⠀⠀⠀⠀⠑⡶⠖⠛⠦⠤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⡏⠀⠀⠈⠉⠀⠀⠀⠀⣀⡈⠢⡀⠘⡆⠀⡇⠀⡠⠃⠀⠰⠂⠀⠀⠀⢹⡈⢳⡎⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠙⢦⣱⢀⢃⡔⠁⢤⠀⠀⠀⠀⠀⠀⠈⣗⠁⠈⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠒⠢⣄⠀⠀⠀⠐⠋⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⢿⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⡀⠀⠀⠀⢠⠃⢀⠂⠀⠀⠰⠀⡏⢳⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⠀⢠⡷⡄⠀⢳⠏⠀⢠⣶
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠈⢉⡇⠀⠀⣸⠀⠸⠀⠀⠀⠂⠀⠀⢀⡇⠀⠀⠀⠀⠀⣀⡿⣿⡟⠁⠀⠀⢷⡀⠀⢸⠁⠘⢆⠀⠀⣰⠃⡏
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣼⠀⠀⠀⠈⢯⡉⠒⠒⠒⠶⠶⢊⠝⠀⠀⠀⣀⣴⠞⠙⠃⠉⠀⠀⠀⠀⠘⡟⢦⣸⠀⠀⠈⢧⡞⢹⢰⠇
⠀⠀⢾⣉⣹⠏⢠⡖⠲⢤⡇⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠊⠁⠀⠀⠀⠈⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠏⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠰⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠈⣧⣴⠚⠛⠛⠟⠙⠒⠋
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠔⠺⠍⠩⠶⢄⠀⠀⠀⠀⡿⣿⣦⡀⠀⢀⣤⣾⣿⣿⠁⠀⠀⠀⡇
⠀⠀⠀⡴⠋⠁⠀⠋⡟⠢⡀⠀⠀⡞⣁⠀⡤⠀⢄⣰⠀⢣⠀⠀⠀⡇⠸⢿⣿⣶⣿⣿⠟⠛⠁⠀⠀⠀⠀⢠⣤⣄⠀⡆⠀⣴⠀⣷
⠀⠀⡜⠀⣾⠂⢰⡇⠃⠀⢱⠀⢸⡁⠀⡽⠓⠀⢰⠋⠉⠸⡇⠀⠀⠀⢀⣴⣿⣿⣿⣌⠀⠀⠀⠀⠀⠀⠀⣿⠁⢻⠀⣇⣼⡇⠀⢻
⠀⠀⢳⠀⣉⠀⠀⢠⡆⠀⡼⠀⠀⢧⢀⣠⠤⠔⠒⠒⠒⠒⡧⠀⢠⣶⡿⠟⠁⠈⠻⣿⣧⡀⠀⠀⠀⠀⠀⣿⠀⢸⠂⣿⡏⠃⠀⢸⡇
⠀⠀⠈⠳⢌⡉⠉⢙⣣⣜⠇⠀⠰⠊⢹⠦⢄⣀⣀⣀⡤⡞⠀⠀⠛⠋⠀⠀⠀⠀⠀⠟⢿⣿⡆⠀⠀⠀⠀⣿⠀⣿⠀⣿⠻⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⣦⡿⠀⣿⠀⠹⡆⠠⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢉⡏⣙⠹⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣶⣶⣦⣬⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠿⠿⠿⢿⡿⠛⠋⠉⠙⢿⣿⣿⣿⣧⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠹⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣿⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠀⠀⢠⡿⠂






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

⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠭⡿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣶⣿⣿⡾⣿⡟⣟⠿⢿⣿⣿⣿⡿⢿⡿⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿
⠀⠀⢸⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣭⡷⢶⡬⣿⢌⡝⢀⠰⠆⣰⡘⣧⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⣾⠋⠀⠀⠀⠀⠀⠈⢻⣿⠿⢿⣽⣿⣯⢟⣧⣵⣆⣿⣽⢃⡇⢰⠀⠀⣿⠽⢽⠃⠀⠴⠦⠶⢤⢴⠆⠀⣿⣿⣿
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⣛⣾⡟⣛⠿⣿⣿⣿⣿⣏⣿⣰⡤⠀⢆⡿⣰⣾⡀⠒⣻⣓⣺⣋⣻⠃⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠶⠦⠦⠤⠆⠀⢸⣹⣏⣿⠶⠷⠾⣷⣷⣷⣿⠴⠶⠶⠲⣿⢯⣷⣹⣿⡇⠈⠉⠈⠁⠉⠉⠁⢀⣿⣿⣿
⠀⠀⢰⠀⢀⣛⣚⣓⣓⠀⠀⢸⠿⠿⠛⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣯⣿⡀⠀⠀⠀⠀⠀⠀⣸⠻⣻⣿
⠀⠀⢸⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣤⡆⠀⠀⠀⠛⢋⣠⡀⠀⠹⣿⣿⣳⣾⣷⣤⣀⣀⡀⡀⣴⣏⣧⣿⣿
⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠠⠿⠛⠀⠀⠀⠘⣩⣅⡀⠀⠀⢘⣭⣍⠃⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣿⣶⣿⡟
⠀⠀⢨⠀⠀⣀⣀⡀⠄⠀⠀⠴⢒⡻⠆⠀⠀⢻⣌⣁⠀⠀⠈⣻⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣁⣛⡽⢿⣿⣿⢿
⠀⠀⠸⠀⠀⢠⣿⣟⠂⠀⠀⠘⣿⣷⠀⠀⠀⣨⡽⣷⠦⠀⠀⢸⣿⣷⡄⠀⠀⠸⣿⢹⣿⠿⣿⣿⣿⣿⣿⣿⣶⣭⣻
⠀⠀⢸⠀⠀⠀⣹⣿⣀⠀⠀⢀⣥⢾⡤⡀⠀⠀⠀⣻⡀⠀⠀⠀⠿⣥⠄⠀⠀⠀⠸⣿⣷⣶⠿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠀⠀⠀⢸⣟⣏⠀⠀⠀⠐⣿⣿⠁⠀⠀⠀⣿⣿⡄⠀⠀⢀⣼⣦⣤⠀⠀⠀⣼⡻⣾⡿⣛⣿⢟⣿⣿⣿⣿⣿
⠀⠀⠘⠀⠀⠀⢼⣯⣿⡆⠀⠀⠀⠁⢿⠀⡀⠀⠀⠟⠙⠀⠀⠀⠀⣿⣽⡅⠀⠀⠀⢹⣿⣌⢻⡿⢯⣞⡁⡾⣿⣿⣿
⠀⠀⢘⠀⠀⠀⠀⠀⢺⣅⣀⠀⠀⠐⢛⣛⡻⠀⠀⠀⠀⠀⠀⠀⠀⠬⠿⠶⠂⠀⠀⠈⢻⣿⣷⣮⣻⣿⣿⣿⣻⣩⣞
⠀⠀⠸⠀⠀⠀⠀⠀⣿⣻⡟⠀⠀⢰⣏⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠻⣛⢿⣿⣿⣿⣾⣿⣿⡾
⠀⠀⣠⣀⣀⣀⣀⡀⣻⣿⣷⢂⣀⣀⣘⣿⣿⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣞⣠⣿⣿⣿⣾⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠽⠉⠉⠉⠉⠉⠉⠉⠙⣿⠿⠛⠛⠛⠛⠛⠇⠉⠉⠉⠉⠉⠉⠉⣉⡉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠶⡒⢐⢶⡶
⠀⠀⢸⠀⠀⣛⣛⣛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠸⢯⣿⠒⠂⠚⠃⠀⠀⣬⣽⡭⣭⠥
⠀⠀⠸⠀⠈⠉⠉⠉⠉⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢀⠀⢋⠀⠀⠀⠀⠀⠀⠘⢷⣦⡔⠚
⠀⠀⣶⡀⠀⠀⠀⠀⠀⢀⢐⠃⠐⣉⢤⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠠⢈⠃⠱⠀⠀⠀⠀⡶⠶⠾⠷⡶⠄
⠀⠀⣿⣿⣶⣤⡤⠐⠂⠉⠘⠶⠀⠙⠂⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠉⢩⣭⡅⠙⠀⢀
⠀⠀⣿⣿⣿⣿⣏⠀⠀⠀⠀⢁⠀⠀⠀⢀⠀⠰⠀⠀⠀⠠⠤⣤⠤⡄⠀⠈⠆⠉⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠
⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠒⠂⠀⠀⡀⠈⠀⣲⣲⣛⣻⣂⣓⣻⠀⣇⠰⠀⠀⢄
⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⡄⠀⠀⠠⠠⣾⣰⣧⠈⠀⠡⠬⠿⠿⠦⠭⠄⢀⣣⠐⠀⠴⠀⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣾⣿⣿⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠀⢀⠠
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⣿⣿⡃⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⢀⣤⣤⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢤⣴⣤⣴⣄⠀⠀⠀⣤⣤⣤⣤
⠀⢀⣴⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⡀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⣿⡿⠟⠛⠿⣿⣿⣿⣷⡀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠻⣿⣿⣿⣆⠀⣼⣿⣿⣿⠋
⠀⣼⣿⣿⣿⠋⠀⠀⠀⠙⣿⣿⣿⣷⠀⠀⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠃
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡁
⠀⣿⣿⣿⣿⡀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣷⡀⠀⠀⠀⣰⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣴⣿⣿⣿⢿⣿⣿⣿⣄
⠀⠹⣿⣿⣿⣷⣦⣤⣴⣾⣿⣿⣿⠏⠀⠀⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⣿⣿⡇⠀⠀⢀⣾⣿⣿⣿⠃⠀⠻⣿⣿⣿⣦
⠀⠀⠈⠛⠿⢿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠿⠿⠿⠟⠁⠀⠀⠀⠙⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⢻⣿⣿⣿⣆⠀⢠⣾⣿⣿⡿⠁⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⣿⣿⣿⠟⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⠁⢻⣿⣿⣿⣆⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣴⣿⣿⣿⠟⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠈⢿⣿⣿⣿⡄⠀⠀⣰⣿⣿⣿⠟⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠻⣿⣿⣿⣆⣼⣿⣿⣿⠋⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⠟⠛⠛⢿⣿⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⢻⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⣴⣿⣿⣿⠏⢿⣿⣿⣿⣄⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢹⣿⣿⣿⣇⠀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⢀⣾⣿⣿⣿⠃⠀⠀⠻⣿⣿⣿⣦⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿⣿⡇
⣉⣉⣉⣉⣁⣀⣀⣀⣀⣉⣉⣉⣉⣁⣀⣀⣿⣿⣇⣀⣀⣀⣀⣀⣈⣙⣛⣛⣻⣟⣛⣛⣁⣀⣀⣀⣀⣀⣀⣸⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⢀⡎⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢱⡀
⠀⠀⠀⠀⠀⢸⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇
⠀⠀⠀⠀⠀⠸⡇⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠙⠓⠒⠀⠀⠀⠀⠈⠓⠒⠒⠚⠁⠀⠀⠀⠀⠒⠚⠋⠁⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⡠⠚⠛⠳⠶⢤⣀⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⣀⡤⠶⠞⠛⠓⢄
⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢰⣿⢶⡄⠀⠀⠀⠀⠀⠀⢠⡶⣿⡆⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸⡆⠀⠘⠇⠀⠀⠀⠀⠸⠃⠀⢠⠇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢻⡀⠀⣠⣤⣤⣤⣤⣄⠀⢀⡟⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠈⢦⠀⠀⠀⠀⡴⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠘⠀⠈⠃⠀⠀⠘⠁⠀⠗⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⢭⣭⣭⡭⠁⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡼⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣇⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⡼⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⢦⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⠞⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⡰⠃⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠙⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡇⡜⠁⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢦⢻⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡿⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⣿⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢳⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡏⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⢳⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⠞⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠲⠤⣍⣻⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣾⣟⣡⠤⠖⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⢉⡏⠉⠛⠫⣍⡀⠀⠀⠈⢉⡭⠟⠋⠉⢹⠉⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣭⣉⣉⠉⣹⠉⠉⠉⠓⢤⠙⢦⢀⡴⢋⡴⠚⠉⠉⢙⡏⢉⣉⣉⡭⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣌⣉⡟⢹⢿⡒⢤⡀⢧⠈⣿⢀⡞⢀⠴⣺⢿⠍⣏⣉⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠳⠽⠆⢳⢸⡆⠀⢸⢡⠏⠼⠥⠋⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢳⡀⠀⢸⠀⡇⠀⣼⢸⠀⠀⢠⠞⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⡄⢸⠀⡇⠀⡏⢸⡄⢰⠋⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢳⡟⠀⣷⠀⡇⠀⣇⡏⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢸⠃⠀⣿⠀⡇⠀⢹⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠤⠒⠛⠚⠓⠲⢬⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⠷⠀⠀⠀⠿⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣒⣒⣒⣒⣲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⣿⣿
⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠖⠚⢈⣿⣿⣿⣿⣷⣶⣶⡖⠛⠛⣉⣁⣴⣿⣿⠏
⠀⠀⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠈⢿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣄⡂⠀⠀⠀⠀⠀⢀⣶⣆⣰⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠋⣉⣭⣴⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⣉⣉⠉⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠔⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣴⣶⣶⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁⠈⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣠⡄⠀⠀⠀⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡞⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣾⠁⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠁⣼⡟⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⣿⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⡿
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠁
⣏⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡅
⣿⣷⣄⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⠾⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⡀
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠘⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⣿⡀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣇⣤⣶⠿⠛⠋⠉⠉⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠙⠛⠿⣶⣤⣸⣇⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠘⣿⠀⢀⣠⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄⡀⠀⣾⠃
⠀⠀⠀⠀⣿⣷⣿⠿⠋⣀⣤⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣦⣤⣀⠙⠻⣿⣶⣿
⠀⢀⣤⣾⠿⠋⠁⣠⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡴⠃⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣄⠈⠙⢿⣷⣦⡀
⢠⣿⡟⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣼⠀⢀⣼⡿⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠈⢻⣿
⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣠⣾⠏⠀⢸⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢻
⠹⣿⡄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢠⣿
⠀⠈⠻⣦⣄⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣠⣴⠟⠁
⠀⠀⠀⠀⢙⣿⣦⣄⡀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁⠀⣠⣴⣾⡋
⠀⠀⠀⠀⢸⣿⠃⠉⠛⠶⣤⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⣀⣤⠶⠛⠉⠘⣿⡇
⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠉⠻⣦⡄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⢀⡇⠙⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠊⢸⡀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⣦⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣴⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣠⠞⠉⠓⣦⣮⣤⣀⠀⣠⠀⠀⠀⢀⡄⠀⠀⠀⣄⠀⣀⣠⣵⣴⠞⠉⠻⣦⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣞⠁⠀⢀⡼⠋⠀⠀⠈⣹⠟⠛⠉⠛⢻⡟⠛⠙⠛⠻⣏⠁⠀⠀⠙⢧⡀⠀⠈⣳⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣾⠁⠀⠀⠀⣼⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣧⠀⠀⠀⠈⣷⡴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣶⠶⣿⣄⣀⣀⣀⣀⣼⣧⣀⣀⣀⣀⣠⣿⠶⣶⣶⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⣞⡀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣄⣀⠀⠀⠀⠀⢀⡤
⠀⠈⠿⣿⣆⣀⡤⠖⠋⠉⠉⢉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠀⢛⣫⣿⡿⠿⠛⠋
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠈⠙⠛⠛⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠟⠋⠁⢸⣿⡟⠋⠉⠀⠀⠙⣿⡿⠿⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⡛⠁
⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⢻⣧⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠠⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣨⣿⣿⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠈⢉⣻⣈⡗⠒⠠⣠⠤⣀⠀⣀⡠⠜⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣦⣤⣸⡯⣍⡉⢢⡀⢄⠻⡁⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣭⣿⣿⡿⠿⢷⢾⣧⣄⡁⠛⢎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⣧⡀⣭⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⣛⡯⠒⣚⠛⠀⠀⠀⠀⠀⠀⠳⣦⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⣶⣍⠛⠦⠄⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣷⣶⡶⠶⠶⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⡿⠋⠀⠀⠀⠹⣦⣤⣄⣀⠀⠀⠀⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣾⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣾⣿⣿⠿⠛⠛⣿⣶⣶⣤⣀⣀⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠰⢦⡀⠀⠀⠀⠐⢦⣧⡀⠀⠀⠀⠀⢸⣿⣿⣍⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣯⣴⣦⣄⠀⠀⢸⣷⠀⠀⠀⠀⠀⠙⣇⠀⢀⣶⡆⠈⣿⣿⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣼⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⣿⣷⣼⣿⡇⠀⢹⣿⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⠀⠀⠐⣿⣿⣿⣶⣄⣀⣠⣾⣿⣿⣿⣧⠀⢾⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣸⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣧⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⣾⣿⣿⣿⠟⠋⠉⠉⠉⠙⠻⣿⣿⣷⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⡿⠛⠿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠈⠛⠿⠿⠿⠿⠛⢿⣿⣿⣿⠇⠀⠀⠘⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⣹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⢀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣿⣷⣶⣶⣶⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣦⣶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣾⣿⣿⣿⣿
⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⣿⣿⣿⣿⣿⠇⠈
⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⠿⣿⠿⠏
⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠙⠓⠦⣀
⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⢀⡿⠛⠀⠁⢢⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠯⠀⠤⠤⠔⢊⠆⡠⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢏⠓⠤⣀⡀⠀⣀⣀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡠⢂⡥⠖⠒⠉⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠈⠙⠒⡎⠙⠉⠉⠉⠉⠉⠑⠒⠖⠒⠒⠒⠒⢊⡝
⠀⡇⡎⠀⢀⢔⢤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠉⢐⡒⡤⠇⠐⠦⣄⣀⠂⣐⣂⡀⠘⣒⡶⠶⣆⡉
⠀⠱⣘⣒⠕⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⣠⠞⠁⠀⠀⠈⠃⠀⢀⠊⣭⡙⡅⢈⣭⢱⠀⠐⢤⣀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠑⠒⠒⢒⠟⢀⡀⠈⠯⣯⠵⠁⠈⠓⠋
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢠⠏⢀⣿⡅⠀⠀⢀⡴⠤⠤⠀⠤⠵⢤⣀
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡄⠀⠀⠀⠀⣾⢠⡸⠤⠽⢤⣀⡏⠀⠐⢦⠀⠀⠀⢀⡀⢱
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣠⠃⠀⠀⠀⠀⢳⠀⠀⠀⠀⠘⠲⠛⠒⠒⠂⠈⠳⣤⠊⠌⠳⡀⢀⡟⢤⠎
⠀⠀⠀⠀⠀⠀⢀⡼⠀⣠⠴⠊⠳⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⠢⣑⠢⠬⢑⡡⠂⠘
⠀⠀⠀⠀⠀⠀⠯⡀⡎⠀⠀⠀⠀⢹⠲⣄⡀⠀⠀⢻⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⠄⡰⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠈⠁⠢
⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀⠀⠀⠀⢜⡐⢁⠈⠑⠲⠬⣇⠀⠀⠀⡴⠁⠀⠀⠀⣠⠃⠀⡇⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠉⣦⠀⠀⠀⠈⣇⣑⣄⠀⠀⠘⡆⠀⡼⣀⠀⠀⣠⠞⠁⠑⠀⡳⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠘⢯⡭⢄⡄⠀⠀⠀⠫⡴⠪⠇⠀⠀⢳⢾⠁⠈⠛⠋⠁⠀⠀⠀⡁⠑⢤⡀⠀⠀⠀⠀⠈⠒⠐⠠⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠘⢤⡀⠀⠀⠀⠀⠀⠀⡷⠖⣒⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡓⠒⣒⣻⠀⠀⠀⠀⠀⠀⠙⠎⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣧⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⢻⣧⡀⠀⣀⣀⣀⣀⠀⠀⣰⡿⠀⠘⣿⣄⠀⠀⣀⣀⣀⡀⠀⢀⣼⡟⠀⠸⣷⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠻⣿⣿⡿⣯⡿⣿⣿⡿⠋⠀⠀⠀⠙⠿⣷⣶⣾⣿⣷⣶⣿⠟⠁
⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠛⢝⢿⣿⣌⣾⣿⢟⠝⠃⠀⠀⠀⠀⠘⢻⣿⣯⢉⣿⣿⡟⠃
⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠏⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠿⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣠⠖⠉⣩⠭⢵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡯⠭⣍⠉⠲⣄
⠀⠀⠀⠀⠀⡼⠁⠀⡞⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢳⠀⠈⢧
⠀⠀⠀⠀⢸⠁⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠘⡇
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠙⠲⠤⢤⣤⣄⣀⣀⣀⣀⠤⠶⠦⢭⣁⣀⡤⠬⠭⠷⣦⣀⣀⣀⣠⣤⡤⠤⠖⠋⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠉⠂⠄⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠒⠶⠤⢤⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠤⠶⠒⠛⠓⠦⣄
⠀⠀⠀⠀⢀⠞⢁⣤⣀⣀⠀⠀⠀⠀⢘⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⣀⣀⣤⡈⠳⡄
⠀⠀⢀⡠⠋⠔⠁⣀⠾⢿⣿⣷⠦⠄⡎⠀⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⢢⠠⠴⣾⣿⡿⠧⡀⠈⠢⠙⢆⡀
⠀⠐⠛⠤⠤⠖⠚⠢⢤⣀⣀⣀⡤⢾⢥⡴⢶⣄⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣠⡶⣦⢼⡷⢤⣀⣀⣀⡤⠜⠓⠲⠤⠤⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠻⣿⡿⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⢿⣿⠟⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡼⠁⠀⡐⠀⠀⠀⠀⢄⠀⠈⢇⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⠞⡠⠒⠉⠀⠀⠀⠀⠀⠀⠙⠢⠌⢣⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡞⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣵⣶⣤⡀⠀⠀⠀⠀⣀⣴⣶⡮⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠉⠉⠆⠀⠀⠰⠉⠉⠉⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⢶⣦⡤⣒⣒⣒⡒⢤⣶⡶⠋⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⠀⠉⠁⠀⠀⠉⠉⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⡄⠀⠀⠠⠄⠀⠀⢰⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠳⣄⡠⠻⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀⣠⣤⣤⣤⣤⣀
⠀⢀⣠⣴⠶⢦⣤⣀⡀⠀⠀⠀⣀⣠⣶⣿⣿⠿⠟⠛⠻⢿⠿⠟⠛⠻⢿⣿⣿⣦⣄⡀⠀⠀⠀⢀⣀⣤⡴⢶⣤⣄
⠀⠈⠉⠉⠛⢶⣌⡙⠻⠿⢿⣿⡿⠿⠛⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠙⠻⠿⣿⣿⡿⠿⠛⢉⣴⠞⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠻⣶⣤⣤⣤⣤⣤⣤⣤⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⣤⣤⣤⣤⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣿⠇⠈⠂⠀⠀⠀⠀⠀⠀⠘⠁⢻⣟⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⡀⡀⠀⢰⣦⣴⡄⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢻⣧⠈⠲⣿⢸⡅⣷⠞⠁⣿⣏⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⡏⢸⡇⣿⠀⢸⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⣼⡇⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣾⣿⣿⣿⣿⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢻⣿⣿⣿⣿⢏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠿⠿⠿⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣶⣶⣿⣿⣿⣷⣾⣿⣿⣿⣶⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⣿⣿⣿⡻⠿⠿⢟⣿⣿⣿⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⣠⠀⢜⡿⠬⠛⠿⣿⣿⣿⣿⠿⠛⠥⢿⡣⠀⣄⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠋⠁⠀⠈⢿⣷⢸⣿⣿⣿⣿⣿⣿⡇⣾⡿⠁⠀⠈⠙⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣾⣿⣿⣿⣿⣿⣿⣷⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣡⠙⠿⠿⠋⣌⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠛⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⣠⣿⣇⣀⣀⠀⢀⣀⣀⣀⣠⣦⠀⣿⢳⣟⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⣿⣾
⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣶⣶⣶⣶⣶⣷⡶⣶⣶⣶⣺⣷⣾⣟⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⢻⡆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⡞⢻⣿⣼⣧
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⣷⡾⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣇⢻⣿⣿⡇
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢻⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠘⣿⣯⣽⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢂⡀⠈⠉⠈⠁⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠂⠠⠤⠤⢄⠖⠊⠁⠀⠀⠈⢣⠡⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⡆⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢦⡑⠶⠶⠶⢶⠒⠶⡶⠼⠲⢤⡤⠒⢁⠇
⠀⠀⠀⠀⠀⠀⠀⡠⢪⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠻⣶⠶⢰⡄⠀⢀⠀⠀⠀⠀⠸⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠔⣠⣼⡿⠑⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⢀⠀⠀⠀⠀⠊⠀⠀⠐⠋⠙⠛⡀⠀⠘⣦⣦⠸⠚⢤⡇
⠀⠀⠀⠀⠀⠠⠀⣿⣿⡉⠀⠀⠀⡀⠀⠀⠀⠀⠈⠺⣧⣶⣷⡴⠞⠂⠀⠠⢀⢀⣀⢀⣀⣷⠀⢠⡀⠘⠃⠀⠘⣆
⠀⠀⠀⠀⠀⠀⡀⣿⣿⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠁⠁⠈⢙⣿⣿⣤⣙⠀⠚⠶⠀⠨⠇
⠀⠀⠀⠀⠀⠀⠐⠙⣿⣇⢂⠀⠀⠀⠀⠀⣾⢿⣟⢄⠀⠀⡀⠀⠈⠃⠀⠀⠀⠀⠀⠄⠀⠉⣽⣿⣿⣷⣠⢖⣒⣲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡅⢿⣀⠢⡠⡶⠀⢾⣿⣿⣙⠻⢦⢀⣿⡄⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⣽⣿⣿⣿⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣴⠇⠈⢿⡂⠈⠁⣡⣿⣿⣿⣿⣿⣿⣶⣿⣏⠃⠀⠀⣏⢀⣀⣠⣤⣾⣿⣿⣿⣿⡏
⠀⠀⠔⢈⡟⠛⠛⠁⠀⠀⠘⣷⡔⢰⡟⢻⣿⣿⣿⠛⢿⣿⠛⢻⡕⠀⢠⣿⣿⣿⣿⣿⠿⣿⣏⠀⡟
⠀⠰⢀⣿⠃⠀⠀⠀⠀⠀⢀⠋⠃⣾⠀⠈⣿⣿⡇⠀⠀⠙⠀⢸⣷⠁⢨⡄⠉⠛⠻⠋⠀⢿⣿⠀⡇
⠀⠴⠛⠁⠀⠀⠀⠀⠀⠀⠈⢴⡄⣿⠀⠐⣿⣿⡇⠀⠀⠀⠀⠈⣧⢀⢠⠃⠀⠀⠀⠀⠀⢸⣿⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⠀⠀⠸⣿⢿⠀⠀⠀⠀⠀⢹⡇⢸⠀⠀⠀⠀⠀⠀⠈⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠘⡇⠀⠀⡿⣾⡄⠀⠀⠀⠀⢸⣗⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣶⠀⠀⢹⣿⣷⠀⠀⠀⠀⢸⠟⢸⠀⠀⠀⠀⠀⠀⠀⣿⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠈⠉⠁⠀⠀⠀⠀⠿⡶⠼⠀⠀⠀⠀⠀⠀⠈⠿⠶⠇



⣟⠉⣻⣛⣛⣛⣛⣻⣛⣛⣛⣛⡛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣿⣛⣿⣟⣛⣻

⠀⠀⠀⢀⣀⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠠⠤⠤⣀⣀
⠀⠀⣰⠋⠀⠀⣀⡤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣤⣄⡀⠉⠢⡀
⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢹
⠀⠀⣧⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⣸
⠀⠀⠘⢧⡀⠀⠀⠙⠓⢤⡤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠤⣄⣠⡴⠞⠋⠀⠀⣴⠏
⠀⠀⠀⠀⠙⠳⢦⣄⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⢀⣠⣴⠟⠁
⠀⠀⣀⠤⠐⠒⠂⠬⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⡄⠀⠀⠀⠀⠀⡿⠛⠛⠛⠒⠲⠤⣄
⠀⢾⡀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠐⠋⠁⠀⠀⠀⢀⣀⡤⢤⣀⠉⠀⠀⠀⠀⣸⠧⣀⣀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠉⠙⠛⠛⠛⠋⠉⢹⠀⠀⡴⠋⠉⢉⣳⣤⠏⠁⣀⡤⡈⠳⡄⠀⠀⣰⠛⢦⡀⠉⠉⠙⠛⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡀⠀⠰⣿⣴⣿⠀⠰⣿⣷⡾⠀⡇⠀⠷⡌⢢⡀⠙⣦
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢳⠀⠓⠦⣄⡤⠟⠙⢦⣀⠈⠉⣀⡼⠃⣼⠀⢸⡄⠳⡄⠈⢳⡀
⢠⡤⡆⢱⡀⠀⠀⠀⠀⢀⡼⡆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⢰⠇⠀⣿⠿⡄⠹⡄⠀⢹⡤⢄
⠀⠳⡰⠋⢳⠀⠀⢀⡴⠋⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣸⣟⠀⢷⠀⢹⠀⠈⡇⠘⡆
⠀⠀⠱⡀⠀⢧⣠⠊⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣴⣿⣿⣁⠼⠀⣸⠁⢰⠃⠀⢧
⠀⠀⠀⠹⣄⡸⠁⠀⠀⠀⠀⠀⢳⠄⠒⠂⠀⠀⠀⠀⠐⠀⢴⡷⠤⠬⠤⠤⠤⠔⠚⠁⠀⢸⠀⢸⠈⣇
⠀⠀⠀⢀⡟⠀⠀⠀⢀⡤⠖⠒⠺⠦⠀⠀⠀⠀⠀⠀⠐⠒⠛⠋⠙⠓⠲⣤⠀⠀⠀⠀⢀⡇⠀⡏⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄
⠀⠀⢀⡞⠀⠀⠀⢠⡏⠀⢀⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠈⣇⠀⠀⢠⣾⣀⢰⠃⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⢀⡞⠀⠀⠀⠀⠘⣇⠀⠈⠻⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠀⢀⡿⠀⢀⢏⣀⠈⡏⠀⠀⠀⢹⠀⠀⡠⠄⢒⣂⡀⠽⣄⠀⣀⠔
⠀⣼⠁⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠶⠒⠒⠋⠉⠙⠲⢤⣀⣀⣀⣠⡾⠁⠀⠘⠋⠈⡾⠀⠀⠀⠀⢸⡆⡈⠀⡎⠁⠀⠀⠀⠈⠉⠁
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢡⠀⡇
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⢹
⡇⠀⠀⠀⠀⠀⠀⢀⠠⠔⠂⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡰⢀⡞
⣷⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠊⣠⠞
⠸⣇⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠊⠁
⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⠶⠚⣇
⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⢻⡖⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⠤⠴⠶⠖⠒⠋⠉⠉⢻⡀⠀⠘⣦
⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⣧
⠀⠀⣀⡴⠛⢧⡀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠤⠴⠚⢳⡀
⠀⣴⣋⣀⣀⠀⠉⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢀⡀⠀⣷
⠀⠀⠉⠀⣿⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠴⠋⠙⠦⡿
⣀⣀⣀⣀⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣤⡤⠤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⣫⢴⣿⣿⣿⣿⢫⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣆⠀⠀⠀⠀⣠⣴⣞⣾⣿⣿⣯⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⣀⣀⠀⠀⢀⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣾⣶⣶⣿⣿⣿⣿⣿⢹⣿⣿⢻⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣆⠀⠻⣿⣿⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣀⡙⠻⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣠⣴⣶⣲⣶⠞⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣤⣤⣤⡤⡀⠱⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠻⠦⠀⠤⠀⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢘⢿⣿⣿⣿⣿⣿⣿⠟⣹⢟⠇⣴⣤⣉⣉⣉⣉⣽⣿⣽⣿⣝⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⠿⠿⢛⣿⣟⡻⠋⣴⣿⡿⣿⣿⣿⣿⣟⣽⣏⢻⣿⣷⣿⣿⣛⣿⠏⢸⣿⢿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⡏⢠⠟⠁⠀⠸⣟⣿⢷⣌⢾⣿⣿⣮⣿⢿⠄⣿⣿⣿⣿⢟⡟⠀⢸⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡧⣿⠀⠀⠘⢄⢻⣷⣿⣿⡾⣿⣿⣿⣿⣿⣄⣿⡿⢿⡿⠋⡀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣭⣭⣥⣵⡶⣶⣿⣎⠻⣽⣿⣿⣷⡙⢿⣿⣿⣿⠼⠿⠟⠋⠀⣼⡄⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⢙⣿⣿⠏⠀⠈⠉⠙⠛⠋⠀⢻⣿⣿⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡆⠀⠀⢀⣼⣻⠟⠀⠀⠀⠐⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡟⠀⠀⠀⣼⢻⡆⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠉⠋⠀⠀⠀⢀⣄⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⡾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣮⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠈⠲⠶⠶⠄
⠀⠀⠔⠛⠒⠢⢄⡒⠂⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃
⠀⠈⢧⡷⢤⣀⠀⠉⠳⣫⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠉⡚⣳⣤⡀⠙⣇⠀⠀⣤⠄⠀⠀⠀⠀⠐⠂⠐⠁
⠀⠀⣠⢺⡞⠉⠀⠈⠻⢦⣈⢷⡉⠀⢠
⠀⠸⡐⠾⢽⣆⡆⠴⠶⠖⠻⣷⣤⡀⢋⣭
⠀⠀⠀⠲⢤⣠⠀⠀⠀⣈⡾⠿⢿⣯⡯⣉⠋⣦
⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠀⠉⠉⠛⠛⠖⠲⠆⢰⡦⡀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠂⠐⠂⠉⠙⠢
⣶⣶⡶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣄⠠⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⣄⣀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀
⠀⠀⠘⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣩⠔⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠈⣰⠒⠉⠉⠉⠉⠉⠉⠻⣀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠑⠄⣀⡠⠞⠁⡀⠀⠀⠀⠀⢀⣄⡀⠈⠉⠒⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⢀⡤⠀⠒⠒⠒⠁⠀⠰⠋⠁⠀⠀⠀⠀⡀⢀⡙⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⣏⠀⠀⢀⣠⢶⠀⠀⡰⢉⡬⠑⡄⠀⡎⣤⣂⠘⡄⠀⠀⠰⡄⠈⠁⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠈⠙⠛⠉⠀⢸⠀⠀⠣⡈⠛⣡⠇⠀⠣⣈⣉⠔⠁⠀⠀⠀⠹⣄⡀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⢘⣄⣠⠤⠤⠥⠤⠤⣤⠴⠒⠚⠉⠉⠉⠙⠓⠒⠦⣉⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡞⠀⠀⡔⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⡜⢀⡴⢶⢦⡈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢸⠀⠀⠀⠙⠆⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⡼⠣⠔⠊⠀⠀⠀⢸⠀⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⡀⠀⠀⠀⠀⠀⠈⠙⠃⠀⡅⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡸⢀⡀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸
⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠘⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⢸
⠀⠀⠀⠈⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡇⢸
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⡶⠶⠶⠶⠶⠖⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢐⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⠀⢀⡴⠊⠀⠀⢠⠖⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠈⢆⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣴⠞⠉⠀⠙⠢⣄⠀⠀⢏⠀⠀⠀⠀⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠛⡶⠾⢆⠀⢹⠤⠤⠤⠤⠴⠶⠶⠒⠚⠋⠉⠝⡰⠁⠀⠀⠀⠀⢀⡴⠃⢀⡼⠃⠀⠀⠀⠸⡀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⡆⠈⡇⠀⠀⠀⢀⠠⠀⠀⠀⠀⠘⠘⠁⠀⠀⠀⣀⠤⣏⠀⣠⠋⠀⠀⠀⠀⠀⢀⡇⡰⣁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠂⠑⠇⠀⢀⡵⠤⣿⠀⠀⠘⢳⠶⠊⣁⣀⡤⢴⡴⠥⠖⢊⣩⠄⠀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠼⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠥⠄⠴⠠⠤⣤⠶⠊⣁⠀⠀⠈⡆⠀⠀⠉⠉⠉⠀⠀⠀⠀⢙⡶⠖⠉⢀⣀⣠⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⡶⠊⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣶⣶⣶⣾⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠈⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⢸⡿⠃⠀⠀⣀⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠿⣿⣿⣀⣀⡀⠴⠿⠄⠄⣢⠕⠋⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀
⠀⢀⣀⠤⠤⠤⠤⣀⣀⠤⠤⠒⣿⡅⠀⣠⣤⠖⢀⣤⣤⣤⣴⣶⣞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⣀⡀⢀⡀
⠀⠈⠉⠉⠙⠓⠶⣤⣤⣴⡶⣾⣿⣿⣿⠋⢀⣴⣿⣿⣿⢿⣿⣋⣝⣁⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⢲⡄⠙⠃⣀⣀⡀⠀⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⡿⠿⣷⣅⣴⣿⡏⣰⠏⠁⢸⣿⣿⣿⠋⣠⣆⠀⢀⣠⡄⠀⠀⠻⣧⡀⠀⠀⠈⠁⠀⠀⠹⡀⠀⠈⠑⢆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⡟⠉⢀⠀⣼⣿⣯⣴⣾⣿⣿⠛⠋⠁⠀⠀⣀⠀⢿⣷⡀⠀⣠⣶⡀⠀⠀⢱⠀⠀⢀⢆⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⢀⠏⣿⣿⡿⣿⣿⣿⣿⣿⠀⢶⠞⠛⠋⠁⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠃⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣤⣷⠈⣾⣿⣇⠀⠈⠙⠛⢿⣧⠀⣸⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠘⢿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⠖⢻⣿⣿⣄⣤⣶⣄⠀⣿⡆⢿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠆⢻⣇⠈⣴⣷⣦⣿⣿⣿⣿⡿⠟⠋⠛⠻⣿⣿⣷⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠈⣿⣾⠟⠁⢻⣿⣿⣿⣿⣁⣀⠀⠀⠀⠈⢿⣿⡇⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⡇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⢹⣿⣇⣀⠀⠀⠀⢿⣷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠈⣿⣇⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⠀⠈⢿⣿⡿⠂