Generous Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠟⣛⣮⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣂⠀⠉⡁⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡉⢀⠞⣠⠞⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠷⠊⠁⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠍⡡⡄⠇⢰⠀⠀⡴⠂⠒⠀⠀⡀⠀⣀⡴⠋
⠀⠀⠀⠀⠠⣆⣹⣤⣿⣧⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡑⡸⠀⠇⠀⠀⣿⢀⣾⣧⣀⣀⣴⡾⣕⠋⠁
⠀⠀⠀⠀⠀⡽⡻⣿⡿⣟⣷⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠔⣰⠁⠀⠀⠀⣀⣽⣿⣿⠏⠀⠀⣿⢹⠳⣅⠀⠘⡄
⠀⢠⡖⠒⢢⣴⣷⣹⡿⣏⣿⣅⠀⣰⠆⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠁⣠⠎⣠⠃⠀⠀⢀⢜⡡⢂⣿⣯⠀⠀⠁⠀⠘⠓⠀⠀⠀⠇
⢀⠼⢃⣤⡚⡯⡟⣷⣿⣎⣿⣛⣷⢿⣟⣝⣋⡇⠀⠀⠀⠀⢀⣀⣠⠤⠖⠈⠁⠀⡀⠞⠁⡤⢁⢀⡤⢊⡥⠊⣤⠟⠁⢿⣧⡒
⠈⡖⠦⢄⡀⠻⡟⠾⣿⡟⢷⣽⣿⣆⠴⣹⣉⡉⠉⠉⠉⠉⠉⠀⠀⠀⣀⣤⠖⠋⢀⣠⡾⠛⠉⣡⠖⢉⣤⠾⠃⠀⠀⠈⠻⢿⣿⠧
⠀⠃⠀⠀⠀⠀⠉⠈⠋⠳⣦⣿⣿⣿⣝⣶⣬⠁⠚⠒⠦⠦⠴⠾⠿⠿⠽⢒⣳⠾⠋⠁⣠⠴⣋⣵⣾⠟⠁⠀⠀⠀⠀⠀⠀⢠⢿⡇⠦
⠀⢰⡀⡄⠐⠀⠀⠅⠁⠀⠈⠹⣷⣾⣿⣿⣿⣿⣶⣶⣦⣦⠶⠶⠶⠒⠋⢁⣠⣤⣶⠾⢗⣯⠿⠻⠴⠶⣶⣶⡶⣶⣶⣶⣶⡧⣾⠁⠄⠀⢀
⠀⠘⠃⠈⠁⠀⠁⠀⠠⠤⠄⠀⢿⣟⣿⡎⠉⢉⠛⠛⠛⠻⠿⠿⠷⣿⣿⡭⠭⠷⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⢉⣠⣴⠛⢙⣃⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠒⠈⠁⠀⠈⠁⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡄⢠⣄⣠⢀⢀⡀⢀⢀⡀⠀⡆⢀⢀⢀⣀⡀⡀⣀⣀⠀⡄⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣈⣁⣈⣀⣁⣀⣀⣁⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⠻⢦⡀⠀⠀⢀⣤⠶⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠈⢷⡀⠀⡾⠁⣾⣿⠆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠘⣦⠈⢷⡈⣷⡐⢾⣿⠇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠸⣆⣸⡇⠈⠛⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠈⠻⣄⠀⠀⣴⠞⣫⣝⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠙⢷⣄⣈⡷⢸⡇⢘⢿⣯⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢡⣾⣦⡈⢳⡄⠈⠉⠁⠈⢷⣌⣻⢟⣡⡞⠁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣘⢻⣦⡀⢸⡇⠀⣠⡤⠦⣤⣌⠉⠉⢁⡴⠛⢉⣭⡉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⣻⣛⣠⠞⢁⡾⠁⣶⣿⢦⠙⣷⠀⣾⠁⠐⣯⣿⠙⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⢸⡇⠀⣝⣿⣦⠀⣸⠀⢿⡀⠠⣌⣿⣿⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣉⣛⣡⡴⠋⠀⠈⠳⣤⣈⣋⣡⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠯⠀⡞⠉⠙⠴⠋⢹⠀⠐⣶⠂⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠾⠀⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠱⡄⠀⠀⠀⡎⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠢⡀⢠⠏⠸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⡸⠀⠀⠀⠀⢀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠏⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠇⠀⠀⠀⢀⠔⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⡠⠚⢁⠔⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠀⠀⠀⢐⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⡠⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠁⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠠⡏⢶⠁
⠀⠀⠀⠀⠀⠀⠎⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⢰⢺⡀⡦⣠⢄⠀⠀⠀⠀⠀⠀⠀⠁⢣⠈⢦⠒⡤⣀⢄
⠀⠀⠀⠀⠀⡀⠣⣤⡤⠃⠀⠀⠤⡤⡄⠀⠀⠳⢤⠴⠓⡄⠀⠀⠀⠀⠀⠀⢸⠀⡆⢡⠙⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⡰⠁⠀⠀⢆⠀⠀⠀⠁⡠⠁⠀⠀⢀⠆⠀⠀⠘⡀⠀⠀⠀⢀⡀⠸⠀⣅⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠀⠀⠀⠀⠀⠼
⠀⠀⠀⢠⠃⠀⠀⢠⣈⠲⢤⣄⡀⠀⣠⡤⠔⢋⡴⠀⠀⠀⢡⠀⠀⠀⠸⡈⠲⡄⢆⠹⢁⢿⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢀⣠⣶⡷⠊
⠀⠀⠀⢸⠀⠀⠀⢀⠈⠑⠦⢄⠽⠚⠤⠴⠚⠉⠀⠇⠀⠀⢸⠀⠀⠀⠀⠑⢄⠀⠈⠒⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁
⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⢸⠀⢸⡆⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣾⠀⢸
⠀⠀⠀⡼⠀⢸⣧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡇⢀⡇⡏⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡇⢇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠈⢣⠞⢉⠇
⠀⠀⠀⠙⠋⠀⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢈⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠢⡀⢀⠔⠒⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠂⠤⠤⠤⠤⢤⠀⢀⡰⢋⡜
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡀⠀⠀⠀⠣⡀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⡴⢦⠀⠀⠀⠐⠀⠀⠀⠀⠀⠴⠒⠒⠛⠒⢉⡴⠋
⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⢀⡀⠀⠉⠲⣄⣠⠖⠉⠀⠀⢀⡀⠀⠀⡇⡘⠀⠀⠐⢦⣀⡀⠀⢀⣠⠤⠤⠴⠖⠒⠉
⠀⠀⠀⠀⠀⠑⢄⠈⡙⠣⢄⠀⠀⣴⢏⡦⠀⠀⠈⡆⠱⣄⢪⠘⣆⠀⠀⠈⠁⠀⠀⠀⢠⠎⢨⢀⠜⢀⠇⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠎⡨⢉⠊⢠⠴⣍⠏⠀⠀⠀⠀⠘⢄⠀⠙⠆⠈⠑⢢⡀⠀⠀⡴⠚⠁⠠⠛⠁⣠⠎
⠀⠀⠀⠀⠀⢠⢃⡔⣡⢋⠔⡰⠓⠁⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠉⠯⡾⣡⣎⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⠠⠤⠤⠀⠠⠤⠄⣀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢮⢠⠖⠢⡠⠒⡄⢤⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣤⠉⠑
⠀⠀⠀⠀⠀⠘⠃⠀⠈⢆⠀⠀⠰⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠄⡠⠃⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣠⠃⠀⠀⠀⣄⢀⣄⠤⠀⠀⠀⠀⠀⢀⡤⢊⠔⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠉⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡔⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡤⣀⠂
⠀⠀⠠⠂⠒⠀⠀⠀⠀⠀⢀⡎⠁⠆⠀⠀⠀⠀⠀⢀⢴⠰⣄⡀⠀⠀⠀⠀⠀⠑⢣⠐⡄⣄⣀⡀
⠀⢀⠒⢤⠔⠀⠀⠠⢤⠀⠀⠳⠴⠛⠀⠀⠀⠀⠀⢨⠈⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠘
⠀⠂⠀⠀⢢⡀⠀⠀⠂⠀⢀⠔⠀⠀⠘⠀⠀⠀⡄⢄⡠⠢⡤⢤⠀⠀⠀⠀⠀⠀⢌⠁⠀⠀⠀⣠⣀
⠸⠀⠀⠈⠒⢬⣁⡢⢖⣉⠥⠚⠁⠀⠀⡇⠀⠀⠑⢄⠁⠁⠕⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠋⠁
⢈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠈
⠈⠀⣧⠀⡆⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⡇
⡆⢀⢻⢀⠇⠀⠀⠀⠀⠀⠀⠘⡄⡏⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡠⢤
⠓⠊⠐⠜⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠢⣀⠖⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢇⠀⠀⠁⠀⠀⡜⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⠉⣀⣈⣵⠠⠔⡡
⠀⡀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠈⠓⢄⢀⡤⠊⠀⠀⠀⠀⡈⠸⠀⠀⢄⡀⠀⠀⣀⡀⠤⠤⠔⠊
⠀⠈⠢⣀⣘⠠⠀⢀⡶⡥⠀⠀⢲⠑⢄⡉⢣⡀⠀⠉⠀⠀⢀⠞⣙⡤⢃⠇⠀⠀⠀⠈⠉⠉
⠀⠀⠠⢂⢀⢌⠀⣙⠞⠀⠀⠀⠀⠳⡄⠉⠀⠈⠳⠀⢀⠊⠁⠀⠉⡠⠊
⠀⠀⢳⣣⢫⢊⠊⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⣀⣀⠀⢀⣀⡀⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⢀⣤⣤⡀⠀⠀⡏⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆⢺⣿⣿⣿⣿⣷⣠⣇⣠⡶⢀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁⠀⠙⢿⣭⣭⣭⣭⣥⣤⣤⡾⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠚⠻⠿⣿⣁⢀⣴⠟⢳⡄⠀⠀⠈⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣀⡀⠉⠛⢁⡴⠋⠀⠀⠀⠀⣷⠶⠶⠶⠶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠈⣽⠛⠲⠋⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⣠⡴⠶⠛⠳⣤⡀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠋⠁⠀⠀⢀⡀⠀⠙⠷⣤⡀⠀⠀⠀⢸⡀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢀⣠⡴⠞⠋⠙⠳⠶⣤⣈⢙⣦⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠘⠀⡎⠀⠑⠊⢀⠃⠈⡟⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠻⠀⡸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⢆⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠈⠂⡴⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡰⠁⠀⠀⠀⢰⠄⠰⠤⠂⠀⠀⠀⠀⠀⠀⡠⠊⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⡠⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣁⠀⡘
⠀⠀⠀⠀⠀⢀⠤⢤⠀⠀⠀⠀⠀⠀⢀⠤⠠⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠑⢄⣇⠻
⠀⠀⠀⠀⠀⣆⠀⢀⠇⠀⢀⠀⠀⠀⠸⡀⠀⣼⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠘⠄⠛⠘⠓⠒⠄
⠀⠀⠀⠀⠎⠀⠙⡉⠀⠀⠀⠁⠇⠀⠀⢈⠉⠀⠡⠀⠀⠀⠀⠀⢸⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠀⠀⠀⢘
⠀⠀⠀⠸⠀⠀⢀⡑⢄⣀⠀⠈⢀⣀⠴⢃⡄⠀⠀⠂⠀⠀⠰⡉⢢⡨⡉⣶⢲⡄⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢀⣠⣾⠖
⠀⠀⠀⡇⠀⠀⠀⠉⠲⢤⡩⠒⠥⠴⠚⠁⠂⠀⠀⠉⠀⠀⠀⠑⢄⠁⠀⠂⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⠋
⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⡇⠈⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠀⢀
⠀⠀⢠⠁⢰⣳⢀⠇⠀⠀⠀⠀⠀⠀⠀⠸⡀⡏⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡀⣠⢤
⠀⠀⠘⠤⠃⠦⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠄⠑⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⡀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠘⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⠤⠄⠀⠤⢄⠀⢀⠔⡩
⠀⠀⠀⠀⠀⠀⠑⠂⢄⡀⠀⠀⠀⠀⠀⠀⡔⡄⠀⠀⠘⢆⡀⠀⠀⠀⣠⠎⠀⠀⠀⠀⡔⠄⠀⠀⠀⠁⠀⠀⠀⠀⠒⠒⠃⠈⡡⠞
⠀⠀⠀⠀⢄⠀⠀⣀⠌⠀⠀⠀⣀⠀⠀⠀⡇⢃⠀⡠⡀⠀⠙⠦⡤⠊⠁⠀⢀⣠⠀⢀⠂⡆⠀⠈⠳⠤⣄⣀⡤⠔⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠑⠤⠬⢐⡄⢀⢼⡱⠃⠀⠀⠹⡈⠢⣅⠘⢄⡀⠀⠀⠀⠀⡠⠎⣠⠵⠋⡸
⠀⠀⠀⠀⠀⡐⢡⢂⠄⠀⡋⠓⠁⠀⠀⠀⠀⠙⢄⠈⠁⠀⠈⠂⠀⠀⠋⠀⠀⠁⢀⠞
⠀⠀⠀⠀⠰⢴⣵⢋⢎⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣀⣀⠀⢀⣀⡀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠗⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁
⡶⠶⣶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⠶⣶⣶⡶⢶⣶⢶⣶⡶⢶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⢦⣠⠖⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠃⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⡀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⡴⠋⠸⡇⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢰⠁⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣠⣤⡀⠀⡇⡆⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠼⠁
⠀⠀⠀⢀⣤⡀⠀⣿⣿⣾⣿⣿⡇⠀⡇⡇⠀⡇⢸⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⢷⠋⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⠶⡄
⣠⣴⠋⣿⠀⡟⢣⠹⣿⣿⣿⡟⠀⠀⣇⣧⠀⣿⢸⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠃⠀⣷⢦⣀⡀
⡇⢹⠀⣿⠀⡇⢸⠀⠹⣿⠟⠀⠀⠀⠈⠙⠚⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⡠⢆⠀⠀⠀⠀⠰⡇⠃⠀⠀⠀⠁⠈⠉⣿
⣇⢸⡄⣿⠀⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⠇⡜⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡏
⢿⠘⠃⠉⠀⠁⢿⠀⣠⠖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⣸⡧⠤⠒⠉⣰⠃⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⣧
⢸⠀⠀⠀⠀⠀⠘⠚⠁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠞⠁⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⠶⣾⠇
⠘⠀⠀⠀⠀⠀⠀⡤⠊⠀⢀⡤⠶⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⢸⠀⣀⠴⠋⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠶⢦⣤⣴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠒⠒⠉⠁⠀⠀⠀⠀⣠⠤⡶⠶⠤⣄⣈⡙⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⡏⣇⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣡⠔⠁⠀⠀⠀⠈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡇⢸⡇⣿⢹⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⣭⣽⣆
⠀⠀⠀⠀⠀⠀⠚⠉⠉⠙⠒⠲⠚⠁⢾⠑⢆⢠⠖⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡇⣿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⠤⠤⠬⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⠇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠸⠇⢿⠘⡧⠶⢶⠤⣄⠀⠀⠀⠀⠀⠤⠴⠶⠶⠒⠒⣶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡜⠁⠀⠀⢀⡤⠤⣤⣀⠀⢀⡄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠀⠷⠜⣹⠃⠈⠳⣄⡀⠀⡶⠒⠒⢛⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀⠀⠀⠀⠀⢠⠔⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠙⠲⢌⡦⠀⢸⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⢦⣌⣉⠙⠓⠶⠤⠷⠒⠋⠀⠀⠀⠀⢀⣀⠀⠤⠤⠀⠀⠀⡇⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸
⠀⠀⠑⠤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢰⡲⢤⡀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣀⣠⣽⠄⠉⠳⣄⠀⠀⡰⢃⢤⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠑⠒⠤⢴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣭⣭⣭⡥⠤⠤⠀⠀⠀⠀⠈⠳⠤⣽⣅⡀⡇⢸⡀⣦⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣠⣞⠆⠀⠀⣠⣄⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠶⠶⠶⠶⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡇⢿⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣀⠀⠹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣛⣛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢸⠀⣗⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠛⣇⣼⡴⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⣀⣀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡎⠀⠀⠱⡞⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢠⡋⢠⡄⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⢠⠤⣼⠀⢇⠀⢣⡀⠀⠀⠀⠀⡼⠀⠀⠀⠉⠶⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠙⠃⠘⢦⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⣄⠈⠁⢸⠀⠀⠱⣄⠀⢀⡜⠁⠀⠀⠰⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣭⠖⣦⣀⣀⣠⠼⠇⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣤⣀⠀⠀⠀⠀⠀⠈⠑⠲⠼⠀⠀⠀⠈⠑⠎⠀⠀⠀⠀⢀⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠆⢋⠫⡳⣄⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢀⠔⠀⠀⠀⣤⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡂⢱⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠋⠙⡏⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠕⠠⠀⢀⠄
⠀⠀⠀⠀⠀⡠⠀⢀⣀⡀⣦⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡤⠞⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡜⠁
⠀⠀⠀⠀⡜⠀⣤⠸⣿⡏⢹⠀⠀⠀⠀⠀⢰⠋⠉⢳⢀⡠⠖⢉⣠⠴⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠉⠋⠑⠄
⠀⠀⠀⠘⠶⠊⢹⢀⠉⠀⣸⡄⠀⠀⠀⣠⠜⠳⠤⠞⢉⡤⠞⠉⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡏⢸⢸⡇⡇⡇⠀⠀⠀⡇⠀⠀⠀⢰⠋⠀⠀⠀⠀⢰⡁⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂
⠀⠀⠀⠀⠀⠀⣇⣿⠘⣇⠷⠃⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢰⠁⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠘⠹⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠹⣄⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⡀⠀⠀⢀⠴⢊⡉⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢹⡀⠟⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⡦⠒⣲⠄⢦⠀⣇⠘⠏⡽⠀⠀⢛⠖⠋⠀⠀⠀⠀⠸⠀⠀⠀⠀⢠⢲⡄⠀⠳⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠉⡴⠊⣡⠖⠁⠀⠘⠦⠊⠀⠀⢀⡅⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠸⠀⢧⣀⠀⠀⣎⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢚⠁⠀⠀⠀⠀⠀⠀⢀⠶⢋⡠⠖⢀⠄⢀⠔⠁⠀⠀⠀⠀⠀⠀⠣⣄⣈⠉⠉⠛⠆⠹⡀
⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⣀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠇⠰⡋⠤⢞⠡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠘⣡⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠚⠉

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⢀⡀
⠀⠀⠀⠀⢀⡀⠈⠒⢠⠄⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⠀⡃⠀⠙⠁⢨⠀⠀⠤⣀⡀
⠀⠀⠀⠀⠀⢱⡀⠀⠐⠎⠳⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠑⠘⠀⠑⢄⠀⣠⠃⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠈⠻⣗⣦⠤⠔⠋⠀⠀⠀⠀⠀⠀⠰⢦⣄⡀⠀⠀⠀⠉⠒⠀⠀⠀⠙⠁⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠊⢻⠉⠃⠙⠁⠀⠀⠀⠀⠀⠀⠕
⠀⠀⠀⡀⠀⢰⣤⠅⠀⠀⠀⠀⡠⢤⡀⢀⡠⠒⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠴⠁
⠀⠀⠐⠤⢻⠈⠃⢸⠀⠀⠀⡠⠧⡠⢓⡡⠖⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠸⡄⡇⣛⠀⠀⢰⠀⠀⠀⡏⠀⠀⠀⠤⢨⡀
⠀⠀⠀⠀⠸⢇⡐⠉⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠤⠒⢦⠀⠀⠀⠀⠀⠸⡉⡶⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡤⠄⢸⠙⢖⠆⠀⠢⠖⠀⠀⠀⡁⠀⠀⠀⢤⠀⠣⡠⠊
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠐⣉⠔⠁⠀⠓⠋⠀⡠⠀⠀⠀⠀⠐⠀⠀⠀⠀⡈⠢⢄⣰⡝⢄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⢊⡔⣩⠔⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⢄⠀⠈⡀
⠀⠀⠀⠀⠀⢆⡀⠀⣈⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉
⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉⠻⣦⣴⠟⠉⠉⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡆⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢸⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣇⣀⠀⠀⠀⠀⠀⠙⢦⡀⣠⡾⠋⠀⠀⠀⠀⠀⣀⣼⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡟⠉⠳⣤⡀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢀⣴⠟⠉⣻⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠛⢦⡀⠈⠛⠶⣦⡀⠀⠀⠀⠀⢀⣴⠾⠛⠁⣠⡾⠋⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠉⠀⠀⠀⠈⢻⡄⠀⠀⣰⡟⠁⠀⠀⠀⠉⠀⣠⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⢻⡆⢰⡟⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠁⠈⠛⠛⠛⠛⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⣠⠶⣄⣴⡀⣤⢰⠶⡄⣰⡄⢠⡶⠠⣶⠆⡄⡴⠶⡄⣦⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠙⠶⠋⠛⠙⠘⠘⠉⠐⠛⠛⠐⠞⠓⠾⠃⠃⠓⠖⠃⠃⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣒⢰⣚⠀⣖⡂⣶⠀⣶⢰⣄⡆⡔⠒⡀⢖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠒⠀⠓⠂⠙⠂⠉⠘⠈⠃⠙⠚⠁⠓⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠉⠙⠓⠲⣤⡀⠀⠀⣠⡴⠒⠛⠉⠙⠛⠲⢤⡀
⠀⠠⡶⠦⢤⣤⣀⣀⠀⠀⠀⢠⠞⢁⣤⠖⠋⠁⠀⠓⢦⡀⠙⣦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⣀⣠⣤⠤⠴
⠀⠀⣷⠤⠀⠀⠀⠉⠙⠲⣤⡟⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡴⠚⠉⠁⠀⠀⠠⢴
⠀⠀⠘⣷⠆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠶⢋⣩⠭⣭⡙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢴⡞
⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣰⢃⡞⠉⢀⡼⠄⠀⠳⡌⢧⠀⠀⠀⠀⢀⡀⠀⡏⠀⠀⠀⠀⠀⢐⡟⠁
⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⢻⡀⠀⠀⠀⠀⠀⡏⢸⠁⠀⠸⠦⣄⠀⠀⣿⢸⡆⠀⠀⠀⣼⠁⣸⠓⠒⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢻⡈⢁⠀⠀⢤⠞⠀⠠⠏⣼⠀⠀⢀⡀⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠳⣌⡛⠶⠬⠴⢚⣡⠞⠁⢀⣠⠞⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠉⠛⠛⠛⠉⢁⣠⠴⠋⢁⡴⠋
⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣄⡀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠚⠉⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠉⠓⠲⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠙⢦⡀
⠀⠤⠤⠤⠤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⣦⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣞⣀⣀⣀⣀⣀⣀⣀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠞


⡆⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠐
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢰⠅⠐⠁⢱⠀⠈⡇⢠⣣⢸⠭⡂⡇⠀⡥⠀⠐⢤⠀⡏⠠⢥⠂⡊⠘⡄⢴⠁⡧
⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣧⣶⣤⣧⣧⣼⣶⣶⣴⣷⣶⣶⣦⣴⣾⣤⣷⣴⣦⣧⣷⣾⣴⣦⣷⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠀⠠⠀⠂⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠂⠀⠉
⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠸⣿⣿⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠘⠿⠟⠁⢠⣶⠄⣤⢼⣿⣦⣄⠀⠀⠀⠀⠀⠀⣀⡻⣿⠇⠀⢠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⢿⣿⠏
⠀⠀⠀⠀⣴⣿⣿⣿⣷⣬⡿⢸⣿⣷⣝⣿⣿⡄⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠘⢿⣿⣿⣿⠉⠻⢶⡄⠀⠀⠀⠀⢠⣾⣿⣿⣷⣆⠀⢀⣾⣿⣿⣿⣦
⠀⠀⠀⠀⣿⣿⣿⣿⡇⠷⠶⠿⠾⠿⠿⠷⠾⠷⠀⠀⠀⢸⣿⡟⠳⡶⢄⠀⠀⢐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠀⠸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣆⠀⢸⠀⠀⢸⣿⣿⣿⢀⣤⣀⣀⣀⣀⣄⢀⣰⣿⣿⣿⣿⣧⡀⣰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⢸⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢀⣿⡏⠀⠘⣿⡇⠀⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠸⠀⠀⠘⠿⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠸⠿⠁⠀⠀⠹⠿⠐⠿⠃
⠀⠀⠀⠀⠀⠙⠋⠛⠃
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡆
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠸⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡏
⠀⠀⠀⣴⣾⠁⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣟⡀⠀⠀⠀⢠⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⢸⣿⣿⠳⠶⠶⠂⠄⠀⠀⣸⣿⣷⡀⠀⠀⠀⠀⣼⢛⣿⣿⣿⣿⣄⠀⠀⣿⢻⣿⣿⡟⣷
⠀⠀⠸⣿⣿⠀⢰⣾⣿⡾⢀⣴⡿⢻⣿⡇⠀⠀⠀⡼⠻⢼⣿⣿⣿⣇⠻⠀⢠⣿⢸⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡦⠶⠒⠶⠶
⠀⠀⠀⢸⣿⡆⠀⠙⠛⠁⢸⠀⢁⣾⡿⠃⠀⠀⠀⡇⠀⣾⣿⣿⣿⣿⣰⣿⣿⣧⣼⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⣴⠟⠁
⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⢸⠀⠸⣿⡄⠀⠀⠠⠄⣧⠤⠈⣿⠛⣿⡏⠘⢿⣿⡿⣿⣿⣿⣿⡿⠿⠛⠤⠄⠀⠀⠀⠀⠀⢻⣿⡿⠋
⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠸⠀⠀⠹⣿⠀⠀⠀⠀⠇⠀⠘⡿⠀⢿⠇⠀⠀⠀⠀⠸⠟⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀⣀⠀⠀⠀⠀⠀⢪⣭⡽⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠸⣿⣿⠆⠀⠀⢀⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠸⣿⣿⠏⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⣶⣶⣦⣄⣸⡇⣿⣿⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⣠⣠⣾⣧⣤⣤⣴⣌⣿⣦⣀⣀⣀⠀⠀⠀⠀⠤⠄⠀⠤⠀⢸⣿⣿⣿⠀⠙⢿⠀⣿⣿⣿⢸⡇⠠⠄⠀⠀⠀⠀⣀⣘⣛⣀⠀⠀⠰⣄⣾⣿⠆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⣽⠉⠉⢹⠀⠀⠀⠈⢻⣶
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣩⣝⡻⣿⣿⡏⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡧⠤⠤⢼⡆⠀⠀⠀⢸⡿
⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢱⣿⠛⢻⣿⡌⠿⠿⠃⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠉⠘⠛⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣧⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⠸⠿⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢇⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠑⠀⢀⣴⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⠹⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠈⣱⣶⣶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⡸⢋⡼⡻⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣙⣿⡄⠀⢸⠃⠁⣠⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄⢸⢀⠴⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⡿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠠⠤⢤⣀⣿⣠⠴⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠦⠠⠤
⣶⣶⣶⣦⣤⣤⣤⣤⣤⠀⠔⠊⠀⠀⠀⠀⠀⠀⣀⣤⡍⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⠴⠖⢲⣎⣁⣤⣾⡿⡿⢿⣟⠋
⠛⠻⠟⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠀⠀⠐⠊⠉⠙⠛⠁⠀⠀⠀⠉⠙⠛⣀⣠⣴⣾⠏⢹⡿⠝⢇⠀⠀⠈⠛⠶⢦⠄⠀⠂⠔⠤⠀⠀⡀
⠀⠀⢀⣄⠀⡀⠀⡈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡈⠑⠦⠤⠤⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⣰⡄⢠⠀⠀⠀⠀⢀⣠⠖⠒⠒⡶⠤⢀⣀
⣿⣷⣿⣿⣿⣧⣷⡇⢠⡇⠀⠀⠀⠀⠠⢤⣀⠈⠙⠶⠤⠤⠤⣤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⢀⡾⢻⣤⡀⠀⠴⠒⠚⠉⠀⠀⠀⠀⢻⠀⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⡇⢸⠒⠒⠒⠤⢄⣀⣀⣿⢤⣤⣤⡤⠔⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗
⣿⣿⣿⣿⣿⣿⣿⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣴⣵
⠉⠉⠉⠉⠙⠛⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣷⣦⣀
⣿⣿⣿⣿⣷⣦⣄
⢿⣿⣿⣿⣿⡿⠻⣷⡢⣀
⢈⡋⠿⠝⠯⠀⠃⠀⠀⠙⠷⣦⣄
⠀⠀⠀⠐⠃⠀⠀⣀⣣⣶⣤⣽⣿⣿⣶⣄⠀⠀⠀⠀⣠⠤⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠀⠀⠀⠀⠚⣹⣿⣿⣿⣿⡾⣿⡿⠁⠀⠀⠀⠇⣤⣴⣀⣤⣀⠈⠑⢦⣀⡀⠀⣀⣴⠾⠿⠦⡀
⠀⢀⠃⣀⠠⠤⠐⠒⡿⠟⠛⣋⣉⣽⠟⠀⠀⠀⠀⠀⠾⠿⠋⠈⠉⠉⠓⢶⡏⠛⠻⣿⣿⡏⠀⠀⠀⣹⣦⡀
⣼⣿⣿⣾⣿⣷⣶⣦⣶⣾⣿⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣠⣿⣿⣿⣦⣤⣾⡿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠈⣿⣿⣿⡿⡟⠉⠀⠘⠃
⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⡟⠁⣠⠶⢄⣠⠒⢁⡼⠒⣆⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠋⢀⡾⠛⠁⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠙⠛⠿⣿⡿⠀⠀⠻⠞⠋⢿⣷⡏⢁⣴⢋⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡟⢁⣼⠃⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠀⢲⡄⠀⠀⠀⠀⠀⣤⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢸⡟⠛⠛⠛⠋⢰⡏⢰⠇⣰⢂⠀⣠⡄
⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⢹⠀⠀⠀⢰⠇⡼⢣⡾⢱⡆⢀⣿⢀⡿⢸⢳⡟⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠈⠀⠸⡷⠊⠀⢸⡇⣼⢁⡟⠘⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠿⠀⠈⣷⣀⡀⠀⠀⠀⠁⠘⠇⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠀⠠⠟⣸⡤⠖⠚⠛⠛⠛⠶⢼⢃⡶⢰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣯⠁⡀⡆⢸⡆⢰⡆⢀⡿⠃⠛⠛⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⣴⠃⢸⡆⣿⠸⣇⠇⣿⠀⣾⠁⣾⠃⣼⢠⡆⢀⣤⢀⣀
⠀⠀⠀⠀⠀⠀⢠⡆⠁⠲⣄⠀⠘⣇⣿⠀⣼⠀⢿⠘⣇⠰⠏⢸⡇⣾⡃⣼⠃⣼⠋⣹⢻⡏⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⢷⡄⡏⢿⡄⠙⠂⠘⠇⠉⠀⠀⠀⠀⠈⠁⠙⠺⠃⣼⠃⣼⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⠇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠓
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⢿⣦⡀⠀⠀⠉⠒⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠀⠀⠈⠉⠻⣶⣄⠀⠀⠀⠀⠁⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢸⣦⣶⣤⠞⠉⠙⠃⠀⠀⠐⠀⠺⢿⣷⣄⠤⠒⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⣿⢻⣷⣤⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢐⣦⣄⠀⠀⠀⠀⠀⠀⢶⣶⣤⣿⣮⣿⣿⣿⣦⡀⠀⠀⡆⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠀⠀⠀⢀⡀⣀⠀⢉⠤⠈⠁⠈⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠘⠀⡀⠀⠀⠄⠨⣿⣨⠶⠆⠂⠀⠀⠀⠀⢸⣿⣦⣅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠘⢿⠀⡄⠀⢀⡄⠁⠰⣞⣏⡻⠀⠀⠀⡾⠻⠝⢿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠘⣷⠀⣤⠁⣀⠀⠀⢺⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠐⠆⠀⠀⠰⠏⠀⠀⠄⣸⣆⠀⠘⠏⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠰⣿⣏⠈⠙⠲⣦⢄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠷⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⢀⣀⡀⡿
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠐⢠⣤⡴⠂⠀⠀⣿⠁⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠃⠀⠋⠀⠀⠰⡾⠏
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⡗⢚⠉⢋⡡⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠰⠞⠂
⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣀⣠⣴⣶⣾⣿⣶⣶⡶⠄⠀⠀⠀⣠⠀⠀⠀⡠⠉
⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠒⣀⣠⣤⣤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠙⠚⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠘⢲⡻⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣟⠳⠀⣉⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠸⡓⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⢃⠀⠀⠝⠀⢐⣤⠌⠀⠀⠀⠀⠀⠀⠈⠄⢤⢀⠙⠛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠐⠂⠀⠘⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠂⣬⡟⡀⢁⠙⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠦⠰⣪⠄⠀⠀⠀⠀⠀⠱⠇⢰⢹⠻⡆⠀⠀⠐⠄
⠀⠀⠀⠀⠀⢀⠔⠁⠢⣄⠀⣆⠀⠀⣤⠀⠀⠉⠉⠍⠀⠀⠀⡀⠀⢀⡅⠈⠀⠀⠈⢁⢠⡀
⠀⠀⠀⣀⡔⠁⠀⣀⠤⠘⠂⠨⡓⠀⠘⠃⠀⠀⠀⠁⠐⠒⠫⠄⠀⢈⡇⠰⡀⠀⠀⢘⣀⠱⣄
⢀⡴⣿⣿⡧⠐⠉⠀⠀⠀⠀⠰⠵⠀⠀⠀⠀⠀⠈⠀⠈⠂⠈⠀⠀⠈⢻⠀⠁⠀⠀⠀⡅⠀⠈⠂⠀⠀⠀⠀⢈
⣮⡆⡙⣻⣄⠀⠀⠀⠀⠀⠠⡀⢀⠠⠀⠀⠠⠄⠈⠯⠄⠈⠃⠤⠪⠀⠘⠦⣄⠀⢀⣀⠈⠓⠒⠀⠀⢀⠆⢀⣈
⠙⠻⣂⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠏⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⡐⠠⠄
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠋⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣽⣇⡀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⠶⢶⣤⣤⣄⣀⣤⣿⡄⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣼⠏⠉⠛⠻⠿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⣷⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⢠⡾⠋⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠸⣧
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠃⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠘⠳⢦⣄⡀⢀⣼⡃⠀⠀⠀⠀⠀⠀⠸⣧⣤⡾⠃⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠻⣦⣠⡶⢶⣄⣀⣀⡀⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⢷⣄⠀⠀⢀⣴⠟⠻⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡋⠀⣠⡿⠋⠉⣻⡄⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠉⠻⣾⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⡟⠁⢀⣴⠟⠛⣷⠀⠀⠀⠻⣶⣄⠀⠀⠙⢷⣦⡀⠀⢘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣴⡿⠁⠀⣰⣿⣀⢶⣄⠀⠀⠙⢷⣄⡀⠀⢹⡿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣾⠋⠈⣹⡆⠙⢷⣤⡀⠀⢹⡿⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠿⣦⣀⣀⣿⠛⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣦⣄⣀⠀⢀⣤⣄⠀⠀⠉⠛⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠋⠉⠀⢠⣿⣿⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⠀⠀⠀⣩⣿⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠴⠾⠟⠋⠀⠀⠀⠘⣧⠀⠀⠀⢨⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠘⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢻⣿⣿⣿⣿⣸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⠏⠹⡇⢸⣿⣿⣿⣿⣿⡇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⡉⢄⠀⢷⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⠻⣿
⠀⣶⠀⠀⠀⠀⠀⠻⣼⢦⠘⡇⠀⠒⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⣠⣿⣿⣿⣿⣿⣿⣿⡁⠁⠀⠀⢸⣿⣇⠈⠉⢛⣫⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣿
⠀⠘⣧⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⠀⠘⣿⣿⣆⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠹⣷⣦⣀⠀⠀⠀⠀⠈⢻⣿⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠋⠉⠀⠉⢻⣿⣿⣿⣿⡆⠀⢿⣿⣿⡆⠀⠈⢿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿
⠀⠀⠀⠹⣿⣿⠁⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠱⢦⣀⣠⣤⠀⠀⠈⠛⠿⣿⣿⣇⡜⡘⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿
⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⣄⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⠀⠀⠩⠿⢿⣶⣶⡇⠀⠀⠀⠙⢿⡼⣷⢿⣿⣿⡷⠀⠀⠘⣿⣿⣿⣿⡿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣧⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠈⣿⣹⣤⣿⠿⡥⠃⠀⠀⠸⣿⣿⣿⡇⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣤⣤⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣈⠻⣷⣦⣤⣴⣿⣿⠟⠁⡆⠀⠀⠀⠀⣰⣿⣿⣿⡇⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⠀⠀⠄⠀⠀⠀⠀⢸⣿⣿⣿⣧⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣼⣷⣼⣿⣿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿⡇⠀⢰⣾⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣼⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣰⣦⡀⠀⠀⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⡴⠛⠿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣦⠀⠀⠀⠀⠀⠲⣌⠻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⢻⣿⣿
⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠹⢶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣵⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢿⣻⣿⣿⣿⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠋⣀⡼⣿⣹⣿⣿⡆⠀⠈⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣧⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣿⣿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡝⣯⡇⠈⢻⣿⣿⣿⣿⣿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣼⣷⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣱⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠎⠿⣦⡀
⠀⢀⡀⠐⠛⠃⢠⡄⢶⣦
⠀⠸⡷⠰⣦⡄⠀⠿⣤⡉⠈⢳⣄
⠀⠀⠁⠐⠘⢧⡀⢀⠀⠹⣶⣄⠈⠳⣦⡀
⠀⠐⠀⣄⠀⠈⠓⡀⠁⠀⠘⣿⣿⣄⠀⠻⢶⣄⠀⠀⠀⠀⠀⠀⣀
⠀⡇⣼⡏⢠⣶⣄⠙⢦⡢⡀⠈⠻⣿⣷⣦⡀⠹⣷⣄⠀⠀⠠⠊⢹⡇
⠀⠀⣿⡇⠀⢩⣽⣷⠀⠹⣟⢦⡀⠈⢻⣿⡷⡄⠸⣿⣆⢸⡇⠀⠘⣄
⠀⢸⠁⢀⡀⠘⢿⡽⣷⣄⢈⣷⣝⢶⡀⠹⣿⣆⠀⠉⢿⣿⡗⡗⢄⣻⡄
⠀⠇⠀⣾⣿⡄⠈⢿⣜⢿⡁⠀⢙⣿⣧⠀⠘⢿⣦⡀⠀⠿⠇⠹⡈⠹⣷⡀
⢸⠀⢰⣿⢿⣷⡀⠈⠹⣿⣿⣆⠀⠈⢻⣧⠀⠈⣿⣿⣦⠀⠀⠠⣵⣄⠙⢷⡀
⡇⠀⠈⣿⠘⠿⣷⡀⠀⠘⢻⣞⣿⣄⠈⢿⡆⠀⠘⠉⢦⣄⠀⠀⣿⣿⣕⢾⣿⣄
⡇⠀⠀⣿⠃⠈⣿⣷⡀⠀⠈⢻⡞⢿⡄⠀⠁⠀⠀⠀⠀⠙⠗⢀⠘⣧⠙⣷⠈⠛⢷⡄
⠀⠀⢀⣿⣇⠀⠸⣿⣧⠀⠀⠈⣿⣎⠟⠀⠀⠀⠀⠀⠀⢀⡀⢸⣇⠈⢳⣟⠀⠀⢰⣍
⠐⡄⠀⠀⠻⣧⠀⢻⣿⡆⠀⠀⠸⣿⡆⠀⠀⢀⠀⠀⠀⢸⣷⡀⢿⠀⠀⢹⡜⣆⠈⣿⣆
⠀⠘⢇⠀⠀⠘⣷⣼⣿⣿⡀⠀⠀⣿⡃⠀⠀⢸⣦⣤⡀⠈⢿⣿⡄⠀⢀⡀⠠⣿⣆⢹⣿⡄
⠀⠀⠈⢧⠀⠀⠘⣱⣿⣿⣇⠀⠀⢸⣿⣦⣆⠈⢻⣿⣿⣆⠸⣿⣿⡀⢸⣧⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠈⣷⠀⠀⠀⠹⣿⣿⠀⠀⠈⠉⠻⣿⠃⠀⢻⣿⣿⣦⠹⣿⣷⡌⢿⣷⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠉⠀⠀⠈⢿⣿⣿⡇⠙⣿⡛⠀⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠙⣄⠀⠀⠈⠙⣿⣆⠀⠀⠀⠀⢾⣷⣦⣀⠻⣿⡇⠀⣾⣇⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠉⠙⢻⣧⡈⠀⠀⠿⠛⢃⡀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⣿⣿⣆⢠⣄⠀⠀⠈⠛⠃⠀⠀⢀⣴⣿⡇⠀⠈⠙⠻⠿⣿⣿⣷⠆⣴⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠐⣿⣿⣿⣷⡛⢿⣖⠀⠀⠀⠀⠀⠉⠙⠛⣧⣄⡀⡀⢸⣶⣤⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⠟⣋⣙⡻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣯⣄⣀⠀⠀⠹⢿⣿⣿⣿⣦⣝⢿⣦⣀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⠋⠁⠀⠀⢾⣿⣿⣿⣿⣿⡿⢡⣾⣿⣿⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣸⡟⣿⣿⣿⣷⣀⠀⠉⠹⣿⣿⣿⣿⣷⣌⡛⢿⣶⣶⣿⣛⠀⠀⠀⠀⠁⢀⠀⠀⠀⢨⣿⣿⣿⣿⠟⣱⣿⠟⠉⠀⢀⣴⠶⠃
⠀⠀⠀⠀⠀⠀⡟⠈⢿⣿⣿⠻⣷⠂⠀⣤⣿⠿⠋⢋⣵⣿⡿⣶⣮⠻⣿⣿⣤⡀⠀⠀⠀⠘⠓⠶⠶⣄⢙⣿⣿⢃⣴⣿⠃⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⡇⢀⣿⣿⣿⠀⠈⠓⠀⠀⠠⠴⣾⣿⡿⠏⣰⣿⣿⣷⣮⡛⡿⠃⠀⠀⠀⠀⠀⠀⠀⢻⡌⠀⢡⣿⠟⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠆⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⣰⣿⣿⣿⣿⣿⠃⡀⠀⠀⢠⣆⠀⠀⠀⠀⠈⢷⠀⣾⠇⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢊⣠⡖⠀⢩⣿⣿⣿⣿⠟⢸⡇⠀⠀⣿⣿⣦⡔⢀⡀⠀⠈⢠⣿⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⡘⠻⠿⠿⣿⣟⣛⣤⣴⣾⣿⠏⢀⣴⣿⣿⣿⣿⠏⠀⢸⡇⠀⠀⠻⣿⣿⣿⣿⢖⣠⡀⣸⡇⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣴⣿⣿⣿⣿⣿⣿⡿⠃⠀⠈⢡⣿⡿⠟⠁⠀⠀⠈⢷⣤⣤⣼⠶⠍⠉⠁⠉⠀⢀⣿⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣭⣀⣀⣠⠄⠀⠀⡴⠟⠋⠀⠀⠀⠀⠀⣠⣸⣯⣵⡖⠂⠀⠀⠀⠀⢀⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⣿⣿⣥⠤⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠒⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣤⣤⠤⠔⠊

⠀⣠⠖⠛⠛⠒⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠓⢦
⡾⠁⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⡅⠀⠀⠀⠀⠀⠀⢀⡼
⠈⠳⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⠀⠻⡄⠀⠀⠀⠀⣠⠞
⠀⠀⠈⠓⠦⣤⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⢷⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠙⠒⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢀⡞⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣀⡴⠞⠋⠉⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡾⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠛⠒⠦⢤⣄⣀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⣤⠖⠋⠁⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣄⡀⠈⢳⡀
⠀⠀⣰⠊⢁⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠓⠲⢤⡀⢀⡴⠚⠉⠉⠛⠳⢤⣀⡀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢻⡀
⠀⡾⠡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⠀⠙⣾⣀⣀⣄⡀⠀⠀⠀⠈⠉⠓⠦⣄⠀⠉⠓⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠛⠦⣀⣠⡴⠖⠋⠉⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⢤⣤⣟⡛⠲⠦⣤⣀⡀⠀⠀⠀⣠⠞
⠘⢧⡀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠟⣧⣀⣀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⠏⠉⠛⠛⠁
⠀⠀⠙⠲⠶⠤⠶⠖⠛⠉⢹⠂⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠙⠛⠓⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⢘⡧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⡵⠞⠛⠳⢦⣤⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠛⠋⠀⠀⠀⢀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⠖⠛⣇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣞⣉⣀⣀⣀⣤⣤⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣤⠴⠚⠉

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡠⡜⢄⠉⣇⣀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⡠⠚⠢⢠⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⠳⢘⡌⠢⡨⠊⠀⠀⠰⠝⠊⢀⠀⠀⠀⠊⠈⢃⡀⠤⠆⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⡏⡭⡢⡀⠙⠢⠅⢀⡴⢊⠁⠀⠰⠇⠀⠈⠀⠀⠀⠀⣠⡄⠀⠀⢀⠀⠀⠀⠰⡹⠁⠹⠂
⠀⠀⠀⡖⠢⡀⠘⢄⡚⡀⠀⢀⠔⠉⠀⠣⠐⠀⠀⠀⠀⠀⠐⠂⠀⠄⠻⢀⡠⠐⠁⠀⠀⠀⠛⠃⢀
⠀⠀⠀⠘⢄⠑⠄⡄⠀⠀⠘⠀⠀⠀⠀⠀⣠⡾⢑⢄⠀⠈⠐⠁⠀⠐⠂⠁⠀⠎⠐⠀⠀⠀⠀⠀⠃⠁
⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⣰⢯⣲⠃⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠞⢛⣼⠃⡐⠀⢀⠀⡰⡡⢀⠀⠀⠀⠀⠛⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠂⠀⠀⠀⠀⠠⡒⠁⠀⠀⠀⠉⢤⡾⠋⢉⠔⠀⠀⠈⢄⣔⢲⠇⠀⠂⠂⠢⡀⢀⠾⠆⠀⠞⢂⠄
⠀⠀⠀⠀⡀⢄⡀⠀⠈⠈⠁⠀⠀⢀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠁⠀⠀⢀
⠀⠀⡄⠊⡤⠊⠑⡀⠀⠀⠀⠀⠒⠀⠆⠀⠀⠀⠰⡟⠋⡰⠀⠀⠈⢇⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠈⢎⠀⠀⠅⠅⠀⠀⠀⠀⠀⠆⠐⠄⠀⠀⠀⣵⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡁
⠀⠀⠀⣊⡇⠀⠀⠀⠀⢀⠊⠀⠉⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠊⠉⠉⠀⠀⢠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣀⠀⠀⣈⠀⠀⠀⠀⠂⠀⠀⠠⠒⢀⠀⠀⠒⠀⠀⠠⡀⠀⠀⠀⡞⠂⠂
⠀⠀⠀⠀⠀⢶⣀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠐⠀⠀⠐⢄⣀⣶⣠⡔⠁⠀⠀⠀⠁⠀⠘
⠀⠀⠀⠀⠄⠄⠌⠍⡒⠀⠀⠀⠀⢀⠴⣗⢄⠀⠀⠀⠀⠀⠀⠀⠌⠍⢿⡇⠀⡃⠀⠀⠀⣀⣀⡀
⠀⠀⢠⢆⠀⢊⠈⠌⠀⠀⠀⠀⣰⡁⠀⠈⡪⠇⠀⠀⠄⠀⠀⠀⡂⠀⢺⣇⣀⡁⠀⠸⠛⠃⠉⢁
⠠⠀⠀⠀⠁⠆⠆⠀⠀⠀⠀⠀⠀⠀⢠⢊⠀⠀⠀⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢆⠄⠀⠘⠃⠀⠀⠂
⠀⠀⠄⠆⠶⠲⠀⠀⠀⢀⠀⠀⠀⠀⠈⠈⠀⠀⠘⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠋⠉⠀⠀⠀⠀⠀⠉⠉⠓⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠤⡀⠀⠀⠀⢀⣀⣀⠀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠞⠁⡠⠖⠚⠛⠋⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠊⠁⠀⠀⠈⠉⠒⠒⠐⠓⠢⠤⠒⡖⠋⢰⠁⠀⠈⠀⠀⠀⢀⡴⠁⡄⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡎⠀⠀⣀⠤⡄⢀⠏⣀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠚⣅⠀⢇⠘⡰⠉⢆⢳⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⢸⠀⢀⢳
⠀⠀⠀⠀⠀⠘⡄⠀⢰⠁⠠⠏⠋⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢰⠀⡄⠀⢹⠁⠀⡅⠀⠘⢤⠁⠀⠘⢦⡁⠀⠀⠀⠀⠀⠀⠈⢧⠀⡜⣠⠇⡌
⠀⠀⠀⠀⠀⠀⠘⡄⠈⢢⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠉⢠⠃⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⣀⣠⣈⣀⡐⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣋⠀⠀⠀⠀⠀⢐⠀⠘⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⢃⢷⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠉⠉⠽⠀⢸⠁⡑⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠦⠄⣀⡀⠀⠊⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⢠⠇⠸⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠻⣦⣀⡀⠀⡹⡌⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢄⠀⠀⠋⡸⠀⠀⠀⠀⠀⠀⠀⢠⠇⠈⠀⠀⡇
⠀⠀⠀⠀⠀⡰⠋⡄⠀⠀⠈⠙⠛⠛⠋⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣚⢁⣀⠉⠑⡲⠤⠤⡤⠤⠄⢶⡒⣩⠋⠀⠀⢀⠀⡇
⠀⠀⠀⠀⡜⠀⠀⢸⠀⠀⠀⡠⠀⠄⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠊⠁⠀⡜⠑⠤⠴⠚⢫⡀⠀⣀⠤⢃⠜
⠀⠀⠀⣸⢄⠀⢠⠀⠀⠀⠠⠠⠀⠀⠀⣸⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⢤⠴⠒⢲⣦⣄⣀⣠⢎⡀⢀⡀⣠⠋⢱⡈⢻⠃⠉⠁
⠀⠀⠀⠇⠀⢹⢛⠀⠀⠀⠀⠀⠀⢀⠀⢻⠀⠙⠒⠳⡄⠀⠀⠀⠀⠀⢸⠬⡑⢠⡄⢐⠋⠀⣀⢀⢉⠤⢷⣀⠀⠁⣀⡸⠁⠘⡆
⠀⠀⣸⠀⠀⠇⢸⡀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠑⠂⠈⠀⠀⠀⠀⠀⠀⢸⠀⠂⠰⠮⣒⣂⠤⠬⠀⡲⠗⠋⢁⡠⠔⠋⠁⠀⣀⣽⡀
⠀⠘⢤⣀⠘⠄⠘⡛⠭⠭⠭⠭⠭⠭⢭⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠀⠘⠐⠀⠀⠀⠀⠀⠓⠀⢹⡏⠉⠉⠈⠋⠉⠑⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⢀⣄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣘⠀⠄⠠⠀⠁⠀⣀⣀⡿⠀⠘⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⢸⡗⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠔⠒⠊⠉⠁⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⠾⣀⠠⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠤⠤⡤⡤⠤⢴⠊⠁
⠀⠀⠀⠀⠀⠀⠘⠦⣤⣔⡏⠀⠈⠙⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡒⠒⣇⢡⣀⣈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⢨⣇⣀⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⡀⠙⢆⠱⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠑⠀⠀⠑⠄⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠄⠀⢦⡀⠀⡆⢠⠄⢦⡎⡏⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣦⠊⠢⠀⢵⠀⠀⠈⢆⠀⠑⣇⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⡆⠀⡁⠀⠀⢄⠀⠀⠣⠀⡈⠦⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠈⢆⠈⠰⠀⠀⢣⠀⠄⠁⠀⢄⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠱⡄⢺⡀⠀⠁⠀⠇⠀⠀⠀⠀⠈⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠈⡞⠃⠀⠔⠀⠀⠀⠀⠀⠀⡀⠀⢣⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣧⠀⠀⠀⠀⢆⠀⠂⠀⠘⡀⠀⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠀⠀⠈⠢⠀⠀⠀⠐⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠂⠤⠀⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠈⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠚⠛⠑⣄⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠈⠳⣄⡰⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⢻⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⠏⠀⠀⠀⠀⠀⢀⣾⠣⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡄⠀⠀⠀⢀⠔⠃⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠴⠶⠦⠤⠤⣀⡀
⠀⠀⠀⠀⠀⢀⡄⢀⡤⠖⠋⠉⠀⢀⡠⠴⠒⠚⠛⠛⠛⠒⠀⠉⠛⢦⣄
⠀⠀⣀⢶⡒⠈⡷⠉⠀⢀⠀⠀⠈⠁⠀⠀⣀⣠⠤⠴⠒⠒⠀⢀⣀⣠⠽⢚⡥⣰⠃
⠀⠀⠉⢢⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⡤⠶⠊⠉⠠⠒⣩⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣤⣤⡤⢤⣤⣀
⠀⠀⣰⠃⠄⠀⠀⠀⠀⠐⠂⠤⠤⠤⠤⠤⠤⠐⠒⠉⠀⠀⠀⠀⣠⡴⠞⠁⡌⣇⠀⠀⠀⠀⢀⣀⡠⠔⣉⠤⠒⠉⠀⠀⠀⠀⠀⠀⢉⣲⣄⣀⡴⠋⡙⠛⢶⡄
⠀⢰⠃⡜⠀⢠⠂⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣤⣤⠤⠶⠚⠋⠁⠀⠀⠀⢱⣹⠀⠀⣠⠞⠩⠎⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⢿⣀⡠⠀⠹⡄
⠀⡟⢠⡆⠀⡇⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼⠃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣆⢷⡠⠔⠀⡇
⠀⡇⢸⡇⠀⢀⠄⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠆⠀⡇⢰⢻⠸⡀⠀⠀⠀⠀⠀⢈⣉⣉⠉⠑⠒⠤⣄⡀⠀⠀⠀⠀⣠⠟⡾⠇⠀⣸⠃
⠀⣧⠀⠇⣠⠋⣠⣤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠸⢀⠓⢦⣄⣦⠤⣄⣀⠀⠈⠙⠲⢤⡀⠀⠉⠓⠶⠶⠾⣫⠀⡷⠴⠚⠁
⠀⢻⡀⠀⡏⣾⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⠀⠀⣸⠁⠘⣇⣸⠰⠀⠀⣤⠀⠀⠉⠙⢦⡀⠀⢀⠙⣆⠐⠀⠀⠀⠊⠁⡔⡇
⠀⠘⣷⠀⠀⠙⢮⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡼⣰⠃⠀⠀⠈⢿⠀⡄⠀⠈⠀⠀⠀⠀⠀⠃⢸⡍⢱⡈⠀⠀⠀⢀⠄⡼⢡⡇
⠀⠀⠸⣧⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⢞⡵⠃⠀⠀⠀⠀⢸⡘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⠼⠁⠀⠀⠂⣡⠞⢁⣾⠁
⠀⠀⠀⠈⠳⣄⡙⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⢉⡴⠋⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⡤⠎⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠒⠉⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠈⠙⠓⠺⠷⠦⢄⣀⡀⠀⠀⠀⠀⢀⣀⣀⡤⠖⠋⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠂⠤⠤⠖⣋⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡄⠈⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠙⠒⠤⣄⡀⠀⠀⣀⣠⣷⣤⠤⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠓⠤⠍⠓⢤⡀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦⣀⣀⠀⠀⠀⠀⠀⢀⡨⢆⣀⢎⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠰⡄⠀⠀⠀⠀⢡⠈⠧⠀⠀⠀⠀⠹⣿⣿⣿⡿⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⠀⡆⠣⢄⡀⠀⠀⠀⠀⠙⠛⠁⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠛⠃⠀⠀⠀⠀⠀⠀⢡⠀⠀⠉⠙⠒⠶⡶⠜⣦⠤⠴⠚⠉⢈⠀⠀⠀⠀⠀⠀⠸⣄⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡞⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⠲⡄⠀⠀⢀⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢠⠇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠦⠤⠿⠗⢒⠋⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢑⣠⠀⡀⠀⣀⣠⣼⢰⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⠞⠉⢂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠾⢦⢰⣴⡉⡿⠾⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠃⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠤⣤⣄⣀⣤⡤⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣄⡤⡀⠀⣏⠋⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢮⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠜⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠈⠰⠩⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣳⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣄⣀⣤⣶⠟⠃⠀⢿⣿⣿⣿⢻⠻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⣿⣿⣿⣇⣷⢹⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⢹⣿⣦⢠⣤⢀⡀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢠⠦⣠⣶⣦⣤⣤⣤⣵⣦⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣏⣾⣿⣿⣶⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⢠⣀⣴⣿⣿⣿⡿⢉⠀⠀⠀⠰⣦⣤⣄⣀⠀⠀⠉⢩⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣧⣶⣴⣸⣿⣿⣿⣿⣶⣶⣿⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠛⠇⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠
⡄⠀⠀⠀⠀⠀⠀⠀⠉⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣼⣿⣿⣿⣿⣿⣿⡇⣹⡁⢀⣴⡀
⣿⣶⣄⣠⣾⣿⣿⣿⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠛⠉⠿⣿⣏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣯⣷⣿⣷⣿⣿⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠐⠂⠈⠉⠀⠉⠉⠉⠀⠐⠒⠒⠂⠠⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢀⠴⠒⠒⢤⡀⠀⡠⠖⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⢄⡀⠀⡠⠖⠒⠒⠦⡀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⡌⠀⢐⠄⠀⠈⠎⠀⠤⡊⠔⢀⠈⡆⠀⠀⠀⠀⠠⠁⠀⡂⠄⡀⠑⣜⠀⢔⠡⢊⠀⢹⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠰⡀⠀⠄⢊⠔⡀⠔⣊⠔⡊⠅⢠⠁⠀⠀⠀⠀⢸⠀⠨⢔⠩⢂⠀⡠⢐⠡⢂⠕⠀⠐⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠄⢊⠔⡪⢔⠨⠐⡠⠁⠀⠀⠀⠀⠀⠀⢆⠀⠠⢊⠄⡨⢐⠡⢊⠅⠊⡠⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠩⢂⠥⣢⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠁⡨⢂⠥⢊⠕⢊⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠡⠊⢔⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠃⠌
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⡀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⠀⡀⠆⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡠⠀⠠⠁⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣁⣈⠀⠂⠐⠂⠀⠁⠀⠊⠐⠁⠊⠀⢁⣠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠐⠊⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⡀⠄⠀⡀⠄⠂⢀⠠⠀⣀⠤⠀⡀⠄⢀⠠⠀⣀⠄⠀⢀⠄⠀⡀⠄⢂⠠⠒⡀⠄⢂⠀⠀⡀⠄⢈⠡⠀⡁⠀⡠⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠡⠔⠊⢁⠄⠒⢈⠤⠒⣁⠠⠒⡁⠄⠊⡁⠔⢊⡠⠔⢈⠠⠐⡉⠔⢈⠠⠐⢉⠤⠂⡡⠐⠉⠠⠂⠁⠔⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠉⠀⠂⠁⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣠⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡛⠛⠿⠟⠛⠛⡉⠩⠍⣛⡻⠯⢝⣫⢿⣿⣿⣟⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠪⠓⠢⢍⠉⠉⠩⠤⠔⠂⠀⠥⢒⠈⠁⠈⠁⠊⠙⠙⢚⢗⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⡬⠷⡶⠖⡷⠿⢷⠥⠀⠈⠰⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠒⠀⠉⠛⠻⠿⣿⡿⢿⣿⡟⠛
⠀⠀⠀⠀⠈⢿⣿⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣫⣤⣦⣶⣶⣼⣿⣧⣦⣶⣡⡂⠘⠙⠰⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠐⠁⢀
⠀⠀⠀⠀⠀⡩⠳⠚⠐⠻⢻⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣶⣤⣈⠙⠻⢶⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣤⣶⣖⢠⣠⡖⢶⣦⡀⠉⠛⠻⢿⣿⣿⣷⣀⣉⠛⢿⣴⣤⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠿⠿⣿⣿⣿⣿⠅⠀⠀⠀⠀⢸⣿⢯⢙⣼⣤⣀⣽⣿⡇⠀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣦⡉⠻⢿⣮⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠋⢀⣾⠟⠛⡩⣷⠀⠀⠀⠀⠀⠀⠻⠉⠛⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠐⢹⣿⣿⣿⣿⡆⠄⠙⠻⣯⣢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠈⠻⠒⠒⠚⠻⠧⣀⠀⠀⠀⠀⠀⠑⢿⣶⣶⣾⠟⠀⠀⠀⠀⠀⠀⠁⠡⠈⠀⠁⠞⠉⠻⠿⡷⠂⠀⠀⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⣲⠦⣄⣈⠃⠶⡠⠄⢀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣦⣤⣀⣁⣀⣀⢀⡀⣀⣀⣀⣠⣤⡴⡾⢯⠣⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠜⠉⢉⡛⠛⠛⠛⠿⠛⠋⠙⠙⠩⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⣄⣼⡿⣷⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣶⡏⢟⡇⠻⢿⠿⣇⠊⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠁⠋⡿⡿⠿⢶⢷⡌⢀⣼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠶⠿⡏⠂⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠙⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⡿⠂⠁⠀⠀⢠⡀⣀⢂⠀⠀⠀⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣬⡙⢋⠇
⠀⠀⠀⠀⠀⣠⣶⡿⠿⢫⠙⠿⠶⠒⢂⠐⠛⣄⠻⠿⠻⠁⠀⠀⠠⣆⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⢦⢢⣦⣼⡇
⠀⠀⠀⢀⣾⣷⣮⡀⣠⣀⡄⣀⠀⣰⣺⣠⣀⢿⠋⠀⠀⠡⠀⠀⠀⠈⠝⠀⠀⣿⡛⠶⣤⠀⠀⠀⣼⣿⣶⣿⣿⡿
⠀⠀⢀⣾⣿⣿⡟⠁⠈⣛⣾⡿⠒⠺⣿⣿⣉⠀⠀⠀⢀⡴⡆⠇⢠⠀⠀⠀⠀⠈⠛⠂⠈⠁⠀⢸⡿⢿⠿⢿⡿⠁
⠀⠀⢸⣿⣿⣟⠂⠀⣼⢿⡿⠁⠀⢠⣿⣿⡇⠀⠀⢀⠊⢠⠇⠀⠠⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⢹
⠀⠀⢸⣾⣿⡏⠀⠀⢻⣸⣄⣼⠯⠛⢱⣶⣬⠀⠀⠈⠀⠁⠀⠀⠄⠈⠒⠦⠄⠀⡀⠀⠀⠀⠀⠀⠒⠃⡠⠂
⠀⠀⠘⣿⡜⡿⠁⠀⠸⣷⡜⣿⡷⠀⢘⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠁
⠀⠀⠀⢿⡷⣘⣴⣦⢀⠀⣤⣿⠃⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠘⠬⢀⡀⣀⢀⡀⠤⠄⠐⠊
⠀⠀⠀⠘⣿⡹⣿⣷⢀⢸⣿⣇⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⢀⠈⢀⠘⣿⣷⡍⢻⣶⣿⡉⠑⠚⠁⠀⠀⢀⣠⠤⠖⠊⠁
⠀⠀⠈⣾⣶⣿⡇⣾⣿⣽⣿⣿⡿⠇
⠀⠀⠀⠙⠝⠟⠿⣿⣦⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡽⠆⢀⡤⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⢧⣤⣄⠳⣿⠟⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢠⡁⣄⡹⣶⠀⠈⢛⠒⠊⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠧⣉⠁⠈⢷⡀⠀⠙⢆⠀⠈⢦⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠙⡄⡄⡘⡧⠀⠀⢻⡆⠀⠀⠹⣄⠀⠳⣤⠯⣠⠟⠁⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⣷⡈⠁⠘⣦⠀⠀⠻⣦⠀⠁⠘⣄⣀⠈⢷⣿⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢺⣿⣷⡀⣀⠘⣆⠀⠀⠘⣷⠀⠀⠘⢯⠀⠈⠻⡖⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⢾⣿⣷⡹⠀⠙⢦⠀⠀⠘⣧⡄⠀⠈⢳⣄⠀⠹⣆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⢀⢸⡿⣿⣧⠃⢀⠙⣾⡄⠀⠈⢿⡆⠀⠀⠁⠀⠀⠹⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢻⣷⣴⣿⣯⢻⡁⠘⢷⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⢸⣿⣿⣿⣿⣆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠽⠿⠟⠉⡉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⡘⠟⢿⣿⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣜⡖⠁⢁⠀⢺⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣦⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠨⠅⣙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣀⠈⠩⢭⣿⣷⣔⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡾⢁⣿⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢫⠛⣿⣿⣅⢦⡀⠀⠀⠀⠢⠀⣤⢾⠔⠟⠛⠋⢠⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣲⣿⠀⠀⠀⣀⠿⣿⣿⣯⢻⣞⣿⣦⠴⠋⠀⠀⠀⠀⢀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡺⠿⢿⣤⣴⠞⠁⣸⣿⡟⢛⣿⠿⠋⠀⠀⠀⣀⠄⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣄⢠⣴⠟⢁⡴⡋⣾⢃⠀⠀⢀⣀⣀⣤⠶⡛⠁⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣦⡼⠟⠋⠀⠀⠀⠺⣿⣿⠀⣓⠚⠛⠍⢀⣠⠖⠚⠉⠩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢁⡀⠀⠠⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠙⠳⢦⡀⠀⠀⣀⡤⠶⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠹⣦⠞⠁⠀⠀⠀⢀⣀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣰⡟⢻⣿⣶⣿⣿⣿⣷⠀⢸⣤⡞⣹⣿⣾⣿⣿⣷⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⠇⢀⡿⣿⣾⣿⣿⣿⣿⣿⠁⠀⢸⠇⠀⠀⠀⢀⡀
⠀⢀⣤⣤⣤⡀⠀⠀⢳⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠋⠀⣸⠃⠈⠻⢿⣿⡿⠛⠁⠀⢀⡾⠀⠀⠀⠛⠉⡟⠷⡄
⠀⠋⠉⣧⠀⠀⠀⠀⠈⢳⡀⠀⠀⠈⠙⠻⠛⠉⠀⢀⣴⠋⠳⣄⡀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⢰⠃
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠉⠳⢤⣄⣀⣀⣀⣤⠶⠋⠁⠀⠀⠈⠙⠓⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⠤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣤⡤⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣷⣶⣄⡀⠀⠈⢻⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠉⠛⠉⠀⢀⣤⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣏⠻⠟⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣛⣛⠛⠛⠉⠁⠀⣴⣶⣦⣤⣤⣄⣀⣠⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣄⠀⠀⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⡙⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢛⣥⣶⣦⡙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⢿⣿⣿⣿⣶⣝⠻⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⢇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣝⠿⣿⣿⣿⣷⣄⡀⠀⢀⣴⣾⣿⣿⣿⠿⣋⣴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡙⢿⣿⣿⣿⠟⠋⣴⣿⣿⠿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠉⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢿⠟⢿⠟⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣽⣤⣽⣳⢿⣷⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⡿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⡀⠀⠀⢿⣿⣤⣿⣦⣾⣦⣾⣿⠇⠈⠉⢀⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⣿⣿⡿⢿⠿⠿⠛⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠀⢀⣀⣀⢀⣀⣀⢀⠀⣀⢀⣀⡀⢀⣀⡀⠀⢀⡀⢀⣀⡀⢀⣠⣀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠟⣰⠿⣇⢸⡶⣟⠀⣿⠀⣿⠳⣿⢸⣗⡂⢸⠶⡟⠀⢸⡇⢯⣀⡀⢿⣀⣽⢸⡟⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⠈⠀⠈⠀⠈⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⢣⠀⠀⠀⠌⠙⠃⡄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠚⢒⠀⠀⡀⠀⠐⠉⢃
⠀⠀⠀⢠⣀⡀⠀⡄⠓⠂⠋⠈⠈⠈⠈⠒⠒⠄⡀⠀⢀⡤⡀⠀⠀⢀⠱⠆⠊⠁⠁⠁⠉⠑⠐⠢⢀⠀⢀⡀⡀
⠀⠀⠀⡄⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣀⢀⠂⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢀⢀
⠀⠀⠀⣐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⡀⠠⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠠⠦
⠱⣀⢴⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠
⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⠀⠱⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠔⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⢻⡆⠀⠀⠀⠀⠀⠀⢰⠟⠛⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡜⠒⡀
⠀⠡⠠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⡀⠃⠀⠀⠀⠀⠀⠀⠸⣄⠀⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⠄⠄⠆
⠀⠀⠀⠠⢥⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡀
⠀⠀⠠⠁⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡋⠀⡐
⠀⠀⠀⠈⠈⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠁
⠀⠀⠀⠀⠀⠠⠈⠙⢢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⠉⢃
⠀⠀⠀⠀⠀⠀⠁⠂⠃⠈⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠑⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠈⠑⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠋⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠨⠓⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠓⡀⠈⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠄⠁⡱⠦⢤⠀⣀⠤⠶⠌⠐⠠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠌⠉⠐⢀⠠⠂
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⢁⡈⠉⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣴⠛⠛⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⠛⢻⡆⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢻⣄⠀⠚⢻⠟⠁⠀⣠⡟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠙⢷⡀⠀⠀⠙⠀⢤⣤⣤⡴⠞⠋⠀⠀⢀⡾⠋⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢠⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣸⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣇⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣯⢴⣰⢰⣇⢴⢰⢲⢴⣶⠔⣦⢰⣆⠀⣶⣲⢠⣶⢰⡆⣟⢰⢲⣤⣶⠆⡖⢲⡆⡟⣦⣾⢰⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠁⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠉⠁⠀⠈⠀⠀⠈⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡰⠁⠀⠀⠈⡇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠃⠀⠀⠀⠀⠸⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡰⡄⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡰⠁⠹⡄⠀⠀⠀⠀⠈⡇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠚⠀⠀⡰⠁⠀⠀⠱⡀⠀⠀⠀⠀⠋⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢠⠃⢰⠛⢄⠀⠱⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⡇⠀⠀⠳⡀⢳⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⡆⢇⠀⠀⠀⢳⠀⡇⣀⣀⣀⣸⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡤⠒⠉⠉⠙⠒⢤⡀⢈⡷⠋⠉⠀⠈⠉⠓⣌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⡴⢦⠀⠀⠀⠀⠙⠞⠀⠀⠀⠠⣎⣳⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠉⠁⠀⡀⢠⣀⣇⣰⣁⡆⡄⠀⠈⠀⠀⡜⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢄⠀⠀⠀⢸⡊⠉⣞⣳⠀⢉⡽⠀⠀⢀⠞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⠳⣄⠀⠀⠏⡲⢪⠥⡖⠻⠈⢀⡴⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⡑⢦⡀⠀⠈⠀⠀⠀⡠⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⡓⢄⡀⣠⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠒⠀⠠⠤⠀⣀⠠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⡉⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⡀⠀⣀⠔⠃⠀⠤⣄⠀⠀⠀⠀⠀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⢀⡠⠤⠒
⠀⠀⠀⠀⠀⠀⠈⠀⠎⠈⠠⣀⠀⠀⠀⠂⠁⠢⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠐⢄⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡜⠀⠀⢠⠦⡄⠀⠀⠀⠠⡀⠀⠀⠑⠀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡃⠀⠀⡆⠀⢨⠀⢄⠀⠀⠘⢄⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⡀⠀⠠⡀⠀⠑⢄⠀⠀⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠋⠲⣂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⠝⠁⣰⠉⡙⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠊⠀⢠⠂⠀⡔⠀⢉⣴⠇⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡅⡸⠁⠀⡔⠁⢀⠄⠀⠠⢆⡀⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠟⠁⠀⡸⠀⠠⠋⠀⠰⠁⠀⢠⠃⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠔⠀⢠⠁⠀⡴⠁⠀⠀⡿⣄⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠖⠁⠀⠀⠘⠀⣠⠂⠀⠘⠀⠀⠀⠜⠀⡇⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⢀⠷⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠴⡃⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⡠⠊⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢠⠞⠓⠆⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⣧⣀⠄⠈⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠠⣖⠉⣉⣉⡳⣽⣗⠈⠂⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠈⠙⢏⠀⠈⠻⣦⣄⠀⠀⠀⠀⣰⣿⣤⣤⣴⣢⣤⣶⡋⠀⠀⠀⠀⢀⡔⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⠜⣧⠤⠂⠀⠉⠉⣿⡯⠊⢻⣮⡛⠿⣿⠛⠷⠒⠉⡷⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣅⣌⡄⣿⡌⠄⠈⠳⣤⣬⡿⣡⣿⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⢻⣛⡓⠢⣀⣤⡇⢹⣿⣧⠹⣧⠀⠀⠀⠈⠻⠿⠿⠏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⡻⠋⠁⠀⠙⠓⢾⠉⡛⢦⡙⠂⠈⠢⠵⠖⡶⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⢀⣠⠇⠀⠀⠉⠉⠐⠒⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠴

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠄⠠⠄⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠁⠀⠀⠀⠂⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡐⠁⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⢠⢜⡂⠈⠁⠆⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⡖⠠⡆⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⣄⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠐⠢⣠⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣰⠋⠁⠀⠀⢴⡤⠖⠓⡦⠄⠠⠀⠘⠀⠚⢃⡤⣀⣀⠀⠀⠀⣟⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠤⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⣀⠒⡐⢆⠀⠀⠀⠈⡆⠀⠀⠀⠘⡶⠁⠀⠀⢀⠀⣄
⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠂⠀⠸⣇⠀⠀⠀⢸⡇⠀⡄⠀⠐⠀⢁⠈⠀⠀⠀⠀⠀⠉⣇⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠠⠀⠀⠐⠂⣨⡀⠀⠀⡄⣿⠀⠀⠉⠘⣷⡐⠁⠆⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡏⠰⠀⢀⠀⢸⡁⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡂⠐⢼⡇⠀⠀⠐⠚⣧⠀⠀⠀⢀⠀⠀⠀⠠⠀⠂⠘⡾⠀⠀⠀⠀⢬⢸⡗⠁⠰⠉⢐⠁
⠀⠀⠀⠀⠀⠂⡀⠈⠹⣤⠀⠀⠉⢀⢸⡏⠀⠀⠀⠀⠘⠳⣅⠀⣸⠀⠀⠀⠀⠆⣠⠞⠁⠀⠀⠀⠀⠀⣻⣠⠀⠀⡷⠈
⠀⠀⠀⠀⠐⠀⠀⠂⠠⠷⢄⣀⡀⠈⢹⡷⠀⠀⠀⢈⠀⠄⣌⠿⢧⣄⡀⢤⣰⠟⠁⠀⠀⠀⠂⠠⠂⠈⣿⠀⣠⡾⢁⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠉⢳⣼⡇⠀⠀⠀⠀⠁⠀⠁⢀⢀⣩⡿⠿⣧⣞⣀⠀⠀⠼⠀⠁⠀⣸⣧⡾⠋⠀⠈⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢹⡇⠀⡀⠃⠀⠀⢦⣠⡾⠛⠁⠄⠀⠀⠉⠙⠿⢿⣦⣀⡄⠀⣿⣿⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢙⢀⣴⠟⠛⠲⠀⠂⠰⠁⠀⠀⠀⢤⡀⡬⠻⢿⣦⣿⣿⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠘⠀⠀⢻⣆⠐⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⠀⠈⠀⠀⠹⣿⣿⡇⠀⢀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠁⢀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡅⠀⠀⠰⠀⢠⣿⣿⡇⠀⠀⠀⡵⠚
⠀⠀⠀⠀⠀⠀⠉⠑⣮⡀⠀⠀⣜⠁⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⣿⣿⠋⠠⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠹⣆⡀⣾⠁⡄⠀⠀⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠾⣯⠈⠀⠈⡄⠂
⠀⠀⠀⠀⠀⠀⢀⡰⠆⠈⠉⠛⠛⠻⢷⣤⣿⣿⡇⠀⠀⠀⠀⠀⣠⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⣼⣿⣧⠈⣿⣿⡆⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⢸⣿⡿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢻⣿⡟⠁⢹⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡀⠀⠀⠀⣿⣿⡇⠀⠈⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⡇⠀⠀⢀⣿⢸⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⡆
⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠶⠾⠿⠿⠷⠶⠶⠶⠾⠿⠿⠟⠛⠛⠿⠿⠛⠛⠳⠶⠶⠤⣤⣄⣀
⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣶⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣟⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⠀⠈⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⠗⣶⣹⠇⠀⢠⣤⡈⡆⠀⢠⡇⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣿⡁⣺⠗⠋⠀⠀⠀⠈⠙⣷⠀⠈⠀⠑⠀⠀⢀⠱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢹⣿⢿⢇⠈⠀⠀⠀⠀⠀⠀⢄⠉⠀⠀⣀⣀⡀⠀⠈⠀⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠀⠀⠹⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌⠍⢻⣖⠂⣭⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠑⠀⢸⡯⢳⣻⣿⡄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠣⠀⠀⠀⠀⠀⢀⣬⠁⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠔⠋⣀⠞⣡⢿⣇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠤⠐⢚⡡⠞⣵⣿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡩⢒⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⢶⣿⠻⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠔⠐⠋⠁⠃⠀⣏⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢀
⣿⣟⣟⣿⣟⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣛⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠛⠛⠻⠷⣶⣤⡀⠀⠀⠀⢀⣤⣶⡾⠿⠟⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠀⠀⠀⠀⡘⣄⠳⣀⣀⠀⠀⠐⠘⣟⣿⣿⠋⠐⠲⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣌⠑⢦⣤⣤⡼⢾⣿⣽⣷⣭⣶⣾⣿⣿⣿⣿⣷⣶⣤⣧⣀⠈⠙⣷⣆⡀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⢤⣬⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡀⠙⢦⡀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢒⣲⣿⣿⣻⣿⣿⠋⢯⣹⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⠀⠙⠦⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠹⣿⣿⡇⠀⠘⢻⣿⣿⠟⠺⢷⣿⣿⣿⣿⣿⡟⣿⣿⡿⠻⣿⣮⣳⡀⠈⠂⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠁⢻⣻⣿⠀⠀⠈⣿⣂⠖⠈⠙⢿⣿⣿⣿⡿⠫⢿⡟⠀⠀⢿⢻⣷⡝⠄⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢰⢿⢿⣧⣄⠀⠘⢿⡀⠀⠀⠀⠏⠉⣉⣋⢠⠟⠀⠀⠀⠈⠀⢙⣿⡄⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠁⠀⠹⣿⣿⣷⣄⠀⠙⠦⣴⡊⡈⠸⡁⢘⠁⠀⠀⠀⠀⠀⢀⣄⢉⡿⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⢹⡗⣿⣿⣿⣷⣶⣤⡀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠊⠉⣿⠛⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢸⣿⠇⠙⣿⢶⡟⠉⠙⣏⡉⠉⠉⠉⠀⠠⠤⠐⠀⠀⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⣿⠀⠀⠘⣿⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⢀⣤⡾⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢿⣦⣿⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⢻⢷⣦⣀⣀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⠉⢉⠱⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣆⢴⣏⣀⣀⣠⣨⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡾⢿⠻⠿⠃⠁⠝⠉⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⠻⠟⠁⠀⠀⠀⠀⠑⢐⠀⠀⠀⢀⡈⠳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣆⠦⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡗⣤⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⢿⣿⡁⠬⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⠲⣶⡀⠀⠀⠀⠀⠀⠀⢺⣤⠋⠀⠀⢀⢻⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣦⡝⠁⠀⠀⠀⠸⠘⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡇⠀⠀⠀⠀⠀⠢⢹⡄⠀⠀⢀⡤⠃⠀⠀⠀⠀⠀⢀⣾⡄⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠁⠀⠀⠀⠀⠀⠀⠻⢷⡀⠀⠉⠀⠀⢀⠀⠀⠀⠀⣠⠟⠛⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠖⠛⣳⠀⡤⠀⠀⠀⡀⡤⠔⣻⠟⠰⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⣤⠶⠉⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢄⣀⠀⠀⠀⠀⢀⣀⣤⣾⠃⠀⠠⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠛⠛⠛⠋⢻⣿⣏⠀⠀⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣾⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁⡄⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠿⠉⠛⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠏⠁⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⣼⣧⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⡀⢸⣿⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢻⣿⡀⣿⠀⠀⠀⠀⠀⠀⢸⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢩⠂⠀⠀⠀⠀⠘⢻⡇⣿⠀⠀⠀⠀⠀⠀⢸⠃⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⡇⠿⠀⠀⠀⠀⠃⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⢻⠿⡿⠟⣿⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⡟⢻⡶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⠙⣿⢸⡇⢸⡇⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠾⠷⠶⠶⠶⠶⠶⠾⠷⢦⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠋⠈⠁⠈⠁⢸⡇⢀⣤⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢸⡿⠋⢉⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠗⠲⠶⠶⠶⠶⠶⠿⠶⢶⣄⠀⠀⠲⣶⠶⢶⡶⠲⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⢦⣄⠈⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠛⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡀⠙⠳⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠷⠶⠿⠦⠀⠀⠙⠷⢶⣶⠶⠶⠶⠶⠶⠶⢶⣴⠟⠁⠀⠀⠀⠀⠀⠀⣿⣶⡄
⠀⠀⠀⠀⠀⢀⡿⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠘⢷⡶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣁⣠⣾⡇⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢷⡶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠛⠉⢸⡇⢀⡀⢀⡀⣠⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⡶⠶⠶⠶⠶⠶⢶⡶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⢸⡇⢸⡇⣿⣄⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⣧⣼⠗⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠚⠉⠉⠉⠉⠓⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠉⠉⠉⠉⠑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠘⢆⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠉⠓⢴⡁⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠈⠙⠢⣄⠈⡆⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡼⠁⠀⠀⠀⠠⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘
⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢣⡜⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⣌⡉⠛⠻⠂
⡞⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣴⡟⠻⣿
⡇⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣴⣿⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣿⣧⣴⣿
⣇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⡀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠻⢷⣄⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠘⣷⡀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⣞⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠘⢷⣄⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠙⠻⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠶⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠒⠉⠀⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⢀⢔⣿⠇
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⣾⣵⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢗⣦⣀⠀⠀⠀⠀⠀⠀⣀⠤⢺⣧⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⣿⠀⠉⠲⢤⣤⠒⠉⠀⠀⠀⢿⡄⠀⠀⢸⡟⠢⠤⠖⠊⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠁⢸⡿⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⢸⣷⣄⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠼⣿⡿⠷⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠰⠿⣿⡿⠄⢸⡇⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠷⡄⠀⠈⠻⣦⡖⠤⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠸⣀⠀⠈⠲⢄⠀⠀⠀⡠⠼⠧⡄⠀⠀⠀⢀⡤⠿⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⠃⠀⠀⠀⠀⠀⠉⠢⡄⠀⢱⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⠃⠀⠀⢣⠀⠀⠇⠀⠀⠀⠀⠸⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⡤⢃⡤⠒⠒⠋⠀⢸⠀⠀⠀⡠⠔⠒⠁⠀⢸

⣟⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⠟⠁⠀⠀⣠⣿⢃⣾⠃⠀⣾⠇⠀⣠⡿⣧⣼⡿⠛⠁⠀⠀⠀⠀⠀⠀⢠⡾⠃⢀⡜⠁⠀⠀⢠⣿⣧⣃⣤⢾⡟⠀⢀⡆
⠛⠁⠀⢀⡴⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⣠⡾⠛⠻⠟⠁⠀⠀⢿⣤⡾⠋⠀⠀⢻⣀⣀⣴⠖⠀⠀⠀⠀⢀⡿⢁⡴⠋⠀⠀⢀⡴⣿⣏⡟⠉⠀⣾⠁⢠⠚⠷⣴
⣀⣀⡴⠞⠀⠀⠀⠀⠀⠀⠸⣿⡀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣾⡷⠋⠀⠀⢀⡴⠋⠀⠻⠛⠀⠀⠀⠿⠖⠁⠀⠀⠸
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄
⠀⠀⠀⢠⡾⠛⣦⠀⢀⣠⣾⠆⠀⠀⠀⠀⠀⣴⡶⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⢀⣿⣿⣰⣿⡶⠿⣿⡏⠀⢀⡄⠀⢠⣾⠋⢠⣾⠇⠀⠀⠀⠀⠀⠀⣠⡟⠁⢀⡼⠁⢀⠄⠀⠀⠀⢴⡆⠀⠀⢀⡀⠀⢀⣴⢿⡆⢠⡀⠀⠀⠀⣠⣿⣿⠁
⠀⣴⣿⣿⢋⣿⠋⠀⣸⣿⠀⢀⣾⡇⢠⣿⣃⣴⠿⠃⠀⠀⠀⠀⠀⢀⣾⡿⠁⣰⠟⠀⣠⢷⣠⡤⠊⣠⠏⠀⠀⠀⣼⠃⣠⣞⡡⠋⠀⢸⠃⠀⠀⣰⣿⡿⠁
⡾⠋⢿⣇⡼⠃⠀⠀⣿⡇⢠⣾⠉⠓⢾⡿⠋⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣡⡞⠁⢠⢾⣏⡠⠀⠀⢠⠏⠀⠀⣠⠎⠻⣴⣿⠋⠀⠀⣠⡟⠀⠀⢠⣿⡿⠁
⠁⠀⠀⠉⠀⠀⠀⠀⠻⢷⠿⠁⠀⠀⠸⣇⣀⣀⣤⡴⠂⠀⠀⢀⣿⣿⠟⠃⢀⡴⠋⠈⠉⠀⠀⠀⣏⠀⢀⡼⠁⠀⠀⢾⣿⣀⣠⣾⠏⠀⠀⢠⣿⡿⠁⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⣿⣟⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠿⠶⠋⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⣼⡿⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⣠⠟⠀⠀⡠⣄⠀⠀⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀
⡆⠀⢸⡆⠀⠀⠀⠀⠀⢀⣾⣿⢃⡾⠋⠀⣠⣶⡇⠏⣠⣾⠏⠀⣴⠂⢠⣾⣿⣿⠋⠀⠀⡀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⢠⣿⢻⡆⠀⣠⣿⠃⢀⡀⠀⣠⡿⠋
⠀⠀⣸⠇⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⢀⣴⣿⣏⡟⠋⣹⡟⠀⡴⣿⣤⣿⡿⠋⠁⠀⣠⠎⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣼⡿⣿⣧⣾⣿⠃⠀⣸⡇⢰⣿⣡⣾
⠀⣴⠏⠀⠀⠀⠀⠶⣾⣿⡏⠀⢀⣴⠏⠸⠿⠟⠀⠀⢿⣧⠞⠁⠀⠹⣿⣄⣀⣤⡾⠃⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠛⢹⡏⠀⣰⡟⠙⣿⠟⠋⠁
⡾⠋⠀⠀⠀⠀⠀⠀⣿⣿⣥⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⣠⣿⠛⢿⠟⠀⠀⣿⡇⣴⡿⠁⠀⢿⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠙⠻⠛
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⣠⡖⢶⡄⠀⢀⣴⡷⠀⠀⠀⠀⠀⣠⣾⠟⣶⡀⢠⡏⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⢠⣿⡇⣼⣧⣶⢿⣿⠃⠀⣀⠀⠀⣴⡟⠁⣴⡿⠀⢸⡇⠀⠀⣠⣾⠟⠀⠀⠀⣠⠖⠀⣸⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀
⠁⣶⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⣰⣿⡿⢻⣿⠉⠀⣾⡏⠀⢠⣿⠀⢰⣿⣠⡾⠛⠀⠀⠀⠳⠶⠿⠋⠁⠀⠀⢠⣾⠏⢀⡼⠁⣠⠈⠀⡠⢀⡟⠁⠀⢀⡖⠀⣰⢏
⢠⠇⠀⢠⣿⡿⠃⠀⠀⠀⢠⣾⠟⣿⢃⣾⠃⠀⢰⣿⠃⢠⡿⠉⠳⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢏⣴⠋⢀⣴⠋⡙⠋⢀⡞⠀⠀⣠⢾⣇⣼⡿⠋
⠏⠀⢀⣾⣿⠃⠀⠀⢀⣴⡟⠁⠀⠉⠛⠁⠀⠀⠈⢿⣴⠟⠁⠀⠀⠹⣇⣀⣠⣴⠖⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠁⣴⠋⠘⠛⠀⠀⡼⠀⢀⡴⠁⠀⢸⣿⠀⣀
⠀⠀⣸⣿⠃⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢁⣠⠟⠁⠀⠀⠀⠀⠀⠿⡶⠋⠀⠀⠀⠈⠛⠛⠋
⠀⠀⣿⡏⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠁
⠀⠀⠻⢧⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⣠⣾⠟⠁⢠⣤⣄⠀⠀⣠⣶⠄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⢷⠀⠀⠀⢀⡀
⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣠⣾⡟⠁⠀⢠⣿⣧⣿⠶⢻⣿⠃⠀⢠⡄⢀⣔⢋⣼⠃⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⣠⠏⠀⠀⢠⡏⡃⠀⠀⣠⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠁⢴⡅
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠤⠠⠤⠄⠀⠀⠀⢲⠸⠇
⠀⠀⠀⠀⠀⠀⠀⣀⠗⠁⠒⠀⠀⠀⠀⠒⠐⠢⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢘⠂⢀⢀⠀⠈⠁⢄⣀⡀⠰⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠂⠔⠚⠃⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠂⠀⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡠⢄⡀⠀⠠⠀⠀⠀⠀⣀⠠⠀⠀⡉⡁⠀⠠⢠⠀⢀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⢛⠀⠀⠀⠀⠀⠀⠀⠁⠠⣂⢋⠃⢁⢑⡤⠐⣇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢈⡄⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠈⠀⠉⢠⠰⠅⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠓⠄⡄⠀⠀⠀⡀⠁⠀⡀⠀⠀⠀⢀⠨⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠓⠂⠀⢀⢀⢈⡀⢀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠂⠀⠂
⠀⠀⠀⠀⠀⢀⠤⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣾⢟⣿⣖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠩⠍⠉⠉⠍⢹⣿⡀
⠀⠀⠀⠀⢀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠙⠛⢿⡷⢞⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢰⠇⠀⠀⠁⠀⠀⠀⡩⠇
⠀⢠⣴⣿⣿⣽⣿⣏⣶⣲⠄⠀⣀⣤⣄⠤⣯⡔⠉⠖⣂⠉⠉⢠⢸⡗⠀⠀⠀⠀⠀⠀⢠⠾⠿⠛⠉⠀⣄⠀⠠⡀⠈⢁⠀⢠⠞
⣼⣿⡏⠀⠉⠋⠉⠉⢫⣉⣧⣿⣿⡿⣿⡿⠿⢿⣶⠄⠉⢠⡴⠀⡤⠀⠀⠀⠀⠀⠀⠀⡾⢲⣄⢒⡀⣼⠿⠃⠀⣱⢄⣤⣴⠃
⢿⣿⡆⠀⠈⠁⣀⣠⠄⣾⢽⣯⠁⣀⣀⠀⠀⠤⢿⠒⢢⣄⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡅⠈⠊⠀⠀⠀⠁⡄⠁⠘⡿⡇⢱⡄
⠀⠉⠛⠓⠒⢤⡀⢡⡞⠹⢮⣿⡄⠀⠀⢄⡰⠀⡘⣰⣧⡼⣿⡗⢤⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⣶⠀⠂⢀⡞⣧⠀⠆⠈⠢⡀
⠀⠀⠀⢠⣴⡊⣭⡉⠑⣲⠄⠙⠛⠒⡄⢀⠔⠈⠐⢋⠃⡠⠋⣀⣌⢡⠀⢠⡄⠒⠀⠄⠀⠘⠒⠒⡤⠘⠑⡀⠀⠀⣧⠛⠄⣂⠀⠀⡇
⠀⠀⢰⠟⠿⢀⡉⣵⢄⣿⠀⢆⡠⠘⠓⠚⣁⡀⠇⠈⠀⠀⣀⣿⠿⠀⢂⣠⣜⣶⡶⠂⠖⠂⠐⢾⡏⠀⠀⠐⠀⠀⢹⠁⠁⠀⠀⠀⡇
⠀⢠⡟⠀⠀⠀⢁⡝⡠⠊⢰⡌⠉⠈⠁⠉⠉⠱⠀⠀⡇⡔⠉⡾⠒⠓⠀⢤⠾⠉⢁⠁⠊⡇⠀⠸⡀⠀⠀⠀⠀⢆⠸⢀⠄⠀⡄⢸
⠀⣸⠃⠀⢀⡤⡸⠾⣀⠀⣾⠁⠀⠀⠀⠀⠀⠱⠡⠀⠳⡎⠀⣥⠀⠀⠀⠐⠀⠀⠀⠀⠀⠆⠀⢸⠁⠀⠀⠀⠀⠈⢷⡈⠀⠀⠀⡜
⠀⣾⠀⡞⠩⠶⣄⠸⠋⣼⠇⢀⡀⠀⠀⠀⠀⣰⡀⠐⠇⢠⡎⠠⣄⣀⣀⡰⡀⠀⢀⡄⢠⠙⠃⠆⠀⢀⠀⠀⠀⠀⠈⢆⠀⠀⢠⡇
⠀⢿⡄⠋⠀⠀⣈⣶⣷⣼⡀⠾⢄⠀⠀⠀⡔⠁⡰⢄⡀⡀⠈⠆⠸⣦⠄⠃⠈⢄⣀⠀⠀⠑⠴⠀⠀⡌⠀⠀⠀⠀⠀⠀⢷⡀⠈⡇
⠀⠀⠀⠀⣰⣿⡋⠉⠁⢸⣧⠀⠈⠳⢤⣴⠁⠐⠀⠀⠈⠁⠀⠰⡆⠙⠒⠈⠉⢢⠀⠈⠀⠀⠃⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠁⡀⠃
⠀⠀⢀⡞⢻⠀⠈⠉⠐⠺⠇⠱⣤⣾⣿⣿⣧⡇⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠉⠓⠤⢄⣀⢀⠀⢰⣀⡀⠀⠀⢀⡠⢄⠀⠙⡄
⠀⠀⣾⡀⠛⠂⣴⣦⡀⢀⠇⣸⣿⣽⣿⣟⣿⣷⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠖⠁⠀⠀⠂⠀⠘⡄
⠀⠀⠈⠉⡷⠒⠂⠤⠤⠃⢸⣿⣿⣿⣧⢉⠟⣹⣧⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⡾⣻⣷⢻⠈⠉⢰⠟⣻⡆⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠀⢀⠀⠐⠒⠂⠁⠀⠂⢹
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡗⣾⣿⣼⠇⠇⠁⣸⠋⣿⡄⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠑⠴⠄⠛⠀⠀⠀⢸
⠀⠀⢀⡿⠀⠀⠀⠀⠀⣾⢻⢸⡃⡏⠰⠀⡞⠁⠇⠈⢷⡀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⢠⣿⡌⢸⡇⠃⢠⡼⠀⠀⠸⡆⠘⣷⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⠀⠀⠀⠀⢸
⠀⠀⢸⠁⠀⠀⠀⠀⡾⠹⠇⢸⡟⠀⠘⠀⠰⠀⠆⠁⠀⠸⣧⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⣼⠀⠀⠀⠀⢠⡏⣿⠈⠄⠃⠃⠀⡀⠀⡀⢀⠀⠀⠠⠛⡆⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢈⠀⠀⠀⠀⡆
⠀⠀⡟⠀⠀⠀⠀⣾⠓⠈⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠘⢹⡄⠀⠀⡀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⠀⠀⠀⠀⡇
⠀⢀⣇⣀⡀⠀⢀⡿⡄⠀⠀⠀⠀⡀⠀⢀⠁⡈⢀⠀⠀⠈⡠⢁⡴⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠉⠈⠷⢶⡾⠀⣷⠄⠀⠐⢠⡇⠀⠃⡸⠃⠀⡐⢀⠀⢁⠌⠀⢠⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠶⠈⠃⢀⠀⢸⠀⠈⢀⠁⠘⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢸⡄⠄⠀⠀⠀⢰⠈⢰⠀⢀⢠⠀⠀⠀⢇⠘⠀⠀⠀⠀⡘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⠄⠀⠀⢰
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⡜⠀⠀⠎⠄⠀⠀⠈⡀⠀⠀⠀⠀⢶⠂⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡦⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣼⢃⠀⠀⠀⠀⠘⢀⠃⠀⠀⠢⠚⠳⠶⠄⠀⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢨⡀⠀⠀⠸
⠀⠀⠀⣀⡴⠓⠚⠁⠀⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⢁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠡⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡷⠚⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣅⠄⠚⠋⠉
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠠⠀⠄⠖⠰⠂⠴⠠⠤⢀⡀⠀⠀⠀⠀⢀⣀⠠⠄⠤⠰⠄⠦⠠⠄⡄⣀
⠀⢀⡠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠐⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⡀
⠀⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⠀⠀⢀⠆⠀⠄⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⢀⢸⠀⠀⡈⠀⠸⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣼⠀⠀⢸⣠⢸⢹⠀⠀⠀⠀⠀⠈⢻⠀⠀⠠⠀⢠⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠙⣆⠀⡼⢸⠸⣘⡇⢸⡀⠀⠀⠀⠸⡄⠀⠈⡇⠸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠈⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢀⠐⠠⢠⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡏⡆⢰⡇⠀⡷⠈⡦⡇⠀⠹⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠀⠠⠇⠀⠠⠊⠀⠣⠘⠣⠀⣷⠠⠇⠣⠀⠠⠇⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⢠⠎⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣸⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡆⠀⡆⢰⠐⡄⢰⢰⠀⢸⢰⡇⣸⢸⢸⡀⢢⢰⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠑⠀⡇⠘⠀⠁⠘⢸⠂⠋⠀⠁⠹⠘⠘⠁⠈⠀⠃⢸⡆
⠀⠁⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⠠⡐⠊
⠀⠀⠀⠈⠁⠂⠤⢠⢀⡀⣀⢀⡀⣀⠠⡤⠴⠒⠋⠀⠀⠙⠑⠖⢤⠄⣠⢀⡀⣀⢀⡀⠄⠰⠘⠊⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡶⠶⠿⠿⠿⠿⠿⠿⠿⠷⠶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⣴⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠈⠛⠃⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠈⠛⠋⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣤⣀⣀⣀⣀⣀⣠⣤⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠷⢶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠟⠛⠋⠉
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⣀⠀⠀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⡀⣸⠀⣆⠀⡇⠀⢰⡇⡒⢰⡧⣴⡇⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⣇⢠⡘⡦⣇⣻⣇⣿⣄⣷⣷⣿⠃⣇⣾⢣⣿⣼⣿⡿⣄⡴⣴⡆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⢌⣿⣧⣿⣿⣼⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⢟⣿⣿⣾⢋⣿⣯⣾⣣⡞⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠦⣀⣻⣖⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣡⡾⣃⡴⠋⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣨⣿⣿⣿⣿⣿⣿⡿⠿⠛⢋⠉⠉⣍⣉⡟⢩⢋⠻⡿⣿⣿⠿⣿⣿⣿⣿⣿⣻⣵⣿⣿⣾⣯⡴⢟⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠉⢹⣿⠁⠀⠀⠀⣶⡿⠻⠟⠛⣷⠎⠀⠘⣿⣿⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣫⣴⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣿⣿⡿⠋⠀⠀⠀⠀⠸⣿⡛⠀⠀⢘⣿⣧⡀⠀⣠⣿⣖⡁⠀⢿⡿⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣭⣥⣤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⢿⣧⡠⢔⡩⣻⠿⠷⠾⠿⡃⠀⠀⢀⣾⠁⢀⣠⣤⢤⣶⣴⣾⣿⠿⡿⣍⡛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢚⣋⡙⠻⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⠻⣿⣧⠞⡅⠀⠀⠀⠀⢳⡌⣠⠞⠁⣀⣢⣤⣶⣿⣿⣯⠛⢫⢔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢩⠟⢿⣶⣶⣿⣯⣭⣤⣌⣙⣿⣶⣦⣤⣤⣤⣶⣿⣷⣾⣿⣿⡿⢿⠛⠓⠀⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀⠈⠈⠁⠋⠉⠋⠙⠛⢛⠛⠟⠛⢿⠛⣿⠛⠏⠙⡟⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠁









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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋⠀⠀⠀⢀⣴⣶⣄⠀⣀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣦⣄⠀⠀⠙⠛⠋⠁⣠⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⣄⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣠⣤⣴⣦⣤⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣿⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣀⣀⣹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡟⠋⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠙⢻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠐
⠀⠀⠀⠀⢀⣀⣀⠌⠀⠀⣀⣀⡀⠉⠙⠛⠿⣿⣿⣿⣟⡁⢄⠀⣾⣷⠀⠀⠀⠀⠀⠀⣾⣷⠀⡠⢊⣻⣿⣿⣿⠿⠟⠋⠉⢀⣀⣀⠀⠀⠡⣀⣀⡀
⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠂⢸⣿⡷⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⢸⣿⡇⠔⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⠴⣄⠀⠀⠀⠀⢀⣤⠴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣠⠏⠀⠀⠀⣰⠋⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢀⡾⠀⠀⢠⠶⠒⢦⠖⠲⢦⡀⠈⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠶⠖⠋⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠈⠷⣄⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠛⠶⣄⠀⠈⠳⣄⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣤⠀⢹⡆⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣠⡴⠚⠛⠛⠛⠛⠛⠛⠛⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣳⣼⡟⠓⠒⠶⠶⢦⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢦⣤⡟⠁⠈⠳⢦⣄⣀⣀⡼⠃⠀⠀⠀⠀⡇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠳⢤⣀⠀⠀⠈⠉⠙⣦⠀⠀⠀⠀⠀⡿⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹⡟⠓⠶⠦⠶⠃⠀⠀⠀⠀⠀⡇⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣧⣄⣠⣤⣤⣠⣤⣀⣠⡇⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⡗⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⢠⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⢸⡇⠀⣀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠙⣟⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠛⢾⡷⠛⠉⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⠶⠿⠶⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠾




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠄⠀⢠⠀⢀⠀⡠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⡀⠂⠒⠀⡆⠒⠀⠂⠀⠸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⡤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤
⠧⠤⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⢀⣀⣀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⢀⣀⡀⢀⣀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠐⠁⠈⠃⢸⠀⢸⠀⢰⢻⡀⢸⠤⡇⢸⠀⠀⡇⠀⠀⠀⠧⡀⠀⢸⠀⢸⣀⠇⡜⠀⠈⡆⣧⡜⠀⡧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣥⣤⣈⣤⣤⣃⣸⣀⣸⣀⣏⣈⣁⣸⣤⣏⣸⣤⣄⣧⣤⣀⣀⣤⣜⣀⣸⣀⣸⣀⣇⣘⣤⣔⣁⣇⣈⣆⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡞⡟⠳⣄⠀⣀⡀
⠀⢰⢻⣌⣖⠋⠻⣄⣹⠶⣄
⠀⠘⣟⢸⡉⣳⡀⠈⠳⣄⠈⠳⣄
⠀⢰⠻⡄⠉⠉⢳⡀⠀⠀⠙⢆⠈⠙⢦⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣿⠛⢦⠀⠈⠲⣄⠀⠀⣱⣦⡀⣙⢷⣄⢀⣠⠞⡭⣷
⠀⢀⣴⣇⠒⠚⢇⠀⠀⠈⢧⡀⠈⡠⢷⡄⠋⠙⣯⡟⠀⠳⢿⡀
⠀⢸⠁⠙⡄⠀⠈⢆⠀⡠⣢⡙⣦⠀⠀⢻⡄⠀⠘⣧⡀⠀⠸⡇
⠀⡆⠀⢀⠿⣄⠀⠈⠣⡀⠀⠘⠉⣧⠀⠀⠙⣆⠀⠈⠳⣤⡀⠙⣆
⢰⠇⠀⢹⠀⢹⡆⠀⡴⡟⢦⠀⠀⠘⣧⠀⠀⠘⢦⠀⠀⠈⠻⣄⠙⣆
⣼⠀⠀⢸⡀⠉⢻⡄⠀⠀⠞⢦⡀⠀⠘⢷⡀⠀⠀⠑⠀⠀⠀⠈⠓⠮⠳⣄
⡇⠀⠀⠜⢧⡀⠈⣧⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠹⡄⠀⠀⠀⠱⣄⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠙⣆⠀⠀⠀⣹⠁⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠈⢦⠀⠀⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⡆⠀⠀⠀⠀⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣧⠈⣧
⠀⠀⠀⠀⢳⡀⠀⠀⠔⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠸⡄⢸⡄
⠀⠀⠀⠀⠀⢹⡗⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣳⡈⢧⣀⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⣨⠷⠋⠁⠀⠀⠈⠛⢳⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⡸⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠙⠳⠦
⠀⠀⠀⠀⠀⠀⡈⠀⠙⣆⠀⠀⠀⠁⠀⣸⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⣇⠀⠀⠀⢀⠼⠁⣠⣿⣿⠳⣄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢸⠀⠀⢀⡏⠈⠓⠦⠀⠀⠒⠾⢡⠞⠁⢀⠈⠻⢦⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⢸⠀⠀⠸⣧⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⣸⠀⠀⡞⡽⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠨⢸⣄⣇⣿⣆⣀⣤⠤⠞⠁⠀⠀⠁⠀⡏⢀⡾⠁⡇⠀⣄⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠳⡉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣧⠀⠙⣦⣀⣀⣠⠤⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢲⠒⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⣀⣼⠷⠒⠊⠉⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣀⡴⠋

⠀⠀⠀⠀⠀⠈⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠦⠤⠤⣄⡀
⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠶⣤⣄⡀
⠀⠙⠦⣀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠈⠓⠾⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡹⠓⡶⢤⣀⣤⡤⠀⠠⢤⡀⠀⠀⠀⠀⠹⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⠤⠤⢤⣤⣀⠀⠀⠀⡗⠸⠁⠀⡩⠿⣦⣀⣀⡀⠉⠓⠶⠶⢆⡀⠑⢌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣄⡀⠘⢄⠀⢈⣿⢤⣉⠓⠦⠤⢤⡀⠙⠒⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠛⠛⠉⠓⠮⠭⠿⠦⣄⡉⢲⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁⠀⠀⠀⠀⠜⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡒⠀⠁⠀⡠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡳⡈⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠻⢄⡉⠓⢴⣖⣒⠦⣄⣠⣤⡤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠣⢤⡀⠙⠒⠒⠦⣌⡙⢺⣏⠀⠈⢢⠀⠉⠙⠓⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠢⡀⠉⠶⠶⠦⣄⠀⠉⠈⠙⢷⣖⡁⠀⣰⠠⡇⠀⠀⠈⠙⠛⠒⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⡀⠀⠀⠀⠀⠙⠒⠀⠐⠛⠋⠙⠲⠧⣼⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠤⣄⠀⠀⠀⠀⠀⠀⠈⠁⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠶⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠰⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠉⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⢠⣾⣿⣿⣿⣷⣄⣶⣿⣽⣯⣕⣆⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢧⡿⣀⣀⣀⣀⣀⡀⠙⢿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣼⣧⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⢀⣠⣤⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⢸⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣫⣴⣶⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢻⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠎⠷⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠈⠙⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣞⡽⠓⠲⠤⣤⠞⠉⠉⠉⠉⠙⠛⠒⠒⠒⠚⢯⢳⡀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢠⢎⡞⠀⠀⠀⡰⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡳⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣠⢫⠏⠀⠀⠀⡞⠁⢀⣠⠞⠉⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⢳⡹⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣰⣣⠋⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⡹⣽⣄⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⡜⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡴⠁⠀⣳⠊⠉⢢⠀⠀⠀⠀⠀⠓⢤⡀⠀⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡜⠁⠀⡰⠋⠙⡆⠀⠓⢦⡀⠀⠉⢳⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣠⡞⠁⠀⡴⠛⠲⣄⠀⠈⢳⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣞⠁⠀⡴⠋⣳⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠙⠒⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠦⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠙⢦⡀⠀⠀⢀⣴⣋⢀⡴⠚⠛⠲⣄⠀⣠⠖⠛⠛⢶⡄⣈⣷⡀
⠀⠀⠀⢠⡏⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠛⠻⢿⣷⡄⠈⢷⠀⡼⠋⠁⠉⠛⠀⠀⠀⠀⢸⠆⡏⠀⠀⠀⠀⠛⠉⠉⠉⢳⡀
⠀⠀⠀⡿⠀⢰⣿⠟⣋⣉⣀⠀⠀⠀⠀⢠⣾⣷⡆⠙⠟⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠈⠁⣾⠿⠿⠿⣷⠀⠀⠀⠸⣿⣿⡷⠤⢤⡀⢀⣿⠿⢶⣤⣀⠀⠀⠀⢀⡞⠀⠀⠀⢻⡄⠀⠀⠀⢀⣠⠶⢿
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠹⠏⠀⠀⠀⠘⡏⠓⢦⣤⡞⠀⠀⠀⠀⠀⠻⣄⡴⠞⠉⠀⠀⣿
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠚⢙⠆⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠠⣦⣄⣀⡀⠀⠀⣀⣠⣴⡄⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠤⠬⠃⠀⠈⠳⣄⠀⠀⠀⠀⣠⠞⢧⡀⠀⠀⠙⢦⣉⠉⠉⠉⣉⡴⠏⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠷⣄⣠⠞⠁⠀⠀⠛⢦⣀⠀⠀⠈⠉⠛⠋⠉⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⡤⠤⠴⠖⠛⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⣤⠤⠴⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠚⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⡄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡊⠠⠞⢁⡿⢀⡄⠀⠀⠟⠃⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠛⠂⢠⣀⡶⣴⢢⣆⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠀⠁⠘⢻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⣀⣤⡄⢀⣠⣤⠿⣤⣤⣄⣀
⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⢉⣙⡓⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⣾⣯⠵⠖⠛⠉⠁⠈⠙⠛⠶⣤⣀
⠀⠀⠀⠀⠀⡾⠀⣾⣷⣶⣿⠻⡿⣿⠙⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⡀
⠀⠀⠀⠀⠀⣧⢸⣇⠉⠈⠁⣀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢻⡄
⠀⠀⠀⠀⠀⢹⣾⡟⠃⠀⠀⠉⠀⠠⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠃⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠻⢴⣂⣀⠀⠀⠰⠲⠟⣸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠓⠦⢤⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡼⣇⠀⢠⡶⠓⠚⠉⢹⡄⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡟⣧⠀⠀⠀⠹⣧
⠀⠀⠀⢠⣴⡞⠛⠳⠶⠾⠿⠚⣧⠀⡀⠀⠘⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⠁⠘⢷⡀⠀⠀⢿⡆
⠀⠀⢰⡿⠴⠤⠀⣸⣟⢙⡿⢿⣿⣆⠉⠳⢤⡽⠻⣄⠀⠀⠀⣿⠁⠀⠀⠀⣾⢷⣄⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣿⠀⠀⠈⣷⠀⠀⢸⣗
⠀⠀⡾⢿⣟⠓⢀⡟⠈⠁⠀⢸⠋⣿⣧⡀⠀⠀⠀⠈⠛⣶⠶⠋⠀⠀⠀⢰⡇⠀⠉⢷⡄⠀⠀⠀⠀⠰⠇⠀⢸⡇⠀⠀⠀⠸⣇⡀⠈⣿
⠀⣸⠃⠀⣀⡴⠛⠀⠀⢀⣀⣟⣙⣻⡎⠛⠶⢦⣶⣤⣀⣿⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣿⣦⣤⣤⣤⣤⣴⣾⠯⠜⢷⡀⠀⠀⢹⡟⠶⣿⡇
⠀⡿⠰⠚⠁⠀⢀⣴⣋⣉⣁⣀⣀⣈⣽⡶⢻⠋⣿⠀⠙⠿⣧⣄⣀⣀⡾⠀⠀⠀⢀⡾⠁⣸⢿⠉⠁⠀⢻⡜⣆⠀⠈⢻⣄⠀⠘⣇⠀⢸⡇
⢸⡇⠀⠀⠀⣠⢾⣅⣄⣀⣀⣀⣀⣀⣤⣤⣼⠀⢻⠀⠀⠀⠈⠉⠙⠋⠀⠀⠀⢀⡞⠁⢠⡏⡞⠀⠀⠀⠀⢷⠘⠆⠀⠀⢻⣆⠀⢿⡀⠘⣿
⠀⠻⠤⠤⠞⠁⢸⡧⠤⠤⠤⠤⠤⠤⠴⠶⣾⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠸⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⢸⣯⣻⣶⣶⣒⣚⣛⣛⣩⣿⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⣇⢠⢸
⠀⠀⠀⠀⠀⠀⣸⣿⣶⣶⣶⣶⣶⣲⣾⡥⢿⡀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠙⣿⣸
⠀⠀⠀⠀⠀⢀⡟⠓⠻⠷⠶⠾⠶⠖⠚⠛⠛⣇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠟
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠒⢤⡤⠤⠴⠤⣤⠔⠚⠉⠉⠙⠲⣄
⠀⠀⢀⣠⠴⠒⠒⠦⣤⠋⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⢣⠤⠖⠒⠲⢤⣀
⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⣸⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢹⡀
⠀⢀⡇⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣀⣠⠤⠒⠉⠀⠀⠀⣇
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠉⠉⠙⠛⠓⠒⠒⠒⠛⠛⠉⠉⠉⠀⠈⣧⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⣀⠀⠀⢠⡅⠐⠀⠉⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠈⠑⢤⠀⠀⠠⢊⠀⠀⠀⠀⠀⠀⠑⢄⢀⡠⣛⡆⢀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⡢⡀⠱⡀⠁⠀⠀⠀⠀⠀⠀⢀⢔⣫⣵⠊⢁⢁⠅⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⠀⠀⠀⠀⡀⢔⡬⠒⠉⢁⠸⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡥⠚⠁⠀⠀⠀⢐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠈⢆⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢆⠆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⢁⠊
⠀⠀⢀⠄⣒⠃⣿⣶⣤⠄⡪⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠶⠁⠀⠀⠀⠐
⢔⣪⣿⣛⣽⡿⢞⣩⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⡀⢈⠕⠉
⠀⠈⠉⠻⣵⠚⠅⠀⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢑⠠⠂⠁
⠀⠀⠀⠀⢱⠆⣆⠠⠊⠁
⠀⠀⠀⠠⠌⠊
⠀⠀⠀⠀⠀⢠⠀⠀⡠⠠⡀⡠⡢⠀⠀⡰⠂⠀⢀⢄⠀⠀⡀⠀⢀⠐⢀⠀⠀⡀⡀⡀⢀⢀⢀⢀⠀⢀⠀⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠦⣄⣀⣀⣀⣀⣀⣀⣀⣠⠶⠒⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⢀⣀⣀⣀⡀
⠀⠀⢀⡴⠛⠉⠁⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠈⠉⠓⢦⡀
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⣽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏
⠀⠀⢰⠇⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⢹⡀
⠀⠀⣾⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠶⠦⠤⢤⣤⣤⣀⣀⣀⣀⣠⣤⣤⡤⠤⠶⠖⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⣀⣀⣀⣤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⡀⠀⢀⣀⣀⣀⡤⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⢀⡠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠔⠂⠠⢄⡀
⠀⠀⢀⠃⠀⠀⠀⠱⡀⡠⠔⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⡍⢩⣭⡖⠢⣝⢆
⠀⠀⠘⡆⠀⠀⠀⠀⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠈⠢⢉⠠⠊⠀⣹⠂
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠒⠒⠂⠈⠉
⠀⠀⠀⠀⠈⠢⡄⢀⡀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠠⠤⠐⠀⠀⠉⠉⠉⠉⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢁⡀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⢗⡒⠂⢐⣂⣉⡀⠀⠤⠔⠙⣄⡀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠠⡤⠒⠒⠐⠈⠐⠒⢄⠀⡼
⠀⠀⠀⠑⡶⢇⠀⠀⢀⡠⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠉⠁⠀⠀⢇⡔⠒⠒⠒⠤⠤⠤⠞⢧
⠀⠀⠀⠀⡇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠂⠈⠉⠁⠒⠠⣡
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⠀⢀⠀⠠⠐⠀⠀⠡
⠀⠀⠀⠀⠀⠁⠒⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢢⠀⠀⡠⠄⠂⠄⡀
⠀⠀⠀⠀⠀⢠⡶⢳⠒⢒⣀⠤⣉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡠⢣⠊⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠙⢤⠀⠣⣈⠋⣀⠜⠑⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⡔⡠⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠄⠀⢙⠂⠤⠭⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⢇⠀⣀⡤⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠛⠛⠛⠻⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠛⠛⠛⠻⠷⣶⣄⡀
⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣄
⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣇⣶⣶⡀
⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⣿⣷⣿⣯⣥⠄
⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡾⠽⢿⣿⣿⣥⠄
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⣀⣹⣶⣦⣿⣿⣿⣟⡥
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣐⣾⡞⠁⢹⣿⣿⣿⣯⢾⠏
⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣩⠴⢿⣾⠞⣯⢻⣸⣧⣯⣤⠄
⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢤⣰⣿⢻⣀⣰⣿⣋⠙⣿⣿⢿⣿⡿⠆
⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⢁⣏⣓⣿⣿⣿⣿⠛⢾⡿⠴⣿⠋⣩⠽⣇
⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠳⢾⣍⡿⠛⢿⣽⣿⣌⣿⡛⠙⣧⣾⣷⣤⠈
⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣤⣴⡷⢋⣹⣯⡤⠿⢶⠤⣿⡙⠛⣿⢧⢞⣙⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⡏⣸⣿⡁⡞⢽⠳⡍⢳⣸⡄⢸⡝⢦⢸⣟⣉⠉⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠻⣿⣴⠻⣄⡙⣷⣮⠷⢛⡫⠋⡇⢸⠀⢨⢿⡁⣠⣾⡅⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⡏⠙⠒⡾⠻⣄⠈⠫⣭⣭⣟⣋⡤⣾⣴⠋⡰⠋⢀⣿⢁⣀⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⡞⢆⢀⡄⣈⠉⢳⣙⣤⣹⣇⠀⢻⡓⠲⠾⠶⠤⠴⠚⢁⡼⠋⢀⣤⠿⣿⠈⣉⣠⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠶⣷⣿⣸⣇⣿⢶⣋⣉⠤⢬⡟⠓⣤⣉⣓⣲⣶⡦⠴⠒⠛⢒⣫⢿⣯⣅⠘⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠒⠶⢿⣿⣿⡛⠛⢉⢩⣿⡿⠂⢲⠞⢳⠤⠽⠓⢤⣤⣤⣔⣋⣁⣀⣁⣈⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣉⠛⣹⠿⣿⠗⣺⣿⢻⡇⠀⠉⠀⣀⣠⠴⣾⡉⠉⠗⣨⠏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⣰⠃⢈⡵⢿⣨⡷⠛⢛⣯⠤⠶⠛⠁⠉⠳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡾⠋⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⣴⣾⣶⣶⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠰⣿⣿⣿⡿⠿⢿⠻⠿⠿⠿⢿⣷⡀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⡿⠁⠀⣀⡀⠀⢀⣤⠀⠈⠃⠈⠀⢀⣀⣠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⣿⣀⠀⠀⠀⠙⠁⠀⠀⠉⠀⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⣿⢿⡦⠀⠀⠀⠶⠶⠆⠀⠀⠀⠀⠸⠶⠶⣿⡷⠶⠶⠀⠀⠀⠀⠀⣠⣄⠀⠀⠶⠶⠖⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣤⣤⣴⣶⣤⡀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄⡀⢀⢀⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠠⣿⢿⣿⣿⣿⣿⣿⡟⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢾⣿⣶⣄⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⢠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠟⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠘⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡘
⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠑⠄⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⡀⠀⠈⢆⠀⡔⠁⠀⠀⠀⠀⢀⠠⠒⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠻⡀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢤⡀⠀⠀⠀⠑⢄⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⢳⣄⠀⠀⠀⠈⣇⠐⠄⣠⠤⣄⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠐⡂⢄⠀⠀⠀⠈⢆⠀⠀⠀⢫⠢⡄⠀⠁⠘⡄⠀⠁⠀⡄⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠘⡄⠣⡀⠀⠀⠈⢆⠀⠀⠀⢣⠻⣄⠀⠀⠰⠀⡇⢠⠁⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠘⡀⠐⡄⠀⠀⣸⡄⠀⠐⠉⡄⠈⢂⢀⣀⡇⠈⠪⢄⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⡇⠀⡇⠀⠀⠸⠱⠀⠀⠀⣱⢖⢜⠀⢀⡦⣀⠀⠀⠑⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠘⠁⠰⠁⠀⢀⠃⠀⠃⠀⠀⢸⡌⠛⣼⣡⠃⢸⠑⠤⣀⡀⣈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡌⠀⠀⡆⠀⠀⢀⠘⢄⠀⠀⠑⢄⢆⡰⠁⠉⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠘⠀⠀⠀⠇⠤⠤⡎⢢⣈⠑⢤⡤⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠀⠀⢣⣣⡸⠑⠮⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⣾⣿⣿⡆⠀⢀⣾⣿⣺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢀⣦⣆⡀⠀⣰⣦⡄⠀⠀⠐
⠀⠀⠻⠿⠿⠃⠀⢈⠿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡤⢠⣿⣿⣿
⠀⠀⣾⣿⣿⣷⡀⢿⣷⢠⡾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⣷⣿⣿⣿⠃⠈⣿⣿⣿⣿⡇⢀
⠀⠀⣿⣿⣿⣯⣅⡘⣿⣾⣞⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠸⣿⣿⣿⡇
⠀⠀⣿⣿⣶⣿⠻⠿⢿⣿⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠃⠸⣿⣿⣇
⠀⠀⣿⣿⣿⣿⠃⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠿⠿⠿⠿⠃⠀⠀⢸⣿⣿⣿⡿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⣠⣿

⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠠⣶⠄⠀⢀
⠀⠀⡀⠀⣾⣻⣵⡂⢐⣾⢟⣧⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡃⠰⣷⡷⠀⠀⠰⣿⡗
⠀⠀⠤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⢤⣤⠀⠀⣴⣶⣶⣶⣶
⠀⢠⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠛⠓⠀⠀⣶⣿⣿⣿⣾
⠀⢠⣿⣿⣿⡟⠿⣿⣿⠿⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠛⣾⣿⣷⠚
⣠⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣯⣿
⠙⠻⠿⠿⠿⠀⠀⠀⠀⠀⠀⢿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠀⠀⠀⠀⠀⠛⠃⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⢠⠔⠰⢠⡆⢊⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢠⣤⣮⣭⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠑⢈⡊⠒⠘⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠀⠛⠛⠃
⠀⠀⠀⢲⢤⡠⠆⠀⢦⢠⡄⠆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡰⢸⠰⠲⠆⠞⠈⠀⠆⠳
⠀⠀⠀⠛⠈⠃⠘⠁⠈⠉⣂⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⢻⡏⣿⢩⡏⡻⡟⣻⡏⣿⢽⢿⣿⡯⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣵⣷⣾⣵⣽⣼⣧⣧⣷⣿⣷⣿⣾⣶⣿⣧⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⢰⣶⣶⡶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠛⠛⠋⠛⠿⠿⠛⠋⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣽⣿⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⡼⣠⠤⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⢛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠗⢦⡣⠤⠏
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⢠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠒⠚
⠀⠀⠀⠀⠀⠀⠈⠘⠐⠑⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⢸⡆⡷⠴⡄⣾⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠋⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠘⠀⠁⠈⠀⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢤⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠙⠂⢙⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⢰⡘⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠁⠈⠀⠁

⠀⠀⠀⢀⠞⠙⢦
⠀⠀⠀⠘⡄⠀⠤⠳⣤⠤⢤⡀
⠀⠀⠀⢠⠫⠿⡄⠀⠙⣆⠒⠉⠲⣄
⠀⠀⢠⣷⠘⠤⠽⣦⠀⠈⢳⣄⠀⠈⠳⣄
⠀⠀⢻⠙⣧⠀⠀⠈⢷⡀⠀⠙⢦⡀⠀⠈⠻⣦⡀
⠀⠀⢸⠀⣆⠘⣆⠀⠈⢳⡄⠀⠀⠙⢦⡀⠀⠈⣙⣦⡀⠀⠀⠀⠀⡴⢋⡙
⠀⠀⣼⢶⡏⠛⠹⣆⠀⠀⠹⣄⠀⠀⣤⠻⣄⠈⠵⠊⢳⣄⣠⠶⠻⠁⠈⠳⡆
⠀⢸⠃⠀⢇⠀⠀⢻⡄⠀⠀⠘⢧⡀⠘⠁⠹⣆⠀⠀⠀⠹⣯⠀⣴⠀⠀⠀⡇
⠀⣼⣄⠀⡸⣆⠀⠀⢻⡄⠠⢶⡨⣷⡀⠀⠀⠘⣧⡀⠀⠀⠸⣷⡇⠀⠸⣿⡇
⠘⠇⠘⣇⡇⠹⣆⠀⠀⠙⣦⠀⠁⠀⢱⡀⠀⠀⠹⣷⡄⠀⠀⢻⣷⠀⠀⠸⣇
⢸⠀⠀⣿⠀⠀⠹⣧⠀⢻⠊⣧⠀⠀⠀⢷⣀⠀⠀⠙⠛⠂⠀⠈⢿⣦⠀⠀⢿⡄
⡆⠀⢀⣿⠀⢧⠀⢹⡄⠀⠀⠘⣦⠀⠀⠀⠙⠗⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠈⢷⡀
⢧⠀⢸⠟⢳⡈⣆⠀⣷⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠘⣧
⠈⢧⠀⠀⠈⢧⠸⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠈⣧⠀⠀⠈⢣⡹⠖⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⣇⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠠⢋⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠉⠉⠉⣙⣿⣿⣿⣶⣦⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠰⠹⣿⣿⣿⣶⣶⣤⣤⣶⠖⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⠟⠋⠹⣷⣄⡀
⠀⠀⠀⠀⠀⣴⠟⢿⣦⣀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⢿⣿⡉⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⢴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⢠⡾⠁⠀⠈⣿⠿⠿⠂⠀⠀⠀⠀⠉⠁⠀⠀⢀⣿⣿⣄⠀⠀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢈⢵⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⣰⠟⠀⠀⠀⣼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⣿⣿⣿⣦⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⢸⡏⠀⠀⠀⠀⣿⣿⣄⣀⠀⠀⢀⣠⣤⣶⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⣿⠃⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠃⠀⠀⠀⠀⠀⢀⣴⣿⠟
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢋⣁⣴⡾⠁⠿⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⢀⣴⣿⠟⠁
⢸⡟⠀⠀⠀⣠⣤⣤⣤⣤⣶⣶⣿⣿⡿⠋⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⢀⣴⣿⡿⠋
⣸⡇⢰⣶⣤⣽⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⣠⣿⣿⠟
⣿⣿⡈⠛⢿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠇⢀⣠⣾⣿⡿⠃
⠘⢿⣿⣦⣄⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣶⣿⡿⠟⠋
⠀⠀⠙⠻⣿⣿⠶⠄⠀⠀⢠⣾⣿⣿⣶⣿⣷⣴⠟⠋⠁⠀⠻⣿⣅⡠⠴⣾⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠈⠻⣶⣶⣶⣶⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠉⠁