Drums Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠄⢴⣧⣤⣤⠄⠀⠈⢓⣒⣶⡖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠐⠀⠀⣵⣟⡈⠀⠀⠀⣀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⠄⠀⠀⠀⡘⠀⣰⣿⠟⠀⠀⠀⠀⡅⠀⢸⣶⣖⠂⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⣹⠁⠀⠙⢻⠗⠁⠀⠀⢰⡄⣤⣿⣿⠂⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⢹⡟⠒⠀⠀⠀⠐⠉⢹⢃⠀⠀⣸⣶⢲⣶⠶⠆⠀⠀⠀⣹⣿⣃⡀⠀⠀⠀⢰⣾⠃
⠀⠀⠀⠀⠀⢀⣬⠇⠀⠀⠀⠀⠀⢀⣼⠴⠒⠟⠿⢿⡞⠃⠀⠀⠀⠀⣤⢿⠉⠛⠒⣶⠶⠶⠾⠛
⠀⠀⠀⠀⠀⠐⣾⡆⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠘⡦⡄⡀⠴⠂⠀⠈⢄⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠻⢧⠄⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢀⠀⠀⠀⠀⢸⣆⠀⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⣯⠈⢂⣦
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣠⣾⣿⢉⡤⡾⡿⠲⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠣⠏⢱⢳⣇⡴⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⡋⠀⠀⠀⠀⠾⠀⠁⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠛⠉⠁⠀⠙⠦⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⠝⠋⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⠵⠊⠁⠀⠀⠀⠀⠀⣠⠔⡫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡽⠚⠁⠀⠀⠀⠀⠀⢀⡤⣊⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⣤⢖⣩⢾⣥⣀⣀⣀⣀⣀⢀⡠⢖⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⢉⠔⠛⠓⣯⠔⠋⠀⠀⠀⠀⣀⣠⣠⠝⢋⠴⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠸⣄⠀⣀⡜⠀⠀⠀⠀⠀⣼⠁⠀⠈⡟⠁⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠘⠦⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢸⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⡇
⠀⠀⠀⠀⢸⡷⣌⠙⠲⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠖⠚⠉⢀⣠⠔⡇
⠀⠀⠀⠀⠀⡇⠈⢑⡦⢄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⡠⠤⡖⠊⠉⠀⢠⡇
⠀⠀⠀⠀⠀⡇⠀⡆⢡⠀⠀⠉⠉⢉⠗⠒⠒⠒⠒⠒⠒⡚⢏⠉⠉⠉⠉⠀⠀⠂⢱⠀⠀⠀⡎⡇
⠀⠀⠀⠀⠀⡇⣸⠀⠘⡆⠀⠀⢀⠎⠀⡇⠀⠀⠀⠀⡰⠁⠘⡄⠀⠀⠀⠀⢸⠀⠀⢣⠀⠀⣇⡇
⠀⠀⠀⠀⠀⣧⠇⠀⠀⢣⠀⠀⡌⠀⠀⢃⠀⠀⠀⡰⠁⠀⠀⠱⡀⠀⠀⢀⠇⠀⠀⠈⡇⢰⢹⡇
⠀⠀⠀⠀⠀⢻⢦⡀⠀⠘⡄⡸⠀⠀⠀⢸⠀⠀⢰⠁⠀⠀⠀⠀⢳⠀⠀⡘⠀⠀⠀⠀⠸⣸⢸⠁
⠀⠀⠀⠀⠀⠘⢦⡙⠢⣄⣁⠁⠀⠀⠀⠀⠇⢠⠃⠀⠀⠀⠀⠀⠈⡆⢠⠃⠀⠀⠀⠀⣀⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣉⠑⠲⠤⢄⣀⣀⡈⠀⠀⠀⠀⠀⠀⠀⢠⣌⣠⠤⠤⠒⣉⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣌⣉⣉⣉⣉⣉⣉⣉⣉⣥⠤⠴⠖⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣿⢰⣿⣿⣦⣿⣿⢰⣿⣿⣾⣿⣿⢹⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣿⣿⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿⣶⣶⡆
⠀⠀⠀⢳⣶⣿⣎⣁⡄⣤⣤⣥⣤⢿⣿⢸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣸⣿⡟⠿⠿⢿⡿⣿⠿⠿⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠻⠿⠿⢿⡿⡿⢿⣿⣿⡛⠿⠛⠋⠉⠉⠀⠐⣾⢾⠿⢶⠶
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠁⣰⣤⣿⠿⠿⣧⣴⣀⠀⡀⠀⠀⠀⡇⣸⡀⢸
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡃⢰⡶⠛⠉⠀⠀⠀⠀⠀⠀⠉⠳⣶⠀⠀⠀⣿⣛⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⡿⠋⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⣴⡿⣷⠛⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡷⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡯⣿⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠫⢀⣀⣁⢜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢜⡇⢀⡙⠆⢀⠀⠁
⠀⠀⠀⠀⣀⠴⢊⠁⠀⠀⠀⠀⠁⢸⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⡏⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠁⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⡄⠉⠛⠶⠶⠶⠶⠶⠾⠋⢁⡄⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠟
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠐⣧⡈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠈⠛⢦⣄⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣌⠙⢦⣤⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠓⢲⣶⠋⣠⣾⣅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠑⠦⣉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣡⠞⠁⠀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠑⠦⣙⠲⣄⡀⠀⠀⠀⢰⠟⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡙⠓⣆⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⠉⠛⠓⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠉⠀⣠⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⠙⠳⢦⣄⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⣤⠶⠟⢩⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡄⠀⢀⠌⠉⢟⠛⠒⠶⠶⠶⣶⠶⢶⡶⠶⠶⠶⠶⠶⢻⠟⠛⢍⠀⠀⢀⡎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢱⡴⠁⠀⠀⠈⢦⠀⠀⢀⡔⠁⠀⠀⠑⢄⠀⠀⠀⡰⠃⠀⠀⠈⠳⣄⡼⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⠊⢣⠀⠀⠀⠀⠈⢧⡠⠋⠀⠀⠀⠀⠀⠀⠱⣄⡜⠁⠀⠀⠀⠀⠀⣸⢧⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⣀⠀⢧⠀⠀⠀⣠⠞⢣⡀⠀⠀⠀⠀⠀⠀⣠⠞⢦⡀⠀⠀⠀⠀⣰⠃⠀⢙⡶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⠳⢦⣧⣴⠊⠀⠀⠀⠱⣄⠀⠀⠀⢀⡴⠁⠀⠀⠙⢆⠀⠀⣰⣁⣤⠶⠋⠀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠉⠙⠳⠶⢦⣤⣬⣦⣀⣠⣎⣀⣀⣀⣤⣤⡤⠷⠞⠛⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢤⣄⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠲⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠉

⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠⢈⠉⠉⢓⣶⣶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⢾⠷⠐⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡾⠁⠁⠀⠉⠢⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣈⡑⠒⠒⢺⠒⠒⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠉⠀⠀⠀⠀⢠⠀⣾⡹⠀⠤⠄⢻⢈⡆⠀⢸⣀⠠⠤⠧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣿⣧⣤⣤⣾⣼⣧⠀⡚⡂⠀⠘⠋
⠀⠀⠀⠀⠀⠀⣤⢤⡮⢤⡤⣤⣤⣄⠀⠀⠀⠀⠀⣸⠀⠹⡁⠀⠀⠒⣿⣿⠟⠈⣯⡇
⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⡟⣸⡗⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣾⣏⢙⡃⠀⠰⣿⣿⠀⢀⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⣄⣿⡇⠀⢘⠿⠫⠉⠀⠉⠹⢆⠀⠿⠿⢿⣿⣿⡟⣿⠅⠀⠙⡂
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⣿⣿⣧⡾⠉⠀⠀⠀⡀⠀⠀⠀⠈⢲⠀⠀⢉⢿⠀⢹⠀⠀⣴⣧
⠀⠀⠀⠀⠀⠀⢹⣟⠋⢺⠋⠋⠛⡿⠁⠀⠀⠐⢧⡀⠠⡂⠀⠀⠀⢃⠀⢀⣏⠀⣿⢆⡔⣾⠂⠱⡀
⠀⠀⠀⠀⣠⠴⢻⢻⢧⣸⠀⠀⠀⣇⠀⠀⠀⠐⠉⠀⠀⠃⠀⠀⠀⣸⠀⠀⠏⡸⢹⠞⢳⡟⠂⠒⠙⡄
⠀⢀⡤⠚⠣⣠⠎⢸⡸⣿⣄⠀⠰⢅⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢉⡦⠤⣶⣤⣸⣀⡼⣿⠁⠀⠀⠈⢄
⡴⠉⠀⠀⠘⠀⠉⠙⠀⠈⠾⢢⠃⠀⠙⠈⠂⢄⣀⣀⣀⣤⡄⠘⠋⠀⢱⠈⣹⠛⠼⠛⠒⠛⡆⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠒⠀⠤⠴⠒⠒⠋⢀⣀⣀⣠⡄⣾⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡍⠉⠋⠋⠉⠑⠤⢄⣀⠠⡄⠀⠰⠤⠤⢤⡤⠖⠒⠀⣀⠀⠀⠈⢻⣷⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠉⠀⢀⣤⣇⡀⠀⠀⠀⠀⠈⠰⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡴⠛⠁⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⣤⣀⣀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⣿⣄⠀⠀⠈⠸⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠺⠀⢹⠀⠙⠢⢄⠀⢂⡤⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣤⢒⡟⣉⣧⣄
⠀⠀⠀⠀⢀⡠⠚⠁⠃⠀⠀⠙⠁⠀⠀⠀⠉⠺⠇⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⡨⢠⣃⣤⠟⠫⠄⡇⠝⠦⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠑⠳⢦⣀⣀⣀⣸⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢰⣆⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⡿⠯⠽⠿⠛⠿⠿⠿⠿⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⠁⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣬⡙⠛⠿⣿⡷⣶⢶⣦⣤⣤⣤⣤⣤⣤⠴⠶⠚⠛⠉⠁⠀⣡⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣶⣄⣒⣘⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠿⣿⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡏⢹⠛⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⢿⣿⣿⣿⣿⣿⣗⡋⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠘⣿⣿⣿⣿⣟⡺⠀⠀⠀⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢻⡿⣿⡏⠀⠀⠄⠸⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣟⠃⠀⠀⠀⠸⣿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣹⡇⠊⠀⣿⣍⠐⠂⠠⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⡏⠀⠀⠀⠀⢀⣿⠀⠉⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠀⠀⠀⠘⣿⣷⢶⣤⠄⠈⣿⣿⠁⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⣀⡴⠛⡅⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠸⣤⣉⠛⠿⣾⣤⣽⣧⣄⣀⣀⣀⣀⣀⣀⣹⣧⡴⠶⠛⠉⢀⣴⠇⠀⠀⠀⠀⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠘⠷⣧⣀⡑⠈⠉⠙⠩⠍⢉⡉⠉⠉⠉⠀⠀⠀⢀⣤⠶⠛⠁⠀⠀⠀⣠⢄⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠶⠤⠤⠶⠴⠶⠶⠖⠒⠋⠉⠀⠀⠀⠀⠀⣠⣾⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⠦⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠃⢹⣿⣿⣿⣿⣿⣿⣿⣧⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢧⠠⢀⣠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣄⣰⣷⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⢀⠀⠄⠀⠀⢤
⠀⠀⠈⠀⠢⡀⠀⠀⠒⠀⠀⠩⢂⠐
⠀⠀⠀⠀⠀⠈⠙⠋⣩⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⠒⢲⣾⣿⣿⡟⢻⡆⠠⢤⣶⣶⣾⡏⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣌⠀⠀⢸⣿⣿⠛⡇⠘⢓⣢⠄⠠⠻⣿⡇⡆⣿⣿⠀⠀⠀⠀⢠⣶⣒⣚⠡⠤⢤⣼⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢹⠀⠀⠈⣿⣿⠀⡿⢸⣾⠁⣾⢠⣿⣿⣇⣧⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⢀⣀⣸⡿⣧⠿⠼⢿⡇⣿⣠⣿⠿⢟⣚⣾⡿⢃⣀⣀⣀⣀⣀⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠀⢸⡀⣀⣈⣃⣈⡑⠛⠒⠚⠓⠚⠉⠀⣿⣈⣯⡉⠀⠀⢸⠷⣷⡶⠶⠶⠶⢾⢿⣿⣾⡇
⠀⠀⠀⢸⣿⣿⡟⣿⢉⡏⠩⢭⣭⣿⡇⠀⠀⣠⠤⠒⠒⢚⡷⠞⠛⠛⠒⠲⢤⡀⣿⡇⠀⠀⣇⠀⣾⣿⣿⡇
⠀⠀⠀⢸⣿⣿⡇⣿⢸⣇⣸⠘⠻⣿⣇⣴⣯⣄⠮⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⢿⣻⣻⠟⠛⠋⢉⡧
⠀⠀⠀⢸⣿⣿⡇⢿⢸⣿⣿⣰⠀⠏⣿⣿⣿⣿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡅⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣿⣿⣷⠀⢸⣿⣷⣼⡆⣸⣿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⢀⠴⡷⣄
⠀⠀⠀⠈⣿⣿⣿⣀⣸⣿⣿⣿⣇⣿⣿⣿⣿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡇⠀⢠⠏⠀⡇⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⣰⠯⢤⣀⣇⣠⠼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⣷⠃⠀⠀⠈⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡼⢧⡀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡼⠁⠀⠀⠀⣴⢃⡄⠀⠀⠈⣆
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⠤⣀⣀⡠⠤⠖⠋⠀⠀⠈⠁⠀⠀⠀⠀⡏⢸⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⠒⢶⣴⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⣀⣸⠁⢈⣀⣉⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡝⢁⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠁⠀⠀⠀⠀⢱⡞⠁⠂⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⡾⡯⠀⠻⠛⡳⠆⠉⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠤⠤⢼⡇⠀⠐⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⡴⣿⣅⣀⣀⣠⢤⡶⠞⣍⠉⠉⠛⠻⢲⡧⣭⠤⠀⣀⣐⣒⣒⣤⣤⡄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣸⠀⠀⢠⠀⠀⡇⠀⣿⠀⣀⠄⠀⠨⡇⠀⠀⢨⠀⠀⠀⠀⠉⢹⠇⢠⣼⣃⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠄⠀⢸⠀⢠⡅⠀⢉⣀⣻⣀⣟⢸⠋⠉⠀⣸⠀⠀⠀⠀⡇⢸⡧⠤⠼⡧⠀⠀⢒⣛⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⢸⠀⠀⢠⡆⠀⠙⠿⣷⠿⢿⣿⣂⣀⣀⣤⣤⣀⣠⣤⣼⠁⠀⠀⣇⠀⠀⠁⢘⡂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⣄⠀⢸⣇⡀⠀⠀⣇⢀⣀⣘⣀⣠⡇⠀⠀⠉⠉⠙⠁⠉⠀⠈⡬⠒⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡄⢀⡾⡿⡄⠀⠉⠙⠛⣸⢿⡍⠉⠁⠀⡨⠙⠓⣀⠀⠀⢠⠀⠆⠀⠀⡇⠀⠀⠀⢀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣼⣁⣧⣨⣆⣀⡀⠀⠑⠚⠷⣂⠀⠀⠀⠀⠀⠙⢿⣆⠰⡄⣇⠀⢀⣟⠀⠀⠀⢸⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠋⢉⣿⡷⠾⠟⠯⠭⣤⣀⠈⠁⠀⠀⠀⠀⠀⣠⠆⢻⣇⠢⣿⡀⢈⣉⣀⠀⠀⠸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢰⡿⠉⠀⠀⠀⠀⠀⠀⠉⠙⢦⡑⣄⢀⡤⠀⠀⠀⠈⣿⡄⠬⠀⠈⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⠹⣯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⠎⠀⠀⠀⠀⠀⡿⠃⠀⠀⠴⡆⢀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠍⢐⠃⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢃⠀⣀⠴⢁⡟⠀⠂⣠⣘⣻⠈⠉⠁⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢀⡄⠉⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⡀⠉⠀⠈⠃⢠⠚⠁⡹⣷⠤⠧⠂⢀⡀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡾⠏⠀⢠⠿⠛⠒⠶⠫⠾⠒⠀⠀⠉⠓⠤⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⢍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠃⢀⠴⠁⠀⠸⠆⠀⠀⢠⣦⠀⠀⠀⠀⠀⠈⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠱⢿⣶⠀⠀⠀⡀⡀⢀⣴⠟⠉⢀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⠉⠐⠈⠛⠊⠈⠀⠀⠀⠘⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡀




⣿⣟⣛⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣤⠤⠴⠶⠶⠶⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠶⠶⠶⠤⠤⣤⣀
⠀⠀⠠⣽⡉⠀⠀⣠⢀⣀⠤⢾⣥⡤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣬⡳⠤⣀⡀⣄⠀⠀⢉⣿
⠀⠀⠀⠀⠈⢉⡟⠁⠀⠀⠀⡌⠀⠀⠀⠸⠉⠙⠓⢦⡀⢠⡴⠚⠋⠉⡅⠀⠀⠀⢡⠀⠀⠀⠈⢻⡉⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠤⠄⠀⣀⡇⠀⠀⣾⣆⠃⠘⣸⡇⠀⠀⢰⣀⠀⠠⠬⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡸⡄⠀⡘⠛⠛⠒⢦⣤⣀⠈⠁⠒⣯⡟⠀⠀⢯⡵⠒⠉⠀⣀⣤⡴⠚⠛⠛⢂⠀⢠⢇
⠀⠀⠀⠀⠀⡇⡇⠐⣃⠐⠂⠠⢎⡀⠈⠙⢯⡀⢻⡁⠀⠀⢸⡇⢀⡽⠋⠁⢀⡡⠄⠐⠂⡸⠂⢸⢸
⠀⠀⠀⠀⠀⢹⠁⠀⠀⠑⠂⢴⣆⡈⠑⠢⡜⠀⢸⡃⠀⠀⢸⡇⠀⢢⠔⠊⢁⣰⡦⠐⠈⠀⠀⠈⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠀⣠⠞⠛⠉⠉⠙⠳⣤⡀⠉⠉⠁⠘⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡜⡆⠀⠀⠀⠀⢸⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠸⡇⠀⠀⠀⠀⠰⢣
⠀⠀⠀⢀⡴⢳⠷⣄⠀⢀⡴⢿⠷⣄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⢶⡿⣆⠀⠀⣠⠾⡞⢦⡀
⠀⢀⡴⠋⠀⠘⠀⠈⣷⡞⠁⠘⠃⠈⢳⣄⠀⠙⢦⣄⣀⣀⣀⡤⠞⠁⣠⠞⠁⠘⠃⠈⢳⣾⠁⠀⠃⠀⠙⢦⡀
⠀⠈⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢉⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣋⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢋⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⣁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢋⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⡴⠋⣡⣴⣿⣅⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⢋⣭⣭⣤⣄⡴⠛⢁⣤⠞⠉⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⣻⡶⠛⠁⣠⣾⣉⡀
⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠀⠀⠀⠀⣴⠋⠀⠀⠀⠘⣷⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⠞⠁⣀⣴⠟⠁⠀⠉⠉⠛⠶⢦⣄⡀
⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠙⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣿
⣿⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⣿
⣿⣆⠈⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠛⠁⠀⠀⢀⣴⣿
⣿⠈⠳⣄⠀⠈⠉⠛⠳⠶⠶⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠋⠉⠀⠀⠀⠀⢀⣠⠴⠋⢸⣿
⢸⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠉⠀⠀⠀⡾⣿
⢸⠀⠀⠀⢀⡏⣇⠉⠛⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠚⠋⠍⢦⠀⠀⠀⠀⠀⠀⢠⡇⡿
⢸⠀⠀⠀⣼⠀⢸⡀⠀⠀⠀⠀⠀⠉⠉⠉⣹⢻⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⢯⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⣆⠀⠀⠀⠀⠀⢸⢰⡇
⢸⡄⠀⢰⠇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣰⠃⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠸⣆⠀⠀⠀⠀⡟⢸⡇
⢸⡇⠀⡟⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢹⡄⠀⠀⠀⡇⢸⡇
⠸⡇⢸⠃⠀⠀⠀⠈⣇⠀⠀⠀⠀⣰⠃⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢳⠀⠀⢸⡇⢸⠇
⠀⣇⡏⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⠏⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⣇⠀⢸⠁⣿
⠀⣿⣦⠀⠀⠀⠀⠀⠘⡇⠀⢀⡞⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿⠀⣿
⠀⢻⡌⠳⢤⡀⠀⠀⠀⢹⠀⡼⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡿⢀⡿
⠀⠈⢳⣄⠀⠙⠲⣄⡀⠘⢀⠇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⣻⠇
⠀⠀⠀⠈⠳⢤⣀⠀⠙⠓⠾⣤⣀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⡏⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠈⠙⠓⠶⠦⣤⣀⣀⣈⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣼⣁⣤⡤⠴⠶⠚⠉⠁⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣄⣀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠒⠒⠓⠒⠒⠒⠚⠛⠛⠛⠉⠉⠁⠀⢀⣀⣠⣤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠖⠛⠛⠋⠉⠁

⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣶⣾⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠿⣿⣿⣿⡿⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠴⢿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡆⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⣤⣴⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡦⡷⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⡷⠀⠀⠀⣿⡂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⡷⢶⣶⣶⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠛⠛⠿⣯⠀⡇⠀⠀⠀⣿⣿⣿⠿⠟⠛⠛⠿⣧⠀⡇⠀⠀⠀⣽⠋⠁
⠀⢠⣶⣿⣁⣀⣀⣿⣿⣿⣷⣾⣿⣶⣶⣿⣤⣷⣶⣶⣴⣿⣿⣿⣷⣾⣿⣶⣶⣷⣤⡷⠀⠀⢠⠿
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⠿⠟⠛⠉⠈⠁⠉⠉⠛⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠺
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⢟⠝⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠪⣻⣿⣿⣿⠿⠿⠿⠿⢇⡀⢸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⠃⠀⠀⠀⠀⢀⣴⣿⣶⡆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶⣶⣶⣾⣧⡼
⠀⣿⣿⢿⠛⠛⠛⠛⢻⣿⠃⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠰⣿⡉⠉⢹⡟⠉⠉⣿
⠠⣿⣿⣿⣷⣷⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠰⣿⡇⠀⢾⡷⠀⡰⡿⣆
⠀⠙⠿⣿⣿⣿⣿⡿⢻⣿⡄⠀⠀⠀⠀⠀⠘⠠⣟⣿⠠⠇⠀⠀⠀⠀⠀⢠⣿⡧⡀⣸⣧⠞⠁⠿⠈⠳
⠀⠀⠀⠀⠀⠈⠉⡄⠀⢻⣶⡀⠀⠀⠀⠀⠀⠐⠿⠿⠀⠀⠀⠀⠀⠀⢀⣷⡟⢈⣮⠛⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⣞⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⠟⣴⠟⢉⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠀⠈⠻⣷⣢⠄⣀⠀⠀⠀⠀⢀⢠⣰⣽⠟⠁⠚⠋⠀⣰⣦⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠾⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠠⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠻⠛⠓⠚⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠊⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣶⣾⣶⣶⣶⣶⣶⣦⣤⣤⣶⣶⣦⣤⣤⣴⣶⣶⣴⣶⣿⡷⠾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣷⣶⣿⣷⣶⣦⣿⣿⣤⣤⣤⣴⣿⡷⣤⣾⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⣷⣿⣿⣿⣿⣿⣿⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⣰⣿⡏⠀⠻⣿⠏⣩⠝⠉⢹⣿⠉⠙⢻⣿⣙⣿⣿⣿⣿⣿⣬⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠃⢹⣿⡇⠀⠘⣿⠇⠁⠀⠀⢸⣿⠀⠀⠽⢿⣩⡿⠟⢹⣿⣿⣯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠸⣿⡇⠀⠈⣿⠀⠀⠀⠀⢸⣿⠀⠀⢼⣿⣤⠴⠚⢹⣿⣿⢋⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠄⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⢸⡿⠀⠀⣸⡟⠀⠶⡶⢼⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡤⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⢸⣇⠀⠀⢹⣿⠒⢶⡦⢸⣿⡿⣦⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣧⠀⣿⡀⠀⠀⠿⡇⠀⠀⠀⢹⡿⠀⠀⢸⣿⢰⠞⠀⣼⣿⡇⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢉⣿⠀⠀⢰⡇⠀⠀⠀⢸⡇⠀⠀⢸⡟⢀⣿⣴⣿⣿⣤⣮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠸⣿⡄⠀⢸⣧⠀⠀⠀⣾⡏⠀⢀⣿⣿⣋⣿⣿⣿⣿⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢹⣷⠀⢹⣷⠀⠀⠀⣿⡇⣤⣸⣿⠏⣛⣩⣿⡟⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠸⣿⡄⢨⣿⡅⠀⠀⣿⣿⡂⣾⣿⣰⣿⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣹⣷⣤⣽⣧⣤⣤⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡟⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠻⢶⣿⡏⠹⢿⣿⣿⣿⣿⣏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠤⠀⠉⡿⠟⢿⣿⣿⣟⡵⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⠀⠀⠀⠉⠀⣛⣛⣿⣿⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠳⠛⠶⠶⠶⠦⠾⣿⣿⣿⣿⣿⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡤⠰⠿⣿⣿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⣃⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⠟⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣿⡿⢂⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠺⣟⣿⣏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠻⣶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣧⣶⣿⣿⡿⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠙⠯⣹⣿⣿⣶⡁⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢲⣤⣿⣯⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠈⠋⠐⠲⠛⠋⠹⠟⣻⣧⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠶⢦⣤⣬⡥⠾⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⣄⣐⣲⣶⣶⣶⣶⣶⣶⣖⣂⣀⣤⣄⠀⣀⡀
⠀⠀⢀⠔⢀⠀⡔⠢⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣢⣬⣷⣾⣿⠿⠿⠿⠛⠛⢛⣛⣛⣛⣛⡛⠛⠛⠻⠿⠿⢿⣿⣾⣯⣴⣢⢄⡀
⠀⠀⠈⡗⠜⠀⢻⠐⠀⠀⠀⠀⠀⡠⢒⣵⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⢸⡿⠿⢿⣿⠿⠿⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣮⣕⢄
⠀⠀⠀⡇⠀⠀⣼⢀⠀⠀⠀⠀⡜⣵⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠈⠙⠻⠿⠬⠿⠛⠋⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠉⠻⣧⢱
⠀⠀⠀⡇⠀⠀⣿⢸⠀⠀⠀⢠⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⠦⣤⣀⣤⡴⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢆
⠀⠀⠀⡇⠀⠀⣿⢸⠀⠀⠀⡇⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡶⠶⠛⠋⠉⠈⠉⠛⠳⠶⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⢘
⠀⠀⠀⣇⠀⠀⣿⢸⠀⠀⠀⠸⡉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠋⢉⡏
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⢰⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡆⠀⠀⠀⢀⣴⣿⡄
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⢸⢸⣿⣿⣷⣦⣤⣀⡀⠉⢹⣯⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⢵⡏⠉⠀⣀⣠⣴⣾⣿⣿⡇⡇
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⠘⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣦⣤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃⡇
⠀⠀⠀⣿⢀⠀⣿⢸⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣟⣗⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⠱⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡜
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⠀⠑⢄⡀⠙⠻⢿⣿⣿⣿⣿⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⡿⠟⠋⠁⡠⠊
⠀⠀⠀⣿⠸⠀⣿⢘⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠈⠉⠛⣻⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⠛⠉⠁⣀⡠⠔⠋
⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢭⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡭⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠲⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠺⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⢿⣤⠤⠤⣤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣤⣖⡺⠭⠷⠶⠟⠉⠻⣏⠉⠉⠉⠉⠍⠛⢛⠚⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠛⠿⢦⣄⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠈⠙⠲⢄⣀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣦⣬⣑⠲⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⢀⣠⣴⣾⠟⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢯⣾⠛⠿⢯⣿⣒⣲⠦⠤⠤⢤⣤⠤⠤⠤⠤⣖⣒⣾⡭⢷⡏⣿⡴⢛⣥⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢘⡏⠙⠳⠶⣤⣌⣉⣉⣉⠙⢻⢸⡛⠋⣉⣉⣉⣤⠧⠿⠛⢻⡃⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠉⠩⠭⢹⢻⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢞⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠖⣦⠀⢐⢂⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣽⣠⠴⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠶⣏⡛⠲⠦⣭⣔⣀⡀⠀⠀⣠⣄⠀⠀⠀⠀⣀⣐⣲⠭⠷⢿⣀⡷⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠥⣭⣩⣿⣿⣇⣸⢟⣛⣋⣉⣉⣤⠤⠴⠒⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠋⠉⠉⠉

⠀⠀⠀⠀⡤⠤⣄⣀
⠀⠀⠀⠀⠓⠢⠤⣌⣉⠓⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⠬⣉⣓⠲⠤⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠛⠉⠑⠒⠮⢭⣛⡾⠛⢯⡉⠉⠉⠓⠒⠲⠤⢤⣀⠀⣀⣠⠤⠴⠒⠊⣱
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⣀⣀⣠⠤⠤⠴⠖⠒⠒⠒⠛⠒⠚⠳⠦⠤⣤⣤⠤⠴⠒⣋⣉⡥⣤⠖⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⢰⡇⣠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢔⣒⣪⠭⠴⠒⠒⠉⠉⠓⠦⣄⢸⡆
⠀⠀⠀⠀⠀⠀⢠⡟⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢻⡇
⠀⠀⠀⠀⠀⠀⢨⡗⢄⡈⠙⢒⣦⠤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⣴⡒⠋⢁⡠⢺⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠑⢻⣬⢧⣀⣀⣀⠈⠉⠉⠉⡽⢯⠉⠉⠉⠁⢀⣀⣀⡼⣥⡟⠊⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠈⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠁⠀⠀⠀⡇⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠧⡀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⢀⢼⡇
⠀⠀⠀⠀⠀⠀⠈⢧⡈⠓⠢⣼⢼⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡧⣧⠴⠚⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣽⣴⡋⠉⠉⠑⠒⠒⠒⠒⡟⢻⠒⠒⠒⠒⠚⠉⠉⢉⣦⡯⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠲⠶⠿⠿⠶⠖⠒⠒⠒⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣴⠋
⠰⣟⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⡾⠁
⠀⠈⠳⣄⡈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠟⠁
⠀⠀⠀⠈⠛⢦⣄⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣳⣦⣙⠷⣄⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⡾⣡⣞⣁⡀
⠀⠀⠀⢀⣀⣤⠤⠴⠒⣚⣛⣻⣯⣭⠭⠭⠿⠷⠿⠷⣿⠉⠙⠳⣶⠲⠶⠶⢶⡾⠛⢻⡿⠿⠭⢭⣭⣽⣛⣛⣒⠲⠤⢤⣀⣀
⣠⡶⠛⠉⣡⠶⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⣷⠀⢰⠏⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠶⢦⡈⠙⠳⣦
⣿⣄⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⢻⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠿⠀⢀⣼
⣿⠈⠛⠶⢦⣭⣝⣛⣲⠶⠦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠶⢶⣚⣛⣭⣥⠶⠞⠋⢸
⣿⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠲⠶⠿⠿⠿⠯⠭⢭⣭⣭⣭⣭⣭⣭⣭⣭⠭⠭⠿⠿⠿⠶⠶⠒⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢸
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿
⣿⠀⢿⡙⠛⠲⠶⢤⣤⣤⣀⣀⣀⣀⢠⡟⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢷⡀⢀⣀⣀⣀⣠⣤⣤⠴⠶⠚⠛⢉⡿⢸
⡟⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡟⠙⠛⠛⢿⡛⠛⠛⠒⠒⠒⠒⠒⠚⠛⠛⣿⠛⠛⠛⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸
⣷⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸
⣿⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸
⣿⠶⣄⣀⠀⠹⡄⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣸⠃⢀⣀⡴⢾
⣿⠀⠀⠉⠙⠛⢿⡶⠤⣤⣾⣅⣀⣀⣀⣀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣧⣤⣤⠴⣶⠟⠛⠉⠁⠀⢸
⣿⠀⠀⠀⠀⠀⠀⢷⣀⡾⠁⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⢻⡟⠛⢻⡟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠹⣆⠀⣰⠏⠀⠀⠀⠀⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡏⠀⠀⠀⠀⠀⠀⣸
⠈⠛⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠶⠶⠦⠤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠶⠶⠖⠚⠛⠋⠉⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣀⡤⠴⠒⠋⢉⣀⣤⠤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠙⠢⣄⠀⠀⠀⠀⠀⢸⢡⠞⠁⠀⠙⢶⣋⣉⣤⠤⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣌⠑⢦⡀⣀⣀⠀⢸⡀⠀⠀⢀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⠏⠁⠈⠙⣆⠙⠒⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠒⢻⡀⠀⠀⢀⡾⠉⠉⠉⠉⠉⠛⠒⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠙⠶⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡈⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠋⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠲⢤⣀⡈⠉⠙⠓⠚⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠴⠶⠒⠚⠋⠉⢁⣀⡴⠞⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢧⠀⠈⠉⠛⠒⣶⡤⢤⣤⣤⣀⣀⣀⣀⣀⣠⣄⣤⠤⠶⢶⠾⣏⡉⠁⠀⢠⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧⠀⠀⣠⠞⠁⠹⣆⠀⠀⠀⠀⣰⠏⠙⢦⡀⠀⠀⢠⠏⠀⠀⠙⢦⣀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣧⠞⠁⠀⠀⠀⠈⢧⡀⢀⡾⠁⠀⠀⠀⠙⢷⣴⠏⠀⠀⠀⠀⢀⡿⣦⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠴⠋⠙⣇⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⣴⠛⢦⡀⠀⠀⠀⡜⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⡀⠀⠀⠸⡆⠀⠀⢀⡴⠋⠀⠹⣄⠀⠀⠀⢠⡞⠁⠀⠈⢻⣄⠀⡼⠁⠀⣀⠜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡈⠓⠲⠤⣿⣤⡞⠉⠀⠀⠀⠀⠸⣆⢀⡴⠋⠀⠀⠀⠀⢀⣹⣾⠥⠖⠋⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⡀⠉⠉⠙⠓⠒⠒⠒⠛⠛⠒⠒⠒⠛⠉⠉⠉⢀⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠶⠶⠶⠶⠶⠤⠤⠶⠒⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⣈⡀⣖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣿⣿⣧⣾⣿⡣⣿⣿⣾⣿⣿⢸⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⡀⢸⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣻⣶⣿
⠀⠀⠀⢻⣿⣷⣤⣤⡠⣶⣶⣶⣾⣿⡇⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣼⣿⠏⠛⣛⢻⡟⠿⠙⠁
⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⠉⠋⠙⠛⠟⡻⢻⣿⣷⡍⠛⠃⠀⠀⠀⠀⠈⡟⢻⠛⡟⠃
⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣷⠀⠀⣀⡴⠷⠛⠛⠛⠛⠻⢦⣄⠀⠀⠀⠀⣇⡾⠀⡇
⠀⠀⠀⠘⠛⠟⢿⣿⣿⣿⣿⣿⣿⡗⢀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⣿⡥⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⣹⡁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣾⡇⡟⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠈⠁⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡟⠓⢄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠐⠂⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠈
⠀⠀⠀⠰⠊⠁⠐⠀⠀⠀⠀⠀⢠⠃⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢧⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠙⠶⣤⣄⣀⣀⣀⣤⠴⠚⠁⠀⠀⠸⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣀⣀⣀⣀⣈⣁⣀⣀⣀⣐⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⠄
⠀⠀⠀⠀⠠⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠄
⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⡌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢄⠀⠀⠀⠀⠀⠘⡄⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠀⠘⡄⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⠠⡀⠀⠀⠘⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠢⡀⠀⠘⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠦⡈⠢⢤⣘⡄⠑⣖⣒⣂⣀⣀⣀⡀⠤⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⠨⠑⠒⠀⠉⠁⠈⢆⠀⠾⢧⡐⢿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠊⠥⡒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠣⣀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⠒⠈⠥⠖⢂⣀⢀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⣀⣐⠒⠤⠍⠒⠈⣠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠡⢒⡀⠤⠀⠀⣀⣀⡀⡉⠉⠉⠉⠉⠉⠁⠁⠀⣀⣀⠀⠀⠤⢀⡐⠠⠑⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠜⢂⢀⠔⢄⠉⠰⢆⠐⢐⠶⠂⠂⢐⡆⠒⠐⣀⠒⠀⢀⢀⠉⠀⡠⢄⢀⠒⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠈⠁⠀⠀⠑⠁⠀⠑⠐⠁⠈⠢⠊⠀⠀⠒⠁⠀⠑⠊⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡟⡀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣇⠀⠀⠀⠀⠀⠀⠀⣸⣦⠀⠀⠀⠀⠀⠀⠀⢰⣾⡆⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⡇⢿⠀⠀⠀⠀⠀⠀⠀⣿⠸⡀⠀⠀⠀⠀⠀⠀⣾⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠇⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢰⡏⠀⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣸⠀⠀⣿⠀⠀⠀⠀⠀⣾⠁⠀⢱⠀⠀⠀⠀⠀⣼⠃⠀⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢹⡆⠀⠀⠀⢰⡏⠀⠀⠘⡄⠀⠀⠀⢠⡟⠀⠀⢸⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣷⠀⠀⠀⣼⠃⠀⠀⠀⡇⠀⠀⠀⣸⠃⠀⠀⢸⡆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⡇⠀⠀⠀⢻⡄⠀⢀⡿⠀⠀⠀⠀⢸⠀⠀⢀⡿⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠃⠀⠀⠀⠘⣧⠀⢸⠇⠀⠀⠀⠀⠈⡆⠀⣸⠇⠀⠀⠀⠀⣧⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⢿⡀⣿⠀⠀⠀⠀⠀⠀⢃⠀⡿⠀⠀⠀⠀⠀⢿⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⠀⠀⠀⠀⠀⠸⠛⡇⠀⠀⠀⠀⠀⠀⢸⠊⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠳⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠉⠀⠄⣀⠀⠀⢀⠠⠐⠉⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣷⣼⣼⣧⣿⣤⣼⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣶⣿⣶⣿⣿⣶⣶⣶⣶⣾⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠂⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠐⠒⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡂
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢘
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠒⠤⡀⠈⠀⠐⠢⠀⠀⠀⠤⠤⠀⠀⠀⠔⠂⠀⠀⢀⠤⡒⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠄⢃⠃⠀⠀⠀⠤⠤⠀⠀⠀⠘⡘⠨⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡐⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠅⢐⠐⠂⠀⠀⠐⢐⡈⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢁⠀⠁⠈⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢂⠀⠀⡈⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡐⠀⠀⠀⠀⢂⡐⠀⠀⠀⠔⢀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠄⠁⢈⠁⠡⠄⠀⢈⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣂⠄⡀⠀⠠⠀⠀⢐⠠⡄⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠐⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣠⡴⠶⠒⠛⠛⠛⠛⢷⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⠟⠛⠛⠛⠓⠲⠶⣤⣀
⠀⠀⠀⢧⣅⣀⠀⣠⣖⣀⠤⠔⢺⠥⠤⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠦⢽⠒⠤⢄⣐⣦⡀⢀⣀⣥⠇
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⡆⠀⠀⠀⢠⠃⠉⠙⠳⢤⡀⣠⠶⠛⠉⠁⠇⠀⠀⠀⠈⡄⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠠⣄⣀⣀⡜⠀⠀⠰⣷⢸⠁⢡⢼⡧⠀⠀⠘⣀⣀⣀⡤⣴⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⣇⠀⠀⡔⠒⠺⠦⢤⣄⡀⠉⠑⠒⢼⣹⠇⠀⠀⣏⡧⠔⠂⠉⠁⣀⣤⠴⠾⠒⠒⡄⠀⢀⣇
⠀⠀⠀⠀⠀⢸⢸⠀⢸⠠⠤⠄⣀⡜⠈⠙⠳⢶⡒⠒⣿⠀⠀⠀⢸⡗⠒⣲⠶⠋⠉⠘⣀⡠⠤⠤⢰⠀⢸⢸
⠀⠀⠀⠀⠀⠘⡾⠀⠀⠑⠤⣀⣠⡌⠑⠢⢄⢀⠏⠀⣿⠀⠀⠀⢸⡇⠈⠇⢀⠤⠒⠉⣤⣀⡠⠒⠁⠀⠸⡞
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡗⠒⠢⠤⠛⠀⢀⣿⠴⠶⠶⢾⣇⡀⠘⠧⠤⠒⠚⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣸⢹⠀⠀⠀⠀⠀⡘⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⢻⡀⠀⠀⠀⠀⢸⢹⡀
⠀⠀⠀⢀⡾⠋⡟⠳⣄⠀⣠⠞⢳⡟⢦⡀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠞⣿⠛⣦⡀⢀⡴⠛⡏⠻⣆
⠀⢀⡴⠋⠀⠀⠁⠀⢘⣿⡏⠀⠘⠁⠀⠹⣦⡀⠈⠻⢦⣄⣀⣀⣤⡴⠛⠁⣠⡞⠁⠀⠉⠀⠈⣿⣏⠀⠀⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⡶⠶⠶⠶⠶⠛⠛⠿⠿⠻⠿⠟⠛⠛⠛⠛⠿⠶⠶⠶⠶⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⢀⣠⣤⠶⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠶⢦⣤⣀
⠀⢀⣴⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄
⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠸⣿⡄⠀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⣸⡿
⠀⠘⣿⡦⠀⠀⠀⠀⠀⣀⠉⠁⠒⠒⠒⠀⠤⠤⠤⠄⢀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠤⠤⠄⠐⠒⠒⠈⠉⢁⣀⠄⠀⠀⠀⠠⣾⡟
⠀⠀⣿⡇⠀⠀⠀⠠⠄⠀⣈⠉⠑⠒⠒⠂⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠠⠤⠤⠤⠀⠒⠒⠒⠉⢉⡀⠀⠠⠀⠀⠀⠀⣿⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠐⠒⠒⠤⠄⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠔⠒⠒⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿⡇
⠀⠀⣿⣇⠀⢘⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠘⣄⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⢀⣿⡇
⠀⠀⢿⡏⠢⣘⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⠀⠀⠀⢸⡠⠊⣿⠇
⠀⠀⠘⢿⣆⡘⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⡠⠔⠈⣀⣾⠟
⠀⠀⠀⠀⠙⠷⣶⣄⣀⢸⠃⠀⠠⠄⢀⡄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣇⠀⠠⠐⠂⢳⠀⣀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢷⣤⣤⣀⣸⣀⠀⠀⠉⠁⠀⠐⠂⠐⠒⠒⠀⠒⠂⠀⠘⠃⠉⠀⠀⠀⣸⣀⣠⣤⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠶⠶⣶⡶⢶⣶⣶⣤⣶⡶⢶⣾⠷⠾⠟⠻⠛⠛⠋⠉⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠛⠊⠄⠀⠉⠢⣄⠄⣀⣀⡀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⢀⢀⡉⡳⠆⣈⣿⣧⣭⣿⡷⠶⣫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣸⡙⠀⠐⡆⠀⠀⠉⠉⠩⠥⡤⣴⣶⣾⣏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠙⣶⢴⢧⣀⡀⠀⣀⣀⣼⣧⣿⣿⣿⡷⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠅⡿⡎⠋⠛⠛⠛⢻⢻⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⡆⢸⠃⢣⠀⠀⠀⢀⠇⠈⣿⡧⢻⣿⣿⣞⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣃⡞⠀⠸⡀⠀⠀⡼⠀⠀⣹⣿⣎⢿⣿⣿⢿⣿⡄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⢇⠀⠰⠃⠀⠀⢹⣿⣿⣎⠻⣿⠘⢿⣿⡆⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠸⡀⡎⠀⠀⠀⢸⣿⣿⣿⣷⡝⢶⣤⣻⣿⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⣿⡏⣽⠒⠂⠀⠠⢿⡄⠀⠀⠀⠴⣿⣟⣽⣿⣿⣯⣿⣟⣻⣿⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠻⠿⢁⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⣡⣤⣷⣛⣯⣭⣭⣭⣭⣿⣽⠾⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢀⡄⡄⣀⡠⠀⢠⣀⣄⣀⠄⢀⣄⣀⢀⢀⠄⠀⣤⡀⣀⢀⡄⢀⣄⣠⣀⣠⠀⣀⣠⣀⡤
⠀⠀⠀⠀⠐⠓⠙⠉⠛⠃⠘⠓⠋⠘⠃⠘⠉⠋⠛⠙⠛⠀⠓⠓⠛⠘⠁⠘⠋⠙⠋⠋⠀⠟⠋⠉⠛⠃
⠀⠀⠀⠀⠀⡴⠤⢄⣀
⠀⠀⠀⠀⠀⠉⠒⠢⠤⣉⣑⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣒⣮⠭⣒⡢⠤⣤⡤⣤⠤⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⠀⢀⣀⣀⣈⣉⣑⣯⣀⣵⣀⣀⣀⠀⣈⣉⠷⠖⢒⣊⣉⠤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⢰⠁⡠⠴⠒⠋⠉⠁⠀⠀⠀⠀⣀⢤⣀⠤⣔⣒⡩⠭⠔⠒⠺⢭⡉⢱
⠀⠀⠀⠀⠀⠀⠀⢸⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⣀⡼⢻
⠀⠀⠀⠀⠀⠀⠀⢸⠳⢄⡉⡱⢶⠢⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⢲⢖⣉⣁⠔⢻
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢱⡞⠒⠦⠤⠤⠤⠤⢼⣱⠤⠤⠤⠤⠤⠖⢺⢺⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠘⣌⠑⠢⣼⢇⣀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⣸⢼⡤⠒⢉⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠵⢎⣀⣉⠉⠉⠓⠒⢾⣻⠒⠒⠋⠉⢉⣁⡙⠾⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠛⠂⠒⠒⠋⠉⠁⠠⠤⠤⠴⠆⢿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢤⣀⣀⣠⣴⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢾⠀⠈⠈⠀⠀⠀⠒⣶⡆⠶⠀⢄⡓⠒⠒⡖⠉⠉⠀⠀⠀⠀⢠⣽⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠐⠁⠀⠁⠀⠀⣀⡤⠶⠓⠠⣀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠁⠀⠀⠀⠀⠈⠣⠀⠀⠀⢸⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢺⡆⠒⠒⠒⠆⡄⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣎⡴⠚⡟⠢⣄⠀⠀⠁⠀⠀⢁⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⡀⣼⢾⡇
⠀⠀⠀⠀⠀⢀⡠⠚⠇⠀⠢⠧⠀⠀⠉⠲⢼⣾⠁⠈⢦⠀⠀⠀⠀⠀⠀⠀⢘⠀⠞⢋⣴⡗⢺⡟⣇⡠
⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠨⣧⡀⠀⠀⠀⢠⣦⡋⠊⠘⠋⠀⠈⠘⠉⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠓⠒⠒⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⢤⡤⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣤⣤⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⡀⢰⣿⣿⣿⢿⣿⢎⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⣼⣿⢼⣻⠝⣿⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣧
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⣿⡟⢸⣿⠀⢿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⡆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠀⣿⠇⠸⣿⠀⣼⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣟⣿⣍⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⡿⠀⠄⣽⠀⠀⠿⢀⢸⢿⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣦⣴⣶⣤⣴⣶⣦⣾⣷⣿⣿⣿⣿⢷⣿⣇
⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⢠⡇⠀⠀⠀⠀⠺⢸⣿⣿⣿⣿⣿⠻⣿⡿⠛⠛⠛⣿⡿⠏⠀⠉⠋⠙⠉⠉⠉⠉⠉⠁⠀⠀⠉⣿⠸⣿⣟⣻⣿
⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⣆⠀⠀⠀⠀⠀⠀⡈⢿⡿⣧⠰⣿⡇⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠎⠛⡷⢿⡇⢸⣿⡆
⠀⠀⠀⣸⣿⣿⣿⡏⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠰⠸⣷⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⢻⣇⠀⢹⣇
⠀⠀⣶⣿⣿⣿⣿⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⣶⣛⣙⣷⣶⣾⣈⣿⣦⣔⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣙⣘⣿⣾⣾⣿
⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣥⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⢿⡇
⠀⠀⢸⣿⣿⣿⡷⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡇⢸⣿⠉⠉⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡉⠁⠸⣇
⠀⣠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠈⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⢰⣿⢻⣟⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢠⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢻
⣿⣷⣾⣟⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠈⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⢸
⣿⣿⣿⣇⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠀⠀⢸
⢿⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⣿⡄⠀⢸
⠈⢿⣿⣿⣏⣹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡀⠀⠸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⢻⡇⠀⢸
⠀⢸⣿⠏⠐⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣗⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢿⡇⠀⢸
⠀⣸⡟⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣠⡄⣿⠸⣿⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠹⠛⠺⣿⢾⣧⣬⣬
⠴⣿⡏⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⣴⣿⣿⣷⣿⣴⣿⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣆⣛⣿⣟⣿⣿
⠀⠙⣇⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣿⣿⣿⣿⣶⣶⣶⣴⣶⣶⣶⣶⣶⣤⣤⣤⣤⣴⣶⣶⣶⣾⣤⣬⣿⢿⣿⣿⡿
⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣏⢩⡀⣼⣴⣿⠀⠀⠀⠀⠈⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡷⢾⣿⣬⣴
⠀⠀⣿⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣹⣾⡇⢿⢼⣿⠀⠀⢀⠀⠀⡤⠆⠀⠠⠄⢀⡀⠀⣠⠄⢀⡀⠀⠀⠀⢻⢀⣿⠀⢻
⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⣾⣟⢋⣠⣿⣼⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⢀⣤⡀⠠⠀⠀⣿⣹⣿⣘⣿
⠀⠀⣿⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⡧⢤⣿⡿⣿⣿⠠⢄⣤⠤⠶⡤⠤⢤⣤⣤⡤⢤⣤⣴⣿⣧⣤⠀⢄⠀⠾⢼⣿⢻⣿
⠀⠀⣿⡀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⠿⢶⣾⣷⣿⣿⣄⢻⣁⣶⣶⣶⣶⣶⣤⣬⣥⣤⣤⣄⣉⣿⡏⢀⢈⡀⣷⣾⣿⣶⣿
⠀⠀⢸⡇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻⣿⣿⠛⢿⣤⣿⣿⣿⣿⣻⣿⣿⣿⠟⢛⣻⠿⣿⣟⣛⣭⣽⣿⣿⡿⠿⠛⡵⣧⣾⣿⣾⣿
⠀⠀⢸⣇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⣿⣿⣿⣯⡗⣿⣿⣟⣛⣹⣿⣿⣿⣛⣛⣛⣛⣋⣙⣛⣛⣻⣿⣿⠿⠿⠺⣦⣦⣿⣯⣾⣿
⠀⠀⢈⣯⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢴⣿⣧⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣿⣽⣿⢽⣿⣿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡯⣼⣿⣿
⠀⠀⠙⢻⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⣿⣛⣿⣿⣷⣶⣶⣶⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣟⣿⣿⣿⣶⣿⣿⣿
⠀⠀⠀⢸⣧⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⡇
⠀⠀⠀⠀⣿⡇⠀⢸⡆⠀⠀⠈⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢹⣧⠀⡈⣧⠀⠀⠘⠇⠀⠀⠀⠀⠀⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢿⣧⣥⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢰⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢠⡟⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠐⢻⣧⣿⣿⣿⣿⡆⠀⠀⢀⡴⢟⡩⣍⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⡿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣥⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠉⠈⣿⣿⣿⣿⣄⠀⠀⣵⣎⣷⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣫⠉⣽⠟⠛⠛⠛⠛⠻⠟⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡷⠟⠛⠛⠛⠉⠛⠂⠈⠛⠛⠛⠛⠋⠉⠉⠉⠛⠋⠙⠋⠉⠉⠉⠛⠉⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⠁⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠴⠄⢀⣀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⡀⠀⠀⠀⠀⣅⠀⠀⠀⠒⠂⠀⣀⡀⠐⠂⠀⠀⠀⠀⠈⣿⣟⠃⠀⠐⠈⢉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⣿⣷⣶⠴⠒⠛⠛⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠴⠗⠂⠀⠀⠀⢀⣀⣉⣈⣿⡁⠀⠀⢀⡴⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⢐⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠋⠉⠉⠙⠫⡲⠛⠉⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⣿⣶⠀⠀⠀⠀⠨⢻⡆
⠀⠀⠰⣦⣄⠀⣀⡀⠐⠗⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠿⢷⡟⠀⠀⠀⠶⠚⡛⠁
⠀⠀⠀⠉⢙⡭⢉⢳⣶⡤⡀⠀⠀⠀⠀⠀⣼⢣⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠌⠀⠙⠻⢷⣄⣀⣀⣠⠞⠁⠀⠱⠀⠀⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⡾⠋⢀⡠⢊⡵⠚⠛⠛⠓⠲⠶⠿⠷⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠬⠬⠥⢽⢭⣥⠟⡹⠋⠉⢉⡽⠿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠿⠿⣿⠟⣷⠛⠖⢺⡿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠴⠶⢶⣶⢦⢤⣤⣤⣴⣧⣾⣥⣤⣴⣿⢷⡀⠈⢿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣰⢃⡼⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⣼⢁⡟⠀⠀⢸⡿⠋⠀⢷⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⡸⠁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣼⠁⠀⢀⡟⠀⠀⠀⠘⡇⠀⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢰⠇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡇⠀⠀⣼⠁⠀⠀⠀⠀⣿⠀⠸⡇
⠀⠀⠀⠀⠀⢀⡀⢰⡏⠀⠀⡏⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⠀⠀⢠⡟⠀⠀⠀⠀⠀⢸⠀⠀⣷
⠀⠀⠀⠀⠀⣾⣿⣿⣀⡀⣸⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡇⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⢸⣿⣿⡿⠿⢻⡟⢳⠖⠒⠒⢿⣿⠿⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣶⣄⣀⣀⣀⣸⣃⣼⡤⠀⢀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⢿⣷⡄
⠀⠀⠀⠀⠘⠛⣿⠀⠀⢸⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⣿⢩⡟⠉⠛⢿⣇⡿⠀⠀⠀⠀⠀⠐⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⡟⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠇⠀⠀⣼⠿⠃⠀⠀⠀⠀⠀⠀⡇⠀⢸⡟⠁
⠀⠀⠀⠀⠀⣼⠃⠀⢀⠇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⡿⠀⠀⢸⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⢰⡇⠀⠀⢸⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡇
⠀⠀⠀⠀⢸⠃⠀⠀⡆⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⡇⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⢀⣦⣾⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⣿⡆
⠀⠀⢸⣿⣿⣦⣤⣀⡁⠸⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇
⠀⠀⢸⣿⣿⠛⠛⢙⠉⢻⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷⣶⣶⣶⣶⣶⣤⣤⣴⣶⣀⣀⣀⣘⣃⣼⣤⣤⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣿⠛⠁
⠀⠀⠈⠁⣿⠀⠀⢸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⢹⡏⣿⠛⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⠀⠀⢸⡿⠟⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡟
⠀⠀⠀⠀⢿⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠈⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠇
⠀⠀⠀⠀⢸⡆⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣾⣦
⠀⠀⠀⠀⠸⡇⠀⠀⠇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⣿⣿
⠀⠀⠀⢸⣿⣧⣀⡀⢠⠀⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡿⠟
⠀⠀⠀⠸⣿⣿⠭⠿⠿⡷⢷⠦⠤⢴⣿⣿⣦⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⢀⣤⡀⠀⠀⢸⡇⣿⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⣸⠁⠀⣿
⠀⠀⠀⠀⠙⠛⣇⠀⠀⣇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠷⠤⠤⢼⣧⣿⣶⣶⣾⣿⣿⠀⠀⠀⠀⢀⡏⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡍⠉⠙⣿⣿⠂⠀⠀⠀⣼⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢷⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⡇⠀⠀⣿⠁⠀⠀⠀⢠⠇⠀⢰⣧⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⢆⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢿⠀⠀⢹⡆⠀⠀⢀⡟⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⣼⣦⣀⣀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⡆⠀⠈⣧⠀⠀⣼⠁⠀⡼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠓⠚⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣷⣶⣶⣶⣶⣦⣤⣤⣤⣠⣿⣟⠀⠀⠘⠇⢹⡀⢀⣹⣦⣴⠃⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠰⠠⠔⢀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠂⠃⠘⠁⠈⠠⠃⠀⠀⠢⢂⠠⢦⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠌⠠⠀⠀⠀⠀⠀⠀⠀⠠⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠒⠀⠐⠤⠆⠄⠀⠀⢀⡐⠀⠀⡀⠀⢸⠗⣢⡡⠘⣀
⠀⠀⠀⠀⠒⢀⠠⠈⡀⠀⠀⠀⠚⠧⠰⡞⠀⠀⠀⠀⠱⠪⡄⡢⠀⠐⠐⠂⠀⠀⢈⣱⠈⠊⠤⠀⠭⣲⡠⠠⢴⠀⠀⠀⢬⠇⡡⡈⠈
⠀⠀⠀⠈⠀⠀⠀⢠⣋⣤⢠⣴⠀⢄⠘⠊⠍⠰⢀⣠⡄⠀⠀⠳⠁⠰⠩⠤⢆⡴⠤⠁⡀⠀⠀⢀⣒⠁⠀⡤⡀⣰⠐⠀⡂⠀⠀⢂⠂
⠀⠀⠀⠉⠄⠁⠀⠀⠼⠄⠙⠀⠉⠀⠉⠀⠀⠀⠀⠈⠆⠑⠆⢰⢣⠄⠀⠀⠄⠀⠀⠀⢓⠘⠂⠊⠀⠀⡈⠅⠈⠁⠀⠀⡤⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠄⢀⡤⠈⢐⡀⠀⠀⢀⡀⢠⠀⠔⠲⠀⠈⢠⢠⠉⠈⢁⠀⠀⣈⠰⠀⡐⠤⡍⡒⣱⠄⠒⠤⠑⠠⠶⡀⠋⡀⠉⡀⠠⣉⡧⠐⠂
⠀⠀⠀⠀⠐⠊⢱⠂⠀⠀⠅⠈⡁⠈⠁⠉⠀⣰⡀⠖⠀⠧⡴⠃⠚⠈⠠⠘⠂⡧⠒⠈⠀⠀⢆⠠⠀⠒⠀⠢⠀⠔⠠⡀⣀⠈⠊⢀⡨⠃
⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠦⢈⠁⠘⠀⠀⠙⠀⠋⠀⠑⠄⠀⠀⠀⠀⠀⠀⠐⠃⠐⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠀⠨⠒⡠⠢⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠄⠠⠀⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡀⢀⠀⠀⠐⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠢⣄⠀⠂⢄⠀⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⡀⠘⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⠄⠀⠁⠀⡐⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⢃⠀⠀⠀⢀⡀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠁⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⠀⠀⠀⠀⠀⠀⠀⠀⢆⠠⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠠⠀⢢⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠉⠄⡀⠐⠀⢡⠀⠒⢆⠀⢄⠈⠑⠌⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣠⠎
⡀⠒⠠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣠⠞⠁
⠑⠤⣄⣀⡀⠀⠀⠉⠉⠒⠲⠦⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢠⠞⠁
⠀⠀⠀⠀⠈⠉⠉⠑⠒⠒⠲⠤⠤⣤⣄⣉⣉⠒⠒⠦⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠓⠒⠂⠤⠬⣉⣒⠒⠒⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠊⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡠⠊⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠤⠶⠶⠶⠶⠶⠶⠤⠤⣤⠞⣡⣞⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠋⠉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⡶⢃⡼⠧⠤⣭⣉⡛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠛⠛⠒⠂⠀⠀⠀⠀⢠⡴⠋⡡⣐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⢠⠋⠀⠀⠀⠀⠈⠉⠗⣦⣈⠳⣄⠀⠀⠀⠒⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡰⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⣄⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢸⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠻⠿⠿⠶⠶⠆⠀⠀⠀⣿⡀⠫⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢃⣼⡇⠀⠀⠴⠶⠶⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⢦⣄⣁⡀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⣤⣤⡤⠤⠤⠤⠤⠶⠖⠒⢋⣉⣤⠴⠛⢹⡇
⠀⠰⠶⠶⠶⠶⠶⠀⠀⠀⠀⠳⣤⡀⠀⠉⠉⠛⠓⠒⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠖⠚⠛⠉⠁⠀⠀⣠⡾⠃⠀⠀⠠⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⡗⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⢹⡇
⠀⠀⠀⠀⣠⣤⣤⣤⠄⠀⠀⠀⢸⣇⢹⡄⠀⠀⠀⠉⠉⠉⠁⣶⠒⠒⠲⠒⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⡄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠤⣤⣤⣤⡀
⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⢸⡿⡄⢣⠀⠀⠀⠀⠀⠀⢰⠇⢠⡆⠀⣇⠀⠀⠀⠀⠀⠀⣸⠁⢠⡄⢹⡀⠀⠀⠐⡇⡇⠀⠀⠀⠀⠀⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠆⠀⠀⢸⡇⢳⠈⣇⠀⠀⠀⠀⠀⡿⠀⡾⢸⡀⢹⡀⠀⠀⠀⠀⢠⠇⢀⡞⣧⠀⠇⠀⠀⢸⡇⡇⠀⠀⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠈⣇⢸⡄⠀⠀⠀⢸⠃⢰⠃⠀⡇⠈⡇⠀⠀⠀⠀⡾⠀⡼⠁⠘⡄⠘⡀⠀⢸⠃⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠸⡄⣧⠀⠀⠀⡟⠀⡟⠀⠀⢹⡀⢹⠀⠀⠀⢰⠇⢰⠃⠀⠀⢻⠀⢷⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣷⠸⡄⠀⢰⠇⣸⠁⠀⠀⠈⣇⠘⡆⠀⠀⡞⢀⡏⠀⠀⠀⠘⡇⠸⡇⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡆⢧⠀⡾⢰⠇⠀⠀⠀⠀⢹⡀⢳⠀⣸⠁⣼⠀⠀⠀⠀⠀⢹⡀⢻⡇⢸⢿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡇⠀⠀⠀⢳⠘⣶⠃⡼⠀⠀⠀⠀⠀⠈⣇⠸⣇⠇⢰⠇⠀⠀⠀⠀⠀⠀⢷⠀⢀⣽⠞⠁⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⡙⠲⠦⣤⣬⣇⣉⣰⡃⠀⠀⠀⠀⠀⠀⢹⡀⠉⠀⡞⠀⠀⠀⢀⣀⣀⣤⠼⠗⠋⢀⣤⠞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠉⠙⠒⠂⠄⠈⠉⠉⠉⠙⠛⠛⠛⠒⠒⠚⠓⠒⠚⠛⠛⠛⠉⠉⢉⡁⠠⠤⠖⠚⢉⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠶⠞⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣽⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠉⠲⣄⢀⡤⣄⠀⠀⣀⣀⣀⣀⣀⣰⣲⣄⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣹⠶⣄⡀⠙⠿⣍⠉⠉⠁⠀⠀⠀⠀⡴⠛⡌⠉⠉⠉⠑⠚⣳⣾⢤⣀⣀⣠⠤⠶⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣍⠁⠀⢀⣀⣉⡶⢤⡈⠓⢶⡒⠒⠒⠒⠒⠛⠛⠓⠶⠶⠤⢤⣼⡧⠼⠒⠉⠁⠀⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣧⠼⠒⠋⠉⠀⠀⠀⠀⠈⠓⢦⣉⠳⢤⡀⠀⠀⣀⡤⠴⠖⠋⢉⣀⣠⠤⠴⠒⠛⠿⣥⠟⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠳⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢮⡵⠶⠛⣋⣉⣤⠤⣖⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢠⠖⠚⠉⠙⣆⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⡇⠀⠀⠈⠙⠒⠶⠤⡤⢤⣀⣈⡓⠚⠉⠀⠀⠀⠀⠘⠦⣤⡔⣣⣄⣀⡠⠤⠴⠒⠋⠉⠀⠀⠀⣇⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠙⣆⠀⠀⠀⠀⠀⠀⡇⢸⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣰⠋⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⡏⠉⠒⠦⢄⣀⣀⣇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⢀⣀⡤⠔⠚⠉⢸⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢷⠀⠀⠀⠀⠀⢻⠀⠀⣹⠦⠤⠤⠤⢤⡤⠤⣤⠤⠤⠤⡞⠉⠉⡗⠚⠉⠁⠀⠀⠀⠀⠀⡞⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡀⠀⠀⠀⠀⡸⢦⣴⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣀⡴⢣⠀⠀⠀⠀⠀⠀⠀⢠⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡇⠀⠀⠀⢠⠇⡼⠹⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠏⠹⡄⢳⡀⠀⠀⠀⠀⠀⣸⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⢹⠀⠀⠀⡞⢰⠁⠀⢣⠘⡆⠀⠀⠀⠀⠀⠀⠀⡸⢁⡏⠀⠀⠘⣄⢳⡀⠀⠀⠀⠀⡇⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢷⠸⡆⠀⣸⢁⡏⠀⠀⠘⡆⢳⠀⠀⠀⠀⠀⠀⣰⠃⡞⠀⠀⠀⠀⠘⣄⠳⡀⠀⠀⢸⠃⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⣇⢠⠇⡼⠀⠀⠀⠀⢹⡈⣇⠀⠀⠀⠀⢠⠇⡼⠀⠀⠀⠀⠀⠀⠘⣆⠱⡀⠀⡼⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢹⡞⢰⠃⠀⠀⠀⠀⠀⢧⠸⡄⠀⠀⢀⠏⣸⠁⠀⠀⠀⠀⠀⠀⠀⠘⢆⠹⣄⡇⣸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢹⣸⢁⡏⠀⠀⠀⠀⠀⠀⠸⡆⢳⠀⢀⡞⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠹⣄⡇⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢾⡋⣻⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣇⡼⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⣉⠷⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠉⠛⠦⢄⣀⠀⠀⠀⠀⠀⠈⣇⣸⢁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠊⠁⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠈⠉⠓⠲⠦⠤⠤⣼⣉⣿⣀⣀⣀⣀⣤⠤⠤⠴⠒⠚⠉⠁⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢄⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⢀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⢀⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣂⣒⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣯⣿⣿⣿⣽⣟⢿⣿⣿⣿⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⣿⣿⣿⣿⣿⡿⣫⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⢈⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠆⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠐⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⡍⢭⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⢀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣯⡤⠤⠀⠀⠤⠤⠀⠀⣀⡉⠛⠛⠒⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⢹⡿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⢶⣤⣈⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣠⣶⣿⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠘⠍⠟⠒⢤⣿⢢⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠐⣥⠃⠀⠀⠀⠉⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢀⢁⠕⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⣶⡾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣠⡀
⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠃
⠀⠀⠀⢀⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⣿⣿⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⢻⣿⡆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⢯⣿⣿⣘⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡅
⠀⠀⠀⠀⣼⣿⣿⣿⣮⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢹⡏
⠀⠀⠀⠀⢹⣇⢿⣿⣯⡟⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⣭⠟
⠀⠀⠀⠀⠀⢹⡘⣿⣿⣿⣝⢿⡏⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢁⡴⣿⢠
⠀⠀⠀⠀⠀⠀⡁⡸⣧⣿⣿⣷⣕⠄⡀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⣡⢜⣵⡿⠋⡏
⠀⠀⠀⠀⠀⠀⠡⣰⡄⣿⣿⣿⣿⣿⣏⠲⣤⣀⠀⠉⠒⠤⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠐⣐⣾⣿⣼⠿⣫⣷⣿⣿⢳⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠹⡧⣿⡻⣿⣿⣿⣬⣷⡧⣬⣯⣔⡂⢼⣿⡏⠀⠀⠈⠀⠀⠀⠀⠒⠀⠀⠈⣽⣿⣿⠸⢾⣟⣭⣶⣿⣿⣿⣟⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣽⣿⣌⡻⣿⡟⡟⢱⣦⣽⣿⣿⣿⣼⣿⣭⣤⣤⣄⣀⣀⣀⣀⣠⣤⣬⣭⣵⡾⠗⣾⣿⣿⣿⣿⡿⡫⠊⣾⣞⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠛⢿⡍⠎⢷⠷⠸⣛⣿⣿⣿⣗⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣁⣿⣿⡿⢟⡵⠊⡡⠊⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡈⠻⢿⡟⠻⢿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⢛⡩⣶⣯⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⢀⠀⠁⠂⣍⣼⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣟⣛⣫⣭⣏⣼⣿⡇⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠐⠢⣬⣱⡄⠀⠉⠁⠈⠉⠀⠀⠀⠀⢠⣸⣿⡿⢷⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠲⣢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠺⢵⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⢷⣤⠤⠤⠤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣤⣖⡺⠭⠷⠲⠟⠉⠹⣏⠉⠉⠉⠉⠍⠛⢛⠛⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠛⠷⢧⣄⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠈⠙⠲⢄⣀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⣦⣬⣑⠲⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⢀⣠⣴⣾⠟⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢯⣾⠛⠿⢯⣿⣒⣲⠦⠤⠤⢤⣤⠤⠤⠤⠤⣖⣒⣮⡭⢷⡏⣿⡴⢚⣥⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢀⡏⠙⠳⠶⣤⣌⣉⣉⣉⠙⢻⢸⡟⠋⣉⣉⣉⣤⠤⠿⠛⢻⡃⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡇⠀⠀⠀⠀⠐⠀⠉⠉⠭⢹⢻⠉⠉⠉⠁⠀⠀⠀⠀⠀⠽⠇⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠰⢖⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠖⣦⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣵⣠⠴⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠶⣏⡛⠲⠦⣭⣐⣀⡀⠀⠀⣠⣄⠀⠀⠀⠀⣀⣐⣲⡭⠷⢿⣀⡷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣭⣩⣿⣿⣇⣸⢟⢛⣋⣉⣉⣥⠤⠴⠒⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠄⠰⠀⠂⢦⠄⠀⠀⠀⠀⠀⡘⠃⢀⠐⠃⡀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠀⠀⠀⠀⠀⠂⠀⠘⠀⠃⠀⠀⠂⠀⠀⠘⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⠐⠄⠠⠀⠀⠀⠚⠤⠀⡄⡄⠀⠀⠀⠛⠐⠁⠀⠄⠀⠀⠀⣀⡀⠀⠀⠀⡀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠁⠀⠀⠀⠀⡀⡐⠀⠀⠀⠀⠀⢄⠀⢀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠄⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠀⠀⠿⠃⠀⠀⠙⠀⢀⠃⠂⡀⠀⠈⢀⠂⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠢⣄⡀⠀⣄⠤⠀⠁⠠⠀⠀⠀⠈⣧⢀⡄⠀⠀⠀⡀⠰⡶⠀⢠⠄⡐
⠀⠀⠀⠀⠀⠀⠐⡀⠃⡄⠃⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⡀⠀⠠⢄⡀⢸⡇⠀⣤⡄⠀⠀⠀⢀⣰⣷⠦⢀⠀⠀⠀⠀⠀⠀⠘⠰⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⠌⠀⠘⠀⠈⠒⢷⡀⠙⠁⠀⣠⣴⠟⠁⠀⠐⠃⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⣞⡿⠁⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⣿⣾⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠚⠛⠛⠋⠉⠉⠉⠙⠛⠛⠒⠒⠒⠤⠄


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⠤⠐⠒⠂⠀⣀⢀⣀⡠⠖⢊⣁⣤⣒⣻⣤⠤⠔⢒⣋⠁⠤⠤⠀⢀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⢀⣼⡁⠤⠒⠒⠘⠉⠉⠉⠀⠀⠀⡠⠋⠁⠀⠈⠀⡐⠊⠩⠁⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠂⠀⠈⠉⢉⣓⣄
⠀⠀⠀⠀⠈⣷⠨⠥⢤⣀⣀⣀⠀⠀⢠⡤⠬⠀⠀⣀⣀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⠀⠀⠀⠨⠭⣑⣺⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠒⠲⠤⠭⠭⣿⣖⣒⣀⣀⣀⣀⣀⣠⡤⠤⠿⡧⠤⣤⣤⣄⣤⣤⣤⣴⣶⣾⣷⣖⣒⡚⠋⠭⠭⠤⢺⡅
⠀⠀⠀⠀⠔⢻⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣸⡁⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⡇⠀⠀⠐⠂⠀⠀⠀⠸⠇
⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⠀⠀⠀⠀⠀⠀⠶⠐
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠰⣆⡉⠁⢀⡀⢰
⠀⠀⠀⠀⢧⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠂⡇⠈
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠇⠀⠀⠀⠀⡄⠀⢱⠁
⠀⠀⠀⠀⢼⡧⢄⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢐⣶⣺⣽⣺⠄
⠀⠀⠀⠀⠈⠉⠁⠟⣛⣻⠶⠶⣶⣤⣿⣥⣒⣒⣒⣒⣒⡒⠒⠂⢠⣷⣤⣴⣶⡒⠀⠀⠐⠲⠶⠶⢾⡷⠶⠯⠭⢹⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠀⠀⠀⠈⠭⠉⠩⠭⠭⠭⠿⠀⠠⠤⠀⠠⠤⠄⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢦⠀⠀⠀⠀⡄⠀⢠⠀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣠⣯⣷⠀⢀⣌⠃⠀⠈⢧⠐⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠻⣷⡆⠜⡄⠀⠀⢀⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠄⠰⠆⠐⢫⣧⡀⠈⡆⠀⠈⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡘⠄⠈⢿⣷⡀⠚⣆⠀⡃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠐⠀⣍⢷⡀⢹⢠⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢢⠀⡀⢹⠀⠀⠉⠊⠙⣾⡿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠲⣧⡀⠀⢈⣠⣿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠚⠛⠦⠌⠉⣿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠐⡄⢀⣞⣿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠈⢸⣿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠷⠀⠰⡇⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⣤⣄⠀⠀⡀⣀⡀⢀⣄⢀⣠⡤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡷⢒⣵⣾⣿⣿⣿⣧⣤⣤⣶⣦⣾⣿⣿⠟⣫⣾⣯⣾⣿⣿⣿⣷⣦⣌⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣼⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠾⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⡀⠀⣜⢃⣺⣿⣿⣿⠉⠉⠻⠛⢿⣿⣿⣿⣿⣿⠟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠤⣴⠇⣸⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⠇⣤⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⣆
⠀⠀⠀⠀⠀⠄⢸⡟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢻⡀
⠀⠀⠀⠀⠀⠀⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⣧
⠀⠀⠀⠀⠀⢸⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢻
⠀⠀⠀⠀⠀⣼⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠸⡇
⠀⠀⠀⢀⠀⢿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠛⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣇
⠀⠀⠀⠰⠀⣸⠆⢾⣿⣿⣅⣀⣉⡁⠸⣯⣭⣭⣭⣭⣽⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿
⠀⠀⠀⠘⠀⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢠⡇
⠀⠀⠀⠀⠀⢹⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣼
⠀⠀⠀⠀⠀⡀⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⡟
⠀⠀⠀⠀⠀⠡⠘⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘⣛⣛⣻⡏⠉⠉⠉⢩⡿⣿⣿⣿⣿⣆⠛⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠄⢻⣧⠹⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣶⣾⣆⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⣧⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢾⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢻⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣌⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⠛⠯⣈⡉⡉⠛⢹⡟⠛⠛⠿⠮⣻⡎⠙⢿⣿⣿⣿⣿⠿⢋⣡⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠛⠛⠛⠛⠓⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⡄⠀⠀⢙⡾⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⡆⡹⣄⣧⠀⢰⠂⡱⠶⠄⠀⠀⠀⡀⠖⠂⡈⢁⠀⠈⠈⠀⡂⠠⡀⠀⠐
⠀⠀⠀⠀⠤⡁⠘⣇⠺⡏⠷⠴⢩⠎⢴⡄⠀⠀⡠⠉⠀⠀⠀⠀⡄⠀⠠⠘⠀⠀⠀⠀⠀⠂⠀⠒
⠀⠀⠀⢀⠻⣏⣣⣈⡅⠉⠃⠸⠃⠀⠎⠀⠁⢀⠁⠀⠀⢀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠁⠲⠆⢀⠤⠘
⠀⠀⠀⠀⠀⠀⣨⠚⠥⡀⠀⠀⠆⡜⠀⠀⢀⠀⠀⢒⠂⠀⠀⠈⠉⠑⠒⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠉⠘⠀⠀⠀⠁⠈⣏⠀⠀⠀⠀⠠⠀⡐⠃⠁⢒⢠⠈⢀⠉⠠⠄⠀⢀⢀⠠⠀⠅⠀⠀⠀⣨⢃⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠞⠣⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠐⠄⠀⠈⠄⠀⠀⠂⣀⣤⡴⠲⠉⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠐⢤⣔⣲⡍⠁⠀⠁⠉⠁⠁⠀⠂⠂⠂⠉⠉⠀⠈⠀⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠐⠍⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⢀⠂⠀⠀⠠⠈⠒⠠⢀⡀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⡀⠄⠐⠀⠁⠀⠀⠁⠐⠀⠂⠀⢀⠉⠉⢀⠁⠁⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠆⠀⠁⠐⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠠⠄⣀⠀⠀⠐⠌⠀⠀⠀⠐⠠⡀⠀⠀⠀⠂⠀⠀⠀⠀⡠⠄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢴⠲⣄⡀
⠀⠀⠀⠀⠈⠙⠦⣝⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⡳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢍⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢌⠐⠤⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢚⡩⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡌⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⣊⡡⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡤⠞⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠂⠒⣋⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣈⣁⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⢰⢆⠀⠀⠀⠀⠆⠀⠀⠀⢀⡶⠀⠀⠀⢀⢶⡀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠘⡆⠀⢀⠇⠈⠆⠀⠀⠆⠀⠀⠀⠀⡞⠀⢀⠀⠀⡜⠀⢣⠀⠀⡼⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠀⠌⠀⠀⠀⠀⡌⠀⠀⠀⡆⡼⠀⠀⠀⠄⠐⠀⠀⠀⠆⠠⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠓⠒⠒⠒⠚⠛⠓⠚⠓⠒⠒⠚⠚⠚⠒⠒⠒⠛⠛⠛⠒⠒⠒⠚⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢀⣨⠦⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠴⣅⡐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢋⡂⠔⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠦⢐⠝⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠐⠠⠜⢠⠖⢢⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⡔⠲⡄⠣⠄⠂⠁⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠐⣡⡜⠋⠀⢑⣦⢀⠀⠀⠀⠀⡀⣴⡈⠀⠙⢷⣌⠢⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣔⡁⡎⠀⠀⠀⠀⠀⠀⠀⠐⠋⠋⠻⡀⣀⠴⠋⠀⢁⣥⠀⠀⣨⡁⠀⠙⢦⣀⢠⠟⠋⠙⠆⠀⠀⠀⠀⠀⠀⠀⢱⢈⣢
⠀⠀⠀⠀⠀⠀⠘⠁⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⠀⠀⠊⢄⢀⣴⣏⡱⠂⠐⢎⣹⣦⡀⣀⠑⠁⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠳⣔⠠⡀⠙⣋⡠⠓⢱⠀⠀⡮⠚⢄⣹⠋⢀⠄⣪⣊⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠤⠀⠀⠀⣀⢀⠀⠀⠠⠔⢢⡝⠋⡀⠄⢾⠒⠒⡷⠠⢀⠙⠌⠐⠟⣀⠤⠤⠤⠤⠤⣄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠂⠀⠀⠖⠂⠀⠀⠛⠉⠀⠖⣡⡴⠖⠒⠉⠉⠒⠲⢆⢌⠢⢀⠀⢸⠀⠀⢹⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⡇⠀⠀⠀⠀⡴⣡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣌⢧⠞⢂⣀⣚⣀⣀⠺⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⣮⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⡄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠸⢅⡓⠂⠀⠀⠓⠂⠀⠀⡛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠶⠓⠀⠀⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠘⡛⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⡐⡀⠀⠁⠀⠀⠀⠀⠀⠠⢛⢛⢃
⠀⠀⠀⠀⠀⠀⡐⡐⠀⠘⠸⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡁⢰⡔⠀⠀⠀⢠⡄⠀⠀⠀⠀⠃⠀⢂⠂
⠀⠀⠀⠀⠀⠰⡦⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠟⢦⡄⡀⠀⠀⠀⠀⢀⣠⠴⠋⠑⡐⡈⠀⠀⠀⠀⠀⠁⠀⠀⣴⠀⠀⠀⠀⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠁⠊⠉⠉⠙⠈⠁⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡄⢠⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢠
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠇⢸⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡄⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢈⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸
⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠰
⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠌
⠀⠀⠀⠀⠀⠰⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⡄⠰⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣆⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢠
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠠⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠄⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢸
⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠟⠛⠋⠉⠉⠉⠀⠀⠈⠉⠉⠉⠛⠻⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⣀⣤⣄⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣰⣿⡟⣩⠔⠲⣶⠀⣸⡟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣀⠋⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣧⣤⣀⠀⠀⠀⢿⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⢸⣿⣿⣿⣿⣷⡀⢠⣼⠇⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⢸⣿⣿⣿⣿⢋⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇
⢸⣿⣿⣿⡇⠸⣿⣿⣿⡿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿
⠈⠻⣿⣿⣿⣦⣾⣿⣿⣿⡜⣟⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠈⠀⠙⣿⣿⣿⣿⣿⣿⡘⡆⢈⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣷⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠗⢹⣾⠏⠈⢹⠿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⣾⡟⠀⡰⠋⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⢸⣧⢾⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣯⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣭⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⢰⠛⠙⠒⢦⣀
⠀⠀⠀⠘⢦⡀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⠀⠉⠳⣄⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠙⠦⣼⣭⣽⠦⠴⠶⠶⠶⠶⠒⠒⠒⠒⠚⢓⣶⠿⠶⠶⠤⢤⣤⣄⣀⣠⡒⢢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⢤⣀⣤⠴⠟⠓⢦⣄⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠉⣷⠀⠀⠀⠀⠀⠀⠈⢉⡽⣿⠓⠶⣤⣀⣤⠶⠚⠛⠉⠉⠙⣧
⠀⠀⠀⠀⢀⡼⣗⣋⠉⠀⠀⠀⠀⠀⣈⣙⣶⣄⡀⠈⠳⢶⡶⠶⠒⠒⠒⠒⠒⠒⠛⠛⠳⠶⠶⠶⠤⢤⣤⣄⣿⣤⠼⠗⠛⠉⠀⠀⠀⠀⢀⣀⣤⠾⠃
⠀⠀⢠⡾⠋⢠⡏⠈⣷⣤⠴⠖⠛⠉⠉⠁⠀⠀⠙⠳⢤⣀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠉⠁⠀⠀⣀⣀⣤⣤⣶⣶⠋⢹⠿⣦
⠀⢰⡏⠀⢀⣤⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀⠙⠲⣄⣀⣠⡤⠶⠚⠋⠉⢀⣀⣤⠤⠶⠒⠛⠉⠉⠀⠀⠀⠈⠙⠻⣏⡀⠈⣷
⢰⠏⠙⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠉⠓⠦⠼⠗⠚⠉⢉⣀⣤⣴⠶⢾⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⠉⣧
⢸⠀⠀⡟⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⢀⡤⠶⠚⠋⠛⢦⡀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⡇⠀⣽
⢸⠀⠀⡇⠀⠀⠉⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠘⠧⣤⣠⡴⠟⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣀⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠋⠁⠀⠀⡇⠀⣿
⣸⡤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣶⠛⢳⡤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣚⣭⡴⠒⣦⠶⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⢾
⢧⠀⠀⠈⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠁
⠘⣇⠀⠀⢸⠋⠳⠦⣤⣀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠙⡇⠀⠀⣸
⠀⠘⢧⣀⣾⡀⠀⠀⠀⠉⠙⠓⠶⣶⠛⠛⠺⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣤⢼⣀⣀⣠⣤⠴⠖⠛⠉⠁⠀⠀⠀⢰⣷⣀⡼⠃
⠀⠀⠘⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣼⠛⠛⠓⠒⠒⠶⠶⠦⠤⠤⠤⠄⠴⠶⠶⢺⡇⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣹
⠀⠀⠀⣿⠀⢿⠀⠀⠀⠀⠀⠀⠀⣸⢧⣀⡴⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⠀⢀⡾⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣿
⠀⠀⠀⣿⡄⢸⡆⠀⠀⠀⠀⠀⢰⡇⢠⡏⢷⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢈⡿⠛⣆⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⣿
⠀⠀⠀⡟⣇⠈⣧⠀⠀⠀⠀⢀⡿⠀⣾⠀⠸⡆⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡼⠁⠀⠘⣦⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼⢻
⠀⠀⠀⡇⢻⠀⢻⠀⠀⠀⠀⣼⠃⣸⠃⠀⠀⢻⡀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣼⠃⠀⠀⠀⠘⢧⠈⢷⡀⠀⠀⠀⠀⠀⢠⡏⢀⡇⢸
⠀⠀⠀⡇⢸⡇⢸⡇⠀⠀⢰⡇⢠⡏⠀⠀⠀⠈⣧⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠇⠀⠀⠀⠀⠀⠈⢧⠀⢳⡀⠀⠀⠀⠀⣸⠁⢸⠃⢸
⠀⠀⠀⡇⠀⣧⠀⣷⠀⢀⡿⠀⣾⠀⠀⠀⠀⠀⠸⡄⠸⡆⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠳⡀⠀⠀⠀⡿⠀⡿⠀⢸
⠀⠀⠀⡇⠀⢹⡀⢹⡄⣼⠃⣸⠃⠀⠀⠀⠀⠀⠀⢳⠀⢹⡀⠀⠀⠀⠀⠀⢰⠏⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠱⡀⠀⢸⡇⢰⡇⠀⢸
⠀⠀⠀⡇⠀⠸⡇⠘⣷⠏⢠⡏⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⣧⠀⠀⠀⠀⢠⡟⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠹⡄⣼⠁⣸⠁⠀⢸
⠀⠀⠀⣇⠀⠀⣷⢀⡿⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⡆⠀⠀⢀⡞⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠹⣟⠀⡿⠀⠀⣼
⠀⠀⠀⡟⠳⣄⢸⣾⣇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢻⡀⠀⡼⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡽⢾⡃⣠⠞⢻
⠀⠀⠀⡇⠀⠈⢻⣅⢈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⣇⣸⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣼⠟⠁⠀⢸
⠀⠀⠀⢷⡀⠀⠀⠉⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣸⠃⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠈⠉⠓⠶⠦⣤⣄⣀⡀⠀⠀⠈⣷⠟⢦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⠊⠉⠀⠀⠀⠀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠳⣤⠾⠷⠶⠶⠶⠶⠖⠒⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠚⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣛⣒⣒⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⣒⣒⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠒⣈⣉⣉⡉⠭⠥⠤⠤⠉⣉⣉⣁⣒⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡡⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⢍⡢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢹
⠀⠀⠀⠀⠀⠀⢀⠏⢸⡈⠓⠮⣓⡢⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢚⡵⣿
⠀⠀⠀⠀⠀⠀⢸⠀⡾⡏⠓⠤⣀⡭⠵⣖⠒⠤⠬⢭⣭⣉⣉⣉⣉⣭⡭⠭⠤⠒⠒⢻⠟⠹⡿
⠀⠀⠀⠀⠀⠀⢸⠀⢧⢱⠀⠀⢸⠀⠀⣿⠒⠒⠲⠤⠤⠄⠀⢾⡏⠉⢳⠤⠄⠒⠂⡏⢸⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⡸⡆⠀⠘⡆⠈⡇⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠀⢠⠃⡟⢀⠃
⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⣏⠢⡀⢳⠀⡇⢣⠀⠀⠀⠀⠀⠀⢸⠆⠀⢸⠀⠀⠀⢸⢀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡄⢱⢸⡀⠙⡾⡆⢰⢸⡀⠀⠀⠀⠀⠀⢸⢰⠀⢸⠀⠀⠀⡇⢸⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⡆⢇⠀⢳⢳⠀⡆⣇⠀⠀⠀⠀⠀⢸⢸⠀⢸⠀⢀⠴⠇⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⡸⡄⠸⡈⡆⢃⢫⠙⠲⡤⣤⡤⢼⠈⠀⡴⢶⢹⡘⡸⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣱⣳⠀⡇⢠⢸⣸⡀⣀⣀⣰⣀⢀⢸⠀⡇⡌⢇⢧⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⡼⠚⡇⡂⡍⠀⠀⠀⠀⢨⡎⢸⠛⢾⣾⢧⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠤⣀⣰⠁⡇⠀⠀⠀⠀⢸⠇⡎⢀⣼⢯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⣧⡞⠒⠒⠒⠒⠺⠤⠉⠙⠛⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢉⡰⠒⡶⠤⠤⢤⡤⢤⠤⠶⠲⣎⠻⣀⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣁⣀⣔⣋⠴⠋⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠢⣄⣑⣢⡬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣻⣿⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⡉⢢
⠀⠀⣀⠤⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣡⣇⣼⣜⡀⡀
⠀⡰⢁⡾⠥⠤⠤⡔⣖⡒⠒⠈⠉⠉⠀⢰⠋⠂⠀⠈⢢
⢠⠁⡾⠀⠀⠀⠀⠀⠀⠉⠙⠒⢤⣤⣠⡏⠀⠀⠀⠀⠀⡆
⡜⢠⣧⠀⠀⢀⡄⡤⠐⠒⠋⠉⠉⠀⢸⢁⠀⠀⠀⠀⠀⢰
⡇⢸⠀⠀⠀⠀⠈⠉⠙⠒⠢⠤⣀⣀⣞⠈⠀⠀⢰⠁⠀⠸
⡇⢸⣀⠀⠀⣀⡄⡤⠤⠒⠒⠈⠉⠉⣿⠀⠀⠀⢀
⡇⢸⠀⠀⠀⠀⠉⠉⠑⠒⠦⠤⣀⣀⣟⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⢠⡀
⠇⢸⡆⠀⠀⠀⠀⡀⣀⣠⠤⠔⠊⠁⢹⠠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠅⠂⠄⠀⠀⠀⠀⠀⢔⠀⡂
⢸⠀⣷⠒⠒⠒⠘⠼⠠⠤⠤⠤⢀⡀⢸⡄⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠠⢅⡒⠥⠂
⠀⢀⠘⣄⠀⠀⠀⣀⢤⠤⠔⠐⠋⠉⠀⢳⡀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢈⠐⠠⢀
⠀⠀⠣⠝⠯⠭⠭⠬⠬⠭⠭⠶⠶⠶⠶⠤⠷⠦⠄⠔⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣷⣤⣤⠀⠠⣦⠀⠰⠀⠀⢠⡑⠀⢰⡀⠀⠁⠠⠄⠀⠀⢰⠀⡄⠀⠀⢠⡋⠙⠀⠀⠀⠀⠀⣘⠂⠀⠀⣋⠀⣸
⣿⣿⣿⠀⣼⣿⠀⢸⣗⠲⣿⣿⠀⠠⣃⣴⡠⢿⠀⠀⠀⢰⡶⠇⠀⠀⢈⡇⢠⠀⠀⣿⣿⠙⣿⠃⠀⢸⣿⠀⣿
⣿⣿⣿⡆⢹⣿⠀⠘⣿⡟⣿⡿⠀⠀⣿⣿⣧⣄⣶⠀⠀⣾⣿⡇⠀⠀⢸⡇⣿⠀⠀⢹⣿⢀⣿⡆⠀⢸⣿⠀⣿
⣿⣿⣿⠃⢸⣿⡀⠀⣿⡇⢻⣷⠀⠀⣿⣿⣿⡟⣿⠀⠀⣿⣿⡇⠀⠀⢸⡇⣿⠀⠀⠀⣿⢸⣿⠁⠀⢸⣿⠀⣿
⣿⣿⣿⠀⠸⣿⡇⠀⣿⢧⢸⣿⠀⠀⣿⡿⣿⣇⣿⠀⠀⢾⣿⠇⠀⠀⢸⡇⣿⠀⠀⠀⣿⢸⣿⠀⠀⢸⣿⠀⣿
⣿⣿⣿⠀⠀⣿⡇⠀⣿⣿⢸⣿⡄⠀⣿⣿⣿⡇⣿⠀⠀⣿⣿⡇⠀⠀⢸⡗⣿⠀⠀⠀⠁⢸⣿⠀⠀⢸⣿⡀⣿
⣿⣿⣿⡇⢀⢹⡇⠀⣿⣿⢸⣿⡇⠀⢿⡏⣿⠁⡏⠀⠀⢻⣿⡇⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⣿
⣿⣿⣿⡇⢸⠘⣇⠀⢿⣿⢸⣿⡇⠀⢸⡇⣿⠀⡇⠀⠀⢸⣿⡇⠀⠀⠘⡇⣿⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⣿
⣿⣿⣿⡇⢸⡇⣿⠀⢠⣿⢸⣿⣿⠀⠘⣯⣿⢸⡇⠀⠀⢸⣿⡇⠀⠀⠀⡇⣿⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⣿
⣿⣿⣿⡇⢸⡇⢿⠀⢸⣿⢸⣿⣿⠀⠀⠟⣿⢸⠇⠀⡄⢸⣿⡇⠀⠀⠀⡇⣿⠀⠀⣄⠀⢸⣿⠀⠀⢸⣿⡇⣿
⣿⣿⣿⠀⠘⠃⠈⠀⠸⠿⢸⡿⠛⠀⠀⠀⠈⠉⠀⠀⡄⠈⠿⡇⠀⠀⠀⣷⣿⠀⠀⣿⠀⢸⣿⠀⠀⢸⣿⡇⣿
⠛⠛⠛⠀⠀⠀⠀⠀⢀⡀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠈⠉⠀⠀⠘⠿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢘⠀⠀⢐⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠄⠀⠰⠀⠀⠀⠠
⠀⠌⠛⠀⠀⠉⠁⠀⠀⠟⠿⠷⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠠⣿⠉⠙⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠉⣿
⠀⠀⠀⠀⠉⠛⢶⣤⣀⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⣀⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡈⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⢁⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀⠉⠻⢶⣤⣠⡶⠟⠉⣀⣤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠽⣿⣶⠾⠋⢁⣠⣶⣿⠯⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⢀⣤⠴⢒⣋⣭⣭⣶⣶⣾⡿⠟⠉⣠⣴⣾⣿⣧⣄⠉⠻⢿⣷⣶⣶⣭⣭⣙⡒⠦⣤⣀
⠀⠀⣠⠞⣩⣶⣿⣿⡿⢿⣿⠟⠛⣉⣤⡶⠛⠉⠉⠉⠉⠉⠉⠛⢶⣤⣈⠛⠻⣿⡿⢿⣿⣿⣶⣍⠳⣤
⠀⣼⠁⣾⡿⠛⠉⠀⠀⠸⣧⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣼⠇⠀⠀⠉⠛⢿⣷⠈⣧
⠀⣿⣄⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⣠⣿
⠀⢿⡈⣳⣮⣙⡲⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⢖⣋⣵⣞⢁⡿
⠀⠀⠙⡿⠛⢻⡏⠓⠲⠮⢭⣝⣛⣒⣒⠶⠶⠶⠦⠦⠴⠶⠶⠶⢶⣒⣒⣛⣫⡭⠵⠖⠚⢹⡟⠛⢿⠋
⠀⠀⠀⣿⠛⣿⣷⣦⣤⣄⣀⣀⢸⡿⠿⢿⠉⠉⠉⠉⠉⠉⠉⠉⡿⠿⢿⡇⣀⣀⣠⣤⣴⣾⣿⠛⣿
⠀⢀⡴⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⢦⡀
⠀⣾⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⣷
⠀⣿⢦⣹⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣏⡴⣿
⠀⠻⣄⣽⡀⣿⣙⡻⠿⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡿⠿⢟⣋⣿⢀⣯⣠⠟
⠀⠀⠙⣟⡛⣹⡏⠉⠙⠒⠲⠮⢭⣿⠀⣟⣛⣛⣛⣛⣛⣛⣛⣛⣻⠀⣿⡭⠵⠶⠒⠋⠉⢹⣏⢛⣻⠉
⠀⠀⠀⠈⠉⠉⠙⠓⠶⠦⣤⣤⣸⡟⠛⢻⣀⠀⠀⠈⠁⠁⢀⣀⡟⠛⢻⣇⣤⡤⠴⠶⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⢰⡏⡎⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⠀⠀⢸⣷⣷⣾⡇⠀⠀⣰⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⡆⠀⢸⣿⣿⣿⡇⠀⢀⣿⣉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣦⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⢸⣿⣿⣿⡇⠀⣿⣿⢹⣿⡇⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⢸⣿⠨⣿⡇⠀⣿⣿⢸⣿⡇⠀⣿⣿⢻⣿⡇⠀⠀⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣿⠀⢸⣿⣶⣿⣿⢸⣿⣿⣿⣿⡇⢠⣿⣿⣸⣿⡇⠀⢀⡇⢰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⡄⢸⣿⣿⢿⣿⢸⣿⡟⠛⣻⣿⣸⣿⠿⢿⣿⡁⠀⢸⣿⢈⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣇⢸⣷⣤⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣧⣾⣿⡇⠀⠛⢻⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⢛⢿⡟⣻⣿⡇⠀⠀⣿⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⢿⣿⢹⣿⠘⣷⣆⣾⣇⣿⣧⣿⠸⢻⡟⠿⣿⡇⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡶⣌⢿⠸⣟⣶⢻⣿⣻⣇⢻⡇⣿⢠⣿⡇⣿⣿⣷⠀⣾⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣌⠀⡧⣽⡬⣿⡘⣿⡘⢃⣿⣸⣍⣧⣙⣿⣟⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⡿⢿⣿⡿⠿⠻⣿⣿⣿⣿⣿⣿⣈⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⣿⣛⣉⣿⠟⠀⠀⢙⢶⣶⣶⡾⠋⠻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣃⣀⣈⣻⣿⣿⣿⣷⣾⣿⣿⣿⣻⣷⣦⣶⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⣿⡾⣻⠻⡙⣿⢩⡇⣶⣿⢰⣽⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⣿⣇⣿⣧⣷⢿⣾⡇⣻⣿⡀⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣿⢿⡇⣿⡞⣿⡏⠿⣿⣿⣎⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣷⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠌⣿⣿⡫⢻⣿⣿⡏⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⢨⣿⣷⣇⣺⣼⣿⣇⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠬⠿⠷⡵⠾⠻⠿⠛⠈⠁

⠀⠀⠰⠂⠉⠃⠄
⠀⢲⠁⠀⠀⠀⠀⢥
⠀⠈⢻⠀⠀⠀⡠⡏
⠀⠀⠀⠛⣢⡮⠂⠱
⠀⠀⠐⢮⠋⡔⣐⡠⢃⠴⢀⣀⠀⡠⠂⠐⡄
⠀⠀⠀⠀⢹⠂⠑⢙⠜⢴⡅⠈⠙⢾⠃⢀⠇⠀⢀⠠⠔⠂⠉⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄
⠀⠀⠀⠀⠈⠵⠆⠀⠰⢦⡻⣄⡀⠞⡷⠚⡀⠉⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⠀⠀⠀⠈⠀⠈⠐⠒⠤⣄⠂⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⡻⠗⢙⡫⠁⠀⠀⠀⠀⠀⠲⡒⠋⠉⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠂
⠀⠀⠀⠂⠀⠒⠄⠘⠤⠁⠐⠂⠈⢿⣄⠡⢂⠄⠀⠀⠀⠱⡀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡄
⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠀⠈⡄⠀⢳⠓⠀⠀⠁⠘⠁⠀⠝⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢠
⠀⠀⠀⠁⠀⠚⠀⠀⠀⠀⠀⠲⣵⣀⣀⠀⠀⢀⡀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⣚⣏⠐⠖⠒⠛⠂⠉⠈⠉⠒⠚⠲⠄⠀⠮⠄⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘
⠀⠀⠀⠀⠀⠱⠸⠖⠢⠤⠤⠄⠀⡀⠀⠀⠀⠀⠀⠀⠈⠃⢚⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠄⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠔⣂⢄⣀⣀⣀⠀⠀⠀⢀⢂⢈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠉⠉⠉⠁⠁⠀⠀⢳⡞⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣊⠔⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢂⣀⣀⠀⠀⠀⠈⠙⠃⡠⠂⠈⠐⠠⠤⢀⡒⠊⠩⠉⠒⠂⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠫⠁⡈⠉⠓⠶⢶⢰⠾⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⢀⣈⣦⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣦⣴⣾⠛⠓⠛⠉⠹⡍⠉⠉⠉⠉⠙⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠈⠉⠓⠤⣀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣄⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠻⠶⣶⣤⣤⣤⣀⣀⡀⡠⣤⣤⣴⡶⣾⣿⠵⣋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⣯⠉⠓⠲⠤⢭⣭⣩⣭⣍⡩⠭⠵⠗⠛⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠁⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⡇⠀⢐⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣹⣤⣀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⣰⣠⠓⣞⡹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠪⠭⢿⣒⣶⣾⣹⡶⣖⣒⣚⡉⠥⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⢠⣾⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠆
⠀⠀⢰⡟⣿⣿⣿⣿⡄⢠⣿⣿⣿⣿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⣿⣾⣿⣿⣿⣿⣻⣿⣟⣶⣿⣿⣿⣿⣽⣿⣾⣿⣷⠦
⠀⠀⠸⣧⢿⣿⣿⣿⡇⢸⣿⣿⣿⠟⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠹⣎⢿⣿⣿⠇⢸⣿⣿⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠈⠻⣿⣿⠀⣸⣿⠏
⠀⠀⠀⠀⠀⠀⢹⣿⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⠲⠶⠤⠤⠤⠤⠴⠶⠚⠃
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣶⣶⣶⣶⣾⣿⠃⢻⣷⣶⣶⣶⣶⣶⣾⡟⠀⠀⠀⠀⠈⠃⠶⠴⠦⠆⠀⠤⠆⠲⡆⠙
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣴⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠚⠛⢾⡷⢰⣷⢰⢿⠚⡿⠈
⠀⠀⠀⠀⠀⠀⠉⠁⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣾⣯⣾⣽⢬⣿⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠲⠆⠶⢴⠠⠴⠆⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣽⣻⣍⣣
⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⡀⣀⠀⣀⢀⡀⣀⡀⣀⢀⡀⠀⠀⠀⠀⣨⣟⣿⣿⢿⣿⣟⣯⡀
⠀⠀⠀⠀⠀⣈⡦⡤⠤⠒⠂⠠⠴⣄⡀⠀⠀⠀⠀⠰⣿⣾⣷⣿⣷⣿⡾⢁⣨⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠛⠿⢿⣿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⡍⠀⠀⠈⠂⡀⢀⠀⠀⠀⠈⣃⠀⠀⠀⠀⢻⣿⣿⣿⣿⡞⣥⣾⡿⢿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⣿⣷⣦⣤⣤⣤⣤⣤⣤⣶⣾⡇⠀⠀⠀⠀⠘⣿⣿⣿⢻⣾⡿⣫⣷⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢹⡿⣵⢟⣽⣾⣿⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣛⣀⣀⡀⠀⠀⠀⠀⠛⠻⠿⠿⠛⠋
⠀⠀⣶⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⣴⣿⣿⣷⣦⠀⠀⣀⣀⣀⣄⣀⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣿⣿⣿⣿⣿⢷⣴⣿⣿⣿⣿⣿⡄⢈⠁⠉⠉⠋⠉
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣼⡏⠻⢿⣿⣋⣠⣤⣤⡄
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣠⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⠜⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡟⠛⢻⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢘⠂⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⢠⠆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠾⢰⣶⡆⠧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢈⡃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡽⣦⡐⢈⡉
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠘⣿⠃⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠀⠀⠈⠀⠈⠻⠿⣿⣿⡿⠿⠋⠁⠐⠆⠘⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣦⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡶⠾⢇
⠀⠠⣄⡤⠿⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢸⣿⠇⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣁⡤⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣶⡶⠴⡈⡁⠀⠈⠂⠰⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠍⠀⣉⣾⠁⠀⠀⠀⠀⠈⢷⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠿⣿⣦⡀⠁⠀⡀⢰⣿⠦⡀⢸⣿⣿⠀⠲⠆⢠⣿⡇⠀⡀⠤⢄⠀⣾⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣄⣄⠈⠺⣻⣥⡹⢶⣿⢏⡀⠀⢰⣿⣿⡷⠋⠀⠀⠀⢱⡿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣄⡇⠘⣵⡄⣏⠿⣿⣿⠟⠫⠋⠀⠀⠀⠀⠀⢸⣷⡀⢈⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⣾⠀⠈⠋⠢⠅⡹⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣻⣴⣶⢦⡀⠲⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢽⡿⠃⠀⢙⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⡆⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠐⠠⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣧⣀⠐⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠈⠛⠀⠀⠀⠀⠀⠀⣸⡿⠣⢄⡀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⠀⠀⠀⠀⠀⠀⠤⠀⠀⢠⣾⠟⠁⠀⠀⠉⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣰⣿⡇⠀⠀⢀⠄⠀⠀⢀⠇⠀⣚⠟⠉⠐⠠⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢿⣷⡀⠀⠈⠀⠀⠀⢠⠀⠈⡄⠁⠒⢤⡀⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠁⠃⠀⠀⠀⠆⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢧⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡎⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⡆⠀⠀⠀⠀⠀⣼⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⢧⠀⠀⠀⠀⠀⡼⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠹⡀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢸⠀⠀⠀⠀⢰⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⠀⠀⠀⠀⢹⡀⠀⡀⠹⡄⠀⠀⣀⡀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡼⠀⠀⠀⢠⠇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⢧⠀⠀⠀⠀⢣⡀⠀⠀⣹⢒⠛⠅⠜⠒⢄⣠⠎⠀⠈⠀⢀⠞⠁⠀⠀⢀⡞⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣶⡒⣄⠀⠀⢙⡦⠘⢧⣂⠴⠠⠀⠀⣨⡇⠀⠀⣀⡴⠋⠀⠀⢀⣀⡼⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣆⠸⡁⢘⣦⣀⡞⢠⡀⠀⠈⠉⠉⠉⠉⠁⠀⠀⣞⠁⠀⠀⠀⢠⠏⣠⣹⡞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠚⣡⠊⠀⡞⠀⠀⠁⢤⣤⠀⠀⠞⠉⠀⠀⠀⢸⡄⠀⣠⠞⠻⣄⣇⡸⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠁⣠⠞⡇⠀⠀⠀⡐⠐⠒⠀⠀⠀⠀⠀⠀⢀⣧⠾⡅⠀⠀⠈⠳⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡜⠁⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠱⡄⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠉⠳⢤⣀⡀⠀⠀⠀⢀⣀⣤⠞⠀⠀⠀⠀⠘⢆⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡶⠀⠉⢹⡛⠛⠉⠉⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠤⠶⠶⠶⠿⠶⠤⠤⣤⣍⣐⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⡇
⡠⠤⣧⠙⠲⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠚⢡⡷⠤⡀⠀⠀⠀⢻
⢇⣠⠋⢳⡦⣀⣀⠈⠉⠉⠋⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⣠⢠⡴⠋⢣⣀⠏⠀⠀⠀⢸⡄
⢰⠟⠶⣆⢀⡎⠉⠉⡟⠓⡖⠶⢶⠖⠲⠶⠶⢶⠂⢲⠚⠋⠉⢹⡀⣸⠴⠚⢳⠿⡆⠂⠀⠸⡇
⢸⠀⠀⠀⠈⠉⠛⠛⠓⠰⠷⠦⠾⠤⠴⠤⠤⠼⠦⠾⠶⠒⠛⠉⠉⠀⠀⠀⢸⠀⠛⠲⡞⠛⠁
⣿⠀⢠⢂⣭⡐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣨⣅⢢⠀⢸⡆⠀⠀⣧
⢻⠀⡞⡼⠼⢯⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢊⠿⠼⠝⠇⢸⣁⣀⣀⢸
⢸⡄⠘⠓⠠⠐⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡗⠄⠔⠻⠘⣸⠁⠀⠉⠻⡄
⠈⣧⠡⡑⠤⠴⣣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡳⠤⠔⡡⢃⡏⠀⠀⠀⢠⡇
⠀⠸⡄⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⡼⠓⠒⠒⠒⠉
⠀⠀⡟⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⡇
⠀⠀⠳⣄⠈⠛⠒⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⢤⠴⠖⠚⠉⢁⡴⠃
⠀⠀⠀⠈⠙⠲⠦⣤⣄⣀⣂⡀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣠⡤⠶⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠉⠀⠀⠘⠿⠃⠀⠀⠀⣀⣀⠀⠰⠖⠀⠀⠻⣿⠇⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠙⠟⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⢠⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃⠀⠐⢿⣿⠗⠀⠀⠘⣿⠟⠀⠀⢀⣀⣀⠀⠈⠁⠀⢠⣤⣦⠀⣰⠷⡶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠡⠄⠰⠆⠀⠀⠀⠀⢀⠈⠻⠋⠀⣠⣀⣄⠀⠙⠁⣰⠏⠀⠀⠀⠰⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠙⠷⣤⡀⠀⠀⣴⣴⡦⠀⠈⢧⠀⠀⠀⠙⠿⠋⠀⣠⣾⠋⠰⣶⠀⠠⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣤⣤⡄⠈⠙⢷⣄⠈⠛⠁⠀⠀⠘⣧⠀⠀⠀⣠⣴⡿⠛⢁⣀⡀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋⠀⢠⣤⠄⠹⣧⡀⢰⣶⣶⠀⢻⣇⣴⣿⠟⠉⢀⡀⡈⠛⠁⢿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣷⡀⠉⠁⠀⣸⣿⣿⠃⠀⠀⠙⢿⠏⠀⠀⠀⠙⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣻⣷⣄⠀⠀⣿⣿⠇⠀⢙⣡⣤⡴⠶⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣼⣿⣿⡀⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⡻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣩⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣫⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢋⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣫⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠴⢋⣵⣚⣁⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣩⠴⠋
⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⡩⠭⠭⣴⢋⣡⠞⠉⠀⠉⠉⠉⠉⠉⠙⠛⣻⠖⢋⣠⢾⣅⣀
⠀⢀⡤⠖⠛⠉⠀⠀⠀⠀⡎⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢠⠔⠚⠓⢾⣁⠴⠋⠀⠀⠀⠈⠙⠲⢤⡀
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧
⡟⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠃⣽
⣷⡈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠁⠀⣠⣾
⣧⠘⠢⣀⠀⠉⠙⠛⠒⠲⠦⠤⠤⠤⣤⣤⣤⣤⣤⣠⣤⣤⣤⠤⠤⠶⠖⠒⠛⠉⠁⠀⠀⣀⡠⠔⠊⢁⣿
⣿⠀⠀⢨⢛⠒⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡤⠤⠖⡞⡉⠁⠀⠀⠀⢸⣿
⣿⠀⢀⡏⠸⡀⠀⠀⠀⠉⠉⢩⢻⠉⠉⠉⠉⠉⠉⠉⢉⠟⡍⠉⠉⠀⠀⠀⠀⠀⣸⠀⢱⡀⠀⠀⠀⡇⣿
⢹⠀⡸⠀⠀⣇⠀⠀⠀⠀⢠⠃⠈⡆⠀⠀⠀⠀⠀⢀⡎⠀⠸⡀⠀⠀⠀⠀⠀⠀⠏⠀⠈⢷⠀⠀⢸⣧⡟
⢸⢀⠇⠀⠀⢸⡀⠀⠀⢠⠇⠀⠀⣧⠀⠀⠀⠀⢀⠞⠀⠀⠀⢣⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⢇⠀⢸⢁⡇
⢸⡞⠀⠀⠀⠀⣇⠀⢀⡎⠀⠀⠀⢸⠀⠀⠀⠀⡎⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⡇⠀⠀⠀⠀⠸⡄⣸⢸⡇
⠘⡟⢦⡀⠀⠀⢸⠀⡼⠀⠀⠀⠀⠘⡄⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡸⠀⠀⠀⠀⠀⠀⢣⡟⢸
⠀⠹⣄⠈⠳⢄⡀⢠⠃⠀⠀⠀⠀⠀⢣⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⠃⠀⠀⠀⠀⠀⠀⣈⠷⡿
⠀⠀⠈⠙⠦⣄⡈⠙⠲⠤⣄⣀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡌⠀⢀⣀⣠⠤⠒⢋⡡⠚
⠀⠀⠀⠀⠀⠀⠉⠳⠤⣄⣀⡈⠉⠙⠒⠓⠲⠦⠤⠤⠤⠤⠤⠶⠒⠒⠚⠋⠉⣉⣠⡤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣤⣤⣤⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡷⣿⣯⣉⡉⠉⠉⠙⠳⠤⠿⠛⣒⣆⣶⢤⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⣴⡿⠋⠀⠈⠙⠙⠛⠻⢷⠶⢶⣦⣄⣀⣤⣼⣙⣩⠵⠚⠛⠻⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠈⢠⡾⠋⠀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠈⢛⣿⠟⣿⠟⠉⠀⠀⠀⠀⠀⠀⠑⡕⣄⠀⠀⠀⠀⠀⠀⠀⣏⣉⠉
⠀⠀⠀⠀⠀⢠⣆⣰⡟⠁⠀⠀⠙⠛⠻⠯⢼⣟⣷⠂⠀⣴⠟⢁⠜⠁⠀⠀⣀⣀⣄⣶⣾⠀⠀⠘⣻⡇⠀⠀⠀⠀⠀⠀⣥⢠⣤
⠀⠀⠀⠀⢠⢿⣻⡿⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣠⣪⣤⢰⣖⣷⣿⡿⠿⠙⠉⠀⠀⠀⢀⠘⡀⠀⠀⠀⠀⠀⠇⠘⢰
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠈⠉⠙⠛⠲⢶⣤⣄⣀⣾⣧⡴⠡⣯⣿⠸⠟⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡖⢲⢓
⠀⠀⠀⠠⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡿⣿⠃⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠁⠈⠘
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣰⣶⣤⠀⠀⠀⠀⢘⠌⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡦⣼⡏⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠦⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠠⠤
⠀⠀⠀⡟⢺⣧⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⡵⠒⠒⠦
⠀⠀⠀⠀⠈⢱⠉⠐⠛⠳⠖⣤⣄⣀⣀⡀⠀⣀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠇⠀⠀⠀⠀⠀⠀⠏⡁⠌
⠀⠀⠀⠀⠀⡸⡇⠀⠀⠀⠀⠀⠈⠐⠛⠳⠶⢤⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡟⠀⠀⠀⠀⠀⠀⠀⡞⢉⣉⣉
⠀⠀⠀⠀⠐⣬⣿⡄⠀⠀⠀⠰⠀⠀⠀⠀⠀⠒⢭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡸⠁⠀⠀⠀⠀⠀⠀⠀⣧⠤⡄
⠀⠀⠀⠀⠀⠘⠻⣿⣦⡶⠠⣀⣀⡀⠀⠀⠀⣁⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⣗⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢦⣄⣐⡆⢀⣹⡄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠱⠷⢿⣿⢶⣃⡀⠀⠀⠀⠀⠀⠀⢀⡠⢪⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠶⠦⣄⣀⣠⣤⣒⠡⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠒⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠤⡀⠀⠀⠀⠀⢀⠀⠄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠠⠤⠌⠀⡒⠁⠂⠤⣁⠂⠠⠀⢐⢀⡉⡡⠄⠐⠒⠐⠊⠓⠦⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠄⠐⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠒⢀⠀⠀⠀⠀⠀⠀⠺⠤⣀⠐⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠈⠀⠀⠀⠤⠤⠭⠐⠒⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠠⣥⣁⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⡘⡀⠤⠀⠀⠒⠀⠡⠤⠀⠀⠀⠈⠐⠲⠄⢠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠐⠈⠉⠓⠒⢠⠇⠀⠀⠤⠄⠀⡥⠐⠒⠒⣂⠈⠉⠡⠐⠃⠀⠀⠀⠀⠀⡀⢸⡀
⠀⠀⠀⠀⠀⠀⢠⣤⣀⣄⣀⣀⡀⠁⠒⠠⡄⠤⠐⠂⠀⠒⠒⠒⠢⠀⡀⠀⢀⠀⠀⠀⠀⡀⠄⠂⠂⡃⣀⠠⡥⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠞⠁⠠⠖⠋⠁⠀⠈⠁⠐⠠⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠤⠘⠢⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⡂⠩⠠⠀⢐⠀⠈⠅⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⢀⢤⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠈⠀⠒⠒⠀⠀⠌⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⡄⢁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠔⢀⣤⡄⠆⡀⠀⢠⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⢀⠀⠊⢀⠎⠐⠀⠀⠀⠀⠈⠂⣉⠀⢖⠁⠀⠁⠘⠁
⠀⠀⠀⠀⠀⠉⠁⠀⢐⡢⠀⠀⢚⠀⡀⣉⠂⠀⠍⠁⢀⣀⣀⠈⠤⠶⠚⠋⠀⣀⠀⠠⡀⠀⠀⠠⠤⡐⡄⠤⠄⠠⠆⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠠⠄⠃⠂⠐⠀⠀⠀⠀⠀⠀⠂⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⡾⠁⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠹⡆⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡞⣠⡟⠀⠀⠀⠀⠀⢀⡴⢟⢡⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣴⠟⠀⠀⠀⠀⢀⡴⠋⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣱⠏⠀⠀⠀⣠⠾⣋⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡾⣵⠃⠀⠀⣠⢞⣱⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠖⠚⠛⣛⣋⣉⣉⣉⣉⣿⣽⣋⣙⣿⣿⡵⠿⢥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢡⣤⡤⠶⠚⠛⠉⠉⠉⠉⠉⢹⡏⣸⢯⡟⢉⡿⠛⠓⠶⣤⣌⣙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠛⠶⢤⣤⣀⣀⣀⣀⡀⠈⠛⠋⠈⣙⣍⣰⣠⣦⣾⠷⠞⠉⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢰⣤⣄⣀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢶⡿⢞⣥⠟⣩⠟⣻⡷⢲⡶⢶⠀⢰⡶⢲⠞⢛⡿⠛⣹⠋⣹⠿⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⡴⢋⣵⠞⣡⠞⢁⡴⠋⢀⡾⠳⢞⡵⠋⣰⠟⢀⡼⢁⡼⠃⣰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡴⢋⣠⠞⢁⡴⠋⢀⡴⠋⢀⣴⠋⢀⡾⠁⣠⠟⣠⠞⠁⣴⠃⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⣶⠛⣡⡴⠋⢀⡴⠋⢀⣴⠟⢁⡴⠋⢀⡼⢃⡴⠋⢀⡾⢁⣼⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠿⣾⣉⣠⡶⠋⢀⣴⠟⢁⣴⠿⣤⣰⠏⣠⠞⠁⣠⠟⢠⣞⡿⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⡉⠙⠛⠒⠿⠷⠴⢿⡿⠀⠸⠧⠾⠧⠶⠾⠟⠚⠛⠉⣁⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⠶⠦⢤⣤⣤⣤⣤⠤⠤⠶⠶⠶⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢃⡴⠋⣠⡾⠃⣠⠞⢀⡾⠁⣠⠟⢀⡴⠃⣠⠟⠀⣰⠏⢀⡾⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠚⠋⠀⠞⠁⠰⠋⠀⠞⠁⠠⠟⠀⠚⠁⠀⠚⠁⠀⠋⠀⠘⠁

⣿⣿⣿⢿⣿⣿⠿⠟⠿⢿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣸⣿⣟⠀⠀⠀⢘⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢻⣿⣧⣀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⠟⠛⠛⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⠁⠀⠀⠀⠀⠘⣿
⣿⣿⣿⠿⠿⢿⣿⣿⣟⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⢟⠿⠿⠿⣿⣿⣿⡇⠀⢀⣿⣿⣄⡀⠀⠀⠀⣠⣿
⣿⣿⣿⣀⠀⣸⣿⢿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⡹⠟⠀⠀⣀⣈⣉⡛⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠠⡀⢹⣄⣨⢞⠐⠀⠀⣘⠛⢶⡍⠙⣶⡈⠻⣿⣿⣽⣿⣿⣿⠋⠉
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⡀⠐⢦⡓⣼⠃⣼⣿⠁⢀⣰⣿⣋⡭⡟⠛⠀⠀⠙⢧⠀⠈⢻⣿⣿⣿⣿⣤⣀
⣿⣷⣾⣿⣿⡟⠀⠀⠀⠠⠵⣤⣀⠙⣿⠀⢙⣿⠀⣩⠽⣛⠁⠀⣰⣦⠀⠀⣠⣾⠿⠿⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢈⠛⣞⣷⣌⣿⣏⡵⢯⠥⠄⢸⣿⣍⢉⡽⠻⣿⢷⣄⠀⠀⠹⣿⣿⣯⣽
⣿⣿⣿⡿⠀⠀⠀⢠⡴⠀⠀⢤⣾⠀⠀⠀⢈⡿⠁⠀⠀⠀⠀⢼⣿⣿⠏⠀⢰⣏⡠⡿⠀⠀⠀⢿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⣽⣿⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣰⣤⣆⣤⣤⣰⣏⣽⣄⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠙
⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣄⣠
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣾⡿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⠉
⡿⠋⠉⠉⢻⣿⣦⡀⠀⠀⣸⣿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣿⡃
⣇⠀⢀⣀⣼⣿⣟⣿⣦⡀⢹⣿⣿⣿⣿⣾⣶⣶⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⣠⣾⣿⣯⣿⣿⣿⣷⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⠾⣿⣿⣿⣿⣻⣿⣿⣿⣿
⣿⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⢈⣿⣿⠟⠛⠛⢿⣿⣷
⠈⠙⢿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠁⠀⠀⠀⠀⢹⣿
⠀⠀⠀⣿⣿⣿⣯⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣼⣿
⣀⣤⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⡿⠿⢿
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⣿⡀⠀⣠
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣻⣿⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⢉⣉⣉⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠛⠉⣀⣤⡶⠶⠛⠛⠛⠉⠉⠉⠙⠛⠛⠳⠶⣦⣄⡈⠙⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⢁⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠋⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠈⠻⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠘⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⢻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⢰⣿⣿⠇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⡟⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠘⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⡇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣿⣿⣷
⠀⠀⠀⢀⣼⣿⣿⠃⢸⡇⠀⠀⠀⠀⢀⣴⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣄⠀⠀⠀⠀⠀⣿⠀⢻⣿⣿⣄
⠀⠀⠀⠘⢿⣿⣿⡆⢸⡇⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣿⠀⣾⣿⣿⠟
⠀⠀⠀⠀⠸⣿⣿⡇⠸⣇⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢀⡿⠀⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣧⠀⢿⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣸⠇⢠⣿⣿⡇
⠀⠀⠀⠀⠀⢹⣿⣿⡆⠘⣧⠀⠀⠀⠀⠀⠈⠛⠛⠉⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠙⠛⠋⠀⠀⠀⠀⠀⢠⡟⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣦⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣌⡙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢠⣾⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣷⣄⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡌⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢀⣴⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⡈⠙⠷⣤⣀⠀⠀⣠⣾⣿⣿⠟⠀⠙⢿⣿⣿⣦⡀⠀⢀⣠⡴⠟⠉⣠⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣤⣀⠉⣡⣾⣿⣿⠟⣁⣀⣀⣀⣠⡙⢿⣿⣿⣦⡉⢁⣠⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠟⣡⣾⣿⣿⠟⣡⣀⣀⣉⣉⣉⣀⣀⣠⡙⢿⣿⣿⣦⡙⢿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣿⠟⠡⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠙⢿⣿⣿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠂⠁⠀⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣻⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠙⠛⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⡟⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠴⣶⡞⣲⣶⠄⠀⠠⣶⣶⢲⣶⠦⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡿⣿⡟⣿⣿⢶⣀⡴⣿⣿⢻⣿⢿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢶⣶⣶⣿⠿⡃⠀⡇⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠒⢹⠐⠉⠉⠉⠙⠳⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣴⣿⣷⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢬⣛⣟⣙⣟⣧⠄⠀⣇⠀⠀⣀⣤⣶⠶⠾⠷⢶⣶⣤⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠉⠁⢀⡼⡟⣦⠞⠛⠁⠀⠀⠀⠀⠀⠀⠙⠻⣿⣄⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡞⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⢠⠏⡏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡞⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢠⠏⠀⡇⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⡇⠀⣰⠏⠀⠀⣇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⡀⠀⠀⢿⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⣾⠧⣄⡀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠴⠖⠋⢹⠃⠉⠓⠦⡌⢷⡀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣴⠏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠻⣦⣤⠉⠛⠒⠒⠒⠋⠀⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠙⠛⠶⠶⠤⠶⠶⠞⠋⠉⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⣾⣤⣤⣄⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡴⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢳⣿⡤⣤⣭⣉⣉⣉⢹⡟⠛⠋⠉⠉⣉⣀⣽⣶⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⡇⣿⠋⠁⠉⠉⠙⣿⢹⡿⠛⠋⠉⢉⣭⣇⡿⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⢻⠀⠀⠀⠀⠀⣿⣾⠀⠀⠀⠀⠀⢸⠇⠇⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣸⡄⠀⠀⠀⠀⢹⢻⠀⠀⠀⠀⠀⢸⢸⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡏⡧⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣸⠸⠀⢠⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢷⢣⡄⠀⠀⠀⢸⣸⠆⠀⠀⠀⠀⣿⡇⠀⣸⣿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠸⣼⠀⠀⠀⠠⠸⣿⠀⠀⠀⠀⠀⣿⠃⠀⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡄⢷⡧⢀⠀⢀⠀⣿⠀⠀⠀⠀⠀⣿⡀⣠⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣧⡸⣿⠘⠐⠶⠀⣿⡁⠀⠀⠀⠀⡿⣠⣿⡿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣄⣀⣀⣀⣻⣇⣀⣀⣀⣸⣿⣿⠟⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣩⣉⣉⣉⣙⣛⣋⣋⣉⣩⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⣴⣿⠉⣈⣅⡉⢩⣿⡉⣿⣿⡀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⡀⢛⢋⣾⡷⠺⣿⡌⣯⣸⣿⣿⢷⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⢸⣇⠛⣸⣿⠀⡛⢸⣏⠁⣿⣿⣿⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⣿⠀⣿⣿⢸⣿⠸⢸⢀⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣡⣘⠛⠀⡛⠋⢘⣋⣘⢛⣠⣟⣁⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠻⠻⠿⠏⠿⠿⠿⠿⠻⠟⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣶⣤⣠⢄⣄⣤⣴⣀⣠⣤⣤⣤⣤⡴⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢙⣋⣵⡿⣽⣿⣻⡯⣿⣽⣿⣿⠛⢿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣷⣻⣗⣿⣿⣿⣿⣿⣾⡿⣳⣿⣦⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣯⣁⣉⣉⣉⣉⢉⣉⣉⣉⣉⣥⣥⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⡼⠀⠀⠀⢀
⠀⠀⠀⣀⠀⠀⠀⠀⠛⢀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠙⠦⠀⠀⣰⠖⡿⡫⠋⠙⠳⢤⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣠⠞⡜⠸⠁⠃⣀⠀⣠⣄⠹⡄⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣠⠀⢀
⠘⠓⠂⠀⢠⠇⠀⠀⠄⠀⡀⠀⠀⣉⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡶⠟⠉⠉⠛⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠡⠎⡇⢘⡰⢉⣀⡀
⠀⠀⠀⠀⢸⠀⢀⣤⡄⠈⠁⠀⣠⠇⠀⠀⡝⠀⠀⠐⠚⠛⣠⢞⠟⢁⠞⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢸⣾⠕⠉⢂⡝
⠀⣀⣠⠀⠘⣇⠈⠛⠁⠘⠒⠚⠁⠀⠀⣰⠃⠀⠀⠀⠀⣰⢁⡏⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢸⡦⣼⣧⣀⣀⠜
⠈⠉⠀⠀⠀⠈⠳⣤⣀⠀⠀⢀⣀⡤⠞⠁⠀⠰⣄⠀⢠⡇⠸⢃⡀⠁⠀⠉⠉⡀⠀⠀⢠⢶⠀⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠁
⠀⠀⢀⡴⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⣄⠀⠀⠈⠁⠘⡃⠐⠋⢀⡀⠀⠚⣿⣁⠀⠀⠈⠉⠀⠀⡼
⠀⠀⠉⠀⠀⢀⡞⠀⠀⢰⠂⠀⢸⠀⠀⠘⠆⠀⠀⠀⠀⣧⠀⠲⣿⡁⠀⠈⠟⠁⠀⠀⡄⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠘⠛⠁⢾⡅⠀⠀⢠⡤⢃⡠⠊⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢰⣔⡇⢠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣤⣄⣤⣤⠤⣶⠒⠉⠀⣰⡷⠀⠙⣆⠀⠀⠀⠀⠀⢀⡤⢼⣧⡏⠐⢧
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣧⠇⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⡾⢿⣠⠖⠚⣹⡅⠀⠀⢸⡄⠀⠀⠀⠀⢾⣐⡀⠙⣷⣶⡞
⠀⠀⠀⠀⠀⠀⠠⣯⡍⠙⢾⣦⡤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⢿⣿⠛⠺⣿⣾⡿⢿⣧⣀⡴⠿⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠴⣮⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⠏⢀⡴⣄⡈⠙⠻⢿⣅⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⠋⠀⡯⢰⣮⡏⠀⠀⠀⣸⠇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⣄⡀⠙⠙⠛⠁⠀⠀⣠⣇⣤⠼⠲⣦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⣤⣶⠒⣚⣙⣓⣶⣤⣄⣠⡖⠁⢠⡀⠀⡀⠀⠀⠀⠉⡓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢘⡅⣼⢸⣿⣷⡎⠉⠉⠉⠉⠉⠁⠀⠀⠀⣄⢠⡄⠁⠀⠻⠀⠇⠠⠂⠋⢀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠥⢸⢅⡘⢻⠃⠫⠨⠛⠀⠲⡌⠃⣀⠐⠇⠀⠀⠀⠙⠘⠣⠄⠀⠀⢀⠀⠀⠀⠀⠈⣿⡥⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠻⣆⡤⢀⠘⠁⠂⠴⠊⡴⠋⠘⢰⣦⣤⣠⠖⠛⠳⡏⠀⠀⡴⠛⢶⠖⠚⢶⠀⠀⠀⠺⠀⠀⠊⠀⠀⠀⠀⠀⠀⢰⠀⡙⢦⡀
⠀⠀⠀⢀⡖⠒⠒⢋⣤⣄⡛⠷⢧⡘⠀⢋⡤⢾⣄⡟⠀⣿⡦⢸⡧⣶⣄⠸⣧⠀⢀⣷⡄⣨⢤⡀⠾⢦⠀⠀⠸⠆⠀⠀⡴⠀⡜⢀⠀⢀⢀⢀⠃⠀⢳
⠀⠀⠀⠘⢷⡤⢈⠿⢶⠋⢀⣀⡼⢛⣥⡟⣀⣀⠟⠳⣾⣏⣀⡘⠛⠃⠀⠀⣸⠇⢯⣀⣀⠹⠾⠃⠀⣸⠂⠀⡄⠀⡄⠘⠁⠀⠀⠋⠀⠸⣧⠎⠀⢀
⠀⠀⠀⠀⠸⣄⣀⡴⣆⣀⣸⣭⡀⢻⣤⢸⣿⠛⣤⡼⠁⠈⢹⣅⣀⡾⡟⣻⡥⠀⠀⠈⣯⡀⣠⣿⢛⡵⠄⠙⠃⠀⠀⠐⣆⣦⡄⠀⠀⢀⠀⣤⠀⠎
⠀⠀⠀⠀⠀⣿⠳⢦⣸⣧⣫⡄⠁⣾⠀⣠⣦⣤⣼⡧⠲⠆⠀⣉⣉⢿⣷⡧⢲⠆⠀⠀⢈⣍⡁⢸⣏⡴⣦⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠳⣤⣿⣟⡵⠃⠀⠈⢻⣥⣼⣶⡯⢶⡆⠀⠀⢳⣝⣿⣋⡴⠋⠀⠀⠀⠘⢧⣽⣿⣯⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⣁⣠⣤⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣈⡉⠛⠶⣤⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⣁⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢦⣄⡉⠻⣿⣝⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢁⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠙⢿⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡙⢷⡄
⠀⠀⠀⠀⠀⣠⠟⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⣦
⠀⠀⠀⠀⣴⠋⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⢷⡀⢀⡀
⠀⠀⠀⣼⠃⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢿⣿⠻⣄
⠀⠀⣼⠃⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢿⡶⠛
⠀⢰⡏⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⣇
⠀⣾⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢻⡀
⢠⡏⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇
⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣇
⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿
⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⣶
⠈⣧⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⣇⣸⡆
⠀⢻⡀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾⠋⠉
⠀⠈⣧⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⠇
⠀⠀⠹⣆⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡟
⠀⠀⠀⠹⣆⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡳⣟
⠀⠀⠀⠀⠙⣦⣨⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣨⠿⣿⣧⡀
⠀⠀⠀⠀⢀⣼⣿⡏⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⢃⡴⠋⠀⠈⠻⣿⣆
⠀⠀⠀⣰⣿⠋⠀⠙⢦⡈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠟⠁⠀⠀⠀⠀⠈⠻⣷⣄
⠀⢠⣾⠏⠁⠀⠀⠀⠀⠙⢶⣄⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣁⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣌⡙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⣉⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣄⣉⡙⠛⠲⠶⠶⠶⠤⡤⠤⠶⠶⠶⠚⠛⣉⣡⡤⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⢀⣀⣀⠀⠀⣀⣀⣀⢀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣛⣩⣵⣾⡿⠿⢿⣿⣿⣿⠿⠿⣿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⠉⢹⠛⠒⠦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⡄⠀⠀⢸⣿⣷⣶⣤⣭⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣁⣃⡉⠁⣶⣶⣶⣶⣾⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠿⠷⣾⡏⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣴⣤⣤⣤⣠⣾⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⠙⠛⠛⡻⠋⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣇⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢼⣦⣷⣄⠘⣿⣿⣿⣿⣿⣿⠋⠀⣰⣧⣼⢄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⠈⣀⠩⣷⡄⠀⠀⠀⠀⠀⢠⣿⠿⡙⠟⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠊⠈⠏⢨⡟⠁⠈⠉⠉⠁⠈⠳⡎⠘⠋⠈⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣶⣦⣶⢊⣾⣿⣷⣶⣾⣿⣦⡹⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⢋⡩⠅⠚⠉⠙⠿⠖⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢙⠶⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⡿⣶⣤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢠⠂⠁⠀⠀⠀⠀⠀⠀⢀⡀⢀⣴⡖⢫⡯⠟⠓⠉⠀⠁⠈⠉⠙⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠋⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⣝⣶⡝⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣡⣀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠡⣡⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡏⠩⠂⠀⠀⠀⠀⠀⠀⢀⠌⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢺⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡐⡇⠀⠀⠀⠀⠀⠀⠀⠻⠉⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣧⢀⠀⠀⠀⠀⠀⠀⠀⠁⢱⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣏⠁⠀⡀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⣼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⠑⡆⠈⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⢰⣍⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣀⠀⠀⠀⠀⠰⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⣀⠀⣠⣿⣉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠏⠙⠲⢤⡀⠀⠸⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⠀⠀⠠⢈⡷⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠈⠓⠦⣍⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣷⠀⠀⣠⢨⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡏⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢄⠋⠀⠚⠟⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣄⡀⠀⠀⠀⠀⠀⠀⢀⠸⠇⠀⠐⠒⢮⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠻⣄⡀⠀⠈⠦⡉⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣌⡉⠑⢤⣐⡿⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣍⣂⡀⢠⡝⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⢿⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣶⣾⣄⠀⢀⠀⢰⣾⡇⠀⠀⠀⣾⣿⡀⠸⠿⠿⣿⡿⠿⠿⠀⣿⡇⠀⠀⠰⣿⡇⠀⢸⣿⠙⢿⡆⠀⢰⢸⠟⠛⠓
⠀⠀⠀⠀⣿⢸⣿⡄⠻⠤⢸⣿⡇⠀⠀⢸⣿⣿⡇⠀⠀⠀⠿⡧⠀⠀⠀⣿⡇⠰⡶⠀⣿⡇⠀⢺⣿⠀⠨⠧⠀⢸⢸
⠀⠀⠀⠀⣿⢸⢹⣿⡀⠀⢸⣿⡇⠀⠀⢾⡏⢹⡿⠀⠀⠀⢛⣃⠀⠀⠀⣚⡂⠀⠀⠀⣒⡀⠀⣿⡿⢠⣿⡇⠀⢸⢸
⠀⠀⠀⠀⣿⢸⠀⢿⣧⠀⢸⣿⡇⠀⠀⣦⠃⢨⡾⡀⠀⠀⢰⣭⠀⠀⠀⣭⡅⢀⣄⢰⣶⡆⠀⣿⡿⣿⠋⠀⠀⢸⢸⠿⠟⠃
⠀⠀⠀⠀⣿⢸⠀⠈⣿⡆⢸⣿⡇⠀⢰⣎⠁⠀⣾⡇⠀⠀⢀⣒⠀⠀⠀⣿⡗⠉⠉⢸⣿⠀⠀⣿⡇⢿⡆⠀⠀⢸⢸
⠀⠀⠀⠀⣿⢸⠀⠀⠘⣿⣸⣿⡇⠀⣾⣿⣤⣤⣿⣷⠀⠀⠨⣷⡆⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⣿⡇⠸⠇⠀⠀⣿⣼
⠀⠀⠀⠀⢻⢸⠀⠀⠀⢹⣿⣿⡇⠀⣌⠋⠀⠀⢙⣭⠀⠀⠀⣷⡆⠀⠀⢹⣧⠀⠀⣿⡇⠀⠀⣿⡇⠀⣶⡄⠀⢿⢾
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠿⠟⠇⠀⢿⠇⠀⠀⠘⠿⠂⠀⠀⠿⠇⠀⠀⠈⠻⢷⠿⠟⠀⠀⠀⠿⠇⠀⠙⠷⠀⠸⢸⣶⣶⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⠀⠀⠀⠀⢀⣴⠐⣶⡄⠀⢰⣦⠀⠀⠀⢸⣿⠀⣸⣿⠿⠿⠗⠀⡆⡶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⣾⠇⠀⠘⣿⡀⠘⣛⡂⠀⠀⣙⡛⠀⣿⡿⠀⠀⠀⠀⣇⡇⣤⠈⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⢰⣿⠠⣤⡄⣿⡇⠀⢿⣷⠀⠀⣿⡇⠀⢼⠖⠀⠀⠀⠀⢨⣇⣠⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠸⣿⠀⠉⠀⢻⠇⠀⢰⢰⡀⢰⣴⠀⠀⣺⣿⠶⠆⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠠⣤⠀⢀⠀⣤⡄⠀⠘⡟⡇⣸⡘⠀⠀⣘⠿⠀⠀⠀⠀⠀⡟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⣛⠀⣼⠀⣨⡅⠀⠀⢳⢃⢀⡇⠀⠀⢨⡓⠀⠀⠀⠀⡇⡇⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⡀⠀⠻⣧⢻⢣⡿⠁⠀⠀⠸⡞⢸⠀⠀⠀⢸⣿⣀⣇⠀⠀⡇⡇⠀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠛⠛⠀⠀⠉⠛⠊⠁⠀⠀⠀⠀⠙⠃⠀⠀⠀⠘⠛⠛⠛⠃⠀⠉⠁⠀⠀⠈⠛

⣶⣶⣶⣶⣶⣶⡤⣶⣶⣶⣶⣶⣶⠀⣿⣶⣾⣿⣾⣾⣿⣶⣷⣷⣷⣾⣾⣶⣾⣿⣿⣶⣾⣿⣶⣶⣷⣷⣷⣶⣾⣾⣶⣶⣶⣿⣾⣷⣶⣶⣿⣷⣾
⢸⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⡏⠀⢹⠈⠉⠉⠙⠛⠟⢿⢿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁⢸
⠀⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⢻⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸
⠀⠙⠛⠛⠛⠀⠀⠀⠙⠛⠛⠋⠀⠀⣼⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠻⠿⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠒⢾
⠀⠀⠀⠀⢠⠞⢧⡀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠙⠋⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸
⠀⠀⠀⢠⠏⠀⠀⠳⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠟⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⢸
⢨⠀⢠⠏⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⣶⣦⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠻⠇⠀⠀⢸
⢸⢠⣏⣀⣀⣀⣀⣀⣀⣘⠃⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠟⠉⠁⠀⠀⠉⠛⢿⣦⡀⠀⠀⡶⢶⠀⠀⠀⣀⣀⣀⣤⣤⣤⣀⣀⣀⠛
⠀⠀⠀⠀⢈⣉⣭⣭⣍⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠆⠀⡄⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠀⠀⠸⣿⠆⠀⣠⣴⣶⣶⣶⣦⣄⠀⠀⣿⡇⠀⡇⢠⠀⠰⣦⣭⣛⣛⡛⠛⠛⣛⣋⣭⣴
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠸⣿⣲⣿⠇⠈⠉⠉⠉⠁⠸⢿⣖⣿⠇⠀⡇⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠈⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⡏⠀⣿⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢐⢒⣒⣒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢒⣒⣛⠒⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⡇⠀⣿⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣮⠓⢄⠀⠀⠀⠀⢀⡴⢫⣶⣿⣿⣿⣿⣾⣿⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃
⠀⡇⠀⣿⠀⠀⢀⣤⣾⠀⠀⠀⠀⠀⠀⢠⡿⠻⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀⠀⣀⣀⣾⣿⣿⣿⣿⣿⡿⠿⠻⣷⠀⡦⠀⠀⠀⠉⠹⠉⠀⠀⠀⠰
⠀⣷⠀⣿⠀⠀⢀⢿⠃⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣬⣙⡋⣿⡿⠁⢹⣧⣠⣿⠁⠘⣿⡟⢛⣩⣴⣶⣶⣷⣶⣦⣀⣤⣀⣀⣀⣀⣀
⠀⣿⠀⣿⠀⠠⠃⢸⠀⢀⣀⣤⣤⣤⣄⣙⣿⣿⣿⣿⣿⣿⣿⣮⠁⠀⠈⠁⣿⠉⠀⠀⢩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢀⣒⣂
⠀⣿⠀⣿⠀⠁⠀⢸⠀⣬⡍⣛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠀⣀⣤⣤⣿⣤⣤⣀⠀⠙⠛⠛⣛⣉⣉⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠐⡿⠀⣿⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣿⠀⣠⣶⣿⠿⠿⠿⠿⠿⠿⣿⣿⣶⣟⢻⣿⡿⠿⠿⢿⠿⣿⡧⠀⠀⠀⠀⠀⢰⣿⣿⣿
⢠⣷⠀⣿⠀⠀⠀⣸⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡾⢟⣩⣴⣶⣿⣿⣿⣿⣿⣶⣮⣝⠿⣿⣿⣿⣿⣿⣟⣣⡿⣗⠀⠀⠀⠀⠀⠸⣿⣿⣿
⢸⣿⠀⣿⠀⠀⠀⢸⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢿⣈⣉⡥⠚⠁⠀⣿⠀⠀⠀⠀⣤
⢸⣿⠀⣿⠀⠀⠀⢸⠛⡶⢬⣝⣛⣛⠛⠛⣛⣛⣋⣍⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢠⣶⣶⣶
⠈⠿⠀⠿⠀⠀⠀⣼⡆⡗⠀⠈⠉⢹⠉⡋⠉⠉⢹⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣠⠊⢸⡏⣧⠀⠀⠀⢸⠀⠀⠀⠈⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⢀⠞⣿⠳⡀⠀⠀⣿⠈⣿⣿⣿
⠀⠀⠀⢀⠜⠁⠙⠺⢟⠯⠱⡄⠀⠈⠀⡀⠀⢀⡿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣧⢣⣠⠊⠀⣿⠀⠙⢆⠀⠛⠀⣿⣿⣿
⠀⠀⡠⠋⠀⠀⠀⠀⠙⠀⠀⠘⢆⠀⠀⠀⠀⡞⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡛⠛⣷⠙⢄⠀⠛⠀⠀⠈⠣⡀⠀⠻⠿⠿
⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⣼⠁⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠋⠀⠀⢀⠞⠹⡄⠈⠣⡀⠀⠀⠀⠀⠙⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢧⣀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⢀⡽
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢⣄⠁⠢⣄⣐⠒⠤⠤⠤⠤⠼⠦⠤⠤⠤⠔⢒⣂⣤⠔⠀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣂⠄⠋⣈⡹⠶⢄⡑⠶⡚⠉⠉⠉⠉⠉⠉⠉⠁⢀⠖⢈⡠⠶⣉⡉⠉⠕⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⢊⡴⠚⠉⠁⠀⠀⠀⠉⠲⢬⣒⣄⡀⠀⠀⢀⡤⣪⠴⠚⠉⠀⠀⠀⠈⠑⠢⢌⠢⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠼⠂⠰⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠃⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠦⣉⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣂⣥⠞⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠉⠒⠢⠾⢍⣐⣒⡀⠠⠤⠤⠀⠀⢀⡀⠀⠀⠤⠤⢀⣀⣒⣠⡽⠶⠖⠛⠉⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠒⠤⢄⣀⡀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⢀⣀⣠⡤⠶⠚⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⠉⠉⠙⠓⠒⠒⠲⠶⠶⢶⡶⠶⠶⠒⠒⠚⠛⠋⠉⢹⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⢇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡈⠢⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡤⠋⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⠲⢬⣙⡒⢾⠄⣀⡀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⣀⣠⣼⡗⣚⣩⠴⠛⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠉⠙⠓⠶⠮⠭⠭⣭⣿⣛⣛⣻⣟⣛⣻⣯⣭⠭⠭⠶⠖⠛⠉⠉⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⢀⡠⠆⠀⠀⢀⣀⡤⠔⠒⠋⠱
⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠒⢓⣚⣋⣉⡩⠭⠭⠤⠤⣤⡤⠔⠋⡡⠤⠶⣮⣭⠭⠟⢛⣋⠩⠤⠤⣐⣛⣓⡒⣒⡒⠂⠠⠤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⣴⣯⣐⠒⠍⠉⠀⠀⠀⠀⠀⠀⠠⣒⠼⠁⠀⡄⠀⡀⠠⠍⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠂⢀⣠⣭⢦⡀
⠀⠀⠀⠀⠀⢸⣗⠊⠥⠤⠤⣄⣒⣒⡒⠂⣤⠄⠀⠠⠤⠤⢤⣁⣀⣀⣀⣀⣀⣒⣒⣒⠀⠀⠀⠀⠐⠒⠒⠂⢀⣤⠉⠭⠭⠭⠥⠤⣖⣒⣿⡟
⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠒⠲⠦⠬⠭⢽⣿⣒⣒⣀⣀⣀⣀⠀⠠⠤⠤⠴⢿⡧⠤⠤⠤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣖⣒⣛⡋⠩⠭⠥⠴⢺⡇
⠀⠀⠀⠀⢠⠞⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⣽⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⡇⠀⠀⠀⠒⠀⠀⠀⠀⢸⢇
⠀⠀⠀⠀⢸⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⡄⠀⠀⠀⡆⠀⠀⠀⣶
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡇⠀⠀⣰⠠⠖⠂⠂⠉⢀
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠙⠛⠀⣀⡃⡇⢰
⠀⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡐⠀⠀⠀⠈⠉⡉⠀⣇⡄
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠃⠀⠀⠀⠀⠘⣃⣤⢸
⠀⠀⠀⠀⠸⣿⣓⠦⣤⣀⣀⠀⠀⠀⠀⢀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣀⣀⣹⣿⣶⣿⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠩⢽⣛⣛⡻⠷⠶⢾⣿⣬⣭⣭⣽⣭⣭⣍⣉⣒⣒⣺⣷⠶⢶⣖⣒⡒⠐⠒⠚⠛⠻⠯⢽⣿⠿⠿⠒⠲⢾⣿⠿⠿⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠐⠒⠂⠈⠉⠩⠥⠬⠿⠬⠅⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠰⠶⠒⠒⡖⠒⠒⠒⠲⢂⡤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⡦⠀⠈⠁⠀⠀⠀⠀⠉⠙⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⠉⠑⢄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢉⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⣀⠀⡠⠔⠊⠀⠀⠈⠉⠀⠀⠀⢙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠑⠧⡀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠎⡌⢢⡀
⠀⠀⠀⠀⠀⠀⣠⠋⠆⠀⡀⢂⠲⠂⠀⠀⠀⠀⡠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢒⡄⠀⠀⠈⠒⢄⡀⠀⠟⡄⡄⠀⠀⠘⠰⠀⠙⢦
⠀⠀⠀⠀⢀⡼⡃⠞⡈⠋⠌⠀⠀⠀⠀⠀⣀⣰⡀⠒⣄⠠⢄⢀⠀⠀⠀⠘⠇⠀⢀⠈⡜⡀⠀⠀⠀⠀⠘⡀⠘⡘⢃⠀⠀⢀⠀⡆⢀⠨⢷⡀
⠀⠀⠀⢀⡾⢁⢙⡠⡇⠈⠀⠀⠀⡘⠆⠎⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⡐⠀⠈⢧⣰⠐⢄⠀⢠⠀⠀⢁⠀⢡⠀⠀⠀⡘⢀⡥⢂⠊⡀⠳⡄
⠀⠀⢀⡞⠁⠀⠘⠁⢀⠘⣀⠐⢁⠄⡈⡄⢠⠀⠀⢀⡤⢀⠄⠀⡜⠀⠀⣸⠀⡀⠀⠀⠙⢇⠀⠙⠆⢠⡀⢸⠀⠈⡀⠁⣰⡇⢸⠁⡆⢀⠁⠀⢻⡀
⠀⠀⣸⠃⠀⢸⠀⡆⠈⡆⠈⡰⠁⠀⡝⠀⠳⠀⢈⡡⠐⠁⢠⠎⡠⠖⠁⠀⠀⠈⠶⠦⣄⠀⠙⠂⠄⡀⠙⢧⡀⠀⠇⠀⠿⣠⢸⠀⠀⠸⡄⠀⠜⣧
⠀⢠⠉⠀⠀⠸⠀⠡⢀⠁⠰⠗⡲⠋⠀⣀⠄⠒⠥⠔⢈⡠⠂⠈⢀⣴⣿⣶⡄⠀⠀⠀⠀⢍⠐⠤⢀⡈⠒⣄⠈⠢⢠⡀⠀⢡⡆⡀⡀⠀⡖⠐⢡⣞⡆
⠀⡘⢰⠀⠄⠀⢃⠀⡏⠀⣷⠈⢀⣔⠩⠄⠐⠉⠉⠁⠀⠀⡠⢖⢯⢣⠈⠨⢟⡦⠀⠀⢀⠔⡣⢄⠀⠀⠉⠀⠑⢄⠀⠳⡀⠀⣷⢠⠃⠀⠸⠊⣹⠈⣧
⠀⡇⠀⢠⠅⠢⠈⠄⡇⠀⠋⠠⠊⠀⠀⢀⣠⣤⡄⢀⡤⠊⡠⢁⠂⡌⠀⡆⠈⠈⡑⠎⠁⡐⠁⠎⠈⡋⠉⠉⢰⠺⢧⠀⠑⡀⢩⠀⡞⡀⠀⢞⣻⠀⣹
⠀⡇⢱⡏⠂⠑⡘⢰⢁⠀⢠⠀⠀⡠⢰⡿⡛⠻⠑⠏⣀⠎⢀⠆⢀⠁⠀⠀⠀⢡⠐⡈⢶⠀⠘⠀⢀⠀⢸⠀⢸⠀⢁⠂⠀⠅⢸⡇⢀⢡⠀⢈⣿⠀⣿
⠀⣿⢀⠁⡆⠀⠇⢃⠘⠀⡇⡠⠊⠄⡙⠠⠁⠀⢃⡜⠁⢀⠊⠀⠸⠀⠀⢀⡀⠀⢆⠐⡀⠳⡇⠀⠀⠀⢸⠀⢸⠀⠀⠄⡁⢰⢸⡇⢸⠀⠆⠀⠴⡃⣿
⠀⢟⠘⢧⡐⠥⠈⠸⠀⠀⣾⠴⠥⠞⠀⠓⠚⠒⠚⠓⠀⠉⠀⠀⠃⠐⠀⠈⠁⠀⠉⠀⠉⠀⠉⠀⢁⣀⣼⣶⣾⢛⡴⠟⠛⣸⢸⡇⢸⠀⢰⠀⢚⡁⣿
⠀⢸⠘⡀⠑⢄⠀⡆⠀⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢀⠀⢄⡰⣖⠨⣉⣀⡉⠜⠛⠉⠀⠀⠀⠀⡇⢸⡇⠸⠀⠘⠀⡋⣼⡇
⠀⠀⢇⠘⢄⠈⢢⠃⠀⢁⢠⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⡈⠒⠉⡀⠒⡩⠊⡄⠊⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇⠸⡃⠆⠀⠘⠀⢃⣿
⠀⠀⠘⣧⠈⢢⠀⢿⡀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣐⠊⡁⢴⠂⡁⠄⠢⢐⠔⠋⠀⢊⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠎⠀⡆⡷⠀⠀⠆⠜⣩⠇
⠀⠀⠀⠹⡗⠀⢣⠈⠌⢄⠈⢦⠑⢄⠀⠀⠐⠀⠀⠃⢘⠠⠁⢌⢠⠊⠀⡄⠀⢠⠀⠣⠌⣀⣊⡁⠐⢂⡤⢀⠀⣀⠔⡩⠀⡰⢀⠃⠀⠰⠠⢶⠏
⠀⠀⠀⠀⠹⣌⢄⢣⠈⠄⠑⢄⡑⠤⣈⠉⢁⠜⠀⠈⠒⠠⠌⡐⠠⣉⠂⠌⡁⠑⢋⠀⠲⡖⢠⠐⡏⢡⢠⠀⠏⢁⠞⠀⠠⠁⡜⠀⠠⢁⣠⠋
⠀⠀⠀⠀⠀⠘⢮⣂⠡⡈⠄⠀⠙⠰⢄⠈⠋⠒⠀⣄⣀⣀⣠⣤⠑⢄⠁⡀⠀⡆⢀⠜⢊⠐⡁⢊⢔⠕⣡⣮⠤⠁⠖⡘⠀⢠⠃⡰⢁⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠰⠈⢄⠀⢢⡀⠀⠐⠛⠂⢀⣀⣀⠀⠀⡠⢊⠔⢀⠂⠀⣡⠔⠁⡠⢂⠀⡁⠚⠀⠀⠀⢀⠔⠀⠰⣋⠌⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠈⠗⡀⢣⠈⠉⠉⠀⠁⣈⠭⠓⡁⠔⢁⢔⠅⠨⠂⡁⠀⡈⠀⡡⠊⠀⠀⠀⠀⣠⢂⠴⢊⣴⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣈⠢⣄⡀⢄⡢⢍⡠⠔⢈⠀⠀⠔⠁⠠⠔⣡⠔⢁⠔⠀⠀⠀⣄⠀⠀⠈⠁⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢌⣁⡨⠒⡠⠊⠡⠊⡰⠊⠀⠀⢚⡬⠒⠁⠀⠀⠀⠀⠉⠁⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⢠⣤⣀⡤⢂⣤⡒⣀⣀⠀⠀⣀⣤⡀⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠆
⠀⠀⠀⠀⢀⠰⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣕⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢵⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣴⠄⠀⠀⢄⣀⣀⣀⣀⣀⣀⣀⠌⠁⠀⠀⣈⠣⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡇⢀⡄⠐⡆⡿⠿⠿⠿⠟⠻⠿⡄⠀⠀⢈⠞⣰⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣷⡛⠉⠉⠑⠤⡁⠨⠜⠀⠀⠀⠀⠀⠀⠀⠈⢐⡀⠀⠊⠀⠀⠉⠑⢽⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣙⠧⢄⣀⠀⠉⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⢠⠤⢒⣡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠉⠒⠀⠈⠉⠁⢀⣒⣒⣒⣒⣒⣒⡒⠀⠀⠉⠉⠉⠀⠀⠂⠉⠈⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⣠⠴⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡀⠉⠉⠒⠒⠦⢤⠤⠤⠠⠀⣀⡀⠠⠥⠤⠤⠤⠒⠚⡉⠀⡘⢥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠁⢣⠀⠀⠀⠀⢀⡎⠆⠀⠀⠀⠀⠀⠀⠀⡚⡀⢕⠁⡨⠐⠀⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢄⠐⠀⠀⠀⢀⠊⠠⠈⡄⠀⠀⠀⠀⠀⠌⠠⠲⡀⢊⢔⠁⠘⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠐⠄⠄⠀⠀⢎⠴⠀⠂⠠⠀⠀⠀⠀⠌⠰⢄⠀⢡⠀⡴⠀⠧⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢐⠄⢰⠀⠌⣐⠄⡀⠀⡀⠄⠀⠀⠀⢀⠢⢑⠢⡀⠵⡀⢀⠡⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠷⣤⣈⠖⠠⠐⠠⠀⠄⡠⠘⡄⢠⠀⢁⠪⠐⠡⣐⠄⢣⡌⡈⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡈⠙⠒⠻⠶⣤⣆⣥⣠⣁⡰⠆⠈⡀⠨⣒⣀⣠⣷⢿⣶⢟⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⢀⡀⠀⠀⠈⠉⠉⠋⠛⠛⠛⠉⠉⠉⠥⠄⣼⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠀⠠⠄⠀⠀⠀⠐⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠁⠐⠁⠀⠀⢐⠠⡂⠔⡠⢒⠠⢂⠄⠀⠔⠀⠄⠠⠂⠀⠂⠀⠀⢂⠔⡠⢂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠊⠔⠀⠐⡡⢀⠡⢊⠄⠈⠀⠠⠀⠡⠂⠄⠠⠈⠐⠊⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣄⠀⣠⠔⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⣒⣒⡿⠿⢟⣻⠿⠗⣚⣩⡭⢶⣖⡫⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣫⣵⡶⠿⠛⠛⢻⡀⠀⣸⠗⠛⠙⠛⠛⢻⡷⢋⣴⣞⠷⣄
⠀⠀⠀⠀⠀⠀⢀⡞⢱⡿⠋⠁⠀⠀⠀⠀⠀⠙⠛⠉⠀⢀⡴⠒⠲⣏⡴⠋⠈⠙⣷⠘⣇
⠀⠀⠀⠀⠀⠀⢀⣧⡈⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⡼⠀⠀⠀⣠⠞⣠⣿
⠀⠀⠀⠀⠀⠀⠈⣇⠙⠦⣌⣙⠒⠦⠤⢤⣀⣀⣀⣀⣀⣀⣈⣻⠯⠤⠖⢒⣉⡤⠞⢁⡟
⠀⠀⠀⠀⠀⠀⠀⢾⣳⣤⣀⠈⠉⠓⠒⠶⠦⠤⠤⠤⠤⠤⠤⠶⠖⠒⠛⠉⢁⣠⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠉⢻⣲⠶⠤⢤⣤⣄⣀⣀⣀⣀⣤⣤⡤⢴⡖⣾⠉⠁⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢀⡼⠹⡄⠀⠀⠀⠀⢘⣦⡇⠀⠀⠀⠀⢀⡟⢇⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣧⣷⡞⠀⠀⠹⡄⠀⠀⣠⠞⠀⠹⣄⠀⠀⢀⡞⠀⠈⢦⡞⢸⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣜⢧⣇⠀⠀⠀⢱⣀⡴⠁⠀⠀⠀⠈⢦⣠⠞⠀⠀⠀⣞⣹⢋⣼⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡌⠳⢮⣙⡲⠦⣼⣙⣆⣀⡀⠀⠀⣀⣸⣉⣧⡤⠶⣚⣩⠿⠋⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡈⠉⠛⠲⠿⠬⠭⢭⣭⡭⠭⠭⠵⠖⠚⠋⠉⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⢤⣤⣄⣀⣀⣀⣠⣤⣤⠤⠴⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠳⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⣦⣄⣈⠙⠛⠶⣤⣄⡀⠀⣀⣠⡴⠶⠛⢉⣁⣤⡴⠞⠁
⠀⠀⢀⣀⣤⡴⠶⠖⠛⠛⠛⣉⣉⣉⠛⠳⣦⣤⣈⠙⠛⠿⢤⣤⣶⣾⣯⣭⣍⠛⠛⠓⠶⠶⢤⣄⣀
⣠⠞⣋⣥⡴⠶⠚⠛⠋⢁⡾⠋⠉⠉⠻⣟⢉⣠⣭⠿⠳⠦⣤⣈⣿⠋⠀⠀⠈⢷⡀⠙⠛⠲⠶⣤⣍⡛⢦
⣿⣄⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡿⠋⠁⠀⠀⠀⠀⠀⠉⢿⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⢁⣼
⣿⠉⠛⠶⣤⣤⣀⣀⠀⠀⠛⠦⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠞⠁⠀⢀⣀⣠⣤⡴⠞⠋⢹
⣿⢦⣄⡀⠀⠀⠉⠉⠉⠛⠛⠛⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠚⠛⠛⠋⠉⠉⠁⠀⠀⣀⣤⢾
⣿⠀⢈⣽⣿⠳⠶⠦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠶⠶⣿⣿⣍⠀⢸
⣿⠀⢸⣇⣸⡇⠀⣀⡀⠀⠀⠈⣩⣭⣍⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣭⣍⡉⠀⠀⠀⣀⠀⠀⣿⣀⣿⠀⢸
⣿⠀⠀⠉⠉⠀⣰⠟⢷⡀⠀⠰⣏⠀⣹⠇⠀⠀⣠⠟⢧⡀⠀⠀⣿⡀⢈⡗⠀⠀⣰⠟⢷⡀⠈⠉⠁⠀⢸
⣿⣤⠀⠀⠀⣼⠃⠀⠈⢷⡀⠀⠉⠛⠉⠀⢀⡼⠋⠀⠈⢻⣄⠀⠈⠙⠋⠀⠀⣰⠏⠀⠈⢳⡄⠀⠀⢰⢾
⣿⠸⡇⢀⡾⠃⣴⠛⣶⠈⢷⡀⠀⠀⠀⣠⠟⠁⣠⡴⣤⡀⠙⣦⡀⠀⠀⠀⣰⠏⣰⠟⢷⡀⢻⡄⠀⡿⢸
⣿⠳⢦⣾⣁⠀⠻⣤⠟⠀⠈⢷⠀⢀⣼⠋⠀⠀⢿⣀⣸⡇⠀⠈⢷⡄⠀⢰⠏⠀⠹⣦⠾⠁⢀⣀⣤⠶⢻
⢿⣀⠀⠈⠉⠛⠛⠲⠶⠶⢤⣤⣤⣤⣥⣤⣄⣀⣀⣉⣁⣀⣀⣤⣤⣤⣤⣤⡤⠴⠶⠶⠚⠛⠋⠉⠀⢀⣸
⠀⠙⠳⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠾⠛⠛⠛⠛⠛⠟⠻⠿⢿⣛⢓⠶⡦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⣤⣉⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣚⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡕⢀⠹⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡲⡉⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠣⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡔⡀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡐⡀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢺⠓⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⡄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣘⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡍⠘⣿
⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠓⢲⣿
⠀⠀⣾⣿⣿⣿⣿⣶⣄⠀⠀⣻⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠘⠀⣿⠃
⠀⠸⣿⣿⣿⣿⣿⣿⣿⡗⢠⣾⣿⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠒⣼⡟
⠀⠀⠈⠻⢿⣿⣿⣿⣿⣶⠞⠋⣡⣳⡜⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠡⢁⣼⠟
⡠⠒⢦⠀⠀⠈⠛⠻⢷⡴⣖⣁⢨⡇⠹⣎⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⢠⣾⠏
⢧⣂⡰⠀⠀⠀⠀⠀⠀⣀⣪⣷⡞⢦⣼⣉⠷⣵⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠒⣤⠟⠁
⠀⠀⠙⢦⣀⡀⠀⠀⢰⣿⡿⠟⢻⡆⡅⣹⣦⡙⣷⣵⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⣩⡴⠛⠁
⠀⠀⠀⠀⣏⣳⡠⠄⠠⠈⢀⡐⣹⣿⣉⣩⠄⠻⣆⠠⠙⣮⣵⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡯⠖⠋⠁
⠀⠀⠀⠀⠀⠉⠚⠳⢴⡖⠪⠽⡿⢿⠼⠒⠃⠀⣏⡳⠇⠀⠀⠉⠛⠛⢷⣶⠶⠤⣤⣤⣤⣤⣤⠶⠖⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠠⢤⣼⡶⣿⡶⠛⠋⠁⠲⠤⡀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⡆⠀⠉⠒⠦⣀⠀⠈⣳⣄⡀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠎⠀⠀⠈⠑⠶⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⡌⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⢀⡀⢀⣀⠀⠀⠠⠤⠤⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⣿⣿⣶⣶⣶⠄⠉⠠⠤⠂⠉⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣾⠛⡷⢦⣄⢀⣠⡔⠳⠟⢿⡀⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⣀⣠⣀⠀⠀⠀⠀⠀⢀⢾⡟⠾⣠⠟⠀⣬⣿⣿⣿⣽⠐⣆⢈⡅⣼⡏⣹⣶⣄⡀⢀
⠀⠀⠀⠀⢰⣤⣶⣹⡟⢸⣿⣿⣿⣿⣿⣗⣠⣤⡂⠀⣯⠜⠒⠂⠿⠆⣤⣼⣿⣿⣿⣷⣿⣿⣦⣾⣿⣿⣯⣽⣿⣿⡿⢀⡀⡀
⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⣏⠹⣧⡈⠰⣇⣬⣤⡁⣭⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣻⠿⠟⠻⢿⣯⠁
⠀⠀⠸⠙⠿⣿⣿⣿⡟⣟⡙⣿⢻⢑⢿⣨⣷⡮⢀⣆⣾⣿⣿⣷⣮⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣷⡿⣟⣶⣧⠄⠈⠉⣁⣀
⠀⠀⢠⣤⣦⣼⣯⣽⣅⢻⣿⣤⣿⡛⢿⡛⣏⢁⣼⠻⢉⡿⢻⣿⣼⠋⣿⣿⣿⣯⣿⣿⣿⣡⡟⣿⣿⣿⣿⣿⣿⡇⠀⢠⡿⣿⡿
⠀⠈⣼⣿⢯⡝⢷⣿⠟⠛⣿⣿⠿⡿⣿⣷⠘⡓⡤⣵⠀⠂⡿⠏⢻⣠⠻⣿⣿⣿⣿⣿⡿⢸⣷⣿⣿⣿⡿⣿⢇⢴⣿⣿⣷⠛⠻
⠀⠈⠉⠛⡘⠃⠀⠁⠀⠀⠀⢻⢶⡄⡽⠉⢰⠟⠁⠂⠈⠋⠘⢠⢛⡋⠓⠙⣿⣿⣿⣿⢳⣾⡿⢿⣿⠈⢳⡤⠂⣡⠞⡟⠉⣻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢷⠇⣠⡏⠀⢦⠁⠀⠀⠀⠘⠈⠁⢢⠀⡼⣿⣿⡇⠀⣿⡿⠀⠗⣠⡾⢁⣼⣿⠖⠚⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢷⣿⡆⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⢻⣿⣧⠀⣿⣷⣤⣤⣿⣿⣿⣿⣿⣶⣻⣿⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡁⠀⢹⡇⠀⠀⠀⣀⣠⣶⡆⠀⠀⠀⠀⢀⠾⢧⠞⣤⠫⣷⣶⣿⣿⣿⣿⣿⣿⢏⢁⣞⣛⣋
⠀⠀⢀⡀⠀⠀⡿⠓⠚⢣⣄⠀⣿⣇⠀⠴⠀⠀⢀⣰⣿⣿⣿⣷⣾⣿⣶⣷⣾⠓⡤⣼⣷⣶⣦⣍⣠⣿⣿⣿⣿⣿⢛⣉⣉⡉⠉
⠀⡰⠦⠉⠛⠛⠁⠀⢲⣿⠶⢷⡇⠼⠀⠀⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢙⡹⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡯⠉⠉⠁⠀⢀
⠚⠟⠒⠂⠀⠀⠀⠀⠈⠉⠀⠈⣷⣶⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢀⣾⣧⠠⢿⣿⣿⣷⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⣿⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣜⣾⣷⠜⢻⣿⣿⡇⠈⣿⣿⣿⣯⣿⣿⣿⡧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣛⣦⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣀⣮⣿⣟⣿⠀⠀⣿⣿⡇⠀⣟⣻⣭⣿⠿⣿⢿⡯⢤⠤
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣟⣻⡿⠀⠀⢘⣿⣽⣿⣿⣿⣿⣿⡗⠈⣿⢿⣿⣿⣿⠀⠀⣿⣿⡀⠀⢿⣿⣟⣛⣛⣛⣚⣃
⠀⠀⠠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⢀⢻⣾⣿⣿⣿⠀⠀⣿⡿⡧⠀⠻⠿⠿⠟⠛⠛⠛⠋⠁
⠀⠀⠀⠈⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⠋⣽⠛⡿⠃⢀⣬⠛⡟⠉⠉⠏⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠛⠛⡇⠀⠀⠀⡿⠃⢸⡿⣶⠷⠞⣉⣽⠿⠀⠀⠀⠀⠀⠀⢻⠀⢠⠄⠀⡄
⡴⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⡇⠀⠀⢰⠁⢀⣾⠟⢋⠀⢼⣟⣿⠁⠀⠀⠀⡇⠀⠀⠸⠒⠀⠀⠀⣠⠀⠀⠀⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⣧⠀⠀⡨⠗⣸⣿⠟⠻⡏⠋⣈⡿⢀⣠⣾⣾⡇⠀⠀⠀⣄⠀⠀⠀⠿⠾⠦⣴⠦⠄⠠
⠀⠀⠉⠉⠀⣀⡀⠀⠀⠘⠀⠀⠀⡖⠀⣆⠵⣾⡛⣿⠀⠚⣿⣥⣼⣵⣿⡿⠿⢿⡇⠀⠀⠀⣿⠿⠷⣦⠄⠀⠀⠀⢀⣤⣶⠆
⠀⢀⣤⣤⣿⣿⣿⣦⣴⣦⣤⣀⣠⠷⠀⢱⣄⣿⡇⢹⠀⢰⠙⠿⣿⡿⢯⠣⠔⢻⠀⠀⠀⠀⢹⠐⠰⡟⠲⡀⢤⡖⠈⠁⢠⠀⢀
⠀⡀⠛⠛⠟⠛⠻⠛⠉⠛⠋⠙⣯⠀⠀⠘⢿⣿⡇⢸⠀⡨⣷⢤⣼⣧⠀⠐⣧⠟⠀⠀⠀⠀⠈⠇⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⣄⢀⡀⢠⣾⣿⡀⠆⠀⡇⠋⠀⢻⡃⠀⠁
⠀⠀⠀⡈⠀⠀⠐⠀⠀⠀⠀⠀⡏⢀⡄⡌⣿⣦⢃⠀⠀⠉⡀⣠⠾⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⣿⢁⠘⣇⣀⠓⠁⠘⣷⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⡾⠉⠙⠤⠜⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠚⣶⣣⡜⠈⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⢻⣸⡇⢲⠟⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⣿⢧⡼⠧⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢒⣋⣉⣉⣉⣻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣿⣖⠋⠉⠉⠀⠀⠀⠁⣽⡟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠍⢊⠇⠘⠶⠆⠀⠀⠀⠀⠀⠀⠈⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣄⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢷⡘⣦⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣯⠿⣿⡾⡗⠐⣤⡆⠀⢀⡀⣤⣤⣶⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⠚⢻⣾⡷⣿⣾⣿⣿⣿⣪⡿⠛⠋⠉⠉⠉⠉⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣼⢻⡟⢿⣻⣯⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠠⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⠷⠶⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣻⣿⠏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⠾⢷⠄⠆⠀⠀⢀⠀⠀⠉⠉⠛⠒⠒⠶⠶⢤⣤⣤⣦⣼⣦⣤⣿⣧⣁⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢠⠋⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠲⣶⣾⣿⠿⠿⠛⠋⠉⠉⠙⣛⣮⡙⢦⣀⣠⡤⢶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⢠⡽⢦⡙⢶⡊⠉
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⡹⡄⢳
⠀⠀⠀⠀⠀⠀⠀⡇⣰⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⡌⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣬⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢡
⠀⠀⠀⠀⠀⠀⠀⣇⣟⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠖⣾⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡷⣷⢉⠐⠦⣴⢠⣀⣀⣀⠀⠀⠈⠙⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣷⣵⣟⠗⣶⣿⢾⣿⣿⢈⡖⡀⢠⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣗⣦⣌⡃⠈⡟⢊⣷⣿⣮⣷⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡼⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠮⠷⠀⢉⡾⠠⣬⡻⢿⣿⣿⣆⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⡼⠁⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠙⠓⢿⣿⣿⣗⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣣⠟⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣶⣤⣤⣀⣠⣤⡴⣟⡭⠞⠁⠀⠀⠀⠀⠀⠀⢳⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣄⡜