Star pretty astrology Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠰⠂⠀⠚⠷⠾⣭⠽⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣀⡘⠡⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠔⠢⢎⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⠛⠃⠀⠀⢀⠀⠀⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠁⠀⠛⠙
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠐⠉⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠊⠀⠀⠀⠠⠕⢁⡴⠖⠀⠈⠁⠠⠅⣀⠠⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠰⢶⣄⠑⠕⠀⠀⠀⠈⠈
⠀⠀⠀⠀⣠⣤⢀⣶⠠⠾⠖⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣰⣶⣾⣿⣶⣠⡀⠀⠀⠀⠀⠁⠻⠡⣶⠀⠀⠐⠾⠦⢰⣆⢠⣤⡀
⠀⠀⠀⠀⠉⡉⠸⣃⡤⢐⠀⢠⣶⡄⠀⢀⣀⡀⠀⣤⠟⠛⠛⠛⢿⣿⣿⣇⠀⠀⢀⠀⠀⠀⠀⠉⠑⠀⠀⠔⠠⣌⡻⠈⡋⠁
⠀⠀⠀⠀⢰⡇⠛⠉⠀⣤⠀⣀⡄⠀⠈⣏⢭⡁⢠⠃⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠈⠀⢀⣆⡀⠀⣴⡇⠀⠀⠀⠈⠙⠃⡷
⠀⠀⠀⠀⡀⢡⡿⠃⠀⣇⣮⠅⠀⠀⠀⠈⠀⠙⠈⢠⠄⠘⠃⠀⢀⣿⣿⡿⠆⠀⠀⠀⣸⠟⠁⠰⣼⡗⣘⣇⡤⠀⠻⣦⠁⡀
⠀⠀⠀⠈⠳⡟⢁⠀⠀⠀⠀⠀⠀⠗⣠⠀⠀⠀⠀⠈⣽⣶⣦⣶⣿⠿⡽⠇⠀⠀⠀⢠⠁⠀⠀⠀⠛⠋⠉⠈⠀⠀⢀⠙⡷⠋
⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⠙⣦⡐⢀⡾⠀⢠⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⢹⡄⠀⠀⠀⠘⠛⠁⠀⠸⣆⠐⣠⠞⢀
⠀⠀⠀⢀⣀⠈⢃⡾⠃⣴⡆⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⣴⠟⠀⠆⠀⢀⠀⠀⢀⣦⡀⠻⣆⠁⢀⣀
⠀⠀⠀⠛⠟⠀⠈⠀⠀⠿⠇⢀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⡬⠀⠁⠀⠀⠀⠀⢘⡇⣀⠈⢿⠇⠀⠈⠀⠘⠟⠂
⢠⢢⠀⠀⠀⢀⢄⠀⡀⣀⠀⠈⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢠⡀⢄⠀⠀⠘⠷⠋⠀⢠⠀⡀⢀⢄⠀⠀⠀⠠⠢
⠀⠀⠀⠀⠀⠀⠀⢠⢽⣿⣞⠙⢦⡞⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠹⣦⠞⢙⣾⣿⢥⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠋⠀⠀⢶⡄⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⢰⢡⢠⠈⡆⠀⠀⠀⠀⠀⠁⢀⣶⠆⠀⠋⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⠘⠲⢔⣦⣠⢤⡰⣄⢀⠀⠀⠀⠈⠛⠋⣛⠁⠐⠀⠀⢀⢀⡆⣠⢦⣤⣆⠴⠚⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣻⣿⠿⣿⣜⣧⠄⠀⠀⠒⠶⠯⣹⠂⠀⣤⣏⣾⡾⢿⣿⣛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣛⣿⣿⠿⢶⣤⣄⡀⣚⣥⣴⠾⢿⣿⣟⡛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⠿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⣾⣿⡿⠋⠉⠀⠀⠀⠀⠀⢈⠛⠋⠉⠙⢿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠓⢦⣤⠀⢹⣿⣿⣿⣿⣿⣦⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢿⣿⡿⢀⡆⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠈⠳⢌⠻⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠁⡈⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠙⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠃⠀⢷⢃⠀⢠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣦⡙⠿⣿⡟⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠟⠀⢠⠀⢸⡀⢂⠸⡆⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⠏⠂⠈⠙⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠐⢠⣿⣿⣿⣿⣿⠟⠁⢀⠔⠔⠠⠻⠁⠀⠁⠜⠀⠘⠌⢇⠀⣀⣀⠀⠀⠀⠀⣠⣤⢤⡄⠈⢂⡀⠈⠁⢀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⠀⡴⠚⠁⠀⠄⠀⠠⠀⠀⠀⠀⠸⣄⠀⠸⣿⡙⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢹⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⡔⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠃⠈⠌⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢁⠀⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⢸⡀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠁⢄⡀⠀⢢⠀⢀⡃⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠎⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡀⠆⢎⢏⡠⠂⠀⠀⠀⠀⠀⠀⠀⢀⡴⠠⠊⠀⠀⠀⢀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠂⠀⠀⠀⢀⣳⣼⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⢀⣴⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡘⠲⠦⠤⠄⠀⠠⠀⠀⣁⣠⠶⠶⣿⣿⢷⡄⠀⠀⢀⠀⠁⠀⠀⠒⠁⣀⣲⣭⣶⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣦⣤⣤⣤⣶⣿⣿⣿⣿⣿⣶⣦⡹⢆⣳⡀⠀⢈⣀⠀⡄⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⢢⣟⣹⡄⠸⣿⣿⣿⢧⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣷⣤⡽⣧⡀⣿⣿⣿⠻⠟⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠋⡟⠛⣿⣿⣶⡄⢠⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⢿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣇⠀⠁⠀⠘⣿⣶⣶⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠀⣆⢿⢛⡋⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠛⠙⠊⠁⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡞⣸⢰⡎⢸⡇⡇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠙⠉⠈⠁⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠖⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡴⢦⠠⡆⢠⣤⡶⠆⡦⠀⡆⡴⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢠⡔⣾⠏⡇⣸⢸⡇⠀⣏⢀⡇⡙⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠁⠉⠉⠋⠀⠀⠈⠁⠁⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠀⡄⢀⠀⠀⠀⠀⠀⢠⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣅⣀⡈⢰⡄⠀⠀⠀⢘⡛⠞⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣿⣦⠀⠀⠀⠈⣿⣶⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣾⣿⡿⠟⠃⠀⠀⠀⢹⣿⣿⣿⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣸⣿⡏⠙⠉⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⡀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠙⠶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⠀⠀⠀⠀⠀⢠⣤⢠⣶⣆⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⢸⣿⡌⡟⠃⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠙⢂⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡿⠏⠀⠀⠀⢀⣿⣿⠙⠀⢠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠳⢊⡼⢃⠀⢀⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⣤⡏⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⠀⠀⣈⡓⠞⣱⠏⣀⠘⣿⣿⣿⣿⣦⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣴⣿⡌⡆⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⡇⠀⡀⠈⣙⠲⠇⢰⠏⡀⠈⠙⠻⣿⣿⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡿⠁⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠁⠀⣈⡙⠲⠏⣼⠃⢀⣤⡾⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣟⣸⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠉⢛⣛⠲⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠋⢀⣀⠀⠠⠀⠀⠀⣰⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣈⣽⣿⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⡿⠀⠀⠀⠀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣠⠾⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢸⣿⣿⣿⣿⣿⣸⠋⠋⠁⠀⠀⢠⣞⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⡄⣠⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠀⣠⠀⠸⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⣹⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣼⣿⣿⣿⣾⣿⠁⡀⣴⣿⣡⣾⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣴⣿⣦⣽⣿⣶⡟⢻⣿⣿⣿⣿⣿⣿⣧⣼⣿⠀⠀⠀⠀⣿⣿⣧⣽⣿⣿⣎⣀⣋⡈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠀⠀⠀⠀⣿⢿⣿⣿⣧⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠙⠛⠟⠿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣿⣿⠟⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣶⣾⣿⣿⣯⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⣤⣿⠿⠟⠋⠉⠀⠈⠙⠛⠛⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢸⠀⢀⣤⣀⢠⣆⣤⡀⣄⢤⢀⠁⣠⣤⡀⢀⡀⣄⢀⡄⣄⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠉⣀⢘⠤⢼⢸⡏⠀⣧⡏⠀⢸⠀⡏⠸⠃⢸⠀⢸⣾⡇⠀⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢦⠴⠃⠓⠤⠸⢸⠗⠴⠃⠃⠀⠘⠀⠳⠶⠋⠘⠦⠞⠘⠃⠀⠸⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣷⠸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠐⠄⠀⠀⠀⠀⣰⠀⣨⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⣠⠹⠎⠉⡠⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⡀⠈⠀⡀⠀⠀⢀⣠⣤⣄⣀⠀⠀⣠⠹⣶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⣄⠀⠀⣀⣴⣶⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⠇⣀⠀⠁⠀⠀⠀⠀⠌⢁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣄⡁
⠀⠐⠀⠀⠀⢀⣈⣽⠩⠞⠉⠒⢂⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢴⠄⠀⠀⠀⢸⡄⠐⠄⠀⠀⠀⢀⣤⣶⣿⣿⡏⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠰⠈⣎⢁⠄⠀⠀⠁⣤⠀⠈⢸⣿⣿⡿⠉⠉⠀⠉⠉⠙⢿⣿⡷⣀⠀⠀⠀⠀⠉⠀⢾⣾⠃⠀⠀⢠⣾⣦⡽⢋⣿⡿⠁
⠀⠀⠀⠀⠘⢡⡀⡄⠀⠀⠀⠀⢻⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣌⠳⣦⡀⠀⠀⠱⣌⣿⣼⠀⡀⢿⣿⣿⣿⣾⣿⠃⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠘⣴⠀⠠⠀⠀⠀⡠⠀⢹⡿⠋⡀⠈⠁⠀⠀⠀⠀⠀⢰⣶⡆⡀⠈⠛⠿⠃⠀⠀⢲⣌⢿⣏⡾⣿⣾⣿⣟⣿⣿⡟⠀⠀⠀⢠⣿⡇⠀⠆
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠸⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⢀⣠⣶⣿⡿⠀⢦⡹⣾⣏⣴⣿⣿⣿⣿⣿⠟⠁⠀⣠⠀⠸⢋⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⡀⠀⠀⢿⡷⠀⠀⠀⠁⠀⠀⡀⠀⠻⣿⣿⣿⠿⠋⠀⠀⠀⠙⣾⣿⣍⣿⣿⡿⢿⣿⡇⠀⠀⣿⠐⠛⠉⠁⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⠀⣤⡀⠁⠀⠀⠀⣙⣭⣶⣶⣶⣶⣶⣶⣾⣿⣟⣿⣿⣿⣥⣿⡿⢀⡆⢠⡴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⢻⣿⣿⣶⣥⣀⣐⣆⣼⣿⣿⣎⣻⣷⣿⣿⣏⣭⣿⣿⣿⣿⣿⣿⢿⠃⠘⠷⠶⢂⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠓⠶⣽⣶⣄⣷⣌⢷⡉⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡟⢻⣿⣄⣼⣿⡿⠐⠛⠛⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠭⠿⠿⢛⣛⣿⣦⠆⢀⠀⣠⣿⣿⣿⣟⠻⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠭⠽⠖⠓⠈⢱⣿⣿⣙⣿⡿⢶⣿⡿⣿⣿⣿⣇⣼⣿⣿⡿⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠛⠹⣿⣷⣶⣿⣧⣬⣿⣿⣿⡿⠟⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣦⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣙⣿⡇⠀⠀⠙⠸⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣽⡿⢏⢋⠰⠀⣴⣄⣌⣿⠿⣻⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠁⠀⠀⡉⣢⢢⢇⠸⠀⠃⠀⠂⡍⢻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠁⢀⡀⠀⠹⢴⣆⣥⣀⣤⠐⢀⣠⡥⣼⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠁⠀⠀⠈⠃⠀⠀⠀⠉⣼⡾⠋⠟⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡆
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣠⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄
⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⣄⠀⢀⣠⣾⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠐⠿⠿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠈⣿⣿⣿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⣿⡇⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⡝⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣌⣀⢀⣠⣦⠈⠓⣰⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣬⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣯⣉⣀⣼⣿⡍⠋⠀⠀⠀⣀⣀⣤⣤⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣶⣶⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡍⠇⢸⢸⠸⡧⡇⡗⠿⡄⢣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠶⢠⡄⠀⡄⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠚⠀⡴⠛⢦⢠⡟⢳⠀⡦⢼⠧⢴⡧⡼⠛⣆⢸⠟⠂⡶⢸⠀⢰⠠⡟⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠈⢱⡤⣖⠒⣿⢺⣀⣸⠀⡇⢸⡆⢸⣧⡖⠂⣻⢸⠀⠀⡇⢸⣄⣼⢠⡙⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⢥⣀⣉⣹⠀⠀⠀⠁⠀⠁⠉⠉⠁⠀⠀⠀⠁⠀⠉⠈⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⢀⣀⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠋⠛⠉⠀⠉⠉⠛⠛⠻⢍⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡁⠀⣀⡀⠀⠀⠀⠈⠋⣠⣦⡀⢀⣩⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣄⣄⠀⠀⠀⠀⠀⠀⠛⠛⣱⣿⠋⠀⣸⠃⢠⣤⠄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⠁⠀⠀⠀⠀⠀⢀⣼⡿⣁⣀⡠⠿⣌⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠄⠀⠀⠀⠀⠀⢀⠫⢶⣿⡇⠀⠀⠀⠹⣆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣴⣿⡿⠁⠐⠀⢀⣴⠟⢠⣶⡄⠀⣴⣦⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⡀⠀⣤⣀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⢸⠟⠁⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠙⠿⣿⠿⣠⡆⠀⠛⠀⠀⠀⠀⠢⠄⠈⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠀⠀⠀⢰⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⢾⣷⣦⡈⠀⠀⠀⠀⠀⣿⠇⣠⣄⠀⣠⣤⣠⣶⣦⣤⣴⣶⣦⣶⣶⡄⠀⣶⣄⣶⠈⠁⠀⠀⠀⠀⠀⠘⠹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠸⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢆⠀⠸⣿⣿⣿⣇⡀⠀⠀⠀⠠⢶⠿⠟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣦⠖⠀⠀⠀⠀⠀⠀⠀⠶⠸⣿⠟⠛⠋⠉⠉⠉⠉⢹⣆⢠⣼⣧⡀⠘⢿⣿⣿⣿⡀⠀⠀⠀⣠⣶⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠘⣩⢿⢟⠂⠀⠀⠲⠀⠀⠐⣷⠁⠀⣼⠀⠀⡀⠀⠀⠀⠰⣦⡉⠀⠉⢰⠟⣧⣀⠻⣿⣿⣧⡀⣤⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠈⠒⠀⠘⣾⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⢻⣽⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⣻⣿⡿⣧⣶⣦⡄⠀⠀⠀⠈⠀⠰⡆⠀⠈⠿⢿⣿⣿⡀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣾⣦⠒⠀⠀⣿⣿⠛⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⢰⣿⡏⠱⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣿⡿⠀⣼⣿⡿⣀⠀⠠⠀⠀⠀⠀⣀⣀⣀⢀⡈⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠃⣼⣿⣿⠿⠟⢓⣦⣴⣶⣿⣿⣿⣧⠉⠸⣫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠺⠿⠿⠃⠀⠸⠟⠛⠛⠋⠛⠛⠓⠒⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣆⠀⠀⠀⠀⠉⠛⠂⣝⡂⣦
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠳⣬⣡⡄
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢺⣤⠛⣠
⠀⠀⠀⠀⢻⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡃⣿
⠀⠀⠀⠀⣤⠛⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡌⣴
⠀⠀⠀⢠⣿⣷⣸⣿⡿⣣⣿⣿⣿⣿⣿⣿⢳⣿⣿⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡔⡿
⠀⠀⠀⣿⣿⣿⣧⠋⣴⣿⣿⣿⢻⣿⡿⢣⣿⣿⡟⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣡
⠀⠀⠀⢿⣿⣿⣿⣼⣿⣿⣿⣏⡾⢋⣴⣿⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢸⡿
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⢏⣤⣦⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡿⣨⣴
⠀⠀⠀⠀⣎⠻⣿⣿⣿⠃⣾⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣰⡿⠃
⠀⠀⠀⠀⢿⣷⣤⣝⠃⢸⣿⣿⣿⢈⢻⣿⣷⣶⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣰⡿⠻⣋⣭⡄
⠀⠀⠀⠀⠘⢿⣿⣿⣷⣼⣿⣿⡇⣿⣷⡹⡟⡻⣿⣿⣿⣿⣷⠀⢀⠀⣠⣶⢰⣿⢧⣾⡿⠟
⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⢸⣿⣿⣧⠀⣿⣎⠇⣝⢿⠟⣴⣿⣧⣿⣏⣬⣥⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣙⡀⣿⣿⡿⢸⣿⣿⢠⣿⣧⢸⣿⣿⣀⣚⣛⡛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⢇⣸⣿⣿⠘⢿⣿⠘⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠿⠛⣩⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⡰⠟⠙⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡄⠀⠀⢈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠈⣿⣦⠠⠀⣼⡿⠀⠀⠀⠀⠀⣴⠶⢦⢸⡇⠀⠀⡀⣤⣴⣶⣤⡄⡀⠀⠀⣤⢶⣿⣿⣿⢶⡄
⠀⠀⠀⠈⠛⠶⠾⠛⠁⠀⠀⠀⠘⢦⣷⣤⡼⠟⠀⠀⠀⠀⠤⣼⣿⡤⠄⠀⠀⠀⠀⠀⢉⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⣾⣿⡄⠀⠀⠀⠀⠀⡀⠀⠘⠒⠀⠀⠀⠀⠀⢀⠛⠷⢿⢿⣄⠀⠀⠀⠀⣔⢂⠀⢒
⠀⠀⠀⠁⠈⡀⠻⣿⠆⠀⠀⠀⠁⣶⣦⣴⣄⠀⠀⠀⠀⠿⠀⢀⣀⠀⢈⠁⠀⠀⠘⠉⠏⢀⣾⡀
⠀⠀⠀⠐⠲⡝⠖⠁⠀⠀⠀⠀⠀⢌⠀⢉⠁⠂⠀⠀⠀⠀⠀⠂⣤⡀⠈⠀⠀⠀⠀⠀⠀⠋⡟⠆

⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠠⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠤⠚⡟⠢⠤
⠀⠀⠀⢠⡄⠀⠀⢩⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠸⠋⠿⣿⠇⠂⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠘⠻⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠺⡿⠂⠀⠀⠀⠀⠘⠛⠀⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣰⢂⡴⣰⢋⡞⣡⠞⣡⠴⠀⢀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠣⣏⡚⢤⣉⠳⣝⢮⡳⠄⠲⠄⠨⠿⠟⠃⢴⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⠴⠂⠧⢤⣄⠉⠃⠈⠁⢀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣀⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡋⠩⣤⣤⠀⠀⠐⠀⠀⠂⠀⠀⠀⠠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡀⠀⡄⠀⠀⠘⢿⣷⣦⠀⠁⠀⠀⠰⠈⠙⠃⠈⠁⠀⢀⡀⠀⠀⢠⣄⡀⠠⠅
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⠀⠀⣀⣤⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠈⢿⣿⡄⠀⠚⠂⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠠⣿⡞⠀⠀⠀⠀⠿⠟⠁⠀⠀⠠⠀⠀⠀⠀⢒⣀⣀⡀⠐⢾⣿⣿⣿⣿⡗⠀⠀⢰⡄⠀⡀⠀⢱⣿⠅
⠀⠀⠀⠀⡀⡴⠀⠀⠀⠀⠀⠀⠀⠰⠘⠚⣠⠀⠀⠀⠀⠀⡀⠀⢀⠦⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⠈⠻⠠⠇⠀⠀⠀⠀⢲⢀⡀
⠀⠀⠠⠀⠙⠜⢡⢀⡴⠂⠀⠀⠀⠀⢠⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣼⣿⠿⣿⣿⣿⣿⠧⠉⠉⠉⠀⠀⠦⡀⣄⠳⠞
⠀⠀⠃⢃⠀⣄⠘⠛⠁⠀⠠⣄⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣤⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠈⠛⡃⢠⠀⣘
⠀⠀⡀⠉⢠⡝⠋⠁⢀⠀⠒⣿⣿⠅⠀⣀⣀⠀⠀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁⠀⢁⠀⠀⠂⠉⠛⣤⠈⠐⡀
⠀⠀⣷⡀⣌⡁⠴⠆⠈⠀⠀⠀⠀⠀⡞⠁⠙⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠐⠀⠸⣤⠋⡖⠈⠀⠶⢀⢁⡄⣼⡇
⠀⠀⢻⣇⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢠⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀⣾⣰⡿⠁
⠀⠘⠳⣦⣛⣹⡄⡁⠀⠀⠀⠙⠛⠁⠀⠀⠈⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⠻⡇⠀⠀⢀⠀⠀⠀⠀⠀⢠⡁⣿⢹⣥⡶⠃
⠀⠀⠀⢦⣔⡘⠅⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⣿⣿⣿⣿⣿⣿⣻⣿⠶⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠋⣢⡤⠆
⠀⠀⠀⠀⠾⣝⢳⢏⡇⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⠀⠀⢸⡟⣧⢻⠵⠂
⠀⠀⠀⠀⠂⠀⠙⠼⠋⠁⠠⢾⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠄⠀⠉⠷⠚⠁
⠀⠀⠀⠀⠐⠦⢢⡀⠠⠀⠀⢠⣀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠒⢀⠀⢀⡀⠀⢀⠀⣰⣶⣆⠀⠀⠀⠀⢀⣄⠀⠀⠄⠀⡴⠦⠖
⠀⠀⠀⠀⠀⠘⠉⠀⣠⡀⠰⣾⣿⠆⣶⠀⣀⠈⠻⠟⠃⠀⠀⠖⠃⠸⠀⠑⠂⠀⠘⠛⠉⣀⡀⣴⡄⣿⣷⠖⢀⣀⠀⠈⠙
⠀⠀⠀⠀⠀⢀⡐⠠⠈⠻⠆⠀⠀⠀⠙⠀⠛⢧⡀⠀⣰⣦⡀⠀⠀⠀⠀⠀⣠⣶⠀⢄⣰⠟⠁⠛⠀⠀⠀⠰⠟⠋⠠⠐⡀
⠀⠀⠀⠀⠀⠑⠐⠐⠿⠿⠖⠀⠀⠀⠀⠀⠀⣈⠃⢘⡀⠙⠃⠀⠀⠀⠂⠘⠛⠁⡈⠀⢃⡀⠀⠀⠀⠀⠀⠐⠿⠿⠃⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠰⠀⠀⠙⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠋⠀⠀⠒⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⣧⢰⣤⢠⡅⡠⢄⢠⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠶⢿⢸⠀⢸⡿⡡⣾⣜⠧⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠈⠀⠈⠁⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⢠⡄⠈⢁⡀⢀⣴⣾⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠈⠉⠛⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⢠⡀⢠⡀⠙⠛⠉⠀⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠁⠘⠃⣤⣅⠈⠄⠚⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⣠⢆⡴⢠⠴⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣰⣶⣿⣿⡀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⢸⠇⡞⠼⣅⡣⣬⡲⣅⠒⠦
⠀⠀⠀⠀⠀⠀⣈⣾⣴⠊⢉⣛⣟⣿⣿⣇⠀⠘⠿⠀⠀⠀⠀⠀⠀⡟⠼⢬⣕⡲⠦⠉⠀⠉
⠀⠀⢶⣄⠀⠀⠀⠛⠉⠀⠉⢈⡝⠻⠿⠿⠞⠋⠀⣀⣰⣆⣤⣶⣶⣧⣶⣿⣿⣿⣿⣶⣦⣄⠀⣤
⠀⠀⠈⠻⠇⠀⣦⢀⡀⠀⢀⢤⣀⡘⠃⠈⢉⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣧⠖⣀⣤⠄
⠀⠀⠀⠈⠉⢁⣙⣀⠱⡄⠸⣿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣁⢲⣄⠚⣭⣄
⠀⠀⠀⠀⠀⠀⠉⠩⠶⣟⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠃⠀⠉⠀⠈⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠀⢠⡌⠁⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣹⣿⣿⣿⣏⣽⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣄⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣯⣹⡿⠀⠀⠀⠀⠀⠒⣺⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠹⠙⠛⡛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⣶⣶⢤⡀⠀⠀⠀⠀⢠⡾⠷⠀⠀⠾⣷⡀
⠀⠀⠀⠀⠀⣀⢠⡄⢴⡆⠀⠀⠀⠀⢦⣀⣀⣀⣠⠀⠠⡶⢄⠀⠀⡼⠟⣿⣿⣿⣿⣿⡿⠗⠀⠀⠀⠀⢾⡁⢠⣤⣤⡄⢈⡷⢀
⠀⠀⠀⢸⡿⠿⠀⠓⠈⠁⠀⠀⠀⠀⠛⠟⠿⢿⣷⠖⠢⠈⠀⠀⠘⠁⠀⣿⣿⣿⠷⡄⠀⣼⣇⠀⠠⠰⢄⡙⢿⣿⣿⡿⢋⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣈⡙⠒⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⡤⠟⠛⠋⠀⠈⠀⠁⠈⠙⠋⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠁
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠆⠀⠴⣾⡷⢄⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣦⠀⢰⣾⣿⣷⣤⠀⠀⠀⢀⣶⠟⠉⠀⣈⡑⢦
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⣼⣿⣿⣿⣿⡇⠀⢠⣿⠁⣾⣿⣿⡏⠳⠀⠁
⠀⠀⢸⣶⠾⣷⢸⢰⡶⣦⣴⠀⠀⠀⠀⠀⠰⠿⠛⠃⠀⠀⠀⠀⢻⣿⣿⣿⠿⠃⠿⢿⣿⣿⣿⠃⠀⠸⣿⠀⠻⠿⠿⠷⣽⡆
⠀⠀⠈⠙⠛⣡⣿⣬⣷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣤⡀⣠⣼⡿⠟⠁⠀⠀⠀⠙⢷⣄⣀⣀⣤⡿⠃
⠀⠀⠀⠀⠘⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢠⣿⣿⣿⣿⠂⠀⠀⡄⠀⠀⠀⢀⡴⠛⠻⣟⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣯⡙⠛⠒
⠀⠀⢠⡿⣿⣿⡿⠀⠀⠀⢸⠀⠀⠀⡟⠀⢀⣾⣿⣿⣻⣇⠀⠀⠀⠀⠀⣀⢀⢠⡤⡖⣶⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣽⣆
⠀⠀⠈⣿⣿⣿⣀⡀⠀⢀⡼⠀⠀⠀⡇⠀⣾⣿⣿⣿⠿⣿⠗⠀⠀⠀⠀⡇⠜⠛⠃⠁⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠙⣿⣿⣿⣿⣶⠿⠁⠀⠀⠀⠀⠀⠘⢿⣿⣇⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣀⠀⠞⠉⠉⠳⡄⢠⡤⠀⠀⢀⣴⣿⣿⣿⣿⡧⣄⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣶⣦
⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⢿⡆⠀⠀⠈⠸⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⡤⣤⣶⡆
⠀⢠⣤⠖⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠘⢿⣿⣿⣿⠙⠛⢲⠆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⡀⠀⠀⠀⠀⠀⠃⠛⠉⠁
⠀⠀⠀⠀⠀⢐⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠆⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⠿⠿⢿⠿

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⣀⡀
⠀⠀⠀⠀⣞⣉⢦⠀⣶⢫⣙⡇⠀⠀⠀⠀⠸⡏⠳⣌⣡⡏⢹⠇⠀⠀⠀⠀⠈⠙⢻⡶⠶⡶⠟⠁⠀⠀⠀⢀⡾⠛⣯⠉⢙⣹⡀
⠀⠀⠀⠀⠉⠉⠘⣧⡇⣈⡉⠁⠀⠀⠀⠀⢀⢤⡞⠋⠙⣦⢢⠀⠀⠀⠀⠀⠀⣤⢸⠇⠀⡇⠀⠀⠀⠀⠀⠈⠻⠶⣏⠀⢯⡀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⢻⠁⠙⠁⠀⠀⠀⠀⠀⠀⠸⣧⣀⣠⡿⡘⠀⠀⠀⠀⠀⠀⢶⡾⠟⠛⠳⢦⡀⠀⠀⠀⠀⠨⢓⠶⠶⠾⢟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣤⡠⡀⠀⠀⠀⠀⠀⢀⢀⣤⣤⣄⢠⡄⠀⠀⠀⠀⠀⠀⣶⣀⠠⢤⡀
⠀⠀⠀⠀⠀⣄⢰⡟⠉⠙⡮⠆⠀⠀⠀⠀⠻⠩⢻⠀⣿⠈⣗⡁⠀⠀⠀⠀⠀⢠⡟⠁⠀⠝⣇⠀⠀⠀⠀⢰⡟⢻⡋⢹⡏⣷⢀⠂
⠀⠀⠀⠀⠀⢩⠼⢧⢠⣴⠃⠁⠀⠀⠀⠀⢠⣤⢹⠀⣿⢰⡏⡿⠀⠀⠀⠀⠐⢈⣧⡀⠀⣡⣏⡀⠀⠀⠀⠈⣤⢸⠃⢸⠁⡿⣨⣄
⠀⠀⠀⠀⠀⠹⠤⠞⠰⡇⡃⠀⠀⠀⠀⠀⠈⠁⠈⡄⠁⣸⣞⠑⠂⠀⠀⠀⠀⠭⠭⠥⠤⠬⠭⠅⠀⠀⠀⠀⠀⠸⠀⠚⠸⢧⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣄⣠⣤⣤⣄⠀⠀⠀⠀⠀⢀⣤⣄⣠⣴⡢⠀⠀⠀⠀⠀⠀⢠⣀⠀⣀⡀⣄⣀⠀⠀⠀⠀⠀⠀⢄⡀⣠⡀⢂⡄
⠀⠀⠀⠀⢠⡀⢤⣀⡼⠛⡏⠀⠀⠀⠀⠀⠘⠀⣹⡅⢸⡇⠶⠀⠀⠀⠀⠠⠞⠙⠞⠉⠳⠞⠙⠆⠀⠀⠀⠀⠰⠎⣷⣈⣁⣟⣀⠆
⠀⠀⠀⠀⠈⠁⣠⣟⠳⠄⠁⠀⠀⠀⠀⠀⢀⣦⠹⠃⢸⣟⢉⡇⠀⠀⠀⠺⢂⣴⣄⡴⣦⣠⢶⣘⠋⠀⠀⠀⠈⡉⣿⠉⠹⣏⢍
⠀⠀⠀⠀⠀⠐⠋⠈⠀⠘⠁⠀⠀⠀⠀⠀⠀⠁⠠⠴⠋⠈⠉⠀⠀⠀⠀⠐⠁⠀⠙⠁⠈⠑⠀⠁⠀⠀⠀⠀⠘⠚⠃⠀⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠃⠀⠀⠈⠙⠻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠿⠁⠀⠀⠀⠀⠄⠀⠙⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣯⡝⠀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠟⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢿⣿⣿⣧⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⣀⣀⣠⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢻⣿⣿⣿⣿⣿⠟⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠂⠀⠀⠀⣠⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠄⡀⢀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠙⣦⡇⢸⠈⣆


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣯⣽⣵⣯⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣷⣾⣿⣿⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⢶⣖⣒⣚⣛⣛⣓⣒⣲⡶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣻⡭⣷⣿⣿⣿⣿⣿⣿⣿⠟⢻⣯⣽⣿⣿⣷⣾⣭⣟⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⣭⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣝⢶⣄
⠀⠀⠀⠀⠀⢀⡼⣫⣿⣿⣿⣿⣿⣛⣿⣿⢿⣿⣿⣿⣥⣿⣿⡷⣿⣿⣿⣿⣿⢷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⢧⡀
⠀⠀⠀⠀⣰⢯⣾⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⠷⠿⠿⠿⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣿⣿⣿⣷⡙⣆
⠀⠀⠀⣴⣳⣿⣿⣿⣿⠿⣿⣿⣷⢟⣿⣿⣧⣿⠟⡋⠉⠀⠀⠸⢳⠟⠀⠀⠀⠉⡙⠻⢿⣿⣿⡿⡫⣾⣾⣿⣿⣿⣿⣻⣿⣞⢦
⠀⠀⣼⣳⣿⣿⣿⣿⣻⣿⣿⣿⣿⣷⣭⠛⠋⠀⢳⡿⠄⢀⣠⣤⣬⣤⣤⣄⡀⠰⣻⠟⠀⠙⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣎⢧
⠀⢰⢧⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⣀⣵⣶⠿⣿⣾⠿⢿⣼⣿⣛⣷⣦⣀⠀⠀⠀⡙⣷⡿⢿⣿⣿⣿⣿⣿⣷⣾⣿⡜⡆
⠀⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⡟⡼⢀⣼⣿⣽⢟⣛⠿⠗⠒⣆⠀⠈⠻⣿⣼⡿⣧⡰⣾⡷⠌⢿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣿⢻
⢸⣷⣿⣿⣿⣿⣿⣭⣾⣯⣛⠛⠃⠀⠈⠁⣾⣿⡾⠿⣿⠿⠶⡔⠀⢸⠀⢀⠀⠘⣿⡷⣿⣷⡈⠀⠀⠘⣛⣋⣭⣬⣷⣾⣿⣿⣿⣿⡞
⣸⢸⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⠀⠀⠀⢼⣿⣿⣿⣷⡀⢀⣺⠀⠐⢂⠇⠒⠀⠀⢸⣿⣿⢿⡷⢂⢀⡀⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣾⢸⣾⣿⣿⣯⣽⣿⣿⣿⣿⡏⣤⠞⠆⣹⢷⣾⣿⠿⠟⠉⠙⡄⠀⠈⣟⠀⠀⠀⣼⣻⣿⣿⣿⢻⣝⡳⢸⣿⣿⣿⣿⣿⣩⣿⣿⣿⡇
⢹⢸⣿⣿⣿⣿⣽⣿⣿⣭⣿⣿⠀⣀⠠⢼⣿⠛⣻⣷⣿⣦⡤⠾⣂⢠⠆⠀⢀⣼⣿⣿⢿⣿⡿⠀⠀⠀⣼⣿⣿⣽⣿⣿⣿⣿⣿⣿⡇
⢸⡾⣿⣿⣿⣿⣿⣿⣿⢯⣟⣭⡏⢀⡀⠀⢿⣿⣿⡿⢿⣭⣾⣿⣧⣤⣤⣶⣿⣿⣿⣷⣾⡿⢁⡀⠀⢠⣼⣭⣻⡿⣻⣿⣿⣿⣿⣿⢧
⠀⣷⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡜⠛⠟⠈⢻⣿⡿⢿⣿⣿⣯⣽⣿⣶⣿⣿⠟⣿⣿⡟⠑⠻⠿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾
⠀⠸⡞⣿⣿⣿⣿⡏⢙⣿⣿⣿⣿⣿⣆⠀⠀⠀⠉⡛⢿⣿⣹⣿⡟⣿⣿⣭⣿⡿⠟⠉⠀⠀⠀⣠⡿⣿⣿⣤⣿⣿⣿⣟⣿⣿⣿⢣⠇
⠀⠀⢻⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡟⣡⣄⡀⢀⣿⡄⠀⠉⠋⣭⣭⠉⠛⠀⠘⣷⣿⠄⣠⣦⣹⢷⣿⣿⣿⣿⣿⣿⣿⢿⣿⢏⡞
⠀⠀⠀⢻⡝⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣯⣿⣿⣶⣭⣁⠀⠀⠀⠻⠼⠁⠀⡀⣀⣤⣶⠿⣿⣿⣿⣗⢿⣿⣿⣿⣿⣿⣿⣾⢋⡞
⠀⠀⠀⠀⠹⣞⢿⣿⣿⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣷⣶⣿⣿⣧⢻⣿⣿⣶⣿⣿⣏⣹⣿⣝⣿⣿⣿⣿⡿⣣⠏
⠀⠀⠀⠀⠀⠈⢳⣝⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⣷⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣝⢿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⣋⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣻⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⢟⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠮⣽⣛⠿⢿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣹⣿⣿⣿⠿⣛⣫⠵⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠾⠭⠭⣭⣭⣭⣭⠭⠭⠷⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡷⠷⠟⠛⠛⠛⠛⠛⠛⠻⠷⢿⣿⣿⣷⣦⣄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣴⣿⣿⡿⠟⠉⠁⢀⠠⡠⠀⠀⠀⠀⠀⠠⡀⡠⠀⠈⠉⠻⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠈⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠉⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠤⠤⠀⠀⠀⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣿⣽⡏⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⡄⢠⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⠀⠀⠀⢹⣟⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠨⠩⠍⠭⠀⠀⠀⠀⠔⠀⠀⠀⠲⣌⣷⣿⣿⣾⣡⠖⠀⠀⠀⠢⠀⠀⠄⠀⠖⠁⣭⠂⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣈⣹⣿⣿⣿⣿⣿⣿⣏⡁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⠴⢞⣿⣿⣿⣿⣿⣿⡳⠦⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣷⣿⡀⠀⢸⠁⡆⠀⠀⠀⠀⠢⠀⠀⠐⠉⡽⢻⢿⡿⡟⢯⠉⠂⠀⠀⠔⠀⠀⠀⢀⡌⠈⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣽⣇⠀⠈⠠⠃⠂⠀⠠⠀⠀⠈⠂⠀⠀⠀⠊⠈⠁⠙⠀⠀⠀⠄⠀⠀⠀⠄⠀⠈⠀⠦⠀⠀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣯⣿⡆⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⠔⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⡠⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠒⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣽⣧⡀⠠⠛⠀⠀⠀⠀⠀⣀⣀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⠂⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠁⠁⡇⠀⠀⠀⠀⣸⠉⣁⠀⠀⠀⠀⠀⣀⣴⣟⣿⠟⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣿⣿⣷⣦⣀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣴⣾⣿⣿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⢶⢦⣤⣤⣤⣤⣤⣤⣴⠶⣾⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⠿⠿⣿⣿⡿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡄⠀⣤⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠿⠀⡴⠒⣄⢠⡾⢶⠀⣦⣴⡧⢠⣿⣠⠖⢦⠀⡶⠲⢰⡄⡆⠀⣶⢠⡞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠙⢦⢠⡶⠰⡯⣿⠀⣸⠀⡇⢸⡇⠀⣧⣴⠆⠺⠄⡇⠀⢸⠀⣧⠀⣿⢈⠣⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠋⠈⠙⢚⣋⠈⠛⢹⠀⠁⠀⠃⠀⠉⠙⠚⠛⠁⠁⠀⠈⠁⠈⠉⠉⠈⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣶⣿⣿⣶⣄⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⡿⢿⠿⠛⠿⠿⢿⣿⣿⡿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⣀⠄⠀⠉⠙⠢⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣦⠀⠢⠖⠀⠀⠀⠀⠀⢰⣿⣷⣀⣴⡟⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠉⣰⣿⠏⠀⢀⡟⡀⢿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⣾⡟⢡⣦⠄⠚⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣷
⠀⠀⠀⠀⠐⠀⠀⠀⠉⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠂⠀⠀⠀⠂⠁⠀⡿⢚⣿⣿⡀⠀⠀⠀⢹⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣀⣠⣷⣿⣿⠃⠀⠁⠒⠤⠾⠃⢰⣿⡆⢀⣶⣷⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠄⠀⢰⣦⡀⠠⣦⣼⣿⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠸⠟⠁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠙⠿⡟⠇⣴⡆⠈⠛⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⠀⠀⠀⢸⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠈⠀⠀⠀⠀⢘⡿⠃⣴⣄⠀⣠⣤⣤⣶⣶⣦⣴⣶⣶⣶⣾⣦⠀⠰⣶⣴⣧⠈⠀⠀⠀⠀⠀⠀⢈⠹⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠹⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢻⣿⣿⣾⣆⡀⠀⠀⠀⠢⠶⠿⠟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣷⣤⠆⠀⠀⠀⠐⠀⠀⠀⠶⠺⣿⠟⠛⠛⠉⠉⠉⠉⠉⣧⠀⣤⣿⣄⠀⠹⣿⣿⣿⣷⠀⠁⠀⠀⣠⣶⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠐⢛⣽⣿⡳⠀⠀⠀⠆⠀⠀⠰⣯⠐⠀⣸⠀⠀⠀⠀⠀⠀⠠⣶⣌⠃⠈⠉⣼⢛⣤⠘⢿⣿⣿⣧⠀⣠⠼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠸⣾⣿⣿⣦⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠟⠙⢺⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣴⣤⡄⠀⠀⠀⠀⠂⠀⢴⠀⠁⠸⡿⣿⣿⣿⡀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣾⣄⠦⠀⠀⣸⣿⡗⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣱⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⢀⣿⡿⠣⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⣿⠃⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠏⢠⣿⣿⣯⣿⡦⢄⣁⣠⣤⣶⣶⣶⡛⠇⣶⢶⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠋⣠⣿⣿⡟⠀⢀⣴⣾⠿⠿⠿⠿⢿⣏⡭⠀⢀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣿⣿⣿⠿⢿⣶⣶⣒⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠛⠉⠛⣿⣿⣷⣿⣾⣿⣷⣾⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢶⣶⠞⠫⠀⢿⠟⠃⠀⢿⢹⡏⣿⣿⣻⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡿⣦⣿⣿⣽⣿⣇⣿⣿⢹⣷⢹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠲⢶⣾⣦⣄⠀⠀⠀⠀⠀⣃⣴⣿⢣⣿⣿⠏⣿⣿⣿⣾⡟⢸⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⣿⣿⣷⠀⠀⠀⢰⣿⣿⣷⣿⣿⣥⣸⡿⣿⣽⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣏⣀⡞⠀⣠⣠⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣻⣿⣧⢏⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⢻⡟⠁⢀⣶⣿⠋⣿⣿⣿⣿⡿⢝⠋⢻⡯⠻⣷⣹⣼⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣥⣿⣷⣿⣿⣿⣿⢿⣿⣿⣟⠿⠀⣤⣀⢻⣄⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣙⣿⣿⣿⣿⣿⢻⣿⣯⣙⠛⢦⣾⣿⠘⣿⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢰⡊⢉⡉⣿⣿⣿⣿⣿⣿⣿⣙⠁⠀⢀⣹⣟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠋⢁⠀⣷⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⠏⣼⣼⣿⣿⣿⣿⣿⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠉⠀⠀⠙⠻⣿⣿⣿⣿⣿⡜⢻⣿⣿⣼⣿⣿⣿⣿⣿⣿⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢠⡾⢫⣿⣿⣿⣿⠈⠙⣿⣿⣿⣿⣿⣿⣿⢻⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠘⠀⣸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⣤⣤⣴⣾⣿⣿⣿⣿⠇⠀⠀⣾⣿⣿⣿⣿⣿⣆⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⠿⣿⠛⣒⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣧⡤⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡶⠟⠛⠛⠛⠛⠛⠛⠷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⣉⠀⠀⢧⠀⢹⡿⣦⣾⣿⠂⢸⠃⢠⡌⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢁⣤⣀⣿⣴⠀⠘⡄⢀⠁⢿⠃⠁⠀⡏⠀⣸⣷⣦⣀⣀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⢀⣾⠟⠳⣄⠀⢨⡽⣿⠛⠶⠀⢡⠀⣀⣈⣀⡀⠰⠁⠸⣿⠀⣹⡟⠉⢀⠔⠉⢻⣄
⠀⠀⠀⠀⣠⡿⠁⣶⢆⣈⠳⣄⠀⠛⣀⣤⠶⣛⣫⣭⣭⣭⣭⣛⡳⢦⣍⠋⠛⢁⠔⠁⠀⢴⣤⡙⣧⡀
⠀⠀⠀⢰⡟⢀⣼⠛⣼⠏⠀⠀⣱⡾⣫⣶⣯⣿⣿⢿⣯⣽⣿⣿⣿⣷⣮⡻⣶⡁⠲⣦⡾⢋⣹⣇⠘⣷
⠀⠀⢠⡿⠁⠈⠃⢿⠃⠀⠀⣼⢫⣾⣿⣿⣽⣷⣿⢋⣉⡉⠉⠛⠻⣿⡟⣿⣮⢳⡄⠘⣷⠋⠁⠉⠁⢸⣧
⠀⠀⣼⠟⠓⠲⠤⠌⣀⡀⣼⢣⣿⣾⣿⣿⡁⣀⠀⠉⠀⠙⡆⠀⠀⠉⣻⣿⣻⣧⢻⣄⡈⠥⠤⠒⠛⠉⢿⡄
⠀⠀⣿⠀⡀⢀⣤⡄⠀⢨⡟⣾⡷⣿⠋⠉⠉⠉⠀⠖⠀⣐⢇⢰⠂⠀⠐⢿⣿⣿⡏⣧⠀⢠⡆⣶⣳⣄⢸⡇
⠀⠀⣿⠀⣴⡾⠷⠿⢦⢸⡇⣿⢷⣿⣤⠄⠀⠀⢨⠀⠈⠉⢇⡀⠉⠀⠀⢸⣷⠻⡇⣿⠀⢸⣧⣭⡉⣿⢸⡇
⠀⠀⣿⠠⠟⠛⠁⠀⠀⢘⣧⢿⣾⣿⣇⠀⠀⠀⠀⢣⡀⠐⢺⠕⠀⠀⢀⣿⣿⣿⢇⣟⠀⠈⠻⠾⠓⠋⢸⡇
⠀⠀⢹⣧⣤⠤⠖⠚⠉⠁⢻⣞⣿⣿⣿⣿⡟⠀⠀⠀⡠⠀⠘⠀⠀⣠⣾⡷⣾⡟⣼⠃⠀⣉⠓⠒⠦⢤⣾⠁
⠀⠀⠈⣿⡀⣀⣠⣖⣰⣆⠀⠻⣮⢿⣽⣿⣿⣶⣶⣻⣷⣦⣤⡶⣿⡿⢿⣿⢏⡾⢃⢀⣾⣽⣇⡁⠀⣰⡏
⠀⠀⠀⠘⣧⡈⠛⠏⢹⠏⠓⠀⡙⠷⣭⡻⢿⣽⣿⣛⣿⣿⣿⣷⣿⠿⣻⡵⠟⢄⠛⡉⢳⣌⣉⣷⢰⡟
⠀⠀⠀⠀⠘⢷⡀⠀⠀⢂⡴⠊⠶⣦⠈⠙⠳⢾⣭⣽⣿⣿⣯⣽⠶⢛⠉⠀⠀⠀⠑⢄⠀⠈⠉⣱⠟
⠀⠀⠀⠀⠀⠈⠻⣦⡴⠋⠀⢀⡴⢋⠀⡀⠀⡐⢀⡄⢀⡀⢀⡘⡔⢿⣳⡰⣄⢹⣤⡈⠳⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠘⠷⣯⣰⠇⢠⠇⠽⣷⠊⢻⣯⠆⢳⠈⠉⢷⡽⠛⠋⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣌⡛⠀⣾⠀⠀⢿⣤⡜⠃⠀⠸⡆⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⣦⣤⣤⣤⣴⠶⠶⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠰⠀⠰⠀⠃⠀⠀⠘⠀⠆⠀⠆⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠑⠀⠀⢀⣀⠠⠤⠤⠄⢀⠀⠀⠀⠊⠠⠆⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠈⠂⠀⣠⣴⣾⣿⣿⣷⣦⡀⠀⠀⠈⠳⢄⣀⠀⠐⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⢠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠙⠓⡀⠀⠠⠔⠊⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣰⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⡄⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠀⠀⠉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⡇⠀⣀⣀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⢸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⠇⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠐⣠⡿⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠲⣆⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⢀⢴⡿⠁⢀⣰⠖⢀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠁⠈⠧⡀⠀⠻⢿⣿⠿⠿⠟⠛⠋⠀⠀⠀⢀⣺⣵⡿⠋⠀⢀⠼⠁⠈⠩⠁⠀⡀
⠀⠀⠠⣥⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠐⠃⡀⠀⠉⠒⠦⣤⣤⣤⣤⣶⡾⠿⠛⠉⠀⢀⠐⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄
⠠⢤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠄⢀⡀⠁⠈⡀⠀⠀⠓⠎⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢⠂⠀⠀⢀⡐⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⡠⠄⠀⠀⡤
⠀⠀⠈⡂⠀⢀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠈⢊⠀⡌⠀⠆⠀⠀⢰⠀⢃⠀⡱⠁⠀⠀⠀⠈⠋⠉⠀⠀⠀⢀⠀⠀⠀⠀⠁⡀⠀⢐⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠂⢀⣀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⣀⡀⠐⠂
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠐⠀⠪⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠕⠀⠂⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠌⠀⢪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣷⣽⣿⣿⡿⢟⣿⣽⣿⣿⣿⣿⣟⣽⣿⣿⣿⣧⣄⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⢻⡉⠀⠙⣿⣿⣿⣿⣿⣅⠘⠻⣶⣄⠉⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣷⣶⣽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣿⣿⣿⣿⡿⠀⢾⣿⣿⣿⣿⣿⣿⣿⣟⣻⣧⣾⣿⣿⣿⣿⣿⡖⠙⠻⠿⣿⣿⣿⣿⣷⣍⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣿⣿⣿⡟⣴⣿⣾⣿⣿⣿⠿⠿⠿⣟⣛⣻⣻⡿⡿⠽⣷⣿⣿⣿⣤⣶⣴⣿⣿⣫⣾⣿⣿⣧⡙⣦
⠀⠀⠀⠀⠀⢀⣼⣿⠛⠟⣿⣿⣯⣽⣿⣿⣿⣿⡿⢟⣭⣵⣾⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢳⣿⣿⣿⣿⣿⣿⣌⢳⡀
⠀⠀⠀⠀⢀⣾⣿⣿⡅⠈⠿⣿⣿⣿⣿⡿⢟⣩⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣵⣿⠟⠋⠀⠈⠻⢿⣿⣧⢻⡄
⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⢀⣿⡿⣱⣿⣯⣿⣿⣿⣿⣟⣿⣿⣿⡗⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⣤⣤⣀⣸⣿⣿⣆⢻⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⣿⣿⣿⣟⣼⣿⣝⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⣻⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡌⣧
⠀⠀⢠⣿⣿⣿⠿⠻⠟⠛⠻⣿⡟⣾⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣩⣶⣿⣿⣿⣧⢸⡆
⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣾⡿⠋⢿⡏⠙⢿⣿⠀⣧
⠀⠀⣿⣿⡿⣿⣟⡁⠹⣿⣿⡟⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣷⣤⠹⠇⣤⣾⣿⡆⣿
⠀⠀⣽⣿⣷⣀⣈⠁⢀⣿⣿⣇⣿⣯⣴⣿⣿⣧⣿⣿⣯⣿⣿⣿⣿⣽⣀⣀⣿⣿⣿⣿⣿⣽⣿⣧⣼⣿⣿⣹⣿⣿⢸⣿⣿⣇⣀⣀⣹⣿⣿⡇⣽
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣼⣻⣿⣿⣿⣿⣿⣿⣿⠁⣿
⠀⠀⢸⣿⣿⣿⣿⠿⣫⣵⣾⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣯⣭⣭⣽⣿⣿⣿⣿⢠⡏
⠀⠀⠈⣿⣿⣿⣶⡿⣿⣿⣿⡿⣿⡽⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⣸⠃
⠀⠀⠀⠸⣿⣿⣿⣷⣿⡟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣟⣁⣉⣿⣾⣿⡟⢠⡟
⠀⠀⠀⠀⢹⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣯⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⠫⣽⣿⣿⣿⣿⣿⣿⣿⡿⢀⡾
⠀⠀⠀⠀⢠⡻⣿⣿⣿⣷⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣵⣿⣿⣿⣮⣽⣿⣿⣿⣿⣿⡟⢁⡾⠁
⠀⠀⠀⠀⠀⠙⡿⣿⣿⣿⣿⣿⣿⢏⣾⣿⠿⠿⠿⣿⣿⣿⣿⣟⣿⣿⠿⠿⢿⣿⣛⣿⣭⢶⣿⣿⠃⠈⠿⢻⣯⣿⣿⣿⣿⣿⠏⣠⠟
⠀⠀⠀⠀⠀⠀⠈⢮⣙⣿⣿⣿⣵⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⠿⠋⣿⣿⣿⡿⣿⣿⣿⣇⠀⠀⣈⣿⣿⣿⣿⠟⢁⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢿⣿⣿⡿⠿⠿⣿⣿⠿⠿⣿⣏⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⠟⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⣝⡛⠿⣶⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣮⣙⠛⠿⢿⣿⣿⣾⣿⣿⣿⣿⣴⣤⣿⣿⣿⣿⣿⠿⠟⢛⣉⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢤⣭⣍⣉⣙⣛⣛⣛⣛⣉⣉⣉⣥⣤⠶⠞⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠾⣿⠃⠀⠀⠀⠀⢻⡿⢶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠋⢉⣀⣠⡄⢻⣄⠀⡄⡄⠀⣼⠃⣤⣀⣈⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⠴⣚⠉⠁⣸⠁⠀⠙⠛⠁⠙⠛⠁⠀⢹⡀⠉⢙⡲⢄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣼⢟⡳⠄⢿⠀⠀⠀⢠⡄⣤⠀⠀⠀⢸⡇⠴⣛⢿⡄⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣸⡏⠇⠙⡦⢸⡄⠐⠋⢹⣧⣿⠉⠓⠀⣼⠠⡞⠁⠏⣿⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠁⣀⡄⠘⢧⠀⠈⠰⠋⠀⠀⠀⣸⡿⣿⡀⠀⠀⠈⠳⠈⠀⢠⠟⠀⣄⡀⠉⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⠟⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠙⢷⣌⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⣴⠀⠁⠀⠀⢰⣮⢠⣀⣀⡀⠀⠘⡇⡟⠀⠀⣀⣀⣠⢨⣦⠀⠀⠀⠁⣰⣄⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣾⠇⡼⠉⠳⣄⡀⠀⢸⡏⠢⣈⡫⠙⠦⠀⠀⠀⠠⠞⠩⣋⡠⠊⣿⠀⠀⣀⡴⠋⠹⡄⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⣀⠀⠉⠀⠘⣷⣰⠁⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⢱⣰⡟⠀⠈⠁⢀⣀⣀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣷⠏⠁⢀⠀⠀⠀⠀⠈⠻⡄⢀⡀⠼⠀⠀⠀⠹⠄⣀⠀⡾⠋⠀⠀⠀⠀⢁⠀⠉⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⣾⡆⠀⠀⠀⢠⠀⢀⣞⡧⣄⣀⣀⣀⣀⡤⣟⣆⠀⢠⠀⠀⠀⠀⣾⣆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⡀⠀⠀⣸⠀⢸⣿⠀⠻⢦⡀⡠⠶⠃⢸⣿⠀⢸⡀⠀⠀⣰⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠛⠛⢁⣴⡄⠙⠶⣤⠤⠀⠠⣤⡴⠞⠁⣴⣄⠙⠛⠛⠁⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠉⠒⠒⠋⠁⠙⠦⣄⠙⠶⠤⠴⠞⢁⡤⠞⠁⠉⠓⠒⠊⠁⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⢆⠀⠀⣀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢀⡀⠀⢀⢆⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⠦⣌⡙⠓⠒⠾⠷⠶⠤⠴⠶⠿⠖⠒⠛⣉⡤⠖⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣉⡛⠶⢤⣀⠀⡀⡀⢀⣠⠴⠞⣋⣡⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⡈⠳⡇⡷⢋⣠⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠠⠊


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⢰⠂⠒⠛⠲⢦⠀⠠⡶⠚⠋⠃⢰⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠚⠙⠃⠀⠀⠀⠀⠂⠛⠛⠀⠀⠀⠀⢿⣿⣽⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠛⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⡈⠻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢹⣧⢻⢠⣏


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡶⡄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⠁⢻⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡏⢰⡆⢷⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⠁⣿⣷⠈⣧⠘⣇
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣸⢡⡏⢸⠃⡎⣇⠘⣆⠹⣄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠙⢯⡉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣼⢀⡟⠀⡇⢹⡀⠸⣆⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⣩⠽⠃
⠀⠀⠀⠀⠈⠳⢜⡻⢬⣩⣿⣭⡤⠤⠶⠶⠖⠛⢧⡀⡇⢀⡷⠤⠭⠭⠤⠤⣤⣤⢀⣠⠾⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠶⣍⠳⢭⡛⠿⣝⡓⠒⠒⠶⢽⣷⣯⠖⠒⠒⠒⣊⠜⣫⠖⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣌⠓⢦⣝⢶⡤⠒⣹⢻⠿⡙⢶⠞⢉⡴⠚⢁⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠆⣼⢸⠃⣰⠃⢸⡀⠙⣌⣧⠸⡄⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⡇⡞⡴⢃⡴⢋⡛⢦⣈⢻⣆⢳⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡞⣸⡿⠞⢁⡴⠋⠙⠳⣌⡛⢿⡆⢧⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡼⠐⢋⡤⠞⣉⡤⠚⠛⠳⣌⡙⢦⣄⠘⡆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣼⠷⢚⣩⠖⠋⠁⠀⠀⠀⠀⠈⠙⠦⣌⠻⠿⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⡆⠰⠖⠶⢦⣤⡀⠀⣠⣴⠶⠒⠆⢠⡄⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠘⠀⣴⠶⣤⠄⠉⠁⠀⠉⠀⠴⡶⢦⠀⠃⠈⠀⠘⣿⣿⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣿⣿⠟⠁⠀⠎⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠁⠸⣿⣿⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠶⣦⣤⣤⣄⣀⣀⣠⣤⣤⣤⡶⠾⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⣏⢶⠃⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠙⠀⠁⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⢀⣨⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣬⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣾⣿⣿⣿⠿⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⣴⣾⠿⠛⠉⠁⠀⠀⠤⠋⠁⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣶⣄⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠁⠀⠁⢀⡴⠮⢀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣾⣿⡟⠁⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣴⣿⣿⠟⠀⠀⠀⠀⠐⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠁⠀⠀⠀⠐⠇⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠸⠀⠀⠀⠡⡀⠀⠤⠄⠀⠀⠀⠀⠐⠀⠘⣿⣿⣿⣻⣿⣿⣿⣿⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⢦⡀⠀⡀⠠⡀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡁⠂⠀⠀⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣄⠀⠀⢄⠀⢀⣤⣄⠀⠀⠀⠀⠀⠁⠙⠥⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠂⢢⡈⢂⠓⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⠏⠀⠀⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣴⣿⣿⠋⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠈⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣽⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠈⢁⠀⠈⠀⠀⠿⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣦⣄⣀⣤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⣿⣿⣷⣤⣤⣶⣶⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⢿⣿⣿⣿⣿⡟⠀⢀⠉⠀⠠⠀⣀⣤⣦⣤⣀⣀⣉⣤⣶⡄⠀⠐⠄⠀⠀⠀⠻⣿⡿⠘⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⠻⢿⣿⡄⠀⢠⣶⣶⣾⣿⣿⣿⣿⡋⣿⣿⡟⢻⣿⣿⡀⠀⢸⣦⣀⠀⣀⡀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠓⠲⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠉⠉⠁⣠⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠁⠙⠛⠛⠻⠿⠿⠿⠿⠿⠇⠀⠀⠀⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠒⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠈⠓
⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠠⠤⠴⠄⠂⠒⠲⠀⠠⠶⠤⠄⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠣⡰⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⠀⠀⠀⠀⠀⠀⠓⠈⣷⠉⠅⠀⠀⠀⠀⠀⠀⠀⠐⣤⠊⠘⠃⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡐⢣⣴⠶⠛⣛⠛⠒⢤⡠⠄⣈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢫⣽⣿⣿⣿⣷⡀⠀⠐⠂⠀⢰⡛⣿⡘⢣⠉⡆⠀⡀⠱⠀⠀⠀⠀⠂⠄⠀⡀⠀⠀⠀⠀⠀⠀⢈⠳⢄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢩⣼⡟⣃⣾⡿⠟⠅⠈⠀⠀⠀⠀⣼⡟⣻⣷⣾⠟⠓⠪⠁⢾⡃⠀⠀⠀⠀⠀⠁⠪⢀⢄⠀⠀⢀⡴⢋⠟⡮⠳⡄
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣧⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡔⣿⠃⠒⢀⣄⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢡⠎⠀⠀⠘⢆
⠀⠀⠀⠀⡴⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⡀⣀⣠⢼⡟⠻⡰⠀⡈⠙⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⠄⠀⠀⠀⠈⢧
⠀⠀⠀⡼⠀⠀⠀⠉⠛⢿⠟⠅⠀⠀⠀⠀⣀⣄⠚⢋⢥⣌⣿⡘⣿⠀⠣⢔⣓⣤⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠀⠀⠀⠀⠀⠀⢣
⠀⠀⡸⠁⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣀⣜⣯⠐⣀⣠⢌⣣⣿⣿⣻⣷⡄⠀⠐⣸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠀⠀⠀⠀⠀⠀⢧
⠀⢠⠇⠘⠢⣄⣀⠀⠠⠀⠀⠀⠀⢰⣿⣅⡤⠦⣤⣹⣿⡿⣻⣫⣾⠫⣽⣿⡿⣛⣽⣿⢿⣿⡕⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⣀⣀⠘⡆
⠀⡼⠀⠒⠺⠢⡀⢠⠃⠀⠀⠀⢠⣾⡿⠉⠀⠀⠀⠈⠿⡽⣱⡟⢦⣿⣿⢏⣴⣿⡿⢍⣶⠿⠉⢢⢴⡖⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡇⢁⣀⠇⢛
⢀⡇⠀⠀⠀⠀⠀⡌⠀⠀⠀⢀⡟⣽⠀⠀⠀⠀⠀⠀⢸⣇⡟⢬⣿⡿⣱⣿⣿⠯⣹⠞⡗⡀⣪⣾⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠘⠀⠀⢸
⠈⠃⠀⠀⠀⠀⠀⢁⠀⠀⠀⣸⡇⣿⡀⠀⠀⠀⠀⠀⢸⠟⣿⣷⣿⢳⣿⡿⢍⣾⣫⠁⢄⢾⣽⢿⠅⣸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠰⠄⠄⠀⠀⠀⠠⠐⠀⠀⢀⣿⢿⣿⣧⠠⢄⠀⠀⠀⢸⣿⣿⣿⢇⣿⣿⣝⣲⣯⠅⢣⣟⣿⣃⢨⡴⠇⡼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐
⠈⡀⠀⠀⠀⠀⠀⡈⠀⢠⣾⠙⣸⡟⣿⠀⠀⠀⠀⠀⠸⡇⢿⡾⡿⣿⣟⢢⡟⡔⡈⣞⣾⠋⢃⡾⡄⡟⠭⣰⠏⠁⡗⠂⣴⡄⠀⠀⠀⠀⡀
⠘⡇⡠⠄⠀⠀⠀⢃⠀⠀⣿⣦⣿⠃⡻⡆⠀⢀⠀⠀⠀⣷⣤⡝⢾⣺⣷⣾⣱⠣⢹⢿⡟⢎⢸⣧⣽⣖⣹⠃⠀⡘⠀⣾⣿⠃⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢰
⠀⢼⠒⠤⢤⡤⠄⠘⡀⠀⠀⢸⣿⡀⢳⣷⠀⠀⠳⠀⠀⠸⠦⠿⠷⠽⠿⠿⠟⠛⠛⠉⠁⠀⠀⢤⡤⠋⣱⣄⠀⡧⢰⣿⠏⠀⠀⠀⠀⠀⠆⡀⠬⢍⣓⣶⡘
⠀⠸⣠⠊⠁⠀⠀⠀⠰⠀⠀⢠⢹⣿⣅⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣴⣿⣿⣗⣁⣾⡄⠀⠀⠀⠀⠀⡌⠈⠙⢖⢲⡬⢧⠇
⠀⠀⢫⠉⠁⠀⠀⠀⠀⠑⠄⠘⠻⣿⣿⣸⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠍⠁⠿⣿⣿⡏⠀⠁⠰⠄⠀⢀⢎⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠪⠀⠀⠙⠛⠿⢿⣿⣦⣤⣴⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣀⢄⣪⠟⠏⠐⠀⠐⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠻⠿⠟⠋⠉⣙⠛⠻⠿⠿⠟⠛⠛⠉⠉⠉⠉⡉⠉⠈⠁⣀⡀⢀⠀⠒⡀⠄⠀⠊⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⠤⠤⡙⠄⠀⠀⠀⠀⠀⠀⡀⣻⠀⠀⡦⢠⢠⠄⠀⠤⢠⠄⢦⠠⠄⡄⢀⡀⠀⠈⠐⠘⡛⠀⠀⡄⢐⠄⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠘⢦⠠⠤⠼⡀⢀⡄⠀⠁⠐⡄⡀⢀⠀⠀⠃⠘⢒⠃⠚⠚⠘⠒⠃⠈⠀⠂⠘⠒⠃⠂⠁⢀⡠⡀⠊⠀⠊⠑⢄⠈⠉⢆⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡍⠁⠀⠀⠀⠀⠀⠐⠐⠤⣀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠑⣲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⠀⠒⠀⠀⠴⠔⠲⠤⠄⠉⠓⠀⣩⡤⠤⠄⡀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⢀⠇⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠃⠠⠀⡄⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡠⠾⢀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚⣋⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠒⠛⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⡶⠶⡶⢶⣶⣶⠶⣶

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠴⠶⠒⠒⠲⠒⠶⠦⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠉⢦⡜⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⡙⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡏⠀⠀⠀⠀⠀⠈⠇⠘⣖⢉⡆⠀⠀⠀⠀⠀⠀⠀⠀⢦⡔⠁⠃⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⡴⢃⣤⣶⠖⣓⠚⠢⢤⡀⠀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⠟⣹⣿⣿⣿⣿⣆⢀⠠⠒⠈⠐⣴⠻⣿⠸⢂⠙⡖⠀⠀⠱⡈⠀⠒⠩⢒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⣡⣾⠟⣡⣿⡿⠟⠉⠀⠁⠀⠀⠀⣾⡾⢿⣧⣽⠾⠷⠼⠇⢼⡇⠀⠀⠀⠀⠈⠀⠉⢁⢄⠀⠀⠀⢀⡔⢫⢧⡝⢦
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⣥⣿⡟⡫⠊⠀⠀⠀⠀⠀⠀⠈⢿⠰⣿⠋⠿⢀⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⡠⡀⠊⠔⣡⠎⠀⠀⠳⡄
⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⡿⡃⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠗⢿⢠⠀⠘⠻⢢⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⡀⠐⣇⠀⠀⠀⠀⠙⣄
⠀⠀⠀⢠⠋⠀⠈⠛⠻⢿⣿⠏⠊⠀⠀⠀⠀⢀⡠⡶⣛⣈⡛⣏⢹⣇⠈⣌⡠⢧⣀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠕⡀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⡶⠟⠩⠁⣁⠻⢝⣿⣶⢿⣦⠀⠈⠩⢙⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⡆
⠀⠀⡎⠀⣄⡀⠀⠀⢀⠢⠀⠀⠀⠀⣴⢟⠛⣉⣈⡙⣷⣿⣿⣯⢟⣿⣿⢷⣶⣶⡾⣻⣿⣿⣿⣣⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢀⠀⠀⢹⡀
⠀⣸⠁⠤⢤⣍⠉⠁⢆⠂⠀⠀⠀⣠⣿⡿⠋⠉⠉⠙⠻⣿⣽⢣⣿⣓⣮⣿⡿⣫⣾⣿⡟⣭⣷⠿⡄⢀⣀⡥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠀⠸⠀⠋⠙⡆⣧
⠀⡏⠀⠀⠀⠀⠑⠈⡌⠀⠀⠀⣰⢿⡟⠀⠀⠀⠀⠀⠀⣿⢣⣿⡳⣾⣿⢏⣾⣿⣟⢣⣼⢯⠝⠩⣈⢷⣷⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠇⠻⠉⠀⢸
⠰⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢿⣾⣧⣽⣿⢯⣾⣿⡻⢬⡿⡗⡂⣫⢾⣽⠟⠉⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘
⢰⡃⠀⠀⠀⠀⠀⣸⠀⠀⠀⣼⣿⣼⣷⡀⠀⠀⠀⠀⠀⣸⣼⣿⣿⣏⣾⣿⠯⣹⡟⣟⡅⡰⣯⡿⠉⣏⡽⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀⣀⣀⢀
⠸⡅⠀⠀⠀⠀⠀⢸⠀⠀⢠⡿⡇⣿⣿⡇⠈⠁⠀⠀⠀⣿⠇⣿⣿⣼⣿⣿⣓⣾⢻⠱⣸⣻⡟⠖⢸⡏⢤⠾⣝⣻⠆⢠⣤⠀⡀⠀⠀⠀⠀⠀⠰
⠸⡇⠀⠀⠀⠀⠀⠸⡀⠀⣿⡅⢸⡿⢸⣧⠀⠀⠀⠀⠀⢸⣎⠙⢎⣯⣻⣗⣾⢏⡇⢢⣷⣿⡠⢀⣏⢱⣞⠬⣾⠃⢀⡉⢁⣼⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠
⠀⣧⣚⠁⠀⠀⠀⠀⢃⠀⠙⢻⣿⡇⢸⣿⡀⠀⠣⣀⠀⠀⣿⠻⣦⡷⢾⣿⣿⣾⣼⣼⡿⠏⠑⠚⠛⠚⢷⠿⡃⠀⡸⡄⣸⣿⡏⠀⠀⠀⠀⢰⠁⠀⠠⠤⣀⣀⣸
⠀⢹⠈⣉⠝⠛⠉⠀⠞⡄⠀⠀⣿⣷⡤⢻⣧⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠁⣴⣿⣶⡱⢁⣿⠛⠀⠀⠀⠀⢀⠣⢠⣤⣑⡒⠶⠷⡞
⠀⠈⣶⢥⠤⠀⠀⠀⠈⢔⡀⠀⣧⣿⣿⡎⣿⣆⢀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣤⣤⣴⡶⠿⢿⣿⣿⣴⠞⡿⠃⠀⠀⠀⠀⢊⠃⠀⠈⠒⠽⠋⢹⠁
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⣂⠀⠈⠻⣿⣷⣿⣿⣆⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠣⠀⡚⣻⣿⠟⡌⠀⣀⠛⠀⠀⠤⠃⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⢲⢄⠀⠀⠈⠙⢻⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣥⡤⠶⠾⠽⠋⠍⠐⢀⡀⠁⡀⠠⡪⠀⠀⠀⠈⠀⠀⢠⠎
⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠙⠛⠉⠀⠀⡤⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠂⠆⠈⣡⡄⠹⠈⣀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢀⠖⠛⢲⡙⢄⢄⠀⠀⠀⠀⠀⠄⡇⢠⢀⡇⡸⢰⢀⠀⡗⢰⠋⣹⠀⡇⡴⠀⣆⠀⠀⠀⡉⡭⠁⣀⡘⠄⠤⣀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠱⣔⠒⠚⢢⡀⠜⠀⠀⠀⠣⠠⡀⠄⠀⠀⠁⠈⣸⠁⠉⠉⠈⠉⠁⠈⠀⠉⠈⠉⠁⠁⠀⢀⡰⠮⠊⠀⠈⠀⠙⢦⡀⢈⣣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡸⠀⠀⠀⠀⠀⠀⠀⠉⠐⠰⢂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡐⠎⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⢀⣀⠉⠉⠉⠉⠉⠏⠉⠉⠀⠀⠤⠖⠒⠒⠤⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠀⠀⣸⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⣊⡁⠀⠀⣀⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⣒⠓⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⣉⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠤⠤⠤⠤⠦⠄⠲⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⠀⠀⠀⠄⢶⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠉⠀⠠⠂⠁⡀⠀⠀⠀⠀⢀⠀⠀⠴⠀⠀⠈⢻⣦⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠆⢀⡀⠂⠀⢂⠐⠃⠀⠀⠀⠀⣀⣼⡿⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠔⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠡⠄⠁⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢦⣝⣦⡇⡀⣶⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⡀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠙⠁⠀⠐⠲⣾⣽⡁⣷⣿⢂⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡀⠀⠂⠀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠐⣄⠰⠤⢾⣿⡽⡏⡟⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⢀⢀⠀⠙⠳⢿⣭⡳⢷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠖⡠⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⡠⠈⠁⠀⠀⠀⠈⠉⢻⠀⠀⠀⠀⠀⠀⡤⠀⡠⠂⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⠀⣀⡀⠀⠑⠀⠂⡘⡄⠀⠀⠀⠀⡔⠀⠊⠀⠐⢀⡄⢀⣤⠤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠙⠶⢨⣴⡶⣳⣷⠞⣵⡾⢋⣾⣴⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠓⠀⠀⣰⠖⠀⠀⢿⡄⣿⣄⢾⣛⢶⣶⣯⣍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂⠐⠂⠉⠗⠀⠀⠀⠀⠉⠘⠿⣷⡉⠳⠈⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠿⠿⠿⠛⠛⠿⠿⠿⢷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠀⠀⠀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⢺⠷⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣴⡄⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢀⠀⠠⠦⠀⢀⠀⠀⠀⠘⠀⠐⠂⢀⠀⠀⠈⣴⡆⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠁⠂⠀⠀⠀⡀⠀⠀⢉⣴⣶⠿⠿⠿⢷⣶⣄⠘⢆⠀⠘⠇⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠰⠆⠀⠀⠀⢀⡀⠁⠀⢠⣿⠏⠀⣀⣀⣀⠀⠙⢿⣆⠀⣶⡄⠀⠰⠀⣴⣶⡄⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢠⣄⠀⠀⣴⣤⠀⢀⣡⣴⣾⣿⠀⢰⣿⣿⣿⡇⠀⢸⣿⡀⢉⡀⠀⠀⠀⠈⠉⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠉⠀⠀⠉⠁⣰⡿⠋⠁⠀⠀⠀⠀⢿⣿⣿⠇⠀⣼⡿⠀⠻⠃⠀⠀⠀⡀⢀⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣴⣤⠀⠀⢰⣿⠃⠀⣴⣿⣿⣦⠀⠘⣿⠏⠀⣰⣿⠃⠐⠂⠀⠀⠀⢈⡁⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠁⠀⠀⠘⣿⡄⠀⠻⣿⣿⠟⠀⢠⠏⠀⣰⣿⠃⠀⠀⠀⣀⠈⠁⠈⢡⣄⢀⡀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⣄⠀⠀⠀⠀⢹⣿⣦⣀⡀⢀⣀⣴⡏⠀⢰⣿⣷⣶⣶⣦⡌⠛⠀⠀⠀⠈⠁⣬⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠙⠁⠀⠶⠀⠀⠀⠀⠙⠛⠻⠟⠛⣿⡇⠀⢾⣿⣿⠀⠀⣿⡇⠀⠀⡀⠀⣤⠀⠉⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠈⠀⢰⡦⠀⠆⠠⣶⠄⠀⠂⢻⣧⠀⠈⠛⠋⠀⣠⣿⠃⠀⠈⠉⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠃⠀⢰⣶⠀⠀⠀⠀⠤⠀⠰⠀⠻⣷⣦⣤⣤⣾⡿⠁⠀⠀⣀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠈⠉⠉⠉⠐⠂⠀⠀⠀⠉⠁⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⢀⠸⠆⠒⠀⠉⠀⡙⠡⠄⠀⢰⣶⡆⠀⠀⢀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠀⠶⠀⠀⠁⠀⠀⠀⠀⢉⣠⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢷⣶⣶⣶⣶⣶⣶⡾⠿⠟⠛⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⣄⠀⠀⠀⠀⣿⠀⠀⠀⢿⡄⠀⠀⠀⠀⣠⡟⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢻⡆⠀⠀⢸⡇⠀⠀⠀⠸⣇⠀⠀⠀⢠⡿⢁⣼⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡈⠻⠀⢀⡿⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⠁⠛⠁⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⣦⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⣀⣀⣠⣤⣤⣤⣤⣶⠶⠶⠶⠾⠿⠛⠛⣻⣿⠿⠋
⠀⠀⠀⣤⣤⣤⣤⣤⣴⣶⠶⠶⠾⠿⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠙⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⠙⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠐⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣙⣛⣀⠀⠈⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠿⠶⠶⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⠉⢉⡄⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣈⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣸⠇⠀⠀⠀⠀⠀⠀⢀⡴⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠙⠀⠀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢠⡟⠀⠀⠀⠀⠀⣠⡾⢋⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⣾⠃⠀⠀⠀⣠⡾⢋⣴⢏⣶⠀⢀⡀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣠⡾⠋⣴⠟⠁⠘⠃⣿⠈⢿⡀⠀⠀⠈⠙⢷⣤⡀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣠⡾⠋⠀⠘⠁⠀⠀⠀⢰⣿⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣤⣶⣶⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠁⣠⣾⣿⠿⠋⠁⣀⣤⣶⣾⠿⠿⢿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣼⣿⠟⠁⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢰⣿⡇⠀⠀⡴⠋⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⢿⣇⠀⠀⠁⠀⢀⠴⠛⠉⠁⠀⠀⠉⠙⠻⢷⣦⡀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⡀⠙⠳⠄⠀⠀⠁⠀⠤⠶⣶⣶⣤⣄⡀⠀⠀⠙⢿⣦⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⠙⠻⣿⣷⣄⠀⠈⢿⣧⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⡌⠻⣿⣆⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠠⢒⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⡄⠘⣿⡄⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⢸⠀⠈⠃⠀⠘⠇⠀⢸⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢠⡟⠀⢀⠀⠀⠀⠀⠀⣸⡿⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠀⠀⢀⡞⠀⠀⢠⠀⣸⠃⠀⠀⢀⣴⡟⠁⢀⡞⠀⠀⠀⠀⢀⣿⠃⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠶⠿⠄⠀⠀⠈⠀⢁⠀⢀⣴⡿⠋⠀⣰⡟⠁⠀⠀⠀⢀⣾⠏⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠬⠅⣴⣿⠟⠁⣠⣾⠏⠀⠀⠀⠀⢀⣾⠋⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⢠⣾⡟⠁⠀⠀⠀⠀⣴⠟⠁⢠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣰⣿⠏⠀⠀⠀⢀⡴⠚⠁⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣿⡏⠀⠀⠀⠀⠁⠀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⢀⣠⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⣿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠉⢻⡇⠀⢠⣶⣿⠃⠀⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣄⣤⠃⠀⠁⠀⣿⣿⣿⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢀⣾⠀⢸⣿⣿⠀⠈⠂⠀⣴⠀⢿⣿⣿⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢀⣴⣿⡿⠀⣸⣿⣿⡄⠀⣠⣾⡿⠀⠈⠻⢿⣄⠀⣼⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠌⢀⠀⠀⠀⠀⠀⠀⠀⡜⢃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⠈⢀
⠀⠐⠀⠀⠀⠀⢀⠌⠀⠀⠈⠫⡉⠁⠈⣉⠟⠁⠀⠀⠀⠐⠀⠀⠀⠂⠈⠀⠀⠒⢶⣾⣿⡶⠊⠁⠀⠀⠀⢈⠀⢈⠀⠐
⠀⠀⠀⡁⠠⠀⢐⠀⠀⠀⠀⢀⢎⠠⢄⣡⠀⠀⠀⠀⠀⢀⠒⠀⠂⡀⠀⠀⠀⠀⢠⠟⠛⢧⠀⠀⠀⠀⠀⠉⢀⠈⠁
⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣤⣴⠪⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⡵⣿⣿⠁
⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠈⠉⠈⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢠⡇⣀⣀⡀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⣀⣀⡁⣀⢀⠀⠀⠀⡀⠀⠈⢧⠀⣀
⠀⠀⠒⠒⠒⡵⠐⠒⠒⠂⠀⠉⠳⢶⡿⣭⡉⠉⠀⠀⠀⠐⠀⡀⠆⠀⠀⠀⠀⠀⠉⡿⣷⣎⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣠⠟⠁⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠙⠂⠀⠀⠀⠀⠖⠉⠀⠷
⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢼⣀⣀⣀⠀⠀⠀⣀⣀⡞⢧⣀⣀⠀⠀⠀⣔⠛⠁⠀⠉⣃
⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠛⠋⡇⠀⠀⠁⠀⠀⠈⢻⡤⢼⡏⠀⠀⠀⠀⠸⠇⣀⣄⡘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈⠈⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⣀⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣹⣿⣿⣮⡻⣷⡀⠀⠀⠀⠀⢀⣤⣾⣿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⣄⡀⠀⠀⠀⢀⣠⣤⣦⣤⡀
⠀⠀⠀⠀⠀⣀⣤⠀⠀⣠⣿⣿⣿⣿⣿⣧⢻⡇⠀⠀⢀⣴⣿⡻⣿⣿⣿⣿⣿⣿⣦⠄⠀⠀⠀⢰⡿⠉⠉⢙⢿⣿⡄⠀⣰⣿⡿⡉⠉⠉⣿
⠀⠀⠀⠀⢰⣯⣀⣠⣾⣿⣿⣿⣿⣿⡿⠿⠙⠀⠀⠀⢸⣿⣿⠟⠙⠛⠂⠀⠈⠻⢿⣷⣶⡆⠀⠸⣧⣄⣠⡾⢈⣿⣿⣶⣿⣿⡘⣧⣄⣠⣿
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠋⠀⠿⠄⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠀⠀⠀⠀⠈⠉⠉⢰⣿⣿⣿⣿⣿⣿⣿⡌⠉⠉⠁
⠀⠀⢀⡀⣰⣿⣿⣿⣿⣿⡇⠀⣴⣶⣦⡀⠀⠀⠠⣶⣿⣿⣤⠀⠀⠀⣠⣀⢠⣶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠸⣯⣹⣿⣿⣿⣉⣽⠃
⠀⢰⠯⣽⣿⣿⣿⣿⠛⣿⣷⣄⣳⣼⣿⠇⠀⠀⠀⠀⠀⠿⣿⣷⣶⣶⣿⣿⣾⡻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠁
⠀⠘⢿⣿⠟⠛⣯⣛⣣⡿⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⡏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⢀⠀⡀⠀⢀⣤⣶⣶⣦⣄⠀⠀⡀⢀⠀⠀⠀⠀⢤⣄⠀⣦⣰⣀⣦⣠⣆⣴⠀⣠⡄⠀⠀⠀⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⢰⡇⣄⡼⣠⣿⣿⣿⣿⣿⣿⣧⡸⣄⠄⡷⠀⠠⣤⣼⣿⣿⣿⣿⢻⣿⡏⣿⣿⣿⣿⣥⣤⠄⠀⣿⣷⣄⣀⡀⣀⣀⣀⣀⡀⣀⣀⣠⣿⡇
⠀⠀⠛⠶⠾⠿⣿⣿⣿⣿⣿⣿⡿⠿⠶⠟⠁⠀⢀⣈⣿⣿⣿⣶⣿⣿⣿⣿⣿⣾⣿⣿⣯⣁⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣈⣽⣿⣿⣿⠿⣿⣿⣿⠿⣿⣿⣿⣯⣁⡀⠀⢀⣴⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⡋⠁⠀⠿⠟⠀⢿⡛⢿⣿⣿⣿⡿⢻⡏⠈⠻⠇
⠀⠀⠀⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⣹⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠀⠀⠀⠀⠀⠈⢻⡿⢿⣿⣿⣿⡏
⠀⠐⠤⣁⣤⣼⡿⢭⢹⣿⠩⠿⣿⣤⣀⡠⠆⠀⠀⠘⠛⣻⣿⣿⡿⢿⣿⡿⣿⣿⣿⣟⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠅
⠀⠀⠀⠀⠀⠀⠣⠠⠜⠙⠤⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⠉⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⣠⡎⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⢲⡄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣶⢲⣶⣤
⠀⠀⣿⡇⣨⡇⠀⢠⠀⠀⠰⡀⠀⢸⡄⢸⣿⠀⠀⠀⢀⣠⠶⠛⠉⠀⣇⠈⠉⠛⠦⣄⠀⠀⠀⠀⠀⡾⢿⠋⠁⠀⠉⣷⡦
⠀⠀⠈⢷⠍⠀⠀⣤⣄⣀⣤⣅⠀⠀⢹⡎⠁⠀⠀⢠⠋⢀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠑⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⡹⠇
⠀⠀⠀⠸⣧⠀⢀⣽⣿⣿⣿⣇⠀⢀⣾⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠀⠀⠀⠀⠂⠈⠀⠀⠀⢛⣭⣷⣤⣀⡀
⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⠷⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣶⠶⠶⡦⣤⡀
⠀⠀⠀⠀⡠⠔⣁⠝⢿⣿⠟⠫⡑⠢⡀⠀⠀⠀⠀⢶⣶⣶⠇⠀⠀⢠⣿⡀⠀⠀⢳⣶⣶⠗⠀⠀⢀⠞⢛⡿⣿⣿⣿⠃⠀⠀⠓⠈⠇
⠀⠀⠀⠈⠀⠈⢀⠔⠁⠀⠁⠢⠈⠢⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⡄⠀⠀⠈⠀⠀⠀⠀⠈⠀⠎⠀⠞⠁⠙⢷⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⡶⠟
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⣾⣇⣈⠀⢉⣨⣿⣿⣦⠀⠀⠀⠀⠀⠘⢷⣤⠀⠀⠀⠀⠀⣤⡿⠁⠀⠀⠀⠊⠉⠉⠉⠉⠉⠛⠿⣿⣶⣄⠀⢀⣤⡶
⠀⠀⠀⠀⢠⣿⣿⣇⣠⠀⣤⣨⣿⣿⣿⣷⡀⠀⠀⠀⠀⠘⣿⣿⣶⣶⣶⣿⣿⠁⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠃⣀⠋
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣿⡛⠙⣿⣿⣿⡇⠀⠀⠀⣠⠲⣿⣿⣿⣿⣿⣿⣿⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⠁⠈⡇⠀⠀⣾⣿⠴⣿⣏⣻⣿⣟⣹⡿⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⢀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠛⠋⠀⠈⣿⣿⣿⣿⡟⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⢰⡁⠀⠁⠺⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠈⠳⣦⣤⣤⣴⣶⣾⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠃

⠀⠀⠀⢹⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠶⠶⣤⠀⠀⣀⣤⣤⣽⣦⡀⠀⠀⠐⠛⠻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⡿⠟⠷⠀⠀⠀⠀⡀
⡶⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠶⣴⠞⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣠
⠃⠀⠀⠀⠀⠀⠀⠀⠠⣼⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿
⣷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣄⢀⣀⠀⠀⠀⠀⠀⣰⣶⡀⠀⠀⠀⠀⢰⣄⣠⡤⠀⠀⢠⣾⣁⣀⡀⠀⣤⠶⠞⠛⠀⠀⠀⠀⠀⠐⠻⠿⣿⣿⣿⣿
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢴⣿⣿⣿⣇⠀⠀⠉⠉⠉⠉⠻⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠋
⠀⠀⠀⠀⠀⢸⡷⣄⡀⠀⠀⢀⣀⠀⠀⠀⠹⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⠈⠻⠞⠛⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⢿⣦⣄
⠀⠀⢀⣤⠶⠋⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡿⢻⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣸⡷⣾⡟⠁⠀⠀⠀⢀⣷⣧
⠀⠀⠉⠛⠶⢤⡄⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⢀⠀⣼⠿⠋⠀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣷⢀⣴⠛⠛⠛⠛⠃⠀⠀⢠⣄⠀⠀⠀⠀⡾⠿⠛⢿⣿⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⠀⠀⢹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⢀
⣷⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⣶⣶⣶⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣼
⣀⣿⣏⠀⠀⠀⠀⠀⠀⠀⢤⣼⣷⡶⠀⠀⠀⠀⠀⠀⢀⣴⣷⣤⠀⠀⠀⠀⠀⠀⢸⡏⠻⢦⣤⣴⠶⢿⡇⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠛⠿⣿
⠟⠻⠿⠄⠀⠀⠀⠀⠀⠀⠀⠿⠛⠇⠀⠀⠀⠀⠀⠀⠀⠽⠟⠷⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢠⣿⣿⠿⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠈⠁⠀⠀⠈⠙⠃⠀⠀⠀⢲⣿⡷⠀⠀⠀⠉
⠀⠀⠀⠀⢰⣤⣆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢶⡄⠀⠀⣠⣤⣤⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣶⣦⡄⠀⠀⢰⣷⡆⠀⠀⠀⠀⠈⢷⣀⣼⠋⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠽⢿⠃⠀⠀⠀⠠⣤⣴⢿⣆⡀
⢀⡾⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⡿⠋
⠉⠁⠀⠙⠛⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡾⣷⣀⠀⠀⠀⠀⠀⢤⣾⣿⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠃⠀⠀⠀⠀⠘⢿
⠂⠀⠀⠠⡟⠁⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⢠⣄⣀⡄⠀⠀⠀⠀⢸⣷⣀⣼⡟⠋⠀⠀⠀⠀⠀⡿⠟⠿
⠶⠛⠳⢶⣧⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⡄⠀⠀⠀⠛⠉⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⡀⡀⠀⢀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⣤⣶⣿⡇⠀⠀⠀⠀⠀⠼⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢶⠆⠀⠀⠀⠀⢸⣿⠉⠁⠻⣷⣤⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠿⠆⠀⠀⠀⢸⣿⠿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⢠⣤⣤⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣶⣤⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣄⣴⣿⣿⠦⠀⠀⠀⠀⠀⠘⠻⠋⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏
⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⢀⣠
⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⠷⣤⣀⣀⣤⣶⡇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⢻
⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⣿⠀⠈⠉⠉⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠿⠿⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⣀⣤⣿⣦⣤⠀⠀⠀⠐⠛⡟⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⣻⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠃
⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠖⠒⠒⠒⠒⠒⠒⠲⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠑⣄⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠉⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠘⠀⢹⡇⡸⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡊⠈⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠤⠀⣐⠒⠒⠒⠂⣒⡀⠤⠈⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⠜⠀⡀⠀⠀⠀⠤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣥⣀⠁⠠⢀⠄⣀⠂⠀⠀⠀⠀⠀⢠⠑⢄
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠞⢁⣠⠞⠁⠠⡢⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣫⣭⣿⠿⠿⠛⠻⣂⣀⠀⠀⠀⠐⠀⡀⠀⢀⡴⣩⠞⡶⠱⣄
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠈⢁⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢅⣤⠾⢿⠃⠒⠈⠙⠳⠄⠉⣻⡄⠀⠀⠀⠀⠢⡈⠐⣡⠊⠀⠀⠈⢦
⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢡⣾⡥⠟⣼⠃⠀⠀⢼⣷⣏⠿⡿⠁⠀⠀⠀⠀⠀⠈⠢⠱⠄⠀⠀⠀⠀⢷⡀
⠀⠀⠀⡜⠀⠀⠀⠀⠐⢀⠄⠂⠀⠀⢠⠇⠀⢀⡐⠆⡀⢲⠀⠀⢸⠇⠈⣿⣿⠿⣿⣦⠀⠀⣾⠿⢯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠡⡀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⡸⠀⠀⠀⠀⠀⠀⢊⠂⠀⠀⠀⠾⠤⠄⠘⠇⠈⠧⠈⠀⠀⣿⣆⠀⡸⣿⣶⣮⣧⣄⣀⠘⠛⠒⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢣
⠀⢠⠁⠈⠢⠦⠤⠄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⣿⣦⣙⣉⣙⠝⠻⣿⣷⣶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢠⡀⣆⢤⡈⡆
⠀⡞⠀⠈⠙⠲⡀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠛⠻⢿⣿⣯⣻⣿⣎⢺⣿⣯⡉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡇⢠⠤⠇⢸
⢠⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠳⡄⠀⠀⢬⡛⠻⠿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠈⠀⠀⠘
⢸⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠉⠀⠀⠀⠈⠣⡀⠁⢯⣿⣿⡇⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠆
⢸⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠑⠀⠐⢆⠀⠀⠀⠀⡀⠠⢲⣾⣿⠇⣼⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⣀⣀⡀⠄⠖⠋⠀⢀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡀⠁⢠⣿⣷⣧⣿⣋⢀⡀⠈⠂⠄⡀⠀⠀⠀⠀⠀⡄⠃
⠘⣇⠤⠀⠀⠀⠀⢃⣀⣴⠾⠛⢛⢛⠄⠀⠠⠟⠀⠈⠳⣄⢶⣤⣦⣤⠄⠀⠈⣶⠀⠁⢈⡈⠙⢿⠛⠛⠒⠦⠼⣿⡆⠀⢘⠀⠀⠀⠀⠀⢡⠀⢀⠀⠀⠀⢰
⠀⢯⠒⠲⡦⠴⠖⣾⢟⠁⣀⡰⠁⠀⠀⢀⠀⣢⣶⣶⣤⣤⣄⣀⣀⣠⣤⣼⡿⢿⡇⠱⣄⡀⠀⠀⢣⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠤⠭⣙⣒⣺
⠀⠘⣴⡋⡀⠀⢸⡟⠄⠈⠀⠀⠀⠀⠀⣡⡿⣿⢿⠟⠛⠋⠙⠛⡹⠋⠉⠀⠀⠀⠈⠳⢄⡉⢁⠀⠀⠣⡀⠀⠀⠮⢀⠀⢠⠀⠀⠀⢀⠀⠈⠙⢭⣲⡆⢣⠇
⠀⠀⢱⡀⠀⠀⢸⣿⣀⣠⣶⣶⣶⣾⣿⠁⠀⠻⢄⣀⠀⠪⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡕⠀⠀⠈⠆⠀⠙⢧⣬⡸⠀⠀⠀⣢⡁⠀⠀⠀⠀⠀⡜
⠀⠀⠀⢣⠀⠀⢸⠉⡉⢉⠫⣅⠉⢻⠏⠀⠀⠀⠀⠙⢦⡀⠀⠢⣙⣕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡈⠀⠈⠢⠀⠀⠀⠀⠀⡀⣱⣿⣿⣷⣦⣄⠀⡼⠁
⠀⠀⠀⠀⢣⡀⠈⠲⠷⠼⠣⠞⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⣀⠀⠀⠑⠔⠂⣢⣾⢫⣝⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⢠⠔⠢⡕⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠬⡷⣾⣿⣫⣝⢾⣥⣭⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠈⢦⡠⠄⠺⣀⣀⠇⠀⠁⠃⢆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣾⣿⣿⣿⣿⣷⣝⣫⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⡇⠀⠀⠀⠀⠀⠀⠙⠊⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠑⠈⠀⠙⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣈⡁⠒⠂⠈⠉⠩⠍⠐⠒⠈⢉⣀⣄⣀⡀⠀⠀⠀⠈⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⡈⠀⠀⠀⠀⢠⠃⢠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⣉⠇⢀⠀⠅⠀⠀⠀⣀⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢄⣐⠚⠤⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⠄⢒⡡⠤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠦⠤⠤⠤⠦⠴⠶⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⢀⡀⡀⠀⡀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠐⠓⠀⠈⠁⠃⠘⠀⠃⠐⠁⠁⠀⠀⠉
⢀⣀⣀⡀⠀⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡐⠀⡐⠀⠒⠀⠐⠒⠒⠒⠒⠛⠓⠐⠓⠒⠒⠒⠒⠂
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⣁⢀⡉⡁⢈⣀⡈⠀⠁⠀⠈⣉⣉⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⠄⣀⠀⠀⠀⠀⡠⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠴⣄⠈⡀⣀⡀⠁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠁⠀⠃⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡡⠈⢠⡀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⣼⠋⡀⢀⠸⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⢈⠁⠈⠁⠁⢀⢈⠀⢿⡉⢡⡌⠀⣿⣧⡐⠄⠠⠤⢤⠰⠆⣠⢠⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⢀⡀⠀⠀⠈⠾⠛⠀⠠⠌⠋⢁⠁⠰⠀⠄⠤⠤⠄⠀⠤⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⡀⡁⠀⠁⠈⠉⠊⠉⠁⠀⠰⣄⡀⣀⣶⠆⠈⡀⠢⡀⠀⠄⠐⠂⠤⠴⠶⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠨⠁⠀⠀⣀⣔⡆⠀⢀⡄⢀⡁⢟⡛⢫⢣⡀⠀⡀⠀⠀⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠐⠀⡍⢿⠰⠈⢩⠁⡿⠛⣚⠛⢿⠀⣤⢸⠀⡈⡑⠄⠀⠘⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠠⠀⢰⠓⠰⠂⠀⠀⠂⠱⣦⣈⣳⣤⣶⡓⡞⠀⠠⠤⠀⠂⠀⠐⣪⡿⠀⠀⢀⡀⠀⢀⠀⢠⠀⢠⡄⠀⢰
⠀⠀⠀⠘⡃⠈⢩⠀⣁⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢛⠹⠀⠏⣽⡙⣇⠀⠐⠐⠒⠀⠀⣰⠟⠀⠀⠀⠀⠁⠀⢍⡀⠂⣀⠀⠀⠀⢸
⠀⠀⠀⡈⢉⢀⠈⠀⡀⠀⠀⠁⠀⠀⠂⠀⠀⢤⠄⠰⠄⢀⣿⣶⠿⢾⣿⣷⢿⡤⠄⠀⠄⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠆⠸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⡤⠤⠠⠀⠤⠀⢰⡾⣧⢀⣀⣶⠛⢡⡟⢦⠤⠐⠁⠀⠀⣀⠀⠀⠀⠀⠒⠀⢀⠀⣎⠀⠀⠀⠀⢸
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠈⠃⠒⠒⠀⠒⠀⠀⢷⣼⡏⢛⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⡦⠀⠀⣸⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠙⠆⠀⠀⠁⢒⠐⠂⠀⠁⠀⠀⠀⢉⠙⢉⠁⠙⢿⣯⠀⠀⡀⠀⠀⠀⠀⠀⠘⠈⠆⠘⠀⠚⠿⠟⠀⠀⠀⠀⢸
⠀⠆⠀⢀⣀⠈⠠⠤⠤⠠⠄⠀⠀⠀⠠⣈⣨⠉⡉⠀⠀⠀⠀⠀⡀⠘⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠠⠤⠤⠀⠀⠠⠤⠇⢸
⠀⡀⠀⢠⠄⠤⠄⠼⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⡘⠒⡀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡖⡀⠐⠐⠀⠀⢰
⠀⡇⠀⢠⠀⠀⠀⠒⠀⠐⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣄⠀⠀⠈⡀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣇⠀⠀⠀⠀⢸
⠀⠂⠀⠈⠀⢀⠀⠐⠈⠀⠤⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠹⡆⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠢⠄⣼⠀⢀⡀⠀⢸
⠀⡁⠀⣀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢰⠇⠀⠀⠀⠀⠀⠀⢀⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸
⠀⡁⠀⠀⠠⠚⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀⠑⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⠆⠀⠘⠶⠂⠂⢸
⠀⠁⠀⢰⣤⠦⠰⠤⠦⠀⣤⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠠⠠⠤⠀⠀⠀⢸
⠀⠂⠀⠀⠤⠤⠤⠀⠶⠀⠓⠀⠀⠀⠀⠀⣀⠠⠂⠀⢠⠃⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠁⠡⠤⠤⠤⠀⢸
⠀⠀⠀⠀⠂⠄⠐⠀⠲⠀⠂⠂⠀⠀⠀⠉⠀⠠⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡄⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠶⡖⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⣀⠀⠀⠀⠀⡀⢠⠀⠁⠀⠁⢀⡷⢸
⠀⠀⠀⠀⠀⠀⠘⣇⢔⠁⡖⠀⠃⠁⢍⠐⠂⠀⣼⣀⠀⠀⠀⡆⠦⠠⠄⡠⠥⠤⠴⠔⢀⡀⠀⠀⠉⠛⢦⡈⠠⠄⠀⠁⠠⠼⠷⠞⠋⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⢀⡿⠀⣀⠘⠛⠛⠻⣄⠀⠀⠀⠀⡂⠀⠄⠒⠂⢀⠔⠈⠠⢁⡽⠖⠠⠀⠇⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠀⠈⠁⠀⠀⠈⠷⠀⠀⣀⠉⠀⠀⠀⣴⠦⡄⠀⠀⠀⠤⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢸
⠀⠀⠁⠒⠛⠒⠒⠒⠒⠒⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠼⠾⠻⠤⠤⠾⠿⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠋

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⣈⣉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⠿⣿⣟⠛⢻⣿⡟⠿⢶⣶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠿⠛⠉⣿⡂⠀⠀⠰⠀⣄⠑⢡⡄⡀⠀⠀⠀⢸⡟⠉⠛⠿⣳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠛⠀⠀⠀⠀⠸⣧⠀⠀⠀⠁⠙⠀⠈⠁⠀⠀⠀⠀⣿⠃⠀⠀⠐⠍⠍⠙⠿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠫⡐⠨⠄⠓⡆⠀⠀⠀⢻⡄⢀⣀⣀⣀⣦⣇⣀⣀⣀⡀⣸⡏⠀⠀⠀⠔⡉⢈⢀⡄⠽⠙⠯⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡛⠁⠀⠀⠀⠮⠄⠊⠂⢀⣦⡤⠶⠚⠛⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠙⠓⠶⢤⣀⣃⠂⠐⡠⠀⠀⠀⠀⠈⣫⣿⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠗⠙⢿⣦⡀⠀⠀⠀⢠⣦⠶⠛⠁⠀⠀⢃⠀⠀⠀⠦⠀⠀⢠⡠⠀⠺⠀⠀⠀⠀⠀⠈⠙⠻⢥⡀⠀⠀⠀⢀⣼⠟⠁⠻⡷⣄
⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠙⠻⣄⣀⣴⠋⠀⠀⡀⠀⠀⠈⠀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠰⠀⠙⢲⣄⣴⠟⠁⠀⠀⠀⣈⢞⢧⡀
⠀⠀⠀⢀⣾⡟⡉⠹⢂⠄⣀⠀⢀⡼⠋⠀⠀⠀⠘⠛⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡘⢳⡄⠀⠀⠀⡠⠃⢢⠫⣳⡀
⠀⠀⢀⣾⠏⠜⢀⠂⠘⣀⠁⣰⠋⠀⠀⠓⠀⠀⠀⡀⢠⡄⠀⠀⡄⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠁⠀⠹⢿⣌⠡⡂⠜⢀⡑⣽⣳⡀
⠀⠀⣾⡏⠀⢄⡍⠊⡀⠃⣼⠋⠀⠀⠐⠠⢠⡂⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠠⠄⠀⠀⣀⡀⠀⠀⠀⠀⠶⠀⠀⠘⢧⣆⢠⠊⠀⠈⠀⢡⢳⡀
⠀⣼⡿⠀⠀⠀⠀⠐⠂⣼⠁⠀⠠⠄⠀⢠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠠⠆⠀⠀⠀⣤⡀⠈⣿⠁⠀⠀⠀⠀⠀⢛⣧
⢠⡿⠷⠶⣦⣤⣀⡀⣰⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡥⠲⣄⠀⠈⠢⣄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠸⣇⣠⣤⠶⠶⠛⠛⣻
⢸⣿⠀⠀⠀⠀⠉⠉⣿⠀⠀⣄⠁⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠘⠃⠑⢤⡀⠀⠙⡆⠀⠀⠈⢆⠀⠾⠆⠀⠀⠀⠀⢠⠄⠀⡀⠀⢠⡤⠀⢻⡁⠀⠀⠀⠀⠀⣷
⣿⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠄⠀⠀⣸⠁⢠⡌⡇⠠⡀⠀⠈⡆⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⣤⠄⠠⡀⢠
⣿⠁⣀⣀⠇⠏⡆⢸⡇⠀⠀⠠⠀⠀⠀⠛⠀⠀⠀⠶⠀⠀⠀⣁⡌⡀⠀⢾⠀⠀⠀⣧⠀⠀⠀⠀⠇⠀⣄⡀⠀⠀⠀⠀⠀⠠⡴⠀⠀⣶⠀⠈⣿⠀⠁⠌⠡⡈⢆
⣿⣻⠀⠒⡆⠁⠒⢿⡇⠀⠀⠀⢠⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠣⠁⢀⠀⠈⣧⡀⠀⢹⠀⠀⠀⡘⠀⠀⠉⠀⠀⠀⢠⠀⠀⠀⠀⠄⠀⠀⠀⢀⡟⢄⠩⡍⣂⠃⡾
⣿⡏⠁⠀⠀⠀⠀⢸⡇⠀⠋⠀⠀⠁⠀⠈⠀⠀⠀⠀⠻⠃⠀⠀⠀⠶⠈⠓⠨⡞⠀⠋⠀⠀⠊⢀⣤⠀⠀⠠⣞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠑⠂⠉⠈⠘
⢹⣟⠀⠀⠀⠀⠀⢀⣿⠀⠀⡀⠀⠀⢀⡔⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⣾⣅⣀⠀⠀⠀⠀⣞
⠘⣿⣤⣦⡶⠶⠛⠋⠹⡇⠴⠀⠀⠀⣀⠉⠀⠀⠆⠀⠁⠀⠀⠀⠀⠙⠁⠂⠀⠐⠂⠀⠀⠈⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⢰⡏⠉⠙⠛⠿⠶⣶⣾
⠀⢻⣯⠀⠀⠀⣀⠀⠀⢹⡄⠀⢀⠁⠀⠀⢀⡀⠀⠤⢀⣀⠈⠄⠀⠀⠐⠂⠀⠀⠄⠀⡔⢦⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⡈⠀⠰⠀⢀⡟⠀⠀⠀⠀⠀⠀⣆⡟
⠀⠈⢿⣧⠠⠊⠰⠁⡔⡂⠻⣄⠈⠐⠀⠀⠙⠃⠀⠀⠈⠉⠀⠠⡆⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠒⠀⠈⠁⠀⠀⢠⣞⣀⠃⠌⢁⠀⠀⡜⡼⠁
⠀⠀⠈⢿⣦⣀⠇⢲⢀⠌⠵⠙⣦⠀⠀⠀⢀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡶⠀⠀⠀⣀⠀⠀⠐⠇⠀⣰⠟⣁⠤⡑⢆⡢⢑⡴⡽⠁
⠀⠀⠀⠈⢿⣧⠀⠀⠉⠀⠀⠀⠈⢳⣄⠀⠛⠁⠀⠀⠀⠀⢀⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢠⠀⠀⠀⠙⠁⠀⠀⣠⡞⠁⠀⠀⠀⠈⠐⠐⡞⡽⠁
⠀⠀⠀⠀⠈⢳⣵⡄⠀⠀⠀⠀⣠⠞⠉⠳⣤⡀⠀⠀⠈⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠲⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⢀⣤⠞⠉⠹⣦⡀⠀⠀⠀⢠⢎⡞⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⢀⣴⡿⠋⠀⠀⠀⣶⡿⠷⢤⣀⠀⠒⠀⠀⠀⠀⠉⠀⢀⡄⠀⠀⠀⠈⠋⠀⠀⠀⢀⣤⣶⠋⠁⡀⠀⠀⠈⠻⣷⣄⠴⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣫⡀⠀⠀⠀⠀⠀⢃⡦⣀⠈⠙⠳⢶⣤⣄⣀⣀⣀⣀⣁⣀⣀⣀⣠⣤⡶⠖⠛⠩⣯⡍⢶⡻⡌⢄⠀⠀⢀⡬⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣶⣄⠀⠀⠎⡼⢱⢏⠇⠀⠀⣸⠇⠀⠉⢩⠛⠛⠛⠋⢩⠉⠀⠘⣷⠀⠀⠀⠠⡸⣆⠳⡜⣦⡷⢖⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢿⣶⣾⣅⢏⣈⠀⠀⢠⡿⠀⠀⠀⠸⡉⠹⠸⠉⡹⠀⠀⠀⢻⡆⠀⠀⠀⠀⣈⡵⣟⡭⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢯⣶⣤⢀⣼⡇⠀⠀⠀⠀⠠⠑⠐⠡⠂⠀⠀⠀⢸⣿⡀⢤⣒⡭⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠿⢿⣶⣶⣶⣘⣯⣠⣶⣦⡿⠿⠷⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣤⠟⣹⣿⣿⣿⣿⡷⠶⢾⣿⣿⣶⣾⣛⡋⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡿⢿⣿⣿⣿⣿⣿⣿⣋⣷⣶⠈⣿⣿⣟⢛⣉⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣿⣯⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣾⡟⠉⠀⢹⣿⣏⠻⠦⠀⠀⣾⣿⢹⡿⠉⢿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢠⣾⣛⣥⡉⠙⠻⣿⣧⠀⠈⠻⡁⠘⠛⠁⠀⠀⠀⠀⠘⢇⣼⡷⠘⢿⣿⣿⡗⣾⣿⣿⡏⢿⣯⣿⡿⣿⣷⡿⣿⣾⡻⣦
⠀⠀⠀⠀⠀⡝⣿⠋⠉⠀⠀⠀⢹⣷⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢈⡿⢷⣤⣾⡿⣿⣥⣿⣿⣿⣧⢸⣿⣿⡇⠸⣿⣷⡘⣿⣷⢻⣷⣄
⠀⠀⠀⠀⠀⠳⣦⣀⣀⡀⣀⣀⣴⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡁⣠⣿⡿⠡⢿⣿⣿⣿⢹⣿⣸⣿⣿⡇⢸⣿⣿⠀⣿⣿⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠉⢿⡉⠻⢻⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠘⢁⣴⣿⣿⡿⢃⣴⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⡀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠹⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣠⡾⣻⣿⠟⡁⠰⣾⣿⣿⣿⢩⣿⢿⣿⡇⣾⣿⣿⣧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣮⣿⠿⠟⣿⡇⠀⠀⠀⠀⢀⣿⣿⢸⣿⣿⣿⣿⣿⣧⣾⣿⣾⣿⣿⢸⣿⣿⣿⣿⣿⢁⣿⣿⣿⢿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣿⣿⠃⠀⢠⡿⠀⠀⢸⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡇⢸⣿⣿⣿⣿⣟⢸⡿⣿⣧⣼⠇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣻⣷⣿⠶⠛⠀⢀⣤⠸⣿⡾⢟⣿⣿⣿⣿⣿⣿⣿⣏⢻⣿⢿⣿⣮⡿⢮⠻⣯⡛⢿⡉⠟⣧⢸⣿⢇⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣤⢩⣿⡇⠀⠀⠀⣸⣿⣾⡏⡐⣈⣿⣿⣿⣿⣿⣿⣿⠻⢦⡙⢢⡙⢷⡝⢦⠹⣿⣷⡈⣿⢠⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡛⢀⢻⣷⣶⣼⣿⣿⣿⣿⢹⣷⡟⣿⣿⣿⣿⣿⢿⣿⣦⣀⠈⢠⠙⡌⢿⣆⠀⢹⣿⣷⣿⢸⣿⣿⣦⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢙⡿⢾⣿⣿⣿⣿⣿⣿⣿⣼⣿⣇⡙⣿⣿⢻⣿⣦⣈⠑⠀⡙⢿⣿⡆⢸⣿⣧⠈⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡎⠹⢻⣿⣿⣿⣿⣿⣿⣿⠯⢳⣿⣿⣧⣉⠙⠛⠻⣦⡹⣮⣿⡇⠈⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠿⣷⣿⣿⣿⣿⣿⣿⣷⣿⣦⣾⣧⠻⣿⣟⠛⠒⠄⢈⠹⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⡶⠐⢿⡇⠀⢲⣄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡑⠀⠀⣄⠀⣱⣸⡿⣡⡾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣶⠞⠻⠃⠀⠀⢻⣄⠠⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣭⡟⣿⣿⣿⣷⣶⣌⣻⣿⠟⢰⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⡏⠀⠀⢻⡄⢀⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠿⣿⣿⡻⡏⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠈⣷
⠀⠀⠀⠀⢰⢀⣤⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦⣰⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡟
⠀⠀⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠈⣿⣿⣿⣿⣿⡟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣾
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣞⠁⠸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⢀⡾⠃
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣻⣿⣿⣿⣿⣿⣿⠈⠙⢶⡟⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣥⣴⡟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⣰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢿⡟⣿⡟
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠐⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣲⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⣤⣦⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⢿⡿⠋⠙⢿⠀⣀⣾⣤⢾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣸⣇⣠⣤⣼⣿⣷⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⠟⠋⠁⠀⠀⠉⠛⠿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠟⠋⠁⠀⠀⠉⠛⠿⣿⡿⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠶⠶⠶⠶⠶⠦⣄⣠⠴⠶⠶⠶⠶⠦⢤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⠁⠀⠀⠀⠀⣀⠀⠠⡄⠀⣿⣿⠀⢠⠄⠀⣀⠀⠀⠀⠀⠈⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣾⠋⢀⣀⠀⣰⣋⣀⣀⣀⠀⠀⣿⣿⠀⠀⣀⣀⣀⣙⣆⠀⣀⡀⠙⣷⣽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡿⠛⠁⠀⣸⠏⠻⣦⠀⠀⠀⠈⠙⠲⣿⣿⠖⠋⠁⠀⠀⠀⣴⠛⠻⣇⠀⠈⠻⢿⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣶⡾⠋⢠⡏⠀⠈⠀⠀⠀⣠⣤⠤⠴⣾⣷⠦⠤⣤⣄⠀⠀⠈⠁⠀⢹⡆⠙⢷⣶⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢫⡿⠋⠀⠀⠀⠳⠀⢀⡴⠶⠾⠋⠀⠀⢀⡿⢿⡀⠀⠀⠘⠷⠶⢦⡀⠀⠎⠀⠀⠀⠙⢿⡝⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⠃⣰⡟⡡⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠹⡀⢀⠏⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢌⢻⣆⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡸⠁⠀⣀⡀⠀⢰⡏⠀⠀⣀⡀⠀⠀⠀⠀⢁⡈⠀⠀⠀⠀⢀⣀⠀⠀⢹⡆⠀⢀⣀⠀⠈⢇⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣇⡴⠋⠉⠇⠀⢸⡇⠀⠀⢹⣙⢻⡶⠀⠀⠀⠀⠀⠀⢶⡟⣋⡍⠀⠀⢸⡇⠀⠸⠉⠙⢦⣸⡘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠏⠀⡀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠘⢁⠀⠀⠀⠀⠀⢀⡈⠃⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⠀⠹⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣠⡾⠁⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⢷⣄⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⡇⠀⠀⠀⠀⠀⠙⠲⣦⣀⠀⠀⣿⡀⠀⠀⠀⠀⢀⣿⠀⠀⣀⡴⠖⠋⠀⠀⠀⠀⠀⢸⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⡆⠀⢀⣀⠀⠀⠀⠹⣦⡄⠘⢯⣙⢉⡉⣉⡽⠃⢠⣴⠏⠀⠀⠀⣀⡀⠀⢰⠀⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢿⡄⢱⢀⡏⢿⠀⠀⠀⠀⢸⡀⠀⠀⠈⠛⠛⠁⠀⠀⢀⡇⠀⠀⠀⠀⡿⢹⡀⡎⢠⡿⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡈⢿⠁⠈⢳⡀⠀⠀⢸⣆⠀⢀⣠⡴⢦⣄⡀⠀⣰⡇⠀⠀⢀⡞⠁⠈⡿⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣠⠀⠀⠱⡀⠀⠀⠙⡛⠋⠁⠀⠀⠈⠙⢛⠋⠀⠀⢀⠎⠀⠀⣄⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣴⠟⢷⡀⠀⠓⠀⠀⠈⢳⣦⡀⠀⠀⢀⣴⡞⠁⠀⠀⠚⠀⢀⡾⠻⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠁⠀⠈⣷⠀⠘⠀⠀⠀⠀⠀⠻⢶⡶⠟⠀⠀⠀⠀⠀⠁⠀⣾⠁⠀⠈⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣄⢣⡀⠀⠀⠀⠀⠀⠀⢀⡜⣠⠀⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠤⠶⠋⠙⣆⠱⣄⠀⡄⢠⠀⣠⠞⣰⠋⠙⠶⢤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣌⠳⣧⣸⠞⣡⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣸⣇⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠈⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢧⠀⠀⡞⠀⣩⠿⢯⣄⣻⣀⡜⠀⢠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠂
⠀⠀⠀⠀⠀⣼⠛⣿⣈⣧⠀⠀⠈⠀⠀⠘⠀⠈⠐⠒⠀⠀⠀⠈⠁⠀⠀⠳⢼⣧⠚⠁⠀⠀⣈⡽⠋
⠀⠀⠀⠀⠀⠙⢿⣅⣽⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⣆⠴⠂⠀⠈⠉⠛⠋⠉⠁⠀⠀⢀⣠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠰⠿⢧⠀⠀⢀⠐⣆⡓⠎⠁⢀⡄⠀⠀⠀⠀⠀⠖⠀⠀⠀⠈⠇⠉⠁⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠉⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠰⠤⢤⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⡄
⠀⡀⠀⠠⠀⠀⠬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠆⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⠘⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⠉⠉⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀
⠀⠀⢠⠀⠀⠀⠁⠀⠀⢠⣶⡄⢿⡇⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠐⠕⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠃⠀⠠⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠖⠀⠀⠀⠀⠂⣠⣴⣾⠟⠋
⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠈⠁⢠⣾⣿⣿⠃
⠀⢀⠀⣀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠤⡄⣼⣿⣿⡏
⠀⠀⠈⠁⢠⡀⢓⡀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡆⠀⠀⠀⠀⢹⣿⣿⣷
⠈⠠⠀⢠⠩⢸⡋⠱⠀⠀⢠⣦⠀⠀⠀⠀⠠⠃⠁⠄⠀⠀⠀⠀⠀⢶⣿⡾⠀⠀⠀⠀⠀⠀⠀⠐⣰⣦⠀⠀⠀⠀⢉⣤⡀⠀⠀⠂⠀⢻⡿⣿⣷⡄
⠀⠁⠀⠀⠁⠰⠀⠒⠀⢀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⣿⣍⡿⣄
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠐⡃⠠⡀⢀⠄⠌⠤⠀⠀⠀⠀⠀⠀⠈⠟⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢿⣼⣳⡿⠀⠀⢄⡀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⢦⡄⠀⠀⠀⠀⠒⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡆⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⠀⠙⠿⠃
⠀⢀⠀⠈⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⡄⠀⠀⠹⠿⠋⠁⠀⢀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠈⠋⢀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⠃⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⣾⢻⣏⣹⡆⠀⢠⢟⠒⠀⠀⠀⢄⠀⠀⠘⠿⠋
⠀⠀⠘⡤⠀⠀⣾⣧⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠀⠀⢰⣶⣶⡖⠀⠈⣿⡿⠛⠀⠀⠻⣿⣀⣿⠃⠀⠀⠘⠆⠀⠀⠀⠁
⠀⠀⠙⣠⠄⠀⢷⣞⣿⠇⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠒⣀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠂⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⢀⠀⠉⠡
⠀⠀⠀⠀⠀⠀⡄⢠⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠈⢁⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⡀
⢀⠀⠀⠀⠀⠸⠟⠆⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢴⠀⠀⢰⠆⠀⠠⠆⠀⠀⠀⠀⠃⠀⠀⣀⠴⠒⣶⡟⠓⠲⢤⡀⠀⠀⠀⠀⠀⠠⠆⠀⠶⠆⠀⠀⠀⠀⠀⠠
⢀⣠⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡤⠀⠀⠀⢀⠞⠁⠀⢠⠇⠹⡄⠀⠀⠘⢦⠀⢐⠀⠀⠀⠚⠀⢀⠈⠠⢈⡁⠀⠄⢀
⠴⢠⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣖⡒⠒⡿⠛⠛⢻⡟⠛⣻⠟⡆⠀⠀⢀⡋⠀⢠⠃⢀⡂⠀⠁⡀⠈⢰⠉⠡
⡂⢲⡐⠐⠃⠀⠀⢻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠈⠙⢲⣧⣀⠀⢀⣿⡎⠁⠀⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⡄⣣⣋⡀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠴⠶⠒⠒⠲⠒⠶⠦⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠉⠑⣤⠻⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣍⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠘⠀⢻⠅⠅⠀⠀⠀⠀⠀⠀⠀⢐⣴⠃⠈⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⢃⠀⠀⠠⠀⠀⠤⠀⠀⣀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡤⠃⢀⠀⠀⠀⠠⠀⠀⠁⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⣉⠢⡀
⠀⠀⠀⠀⠀⠀⡠⠋⠀⠠⠉⣀⡔⠃⡀⠀⠑⠀⠀⠀⢸⡁⢠⠀⡄⠁⡸⡸⢹⢀⡇⡘⢸⠁⡇⠀⠀⠀⠀⠀⠐⢄⠀⠀⡴⢫⢞⡕⠙⢆
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠁⡠⠊⠊⠀⠀⠀⣀⡀⢀⡅⣝⠀⠉⠀⠀⠀⠈⠀⠀⠀⠈⢀⡁⠀⡀⣀⡀⠀⠀⠐⠤⢀⠀⢡⠊⠀⠀⠀⠳⡀
⠀⠀⠀⢀⠞⠀⠀⠀⢀⣀⠤⠶⠖⠀⠀⠀⢠⣧⣀⠀⠀⠀⠈⠉⢦⠔⠨⢶⠶⠦⢄⢠⠞⠉⠀⠀⠀⢀⣨⣱⠖⠐⠀⠑⠲⠬⣂⠀⠀⠀⠀⠳⡄
⠀⠀⢀⡎⠀⠀⠀⠀⣇⠀⠐⠂⠀⠀⣀⣠⣤⡭⡍⠓⣄⣀⠀⠀⢨⡿⡋⠒⣂⡠⠴⣾⠄⠀⢀⣀⡴⠛⣯⢥⣤⣀⣀⣀⣈⣡⠀⢹⠀⠀⠀⠀⠹⡄
⠀⠀⡜⢀⠀⠀⠀⠀⠸⣆⠀⠚⠺⢿⡉⠁⠀⡁⠉⢀⣽⡿⣩⣃⣾⠁⠀⠀⠀⠀⠀⢸⣦⣩⡝⣿⣇⠀⠁⡀⠀⠉⣩⡿⠟⠁⢠⠏⠀⠀⠀⠀⠀⢱
⠀⢰⠁⣀⣙⡃⠐⢀⠂⠈⠳⣄⠀⠀⠙⠲⢄⠒⠀⣹⣿⢻⠻⡟⠃⠀⠀⠀⠀⠀⠀⠀⠙⢻⡿⣿⣿⡃⠀⠇⣀⡼⠋⠀⠀⡰⠋⢈⠄⠀⡆⠖⠒⡄⠇
⠀⡎⠀⠀⠈⠈⠂⠀⠂⠀⠀⠈⠳⣄⡀⠀⡌⠿⣽⢿⠋⠘⠀⠙⠶⣤⣤⡀⠀⢀⣤⣤⡴⠋⡳⢫⢻⣏⠟⡟⢿⠀⠀⣠⠞⠁⠀⠈⡄⠀⢡⢰⠒⠁⢸
⢀⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢷⣴⠇⠀⢾⠀⠀⠠⢈⠒⠐⠛⣻⣿⣤⣪⠭⡉⠓⢌⡁⠹⠀⠈⠄⡌⡜⣧⡴⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⢼⠐⢀⠔⠁⢠⢦⠞⠁⠀⠘⣇⠀⠈⢢⠀⠻⣦⡀⠀⢸⡄⠁⠻⠁⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⢀
⢸⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⢀⠀⡄⣸⡔⠁⠀⠀⣟⣿⠀⡀⠀⠀⢹⠀⢀⣾⣆⢱⡘⢿⣦⣼⣿⣄⣸⠀⠀⠀⠀⠀⠀⢰
⠈⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⡎⢀⣿⠫⠋⠀⠀⢠⡿⣿⣇⠉⠂⠀⠻⣦⣾⣿⠏⢸⡇⠈⢿⠻⣼⣩⣿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸
⠀⢷⠯⣀⣀⣀⡀⢀⠄⠀⠀⠀⠀⠀⠀⣷⡿⠁⠀⠀⠀⢀⠈⠷⡘⣾⡇⠀⠀⠀⠸⡿⠋⠀⠚⢱⠀⠀⢀⠙⢿⡿⠀⠀⠀⠀⠀⢀⠣⠀⣐⠂⠤⣤⢼
⠀⠸⣀⠔⠋⠁⠀⠈⢄⠀⠀⠀⠀⠀⢀⣿⣠⠂⠀⠀⠀⠸⡌⢤⠀⡿⠁⢆⠀⠀⠄⣷⠘⠁⢀⡾⠀⠀⠸⣔⢮⣧⠀⠀⠀⠀⠀⢀⠂⠳⡄⢭⣙⢏⠇
⠀⠀⢻⠒⠂⠀⠀⠀⠈⣂⠀⠀⠀⠀⢺⣿⡇⠀⠀⠀⠀⢷⡈⠀⣼⠁⡀⠈⢦⡀⣠⣾⣆⠠⠋⢁⡶⢀⠆⣈⡻⠁⠀⠀⠀⠀⡠⠁⠀⠀⠈⠉⠁⡼
⠀⠀⠈⢇⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⠀⠻⣿⣄⣘⣄⣬⠶⣧⠤⣾⠀⠀⠀⢨⡈⠀⢀⣿⢤⡤⠽⠴⠡⢦⡟⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⢩⣭⡥⢤⣤⡤⠴⠶⢻⣇⠀⠀⢸⡇⢀⣾⣿⣶⣶⡶⠖⣰⣿⣷⡀⠀⠀⠠⡈⠁⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⢠⠤⢄⠀⡄⣠⣾⣻⣟⠲⠤⠷⠀⠉⠛⢻⣆⠀⠀⣷⠋⠈⠉⠉⠁⠐⣫⣽⣿⣿⣷⡄⡰⠊⢠⠒⡄⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠙⢄⠤⠤⢇⡀⣠⠇⠈⠉⢻⢿⡳⢭⣂⠀⠀⡀⢀⠋⠉⣧⠀⢻⡀⠀⠀⠐⣐⣫⠾⢋⡞⣙⠏⠀⠐⠉⢢⡉⠉⠢⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢠⠍⠀⠀⠀⠀⠀⢠⣷⣦⣍⡛⢋⠁⢈⠀⠘⣯⠇⠈⡃⠀⠀⣛⠉⡁⠔⠈⠈⠀⠀⠀⠀⠀⠀⠉⣲⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡀⠀⠀⠀⢠⣿⣿⣿⣿⡻⠿⣶⣿⣶⣤⣭⣤⠆⠑⠉⠁⠀⣥⠄⠤⠄⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣠⣿⣿⣿⣿⣿⢹⣿⢰⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣥⣟⡃⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠑⢒⣈⠥⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠶⠖⠒⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⢀⠀⠀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⢀⠠⠐⠛⢿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠄⠂⠁⠀⠀⠀⠀⠈⠙⠿⠿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢉⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣴⣄⠀⣴⣀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⡛⣦⣬⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣾⣿⣿⠏⢑⢫⣅⣿⢿⣿⡸⡟⣿⣿⣿⣦⡀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⣼⣿⣿⡟⠁⠀⢾⡚⠛⣿⣿⡿⠗⠋⠋⠙⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣠⠼⠛⠛⠉⠤⢤⠀⡀⠀⠀⣽⣿⣷⠀⠀⢀
⠀⠀⠀⣀⡀⠀⠀⠀⣀⢀⠀⠠⠅⠀⠀⣿⣿⣿⣿⡄⠀⢸⠁⠀⢄⠀⠐⠘⠃⠀⠘⠷⢾⣿⣿⣿⡄⠀⠀⠇⠀⢀⢀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢿⣿⣿⣿⠛⠢⡼⢦⠄⠀⠁⢐⠀⡀⠀⠀⠰⠀⠀⢹⣿⠇⠀⢀⠧⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⠃⠀⠀⠀⡡⠖⠒⠠⠚⠀⠀⠀⣠⣤⣤⣼⡟⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠀⠀⠹⣿⣿⣷⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⡿⠁⠀⢀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣍⣀⣴⠑⣾⣶⡆⠀⠀⠀⠀⣀⣽⡿⠟⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠻⠷⣶⣿⣯⣤⣤⣴⣶⡿⠟⠋⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠄⠀⠰⠖⠆⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠰⠦⠙⠃⣛⣚⣓⠛⠈⠋⠰⠆⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠺⡃⡉⠀⠀⠀⠀⠀⠨⠸⢸⠝⠀⠀⠀⠀⠀⠉⢃⠗⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡺⠃⠁⠘⡄⣘⠠⠀⠀⡄⠠⠀⠀⡚⠀⠄⢠⠀⠀⠊⠇⡠⠃⠈⠙⢗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠊⠁⠀⠀⠀⠈⠁⠲⡀⠤⠒⠂⠀⠉⠉⠀⠒⠂⠤⢀⠀⠀⠓⠀⠀⠠⠊⠑⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡑⣁⠀⠀⠀⠀⢄⡀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡠⠈⠀⠀⡠⣬⣋⣄
⠀⠀⠀⠀⠀⠀⠀⠰⡟⣰⠃⠌⠙⠀⣤⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠆⠐⣈⢥⠈⢸⢆
⠀⠀⠀⠀⠀⠀⢀⡛⠀⠀⠉⠀⠄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠀⠈⠉⠀⠀⢛⡀
⠀⠀⠀⠀⠀⠀⣌⠃⠀⠀⠀⠤⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠤⠀⠀⠀⠈⣡
⠀⠀⠀⠀⠀⠀⢭⢀⡀⠀⠀⠂⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢀⣬
⠀⠀⠀⠀⠀⠐⢾⠈⠑⢄⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠧⠼⠄⡷
⠀⠀⠀⠀⠀⠀⡛⠀⠀⠈⠁⠆⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣛
⠀⠀⠀⠀⠀⠀⢋⡀⠀⠀⠀⠒⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠒⠀⠀⠀⢠⡙
⠀⠀⠀⠀⠀⠀⠈⣡⠀⠀⣔⣄⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⡐⣀⠀⢀⠀⣬⢃
⠀⠀⠀⠀⠀⠀⠀⠱⢧⣔⡩⠄⠀⠐⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢉⠈⠸⡤⡗⠚⣾⠏⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣱⡃⠀⠀⠀⠀⠊⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠐⠂⠀⠀⠀⢚⣾⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢄⡠⠂⠀⢀⠀⠈⠘⠇⠒⠠⠤⢀⣀⣀⠀⠤⠄⠒⠈⠁⣈⠀⠀⠀⠐⢀⡠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⡤⡀⠚⠭⣒⠀⠀⠀⠃⠐⠀⡄⢐⠀⠂⠘⠀⠀⠀⢸⠲⠤⢂⢠⡮⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢴⣤⣀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠀⠀⠀⠈⣊⢤⡢⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠰⠆⣠⡄⣤⢤⡬⣤⠠⣄⠰⠆⠛⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⡿⠀⠀⢷⡀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢰⡇⠀⠀⠘⣧⠀⢠⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣍⣽⠀⣼⠀⢰⣦⠀⠸⣇⠹⣦⣤⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⠁⠀⡟⠀⣸⠛⣧⠀⠹⡆⠀⢀⣴⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⢰⡟⠉⠉⣷⠀⢸⡇⠀⡿⠀⠘⣧⠀⢻⡄⠸⣧⣀⣀⡟⢠⡞⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠶⠀⠻⣤⣼⠇⠘⠷⠶⠶⠏⠀⣼⠁⢰⡇⠀⠀⠘⣧⠀⢿⡀⠈⠉⠉⠀⠈⠛⠞⠃⠀⠛
⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⠤⠤⠶⠶⠶⠖⠒⠚⠛⠀⣾⠀⠀⠀⠀⢸⣆⠈⠙⠛⠒⠲⠶⠶⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠶⣿⣋⠉⠉⠁⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⠤⠴⠾⠏⠀⠀⠀⠀⠀⠻⠛⠛⠛⠛⠛⠛⠛⠛⢒⣿⠗⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠈⠉⠛⠶⢦⣄⡀⠉⠛⠿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⢀⣤⠾⠋⠁⣀
⠀⠀⠀⠀⠀⣤⡀⠀⢈⡙⠛⠶⣤⣄⣈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⠀⣀⣤⠞⢋⣥⣤⣄⠈⠉
⠀⠀⠀⠀⠀⠉⠀⢼⡋⣿⠀⠀⣀⣈⠉⠛⠲⣦⡈⢹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⢀⣴⡶⠛⠉⣀⡀⠘⢷⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⣾⠉⠉⣷⠀⢠⡟⠁⣾⠃⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢷⡀⢻⡄⢠⣟⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⢠⡟⠀⣼⠃⠀⢀⣠⠾⠋⠉⠛⢷⣄⡀⠀⠈⣷⡀⢻⡄⠙⠳⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⡆⢰⡟⠀⣰⣏⣠⠶⠋⢁⣠⡾⠛⠷⣄⠈⠛⢦⣄⠘⣧⠈⣷⠀⣴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⠋⢠⡟⠀⠰⠿⢋⣡⡴⠞⠋⠁⣠⣤⣤⡈⠻⢦⣀⠈⠻⢿⡆⠸⣇⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡟⠀⣠⡴⠞⠋⠁⣠⣤⣄⠰⣯⣀⣠⡟⠀⣀⠉⠳⢦⣀⠀⠀⢹⡄⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠷⠛⠁⠀⢠⡄⠀⠻⣦⡼⠇⠈⠉⠉⠀⣿⣛⣷⠀⠀⠉⠳⣤⡀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠶⠆⠀⠈⠙⠿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠁⠀⠳⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠔⠉⡐⠀⠠⠲⠀⢹⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⣤⣤⡀⢀⠦⣄⠩⠉⡛⢿⣆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⡟⠀⠀⠀⠉⠀⡀⠀⢀⣼⣿⡀⠀⢀
⠀⠀⠐⠂⠀⠀⠐⠐⠀⢨⠀⠀⢿⣿⣿⡀⡆⠈⠀⠀⠀⠀⠀⠀⠀⠠⢿⠹⣿⡇⠀⢨⠀⠐⠐⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡣⣌⣷⠻⡷⢦⠀⠀⠰⠊⠀⠠⠀⠀⠀⢬⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠉⠁⠄⠀⠀⠀⠀⠄⠀⢘⠂⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⡈⠦⠄⢀⡠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠲⠶⠶⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠔⠖⠒

⠀⠀⠀⠀⡀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⡈⠀⠀⠁⡐⠀⠀⠀⠒⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰⣾⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣾⣿⣷⠟⠃
⠀⠀⠀⢀⠀⠄⢈⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠋⠙⢂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠻⡆

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⢠⣤⣼⣿⣤⣄⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⠀⠀⠀⠠⢴⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠁⠑⠤⣄
⠀⠀⠀⢹⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠈⠁⠀⠙⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⠙⠛

⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⠀⠀⠀⠀⠜⠂
⠀⠀⠙⢦⠀⠀⢉⠉⠁⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⠀⠣⠀⠀⠀⠀⠋⠀⠀⠀⠐⠀⠀⠀⠐
⠀⠀⠀⢠⠴⠂⠸⡀⠀⠀⠀⠀⣀⠤⠢⢸⡀⠀⠀⠀⠀⡼⠛⠻⢧⠀⠀⠀⠀⠀⡤⠄⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⢤⡶⠋⠈⠓⣢⠄⠀⠀⠄⠐⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⣷⣤⡤⠀⠀⠠⣄⠘⠁⠐⠒⠦
⠀⠀⠀⣹⢀⣀⢸⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠈⠃⣀⠀⢆
⠀⠀⠀⠋⠁⠀⠙⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠋⠁⠀⠙

⠀⠀⠀⠀⠐⠱⡀⠀⠀⠀⠀⠀⣀⢰⣧⢀⠀⠀⠀⠀⠀⣀⡼⢣⣀⡀⠀⠀⠀⢀⣠⠎⢠⡀⠀⠀⠀⠀⠀⢀⣼⣄
⠀⠀⠉⢢⠀⠀⢠⠎⠀⠀⠀⠙⢻⣃⢑⡻⠋⠀⠀⠀⠙⢦⠀⠀⣼⠋⠀⠀⠀⠑⡄⠀⠀⡰⠁⠀⠀⠀⠙⣿⣿⣿⡿⠃
⠀⠀⠀⠼⠖⠒⠾⠀⠀⠀⠀⠀⠼⠛⠘⠿⠀⠀⠀⠀⠀⠿⠚⠓⠾⠀⠀⠀⠀⠀⠄⠒⠂⠠⠀⠀⠀⠀⠀⠟⠋⠛⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠛⠉⠉⠈⠉⠉⠙⠿⠿⠛⠛⠿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠦⣤⣀⠉⠻⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠈⠑⢷⣄⠹⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⣿⡟⠀⠏⠀⠀⠠⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢿⣿⣿⣿⣿⣿⣷⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⡦⠀⠀⡄⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⠙⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢸⠃⢆⠀⢇⠀⠐⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⢄⠙⠻⣯⣽⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠋⠀⠠⠃⢀⣧⠈⠂⠘⡄⠀⠄⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠃⠀⠢⠀⠉⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠟⠋⢀⣠⠆⠔⠡⠚⠃⠀⠀⠐⠘⡄⠈⢄⢧⣀⣤⣤⣄⠀⠀⠀⠠⠾⠟⠳⠖⠀⠡⡀⠀⠀⠰⡄⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠁⠀⠞⠉⠀⠀⠔⠀⠀⠆⠀⠀⠀⠀⠻⣦⠀⢊⢯⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢂⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⢁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⠀⠀⠀⡟⢸⣿
⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠢⣀⠀⠈⢄⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢦⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠈⡀⠈⡜⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⠜⠀⠀⠀⠀⢠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠀⠀⠀⢇⣶⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀⠲⢤⣄⣀⠀⠈⠉⠛⡀⠠⠤⢤⣤⣼⣾⣿⣧⡀⠀⠀⠀⠀⠒⠀⠀⢁⠔⠁⠠⠴⣖⣥⣴⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣤⣶⣶⣿⣿⣮⣦⡉⢾⡌⢷⠀⠀⠀⢣⠀⣅⠀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣚⣾⢿⡄⠀⣷⣷⣾⣾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⠓⢾⣇⠀⢻⣿⣿⣧⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣽⢷⣸⣿⣿⣇⡙⡄⠙⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠙⠉⠘⣿⣿⠟⢀⣼⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠀⠀⢹⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣷⠄⣑⢸⡼⢏⠩⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⡿⢸⢸⡘⠀⡇⣿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠙⠙⠀⠃⠙

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣭⣽⡟⠻⠿⠿⠛⠿⠻⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠋⣾⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠀⠻⢏⣀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠻⣉⠉⠃⠈⠻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⣀⠀⠀⠀⠁⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⢸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠠⠺⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠿⠋⡀⠀⠀⠀⡀⠀⣤⣤⣄⣀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⢀⢰⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠁⠀⠈⠁⠀⢀⠂⠄⠈⠀⠀⢀⣈⠙⠃⠀⠀⠘⠋⣉⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈⠛⠁⠉⠙⠀⠀⠀⠀⠚⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣶⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣯⣀⣤⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣴⣷⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡀⢀⣴⣾⣿⣿⣯⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠚⠋⠩


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⢸⠃⣿⡳⡇⡙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠈⠁⠉⠀⠉

⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣒⣿⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢤⡀⠈⢄⠠⠀⠈⠄⠢⠀⢀⡤⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢀⠨⡉⡰⠐⠠⠀⠀⠄⢂⢆⢉⠅⡀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠢⡀⢀⠈⠘⠀⠯⠙⠁⠑⡄⢠⠊⠈⠋⠽⠀⠃⠁⡀⢀⠔⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢭⠀⢪⠆⠁⠄⡄⠸⡀⠀⠀⢀⠃⢠⠠⠈⠰⡕⢀⡬⠃⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢀⡀⢋⠳⠘⠄⠠⣆⠙⡀⢀⠏⣰⠄⠀⠃⠞⡙⢈⡄⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⠀⠐⠢⠢⠄⠀⢀⠢⢄⡑⠈⠃⠑⢊⠘⠁⢚⡡⠔⡀⠀⠠⠔⠰⢂⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⠁⢀⢀⠄⡠⣀⣁⠒⠀⠀⢀⣺⣗⡀⠀⠁⠒⣈⣁⢄⠠⡀⡀⠄⡁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⡹⠃⢁⢜⠙⠀⠁⠖⠀⣈⠁⠁⠀⠀⠉⠈⣀⠀⠲⠘⠀⠋⠃⡈⠐⢯⡁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠁⢰⠂⡈⡆⡈⠀⠂⠄⠤⣄⣉⡁⠀⠀⢉⣉⣠⠤⠠⠰⠀⢁⢰⢁⠐⡆⠈⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣦⣁⠑⠴⡁⠀⠐⡄⠈⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⣈⠦⠋⣈⣴⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠃⠟⣣⡈⢠⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡄⢁⣜⠻⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣜⠳⢈⡁⠀⢘⡅⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⢸⠇⢸⡂⠀⢈⡁⠜⣣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣄⠁⠒⠂⠔⡣⠀⠰⠻⣄⠀⠚⠀⠀⠓⠀⣠⠟⠆⠀⢝⠢⠐⠂⠈⣠⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⢀⠀⠹⠃⠄⠀⠀⠠⢐⠏⠀⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡗⠓⠀⡇⠈⠀⠐⠈⠀⠀⠀⠀⠁⠂⠀⠁⢸⠀⠚⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣄⠰⢄⠐⢉⠀⣠⠀⠀⠀⠀⣄⠠⡉⠂⡠⠄⢠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠈⠟⢦⡔⠂⠀⠀⠀⠀⠰⢦⡴⠻⠃⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⢀⠒⠉⠑⠄⠂⠐⠠⠊⠉⠐⡀⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⢁⡘⣀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠐⠂⢀⣈⣉⣉⣉⣉⣉⣁⣀⡀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣈⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⣠⣴⣾⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣠⣶⣿⣿⣿⡟⠁⠀⠀⠀⠀⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡛⠛⠛⠛⠉⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠾⣿⣿⣿⣿⠿⠁⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣤⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⢀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠀⣴⠋⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠡
⠀⠀⠀⠆⠀⠀⠀⠰⠆⠀⠀⢠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⢀⡀⠴⠀⠀⠀⠈⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆
⠀⢀⠂⢰⣆⢨⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢬⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠅⡀⠀⠀⠀⠀⠀⠄⠀⢠⠁⢠
⠀⡌⢀⣿⣿⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠄⠀⠀⠠⡉⠑⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⢰⣴⣶⣤⡀⠀⡾⣠⢠⣶⠈⡀
⢀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⠈⠀⠀⠉⠁⠰⣿⠀⠀⣿⡆⢀
⠀⠀⣿⣿⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠰⠤⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⡰⠀⠀⣤⣀⡤⠄⠀⠀⠀⠀⠀⢻⣷
⡇⢸⣿⣿⡅⠀⠟⠛⠛⠷⣦⡀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢤⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⣿
⡇⢸⣿⠏⠀⠀⣀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠁⠀⠂⡀⠉⠐⠂⠀⠀⠑⡀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣿⣿
⡇⠘⠁⠀⠀⠀⠈⠛⠒⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠈⠒⠤⡀⢠⠴⠀⠀⢠⡀⠴⠀⢈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠴⠶⠸⠦⠤⣀⡀⣀⠀⠀⠘⢀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⢠
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⡄
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⢀⣀⡤⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⠀⠀⠀⠀⢿⣿⣿⡏⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢈⡉⠙⠛⠁⠀⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⢀⠁
⠀⠀⠀⠀⠀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢠⡆⠉⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡟⢀⠂
⠀⠀⠀⠀⠂⠈⢿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢯⢨⣁⡀⠀⠀⠀⠀⠀⢳⡀⠀⡆⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠠
⠀⠀⠀⠀⠀⠀⡀⠹⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠒⠀⠛⠉⠈⠠⢄⠀⠀⠀⠀⠀⢣⠀⡄⠀⠀⠀⠀⠀⢀⣤⣄⣐⣚⡡⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣿⣿⣿⣿⣦⡀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠈⠀⢸⠀⠀⠈⠈⡧⢀⠀⢿⣿⣿⣿⠟⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠻⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠡⠀⢠⠀⠌⠀⠀⠈⡀⠀⠀⠀⠀⢹⠀⠘⡿⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠙⠛⠿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠂⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠉⠙⠛⠻⠿⠷⢦⣤⣤⣀⣀⣠⣤⡤⠶⠿⠆⠀⣀⠀⠀⠒⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣿⢸⡇⢸⣿⡷⢰⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡄⢿⠘⠗⢻⠞⠻⢾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠕⠀⠀⠀⠐⣊⡒⠀⠀⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⠲⠀⠀⠱⠊⠀⠰⠂⠀⠨⢀
⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⢀⠔⢁⠠⠀⠀⠀⠢⡀⠀⠀⡠⠀⠀⠀⠀⢄⠑⠄⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠀⢄⠔⠀⠀⢠⠈⡠⠒⠈⠈⢈⠐⠪⠀⢡⠀⠀⠑⠄⠄⠀⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⡀⢠⢊⣤⣄⠀⢀⠀⢀⣤⣌⢢⠀⡀⠀⠢⢷
⠀⠀⠀⠀⠀⠀⡀⢆⡀⠀⠀⠁⡐⠁⠀⠀⠀⠉⠉⡆⠠⠀⠎⠉⠁⠀⠀⠀⠑⡀⠁⠀⠀⣀⠆⡀
⠀⠀⠀⠀⠀⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠡⠐⢄⠀⠀⠀⠀⠀⢀⠔⠠⠁⠀⠀⠀⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢈⠁⠠⠀⠁⠈⠃⠓⢄⠐⠃⠋⠀⠁⠠⠀⢉⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠘⠸⢀⠁⠀⠀⠢⡀⢀⠜⢄⠀⡠⠂⠀⠀⢁⠸⠈⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠀⠀⠀⢆⠠⠀⠀⠠⠃⠀⢀⠀⠀⠣⠀⠀⠠⢀⡆⠀⠀⠘⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠇⠀⠀⠀⠀⠀⠀⠇⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠑⠄⣄⠀⠀⠀⢀⡌⠔⠁⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠉⢈⠈⠁⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡊⣔


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠄⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠑⠄⠀⠀⠀⠀⠀⠀⠀⠁⠡⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠰⠁⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⢆⠀⠀⠀⠀⠆⠀⠐⡀⠁⠀⠆⠀⢸⠀⠀⠀⢀⠴⠃⣠⠃⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠣⢀⠁⠀⠀⠀⠈⠠⡄⠀⡆⠀⠀⢀⠈⠀⠆⠈⠁⢀⠈⡀⠄⢊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⢄⠀⠠⠀⠀⡄⠠⠀⠀⢦⣁⠀⠀⠀⠀⠐⠋⠁⣀⠔⠁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠸⡂⠀⠀⠀⠤⣈⠀⠀⠀⠈⠑⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⢀⠔⠁⢀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠠⡀⠑⠠⠈⠢⡀⢐⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠁⠤⠔⠓⠈⠉⠉⠀⠀⢈⣡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠉⡍⠀⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⢀⠀⠀⠀⠐⡄⠀⢀⡴⠚⠩⢍⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢀⠀⠒⠒⠂⠉⠀⠈⠁⠀⠐⢻⣿⡖⠤⠆⠠⠧⠰⡿⢉⡁⠀⠀⠀⢁⠈⠀⠤⢤⡀⢀⣈⠀⠘⠁
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠀⣀⠈⠂⢀⣠⡤⠤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢘⠁⠀⠀⢀⡁⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢀⠀⠀⠀⠀⢀⡠⠤⠤⣇⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠉⠉⠉⠉⣉⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠝⠉⠀⠀⣠⠼⣄⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⡀⠀⠙⠒⠒⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡨⠔⠒⠉⠉⠁⢀⡸⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⢀⠠⣀⠈⠢⣄⠱⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⢁⠠⠒⠀⣠⠖⠋⠀⢀⠌⠑⢢⠠⢀⠀⠀⡤⠀⠐⡁⠀⠀⣄⢀⠀⡀⡄⢿⡙⠶⠬⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡎⠀⠀⡠⠂⡋⠀⠀⢸⠀⠀⡂⠀⢸⢄⠀⠈⠻⡄⠄⣦⠀⠰⡄⠠⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠊⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⡞⠈⡄⠀⠀⠀⠀⠌⠢⡀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⠴⠃⠀⣀⠀⡰⠃⢸⠀⠈⢀⠃⢠⠁⠀⠀⠀⠀⠘⠤⣀⠀⠈⠒⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⡆⠘⠀⠘⠀⠀⠐⢄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢡⠀⢣⠀⠀⠀⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣼⣧⣀⠀⠀⠀⠀⠀⠀⣀⣣⡀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣀
⠀⠀⠈⠉⢻⡟⠉⠁⠀⠀⠀⠀⠀⢑⡠⠫⡀⠀⠀⠀⠀⠑⠆⣀⣀⠚⠁⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠁⢱⢿⡏⠁
⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠉⠃⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠐⠋⠀⠑⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠄⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣦⠀⢠⠆⠩⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⣤⡀⢛⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠢⣤⡀
⠀⠀⠀⠀⠐⠐⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡀⠒⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄
⠀⠀⠁⠘⢃⠂⡈⠀⠀⠀⠀⠄⠀⠀⠑⡒⠀⠀⠀⠀⠀⠤⢠⠟⢲⡶⠀⠀⠀⠲⣿⣿⣿⣤⡤⠞⠁⠀⠀⠐⢀⠈⡠⠂
⠀⠀⢠⡄⠈⠚⢠⠶⠀⠀⠀⠀⠀⠀⢈⠉⠉⠀⠀⠀⠀⠀⣙⠶⢭⡆⠀⠀⠀⠸⠋⠉⠃⠀⠀⠀⠀⠀⠀⢠⠄⠁⠰
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⠀⠐⠀⡀⠀⡀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣀⡄
⠀⠠⢴⣿⡶⠂⠀⠀⠀⠀⠀⠀⠉⠈⠀⢉⠐⠁⠀⠀⠀⠒⠿⣭⢭⡉⠀⠀⠀⠠⢤⣥⡤⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡍
⠀⠀⠞⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠛⠀⠈⠑⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠟⠁⠉⠓
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠰⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⡁⠀⠀⠀⠀⣀⡀⣿⢄⣠⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⡟⣿⡉⠁⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡀
⠀⠀⠀⢠⠀⠤⠀⠀⠐⠀⠀⠀⢄⡄⠀⡷⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⠒⠀⠀⠀⠀⢺⠋⠀⠰⠛
⠀⠠⠁⡐⠀⠰⠖⠀⠀⠀⠀⠀⠀⣑⡤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡀⠀⠀⠀⠐⠿⠈⠈⠳⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁

⠀⠀⠀⣠⢄⠀⢀⣀⠄⠀⠀⠀⠠⢀⡀⠀⢀⣄⢀⣀⠤⠶⠒⣚⡻⣿⢿⠛⠷⡒⠶⠤⣀⡀
⠀⠀⠀⠿⠶⠾⠟⠛⢫⡏⣾⠟⠉⠉⢛⣿⡿⠟⠉⠀⠀⠀⢞⠣⠀⠸⠀⠀⠀⢨⢳⣀⠀⠍⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠷⠟⠻⢹⣿⣥⡂⠀⠀⠀⠀⠀⠀⠁⣀⣐⠀⠀⠀⠀⡡⢎⠀⠀⠀⠐⠌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⢧⡴⣂⠠⣤⠌⠀⠙⣷⡄⠀⠀⠀⠀⠀⡸⠸⠡⠛⢧⠀⠈⢐⡣⠀⠀⠀⠀⠀⠀⢌⠳⡀
⠀⠀⠀⠀⠀⢠⣮⠄⢢⠀⠅⠀⢹⠀⠀⠀⠘⣷⡀⠀⠀⢀⢔⣴⡣⠊⢀⢽⠀⠂⢩⠇⠀⠀⠀⠀⠀⠀⠀⠡⠙⣄
⠀⠀⠀⠀⠀⠀⠙⠒⢋⣹⠤⢀⠞⠀⠀⠀⠀⠹⣷⡄⢤⣗⡻⢸⠄⠀⣠⠊⠀⢤⠟⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠖⢻⠀⠀⠀⡠⣤⠫⣫⠽⠋⢁⡐⠃⢀⠐⠁⣉⡽⠛⠛⠛⠋⠉⠉⠉⠙⠛⠷⢀⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡌⢀⣰⡼⣋⠔⠈⢀⠴⠃⢉⡀⠚⣁⡠⠟⠁⠀⠀⠀⠀⠀⢠⠔⢦⡀⠀⣠⠶⣌⠲⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠁⣦⡯⠞⢁⡰⠊⢁⡀⠀⢉⣠⠔⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⢡⠀⡅⠀⡇⠀⣯⠙⣎⢇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀⣸⣓⡃⡞⢀⡞⠉⢠⠔⠛⠧⠄⢀⠠⠤⠊⠁⠀⠆⠀⠀⢰⣁⠴⠁⡠⠃⣰⠀⣇⠉⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢧⣸⠾⢽⣠⢎⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠛⡆⠀⠀⡠⠊⢀⠔⠁⠀⢻⢀⡟⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⠃⣠⠀⢸⠶⡟⡉⠉⠙⡟⢳⠒⠚⠛⠉⠁⠀⠀⠀⣆⡖⠆⣸⠶⠲⣍⠀⡖⠢⣘⡾⣀⠙⡄⠀⠀⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠆⢠⡇⢀⠞⠀⠘⠯⡢⡀⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⢡⣗⣤⡇⠀⠀⠈⠉⠙⡶⣧⣔⣨⠃⠈⠢⢄⡲⣟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⣠⣾⠔⠉⠀⠀⠀⠀⠈⠉⠺⣤⠷⣀⠀⠀⠀⠀⠀⢀⠎⣘⠝⠁⠀⠀⠀⠀⡀⣰⠋⢩⠁⢣
⠀⠀⠀⠀⠀⠀⢀⣀⡰⢋⣮⣴⢣⡀⠀⠀⠀⠀⠀⠀⢀⣀⠎⢲⣭⡥⠀⠀⠀⢀⠎⣌⡏⠀⠀⠀⠀⡠⢂⠜⠁⠀⠀⢱⠀⢧
⠀⠀⠀⠀⠀⣐⣟⣡⣄⣤⡟⠒⠀⠙⢦⡀⠄⠀⠀⢰⣷⣄⣷⣦⢇⣶⣎⣩⣴⢪⣔⣥⣻⣉⣷⡤⣈⠔⠋⣰⣴⣠⣔⣧⣀⣤⣩⣉⣷
⠀⠀⠀⠀⠀⠁⠀⠁⠉⠁⠀⠀⠀⠀⠀⠈⠓⢬⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠈⠀⠀⣂⠴⠚⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⢤⡤⠤⠥⠴⠖⠚⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡯⢰⡋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠘⠓⠈⠓⠚⠁


⡿⢿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⡿⣿

⢀⣀⡞⣆⣀⠀⠀⠀⠀⠀⡼⠆⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⢠⡆
⠙⡦⢁⠡⡞⠁⠀⠀⠉⠃⠀⠀⡜⠁⠀⠀⠈⢛⡽⣻⠛⠁⠀⠀⠀⠡⠐⠂⠌⠀⠀⠀⠈⠑⠀⠀⡉⠁⠀⠀⠀⠙⣯⠶⠎⠁⠀⠀⠀⠙⣯⠀⠈
⠀⠓⠉⠑⠃⠀⠀⠀⠀⠛⠉⠘⠃⠀⠀⠀⠀⠚⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠃⠀⠀⠀⠀⠘⠃⠈⠓⠀⠀⠀⠀⠐⠃⠀⠁

⠀⠀⣠⡀⠀⠀⠀⠀⠀⢀⣰⡄⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⢀⣀⡜⢣⣀⡀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⣀⡰⡱⣀
⠀⢪⡱⣯⠂⠀⠀⠀⠈⢹⣅⡹⠟⠀⠀⠀⠈⠃⠀⠀⡉⠁⠀⠀⠈⣧⠀⠀⣜⠁⠀⠀⠀⠙⣃⢀⠛⠁⠀⠀⠈⠛⢀⢀⡋⠁⠀⠀⠈⠲⡀⠀⣰
⠀⠈⠀⠉⠀⠀⠀⠀⠀⠘⠈⠙⠂⠀⠀⠀⠀⠘⠁⠀⠃⠀⠀⠀⠀⠙⠋⠙⠋⠀⠀⠀⠀⠘⠁⠈⠃⠀⠀⠀⠀⠐⠋⠀⠓⠀⠀⠀⠀⠘⠊⠉⠒

⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣀⠆⢠⣀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⢰⠇⢀
⠈⢩⣭⣭⠉⠀⠀⠀⠈⢻⣿⣿⠟⠀⠀⠀⠐⡆⠀⠀⡔⠀⠀⠀⠈⠢⠀⠀⡌⠁⠀⠀⠀⠘⣏⣹⠃⠀⠀⠀⠀⢻⡅⡀⡟⠁⠀⠀⠈⠓⠀⠀⡉
⠀⠈⠀⠈⠀⠀⠀⠀⠀⠙⠉⠙⠃⠀⠀⠀⠀⠀⠊⠑⠀⠀⠀⠀⠀⠐⠊⠀⠓⠀⠀⠀⠀⠐⠁⠈⠂⠀⠀⠀⠀⠈⠋⠉⠃⠀⠀⠀⠀⠐⠋⠀⠛

⠀⠀⡰⡀⠀⠀⠀⠀⢀⠀⠜⠆⠀⠀⠀⠀⠀⢀⡴⢆⡀⠀⠀⠀⠀⣴⡶⢶⣦⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⢦
⠘⢧⣀⠠⠛⠀⠀⠀⠙⠀⠀⠀⠘⠁⠀⠀⠈⢣⠀⠀⡜⠁⠀⠀⢸⣷⠀⠀⣾⡇⠀⠀⠀⠛⠀⢀⡙⠁⠀⠀⠈⠡⢀⣴⣤⠁⠀⠀⠀⠛⠁⠀⠙
⠀⠚⠉⠙⠂⠀⠀⠀⠀⠓⠀⠐⠂⠀⠀⠀⠀⠓⠋⠑⠃⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠐⠋⠈⠃⠀⠀⠀⠀⠐⠀⠛⠓⠀⠀⠀⠀⠐⠋⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢀⡰⢆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡴⠰⣄⡀⠀⠀⠀⠀⣰⢂⡀⠀⠀⠀⠀⠰⠆⠂⠶
⠀⢒⢾⡇⠂⠀⠀⠀⠈⢛⡀⣘⠋⠀⠀⠀⠈⢥⠀⠀⡜⠁⠀⠀⠈⢻⣿⣦⠀⠀⠀⠀⠐⢦⠀⠀⢠⠂⠀⠀⠈⠢⠁⠈⣍⠁⠀⠀⠠⡅
⠀⠈⠀⠈⠀⠀⠀⠀⠀⠘⠁⠙⠀⠀⠀⠀⠀⠐⠈⠁⠂⠀⠀⠀⠀⠘⠋⠉⠃⠀⠀⠀⠀⠉⠖⠓⠈⠀⠀⠀⠀⠐⠚⠁⠘⠀⠀⠀⠀⠀⠐⠶⠁

⢀⣠⠖⢄⡀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⣀⡔⢢⣀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⣰⠂⢀⠀⠀⠀⠀⠀⢀⡆
⠑⡄⠀⠀⡔⠁⠀⠀⠘⢫⠁⢩⠟⠀⠀⠀⠈⢻⣏⣻⡟⠀⠀⠀⠐⢦⠀⠀⡴⠂⠀⠀⠈⠛⢊⢀⡛⠁⠀⠀⠈⢳⡄⠀⠰⠋⠀⠀⠀⠹⠞⡳⡋
⠀⠗⠚⠒⠇⠀⠀⠀⠀⠚⠉⠙⠃⠀⠀⠀⠀⠘⠋⠙⠃⠀⠀⠀⠀⠙⠚⠓⠋⠀⠀⠀⠀⠘⠋⠈⠃⠀⠀⠀⠀⠘⠛⠙⠛⠀⠀⠀⠀⠀⠋⠁⠑

⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⢀⡴⢀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⢶⡀
⠀⠬⠀⢩⠄⠀⠀⠀⢠⠘⣿⣟⢀⠀⠀⠀⠐⠄⠀⠀⡈⠁⠀⠀⠈⢻⠁⢈⡟⠁⠀⠀⠀⠺⣿⣿⡗⠀⠀⠀⠈⠃⠀⠀⠀⠃⠀⠀⠈⢫⡰⠶⢨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠰⠆⠙⠃⣛⣚⣓⠛⠈⠋⠰⠆⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠺⡃⡉⠀⠀⠀⠀⠀⠨⠸⢸⠝⠀⠀⠀⠀⠀⢉⢋⠗⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡺⠃⠁⠘⡄⢘⠠⠀⠀⡄⠠⠀⠄⡚⠀⠄⢠⠀⠀⠊⠇⡠⠃⠈⠘⢗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠊⠁⢄⠀⠀⠈⠁⠲⡀⠤⠒⠂⠀⠉⠉⠀⠒⠒⠤⢀⠀⠀⠓⠀⠀⠠⠊⠑⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡙⣁⠀⠀⠀⠀⠄⡀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡠⠈⠀⠀⡠⣬⣛⣄
⠀⠀⠀⠀⠀⠀⠀⡰⡟⣰⠃⠌⠙⠀⣄⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀⠆⠐⣈⢥⠈⢸⢆
⠀⠀⠀⠀⠀⠀⢀⡛⠀⠀⠉⠀⠄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠀⠈⠉⠀⠀⢛⡀
⠀⠀⠀⠀⠀⠀⣌⠃⠀⠀⠀⠤⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠤⠀⠀⠀⠈⣡
⠀⠀⠀⠀⠀⠀⣭⢀⡀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢀⠀⠀⠀⣬
⠀⠀⠀⠀⠀⠀⣾⠈⠑⢄⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠧⠼⠄⡷
⠀⠀⠀⠀⠀⠀⡛⠀⠀⠈⠁⠂⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣛
⠀⠀⠀⠀⠀⠀⢋⡀⠀⠀⠀⠒⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠐⠀⠀⠀⢠⡙
⠀⠀⠀⠀⠀⠀⠈⣱⠀⠀⣄⣄⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡔⣀⠀⢀⢀⣬⢁
⠀⠀⠀⠀⠀⠀⠀⠱⢇⣔⡩⠄⠀⠐⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢛⠟⠹⢭⡗⠚⣿⡏⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣱⡃⠀⠀⠀⠀⠊⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠔⠊⠱⠆⠀⠀⠀⢒⣶⠆⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⢄⡠⠂⠀⢀⠀⠈⠘⠇⠒⠠⠤⢀⣀⣀⠀⠤⠄⠒⠈⠀⢈⠀⠀⠀⠑⢀⡠⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⡤⡀⠚⠭⣒⠀⠀⠀⠃⠐⠀⡀⢐⠀⠀⠘⠀⠀⠀⢸⠲⠤⢂⢠⡮⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢴⣤⣀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠀⠀⠀⠈⣊⢤⡢⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠰⠆⣠⡄⣤⢥⡬⣤⠠⣄⠰⠆⠛⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⣠⠤⠖⠶⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⠶⠔⡀
⠀⢾⣤⠀⡠⢒⠠⢹⣆⠀⠀⠀⠀⠀⠀⢨⣠⠈⠚⢽⣍⣀⢢⣈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢠⣆⢈⣑⡣⢀⢀⣬⣿⠖⠀⠀⠀⠀⠀⠀⡌⠘⣄⣤⢠⢹⠁⣛⣼⡄⠀⠀⠀⠀⢀⡶⠆⠀⢀⡀
⠈⠛⠁⠀⢀⢴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠺⣡⠇⣈⡿⡠⡿⠀⠀⠀⠀⠀⠀⠻⠶⠶⠛⢀⠀⠄⠴
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⢴⣌⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠘⠷⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣶⢶⠶⣄
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠆⠒⠛⠒⠳⢤⣄⠄⠀⠀⠀⠀⠀⠀⠮⠍⣘⢁⣀⣃⡉⡽
⠀⠂⠀⠀⠐⣗⠀⠀⠪⠁⠘⠀⠀⠀⠀⠑⢀⠐⠂⠀⢀⠀⠋⢹⣥⡀⠀⠀⠀⠀⠀⡆⣎⡉⠈⡁⠌⣱⡐
⠀⠀⠀⠀⠀⠙⢦⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠫⠁⠀⠒⡼⠟⠃⠀⠀⠀⠀⠀⠉⠱⣛⠖⠓⢞⠯⠉
⠀⠀⠀⠀⠀⠐⠁⠈⠉⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⡀
⠀⡐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⡐⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⡷⠃⠀⠀⠰⠹
⠀⢳⢿⠂⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠖⠐⢒⠳⣄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡂⣠⢃⣟⣀⢒
⠀⠀⠁⡠⠂⡉⢉⣼⡒⢭⠲⢤⡀⠀⠀⠀⣄⠐⠀⢀⠀⢙⠁⢀⠀⣀⠨⠃⠀⠀⠀⠀⠀⢒⡾⣄⠤⣹⢾
⠀⠀⠀⠉⠃⣀⡘⠀⠶⢐⡻⠶⢦⠀⠀⠀⠹⢶⣀⣌⠓⠦⣜⣛⣛⣌⠁⠀⠀⠀⠀⠀⠀⠘⠁⢛⠑⡄⠈
⠀⠀⠾⠅⠈⠀⡤⠒⠋⠉⢨⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠌
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⡁⠀⠀⠀⡃⠼⠃⠀⠀⠀⠀⠀⢠⠶⣶⢖⢦⣀
⠀⠀⠀⠀⠀⠃⣡⡔⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠸⣏⣃⢂⣋⡽⠂⡂⠀⠀⠀⠀⠀⢸⡟⠀⠳⣼⣿⡀
⠀⠀⠀⠀⠘⢻⣴⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢬⠞⠋⣳⣴⣋⠙⠪⠅⠀⠀⠀⠀⠀⢹⡿⠦⠞⢛⠙
⠀⠀⠀⠀⠀⠘⡤⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⣲⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠦⡤⣾
⠀⠀⠀⠀⠀⠀⠐⠈⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠼⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⡁⠚
⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣈⣠⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠑⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⡀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠈⠙⢷⣦⣀⡀⠀⣠⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢀⣭⣶⡆⠹⣿⣿⣿⣿⠿⠛⠛⢿⣿⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⢠⣾⣿⣿⢇⡆⣿⣿⠿⣣⣾⣿⣿⣷⣮⣀⣤⣤⡀
⠀⠀⠀⠀⣰⣿⣿⣿⢏⣼⠃⣫⣤⣤⣿⢛⣽⠿⣋⡍⣀⣤⡄⢀⡀⢤⣤⣤⣤⡀
⠀⠀⠀⢰⣿⣿⡿⠣⣾⡿⣰⣿⣿⡿⢱⣿⠃⣼⡏⢸⣿⣿⣦⣻⢿⡿⣿⢿⣿⣧⣄⡀
⠀⠀⢸⡘⢛⠉⠀⠀⠈⢡⣿⣿⡿⣡⣾⣿⡀⢻⣷⣌⣛⣛⣛⣟⣛⡛⢿⢿⣷⣿⣿⣿⣷⡄
⠀⠀⢸⣿⡏⠀⠀⠀⠀⣼⣿⣿⢱⡿⢛⣛⠻⣦⣬⣙⣛⣛⣛⣛⣿⣯⡟⣷⠻⡿⢿⣿⣿⠃
⠀⠀⣾⣿⠀⠀⠀⠀⠀⣿⣿⠇⣾⠀⣿⣿⣦⣿⢿⣟⣻⣿⣿⠀⣶⣿⣿⣶⣶⣶⣶⣬⡁
⢀⡘⠿⠛⠀⠀⠀⠀⠀⣿⣿⠀⣿⣧⣬⣩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡆
⠿⠿⢿⣿⠘⣿⠄⠀⡀⢿⡿⢀⡈⠛⠛⣛⣭⣿⣿⣿⠟⣽⣿⣿⣿⣿⣿⣿⣶⡟⠛⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣷⣤⣄⢸⣿⠟⢉⣍⠉⣿⡿⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡜⣿⠀⢿⣿⣷⠟⣱⡿⣻⣿⣿⣿⡉⣿⣿⢋⡛⠻⠿⠃⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠘⢦⣤⣤⣴⣾⠟⣰⣿⣿⣿⣿⠇⢹⣿⠸⣿⣾⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣛⠃⠀⠀⠀⠈⠙⣉⣥⣴⡿⠟⣫⡞⣱⢋⡤⠙⠷⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⡰⠿⠿⠷⠄⠀⠀⠀⠀⠠⣤⣴⣶⣿⡿⢸⡇⣸⢳⣿⢃⣴⣿⣦
⠀⠀⠀⠀⠀⠀⢿⣿⢲⣿⢶⣿⡆⠀⠀⠀⠀⠀⠉⠙⠛⠃⠘⠏⠙⠈⠛⠘⢿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣾⣶⣍⢾⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡿⢿⠅⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⣵⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠾⢿⣷⣶⠀⠀⠀⠈⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠛⠁⠀⠀⠀⠀⢀⣠⢦⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⢠⡄⠀⡈⠁⠀⠀⠂⠀⠀⠀⠀⢀⢀⣆⣠⣀⡀⠀⠀⠁⠀⠀⣤⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄⠀⠀⢲⣼⣿⣿⣿⣿⣢⠄⠀⠀⠴⠶⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⢽⣷⡿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠲⣽⣿⣿⣿⣿⡦⡕⠀⠀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠤⠰⠖⠠⠀⣀⠀⠉⠀⣤⠀⠀⠰⣁⣀⠰⠀⠀⠈⠻⣿⣿⣿⣿⢛⠈⠀⠸⣄⣀⡏⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠖⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠐⠁⠃⠀⠀⣀⠆⠈⠁⠀⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠃⣴⣦⠠⠀⡠⢄⠀⡀⠀⠈⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠣⡘⢁⡨⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠛⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢀⣤⡄⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⣿⡿⢺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣷⣀⡀⠀⠀⠀⠀⠀⢠⣎⣠⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠙⠓⠀⠀⠀⠀⠀⠨⡿⠟⠿⡯

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣴⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢙⡆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡼⠁⠹⡄⠀⠀⠀⣠⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢠⡇⠹⡄⠀⠹⡄⠀⢀⡽⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣼⣄⠀⠀⠀⠀⠀⣼⠀⠀⠹⡄⠀⢹⡴⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠉⠓⠦⣄⡼⡿⠀⠀⠀⠹⡴⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣤⠤⠤⠶⠒⠒⠚⠉⠉⠉⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀
⢀⣤⡤⠤⠴⠖⠒⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠒⠦⠤⠤⣤⣄
⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣷⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⢸⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣹⣶⣞⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣶⣾⣁⣄⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡀⣄⢤⣆⢰⣶⣤⣀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠌⠫⠡⣄⣙⠋⢡⣿⣿⡈⠝⣃⣠⠌⠝⠁⢤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⠉⢉⣡⠤⠤⠀⠀⢤⡀⠀⡛⣹⠃⣠⠔⣀⣠⡀⠤⣄⣉⠈⣳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠤⠞⠁⢀⠄⢂⣠⠤⣤⣿⣲⠀⠋⠘⣩⠋⠀⠀⠉⠣⡀⠈⠱⠤⡔⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣀⠄⣸⠋⠀⢀⣴⡾⠿⠿⢿⡶⢲⡿⠛⠛⠿⢦⡀⠠⡙⢇⠠⢤⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢃⠔⠁⠤⠒⠁⢀⣀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢀⣙⠮⢓⠤⠈⠑⡘⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡵⠀⡠⠞⠐⣨⠟⠁⠰⣤⣦⢀⠀⣀⣀⣀⣀⠀⡀⣶⣬⠆⠈⣿⣷⠂⠑⢄⠀⢮⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣻⡕⠀⠎⠀⠀⣰⠏⠸⠀⠠⠋⠉⠀⠀⠻⣝⣫⠟⠀⠀⠉⠙⠄⠀⠋⠻⣦⡄⠀⠱⠀⢪⣛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠂⢚⣿⠴⠁⠀⠉⠈⠀⠀⠀⠀⣨⣅⠀⠀⠀⠀⠁⠉⠀⠈⠤⣿⡆⠐⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣝⡤⠂⠠⠊⠀⠈⡟⠀⠀⠀⠀⠀⠀⢠⠔⡞⢋⡉⢳⠲⡄⠀⠀⠀⠀⠀⠀⢿⠇⠀⠑⠀⠐⠤⣯⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢋⠀⠀⠀⢠⡄⣾⢇⣴⠀⠀⠀⠀⣴⡾⠀⢀⣾⣷⡀⠀⢷⣇⠀⠀⠀⠀⠀⣿⣦⢨⡄⠀⠀⠠⡉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡌⠀⡆⠀⡆⠻⠙⣿⢻⣷⡄⠀⡠⠋⠀⠀⠸⠿⠿⠇⠀⠀⠙⢤⠀⢀⡼⣱⡿⠁⠌⢰⠀⢀⠀⢡⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⢃⠀⠀⠸⣿⠽⢣⣿⣿⣦⢀⡀⠀⠀⠀⠀⢀⡀⣴⣿⣿⡞⠀⣷⠁⠀⠀⡘⠀⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣾⡇⡀⢸⠀⠘⠀⠛⡄⣾⣿⠹⣿⡤⡂⣂⠐⠢⣐⢐⢬⣿⠏⣿⣷⢠⠁⠀⠀⠀⣇⢀⢸⣶⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣇⠁⢧⢰⠀⢀⠃⠘⠹⣿⡄⠉⠁⠀⠁⠀⠀⠀⠀⠈⠁⢠⣿⢏⠇⠘⡀⠀⡆⡜⠀⣸⡃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣀⣿⡀⠘⠀⣄⠇⣷⡁⠀⠀⢀⣠⣤⣦⣄⡀⠀⠀⣈⣾⠸⣠⠀⠃⢂⡿⣀⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢣⣻⡇⠄⢠⠹⣀⣌⠃⠙⣿⣿⡿⢋⣙⢿⣿⡿⠫⠗⣠⣀⢏⡄⡠⢸⢟⡜⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⡿⣿⣺⣄⢱⢌⠫⠀⡀⢬⣿⣇⢻⡿⣸⣿⠗⣣⠊⢘⡡⡎⣠⣧⣿⠿⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣻⢻⣧⣿⢿⣦⣦⠸⣿⡛⢊⣧⣿⡿⢃⣿⣴⡿⣿⣼⡟⣟⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣟⣛⣿⣿⣿⣿⡿⠿⠿⣿⣴⣿⣿⣷⢛⣻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⢈⠋⢿⡟⢻⣟⠋⠅⠸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣾⣿⣿⣿⣿⣷⣦⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢯⡛⢩⡽⠟⠋⠁