Chew Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠒⠚⠋⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠖⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠋⠉⠁⠀⠀⠀⣠⣄⠀⠘⢵⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠚⠉⠁⠀⠀⠀⢀⣀⠠⡒⠄⢋⢉⣁⠤⣉⡖⠀⠺⠂⠀⠀⠀⢀⣀⠤⠤⠒⠚⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠉⠀⠀⠀⠀⢀⠔⡲⣂⠤⣼⣄⠓⠁⣰⠸⠼⠉⠀⠀⠀⣀⣠⠤⠖⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠓⣣⣌⡣⠈⠉⠋⠁⢀⣀⠤⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⣀⣨⠿⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣀⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠚⠛⠉⠉
⠀⠀⠀⠀⣰⡯⢍⡀⡀⠀⠀⠀⠈⠳⠤⠐⢲⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉
⠀⣀⡴⠚⠁⠀⠀⠀⠙⠤⠤⢶⠀⠀⠀⠀⠸⢤⡀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉
⠘⢧⡀⠢⢀⠀⠀⠀⠀⠀⠀⠘⠖⢲⠄⠁⠀⠠⠧⢾⠁⠀⠄⠂⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁
⣠⠖⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱⠀⢀⠀⣸⠀⠀⠀⠀⠀⢀⣠⠔⠊⠉
⠙⠷⣄⠀⠈⠳⢄⡀⠠⠀⠀⠔⢀⡤⠚⢧⠈⠁⠀⡟⠀⠀⢀⣠⠖⠋
⠀⠀⣸⠓⢄⡀⠀⠙⠢⣄⡢⠔⠁⣀⢶⡁⠀⠀⡔⢃⣠⠖⠋
⠀⣰⠃⠀⡀⠉⠢⢄⡀⠀⠀⡠⠊⢁⣮⣀⣠⣴⠿⠋
⠀⠈⠻⢦⣈⠢⣄⠀⠙⠒⠁⣠⡾⠳⡴⡿⣟⠁
⠀⠀⠀⠀⠈⠓⢦⣁⠀⢠⠞⢹⣥⡐⠀⠃⠈⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠈⠛⠋⢷⣠⠼⠁
⠀⠀⠀⠀⠀⠀⢰⠶⢒⡶⠀⠀⢰⡶⡄
⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⢀⡷⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠙⠻⡆
⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣶⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣏⠀⠘⣟⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡁⣾⣿⡀⡌⢻⣿
⠀⠀⠀⢠⡴⠶⣦⠶⠚⠋⢁⣀⣽⣆⣠⣽⠋⠀⠀⠀⣿⣿⣿⣿⢛⣉⡙⢋⣩⣴⣶⡿⠗⠘⠿⠓⣰⣿⣿
⠀⠀⠸⣏⣧⠀⠘⣧⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡐⠎⢿⣿⡌⢛⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠛⠧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣌⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣀⣤⣤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⠉⠛⠛⡒⠒⠂⠀⠀⠀⡀⠀⠀⠀⠤⠿⠓⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡤⠀⠀⠀⠁⠀⠀⡇⠀⠀⣿⠀⠀⠁⠀⠀⠀⣀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⡆⠀⠳⠤⠤⠃⣠⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⡇⠀⠑⠉⠛⠀⣇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢒⣉⠀⠈⠉⠒⠒⠒⠂⠀⠀⠐⠒⠒⠂⠈⠉⠀⠀⠈⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢰⠛⠏⢙⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⣶⠋⠁⠉⠙⠻⣿⣦⡀⠀⠀⠀⢠⣤⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⣀⠴⢒⡾⠛⣿⡄⠀⠀⠀⠹⠴⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢠⠞⢙⡦⠊⠁⠀⠘⡲⣾⠋⣿⣆⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢈⡗⠒⠒⠒⠒⠒⠒⣟⠀⠉⠀⠀⣠⠔⠉⠀⢙⣿⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⢷⠒⠒⠒⠒⠒⠒⠒⠛⠛⡇
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⣶⠻⣿⡄⠀⠤⠶⡄⠀⠀⠸⣄⣀⣀⣀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢸⡇⠀⠰⠋⠉⡆⢠⠞⠉⠀⠂⠀⠉⠓⢦⡀⠀⠸⡆
⠀⠀⠀⠀⢸⠃⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣴⠊⠁⠀⠀⢀⡸⣷⣶⣶⠿⠿⠶⣏⣀⣀⣤⠤⠤⠤⠤⠤⢵⠀⠀⣇
⠀⠀⠀⢀⡏⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⠙⠒⢖⡊⣉⠝⡇⠤⠤⠤⠤⠲⣧⣤⣶⡶⣶⣶⣶⣶⣶⣾⡇⠀⢹
⠀⠀⠀⣸⠁⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⣶⡄⢀⣄⢸⠧⠴⠤⠤⠀⠐⠃⠙⠓⠒⣒⣀⣀⣁⡒⠘⡧⠀⢸⡄
⠀⠀⠀⡏⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠢⡰⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠁⠀⢸⡇
⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠥⠤⠤⠀⠀⠒⠒⠀⠁⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⢣⣶⣾⣿⣿⣿⣿⣦⠀⣀⣤⣶⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣘⠛⠛⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠯⢉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠁⠈⠛⠻⣿⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⣿⣿⣿⣿⣿⡿⣫⣵⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⢿⣿⣿⣿⠿⠾⠛⠉⠁⠀⣠⠶⠶⠆⠀⠀⠀⠀⡠⠛⠉⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠨⢯⣷⣿⣧⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⢰⡇⠠⡆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢛⠺⠏⠹⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⢧⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠠⠏⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⢸⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣶⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠁⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⡼⠓⠒⠋⠉⠀⠀⢰⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢷⡀⠈⠂⠀⢀⠀⠰⠚⠀⠀⠚⠉⠀⠀⠀⣼⡋⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣿⣷⡀⠀⠀⠈⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠘⣦⣦⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠘⢿⣷⣄⠀⠀⠈⢻⣿⣿⣷⣦⣤⣄⣀⣀⣀⣀⡤⠞⢇⠀⠀⠀⢻⡟⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢋⡥⠞⠛⢦⠀⠀⠀⠙⢿⣷⣤⣀⡀⠙⠻⣿⣿⣿⡿⠟⠙⣏⠉⠀⠀⠈⢆⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⢀⣴⠟⣡⡶⡋⠁⠀⠀⠀⠱⡀⠀⠀⠀⠘⢿⠻⠿⠿⠿⡿⠛⣏⣤⡄⠀⠘⣆⠀⠀⠀⠈⠆⠀⠀⠀⠱
⠀⠀⠀⠀⣴⠟⢁⠚⠁⠀⠘⣆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⢇⠀⠀⠀⠘⡀⠘⣏⠀⠀⠀⠘⡄⠀⠀⠀⠈
⠀⠀⢠⠞⠁⠚⢣⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⣆⠀⠀⠀⠸⠀⢸⡄⠸⠵⠀⠱
⠀⠴⠃⠀⠃⠀⠀⢣⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⡀⠀⠀⠀⠂⠀⢷⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠶⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢞⣀⠄⠀⠈⠛⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠈⠲⠊⡀⠀⠙⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠗⢦⣀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⡆⠠⠀⠀⠀⠘⣤⠆⣀⠼⠁⠀⠀⠀⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠾⠀⠀⢐⡝⠛⢦⣀⡴⠚⠁⢀⣀⠀⢠⠤⠇⠀⠀⠀⠀⢀⡼⠛⠁⠀⠀⠀⠀⢀⡞⠉⠛⠉⢳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢈⣧⠀⠀⢈⡼⢁⡠⠞⠉⢀⠠⢲⠍⠠⠉⠉⢠⠤⠔⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⣠⣞⢀⡤⠀⠀⠀⠀⢸⣤⡄
⠀⠀⠀⠀⠀⠀⠘⣏⠁⢀⣈⣿⠔⠋⠀⢠⢄⠳⡐⢟⣆⠤⡀⠑⠊⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠉⠑⠴⠏⢀⠀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⢰⣋⢁⡤⠋⠁⢀⡤⢄⠴⠀⡑⠈⢄⠤⠊⠁⣠⠴⠛⠳⢄⡀⠀⠀⠀⠀⢠⠞⠁⠀⢀⠀⠀⠀⠀⠘⠧⣠⠂⣠⠞
⠀⠀⠀⠀⠀⠀⢀⡽⠋⠀⡰⠚⡦⠑⠄⢐⠉⠑⠊⠀⢀⡴⠾⢉⡆⠀⠀⠀⠉⠓⠦⣀⡴⠋⠀⠀⠘⡉⢱⠀⠀⠀⠀⠀⢨⡟⠁
⠀⠀⠀⣠⠤⠚⠁⠀⠀⠜⠀⢨⡠⠔⣪⠊⠀⢀⡴⣞⡻⡐⢁⣀⠑⠶⠊⠇⡤⢄⡀⠈⠙⠲⣄⣀⣀⡧⠀⢣⠀⠀⠀⣠⠏
⣠⡤⠚⠵⣄⠀⠀⠀⠀⠘⢄⣀⠠⠚⢁⣤⠾⢧⣄⠈⠀⠙⠃⠈⡱⠀⠌⡔⠡⡶⢌⡇⣀⡀⠀⠉⠳⢤⣤⡞⠀⢀⡼⠃
⠈⢧⡀⠐⠈⣢⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠈⠙⠲⢤⣀⠀⠙⠊⢴⡀⠶⣚⠀⢀⠀⡇⠀⠀⠀⠀⠈⠙⠲⢯⡀
⠀⠰⣇⣀⠀⢀⠷⣀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⠀⠈⠒⠜⠀⡘⠀⠐⠒⣆⡠⠔⢢⠀⠀⠀⠙⠦⣤⣀⡀⠀⠀⢀⣠⢴⡄
⠀⠀⠀⠸⠧⢤⠀⣡⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠈⠙⠶⣤⡀⠀⠙⠒⢒⡀⠈⣀⠔⠊⠀⠀⠀⡸⠦⠄⣀⣽⣧⡴⠛⠁⠀⠛⠋⣻
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡤⢤⠔⠁⡨⢗⣦⣄⡀⠉⠉⠀⠀⠀⠀⠀⣀⡵⠖⠛⠉⠉⠉⢦⡴⠂⠀⠀⠀⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢀⡤⢄⠑⢄⡀⢎⠀⠈⠁⠈⣹⠷⢦⣄⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠡⣤⠴⠂⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡌⢀⠞⠀⡠⢭⡚⠁⠀⣠⣿⡥⠶⠛⠉⠀⠀⠀⠀⠀⠀⡖⠒⡆⠀⠀⠀⠀⠀⣦⠴⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠸⡄⠈⠳⠊⢁⣾⡥⠶⠛⠉⠁⠀⠀⢀⣠⣀⠀⠀⢀⡀⢠⠃⠀⡇⠀⠀⠀⠀⠀⢣⣠⠖⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⢑⣤⠴⠚⠉⠁⠀⠀⢀⣠⠴⠚⣹⡌⠣⡈⠓⠤⠎⠈⠙⠀⠀⡇⠀⠀⠀⠀⠀⢀⣷⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠗⢦⡄⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⡶⢆⠙⡆⠰⡯⢥⡀⠀⠈⢢⡀⠀⡼⠓⠴⠚⠁⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣦⡀⣀⡥⠞⠋⠁⠀⢀⠀⢠⠖⢢⣀⣤⠇⠈⢦⠘⡀⠰⠯⠖⢒⡄⠀⠙⠈⠁⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠟⠋⠉⠀⠀⢀⡴⠋⢉⣳⠈⢧⠀⠁⣠⠤⢦⠈⢧⠱⡤⠔⠚⠉⠁⠀⢀⣠⡴⠖⠛⠉
⠀⠀⠀⠀⣀⣤⣤⠴⠾⡏⠁⠀⠀⠀⠀⠀⠀⢰⠏⠀⣴⠋⠀⢀⣈⢦⢸⡄⠀⠈⠓⠉⠀⠀⠀⣀⣤⠶⠟⠋⠁
⠀⠀⠀⠀⠙⣷⠀⠀⠁⢱⣄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠙⠒⠊⢁⣸⠎⠉⠀⠀⠀⣀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠘⠳⡆⠀⠈⠁⢸⡀⠀⠀⠀⠀⠀⠀⠳⠦⠤⠴⠖⠋⠀⢀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⣼⡁⠀⠀⠐⠊⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠈⠉⢻⡆⠀⠶⠚⢣⡀⠀⠀⠀⢀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠻⠶⢶⡆⠀⢋⣩⡶⠒⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⠆
⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠⣤⠖⠶⠇⠂⠤⠤⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣁⠀⡀⠀⠈⠙⠿⣿⣿⠿⠿⠟⠛⠁
⠀⠀⢸⠙⢋⡷⠀⠀⠺⡍⢉⠀⣄⠀⠀⠀⠀⠈⠁⠀⠒⠀⠀⡀⠀⠀⣠⠏⠴⠏⠂⠭⠁⠀⠀⠀⠀⠈⢦⡀
⢰⡋⣉⠀⠈⣆⢀⣠⠞⠱⠏⠈⠉⠀⠀⠀⠀⠀⠔⠂⠀⠈⣹⠃⢠⠞⢁⣀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠘⣎⠁
⠀⠉⠉⢧⠐⢟⡿⡿⠀⣀⠈⢡⣤⢢⡀⠀⠀⠀⠀⠠⠒⠏⠁⢀⠃⠔⠫⠒⠉⠱⠄⠀⠰⠄⠀⠀⠀⠀⠀⠈⡯
⠀⠀⠀⠈⢧⠀⣷⣴⡆⠘⠃⠹⠉⢨⢸⢸⠀⠀⠒⠂⢠⠅⢠⠃⠠⠒⠉⠉⠀⠀⠀⠰⡢⠁⡾⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠘⡆⠋⡉⣿⠆⠀⠀⠀⢂⠏⡾⠀⠀⠀⠀⠉⢻⢿⠀⠆⢄⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢸⡷⡄
⠀⠀⠀⠀⠀⠘⡄⠀⠳⡶⢢⡀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⣾⠕⢁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠹⡌⠛⣣⡶⡵⠄⠀⠀⠔⡙⡄⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡆⢽⠀⠀⠀⠀⠀⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠙⡲⣼⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡄⢠⢧⠏⠘⢦⣀⣀⣀⠔⠉⡇⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢮⣀⣲⠬⠭⢵⠟⢸⣆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⡱⠀⠀⣾⠀⡹⠋⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢔⣿⠀⠀⡟⣢⡂⠀⠀⠀⠀⡰⠘⠀⠠⡪⠄⠀⠀⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡒⢺⣥⠶
⠀⢀⣀⣀⡰⠛⠉⠀⠀⠀⢠⣣⣾⠀⠀⡀⠀⣠⢃⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣮⢅⠄⠀⠀⣀⡤⠤⠤⠴⠒⠛⠑⠓⠋⣠⠿⠒⠈⠉⠁⠀⠀⢀⣀⣀⣠⣄⣠⡚
⠀⠉⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠐⠒⠒⠂⢘⣛⣿⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠈⠉⠉⠉⠒⠃⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡜⡀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠎⠀⠲⠌⠆⢠⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠞⣀⠔⠏⠊⠁⠀⠀⠄⠁⢸⣎⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⢧⠖⠈⠛⠚⠂⠀⠀⠀⠀⢀⣀⡠⠀⠈⢉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢨⠀⠀⠀⠤⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠂⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⢸⠀⠀⡄⢀⠀⢀⢀⡄⣰⣄⠷⣼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⣀⡁⣀⣀⠠⠼⡀⡘⡀⠀⢗⢺⠰⠎⠾⠙⠃⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⠔⠊⠙⡒⠀⠉⠀⠀⠄⢸⡇⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠥⣴⡻⠤⠀⠒⠂⠀⠄⢀⢤⣤⠄⠛⢂⣈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠈⠉⠁⡰⠊⡁⠀⠠⠀⠂⠐⠂⠴⠎⣊⡔⠀⠁⠀⠀⠙⠻⣯⡀⠒⠂⠤⠤⠄⣀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡀⠴⠒⠒⠂⠤⠤⠛⢀⡶⠛⠛⠙⠓⠤⡀⠘⡄⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠲⣎⡠⢠⣖⣶⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠀⠀⠈⠁⠀⠈⠛⠻⠿⣟⣛⣲⠒⣉⣀⠀⠈⠁⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠂⠀⠀⡼⠀⠀⠀⠀⢠⠒⠀⠉⠉⠁⠈⠁⠀⠈⠁⠀⠀⠙⡏⠀⠀⠀⠈⡆⠀⡃⠀⠀⠀⠀⠀⠸⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⡴⠁⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⠀⠀⠀⠀⠀⠀⣸⠎⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢻⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⡇⠀⠙⢶⣄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣠⠞⠁⠀⠘⣧⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣰⣇⡾⠁⠀⢀⣠⡽⠶⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠷⠦⣤⣀⠀⠘⣾⣧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⢁⡴⠿⢿⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣾⠋⢙⣇⡉⠛⢦⣄
⠀⠀⠀⠀⠀⢀⣾⣃⣤⡴⢶⣾⣧⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡞⠻⣯⣛⠲⠾⣷⡀
⠀⠀⠀⠀⠀⡾⢿⡯⠶⠞⠋⠁⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠉⠛⢛⡿⣷
⠀⠀⠀⠀⢰⡇⠈⢷⡄⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⣠⠞⠁⢹
⠀⠀⠀⠀⠸⡇⠀⠀⠙⢦⣀⣾⠁⠀⠀⠀⠀⠀⠀⣿⢷⡀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⠁⠀⠀⣸
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣽⠃⠀⠀⠀⠀⠀⠀⢀⡿⠀⠙⠳⢦⣤⣀⣈⣿⠶⣤⣀⣀⣀⢀⣀⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⠀⣿
⠀⠀⠀⠀⠀⢸⡆⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⢀⣭⣍⠁⠀⠀⣭⣍⠉⠉⠉⠉⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⣀⣴⠏⢰⡇
⠀⠀⠀⠀⠀⠈⣷⠈⠳⣤⣄⣀⣠⣤⡤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠘⢁⣀⠀⠀⠀⢀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢿⡉⣧⠀⣾
⠀⠀⠀⠀⠀⠀⠸⣆⡾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⣀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⣷⠇
⠀⠀⠀⠀⠀⠀⠀⣿⢧⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠉⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⢀⡀⠀⢀⣰⠏⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢧⣀⣀⣀⣤⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠹⣆⠀⠀⢀⡀
⠀⠀⠈⢻⣟⠋⠁⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠈⠙⣻⡟⠁
⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣻⣭⣽⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⡽⢦⣄⠀⠀⠀⠀⠀⠀⠀⢸⡿⠭⠿⠿⠿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⣥⣤⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⡾⢿⣿⣛⣛⣫⣽⡇⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠶⠶⣷⠈⠉⠉⠉⠉⠁⣧⣤⣤⡤⠶⠞⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢾⡿⠶⣤⣌⣷⡄⠀⠀⣠⠞⠁⣀⣀⣠⣤⡶⢿⡷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⣸⠇⠀⠀⠈⠉⠙⡆⢼⠛⠛⠛⠉⠉⠁⠀⠀⠀⣿⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣀⣸⡇⢀⣤⡴⠶⠞⢻⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠉⠙⢛⡟⠁⠀⠀⠀⠘⠷⠾⠷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠴⠾⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣴⣾⣋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⣘⣧⣀⣀⣠⣀⣀⣀⣾⠉⠉⠙⣷⡀⠀⠀⣀⣀⠀⠀⠀⣿⠀⢀⣴⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⣿⠉⣉⣉⣩⣿⠋⠉⠉⢻⡍⢻⣤⣤⣤⠟⠙⢛⡿⠿⢿⣛⠳⣦⣼⣷⡟⠁⣸⠛⣧
⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⣾⣧⣉⣿⡿⣯⣭⠽⠋⠘⢧⣤⣤⠾⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠻⣯⣿⣾⡷⢶⣿⡶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⢠⡟⠀⠀⣠⡶⠛⠉⠈⢻⣏⣻⣿⡁⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⠋⠀⣀⣠⡟⠁⣼⢻⣆⣈⠙⠳⣤⡀⠀⠀⠹⣆
⠀⠀⠀⠀⢠⡟⠀⠀⣾⠋⠀⠀⠀⣿⣿⣿⣥⣼⡿⠶⠞⣋⣩⣇⠀⠀⢀⡶⠾⠿⣶⣄⡀⢀⣠⡾⠿⠷⡶⠏⢀⣠⠾⣿⠟⠉⣹⠇⠀⠈⢻⡄⠀⠀⢻⡆
⠀⠀⠀⢀⡾⠁⠀⠀⣧⠀⠀⠀⠀⢿⣈⠙⠳⢬⣍⣛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠙⠋⠁⢀⣀⣤⡴⠾⠛⢁⣴⠛⣷⠛⠉⠀⠀⠀⣸⡇⠀⠀⠀⢿⡄
⠀⠀⠀⣾⠃⠀⠀⠀⠈⠳⣤⡀⠀⠀⠙⠳⣦⣄⡈⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠋⠉⣁⣠⡴⠞⠋⠉⠳⠾⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠈⢷⡀
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⠈⠉⠛⠳⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠞⠛⠉⠁⠀⠀⠀⢀⣀⣤⡴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠶⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠖⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠈⠉⠠⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡠⠆⠘⠋⠁⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠁⠀⠈⠑⢄⡀⠐⠊⠉⠀⠀⠀⠀⠀⠉⠉⠒⢦⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠱⣀⠀⠀⣀⡀⠀⢀⠜⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠠⣠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠗⠀⠀⠀⠀⠀⠀⢠⣄⡀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣏⡐⠈⢳⣀⠀⣀⠤⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠀⠀⡁⠄⠀⠛⠒⠫⠤⣉⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠲⡀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠛⠉⠉⣹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠑⠤⢀⡀⢀⡀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⢉⠋⠀⠷⣀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢿⣿⠟⢀⠌⠀⠀⠀⠀⠉⠢⡀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢇⠀⠀⠂⠀⠀⠀⡠⠒⠒⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂⢀⡟⠁⡠⠂⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠘⠀⠤⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⢇⠀⠤⠀⣀⣀⡀⠀⠀⠃⡠⠎⠁⠈⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⢣
⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠈⠁⢣⡀⠀⠀⠀⠑⢦⡀⠀⢀⣠⣏⠋⢻⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢱⠀⠘
⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠒⠂⠀⠛⢦⡀⠀⠀⠀⠈⠫⣽⠤⠘⢀⠜⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠐⢠
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⠢⣄⠃⠀⠀⢀⠀⠤⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠚⠔⠣⢄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠀⠀⠀⠀⠀⠑⢤⡜⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠓⠚⠃⠒⠒⢠⡀⠀⠀⠀⠆⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠣⡀⠀⠁⢦⠀⢰⠀⠀⠀⣀⣀⣀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠁⠈⠢⣀⡐⠁⠀⠈⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⢀⠔⠁⠁⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠂⠠⠤⠤⠤⠤⠤⠤⠄⠠⠤⠤⠤⠤⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⡠⠄⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡼⠀⡀⡀
⠀⠀⠎⠀⠀⠐⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡊⠀⠀⠈⠂⠀⠐⠃⠀⠀⠀⠀⠀⠀⣸⠃⢀⠜
⢀⣈⠰⠈⠉⠀⠉⠁⠈⠻⣿⣛⡶⣄⡀⠀⠀⡄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠚⠃⠀⢤⣄⡀⠀⠀⠀⠀⠀⠉⠑⠛⠛⠓⢄⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⠀⠂
⠀⠀⠐⠐⠀⣉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⢗⡒⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣬⣳⣄⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢⡑⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡄⠀⠀⠀⠀⣀⣤⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠤⠤⠤⠤⣤⣀⣀⡀⠈⠻⠀⢀⡤⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠶⠶⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣄⣤⣄⣫⣤⣶⣀⣨⣷⣁⣴⣞⣱⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠙⢦
⠀⠀⠀⠀⠀⢀⠔⠉⣀⣠⣄⣀⡀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠛⠲⣄
⠀⠀⠀⠀⡴⠁⢠⠞⠁⠀⠀⠈⠙⢦⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡤⠒⠒⠦⣄⠀⠱⣄
⠀⠀⠀⢰⡇⠀⣿⠀⠀⠀⠀⠀⠀⠈⡇⢸⣀⡤⠴⠶⠒⠒⠒⠒⠒⠶⠦⢤⣄⡾⠻⣴⡵⠋⠀⠀⠀⠀⠈⠓⢄⠈⢧
⠀⠀⠀⠈⣧⡀⠸⣆⠀⠀⠀⠀⢀⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⠼⣷⠶⢦⣄⠀⠀⠀⠀⠈⠂⢸⡆
⠀⠀⠀⠀⠈⠻⣦⡈⠳⣄⢀⠞⠁⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡂⠀⠀⠀⠀⠸⣤⣴⠋⠀⠀⠀⣠⠔⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣨⠋⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⣶⠖⠚⣋⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠸⣏⠛⣻⠆⠀⠀⠀⠉⠄⠀⠈⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣠⠼⡻⢥⣤⣤⣖⣀⣠⠤⠤⡤⣄⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢀⣖⠻⠛⠛⠛⠛⢻⣇⣸⠳⣼⠇⠀⠀⠀⠉⠀⠂⠁⣀⡇⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡇⠀⠐⠒⠒⠶⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⡀⣾⠁⠙⠛⠒⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠳⣤⣀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⡆⠀⠀⠀⠉⠉⠀⠈⠳⣤⣤⣤⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⢳⡤⢤⡀⠀⠀⠀⠀⠀⠈⠘⠉⠀⠀⠀⠀⣀⣀⠀⠀⢀⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠙⠦⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⣙⡶⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠓⠒⠒⠒⠒⠒⠛⠛⠉⠁⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⢤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⡹⠁⠀⠀⠀⠀⠀⠀⢸⣡⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠲⠶⠶⠒⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⣤⣶⠋⠉⣘⣿⣇⠈⠉⢳⣦⣄⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢀⣴⣏⣤⠾⣻⠟⠉⠁⠉⢉⠛⠻⣦⣍⣷⣄⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡴⢯⡴⠋⠀⢠⡏⠀⠀⠀⣾⠋⢻⠄⢹⠈⠳⣬⠷⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⣄⡀⠈⣧⠀⠀⠀⠈⠛⠋⠀⣼⣀⡴⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠙⠿⣟⣾⣷⣄⣀⣀⣀⣤⣾⡿⠟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠛⠶⢬⠿⣯⠴⠞⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠶⠶⠶⠖⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠒⠀⠀⠀⢈⡡⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠰⠄⣀⠤⠄⠀⠀⠁⠀⠞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⠀⠀⠀⠀⠁⠒⠊⠀⠂⢰⣾⠀⢳⣄⡀⣴⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⡄⢀⡀⠀⠈⠤⠀⠒⢀⣀⣈⠥⠀⢊⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣝⣻⠛⠓⠒⠚⢋⣉⡤⠄⠀⣠⠿⠈⣻⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣦⣙⠛⠛⠉⠠⣶⣾⣧⣠⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠠⠜⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⢰⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠸⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣆⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠛⠛⠛⠿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠏⠉⠙⣿⡛⠋⠉⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠛⠃⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠳⣄⢀⣀⡤⠤⠤⠤⣄⣀⠀⢀⠤⠒⠉⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣇⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣀⡤⠬⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⠋⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠓⠦⣤⣼⠀⠀⠀⠀⡈⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠃⠀⠀⠀⠀⢸⡄⠀⠀⣤⣽⠄⡴⠛⠟⠓⠒⠶⠤⠤⠤⠴⠒⠚⠉⢹⠀⠀⠀⠰⡦⠤⠚⠓⠊⠀⠈⠀⠀⠈⠣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠷⢤⣀⣀⣀⣀⡤⠞⣇⠀⡠⠼⠆⠀⠀⠀⣀⣀⠀⠀⠀⠀⢠⣤⣤⠀⠀⣾⠀⠀⠀⡸⠀⠀⠀⢤⡀⠀⠰⠀⠀⠀⠀⠀⢉⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⢀⠀⠈⢿⡀⠀⢈⡤⢤⡀⠀⠀⠀⠺⢿⣿⡆⠀⠀⠀⢉⣹⣿⠀⠀⡇⠀⠀⠺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠒⠀⠀⠈⠀⠀⠀⡹⠀⡏⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⢀⠀⠈⠉⠁⠀⠀⢳⡀⠀⠀⠘⡆⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⡠⠀⠀⠀⠀⠉⠀⢸⠦⣙⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⡇⢀⡤⢾⡇⠀⠀⠀⠰⠃⠀⠠⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠾⣄⠀⠊⠀⠀⠀⠀⠀⠀⢰⡁⠀⠈⡽⠉⠉⣇⠀⠀⠀⠀⠀⢀⠞⢓⡇⠀⠀⠀⠀⠀⣰⠛⢧⠀⠀⠈⣙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢢⡜⠁⠀⠀⠈⠳⣄⡀⠀⠀⠈⠒⠋⠀⠀⠀⣀⡤⠞⠁⠀⠀⠱⣖⠉⠀⠠⡧⠤⠤⣀⢀⡤⠒⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠑⣄⡤⠤⢤⡎⠀⠀⠀⠉⠓⢤⣀⠀⠀⠈⠉⠛⠓⠒⠒⠒⠛⠋⠉⠁⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⠓⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠙⠛⠛⠚⠋⠉⠉⠉⠀⠀⠀⠘⢆⠀⠀⠀⠉⡱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠉⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠖⠈⠉⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠈⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⡠⠊⠀⠌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠠⠴⠤⠤⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢎⡀⠀⠀⠀⠀⠙⣦⠤⠐⠒⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⢀⠞⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠉⠁⠒⠢⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⡜⠉⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡼⠁⠀⠀⡇⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣤⣀⣀⠀⠀⠀⠀⠐⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣿⣿⣶⣤⣀⡠⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡷⠛⠉⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠚⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢠⠊⣩⣙⢆⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⢿⣿⣿⣿⠋⠀⠀⠀⠘⢻⣿⡎
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠾⣿⡟⠀⠀⣼⡿⠋⠀⠀⠀⠀⢰⡘⢿⡿⡱⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠜⠁⠀⠠⣤⢤⡤⠀⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠛⠒⡖⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡊⠁⠀⠀⠀⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣻⣟⣿⣻⣏⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣼⣯⣿⣽⣯⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⢾⡷⣿⢾⡷⣿⡾⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⣿⣿⢺⡟⣿⢻⣟⣿⣟⢁⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⢘⠛⠛⠛⠋⠉⠉⠉⠉⠙⡇⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⡀⠀⠀⠀⡀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠒⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⣿⣿⣿⣿⣿⣿⣿⣾⣿⣦⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⠁⢀⣠⣤⣧⣽⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠃⠐⠟⢩⣿⡟⠟⢻⡇⠸⠉⣿⣿⣿⣿⡎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠘⠃⠀⠀⠘⠇⠀⠀⣿⣿⣿⣿⡇⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⢸⡄⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⡴⠋⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⢻⣿⣿⣿⣿⣿⣿⣤⠴⠊⠀⠀⢀⣴⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⢀⣾⣿⣿⣿⣿⣋⠉⣠⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣉⣭⣾⣿⣯⠙⣿⣿⡿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠘⣿⣷⠈⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⢿⡆⠀⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⣦⣄⡀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠤⠀⣀⣀⣀⠀⠀⠀⢠⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠉⠁⠀⠀⠀⠀⢀⠀⠀⠤⠤⠀⠉⠡⠶⣤⣍⡉⠙⠳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠀⠀⡀⠄⠂⠈⠀⠀⣤⣀⡀⠀⢀⡠⠂⠁⣘⡉⠛⠕⠤⡀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⣀⣀⣡⣤⣴⣶⣶⣾⠿⠿⣿⣿⡟⠛⠛⠛⢿⣿⢹⣦⡀⠐⠈⠳⢄⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢀⣠⣤⣶⣿⠿⠿⠛⠋⠉⠁⢀⣀⣤⣄⠙⣿⣷⡀⠁⠀⢹⣿⢸⣿⡇⣄⠀⠢⡀⠑⣄⠘⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⢰⣿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⠙⢷⡌⢿⣧⠘⠃⠘⣿⡞⣿⣷⣿⣀⠀⠀⠀⠈⠳⡀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⡘⢿⣿⣤⡀⢀⣀⠈⠓⠄⣀⠜⡇⢻⡌⣷⡄⢻⡄⠙⠃⠀⠀⠘⣿⣼⣿⣟⢯⠳⡀⠀⠀⠀⠙⡄⠈⣷⡀
⠀⠀⠀⠀⠀⣀⡜⠀⠀⠀⠀⠀⠔⠀⠀⠉⠻⢿⣦⡉⠃⠀⠐⠇⠀⠀⠀⠳⣝⠛⢶⣿⠄⠀⠀⠀⠀⠈⠻⣯⡻⣮⡳⣄⠀⠈⡀⠀⠘⡄⠘⣷
⠀⠀⠀⠀⠀⡸⠉⠐⠢⠄⣀⠌⠀⠀⠀⠀⣠⣾⡿⠋⠀⠾⠀⣸⠀⠀⠀⠐⠈⡛⢛⡛⠀⠀⠀⠀⠀⠀⠀⠉⣻⢮⣻⢮⣅⠀⢠⠀⠀⠘⡄⠸⡧
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡈⠉⠒⠠⢀⣾⡟⠋⠀⢀⠂⠀⠙⠁⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⣟⣋⠩⣆⠀⠀⠀⠀⢱⠀⢿⡄
⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣤⠀⠀⠀⠀⡀⣤⡲⣝⣻⡿⢭⣛⠶⠀⠀⠀⠀⠈⡄⠸⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠓⠶⣟⡉⢙⡷⠶⢦⣤⣤⣤⣤⣴⠶⣶⣿⠃⠀⠀⠀⠙⠉⡿⢽⠷⠉⠉⠳⣮⣝⠆⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠴⣿⣥⡾⢣⣄⣴⡿⠃⠀⠀⠀⠀⠀⠀⠉⢠⡴⠚⠛⢢⠘⣮⡃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠁⠰⠠⡀⠀⠀⠀⠀⣤⠀⣨⠂⠀⡼⢀⡷⡄⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⢣⠀⠀⠀⠀⡀⠀⠀⣶⣦⣤⣀⣠⣀⣀⣀⣀⣀⣤⣴⣾⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣟⠶⠀⠈⠛⠁⠀⠀⢀⣞⢷⠀⠀⠀⠀⠀⡇⢰⡟
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠇⠀⢰⣿⡟⠛⠛⠻⠿⠟⠛⠛⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⠀⠀⠂⣠⣴⣿⣿⠆⠄⠀⠀⠀⢸⠀⣸⠇
⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠙⠛⠛⣹⣿⣿⣿⠲⠄⣀⠀⢀⠇⢀⡿
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠆⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣶⣦⣹⣿⣿⡟⠀⠀⠀⠁⡟⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠁⠀⢬⡟⣽⣿⣿⠀⠀⠀⢀⡜⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢸⣿⣯⣀⡀⠀⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡴⣿⡟⠀⠀⣠⠎⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡙⠛⢿⠻⠿⢿⣷⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣈⣀⣽⠃⢀⡔⠁⣰⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠳⡀⠀⢁⠀⠀⡈⠉⠉⠉⡉⠝⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⢉⠴⠋⣀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⣢⡈⠀⡐⠀⡀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠀⠀⠀⠀⢀⡠⠞⠁⣠⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢳⢿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⣀⡠⠔⠚⠁⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣆⣈⡉⠑⠒⠒⠠⠤⠤⠤⠤⠀⠒⠒⠊⢉⣀⣠⡤⠾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠏⠈⠈⠛⠙⠷⠶⢶⣷⣿⣿⣿⣿⠶⠺⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣾⣿⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢉⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⢥⣤⣤⣿⡿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣾⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣤⣦⣤⣀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣉⠉⠉⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⢿
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⡏
⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢉⡃⡉⢃⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣀⠠
⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣁⡀⠁⠀⠀⠀⠀⠀⢠⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠘⣿⡏⠀⠄⠀⠀⠀⠀⠠⣄⠀⡀⠀⠀⢀⣠
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠸⢷⣤⣤⣤⠄⢸⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠠
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣄⣘⣉⣁⣠⣾⣷⣄⣀⡙⠿⣿⡇
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⢻⣿⣿⢿⠿⠛⠊⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⣿⣷⣘⡛⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⡀
⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣤⣰⢄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⣤⣶⣶⣆⠀⠀⠀⠀⠀⢶⣿
⣿⣿⣿⣆⠀⠀⠀⠀⠀⣸⢣⡏⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⢀⡠⠋⣿⣿⣿⣿⡿⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⣄
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠁⠀⠀⠛⠛⢿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣀
⣿⣿⣿⣟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⠑⣀⣸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠙⢢⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠐⡄⠀⠀⢤⡖⠁⠀⠀⠙⢶⣤⣀⠀⣀⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⢁⡀⡄⠻⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⡿⢁⠀⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠀⠀⠀⣾⡨⣤⣤⠆⣿⠁⢰⣿⠆⠀⢾⡇⠸⣿⡑⢦⣤⡄⢠⣷⡂⠀⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠥⢄⡇⣤⣿⣿⣶⣶⣿⡟⠀⣀⠀⠐⠃⠀⢀⠀⢻⣷⣦⣤⣶⣿⣿⣅⣠⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⣩⠎⠘⢿⣿⠟⠋⠉⠁⠀⠘⠷⢶⡷⣾⠏⠀⠀⠛⠛⢿⣿⠿⠡⢿⣿⣶⣒⠴⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠁⢤⡠⣴⡟⠰⠐⠂⣿⣾⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣟⠬⣤⠀⣿⣷⣶⣀⡔⠡⢀⠠
⠀⠀⠀⠀⠀⠀⢀⡇⠀⢂⡽⣿⠳⣬⣉⣥⣿⡟⠡⣿⡏⣿⡷⣶⣤⣤⡴⣶⣿⢻⣿⠉⢾⣿⣷⣶⠞⠋⠘⣿⣿⣮⣓⣐⡜
⠀⠀⠀⠀⠀⢾⠟⢷⣍⣭⣼⠡⠠⠈⣿⡿⠋⠀⠀⢸⠇⠸⡇⢹⠀⠀⠁⠙⢳⢸⡇⠀⠈⠻⣿⣿⢠⣃⠒⢸⣿⠟⠑⢾⡇
⠀⠀⠀⠀⢀⣾⢄⢺⣭⣍⡛⠒⠒⠛⠉⠀⠀⠀⠀⣼⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⣧⠀⠀⠀⠈⠛⠷⠭⠵⠛⣛⣦⠤⠈⡇
⠀⠀⠀⠀⠚⢿⠇⠛⢿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠏⣿⠆
⠀⠀⠀⠀⠀⣽⠰⡐⣂⣿⣿⣿⣿⠇⣠⠆⠀⠀⢀⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢣⡀⣶⣿⣿⣿⣿⡟⠠⠀⠘⡄
⠀⠀⠀⠀⠀⣿⣦⣭⣱⠿⠿⠛⠥⢚⡁⠀⢀⠌⠘⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢙⠮⠻⢿⡿⢟⣋⣷⣖⣠⠇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡶⠶⠾⣯⣤⣶⣭⠋⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣶⣿⣶⠶⠒⠛⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣐⣀⡁⣽⣿⣿⣥⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡠⠤⠥⣼⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣾⣿⣿⣟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⣿⣿⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⣿⣶⣿⣷⣦⣀⠀⠐⠄⠀⠄⠊⢀⣠⣾⣿⣿⣿⡿⠿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠿⠿⠶⠶⣶⠶⠿⠿⠟⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⠟⠉⠉⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠖⠋⠀⣀⠤⠔⠒⠒⠐⠒⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⢀⣀⠤⣄⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠼⠁⢀⠔⠊⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢁⣀⠤⠤⠤⢌⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠲⡄⠀⠀⠀⠀⠀⠘⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠒⠉⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠳⠜⠁⠀⠀⠀⠀⠀⢠⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠖⠞⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⣉⡠⠤⠤⠀⡀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⣰⠏
⠀⠀⠀⠀⠀⠀⢀⠞⢡⠔⠉⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠰⠋⠓⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⢀⡴⠃
⠀⠀⠀⠀⣠⠶⠏⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢖⠉⠁⣠⠤⠖⠋⣀⡀
⠀⠀⢀⡴⠃⢀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢃⠀⠀⠀⠀⠀⠈⢆⠀⢿⣠⡴⠋⠁⠉⠓⠄⣀
⠀⣠⠎⠀⡠⠊⠈⠀⠀⠀⠀⠀⢰⠋⠑⡄⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⢂⠀⠀⠀⠀⠀⣈⣠⠞⠁⠘⡄⠀⠀⠀⠀⠈⠓⠦⣄⡀
⣰⠃⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢠⠓⢄⡀⠀⠀⠀⢀⡤⠒⠁⣠⠟⣆⠀⢆⢀⡠⠔⢊⡴⠋⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠙⢲
⠹⣄⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⡀⠀⠉⠁⠉⠁⣀⠤⠚⠁⠀⠘⣆⠀⣁⣤⠶⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺
⠀⠈⠛⠦⣀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠺⡉⠓⠲⠖⠒⠉⠁⠀⠀⠀⠀⠀⠈⣋⠝⠁⠀⠈⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⢀⠴⠊⢀⡏
⠀⠀⠀⠀⠈⠳⣄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠘⢦⡀⠀⠀⠀⠀⠀⠀⢠⢦⣤⠞⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠇⠀⠀⢀⠔⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⠹⣄⠀⠀⠀⠀⢀⡼⠒⠒⡀⢀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑⢦⡀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣠⠞⠁⠀⠀⣀⡴⠋⠀⠀⠀⠉⠀⠣⣀⡀⠀⡇⠀⠀⠀⠀⠀⡸⠃⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡈⠢⡀⠀⠀⣠⠎⠀⣠⠞⠀⠀⠀⠀⣞⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢧⠀⠀⠀⠀⣼⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠳⠚⢁⡠⠞⠁⠀⠀⠀⠀⢰⠋⠑⠤⣙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠈⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⠋⠀⠀⠀⠀⠀⠀⠀⣏⠑⠢⣀⠈⠳⢌⠑⢄⡀⠀⠀⠀⠀⠀⣠⢞⣽⡖⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠑⢄⠈⠑⢤⡀⠑⠦⡉⠢⣀⠀⢀⣔⣵⠟⢤⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠑⢤⡀⠈⠢⣀⠈⠒⢬⡷⢞⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠉⠢⣀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠁⢒⣾⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠈⠲⢄⠀⠀⠀⠀⠀⠀⠈⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠑⢄⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⢑⡴⠋
⣶⣶⡶⢶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣿⣿⣿⣷⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⡿⠿⠿⠿⠿⠛⠋⠉⠙⠻⢿⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⢿⣿⣿⣦⣤⣴⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⡾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⣉⠛⠻⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣴⣿⣿⠿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣧⠀⠈⠻⣿⣿⣷⡄
⠀⠀⢀⣾⣿⡟⠁⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿
⠀⠀⠈⢿⣿⣆⠀⠀⠀⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⠟⠁⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠙⢿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣾⣿⣿⣿⡿⠋⠀⠀⠀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⢀⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣀⣤⣴⣶⣶⡄⠀⣶⣦⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠇⣰⣿⡿⠛⠉⠉⠁⠀⠀⣿⡏⠀⠀⠀⠀⣿⡇⠀⢀⣴⣶⣿⣿⣷⠀⠀⠀⠀⢀⡄⠀⠀⠀⣶⣶⣶⣄⡀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⠀⣿⣏⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⣿⡇⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⣿⣿⠙⢿⣷⡀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⠀⠹⢿⣷⣤⣀⠀⠀⠀⠘⣿⡇⠀⠀⠀⢸⣿⡇⣸⣿⠇⠀⣀⣀⣀⠀⠀⠀⣼⣿⣿⡇⠀⠀⣿⡏⠀⢸⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠉⠛⢿⣿⣦⠀⠀⣿⣷⠀⠀⠀⣸⣿⠃⣿⣿⠀⠀⠛⠛⢿⣷⠀⣰⣿⣏⣿⣿⡀⢸⣿⣇⢀⣼⣿⠇⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⢹⣿⣆⠀⣠⣿⡟⠀⢿⣿⡄⠀⠀⢀⣾⡿⢠⣿⡿⠿⠿⣿⣧⢸⣿⣿⣿⣿⠋⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠇⠀⢿⣿⣿⣶⣿⡿⠟⠀⠀⠀⠙⠿⣿⡿⠟⠀⠀⠈⠿⣿⣷⣾⣿⠟⠁⠻⠟⠁⠀⠀⠹⣿⢾⣿⡏⠻⣿⣷⡀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠈⠻⠿⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⢠⣿⣿⣇⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⠿⠿⠿⠛⠋⠉⠙⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⠉⠉⠉⠛⠿⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿
⠀⣼⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃
⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠹⣿⣿⣷⣦⣤⣤⣤⣴⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄
⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣦⡀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⣀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠿⠟⢁⣿⣿⣿⣿⣿⣇⠙⠿⠿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣁⣤⣶⣤⣈⣿⣿⣿⣿⣿⣏⣠⣴⣦⣄⣙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠹⣿⣿⠏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡄⢹⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣄⣠⣴⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠋⣠⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣗⠀⣿⣿⣿⣿⠀⠀⣈⡉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⢉⡀⠀⢸⣿⣿⣿⣿⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣿⣿⣿⣿⣆⠀⠹⣿⣿⣶⣶⣦⣤⣤⣤⣴⣶⣶⣾⣿⡿⠁⢀⣿⣿⣿⣿⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⡀⠀⠉⠙⠛⠛⠻⠿⠿⠛⠛⠛⠉⠁⠀⣠⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆






⡿⢿⠿⡿⠿⢻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣟⣛⣻⣿⣿⣿⣿

⣠⠚⠉⠓⢦⡀
⣇⠀⠀⠀⠀⠙⢄
⠈⢤⡀⠀⠀⠀⠀⠳⡀
⠀⠀⠙⢦⡀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠛⠛⠉⠑⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⢦⡴⠒⠊⠙⠶⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⠲⣄⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠐⠉⠀⠀⠀⠀⠈⣦⣀⣳⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢀⣀⣠⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣤
⠀⡞⠉⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀
⠀⡞⠃⠀⠀⠑⠤⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠒⠉⠀⠀⠈
⠀⠑⡶⠠⢄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠐⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚
⠀⢰⠇⠀⠀⠀⠀⠀⠐⠒⠀⠠
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠐⣇⠈⠁⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡟
⠀⠀⠀⠀⠈⠓⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣓⣒⣲⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣖⣒⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⣤⣖⡉⠉⠉⠀⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡫⢞⡉⠴⠿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢛⡽⠒⢉⡤⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠈⠁⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡰⠁⠀⠀⠀⠈⣉⣉⣉⣉⡩⠿⠂⠀⠀⠀⠀⣀⣴⣿⠟⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⣰⣿⣿⣿⣿⣶⣦⣶⣤⣴⣶⣿⣿⣿⠿⠛⠀⠀⠀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠈⠙⠟⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⣀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠚⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣿⣿⣿⣿⣷⣦⣴⣶⣶⣶⣶⣖⣛⢋⠽⣿⡿⣿⣯⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⢻⣿⡟⠛⠛⠻⣿⠉⠉⠙⠀⠛⠁⣿⡕⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠛⢁⠀⠀⠀⢀⠞⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠒⠂⣀⡴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢤⡆⠈⣿⣿⣷⣾⣿⠃⢧⡠⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢀⣤⣶⠞⠁⣦⠘⣿⡿⢻⡟⡀⠈⠳⣤⣤⣤⡀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣈⣦⣀⣸⡇⠸⡇⡾⢱⢇⣠⣤⣾⣭⣭⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠉⠉⠻⠀⠳⠤⠟⠊⠁⠞⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠙⢿⣆⠀⠀⠀⠀⣰⡿⠋⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠸⣿⠀⠀⠀⠀⣿⠇⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠛⠀⠀⠀⠀⠛⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⣠⣶⡾⣶⣄⣴⡿⢷⣦⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⢸⣿⠀⠀⠀⠀⣰⡾⠟⢿⣦⠈⠛⣿⣾⠟⠁⢀⣼⡿⠋⢀⣼⣿⣤⣤⣴⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⣿⡆⠀⠀⠀⣿⡇⠀⠀⣿⣧⣾⠟⠁⢀⣴⡿⠋⢀⣴⡿⠛⢻⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠸⣿⡀⠀⠀⣿⡇⠀⠀⣿⠟⠁⠀⠰⠿⠋⢀⣴⡿⠋⠀⣠⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠹⣷⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠀⣠⣾⠟⠁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠙⣿⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠛⠁⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⣀⣀⣤⣾⣿⣧⣤⣤⣤⣤⣤⣤⣶⣶⠾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠂⢊⠀⢫⣟⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⣿⣷⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢈⣶⣿⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣀⣲⡶⠒⠒⠒⠒⠀⠙⢻⠇⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠹⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣼⠀⡸⢠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠨⠛⣹⣿⡆⡄⠀⠁⠇⠀⣰⣿⠃⣀⣀⡀⠀⠇⢸⠇⠀⢰⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠘⣿⣆⠐⣯⡃⠀⢠⣤⢾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⡅⠤⠈⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⣷⣬⠀⢀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠬⠄⢀⡀⠀⠀⠀⠔⠀⢀⣿⣿⢰⠃⠏⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⡀⠠⢤⠀⠀⠀⠀⠀⠀⠀⣠⣿⡷⠃⡸⠀⠀⡀⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢋⠿⠇⠀⠀⠀⠀⠀⢀⣠⣴⠾⠿⠋⠠⠞⣁⠀⣰⠁⢀⠄⠁⠂⠀⠸⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡟⢻⡟⠫⠆⠀⢀⠉⠀⠀⠊⠁⠊⠑⠀⠈⠜⡸⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠁⣰⣶⠖⠂⠉⢀⡀⠀⠀⠀⢅⠋⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⠋⠀⠀⠀⠀⢀⡠⠋⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠔⠋⠀⠀⠀⡀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠉⠁⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢠⣿⢏⣾⣿⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⡾⢃⣾⣿⣧⣿⣿⣿⣿⡿⠟⢉⣾⣿⣿⣿⣿⣿⣿⣿⡿⣫⣿⣿⣿⡿⢟⣿⡿⠋⣰⢿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠁⢸⣿⢿⡿⠟⢋⡉⠀⠀⠒⠛⣛⣻⠭⠽⠟⠋⠉⢡⣼⠿⠛⠋⡁⠴⠛⢉⡴⢚⠵⠋⠀⠀⠸⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠸⠀⠀⡇⠀⠉⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠹⡝⠝⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉⠉⠉⠙⢆⠀⠀⠀⠀⠀⠀⠈⢌⠳⣝⢿⣿⣿⣿⣿⠁
⠀⠀⠀⢀⡤⠤⣀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⣿⣿⣿⣿⡏
⠀⠀⠀⠀⡏⠐⢄⠉⡇⠀⠀⠀⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠩⢹⡟
⠀⠀⠀⢐⡇⠀⢠⠙⡇⠀⠀⠀⢠⣦⣿⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣆⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣃⠤⠟⠀⠠
⠀⠀⠀⠀⡇⠀⢦⢠⠀⠀⠀⠀⠀⠉⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢉⣠⠤⠒⠀⡀⠇
⠀⠀⠀⠀⢿⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡔⠉⣀⠀⠀⠀⠁⡆
⠀⠀⠀⠀⠘⣇⠀⡈⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⢈⡗⠀⢀⡞
⠀⠀⠀⠀⠀⠘⢧⡅⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡐⠂⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠵⠋⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠈⠉⢻⡉⠉⠉⠉⠛⠓⠒⠒⠒⢒⣲⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢻⣶⣶⣶⣶⣶⣶⣶⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⡉⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠾⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⠋⠉⠀⠀⠀⠉⠙⠒⠢⠤⣄⡀⠀⠹⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠏⠁⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠉⠓⠒⠶⣤⡀
⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣄⣀
⠀⠀⠀⣠⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀
⠀⠀⠀⢧⡀⠉⠒⠀⠀⠤⠀⢀⣀⠀⠀⠀⠀⠀⠈⢀⣁⣀⣀⣀⣀⣀⣀⣀⡀⠠⠤⠤⠤⠴⠒⠉⣹⠆
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⢠⡇⠀⠀⠀⣀⡤⣄⠀⢀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⠀⠀⢀⡠⠤⠤⡀⠀⠀⠀⠀⠘⡇
⠀⠀⢸⠃⠀⠀⡜⠁⡔⠊⠀⢸⢸⠀⢸⢡⢠⠊⡠⠔⠁⠀⠀⡏⠈⡇⠀⡇⢀⣄⡀⠘⡀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠱⢤⣓⡒⠢⣼⠸⠀⠀⢸⡏⢸⠁⠀⠀⠀⢸⢱⡇⢹⠀⢇⠀⠓⢁⡰⠁⠀⠀⠀⢠⡇
⠀⠀⢸⡆⠀⠀⠀⠀⠀⢘⡆⢹⠀⠧⠃⡼⡇⠸⡀⡯⠉⣑⣼⠈⢣⠀⠇⢸⠀⣄⠘⡄⠀⠀⠀⠀⣸⠁
⠀⠀⠀⢷⡀⠀⢠⡖⠒⣉⡠⠛⣄⢀⡠⠃⠙⢄⣈⣴⣀⡇⡆⢠⠀⢣⠸⡘⠀⡏⡆⠘⡆⠀⠀⣰⠃
⠀⠀⠀⠀⠳⡄⠀⠉⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠈⠒⠃⠓⠁⠈⠋⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠳⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣤⣴⠏



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⡀⠀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠶⠚⠋⠉⠉⠀⠀⠉⠀⠀⠈⠉⠉⠒⠒⠶⠤⠤⠤⠤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣰⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⣑⠁⠀⠀⡀⠀⠙⠲⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢇⠀⠀⠀⠀⠀⠀⠿⡠⢄⠀⠀⢀⣀⣀⣀⣈⣹⣷⣤⣼⡀⢀⣧⣀⣴⣧⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠰⡔⣃⣐⣀⣂⣐⣊⣐⣿⣬⣅⣉⣈⣡⣍⡤⣴⣶⡴⠶⡶⠋⣩⣴⣶⣶⣶⡚⠳⢯⣛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⠉⡙⠉⠉⠉⠉⠀⠍⡸⠁⠈⠉⠉⠉⠅⠀⡢⢊⣩⠾⢻⠶⠛⣍⢣⠉⠀⠀⣿⠀⠀⠘⡆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⣀⣼⡀⢀⣴⡀⢀⡿⢤⣀⠀⠀⠀⣨⠀⢠⡚⠾⡏⡀⠀⠈⣳⣄⠈⣆⢱⠀⠀⠀⠃⠀⠀⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠉⢀⠈⠉⠀⠉⠉⠀⠀⠈⠉⠙⠉⢉⣟⣟⣣⡤⠾⢦⣤⠞⠙⠳⠶⢾⡄⢟⡲⠰⠧⠤⢂⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡞⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⣒⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⡘⣽⡗⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘⣦⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⡀⠠⡠⠒⠉⠀⠀⠀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⢄⠅⠐⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⢀⡀⠞⠀⠀⠀⠀⠀⠖⠀⠀⠀⡇⠐⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠙⠧⡀⠈⢢⡀⠀⠀⣐⡀⠀⠈⠛⠀⠀⠂⠀⠀⠰⠀⡌⠀⠀⠀⢹⠐⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⠑⠤⠚⠓⠀⠀⠀⠤⠀⠀⡃⠀⠀⢀⡀⠈⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠛⠉⠉⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⢀⣻⠴⠶⠶⠒⠒⣦⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⢀⣄⠀⠀⠀⢸⡧⡀⢀⡤⠖⠉⡟⠀⠀⠀⠀⠀⣘⣦⡴⠛⠓⢦⡴⠚⠁⠀⠀⠀⠀⢀⣀⡤⠶⠚⠉
⠀⠀⠀⠈⢋⣀⣀⣀⣬⣧⣉⡷⠖⠒⠒⠓⢲⣤⡖⠉⠉⠁⠈⢦⢀⡴⠋⠀⠀⠀⢀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⣿⠉⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⡄⡟⣆⠀⠀⢀⡼⠃⠀⠀⣠⣴⠚⠉
⠀⠀⠀⠀⣿⠀⠀⠐⠀⠸⡄⠷⠀⠀⠀⠀⢸⡇⢱⠘⢆⡰⠋⠀⣠⠔⠋⠀⢸⠀⠀⠀⢀⡀
⠀⣀⣠⡤⠿⠒⢶⠒⢲⠚⠛⠋⠉⠉⠹⡗⢶⡻⠛⠒⠚⠳⢴⣏⣀⣀⣀⣠⠼⠤⢤⠀⠙⠃
⠀⢿⠀⠀⠀⠀⠘⡆⠀⢧⠀⠀⠀⠀⠀⣇⠘⡇⠀⠀⠀⠀⢸⠉⠀⠀⠀⢹⠀⠀⠘⡇
⣠⡼⠦⠤⠤⠤⠤⠿⠴⠼⠶⠶⠶⠦⠤⠾⠤⠷⠤⠤⠤⠤⠾⠤⠤⠤⠤⠼⠤⠤⠤⠷⠶⠶⢶⡄
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⢤⡤⠴⠂⠀⠀⠀⠀⠀⠐⠲⠦⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠂⠲⠥⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⣶⢤⣀
⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⠇⠀⠀⠀⠀⠀⠀⢀⡴⢃⣀⣠⡤⠶⣦⣤⡏⠀⠀⠀⠀⢸⠀⠈⡇
⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⢸⠁⠀⠀⠀⢸⠀⠉⢳⡀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢼⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠚⣻⠀⠀⠀⠀⢸⡄⠀⠄⠀⠸⡄⠀⢸⣇⣀⣀⣸⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠖⠚⠛⠁⠀⠀⠀⢠⣼⠓⠒⣶⣖⠚⠓⠒⣏⣹⠏⠉⠀⠉⣩⢤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣯⣤⣬⣷⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭
⣿⣿⣄⣭⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡾⠋⠉⠉⠉⠉⠻⣿⣿⣿⣇⠀⠀⠀⠀⣾⡟⠋⠉⠉⠉⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⢿⣿⠟⠛⠻⠿⣿⠀⠀⠀⠀⠀⠠⠴⠄⠠⠦⠤⠀⠸⣿⣿⣿⣧⠀⠀⠀⠻⣤⣤⠀⣤⣤⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠃⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠘⠃⠀⠘⠃⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣆⠀⠀⠀⠀⢸⣿⠿⠛⠛⠛⠛⠛⣿⣿⣷⠀⠀⠀⠀⢈⣿⡿⠟⠛⠛⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣯⣤⡄⠀⣤⣄⠀⢻⣿⣿⠀⠀⠀⠀⠈⢃⣀⣀⠀⣀⣀⠀⢻⣿⣿⠀⠀⠀⠀⠸⣿⠤⠴⠄⠠⠤⡤⠀⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⣀⣀⡴⠶⢶⣄⣀⣀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣀⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣦⣴⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠀⣿⣿⣿⣿⣿⢟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠛⠛⠋⠉⠉⠉⠙⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠒⢦⡀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠂⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠢⢀⣠⠞
⠀⠀⢀⣠⣤⣽⣦⣀⠀⠀⠀⠸⣧⠀⠀⠀⢀⡤⠤⣤⡀
⠀⡴⠋⠁⠀⠀⠀⠉⠓⠦⣄⠀⢻⣆⠀⠀⣿⠀⡀⢈⡇
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠻⣆⠀⠹⡄⢸⡋
⣧⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠈⠳⡹⣆⠀⠙⠾⣧
⠹⣆⠀⠀⠀⠰⣁⡀⠀⡯⠙⠳⢤⡀⠀⠘⢧⡀⠀⠀⡀
⠀⠈⠳⢤⡀⠀⠈⣀⡼⠃⠀⠀⠀⠙⢷⣌⠋⠳⣴⠋⡉⠳⣄
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠢⠽⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⢼⠷⣄⠙⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣴⠇⠀⡘⠀⠏⠳⣌⠻⡎⠳⣤⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⠀⠸⠀⠀⠃⠀⠀⠀⠀⢠⠟⠀⠘⡄⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡌⠀⡀⠀⠸⠀⠀⠀⠀⠀⠀⢢⠀⠀⠃⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢰⠃⠀⡄⠀⢀⠀⠀⠀⠨⡀⠀⠀⢃⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⢇⠀⠻⣄⣀⠈⠃⠀⣀⠁⠒⠄⣀⠐⠮⠁⠶⣠⡴⠗⠀⠀⠄⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⢰⠛⠓⡆⢤⠭⠖⠠⠞⣁⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢸⡀⠄⢳⣾⣄⡀⢀⡀⢈⣁⡄⠈⠉⠠⣄⠀⠀⠀⠀⡉⢀⣈⠁⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢉⡛⡱⠋⠀⣠⠗⠀⣿⠀⠉⠛⠧⣌⠉⠀⠀⠀⠘⠢⠈⠙⠲⣀⠀⠙⢗⡸⢿⠖⠦⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⠟⠁⠀⠀⠈⠳⠒⠃⠀⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠠⠀⠂⠁⢀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠑⠀⡀⠉⠡⣂⡀⠢⢀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠈⠢⡀⠀⠙⣂⡈⠀⣄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠈⠢⡀⠈⢁⡌⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠈⢦⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⣄⣷⠀⠀⠀⠀⠀⣴⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠉⠛⠛⢻⡄⠀⠀⠀⣼⣿⢿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⠿⠿⣿⣷⣤⠈⣧⠀⠀⠀⢿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⠿⠛⢺⣷⣤⣄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣀⣀⣀⣸⣟⠀⠉⢻⣶⣖⢛⣋⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢁⡠⠦⣼⡏⠳⣶⠋⣨⣿⣯⣷⣿⣿⣷⣦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣴⣿⣿⣿⣿⣿⣻⣻⠇⢠⣷⡇⣿⣿⣿⣿⣿⣿⣿⡎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⣿⣿⣿⣿⣿⣿⡏⣿⡀⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⡇⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣿⣿⣿⣿⣿⣿⡿⣹⣿⣧⠹⣿⣿⣷⣝⣛⣛⣫⣿⣿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡙⣿⣿⣭⣯⣭⣾⣿⡿⢋⣷⣌⠻⢿⣿⣿⣿⣿⠿⢡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⠻⢿⣿⣿⠿⢋⣠⠞⠁⠉⠓⠦⣬⣭⣭⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠚⠋⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣾⣶⣶⣤⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⣀⣴⣾⣿⡛⣏⢻⣟⣿⣯⡻⣿⣿⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡿⠉⡂⠤⠜⢛⡿⢿⣿⣿⣿⣿⣿⡒⠒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣤⠾⠻⠯⣷⣄⠀⠀⢠⠎⠔⠁⢰⣾⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠘⢿⣟⣻⠀⠈⣿⢀⢠⠃⣀⣰⡧⣀⣶⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠈⡟⠲⠶⠶⠊⢉⣿⣿⡄⠀⠀⠘⢿⣧⢷⡄⣸⣶⣽⣰⣿⣿⣿⣿⣷⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠘⡃⠀⠀⢀⠠⠀⠀⡀⠽⣩⡥⠀⠀⠀⠀⢹⠾⠥⠟⠿⣿⣶⣿⣿⣿⠟⠻⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢠⡇⠈⣹⡀⠀⢀⡠⠔⠀⡠⠀⠿⠾⢫⣤⠀⠀⠚⠡⠂⠀⢀⡼⠿⠛⠋⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣸⣿⢇⠀⢧⠀⠉⣧⠀⠄⠀⠄⠂⠀⢠⡄⡉⠋⠁⠀⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⡼⣸⣿⡆⣆⠈⢄⠀⠀⠳⡀⠁⢄⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠂⠀⠈⣿⡇⠀⠀⠘⠃
⠀⠀⠀⠀⠀⢸⣧⠎⠹⣾⣎⢢⡀⢂⠀⠂⠈⠢⢄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣞⠋⠈⢢⠹⢛⣧⣕⣄⠁⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⣿⣆⠀⠀⠁⢎⡹⣿⣷⣵⣤⣘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠿⠿⠷⠶⠾⠷⠿⠾⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠀⠀⠀⠀⠉⠀⠀⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠀⠀⠀⠉⠐⢳⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠰⠁⠀⠀⢈⠂⠀⠀⠀⠀⠴⠒⠀⠀⠀⠒⠀⢳⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠰⠆⠠⠀⠂⠒⠉⢸⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡠⢊⡠⠊⠁⠒⠤⠤⠤⠤⠤⠤⠤⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠆⠀⠀⠀⠀⠀⠐⠀⠠⠁⠀⡇⠀⠄⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠬⠙⡆⠀⠀⠀⠠⣤⣤⠤⠴⠋⡼⢄⡀⠀⡀⣀
⠀⠀⠀⠉⠉⠙⠁⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠰⠀⠆⢀⠏⠀⠈⠓⠄
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠕⡦⢀⠀⢀⣠⠆⠀⠀⢸⠀⠀⠀⡰⠈
⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡌⠀⣠⡈⠓⠢⢸⡀⠀⠰⠁⢱
⠀⠀⠌⠀⠀⠑⡄⠀⠀⠀⠜⠀⠀⠀⠀⠠⣀⠘⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⡸⠀⠀⠀⢦⡇⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠑⠢⢄⢰⡀⣀⠄⠀⣸
⠀⠁⠀⠀⠀⠤⣙⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢦⠀⠀⣀⣀⡴⠃⠀⢠⠃
⡄⠀⠀⠀⠀⢀⠘⠐⠂⠤⠀⠀⠀⠀⠀⠉⠁⢠⠀⠘⡀⠀⠀⠀⠀⢸⠀⢸
⢷⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⢀⡇⠀⢸
⠀⠳⡇⠀⠀⢸⡤⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⡾⠀⠀⢸
⠀⠀⠹⠄⠠⠒⠀⠐⡄⠀⠀⠈⠑⠢⠤⠤⠤⠜⠀⠦⡀⠀⠀⢠⠇⠤⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠁
⠀⣁⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⢀⠄⠀⠀⣀⠐⠈⠀⠀⠀⠀⠀⠀⠁⠒⣅⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣦⣤⣄⣠⣶⣶⣶⣄
⠀⠀⠀⠐⠁⠀⠀⠀⠨⠆⠀⠀⠀⢀⠀⠀⠸⠀⠀⠀⠌⢂⠀⠀⠀⢠⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⣿⡇
⠀⠀⠀⡇⠀⠀⠚⠀⠀⠀⠀⠀⠀⠜⠂⠒⠂⠒⠒⠀⠤⢈⡔⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡆⢹⣿⠃
⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⢄⢘⠀⠀⠀⠀⣿⣿⡏⣿⣿⣿⣿⣿⣿⠉⠁⢀⠀⠀⠀⠀⠐⢡⠘⠁
⠀⠀⠀⠀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⠘⠑⠀⠀⠀⠈⢙⠀⠀⠀⠀⠘⠿⣇⢿⣿⣿⣿⡿⠃⠀⠠⣿⡆⠀⠀⠘⠿⢾
⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⠀⠀⠀⠈⠀⠁⠀⠔⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢀⡀⠀⠀⢀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⡀⠀⠀⠀⠀⠐⠂⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⣈⠣⡀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣒⠖⠲⡒⠒⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡀⠀⠀⠀⠀⡈⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣶⣿⣦⡀⣱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⣿⣿⣿⣿⣿⠣⢌⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠒⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠘⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡌⢸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠈⠉⠁

⠀⠀⠀⠀⠀⠹⣿⣿⡀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⢀⣀⣠⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⢸⣿⣿⠀⠀⠀⠀⠀⢀⣴⠟⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⢸⣿⣿⣧⣤⣤⣴⠾⠋⠁⠀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⢀⣴⣾⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⣿⣿⣿⡋⠀⠀⣀⣴⣾⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣾⣿⠟⠉⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡶⠶⠖⠛⠛⠛⠉⠉⠉⣹⠟⠁⠀⠈
⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣀⣀⣀⠀⢀⣀⣴⠃
⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⢀⣾⣿⣿⡏⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠻⢿⣿⣿⡀
⣤⣀⡀⠀⢹⣿⣶⣤⣤⣶⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⣤⣀
⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣾⣿⣿⣿⡿⠛⠉⠁⠀⠈⠙⠻⣿⣷
⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣀⡀⠈⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⡿⠋⠉⠉⠛⠛⠛⠿⠿⢿⣿⣷
⠀⠀⠀⠀⢀⣴⠿⠋⣰⣿⠟⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⣠⡾⠛⠁⣠⣾⠟⠁⠀⠀⢰⣿⣿⠟⠋⠉⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣄
⣴⠟⠉⠀⢠⣾⡟⠁⠀⠀⠀⢠⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⣰⣿⠟⠁⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄
⣾⡿⠋⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⣄

⠀⠀⠀⢀⡤⠒⠚⠉⠉⠙⠒⠲⢤⡀
⠀⢀⡜⠁⠀⠀⠀⢀⡤⢤⡀⠀⠐⠈⢣⡀
⠀⢸⠀⠀⠀⠁⣰⠃⠀⠀⠙⢦⠀⠈⠆⢻⡀
⠀⢸⠀⠀⠓⠒⠉⠙⡄⠀⠀⠘⡆⠀⠀⠀⢧
⠀⠘⢆⠀⠀⠀⠀⣀⠟⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠈⠙⠊⠉⠉⠀⠀⠀⠀⢀⡇⠀⡀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⡂⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⡠⠖⠋⠓⢦⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢱⠀⠀⠀⠈⠧⠐⠒⠒⠉⠋⠀⠈⠉⠉⢙⠒⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠫⠭⠍⠉⠀⠀⢀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠀⠀⠀⠀⢀⠀⠈⠈⠓⢤
⠀⠀⠀⠠⠖⠒⠊⠀⠀⠁⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⠀⠀⠀⠱⣄
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠔⠣
⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠳⡀
⠀⠀⣀⠞⠀⠀⠀⠠⠠⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡
⠀⠀⠣⠶⣈⣩⡅⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⢤⡚⢢⡀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⡖⠚⢀⠀⠀⠙⢯⠁⠒⠒⠒⠒⠒⠒⠒⠹⠁⠀⠱⠀⢀⠼⠀⠙⣆⠀⠀⠘⡆⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⣀⣤⡴⢥⣤⠀⠀⣔⠁⠀⠀⢀⢸⡆⠀⠃⠘⣉⣁⣀⣀⣀⣀⠈⠑⠀⠄⠠⣄⣀
⡠⣀⣠⠖⠉⠒⠤⠤⠔⠀⠂⠤⠇⠄⠈⠀⠠⠞⠒⠒⠒⠒⠶⠶⠒⠋⢀⣠⣴⣆⠀⢀⠀⡥⢀⣈⣉⡡⠤⠻⠀⠀⠠⠇⠀⠀⠀⠀⠀⠉⠒⠤⣀⣀⣀⡀⠉
⢧⠄⠒⠒⠉⠁⠒⠒⠒⠀⠒⠀⢖⣦⣦⠤⠜⠈⠁⠂⠉⠑⢲⠄⡠⠚⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡤⢂⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡴⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⡰⠉⢀⣀⡀⠈⠄⠀⠀⠀⠀⠘⠀⢀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢰⠁⢀⠔⢀⡆⠀⠀⠀⢀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⡄⠈⠉⠁⠀⠀⢀⡀⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠈⠑⠂⠀⠀⠂⠁⠀⢀⢠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣄⣄⣀⣀⡦⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠢⢀⠀⡁⠀⡀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣻⣿⢷⡤⣀⠀⠀⠀⠀⠀⠐⢄⡀⢠⣹⣥⣔⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⡶⣦⢳⣱⣿⠿⠿⢯⡀⠀⠀⠀⠀⠀⠁⠀⠀⠏⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡳⣿⣮⡷⠉⠁⠐⢆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⡃⣿⢿⣿⡿⣃⡈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢀⡜⠀⡄⠐⢇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠁⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣄⢠⣤⡄⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣠⣄⠀⣭⡈⠀⣰⡈⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠚⠛⠃
⠀⠀⠀⢀⣀⠀⢈⣀⠁⢉⡀⠉⣁⡉⢉⣩⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢄⡀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠶⠚⠉⠁
⠀⠀⢠⡋⠉⠁⠁⢉⡀⠉⣹⠚⠉⢓⠋⢉⡋⠁⠀⠀⠀⣠⣼⠾⣏⠙⣧⠈⣧⢈⣠⠤⠔⠚⠉⠉
⠀⠀⠈⠇⠙⠈⡃⠉⠛⠐⠊⠑⠘⠛⠃⢘⠋⠀⠀⠀⠀⡏⢹⡆⢹⣆⢽⣶⣾⡿⠃
⠀⠀⠀⠘⠃⡈⠛⠆⠛⠛⠀⠞⠛⢆⠘⠀⠶⠀⠀⠀⠀⠸⣿⣿⣷⣿⣾⣿⠛⠁
⠀⠀⠀⠈⠳⠶⡄⠜⠶⠤⠋⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⠨⡝⠃⠀⣚⠠⠌⢹⡻⠦⣥⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⡀⠈⢘⠷⢤⣀⣈⣵⡒⠜⢃⡴⠋⣛⣻⣿⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡉⠀⠀⠈⠐⠯⢁⠵⠔⣩⡿⠙⠈⠉⠒⢊
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⢶⣴⠀⢳⢉⣶⢻⡇⢠⡴⠖⠚⠩
⠀⠀⠀⠀⠀⠀⠀⠶⣤⣀⣀⠀⠀⠐⢀⣺⣖⢂⣸⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⣦⡙⠛⠟⠀⢸⡏⠀⠀⣠⢤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢉⠛⠳⠛⠛⢻⡯⡹⠁⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡤⡼⠧⠤⠞⠛⠉⡇
⠀⠀⠀⠀⠀⠤⠀⠐⣷⠀⠁⠀⠒⢄⢚⡄⣿⡝⠯⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣤⣄⠠⢀⡀⠀⠀⠀⠀⠄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠠⠝⠻⣦⡵⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡛⠊⠿⢶⣥⠴⡧⠤⠆⠂⠈⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠓⠀⠀⡠⠀⠳⡟⠃⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠅⠀⠀⠈⣀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣦⡇⢀⢓⣶⣾⠿⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⡤⠀⡶⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣽⣿⣿⣷⣿⣶⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⠀⢀⣀⠀⢀⡀⠀⢀⣀
⠀⠠⠁⠀⠲⡄⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠈⠉⠀⢫⡀⠀⢀⡈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫
⢀⠀⠀⠀⢀⠀⠉⠀⠂⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙
⡎⡀⠀⢀⠂⡀⠀⠀⠀⡌⠀⢸⡀⢠⠤⡴⡂⠉⠀⠀⠂⠀⠒⠀⢒⣀⣾⣷⡶⢠⣐⡀⠀⠀⠀⠀⠀⠂⠐⢻⡂⠀⠀⠀⠀⢧⠙⡄
⢠⠁⠠⠃⢠⠁⠀⠀⢀⠆⠀⠈⠀⣼⢀⠀⢉⠄⣀⠀⠀⣀⡀⣺⡟⡉⢡⢔⠀⠀⠈⡕⠒⢤⡆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠐
⠇⡰⠃⢠⡇⠀⠊⡁⠀⠀⠀⠀⠀⡄⠘⢀⢸⠂⣿⠴⢋⢹⢭⠉⠉⣡⠱⠛⠓⢤⣤⣄⠠⣠⣶⠀⠃⠀⠀⣴⡈⡀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢡⢀
⠐⠋⣥⡄⠀⠰⠍⠠⢄⠠⠄⠠⢈⣧⠟⣬⡇⠨⠄⠠⢘⣼⠖⣏⠚⣧⡟⢿⡉⢸⡆⡿⠒⢘⣿⣄⠀⡴⠀⠄⡡⠶⠀⠔⠀⡐⠀⠀⠀⠀⠀⡀
⠢⢄⠈⠑⠢⣄⠀⠑⠠⡐⠀⠑⢼⠉⡖⢦⢄⡑⠦⣀⢸⣛⠲⠦⣌⣧⠥⡤⠦⢤⠭⡿⠆⠹⣿⣧⣾⡢⠄⢠⠀⠀⢂⢀⣌⠄⠊⡀⠄⠂⡀
⢸⠀⠹⢢⣄⡀⠙⠢⣀⠈⠐⠄⡆⠀⡇⠸⠨⠘⡑⠠⢙⣎⣂⣁⣚⣛⡛⣶⣯⡴⣊⢀⡤⠤⣭⠿⡏⠀⠰⠈⡄⡈⠌⢋⡠⠐⢀⠠⠂⠁⣀⠴⠞
⠇⢀⠇⡸⢘⡏⢰⠤⠀⠙⠢⢀⠃⢀⠇⡆⠈⠦⠥⠀⠀⠉⠁⠦⠄⣀⠀⠀⠀⠞⣁⣩⠜⠋⠀⠀⠀⠀⠀⠀⠉⢡⠈⢌⡄⠊⢀⣠⠔⠊⢅⠘⡄
⠀⡼⢠⠇⢸⠁⢸⡂⠀⠁⠂⣼⣄⠈⠠⠄⠀⠀⠔⠀⠀⢄⠀⠠⠀⠈⠙⠒⠶⣶⠿⠀⢀⣤⣄⣠⡄⢀⣠⡄⢂⡔⠀⣪⡼⡖⠉⠀⠀⠀⠈⢢⠐
⠀⣇⣞⠀⠛⠀⠈⠁⠐⠂⠈⠉⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⣴⡿⢶⠐⢂⡁⠲⠀⠀⠀⠀⠀⣠⡴⠚⠉⠒⠒⠶⠶⠶⠶⠶⠶⠷
⣸⣿⣟⣉⣙⣑⣀⡒⠒⠂⠒⠒⠖⠖⠀⠈⠳⢦⣄⠀⠀⠀⠀⠀⠠⢀⠑⠦⠖⣊⠴⠊⠁⠀⠀⠀⢀⣠⡖⠛⠁
⣍⣁⣀⣀⣀⣉⣉⣉⣉⣙⣛⣛⠛⠒⠒⠒⠒⠖⠚⠻⠦⣄⡀⠁⠂⢀⠉⠒⠋⣀⡤⠊⠁⣀⣴⠾⠉⢁⣀⣀⣀⣀⣀⣀⡀⠀⠈⠁
⣿⣽⣯⣭⣭⣭⣭⣭⣭⣭⣙⣛⣛⣛⣛⡛⠛⠛⠓⠶⠶⠾⠿⢶⣄⡀⠑⠲⠟⠁⢀⣴⠾⠯⠤⠤⠤⠤⠤⠤⠄⠀⠠⠤⠤⠤⠤⠤⠄⠤
⣿⣭⣽⣯⣭⣯⣭⣭⣭⣩⣍⣩⣉⣉⣉⣛⣛⣛⣛⣛⣓⣶⣶⣶⣮⣿⣶⣤⣤⣾⡛⡒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠄
⣿⣯⡛⣿⣯⣝⣯⣿⢿⣭⡋⠩⠻⣿⣍⠉⠙⠻⣯⣍⡉⠛⢿⣍⠉⠉⠉⠙⠛⠉⠉⠉⣉⡿⠛⠉⠙⣻⡿⠛⠛⠛⣓⡿⠛⠒⢒⡶⠛⠒⠒⣲⡶
⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣖⢹⣿⢷⣌⡠⠐⠉⠻⣶⣄⡙⢷⣄⠀⠀⠀⢀⣤⠾⠋⠀⣀⣴⠟⠉⠀⢀⣤⠞⠉⠀⣠⠶⠋⠀⣀⡴⠛⠁
⣿⣿⣿⣽⣿⣿⣾⣿⣿⣿⣽⡯⣿⢷⣧⡶⣉⠻⣦⣔⠉⡈⠙⢿⣦⡙⢷⣶⡾⠛⠁⢀⣤⠞⠋⠀⢀⣤⠾⠋⠁⣀⣴⠟⠁⢀⣤⠞⠉⠀⠀⣠⠴
⣿⣻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⡻⣿⣿⣦⣗⡬⣻⢷⣤⡀⠀⠙⢻⣦⡉⠀⣀⣴⠟⠁⢀⣤⠾⠋⠁⣠⡴⠟⠉⢀⣠⡶⠋⠁⠀⣠⠴⠋⢁⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣮⣿⣷⣤⡷⣝⡿⣦⣄⠀⡈⢿⣿⡟⠁⢀⣴⠟⠁⣀⣴⠟⠉⠀⢀⣴⠟⠁⠀⢀⡴⠋⠁⣠⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⡄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⡀
⠀⠀⠀⠉⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠢⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠑⠄⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠂⠀⠀⠁⠀⠘⠐⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑
⠖⢺⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠠⢀⠆⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠐⡁
⠀⠐⠅⠀⡄⠐⠈⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⡍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈⠉⣁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠈⠁⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠀⢄⠔⠒⠢⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠈⠉⠂⢀⠀⠀⠈⠂⠀⣀⡀⣀⠤⠄⠊⠀⠀⠀⠀⡀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄
⣤⡒⠁⠁⠀⠈⠈⠝⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠚⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡎
⠑⠈⠦⣄⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠠⠁⠀⠈⠀⠀⡤⠄⠚⢡
⠀⠀⠀⠈⠢⡀⠉⡀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠈⠁⠙⡄⠀⠀⠙⠄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠄⠠⠀⠰⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠈⠐⢲⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠁⠒⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⢄⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠁⠺⠀⠀⠱⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠑⢄⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⣀⣄⠐⠂⠀⠠⢄⠀⢤⠃⠀⠀⠀⠀⠠⠄⠒⠂⠒⡠⠴⢤⡀⠀⠀⠂⠀⠀⠁⠢⡀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⢏⠙⢳⣻⠁⠀⠀⠀⠀⠀⠀⣀⡘⠀⠀⠀⠀⠁⠀⠀⠀⡠⢡⡀⠤⢈⠀⠀⠀⠰⡀⠀⠀⠈⠑⠠⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡋⠀⠄⡀⢐⣠⢜⠤⠉⠀⠀⠀⢄⠀⢠⡄⠀⠀⠁⠀⣀⣀⠚⠁⠀⠀⠀⠈⠐⠢⢀⡀⠀⠀⠈⢛⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠍⢐⡀⠀⠀⢄⠀⠈⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠁⠁⠀⠄⢛⣒⣐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⢀⢀⡀⠀⠀⠘⠁
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣏⣁⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⠷⢶⣄⠀⠀⢀⠀⢹⢿⣿⢇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⠀⠉⠙⠳⠶⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⣀⣬⡷⠾⡋⠁⠟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣠⠀⠀⠀⢤⠀⠀⠀⠉⠛⠒⠶⠚⠲⠶⢶⢲⣾⣺⢗⡪⣽⡿⠚⡂⠀⢀⣴⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡀⠉⠛⠛⠻⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠹⠻⠻⠁⠐⡿⠙⠙⠁⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣀⡠⣄⡀⠈⠛⠻⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠻⣂⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢟⠛⠉⠛⠷⣷⡤⣤⣄⡀⠀⠀⠀⣀⣴⣖⣳⠤⠀⠀⠀⠀⠐⠷⣶⣢⣤⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠉⠛⢿⠛⠛⣛⠷⠶⠿⠿⠟⠛⠛⠿⣟⣻⡿⠛⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡛⠁⠀⠀⠀⠀⢋⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠣⡈⠿⠛⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡖⠄⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠸⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡐⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⠙⠓⣦⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡭⣽⣯⡍⠁⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⡶⠞⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠩⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡍⠑⠿⡷⡀
⠀⠀⠀⠀⠀⠀⠸⣵⣀⣀⣤⣦⡄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣌⣻⣷⣶⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠧⡀⢠⣄⣀⣀⣀⣀⣀⣀⣠⠈⠉⠬⣅⣀⣠⣤⣠⣤⣤⣠⣼⡿⠿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠁⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⣄⣀⣈⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⠉⠹⣄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠚⠃⠉⠉⠉⠉⠙⠳⢆⣤⡠⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⡄⠀⡦⢘⡇⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⡩⠒⠙⢦⡀⠘⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⡇⠀⡷⣾⠀⠀⠀⠀⢠⠃⢀⣤⠀⠀⠀⠀⠠⠤⣄⠀⠀⠀⣎⣀⣤⡄⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⡀⠀⠀⢀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢄⠀⢧⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡇⠈⡇⠀⠀⢸⠀⠀⠀⢰⣶⡀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠁⠀⢻⡀⠀⢸⠀⠀⠀⠸⠿⠃⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣧⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠍⠁⠀⠈⣧⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠂⠀⠀⠀⠀⠀⢠⠃⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠙⣦⠀⠀⠀⣀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣔⡒⢶⣶⣤⡄⢹⡄⠀⠈⠓⢦⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⣶⣿⣿⣿⡶⠚⣷⡗⠂⣄⠀⠀⠀⠀⠈⠙⠲⠦⢤⠤⠤⠤⠶⠚⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠜⠛⠿⠿⠿⠧⠤⠿⠛⠀⠈⢧⡀⠶⠶⠶⠶⠶⠶⣾⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡧⠀⠀⠀⠀⠀⠀⠗⠀⠈⠛⠲⠤⣄⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⡛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⣤⡀⠀⣠⡀⣠⣄⠀⢠⣄⢀⣤⣤⣄⠀⣤⣄⠀⣤⡄⠀⣀⡀⠀⠀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠉⣿⠆⣿⡇⠀⣿⡇⣿⣿⠀⢸⣿⣾⣿⢙⣿⡂⣿⡏⣸⣿⠁⢸⣿⡇⠀⠀⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣇⣀⣿⡇⣿⣿⠀⣤⣥⢿⣿⡀⠉⢠⣿⣿⣿⡇⠀⢸⣿⡇⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡿⢿⣿⡇⣿⣿⠀⢼⣿⢸⣟⠉⣤⡀⣿⣿⢻⣿⠀⢈⣿⡇⠀⠀⣿⠟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣠⡀⣿⡇⠀⣿⡇⣿⣿⠀⣼⣿⢼⣿⣶⣿⡇⣿⣧⠘⣿⣆⠸⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣴⣿⠃⣿⣷⣶⣿⡇⢹⣿⣤⣿⡟⢻⣿⣿⡿⠡⡿⠿⠀⢻⣿⣼⣿⣷⣦⡀⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠄⢛⡻⣿⡛⠋⠁⠉⠙⠁⢀⣾⣿⣷⠀⠀⠖⠀⠐⢚⣿⡿⠯⠉⣭⠀⣉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡾⠐⣿⣿⣦⡀⠀⠀⢠⣾⠟⢻⣿⣷⡀⠀⢀⣤⣿⣿⠃⠀⠀⠀⠀⣤⣦⡀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠃⢠⣿⣿⣿⣿⣷⣶⣿⣇⠀⣿⣿⣿⣿⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠙⢻⣿⣧⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⣿⡆⢸⢿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠈⠛⠛⠛⣿⣿⣿⣿⣿⣿⡿⠏⢹⣿⣿⣿⣿⣿⣿⣿⠟⠙⠛⠃⠀⣠⣿⣿⡇⠛⠰⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣰⣿⣿⣿⣷⣶⣿⣃⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⡀⠀⢠⣿⣿⣿⣿⠟⣡⣼⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣻⣿⣿⣿⣿⣿⣿⡷⣿⠿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣼⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣙⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣯⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣼⣿⢿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠶⡶⠶⠶⠶⠶⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣿⣿⣿⠿⠟⢋⣩⣍⣁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣬⣭⣉⠛⠻⢿⣿⣿⣿⡜⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠇⠀⠀⣉⣠⣬⣙⣿⣿⣦⠀⠀⠀⠀⣴⣿⢏⣡⣤⣤⣄⣉⠀⠀⢻⣿⣿⡿⠛⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣼⡟⣿⡟⠀⠀⠘⠻⠯⣠⠈⠙⠿⠛⠀⠀⠀⠀⠉⠑⠋⠁⢀⡤⠟⠛⠀⠀⠈⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣷⣿⠃⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⠀⢸⣿⡇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣿⣿⡄⣠⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⣿⣿⣷⡀⢨⣿⣇⡿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢸⣿⣿⣿⡿⣿⣿⣄⠀⠀⠀⠀⠰⠾⣶⣶⡶⠷⠀⠀⠀⠀⢀⣼⣿⡿⣿⣿⣿⣿⣿⡷⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡅⢹⣿⣿⣷⣤⣄⣀⣀⡀⠀⠀⢀⣀⣀⣤⣤⣾⣿⣿⣿⠃⢸⣿⣿⣿⢹⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢸⣿⣿⡗⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⡼⣹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣾⣿⣿⠲⠀⠀⠹⢿⣷⣾⣿⡛⠀⠁⠀⠈⢘⣹⣿⣾⣿⡿⠁⡀⠀⠷⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠉⠀⠇⣿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⡿⠦⠀⣀⣀⠀⠀⣘⣟⠹⠟⢻⡿⠟⣻⣋⣻⠀⠀⢀⣠⣤⣐⢷⣿⣿⣷⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⢻⣷⣿⣿⡄⠀⣿⡿⢰⣿⠿⣷⡌⣿⠟⢿⣿⠀⣿⡟⠻⣿⡎⣾⡷⢾⡄⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡧⣿⣾⣟⣿⣿⣿⣀⣿⡏⢺⣿⠀⣿⠇⣿⡄⢠⣿⣠⣿⡇⠀⣿⣧⣿⣷⣼⢳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣤⣶⡄⣿⡃⠀⣿⣿⣿⡇⠀⣿⣿⣿⣏⣿⣼⡟⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢾⣿⡟⢙⣿⡉⢹⣿⣿⣿⠙⢻⣿⣿⣿⣿⣽⣿⣇⣿⡀⠀⣿⢿⣿⡇⠀⣿⡇⣽⡇⣿⣿⠛⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⣿⡇⠀⢿⣧⢸⣿⣿⣿⠀⢻⣿⡿⢿⣿⣻⣿⣿⣿⣦⣼⣿⠀⢿⣿⣾⡿⠃⣾⡇⢸⡿⠀⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠛⠀⠀⠛⠃⠀⠘⢻⣿⡟⠛⠛⠛⠁⠀⠀⠈⠁⠀⠀⠻⠃⠘⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠒⠈⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠉⠀⠀⠀⠀⠀⠂⡀⠀⠀⠈⠐⣄
⠀⠀⠀⠀⠀⡀⠄⠂⠁⠠⠀⢀⠀⠀⠀⡐⠁⠀⣁⠤⠔⠒⠈⠉⠀⠈⠢⢄
⠀⢀⠠⠂⠁⠀⠀⠀⠀⠐⠂⠀⠀⣀⠤⠐⠊⠁⠀⠄⠀⠀⠀⠃⡀⠀⠀⠀⠑⣠⡀
⠀⠘⠂⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⢠⠐⠀⠀⠈⠀⢀⡀⠤⠐⠂⠉⠁⠀⠈⠒⢄
⠀⡆⠀⠀⠈⠀⣀⠀⠊⠁⠀⠀⠀⠀⠀⠑⠀⢀⣡⠴⠒⠉⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠁⣀
⠘⠀⠑⢅⠀⠁⡈⠁⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠐⠀⠠⠐⠀⠂⠈⠀⣀⡠⠤⠐⠊⠉⠁⠀⠈⠓⢄
⣅⠀⠀⠀⠑⣲⣇⠀⠀⠀⢀⣤⠔⠉⠀⠀⠀⠀⠀⠀⠁⠀⢀⡠⠴⠚⠉⠀⠀⠀⠀⡀⠀⠁⠄⠀⠀⠀⠈
⠈⠢⢔⠀⠊⠈⠁⠑⢄⡀⠁⠈⠁⠀⢀⠀⠀⠀⢀⣠⠴⠊⠁⠀⠐⠀⠠⠐⠀⠀⠌⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠑⠠⢦⠐⠀⠀⠙⢲⣇⠀⠀⠀⡁⢤⠖⠉⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠹
⠀⠀⠀⠀⠀⠀⠉⠪⣀⠀⠀⠁⠙⠢⡀⠀⡌⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠉⠀⡀⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⣀⢀⠀⢈⠴⣧⡀⡀⠀⡁⠤⠀⠀⠀⠀⡠⠄⠀⠁⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⢠⠀⠈⠢⡀⠀⠀⡀⠄⠂⠁⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡁⡀⠈⠒⠋⠀⠀⡀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⢀⡤⠐⠉
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣴⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢿⣿⡿⠟⠉⢀⣠⣴⣶⣾⣿⣿⡿⠋⠉⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⢀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠟⢉⣤⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠸⣿⣿⡏⠀⣴⣿⣿⣦⣤⣼⣿⣿⡿⠿⠛⠁⣶⣿⣿⣿⣿⡿⠿⢿⡿⠉⢻⣿⡄
⠀⠀⠀⠀⠀⢈⣿⣿⡶⠊⠻⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠈⠉⠉⣀⣠⣤⣤⣤⣤⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⡿⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣿⣿⡿⢿⣷⣦⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⠿⣿⣦⣄⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡀
⠀⠀⠀⣿⣿⣿⣤⡤⠈⠉⣠⡀⠀⠀⢀⣍⠉⠉⠀⠀⠀⣤⣄⠙⠛⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⢰⣿⣿⣿⣿⠶⣿⣷⣿⣧⠀⢀⣾⠟⣠⣾⣿⣿⣿⣿⣿⠃⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠘⣿⣿⣿⡿⠶⠿⠋⣸⡿⠀⠈⠋⠀⠿⠃⠐⠿⠿⠟⠀⠀⠀⠀⠙⠋⠉⣿⠟⠛⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢹⣿⣿⠁⠀⠀⢰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢰⡇⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⡏⠀⠀⠀⣿⣷⡀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠘⣿⣧⠀⠀⠀⠙⠛⢛⣈⣉⣉⣀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠙⣿⣧⠀⠿⣿⡿⠟⠛⠛⠿⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠘⣿⣧⠀⠘⣷⡶⠶⠶⠒⣿⣿⠃⠉⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠻⣿⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⣴⢶⡦⠀⠀⠀⠀⡀⡀⣄⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⢰⣌⠉⠉⢹⡥⠄
⠀⠀⠀⢀⢴⡿⠻⣆⡄⠀⠀⠀⠐⣿⣾⠗⠀⠀⢠⣿⣷⣼⣲⣶⣴⣶⣆⠀⠀⠀⠀⠰⠀⠀⠠⠞⠛⠒⠂⠚⠓
⠀⠀⠀⠀⠆⢸⣼⠙⠆⠀⠀⠠⠀⡧⣼⠹⡀⠀⠐⠛⠛⠉⡟⠀⠈⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⠛⠒⠞⠋
⠀⢀⠄⠉⣠⣀⠔⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⢀⣈⣼⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠘⠀⠀⠀⠈⠀⠀⢸⠀⠤⠤⠄⠀⠀⠀⠠⢤⡿⡿⣿⡿⣧⣀
⠀⠀⠛⠿⢾⣿⣿⠮⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠊
⠀⠀⠀⠀⣤⡀⠀⠀⣾⣿⣿⡇⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⣀⠘⢿⣧⡀⠙⠛⠛⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⣀⡵⠃⠀⠀⠀⠈⠁
⠀⡚⠛⢛⣷⡘⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠛⢦⡀
⠀⠀⠀⠀⠀⢀⣁⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡄⠀⢰⡄⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠸⣷
⠀⠀⠀⣀⣠⣏⣉⡉⣙⣀⠀⠀⠀⠀⠀⠈⠁⠀⢘⠈⠁⠀⠀⠀⠈⢸⠧⠆⡄⠇⠀⠀⠀⠀⢸⣅⡀⠀⠀⠀⣂⣿
⠀⠀⢈⡉⠙⠉⠉⢛⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⢀⣃⢀⡀⠀⠀⣸⣿⣶⡀⢐⣲⣯⣿⡇
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠁⠀⠀⠸⠃⠃⠻⠛⠼⠇⠀⠀⠿⠿⢿⣿⣿⣿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠤⡤⢤⠤⡤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⠀⠃⠘⠀⠃⠘⠀⠃⠘⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠉⠉⠋⠙⠉⠉⠉⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠤⠒⣤⣤⣒⣒⣦⣤⡒⣲⡇⠀⢀⣀⣀⡀⠤⠤⠐⢒⣒⣒⡒⠒⢊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢆⠀⠒⠊⠉⠉⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⡴⣂⠉⠉⠒⣤⠔⠒⠒⣲⢆⠀⠀⠀⠀⠣⡀⠀⠀⣀⡀⣀⢀⡄⢸⣾⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢰⠀⠀⠙⠎⣿⣁⣀⣠⠞⣁⠈⢇⠀⡀⠀⠙⠁⠓⢃⠛⡏⠛⠉⠀⢸⣿⠀⠀⢸⠀⠀⠧⣀⠀⡀⣤⣀⣶⢰⠀⢸
⠀⠀⠀⠀⠓⡤⠜⠋⠉⠒⡢⡄⢣⠀⢁⠜⠀⣇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣘⡏⠀⠀⢰⠀⠀⠊⠁⠓⠃⣉⡇⠉⠀⠀⢸
⠀⠀⠀⠀⢸⠁⠒⠲⡔⢉⠀⢸⠤⠷⠋⠀⠀⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⣄⠀⠀⢃⠀⢁⡜⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⡇⠀⣀⣀⣀⣀⣀⣀⠀⠠⠄⠀⡇
⠀⠀⠀⠀⠀⠈⠉⠒⠸⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⡇⡠⠤⠤⠤⠄⠒⠒⠒⠈⠉⠓⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡚⢦⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡰⠚⠙⠒⠐⢦⡀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⢀⠀⠑⡄⢣
⠀⠀⠀⠀⠀⠀⣰⣋⣀⣀⣀⣀⣀⣀⣹⣄⠀⠀⣠⠔⡋⠀⠀⢰⡜⠋⠀⣀⠜⠚
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡝⡅⠀⠀⠇⠁⢀⡪⠔⠊⠁
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠈⠀⠀⠀⠀⡇⠀⠳⡘⢆⡠⠴⠒⠁⠀⠀⡀⠴⠔⠒⠢⣀⠀⠀⠀⠀⢀⣀⣀⠀⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⠀⢀⡀⢀⡰⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸⡙⠒⠒⠒⠒⠚⠃⢐⣀⠭⠤⠤⠀⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠤⠔⠒⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⣄⠀⠀⠀⠀⠀⣠⡖⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠁⠉⠉⠉⠉⠉⠊⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢽⣄⠀⠤⠄⠀⣤⡀⠀⠀⡼⠑⠒⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢍⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⣳⡄⢆⣰⣆⡠⠀⣒⡾⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡌⢢⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⣠⠓⢤⣀⣀⣠⠴⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢱⡀
⠀⠀⠀⠫⡉⠈⠀⠀⠀⠀⢹⡀⠀⠈⠉⠀⠀⠀⠀⠀⡔⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⣠⣷⡀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠈⡇⠀⠀⠀⣞⠒⠒⠛⠒⠒⠲⠤⠤⠤⠈⢧⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠳⣄
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠻⠦⠴⠒⠒⠒⠒⠒⠒⠒⠒⠮⠷⣄⡀⠀⠉⠒⠦⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠄⠐⠒⠒⠀⠠⠤⠤⠤⠤⠄⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠔⠒⠊⠉⠉⠉⠉⠒⠒⠲⠤⠤⠤⠖⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⡞⠁⠽⠿⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⣀⣠⠴⠛⠛⠳⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣵⣋⣉⡝⠁⠀⠀⠀⣸⣥⣤⡧⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⢀⣠⣾⠿⠛⢉⣤⣤⡤⡔⠛⠓⠒⠓⠶⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠉⠀⣀⠴⠛⠁⢀⣠⠞⣿⣚⣯⡀⠧⡀⠀⠀⠀⢀⡧⣤⡼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⢀⡤⠖⠋⠁⢀⣠⠖⡏⠀⠀⡇⡇⠀⠀⠀⢸⣄⣀⣀⣸⣇⠘⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⢀⣠⡴⠚⠉⠀⢀⡤⡞⠉⠉⠉⢱⣤⣴⣇⡟⠉⠉⠙⠛⣆⠀⠀⠀⢸⣉⡩⠭⠛⠛⡶⢤⡀
⠀⠀⠀⠀⠀⡠⠞⠉⣀⡤⠖⠋⠁⠀⣀⡴⢚⡇⣠⠦⣄⣠⠤⠾⣭⣤⣸⠀⠀⠀⠀⡏⠉⠉⠉⠉⢹⡿⠤⠤⣄⠀⡇⠀⠙⣦
⠀⠀⠀⠀⣮⣤⠴⠚⠉⠀⠀⣀⠴⠊⢷⡀⠈⢷⠃⠀⠀⠈⠉⢲⠀⠈⠉⠉⠙⠛⡻⣇⣀⣀⣀⣀⣸⠛⠛⠓⠚⢳⠟⠀⢀⡟
⠀⠀⠀⠀⠘⣆⠀⠀⣠⠴⠋⠁⠀⠀⠀⠙⠶⣇⣀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠋⠀⠉⠉⠉⠉⡏⠀⠀⠀⢀⣸⡤⠞⠋
⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣶⣧⣤⣤⣤⣄⣀⣸⣇⣤⣤⣤⣤⣤⣿⣶⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠚⠚⠛⠛⠛⠚⠛⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⣻⣷⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣙⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣿⣿⢿⣿⠾⠟⠋⢿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣶⣶⣙⣻⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣉⠿⠟⣛⣯⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣯⣿⣿⣿⣿⡿⢿⠿⣿⣿⢿⣉⢛⠻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢹⡆⣿⣿⣷⣾⣿⣿⣻⣿⠿⣿⣿⣷⣯⡴⠀⢠⡽⣿⣿⣿⣦⣹⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⣽⣿⣿⣿⣿⣿⣿⡿⠙⣩⠷⠀⠉⣽⣿⡏⠙⢿⣿⣿⡟⠁⠀⣿⣿⡈⠙⠷⣾⡉⠀⠀⠉⢿⣿⣷⣾⣿⣯⡀
⠀⠀⠀⠀⠀⣿⣿⠿⠛⠛⠋⢙⣋⣉⣠⣤⢁⣀⣀⣀⡛⠿⠿⣄⠀⣿⣿⡀⠀⠀⢛⣋⣡⣤⣤⡤⠤⢤⣤⣄⣀⠉⠉⣙⣻⣿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣷⢸⣿⣿⣿⣿⠋⣰⣿⣿⣿⣿⣿⣿⣦⣄⣀⣉⣉⣀⡠⠞⣡⣾⣿⣿⣿⣿⡀⠀⣰⣿⣿⣿⡜⣿⣿⣽⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⡇⢼⣿⣿⣿⠿⠿⣿⣿⣿⣿⡟⢻⣿⣧⣿⣿⣿⣿⣿⡇⢸⣿⣯⣀⣼⣯⢻⡟⠀⢸⣿⣿⠟⠃⣸⣿⣿⠟
⠀⠀⠀⠀⠀⠸⣿⣿⡇⢸⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣼⣿⣿⣿⣿⣯⡾⠃⠀⣰⣿⣿⡇⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡸⣿⣿⣿⣂⠀⠘⠿⣿⣿⡿⠿⠛⣩⣟⠁⠀⠀⠄⠹⣧⠉⠉⠛⠋⠉⠁⠀⣠⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣏⡹⣿⣿⣶⣤⡄⠀⠀⠀⣠⡼⠻⣇⣤⡄⣀⢠⣤⡼⠷⣄⡀⠀⠀⠀⢾⣿⣿⠿⣡⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡷⠹⣾⣭⣭⣤⡤⠶⠒⠋⠁⠀⠀⠀⠙⢧⣿⣼⠃⠀⠀⠀⠙⠛⢶⣶⣾⣷⣶⣋⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⢿⣿⣿⣿⡆⠀⠒⠀⠂⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢀⠠⠍⣠⣿⣿⣿⡿⣟⣾⣿⣾⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣷⣂⣱⡶⠊⠁⠀⠀⢠⣿⣧⡀⠀⠀⠀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⣿⣿⣿⡿⠿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣷⣄⠁⠈⠙⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠻⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣿⣿⣷⣦⣤⣤⣴⣦⣤⣶⣿⣿⣿⣿⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⠛⣿⣿⡿⠛⠿⠋⢈⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠀⢸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣄⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠾⠿⠏⠁⠂⠀⡀⠁⠀⠀⠳⠔⠠⢀⠀⠀⣴⣿⠿⡿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠡⠆⠀⡄⠀⠀⠀⠀⠁⠠⠀⠀⠀⢈⡿⡍⠛⣶⣿⣯⠴⡀⢙⣿⣻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢀⣤⡆⠆⡗⠾⡆⠢⣕⣄⠀⡄⠐⣄⠈⠳⠩⡀⠙⣿⣿⡆⠃⠸⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣴⣼⠇⢳⡐⠁⠾⣿⡖⢜⠩⠕⠮⢆⡈⣦⢐⠀⠁⢀⣨⣿⡀⠐⠂⢋⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣼⠀⠀⠳⢌⠋⠙⢷⣈⠁⠀⠀⠉⠛⢬⢣⠐⠀⠈⣿⣿⣿⠻⠞⠋⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⢄⡀⠀⠀⠙⠳⢤⠀⣀⣤⣤⡄⠠⣸⢼⡄⠀⠰⢚⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠇⠹⠀⠈⡵⢶⣾⡆⠀⠀⠈⠉⠁⠀⡿⠀⠈⠀⣿⠀⢠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⣷⠀⠀⠀⣿⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡿⣤⠐⢸⣇⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⣿⢃⣿⣿⣿⣿⣿⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡰⠚⠁⠀⠀⠀⠀⣠⡔⢸⣧⡾⠿⠟⠩⠽⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⣄⣤⡶⠞⠋⠛⢿⡄⠈⣠⢄⡠⠴⠛⠹⢿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣷⠛⠋⠁⠀⠀⠀⠀⢀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠀⠀⠀⣀⣤⠴⠚⠁⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢚⡹⠛⠒⠉⠊⠉⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⣟⡽⠋⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠴⢦⣎⡀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣶⢟⡛⢴⣦⡹⣿⡅⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡿⠟⣠⣌⠻⣷⣿⣧⣸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠄⠙⣿⣿⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⢄⣀⣡⡽⡻⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠚⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⣍⣠⣈⠁⠈⣅⣀⣉⠉⠈⠉⠁⠐⠒⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢡⣞⣛⣉⣋⣽⢨⣧⠀⠁⢨⣥⣤⣤⣄⣍⡉⠘⠦⡀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠐⠁⢰⠾⣭⣉⠁⢠⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⡀⡤⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣀⡬⠁⣀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⢆⢀⠀⠄⢈⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠝⢣⢜⢿⠀⠀⡀⢸⣷⣬⣻⣱⣤⣴⠼⠄⡖⡏⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠢⢀⢴⠋⠀⠀⠀⠉⠀⠀⠄⢸⠐⠌⠈⠈⠀⠀⠀⠠⠠⢀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⣈⠀⣾⣦⣤⢌⠀⠁⠀⠀⠂⠃⡈⠘⠈⠳⡀
⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢀⣴⠞⠉⠑⠤⠚⣆⣀⡀⠘⣆⠀⠀⠄⡐⢀⠃⠂⠄⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠋⡀⠀⠄⠀⠠⣼⣷⣬⠁⠄⠈⢦⡀⠀⠁⡘⢰⠊⠑⠄⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣐⡿⣷⣼⡏⠀⠀⠔⠈⡷⠄⠀⠁⣎⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠄⠈⡑⣦⣭⠻⣾⣏⡧⠀⠊⡀⠀⠜⠀⢰⢠⠈⡈⠢⢀⡀⠈⢸
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢡⠲⡌⡤⢧⡹⠟⣝⠒⡀⢀⠊⢀⠊⠀⠀⠄⠆⠘⠘⠙⢠⠀⢀⠸
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠣⠡⠢⡠⡀⠊⠀⠀⠠⠀⡰⠃⠀⠀⠐⠐⠠⢣⠡⢁⠂⠀⠊⠃
⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠐⠫⠶⠶⠖⣠⣤⣠⠞⠀⠀⠀⠀⠠⠠⢡⢡⠂⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠡⢃⢃⠎⠈⣠⠊
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠒⠦⣠⡅⣂⠐⠀⠀⠁⠀⠐⠐⠀⠈⠀⠀⢁⣠⠴⠛⠕⡡⠈⠔⡠⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡥⢊⠁⢋⠐⠃⠀⠀⠒⠐⢀⠛⠀⡁⢀⠄⠀⡠⢊⠄⡡⠊⠔⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠘⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠂⠐⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⣶⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⢧⣤⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠻⣧⠉⣹⠟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⠉⠛⠛⠷⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠶⠟⠛⠋⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡷⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡴⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠉⠉⠙⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠋⠉⠉⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⡆⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣶⠤⣤⣤⣀⣀⣀⣀⣤⣤⡤⠴⠶⠒⠛⠉⠉⣁⣀⡀⠀⢻⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⣀⣄⡀⣀⠀⣀⢀⣤⣤⡀⢀⣶⡄⠸⣯⣼⡇⠀⠘⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⢀⡻⢦⡅⣿⠀⣿⢸⡅⠰⣦⢸⡷⠻⡆⠿⠀⠙⠂⠀⣿⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡀⠈⠓⠚⠃⠙⠶⠛⠀⠙⠛⠉⠈⠀⣀⣀⣠⡤⠶⠒⠛⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠀⠀⠀⠈⠉⠛⠒⠒⠶⠶⠶⠶⠖⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠷⠶⠶⢦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡶⠶⠶⠞⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠋⠀⠈⠃⠀⠀⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠘⡄⠀⠀⠀⣀⣀⣀⠀⠂⠀⡰⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠈⠢⣄⡀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠉⣭⡉⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⡟⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⠀⠀⠀⢤⡀⢀⠀⢨⡄⠀⠀⢂⢀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡇⠀⠆⢸⠁⠀⠀⠰⣿⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠐⠒⠤⣐⢕⠠⡀
⠀⠀⢠⠠⠤⠤⠤⠤⠤⠄⠀⠐⠒⠀⠒⠒⠒⡄⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀⠈⠑⢕⡌⢂
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢻⣿⣿⣿⡄⠀⠀⠀⠀⠁⠀⠀⠈⢫⠀⠃
⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢁⠀⠀⢀⣀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠈⠐⠢⠤⣀⡀
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⠜⠀⠀⣖⡄⠁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠠⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢀⣀⡀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢎⠆⠀⠀⠀⠉⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠉

⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⡀
⠀⠀⢀⣠⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣦
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠉⠋⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣦⠶⠾⠛⠛⠋⠉⠙⠛⠛⠳⠶⣤⣤⣾⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡿⠋⠀⠀⠀⣀⣤⣶⣾⣿⣷⡄⠀⠀⠀⢀⣶⣿⣶⣦⣤⣀⠀⠀⠀⠀⠉⠙⢿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⣿⣿⣿⡿⠟⢁⣀⡀⠀⠀⠀⠀⠀⠀⣀⠠⠄⡉⠻⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢰⠁⡌⠀⡝⠁⠀⠂⠉⠀⠀⢀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠰⣼⠓⢰⠤⠀⣠⢀⣀⡇⠤⠐⡆⡡⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠁⠒⠀⠤⠼⠤⠤⠧⠀⠒⠉⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⡠⠊⠉⠓⢄
⣇⠀⠀⠀⠀⠳⣄
⠘⢦⡀⠀⠀⠀⠈⢢⡀
⠀⠀⠉⠢⡀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠳⡀⠀⠀⣀⣀⣀⣀⣠⡴⠾⠁⠀⠀⠀⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠹⡞⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⢄⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠚⠉⠳⡄⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠘⠒⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄
⠀⠀⠀⠀⢀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⡴⠒⠒⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱
⠀⡷⠄⠀⠐⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⠀⠀⠛
⠈⠧⡤⣀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠠⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠄⠒⠒⠂⠉⠉⠉⠀⠀⠀⠀⠄⢾
⠀⢰⠃⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠸⣄⠈⠁⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠈⠙⠒⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠌⠛⠛⠛⠻⠿⠿⠷⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⣀⡀⠒⠂⣠⣤⠿⠿⠿⣷⣄⠀⠀⠀⠀⠣⡀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠴⠉⠀⢀⣶⠏⠉⠀⠀⠀⠈⠹⣷⡀⠀⠀⠀⠉⣄⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣶⡀⠀⢀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠛⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠔⠛⠻⠟⠉⠙⠛⢻⡆⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠻⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣠⣤⣾⣤⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣰⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠉⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠁⠀⠀⠀⢀⠈⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠁⠀⠀⠀⠐⠁⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⣀⠀⠰⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠁⢠⣤⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠘⠁⠀⢰⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣘⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠈⠀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣀⣀⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠙⠉
⠀⠀⠀⠀⠀⠀⢀⡿⢷⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠛⠛⠠⣤⡀⠀⠀⠀⣤⣤⣶⡆⠀⢤⣤⣤⣤⠀⣠⡾⠋⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠶⠛⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢳⡆⠀⠰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⠿⣄
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠘⠀⠀⠀⠀⠈⠁⢰⣿⣿⣷⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⡿⠀⢠⠀⠀⠀⠀⠀⠀⠈⠻⠿⠏⠀⣀⠀⠀⠀⣿
⠀⠀⠀⢴⣶⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠉⠉⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠁⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⢉⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⢀⡀⠤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠁
⠀⠀⠀⠀⠙⢿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⠃⣀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣤⡿⠃
⠀⠀⠀⠀⢰⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠛⠛⢠⠀⠀⠀⠀⠄⠀⠀⠀⣠⣤⡿
⠀⠀⠀⠀⠀⠀⠙⠛⠳⣤⣄⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠗⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠘⠻⣦⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⣼⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⣄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠀⠀⠀⠀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⢀⣴⡞⠁⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣴⠟⠁⠀⠀⣠⡶⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⣴⠟⠁⠀⢠⡴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⣾⠁⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣦⠉⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⣿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⣿⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⡀
⠀⣴⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⠙⣤⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠉⢷⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠙⢳⣤⡀⠀⠈⠛⢦⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠡⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣤⠐⠛⠓⠚⠛⠛⢤
⠀⠀⠀⠀⠀⣀⠾⠃⠈⠁⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢾⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⠉⠁⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⢠⡌⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⢶⡟⠁
⠀⠀⠀⠀⠀⠀⠘⠇⣀⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣴⣦⣤⣀⣠⡟⠛⠀⠛⠛⠁⠀⠘⢻⣤⣤⣤⣤⣤⣤⣤⡟

⠄⢣⡶⢦⡀⠀⢠⡾⣷⠀⠀⠀⠀⠀⠘⠃⠈⣾⠻⡆⠀⠀⣾⢿⡆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡜
⠀⠈⣓⡋⠀⠀⠈⣻⠛⠀⠀⠀⠀⠀⢰⣿⣿⣝⣟⠁⠀⠀⢨⡟⠁⠀⠀⠀⠀⠀⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⠀⠘⠁⠀⣠⣤⡸⣧⡰⠂⠀⠀⠀⠀⠀⠉⠉⠉⠀⢠⠶⣦⠳⣤⠇⣤⣤⡀⠀⠀⠈
⠀⠀⠀⠀⠳⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠞⠀⠀⠈⠻⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡙⠈⢳⣄⣀⣀⣀⣀⣀⣀⡀⢀⡀⣠⣀⡀
⠀⠀⠀⢆⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣠⣄⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠒⠛⠓⡉⠤⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣧⣀⣀⣀⣀⣀⡀⣀⣤
⠀⠀⠀⠀⠰⣯⣄⠀⠘⡃⠀⠀⠀⠈⠉⠑⠒⢤⣄⠀⠊⠁⠀⠆⣀⠀⠀⠀⠀⠢⣟⡇⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⢀⠜⠁⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠁⠀⠀⢈⠁⠀⠀⠀⠁⠉⠿⠂⠀⣀⠀⠀⠀⠊⠁
⠀⠀⠀⢸⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠐
⠀⠀⠀⢠⠄⠀⠀⢻⡥⡀⣄⠀⠀⠀⣷⡏⢹⡄⠀⠀⠀⠀⠀⢀⣮⣼⠗⠀⠂⠀⠀⢀⠀⠀⠀⠀⣰⣿⣿⣼⣀⡀
⠀⠀⠀⢰⡄⠀⠀⠀⠉⠁⢹⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢐⣞⡞⠀⠀⠀⠠⢫⢿⢝⣯⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠀⠘⠀⠙⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡒⡳⠓⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠅⠚⠙⠣⠴⣶⠪⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢗⠉⠑⠿⠤⣄⢈⢀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠸⠛⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠂⢀⡔⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⡠⠞⠉⡅⠀⣠⠞⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⠀⠀⠀⠈⠙⠒⠤⣀⣀⡾⠀⠀⠀⠈⠙⠁⠀⠀⡠⠕⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⣠⣶⡟⠓⢄⡀⠀⠀⠀⠀⠀⠀⡍⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⢄
⠀⠀⠀⠀⣀⣤⠶⠋⠁⢀⢤⠂⠀⠙⠋⠐⠠⡎⠈⠉⠉⠁⠀⠀⠾⢁⠃⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠈⢦
⢀⣠⠴⠚⠉⠀⠀⠀⠀⠈⢈⠢⡀⠀⠀⠀⠀⠈⢣⡒⠤⣀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠑⢌
⠻⣧⡀⠀⠀⠀⠀⠀⠀⠠⡊⠑⢌⠢⡀⠀⠀⡀⢸⡁⠀⠈⣡⠾⢻⢀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡇
⠀⣿⠙⠷⣄⠀⠀⢠⠀⠀⠘⣄⠈⠢⠟⣖⠭⠒⢉⣠⣶⣋⡥⠴⡇⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⡸⠁
⠀⣿⠀⠀⠈⠻⢦⡈⠘⡄⠀⠈⠢⠖⠉⣀⡤⠞⠋⠁⠀⠀⢀⡄⠈⠒⠤⣀⠀⠀⠀⠀⢀⠀⠀⢀⠞⠁⠐⢀⠞
⠀⣿⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠀⣰⠟⢴⢡⠄⠈⠧⠚⠁
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠶⠋⢹⡆⠀⠀⠐⢼⡶⠋⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠋⠁⢀⡸⣃⡠⠎
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣸⠀⠀⠀⢀⣤⡤⠖⣾⠛⠒⠒⠒⢶⠚⢻⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⢹⡀⠀⠀⠀⠙⡆⠀⣿⠀⠀⠀⠀⢸⡇⠸⣿
⠀⢹⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⡇⠀⠀⠀⡏⢳⡀⠀⠀⠀⠹⡀⣿⠀⠀⠀⠀⢸⡆⠀⣿
⠀⠘⢧⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⢳⡀⠀⠀⠀⢳⡟⠀⠀⠀⠀⢸⠁⠀⣿
⠀⠀⠀⠻⣄⠀⠀⠀⠀⢸⠀⣀⣠⡷⠤⠴⡖⠃⠀⠀⢳⡀⠀⠀⠈⡇⠀⠀⠀⠀⠚⠂⠀⢹⡄
⠀⠀⠀⠀⠈⢷⡀⠀⠀⢸⠋⠉⠀⣀⣀⣀⡗⠀⠀⠀⠀⢳⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠸⠷
⠀⠀⠀⠀⠀⠀⠙⣆⠀⢸⡛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⠤⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣰⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣶⣶⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠠⢌⡑⠂⢄⠀⠀⠀⣀⣤⣤⣶⣿⣿
⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⣂⡐⠲⠿⣓⡀⢄⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢿⣾⡿⣻⣡⣿⣾⣿⣿⣿⣿⣟⠿⣏⠟⠻⠛⠛
⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠂⠀⠐⠀⠀⣠⠀⠀⠀⠀⠀⠀⢺⢷⡶⠿⠻⢛⠟⣹⣟⢿⡻⣌⢣⠙⣄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠅⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢠⣾⣿⣿⢦⠳⡈⢀⢡⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠠⠀⠀⠀⠰⣟⡻⠯⡹⠽⠖⢱⡀⠀⠃⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣐⠀⢀⡀⠄⠀⠀⠀⠂⢀⠀⠀⠀⠀⠒⠉⠉⠁⠂⠀⠀⠀⠀⠀⠚⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣨⣷⣄⡀⢤⡁⠀⠁⠀⠀⢈⢲⣴⣋⠀⣀⠀⢀⠀⣀⠀⣀⣀⣀⣀⡀⡉⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠠⠾⠻⠙⠋⠦⠈⠑⠈⠙⠭⡙⠽⡿⣿⢯⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠈⠑⠀⠀⠀⠀⢸⢦⡀⠀⠑⠤⠑⢌⠛⡾⠁⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⢀⡀⠠⠄⡒⠈⠁⠀⠀⡀⠀⠀⠀⠀⠏⡙⢦⣀⠠⠈⠢⠈⠃⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠌⠷⣶⡾⠁⠀⠀⠀⠔⠀⢞⠀⠀⠀⠀⠁⠈⢻⣷⣄⠐⠄⠐⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠋⠀⠀⠀⢀⠄⡠⢁⣾⠀⠀⠀⢸⠀⠀⣛⣷⡍⠳⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢁⡴⠠⣿⣿⠁⠀⠀⠈⠀⠛⡡⢋⠔⢁⠄⠉⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠁⠄⠀⠒⠀⠈⠒⠀⠉⠑⠀⠀⠀⠀⠘⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠃⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠄⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⡀⣠⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⠳⢤
⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⠂
⡰⠋⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠙⠉⠀⠀⠀⣠⠶⠖⠲⠚⠲⡄
⣧⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣿
⠘⢦⡀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠦⢤⡀⠀⠀⢰⠇
⠀⠀⠙⢦⡀⠀⠂⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⣇⣀⣤⣀⡟
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡞⠉⠉⠉⠀⠉⠷⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢳⣤⠴⠒⠋⠉⠳⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠦⣄⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠛⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠉⠉⠀⠀⠙⢦⡀⠘⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠎⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄
⠀⠀⠀⠀⠀⠀⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄
⠀⠀⣀⣤⡤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠲⢦
⠀⣾⠁⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣰
⠀⡼⠋⠀⠀⠈⠑⠤⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠋⠀⠀⠀⠈
⠀⠻⢦⡤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠒⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠔⢺
⠀⠀⡾⠀⠀⠈⠁⠒⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠐⠁⡆⠀⠀⠀⠀⠀⠀⡾
⠀⠘⣇⠀⠀⠀⠀⠀⠀⡇⠀⣎⠉⠀⠀⡏⠓⠀⡖⡆⠀⡠⠒⢂⣉⣐⡆⠀⠀⡠⠄⠂⢦⠀⠀⢠⠞⢉⡀⠀⠈⣆⠀⡇⢀⠃⠀⠀⠀⠀⠀⣸⠃
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠙⠒⢬⡉⢢⠀⠀⠂⢠⠁⡇⠚⠀⡴⠉⠀⠉⠀⠀⢀⠇⢀⣄⠘⡀⠀⠘⠀⠸⣉⠇⣀⠄⠀⣇⡜⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣠⠅⢰⠀⡆⠰⠞⢠⠃⡇⠀⣇⢰⡆⠀⠤⠄⡸⠀⢘⠁⠀⢃⠀⠀⠀⣠⡀⠘⠣⡀⢀⣈⠁⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠾⠤⠴⠖⠃⠀⠳⠤⠤⠎⠀⠘⢤⣈⣩⡀⠀⡗⢊⡇⠀⠏⢣⠀⠘⡀⠀⣀⣀⠰⣄⡴⠋⠘⠊⠀⠀⢀⡴⠏
⠀⠀⠀⠀⢰⠛⠒⠤⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠑⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠴⡿
⠀⠀⠀⠀⠘⢧⣄⡀⠀⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠦⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣛⣛⣓⣲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣒⣚⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠴⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⣤⠤⠤⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠠⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠢⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠤⠤⣤⣤⣀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠇⠀⠀⠀⠀⠀⠢⠤⠀⠀⠀⠀⠤⠤⠀⠐⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢀⡀⠤⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠠⠤⠁⠀⠀⠀⠀⠁⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠑⢥⣀⠀⡀⠀⢠⡄⠀⠀⠈⠠⠐⠀⠀⠄⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠋⠐⠒
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⢀⠀⠀⡄⠐⡶⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡱⠀⠀⠀⠀⠘⠃⠐⠀⠀⠀⠁⠀⠀⠀⡇⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠰⠆⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⠖⢦⣤⡄⠀⠇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠆⠀⠀⠀⠀⠀⠀⠔⢀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠄⠀⠈⠃⠀⠀⠚⠃⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠀⠠⠄⠀⠀⠂⠀⠀⠀⠊⡀⠉⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠎⠀⠀⠠⠀⠄⠠⠤⠄⠀⠀⠀⣧⡤⠤⢤⡄⠀⠀⠀⠀⠈⡀⢀⠞⠁
⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠒⠀⠀⠀⢀⠀⠀⣅⠀⠠⠴⠾⠶⢤⠎⠘⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠤⡀⠀⢠⢆⡞⢾⡝
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⡰⠀⠀⠀⡆⣄⡜⠀⠀⣀⠤⠀⠈⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠔⣁⠒⠒⠀⠒⠁⠂⣂⠀⠐⠔
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⡄⠐⠈⠀⠀⠀⠀⠀⠀⡴⠆⠁⢀⡀⠀⠁⠀⠀⠈⠀⠀⠀⢡⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⢰⠀⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠉⠀⠀⠀⠀⠀⠠⠐⢁⡀⠴⠃⠀⠀⠀⠀⠀⢀⠀⠀⠈⢀⠈⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢐⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⡠⠤⠒⠉⠀⠀⠀⠁⠀⠀⢀⠀⠀⢣⣀⠀⠀⢀⡀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠈⢆⠀⠀⠀⠀⠀⠀⣠⠶⠚⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠃⠀⠃⡀⠀⡀⢠⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⡀⣀⣆⠀⠈⠀⢸⡇⠀⠈⡶⠀⡀⠤⠒⠁⠀⠠⢂⠉⡙⠢⣄⠀⠐⡀⠀⠀⠰⠁⠀⠀⢰⠀⠀⣀⡀⢀⣀⠈⠀⠀⠀⠰⠄⠌
⠀⠀⠀⠀⠀⠀⠉⠁⡀⠀⠀⡄⠉⢿⡁⠀⠀⠈⠁⠀⠀⠃⠀⣸⠄⠀⠀⠀⠂⠘⠆⠀⠀⣁⠒⠒⠃⠀⠀⠘⠃⠀⢀⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⢵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣏⣉⣠⣖⠈⣀⠘⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣝⢻⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣛⠉⠙⢿⣿⠿⢛⣙⣛⣿⣿⣿⣡⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⡿⠀⠀⠀⠀⠘⠋⠙⠋⠙⠋⠀⠉⠉⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⠀⠀⠛⠛⠋⠉⠙⣆⠀⠀⣴⠛⠛⠓⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⡘⣿⠀⠀⠰⠶⠿⠟⠂⠘⠀⢸⠁⠴⣶⣤⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣰⡟⠀⠀⠀⠀⠠⠶⠛⠁⠀⠀⠘⠃⠑⠢⠁⠀⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣹⠄⠀⠀⠀⣿⣿⡆⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡄⠀⠀⣠⡄⠀⣀⣠⣤⣭⣤⣁⡀⢠⡀⢸⣿⣿⢃⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⢿⠃⠙⢷⣤⣤⣤⣤⡴⠋⢀⠏⣿⣿⣿⡿⠟⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⠀⠘⠆⠀⠀⠈⠹⣿⣏⣁⡀⢚⡿⠭⣙⢦⡈⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡘⣇⠀⠀⠀⢀⣤⣶⠿⣿⣿⣿⡶⠚⠓⢌⡳⣍⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡏⢳⡈⠳⣄⠰⣿⣿⣵⠿⣿⣿⣿⣷⣿⣶⠀⠃⠏⠀⣽
⠀⠀⠀⣀⣠⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⡇⠀⠳⣄⠈⠻⠿⠻⢿⣤⣨⣿⣿⠿⣺⠇⠀⠀⠀⣴⠃
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⠳⣄⡀⣠⠴⠋⠉⠉⣤⡴⠁⠀⠀⢀⣼⣧⣀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⣿⣇⠀⠀⠀⢼⣿⣿⣿⣷⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣰⡟⢻⣿⣿⣧⠀⣠⠴⣿⣿⣿⡖⠀⠀⠹⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⠋⠀⢸⣿⣯⠙⠛⠁⢰⣿⣿⣿⡷⠴⢒⣫⣭⣶⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⡄⠀⠀⣸⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣧⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⠿⠿⠛⠿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠁⠀⠀⠘⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⡀
⠀⢀⣀⡀⠀⠀⠈⠻⢶⡤⠒⠀⠀⠀⠀⠀⠘⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠉⢿⣿⣦⠀⠀⠀⠀⡷⠷⠆⢠⡀⠀⠰⠀⠀⠀⠀⠉⠀⢀⣀⡀⠠⢾⠏
⠀⠀⠀⠙⠿⣦⠀⠀⠆⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⠷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠆⠙⠛⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠠⣄⣸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣿⣿⠟⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡇⠀⠀⠀⠀⠀⠀⣀⠀⠙⠓⠒⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢜⠛⠢⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠓⡄⠾⡀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⢿⣼⣜⢦⡩⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⣀⠂⠀⡀⠀⠀⠀⠀⠔⠏⠀⠀⠠⢀⠀⠀⢈⠁⠀⠁⠄⢀⠄⡹⢶⠴⡧⠄⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠑⠓⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠢⡈⠀⠀⠀⠀⠁⠘⠀⠀⠁⠈
⠀⠀⠀⠅⠒⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠴⠴⠐⠐⠢⢁⡀⠅⠒⢀⣀⢀⡐⠾⣓⢸⠀⠀⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⡀
⠀⠂⠀⠀⢀⢠⡦⢈⠁⠁⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠈⠈⠉⠡⠐⠛⠉⠁⠀⠋⠀⠈⠃⠥⠀⠂⠀⠐⠒⠀⠅
⠀⠀⠀⠀⠀⠐⠊⠰⠪⠤⠉⠀⢠⣒⡪⠵⠊⠀⠀⠀⠠⢤⠠⢔⠀⠄⣀⣀⡀⠠⠀⢀⠠⡀
⠐⠤⣀⠨⠀⠀⠀⠤⠤⠄⣀⠀⠻⣗⡢⠄⠀⠠⠤⠤⠀⣀⠈⡱⠞⠉⠤⠦⠤⠤⠀⠒⠫⣁⠀⠀⠀⠠⠀⠄⡀
⠀⠀⠀⠈⠁⠀⠐⠒⠒⠀⠀⠀⠉⢀⠄⢀⠀⠀⠈⠉⠉⠀⠁⡀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⡈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡘⠶⠬⢐⠀⠩⠀⠀⠬⢥⣐⡨⠄⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⣐⠬⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⢀⡀⢀⠤⠍⠉⠉⠉⠁⠀⠀⠀⠐⠀⠍⣤⣌⣀⠀⠩⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄⣀⣄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣄⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⣿⣇⣀⣀⣀⣴⡄⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠛⠈⠻⢿⣿⣿⡏⠉⠉⠀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣄⣐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣽⣿⣿⣿⣿⣿⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠽⣿⣿⢿⣽⣿⡽⠟⣿⣟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠈⢹⣿⣿⠉⣵⠗⣀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣶⣎⣍⣙⣿⣧⣄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣴⠋⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡽⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⡻⠿⠛⠿⣿⣿⣿⣯⠉⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⠗⠀⠀⠀⠘⠏⠈⠏⠛⠋⢹⣿⣿⣿⡿⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢙⣿⣿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠀⠀⠀⠈⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⣿⣴⣦⡄⠀⠀⠀⠃⠀⠀⠀⠀⣿⣿⣯⠀⠀⠀⠐⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠻⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⠀⠀⠀⠀⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠙⢿⣦⡀⠀⠀⢀⣤⣤⣶⠀⠀⢸⣿⡀⠀⠀⠀⢉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣴⣿⠿⣿⡆⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠉⠉⠀⠈⠁⠈⠛⢿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠿⠋⠚⡘⠄⠀⠀⠀⠀⢠⠈⢉⠋⠛⡿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠋⠁⢀⡀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠈⣠⣀⣀⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⣬⣤⡄⠀⡀⠀⠀⠀⠀⠀⡀⠀⢠⢎⣥⣽⣿⣿⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣶⣿⣯⣿⢟⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣻⢼⡿⠛⢻⡆⠀⠀⠄⠐⡖⢀⢸⣟⣷⣴⣾⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⠓⠴⠞⡉⠀⠁⠀⠀⠀⠠⢦⣾⣿⠿⢿⣿⣿⣿⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⡵⠶⣽⣤⠁⠰⠆⠀⠀⠸⠟⣣⡾⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣶⠆⠏⠹⢮⡀⠀⠀⠀⣠⢎⣥⣶⣍⠫⣿⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⢼⣿⣿⡤⣤⣴⣿⣿⣷⣭⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠻⢿⠿⠟⣟⣠⣧⠽⢽⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠋⠝⣽⣯⣰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀
⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣼⣷⣭⣿⣥⣥⣯⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣾⣷⣿⣾⣥⣾⣿⣷⣞⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣾⣿⣷⣾⣿⣿⡿⠿⠟⠛⠛⠛⠻⠿⣿⣯⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢟⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠻⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣥⡾⣻⣿⠟⠁⠀⠀⡀⠀⠀⠀⡀⠐⠋⠉⠀⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡿⠃⠀⠀⠐⠋⡉⢁⡀⠋⠉⠀⢠⠟⠁⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠁⠀⠀⠀⠀⠠⠛⠾⠿⠤⠴⠞⠉⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⠉⠀⠀⠀⠀⣠⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣄⡀⠰⠋⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⡄⠫⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣴⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⣀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠒⠚⠋⢰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⠀⠀⠀⠦⣝⣳⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⠶⠶⣬⣓⣶⠋⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⢌⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠙⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠿⠆⠀⠀⣶⡇⠀⢿⣧⡤⠴⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣁⠀⠀⠀⠀⢀⣸⣿⡇⠀⠀⠀⢀⣀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠙⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠉⠉⠉⠉⠙⠋⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣷⡄⠀⠀⠾⠂⠀⠀⠀⠛⠆⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⢀⡀⠀⢁⢄⣀⠀⠀⢧⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢛⢻⣿⡀⠀⠀⠈⠉⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⣧⡙⠃⠀⠀⢰⠀⠘⠲⠾⠀⠀⠀⠀⢸⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⠙⣧⠀⠀⠈⠙⠦⣀⣀⣀⣀⠴⠀⠀⡟⠀⠀⠀⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠚⠃⣤⣀⠉⣹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⣸⣯⢳
⠀⠀⠀⠀⠀⠀⠀⡴⠯⣀⣨⠟⠃⡆⠙⢦⣄⣀⡈⢂⣓⣃⡤⢾⣅⠀⠀⠀⠁⣽⢛⠿⣀⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢩⣀⡀⠇⠀⠀⣼⠋⡏⠉⢫⠀⠱⠸⣏⢦⡀⠀⠸⣌⠷⢂⣩⠞
⠀⠀⠀⠀⠀⠀⠀⣰⠯⢿⢩⢩⡟⣧⣐⣼⠃⠀⠁⠀⠘⡀⣄⢠⡹⡆⡟⣦⠀⢸⠀⡎
⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⢸⢸⡇⡌⢹⡃⠀⠀⢾⢦⣠⠓⠟⢌⡇⠘⢟⠁⠀⡸⠀⡇
⠀⠀⠀⠀⠀⢠⡏⡇⣧⠀⢸⢸⡇⡇⣨⡧⠀⢠⠈⡆⠁⠀⢀⣈⡵⠁⢈⡇⢠⠃⠀⢹
⠀⠀⠀⠀⠀⣸⡃⡇⢹⣧⢸⣸⡷⠟⢻⡀⠀⣸⣁⡇⢠⠖⠉⣡⣤⣤⡜⠱⡾⠀⠀⡽
⠀⠀⠀⠀⠀⣿⡇⡇⠀⢹⠻⣿⡄⠀⠀⢿⡛⢫⠟⢠⢻⢳⢻⣽⠀⠳⣄⢀⡠⠔⠰⠳⢄⡄⡀
⠀⠀⠀⠀⠀⣿⠁⠇⠀⣿⢰⠀⢿⡄⠀⠘⠷⠉⠀⣎⢸⢸⣽⡇⠀⠀⢠⠋⠐⠂⠄⡀⣠⠤⠈⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⣿⠷⢾⣦⣼⣬⣀⣆⣻⣄⠀⠀⠀⣸⣋⣸⣴⣿⣡⣤⣶⠾⡎⠁⠑⠁⠀⠀⠀⣠⠏⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠉⠉⠙⠛⢧⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠒⠤⣤⡤⠖⠚⢁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣻⢿⣿⣿⣿⡿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠠⠴⠶⢦⣬⣍⣒⣨⣔⣕⣪⣐⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⢠⣿⠄
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡛⠛⠛⠛⠻⣿⣿⣏⠟⢻⡿⠻⢛⣛⠛⡛⢟⢻⡿⠟⠋⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⠻⣮⠙⠻⣿⣿⣿⣿⣿⣿⣿⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣤⣤⣤⣤⣤⣼⣿⣀⡀⠀⠀⠈⢷⡀⢿⢀⣩⣿⣿⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠉⠙⢦⡀⠀⠀⢳⣾⣿⣿⣿⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⣄⣴⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⣿⡿⣫⣉⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠘⠿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣀⡀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⢠⣾⡿⠁⠀⣼⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⢠⣿⠟⠁⣠⣾⠟⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠈⠁⢀⣴⡿⢁⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⣴⡿⠋⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠛⠛⠛⠿⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⣠⡄⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠸⠗⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⠁⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠸⠗⠀⠀⠀⠙⠻⢿⣦⣄⡀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠀⠰⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣤⡀⠀⡀⠀⠈⠙⠿⣶⣤⣿⠃⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠋⠀⠀⠀⠾⠎⠛⢿⣦⡀⠀⠀⣠⣿⠏
⠀⠀⠀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢠⣤⠀⠀⠠⣦⠀⠀⠀⢀⡀⠀⠀⣤⡄⠙⢿⣦⣼⡟⠁
⣰⣾⣿⣻⡿⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢠⣄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠋⠀⢀⣤⠀⣤⡄⠹⣿⣿⣶
⣿⡏⠙⠻⠿⣶⣦⣄⣀⡀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣤⠀⠀⠀⢠⣦⠀⠈⠋⠀⠀⠀⠀⠉⣀⣬⣶⡿⠟⠉⣿⡆
⣿⡇⠀⠀⠀⠀⠈⠙⠛⠻⠿⣶⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⡶⠿⠛⠋⠁⠀⠀⠀⢻⣇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢀⣤⣾⢿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠋⠁⢸⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢰⣶⣶⡾⠿⠟⠛⠉⠁⠀⠀⠀⠀⣾⡇
⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠃
⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⠿⢛⣿⡿⠁
⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠻⠿⠿⠛⠛⠛⠋⠉⢁⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠿⠿⠛⠛⠉⠁