Ib Emojis & Symbols

⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⡤⠶⠦⢤⣀⠤⠀⠤⡀⠀⡠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⠔⠁⠀⠀⠉⠓⠤⡀⠀⡇⠀⠪⢐⢄⠙⡖⠂⠀⠘⡎⠀⠴⡠⠞⠀⠀⠀⠀⠀⣀⣠⡴⠊⢉⡑⠠⡀⠉⠑⠢⣄⡘⢀⠁⡠⢤⣤⠄⢂⠶⠤
⠀⢠⡏⠀⠀⢀⠴⠤⠀⠀⠈⠑⢬⣒⠚⠁⠈⠑⡿⠀⠀⠀⠱⣢⣀⣀⣀⡀⠠⠤⠒⠋⠁⢸⠁⠀⠀⠀⡠⠃⠀⠀⠀⠀⠙⠒⢾⠒⠣⠂⡠⠁⠀⠀⢱
⠀⢸⣆⠀⠀⠀⠉⢀⠁⠀⠀⠀⠀⠈⠉⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠚⠁⠀⠀⠀⢀
⠀⠀⠙⣗⠒⡒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡚⢉⠕⣸
⠀⠰⢲⠞⡏⠁⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠙⠲⡞⠁
⠀⠈⡟⢸⠓⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣈⡷⠀⠀⡇
⠀⠀⢇⠈⠒⠲⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠁
⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠙⠀⣰⠏
⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣤⣠⣤⡀⢀⣠⣤⣤⢠⣤⢠⣤⡀⣽⡁⣤⣄⣄⣀⣤⣤⣀⢠⣤⣠⠀⠀⠀⠀⣇⠸⠥⢾⣾⠏
⠀⢀⡔⣲⡶⠒⠺⡆⠀⠀⠀⠀⠀⠀⢸⣿⢀⣿⡇⣿⡇⢿⣏⢸⣿⢸⣿⣘⣿⡇⣿⡇⣿⣿⠀⣿⣗⠚⣛⢸⣿⠀⠀⠀⠀⠀⠈⠢⢰⡏⢁⠤⠰⣄
⠀⠘⣶⣉⣠⡠⠞⢸⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠉⠁⠀⠉⢽⠿⠀⢉⠉⠉⡁⠉⠉⠉⠉⠀⠀⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⡁⠘⠄⠀⢈⠂
⠀⠀⢈⠀⠙⠦⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⡆⢰⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣀⡴⠊
⠀⠀⠈⢄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣇⣸⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⡈⢣⡀⢀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠢⡈⢂
⠀⣠⠊⠀⠀⣹⡏⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡲⡖⠀⢀⡘
⢠⣡⠒⠀⢸⣠⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠰⠦⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠈⠒⢻
⢸⡌⠲⠴⠚⠁⠀⠠⣆⢀⣀⠴⢁⣠⠤⠒⠈⠉⠉⠉⣓⣦⣄⣀⣀⠀⣠⣴⣦⣄⠐⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠈⢳⡀⢀⠇
⠀⠳⣀⠀⠀⢢⠀⠀⠈⣩⠵⠚⠉⠀⠀⠀⠀⣰⠛⢙⡅⠀⢰⠀⢈⢿⣿⡛⢻⣿⣿⠏⢉⡭⠖⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⡧⠂
⠀⠀⠀⠉⠓⠚⠻⡉⠁⠀⠀⠀⠀⠀⣀⡴⠊⠈⠦⣬⣤⠴⡟⠙⢺⢾⣿⣿⣿⡿⢻⡴⠃⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠋⠀⠈⠉⠉⠉⠛⠒⠀⠀⠀⠀⠀⠀⠈⠙⠒⠗⠒⠒⠐⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⣶⠶⠶⠦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⣿⠉⠉⠙⠛⢶⣄⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣀⣠⡴⠟⠁⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠉⠙⠳⣦⡀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⣿⣄⣠⣤⣤⠶⠟⠉⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⢠⣤⣤⣤⣴⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣤⣄⠀⠀⢀⣤⣴⣿⠟




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣶⣶⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⠛⠛⠁⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣶⡀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣷⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠿⠿⠟⠁⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠆⠀⢰⠀⠀⢲⠀⠀⠂⠀⠀⠀⢰⠀⠀⠆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⣿⣟⣟⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣿⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠈⠛⠀⠉⠉⠙⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⡆⣰⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠀⠛⠛⢻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⠀⣀⣀⣀⣠⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⣠⣤⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⠈⠉⠉⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠰⠿⠿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⣀⣀⣀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠓⠀⠖⠶⠶⠛⠛⠁



⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠘⠛⠛⠿⣿⣷⡀
⠀⠀⠀⠀⣿⣿⡟⠉⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣴⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⡟⠛⢻⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⣤⣴⣶⣿⡋
⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⠉⠉⠉⠻⣿⣦
⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⣶⣶⣶⣿⠿⠋
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣶⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⢀⠀⠄⢀⠀⠀⢀⠔⠒⠢⣤⠒⠂⢄⡠⠐⠒⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⣀
⠀⡰⠃⠀⢀⡀⠉⠂⢐⠠⠤⠊⠨⡇⠀⠘⢄⠈⠈⠀⠀⠠⠤⠒⠟⠀⠀⢀⠀⠀⠉⠒⠀⡤⠝⠃⡀⠉⠁
⠀⢧⠀⠀⠁⡀⠀⠀⠀⠈⠐⠒⠈⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⣀
⠀⣈⣫⡥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠨⣕⡱
⠀⠰⠳⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢐⠄⠈⠇
⠀⠠⡈⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡴
⠀⠀⠘⠂⡀⠀⠀⢰⣶⢲⣦⠀⠀⠶⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⣶⠀⢀⡀⠀⠀⣶⡄⡊⣀⣤⡼⠁
⠀⡀⣠⠖⢺⡄⠀⢸⣿⠚⠉⣿⡉⣿⣸⣿⣿⡷⣿⡀⣿⡏⣿⣹⠆⣿⢸⣗⣛⠲⣿⣿⡇⠑⢠⠏⡀⠠
⠀⠹⠥⢖⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⢶⣄⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠁⢀⠄
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⢀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡁
⠀⢀⠘⢣⣀⠴⠒⠠⠀⠀⠀⠀⠀⠀⠀⠘⠒⠶⣿⡿⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⡀⠈
⢠⡣⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣺⡇⠙⣧⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠦⠴⠁
⢸⠑⠄⠊⠀⠰⣀⡀⢌⡠⠄⠀⠉⠉⡐⢮⡍⣀⢳⣞⢩⠗⡖⢢⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠉⢢⠀⠄
⠀⠃⠤⠀⣄⠤⠀⠊⠁⠀⠀⡀⠸⡁⠇⠠⢧⢈⣾⡿⠓⠞⢓⠜⠋⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠓⠤⠤⠄⠐⠊⠀⠀⠉⠙⠢⢤⡽⠋⠁⠀⠒⠁⠀⠀⠀⠀⠈⠒⠠⠤⠤⠠⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠤⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠁⠡⠌⡀
⠀⠀⠀⠀⢠⡏⠁⠉⠓⠒⠖⠋⠓⠊⠙⠁⠀⡀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⠈⠉⠒⠲⠞⠙⠒⠉⠋⠀⢀⠀⠈⠈⠈
⠀⠀⠀⠀⡜⠁⠀⣦⣀⣀⣤⣀⣠⠤⠴⠒⢊⡗⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢰⣄⣀⣠⣄⣀⡤⠤⠖⠒⣹⠃⢠⡇
⠀⠀⠀⠀⡇⠀⢸⠓⠢⠀⣸⢤⡀⠀⠒⠒⠋⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⠒⠄⠀⡧⣄⠀⠐⠒⠚⢹⠀⢠⡇
⠀⠀⠀⠀⡕⠀⠘⡟⠒⠋⡁⠀⣧⠤⠤⠖⠊⡗⠀⢸⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⢻⠓⠒⢉⡀⢸⡤⠤⠴⠒⢹⠂⠀⡇
⠀⠀⠀⠀⢹⠀⠈⡏⠉⢉⣁⣴⠃⠀⠶⠶⢲⡇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⠉⠉⢉⣠⡞⠀⠰⠴⠖⣾⠀⢀⠇
⠀⠀⠀⠀⠘⣆⠀⠉⢻⠉⢁⡤⠉⠛⠛⢻⠟⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠙⡏⠉⣠⠌⠙⠛⠚⡿⠋⠀⣸⠃
⠀⠀⢀⣀⡀⠸⣀⢀⡼⠤⣏⡔⠋⠉⠛⠉⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⣠⠧⢼⣡⠞⠋⠙⠋⠁⢀⡸⠁
⠀⠀⠀⢀⠄⠀⠈⠉⢧⠀⠀⣧⠀⠀⡦⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡄⠀⠸⡇⠀⢰⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁


⣿⣿⢛⡻⣛⡛⣛⣛⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⢀⣀⠀⣀⣀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⡙⠿⣶⢈⣛⠿⣾⣭⣷⡿⣟⣩⣴⠾⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣙⠻⣶⢈⣛⠿⣾⣭⡗⠛⠛⣭⣴⠾⢛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⣙⠻⢶⢈⣛⠿⣶⣽⡇⠀⠀⢨⣶⠾⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⣙⠻⢶⢈⣛⠿⣶⣭⣷⠶⣒⣭⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⣙⠻⢶⢈⣛⡿⣶⣿⣷⠿⣛⣭⣶⠿⣛⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣙⠻⢶⢀⣝⡻⣷⣿⡇⠀⠀⠉⠰⠟⣋⣥⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣙⠻⢶⢀⣝⡻⣷⣿⡆⠀⠀⢀⡰⠟⣋⣥⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣙⡻⢷⢀⣝⡻⢷⣿⣾⢿⣛⣭⡶⠟⣋⣥⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠍⠻⠷⠀⠝⠻⠷⠿⠾⠿⠫⠥⠶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠯⠿⠿⠿⣽⠽⢿⡿⣿⢿⠷⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⢲⠀⠃⢀⣿⡃⣿⠊⠁⡲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠟⠀⡇⠀⠀⢸⠀⢸⢱⢸⢋⡶⢞⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠀⠀⡇⠀⣷⢸⠀⠘⢊⡾⡩⢊⠥⢊⠝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⠆⠱⡐⣄⢆⢁⠀⡯⡸⠀⣰⣏⡤⠭⠵⣊⣥⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡘⡘⡄⠱⡀⠙⢶⠀⠘⢭⡶⣋⣠⣒⣒⠬⣛⠒⠜⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠑⢌⠺⢦⠇⠀⠈⢏⣄⡾⠓⡓⠉⡉⠉⣁⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢅⠢⢁⣆⠈⠫⢵⣠⣿⢒⣛⠚⠻⠐⠠⠬⠈⠤⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣕⡂⠬⣀⠉⣩⠓⣦⣟⣁⠉⠒⠓⡔⠮⠴⠦⢄⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠢⠭⠬⠋⠉⣩⡟⢀⠇⠑⡄⢖⠨⠰⠶⠦⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠊⢡⡿⠇⢸⠀⡖⢌⠒⠭⠒⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠃⠐⢈⠢⣡⠀⠑⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡜⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠉⠉⠙⣆⠀⠀⠀⠀⠀⠀⣠⡤⣤⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⣄⠀⠀⢀⡄⠈⠙⣆⠀⠀⣠⡼⠁⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠈⠃⢀⡞⠀⢀⣠⠏⠀⢰⡇⢀⠐⣆⠀⣠⠖⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀⢀⡀⠀⣀⡟⠁⠀⠀⠀⠙⢾⡀⠘⠂⠀⠀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠋⠉⣿⢹⡏⠉⡇⠀⠀⠀⠀⠀⢸⠓⢶⡖⣶⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣀⣀⠀⣿⢸⡇⠀⡇⠀⠀⠀⠀⠤⠞⠀⢸⡇⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠉⠀⠀⠀⢀⣠⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡇⠀⠀⠀⠀⢀⣀⣀⠀⠀⢸⡇⣿⠀⠀⢀⡴⠒⠛⠳⢦⡀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣦⡀⠀⣿⠀⡇⠀⠀⣀⡴⠋⠁⢉⠳⣄⣸⠇⣿⡤⠶⠋⢠⡀⠀⠀⠀⠳⠾⣄⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢸⠁⢻⡀⢿⣤⣧⣤⣾⣥⣤⣴⣤⣿⣤⣤⣽⣦⣿⣤⣤⣤⣤⣷⣤⣾⣤⣤⣤⣼⣧⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡿⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⡇⠀⣀⣽⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣲⣄⢰⡇⠘⠛⢶⣀⣠⣄⢸⡏⠁⠀⠀⠀⣀⣠⡽⣷⠀⠀⠀⠀⡼⠋⠁⠀⣄⠀⣰⠃⠈⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠀⢸⡇⠀⠀⢸⠋⠀⠈⠛⢁⣠⠴⠖⠋⠉⢀⡴⠃⠀⠀⠀⠈⠃⠀⠀⠀⠙⠀⠃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠘⡇⢠⣴⡚⠀⣀⡤⠞⠋⠀⢶⣤⣀⡴⠋⠀⠀⠀⠀⠀⣠⠖⠚⠒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⣄⠀⡇⠀⠀⣿⠚⠩⣄⡀⠀⣀⡼⠃⠉⠀⠀⠀⠀⢠⠴⠞⠁⠀⠀⢀⡄⠙⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⣿⠀⠀⡏⠀⣀⡿⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠳⠀⠾⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢸⡄⠠⡟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⡀⠀⠂⢀⢆⠀⢂⠀⠀⠐⠀⠠⡐⠀⠖⢠⠂⠠⠀⠄⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⠀⠁⠀⠀⠈⠈⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠁⠂⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢶⣿⣶⣤⣤⣀⣾⣿⣿⡷⣁⣤⣴⣒⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛⠛⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⡩⠿⠛⣩⣿⣿⡿⠋⠘⣷⠈⠛⢿⣿⣿⣿⣿⣟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⣿⡆⠀⠀⠻⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⢻⣷⠀⠀⠀⠙⣏⢻⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢻⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠈⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⢘⣿⣧⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠿⣄⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢸⣿⡿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⢸⣿⣷⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣯⣭⣭⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢡⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⣿⡇⠀⠀⠀⢸⣿⣟⣛⣛⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠉⠉⠉⣉⣉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣿⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⠀⢴⡄⢸⣇⢀⣶⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣹⡷⠿⠛⠛⠛⠛⠻⢶⣿⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣤⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⠻⠷⣦⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢀⣴⠟⠛⠛⠷⣦⡀⠀⣼⣏⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⣿⡿⠁⠀⠀⠀⠀⠈⣿⣾⣏⠙⠻⠦
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣛
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⢀⣴⣄⠀⠀⠀⢀⣴⣄⠀⠀⠀⢀⣴⣄⠀⠀⠀⢀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠹⣷⠀⢠⡿⠁⠹⣷⠀⢰⡿⠁⠹⣧⠀⢰⡟⠁⠹⣧⠀⢰⡟⠁⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⠟⠀⠘⢷⣤⣴⠟⠀⠘⢷⣤⣴⠟⠀⠘⢷⣤⣴⠟⠀⠘⢷⣤⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⣆⠀⢀⣾⠟⢿⣆⠀⢀⣾⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣠⡿⠀⠸⣧⣀⣠⡿⠀⠸⣧⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠈⠛⠉⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⠻⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⣿⠈⠻⣦⣈⣻⣶⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠀⣷⣄⠙⢷⠌⠉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠻⣯⡙⠃⣿⡏⠻⠿⠛⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣄⠰⣶⣾⡄⢻⣤⣤⣤⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠻⣦⡈⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢶⣄⠙⢷⣌⠻⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡈⠻⣦⣨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⢷⣄⠙⠷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣤⣶⣤⣸⣿⢠⣌⣻⣦⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠛⠛⣧⠘⡿⠿⠆⠙⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠿⣯⡉⠻⣦⡀⣿⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣀⡐⢷⣄⠙⢿⠀⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡙⢷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡑⠀⠘⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⠀⠀⡠⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⠤⢶⠊⣠⢼
⠀⠀⠀⠀⠀⠀⢠⠖⡉⠂⠄⡻⣀⠀⠀⠀⠀⠀⠀⢠⣮⡤⢤⡈⣙⣿⣯⡫⠠⣌⣷⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⡟⠘⣧⠀⠉⠀⢸⠄⠀⠀⠀⠀⠀⣿⣿⡇⢀⢼⠋⢩⡇⠳⣸⣃⢀⣼⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠙⣦⡙⡒⠤⠤⣤⠇⠀⠀⠀⣀⣼⣿⣿⢷⢃⡏⡄⠀⠙⡲⣿⡻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢀⠠⢤⣸⠙⠈⣦⣄⣚⣻⣷⣴⣿⣿⣿⡿⠋⢸⡾⠀⠀⠀⢠⠗⣻⣷⣿⣿⠟⠛⠋⠉⠁
⠀⠀⠀⢀⣏⠿⠼⠃⠀⠀⣿⣿⣟⣿⣿⣿⣿⣿⣿⡄⠀⢸⡇⠐⠀⠀⡞⣼⣿⣿⣿⠋
⠀⠀⠀⠘⢧⣤⠀⢀⣠⠞⠙⢿⣾⣿⣿⣿⣿⣿⣯⢀⣤⣿⣿⡀⠀⠀⣷⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣻⡿⠿⡿⢷⣤⡾⠿⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⣛⡿⡟⠁⠀⠀⠀⢸⡛⠂⠀⢀⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⣿⣿⡇⠀⠀⠀⠀⢸⣧⣿⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⣿⣿⣿⣿⡿⠀⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡛⠕⠉⠈⢳⡀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠈⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢿⣷⣄⠀⠀⠀⢳⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠀⢀⡟⠉⠙⢆⠀⠈⢧⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⢀⡤⠊⠀⠀⠀⠈⣇⠀⠈⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⢀⣤⠤⠴⠊⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄
⣤⣾⠏⠉⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠛⢆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠙⣶
⢸⣧⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⣏⠀⠀⠀⢣⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⣆⣿
⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣤⠞⠁⠀⠀⠀⢿⣧⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠁⠀⠀⠀⠀⠀⠀⠻⠳⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

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

⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄
⠀⠀⠀⠀⠀⡰⢿⡿⠤⢿⡿⢄⠀⠀⠀⠀⠀⠀⠀⠖⠐⠀⠉⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⢿⣿⣿⣶
⠀⠀⠀⢸⠭⠤⠔⢀⣀⡌⠢⠤⠯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠁⠈⠉⠙⣿⡇
⠀⠀⠀⢸⠀⠀⢻⣿⣿⣿⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠛⠋⢸⣿⠠⠁⠀⠀⠀⠀⠀⠀⢸⣿⣤⣶⣶⣷⣶⣦⣿⡇
⠀⠀⠀⢸⢤⣠⣾⣿⡿⠿⠟⠛⠀⢃⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⣿⣿⣿⠟⣿⡇
⠀⠀⠀⢉⠀⠉⠉⠀⠐⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠾⠆⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠿⣄⢤⣥⣌⣤⣤⠿⡇
⠀⠀⠀⠀⠀⠣⡀⢠⠒⡀⠀⠊⠀⠀⠀⠀⢀⣀⣰⣶⡶⠄⣀⠄⢄⣤⣶⢄⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⠉⠁⠀⠀⠙⠛⠋⠉⠉⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣤⡤⠤⣾⣿⡧⠤⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢞⣷⣿⡳⣤⣄
⠀⠀⠀⠀⣿⣿⣿⡏⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⣿⣶⣬⠛⠛⣵⣶⣿
⠀⠀⠀⠀⡿⣿⡿⢀⣴⠛⢿⣿⠿⠀⠀⠀⠀⠀⠀⡠⡔⢶⢾⡣⡣⡦⢄⡇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⣿⡟⠁⣾⣯⣶⡈⣿⣿⠀⠀⠀⠀⢠⡶⣭⣵⣷⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣤⣿⣷⢠⣯⠈⠁⠀⣿⣿⣤⠀⠀⠀⠀⠁⠘⣿⠿⢿⣯⣿⠏⠀⠀⠀⠀⠀⠀⢹⣿⠻⣟⣻⣷⡿⠃⣿⡏
⠀⠀⠈⢿⣯⣥⣤⣶⡴⢶⣦⣭⣿⡟⠀⠀⠀⠀⠀⢀⣈⣻⣟⣉⣁⣀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣬⣥⣴⡾⠿⣿⠇
⠀⠀⠀⠀⢅⡐⠋⠛⠙⠀⠐⣀⠍⠀⠀⠀⠀⠨⠿⠛⡛⠛⠃⠑⠘⡛⠻⠯⠀⠀⠀⠀⠀⠙⠛⣏⠉⠀⣰⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡿⠿⢿⠷⠄⣴⣿⣿⣿⣿⣷⠶⣦⠀⠀⣴⣾⡿⢿⣿⢦⠄⠀⠀⢀⣤⣶⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣦⢠⢸⡤⠄⠉⠀⡏⠀⠉⠀⠀⢸⠀⠀⢀⣅⣄⣀⣇⣀⠀⠀⠀⠀⡏⡏⠀⢹⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠾⠾⠛⠁⠓⠲⠿⠒⠛⠒⠒⣾⣤⣤⣼⣿⣿⣿⣿⣿⣤⣤⣤⢀⣷⣧⣤⣼⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⣀⣤⣶⣤⣀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⢀⣠⡤⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⡿⠿⠿⣿⣷⡄⢺⠟⣿⠛⠛⠛⣿⠻⡆⠠⠾⠿⠿⠿⠷⠐⠚⠛⠛⡟⠒⠚⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣷⣶⣶⣷⣾⡆⢸⣀⣿⠀⠀⠀⣇⣀⡇⠀⣀⣀⢀⣀⡀⠀⠀⠀⠀⡇⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠀⠘⠛⠋⠀⠀⠀⠛⠛⠃⠀⠤⠦⠤⠤⠆⠀⠀⠀⠀⠧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣶⣄⡀⢠⠴⠶⠟⠛⠷⠶⢤⠀⠀⠀⣴⣿⣿⣦⡀⠀⠀⣴⣾⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠉⡇⢈⢸⡇⠀⠀⠀⠀⠂⠀⠐⠀⠀⠉⠉⡟⠉⠉⢹⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠀⠈⡇⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⣷⣤⣤⣾⣏⠄⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠈⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠈⠉⠉
⣿⣿⡟⠻⣿⡏⢹⠛⡏⣿⡏⡍⢙⠹⣿⡏⡋⠉⢹⣿⠋⡟⠙⠹⣿⡏⡋⠛⠩⣿⣿⠋⢙⠉⣿⢩⠉⡟⢹⡏⡏⠙⠙⠙⠉⣽⢹⣿⣿
⣿⣿⣷⣶⣾⣷⣾⣾⣿⣿⣷⣶⣿⣶⣿⣿⣷⣾⣿⠛⠓⠟⠛⠖⢻⡟⠛⢻⡖⠻⣷⣶⣿⣶⣿⣿⣷⣶⣾⣿⣷⣾⣷⣶⣶⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠠⠀⠠⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠂⠀⠄⠀⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠤⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⢀⡀⠀⠈
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢐⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢃⠀⠂⠐⠂⠐⠀⠂⠐⠂⠐⠀⠒⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠃⠚⠀⠐⠀⠂⠀⠂⠐⠀⠐⡞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡁⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇⠀⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠈⢁⠀⠀⢸
⠀⠀⠀⠀⣀⡀⣀⣀⢀⣀⡀⠀⠀⣀⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⢋⣉⣉⣉⣉⣉⡉⣫⣉⡀⠀⠀⣀⠀⢀⡀⢀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡡⠊⢑⡄⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⢐⢑⠔⠁⠈⢐⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠣⠔⠀⠀⠀⢠⣿⣿⣿⣿⠛⠛⠛⠛⠉⠁⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠅⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢅⠕⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠠⠄⠀⠀⠀⠀⢸⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁
⠠⢁⠓⢄
⠱⣆⠀⢎⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣤⣶⣶⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢨⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠻⠿⠿⠉⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡨⠖⠋⠑⢌⣳
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠠⠈⠒⢿⡷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣩⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡽⠚⠈⠀⠀⠀⠂⠀⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢄⠀⠈⠀⠠⣿⡄⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⢆⠀⠀⠀⠀⠀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠈⠙⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠛⠛⠿⡇⠀⠀⢸⣿⡿⠿⠛⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⣷⣦⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⢀⣴⣿⣿⣿⣿⣿⣟⠛⣻⣿⣿⣿⣿⣷⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣿⣿⣿⣷⠇⣸⣿⣿⣿⣉⣉⣻⣿⡿⣿⣿⢿⣿⣿⣿⣇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠿⠟⢾⡇⠸⣿⣿⣿⣿⣿⣿⣇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⣿⣿⣿⡀⠀⠀⠀⢸⣧⡀⢘⣿⣿⣿⣿⣿⣿⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣦⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣧⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿⣟⣯⣭⣹⣿⣿⣿⣿⣿⣿⣿⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡏⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣷⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣿⣿⣿⣿⣿⣿⠏⢀⡄⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⣤⠊⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡉⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⣿⠻⣿⡿⣿⢿⣿⣿⠛⠃⠈⠻⢿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⣥⣤⣌⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠍
⠀⠀⠀⠀⢴⣶⣶⣾⣿⣿⣿⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠏⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣧
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⣿⣵⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣜⡻⠟⠷⣞⣧
⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣬⣭⣤⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣿⣿⣿⣶⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⢉⡿⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢀⡿
⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⡾⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣅⡀⠀⢀⡾⠁
⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠸⣧⠀⣰⡟⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣤⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠃


⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣠⠞⠁⠉⢓⡴⠟⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣾⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⡀⠀⢠⣄⠈⡆
⠀⠀⠀⢸⡁⠀⢀⣴⠏⡱⣆⠀⠀⠀⣹⠂⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤⣤⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣉⡁⠀⢈⣉⠀⡇
⠀⠀⠀⠀⢙⡴⣏⠡⠀⡅⢨⠳⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢉⣉⣉⡉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠃⠀⠘⠋⠀⡇
⠀⠀⠀⣰⠏⠁⠈⠳⣄⠂⣠⠞⠁⠘⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⠤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⠃⠀⠘⠟⠀⡇
⠀⠀⠀⠘⢦⡀⠀⠀⣨⢟⡁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣤⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠾⠆⠀⠰⠷⠀⡇
⠀⠀⠀⠀⠀⠙⠳⠞⠁⠀⠙⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⡀⠀⢀⠀⡀⢀⠀⠀⢀
⣿⣯⢽⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢦⠀⠀⠀⠑⢦⣁⠚⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⠀⠘⡐⠓⡀⠀⠀⠀⢻⠻⣄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠂⠀⠂⠀⠋⠀⠀⠀⠀⠀⢿⠏⠳⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡔⠈⠂⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠠⠀⠀⠀⠁⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢰⣄⠀⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⣱⡀⠜⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠓⢀⠀⠀⠈⠉⠰⠆
⠀⠀⠀⠀⠀⠀⢉⠀⢄⠀⠀⠀⠀⠀⠀⠉⢀⡀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⢸⡀⢈⡆⠀⠀⠀⠀⠠⡄⠀⢛⣆⠀⠸⣅⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠇⠈⠁⠀⠀⠀⠀⠈⢠⠄⠘⢀⠱⡀⠀⢰⠄⠀⠀⠀⠀⢐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠙⣷⠀⠀⠐⠁⠀⠁⠈⠀⠀⠄⠀⠀⠲⣄⠀⠀⢐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠳⣮⣾⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠙⣆⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣴⠏⠳⣄⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣈⣁⣀⣀⣀⣀⣠⣚⣁⣀⣀⣌⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⡉⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢀⠰⠀⠒⠄⠀⠀⠀⠀⠀⠠⠰⠀⠀⠄⠆⢀
⠀⠀⠀⠀⠀⠀⠸⠤⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⢃⢀⠞⡍⠌⣂⠄⡀⠀⠀⠀⠀⠐⢯⢨⠭⠅⠭⠝⠹⡆
⠀⠀⠀⠀⠀⠓⠖⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⡤⢁⡀⣤⠀⠀⡖⠒⠀⠀⠀⠀⠀⢀⠉⠲⡠⠔⣇⡄⠴⠀⠀⠀⠀⠀⠇⡀⠀⢆⠂⠀⠠⠇
⠀⠀⠀⠀⠀⠰⠉⠈⠉⠡⠀⠀⠀⠀⠀⠀⠀⠨⠀⠨⠤⠄⠅⠀⠑⠀⠀⠀⠀⠀⠀⠈⠚⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠱

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠀⠀⠈⠀⠠⢀⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡉⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠄⠀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠊⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠰⣤⠚⡖⣰⠒⠀⠀⠀⠀⠀⠀⢠⣾⠒⢠⡘⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠠⡢⠀⣠⠀⡔⠄⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠀⠀⠈⢬⡱⡀⠰⠹⠀⠀⠀⠀⡔⠁⠀⠈⣤⡀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣈⠉⠀⡀⠡⠂⠀⠀⠀⠀⡄⠀⣠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⢀⠁⠐⢄⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢦⠀⢤⠂⠀⠀⠀⠀⠀⠀⢀⡔⠓⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠤⠤⠴⠤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⣾⡃⢶⠇⠻⢸⡇⢸⠇⢺⠃⢾⡦⣤⣤⡆⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣁⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⢀⣄⠀⠛⠛⠻⡧⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⢸⠀⠳⢤⣀⣀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠲⣄⠀⠀⠀⢳⢸⠀⠀⠀⠀⠉⢳⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⡤⢿⡆⠀⣾⡏⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣠⠖⢉⣀⣸⢃⡼⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣠⠞⠁⠀⠀⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣠⠞⠁⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡶⢫⡙⠲⡄⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠁⠀⠀⠀⠀⢀⣴⡾⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠀⠀⢻⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠙⢦⡀⠀⠀⠀⢀⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣾⠁⠀⢀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠉⠛⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⠾⠛⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣌⢻⣿⡟⣰⣿⣿⡇⠀⠀⠀⣠⣶⣿⢿⣿⣿⡿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⣿⣿⣴⣿⡟⠋⠀⠀⠀⠀⠻⠟⣿⣎⢻⣯⣴⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠛⠛⣛⠻⠛⢵⠀⠀⠀⠀⠀⠀⢠⠻⣿⡿⢿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠟⠀⠀⢠⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣄⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠀⠀⠀⠀⠠⠴⠞⠀⠀⢸⡇⠀⠘⠃⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣤⣦⣬⡙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣠⣶⣿⣷⣄⠀⠀⣸⣇⠀⣀⣠⣾⡟⣿⣿⣿⣿⣮⣁⡸⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⣶⡀⠀⠸⠿⠃⠀⠀⠰⠿⠿⠙⠟⠐⠿⠿⠆⠻⠟⠸⠿⠿⠿⠷⠘⠿⠃⠿⠿⠿⠿⠘⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢸⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⣸⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣴⣶⣶⣌⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡇⣿⠛⠉⢴⣿⣿⣿⣟⢋⣭⣴⣶⣶⣶⣶⣦⡍⠻⣿⣿⣿⣿⣿⢋⣭⣴⣿⣿⣿⡿⢻⣷⣬⣝⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⡇⣿⣿⣿⡆⢉⣵⣦⡙⢸⣿⣿⡿⠟⣋⣩⣤⡶⢠⣿⣿⣿⣿⡇⠛⠛⠛⠛⠈⠛⠃⠛⠛⠛⠛⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⠇⣿⣿⣿⡿⠟⣉⡵⢶⣿⣿⡿⠟⣰⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣉⣀⠶⡿⢛⣥⣶⣿⣿⣷⡀⣌⣋⣴⣾⣿⣿⣿⣿⣿⢋⣴⣾⣿⣷⣦⡙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⢻⣿⣿⠀⣾⣿⡉⠻⡿⠟⣋⣴⣿⣿⣿⣿⣿⣿⡟⣡⣶⣾⡿⢿⣿⠏⣽⣷⣶⣌⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣷⢸⣿⣿⢸⡿⠟⣁⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣉⣉⣉⣉⣈⣉⣀⣉⣉⣉⣉⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠈⠿⠟⠰⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⡏⣬⡍⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⡏⣭⣥⣿⣧⣭⣍⢹⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣧⣭⡍⣿⡏⣩⣥⣼⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣇⣙⣁⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⡴⠟⠛⠛⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠉⠉⠉⠉⠉⠋⠙⠉⠉⠉⠉⠉⢁⣤⣶⠛⠉⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣷⡀⣀⣀⣀⣀⣀⣀⣾⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠛⠷⢶⣦⣤⣤⣤⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣶⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⣝⣛⣛⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢫⣭⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠈⠉⣿⣿⣿⣿⣿⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⢤⣤⡀⣀⣷⠀⠀⢾⣿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠧⣤⣤⣤⣬⣿⣿⣄⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣩⣄⣬⣉⣟⣛⣛⣋⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⠁⠈⣿⣿⣿⣿⣿⡿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡟⢻⣿⠋⠀⢀⣹⣿⣿⣿⣿⡆⠘⢿⣿⣷⣆⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠿⠛⠛⠛⠛⠋⠀⠀⠙⣿⣿⣿⢿⣏⠙⢧⡀⠉⠻⠎⠁⠙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠉⣼⠉⠟⠧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⡶⢿⣿⣶⣿⣿⠿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⣿⣣⣞⢻⣾⣿⣿⣿⣿⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢻⣿⡛⠿⢛⣿⣋⣽⣿⣿⣷⠆⠀⠀⠀⠀⠀⢠⣦⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⣽⣿⣄⢋⣽⣿⢟⣶⢿⣿⣿⡧⠀⠀⠀⠀⢸⣍⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡉⣿⣿⣯⢉⣽⣿⣩⣾⢿⣿⡿⠀⠀⠀⠀⠀⠹⠾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠰⣞⣿⣷⡈⣽⣿⣟⢁⣼⣿⣷⣾⣿⣿⡆⠀⠀⠀⠀⢐⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⣿⣄⣼⣿⣿⣯⣿⣯⣴⣿⣿⡇⠀⠀⠀⠀⠈⢹⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣦⣤⣀⣶⣦⣆⣹⣿⣧⣾⢿⣷⡿⣿⣷⣿⣿⣿⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⣿⣿⣿⣼⣿⣽⣿⣿⣿⣿⣧⡻⢿⣶⣿⣵⣿⣿⣿⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡗⢿⣷⠞⣻⣿⣿⡟⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠾⠟⢿⣿⣿⣿⠃⠀⣿⣹⣿⣿⣿⣼⣿⣾⣿⣶⣷⣿⣿⣧⣤⣤⣤⣤⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡀⣼⣿⣿⡿⣾⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⣿⣿⣿⣇⣿⣷⡎⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⢹⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢾⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡛⢻⡋⠈⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⡷⣾⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣁⣼⣿⣿⣿⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡗⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡅⢸⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⠛⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⢸⣿⣟⣿⣯⣝⣿⣦⣤⣋⣁⣀⠛⣩⣿⣿⣿⣿⣿⣿⣹⣷⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣧⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣹⣿⣇⣀⣠⣤⡄⢀⣴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠓⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣻⣿⢿⣭⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠻⠛⡁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⡀⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢉⣠⣾⣿⣯⣿⣿⠟⠛⡟⢿⠿⢿⡿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⢦⣤⣤⣤⣤⣉⠉⠙⠛⠛⠉⢈⣩⣉⣭⣤⣬⣤⣴⣶⣾⣿⣿⣿⣿⠟⠛⠁⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⠀⠉⠛⠋⠉⠈⠛⠛⢻⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠉⠉⠋⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠤⠤⠄⣠⠤⠤⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠀⣀⣤⡟⠛⠛⢶⡀⠀⠀⠀⠸⣄⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣾⠁⠀⠀⠀⠀⠀⢻⠀⠴⠞⠁⠙⠳⢤⣤⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠂⠀⠖⠳⠶⠶⠋⠀⠀⠀⠀⢀⡞⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢶⣶⣤⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠖⢛⠲⣼⠑⣞⡿⢭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣤⡞⢉⣽⡷⢻⢟⡵⡆⢸⡏⠉⠳⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣿⣾⠘⡟⠋⠀⣾⣼⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣾⣘⡇⠀⠀⠹⡏⡼⠓⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠃⢸⡇⠀⠀⠀⡼⠃⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠾⠇⠀⠀⠾⠷⠶⠶⠾⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠛⠉⠉⠉⠉⠙⠛⠻⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⣶⡄⢀⣴⠇
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⢀⣤⣴⡶⠿⠿⠿⠶⢶⣤⡀⠀⠀⠙⣧⠀⣿⢃⡾⠋⠞⠁
⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⢠⣾⠟⠋⠁⣀⣴⡶⠟⠛⠀⠈⠻⣆⠀⠀⣿⠀⠛⠘⠁⢀⣤⣶⠶⢶⣤⣄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢰⡿⠁⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠟⠋⠀⠀⠀⠀⠙⢷⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⣿⠇⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠋⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⢻⣆⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣾⠏⢀⣀⠀⠀⣀⠀⢀⠀⢀⡤⠀⢀⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⡇⢠⣿⡟⡠⠋⢸⢇⣾⠽⢡⣟⣴⣿⠁⣠⡿⠁⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⣿⡟⠓⠦⠤⠤⠔⠒⠊⠁⠀⢠⡟⢀⣿⡿⡐⢀⣤⡟⢻⣇⡴⢿⠏⠘⠿⠾⠋⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣾⣿⠁⠀⠿⠞⠀⠈⠉⠀⠀⠀⣠⣄⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⢸⣿⢇⠂⠀⣀⠀⢀⣤⢀⣴⡆⣼⣟⠍⠀⠀⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢿⣷⡀⠀⠀⠀⢀⣤⡿⠃⠀⠀⣿⣿⡀⠀⢀⣿⣠⢻⣿⠎⠿⠱⡿⠏⠀⢀⣼⠇⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠈⠻⣿⣦⣤⣶⠟⠋⠀⣀⠀⠀⣿⣿⠀⠀⣼⣿⠃⢾⡏⠰⠶⠒⠷⠷⠾⠟⠁⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣴⣿⠀⠀⠿⠃⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⣄⡀⠀⠀⢀⣠⣴⠟⠉⣿⡄⠀⠀⠀⣠⣾⣆⣀⣤⡾⠃⢠⡇⣶⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⠋⠁⠀⠀⠙⢿⣶⣶⡾⠛⠉⠙⠛⠉⠀⣠⡿⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣘⣧⣰⣿⢠⣇⣀⠀⠀⠀⢠⡞⢱⡜⣧⡉⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠖⢓⡞⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢘⡃⠀⠀⠜⠉⠀⠀⠀⠀⢉⡿⡽⣮⢯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡀⠀⠻⠶⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⠃⠘⡎⣇⠀⠚⠉⠀⠀⠀⠀⠀⠠⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⣓⣊⡁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣘⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢸⣿⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠧⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠸⢻⠀⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⡷⠒⠲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⣸⠀⢳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⠁⢀⠀⢸⠀⢠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⡀⠘⠆⢀⢔⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⢸⠘⢻⠛⢸⢰⢋⣌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡰⢣⡈⠊⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠁⠀⠀⠀⠀⠀⢸⣀⠀⢀⣸⠘⠦⠥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡔⢃⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠊⠀⠘⠲⡄⠀⠀⠀⠀⠀⠀⡸⠀⢞⠁⠀⢣⡀⠀⠀⠀⠀⡀⠘⠀⢰⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢸⡀⠀⠀⠀⣼⡿⠀⠀⠀⠀⢀⠞⠀⠀⠈⠣⠀⠀⠑⠄⠀⠀⣌⠀⠀⣀⣤⠵⣤⣀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠟⠀⠴⠾⠋⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢤⡆⠀⠐⣦⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⢻⠏⠀⠙⢯⠀⠀⠻⠀⠀⠀⠀⠀⠐⣦⡤⠄⣀⠇⠀⠀⠀⠀⠀⠐⠰⠀⠐⠰⠒⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢠⡄⢶⢼⠀⠀⠀⡄⠀⠀⠀⠀⣰⠀⠀⠀⠀⣶⠀⠀⠀⢀⡤⢶⠰⢖⢀⢀⣐⢶⢐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢰⣦⣶⢸⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⡀⣀⡀⠿⡀⠀⠀⣿⢸⡁⠀⠈⠀⠀⠀⠠⡀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠖⢿⣀⣀⣀⣸⠗⠲⠚⠁⠀⠀⠀⢰⠃⠉⠗⠘⡏⠈⠇⠀⠀⠈⠒⠦⠴⠉⠉⠉⠙⠠⠭⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⣫⡍⢴⣲⡆⣿⡇⣼⣻⡆⡶⣮⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣋⢵⡆⠀⠈⢧⠘⣋⣃⣛⣃⣋⣙⣙⡷⠟⠀⣰⢛⡶⣉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⣁⣲⣌⠀⢙⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠲⢭⣛⠴⣻⡾⠀⠙⢦
⠀⠀⠀⠀⠀⠀⡴⢣⠶⣌⡻⢉⡴⠋⣀⡀⠁⣀⠂⠠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠀⣠⣾⣂⡳⡄
⠀⠀⠀⠀⢀⡞⣡⢬⣛⢞⡵⠋⠀⠀⠀⠀⠈⠁⠀⠁⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠽⣛⣥⡙⣆
⠀⠀⠀⠀⡼⠀⠙⠲⢏⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠯⠃⠉⠸⡄
⠀⠀⠀⢰⠃⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣺⣿⠋⣶⣟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢻
⠀⠀⠀⡾⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⣀⡀⣭⡹⣿⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠘⡆
⠀⠀⠀⡇⠀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢪⣿⣯⡿⢿⣷⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡀⠀⡇
⠀⠀⠀⡇⠨⠛⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣬⣀⣀⣀⡀⢈⣿⣌⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠋⠀⡇
⠀⠀⠀⢻⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢀⣶⣟⣧⣽⣦⡛⣩⣯⣸⣏⣹⣵⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢰⠇
⠀⠀⠀⠸⡆⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢠⣍⣍⣀⣈⠉⠁⠉⠉⢉⠟⠿⣿⡀⠀⠀⠀⠀⡀⣰⠃⠀⠀⠀⡾
⠀⠀⠀⠀⢹⡀⠀⣀⣤⢳⣤⣴⡶⡶⢶⠿⣿⢭⣽⣿⠿⢾⣟⠛⣿⣿⣞⡫⢹⣦⡤⣄⡾⣳⢣⣆⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠳⡘⣯⠘⣣⡙⢿⠓⠛⠋⠙⠛⠻⠿⠯⣤⣿⣿⡵⣿⡿⠿⠿⠯⣭⣭⣹⠟⠑⠿⣌⠛⡰⠃
⠀⠀⠀⠀⠀⠀⠙⣆⢺⣥⠟⣠⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⡁⢾⡴⡀⢈⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠾⠻⣾⢃⡬⣝⠲⠤⣄⣀⣀⣀⣀⣀⣀⡤⠴⢚⡭⣄⢷⡷⠤⢙⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⣁⠾⣝⠀⢠⣟⡷⢰⣄⡎⢱⢶⡄⠘⢿⠈⢷⠻⢆⣡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣌⣙⣃⠸⠹⠇⠸⠾⢃⣀⣈⡥⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠗⣠⡂⠶⠀⡏⢤⠸⠰⠇⠰⢲⣦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⡀⢘⣥⠬⠗⠚⠛⠛⠛⠛⠛⠛⠳⠦⣮⣸⡕⢃⢗⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢄⣬⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣾⣇⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⣹⠞⠁⠰⣆⣀⠀⠀⠀⠐⣿⣆⣀⣤⠰⣶⡀⠀⠀⠀⠀⠀⠉⠳⣤⡋⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠏⠀⠀⠀⢇⠉⠁⠀⠀⠀⠀⠉⣁⡀⠀⠹⣷⣄⠀⢀⠀⣀⠀⠀⠈⠻⣄⡴⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣛⡇⠀⠀⢀⣤⣾⡆⠀⠀⠀⠀⠀⠀⠙⠿⠦⠐⠋⠉⠀⢘⣧⡿⣁⠀⠀⠀⠈⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡥⢸⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⡀⠈⢳⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣠⣤⣀⠀⣀⣠⡄⢰⣾⣿⣿⣶⡶⠂⠀⠀⠻⠿⠃⠀⠈⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⣧⣶⣿⣿⣿⣿⣷⠌⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⡿⣆⠀⠘⣟⠛⠉⢻⠁⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢺⡟⠹⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⡄⠀⠒⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠈⢧⠀⠈⠳⠶⠟⢠⡟⠁⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⣨⡅⢴⠖⠀⠸
⠀⠀⠀⣠⡴⠶⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⡾⠀⠀⠀⠸⡇⠀⠀⢸⣦⡀⠀⣀⡀⡈⠉⠀⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠻⣤⡀⠀⣴⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⠞⠁⠀⠀⠀⠀⢷⠀⢀⠸⠆⠀⠀⠉⠻⠉⠀⠀⠀⠀⠀⠀⢰⠁
⠀⢴⡞⠛⠋⠀⠀⠈⣳⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣤⣶⣄⠀⠀⠸⣄⠿⠗⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⣠⠏
⠀⠀⠙⠳⣦⠀⠀⢼⡁⠀⠀⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⡴⠃
⠀⢸⣿⠉⠉⠀⠀⠀⠉⠉⠒⢦⡙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠙⢶⣄⣀⠀⠀⠀⢀⣀⣤⠶⠋
⠻⣏⣉⠀⠀⠀⠀⠀⢀⣔⠒⠉⠀⠀⠙⢿⣿⣷⣶⣤⣀⡀⠀⠀⠈⠉⢻⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉
⢠⡏⢍⠙⠛⠲⢲⡆⠀⠉⠛⠓⠒⠠⣤⡀⠈⠙⠻⢿⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⣶⣦⣤⣀
⡼⠃⠉⠀⠘⠆⣾⠁⠀⠀⢈⣿⡏⠉⠁⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⡀
⢧⠺⣤⣂⣠⣶⠛⠚⠛⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠉⠉⠛⠛⠿⢿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣀
⠘⢧⣈⣉⡴⢻⣶⠏⠀⠶⠆⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⢿⣷⣦⣄
⠀⠀⠉⠉⠀⠈⣇⠀⠧⠄⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠈
⠀⠀⠀⠀⠀⠀⠻⣌⡓⣋⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⡀⣀⣀⣤⣤⣴⣶⣶⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⣶⣶⣶⣶⣶⣶⣶⣿⣿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠟⠛⠉⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢶⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⢇⠈⢧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣧⠈⢇⠈⢆⠈⢇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠠⣦⠘⢇⠘⣇⠘⣇⠙⢧⠈⢆⠈⢆⠈⢆⠈⣦⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣈⣿⣿⣿⣿⣿⣿⡙⡇⠘⣄⠈⣆⠘⣆⠈⣆⠸⡧⠸⠇⠸⠇⠸⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠛⠋⠈⠛⠛⠿⣿⣿⣿⣿⣧⠘⡄⠈⡄⠘⡄⠈⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢋⣬⠐⡄⠘⣄⠸⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⡷⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡠⠞⠋⠁⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⣀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⢶⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠁




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠡⠴⠂⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⡀
⠀⠀⠐⠉⠀⠀⠠⠀⢲⡇⠀⠀⢀⠀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡡⠈⠁
⠀⠨⠀⠀⠐⠀⠀⠀⠈⠀⠄⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠁⠀⠀⠈⠀⠀⡀⠌⠁
⠀⠠⡄⠀⣀⠋⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠐⠀⠀⠂⠀⠀⣰⠣⠈
⠀⠀⡍⠀⠈⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢨⠀⠀⠠⠁⠈
⠀⠀⣅⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠸⡄⢀⣇⣀⣀⣀
⢀⣀⢸⣀⣀⡀⣿⠇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠑⠀⠓⠋⠉⣉⠉⠉
⢠⢯⣼⣟⡌⣁⣿⡀⠀⠀⢀⠀⠈⠉⠁⠀⠀⠀⠰⠔⣠⡄⠀⠀⠀⠀⠐⣂
⡧⣿⠿⣿⣿⣿⡿⢸⣗⣠⣄⠀⠀⠀⠉⠓⠓⠒⠦⠼⠿⠿⣿⣦⡚⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣷⣶⣶⡿⠟⢿⡇⠸⠿⠛⠁
⠉⠀⠉⠀⠀⣿⠆⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⡈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⡀⠀⠙⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⢘⡇⣛⢸⡃⡗⠀⠃⠒⠒⠒⠒⠒⠒⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠙⠈⠃⠛⠘⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡄⠀⣀⠂⠀⠀⠂⠀⠀⠀⢀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠴⠀⠀⠀⠉⡄⠀⠒⠀⠀⠀⠀⠁⠈⠉⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠀⠀⠀⠠⠆⠀⢀⣀⡀⠀⡀⠀⠀⠀⠀⡠⢴⡀
⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠰⠀⠀⠆⠀⠀⠈⠉⠁⠀⠉⠀⠀⠀⠀⠉
⠀⠰⠗⢀⠀⠀⠀⠀⠀⠀⠆⣀⡤⠄⠀⠀⡠⠀⢀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠘⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⣤⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠈
⠀⠸⠇⠀⠐⠗⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠉⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠶⠤⠶
⠀⠀⠀⢀⣀⣀⠀⢀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇
⠀⠀⠀⢸⠁⠈⢷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢺⠶⠶⠋⢸⠶⢦⠀⢀⣤⣄⠀⣤⣀⡠⣄⣄⣄⠀⢀⣀⠀⠀⣠⡄⠀⣤⢤⠀⠀⠀⣀⠛⣿⠋⠄⢀⣀⠀⠀⣀⡀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⡆⣏⠀⣿⣀⣿⠁⢀⡏⡏⢸⠰⡏⣹⣄⢸⡁⣀⢸⡟⢚⡁⢿⢀⣿⡀⣿⠸⡇⣏⢀⡀⣞⢩⣇⠀⡇
⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠉⠁⠁⠉⠀⠀⠁⠁⠈⠁⠉⠁⠈⠀⠉⠉⠀⠉⠉⠁⠈⠉⠁⠀⠈⠀⠁⠈⠉⠁⠈⠉⠀⠀⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣶⣄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢿⣿⡿⠀⣠⣼⣿⣧⣤⠀⠐⢂⡀⠀⠀⣠⡖⢣⣄⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈⠁⠀⠻⢿⣿⡿⠿⠀⠀⠀⠀⠂⠀⠻⢤⣴⠋⠀⠐⠈
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠰⠦⠴⠶⠀⢱⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠺⡄⠘⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠰⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠁⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠘⠋⣡⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢈⣤⡀⠈⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⠟⠿⣿⡀⠈⠑
⠀⠀⠀⠀⠀⠀⡛⠁⠀⠀⢠⣤⣤⡌⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠁⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣈⣥⡦
⠀⠀⠀⠀⢀⡀⡁⠀⠀⠠⠿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣄⠋⠁⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⡂⠀⠀⠀⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⢰⣶



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣋⣍⣹⣟⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⣦⣠⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠜⠁⠀⠀⢀⣀⣤⣾⣿⣷⡚⠓⠂⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢈⡤⠊⠁⢈⠂⠀⠀⠙⠒⢲⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⣴⡧
⠀⠀⠀⢀⠀⠉⠘⠒⠤⠀⡤⠀⡀⠴⠦⠀⠀⠀⠀⠀⠀⠸⣄⠀⡇⠀⠀⠈⠳⡄⠀⣀⡴⠋⠀⠀⠀⢀⠀⠀⠘⢛⣟⠃⠟⠙⠛⠛⢻⣿⣦⣄⣀⣀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠷⠶⠶⠶⠓⠚⠉⠀⠀⠀⠀⠚⠰⣦⠀⠀⢹⡶⣤⣀⣀⢢⣄⣮⣵⣿⣷⠶⠶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣀⣀⣠⣀⣾⣿⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠉⠛⠆
⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⢿⣿⣯⡉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢠⣇⢁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠌⢻⢠⣏⠂⡴⢐⢸⡇⣴⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠋⠀⠙⠊⠋⠉⠉⣇⠘⢺⡗⠊⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡾⠋⠡⠊⡀⠀⢸⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠘⢷⣶⣆⡘⠁⠀⠸⠀⠈⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠻⢿⣤⣴⣿⣾⣶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢤⣰⡄⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣂⣩⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠶⠚⠙
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⢿⢻⣽⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⣀
⠑⠃⠀⡰⠁⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢸⡞⠋⠁⡢
⠀⠀⣸⡇⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⢰⡞⠂
⠀⠀⢿⡇⠀⡞⠀⢀⡔⠀⠀⢠⡄⠀⢡⠀⣧⠀⢠⠰⡟⣁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠁⠀⣻⡇⢠⡇⢀⡿⠁⠀⠀⠀⠀⠘⡉⣇⣻⠀⡆⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠄⠀⣿⣇⡼⠁⣼⠇⠀⠀⠀⠀⠀⠀⢹⣿⡟⣸⣏⣴⣿⢃⡀⠀⠀⣀⣀⡤⠘⢿⣼⣸⠟⠶⣶⣶⣦⣤⣀⠐
⠀⢠⣿⡟⢁⣼⠇⠀⠄⠀⠀⠒⠀⠀⠀⠙⠷⠟⠿⠿⠛⠚⣠⣴⣾⣷⣶⣶⣆⣀⣿⡗⠇⢤⣼⡾⡿⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣠⣤⣀
⠀⣸⠟⡴⠟⢁⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⡏⢀⣹⣧⡷⣀⢷⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⣰⠟⣭⣷⣦⣽⣧
⢰⣿⣫⠔⠚⠋⠉⠀⠈⠉⠙⢶⡀⠀⡀⠉⠉⠉⠉⠁⠀⠀⠈⠉⠛⠒⣶⣿⣶⣿⡿⠙⢿⣮⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⣘⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⡖⠔⡄⠀⠀⠐⠄⠀⠀⠀⠈⠀⠀⠐⠀⠀⠠⠀⣾⣿⣿⠟⠫⠁⠈⠲⡟⢿⣿⣷⡄⠀⠀⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠿⠿⠋⠈⠉⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⢀⣀⣀⡀⠀⢤⢀⣀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⠐⠀⠀⢀⣀⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣛⣋⣭⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠙⠻⣭⠭⠿⠿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣷⢂⠀⠀⣠⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢿⢷⣾⣦⡀⠀⣸⣿⡤⠄⢠⣿⣿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠠⢾⣿⣿⠳⣄⣨⣿⣷⣤⣾⠋⠉⠀⢀⣤⣶⣀⣀⣰⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣷⣶⣾⣿⣯⣀⣾⣿⣿⣿⣿⣾⡿⢿⣶⣶⣿⣿⣿⣿⣿⣿⣿⠇⢀⣠⣶⡶⠖
⠀⠀⠲⢶⣶⣵⣄⡀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣤⣽⣿⣭⣴⣿⣿⣿⣟⣄⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⣀⣉⣙⣻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠃
⠀⣴⡾⠟⠋⠉⠋⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣶⢿⣿⣹⠿⠟⣻⣿⣿⣿⣿⡿⣿⣿⣧⣤⣄⣀
⠀⠁⠀⣀⣤⡴⣶⠿⢛⣿⣿⢻⣿⠟⠻⣿⣿⣿⣿⠀⢹⠀⠀⢈⣿⣾⣿⡿⠛⢻⣿⣍⢻⣮⣝⠋⠛⠋⠻⢿⡆⠀⠀⠀⡀⡀⠀⢠⡄
⠀⠀⢠⣿⠻⠃⠈⠀⣸⣿⡯⠉⢿⡀⠀⠈⢿⡿⠓⠀⢸⡄⠀⢸⣿⠇⠈⠀⠀⢿⣿⡿⠀⠙⢿⣦⠀⠀⠀⠈⠁⠀⠀⢐⢌⠂⢀⢹⣧
⠀⠀⠈⠁⠀⠀⠀⠀⢘⣿⡝⠀⠸⣧⠀⠀⠈⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣅⠠⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣰⡟⠿⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⢙⠿⡐⡅⡿⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⢸⣛⣿⣇⡟⣿⣀⠀⠀⠀⠀⠠⠭⠥⠱⡐⡸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢸⣿⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⣙⡾⢿⣿⣿⣷⣫⠀⠀⠀⠀⠈⠭⠤⠄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣷⣤⣤⣼⣿⣤⣤⣾⣟⠁⠀⠀⠀⠀⠀⠀⠈⢩⣭⢿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠦⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⠐⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⢉⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢷⠸⡇⠉⢠⡇⢀⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣮⡛⣦⣿⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠛⠄⠁⠀⠀⠀⣫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣚⣶⣿⣷⣿⢿⡻⠇⠀⠀⠀⠀⠀⡀⠀⠂⠃⠘⣱⣲⣥⠀⠀⠘⠁⠀⠀⠀⠒⠀⠘⠁
⠀⠀⠀⠀⠀⠀⡀⠠⢀⠠⡀⠀⠀⠀⠀⠀⢸⣿⣿⢩⡿⢸⡜⡗⠀⠀⠀⢀⠰⠂⠁⠀⠀⠀⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠡⠁⠃⠀⡀⠀⠀⠉⠁⠘⠁⠋⠁⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⡶⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠋⠠⡄⠐⠀⡀⠁⢿⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⣿⣿⢻⣇⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣒⣒⠈⠂⠘⢼⢷⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⢠⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣻⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⣒⣂⠐⢌⢃⠇⡄⠀⠀⠀⠀⠀⠀⢤⣿⣿⣾⣿⣿⣄⡤⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡷⠾⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠿⠥⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠝⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋

⠀⠀⠀⠀⠀⢠⠐⠀⢀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣶⡂⣄⠀⠀⣸⠀⡈⠁⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠁⠀⠀⣀⠀⠀⠀⢀⡶⠀⠀⡄
⣀⡄⠈⣿⠀⠀⠀⠀⠀⠀⠤⠀⡰⣄⠀⠸⠇⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠈⠐⢶⠗⠊⠁⠀⠀⢸⠀⠀⠀⠀⠀⢀⡀⠀⠰
⠶⠦⠛⠉⠹⣋⢛⣴⣦⣤⡆⠄⣐⠀⢣⠰⠛⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠔⠊⢁⠀⠀⡀⠀⠀⠈⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠙⠂⠘⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠮⠡⡀⠀⠀⠈⠉⠉⠻⠂⠀⢠⣄
⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠢⠄⣀⣀⡄⠔⠂⠈⠀⠂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⣀⢀⣀⡀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠁⠉⠑⠖⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠠⠠⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠰⠿⡓⠡⠆⠂⠤⣥⣖⡂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠑⠦⡖⠀⠀⠀⠀⠀⠀⠀⠃⠐⢠⡼⠉⢝⠲⠀⢀⣶⠠⠔⠀⠁⠀⠀⠆⠈⠁
⠀⠀⢠⢤⡤⣶⢏⡌⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⠱⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠙⢤⠤⣶⢺⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠂⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⢄⠚⣶⡄⠀⠀⠀⣆⡁⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠙⢤⡆⡀
⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⣀⠌⠠⠋⠀⠉⠀⠀⣰⢃⡃⠀⠀⠁⠀⠀⠀⠁⠀⡀⠀⠀⠀⠠⠀⠀⠀⠘
⣤⣤⣦⣜⣀⡛⠀⠈⠈⠀⠐⢀⠊⠀⠀⠀⢰⡾⠞⠁⠎⠀⠀⠐⠀⠀⠀⡆⠐⠀⠈⠐⠁⠒⠀⠉⠀⢀⣠⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠤
⠈⠁⡉⠀⠐⠃⡤⠀⠤⠠⠞⠁⠀⠀⠀⠀⠛⠁⠈⠂⠀⠀⠀⠀⠀⠀⠍⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⣀
⠠⠐⢀⠄⠀⠀⡡⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⣠⡀⢄⠀⠠⠁⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⢰⡆⡀⠀⠘⠿⠃⠂⠀⢈
⠟⠀⢁⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⠀⠈⠀⠀⠀⠀⠈⡴⠀⢀⡀⠀⠀⠀⠀⠀⠲⣀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⢀⡀⠢⠠⠈⠓⠠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣹⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡄⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⡄⠀⡀⢄⠀⠀⠀⠀⠀⠀⡂⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢡⠆⠀⠀⠀⠀⠀⠘⡂⠀⠀⠠⢾⡿⠈
⠀⠀⠀⠀⡨⠣⠄⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⢣⠹⠌⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠁⠈⠀⠀⣠
⠀⠀⠀⠘⠀⠀⠂⠀⠄⠀⠀⠀⢀⣰⠆⠀⠀⠀⣀⠀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠄⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⣀⡔⠁
⠈⠀⠀⠀⠀⠄⠤⠊⠈⠀⢣⡀⠠⠁⡆⣀⠖⠉⠀⠁⠀⠈⢋⡋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠔⠈⠀⢀
⠀⠈⠐⣄⠀⠀⠀⠀⠀⢈⠄⠈⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⢰⡵⠂⠉
⠺⠃⠀⠁⠐⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠠⠀⠉⠊⠉⢳⠂⠀⢄⡰⢾⠀⠀⠚
⢂⠒⠀⠀⠀⠀⠀⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠈⠺⠻⠖
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⡆⡴⡄⡄⡄⡶⢰⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠠⡄⠂⠀⡟⠛⡇⣇⡇⢻⠃⣇⢸⠀⠛⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠂⠀⢀⠀⡀
⠀⢠⣤⣶⣶⠆⠆⠀⠀⠀⠁⠀⠁⠈⠀⠈⠀⠉⠈⡁⠀⠀⠀⠀⠀⠄⠀⠈⠀⠐⠀⠀⠀⠀⠀⠊⠀⠁⠀⠀⠐⠀⠂⠀⠐
⠈⠀⢀⣤⣤⠀⢠⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠘⠊⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠠⠀⠀⠀⠄⡀
⠀⠠⠈⠻⠋⡀⠊⠀⢀⣀⣀⣈⣁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠖⠀⠀⠁⠐⠀⠈⠡⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⡀⡀⠀⠀⠀⣿⣿⣭⣿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠰⢴⢢⠀⢀⠀⡄⠀⠰⠁⠸⠀⠰⠆⠤⠀⠀⠀⠃⠑
⠀⠀⠀⠀⠈⠂⠀⠂⣿⣿⠿⣟⡻⢿⣿⠀⠀⠀⠀⢀⡠⠀⠄⠤⠤⠤⠤⠄⠀⠀⠂⠀⠀⢰⠀⠘⠀⠀⠈⠀⠀⢀⣴⡆
⠠⣀⢀⡀⣀⡐⠀⠀⠛⠿⠿⠿⠿⠿⠛⠀⠀⠀⠀⠈⠁⠀⡀⠄⠀⠈⠍⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠁⠀⠙⣆⠀⠰
⠀⣿⠘⠃⠏⠀⠀⠐⠋⠀⠈⡁⠁⠀⠀⠉⠹⠀⠀⠀⠠⠄⡠⠀⠀⢀⠄⠈⡅⠠⠀⠀⠀⠔⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠿⠛
⠀⣿⢠⠐⡇⠀⠀⠀⢈⠀⠀⠀⢴⡀⠀⠀⠀⠌⠀⢠⠀⢀⠀⢀⠀⢈⠀⡠⠁⠈⠀⠢⠀⠀⠀⠰⠰⠆
⠀⠉⠘⠊⠁⠀⠀⠀⠀⡄⠀⠐⠈⠁⠀⠀⠀⠀⡀⠁⠈⠀⠆⠠⠀⠀⠀⠄⠨⠈⡠⠀⠀⠑⠀⠀⠒⠂⠀⣀⣠⣠⣤⣤⣄⣀⣀
⠀⠀⢠⡶⢿⣛⠿⣦⡀⢀⣀⠈⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⣿⣼⣤⠝⠘⢯⣤⣿
⠀⢰⣏⠞⠋⠙⠿⣮⣧⠀⢀⣀⣀⣀⣀⣀⣀⠀⠲⠚⠴⠀⠀⠀⠀⡠⠂⠉⠁⠠⠄⡄⠀⠀⠀⠀⠀⠀⡆
⠀⠄⠈⠀⠀⠀⠀⠈⠈⠀⢼⠛⠛⡏⣿⢉⢻⡇⢀⢩⣀⠠⠋⠃⠀⠀⠾⠗⠀⠤⢀
⠀⠈⢺⣆⠀⠀⣠⣿⠀⠀⢸⡇⠀⢀⢸⢠⢾⡇⠙⠁⠁⣤⠛⠂⠀⠐⠤⠔⠀⠀⠀⠁⠁⠀⠀⠀⡤⠄
⠀⠀⠚⢛⠇⠈⠛⡙⠀⠀⢸⣧⣷⣼⣴⣤⣾⡇⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠦⠀⠠⠀⠀⠠⠄⠐⠂⠀⢐⣒
⠀⠀⣌⡁⠀⠀⠀⣈⡄⢀⣼⣇⣾⠸⢀⠸⢸⣧⠀⠘⠒⠚⠓⠲⠄⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⠛⠁⠀⠾⠿⠟⠃⠀⠘⠿⠿⠾⠿⠟⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠱⠀⠂⠀⡄⠲⡀⠀⠀⢀⠀⠀⠀⡀⠀⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⠂⠀⠀⠳⡀⠀⠀⠀⠀⡀⢀⠇⡀⠀⠈⣤⣤⣄⠀⠀⠀⠈
⢀⠀⠂⠀⠀⠀⠄⠀⠀⣀⠄⠀⠀⠀⢀⠀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠘⢻⠿⠟⠁⠐
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢴⣦⠀⠀⠃⠈⠀⠀⠀⠀⠀⢃⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⢸⠀⠀⠀⠇
⠀⠀⠀⢠⡤⠀⠰⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⢀⠄⣨⠀⠁⠀⠀⠀⢀⣴⣦⡐⠀⠀⠀⠠⡀⠀⡀⢰⠄⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠁⠀⠈⠅⠀⠀⠀⠰⠖⠅⠇⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠘⢯⠞⠁⠀⠀⠀⠀⢀⠀⠀⠀⣠⡄⠀⢀
⠀⠒⠒⠒⠒⠒⠚⠀⠀⠰⠀⠀⠀⠀⠀⠐⠂⠈⠀⠒⠀⠀⠤⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠒⠐⠉⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠰⢦⣄⣀⣛⣙⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣠⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣭⣶⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢸⣿⣿⣟⡽⠚⠋⠉⠀⠈⠁⠀⠀⠀⠠⣄⠀⠰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣇⣠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⣶⣶⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣶⣾⠖
⠀⠀⠀⠀⠀⠸⣷⣶⣦⣤⢠⡀⠀⣀⣾⣿⢿⣿⠿
⠀⠀⡀⣀⣤⣤⣤⡘⢿⣿⣿⣿⣾⣿⣿⣣⢿⣿⣱⣾⣿⣿⣿⣦⣤
⠀⠸⢟⣿⣿⡿⣿⣷⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡏⠉
⠀⠀⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⠻⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠹⢳⢫⣟⡿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠙⣴⣆⢀⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣄⣶⣶⠀⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣘⣿⣿⣿⣿⣿⣿⣤⣀⣀⣤⣾⡄⠀⠰⢶⣯⣵⣾⣿⣿⡀⠀⠹⣶⣾⢀⢀⣼⣖⡒⠄⠀⠰⣷⣦⣤⣤⡄
⠀⠀⠀⠀⠀⢠⣤⡤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⣿⣷⣧⣼⣿⣿⣴⣶⣿⣿⣷⣿⣿⣤⣤⡀
⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢦⡓⣆⢀⡤⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⣀⢣
⠀⠀⠈⣄⠀⠀⢷⠀⠀⠄⠀⠀⣠⠂⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠊⠁⠒⢠⡻⠴⢰⡂⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⡐⠁
⡀⢐⠀⠘⡆⠀⢸⠀⠀⠐⠀⡜⠁⠀⠀⠀⠀⡛⢼⠄⠀⠓⠀⢀⢠⠔⠀⠁⡙⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡅
⡁⠌⠄⠀⠸⡀⠈⣷⢤⣠⡎⠁⠀⠀⠀⠀⠀⠘⢿⠂⠀⠈⠁⣄⠀⠀⠘⠀⠁⠀⠒⠆⠀⢄⠀⠀⠀⠂⠐⢠⠘⠀⠉⠳⢤⡄⠠
⢘⠇⢰⠊⠀⢗⡀⠉⠀⣞⠀⠀⢈⡡⠖⠒⠉⠉⢻⡄⠀⠀⠀⠈⠂⠀⠀⠀⢹⡃⠐⠚⠚⠈⣴⢞⠠⣦⡲⠂⣤⣶⡶⣴⣿⣿⣾
⡮⡀⠻⢄⡀⠈⢧⡄⢠⠋⢀⠔⠉⠁⠀⠀⠀⢀⠠⠣⠀⠀⠀⠀⡀⠀⠀⠃⠼⣽⠉⣉⡯⢠⠉⠉⠁⠉⠡⣾⣿⣿⣿⡿⣿⣿⣿
⡇⡃⡀⠀⠒⣷⠀⠸⠚⠠⢇⠀⠀⢀⡠⠌⠺⡋⡀⠠⠀⠤⠄⠀⠁⠀⠀⠀⠀⢸⠀⠀⠁⠀⠦⠀⠀⠀⠀⠀⠊⡌⠃⠀⠈⡇
⣧⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢉⠁⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠐⠄⠀⠀⠐⠀⠹⣏⠤⠂⡞⠤
⣼⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠠⠀⠀⠈⠀⠀⡀⠁⡀⠀⠀⠀⠐⠄⠀⠀⢸⢰⠆⡀⡅⠀⠀⠀⠀⠂⠔⠀⠀⡀⠀⢀⡅

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⠀⠀⠀⢠⢒⣶⣶⣶⣶⣖⢆
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠙⡷⠖⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣿⠀⠀⠀⢰⠚⠸⠤⠔⠲⠤⠼⠘⢦
⠀⠀⠀⠀⠀⠀⢰⠷⠦⠄⢸⣇⡀⠀⠀⢠⢳⡀⠀⠀⣠⣤⠞⠁⠀⡀⢑⣤⠟⠀⠀⠀⢸⠀⠀⠀⡤⢤⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡟⠀⠈⠙⠢⡜⢸⠃⠀⢈⢳⡄⠀⠀⢆⡾⠋⠀⠀⠀⠀⠀⠘⠀⠀⣋⡁⠈⣙⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠷⣀⣀⠜⠓⢤⣀⣐⣈⠴⠃⠀⠀⠙⢅⣴⢦⣙⡏⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⡄⠀⡼⠀⠀⠀⠀⠀⠀⠀⠎⠔⢌⢀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠤⠤⠀⠠⠤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣠⠏⢀⡞⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⢘⠛⠛⡛⠛⠛⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠜⠀⣰⠋⣠⠏⠀⠀⠀⠀⣴⣯⠇⠀⠀⠀⢀⡔⣵⡧⠒⡄⠀⠀⠀⠀⠀⠀⠀⣷⡛⣵⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠈⠫⣁⠄⠠⢔⠕⢬⡯⠀⡈⢽⠒⠢⠀⠀⠀⠀⠐⠗⠌
⠀⠀⠀⠀⠀⢀⣌⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠃⠊⠀⠐⠤⠔⠀⠀⠰⠀⠀⢀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠈⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉⠉⢳⡄⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣴⣏⣩⠿⣍⣹⣼⠁⠀⠀⠀⢀⣏⣠⣒⣒⣂⣲⣬⣀⠀⠀⠀⠀⣃⠀⢹⠁⠀⠀⡜⣰⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠶⣄⠃⣤⠶⣟⠀⠀⠀⠀⠸⣏⠉⠉⠉⠉⠉⢀⣸⠃⠀⠀⠀⣿⠐⢸⠀⢀⣜⡰⢿⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⣴⠋⣀⠙⠺⢯⣀⣽⡆⠀⠀⠀⠀⢈⣉⣓⣆⣰⣚⡏⣹⠀⠀⠀⠀⣧⠘⢸⠀⡜⢠⠟⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠻⡍⠉⣿⠶⣄⠁⣬⠇⠀⠀⠀⢰⣉⣤⢤⣤⣤⣤⡖⠁⠀⠀⠀⠀⢷⣚⣸⡜⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠃⠀⠙⠶⠋⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠐

⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⢶⣶⡶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣤⣤⣬⣥⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⣉⣩⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣉⣩⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠋⠛⠛⠋⣛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⡀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⣤⠀⠐⡀⠀⠀⡄⠀⢠⡀⣰⠛⣿⠋⠉⢍⠀⣼⢟⣉⣽⣿⠁⠀⠘⠀⠀⠞⠀⠀⠀⠿⠀⠁⠦⠀⢀⠀⠀⡿⠀⠀⠀⠀⠀⢀⠀⢀⡀⠈⢶⣾⡿⢷
⠀⢀⠀⠀⠿⠀⠈⠃⠀⢈⠙⢿⢸⠇⠀⠀⠀⠀⠈⣿⣿⢆⡿⣠⣀⡀⠀⢦⣴⠆⠀⠀⢐⠀⠀⠇⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠋⡀⠠⠷⠂⠀⢻⡄⠄
⢰⣮⠛⠣⣈⠳⡄⠰⠶⢾⠁⠈⠈⣦⠀⠀⠀⠰⠀⢻⡟⠰⠛⠟⠋⠁⣴⠀⡍⠀⢾⣤⠴⠖⢶⠦⠶⠒⡤⣶⡶⡴⠆⣀⠀⠤⣀⠀⠀⠀⠀⠀⠰⢧⣴
⢀⡀⢦⡀⡏⢾⡟⡷⢀⣸⣟⣀⣫⣿⣶⡆⢠⣁⠜⠛⠧⠰⣆⠀⢰⠆⠀⢀⡄⠀⠓⠠⠟⣷⠄⠈⢠⡄⠄⠈⡁⠀⢀⣯⡀⠀⠉⠀⠐⠀⠀⠀⠀⠀⢾
⠀⠁⢠⡜⠃⠀⣿⣧⢻⡟⠉⠁⠀⠉⣸⣇⣿⠁⠀⠀⠄⡴⠂⠀⢃⣴⣦⠼⡇⣀⠣⠀⠂⢿⠀⠛⠀⠛⠀⠀⠀⠀⠄⠛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡝
⠀⠰⠄⠇⢸⠄⢸⣏⡮⢁⡀⠀⢸⣦⠸⠿⠀⣀⠀⣻⡄⡁⠀⢀⣿⠛⠿⠀⠀⣘⡂⠀⠀⠈⢁⣠⠀⢻⣇⠀⣀⡀⠀⠈⠠⠈⠀⠀⠄⢀⡄⠀⠁⠀⣸
⠀⣠⠀⠀⢠⠀⡰⣿⡗⠘⠃⣈⠨⡯⢰⡶⠸⡿⢢⡏⣡⣿⡆⠈⠉⠀⡄⠰⠀⠈⠁⢤⡄⢺⣿⡿⡄⠈⠁⠰⠛⠻⠄⡈⠃⢀⠘⠁⠀⠂⠀⠀⢠⠀⠉
⠀⠋⠘⢣⠇⠀⠧⠘⣣⢀⡀⢻⣾⣷⣴⢦⡟⠛⠛⠁⠈⠈⠀⠀⠀⠀⢀⠀⠀⢀⠀⠠⠑⠀⠀⠐⠀⠠⡀⠀⠂⢶⠴⠄⠃⠀⠃⢀⣼⠀⠀⠀⠀⠃
⡀⠉⡀⠀⡀⠀⠠⣈⠹⢾⠏⢀⠉⢀⡛⠀⢷⠀⠁⠀⠀⠀⠀⠀⠀⡀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠤⠆⠙⠀⣶⠀⠀⠀⢠⠀⠀⠠⢀⠀⢀⠐
⣷⣘⠓⢅⠀⣀⠀⢹⣿⣟⠀⠈⣀⡀⠊⡀⠉⠑⠉⠂⢀⣀⣀⠀⠀⣑⣿⣶⢴⣻⡄⠀⠀⠘⠿⠃⠀⠀⠀⢨⡿⣠⠶⣶⠶⡄⠀⠀⢠⠀⠀⠀⠀⣴⡦
⣏⣿⡁⠀⠀⠁⢰⠀⢿⡇⠠⢤⣿⡗⢠⠟⠠⠶⠀⠀⠻⣿⣿⡧⠸⢿⡍⠛⠀⠀⣹⠂⢀⣿⣇⠀⠀⠀⢠⣿⣵⡏⠀⠊⠧⠀⠰⠆⠠
⠈⣿⡆⣡⢀⣄⠀⠀⠹⣥⡄⡾⣇⡀⣼⢷⠦⣀⡁⠐⢰⡬⣼⣃⢀⣘⡇⠀⠀⠠⣾⣿⡞⣻⣏⢻⡄⢀⣾⣿⢋⡿⠀⢀⠀⠀⠀⠀⡧⠀⠴⢦⡀⢀⡀
⠀⠀⠀⢏⡰⣿⠃⢼⣿⣽⡗⡁⠸⣤⠷⠀⢻⣿⡆⠀⠰⠌⣫⣙⠊⠩⠄⣤⡄⠀⣯⣴⣶⣶⣿⣾⡇⢸⣿⣿⣿⣶⠀⠀⠐⣿⡷⠄⠀⣠⣴⣆⠁⢿⢀
⣠⣼⣄⠸⣏⣠⠀⢸⡟⠉⣿⣀⠃⠉⠀⠀⢀⡙⠛⢷⠶⠀⣳⡞⠈⡳⣄⠹⣿⡿⢻⠿⣭⣸⣿⣉⣟⣿⡿⠼⠛⣿⣿⣿⠛⠃⠤⠘⠀⣿⢿⡏⠀⠛⠉
⠟⠉⠉⠐⠉⣄⢰⣀⣧⠀⢿⡏⡠⠀⣀⣠⡀⢀⣦⣬⠜⠉⠁⠛⢸⣛⡟⠛⠋⢰⣿⣾⢃⣽⣀⣿⡟⣿⠃⠀⢀⣿⠋⠉⠀⠀⠈⢶⡷⣿⣼⣿⠖
⠀⠐⠀⠀⠀⣤⣤⢉⣼⢢⣄⣷⣠⡄⢿⣿⠁⠈⠃⠀⠀⣴⣦⣤⣼⠉⠉⠹⣷⡈⣻⠿⣿⡿⠛⢻⣿⣿⠀⠀⠈⠀⠀⣀⣤⣤⣤⣤⣤⠀⠀⣻⠀⠠⠶
⢸⠃⠷⢀⡀⠈⠑⠟⣹⣆⣙⣽⠟⠹⠦⣽⡆⢀⠀⢀⣾⣄⢀⣿⡏⠀⠀⠀⢹⡆⣹⡗⠯⠇⠀⣼⡏⢀⡀⣀⠀⠀⢸⣿⠏⠁⣐⠛⣿⣁⣤⣉⣁⣀
⢸⡄⢺⣯⠓⡄⠒⠀⠘⣿⣿⠈⠐⠀⠀⣼⣻⠋⢹⣿⢸⣟⣿⣿⠃⠀⢀⣀⢾⠈⡯⠁⠂⠰⢀⣼⣧⢸⣷⣏⣀⡀⠀⠾⠆⠈⠙⠃⡸⠋⠉⠉⠈⠉⢀
⠘⠆⢘⠙⢄⠀⠀⠀⠂⣽⣿⢠⣀⠀⢘⣿⣽⢰⡏⠈⢸⡏⠻⣿⠀⣀⠈⠉⢠⣞⢡⠀⠀⢀⣿⣿⡟⠀⠀⠉⠉⠁⡂⢀⠀⠀⠀⠀⡗
⠰⠂⣼⠟⠃⠐⣄⠾⢀⣯⣿⣬⢸⡁⠛⣿⡋⡞⠃⠀⣰⢿⡖⢻⠀⢀⠀⢰⠈⣿⠀⠐⢀⣘⡃⢻⠇⠀⠀⢀⣀⡀⠛⠙⡇⠀⠀⠀⣿
⠀⡈⢹⢰⣿⢲⡄⢠⣤⣶⠾⠗⠾⠆⢨⡍⣗⢷⣴⣿⣥⣄⣠⠛⣶⣿⣿⡸⢀⡿⢓⣺⡿⠋⠙⠃⠀⠀⣾⡿⠿⠻⣆⠈⢁⠀⠀⢠⠁⠀⠀⠀⠀⠐
⠸⠟⠿⢰⣿⡄⠀⠀⠃⠀⣤⡾⠂⣤⠀⣼⣿⡞⠋⢿⣧⠈⠋⣾⣏⡿⣟⢳⢤⡶⠟⠉⢀⣴⣟⠀⣀⣰⡎⣁⡓⠀⠹⠆⠀⣀⣠⡞⠀⠀⠀⡄⠀⠠
⠀⢠⠀⠌⠁⠀⠀⠂⡠⠀⢿⣧⠀⠎⠱⣼⣻⣇⠀⣾⠉⠁⠀⠋⠏⠁⢙⣰⡾⣿⣤⣾⡏⣿⡿⠻⣩⣿⣿⠻⣿⢦⣴⡄⠀⠺⣿⣶⠀⠀⠰⣱⡆⠀⣀
⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⡿⢣⠴⢔⠀⢿⡿⢿⣾⣧⣘⡀⠶⣴⣶⢀⠘⠻⡛⢉⡁⠜⡇⠙⠷⣤⣿⣖⣴⠛⡿⠀⣼⡇⣸⣧⣼⡏⠀⢀⡀⣡⢿⣿⣯
⠠⠀⣼⠛⠉⠀⢰⣦⠐⢈⠛⠀⣠⡀⠀⠋⡘⣁⢿⣳⣤⣉⣆⣠⣄⣌⣁⠀⢋⣼⣏⣺⠞⠀⠖⠻⣷⢿⡇⠀⠀⠀⢘⣷⠹⡀⣼⠛⠿⠦⢠⣬⣲⡏⣿
⢰⣦⠀⠀⠀⠀⣌⠃⠉⠸⡄⠘⡍⠓⢮⢀⣿⠟⠉⠛⣀⡜⠣⡿⡍⠹⠏⠀⣀⠹⣿⡻⡆⠀⢶⣄⢻⡾⢣⣤⠀⡁⠘⢋⣀⣡⣾⠀⣰⡾⢹⣿⡟⢡⣿
⠘⣿⡄⡆⠘⠁⠐⢠⠚⠠⡄⠀⢈⣹⣷⠀⠅⠀⢴⣰⠟⠃⠠⠢⠄⢿⣿⣧⠻⣯⣿⡟⠁⣶⡀⠍⢻⣿⣹⡏⠉⠀⢀⡽⠉⡽⣿⠀⣹⡴⣣⡞⠈
⢸⢿⣇⣿⡆⠀⣤⠖⣤⠀⡀⠰⣟⠛⢣⠚⠀⠐⠈⢻⠠⠀⠀⠀⠐⠀⢻⡛⢋⣙⢻⠖⠀⠀⡅⠀⢀⣽⣧⣿⡷⠀⠿⢿⣦⣴⠋⠀⠿⠁⣿⣻⣦⢹
⢸⡊⣿⡿⡆⢠⡟⠄⠙⢈⣉⡀⣏⠩⠽⠶⠄⠀⢠⡚⠀⠄⢀⣠⡄⠀⣿⡿⠿⠋⠘⠇⠀⢠⣇⠀⢈⠾⣿⣼⣧⣶⣷⣸⠿⢡⠶⠈⠀⠀⡿⢀⠳⠀⠈
⢸⡗⢸⣇⠀⢈⣷⣴⣦⠸⡟⢀⣙⠀⠀⠓⣼⢶⣾⠁⣴⣷⠀⢐⣳⠂⠌⠻⠾⠛⠙⢠⠇⠀⠁⠀⠀⠀⢻⠉⢻⠛⠁⣁⢄⡠⣶⣌⡀⢀⢿⣾⠗⠀⡤
⠸⡇⢸⡟⣨⢸⡿⠃⡙⠂⡀⢈⠀⠈⠀⠀⢹⡌⣿⠿⣯⢿⣦⠀⢸⠀⠀⠀⠊⠀⠐⠀⡀⢀⡀⣀⢀⣤⣨⣀⣼⡛⠃⠁⠀⠀⡉⠉⠐⠺⡇⠈⣧⣀
⢀⠃⣼⡷⠟⡌⠁⢰⡟⠿⠋⠎⠀⠀⠀⢠⣼⣿⠈⠀⠉⠀⣿⣧⠘⠀⠀⡜⠁⠀⠦⠀⢉⡬⠹⣿⣯⡇⢀⣈⡈⠃⠀⠠⣀⠈⠀⠠⠀⡀⠀⢀⣛⠙⣿
⢸⣉⡋⢸⠀⠀⠀⠀⠐⠦⠀⠙⡇⠀⠀⠈⢬⣿⣄⠀⠀⠀⠛⠛⢠⡀⠀⠁⠀⠀⠀⠈⠉⣀⠰⢚⠣⠖⠀⢘⠁⠀⣦⡄⠀⠀⣠⣀⠔⠀⠀⢹⠟⠀⠋
⠈⠙⠦⠄⠀⠀⠀⠐⠀⠀⠀⣀⣵⠀⣠⣼⣸⣇⣿⡆⠠⠀⠀⠀⢼⣦⠀⠀⠀⠰⠀⠋⠖⠉⣰⡯⠀⠀⣤⣼⢡⡴⣆⠸⡀⡼⠁⠈⠠⣀⣿⣴⠖
⠀⠀⢸⣶⣀⣤⣄⢀⡀⠀⠈⢉⡅⠀⠀⠁⠈⢻⡿⣶⣆⠀⠀⠀⡈⣇⣀⠀⠀⢠⣤⣴⡆⠀⠉⠀⢠⡄⠁⢰⠈⠁⠨⠖⠺⠏⠀⢰⡾⠏⠁⠈⢠⣀
⠀⠰⡇⠀⠀⠀⠀⠀⠁⠀⢠⡤⢶⣶⣿⡇⠀⠈⠃⠈⢹⡷⠂⢼⠃⠸⣿⣤⢐⣦⡙⠋⠠⠄⠀⠀⠀⠙⢳⡅⣀⣠⡀⣄⠀⣀⡀⠀⠀⠀⠠⣤⣶⡿
⠀⠀⢧⠀⠀⠀⠀⠀⠆⠀⠰⠀⠀⠙⢿⠋⠉⠁⠀⠀⠀⢠⣾⣶⠀⠀⣾⣿⣿⡋⠉⠒⠀⠀⢀⣀⣀⡀⠘⡁⠙⣀⠃⠀⣁⣭⠹⠳⣦⣶⠀⠿⡿⠦⠳
⢐⣷⣾⠇⢀⣄⠀⢀⣀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠛⠣⣾⡉⣥⠀⠀⠁⣹⣿⡃⠒⡠⠀⠀⠻⣿⢿⣿⠀⠁⢸⣿⠀⠖⢉⣉⠀⠀⠂⣽⠀⣤⡖⢶⠂
⠘⡿⠀⠀⠉⠁⠀⠀⠁⠀⠁⠀⠄⠠⠆⢀⠀⢘⡅⠀⠀⣰⣄⠉⠀⠀⠄⠹⠏⠀⠀⠀⠀⠀⠐⠊⠑⠉⣿⠀⠈⠓⢀⠐⠁⠉⢀⣦⠀⣤⡄⢹⡦
⠀⠀⠀⠈⠁⠀⠀⡄⠀⠰⠀⡀⠀⡀⠀⣈⡞⠈⠁⠀⠠⠟⠛⠛⠛⠆⠘⠘⠃⣿⣿⣿⣶⣀⣀⠀⣠⣜⣃⣀⣀⣀⣀⣠⣧⠀⡀⣠⣿⣿⣁⣈⣻⣣⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠁⠠⠈⡊⠀⠀⠈⠓⠂
⠀⠀⠀⠀⠀⣉⣉⡀⠂⠀⠀⡀⢀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢄⡠⠶⠤⡀⡀⠀⠀⠉⠉⠙⠀⠀⠀⠌⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢴⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⢀
⠀⠀⠀⠀⣦⣀⢀⠀⣸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠂⠀⠀⠣⡀⠠⠑⠀⠀⠐⠣⠀⠀⠀⠐⠀⠂⠀⠈
⠀⠀⠒⠀⠀⠀⠊⠉⠹⠀⠀⠀⠘⠄⣀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇⠀⠀⠀⠀⠉⠀⠅⠀⠀⠀⠀⠤⠀⠆
⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⡠⠀⠠⡀⠀⠀⡠⠠⢀⠀⠀⠀⠀⠀⠀⡑⠉⠀⠀⠀⡜⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢠⢁⡀
⠀⠀⡀⠀⠀⠈⡢⠊⠀⠀⠀⡀⠀⠀⠀⠰⡄⣀⠀⣰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⠠⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠈⢂⡠⠊⠀⠀⠀⣠⠊⠀⠀⠀⠂⠀⠀⠘⠁⠀⠀⠀⠀⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢢⣀⢀⡀⢠⠇
⠀⠀⡠⠊⠀⠃⠠⡢⠚⠉⠢⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⠁⠁⠀⠐⠆
⠀⠀⠀⠀⠀⡠⠚⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠐
⠀⠀⠀⠐⠈⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢲⠀⠀⡆⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠁⠀⠀⠀⢀⡌⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⣐⢫⡕⢀⠀⠀⠀⠀⠀⠐⠂⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢶⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠂⠀⠀⠀⠀⠈⠈⠂⠀⠀⠀⠘⢭⡅⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡀⠋⠰⠉⠇⣰⠀⠀⠀⠈⠀⠁
⠀⠀⠀⣤⡀⢀⢀⣠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⡉⠈⠛⠰⠃⠀⠀⣂⠈⠉⡁⡠
⠀⠀⢠⠀⠉⠹⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠚⠃⠁⠁⠀⠄
⠀⠀⠀⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠂⠀⢀⠀⠀⠀⠀⠀⠘⢣⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠉⠉⠀⠂⠀⠀⠠⢄⣀⠤⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢤⣀⠡⠄⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠒⠛⠂⠐⢢⣀⢈⢁⡒⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠭⠀⠀⠀⠀⠄⠐⠀⠒⠐⠶⢶⢎⠀⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡉⠉⠙⣫⣍⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⡦⠄⠠⠠⢠⣄⡀⠀⠀⠀⠀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣔⣤⣀⣩⠥⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠦⠉⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠚⠋⠁⠐⠶⣀⠂⢀⢙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⠥⠌⠀⠀⠀⠸⠀⠀⠈⠠⡥⡍⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡠⠟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢹⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⣃⣥⡠⠤⠤⠀⠀⢀⡤⠔⠋⢠⠤⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⣔⣓⡶⠒⠒⠀⠀⡄⠄⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⢀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠰⣯⡽⠇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠓⠒⠐⢊⠔⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢉⢣⡀⠀⠀⠀⠀⢀⠈⠒⠄⠠⠀⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠒⢆⠀⠀⠀⠀⠀⠀⠁⠀⠲⢜⡀⡃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣚⡂⣀⣀⢀⣀⡀⠀⠀⠀⢙⡓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠕⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⡶⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⢫⣅⠀⣧⠘⣆⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡋⠰⣔⠆⠀⡼⠇⣸⡄⠛⠰⠇⠈⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠖⠋⣉⠲⡜⢄⡙⣆⠒⠀⣀⣥⣶⣶⣶⠆⠀⠀⠀⢈⣙⣷⠆
⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⣊⡤⠘⠉⡀⠐⠚⢧⡀⠙⠦⢉⣤⣶⣿⣿⣿⡿⠿⣶⡤⣀⢀⣀⣠⣽⣿⠇
⠀⢸⣇⠈⠙⡒⡚⠓⢊⠉⣉⠁⠠⠒⠻⡅⠀⠀⠀⢱⡤⠞⣿⣿⣦⢩⣥⣬⡀⡀⡊⢀⣥⠘⡏⠉
⠀⣸⠋⠀⢸⠇⣹⣾⠀⣇⠙⢦⡀⠀⠀⠉⡄⣀⣴⢿⣴⡿⢿⣼⣿⣼⡟⣿⣧⣵⠃⢼⣯⠀⡇
⠘⣧⣄⡀⠾⠀⠋⠙⠀⠃⠐⠀⣃⣠⡴⣶⣿⣿⡧⣜⡦⣟⡶⢤⣛⡏⢸⣿⠿⣮⠱⣿⣿⠀⡇⡀⠀⡄
⠀⠈⣿⣿⣗⠒⠒⠒⣾⣶⠉⣿⣿⢻⢶⠿⠻⢛⣶⢸⣷⣯⣁⣶⣼⣷⢻⣯⠰⠀⠀⢉⣿⠀⡇⣧⠸⡃
⢰⣿⣿⢿⣧⣤⣶⠿⠛⠻⠀⣟⣿⣾⣛⣿⣷⡈⢛⣂⣹⡿⠛⠻⢿⠆⡾⣿⠂⠂⣤⣿⣿⠀⡇⣿⣦⠀⡀
⠸⣆⡈⠛⠋⠀⢿⣆⠀⢠⠀⣿⣿⣹⣿⣯⣹⣿⣷⣽⣿⣿⣦⡄⣸⠟⣿⣿⢡⣴⣿⣿⣿⠀⡇⣿⣿⠀⠁
⠀⠉⠻⣦⠀⠀⢀⣿⡶⣻⡆⣭⣿⣿⣥⣿⣿⣯⣿⢿⡭⠻⢽⡯⢠⡗⢸⣿⣾⠟⠁⣿⣿⠀⡇⣿⣿⠀⡄
⠀⠀⠀⠻⣷⣶⣿⣇⠀⠉⠃⡿⣿⣼⡿⣷⣼⣿⠟⠅⢈⠄⣾⡇⢸⡇⣽⣿⡿⡷⣄⣿⣿⠀⡇⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⢹⡏⠀⠰⣄⣶⣿⣿⣷⣿⡿⢁⠃⡄⣡⢆⣿⢁⣿⠃⣿⣿⡛⣽⣟⣿⣿⠀⡇⣿⣿⠀⡀
⠀⠀⠀⠀⠀⣀⣼⣗⠀⣀⡸⣿⡇⣄⢸⣿⣃⡈⡰⣴⣏⣹⣿⣾⣿⢹⣿⣽⣂⡿⢹⠟⣿⠀⡇⣿⣿
⠀⢀⣶⡟⢿⣭⣀⣼⠟⠿⠁⣿⡇⠇⡜⣸⣁⣀⡴⣿⣯⣿⣿⣿⡇⣼⣿⣟⠙⢀⠃⠀⣿⠀⡇⣿⣿
⠀⠸⣇⡀⠀⠙⢿⣁⣀⣼⠀⣿⡇⠀⡇⢹⣿⣀⠔⠛⠻⢯⡟⢿⣿⣿⠟⠯⣄⣿⣿⣼⣿⣬⡃⣿⣿
⠀⠀⠈⢻⣄⠀⠈⢻⠏⣹⢰⣿⡇⠀⡇⠀⣛⢷⣴⠃⣀⠀⠀⠐⣾⡿⣷⣶⡈⣶⡟⠉⠀⠸⡇⣿⣿⠀⠄
⠀⠀⠀⠀⠙⠳⠶⠾⠶⣿⢸⣿⡇⡄⢱⠚⣘⣧⡙⢻⣷⣶⣴⣦⣧⣴⠿⣼⡿⠟⠀⣀⣀⣀⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⡇⢸⣧⡽⠿⡟⠦⣌⡻⠛⢋⣁⣤⡾⠋⣿⠛⠛⣿⣿⠉⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⡇⡇⢸⣿⡟⢦⣉⣴⣋⠶⢋⣿⣏⠻⣧⣴⣏⡀⠀⣿⣿⠀⡇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣿⡇⡇⢸⣿⡇⠀⡼⠠⣽⣷⣿⣯⣮⣽⣤⢻⣟⠻⣤⣿⣿⠀⡇⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⡇⢸⣿⡇⢨⠃⢶⣻⣏⠛⢻⢿⣿⣿⣇⢹⣿⡗⣿⣿⠀⡇⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢸⣿⠀⢸⠒⢻⣿⣿⢐⣾⡿⣧⣘⣋⠈⣷⡼⣿⣿⠀⣧⡈⣀⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇⢸⣿⡆⠸⡆⠘⣿⡏⣼⡏⣰⣿⡏⡥⣰⠇⠀⣿⣿⠀⣿⠟⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠇⠸⣿⡇⣤⠙⣾⠇⠃⣿⡇⢘⣿⣏⣤⣯⢰⡇⣿⡿⢆⠁⠀⠀⠀⠀⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡇⢿⢠⡟⠰⢰⣿⣷⣿⣿⠋⢷⡟⢸⣧⠘⣑⢾⣅⠠⠄⣠⣾⣿⣉⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡇⢺⠀⢧⣤⣜⣋⣿⣿⣿⡧⠈⣁⠞⢡⣄⣙⠒⡈⢃⣾⣉⣉⣿⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⣹⡀⣀⣀⡉⡍⠩⢭⡝⡒⣯⡐⠀⠀⠈⢬⣁⣠⠟⠉⠉⢻⡇⠀⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣼⠿⢷⠀⢸⠀⢹⠈⠙⢦⠀⣀⡴⠋⠁⡖⠲⠶⢿⣄⣀⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠒⠋⠉⠀⠀⣬⣀⡈⢃⣈⣠⣴⡶⠟⠉⠀⠀⢠⡟⠀⠀⣿⡿⠋⠉⣉⣿⢿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⠃⠀⠀⠀⢀⣉⣻⣿⣯⡾⠁⠀⠀⠀⠀⠀⠈⢧⣀⣤⣽⣷⣶⠞⠁⠀⠀⠉⢿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⣙⣦⣤⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠠⠄⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡀⢴⡠⢖⠒⢦⣄⡆⡄⠉⠐⠠⠄⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠃⠒⢸⠀⠀⠉⠀⠀⠙⠀⢀⠀⡀⠀⠀⠀⠀⠀⢸
⠤⠄⠒⠒⠂⠀⠀⠒⠒⠬⡢⣀⠀⡼⠐⠒⠠⡀⠀⠜⠀⠓⠒⢲⠄⠴⠭⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠹⣀⠤⠀⠤⠜⡼⠤⠤⣀⠠⠊⠀⠀⠀⠀⠀⠋⠒⢄⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⡔⠀⠡⡀⠇⠑⠂⠤⠄⡲⠀⢀⠆⠀⠀⠈⠑⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢡⠈⠢⣀⡠⠔⢡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⣀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢭⢑⡆⠀⠀⠀⢠⠒⣯⠑⡄⠀⠀⢠⠖⢿⠙⣦⠀⠀⠀⡰⢻⡗⡆⠀⠀⢠⡖⣿⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠯⢁⡆⠀⠀⠀⠠⠚⣯⢑⡄⠀⠀⠠⠒⣿⢑⡄⠀⠀⢠⠚⣯⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠉⠛⠋⣀⣀⡀⠀⠉⠛⠉⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢿⢑⡄⠀⠀⠀⢠⡞⣿⢱⡄⣠⢶⡟⢿⡿⡛⣷⣤⠀⠀⡖⢻⡏⣢⠀⠀⠀⡖⢿⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠉⠘⠋⢸⠇⢹⡗⠀⡃⠀⢸⡿⢣⡀⠈⠙⠛⠁⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢿⣱⠄⠀⠀⢸⣿⣀⣾⡧⠓⡇⠀⢸⠀⠈⣇⠀⠀⠀⢠⡒⣾⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠁⠀⠀⠀⢀⡿⣿⢯⣇⢀⡗⠀⠈⠀⠁⢿⡀⠀⠀⠈⠙⠛⠉⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⢀⡇⠀⠀⠀⠀⠋⠶⣼⢃⣿⡤⠁⠞⢠⠂⠀⠀⢹⠛⣷⠹⠎⣠⠀⠀⠀⡋⢿⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀⡀⠀⠀⠉⣹⠐⣏⠔⣰⣿⡟⢠⣼⣮⢸⡆⢧⡇⠈⠁⠀⣀⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⣿⣈⡆⠀⠀⣿⡾⠛⡹⢻⡜⢃⣾⡿⡇⢻⢿⡼⣇⠀⢰⣉⢿⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠉⠁⠀⢀⡴⠋⠁⠀⠁⢸⡇⢸⠘⠁⡏⣿⠉⠧⣹⡀⠀⠉⠉⠁⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣻⣬⠇⠀⠀⠀⠰⣍⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡷⠁⠀⠀⠀⠙⢦⠀⠀⠀⣁⣺⣢⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣁⣀⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣿⣩⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⣷⣨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠉⠉⠁⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⠉⠁⢀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣛⣢⠗⠀⠀⡀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠞⠀⠀⠐⢅⣻⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠇⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣿⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⣛⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣠⡤⡀⠀⠀⠀⣠⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢘⡴⠃⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠫⣘⣧⠟⠀⠀⠘⢄⣛⡵⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⣶⢀⣾⡞⣠⠋⢠⣆⢄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡇
⠀⠀⠀⢀⠀⢸⣿⡇⣼⣿⢱⠏⣴⣿⣿⣿⣵⣿⣾⣷⣞⡤⠄⠀⠀⠀⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⠇⠀⠀⠘⢷⡀⣆⢀⡀
⠀⠀⡀⢸⠀⣿⢻⣳⣿⠇⢏⣾⢿⢿⡻⢙⠙⣻⣿⣿⡖⠂⠀⠀⠀⠀⣻⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡷⠀⠀⠀⠹⢆⠀⠘⢼⣇⣾⡀
⠀⠀⣘⣮⡇⠻⢈⡡⠳⢬⣿⣾⣶⣿⣿⣿⣿⡻⢿⡿⠍⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠿⠁⠀⠀⠈⢥⣒⣶⡄⠀⢸⡷⣜⡆
⠀⠀⠈⣮⠲⡊⣤⡰⣝⠲⣬⡛⠿⠿⣿⡻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣐⣒⣊⠀⠀⠈⡿⡁
⠀⠀⠁⠀⢧⣟⢾⣿⣮⡳⢄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠚⣛⣃⠀⠀⠈
⠀⠀⠀⠀⠘⢟⢆⠻⢧⠉⠀⠀⢀⣀⣀⣤⢖⣤⣄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡄⢀⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠭⢩⠿⠅⣙⣻⣟⡿⣿⣿⡿⣖⣢⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠙⠓⠺⠿⣿⣿⢿⣷⢶⣦⣶⣌⣭⣿⣖⡂⠀⠀⠀⠉⠛⠿⣿⠃⠀⠻⣿⣿⠿⠋⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠙⠛⠋⠊⠻⠛⠗⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡏⢀⣠⣾⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢁⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⢄⠘⡄⢆⢰⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⠀⢹⣿⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢠⣷⣷⣆⣦
⠀⠀⠀⡀⠐⣌⢦⢹⡸⣜⡆⡇⢰⠇⡠⠀⠀⠀⠀⠸⣷⣬⣿⣿⣧⣼⣿⠀⣼⣿⡟⢰⣆⠀⠀⠀⠀⢹⠟⠁⠀⣀⣄⠀⠀⠀⠀⠀⠘⣻⣿⣟⣿⡦⡀⡄
⠀⠀⠀⢌⡳⣌⢳⣷⣷⣿⣧⡇⣾⣼⠃⡠⠂⠀⠀⢠⣈⠉⠙⠻⣿⣿⡇⢠⣿⡿⣡⣿⠏⡀⠀⠀⠰⠁⢠⣶⠸⣿⣿⢟⣤⠀⠀⠀⠀⠚⣿⣿⡏⢹⣿⣧⡄
⠀⠀⠀⠤⣍⣲⢵⣿⣿⣿⣿⣷⣿⣣⢞⣡⠖⠁⠀⠸⢿⣿⣶⣶⣷⣾⢿⣿⣯⣴⡿⣣⣾⡷⠀⠀⢰⣿⡎⢻⣧⣿⣷⡾⢋⣴⣆⠀⠀⠀⢹⣿⣷⡈⠛⣿⣇
⠀⠀⠈⣙⣒⣻⠿⣿⣿⣿⣿⣿⣿⣿⡫⠔⠒⠁⠀⢰⣶⣦⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⣾⣌⢿⣿⣤⣿⣿⣿⣶⣿⠿⠋⡀⠀⠀⠀⢻⣿⣧⠀⢻⣿⡇
⠀⠀⣉⣉⣩⣭⠭⣽⣿⡿⠛⠋⠛⠛⠛⠉⠉⠀⠀⠀⠉⣛⣛⣋⣭⣴⡿⣿⣿⠀⠈⠁⠀⠀⢈⠿⣿⣦⣽⣿⣿⣿⣿⣥⣤⣴⣾⣷⠀⠀⠀⠀⢙⣿⣋⡤⣿⡷
⠀⠀⠀⠀⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣛⣭⣾⡿⠋⠀⢠⡀⠀⠀⢻⣷⣌⠙⣿⣿⣿⣿⣿⠛⠛⠛⢛⣡⠀⠀⠀⠀⠀⠺⣧⢼⣿⠅
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣼⣿⣦⡄⠀⣭⣛⣻⣿⣿⣿⣿⣿⡿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣿⣾⡥⠈⠛⠿⠿⠟⠁⠙⢿⣿⣷⡶⠆
⠀⠀⠀⠀⢀⡔⡆⠲⣶⣶⣶⣦⡀⠀⠀⠀⠀⣀⣠⣤⣄⣤⢠⣤⡀⢀⠀⠀⠀⠀⠸⠿⠷⡾⣿⢃⡄
⠀⠀⢀⣴⣿⠆⠘⠀⢽⣿⣷⣶⣤⠄⠀⠀⠀⠹⣿⣿⣿⣽⣫⡿⣷⢽⣾⣧⡆⠀⠈⣉⣛⣻⣣⣿⣴⢂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⣸⠀⡆⡄⡠
⠀⠀⡰⠫⢣⣞⢄⢡⠀⠠⢭⣫⢟⣿⡀⠀⠀⠀⠀⠉⠛⠿⣿⢿⣵⢏⡼⢿⡟⣿⠀⠀⠴⢿⣾⣾⣣⡟⣴⠀⠀⠀⠀⠠⡈⢦⣢⣆⣶⣿⣾⡸⡱⢃⡔⠁⡀
⠀⠀⠀⢁⠏⣾⢧⠀⠱⡈⠭⠾⣿⣾⣝⠂⠀⠀⠀⠀⠀⠀⠀⠉⠘⢻⠷⣿⡷⢛⣼⠃⠀⠉⠽⠿⣿⣾⡷⠀⠀⠀⠀⠢⣌⡳⣽⣿⣿⣿⣿⣿⡵⣫⠴⢋
⠀⠀⠀⠀⠁⠘⣾⠀⣴⠐⢀⠉⣽⣿⣿⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠋⠀⠀⠀⠀⠀⠀⠉⠀⠑⠀⠀⠀⠀⠐⠢⠭⢾⣿⣿⣿⣿⣿⣿⣓⣋⣁⡀
⠀⠀⠀⠀⠀⠀⢸⢠⣿⡎⣴⣐⠄⡛⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣟⠳⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠘⠀⠸⠣⠃⢿⣿⣕⢶⣽⣖⡀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠈⠓⠌⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠂⠛⠛⠓⠀⢠⡶⠿⠿⠿⠋⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠛⠿⠻⠿⠟⠙⠦

⠠⠱⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢠⡶⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⠋⠃⠀⠠
⠀⠂⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠄⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢧
⠀⠀⠀⠀⠀⢣⣥⡆⠀⠀⠀⠀⠀⠀⠄⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠀⠈⠀⠲⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠘⡃⡔⠀⠀⢸⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠄⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠃⠢⢞⠄⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠘⠁⢐⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡀⠁⠐⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢁⡀⢀⠀⠀⠀⡀⠀⠐⠋⠘⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠠⠀⠀⠀⠀⠀⠀⠊⠉⠑⠀⠀⠈⠃⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠖⢰⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠰⠀⢀⡂⠀⠀⠀⣀⣀⠔⢠⠀⠀⠀⠀⠀⠀⠀⠀⣅⠉⠀⢀⠛⡔⠀⠁⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣶⣌⣠⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠘⠃⢋⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠂⠀⠀⢠⢷⡾⠋⠁⠄⠸⡷⠆⠀⠐⠃⠀⠰⠆⠈⠁⠀⣄⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠤⢍⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣤⡴⠶⣧⠀⠀⠀⠀⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠠⣌⣙⡛⠲⢦⠀⠀⠦⣤⢶⡾⢋⠙⠳⠤⠀⢂⠀⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⡄⢿⡀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡀⠀⢸⡏⣷⠚⠁⠀⠀⠉⠀⢤⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣿⠀⠀⠀⠀⠐⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠉⣿⠁⢯⡀⠀⠀⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠟⠀⠀⠀⠀⠀⢀⡵⡋⠤⢤⣤⣤⡀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢠⡟⠀⠀⣤⡀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢤⣤⠶⢋⠆⠀⠄⠈⠋⠉⠄⠀⡄⠀⢀⣾⢁⠀⠠⠶⠟⠁⠀⠀⡠⠈⠃⢀⠆⠠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢄⡀⠘⠲⠟⠱⠇⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡾⠋⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣶⡂⠀⠀⠀⢈⡓⡀⠀⠠⠄⠠⠀⠀⠄⣈⣳⣥⡄⢠⣴⡎⠻⣄⡀⠀⠀⠀⠀⠐⢀⡠⠁
⠀⠀⠀⠀⠁⠀⠂⠀⠀⣸⡇⠀⠀⠀⠀⠀⢰⣁⡠⠘⠛⠸⣟⠲⣦⣤⣤⠆⣹⣇⠀⢠⠀⠀⠀⠀⠀⡿⠛⠙⠏⠉⠀⢰⡞⠛⠦⠀⢧⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠞⠀⠀⠀⠀⠀⠀⢸⡟⠇⠀⠀⠀⠹⣇⣠⣽⠃⠀⠀⠙⢦⠀⣿⣿⣤⣀⢠⣆⡀⠀⡀⣐⣤⣟⣧⠉⠒
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡈⢢⣤⣴⡟⠻⠟⠀⠀⠀⠀⡈⢿⡿⠛⠛⣛⣺⠿⠿⠿⣟⠁⠁⠘⡆⠑⠤⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠩⠄⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠉⠉⠉⠙⠉⠀⠀⠀⠄⠐⠂⠁⠀⠘⢧⣤⣥⣴⣾⣾⣿⣿⣿⠃⣀⠀⠸⣄⡴⠋⠀⠀⠤
⠀⠠⠄⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠂⠢⠀⣀⣠⠐⠀⠀⠀⢀⣶⣄⣹⣧⣀⠀⠌⠛⠛⢻⣏⠀⠀⠀⠈⠑⠂⠀⠀⢀⡅
⠀⠀⠀⠠⠤⢦⠼⠉⠀⠀⠀⠀⢀⡠⠘⢃⣠⡀⠀⠀⠄⠀⠀⠀⠈⠳⡀⠀⠀⢲⡞⢿⣣⣰⣧⣽⡿⡄⡀⠀⠀⠈⣧⠀⠀⠠⣄⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠹⡇⠈⠙⠓⠠⠤⢤⣤⠆⠈⠛⠒⠋⠠⠋⠀⠈⠀⠹⣤⣆⣔⣷⡶⠞⠁⠙⠒⠀⠉⠓⠂⢈⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡜⡠⠀⢠⡄⠀⠀⠀⠆⠀⠔⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣀⠦⠀⠒⠂⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⣴⠏⠀⢰⠏⡇⠀⠀⠀⠀⡐⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠈⣦⠎⠉⠀⠀⠀⡀⠐
⠀⠀⠀⠀⢀⠵⠂⠀⠀⠀⠀⠹⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠃⡆⠀⠉⡀⠁⠀⠀⠀⠀⢀⣄⡼⠀⠀⠀⠀⠴⠒⠚⠂⠀⠀⠀⠀⠰⠋⠀⠀⠰⣄⠐⠊
⠀⠠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠋⠀⠀⠀⠀⡀⢀⡿⠋⡤⠖⠀⠈⠂⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⡘⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣸⠦⡄⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡿⠟⢻⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣷⣶⠿⠟⠋⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠋⠁⠀⠀⠀⠀⣿⠘⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠛⠛⠛⠛⠓⠒⠶⠤⠤⣤⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣼⣿⠏⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣰⣶⣶⠀⠀⠀⣀⠀⣰⣿⡏⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⡿⠀⠀⢀⡏⠀⢀⣠⣤⣀⠀⠀⠀⣠⣴⣿⡟⠉⢳⠀⠀⠀⣿⣿⠃⠀⠀⣰⣿⡇⢿⣿⣧⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⣿⡇⢀⡴⠋⠀⣸⣿⣦⣄⢈⣳⡴⠚⢡⣿⣿⠀⠀⠈⠀⠀⣸⣿⡏⠀⠀⡴⢻⣿⠀⠘⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⣼⣿⣟⠉⢀⡴⠋⢹⣿⣿⠏⠉⠈⡇⠀⠈⣿⣿⠀⠀⠀⠀⣰⠃⠹⣇⣠⠞⠁⢸⣿⠀⠀⠀⠙⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡴⠋⠀⠀⢸⣿⣿⠀⠀⠀⡇⠀⠀⠹⣿⣆⠀⢀⡴⠃⢀⣀⡬⠵⠒⡀⢸⣿⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠵⠻⣿⡙⢦⠀⠀⠀⢿⣧⠀⠀⣸⠃⠀⠀⠀⠈⠛⠖⠋⠀⢠⡤⣄⠀⣶⣄⢷⠀⢿⡄⠀⣄⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢸⡆⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⢀⣠⣤⣤⡀⠀⣿⣇⠈⡇⣏⠻⣿⡆⠈⢷⡀⠈⠳⢤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⢀⡴⣶⣦⡀⠀⢀⡴⠋⠀⠀⣿⣿⡆⠈⠻⠗⠃⢈⣀⡤⠶⠀⠀⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⣶⣤⣄⡀⠀⠻⠁⠀⣰⠋⠀⠘⣿⣿⡆⡾⠁⠀⠀⢠⣿⣿⡇⠀⠀⠖⣋⣩⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠙⠿⣿⣦⣀⠀⠀⡏⠀⠀⠀⠸⣿⣿⡇⠀⠀⢀⣾⣿⣿⠁⠀⢠⣾⣿⡟⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡄⠀⠀⠙⢻⣿⣦⡀⡇⠀⠀⠀⠀⢻⣿⠇⠀⠀⣼⣿⣿⠃⠀⠀⣾⣿⡿⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠙⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⣿⣿⣿⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠗⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⢸⠋⠻⣿⣆⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢠⡏⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡒⡈⠀⠒⠲⢂⢤⣀
⠀⠀⠀⠀⠀⠀⠊⢅⠠⣀⣲⣁⠀⠐⠂⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⠲⢀⣈⣠⠊⠑⢭⠠⡀⠀⠀⠀⢀⠤⠒⠂⠉⠉⠐⠢⡄
⠀⠀⠀⠀⠀⠐⠁⠔⠉⠀⠀⠀⠀⠉⠐⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡖⢁⠱⣥⠀⡠⢚⢁⠖⠀⠐⠗⠟⠛⠣⠘
⠀⠀⠀⠀⢁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠁⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠒⠒⡂⠠⢤⣄⡀⠹⡄⢻⠟⡱⡍⢸⣤⠄⠋⣲⣀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠠⠊⡀⣰⡢⡆⠀⣠⡆⡤⠈⢊⠓⣼⡾⢘⣿⠿⠑⠒⠒⠚⠛⠛⠒⠂⠤⢀⡀
⠀⠀⠀⠀⠄⠄⠱⡀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠂⠀⠀⠀⠀⠀⠀⠦⠈⠛⠈⠁⣁⢨⣼⣷⢷⣺⡟⣦⣾⣏⢁⣠⣦⣤⣰⢖⣤⡤⡔⣄⠀⠄⠀⠈⠁⠢⡀
⠀⠀⠀⠀⠀⠢⠂⠘⠳⠤⠤⠀⡀⠔⠊⠀⢕⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢫⣽⣻⡭⣼⣞⡵⢟⡿⣿⠋⣧⠈⠢⡀⠉⠦⣼⠠⠑⠌⢣⠹⢇⣦⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠠⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠔⡉⠚⠏⢚⠕⣩⠶⢯⢼⡇⡩⢿⡞⣠⠈⢂⠀⠈⠢⠀⠀⠀⠉⠈⠁⠂⠂⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠡⠊⣮⠓⠂⡰⢡⠞⠁⣛⡀⣼⠂⢹⠛⠰⠻⠀⠂⢧⠀⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠒⠁⢀⠜⡰⠃⠀⠀⢹⠁⣿⢰⣼⠀⠀⠱⣀⣆⡁⠁⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⡼⠁⠀⠀⠀⠈⡋⣷⡼⠀⠀⠀⠀⠀⢛⠁⠬⣆⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⣰⠃⠀⠀⠀⠀⠀⠈⠯⠁⠀⠀⠀⠀⠀⠀⠑⢡⡸⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣿
⠀⠀⠀⠀⠤⠤⠠⠤⠄⠐⠠⠤⠀⠐⠂⠀⠒⠀⠂⠀⠒⠢⠤⠤⠤⠇⠠⡦⠤⠤⠤⠀⠤⠄⠀⠀⢀⠀⠠⡀⢀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⡀⠀⢀⣠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠚⠁⠭⠭⠭⢩⡉⡒⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣀⠄⠂⠈⠉⠉⡁⠻⡀⣠⣯⠕⠁⡀⠐⠦⠤⢀⡀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠔⠢⠄⠄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠁⡀⠡⠂⠀⢁⠀⡉⠀⡠⡢⠂⠈⠉⠓⠦⣀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣦⠞⠀⠀⠁⠀⠀⠤⠀⠀⠀⠀⡀⢀⠡⡂⠑⠈⡀⠕⠈⢀⠠⡊⡈⢀⠔⠀⠙⢶⡖⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠘⠀⡁⠐⡁⠑⠌⡚⠪⣀⠀⠀⠀⠠⣀⡰⠆⢀⠐⡁⢁⠀⠒⢀⠀⡡⠒⡡⢐⠨⡂⠅⠠⠀⡢⠸⠈⠀⠈⢢⠐⡄⠄⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠤⠀⢉⠐⠠⠁⠒⠀⠤⠀⡁⠉⠐⠓⠊⠭⠥⠈⠔⠀⠔⠠⠎⠴⠪⠴⠤⠒⠓⠋⠭⠑⢀⡡⠔⠈⠁⡠⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⣒⠀⠀⠉⠉⠁⠀⠀⠀⠂⠀⠠⠠⠌⠠⠠⠄⠀⠉⠩⠭⠥⠀⠀⠂⢉⡀⠠⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠉⠉⠀⠂⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠉⠙⠛⠛⠻⠿⢷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠉⠛⠳⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⠶⣦⣤⣀⣀⠀⣿⣧⣀⠀⠀⠀⠀⣨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⣿⣿⠿⢿⣿⢟⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠊⢏⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢈⣀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⠀⠈⢿⡿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠘⠲⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣤⣤⣄⣀⠀⣀⣠⠀⠀⠀⠀⣤⡲⢆⠀⣄⡀⠉⢻⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠾⠛⠋⢉⠙⠿⣿⢿⣿⣿⣿⣧⠠⡤⠀⣠⣿⣿⣿⣾⣿⣿⢿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠘⢿⠋⢀⣠⡴⠖⠛⢛⣚⣛⠳⣎⣿⣿⣿⣦⠾⢆⣿⣿⣷⠿⠗⠀⠝⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠚⠓⢁⣈⣥⣴⣶⣿⣿⣿⣿⣿⣾⣿⣷⡅⠉⠓⠑⣿⣿⣿⣾⣿⣿⣷⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⢻⠃⠀⣿⣿⣿⣿⣿⣿⠟⢡⠄⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣭⡉⠳⡀⠀⠀⢳⠀⠀⠀⠀⠀⡠⠐⠁⢀⠜⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣥⣴⣶⣾⡇
⠀⠀⠀⠀⠀⢸⣿⡀⠁⠀⠁⠀⢡⠀⠀⢸⠀⠀⠀⠒⠉⠀⣀⡴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣗⡨⠚⢿⣿
⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⢠⠀⠸⠀⢀⠜⠋⠀⢀⡠⠐⠚⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣂⢽⣿⡄
⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⢸⣄⣸⠖⢁⣠⡔⠈⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣀⢀⣠⣤⣤⣀⠀⢀⣿⣯⣹⢿⣿⠃
⠀⠀⠀⠀⣠⣯⡈⠙⠣⡀⠸⣿⠁⡴⠋⢠⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠙⠿⠿⠛⠿⢿⣿⣿⣼⣿⣿⣯⣻⣿
⠀⠀⠀⢠⣿⢻⠏⠳⠀⠘⢆⡏⢸⣇⣀⣼⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠿⣿⣿⢻⣿⣿⠇
⠀⠀⠀⢸⣎⣼⡄⠀⠀⠐⢮⣿⢟⣿⣿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⢶⢈⣙⢶⡶⢾⣿⣀⡍⢻⣦⣿⠏
⠀⠀⠀⠀⣿⣏⡇⠀⠀⠀⠀⣷⣬⣿⣤⡀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠶⣿⠿⠿⠛⠛⠿⣷⣶⠿⢿⣿⡿⢶⡿⠁
⠀⠀⠀⠀⠸⡿⢷⡀⠀⠀⠸⠀⠙⡿⣿⡿⡄⠱⡄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣿⣀⣴⣿⣇⡿
⠀⠀⠀⠀⠀⢿⡿⢷⠀⠀⠀⡇⠀⢹⢸⣿⡟⠀⠘⢦⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣶⣾⣿⣿⣿⣷⡟⠁
⠀⠀⠀⠀⠀⠸⣿⣪⣓⠀⣴⠄⢠⡟⠈⣿⣇⠘⢄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠙⠛⠏⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠞⠉⠀⠀⠹⣯⠻⣦⣷⡦⠀⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣶⣤⣄⣀⣀⣀⣀⣤⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠥⠤⠤⣀⡀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠏⠀⠀⠀⠀⠀⠉⠓⣄⠀⢀⡼⠁
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡞
⢀⡴⢏⠉⠋⠉⠙⠓⠦⣄⠀⣷⣀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠒⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⣾⡇⠀⠑⢦⡀⠢⡀⠀⠈⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⣯⠱⡀⠀⠀⠙⢦⡈⠢⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠹⡄⠱⣄⠀⠄⣀⠙⠢⡘⢦⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠐⠀⢀⡾⠁
⠀⠙⣆⠈⠣⣄⠀⠀⠀⠈⠢⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠈⠛⢦⣀⠉⠒⠦⠤⠤⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠴⠶⠚⠉⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⢀⡠⠞⠋⣻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⢀⣴⠋⠁⠀⠈⠁⡠⠖⠉⣠⠔⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠈⠛⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⣼⠃⠀⠀⣠⠔⠁⢀⡴⠊⠡⠀⠀⢀⢼
⠀⠀⠀⠀⣠⡞⠻⢭⣀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⣴⠋⠀⣀⠔⠁⠀⢀⣀⠤⠞⢁⡾
⠀⠀⢀⡾⠉⠙⠲⢤⣈⠉⠒⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡉⠉⠉⠋⠉⣉⣉⣀⡤⠶⠋
⠀⠀⢸⣧⣀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠒⠤⣄⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢿⡈⠓⠦⢄⡀⠀⠀⠀⠈⠙⠲⢤⣀⣉⣳⠖⢻⡆
⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⠑⠒⠒⠒⠒⠚⠋⠉⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣶⣶⣶⣶⣾⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠸⢼⠇⠀⠀⠀⠀⢠⣶⣦⡀⠀⢰⠀⠀⠁⠀⠀⠀⢘⣈⠁⠠⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧⠀⠀⠀⢸⡇⠈⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹
⠀⠀⢠⠀⠀⠀⠀⠀⠚⡀⠁⠀⢠⡥⢿⣄⠀⢻⣶⡄⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⢸⡀⠀⠀⠀⠀⠠⠁⡄⠀⠈⠀⠈⠙⣧⢸⡆⠀⠤⣀⢒⣾⣿⣀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⢀⣀⣀
⡿⠋⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣈⠈⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⢀⣿⡥⣾⡿
⠃⠀⠀⣸⠏⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣧⠀⢀⡀⠀⠀⠰⣶⣦⣼⡄⠀⠀⠀⠀⠈⢧⠀⠈⢿⡇⠀⠃
⠀⠀⠀⢫⠾⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡏⠁⠐⠂⠀⠀⠀⠀⠈⠐⠀⠀⠀⡀⠈⢇⠀⠀⠈⢀
⠀⡠⠀⠀⠀⠀⠉⠀⣀⣀⣤⣶⠛⠛⡐⠂⠀⠀⢤⣀⣉⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡼⠶⣶⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠽⠏⠉⢛⡷⣾⣛⠉⠉⠀⠀⠀⠈⠈
⠀⠀⢤⡀⢀⠠⠖⠉⠀⢀⠈⠉⠀⠀⡠⠀⢤⡀⠀⠀⠀⠁⠀⠀⠱⡍⠢⢭⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠐⠓⠀⠀⠐⠒⠀⠀⠀⠀⠀⢰⠀⠀⠀⠁⠀⠀⠀⠀⠀⡶⣧⡇⠀⠀⠀⠉⠉⠛⠛⢷⡴⠞⠁⠀⠀⢠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠇⠀⠀⠀⠀⡀⠀⠀⠸⠇⠀⠀⠀⣀⡘
⢙⣀⢀⠀⠀⠀⢀⡦⠀⠀⠀⢀⣀⣀⡦⠀⠀⢀⠀⠀⠀⠀⢀⠎⠲⣀⠀⠀⠀⠀⠹⡄⠀⠀⢀⠐⣾⠐⠉⠳⠒
⠋⡏⣸⠧⠤⠖⠋⢀⣶⢟⡺⢯⣍⣭⡄⠀⠀⠠⠀⠀⠀⢠⡟⣗⠀⠈⠑⠂⠀⠀⠀⢻⠀⡔⠻⠆⡠⠚⠐⠟
⢀⡁⠀⠀⠀⠀⠀⢸⡟⢸⣋⣐⡤⠄⠀⠀⠀⠀⠳⡀⠀⠈⡤⠚⣀⠀⠀⠀⠀⠀⠀⢈⣾⣱⡁⢀⡷⠤⣴⠋⢰
⠘⠀⠀⠀⠀⠀⠀⣠⡿⢛⠿⡿⣿⢯⣧⣠⡀⣰⠀⡗⠀⠀⣹⡉⠱⠆⠀⠀⠀⠀⠀⢸⣬⡉⠙⣿⡀⠠⣟⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠏⠐⣾⠚⠇⠈⣷⡦⣤⣴⡇⠘⠀⠀⠀⢹⡖⠆⢛⠀⠀⠀⢀⡜⡀⠠⠿⣤⣯⣤⣟⣁⣠⣀
⢶⡄⠀⠀⠀⢀⡀⠀⠀⢩⠀⠀⠀⣿⣼⠛⢿⡸⡆⠀⠀⠀⠘⣷⠬⠋⠀⠀⠀⠘⠀⠐⠀⠀⠀⠀⠉⠈⠉⠉⠉
⠉⠀⢀⡴⡞⢯⠉⠳⠄⠈⠀⠀⢀⣿⡇⠀⠰⣇⠁⠀⠀⠀⠀⠿⡂⠴⣢
⠀⠀⢸⠀⠀⠀⠀⣄⡗⠀⠀⠠⠞⡿⠁⠠⠴⠋⠀⠀⠀⠀⠀⠈⠣⠿⠟⠀⠀⠀⠀⠀⢀⡔⠛⠛⠐⠒⢠⡀
⣀⣀⣰⣛⣆⣀⣂⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣐⣒⣄⣀⣀⣀⣀⣄⣀⣀⣀⣀⣸⣀⣘⣛⣀⣀⣀⣘⣄
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⠄⠰⠂⠐⠶⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠉⠉⡍⠉⠉⠉⠹⡄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢀⣤⣤⣤⣤⣤⣤⣀⠀⣴⣦⠀⡁⢠⣶⡄⠀⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣠⢬⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣹⠀⠈⠁⠀⠃⠀⠉⠁⠀⡇⠀⠀⢀⣥⣀
⠀⠀⠀⠀⠀⠀⠳⠤⠏⠀⠀⣈⣿⣉⣉⣉⣉⣻⣉⠀⢾⡷⠀⠇⠰⣿⠆⠀⡇⠀⠀⠸⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠦⠀⢠⡞⠉⠉⠉⠉⠉⠉⠉⠉⠹⡆⠀⠀⡆⠀⣀⠀⠀⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢰⡀⢸⡇⠲⠶⠶⠶⠶⢶⠀⠀⠀⡇⠀⠀⡄⠘⠿⠃⠀⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⡇⠀⠀⠀⠀⠀⢸⠀⢀⡤⢧⡀⠀⠁⠀⠀⠀⣠⠇⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⣷⠈⠉⠉⣉⠉⠁⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠛⠛⠛⠛⠛⠀⣿⠉⠉⣿⢀⡾⣏⠉⠙⢦⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠹⣤⡤⠏⢸⡅⠈⠳⣄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠙⠶⠤⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⣀⣤⣤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡶⠟⠛⠉⠁⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⡿⣿⣯⡻⢿⠿⠿⠿⣿⡏⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣎
⠀⠀⣠⣴⣾⠿⠟⠛⠛⠻⣿⣿⣿⠿⢟⣱⣿⣿⣿⣶⠶⢿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣷⣄
⠀⣾⡟⠉⠀⠀⠀⠀⠀⠀⢹⣿⣶⠛⠉⠛⣿⣿⣿⡁⣀⣀⡈⢻⣧⠀⢠⡶⠛⠉⠀⢈⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣇⡀⠀⢠⣿⣿⣷⣮⡝⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠙⠻⠶⡶⠶⠶⠶⠖⠶⠦⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢛⣳⣾⣟⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣴⣿⣿⣿⣿⣷⣦⡸⣿⣿⣿⣿⣿⣷⣶⣝⣿⣿⣿⣿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣽⣵⣾⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⣿⣿⣿⡟⠛⠉⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠟⠛⠛⠋⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⢸⣿⠟⢁⣀⣠⣤⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠉⠻⣿⣇⠀⢀⣤⣴⣶⣦⣤⣿⣿⣿⡏⢹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣠⣾⠟⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⢀⣠⣤⣤⣜⣿⣶⣿⣯⣶⣶⣿⣿⣿⢸⣿⡇⠀⠻⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠉⠀⠀⠉⣿⠻⣿⣿⣿⣿⣿⣿⠏⢸⣿⡇⠀⠀⠈⠛⠿⠿⠿⠟⢛⣿⣿⡿⠋⠀⠻⣧⣀⣀⣠⡴⠶⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠈⠛⠛⠛⠛⠁⠀⣼⣿⣿⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣤⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⢺⠓⡀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠊⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡉⢀⢬⠝⠀⠀⠀⠀⠀⠀⠤⠤⡀⣃⠀⢠⠔⢓⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠾⡄⠀⠀⠀⠀⠀⠀⠈⠢⢻⣟⡔⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡚⣇⠴⠃⠀⠀⠀⠀⠀⠀⢀⣧⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⢏⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣖⣲⠖⢒⣒⡒⠲⣲⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢒⠲⣆⠀⠘⢆⠹⣤⠤⠤⠤⢤⣤⠋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠿⠃⡸⢀⣠⡈⠓⢌⡻⣦⠼⢛⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡒⠛⢧⡏⡹⠁⠀⡴⢛⡴⢎⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠤⣿⠃⢠⠞⣴⡯⠠⢤⣵⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⢼⣤⠋⣼⢉⠉⠉⢉⣉⣙⣆⢹⡀⠀⠀⢀⣨⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢨⠓⠛⠛⠒⠒⠒⠚⢻⡄⡗⢦⣀⠜⣡⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⡇⠤⠒⠨⠀⠀⠲⣾⠃⡇⠸⢡⣾⣩⣾⢄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠱⣤⠤⠤⠤⢤⣴⠇⡼⡡⠖⠣⡀⠳⠵⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⡙⢮⠽⢋⡤⠞⠊⢱⡂⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡥⢆⠛⢄⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣠⠿⠅⠨⠷⣌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣼⣓⣲⠖⠒⢲⡾⣆⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣸⠁⠀⠀⠀⠀⠀⠀⠘⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢲⠾⡀⠀⠀⠀⠀⠀⠀⠀⠰⠟⡴⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣦⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠊⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠦⠦⡍⠃⠀⠀⠀⠀⠀⠀⠀⠘⢢⠤⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣂⠓⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⠓⢄⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢇⠉⣐⠀⠈⠘⡄⡴⠊⠗⡙⢉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣏⠂⢄⠉⠰⠃⡰⢧⢇⡁⢸⠆⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡕⠒⠒⠪⢕⠤⡀⠀⢿⡟⡇⠀⠀⠀⠈⠫⣍⠉⡴⠊⢨⠄⠀⡵⠂⠸⠥⢲⡒⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⡎⠙⠻⣶⡀⠀⠉⠪⡳⢌⢯⣿⠀⠀⣀⡤⠴⠞⡗⢳⡒⠛⠚⠳⠠⠆⣀⣤⣊⠀⠠⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⡇⡁⠀⠀⠘⣵⠀⠀⠀⠈⠳⣽⣝⣷⠋⠁⠀⡠⠊⠁⠘⡀⠁⣬⣀⠃⠠⠀⡶⢭⣀⣐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠱⡳⣄⣀⢴⡝⠀⠀⠀⠀⠀⠈⣯⠞⣆⠀⠀⣅⠄⠈⠐⢙⠀⢧⢸⡀⢒⠀⠈⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸⣝⣛⡟⣆⠀⠈⢆⣰⡀⠚⢢⣈⠎⢦⠮⣡⣹⠁
⡰⠋⠑⠂⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠫⡏⣀⠹⠊⢧⡀⠀⠁⠙⠢⠇
⣇⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⣠⣤⡶⠶⠶⣤⣴⠾⣾⣷⢦⣤⠿⠛⠷⠻⠿⣦⣤⣤⣤⣤⣀
⠘⢦⣀⣀⡤⠞⠁⠀⠀⠀⢀⡴⣛⡉⢢⡀⠄⢡⢬⡸⣌⣛⡿⠀⠁⠔⡠⣒⡪⡔⠘⡆⢉⢰⠋⢨⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡏⣿⣷⠄⡇⣰⠿⢀⠅⣂⢀⢀⡈⢿⡜⣇⠳⠟⢣⣴⠣⠘⣾⡀⠻⡿⢃⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢄⠑⠬⠵⠚⣰⠯⡡⢀⡈⠄⠀⠃⠀⢸⡇⡈⠙⠋⠉⡲⠓⠓⢾⡝⠲⠒⠋⠂⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡈⠂⡬⠀⡀⣾⡟⢢⠐⠀⡠⠚⠉⢉⢉⢺⡇⢀⢄⠰⠶⢇⠿⣿⠀⣷⡠⠖⢛⠓⣅⠾⣇
⠀⠀⠀⠀⠀⠀⠀⣼⠋⢅⣁⡤⢴⡀⣿⠿⠠⠀⠀⡇⢴⣫⢯⢣⢸⡇⠁⠣⡠⠄⡀⢝⠩⡀⣿⠡⢸⣟⡿⢸⢆⢻
⠀⠀⠀⠀⠀⠀⠀⢻⠇⡔⠁⣴⠲⡌⣿⠦⠴⣥⠤⡱⣤⣉⣩⢼⢸⣇⠔⠋⡉⠑⣔⣘⠀⢈⣿⠀⠘⠲⡖⠁⠜⢻⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢧⠀⡳⢭⠖⣿⣤⣦⡄⢱⠈⠂⢠⣴⡅⣿⡇⠀⡎⣟⡇⣸⣀⣪⣦⣿⡢⠂⡻⠋⢋⡄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢙⠲⠤⠤⢿⣜⣚⣃⠜⢀⠄⢀⢠⠡⣺⠇⡁⡑⣢⢒⡏⣡⣤⣨⡏⣴⣕⠐⢈⠚⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣅⠖⡡⢌⡘⡇⢩⠝⡋⠉⠑⢦⡌⡄⣿⢀⠂⠐⠙⠉⢣⣙⣛⣼⠏⣤⣌⠑⣬⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣴⣶⣁⣹⢏⡀⡔⡭⣷⢈⡗⡈⠃⡸⣋⡙⢦⡕⣠⠀⡸⠁⡧⣝⣝⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢤⣧⣉⣡⣞⣀⣤⡾⢧⣿⣿⣀⣏⣥⡶⠶⠶⠶⠟⠋

⠀⠀⠀⢀⢠⠄⠐⠶⠀⠈⢈⡠⠊⠁⠀⠐⠃⠸⠃⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢟⠏⠉⠀⠈
⠀⠀⢰⢁⡾⠀⠀⠀⣀⡼⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠂⡔
⠀⠀⠈⠠⢤⣀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⢠⠄⠁⠀⠶⠘⢲⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⡤⠀⠀⠀⠶⠁⢀⡴⡁⠈⠀⠉⠀⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢶⣶⢶⢲⠒⡆
⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠐⡷⣮⡏⠿⠘⢰⠃⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠀⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣤⣴⡖⡖⡆⠀⠀⠀⠚⠳⠧⠆⠎
⠀⠀⠀⠾⠀⢠⠆⠀⠀⠰⣄⠀⢸⣴⡇⠀⠀⠀⠀⠤⠔⠀⠀⠀⢲⣾⡏⡇⠀⡇
⠀⠀⡀⠀⠰⠈⠀⠀⠀⠐⢂⢼⣿⣷⠯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⣰⡾⣠⠀⠀⠀⠀⡤⣡⠻⡟⠻⡕⠀⠀⠀⠀⢠⢿⣷⠀⣠⡤⣶⣶⡶⠀⠀⠀⣄⠒⣄⠀⢠⠀⠀⡆⠂
⢠⠀⠀⢐⣿⠁⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠩⠬⠱⡰⠉⠁⠀⠀⠀⠀⠈⠁⠘⢀
⢸⡄⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡽⣆⠁⠀⠀⠀⠑⣃⠀⠀⠀⠀⠀⠀⣀⡝⠀⠄⠀⣤
⠀⢳⡄⢰⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢸⣿⠁⠝⠀⠠⠄⠀⠄⡑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠃
⠇⣺⡄⣾⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⡄⠚⠃⠀⠈⠠⠢⠁⡀⠒⠬⣆⠀⠄⠀⠀⠀⠶⠄⠀⠁⠈
⣇⣟⣼⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠀⠀⠀⠀⠀⠸⠾⠛⠋⠈⠙⠀⣆⠀⠀⠀⠀⠀⡀
⠘⠋⠁⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣺⡇⠀⠀⠈⠢
⣤⠶⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠀⠈⣆⠘⡇⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢄⡀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠨⡇⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡯⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢆⣰⡂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⢀⠀⢀⠄⠀⠀⢸⠀⠀⠀⠀⠀⢰⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⢀⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⠉⠉⠉⢩⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⣶⣶⣶⡶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠁⠀⠀⠀⠈⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢁⣴⣾⣿⣿⣿⣿⣶⣦⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠀⠀⠀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠸⠿⣿⣿⣿⡿⠿⠿⠿⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⡉⠠⣿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠷⣿⠃⣀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⣿⡆⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⡿⡟⠀⠀⠛⠃⠹⣿⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠟⠀⠆⠈⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡷⡷⢠⡤⡤⣤⣾⢿⣿⣦⣤⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠶⠶⠶⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠀⣀⠀⠀⠀⣠⣤⠀⠀⣠⡄⢀⣠⡀⠀⠀⡤⠠⠄⠀⠠⠀⢀⣠⣄⣀⡀⠀⠀⣀⣀⣀⡀
⠀⠸⠀⠉⡜⠀⠀⠙⠿⠃⠐⠿⠀⠐⠛⢋⣤⣤⣤⣶⣶⣤⣤⣄⣈⠙⠉⠉⠁⠀⠻⢿⣿⣿⡿⠂
⠀⢠⠠⠊⠀⠴⢶⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡈⠁⣀⣀⠀⠀⠀⠀⡀⠄⠀⢸⢰⢺⡇
⠀⡄⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠘⠿⣿⣿⣿⣿⣿⡿⡛⠛⠛⠛⠛⢻⣿⠛⠁⠀⢤⢠⣿⠇⠀⠀⠈⠈⠈⠁
⠀⠈⠑⠀⠀⣾⣖⠀⠀⢠⣤⠀⠀⠉⠉⠀⠀⠀⢡⠄⠀⣀⠀⠀⢠⣶⣆⠀⠈⠻⠀⠀⠀⠈⠛⠀⠀⠀⠀⣶⢠⡄⡆
⠀⢰⣿⣷⡄⠙⠇⠀⠀⠘⠋⢠⠀⠀⢠⠀⠀⠀⠉⠀⠀⢉⠀⠀⠀⠉⠁⢠⣴⡶⠿⡇⠀⠀⠀⠀⣍⡀⠀⠛⠘⠃⠃
⠀⠘⣿⠀⠀⠀⠀⠘⠃⠀⡀⢸⡇⠀⠈⠉⠡⢠⢀⣤⢠⠉⠉⠀⡤⠀⢀⠘⠻⢷⣦⡇⠀⠀⠀⢸⣿⠄⠀⢠⢀⣀⡄
⠀⠀⣿⠀⠀⠀⠀⢸⣦⠀⣿⢸⣷⠀⠑⠀⠀⠀⠄⠀⠀⠀⢀⠠⠈⠀⠈⣀⣀⠀⠀⠀⠀⠀⢠⠀⠋⠀⠀⢸⠸⠯⡇
⠀⠂⠻⠤⢝⣤⡄⠀⣿⣆⣿⣿⡿⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⢿⡇⠀⠐⠄⠀⠀⢰⡀
⠀⠘⢷⣲⣼⡽⠀⠀⠈⠿⣿⠟⢁⣀⡈⠉⠀⠀⡆⢰⢶⡶⠂⠀⠀⡄⠈⠀⠀⠀⣰⣿⣿⣿⣦⠀⣳⠀⠀⢸⣶⣴⡇
⠀⠀⠀⠉⣿⣇⠀⠐⠀⢀⣴⣾⣿⡟⠓⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠰⣿⡄⠀⠈⠈⠉⠁
⠀⠠⡀⠀⣿⣿⣿⣶⣶⣿⣿⣿⣿⠃⠀⠀⢰⣿⣷⡄⠐⠐⠄⠀⠄⠀⠀⠀⠀⣀⣸⣿⣿⣿⣥⣄⢼⠃⠀⢀⢤⣤⡆
⠀⠀⠀⠀⠘⣿⠿⡿⢛⡉⠿⠛⠁⠀⠄⠀⠀⠉⣿⣷⣄⠂⠀⠀⠀⠀⠀⠀⢀⣿⡟⣿⢿⠿⣿⢻⡆⢀⠀⠰⠛⠹⠇
⠀⠀⠀⣶⣤⠈⠓⠁⠋⡀⠀⠀⣴⠀⠘⡶⠀⠸⠏⢹⡏⠀⠀⣿⠘⣿⡄⠙⢛⣱⢇⡇⣿⢸⣇⡀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣶⣇⢹⠺⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣿⣿⣿⣿⣄⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⣼⣿⣿⣿⣿⣿⣟⡿⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⢠⣤⣴⣢⣆⣾⣶⣶⣾⣿⣏⣸⡁⠈⣙⢻⠿⢃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⢿⠟⠟⢛⠈⠋⠙⠁⠀⢙⠁⣿⣿⣿⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣟⣿⣴⣬⣶⣷⣿⣷⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣀⣠⣴⣿
⠠⠄⢠⣤⣿⣿⣿⣿⣿⣿⡿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⢣⣤⣿⣿⣿⣿⣿⣿⣿⣷⠴⠾⢿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠈⠉⢉⣿⡀⠘⠛⠓⠰⣶⣶⣒⡂⢀⣀⣠⣍⣻⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣉⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠉⠛⠟⣛⢿⣛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃⠂⠀⠀⠀⠀⠀⠀⣾⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠗⠨⠩⠟⠯⠿⡿⢿⣿⠿⣿⣿⣿⣿⠿⠿⠭⠑⠀⠂⠐⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣶⠃⠀⠔⠋⠉⠙⠻⢶⣤⡀⠀⠀⠀⣀⣤⢴⠀⠀⡇⣼⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⣤⣤⣄⡀⠈⠀⠀⠀⠈⠉⢀⡸⠀⢸⣿⡸⣿⡇
⠀⠀⠀⠀⠀⢀⣿⠟⢻⠀⠀⠚⠋⠁⠀⠟⠓⠀⠀⠀⠀⠀⠰⠛⡄⠀⠸⣿⠀⣿⣯
⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⣿⢘⢹⣿
⠀⠀⠀⠀⠀⢸⠇⡄⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⡇⠸⢸⣿
⠀⠀⠀⠀⠀⢿⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠂⠀⠀⡄⠀⢸⡇⠀⢸⡏
⠀⠀⠀⠀⢀⠀⠀⢹⢸⣿⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠃⠀⢸⣧⠀⣸⡇
⠀⠀⠀⠀⠈⠀⠀⣼⢸⣿⣧⡀⠋⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠁⠀⠀⢨⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣹⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣸⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⡇⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠇⠸⢺
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⢸⡄
⠀⠀⠀⠀⠀⢀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⣼⢸⠰⡀
⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢠⠀⠀⣧⣅⡳⠾⢿⣾⣶⣦⣄⡀
⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⡆⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠾⠿⠿⣶⣶⡾⠿⠿⡇⠀⠀⠀⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠙⠻⣷⣾⡇⡆⠀⢠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣤⣀⠀⢀⣠⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⡇⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡾⠿⠟⢻⣿⣿⣿⡇⡇⠀⡞⠀⢸⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠸⠿⠛⠿⠂⠀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝
⣻⣶⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠄⠀⠀⠀⠀⠀⠀⠑⠓⠀⠀⠀⠀⢸
⠛⡻⠀⠀⠀⠀⣀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⣠⣴⡄⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠸⣵⣠⣶⣶⡶⢶⣖⠒⠀⠀⠀⠀⠀⠐⠛⠀⠺⠿⠿⠀⠀⠀⠀⠀⡏⡀⠀⠠⣧⣷⣾⣷⣾⢿⠇
⠀⠀⠰⠶⡀⠳⣿⣜⢿⣛⠛⢛⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⣼⣿⠿⣟⡆
⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣳⠫⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣤⣤⣄⡀⠀⠀⠀⠀⠘⣯⣾⡿⠁
⡀⠀⠀⠀⠀⠀⠣⡘⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⣠⡴⠾⠟⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⣀⣸⣾⠆⠀⠀⠀⢰⣶⣶
⠁⠀⠀⠀⠀⠀⠀⠈⠁⠐⠈⣠⠀⣄⠄⠄⠀⠀⠀⡟⠉⠀⠀⠘⠃⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠀⠀⠀⠀⠘⠉⠉
⠀⠀⠀⠀⠠⢤⣀⣄⣠⠎⣽⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀
⠀⠀⠀⠑⢤⣀⢹⣷⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⣀⣀⣀⣺⠩⠁⣀⣬⣤⣤⣒⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠙⠛⢷⣿⡟⠢⠀⠀⠀⡀⢀⣿⣿⣶⡎⣀⠀⢈⣀⣠⣤⣴⡿⣽⣿⢾⡷⣷⡷⠟⢿⣿⣿⣿⣿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⡀⠀⠀⢀⣤⣾⣿⣿⣷⣿⣶⣿⣿⣛⣿⢿⣟⠿⣯⡭⠾⠃⠀⠀⠈⠙⢻⡿⠟⠉
⣤⣄⣤⣴⣆⠀⠀⠀⠀⠀⠰⠝⠿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠁⠐⠀⣠⠬⠟⠚⡻⠟⠀⠀⠀⢀⡠⠔⠋⠁⠀⢀⣀
⠉⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⣽⣿⣀⡀⡠⠄⠒⠀⠠⠽⠖⣚⣋⣉⠠⠄⠀⠶⠿⡿⠿⠂⠀⠀⠀⠛⠛⠃
⠀⠹⣿⠏⠀⠀⠰⠶⠶⠀⠀⠀⠀⣠⠒⠛⣿⣿⠋⠐⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠬⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠟⠛⡉⠉⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⡿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⢀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⣛⣿⡿⠂⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣶⣶⣭⣵⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠄
⠀⠀⠀⠀⠀⠀⠰⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠟⠯⠷⠿⠃⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⢀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠃⢈⣿⡷⠂
⠿⠟⠛⠒⠤⠀⠀⢸⡁⡠⠄⠀⠀⢠⣶⣴⣤⣷⠄⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⢺⠟⠛⠁⠀⢺⠟⠁⠀⠀⢰⠤⠤⠴
⠀⠐⠂⠀⠀⠀⠀⠀⠑⠚⠃⠀⠀⠀⢸⣿⣿⡆⠀⠀⢀⡀⢠⣶⣾⣿⣯⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠄
⠀⠀⠀⠀⠀⠀⢀⣠⢄⣀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠁⠀⠿⣷⣷⡖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀