Short funny bio ideas Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠠⠂⠀⠰⠀⡀⠀⠀⠀⠁⠐⠁⠘⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠤⠀⠀⠀⡀⢀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠁⠀⠁⠘⠂⠙⠂⠀⠀⠁⠐⠀⠀⠄⠘⠐⠀⠂⠠
⠀⠀⠀⢀⠀⠀⠈⠂⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⢸⡄⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⢀⡀⢀⠀⠀⠄⠀⠀⠀⡠
⠀⠀⠐⠼⡄⠀⠀⠠⠊⠀⠀⠀⢔⡀⡠⠀⠐⠤⠀⢘⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠗⠶⠺⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠑⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠎⠔⠀⠀⠀⠈⠀⠠⠂⠠⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠙⢄⡀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠤⠄⡼⠡⡴⡿⠿⠻⢧⡗⢢⠄⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠠⠄⡤
⠀⠀⠀⠀⠀⠰⡀⠠⠊⠀⠁⠀⢂⠀⠀⠀⠈⠀⠐⠏⠀⠀⠘⠸⠀⠀⠀⠀⠀⠀⠑⠦⡤⠙⡤⠛⡤⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠃⢀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠐⠐⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⠗⠛⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⢸⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢠⡾⠓⠀⠀⣀⣠⣴⣶⡆⠔⠀⢸⠀⠀⠀⣴⣤⣾⣦⣴⡆
⠀⠀⢀⠔⠂⢲⡉⠐⠄⠀⠀⠀⡇⢠⠀⠀⠀⠘⢡⣿⣶⣿⣿⡿⠋⠁⠀⠀⠀⢸⠀⠀⢀⡟⠉⣿⡇⣿
⠀⢠⣟⠀⠀⠀⡆⠀⠈⢄⠀⠀⡿⣿⣿⡆⠀⠀⣹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⠀⠀⣸⠁⢸⡇⣷⡿
⠀⣾⣿⣦⠀⠀⠷⠠⠀⠀⢢⠘⣯⢸⣿⠄⠠⣶⣭⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⢠⡏⠀⣾⠀⢸⡇
⢰⣿⣵⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡮⣿⣿⣿⠟⢻⣿⣿⣿⣿⣧⠀⡐⠒⠒⠂⠼⠤⠼⠤⠤⠿⣀⣘⣇⣀⣀⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣀⠀⢠⡿⠛⣿⣿⠿⢦⠃⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠈
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣠⡿⢀⣴⠟⠙⣇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⣦⣜⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠉⢿⣿⣿⣶⣿⣥⣤⣤⣤⠇⠀⠀⠠⣦⢂⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠐⠀⠚⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣧⠁⠀⣠⣤⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢸⣷⠀⢀⠁⠀⠈⠉⠉⠉⡅⠀⠀⠀⢸⣿⣧⣰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⣿⣀⡸⠀⠀⠀⡀⠀⠀⠧⠤⢤⣀⣸⡃⢠⣿⣿⣿⣿⣿
⣿⣿⡿⠿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣾⣿⣿⣶⣄⠀⠐⠀⠠⠁⢉⠀⠁⢠⢸⠀⢸⣿⣿⣿⣿⣿⣶⠦⠤⠤⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⣡⣯⠎⠀⠀⠀⠀⢀⣟⡔⠀⠀⠀⠀⡀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣿⢱⠇⣴⠷⡀⠀⣼⡫⣄⣰⡶⣠⡆⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡡⠊⠃⠺⠞⠧⠊⠀⠀⠟⠶⠋⠯⢎⣴⡏⣿⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⠐⡧⢠⡞⣷⠀⠀⠛⢻⡟⢓⡟⠓⠀⣴⡆⢠⡟⢿⠐⣿⠀⡇⣴⠓⠂⣶⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⢀⡇⢸⡇⣿⠀⠀⠀⢸⡇⠈⣧⡤⢠⡿⡇⢸⡇⠙⠀⣿⣴⡇⢹⣤⠀⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣻⢀⡇⢸⣇⣿⠀⠀⠀⢸⡇⢰⣇⠀⢼⣷⣧⢸⡇⣸⠂⣿⠀⡇⣾⠀⠐⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠈⠃⠀⠉⠁⠀⠀⠀⠈⠁⠈⠛⠁⠘⠀⠛⢈⣿⡋⠀⠉⠀⠃⠙⠋⠀⠛⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⢀⡀⠀⠀⣾⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣼⢇⡿⣾⣡⡟⣻⠋⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠋⠀⠀⠈⠋⠻⠎⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⢿⡆⣼⢻⡏⠁⣿⡀⣿⠀⠀⣿⢙⡇⣿⠉⠙⢻⠉⠉⢻⠉⢹⡏⠁⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠂⢸⣇⡏⢸⡗⠂⣿⣧⣿⠀⠀⣿⢾⡅⣿⠒⠀⢸⠀⠀⢸⠀⢸⡗⠂⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠄⠈⣿⠀⠸⣧⣀⡿⠘⣿⠀⠀⢿⣼⠇⢿⠄⠀⣺⠆⠀⢸⠇⠸⡧⠀⣯⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⢹⡧⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠰⠟⢰⡦⠀⠀⠠⣤⡤⣤⣿⠧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⢀⡛⠀⢠⣶⠀⠀⢠⣿⠃⠀⠀⣠⡶⣦⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢻⣧⠀⣾⠏⣠⠾⣿⢀⣠⣾⠏⠀⣠⣾⣿⠶⠋⡴⠛⣶⡦⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣸⡿⠀⠙⠞⢁⡴⣿⡏⢹⡿⢀⡴⠃⢾⡇⣠⠞⠁⢰⡿⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡶⠞⠋⠀⠀⠀⠰⣏⣤⣿⠃⠻⠷⠋⠀⠀⠈⠉⠁⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠈⠀⠁⠊⠑⠀⠀⠙⠐⠠⠃⠤⣌⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣀⠸⠀⡆
⠐⠠⠀⠤⠀⠀⡀⠄⠀⠤⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣉⠢⡀⠇⢠⠒⢂⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡀⡏⡆⠀⠀⡇⡩⠠⡱⠔⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢃⠚⣁⠣⢠⠐⡂⠗⠓⠁
⠀⠀⠀⠀⣠⡤⠰⠆⣠⣄⠀⠀⠀⠀⢀⠼⣿⣿⣿⡄⠸⣘⡪⠚⠈⠁⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⣾⣿⣿⠾⠛⠛⢹⡇⠀⠀⠀⠀⠀⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢼⠁
⠀⠀⠀⢿⣿⣿⡇⠐⠀⣅⠀⢄⠈⣘
⠀⠀⢀⠀⠉⢯⣀⢀⣀⣤⡆⠎⠉⠹
⠀⠺⡄⢀⣀⣠⣿⣿⣿⣿⣿⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠚⠄⠀⠀⠀⠀⡤
⠂⠀⠈⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠀⠀⠈⠢⢀⠴⠒⢄⠞⠉⠳⠎
⠀⠀⠀⠀⠋⠉⠛⠛⠛⠋⠉⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠠⠠⠤⠤⠀⠤⠠
⢰⠋⠐⠲⠀⠀⠀⢀⠠⠄⠀⢦⠀⢀⡶⠒⡄
⢿⣄⠆⠄⡀⠠⠊⠀⠀⠀⠀⠀⢱⠚⢒⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠿⣿⣿⡿⠿⣿⢸⣿⣿⠁⠀⢻⣿⡏⠀⣿⣿⡇⠀⣴⣾⡿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⣤⣤⣼⣿⡇⠀⣿⣿⡇⠘⢿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⠛⠛⢻⣿⡇⠀⣿⣿⡇⢀⠀⠈⠙⢻⣿⣷
⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⣿⣿⣶⡄⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⣸⣿⡇⠀⣿⣿⡇⢸⣷⣦⣤⣴⣿⡿⠀⠸⣿⣿⣿⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⢸⣿⣿⡇⠀⣤⣉⣍⣡⣤⣤⣌⠉⠉⣡⣤⣉⣩⣉⣤⠉⠉⢡⣬⣩⣍⣛⣋⣁⡀⠀⢰⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣥⣾⣿⡿⠃⠀⣽⣿⣿⡏⠉⠿⠿⠀⠀⣸⣿⡟⣿⣿⣿⠀⠀⢸⣿⣿⣟⠉⣿⣿⣿⡆⠀⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡋⢻⣿⣿⣆⠀⣿⣿⣿⣧⣴⣶⠀⠀⠀⣿⣿⠁⢹⣿⣿⡆⠀⢸⣿⣿⣿⣀⣿⣿⡿⠃⠈⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃⢸⣿⣿⣿⠀⣿⣿⣿⡏⠉⠋⠀⠀⢸⣿⣿⣦⣾⣿⣿⣧⠀⢸⣿⣿⣿⠙⣿⣿⣷⡄⢸⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⣾⣿⣿⠟⠀⣽⣿⣿⣇⣠⣿⣿⠀⣼⣿⣏⠀⢈⣿⣿⣿⡀⢸⣿⣿⣿⠀⣿⣿⣿⡆⢸⣿⣿⣿⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⢠⣶⠞⣋⣉⠑⠢⣄⣀⣀⡤⠒⢉⣉⡛⢶⣤⠀⠀⢠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣕⢦⡈⢿⠀⠡⠰⠏⠀⡟⠉⠹⡁⠈⠗⠠⠁⢸⠋⣠⢖⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣇⠈⠦⣀⣀⣠⠔⠁⠀⠀⠑⢤⣀⣀⡠⠃⢨⣷⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣀⣠⣶⣶⣶⣦⣀⣠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⠘⠿⠛⠋⠉⠙⠛⠻⠟⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣀⣠⣤⣴⣤⣤⣀⣠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠋⠻⠋⠻⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠈⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢠⡄⢠⣤⣤⣤⢠⣤⣤⣤⠀⣤⣤⣄⡀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿⣦⣸⡇⢸⣿⣉⡉⢀⣿⣉⡉⠀⣿⡏⠙⣿⡄⢿⣇⡈⠁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡟⢻⣿⣿⣿⣆⢸⣿⠙⣿⡇⢸⣿⠛⠃⠀⣿⠛⠃⠀⣿⡇⢀⣿⠇⡀⠙⢿⣆⢹⣿⣿⣿⡿⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⠸⠿⠀⠸⠇⠸⠿⠶⠶⠸⠿⠷⠶⠂⠿⠷⠿⠋⠀⠿⠶⠿⠃⢸⣿⣿⣿⡇⠀⢾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣰⣿⣿⣿⡟⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⢸⣿⣿⣿⡇⠀⣺⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⣿⣿⣿⣿⠁⠀⠛⠛⠛⠀⠀⣿⣿⣿⣿⠀⠈⠛⠛⠃⠀⢸⣿⣿⣿⣷⣶⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣦⣴⣶⡆⠀⠀⠀⣿⣿⣿⣿⣤⣴⣶⡄⠀⠀⢸⣿⣿⣿⡟⠙⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⠉⠉⠛⠃⠀⠀⠀⣿⣿⣿⣿⠉⠙⠛⠁⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣴⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣀⣀⣾⣿⡇⠀⢀⣿⣿⣿⣿⣀⣐⣿⣿⡆⠀⣸⣿⣿⣿⣧⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠁⠉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠍⠩⠙⠍⡛⣿⠉⠛⠛⠛⢹⡏⠛⠛⠋⠛⠛⠛⢛⠛⠛⠛⢻⠋⠉⠛⢻⠉⠛⠛⡟⠉⠋⢋⠙⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣾⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⡖⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣿⣷⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠿⢿⡇⠯⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡆⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠈⠀⠈⠀⠀⠁⠁⠈⠀⠀⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⢀⢄⢀⠠⠤⠤⠤⠤⠤⢄⠀⠀⠉⠀⠀⠀⠔⠩⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⠀⡀⡠⠀⠘⠒⠈⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⠚⠛⠑⠂⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⠀⢀⠤⢒⣀⠑⠲⢄⠀⠀⣀⣀⡀⣀⣀⡀⣀⣀⡀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⢸⠀⠀⠎⢴⠁⠀⢰⠀⠀⢣⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⡠⠧⢼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⢸⠀⠘⡀⠈⢄⡀⠆⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⢴⣶⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠑⢤⣔⣗⣲⣄⡴⠁⢠⡶⠀⠀⠀⠀⠀⠀⠀⠘⠠⣄⢉⡀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⣈⣡⡈⢁⠀⣀⣀⣈⣉⣉⣀⣉⡹⢿⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠛⠟⢁⢺⠀⡁⠀⠀⠀⠀⠐⠐⠒⢂⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢰⠖⢹⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⠈⠉⠀⠁⠀⠀⣀⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠀⠀⢿⣿⠃⠀⠀⠀⠀⢠⣶⠅⠀⠀⠀⠀⠨⡿⠅⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠠⠀⠀⠦⠀⠠⡦⠀⠠⠄⠄⠠⡦⠀⠀⠀⠐⠠⡄⠀⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠅⢸

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡠⠴⡆⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡆⠙⠋⠁⢸⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠘⣃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠄⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠉⣉⡉⠉⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⠛⢋⣀⣤⣤⣤⣀⡐⠶⣤⣍⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⢫⠿⡤⠶⣿⣯⣿⣿⣿⣿⣿⣿⣷⣿⡻⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠋⣡⠶⠿⠿⠿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣀⣶⡿⠋⠉⠉⠉⠻⣿⣿⣿⣿⣿⣯⠈⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢟⣠⡤⠶⣞⣛⣛⣒⡦⠌⠛⢷⣄⠀⠀⢀⠀⠀⠀⠀⢸⠐⣿⣿⣶⣶⣄⣴⣮⣿⣽⣿⣿⣿⣿⣿⣷⣬⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣟⣥⣾⡿⢛⣹⣿⣿⣿⡷⣶⡌⢻⠀⠀⠀⠀⠀⠀⠀⢸⡔⠋⡿⢿⣿⠟⠈⠻⣛⡟⣿⢱⣿⣿⣏⣿⡟⣴⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⢡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣗⠀⣠⠶⠒⠶⠄⠀⣿⣾⣿⢟⣽⢋⣴⠏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢴⣶⣻⣿⣇⠀⠀⠀⠀⠀⠈⣿⡿⠟⢋⣠⣾⣏⡀⠀⠀⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢠⣄⢀⣤⠞⣫⡿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠲⠋⢀⣾⣿⣿⡷⡄⠀⠀⠀⠀⠈⠘⢿⣿⣿⣿⣿
⣿⡋⢹⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢤⣨⠟⣋⣴⠞⣩⣴⣿⣋⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣹⣿⠀⢱⡀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿
⣿⣷⣤⣿⣿⣿⣿⣿⡿⠟⠃⠀⣀⡴⠃⠏⣡⡞⢋⣴⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⡏⠀⠀⢧⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿
⣷⣬⣍⣉⣽⣛⣁⣤⡴⠂⠀⠀⠀⠀⠀⠀⢩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢁⣀⣀⣀⣿⣿⣿⣿⣿⣦⠀⢘⣧⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠋⠀⢆⡦⠀⠀⠀⠀⢾⣼⡟⠿⠁⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣄⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿
⢭⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣾⣀⣀⠀⢈⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠷⣷⣶⣶⣾⣿⠻⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿
⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡏⠀⢸⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣄⣀⣈⣹⣿⠘⣇⣀⣹⣾⣿⣿⣿⣷⣤⣴⣿⣾⣿⣿⣿⣿⣿
⣧⠙⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣦⠀⠈⠻⠿⠀⠻⣿⣿⣿⣿⣿⣿⡏⠉⠙⠻⢿⣿⣿⢷⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿
⣿⣆⠀⠀⣠⣴⠉⡿⠄⠹⢿⣿⣿⣿⡗⢠⣤⣤⡄⠀⣻⣿⡏⣿⣿⢿⠿⢷⣶⣷⣶⣯⣩⣾⣿⣩⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠿⣧
⣿⣿⣷⣾⣿⣿⠈⡇⢀⣄⡘⠿⠟⠋⠀⠀⠙⢿⣿⣿⣿⣿⣇⣿⣿⣧⡀⢀⡉⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢿⡟⠁⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣷⣶⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠉⠻⢿⣿⣤⣼⣿⣿⣯⣿⣿⣶⣶⣿⣿⣿⣷⡀⠀⠘⠃
⣿⡏⠉⠉⠻⣿⡇⠀⠈⠉⠙⣿⠿⣿⣦⣀⠀⠀⣹⣿⣿⣿⣿⠿⣷⣴⣤⣦⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠤
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣷⣶⣿⣿⣿⡻⢿⣦⣴⣿⡿⠿⠟⣛⣛⣉⣭⡉⠉⠈⠹⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠉⠀⣆⡠⠙⠻⠏⠀⠰⠿⠿⠿⠛⠛⠁⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⣷⣷⣄⠀⠀⠀⣶⣶⣿⣿⠿⠇⠀⣀⣤⣶⣶⡿⣿⣿⣿⣿⡄⠀⠀⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣦⣤⣤⣶⣶⣶⣾⡟⠋⣯⠀⣾⣿⠀⢾⢉⣿⣿⣿⣦⠀⣸⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⢀⣿⣿⡀⢸⠘⢿⣿⣿⣿⣷⣿⣿⣭⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⢸⣿⡿⠛⢾⠀⣼⣿⣿⣿⣿⡟⣿⣷⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣥⢷⣼⣸⣿⣿⣿⣿⣿⡏⣿⣟⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠰⠄⢼⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣙⣿⣿⣿⣿⣿⣿⣿⡿⠏⠈
⠀⠀⠀⣀⠀⠀⠀⠸⣿⣶⣷⠀⢠⣄⠀⠈⠉⠀⠀⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⣼⣧⣀⠉⠢⠀⠀⠁⠀⠈⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⣴⣿⣷⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡒⠉⣶⣲⣿⣿⡧⠀⣾⠟⠻⠿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣟⣛⣟⠃⠉⠋⠁⣸⡏⡔⠒⣄⡼⠛⣍⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⢼⣧⠓⠖⠃⠑⠶⢃⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⣀⣀⣀⣈⣓⠤⢤⠤⠔⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣴⣾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⣽⠇⠀⠈⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠸⣿⣦⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣛⣻⠇⠀⢿⣛⣻⠇⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⡄⠀⣤⣿⣧⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢹⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠘⠛⠛⠛⠓⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠸⣤⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣬⣴⣾⣯⣵⣦⣴⣤⣀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣖⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠿⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠠⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠈⠄⠀⠀⣼⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡀⠄⠀⠀⠀⠀⠀⡀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠉⠀⠀⠁⠈⠿⠒⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⢄⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣀⠀⠄⡀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⢸⣿⣷⣦⣈⠁⠂⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠿⠛⠿⣿⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠙⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣽⠛⠶⢦⠀⠀⠀⡀⠀⢀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠉⠀⠀⠀⠿⣤⠀⣀⠀⠀⠈⠉⠉⢙⡛⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠉⠐⠒⠒⠒⢉⣉⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠈⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣆
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⣼⣃⣀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠈⢻⠿⠿⠿⠿⣷⣦⣤⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣶⡖⠉
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠉⢻⣿⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢰⡇
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⡇
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⢉⠃⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⣆
⠀⠀⠀⠹⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢄⣀⠀⠀⠀⠀⠀⠈⣂⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉
⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⠖⠁⠈⠀⠀⠀⠀⢣⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠃⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠙⠛⠙⠓⡖⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡤⢶⠁⠀⠀⠀⠢⡄⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⣀⣀⡸⣠⣤⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠛⢆⠀⢠⣌⣡⠟⠰⡌⠙⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⣀⣀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⢦⣤⣿⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢀⣤⢠⣶⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣤⣿⡇⠀⢸⣿⢸⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⢰⣿⡀⣴⢶⣦⢸⣿⢻⡇⢻⡇⣿⡇⠀⢸⣿⢸⣿⢸⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣬⡉⢹⣿⠈⣭⢾⣿⢸⣿⢸⡇⣿⣇⣿⡇⠀⢈⠿⠿⠋⢸⡿⠉
⠀⠀⠀⠀⠀⠀⠀⣬⢻⣿⢸⣿⢸⣿⣼⣿⠸⠟⠘⠃⠈⠋⠉⠀⠀⣼⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠋⠈⠛⠁⠉⠀⠀⠀⢀⣀⣤⢀⣴⡆⣰⢛⡏⣸⡷⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⣴⢻⣦⣾⢩⠏⣾⠼⠿⠞⠛⠋⠉⠉⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⠛⠋⠀⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠛⠓⠶⠋⠉⠉⠉⠉⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠈⢷⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠃⠀⠘⠀⠀⠀⠀⡀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠈⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠊⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⣠⠴⠆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰⠋⣠⠊⢸⠁⢠⠚⠉⢧⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⡀⠀⠀⠀⢸⠦⠽⠦⠘⣆⣹⣀⠀⠈⠲⠤⠖⠋⢻⣀⡀⠸⠤⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠉⠀⣀⡴⠃⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠠⠜⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⢹⢠⣾⠏⠙⠳⠦⣄⠀⠀⠀⢀⡤⠖⠚⠉⠛⠲⣌⠓⡶⠶⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠒⠀⠀⡀⢸⣽⢋⣀⣤⣤⣤⡼⠃⠀⠰⣅⣤⣤⠤⠤⢤⣄⣸⢸⠇⠀⠠⣀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡀⠀⡏⠉⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⢤⣤⡀⠀⠀⠀⢸⠀⡄⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣀⣧⣀⡇⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⢸⡼⠓⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣏⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⢻⠀⠀⠀⠰⠤⠤⢄⣀⣀⠤⠤⠼⠃⠀⠀⠀⠀⡿⠈⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠾⡇⠀⢀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⠀⠀⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠀⠀⠀⠀⠨⢍⣥⠀⠀⠀⠀⠉⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⡏⠉⠓⠲⠶⠶⠶⠚⠋⠁⣸⠒⠒⣒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠠⢤⣇⡀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠚⠉⣉⣀⣤⠬⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠉⠉⠉⠛⠓⠲⢦⣍⡲⢤⡀⠀⣠⣖⣭⠴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠀⣠⡔
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠦⣤⣤⣠⠞⠛⡅
⠀⠀⠀⠠⣄⣎⡾⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠘⠒⠛⠛⠃⠀⠀⠱⡀
⠀⠀⠀⠠⠋⠠⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠌⣤⡤⣄⡀⠀⠀⠀⠑⢄⢀
⠀⠀⠀⢤⡄⢶⢹⡙⣿⡷⠤⠔⠂⠒⠐⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⠘⢙⢆⠀⠀⡀⠠⠁⠄
⠀⠀⠀⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠉⠛⢧⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡁⠀⠀⠀⠉⠉⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠀⣀⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⢶⡆⠀⠀⢲⣶⣿⣶⣶⣶⣶⣶⣶⣶⠒⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡀⠀⠀⣲
⠈⠳⣦⣶⣾⣿⣿⣿⣿⡛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣿⡿⠟⠓⢲⡶⠋
⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣶⣶⣤⣄⣀⡀⠀⠀⠀⣀⣠⣤⣶⣾⣷⣦⣀⠀⠀⠀⠀⢀⣀⣤⣤⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠾⠟⠛⠛⠉⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⢰⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢶⡶⢤⡄⠀⠀⠀⠀⠀⠀⢸⡷⠈⠀⢈⡠⣾
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠳⠘⢳⣶⡀⠀⠀⠀⠀⠈⣴⣷⡶⣿⡿⢿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠠⣟⠷⠄⠀⠀⠀⠀⠋⠉⠤⠄⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⣠⡀⠀⠀⣀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⣠⠇⠘⢷⣤⠏⠁⠀⠀⠠⠠⣤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣾⣿⠇⠀⢻⣗⣿⡛⠠⠶⠦⢴⡧⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠄⣿⡦⠀⣸⣿⣟⢳⣄⠈⣄⡆⢹⣧⡉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣡⣾⣿⡟⢻⠈⠛⠛⣃⡹⠾⣟⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣤⡖⠈⠙⢁⠰⣯
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣟⣛⠋⠲⣿⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣋⣘⣻⣿⣿⡗⠢⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠟⣀⣠⢶⣤⣀⣀⣀⣼⡟⠧⢀⠀⣠⣸⠆⣀⣠⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⢿⡟⠀⠁⠀⠀⠉⠻⣷⠀⠀⠀⠛⠿⣶⣽⠉⢶⢂⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣠⡄⢉⡟⠀⢣⣼⣿⣶⣦⢀⠾⠙⢂⡀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⠇⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⡉⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⣸⠀⢀⣿⡿⣿⣿⣿⣿⡄⢰⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣤⣾⣿⠁⠸⣿⣿⣿⣷⣾⣿⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⣟⡙⢻⣿⠏⠀⠀⠙⢿⣿⡿⢻⣯⡿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠀⠀⠐⠟⠛⣿⣿⣿⣿⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠛⠻⠏⠀⠀⠀⠀⠀⠀⣿⣿⡋⠈⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣤⣠⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⢉⣉⡛⡗⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣰⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⢀⡈⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⣤⣹⣿⡀⠇⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠿⠃⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣶⣄⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣴⣬⡋⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠍⠻⢿⣯⣤⣙⣂⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠆⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣼⣿⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⡇
⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⡈⠑⢄⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⢀⢀⡀⠀⢀⡤⠀⠀⠑⠄⠂⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⡄⠀⢠⣠⠐⠒⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠃⢈⠉⠻⠿⠁⠀⢀⡀⢠⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠛⠀⠀⠀⠀⠦⡕⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠡⡀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣀⣀⡀⠈⠀⠀⢾⠀⠅⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢀⠀⠀⠄⢠⡗⠤⣠⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠢⠀⢆⠣⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠘⠠⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⠀⢜⠸⣤⣀⠀⠀⠀⠘⠸⠇
⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠑⠀⠀⠀⠀⠁⢢⠀⡀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⡀⠀⡇
⠀⢠⡜⠙⠘⢄⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⢇⣿⠤⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁
⠀⢸⡆⠀⠀⡇⢳⠀⠀⠀⡇⠀⠀⠀⠀⠀⢢⢸⠀⠀⠀⠀⠈⠷⠠⠤⣄⠈⠢⣄⠀⠀⢠
⠀⠈⡇⠂⠀⢹⡸⡆⠀⠀⡇⠰⡀⠀⠀⠀⠈⠆⠀⡀⠀⠀⠀⢸⠦⣄⠀⠀⠀⠈⠋⠑⠦⢷⡀
⠀⠀⣇⢸⠀⠘⡇⡇⠀⠀⢡⠐⢿⣄⠀⠠⠀⠈⢎⠁⡠⠂⢀⠄⢀⡠⠓⢦⣄⡂⠀⠀⠀⠀⢻⡍⠋⠐⠒⠒⡀⠀⢤⢀⠀⠀⢀⠴⠀⠀⠄
⠀⠀⢻⠘⠀⠀⣇⢸⠀⠀⢸⠀⠀⠙⢆⠀⠀⠀⠈⢧⡀⣴⠡⠒⠁⠀⠀⠀⠀⠉⠉⢁⠖⠛⠚⠛⢍⠀⠐⠒⠂⠀⠔⠋⠓⠴⡒⠒⠦⠤⠦⢤⣤⣤⡀
⠀⠀⢸⡀⠄⠇⢹⠸⡄⠀⢠⡇⠀⠀⠈⢣⡀⠀⠀⣰⠯⠇⢤⣀⣀⣠⣤⠤⠀⡶⠖⠁⠀⠀⠀⠖⠌⣃⠀⠀⠀⠀⠀⢀⠤⠶⠍⡀⠀⣀⡠⠖⠀⠀⠞⠃
⠀⠀⠀⣇⠀⠀⢸⠀⡇⠀⣸⠉⡁⠒⠒⠒⠓⣄⠀⠷⠀⠀⠀⢀⠀⠀⢠⡄⠀⡴⠤⠤⠒⠃⡚⠁⠀⠀⠀⢀⡀⢄⡨⡀⠄⣂⡥⠖⠋⠀⠀⠀⠀⢠
⠀⠀⠀⠸⡀⠀⠀⡇⣿⢠⠇⠀⠀⠈⠉⠀⠀⠈⠻⣦⠤⠤⠴⠷⠒⠛⢛⣗⡈⠀⠀⠂⠠⠄⢉⡑⠄⠤⣚⠵⠚⣉⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢧⠀⠀⠇⢸⡏⢀⠔⠁⠀⠀⠀⠀⠀⠀⡟⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠚⡁⠄⠒⡀⢈⣉⠵⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣧⠀⠀⢸⡇⠄⠀⣀⠤⠔⠒⠒⠤⣀⠁⠀⠉⠉⠙⠓⠲⠶⠤⣤⣄⣀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠄⠁
⠀⠀⠀⠀⠸⡏⣷⠼⡌⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠭⠑⠒⠲⠦⠄⣄⠠⠤⠶⠖⠒⠛⠋⠉⠉
⠀⠀⠀⠀⠀⡇⡇⠀⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⡇⣷⠚⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡇⣅⠠⣿⢸⡀⠀⠤⢤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢠⠃⠁⠀⡇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈
⠀⠀⠀⠀⢠⡇⣿⠀⠿⢸⠿⠿⠷⢶⣴⣄⣤⣤⣤⣭⣭⢭⠼⠲⢤⣀⠀⠦⣤⠀⢀⡤⠚⠁⣰⠃⠀⡀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠐⠋⠹⠀⢨⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠨⣗⣀⡤⠞⠉⠀⣠⠞⠁⠀⠸⡇⢀⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠀⡇⠰⠋⠉⠉⠓⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⣶⠒⠂⢈⠁⠀⠀⣀⠞⠁⠀⠀⠀⠀⡇⢸⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠸⠀⠀⠀⠁⠀⠀⠀⠀⡞⢠⡇⢸⢀⣀⠤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠺⣤⣀⣀⣀⣀⣀⣀⣰⠷⠘⡇⠟⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⠀⠀⡇⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠀⠈⠉⠉⠉⠉⢍⠀⠒⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⡤⣤⠄⢀⡀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠀⠐⠂⣠⠤⢤⣤⠤⠀⢤⠀⠀⢶⣀⠈⢆⠀⢣⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠉⠒⠂⠀⠀⠁⠀⡄⠀⠀⡀⡀⠁⠀⢀⣀⠀⠀⠀⠁⣂⠤⣵⠌⠀⠓⠮⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣵⠀⠀⠀⠀⠀⠀⠀⠰⠣⡄⡏⠃⠀⡔⠁⠀⠙⡄⠀⠈⠁⠀⠈⠂⠀⠀⠀⢀⣰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠀⠀⠱⢿⣆⠔⠀⠀⠘⠿⠆⠔⠁⠀⠀⠀⠘⢡⠀⡔⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⢦⠐⠒⠀⠐⠂⠡⡈⠆⢇⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⢠⠀⠀⡂⢁⠜⠀⠀⠀⠀⠀⠀⠁⢐⣄⡀
⠀⠀⠀⢀⣀⠀⠀⠀⠤⠤⠦⡀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠉⠂⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢉⠀⡰⠉⠀⠀⡀⠂⠀⡠⠂⣈⠄⠀⠀⠈⠑⢄
⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⣂⠀⠡⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠈⠀⠒⠒⠀⠉⠀⠀⠀⠁⠔⠃⠀⠀⠈⠉⠉⠉⠁⠈⠉⠈⠉⠉⠉⠉⠉⠂⡀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠁⠒⠷⣵⣀⡀⠀⠀⠀⠀⡀⠄⠔⢊⠳⣄⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⢄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠐⠂⠤⢀⣑⠈⡆⠈⠉⠈⠰⠆⠉⠀⠀⣤⡨⠶⠤⠤⠤⠒⠉⠉⠉⠁⠒⠒⠠⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠁⠒⠒⠒⢄⠀⢀⠂⠀⠀⠠⠄⠀⠀⡠⣿⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⡀⠀⠀⠀⠀⠀⢀⠠⠀⠂⢀⣉⠉⠉⠉⠉⠉⣛⡒⠲⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⡖⠁⠐⢼⠀⣀⣀⡀⠀⡤⠔⠐⠒⠒⠀⠉⠉⠀⠀⠁⠒⢤⣠⣤⣤⣤⣤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠠⠆⠀⠀⠀⠀⠀⠈⢁⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡍⣒⠀⣈⣉⡀⣀⣀⣀⠤⠤⠤⠔⠒⠒⠊⠉⠙⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠊⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠠⢟⡂⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⠁⠀⠀⠘⠻⠆⠀⠀⠀⠀⠀⠀⠀⠠⠖⠃⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠚⠀⡄⠛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⡄⠚⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠶⡀⠀⠀⠠
⠀⠐⢫⡀⠁⠀⠐⠀⠀⠁⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠈⢀⡁⠀⠁⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⢉⠠⠄⢀⠀⢀⠀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⠀⠀⠀⠤⠐⠈
⠀⠀⠀⠀⠣⡤⠐⠄⠐⠤⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠃⢀
⠀⠀⠀⠀⠀⠈⠀⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠈⠉⠉⠉⠈

⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠉⡓⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⠀⠀⠀⠉⠃
⠀⠀⡤⠀⠈⠉⠁⠀⠀⠀⠘⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡑⣢⣤⣤
⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢆⠀⠀⢀⠄⠀⠀⠀⠠⠄⠀⢀⠙⠿⠟⠡⠍⠛⠋⡀⠀⠐⠂
⠀⠀⠀⠈⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣀⡤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠘⠦
⠀⠀⠈⠙⠛⠻⣶⢦⣴⣤⣶⡞⠛⠉⡏⠀⠀⠀⠀⢖⠉⢁⡄⠈⠇⣀⡉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠂
⠀⠀⠀⠈⠆⠀⢀⠠⠂⠐⠄⣀⠀⠔⠀⠀⠀⠀⠀⠐⠀⡀⠀⠠⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⠛⠁⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠁⠁⠒⠒⠁
⠀⠀⠀⠀⡤⠪⢀⡠⠄⢠⠤⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣀⡀
⠀⠀⠀⠸⣡⠀⠴⠂⠀⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⠆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣴⡶⠰⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⣴⣿⡀⠀⣤⠀⠀⠀⠀⢀⣠⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣼⣿⣿⣿⣿⡀⠶⠂⠀⢀⣼⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣦⣶⣠⣾⣿⣿⣿⣿⣷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠛⠙⠁⠘⢁⡞⠛⢿⣿⠿⣿⡏⢻⠇⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢹⡒⠂⡷⠀⠸⠧⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠆⢀⣄⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢸⠃⠀⠀⠀⠀⠹⠦⠴⠟⠁⠀⠀⠀⠀⣀⡀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⣼⢿⠀⡏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠉⠈⡇⢠⣴⡇⠀⢀⣴⠟⠛⢷⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠈⣯⡀⠀⣸⡄⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢠⡄⢸⡇⢸⡏⠀⠀⠀⠙⠻⠟⠉⠀⢠⡀⢠⣀⠀⢸⡀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠃⠸⢃⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⣻⠇⢸⡇⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⡞⠀⠀⣠⣴⢦⣤⡀⠀⠀⠢⡈⠏⠷⠄⣿⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄⢰⡄⠀⠰⣿⠁⠀⢹⡗⠀⠀⠀⠈⠂⣦⡀⢿⡆⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠛⢳⡶⠟⠃⠀⠀⠀⠀⠀⠈⠃⢸⡇⠀⠿⣁⠼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢠⣴⠟⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡞⠁⠀⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⣼⠇⠀⠀⠘⠻⣷⣄⡀⠀⠀⠀⠀⠀⢸⣷⣌⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⢀⣿⠀⠀⠀⣀⠀⠈⠛⢿⣄⠢⣄⠀⠀⠈⣯⣿⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣾⡏⠀⠀⠈⢛⠀⠀⠀⠀⠙⢷⣌⠳⣤⡀⢸⡿⠗⠀⣔⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⣿⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⣬⣻⣾⠀⠠⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢠⡾⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢿⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠘⣷⢌⠻⣆⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⠀⠀⠀⠀⠀⢻⡜⣷⣮⠳⡄⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠉⠀⠘⢦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣄⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣥⡀⠀⠀⠀⠀⢀⠀⣠⠀⢀⠐⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠛⠋⠉⠁⠒⢒⣒⣦⣴⣿⣿⠿⠿⢧⡀⠀⣠⣶⡉⠀⠀⠀⠈⢱⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⢶⣄⡐⢂⣈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣱⠖⢾⣏⣉⣉⣉⣉⣉⣉⣉⣁⣍⣉⣉⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⠀⠀⡀⠀⢀⡿⡆⣠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠸⣷⣄⠀⢳⣰⡇⢠⣸⠁⣷⠇⢸⠁⣤⣴⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢦⡀⠢⡈⠲⣿⡙⢶⡀⢻⣇⠀⡏⠀⠙⠀⣿⠀⠁⣾⠞⣰⢁⣤⢔⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠢⣌⡃⠙⢦⡈⡦⠈⢅⡄⠀⢧⠉⢸⠀⠀⡤⠀⠀⢰⠆⠏⢀⠀⠚⠁⣽⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣥⣀⣬⠿⠶⠱⣦⠙⠆⠀⠉⠀⠠⠄⠈⠀⠇⠀⠸⠀⠈⡠⠀⠋⢠⠀⢘⡿⠋⣀⠄
⠀⠀⠀⠀⢤⣀⠀⠀⠒⠮⣿⡌⢍⠁⠠⡀⠈⠀⢢⡀⠁⠀⠀⠄⠀⢀⠀⠂⠀⡀⢀⠁⢠⠀⠁⠀⠠⠴⠯⠗⣶⠃
⠀⠀⠀⠀⠀⠈⠙⠳⠦⠍⠒⠦⣄⠁⠀⠉⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣷⡀⠀⣾⡴⠀⠀⠀⠀⠀⠬⣭⣤⠄
⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠀⠠⡀⠀⠀⠙⠂⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠠⣄⣿⠳⠀⠻⠁⠀⠀⠀⠀⠐⢻⠯
⠀⠀⠀⣤⣤⣽⠿⠧⣤⠀⠒⠚⠛⠂⠈⠁⣀⠀⠀⠀⠈⠀⠀⠀⠠⣤⣬⣆⠈⠛⠀⠀⢀⣄⠀⠀⢠⣀⢸⡇
⠀⠀⠀⠀⢀⣀⣠⡤⠀⢉⠿⠒⠀⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢙⣦⠀⠀⠀⠀⠀⠋⠥⡀⠀⠀⠪⢸⡇
⠀⠀⠀⠀⠀⠰⠶⣤⣤⡄⠐⣋⡀⠀⡴⠀⠠⠀⠀⠀⠒⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠰⠇⡇⠀⢞⢣⢺⠇
⠀⠀⠀⠀⠀⠐⠋⠉⠐⠚⠉⡉⠁⠊⠀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⡞⠀⠀⢀⣀⣀⡀⠈⠒⠃⢀⠬⠛⠛⠛⠻⢶⡞⢻⣦
⠀⠀⠀⠀⠒⠚⣋⣴⡾⠀⠚⠇⠀⠚⠀⡄⠀⠆⠀⠀⠂⠀⠀⠀⠈⠃⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠤⠚⢩⣟⣥⡶⢀⣠⣀⣄⡀⡁⠰⠀⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⣀⠀⠀⢀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠠⠟⠋⢰⡷⠛⢛⠁⠀⠉⠀⠀⣄⣀⣀⣀⡀⣶⠀⠀⠀⠀⠀⠉⠉⠽⣦⣠⣶⣭⡽⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣴⣤⣤⡼⠶⠏⠈⠉⠉⠁⠙⠳⠶⠶⢶⣦⣀⣠⣤⣾⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣰⣦⣻⣷⣾⣷⣶⣦⣠⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⣼⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣵⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⡟⠀⠀⣀⣀⣀⠀⠀⠀⢀⡤⠴⠒⠒⠲⣤⣀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⡏⢠⣶⣿⠟⠉⠉⠉⠑⠋⠁⠀⣀⣀⣀⠀⠈⣿⣧⠀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡟⢠⣿⡟⠁⣠⡴⠞⠛⠓⢦⣴⠋⠁⠀⠈⠻⣄⠈⠋⠀⠀⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠃⠀⠉⠀⣼⠋⠀⢠⣦⠀⠈⣧⠀⠀⠿⠆⠀⣿⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡿⣦⣀⠀⠀⢀⣿⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠘⢷⣤⣤⣤⣾⠏⠀⠈⢻⣿⣿⣛⣁⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣺⠟⢱⠀⠀⠀⠀⢻⣏⡓⠂⠀⠀⠀⠀⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⣴⢺⡇⠀⠀⠀⢸⣿⠹⡆⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷⣬⣷⣀⣀⣠⣾⢿⡾⠟⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠈⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠐⡀⠀⠀⠀⠀⠀⠂⠙⠻⠿⣇⠀⠀⠀⡀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣥⣀⣀⣀⣀⠀⠀⠀⠀⠲⠈⣷⣀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁


⠀⣁⡈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠴⠆⠶⠞⠶⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⢙⢁⡼⢻⢏⡀⣄⣥⣜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣆⡩⡤⣀⣴⠈⠉⠉⢇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣤⢀⠀⠉⠛⠲⢆⢰⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠋⠹⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⢀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠒⡊⠁⠀⢀⡼⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠉⠀⠉⠙⠳⠦⣄⡀⠀⠀⠀⢠⡞⠁⠀⠀⠑⠦⡴⠋⠀⠈⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉⢀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣴⢫⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠋⠁⢀⣠⣞⡽⢏⡙⢦⣀⣀⠀⠀⠀⠀⠀⠀⢀⡆⠘⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠳⡀
⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⣠⢔⠊⠀⠡⠛⠀⠣⢀⡼⠐⠉⠉⠉⡉⠀⠂⠠⠞⠁⠇⠀⡀⠀⠀⠀⠀⠀⠀⠄⠐⣦⡀⠂⠀⠀⠀⠹⣄
⠀⠀⣀⡤⠾⠛⠁⠀⠀⠀⠀⠘⠇⠳⣄⠀⠀⠀⠀⠀⠀⠘⢻⡒⢤⣀⠀⠀⠀⠄⠀⡼⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⢠⠁⠙⢦⡘⠀⠀⠀⠘
⣴⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⣌⠳⡄⠀⠀⠀⠀⠀⠀⡿⠀⠈⠙⠢⢁⣴⣾⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠙⡆⠀⠀⢀
⠀⢹⣷⣤⡀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⢣⡈⢢⡀⢀⡠⠖⣀⠷⠀⢀⣠⡴⠛⢁⡞⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⢀⡇
⠀⢸⡇⠉⠻⣦⣀⠀⠀⢴⢀⠀⠀⠙⣄⠀⠙⠚⣙⠭⠖⠋⢁⣤⣶⠿⠥⠶⠖⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⢀⡞⠀⠀⠠⢠
⠀⢸⡇⠀⠀⠀⠙⢷⣄⠘⠈⣦⠀⠀⠈⠢⠖⠉⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⣠⠄⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠋⠀⠐⠀⣰⠋
⠀⢸⡇⠀⠀⠀⠀⠀⠈⠳⣤⡈⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠉⠙⠒⠂⠄⣴⠟⣠⢒⣈⠉⠙⣄⠶⠊⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣤⡶⢻⡇⠀⠀⠀⢀⣐⣤⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⠀⢹⡖⠁⢀⠌
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⠁⠀⢸⡇⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣔⣛⠉⠀⣀⣀⣠⣴⡟⠓⠒⠋
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢸⡁⠀⠀⠀⠤⣱⡶⠚⠉⣿⠇⠉⠉⠉⠉⢹⡏⠀⢻⣇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢳⡀⠀⣿⠀⠀⠀⠀⠀⢸⡿⠀⢸⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⡾⢹⡄⠀⠀⠀⠀⠈⣧⠀⣿⠀⠀⠀⠀⠀⢸⣧⠀⢸⣿
⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠻⡄⠀⠀⠀⠀⠸⣆⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢀⡇⠀⠀⠹⡄⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡆
⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢸⣇⣀⣤⡾⠟⠛⠛⢻⡏⠁⠀⠀⠀⠹⡄⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⢸⡏⠁⠀⠀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⢸⡟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣀⣠⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢀⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⣠⡟⠁
⣶⣶⣦⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣼⣿⣿⣿⣿⣥⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠠⠀⠠⠄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣄⢠⣤⣤⣾⣿⣿⣶⣶⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣟⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣁⠀⣭⣽⣿⣿⡿⠉⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⣇⣹⣿⣿⣿⣷⠀⠈⢿⣿⣿⣿⣏⣤⠔⠀⠈⢿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⣀⣍⢻⣿⣿⣿⣿⣿⣿⢀⢡⡃⡘⢿⣿⢿⣿⡀⣦⢀⡄⣠⣤⡴
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⢷⣄⣤⡀⠄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠘⣿⣤⠇⠉⠈⠀⣴⣿⣿⡎⠁⣱⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⠂⠸⠉⠉⣿⣤⢰⣄⣿⣿⣿⣿⣿⡿⢉⡇⠀⣿⣄⠀⡘⠁⠀⠀⠘⣿⣿⣿⣻⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⡉⢀⣾⣖⣠⢆⣀⠞⣠⣿⣿⡿⠛⠋⢹⠇⠀⠀⢻⣎⠻⣿⡄⢻⡀⢀⡄⢸⢿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⢎⢥⡀⠙⠛⠁⠈⠉⣴⣿⣿⣧⣿⣤⠀⠀⠀⠀⣴⣾⠁⠀⠘⠃⣸⡇⠀⢁⣀⠈⠻⣿⣾⠏⠉
⠀⠀⠀⠀⠀⠀⠸⣆⢓⠀⠂⢊⣸⣹⣿⣿⠿⢋⣿⣶⠃⢀⠏⠀⠿⣿⣶⡄⠀⠀⣿⣿⣶⠋⠀⠀⣠⠽
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⢃⣜⢛⣧⠀⠀⢹⣿⡆⠀⠘⠿⠟⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⣿⣿⣿⣹⣿⣯⣾⣿⣴⡟⠀⠀⠀⣿⣅
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣒⠘⠻⢿⣿⣿⣿⣿⣿⡿⠏⠀⠀⣠⣬⠍⠁⠀⠤⠀⣶⣿⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣀⣹⢻⣞⡃⣠⡝⢹⣿⣿⣿⣿⣿⣾⣿⠛⠻⠀⣼⠂⣠⡄⡴⢧⣼⣿⣧⣴⢶⣿⣿⡟⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠙⠿⠟⠿⣿⣿⡄⠀⠟⣿⣿⠟⢿⠿⣿⣷⣾⣟⣡⡴⢟⣿⣿⢽⡿⣯⡟⠛⣿⣿⠙⠁
⠀⠀⠙⠄⠀⠀⠀⠀⢀⣄⣙⣶⣿⣿⣷⡄⠀⡜⢿⣶⣷⣦⣄⠖⢰⣿⢓⣼⣾⣷⣶⣶⠶⢁⣀⣤⣴⡴
⠀⠀⠀⠈⢷⣦⡀⠸⣿⣿⡟⠻⣿⣿⣿⣷⡄⢸⣸⣿⣿⣿⣿⠶⣪⣿⣿⠀⠒⢽⣿⣿⡀⢹⣿⣿⣿⣁⠀⣰⣶⡆
⠀⠀⢠⣤⣌⣻⣿⣶⣾⣿⣷⣤⣤⣿⣿⣛⢿⣿⣿⣿⣿⣟⣴⣿⣿⣿⣿⣄⠀⣠⣽⣿⡟⣿⣿⣿⣿⡅⠀⠀⠀⠁
⠀⢺⣿⣿⣿⣿⣿⠻⠛⢿⣿⣿⣿⢛⣯⡀⠜⢿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡷⠈⠀⠋⠓⠻⢿⣿⣿⠏
⠀⢸⣿⣿⠿⠻⣷⣀⣠⣄⡿⣿⣷⢘⣷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣦⡁⠀⢸⣿⣶⡄⠀⠀⠀⣀⢀⡀⢀⣀
⠀⢠⣤⣤⡿⠀⣈⣿⡍⢀⠋⠙⠃⢹⣎⠉⠀⠙⠻⡟⡛⠋⠙⢿⣿⣿⣿⣶⣿⠿⠃⠀⠐⠶⠋⣼⠿⠿⠁
⠀⠀⣋⠉⢸⡧⢘⣿⡟⠉⠐⢤⡀⠙⣄⢀⠀⠠⣾⡿⠁⠀⠀⢠⣌⡟⠉⠉⠁⠀⠀⠀⠀⠀⠈⠈
⠀⠈⠿⠛⣼⡇⣿⣏⠉⠛⠁⠀⠀⢰⣿⣧⡷⣀⠀⢠⡄⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠐⢿⡧⠛⣿⠷⣄⣀⢀⣾⡗⠿⣟⠚⢛⡷⠦⣤⡄⣲⣿⣿⣧⡄⠀⠀⠀⠀⡀⠀⢀⡠⠀⠀⣠⣾⣿⣿⠿⠿⢶⣦
⠀⠰⣶⣦⣄⣀⣀⣠⣤⡘⢛⣛⠉⣀⡴⣿⣧⣼⣷⣶⡆⣼⣿⣿⣿⣿⡟⢦⣶⠇⠀⠈⠛⠉⠀⠀⣾⣿⡿⠋⠁
⠀⠀⠈⠻⠿⠋⠀⣈⣿⣿⣿⣿⣿⣯⡈⠻⣫⣍⣿⣿⣅⣽⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⢤⣼⣿⠁
⠀⠀⠀⠀⢀⣠⣟⡻⣿⣧⣄⣠⣍⠻⣷⡄⠈⠛⠙⢻⣿⣿⣿⣿⣿⣿⡿⠗⠀⠀⠀⠀⠀⠈⠋⠉
⠀⢰⠿⠟⠛⠉⠉⠁⠉⠙⠛⠛⠉⠛⠈⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⡀⢙⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡅⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⡿⠟⠁⠀⠀⠀⠀⠀⣀⣀⣀⠀⠶⠘⠋
⠀⠘⢷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉
⠀⠀⠈⠛⠛⠛⢿⣿⡭⣏⣭⣻⡷⣶⡀⢰⣾⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠄
⠀⠀⠛⠒⠻⠿⠿⠛⠟⠛⠛⠉⠚⠻⠗⠘⠛⠿⠓⠀⠀⠀⠀⠀⠤⠐⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢚⠉⢑⠰⠻⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠉⠁⠒⠒⠀⠤⠄⠠⣧⠬⠄⠚⡂⡤⢏⡗⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠄⠐⠂⠒⣿⢸⠀⠀⠠⠒⢧⡟⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⣍⢠⠱⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢈⠉⠢⠀⠀⡊⠀⠀⢻⠀⠀⢀⠄⠀⠀⡌⠁⢱⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⣀⠙⣑⠀⣇⠄⢌⠁⡟⠀⠁⠉⢋⡠⠠⠀⢱⢐⡑⢁⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠴⠀⠀⠯⢔⠖⠉⠀⠀⠀⠀⠎⢐⣓⠤⠎⠈⠐⡂⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠔⠹⢀⢀⢠⡎⠀⠀⠀⠀⠀⠀⡄⠑⢄⢑⠀⠀⠘⠈⠢⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠣⡀⠀⠀⠀⢀⠔⠁⡘⠀⢫⠃⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⣉⣔⠉⢁⡀⢠⠃⡰⠃⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⠊⠀⠀⠘⠓⠁⠈⡡⠊⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡑⡈⠢⠔⠁⢰⠀⠀⡔⠁⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠓⢄⣀⡘⢀⡊⣀⠴⠶⣒⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠈⠣⠘⠁⠁⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⡀⠀⠀⠀⡩⣀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠠⡤⢷
⠀⠀⠀⠀⠀⠀⠀⠪⡑⢄⣈⣦⡀⡸⠀⠀⠑⢤⣁⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⢘⠻⣅⠀⠀⠀⠈⠀⠉⠒⠦⡤⠌⠡⠤⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⡦⡀⠀⠦⢄⠀⢀⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢕⢝⣶⡙⠀⠉⡟⠛⠂⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠛⠀⠀⠀⢱⣀⣀⣀⣀⣀⣐⣦⣤⣤⣤⠠⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠙⠊⠉⠋⠉⠉⠉⠋⠛⠋⠉⠀⠈⠋⠉⠉⠭⠟⠓⠒⠓⠒⠘⠓⠢⠝⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠶⠖⠒⠛⠋⠉⢙⣦⠤⠶⠶⠶⢤⣀⣤⠶⠛⣿⡤⠟
⠀⢠⡤⠶⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⢀⢀⣠⠶⠛⣩⣴⡖⠋⠁
⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⠀⣀⣤⠶⠛⣉⣤⡶⠋⠁⠀⠹⣆
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣛⣉⣤⢶⢾⣿⡟⠀⠀⠀⠀⠀⠈⠷⣄
⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣠⣯⣀⡿⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠸⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠉⠉⠓⢦⣤⣀⣀⣤⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⣀⡤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠉⢀⣀⠀⠈⠻⣦⠀⠀⠀⡠⠒⠤⣄⡀⠀⠀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⣾⠫⡉⠙⢆⠀⠈⢷⠀⣰⠁⠁⠀⠀⠈⢉⠖⠁⠂⠈⠳⣄
⠀⠀⠀⠀⠀⣴⠋⢀⡴⠶⣄⠀⠀⠀⢻⡀⢹⣤⣬⠄⠀⠘⣧⠃⠒⠀⠀⢀⡴⠋⠒⠀⠀⠀⠀⢈⡱⠤⠤⡀
⠀⠀⠀⠀⢰⡇⠀⡟⠠⡄⠸⡆⠀⠀⠀⠑⠟⣁⠤⠀⠀⠀⠸⣆⠀⠀⡠⠊⠀⠀⣀⡠⠤⠒⠉⠥⠀⠈⠀⢱⡀
⠀⠀⠀⠀⠸⡇⠀⠃⣠⡷⠶⠇⠀⠀⠀⠀⠘⣴⠿⠛⠀⠀⠀⠘⢦⣊⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢦⡀
⠀⠀⠀⠀⠀⢻⡀⠈⠁⠠⢒⣓⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠁
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡡⢠⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣢⠞⠀⣼⡏⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡠⠄⠀⠠⠀⠉⠑⠊⣀⣴⣿⠇⠀⣰⣿⠁⠀⠀⢀⣀⣀⣈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⢠⠊⠉⢣⡀⢇⡀⠒⠲⠲⣶⣶⣿⡿⠟⠁⣠⡾⠃⠇⢀⡴⠚⠉⠉⠀⡉⠉⠛⠶⣦⣄⣀⡠⠴⠒⠉⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠒⠭⣓⠦⢤⣤⣤⣤⡴⠶⠛⠉⠀⡸⢠⠏⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⣠⢳⡟⠀⠀⠠⣤⣭⣤⠄⠀⠀⠀⠀⠀⡆⠀⠀⠙⣧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡍⠐⠂⠐⠊⢁⡜⠀⠀⠀⠤⠤⡀⠀⠀⠀⠀⢀⡀⢸⡇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⣀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⣠⠎⠀⠀⠀⠚⠛⠷⠆⠀⠀⠀⠀⠈⠳⣌⡀⠀⠀⠀⣾
⠀⠀⢠⠊⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣴⠒⠂⠀⠀⠀⠀⠀⠀⠀⣤⣐⢤⠁⠀⠀⣸⠃
⠀⠀⠘⢆⠀⢸⡀⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⣼⠃
⠀⠀⠀⠈⢣⡀⠉⠉⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢘⡄⠀⣀⣀⣀⣀⡐⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⢱⠘⠁⣤⣶⣶⣭⣝⠲⢄⡀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⣀⠀⢀⡠⠔⠉⠀⠀⠀⢨⡎⠒⠤⢤⣉⠙⠻⢿⣆⠙⢦⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢀⡩⣖⠉⠀⠀⠀⠠⠆⡴⠁⢸⠀⠀⠀⠈⠙⠲⣄⣉⣀⡼⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⠀⠈⠑⢤⡀⠀⢠⠎⠀⠀⠀⢣⠀⠀⠀⠀⠀⣀⠬⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠁⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠊⠉⠉⠑⢆⢀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠠⠼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⡄⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠲⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠖⡏⠀⠀⠀⢀⡤⠶⢤⡀⠀⠀⠄⠀⢳⡄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⠀⠀⠀⠀⠀⠳⣄⠀⣰⠋⠀⠀⠀⢳⡠⠤⣄⡠⡞⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⣧⠀⠀⠀⠀⠀⣀⠜⠁⡏⠀⠀⠀⠀⢸⠀⠀⢸⡇⢧⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡈⠙⠒⠒⠒⠋⠁⠀⣠⠷⡀⠀⠀⣰⢻⡙⠒⣏⡀⠈⢧⡀⠀⠀⢀⡴⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠈⠉⠉⠁⠀⢣⡀⠀⠙⠢⣄⠉⠓⠚⠉⣤⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢉⡠⠚⡡⢲⠟⢏⠁⠀⠀⠀⠀⠀⠀⠳⡀⠀⢐⠎⠳⣄⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⡄⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⠉⠒⠻⠤⢤⠀⠀⠀⠀⠦⣀⠱⣄⠘⣄⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⡤⠒⠁⠀⠀⠀⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣀⣈⣙⣚⠂⠘⣆⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠂⠅⠘⣄⡀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣀⣀⣀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⢻⡳⢾⣂⣠⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣫⠔⡢⢀⡤⠚⠒⠧⣍⡓⢦⣀⠀⠀⠀⠀⢀⡠⠤⢀⡴⠻⡿⠋⠉⠉⠑⠒⠒⠛⠉⠉⠉⠉⢹⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣻⠎⡡⠋⣠⠋⠀⠀⠀⠀⠀⠙⠲⢾⣙⠢⣄⡀⢹⠆⠐⠾⣤⣼⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠧⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡉⠛⠣⣀⠀⠀⢀⡶⠉⠀⠀⠀⠀⠀⢠⠊⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠤⠀⠀⠉⠈⢳⡄⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⠀⠎⠀⠀⣠⠘⣆⠀⠀⠀⠀⠀⡴⠉⢮⣛⡶⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⡰⠁⢣⠸⣄⠀⠀⣀⡼⢁⡄⢈⣅⣀⣠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣧⠉⠛⢉⡨⢤⡀⠓⠻⣤⣀⣀⡦⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡫⠕⡾⠁⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠸⠖⠋⠁⠀⠀⠓⠒⠚⡇⢘⠑⠲⢤⣨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠊⡱⢋⢤⠀⠇⢀⡾⣠⡛⠒⠤⢤⣀⣀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠈⢳⠏⡸⢸⢠⣧⣜⠉⡱⠛⢶⡶⣥⣬⠝⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣠⠋⠀⡗⣸⣷⣮⣉⡟⠓⡲⢶⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠉
⠀⠀⠀⠀⠀⠀⠸⢤⡤⠞⠁⠀⠀⠿⠛⠉⠒⢯⣽⡿⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉⠉⠉⠉⢳⡒⠇⠱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠄⠀⠠⠤⠤⠤⠭⠭⠯⠭⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢹⠀⠀⠀⠀⢠⣞⣙⡷⠤⠤⠶⠾⠾⠶⠦⠤⣤⣤⣒⠙⢦⠀⠀⠀⢀⡔⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡼⣧⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⠿⣄⡀⢀⡜⣀⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠀⠀⡴⠚⢶⠀⠀⢀⣠⣀⠀⠀⠀⠠⠊⠉⠇⠀⠀⠀⠈⠙⢿⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠞⠁⠀⠀⠀⠀⠉⠀⠀⠀⠸⣇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⢳⡄⠀⡹⠃⠙⣦
⠀⠀⣀⡀⠀⠀⠀⠐⣧⠚⠁⣠⠴⢤⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠞⠀⠚⠁⣠⠖⢷⡒⠋⢻⠆
⠀⣼⠁⢹⣀⠀⠀⡜⠁⠀⠸⣥⣀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⠋
⠀⠘⠷⠶⢽⣷⡞⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⡌⢷⡀⠀⠀⣠⠶⠶⣄
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡸⠀⠀⠀⠀⠀⠀⠒⢉⡀⢀⡿⠘⣧⢴⣊⣥⣤⣄⡏
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣿⣷⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⢹⡏⠁⠀⠀⠈
⠀⠀⠀⠠⣯⡇⠀⠰⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⣿⣿⠇⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⠋⢹⠆
⢀⣀⣀⣀⣙⠁⠀⠀⢳⡈⠉⠛⢶⣶⣦⠤⣤⠄⠀⠈⠈⠙⠒⠦⠤⠤⠤⠤⠖⢉⡤⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⣦⠀⠀⠉⣿⢉⣀
⣎⡀⢈⠥⢼⡆⠀⠀⠀⠳⣄⡀⢈⣛⠯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠀⠀⠐⢿⣀⢹⠆
⠈⠉⠀⢀⠔⡇⠀⠀⠘⠦⣌⣉⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠘⠦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠸⡆⢠⡏⠀⠀⠀⠀⠀⠀⠀⠰⣏⠈⢧⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠜⠀⠀⢠⡀⢹⡶⢤⡀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⡔⠹⡌⠙⠛⠀⠀⢠⡔⠹⣆⢀⡹⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢃⡟⠀⠀⢧⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠳⠤⠿⣆⠀⠀⠀⠀⢹⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣦⠀⠈⠳⢤⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣦⠰⠽⢦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⠛⣧⠰⠋⠀⠀⠀⢀⡟⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⢹⠀⣠⠜⣻⣦⡀⠀⠀⠀⠀⡟⠀⣷⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠦⠴⠃⠀⠀⣠⡞⢥⡀⠸⡆⠀⠀⠀⠀⢸⠀⣤⡀⢰⡈⡇
⠀⠀⠀⠀⠀⣴⠊⠉⢀⡞⠁⠀⢙⡶⣄⡀⠀⠻⠖⠋⠀⠀⠀⠀⢸⣄⣸⠄⠀⠀⠀⠀⠀⠀⣀⣠⠴⠊⠁⢹⡄⠙⡆⢿⠀⠀⠀⠀⢸⡼⠁⠹⣄⡿⡗
⠀⠀⠀⠀⠀⠘⠳⠶⢞⠁⠀⠀⠸⡇⠀⠉⢳⠶⣦⣤⣤⣠⣀⣀⣀⣉⣡⣤⣤⣤⠴⠶⡖⡟⠧⠙⢦⡀⠀⠈⠛⠊⠁⢘⡀⠀⠀⠀⠈⠁⠀⠀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⡞⢀⡎⠀⠀⠀⠀⣷⠀⡴⠋⣼⠁⠀⣰⠁⠜⠁⠹⢡⠀⠘⠒⠂⠀⠀⡇⠙⢶⡄⠀⠙⢦⡀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠇⠈⢶⠲⡄⠀⠀⢻⡘⠷⠤⠟⠀⣰⠃⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⠙⠳⡶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⣆⢸⣇⡸⠀⠀⠘⣧⠀⠀⢀⡴⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣄⠹⠦⣬⣷⡼⠟⠀⠀⠀⣽⣂⠀⠈⠳⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠳⢦⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⡵⠞⠛⠶⠲⠦⠟⣠⠼⠦⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠇⠀⠀⠤⠍⡁⠶⠂⠴⣤⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢼⡟⣥⠶⣄⣀⡴⠒⠦⣤⡤⠛⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠭⠔⠒⢦⣀⣠⠔⠒⢠⡀⠀⢀⡨⣷⣄⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠿⠊⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢧⣽⠙
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶
⣿⣿⣇⣀⣀⣀⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⠊⠁⠀⠤⠤⡞⣾⠓⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⠃⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡆⢰⣏⠁⠀⠀⠀⠀⡼⢀⣿⠃⠀⣀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣡⢏⡴⣻⠀⠀⠀⠠⣇⡜⡇⠀⢠⠇⠀⠀⣄⠊⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠥⠼⠛⠓⠉⠉⠁⠀⠉⠀⠻⠴⠏⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣠⠤⠤⠀⠠⢤⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡠⠔⠋⢀⠆⠀⠀⠀⠀⠘⠁⠀⠀⡜⣹⠁⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠏⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢣⠀⠀⠀⠀⢰⠇⠀⠀⡀⣠⡀⡄⠀⣀⣿⠏⣠⣤⠀⠀⢠⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠠⣾⠃⠀⢸⣼⠁⢰⠁⡜⢠⡏⢰⢣⣿⢰⢂⠞⣇⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡏⠀⠀⠸⠇⠀⢸⠞⠛⠉⣇⠹⠂⢿⠘⣫⣖⣏⣡⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠠⠏⠈⠂⠀⡇⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠐⠂⣨⠋⠁⢸⠇⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠁⠀⠀⠀⠹⠀⠀⠀⣀⣠⣧⣾⣤⣤⠀⠀⡼⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡟⡸⠁⠀⠀⠀⠀⡇⠀⢀⡜⠁⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⣆⠀⠀⠀⣤⠀⡴⡦⢸⡷⠃⠀⠀⠀⠀⠀⠉⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡏⡼⢹⣼⠋⢁⣾⠁⠀⣰⠃⠀⢀⠔⠉⢀⡇⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠖⢹⠉⠓⠋⢸⡄⣰⠃⠀⢠⠏⠀⣠⠞⠀⠇⠀⠀⠈⢲⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⡿⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⡼⠁⠀⠀⢧⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣰⠧⠤⠒⠒⠒⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⢂⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⢛⣉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠈⠳⡄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣀⣀⣀⡠⠄⠀⠈⠛⢶⡶⢄⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡴⠊⢿⡏⠀⠀⠀⠀⢠⣴⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⡤⢾⠀⠌⠀⠀⠀⠀⠀⢤⡀⠐⡶⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡞⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠛⢂⡔⠊
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠾⢯⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸
⠀⠀⠀⠀⠀⠐⠋⢀⡀⠀⠀⠸⡆⠀⠀⠹⡀⠀⢈⣦⡀⠀⠀⡾⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⣧⠀⠀⠀⢳⡔⢯⠈⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⡀⠀⣼⠻⢳⣼⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢧⡸⣇⣠⠞⠿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠀⢀⡔⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡎⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⡞⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⠾⡇⠀⠐⠶⠀⢸⡇⠀⠀⠀⠀⢶⣄⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢘⣧⣀⣤⣿⣎⣀⣀⣀⣤⠶⠾⠷⣤⠤⠒⠒⠛⠽⣿⣦⡀⠀⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠐⠒⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⢹⠀⠀⠀⠀⠀⠀⠦⣝⠢⣄⠀⠀⠙⠳⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⢀⣼⣥⣀⣀⣀⠀⢬⣽⠞⠉⠛⡦⠠⠀⣀
⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣤⡤⠤⠖⠚⠛⠛⠛⠉⠁⠀⠈⠉⠉⠉⠉⠀⠀⠀⢀⡩⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⢠⣴⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⣤⣤⡤⠄
⠀⠀⠀⠀⠀⢰⡆⣿⡏⣿⣇⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡴⠋⠁⣀⣀⣀
⠀⠀⠀⠀⠀⢹⣿⠻⢟⣙⣫⣴⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣿⣵⣾⠿⠛⠛⠻⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢻⣸⣿⡿⢁⠔⠛⠛⠛⠦⣌⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⣤⣀⠘⣿⣿⣧⡇⡴⣊⣉⣑⢦⠈⠳⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣾⣿⣿⣷⣾⣿⣿⣿⣶⣅⣀⡉⢢⠳⡄⠙⢦⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⡿⠿⠛⠛⠉⠉⠉⠀⠀⠀⢠⣾⠋⠉⠀⠉⠉⠻⣿⡉⠛⠛⠿⣿⣷⣬⣢⡀⠳⣄⠙⢿⣷⡀
⠀⠀⠀⠀⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣟⡳⢸⣧⠀⠀⠀⠀⠉⠻⣿⣯⢦⡈⢳⡈⣿⣷
⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⠀⠉⣁⣼⠇⠀⠀⠀⠀⠀⠀⠈⢻⣷⡝⡀⡇⣿⣿
⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡇⣿⡿⣇⣀⡄
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⢻⣧⠈⠉
⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣤⣀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣽⣷⣤⢀
⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣏⠙⠂⠀⠀⢀⣴⡾⠿⣿⣿⣿⣿⡵⠃
⠀⠀⠀⠈⠻⢿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⢀⣾⠋⠀⢿⡄⠀⠀⠀⠈⠉⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠉⠉⠁⠀⠀⠀⢀⣴⠟⠁⠀⠀⢸⡇⠀⠀⠀⢀⣄⡀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⢤⣤⣤⣴⠶⠞⠋⠁⠀⠀⢀⣴⣿⡇⠀⠀⠀⠈⠿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿⡄⠀⠀⢀⣀⣴⣿⠿⣿⠃⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⣠⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠋⠀⣴⡟⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⢀⣤⣶⣤⣠⣼⣿⣤⡾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠈⠛⢿⣤⣤⣤⡾⠋⠀⣰⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣤⡾⠟⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⣠⣴⡿⠿⠿⠟⠁⠙⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢧⣤⣤⣬⣉⣉⣉⣛⣻⣦⣀⣴⠟⠛⠓⠛⠛⠛⠛⠋⢹⣿⣇⣠⣴⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠒⠶⠶⠶⢾⣿⠛⠋⠉
⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⢀⣾⠋⠈⣿⠄⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣸⡏⠐⠀⣿⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⢿⡇⠀⢰⡏⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣴⠾⠗⢠⣟⣀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢿⡴⠶⠛⠛⠻⠿⢿⣶⣦⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠷⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠶⢦⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⡆⠈⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⡾⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⡇⡄⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⣿⣿⡟⠙⢿⣿⣿⣿⣹⡏⠀⠀⠀⢸⣿
⣿⡇⣧⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣅⢀⡀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⠁⠀⢺⣿⣿⣿⡌⠃⠀⠀⠀⢸⣿
⣿⡇⣿⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⣾⡟⠉⠁⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⢸⣿⢸⣿⣿⡄⠀⣾⣿⣿⣿⢳⡇⠀⣠⠄⢸⣿
⣿⡇⣿⠀⠀⠀⢠⡞⠃⠀⠘⣦⠀⢀⣿⡇⢀⣤⣴⣶⣶⣶⣶⣴⣶⣿⣿⣦⡀⢸⣿⢸⣿⣿⡇⢀⣿⣿⣿⣿⣾⡇⢼⣷⡆⠀⣿
⣿⡇⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⣇⠀⣿⡷⢠⣿⣿⣿⡿⠿⠟⠛⠛⠻⠛⠿⢻⡎⣿⢸⠿⠿⠀⣿⣿⣿⣿⡿⣿⣷⣦⣹⣷⡆⠿
⣿⡇⣿⠀⠰⠖⣿⠀⠀⠀⠀⢸⣿⣠⣿⣯⣘⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣘⠀⠀⠘⢿⣿⣿⣿⡿⠛⠛⢿⢿⡿⠟⣤
⣿⣿⣿⢠⠀⢦⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣀⣀⣠⠀⠠⣾⠀⣷⣿⣿⠀⠀⠀⠀⢻⠛⠿⣏⠀⣀⣀⣈⣤⣤⣴
⣿⣿⣿⡞⠁⠻⢿⣿⡶⠶⠶⣿⣿⣿⢿⣿⣿⡛⢻⠂⠀⠀⠘⠛⠋⠀⠀⠀⣤⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢣⣽⣿⣿⣿⣿⣿
⣿⣿⣿⠃⠀⠀⠐⢿⡀⠀⠀⠈⢿⠁⠚⣿⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⠀⠀⠀⠀⠀⢶⣤⣀⣼⣯⣿⡿⢿⣿⣿
⣿⣿⣇⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡿⠛⠻⢿⠀⠀⠀⠀⠀⢸⣿⣯⣿⣿⡿⣿⣿⣿⠫
⢿⢹⣿⣆⠀⠀⠀⠘⣧⠀⠀⠀⠱⣀⣼⣿⠏⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⡷⠋⠉⠉
⢸⣸⣿⣿⠂⡀⠀⠀⠈⠀⠀⠀⣰⡿⣿⡟⠀⠀⠀⠈⢿⣶⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁
⢸⣿⣿⣿⠀⠹⣄⠀⠀⠀⢀⣾⣿⡁⣿⡇⠀⠀⠀⠀⠀⠈⠙⠿⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠓⠶⠖⠂⠀⠤⠄
⣿⣿⣿⡏⠀⠀⠈⠳⡦⠶⠟⠛⣩⡇⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⠁⠀⠀⠀⠀⣦⣄⣼⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡿⠿⠶⠶⡶⠶⠶⠶⠿⡿⣿⣿⣶⣿⣷⣶⠶⠶⢶⡶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⠶⢶⢶⣶⣶
⣿⡇⠰⠀⠀⠉⠀⣴⠆⣀⢀⣿⣿⣿⣿⣿⡏⢀⣤⡯⠀⠀⡗⠀⣸⣿⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⢰⣿⣿⣿⣿
⣿⡃⠀⢀⡀⠀⣰⡿⢠⡏⣼⣿⣿⣿⣿⣿⠃⣠⣾⠁⠀⡀⠁⢰⣿⡏⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⣸⣿⣿⣿⣿
⣿⠄⣠⢟⣿⢠⣿⡇⣾⠁⣿⣿⣿⣿⣿⣿⢀⣿⣾⣇⣀⣿⡄⣼⣿⠀⠀⠀⣀⣿⣿⣧⠀⠀⠀⠀⠀⠀⣸⣿⡿⢠⣿⣿⣏⡿⣿
⣿⣼⣿⣿⣿⣼⣿⣻⠃⣰⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⢿⣶⣿⣿⣠⣶⣾⣿⣿⣿⣿⡆⡀⠋⠀⠒⢛⣯⣿⣇⣾⣿⣟⡜⣰⣿
⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠉⠉⣻⣿⣿⣟⣿⠉⠛⠉⣿⣿⣿⣟⠙⢻⣿⣿⠛⠉⠉⠉⠉⠉⠉⠉⢻⣿
⣿⣿⣿⡜⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⣿⣸⡟⢸⣿⣿⣿⣿⣦⢸⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⠃⡿⣿⡇⢸⣿⣿⣿⣿⣿⣬⣷⣿⡿⠿⠆⠀⠂⣄⡀⠀⢸⣿
⣿⡿⠿⠛⠿⠟⠋⠉⠁⠀⡛⠑⠋⠈⠉⢸⣿⡇⠀⠀⠀⢸⣿⡟⠀⡇⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⢦⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⢄⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⣿⣿⠁⢸⣿⣿⣿⣿⣿⡿⢏⣴⠀⠀⠀⠀⠀⠀⠀⠀⠦⣧
⠀⠀⠀⠀⠀⠀⠀⢐⣶⡶⣶⣶⠀⠀⠈⢸⣿⣧⠀⠀⠀⢸⣏⠁⠀⠙⠿⠀⢸⣿⣿⣿⣿⣿⡾⢋⡗⠀⠀⠀⠀⢀⠰⠀⢀⡁⣿
⠀⠀⢀⡀⠀⠀⠀⡈⣽⢀⣻⣿⢆⠀⠈⢸⣿⣿⠀⠀⠀⢸⣿⢀⠀⠘⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠁⠀⠀⠀⣸
⠀⠀⠀⠐⠂⠰⠂⠴⠁⠀⠉⠋⠘⢁⣴⣾⣯⣿⡆⠀⠀⠀⠻⠄⠀⢸⡏⠀⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣦⣴⣶⣦⣤⣶⣶⣧⡀⠀⠀⢰⣄⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣀⠸⠾⠇⠀⣿⣿⣿⣿⣿⣿⡿⣇⡀⠀⠀⠀⠀⠀⣠⣤⡄⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⡇⢸⡿⣿⣿⣿⣿⣇⣦⣀⢸⣿⣀⢁⠀⣸⣿⣿⣿⢿⣿⡽⣧⠈⢣⣄⠀⠀⠀⠀⠃⠀⠀⣿⣿
⣿⣿⠛⢻⠿⠟⢿⠿⠛⣿⣿⠛⠁⠀⠀⣸⣿⣿⣿⣿⠿⠯⡿⣿⣴⣿⢠⣿⣿⣟⠹⣿⣉⡷⣾⣳⣄⡙⠷⠶⠦⠀⠀⢠⣆⣿⣿
⠉⠁⢀⡀⠀⠒⣦⣠⣂⣿⡇⢀⣀⣀⣨⣿⠿⠛⠀⠉⠁⠁⠈⢁⣈⣠⣶⣿⣿⣿⠀⠀⠉⠁⠉⠉⠀⣉⣀⡀⠀⠀⣀⢼⣿⣿⣿
⢀⠁⠘⠇⠃⠀⠀⣬⣬⠛⣿⡿⠿⣿⣿⣿⠀⢀⣠⣤⣤⣿⣿⣯⣿⢿⣿⣿⣿⣿⣌⣁⣠⣤⣴⣶⣫⡤⠛⠛⠗⠸⣿⣿⣿⣿⣿
⠈⠀⠀⠀⠀⠀⠤⠀⠀⢀⣬⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉
⡀⠀⠀⠐⡦⠄⠀⠀⠑⠛⡉⣹⣍⣭⣿⣿⢿⣿⣿⣿⠿⡿⠿⣿⣿⠿⠟⠿⠿⣟⠉⠉⠉⠁
⣡⣴⣶⣦⣿⣶⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣄⠐⠏⠁⠀⠀⠀⢀⣁⢠⠀⠀⠠⣞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠯⠉⠉⣙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠔⠒⠒⠒⠒⠲⢤⣀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡀⠀⠀⠀⠀⠀⠀⣀⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣍⠂⠲⠤⠖⠒⢋⡍⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢿⣷⣶⣤⣶⣶⡟⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠛⠿⠿⠟⠋⠀⠀⠀⢠⡏⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢄⡀⠈⠉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠓⠲⠤⠤⠤⢴⠞⠫⢄⠀⠀⠉⠲⡀⠀⠀⠀⠙⢦
⠀⠀⢀⠤⠐⠒⠒⠒⠤⢄⣀⣀⣀⣠⡤⡾⠁⠀⠀⣀⣠⠴⠚⠁⠀⠀⢠⠇⠀⠀⠀⠙⣆⠀⠀⠀⠀⢳⡄
⡖⢶⠡⣆⣀⡹⢤⣼⣃⡀⢠⠤⠤⠤⠴⠷⠒⠚⠙⠍⠀⠀⠀⠀⢀⡴⠋⢀⠠⠂⠀⠀⠘⡆⠀⠀⠤⠊⢹
⠓⠒⠶⠮⠭⠥⠤⢤⣀⠉⠙⢧⡔⠁⢀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠂⠉⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⡗⠺⢿⡹⣄⣀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠸⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠈⡗⢬⣃⡇⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠀⠀⠀⡇⢰⠀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⡇⠀⠀⠀⢧⠀⠈⠀⠀⠀⠐⠂⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⢠⡇⡸⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⣸⢁⡇⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡷⠀⠀⠀⠀⠈⣧⣄⠀⠀⠤⠤⠤⠤⠤⠶⠒⠉⠀⠀⢀⡴⢧⣼⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢀⡞⠀⠈⠙⠓⠦⢤⣄⣀⣀⣀⣀⣠⣤⠖⠋⠀⠀⠈⠉⠛⠛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⣇⣇⣠⠤⢤⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣠⡤⠖⠁
⠀⠀⠀⠀⠀⡜⠁⣜⣆⠀⢀⡞⢁⣤⡀⢻⠒⠾⠦⢤⠼⠤⠼⠶⠤⣤⣄⣀⣀⠤⠤⢄⡈⠉⠉
⠀⠀⠀⠀⠀⢧⠀⠛⢉⡗⢺⠀⢸⡠⠃⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡔⢢⠀⢷
⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠘⣆⠀⠀⣠⠏⠀⠈⠉⠉⠉⠉⠉⠙⠒⢶⠒⢿⠀⠳⠁⢀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠣⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡃⠘⢦⡀⠀⣀⠾⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠐⠀⠈⠉⠀⡄⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⠄⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣏⡀⠐⠂⠀⠀⠀⣀⠠⠀⠀⠀⠳⠤⠤⢶⡞⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⣠⠋⢫⠀⠀⢀⣀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠈⢲⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣰⠏⢃⠀⢀⠜⠁⠀⠀⠉⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠡⠀⠀⠀⠀⢀⣃⣀⣀⠀⠀⠀⡰⠁⠀⠀⠑⠁⠀⢀⠁⠀⠈⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡋⠀⠀⠀⠀⢀⣠⠤⢄⠀⡞⠁⠁⠀⠙⣆⡜⣁⣄⠀⠀⢀⠀⠂⠀⠁⠐⠠⢈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⢤⣸⠀⠀⠀⡽⣇⠀⠀⠀⠀⡼⡏⠀⠀⢱⠊⠀⠀⠀⡠⠖⠊⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⠒⠊⠀⠈⠓⠤⠤⠚⠁⠑⠤⠤⠊⠑⢤⣀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⣀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⡠⠖⠁⠀⠀⠈⠳⣄⢀⡠⠞⠁⠀⠀⠈⢦⠀⠀⡠⠤⢄⡀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠉⠉⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠋⠁⠀⠀⠉⠒⠶⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠸⠿⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⡄⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⢶⠆⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠾⠂⠀⠀⠀⠀⠶⠄⠀⠀⠰⠆⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⡀⣀⡇⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠙⠓⡲
⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠚⠛⠋⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣽⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣶⣾⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣟⠲⠻⠤⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠀⠈⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡤⠺⡅⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣾⠞⣉⡉⢦⢀⡴⠦⣸⠃
⠀⠀⠀⠀⠀⢡⠀⢰⠀⠀⠀⠀⢰⣿⠉⠀⠉⠻⣿⠀⢓⡟
⠀⠀⠀⠀⠀⠈⡀⢸⠀⠀⠀⠀⢸⠘⡄⣸⠷⠒⠙⣀⠀⡇
⠀⠀⠀⠀⠀⢠⣇⡸⣆⠰⣄⠀⠈⢇⡼⠁⠀⠀⢆⡿⢸⠗⠒⠒⠒⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠹⢇⣤⡙⠒⠚⠓⠋⠀⢀⣪⠵⣻⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠉⠙⠒⠚⠛⠛⠭⠤⠚⠁⠀⠀⢀⣀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠚⠉⠸⡇⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣰⡀⠀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⡏⢦⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠲⣠⣅⣧⡀⠱⣄⠀⠘⡦⠀⠀⣀⣀⣤⠇⠀⢱⡄⠀⠀⠀⢧
⠀⠀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠛⣦⣤⢷⡱⡯⢭⠤⣤⠤⠶⢾⡷⣶⠆⠀⠸⡦⠤⢴⠎⠉⡧⠒⠒⠒⠶⠶⡶⠶⠂
⠀⠈⠉⠉⠀⠈⠉⠹⣄⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠠⢱⣝⡄⠀⠀⢀⠔⠋⠸⠃⠀⠀⢀⡇⣾⢧⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣤⣀⡀⣀⣸⣀⠀⠀⠶⠿⣿⡄⢠⣃⣀⠀⠀⠀⠀⣠⠞⠀⠙⣿⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠋⠈⠉⠓⠂⠉⠈⣟⠍⠀⠀⠉⠑⢦⠞⠁⠀⡴⠶⢼⡿⠴⠻⡆
⠀⠀⠀⢀⡠⠔⠒⠒⠒⢦⡄⠀⠀⠀⣰⣃⣀⠀⣀⣀⣀⣀⣀⣼⡀⣆⢀⠀⢺⡇⠀⠀⠀⢹⠉⠉⠙⣧⠀⡇
⠀⠐⠶⠥⣤⠤⠄⠀⣠⠞⠤⣀⣀⠀⠀⠀⠁⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉⠛⠉⠀⠀⠀⠀⠈⣗⠒⠒⠃⢀⡇
⠀⠀⠀⠀⠓⠤⠤⠤⠤⣤⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⢰⣶⡄⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶
⠀⠀⠀⢠⣶⣿⣶⡌⢢⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣠⡄⣠⡀⣤⣀
⠀⠀⠀⢸⣿⣿⣿⡇⠨⠁⠀⠀⠀⠀⠈⣿⡿⠟⠀⠀⠀⢸⣿⣿⣿⣿⠿⢿⣿⣿⣿⡏⠀⢿⣏⢺⣛⣻⢊⣽⠇
⠀⠀⠀⠀⢸⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⢩⣇⠀⠀⠀⠀⢸⣿⠟⡃⠘⢰⠐⠐⡛⢿⡏⠀⠀⠙⣿⣾⣾⡟⠁
⠀⠀⠀⠀⠸⠿⠇⠀⣼⡄⠀⠀⠀⠀⠀⢲⠟⠀⠀⠀⠀⠸⣯⣍⣀⣩⣀⣨⣁⣀⣭⡧⠀⠀⠀⢿⣿⣿⠇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣾⢟⣵⡄⠀⠀⠀⠀⠀⠀⢀⡴⡚⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡇⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠛⠋⠿⣀⡀⠀⠀⠀⠀⡞⠀⠜⠀⠚⠄⠸⡆⠀⠀⠀⠖⠲⠂⠀⠀⠀⠀⠀⠀⢰⡃⠛⠁⣧⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⠀⠀⠀⢳⡘⢿⣿⣿⠟⣸⠃⠀⠀⠀⠆⠠⠄⠀⠀⠀⠀⠀⠀⠀⣇⠙⠺⣝⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠀⠀⠀⠀⠙⠒⠶⠖⠊⠁⠀⠀⠀⠘⠀⠀⠃⠰⠷⠆⠀⠀⠀⠀⠈⠓⠶⠛⠋
⠀⠀⠀⡤⠴⢄⣀
⠀⠀⠀⠀⠀⣼⣿⣍⠙⠛⠀⠀⠀⣶⣶⢤⣶⣦⣴⣶⡆⠀⠀⠀⣠⠴⠖⠶⢤⡀
⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠐⠿⣿⣺⠿⣗⣿⢿⣺⡿⠗⠀⡼⠁⢄⠄⢄⢀⡹⡄⠀⢀⣠⡀⢠⣶⣦⠀⣀⣀
⠀⠀⠀⠀⠀⣼⢹⡇⠀⠀⠀⢀⣤⣿⣻⣽⣟⣿⣽⣻⣧⣄⠀⢧⠰⣶⣤⣶⡶⢠⠇⠀⠸⣿⡇⣿⣟⣿⡇⣿⣿
⠀⠀⠀⠀⠀⢻⢸⡇⠀⠀⠀⠀⠉⣿⣾⢽⣷⣿⣽⣾⡏⠁⠀⠈⠳⢬⣉⣩⠴⠋⠀⢠⣤⣿⢢⣴⣿⣷⣤⢺⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣀⡀
⠀⠀⣀⣀⣀⣀⣠⣶⣶⣶⣶⡄⠀⢀⣬⣿⣥⡙⢄⠀⠀⠀⠀⠀⠀⢉⣤⣌⠁
⠀⠀⡇⣀⠀⣀⢸⣟⣻⣟⣻⡇⠀⢸⣿⣿⣿⡇⢘⠃⠀⠀⠀⠀⠀⠘⣿⠟
⠀⠀⡇⣍⣀⡩⢸⣿⣛⣛⣿⠁⠀⠈⢹⣿⡏⠁⢸⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⢺⠉⣇⡞⣭⡙⣏⢹⠃
⠀⠀⠰⢄⣉⡠⠎⠙⠛⠛⠋⠀⠀⠀⢸⢿⡇⠀⣈⡀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⢼⣀⣛⣥⣛⣡⣟⣘⣻⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿⠀⠀⠀⠀⠀⢶⠋⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⣀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⢸⡇⢀⡇⠀⠀⠀⠀⠀⠀⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠁⠁⢼⠀⠀⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⢸⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡞⠈⠀⠀⠀⠀⢸⣿⠙⢿⣿⣦⣄⠀⢸⠀⠈⠀⣠⣾⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠻⢿⣿⣿⣶⣶⡿⠿⢿⣷⣦⣤⣶⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⢠⣿⡏⠀⠀⣶⠚⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠾⠿⠷⣄⣀⣸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⢳⣠⡞⠁⠀⣀⣀⡀⠀⠀⣹⠟⣧⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠉⠀⢀⣼⣿⣀⣀⣀⠀⣶⠞⢧⡀⠀⠀⠀⡟⠀⠀⢿⣮⢠⡾⢟⣛⣿⠿⠾⠟⢲⣿⡀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢶⣶⣶⡾⠛⠉⠀⠀⠀⠉⠓⢿⣿⡧⠹⣦⠀⢰⡇⠀⠀⠈⣿⣿⡁⠘⣧⠀⠀⠀⠀⢸⣿⣇⣀⣸⣏⠒
⠀⠀⠰⡦⠞⡆⠀⠀⠀⠀⠀⠀⢹⣿⡠⠀⣠⣤⣦⣤⡀⠀⠀⢻⡏⢄⠈⠳⣼⠁⠀⠀⠀⠘⣿⠳⣤⣿⡾⠟⠀⠀⣼⠙⣏⠁⠀⢿⣤⣤⣀⣀
⠀⠀⠰⠧⢄⡶⠀⠀⠀⠀⠀⠀⣿⣿⣟⣾⠏⠁⠀⠉⢻⡆⠀⠀⢿⣡⡄⠀⠉⠀⠀⣿⡀⠀⠸⣇⠀⠀⠀⢀⣠⡾⣫⡴⣿⣶⠶⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣯⣿⠀⠀⠀⠀⠀⡿⠀⠀⣸⡟⣿⣦⡀⠀⢸⡟⣷⣀⣀⣽⡿⠿⠿⠟⠋⢰⡟
⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣠⠞⠁⠀⢀⣿⣿⣿⠙⢿⣤⣾⠀⠘⠛⠛⠉⠁⠀⢄⡀⠀⠀⢳⡀⠀⠀⠀⠀⣨⠁⠸⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡤⠊⢻⣿⣿⣷⠀⠀⠀⠀⠀⣠⣾⡿⠿⠟⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠵⣎⠹⣍⠉⠉⠉⠁⠀⠀⠀⠀⠱⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠈⠛⣿⣷⣶⣶⡶⠟⠛⠻⣷⡄⠀⠀⠀⠀⢀⣠⣤⢤⣄⡀⢸⠀⠈⢧⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠟⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢀⡴⠋⠁⠀⣀⡀⠈⠲⡄⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠘⡇⢀⡞⠁⢠⡞⠋⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠋⠀⢲⠅⠀⠀⠀⠀⠀⣿⡞⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⢤⣰⠇⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠐⢎⣀⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣤⡄⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⢸⠇⢰⠇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢏⠀⡐⡅⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⢸⠀⢸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⣿⡇⠙⢿⣷⣦⣄⣘⠀⣂⣀⣴⣷⡀⠀⠀⠀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠻⣿⣿⣿⠿⢿⣿⡟⠀⠀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⣠⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠸⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⢿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠖⠛⠉⠉⠉⠓⠶⣤⣿⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⢸⣿⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠁⠹⣾⡣⠀⣠⣴⣶⣤⣀⣰⣟⣹⡆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣀⣀⣤⡾⠟⠓⠒⠶⣄⣹⣯⠝⢦⡀⠀⢀⡇⠀⠀⢻⡗⣾⠋⢰⡖⠛⠉⠉⠉⣿⣷⠀⠀⣧⣀⡀
⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⠀⣀⠀⠀⠀⠙⢿⡯⡈⠻⣄⢸⠃⠀⠀⠀⢻⣻⣄⠸⣇⣤⡄⠀⠀⡿⣿⠖⠒⢻⡀
⠀⠀⢈⡏⠀⡣⠀⠀⠀⠀⠀⢠⣿⣷⢅⣴⠿⠛⠻⢷⡄⠀⠈⣿⣔⠀⠈⠿⠀⠀⣆⠀⠈⢷⡉⠛⠛⠉⠀⢀⣼⢃⣸⣄⣀⣼⡿⠛⠓⠒⠂
⠀⠀⠈⠁⠙⠃⠀⠀⠀⠀⢀⣼⣿⣿⣺⡇⠀⠀⠀⠈⢿⠀⠀⢸⣿⣷⡄⠀⠀⢰⣿⡆⠀⠘⣷⣤⣤⣴⡶⠟⣱⠟⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⡞⠀⠀⣼⣷⣿⠻⣦⣀⣸⠀⢻⡶⠾⠿⠉⢉⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡆⠠⡄
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠉⠉⣹⠿⣿⣿⣾⠉⠛⠛⠁⠀⠀⣰⣿⣿⣿⠄⠘⠛⠋⠀⠀⠀⠀⠀⠐⣤⠷⣖⠒⠒⠛⠦⠀⠀⠀⠈⢠⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⡰⠛⠻⣷⣧⣤⣤⣤⣴⡾⢿⣯⡁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡎⠈⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⠞⠁⠀⠀⠜⠋⠉⠉⠉⠀⠀⠀⠙⣷⡀⠀⠀⣠⠞⠋⠉⠉⠉⠲⣇⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡼⠃⢀⡴⠞⠛⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠒⠈⠦⡄⠀⠀⠀⠀⠀⣿⡼⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣠⣀⡎⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠠⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠲⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⣄⠀⠀⠀⠀⠀⠀⠀⢠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠉⢴⣷⣶⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠿⣿⣿⢼⣿⣿⡆⠀⠀⠀⠀⢀⣄⢀⣀⣠⡄⢠⣠⣄⣀⣀⣶⣾⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⢻⡛⢿⡷⣿⠟⢛⣿⠗⢶⡄⠀⠀⠀⠘⠛⠋⠛⣿⡟⢺⣋⡙⣛⠿⣋⡻⣋
⣿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢨⡇⠀⢷⣿⠀⠀⢻⡆⠀⢻⣆⠀⠀⢴⣾⡇⢷⣾⣿⣾⣿⣧⣿⣷⣾⣶⣿
⣿⡇⠀⠀⠙⠁⠛⠁⠀⠀⠀⠈⠉⠉⠙⠃⠀⣘⠁⠀⠀⠸⡇⠀⢸⡇⠀⠀⣼⠇⣀⠀⢿⡄⠀⠺⠷⠷⠶⣶⣾⡶⣶⠶⢶⣶⡾⠦⠈
⣿⠁⢀⣄⣉⠋⣉⣍⠉⠉⠉⠈⠉⢉⠉⠀⢸⣯⣳⣶⣿⣤⠀⠀⢸⣷⠀⠀⢱⢆⣿⣦⣠⣽⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂
⣿⣧⣾⣧⠴⠷⠤⣤⣤⣤⣤⡤⠌⣉⠀⡄⠀⡙⢿⣿⡉⢿⣇⣠⢾⣿⡄⠀⣸⣿⢿⣿⡿⠃
⣿⣿⠛⠛⠒⠶⠶⡦⠴⠍⠀⠀⠦⠠⠦⠈⠀⠀⠀⠙⣿⣴⣿⣯⣼⣿⣷⣤⣿⢣⣼⠏
⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⡟⠀⠸⠁⠿⢿⡿⣾⡁⠀⠀⢀⡀⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⣤⣾
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠐⠀⠀⠀⠀⠀⠐⠸⣿⡀⠿⠄⣾⣶⣤⣤⣴⣏⣿⣧⣤⣤⣶⣾⣷⣶⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠐⢀⣶⡆⠀⠀⠀⢹⡇⠀⢼⡯⣴⠿⣷⣶⣴⣿⢷⣤⣤⣭⣭⣧⣤⣔
⣿⠀⠀⠀⠀⠐⠒⠦⠤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⣼⡏⢻⡆⠀⠀⢈⣿⠴⠰⠷⢾⡷⣾⣷⣼⣶⣤⣤⣤⣬⣿⣿⣿⣾
⣇⠀⠀⠀⠀⠀⠀⠄⠀⠤⠀⠠⠀⢠⣤⢀⣤⠀⠀⢰⡇⠀⠀⣌⣿⠁⠈⣿⡄⢀⠀⣿⡆⢘⠛⣸⡷⣾⡿⢨⠍⠉⠃⢈⣉⣿⣿⣭⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠰⠤⠤⢤⠄⠀⠀⣼⠀⠀⣠⣿⡇⡀⠈⣇⣿⠘⢠⢸⣧⠏⡄⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠠⠰⢿⣿⠁⡷⠀⣿⣿⡆⡉⣞⢿⡞⣹⣿⣄⡀⣿⠇⣀⠀⣀⠀⣸⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣿⢻⣄⠁⢠⠀⣻⣷⢻⣏⡾⣿⡛⣻⣿⡿⣿⣿⣿⣶⠿⣿⣿⣿⣿⣿⣿
⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡿⠠⢠⣾⢿⠄⠀⠀⠘⢠⡏⣿⣇⡿⢱⣿⣏⢿⠛⣿⣻⣷⣆⣰⣾⣟⣿⣿⣿⣿⣿
⣷⣶⢰⣶⠠⠶⠶⣶⡦⢤⣶⢤⡀⠀⠀⢀⠀⣿⠇⡄⢈⣿⡾⠀⠀⠀⠀⢸⣷⣿⣿⣿⡟⣏⣿⣾⡾⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⠿⣿⠗⠒⠲⠿⢶⠾⠗⢸⠿⠛⠃⣀⣼⣯⠷⣸⣸⡧⢃⠀⠀⣀⣰⣯⣼⣿⣿⣿⣵⣯⣽⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠟⢻⠗⠰⠂⠐⠛⠒⠒⢶⠆⠀⢀⣭⣿⣿⣶⣇⣿⣧⡍⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣻⠿⡿⢓⡛⠂⠚⠛⠛⡒⠛⠓⠀⢸⡿⠾⠿⠛⠙⠓⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡷⢴⠷⣤⣭⣿⢿⣤⠾⠿⢷⣶⢀⡄⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣟⡛⢛⣧⠟⠛⠛⠛⠛⢻⣿⠟⠛⢻⣖⣮⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢿⣷⣿⣿⣶⡦⠴⣶⣤⣤⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣿⣾⣿⣶⣶⠶⠶⠿⠷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢶⣤⣀⠀⠀⢀⣤⣤⣤⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠈⠙⠷⣴⡟⠁⠀⠀⠙⢿⣮⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣿⠶⠶⠶⠶⠶⣾⣷⣀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢞⠋⠁⠀⠀⠻⢷⣦⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⢰⠟⢉⡙⢷⣴⠞⠛⠳⣌⠁⠀⠀⠀⠀⠀⠹⣧
⠀⣀⣤⣀⠀⢀⡞⢉⡗⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢹⣄⠘⢋⣼⣧⠀⠻⢇⡿⠀⠀⡄⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠲⣆
⠀⠻⣄⣈⡙⠾⡄⠾⢦⣄⣠⣤⣤⣤⣄⣸⡿⠀⠀⠀⠀⠀⠀⠙⠋⠉⣹⠏⠻⠟⠛⠁⠀⣸⣧⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣿
⡴⠒⠒⣺⡇⠀⣇⠀⠀⢉⣀⣠⡤⠤⠤⢼⡇⠀⠀⠀⠀⣴⠟⠛⠛⠛⠁⠀⡀⠀⠀⠀⣴⠋⣿⠀⠀⠀⠀⠀⣿⣟⣛⣛⣛⠛⠒⠶⡞⠋⠛⣰⣇⣀⣀⡀
⠉⢛⡩⠽⢃⣤⣀⣀⡴⠋⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠙⠷⣦⣤⣀⣀⣠⡿⠀⣠⡾⠁⠀⣿⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠙⠓⣦⠀⠀⠀⣿⠀⢦⣈⠉⠳
⠀⠘⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⣥⡴⣾⠋⠀⣠⡞⢹⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠘⣦⡀⠐⠋⠀⣼⠙⠻⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⠉⠀⣼⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡄⢧⠙⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠋⢠⣇⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⣀⡆⢠⠄⠀⢀⠀⢀⠀⢀⣐⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠁⣿⠻⠶⢿⡄⠸⡟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⢿⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⠾⠦⠸⣇⠀⢸⡇⠸⠿⠶⠶⢶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣛⣋⣉⣁⣀⣀⣀⣀⣀⣤⠟⠀⠸⢧⣤⣤⣄⣀⣀⣀⣀⣉⣹⣷⡄
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢠⣏⠂⠀⠀⠀⠀⡞⡐⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠁⡿⣱⢀⣴⠀⠀⣸⢋⡀⣤⠆⣴⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠜⠃⣿⢾⡡⠀⠀⠿⠰⠛⣏⢀⢼⢻⣇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡶⣄⢰⠀⣼⣧⠀⠀⠦⣶⠶⣴⠦⠀⣠⡄⢠⠾⡏⢼⠁⣶⢰⠶⠀⡶⢦⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⢃⡿⢸⢀⡇⣿⠀⠀⠀⣿⠀⢿⣀⠀⣿⡇⣿⠀⠗⢸⣀⣿⢸⣀⠀⡇⣾⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠛⣧⢸⠸⡇⣼⠀⠀⠀⣿⠀⣾⠉⢸⣇⡇⢹⠀⡤⢸⠁⣿⢸⠉⢠⣿⡅⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠦⠏⠸⠀⠻⠟⠀⠀⠀⠿⠀⠻⠖⠸⠋⠧⠸⡶⠃⠘⠀⠿⠹⠶⠀⠇⢳⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣤⢰⣳⠆⡤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⣧⠊⠘⠺⡇⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡤⢤⡄⢀⣤⡤⠄⣄⠀⡄⠀⢠⡤⡄⣤⠬⣤⣤⢦⣤⡴⢦⣤⠄⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⣷⢸⠹⣇⠀⣿⡀⡇⠀⢸⣀⡟⣷⡀⠀⡇⠀⠀⡇⠸⣇⠀⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡏⠁⢻⡟⢰⡏⠀⡏⣷⡇⠀⢸⠛⡇⡟⠁⠀⣿⠀⠀⡇⢨⡏⠁⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠀⠸⠇⢘⣷⣤⣇⠘⠇⠀⠘⠶⠃⠻⠂⣀⡟⠀⠘⠇⠘⠷⠀⠇⠳⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⣸⠇⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠁⣰⠆⠀⠀⠤⠤⢴⡿⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢀⡋⠀⣴⠆⠀⢀⣿⠁⠀⢀⣴⢲⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠸⣷⠀⣿⠁⡴⢻⡆⣀⣼⠇⠀⣠⣾⡧⠞⢠⠞⢷⡦⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣠⡿⠀⠻⠞⣠⢼⣟⢩⣿⠀⡴⠃⣿⠀⡴⠃⢀⣿⢁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠞⠛⠁⠀⠀⢸⣁⣼⠏⢸⣧⠞⠁⠀⠙⠋⠀⠀⠘⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠉⠉⠁⠉⠉⠉⠉⠁⠀⠉⠁⠀⠉⠉⠉⠉⠛⠒⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠠⠄⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢉⠑⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢔⠐⠃⠀⠀⠀⠀⠀⠂⠀⠠⢄⡀⠀⠀⠈⠑⠑⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠚⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⡤⠄⠒⠈⠁⠀⠀⠀⢀⡠⠔⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠎⢡⣀⣀⣀⣀⣤⡤⢰⠊⢉⠅⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠸⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠰⠤⠒⣂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠂⠀⡀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠡⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⢀⠴⠥⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡆⠀⠀⠀⠀⠐⠠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⢠⡒⠒⢣⠤⠀⡀⠀⠀⠀⠀⠀⠛⠛⠉⠡⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠇
⠀⠀⠀⠩⣉⡩⠄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠤⠤⠤⠤⠤⢀⣀⣀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠦⣀⡠⠛
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⣠⣠⣄⢤⣄⣄⠀⠈⠣⡄⢀⡠
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠠⠶⠢⠀⠉⠃⠑⡿⢆⠙⠀⠁⠀⠁⠈⠿⢆⠀⠘⠤⠀⠤⠄
⠀⠀⠀⠀⠀⠳⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⠀⠀⢁⡀⠀⠀⠀⠠⠈⠀⠉⠉⠀⠠⢄⡀⢀
⠀⣀⠔⠊⠃⠀⠀⠀⣀⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣠⠤⢀⡄⠄⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⢠
⠘⠂⠀⠛⠐⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠞⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠐⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⣀⣤⣴⡶⠶⠿⠿⠿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣤⣴⣶⣶⣿⣿⣿⣿⣧⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣯⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⢀⣀⣀⣀⣿⣿⣫⠷⠋⣿⣿⣍⠙⠛⠳⣿⡄⢰⣿⣿⣿⣿⣶⣶⣦⣶⣯⣥⣤⣤⡀
⠀⣴⠋⠁⠀⠀⢉⣿⣯⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠈⢿⣦⣄⣀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣭⣄⣈⢿⣿⡿⠋⠉⠻⣿⡏⠀⣀⣀⣀⣀⡀
⠀⠀⠉⠙⠛⠛⠻⠿⣿⣿⣿⡏⠀⠀⠀⠀⠙⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠛⠉⠁⠀⠈⣹
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⣿⣤⣴⣾⣿⣿⣷⡾⠁⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣴⠶⠛⠁
⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠈⠻⢿⣿⣿⣿⡿⠃⠀⠈⢿⣿⡇⠀⠉⠙⠛⠛⠉⣽⣿⣿⢯⡾⠋
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⣀⣤⣭⣯⣍⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⣠⣼⣿⠟⢱⡿⠁
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁⠀⢀⣽⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠁⠀⢸⣇⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠋⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠐⣶⣶⠀⠀⠒⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⣀⠀⠀⢤⣄⡀⠀⠀⣠⣤⡄⡀⣤⣤⡀⢀⡀⠀⠠⣤⣀⠀⠀⠀⢀⣠⠤⢀⢸⣿⡇⠀⠀⠀⣠⡤⠀⢤⣀⠀⠀⠀⢸
⠀⠀⣿⣿⠀⠀⠀⢠⣿⡿⠀⠸⣿⡿⠀⠀⣿⣿⡀⠀⢸⣿⡟⠀⠻⠿⠃⣿⣿⠆⠀⢸⣿⡇⠀⢰⣿⡏⠀⠀⢺⣿⡇⠀⢀⣾⡟⠀⠀⠸⣿⣧⠀⠀⢸
⠀⠀⣿⣿⠀⠀⠚⠋⠉⠀⠀⠀⣠⡤⠂⠀⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⣀⣤⠄⠀⢸⣿⡇⠀⣿⣿⡇⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⣉⡉⠀⠀⢸
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⣸⣿⡇⠀⢻⣿⡇⠀⠀⣼⣿⡇⠀⠸⣿⣧⠀⠀⠸⣿⣿⠀⠀⢸
⠀⠒⠛⠛⠓⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⠂⠙⠿⠧⠀⠾⠿⠷⠀⠀⠀⠈⠻⠿⠶⠖⠙⠻⠿⠄⠀⠙⠿⠦⠂⠘⠿⠷⠀⠀⠈⠛⠳⠆⠀⠀⠁⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣰⣶⡄⢰⣶⢶⣤⢀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡏⢻⣧⣿⣿⣿⣸⣿⣰⡿⠈⠀⢿⡝⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠾⢏⣙⠁⡍⢡⡍⢭⠡⣀⡸⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢺⡁⣧⠄⡷⣜⠟⢸⠀⢰⣌⡓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⣼⡁⡽⢤⡧⢸⠁⣿⣀⡼⢹⡧⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠶⠯⣭⡴⠳⠾⠦⣚⣠⣭⠵⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⣄⠀⢣⠀⢠⡄⠈⠟⠀⡔⡆⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠎⣠⢻⠀⠸⠃⣠⡀⢸⢀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠄⢙⢼⠀⣴⠀⢱⠣⡀⠁⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣲⠖⠉⠘⠒⢲⡖⠁⠀⠈⢻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣦⠉⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡙⠄⠀⠀⠀⠀⠀⢀⠘⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⢲⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠻⣗⢦⡀⠀⠀⠙⣷⣤⣀⣀⡈
⠀⠀⠀⠀⠀⠀⡴⠋⠹⣆⠈⠣⡈⠀⠀⢀⡏⠀⠈⠁
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠘⣆⠀⠈⠢⡀⠸⣿⠂
⠀⠀⠀⢠⣾⡥⠂⠀⠀⠀⠀⡀⠠⠶⠈⠲⢻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠳
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠻⣀⢀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠟⠁
⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠢⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡿⠃
⠀⠀⠈⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢹⣯⠓
⠀⠀⠸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠍
⠀⠀⠀⢇⠘⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠿⠃
⠀⠀⠀⢸⡆⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⢀⣀⡤⠂⠀⠀⠀⠐⡄⢀⠐⣩⠉
⠀⠀⠀⠀⢿⡀⠈⣿⡀⠀⠀⠀⠀⠀⠀⣠⡼⣿⠉⠉⠀⠀⠀⠀⠀⠀⠈⡀⠀⠁
⠀⠀⠀⠀⠘⣇⠀⠘⢿⡄⠀⠀⠀⠀⠚⢩⡇⣻⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⢻⡄⠀⢸⣿⡄⠀⠀⠀⢀⡾⠀⢻⡇⠀⠀⠀⣀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠈⢷⠀⠀⠳⣝⠀⠀⠀⡸⠁⠀⢸⢃⣠⡶⢿⠉
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢈⡧⣤⣀⡁⠀⣀⣼⠟⠋⠀⠈⡆
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠉⠛⠾⠟⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣾⡦⠖⠚⠛⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠘⠉⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⣋⣀⣠⠤⠴⠶⠶⣤⣄⠀⠀⠠⣬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠤⠀⠀⣸⡿⠻⣿⣷⠀⢠⠎⠻⠿⡷⡆⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⡼⢻⠀⠀⢀⡜⠀⠼⢦⣀⡼⠁⢳⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡖⠀⠀⡼⠁⠘⠲⠖⠋⠀⠀⠀⠀⠀⢱⡄⢸⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡇⠀⠀⠀⠀⠀⠐⠶⣤⣴⡾⠟⠀⢸⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀⠘⠳⣆⢸⡇⠀⣠⢶⣀⠀⠀⠀⠀⠀⠀⡼⠛⠘⢏⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣲⣶⣧⠀⢠⡞⠿⠇⠀⠀⠀⠈⠙⠦⠄⠀⢀⠞⠁⣀⡀⢯⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣈⣹⠀⠘⡏⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⢡⡖⠉⣉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢭⢶⣾⡆⠀⠙⠦⠩⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⢩⠟⠛⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⡇⢠⠖⠒⠀⠛⠛⠓⢤⡀⠀⠀⠀⠀⣠⣿⠟⢋⠁⢸⠉⠉⠉⠉⠉⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⡇⡟⠀⠀⠀⠀⣠⠆⠈⣿⡇⠀⠀⢸⣧⣿⠾⠋⠁⣼⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⡇⢀⣤⠤⣾⠁⠀⠀⠙⠧⢄⡤⠿⠃⢹⣤⡾⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⠈⣿⣧⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠞⠉⢷⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⢿⡇⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠈⢳⡿⠀⣷⢀⣎⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢻⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏⠙⢦⣀⣼⣧⡴⠇⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠈⣧⠀⠀⢸⠇
⠠⠤⠤⢤⣤⣤⡟⢹⠀⢧⠀⠈⣿⣧⣿⣧⣠⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣀⣀⣠⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠻⠶⠾⢦⣨⠷⣴⠟⢷⣾⠷⠤⢤⣍⣉⣉⣛⠛⠛⠛⠓⠒⠒⠶⠤⠤⠤⣤⣤⣼⣿⣯⣭⣥⣤⣄⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠈⠿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠉⠋⡿⠃
⠀⠀⠀⢀⣤⣔⣚⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠒⠶⠤⠤⠤⠤⠤⠤⣤⣤⣤⣀⣀⣀⣀⣠⣤⠤⠤⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠔⠃⠉⠘⠁⢄⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠨⠬⠥⠄⢴⠀⠀⠀⠀⠀⠀⠀⠀⣘⣀⠤⠀⠀⠀⠀⠉⠀⠒⠒⠂⠀⠀⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠔⠀⠐⠐⠀⡜⠁⠈⠀⢸⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠠⠴⠶⠍⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢱⠀⠀⠠⣶⣿⣤⠀⠀⠀⣸⠀⠀⡔⣐⠂⣇⠀⠐⢠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡑⠀⠀⠚⠉⠂⠠⠔⠁⠀⠀⠊⢁⠂⣀⡀⢁⠉⠚⠊⠊
⠀⠀⢀⠤⣤⣄⠸⡀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⢀⠴⡂⠠
⠀⠀⠇⢠⣿⣿⡆⠈⠒⠢⠤⠄⣀⣀⣀⣀⡀⠀⡤⢴⠒⠻⡀⠀⠀⠀⠠⣨⠂⠒⠁
⠀⠀⠰⡟⢻⠋⢢⠤⠀⠠⠄⠂⢩⠈⠀⣀⠄⡈⠠⠃⠐⢄⠀⠀⠜⢡⡴⢠⠃
⠀⠀⡀⠣⡄⣇⡴⠀⢀⡠⡴⠂⢡⠴⠊⠀⠀⠃⠀⠀⠀⠘⠑⠠⠖⣮⠀⠋
⠀⠀⠀⠀⠠⠄⠉⠈⢀⠊⠀⢠⡶⠂⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⠃
⠀⠀⠀⠀⠈⠁⠀⠰⣁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠀⠂⠄⠀⢀⣀⣀⡀⠀⠀⠠⠂⠁⢠⠇
⠤⠀⠠⠤⠆⣠⠂⠰⠷⠄⠀⠙⡓⠂⠀⠀⠀⢀⣠⣴⡗⠀⡄⢸
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠘⠛⠉⠒⠒⠦⠤⠤⠀⠀⢂⣀⡷⠟⠧⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠒⠀⠈

⠀⠀⠀⠀⠀⠀⢀⠠⠀⣄⠀⠀⠀⠀⡀⢀⠀⠀⠀⢀⠀⠄⢀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⡀⠀⡀⡀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠤⠐⠀⠀⠐⠄⠀⠄⠆⠀⠀⠘⠀⠀⠀⠄⠀⠰⠄⠂⠄⠠⠄⠀⠐⠠⠀⠀⠐⠄⠠⠄⠀⠀⠀⠠
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠑⠃⠀⠉⠀⠃⠀⠁⠈⠐⠂⠀⠈⠉⠈⠀⠁⠀⠈⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠠⢠⠠⠀⠀⠀⢠⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⣀⢀⢀⠄⠀⠀⠀⠄⣤⠀⠀⢀⠀⠀⡀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠂⠐⠂⠂⠀⠀⠂⠐⠀⠂⠀⠂⠀⠐⠀⠂⠐⠂⠂⠀⠢⠀⠲⠀⠀⠘⠐⠠⠃⠂⠀⠀⠂⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⡀⣄⡀⡀⢀⢀⡀⠀⠀⡀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠀⠈⠀⠁⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠀⠁⠀⠈⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠤⠠⠀⢐⠄⠄⠀⠠⠠⠀⠄⠆⠤⠠⠂⠀⠄⠐⠀⠀⠠⠀⠠⠀⠀⠠⢀⠤⠀⠆⠠⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠁⠀⠘⠀⠁⠁⠀⠁⠀⠀⠀⠁⠀⠀⠑⠃⠃⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠃


⠀⠀⠀⠰⠀⠰⠀⠆⡄⠀⠀⠰⠆⠤⠠⠀⠀⠆⠢⠄⠄⠆⠤⠂⠀⠠⠆⠠⠠⠄⠠⢀⠆⠐⠠⠄⠄⠀⠂⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⡀⢀
⠀⠀⠀⠃⠈⠀⠀⠀⠁⠈⠀⠀⠁⠈⠋⠀⠀⠀⠁⠀⠊⠀⠁⠈⠀⠁⠛⠈⠁⠈⠂⠀⠀⠈⠀⠙⠀⠈⠈⠀⠁⠀⠙⠀⠀⠁⠘
⠀⠀⠀⠠⠀⣀⠄⠀⢀⡀⢀⢠⠀⠀⠀⡀⠀⢀⢀⢀⠀⡀⢀⠀⠀⠀⠀⠀⢀⣀⢄⠀⡄⠀⠠⢀⡀⡠⠀⠀⡠⡀⠀⡀⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠂⠰⠂⠰⠆⠐⠀⠲⠀⠐⠚⠐⠀⠂⠀⠀⠒⠀⠂⠐⠀⠀⠂⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⡀⡀⠀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠴⠐⠃⠲⠀⠇⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠁⠀⡀⠈⣀⢁⠀⠀⠀⡀⠉⠀⠈
⠂⠛⠉⠉⠉⠁⠀⠈⠉⠉⠁⠒⠒⠢⢄
⠀⢠⢀⣠⣀⢤⠀⢠⣤⠄⠀⣄⠀⠀⠀⠑⡄⠀⠀⠀⠀⠐⠲⠟⠟⣷⠀⠀⠀⠀⠴⢺⣵⣄
⡀⠀⢠⣠⡀⢤⠀⠀⠀⡁⢠⡀⠀⠀⠀⠀⡅⠀⠀⠀⠠⠀⠒⠀⠠⠋⠀⠀⠀⡠⢀⡁⠀⢿
⠀⠉⠀⠁⠈⠀⠀⠁⠀⠁⠀⠀⠀⠀⢀⠤⠬⠀⠢⠤⠀⠀⠘⢁⠀⣜⡷⣄⠀⠀⠀⠁⡄⠈⠢⣄
⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠟⠀⣘⡇⠀⠀⠀⠈⢳⣴⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡖⣺⣧⢻⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⣿⣽⣿⡇⠀⠀⠀⢘⣿⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣀⣤⢀⠴⣿⠀⠀⢸⣿⣿⣿⡇⠀⠘⢿⡺⣿⣿⣯⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠛⢿⠗⠉⡷⠀⠀⢸⡏⣹⣿⡇⠀⠀⠀⠀⣼⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣆⣤⣼⡧⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣿⣿⣤⣿⣾⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⣼⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯
⣶⣶⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⠿⠿⠿⠿⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠿⣿⣿⣿⡦⠤⠤⠤⠤⠤⠴⠶⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠊⠀⢀⡀⣣⠴⠂⠉⠁⠺⠎⢅⠀⠀⠈⠢⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠖⠚⠉⠆⣥⡄⣤⡀⠀⠀⠀⢎⠲⠤⠤⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠃⠦⡟⠁⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣤⠀⠀⠀⠀⢰⠆⠀⣠⠞⠁⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠳⢼⣄⣀⣀⣠⣦⠴⠚⠁⠀⠀⠀⠀⠀⠱⣄⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠉⠐⢤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠁⠀⠀⠉⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢳⡀⠀⠹⣄⠀⠀⠀⠀⠀⠸⣄⠀⢀⠀⠀⠀⠀⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⡀⠀⠈⢧⡀⠀⠀⠀⣰⠋⠁⠀⠁⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠈⢳⠀⠀⠀⠹⣄⣠⠞⠁⠀⣠⠟⠢⢄⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠑⣤⠞⢀⡿⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡠⠖⠋⠀⠀⠉⠀⠀⠈⢀⡴⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢀⣤⠎⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠙⠒⠒⠉⠓⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⠞⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢠⠚⠒⠒⠦⠤⢄⣀⣀⡇⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⢦⣀⡀⣀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⡴⠓⠦⠤⠤⠴⠒⠚⠉⠀⠀⠀⡼
⢌⣀⠀⢀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢠⠖⠉⠀⠀⠀⠉⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠁⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⢸⣉⣉⣉⣰⠒⠒⠒⠒⠒⠒⠛⠓⠇⠀⡆⠀⠀⠀⡆⠀⡿⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠤⢾⠀⡆⠀⠀⢀⠀⠀⡀⡋⠛⠒⠒⠶⠶⠶⠒⠛⠉
⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⢇⣀⣀⣤⠗⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢧⣄⣀⣴⠤⠤⠋⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣄⡀⠀⣩⠀⢀⣤⣄⡀⠀⠀⠀⣽⠀⢀⣤⣄⣿⠀⢀⣤⣤⡀⠀⣠⣄⣠⡄⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⢀⣿⠀⣿⠀⣿⡀⢈⣿⠀⠀⠀⣿⠀⢿⡁⠀⣿⠀⢿⡇⠸⠇⢸⣏⠀⣸⡇⠈⡧⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠀⠛⠀⠈⠙⠛⠁⠀⠀⠀⠛⠀⠈⠛⠛⠛⠀⠈⠛⠛⠁⠀⠛⠛⠛⠃⠈⠛⠚⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⣿⠿⡿⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⣰⣿⡿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢨⣿⢋⡁⠒⣿⡿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠉⠐⣨⣷⣟⢳⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠀⠀⠉⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠶⠶⠤⠶⠶⠦⠀⠆⠰⠆⠶⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢰⠀⠛⠂⠚⠒⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠘⠛⠓⠒⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣾⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⢹⠛⠛⢻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢁⣀⢨⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡅⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠐⡄⠠⡀⠁
⠀⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠄
⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⢁⡠⠜⠀⠀⠀⠠⠁⡰⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠂⠤⠐⠈⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠇⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠄⠀⢀⠀⡈⠀⠀⠀⠀⠀⠀⢰⠠⢀⠌⠀⠀⠀⠀⠀⠀⠀⡇⡠⠂
⠐⠀⠁⠀⠌⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⡇⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠉
⢠⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⢤⠊
⠀⠀⠐⠁⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠄⠒⣆⠁⠀⢀⠀⠀⠀⠒⠂⠀⠀⠀⠐⠒⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⢀⣾⠋⢀⣙⣛⡛⠁⣾⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⢻⣷⣿⣡⣤⡌⣻⡏⠁⣼⣿⣰⣿⠰⣦⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠿⠋⠀⠀⠻⠉⢿⣼⠟⣿⠁⣼⡟⢹⡿⢻⡇⢈⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⣶⡄⣠⡀⣠⣤⣄⡀⠀⠈⠋⡀⠘⠁⠈⠁⢘⡋⣱⡾⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⣉⠁⣿⠉⠉⣿⡄⣴⠟⠛⠿⠚⣿⠀⢸⣷⠈⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⢠⣿⢀⣿⣶⣾⠟⠰⣿⠀⠀⠀⠀⢿⡆⠀⣿⢰⣇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠆⢸⡟⢸⣿⠘⣿⣄⣀⠻⣧⣤⣼⠟⠈⠻⣾⠟⠀⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣴⠆⠀⣠⣀⠈⣡⡀⢈⣹⣛⢁⣈⡉⣁⣀⠀⣴⠶⣦⡀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣶⡟⣴⣶⣍⠙⣿⡿⣶⠀⠉⣿⡟⠙⣿⠟⢻⡇⠉⠀⠙⠿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠹⠁⠿⣶⠟⣼⡟⠀⣀⠀⠀⢻⡇⠀⣿⠀⢸⣇⠀⢠⣾⣛⣿⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠙⠿⠿⠋⠀⠀⠸⠇⠀⠛⠃⢀⠛⠿⠀⠙⢛⡍⢠⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣰⣿⢁⣾⠻⣷⡀⣾⡀⠀⢠⣷⢰⡆⢀⣾⠛⠷⠐⣿⡀⣸⣷⠸⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⠟⣿⠘⢷⣴⡿⢰⣿⣷⠀⢸⡇⢸⣷⡿⢡⡾⢛⣷⠙⠛⣋⣿⣦⣤⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠋⠀⣿⡄⠀⠀⠀⣾⠇⢹⣧⣿⠃⣸⣿⣧⠘⣿⣯⣥⡆⣼⡟⢹⡇⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠓⢰⣿⠀⠀⢻⣿⠀⣿⡇⠹⣧⣀⣭⠁⠀⠙⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠿⠃⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡆⡄⣀⢰⠆⠀⢠⣀⠘⡄⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠠⢧⠃⠉⠈⠀⠀⠈⠉⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⣠⢰⢀⠀⠀⡀⢠⡇⠀⠀⡀⠀⠀⡚⢀⠀⡄⠀⠀⢀⠀⠀⡄
⠀⠀⡄⠀⠀⠀⠐⠓⠃⠀⠘⠀⠛⠩⢸⠜⠀⠀⠃⢻⡄⠀⠘⠛⠀⠀⢣⠙⠘⠺⡀⠀⠈⠀⠁⠂
⠀⠀⠳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢄⠁⠀⣠⢺⠀⠀⠀⡇⠀⠀⢸⡇⡄⢀⠀⠀⡀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢼⠇⠓⠫⢸⠀⠀⠄⠃⠀⠰⠹⠄⢸⠸⠐⠂⠇⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠀⠀⡀
⠀⠀⠀⠀⠐⠒⠢⣸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⡔⣾⢠⢀⣆⢠⠀⣆⠀⡇⣤⣆⢠⣀⢠⡄⡆⡆⠀⠀⢤⢰⢠⢂⡤⡗⠂
⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠈⠀⠉⠁⡏⢉⠉⠈⢸⠀⠀⠃⠁⠀⠀⠀⠈⠈⠀⠁⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠀⣨⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠔⠲⠏⠉⠉⠍⠐⢢⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠁⡀⠀⠀⠗⠀⠀⠂⠀⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⣀⣠⣀⣰⣆⣀⣰⣆⣨⡀⡇⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡊⠀⢠⡀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠑⡆⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠈⠳⣄⠀⠉⠀⠀⠀⠀⠀⠈⠉⢀⣠⠞⠑⢆⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠈⠳⣤⣀⠀⢀⣀⣤⠴⠚⠋⠀⠀⠀⠸⡄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⠀⠉⢙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢀⠀⠀⠀⣠⠞⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢸⠀⠀⢸⠀⢀⠞⠁⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠘⡄⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⡘⢠⣿⣿⣿⣿⣿⣿⣿⣆⠰⡄⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠀⠀⡇⢸⣿⡿⠛⠛⠋⠙⢿⣿⠀⡇⠀⠀⠀⠀⠀⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⢻⠸⡿⠶⠶⠶⠶⠶⠾⡿⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠀⠀⠘⡄⠻⣄⠀⠀⠀⠀⡴⠃⡀⠀⠀⠘⡆⠀⠀⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠈⠛⠶⠶⠾⠗⠋⠀⠀⠀⢀⠇⠀⢠⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣤⡴⠃⠀⠀⠚⠀⠀⡾⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⡼⠉⠉⠈⠳⣄⠀⠀⠀⢀⡼⠃⠀⠀⠀⡇⠀⠀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡼⢆⡀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠙⠶⠤⠶⢻⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠃⠈⠙⠲⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡄⠀⢀⡞⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠸⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣧⠔⠉⠀⢀⡞
⠀⡠⠒⠂⠒⠒⠒⠀⠀⢼⠁⠀⠀⠀⠀⣇⢸⠀⠀⢰⡄⠀⠀⠠⠶⠂⠀⠀⠘⡄⠀⠀⣧⠀⠀⠀⠀⠀⠸⡦⠤⠖⠛⠉⣉⠱⡄
⡜⢀⠎⠀⡰⠒⠁⠀⢠⡏⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⣧⠀⠀⠀⠀⡇⠀⠀⠀⠹⡄⠀⡟⠀⠀⠀⠀⠀⠀⠘⢦⣀⠐⡄⠈⢧⡱
⠙⠺⢤⣤⣇⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢹⠀⠀⠀⠀⣧⠀⠀⠀⠀⠹⣆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢾⡀
⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⢀⣠⠤⠤⠏⠀⠀⠀⠀⢸⣦⣤⣤⣤⣷⠤⠀⠀⠀⠀⠈⠛⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠸⣄⠀⠀⠀⠀⠀⣀⡴⠋⡀⠀⠀⢀⠀⠀⠀⢀⠈⠳⡄⠀⡞⠁⠀⢰⠂⠀⠀⠀⡀⠀⠀⡄⠘⣆⠀⠀⠀⠀⠀⣀⣠⡴⠟
⠀⠀⠀⠈⠛⠛⠚⠛⠛⠙⠦⢼⣃⠀⠀⢸⡀⠀⠀⢸⣄⣀⣿⠀⠳⠤⠴⢿⣄⣀⣀⣀⣇⣀⣀⡽⠒⠋⠙⠛⠛⠛⠛⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠠⡇⢸⠀⠀⠀⠐⠣⠌⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠑⠟⠀⠀⠀⣿⣦⡀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⣿⠙⢿⣶⣄⣈⣀⣀⣴⣇⠀⠀⢀⣠⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠈⠙⠛⠛⠋⠉⠛⠻⠟⠛⣿⡏⠀⣤⠤⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠤⠶⠻⠦⣄⣻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⢸⣧⠀⠀⠀⣀⡀⠀⠀⢀⡏⠹⣞⠁⣠⣤⣤⣄⣼⣻⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⡶⠛⠉⠉⠓⠾⣿⡛⢦⡀⢸⠀⠀⢹⣼⠃⢾⠉⠉⠀⢸⣷⣀⣹⠂
⠀⠀⡖⠒⠀⠀⠀⠀⢈⣿⡄⣠⡶⠶⣤⠀⠘⣟⠀⠙⠾⠀⣀⠀⢻⡳⠾⠟⠃⢀⡞⣹⠀⣘⡶⠤⠤
⠀⠀⠉⠘⠀⠀⠀⢀⣼⣿⣿⡏⠀⠀⠈⡇⠀⢹⣻⣄⠀⠀⡿⡄⠈⣷⣤⣤⡶⢛⡞⠙⠛⠉⠁
⠀⠀⠀⢀⣠⣴⣾⠿⢿⣿⣿⡷⣤⣤⠴⠁⢀⠛⠻⠟⠿⠾⠃⠻⠛⠛⢄⢦⣀⣈⣧⡀⠀⠀⢼⡓⣆
⠀⠀⠀⠀⠀⠀⢀⡠⠊⡸⠿⣯⣄⣀⣀⣤⣿⣞⠚⠀⠀⠀⠀⠀⠀⠀⠞⠿⠹⠆⠈⠉⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠋⢠⠞⠀⠀⠘⠉⠉⠉⠀⠚⠻⣧⠀⢀⡶⠻⢚⡒⠲⠦⠂⠻⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⢹⢠⠏⢠⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⣀⡠⠁⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠛⠎⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⡛⠍⠀⢠⣬⡷⣿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡿⢿⣿⣶⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠉⠛⠛⠉⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⣋⠓⢆⠀⠀⣤⡶⣤⠀⠙⠛⠁⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡇⠉⠛⠋⠀⠐⢤⣤⠀⠀⠀⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⣄⢀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢴⣶⡦⠄⠀⠀⠀⣠⠎⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣆⠀⠀⢀⣀⡴⠞⠁⠻⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠺⠭⠀⢾⣧⡻⢿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⡟⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠈⡆⠀⠀⢸⠉⠁⢀⡹⠀⠀⠀⢀⣀⠀⠀⠀⢠⠄⠀⠈⠙⠋⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡇⠀⠀⠈⠀⣴⠏⠀⠀⣀⠀⠈⠿⢆⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠃⣀⠊⢀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣿⣇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⡄⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⠞⢤⣤⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡆⠣⠀⠂⢤⡶⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⠶⣬⣿⡟⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣇⡠⠒⠒⠂⠠⢍⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣿⣿⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡟⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⣀⣄⠀⠀⠁⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠐⡄⠂⠐⠂⢄⠀⠘⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣄⠀⠀⠀⢠⡎⠠⢀⠒⢣⠀⠇⠀⠚⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢮⣇⡀⠰⣿⠀⠀⠀⠀⠰⡘⠀⠁⠶⡦⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⠘⠛⠙⢄⠀⠀⡀⣴⣄⠀⠈⠈⣷⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡄⠀⠁⢨⣤⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠠
⠀⠈⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⢡⡀⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⣠⠶⣋⡉⠁⢠⣤⣬⣉⣉⣹⣿⣿⣋⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⢿⡷⠶⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢢⠄⢠⣀⣀⣀⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⠛⠛⠋
⠀⠘⠀⠀⠀⠀⠀⠘⠿⠟⠛⠛⠛⠉⠉⠉⠁
⠀⢠
⠀⠘

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣝⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣻⣅⣌⢧⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣿⡉⠈⠉⠛⠷⣿⣌⣩⡉⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠖⢉⣴⣾⣿⣿⣲⠁⠀⠀⠀⠉⠛⠳⢤⡀⠙⠄
⠀⠀⠀⠤⠴⠶⠖⠒⠒⠒⠉⠉⠀⠀⠀⣩⡇⠀⠀⠈⠙⠀⣶⣤⣀⢀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣶⣿⣷⣤⠀⠀⠀⠀⠀⠙⢿⣾⣵⡗⠐⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣍⠉⠉⠀⠀⡀⠀⠀⠀⠀⠉⠻⣍⡙⢶⣄⡀⠀⢀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⠀⠼⣷⣦⣄⠀⠀⠀⠀⠉⠛⠻⣿⣦⣄⡀⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡉⡛⠻⠯⢳⣄⢠⡁⠈⢹⡿⢾⣕⣀⡀⠀⠀⡽⠛⢿⡉⠁
⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣴⣿⢰⠿⣿⣧⣀⠀⠀⢛⣿⣿⣧⣾⣀⠤⣿⣿⣿⣿⣿⣶⡶⠾⠻⣶⡀
⠀⠀⢀⣶⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣮⣿⣻⣷⣖⣬⡈⡹⠿⢻⣿⣿⣿⡿⠛⠉⠉⠁⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠘⠛⣛⣛⣛⣹⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠿⠛⠛⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠪⠭⠩⠍⠉⠉⢉⣒⣒⣒⣒⠒⠒⠤⠤⠤⠤⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⣷⡾⠯⠍⢉⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⣁⣐⡒⢒⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠛⠓⠒⠒⠒⠒⠲⠦⠤⠤⠤⠤⣤⣉⣉⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠐⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠷⠟⢿⡀⢠⠞⠉⣉⡻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇
⠀⠀⢔⢢⡀⢀⠖⠆⠀⠀⠀⠀⢸⠀⢾⣿⠇⢸⠀⢸⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⡇
⢯⡙⠺⠆⠱⣼⠐⢦⠀⠀⠀⠀⢀⡟⠚⠉⠀⠀⠑⠲⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⡇
⠪⠭⣭⠄⠀⠧⠀⣘⠤⠤⠔⣂⠭⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⡇
⠀⠰⠥⠊⠑⠒⠚⠁⠉⠉⠉⠀⠀⡇⠀⢢⡀⠀⠀⠀⣈⢷⢦⠀⠀⠀⠀⠀⠀⢣⠘⢄⠀⣿⠀⠀⡇⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢏⠂⠉⢁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⡑⠿⣤⣠⠗⢁⡜⣀⡠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠼⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⢦⠀⠀⠀⡇⠀⠒⠚⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⢳⣄⣀⣄⠠⣍⡙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⣎⡇⠀⠈⠦⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠉⡟⠙⠛⠓⠒⠒⢶⠲⡶⠶⠦⠤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡙⣄⠀⠀⠀⠀⢰⠃⡜⠀⠀⠀⠀⠀⠀⠀⢣⡘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠓⠤⣀⣠⠃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⠇⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁⣠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣤⣴⣾⣷⣦⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⡿⠟⠋⣅⠀⣴⣄⣰⣧⣀⣶⡀⣠⡌⠙⠻⢿⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢟⡁⠐⣷⣤⣿⣿⠿⣿⡿⠛⠿⣿⠿⣿⣧⣶⡏⢀⣈⠻⣿⣿⣧⣠⣤⣴⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⣁⡈⣿⣿⣿⡋⢹⣿⣤⣾⠿⠶⠴⣷⣴⣿⣏⣹⣿⣿⣷⣶⣿⠿⠿⠛⠛⠉⠁⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⢃⣀⣹⣿⣿⣉⣹⡿⠛⠉⠀⣀⣀⣤⣴⣶⣿⡿⠿⠟⠛⠋⠉⠁⢀⣤⠰⣦⠀⣿⡇⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠈⣻⣿⣋⣹⣿⣯⣴⣶⣿⡿⠿⠛⠛⠉⣉⠀⣠⡀⠀⣶⡀⢿⣧⡀⢿⡆⠹⣷⣿⡇⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣼⣿⣿⡿⠿⠿⠛⠋⠉⠁⣠⣤⣤⢰⣶⠀⣿⡇⢸⣿⣆⢹⣧⠸⣿⢷⣼⣷⠀⠙⣿⡇⠀⠀⢻⣿⡀
⠀⠀⢠⣤⣶⣶⣿⡿⠿⠛⠛⠋⢉⣀⣀⠠⣶⣶⡿⠀⠀⢿⣇⠀⠈⣿⡆⠸⣿⠀⣿⡿⣦⣿⡄⢻⣇⠻⣿⡇⠀⢹⣷⠀⠀⠘⣿⡇
⠀⠀⠀⣿⣯⠁⢀⣤⡀⢰⣦⠀⣿⡏⢻⣷⠀⢻⣇⠀⠀⠸⣿⡶⠇⢸⣷⠀⢿⡇⢸⣷⠘⢿⣧⠘⣿⡀⠙⠿⠀⠈⠛⠀⢀⣀⣿⣧⡀
⠀⠀⠀⢸⣿⡄⠀⢿⣷⡄⢿⡆⢻⣧⠈⣿⡆⠘⣿⡄⠀⠀⢻⣇⠀⠀⣿⣇⣸⣿⠀⢿⠆⠈⠛⠀⠉⣀⣀⣤⣤⣶⣾⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⣿⣧⠀⠸⣿⢻⣾⣷⠘⣿⡄⢹⣷⠀⢻⣧⠀⠀⠘⣿⠄⠀⠈⠛⠋⢁⣀⣠⣤⣶⣶⣾⣿⣿⣿⡟⣿⣿⡿⠁
⠀⠀⠀⠀⢸⣿⡄⠀⢻⡇⠹⣿⡆⠻⣷⣴⡿⠀⠘⠛⠀⠀⣀⣀⣤⣤⣶⣾⣿⠿⠿⢛⣿⣿⣧⣴⣿⣧⡀⣰⣿⣿⣧
⠀⠀⠀⠀⠀⢿⣷⠀⠸⡿⠀⠘⠛⠀⢈⣁⣠⣤⣴⣶⣾⣿⣿⠟⠛⠉⠉⠀⣀⣠⣴⣿⣯⣀⣿⣿⣇⠀⣰⣿⣿⠿⠋
⠀⠀⠀⠀⠀⢸⣿⣆⣀⣤⣤⣶⣶⣿⣿⠿⣿⣿⣿⣧⣼⣿⠛⢿⣿⠿⣿⡟⠛⣿⣧⣼⣿⠿⣿⡀⢈⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢿⠿⠟⠛⠋⠉⢿⣿⣿⣦⣍⠀⣾⠟⢻⣿⢷⣿⣿⣶⣿⣿⣾⢿⣿⠛⠻⠄⣠⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣶⣤⣘⡁⠘⠟⠀⠻⠃⠘⠇⢀⣉⣤⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠛⠿⠿⠛⠋⠛⠻⠟⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡰⢆⡆⢠⠤⡤⢀⠀⢰⠶⡆⠠⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠀⠁⠘⠀⠘⠋⠉⠈⠃⠉⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⠀⡀⠀⢀⠀⣀⠀⠀⢀⡀⣀⠀⣀⠀⣀⠀⡀⠀⣀⡀⣀⠀⢠⡀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠂⠀⠂⠀⠂⠐⠀⠀⠀⠀⠒⠀⠀⠐⠀⠒⠀⢠⡄⠐⠀⠀⠀⠀⠀⠀⡖⠀⠂⠀⠂⠀⠀⡤⠀⠀⠐⠀⠀⠀⠐⠒⠀⢤
⠀⠀⠀⡀⠀⠀⣀⣀⠀⣀⡀⠀⠀⢀⣀⢀⡀⡀⢀⠀⣀⢀⡀⠀⠀⣀⢀⠀⣀⠀⣠⠀⠀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠃⠀⠀⠐⠀⠀⠒⠂⠀⠒⠒⠂⠐⠀⠀⠒⠂⠀⠐⠒⠀⠐⠒⠒⠒⠂⠀⠒⠒⠖⠒⠀⠂⠂⠐⠲⠖⠐⠂⠀⠂⠲⠂⠀⠐⠒⠒⠒
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀

⠀⠀⠀⠃⠀⠀⠒⠀⠂⠑⠂⠀⠐⠀⠀⠐⠒⠀⠐⠀⠀⠐⠂⠀⠀⠒⠂
⠀⠀⠀⡄⠀⡄⠀⣀⣀⡀⠀⢀⡀⠀⢠⡀⢀⠀⠀⡀⠀⡀⢀⠀⢀⡀⣀⠀⡤⢀⡀⠀⠀⠀⡀⠀⢀⢀⡀⣀⠀⢄⣀

⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠓⠀⠀⠀⠀⠛⠀⠂⠀⠐⠀⠂⠀⠐⠂⠂⠀⠀⠚⠒⠂⠀⠀⠀⠀⠐⠃⠂⠀⠀⠀⠐⠓
⠀⠀⠀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠄⠀⠄⠀⢄⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⣀⠀⠠⠀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠉⠐⠀⠐⠀⠂⠀⠀⠘⠁⠀⠀⠂⠀⠈⠁⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⠀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠀⠐⠆
⠀⠀⠀⠄⠀⠠⢤⠀⠀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠉⠀⠀⠀⠂⠀⠀⠁⠀⠀⠈⠈⠁⠁⠀⠈⠈⠈⠁
⠀⠀⠀⠄⠀⠀⠠⠄⠀⠀⠤⡄⠀⠄⠀⠠⠀⠀⠀⠠⠂⠠⠄⠰⠤⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠈⠀⠀⠈⠁⠈⠉⠀⠁⠈⠀⠀⠈⠁⠀⠁⠈⠀⠁⠀⠈⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠤⠄⣠⠴⠀⠠⠄⢠⠄⠠⠄⠠⠤⠤⠀⠀⠠⠤⠀⠀⠤⠄⠀⠀⠂⠠⣤⠄⣄⠢⠀⠤⠀⠀⠠⠤
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠉⠀⠁⠀⠈⠉⠉⠀⠉⠀⠉⠁⠈⠁⠈⠉⠉⠉⠉⠉⠁⠈⠀⠉⠈⠁⠀⠀⠈⠁⠁⠀⠉⠉⠁⠀⠀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠔⠂⠤⠄
⠀⠀⠀⡀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠉⠀⠉⠈⠉⠁⠀⠁⠀⠁⠀⠉⠀⠀⠈⠁⠈⠈⠁⠀⠀⠛⠀⠁⠉
⠀⠀⠀⠂⠀⠀⠐⠦⠀⠄⠀⠠⠄⠀⠠⠀⠦⠤⠠⠀⠆⠰⠦⠀⠰⠀⠀⠀⠄⠀⠀⠠⠤⠀⠠⠄⠠⠄⠠⠆⠠⠠⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠒⠀⠀⠀⠐⠀⠀⠲⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠲⠀⢦⠀⠆⠆⠀⠦⠄⠀⠐⠀⠀⠠⠄⠀⠀⠐⠤
⠀⠀⠀⡀⠀⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠂⠒⠒⠂⠀⠀⠚⠂⠐⠁⠐⠀⠂⠐⠀⠐⠀⠀⠖
⠀⠀⠀⡀⠀⢀⢀⡀⠀⣀⢀⣀⢀⠀⠀⠀⠀⢀⡀⢀⠀⢀⠀⡀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⢤⣠⠄⣀⠀⠀⠀⠀⠠⠄⠀⢀⡀⠀⢀⣀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣾⣩⣟⣿⣿⣾⢿⣟⣣⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣾⠿⣻⠟⠟⠻⣻⣟⣿⣷⣿⣯⣍⣟⡽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣊⣿⣿⣿⡵⢿⠃⠀⠀⠀⠀⠘⠉⠋⣼⢛⢋⣴⣻⠿⡾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣟⣾⠛⣷⡙⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⠉⣿⣿⣿⣿⣵⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢿⣝⣀⠉⠃⠀⣋⢉⠃⠀⠀⠀⠈⣉⠁⢠⡚⣩⣷⣿⣗⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⡏⣃⠀⠀⠀⠰⠦⠇⠀⠀⠀⢨⡦⠅⢸⡾⣯⣷⣏⣰⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢣⣧⡙⠆⠀⠀⠀⠀⠀⣀⠒⢀⡀⠀⠀⠈⡋⠉⢿⣯⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⢷⣦⡄⠀⠀⠀⢤⣈⠀⣉⣠⠀⠀⡸⠃⠀⣈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠣⣀⠀⠀⠉⠻⠉⠁⣀⠔⠁⢀⠄⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠈⠁⠒⠒⠒⠏⠽⠀⠀⠘⢒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⡀⠀⢀⠇⠀⠑⡀⡠⠋⠡⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠁⣠⠞⠀⠀⢠⡿⠁⠀⠀⠘⠄⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢢⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠳⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⣀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠱⢤⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠖⠊⠉⠉⠁⠒⠂⠼⣀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠳⠦⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠘⠫⣵⠀⠀⠠⠤⠤⠤⠄⠠⠀⠀⠀⠘⠛⠢⡀⠀⠀⠀⠔⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠉⠉⠉⠉⠙⡄⠀⠀⠀⠀⠀⢀⣀⡀⣼⠗⢄⠀⠀⢀⡸⠳⢌⡀⠉⠉⠉⠉⠉⠈⠉⠉⢱⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠁⠀⠀⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠃⠀⠀⢀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠉⠈⠀⣀⠈⠈⠁⠀⠀⡉⢉⣉⢀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⡀⠀⠀⢀⣀⣀⣈⢀⣁⡀⠐⠁⠀⠘⣁⡁⠋⠘⣀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⢛⢿⢛⡻⠻⣿⢛⣻⡿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠋⠻
⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⣠⠀⣠⣄⠀⠀⣤⣤⣤⢄⣤⡀⠀⣀⠀⢀⣤⡄⣀⠀⣤⢀⣤⡀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⣿⢠⡇⣿⢰⡏⢹⠀⠀⠀⢸⡇⢸⡇⠀⢠⣿⠀⣿⠸⡇⣿⠀⣿⢸⡁⠀⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⡿⢻⡆⣿⢸⡇⣸⠀⠀⠀⢸⡇⢸⡟⠁⣼⣹⡀⣿⠀⡄⣼⠋⣿⢸⠋⠁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠿⠼⠃⠿⠀⠻⠟⠀⠀⠀⠸⠇⠸⠷⠂⠿⠙⠇⠿⠼⠃⠻⠀⠿⠸⠷⠀⠧⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷⡲⠻⣖⡞⠀⠀⠆⡶⠞⠑⠎⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⠆⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣋⠀⠀⠀⠀⣾⠃⠞⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⡞⢰⠆⡦⣾⢢⡏⣴⣩⡟⣰⠃⡀⠀⢿⡠⠀⠀⡾⠀⣶⢠⡟⡰⢳⡇⣾⠜⡷⢠
⠀⠀⠀⠀⠀⠀⠀⠁⠘⠊⠀⠀⠈⠋⠈⠀⠋⠿⠞⠀⢾⡽⠀⠀⠀⢉⣴⠃⠀⠉⠀⠈⠀⠀⠘⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠏⠀⠀⠔⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢀⡄⠀⢠⠶⢀⡆⣠⢀⡆⢀⣴⠆⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡠⢈⣏⠔⢿⡠⠛⠂⠛⡠⣿⠹⣧⠈⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠉⢛⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣗⠀⠸⠹⣃⠤⠴⢶⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢽⡿⠒⠖⠂⠀⠀⠀⠀⣿⠋⠈⠑⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⢱⣤⣤⣀⣀⡴⢿⡶⢿⡄⣰⣄⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣀⢸⣧⣤⣇⠘⠋⠉⠁⠀⠀⠀⠹⣯⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡿⢿⣿⣿⣷⣿⡄⡄⠑⠀⠁⠆⠀⠲⠾⠿⡜⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠷⠋⠛⢿⢻⣷⠹⣶⣶⡿⠀⠀⣦⠀⠀⠠⣉⠀⡇⠀⢀⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠺⠇⠘⣷⣦⣔⣁⡴⠆⠀⠂⠀⠘⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⠐⢠⠀⠀⢸⡄⠀⠀⢀⡀⠀⠀⣿⣟⣿⣿⣿⣦⣀⠀⠀⠀⢀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡐⣄⡀⠠⣿⣧⣀⣀⣼⡛⠽⠿⠿⣿⣿⡄⠀⠀⠈⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⡸⡿⢿⣿⣉⠀⣀⣀⣘⠛⠿⠿⣴⣤⡆⣀⠟⣿⣧⣀⠂⠀⠀⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⠣⣤⠀⠀⠀⠀⠀⣀⣒⠚⣽⣿⡛⣭⣭⠹⠿⠛⣿⣾⣯⣴⠁⠐⠛⠻⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠑⠂⠀⠀⠀⠐⣀⠨⠿⢿⣿⣿⣧⣹⣿⡆⠐⠐⣿⣿⣿⣷⣧⠀⡀⠰⣧⡂
⠀⠀⠀⠀⠀⠀⣐⠂⢠⡘⣷⣤⣀⣀⢠⡆⠀⠀⣤⡄⢀⣽⣿⣷⣽⣿⣿⡿⠁⠀⠀⣠⡵⢿⠰⠃⠀⠘⠀⠀⠀⣠
⠀⠀⠀⠀⠀⢸⠯⣽⣷⠀⠀⠈⢻⣿⣿⣷⡄⠈⢩⣁⠈⠛⠛⢿⣿⣿⣿⣤⣶⣦⣰⡟⠁⠀⠀⠀⠀⢷⣀⠀⣼⣿⣦
⠀⠀⠀⠀⠀⠸⢿⣾⣿⣗⡀⢃⣘⣿⣿⣿⡿⣳⡄⠻⠇⢤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣧⡼⠛⡀⠀⡆⠐⢛⣶⠌⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠺⢾⡟⢿⣿⣿⡟⠻⠋⢐⠻⠶⣤⣤⣤⣈⣉⣛⣿⣿⣿⣯⣿⣿⣿⡴⢋⡤⠂⣤⣶⢿⡁⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⣧⣄⠀⠘⣷⡐⠀⠈⠉⠙⠛⠿⠿⠛⠛⠋⠁⠈⢿⣿⣯⡴⠏⠏⢡⣶⡁⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣌⣄⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⣠⣴⠈⢻⣿⣧⠀⣧⢸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⡂⢨⣭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⡀⠻⣿⢸⣿⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⠈⣻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣻⣿⣿⣆⣿⣿⣿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⢟⣫⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⣿⣿⡟⠁⠀⣦⠀⠀⢸⣇⢺⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠙⠋⠀⠀⣧⠀⠀⢸⡟⣾⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡏⠀⠀⠶⠀⢾⢻⡇⠀⢸⡇⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠉⠀⠖
⡇⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣿⡦⠶⠶⠾⠿⣿⠶⠼⠿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⠖⠒⠤⠤⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃
⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡃
⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣤⣶⣦⣿⣯⣤⣤⣑⣄⢀⠀⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠶
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⡶⣶⣶⣶⣿⣿⣿⣶⣶⣿⣿⣿⡿⣿⢲⡶⡶⠂⠀⣀⠟⠋⢠⣶⡈⠿⠓⣿⠀⠀⠀⠀⠀⠀⠀⡟⠁⠰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⡏⠛⠉⠉⠉⠀⠘⡿⠃⣤⠀⢙⠀⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠠⠄⠀⣀⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠰⠂⠦⠒⠃⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠴⠄⠀⠀⠈⠋⢀⣶⠶⠖⠚⢤⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠊⢱⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣤⣄⣠⣼⠃⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣗⣂⣀⣙⣿⣒⣂⢀⣀⣈⠀⠀⣀⢀⡓⣲⣞⠋⢈⣙⣻⣋⣉⣥⡤⢲⡖⢲⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢩⣿⡏⣽⣿⠉⣿⣿⠉⡯⡏⢩⡛⡏⢩⣿⠉⣿⣿⢘⣯⢸⡇⢸⡇⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢸⣿⠃⣿⢿⠀⣿⣿⠀⡇⡇⢸⠥⡇⢸⢸⠀⡏⢏⢀⣿⠨⠇⠸⠃⠈⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠈⠁⠂⠁⠈⠐⠁⠘⠈⠀⠁⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠂⠐⠂
⡇
⡇
⡇
⡇
⡇
⡇
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡞⠫⣾⣿⠟⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⢉⡴⢋⣾⣷⣬⠏⣀⠄⠀⠀⠈⠉⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢧⢀⡴⢋⣴⣿⣿⣿⣿⣴⣧⠞⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢠⣾⢋⣴⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⡠⠊⣠⠞⠋⡡⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⡄⣠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⡎⠀⢻⠟⢥⠊⠁⠀⠀⠑⠛⣀⣀⣀⣤⣤⣾⣵⣷⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⣿⣿⣿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⠀⢰⠁⣠⡎⠀⢀⡱⠀⣀⣠⣴⣾⣿⣿⣿⡿⢿⣿⣧⣌⠛⠛⣧⣤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠎⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡠
⠀⠀⠀⠀⣅⡼⣡⡿⣶⡶⡾⢿⣿⣿⣿⣿⣿⣿⣿⣏⠸⣿⣿⣿⣷⣶⣤⣄⣙⣳⣂⣤⢀⣠⣾⣷⡶⠀⢰⣿⣿⣿⠛⡿⣿⢿⣿⡿⡏⡇
⠀⠀⠀⣠⣿⡾⠋⠀⠻⣧⡄⢤⠙⢯⣽⣿⣿⣿⣟⡛⠲⢿⣝⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⠯⣼⣿⠃⠀⢸⣿⣿⣿⣇⣁⡇⣿⣿⡇⠇⠃
⠀⠀⠐⠿⠋⠀⠀⠀⠀⠈⠙⢦⣀⣼⠁⠈⠹⣷⠀⠈⠉⣦⠈⠻⠦⣤⣽⣿⣟⣻⣷⢤⣬⣀⣻⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⢸⠟⠒⠚⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠓⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣠⡶⠶⠀⠀⢀⣤⣄⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠞⢩⣿⣦⡔⢹⣿⣆⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠛⠇⠀⠀⠘⣇⠘⢻⡿⢧⡈⣋⡝⠀⠀⠀⣀⣿⣤⣶⡶⠶⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢇⠀⠀⠀⠀⠉⠛⠉⠀⠀⢉⣡⣤⠶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⠂⠀⠀⢤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣾⠀⠈⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡏⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣄⣀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡇⠀⠀⠀⠉⠙⠛⣿⡟⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⠀⠀⠀⠹⣦⣀⣀⣠⠞⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣷⠉⠳⢦⣄⣀⣄⣉⣉⡅⠀⠀⠀⣀⣠⣴⡾⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⣠⠞⣿⠀⠀⠀⠈⠉⠛⠛⠻⠶⠾⠛⠛⠛⠉⠁⠀⠀⣿⡟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢁⣤⠞⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡌⠻⣦⡀
⠀⠀⠘⢍⠓⢤⣠⠴⠶⠶⠋⢀⡴⠋⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢦⡈⠛⢦⡀
⢰⣚⣛⣛⣛⠂⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢦⡀⠙⢦⣀⠀⠀⢀⠔⣶⣀⡀
⠯⣀⣀⠠⠤⠀⠀⣀⣤⡀⢀⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢦⡀⠋⠉⠙⠃⢚⣩⣔⣁
⠀⠯⠤⠤⠖⠚⠉⠀⠀⢹⡀⠳⣄⡀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠤⠔⠋⣠⣄⡀⠀⢀⣬⣥⡤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠞⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠳⠶⠞⠁⠀⠙⠶⣄⣈⠙⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡀⠀⠀⠀⠀⠙⠻⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠾⠇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⢷⡀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠉⠛⠛⠛⢻⡟⢻⠉⠉⠀⠀⠀⠀⠀⣠⠞⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⢿⠳⠶⠶⠶⠶⢶⣼⡇⢸⡀⠀⠀⠀⠀⢀⡇⢸⣀⣀⣀⣀⣀⣤⡞⠁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢷⡀⠀⠀⢰⡟⠁⠉⠉⠀⠀⠀⠀⠈⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠶⣶⣦⣤⣴⡾⠃⠀⠀⠘⠷⢶⡶⠶⠶⠶⠶⠶⠶⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⢠⡿⠿⠀⠀⠀⠰⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⡄⠀⠀⣶⠶⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠰⠶⢤⡀
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠠⠀⠀⠀⠁⣀⠔⠀⠀⠀⡀
⢀⠁⠀⠀⠀⠈⠆⢀⠀⠀⠈⠄⠀⠀⠶⡿⣶⣤⣄⣀⣤⣀⣤⣴⣾⠟⠛⢫⠉⠀⡇⠀⠀⠀⣄⠸⢠⡄⠀⠀⢰⠀⠀⠈⠀⠀⠀⠀⠈⠃⠛
⠸⡀⠀⢿⠂⠀⡇⠸⠃⠀⢀⠆⠀⠀⠀⠀⠀⠀⠘⠋⠙⢍⠁⠈⠁⠀⢀⠎⠀⠀⢇⠀⠀⠘⠿⢠⠘⠏⠀⠀⡌⠀⠀⠠⠀⠀⠀⣠⠀⡀⠀⠀⠀⢠
⠀⠣⡀⠀⠀⡠⠃⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠒⠐⠂⠁⠀⠀⠀⠈⠢⡀⠀⠠⠊⠑⠀⠀⠊⠀⠀⠀⠀⠠⠀⠀⠛⠌⠁⠀⠀⠐
⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣄⠀⠦⣄⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠢⡀⠀⠀⠠⠞⠉⠁⠀⠀⠀⠀⠁
⠀⢀⣠⡴⠛⠉⢠⡐⠉⠳⢤⣀⠀⠀⠀⠤⠴⢤⣤⣀⠀⠀⠌⠀⠀⠀⠁⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠊⠀⠀⠀⠀⠋⠐
⠐⣏⠁⢠⣤⠀⠀⡇⢤⡄⠀⠀⡍⠀⠀⠀⠀⠀⠈⠹⣷⢸⠀⢠⣄⠀⠀⠁⠀⠀⠆⠀⠀⢀⣶⡆⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⢸
⠀⢩⠄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠇⠀⠀⠰⠦⠦⢌⣦⠈⠀⢆⠸⠟⠀⢀⠂⠀⠀⠡⡀⠀⠀⢈⡁⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠃
⠀⠀⠢⣐⠀⢀⠖⠉⠠⠀⠠⠋⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠈⠒⠠⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠙⠂⠀⠂⠀⠀⠀⠀⠑⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⣰⡿⠆⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠘⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡠⠞⠃⣀⠉⠙⠷⢤⣀
⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠰⢡⡴⣦⣤⠘⠁⣴⢷⠶⠐⡀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢰⡆⢶⠀⠀⠀⢸⠁
⠀⠠⠀⠀⠀⢀⡆⠀⠀⠀⠠⡶⠀⠀⠀⠀⡇⣠⣽⣷⡅⢀⠀⣙⣿⣿⠆⡇⠀⠀⡤⠀⠠⠄⠀⡄⠠⠄⠀⠐⠀⠀⠀⠀⠆⠀⠀⠀⣠⠀⠀⠀⠀⠌
⠀⠀⠃⠀⠀⢀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⡀⠉⠋⢁⠊⠡⠉⠈⠁⡐⠁⠀⠀⠐⠀⠀⠀⡸⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠄⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠛⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣴⣿⡆⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠰⠟⠃⠀⠀⠰⢤
⠀⢠⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⢠⠀⠀⠉⠭⢙⠆⠀⠀⠀⢈⡥⠴⠂⣄⠀⢀⠬⠍⣇⠀⠀⠀⠀⢀⠴⠀⠰⠄
⠀⠉⠀⠀⠀⠀⠀⠀⠺⠿⠻⠷⠆⠀⠀⠀⡌⢀⢒⡓⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⣤⣶⠀⠀⠀⣴⡆⠀⠀⠀⠆⠀⠀⠀⢀⡀⠀⠀⠀⠄
⡆⠀⠀⠲⠶⠀⢐⠀⠀⠶⠶⠒⠀⡆⠀⠀⡇⠉⠙⠉⠀⠄⡆⢰⠋⠀⣰⠆⠀⠈⡇⠀⠀⠘⠿⢸⠂⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠈⠀⠀⠀⠀⠀⠈⠁⠀⠠⠄⠴⠚⠀⠀⠀⠹⡄⠬⡭⠔⣴⣎⠛⠯⢚⡟⠀⠀⠀⢳⡄⠀⠀⢀⡼⢄⡀⢀⡴⠇⠀⠀⠀⢃⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠖⠚⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠂⠰⠋⠀⠈⠙⠉⠀⠀⠀⠀⠀⠈⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣎⠁⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡲⢶⡦⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⠙⠺⠗⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⣤⣀⠀⠀⠀⢀⣀⡀⠴⣶⣿⣤⣶⠿⠛⢳⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠃
⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⡿⢿⣿⣤⣤⠛⠛⣥⣤⡋⣁⠼⠟⠀⠀⢸⣿⣶⠦⠀⠀⠀⠀⠀⠀⠀⣇⡆⠀⢺⢠⠆
⠀⠀⠀⠀⠈⠙⠻⢿⣿⠿⢛⣿⣯⢤⣶⣾⣀⣁⡹⠶⢂⡉⠉⠉⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡅⠀⢘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢘⣿⣏⣀⣤⠏⠀⢈⣡⣴⣾⣿⣷⣤⣤⡘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠲⠋⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠟⠛⠉⠀⢉⣠⣴⣾⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⠜
⠀⠀⠀⠀⠀⣀⣴⠌⡁⢡⣴⣾⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠶⣿⣿⣄⣀⠰⠂⠨⠛⣿⡿⠟⠋⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⣤⡆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⢷⣠⡤⠆
⠀⠀⠈⠳⢍⡙⢿⡿⣷⣦⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢂⢻⡀⠀⠀⠀⠀⢨⢹⡇⠀⠀⠀⠀⠀⡴⣦⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠉⠙⠷⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⣮⣉⡤⠆⠀⠀⠰⣮⣬⣵⠆⠀⠀⠰⣬⣜⡡⠀⠀⠀⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣦⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠉⠉⠁⠀⠀⢠⣭⣭⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠄⠀⢰⣤⡀
⠀⠀⠀⢰⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢸⣿⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠟⠁⣀⣤⣤⣼⣻⣿⡆
⠀⠀⢠⣿⢿⣤⠾⠷⠀⠀⠀⠀⠀⠾⣿⣉⡿⣧⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠁⠘⣧⣿⣿
⠀⠀⠸⠟⠈⢻⣤⣶⣶⠶⠶⣶⣶⣤⣠⡟⠠⣿⠂⠀⢸⣿⠀⢸⣿⠀⠸⢦⡀⣀⠀⠀⠀⢀⣤⣾⣿⠀⠀⠀⠀⣠⢧⣿⡏⠀⢰⡎⣹⣷⡄
⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⢸⡿⠶⠾⣿⠀⠀⠈⠉⠘⠆⠀⠀⠀⠙⠳⡻⣦⣠⣶⢿⣻⣿⡿⠁⠀⠈⠙⠛⠉
⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⢸⣯⣭⣭⣿⠀⠀⠀⠀⣄⣀⣀⣠⣤⣶⣾⣿⣾⣿⣾⡿⠿⠋⠀⠀⠀⢀⣀⣀⣀
⠀⢰⣿⠀⠀⠀⠀⠄⡀⠀⠀⢀⠀⠀⠀⠀⠀⢻⣇⠀⣸⣗⣒⣒⣿⠀⠀⠀⠀⠈⠛⠿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠈⠀⠀⠀⠀⣾⢿⣿⠏
⠀⢸⡇⠀⡄⠀⠀⠀⢈⣷⢾⡁⠀⠀⠀⠀⠀⢸⣿⠀⣿⡷⠲⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢻⣽⡙⠶
⠀⠸⣿⠀⠐⠲⣀⠀⠀⠀⠀⠀⠀⡀⢦⠆⠀⣼⡟⠀⣿⡇⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⣻⣷
⠀⠀⢻⣧⡀⠀⢷⠀⠉⠘⡀⡋⠁⠠⠄⠀⣰⡿⠁⠀⣿⡇⠀⣽⣿⡇⠀⠀⠀⠀⠀⠀⠹⢛⡡⠈⠀⢀⣀⣠⣤⣾⣿⠀⠀⠀⣿⣸⢹⢸
⠀⠀⠀⠙⢿⣦⣀⠀⠀⠐⠙⠐⠁⢀⣠⣾⠟⠁⠀⠀⣽⠁⠀⢩⣯⠁⠀⠀⠀⠀⠀⠀⢸⣯⣶⣾⡿⠿⠟⠛⠋⠉⠀⠀⠀⠀⣿⣿⢸⢸
⠀⠀⠀⠀⢠⡝⠻⠿⣶⣶⣶⣶⡾⠿⣻⡁⠀⠀⠀⠀⠙⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢸⠸
⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈

⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⢄⡀⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠈⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠠⡄⣠
⠀⠀⠀⠷⠀⠷⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠃⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢺⡄⠒⠒⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠀⠉⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⣿⣟⣻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠆⢠⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠋⠒⠀⠁⠀⠈⠀⠀⠉⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠤⠤⢀⣀⠀⠀⢀⡀⠠⠤⠀⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣀⣀⣠⣤⣤⣤⣶⣆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⠿⠿⣿⠿⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⣴⡿⠁⢘⣱⠏⣱⠆⠀⠠⡀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀⣼⣿⠆⠀⠀⠘⠦⣄⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡐⠀⠒⠀⠀⠀⡎⠀⠛⠉⠀⠀⠔⠒⠒⠚⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠊⣠⠀⠈⢛⠓⠂⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠆⠀⠛⠁⠀⠛⢈⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠂⠀⠀⣤⡌⠀⠰⣞⡁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⣿⣾⡿⠀⠀⠀⠠⠐⠃⠔⠻⠁⠆⠻⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣤⡚⣻⣿⣧⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣤⣶⡋⠿⠋⠀⠀⠀⠀⠘⠒⠒⠒⢲⡖⠒⢒⡚⠉⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠯⡧⡌⠠⢿⠟⣿⣷⡆⠀⠀⠈⠁⣀⣤⣾⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠹⠃⠀⢠⣼⣇⠀⠈⠀⠐⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡤⣡⠀⡀⠄⣤⣀⡁⢻⣽⣿⡀⢀⣤⣾⢻⡿⠟⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡤⠄⠀⠀⠀⠀⠀⠲⠆⠀⠐⠦
⠀⠀⠀⠀⠀⠀⠀⣸⠇⣛⠒⠳⡦⠶⣼⠇⣆⣿⡿⣻⣿⡻⠟⠋⠀⠀⠀⠀⠀⠀⠠⡞⠃⣀⠀⠀⠀⠀⠀⡀⠀⠀⠒⡤⠀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⢀⣭⡈⣍⣈⠀⡗⠐⢺⠀⠷⠖⢀⡨⠏⠀⠀⠀⠀⠰⠃⠀⠀⢠⡿⢠⢰⡉⠁⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠸⢿⠀⣧⢨⡄⡇⣹⢸⡇⣀⣋⣿⡿⢠⠀⠀⠤⠀⠀⠀⠀⢠⣾⡃⠀⠀⠩⠝⠃⠀⠀⠀⠙⠂⠀⠀⠐⠰⠖⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⢺⡀⣏⢀⠀⠂⠀⣿⠁⢸⠡⣿⣿⢄⡄⠀⠀⠀⠀⠀⠰⡀⠙⠀⠀⠀⡀⠀⠀⠰⠂⠀⠀⠠⣤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡡⢾⠅⣯⢨⢨⣍⠈⣟⢐⣘⣘⣓⣻⠾⣷⡀⠠⣤⠀⠀⠀⠀⠈⠁⠀⠀⣁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⠁⢀
⠀⠀⠀⠀⠀⠀⠀⢸⡒⡗⡖⢲⠶⠠⡷⠸⠨⠨⠭⠭⡿⣟⣳⠀⠀⠀⠀⠀⣀⣤⣀⣀⣊⣁⠀⠀⠀⠀⠀⠁⠀⠐⢦⠤⠀⢀
⠀⠀⠀⠀⠀⠀⢀⢾⣍⣏⡏⢹⢉⠀⡟⢸⠈⣈⣘⢻⡇⠸⣿⢇⡀⠀⠀⣸⣟⡡⠋⠉⡏⠉⢷⡀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠸⣈⣷⠀⠃⡂⢸⠀⡇⠀⠀⡆⠠⠈⣷⠀⣿⡮⢷⡀⢀⡇⠁⠀⢀⡼⠁⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠐⢛⢐⢀⡒⡐⠀⢰⡂⡆⠆⢀⠀⢹⡁⠐⠿⣛⣿⠉⠙⠦⠶⠋⠀⠀⠀⠀⠀⠙⣀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠃⠘⠸⠸⠥⠁⠄⠸⠇⠇⡁⠍⠹⠨⢙⠛⠛⢿⣿⡖⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠐⣚⣂⣤⣶⣶⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⠛⠒⠒⢲⣶⣶⡖⠒⠚⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠛⠀⠚⠛⠓⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⠄⠀⠀⠈⠟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊⠀⠃⠀⠀⠀⠓⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠁⠂⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠋⠈⠁⠬⠀⠈⠁

⠀⠀⣿⡟⠛⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⣠⣟⡧⠀⠀⠀⠀⠀⢀⣛⡠⠀⠀⠀⠀⠀⣴⣶⡷⣤⣴⣶⣤
⠀⠀⡏⠀⢸⠇⠀⢹⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⠀⢿⣿⠳⠁⠀⠀⠀⠀⢻⣾⣛⣿⣿⣿⣿
⠀⠀⠿⠻⠾⠷⠞⠿⠄⠀⠀⠀⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠳⠾⠋


⠀⠀⠸⠿⠿⠿⠿⢿⠀⠀⠀⠀⢠⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠀⠀⠀⠀⣤⣀⣶⡆⣤⡄
⠀⠀⡄⠀⢰⡇⠀⢘⡄⠀⠀⠀⣿⡿⠿⠿⢿⣿⠀⠀⠀⠀⠀⣀⡹⢋⣀⠀⠀⠀⠀⣤⣭⣤⣙⣡⣭⣥
⠀⠀⢰⣶⣶⣷⣶⣶⠀⠀⠀⠀⠘⠿⣶⣶⠿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⠆⠀⠀⠀⠉⠙⠿⠿⠿⠏⠉


⠀⠀⢶⠄⠀⠀⠀⢴⠆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⣀⡀⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⢸⠦⠀⠀⠀⠀⠀⠀⠉⠅⢠⠀⠀⠀⠀⠀⣸⣅⣼⣍⣷⣽⣀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠂⢐⡀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠉⠉


⠀⠀⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⡠⢚⠉⠉⠲⡄⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄
⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣶⣶⣾⣿⣶⡆⠀⠀⠀⠀⡇⠈⠀⠈⠁⢸⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡿⡿⢿⣿⢿⢿⠇⠀⠀⠀⢿⡿⢿⣿⣿⠇⠀⠀⠀⠀⠑⢌⣐⣊⠵⠃⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠴⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠀⠀⢠⠤⣤⣤⣀⠀⣊⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⡇⣿⠄⠀⠀⠀⠀⠀⠈⠳⣶⣭⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⡀⢉⢳⡀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡿⠁⠹⠀⠀⠀⠀⠀⠛⠷⠋⠉⠠⣄⣀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠟⢿⠏⢸⠁⠀⠈⠄⠀⠀⡀⠀⠀⠀⠼⣷⡔⠊⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠀⢀⠖⠂⢸⡀⠀⠀⢀⠀⠀⣽⡲⢈⠫⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣠⠊⢈⣠⣄⢳⠀⢀⣴⡟⠒⠛⠉⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⡀⠁⠀⠉⠀⠀⠙⠇⠸⠟⠋⠁⠀⠚⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡤⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠃⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠠⠋⠀⠀⠀⠉⢢⣔⣢⡤⠖⠀⠂⠀⠀⠢⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣂⣤⣤⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⡄⠛⠛⠀⠂⠀⠀⠀⠘⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠛⠛⠉⠁⠈⠉⠉⠙⠒⠠⠆⠀⢠⠤⠒⠁⠀⠈⠀⠶⣶⣄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢽⣿⡺⠿⠷⠾⣤⣤⣀⣀⣀⣀⣀⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣀⣠⠤⠤⠤⠅⠤⠾⢦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢀⡀⠤⠀⠈⠉⠉⠙⠁⠀⠒⠂⠀⠤⠤⠴⠲⠐⣀⡷⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⣔⠦⠄⣀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣷⣶⣤⣤⣀⣀⣤⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠛⠉


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