Excursion Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠉⠱⡎⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠐⠒⠒⠂⠚⠛⠒⠲⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢧
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠰⠂⢀⡤⠤⠀⠀⢦⣤⣤⠤⠤⠤⠖⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠆⠀⠀⠀⢀⣷⠁⢡⠀⠀⠄⠀⠀⠀⡼⠀⠐⠶⠶⠀⠐⢸⡇⠀⠀⠐⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣀⡀⠳⠤⣤⣤⠤⠞⠁⠀⠀⠀⠀⠀⠀⣈⢓⠀⠀⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠇⠀⠀⠉⠉⠙⠓⢒⠒⠒⠚⠚⠛⠋⠉⠉⠁⠀⢸⠀⣀⢰⠀⠄⡄⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠤⣴⣦⠴⣿⠦⠤⢤⡤⠤⠥⠤⠤⠤⠤⠤⠤⠜⠀⠁⠘⠀⠰⠀⠀⠘⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⢲⡄⠀⠀⠉⠉⢉⣀⣀⣀⣀⣿⡗⢀⣿⡆⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⣶⠀⣸⠇⠀⠀⢀⡾⠟⠛⠛⠛⢻⣿⠗⢾⣿⠥⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢼⣿⣰⣏⡀⣾⡗⣠⣾⠓⠀⣰⡟⢾⡿⠀⣿⡏⠸⠴⣶⠀⣾⠁⣴⡆⡰⣦⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠸⣿⡟⣿⡏⢿⣈⡿⣏⢰⡇⣿⡃⣽⣇⣰⣿⡇⠀⢰⡿⢠⣿⠀⣿⣷⣰⣿⢧⣄⣀⣀⣀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠖⠒⠀⠙⠃⢿⣿⣦⠍⠀⠈⠁⠉⠀⠉⠀⠉⠰⣿⣠⠃⠈⠉⠁⠈⠉⣿⡏⠈⠁⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃



⣿⠿⠿⠿⠿⡻⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢹⠉⢎⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⡶⢟⢛⡚⠻⠿⣎⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣷⠦⢄⡀⢀⣴⠋⠉⠀⠉⠁⠉⠘⠢⢖⣤⡈⠻⣼⠷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠿⠳⣄⢹⡟⠋⠉⠉⠙⠒⠢⢤⡀⠀⠀⠙⢷⣄⠈⢿⣿⡛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠟⠉⠀⠀⠀⣈⣿⠀⠀⠉⠀⡦⠀⠀⠀⠙⢦⡀⠀⠀⢻⣆⠀⢻⢻⡌⢻⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⡟⠋⢀⣠⡤⠶⠚⠉⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣽⡄⠈⣧⢿⡄⠳⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣸⡷⠚⠉⠁⣀⣠⣴⡾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠧⠇⠀⢸⣯⢿⡄⢱⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢀⣤⣶⡿⠝⠋⢡⡾⠛⠛⠛⠿⢭⣍⠙⠳⣄⠀⠀⠀⠀⢸⠀⠀⠸⠥⠀⠈⡏⢿⣿⡀⢷⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠒⠺⠿⡋⠁⠀⠀⠀⣿⠀⠀⠈⠁⠐⠢⡈⠻⣦⠈⢦⠀⠀⠀⢸⡆⠀⠘⠚⠒⠂⢿⢸⣿⡇⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠈⢢⡘⣧⢘⣧⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⣾⢻⡇⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣷⣄⣀⠀⠀⠀⠀⠀⠀⢱⠚⠁⠸⡆⠀⠀⢿⠀⠀⠀⠀⠀⠈⣿⣼⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢿⠉⡝⡛⠛⠛⠛⠛⠛⠛⡇⠀⠀⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢠⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠀⠀⠉⠉⠒⠒⠒⠒⠺⡇⠘⡾⡸⠈⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣦⣤⣶⣿⠿⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢻⡣⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⢀⡾⠿⠛⠋⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢊⠀⠙⠷⣤⣄⣀⣀⣀⣀⣀⣀⣠⣾⣥⣤⣬⣿⡈⢀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠇⠀⡾⠀⠀⠀⢀⣠⠞⠁⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⡀⠈⠙⠳⠦⢤⣤⣤⣤⣯⣀⣤⣾⢤⠾⠧⠴⠖⠛⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⠒⠦⢤⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠁⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠉⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡒⠋⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⢀⠎⠀⠀⠈⠀⠀⠀⠀⢰⠌⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⢀⠀⠀⢘⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢂⠄⠘⡀⠀⠀⠈⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠙⠃⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠱⠃⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠢⠀⠀⠀⠀⠑⠄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠄⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⡀⠀⠀⠀⠀⠃⢀⠐⢰
⠀⠀⠀⠀⠀⠀⠀⢂⠢⣀⠀⠆⠒⠈⠉⡈⠄⠐⣌⠢⠀⠀⠀⠀⢀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠢⠐⡀⠂⠀⠈⠀⠀⠘⠃⠈⠢⠀⠀⠀⢀⠲⢄⠀⠀⠀⠀⠀⢀⠄⠀⣀⡀⠀⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⢶⡀⢄⠀⠀⠄⢄⠈⢀⠠⡘⢢⠀⡁⢡⠀⠀⠀⠀⠀⠀⡶⠚⠋⠉⠡⢰⢀⠐⠠⡀⢀⠀⣀⡀⢀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠢⡑⢄⠢⡀⠈⠂⡱⢄⠑⠬⢦⡘⢔⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠹⣁⠀⠀⠀⠾⠮⠐⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠢⠑⢌⠢⡄⠐⠃⠅⡀⠌⡙⠂⠱⠆⠀⠀⣀⡽⠐⠀⡀⢀⣠⣤⣀⢹⠀⠀⣀⠋⣁⠄⠀⢄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠋⣬⠑⠠⠀⠀⢈⠉⣉⣻⣶⠺⠋⠁⡠⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢑⡲⠆⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠾⣄⣡⠔⠂⣉⡴⠖⠫⣀⡌⠓⣸⠲⠁⣾⠀⠠⠤⢤⣤⣤⣤⣤⣤⡀⠀⣼⢈⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢡⠆⠐⠈⠠⠖⠫⣅⣀⠀⢀⡭⠦⠠⠄⠇⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⡇⠈⠀⠀⠠⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⠄⠂⠀⠀⠀⠀⠢⡈⠦⠀⠀⠀⠀⠀⠀⣰⠬⠧⣀⢀⡤⠤⢤⡀⠀⠀⠐⠀⠀⡇⠀⡀⠀⢀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡠⠀⠀⠀⠀⠘⣇⠀⠀⡸⠿⡀⠀⢀⡿⠷⠶⠶⠶⠶⠁⠴⠄⠀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠈⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠈⠀⠠⠀⠀⢠⠸⠵⠠⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⡠⡄
⠀⠀⠀⠀⢰⣟⣶⡀⢀⢤⢤⠀⢎⣉⡉⢉⡆⠀⠀⡎⠀⠀⠀⠉⠱⠞⢡⠀⢀⠴⠊⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠲⡄
⠀⠀⠀⠀⠚⢟⡷⠁⠘⠞⠐⠀⠐⢀⡘⠃⢀⡀⢀⠄⠐⢀⠌⠋⠰⠚⣉⣠⠄⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⣀⠀⠀⠀⠑⡖⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠀⠉⠀⠈⠁⠀⠈⠁⠉⠉⠛⠁⠀⠈⠀⠀⠀⠀⠉⠑⠁⠀⠀⠛⠈⠀⠈⠙⠃⠁⠀⠁⠙⠈

⠀⠀⠀⠀⣠⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠂⡀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠴⠀⠐⠒⠢⠀⠤⣀⠀⠐⢟⠦
⠀⠀⢠⠐⠁⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⢉⠀⠀⣈⠀⠀⠀⠀⠈⢁⠀⠀⡄⠀⠀⠀⠀⡄⠀⠀⠄⠀⠀⠐⠠⠐⢠⠀⠈⠆⠃
⠀⠀⠈⠐⠄⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠠⡀⠀⠀⡄⠀⠀⠀⠀⠈⠧⠈⠉⠡⠈⠀⠀⠨⠒⡒⠅⣠
⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠐⡀⡀⡐⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠀⠀⠀⠀⠊⠀⠀⢠⣶⡀⠀⠀⣰⣀⣴⠄⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠤⣄⡀
⠀⠀⠀⠁⠀⠠⠀⠀⢢⠀⠀⠀⠀⡄⠀⠀⠀⡖⠚⠒⡄⠀⠀⠀⡀⠀⠃⠘⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠃⠀⠐⠁⠀⠀⠀⠰⡶⠀⠀⠰⠿⠆⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⢀⣠⠾⠁
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⠀⠀⠵⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁
⠀⠀⠀⠀⠰⣟⣙⡧⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⣹⡆⠀⠁⠀⠀⠀⠀⠈⠂⠀⠤⠀⠄⠿⠀⠀⠠⠀⠰⠆⠠⠄⠀⠀⠀⠶⠀⠰⠀⠶⠂⠠
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡻⣱⢴⠄⠀⠀⠀⠀⡰⣸⣴⡆⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡁⢀⣴⡉⠀⠀⠀⢤⣶⣷⣿⣿⣾⢿⡗⠀⣀⣀⣤⣠⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡟⠉⠀⢳⣤⣀⡤⠞⠛⠛⠻⢿⣣⠟⠋⠉⠉⢸⠉⠙⢿⣳⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣖⣶⣿⣯⣍⠉⠉⢧⠀⠀⠸⡴⠋⠀⣀⠀⡤⡀⠄⢹⠒⡆⢀⡴⠛⣦⡤⠶⡛⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⠻⡷⢿⢴⠠⠾⠴⢸⡇⠀⠘⡆⠀⢸⠁⠀⢠⣿⣄⡀⠁⡀⠀⣿⠕⠋⢠⣶⠛⡄⠀⠱⡄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣛⣚⣛⣛⣛⣻⣿⣿⣿⡁⢠⡄⢹⠀⢸⠀⠀⣼⢿⣿⣿⠟⠁⢰⠃⢀⣺⠟⠉⡄⡇⠰⠄⢳⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡞⠁⢏⠀⠀⠀⠘⡇⠀⢸⠁⡼⢻⠀⡇⠈⢇⠀⠸⢤⠼⠋⠀⡰⠃⣀⠾⡋⡆⡈⢣⢹⡜⠀⠸⡇⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⡇⠀⢸⡄⠀⠀⠀⢿⠀⣼⠀⡇⢸⡄⣧⠀⣸⢻⣒⣤⡤⠖⠋⠠⣸⠏⠀⡇⡗⠘⠘⣼⡇⠀⢀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠈⠁⠀⠀⠀⠈⠀⡇⠀⡇⠘⡇⢻⠀⠹⢤⡭⠟⠀⠀⢀⣳⢟⡆⠀⢧⢹⣀⣠⡧⠳⠚⠉⢁⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠀⡀⠂⠀⠀⡇⠀⣷⠀⣧⢨⠀⠀⠀⠀⠀⠀⠀⣲⡏⠘⣧⠤⠞⠈⠁⣀⣠⠤⠖⢚⣉⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡆⠀⠀⠀⠈⠀⠎⠀⠀⡇⠀⢹⠀⢹⢸⠀⠀⠀⠀⣀⣦⠶⠟⠃⠀⢀⣠⠴⠚⠋⢉⣀⡀⡀⢸⣥⠟⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⠀⢸⡘⠶⠖⠒⠋⢉⣀⡤⠴⠒⠋⠁⠀⠶⢶⠃⢸⣴⠃⡟⢸⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠈⠓⠒⠲⠦⠤⠤⣤⣤⣤⠾⠀⠈⠉⠉⠁⣀⡠⠖⠚⠉⠀⠀⣴⠓⢦⠀⠀⠀⢸⠀⠘⠘⠃⣉⡠⠴⠓⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠚⢉⠁⣴⠛⠀⢸⠀⠀⢸⣠⠞⠀⠀⢀⣀⡤⠶⠚⠋⣁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡁⢸⠀⢿⣡⡄⠘⠛⣁⣈⡤⠴⠒⠋⠉⠀⠀⢀⣴⣿⣿⣿⣦⡀⠀⢀⣧⣀
⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢈⣀⣨⠤⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⡏⠀⢸
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠛⢿⣿⣿⢸⡇⣀⡼
⠀⠀⠀⠀⢰⠇⣀⣤⡀⢠⠦⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⡿⣠⠶⡄⢻⠉⠉⠉⠁
⠀⠀⠀⠀⢼⠠⣇⠀⡧⢸⣙⠓⠶⠤⣍⣻⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⡇⣏⢀⡇⢸⠀⣀⠐
⠀⠀⠀⢠⠿⢦⡈⠉⠁⢸⣮⣍⣓⠶⠦⣼⠀⡴⠚⢶⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣠⠴⠖⠚⠉⢻⣷⣖⣆⡷⡈⠋⣡⠏
⠀⠀⠀⢧⠄⠀⠈⠓⠦⣄⣉⠓⠮⢽⣒⣾⡄⢧⣀⡼⠁⠀⠀⢸⣿⣾⠟⣉⠉⢻⡿⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠛⠋⠁
⠀⠀⠀⠈⢾⡴⢀⠀⠀⠀⠈⠙⠓⠲⠦⢭⣄⣀⣀⣀⣀⣀⡀⣸⢠⠇⡞⠉⢳⠈⡇
⠀⠀⠀⠀⠀⠙⠯⣞⡄⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⢸⠈⣇⠂⡾⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠷⢧⣆⣄⠀⡀⠀⠀⠀⠀⢀⣠⢿⣿⣿⣧⡈⠛⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠚⠋⠁⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡄⠀⠀⢀⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠈⣿⡿⠋⠀⠀⠀⡈⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠂⠰⠿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⣠⡀⣠⣤⡄⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠁⠀⠀⠀⠘⠂⠀⠀⠀⠀⢰⡶⣶⣶⣴⣟⣿⣿⣿⠀⣫⣾⣿⣅⣾⡿⣋⡀
⢤⠤⣤⡤⣤⣤⣤⣤⠄⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠉⠉⠙⠛⠁
⠀⠈⠹⣿⣿⡟⠉⠠⠎⠀⢀⢔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣠⠿⠁⠀⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠢⠀⠀⢀⡇⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⢀⠿⠐⢈⡀⠀⠀⠀⠂⠬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠔
⠀⠀⠈⠉⠉⠌
⣀⣀⣁⣀⣀⣀⣀⣘⣓⣒⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⠒⣀⣉⣀⣈⣁⣐⡒⠒⠒⠒⢤⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢖⡲⠖⢋⣴⡶⠟⡛⡛⢻⠛⠙⡟⠛⠟⠻⢻⣿⣷⣄⠑⠶⡏⣹⠵⠆⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠉⠀⠀⢸⣿⣿⣯⣧⣡⣬⣴⣤⣥⣵⣦⣶⣼⣿⣿⣿⠀⠀⠙⠁⠀⠀⠀⠀⡀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠓⠦⣀
⠀⠀⣀⡀⠀⠀⠀⠆⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⡴⠒⠓⢺⢹⣿⡏⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣦⣄⣭⣌⢿⣶⡆⠀⣀⠀⠀⠈⠓⢦⡀
⠰⠊⠘⠢⡀⠀⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⠐⠋⠘⢿⣧⠀⣿⣿⡆⢀⣀⠀⠉⢢
⡇⠀⠀⠀⡇⢀⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⠀⠀⠀⢸⣿⠀⣿⣿⡇⢸⣿⡇⣴⡄
⡇⠀⠀⠀⡇⢸⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⢸⡏⠀⣿⣿⡇⢸⣿⡇⢸⣇
⣇⠀⠀⠀⡇⢸⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿⡇⠀⠀⢸⡇⢠⣿⣿⠃⣾⣿⠇⢸⣿
⠈⠉⠛⠋⠁⣼⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠁⢳⣤⣤⣾⠇⢸⣿⣿⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⡏⠠⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢀⡤⠀⠀⠀⠀⠈⠙⠛⠋⠀⠻⠿⠇⠀⣿⡿⠀⣼⣿
⠀⠀⠀⠀⠀⡇⠀⠈⠑⠲⠤⠤⠀⠤⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠒⠂⠀⠠⠠⠀⠈⠁
⠀⠀⠀⠀⠀⡏⠉⢐⠒⠤⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⡄⢀⣴⣾⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣤⣄
⠀⠀⠀⠀⠀⡇⠀⠸⠀⠆⠈⠑⢦⡉⢧⠉⠉⠉⢭⣭⣭⡭⠭⠭⠭⠭⣭⠙⠀⠀⡠⠊⡅⢸⠀⢸⡄⣰⢸⣿⠟⣋⣭⣭⣷⣶⡶⠶⠶⠶⠶⠶⠾
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠹⡄⢻⡬⣤⣿⣄⣀⣼⣏⣉⣉⡽⠁⠀⢠⠊⠀⠀⠀⠀⠀⠀⢁⡏⠘⠁⠚⠛⠉⠉
⠀⠀⠀⠀⠀⢹⣈⣉⣒⣒⡒⠒⠒⡒⠛⠈⠉⠉⠛⠛⠛⠛⠒⠛⠚⠃⠀⠀⠓⠒⠒⠒⠒⠒⢒⠚⠉⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢸⣏⡙⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠿⠿⠛⠛⠛⠃⣆⣀⡞⡁⢳⠀⠀⠀⠀⠀⠀⢠⠏⠉⢇⣀⣸
⠀⠀⠀⠀⠀⡇⠀⠀⠉⣩⣥⡖⢒⡒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠉⠩⣽⡎⢹⠘⡧⠖⠒⠒⠒⠒⠛⢀⢿⠈⠁⢸
⠀⠀⠀⠀⠀⠙⢤⡀⠀⠙⠒⠒⠒⠋⠀⠀⠀⡎⠉⠉⠉⠉⠉⠉⢹⠀⠀⠐⢯⣍⣉⣩⠴⠀⠀⠀⠀⣺⠁⢸⢰⠁⠀⠀⢀⣀⣀⡄⢸⠸⠀⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠉⡟⠓⠒⠒⢶⣶⣶⣦⠴⣇⣀⠀⠀⠀⠀⠀⢘⡦⠤⠤⣤⣤⣤⣤⡤⠤⠤⠖⢺⠁⡆⡜⢸⠉⠉⠉⢏⠁⠀⢰⠘⠞⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀⠀⠈⢻⣿⡏⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠘⠿⢿⡿⣇⠀⠀⠀⠘⡆⠘⢁⠇⠀⠀⠀⠈⠢⠤⠀⠧⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠤⠵⠤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠰⠶⠶⠰⠶⠶⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⠀⠀⠀⢠⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠀⠀⣠⠈⠻⢉⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠈⠀⠀⢩⡖⠀⠀⢀⡟⢀⡀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⠀⠀⣤⡐⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⡩⠙⢛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⢀⡼⠉⠉⠉⠙⠛⠛⠛⠛⠿⠇⠀⠸⢿⣿⡿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠁⠀⢸⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣤⡀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠿⠇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣤
⠀⠀⠀⢀⡿
⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣟⠑⡂⣰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠄
⠀⠀⣻⣄⣠⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡯
⠀⠘⠛⠣⠤⣼⡿⠇⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣭⣀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠈⠻⣆⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣯⣵⣿⢿⣿⣦⣤⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣴⣾⣄
⠄⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⣷⣿⣿⣟⣿⣿⡿⠛⠁⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠠⣾⠋⢂⣤⢞⣿⠟⠋⠁⠀⠀⠀⠀⠐⠾
⡩⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠁⢛⠁⣏⢼⣿⣟⠋⠁⠀⠀⠀⠀⠘⢧⣤⡀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠛⠳⠷⠶⠀⠀⠠
⠐⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠀⠀⠀⠀⠀⠈⠛⠛⠿⠗⣶⡀⠂⠀⠀⠈⠙⠷⡀⠐⠃⠀⠐
⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠉⠀⠻⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣴⣶⣶⠤⠤⠤⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠉⠙⠿⠛⠛⠿⣯⣍⣀
⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣤⡄
⡁⠁⠁⡠⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡀⠀⠀⠀⠉⠙⠻⠆
⠀⠀⠀⠀⠄⠤⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠻⠄⠠⢀⣠⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠓
⠀⠀⠀⠀⣠⡀⠐⠄⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠐⠒⠐⠒⠀⠀⠀⠀⠀⠀⠛⠛⠀⠂⠉⠀⠀⠀⠉⠉
⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠤⢒⣈⣡⣴⣾⣶⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⢱⠀⠀⠀⠀⠀⠼⠿⠛⠛⢻⠉⠁⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣠⣤⣴⣶⡄⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡜⠻⣿⣿⠀⠀⠀⠀⠀⢤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣄⠘⢿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣼⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠸⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⠜⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠿⠿⠛⠛⠛⠋⢻⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⠋⠉⠀⢀⣠⠤⠈⠁
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⡇⢀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣤⣴⠶⠛⠉
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣃⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣶⣶⣶⣶⡿⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠐⠂
⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⠀⠀⢤
⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⣠⠄⠁⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠁⢀
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠁⠠⠀⠀⠀⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠰⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢀⡆⢐⠱⠀⢂⠀⠀⠀⠀⣀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠉⠀⠄⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠑⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠘⠀⠀⡀⢀⡬⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠐⠆⠉⠀⠀⠡⠀⢮⠂⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠳⡜⠆⠀⠈⠢⠀⠓⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡁⠀⠆⠐⡀⠐⠤⢀⠋⠷⣈⡳⠌⢆⠀⠀⣲⠒⢀⣀⣀⣀⡀⠀⢾⡟⣻⠷⠟⡟⠉⠈⢓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠈⠐⠀⠀⠈⠀⣠⢡⢆⢀⡠⠉⣫⠵⠖⣋⠀⡌⠉⠉⠁⠀⠁⢀⡿⠂⠔⡀⢀⡁⠀⠀⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⢰⠀⠀⢀⡡⠙⠊⡩⠖⢉⢀⠐⠈⠊⠀⡄⣴⠋⠙⡆⠀⣸⡇⢀⣨⠼⠚⢡⣀⣴⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠓⠀⠠⠊⠀⠰⠁⠀⠀⠀⠀⠊⠓⡀⠀⠄⠀⠀⠠⠘⠒⠚⠃⣴⡟⢻⡉⠀⠐⣀⡄⢾⡏⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣈⡠⠶⠋⠉⢀⣾⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⢀⠀⠀⠀⠀⠀⠀⡀⠀⢏⠩⠗⢶⠤⢆⣽⡏⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⡝⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀
⠀⠀⡔⠁⠠⠤⠤⠤⡤⠀⠀⠤⠤⠤⠤⠤⠤⠄⠤⠤⠤⡤⠤⠤⠤⢤⠀⠠⠤⠤⠤⠤⠤⠤⡄⡤⠠⠤⠤⠤⠠⣈⣂
⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠁⠈
⠀⠀⡇⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⡇⡇
⠀⠀⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠧⠀⠀⠀⠀⠀⠄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬
⠀⠀⡷⠀⠴⠄⠀⠀⠀⢠⣾⠟⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⣷⡄⠀⠰⠀⠀⠀⠀⢚
⠀⠀⠈⠒⠒⠒⠒⠒⠒⢻⣧⣀⣁⣿⠗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣿⣌⣈⣼⡟⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⠀⠠⠀⠀⠇⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⣀⣀⠀⠀⠀⢀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣢⡶⣿⣿⣿⣿⣿⡟⢀⣶⣿⣿⣿⣧⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠣⠿⠿⠿⠿⠿⠃⣼⣿⣿⣿⣿⣿⣧⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⠢⠤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠰⢦⣄⡀⠀⢠⣾⣿⠿⣿⣦⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⠀⢠⣴⣾⣷⣦⠀⠀⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⣨⣿⠃⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⣶⠀⢸⡇⠠⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⣰⣿⣥⠶⠾⠿⠿⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⣿⣀⣿⠀⢀⣴⠄⣠⡤⠄⢠⡤⣼⡟⢩⣿⢣⣀⣄⠀⣘⠃⣤⢈⣀⡀
⠀⠀⠀⠀⣤⣀⡀⠀⣈⣀⣀⢿⣿⢻⣿⢸⡏⣾⡟⢠⡆⣿⢁⣿⢁⣾⡏⠀⣰⡟⣸⡏⣸⣿⣊⣿⣇⣀⡀⢀⣀⣤
⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠙⠘⣿⣦⣭⠁⠉⠉⠁⠉⠉⠉⢡⣿⡀⠀⠉⠉⠈⠉⣿⡏⠉⠁⠀⠉⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⣠⣾⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣻⣿⡟⠛⠛⠛⠛⢻⣟⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠒⠓⠛⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢰⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠃⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣰⡶⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡦



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠒⠀⠂⠀⠀⠀⠐⠒⠂⠐⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠂⠐⠂⠀⠀⠀⠀⠀⠀⠒⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠉⢨⠈⠙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⡀⠀⠀⣀⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⡀⠀⡀⡀⠀⠀⠀⠀⠉⢳⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⢠⡀⠀⠦⠀⠠⠀⠁⢖⡏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⡈⢉⠱⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⡔⢻⡠⠀⠋⠠⢀⢀
⠀⠀⠀⠀⠀⠰⢠⢾⠀⠀⠈⡙⠈⠊⠙⢁⣹⠀⠀⠀⣠⠀⠀⠔⠀⠵⠔⠛⠃⠀⠀⠰⠆⠀⠀⠦⠉⠁⣀⠄
⠀⠀⠀⠀⠀⠀⠀⣰⠄⠂⣌⠁⠀⢀⡠⡌⠛⡄⠈⠀⠀⡄⠐⠆⠺⠂⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⣶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⡈⠁⠀⠞⠁⠀⠀⠀⢔⡁⠀⣠⠀⠀⢠⠄⠀⠀⠀⠀⣉⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣑⢶⣦⣬⠀⠀⠀⠀⢨⡀⠀⠉⢧⡐⠦⢄⠀⠀⠀⡤⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⠆⠠⡀⠐⠨⠠⠀⠘⣃⠭⣀⢘⠃⠘⣿⠀⠹⠒⠂⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠔⠃⠀⠀⠀⢹⣤⣤⣥⠀⠁⠀⠀⠀⠈⠀⠈⠀⣀⣀⣄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠊⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠸⠿⠆⠀⢀⣴⠶⠶⠶⢶⣤⡀⠀⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⡄⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⢠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠐⠐⠒⠀⠂⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂


⡿⠉⠙⡟⢛⠛⠟⠟⠻⣟⠻⠟⠿⠿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣿⣟⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡖⠊⠉⠦⠒⠞⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠁⢀⣤⠤⢤⣤⠂⠁⠈⠀⠐⠃⠀⠀⠀⠀⠀⠢⠠⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢨⡏⠀⠀⠀⢹⡆⠀⠀⢀⠀⣎⠀⠀⠀⠀⠀⢢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢧⣀⠀⢀⣼⠧⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⡖⠚⠛⠹⠉⠀⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠢⠀⠀⠠
⠀⠀⠀⠀⠀⡠⠒⠒⠒⠒⠢⢄⠀⠀⢀⠴⠒⠒⠒⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠥⠰⡃⠀⠀⠀⠀⠀⠀⠘⣁⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⢀⡰⠉⠉⠱⣄⠀⠀⠀⠀⢀⡜⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠊⠁⠀⠀⠀⠀⠀⠉⠒⠒⠚⠁⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠐⠀⠩⠞⠋⠉⠀⠀⠘⠛⠋⠉⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡀⢀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠑⣀⠀⠠⠃⠉⠉⠉⠉⠉⠙⠀⠀⢠⠁⠐⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⡀⠀⠀⠐⡀⠀⠖⢀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠉⢁⠔⠀⠀⠀⠀⠀⠀⠀⣀⠔⠚⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠦⠀⠀⠃⠀⠀⢄⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡌⢶⡤⠺⡿⣧⣴⡆⠀⠀⠀⠡⠀⠀⠀⠀⠁⠀⠤⠄⠀⠀⡉⠀⠁⠀⠀⠂⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⡖⣷⡀⣏⡉⣿⠏⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢈⠤⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠠⣆⢁⠀⣿⣿⣿⣀⠉⠀⠀⠀⠀⠀⠃⡀⠀⡴⠄⠀⠀⠀⠀⡌⠀⠟⣧⠠⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢠⠈⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠘⠢⠀⠈⠄⠀⠀⠐⡄⠠⢋⠞⣰⠃⣠⢆⡕
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠀⠈⠀⠀⠀⢤⡀⠀⡜⣕⡡⠾⣁⠜⠀⠋
⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⢘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡀⠀⠀⠀⠀⠀⢼⠟⠺⢅⡲⢼⣯⠆
⠀⠀⠀⠀⠀⠀⠨⣀⣂⠰⣏⠀⣹⣏⠀⢙⣦⣤⡤⣤⡤⠾⠂⠀⠀⠀⠀⠀⠀⠀⠈⠈⠲⠉⠂⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠖⣋⠭⠤⠉⠦⢀⠄⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠂⠀⠀⠀⢸⠁⠃⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄⠀⠀⠀⠀⠀⣐⣲⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠂⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣨⡀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⢇⡔⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠐⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀⠀⠘⠁⠇⠀⠄⠀⠀⠀⠸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠀⠐⠒⠒⠀⢹⣤⠁⠀⠠⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡍⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⢠⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠤⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠚⠘⠂⠀⠰⠀⠰⠀⣢⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡀⣀⡂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡀⡀⠀⠀⠀⠀⠐⢤⠃⠀⣠⠀⠀⢀⡉⠉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⠀⠀⠀⠀⠀⠀⠀⠸⢄⢀⣀⡏⠳⠀⠀⠀⠀⠉⣠⡄⠀⢰⠀⠀⠀⠷⠒⢘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡀⠀⠁⠈⠀⠀⠀⠀⠈⠀⠀⠰⣛⡄⠀⠀⠀⠀⠛⢣⠀⢸⡀⠀⠀⠸⡉⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⡁⡤⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠐⠂⠀⠀⠀⢸⡀⠈⠄⠀⠀⠀⠇⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⢀⢷⠁⠀⠀⠀⠀⠀⠀⡀⠀⢘⡇⠰⡀⠀⠀⠀⠰⡀⡃⠨⡀⠀⠀⠀⡀⠀⣤⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⠈⠁⠀⠀⠀⠑⠀⠀⢁⠀⠀⠧⣠⠇⠀⠀⠀⠀⡍⠇⠀⠂⠀⠀⠀⠇⣠⠃⢣⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠁⢆⠀⠀⠒⠒⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠁⠀⡘⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡰⠁⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢠⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⢿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠉⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠦⠀⠄⠂⠀⠀⠀⠁⠀⠔⠃⠀⠀⠀⠀⠀⡤⠂⠤⡄
⠀⠀⠀⢀⣯⣤⣶⣶⣄⣄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⢠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠆⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠘⠛⠟⠁⠀⠸⡆⠀⢀⡤⢀⣀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⣉⠥⠐⠛⠁⠀⠈⠈⠩⠉⠀⠂⠀⢼⣆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⠁⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠀⣠⠶⠦
⠀⢀⠈⡠⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⠀⠀⠀⠄⠀⠀⡀⢀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠘⠁⠔⠀⠀⠀⠀⠀⣸⠏⠀⠀⢼⣒⠲⠞⢒⣠⣈⡀⣸⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠓⠀⠀⡆⡂
⠀⠀⠀⠰⠖⠀⠒⠀⠀⠞⠁⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠶⠞⠁⠀⠀⠚⠓⠚⠓⠀⠐⠐⠛⠓⠛⠂⠀⠀⠀⠀⠀⠄⠀⢀⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢡⠋⠔⠁⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⣠⢀⡀⢀⣠⣤⣤⣶⣶⡆⢾⠁
⠀⠀⠠⢀⠀⠀⠀⠀⠀⠄⠀⠠⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⠋⡽⠋⠹⠈⠈⢣⡼⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⣀⡀⠂⠀⠀⠤⣤⣤⣤⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠦⠄⠀⠀⠀⡀⠠⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡟⢸
⠀⠀⠀⠀⠀⢀⠴⠶⢾⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠀⠄⠈⠆
⠀⠀⠀⠀⠠⠀⠀⠀⠘⠄⠐⠓⠀⠂⠐⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠤⠤⠖⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡅
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⠀⠠⠤⡄⣤⣤⣤⣠⣀⣤⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡇
⠀⠀⠀⠀⠀⠀⠈⢧⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣷⠟⣋⣹⣾⡷⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⢿⣽⠦⣤⡀⢀⠀⠀⠀⠻⣶⡯⣵⡾⠁⠀⠀⠀⠀⠀⠀⠀⣼⡛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣍⢰⠀⠻⣌⡃⠀⠀⠀⠘⡾⠋⠀⣀⣠⢤⡄⠀⠀⠀⢰⡿⢿⣿⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠙⠈⠁⣄⠀⠀⠀⠀⢀⡞⠁⠀⠀⢷⡀⠀⠀⣼⣿⣾⠃⠀⠈⠙⢷⡄⠀⠀⠀⠀⢀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡄⢠⣿⣦⠀⠀⢀⣸⠁⠠⠄⠆⣶⢿⣟⣞⠻⠷⢯⣀⡀⠀⠀⠀⣿⠀⠀⣠⡴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢀⣿⣯⠷⣾⣱⢿⣤⣠⢠⢠⡀⣘⢷⣸⠳⠂⠀⠀⣷⢤⡀⣀⣬⡶⡛⢭⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡚⠋⢠⡾⢊⢥⣾⠟⠁⠀⠀⠉⠉⠉⠉⠉⠙⠉⠉⠙⠙⠐⣷⣶⡏⠙⣏⣻⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠑⣾⣹⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠴⢿⣖⣈⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠾⣠⡴⢸⣯⣙⡙⠗⢛⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢾⢿⣿⡟⢲⡤⠤⢄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣽⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⢩⠈⠡⣤⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣽⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣨⠀⠸⣷⣿⣿⡀⠈⣌⣤⠶⠶
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⠰⠇⡀⣈⣨⣿⠎⠉
⠀⠀⠀⠈⠒⡄⠀⠀⠀⢀⣁⢰⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⢨⣭⡿⣿⣻⠆
⠀⠀⠀⢀⣙⣛⠐⠀⠙⢛⡃⣀⣀⠠⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠁
⠀⠀⠘⠛⢧⣨⡄⠁⠛⠋⠉⢠⣄⣠⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠶⡿⣡⣦⡴⠂
⠀⠀⠀⠀⠀⢙⡇⠀⠀⠀⣀⣻⡿⠟⢋⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⢤⡀⢠⣮⡟⠁
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠛⠉⠀⡠⢞⣷⢾⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⢿⠉⠻⢑⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⡴⠋⢠⡿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⢿⡿⣷⣿⣯⡄⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠃⣤⢜⡛⣋⡽⢟⡿⣿⠟⣻⡿⣿⡿⣿⣿⡿⠀⣧⡀⠛⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠁⠀⠀⠀⠀⠀⠀⠸⢳⠵⠶⠋⣠⠊⢹⣏⣣⡈⢿⣯⠿⣛⡋⠽⠉⡁⠈⠹⣯⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠈⡡⡤⠀⠀⠏⠐⣭⡥⣕⢺⡟⠂⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣅⣘⢧⣰⠀⠀⠉⠉⣿⢿⡾⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠀⠀⠀⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⣿⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡿⣋⣿⡷⠀⠀⠀⠀⠀⠀⣿⣜⡇⣀⣀⡀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡿⣾⠇⠀⣠⣄⠀⠀⠀⣿⣼⡇⠀⠀⠉⢳⡄⠀⣠⢾⠋
⠀⠀⠑⠦⡀⣴⣲⠤⣄⡀⡀⠀⠀⠙⠇⢠⠋⠀⠈⢣⣤⣤⠿⠿⠧⢤⡀⠀⢀⣷⣞⣍⠃
⠀⠀⠀⠀⠀⠻⢯⣐⠈⢓⠲⢀⠀⠀⠀⢸⠀⠐⠒⠛⢹⢮⣽⡂⢀⣠⢷⣶⣪⣝⡺⠿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⠈⣷⣦⠀⡸⣀⣤⠾⠚⠊⠉⠉⠉⠉⠙⠻⢿⣍⣿⣾⣧⣀⣀⡄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠧⢸⡿⢫⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⡇⡈⠃⠦⣴⣟⠒⢄⣲⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⠤⠤⣾⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣈⠀⠻⠛⠟⣶⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⢤⡾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⠃⢖⣿⣿⡗
⠀⠀⠀⠀⠀⠤⣤⣞⣳⣶⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣯⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣰⠦⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⢀⢿⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⣁⡸⠛⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠙⠓⠘⣿
⠀⠀⠀⠀⠀⠵⢄⠀⢘⣩⡤⠒⠈⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢵⣿⣦⣄⠈
⠀⠀⠀⠀⢰⠮⣹⡤⠈⠁⠀⢸⣿⠟⢩⢙⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣧⢐⣄⣉⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡽⠀⠀⠘⠟⠁⢠⣻⢟⡟⢹⡿⣿⣷⢶⡶⡶⣶⣺⣟⠻⣿⣻⢗⣯⣒⣤⠀⠀⠈⠙⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢡⠎⠀⠉⠰⢪⣕⣪⡼⢡⢏⠹⠂⠃⠸⠓⣯⣗⣯⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠐⣇⡠⡄⠈⠁⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⢲⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡰⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠦⢄⠀⠀⠀⠀⠀⢀⠊⢠⠀⠈⢄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢈⠔⠁⠀⠀⠉⠢⢀⠀⡔⠁⢠⠂⠀⠀⠀⠱⡀⡠⠔⡉⠉⠢⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠁⠤⠃⠀⠀⠀⠀⠀⠀⠠⢋⠠⠄⠋⠀⠀⠀⠀⠀⡼⠀⡨⠃⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⠀⠀⠀⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣄⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠖⡍⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⠔⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠀⠀⣀⠤⠒⠋⠚⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠒⠚⠁⠀⠀⠀⠀⠑⠄
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠒⠒⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡗⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠂⠀⠀⠀⠀⠹⠇⠀⠀⠂⣩⠐⠀⢀⡀⠀⠀⠀⢴⡦
⠀⠀⡀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠐⡗⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠘⡃⠀⠀⠀⣠⣌⠀⠀⢀⣄
⠀⣨⣏⠀⠀⢸⣏⠀⠀⢀⠀⠠⣦⠀⠀⠀⠀⢀⡴⣉⣀⣈⠧⢀⣀⣃⣚⠃⢛⣛⣛⣋⠱⡄⠀⠀⠀⠀⢀⢺⡗⡀⠀⠀⠀⠀⠀⠀⠐⠿
⠀⠉⠏⠀⠀⢼⡷⠀⠀⠛⠃⠀⠁⠁⠀⠀⢠⠞⡮⡥⠀⠠⢇⠸⠠⠤⠤⠇⠨⠤⠧⠤⠵⢰⠀⠀⣶⡀⠀⢼⡧⠀⠀⠀⣴⡆⠀⠀⢐⣷
⠀⠀⠀⣀⣀⣀⣀⠀⣀⡀⠀⢀⣀⣀⣀⡀⠈⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠈⠀⠀⠛⠀⠀⠁⠁⠀⠀⠀⠙⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⢀⣶⠾⡲⡄⠀⠀⠀⠀⡉⣠⡤⢤⢤⠀⢐⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⡀⢹⡦⡾⠁⠀⠀⠀⠀⣉⡘⣧⣴⡅⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣐⡄
⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⠐⠒⠀⠀⠱⠀⢀⠀⠸⣹⣿⢳⡆⠀⣾⣹⣿⡿⢿⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⣤⣦⣤⣦
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣀⣶⠀⠀⠀⢀⣀⣸⣀⣀⠀⠘⠤⠄⢻⠀⠀⠀⠀⠿⠯⠥⠥⠼⠃⠀⠀⢻⡯⣝⠛⠀⠀⠀⠷⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⢠⡶⢶⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠉⣿⣿⣿⣶⡆⠀⠠⣿⢟⣷⡿⠃⠀⠠⠒⠶⠶⠖⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠿⠦⠤⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠙⠟⠁
⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣿⣿⣿⡿⠀⠀⠀⠰⠶⣾⣶⣶⠀⠀⠰⠶⣾⣂⣀⠀⠀⠀⡀⢸⣦⠀⠀⠀⢀⣶⣿⡏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⢺⡿⢷⡦⠀⠀⠀⠰⠴⠿⠿⠿⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⢙⣍⣙⠃⠀⠀⠠⠤⠤⠤⠿⠧⠀⣰⣷⣾⣧⣤⡀⠀⢾⣦⡎⠱⠿⠆
⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠋
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⡥⠙⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⢰⣦⣀⣠⣞⠀⠀⢠⣀⣂⡄⠀⠀⠀⠀⣠⢀⣀⣀⠀⠀⠀⣬⠀⢀⡃⠀⠀⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠈⠁⠈⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠋⠛⠛⠋⠀⠀⠀⠀⠙⠈⠁

⠀⠀⠀⢠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠀⠀⠀⠀⢀⡄⢀⣀⣀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢀⣶⣠⡄⠀⠀⠀⡶⢶⠶⢤⡀
⠀⠀⠤⠀⠀⠙⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠘⠛⠛⠉⠛⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠛⠟⠀⠀⠀⢀⣷⣾⡶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣩⣽⠅⠀⠀⢺⣷⡀⣀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡇⠤⠤⠄⣜⢸⣿⠙⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⡟⠀⠀⢀⣤⡄⠀⢹⣇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⢸⢀⣿⢀⣀⣠⠤⠰⡖⣄⠀⢹⣄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡄⢈⢦⢚⣻⠏⠑⣷⠓⠀⠀⡉⠗⠀⠈⢷⠀⢸⠀⢀⣀⣀⣀⣠⢐⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢆⡻⢸⢃⠤⠐⢻⠒⠂⠉⠙⠛⠛⠒⢺⣷⠋⠡⠀⠀⠀⡀⠳⠃⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣙⡟⣿⣄⠤⡞⢠⠀⠀⠀⢸⠀⠀⠀⢫⡐⠌⠘⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣑⣾⣁⡿⠅⠒⡇⢸⠀⠀⠀⢸⡇⠠⠀⢾⡁⠀⣸⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢘⣽⠏⠈⣇⠀⠀⣁⡸⠤⡖⠒⠊⠉⠉⠀⣈⣿⡉⣉⡁⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⣟⡠⠤⠛⢃⣁⣀⣀⠀⠀⡦⠍⠊⠉⢠⠀⢹⡅⠠⠤⣱⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢓⣯⣥⠴⡴⠋⣿⡵⠾⢻⡀⠀⢹⠁⠀⠀⢀⣴⢺⡷⠙⠂⠌⣇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢾⠟⠟⠚⡇⡄⠆⠁⠀⢸⡃⠀⠈⣇⣒⣚⣫⣅⣸⣿⣀⣠⣀⣞⡄⢃
⠀⠀⠀⠀⠀⠀⠰⢁⣆⣄⣸⠤⠦⠔⠃⠑⠋⠉⠉⣩⣁⣀⣀⣀⣀⡀⠀⢀⣸⣿⢀⣀⣀⠀⠀⠈⣆⡀
⠀⠀⠀⠀⠀⠤⠀⢈⣉⣠⡤⠀⠀⠀⠀⠀⠒⢀⣿⠂⠼⢖⡂⠔⠂⠀⠐⠒⢘⣿⣛⠂⠀⠀⠀⠀⢻⠤⡀
⠀⠀⢀⣰⡷⠚⠛⠛⠃⢤⡀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⢣⠀⢔⠀⠀⠞⠶⡶⠀⠃⠀⠒⠒⠀⢺⡀⢰
⠀⠀⢾⠋⠁⣀⠀⡀⠀⠀⠱⡌⠀⠀⠀⠐⡞⠀⡠⡄⠀⠀⠀⣧⢀⣀⡀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠨⡇⢸
⠀⠂⢾⠀⣰⢇⢴⠘⡄⠀⠀⢳⠀⠀⠀⢺⡇⢰⢃⠿⠀⠀⡀⠛⠀⠘⡇⡟⠉⠀⠀⠀⠈⠘⡆⠗⠒⠣⡜
⠀⠀⠚⠀⢻⠸⣸⠀⠋⠐⡶⠊⠀⠀⠈⠉⠉⢹⠘⣤⠃⠒⠛⢲⠖⠒⠻⠷⠖⠒⡖⠒⠂⠲⠓⠓⡖
⠀⠀⠀⠀⣀⡄⠁⣰⣁⣼⣇⣀⣀⣀⣀⣀⣀⣀⣇⡈⣠⠃⣀⣼⣀⣀⣀⣀⣀⣀⣠⡀⣠⠃⢀⣼⠁
⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⢐
⠀⠐⣎
⠀⢤⣬⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠃
⢤⣦⣜⠀⢠⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠂⣹⣥⡄⠀⠀⠀⠀⠀⢠
⡀⢃⡈⠃⠄⠀⠀⠀⠀⢠⡽⣀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡷⠀⠀⠀⠀⠨⡆⠀⠀⠀⡤⠀⣼
⡯⠟⠓⡄⠘⠃⠀⠀⠀⢶⣽⡡⠀⠀⢸⡇⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠠⢚⠟⠁⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⣘⣿⠂
⡁⠀⠸⡷⠀⠀⠀⠀⣙⠻⣴⡤⠔⠀⣿⣷⣶⢅⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠚⡋⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⣸⣮⢁⠀⠀⣦
⠗⠁⢀⣦⡀⢀⠀⠀⣹⠗⣿⠤⠂⠀⠀⠀⠀⠁⠦⡄⠀⠀⠀⠀⠀⠀⢸⣇⡄⠁⢀⡆⣼⣠⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣃⠀⠀⢸⣿
⢿⡔⠷⠛⠀⠀⠀⢠⣾⡞⡿⠰⣂⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢣⣛⠹⠥⠴⣿⣿
⣤⠦⠄⠀⠀⠀⠈⢩⡯⣹⡿⣨⠇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣭⡟⣖⠀⠀⠀⠈
⠁⠘⠂⠀⠰⠓⠄⠛⠓⠿⠖⠉⣿⣇⣀⡈⠀⠀⠀⣼⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠉⠋
⠖⠺⡟⢒⠀⠀⢠⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣀
⣠⣠⣧⣼⠦⠞⠁⠀⠀⠀⠈⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⣀⢀⠀⠂⠀⡀
⣀⠀⠀⠀⠤⠀⠨⠅
⠀⠤⠄⠉⠀⠀⠀⠀⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⣤⣴⣦⣤⡄⠀⠀⢐⠒⠒⠂⣀⡀⠀⠀⠀⢀⣰⣾⣾⡆⠀⠀⠀⠠⢤⠤⠀⠀⠀⠀⠀⣠⣾⣧⣠⠂
⠀⠘⠿⠿⠿⠿⠇⠀⠀⠘⠀⠀⠣⠀⠀⠀⠀⠻⡿⠛⠻⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⠄⣦⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣾
⠀⠀⢸⣿⣤⣄⠀⠀⠀⠀⠸⠿⠿⠿⠄⠀⠀⠀⠀⠚⠁⠟⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢰⣾⣶⣾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣾⣿⣄⠀⠀⠀⠀⠠⢨⡂⠀⠀⠀⠀⢸⡇⣶⣿⣿⠀⠀⠀⠀⠀⣠⡺⠟⠁
⠀⠀⢹⣿⣾⡿⠀⠀⠀⠀⢠⢏⠷⡄⠀⠀⠀⠈⣇⣻⣿⡟⠀⠀⠀⣠⣦⠟⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣤⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣶⡦
⠀⠀⣾⣿⣿⣿⡄⠀⠀⠈⠉⠉⠉⠈⠁⠀⠀⠀⠀⡇⢥⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⢀⣺⡁⣀⡀⠥⠤⠄⠐⠃⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⢀⡈⠡⠤⠤⡆⠰⠰⠀⣇⣀⣀⣀⡀⠀⠀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⡀⡀⢀⣁⠀⠤⡄⠀⠈⠉⠅⠀⠀⠠⢔⡒⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣉⡁⠩⠤⠤⠤⠷⠤⠄⠀⠒⠒⠒⠒⠒⠒⠒⠒⠂⠾⢄⣀⠀⠀⠀⠀⠀⠀⠠
⠐⢀⠀⠄⠄⢀⡤⢀⣠⢊⠭⠥⡤⢤⣒⠒⠐⢠⠄⢀⠄⠀⠀⠈⢉⠝⡋⠩⠁⣂⠐⠊⢉⣉⠓⠒⠢⣝⠢⡀
⠀⠀⠀⠀⠀⠉⠀⢀⠇⣎⠒⢉⠄⠀⣀⢑⡆⠎⠠⢡⠊⢁⡀⡈⢣⠉⠠⡀⠃⠈⡀⣾⢁⠌⡩⢉⡁⠪⢢⠘⡄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢰⢕⡐⢁⡀⣤⡰⠋⢺⠀⡒⡰⠊⠁⢀⣉⠛⠊⠙⠓⠗⢊⡇⢠⠜⢀⡀⠄⡈⢠⠂⡆⢹
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢹⡾⠦⠜⢣⠄⡂⠀⣶⣄⠀⠉⠀⣡⠋⠀⠐⠠⠎⡡⠇⠀⠀⢱⠀⣨⠽⢑⠊⣻⠎⠒⢦⠀⠀⢀⣀
⠀⠀⠀⠀⢀⣀⠠⠇⠈⠋⠛⠒⠒⠂⠦⠬⠴⠀⠘⠈⡇⠀⠀⠀⠈⠒⠀⠈⠀⣇⣠⡠⠀⠤⡬⠕⠩⠏⠩⠴⡊⠁⠉⠉⠁⠀⠀⠈⢆
⠀⠀⣰⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠓⢳⠂⢀⠂⠀⠉⠀⡡⠄⠠⡀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡇⠨⣍⠉⠉⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠉⠉⠉⠀⠀⡄⠀⠀⠀⠀⠀⢀⠤⢐⣒⡒⠠⣀⠀⠀⠀⠀⠂
⠀⠀⢣⣞⣉⡆⡆⠀⠀⠂⢐⡩⢑⣼⠀⠀⠀⠀⠀⢀⠤⣶⠶⠛⠛⠶⢶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡡⠚⠉⠀⠈⠑⢌⢢⣀⠤⠤⠄⡀
⠀⠀⠀⠇⠈⠀⠃⠀⠀⠀⠘⠔⠽⢹⠀⠀⠀⠀⢰⢡⠊⠀⠀⠀⠀⠀⠀⠙⡽⡄⠀⠀⠐⠀⠀⠀⢠⣸⠁⢠⠖⠒⢢⠀⠘⣼⡆⠀⠀⠀⡀
⠀⠀⠀⠞⠉⠐⠢⠤⠄⣀⣀⡀⣀⠀⣇⣀⡀⢀⡇⡇⠀⢠⠊⠉⠉⠇⠀⠀⣧⡇⠀⠀⢐⠀⠀⣀⣨⣧⠀⠘⢄⣠⠞⠀⢠⡿⠤⠤⠔⠊
⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠇⡇⠀⠈⠢⠤⠚⠀⠀⢀⣧⠋⠉⠉⠉⠉⠁⠀⢀⣫⢦⡀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠉⠐⠒⠠⠤⠄⠀⠨⣄⣀⣀⣀⡠⠤⠒⠙⢄⡀⠀⠀⠀⠀⣠⠞⠓⠒⠒⠒⠉⠉⠉⠀⠀⠀⠀⠈⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠐⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠚⣷⡂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠉⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠡⠻⡀⣸⣆⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠁⠀⠌⠀⠀⠁⡄⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣒⡀⠀⠀⠈⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠸⠗⢋⠀⣘⠛⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⢀⠘⠦⠟⢻⠛⠋⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⣾⠓⣿⢋⣴⣬⡐⠂⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⠤⠀⢡⠀⠀⣐⡀⠀⠀⡶⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣰⣁⠉⠁⡤⠄⠀⠀⠀⠀⠀⠀⠰⠀⠀⢁⡁⠀⠀⢀⠃
⠀⠀⠀⠀⠠⠀⠀⠈⠀⠈⠒⠀⠠⢆⠀⠉⠀⢀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠋⠉⠙⠂⣼⠀⠀⠀⠀⠀⠀⠀⢀⡔⣆⡀⠛⠁⠀⡠⠊⠠⡤⢤⣀⢀
⡀⡄⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠀⠈⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠜⢉⠒⠀⠰⠊⣵⠀⠀⠀⡏⠉⠉⠉⠈⠁⠉⠉⠉⡛⠙⠉⣋⡴⢀⢀⠉⡆
⠇⠠⠄⠇⠀⠀⢀⡀⢤⠴⠄⠀⡐⠀⠀⠀⠀⠀⠀⢀⢠⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡇⠀⠀⠁⠐⠄⠀⠀⠀⡇⠈⠀⠀⠉⠉⠉⠁⡇
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⡈⠆⠎⠾⠁⠀⠀⠀⠀⣰⡖⣀⣄⣀⢀⠀⠀⢰⠇⠀⠀⠀⠀⢡⠀⠂⠀⡤⠠⠐⠒⠒⠁⠀⠀⠀⠀⠀⢤⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠆⠀⠀⠀⠀⠀⠀⢈⠀⠀⠄⠀⠀⠀⠀⣈⠣⢄⢀⠀⣄⠀⠀⣰⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⠁⠂⠀⠀⡀⠀⠀⠈⠛⠁⠀⠀⠀⠰⠀⡗⣒⠋⠸⣿⡿⠀⣞⠒⠤⡆⢀⠀⠀⢠⡊⢨⣽⠀⠀⡀⠀⠄⠀⠸⢀⠄⢀⠄⠌⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠀⠀⠀⢈⠂⠀⠃⠀⠂⠀⢤⠃⣍⠭⠀⠈⠀⠀⢲⠀⢐⡞⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⠁⠀⠀⠀⠈⠙⠒⠠⠤⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠘⠒⡖⠋⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⡄⠠⠼⠒⠒⠒⠒⠒⠲⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠒⠤⢀⣀⠀⠀⠀⠀⣀⠠⢸⠀⠀⡐⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠀⠀⠐⠒⠆⠀⠀⠀⠠⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⣀⡀⠀⢀⣀⣀⡤⠔⠊⠀⠐⠦⣀⡀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⣀⣀⠀⠀⡀⠀⠳⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⡟⠉⢹⠉⠁⠇⠀⠀⠀⠀⠀⠈⠈⢱⠒⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠀⠀⠘⠒⠖⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠸⠀⢠⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡍⠈⢙⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⡆⠀⠀⠀⠀⢠⠦⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡌⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢠⠃⠀⠀⠀⢰⢀⡇⠀⠀⠀⠀⠤⡀⠈⠀⠁⠀⠀⠀⠀⠃⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠐⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠞⠈⡇⠀⡰⠍⠀⠀⢐⡄⠀⠀⠀⠀⠀⠀⠂⠸⢄⠀⠀⠀⠀⠀⠀⡀⠀⢀⠠⠆⠼⡀⠀⠀⡀⠄⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣠⡀⠜⠁⠀⠀⠀⠀⠀⡞⢐⡐⡘⡒⡄⠀⠀⠀⠀⠄⡀⠀⠀⠂⡠⡀⢠⠀⠸⠀⠀⠀⢀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡠⠐⢈⠔⠀⠀⠈⠐⢄⠀⠀⡐⠁⢀⠀⠀⠀⠀⠢⡀⢀⠄⢈⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⡠⠔⠁⠀⠀⠀⠀⠀⠀⢈⠊⡀⠐⠂⠀⠀⠀⠀⠀⠸⠁⢀⠇⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠆⠂⠁⠀⠀⠀⠀⠀⠀⡀⠀⠰⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⡐⠁⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠪⠁⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡠⠑⠄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠒⠒⠒
⠀⠀⠀⠀⠀⢀⠠⠎⠠⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠈⠴⠋⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡂⠀⠀⠄⠻⠠⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠰⡗⠀⠀⠀⠐⡛
⠀⠐⡂⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢰⠀⠀⠀⠀⣀⠀⣀⣠⣀⠀⢀⠀⠀⠀⠐⠀⠀⢠⣦⠀⠀⠀⠊⠉⠀⠀⠀⢿⠅
⠀⠼⠷⠀⠀⢨⣿⠀⠀⢰⡆⠀⠺⠃⠀⠀⠀⢠⣞⠱⠀⠈⢰⠁⡌⠈⠀⢰⠀⠀⠀⡆⠀⡄⢁⠀⠀⣠⡀⠀⢸⡯⠀⠀⠀⢀⡄⠀⠀⠀⣹⡁
⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠋⠈⠉⠉⠉⠉⠆⠉⠉⠉⠉⠄⠈⠉⠉⠉⠁⠸⠀⠀⢾⠆⠀⠙⠏⠀⠀⠀⠼⠿⠀⠀⠀⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣠⣤⣀⠒⠀⠀⠀⠀⠆⠃⠀⠀⠀⠀⣀⡄⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠚⣇⢈⡿⠂⠀⠀⠀⠠⠄⢻⠉⢹⠓⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠐⠒⠒⠛⠚⠓⠀⠀⠂⠀⠐⠒⠚⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄






⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣴⢲⡀⠀⠀⠀⠀⠀⢦⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢦⠀⠀⡾⠷⠀⠀⣹⣞⣁⣀⣀⣀⣀⣀⡟⠀⠀⢰⢯⡗⠀⠀⡤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⡀⠀⢈⣓⣯⠤⠖⠒⠚⠉⠉⠁⠀⡀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠲⠥⣤⣐⡃⠀⢠⠯⠷
⠀⠀⠀⠀⠀⠀⠀⣀⡬⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠉⠙⠒⠦⢤⣀
⠀⠀⠀⠀⡴⠚⠉⠁⠀⠀⢰⢳⠀⠀⠀⠀⠀⠀⠀⣀⣠⠎⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠀⢀⣤⡜⠸⢻⣢⡄⠀⠀⠈⠉⠓⢦
⠀⠀⠀⠀⢧⠀⠀⠀⢰⣙⣇⣸⣲⠀⠀⣠⡀⢀⡞⠁⠀⠀⠀⠀⠀⢀⠀⠀⠑⠦⣤⣤⣀⠸⠮⠧⠤⠤⠥⠏⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠙⢾⡀⠀⠀⢀⡼⠿⣲⣯⣀⠀⠀⠀⠀⠀⠈⠑⢤⠀⣀⣠⠤⢤⡀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠙⢲⡞⡹⠁⠀⠈⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⠻⣅⠀⠀⠀⠉⠓⢤⣀⡾
⠀⠀⠀⠀⠀⠸⣤⠞⠓⠲⠖⠁⠀⠀⠀⠀⢀⣴⣶⣶⣿⣗⣶⣶⢲⣶⣶⣶⢒⣿⣿⣿⠒⠒⡆⠀⠈⠀⠀⢀⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⢧⠀⠀⢠⣷⠃⠀⠀⢴⣦⠎⠀⢰⣦⣾⣿⠋⢸⢸⠀⠀⢸⢸⠀⠀⢸⠈⢿⣷⠀⠀⠀⠀⢸⢃⡆⠀⡼
⠀⠀⠀⠀⠀⠀⢸⡄⠹⢄⣿⡸⠇⠀⢸⣿⣶⣶⣾⣿⠯⣿⣶⣾⣼⣷⣶⣾⣬⣷⣶⣾⣤⣤⣽⡄⠀⢸⡇⢸⠛⡁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣻⠀⠀⠀⢸⠯⠇⠐⡯⣯⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠐⠿⠂⠀⡿⠀⠀⠛⣻⠞⠃⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠿⠤⠤⠤⣾⣼⣯⣭⣧⣿⣴⣖⣖⢶⣄⣀⣀⣀⣀⣀⣀⣶⣲⡲⣶⣧⠤⠤⠤⠿⠦⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠻⠭⠽⠉⠉⠈⠫⠯⠟⠈⠉⠉⠙⠭⠽⠃⠸⠽⠽⠃⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⣛⣛⣛⣀⣀⣀⣛⣛⣛⣛⣀⣀⣀⣛⣛⣛⣛⣀⣀⣀⣛⣛⣛⣛⣀⣀⣀⣛⣛⣛⣛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣤⢤⠀⢤⢠⡄⢠⡤⠄⢠⡄⣤⠀⢤⠄⠀⡄⢠⠀⣤⢤⠀⢠⠤⠀⢠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠷⠀⠷⠼⠁⠈⠟⠀⠸⠯⠄⠸⠙⠇⠀⠸⠀⠀⠧⠞⠀⠟⠻⠀⠸⠯⠀⠠⠽⠂




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣦⡄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡇⠈⠀⠈⢸⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠛⠿⠿⠿⠛⠁⠀⠀⠀⠈⠛⠛⠛⠉⠉⠃⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⡀
⠀⠠⠾⠿⡤⢮⡌⠷⠺⠦⠶⣰⢾⡝⠿⠛⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠉⠉⠃⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠂⠀⠀⠀⠐⠚⠡⠋⠀⠉⠁⠁⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠤⠀⢄⣀⢀⠀⠤⠤⠤⠀⠄⠀⣠⣄⣤⢤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⡤⠄⢀⠀⠀⢀⠙⠉⠉⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢽⣯⡳⠀⠀⠀⠀⠂⠒⠀⠉⠀⠛⠃⠀⠈
⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢛⡷⡆⠁⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣆⠀⠀⠀⠀⠀⠈⣃⠀⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣢⣍⡰⠀⠀⠀⠀⠠⠀⠀⠀⢤⣤⠀⠀⠘
⠀⠲⠄⠒⠠⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠉⠁⠈⠙⠀⠁⠀⠀⢀⡤⠤⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠉⠀⠀⠀⠀⠤⠤⠤⠂⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠠⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢦⠀⣀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠲⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠒⠒⠀⠈⢠⡔⠒⣆⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠊⠅⠂⠀⠀⠀⠀⠀⢠⣌⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⣤⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⢀⠀⠀⠄⠠⠠⢤⢠⠀⠀⠀⡀⡒⡤⠀⠀⠀⠐⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠶⠀⠈⠈
⠀⠀⠀⢀⠠⠄⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣀⠀⠴⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒
⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠄⠀⠒⠦⠤⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠰⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠀⠬⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢻⣿⣿⠿⢿⣿⣦⡀⠀⠀⠀⣠⣶⡿⢫⣵⣄⠀⠀⢠⣾⡟⠙⢿⣆⠀⠀⣺⣿⡿⢿⣷⣦⡀
⠀⠀⠀⣾⣿⣿⠀⠈⢿⣿⣧⠀⠀⣼⣿⡿⠁⠈⣿⣿⠀⠀⣾⣿⠀⠀⢸⣿⡄⠀⣿⣿⡇⠀⠹⣿⣷⡀
⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⠀⢸⣿⣿⠃⠀⠀⢹⣿⠇⢸⣿⣿⠀⠀⠀⣿⡇⠀⣿⣿⡇⠀⠀⣿⣿⣧
⠀⠀⠀⣿⣿⣿⠀⢀⣾⣿⡿⠀⢸⣿⣿⠀⠀⠀⣿⣿⠀⢸⣿⡏⠀⠀⠀⣿⡇⠀⣿⣿⡇⠀⠀⣸⣿⣿
⠀⠀⠀⣿⣿⡿⢰⣿⣿⣟⠀⠀⠘⣿⣿⣄⣀⣼⣿⠃⠀⣿⣿⡇⠀⠀⠀⣿⡇⠀⣿⣿⡇⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠙⠿⠿⠷⠂⠀⠈⠙⠛⠛⠛⠁⠀⠀⣿⣿⡇⠀⠀⠀⣿⡇⠀⣿⣿⣇⠀⢀⣿⣿⡟
⠀⠀⠘⣿⣿⠇⠀⣠⡤⠐⠛⠛⠀⠛⠒⢠⣄⡀⠀⠀⠀⣿⣿⡇⣴⣤⣶⣿⡇⠀⢿⣿⣿⠀⣼⣿⡿⠡⣄⡀⡼
⠀⠀⠀⡀⠴⠞⠀⢀⣠⣶⡿⠃⢤⣶⣶⣶⣮⣁⢤⣄⠀⣿⣿⡇⠉⠉⢻⣿⡇⠀⢸⣿⣿⣼⣿⠟⠁⣀⡤⣹⠷⢤⡀
⠀⠐⠞⠁⠀⣠⣶⣿⣿⠁⠀⠀⢸⣿⣿⡟⠛⣿⣷⡌⠰⣬⡻⠇⠀⠀⢸⣿⡇⠀⠈⠿⠟⠛⢁⡴⠆⠉⠀⠋
⠀⠀⢀⣴⣾⢿⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠘⣿⣿⡀⠀⠀⠲⠦⠀⣤⣿⠁⣤⡤⠀⢖⣋⣀⣤⣀⡀
⠀⠀⠿⠟⠁⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣇⠀⠀⢴⣶⣶⣶⣶⣤⠀⠲⣿⣿⡟⠛⠛⢿⣿⣦
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⢀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⢇⣤⣾⣿⠿⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⣷⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⢿⣿⡿⠀⠀⠙⠿⣷⣶⣦⡄⣴⣦⣾⣿⣿⣶⡿⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⢾⠊⠙⡂⠂⠤⠀⢰⠚⠂⠐⢳⠂⠀⠄⠚⠃⡆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⡄⠀⠀⡌⠀⠀⠀⠀⢃⠀⠀⠃⠀⠰⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣼⣦⣄⣤⣤⣼⣄⣴⣥⡄⠀⠀⠀⣼⣤⣴⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣇⡇⠀⠀⠀⠀⢀⣘⠀⠀⠀⠀⠀⠀⢡⡏⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⢀⠀⣼⡉⣿⡈⠃⠀⠀⠀⢀⡈⠋⢀⣀⣀⣀⣀⣀⣘⣓⡀⢀⣀⣰⣛⣱⣀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠐⡡⠀⠀⠒⠀⠀⠀⠠⠀⣤⢤⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠐⠄
⠀⠀⠀⠀⠈⣰⡁⠞⠃⢐⡠⠤⠤⣶⡄⢾⢈⠀⠠⡠⠀⠀⠀⡠⠄⠀⠀⠀⠀⠁⡀⢀⡄⠀⡤⢤⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠌⣰⠂⢠⣯⡄⠈⢧⠀⣄⠈⣏⣿⠑⠀⣦⣀⠀⢀⠂⢲⣀⡠⠆⣀⠀⠀⣵⠂⠁⠷⢄⠰⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠌⠰⠃⠀⠀⣋⣀⣀⡴⠀⠇⣴⠁⣼⡎⢻⠀⣏⡀⠈⡶⠦⢼⡕⠓⣀⡆⢀⣿⡦⠀⣈⡼⠃⠰⠌⡁⠀⠀⠀⠀⠈⢄
⠀⠘⠀⠁⠀⠉⠉⠉⠉⠉⠉⠉⠀⠉⠈⠛⠛⠛⠉⠛⠛⠛⠓⠓⠚⠛⠂⠚⠃⠀⠛⠂⠀⠈⠀⠀⠀⠀⠀⠐⠀⠐⠃⠀⠀⠈⠐⠀⠄⣀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄
⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⠀⠀⠀⢈⣰
⠀⣿⣿⣿⠀⠀⠀⣰⣾⣿⣿⠿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣦⡀⣤⣾⣿⡄
⠀⠛⠻⠿⠀⠀⠐⢹⣿⣿⠀⠀⠀⣿⣿⡗⠀⠀⠤⠠⠤⠤⠤⠤⠀⠀⠀⠀⠠⠦⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⢹⣿⣿⠁⠀⠈⢻⣿⡧⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣤⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣤⣤⣿⣿⠇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠂⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠠⠮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠤⢀⠀⠀⠨⠀⠀⠀⠀⠀⠸⠀⠈⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⡁⠄⠂⠉⠐⠠⠌⡀⠂⠔⢄⡀⠀⠀⠀⠀⠢⢀⠀⠀⠁⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠉⣐⢠⣀⠀⠀⠀⠈⠐⠠⢀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠇⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠀⠄⠀⠠⠄⢀⣀⠀⣉⠀⠤⠏⠒⠦⠄⡀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⢀⠄⠓⠊⡡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠁⡆⠂⠄⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢃⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡅⠂⠀⠀⠀⠁⠘⠀⠂⠒⠒⠒⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⡄⠀⠀⠀⠈⠄⡀⢰⡀⢀⠊⠀⠀⠣⡀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⢀⠤⠶⠊⣋⣀⡀⠠⠀⠀⠀⢀⠄⠊⠀⠘⠀⠑⡀⠀⠀⠈⠄⠀⠀⠃⠀⠉⠀⠐⢄⢰⠅⡤⢆
⠀⠀⠀⠘⡀⠠⡾⠉⠉⢳⡄⠁⠀⠀⠀⠢⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⢀⡦⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠈⠄⠻⣤⣤⠾⠀⠐⠀⠀⠀⠀⠁⡄⠀⠃⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠀⡇⠑⠠⠾⠂
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢀⡵⠒⠚⠛⠒⠛⢶⡀⠀⠀⢀⡤⠒⠒⠛⠓⠲⢤⡀⠀⠀⢀⠠⠦⢀
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢀⡝⠀⠀⠀⠀⠀⠀⠀⢹⠥⢤⡃⠀⠀⠀⠀⠀⠀⠀⡟⣐⡀⠀⠂⠑⠀⠀⡁⠂⠄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢂⣀⣠⠵⡀⠀⠀⠀⠀⠀⢀⡾⠒⠚⢳⡀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠠⠛⡲⠤⠤⠴⠖⠋⠀⠀⠀⠀⠙⠲⠦⠤⠤⢶⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠄⠄⢀
⠀⠀⠀⠀⠀⠀⢀⡔⠈⠀⠀⠀⠀⠀⢀⠠⠀⠒⢦⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⢀⡀⠊⠀⠀⠀⠀⠠⠀⠐⠀⠈⠉⠀⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⡰⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⡄⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠰⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡔⠙⠂⠀⠀⢀⠀⠀⡍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠀⠙⠀⠠⠀⠀⠀⠀⠰⡆⠀⠑⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⠁⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠃⠀⠀⠀⢀⡠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠆⠋⢁⠀⢒⠀⢦⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠧⡙⢆⠀⠀⠣⠀⠙⠆⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠦⠠⡀⠠⡀⠰⠒⠀⠳⢌⢳⡀⠀⠀⣴⠦⠀⠀⠀⠀⠀⠀⢠⣾⠿⣲⢶⠯⡖⠈⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠐⢤⠀⠀⠁⠌⠂⡘⣕⠀⠈⠙⠒⢒⣤⠴⠒⠉⠀⠚⠋⠋⠈⠁⠀⣼⡧⢃⠁⠀⠀⠀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⠀⠀⠀⢀⠀⠀⠀⠜⠐⡬⠗⣋⠥⠚⠉⠀⠢⠗⠄⠀⣠⠶⠶⣄⠀⠠⣿⠀⠀⠡⢦⡼⠇⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⢀⠀⡨⠀⠀⠖⡊⠄⠀⠙⢌⠦⠀⠀⠀⠀⢀⡀⢁⢻⣄⠠⣟⢀⣴⡿⡦⠒⠁⠀⠄⢋⣼⡿⠀⠠⠀⡀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠉⠀⠀⠀⠀⠘⣿⢀⠗⢀⣀⠴⠞⠊⣽⠂⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢴⣦⣋⠁⠀⡀⢴⡟⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⡹⠀⠚⠉⡛⠙⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠀⠀⠀⠀⠀⠀⠐⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠴⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠒⠒⠒⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠉⠁


⣿⢿⠿⡿⡿⢟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⠀⠀⠀⠀⠀⢺⠇⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠀⢾⡲⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⣢⣰⠄⢀⠎⢠⢃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠀⠙⠀⢀⠄⠀⠿⠇⠀⣱⣾⠀⠀⠀⠀⠀⣿⠏⠂⠊⡰⠣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠆⠶⢶⡔⠂⠀⠀⠸⣿⣀⠀⠀⡀⠘⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⡁⠀⠀⡄⠀⠀⠀⠀⠀⠈⡉⠀⠶⠁⠂⠠⠚⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠃⠀⠀⣴⣇⡀⡀⠀⢀⠀⢠⠛⠉⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⣷⡶⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠀⠂⠚⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡑⠒⠫⠤⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠁⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⢀⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣒⠠⠭⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⣰⣖⡲⠥⣀⠠⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠤⠀⢯⣛⣷⠻⠄⠀⠀⠈⡓⠈⠢⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠥⠀⠀⠀⠀⡄⢛⢰⡈⠐⡠⠘⠁⠀⠐⡐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠆⠀⠀⠀⠀⠃⢠⠰⠀⠀⠀⠀⠠⣦⠷
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠷⠾⠀⠀⠁
⠀⠀⠀⡠⠰⠬⠀⠀⠘⠄⠠⡄⠈⠀⠤⢤⠀⡀⠀⠀⠀⠀⠀⠀⢄⠀⢀⠀⠀⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⡜⣈⣠⣀⣀⣠⣤⠤⠤⠤⠤⡀
⠀⠀⠀⠀⢀⣆⠀⠀⠀⣶⠐⠀⠄⠀⠀⠀⠀⠢⠀⠀⣈⠉⠉⠉⠉⠉⠋⠉⠀⠐⡵⠒⢦⠀⠀⠀⠀⠰
⠀⠀⠁⠀⠀⠃⠀⠀⠀⠋⠀⠀⢠⡀⠀⠀⠀⠀⠠⠂⠀⣄⠀⠀⠀⠀⢀⠀⠀⢁⠙⠤⠞⢀⠀⠀⠀⠀⠁
⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠦⡄⠐⠀⠐⠂⢐⠈⡄⠠⠀⠐⠀⠬⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡥
⡇⠈⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠒⠀⠠⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠇⠀⠀⢀⠓⡀⠇⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⢀⠀⠘⡀⠀⠀⠀⠀⢀⠀⠰⠄⢙⡟⠲
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣠⣀⢘⣁⣀⡀⠤⠤⠼⠤⠤⠤⠤⠴

⠀⠀⠀⢘⠛⠛⢛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠐⠐⣐⠀⠂⠀⠀⠀⠀⠀⠀⠀⡰⢡⠔⠉⢉⠁⠙⢊⢂⠀⠀⠀⠀⠀⠀⢀⣀⣤⡆⢤⡶⠚⠳⣄
⠀⠀⠐⠺⠫⣉⣽⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⠊⠈⠓⠦⡀⠀⠀⠀⠀⠀⢠⢡⡅⠀⢀⡈⠀⠀⢨⡄⡄⠀⠀⠀⠀⠰⣏⢀⠽⠑⣼⠷⣞⢶⣈⣷
⠀⠀⢀⢀⣀⣀⣘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⠀⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⡌⢅⠀⠈⠀⠀⠀⢠⢀⠇⠀⠀⠀⠀⠀⠈⠱⢞⡻⠵⣛⣚⡣⢤⠞
⠀⠀⢸⠰⠃⣤⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠢⠇⣔⣙⡷⠐⠼⠝⠀⠀⠀⠀⠀⠀⠈⠪⡑⠂⠒⠐⢊⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶
⠀⠀⠈⠙⠛⠉⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⣀⠀⠀⠀⣀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠒⢒⠚⢵
⢸⢟⢧⠊⠄⢤⠄⣭⢂⠃⡬⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⡀⠀⠈⢣⠀⠀⠀⠀⠀⠀⢀⣴⣗⠒⢀⣀⡀⠲⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢢⠀⢸
⠀⡏⠀⢠⠠⠂⡟⡜⢀⢰⠇⠀⠀⠀⠀⠀⠀⢸⠀⢰⣉⡙⣦⠀⢸⠀⠀⠀⠀⠀⠀⢸⠓⠊⡜⡏⠀⠩⣟⠙⠁⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⢂⣤⠀⠼
⠀⣳⠘⢄⠛⠚⠿⣁⠌⠸⠄⠀⠀⠀⠀⠀⠀⠘⠀⠘⠲⠖⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⣷⣄⣀⣔⠽⣄⡀⡇⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠄⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠐⠛⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⣴⣦⡐⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉


⠀⢀⣀⠀⠀⠐⠈⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢉⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⡀⡒⠔⢓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤
⠀⠄⠀⠈⠉⠛⠫⣭⠕⠈⠃⠀⠀⠀⠀⠀⡠⠠⠀⢀⠠⠀⠒⠁⣔⣂⡄⠀⠀⠀⠀⠀⢰⠀⠀⡐⠁⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⢖⣾⡁⠤⠶⠶⣶⠾⡆
⢰⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⢰⠔⠊⢀⡤⠈⣤⣚⠁⠀⠀⠀⠀⠀⠀⠀⢀⠑⠸⡷⠀⠀⠀⠓⠶⠀⠀⠀⠀⠀⠀⢰⡾⣼⡇⠰⢖⣀⣶⣤⡇
⠈⠂⠄⣀⣀⡀⢀⠖⠒⠚⠁⠀⠀⠀⠀⠀⠶⠞⠀⠮⣀⣐⣀⡀⠩⠆⠀⠀⠀⠀⠀⠀⠘⢦⡀⠓⠄⠔⠀⠴⠀⠀⠀⠀⠀⠀⠀⠘⠓⠻⠯⠤⣼⣿⠇
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡂⣀⣀
⠀⠀⠀⠀⠀⠴⢩⣁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡤⣤⣄⡄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡆⠋⣃⠠⠁
⡖⠈⠍⠉⠉⠉⢉⡉⠀⠒⡎⠀⠀⠀⠀⠀⠀⠐⣶⠆⠖⠠⡀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠣⣀⣁⠀⣇⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠤⠤⠤⠹⢆
⡇⠀⠀⠀⢠⠞⢉⣠⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⢦⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⣍⣀⠑⠀⠀⠀⠀⠀⠀⠀⢘⠀⡀⢀⢀⠀⡨⢈⠂
⠠⣀⣀⣀⣈⣀⣉⣀⣠⠠⠃⠀⠀⠀⠀⠀⢠⡘⣿⣬⣿⡝⠳⢶⣟⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⣄⣀⣠⣴⢉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠻⠟⠻⠛⠻⠛⠞⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣋⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠚⠙⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠆⠉⠉⠉⠉⠉⠉⠰⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠚⠛⠛⠛⠛⠛⠒⠀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣀⣤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠉⠻⠿⣶⣶⣶⣍⡉⠀⠀⠀⢀⣤⣤⣄⠏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠀⠀⢀⣤⣼⣿⣿⡟⠛⠓⠀⠛⠉⢀⠉⠐⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⣠⣾⣿⣿⣿⡄⠀⠀⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠙⡄⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣤⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢰⣾⡙⢣⡀⠀⠀⠈⣿⣿⢀⡿⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢏⣉⣙⡻⢿⡿⠀⠀⠀⠀⠀⠸⣿⡷⢨⡀⠀⠀⠀⠹⣿⣯⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⢏⣿⢿⣆⠁⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⣼⠏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠁⠀⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡤⠚⣓⣶⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠂⠚⢻⣿⠏⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠶⠖⠈⠉⠉⠁⠀⠀⠀⣿⠀⠀⠀⠉⠲⠦⣄⣀⣀⣀⣀⣠⡀⠶⠋
⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡈⠙⣄
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣀⣤⣄⣠⡞⠛⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣈⣳⣀⣤⣤⡀
⠀⠀⣠⠏⠀⠀⠀⠀⢀⣠⠴⣿⣿⣿⣿⣇⡇⠀⠀⠙⣿⣿⣿⣿⣦⣄⣀⡀⢀⣀⡠⢿⣿⣿⠹⣏⠻⣿⣿⣿⣄⡀
⠀⢠⠋⠀⠀⠀⢀⡴⠋⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠈⠻⣿⣿⣿⣧⠉⠉⠀⠉⠀⠀⣿⣿⡀⠘⣦⢹⣿⣿⣿⡙⢦⡀
⢀⡎⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠀⠀⠀⠀⢹⣿⡇⠀⠈⢎⣿⣿⣿⣷⠀⠹⣄
⠘⠃⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⢿⣿⣿⣿⡆⠀⠘⣄
⡇⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⠀⠀⠀⠸⣿⡇⠀⠀⠀⣠⣿⣿⣿⡇⠀⠀⠈⡆
⡇⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⠀⠀⣿⡇⠀⣠⠞⠁⢸⣿⣿⣇⠀⠀⠀⢰⡄
⡇⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠉⠉⠉⠉⠁⠲⠶⠶⢠⡀⠀⠹⣿⣇⠀⠀⣿⠃⢸⠁⠀⠀⠀⣿⣿⡿⠀⠀⠀⠈⣧
⡇⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀⣀⠿⢿⣶⢼⣿⣋⣈⣿⣄⠀⠀⣿⣿⡇⠀⠀⠀⠀⠙⡀
⢁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⣁⣠⣤⡔⠚⠛⠃⠉⠀⠀⠀⠘⠃⠀⢹⣿⡇⠀⠀⠀⠀⠀⡇
⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⡅⠀⡾⢿⡿⠿⠓⠀⠀⣀⣀⣀⠀⠀⠀⣧⠀⢸⣿⠇⠀⠀⠀⠀⠀⢣
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⢠⡀⠀⠀⠀⢰⣿⣿⣿⣿⣧⡀⠀⡉⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⠇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠘
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⣼⣤⣤⠤⠒⢿⣿⠋⠀⠀⠀⢹⣷⠀⣿
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⢀⣼⡟⢠⡏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠉⠀⢸⠇
⠈⠓⣶⡦⢀⣀⣀⡠⠴⡶⠚⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠆⢀⣀⣀⡀⢴⠶⠊
⠀⠀⣿⡇⠀⠀⠀⠀⠀⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢸
⠀⠀⢹⡇⠀⠀⠀⠀⠀⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢸
⠀⠀⠘⡇⠀⠀⠀⠀⠀⣷⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⣟⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢀⠃
⠀⠀⢸⡇⠀⠀⠀⠀⠀⣇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⢀⠋⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡆⠀⠀⠀⠀⢸
⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⣶
⣿⣿⣿⡟⠋⠉⠛⠋⠙⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣴⣦⣴⣶⣶⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡛⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠠⡀
⠀⠀⠀⣧⣤⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠠⠀⠉⡌⠄⠀⠀⠀⠀⢀⡤⢒⡆⢲⡞⡉⠳⣄
⠀⠀⠁⠁⠒⠺⠁⡃⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠔⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠑⢤⣊⣤⢞⣝⣭⠓⢾
⠀⠀⡂⡰⢒⠒⡀⠃⠀⠀⠀⠀⠀⠀⠀⢀⢡⠠⠶⣄⠈⡢⠄⠀⠀⠀⠀⠈⢌⠠⡀⣀⣀⠄⠣⠂⠀⠀⠀⠀⠀⠀⠙⢦⣉⠐⠚⠉⠁
⠀⠀⠐⠶⠞⠶⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠒⠈


⢠⣤⠀⡀⠤⠠⢀⠀⠠⡀⠀⠀⠀⠀⠀⡤⠶⠶⠾⠯⢉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠂⠋⡆
⠈⡙⠊⠀⠐⢆⢶⠂⢠⠁⠀⠀⠀⠀⠀⡇⠀⡤⢤⡀⠀⡇⠀⠀⠀⠀⠀⢰⣛⠉⣡⠶⣄⣛⡻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡃
⠀⡅⢄⠣⣥⢧⠇⡀⢸⠀⠀⠀⠀⠀⠀⠇⠀⣖⣂⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠐⠓⠀⡁
⠀⠃⠀⠑⠒⠂⠈⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠄⠈⠉⠮⠁⠘⠓⠃⠀⠀⠀⠀⠀⠐⠀⠀⣀⣀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠤⠀⠄⣀⣀⡬⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠀⡀⠈⠄⡍⠉⢹⢆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣒⣐⡄
⠀⠀⠀⠀⠀⠁⢲⠈⠀⠀⠀⠀⠀⠀⢂⡀⠄⠀⠀⢀⡴⠃⠃⠀⠀⠀⠀⠀⠣⡀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠖⠆⠉⠉⠙⠋⠁
⠠⢀⠀⠀⠀⠀⠄⠀⠤⠀⠀⠀⠀⢀⣬⡄⢰⠉⠉⠉⠒⢀⠀⠀⠀⠀⠀⠀⢦⠀⠈⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠢⠤⢀⡀⠠⡄⠈⠁
⠀⠀⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠄⢄
⡀⠠⠤⠤⠜⠶⠖⢁⢐⠄⠀⠀⠀⠀⠀⣀⡐⢢⠀⠊⠱⣵⠀⠀⠀⠀⠀⢀⠔⠲⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⡇⠀⠀⠀⣠⠔⢂⠀⠈⠀⠀⠀⠀⠀⠀⣼⡇⠀⠈⣁⣀⢉⠀⠀⠀⠀⠀⠀⠁⠈⠀⡋⠳⠆⠢⡄⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠀⡂⡄
⢁⠀⠀⠀⠁⠚⠉⣀⡸⠀⠀⠀⠀⠀⠀⠻⣗⣶⡶⠥⣭⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⡄⢣⣀⠀⣀⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⠿⠻⠟⠶⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⢤⣋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⢴⠉⠀⠀⢋⣁⣀⡀⠸⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠤⠀⠀⠸⠀⠀⢀⠌⣉⢉⠉⡏⠉⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠐⠒⠃⠀⠂⠈⠉⠁⠘⠂⠠⠦⠆⠒⠒⠒⠒⣒⡀⠀⠈⠉⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠤⠤⢤⠒⠂⠀⢙⠀⠀⠀⠀⡇⠀⠀⣀⣑⣒⣈⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠴⠀⠀⠀⠐⠒⠒⠒⠊⠁⠉⣉⣉⣉⣁⣀⣀⡀⠀⢀⣀⣀⣉⣉⠒⠢⢄⡀⠀⠀⠀⠀⠀⠐⠀⠈
⠈⠡⠀⠐⠘⠀⡠⠂⠤⡠⢁⠔⢒⡲⠒⢭⢍⠉⠉⢰⠂⢠⠔⢈⠁⠀⣀⡠⣊⠄⠀⠄⢭⡄⢡⢀⢠⢄⠉⣉⠙⠢⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠢⠤⢤⠃⡇⠌⢀⠔⠠⠤⣀⣉⢆⠇⢠⠃⠎⢁⡀⢀⡈⢶⠀⣀⣀⢸⠀⣁⠘⡏⠠⢊⠠⠄⢌⠑⠜⡄⢱⡄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡈⢸⢴⡀⢁⣀⢠⡆⡜⠃⢺⠀⠘⢠⠊⠁⠀⣀⡉⢛⠉⠉⠛⢺⠚⣹⠀⣤⠻⢁⡀⠄⣀⠡⡐⠸⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢙⣖⠵⠄⠺⢄⠤⡆⠀⠰⢀⠐⢌⠃⠈⡱⠃⠀⠠⠄⡼⣉⣼⡄⢀⢀⢁⠀⢘⣱⠯⣒⠨⣷⠛⠲⣆
⠀⠀⠀⠀⠀⠀⢀⣀⠇⠘⠒⠑⠣⠤⠤⠤⣧⣎⣱⠀⢸⢊⣷⠋⠀⠀⢉⠱⠦⠄⢰⠀⢸⠘⡂⢸⣀⣠⣒⡩⢒⣟⢒⣊⣥⠒⠐⠚⠛⠉⠉⠉⠉⠳⡀
⠀⠀⢀⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠽⣤⠀⡠⠤⠤⠒⠐⠚⣂⣉⡉⠉⠅⠀⠀⠀⠀⠀⠀⡄⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢓
⠀⠀⡞⠒⠒⠒⠒⠠⠤⠄⠀⠀⠀⠀⠀⡄⠀⠀⠈⠈⠁⠀⠀⠧⠀⠲⠀⠀⠀⠀⠢⠤⠤⠜⠀⠐⠀⠀⠀⠀⠀⠀⠁⠉⠀⣇⣀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⢡⢐⡭⢆⢀⢀⠀⠀⡀⠀⡠⠤⡀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⢊⡥⠖⠒⠲⢬⡑⢄⠀⠀⠀⠘
⠀⠀⠀⡧⢂⡾⢈⠁⠀⠀⠀⠈⠞⢁⠸⡇⠀⠀⠀⠀⠀⡠⢚⡽⠋⠉⠉⠉⠉⠓⢭⢢⡀⠀⠀⠀⡇⠀⠀⠀⠀⠌⡴⠁⠀⢀⣀⡀⠀⠱⡘⡖⠒⠉⠉⠑⠆
⠀⠀⠀⠘⣀⣀⠈⠀⠀⠂⠀⠀⠐⠐⠁⢻⠀⠀⠀⠀⠰⢱⠋⠀⠀⡀⠀⣄⠀⠀⠀⢣⢳⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡜⠁⠀⢱⠀⠀⡇⡇⠀⠀⠀⡠⠂
⠀⠀⠀⠞⠁⠀⠉⠑⠒⠢⠤⠄⢀⣀⣀⠸⠤⠄⢀⠀⡇⡏⠀⠀⡜⠀⠀⢀⠃⠀⠀⢸⢸⠀⣀⣀⣰⣀⣀⠀⠤⡼⡄⠀⠙⠦⠴⠋⠀⠀⡗⠷⠖⠒⠊
⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠁⢧⠀⠀⠘⠦⠄⠊⠀⠀⠀⡞⡎⠉⠈⠁⠀⠀⠀⢀⣀⣰⠽⢤⡀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠁⠐⠒⠤⠤⠄⠀⣀⣧⣀⡀⣀⣀⣠⠤⠚⠉⢦⡀⠀⠀⠀⠀⠀⣠⠞⠛⠒⠒⠒⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄⢀⠊⠀⠐⠦⡀⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣈⠉⠙⠓⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠤⠀⠀⠛⡀⢠⠇⠀⠀⠂⠂⠼⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠁⠀⠈⠉⠙⠓⠲⠶⢤⣄⣀⡉⠉⠛⠒⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠤⠤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡠⠁⠀⠐⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠱⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢸⡤⣤⢀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⡈⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠋⠔⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠈⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠐⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠠⠀⡀⠀⠀⠓⠐⠁⠀⠀⠀⠀⠀⠒⡌⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣴⠟⢻⣧⢀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠙⠷⠾⣿⡾⠃⢠⡞⢻⡗⣴⠟⠷⣰⠟⣿⠇⣺⠟⢃⡾⢠⡞⢻⡆⣸⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠟⠁⠀⠘⠛⠛⠓⠙⠓⠒⠛⠛⠛⠒⠛⠓⠘⠓⠚⠛⠛⠑⠋⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠄⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡂⠀⠀⣀⠊⠀⠈⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠀⠀⠇⠓⠠⠄⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠦⠀⠀⠰⠀⠀⠀⠒⠀⢤⡄
⠀⠀⠀⠀⠀⣀⣉⡂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠟⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠈⠇
⠀⠀⠀⠘⠿⠟⠛⠃⠐⠿⡷⠆⠀⠀⠁⠀⠀⡀⠀⠀⠀⢀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⣴⣶⣄⡀⠀⠀⢀⣠⣶⣦⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⢀
⠀⠀⠈⠛⠿⠋⠀⠀⠈⠿⠟⠁⠠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⣿⠟⠃⣠⣈
⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠲⢠⡶⠀⡰⠠⡀⠀⠀⠠⠼⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡀⢸⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⠐⠚⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠃⠀⠠⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠇
⠀⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⣀⣍⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢐⣟⡀⠀⠀⠀⠀⢀
⡀⣁⠑⠀⠀⠀⠀⢠⣽⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠸⣅⠀⠀⢠⠀⣼
⠟⠋⣢⠈⠁⠀⠀⣘⠾⠑⠀⢀⣿⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠐⠟⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⢨⣿⠄
⣤⠄⢹⠃⠀⠀⠠⣍⣲⠛⡀⠘⠛⠛⠄⡀⠀⠀⠀⠀⠀⠀⡆⠉⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢲⡗⠀⠀⣾
⣧⣠⣾⠗⠘⠀⣘⣧⣿⢉⢄⠐⠀⠀⠀⠉⠀⠀⠀⠄⠀⠈⠛⠃⠀⠚⠚⠚⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⢶⡈⣠⣿
⣬⣤⠀⠀⠀⠐⣻⢃⣿⣸⠉⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣞⡿⢦⠀⠈⠉
⠁⢘⡀⠐⠒⠈⠛⠺⠒⠙⣿⣄⣀⣁⡀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁
⠋⢻⠉⡀⣀⡄⠁⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣥⡄
⠒⠚⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈

⣀⠀⠁⠠⠀⢨
⠀⠀⠀⠀⠀⠀⠰⢦⠄⠀⠀⠀⠀⠀⠀⠀⠠⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⢤⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣀⣠⣄⣀⣀⡀⠈⠙⣦⠡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⡀⠀⢂⣠⣠⣤⣐⣲⣤⣤⣤⣤⣀⣀⣘⣷⣿⢉⣀⠀⠈⠩⠙⣳⣶⠋⠄⢂⣠⣤⠒⣠⣐⡠⠠⠀⠄⣒⣄⣂⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢐⠪⣢⠶⠶⠶⠿⠛⣫⣩⠟⠛⣲⣶⡿⠟⠉⠉⠭⠿⠿⠍⠩⠭⠭⠿⠿⠿⠿⣿⡿⠟⠓⣲⡮⠽⠟⢛⣛⣛⣛⠛⠫⢉⢳⡄
⠀⠀⠀⠀⠀⠀⢈⠰⣿⡀⡀⢀⡠⠞⠁⠀⠀⢸⡏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡏⠀⠀⢎⠀⠀⠀⡀⠈⡆⣯⠀⡗⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠁⣿⠀⡴⠁⠀⠀⠀⠇⠀⠘⢣⠀⠀⠀⢈⡄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡇⠀⢀⠀⠙⢆⡒⠋⠀⢷⡇⢰⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠰⡅⢻⠜⠁⠀⠀⠀⠀⠛⠛⠛⡟⠀⠀⠀⢀⠈⠁⠁⡰⠖⠀⢀⠀⠀⠀⠀⠀⠟⠉⠉⢺⡇⠀⠀⠀⠀⠀⠙⢄⣀⡜⣇⠀⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠠⢺⠀⡀⠂⠁⠂⠘⡀⠀⡠⠇⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠘⢄⠀⡠⠛⠑⠀⠀⠀⠀⠠⡐⠦⠀⠇⢻⠀⣿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠁⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠐⡝⠀⠀⠀⢸⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⢠⡀⣿⠀⠀⠁⣤⠆⠀⣼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠒⠚⠈⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣿⠀⠀⠈⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⣿⠠⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠄⠠⠤⠄⡀⢨⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢨⡀⠙⠊⠀⠂⠄⡀⡇⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⠁⠀⠀⠂⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠅⣁⣀⠌⠙⢺⠃⠃⡆⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣄⠀⠀⠀⠠⡀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠇⠀⠀⠓⠠⠀⠈⠁⠒⢿⠆⢀⠀⡇⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣯⠀⠣⡀⠀⣠⠶⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠠⠀⠀⠀⡀⠀⠀⡀⠀⢡⠀⠀⠀⠀⢀⢸⠀⡇⠈⡇⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡀⣿⢀⣰⠈⢦⡀⠀⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣠⡄⢣⡘⠆⠄⠈⡇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠻⠶⠶⠶⣶⡿⠶⠶⣤⢤⣼⠷⠤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⠀⠀⡠⠂⢐⡦⠀⣠⣿⡶⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠻⠁⠦⠄⠐⠒⠂⠉⠀⠘⠆⠠⠁⠀⠈⠉⠉⠉⠩⠉⠙⠛⠛⠛⠛⠙⠉⠝⠓⠲⠶⠟⠛⠓⠿⠤⣤⣤⣤⡼⠋⠂⠈⠗⠀⡇
⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠀⠀⠠⠬⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠀⠈⠉⠉⠉⣩⠸⣶⣶⡤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⡀⠠⠶⠀⠀⢤⣰⠞⠙⠫⢴⣷⣶⣿⣖⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣁⣀⣀⠀⣀⡀⢸⣿⣶⠆⠒⢀⣴⣟⣜⢢⡀⠀⠈⠙⠛⠿⣷⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⠛⠫⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣠⣾⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣿⡤⠀⢈⣿⡁⢉⠢⡀⠀⠀⠀⠀⠉⠐⠬⢦⡀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠂⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠂⣀⠛⠿⠋⠀⠰⣎⠢⡀⠀⠀⠀⠀⠀⠀⠈⢑⡶⠈⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⢰⠡⠛⠉⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⣘⣿⣧⣴⡀⠀⠀⠙⠓⠈⠢⡀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⡿⠿⢷⡀⠀⠀⠀⠀⠐⢌⠢⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠤⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⢁⠎⠀⠀⠀⠀⠀⠀⢀⠔⠁⠰⡀
⠀⠀⠀⠀⡀⠤⠃⠉⠀⠀⠐⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠱
⣠⠤⠂⠁⠤⠤⠤⠤⠴⠶⠤⠰⠄⠹⢿⣿⣿⣿⠟⠉⠉⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⣠⣦⡡⡀⠀⠀⠀⠀⠀⢡
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡟⢿⣯⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠒⠠⠔⢁⡀⠀⠀⢀⣴⣾⣿⣿⣿⣬⠢⡀⠀⠀⠀⠀⢣
⡇⠀⠀⠀⢀⣤⡴⢶⢶⣤⣀⠀⠀⠀⡇⡇⠀⠀⠉⠙⢶⡀⠀⣀⣀⣀⠔⠁⠀⠀⢀⢎⣴⡿⠃⡠⠒⠉⠻⣿⠿⠿⢿⡇⠀⢈⠢⡀⠀⠀⠀⢃
⡇⠀⠀⡴⠯⣾⠥⣾⠤⢷⡬⢷⡀⠀⡇⡇⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣄⢒⠒⢿⡫⠒⠉⠇⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠑⠌⠢⡀⠀⠀⢂
⡇⠀⢸⣇⣀⣇⣀⣿⣀⣸⣇⣀⡇⠀⡇⡗⡄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡠⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠐⢌⡢⡀
⡇⠀⠘⣇⠀⣇⠀⣿⠀⢸⡇⢀⡇⠀⡇⣇⠘⡄⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⡿⡀⡠⠊⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡠
⡇⠀⠀⠙⢯⣹⣍⣿⢉⣟⣩⠏⠀⠀⡇⣟⣠⠴⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⠀⠉⠀⠀⠀⠀⠀⢀⠔⠁
⡇⠀⢀⣀⣀⣈⣉⣛⣋⣉⣀⣀⡀⠀⡇⡏⠔⠢⡘⡄⠀⠀⠀⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁
⡇⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⡇⣿⡌⠄⠡⠘⡄⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡜⠤⣃⠬⢀⣀⢀⠘⣿⢀⣀⡀⠤⠄⠒⠉
⡇⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠇⣏⠥⠒⠉
⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⡄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⢀⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡃⠀⠀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⠀⠀⠀⠸⠀⠀⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠄⠠⠤⠤⠄⠠⠤⠤⢄⡂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠨⡴⠁⢀⣀⡀⠀⢀⣈⣁⣀⣀⣀⠇⢣
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠘⠀⠐⠀⠀⠀⢠⡇⠓⡆⠀⠃⠀⠀⠀⢸⠀⠀⠤⠤⠀⠐⣸⡀⠀⠀⠆⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⡘⣅⣄⡘⠦⢤⣤⡤⠖⠃⠀⠀⠀⠀⠀⢀⠋⡇⠀⢀⠀⠀⠀⠃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠈⠀⠐⢠⢂⠀⡇⠀⠀⠉⠉⠉⠓⡖⠒⠚⢛⠋⠉⠉⠉⠀⠀⠰⠀⡀⡆⠀⡠⠀⢄⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠐⠲⠶⠶⠾⠷⠶⠶⠶⠶⠶⠒⠶⠖⠶⠒⠖⠃⠀⠀⠃⠀⠁⠀⠈⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⣀⣀⡞⠀⠀⠀⢀⣀⠀⣀⡀⢆⣀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⢹⡄⠀⠀⠀⣀⣤⣤⣦⣤⣾⣟⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢱⣿⠀⣾⠀⣠⡀⠸⣋⡉⠀⣀⣀⣿⠏⢹⣿⣋⠀⡀⠀⠶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⢸⣿⣾⣷⡶⣿⢡⣾⡏⢠⢸⡟⢹⣿⠀⣿⡇⢘⢩⣿⢸⡿⢠⣿⡴⣹⡇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠰⣦⣄⣀⣀⣠⠤⠿⡇⣿⣧⡛⠾⠙⠧⠻⠜⠳⠞⠿⠚⣿⠇⠄⠸⠧⠚⠳⢼⣿⠗⠿⠋⠛⠒⠶⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡠⠂⢤⡀⠀⠀⠀⠀⢀⡠⠔⠂⢉⠀⠀⠉⠑⢢⡀⠀⠀⠀⠀⠀⢀⠔⠊⠉⡁⠀⠡⡐⢀⠀⠀⢀⡀⠀⠀⢀⠔⠈⠣⠄⠠⠜⠁⠢⡀
⠀⠀⠣⡀⠀⠈⠐⠂⢀⠍⢠⠀⢁⠚⠥⠈⠀⠁⠈⠁⠀⠕⠀⠀⠀⢀⠃⠀⠀⠀⠁⠀⠀⠀⠑⢆⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⢀⠜⠂⣈⠕⠀⠈⠀⡡⠐⠒⠢⢄⠀⠀⠀⠀⡀⣀⠜⠁⠀⠀⢸⠠⠈⢈⡈⡀⠀⣠⠀⠔⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠘⠒⠫⠤⠄⠄⡤⠊⢀⠀⢀⡠⠚⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⢳⠀⠀⡆⠁⢀⠀⡘⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠮⠤⠴⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣢⡈⢲⡰⠁⠀⢘⠖⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⣀⡀⡠⠔⠢⢄⠀⠀⠀⠀⠁⠈⠀⠀⠈⠀⣀⠀⠀⠀⣀⠀⠀⠈⠒⣄⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠇⢯⡱⢸⠒⠀⠀⠆⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠠⡤⠴⠎⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⢄⡠⠃⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⢀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠛⠁⠀⣠⠄⠒⠒⡤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠆⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠠⠁⣠⡀⡄⠀⢡⣧⠀⠀⠀⢨⠉⠉⡇
⠀⠀⠀⡀⠀⠐⠚⠀⠀⠐⠻⣄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⢰⠃⠀⣙⡷⠃⠀⣿⠇⠀⠀⠀⠈⠳⠞
⠀⠀⢰⠁⠀⠀⠀⠀⠠⡄⢠⠀⠃⠀⠑⠐⠊⠉⠉⠐⠢⠤⠀⠤⠔⠊⠀⠀⢀⠀⠀⠀⠇⠀⠐⠇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠁⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⢸⠂⣠⣀⡀⠀⠀⣼⠇⠀⠀⠀⠈⠁⠀⣀
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠲⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡈⠈⠁⢐⡏⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⡧⠀⠐⠒⠒⠀⢤⢄⠈⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠂⠚⠉⠢⣄⠀⠀⠀⠀⠀⠙⠶⠶⢥⣤⣤⡾⠁⠀⠀⠀⠀⡚⠓
⠀⠀⣇⠀⠀⠈⠀⠂⠈⣾⠘⠃⡅⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⣄⣴⠟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⡀⣠⣖⠠⠀⠐⠀⣷
⠀⠀⠉⡄⠀⠀⠀⠀⠀⠙⠐⡀⡇⠀⠀⠀⠊⠀⠀⠀⠀⢰⠃⠙⠋⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⢠⡄⠀⠀⠀⠀⠄⠀⢀⢸⣿⣿⡇⠀⠀⢸⡅
⠀⠀⠀⠲⠶⢤⠤⠤⡾⠡⡾⠐⠁⠀⠀⠀⠀⠀⠀⣀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠈⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⠛⠶⠶⠶⢴⣄⡺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⣉⠀⢀⡂⠀⠀⠀⠁
⠀⠀⡀⠀⣀⠠⠖⠒⠊⠱⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠁⠀⠠⠔⠒⠂⡄⠀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⣤⣀⣀⣀⡀⢈⠀⠀⠀⠀⠀⠀⠀⢀⢄⠤⠄⠀⠀⠄⠀⠀⠰⠂
⠀⠀⢸⡀⠀⠀⠒⠡⡐⠀⠰⡀⠀⠀⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⡄⠕⠁⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠈⡃⠀⠐⠂⠚⠁⠀⠀⡁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⡈⠀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡀⡀⠀⠀⠀⡧⡀
⠀⠀⠀⢣⠀⣤⠴⠖⠚⠀⠀⡁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣄⢣⣤⠇⠀⠀⠀⠀⠀⡀⠀⠁⠀⠰⡀⠀⠀⠀⠀⠂⠀⠒⠊⠁⠈⠒⢄
⠀⠀⠀⠈⠆⡀⠀⠠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠙⠋⠘⠻⠂⠀⠀⠀⠈⠛⠀⠀⠀⠀⠈⠀⠀⠉⠉⠐⠲⠀⡀⠤⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠚⣷⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⠙⢆⣸⠆⣰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⢀⣁⣀⣤⣰⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣄⢀⡀⢀⢀⢂⠀⠀⠀⠀⠀⠀⠙⠁⠂⠈⠉⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠠⠂⠋⠁⠈⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠃⠘⠱⠟⠛⡀⠀⠀⠀⠀⠀⢀⠀⠑⠀⠊⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠐⠁⠺⡄⠀⣄⠀⠛⠀⢀⡌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢋⠩⠛⠻⠆⣰⠉⠀⠀⠀⠀⠀⠀⠠⡀⠰⠇⠀⢀⠜⣀⢀
⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢨⠓⠂⠒⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⠠⠀⣧⠀⠀⢰⠒⠂⠈⠉⠉⠉⠐⠒⠚⠓⣆⠌⠀⠙⡦
⠇⡤⠘⡀⠀⣀⡀⠤⠴⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠄⡒⠚⠚⠀⠀⠀⠀⣰⠞⠠⠀⡀⠄⠀⠠⠃⠀⠀⠀⠈⡀⠂⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠐⡀⡉⠀⠀⠀⠀⠀⠀⢄⡀⠚⠀⠀⠀⢠⠈⣢⢠⣀⣦⠀⣀⠃⢄⠀⠀⠀⠀⠅⢄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠤⠤⣠⠀⢠⠀⠀⠰⢼⢠⡆⣈⣛⠏⣸⠦⢽⠃⣴⠀⠀⡗⢒⣻⠀⠐⠀⠀⠀⡜⠐⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠄⢧⡜⠀⠀⠀⠇⠤⣀⣀⡀⠀⠑⠚⠉⠀⠈⠀⠀⠇⡰⠿⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⣆⠀⠀⠀⣀⣀⡀⠤⠤⠤⠤⠤⢤⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠑⠠⠄⣀⡀⠀⠀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢄⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣄⣀⣀⣀⣠⣤⠐⠈⠀⠐⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢀⣀⣀⠀⢸⠀⠀⠀⠀⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠘⠀⠀⠀⠀⠀⠄⠀⠉⠈⠀⠄⣀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⡋⢉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠠⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⢘⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠜⢲⠀⠀⢀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡘⠀⠀⠀⢠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⢂⢀⠐⠈⠀⠀⠀⠀⡔⢒⢀⢒⣤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠁

⠀⠀⠀⡄⠀⠀⢠⠀⡀⣀⣅⠀⠀⠠⠀⡐⠤⠴⠘⠀⠄⠙⡀⠁⠐⠸⠤⠤⠀⠷⠩⠔⠁⠸⠄⠀⠘⡄⠀⠀⠂⠐⠈⠁⡄
⠁⠀⢠⠠⠉⢄⠄⢂⠀⠻⠅⠂⠈⠂⠀⠂⠀⠀⠀⠀⣔⡀⡡⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⡄⠀⠀⠠⠐⠂
⠀⡀⠀⠀⠀⠈⠀⠀⠰⠰⡂⠈⠀⠀⠂⠛⡍⠈⢠⠀⠀⠀⠣⢀⠀⠀⢃⠘⠀⠢⠀⠀⠀⠀⣀⡀⠇⠀⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⠠⠈⠣⠐⠀⠀⠒⠀⠀⠉⣀⠀⠠⠉⠉⠤⠄⠠⠊⠀⠁⠂⠀⢈⠤⠀⠀⠀⠀⠀⠨⠀⢃⠐
⠀⡀⠂⠀⠂⠐⠀⠃⠀⠀⢀⠀⠒⠀⢊⠀⢀⢀⣾⠆⢧⣀⣒⠠⠤⠄⠄⠀⠠⠀⡈⣀⠀⠁⠨⠂⠀⠠⠈⠀⣀⣀⠀⠂
⠸⠀⠀⠀⠀⠀⠀⠲⡄⠈⠀⠁⡀⠀⠀⠀⠀⠈⠉⠁⠀⠠⠄⠐⠀⠐⠂⠀⠀⠄⠐⠐⠂⠀⡀⠀⠀⠀⠀⠀⡷⠐⠃⣴⠖
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠐⠄⠃⠀⠴⢂⡈⠀⠀⣆⠀⠀⡡⢔⠀⠁⠀⠀⠀⡰⠁⠀⠄⠛⡄⠂⠀⠤⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠁⡔⠐⠒⠁⠀⠁⣲⠀⠀⠀⠀⡘⠟⠁⠠⡔⠀⠀⢹⠀⠀⠀⠀⠀⠁⠐⠠⠭⣠⣀⠈⠁⢠⠂⢚⡆⠢
⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠑⡀⠀⠄⠑⣁⠀⡀⠀⢂⠄⠈⠄⠀⠠⠂⠀⣀⠀⠈⠀⠀⠉⠘⠄
⠈⠤⠄⠀⠰⠄⠀⠀⡆⠀⡀⠀⠀⢀⠀⠀⠀⠀⠉⠰⡀⠀⠀⢀⡀⠔⠘⡀⠀⠀⡦⠔⠀⠁⠀⡀⠀⠀⠀⠠⠠⢆⠃
⠂⠀⡄⠀⠀⠠⠀⠀⣤⠀⢁⢀⡀⠈⠉⠀⠀⠀⠉⢀⠀⠀⣀⠀⠀⠀⠀⡀⠀⢀⠁⠈⠀⠀⠀⡄⢀⠀⠈⠀⠀⢠⠖⡄
⠄⠀⠂⠀⠆⠠⠀⠸⠾⠀⠀⠂⠠⠀⣇⣀⠀⣶⠐⡠⠂⠀⡀⠀⠀⠢⠤⠌⡆⠀⢉⡄⣉⢀⡠⠐⡔⢈⡁⠀⠀⠨⠅
⠀⢀⡀⢀⠀⠀⡠⠄⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠁⢠⡉⠀⠀⠀⠀⠠⠠⢀
⠀⠄⠉⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠂⠀⠀⠀⣀⠔⡄⠀⡠⠰⠀⠀⢀⠀⠀⠀⠀⠐⠇⠠⡄⠀⠆⠘⠀⠀⡇⠈⢀
⣐⡀⠤⠤⠤⠀⠐⠢⢈⠐⠀⠀⡀⠀⠀⡀⠀⠂⣀⠁⠘⡇⠑⢀⡀⠀⠔⠦⠀⢀⠄⠀⠀⢄⠀⡀⠃⠠⠀⠀⠰⠄⠁⠀⠘
⡀⠀⠀⠄⠀⠤⠀⠁⠀⠂⠀⠀⠛⠀⠐⠀⠀⢀⠛⠋⠀⠀⠀⠀⡑⠀⢀⡀⠄⠎⠀⠀⠀⠀⠀⠁⡀⠂⠀⠀⠨⠀⠃⠄
⡀⠀⠀⢐⠄⠀⠈⠀⢀⡀⠁⠀⠂⠂⠀⠈⠁⠈⠀⢀⠀⠀⠀⠀⠀⣄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠇⠁⡀⠂
⡀⠀⠔⠁⠀⡀⠀⠸⡅⢂⠀⠀⠀⠠⠀⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠈⢠
⠄⠀⠀⠀⠐⠀⡀⠀⠁⢠⠀⠈⠄⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠑⠈⡄⢀⡤⡀
⠀⠐⠊⠁⠀⠀⠀⠘⡄⠀⠉⢀⠈⡀⠠⡒⢈⡂⠀⢁⠐⠀⠀⠀⡀⠃⠀⠀⣀⠡⡤⠀⠀⠤⠀⠐⠀⠀⠀⠀⠑⠀⠘⠥⠂
⣀⠀⠀⠀⠀⠚⠄⠀⠐⠀⠀⠘⠉⠂⢀⠈⠀⠇⠀⠀⠉⠀⠀⢈⣄⠀⠀⠀⠒⠀⠀⠀⢂⡀⠀⠀⠀⠫⠄⠀⣰⡄⠀⢀⠰
⠀⡀⠀⡀⠠⠠⠄⠀⢸⠀⠸⢉⠀⠸⡃⠅⠀⠀⠀⣴⡧⠀⠰⠈⢋⠁⠀⠒⡄⠈⠀⠇⠈⠀⠀⣀⠀⠠⠤⠀⢄⡀⠘⠢
⠀⣂⡀⣀⣀⣀⣀⠀⢀⢀⣀⢀⣀⠀⢀⣀⣈⣀⣀⢉⡉⣀⢀⣢⡀⢀⣀⠀⢀⣀⣀⣠⠀⣐⣐⡃⠀⣀⢀⣀⠀⢐⣈⢂⡀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠀⠀⠀⣠⠎⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡙⢦
⠀⠀⠀⠀⠀⣤⡤⠤⠤⣄⠀⠀⠀⣴⠋⠁⠈⠙⢦⠀⠀⣴⡋⣋⣀⣈⣋⣀⠗⠀⠀⠿⠊⠀⡠⠦⣄⡀⢠⠞⣙⡒⢦⡀⠀⠀⣹⢾⣧⡀
⠀⠀⠀⠀⢸⢸⡇⠀⢸⠈⣧⠀⢸⡇⠄⠀⠀⠀⠈⡇⠀⠈⠉⠁⢀⠀⠀⠀⠀⠠⠀⠀⠀⢾⠀⣠⢤⠙⣫⠔⢃⡉⠛⡇⠀⠀⡇⠻⠃⡇
⠀⠀⠀⠀⢸⠞⡗⠀⢚⠐⣿⠀⠈⣷⠀⠀⠀⠀⣰⠃⠈⣀⣄⣀⡈⠀⠀⢀⣀⣠⠚⡟⠀⢸⡖⢁⣬⡷⠳⠾⡿⢉⠉⢷⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠈⡇⢷⠀⠘⠀⢻⠀⠀⠘⣗⠒⠒⣺⠃⠀⣞⣧⣿⠖⠛⡆⠀⠛⠶⣤⢸⠁⠀⠀⢷⠞⠀⡀⠀⢀⡴⠋⠉⠛⠀⠈⢧⣤⣬⠇
⠀⠀⠀⠀⠀⣇⢸⠀⠀⣀⢸⠀⠀⠀⠘⣦⢴⡇⠀⠀⢷⡛⡌⠉⢳⠇⠀⡀⠀⠈⠛⠀⠀⠀⠘⠶⠚⠉⠉⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢻⠈⡇⠘⠉⠸⡇⠈⠀⠀⣷⠴⡇⠀⠂⠀⠙⠛⠚⠁⠀⠀⠀⢀⣀⡀⠀⡀⠀⠀⣀⡀⠀⠀⡀⠀⠀⣠⣄⡜⠙⡀⠀⠐⣥⠇
⠀⠀⠀⠀⠀⢸⡇⢣⢈⣉⣁⡇⠀⠀⠀⠙⢚⣁⡀⣠⣴⣶⣤⡀⠀⣷⣄⠀⢹⣿⣧⢸⣿⣿⢶⣿⢿⡏⣼⣿⡇⠀⢷⣄⠀⢀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⣠⣤⣾⣿⣿⣷⣿⣿⣩⣿⡇⢸⣿⣿⣦⡈⢿⣿⣿⣿⠏⣾⣿⣦⢠⣿⣿⠀⠀⠀⠉⠓⠾⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⡤⢄⡀⠀⠰⣿⣿⣿⣿⣿⠉⠁⢿⣿⣿⡏⠀⣼⣿⣻⣿⣧⡘⣿⣿⠋⣸⣿⣯⡅⢸⣿⣏⠀⠀⠀⠀⣀⠀⠀⢀⡤⢤
⠀⠀⠀⠀⢀⠞⢁⣠⢤⡙⡄⠀⠉⠁⢸⣿⣿⡀⠀⢺⣿⣿⣿⡄⣿⣿⠟⢿⡿⠟⠘⠃⠀⠛⠛⠛⠃⠛⠿⠿⠿⠇⠀⠀⠀⠀⠀⠘⠲⠞⠁
⠀⠀⠀⠀⢸⡄⢸⡀⢀⡇⢹⠀⠀⠀⢸⣿⣿⡇⠀⠸⠿⠈⠟⠋⠛⠛⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡤⠤⠤⢤⣤⣤⣀⣿⡉⣉⠉⣳
⠀⠀⠀⠀⠀⢳⡀⠉⠉⠀⣼⠀⠀⠀⠸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⢿⠛⠀⠀⠀⣀⣼⠦⠤⢤⣤⣀⣀⠴⠒⠒⠻⣍⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠳⡄⠀⢀⠇⠀⠂⠀⣠⢚⡆⠀⠀⠀⢀⣠⣤⣄⣠⡤⢾⡄⠀⠀⢸⠆⠀⠀⠀⡇⣿⠁⠀⠀⠀⠀⡏⢠⠎⠉⠙⡌⣷⠤⡟
⠀⠀⠀⠀⠀⠀⠀⠱⣄⡞⠀⠀⠀⠀⠧⠚⠀⠀⡤⢴⡏⠀⠀⠀⢳⡀⠈⠛⠤⠔⠋⠀⠀⠀⠀⠳⣏⠀⠀⠐⠏⠇⢳⡘⠦⣤⠴⣡⠏⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠤⠤⣄⠀⠀⠀⠙⠉⢷⡀⠀⣀⡜⠀⠀⠀⠀⢠⠀⢀⣀⠀⠀⠀⠯⠬⢭⣙⣒⣒⣲⠽⠷⢶⣞⣁⣀⣸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣰⠏⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠉⠉⠁⠀⣶⣄⠀⠀⠀⠀⠀⠹⡷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠲⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠀⢰⠋⢉⡗⠀⢸⠀⠀⢠⠒⢦⠀⠀⠀⢹⠈⠳⠤⠤⢤⣤⣤⣹⣄⣑⣄⡀⠀⠀⠀⠀⠀⢠⠶⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⢀⡇⠸⣄⣠⠏⠀⠀⠈⠛⠋⠀⠀⢠⡯⠤⠶⠒⠶⠶⠦⠤⣤⠤⢤⣈⣉⣙⡷⠀⠀⠀⢻⡀⣀⡟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡜⣁⡤⠚⠉⠀⠀⠀⠤⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢠⣤⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⡄
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡤⠀⠀⠀⢠⠀⠀⠀⠀⠿⠿⠀⠸⠿⠀⠸⠿⠇⠀⠿⠇⠀⠿⠷⠀⠀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣶⡆⢰⣶
⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠸⠡⢤⠀⢰⣶⠀⢰⡖⠂⠀⣶⡆⠀⣾⣷⢴⡄⢀⠀⠀⢀⠀⢁⡀⠀⠀⠀⠀⠉⠀⠿⠟⠈⠉
⠀⠀⠀⢸⣿⠀⣿⡇⠀⢀⣠⠀⢚⣿⠇⢸⡿⠀⠒⠀⠙⠀⠈⠉⠀⠈⠉⠁⠀⠉⠁⠀⢹⣷⢰⡇⠰⡂⣠⣦⡰⣿⣿⣷⠀⠀⠀⣶⠀⢴⡆⢰⣦
⠀⠀⠀⠤⠤⠤⠤⠥⠄⢈⣿⠁⠸⡏⠀⠈⠀⠀⠰⠀⠀⠋⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢾⡗⢸⠇⠈⠀⠿⣿⣻⣿⣿⠛⠃⠀⠀⠉⠀⠈⠁⠈⠉
⠀⠀⠀⣤⣤⣤⣠⣤⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠃⠀⠀⡄⠰⠀⠠⠀⠀⠁⠀⢤⡀⣄⠀⠀⠁⠃⠂⢠⣿⣿⣿⠇⠐⠀⠀⣿⡇⢰⠂⣿⡇
⠀⠀⠀⠿⠿⠿⠿⠿⠠⢤⣧⣤⣀⣠⡹⠿⠆⠀⠀⠀⢀⡀⠀⠂⠚⡤⠀⠀⠀⢀⡀⠾⠇⠀⠀⠀⠒⠀⠀⢺⣿⣿⠿⠂⠀⠀⠀⠛⠃⠘⠃⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠐⣷⣾⣾⣿⣿⠁⠀⠀⠆⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⡇⠀⠀⣿⣿⢹⣿⡟⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣀⠸⠛⠃⠀⠛⠛⠀⠀⠀⠀⠀⡀⠄⠀⠘⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣀⣀⠀⠀⠠⠀⠀⠀⠂⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⣀⡄⠂⠀⠀⠐⠀⠀⠀⡊⠀⠀⢆⡄⠀⠀⡇⠀⠢⠤
⠀⠀⠀⠀⠀⠄⠈⠀⠁⡠⡀⠀⠋⠡⠠⠂⠀⡀⠂⠈⠈⠁⠐⢀⣀⡝⠥⠄⣠⣇⠀⠀⠠⠆
⠀⠀⠀⠀⠀⢀⠁⠶⠩⠀⣰⠈⠀⠐⢈⡁⠀⠄⠀⠈⠀⠀⠀⡉⣛⠚⢂⠀⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠠⠀⠈⢁⡈⠁⠨⠐⠂⠈⠍⠉⠭⠅⠩⠤⠀⡽⠁⠄⡠⡎⡅⠺⠏⢛⠓⠀⠔⠀⡀⠠⣠⡀
⠀⠀⠀⢠⠀⠀⠄⠈⢁⣠⠤⠐⢀⠑⠀⠁⠀⣠⡀⠀⠀⠀⠀⡄⢰⠅⢙⣋⠊⠉⠉⠫⡀⠈⠀⠉⠀⠉⠋
⠀⠀⢁⠀⢦⡀⠂⠡⠠⠧⠤⠄⠀⠁⠀⢁⡄⠉⠁⠀⠀⠀⠀⠁⢈⠀⠅⠿⠀⣀⠀⠱⡀⠀⢁
⠀⠀⠀⠐⠀⠈⢀⢰⠛⣻⣧⣃⢀⣚⢁⢈⠁⠐⠲⡉⠀⠈⠙⢓⣂⡖⢀⠀⠐⣩⠀⡲⠚⠊⠀⣀⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠠⣀⠀⢀⠀⠀⢨⠀⠜⠨⠄⢣⢠⠁⠈⠈⠁⢆⠀⠠⢰⠅⠀⠘⣤⣧⠃⢡
⠀⠀⠀⡀⠆⠀⠀⠆⠈⠂⠐⠠⡴⠢⠀⢀⡄⠂⠐⠘⠆⢄⠐⠨⠀⠀⠡⣤⠦⠴⠆⠋⠉⣡⣼⣻⡀
⠀⠀⡜⠀⠑⡀⠀⠑⢀⢀⠀⢸⠂⠀⠀⠈⠉⠱⡠⢀⢀⣼⡆⣁⣤⣶⣦⠹⡄⠐⠠⠄⢪⣄⠀⠀⠅
⠀⠀⠇⠀⠀⡏⢰⠌⠆⡚⣎⠠⠀⠀⠀⠀⠀⢀⡀⠐⠈⠊⠑⠘⣷⣙⡼⠀⠹⣄⠒⠀⢀⠉⠵⠖
⠀⠀⠀⠀⠀⠀⠘⡆⠆⢱⡶⢴⠀⣠⡄⣀⠈⠀⡀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠉⢆⢠⣀⣬⣱⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⢁⡄⢠⣯⠀⡇⠀⠂⠈⠰⠤⢀⠀⠀⠁⢀⣀⣠⡤⠴⠀⠉⠂⠁⠃⠃
⠀⠀⠀⠀⠄⠠⡞⠉⠁⠀⠻⡀⣸⠇⠂⠀⠀⡰⢀⠀⠀⠈⠀⠐⠀⠭⠁⠀⠀⢀⡀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠈⠀⠎⢘⡈⠉⠠⡤⠂⠀⠀⠄⠀⠀⠀⠀⠓⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠠⠤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⣀⣀⡀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡞⠉⠉⠈⠉⠉⠙⢲⡴⠁
⠀⠀⠀⠀⠀⢀⡠⠤⠤⠶⠽⣛⣹⠗⢛⣦⣼⡿⠛⠓⢶⣿⣷⡖⠒⠲⠶⣶⣶⣾⣷⣶⣶⠺⠥⣤⣼⠿⠷⠒⠒⠶⠲⠴⡒⠢⡀
⠀⠀⠀⠀⠀⢸⡀⡀⢀⡠⠞⢉⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣼⠋⠀⠰⡀⠀⠀⡀⠉⠆⣱⠀⡇
⠀⠀⠀⠀⠀⢸⠈⡔⠁⠀⠀⢨⠄⠀⠸⠆⠀⠀⠐⣀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠻⠀⠀⠀⠈⠢⡐⠜⠀⢠⣯⠠⡇
⠀⠀⠀⠀⠀⢸⠞⠁⠀⠀⠀⠘⠓⠒⢺⡇⠀⠀⢀⠀⠉⠉⡀⠖⠀⡀⠀⠀⠀⠀⠀⠋⠉⠉⣿⠀⠀⠀⠀⠀⠈⢦⡀⣸⢻⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠠⠀⢀⠼⠁⠀⠀⠀⠁⠀⠀⠀⠂⠀⠠⠀⠀⡀⠀⠘⡄⢀⠔⠋⠀⠀⠀⠀⡁⢎⠓⠆⢰⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠫⠃⡀⢸⢸⠀⢻
⠀⠀⠀⠀⠀⣼⠀⠀⠐⠶⠃⠀⡇⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠓⠁⠁⢸⢸⠀⠈
⠀⠀⠀⠀⠀⢿⠀⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠄⠀⠀⠀⠀⠈⠈⠀⡄
⠀⠀⠀⠀⠀⢸⠀⡂⠂⠀⠢⠀⠁⠀⠀⠀⠐⠀⠠⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⢀⢢⣄⠋⠀⡁⠂⣀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠁⠐⡀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠰⢥⠒⢀⡉⠃⡄⡇⠀⡇
⠀⠀⠀⠀⠀⠸⠦⡀⠀⠀⠑⡀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠲⠀⠀⠀⠀⠐⠠⠀⠀⠀⠈⠀⠀⢈⠈⠂⠀⠀⠈⠐⠇⠀⠁⡁⢠⡁
⠀⠀⠀⠀⠀⢰⠀⠐⢄⠠⠂⠉⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠰⢀⠀⢡⡄⠀⢸⠀⠀⠇⠀⠀⠀⠠⠐⠂⡇⠀⡇
⠀⠀⠀⠀⠀⢸⡀⢘⣀⢑⣄⣀⡃⠀⠀⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠜⠆⠈⠓⠀⠀⠀⡇⠀⢸⠄⠂⠀⠀⠀⠀⠀⠀⠀⣧⣀⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠈⠉⠁⠁⠐⠒⠂⠀⠠⠤⠄⠀⠀⠀⠠⠄⠠⠰⣇⣀⢸⣤⠄⢠⡔⠁⠀⠜⢀⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠁⢒⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⠀⢠⣎⣀⠀⠀⠀⠀⢀⠔⠉⠀⢛⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠀⢝⠂⠀⠀⠈⠙⢲⡀⢠⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡉⠉⠋⠀⠀⠀⠀⠀⠀⢱⣇⣀⡀⠀⠀⠀⡇⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⡀⡰⣖⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⠀⣸⠋⠑⢄⠀⡇⢀⠀⡾⠀⠀⠀⠀⠀⠀⠠⣀⡳⠜⠀⡏⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠈⢀⡀⠀⠀⠀⣰⠃⠀⠀⠈⠷⠋⡠⠈⠴⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣴⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⢤⠠⠒⣓⡄⢠⢀⠤⢡⠒⠢⢴⠃⠀⡠⠀⢀⡠⠊⡱⠷⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠈⠖⢥⠀⠀⢠⠃⠀⠀⠀⠑⡏⣀⠴⢋⠔⠙⠦⠤⠊⢀⡀⠀⠀⠀⢀⠔⢲⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢽⡖⠨⡷⠔⠂⠄⠀⠀⡾⠀⠀⠀⢀⣠⢼⡥⢚⡁⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⡠⠎⢳⣀⣠⠗⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠃⠠⡀⠔⠀⠐⢦⣀⣇⠀⡠⠞⠉⠀⢀⡟⡉⢁⡀⠀⠀⠒⠀⠀⠀⠢⡀
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡿⠀⠄⠉⠋⠀⠑⠀⡇⠀⣏⠀⠀⠀⠀⣼⠁⡀⠄⠩⡒⠀⠒⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠒
⠀⢀⡗⡁⠀⠀⠀⣀⣂⣀⣃⣤⣼⢥⣿⣿⣱⠁⣠⣨⣁⡖⡀⣉⣞⣀⣑⣤⡠⠾⠷⠏⠘⠚⣢⠈⠂⠀⢄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣳⡜⢻⡄
⠀⡌⠁⠁⢠⠀⣀⡤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠙⡖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⠉⠂
⠀⡃⠀⠘⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠇
⠀⠿⠲⢶⠃⠀⠀⣠⠔⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏
⡴⠛⠈⠚⠲⡀⢸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣠⣇⣀
⡇⠀⠀⠀⠀⡇⠀⠀⢠⠁⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡆⡴⠁⠀⠀⠈⡷⠀⠀⠀⠀⠀⢈⠀⠀⠃
⠘⠠⠄⣀⠴⣧⠃⠀⠘⢀⡀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⠀⡠⠀⢁⢧⠀⠀⠀⡰⠃⠰⠄⠉⢹⡂⠘⠆⠁⢀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠙⡤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠈⡏⠍⠉⣉⢱⠀⡔⠀⠀⠠⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠈⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣾⣳⣄⠀⠀⠀⠀⠀⠀⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣍⡿⢇⣼⢿⣟⠀⠀⠀⠀⠀⠀⢿⣶⣿⡆⠤⠤⢤⣄⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⢳⡺⠋⠀⢀⣀⠀⠀⠀⢸⣷⣽⡇⠀⠀⠀⠈⢳⡄⢠⣞⣱⠁
⠀⠀⠀⠀⣄⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢻⠁⢀⡔⠋⠙⢦⣀⣀⡸⢿⠾⠧⠤⣄⣀⡀⢀⣿⣏⣬⡍
⠀⠀⠀⠀⠈⠙⠳⠀⢿⣮⡛⠛⢶⣌⣃⠀⠀⠀⠀⠀⠀⢸⠀⠀⣁⡠⡝⢧⣞⢲⠖⠀⢀⣸⣯⡿⢋⡙⢼⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣵⡀⠒⠉⠑⢦⡀⠀⠀⠀⣼⡄⠃⠁⣀⣤⣰⠼⠿⠋⠈⢉⣠⢿⡥⠿⣿⣕⣦⡄⢀⠤⠒⠒⢿⡆⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢦⣄⣄⠸⣿⡷⢤⢾⣹⠗⠚⠘⠉⠉⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠻⠛⣿⡆⡜⠂⠶⣾⣯⣉⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⣚⡻⣡⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠇⢘⠀⠙⠛⢋⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠹⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡟⠰⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢀⣰⢢⣾⠃⠀⠀⠀⠀⢸⣿⢸⠸⣆⠐⢘⡉⠀⢸⣿⠠⠁⠀⠀⠀⠀⠀⢸⣤⡬⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⡨⣿⠀⠀⠀⠀⠀⠘⠋⠘⠀⠈⠀⠘⠓⠀⠈⠛⠀⠐⠀⠀⠀⠀⠀⠸⣯⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣻⠃⣿⠀⠀⠀⠀⠀⠐⣶⠀⣶⡄⢠⡀⠀⠀⠰⠖⢰⡆⠀⠀⠀⠀⠀⢸⣿⢩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣏⡦⢻⣆⠀⠀⠀⠀⠘⠿⠀⠟⠆⠋⠑⠀⠂⠸⠧⠸⠇⠀⠀⠀⠀⠀⣼⢿⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠏⠀⣀⣤⠠⢬⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡦⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣍⠉⠀⡠⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡼⠓⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀⠈⢁⢴⢛⢀⡠⠊⠀⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡻⣆⢹⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⡄⠁⢰⠞⠁⠀⣄⣴⡿⠛⡛⢷⣦⣄⣀⣀⡀⠀⢀⣀⣠⣴⣾⣿⡇⣸⢣⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠧⣬⣶⠀⠀⠀⢀⣹⡟⠁⢸⣳⠟⠀⠈⠉⠉⣭⡉⠉⠉⠙⢿⠉⣩⠴⣧⡞⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⠀⠘⠛⠀⠀⠳⠏⠀⠀⢀⡄⣠⡿⠑⠄⢀⠀⠘⠂⠈⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠃⡶⡒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠈⠉⠉⠉⠉⠁⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⡠⠤⠄⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣄⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠄⢀⣀⡀⠀⢰⣄
⠀⠀⠀⠀⠀⡜⠀⢸⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣬⠀⠳
⠀⠀⠀⠀⠀⢡⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡔
⠀⠀⠀⠀⠀⠀⠟⠋⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣤⢀⣀⡀⢤⠒⠉⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⡇⠀⠀⠀⢸⠉⠉⠲⣄⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⡆⠀⠀⠑⣄⠀⣠⠎⠀⠀⠀⡏⢢⠀⢨
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠑⣤
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⡀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⣌⠀⠀⣠⠞⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⢦⡀⢀⠱⡄
⠀⠀⠀⠀⠀⡄⠉⡟⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠹⡀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠹⡄⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠐⡀⠘⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠱⡀⢡
⠀⠤⢀⡀⢠⠂⠘⢢⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠀⢠⡞⠁⠈⢢
⢰⣄⠀⠘⣷⠀⠀⠀⡿⢶⠚⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣾⣾⣅⠀⢀⣬⡀
⢸⣿⣿⣾⠁⣷⣶⣎⡀⠙⠀⠀⣀⢀⡀⣠⣀⣄⣀⣀⠀⢀⣬⣀⣀⣾⣟⣁⣈⣁⣉⣹⣿⣄⠀⠀⠀⠀⣀⣀⣀
⢈⠀⠉⠓⠸⠿⠿⢿⠇⠀⠀⠀⡧⠤⢤⡬⣷⣤⡤⠤⠤⣤⠼⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠐⣿⣿⣿
⣼⣿⣷⡎⢠⣤⣤⣤⢰⡇⠀⠀⡇⠀⢰⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⣼⠋⠀⠈⠻⣆⠀⠀⠙⢿⣿
⠧⣉⠛⠃⠸⠿⠿⠿⠀⠀⠀⢀⠀⠀⢸⠀⠨⠀⡇⠀⠀⠀⠠⣞⠉⣀⡿⠆⠀⠀⢻⣄⠀⢀⣰⠏
⠀⠀⠉⠁⠒⠒⠒⠒⠚⠒⠂⠙⠀⠀⢸⠀⠀⠀⡇⠀⠀⠇⠀⠙⠋⠁⠀⠀⠀⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢠⠀⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢨⠀⣸⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⢽⣿⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠘⠿⡇⠀⣧⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠒⠚⠂⠚⣄⠀⢠⠓⠐⠛⠐⠒⠠⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣶⣶⣦⣶⣶⣦⣤⣬⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⣤⣦⡄
⠀⠀⠀⠙⢻⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣭⣥⣄⠀⠀⣘⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣭⣥⣴⣿⣿⣥⡀
⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣿⣿⣿⣻⣆⣾⣿⣿⣿⠻⡄⠀⠀⠀⠀⠀⠀⢠⡟⢹⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠚⢛⣿⡿⠸⡇⠀⢀⠀⠀⠀⠀⠀⠀⠰⠻⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠟⠲⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣹⡆
⠀⠀⠀⢀⣿⣿⡅⠀⢼⣾⣿⣷⡀⠀⠀⠀⠀⣾⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡆⠀⠀⠀⠀⣾⣦⢸⣿⣿⡿⣿⣿⡿⢧⣧
⠀⠀⠀⣸⡟⢻⣧⡀⠘⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⣿⡟⣽⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⢸⣿⣿⡇⣿⣿⡇⢸⣿
⠀⠀⠀⠻⠃⠀⠙⠟⠀⠨⠛⠉⠀⠀⠀⠀⠀⠛⠛⠛⠻⠛⠿⠛⠃⠻⠻⠛⠟⠛⠛⠛⠃⠀⠀⠀⠀⠉⠉⠸⠏⠻⠃⠻⠻⠃⠈⠉
⠀⠀⠀⣠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢠⡾⢻⣿⡟⣿⣷⡆⣀⠀⠀⠀⢀⡾⣽⣿⠇⣼⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣤⣀⣿⣿⠇
⠀⢻⣶⣶⣶⣾⣿⣷⣼⠇⠀⠀⣾⣷⣶⣶⡶⠋⠀⠀⠀⠀⠀⣀⣸⣿⡟⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢸⣿⣿⣿⣿⢿⣄⡀
⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠉⢸⣿⣿⡇⠀⠀⠀⠀⢠⣾⣿⣿⣦⣀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⢥⣿⣿⣧⣀⣈⣙⡿⠁
⠀⢸⣿⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⢿⣷⠀⠀⠀⠀⢿⣿⣷⣾⣻⣯⣅⣤⣤⣀⣼⠋⠀⠀⣠⠴⠒⢲⢮⠈⣿⣏⣿⣿⡽⡖⠒⢦
⠀⢸⡿⢸⡿⢿⣿⣿⠛⠀⠀⠀⠀⣸⡟⢸⣿⠀⠀⠀⠀⠀⣼⡟⠁⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⡏⠀⠰⠧⠬⡧⣿⡯⠏⣿⠀⠸⠄
⠀⣿⡇⢸⡇⢸⣿⣿⠀⠀⠀⠀⠀⣿⡇⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣠⡼⠃⠈⠀⠀⠘⢦⣀⣀⡴
⠀⠛⠁⠘⠃⠘⠋⠋⠀⠀⠀⠀⠀⠛⠁⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣼⣷⣮⣻⢿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣟⣾⣿⣿⣿⣿⣷⣝⣿⡿⠟⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠽⠟⠋⠙⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣹⣿⣯⠉⠉⣿⢽⣿⣿⣿⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢿⣿⠃
⠀⠀⠀⠀⣀⣀⣴⣏⣀⣠⣾⣿⣉⣀⣀⣿⣨⣿⣿⣅⣀⣠⣄⣹⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠺⣆⢺⣷⣄⣀⣀
⢶⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⠀⠀⢸⡄⠹⣮⠻⢿⣿⡿⠿⠿⢶⣶⣄
⢘⣿⣿⣿⡿⣫⣭⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣭⣭⡛⢿⣿⣿⣏⠀⠀⠀⣧⠀⠘⣷⠶⠷⠞⠻⣟⠷⣮⣅
⠛⠛⠛⠛⢸⣿⠉⣻⣷⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⣿⡏⢙⣿⠌⠛⠛⠛⠃⠀⠀⠹⠀⠼⠃⠀⠀⠀⠀⠘⠦⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠱⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⣤⠴⣤⡁⠂⢄⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⡤⢾⣦⡰⡟⠻⡉⠉⠙⢦⢌⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡕⠁⠈⠁⠈⠛⠒⣻⢏⠎⣂⢰⠀⠀⣼⠀⠑⡈⢂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠒⡄⠀⠀⠀⠀⠎⢀⠇⠯⡀⠘⠔⠁⠀⠀⠀⠈⡄⢂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡔⠀⠀⠀⠀⠀⠉⠀⠖⠺⠀⠀⠀⠀⠀⠸⣮⠿⡀⡀⢠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⢀⠀⠀⢄⠀⠀⢀⣀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢐⠵⠾⡇⠁
⠀⠀⠀⠀⡀⠨⠰⠉⠉⠉⣷⠨⠀⡀⠈⠓⢄⣇⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡄⡀⢀
⠀⠀⠀⠎⠇⠀⠀⠀⠀⠀⠀⠀⠀⡉⡄⠀⠀⠀⠙⠶⠒⠂⠀⢀⠀⠀⠀⠀⠀⠑⠂⢲⢁⠁⠈
⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⡄⠵⠤⠤⠤⠤⠤⡧⠾⠴⠓⠲⢟⣚⡛⢍⠉⠙⠛⠛⢋⠆
⠀⠀⢰⠀⠘⠠⡷⠀⠀⠀⠰⡧⠈⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡏⠈⢻⡆⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠘⣄⡀⠀⠀⠀⠀⠀⠄⠁⠰⢢⢀⠀⠀⠀⠀⠰⠀⠀⡠⢄⠙⠒⠚⠁⠃⠀⠀⠀⠀⡔
⠀⠀⡄⠀⡠⠐⠒⠒⠒⠒⠒⠒⠠⠀⠀⡷⢤⡁⡐⠂⠀⠒⠀⠒⣘⡜⡠⠀⠡⠄⠀⠬⠅⠀⠀⠈⢆⠀⠀⠠⠀⠀⠀⠀⠀⡠⠂
⣆⠀⠀⠀⠃⠤⠤⠤⠤⠤⠤⠤⠄⡆⠀⠂⢀⠧⠀⢀⡀⠀⠀⡀⠠⠖⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⢀⠦⠀⡖⠐⠀⠂⠈⠀⠠
⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠰⠠⠀⣠⠑⠶⣷⣴⡀⠀⠀⠀⠀⠀⡔
⠣⣘⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⡠⠂⠐⠒⠓⠒⠒⠓⠒⠀⢀⠀⠀⠀⠀⠀⢀⡤⠊⠈⠁⠐⠃⠀⡀
⣀⣈⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣌⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀⣎⣶⣶⣶⣦⣬⣤⣦⣤⣖⣒⣒⣚⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⣋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⠤⢄⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⢀⣦⣶⡀⠀⠀⠀⠠⠆
⠀⠀⠀⢁⢀⠘⠀⠀⠀⠀⣀⠓⢈⡀⢊⣀⠀⠀⠀⠀⡐⠀⠆⠀⠀⠀⠀⠠⠗⢀⠈⠑⠀⠀⠀⡰⠄⠀⠰
⠀⠘⠄⣀⣠⠔⠀⠀⠀⠀⠀⡠⢀⠁⣌⠀⠀⠀⠀⢌⣀⣤⣀⠀⠀⠀⠀⠀⢆⣼⠀⠀⠀⠀⠀⠨⠈⠈⡀
⠀⠀⠐⠛⠓⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢰⠙⠩⠉⠛⣄⠀⠀⠀⠀⢀⡡⠠⠆⠒⠁⠀⠀⠐⡈⠉⠉⠈⠂⠀⠀⠐⠀⡀⢉⠷⠂⠀⠀⠀⡄⠆⠐⢀
⠈⠑⠐⠘⡈⠘⠣⠀⠀⠀⢠⡀⣀⠐⠀⠀⠀⠀⢀⡆⠀⠀⣀⡐⠀⠀⢀⠀⢄⡠⠀⡀⠀⠀⠐⠀⠁⠛
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠚⠂
⠀⠀⠀⡀⡠⡀⠀⠀⠀⠀⠀⠄⠂⠀⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⢠
⠀⠰⠈⡈⠂⠌⠇⠀⠀⠀⠈⢀⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠑⢒⡢⠀⢠⠀⠄⡈⠀⡠⠀⠀⠀⢀⠂⠁⢡
⠀⡆⠄⠡⠀⢠⠎⠀⠀⠀⠀⠑⡦⡖⠁⠀⠀⠀⠼⣒⠶⠂⠁⠀⠀⠀⠀⢨⠁⠥⢈⡂⠀⠀⠀⠌⠄⡀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢄⠀⣀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠄⠠⡀⢠⠄⠀⠀⠀⠄⠀⠠⡀⠀⠀⠀⡀
⠄⠀⢘⢘⡁⠈⠀⠀⢰⠞⠒⠒⠒⠚⡆⠀⠀⠐⠘⠀⠀⠀⡎⠶⠆⠀⠀⠀⠀⣀⡄⠁⠀⠀⠐⠉⠉⠉⠑
⠀⠀⢰⠘⠒⠁⠀⠀⠈⢳⠒⠒⠒⢺⠃⠀⠀⠀⠀⠢⢀⡀⠔⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠐⠀⠀⠀⠘
⠀⠀⠈⠊
⠀⢀⡤⡄⢤⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠂⠀⠒⢄⠀⠀⠀⠀⡠⡖⠐⠂⢄⠀⠀⠀⢠⠊⠉⠑
⠀⣩⡁⣍⢈⣉⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠈⢢⠀⠀⡆⠀⠀⠸⠄⠁⠀⠂⠘⠀⠀⠀⠐⡄⠀⢀
⠀⠲⡄⡦⢰⠖⠀⠀⠀⠤⠤⠤⠤⠤⠃⠀⠀⠈⠢⠀⣀⠀⠔⠀⠀⠀⠈⠢⡀⠀⢀⠔⠀⠀⠀⠀⠈⢦⠃
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⡠⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠂
⠀⡤⢊⠄⠀⠀⠀⡀⠠⠀⠀⠀⡡
⠀⠄⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⠀⡆⠄
⠀⠠⠀⠀⠀⠀⠀⠈⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⣿⣿⣿⣿⣿⡇⡄⠀⢀⣶⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣿⣿⣿⣷⠁⠀⠀⢿⣿⣿⣿⣿⣿⡾
⠀⠀⠀⠙⠿⠿⠿⠛⠁⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡾⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠟⠃⠢⡤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⣀⣀⡀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⠉⠈⠉⠉⠙⢲⡴⠁
⠀⠀⠀⠀⠀⢀⡠⠤⠤⠾⠽⣛⣹⠗⢛⣦⣶⡿⠛⠓⢶⣾⣿⡖⠒⠲⠶⣶⣶⣾⣷⣶⣶⠺⠥⣤⣼⠿⠷⠒⠒⠶⠶⠶⡒⠢⡀
⠀⠀⠀⠀⠀⢸⡀⡀⢀⡠⠞⢉⠀⠀⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣼⠋⠀⠰⡀⠀⠀⣀⠉⡆⢱⠀⡇
⠀⠀⠀⠀⠀⢸⠈⡔⠁⠀⠀⢸⠀⠠⠸⠆⠀⠀⠐⣃⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠻⠆⠀⠀⠈⠢⡐⠜⠀⢰⣟⠠⡇
⠀⠀⠀⠀⠀⢸⠜⠁⠀⠀⠀⠘⠒⠒⢺⡇⠀⠀⢀⡀⠉⠉⡀⠒⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⣿⠀⠀⠀⠀⠀⠈⢦⡀⣸⢻⠀⡇
⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠂⠀⠀⢀⠼⠁⠀⠀⠀⠁⠈⠀⠀⠂⠀⠠⠀⠀⡀⠀⠘⡄⢀⠔⠋⠀⠀⠀⠀⡁⢂⠓⠂⢰⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠑⠫⠃⡀⢸⢸⠀⠋
⠀⠀⠀⠀⠀⡼⠀⠀⠘⠶⠃⠠⡇⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠁⢸⢸⠀⠸
⠀⠀⠀⠀⠀⢶⠀⠆⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⢠⠄⠀⠀⠀⠀⠘⠈⠀⡀
⠀⠀⠀⠀⠀⢸⠀⡂⠐⠂⢢⠀⠁⠀⠀⠀⠐⠂⠠⢁⠁⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢘⠀⠀⠸⢀⢢⣄⠋⠀⣁⠂⣀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢁⠐⡀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⠀⠀⢸⠀⠐⠲⢅⡒⣀⡙⠃⡄⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠦⡀⠀⠀⠑⡀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠲⠀⠀⠀⠀⠐⠠⠀⠀⠀⠈⠀⠀⢈⠉⢂⠀⠀⠈⠐⠇⠀⠀⡁⢠⡁
⠀⠀⠀⠀⠀⢰⠀⠐⢄⠠⠂⠉⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠄⠰⡀⠀⢠⡀⠀⢸⠀⠀⠇⠀⠀⠀⠠⠐⠆⡇⠀⡇
⠀⠀⠀⠀⠀⢸⡀⢘⣀⢑⣄⣀⡃⠀⠀⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠜⠆⠈⠛⠀⠀⠀⡇⠀⢸⠄⠂⠀⠀⠀⠐⠀⠀⠐⣧⣀⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠁⠐⠒⠒⠀⠀⠤⠤⠀⠀⠀⠠⠄⠠⠰⣇⣀⣸⣤⠤⢠⡔⠁⠀⠜⢀⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣀⠐⠂⠀⠀⠀⠀⠀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣁⠈⠀⠀⠀⠀⢀⠉⢀⠀⠀⠀⠀⠚⠁⠀⠐⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠆⠀⠐⠐⠒⠈⠁⠀⠈⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠐⠀⠰⠀⡆
⠀⠀⠀⡠⠒⠒⠒⠒⠒⠃⠀⠐⠒⠒⠒⠚⢒⣒⣒⣚⣒⣒⡂⠀⠈⠉⠐⠒⠒⠒⠒⠒⠂⠠⠤⢄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⢠⠁⠀⠀⠀⠀⠔⠒⠂⢠⣴⣦⣄⠀⠒⠂⠒⠦⡒⠒⠒⡄⠀⡔⢒⣲⣶⣒⠒⠒⠒⠒⠢⠀⠀⠙⡄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⠅⢸⡄⠠⠀⠀⠄⢤⡄⠀⡇⠀⡇⢾⡿⠟⠛⠀⠀⠀⠀⠀⢣⠀⠀⠘⣄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠠⣾⡆⠀⠀⠀⠀⠐⠀⠀⡇⠀⡇⠈⡁⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⢆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⡟⠁⠀⠀⢸⡷⡄⠀⠀⠀⠀⠂⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡀⠀⠀⠈⢦
⠀⠀⢸⠀⠀⠀⠀⠀⠃⠤⠄⢀⡠⠄⡰⠤⠷⠄⠄⠀⠲⠶⠖⠁⠀⠳⠤⠤⠤⠤⠴⠦⠤⠦⠀⠰⠂⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈
⠀⠀⠘⠀⠀⠀⡠⣀⠄⠒⠀⠉⠁⠐⠢⢅⠢⡀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠛⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡖⠢⠀⠀⠀⢈⠞⠁⣠⣶⣿⣿⣿⣷⣦⡀⠑⠌⠄⠀⠀⠀⡠⠊⢀⣴⣾⣿⣿⣿⣶⣄⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢐⢖⡁
⡇⠘⡃⠀⢀⠎⠀⣼⣿⣿⠟⢛⠛⢿⣿⣿⡄⠘⡀⠀⠀⢢⠁⢠⣿⣿⡿⠛⠛⠻⣿⣿⣧⠀⠱⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠁⠉⠉⢹⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⡖⠒⠒⠞⠀⢸⣿⣿⡇⠀⡀⠀⣿⣿⣿⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠈⠁⠀⠹⣿⣿⣷⣶⣶⣿⣿⡿⠁⠉⠁⠀⠀⠈⠂⠈⢿⣿⣿⣶⣶⣾⣿⣿⠟⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠤⠤⠴⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠀⢀⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠠⠂⠀⠀⠸⠀⣰⠀⠀⠃⠀⠰⠀⠆⠠⠄⠀⠀⠀⠀⠤⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣚⣛⣿⣭⣍⣿⣿⣭⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⠞⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠡⠾⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⢤⣀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠁⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡉⠙⡷⢴⣿⣿⣿⣿⣷⠀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⡌⣀⡀⢀⣾⣿⣿⣿⣿⣶⠔⢂⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢿⣿⠁⠀⣿⣿⠲⠿⠿⠶⠶⠿⠿⠿⠿⠗⠒⠒⠒⠒⠺⣿⣏⠀⢈⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⣡⠀⠀⣱⠀⠀⠀⢀⠀⠀⠀⠀⣼⠏⠴⠶⠶⠶⠶⣾⠷⠶⠖⠖⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⣿⠶⠛⣱⡟⢩⡾⢋⡿⣵⢟⣿⠃⢀⠆⠀⠀⠀⣼⢯⡷⢟⡟⣴⡯⠊⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠓⢤⣝⣀⣘⣣⡄⠁⠙⠁⡏⡠⠋⠀⠀⠀⢰⠃⠉⠀⠘⣱⡟⠑⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⣆
⠀⢀⣴⡆⣀⡐⣞⣇⣀⡀⣠⣤⣶⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠠⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⢆
⠄⠀⠉⠉⠤⠋⠖⠘⠚⠙⠉⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠀⣀⡀⡀⣄⠉⠠⠠⠀⢀⣀⣤⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠠⠤⠀⢀⠀⠀⣈⠉⠉⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⣀⡀⠀⠀⢀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⡑⠀⠀⠀⠁⠀⠀⠈⠈⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢹⡐⠠⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⡪⡟⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠄
⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠤⠄⠤⠂⠂⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠗⠁⠀⠀⢀⣀⡀⠀⠈⠛
⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠉⠁⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠦⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠁⠀⠀⢠⡎⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⣤⠀⠀⠀⢠⠀⢶⠴⣶⡂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⠂⠀⠀⡇⠠⢤⣄⣀⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠦⠴⠶⠾⠶⠦⠀⠚⠓⠚⠛⠛⠋⠉⠁⠀⠀⠧⠤⣄⣀⡀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣔⣒⣓⡖⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠁
⠀⠀⠀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣉⢀⠉⠁⠐⠒⠀⠀⠀⠠⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⣀⠀⠩⠿⠷⣤⣀⠀⠀⠴⣾⣿⣿⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⢻⣾⣧
⠀⠀⣸⠁⠀⠀⠀⠘⠉⠙⠓⠀⠀⠀⠻⠿⣻⣍⡉⠁⢀⣀⣈⣁⣀⣀⠀⠀⢠⡄⢰⣿⣖⠀⣦⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢿⣦⠀⠀⠀⠰⡀⠀⢀⠀⠀⠀⠈⠀⠈⠁⠀⢀⢀⡀⠀⠀⠉⢛⡉⠉⣺⣗⣿⣿⣿⢠⣿⣦⠀⠀⠀⢀⣠⣤⣤⣀
⠀⢀⣼⣿⠀⢰⠀⠀⣀⣰⣤⠴⠤⠀⠄⠀⠀⣀⣠⣼⣿⣿⠄⠀⠀⠀⠀⠉⣉⡙⠛⣛⣓⣺⠿⢿⡆⢴⣴⢿⣆⠸⠀⠈⠁
⠀⣿⣿⣿⡄⠼⠓⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠜⠿⣿⣃⠀⠀⠈⠈⠀⠉⠁⣀⣤⣤⡀⠁⠀⠉⠉⠙⢒⢛⠼⠇
⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣀⠀⠤⠶⢦⣙⣛⠟⠻⠷⣦⡄⠀⠀⢾⢿⠿⠿⣷⡄
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠭⠵⣶⠀⠀⠀⠉⠙⠲⢦⣄⡉⠋⢄⢰⠄⢠⠄⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣎⣠⠀⠠⢐⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⣂⣀⡀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⢀⢠⣤⣄⣀⡀⠀⠀⠀⠀⢱⡀⢀⡠⠸⠃⠀⣀⣀⣀⣀⣀⣀⣀⢤⣴⡿⠷⡴⢫⠟⠄⠨⠙⠂⡀⠠⠀⠒⠐⢆
⠀⣴⡎⠀⠀⠄⠀⠀⢡⠀⠤⠤⠐⣴⣿⠆⢀⣨⠂⠉⠉⢉⠀⠀⣰⠂⡠⢶⠻⠖⠛⠈⣈⠠⠄⠐⠀⡈⠄⠀⠀⠀⠀⠈
⠸⠿⢡⣦⣠⣔⡒⠀⣮⠀⠀⣠⡶⢏⠋⠐⠿⠟⢱⣡⣤⣤⣴⣤⣝⠀⣈⠀⠀⠒⠊⠡⢆⣤⠴⠖⡒⡗⠻⣆⣀⡆⠄
⠀⡠⢺⡏⢳⠉⠀⢀⣉⠀⠄⠀⠄⠠⠤⠀⠀⠐⠀⠉⠉⠉⠛⠉⣿⡈⠠⣼⠁⠀⠀⣘⡏⠐⢊⠅⠄⡀⠀⢩⠇⢷⠒⣀⠄
⠞⠀⠾⢰⣸⡄⠈⢉⡟⡀⢀⠈⠀⠀⠀⠀⠴⠷⠀⠀⠀⠻⡾⢇⣤⣡⠰⠚⠂⢐⡀⡸⠁⠀⠀⠴⠾⢂⣀⠴⠒⠋⠉
⠉⡀⠀⠘⠸⡇⠀⣘⠁⠀⣿⣀⢘⠢⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢹⡀⡆⠄⠚⠁⠉⠁⣀⠠⠄⠒⠉⠁
⠀⠀⠀⠀⠀⣧⠀⢂⠀⠋⢩⣤⡅⢀⣷⡶⠶⠚⠛⠓⠃⠀⣁⣀⣀⣀⡸⠄⠤⠒⠈⠁
⠀⠀⠀⠠⡀⢸⠀⠀⠒⠒⠀⠀⠉⠉⠉⠈⠉⠉⠁⢀
⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢠⠀⠀⡄⠀⠀⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠄⠀⠀⢀⠀⢸⠀⠀⠇⠀⠀⡇⣀⠀⢙⠀⠀⠀⠀⣀⡀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠾⠶⣶⡶⣶⣶⣶⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡤⢚⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠒⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⣤⡀⢀⣴⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣀⠀⣀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠜⠱⣿⡿⠛⠻⣿⣿⣀⣴⣶⣄⣀⣰⣶⣶⣶⣶⣶⣒⣐⣒⣂⣼⣿⡿⠛⢻⣿⡿⣥⠴⠦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣠⣿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣦⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⢠⡶⢀⣠⡞⣀⡀⢀⣀⣀⡀⣀⣀⣼⠏⠐⠛⠛⠛⠛⢻⣟⡛⣛⣉⢃⡄⢀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣟⠋⠁⣾⠋⢴⡿⠡⡟⣽⡏⣾⠇⢀⠎⠀⠀⠀⢠⡟⡾⠻⢻⢧⣾⠗⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠙⠦⣥⣤⣬⣭⠆⠁⠈⠀⡏⡠⠋⠀⠀⠀⢀⡟⠈⠁⠀⠈⣼⠋⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣭⣷⣶⣴⣶⣶⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠂⠱⠟⢰⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠋⠛⠁⠀⠁⠀⠀⠀⠀⠀⠀⠚⠛⠸⠣⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠸⠄⠠⠀⠚⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⡋⠓⠓⠆⡌⠁⠀⠀⠀⢀⣠⡀⠒⠀⡠⠂⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⢨⠉⠁⠀⠀⠉⠈⠉⠉⠄⠄⠌
⠀⠁⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⢠⠄⡀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⢠⡀⠀⠀⠀⢠⠑⣄⣀⡄⢀⠖⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠗⡠⠁⠂⠀⡄⢀⡅⠀⠀⠀⠨⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⡆⠁⠀⠀⠀⠀⢀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠠⠁⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠐⠤⢤⡄⡠⠤⠄⠀⠀⠐⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠰⠁⠀⢰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠄
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶