Sac Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⣤⣤⣤⣶⠷⠀⣶⣄⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠿⡀⠻⠀⠀⠈⠀⣹⠷⠷⠶⣦⡀⠉⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⣟⠻⣿⡃⠀⠀⠤⠀⠀⠰⠂⠈⠅⠀⠀⠀⢽⣷⠆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⠀⠁⠂⠡⡄⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⢀⡴⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠯⠶⣿⣆⡤⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠘⠀⡀⠈⠡⢤⣄⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣴⡂⠀⠐⠈⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠄⠀⠀⠀⣀⡤⢾⣿⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠒⠠⢼⢠⠀⠀⠀⠀⠀⠉⠛⠀⠀⠘⠂⠈⡀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⠋⡀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⢀⡀⠾⠁
⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⡀⠀⠀⠰⣾⣿⣤⠀⠠⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠲⣴⠛
⠀⠀⠀⠸⠃⠲⠛⠀⠋⠘⠓⠓⠐⠚⠊⠚⠛⠃⠓⠛⠈⠁⠀⣰⣿⡿⠁⠀⠀⠆⠀⡀⠀⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⣸⠇
⠀⠀⠀⠀⠀⠀⣠⠠⡆⢠⣔⡐⢲⠣⢦⡔⢲⠄⠀⠀⠀⠀⢀⠻⣫⣴⣀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢦⢀⣤⠸⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⢀⡀⠀⢠⡿⠋⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⢠⡿⠁⠀⢀⡤⢤⣄⡀⠀⣸⣦⣀⣀⣀⣸⠟⠛⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⣶⢶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠛⠀⢀⡴⠋⠀⠀⠈⠉⠁⠉⠉⠛⢙⣉⡁⠀⠀⣠⣤⣶⣶⢶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⡠⠀⠀⠠⡀⣀⠀⠀⠀⠀⠀⢠⠋⡠⠀⠀⠀⣠⣾⠿⣿⣷⣾⡿⠟⠛⠙⣿⣯⠁⠀⠀⠘⡿⠠⠉⢿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈⠀⠀⠁⠉⠁⠁⠈⣁⣀⣀⠀⠀⠀⠘⢷⣄⣠⣶⠿⠻⣏⠀⢀⣾⠉⠀⣴⠂⢠⡟⠀⠀⠀⣀⡬⠇⠀⡠⠇⠀⠈⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢶⣄⡀⠀⠉⢻⣯⠀⠀⢿⠀⠸⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠐⠠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠆⠒⠦⠀⡀⠀⠀⢴⠉⠙⢶⣄⠀⠻⠀⠀⠀⣀⠂⠀⢀⠀⡀⠰⣷⡆⠀⠀⠀⢀⣀⡀⠀⠀⣀⣤⠴⠶⠛⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡄⡤⠠⠠⢤⠀⣤⡤⡠⠐⢸⠀⠀⠀⠃⠀⠀⣿⠗⠀⠀⠀⣴⡇⠀⠀⠀⠈⠧⠀⢻⡁⠀⢠⡾⠟⠉⠛⠄⠚⠁⠀⠀⠀⠀⣠⣤⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⠠⡤⣤⡄⡀⢄⢤⠀⠀⠀⠀⠀⠀⠘⣿⣤⣄⠀⠀⠹⣧⡀⠀⠀⠀⣀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠐⠶⢶⠶⣿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠀⠀⠀⠀⠈⠉⠁⠐⢺⠐⠛⠛⢢⣄⠈⠳⠦⠤⠖⠃⠀⠀⠀⠀⠀⠀⠀⢻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⣤⠤⠄⠀⠀⠀⠀⢶⡂⠀⠀⠀⠀⠈⣱⠀⢀⣀⣀⡀⠀⠀⡀⣀⣤⢨⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠠⠄⣠⠌⡾⠃⠀⠈⠉⠳⣆⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠐⠠⠼⠁⠀⢼⠀⠀⠀⠀⢀⣸⠇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣟⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⠴⠊⠁⣶⡄⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢗⠄⠻⣦⣀⡀⠀⣠⡞⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⠌⠉⠛⠛⣗⠀⠀⠀⠀⠀⢬⡀⠀⠀⠙⢆⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠄⠀⠈⠓⢦⣤⣤⣀⣼⣅⠀⠀⠀⢨⡷⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⡍⣽⠟⠻⠿⠿⠛⠟⠻⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢒⣓⠺⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣆⡉⠉⢹⠃⡻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣘⠃⠁⠽⡟⠠⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⢁⠀⠄⠂⣀⠃⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠁⠀⠀⠂⠂⣀⣀⠟⠻⢰⠀⣨⡁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴⠁⠀⠀⠀⠀⠀⠒⠋⠉⠁⠀⠃⡆⣴⠏⠐⢤⡉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠟⠁⠀⠀⠀⢸⣇⠄⣠⣷
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠞⠁⠐⢙⣿
⠀⠀⠀⠀⠀⠀⢸⡘⠀⠀⠀⠀⠈⢉⠐⠒⠒⠤⠤⠤⠤⠤⠤⠤⠤⠒⠊⢉⢡⡀⠂⠀⣮⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⠀⠀⠀⠈⠆⠀⠀⠀⢀⠠⢄⠤⡀⠰⠀⠀⠀⠀⠀⠃⢠⣼⣯⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠉⠓⢮⣄⡀⠁⠈⠀⠀⠀⠀⠈⠀⡁⠀⢣⠀⢀⣐⡾⢾⢛⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⢶⡦⢤⢤⠴⠶⠚⠛⠋⠁⠀⣠⠼⣝⢻⡇
⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢛⣷⡏⣿⡇
⠀⠀⠀⠀⢀⡼⠍⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⣀⠋⣞⣻⣟⡀
⠀⠀⠀⠀⠘⢧⣀⣀⠀⢉⠒⠤⣄⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠶⠋⠩⣩⣹⠗
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠶⠤⠤⠤⣤⣄⣈⣉⣀⡀⣈⣀⣠⣤⣤⣤⣬⣤⠤⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢸⠙⠦⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⠤⠤⠔⠒⠒⠒⠒⠀⠀⠈⠉⠀⡸⠤⠴⠖⠒⠖⠲⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣦⣄
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⠳⣄⡀
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡍⠑⣦
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠒⣒⠒⣒⣒⣒⣚⣻⠆⠀⠀⠀⠀⠀⠀⠀⡇⠈⣧⠞⠁
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠉⠉⠉⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼⠁
⠀⠀⢰⠃⠀⠀⠀⠀⣼⢉⡷⢀⡆⢠⠴⠆⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⢀⡏⠀⠀⠀⠀⠐⠋⠣⠀⡼⠀⢿⣀⠀⢸⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠞⠳⢖⡒⠲⠦⣄⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠗⠤⠤⠤⠤⠶⠒⠒⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠈⠓⢦⡈⠻⣝⠲
⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⡀⠀⠀⠀⠀⠀⠀⣾⡆⣿
⠀⣷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⠿⠯⣽⣶⣒⣶⣒⣒⣒⣲⣒⡒⢒⡋⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⡽⠃
⠀⡏⠑⠬⢑⡢⠤⡄⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⢠⠿⢷⣦⠤⣈⡉⠒⠒⠚⠋⠏⠨⠭⠭⠭⠭⠭⠯⠽⢉⣋⣉⠩⠭⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⢸⠀⠀⠈⠑⠒⠊⠬⠭⠴⢒⣖⣴⢦⠄⠤⠠⠄⠠⠀⠄⢠⢀⣤⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⢎⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠁⠀⠉⠁⠈⣏⠉⠓⠒⠶⠤⠤⠴⠤⠤⠤⠶⠖⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀⣀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠤⠤⣀⣀⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠙⢦⡀⠀⠀⢰⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⣆⡴⠋⠁⠀⠀⠈⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠤⣀⠀⠀⠰⣄⠀⠈⣇⠀⡏⠀⠀⠀⣀⠤⠂⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⢆⠀⠈⠳⡀⠀⠈⠳⢄⠘⡄⢠⠀⡠⠚⠁⢀⡠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⢄⡀⠙⠒⠒⠚⠓⠦⢤⣀⡱⠧⠼⣎⡀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠔⠒⣒⣢⡿⣤⠤⣄⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠝⠋⢁⡔⠁⠀⠸⡈⠙⠫⢍⡓⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠭⠒⠉⠀⠀⢠⠊⠀⠀⠀⠀⠱⡀⠀⠀⠈⠓⢤⣀⠉⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠈⠑⠦⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠈⠳⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⠒⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢤⡀
⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠈⠲
⠀⢀⡤⠒⠉⠀⠙⢄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠴
⡔⠁⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢃⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠔⠊⠁
⠈⠑⠒⠢⠤⠤⣤⣀⣀⣀⡑⢄⡀⠀⠀⣀⣀⣀⣤⠤⠤⠔⠒⠒⠊⠉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣾⣿⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⢶⣶⡦⢤⡀⠀⠀⠀⠀⣠⠞⣋⣉⣉⡛⠓⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣤⣤⣤⣀⡀⠀⠀⠘⠫⡳⣄⠀⣰⠿⡶⠋⠉⠛⠛⣷⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣴⣋⡑⠛⠋⠉⠓⠉⣿⠆⠀⠀⠀⠈⠪⡳⣿⡟⠁⠀⠀⢠⣾⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⡶⣿⠚⠳⠮⢭⣉⡙⠒⠲⠾⣏⡀⢀⡀⠀⠀⠀⠙⣜⣧⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢛⡼⠁⠠⠄⠒⠒⠤⠉⣻⡶⣄⡀⠉⢻⠍⢳⡄⠀⠀⢈⣽⣶⣾⣽⠏
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠹⣦⡞⣠⠊⠀⠀⣰⢫⠟⠹⡻⡇
⠀⠀⠀⠀⠀⠀⢸⢧⣄⣴⠋⠀⠀⠀⠀⠀⠀⢀⣾⢿⡀⠀⠀⣴⡷⠙⠁⠀⢀⣼⣿⠋⠀⠀⠘⣽⡄
⠀⠀⠀⠀⠀⢠⡟⣸⡭⣯⢧⠀⠀⠀⠀⠀⢠⡾⠉⢿⡧⢀⣞⠟⠀⠀⠀⢀⣞⡼⠃⠀⠀⠀⠀⢻⡷
⠀⠀⠀⠀⠀⣾⢠⡿⠁⢸⡌⢳⡀⠀⠀⢠⡿⠁⠀⠀⢠⣮⠏⠀⠀⠀⢠⣿⡾⠁⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⣸⡷⠋⠀⠀⠀⢳⡀⠳⣄⣠⡟⠀⠀⠀⡰⡿⠁⠀⠀⠀⣰⣛⠟⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⣰⠋⠐⢒⣆⡀⠀⠀⢳⡀⠘⢿⡀⠀⢀⣜⡟⠀⠀⠀⠀⣴⣳⠋⠀⠀⠀⠀⢠⣾⠋
⠀⠀⣼⣧⠶⠶⢦⣤⣬⣥⣤⣴⠿⣄⠀⠙⡟⢟⢻⣹⡄⠀⢀⡼⣿⠃⠀⠀⠀⠀⣰⡞⠁
⢀⡞⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣰⢃⡜⠈⠓⠁⢠⣞⡿⠁⠀⠀⠀⢀⣾⠋
⣿⠟⠋⠋⠉⠛⠻⠷⣦⣀⠀⠀⠀⠈⠣⡴⠋⠀⠀⠀⣰⣿⡞⠁⠀⠀⠀⣴⡟⠁
⠈⢿⣷⣄⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⢠⡾⠋
⠀⠀⠙⣿⣷⠀⣀⡀⠀⠀⠀⠙⣷⣄⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⣀⡞⠁
⠀⠀⠀⠈⠻⣿⣿⣿⣦⣴⣖⣒⣚⣻⣷⣤⣴⡿⠋⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠈⢫⣻⣽⣟⢿⡟⡟⡛⡟⣟⢿⡄⣀⡠⡀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡷⣏⣾⣞⣎⣮⣣⡽⣞⢏⣀⣨⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠉⠉⠉⠙⢿⣎⣻⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⣀⣠⠄⠀⢠⢷⡠⠞⣫⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠂⠾⠖⠘⠋⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣠⡤⢄⠀⠀⠀⠀⠀⠀⢀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡏⡰⣆⡠⣄⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠿⠧⠹⠽⠇⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠉⠀⠀⠒⠒⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⣚⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣶⣶⠖⠒⡖⠖⠶⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢰⡟⠙⠁⠀⠀⠀⠀⠀⠀⠙⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⣤⣤⣤⠤⠤⠤⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⠄⠒⠀⠀⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⢶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠙⠋⠀⢻⣿⣿⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⡀⠀⠀⢹⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⢀⣈⣸⠟⠁⠘⢻⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⢥⠘⡇⠀⠀⢀⣿⣿⣿⠿⠓⢎⠉⠉⠘⢉⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡇⠙⠶⣶⣿⡿⠋⠀⠀⠀⠸⠁⠀⠀⠀⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠻⣿⣿⣶⣀⣠⣶⠀⠀⠀⠐⠀⠀⣠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢸⡇⠀⠀⣾⣿⠛⠻⢿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣻⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠾⠛⠙⠋⣉⣩⡶⣠⣄⠀⠀⠀⢘⡇⠀⠀⣿⣷⣶⠠⣄⣀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣤⠾⠛⠛⠉⠀⠀⠀⠀⠀⠀⠚⣉⣾⣿⣿⣿⣷⣿⣇⢀⠄⠻⣿⣿⣷⣿⣿⡆⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⣴⡟⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠉⠉⠉⣳⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⡿⡏⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡏
⠀⢰⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠈⠉⠉⠙⣿⣿⠿⠿⠟⠋⠁⠔⠁⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⡇
⠀⠀⠻⣦⣄⣀⣠⡄⡀⣀⠠⣀⡠⢈⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣟⣯⣴⣾⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣶⣶⡖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠿⠛⠛⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠒⠒⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡠⠔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠉⠉⠑⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠛⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⢀⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⣻⠀⣷⢀⡽
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠭⠤⠥⠼⠥⠾⠴⢞⠁⠀⠀⠀⠀⠀⠀⢹⠀⡿⠋
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⣠⢤⡀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠞⠁
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣸⢧⠖⠁⡞⢠⡏⠉⠀⣞⡓⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣀⣀⣀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠓⠀⠐⠧⡅⠀⠀⠀⠀⠀⠀⠀⠀⣏⡀⠀⢀⣀⣀⣀⣠⣤⣤⠤⠤⠴⠖⠚⠉⠈⠙⠲⢤⣉⠻⣶⡤⣀⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣄⠱⡄⢹
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢻⡞
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣲⡦⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟
⠀⠀⠀⠀⢀⠯⢝⡢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠉⠉⠉⠁⠛⠒⠚⠛⠒⠚⠓⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⢸⣦⣄⡈⠑⠂⠭⠩⣒⣶⣲⠖⣲⢒⠄⢀⠀⡤⢤⠤⠤⠄⠔⣲⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⢸⠀⠉⠒⠭⢋⡒⣢⢤⡄⣀⢀⡀⣀⢀⣉⣈⢉⣉⣉⣉⡉⣉⣁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠐⠓⠂⠐⠂⠒⠐⠒⠚⠒⠛⠚⡗⠲⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⠠⠤⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠉⠉⠉⠉⠓⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠒⠛⠉⠉⠉⠀⠀⠉⠉⠉⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣼⣽⣥⣮⣭⣿⣵⣾⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⢸⣿⣿⡇⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣿⠀⠀⣸⣿⡀⣿⣿⣿⣿⠀⢸⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⢸⣿⣦⠀⣿⣿⡇⣿⣿⣿⣿⠀⢸⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣿⣿⣧⣾⣿⣿⣤⣿⣿⡇⣿⣿⣿⣿⡇⢸⣿⣿⡿⢸⣿⠀⠀⠂⠐⠀⠐⠒⠖⠒⠀⠀⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣷⣿⣿⣿⡗⢺⣿⠤⠤⠦⠤⠴⠤⠦⢤⣤⣤⣠⣿⡇⣿⣿⣿⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣶⣶⣶⣿⣿⣤⣤⣤⣤⣤⣤⣿⣧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣿⣇⣀⠘⠿⠿⠉⠙⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠐⠚⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡟⣿⣿⡇⠿⠿⢿⣿⠋⢹⣿⣿⡏⠈⡿⠁⠀⠀⠈⠀⠉⠁⠀⠀⠀⠒⢺⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠙⠉⠛⠛⠛⠋⠙⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢸⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡲⠠⠸⠀⢎⠀⠸⠀⠄⠇⠸⠹⠈⠀⢆⠁⠸⠉⠀⢩⠀⢇⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣻⣛⣟⣙⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⣫⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢉⠙⠒⠂⠉⠉⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠋⢉⡵⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡁⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⠀⠀⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⠀⠘⠶⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠎⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠉⠻⠦⢤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠁⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠉⢙⡻⠟⠋⠀⠀⠀⠀⠀⠀⢀⡴⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡠⡋⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠀⠀⠀⠀⠙⠒⠠⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠂⠀⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠐⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠢⢀⢀⣴⠃⠡⠀⠀⠀⠀⠀⠀⠀⡀⢠⠄⢀⠄⠠⠆⡰⢊⡴⢁⣌⣁⡚⠁⠒⡢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠛⠷⠼⣄⡠⠀⠀⠀⠰⠐⢧⠐⢯⠐⠳⡈⠓⣌⣑⢄⠉⠂⡬⠟⣉⣙⢉⠀⠀⠀⠀⠀⢀⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⠀⡀⠀⠀⠀⠉⠈⠑⢌⡑⠖⢀⡦⢁⣈⠝⠙⡶⠾⡇⢰⠊⢤⡎⣡⡌⠃⡰⠆⢠⠶⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠒⠒⠀⡛⢠⢐⡀⡅⠘⡏⢤⡖⣉⣀⠷⢄⣧⣈⣄⠙⡅⠻⡁⢴⠂⡤⠔⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠟⣆⠻⢀⣆⠹⡁⢶⢈⣠⠘⠋⠀⠉⠀⠓⠀⠂⠈⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠘⠡⠠⢟⠓⠸⠆⠩⠘⣍⡉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠲⢆⠠⠠⡴⢦⠶⣶⣦⣤⣤⣤⡄⢤⠀⣤⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⡁⠀⠀⠈⣿⡾⠋⠉⠀⠸⡇⠈⠛⠛⣿⣷⡕⠢⣿⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣠⡴⠶⠍⣤⠂⠀⠀⢾⠇⠀⠀⠀⠀⡿⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⢤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣠⠄⠰⣟⣶⡎⣴⡆⠀⠀⣀⠀⠀⢀⣀⣲⣖⠉⠟⢉⡄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠐⠛⠹⠟⠁⠉⠀⠀⠀⠛⢀⣤⡿⠿⢻⣯⣹⣿⡏⠁⡉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠨⣍⣠⠀⠀⠀⡈⠀⣀⡐⠀⣵⠾⣇⣴⡦⡀⠜⢙⣷⣶⣼⣿⣿⣿⣿⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣴⣶⣿⣿⡇⠐⠂⣴⡴⠌⢱⡄⠘⠀⠈⠙⢿⣿⠀⠀⠀⠀⠀⠁⠈⠙⡃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡛⣅⠉⣽⣷⣈⣀⣈⣭⣿⣻⠟⣼⣷⣿⣿⣞⢿⣄⣀⡀⣀⣌⠀⠀⠀⠀⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡤⢩⡯⣾⣿⣿⡿⠋⠛⢿⣯⠈⢰⣿⣿⣿⣿⣿⣿⣿⣿⠋⢉⣾⠣⠄⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡚⣯⡇⠀⠙⣿⣟⠀⡀⠸⠃⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⢱⠈⣥⣶⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠟⠉⠀⠀⠀⠈⠈⠁⣀⠀⠀⠰⠾⠟⠻⠛⢿⣿⣿⣿⡿⣾⣵⣄⠶⣀⡈⠹⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢐⡋⠀⣀⢀⣸⣇⠁⠦⠴⠄⠀⠈⠀⢄⣼⣿⣿⣿⡟⠀⣿⣿⠛⠛⠿⣿⠿⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢲⠚⠁⠀⠈⠀⢤⢽⠆⠨⠀⠀⢠⠀⠀⢈⣇⢸⣿⡏⠽⠒⢻⣛⠃⠀⠂⠙⢂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠎⠲⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠟⣤⡀⣸⠯⠜⠻⣇⢀⠚⠈⠻⣬⠁⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠠⡇⠄⠀⠀⢏⡉⠀⠀⠠⣏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠒⠂⠘⠃⠀⠋⠀⠙⠋⠉⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⢰⣿⣀⡀⢠⡆⢰⣆⣀⣦⡄⢀⡀⣀⣿⣀⠀⢀⣿⣿⡀⢉⡇⣤⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⡶
⣤⣼⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣷⣶⣿⣶⣆⣀⣀⣀⣀⣷⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠛
⠛⠛⠻⣿⣿⣿⣿⣿⡿⢟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢼⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣶⠃
⠀⠀⠀⠘⡿⣿⡟⠹⠗⠀⠢⠓⠋⠉⠈⠹⠏⠙⠋⠿⠿⠛⠛⠈⠉⠟⠁⠉⠁⠘⠛⠛⠃⠈⠸⠷⢾⡧
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠆





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⡄⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⣿⣿⡇⣠⣤⡀
⠀⠀⠀⡇⠀⠀⣿⣿⡇⠀⠀⠀⣾⡆⣿⣿⣠⣠⣿⡆⣀⠀⠀⠀⢸⣿⣦⣦⡀⣀⠻⣭⡇⣿⣿⣧⣶⣶⣆⡄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢰⡆
⠀⠀⠀⣧⣄⣴⣿⣿⣷⣾⣿⣦⣿⣷⣿⣿⣷⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿⣿⣴⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣫⡁⠀⠀⠀⣿⠀⠀⠀⠀⣸⣧⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⢛⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣼⣿⣦⣤⣤⣤⣿⣿⡇
⠀⠀⠀⡿⣟⣿⡿⢿⣿⠿⢿⣿⣿⠙⠟⠳⡌⠉⠁⠛⠛⠋⠉⠛⠛⣿⣿⠟⠙⢩⣿⣛⣿⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠿⠿⣿⣿⢿⣿⡿⠿⠇
⠀⠀⠀⠇⠀⠀⠀⠀⠀⢰⣤⣿⡯⠠⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠋⠛⠛⠛⠁⠀⠈⠋⠁⠐⠀⠀⠀⠀⠁
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡇
⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⢄⡀
⠀⠀⠀⠀⠀⠻⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢃⣀⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣶⣰⣷
⠀⠀⠀⠀⠀⠀⠈⠢⠹⢦⡀⠀⠀⠀⠀⠀⠀⠰⣿⠋⠛⠉⢻⠆⠀⠀⢀⡀⠀⠀⠰⣟⠉⡉⢉⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣀⡀⣀⣀⣀⣤⣠⣀
⠀⠀⠀⠀⠀⠀⣤⠀⣨⡾⠟⢦⡀⠀⠀⠀⠀⠀⢰⣼⡇⣶⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⡟⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⡿⠻⠿⣿⣿⣿⣿⣿⢿⣿⡻⣽⣿⣳
⠀⠀⠀⠀⠀⠰⣿⣿⣷⣅⣀⠈⠙⢦⡀⠀⠀⠀⠸⠋⠃⠛⠀⠀⠀⣠⠟⠛⢦⡄⠀⣀⣁⣥⣤⣦⠀⠀⠀⠀⠀⠀⣾⡁⢰⡆⠀⠈⠙⠙⢻⡟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢈⡙⠿⢸⡿⠀⠀⢈⣿⣦⡀⠐⠟⠾⠛⠛⠻⡧⠚⠁⢠⣆⠀⠈⠓⢿⠀⠀⠀⢰⠀⠀⠀⠀⠀⣻⢹⡇⠘⠃⠐⢛⡀⠀⢸⡃⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢹⣃⠀⠀⠀⠀⢰⣿⠙⡏⢀⣀⣀⣀⣀⣀⣀⣶⣆⣸⣧⣠⣶⣦⣼⣶⠶⠶⠶⠆⠀⠀⠀⣰⠃⠈⢠⠀⣤⢸⣿⠀⠀⢸⡇⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣮⠁⠀⠀⠀⠸⠏⠀⠀⠈⠉⠉⠉⠀⠈⡿⠉⠉⠁⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⢰⡏⣄⣠⢸⡇⣿⢸⣿⠀⠀⢸⡇⠜⠉⠥
⠀⠀⠀⠀⠀⢀⡀⣿⣿⢀⠀⠀⠀⢲⠦⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡶⡆⠀⠋⠁⣿⣿⢸⡇⠟⠈⠁⠀⠀⢸⡇⢠⣀
⠀⠀⠀⠀⠀⢸⠶⠿⡿⢾⡏⠀⠀⣸⢹⡇⡀⠀⠀⠀⣠⠀⠀⠂⠀⢰⣿⢿⣷⡄⠀⣀⠀⣰⡆⠀⡟⠉⠁⢲⠆⡿⠋⠈⠄⠀⢺⡇⠀⠀⢸⡇⢘⣿
⠀⠀⠀⠀⠀⠈⠒⠌⡳⠦⠀⠀⠀⣾⣸⠃⠇⠀⠀⠀⣿⠇⠀⠀⠀⠸⣿⣼⣟⠁⠀⡏⠀⢿⡇⠀⣷⣶⢻⣿⠀⢀⠀⡶⡇⠀⢸⠟⠀⠀⢸⡇⠰⠿
⠀⠀⠀⠀⠀⠀⠈⠛⠤⠀⠀⠀⠀⡀⢁⠀⠃⠀⠀⠀⠿⠆⢀⡀⣠⡖⠉⣍⠈⠓⡤⣡⡀⠈⠁⠀⠼⢿⣦⢻⡌⣿⠀⡯⠟⠀⠻⣏⠁⠀⠈⣇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡂⠀⠀⠀⢹⡯⠀⢀⣴⠒⠚⠛⠛⣻⡿⠷⠶⠐⠛⠒⠒⠓⣘⡿⠁⠀⠀⠀⢸⡏⢻⡄⢋⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢸⡟⠀⢸⣿⡆⠀⣀⡀⠀⠏⠀⢉⣽⠷⠷⣦⡀⠀⣇⠸⠥⠃⠀⢰⣧⣾⣇⣤⣿⠀⣄⠀⠐⠜⠿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠠⠚⠛⠓⠲⠂⠀⠘⠉⠀⢾⣿⡇⠀⠀⠀⠀⠃⠀⣭⢳⣿⣿⡌⣷⠀⣇⠀⠀⠀⠀⠸⣿⣿⡇⣿⣿⠀⠀⢠⣤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⣴⣿⣶⡷⠸⠿⠃⠀⠀⠀⠀⠱⣤⣿⡆⣭⣭⢸⡿⢶⣇⠀⡇⠀⠀⠀⠉⠙⠃⣿⣿⠀⠀⢸⣿⠄⠀⠀⠈⣯
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⣿⣟⣸⠇⠀⢀⡾⠀⠀⢰⠆⠀⣿⣶⡇⣿⣿⠠⣷⡾⢿⠚⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠛⢤⡠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣫⡇⠀⠀⠙⠉⠀⠀⠀⡈⠀⡇⠀⢸⣶⠀⣯⡻⠇⠉⠉⠁⢻⡕⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠐⠿⠀⠰⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⠀⠀⠀⠀⢀⢼⠟⢻⡇⠀⠘⠟⠀⢿⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡽⠧⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⢿⢦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⣸⠳⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⣀⣴⣄⣾⣶⡴⣿⣿⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⡀⢀⣸⣿⢹⣿⣿⣿⣾⡅⣿⣷⠿⢿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠓⠖⠒⠒⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⡟⣿⣿⣯⣿⠻⣿⣿⣷⣶⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣛⣷⣿⣿⣯⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢳⣿⣿⣿⡿⢷⣹⣿⣿⡙⢿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠥⠤⠾⣿⣿⣿⣟⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣼⣟⣿⣿⣿⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣃⣀⣉⣉⣉⣤⣤⣭⣭⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣏⣩⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠷⠶⠶⠶⠖⠒⠒⠒⢒⡒⣒⣒⠒⠒⢻⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⢷⣄⣿⠛⢻⣿⣷⡿⡿⠿
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⢉⣩⣯⣽⢿⠻⣿⣭⣭⣭⣭⣭⣥⣤⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠘⣿⣄⡿⠐⠋⠞
⠀⠀⠀⠀⠀⠀⠉⠉⣿⡶⠲⢶⡾⣡⣿⣦⠘⢿⣿⣶⠶⠶⠶⠦⠀⣾⡇⠀⢰⣿⡆⠀⣸⣿⡀⠀⣼⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣩⣿⡽⣣⣿⣿⣷⡈⢿⣯⣭⣍⣉⣉⣛⣿⡇⠀⢸⣿⣯⠀⣿⣿⡇⢸⣿⡗⠀⠀⢸⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢿⡟⣰⣿⡟⣉⠉⣷⡈⣿⣛⣲⣶⣒⣒⣿⡇⠀⢸⣿⣿⠀⣿⣿⡇⠘⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣴⣆⣰⣄⣿⣿⣾⣧⣿⡟⣿⣽⣷⣿⡇⣿⠤⠤⠤⠤⢤⣿⠇⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣧⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣸⣿⣿⣿⣯⠿⣿⣿⣿⣿⢹⣿⡇⣿⣿⡏⣿⠄⣿⢭⣭⠛⠋⢙⣿⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣯⣿⣿⣶⣾⣷⣿⣿⠧⣿⣾⣿⣇⣿⣿⢧⣿⠂⣿⣿⣿⣛⠛⢻⣿⠀⠀⠸⠿⠿⠀⠿⠿⠇⠀⠿⠟⠀⠀⢸⣇⠀⠀⠀⢸⡇
⠀⣉⣿⡿⠿⣬⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣷⣿⣒⡒⠶⠖⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣀⢀⣾⡟
⠐⠛⠻⠿⠿⠿⠿⣿⣿⣟⣛⡒⢒⡀⢠⣤⣴⣿⣿⣿⣶⣿⣿⣿⣿⣿⣾⣿⣿⣶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣾⣶⡶⠿⠶⠾
⣀⣀⣢⣤⡄⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠃⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⡀⠐⠒⠂⠒⠈⠉⠉⠉⠁
⠀⠀⠑⢀⠶⠊⠀⠀⠀⠀⠀⣤⣀⡁⡀⢀⠀⢤⣍⣀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣀⣨⠽⠵⠀⠀⠀⠀⠀⠲⠂⠠⠄⠤⠄⠀⠄⣀⡀⠀⠤⠰⠂⠉⠉⠉⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⠤⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠒⠒⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⣀⣀⣀⣀⣤⣄⣠⣤⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠿⠻⣿⠟⢻⠿⢿⠿⢻⢿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣷⣾⣶⣿⣶⣿⣷⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡄⢀⣀⣴⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠻⠟⠙⠁⠀⠀⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⢘⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣋⣤⣷⣄⣈⣉⣃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠙⣤⣠⠌⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠔⠀⠐⠧
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠈⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⡿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⡛⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠴⢆⣾⡿⣫⡌⢻⣦⡀⠀⠀⠀⠀⣸⣶⡙⠻⡇⠀⠀⢀⣴⣦⠀⢀⠀⠀⢰⣿⡀⢀⡀⢀⣴⣿⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠍⠀⣮⡀⣥⠙⢿⣶⡆⠀⠀⢸⣿⡇⠀⠅⢨⠀⢸⣿⣿⠀⠘⠓⠀⣿⣿⡇⠈⠉⠈⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠙⠃⠻⠁⠨⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠈⢩⡅⠀⠀⢀⣶⠀⡀⢠⣿⢺⣇
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢸⡜⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡃⠀⠀⠀⣿⣿⠀⠀⠀⢀⣾⡟⠀⠒⢸⣿⠀⠉⢸⣷⢸
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢱⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢸⠀⠀⠀⢸⣿⣿⠀⠀⢠⠀⢿⣿⡇⠀⠰⣆⢹⣿⠀⠀⠀⠈⢻⣏⠀⠀⢸⣿⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠁⠀⠀⠀⡆⠀⠸⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢘⠁⠈⠀⢸⣿⣿⠀⠀⠘⠓⢸⣿⡆⠀⠀⠀⠀⣿⠄⠀⠀⠀⢸⡏⠀⠀⢈⣿⠀⠀⡠⣿⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⡿⢿⡇⣸⠀⠀⠀⢀⣼⣿⡇⠈⠀⢘⠀⢸⣿⣿⡄⠀⠀⠀⠈⠏⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢠⡰⠆⣻⠀⡆⠁⢂⣸
⠀⠀⠀⠀⣄⠀⠀⠀⢀⡀⠐⠇⠀⠀⢹⠀⠀⠀⠀⠀⠉⠉⢰⠀⢠⠀⠉⠉⠁⠀⢀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠆⠂⢸⣿⢸⡇⠀⠹⠂⣿⡄⢈⠱⠂
⠀⢰⡆⡆⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢸⠀⠀⢘⢁⠰⠿⣦⡄⠀⠉⠙⠃
⠀⠀⠁⠀⣠⠞⣻⣄⠅⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠀⣿⠀⢘⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⣿⣿⠀⠀⠐⠚⠀⠐⠂
⠀⠀⢤⣴⠗⣫⠭⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣄⠀⠀⠀⠀⣿⠀⠠⠀⠀⠀⠀⠀⣸⣿⣷⠀⠂⠀⠀⠉⠛⠊
⠀⠀⠀⠀⠉⠓⠯⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠒⠿⠀⢠⡄⠀⠐⠀⠉⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⣯⣋⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣶⠞⠝⢶⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠅⠄⠀⠤⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠠⣏⢘⠃⢱⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠉⡈⠔⠃⠀⠀⠀⠙⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⢀⣿⣆⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣾⣏⣿⡀⠀⠉⢱⠀⣠⣤⣤⠠⣾⠿⢽⠿⣿⢿⣿⣷⠄⢠⣄⣀⠀⢰⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⢠⣿⣏⣿⣷⡀⠀⢸⠀⣿⣿⠉⠀⠀⢀⡔⢂⢐⡒⣄⠀⡇⢸⣿⡿⡄⢸⠀⣤⣤⠄⢠⣤⣤⠄⢸
⠀⠀⢰⡿⣟⣯⣿⣯⡁⠀⢸⠀⣿⠋⠰⠀⠀⣮⠛⠛⠛⠛⠋⠀⣧⢸⠟⠀⠇⢸⠀⣿⣃⠀⠘⣇⣋⡆⢸⠀⠀⢀⡆
⠀⠀⠀⣴⣿⣷⣿⢿⣷⡄⣸⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠂⢸⠀⡿⠀⠀⠠⡇⠀⠇⢸⠀⠀⣼⣿
⠀⢀⣾⣿⣿⣏⣿⣿⣿⡄⢿⠀⠋⣶⠐⣠⠖⣯⠀⠀⠀⠀⠀⢠⣿⡎⢠⠆⠀⢘⡀⠃⠀⠀⠀⠃⠀⠠⢼⠀⣸⣿⣿⣧
⣠⣿⣿⣿⣿⣧⣷⣿⣿⣿⣿⠀⠀⢼⠋⢠⠤⠯⠤⠤⠤⠤⠤⠴⣿⡿⠦⣄⣀⣸⣁⣉⣉⣙⣉⣿⣿⣿⣏⣰⣿⣿⣿⣿⣆
⠉⠉⠉⠉⠉⠱⢿⣿⣿⣿⣿⣶⣶⣸⣦⣭⣭⣭⣭⣭⣭⣭⣭⣭⡘⣧⣀⣻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠹⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡿⠿⠛⠉⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠃⠀⠓⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⡇⠀⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣀⠤⢤⡈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⡤⠚⠁⠀⠀⠘⠢⣄⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢉⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣠⡔⠋⠀⠀⢀⡴⣺⠿⢭⠿⢿⣦⡀⠀⠀⠙⣦⡈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⢹⡆⠀⠀⢠⣯⡟⠁⠀⢐⠀⠀⢹⣿⡀⠀⠠⡇⡿⠦⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⡇⠀⠀⠸⡿⡇⠉⠉⠙⠀⠉⣨⣿⠃⠀⢸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣩⢼⠀⡇⠀⠀⠀⠙⠮⣒⡶⠶⠶⣚⡿⠋⠀⠀⢸⢸⣯⡓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣉⠤⠒⠉⠁⢸⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢸⢸⠀⠈⠓⠦⣍⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⡥⠖⠋⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠀⠀⠀⣀⡀⠙⠳⢬⣙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⢺⣥⢴⡛⡇⠀⠀⣠⡶⠿⢷⣦⡀⢸⡄⣇⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⡿⣿⠀⣰⡿⠻⠛⢷⡄⠀⠈⢳⢦⣉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣧⠀⠐⣿⠀⠲⠆⣿⡇⢸⡇⣿⠀⠀⢠⢞⡿⠋⢹⠉⠻⣿⣦⠀⠀⠀⡇⣿⠀⢿⡇⢙⠃⣻⡗⠀⢀⡏⡟⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡀⠀⠙⠧⣴⣶⠿⠁⠸⡇⢸⠀⠀⡟⣸⠁⠀⢸⠀⠀⠘⣿⣧⠀⢠⡇⡏⠀⠈⠛⠾⠟⠋⠀⠀⣸⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢠⡇⣿⠤⠤⢼⠤⠤⠄⢿⣿⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢷⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⡇⡗⠀⠀⢸⠀⠀⠀⢸⣿⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡄⠀⠀⠀⠀⠀⠀⠀⣧⢸⡄⢸⡇⡧⠤⠤⢼⠤⠤⠤⢼⣿⠀⡿⢸⠇⠀⠀⠀⠀⠀⠀⢰⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⣴⠒⠒⡖⠛⠛⡆⣿⢸⡇⢸⡇⡇⠀⠀⠸⠀⠀⠀⢸⣿⠀⡇⢸⢰⠖⠒⠶⠒⠲⡆⣼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣹⡀⠀⠁⠀⠀⡇⢿⢸⡇⢸⡇⡇⠀⠀⢀⡀⠀⠀⢸⣿⢰⡇⢸⢸⠀⠀⠸⠀⢰⣷⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣏⠛⠛⠛⠛⠛⠁⢸⢸⡇⢸⡇⡏⠉⠉⠉⠉⠉⠉⢹⣿⢸⠃⢸⠈⠙⠛⠛⠛⠋⡟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡇⡇⠀⠀⠀⡀⠀⠀⢸⡏⢸⠀⢺⠀⠀⠀⠀⠀⢰⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

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

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠒⠠⡀⠀⠀⠀⠀⢀⡤⠒⠠⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⢀⣀⠄⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⡀⠠⠀⠈⢀⠀⠈⠆⠀⠠⠀⡞⠠⠀⠈⠀⠀⢉⡉⣠⣅⢠⡄⠀⠀⡔⢎⠀⢸⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠅⣾⠀⠀⠀⠀⠀⠀⠍⢠⡌⢡⡄⠀⠀⠀⠻⣬⠑⡯⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠅⠀⢠⠀⠀⠡⡈⠀⠘⠀⠄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢁⠉⠀⠀⠀⠈⠀⠀⠀⠒⠊⠀⠀⠀⠀⢀⠐⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠚⡆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⢀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠐⠀⠀⠀⠁⢸⣏⠀⠀⠀⡀⢀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠆⠐⠤⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⢰⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠀⠐⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⢀⡁⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠈⠃⠀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠠⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠱⣀⢀⣀⣤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⠂⠉⣩⠉⠑⠠⡀⠀⠀⠀⠰⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⢄⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⣐⣀⠔⠁⠀⠀⠠⢄⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠔⠉⡑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣃⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠑⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠃⠂⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠊⡀⠀⣀⠈⠳⣄⠀⠰⠌⡤⠂⡐⡄⠀⠀⠛⠀⢠⡷⢾⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠉⠁⠢⠀⡇⠈⠋⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠇⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠁
⠀⠀⠀⢠⡀⠢⡈⠀⠀⠰⠂⠀⠀⠀⠀⠐⢁⡄⠀⠑⡈⠀⡠⠁⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠈⠒⠴⣦⣿⣔⡾⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡷⠈⠀⡌⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⢀⣤⠖⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⣤
⠀⠀⠀⠀⢠⣿⡇⠐⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⢀⢠⡀⡄⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠙⠶⠞⠃
⠀⠀⠀⠀⣾⣿⠓⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣷
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀⠀⠀⠀⠀⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣸⢀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡄⢸⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⡀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣶⡿⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠋⠉⢷⣄⠀⠀⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⡟⠁⠀⠀⠀⠙⢧⡀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡇⠀⠀⠀⠀⠀⠈⠗⠲⠶⠶⠦⢽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣄⠀⢰⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠔⡩⢾⣿⣦⣸⠀⠀⠀⠀⢸⣯⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠊⠀⠀⠈⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠷⠶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢣⠞⠁⠀⣀⣄⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢁⡴⠃⠀⠀⢸⡅⠀⢱⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⢀⣴⣟⣋⣠⡏⠀⠀⠀⠀⠘⢧⣀⠞⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠊⠉⠉⢦⠀⠀⠀⠀⢸⠀⠀⢀⠤⣄⠀⠈⠉⠉⢉⡙⠛⠒⠒⠶⠤⢤⣤⣙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡟⠛⢻⠛⠻⡀⠀⠀⠀⡜⠀⢠⠃⠀⢸⠄⠀⠀⡰⠉⢹⡆⠀⠀⠀⢀⡾⠋⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠁⠀⢸⠀⠀⡇⠀⠀⠀⡇⠠⠞⠒⠄⠾⡄⠀⡤⢇⣀⣼⡇⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡀⠀⠘⡇⠀⢸⠀⠀⡇⠀⠀⢸⠁⠈⠛⠓⠒⠒⠁⠀⠓⠂⠤⠤⠜⠀⣰⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠘⠷⠶⢦⡷⠤⠾⠤⣤⣇⣀⣀⣼⣀⣠⣤⣤⣤⣤⡤⠤⠤⠤⠤⣤⣤⣤⠋⠉⠉⠛⠻⢷⣄
⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠫⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢺⣯⣀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⣶⣦⣾⣋⠁⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⣴⡾⠿⠟⠛⠛⠛⠛⠻⠿⠿⠷⠶⠶⠶⠶⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⠶⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⢀⡤⠒⠳⣄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠢⠤⠤⠴⠊⣀⠀⠉⠛⠉⠀⡄⠀⠈⠛⢋⠍⢩⣀
⠀⠀⠀⢠⡤⣀⠤⠜⠃⢠⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠙⢦⣀⢤⡄
⠀⠀⠀⢸⢠⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⢐⠃⠀⡀⠀⢳
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠌⠀⣀⠉⠀⢸⠀⠀⠈⡆
⠀⠀⠀⠸⠈⠀⣠⠴⠶⠶⠤⠄⣠⣤⡴⠋⠈⠙⠶⠛⠁⠈⠙⠶⠴⠖⠋⠁⠙⠓⠦⣄⠀⠀⢡
⠀⠀⣀⡆⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⣤⡀
⠀⡘⢩⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢷⠸⡆
⢀⠇⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢲
⡎⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡃
⠱⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠃
⠀⠈⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣖⠒⠤⠤⠤⠀⠲⠶⠦⠤⠤⠐⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠙⠶⠤⣤⠄⢀⣀⣀⣤⠴⠊⣹⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣀⠴⠒⠒⠒⠢⣄⠀⠀⣠⠎⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢐⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢰⠁⠀⠀⠀⠀⠀⠈⢆⡼⡷⢶⠒⠒⠒⠢⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣄⡀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣼⡄⠄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣸⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠸⡀⠀⣀⣀⠠⠤⠄⠒⢻⡇⢠⠁⠀⠀⠀⠀⣀⠤⢤⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠘⢧⣠⣄⣀⡠⠴⠚⠁⠀⣞⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠽⣍⠉⠉⠉⠉⠉⠛⠛⡛⠛⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⡎⠀⠀⠀⠀⠀⠀⠀⡇⠃⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠶⠶⠒⠒⠒⠒⠒⠶⣤⣀⡀⠀⠘⡳⠛⠛⠚⠒⠒⠀⠀⠒⠐⠓⠚⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠤⠤⠤⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠤⠤⠿⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣼⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣇
⠀⠀⠀⠀⠀⢸⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣄⢰⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠓⠦⡀
⠀⠀⠀⠀⠀⣇⣀⣀⣀⡠⠤⠤⠄⢀⣀⣀⣀⣿⡀⠀⢃⠀⠀⠀⠀⢀⣠⡤⠒⠊⠉⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⣏⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠹⡆⠹⡀⠀⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠙⠷⣢⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⢱⠊⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣔⡪⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠾⣦⣤⣄⣀⣤⣤⡤⣾⠀⠀⠈⠻⠶⠶⠶⠶⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠤⠤⡀⠀⠀⢀⡀⠀⠀⠀⠀⡰⠁⢷⡀⠌⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⢀⣠⣦⠵⠒⠀⢢⡀⢠⣄⣩⡽⠁⢳⡿⢿⣿⣾⠇⡄⣁
⠀⠀⠀⠀⢀⠠⠀⠨⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⢣⢬⣬⣽⠃⣾⡟⠀⠀⣼⣿⣾⣴⣈⣩⣶⣤⣀⠀⡀
⠀⠀⡤⣊⠐⣖⡂⠙⠂⠀⠀⡖⠨⠆⠀⠀⠀⠀⠀⠈⢯⣽⢇⣶⢻⣿⠶⠙⠫⠟⢿⡟⠙⠻⣿⠟⠻⠇
⠀⠀⢧⠉⡘⡎⢉⠀⠀⠀⠀⡀⠀⠂⠔⠈⠀⠐⠀⠀⠘⡧⠿⢿⡞⠁⣀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⡶
⠀⠀⠈⢆⠁⣽⡌⠀⠀⢀⠠⡀⠀⠄⠓⠐⠂⠔⠁⠈⠀⠹⠀⢰⡆⠀⠀⠁⠀⢤⢄⠀⣿⣶⣄⡀⡼⢁⡀
⠀⠀⠀⠈⠆⢟⣷⠠⠀⠀⠀⣈⠁⠀⠀⠀⠀⠄⢉⡀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⡀⠺⣿⣿⣿⣿⠿⣿⣆⡀
⠀⠀⠀⠀⠘⣤⢿⣧⠀⠢⠄⠤⠀⠀⠀⢀⠀⠀⠠⠄⠁⠂⠈⢦⣀⠀⠂⠄⢠⣤⣾⡅⣀⡞⣽⠿⢿⣷⣿⣿⠷
⠀⠀⠀⠀⠀⠘⡜⣿⣆⠀⠀⢐⢃⡀⠀⠀⠀⡀⠓⢄⠠⠀⠀⠘⣟⡳⣦⣤⣸⣟⣹⣿⠏⣼⠳⣶⣾⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠘⡌⣿⡄⠀⠀⠂⠀⠉⠈⡁⠀⠀⠄⠀⠀⠀⢀⡹⣠⠏⠛⠻⠻⢻⡿⣼⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡘⣷⡀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣠⠀⠀⢈⣀⠡⠈⠉⠉⢈⡁⣹⣏⡍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣧⠀⠀⠀⠀⠤⠰⠀⠀⠈⠀⠀⠰⢀⣩⣄⣠⠄⣤⡈⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢧⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠛⡯⠙⡛⠹⢗⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⠢⠀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠈⠨⠋⠔⠉

⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣾⣿⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣿⣧⠐⢠⠀⠀⠀⠊⠉⠉⠀⡀
⠀⠀⠹⣿⣿⣿⡷⠀⠀⠄⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⢶⡾⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣿⣿⠂⠀⢀⡔⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠙⠦⠀⠀⠀⠼⣿⡆⣠⣿⢉⢓⣀⠀⠀⠀⠀⠀⢠⣴⡦⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠒⠋⠀⠀⠀⠀⣸⣧⢋⢇⣼⣿⠇⠀⡀⠠⠤⠶⠛⠉⠀⠀⠀⠀⠁⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡿⣿⣿⣿⣾⡏⠁⠀⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⢻⡿⣿⣿⠃⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⣇⣹⣾⣿⣿⡀⠀⠀⠉⣀⠀⠀⠀⡀⠀⠀⠀⢀⡴⢒⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⢘⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⣶⣄⣀⣠⣀⡀⠀⠀⠀⠑⠾⠋
⠀⠀⠀⡇⠂⡒⢤⣄⣀⣠⡀⠀⠀⠈⠙⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁
⠀⠀⠀⢿⣤⣮⣿⣿⣿⣿⡷⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣠⠂
⠀⠀⠀⢸⡉⠉⢉⠻⠋⠁⠀⢀⣤⢤⣖⣋⠉⠀⠀⠀⠀⣀⢐⠀⢠⣾⡿⠄⢠⣾⣏⣀⣀⢀⢲⢩⣵⡗
⠀⠀⠘⢾⣷⣄⠘⢿⣷⣤⣾⣿⠿⠿⣿⠋⠀⠀⠀⣀⡐⠉⠸⢀⣄⣿⣿⣗⡟⠘⣿⠁⣹⡿⠛⢷⣆
⠀⠀⠀⣺⣷⣿⣿⡺⢿⣿⣿⠋⢾⣿⣿⣦⣴⡂⠀⣿⣛⠛⠂⠀⣿⣟⣏⡙⣿⣽⣿⣿⣴⣧⣤⢶⡌
⠀⠀⢀⣡⡼⣿⣿⣿⣾⣿⡿⢿⣾⣿⣟⣿⠋⠀⠀⠻⢿⣷⣌⣾⣿⣿⣿⣿⣋⣙⣿⡮⣤⣯⡶⠻⠁
⠀⠀⠀⠀⠀⠀⠈⠛⣿⠷⠟⠛⠓⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠀⠉⠁

⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠿⠿⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣯⣽⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⣿⣷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⡾⠿⠿⠶⡿⠷⣶⡿⠷⢾⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠐⠒⠂⠀⠰⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠦⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠇⠠⢸⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡗⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢸⠀⠠⠦⠴⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡇⡀⠠⠞⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⠀⣀⠀⠀⠀⢸⡇⠀⠀⣿⠡⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡇⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⣿⢏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⢀⣀⣸⣇⠀⠀⠹⠴

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠀⠀⢄⠀⠀⡇⠀⡇⠀⡇⠀⢰⠀⠀⡞⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⣆⠈⢆⠀⠸⡀⠀⡇⠀⡇⢀⡇⠀⡇⠀⡼⠀⢠⠂⠀⡠⠋
⠀⠀⠀⠀⠠⡀⠀⠀⣄⠈⢣⡀⠈⢆⠘⡆⠀⢇⠀⢧⠀⡇⢸⠀⢸⠀⣰⠁⣰⠃⢀⡜⠁⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠈⠣⡀⠑⣄⠈⢆⠘⡄⠸⡀⢸⠀⡇⣸⠀⡇⢠⠃⡰⠃⣠⠊⢀⠔⠁⢀⡴⠊
⠀⢀⡀⠀⠀⠤⡀⠈⠳⢄⠈⢢⡈⢢⡈⢧⠹⡀⢇⢸⠀⡇⡇⢸⢀⠏⡰⠁⡴⢁⡴⠋⣀⠔⠉⢀⡤⠂
⠀⠀⠉⠒⠤⣀⠈⠑⢤⡀⠑⢤⡙⢦⠑⣌⢧⠱⠘⠈⠀⠁⠃⠃⠞⡰⢡⠎⡠⠋⡠⠞⢁⡤⠚⠁⣀⠤⠒⠉
⠀⠀⠠⠤⣀⡀⠉⠒⠤⣈⠓⢤⡙⠦⡑⠈⣠⠔⠚⠉⠉⠉⠉⠑⠲⢄⡁⠊⡴⢊⡤⠚⣁⡤⠖⠋⢁⣀⡤⠄
⠤⢤⣀⣀⠀⠉⠙⠒⠤⢄⣙⠲⢬⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⡡⠖⣋⡡⠤⠒⠊⢉⣀⣀⠤⠤
⠀⢀⣀⣀⣉⠉⠑⠒⠒⠤⠬⢍⠃⢰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢐⡩⠥⠴⠒⠚⠉⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠉⢉⣉⣑⣒⣲⣤⣿⣤⣤⣤⣿⣿⣿⣾⣶⣷⣴⣶⣤⣶⣦⣼⣿⣒⣒⡚⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⢒⠺⠓⠢⠒⠘⠏⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⣀⡀⠀⠀⡰⣴⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⡤⠤⠀⠤⡴⠀⠀⢠⠂⠁⠀⢀⠔⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡛⠚⠒⠐⠢⠐⠚⠁⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠠⠇⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠀⠀⠷⠀⠀⠀⢀⣠⠖⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠂⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢳⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠠⡴⠿⠁⠙⠉⠁⠀⠀⠀⠩⠆⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠙⠉⠃
⠀⠀⠀⠀⠀⠀⢐⠀⢓⠀⠀⠀⠀⠀⠘⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠜⡁⡐⠲⠂⠀⠀⠀⠈⠇⢀⣄⠀⠀⠀⠀⠀⠀⢰⡶⣶
⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⡀⣈⠙⠀⠀⢐⣖⣶⡞⠆⠀⠀⠀⠀⢤⠚⣁⡚⣠⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠌⣄⡀⠀⠈⠉⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠯⣣⠅⠀⠀⠀⠀⠀⠀⠀⣩⠃
⠀⠀⠀⠀⠀⠈⠛⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠄⢺⠠⠄⠀⠀⠀⠁⢀⣦⢤⣤⡤⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣠⠰⠿⠒⠛⠋⡉⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢤⡘⡛⡀⠜⠁
⠀⠀⠀⠀⠤⠴⠦⣐⡀⠁⣠⠦⠤⢀⣀⡸⠂⠘⠀⠙⠤⠖⠀⠀⠀⠛⠋⠙⠁⠀⠀⠀⢀⡀⠀⣴⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⡀⠀⠀⠆⣄⠈⠀⣠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⡀⠀⠀⠀⠀⠙⣃⠅⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠑⡐⠋⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠄⠀⠀⠀⠀⣤⢘⠃⠀⡀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⠀⠀⠀⠀⠠⣶⢈⠀⣠⢫⠄⠀⠀⠀⠀⠒⠤⢄⣀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⡌⠰⠀⠀⠀⠀⠀⠲⢿⡆⣀⡌⠀⠀⠀⠀⠀⠀⠐⠘⠛⡀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⣁⢴⡂⠀⠀⠀⠀⠀⢭⣿⠋⠁⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⡀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠄⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣘⠉⠀⣚⠀⠛⣄⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠰⡆⢁⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠜⠋⠁⣠⣄⣧⠤⠀⢄⠁⠈⣉⢐⣂⡞⠰⣴⣾⣷⣶⣶⡀⡉⠂
⠀⠀⠀⠀⠀⠀⡀⠀⡂⠁⣀⡴⠶⠛⠉⠁⠀⠀⠀⠀⢇⢘⣉⣻⡟⢠⣶⠏⠀⠉⣷⣧⣞⣨⢛⢂⣤⣀⡀
⠀⠀⢀⠤⠂⣥⠤⠐⡎⠉⠀⢀⠤⢰⡀⠀⠀⠀⠀⠀⠈⣞⢻⡟⢀⡼⣿⣦⡄⠾⢿⡿⣽⡿⠻⣿⣿⣿⣿⡦
⠀⠀⢇⠛⠂⣇⠺⠂⠀⠀⠀⠈⠁⠀⠀⡠⠐⠀⢀⠀⠀⠘⡟⣥⣿⣥⠟⠁⠀⠀⠁⠀⠑⠁⠀⠈⠛⠀⠀⢡
⠀⠀⠘⡆⠳⢸⡄⠆⠀⠀⠀⠐⠀⢀⠀⣅⢀⡀⣁⠆⠠⠄⢻⡀⠈⡍⠀⠙⠳⠀⠀⠀⠀⠐⣤⣀⠀⠀⢠⠯⠀⢀
⠀⠀⠀⠘⡄⣻⣷⡀⠀⠀⠂⠉⢠⠊⠁⠀⠀⠀⠑⠤⠀⠀⠄⢆⠀⠱⠀⢀⠀⠀⠀⠙⠂⠀⣿⣿⣷⣦⣯⣴⣷⣄
⠀⠀⠀⠀⠸⡜⢺⣧⠁⢀⠀⠀⡂⢀⠀⠀⠀⠀⠁⠐⠂⠈⠀⠈⣄⠀⠀⢈⠀⠀⠀⠀⣠⠂⠈⣻⢿⣿⣿⣏⣻⣿⣶⡠
⠀⠀⠀⠀⠀⠹⡝⣿⣗⢀⠉⠈⢣⠀⠀⠀⠠⠀⠀⣈⠁⠀⠀⠀⡘⣷⣄⡀⠈⠀⢶⣿⣿⣴⡴⢣⣟⣉⣹⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠱⡘⣿⡄⠀⡀⢂⡰⠂⡄⢤⠄⠂⠀⠱⠈⠀⠀⠀⢹⡦⠹⣿⣶⣿⣦⣾⡿⢡⣿⣨⣹⣿⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠱⡸⣷⡀⠀⠁⠄⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⡆⢣⠘⡄⢄⠉⠩⠿⢧⣿⡻⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⢴⠌⠀⠀⠀⠀⢴⢀⣞⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⣇⠀⠀⠀⠀⠐⠘⠀⠈⠀⠀⠀⢐⣃⣤⣶⣤⡖⠀⣶⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⢝⠀⣩⠁⠛⡻⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠢⠄⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⠍⠀⠉

⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣴⠇⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣻⣏⣾⣧⣀⠉⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢆⠛⡟⢯⡛⡛⢏⠂⠀⠀⠀⠀⠀⠀⠀⠘⠙⣾⣷⡋⣿⠀⠀⠁⠀⠠⡰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⣶⣨⣆⡓⠄⠸⠀⠀⢠⠀⠀⠀⠀⢀⢰⣶⣏⣿⡷⣶⡤⣄⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⣼⣿⣿⡿⢩⡇⠀⠀⠀⠀⠘⠆⠀⠀⠀⢸⣆⢻⣿⡿⠁⣾⣧⡹⢧⡀⠀⠠⠀⠠⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⡇⢙⡿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⢿⣿⣿⣶⣿⠙⣿⣿⡷⠀⠀⠀⠀⠈⢹⢷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢨⣿⣿⣿⣷⣤⣴⡼⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢲⣶⣤⣿⣿⣏⠀⠈⢻⣷⡀⠀⠀⠀⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡀⠈⠀⣿⣿⣿⣿⡟⣅⣾⣿⡇⠀⠀⢀⣀⣄⠀⠈⡘⡆⠀⡸⠞⢿⣿⣽⣿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠚⠻⣷
⠀⠀⠀⠀⢠⣀⡄⠀⠠⠀⠀⠉⠀⠀⠠⠟⠉⠀⠡⢂⣽⣿⣿⡇⠀⠀⢸⣉⠉⠄⠀⠸⡀⢀⠃⡘⠚⠉⢩⢿⡟⣷⠀⠀⠘⢿⠄⠀⠀⠀⣠⡍
⠀⠀⠀⠀⠈⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣿⣿⠉⣿⣷⠀⠀⠀⠇⠀⣶⡆⠀⣅⢨⠀⣵⠈⢰⢘⡭⡇⡟⠀⠀⡉⠈⢻⠂⠀⠀⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠞⢩⡿⠿⡿⡟⠀⠹⣷⣄⠀⠀⡀⠀⠀⢠⡀⠹⡇⠀⠹⠼⣒⡉⣈⠇⡇⠀⠀⣉⣹⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣴⡀⠱⣄⠱⣾⣦⠀⠘⠻⣴⣄⣁⠈⠀⣾⠳⠀⣿⡄⡀⢸⠖⢹⠉⢸⠁⠀⠀⢹⣿⢿⠒⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⠈⠳⠿⣿⡇⠐⠂⠘⢿⣿⣿⠒⠻⡿⣶⢿⣿⣴⣾⠀⠘⠴⠇⠀⢸⡀⠉⠇⢈⠀⠉⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢻⣷⣿⣿⠷⣄⡀⠈⣿⡀⠀⠀⠀⣷⡄⡀⠀⢱⡜⣾⠛⠛⠃⠀⠀⠀⢸⣦⡼⠏⢠⠀⢈⡀⢠⣷⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢘⠁⢹⡟⠀⠈⠙⣦⢨⣷⠀⠀⠀⢹⡻⣷⡄⠘⣿⣏⢰⡀⠀⠀⠀⠀⠈⢿⣿⣦⣶⡤⠾⠀⠈⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠊⠀⡞⠀⠀⠀⠀⣘⡃⠏⠀⠀⠀⠀⣿⣻⣷⠀⠉⢉⡜⣿⡌⠁⠀⠈⠀⠈⣿⣿⠁⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣡⡤⠖⠚⠉⠀⠀⠀⠀⠀⢸⠃⣷⠈⠐⠛⠚⠋⠻⢁⣀⡴⢿⡿⠃⣯⣦⣬⠉⣷⣄⠸⣽⣿⣇⣀⡀⣿⣧⣶⣷⡆
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠶⠿⠿⠛⠻⠿⠏⠀⣿⣿⣿⣷⣧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠙⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀

⣠⣤⣤⣤⣤⣀⣀⣄⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣄⣠⣤⣤⣀⣀⣄⣀⣀⣀⣀⣤⣤⣤⣀⣠⣤⣤⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣠⣀⣀⣀
⣿⣿⣏⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣯⣭⣽⣗⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣀⣤⣤⣤⣤⣄⣄⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣩⡿⠟⠛⠛⠛⠛⢿⣯⡛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⢰⢁⡏⠀⠂⠀⠀⠀⠀⠀⣠⠔⠀⠀⠙⡆⢡
⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⠰⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠒⣦⡎⠁⣴⣲⠀⠀⣽⠘⣯⡇
⠀⠀⠀⠀⠀⠀⣼⣿⡦⢟⡿⣿⣿⣾⣷⣶⣿⣯⢟⡻⣼⡿⡇⠀⠀⠀⠀⠀⢠⢸⡄⠀⠀⠀⢀⠀⠀⢀⡀⠈⠀⠀⠀⡾⢸⠛⠃
⠀⠀⠀⠀⠀⢠⡗⠋⣪⠗⠋⠁⠀⠀⠀⠀⠀⠉⠑⠪⣗⠝⢿⡀⠀⠀⠀⠀⠈⢎⢷⡄⠀⠀⠈⠁⠈⠉⠉⠀⠀⢀⣾⢇⣿⠇
⠀⠀⠀⠀⣠⠋⣠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠳⣖⣦⠀⠀⠀⠈⢯⣿⣦⣄⡀⠀⠀⠀⠀⣠⣴⣿⣷⠋⠈
⠀⠀⠀⢠⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠹⡄⢹⡿⠂⠀⠀⠀⠘⣿⢻⠿⢿⣿⣿⣿⡿⠿⠟⢻⡏
⠀⠀⠀⡞⢀⡏⠀⠀⠘⠲⠤⣀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣷⣤⡄⠀⠀⠀⠻⠼⠀⠀⠀⠀⠀⠀⠀⢠⠼⠁
⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⣀⣀⣉⠻⣷⡄⠀⠀⣏⣹⠀⠀⠀⢸⠀⣿⣷⡇
⠀⠀⠀⢧⢸⡇⠀⠀⠀⣾⣿⣿⣿⣧⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿⠛⠃⠀⠀⠀⣴⢲⠀⠀⠀⠀⠀⠀⠀⠐⢲⡀
⠀⠀⠀⠘⡌⣿⡄⠀⠀⠻⣿⣿⣿⡟⠀⠀⠁⠀⠀⠀⠐⠀⣰⡿⣰⢷⠄⠀⠀⠀⢠⣿⡼⣶⣾⣷⣿⢿⣿⣶⡦⣾⣇
⠀⠀⠀⠀⠙⣜⢿⣄⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡱⠿⠏⠀⠀⠀⢀⡞⢡⠞⠉⠀⠀⠀⠀⠀⠉⠛⢆⠹⡄
⠀⠀⠀⠀⠀⠘⡦⣟⢿⣦⣄⡀⠀⠀⠀⠀⠀⣀⣠⣴⡿⣫⣾⠁⠀⠀⠀⠀⢀⠎⡴⠁⠀⠄⠂⢁⠀⠀⠈⠂⠀⠀⠳⡘⡆
⠀⠀⠀⠀⠀⠀⢻⣿⠗⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣷⡇⠀⠀⠀⠀⠀⠸⢰⠁⠀⠈⠀⠀⠈⡄⠀⠀⠀⠈⠀⠀⢣⢸⣤⡄
⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⣿⣿⣯⣭⣭⣭⣽⣿⣿⣿⠰⡿⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠻⠃⠐⠐⠀⠂⠀⢸⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠊⠁⠀⠀⠀⠀⠀⠀⠸⡸⣆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡟⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡻⣦⡀⠉⠂⠀⠀⠀⠐⠀⠀⣠⡾⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⣤⣄⣀⣀⣤⣴⣾⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣽⣟⣛⣻⡯⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢨⠉⠙⠛⠛⠛⠛⠉⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⣿⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣶⠶⠿⠿⠿⠶⣾⣿⣿⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾⠛⣁⣤⣴⣶⣶⣶⣦⣤⡉⠙⢶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣶⣿⣿⣿⣤⣽⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣛⣛⣛⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⠍⠉⠉⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣭⣽⣯⣬⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣯⣿⣿⣭⣿⣯⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡛⢝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠊⣏⣸⢀⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠉⣹⡍⢹⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⠭⠤⠼⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⡐⢀⡠⠾⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣴⣿⣷⣦⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢫⠟⠱⢽⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⠔⠀⠀⠀⠄⠂⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢟⠅⠉⠁⠀⠀⠀⠀⠀⠊⠁⡝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢟⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢋⠅⠁⣂⣀⠀⠀⢀⢀⣠⣄⡀⠐⠀⠈⠀⠀⣨⣙⢝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⠕⠑⠂⠀⠀⠀⠀⠀⣠⣶⡫⠟⢧⣄⠀⠀⠀⠀⠀⠀⠀⠐⢙⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠕⠋⠀⠀⠀⠀⠀⠰⣤⡾⠫⠊⠀⠀⠑⢝⢷⣄⠀⠀⠀⠒⠀⠀⠀⠙⢿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣷⠁⠀⠀⠈⡀⠀⠉⣩⡾⡡⠊⠀⠀⠀⠀⠀⠂⠑⢝⢷⣅⠀⠀⠀⠀⠀⠀⠀⠗⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠄⠀⠀⣴⣶⡤⠘⠛⢺⡧⠭⠥⠍⠁⠀⠀⠀⠀⢽⡿⠛⠁⠀⣢⣖⠀⠐⠂⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢹⣽⣟⠀⠀⢸⣟⠀⢠⡤⠤⠬⠥⣄⠀⣿⡇⠀⠀⢘⣿⣿⣃⠀⠁⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠠⠴⢺⣿⡷⠀⠀⢹⡷⠀⢸⡅⢠⡄⡂⡇⠀⣾⡇⠀⠀⠰⣿⣿⠂⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⠂⣀⣸⣿⣟⠀⠀⢸⣗⠀⠘⡄⢸⡇⡇⡇⢀⣻⡇⠀⠀⢺⣿⣿⡂⠐⠆⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠄⠀⠠⣾⣿⣯⠀⢀⢸⡯⠀⠠⠀⠸⠇⡈⡧⠄⣽⡇⠀⠠⢴⣿⣿⡇⠀⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡂⠀⠀⠛⠛⠛⠂⠀⢸⡷⠀⢰⡇⠐⠐⠆⡇⠀⣿⡇⠀⠀⠘⠉⠛⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣁⡀⠈⡉⠉⣉⣁⣹⣹⡶⠈⠉⢉⠩⠿⢽⣿⣏⣏⣀⣁⣉⣀⣉⣉⣀⣁⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢰⣿⣿⠀⡀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢠⡀⣠⠀⣿⡄⠀⠀⣠⣼⣿⣿⣰⣿⡄⣿⣿⡇⠀⠀⠀⠀⢰⡆
⢠⣄⡀⠀⠀⡷⣇⣄⠀⣀⢀⡄⠀⠀⣀⢠⣄⣼⣿⣿⣧⣼⢸⣿⣿⣿⣿⡇⢼⣷⣿⣿⣿⣿⣿⣿⣧⣿⣿⣧⢸⣷⠀⠀⢸⡇⢀⡀
⠈⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣧⣿⣷⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⡷⠗
⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⠿⢃
⠀⠀⠀⠀⠀⠻⠿⠛⠛⠁⠀⢸⣿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠌⠁⠀⠀⠘⠻⠿⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠁⠀⠀⠀⠈⢹⡿⡟⠻⢿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠀⠀⠉⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠒⠉⠀⠀⠀⢦⠾⠃⠀⠉⠁⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣴⡀⠀⣀⣀⢀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠻⠋⡏⡏⡏⠍⠋⢀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣤⣤⣤⣤⣤⠤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣂⣀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣟⣚⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠻⠛⠛⠛⠿⠶⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠁⠀⠀⠀⠀⢀⠰⠂⠀⠐⠄⠀⠰⠖⠄⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⣏⣸⡃⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠇⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⡀⠀⠂⣶⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⡄⠠⠀⡠⠀⠀⠀⢡
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠀⠀⠀⠛⠑⠊⠀⠀⠀⠄⢀⡤⢸⢁⠠⡀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠀⣥
⠀⠀⠀⠀⢸⣿⠀⡠⢄⠀⠄⠺⢸⠀⠀⠀⠀⠠⠖⢡⠆⣤⣞⣻⣆⠀⣱⣤⡀⠀⠀⠀⠀⠈⢤⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⣿⠀⠁⢀⠀⠀⠀⠀⠐⠒⠚⠛⠙⠀⠂⢀⠀⠈⠉⠀⠸⠷⡽⡧⡀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⣿
⠀⠀⠀⠀⠀⣿⡇⢰⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠁⠛⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠃⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠹⣿⡄⠆⠀⠠⠀⠀⣼⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠌⣯⠀⡀⠠⠘⠂⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢻⣷⡀⢂⠀⠤⠀⠈⠀⠀⠀⠤⠄⠀⠀⠀⠰⡆⠀⠀⠀⠀⠤⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠐⡀⠀⠀⠀⠀⠀⢀⡎⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⡈⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠙⠢⣄⡀⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⠀⡀⠤⠚⠁⣠⡴⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠈⠉⠒⠒⠂⠘⠧⠒⠐⠒⠈⠁⣀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠴⠴⠤⠤⠴⠶⠶⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠛⣋⣉⣡⣦⣴⠍⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⡇⠈⠣⡈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣯⣯⣽⣿⣿⣾⣷⠄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣷⣌⠳⣄
⠀⠀⠀⠀⠀⠀⣰⡿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣷⡌⠳⣄
⠀⠀⠀⠀⢀⣼⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣦⡙⠦⡀
⠀⠀⠀⣠⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢦⡀
⠀⠀⣼⡯⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠻⣆⠀⠀⣀⣀⠄
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⠛⠋⠉⠉⠀⠀⠀⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⡄⣻⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣿⠂⠀⣀⣠⣾⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣶⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⢹⣿⣿⣿⣿⣿⣧⠀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⡿⠿⠿⠿⠿⠷⠤⠤
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇⣾⠁
⠀⠀⠀⠈⠉⠛⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⠃
⠀⠀⠀⠀⣀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠆⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠙⠛⠛⠶⠶⢶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠷⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣥⣀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣘⣿⣁⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣾⡟⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⡛⣷⠶⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠹⣇⠿⢤⣸⠓⠒⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣤⡀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⣿⣿⡧⠀⠀⠈⢳⣴⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⠲⠷⠶⢤⣤⡔⣂⣀⣀⡐⠲⣶⣾⣦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⢸⠀⠀⢰⣿⠀⠯⢭⣭⡭⠭⢽⡿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠿⢿⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡟⡇⠀⢸⣿⠀⠘⡈⠳⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡯⡇⠀⢸⣿⠀⠀⣏⡙⠛
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⢤⣇⣶⣾⠀⠀⠀⠀⢸⠀⠀⠘⠓⠀⠀⢸⣿⠇⠀⢸⣻⠂⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣴⡏⣇⣾⢻⣿⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠸⣼⣟⡁⣿⡞⠁⢰⠀⠀⠀⠀⠘⠒⠒⠂⠠⠤⠤⠤⠤⠤⠀⣀⡀⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⠉⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠀⣀⠀⠀⠀⠀⠁⠀⣠⡤⢤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠸⡀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡄⡜⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠘⠈⠒⠠⠄⠠⢄⡤⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠰⡀
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠠⠉⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠀⠀⣠⠔⡋⠗
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⡁⠀⠈⠠⢬⣷⣄⡀⠀⣠⣤⣀⣀⣀⣀⣀⣀⣸⣇⣀⣠⡴⡋⠐⠀⠀⣅⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠐⢄⡀⠀⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠀⢀⡠⠖⠍⠋
⠀⠀⢰⡇⠈⡄⠀⠀⠀⡇⠀⣦⠀⠀⠀⢀⡴⠇⠀⠑⢠⠟⠌⢱⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⢧⡤⠤⠶⡏⠊⠀⠀⣦
⠀⠀⠸⣧⠀⢳⡀⠀⠀⠃⠀⠄⠉⠒⠒⠿⢁⡀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⣀⠔⠋
⠀⠀⠀⢿⣷⣄⡑⠤⣠⠎⠀⠠⠤⠀⣀⣀⣀⣈⡉⠤⠒⠤⣀⣐⣒⡲⠊⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠋⠁
⠀⠄⠀⠀⠙⢻⢿⠿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄
⠀⠀⠀⢀⡼⠃⢈⢆⡈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⢀⡾⠁⢠⠃⠀⠈⠁⠈⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⢀⡾⠁⣠⢿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⡀⠀⠀⠀⠀⠀⠈⢢⡀
⢀⣾⠁⡴⣵⠋⠉⠒⠂⠤⠀⠄⠀⠀⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⡱⢤⣴⣾⠼⠁⠀⡵
⣼⠇⢸⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠚⠊⠉⠁⣀⠠⠐⠉
⠻⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⣀⡠⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⣀⡀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣐⡀⢀⣿⣯⣽⡆⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢻⣿⣿⣿⣿⣿⠇⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣸⡇⡃⠘⣿⣗⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣥⣥⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠇⠤⠭⠿⠧⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⡀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⢀⠔⢈⢠⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡆⠔⠉⠀⠀⠙⣧⠀⠀⠘⠛⠀⠀⠀⠎⠀⠀⠀⠈⠿⣷⣄⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠋⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣢⡤⠀⠻⣿⣿⡌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠈⠁⠀⠀⠀⢻⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⢤⡔⠋⠀⠀⠀⠀⢀⡀⣀⡀⢸⣿⣿⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠻⡾⠃⠀⠀⠀⠀⠀⠈⠉⠉⠁⢸⣿⢼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣧⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣼⣿⣟⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⣟⣆⠀⠾⡫⠋⠀⠀⠀⠀⠂⢰⡇⠀⠠⠀⠀⠀⠙⠻⠿⠆⣰⡿⡿⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣟⢟⣈⠁⠀⠀⣰⡗⠀⢠⢸⡦⠀⠐⠂⢄⠀⠀⠀⢀⣴⣟⡕⠈⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⢿⣷⣿⣷⣄⠘⠃⠀⠀⠀⠀⠀⠀⠠⠂⠈⠁⣠⣼⣿⣫⠞⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⣿⣿⣯⣳⣖⠂⠀⠀⢀⣉⣀⣠⣤⣶⠿⣛⣥⠞⢁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣶⣦⣬⣭⣭⣽⣷⣶⣿⣫⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠛⠻⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⢠⣖⠂⠀⠀⣀⠀⢀⡄⠐⢦⡀⢠⠂⡄⢰⠘⡆⢰⠀⡔⠐⢀⡀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣰⡇⣿⢸⣇⣀⠸⣄⠘⠚⠇⠛⠈⠁⠣⠐⠅⠀⠘⠀⠇⢠⡏⢠⡖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡼⠃⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠻⠬⠖


⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⢀⣤⢠⣿⡄⡀⢠⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⢀⣠⣿⣷
⠀⠀⢸⣿⡇⢠⣤⢸⣿⣸⣿⣿⣷⣿⣧⣶⠀⠀⠀⣸⣿⣿⣷⣀⣠⣸⣿⡇⢰⣿⣿⣿⣿⢀⣤⣄⠀⠀⠀⢠
⠀⣀⣸⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣼⣿⣿⠀⠀⠀⢸
⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣿⣦⣤⣤⣤⣼
⠈⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠛⠛⢿⣿⣿⣿⣿⡿⠛⠋⠋⠻⣿⣿⣿⠟⠋⠉⠋⠁⠀⠀⠀⠻⠿⠟⠉⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠈⠻⠟⠋⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠟⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠳⠃⠺⢺⠘⠐⠟⠃⠚⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠠⠀⢀⡀⠀⡀
⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠠⠋⠀⠀⢠⡀⠀⠉⠀⠀⠀⠀⢀⡄⠀⠀⠉⠀⠊⠀⠀⠀⠀⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⣗⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⠻⡉⠀⣸
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡄⠀⠄⠀⠀⠀⠀⠀⠁⠂⠀⠀⢶⠃⠀⡇⠀⢀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠐⢀⢐⠀⠀⠀⣴⡦⠀⠀⠸⠂⠀⠇⠀⠈⠀⠀⠀⢈⣁⠄⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡸⢬⠂⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠾⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⢫⣧⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⣠⡇⠐⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⣴⣾⠋⠀⠰⠷⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣋⣭⣵⣾⣯⡐⠷⠀⠀⡀⠀⣿⣿⣿⣿⣦⡀⠀⠘⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠛⠉⣿⣦⣄⠈⠁⠀⠀⠉⠛⠿⣿⣿⣶⣶⢟⣿⠇⠀⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠅⠈⠀⠃⡸⠋⡣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣏⣀⣠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣫⣿⠃⢀⠀⠐⠜⠂⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣽⣿⡿⢛⡛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠠⠀⠤⠜⠀⢀⡠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢹⡿⣷⣌⢻⣿⣿⣿⠿⠿⢿⣦⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⡀⠀⠀⠐⠀⠀⠠⠄⠅⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣾⣿⡟⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠵⡀⢄⣴⣶⣿⣿⠥⢀⠀⠄⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⣿⡿⣏⣿⡇⠀⠀⣀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣴⢬⣿⣿⣝⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⢰⣿⣿⠀⠻⣾⣿⣦⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠋⠈⢿⣝⡻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⣿⣟⢇⠀⠀⠉⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠉⠛⣿⣷⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠿⠟⢻⣮⡓⠦⡤⣶⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⣴⡿⠟⠿⣶⡀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣾⣿⠃⣴⢿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠹⠦⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⢠⣾⡿⠙⣷⠀⣀⣤⣤⡴⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⣴⣿⠟⠀⢀⣿⣿⣿⠟⠁⢀⣤⣾⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⡲⠶⢶⣶⣶⠿⠟⠋⠀⢀⣠⣾⡿⠋⠀⣠⣾⣿⠟⠁⣠⣶⣿⣿⡿⠛⠁⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠳⣝⡶⣤⣤⣤⣤⣤⣴⣶⡿⠟⠋⣠⣴⣿⠿⠋⢁⣴⣾⣿⣿⠟⠋⠀⢀⣴⠶⠶⢤⡈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣛⠶⣯⣭⣍⣉⣩⣥⣤⡶⠿⠛⠉⢁⣤⣾⣿⡿⠟⠉⠀⠀⠀⠀⣾⡇⢰⣆⣼⡇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣍⣉⣉⣁⣀⣤⣤⣶⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣉⣉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

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

⠀⠀⠀⠀⠀⣇⡴⠪⡀⣨⡶⢋⡵⠊
⠀⠀⠀⠀⠀⣿⠗⣠⢟⡡⢞⠍⢀⠂⠀⠀⢈⠤⠀⢀⣤⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢺⣷⢷⣷
⠀⠀⠀⠀⠀⣿⣾⡫⠋⠈⣀⣤⡶⠞⢁⠆⢀⣤⠔⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢾⣿
⠀⠀⠀⠀⠀⣿⣿⣴⠞⢛⣉⣵⣶⣤⠾⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣿
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⢻⣿⣼⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠀⢹⣿⢺⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⣿
⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡏⢣⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣬⣭⣥⣴⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠿⠟⠋⠉⠉⠉⠉⠉⠛⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣀⣼⣧⣴⣿⣤⣀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⢸⣷⣷⣷⣧⣿⣿⣿⡀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⣏⣉⣻⣿⣿⣿⣟⣥⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⡿⠿⠛⢛⣛⣛⣛⣛⣛⡛⠉⢿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⣉⣥⠶⠞⠛⠋⠉⠉⣉⡉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⣡⡴⠟⢉⡀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⡀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠁⠀⠀⠀⢠⠐⠁⣀⠀⠙⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣾⣿⡟⢁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠀⠀⠀⠀⢸⠀⣠⠊⠁⡀⠀⠙⢿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣾⣿⡟⢀⡾⠁⠰⡄⢁⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠞⠀⠀⠀⠀⢂⡠⠈⢻⣿⣿⣿⣷
⠀⠀⠀⣾⣿⡿⢁⡾⠁⠀⡐⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⣰⡿⠃⠀⠀⠈⠠⠇⠀⢁⠀⠀⢿⣿⣿⣽⡇
⠀⠀⢠⣿⣿⠇⣸⠃⠀⢰⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠒⠂⠀⠀⠂⠀⠘⣿⣿⣿⣿
⠀⠀⢸⣿⣿⠀⣿⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡺⠋⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢸⠀⠀⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠀⣿⠀⠰⠆⠘⠇⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡤⠀⢸⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠀⣿⠀⠀⣇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠈⣿⣿⡄⢹⡄⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡤⠔⣶⠾⠿⡋⠛⠟⢦⡀⠀⠀⢀⡀⠀⢀⠄⠀⢀⣿⣿⣿⣽⠁
⠀⠀⠀⢻⣿⣷⣌⣷⠀⠀⢁⡀⡘⢦⠀⠀⠀⠀⠀⣼⡁⠀⠀⢉⠀⠀⢈⣦⠀⠀⠈⠑⠄⠈⡇⠀⠄⠀⠀⣼⣿⣿⣯⡏
⠀⠀⠀⠈⢿⣿⣿⣿⣧⡀⠋⠡⡁⠈⠀⠀⠀⡀⢸⠈⠀⠀⠀⣨⠀⠀⠀⠈⣇⣀⠀⠀⠀⠀⢁⠸⠓⢀⣼⣿⣿⢿⡿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄⠀⠈⠀⠀⠀⠀⡩⠻⣀⡀⠀⠀⠉⠀⠀⢀⣀⠏⢠⠀⠀⢀⠐⠃⠀⢠⣾⣿⣿⢫⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠃⠀⠙⠄⡀⠀⣀⠀⢀⡨⠉⠁⠈⣀⠐⠀⠀⣀⣴⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠊⡏⠂⠀⠀⠀⠀⠀⠈⣁⣤⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣈⣁⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠰⣶⣶⣶⣤⣤⡤⠆⠄⠀⠀⠛⠛⠛⠛⠛⠛⠛⢿⣿⣟⡛⣛⠛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣤⠤⠤⠀⢠⠀⠀⠀⠀⠈⠭⢉⣠⣦⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣾⣾⣾⣶⣶⣶⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠘⢦⡀⠈⢧⡀⢀⡀⢳⡀⢶⠀⡇⠺⢰⠇⡔⠀⡴⠃⠀⠀⣠⠞⠃⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠙⢦⡀⡑⣄⠱⡄⢣⠸⠀⡇⠆⡌⡼⢁⡞⢠⠜⣠⠞⠁⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠢⡙⠢⡈⠣⡀⠀⢀⡀⠇⡀⠈⣠⠋⠔⡡⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⣤⣀⠀⢬⡑⠬⡂⠀⠠⠂⣠⠤⠄⠢⠄⡁⠐⠄⢀⡠⠖⣉⠤⠒⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣤⣀⣀⣀⠀⠈⢉⡒⠈⠑⢈⠟⢠⠊⠀⠀⠀⠀⠀⠈⠳⡔⢅⠀⠪⠄⠒⠊⠁⢀⣀⣠⡀⣀⣀⣀⣴⠖⠶⠤⠤⠤⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠑⠒⠒⠤⠌⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⠀⠐⠒⠊⠉⢉⡝⠒⠛⠁⠈⠉
⠠⡤⠶⠒⠒⠒⠚⠋⠉⠀⠀⠀⠉⠉⠳⢄⣀⡉⠉⠉⠀⠐⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⡧⠬⠥⠤⠬⠭⣱⠖⠋⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⣄⣠⠤⠸⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⢀⣠⠼⠃⣀⠖⠋⠀⠀⠈⠉⢁⣠⠤⠤⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠐⠂⠁⠁⠐⢦⡀⣀⡀⠀⠀⠀⠤⡀⠀⡀⢠⣳⣖⡒⠒⠠⠤⠤⠤⠤⠤⣴⠞⠉⢉⡀⠐⠬⠈⠀⠴⡆⡀⠾⣉⣉⠁
⠀⠀⢀⣄⠀⣤⠄⢤⠄⠀⣠⠀⠤⠤⠬⠭⠥⠴⠦⠤⠤⠤⠴⠶⠿⠖⠂⠀⠀⠀⣤⠼⢃⢰⣀⠉⡤⡀⣀⠠⠀⣠⠔⢁⡽⠄⠈⢫⣉⣙⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡼⣃⣤⠾⠷⡼⠖⠛⠶⢅⣀⣀⣀⣜⣠⣤⣅⣐⣀⣈⣉⣄⣁⢇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡖⠒⣒⠲⠤⠶⠻⢛⣛⣋⢀⣤⣄⠤⠤⠤⠤⠤⠶⠤⠭⠭⠁⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣴⠶⠒⠒⠚⠛⠚⠛⠛⠒⣾⣉⡉⠀⠀⠀⠈⣡⣤⣵⣐⠒⣒⣒⣃⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⠤⠤⠔⠀⠀⠦⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠀⠈⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠤⠀⠒⠀⠉⠉⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⡠⠒⠉⠀⠀⠀⠀⠠⠤⠤⠄⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠁⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠙⠲⡶⠄⠀⠀⠀⢤⣴⣚⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠰⠆⠀⠃⠄⣀⡀⡀⣀⣧⠀⠞⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠜⠋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠖⠀⠀⠀⢀⠵⢒⡄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⡀⠀⠐⡀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠡⠂⠠
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠞⠃⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠒⠛⠡⠀⠁⠀⠀⣈⢶⡦
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡇⢀⠀⠀⠀⠀⠀⠈⠢⣀⠀⢨⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⡴⠚⠑⠈
⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⢸⡇⠀⠣⡀⠀⠀⠀⢀⠔⠃⠀⠀⠀⡲⠟⠅⠈⡶⠒⠒⠒⠲⠶⠶⠶⠤⠤⠤⠴⠶⠤⠤⠴⠞⡅⠈⠀⠀⢀⡦
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠘⠃⢀⡠⠈⠒⠐⠺⢁⡀⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⠎⠀⠀⢀⡀⠀⠀⠀⠈⠁⠐⢂⣠⠤⣀⠀⠠⠤⢤⣠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠐⠶⠀⠀⠀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⠉⠈⠀⠀⠈⠉⠁⠀⠈⠁
⠀⠀⠀⠀⠀⢀⡴⠀⡄⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⣰⠏⠀⢀⠗⠤⢀⠑⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣼⠏⠀⢠⠁⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⣼⠏⠀⡠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⢠⡛⢀⣼⣱⠏⠑⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢘⣆⡀⠤⣐⣠⠏⠀⠉⡦
⢰⡷⠀⡎⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢀⡤⠦⠔⠓⠉⠉⠀⢀⠤⠊
⣗⠃⣀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠐⠀⠀⠀⠀⠀⢀⠠⠐⠈
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣠⠄⠒⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠈⢣⡄⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⡰⠁⠠
⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠈⠀⠀⡂⢀⠢⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠂⠀⠀⠁⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠑⠠⢐⠠⠂⢁⠠⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠔⡀⠀⠈⠀⠀⠀⡐⠀⠀⢀⠔⠁
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠂⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠠⠀⠀⡀⠐⠀⠀⢀⠔⠁
⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠌⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠐⠠⠀⠑⠈⠀⢀⠔⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠐⠀⠈⠀⡠⠊⠀⠀⠀⠀⠀⠁⠐⠀⢀⠠⠐⠈
⠀⠀⠁⠂⢀⠀⠀⠀⠠⠀⠀⠁⢄⠀⠀⠀⠀⠡⡂⠀⠁⠀⠀⠁⠀⠘⠀⡄⠀⡈⠀⠀⢀⠇⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠘⢆⠠⠄⠀⠄⠠⠠⠄⠁⠠⠀⠀⣠⠃⠀⠀⠀⡠⠈⠀⠀⠀⠀⠂⢀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⢠⠄⣑⢄⠠⠀⠈⠁⠀⢀⠀⠁⠀⠀⠉⠀⠒⠤⢄⠀⠜⡢⠈⠀⠀⠠⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠐⠀⠀⠤⢀⠀⠀⠂⠀⠀⠀⠀⠂⢄⠈⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡢⡀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠤
⠀⠄⠀⠀⠀⠀⠈⠐⠠⠄⣈⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⢀⣀⠀⠀⠀⠐
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠂⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠠⠠⠤⠄⢄⢤⢠⢀⢠⢠⠠⣠⠀⡄⣤⣦⠧⠤⠤⠤⠤⠤⠤⠸⠠⠠⠤⠤⠤⠦⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠒⠒⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠈⠁
⢀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀
⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⢠⡆⠀⠀⢸⣿⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⢀⣤⣿⣦⡿⢀⢀⣀⣸⣿⣷⣤⣼⣿⣰⣿⠟⢠⣶⡄⠀⢸⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠛⠻⠟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠃⠂⠚⠢⠦⠤⠞⠒⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⢠⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠠⣀⡄
⠀⠀⠸⡆⢸⡆⠀⠀⢠⡇⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠈
⠀⠀⠀⢷⠸⡇⠑⠀⢸⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠰⠀⠄
⠀⠀⠀⠘⠀⠇⢀⡇⠈⢸⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣧⢲⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠛⠳⣀⠃
⠀⠀⠀⠀⡰⠀⠀⢠⠀⠸⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠈⡇⠈⠠⡆
⠀⠀⠀⡀⢃⠀⢰⡸⡄⢻⠇⠈⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠲⡿⢀⣀⠄
⠀⠀⠀⠇⡞⢰⠀⠷⣧⠋⡆⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⣳⡀
⠀⠀⠀⢸⠃⠸⠀⠀⣿⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠹⠉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⡿⠘⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣧⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣤⡰⠦⠀⣙⡛⠀⠀⠀⠀⠋⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠖⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣾⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢴⣦⣃⡀
⠂⡐⠀⠈⠀⠀⠈⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⣿⡛⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⠀⡈
⠀⠀⠀⠀⠀⢤⢠⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⡞⠅
⠀⠀⠀⠀⠀⠈⢀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠇⢼
⠀⠀⠀⠀⠀⠢⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡽⣾⣉⣈⣆
⠀⠀⠀⠀⠀⢀⡀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⡾⠟
⠀⠀⠀⠀⠀⠈⠁⠀⢡⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠆⠀⣿⣟
⠀⠀⠀⠀⠀⠀⠐⠢⣬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢾⣶⣤⢏⣶
⣤⣤⣤⣤⣤⣤⣤⣤⣴⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣬⣯⣤⣿⣬
⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⢄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣟⣻⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠋⠧⠽⠈⢪⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢵⣿⠀⠀⣿⠀⢿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣌⣛⣀⣀⣛⣃⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⡏⠉⠉⠉⠉⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠇⢰⣯⣿⣦⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢒⡭⠚⣿⡀⠘⠾⠽⠋⠀⣿⡓⢯⣒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡮⠕⠊⠁⠀⠀⣹⣧⣦⡀⢀⣴⣄⢿⡇⠀⠈⠑⠺⢵⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣪⡵⠚⠁⠀⢀⡀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣸⡇⠀⠀⢀⡀⠀⠈⠓⠮⣕⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢹⡅⠀⡴⢻⣿⣯⠳⡄⠀⡏⣿⣿⡇⢸⣿⣿⢹⡇⢠⠞⣹⣾⡝⢦⡀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⡇⢸⢁⣀⠉⢁⡀⢸⠀⢧⡟⠉⢃⡘⠉⠛⣿⡇⡇⢀⡈⠉⣀⡀⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⢸⣾⣿⣧⣿⣽⢸⠀⣼⠇⣴⠋⡏⢦⠀⣿⡇⡇⣿⣿⣴⣯⣷⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢸⣷⣿⣦⣾⣿⣼⠀⢿⠠⡇⢰⡇⢸⡆⣾⡇⡇⣷⣗⣽⣻⣆⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣺⡇⢸⣿⣿⡟⣿⣽⢻⢰⣿⡶⡇⢸⡇⢸⢷⣿⡧⡇⣯⣯⣿⢽⡏⡇⢰⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠸⠽⠿⠥⠿⠯⠼⠀⡛⣶⡇⢸⡇⢸⡆⣻⡇⠧⠿⠿⠭⠿⠯⠇⠀⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⣤⣧⣤⡤⣤⣤⣤⣤⣤⣴⣷⣿⣿⣿⣿⣿⣷⣾⣧⣤⣤⣤⣤⣤⢤⣤⣴⡧⢧⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠠⠀⠀⠀⠀⡠⠐⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⣀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⠊⠀⠀⠀⠒⠲⣄⠀⠀⠀⠀⠀⠀⡀⠀⠓⠠⠀⠨⠄⠂
⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⡠⠒⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡀⠤⠀⠀⠈⠐⠀⠀⠈⠉⠉⠀⣀⡀
⠀⠀⠀⠀⠂⠀⠁⠉⠉⠀⡀⠠⠔⣈⡀⠀⡀⠀⠠⠄⢂⢠⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⠤⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⢾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠨⠤⢄⣤⣬⣥⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠠⢔⣽⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠄⣀⠀⠉⠐⠒⠀⠄⠠⠤⠀⠀⠀⠀⠤⠄⠐⠷⢤⣤⣀⣀⣀⣀⣐⣐⣨⡧⢀⠠⣊⠔⠒⠉⠉⠉⠙⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⠒⠠⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢁⠀⡀⠀⡴⠃⢀⠠⠀⠀⠄⠠⢌⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⢧⡔⡥⠚⠉⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⠀⠀⠀⠀⠠⠀⠐⡀⠔⡯⠥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣟⠒⠤⠤⣤⠭⢤⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌⡀⠉⠉⠙⠛⠛⠛⠛⠛⢑⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠶⠖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠄⠠⠈⠠⠀⠀⠠⠂⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡳⠀⠀⠇⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠂⠀⠀⠁⠐⢀⠉⠃⠀⠉⣉⡀⠀⠀⠀⠐⠈⠠⢶⡷⠆⠀⠀⠉⠁⠉⠉⠏⠉⠛⠳⣶⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠦⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡎⡀
⠀⠀⠀⠁⠀⢀⡤⠒⠩⠀⠀⠀⠒⠀⠀⡀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⡅
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⡄⠀⠀⠀⠈⢮⣐⢄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠘⠁⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡆⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢠⡤⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠖⢻⠇⠀⠀⠀⠐⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠐⠋⡁
⠀⠀⠀⠀⠤⣾⠁⠀⠀⠀⢀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠘⣓
⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⣘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠠⠋⠀⠀⠁⠀⠀⠀⠈⢹⡄⠀⠀⠀⠀⣉
⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⢸⣹⡄⠀⠀⠀⠀⠀⠀⠀⢐⠀⠈⠀⠀⠀⠀⠠⡀⠠⠀⠀⠀⠂⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⢉⠁
⠀⠀⠀⢠⠼⠀⠀⠀⠀⢀⣸⡌⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠀⠀⠀⠐⣀⣀⡀⠀⠀⢀⡞⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠄⠰⡇⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠆⣤⣶⠟⠋⠉⠉⡙⠛⣦⡞⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⢈⣭⣶⣶⣦⣤⣄⠁⠀⠀⠀⠀⢀⡇⠀⠀⣀⢀⣰⠏⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢨⣄⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢧⠀⠠⠴⠀⠐⠏⠋⣈⡠⠟⠋⠀⠀⠀⠀⣼⠀⠀⢐⣭⠞⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠷⣼⣶⡄⠀⣀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡇⣠⣶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣾⡇⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢻⡿⣽⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⡟⢇⢏⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣅⣅⠙⢃⠰⡄⡄⠀⠀⡀⢠⣾⡏⠀⣁⣼⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠀⣒⣀⠡⢘⣄⣠⠁⢃⣙⡀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠋⠁⠈⠛⠲⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⠀⠀⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦
⠀⠀⠀⠀⣠⡄⢠⣿⣿⡄⣿⣿⡇⣠⣠⣠⣿⣿⣿⠶⠿⠷⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿⣿⣿⣿⣀⣀⡀⠀⠀⠀⠀⣴⣶⣶⠀⠀⠀⢰⣶⡀
⠀⠀⠀⠀⣿⣿⠸⣿⣛⡃⢿⣿⡇⠿⢿⢼⣿⣿⣿⣐⡀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⣿⣿⣿⣿⣿⣟⡇⠀⠀⠀⠀⣾⣾⣿⢀⣿⡀⣾⣿⡇⢀⣀
⢠⣾⣷⡄⣿⣿⢰⣿⣿⡇⣿⣿⡇⣠⣆⣹⣿⣿⣿⡿⢶⠶⠤⣤⣶⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣏⣹⡇⠀⠀⢸⢿⢿⣿⣸⡿⡇⣿⣿⡷⣿⣿
⢸⡿⢿⣇⣿⣿⢸⣿⣿⣇⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⠿⠥⢿⣿⣿⣿⣿⡿⣿⣿⣿⠿⣿⣧⣴⣤⣶⣿⣾⣿⣿⡿⠛⣿⣿⣷⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠛⣿⣿⣿⣿⠿⣻⠟⢻⠿⣿⣿⡟⣿⣿⣿⠟⣿⠿⣿⠿⠛⠻⢛⣿⣿⣏⣻⣤⡿⠛⠿⣿⡿⢿⣿⣿⣿⣿⣿⡽⢿⢿⣿⣿⣿⣿⣿⣿
⠽⠁⠈⠀⠀⠛⠟⠛⠟⠀⠀⠀⠀⢀⣿⣿⠛⠿⣿⣿⠀⠘⢿⢯⠀⠀⣴⠏⠘⠻⣿⣿⣿⣧⠀⠀⠛⠀⠀⠀⠙⠻⣿⠟⠀⠚⠀⢿⠿⣿⡿⢿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠙⢿⡄⠀⠘⠀⠀⠘⠁⠀⠀⠀⠋⠋⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⣼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠠⠀⡄⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠈⠁⠈⠁⠘⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠨⡼⣌⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠅⡇⠀⠀⠀⡇⠘⡩⡇⠀⠀⠀⢀⠂⠉⠠⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠠⠄⢀⠀⠄⠆⠁⠀⠁⡇⠂⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠠⢠⠀⠄⠄⠀⠀⡇⠀⡔⠓⠀⠀⠤⠀⠀⠀⠀⢛⠧⡀⠀⡇⠋⠀⠀⠁⡌⠀⠀⠀⢘
⠁⠏⡇⠀⠀⠀⢸⣹⠧⠇⠶⢄⠀⡇⠀⡇⠀⠐⠗⠾⣷⡄⠀⠀⢸⠠⣧⠀⡇⣁⠀⠀⠀⣇⣘⣿⠀⢸
⠀⠚⡗⠤⠀⠀⢸⡇⠀⠀⠀⢸⣀⢧⣬⡇⠀⠀⠀⡃⡸⠧⣄⡀⢸⡴⠓⠒⢻⠷⣀⠀⠀⠁⠸⠙⢳⢸
⠀⠀⠃⠠⡆⠀⠀⡇⠀⠀⠀⢸⡇⢸⠀⡇⠀⠀⠀⠄⠇⠀⡷⡇⠘⠆⠀⠀⢨⠀⣹⣇⣰⣀⣨⠀⣾⣸
⠨⠀⠠⠘⠇⠀⠸⠷⠀⠀⠀⠤⠅⠠⠀⠆⠀⠀⠠⠢⡆⠀⡇⡗⡄⡆⠀⠀⢠⠠⠀⠀⠀⢰⠉⣷⣾⠁⠀⡀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠠⠦⣄⣉⣉⣛⣉⣉⣉⣉⠀⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣌⣷⣼⡿⠿⠟⠙⠋⠉⠉⠉⠉⠿⢿⣿⣦⣄⠉⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣑⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣄⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠙⡆⠀⠀⠀⠀⢠⣶⠾⢿⣷⣶⣶⣶⣤⠀⠀⠀⠀⡀⢃⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⡈⠳⡄⠀⢀⣼⡇⠀⢸⣸⣿⣿⣿⣛⢠⡀⣀⠚⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣼⣯⣀⣸⣷⣤⣼⣿⢿⡿⢿⣛⣀⣤⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⡿⠶⠶⠶⠶⠶⠾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠒⣶⡶⣾⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢈⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⣬⣿⣿⣶⠖⠻⠓⠛⠷⠄⣨⡿⠿⠿⠟⠛⠳⣶⣤⣤⣤⣀⢠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣀⠤⠘⠛⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠁⠐⠒⠉⠽⢛⡻⢶⣤⣌⡓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⡋⠁⠊⠁⡀⢠⡀⠢⠈⠉⠒⠀⠄⠀⠠⠤⠠⠤⠔⠒⠋⣉⠤⠖⣀⡤⠘⡻⠶⣹⢿⣶⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡉⠀⠀⢀⡠⣔⠲⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣴⡆⠀⠀⠀⠀⠀⠉⠁⠚⣣⢄⡈⠸⣋⢜⡽⣿⣶⣅⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠑⠋⠀⣠⠶⠓⠉⠀⣀⢶⣄⠀⠀⠀⠀⠀⢀⠀⢠⠀⢠⡄⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⡉⠙⠫⣦⡉⠺⣿⣽⡽⢿⣷⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢀⡨⠂⠀⠁⠀⠀⠉⠆⣈⠀⠠⡄⠘⠂⠈⠀⢀⣀⣈⣁⡀⠀⠀⠁⠘⠃⠰⠆⢀⡘⠃⠀⠀⠈⠻⣷⡌⠻⣿⣷⡽⣿⣧
⠀⠀⠀⠀⣰⢋⠄⠀⠀⡰⠋⠠⠀⠀⠀⠀⠀⢠⡀⠙⠃⣀⡀⠀⠀⠀⠀⠀⢻⠈⣿⠀⠀⠀⠀⠀⠀⠀⢀⠙⠃⢀⠀⠀⠀⠀⠈⠻⣷⡘⢻⢿⣞⣿⣧
⠀⠀⢀⡴⢡⠆⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠻⣿⢧⠀⠀⠀⠀⠀⣞⣦⣿⠀⠀⠀⢠⠀⠀⢠⠟⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠻⣦⢫⢻⡾⣻⣧⡄
⠀⠀⣼⠃⠀⠀⣠⡞⠀⠠⣆⡄⠀⢠⠈⠃⠀⠀⠀⠀⠀⠙⣧⡦⠄⠀⠀⠀⠉⠁⠀⠀⠀⢀⡞⠀⢴⠏⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣀⠙⢷⡋⠾⣣⢿⣎⡄
⠀⣾⡇⠁⠀⢰⡿⠀⠐⠶⠷⠀⣤⡄⢀⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠆⠘⠋⠀⠈⣿⡜⢩⡞⣿⣤
⢠⣿⠀⠀⢀⠻⡄⠀⠀⠀⠀⢤⡀⠀⠾⠷⢦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣀⡴⢊⡽⠆⠀⠀⠂⠀⠀⠀⢸⣷⠋⣿⢹⣿⡆
⢸⡇⠀⠀⡘⢠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢙⡿⢂⠀⠀⠀⠀⠀⠀⠀⠻⠞⠉⠀⠀⠀⠀⠒⠂⠀⠀⠀⣿⡞⠛⣿⣿⣧
⣾⠃⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢸⣷⢸⣿⣿⣿
⣿⠀⠀⠀⢿⠀⣄⣸⠏⠉⠉⠙⣶⣶⢶⣶⣶⣶⣦⣤⣼⣯⣁⣀⣀⣀⣀⠀⢀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⣿⡄⣿⣿⣿
⣿⠀⠀⠀⢳⠀⠐⠲⠆⠙⠃⠀⣿⠭⠭⠷⠆⠀⠁⠀⠸⣏⡍⠙⠻⠿⣿⣿⣯⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣓⣾⡇⠀⠒⠀⣭⠿⠀⣿⡇⢿⣿⣿
⣟⣹⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⣿⡇⣾⣿⣿
⢻⡏⠀⢀⡿⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣿⢷⣼⣿⣿
⢹⡇⡆⠀⢡⡶⠀⠀⠀⠀⣠⡄⠀⣿⣻⡞⣛⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢆⣤⢀⡀⠀⠐⠂⠀⠀⠀⢸⣇⣸⣿⢿⣿
⠈⣿⠀⠀⠈⣰⠆⠀⠀⢠⡀⣠⠄⠈⣧⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⣈⣔⡿⠁⠀⠀⠀⠀⠀⢀⣿⣉⣿⡛⣾⡿
⠀⢹⣇⠀⠀⢡⠞⡀⠉⣩⠆⠀⢠⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠸⠷⢀⣛⡦⠀⣼⣏⢻⣞⣿⣿⠃
⠀⠀⢻⣆⠀⡀⠚⡱⠂⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⣼⠍⣩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠈⠐⠀⣼⠇⣼⣿⣽⣿⠏
⠀⠀⠀⢻⣦⣉⠄⠰⣫⢆⠀⠀⠀⠀⠰⠆⠀⠀⠻⢮⣰⠋⠀⠀⠀⠀⠀⠀⡦⢠⡀⠀⠀⠀⠀⠀⠘⡷⣀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢀⣾⢃⣰⣾⣿⣿⠛
⠀⠀⠀⠀⢻⣷⣻⢆⡀⢀⢁⠀⠀⠀⠀⠀⠋⢀⣤⠀⠋⠀⠀⠀⠀⠀⠀⠀⣷⡸⠃⠀⠀⠀⠀⠀⠀⠰⠍⠃⠀⠠⠄⠈⠀⠀⠀⢀⣴⡿⢡⣿⣟⣿⣿⠃
⠀⠀⠀⠀⠀⠹⣷⣾⢦⡄⠲⡝⢦⡀⠀⢸⡅⡠⠀⠀⠆⢀⡀⠀⠀⠀⠀⠀⠷⠥⠶⠀⠀⠀⠀⠀⠀⢀⡀⠸⠆⢀⠀⠀⠀⢀⣤⡾⢋⢔⣵⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⢻⣷⣽⣦⡈⠀⣙⠔⣈⡐⠁⠀⠀⠀⠈⠁⠀⠿⠀⢀⡄⠀⣤⠀⢀⠀⠀⡀⠐⠆⠀⠁⠀⠶⣬⠀⢀⣴⠿⠋⠔⣵⣿⣿⢟⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣽⣦⡠⠂⢈⠟⢢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⢏⠉⣢⣴⣿⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣮⣕⡣⣎⡳⢮⡽⠻⣶⠶⣀⣤⣀⠀⠀⠈⠉⠓⠀⠀⣀⣠⣤⣴⡶⢿⠛⢏⡡⣜⣶⣿⣯⣾⠛⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠩⣿⣿⣿⣣⡶⣍⡐⠚⠋⠥⠿⠟⠛⠛⠛⠛⠛⠛⠛⠶⢄⣉⣓⣦⣽⣾⣻⣿⣿⡿⠯⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⠟⢿⣿⣷⣿⣿⣶⣶⡾⣿⣿⣿⣭⠷⠶⠿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣩⣵⡶⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⠀⣤⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣽⠀⢿⡤⠠⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡼⠹⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢰⠁⠀⢰⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡰⠃⠀⠀⠀⣦⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⣀⠃⣀⣀⡀⠀⠀⡆⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠁⠀⠈⠉⠀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡀⠁⠀⠒⠂⠀⠀⠋⠀⠀⢳⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣠⠁⢀⣀⣀⣀⡀⢀⣀⡀⠀⠀⣦⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣰⠁⠀⠈⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⡄⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⠸⠇⠀⠲⠖⠒⠐⠒⠒⠒⠲⠶⠶⠃⠀⠰⠆⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣠⡷⠀⠤⠤⠴⠦⠤⠤⠦⠤⠤⠴⠆⠠⠶⠶⠾⣀⣸⠧⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣀⠠⢠⡄⢀⣀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡋⠉⠉⠉⠀⠀⠀⠈⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⣉⣉⣉⣉⣀⣁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠀⠀⣾⠁⠀⠀⠀⣨⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⡀⠀⢀⡀⢁⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠤⠐⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⣠⠶⠋⠉⡄⠑⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⣠⡴⠋⠀⠀⠀⢀⡇⠀⠈⠛⢤⡀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠊⠉⠀⠀⠀⢀⡴⠞⠻⡄⠀⠀⠀⠙⢦⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢀⡰⣿⠛⠋⠉⠉⢻⠶⢆⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠠⠔⠉⠀⠀⠀⠀⠀⢀⠏⠀⠉⠀⠀⠀⢰⡆⠀⠀⢳⠀⠀⠀⠈⠣⡄⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⢠⠀⠀⠀⠟⠀⠀⠀⡇⠀⠀⡌⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⡠⠀⠀⠀⣴⠚⠻⢆⡰⠏⠀⠀⢸⠶⡀⠀⠀⠃⠀⠀⠇⠀⠀⢀⣴⣿⠀⡆⠀⠀⣀⡀⠙⢆⡀⠀⠀⠀⠀⠙⣤
⠀⠀⠀⠀⠀⠘⣦⣠⠴⡟⠀⠀⠀⠀⠛⠠⠖⠋⠁⠀⠀⠀⠈⠀⠹⣄⠀⢸⡆⣨⠀⢀⡴⠋⠀⠀⠀⠹⢀⠘⠋⣁⠀⠀⠙⢤⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠳⣤⣧⣇⡴⠋⠀⠀⠀⢀⡄⠀⠈⠁⠀⠉⠀⠀⠀⠀⠙⣤⡀⠀⠀⡰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⡆⠙⠋⠙⠑⡆⠀⠀⠈⠛⠀⠀⠀⠀⠙⠛⠛⠤⠦⠄⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⠙⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀⣀⡀⢀⣀⣀⣀⠀⠀⢸⡏⠀⠀⠈⠉⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⡾⠉⠉⠉⠉⠉⠉⢹⠋⠉⠉⠉⠉⠉⠉⢹⡇⠀⢸⠃⠀⠀⠀⠀⡇⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢨⡇⠀⢸⡀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡄⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⣠⡄⣤⠀⣿⠀⢀⠀⡀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⠷⠃⠀⣯⠐⢯⣤⠃⠀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠳⠆⠀⠀⠘⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀⣀⣀⣀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈
⠀⠀⠀⠐⣷⣄⣸⡇⠀⢠⣦⣀⡀⠀⠒⠒⠒⠒⠒⠂⠘⣿⠀⠐⠒⠒⠒⠒⠒⠛⠶⠖⠀⠀⣠⣄⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⠀⠤⠀⠉⠁⠉⠉⠋⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠉⠉⠛⠋⠀⠀⠠⠶⠦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠒⠛⠉⢹⡆⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣤⣤⣤⡤⠀⣤⠉⠑⠒⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣤⡀⣴⠊⢳⠀⠀⠉⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⣤⢀⣈⣄⠀⣼⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠛⠋⢀⡴⠞⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢰⡏⠀⢀⡀⠴⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠓⠄⠺⢆⡀⡨⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠘⠛⡆⠀⠀⡶⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠘⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡀⠀⣠⣤⣤⣄⡀⠙⠏⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢀⣸⣧⡀⠙⠛⠛⠛⠁⠈⠋⠀⣀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠈⢻⡟⠉⠀⠀⠀⠰⠶⠀⠀⠀⠙⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠇⠀⠃⠀⠈⠁⢀⣤⣶⣷⣦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⡀⠀⠀⠉⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⡀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⠋⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠉⠁⠉⠀⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣤⠀⠈⠋⠀⢰⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⣿⣿⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⡿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡀⣀⣤⣤⣄⣤⠤⠀⠀⠀⣀⠄⠘⠛⠉⠹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠈⠛⠟⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⢤⣶⣤⣤⣼⣦⣤⣤⣤⣤
⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡆⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣹⠀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠿⠶⠶⠶⠤⠶⠶⠶⠶⠶⠶⠶⠶⠿
⠈⢅⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⡄
⠀⠀⠑⠢⠤⠤⠤⠆⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠉⢉⣀⠀⠀⠀⠀⠀⠀⢀⡴⢿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠤⠀⢀⡀⠀⠀⠀⠀⣀⣈⣀⣼⠁⠀⠀⠀⠀⢀⡤⠒⠉⢸⡀⠈⠙⢆
⠀⠀⠀⠀⠀⣇⢠⢄⡠⠀⠘⠀⠀⠈⠦⡄⠀⠀⠈⡇⠀⠀⠉⠀⠀⠀⠀⠀⠜⠁⠀⠀⠘⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠉⢠⠋⢰⣄⠀⢀⡀⠀⠀⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠘⠁⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⢺⠐⠀⠈⠓⠋⠀⠐⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠀⠀⠀⠀⢀⣀⣔⠇
⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠈⠀⢀⠀⠀⠀⠀⠀⠈⠚⠶⢾⢿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠑⠀⠀⣀⡀⠀⠀⠀⠀⠀⣆⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄
⢠⣠⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜⠃⠈⠉⠳⡄⠀⠀⢠⣿⠋⠀⠀⠀⣀⣤⣬⣤⣾⣜⣂⣐⣒⣢⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠘⣆⢀⣾⡏⠀⠀⠀⠀⠈⠥⢤⣤⣤⣤⣤⣤⣤⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⢹⣾⠋⠀⠀⠀⠀⠉⡉⠡⠉⠉⠁⠀⠉⠉⠙⠉
⠀⠀⠀⠀⣀⠄⢂⠄⠀⠀⠀⠀⠀⠀⢴⡈⣿⣷⣄⣠⠾⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠂
⠀⠀⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⡠⠉⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠁⠈⠁
⠀⢈⠀⠀⠀⠈⠀⡂⠀⠀⠀⠀⡜⠀⠀⢴⠀⠐⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣀⣂⣀⠂
⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡇⠀⠀⢦⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⠀⢃
⢠⠆⣶⣔⡐⠚⠒⠐⡆⠀⠀⠀⠁⠌⣀⣤⠶⠗⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠿⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠙⠋⠠⠀⠈⠁⠀⡖⠀⠈⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⡙⠋⠁⠀⡇⠀⠀⠀⠐⠀⠀⠀⠀⠅⠀⠀⠈⠀⠀⡀⠙⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⡇⠀⠀⠀⠃⠀⠀⠀⠘⣀⠀⠀⠀⣀⠀⠀⠚⠀⡘⠃⢨⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⢀⡠⠖⠒⠒⠀⠤⠤⠊⠀⠑⠊⠉⠀⠑⠒⠂⠉⠈⠑⠢⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢿⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡤⣀⣀⠀⠀⠀⠀⠀⠀⣀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢤⣀⠀⠀⠉⠈⠉⠁⣀⡡⠔⢻⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⡉⠛⠋⠛⠋⠉⢀⡠⠞⠛⠛⢟⣻⡿⠟⠛⠛⠛⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣡⠞⠉⠙⣦⡀⠀⠀⡠⠋⠀⠀⠀⢠⠞⠁⠀⢀⡤⣦⠛⠒⠲⠶⠚⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⣡⠞⠁⠀⠀⠀⠈⠻⣦⠊⠀⠀⠀⠀⢠⠏⠀⢀⢔⡵⢊⡤⠚⠉⠁⠀⠀⠈⠁⠈⠛⠻⢿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡹⠶⢤⣀⠀⢠⠃⢀⣔⡡⠋⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⢙⠃⠀⠀⠉⠀⡴⠁
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⡸⠁
⣿⣿⣿⡿⢷⣄⠀⠀⢔⠒⠁⠉⠀⠀⠀⠀⠀⢀⠔⣯⢠⡀⠀⠀⢰⠃⠀⠀⠀⠀⡀⢀⠀⡀⢀⡀
⣿⣿⣿⣷⡟⢻⣧⡀⠈⡱⠂⠀⠀⠀⠀⢀⡔⢡⡞⠁⠀⠙⣄⢀⡍⠀⠀⡠⠖⠈⠉⠀⠀⠀⠈⠉⠓⠦⡀⠀⢰
⣿⣿⣿⣿⣿⣿⣤⣿⣾⡴⠒⣀⠀⠀⢀⠎⠀⢸⡇⠤⠒⠉⠙⣨⠁⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠈⠳⣞⣡⢖⢀⠎⠀⠀⢸⡇⠀⠀⢠⠁⡎⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⢿⣦⡁⠀⠀⢀⡼⠻⣄⠀⣠⠶⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿
⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠙⢿⣦⠔⠁⠀⠀⠈⠳⣬⠃⢀⡽⠤⠄⠒⠒⠒⠒⠒⠂⠤⠄⣀⣠⠎⢰⣿⣿
⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠉⠙⢲⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣭⣶⣾⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠁⠀⠙⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⢀⡰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣧⡓⢄⡀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠀⠀⠉⠉⠁⠐⠒⠢⠄⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣶⣶⣦⣤⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⠤⠀⢀⣀⣀⣀⣤⣤⣔⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⢠⢠⡀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡾⠁⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣴⡶⡀⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠷⠏⠀⠁⠀⠀⢀⣨⣴⢙⣿⡿⣿⣟⣿⣾⢿⣿⡇⠀⠑⠳⠽⠇⢲⡿⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⢯⠠⠑⢶⣶⣬⠭⣤⣥⣭⣤⣼⣿⣿⣣⣬⣛⢿⣿⣇⣐⣀⣖⣚⣿⣷⣾⣷⣿⣿⣦⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠶⣛⣽⣒⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡭⢙⣮⣡⣌⣿⣿⣿⣿⡯⠿⢿⣏⡷⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠁⠀⠈⠉⠋⠉⠋⠉⠚⠋⠉⠙⠛⠋⠋⠉⠁⠉⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣦
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣧⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤
⠀⠀⠀⢠⣿⡆⠀⣠⣀⣄⠀⣾⣿⣿⡄⢸⣿⣿⣿⣿⠀⠀⣠⣄⠀⢸⣿⣿⣿⠁⣴⣶⣄⠀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⣿⣿⣿⢰⣿⣿⣿⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⢰⣿⣿⡆⢸⣿⣿⣿⠀⣿⣿⣿⢰⣶⣶⠀⣿⣿⣿⡇⣠⣴⡆
⠀⠀⢀⣿⣿⣿⣰⣿⣿⣿⣠⣿⣿⣿⡇⣸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣾⣿⣿⣿⣧⣿⣿⣿⣸⣿⣿⠀⣿⣿⣿⡇⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣧⣹⣿⣧⣿⣧⣿⣼⣧⣿⣷⣟⣻⣟⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡾⠛⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠤⣾⣏⣠⣤⣤⡄⠀⠀⠀⣀⠹⣷⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣽⣿⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠟⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠿⣿⡟⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⠿⠟⠉⠉⠉⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠦⠴⠶⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠉⠉⠉⠙⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⢐⣾⠂⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⢀⣼⣻⡀⠟⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣷⣼⣯⣿⣷⣶⣼⣦⣤⣶⣶⣴⣧⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠁⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⣠⠊⠙⢂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣀⣀⣀⡀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠠⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣷⣶⣄⣀⠀⠀⠀⠀⠴⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢻⣿⣿⣿⣶⡶⢤⡤⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠚⠿⠿⠿⠿⠟⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⡀⠋⠶⠴⠴⣄⡄⠀⠀⠧⢀⠀⡀⠑⢖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠒⠂⢠⢀⣤⠀⠀⣰⠀⠘⣧⡄⠐⠛⠁⠛⠄⠀⣅⠂⠬⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠠⡔⡖⢻⢀⠠⠒⣄⢼⠝⠏⠋⠀⠀⠶⠋⠑⠒⠚⠷⠂⢀⣀⠀⠂⠰⣌⣆⠀⡀⠈⢹⣄⡄
⠀⠀⠀⠀⠀⠀⢠⠹⣦⣼⣧⣶⡿⠟⠉⠀⢓⡳⠂⠡⣄⠀⠀⠀⠀⠘⣧⠀⠀⢉⡧⢀⠀⢹⡦⣀⠀⠈⢿⡇⠀⡀
⠀⠀⠀⡀⢸⠀⠎⢀⣸⣿⢳⣏⣤⠤⠚⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠲⢇⣆⣷⢀⠁⠘⡧⢸⣦⡀⠎⢷⡴⢻⡆
⠀⠀⠀⣧⠀⡆⠀⡸⢋⡿⣾⣉⣆⢠⢠⡠⣴⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢿⠳⠷⣧⡄⣣⡑⡄⢼⣷⣾⠁
⠀⠀⢀⠸⡔⡉⢶⡟⣹⣱⡟⠓⠁⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠂⠳⣄⣏⠘⠻⢷⣿⣀⣿⡃⣨
⠀⢳⠈⡦⢻⣦⡟⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢸⣿⡝⣿⡷⡴⠂
⠀⠈⢷⡄⠈⢿⣾⢃⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠳⣿⡇⢻⣇⢽⠄
⣄⠀⣸⢿⣆⣼⣟⣱⣿⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢻⡷⣼⣿⣷
⠘⣧⡽⣎⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⣸⣿⣼⣿⠏
⠀⠘⣧⣿⢸⣿⣿⢀⣄⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⡏
⠀⢀⢹⣾⢾⣿⣿⣨⠋⠡⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣀⠀⡓⢾⣿⣿⢳
⠐⢾⡤⣿⣤⣿⣿⡄⠀⢸⣿⣫⡀⠀⢠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⡻⡟⣾⣿⣿⢸⠇
⠀⠸⣿⢏⢿⣿⣿⣧⠀⢒⢸⣟⡆⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡇⢈⣿⣿⡇⢸⠀⠠
⠀⠀⢹⡥⡂⢻⣿⣿⣧⠀⢖⣿⢩⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣿⠥⣼⣿⣿⣇⡏⠀⠏
⠀⠀⠀⠘⢷⡈⣿⣿⣿⣷⡨⣯⣾⣿⢘⣠⣀⡆⠀⠀⠀⠀⠀⠀⠀⢶⣷⣆⡀⠀⠈⢀⣤⡸⣿⣏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⣿⣛⠛⣿⣻⣿⣶⣄⢀⠀⢀⠀⠀⠘⢝⢯⣇⢆⣇⡞⣿⠩⢿⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠛⠻⣿⣷⣾⠼⠿⠿⠿⠯⣄⣠⣤⡄⢠⠴⠓⢉⣩⣍⠙⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠦⠤⣼⣿⣥⠤⠄⣨⡤⠐⠀⠀⢈⠗⣅⠀⣲⣊⣹⣥⣾⣿⣿⣿⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡏⠉⢻⣿⣟⣩⣹⣷⣶⣊⣠⠶⢻⣯⠟⠉⠀⠀⢤⣤⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠿⠏⠀⠊⠁⠐⢤⠞⠂⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⣄⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⢦⣄⡀⠀⠀⠀⠀⣻⠀⠀⠀⠀⣸⠇⠀⠀⠀⢠⡴⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠠⠔⠃⠀⠀⢀⠴⠋⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡆⠀⠀⠀⠉⠉⠛⠛⠛⠶⠀⠀⠈⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠚⠛⠛⠛⠳⠀⢀⡤⠶⠛⠋⠉⠉⠉⠉⠙⠛⠷⣦⣀⠀⠘⠉⠀⢀⡴⠋⠀⠙⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠀⠀⢠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠰⠋⢀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⠀⠀⣠⠞⠁⠀⠰⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡴⠋⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠓⠒⠛⠒⠒⠒⠲⠶⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠚⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⣀⣤⣤⠤⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⢛⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠈⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⢤⠄⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣏⠉⠹⠉⠀⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠘⠂⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣷⣶⣦⢰⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡏⣝⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠙⢻⣿⣶⣶⣿⠏⠋⠉⠀⣹⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠒⠒⠀⠈⠉⠉⠀⠀⠀⠴⠦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠋⠉⠀⠄⠀⡄⠀⠀⠀⠈⡀⠉⡀⠀⠈⠙⠻⠿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡄⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠉⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⡶⠂⠀⠀⠀⠉⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣈⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣾⠀⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⠁⣴⠂⠀⠀⠀⠀⣶⣄⠀⠐⠂⠀⢰⣿⣿⣿⡇⠀⣽⠂⠀⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠘⡟⠀⠀⠀⠀⣾⣿⣿⣷⡀⠀⣿⣿⣿⣧⣽⣷⣶⣄⡀⠀⠀⠀⠀⠀⠘⢿⠄⢀⣤
⠀⠀⠈⠙⠂⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠄⠀⠀⢀⣀⡀⠀⣀⢿⣿⣷⡦⠤⠤⠤
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠐⠆⠀⠀⠉⠉⠑⠒⠒⠒⠐⠶⠶⠶⠴⠾⠦⠤⠶⣶⣶⣶⣶⣶⣤⣤⣼⣿⣤⣤⣤⣤⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⢠⣤⣤⣀⣀⣀⣀⣀⣀⢀⣈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⣤⡤⠤⢤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⣉⣙⣛⣛⣛⡛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⢀⣀⣀⡀⠠⠤⠤⠤⠄⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣃⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⠲⠶⠶⠶⠾⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣄⠐⠒⠒⠚⠷⠶⠶⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣛⣿⡿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣠⣤⣤⣴⣶⡾⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣷⠀⠀⠈⠉⠉⠙⠛⠛⢿⣿⡟⠛⢻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣴⣿⣷⣄⠀⢀⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⣾⡟⠿⣿⣿⣿⣿⠏⢸⡶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣷⣦⣬⣿⣿⣿⣿⠀⠐⠀⠁⠀⠲⠆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠸⠀⠉⠟⠻⣿⣿⡿⠀⠀⠛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⢀⣀⣤⡄⠀⠘⠃⣀⡀⣸⣿⠀⠈⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖⠀⠀⠀⠀⠀⠄⠀⠀⠀⣤⡔⠂⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡐⠻⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠃⡀⠀⣀⠾⠋⠁
⠀⠀⠀⠀⠀⠀⢘⠿⠟⠃⢤⠀⠀⠀⠀⠘⣷⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢹⣿⣷⠄⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠙⠃⠀⠹⢶⠄⠀⢠⣾⡿⠂⠀⢰⣿⣿⣧⡀⠙⠛⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡄⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⡄⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠁⠀⠁⢦⣄⡀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡄⠀⣿⣷⣆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣅⠀⠉⢹⣿⡇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠛⢁⣀⠀⠀⣿⣿⠈⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⢿⣿⠇⠀⣿⡟⠀⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢱⣶⣦⠀⠀⣠⣀⠀⢸⣿⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠋⠀⠈⠿⠿⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠀⠀⣾⣿⡷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⢰⣿⣷⠀⠀⠀⠀⢀⣤⡜⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠀⠈⠉⠉⠀⢀⣀⡀⠘⢿⡿⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⡀⣀⣄⣀⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠈⠉⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠃

⣶⣶⣶⣤⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⠿⠟⢻⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⡀
⠀⠀⢸⣿⡄⠀⠀⢀⡄⢰⣶⠀⠀⣤⡀⠀⠀⠀⠀⣿⣿⣀⠀⠀⢸⠿⡏⣾⣿⡆⣠⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡀
⠀⢀⣿⣿⣇⣴⣦⢸⣿⣸⣿⣿⣿⣿⣷⣦⣀⣤⣤⣿⣿⣿⡏⣼⣮⣿⣧⣿⣿⣿⣿⣿⣷⣷⣤⠄⠀⠀⠀⡯⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣐⣠⣿⣀⣀⣀⣀⣺⣿⣿
⢹⢿⣿⣿⣿⣿⣿⣿⣿⢿⢯⣘⠟⢻⠿⠿⠟⠳⠿⢾⣿⣿⠿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠉⠁⠈⠉⣤⣼⣿⣇⣤⣶⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠈⠛⠓⠋⠉⠻⠿⢿⢿⠿⠛⠉⠛⠿⠏⢁⠉⠙⠙⠙⠿⠋⠉⠃
⠀⠀⠀⠀⠀⠀⠹⠟⠀⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢰⠋⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠻⣌⣧⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⢹⡗⠲⢤⠀⠀⠀⠀⣈⣉⣒⡄⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⢼⡇⠀⢸⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣬⡇⢠⠀⡷⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⣹⡇⣸⠀⣷⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣭⡙⢲⣮⡉⠛⢶⣟⠻⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⢹⡏⠹⣎⢿⡙⣆⠀⠙⣦⡀⠈⠙⠓⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⠁⠀⠸⡎⣧⠘⡆⠀⢹⡷⡄⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠈⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠭⣒⣾⠀⠀⠀⡇⢹⡀⢻⠀⠈⣧⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢬⣑⠎⡟⠀⠀⢀⡄⠸⡇⢸⡇⠀⣿⢸⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⣰⠋⠙⠛⢦⣀⠈⢠⡇⠠⢄⣸⡇⠀⣷⠈⡇⠀⢹⣾⣿⠀⠻⢧⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢀⡿⠋⠲⢤⣀⠈⢳⢸⠀⠀⠀⢹⠃⠀⣿⠀⡇⠀⢸⡏⢿⡒⠂⠈⣣⠀⠀⠀⠀⠀⠀⠈⢧⠀⠲⠤⠄
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢹⢿⠀⡾⠀⠀⠀⣼⠀⠀⣿⠀⠁⠠⢼⡇⠈⣷⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠈⣇⠈⠑⠂
⠀⠀⠀⠰⣁⣀⣀⡀⠀⠀⡾⢸⢰⠇⠀⠀⠀⡏⠀⠀⠟⠀⠐⠒⠲⣿⠀⠘⣇⠀⠈⠙⡄⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⣴⠏⠀⠀⠉⢷⣴⢃⣿⡟⠀⠀⠀⢰⠇⠀⠀⠂⠀⠀⠀⠀⣿⠀⠀⢻⡀⠤⠤⣥⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣰⣿⢀⣀⠀⠀⣼⡟⠈⣿⠃⠀⠀⠀⣼⣶⣶⣞⣒⣲⣶⣤⣀⣿⠀⠀⢸⡇⠀⠀⢸
⠀⠀⢻⣧⡉⠀⠀⢰⣿⠁⣸⠇⠀⠀⠀⢠⠏⢻⣭⣍⣉⡉⠉⠉⠋⢹⡄⠀⠀⡇⠀⢤⣼⡄
⠀⢀⡟⠻⠿⢷⣤⠟⢹⢀⡿⠀⠠⡀⢠⡟⢠⠏⣸⠏⡝⠙⠛⠛⢻⠟⠀⠀⠀⡇⠀⣀⢨⡇
⠀⣼⠃⠈⠙⠉⠁⠀⡇⢸⡇⠀⠀⢀⣞⣴⠏⢠⡟⠌⠀⠀⠀⠀⡞⠀⠀⠀⢀⡇⠀⠀⢸⡇
⢰⠇⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⡀⣼⢹⠋⠀⡾⠁⠀⠀⠀⠀⣸⠃⠀⠀⠀⢸⠃⠀⠀⣸
⣞⠀⠀⠀⠀⠀⣰⠁⢠⡏⠀⣼⢣⠇⠊⠀⣼⠁⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡾⠀⠀⢀⡏
⠈⣿⠦⣄⣐⠄⠁⣀⠞⠀⢰⡏⢘⠀⡄⢰⠃⠀⠀⠀⠀⢠⣿⣿⣦⡀⠀⢰⠇⠀⠀⡼
⠀⠙⠳⢤⣉⠉⠉⠀⠀⠀⡞⠀⠀⠉⠛⠦⠤⣤⣀⣀⣀⣾⣏⣿⣿⡿⣤⡟⠀⢀⡼⠃
⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⣧⣤⣀⡀⠀⠂⠀⠰⠶⠬⣼⠇⠘⢿⣏⡙⠉⢀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠁⠀⠐⠇
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠀⡴⠺⠓⠆
⠀⠠⡇⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠺⣀⡁
⠀⠀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⢱⣶⣶⡾⢿⠷⢶⣶⣤⡀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠐⠀⠐
⣆⠠⠤⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠋⢀⣄⣀⠀⠈⠛⠁⠀⠀⠐
⣯⠙⠲⡄⡈⠀⠀⡆⠀⠁⠀⠀⠀⠀⠀⢀⡄⡀⢀⠀⠀⠉⠀⠀⠀⠸⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡔
⠙⡤⠀⠀⠀⠀⠀⡇⠰⠄⠀⠀⠀⠀⠀⠘⢿⡿⡇⠀⢀⠀⢀⢠⠂⠀⠀⠀⠘⠀⣠⠀⠀⠀⠂⠀⠈⡚⠀⠀⡄
⠄⠗⠓⡄⢀⡄⠈⡗⢠⡌⠂⠀⠀⠀⠀⠀⠀⢿⠁⢂⠏⢠⠀⠈⠀⣠⠄⢃⠄⠳⠟⠂⠀⠀⠀⠑⣌⠱⠚⠂⢀
⠀⣰⠲⡄⠀⡅⢀⡟⢾⣇⠀⠀⠀⠀⠀⠈⠒⠠⠃⠤⠆⢰⠂⣤⡰⠇⢸⠀⡄⣸⡞⠀⡀⠀⠀⠀⠀⠙⠖⠀⠈
⠀⠁⠀⢱⢀⣤⢨⡇⢸⢻⣎⠁⠀⠀⡄⣼⠆⡷⠀⠀⢌⢙⠂⠃⢍⡁⠻⣤⠳⠝⠃⠀⠑⣤⢀⠀⢀⠀⡀⠁⠆
⠔⠄⠈⠒⠋⢗⣠⡇⢂⠉⣿⠳⠀⠄⠀⡏⠀⠈⠳⣔⡀⢈⣀⡀⠀⠀⢾⣾⡧⢤⠀⠐⣤⠒⡖⣆⡓⠀⠈
⠀⢾⠾⠔⣶⣟⢙⡟⠋⡐⠿⠀⠇⢐⢟⡟⢠⣄⡒⠊⠭⣹⠀⠀⡴⢀⠈⣽⣦⡔⡀⢠⣤⢰⡿⣄⡁⡀⠀⠘⢣
⣥⢰⣿⡦⡰⠿⢾⣷⠄⠀⢣⠐⣌⣩⣿⣤⣜⣀⠀⠀⣀⡽⢒⣟⢠⣽⠈⡄⠤⢼⠃⠆⠀⠀⠀⠀⢳⠏⣔⣿⣠
⡤⢺⢿⣷⠟⠻⡖⠃⠀⢀⡈⢸⣫⣿⣿⣿⡿⠉⢄⡀⠹⡳⡷⠆⠻⣿⣿⠗⠂⡬⠜⢓⠠⢀⣯⣾⣁⣴⣣⣄⠈
⠁⢼⣗⣤⣄⢀⣥⣺⣶⡦⣦⣻⣉⢡⣶⠚⣗⣀⡴⠀⣤⣟⣆⡁⡘⠻⡞⠂⠀⢃⠰⢺⠐⠿⡝⣜⣧⢿⢷⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⢈⣵⣶⣶⡶⢒⡶⠦⣴⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⣞⡻⠯⢩⠩⠍⠭⠉⠛⠛⠒⠶⠤⣄⡀⠀⠀⠀⢻⣶⣬⡿⢼⣷⣇⣼⡇⢺⠉⣽⣶⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⢉⠋⠓⠒⠦⢄⣀⠉⠛⠢⣄⣀⠿⡛⢦⡀⡇⠈⠻⣿⣿⣾⣡⡿⢹⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢪⣿⡷⠛⠁⠀⠀⠀⢀⡀⠐⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠉⠒⢄⡈⠙⢆⢨⣾⣷⣿⣾⣿⢋⣛⣻⣟⡃⣀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠫⢱⡿⠋⠀⠀⠀⠀⠳⣦⢄⠙⢶⣄⡀⠀⠀⠈⠁⠀⠀⠀⠄⠴⠖⠀⠀⠀⠙⢆⢸⣿⣿⣿⠛⠻⠿⠿⠯⣿⣿⣯⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢇⣿⠁⠀⠀⠦⢤⠤⠤⠀⠀⠀⠀⠹⢯⣿⣄⣀⡀⣀⣴⢶⡾⠋⠀⠀⣀⡀⠀⠀⢹⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⡀⣇⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠨⣿⣿⣿⡿⣯⠁⠈⠁⠀⠉⠁⠀⠀⠀⠀⠈⣿⡇⣿⣏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⣇⠹⡄⠀⢀⡖⠒⠀⠀⠐⠂⠀⡲⠌⠁⠐⠶⠖⠛⠁⠀⠀⠀⠀⠂⠐⠲⡆⠀⠀⢀⡿⢰⣿⠹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢤⢱⠀⢀⠙⣦⡀⠀⠄⣀⠀⠀⠀⢠⡤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡐⢦⣀⠀⠀⠀⠀⣠⠞⣱⡿⢧⠘⡌⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⢣⡘⢀⢈⠛⢶⣤⣀⠑⠢⢀⣰⠀⠀⠀⢀⡜⠀⠀⠐⣦⠒⡆⠀⠀⠐⢋⡁⠤⠒⣋⣥⡾⠋⡅⢈⡜⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡙⢆⠱⣄⠈⠀⠀⠈⠛⠿⣶⣤⣈⡁⠒⠒⠠⠧⠤⠤⠤⠮⠦⠛⠒⠂⣉⣡⣤⡶⠟⠋⠀⠠⠞⣠⠎⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠳⢬⡓⢤⣀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠶⠶⢶⣶⣶⣶⠶⠾⠿⠛⠋⠉⠰⢀⡴⠓⢈⡴⢚⣡⠞⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣄⡙⠲⢬⣕⡲⠤⣀⡀⠀⠀⠀⠀⠄⠰⢠⣖⠀⣤⡔⡀⠺⠢⠘⠶⢃⣠⣴⣚⣽⡾⠋⣡⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣀⠀⠉⠙⠷⢿⣭⣍⣙⡓⠒⢶⡶⠶⢶⡶⠶⢶⡶⣶⣶⣾⣿⣿⠿⠟⣋⠁⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⠀⠀⠈⠉⠙⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⣡⡴⢖⣫⣵⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⣶⣭⣭⣭⣄⣀⣀⣒⣒⣂⣛⣹⣭⣭⣶⣾⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠈⠁





⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡟⣉⣿⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⣿⣿⣤⣄⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣰⣿⡇⢴⣿⡦⣿⣿⣿⣿⣿⣿⣦⣤⣤⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣾⣿⡄⢸⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢠⡄⣿⣿⣿⣴⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠀⢸⣇⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠚⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⡿⠉⢿⠀⠀⠀⠀⠉⠙⢿⣿⣿⠋⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠘⣿⡟⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠛⠉⠉⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠛⠛⠋⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠟⠘⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠐⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠑⢤⡀⠙⢄⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡙⢮⠢⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣀⣈⣈⠀⠐⢅⠈⠀⢁⡤⠞⢁⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠍⢢⠔⢁⠔⠉⠀⠀⠀⠈⠒⢄⠡⡒⠁⠐⢊⠡⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠥⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠠⠀⢈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠚⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⡀⠀⠀⠈⠀⢣⣀⣤⣤⣄⣀⡤⠴⠚⠉⠉⠉⠉⠉⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠚⠫⣉⢀⣀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠤⠴⠚⢉⠰⠲⠁⢀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⡻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢮⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢰⣐⣋⡂⠆⠀⠀⠐⢉⠀⠜⠅⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠦⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠍⠭⠽⡟⠉⠉⢰⡄⠒⠐⠒⠒⠋⠉⠒⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠻⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⣐⣂⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⠿⠿⠿⠿⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣲⠀⠒⠒⠒⠒⠒⢲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣤⡶⠶⣦⣤⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠉⠀⠀⠄⡀⠈⠉⠛⠳⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠰⢞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣠⡴⠶⣤⣀⡀⠀⠀⠀⢀⠤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠈⠀⠀⠀⣠⠶⠛⠋⠁⠀⣀⠀⠉⠛⠳⡦⠀⡀⠀⢠⠀⠀⢠⣦⣀⡀
⠀⠀⠀⠀⢴⠖⠛⠉⢁⠀⠀⠉⠀⠈⡇⠀⠀⠀⢠⠊⠀⠁⠀⠀⡀⠀⡄⠀⠁⠀⠈⠀⠀⠘⠀⠈⠉⡟
⠀⠀⠀⠀⠸⡀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⢰⠀⠀⡆⠀⠀⠀⡄⠀⠐⠀⠂⠀⢰⠀⡇⠀⠇⠀⠰⠂⢠⠁⠀⠀⠀⡇
⠀⠀⠂⠀⠀⢇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⠇⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⠀⠀⠀⢸⣀⣀⣀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠒⠒⠒⠲⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣲⠶⠖⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣹⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣀⣀⣀⠀⢀⡀⠀⢸⣷⢸⣟⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⢺⡗⣿⢸⣿⡿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢿⠿⣏⣿⣹⡿⣿⢹⡆⢸⣷⣺⣗⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠛⣿⣿⡟⢻⣿⣧⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠠⣿⣿⠿⡏⣿⣽⡿⣿⢿⡇⢸⡯⢼⡧⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⡿⢾⣿⣿⣿⢸⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⣐⣯⣽⣭⣿⣿⣿⡟⣯⣽⡇⢸⣯⣹⣯⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠽⣿⣿⡯⢽⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⣿⣿⣿⣟⣻⣛⣇⣿⣻⣟⣿⣻⡇⣸⣗⣺⣗⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⢻⣟⢻⡖⢲⡞⣿⠛⣿⣿⣿⣶⣷⣿⣻⣟⣷⣾⣿⣿⣿⣾⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣾⣿⣿⠭⣿⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⣿⠭⡯⢽⡯⢽⡯⢽⠭⣿⣯⣽⣬⡷⣿⢸⡷⣿⢿⣿⣿⣿⣽⣿⣿⡇⠀⢴⣶⢶⣶
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠭⣿⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⣿⠭⡯⢽⡯⢽⡭⢽⠭⣿⡯⢽⣭⣿⣿⢿⣯⡿⢿⣿⡟⣿⣾⣿⣿⣧⣤⣾⣽⢿⣿
⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⠭⢽⣿⡯⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⣿⠭⡯⢽⡯⢽⠭⢽⠭⣿⣟⣻⣽⣯⣿⣻⣏⣿⣿⣿⡗⣿⣿⣿⢿⣿⣿⣿⣽⣿⣽
⠀⠀⢸⣿⣿⣿⡟⣿⣿⢻⣿⣿⠭⢿⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⣿⢭⡯⢽⡯⢽⡯⣽⠭⣿⣷⣾⣶⣿⣿⣿⣿⣷⣾⣿⡗⣿⣿⡯⣿⢽⡯⣿⣽⢿⣿
⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⠭⢽⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠩⣿⢹⣿⢹⡏⢹⡇⣿⠁⣿⣟⢻⣶⡇⣿⣻⣟⣷⣾⣿⡧⣿⣿⡇⣿⢸⡇⣿⣿⢿⣿
⣾⣿⣿⣿⣿⣿⣇⣿⣿⣼⣿⣿⠭⣿⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣺⣟⣺⣗⣻⣗⣿⣓⣿⡿⢿⠿⣯⣿⢽⡿⣿⢽⣿⡇⣿⣿⡇⣿⢺⡇⣿⣿⣻⣿
⣟⣿⣿⣟⣿⣿⡷⣿⣿⢸⣿⣿⠭⢽⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣒⣿⣚⣗⣺⣗⣺⣗⣻⣒⣿⣿⣿⠿⣯⣿⢹⡿⣿⢿⣿⣇⣿⣿⡧⣿⢼⡧⣿⢾⣻⣿
⣯⣷⣿⣧⣿⣿⣷⣿⣿⣻⣿⣿⠭⣿⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣒⣿⣒⣗⣺⣗⣺⣓⣺⣒⣿⣿⣾⣾⣿⣿⣿⣟⣿⣾⣿⡇⣿⣿⣇⣿⣸⣇⣿⣾⣻⣿
⣟⣟⣟⣿⣿⣿⣿⣿⣿⣻⣿⣿⠭⣽⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣒⣿⣒⣗⣺⣗⣺⣓⣺⣒⣿⣟⣻⣿⣗⣿⣻⣟⣿⣻⣿⡇⣿⣿⡗⣿⢸⡇⣿⣿⣻⣿
⣟⣿⣟⣷⣿⣿⣏⣿⣿⢿⣿⣿⣭⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣷⣾⣷⣾⣷⣾⣶⣿⣿⣿⣶⣧⣿⣹⣏⣿⣾⣿⣏⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⠐⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢁⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⢠⡄⠀⠈⣿⡇⠀⠀⠀⣿⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦
⡀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⠀⠀⠀⠀⣾⣷⠀⠀⣿⡇⡇⠀⢠⣿⣿⡇⢨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠄⠀⣷
⣧⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠁⠀⠀⣀⣿⣿⠀⢺⣻⡇⡇⠀⣺⣿⢿⡇⢸⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠠
⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⠀⣧⣿⣿⠀⣼⣿⣷⣿⠀⣯⣿⣰⣿⣼⣿⣤⠾⣧⣾⣿⣆⢀⣄⠀⠀⠀⢀⡄⠀⠀⠀⠀⣿⠀⢸⣿⠂
⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣯⣜⣣⣤⠌⣧⣼⡇⠀⠀⡇⠀⣿⡄⣼⣿⡄
⣿⡇⠀⠀⣿⣿⢚⣿⣿⣿⣿⣿⣇⣀⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣦⣷⣴⣽⣷⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠿⠋⢻⡿⣥⣮⣿⣿⣿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣍⣉⣿⣯⣿⣿⣿⣿⣿⣿⡷⠝⣿⣻⣿⣿⣿⣿⣟⣛⠿⠿⠿⠿⠿⠿⣿⣿⡿
⠀⠀⠀⠈⠀⠀⠛⠇⠿⣿⣿⠋⠁⠀⢹⣿⣿⣿⣿⣿⣿⡏⠉⠉⠛⢉⣟⣻⣏⠉⠉⠻⣷⣿⡧⢤⣾⣿⡿⠛⠻⠿⠛⠻⠿⠿⠲⠷⠾⠾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠂⠀⠀⠈⠉⠉⠛⠙⠛⠙⠃⠀⠀⠀⠀⠀⠋⠁⠀⠀⠈⠀⠀⠀⠀⠨⢽⠿⠛⠙⠶⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣸⡷
⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠂⠀⠛⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣧⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠧⠾⠿⠀⠹⠦

⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⣾⣿⣿⣇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠿⠛⡿⠿⠿⠿⠿⠟⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⠀⠐⠉⠘⠛⣿⣿⣿⡟⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⡹⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠒⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣷⡼⠂
⠀⠀⠀⠀⠀⠀⠀⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣶⣒⣛⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠸⣿⠿⣦⣄⠂⠄⠴⡤⢄⠀⠀⠀⠀⠀⢀⣾⠋⡀⠠⠀⠄⠉⠑⡍
⠀⠀⠀⠀⠀⠀⢰⡇⣰⣯⢁⢛⣤⣤⣌⢘⣷⠀⠀⠀⠀⣾⡇⢀⠉⢲⡄⠀⣀⡤⠈
⠀⠀⠀⠀⠀⠀⣸⡃⣧⣽⡀⠨⠤⢽⠈⢧⠄⢌⡀⠀⠀⣿⡇⠰⠀⠀⠛⠉⠁⠐⠀⠄
⠀⠀⠀⠀⠀⠀⢹⡇⢻⣧⣀⡠⠀⡠⠤⠞⡀⣼⡇⠀⠀⠘⣷⣄⠘⠠⠄⠠⠈⠀⠈⠀⠀⠀⠀⣀⢀⠀⠀⠀⣼⣙⠢⡀
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠻⣿⣿⣾⡥⣬⣤⣾⡷⠃⠀⠀⠀⠈⠙⠿⢶⣶⣶⠶⠏⠀⠀⣠⠖⠉⢡⠀⡤⠁⠀⣿⠿⣶⡃
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠈⠙⠛⠿⠿⠟⠋⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⣌⠃⠀⡄⠀⠘⠃⢉⢧⠙⣧
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠖⠛⠋⠉⠉⠙⠛⣦⠀⠀⠀⢀⣄⠀⠀⣯⢈⡁⠀⠀⠷⠄⣀⣈⣈⢸⠀⢸⡆
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⣠⠴⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⠟⠀⠀⠀⠸⣿⣿⣄⣿⣎⢡⡀⠀⠀⠀⢠⣄⡹⡾⢀⡞⠁⢀⡴⢦
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢉⣰⠖⠛⠛⠛⠛⠛⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⢌⠻⣯⣻⣶⣄⣀⣈⣛⣀⣼⠟⠁⢼⣁⣀⡜⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⢐⡋⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⣻⣿⣍⠉⠛⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠶⠚⠒⠲⠦⠤⠦⠤⠤⠴⠶⠖⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⣀⣼⣍⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣝⣫⣍⣉⣏⣹⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠙⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⣼⣿⠀⠀⠀⠀⢳⡄
⠀⠀⠀⢀⡀⠀⠀⠀⠠⢤⣀⠀⠀⣿⡟⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠈⠑⢞⡶⢦⣄⠉⠁⠀⠘⠁⠀⠀⣠⠀⠐⣿⡇⠀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠙⠷⠄⠀⠀⠀⠀⠀⠟⠀⠀⠹⡇⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⢦⣄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠹⠆⠀⠠⡀⢀⡇⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣶⠶⠖⠀⠀⢀⡤⠀⠀⠀⠀⠈⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠁⠄⠀⠙⢿⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠒⠛⠛⠻⠷⠶⠤⢀⠀⠂⠀⠁⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠈⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣤⠀⠀⠀⠀⣠⣤⡄
⠀⢸⣿⡇⢀⣀⢰⣿⣿⣿⡆⣰⣶⡀⠀⠀⣀⡀
⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣷⣾⣶⣿⣷⣰⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣦⠀⠀⠀⢸
⣄⣾⣿⣿⣿⣿⣾⣿⣿⣿⣧⣿⡿⠿⣏⣻⢿⣿⣿⣿⣷⣼⣤⣤⣿⣿⣿⣦⣿⣿⣿⡇⢀⣀⣿⣀⢀⣀⢸
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣯⣭⣭⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⣿⣿⠿⠯⠙⠿⠿⡿⠟⠟⠛⠛⠟⠿⠛⣿⣯⣟⠋⢹⣟⠟⠛⢿⣿⣿⣿⣿⣿⣻⡟⢟⠛⠻⠿⠿
⠏⠉⠉⠀⠁⠀⠀⠤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠓⠈⠉⠉⠀⠀⡈⠁⠀⠙⣍⢩⠝⣠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠂

⢦⣼⠀⡌⣿⣿⡆⢻⡰⣴⣃⡶⠋⠹⢲⡼⠀⢠⡸⠀⠀⢸⡄⢴⠥⣰⣷⡞⢻⡈⢸⣿⣿⣧⢸⣿⣀⢼⢿
⣛⣿⣇⡿⣿⣿⣷⠋⣷⠋⢸⠀⣥⠟⡼⠓⠂⢩⡤⣰⠀⠈⢷⡘⠮⢼⡞⣷⠀⢻⣾⣿⣿⣿⣾⣿⣸⣽⣿
⠘⢧⣸⣧⣿⣿⣿⣼⠉⣧⡯⣂⡴⠚⠁⠀⢐⠘⠆⣬⡳⢦⡈⢷⡤⣉⣿⡜⣄⡄⠹⣿⣿⣿⣿⣿⢉⣿⣿
⡦⣹⡟⠈⣿⣿⣿⣇⣾⣿⡞⠉⠀⢬⠌⣠⠀⠤⠄⠰⠭⠵⢿⣿⣷⣌⣐⠿⣎⠻⣄⣿⣿⣿⣿⣿⣼⣿⡷
⣷⣼⣷⣞⣿⣿⣿⠞⢡⡿⢀⣤⣉⣤⣞⠚⠯⠀⠀⠉⠊⠀⠀⠀⠉⠻⢾⣝⠾⣿⡟⣿⣿⣿⣿⣿⣿⣯⣏
⠸⣏⣿⣿⣿⣿⣿⢸⡿⣷⣩⣠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣿⣿⣿⣿⣿⣿⣦⣾
⠀⢹⣏⣸⣿⣿⣿⣿⣋⡽⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢹
⢀⡾⠙⣿⣿⣿⣿⣏⠿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⢸
⣼⠁⢰⣿⣿⣿⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⡇⢸
⣏⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⣿⣿⣿⣿⣧⢠
⣿⡟⢰⣿⣿⣿⣿⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠤⢻⣿⣿⣿⣿⣿⣿
⣿⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⡏
⣿⣇⢸⣿⣿⣿⣿⣧⡀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡇⠉⣿⣿⣿⣿⣿⣿⡇
⠘⣿⣾⣿⣿⣿⣿⣿⡘⣌⠂⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡇⣴⣿⣿⣿⣿⣿⣿⣇
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⡿⣝⡞⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⢸⡷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢹⣸⣿⣿⣿⣿⢿⣿⣿⡏⣿⣿
⢋⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡕⠀⠀⡀⢀⣀⣀⣠⣤⣤⡤⣶⣷⣿⣿⡟⢻⣿⣿⣆⣿⣿⣧⢻⣿
⣾⠃⣸⣾⢻⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣦⣴⡷⠛⢋⣭⢿⣉⣴⢿⣿⣿⣿⡿⣠⣾⣧⢻⣇⠟⣿⣞⣷⣻
⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣷⡟⢣⣾⣱⣷⣟⣿⣿⣿⢇⣿⣿⣿⣾⣿⣷⣻⣿⡟⣿
⣷⠀⣿⡟⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣷⣿⣿⣿⣿⡏⣽⣿⣿⠸⣿⣿⣿⢣⣿⢹⣿⣿⣿⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇
⠀⠀⠀⠀⢠⣄⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣭⠁
⠀⠀⠀⠀⠈⢁⡙⠛⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡛
⠀⠀⣤⠐⠻⠿⣿⣷⡦⢤⣤⠀⠀⠀⠀⠀⠀⣀⠠⢤⣴⣶⣿⣿⣿⡇⠀⠉⠀⠀⠀⠀⢀⣁⣤⠄
⠀⢀⣼⣴⣶⡄⣀⠈⠻⣏⡀⠀⢀⡀⠂⠉⢁⣤⣖⣋⣩⠄⠀⣿⣿⡇⠸⠀⠀⠐⢤⣾⣟⠋⣽⠷⠉
⠀⠈⠉⣹⣵⣄⠄⣭⢀⡈⢉⣰⣾⣶⣶⣿⣿⣿⣿⣿⡟⠀⠀⣼⣯⡄⢰⡄⠓⠒⠛⣋⡉⣿⠨⠟⢲⣮⣿
⠀⠀⡽⢿⣿⣿⣿⣿⠁⢠⣆⣿⣿⣿⢸⣿⣿⡇⢶⣿⡇⠀⣤⢸⡿⠇⢸⠀⢸⠀⣾⣇⣸⣧⣤⣌⣺⣿⣿⣀
⠀⢶⣷⣾⣿⣿⣿⢋⣰⣿⣿⡿⠟⣻⣿⡿⣿⣿⣿⣿⣧⠀⣉⣼⣿⣷⡘⠶⠌⣴⣝⢿⣿⣟⣟⣿⣿⣿⣿⣿⡇
⠀⣭⣭⣭⣥⣿⣿⡿⠏⠉⠉⠈⠛⠛⠂⠒⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⠂⡸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⡝⠻⢧⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⡏⢠⡎⠹⡆⠀⠀⠋⠒⢤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠡⡘⢷⣠⡇⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⠸⣿⡀⠀⠈⡢⣝⠛⠧⢀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠷⠾⠮⣅⣉⡒⣤⣀⠁⠓⠺⢅⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⠞⠟⠓⣲⣦⣤⣄⡀⠈⠙⠺⣟⡲⣦⡂⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⢀⣽⡿⠛⠉⠈⠹⣆⠀⠀⠀⠹⡄⠙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⣸⣟⠂⠀⠀⠀⢠⡿⠀⠀⠀⠀⡇⠈⠈⢷⣁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡋⠉⠙⢻⠒⢤⣾⡟⠁⠀⠀⣠⠞⠀⢀⠀⢸⣿⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⢄⡈⢿⡻⠶⠃⠀⠃⢈⢿⠿⢷⣿⣅⡀⠀⠈⢀⣾⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⢦⢹⡿⢽⣶⣄⣀⠀⠈⠁⠀⠀⠀⠈⠉⠓⠶⢮⣈⡴⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠙⣿⠁⠄⠈⠹⣿⣭⡒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣦⡀⠀⠀⠻⣷⡙⢦⡉⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⠀⠀⠀⠀⠀⢈⡄⠀⠈⡆⠉⢳⠢⠄⡘⢷⡼⠟⠋⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢤⣝⢦⠈⠳⢤⣀⠀⠀⠀⢇⡀⠀⢱⣀⣨⠦⠖⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡀⠀⠀⠈⠓⢄⣀⠼⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⢀⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠄⠀⠀⠀⠀⠀⠀⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⡗⣊⣉⣦⢠⠤⠤⠤⠀⠛⠚⠉⠙⠛⠓⠒⠠⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣭⣭⡵⠤⠤⠤⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠐⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢀⣀⠀⠀⠀⠀⡴⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠺⢍⠀⠀⡄⠀⣀⡈⠃⠀⢀⢿⣦⡆⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢇⣀⠙⠓⠀⠉⠀⠛⠀⣤⡀⡩⡏⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣾⠃⠼⠟⠀⠀⠀⣼⠀⠀⠀⠉⠁⢑⢽⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⢄⠐⣿⡰⣶⠂⠀⠀⠀⠟⠦⣀⡠⠈⠉⢹⢸⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢿⣧⡱⣿⣧⠁⢔⠀⠀⠀⠀⠀⠈⠉⠺⠗⡌⣸⠀⠀⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢿⣷⣜⢿⣷⣍⠀⢴⠀⣤⣤⠐⠿⣢⠞⣹⠃⠀⡶⠋⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠻⣿⣦⣻⠷⣵⣢⡤⣭⣍⣐⣬⡴⠛⠁⢀⡿⠁⠀⠀⣴⠏⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡿⣿⣦⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠈⣷⣦⣤⡾⠁⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⢠⣤⠿⠝⠛⠛⠺⢶⣦⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⢠⣿⠁⠀⠀⣖⠛⠓⢦⡀⠉⠉⠙⠓⠿⠿⠷⠧⠶⠶⠶⢶⡶⣶⠶⠽⠟⠛⠋⠁
⠀⠀⠀⠀⠻⠷⣾⣿⠿⣷⣀⣼⠇

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠐⢲⣦⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣤⡖⠂⠀⠑⣲⣆⣀⢀⣀⣀⣀⡀
⠀⠀⣯⣉⣉⣛⠛⠿⡿⠿⢛⣛⣛⣛⣻⣯⣟⣻⣛⣯⣿⡷⠀⢠⣯⣉⣉⣛⠛⠿⡿⠿⣛⣟⣛⣛⣻⣭⣟⣛⣻⣽⣿⡇
⠀⠀⣭⣭⣭⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠛⠋⠉⠉⠉⢛⠛⠛⠛⠙⠛⢛⡉⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠉⠉⠉⠙⣟⠛⠛⠋⠉⠉⢙⠉⠛⠛⠙⠛⠛⠃
⠀⠀⠀⠀⠀⣦⠀⣠⣾⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣿⠀⠀⠀⠀⠀⣿⣄⢰⡇
⠀⠀⠀⠀⠀⣹⡞⣫⣾⣇⠠⠤⢤⡀⢸⣯⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⣡⣿⣄⠤⠤⣤⡄⢿⣯⣻⣧
⠀⠀⠀⢀⠔⢻⣾⣿⠧⠀⢴⡶⠾⠯⠤⢿⣿⣿⠳⣄⠀⠀⠀⠀⠀⡠⠞⣿⣾⡿⠅⠀⠤⡴⠯⠭⠌⢿⣿⣏⠳⣄
⠀⠀⠖⠁⣴⣿⡿⠁⠀⠀⠀⠠⠤⠤⠾⠯⣿⣿⣷⡎⠳⡄⠀⠠⠊⢠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⢯⣿⣿⣷⡌⢳⡄
⠀⠀⣴⣿⡿⣿⡉⠉⠉⣉⢉⡉⢉⢉⠉⣉⣛⣻⣿⣿⣇⡈⠀⠀⣶⣿⣟⣟⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣻⣿⣿⣧⡁
⠀⠀⣿⣟⣼⠩⠤⠤⠄⠀⢀⣀⣀⣀⣀⡈⣈⣸⣿⣿⣿⣷⠀⠀⣿⢏⡾⠥⢤⡤⠤⠄⢀⣀⡀⣤⡀⠀⣤⣼⡿⣿⣿⡇
⠀⠀⣟⡾⣉⣀⣀⣀⣀⠀⠀⠐⠀⠀⢀⣤⣄⡍⢭⣿⣿⡿⠀⠠⢏⡟⠁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠭⢭⣽⣿⡇