Carrot Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠀⠀⠀⢦⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⡀⠠⢀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⡦⠀⠀⢄⡤⠀⣈⡂⠀⠀⣀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠛⠀⣀⠗⢊⡤⠀⢀⢌⡆⠈⢹⣦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠤⠚⠃⠀⣠⠞⢁⡴⠟⠀⢠⣷⠂⣧⡀⠀⠈⠙⢦⠈⢧⡀
⠀⠀⠀⠀⠀⣠⠶⠡⠘⢁⣠⡶⠿⡿⠋⣠⡏⠀⠀⣠⣿⠟⠀⠈⠛⠷⣤⣀⡈⠑⠢⡙⢆
⠀⠀⠀⠀⠞⠑⣁⡰⠾⠋⠀⣠⠊⡱⠋⢀⣤⣶⠾⠋⠁⠀⠀⠀⠀⠀⠉⠛⠻⣦⣄⠘⠠⠽⢦⡀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⢠⠞⠑⠂⣠⣶⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠷⣦⡀⠂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢊⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠒⢢⢦⣤⠠⡖⠶⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡼⢏⠤⣀⣵⠀⠈⠓⢃⠄⢤⠄⠛⢚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠀⢚⠄⠀⠀⠁⠐⠀⠀⠀⠀⢀⠀⠀⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣇⣌⡒⡆⣀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠉⡩⠟⠻⢾⠽⣥⡢⠖⣀⡀⠀⠒⠆⠀⠀⣷⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⢀⠤⠊⠁⠀⠀⠀⠑⣽⠊⠉⠙⢯⣄⠰⡶⣤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉
⠀⠀⣰⣋⣠⣤⠤⠒⠒⣩⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠉⠁⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⢀⣠⠶⠊⠁
⠀⠀⠀⠀⢀⡞⣀⣀⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠖⠒⠀⠉⠉⠉⠁⢐⠚⠓⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⠀⠉⠉⠀⠀⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠁
⠀⠀⣀⠤⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡠⠊⠀⢀⡀⠀⠀⠀⠁⠀⠐⢄
⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠐⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠐
⠈⠢⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠉⠒⠤⢤⣥⡀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠳⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠢⣈⠀⠀⠀⠀⠀⠀⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠤⣄⣀⣀⣠⣤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠈⠑⠦⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣵⡀⣾⣿⣯⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣣⡄⣻⣷⡀⠀⠀⢰⣆⣉⣻⣇⠘⢋⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣤⠟⠃⠉⡟⠁⣴⡀⢰⡈⢻⡿⣿⣴⣿⣿⣷⣤⠦
⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⡶⢶⡄⣠⣾⣿⣟⠢⣁⣴⡏⠀⠘⠛⣿⣁⣸⣿⠋⣀⠙⠙⢿⣾⣷⠆
⠀⠀⠀⠀⠀⢀⣴⣾⠧⠛⣃⣾⢿⡟⠻⣾⣿⠷⡈⢻⡄⠀⢘⣷⣺⣿⣿⣳⣿⣿⣿⣦⣈⡉⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠙⠀⣀⣾⠿⣷⠿⢰⡆⢸⠯⡀⢡⡈⢻⣎⣽⣿⣿⢿⣼⣿⣿⣿⣿⣿⣿⡟⣿⣯⡄
⠀⠀⠀⠀⠀⠀⢀⣸⣿⠋⡠⢹⣷⣿⣿⣾⣄⠱⣎⣷⠀⣿⠀⠟⣱⣎⣩⣿⣯⣿⣟⢻⡟⡛⠾⢻⣷
⠀⠀⠀⠀⢀⣾⣿⣿⡏⣿⣇⠈⢹⣿⣿⢻⡷⣄⠘⣧⢇⠸⣧⣿⣿⣿⢻⣿⣯⣅⢹⣿⣿⣿⣶⣄⡙
⠀⠀⠀⠀⠈⢉⣿⠉⠉⢹⣿⣿⣯⡏⢁⡐⢀⣿⣶⣿⣿⡆⣿⣿⣿⣿⡿⣿⣿⣿⣸⣾⣷⣼⣿⣿⣏⡀
⠀⠀⠀⠀⠀⢿⡿⠀⠀⠸⠏⣼⠿⢠⣼⣧⠀⣰⣿⣄⢹⣿⣟⣿⢹⣿⢣⣿⣿⣿⣿⣺⣿⠿⣻⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣾⣿⡏⠀⢿⡗⠀⡌⢿⣿⣿⢻⣯⣾⢉⣿⣿⣿⣿⡇⢰⢿⣧⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠟⣿⢹⡆⠀⠀⠀⢹⣸⣿⣿⣠⣿⡇⢸⣿⡧⣿⣿⣗⣯⠀⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⣿⠇⠀⠀⠀⢿⣿⣿⣿⣿⠀⠠⡿⠀⣿⠉⠹⠃⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠉⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⣉⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣨⣥⡀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠀⡀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡂⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⡆⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠠⠤⣀⠀⢰⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢦⠀⣴⠀⠸⠆⠞⠀⠀⠀⠀⠀⠀⠀⣴⣖⡀⢀⡀⡀⠷⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠀⠳⠒⢋⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⡀⠀⠉⠂⣆⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠛⠉⢁⣤⠊⠘⠋⠀⠀⠀⠀⠀⠀⠀⠛⠘⠃⠉⠠⠜⠑⠞⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣾⣿⣿⡁⠰⢈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣷⣿⣿⣿⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⢀⣼⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⠘⠁⠀⠀⠀⠀⠀⠀⢸⠉⣀⡆⡖⢲⢲⣲⢴
⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢐⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠄⠐⠠⣄⢁⢘⠃⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⣷⠿⠉⠁⠀⠀⠀⣀⡌⠠⣉⢡⡒⢛⡆
⠀⠀⠠⢐⡂⠀⡀⢃⡎⠰⠇⠀⠀⠀⠰⠂⠀⠀⠀⠀⣀⣀⣀⣤⣾⡏⠁⠀⠀⠀⠀⠀⠊⠙⠃⠡⠤⠥⠘⠁
⠀⠀⠀⠈⢀⣈⢧⡄⡀⠂⠀⠀⠀⠀⠶⠶⠖⠚⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣰⠆
⠀⠀⠀⠁⠂⠅⡉⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⣤⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⣠⣿⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠀⠀⣀⣴⣶⡿⠛⠓⢦⡦⠀⠀⠀⠀⢀⣠⠄⠀⠀⠴⠞⠉⠃
⠀⠀⠀⠀⢐⠊⠉⢷⡄⠀⢀⣀⣴⣾⠟⠉⢀⣠⣴⣶⣿⡟⠋⠁⠀⠀⠀⠠⠃⠀⠀⠀⠀⠈⠑⠀⠁⠂⠀⠀⠀⢀⡄
⠀⠀⠀⠀⡅⠀⠀⢸⡇⠀⠀⠀⢀⡠⠴⠚⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠃⠀⠀⣼⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠘⡄⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠂⠉⠀⠀⠀⣹⣆⠴⠫⠀⠀⠀⠀⠘⠛⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣀⣀⣀⣀⣠⣠⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣤⣤⣤⡄⠂
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠄⠀⠀⠀⠀⠀⣠⠄⢠⡆⢀⡤⠄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠒⣯⣫⡴⠽⠷⠳⠆⠀⠀⢠⣤⡴⠿⠶⠛⠖⠚⠋⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢢⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⣀⡠⢐⡲⠿⣇⠀⠀⢱⡀⠘⡦⣤⠄⠒⠀⠐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⣠⢴⡏⠀⠀⢀⡎⣰⠉⠈⢁⣹⢦⡀⠀⠁⠀⠟⠀⡠⠴⠛⠂⠀⣀⣴⠷⢦⣤⣀⡔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢶⠏⢩⠀⠽⢿⡓⡠⠼⢧⣀⡇⢠⠊⠀⡤⢹⣶⢤⠀⠀⠀⠀⢀⣠⣴⠞⠁⠉⠳⡔⠂⠹⢁⢂⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢀⡈⠀⠈⠀⡈⠸⢯⠀⠀⠀⠙⢢⡁⣠⠋⡠⠊⣌⠜⡑⢛⢟⠟⢻⡶⠋⠀⠀⠐⡄⠇⠈⠃⡞⠘⠃⣻⡳⣄⡠⠀⠄
⠀⠀⠀⠀⠀⢸⠉⠀⠀⠈⠳⣄⠑⣄⠀⢀⣴⣧⡀⠑⣤⠶⠒⢛⡿⣯⣩⡤⠊⡖⠚⠛⠛⢆⡠⠊⠀⠀⠀⢰⠁⠳⠤⢤⡴⠃⠰⣋⠆⠁⡸⠁⢠⠆
⠀⠀⠀⠀⠀⠘⣆⠀⠀⣄⠠⣎⠀⢈⢞⠏⠁⠀⢹⣦⡇⣀⣄⡛⠀⠴⢀⠘⠐⠉⢢⣄⣤⠞⠵⣂⣀⠁⠄⢉⠃⡀⠀⠀⠱⠄⢲⠟⠀⢈⠅⠴⢇
⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠘⠚⢽⡏⡜⠠⢠⡀⠈⣷⡾⠀⠠⠂⠠⠃⣸⡲⡀⠼⡱⠋⠀⠀⠀⠈⢻⣠⠞⠁⠀⠀⣀⡀⠀⣧⢀⡠⠖⠂⢀⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣷⡦⢤⣼⢱⡇⠀⢀⡀⡆⢻⡁⠀⡀⠀⢠⠀⠀⢰⣷⡠⣷⣟⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⣌⠀⠀⣿⠋⠁⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠻⢶⣼⡄⢷⠰⡀⣀⠀⢌⢿⠋⠇⣀⣀⣠⠞⠃⠭⣉⠉⠛⣾⡁⠀⡎⠁⡅⢀⡀⠰⠄⢀⠀⢀⡟⠛⣩⡿⡋⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠉⠛⠾⣧⣱⣀⠠⠀⢻⢻⠉⠁⠑⣾⠀⠀⠀⠈⡗⠉⢩⣷⣦⣙⠦⢤⣤⣤⠤⠔⣉⣤⠿⠖⡋⠕⡊⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠉⠙⠒⠳⠶⠧⣤⣤⣿⣄⣀⣀⣸⣃⣠⣏⣈⣧⡭⠿⠶⠶⣒⣚⡭⠥⢒⡪⠅⠒⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣒⣒⣛⡭⠭⠕⠒⠺⢍⡓⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣑⢄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⡛⠷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡓⠶⢬⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢒⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢌⣉⡙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⣉⣉⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⡾⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⠈⠚⠉⠉⡷
⠀⠀⠀⠀⡼⠈⠓⢲⠤⠖⢚⣄⣥⣀⣱⣴⣶⠒⡷⠲⡔⠒⠈⢉⡀⠀⡷⠦⠒⠶⠚⢧⣶⡦⣤⠾⡄
⠀⠀⠀⠀⣣⠀⠀⠘⡿⢸⠥⠒⠤⠖⠓⡏⢈⠛⣄⢃⠷⠔⢊⣩⡿⢋⢧⡲⢤⣄⡮⢶⣎⡃⣠⠀⡇
⠀⢀⡠⠼⢇⠁⠌⠄⠉⡦⠒⠒⣒⡩⡖⢻⡌⠀⢘⠿⢲⠶⠟⢭⡜⠁⠄⠀⠠⢾⣃⠀⠀⡟⢀⡱⠓⣄
⠀⠸⣄⣈⠉⠓⠒⠒⠠⠿⠭⠥⠀⠹⠷⠾⠭⠤⠮⡴⠓⣈⡀⠠⠜⠈⠀⠉⠉⠉⣉⣉⣉⣤⠤⢀⡴⠃
⠀⠀⠀⠈⢣⡀⠈⠉⠉⠐⠒⠒⠒⠒⠒⠲⠶⠂⢰⡡⠊⠁⠒⠒⠒⠒⠚⠋⠉⠉⠉⠀⢀⠀⢠⠏
⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠲⠦⠤⠤⠤⣤⣀⣀⣀⣀⣀⣠⡤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴⠴⣾⣲⠴⡲⣆
⠀⠀⠀⠀⠀⠀⢠⠚⠳⠵⠲⣆⠀⠈⣥⣼⢏⣿⡇⢉⠌⠉⢡⣿⠐⠦⠀⠀⠀⣠⣤⣴⡤⠿⠿⢆⢀⡀
⠀⠀⠀⢀⡴⠚⠛⣤⣄⣠⡇⠉⢻⣿⡧⢠⡈⢸⣿⣿⣠⠄⠈⢉⣿⣿⣷⣶⡟⠁⠈⠋⡀⢰⣠⡾⢛⠟⣳⠄
⠀⠀⠀⢘⣧⠀⠰⣾⣿⠟⠿⠲⣿⣇⣐⢦⡹⣼⣿⣿⣵⡶⠂⠈⢿⡟⠁⠄⢀⠀⣳⢴⣥⡘⠛⠛⢃⣀⣻
⠀⢀⣴⣿⡏⠀⣶⡏⠉⣀⣦⡆⢸⢿⣿⣤⣿⢟⡿⠙⣿⣿⢁⣰⣿⣁⣦⡼⢿⣾⣅⣾⣿⣿⣷⠦⣼⣿⣯⡄
⢠⡟⣿⣿⣷⡶⠿⢷⣏⣿⣼⡇⣀⣘⣿⠋⡴⠋⣤⢯⣿⣿⡿⢟⣿⣟⣩⠤⣤⡀⠜⠻⠿⢍⣻⣿⣿⣿⣿⡿⣦
⣾⣿⣿⣧⠀⠠⣀⡆⠈⣿⡿⢿⣿⣿⣡⠞⡁⡾⣫⣾⣿⣫⠖⣛⣉⠉⠻⡛⠿⠹⣿⢿⣿⣿⣇⣈⠁⢈⣿⡇⢹
⣿⢿⣿⣿⣤⣍⡛⠳⣴⡇⠀⡄⣿⣿⣧⣬⣼⣾⢿⣿⣿⣶⣾⣿⡿⢷⣦⣶⣾⣶⠈⠣⠈⠛⠿⠏⣠⣾⣿⣇⣼
⣿⠈⠻⢿⣿⣿⠡⠤⢼⣧⣼⠀⣸⡿⠙⠿⣿⣶⡾⣿⣄⣰⣋⣀⣤⣾⣯⣽⡿⣿⣿⣶⣾⡟⠻⢿⣿⣿⣿⢟⣹
⢸⡈⣷⣤⡉⠻⢷⣦⣄⣹⣿⣿⠿⠞⠲⣾⠛⠻⣖⣻⡿⢷⣦⣇⠀⢣⠉⢿⣧⡘⢿⣻⣿⣿⣦⡽⢟⣫⣴⣿⡿
⠀⢧⠹⣿⣿⣷⡦⠤⠉⠛⠛⠷⠶⣶⣦⣤⣦⣴⣈⣁⣴⣀⣉⣿⣷⣮⣧⣼⣿⠿⢾⣛⣛⣭⣵⣾⣿⣿⡿⣽⠃
⠀⠈⢧⠹⣿⣿⣿⣯⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⠃
⠀⠀⠈⢳⡘⢿⣿⣿⣽⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣟⡽⠃
⠀⠀⠀⠀⠙⢦⡙⢿⣿⣷⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣛⣿⣿⣿⣿⣿⣿⡿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠦⣝⠻⣿⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣽⣶⣿⣿⣿⣿⠿⣫⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣌⣛⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢭⣿⣿⡿⠿⣛⣭⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣭⣭⣵⣶⣾⣿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡄⠀⠀⠀⠀⠀⢀⣴⡶⢶⣄⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠫⠀⠌⠛⠀⠀⠀⠀⠀⣞⠡⠀⠄⢻⡆⠀⠀⠀⢰⡟⠩⠉⠙⢻⠀⠀⠀⠀⠀⠀⠟⡛⠉⠿
⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣴⡄⠀⠀⠀⠀⠀⠘⠂⠀⠀⠋⠀⠀⠀⠀⠀⢛⣦⣠⣴⡋⠀⠀⠀⠀⠀⠀⢠⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣹⡏⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠞⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠠⣤⠀⠀⠀⠀⠀⠀⠈⢋⡥⠀⠀⠀⠀⠀⠀⠀⢠⡄⠿⠀⠀⠀⠀⠀⠀⠋⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⢷⡄⠀⠀⠀⠀⠀⣴⣿⠷⣶⡀⠀⠀⠀⠀⠀⣾⣿⡿⣿⡄⠀⠀⠀⠀⢠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⠀⡍⠀⠐⢈⡅⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⣟⠡⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⣶⣴⣦⠀⠀⠀⠀⠀⠙⠂⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠙⠿⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠈⠁⠀⠀⠀⠀⠀⠠⠖⠰⠆⠀⠀⠀⠀⠀⠀⠈⠃⠶⠀⠀⠀⠀⠀⠲⠖⠁⠀⠀⠀⠺⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠦⣄⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⣇⡀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⢴⠒⢣⡦⠔⢘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠖⠀⠀⠐⠠⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠙⠀⢀⠁⠠⡾⢸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⢻⠊⠀⣠⠀⡠⢰⠀⠈⠀⠈⢲⡀⠀⡠⠀⠀⢉⣽⣷⣶⣴⢂⡴⠒⠲⡄
⠀⠀⠀⠀⢄⠀⢄⠀⢀⡀⡀⡂⠀⠀⠀⠀⠲⠀⡐⠁⠀⠀⠁⠀⡀⡀⠄⠀⠀⠀⠀⠀⣠⣿⣮⡵⢭⣿⡌⢠⠄⣀⡟⠀⢠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠶⠯⠅⠀⠾⠃⠈⣆⡀⠀⠀⠀⠀⠀⠈⠣⡤⠂⠸⡏⠉⠇⠀⠀⠀⠀⠀⠀⠉⠷⣟⣧⣾⡼⡗⠡⠂⠈⢳⣤⣤⢤⡄⠀⠀⠀⠸⢀⣴⣃⡀
⠀⠀⠀⠀⠀⡶⢀⡀⠶⠁⠀⠀⠁⠀⠀⠀⠂⠀⠀⢠⡽⠷⡆⠀⠁⠀⠀⠀⡰⠂⡤⠀⠀⢀⠀⡿⠤⠀⠑⠨⠀⢀⣋⠈⠂⢀⣾⣟⡭⢿⣯⣀
⠀⠀⠀⠀⡀⠀⠀⠀⢉⠵⠀⣠⣶⣾⣷⣤⡀⠀⠀⠀⠁⠐⠀⠀⣈⠂⠀⠀⡁⠄⠁⠄⠁⡄⡞⠀⠀⠀⠀⠀⠈⠁⢀⣤⠤⠾⢏⠣⣀⡔⣻⣷
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣱⣿⡟⢘⢿⢽⡀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣠⣈⡀⢰⠃⢈⠸⠀⠀⠀⠄⠠⠊⠐⠐⠈⡓⢦⠘⣌⠮⠤⣴⣡⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣸⣿⢯⡉⣁⠞⡽⠉⠲⣄⠀⠀⠀⢀⣦⠈⠋⠉⠉⠑⠠⠍⠈⢠⡇⠐⠠⡺⠁⠔⠑⠶⠖⠾⢠⠀⠀⠉⠋⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠉⠓⠋⠀⠰⠞⠛⡟⠷⡒⣁⣀⣀⠰⠀⠀⠀⠊⠀⠀⡆⢁⠀⢀⡞⠒⠠⠌⠫⠄⠀⠘⠈
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠈⠉⠀⠁
⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁
⣀⣀⣀⣀⣘⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣀⡈⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡛⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢟⠀⢘⠧⠤⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⡇⢹⠀⢈⠑⠒⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠠⢰⠈⠀⠀⠀⠀⠀⠀⠄⠀⢲⡧⠬⠈⠁⠀⢰⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠀⠀⡈⡀⠐⢉⠧⠰⠀⠀⠀⠈⠧⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⢠⠁⠀⡠⢃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⣀⡀⠠⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡰⡀⢠⠀⡀⣘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢈⠿⣁⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⠀⠀⠀⣰⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠜⠒⠒⠴⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡀⠀⠀⠀⢠⠟⣽⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠐⠀⠀⢧⠂⠀⣿⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠓⠀⠀⠀⠐⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡁⠀⠐⠉⠓⠉⠈⡈⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⢀⠀⢸⡥⠤⠥⡴⣀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠁⠃⠀⠀⠀⠀⠀⠸⣇⠀⠆⠀⠀⠆⡉⠉⠉⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢸⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠘⢩⠁⠀⠀⠀⠀⠀⢸⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⢈⣷⠋⠁⠀⠀⠀⠑⠄⠀⠀⠀⡍⢶⡇
⠀⠀⠀⠀⠀⠔⠋⢡⠀⠀⡼⢹⠑⠠⠆⡱⠀⣠⣛⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣆⢀⡜⠛⢀⡀⠀⠀⠀⠀⠈⠢⠀⠀⡁⠘⠙⣄
⠀⠀⢀⠔⠁⠀⠁⠁⠀⠀⠈⠈⠀⠈⠐⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠁⠀⠉
⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⠀⠀⠀⠺⡀⢀⣷⠄⢠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠉⠙⠻⢦⡄⠀⢿⢋⢿⠂⠀⡇⠀⠰⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠉⠻⣤⡀⠀⢠⢹⣤⠇⣼⡋⠀⠠⠀⢈⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠈⠻⡀⠀⠀⢻⡄⠹⡧⠂⠀⣶⠉⣠⠿⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⡀⠀⠀⢹⡆⠀⢸⡇⠘⡇⠀⠀⡇⣼⢃⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⣿⡀⠸⣇⢠⡏⠀⠘⢰⡗⠀⠀⠀⠀⠀⠀⠀⠠⠒⡄⠀⣀⠼⠀⠀⢰⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠸⣇⠀⠻⠸⡏⢰⠃⡼⠇⠀⠀⠀⠀⠀⠀⠊⠠⣼⠏⣾⠁⠀⠀⠀⡴⠀⠀⠷⠈⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠘⣇⠀⣦⢡⢏⡼⠃⠀⠀⠀⠀⠀⠀⢘⠀⠐⣿⡽⠃⠀⠀⠀⡠⠀⠀⢀⡞⠀⢘
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢹⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣿⡟⠀⠀⠀⣰⠕⠀⠀⢠⡇⠀⣈⡀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣾⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣯⠀⠀⢀⡟⡀⠀⠀⣸⠀⢠⡟⠁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠹⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣳⣦⡌⣥⡄⣠⠾⢁⣠⠟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢙⠋⠉⠡⠔⢳⡖⠛⡁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠐⠁⠐⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣙⡼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣷⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢻⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⣀⡀⠈⠁⠀⢨⣳⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠳⡄⠀⠀⡌⠀⢀⠀⣾⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠶⠖⣒⣸⠶⢄⠀⠀⠈⢢⣰⡟⠀⢠⣾⣯⠴⠖⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡀⠀⠀⠙⠻⣦⡰⠙⢶⣄⡘⠘⠛⠓⠞⠋⠁⠀⢀⣠⠄⣒⡭⠟⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠀⠀⠙⢻⣄⡂⠈⠻⣯⡑⠯⠠⠤⠛⠷⠧⠚⠈⠁⡠⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⣤⣉⣀⣙⣓⣤⣜⡨⣿⢲⠦⣀⣠⣤⣤⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢀⡀⠤⠟⠋⠀⠉⠻⢊⣁⡀⡀⠀⠀⢀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⠛⠛⠛⠛⠛⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠁⡀⠀⠀⠀⠀⠐⠦⣄⡀⠀⠈⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⣰⣏⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⣾⣋⣉⡉⠉⠉⠛⠛⠛⠻⠶⣤⣄⡀⠢⡀⠙⠄⠀⢄⠀⠀⠻⣄
⠀⠀⠀⠀⠀⢰⡟⠙⠻⣿⣿⣿⣿⣷⣶⣦⣤⣀⡈⠉⠓⠿⣦⡀⠀⠈⢳⡀⠀⢹⡆
⠀⠀⠀⠀⠀⣼⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡈⠙⢦⡀⠀⢿⡀⠀⣷
⠀⠀⠀⠀⢠⡿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⢛⣆⡀⠸⡇⠀⡟
⠀⠀⠀⠀⣸⠇⠀⢰⡞⠛⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⡿⠿⢁⣹⣿⣿⡄⡇⣸⠃
⠀⠀⠀⠀⡟⠀⠀⣼⢀⣀⣤⣤⣄⣀⠀⠀⠀⠈⠙⣯⣤⣶⠃⣼⣿⣿⣿⠁⣵⠋
⠀⠀⠀⢸⡇⠀⠀⢻⡋⠀⢀⣀⣀⡉⠙⠓⠂⠀⠀⣿⣿⡏⢰⣿⣿⣿⣯⠞⠁
⠀⠀⠀⣿⠀⠀⠀⠘⣷⠞⢋⣁⣈⣉⠉⠀⠀⠀⠀⢿⡿⠀⠟⣹⡟⠋
⠀⠀⢸⡏⠀⠀⠀⠀⠸⣿⠉⣉⣉⣉⠉⠀⠀⠀⠀⣸⠃⣤⣾⣿⣇
⠀⠀⣾⡁⠀⠀⠀⠀⠀⠘⢿⣍⣀⣀⣠⣤⣤⣶⣿⠇⢰⣿⣿⣿⣿⣀
⠀⠀⠙⠻⢶⣤⣄⣀⡀⠀⠀⠈⠉⣽⣿⣿⣿⡿⠟⢀⣿⣿⣿⣿⠋⠙⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠉⠙⢛⣻⡿⣿⣥⣤⣤⣤⣤⣶⣾⣿⣿⣿⠟⠁⢀⡾⠁⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⣀⣴⠾⠋⡽⠁⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⣴⠟⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄
⠀⠀⢀⣴⡾⠋⠁⠀⣼⠁⠀⢸⣿⣿⣿⡿⠟⣫⣵⣶⣶⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⣴⡿⠋⠀⠀⠀⢰⡇⠀⠀⠀⠉⠁⢀⣤⠞⠉⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠺⢿⣤⣀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿
⠀⠀⠈⠛⠿⣶⣾⣿⣆⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⣤⣸⡏⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⣶⣾⣯⣤⣴⣶⡶⠾⠟⠛⠉⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⢼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠐⢻⣿⣧⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⣷⠁⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣧⢻⣯⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡀⠀⢺⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡻⣞⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⣿⣿⡄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣯⣿⢾⣿⣷⣼⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⣿⢿⣿⣆⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣾⣟⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⢿⣧⣔⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠉⠉⠉⠉⠉⠉⡉⠉⠉⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣧⢂⡄⠀⠢⠌⠠⠤⠀⠀⠤⠄⠀⠀⠔⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡧⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣄⣐⣲⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣤⡤⠿⠛⠉⠀⠀⠀⠀⠀⠂⣀⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠈⢄⠠⠀⡀⠠⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠚⠀⠀⠀⠈⠀⠅⠀⠀⠀⠈⠀⡄
⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⢀⠄⠔⢁⠀⠀⠀⠐⠠⠀⠒⠉⠢⣀⡑⡀⡀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⣑⠀⠄⠐⡀⠀⠀⠠⠀⠀⢀⠤⠤⣄⡀⠀⠀⠑⠂⠠⡀⠰⡠⠀⠙⠢⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠋⠂⠀⡔
⠀⠀⣀⠀⡀⠀⣦⠀⢀⠀⠀⢈⠀⠀⠀⠈⠣⣀⠀⠄⠀⢈⠼⠁⠀⠀⠀⢲⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⢄⠀⠀⠥
⠀⠘⠁⠘⠀⠀⣸⡰⢠⠀⡈⢂⡀⠀⠀⠀⠀⠀⠑⣦⠊⠁⠀⠰⠊⠀⢀⠂⢂⡉⠃⠀⠀⣠⡬⠐⠀⠁⠀⡈
⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠹⣄⠈⢂⠀⠀⠀⠙⢄⡀⠰⠂⣀⠔⠁⠀⠀⠄⠠⠒⠋⠉⠀⠂⠀⠀⠁⠀⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠹⠂⠈⠉⠐⢂⠀⠉⠢⠴⢄⡀⠠⠀⠝⠚⠉⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄
⠀⡀⠂⠂⠀⠔⠁⠀⠀⠀⠀⠀⠀⠱⢀⠐⡀⠆⠀⡀⠀⠀⠡⠀⡀⠈⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠈
⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⢀⠀⠀⡉⠄⠂⠀⠀⠀⠀⠀⣀⠀⢀⠰⢀⢀⠠⠄⠀⠊⠄⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠑⠂⠊⠀⠱⡀⠀⠀⠀⠀⠀⠠⠁⢸⡆⢸⠀⠈⠀⠀⠀⠤⢴⡇⠀⠀⡇⠀⡏⠀⠀⠀⠀⠀⠁⠀⡀⠁⠀⠘⠠⠀⠀⠆⡀
⠀⠀⠀⠀⢀⡀⠈⠑⠒⠒⠢⢀⠀⠀⠘⡇⣼⣴⡶⢠⣶⣦⣾⢾⢠⣻⡆⡗⡦⡷⣾⠇⢿⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⢰⠀⠀⠀⠻⠇⠙⠒⠊⢻⡇⠉⣈⣂⣉⣣⣯⣅⣀⣈⡃⠚⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⡀⠀⠐⠂⠀⢄⣂⣀⣀⠈⠃⠠⠄⡀⠒⠒⠃⠐⠚⠀⠉⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠰⠣⢀⠘
⠀⠀⠀⠈⠀⠉⠀⠀⢠⢦⣗⢦⠁⠀⠀⠀⠀⠂⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡅⠀⠂
⠀⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠱⣄⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀⠀⡀⠀⠈⠉⠐⠒⠒⠆⠀⠀⠀⣀⣀⣀⣀⣐⡀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠂⠐⠒⠦⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⠂⠠⡀⠄⠀⠀⠔⠐⠀⠀⢔⠒⠁
⠀⠒⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢆⠀⠡⠀⠀⠀⠠⠀⠐⠠⠀⠠⠠⠈⠀⢠⠀⠀⠆⠀⠀⠀⠀⠀⠒⢠⠄
⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠄⠄⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⡀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠀⡠⠄⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠁⠢⠀⠴⠁⢒⠠⠈⢤⠀⠊
⠀⠀⠀⠀⠀⢀⡈⣀⠀⠀⢄⠀⠀⠀⠀⣀⢀⡀⠠⢈⠐⠀⠀⠀⠂⠀⠁⠈⠐⠀⠀⠀⠀⣀⣀⠀⠃⠠⠁⠀⠠⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠅⠀⠀⠘⠑⠀⠀⠈

⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⠀⡌⢳⠪⡿⣥⢯⠧⣘⠉⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠤⢄⠀⠀⠀⢀⡩⢵⣴⣦⠀⡉⢺⢉⢺⠙⡯⠭⠨⠼⠐⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠁⠸⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠰⣤⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢆⠀⠀⠀⠀⠀⠄
⠀⢀⣀⡠⠖⠋⠀⠀⠀⠐⢈⣿⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠿⣾⡁⠀⠀⠀⠀⠠⡀
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⠉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣝⣥⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠀⠀⠀⠀⠀⣀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠁⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⡀⠀⠀⠀⠀⠀⠀⣤⢀⡔⣹⣆⡴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⢀⡤⠀⠐⠒⠒⠀⠤⡀
⣵⣶⡄⠀⠀⠲⠶⣷⡈⣷⣶⠇⠢⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⢠⡀⣀⠀⠀⠈⠁⠀⠀⠒⠒⠈⠁⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠉⣹⡗⠀⠀⠀⠀⢈⡇⠛⠃⠀⠀⢀⢤⣖⡢⣀⣠⡀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢈⠙⠚⠃⠀⠀⡔⢁⡀⡀⠉⠀⢠⡿⣿⠾⡯⣕⣳⠵⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠠
⠈⠀⠀⠀⠀⠰⢀⣻⣯⢧⡏⣛⡷⣜⡧⡧⠝⢒⠓⢡⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡄
⡀⠀⠀⠀⠀⠀⢨⣾⢻⣅⣓⢹⣜⡯⣷⡕⡝⣶⣻⠍⢰⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠈⡄⠣⡟⡬⣗⢝⡝⠔⣳⢟⣏⢟⡺⢁⠃⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠈⠢⡙⢷⣡⣟⣊⢆⣿⡃⠵⣉⠔⠁⠀⠀⠀⠀⠑⢴⡀⡀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠰⡄⢠⠄⠀⠀⠁⠒⠀⠤⠤⠴⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠬⠈⠡⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⡳⣷⡞⠐⠀⠙⠈⠁⠀⠀⢀⣀⡀⠀⢰⡟⢁⡴⠀⣰⠂
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠁⠀⠀⠀⠀⡠⠖⣉⣠⣤⠄⠉⠉⢾⠿⡷⢞⣩⣄⠀⠀⠀⠀⠀⠀⠀⢈
⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡰⠣⣓⣿⡷⢺⠀⠐⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⠂⠄⠳
⠟⠩⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠙⠂⠈⠼⠓⡀⠀⠀⠀⠀⠀⠀⠸⢠⣒⣊⣝⣱⣭⣅⣢⢤⣠⢤⢠⢀⢂⠀⠀⠀⠀⠀⠀⢠⠹⣒
⢶⣾⠧⡰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠁⠀⠀⠀⠀⠀⠀⣸⡓⢻⢭⠷⣟⢼⢾⠾⢺⢿⣯⠏⠀⠄⠀⠀⠀⠀⠠⡀⢭⡿
⣾⡖⣃⠂⢈⣶⣤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⠘⢿⣓⡟⡞⣤⣳⠧⣝⣫⣧⡗⢮⣣⠀⠀⠀⠀⠀⠀⠐⢌⠁
⠂⠐⠅⠐⠘⣿⡿⠃⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡉⠵⡾⢞⣫⠾⣍⡻⡼⣼⡻⢩⠎⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡷⠉⠃⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠳⠿⠹⠣⠋⡋⠔⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⣀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⢒⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠
⠒⠈⠉⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢱⢤⢀⡻⢧⣺⣕⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠫⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠛⠻⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠶⠤⢤⣄⣀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⣠⠴⠚⠋⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣳⡀⣿⠀⠀⠀⣀⡤⠶⠚⣋⣉⣉⣉⣹⣦⣾⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⣰⠏⠀⡰⠂⠀⢀⠔⢀⡴⠾⢓⣚⣋⣭⣭⣭⣭⢽⣦⣾⠷⣒⣫⠭⠭⠒⠒⠲⡖⢶⡞⠁⠀⢀⡠⠴⠒⠚⠉⠉⣉⣩⣭⣷⠆
⠀⠀⠀⢀⡟⠀⡼⠁⢀⣴⠟⣊⣥⠶⠛⣛⣉⠉⢀⡤⣿⠀⠈⣿⠘⡟⠁⢀⣀⣀⠀⢀⣠⡇⢸⡇⣠⣾⠅⠐⠖⠒⠒⠋⣉⢈⡟⠋⠁
⠀⠀⠀⠈⡇⢰⡇⣀⣾⡶⠛⠁⠀⠀⠙⠟⠛⠀⢸⡋⢹⡄⠀⢹⡄⣇⠀⠈⠛⠉⠀⠘⠉⡇⢸⣿⡿⣿⠀⠀⠘⠟⢃⣠⠽⠥⢧⡀
⠀⠀⠀⠀⢻⡘⢸⣋⠻⢿⣖⣦⢤⣤⡴⠖⠛⠋⠉⠉⠙⢻⡀⠈⣇⢿⠦⡄⢠⠄⢀⣀⡆⡇⢸⣿⡰⡎⠓⠯⣶⠚⠉⠀⠒⠒⠒⢳
⠀⠀⠀⠀⠀⠻⣌⣇⠆⠀⢧⠈⣿⠀⠀⣀⠤⠖⠚⠛⠒⢶⡇⠀⢻⡘⣆⢃⣀⢠⡼⣫⠕⢘⣺⡇⢳⣤⣀⠀⡟⠀⠤⣖⣉⣀⣈⣽
⠀⠀⠀⠀⠀⠀⠈⠛⢶⣶⣸⣇⣿⠀⠀⠀⠐⠚⠛⠓⢦⣼⠁⠀⠘⡟⡾⣏⢉⣿⢾⣥⣾⠋⣉⣿⣸⣿⠙⠻⣅⠀⠀⠀⠠⠤⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠉⢻⣿⡀⠀⠀⠀⠀⢉⣉⣻⠃⠀⠀⠀⢻⠃⠈⠟⠀⠀⠙⡎⣯⠹⡌⢻⡈⡆⠀⠀⠙⠲⠤⢤⣤⡞
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁⠀⠀⢿⡙⠛⠶⢶⡶⠴⠞⠁⠀⠀⠀⣀⣼⢦⠀⡇⠀⠀⠀⢸⢸⣀⡟⣸⡧⠻⣆⠀⠀⠀⠀⠘⣇⠙⢶⣤⣀
⠀⠀⠀⢀⣠⡴⠞⢻⡀⠙⢦⡀⠈⠓⠶⠦⣤⡷⣤⣤⡴⠶⠖⠛⠉⠁⠈⡄⠹⣄⠀⢀⢎⠏⠉⠻⣧⡜⠀⠈⠳⣤⣄⣤⣤⡟⠀⣀⣳⣌⠛⢦⣄
⣀⡴⠞⠋⠁⠀⠀⠀⠹⣦⡀⠙⢦⣄⠀⠀⠀⠀⣾⠇⢻⠓⢦⣄⡀⠀⠀⠘⠦⣈⡉⠉⣁⣀⠔⠊⠁⠙⣦⡤⢤⣄⡹⣄⠞⣴⠟⠉⠉⠁⠀⢀⡿
⠙⣧⡀⠀⠀⠀⠀⠀⠀⠈⣻⣶⠶⠾⣝⠳⠶⢦⣿⠀⠈⣇⠀⠈⠻⢦⣄⠀⠀⠀⠉⡈⠁⠀⠀⠀⠀⡆⠹⣆⠀⠀⠙⢾⡞⠁⠀⠀⠀⠀⣠⡿⠁
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⡄⠀⠀⠀⠻⣦⠀⠀⠀⢳⠀⠀⠀⠀⠀⢹⠀⢻⡆⠀⠀⢸⡇⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠘⣧⠀⠀⠀⢈⣇⡀⠀⢠⠀⢹⡆⠀⠀⢸⡄⠀⠀⠀⠀⢸⡀⠀⣷⠀⠀⠈⡇⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⢧⠀⠀⣾⣿⣷⠀⢸⡀⠀⢿⡀⠀⠈⡇⠀⠀⠀⠀⢸⡇⠀⢹⡇⠀⠀⣇⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠈⠳⣄⠀⠘⣧⠀⠈⠛⠁⠀⢸⡇⠀⠸⣧⠀⠀⣷⠀⠀⠀⠀⠈⡇⠀⠸⣷⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⡀⠀⠈⠳⣄⡘⣇⠀⠀⠀⠀⢸⡇⠀⠀⢿⡄⠀⢸⠀⠀⠀⠀⠀⣇⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣤⣬⣋⣛⠀⠀⠀⠀⠀⣿⠀⠀⢸⣇⢀⣸⣄⣤⣴⠶⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣛⣛⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⡐⠉⠠⠀⠀⠀⠀⠀⠀⡠⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡈⠓⠔⡡⠀⠐⡘⠃⢐⡀⣨⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠜⠀⠀⠓⡄⠀⠀⠀⠎⠒⠓⠢⡤⠀⢤⢀⠗⠈⢱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠤⡀⠂⠀⠀⠀⣂⡈⠁⠀⠀⢠⠖⠈⠉⠍⠣⠀⠾⡗⢀⠖⠚⢼⠄⢄⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠬⡄⠄⠘⡀⠻⣤⠴⠆⡙⠛⠓⢄⠀⠀⡇⠀⠀⢀⡠⠄⣦⣈⣁⡉⠀⠀⣰⠀⠈⠁⠀⠥⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⡄⠐⡘⡀⠡⡴⠅⡘⠀⠀⠀⠀⣸⠤⠒⣇⢀⡰⠉⠁⠀⠀⠀⠈⢅⡱⠜⠾⣷⡀⠀⠔⢢⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣄⠔⡪⠉⣛⡛⠣⡀⠈⣦⡤⡕⡼⠁⡀⡴⠛⡐⠅⠀⠀⠀⠀⠀⠀⣃⠔⠁⣠⠏⠱⠎⠀⢸⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣠⣢⣎⡔⠉⠄⠌⢷⠑⡤⠉⠀⡙⢀⣬⡊⢈⡀⢰⣌⠙⡳⠂⣶⠴⢦⠋⢀⡾⢁⣀⠀⠁⠀⠌⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⡀⠀⠱⣀⠜⠙⡬⠆⠗⠠⣀⣡⠊⠀⠈⠁⠉⠁⠘⠷⠉⠀⠈⠑⢦⠔⣡⡤⠤⡈⠻⣀⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⡀⠀⡂⠀⠙⠲⠀⠀⢸⠁⠀⠄⢤⠵⠤⠤⡔⠀⠀⠀⠀⠀⢠⠹⠋⠀⠑⠌⢤⠋⢰⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡮⡣⢄⡇⠀⠀⠀⠄⠀⠸⠰⠁⣰⠁⠀⠀⡰⠇⡄⠀⠀⠄⠀⠈⢀⠃⠘⠐⢢⡸⢔⢵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⠒⠤⢀⡺⠄⢠⣂⡇⠀⠙⡀⠈⠈⠁⢡⡀⢀⣀⣀⣴⢡⠮⠀⡒⠄⠒⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠉⠀⠒⠐⠈⠈⠩⠦⠤⠬⠥⠍⠁⠁⠂⠒⠀⠉⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠔⠊⠁

⠀⠀⠀⠀⢀⠄⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠔⠁⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⠀⠀⠁⠐⡆⠀⠀⠀⠀⠘⠄⠀⠀⠀⠕⠀⡀⠀⠀⠀⠥⢥⡐⠂
⠀⢀⢂⠂⠀⠐⠄⠀⢀⠴⠈⠢⠀⠀⠀⢸⠀⠀⠀⡔⠀⠉⠐⠢⡀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⡐⠀⠀⠀⠚
⠀⠈⠂⠀⠀⠀⡜⢹⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠄⠄⠋⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⠤⠄
⠀⠀⠀⠀⠀⠀⣀⡀⠄⣀⡀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠑⠲⢄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⢤⣦⠀⠀⢀⠄⠃⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⣈⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡪⠂⢀⠎⠀⠀⠘⠍⠁⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⢄
⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⢆⠄⠄⡕⠁⡰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⡁⠡⠀⢊⣔⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⠐⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⡀⠈⠀⠀⠀⢈⡀⠀⠀⠀⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠊⢑⡦⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠊⠀⠐⠂⠀⠀⡀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠁
⠀⠑⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡡⠄
⠀⠀⠀⠈⠁⠀⠐⠁⠀⠀⠈⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠁⠂⠀⠐⠀⢱
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢁⠐⠀⣀⠠
⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠄⠀⠠⡜
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠖⠄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣀⠀⠀⠀⣀⠠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠙⢄⠜⠁⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣻⠀⠈⡔⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⣀⣀⣾⢻⣄⣴⡿⡇⢀⣀⣀⣀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⢸⢿⡜⠀⣧⣾⠁⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠶⠀⠀⠹⡟⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⣸⠧⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠈⠀⠀⠀⠀⡠⠔⠒⣠⡐⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠉⠀⠈⠉⠢⡀⠀⠀⠀⠀⣡⠀⠘⣷⠋⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣀⠴⢄⠸⡀⠀⠀⢰⡏⣉⣉⣁⠩⠛⣵⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡏⠉⠀⠉⠉⠀⠀⠀⠃⡇⠀⢀⣣⠾⠀⠀⠀⠀⠀⠸⢇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡌⠘⡄⠀⠀⠀⠀⠀⢠⠎⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡇⠀⠘⢄⠀⠀⠀⢀⠎⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢀⠜⠀⠀⢰⠀⠀⠀⢸⠑⠂⠒⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠒⠒⠊⠩⣀⠀⠀⠈⢲⠶⠛⠻⢄⣀⣀⠜⠛⠲⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢂⢸⣁⠑⠤⢄⣀⠤⠊⠀⢱⢰⠢⡝⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⢊⠔⠀⠬⣢⡀⠀⠀⢀⡀⠌⡸⠀⠈⠆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⡊⠀⠀⠀⢸⢱⠀⠀⡏⢶⠆⡇⠀⠀⢸⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣏⣢⠀⠀⠈⠉⠀⠀⠈⠒⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣴⣶⡄⠀⠀⢀⣀⣸⣿⠠⠄⢈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣠⠈⢛⣿⣷⣾⣿⣿⣿⣿⣦⣴⣿⣿⣷⣶⣤⡀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣼⣿⠇⠀⠉⢹⡿⣿⣿⣿⣿⣿⡄⠉⢟⣉⠙⠛⠻⣿⣼⣿⣿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⡿⠿⠶⣦⠀⠙⠁⠘⢿⣿⣿⣿⣷⠀⣘⢛⡑⢿⠇⡋⢹⡟⢻⣘⣀⠁
⠀⠀⠀⠀⠀⢠⣶⣾⠿⣿⣷⣫⣤⠀⠶⠾⣷⣿⣿⠟⠈⠀⠀⠉⠉⠀⠻⣿⣽⣶⣾⣿⣿⣷⣿⠿⠋
⠀⠀⠀⠀⠀⠸⣏⣠⣴⣶⣿⢉⣡⣶⣦⡄⠋⠛⠛⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⣘⠟⣅⠈⡙⣿⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠹⣛⣿⡦⣸⣤⣿⣿⡟⠀⠀⠀⠀⠀⠈⠁⠈⢡⣾⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⡇⠀⠈⢹⠇
⠀⠀⠀⠀⠀⠀⣀⣬⣷⣾⡿⠟⠙⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⠁⢹⣿⣈⣥⣤⣀⣿⡇⠈⠀⠈⠁
⠀⠀⠀⠀⠀⣾⡿⠿⣿⣿⣶⠀⠀⠀⢿⣷⠀⠀⢀⣤⣤⣀⠀⠈⠄⠀⠃⠐⢿⠟⠻⠿⠿⣯⡁
⠀⠀⠀⠀⠀⠈⠀⠠⠙⢿⡇⢀⣴⣶⣿⠿⢇⣀⣾⠛⠛⠛⢶⣦⣠⣤⣤⣤⣸⠐⣀⠐⢸⠉⠳⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠁⣼⣿⣿⡇⠀⠐⠭⠇⠄⠀⠤⠸⣿⠛⠛⣿⣿⣿⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⣘⣛⣛⣾⣿⣶⢾⣿⡄⠀⠻⣿⠟⠃⠀⠘⠃⠀⠀⠀⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣀⣭⡭⠭⠍⣉⡉⠿⠿⢿⣶⠀⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⣹⡀⢀⣴⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠢⠄⠀⠀⠀⣀⠂⡀⠀⡄⠀⠀⠀⠀⢀⠎⠁
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠆⠀⠀⢀⡀⠤⠤⠠⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠄⠸⡿⠁⡀⠀⠠⠀⡈⠐⣄⣼
⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠚⠄⣀⡊⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠀⠀⠀⠀⠐⠦⣼⣧
⠀⠀⠀⢐⡀⠒⠄⠀⡔⠠⢶⡀⠀⠀⠙⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠸⠿⠆⡀⠀⠃⣄⡀⠀⠀⠀⠐⡢⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠀⠀⠁⢠⠶⠃⠉⠀⠀⠀⡀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠠⠀⠀⠘⢄⣠⣠⠆⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⢠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⠀⠉⠉⠒⠤⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠢⠢⡤
⠀⠀⠀⠀⠈⠛⠂⠉⠙⠓⠂⢤⣌⠳⡀⠀⠀⢀⠀⠀⡀⠠⠀⠀⠀⠄⠀⡀⠀⠀⠱⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣗⣄⠀⠸⠀⠀⠃⠀⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⢠⠠⠀⠀⡀⠀⠁⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠍⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⡀⡀
⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣀⠀⠀⠀⠀⠀⠀⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠥⠤⠀⠀⠀⠀⠀⡀⠀⡀⠀⡠⠀⠐⠀⠀⠘⠛⠐⠀⣀⠔
⠀⠀⠀⡐⠠⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⡤⠀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⠀⢢⠞⠁⠀⡐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⡀⠀⠘⠀⠀⡁⠀⢱⠖⡘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠁
⠀⠀⡔⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠇⠀⠀⠀⠀⡁⠀⠃⠀⠀⠀⠀⠀⠀⠐⠈⠈
⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣆⠀⣀⠂⠀⢀
⠀⠀⠀⠣⢠⡀⠀⠀⠀⠀⢁⠠⠄⠀⡀⠠⡄⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⠂⠈⡆⠫⠀⠐⠁⢰⠶
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣴⡶⠟⠁⠀⠀⠀⠘⢄⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⢀⣤⣶⣦⣄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⢰⢿⣷⣷⣿⣷⣝⢿⣿⣿⡿⣣⢞⣿⣿⣝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣏⡿⣿⣿⣿⣿⣿⣿⣳⣾⣷⣻⣽⣿⣿⣿⣿⣼⡆⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣲⡶⣭⣿⣻⣭⣞⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣝⠿⠿⢿⣂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⣿⣿⣿⡿⣱⣿⣿⢹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣽⣿⣯⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣧⢹⣿⣿⢰⡟⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠧⠌⣸⣿⣿⢸⡏⣱⠃⣼⣿⠿⠛⣛⣛⣛⣛⣛⢿⣿⣿⣿⣿⡿⣿⣹⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣛⣯⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣈⣡⡾⢋⣴⣾⠟⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣷⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣾⣿⣿⠟⣡⣤⣄⠈⢿⣿⣿⣿⡿⠋⠀⣨⣿⣿⣿⡟⣰⠿⠛⠹⠞⣿⣿⠟⢋⣴⣿⣿⣿⣿⣿⣿⣯⣿⠮⣽⣅⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⡄⠘⣿⣿⡿⠁⣠⣾⣿⣿⣿⣿⡇⠀⠛⠓⠛⠛⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣦
⠀⠀⠀⠀⠀⠀⣙⢯⡻⣿⣿⣿⠀⢿⣿⡿⣿⡇⠀⣿⡿⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟
⠀⠀⠀⠀⢀⣾⣿⣷⣻⣽⣿⣿⣧⣀⣉⣀⣿⡇⢠⡿⠁⣼⡿⠟⠛⠻⣿⣿⠿⠛⠛⠿⢿⣿⣿⠟⠛⠻⠟⣿⣿⠟⣻⠿⠟⢿⣿⣿⣿⣿⡿⣋⡀
⠀⠀⠀⠀⠘⣿⣿⡿⣻⢿⣿⣿⣿⣿⣿⣿⣿⠁⢸⠃⣼⠋⢠⣿⡇⢀⡿⠁⣴⣿⡇⠀⣼⠏⢀⣾⣿⠀⣰⣿⠃⠀⣡⠆⢠⣿⣿⣿⣷⡟⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠈⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣸⣿⠀⠙⣋⣤⡾⠃⠰⣿⡟⠀⣸⠟⠀⢾⣿⠃⢠⣿⠏⢀⣾⡏⠀⣾⢋⣿⣿⣟⣧⣻⣿⡿⠁
⠀⠀⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⣿⣿⣦⣤⣭⣥⣾⣷⣤⣤⠆⢀⣴⣾⣦⣬⣵⣤⣬⣴⣤⣾⣿⣧⣤⣴⣿⣿⣿⣿⣯⡳⣌
⠀⠀⠀⠀⠀⠹⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡿⠋⡡⠀⣼⣿⠟⢛⣛⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢉⣟⡳⣿⣻⣿⣿⣿⣿⣿⠟⣩⢶⡟⢫⠄⣽⣿⡟⠁⣾⡇⢠⣿⣿⡄⢦⡉⣟⣻⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣯⠾⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⢷⣿⣿⣿⣿⢯⣾⣿⣾⠵⢋⣼⣿⣿⣇⠀⠿⢁⣾⣿⣿⣿⣤⡙⠻⠿⣿⣷⣽⣿⣿⣿⣿⣿⢿⡿⣻⣭⣧
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡟⣾⣿⣿⣿⣿⣿⣛⣭⣥⣶⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡜⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣻⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⣶⣶⣶⣝⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⢟⣯⣿⣛⠾⠿⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠾⡿⣿⣿⣿⣿⢿⣯⠿⡿⢯⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⣿⣿⡵⢫⣵⣿⣿⣷⣝⢿⣿⢿⣿⣻⠏⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠙⠻⠿⠛⠁⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠤⠀⢀⠀⠀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠈⠀⠀⠀⠐⠊⠀⠀⠀⠈⠂⠀⠀⠀⣐⠤⠤⠄⠠⠄⠱⣀⡀⠀⠀⢧⡋⠸⢰
⠀⠀⠠⣔⣛⣈⣡⠀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠂⠆⠶⠐⠀⠄⠀⠈⠃⢠⣶⣶⣴⣦⠀⠍⠃⠀⢀⣀⣢⣔⡃⢀
⠀⠀⠄⠙⠛⠛⠋⠀⠀⠀⠀⠓⠀⠀⠉⠀⠀⠂⠀⠀⠀⠀⠐⠀⠐⢀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠋⠀⠀⠀⠀⣆⠐⠷⠶⠂
⡀⠀⠀⠈⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢄
⠀⡀⠀⠤⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣠⡢⠀⠁
⠀⠀⢰⣶⣤⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣬⣤⣵⠆
⠀⠀⠄⠉⠛⠋⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠔
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠐⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣿⠗⢺⣯⡿⠃
⠀⠀⠀⣴⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠂⠀⠀⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⠏
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠄⠀⠄⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂
⠀⠁⠀⡤⢶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠲⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣀⠀⣀⠐⠄⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠡⠞⠶⠁⠀⠀⠀⠀⠀⠀⠴⠲⠆⠀⠀⠀⠀⠈⠀⠀⠳⠞⠉⠀⠁⠀⠀⠀⠒⠒⠐⠒⠂⠀⠀⠀⠀⠀⠐⠂⠀⠒
⠀⠰⠆⠁⠀⠀⠀⠶⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠐⡀⠀⢠⡆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠂⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠁⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⣾⠃⠀⠀⢳⣿⡟⠀⣦⠀⠀⠂⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠈⠀⠀⡀⣠⣞⣧⡄⣴⡟⠀⠀⠀⠸⡹⣷⣄⣷⠃⠀⠀⠸⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⠠⡀⠙⠛⠾⣿⣿⣏⠀⠀⢀⠀⠀⠰⠉⢻⣿⠒⠀⠀⠀⢳⡄
⠀⠀⠀⠀⢀⣤⡶⠊⠈⠛⠷⠀⠀⠀⠀⠀⠀⠀⠙⢯⣻⡀⠀⠀⢀⠀⠀⠾⢿⣦⡀⠀⠀⠈⠻⠄⠀⠀⠀⠀⣀⡀⡤⠀⠀⣴
⠀⠀⠠⢶⣯⣿⡇⠀⠀⠀⠀⠈⠛⠷⣽⡀⡀⠀⠀⠀⠈⠙⠀⠀⠈⠀⠀⢠⡿⢹⣱⣄⡀⠀⠂⠀⠀⠀⣦⡀⠈⠋⠀⢀⣾⡟
⠀⠀⠀⠀⠀⢉⠻⣄⡀⠀⠀⠀⠀⠀⠈⠛⠿⣴⣄⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⣽⢟⡟⣦⠀⠀⠀⠀⠉⠻⠛⠒⠀⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢯⣷⣄⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠈⠁⢈⣿⠃⢎⡆⠀⠠⢄⠀⠀⡀⠴⠄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠁⠀⠀⠀⠀⣾⠀⠈⠁⠴⠿⠃⠀⠀⠀⠀⢀⡈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠀⠌⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀⣌⡀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⣤⠀⠀⠀⢀⣆⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⢾⠏⠀⣘⣴⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢠⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠈⠛⠲⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⣁⠔⠁⠐⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠲⢶⣼⣿⣷⠀⠀⣹⣿⡇⠀⢴⣧⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠖⣋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠾⣇⣿⣿⡇⠀⣿⣷⢠⣿⣇
⠀⠀⠀⠀⠤⢀⣀⣀⡈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢥⣸⣿⣿⢺⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠒⠷⠞⠉⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣾⠛⣆⣀⣀⠀⠀⣠⣤⡇⢳⡀⠀⢧⡀⠀⣀⣀⣠⣄⣀⠀⠀⣀⡀⠤⠔⠚⢩⠹⡇⠀⢀⣀⠀⣀⡾⠃⣀⡀⠀⠀⣠
⠀⠀⠀⠀⠀⢴⠛⠀⠈⠀⣸⢋⡋⢀⡤⠧⡴⠷⠴⠶⢿⣿⢿⡉⡡⠛⡍⢻⡁⠆⠂⣀⣤⣄⣀⣸⠉⠙⠋⠈⠋⠉⣙⠻⣿⣏⠳⠶⠉⢷
⠀⠀⠀⠀⠀⢘⣆⠀⠀⠀⠉⣿⠉⣻⠴⠚⢦⡴⠚⠉⢲⡇⢀⠉⠳⡌⢇⣹⠷⠒⠋⣉⣴⣾⠇⡹⢖⣚⡤⢤⢄⣮⣴⣶⣮⣹⠀⣤⠀⢸
⠀⠀⠀⢀⣀⣿⠉⠃⢈⣆⠀⠙⢶⣁⡤⠴⢤⠴⢚⣥⠾⣧⠐⢇⠀⢻⣿⣤⣤⣴⣮⣗⢋⡵⠋⠀⣈⠀⠀⣩⢽⠗⠀⠀⠈⣿⠊⠈⢧⡾⣤⡀
⠀⠀⡜⠉⠠⢤⣷⣄⣀⣈⠣⠀⢸⣧⠤⠴⠒⠒⢯⣠⡀⣹⢣⡀⢁⡼⢋⣤⣊⡩⠄⠈⠉⢧⣤⣬⣤⣤⠤⣭⠽⠶⠦⠤⠴⠧⠀⠒⠊⠀⢨⡇
⠀⠀⠳⠤⢤⣀⠀⠀⣉⡉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠛⠛⠓⠒⠛⠚⠓⡿⠋⣁⡤⠤⠒⠒⠋⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠤⠶⠖⠋⣠⠔⠋
⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠈⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠶⠶⠖⠂⣼⢠⠞⠁⠀⠐⠒⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⢀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠶⠦⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢤⣼⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣾⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃
⠀⠀⠀⠀⠀⠀⠘⣽⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⡵⠃
⠀⠀⠀⠀⠀⠀⢀⡟⢿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣋⡉⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠱⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⢠⡟⠿⠂⠀⠀⠀⠀⠀⠀⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⢧⠘⣆⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢇⡀
⠀⠀⠀⠀⠀⠠⣤⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢹⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⡀⠈⣿⣷
⠀⠀⠀⠀⠀⢀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣰⡋⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⣦⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠑⢾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⣀⣼⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠋⢛⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠂⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠄

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆
⠀⠀⠀⠀⠙⢣⣁⣸⣇⣹⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠾⢿⡿⠷⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠉⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠏⡇
⠀⠀⠀⠀⠀⠀⠀⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⢷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⣠⠖⠓⠒⢒⡦⠒⠒⠒⠒⠚⠉⠉⠳⠤⣀
⠀⠀⠀⠀⠀⣀⣠⠊⠁⢠⠞⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣆
⠀⠀⠀⠀⡼⠉⠁⠀⠀⢧⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⡔⠒⠚⠉⠉⠛⢤⡀
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⢱⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠓⢢⠀⠀⢰⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠈⡆⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣳⡀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣄⣀⠀⠀⠘⠦⣄⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠲⡄⠀⠉⣶⠚⡄⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⠤⠟⠀⡇⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣠⡄⠀⠀⠄⢀⠀⠀⠐⠀⣶⢐⣡⡔⠁⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢴⡧⣾⣿⡋⣞⣻⣔⣤⣽⣷⡶⢦⣿⣿⣯⣧⢠⡄⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢺⢛⣿⣶⣽⣶⣾⡻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡙⣿⢷⣿⣿⣿⣿⣇⣾⣿⣿⣿⣿⣿⣦⣼⣿⣍⣷⠤⡀⢀⣀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣘⠀⢯⣣⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣆⠛⢉⡤
⠀⠀⠀⠀⠀⠀⣀⣤⣾⣟⣿⡿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⣻⣾⡷⣂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡟⣻⡥⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢃⣻⡏⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⡄⢣⡀⣠
⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠌⣾⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢿
⠀⠀⠀⠀⠀⠀⢪⠿⣿⣿⣽⣥⣾⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⣎⡨⢱⠾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣷⡽⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠱⣻⡿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣁⣟⣯⡉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠻⠷⡿⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣻⡿⡛⠂⠈⠉⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠟⠚⠻⡉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⣼⣿⣿⣿⣿⡿⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠒⢸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣿⣿⢻⡿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠘⠊⠀⠈⠙⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⣶⣿⣿⠁⠀⠀⠀⠹⡟⠂⠀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡇⠀⠀⠀⠐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⡙⠁⠀⠀⠀⠀⠀⢀⣴⠅⠂⠐⠶⣄⠀⠀⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉⠀⠈⣿⡦⢀⠀⡃⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⢀⠆⠰⠀⠙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠀⠀⠀⠀⠀⢠⡛⢱⠀⠈⢱⡶⠢⡀⠀⢁⠀⢀⠤⡄⠀⢡⠄⠊⠀⠀⠑⠤⠄⠈⠈⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⠀⠂⠈⣥⣴⠗⠀⠀⠓⢀⡸⠃⠁⠠⠀⠀⠀⠀⠆⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⠀⠀⠀⠀⢰⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠀⠀⠀⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠡⡀⠀⢠⠊⠀⠑⠄⠀⠀⠀⠀⠀⠀⡰⠀⠃⠀⠀⢀⠜⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠸⠀⠀⠀⢰⢱⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠀⠀⠱⣀⡬⢠⣤⠊⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⣀⠔⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡆⠀⠀⣼⡎⠀⠀⠀⠀⠀⠀⠔⡁⠀⠀⠃⢀⡴⠋⢠⡎⠈⠁⠒⠀⠄⠀⡤⠁⠀⠁⠀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠰⠙⠂⠤⠀⣀⡀⢀⠌⠀⠐⡀⠀⠈⠀⢀⠴⢤⣿⣀⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⡂⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠢⣀⠔⠁⠰⠋⠀⠀⠈⠈⠓⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠟⡒⠂⠠⡤⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠁⠀⠀⠈⠒⠣⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⡀⠀⢅⡂⠤⢀⠀⠀⠀⠈⠁⠁⠂⠄⣀⢀⠎⠀⠀⠀⢀⠎⠈⠉⠀⠒⠒⠒⠒⠂⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡰⡙⠃⠂⢌⣀⡀⠀⠁⠀⢧⠀⠀⠀⠀⠀⠉⠂⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠕⠀⠀⡠⠃⠀⠀⠀⠉⠉⠉⠉⠁⡐⠀⠀⠀⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠤⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢀⠔⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠷⠺⡀
⠀⠀⠀⠀⢀⠂⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠹⠶⠁⢠⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡠⡀⠀⣠⢾⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠭⠂
⠀⢀⣔⣇⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢦⡍⠀⠀⠈⢄
⠴⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⢐⡤⣀⠑⢄
⠈⠃⢑⡄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⣀⣀⣀⣙⣲⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡽⠛⠒⢄⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⢷⡤⠀⣤⠀⠉⠺⠋⠒⢐
⠀⠀⠀⠀⠀⠀⠢⠤⠥⢈⠁⠀⠛⠞⠒⣶⣆⠀⠀⢻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠏⠀⠓⠲⣿
⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠙⠀⠀⢒⠀⠀⠀⠀⠀⢠⢠⠀⣀⡀
⠀⠀⠀⠀⣤⠔⢄⠀⠀⠄⠀⠀⠀⠀⠀⠐⠢⠔⠁⠀⠉⠙⢞⢀⠎⢀⠀⠂⣎⡀⢶⢉⡒⠀⠀⣀⡄
⠀⠀⠀⠀⡇⠀⠘⠳⢤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡳⠛⠀⣐⢋⡉⠀⠛⠠⢁⡀⠈⠉⣀⠤
⠀⠀⠀⠡⡇⠐⠀⠀⠀⠑⠂⠉⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠘⢀⣈⣩⣥⡉⠠⠀⠀⡀⠀⠀⠀⠠⠄⠶⣴⣤⡄
⠀⠀⠀⠀⠈⠀⢀⡀⡀⠀⠀⠀⠀⠀⢶⣿⠀⠂⠀⠀⠀⠀⢀⡤⢹⠓⠘⠃⠀⣠⣬⣍⠉⠄⠀⠁⠀⠬⢿⣿⣧⣀
⠀⠐⠀⠀⠀⠠⣼⢣⡷⠚⠀⠀⠀⡠⠆⠀⠀⠂⠀⠀⠀⠀⠁⠀⣠⣄⠀⢀⣬⡀⠀⠀⢹⠍⢉⠀⠲⠶⠼⠉⠁
⠀⠀⠀⠀⠀⠀⠙⢉⣁⠹⠉⠀⠞⠁⠀⠀⠀⠀⠀⠄⠀⣣⠀⠀⠉⠁⠀⠈⠀⠀⡀⠀⠀⣴⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠉⠁⠀⠂⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠘⠒⠚⠁⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⡿⠿⣿⣿⠟⣿⡿⣿⣿⠟⣿⡟⣿⣿⢿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⢸⣿⣷⢸⣿⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣻⣿⣦⣿⣧⣿⣿⣮⣿⣵⣿⣿⣿⣏⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠈⠉⠻⣧⣨⣿⣞⠿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⡿⢟⣿⣧⣠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⣀⣤⣤⣶⣶⣆⣿⡿⢀⣤⣀⠀⠀⠀⠀⠀⣀⣠⣀⣀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠨⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠉⠙⠛⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⢀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣭⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⡤⠤⠟⠛⠛⠛⢉⠉⠛⠛⠻⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢂⣠⣤⣀⣨⣾⣿⣶⠶⠤⠾⢿⣛⣻⣛⣩⣿⣿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⣤⣤⠔⠊⢀⣤⠿⠛⠋⠉⠀⠀⠀⠀⢀⢼⣯⣟⣽⣿⣿⣿⣷⣼⣃⣸⣎⣿⣻⡿⣿⣿⣿⣟⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠄⠚⠋⠁⠀⠀⠀⠀⠀⢀⣴⣾⣿⡘⣭⣿⣿⣿⣿⣿⣻⣿⡿⢿⡿⠎⠀⠀⣷⡛⠉⠉⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢿⣛⣫⠽⢿⣿⣿⣿⣿⣿⣿⣿⣓⠣⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠦⠦⣴⣶⣿⡿⠛⠊⢡⣾⣿⣷⣾⣯⣽⣿⠟⠁⢹⣿⣿⣿⣯⣶⣿⣾⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠘⠁⠀⠀⡟⣙⣿⣿⣴⣽⣿⣿⣷⠀⠀⠻⢿⣿⣿⣿⠛⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠀⠛⣾⣿⡿⢣⢿⠀⠀⠀⠀⠉⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⠄⠀⠀⣴⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣠⣴⣾⣏⣙⣿⣿⣿⣿⣿⣥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⣿⣿⣿⣿⣏⣀⣿⣶⣄⡀
⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⡏⣽⣿⣿⢻⣿⣿⣷⣶⣶⣶⣶⣤⣤⣶⣦⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⡇⣿⣿⡟⣼⣿⢣⣿⣿⣿⢹⣿⣿⢯⣿⣿⡛⣿⣯⣿⣹⣷⢻⣿⣦⢻⣿⣷⣍⢻⣿⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣷⣿⣿⣱⣿⣿⣸⣿⢩⡿⣼⣿⣿⢺⣿⣿⣿⣹⣿⢸⡟⣿⢸⣿⣿⣧⣹⣿⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣷⣿⣾⣿⣿⣿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠀⠂⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠊⠀⠁⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⢿⣿⣟⣛⣛⣛⣛⣻⣿⢿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣻⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣝⡻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣻⣵⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠙⠓⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣫⣾⣟⠃⠈⢩⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⣵⣿⣿⣿⣇⣴⠼⣛⡭⠖⠒⠒⠢⣄⡴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣝⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣺⠿⠙⠟⣻⣿⢱⠋⠁⠀⠀⡆⠠⣀⢀⢻⣹⣿⣿⣿⣿⡿⢟⣭⣶⣶⣶⣶⣮⢹⣿⣞⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢀⣈⣣⠀⣀⣿⢇⠏⠀⢄⠀⠴⠊⠉⠉⠑⠄⢃⣿⠿⠿⠛⣵⣿⣿⠛⠉⠉⠉⠙⢎⠛⠛⠚⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠧⣤⣤⣤⠾⣪⣿⢸⠀⠀⠀⡌⠀⠀⢠⠀⢠⣾⢢⡀⠀⠀⠀⣿⣿⣿⡄⠀⠀⡶⠀⣾⡂⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⣶⣤⣖⢸⣿⣿⣸⠀⠄⠺⠙⠲⣤⠔⠊⠀⠈⡏⣧⠀⠀⠀⣿⠉⠻⠧⣠⡤⠖⠋⠁⢹⡄⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢨⠂⠹⢛⣯⡳⡄⠐⠦⢀⡇⠀⠈⠁⢀⢇⣿⣶⣶⣿⣜⢦⣁⠀⢰⠀⠊⠀⠀⢸⢰⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠢⣄⣠⢎⡄⣠⣘⣿⣧⡛⣶⠀⠘⢄⡀⠀⣀⡬⣸⣿⣿⣿⣿⣿⢮⡏⠑⢥⡀⠒⠀⣀⡼⣾⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢀⣇⣻⠿⢿⣿⠿⣛⡥⢯⡀⠀⠀⠉⢻⣏⢾⣿⣿⣿⡿⣟⡵⠾⣧⡀⠀⠈⠉⢿⡁⠀⠉⠉⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠾⣠⡉⢉⡵⠖⠋⠉⢷⡀⠙⢦⡀⠀⢸⠉⣿⠶⣭⡵⠞⠉⠙⢦⡈⠳⣦⡀⠀⢸⠏⢳⠲⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⢀⣴⠏⠀⠀⠀⠀⠘⣏⠢⡀⠙⢦⣼⠀⢸⡇⠈⣷⠀⠀⠀⠀⠙⠢⡈⠙⢆⣼⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡃⠀⠀⠀⠀⠀⠀⠘⣧⠈⠀⠀⠉⠀⢀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⡄⠀⠀⠈⢷⡀⠀⠀⠀⣼⣧⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⡇⠀⠀⠀⠀⣿⣿⣷⣴⠁⠈⢧⠀⢸⡇⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣤⣀⠀⠀⠙⠛⠛⢸⡀⠀⢸⠀⣼⠇⠀⢀⣠⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣶⣦⣤⣤⣥⣤⣬⣤⣷⣶⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠉⠉⠁




⣿⢿⠿⡿⡿⡿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣦⣤⣤⣼⣇⠀⢠⡄⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⣠⣶⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣤⠼⢰⠞⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣸⣿⡀⢠⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡆⢾⡀⠀⠀⣀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣸⣩⠥⠀⠊⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠂⠁⠀⢸⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀⠀⠀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠁⣄⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⡿⢰⡄⡀⠀⢀⡇⠰⠄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢀⡀⠀⠀⠀⣰⠀⠀⢤⠄⠀⠀⠀⠠⠤⠠⢤⣴⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣾⡧⠈⣀⣸⣿⠀⠜⣛⠛⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⢸⣇⢀⠐⠲⣟⢀⡀⠀⢀⡿⠦⡴⠒⠊⠹⠦⠌⣿⡿⡿⠛⠕⡹⠋⢙⠿⣉⢀⠀⠐⠻⠿⣀⡀⣾⣿⡄⡶⠃⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⡾⠟⠛⠁⢀⡔⠋⠙⠲⠥⠀⠖⠀⠀⣶⣈⣠⠔⠐⠁⠈⠀⢤⠀⠁⣶⠀⠈⠐⠁⠧⠤⠄⣴⡀⠈⠈⠍⠣⣠⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠀⠥⠀⠀⠐⡉⠛⠀⡆⢀⠀⢢⣀⠀⠀⠉⠀⠀⠀⡀⠔⠊⠁⠈⣄⢰⣿⡀⢈⠲⠀⠐⢀⠀⢛⠹⢇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠉⠀⠠⠀⠄⠁⠐⢸⣧⠼⠄⠀⠀⠀⠀⠀⠀⣸⡀⠀⠠⠀⠁⢰⣿⣸⣿⣧⡠⠤⠔⢂⠤⠂⠁⡀⢸⠀⠀⢠⠃
⠀⠀⢹⠀⢊⠅⣀⠐⠀⠠⠤⣀⠀⢠⣴⣶⡶⣶⣭⠀⠀⠒⠓⠁⠠⢦⠀⣿⣇⠩⠗⠒⡢⠍⠉⠉⠀⠀⡀⠔⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠀⠍⠀⠄⢈⠁⠀⡀⢀⡀⠀⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠂⠬⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠁⠹⣿⣿⣶⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⣲⠆⡤⣤⣠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠒⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠚

⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠀⢀⣀⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣀⣬⣶⣿⣷⣄
⣸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠇⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠸⣿⣿⣿⣿⣿⣿⣦⣄⠀⠸⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠻⣿⡏⠉⠁⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠘⣿⣿⣿⠿⠃⠐⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠘⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⣤⣾⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⢢⣤⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⡿⠟⠁

⠀⠀⡀
⠀⠀⣸⡍⠢⠴⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡳⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡈
⠀⠀⢧⡁⢕⣶⣟⣋⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠁⠈⠉⢢⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣽⣷⣿⣻⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠉⠉⠉⢳⡼⠁
⠀⠀⠀⠉⠙⠛⠛⠻⠶⠀⠀⠀⠀⠀⡈⠃⠀⠀⠀⠀⣠⠏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠷⠀⠀⠀⠀⠀⢸⣻⠁⠴⠶⠖⣢⣞⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠋⠓⣂⡡⠋⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣿⣛⣛⣻⠟⠁⡾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠍⣱⠏⠉⠀⢀⣾⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣻⡟⠉⡕⣀⣼⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣮⣷⣶⣿⡛⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣿⣶⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⢀⢦⠠⣀⣀⣈⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡡⣤⣄⣉⣛⣷⣤⡀
⠀⠀⠀⠀⠰⠁⠀⠳⢀⠀⠈⠉⠁⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢻⣦⡈⠉⠉⠉⣉⢿⣄
⠀⠀⠀⠀⠸⡄⠀⢰⠀⠁⢤⡀⠀⠈⢈⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⣸⡇⠉⢦⣔⡚⠋⢣⣻⣷⡀
⠀⠀⠀⠀⠀⠙⠷⣾⢄⣀⡀⠈⢱⣤⣤⢤⣭⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣠⣶⣉⢻⣶⣿⣿⣯⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠫⠏⠤⢿⣽⣿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⠿⣿⣿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠑⠄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⠀⡀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⢱⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⣓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣄⠩⣽⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠃⠠⣤
⠀⠀⠀⠀⠢⠘⠀⠀⠀⠸⡀⠀⠀⠏⠖⠒⣀⣀⣤⣤⣤⣴⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠖⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠆⣀⣀⣀⣺⣿⣿⣿⣿⣿⠝⡛⢯⡬⣬⣭⡛⢿⣿⡏⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠄⠀⠀⠄⠃⠰⠃⢠⠒⠛⠋⣽⣿⣿⣿⣿⡿⠴⠷⣶⠀⠀⢳⣿⣿⣿⡆⢹⠀⠀⠀⡠⠀⠀⠀⠀⠁⠀⠀⠀⠤⠶⠂
⠀⠀⠀⠀⠄⠀⠰⡄⠀⠃⠀⠀⢸⣿⣿⣿⣿⣾⣿⣄⣸⡫⠀⢀⢘⣄⢠⣿⠃⡎⠀⡠⠀⡥⠊⠀⠐⠦⠄⡰⠈⠙⠒⠂
⠀⠀⠀⠀⢀⠈⠀⠑⡀⠰⠀⢀⣿⣿⣿⣿⡿⠟⠛⠻⣿⣷⣾⡇⣞⣿⣷⠎⣸⣠⠎⢀⠀⠠⠖⠋⠁⠀⡐⠀⢀⡈⠴⠊⠁
⠀⠀⠀⠀⠀⠑⠤⠀⠁⠀⠃⢸⣿⣿⠟⣁⣤⣾⣿⣦⠈⢻⣯⣄⡠⠁⠀⢰⡟⠁⠠⠁⡀⠴⣊⡤⠔⣉⠀⠀⠉⣀⣄⡀
⠀⠀⠀⠀⠐⠀⠐⠋⠵⢶⣄⣿⣿⣡⣾⣿⣿⣿⣿⣿⠇⣼⣿⣿⡅⠀⠀⣿⡇⠀⠠⠊⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡿⠟⠋⠉⢟⣿⡟⣰⣿⣿⣿⢧⠀⠀⣿⡇⠀⠀⠀⢀⠔⠀⠈⠒⢀⠒⠦⠖
⠀⠀⠀⠀⠀⠀⠂⠀⠀⣼⣿⠟⠉⠁⢠⠄⠀⠰⠉⢁⣿⣿⡿⠁⢸⠀⠀⡟⠀⠀⠈⠀⠈⠷⡙⠦⢀⡀⠁⢄⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⢴⣦⣤⣴⡆⠀⠀⣾⣿⣻⣶⢦⠶⣧⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣷⣷⣿⣾⣿⣿⣿⠇⠂⡸⠿⠿⠿⠶⠶⠞⠛⣄⠀⠁⠒⢂⣴⣶⣶⣤⣶⣆⢀⢀
⠀⠀⠀⠀⢸⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⡁⠀⢀⣀⣀⣀⣀⣤⣤⣄⣀⡬⠳⠆⠚⠘⠟⠻⠛⠟⠉⠁⠁⠙⠛⠑
⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠈⠉⠙⠋⠙⠉⠉

⠀⠀⠀⠀⠀⠀⣜⣭⣙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⣨⣭⢂
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣴⣿⣿⣿⣯⡆
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢠⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⠿⠿⠀⠘⠒⣶⡖⣶⣶⡖⣶⡶⠐⠁⠐⠿⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⢿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠛⠁⢻⣿⠇⠘⠃⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡇
⠀⠀⠀⠀⢸⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡇
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠻⠄⠀⠀⠀⠀⠀⠰⠋⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠹⡄⠒⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠒⣰⠃
⠀⠀⠀⠀⠀⠙⢶⡛⠛⠀⠀⠀⠀⠀⠀⠀⢤⣠⣤⠶⣄⣀⠄⠀⠀⠀⠀⠀⠀⠈⠛⣳⡾⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⢣⡀⠀⣠⠎⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⣶⠶⠦⠤⠽⠿⠥⠤⠶⠶⡖⠚⠋⠉⣰⠏⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡆⠀⢠⠀⠀⢠⡀⠀⡀⢱⡀⠀⣰⠏⠈⠁⠀⣿⣟⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢳⣤⢮⣴⣶⣼⣧⣤⠇⠀⡇⢠⠏⠀⠀⣠⠖⢋⣽⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⡿⠃⣀⡴⠋⠁⠀⠈⢀⣤⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠸⣿⣿⣿⣿⡿⠀⣀⡼⠓⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠒⠠⠤⠤⠤⠤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢨⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⢼⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⣙⠀⠀⠀⠀⠀⠀⠀⣰⣶⡆
⠀⢠⣅⠀⣠⣬⡉⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣭⣥⣄⠀⠀⠀⠀⠀⣴⣦⣭⣴⣄⠀⠀⠀⠀⣀⣠⣍⣿⣄⠀⠀⠀⠀⠀⣀⠬⣿⠥⣀⡀
⠀⠀⠻⡿⠟⢻⣿⣿⢿⣦⠄⠀⠠⠀⠀⠀⢀⣿⢿⣿⣿⢿⣇⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⢠⣀⣽⣿⣿⣿⡏⠀⠀⠀⢠⣾⠟⠉⣿⠀⠻⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣶⣴⣖⠀⠀⠀⢸⣿⣾⣿⣿⣾⣿⠀⠀⠀⣿⣯⣿⣿⣿⣿⣿⠀⠀⠉⠉⢩⣿⣿⣧⠀⠀⠀⠀⠻⡷⣶⣿⣶⠻⠟
⠀⠀⠀⠀⠀⣿⣿⣿⠏⢹⢠⡀⣀⠀⠀⠀⢨⡏⣿⣿⣿⡏⠉⠀⠀⢀⣯⣸⣿⣿⣿⡏⠉⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⢀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢹⣿⡿⠀⠸⠚⠑⠘⠀⠀⠀⠸⠷⢸⣿⣿⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⠈⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢈⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠁⢸⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠘⠛⠀⠛⠃⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠙⠋⠀⠙⠃
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⣤⠀⣄⡀⢀⠤⠤⠤⢤⡀
⠀⠀⢈⡩⠷⠀⠀⠀⠀⠀⠀⠀⠷⠶⠀⠀⠁⢤⡾⢷⣬⡶⠄⠀⠀⡇
⠀⠀⣶⣷⣦⣠⡀⠀⠀⢠⣶⣶⣶⣷⣶⣄⠀⠀⠈⠀⠚⠯⠰⠷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⣿⣿⣿⣿⣵⣾⠃⠸⣿⣸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠖⠓⠐⠆⠀⠀⠀⠀⠀⠀⠀⢶⠲⡶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤
⠀⢠⣿⣿⣿⠉⠉⢰⣾⣦⣿⠁⠁⠈⠉⠀⢰⣤⣤⣤⣤⣤⡀⠀⣥⡤⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣴⣦⡄⢸⣇⡂⠀⠀⣐
⠀⠈⣿⣿⣿⠀⠀⣼⣿⣧⣟⣀⣀⣀⣀⣀⣈⣿⣿⣿⣿⣿⣧⣤⠀⡇⠀⠀⠀⢸⣄⢀⣤⠠⠤⠾⢤⣔⣒⣶⣶⣶⣶⣿⡿⠧⣼⡟⠛⠛⠛⠛⡆
⠀⠀⢹⣿⡏⠀⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠓⠒⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⠘⠛⠛⠋⢻⣿⣿⣧⣤⣴⣿⡇⠀⠀⠀⠀⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⠇⠀⠀⠀⢸⠖⠒⠒⠒⠂⠶⠐⠒⠒⠒⠊⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣰⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠹⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷
⠀⠀⢠⣾⣿⣶⣿⣷⡄⠀⠀⢀⣄⠀⠀⠀⠀⣴⣶⡾⣿⣶⡄⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣶⣦
⠀⠀⣼⡟⣿⣿⣿⢹⣷⠀⠰⣿⣟⡷⠀⠀⠸⣿⡜⠁⠙⣹⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⢠⣼⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⠋⣹⣟⢻⡇
⠀⠠⢻⠗⣿⣿⣿⠸⠿⣤⡾⣿⣿⣿⡄⠀⠀⠈⣿⣤⣼⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣏⠻⣷⣦⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⢸⡿⣸⣿⣿
⠀⠀⢸⢸⣿⠛⣿⡇⠀⠀⠀⣿⣿⡟⠇⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡆⠀⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢠⣿⡟⣿⡆
⠀⠀⢘⢸⣿⠀⣿⣷⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⣧⠠⠃⠘⠛⠛⠛⣿⣿⠀⠀⠀⠀⠀⢸⣿⠁⣿⣧
⠀⠀⠀⠸⠟⠀⠿⠟⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⣿⡿⠀⠣⣀⠀⣀⠜⠻⠿⠇⠀⠀⠀⠀⢸⣿⠀⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠢⢀⠌⠀⠨⢤⠄⠄⠀⡄⠚⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡂⠀⠸⡄⠁⠁⢠⡾⠴⠥⣀⠀⢀⠀⡧⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⡀⠀⠀⠀⢤⠐⠛⠀⠀⢀⠆⠂⠀⠚⡀⠸⡞⠒⡤⠦⣥⠠⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠡⠈⠄⠳⡠⠴⢂⠚⠚⠆⠀⠀⡆⠀⠀⣀⠤⢼⣈⢃⡈⠀⠀⡄⠀⠁⠀⠡⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠰⠀⣄⣄⠀⡼⠅⡀⠈⠀⠀⣸⠄⠒⣇⢀⡎⠉⠀⠀⠀⠀⢥⠱⡚⠽⣆⠀⡀⠂⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠧⠲⠔⡨⢂⡩⣍⡑⣄⢸⡶⢒⡘⠁⢔⢟⠉⠝⢄⣀⡀⠀⠀⠀⣡⠊⢀⡞⠉⠒⠀⢸⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠀⠘⠚⢴⠀⢠⣀⢹⠈⡆⠀⠠⢁⠞⠆⡠⢆⠜⢘⣨⡉⠸⢟⡳⢁⠴⠓⠠⢤⡀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⠀⠒⠆⠈⢿⣀⠁⠑⡶⠀⠀⠀⢇⠀⠀⡀⠁⠀⠀⠀⠙⢧⡶⠉⢑⠄⢡⢖⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠧⡀⠀⠀⠀⠘⠀⠀⡃⡐⠁⠜⠉⠉⢉⠅⠀⠀⠀⠀⠀⠈⠆⢠⢀⠈⣃⢘⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠂⠅⡠⠀⣰⠀⣠⢀⠀⠘⡀⠐⠒⡉⡀⠀⠀⠀⢀⠌⣐⠀⡀⠄⠫⠑⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠈⠀⠒⠉⠉⠀⠠⠗⠤⠤⠲⠟⠦⠉⠉⠉⠈⠂⠈⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠤⠄⠀⠀⠀⠀⠠⠤⠐⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠤⠶⠀⠀⠀⠀⠂⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠊⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡊⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⣤⣧⣴⣶⣶⡶⠷⠶⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢟⡿⢏⣠⣶⣿⠟⠁⠀⠀⠀⠀⠀⢀⣠⣾⣇⠜⠉⢂⡇
⠀⠀⠀⢠⠀⠀⠀⠀⠈⠐⠋⢽⢶⢿⠿⠛⠁⡀⠀⣀⡁⠀⠄⠀⠉⠛⠋⠹⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠭⢅⠀⠀⠀⣠⣀⢲⠢⣄⣄⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⢀⠤⠊⠰⠆⠀⠀⠀⠀⠈⠄⢀⡀⠳⠷⠀⠉⠉⠡⠤⠋⠋⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠀⠦⠀⠢⢀
⠀⠈⠒⠄⠀⠀⠀⠀⠀⡇⠀⢠⠀⢀⠀⠀⡉⠉⠀⠀⠀⠈⢉⢁⠀⡀⠠⡀⠰⢄⠀⠀⠀⠀⢀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⢅⠀⠀⠀⠗⠀⡜⠀⡏⢀⠀⡇⠀⢨⠀⠀⢇⠈⡈⡀⡇⠀⠱⡄⠀⠓⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡈⠀⠁⠀⠘⠀⠀⠘⠀⠃⠁⡀⠀⠀⠄⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⢀⡀⣠⣄⣀⣀⠀⠀⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⢋⣵⡉⢹⠈⢉⣹⠋⢉⣿⢉⠉⣿⡁⠀⠄⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣤⠀⠀⠀⠀⢸⡇⠀⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠛⡆⠀⠸⠇⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢀⡀⠀⠀⢙⡆⠀⢰⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⣾⣇⢀⣼⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠉⠉⠉⠁⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡎⠀⠀⠀⠀⢀⣤⣄⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⣴⣿⣿⣿⢢⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠛⠟⠛⢁⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⣀⣠⣤⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣴⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡈⠀⠀⠈⠉⠉⠛⠛⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠅⠀⠀⠀⠀⢀⣀⣀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠘⠃⠀⣀⠖⠊⠉⠉⢠⣿⣱⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠋⠉⠛⣦⣤⡾⠂⢰⡇⠀⢀⡶⠾⢿⣧⣼⣿⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢹⣇⣀⣀⣈⣿⣷⡟⠀⠚⢾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢈⣻⣿⣿⣿⣿⣿⡃⣨⣿⣿⣿⣿⣿⣿⣷⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠣⠀⠘⠃⠀⠎⠀⠀⠀⠀⢔⠋⢩⡍⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⡿⢷⠼⣿⡟⡀⠀⠀⠀⢴⣿⣟⣹⣿⣶
⠀⠀⠀⠀⠀⠀⢘⢀⠖⠤⡀⠤⢌⡇⠀⠀⠰⠈⠀⣀⠀⠀⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠻⠿⠿⠿⡇⠀⠀⣰⣯⣭⣭⣭⣭⣵⣧
⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⢈⠃⠀⠀⡇⢠⠔⠉⠑⠦⢤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⠟⠁⠙⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⡸⠀⠀⠀⠇⠰⡀⠀⠀⠀⡜⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⡠⠁⠀⠀⣿⣿⡀⠀⠀⠀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣲⠤⠠⡊⠀⠀⠀⠸⢄⡀⢑⡤⢤⡞⢀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠄⠠⡖⠁⠀⠀⠸⢿⣿⣷⡤⠤⣾⣿⣿⠗
⠀⠀⠀⢀⡠⠔⠒⠊⠳⣄⡠⠊⠑⠒⠤⣄⠠⠄⠚⢄⣠⠟⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣷⣦⣴⣾⣷⣶⣤⣄⠠⣤⣾⣤⣤⣷⣦⣤⣄
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠇⠀⠀⠀⠀⠀⠐⠂⠈⠇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣷⢻⣿⣿⣿⣿⣿⣭⣿⡇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡆⠀⠀⠀⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠛⠷⣦⣤⣴⠂⠐⢿⣶⣶⠶⠿⠿⢿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠵⠖⠒⠒⠲⣾⣿⣿⠅⠀⢠⡤⠤⠤⢤⣄⠀⠙⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⠀⠀⣴⣶⣿⣿⣿⡂⠀⠠⣀⡀⠀⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠙⣿⣏⣿⡿⠇⠀⠤⠌⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣰⣿⣿⣿⣿⡟⠁⠀⠀⠀⠙⢷⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⡿⠿⢿⣧⡤⢤⣄⠀⠘⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡿⢁⣀⣀⣀⣀⣠⣀⡈⢧⠀⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⠘⣧⣾⣿⣿⣿⣿⣿⣿⣿⣾⡇⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠀⢻⡏⠀⠀⠀⠀⠀⠈⣿⡿⢀⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⣶⡁⢤⣀⡉⢉⣀⣠⠟⠁⠸⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣭⣛⠛⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⠶⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⡀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⠄⠀⠀⠃⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⢤⡄⠰⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⠀⠀⠀⠐⠀⢀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠉⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⢹⡇⣿⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣴⣾⣾⡄⠀⠀⠀⣀⡀⠀⠘⣧⢸⣿⠃⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⣿⣿⣿⡇⠀⢰⠏⠛⣷⠀⠀⣿⣾⣿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢩⣿⣿⡇⠀⠈⠀⠀⣼⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣿⣿⠀⠀⠈⠀⠀⡏⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⠀⠀⢹⡇⠀⠀⠀⠀⠘⡇⠀⢠⣿⠃⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣾⠁⠀⠀⠀⠀⣸⣀⣤⣼⣿⣶⣶⣶⣶⣤⣄⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠛⠿⢿⣿⣿⣿⣿⣿⡿⠟⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠲⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢁⠀⠀⠀⠀⠒⢤⡀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣚⣒⡛⠒⠒⠶⠤⣀⡀⡙⠂⠀⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⣿⣿⣿⣿⣷⣶⣤⣉⠛⢦⡀⠘⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡦⢻⣀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡼⠉⢀⠈⠉⠙⠻⢿⣟⣛⢡⣾⣿⢘⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⢉⣉⣉⡑⠂⠀⠰⣿⠏⣾⣿⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠘⣯⠴⠦⠄⠀⠀⠀⡟⣈⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠘⢮⣁⣤⣤⣴⣶⢡⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⢤⣤⣀⣛⣛⣛⣡⣾⣿⠟⢁⡟⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⡿⠹⣿⣿⣿⣿⣿⠟⢁⣤⠏⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⢸⠃⠀⠻⠛⢋⡽⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠙⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣄⡀⢀⡏⠀⠀⢀⡴⠁⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⣀⣠⠴⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣄⣀⡞⠀⠀⠀⡠⠋⠀⠀⠀⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠿⠶⠶⠶⠚⠛⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣌⢻⣯⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢹⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠙⣿⣿⡀⠹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠀⠀⠀⠀⠀⢀⣶⡿⠛⠉⠀⠀⠈⠁⠀⠀⠈⢻⣷⠀⠈
⠀⠀⢰⢲⠄⡆⠀⢠⢦⠀⡔⠒⢠⣄⠖⠀⠀⠀⣿⣿⣤⣶⣶⣦⠀⠀⠉⠀⢀⠤⣶⠖⣶⠀⠀⠀⠀⠘⣿⣇
⠀⠀⢸⡩⠆⣇⡀⠞⠘⠆⠧⣀⠸⠍⠦⠀⠀⠀⣿⣿⣏⠀⣀⣀⣀⠀⠀⠀⠈⠒⠛⠋⠀⠀⠀⠀⠀⠀⠈⠻⣧
⣿⣿⢿⣿⢻⠛⣿⡟⣿⢻⣿⡟⣿⢻⣿⣿⣿⣿⣿⣿⢻⡿⣋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿
⣿⣯⣾⣹⣏⣿⣹⣧⣿⣼⣿⣇⣿⣸⣿⣿⣿⣿⣿⣿⡌⣷⠀⠀⢠⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡿
⠀⠀⠀⢠⠀⠀⡀⡄⡀⠀⢠⢀⠀⢀⡀⡀⠀⢨⣿⣿⡇⢻⡄⠀⠀⠋⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠈⠐⠂⢀⣁⣁⠁⠀⠈⠐⠊⠁⠁⠀⢸⣿⣿⣿⢸⣿⡄⠀⣼⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣰⢿⣿⠄
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⣿⣿⣆⠈⠻⠿⠿⠁⠀⠀⠀⠀⠀⣠⡾⠁⢘⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⢻⣿⣿⣧⡀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⢠⠋⢸⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⡀⡀⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠂⡀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⠳⢄⠀⡀⠀⠀⠑⠂⠉⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡆⢀⢜⠀⠑⣶⠒⣦⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠘⢇⢸⠁⡰⡀⠎⡇⠐⢂⠀⠀⠀⠀⠂⠱⢄⠀⠀⠁⠀⠸⠀⠰⢖⠢⢤⣀⣀
⠀⠀⠀⡠⣂⠵⡂⡉⢢⠄⠐⠒⠈⠉⠉⠉⡽⠄⡢⠰⢁⠄⠎⠢⣀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠖⡑⠠⡀⢱⣤⡀
⠀⢀⡾⡎⠁⠆⡈⡾⠁⠀⡠⠒⢐⢄⡠⢊⠜⠑⢦⠒⠂⢍⡀⢒⠨⠉⢡⠀⠀⠠⠤⠤⠤⡀⠀⢠⠀⠠⡀⠶⠀⠁⢸⠇⡅
⠀⠰⠀⠛⣈⣁⡜⠀⠀⢊⡀⢳⣼⣾⢗⡃⢅⠢⣡⡄⠧⠎⢥⣀⣠⢀⢠⠢⡀⠀⠀⠀⠀⠈⠢⡇⠀⠈⠠⡐⣤⢞⠍⡰
⠀⠀⢧⠀⠢⡀⠻⠀⠀⠢⣩⢛⡟⣁⠻⠂⠈⠁⢀⠄⣀⡀⠀⢡⠭⢡⠰⠑⢄⢹⢀⠁⢀⢱⢀⠐⢄⠀⢠⠀⡠⠊⡔⡃
⠀⠀⠰⠐⠠⡈⠉⠂⠠⠎⡸⠋⢌⣡⠶⣖⠒⠚⠍⠉⣉⣀⡠⠂⣭⡈⢰⡀⠱⡜⡇⡔⢀⡟⡎⣥⠂⡴⠇⠊⡠⠊⢀⠃
⠀⠀⠀⢀⠀⠈⠐⠢⠄⡀⠀⠁⠀⠐⠀⠈⠰⠖⠀⣀⠀⣅⠒⢛⣐⣲⣂⡱⠅⠤⠌⠙⠋⠐⠃⠉⢀⡠⠔⠊⠀⠀⡜
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠉⠐⠒⠀⠄⠤⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠤⠄⠒⠉⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠄⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠈⠻⣧⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣴⠀⣰⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣇
⠀⠀⠀⢐⠀⠀⢰⠉⠛⡙⠦⡀⠈⣙⣀⣮⠀⡄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡎⣠⡿⢀⢻⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣦⡆⠀⠈⣼⡀⠀⠈⢢⠱⠆
⠀⠀⠄⠈⢳⣄⡼⣄⠀⠐⠄⠸⣄⣸⡟⢉⢫⣧⠀⠠⢀⠀⠀⠀⠀⠀⠈⠃⣻⡇⡾⣧⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢐⣾⠄⠀⢸⣧⣄⠀⣿⣶
⠀⠀⠠⠌⠉⠙⠋⡈⠷⣿⡆⣀⠀⠁⠈⠑⣷⣌⢷⡀⠈⠧⠀⠀⠀⠀⣠⠈⣿⣶⣵⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⣣⣌⠰⡿⢡⠂
⠀⠀⠀⠀⠲⡀⡈⠀⠀⠈⢹⢿⣣⣘⡀⠀⠈⣋⣄⡳⢄⡀⠄⠀⠀⠀⣿⢸⢠⡻⠉⠄⣴⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠢⠘⡦⠀⠉⡉⠰⠃⠁
⠀⠀⠀⠀⠀⠘⢹⣤⣀⠀⠀⠀⢴⠿⠟⣁⠰⢏⡁⠐⠄⡈⢁⠄⠀⠀⡟⣶⡈⠃⣰⢧⡿⠀⠀⠀⠉⠸⢷⡀⠈⠃⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁
⠀⠀⠀⠀⠐⠒⠤⠬⠟⠋⠈⠁⠋⠁⠀⠀⠀⠀⠈⠁⠀⡈⢦⣦⠂⠀⠘⠌⠻⠜⡷⠋⠀⠀⠀⠀⠣⠀⠛⢯⡀⠀⢰⠀⠀⢠⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⢿⣧⠂⠀⢸⣇⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠘⡄⠀⠀⠙⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣷⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠰⣏⠀⢸⡃⢁⠀⣾⠃⠀⡹⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢉⢀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣄⠀⡆⠀⠙⢷⣼⠃⢸⡏⠁⠀⠀⡇⣧⠂⠀⠀⠀⠀⠀⠀⠑⠂⠀⢾⠣⠠⠇⠂⠀⠀⠈⣱⡇⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢄⠖⡔⠈⠂
⠀⠀⠀⢰⣇⠀⠾⠀⠿⠀⣾⣝⠀⠂⡠⢡⣵⠀⠀⠀⠀⠀⠈⠀⠀⡠⠀⢠⡀⠀⠀⠀⠠⠤⠀⢈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦⢀⡼⠃⢆⡤⠄
⠀⠀⢸⠀⣿⠀⠂⢠⡆⢻⣫⠋⣴⠟⢁⡿⡙⠀⠀⠀⠀⠀⠀⢘⡀⠁⠀⡘⠷⠈⣀⣀⣀⠀⣠⣠⣴⡷⢒⠎⠀⠀⠀⠀⠀⢀⢠⣇⣴⠟⠩⢅⡈
⠀⠀⠀⠳⣟⡆⠀⣨⠄⢼⣇⠀⠁⣄⣾⡷⠀⠀⠀⠀⠀⠀⢰⠈⠟⠀⣠⡴⠂⠈⢀⢩⣿⠻⠋⠙⠋⢀⠘⡀⠀⠀⠀⠀⠀⣬⣵⡟⠉⠄⠊⠁
⠀⠀⠀⠀⠸⣧⡤⠀⡠⠀⢈⠀⣴⢿⡟⠁⠀⠀⠀⠀⠀⠀⠘⢠⡙⢴⢫⠁⠀⢰⡏⠀⠀⠀⢀⠀⠀⣸⠦⠀⠀⠀⠀⣰⡾⠋⠈⠁⠀⠰⣄⢀⡀⡄
⠀⠀⠀⠀⠀⣿⠁⣼⣁⠤⡌⠀⡷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢠⡉⣠⣴⣯⡀⠐⣆⠀⠈⠲⣴⡷⠁⠀⠀⢀⡶⠉⠀⠀⠀⠀⠀⠄⠁⠀⢹⠗
⠀⠀⠀⠀⠀⢷⠺⣘⠋⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠂⠒⠲⠽⡟⣛⠡⠾⠛⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠐⣦⣤⠏⠈
⠀⠀⠀⠀⠀⢸⠀⡍⢡⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠤⢸⡼⢖⠸⠀⠄
⠀⠀⠀⠀⠀⠀⠛⠡⠿⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢔⣂⡱⠄⡀⠄⠀⠀⠀⢀⣉⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢏⡅⢨⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠰⠀⢀⣙⡛⠀⣿⢿⣋⡁⣴⠁⡂⠀⠀⠀⠀⠙⣤⢀⣤⠀⠖⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢦⠁⣦⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡾⣷⡀⠀⡉⢀⡌⣁⡤⢹⠟⡬⢤⡄⠀⠀⠀⠀⣀⠘⠐⢿⣛⣾⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⡸⡇⠠⠀⢹⣠⠂⠐
⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⢀⠌⣼⣧⣌⡿⠌⠛⠭⠂⠉⡤⠋⠂⠤⠴⠆⠀⠀⠀⠀⠀⢤⠀⠄⠠⠝⠳⢿⡷⣌⠃⠀⠀⠀⠀⠀⠏⡇⠀⠀⠻⡟⢁⡇
⠀⠀⠀⠀⠀⣨⠁⠀⢠⠾⣋⣀⣡⣬⠒⢀⣾⣌⠙⠃⠉⠁⠲⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡴⣿⣿⠈⠓⢄⠀⠀⠀⠀⠘⣿⡆⣰⢘⡀⡼
⠀⠀⠀⠀⠒⠒⠒⡀⠀⠹⠄⠀⠄⠰⢥⠀⠀⠰⠐⠶⠦⡔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣡⠃⠀⠀⠀⠱⡀⠀⠀⠀⠈⠓⠇⠞⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠲⠀⠘⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣤⠈⠄
⠀⠀⠀⢔⠐⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡱⠀⠀⢠⣿⣿⠿⠛⠉⣇⠸⡀
⠀⠀⠀⣀⣽⣿⠦⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠱⠀⠀⠘⣿⣇⠀⠀⠀⡇⠀⠈⡄
⠀⠀⢠⣿⣿⡁⠀⠈⠢⡬⠒⢄⠀⠀⠀⠀⠀⠀⡄⢣⠀⠀⠈⢻⣷⡖⠈⢳⠐⢄⡁
⠀⠀⠀⠙⢿⣥⣀⣀⠠⠧⡄⠀⠵⢄⠀⠀⠀⠀⠡⠘⠁⠀⠀⠀⡜⠁⠈⠁⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠔⠁⠀⠀⠀⠈⠢⡀⠀⠀⠙⠢⠄⠐⠘⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠔⠁⠐⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠐⡀⢀⠤⠒⠐⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠀⠀⠀⠀⠀⠘⡳⠀⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠃⠂⠈⠉⠢⡀⠀⠀⢀⡠⠄⠂⠁⠀⢡⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠎⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⡘⠀⠀⠀⢀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡌⠀⠀⡠⠊⠉⢄⠀⠀⠃⠀⢀⠁⠀⡠⠂⠁⠈⠢⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠰⠢⠄⠀⠀⡘⠀⢀⠎⠀⠀⠀⠀⠡⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠈⠢⡀⠆
⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⡰⢀⠔⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠐⠦


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣄⠀⢠⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠈⢿⣦⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠈⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣷⡄⠀⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠻⣿⣿⣿⣿⡿⠋⠀⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠃⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠃⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠠⣤⡄⢀⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠸⠁⠘⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡁⠀⢉⡿⠿⢻⡅⠀⣠⣄⡀⡁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠑⠖⠋⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠑⢦⠀⠀⠀⠀⠀⠀⢀⣀⡸⠀⠀⠀⠀⠀⠀⠘⠦⣄
⠀⠀⠀⠀⠀⠀⡠⠤⠔⠂⠁⠀⠀⠀⢈⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠤⠂⠀⠀⠀⣀⣀⡇⠀⠀⠀⠀⠀⢠⠖⠉⢙⡤⣄⢀⡴⣺⣕⢦
⠀⠀⠀⠀⢀⠎⠀⢀⠤⣤⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢸⡄⠀⠔⠁⠀⢀⣤⣀⣘⠅⠀⠀⠀⠀⠀⠀⠀⠈⡳⢒⡽⠊⠹⣾⠏⠀⢸⢸
⠀⠀⠀⠀⢸⠀⠀⣿⠀⢀⡷⠀⢸⠁⠀⠀⠀⠀⠀⢀⡴⠋⢀⣠⣄⣀⣀⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⠀⠀⠀⠙⣄⠀⣸⢸
⠀⠀⠀⠀⠈⢢⡀⠈⠛⠛⠁⢀⠏⠀⠀⠀⠀⠀⠀⠸⣀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⣈⠢⣽⢸
⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠐⡁⠀⠑⠌⣙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⡞⠁⠀⠈⣇⠤⢤⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⢱⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢄⡀⠀⡠⠖⠋⠉⠹⠶⣄⡤⠶⢼⡇⠘⡰⠊⡁⠀⡼⣀⣀⣀⠀⠀⠀⢸⠀⢄⠀⠀⢀⠠⢂⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠸⡟⠁⠀⠂⠀⠂⠀⢘⡏⣀⢉⡇⡞⠠⠊⡠⢾⠟⠁⠉⠈⠦⡄⠀⠘⢆⠀⠉⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡿⠁⠀⠀⡇⠀⠀⠹⡏⢉⣿⡫⢍⡲⡏⢉⣈⡉⠳⢧⣔⡋⣴⠛⠀⠀⠀⠀⢠⠇⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢹⠣⢤⣀⡀⠀⠐⠀⣠⠟⠧⣴⠛⢖⠋⠍⢿⡴⣫⠟⠅⢈⡿⠀⠀⢀⣀⡠⠔⣻
⠀⠀⠀⠀⡼⠻⠷⠈⢳⡀⠀⠈⣇⠀⠀⠉⠉⠓⠒⠓⠲⠶⠷⠦⠤⠷⠤⠼⠶⠾⠾⠖⠚⠓⠚⠉⠉⠀⠀⢀⡇
⠀⠀⠀⠀⠸⡄⢀⢂⠀⢻⡄⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠹⡄⢂⠀⢀⠙⣆⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢀⣠⢴⡲⡄
⠀⠀⠀⠀⠀⠀⠹⡔⠀⠐⣆⠘⣦⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⣠⠴⠚⠉⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠹⡔⠔⢌⢦⡈⢷⡀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠁⠀⠠⡖⣏⠀⠀⠀⠀⠀⡰⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠢⡁⢵⣌⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠮⣑⠲⠤⠖⣊⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠁
⠀⠠⠀⠂⠠⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡼⠀⣤⠈
⢠⠀⠀⡀⡀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢧
⠀⢀⠐⡇⠃⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠁⠀⠂⣀⠐⣀⠐⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⡀
⠀⠈⡄⣐⢁⡼⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠄⡄⢀⡀⠀⠠⠁⠀⠀⠀⠀⠄⠀⢀⢈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢀⢀
⠀⠀⢣⢸⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠
⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠣
⠀⠀⣸⣿⢣⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠀⠀⠐
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠂⠈⠃⠀⡀⠀⠀⠄⠢⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂⠀⡉⠠⠀⠐⠂⠁⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⣜⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡇⢹⡴⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⣇⠀⠀⠀⠀⠀⢠⠌⠳⣊⡐⠻⢄⣮⡁⠀⠀⠀⠀⠀⠀⢀⠀⡈⠤⠀⢠
⠀⠁⠀⠀⠀⢤⠀⠀⢠⠀⠀⠀⠀⠀⠸⡿⠘⠀⠀⠀⠀⠀⠀⠹⠁⡁⠈⡎⠀⠈⠁⠃⠀⠀⠀⡀⠀⠀⠀⠁⠀⢠⠈⢠
⠀⠀⠀⠄⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠋⢀⠀⠶⠊⠀⠀⠀⠀⠄⣀⠁⠀⠠⠖⠀⡀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠈⠠⠀⠀⠀⠀⠀⠀⡀⠀⡀⠈⠂⠂⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢐⠀⡇⡇⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⣻⠀⡁⠀⠀⠀⠀⠀⠄⠁
⡌⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠀⢠⠙⡄⡅⠀⠀⠀⠀⠀⠂
⠀⡀⠐⠁⠀⠀⠀⠀⠠⠀⠀⡎⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠁⠀⠀⠀⠀⠀⠄
⡈⢀⠄⠀⠀⠀⠀⠀⠀⠀⡴⢀⠀⣁⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣩⠔⢉⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠠⠒⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠝⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠈⠀⠀⠀⣘⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠘

⠀⠀⣀⣀⡤⠶⠂⢤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣘⣧⡞⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡤⠒⠒⠲⣶⣠⣤⣤⡤⣤⠤⢤⣀⡀
⣦⡀⠈⠹⡍⠛⠶⣤⡀⠉⠀⠀⠀⠀⠀⣠⠤⠤⠄⠤⣄⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠉⠉⠉⠉⠁⠘⠛⠛⣿⣿
⢻⡷⠀⠀⠑⢀⣀⠀⢙⣷⣄⢀⠀⢀⡞⠋⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢽⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⡜⠳⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⡀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡈⠽⠿⠶⠥⣔⠄⠀⠀⠙⢿⣿⣷⣤⣠⣤⣀
⣶⠇⠀⠀⠀⠀⢠⣤⠶⠞⠂⠂⠠⣀⠀⢠⣠⣴⡯⢉⠹⢿⡛⠀⠀⠀⠀⡰⢡⠊⠀⠀⠀⠀⠀⡸⠿⠲⣦⠀⠀⠑⢿⣿⣿⣿⣿⣿⣶⣀⣴
⠋⠀⠀⠀⡠⠞⢋⢔⣶⠶⠞⢏⢀⣻⡿⠓⠉⠉⠛⢿⢤⡈⠳⡄⠀⠀⠠⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡆⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠁⠀⢀⣽⠏⠀⢀⠞⠉⠀⢸⡀⠀⠀⠀⠀⠱⣵⠀⡷⠀⠀⠀⣸⠀⠀⠀⣴⡿⢀⣤⠀⠀⠈⣧⡅⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⢀⣤⣀⣰⠋⣴⠇⠀⠀⠁⠀⢀⣤⣘⡢⢄⠀⠀⠀⠀⢿⣦⡃⠀⠀⠀⡷⠀⠀⠀⠘⠎⠛⠁⠀⠀⠀⡏⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠁⠀⢀⣿⠀⠀⠀⠀⠀⡼⠋⠈⣽⣮⣆⠀⠀⠀⠀⠀⠁⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠬⠭⠷⣦⣤⣤⣤⠗⡒⠉
⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⢀⠁⠀⠀⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠦⠒⠶⣼⣿⢿⡏⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠂⠀⠸⢿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⢣⣴⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⢀⣀⣠⡤⠞⠋⡁⠈⠀⠀⣧⡀
⣲⡦⠀⢠⣀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⣿⣿⣷⠡⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣉⡉⣀⣄⡤⠔⠀⠀⠀⠀⠙⠯
⠀⠉⠈⠉⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠛⣿⣿⣆⣇⠀⠀⠀⠚⠉⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠀⠀⢀⣼⣄⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠙⢋⣀⣨⣉⣓⣠⠄
⣀⣆⣀⡀⡀⠀⠀⠀⠘⣗⡀⠀⠀⠀⠀⠀⠈⢀⣿⢋⠇⠀⠀⠀⣿⣿⠀⠛⠙⢟⠲⠶⣴⠤⠀⠸⣡⣘⣷⠯⠒⠊⠙⠛⠃
⠛⠋⠉⠉⠀⠐⠀⠀⠀⠙⢜⢶⡲⠴⣶⠾⢾⡿⣧⠊⠀⠀⡄⠈⡿⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠉⠵⠂⠀⠀⠀⠀⠀⢣⠀⢿⠀⠈⡃⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⣻⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣸⣭⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣰⠃⣿⡆⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠉⠉⠛⢀⡿⠁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡀⠀⠀⠀⠀⠈⢈⡀⣾⠁
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢰⣿⣦⠀⠀⠠⠁⠀⠻⣄
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⢖⠹⠛⠍⠉⠉⠉⠍⠛⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢹⠗⠋⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⠀⠐⠃⠰⠀⠀⠁⡀⠀⠂⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠐⠈⠀⠀⡀⠀⠀⠐⠁⠠⠄⠠⢤⡀⠀⠀⠀⢀⠀⠀⠀⠐⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣴⠏⠰⢀⠀⠑⠀⠀⠁⠀⡄⠀⠂⠀⠀⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠲⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠈⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠑⠄⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠠⠀⠀⠀⢀⣿⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠄⢀⡼⢷⡤⢑⡄⠳⠀⠀⠀⠀⠀⠀⠀⠢⠐⠐⠀⠀⠀⠀⠀⠀⣼⣷⣿
⠀⠀⢀⢀⠀⠀⠀⠀⢐⣼⣾⣴⣯⣾⣽⣿⠶⢛⡛⠂⣠⡀⢁⡀⢰⣮⠉⣠⣄⡀⠀⠀⡀⣀⣸⣟⣿⠟⣥⢀⣀⠀⠀⠀⣠⠄
⠀⠒⠓⠚⠃⣐⠚⠚⢟⠋⠙⠛⠛⣻⣿⠿⢝⣿⣿⣿⣿⣳⣛⣶⣾⡷⠾⢿⡾⣿⡾⠾⠏⠷⢿⢿⡧⣞⡥⠖⠐⢛⡛⠛⠃⠀⠒⠦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣐⣁⣀⣈⣂⣉⣀⣁⣀⣀⣀⣙⣀⣀⣀⣄⣀⣀⣀⣁⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⡀⠀⠀⢀⠀⢀⡀⠀⠀⡀⠀⠀⡀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⢸⣷⣤⣴⣷⠀⠇⠳⠢⠎⢡⢰⠉⠦⠴⠉⡆⣸⣷⣤⣾⣿⠀⠰⠹⠖⠴⡍⡄⠀⠀⡘⡡⠤⢼⠱
⠀⢺⣿⣿⣿⣿⠀⠀⡄⠀⣤⢸⢰⢰⡆⢠⡖⡆⣿⣽⣿⣯⢹⠆⢀⢠⠀⢠⡄⡇⠀⠀⠃⣤⠀⣠⠈
⠀⠘⣿⣿⣿⠟⠀⢦⡀⠒⢀⠌⠘⣌⠑⠊⣡⠃⠹⣿⣿⣿⡿⠀⠹⣄⠀⠀⡰⠃⠀⠀⢧⠀⠀⠈⡸
⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠙⠉

⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠟⠛⣉⣡⣤⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⣶⣤⣤⣉⡙⠛⠿⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⠋⣁⣴⣾⣿⣿⣿⣿⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣄⡉⠻⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠛⣁⣴⣿⣿⣿⣿⣿⣿⡿⠟⠈⠉⣽⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋⣠⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⢀⣤⣽⣿⣿⣇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄
⠀⠀⠀⠀⣠⣿⣿⣿⠟⢠⣾⣿⣿⣿⣿⡛⠁⠈⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⡀
⠀⠀⠀⣰⣿⣿⡿⠃⣴⣿⣿⣿⣿⣿⣿⣿⣇⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄⠻⣿⣿⣷⡀
⠀⠀⣰⣿⣿⡿⢁⣼⣿⣿⣿⡏⠻⠛⣿⣿⣿⣿⠿⠟⣛⣛⣛⣛⣩⣭⣝⡛⢿⣧⠀⢠⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣄⠹⣿⣿⣷⡀
⠀⢠⣿⣿⣿⠁⣾⣿⣿⣿⣦⡄⢀⡘⣿⣿⠟⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⡆⣿⠀⣼⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⢹⣿⣿⣷
⠀⣿⣿⣿⠇⣸⣿⣿⣿⣿⣿⣷⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣟⣭⡾⣧⡹⢰⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⡇
⢸⣿⣿⡿⢀⣿⣿⣿⣟⠻⣿⢿⣿⣿⣿⡃⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠀⠀⢹⡇⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠸⣿⣿⣿
⣼⣿⣿⡇⢸⣿⣿⣟⡋⠀⠀⣾⣿⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣀⣄⢸⢁⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿
⣿⣿⣿⠁⣾⣿⣿⣿⣿⣠⣶⣾⣿⣿⣇⢻⣿⣿⡻⣿⣇⠈⢓⣹⣷⣈⠛⠃⣇⢺⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢹⣿⣿
⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣮⡟⠉⠭⠿⣿⡌⠙⠒⣼⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡄⢹⣿⣿⣿⣿⠹⠿⢿⣿⣿⣿⣿⡎⣯⡻⣧⡁⢀⠒⣶⣿⡇⠀⢀⡏⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣸⣿⣿
⢹⣿⣿⡇⢸⣿⣿⣯⣅⠀⠀⢻⣿⣿⣿⣿⣧⢻⣿⣾⠃⠀⠭⣽⣿⠃⠀⡼⣰⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠸⣿⣿⣷⠈⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⡿⢃⣾⣿⣿⣿⣿⣛⠿⠋⠠⣾⠱⡏⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⢿⣿⣿⣇⠸⣿⣿⡿⠿⣛⣫⣭⣵⠶⢶⠿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⡷⣤⣄⣀⣀⡰⠿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇
⠀⠘⢿⣿⣿⡄⣙⣥⡶⠟⣋⣩⣵⣶⣜⢿⣿⣌⢻⣿⣿⣿⣿⠇⠀⢠⡟⠀⣷⠏⠉⠉⠙⠛⢦⣝⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⣿⣿⡏
⠀⠀⠘⣿⣿⣿⣟⣡⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣝⢿⣿⡏⠀⢀⡾⠁⠀⣿⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠁⣼⣿⣿⡟
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣦⡹⣧⠀⣾⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⣠⣴⣿⣿⡟⢁⣾⣿⣿⡟
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠈⠻⠇⠀⠀⣠⣿⡀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠻⣄⢠⣶⣿⣿⣿⡿⠋⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⠋⠙⡇⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠹⣦⡻⣿⡿⠋⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⢷⣄⣀⡀⠀⢀⡟⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢷⣌⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡁⠀⠀⢸⣿⠟⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣶⣬⣟⣛⠋⠀⢸⠃⠀⣼⠀⠀⠀⠀⠀⢸⣿⣤⣴⣶⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⣀⠀⠀⠄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⡀⠐⠰⣦⡀⠀⣁⠈⢧⣀⣉⡈⣴⠋⠘⠀⠀⠉⢰⡢⡄⢸⣁⠒⠧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠷⠽⠾⣆⡙⠿⣷⡤⠉⢀⡉⢉⠓⠔⠐⠒⠂⠀⣌⣴⡷⢎⠺⠷⢶⢸⡏⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠈⠄⣠⣤⡤⢠⣽⣦⡤⣤⠶⠔⠀⠍⠠⠀⠀⡂⠀⢀⠈⠙⠿⣤⡀⣁⣠⠞⠁⠀⠉⠀⡙⠄⠤⡀
⠀⠀⠀⠀⠀⡰⠀⠘⠁⠀⠀⠀⠉⠀⠀⠀⢘⣿⣷⡄⠟⠒⠓⢮⣀⠰⠶⢿⣞⠉⠁⠈⠒⠶⢤⣝⡁⠀⠀⠐⠀⠐⡂⣠⠆⠴
⠀⠀⠀⠀⠀⢆⣠⡈⠐⠒⠚⠓⣤⣀⣀⣤⣾⣿⠟⠁⢀⠀⠀⠈⣿⡄⠀⠈⠻⣷⡄⠀⠀⠐⠀⠙⣿⡆⠀⠑⠒⢖⡧⠃⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠑⠙⠩⡛⠓⠋⠉⣿⠿⠿⢭⣴⣎⠀⢀⣽⡧⣿⠃⠀⠠⣸⣿⠀⡀⠄⠀⠀⠀⣸⢟⣄⣀⡗⢥⢶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠶⠄⠀⠀⡨⠛⡿⠿⠿⣿⣅⣀⣴⠟⣋⣏⣩⣲⡖⡒⠚⣉⡭⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠈⠉⠈

⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⠀⡠⠤⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠈⠀⡀⠀⡇⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⠀⠀⡤⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⢸⠀⢢⢄
⠀⡄⠁⠀⡇⠀⡇⠐⠂⠰⡃⠀⢀⡘⢂⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠘⠒⠀⠀⠀⠰⠁⣀⠀⠀⠡⠄⢸⠀⢸⠸
⠀⠀⠁⠀⡇⠀⡇⢸⠀⠀⠹⡁⠀⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠎⠀⠧⠤⠡⠆⠈⠀⠸⠀⣩
⠀⠀⠘⠀⠃⠀⡇⢸⠀⠀⠀⡁⠀⡜⠀⠀⠀⠀⠀⣀⠀⠀⠢⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡆⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠐⢤⠀⠐⠀⠄⠈⠁⠀⡄⠄⠀⠀⠁⠀⠀⠀⠁⠀⢠⣠⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠱⡀⠀⠁⠀⠀⠀⢆⠀⠀⠐⠀⠀⡰⠀⠀⠤⠀⠀⠠⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠐⠂⠀⢠⠤⡕⠢⠀⠠⡊⠀⡤⡀⡀⠄⠂⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⢀⠎⠀⡇⠀⠀⠀⠂⠈⡅⢩⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠰⢄⡀⠀⡀⠎⢠⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⡆⠀⠀⠈⡀⠠⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⣶⣶⣷⣆⠀⠀⠃⠀⣃⣉⢡⠀⢃⢀⣠⣼⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⢻⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠋⠁⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠖⢪⠉⠻⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⠇⠀⡠⠀⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠑⢄⣹⡿⠿⠟⠛⠛⠛⠛⠿⠿⠿⣿⡿⠦⠊⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡯⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢶⡀
⠀⠀⠀⠀⠀⢀⠊⠁⠈⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⠈⠈⠉⠒⠄
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠀⠀⠀⠐⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡔⠀⠀⣤⠀⠀⠀⠀⠀⠯⡆⠀⠀⢸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⢰⠀⠒⠀⠀⠀⢂⠀⠀⠀⣶⠀⠀⡁⠀⠀⢸⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢆⠂⠀⠈⠉⠇⠀⠀⠀⠀⠀⠀⣌⠞⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⡈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡀⢳
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠰⡌⠀⠀⠀⠊⠀⠀⠑⠀⠀⠀⡄⠂⣠⣆⣀⣀⠀⠀⢁⡆⠀⠀⠃⠈⡧
⠀⠀⠀⢀⠀⠐⠁⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠋⠉⠁⢀⠀⠀⠈⠁⠀⠀⢱⢐⣿⡄⠀⠀⡟⠗
⠀⠀⠀⡰⣿⡿⢤⢼⡁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠰⡎⠁⠀⠀⠀⠀⠀⠀⠀⢬⢾⡷⠀⠀⢻
⠀⠀⠀⠀⠀⡈⣦⣽⠇⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡇⠀⠀⢠⠀⠀⠀⡤⠔
⠀⠀⠀⠀⠀⠐⡻⢿⡇⠀⠀⠀⠀⠀⠠⠑⠁⠀⠀⠀⠀⠀⠀⠄⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠐
⠀⠀⠀⠀⠀⠈⠘⠀⣃⠀⠀⠀⣀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⡿⠄⠀⠀⡁⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣎⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⡀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡸⠥⠄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠘⠈⡳⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⠀⠥⠀⢂⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⢃⠀⠀⠀⠀⠀⠀⠾⣾⠻⠇⢀⡀⠂⡼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⠀⠒⠀⠀⠀⠣⠀⢑⣄⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⢀⡀⠲⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠰⣦⢇⠀⠀⠐⠀⡂⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⢰⡆⠘⢈⠀⠀⢶⡄⠀⣆⢸⡀⠀⠀⠀⠀⣤⠁⠀⢣⡃⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠘⣀⠋⠀⢀⠀⠀⠙⠲⣜⣦⣱⠀⠀⠀⠀⠈⠉⠀⠀⠹⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡅⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡠⠀⢻⣦⡄⠀⠉⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠔⢘⠕⠀⠙⠿⠑⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠢⢀⠄⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀⣠⣶⣿⡆
⠀⠀⠀⠀⡀⠀⣠⠁⠈⡀⡀⠣⣰⡥⠔⢢⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣽⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⡀⠢⣨⠂⢅⠗⡀⠤⢋⢀⠂⠈⡤⠊⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿
⠀⠀⠀⠀⠡⡀⠘⢀⣠⠋⠐⠀⠃⡾⠂⠀⣄⡃⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⢀⠐⠁⠡⡇⠀⣤⣠⠤⣠⠋⢀⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢃⠒⠺⠠⡀⠁⠀⠛⡁⡄⠀⠠⠂⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠑⢄⠳⡈⠰⠁⡀⡰⢂⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢨⣦⠈⢢⣠⠟⠶⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⡷⢷⣖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢷⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡏⢨⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠲⠼⠦⠴⠤⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠉⠁⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡠⠚⠁⠀⡏⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢂⠘⡶⠉⠀⠀⠘⠃⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠑⣤⡟⠲⠀⠀⠀⠀⣀⣀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⠠⠤⠀⢤⣤⠁⠀⠀⠀⠀⡞⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠋⠁⠀⠀⠀⠘⠀⠹⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢩⠁⠒⠛⢛⠲⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⣀⡄⠐⠒⡇⢀⡴⠃⠀⠀⠀⠣⡀⠀⡸⠀⠲⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠏⠉⠡⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢃⠘⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠘⡄⠀⠑⢄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠣⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⡇⠀⠀⠀⠀⠀⠀⡰⠀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⠀⢰⠀⠀⠀⢀⡰⠊⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡄⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠰⡀⠀⠀⠀⡆⣠⠔⠉⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢢⠀⡘⠑⠂⠤⠄⢀⣀⣀⠀⠀⣠⠆⠀⠀⠱⠀⠀⠀⠉⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠱⣄⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠤⣬⡷⢀⣀⣀⣀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠋⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠈⠙⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠒⠒⢲⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢇⠀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠑⢄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠎
⠀⠀⠀⠀⢠⠏⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠾⠶⠻⡆
⠀⠀⣠⠒⣇⣀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠈⢦
⢠⡖⠉⠈⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢨⠴⠂⣀⠀⠀⠑⢤⡀
⠣⣝⠦⡀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠵⠶⣤⣭⠆
⠀⠀⠑⢌⣦⡀⠈⢆
⠀⠀⠀⠀⠉⠻⢦⣌⡧
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠨⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡀⠀⠀⠀⠀⠀⠀⡂⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠃⡀⠦⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⠤⠂⠉⣍⠙⠉⠒⠠⡘⠧⣀⣀⣠⡟⠲⠊⠻⢳⠋⠀⠀⠈⠂⠉⠊⠀⠀⠀⡀⣀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⣠⠃⠀⠀⠀⣿⣿⠾⠋⠀⠈⢦⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⠠⠚⡥⢄⠀⠀⣉⠠⠔⠋⣥⣶⣦⡀⠁
⠀⠀⠀⠀⠃⠀⠴⠟⠿⣿⡿⠛⠋⠃⠀⠈⡄⠀⠀⠨⠆⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⡆
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠷⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⠟
⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣸⠟⡿⠈
⠀⠀⠀⠀⠀⠈⠒⢤⣀⡀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠁⠟⡲
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠡⠜⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⡠⠂⢄⡎⢠⡀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣲⣆⠄⢤⡝⢃
⠀⠀⠀⢀⠀⠀⠀⠈⠻⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢭⢜⢣⠿⠏
⠀⠀⠀⠀⢁⠖⢠⣄⠀⠀⠀⠀⡐⠀⠀⠀⠀⣀⠔⠚⠓⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⡿⢿⠧
⠀⠀⠀⠀⢨⠀⠉⠁⠀⠀⠀⡰⠁⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠩⠉⠀⠤⡕⡢⠀⠀⠔⠁⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠢⠄⡀⠠⠔⠈⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠠⠂⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠐⠁⠀⢀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⢀⣀⣢⠤⠒⠉⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠙⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⣠⣀⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣀⣄⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠋⠀⠀⠙⢦⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡴⠋⠀⠀⠙⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⠳⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠞⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⢱⡈⣇⠀⠀⠀⠀⠀⠀⣸⢁⡎⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠀⢃⠸⡄⠀⠀⠀⠀⢰⠇⡘⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢱⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧⠀⠀⠀⠀⣼⠀⠁⠀⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠐⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⢠⣇⣀⣀⣀⣀⣸⡀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡴⠚⠉⠀⠀⠀⠀⠀⠈⠉⠲⢄⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⠞⠳⣄⠀⠀⠀⠀⠀⢀⡼⠚⢦⡀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣟⢙⣿⣾⠀⠀⠀⠀⠀⢸⡗⢻⣿⣷⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡄⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁⠀⠀⢠⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⢹⡏⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠈⠳⢤⣀⣀⡴⢻⡟⢦⣀⣀⡠⠞⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠈⠧⠼⠧⠼⠁⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣄⣀⣀⣠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⣷⡿⠓⠀⠉⠁⠚⢷⣶⠀⠀⠀⠀⠀⣷⣾⠗⠂⠀⠀⠒⢿⣶⡇⠀⠀⠀⠀⢺⣾⠶⠒⠀⠀⠒⠾⣷⡇
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠟⠁⢀⠀⠀⢀⡀⠀⠙⠇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠆⠀⠻⠟⠀⠘⠿⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠲
⠀⠀⠀⠀⣂⠀⠀⠀⣭⠀⠀⠀⣈⠀⠀⠀⠀⠀⢠⠀⠀⠀⣬⡄⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢈⡁⠀⠀⢀⡂
⠀⠀⠀⠀⠈⠓⠠⢀⣀⡀⠄⠚⠁⠀⠀⠀⠀⠀⠈⠓⠠⢀⣀⣀⠠⠐⠃⠀⠀⠀⠀⠀⠀⠙⠢⢀⣀⢀⡀⠔⠋


⠀⠀⠀⠀⣿⡿⠒⠀⠀⠀⠒⠿⣿⠀⠀⠀⠀⠀⣿⣿⠖⠂⠀⠀⠒⢾⣿⡇⠀⠀⠀⠀⢺⣿⠗⠂⠀⠀⠐⠶⣿⡇
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠥⠿⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠸⠁⢀⠀⠀⠀⡀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠑⠀⠀⠀⠂⠀⠆⠀⠀⠀⠀⠀⠀⠸⡾⠀⠀⢷⠇
⠀⠀⠀⠀⣀⠀⠀⠀⣭⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢬⠄⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢨⡅⠀⠀⢀⠂
⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠈⠓⠠⠀⠀⠀⠠⠐⠋⠀⠀⠀⠀⠀⠀⠙⠢⠀⠈⠁⠀⠔⠋


⠀⠀⠀⠀⣷⡷⠓⠀⠉⠀⠒⢾⣿⠂⠀⠀⠀⠀⣷⣶⠖⠂⠀⠀⠒⢶⣶⡇⠀⠀⠀⠀⢲⣶⠶⠒⠀⠀⠒⠶⣶⡆
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠟⠁⢀⠀⠀⢀⡀⠀⠙⠇⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⡞⠀⠘⡿⠃⠀⡀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠉⠁⡀
⠀⠀⠀⠀⣂⠀⠀⠀⣭⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢈⠀⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢈⡁⠀⠀⢀
⠀⠀⠀⠀⠈⠓⠠⢀⣀⡀⠄⠚⠁⠀⠀⠀⠀⠀⠀⠑⠠⢀⣀⣀⠠⠐⠃⠀⠀⠀⠀⠀⠀⠙⠢⢀⣀⣀⡀⠔⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡢⠀⠀⠀⠀⠀⢀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⠙⠶⡟⠠⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠓⡄
⠀⠀⠀⠀⠸⣷⡀⢀⢲⢴⡷⣧⣵⣄⣷⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡐⡆⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣰⣿⣷⠈⠉⢙⣟⠁⢤⣠⠶⠚⠉⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣙⡂⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⡀
⠀⠀⠀⠀⣿⣿⣿⣇⢷⢞⣩⠈⣻⡛⠒⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢈⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍
⠀⠀⠀⠀⣿⢿⢸⣷⢿⣽⣿⣄⠻⢷⡿⢿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⡀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠘⣀
⠀⠀⠀⠀⠙⣧⠈⠛⠛⣻⣋⢻⢶⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣌⠁⠄⠀⠀⠀⠈⢰⠘⡦⣄⠀⣁⡭
⠀⠀⠀⠀⠀⣽⣧⣀⠀⠻⡙⠁⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣁⠀⠁⢀⠀⠀⣨⣴⠀⣻⡆⢀⠆⣼
⠀⠀⠀⣠⣾⣿⢾⣟⠳⡶⣷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠂⢠⣤⣾⣿⣿⡗⠹⢁⣾⠗⢋⡠
⠀⠀⢸⣿⣿⣿⣆⠈⢹⣶⣿⡄⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡄⡀⢀⣀⡀⠀⠙⠿⠟⠛⠐⠏⡁⠈⠉
⠀⠀⢸⣿⣿⣿⣿⣷⡘⢻⣿⡀⢿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠋⠉⣀⣤⣄⣤⣠⣴⠶⢚⡀⠀⡐
⠀⠀⢸⣿⣿⣿⣿⣿⣷⣼⣿⣷⣼⣿⣿⣿⣿⣿⣿⡵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠙⢳⣿⣿⣦⣄⢉⠛⠿⠃⠒⢈⢀⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠃⠀⠂⠙⠛⠿⠟⠛⠓⠒⠂⠐⠂⠀⠛
⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠈⠳⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠞⠀⢠⠀⠈⠀⠸⠓⠲⣄⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠹⣤⠴⢦⣀⡤⠗⠀⠀⣀⣀⣗⣁⠔⠊⠉⠒⢋⠅⠀⠀⢸⣃⡤⡄
⠀⠀⠀⠀⠀⣠⠶⠾⠁⠀⠀⠀⠀⠀⠉⠑⠀⢀⢶⠻⡈⡇⡟⡆⡶⣪⠕⣏⠉⠉⠻⡀⠀⠀⠛⠖⠒⢲⡀
⠀⠀⠀⠀⢀⣸⠦⠀⠢⠄⣄⡠⠔⢀⡤⢚⠒⣚⢧⠻⠅⣻⠮⠧⣅⡟⢡⠛⠀⢒⠤⡀⠀⠀⠀⠀⠈⢿⠛
⠀⠀⠀⢠⡏⠀⠀⢄⠀⡠⡺⡒⣄⠈⠫⠆⣏⣂⢳⡗⠞⣵⠁⡨⢮⣧⢋⠄⠶⢜⢤⡌⢯⢤⣍⢢⠤⣾⡤⠖⢦⡀
⠀⠀⠀⠀⢙⡇⢠⢶⢻⣌⡓⠞⣌⠖⠊⢁⡄⣀⣉⢳⣇⠱⣝⡍⠡⡛⢋⣮⡢⣘⠀⠀⡰⠒⠹⠶⠒⢋⠄⠀⠀⣷
⠀⠀⢀⣠⣾⡟⠃⠈⣮⡑⠷⠊⡎⠀⢙⡅⣤⣖⢒⣬⡣⣑⠀⡝⡏⠘⣿⢵⣌⣲⣝⡫⠔⠉⠑⢂⠈⣉⠥⡤⡶⠃
⠀⠀⢺⡀⠀⠈⠛⠚⠛⠻⡛⡏⢳⡄⢀⡀⣡⡴⣪⢎⣿⡟⢿⡕⡿⣦⢤⢋⢆⡸⢨⢷⣦⠕⠂⢛⠀⡨⡆⢇⡇
⠀⠀⢰⠃⡠⠒⠉⠀⢳⢤⠸⣒⡭⠋⢠⠖⠁⠠⢸⠚⡷⠾⠭⢊⠗⠒⡽⡀⠑⠵⠁⣪⢆⣀⠀⠉⡳⣘⢝⣾⠃
⠀⠀⢻⣀⣠⡄⠀⣤⣤⡄⡙⠢⡇⠸⠁⠀⠈⠞⣁⠤⠘⠒⠺⡿⣛⢭⠫⠙⡔⢒⡈⣝⣁⡈⠛⠭⢔⡵⠋⣸
⠀⠀⠀⠀⠀⣧⡅⠀⠙⠺⢾⣢⠿⣐⡗⣤⣈⠋⠰⢦⠻⡤⠄⣼⡉⢔⡇⠁⠸⠇⣄⠈⠀⠙⡖⠀⢠⠶⣦⡏
⠀⠀⠀⠀⠀⢹⣄⠀⠫⠂⠀⣈⡉⠓⠺⠯⠽⣴⣒⣛⣀⣰⣊⣚⣛⣁⣲⡭⠬⠿⠓⡤⠀⠰⠇⣴⣆⠗⣹⠁
⠀⠀⠀⠀⠀⠀⢧⠉⠂⠤⣀⡘⠁⠀⠪⠆⠀⢠⡆⠀⢀⢤⠀⠀⣤⡀⠀⢔⡄⠀⢻⡁⠀⠐⠀⡨⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠉⠑⠒⠂⠤⠤⠤⠤⠀⠉⠀⠠⠬⠠⠤⠄⠒⠒⠉⠙⣄⣀⠜⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⣠⣤⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠐⠒⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⢀⠙⠿⠋⠀⠀⠀⠀⠀⣤⣦⠀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠡⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠐⠠⠄⠄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣌⡛⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣿⣿⠁⢠⣤⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣬⣙⠻⣿⣿⣿⣿⣿⣧
⠀⠀⣿⠃⠀⠘⣿⣿⣿⣶⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⠿⣷⣦⡙⢿⣿⣿⣿⣇
⠀⢸⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⣿⣿⡄⢿⣿⣿⣿
⠀⢸⡿⠀⠀⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣷⠘⣿⣿⣿⡆
⠀⢸⡇⠀⠈⠊⠉⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⣿⣿⡆⢹⣿⣿⣷
⢂⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠘⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⣿⣿⡿⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠘⠋⠁⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⡶⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣌⡑⢼⣿⣇⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠈⠙⠢⣦⡼⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⠀⠀⠈⠻⣟⣿⡀⣠⣤⡄⠐⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣯⠐⠀⠴⢶⣼⣿⡉⠥⠋⠀⠀⢠⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢰⣏⣿⣶⠀⠘⠓⠋⠀⠀⠀⠀⠀⠀⠠⠈⠱⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣷⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢃⢿⣿⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡄
⠀⠀⠀⠀⠀⠀⣠⡄⢀⡙⢷⡆⣿⣿⣌⠘⢿⣿⣿⣿⣟⣉⡀⠀⠀⠀⠀⠀⡀⠀⢙⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣄⠙⠡⣿⡟⢩⣍⣿⣆⢹⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠶⢤⠀⠚⠂⠀⠀⠀⠻⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⠛⠒⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⣾⣿⣿⣿⣛⠻⣇⣀⠰⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⣿⣿⣿⡿⣿⡷⠄⠒⠠⠄⠠⠤⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠶⠖⠂⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⣿⣿⣿⣷⣦⡄⠀⠐⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⢸⣿⣿⣿⣦⠉⠀⠈⠀⠂⠀⠤⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠐⠿⡀
⠀⠀⠀⢸⣿⣿⣟⠛⠂⠀⢉⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣷⡤⠄⠀⠀⠀⠊⠐⠃
⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⡁⠀⠉⠛⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠹⣿⣿⣶⡆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠹⣿⣿⣏⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⣿⣿⣿⣿⣮⠀⠀⠀⠀⠠⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣁⣀
⠀⠀⠀⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢻⣿⣿⣥⣤⠤
⠀⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣴⠉⣿⣿⣿⣿⣿⣿⣑⣚⢛⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣧⣄
⠀⠀⠀⣿⣿⣿⣦⠀⠀⠀⠀⠀⣸⠁⠀⢹⣿⣿⣿⣿⣿⣯⣿⡝⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤
⠀⠀⢠⣿⣿⣷⡆⠀⠀⠀⠀⢀⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⢀
⠀⠀⢸⣿⣿⣿⣟⡀⠀⠀⢀⡞⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣽⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣇
⠀⠀⢸⣿⣿⣿⠉⠁⠀⠀⣸⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⢻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠸⣿⣿⣿⣁⠀⠀⢈⠇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⣿⣿⣿⢧⡀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⣿⣿⣿⠄⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⣿⣿⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⢠⣿⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠋⠀⠀⠀⠀⠀⠈⠀⠀⠦⠁⠀⠀⠀⠀⠈⡩⠀⠃
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡒⠒⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠖⠋
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠁⠀⠄
⠀⢀⡀⠚⣴⡞⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠨⡀⡭⢁⡘⠀⠀⠀⠀⠀⢀⣠⣴⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡀
⠀⠀⠧⠀⠈⠱⣄⠀⠀⠀⢠⣿⣿⣿⣿⡦⣄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣄⠀⠀⢠⣴⣦⠀⠀⠈⠀⢠⣤⡐⠋
⠀⠀⠒⠀⠀⣠⣾⣧⠀⠀⣿⣿⣿⣿⡿⢸⡼⠳⠄⠀⢀⣬⣭⣽⣿⣿⣿⣷⣶⣿⣿⣿⠀⠀⠀⢠⣿⣿⠟
⠀⢀⣀⣤⣶⣿⣿⣿⣇⣸⣿⠟⢹⣿⢇⣀⡆⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⠀⠀⢀⣿⣿⣿⡀⠀⠀⠀⡀
⠀⢿⣿⣿⣿⣿⣿⠿⠿⠟⠁⢠⣿⡿⠀⢰⠀⠀⢠⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠉⠙⠀⠀⣾⣿⣿⣿⣿⠆⠀⣰⡇
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠿⢿⡇⢀⣸⠀⠀⣼⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠉⠀⣸⣿⡇
⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠆⠀⠀⠀⠈⠁⠂⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⡼⣹⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡂⠀⠀⠘⠃⠀⡀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⡌⢀⣿⣿⣿
⠀⠀⠀⠠⠗⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠿⢟⣀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⢀⡜⠢⣸⣿⣿⡧
⠀⠀⠀⠀⠀⡄⠘⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣿⣶⣤⣀⠀⠀⠀⣸⣿⣿⠀⢀⣾⣘⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⡇⠠⠔⠦⢁⡀⢤⠀⠀⣀⡀⠀⢀⣤⡤⠂⠀⢀⣴⣿⣿⣿⣿⣿⣦⣤⣿⣿⡇⢀⠎⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⡇⠀⣅⣴⡆⠿⠀⠀⠐⠀⠁⠈⠁⠀⣟⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣞⣢⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠇⢰⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣤⠑⢒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⣰⣷⠚⢿⢿⣿⣿⢸⣿⡟⠛⢿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠻⠿⠿⠅⣰⣿⣿⣿⣿⢸⢿⡏⠀⣾⣿⣿⣿⡿⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢸⠽⠃⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠇⠘⠰⠇⠘⠻⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠯⠻⠿⠿⠿⠶⠼⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡀⠀⢿⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⡄⠀⢀⠘⣇⣠⣤⣀⠀⠀⠀⠀⣇⢼⠆⡇⠀⢸⣇⣀⡾⠋⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠹⡄⢡⠀⠛⠁⠀⢹⡄⢠⡴⠶⠿⠊⠁⡇⠀⠘⢩⡟⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⣰⠖⠲⣄⠀⢀⣬⢷⡄⠹⠇⠹⡄⠀⠀⠀⠀⠸⠛⢻⣤⣄⡀⠀⠀⣴⠟⠳⣾⠁⠀⢀⡇⠀⠚⢻⡆⣀⣠⣀⠀⠀⠀⣀⣤⣤⡀
⠀⢠⡶⠲⢶⣿⢠⠄⢹⡾⠋⢳⣄⣀⠈⠉⠀⠉⣠⠾⠛⠉⠳⣴⠏⠀⠈⢷⡀⢸⡇⠀⠀⡀⣄⠀⡼⠀⠀⠀⣼⠟⠋⠀⠈⣧⠀⣾⠋⠀⠀⢹⣤⠶⠛⠳⡆
⠀⢸⡇⠀⠀⠉⠐⢂⠀⠀⠀⠘⢃⣹⡷⠶⠖⠲⣏⠀⢠⡀⠀⠙⠀⢠⣾⠋⠙⣛⠳⢦⣄⠉⠀⢰⠇⠀⡆⠀⠀⠀⠀⣠⠶⠿⠾⠇⠀⡆⠀⠘⠁⠀⠀⠀⣿
⣴⠶⠿⣦⠀⠀⠀⣁⣀⡀⢀⡴⠛⠁⢠⢒⡆⣠⠿⠆⣀⡙⠢⠈⠀⢸⡎⢣⡀⠣⡵⠀⠉⠳⣄⠈⠀⠈⠀⠀⠀⠀⢰⡇⠀⠀⢤⡐⢠⡇⣘⠀⠀⠀⠀⢠⡏
⣿⠀⠤⡤⠀⣰⠟⠩⡍⢻⡟⢁⣴⡦⠈⠉⣠⢿⣄⠀⠀⠋⠀⠀⢀⡀⠻⣆⠙⢦⡀⠀⢯⡷⠘⣧⣠⡴⠶⢒⡶⠛⠛⠛⢠⢠⡄⠁⢸⠀⢠⡠⠄⠀⠀⠙⠳
⠘⢷⣈⠓⠀⣿⠀⠀⢳⢸⣁⣀⣉⣀⠴⠚⠁⢀⡽⠗⠀⢀⡴⠚⠉⠉⠳⡌⠳⢦⣈⠓⠲⠤⠤⣿⠁⠀⠀⢸⡇⠀⠀⠲⣄⡈⠁⠀⠿⠀⠀⢁⣠⠖
⠀⠰⣟⣓⢀⣻⣄⣀⣘⣎⣿⣭⣭⣤⣤⣤⣾⣿⣤⣤⣤⣾⣤⣤⣤⡤⡤⠿⠤⠤⠭⠿⠷⠶⠶⠿⠤⠤⠴⣯⣿⣦⣤⣤⣤⣝⣂⣀⣀⣀⣐⣋⣀⣀⣀⣀⣠
⠀⠀⠘⣿⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⡿⠃
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠻⣟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠙⣷⡶⠶⠶⠶⠶⠶⠖⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⣐⣲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠐⠦⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣠⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢁⡞⠁⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠐⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠃⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠕⠄⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠲⠀⠀⠀⠁⠀⠀⠀⠀⠇⠀⠀⡀⠂⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠈⠀⠠⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠰⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢡
⠀⠀⠉⠐⣆⠄⠀⠀⠀⠀⠀⠂⠂⠂⠀⠈⠐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⢃
⠀⠀⠀⢠⠇⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠖
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⢰⣾⣦⡀⠀⠀⠀⠀⡀⠤⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⡀⠀⣼⣿⡇⠈⠊⠆⠀⢁⠄⠀⠀⠠⠄⠁⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠉⠻⣧⣴⣿⣿⠛⡀⠀⠳⢠⣤⣴⣤⣶⣶⡇⡀⠀⡀⠈⠀⠀⢠⣶⡶
⠀⠀⠀⠀⠀⠀⠠⠴⠚⢁⡓⠠⣿⡌⢻⣿⡯⠀⣵⡀⢸⡟⢛⣿⣿⣿⠏⠉⠀⡸⠁⠀⣠⣶⣿⡋
⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⢷⠐⣿⣿⣿⣿⡄⣾⡇⠸⠊⣠⣿⣿⠟⣡⠀⠀⣊⢀⡐⢾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠘⣁⠀⠀⠑⠠⠀⢻⣿⣿⠛⢻⣏⣵⠷⣴⠿⣋⠁⠚⠛⠀⠀⠉⢈⠀⣸⣿⣽⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠉⠁⠂⢷⠀⠀⠤⠙⣿⢠⣿⢻⣇⣄⣋⡀⠀⢀⣠⡼⣴⡄⠀⠀⢀⣿⡿⠃⣈⠉⠉⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣴⡄⢿⣿⢩⡼⡍⣾⡇⡜⣿⣿⡟⢁⣿⣧⣼⠀⠈⠐⠁⢀⣧⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣶⣤⣄⠀⠀⡆⢸⡇⣻⣧⡈⣿⡇⣁⢹⣯⡤⢺⣿⠋⣾⡅⠀⢄⠔⠲⠁⣀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⣷⣀⠇⠀⠋⣿⠹⣧⣿⡏⠀⣜⣿⠃⣿⠏⣴⠉⠂⡙⢀⠄⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⢀⣧⠀⠈⠃⡿⠀⢀⣏⡇⢻⡼⢃⠂⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣸⣿⠀⠀⡀⡇⢠⣼⣋⡇⢸⣇⡈⣾⠶⠞⠋⠙⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠇⡘⠀⣾⣯⠅⣸⣿⣿⣷⣼⣿⡋⠀⠀⠀⠀⢠⣴⣿⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠉⠛⡒⠉⠀⠀⠀⠈⢻⣿⣿⡇⠃⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠐⠀⣾⡷⠀⠀⣀⡉⠀⠁⢸⣿⠿⣾⣿⡆⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢰⣿⣧⠤⠄⠀⠀⠀⠀⡈⠟⠦⠼⢿⠀⠀⠀⠀⠀⠈⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⣿⣿⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠄⢹⣿⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠙⣿⡆⠀⠒⠂⠸⣿⣆⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠸⣿⣷⠀⠀⠀⠀⣿⣇⠀⠀⠈⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣿⣷⣠⠄⠀⠰⣻⣿⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⠙⣿⣧⠀⠀⠀⣿⡿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄⣽⣿⡅⠀⠀⢸⡇⢃⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠟⡇⢸⣿⣿⠂⠀⢸⡇⠸⡁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⢡⠀⣿⣿⣷⡀⠀⠃⠀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠘⠀⢸⣿⣿⡟⠢⡅⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠈⡿⡿⠁⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠘⡄⠁⡇⠀⠀⡏⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠸⡶⠁⠀⡼⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢠⠞⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠃⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢸⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⣀⣰⣆⣼⣷⡦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢀⡤⠖⠚⠋⢉⡽⠛⠛⢿⠒⠖⡒⢠⠔⠄⢐⠀⠒⣤⠐⠂⠉⠁⠐⠢⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠠⠃⢀⡶⠋⠀⢀⡴⡾⠁⠀⠀⠀⠀⣳⠀⡌⢅⣠⣠⣄⠲⠒⠧⢤⡀⠂⠁⠀⠀⢈⢡⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⢧⣣⠀⢠⡀⣺⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⣶⡧⡘⠚⠁⠀⠐⠀⠀⠀⠹⣶⡾⢿⠛⠒⠿⣷⣶⡬⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢻⣧⣱⣮⣟⡛⠧⣀⢈⢥⠆⠀⠀⠀⠀⠠⡄⠀⠸⣿⡺⡿⠀⠤⡀⢠⠀⢀⣠⡸⣧⣄⡑⢤⣀⠈⢙⠇⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠛⢲⣿⣏⣻⡏⠑⠶⣤⡈⢺⢾⠀⠀⠀⠀⠀⠀⡶⢰⡀⠘⠓⢞⣷⣤⣅⢤⣶⢧⡿⡏⢋⠁⠈⠓⢶⣤⣬⣷⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⠀⠉⠉⠀⠀⠹⡅⠘⣿⠀⠀⠀⠀⠀⠀⠨⢸⡇⠀⠖⣪⡇⡯⢙⣻⣻⡏⢭⣽⠀⡐⠀⠀⠀⠛⢇⠛⠛⣷⠄
⠀⠀⠀⠀⠰⢿⠎⠀⠀⠘⣴⡴⣿⣇⡀⠀⠀⢀⢀⡀⠀⣿⡀⣿⣿⠀⠀⠀⠀⠀⢀⣿⡇⠀⢠⣿⡏⢠⣾⡿⡽⢿⣆⢀⠀⡄⠈⢠⣀⢀⣷⠀⠀⠘⡆
⠀⠀⠀⠀⡧⠈⣂⣀⢄⣤⠿⠤⠄⠙⡓⢒⣒⠆⣱⣗⣓⡚⢿⣿⣟⣧⠀⠀⠀⢀⡾⣿⣱⣣⣿⢋⣾⡯⠊⣀⡧⠀⠳⣦⣴⢷⣿⣢⠽⠾⣿⣂⣙⡶⠃
⠀⠀⠀⠀⣱⡀⣋⠩⡀⢎⠈⡧⣝⢞⣂⣩⣥⠶⠷⠿⠛⠷⠦⣿⣿⡿⠧⣄⠀⠁⡠⠿⢷⢫⡵⣟⡽⠞⠣⠍⡙⢄⡀⣹⠿⠉⢈⠀⣂⠝⠙⡓⠫⠆
⠀⠀⠀⢈⢯⣯⣶⣷⣿⣿⣟⡭⠟⠋⠉⢀⠠⢄⣠⡶⠶⠶⠲⡚⠝⢃⢖⣴⢇⢜⡑⣆⡉⣧⣐⣟⡱⠶⢲⠶⢍⡑⢡⣧⠀⣶⢊⢠⡷⠶⡶⣿⠴⠃⡀
⠀⠀⠐⠁⠈⠈⠛⢛⡲⠟⠉⠀⠀⠀⣠⠔⠚⠉⢀⡠⠠⠒⡄⣇⣾⢿⡟⣽⣾⢧⡇⠸⣟⠗⡐⡸⢝⣇⡛⢶⠀⢿⡈⣿⢿⡟⠛⢩⡠⠖⠃⢄⣀⢤⠁
⠀⠀⠀⠀⠠⠤⣤⣬⠄⠈⢁⡰⠖⠉⣀⡄⢔⢁⠃⢆⣸⣿⡶⣫⣽⡯⢕⣬⣿⣿⣻⡫⢻⣏⡔⢀⡗⠜⣮⣻⡄⠈⣧⠿⢤⣥⢴⣫⡠⠶⣦⣌⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢁⢏⠀⠄⠚⢤⡄⢆⣣⣰⣶⡶⠛⢉⣉⡀⢽⣶⠿⣽⣿⣿⡏⠿⠢⣿⣮⡉⢉⣵⣇⠈⢻⣿⡅⡤⣼⣧⢞⠵⠃⠀⠀⠈⠳⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠛⣒⣉⠤⠖⣁⣧⣵⡟⢁⣴⣾⣋⡭⠿⣾⣿⣷⣿⢉⠿⠀⠀⠑⡩⢷⡿⠞⠷⢬⣑⣸⡇⠹⠇⢛⣥⠁⠀⠀⠀⠀⠀⠸⢃
⠀⠀⠀⠀⠀⠀⠀⠔⠥⢿⣷⣶⠴⠏⢹⢙⡐⢠⠋⠀⠀⡜⠁⣠⣼⣿⠿⠒⠂⠒⠶⣖⡋⠀⣼⣏⢤⡀⠀⠣⡎⣤⡄⠀⢶⣺⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠸⠐⠉⠀⠀⠀⢸⠀⠀⢸⠁⠃⠀⠀⠸⠁⣾⣯⣙⣷⣿⣿⢶⣄⡀⢠⣷⠀⠹⡓⢼⣄⠀⠐⠻⣄⣉⢣⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⣄⣣⠀⠀⠀⠈⠦⠽⠟⢏⣛⠿⠃⣰⢞⣻⣿⡟⡆⠀⠈⠙⠿⣿⣦⠀⠴⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡌⢿⣿⠿⢾⡈⠁⠀⠀⠀⠀⠈⠛⣾⡄⠙⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠗⠠⢹⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⡶⠶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡬⠐⠋⢠⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⢻⡬⣧⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⣷⠘⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠏⠉⢠⠀⠀⠀⠢⠀⠀⠀⠀⣇⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⣠⠞⠒⠀⠁⠀⠈⠀⠀⣌⠈⣠⣄⠀⢏⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣑⢄⣼⠀⠀⣘⠆⠠⠘⢶⠒⠈⢲⣽⡉⠓⢾⡇⠀⢈⡴⠋⠀⠀⢄
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⠝⠢⠗⠀⠤⠯⣶⣾⠿⠶⢄⢈⢫⠁⠀⠀⠈⠀⠏⠁⠀⠀⠀⠈
⠀⠀⠐⣧⣤⣂⠈⢤⠀⠋⣄⡀⠀⠀⠀⠀⠈⢧⠀⣀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠠⣾⠀⠘⠏⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣴⡆⠈⠁⠁⢀⡀⠀⠀⢀⣠⣤⡤
⠀⠀⠈⢃⡤⠚⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⡇⢠⣤⡤⣾⡿⠀⠀⠻⠷⣇⠀⠀⡘
⠀⠀⠀⠊⠑⠆⠉⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⣧⠘⠿⠃⢿⠗⠀⠀⠐⠈⠉⠃⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⡀⠸⢡⠀⠀⠀⢦⡀⣴⠂⠀⣠⠲⣈⠖⠂⠀⠀⠀⢀⡴⣾⡿⡿⠳⠄⡀⢀⣤⠀⠛⢳⠇
⠀⠀⠀⠘⣄⣷⡴⡰⠀⠀⢈⡧⡿⠂⠸⡁⠀⠁⠀⠀⢀⠀⠀⠊⠉⠈⠀⠀⢀⠎⠀⣼⠈⡀⠐⠠⢰
⠀⠀⠀⠀⣈⣻⡏⡀⠀⢠⡿⠀⢼⠀⠀⠁⠀⠠⠠⠔⠃⠀⠀⠀⠀⠀⣀⠐⠋⠀⠀⢹⣤⠰⣈⠄
⠀⠀⠀⡎⠁⠀⠀⠀⠀⢸⣽⢀⠇⠀⠀⠀⡀⢤⣠⣢⠀⠀⠀⢠⣾⡻⣽⠟⡆⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠈⠒⠠⡄⠀⠀⢸⣏⠌⠀⠀⠀⣾⢆⡀⠘⢰⡄⢰⠀⠩⠐⠷⠹⠋⠇⠀⠀⣠⠄⣾⠀⠠
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⢠⡏⣼⡿⠀⣸⠇⡜⠀⠀⠀⠀⠤⢥⡁⠀⣞⣿⣄⠉⠀⠀⠀⡆
⠀⠀⠀⠀⢀⠰⡛⡵⠃⢪⣷⠛⠞⡀⠈⠧⣿⣯⣴⣯⠜⠀⠀⡔⣩⣀⡄⢠⣷⠀⢿⡁⠙⠓⠄⡀⣠⠃
⠀⠀⠀⡠⠺⠅⡄⠀⠀⠈⣞⣷⣴⠂⠀⠀⠀⠉⠈⠁⠀⠀⠸⡇⠛⠟⢃⡞⠄⠀⠀⠉⠐⠶⠞⠉
⠀⠀⢀⡿⠀⠀⢣⠀⠀⠀⠁⠉⠈⢀⣴⢖⡛⣁⡀⠀⣠⢾⠁⠙⠀⠚⠉⠀⢠⡖⠊⠉⠉⠀⡄
⠀⢀⣾⡇⠀⠀⠀⢱⠀⠀⢀⠠⡤⠽⠿⠟⠛⠉⣁⡺⠋⠌⠀⠀⠀⠀⢠⠀⠻⣛⢄⡤⣂⠀⠀⠀⠀⡀
⠀⠘⣟⢱⣦⡀⠀⡸⠀⢰⢿⢰⣥⠀⠀⠀⠂⠀⠈⠑⢄⠀⠀⠀⢀⡔⣸⡀⢂⠀⠈⠃⠛⡭⠂⡔⣠⠃
⠀⠀⠈⠙⠶⠟⠋⠀⠀⢫⢫⢸⠀⠄⠀⢀⡆⠀⠄⢀⢸⣙⣈⠆⢸⡇⣿⡄⠈⠀⢀⣀⠀⣤⣩⡁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⠫⠧⠤⠴⠊⠀⠳⠦⠜⠀⠉⠁⠀⠀⢧⣽⣵⡖⠀⣀⣹⣮⣿⣼⡁
⠀⠀⠀⢀⡔⣾⠿⡆⠀⠀⠀⡠⢠⡖⠊⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡷⠛⣀⠈⡀
⠀⠀⠀⣞⢿⠛⡦⠷⠀⢀⡞⡆⠈⣷⡴⠡⠐⢈⠑⠀⣰⠾⠟⠈⠀⢆⡀⠀⠀⣴⡟⢱⠜⠚⣬⠘⡄
⠀⠀⠀⠛⢿⣇⡇⢠⠀⠸⡾⣷⠀⡜⠄⡀⠅⣸⢠⠃⣽⠆⠀⢀⠀⠀⠈⠀⠀⡟⣿⠟⢀⡈⠘⢀⡇
⠀⠀⠀⠀⡀⡗⠀⠀⠀⠀⠳⣻⣴⡧⡴⠁⠔⡡⠂⠀⠀⠀⠀⠐⡑⠀⠁⠀⠀⢳⢫⠿⣿⣧⣹⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁