Cute wizard Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢀⠠⠄⠒⠂⠤⣈⠂⠀⠀⠀⠀⠀⠀⠀⣗⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠒⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⡠⠄⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠘⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⠀⡤⠀⠤⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⣷
⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠉⠉⠙⢆⡀⠈⠒⠘⠛⡫⠀⠘⢄⡘⢿⣭⠋⠉⣿⢉⡿⠓⠟
⠀⠀⠀⠈⠳⣤⣀⣀⣀⣤⠖⠋⡀⠀⠀⠀⠀⠀⠀⣨⡇⠀⠰⣊⢉⣀⠀⢀⠀⠈⠁⠀⠀⢀⣟⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠊⠀⠀⠀⠀⠀⡴⠋⠀⢹⡀⠀⠈⠉⠁⠐⣋⠀⠀⠀⠀⢀⡾⠉⣉⡦
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠘⠒⠢⣄⡀⠻⢤⡀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣠⠞⠁⠠⡀
⡄⠀⠀⠀⠀⢀⠠⠀⢀⡀⠀⠌⠂⠀⠀⢀⠊⠑⠊⠉⠉⠢⣄⠉⢓⠒⠒⠲⡚⡹⠋⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⡀⢀⠔⢉⠇
⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⡀⠹⡅⠀⡝⠀⠀⠈⢱⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠜⣂⡁⠋⠦⠤⠀⠔⢁
⠂⠀⠀⠀⠈⠤⠀⠤⠔⠀⡊⡀⠀⠀⠀⡌⡸⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢘⠀⠀⢀⠔⠃⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠘⢄⠘⣁⡎⠀⠀⠀⠑⠈⠄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⢀⢀⠂⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠈⠓⢄⣿⡠⠊⠀⠀⠀⠀⠀⠀⠘⡇⢀⠀⠀⠀⡠⠒⡀⠀⡂⠀⠉⠐⠀⠀⢠⠤⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⣜⠀⡀⡔⢎⠀⠀⠀⡸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⠄⠀⠀⠀⠀⠀⠀⠀⡁⠀⢄⠀⠀⢱⠀⠈⡖⠀⣀⡀⣀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠆⠈⡰⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠙⢄⡀⢀⣹⡄⠀⠙⡍⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⣀⣀⡀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠢⠠⡀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣀⠔⠋⠁⠀⡔⡏⠐⠀⠀⠀⠉⠑⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣸⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣛⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣀⣀⣤⣤⣤⣀⣀⣀⣀⠰⣀⣤⠤⠤⠔⠒⡟⠉⢃⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠁⠀⢰⠁⢀⠀⠀⠉⠉⠙⠋⠀⠀⠀⠀⠀⢡⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⡊⢱⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⡉⠖⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⢁⠜⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠓⢦⠤⢀⣀⣀⡀⢐⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠲⠋⠀⠀⠀⠀⠒⠲⡖⠉⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣀⣀⣀⣀⣀⣠⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⠗⠂⠤⠤⠤⠴⠖⠛⠉
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣟⡆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⢹⢇⠀⠀⠀⠀⠀⡔⠋⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠙⢲⠶⠦⢤⣀⡀
⠀⢸⢸⠀⠀⠀⠀⠀⠱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠒⢲⡀⡞⠉⠙⠲⢤⣀
⢀⡼⢼⡀⠀⠀⠀⠀⢀⡟⢡⣤⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠙⠦⡀
⡏⠀⠀⣸⡀⠀⠀⢀⡞⣀⡼⠀⠉⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢤⣀
⠳⣤⡞⡇⠙⠢⣀⣸⡀⠙⠓⣆⡴⠛⠛⠃⠀⠀⠀⠀⣀⣠⠤⠶⠒⠛⠉⠉⠉⠉⠉⠉⠉⠛⢲⠶⢤⡈⢳
⠀⢸⣇⣷⠀⠀⠈⡇⠙⣆⠀⠋⠀⠀⠀⠀⣀⡤⠖⠋⣁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢸⠀⠀⠀⢸
⠀⠈⠛⠛⠳⣄⣰⠃⠀⠸⡆⠀⠀⢀⡴⠚⠁⠀⠀⠀⠻⢭⡷⠀⠀⠀⠀⠀⠀⢯⠴⠋⠀⠀⢸⠀⠀⢀⡏
⠀⠀⠀⠀⡼⠸⠃⠀⠀⠀⢹⡀⡰⠋⠀⠀⣤⣶⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠶⠈⢱⢄⡞⠀⠀⠀⠀⠀⠀⣰⢷⡀
⠀⠀⠀⠀⣇⣀⡔⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢯⠞⠀⠀⠀⠀⠀⠘⠯⣅⠀⢩⠟⠁⠀⣀
⠀⠀⠀⠀⠈⠁⠘⣆⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣋⡙⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣴⠷⠒⠲⠷⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠥⠂⠀⠀⠀⠀⢀⣏⣀⠤⠤⠶⠖⢢⣄⣀⠀⢻⠇⠀⢴⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈⠙⠒⠦⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠚⠉⠀⠀⠀⠀⠀⡼⠁⠀⠈⠙⡆⠀⠐⢿⣠⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⢲⡖⠒⠒⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢴⠒⠚⠛⠒⠒⠤⣵⠘⠂⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⣰⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠈⠧⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠉⠁⠈⢓⣲⠖⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡶⡄⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠔⠛⠙⣆⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡤⠃⠙⠋⣽⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡄⣀⠤⠬⢷⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢶⠶⡶⠒⠒⠒⠒⠚⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠙⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠶⠶⠶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣀⣤⣤⠶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡟⠀⠀⠰⠋⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠋⠀⠀⠀⢰⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⣝⠮⡽⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⠀⠀⠸⠛⡚⠶⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⠉⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢀⡴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⡄⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣡⡖⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢷⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠉⣀⣤⡶⠿⣷⣄⠀⢠⡄⠀⠀⠀⠂⢁⠇⠸⢿⢿⣦⢸⣧⣄⠀⠀⠀⣠⣦⢀⠂⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣰⣾⠋⠁⠀⠀⢈⠙⢦⡀⠛⠦⠾⠟⠛⢋⣲⣾⣷⣿⠇⠸⣇⠙⢿⣶⣦⠟⣸⣀⡤⠶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣾⣯⠘⠀⠀⠀⠀⢠⠓⣄⡙⠦⠤⠤⠶⣒⣻⠿⣿⣿⠏⠀⠀⠈⢷⡦⠀⠁⠀⢈⣀⡀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⡄⠀⠀⠀⢧⡈⢻⡟⠀⠀⢰⡿⢋⡿⢸⠟⠁⠀⠀⠀⠀⢸⠁⠠⣼⣦⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⢻⣄⢹⡀⠉⢿⡀⠀⠀⠀⠉⠁⠀⠀⠀⠘⠛⠉⠀⣼⠃⠀⣀⣀⣤⣶⠏⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⠁⣀⢙⣾⠟⢦⠈⠻⣦⡀⠀⠀⠚⠢⠀⠀⠀⠀⢀⡾⠁⣀⡾⠩⣉⡉⠙⢦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢠⡞⠁⠀⠈⢳⠬⣧⣤⣼⣷⣦⣄⡀⠀⢀⣀⣤⣶⠿⠖⢻⣯⣤⣤⡀⠙⠆⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠔⠁⠈⠙⡄⢨⠡⠙⠙⠌⢻⣿⣿⠟⢉⡟⠁⢰⠀⣯⡀⠀⠀⠱⠄⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢀⣤⣿⠀⡶⠀⠈⢿⠋⣠⠾⠋⢠⡄⠃⠀⣿⡁⠈⠀⠀⣀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣠⣶⣿⣿⡿⢀⡇⠀⠀⠈⠋⠁⠀⢠⢈⣿⣰⠀⢹⡿⣧⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠇⣼⡷⢄⣀⠀⠀⠀⢐⣡⡾⣧⠻⣷⠞⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠞⢩⣷⢦⣬⡽⡉⣏⣹⣥⣶⣧⣰⠏⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠃⠀⠃⢀⠌⢉⠀⢧⠈⢹⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⢰⠀⡉⠀⠀⢀⡼⡠⢁⡴⠿⣶⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⠿⠏⠀⢠⣀⡀⠉⠀⠀⠀⡴⠞⡱⠿⠂⣀⣈⣀⣙⣉⣉⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠟⠛⠛⠻⠿⢷⣶⡦⢭⣓⠢⣤⡾⣠⡞⠅⠒⠉⣡⠔⠛⠉⠉⠉⠀⠉⠛⢶⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣾⡯⠥⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡌⠑⣾⡵⣋⡴⠊⣠⠞⠀⠀⠀⠀⠀⣀⣀⣤⠀⢀⣻⡷
⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣶⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋
⠀⠀⠀⠘⠻⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⣰⣾⢿⣦⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⣿⢸⣿⡇⠀⢀⣶⡿⣛⣛⣛⣛⣛⣛⣿⠿⠿⠿⢷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⣿⣏⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⠶⢶⡍⣛⣻⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣼⣿⠤⢿⣧⠀⠈⣿⡇⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣹⣿⣿⣿⣶⣭⡻⣿⣦⣀
⠀⠀⠀⠀⠀⣼⣟⣵⣿⡮⣿⣧⣼⡟⣵⠿⣰⣬⡆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⣛⠿⣷⣄
⠀⠀⠀⠀⠀⢸⣷⡹⢯⣴⣮⣝⢿⠹⣷⣬⠹⢋⣭⣤⣿⣿⣿⣿⡿⢟⣛⣫⣭⣽⣶⣶⣶⣶⣾⣭⡍⣟⡻⣮⢻⣷
⠀⠀⠀⠀⠀⠀⢹⣿⡿⠇⣿⣿⡷⣽⣎⢿⣶⣿⣿⣿⠿⣛⣭⣶⠿⠿⣿⣿⣿⣿⣿⣿⢿⡛⢿⣿⡇⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⡮⡻⢣⣿⣿⡜⣿⣿⢟⣵⣾⣿⣿⣿⣦⣽⣼⣿⣿⣿⣿⣿⣭⣵⣿⣿⡇⢿⣿⢇⣿⠏⠀⢀⣴⡿⣿⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢳⣷⣿⣿⣿⣿⡹⣣⣿⣿⣯⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣉⣤⣴⣿⢣⢏⣾⡟⠀⣾⡟⢛⣱⣜⡛⢻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⣭⡜⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣛⣿⣿⣿⣿⣿⡆⣿⡏⠀⣀⣙⣿⡶⠽⣻⠜⣿⣿⣉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣎⢿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢾⣿⣿⣿⣿⣿⢳⡿⠿⢿⣛⡻⢿⣶⣟⠙⣿⡿⢿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡎⣿⣿⣿⣶⣭⣟⣻⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣛⣫⣵⣾⣿⣿⣿⡟⣵⣷⣶⡝⣿⣿⣁⢿⣊⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣭⢩⣭⣵⡿⠻⣿⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⢱⣶⣶⣶⣯⣍⢹⣾⡟⣾⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢘⣛⣳⣆⠿⠿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣇⢿⣧⠀⠈⠻⣷⣭⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠼⢟⣛⣿⢫⣶⣿⣿⣿⣿⣿⣿⢸⣿⣿⡿⣰⣙⣛⢻⡞⣿⡆⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢇⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣚⢿⡿⠷⠝⢿⡼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣨⣾⣿⣿⣿⣷⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣬⡍⣩⢩⣭⣭⣭⣭⣭⣭⣭⣽⣷⣶⣶⣶⣶⣶⣾⣯⢽⣱⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠟⣼⡟⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣜⣣⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⢄⠀⠀⢀⠔⠁⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⢀⠔⠃⡠⠤⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣀⠀⠒⠀⠀⠄⠀⠃⡀⠰⠁⠬⠂⠀⠠⠋⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠉⠁⠒⠂⠀⢀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⡂⢄⢀⠈⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⢀⡣⠀⡄⠀⠀⢴⡒⠂⠀⠀⠀⢈⣀⠀⠀⠀⠆⠀⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠆⠀⠀⠘⠅⠀⢄⣦⠰⡇⠈⠓⠀⠀⠰⠀⠉⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢮⢦⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠄⣀⣀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠢⠝⠂⠀⢀⡀⠀⠤⡄⠐⠊⠁⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠃⡀⠀⠉⠙⠀⠢⠆⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡈⠁⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⡀⠀⠀⠞⠤⠄⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠤⡄⠀⢠⢀⣈⣢⠀⠀⡤⠤⠤⠄⠒⠊⠉⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢿⠂⠀⠀⠀⠀⣠⠖⠒⠚⠛⠓⠒⠒⠒⠲⠶⠤⣤⢄⣀
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠶⡄⡽⠛⠲⠤⣄⡀
⠀⠀⠀⠀⢀⣼⣄⠀⠀⠀⠀⢀⡗⢠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⡏⠀⢀⣇⠀⠀⢀⡞⣠⠜⠈⠁⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡈⠓⠒⠦⣄
⠀⠀⠀⠀⠳⢼⢹⠀⠙⢦⡸⣅⠉⢹⡶⠋⠉⠉⠀⠀⠀⣀⡤⠴⠒⠛⠉⠉⠉⠁⠀⠉⠉⠉⠙⡗⠢⠌⢳
⠀⠀⠀⠀⠀⢸⣼⣆⠀⠀⡝⠈⢧⠀⠀⠀⠀⢀⡤⠞⠉⠠⡤⢄⡀⠀⠀⠀⠀⠀⡠⢴⡄⠀⢀⡇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠉⠀⡷⡴⠁⠀⠈⣆⠀⢠⠞⠁⠀⠀⠀⠀⠈⠚⠁⠀⠀⠀⠀⠀⠓⠋⠀⣀⠸⢥⡀⢰⠃⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢁⠀⠀⠀⠸⡴⠃⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⣴⣣⠃⠀⠀⠀⠀⣴⠒⠉⠒⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠙⣄⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠦⠄⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⣽⡤⢬⣇⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣥⠀⠀⠀⠀⠀⣼⣀⣀⡠⠤⠤⣄⡈⠁⢰⡆⠀⣤⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠙⠒⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠔⠊⠁⠀⠀⠀⢀⡞⠁⠈⠙⢦⠀⠰⢏⡠⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢶⠒⠒⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⣞⠉⠉⠉⠙⠲⢷⠘⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠤⠋⢇⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⡤⣄⢠⠖⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⢦⡤⡄⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠁⠀⠈⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡀⢀⣀⣧⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠋⠁⠈⠁⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣤⢤⠤⠤⠴⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡖⢶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡀⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢀⡇⠀⠀⠀⠐⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣸⣿⡗⠀⠀⠀⠀⢉⣛⣛⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣿⣉⣀⣼⠿⠦⠀⠀⠰⠶⠶⠶⠦⣤⣬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢚⣋⣉⣉⣉⣉⣉⣡⡤⠶⣲⠖⠒⠲⠦⣄⡉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡏⠀⠈⣩⡟⢻⡏⢬⠇⠀⡇⠀⠘⠿⠃⢨⡟⢦⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⣿⡀⠈⢧⣸⠀⠀⣾⣝⣂⣴⣀⡀⣿⠀⠹⡄⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⠟⠳⠦⣤⣸⡀⠀⠘⠦⣤⡤⡲⢍⣿⣦⣀⣹⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⢁⣠⡀⠀⠀⠀⠉⢷⡆⠀⠀⠈⠙⠉⠛⠉⢧⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠘⠛⠃⠀⠀⠰⣾⡦⠘⠟⣄⣄⠀⠀⠀⠀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠲⠶⢤⣀⠁⠀⠀⣈⣅⢳⡄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣈⡛⠀⢻⣀⡾⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠺⠧⠤⠤⠤⠽⠓


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⡀⠀⠀⠈⢀⡠⠦⡈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠘⠀⠀⠀⠀⠀⠀⣄⠄⠂⠁⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠢⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡍⠀⠀⣀⣀⠁⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠐⠀⠀⠁⠀⠁⢠⢎⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠊⠀⠈⠈⠀⠀⠢⠀⠀⠀⠀⠁⠀⠓⠤⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠭⠥⢡⣤⠴⠀⠀⠰⠀⠀⠃⠠⠃⠀⠀⠐⠒⠒⠒⠒⡶⢖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣂⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⢌⠜
⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠔⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠐⠢⢤⠀⠀⠀⠀⠀⣀
⠀⠘⢉⠢⡀⡇⡜⠀⠀⠀⢘⠶⠘⢇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⣱⡀⠀⠀⠘⢅⢱⢀⠔⢉⠝
⢠⠒⠒⡃⠀⠀⢃⠀⠀⢠⠃⠀⠀⠀⠛⠦⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠱⠀⠀⠀⢀⡏⠀⠁⠐⠃⠒⢢
⠀⠉⠝⡡⠦⠔⠦⠉⠐⡇⠀⠀⠀⠀⠐⡄⠏⠀⠰⠀⡰⠀⠨⠇⠀⠀⠀⠀⠀⠀⢣⠔⠊⠫⡖⠢⠤⢌⠛⠄⠉
⠀⠀⠋⠁⠀⠀⠀⠀⠀⠣⠄⠀⠀⠀⠀⠐⢀⠠⠤⠔⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠑⠃
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢁⠸⡀
⠀⠀⠀⢀⠁⠂⠀⠀⠀⣀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢡⣀⡀⠀⠀⢆⠱
⠀⠀⠀⠐⠚⠂⠈⠁⠀⡜⠀⠀⠓⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢸⠀⠈⠉⠙⠚⠲
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⢄⠀⠸⣀⠀⢀⡠⠀⠀⠀⠀⠈⠉⠐⠢⠄⠄⠂⠁⠀⠀⠠⠤⠤⣤⣹⡀⠠⠢
⠀⠀⠀⠀⠀⠀⢶⠔⠉⠉⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠉⠺⢇
⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠴⠒⠚⠓⠒⠠⠤⠤⠄⣀⣄⠀⠤⠤⠔⠒⠒⠉⠙⠒⠒⠒⠲⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣀⠁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠢⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠉⠁⠹⠖⠂⢤⡀⠀⠀⠀⠀⠀⠀⠠⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⢦⣀⡤⠞⠉⠱⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠰⡎⢀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠖⠳⠎⠀⢠⠈⠑⠲⠦⢤⣤⡤⢀⠀⢤⡤⢤⡄⠀⠐⣠⡿⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠐⣇⡠⡏⠀⠀⢸⠃⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡜⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤⣤⡤⠾⣴⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠉⠁⠀⢀⡜⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠙⢧⡀⠀⠒⠒⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠉⢰
⡀⠀⠀⣠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡐⠰⠆
⠀⠀⢠⡼⠆⠀⠀⡀⠀⠀⠀⢀⡀⠂⠀⠠⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠄⢠⡇⠀⠀⠈⠉⠀⠀⢀
⠈⠁⢠⡜⡀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠲⠲⢤⣌⣉⠒⠠⣄⣀⡙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⣀⠠⠤⠒⣊⣤⠖⠁⠙⢦⣀⠀⠀⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠤⣌⣉⠓⠚⠛⡿⠛⠛⠛⣿⠟⠛⠛⠉⢀⣀⣀⣀⠠⠤⠔⠒⠊⠉⠀⠀⠀⠀⠀⠈⠁⠘⠧⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡙⠀⠈⠙⠉⠁⠴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠒⠒⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⢀⣀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠏⠀⢰⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠲⠶⠶⠶⠺⠟⠋⠁⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⢀⡁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⢨
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠲⣄⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡀⠀⢨⡛⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⢄⠀⣧⡉⣷⣾⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⠀⠘⢳⡆⠀⣤⣿⡇⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆⢸⣸⣧⡰⡶⣏⣳⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⣄⣦⢀⡈⢦⣀⣨⣯⣿⣯⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣻⡄⡤⣀⣻⢿⣷⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⡟⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⢈⣷⣠⣿⣴⣤⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣀⠀⣤⣦⣤⣠⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⢦⡈⡷⡄⣾⡰⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠙⡷⣧⠘⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢤⡴⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣛⢦⣠⣾⣮⣭⣭⣚⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⡾⠛⣹⡟⣿⠙⡟⢟⠻⢿⣶⣦⣍⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠴⠊⠀⢀⡠⣄⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣴⣿⣏⣠⠞⢁⢣⠃⡧⠇⡈⠇⢀⣿⠻⣷⣍⠺⣾⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢯⡀⠀⠋⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⢾⡟⣿⡞⢏⡡⠞⠻⠽⠀⠈⠳⠃⠈⠋⣄⠈⠻⣳⣜⢷⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡏⣾⢡⣹⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠙⣿⠻⣧⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⢠⡏⢸⡇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⠁⠀⡴⣂⠀⢹⡀⢻⡄⡇⠀⠀⠀⠀⠀⠀⡴⠉⠙⣦⠤⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⢨⠱⣾⠇⢠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⢸⡇⢸⢷⢸⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⢸⠀⣿⡀⠸⣌⡸⠀⠀⠀⢄⠀⠀⠀⢀⡆⠀⠀⠀⣠⡇⢠⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠘⣠⡿⡏⠲⣀⠀⠀⠀⠀⠈⠓⠶⠖⠋⠀⠀⣠⠞⠁⡇⢸⣿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣦⣿⠁⡇⠀⠀⠉⠒⠤⣤⣀⣀⣀⢠⡤⠲⣿⡁⠀⠀⠘⠮⣷⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡏⠀⠀⠀⢀⠴⠊⣿⠀⢀⡠⠴⠳⣄⣷⠙⠦⣀⠀⠀⠈⡶⢻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡇⣿⡀⠀⠈⠳⠦⣼⣿⡦⠋⠀⠀⠀⠀⢹⡏⠓⠚⠉⠀⠀⡇⢀⠱⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⡇⡇⠀⠀⠀⢀⠇⡇⢥⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⡦⠈⢆⡃⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢸⣵⠃⠇⠀⠀⢀⡌⢸⠀⠸⡄⠀⠀⠀⠀⡟⢨⢣⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡜⢀⠇⠀⠀⠹⣀⣤⣄⣼⠁⢸⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⡜⠀⠀⠀⠀⡟⠟⠻⠿⠆⢸⡀⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⡄⢰⢠⣏⠟⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠰⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠆⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠶⡄⣄⠇⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠖⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠚⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⣠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠈⠤⠵⠮⢤⣶⠦⢤⣀⠀⠀⠀⠀⣀⣀⣀⣀⠾⠤⣀⣁⣠⡇⠀⠀⠀⠀⠀⠀⣀⣧⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠈⠉⠉⠉⠉⠀⠈⠙⠿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠼⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠒⢶⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⣸⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠉⠁⠀⠈⣿⠹⣆⡀⠀⠀⠀⣀⣀⣤⠞⠋⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⡞⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠈⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⠙⢦⣤⣄⣠⣤⡴⠶⠋⠁⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠂⠀⠀⠀⢸⣷⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆⠀⠀⠀⠀⠀⠀⠠⣄⣀⡤⠶⠚⠛⠛⠋⠙⠛⠛⠻⠦⣤⣀⠀⣼⠋⢀⣠⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢠⣀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠉⠻⣧⡀⠀⠈⢉⡴
⠀⠀⠀⠀⠀⠀⠀⣶⡄⠸⣿⡄⠀⠙⣦⠞⠁⣀⣠⣤⣤⣴⠆⠀⠀⠀⠀⠀⠉⠻⣟⠲⠄⠀⠈⠻⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠙⠁⠀⣴⣧⠶⠛⣩⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠸⣇
⠀⠀⢀⣠⡤⠶⠦⣤⡀⠉⠛⠶⣼⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡤⠤⠦⣤⣀
⠀⢠⡟⠁⠀⠀⣀⣀⣽⣦⣄⣀⡟⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣀⡀⠀⠀⠙⢷
⠀⣿⠀⣠⡴⠚⠉⠉⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠳⢦⡀⢸
⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⢀⣀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠙⢿
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠷⢦⣄⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠶⣤⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⡄⠀⠈⠛⢯⣙⠛⠛⠛⠉⢉⣥⠞⠉⠀⠀⣼⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⣴⠏⠀⠀⢻⡄⠀⠀⠀⠙⠷⣄⣀⡴⠋⠁⠀⠀⢀⡾⠉⢳⡄⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡾⠃⠀⠀⠀⠀⠻⣄⠀⠀⠀⢀⡼⢿⡄⠀⠀⠀⢀⡾⠁⠀⠀⢻⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠹⣆⠀⣠⣿⣁⣀⣽⣦⡀⢠⡞⠁⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠁⠈⠉⠉⠀⠈⠳⠟⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢀⣀⣀⣠⣤⣤⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣴⠞⠋⠉⠉⠀⠀⢀⣠⢴⡏⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⣷
⠻⣤⣀⣀⠀⠀⣰⠟⠁⢰⡇⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⢾⣏⣠⡴⠋
⠀⠀⠉⠉⠙⠛⠻⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠈⠉⠛⠒⠲⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣶⣶⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢍⡱⠋⠉⠀⠀⠉⠒⢰⠆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢻⠻⠿⠓⢟⣀⠬⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⣳⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢿⣠⣤⡀⢼⣇⠀⢀⣞⠀⢲⣾⣿⣶⠛⠃⠀⣤⡙⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⣧⣶⡿⠊⣡⣤⣤⡐⢀⣀⡤⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣏⢹⠉⠁⣞⣵⣷⣤⡎⣿⣿⣿⣿⣿⠯⠃⠉
⠀⠀⠀⠠⠤⠤⡄⢸⠀⠀⣆⡖⠀⠁⠃⣸⠘⣿⢻⡋⣰⡏⠻⠛⡌⣽⣿⣿⣿⡅⠎
⠀⠀⠀⠀⠈⠂⠀⢸⣇⠀⣡⠁⠀⠀⠂⠘⠆⢻⣆⣿⣿⣧⢀⣰⢧⣻⣿⠯⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⠛⠗⠀⠀⠀⠀⠐⠂⠹⣦⣽⣿⣮⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠓⠀⡤⠊⠁⠀⠀⠀⠀⣄⣾⣿⣿⣿⠒⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢿⡧⠄⠀⠤⠀⠀⠀⠀⠀⠘⢛⣿⠛⣿⣽⣿⡿⠘⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣤⣠⣼⠷⠶⠓⠒⠀⠀⠀⠀⠀⢀⡟⠙⢯⣜⣥⣿⢥⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠂⠈⠉⠙⠙⠆
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠐⢀⠸⡇⠈⠑⢄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠳⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡈⠃⠀⠀⠤⠀⠀⠈⠀⢀⠀⠱⡀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄
⠀⠀⠀⢀⠐⠂⠐⠂⡃⢸⠀⠀⠈⠀⠀⢀⢀⡀⡀⠀⠀⠙⠢⣽⣀⡌⠀⣀⠶⡤⢤⣀⠀⠀⠀⣶⢰⠟
⠀⠀⠀⠀⡠⠐⠀⠀⠁⢨⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣣⠏⠀⡄⠀⠀⠀⠀⠤⣴⣯⣀⢀⡀⠀⢀⠶⡃
⠀⠀⠀⠀⠀⣠⠖⣃⠀⠀⢀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣯⣭⣭⠛⠀⠁
⠀⠀⠀⠀⠀⠨⠁⠙⡂⠀⠈⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠛⢿⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠂⠀⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠢⣀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆

⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣾⣟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠘⠻⠷⠶⠶⢺⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣠⡤⣄⠀⠀⢀⣴⠄⢀⣴⠆⠀⢠⣾⡟⢀⣤⠀⠀⣠⡀⢀⣾⣏⡼⠁
⠀⢀⡴⠶⠶⣤⡀⠀⠀⢀⣿⡏⣴⣿⡁⠘⡇⢀⣾⡟⠀⣼⡏⠀⣰⣿⠏⢠⣿⠏⠀⣾⡟⠀⣼⡿⠋⠀⡀
⢰⡏⠀⠀⠀⠸⡷⠀⠀⣼⡿⠀⣿⡏⠛⣾⠗⠻⣿⣀⣴⣿⠣⢶⣿⠷⠖⢾⡏⣠⣼⣿⢀⣼⡿⠀⠀⡼⠁
⠸⣇⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠛⠚⠋⠀⠀⠉⠉⣼⡏⢠⣿⠋⣇⠀⠈⠛⠉⠀⠛⠋⠸⣧⣠⠞⠁
⠀⠈⠛⠶⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⣴⠋⠉⢉⣿⠟⠲⢾⡟⠋⣿
⢰⣇⠀⡴⣿⡄⣰⢻⣷⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠸⣧⣠⡏⠀⠀⠀⠀⣼⣄⡀
⢸⣿⣸⠃⣿⣷⠇⢸⣿⠀⢀⣤⡄⣤⡄⠀⠀⢠⢤⠀⠀⠀⠀⣨⠉⠀⣤⠀⠀⢠⣿⣿⡟
⢸⣿⡏⠀⣿⡿⠀⢸⣿⢠⣿⣣⠇⠙⡿⢤⣤⠈⢻⠦⣤⡄⠀⣿⠀⢠⣿⡆⢠⠈⣿⣿⣿⠆
⢸⣿⡁⠀⣿⡇⠀⢸⣿⢸⣿⠁⠀⡰⠁⣼⡿⢀⠏⢠⣿⠇⣸⣿⣄⡞⣿⡷⠋⠘⠻⣿⡏⣤⣀⣤⣦⣄⣀
⠈⠛⠁⠀⠀⠀⠀⠀⠋⠀⠙⠛⠋⠀⠀⣿⣧⠞⠀⢸⣿⡴⠁⢙⣋⢤⣿⡇⠀⠰⣿⠿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⣠⠖⠚⢻⣦⣀⣀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠀⠀⠀⢰⡋⢀⣼⡿⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⡉
⠀⠸⠥⠂⣠⡿⠋⠉⣸⡷⠀⣰⡿⢡⠇⠀⠀⠀⠀⣀⣀⠀⠀⢙⣛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⢀⣀⣼⡟⠁⠶⣾⡋⠀⢰⣿⡵⢃⣴⡾⢛⡆⠸⣿⡀⠁⠀⢿⣄⠉⠀⢠⣴⠞⢳⠀⢀⣴⠶⣾⡏
⡼⠉⢸⣿⠙⣦⠀⠈⣿⢀⣿⠋⠀⣼⡟⠓⠋⢡⢞⣘⣿⣆⡠⣋⣻⣷⣠⣿⠟⠚⠋⣰⣿⠃⣸⡟⢀⡖
⢧⡀⠀⠓⠀⠟⠀⢠⣿⢹⡇⢀⡼⠙⠧⡤⠖⠋⠯⡥⠿⠃⠸⢭⠼⠟⠀⠻⢦⠤⠚⠙⠷⠖⠻⣧⠞
⠈⠛⢶⣤⣤⣤⣾⠟⠃⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⣶⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠛⡙⠟⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⣾⣿⣿⣿⢸⣇⣱⠞⠁
⠀⠀⠀⠈⢦⣀⣤⣤⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴⣯⣯⣴⣿⣿⣿⣿⣷⡼⠃
⠀⠀⠀⠀⠘⢻⡟⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣿⣿⡟⠉⣙⠿⣿⠏
⠀⠀⠀⠀⢰⡏⠁⢾⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡿⢻⣿⣿⣿⣿⣷⣈⣣⢸⠃⠀⠀⣄
⠀⠀⠀⠀⠀⠑⣄⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡀⠛⡀⠶⠀⣿⣿⣿⣯⡹⡷⠃⠀⣰⣾⣿⠗⣢⡤⣄
⠀⠀⠀⠀⠀⠀⠈⡻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣟⠻⠷⡴⢋⣾⣿⣿⣿⡽⠁⠀⢸⣿⣿⣿⣜⡯⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⡼⠿⣿⣿⣿⣿⣿⠿⠿⣿⢟⣿⣿⠁⠀⠀⣌⣙⢾⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⣼⣘⡅⣿⣿⣿⣿⢯⡄⠶⡀⣼⢹⣇⠀⠀⢸⣿⣿⣷⣹⡶⢄⣀⡀⠀⠀⣀⠤⠀⠀⠀⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠘⣯⣽⣿⣿⡿⣿⣿⣦⡛⢷⣧⣠⡾⠈⠑⢢⣼⣯⣿⣿⣿⣿⣯⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⡀⠀⠀⢀⠟⠈⠻⠙⠋⠓⠂⠈⠙⠻⠶⣿⣿⠃⠀⠀⠀⠈⠙⠿⣿⣿⣯⡭⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠀⢈⡟⠀⣠⠟⠁⠀⡄⠁⢀⠀⠀⠀⠶⢲⠀⠀⠀⠀⠀⣀⠀⠈⠉⠁⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠙⠀⢤⠌⠀⢘⠁⠀⠀⠀⠀⠈⠉⠀⠀⠸⣤⢷⡴⠀⣠⡄⣴⣿⡄⢰⣶⡋⢆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣾⠃⠀⠈⢀⠀⠀⠡⠂⠀⠀⠀⠀⢸⡈⠻⣆⣀⣋⣿⡼⠸⣷⣼⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⣹⠻⠟⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠨⣽⣿⠟⢋⡉⠀⠀⡄⢻⠇
⠀⠀⠀⠀⠸⡁⠈⠉⠁⡀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⢄⢀⠳⢾⢺⣮⠀⡷⢴⣖⡠⠒⢠⣤⣀⣀⠺⡄
⠀⠀⠀⠀⠰⠄⢤⠀⠀⠀⡀⠀⠀⠀⠈⠃⢠⠀⠀⠀⠀⠀⠀⠉⣳⠞⣿⣿⣦⣶⣿⣿⢷⣦⣝⡮⢭⣽⡟⠃
⠀⠀⠀⠀⠀⠀⠀⣴⠆⡄⠀⢲⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠛⠉⠀⢿⣿⠿⢿⣿⣧⡿⠿⣛⢉⣿⡻⣗
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡀⠀⠀⢀⣠⣼⣿⣧⢨⡟⣉⡦⠜⢻⣃⢠⣯⣏⠁
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡀⡀⠡⠀⣀⣨⣴⣿⠟⠋⠉⠛⠁⢀⣠⣶⣿⡟⠁⢈⡀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢣⣶⣤⣮⣟⡉⠀⠀⠀⣀⣴⣶⣿⣽⣿⣿⣤⣾⠿⠿⠃
⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡿⢿⣿⣿⣿⣿⣿⣻⣟⣻⣿⣿⣿⠿⠛⠋⢀⠄
⠀⠀⠀⠀⠐⠁⢀⠀⡀⠀⠀⠀⠀⠠⠀⠀⢠⣦⣖⠡⢿⣿⣿⣿⣯⣭⣭⣤⣼⠿⣿⣿⣿⣋⡄⠀⡄⠀⠁
⠀⠀⠀⠀⠀⠰⠘⠰⠀⣤⠃⠀⣀⠀⠠⡔⠋⠀⠀⠙⣿⡶⠦⠄⠀⠀⠈⠁⠉⠉⠁⠉⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠙⠀⠀⠀⠀⡼⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⢻⣛⣘⣧⠀⠀⠀⢀⣀⣠⣐⣿⠉⠈⠉⠛⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣉⣈⣿⣧⢠⣴⡫⠙⠉⠉⠋⣀⣠⣴⣶⣾⣿⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⣿⣿⡿⣿⡄⠀⣀⡄⣾⣿⢿⡻⢿⣽⣿⣿⡀⢸⣭⣿⣋⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣻⣿⣿⣿⣯⠄⠹⠰⣿⣿⣯⣴⣿⣿⡿⠉⠛⠛⢿⣿⣿⣃⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⡋⠙⠙⠉⣹⣿⣿⣿⣡⣴⣶⣄⠻⢿⣿⣿⣿⣿⡇⠐⢂⠀⠈⠘⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⠀⠇⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣩⠥⠄⠚⠁⠀⠈⠉⠛⣋⣉⣿⣿⣻⣿⡇⠀⠰⠆⠰⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣰⡖⠻⢶⣄⣴⢶⣦⣍⡉⠙⣿⣿⣿⣿⡶⠐⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢏⠀⠀⠀⠉⠘⠃⠁⠙⣷⣾⡟⢻⡿⣿⠳⢦⣀⣠⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠐⠀⠀⠀⣼⣿⣷⣿⣿⠁⠈⠀⠙⠧⠀⠉⠁⠀⣀⡠⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⠐⠤⢀⠀⣿⣿⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⡴⠃⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡍⠁⠀⣠⠄⢸⣿⣿⣾⡏⣿⣿⣧⣤⣶⠟⠀⣾⡿⠇⠀⡺⢁⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠄⠀⢸⡏⢀⣼⣿⣿⣽⣿⣿⣿⣿⣁⠀⠀⠀⣿⡧⠀⣸⢡⡮⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣧⠀⠀⣷⣌⢿⣿⣿⣿⣿⣿⣿⣿⠿⣷⡀⠀⣿⣯⠄⣿⠋⠄⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⡛⠀⣿⣖⣧⣌⡿⣶⣿⣿⣿⣿⠋⡿⠙⠧⠹⣷⡀⢻⣧⢨⣿⣹⡆⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣭⣿⣿⣿⣿⣯⣼⣿⠟⠿⠛⠃⠀⠋⠀⣀⠀⣹⣿⡀⢹⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣩⣿⣿⡇⢹⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠁⠀⠀⣴⣖⡒⠶⢤⣀⠉⢻⣿⣿⣿⣧⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣄⠉⠻⡶⣄⠀⢻⠛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⡬⠙⣧⡀⠀⠈⢷⡀⠀⠀⠙⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣅⠈⣿⣇⡀⠈⣧⠀⠀⠀⠈⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⡘⢿⣷⡀⢹⡄⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⣻⢷⣌⣷⡀⠀⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠿⠿⣿⣦⣀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⠒⢆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡀⢸⠀⠀⠀⠀⣀⣀⣀⠤⠤⠖⠒⠚⠉⠉⠁⠀⠀⠸⣄⠐⠀⢸
⠀⠀⠀⠀⠀⠐⠢⢄⠙⠈⠁⡠⢞⠛⠁⠀⢠⡀⢂⡄⠀⠀⠀⠀⠀⠂⠀⠀⠈⠑⠒⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡼⠀⢸⠀⠀⠀⡼⠙⢦⡀⢀⣀⡤⠤⠥⢄⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇⠀⠘⠀⡆⠀⠧⡀⢠⠟⠉⠀⠀⠀⡠⠔⠃⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠇⠀⡇⢃⠀⠀⢸⠁⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⢡⢸⠀⠀⢸⠐⣄⠀⠀⠈⣇⠀⠀⠸⢧⡀⠀⠂⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠀⡆⠀⠈⢧⡁⠀⠀⠀⠈⠳⠴⠧⣄⠙⡢⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢆⠈⡆⢡⠀⠀⠀⠙⠢⢤⣀⣀⣀⡠⠴⡋⢀⠃⠀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠀⡇⢸⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠘⠾⠀⠀⢠⠊⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢷⠘⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠆⠂⠀⠀⠀⡇⠐⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⠐⠊⠁⠉⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡄⣳⡀⠤⠀⠀⡂⠈⠁⠅⠀⠐⠀⠀⠀⠂⠀⠤⣐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡞⡽⢋⠕⠚⠁⠀⠀⠐⢒⣈⣀⠤⠖⠚⠉⠈⠉⠀⠀⠀⠀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⣷⠁⣇⣤⡤⠶⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⢀⣀⣀⣤⠶⢾⣿⠿⠿⣿⣶⣤⡀
⠀⢣⡀⣩⡧⣤⠎⠀⠀⠀⠀⡈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⣠⠞⢉⡳⢤⣀⣠⣤⣤⣤⠤⢤⣀⡤⠾⡉⠙⢦⣀
⠀⡿⣹⣿⢁⡎⠀⠀⣶⣀⣀⢹⡀⠹⣷⢹⠓⢦⡄⠀⢰⠉⠀⢀⣞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣸⣀⠀⠀⢱
⠀⡇⢹⠇⡜⠙⠉⠉⠀⠁⠉⠉⠱⡀⢹⡈⡇⡴⠁⢠⠊⠀⡤⠊⢿⠁⢀⠔⠉⠓⠒⠒⠒⠚⠲⡄⠀⡟⠻⣦⠀⠈⢢
⠀⣇⢼⡴⠁⠚⠙⠀⠀⠀⠚⠙⠂⠳⣤⣷⡏⠀⠀⢹⠄⢠⠏⠀⢸⣤⡎⠀⣀⡀⠀⠀⢀⣀⠀⠸⣤⣇⡴⠛⣦⠀⢞
⣠⠊⠀⣇⡀⠘⢗⡒⠾⠖⠒⡾⠀⣤⡏⠸⡄⠀⠀⠸⡄⢸⠀⠀⢧⡀⠀⠀⣁⣁⠠⠄⢈⣈⠀⠀⠀⡸⠀⠀⣳⠀⡞
⢧⠀⡴⣏⠓⠒⠬⡓⠦⠖⡞⠚⠉⣁⢧⠀⡽⠀⠀⠀⠳⠞⠀⠀⠀⣿⡷⣄⣘⢯⣉⣉⣡⣯⠤⢿⣿⠃⠀⠀⠙⠚⠁
⠀⠹⡀⠈⡷⠀⠀⢻⠛⣻⠀⠀⠸⡅⢀⣟⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⡄⠈⢱⡬⢥⠏⠀⢰⠊⢹
⢀⡴⢳⡀⡇⠀⠀⠘⡦⡇⠀⠀⠀⣇⡞⠈⢧⠀⠀⠀⠀⠀⠀⢀⡜⢇⢠⠇⠀⠀⣇⣸⠀⠀⠸⣆⡎⢣
⠛⠠⠤⠽⣇⠀⠀⣸⣁⣱⡀⠀⣠⣏⣀⣀⣨⠇⠀⠀⠀⠀⠀⠛⠲⠼⢿⡀⠀⢀⣇⣹⡀⠀⣠⣏⣠⡤⠗
⣤⣤⣤⣤⣬⣵⣾⣥⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣮⣤⣤⣭⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⢀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠂⠀⣷⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⣿⣿⡄⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠓⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠄⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠔⠁⠀⢀⡞⠀⣦⠀⢶⡀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⢄⠀⠀⠀⠀⡈⠀⠀⠀⠜⠀⠀⠀⠀⠈⠁⠰⠃⠀⠈⠙⢄⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡅⠀⠀⠀⠁⡀⣠⢂⠊⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡇⠀⠀⢰⣷⣿⡇⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠉⡟⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⣤⠄⠀⠰⠁⠀⠀⠐⣶⣄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠻⢻⣷⡞⠁⠀⠀⡰⡞⠻⠛⣿⠏⠀⢠⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠛⠀⠀⠀⠀⣷⣤⣤⡀⠁⣔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠰⠾⠟⠓⠉⠀⠀⡙⠻⡟⢧⡀⣀⠀⠀⢠⠟⠋⠉⠀⣼⠿⠛⡇⠀⠀⠀⠀⠀⠄⣰⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠈⢻⠟⠀⢀⡟⠀⠀⢀⡜⠀⠀⢠⠀⠀⠀⠀⢀⣼⣿⡿⠿⣶⡀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠀⡿⠀⠀⢀⡾⠁⠀⠀⣾⠀⠀⢀⣶⣾⣿⣿⠇⠀⠈⢹⡆⠀⡠⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿⠀⠀⣼⣿⠏⠑⠈⠀⡄⠀⠘⠃⠀⠀⠀⠘⠑⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠂⠀⠀⠀⣰⣄⠀⠀⠉⠀⡤⠀⣀⠛⠀⠀⠀⠀⢠⢀⡿⠟⠁⠀⠀⠀⠀⠀⢠⣾⡀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠈⠃⠀⠀⠀⢨⡜⣧⡞⠛⣤⡄⠀⠀⠀⡰⡄⠂⠄⠀⠀⠀⠀⠀⠊⡴⣿⡟⠁⠀⠈⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠀⣺⠿⠟⠛⠂⠉
⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠸⢀⡙⣿⠀⠀⢰⣿⡇⢰⡆⠀⠀⡀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠈⠠⡌⠀⠀⠀⠀⠀⠀⠀⢸⠈⠙⠻⠀⠀⠚⠀⠁⢸⡇⠒⣾⣿⣷⠟⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀⠳⣼⡇⠀⠀⡿⢁⡟⢡⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢧⡀⠀⣷⡞⠀⣀⡆⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⢿⡀⡟⠀⣼⣿⣷⡄⠀⠀⠀⠀⠉⢀⣶⣤⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣇⡆⢸⡆⠉⣤⠀⢿⣿⠋⣀⠀⡘⣃⠀⠀⠈⠻⠟⠛⠁⠀⣾⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣿⡇⠘⡇⡀⠛⠀⠘⡇⠀⢿⣴⠿⡿⣹⠂⠀⢀⣠⠔⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣠⠀⠀⠀⠀⠀⠀⠙⣿⠷⠀⣿⠀⠀⠀⠀⠇⢰⣇⣀⡄⠿⢀⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⡀⠀⣰⣴⣿⡿⣿⠿⠘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⡇⢀⣿⣸⣷⣾⣿⠿⢃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠿⠛⠂⠀⠀⣟⣼⢻⡟⠛⣩⣤⣤⣸⣿⡇⢀⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠁⠀⠀⠀⠀⠀⠄⢹⣿⢸⣿⠋⠛⢿⣿⣿⣾⣿⣛⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠓⠲⢤⣄⠀⠀⠀⠀⠀⣴⡏⢸⣿⣡⢆⣼⣿⣿⣿⣟⣛⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⣿⣇⢾⡿⠏⠺⠛⠟⣿⣇⣿⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⣿⣿⣾⡇⠀⠀⠀⠀⠈⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⣻⡏⠹⡇⢀⣀⠴⠶⢤⠄⠁⢠⣤⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣣⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡶⠷⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠛⠻⣶⡄
⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡟
⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡶⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣠⡿⠁
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⢻⡈⠇⠀⣠⢟⡻⣿⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡿⠃
⠘⠛⠛⢿⣷⣦⣤⣀⡀⠀⣠⡴⢻⣧⣾⡇⠀⠀⢠⠟⢻⡿⡇⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠘⣿⣇
⢀⣠⣤⣤⣿⣿⣦⣉⣹⡿⠷⣤⡀⣿⣿⣤⣄⣀⡏⣴⣿⡁⠀⠀⠀⠈⢿⣦⣤⣶⣾⡿⠿⢶⣿⣿⣆
⠀⠀⠀⠀⠀⠈⠻⢿⣿⡋⠛⠛⢿⡻⣦⣬⡿⠉⣽⣛⣛⣛⠛⠛⠂⠀⠾⢋⡵⠋⠁⠀⠀⢠⡇⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠈⢷⣄⣀⢀⣠⣿⣿⢿⣿⣟⡛⠛⠛⠲⠾⣧⣄⡀⠀⣠⡿⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⣩⣿⣿⠿⠁⠈⠁⠀⠉⠛⠻⢶⣾⣤⣭⣥⣶⠿⠋⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠛⢻⣿⠷⣤⣶⣤⣤⣤⣤⣴⣶⡾⠟⠉⠙⠿⣧⣄⠀⠀⠀⠀⠀⢀⣠⣴⡾⠇
⠀⠀⢴⣦⡀⠀⣀⣤⣾⡿⠋⠘⢛⣹⣿⣤⣾⡿⣍⠙⢿⣌⠙⢷⣤⣤⣤⣤⣤⣤⣥⣴⣶⣶⡾⠿⠟⠋⠁
⠀⠀⠈⢳⣌⠻⣍⣈⣻⣿⣿⡟⠉⠀⠀⠀⠙⠻⢶⣤⣤⣿⣷⡄⠈⠛⢯⣍⠉⢉⣵⣿
⠀⠀⢠⡟⣠⠞⠛⠋⠉⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠷⣦⣤⣀⡀⢈⣿⠟⢿⠁
⠀⠀⣿⠞⠁⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠈⠉⠛⠻⢿⣄⢬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣴⣶⣶⣶⣦⣤⣀⣀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠙⢦⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠉⠉⣍⠛⠛⣿⣿⣿⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣽⣆⣰⡟⠀⠙⠻⢿⣶⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡏⢹⠁⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣿⠀⣷⠸⣷⣶⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣰⡾⠋⠉⠉⠉⠉⢉⣛⣁⣀⣏⣀⣀⣀⣀⣀⣀⡀⠀⠈⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⣼⣿⣴⣶⠶⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡕⠆⠀⠀⡀⠀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⢤
⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⢢⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢻⣿⡿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⢁⡄⠦⠉⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢠⠀⢰⡀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⡴⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠂⠀⠈⠳⣼⡥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⣀⠀⠀⠘⠋⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠁⠈⠄⣠⠀⢀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠁⠅⠀⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠂⠈⢦⠀⠀⠀⠀⢀⠉⠀⠀⠀⢰⣶⣶⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⠆⠀⠀⠀⢺⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢠⡀⠀⠃
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉⠀⢀⡤⠉⠀⠀⠀⠀⠀⠉⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠃⠀⠀⠀⠟⠐
⠀⠀⠀⠀⠀⠀⠉⠙⢦⡄⠡⠎⠀⠀⠁⠁⡀⠀⢀⡀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠠⠄⠀⠠⣠⡄⢠⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢁⠠⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠⡄⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⠲⠦⣤⡀⠀⡴⠃⠀⠂⠀⠀⠀⠀⠤⣌⣦⡄⠍⣻⡶⣀⠁⠀⣼⣦⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⡰⠣⠎⢹⠀⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⠀⠀⠀⠀⡠⠠⢴⠃⠀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢁⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⣧⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⣿
⣿⣿⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⢀⠀⢀⣈⣉⣉⣡⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⢠⣬⣶⣿⣿⣿⣿⣉⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠸⣏⣤⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⣿⣿⣿⣿⣬⣿⣿⣿⡟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⣿⣯⣤⣿⣿⡟⣻⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⢸⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣟⣉⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣤⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣃⣹⣿⣿⣿⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⢸⣾⣿⣏⣩⣿⠿⠛⠋⠉⢉⣉⣉⣉⣙⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⢸⣿⣿⠟⣋⡠⠶⠞⠛⠋⠉⠉⣉⣈⣽⣏⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⠘⣿⡷⣿⣟⠠⢔⣶⣶⣷⡀⢨⣴⣯⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⢰⡏⣿⣿⢚⡷⠾⠽⠟⠋⠀⠀⠀⠀⠀⢸⣿⢿⣟⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⢀⣾⢿⡜⢿⡙⢷⡄⠀⠀⠀⢀⣀⣤⣤⣤⣼⣿⣮⣽⣯⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⢸⣩⣿⣿⣾⡿⣾⣇⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⣩⣭⣍⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠁⠀⢠⣶⣆⠀⢼⣿⡷⢠⡿⠛⢿⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣝⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⣠⣴⣾⣿⣦⣤⣾⣿⣿⣷⣌⠉⣡⣿⣧⣤⣼⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣩⣴⣶⣬⣙⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢉⣤⣶⠟⠉⣛⣿⠿⣿⣿⣿⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⡍⠛⡻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⠉⠰⡶⠀⣾⣿⣿⡆⢾⣿⡷⠀⣾⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡟⠉⠉⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⣿⣅⠀⣸⣧⠀⣹⣿⣄⢋⣴⣶⣦⣤⣬⣭⣉⣉⣤⣤⣶⠋⢹⣷⣦⣀⣼⣿⣿⣿⣷⡀⣩⣴⣾⣿⣿⣏⠀⢈⣷⣶⡀⣼⣿⣿⣿⡄⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⣿⣿⣆⠙⢋⣴⣿⣿⣿⣿⣿⣿⡟⠛⣿⡟⠛⠛⠿⠟⠋⠁⠙⣿⣿⣿⣿⠋⣤⡌⣿⣿⣿⣿⣿⡟⠋⢻⣷⣦⣿⠛⣽⣿⡿⢛⡙⢿⡄⣿⠀⠀⣿
⣿⣿⠀⠀⣿⡋⠀⣹⣿⣷⣿⣿⡏⠀⢙⣿⣿⣿⣷⣴⣿⣷⣄⠀⣰⣶⣦⢀⣼⣿⣿⣿⣿⣷⣬⣴⡿⠿⣿⣿⢿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣧⣘⣋⣼⣿⣿⠀⠀⣿
⣿⣿⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿
⣿⣿⣦⣴⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣴⣶⣶⣴⣶⣶⣦⣶⣶⣶⣶⣶⣴⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣦⣶⣶⣶⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠄⠀⠀⢸⡴⠊⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠠⠀⠀⠀⢀⡞⠀⠈⠀⠀⢸⡀⠀⠀⠠⡀⠀⠀⠀⡦⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣶⣾⣤⢤⣾⣄⣀⠀⠀⠀⣸⠁⠀⠀⠀⠱⡄⠀⠀⠇⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⡋⠰⠇⠞⠚⠙⠉⡙⢷⣎⡁⠀⠀⠢⢄⠀⢈⡠⠄⠒⠒⠢⢄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⣠⢞⣽⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⡦⡀⠠⠀⢠⠋⢠⠀⠀⠂⠀⠀⢱⠀⠤
⠀⠀⠀⠀⠀⠀⠺⢣⢊⡇⠶⠀⠀⠈⠀⠀⠄⠀⠈⠁⠀⠠⠦⡿⣿⣜⡄⠀⢸⡀⢫⡀⠀⢀⡀⠀⡸⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⠋⠹⠘⠅⠀⠀⣹⣦⢍⣉⣉⠤⠚⢁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡈⠓⠤⣄⣀⡠⠴⠊⢀⠴⠃⠀⠀⠀⠀⠀⣀⠜⠃⠀⡀⠀⢄⠀⠈⠑
⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⣉⡶⠢⠤⠤⠤⠒⠊⢷⠤⣀⠀⢀⠀⡠⠊⠁⠀⠀⠀⡇⠀⠸⡄
⠀⡴⠉⡩⢅⠈⣇⠀⠀⢤⣴⠊⢁⡇⠀⠀⠀⠀⠀⠀⠘⡄⠀⢹⠿⠋⠀⠀⠀⠀⠀⣈⠥⢄
⠀⢣⣼⣁⣄⡇⠈⡇⠀⠈⠉⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣹⠚⠁⠀⠀⠀⠀⢀⠤⢺⠔⠈⠚⡍⢱
⠀⢮⡤⢽⣉⡜⠉⣀⠀⠀⠀⠀⣀⣀⠀⡇⠠⠤⠄⡀⠀⣤⣄⣀⣀⡀⠀⠀⠘⠤⣌⠪⣶⡷⠙⡇
⠀⠀⠀⠰⣒⣾⣾⠽⠒⠊⠉⠉⠀⢠⠶⠃⠀⠀⠀⠳⠦⠀⠀⠀⠀⠈⠉⠁⠒⠦⢼⣷⣋⠑⠊⠁
⠀⠀⢀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⡄
⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠐⠁⠐⠱⢠⠂⠀⠀⠀⠠⠀⠐⠀⠆⠀⠤⠆⠈⠇⢣⠂⠧⠀⠀⠖⠀⠀⠀⠴⠠⠀⠀⠏⠁⠞⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⣿⠿⡏⠙⣿⢷⣄⠀⠀⢀⡴⣾⣷⣶⣦⡄⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣸⣇⠸⡇⠀⠀⢹⢉⠋⠆⢠⠟⡴⢿⠏⣸⢃⡇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠋⣻⣿⣇⠿⡀⣧⠀⠰⠀⣾⢰⣷⡟⣰⡞⠁⣰⠏⣼⣷⢀⡿⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⣿⣿⣶⠀⠘⣆⠠⠀⢸⡟⣿⣿⣏⣠⣾⢃⣾⣿⡿⠛⠛⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⣿⣿⣿⣿⡁⠀⠈⢣⣧⡀⣷⣿⣿⣹⠟⡡⢾⣿⣿⣡⡇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣙⣿⣿⣿⣶⣦⣻⣾⣿⣷⣿⣿⣻⡟⣸⣿⣿⣟⠫⣭⡀⣠⣾⣿⣦⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣶⠿⠋⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠋⠉⢉⡙⢻⣿⣿⣿⠋⠀⠀⠈⠉⠛⠛⠛⠋⢀⣴⣿⣿⣿⢿⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠻⣟⣲⣄⢀⠀⢠⣞⠃⠀⠙⣿⡁⠀⠀⠀⠀⣠⣀⠀⢀⠰⣿⣿⢹⡁⣿⣿⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠛⢠⡈⢛⣿⣿⠀⠈⠿⣇⠀⠀⠘⠳⣄⠀⢠⣾⣿⠏⠀⠈⢾⣿⣿⣬⣴⣷⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⣛⣳⣬⣿⣿⣿⠶⠶⠶⠿⠿⠷⠶⠶⣼⣷⣾⣿⡀⠀⠈⠑⠒⠛⢻⣿⡿⠛⠗⠉⣤⠶⠘⠀⡼⠁⢀⣍⡀⠀⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⠀⠀⠀⠀⣾⠋⠷⠀⢠⣾⠇⠀⠆⡜⠁⠶⠛⠉⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁⠀⢀⣤⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⢠⠃⠀⠀⣴⣿⢹⣆⠀⣰⠃⠀⠀⠀⠠⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠓⠒⠒⠀⠐⢢⣤⡀⠈⣷⠀⠀⣼⠀⠀⢸⣯⣻⢸⡿⠾⣥⠤⢄⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠈⢠⣶⣶⣶⣷⣶⣷⣦⡀⠀⠀⢰⣿⡏⠀⠀⢱⣷⣄⣿⠀⠀⢿⡃⢿⣾⣇⣠⡤⢀⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⡞⠻⠃⢿⣿⢻⣿⣿⣷⠀⠀⣬⣿⡷⠀⠀⠈⡟⣿⣿⠀⣠⠞⣹⣾⣾⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⢠⣿⡇⠀⣿⠿⣿⡼⣿⣿⣤⠀⣠⣿⣿⠃⠀⠀⠀⣹⣿⠋⣰⣷⠟⠉⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠀⠘⣿⣇⡘⣧⣼⣿⣷⣿⢋⣾⣶⣿⠟⠃⠀⠀⠀⠀⣿⣧⣴⠟⠀⠀⠀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣏⣿⠿⣿⣁⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⣴⡟⢿⡟⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⢶⣦⣤⣤⣞⣉⣿⣿⣿⣿⡟⠉⣠⣾⢋⣾⢷⢦⡄⠀⠀⠀⢸⣿⡇⣠⣴⣿⢱⡈⠇⠐⣯⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⣿⣶⠟⠋⠁⠀⠈⢻⡇⢸⣿⣷⣸⣏⡸⢘⡇⠀⠀⠀⢸⣿⠃⢈⣿⣿⣾⣇⣴⣷⣯⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠻⣿⣿⣿⣿⠿⠃⠀⠀⠀⢸⡿⠀⠀⠙⢻⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣄⠀⠀⢳⡀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣸⠟⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠸⢻⣧⠀⠀⠻⣄⢻⡄⠀⠀⠀⣀⠀⠀⣦⡏⠀⢀⣀⣉⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠈⠻⡇⢸⠋⣆⠀⠀⠀⢧⣿⡀⢠⣛⣥⠀⢠⣿⣁⣴⠛⠛⠉⢷⣆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⡦⠼⣿⡒⣿⣷⣤⣀⡈⢻⣷⣸⠛⠁⠀⣸⢸⠟⠁⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣀⠊⠁⣿⡿⣿⣿⡿⠟⠿⣿⣷⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠙⠛⢶⣟⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡢⡄⢀⣴⡆⣶⣄⠀⡤⣲⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣌⢳⡡⢼⣟⡓⢻⣿⠤⣱⢋⡜⠀⢀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠹⣷⣦⡀⠈⢿⢿⡛⡧⠹⣼⢦⣙⡮⢨⡾⠡⡟⣻⢿⠏⢀⣴⠿⠋⣀⣤⡤
⠀⠀⠀⠀⠀⠀⣤⣙⣿⣷⡤⠜⠛⣲⢼⣶⣽⢰⢂⣶⡝⢨⠐⣴⣆⠲⢸⣵⣾⠻⣍⣁⠒⢾⠿⣷⣶⠆
⠀⠀⠀⠀⠀⣴⣾⠋⣱⣞⣀⣼⣻⣷⣬⣿⡮⢨⣟⢞⡓⠄⠔⣻⢞⣯⠨⣼⣯⣵⣌⡱⠦⣌⢿⣆⠙⢿⣆
⠀⢀⡄⢲⣜⠋⢀⣼⠋⠉⠅⠀⠉⠉⠉⠁⠈⠁⠁⠉⠁⠀⠀⠉⠁⠁⠉⠈⠉⠙⠋⠡⠠⢀⠈⠻⢄⠀⢉⣶⠂⣄
⣠⣬⣌⣿⣿⠊⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠈⢺⣿⣏⣬⣤
⠈⠙⢿⠿⣻⠠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⢸⡿⢾⠟⠉
⠀⠐⠈⢶⢟⣀⡌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠨⣄⡘⢷⠎
⠀⠀⠀⠀⣉⣿⡁⠀⠀⠀⠀⠃⠈⠀⢣⠢⢂⠔⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠈⠐⠂⢀⠀⠀⠀⠀⠀⣹⣏⡁
⠀⠀⢐⠾⣦⠉⢓⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢐⡋⢥⡾⢖⠂
⢀⣤⣾⢖⣿⠒⠟⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠗⢺⣗⢾⣢⣄
⠉⢛⢉⢿⣿⠄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢼⣿⢏⢙⠋
⠈⠈⠃⠘⢏⡀⠈⢲⣄⠈⠀⠐⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠠⠀⠂⡀⢰⠈⠀⠀⡙⠄⠋
⠀⠀⠀⠀⠈⢿⣶⡈⢻⣎⠛⢶⣻⣿⠾⣷⠮⠱⣗⣽⣃⠤⢄⣻⣕⡷⠩⢿⣿⣾⣗⣢⡴⠎⣴⡟⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠾⢛⣷⣾⠦⠌⢉⡳⢿⣿⢾⢨⡙⣾⢇⠰⢀⢿⡞⣩⢸⢾⣿⣋⠟⠛⣀⣼⣭⡶⢛⡍
⠀⠀⠀⠀⠀⠀⠐⠛⠉⢡⣶⡾⠋⠀⣽⣴⣋⣦⢠⢶⡽⢞⡦⠴⢦⠠⣎⣳⣼⡅⠙⢶⣤⡌⠛⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠉⢉⠟⣱⠫⢼⣗⣋⣸⣾⠬⢳⡙⢏⠉⠙⠀⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡣⠇⠘⢿⡇⣿⠟⠀⠧⣻⣼
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⢀⣠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠉⠉⠉⠉⠛⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⡄⢤⡄⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣬⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡷⢂⣨⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⡄⠠⣀⣠⣶⣄⡀⠀⢰⣿⣿⣿⣿⡿⢟⡱⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣾⣠⠤⠀⠉⣤⣶⣿⡿⠛⢺⡇⠀⣿⠿⣿⠟⢥⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡯⠐⣿⢋⣹⣿⣿⣿⣿⣿⢆⣼⢻⣿⣿⣿⣿⣿⣿⡟⣣⣴⣿⡿⠟⠉⠀⠀⢠⡇⠀⣿⡄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣿⣿⣿⣿⣿⢿⣿⡿⡇⢼⢣⣾⡟⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⢀⠀⣿⣀⢀⣈⣵⣾⢿⣿⣿⣿⡷⠆⠄⠀⡾⠿⢉⣿⡿⠿⣿⣿⡟⠹⢷⡾⠃⠀⠀⠀⠀⡼⠁⠘⡇
⠀⠀⠀⠶⠶⠿⠿⠛⠻⡟⠛⣿⣿⣷⣾⣿⣷⣿⠟⠰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣦⡌⣷⣾⠞⠁⠀⠀⣠⠜⠁⠀⠀⢀
⠀⠀⢀⣠⣤⣴⠶⠶⠚⢻⠉⠙⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣯⠛⠶⠿⠛⠉⠀⠀⠀⠀⢈
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠳⡀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣛⠻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⣿⣿⠟⠻⠟⠻⣿⣿⣿⡟⢻⠂⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⢿⡍⣶⣿⣷⣾⣶⣿⣧⣬⣿⡟⣠⣿⣦⣿⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣯⣼⣿⣻⣿⣶⣶⣽⣿⣿⣿⣿⣿⡿⣿⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⣿⣿⣯⡉⠻⣿⣏⣴⡿⢋⣼⣿⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠉⠉⠀⣸⣿⣇⠀⠀⠉⠉⠉⠉⠋⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⢀⣬⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⣿⣥⣶⣶⣶⣿⡿⢿⣿⣿⣄⡀⠀⡄
⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣥⢿⣿⠿⣿⣿⣦⡀⠀⠁
⣻⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠀⠈⠿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢡⣿⣿⣿⣿⡿⢻⣿⣿⣰⣿⡟⠀⠀⠀⠙⢿⣿⣿⣦⡀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⣿⣿⣿⡿⢱⢸⣿⣿⢻⢻⡇⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⣜⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⡀
⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠴⢶⣶⡦⠄⢻⡄⢻⣧⣿⣷⠀⠸⣿⣿⣃⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣥⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣷⣦⡎⠱⠈⢿⣿⣿⣯⠺⣿⣟⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣤⣄⡁⠐⠂⠀⠀⠀⠀⢀⠀⣠⣿
⣿⣇⠀⠀⣀⣿⣿⣿⣟⣛⣿⣿⣇⣼⣿⣀⣤⡐⢠⣿⣿⣿⡆⢿⣿⣿⠋⣀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⣶⣦⣤⣷⣶⣿⡿⠛
⣿⣿⣿⣿⣿⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣤⠋⢿⣿⣷⢼⣿⣯⣄⣈⡑⠂⢀⠀⠀⠀⠈⠀⠚⠁⠀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣋⣵⣶⣿⣿⣿⠺⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀
⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⢆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣯⠓
⣿⣿⣿⣿⣿⡿⣿⡝⢹⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡯⢭⠛⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡇⠀⣰⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢠⡏⠚⠀⠀⠆⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢯⣼⠗⠀⠀⠘⢧⡀⠀⢀⣠⠇⠀⠀⠀⠀⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⣠⠞⠁⢠⡟⠁⠀⣀⡀⠀⠘⠻⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠒⠛⠛⠛⠛⠉⠉⠉⠉⠉⣀⣠⠴⠋⠀⠀⠀⠙⠃⠀⠀⠀⠁⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣶⡀⠤⠶⠶⠖⠒⠚⠛⠛⠋⢉⣀⣠⡤⠴⠶⠖⠒⠒⠒⠶⢀⣠⠾⠁⠹⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠶⣏⠀⣽⠶⠀⢀⣀⣠⡤⠶⠖⠛⠉⠁⠀⣀⣀⣠⣤⣤⣤⣤⣀⡈⠙⢦⡀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠈⠿⠛⠛⠛⠉⠉⢀⣀⣤⠤⠶⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⢳⡄⠻⠋⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣖⠀⠀⣿⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⣤⠀⠀⣀⣀⣀⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⡴⠃⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣤⡞⠉⢳⣯⠉⠁⠀⣀⢀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⣾⢋⡴⠚⠋⠉⢉⡉⠉⠉⠙⠛⠓⠒⠶⠶⠶⠴⠶⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢾⠀⠀⠀⢤⡞⢳⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠘⠳⣤⣄⣀⠙⠏⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⣤⣬⣉⡛⠲⣄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠔⠀⢠⠉⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀
⡆⣧⠀⣾⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⣇⠔⢩⠆
⠇⠀⠀⠈⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠀⠰⠊⣰⠁
⠈⡄⠀⠀⠀⠉⠛⢠⠄⠷⠀⠀⠀⠀⠀⣤⣀⠀⠉⠂⠀⣀⣀⡀⣀⠀⠀⠀⠀⠐⠧⣀⠙⠀⠀⠀⠀⠠⢀
⠀⠘⠦⣀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠈⣆⠘⠃⠀⠉⠀⠠⠋⠀⢓⠀⠀⠀⠀⠀⢸⠀⠉⠦⠀⢠⠒⠂
⠀⠀⠀⠈⠃⠘⣇⠀⠀⠀⠀⠀⠀⢀⠰⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⡦⠀⠀⠈⠳⠀⢀⡠⠋
⠀⠀⠀⠀⢸⠀⠹⡄⠀⠀⠀⠀⢀⡹⠖⠒⠉⠉⠉⠁⠒⠦⣀⠀⠀⠀⠰⡁⠀⠀⠀⡔⠁⢠⡎
⠀⠀⠀⠀⠀⣧⠀⢳⠀⠀⣀⠔⠉⠀⣀⠀⠀⠀⢀⣀⠀⠀⠈⠲⡀⠀⠀⠘⠀⢠⠞⠀⢠⠟
⠀⠀⠀⠀⠀⢀⣄⣠⣧⡴⠁⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⢀⡄⠀⠱⠀⠀⢤⣠⠋⠀⢀⠎
⠀⠀⠀⠀⠀⢯⠁⠀⢸⠁⠀⠀⠀⠀⠈⠀⠀⢀⣀⠤⠚⢩⡆⠀⠀⣛⡀⣸⠃⠀⢠⠎
⠀⠀⠀⠀⠀⠘⣇⠀⡆⠀⠀⠀⣿⡏⠉⠉⠉⢁⣀⣤⣶⡿⠁⠀⢀⠀⢙⠟⢦⣴⠋
⠀⠀⠀⠀⠀⠀⢹⡀⢡⠀⠀⠀⢻⣿⣶⣾⣿⢋⣉⢀⡼⠃⠀⠀⢨⡟⠁⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⢧⠈⣄⠀⠀⠀⠙⢿⠁⢀⢠⡤⠊⠀⠀⠀⣠⠏⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠓⠶⠶⠶⠶⠖⠋⠉⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠡⡤⠒⠒⠒⠒⠀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠰⠀⠀⠈⠉⠉⠙⠓⠒⠲⠤⡶⠤⠼⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠠⠤⠤⠄⢾
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠉⠻
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠣⣀⣀⡠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡖⠒⠲⠶⠖⠊⠻⣤⣤⣀⣀⣀⣀⣀⣠⠤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⠀⠀⠀⠀⠁⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠧⠴⡇⠀⠀⠀⠀⢀⠞⠀⢊
⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⡇⠀⠀⠤⠒⠁⠀⠀⣸
⠀⠰⢂⠀⠀⢋⠀⠀⠀⠀⣀⠦⠚⠣⠁⠀⠀⠀⠀⠆⢨
⠀⠉⠂⠠⣭⣭⡥⠍⠁⠎⠑⠀⠀⠀⠨⡆⠠⠄⠚⠠⠙
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣴⣁⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠠⢀⣀⠔⣁⣠⠔⣀⠀⠀⣘⣻⣕⣤⣤⣄⡘⠄⠀⣃⣀⠀⣀⣀⣀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠈⢦⡘⠂⡉⠻⢧⠀⠈⠳⡈⠈⠷⡀⠙⠓⠢⠄⠀⠀⠈⠂⣞⡁⠀⠀⠀⠨⠓⠨⠀⠀⠀⠘⠃⠀⢸
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠒⠂⠁⠐⠀⠓⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢲⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠐⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣥⣀
⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠒⠂⠂⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠴⠀⡂⠄⠲⢀⡄⠈⠆⣰⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⡀⡀⠐⣄⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠛⠒⠙⠚⠛⠛⠛⠉⠙⠛⠁⠀⢸⢿⠷⠀⢀⠘⠋⠀⠀⢋⠴⢙⠖⠀⠀⠈⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠱⡀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡄⠀⠆
⠀⠀⠀⠀⠙⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠎⠀⠀⠀⠀⠀⠑⠢⣶⣧⠀⠀⠀⠀⠀⠀⣢⣿⡤⡇⠀⠀⠀⠀⠀⢀⣿⠗⠊
⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⢼⣿⡀
⠀⠀⠀⠀⠸⠃⠻⠀⠀⠀⠀⠀⠐⢢⣼⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠸⡜⡇
⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠎⠘⠄⠀⠀⠀⠀⠀⠰⠀⠡⠀⠀⠀⠀⠀⠀⠐⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⣠⣀⠔⠁⠀⠀⠘⡀⠀⠀⢀⠄
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠳⢄⣶⣀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠈⠙⢿⣶⠄⠀⠀⠺⣿⠁⠀⠀⠀⠀⠀⠳⣿⡴⠊
⠀⠀⠀⠓⢲⣾⠆⠒⠈⠀⠀⠀⠀⢿⡏⠑⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣰⢿⡟⠀⠀⠀⠀⠀⠟⢻⠀⠀⠀⠀⠀⠀⠀⡟⢹⠀⠀⠀⠘⠃⣇⠀⠀⠀⠀⠀⢸⠋⣷
⠀⠀⠀⠀⠚⠟⠃⠀⠀⠀⠀⠀⠸⠈⠛⠐⠆⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢊⣠⣴⣾⠥⠖⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿⠃⠴⣦⠔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠉⠤⠀⠀⠀⠀⠀⠠⡔⢄⠀⢀⡴⠛⠉⠢⣀⣀⣠⣤⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡴⠀⠀⠀⣀⠈⠀⠙⠾⢤⣤⠶⢚⣻⣉⣀⣤⣤⣷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⡴⠒⠒⠂⢀⡀⠀⠀⣠⠁⠀⠀⠀⠀⢂⠁⠀⠲⠿⠿⠿⢃⣴⠶⠎⠑⠙⡏⠉⣽⠁⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠁⠀⠀⠀⠀⠀⠉⢫⣀⠉⢁⠠⠀⠀⠘⠛⠀⠀⠀⠀⠀⡠⠜⠀⠀⠀⠀⢰⣇⠶⠃⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠊⠀⠀⠀⢀⡞⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⠟⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠀⠤⠔⠚⠁⠀⠀⠈⠐⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠀⠀⢀⡀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣎⠉⢉⣀⣠⣼⢄⠀⠀⣿⠐⠉⠉⠙⠛⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠔⠊⢀⠴⠋⠛⠝⣡⠞⠉⠉⠈⠀⢢⡤⠤⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⢠⣞⣁⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠈⣷⠀⢨
⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣀⡜⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢉⣡⡜⠡⠄⠒⠒⠀⠀⠂⠀⠒⠂⢤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠤⠤⠤⣀⣀⣀⣀⣀⣠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠇⠀⠀⢠⠤⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⠀⡰⠃⠀⠀⠀⢀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢛⠉⠉⠁⠀⠀⠈⠉⢋⠉⠠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⢈⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠤⠬⢍⣉⡙⠛⠻⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣬⣉⡛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠚⠉⠀⠀⡾⠀⠀⠈⢈⣀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⠀⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⢠⠔⠂⠀⠄⠀⠀⠀⡀⠀⢸⠀⡀⠈⠀⠤⠤⠄⠐⠒⠒⠒⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⣲⣿⣿⠀⠁⠈⠉⠀⠀⠀⠀⠀⠂
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣿⡟⠀⢄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢞⠟⠁⠀⠐⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠
⢠⠀⠁⠀⠀⡀⠀⠀⠀⣤⡒⠂⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⢁⡄⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊⠀⠀⠈⢂⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⣠⣤⣴⣤⣄⣴⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢳⠀⠁⠁⠀⠀⠀⢀⠀⠀⠀⠈⠉⠙⠛⢿⣿⣿⣷⣶
⠓⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠄⠂⠀⠀⢹⡿⠃⠙⠉
⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⢄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠘⡁⠀⠂⠀⡐⢤⣄⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠀⠈⠉⣺⣿⣿⣠⣿⣶⣾⠆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣻⣿⠛⠛⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⣰⠁⠀⠶
⠀⠁⢰⡀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠤⠀⣲⠞⠛⠻⣇⠐⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⢠⠋⠁⠊⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠶⢦⣤⡄
⠀⠁⠀⠀⡀⠀⠀⠀⠠⠤⣀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠉⢘⡄⠀⢀⠀⠀⠀⣠
⠀⠀⠀⠆⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢹⣄⣸⣦⣶⣾⠋
⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠋⠀⠀⣠⣖⠀⠀⠀⠀⠀⠀⢈⣿⣿⡛⠉
⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⢄⡀⠀⠸⣿⣷⣤⡆⢠⣾⡿⠋⠄⠀⠀⠀⠐⠀⠈⠁⠀⠙⠦⣀⡒
⠁⠀⠀⠀⠀⠀⠀⢠⠀⠠⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣌⣙⠇⠀⡸⣿⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠔⠒⠊⠩⠛⠛⠛⡿⡿⢉⣤⢚⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠒⠒⠒⠒⠒⠂⠄⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣤⣠⣾⣿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⣿⣿⣿⣿⣤⣀⡀⠀⠀⣀⣀⢀⣤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣿⣿⣇⣾⣿⣿⣿⣶⡞⡉⠀⠋⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢀⠐⠁
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠸⠏⢘⣿⣿⣿⣿⠟⠛⢿⣅⠀⠀⠀⠑⠀⡀
⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⠀⠀⠘⠿⣏⢻⣷⣶⣦⠝⠻⠃⠂⠀⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣰⡤⠀⠀⠀⠀⠀⠈⣉⣀⠀⠀⠀⠀⠀⣀⡀⠠⠄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⢀⠦⠼⡥⠬⠿⣶⣖⡊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⡄⠀⠀⠠⠄⠀⠀⠄⠈⠂⠀⡐⠀⠀⠀⢀⠀⠐⠠⡀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠠⠄⠠⠀⡀⠁⡀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠤⠤⠤⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⡀⡈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠂⠙⠦⠛⠳⣦⣤⣶⣿⣿⣏⠹⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⣷⣷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠒⠈⠻⣿⠿⣿⣿⣿⣿⣿⠟⢺⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠊⠄⠀⠀⠀⠰⣿⠗⣀⣤⣹⣿⢿⣿⠆⡀⠁⠀⠀⣀⣴⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠁⠈⠁⠀⠀⢀⡀⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠤⢤⣠⣴⠚⠻⣆⣀⠀⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⣀⠀⠀⢀⠀⠀⢀⣴⣿⠏⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠒⠂⠀⠐⠠⠂⠀⠀⠀⠐⠀⠄⠀⠀⠠⠀⠈⠋⢁⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠿⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣤⣤⣤⣄⡈⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣤⣤⣤⣤⣀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣸⣷⡄⠀⠀⠀⠀⢸⣧⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⠛⠟⠃⠀⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⣠⡼⠏⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⢀⣠⡾⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠈⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠶⠶⠶⠊




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠛⠛⠋⠉⠛⠛⠓⠶⠦⣄⡀⠀⠀⠀⢠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⠀⣠⠖⠋⠁⠀⠀⠀⢀⡤⠶⠒⠂⠀⠀⠀⠀⠀⠉⠓⢦⡀⠘⠛⢂⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⠀⠀⠀⠀⣴⢋⡴⠛⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠙⢷⡄⢠⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⡀⠀⠀⠀⢸⠇⡾⠀⠀⠀⠀⠀⡀⢷⠀⠀⠀⠀⠀⠀⠀⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠞⣋⣭⣄⠀⠀⠀⠀⢿⡀⠀⠀⣀⣴⢃⡾⠀⠀⠀⣀⠀⠀⠀⢻⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡏⣸⠁⢀⡈⢳⠀⠀⠀⠈⠳⢦⣤⣭⡴⠚⠁⣠⠞⣋⣭⣤⣀⠀⠸⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠘⢦⣬⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣼⠁⠀⠀⠙⣧⠀⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠈⠀⢷⡀⠠⠾⢃⡟⢠⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⢀⡾⢩⠴⢦⡀⠀⠀⠀⠙⠛⠛⠉⠀⣸⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡴⢋⡥⠶⠶⣤⡀⠀⠀⠘⠁⢿⣀⣀⡟⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⢡⡟⠀⠀⠀⡀⢻⡀⠀⠀⠀⠀⠉⣉⡤⠄⠀⠀⠀⠀⠀⢠⠟⠀⣰⣆⠛⠠⣴⣛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠈⢧⡀⠠⠴⢃⡼⠀⠀⠀⠀⣴⢊⡤⠖⠒⠶⣤⡀⠀⣴⠋⣠⡴⠋⠈⠳⣤⡈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠙⣆⠀⠀⠈⠙⠒⠒⠋⠀⠀⠀⠀⣸⢁⡟⠀⠀⠀⠀⠀⢳⡈⠁⠀⠈⢙⣦⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣇⠀⠀⢀⣠⠏⣼⠁⠀⢀⡴⠋⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣙⠓⠲⠶⠶⠶⠶⠶⠒⠛⠳⢤⣩⡥⠞⢁⣠⠴⠋⣤⡞⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣤⣀⣀⣀⣀⣀⣀⣠⣤⠴⠞⠋⠁⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⢛⣛⣛⡉⣉⠙⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢩⣖⣦⣶⣆⢸⣿⠃⢹⡇⠸⢷⣿⠀⣿⠉⠉⠉⠛⣲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⣿⢸⡟⠁⠟⢻⠈⠋⠳⠈⠁⠒⢚⣁⣀⣯⣀⣀⣀⣘⠛⢭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠷⡶⠀⠀⣻⣄⣡⣤⡤⠴⠶⠚⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠙⠋⠀⣀⡴⠃⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠠⠀⡇⢨⡄⠀⣼⣡⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⡆⠀⠀⠀⣹⣶⢶⡖⢲⢶⡶⡾⢯⣹⡇⠀⠀⣸⡎
⠀⠀⠀⠀⠀⠀⢸⡇⣇⢸⡀⠀⠉⠁⠀⢠⣤⡤⢤⠤⣤⢲⣶⢿⡁⢹⣌⣿⠀⠀⠀⠿⠋⠈⢷⠇⠈⠻⠃⠀⠙⠃⢸⡇⣿⡇⢠⣤⡄
⠀⠀⣴⢒⠊⠉⠙⠋⠛⠙⢓⠒⢦⡀⠀⢸⣿⡇⢸⣧⣽⠸⣿⠾⡇⠀⠋⠈⠀⠀⠀⠀⢸⡇⠀⢰⣢⠀⠀⣴⣦⠀⢸⣧⠻⠇⠈⠛⠉
⠀⠀⡧⢀⣀⢉⣭⡉⣩⣅⢠⣤⡀⢻⠀⠈⠈⠀⠈⠀⡁⠀⠁⠀⠐⣦⣀⢿⣠⢿⠀⢰⢟⣧⠀⣰⠿⡀⠀⢸⣻⣦⠈⠿⠀⢠⣦⣀⡀⠻⠃
⠀⠀⢷⡟⢻⣿⣸⣇⣟⣻⡿⢹⡇⡿⠀⠀⠀⠀⣶⣾⡿⣰⠀⠀⠀⠹⡉⣂⡡⠼⠀⣯⣵⡿⠀⣿⠃⢷⡀⣾⣭⣼⠇⠀⠺⣼⢿⣻⠅
⠀⠀⣸⣗⠀⣤⠙⠿⣃⠉⢠⣌⣻⠧⣄⢠⡖⠒⢻⣿⡏⢉⣳⡄⠀⡶⣄⢠⢺⠂⠀⢀⡀⣀⠀⢙⡓⠛⠁⠀⠀⠀⠀⢀⣀⡙⠋⠁⣀⢀
⠀⢸⠁⠈⠉⠓⠒⠚⣉⠉⠉⠉⠀⠀⠘⡇⢳⠚⠹⣿⡏⠉⣹⠁⣤⣷⣿⣿⣾⣧⠀⡸⠖⢶⣉⡳⣆⡵⠀⡶⠉⢲⣤⠟⠁⢸⡆⠀⠻⠟
⠀⢸⣴⢶⣄⡞⣷⣼⢻⡆⣾⢻⢠⡟⣦⣿⢸⡀⠀⣿⡇⠀⡟⠀⢸⣏⠋⠋⣭⡏⠀⠁⠈⠁⢀⣉⡉⠀⠀⣷⡀⠀⠉⠀⠀⣼⠃⣶⣤⡤
⠀⠀⣏⠈⠿⠃⠻⡿⠈⠷⣏⠈⢿⡃⣸⡅⠀⠣⢀⠛⠓⠚⠁⠀⠈⣿⣀⣴⣉⡇⠀⠀⠀⣰⠋⠀⠙⣦⠀⠈⠙⠲⢤⣠⡼⠃⠀⠻⠟⠁
⠀⠀⢿⠺⡧⢮⡥⣬⢤⡴⠶⢋⣛⣬⠾⠃⠀⣼⢁⢹⣠⡖⢒⡄⠀⠈⠉⠉⠉⠀⢀⡴⠒⣿⠀⠀⠀⣿⠞⠉⣶⠀⠀⣽⠁⠀⢀⡤⠖⠂⢤
⠀⣠⡀⠉⠓⠒⠒⠒⠒⠛⠉⠉⠀⢠⣤⡀⢠⡬⠽⡿⠻⣿⠏⣤⢰⣿⣤⣶⠀⣤⠞⠳⣄⢿⠀⠀⢀⣟⣠⠤⣿⡁⠀⠙⢦⠀⡞
⠲⡋⠹⡃⢤⢤⣶⣶⡾⣹⠀⡤⣄⢨⣿⡁⠸⡇⠀⣇⢠⡧⠀⢼⢻⣿⣿⡅⣸⠃⠀⠀⢹⣮⠧⣴⠻⡿⠁⠀⠈⣿⠀⠀⣼⠀⡇⠀⠀⠀⢠
⠘⠛⠉⠁⠘⠚⠉⢁⡀⠀⣠⢿⡅⠺⠾⠇⠀⣇⠀⣿⢸⠇⣀⣿⣸⣿⣸⠇⢻⡀⠀⢀⣾⡏⠀⢸⣿⡇⠀⠀⢀⡿⠀⠴⠃⠀⣸⠶⠴⠒⠋
⢸⠖⢦⠀⠀⣠⠚⣵⠿⠀⢑⣛⣁⡔⢰⠀⠀⠈⠉⠉⠉⠉⠁⣀⣀⠀⠀⠀⠀⠙⢶⠋⣿⡄⠀⣸⡿⠙⣦⡴⢛⣀⣀⡀⠀⢠⡇⠀⣸⣷⣶
⠉⠻⠁⠐⢊⣲⠗⠁⣴⠰⣿⣌⠿⠃⠘⢡⣾⠀⠀⠀⢀⡴⠺⣅⡽⢦⡀⠀⠀⠀⠀⠀⠈⢳⣤⠟⠠⠀⡄⡴⠋⠀⠀⠈⢢⠈⢇⠈⠛⡽⠋
⠘⠃⣠⡤⣤⣁⣀⣀⡈⠙⠋⠀⠀⢀⠴⢭⠭⠵⢦⠀⢯⢌⣓⣉⣫⣭⡽⠀⣔⠀⠀⠡⣠⠀⡅⢀⠃⠜⠐⡇⠀⠀⠀⠀⣸⠇⢸⡆⢠⣤
⠀⠀⣿⠀⠖⠀⢀⣠⡾⣷⣀⠀⠀⡾⢉⢚⣩⢉⡹⣄⢸⢸⣻⢹⢸⣿⠃⠀⣾⡆⠀⠀⠈⢳⡁⢀⠎⠀⠀⢻⣄⣀⣀⡴⠋⢀⡼⠀⠙⠛⠁
⠀⠀⢻⣆⣤⡾⠛⢋⣠⠾⣹⠃⠈⣶⣦⡭⡶⣤⡖⡟⠀⠉⠛⠛⠛⠁⡔⠀⠉⠙⠒⢄⡀⠀⣿⠀⠀⠀⢠⠞⠋⠁⠀⠀⣶⡄⠀⣶⣤⠄
⠀⠀⠀⠉⣷⣤⠴⢛⡵⠚⢡⣄⡄⠹⢾⣷⣷⣿⠼⠁⡠⣠⣶⣄⠀⢺⡄⠀⠀⠀⡀⣠⡗⠀⣿⡀⠀⠀⠸⡀⠠⣿⣖⠀⣀⣠⡆⠘⠁
⠀⠀⠀⠘⠧⠔⠚⠉⠀⠀⢮⢿⠗⠀⠀⠀⠀⠀⠀⠀⣱⢺⡟⡞⠁⠀⠘⡏⡏⣿⣏⣿⠃⠐⠁⠀⠀⠀⢀⡇⠀⠁⠀⣠⣯⡀⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣈⣁⣀⣀⣀⣷⣿⣟⣛⣛⣀⣀⣀⣀⣀⣠⣏⣀⣀⣀⣀⣈⣀⣈⣃⣀⣀⣀⣀
⣿⣿⠛⡻⢛⡛⠛⣟⢛⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⢻⡟⠛⠷⣦⡀⠀⠀⠀⣠⡾⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⠂⠀⢀⡀⠀⠀⠟⠻⣦⠀⠀⣿⡁⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⢸⣇⣀⣀⠀⢀⣻⡆⠀⠘⢷⣤⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠋⠀⠀⠀⠈⠛⠛⠛⠀⠙⣻⡇⠀⢀⣴⠾⠿⠿⢶⣄⠀⠀⠀⢠⣴⡶⠶⠿⠛⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣠⠀⠀⠀⠀⠀⠀⣤⣰⡟⠀⢰⡟⠁⠀⠀⠀⢀⢹⣟⠛⠿⢿⡇⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣰⣆⣀⣤⡾⠋⠀⠀⢸⡇⠀⠀⠀⠀⠛⠛⠛⠿⠶⠾⢿⣤⡀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⣷⠀⠐⠻⠷⠶⠶⣦⣤⣤⣤⣀⣼⡇⠀⣀⣀⣠⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠘⣿⠀⠉⠹⠿⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⣦⣄⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⢀⣴⠟⠀⠀⣈⣽⠿⣦⣄⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⢀⣠⡾⠛⠁⠀⠈⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⣀⣠⣴⠟⠋⠀⠀⠀⠀⠀⢸⣇⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⠁⢠⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣬⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣴⡶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⡿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣿⣿⣛⣿⣿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⡀
⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡲⣶⣶⣶⣤⣤⡄
⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡻⠷⠮⠻⢿⣿⣮⣯⣍⠉⠉
⣿⣿⣿⣿⣿⣿⣷⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣫⣭⣶⠏⣙⡻⣿⣷⣜⢷⠀⠀⠙⢿⣿⣿⣿⣶⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣮⠻⣿⣧⡀⠀⠀⠀⠀⠀⠈⠁
⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠈⠻⠗
⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣦⡻⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣾⣿⣶⣄⠀⠀⣷⡝⣿⣿⣮⣉⡱⠿⢟⣛⣵⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣙⡻⠿⣿⣷⣄⡲⢶⣮⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠻⠿⠿⠿⢮⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠠⡆⠀⠀⡠⠀⠀⠀⠀⠀⠅⠀⡄⠀⠈⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢡⠂⠀⠀⠀⠁⠀⠀⠠⠐⠀⠐⠒⠀⠀⠁⠐⠀⠁⠀⠀⠙
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⡚⠂⢀⢀⠀⡬⠆⠀⠰⠂⢤⠀⠀⠀⠁⠀⠀⡀⠀⠀⡀⠖⠄⠀⠀⡠⠀⠀⠤
⠀⠀⠀⠀⢉⠘⡀⠀⠃⠐⠉⠁⠀⣒⡄⠢⠀⠀⠠⠐⠀⡀⠀⠀⠰⠄⠤⠐⠀⠀⠐⠃⠀⢀⣨⣀⡁⠀⠂⠃
⠀⠀⠀⠀⠐⠀⠁⠀⠉⢩⠉⠂⠀⢀⠉⡀⠔⠖⠒⠒⢁⡄⠂⠀⡁⠦⠤⢤⠾⠖⠀⠀⠀⠀⢠⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⡠⢊⠭⠣⠀⠀⠂⠀⠐⠂⠇⠀⠀⢂⠀⠀⠀⠀⠀⢔⠂⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠠⢠⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠄⠂⠀⢄⡀⠈⠀⠩⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠹⡀⠰⠀⠀⠐⠀⠀⠀⠰⠂⠀⠀⠇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢈⠀⢀⠀⢰⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣋⣭⣚⣷⣮⣷⡷⠎⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢿⣿⠛⠁⠀⠈⠿⠊⠙⠳⠰⣣⣴⠆⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⢀⡰⣢⣿⣶⣶⣮⣾⣽⠇⠁⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⠀⠠⣼⣛⣛⣿⣉⣩⣭⣿⠿⠿⢛⡵⠛⠉⠀⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⢠⣤⣤⡼⠿⢻⡿⣏⠳⠤⣤⣀⢯⡞⠋⠀⠀⣼⠖⠡⠿⠿⠗⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⠐⠰⡏⠁⠛⠓⢶⣤⣀⠀⢀⡶⠦⠐⢁⢀⣀⣀⣀⡤⠄
⠀⠀⠀⠀⣀⣤⣥⣶⣶⠚⣒⠒⠒⠛⠒⠂⢤⣷⡀⠀⠀⠈⣿⣿⠞⠉⣀⠠⠔⠛⠉⠀⠚⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⠃⠀⢀⣀⣀⣀⣀⣬⡿⠶⣶⡾⠿⠓⠓⠛⠷⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡻⢛⡩⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣁⡤⠟⠉⠀⠀⠀⠀⡈⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠁⠀⠀⠀⠀⠀⠀⢠⡧⠶⠾⠿⠿⠛⠒⠷⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⢀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠲⡄
⠀⠀⠀⠀⠀⠀⢀⣜⢁⣴⣦⣿⣷⠀⠀⠀⠈⠻⣿⣿⠿⠿⠿⣿⡿⠿⢿⣿⣧⠀⠀⢹⡀
⠀⠀⠀⠀⠀⣴⠉⠉⠛⠿⣶⣭⡉⠀⠀⢀⡤⢚⠭⡅⠀⠀⣀⠭⢭⡓⠀⠀⠉⠳⣄⢸⠅
⠀⠀⠀⠀⠀⣿⣶⣦⢀⣾⣿⡟⠀⠀⡠⠋⢰⢁⣤⡇⠀⡔⢁⣄⢠⠇⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⣼⣿⡟⢿⣿⣿⠏⠀⠀⡴⠁⡀⠉⡾⠿⠀⢸⣀⣿⡿⠃⠀⠀⠀⠀⠀⢼⣿⣿⠖⠒⢦
⠀⠀⠀⠘⣽⣿⢧⣿⣿⠏⠀⠀⢰⠃⠚⡁⠀⠓⠄⠀⠀⠁⠀⠀⠀⠀⣠⢄⠀⠀⠀⠙⠁⠮⡀⢸
⠀⠀⠀⣰⣍⡉⡿⢿⡃⠀⠀⠀⢸⠀⠀⠐⠠⣀⠀⠀⠀⠀⢀⣀⠤⠊⣡⡎⠁⠀⠀⠀⠀⣨⡴⠋
⠀⠀⢠⠷⠦⣸⡀⡰⡇⠀⠀⠀⠘⣇⠀⠀⠀⠈⢫⣍⣉⣩⣤⣤⣶⣿⠟⠀⠀⠀⠀⢀⣼⣁
⠀⠀⢸⢷⣄⡼⠁⢼⡀⠀⠀⠀⣀⠼⠣⡀⠀⠐⠫⣉⣛⣛⣿⡿⠟⠁⠀⠀⠀⠀⣠⠊⠀⠈⠉⠓⠦⣄⡀
⠀⢰⢫⣷⡿⣷⡴⠃⢨⠛⠛⠉⠁⠀⠀⣿⢷⣤⣀⠀⠈⠉⠀⠀⠀⠀⢀⣠⣴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀
⢰⣿⣿⡟⠀⠙⢦⠔⠁⠀⠀⠀⠀⠀⠀⣿⢼⣧⣈⡉⠓⠒⠒⠒⠒⠋⢉⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢦
⠈⠛⠋⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⢀⣀⣿⣤⣷⣿⣽⣿⣿⣿⣿⣿⣿⡯⣇⣸⣧⠀⠀⡴⠒⠒⠒⠶⠶⠤⣤⣄⠃⣠⠷⢤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠸⣇⣷⣺⣿⢻⣿⣿⣛⣻⣿⡿⣽⣿⣙⣶⣾⠇⠀⠀⠀⠀⠀⠀⡸⢉⠉⠁⠀⢀⡇⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡼⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⠀⠀⠀⠀⠀⠀⠀⣳⣾⡇⢀⣤⠎⢠⡶⠃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣷⣯⣽⣿⣭⣯⣿⢷⣿⣿⣿⢧⡀⠀⠀⠀⠀⢠⣎⡀⠳⠟⠉⢀⡠⠚⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⢿⢻⣟⣻⣿⡾⣿⣿⣿⣻⣭⣿⣿⡷⠻⣄⠀⠀⠀⢸⣼⡿⣆⠀⢰⡿⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⣿⣿⣯⣿⣿⣮⣭⣿⣽⣽⣽⣿⣟⣲⠿⣮⣷⣽⡄⠀⠀⠸⣯⣻⢿⡖⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣟⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣞⣟⣿⡄⠀⠀⠈⠙⠾⣧⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣯⣭⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣥⣧⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠟⠁⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣯⣭⣭⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⠟⠋⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠏⠠⠀⠀⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠠⣄⡀⠀⣠⡤⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢦⣶⠂⢦⡀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠆⠀⠀⠀⠘⣃⣴⣿⡿⠟⣿⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠠⣤⣴⣶⣿⣿⣿⠋⠀⢀⣿⡄⠀⠀⠀⢸⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢻⣿⣿⣿⡿⠁⠀⠀⣼⣿⡇⠀⠀⠀⣾⡇⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠈⣿⣿⣿⠃⠀⠀⣼⣿⣿⡇⠀⠀⢰⣿⡇⣸⣿⣿⠀⠀⢀⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡄⢹⣿⡏⠀⠀⠘⠛⠛⣉⡁⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠁⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡏⠀⠀⠀⠀⠰⣿⡟⠀⢠⣇⠀⠀⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⣀⠀⠀⠀⣻⡇⠀⣾⣿⣶⣶⣦⣌⠙⠻⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣀⡀⣾⣿⠁⣸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠉⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠰⠟⠉⣾⣿⢠⣿⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⢸⡏⣀⠹⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢿⣿⡀⠻⠘⢁⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠸⣿⣷⣤⡄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠹⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⠿⠿⢿⠿⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠃⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⡆⠁⠈⢀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠉⠀⠀⠀⠀⠑⣄⠀⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⡀⠀⠀⠀⣀⠐⢲⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡤⠜⠓⠔⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⣀⢴⠀⢈⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠄⠀⠇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⣄⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠄⢡⢀⠀⢣⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⠀⠱⠤⠒⠛⠝⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡴⠎⡄⠀⠳⡈⠢⡀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⢐⣂⠤⠔⠒⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠎⡰⢣⠀⠀⠀⠈⢦⠈⢦⠀⠀⠀⠘⠧⠤⢤⣠⡀⢀⠐⠋⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠁⡠⠆⠆⠀⠀⠀⣧⡄⢇⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠃⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡔⢡⢿⠘⡀⠀⠀⣯⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠⠒⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠃⠀⠇⢣⠀⠀⠀⠧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠉⠁⠘⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠁⠸⠘⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠁⣠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢇⢱⠘⡀⠀⠀⠀⡀⠀⠀⠐⢀⠀⠀⠀⠀⣀⣨⠦⠖⠛⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⡄⡏⠀⠀⠀⢡⠀⠀⠂⢀⣠⣴⠶⠚⠉⠁⠀⠀⠀⠀⠀⣣⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢣⣹⡀⡄⡀⠴⠂⠁⠂⠉⠄⠀⠀⠀⠈⠀⠀⢀⠠⠀⠁⠉⠀⠀⠢⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢘⡝⠡⠒⠀⢀⠀⠀⠀⢀⠀⢠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⡇⠀⠀⠉⠥⣀⠤⠘⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠌⠁⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠟⠿⡆⠀⢠⣿⣿⣿⣗⠐⠛⠻⢻⣿⣿⣿⣯⣀⣀⠀⣿⣿⢯⣿⡀⠀⠀⠀⠀⢸⢿⣻⣿⡷⣿⣼⣵⣤⣤⣀⣠⣀⠄⠀⢠⠈
⠀⠀⠈⢲⣿⡟⡍⠂⢈⠀⢀⡀⠈⠻⣿⣿⣏⡣⣈⣿⣿⣿⣿⣿⡇⡀⠀⣀⣠⣿⢛⣿⣿⣴⣿⣿⣿⣿⣿⠻⠙⣷⣶⢖⠳⣴⣐
⣤⣀⡀⢈⢙⣗⣶⣍⡠⠚⡁⢉⠁⣰⣿⣿⣿⣿⣿⠋⠛⠛⢪⣻⣇⠀⠀⠀⠙⣿⣿⣿⣿⣿⡝⠄⠛⢻⠙⠲⣶⣬⣿⣮⠠⠈⢿
⣿⣿⣿⣧⣾⣿⣿⣿⣶⣶⣄⣠⡷⡛⣛⡍⣇⠈⣿⠀⠀⠀⠈⢣⡏⠙⣷⠁⢤⠛⠿⡟⢩⡤⠀⢺⠃⠸⠛⠘⠀⡯⠾⡄⠞⣬⣹
⠛⠛⢉⣻⣿⡿⣿⢿⣿⣮⢿⡛⢿⣿⣍⣴⢿⣴⣿⣆⠀⠀⡸⠋⣁⠒⣬⣄⠊⡃⠀⣈⣄⠠⣄⠘⡄⠐⡄⠀⠘⢥⠋⠓⡶⢻⢾
⠀⠀⡄⢹⠀⠹⠘⣾⢿⣿⣞⣷⡘⡉⠉⢰⡾⡟⢣⡙⠠⠞⠇⠀⢠⣰⠽⠤⠼⡏⠫⣿⠛⡛⢯⣕⢥⣶⣹⣄⣿⣤⣎⣩⡡⢷⣤
⢷⢠⣏⠿⢷⠠⢧⣸⢧⠠⢹⢐⡄⠛⠃⢘⡛⣏⢫⣿⣘⠓⣿⣿⣷⣀⡾⠃⢤⡀⡀⠘⠀⣉⠿⠻⠟⠿⣿⣿⠇⣿⠃⠉⠀⠀⠉
⣀⣀⣹⠖⠿⢷⣶⠟⠋⣛⢮⣶⣴⠒⣀⢸⣦⣁⠸⠿⠛⠖⠳⢾⣿⣇⡀⠘⢳⣦⣗⣀⡶⠤⠀⢀⣤⢀⣀⣽⣿⡇⠀⠀⠀⣠⣄
⠛⠋⠁⠀⠀⠈⠁⠀⠀⠃⠛⢙⣇⠙⢧⡶⡤⠉⠀⠀⠀⠀⠀⠀⠉⠻⣇⣥⣾⣿⣿⣿⣿⠁⠀⢶⣽⣿⡟⣷⠛⣇⠀⠀⢲⠏⢉
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣻⣾⣿⣄⡀⠐⠂⠐⠁⠀⣄⢀⣰⣿⡟⠛⠙⠻⠿⡿⢖⣾⣿⠿⣿⠿⢿⡓⡿⠂⢬⠇⠀⣴
⠓⠦⣄⠀⠀⠀⣰⠀⠆⡀⠶⡾⢫⣀⠠⢹⡇⠀⠀⠀⠀⠀⠀⣿⠏⠁⠀⠀⡀⠂⢀⠀⢸⣿⠟⠀⠀⠙⠀⠀⡄⠀⠀⠘⠀⢐⠏
⠀⠀⠈⠻⣶⣥⣿⣿⡴⠟⠀⠸⠁⠀⣀⢰⣇⠀⠀⠀⠀⠀⠀⡏⢰⡶⠒⠋⠙⠀⠠⠴⠾⢿⡀⢀⠀⢀⠆⣰⣇⠀⠀⠀⢀⡐
⣠⠖⠀⢀⣽⣿⣿⡿⠀⠀⠀⢰⡇⠀⠉⠀⠈⢢⡀⠀⠀⢰⣧⣾⠟⢀⡀⠀⠀⠺⠶⣤⠀⣀⡘⢿⠞⠁⠼⠋⠙⠀⣀⠀⣘⠿⢀
⠛⠀⠀⠍⠁⠀⢀⣃⠀⣠⡴⢿⡇⠀⢄⠁⠛⢛⡻⡢⡲⣖⢵⡯⡻⠋⢀⠀⠀⠀⠀⠀⢀⡀⠀⡦⣠⣾⡿⠂⠀⠀⠘⠛⠋⠆
⡀⠀⠀⢁⣴⣤⣾⡄⠉⠀⠀⠀⠘⢶⣬⣷⡀⠀⠀⠀⠉⡑⠽⣬⡬⡭⠄⠀⢤⣰⡌⠁⠜⠁⠀⣸⣿⣏⠀⠀⠀⣀⣀⡀⠀⠀⠉
⣥⡿⣯⣭⣵⣸⡿⢿⣦⣤⡀⣠⠖⠉⠋⠀⠈⠻⠷⠆⠀⡨⠉⢩⠤⣤⣤⠠⣴⣭⣥⣶⣂⣤⣴⣿⠏⡉⠁⣶⣾⣿⣿⣷⡀⠀⠈
⣾⠏⠹⢟⣷⠏⠀⠀⠙⣿⣿⠇⠀⢀⣀⠀⠀⠀⣀⣠⡤⠷⢄⡀⠀⠀⠀⠀⢀⠄⠈⠹⣟⢝⢩⠈⡝⢧⠀⣿⣿⠉⠙⠻⣽⡖⢠
⠭⠀⡆⠒⠁⠟⢠⡄⠠⠯⡄⢀⡀⠀⠀⢀⣠⣼⡟⠁⠀⠀⠀⠉⠓⢶⡤⠶⠯⠁⠀⠀⠀⣾⣷⣷⣾⣿⣦⣿⠁⠂⢤⢠⡼⠻⠦
⠀⠰⣇⣟⠺⡉⠷⣵⡄⠠⠃⠈⠀⠒⠀⢩⠿⣿⡅⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢿⣮⣧⠉⠙⠿⣿⠃⣄⢰⠎⠛⢦⠃
⢱⣴⠫⠅⣠⠐⣦⣿⡡⠚⢁⣀⠐⠃⠄⣠⣾⣿⠧⠀⠀⠀⠀⠀⢀⣿⣃⠀⠀⠀⠀⠀⣠⠞⣿⣏⣂⣁⣀⠀⠀⡸⣿⠀⣆⠐
⢂⣨⣷⡶⠟⣡⣿⡷⣘⢛⡁⠀⠈⠓⢀⣼⣿⠇⠢⠀⢠⣤⣤⡤⠸⠿⡇⠀⠀⠐⠚⠝⠀⣼⢏⣻⣯⣭⣷⣦⣄⣰⣿⣿⢗⣀⡖
⠁⠀⣈⡿⠺⡟⠇⢀⠠⠎⠉⠀⠐⣶⠋⢸⣿⣶⣤⣤⣾⡿⠉⠀⠀⠠⣧⢀⣀⡤⢀⡴⠈⠉⠀⠈⢩⣽⣿⡿⠋⠀⠀⠁⠀⠍⢣
⠀⠈⠉⠀⣀⡄⠀⠁⢀⠀⠀⠀⠠⠀⠀⢈⣿⣿⣿⣿⣿⣷⣾⣴⠖⢢⣼⣾⠋⢻⣴⣀⠀⠀⠀⢀⢸⣮⣿⡇⠐⣄⠐⠀⠀⠀⠂
⠀⢠⡆⠈⢁⣼⠀⠀⠀⣄⠢⡀⠀⣁⣀⡾⠿⠿⠿⠟⠿⢄⡻⣼⣧⣞⣡⣩⣩⣞⠛⡛⠃⡄⠀⠈⣲⢏⣻⠇⠚⠻⠄⠀⠀⣤⡀

⡀⠀⡀
⣇⡀⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠓⣤⣾
⣿⣿⠙⣆⣄⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢⡀⢱⣾⣿⣿
⣿⣿⣿⣿⠿⡉⣀⠀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⢏⢢⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠺⣥⣷⣿⣁⢀⡴⠂⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡁⢡⢈⡢⣄⣾⣆⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡾⣿⣂⣼⣤⣸⡎⡹⠀⢰⡄⣠⠄⠀⣄⢄⠰⡀⣦⠀⡀⢲⣀⠙⢄⣠⣷⣏⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣯⣤⣷⣿⣴⣿⣿⣴⣄⣼⢸⣾⣶⣧⢿⣴⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢀⣤⣶⡡⠔⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢈⣴⡿⡄⢠⣾⢟⣼⣿⣀⡤⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠴⠾⠟⣡⣾⠁⣿⣿⠿⠟⠛⢩⢰⣟⡀⣀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡀⣠⠞⢀⣤⣸⣿⠀⠈⠓⡂⠤⠿⣿⣿⣶⣏⣥
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣴⣤⣧⠈⣟⢸⡾⣿⠏⡝⠁⠀⠀⠀⠀⠀⢙⠻⡅⠙⢿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣶⣾⡟⠁⢿⣤⣾⣿⣿⣭⢹⣶⣤⡀⠀⠀⢀⣀⢨⠀⠘⠂⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠜⣿⣛⣭⣖⣯⣽⣇⢿⣿⣿⣶⡄⣾⠋⠸⠃⠀⠀⠀⣡⣀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡞⣷⣍⡻⣾⣿⣿⣿⡇⠹⠀⣎⠀⠠⠀⠈⠋⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣜⢿⣿⣿⡿⡿⣷⡇⠀⠸⣿⡢⣴⣶⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣮⣿⣷⣽⢿⣿⡆⠀⠀⡀⠀⠉⠻⣿⣿⣯⣉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣍⡻⣿⡟⣷⠒⠈⠀⠀⠀⠀⠈⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣧⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠀⠀⠐⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⡩⠄⠀⠀⢀⡴⢏⡨⠁⠀⠀⠀⠊⢄⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣃⠀⠀⠀⠀⠀⠀⢠⡮⣕⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠋⠀⠀⠀⠀⠀⢰⠗⠥⠀⠀⠀⢈⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⣀⠀⠀⠀⠀⠀⢀⠢⢀⠄⠀⠀⠀⠀⠀⠐⢠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠑⠀⡀⢀⠀⠀⠀⠀⠀⢠⠀⢃⠀⠂⠀⠀⠈⢀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⠀⠀⠄⠀⡀⠀⠀⣔⢂⣨⠜⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠦⡀⠀⠈⠲⠐⠤⠊⠁⠀⡀⢀⠀⠀⠀⠀⠀⠴⢆⣄⠉⠸⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠆⠌⠀⠀⠔⠁⠀⢈⠦⡀⠚⠋⢄⠀⠀⠀⠄⠠⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠄⠄⠀⡠⠀⠀⡀⠐⠁⠀⢀⠰⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠈⠀⠀⠐⠈⠀⠄⠀⠈⠁
⠀⠀⠀⢠⠂⠔⡡⠀⡀⠄⡂⠠
⠀⠀⢠⣏⣀⡨⠔
⠀⠀⠚⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣖⣯⣗⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⣒⣯⣵⣾⣿⣿⣿⣿⣮⣗⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣶⣿⣿⣿⣿⡿⣻⡯⠿⠺⣝⡿⣿⣿⣮⣓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣾⣿⣿⣿⣿⣿⣇⣇⠀⠀⠀⣠⢟⣂⣙⣻⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣾⣿⣿⣿⣿⣿⣿⣿⡼⡄⠀⠀⣧⣿⢯⣾⢏⣿⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⢷⠀⠀⣟⠿⠷⠾⣿⣻⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣵⡿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⣞⢧⡀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⣵⣿⣵⣶⣶⣶⣦⣤⣤⣴⣶⣾⣿⣿⣿⣷⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣪⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣫⠖⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠛⢏⠉⠁⠈⠉⠉⠻⣞⡆⠀⠀⠀⢀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⢠⢔⣚⣵⠧⡀⠀⠀⠀⠀⡆⠀⠀⠉⠉⠉⠙⠋⠉⢉⡴⠶⠆⠱⡈⣆⠀⠀⠀⠀⢠⡟⡇⠀⠀⠀⣞⡾⠛⠳⠯⣝⢆⡀
⠀⠀⠀⡴⣫⠧⠿⠅⣤⠀⢰⡤⣤⡤⠴⡇⡦⠞⠛⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⡇⣸⢤⣠⣤⠴⣻⠞⠁⠀⢀⡼⣽⠃⠈⠗⢒⡛⠲⣮⢧
⠀⢠⣎⣽⢋⣰⡊⠀⠀⠀⠀⢳⣛⢯⣇⣇⢢⠀⠀⠶⠀⠀⠀⠀⠀⠈⠁⠀⠀⡇⡏⠁⣿⠰⣏⠁⠀⢠⢞⡵⠞⠁⠀⠀⠀⠀⠸⢧⣘⣷⢱
⠀⠀⢳⣹⣏⣄⠀⠀⠀⡀⠀⠀⠈⢳⣾⠴⣯⣦⠀⠀⠀⢰⠇⠀⢀⠀⠀⠀⣠⠃⠗⠾⣅⡵⣯⢧⣀⢜⣼⡅⢰⡤⠤⢄⣀⣀⣀⣦⡽⣱⠃
⠀⠀⠀⠙⢪⣍⡟⠒⢎⠉⠉⢣⣠⡞⢹⣄⠈⡇⠀⠒⠐⠣⢤⠴⣚⣁⣉⠉⠀⠀⠀⠄⠀⠠⠿⢮⣵⣿⣿⡟⠚⠀⠠⠂⠀⠀⢸⣴⠊⠁
⠀⠀⠀⠀⢸⣸⠁⠀⠈⡆⠀⠀⣿⣿⡦⠝⠉⢱⡀⠀⢉⠉⠉⠉⢁⠜⢠⠀⠀⠀⢠⢂⣀⣠⣤⣾⣿⣿⣿⣷⠀⡔⠁⠀⠀⠀⣸⢹
⠀⠀⠀⠀⢸⢸⡄⠀⠀⢸⡀⣸⣿⣿⣷⣶⣤⣬⡁⠀⠀⡄⠀⡬⢥⠀⢸⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⢀⡟⡜
⠀⠀⠀⠀⠀⢧⡳⣄⡀⣀⣷⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠉⠒⠀⠈⠉⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡴⢟⡼⠁
⠀⠀⠀⠀⠀⠀⠙⠪⢝⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⠿⠿⢿⣿⡿⢿⣟⣛⠭⠝⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⢯⣙⣛⡏⢉⠉⠉⠉⡆⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣦⠀⢈⣷⣽⢻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣽⣛⡻⣄⡀⠀⡇⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣆⢈⡿⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡮⣷⣴⣇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⢹⣿⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣻⠿⢿⣿⣿⣷⣽⣿⡿⣿⠿⣟⣻⡭⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠒⠒⠚⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠃⠘⠙⠂⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⢨⠗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠎⠀⠠⠄⠀⠀⠀⠀⠀⠀⢠⠂⣀⠄⠀⠀⡀
⠀⠀⠀⡌⠁⠀⠀⡄⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠁⠀⠀⠀⠀⢠⠜⠋⠀⠐⠀⠀⠁
⠀⠀⠀⠀⡀⠄⠊⠒⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠄⠀⠄⠌⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠲⣍⠳
⢀⠊⢉⠉⢳⡀⡤⡄⠀⠀⠀⢠⠊⠁⢉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠀⠀⠀⠊⠁⠔
⠸⣀⣈⠤⠈⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⢊⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠄
⠀⠀⠀⠀⠖⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠄⢀⣄⠀⠀⡀
⠀⠀⠀⠀⢄⠀⠀⠀⠀⢼⢀⠀⠀⠐⠄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠉⠂⠀⠰
⢠⠂⠛⢦⠀⠉⠀⠀⠀⠈⠀⠉⠒⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠻⣄⣠⡾⡁⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠀⠀⠀⠐⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠙⣷⠀⣀⡠⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⡽⠃⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠈⠰⢧⡤⢣⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣋⠝⢃⠂⡀⠀⣀⣀⢴⡠⠀⠠
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠤⠤⠀⠁⠀⠈⣴⢤⡀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⡠⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠱⡛⠣⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣉⣤⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠊⠁
⠀⠀⠀⠀⢀⠈⠉⠀⣬⠧⠀⠀⢤⡄⠀⠀⣠⡄⠀⢀⡀⠀⡀⠀⢤⡄⡄
⠀⠀⠀⠀⠀⠀⠔⠂⠉⢷⡀⠢⠾⠷⠫⠞⠿⠅⣤⢛⠉⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⣄⣠⠴⠈⡈⠀⠄⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⣼⡇⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠰⣶⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣿⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣷⣀⣀⣻⣀⣀⣀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠟⠻⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⣻⣿⡇⢸⣿⣷⠈⣿⣿⡏⣿⣿⡿⠿⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠻⣿⠃⢸⣿⣿⠀⠻⠟⠁⣿⣿⡧⠀⢸⣿⣿⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠛⠟⠁⠀⠸⣿⡿⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⢭⣍⣉⣉⡀⠀⠀⠀⠀⠀⠀⡄⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⢸⠃⠀⣧⠀⣿
⠀⠀⣀⣠⡤⠴⠶⣶⣶⣿⣿⣿⡿⠿⢿⣿⡿⠿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⠀⠀⣿⣧⠀⣾⣿⣿⣿⣷⡀⢹⣿⣿⣿⡏⠙⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢰⣿⣿⣇⠈⠛⢸⣿⣿⣷⠀⠙⢻⣿⣷⠀⢸⣿⣿⠀⠉⣹⣿⣷⣹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢿⣿⠏⠀⠀⠘⢿⡿⠃⠀⠀⢸⣿⣿⠃⢸⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠋⠀⠀⠻⠿⠟⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠲⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⣿⠏
⠀⠀⠀⠀⠀⢀⠠⠤⠰⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠲⠶⠖⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⣀⣀⣤⣤⣤⣴⡶⠶⠶⠶⠶⠾⠿⠿⠿⠟⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠶⠶⠶⠶⢶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⣿⣷⣶⣶

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣭⣿⣿⣯⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠄⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠀⠐⠦
⠀⠀⠀⣶⡤⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⣾⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⣽⢷⡀
⠀⠀⢠⢽⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⡟⠷⡄⠂⠀⠀⠀⠀⠀⠀⠀⣿⠄⣿⠀⠰⠀⠀⠀⠀⠀⠀⠘⠀⣸⡇⣸⡇
⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠁⠉⣠
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠁
⠀⠀⠀⠀⠀⣽⣿⢿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⠈⣿⣷⠀⠀⠀⠀⠀⠸⢿⣿⣷⣶⡀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⣦
⠀⠀⠀⠀⠀⢿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠾⠁⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣾⡏⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣤⣼⣿⣿⣿⢷⢁⣄⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⣻⢿⣿⣿⣿⢸⣿⣯⣷⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡈⢦⡀⠰⣆⣰⠏⢸⣿⣿⣿⣿⣿⡁⢷⢀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⣌⠓⢄⣀⠳⡀⢸⡟⣿⣿⡀⣿⣿⣿⣿⣿⣿⡿⣼⣿⣇⣿⣀⣄⣸⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⣌⡳⣶⣿⣤⣽⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⣀⠀⠈⠻⣦⡙⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣇⡶⢣⠀⣀⠴⠂
⠀⠀⠀⠀⠀⠀⢈⡓⢤⣰⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡿⣻⠋⠁⣠⠆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⢞⡡⠶
⠀⠀⠀⠀⣄⠀⠈⠹⣦⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡏⣤⠄⠀⣴
⠀⠀⠀⠀⠈⠓⢤⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡷⠤⠴⠃
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠰⣾⢿⠃⢠⡈⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠠⠤⢄
⠀⠀⠀⠀⠀⠂⠀⠀⠇⠀⢧⠀⠀⠠⠈⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣷⠖⠒⠢⠄⣤⡁⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠓⠐⣆⠀⠀⢀⡀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠶⠀⠀⡀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠃⠀⡀
⠀⠐⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡴⠀⠐⠂⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⢄⠊⠄⠚⡑⠀⠀⢤⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢸⡄⢠⡀⢀⣐⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⡒⠑⠤⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⢛⡛⢿⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⡡⠒⠁⠀⠀⠀⠈⢻⡝
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠉⠙⠻⢷⣄⡀⠀⠀⢀⣠⣤⠶⠶⠚⠛⠛⠓⠾⣯⠔⣋⠀⠐⠂⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠐⡠⢀⡾⠋⠀⠀⠀⢀⢀⡀⠀⠑⢍⠛⠳⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠝⠃⠐⠆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠃⢡⠏⠀⠀⠀⠀⠀⠈⠑⠈⠢⣀⠈⢆⠜⠁⢀⠀⣀⠀⡀⠠⠐⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⡾⠀⡘⠀⠀⠈⠀⠀⠀⢀⣀⠤⠤⠐⠒⠈⠉⠉⠉⠙⢳⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠐⠉⠀⠀⠘⠀⢠⠁⠀⠀⣀⡤⠔⠊⠉⠀⠀⠀⠀⠀⣀⣁⠀⠀⠀⣸⢀⣴⠞⠁
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢾⠉⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠉⠛⠲⢶⣟⠋⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠷⠦⣀⣀⣀⡤⠚⠁⠐⠀⠀⢀⠈⠁⠀⠈⠓⢦⣀⣠⣤⣬⡓
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⣤⣤⣤⣤⢤⡖⢹⡄⠀⡜⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⣴⣿⣿⡁⣹⣿⡶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⡇⠀⠀⠀⠀⢋⡠⠒⠢⣀⣀⠀⠀⠐⠒⠀⠀⣀⡿⢿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠠⡄⠀⠀⠀⠀⢸⣤⠤⠤⠌⢁⣈⡏⠀⣀⢹⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣿⢦⡀⠀⠀⢸⡁⣠⠖⢁⣠⠄⣱⣦⣩⡏⠀⠀⠀⠀⠀⠀⠀⣠⢦⡀⠀⢠⠇⣽⢛⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠛⠳⢦⡘⣇⢹⠈⠻⣦⠀⠢⠍⠁⠀⠀⠁⢰⢿⠈⢦⣄⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⢀⡞⢠⠇⡼⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢣⡾⠋⠉⠙⢶⡄⢻⣿⠈⡇⠀⠈⢷⡀⠢⠤⣠⡤⠴⠃⠈⠓⠦⣌⠙⢦⡀⠀⠀⠀⢠⡟⠀⢻⠋⠀⠁⠜⢡⣾
⠀⠀⠀⠀⢀⣤⣄⣤⠴⣋⡴⠋⠀⠀⠀⠀⠀⢻⣄⠏⢸⠃⠀⠀⠈⣷⠀⠀⡏⠀⠀⠀⠀⠀⠀⠸⣇⣀⡇⠀⠀⠀⢸⡇⠀⠀⠃⠀⠀⠐⢁⡟
⠀⠀⠀⠀⠈⠛⠒⠒⠛⠉⠀⠀⠀⠀⣀⣠⡤⠺⣧⡠⣿⠀⢀⡤⠖⢿⠀⠀⣟⡛⠒⠒⠄⠀⠀⢠⢟⣉⢳⠀⠀⠀⢸⡇⠀⠀⠀⢀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠺⠭⢣⡿⠀⣿⡴⠋⠀⠀⠙⠖⡖⠛⢄⠀⠒⢆⠀⠀⣾⡘⠸⠛⠀⠀⢀⡟⠀⢰⠶⠚⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠘⠳⠞⣹⠃⣸⣿⡄⠀⠉⠉⠺⡄⢆⠀⡈⠀⠀⠈⢄⠀⠈⠃⠈⠀⠀⢀⡞⠀⣴⠇
⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠈⠉⠛⠲⢤⡾⠁⢾⠟⣹⠁⠀⡴⠂⠀⠈⠉⠉⠀⠀⢂⠀⡈⢦⠀⠀⡼⢻⢀⡞⠀⣸⠃
⠀⠀⣴⠋⢀⠄⠀⠒⠀⠐⠀⠤⣄⣀⣀⣀⠠⠔⠁⢠⠏⠀⠘⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⠈⠌⠳⣴⢃⣾⠋⠜⣰⠃
⠀⣼⠃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠓⠊⠀⢚⡡⠞⣹⠃
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢨⡇⠀⠀⠀⠀⠀⠋⢀⡖⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠚⠦⠤⠤⠄⠀⠀⣀⣀⣤⡤⠤⣤⠤⠤⠶⠞⠁
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠙⣆⡀⠀⢀⣀⡠⠴⠒⠋⠉⠀⠀⠀⠀⠙⡄
⠸⡆⠀⠀⠀⠀⠀⢀⣴⠞⠋⣹⠀⢾⠋⠛⠒⠒⠚⢋⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠙⢀⡀⢀⣀⠴⠋⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⢀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⡀
⠀⣼⢆⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡰⣧
⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⢸⣿⣿⣇⣴⣿⠿⠿⢿⣶⠀⠀⠠⡆⠀⠀⣿⣷⠀⣿⣿⣿⡧⢰⣿⡷⠀⠀⡶⠀⠄⠀⣶⡿⠿⠿⣿⣦⣘⣿⣿
⠀⣿⣿⣿⡿⠃⠀⢠⣾⡿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠛⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠘⢿⣿⣿⣿
⠀⢿⣿⣿⣧⣀⣀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣀⣀⣼⣿⣿⡿
⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠈⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠟⠁
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠸⣿⣿⠀⢀⣠⠆⠀⣼⣿⣿⣷⠀⠀⠘⢦⡀⠸⣿⣿⠀⣰⠋⠀⠀⢰⣿⣿⠿⠞⠀⡀⠀⠀⣿⣿⣿
⠀⢀⣀⣤⣶⣿⣿⡟⠋⠀⠀⣇⣘⣻⣿⣯⣀⣀⡀⠀⠹⣷⣿⣿⡿⠁⠀⢀⣠⣾⣿⣧⡀⠀⠀⢧⣠⣴⣾⣿⣶⣄⢀⡤
⠀⠈⠀⠀⢸⣿⣿⡇⠀⠀⠀⠈⠉⣿⣿⡇⠈⠉⠁⠀⠀⢸⣿⣿⡇⠀⠉⠉⠁⣿⣿⣏⠉⠓⠆⠀⠀⢸⣿⣿⡏⠉⠉
⠀⠀⢴⣄⣼⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⠂⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⢿⣧
⠀⠀⠀⠈⠉⠁⢹⡇⠀⠀⠀⠀⠀⡿⣠⠞⠁⠀⠀⠀⠀⢸⠏⠸⡇⠀⠀⠀⠀⢧⠸⢦⠀⠀⠀⠀⠀⣿⠃⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠐⠇⠁⠀⠀⠀⠀⠀⠀⡼⠀⠀⠷⠀⠀⠀⠀⠨⠃⠀⠙⠀⠀⠀⠐⠃⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⠀⢤⣀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠒⠒⠒⠒⠲⠶⠤⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀⠀⠀⠀⠈⣁⠰⠖⠐⠚⠉⠉⠉⠈⠉⠀⠀⠒⠒⠒⠒⠀⠉⠛⣠⡶⠖⠛⠉⠉⠉⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣯⡥⠖⠀⡀⠀⠀⠀⣀⠰⠋⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠚⠛⠉⠉⡀⠋⣀⠴⠒⠶⠃⢀⡄⢊⡁⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠐⣠⣄⣀⣀⡤⣄⠾⠋⠁⠀⠀⢰⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⡆⢸⡆⢀⠴⠂⠀⠀⢀⣠⠞⢀⣀⠀⠈⣉⣥⣆⣀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⣿⣄⢻⡾⠅⠈⣀⠤⠄⠀⠀⠀⠀⠀⣸⣴⣿⠿⢛⣿⣿⣿⣷⣥⣤⣴⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠻⣌⠹⣄⢈⠿⢏⠑⠆⣠⡟⡰⠲⡜⢶⡶⠋⠀⠘⣿⣿⣷⣉⣹⣯⣿⣿⡿⠿⢿⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡈⢖⠉⠠⡀⠈⠳⣏⣾⡗⠀⠐⠉⡈⠁⠀⢀⣼⠂⠉⠛⠿⣿⠿⠟⠋⠀⠀⠈⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠣⡀⠉⣢⣴⠶⢶⣤⣥⠀⠲⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⣤⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡟⠛⠉⠉⠉⠀⣸⠉⠉⠓⠛⠳⢶⣟⣈⠀⠲⠀⠀⢀⠀⠀⠀⠀⠀⠐⠺⡅
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠸⠃⠀⢠⠄⠐⠂⠐⠦⠄⠶⠄⠴⢾⣿⣶⣶⡶⠀⠲⠶⠶⣴⣶⣶⣶⣶⣶⠶⠶⠶⠶⠶⠶⢶⠶⣦⠂⠄⠠⠄⠠
⠀⠒⠒⠒⠒⠦⠼⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⣀⠏⠉⠍⠉⠀⠀⠀⠀⠀⠀⠀⠸⢏⠀⠀⠀⠀⠀⠛⠂⠘⠃⠘⠳⠶⢶⡴⢶⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠁⠀⠈⠙⠉⠙⣏⠀⡀⠀⢱⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⢀⣀⣀⠴⣧⠀⠡⠀⠈⢡⡙⠲⣤⣦⣄⣀⡀⠀⠀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⠤⠞⠀⠀⠀⠀⠀⢁⠀⠈⠉⠙⢛⣿⠛⠋⡃
⣀⣀⣀⣀⣠⡴⢦⣤⣀⣀⣀⣀⡀⠤⠛⠻⣿⣿⠀⠀⠀⠀⡀⠀⢆⠀⣆⠀⠀⠛⡿⠋⠀⠈⠣⠘⠛⢠⠶⠀⢀⣀⣀⣀⣀⡤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⠑⠂⠐⠃⠀⠈⠀⠘⢳⣖⠊⣀⠀⠠⠌⠀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⠤⠤⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠄⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠀⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠸⠃⡐⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⢀⠄⠀⣠⠊⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⣠⠃⠀⡠⠃⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡖⠁⢀⠄⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠤⠸⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠙

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠋⠌⠛⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⢠⡟⠛⡆⢀⠈⢩⠆⠀⠀⠀⠀⠀⠀⠀⡋⠉⢙⠀⠀⠂⠀⠀⡀
⠀⣀⣰⣆⠀⠀⠀⠀⠰⡍⠬⠆⠀⠀⠀⠀⣸⠉⠉⠇⠶⠀⢺⣦⣠⠁⣾⣦⣾⡀⠀⢀⠤⣄⠀⠀⢸⡷⢀⡾⠦⠀⠀⠚⠉⠙⡆
⠀⣿⣿⣿⣷⠀⠁⡆⠂⣠⢀⣦⣤⠀⠀⣾⣿⣣⣴⠁⡧⢠⣼⣿⣯⠄⡁⢸⣿⡇⠀⠠⠀⢈⡄⠐⠇⠀⢀⣠⡗⢀⡠⢶⡄⢠⡇⠀⠀⠀⣄
⠀⢸⣿⣿⠏⠀⠀⠀⣼⣧⣿⣿⣿⣷⠀⣿⣿⣿⡏⠀⣧⣌⣛⠙⠃⠀⠘⠈⣿⣦⠀⠴⣧⣼⡃⠀⠀⠺⣿⣧⠈⠀⢡⣾⣷⣿⣿⡀
⠀⣼⣿⣿⠀⡇⠀⠦⢻⣿⣿⣿⣿⠱⠶⣿⣿⡇⠀⣿⣿⣿⣩⣵⣤⡀⠀⢀⠹⠛⣁⣶⣾⣿⡇⠀⠀⢐⢟⡟⣳⣄⣸⣿⣿⣿⡙⠁
⠀⣿⣿⣿⣤⡇⠀⠀⢠⣿⣿⣿⣿⣿⣷⣶⣶⠇⢨⣭⣭⣿⣽⣿⡿⠀⠀⢡⡄⣺⡿⣿⣿⣟⠑⠀⠀⢼⣿⣿⣿⣿⣯⣭⣭⣉⣥
⠀⢹⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⢛⣿⣿⢠⣿⡿⣻⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⡏⢻⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⡇⠀⢸⣿⡟⣿⣿⣿⡇⢸⣿⣧⣿⡟⠁⢿⣿⣿⣿⡄⠀⠀⣸⣿⣿⠀⠙⣿⣿⣧⠀⠀⢸⣿⠁⢸⣿⣿⣿⢿⣿⡁
⠀⢸⣿⣿⣿⡇⠀⢸⣿⡇⢹⣿⣿⡆⠘⣿⡏⢿⣇⠀⠀⠻⣿⣿⣿⡀⣼⡟⣿⡇⠀⣸⣿⡿⠟⠀⠀⢸⣿⠀⣸⡿⠘⣿⡇⣿⡇
⠀⢸⣿⣿⡟⠀⠀⠈⣿⣧⠘⣿⣿⠅⠀⣿⡇⠘⠿⠄⠀⢀⢸⣿⡜⢿⡿⠀⠛⣀⣠⣿⡏⢀⠀⠀⠀⠸⠿⠀⠿⠃⠀⡘⠿⣿⡇
⢀⣾⣿⣿⠀⠀⠀⠀⠾⣿⠃⢿⣿⡄⠀⠉⣑⡀⠀⠀⠀⠀⠀⠉⠑⠂⠠⠄⠀⠈⢄⣘⡀⠀⠀⠀⠀⠐⠀⠠⡀⠀⠀⠀⠀⠘⠛
⢨⣿⣿⡟⠀⠀⠀⠁⠈⠀⠀⣌⣉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢸⡟⠁
⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡀⠀⠀⠀⣀
⠀⠀⢀⡀⠠⠓⡀⠀⠀⠄⠀⠦⠊⠛⢲⠤⠀⢀⣀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⡴⠛⠉⠋⠀⠄⠀⣐⠐⢀
⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠟⠁⠓⠘⠇⠘⠋⢲⠞⠒⠁⠾⠁⠠⠈⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡻⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⣸⠟⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠅⠀⠉⠒⠂⠀⠰⠀⠀⢠⠤⠀⠦⠀⠀⠀⠴⠂⢤⠄⠀⠚⠁⠘⠉⠉⠁⠀⢀⠀⠀⠁⠀⢸⢥
⠀⠀⠀⠀⠀⠐⠀⠀⠚⠠⡬⠑⠀⠸⠗⠀⢈⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⢠⡀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠥
⠀⢀⣀⣤⣄⡤⠀⠂⠀⠈⠀⠀⠀⢀⠀⠀⠀⠉⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠫⡄⠀⠐⠂⠀⠲⠌⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⣤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠌⠀⠀⢄⠘⢁⠀⠀⠘⠁
⠀⠀⠀⡐⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⡐⣒⠀⠀⠂⢀⡀⠀⡄⢸⡆⡀⠀⢄⠈⠉⠀⠀⢀⡴⠀⠨⠀⠀⠀⠀⠇⠀⠀⠀⣠⡄
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠁⡀⠄⠀⡔⡷⠀⠯⠼⣧⢼⢦⠼⠷⠄⠀⠀⡖⠀⠀⠀⠀⠐⠀⠀⠚⠂⢠⠀⠈⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠎⠀⠃⠞⠚⠀⠀⠂⡇⠀⠀⠀⠀⠀⡄⠀⠀⢠⡀⢠⠀⠀⠀⢀⠆
⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢖⡦⢀⢻⠀⢾⡘⠺⠈⢿⡚⠊⠓⡆⠀⠋⠀⠈⠀⠀⠀⠀⠀⠠⠈⠘
⠀⠀⡀⠀⠀⠀⠈⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠃⡀⠛⢀⡓⠀⣉⠀⠋⠁⢂⡑⠷⠀⠀⠆⠀⠂⠀⠀⠀⢀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢨⠅⠀⣤⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠈⠁⡀⣠⠀⠀⠀⢠⡄⠀⠙⣓⠃⣼⠃⠠⣤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⡾⠀⠦⡀⠀⠠⠄⠀⠀⢀⣀⣀⡀⠀⠀⠐⠊⠛⠀⠀⠀⠈⠀⠠⢀⠉⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠂⠀⠀⠂⠀⠚⠋⠀⠀⠀⠀⠈⠀⠝⠲⢤⣉⣀⡀⠀⠉⠀⡀⠀⠄⠐⢊⠠⠸⢀⡀⠀⠀⠀⣀
⠀⠀⠀⣮⢧⢆⣄⠀⠀⠀⠀⠀⠀⠈⠫⠉⠁⠀⠐⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠈⡀⠀⠀⠀⢀⠠⠀⠄
⠀⠀⠀⠁⠁⠀⠁⢐⢄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣼⡃⠠⠠⠀⠀⠀⠀⠀⠘⢞⡛⣭⡴⣄⣶⠀⠀⠄⠀⠨⠀⠀⡆⢸⠀⡆⡆
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠳⣧⣾⣗⣻⡇⡮⠀⠄⠠⠸⠀⠀⠛⠘⠀⠃⠃
⠀⠀⠀⠠⠄⠐⠆⠠⠤⠀⠀⠢⣀⠀⠆⠘⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠷⠍⠗⠀⢦⡤⢲⡀⠠⣆⠀⠀⠄⠲
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠙⡇⠀⠀⢤⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⡤⠀⡀⠀⠀⠀⠛⠀⠀⠲⠄⠀⠀⢀⡤⠘⣷⠀⠀⠀⢀⠀⠀⠠⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⣀⠀⠄⠀⠰⡤⠀⠀⠀⠀⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⠇⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠶⢄⡀⠀⠉⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠰⠌⠋⣶⠆⠀⠈⣦⠀⠀⠀⠀⠀⠀⣴⣦⣤⣤⡿⣧⠞⠛⠶⡀⢠⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢀⡿⠀⠀⠀⠀⠀⢰⠋⠀⠙⢿⣿⡿⢲⡆⠒⠗⠈⠉⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⡎⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹⣿⣿⣿⣦⣤⣆⡔⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢼⣿⣿⣿⣿⣿⣟⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠙⠋⡛⠉⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠈⡆⠀⣠⣾⠀⠁⣦⣄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⣄⠸⠚⠀⠻⠀⠚⠀⠈⠛⠀⠀⠀⣀⣀⣠⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣰⠿⢷⣶⣤⣤⣴⡄⠐⠀⠀⠧⠀⠀⠀⠄⢠⢾⣿⠉⢻⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⣿⡇⠹⣧⣤⠖⠟⣶⡲⠐⠀⣠⠈⠈⡏⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠙⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡟⢿⣄⣘⢿⢇⣀⡔⠋⠀⠀⠃⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢃⢸⡏⢻⣟⠘⢹⡇⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⡀⠀⣸⣯⣍
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⣾⠁⢾⡏⠁⣸⠀⠀⠀⠀⠀⠀⠀⣿⠃⣰⠀⠀⠀⠀⠀⢹⡦⡿⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣨⠀⠀⢸⠟⠛⣿⡇⠀⠛⠿⠇⠀⠀⠀⠀⠀⣿⢠⣿⠀⠀⠀⠀⠀⠈⠉⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⠀⢿⣿⣧⣤⡄⠘⡃⠀⠀⠀⢸⠀⣻⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠀⠀⠀⠀⠀⢠⣦⣾⣿⠀⠀⢸⠀⣈⠙⢛⡭⠳⣿⠏⠀⠀⠀⠈⠀⢿⣺⣿⣆⠀⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠀⠀⠀⠀⠀⣼⡿⠉⢻⠀⠀⢸⠀⢹⡿⠋⢀⡾⢁⡆⠀⠀⠀⠀⠀⠀⠸⣿⣿⢴⡇⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⣿⠀⠀⠀⠀⢠⡿⠀⠀⢸⣀⡇⣾⠀⠈⢀⣬⣋⡉⠉⣁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣎⢻⡄⠀⠀⣸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⣿⠀⠀⠀⠀⣸⠃⠀⠀⣼⣿⣿⡇⠀⣠⣮⣄⡈⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⡇⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢿⠀⠀⠀⢠⠇⠀⠀⠀⣿⣿⣿⡇⠘⠛⠋⠉⠉⠲⠤⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠸⣃⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢤⡚⠀⠀⠀⠀⣿⣿⣿⡧⢠⣤⣶⣶⠆⢴⣶⣶⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣅⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡇⢠⣼⠃⠀⠀⠀⢰⣿⣿⣿⣿⢼⡇⠀⢸⡏⠀⠀⠀⣻⡆⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⠇⠀⣸⡇⠀⠀⢠⣿⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⠀⠠⣿⣵⠆⠀⠘⠛⠋⠀⠀⠀⣧⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠿⠀⠛⠛⢹⡟⠀⠀⠀⠀⠀⠤⠀⢀⠀⢀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣤⣰⡦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡋⢿⡟⢁⣲⣤⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠀⢀⠀⠀⠀⠀⣄⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣄⣀⣠⣴⣿⣿⣿⣟⣻⣯⣤⡯⠈⢙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣼⣀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⣶⠾⢿⣿⣻⣟⠋⠛⠻⠿⠛⠛⠋⠀⠀⠈⠉⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⠊⠁⣀⡀⠈⠉⠈⠀⠀⠴⠂⠀⠈⣤⠀⠒⢀⣤⣤⣿⣧⠀⡄⠀⠀⠀⠁⠀⠀⠀⢰⡆⢀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡙⣥⣤⣿⣿⣿⢟⣃⠄⠀⠀⠀⠀⡴⠆⡀⠀⠾⠟⠛⠻⠿⠿⢧⣯⣶⠀⣘⣢⣆⢰⡀⣾⣿⠛⠛⡛⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠁⠀⠀⠀⠈⠙⠟⠛⢳⣶⣦⢄⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠛⠁

⠀⠀⠀⠘⣧⡀⠀⠀⠀⢠⣤⢲⡴⣲⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣚⣧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠂
⠀⠀⠀⠢⢡⠐⢤⠀⠀⠘⣿⡾⠞⠁⠀⠀⢀⣀⣀⣀⣀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⢄⣀⠀⠀⠈⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆
⠀⠀⠑⢄⠘⠆⠀⠀⢀⡔⢩⠀⠀⠀⠀⠐⠛⢢⡤⠀⠀⢸⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠰⢃⠀⠀⠀⣴⣾⠀⣾⡿
⠀⠀⡀⠈⠁⠀⢀⡴⠣⡈⣿⠀⠀⠀⠀⠀⠚⠃⠒⠒⠀⠤⠅⢀⣤⠀⠉⠁⠀⠒⠠⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠒⢾⣏⣩⠟⠃⠀⠙⠁
⠀⠀⠀⠀⢀⡴⣋⠐⢄⠘⣿⠀⠀⢀⣀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠢⠀⠀⠈⠲⣄⠀⠀⠟⠁⠉⠛⠀⠀⠀⠀⣀⣀⣀⡀⠀⣀⡀
⠀⠀⢀⡴⠋⠠⣌⠱⣄⢳⡏⠀⠀⢙⣞⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢁⠀⣤⠌⣢⡀⠀⠀⠀⠀⡀⠀⢤⣞⠓⠀⠈⡿⠇⠠⠅
⠀⢰⡟⠒⠚⠦⣌⣆⠘⣎⡇⠀⠀⢸⠹⣆⣰⠋⣿⠀⠀⠀⣞⡦⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⡄⠈⠓⠀⢀⡤⠀⢠⠜⠉⠻⣀⣀⣠⠇
⠀⠀⠙⠦⣄⣀⠀⠉⠳⣿⠇⠀⠀⢸⠀⠙⠃⠀⣿⠴⠚⣹⠋⠀⢴⣏⠩⣳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⡤⠖⠉⠄⠀⠀⢀⣀⢀⣀⣬⠁⠐⢄⡀
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠀⠀⠀⣿⠭⠭⢭⣛⠲⢤⣠⠏⠀⠀⠀⠾⠋⡜⠑⢩⡟⢛⢶⠲⡶⢶⡶⡖⡻⣏⢹⠙⣄⢰⠺⠆⠀⠈⠒⠚⠲⠀⠀⠀⢰⡵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠲⣖⣶⣬⡙⢶⠋⢀⡤⠤⣄⠀⠀⠙⠚⠁⣷⠃⢸⣠⣀⡌⢧⠀⡇⠈⠺⠀⠈⠛⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠸⢆⠄
⠀⠀⠀⢴⠶⠒⠒⠒⢒⡶⠖⢦⠀⠀⠉⠒⠾⠭⠿⠃⢠⠏⡴⠶⣌⢳⠀⠀⠀⣀⣤⣄⡈⠁⠈⠁⠈⠉⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠐⡧⠀⣤⣀
⠀⠀⠀⠈⠳⣄⠀⠀⣾⡩⠽⠾⠦⣄⠀⠀⠀⢀⣀⠀⠸⡜⡇⠀⠈⢧⡇⢀⣾⠟⠒⢆⢻⠀⠀⠀⢰⣆⠀⠀⠀⢸⠁⠟⠁⠀⢠⠖⢉⡠⢪⠞⢀⠉⠛
⠀⠀⠀⠀⠀⠈⠳⣄⡿⢤⡠⠔⠒⠚⠻⢦⡀⠈⠻⣍⠲⢵⡹⣄⠰⡘⣷⣾⡣⠂⢀⠌⡼⠀⠠⣠⠤⠙⠀⠀⣀⣠⡤⠤⠄⠀⠈⠀⠉⡜⢁⡤⠃
⠀⠀⡋⢋⠢⡀⠀⡈⠓⠴⠛⢦⣀⠤⠤⠤⢽⣖⠺⠿⣅⣀⡉⠻⣧⣧⣿⡟⠁⣀⣴⡾⠥⣤⠆⠀⠀⣴⡙⠒⠻⣄⠀⠀⢀⠀⠀⠀⠀⢈⡉⠤⠤⢀
⠀⢠⠛⢶⣌⠁⠀⣍⢦⠀⠀⠀⠻⡄⣀⡤⠶⢻⡄⠀⡤⠬⠭⠷⢾⠧⠀⣹⣿⣭⣤⢤⠬⠑⣦⠀⢸⡇⠈⠉⢉⠃⠀⠀⢸⠀⠀⠀⡴⠅⠐⠂⠂⠀⠑⡄
⠀⠀⠶⡒⢆⠀⠀⣈⣟⡀⠀⠀⠀⠻⣥⠴⡖⠋⠀⠀⢷⡀⠀⠀⡸⠀⠀⡀⠀⠀⠀⣸⠀⠀⠀⠀⢸⠀⠀⢠⠏⢧⣀⢀⡼⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠇⠈⠁⠀⠀⡏⢨⠇⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢰⠉⠉⠉⡇⠀⢀⠋⠉⠉⡏⠁⠀⠀⠀⠀⠘⢦⠴⠃⠀⠀⠈⢩⡄⠀⠀⡇⠀⠀⠂⡀⠀⠀⠀⡘
⠀⠀⠀⢀⡤⠶⠶⡇⣼⠠⠤⣀⠀⠀⠀⠀⢸⡄⠀⠀⢸⠀⠀⠀⡇⠀⡘⠀⠀⢠⠇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠁⠎⠀⢀⠠⠀⠀⡴⠁
⠀⠀⠀⢸⠓⠲⠤⠽⠥⠤⠴⢚⠃⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⡇⠀⡇⠀⠀⢸⠀⠀⠀⠀⢀⣤⣿⠀⠀⠀⠀⠀⢀⣠⣿⡤⠄⠀⣆⣠⣁⣤⠶⠊
⠀⠀⠀⢸⣤⠋⠢⠚⣆⡰⠓⡼⠀⠀⢀⡷⢼⣷⡆⠀⠸⠤⠤⠤⠧⢼⣄⣀⣀⡇⠀⠀⠀⠀⢀⣀⡈⠁⠀⣀⣀⡀⠈⠁⠀⠀⠐⢏⡇
⠀⠀⣠⠼⣇⠀⠀⠀⠈⠁⢠⠇⣀⠀⠈⠿⣤⡼⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⢻⠋⠉⢉⣩⣭⣓⠊⠈⢩⣑⢂⠀⠠⠀⠀⠀⠀⢀⣄
⠀⠀⠳⢤⣈⣓⠶⠤⠤⢖⣋⣠⠼⠂⢀⡼⠉⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⡇⡞⠀⡰⢹⢋⣭⣭⠳⡄⣿⣼⡛⢠⠁⠀⠀⠀⠀⣸⠀⠙⣻⠷
⠀⠀⠀⠀⠈⠳⣭⣭⡭⠟⠀⠀⠀⠀⠈⠀⠀⠀⡠⠀⠀⠀⠀⢇⠘⢆⠠⠀⠢⡀⠀⠀⣇⢧⠀⠱⣼⡜⠿⠿⣣⠇⠀⢠⠁⡆⠀⠀⠀⣴⠉⠉⠳⡞⠁
⠀⠀⠀⠀⠀⣠⣴⣖⠲⣄⠀⠀⠀⠀⢀⡀⠀⢘⠱⡄⢄⠈⠢⡀⠑⣄⠑⠳⠀⣘⣶⡦⠙⢾⣦⣀⠈⠉⠓⠚⠁⠀⢠⠃⢀⠀⠀⢀⠾⡈⠑⠦⢤⡞
⠀⠀⠀⠀⡞⢁⠄⠈⣷⢸⣤⡄⠤⠒⢯⢹⡆⠈⠣⡈⠢⢡⡀⠐⣴⣄⠱⣄⠻⠿⠀⠐⠀⠀⠀⠉⠉⠁⠒⠒⠒⠒⠁⠀⢸⠀⠀⢸⡓⢌⡑⣒⠞
⠀⠀⠀⠀⢧⠈⠳⠴⠛⠁⠀⠀⣀⡤⠞⠋⠀⠀⠀⠈⠢⡀⠶⣿⠋⠙⡀⠈⠑⠂⠄⠊⠀⠀⠀⠀⢠⡀⢸⠙⡆⠀⠀⠀⠸⠀⢴⠋⠝⡲⠚⠁
⠀⠀⠀⠀⠀⠙⠒⠒⠶⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠔⠁⠀⠀⠀⢀⡀⣀⡀⣹⢆⣀⠷⣄⢑⡗⠐⠀⠀⠀⠀⠈⠳⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠉⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣇⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣤⣆⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣿⡏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⣾⣿⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⣤⣿⡿⠛⠀⣤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣄⠀⠀⣴⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⠆⠀⠐⠋⠀⢰⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⣿⣷⣄⣿⡀⠀⣨⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⢋⣿⣿⣿⢡⢸⣷⠄⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠈⠀⠀⠀⠀⡼⣷⠰⠞⠀⠀⠀⠀⠀⣿⣿⣿⡟⠻⠻⢿⣿⣿
⣿⣿⣿⣿⣿⣥⣾⣿⣿⣏⡰⠛⠁⢀⣦⠀⠀⠀⠠⠀⣾⠀⠀⠀⠀⣄⠀⠀⠀⢙⡿⠁⢠⡀⠀⠀⠀⠀⠙⣿⣿⡇⡸⣯⣿⣿⣿
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣷⠛⠁⠸⠇⠀⠀⠈⢄⢸⡇⣰⢀⠀⠀⠙⠀⠀⣾⠟⢿⡀⢸⣇⠀⠀⠀⣄⠘⣿⣿⡟⢿⡾⢻⣿⣿
⣿⣿⣿⣿⣿⠉⠂⠛⠿⣿⣷⡶⠀⠀⠀⢀⣠⣷⠀⣿⡷⢡⡄⠀⠀⠀⠀⢠⡞⢀⠀⠁⢸⡇⠀⠀⠀⠘⣖⣻⣿⣿⡟⠀⢸⣿⣿
⣿⣿⣿⣿⣯⡀⠤⠤⠄⢿⢿⣴⣶⡇⢰⡾⣿⣷⠀⣿⣷⠜⠃⠀⠈⠀⣰⣿⠀⢁⡠⠀⠈⢿⣤⡨⣄⢨⣿⣇⣽⣿⣿⣆⣸⣿⣿
⣿⣿⣿⣿⣿⣶⣤⠀⠠⠧⡤⣿⣿⣿⣿⣿⣿⠿⣲⣿⣿⣄⠀⠀⠀⢠⣿⣷⣗⣮⡐⢀⡄⠘⢱⡇⠹⢄⣿⢹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠏⠒⠀⢰⣂⣬⣽⡟⣹⡏⣰⣾⣿⣿⠟⢻⣆⠀⠀⠀⢻⣿⣿⣾⣧⣦⣠⣿⣾⣿⣦⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠈⠁⠉⠉⣿⢸⣇⣼⣿⣿⣿⣿⡟⠐⠛⣟⡀⢠⡈⣧⠙⠻⣿⣿⣿⣯⣬⣿⣷⣋⣿⣿⣿⡿⠃⡀⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣡⠄⠉⠿⣥⣾⣥⣿⣿⣿⣿⣿⣿⣷⢶⡟⢿⣧⡄⠑⢾⡄⠸⣿⣿⣿⣿⣽⣿⣽⣿⣿⡿⠋⠀⠀⠌⢻⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠴⢦⡋⠍⠙⣳⣦⠀⢷⣴⣿⠙⠻⡿⠿⠿⠟⠋⠉⠀⠀⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⣿⡅⠀⠄⠀⠀⢀⣾⣿⣿⣿⣿⣧⠧⠰⠆⠀⣹⣦⣁⠈⢿⣀⠀⣿⣿⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡿⠃⠀⠂⣄⣾⣿⡿⢿⣿⣧⡠⠐⠀⢠⣄⣠⣿⣯⣇⠰⡾⣿⣶⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣘⣿⣿⣿⣵⢺⣿⣿⣿⠁⡀⠀⢸⣿⣿⠈⡇⠀⢐⣤⣿⢻⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿
⣿⣿⣿⣿⣿⠰⠿⠿⣿⣿⣵⣿⣹⣿⣿⡿⢛⡛⢃⣿⣿⣋⣴⡟⠋⠤⣼⣧⣾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡶⠿⢶⡌⠻⠿⡿⠿⣿⣷⣤⣾⣥⣿⣿⣿⣿⠿⣿⠒⢻⣿⣿⣿⣿⣟⣳⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣇⠀⠘⠁⠀⠀⡀⠐⠀⣉⠛⠛⠛⠻⢿⡿⢿⣷⣦⣿⡿⠟⠻⠿⠿⠿⠛⠋⠙⠁⠀⠀⠀⠀⠀⣤⠰⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠃⣠⠤⣈⠁⠉⠐⠀⢀⣤⣤⣤⠀⠀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣧⡀⠀⠀⠚⠋⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣤⣤⡦⣤⡉⠀⠖⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣰⡍⣽⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣷⣤⣤⣴⣆⣀⡈⠙⣀⣤⣄⣀⣤⣤⣤⣤⣧⣾⣶⣾⣤⣴⣇⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣟⣻⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣽⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⣤⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠘⢾⠋⠀⠀⠀⠀⠀⠉⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢰⡀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢀⣄⣀⣴⣾⣿⣎⣵⣾⣿⣷⣄⠀⠠⣄⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢻⣧⡀⠀⣿⡟
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⣿⠀⣿⣿⠇⣤⣿⠇
⠀⠀⠀⠀⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣾⣿⣼⣿⣿⣾⣮⡟
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠈⠛⣽⣿⣿⣿⣿⣿⡿⢋⣙⡻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⠟⣿⣿⠟⠁
⠀⠀⠀⠿⠇⠀⢿⣿⣿⣿⡇⠘⢿⣿⣿⣭⣙⣻⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡟⢡⡿⠿⣦⢀⣃⣾⠛⢿⣄
⠀⠀⠀⢀⣀⣀⣀⡈⠹⣿⣧⠲⣄⣉⡉⠋⠉⡁⢠⣶⣿⣿⣷⣦⣄⢀⣴⣾⣿⣿⣿⢃⢸⡇⠀⠈⣿⣿⣿⣦⣾⣿⣓⣲⣶⣄⡀
⠀⢀⠔⠁⠀⠀⠀⠙⢦⡈⢿⣧⡽⠥⠉⠋⠁⠀⣿⠏⠀⠀⠉⠛⢿⣿⠟⠛⠛⠋⠻⢁⣶⣿⣦⠘⢿⠟⠋⣱⣿⣵⣿⢿⣿⣿⡇
⠀⡘⠀⠀⠀⠀⠀⠀⠈⢧⠈⢿⣿⣦⣄⡀⠀⠀⣿⠀⢀⣤⠒⠀⠂⣙⣦⣀⣐⣢⣴⣿⣿⡗⠀⢀⣶⣾⣿⣿⡿⠟⣡⣾⣿⠟
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⣿⣿⣿⣿⡷⣦⣽⣷⠿⠿⢧⠀⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠛⠿⡿⣿⠿⠛⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣿⣿⣿⣸⣿⣿⣿⡔⣦⣘⣛⣿⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠘⢿⣧⣹⣟⣻⣿⣶⣿⣿⣿⡿⠀⠀⠀⠀⣿⠟⣛⣿⣿⣿⣿⣋⠉⣀⣿⣧⣤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠘⢷⣄⠙⠿⢿⣿⠉⠙⠉⠉⠉⠀⠀⠀⠀⡰⠴⠋⢀⣽⢿⡟⢻⣿⠆⠋⠛⠻⣥⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣤⣙⠛⣷⣶⢷⠀⠀⠀⡀⠀⠀⠀⠀⢠⣷⠶⡾⠟⠁⠸⣇⣠⠋⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣿⣿⣿⣿⣿⡿⠋⠀⠈⢣⡀⠀⠻⣶⣤⣤⣤⣾⠗⠋⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠋⠀⣀⣤⠤⠈⠛⠶⠦⠤⢬⣭⣿⡿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⡀⠀⣉⠀⠀⠀⠀⠀⣀⣤⡉⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡶⠀⠀⠘⢿⣿⣷⣦⣤⠶⠶⠶⠶⣶⣦⣤⡐⢢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡶⠶⠶⢞⣻⣿⣿⣧⣤⡀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣶⣴⣿⡛⠁⠉⠐⠂⠠⠦⢦⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣯⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣠⣄⣀⡉⠽⣿⣿⣿⣿⣿⣿⡿⠶⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠿⠻⠿⣿⠿⠛⠿⠟⠁⠀⠀⠀⠀⠉⠉⠉⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣆⣠⠂⢠⠆
⠀⠀⠀⢠⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢈⣄⠀⠉⠁⡄⠊
⠀⠄⠀⠀⠣⡀⠄⣀⠀⠀⢀⠀⡠⠀⠀⠈⠒⠀⠈⠉⠙⠆⢀
⠀⠀⠢⢈⠒⠵⠈⢁⡀⠆⠀⠘⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠸
⠀⠀⠄⠈⠁⡀⠐⠁⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⡀⠂
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢹⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢂⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⣤⣤⣹⣄⣡⡀⠀⠤⠤⠤⠐⠒⠊⠉⠀⠀⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢂⠀⠀⠀⣀⠴⠃⠀⠉⠀⠀⠀⠀⡈⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⢤⡤⠋⠀⠀⠀⠀⠀⠀⠀⢀⡼⢣⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠐⢀⠀⠀⠀⢀⠀⣠⡤⠖⠋⢠⣾⡇⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⢰⠀⠈⢋⡀⠈⠉⠁⠀⣀⣤⣾⣿⣿⡇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡦⢀⡀⠀⠀⠀⠠⠐⠁⠀⠀⠀⢸⠆⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡸
⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠙⢿⣿⠋⠀⠁⠀⠟⣻⠃⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠑⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠉⠒⠡⠤⠤⠚⡁⠀⠀⢠⡞⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣴⣏⡀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⣀⣀⣀⣀⠤⠶⢋⠀⠎⢉⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⡳⠀⢀⡅⠀⠁⠒⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠄⢀⣀⠀⠠⠖⠋⠀⠀⢁⢡⡞⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠙⡀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢁⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⣠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢦⣀⢀⠜⡄⠀⢸⡄⠀⠀⠀⠀⠀⠀⠠⠐⣊⡤⠾⣍⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⢉⣴⠃⣀⡞⠀⠀⠀⠀⣀⣠⠤⠒⠋⠁⠀⠀⠘⣆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢆⠉⠉⠉⠉⠉⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⠈⢆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⠢⣀⠀⠀⠀⣰⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⡙⠛⠛⢁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠔⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⣞⣛⣿⣿⣿⣿⣿⣛⣒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡇⠀⠙⢦
⠀⠀⠀⣶⢶⡶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠙⣮⣽⡖⠛⠛⢛⣿⣛⠛⠿⢧⣷⡀⠀⠈⡇
⠀⠀⢀⣿⣿⣿⣿⣷⣿⣿⣹⠇⠀⠀⢀⠔⡫⠖⢆⠀⠀⢀⡤⠶⢮⠁⠀⠀⠈⠙⢦⣀⡇
⠀⠀⢸⢹⣿⣿⣿⣿⣿⣿⠏⠀⠀⡴⠃⢰⢁⣤⡜⠀⡰⠃⣠⡀⡼⠀⠀⠀⠀⠀⢸⣿⣧
⠀⢀⣾⣬⣉⡙⠻⢟⡿⡏⠀⢀⡞⠀⠀⠉⡾⠿⠁⠠⡇⣸⣿⡟⠁⠀⠀⠀⠀⠀⢼⣿⣿⠷⠒⠲⡄
⠀⢸⣉⡉⠛⠛⠻⠷⣾⠀⠀⡼⠀⠚⠁⠀⠣⠄⠀⠀⠉⠀⠀⠀⠁⠀⣤⣄⠀⠀⠀⠙⠋⢰⣃⠀⣿
⠀⠙⢾⠺⠽⣶⣶⣤⡏⠀⠀⣇⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⢁⡎⠁⠀⠀⠀⠀⠀⣊⡴⠃
⠀⠀⢸⡀⢀⡴⠚⠉⠀⠀⠀⢹⡀⠀⠀⠀⠉⢖⠒⠒⠒⢊⣉⣡⣴⣾⡟⠀⠀⠀⠀⠀⢠⠏⠁
⣴⢊⠛⡶⠚⢦⠀⠀⠀⠀⠀⠀⣳⡄⠀⠀⠀⣜⠛⠿⠿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⡴⠋⠙⠓⠦⣤⣀
⣏⣁⣉⣧⣠⢟⠆⠀⠀⠀⢀⡴⠁⢸⣶⣄⡀⠈⠉⠒⠒⠚⠋⠉⠀⠀⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⢞⣠⣀⣹⠀⠘⡦⢤⡤⠖⠋⠀⠀⠘⣿⣸⡟⠲⠦⣤⣀⣀⣀⣀⣤⠤⠶⣾⢫⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢈⠇⠀⡽⣶⡼⠁⢠⠀⠀⠀⠀⠀⠀⣿⠙⣿⣶⢤⣤⣄⣀⣀⣀⣠⣤⣴⡟⢳⡇⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⡰⠙⣆
⢸⠀⢠⠇⠙⠳⣴⡃⠀⠀⠀⠀⠀⢀⣿⣷⣿⣾⣿⣿⣾⣷⣶⣾⣿⣿⡿⣷⣾⢿⡀⢀⣿⠉⠉⠉⠉⠉⠙⠛⢲⠮⣤⠴⠋⠙⡶⢦⣀
⠸⠤⠞⠀⠀⠀⠀⠉⠓⠲⠶⠶⠚⠋⠙⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⡯⣿⣿⣿⣽⣻⡇⠀⠀⠀⠀⠀⠀⠀⡞⢠⡀⠀⠀⢀⠇⠀⣙⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡻⢾⣻⣿⣿⣿⣯⡷⣿⣷⣿⢿⣟⣯⣾⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣅⣀⡴⣾⣶⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣾⢷⣾⣿⣿⡿⠶⣿⣟⣻⣽⡶⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⣾⡿⣿⣭⣾⣿⡿⢩⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⡿⣿⣿⣿⣿⣿⣟⣟⣛⣻⣯⣿⣿⣿⣷⢚⣧⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⢿⣿⣿⡾⠟⡠⠊⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣿⣿⢽⣻⣿⣽⣽⣿⣿⣯⣽⣿⣛⣿⢿⣿⣿⢞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⡿⠛⢉⡄⠊⠀⠀⠀⣸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⢿⢏⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⣝⡻⡞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠑⢆⠀⣀⠔⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣿⣟⣋⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣙⣋⣩⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣉⣥⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠟⠛⠛⠉⠙⠛⠻⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠻⠟⠛⠉⠁⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠶⠶⠶⠶⠶⣼⣆⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⢀⣼⣥⠶⠶⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢀⣽⣿⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠻⣿⡿⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⣷⠀⠀⠀⠈⠛⠋⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠿⢿⣖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⢶⣶⡿⠿⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠉⠉⠛⠛⠛⠋⠉⣉⠥⢊⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢀⣀⣀⠤⠤⠶⠚⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⣠⠁⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⣤⠋⠁⠀⠀⠀⠀⠀⠑⠲⢄⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⣀⠀⠂⠁⠁⠀⠀⠀⠑⣄⢠⣯⣤⣆⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠹⣷⣖⡤⠀⠀⠀⠀⢀⣹⣿⣿⣿⣏⠀⠀⠀⠀⢀⠀⠐⠒⠣⡔⠠⠀⠂⠘⢀⣸⠏⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠠⠄⢀⠛⠻⠿⠷⡄⠲⠬⢵⡾⠻⣿⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠱⡀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⢠⠊⠀⠀⠈⠣⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⣮⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠠⠀⠀⣤⠴⠃⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⢀⣵⣿⡞⢿⡟⠀⠂⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡀⣀⡀⠀⠸⢿⣿⣟⣵⣿⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⣶⣘⢠⠀⠀⠄⠃⢘⣀⠈⠙⠛⢲⣦⣀⠤⠒⠀⢉⠆⠀⠀⠀⠀⠀⠀⡨⠟⢻⣿⣟⡒⠂⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠼⠆⠀⠀⠀⢀⡤⠀⠀⠁⠀⠀⡏⢹⠃⠀⠀⠀⠈⡀⠂⠀⠀⠀⠀⠈⠀⠀⠸⠉⠃⠀⠀⠀⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⠃⠀⠀⠀⠀⡀⢇⢾⡄⠀⠀⠀⠀⠁⠸⠀⡠⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠁⠀⠀⠀⢀⡈⢼⣮⣿⡢⣼⣄⠀⣀⠠⠚⠐⠒⠀⣤⠖⠉⠂⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⡤⠂⠀⠀⠀⠽⢟⣿⡡⠚⠛⠒⠲⠬⣈⣙⣛⣿⢿⣿⣟⠁⠀⠀⠀⠀⣀⡘⢠⡀⢸⣷⣤⣄⡀⢦⠆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡥⡿⠁⢀⡀⠀⠠⠖⠒⠚⢻⣷⣿⣿⠷⡷⠶⠶⠿⠉⠉⠀⠉⡿⣿⡊⠿⡏⠁⢀⠠⣀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⢾⠋⡈⠉⣀⠀⠀⠎⢁⠀⠁⠀⡀⠀⠀⣉⣴⠇⠀⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠷⠀⠀⠀⠀⠐⠘⠑⡄
⠀⠀⠀⠀⠀⠐⠰⠀⠁⠄⠐⠀⠉⠀⠠⣾⣃⢀⠛⠁⢀⠀⠀⢀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢀⡀⠆⠀⠀⣷⢶⢤⡄⢰⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠁⠁⠀⠇⠀⣿⡆⠀⠀⠢⣀⠀⠀⠀⠀⠀⢀⠄⠴⠁⠀⠀⣠⠿⣠⡄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⠀⠀⠀⠀⠀⠺⣿⡇⠀⠀⠀⠈⠉⠁⠀⠉⠍⠀⠀⠈⠀⠀⠂⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⣀⠀⢀⠀⡀⠀⠀⠀⢀
⣿⣏⢭⣟⡛⣛⣛⡛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿