Neny Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⡀⠅⠠⠀⠀⠀⠀⠀⠢⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠉⠃⠀⠈⠁⣾⡒⠤⣶⣆⠀⠀⠀⠀⡀⢠⣄⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢁⠛⠀⢤⡔⡒⠉⠐⠛⣧⢀⣿⠀⠀⢸⣽⣅⣀⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡷⢾⠿⠀⠀⠀⠀⢿⣿⣿⠀⢰⣾⣿⠫⣭⣷⠈⡇⢶⠶⣤⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠉⠀⣡⣯⢀⠀⠀⠀⠐⢩⣿⣦⣸⣿⡏⢸⣲⣿⣷⠇⣾⣴⠏⠀⡂⢑⣮⣶⣤⣤⣴⣊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣄⠘⢁⠀⠀⠀⢀⣿⣤⣤⡀⠀⣠⣶⣿⡿⣿⣿⡟⠛⠁⠀⠈⢻⣿⠏⠀⠀⠀⠸⠡⣿⣿⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢖⣡⣄⡀⠹⣿⣾⣷⣼⡿⠧⠖⠈⠀⠀⠙⠶⠚⠋⠉⠉⠛⠛⠋⠁⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⣸⣿⣿⢎⣙⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠣⠌⠛⠂⠈⠙⠻⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⠀⠀⠁⠀⠀⠀⢀⣿⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⣿⣻⣿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠹⣧⣿⡟⠙⣶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⡇⠀⠁⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠠⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠀⠈⠓⠶⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢀⣧⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠠⡆⠀⠴⠈⠉⠠⡄⣄⠀⠛⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠁⠀⢀⣠⡄
⠀⠀⠀⠠⠀⢀⠀⠀⢸⡇⣿⣶⠀⠀⠀⢘⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠃⠐⠠⡬⣭⢤⣀⢀⡀⢀⢀⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢹⣼⠒⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠶⠤⢽⣿⣤⣞⢻⣿⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠙⠂⠉⠁⠻⠷⠂⠀⡤⠀⠀⠀⠀⠀⠀⠘⠿⣥⣤⣠⣄⣤⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⢚⠛⣻⠇⢛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣸⠷⠙⠓⢰⣄⣘⣉⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠊⠁⠀⢀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠒⠶⠒⠒⣠⡴⢀⡵⠄
⠀⠀⠀⠀⠀⠉⡷⠼⠦⠐⢟⣉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⢀⠐⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠊⠁
⠀⠀⠀⠀⠐⠣⠐⠆⡒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⡎⡛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⠏⠀⢀⡴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠘⠠⠁
⠀⠀⠀⠀⠀⠀⠛⢈⡷⢥⡄⢄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣟⣵⡿⢿⣿⣀⣰⣏⠀⣀⣴⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⡐⠴⠓⠀⡄⢀⠰
⠀⠀⠀⠀⠂⠀⣴⣾⢰⡆⣷⠘⣆⠀⠀⠀⠀⠀⠀⣾⡟⣹⣿⠏⠀⠀⢀⡿⠛⠛⠶⢦⡈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠠⡿⢠⠄
⠀⠀⠀⠀⡄⠀⢹⣿⡆⠑⢈⢣⡙⢦⠆⠐⠀⠀⠘⣿⣷⣿⢿⠀⠀⠀⠈⠀⠀⠀⠈⠂⠙⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⡬⠛⠓⠾⠇⠀⢂
⠀⠀⠀⠉⠀⠀⢌⠙⡓⠻⢾⣶⣷⣀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢦⠀⠀⠀⠀⠀⠀⠠⠀⠠⢀⣿⠀⠀⢀⡀⠀⠀⠀⠀⢀⡼⡟⠀⠙⠃⠀⠂⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠊⢿⣦⣶⣤⣴⠀⣀⠄⢠⡾⠗⠶⠆⠻⢧⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢶⣤⣿⣦⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⢋⣴⠿⢋⡴⠊⠀⡌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣦⠀⢹⣎⢿⣎⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢃⣴⡟⠛⣠⠟⠁⣠⠞⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣆⠈⢻⡔⢻⣆⢨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣱⣾⠋⢀⡾⠃⢀⡾⠁⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠹⡀⠘⣧⠀⢻⡇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣠⣤⠀⢈⣷⠞⠋⠀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣀⠙⢦⡏⠀⡈⢻⣼⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡁⠁⠀⣦⡜⠿⣶⢿⣾⡿⠛⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠛⠿⠛⣁⡄⠙⡦⢿⣷
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠛⠀⠤⣨⠛⡀⣠⣶⣿⣧⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣾⣿⣿⡷⠤⣞⡋⣀⠀⡤⣾⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡿⣟⠛⠳⢄⠀⠀⠀⠀⣽⣿⣷⣿⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⣽⣿⣿⣏⠀⠀⡁⠀⣤⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣇⢻⣄⠑⠈⡁⢠⣦⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣶⠂⠄⠨⠀⢀⡼⣯⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣶⣾⣷⣾⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣾⣄⣶⣟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⠀⢨⣭⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣍⠄⢀⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠗⠠⠘⠻⣾⣧⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿⡄⣀⡏⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⣞⣷⠟⠁⠀⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣖⠀⣜⡻⣽⣧⣿⣶⡄⢠⣴⡾⠿⣷⣆⠀⠀⢸⣿⣿⡇⠀⠀⣴⡿⠻⣷⣤⡀⢠⣾⣿⢿⡿⣞⡧⠀⣲⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣬⠛⠻⠟⣻⢿⣿⣿⣏⠀⠀⠘⣿⣦⣤⣾⣿⣿⣷⣤⣼⣿⠃⠀⠀⣹⣿⣿⡿⣟⠿⠛⢛⣥⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⢇⣸⡿⢻⣿⣷⣦⣄⠘⣿⡯⠁⣿⠋⠉⢬⣿⠁⣠⣶⣾⣿⠿⣿⣇⢻⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⣿⣿⣿⣿⣟⣿⣿⣴⢠⡏⠀⠀⢀⣹⣿⣹⣿⣿⣿⢿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⡤⠤⣀⣸⡇⠀⠀⠉⠙⢿⣿⣿⡿⣡⡄⠀⠀⢀⡈⢿⣿⣿⠿⠉⠁⠀⠀⢹⣇⣠⠤⠤⣤⣤⠤⠄
⠀⠀⠀⠀⠀⣠⡾⣿⠷⠶⠾⠟⠛⠿⢷⣶⣾⣿⣳⠄⡀⠀⠀⠀⢸⣿⣷⣏⣠⡆⠀⢈⣿⣿⣿⡇⠀⠀⠀⣀⠠⣾⣿⣷⣾⡿⠟⠛⠻⠶⠷⢿⣟⢦⡀
⠀⠀⠀⢀⣴⡿⠟⠋⠀⠀⣀⣀⣤⣤⢤⣀⣿⡛⠛⠿⢶⣷⣤⣄⡸⣿⡿⣿⣿⣴⡝⣻⣿⣿⣿⢁⣀⣴⣾⡷⠟⠛⢻⣉⣠⣤⣤⣤⣀⣀⡀⠀⠙⠿⣿⣆
⠀⠀⢰⣿⠟⣁⣴⣾⣿⣴⡾⠿⢿⣷⣶⣼⣿⡗⠒⠤⣀⠈⠙⢿⡿⢋⣶⠄⠀⢰⠋⠀⠀⠈⠹⣷⡿⠋⠀⣀⠄⠒⣻⣿⣵⣶⣾⠿⠿⣷⣴⣿⣷⣤⡘⢿⣷⡄
⠀⠀⠀⣁⢐⣡⣾⠟⠁⠀⠀⠀⠀⠈⠉⠛⣾⠿⠿⣶⣯⣷⡤⣬⢃⣼⣿⡆⠀⢸⠀⠀⡀⠀⣆⢩⣄⣴⣿⣵⣾⠿⢻⣿⠋⠉⠀⠀⠀⠀⠀⠈⢻⣧⣜⡢⡀
⠀⠀⣰⠏⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣀⣰⣾⣄⡀⠀⠈⠉⠛⢿⠸⡟⢧⢹⡀⢸⠀⣠⠀⠀⣿⢸⠿⠛⠉⠀⢀⣀⣠⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠙⠿⣯⢻⡆
⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⣴⣪⣵⣾⠿⢿⣿⠷⢿⣿⣿⣷⣶⣿⠀⠁⠀⠈⠃⢸⠀⠈⠀⠀⠁⢸⣷⣶⣾⣿⣿⠷⢾⣿⡿⠿⣷⣮⣱⣦⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡿⠛⠉⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⢸⣏⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠛⢿⣿⣀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⣀⣼⣿⣷⣶⡶⠾⠟⠛⢿⡆⣰⣆⣉⣀⣨⣀⣀⣀⡀⠀⣸⠟⠛⠻⠶⢾⣷⣾⣿⣦⣀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⣠⣞⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣷⡀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⢼⣷⣿⡿⣁⣀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠻⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣄⣤⣐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠛⠉⠉⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⢶⣶⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⣀⣀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⠟⠛⠛⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣤⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣷⣶⢤⣤⣴⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣿⡿⢻⣿⣇⠀⠈⣹⣿⠟⢿⣿⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣾⡟⠀⠀⢻⣿⡄⢠⣿⠏⠀⠈⢻⡇⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣰⣿⣇⣀⣤⣼⣿⣿⣼⣿⣧⣄⣀⣸⣿⣄⣸⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠛⠛⠛⠛⠉⠙⠛⠛⠋⠛⠛⠛⠛⠛⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⣋⣤⠴⠶⣯⢾⣿⣋⡓⠀⢀⡀⠀⠀⡀⢀⡀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠌⢀⡴⠋⠀⣠⡴⠋⠀⠀⠐⡆⢀⢂⢁⢀⡁⡤⢌⣀⠀⠄⠀⠀⠑⢂⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⣕⡀⢠⡀⣎⠀⠀⣼⠃⠀⠀⠀⠀⠀⠁⣾⢀⠳⠉⠁⠀⠀⠈⠳⣦⠶⡗⠢⠾⣦⣬⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠌⣻⣼⣾⡿⠻⠄⡀⢥⠂⠀⠀⠀⠀⠄⠀⠿⣗⢿⡀⠤⠀⠄⠀⣤⣼⣶⣄⣢⠤⠀⣛⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠹⠟⠽⡏⠉⠳⡦⠘⣿⠀⠀⠀⠀⠀⠇⢰⠀⢉⠩⡿⡶⣔⣾⣳⠏⡏⠀⠀⠉⠻⣶⣺⣷⣄⠁
⠀⠀⠀⠀⣤⡤⠀⠀⢄⢀⣴⡇⠀⠀⠀⠀⠀⢸⡄⣿⡆⠀⠀⠀⠀⢀⣾⠀⠈⣽⡇⢃⣴⣷⣧⡐⠃⠐⠁⠀⠀⠈⡎⠀⠺⡅
⠀⠀⠀⢈⠘⠄⢀⢀⣸⣋⠉⠾⠄⠤⡄⢺⠤⠬⢷⣿⣿⡄⠀⠀⠀⣼⣿⣤⣰⡿⣵⡾⠋⢺⠈⢧⣄⣼⣴⣌⣳⣼⡯⠐⣠⡇
⠀⠀⠀⣰⡄⠋⢁⠨⡁⣢⣾⣜⣐⣭⠤⠿⠿⠿⠤⢿⣿⠿⢄⠀⠈⡠⠷⡟⣿⢞⡿⠖⠭⡙⢆⡀⡻⠟⢉⠁⢀⠜⢻⠟⠅
⠀⠀⢀⠹⢯⣾⡾⢿⣯⠭⠛⠉⠁⠀⣄⠴⠶⠒⠒⡛⢛⣱⣦⡿⣤⡕⣦⢹⠄⢻⡐⠚⡖⠢⣄⢰⣇⣰⣎⣠⠷⢲⠾⠗⠃⡄
⠀⠀⠀⠀⠀⠉⠭⠋⠁⠀⣀⡤⠖⠉⡀⡠⠀⣨⣅⣶⢿⡻⡹⢿⣿⣧⣼⣟⡵⠘⡚⣮⢙⡇⠻⡄⡟⢫⠉⣠⠖⢋⠀⢤⡐
⠀⠀⠀⠀⠉⠛⠛⢠⠔⠊⡡⡰⢚⠈⢐⣈⣡⡿⠟⠛⢿⣣⣾⣿⣿⡟⣧⣻⣮⠤⠦⣈⠓⢷⣤⠟⠉⣷⣫⢗⠞⠉⠛⢦⡀⢂
⠀⠀⠀⠀⠀⠀⠠⠴⠦⢐⣀⠽⢒⣘⡿⠋⡡⡴⣚⣶⣼⣿⣼⡛⣿⠃⠁⠙⣻⣷⠼⣭⡓⠄⡏⢳⠟⢟⡑⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠔⠽⣿⣷⠴⠋⡯⡉⢁⠊⠉⢠⠋⣠⣽⡿⠟⠃⠒⠶⣖⠋⢠⣟⢄⡀⠘⣼⢡⡀⠐⣿⡇⠀⠀⠀⠀⠘⠈
⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⢀⠀⠀⡇⠀⠀⠀⠈⣸⣯⣍⣿⣿⠷⢦⣄⣼⡇⠈⢟⣾⣄⠀⠚⢦⢩⡲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠖⠂⠀⠀⠈⠚⠋⠓⢝⠋⣴⣯⣿⣿⠹⠀⠀⠈⠛⢿⣆⠈⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠘⠛⠉⠑⠀⠀⠀⠀⠀⠀⠙⢷⣄⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂
⣶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⢉⣉⡙⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡹⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢄⠠⠀⠚⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠐⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⢔⣂⠠⠔⠁⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠮⣅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠠⠀⢁⣂⣨⠔⠋⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠉⠉⠁⠀⠀⠀⠀⣀⡴⠒⠛⡉⠁⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢠⡴⠋⠉⠁⠉⠁⠒⠠⢤⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡐⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⢉⠁⡀⠠⠀⠘⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠆⡀⠀⢻
⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⣠⠈⠁⠀⠀⠀⠀⠀⢀⡈⠙⢦⣀⠀⠀⠀⠀⠀⢧⠀⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢨⠐⠀⣼
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⠳⣄⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠂⠸⡀⠀⠀⠔⠀⠀⠀⠆⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢧⠀⠐⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠝⣆⠀⠈⠲⣄⠀⠀⠀⠳⣄⠀⠀⠀⠀⠒⠘⠖⠁⠀⠂⣠⠞⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠣⢄⡀⠀⡀⠀⠀⠀⠄⠀⠠⠀⠤⠀⠁⢀⠀⢸⠆⠀⠀⠀⠀⠙⠓⠲⠦⠤⣀⡉⠋⠘⠋⢁⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠠⢄⡀⠉⠀⠉⠛⠉⠁⠉⠁⠉⠁⠀⠉⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣉⠒⠒⠒⠂⠂⢐⣀⣠⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢢⡆
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡜⢿⡟⠀⠀⠈⠲⠶⣦⣄⣀⠀⠀⠀⠙⠷⣦⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣦⡀⠀⠀⠀⠀⠚⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠠⠀⠀⠀⠉⠛⠢⠄⠀⠲⡄⣠⣴⠊⣿⣾⣷⣯⣽⡿⢁⡈⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣾⠿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠤⣀⠀⠀⠑⢠⡜⣿⣿⣧⣻⣿⡟⠻⣿⣿⣾⣤⣏⢶⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠈⠛⠀⣾⣿⣝⣿⣷⣿⡿⠐⠆⠹⣿⣿⢿⣿⣿⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⢾⣧⠀⢨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠀⣿⣿⣿⢿⡿⠋⡁⢠⡄⢴⣾⣵⣿⣍⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣾⡏⠀⣸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡛⠇⣾⠃⣨⣿⢀⣄⢸⣯⣿⣿⣿⡿⡿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⠀⢀⠀⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣉⠐⠛⠁⠰⣯⣾⣿⣿⣿⣿⠜⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣴⣬⣿⠿⢿⣇⢹⣿⢩⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⡇⠀⠀⢰⡆⢰⡆⠀⠐⠒⠂⠀⣆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠻⠌⢃⡻⠿⠀⢿⡟⠾⠃⣾⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠠⣿⣿⣇⠀⠀⠸⠃⡸⠁⠀⠰⠄⠀⠀⠇⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣦⣬⣻⡄⢸⡟⠀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⠀⣤⠾⣿⣿⡆⠀⠠⠘⠁⠄⠀⠀⠀⠄⠀⠇⠈⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⣉⣉⡁⢻⡿⠁⢀⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠉⠉⠛⠛⢷⣀⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠠⠄⢤⣼⣿⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡇
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢸⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⢀⣘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⢸⠀⠀⠀⡧⠀⠀⢰⡄⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢸⠀⠀⠀⣅⢚⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠙⣻⠀⠠⠚⠀⠀⢀⣤⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⢶⣏⠃⠀⠰⣀⣴⣿⠠⣿⣷⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⣤⢿⡷⢣⣶⣹⣿⡟⢸⣿⣿⣷⣷⣤⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠠⣼⣷⣾⣯⣾⣿⣇⢸⣿⣿⣿⡟⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠸⠛⢻⠿⡿⣿⠻⡉⢸⣿⣿⠜⣉⣄⣥⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠈⢡⡀⣦⡄⠘⣋⣉⣉⣋⣙⣟⣛⣋⣉⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠂⠀⠉⢻⠾⣿⣿⡿⢻⣿⣿⣿⣿⠛⢽⡃⠸⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠂⠰⢂⡻⠿⣠⣽⠻⣷⣾⡿⢿⣋⡗⠹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⢀⡈⠰⢂⡈⣟⠃⣹⠏⢹⡟⠀⠀⠂⠁⠈⠑⠂⠺⠗⠒⠒⠰⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠊⠁⠀⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⠤⡤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣞⠛⠋⠉⢀⠬⠉⠙⠿⠒⠢⣄⡀
⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠂⠁⠈⠹⠿⣿⣿⣶⣤⣀⣀⣤⣀⣤⣀⣠⣬⣿⣗⣀
⠀⠀⠀⠒⠩⠛⠋⠑⠐⠲⠶⠒⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡉⠉⠹⠻⢿⣷⣿⣿⣿⣟⠉⠛⠻⢷⣄
⠀⠀⠐⢦⣸⣆⣴⣶⣶⣤⣄⢀⣀⣤⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣦⠀⠀⠈⠛⠻⣿⣷⣤⣤⣀⣀⣼⢧
⠀⠀⠀⠀⢻⣿⣿⡏⠉⠙⠛⠛⠿⠿⣵⣦⠄⣠⣦⣶⡖⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡤⣤⣄⣹⣿⡿⣿⣿⠝⠃
⠀⠀⠀⠀⠀⠻⣟⣿⠄⠀⠀⠀⠀⠀⠀⠉⠙⠳⣾⣉⣂⠤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠈⠙⠋⠁⢀⡄⡄
⠀⠀⠀⠀⠀⠀⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣮⣭⡑⠶⢀⣢⣴⣤⡦⠄⢴⣲⠒⠀⠀⠀⠀⢀⠀⠀⣰⣯⠞
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣤⡂⠀⣭⡙⠳⠶⣶⠤⠔⠟⣟⣫⠽⠛⠁
⠀⢀⢀⣀⡄⠀⠷⠿⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⡄⠀⠀⠀⠀⣀⡀⠀⠈⠉⠻⠿⣿⣿⣿⣿⣾⣿⣷⣿⣭⡉⠀⠀⠀⠀⢀⣀⡀
⠀⢸⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠄⠀⠈⢍⢩⣿⠟⠛⠛⠀⠒⠶⠀⠀⠠⣤⣷⡇
⠀⠈⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢱⡄⠀⠀⠃⠀⠀⠀⠠⠡⠀⠀⠀⢀⣨⣀⠌⠁⠀⢀⡀⠠⠀⠀⠀⢸⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡤⠶⠁⠀⠀⠀⠀⡄⠀⠀⠹⣧⡀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⢨⣄⣠⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⣴⣴⣋⡀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠿⠀⢴⠀⠀⠀⠀⠄⢉⡷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣦⡝⠉⠀⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⡀⠤⠀⠁⠺⢥⡚⡑⠆⠀⠀⠀⠀⠀⢸⠆⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⣜⡪⠁⠀⠀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠁⠀⠀⠀⠈⡈⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠘⠇
⠀⠀⠀⠀⢠⡶⠀⣙⢶⠟⠈⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⡀
⠀⠀⠀⠐⠂⣄⣀⠋⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢡⠀⠚⠈⠁⠀⠀⠀⠀⠉⠀⣸⣷⡀
⠀⠀⠀⠀⠃⠨⡋⠀⢸⡀⠀⠀⢠⠏⠀⠀⠀⠀⠐⠂⡀⠛⠃⠀⠀⠀⠀⠀⡰⢐⠂⠀⠀⠀⠙⠀⣤⣌⣆⢀⣠⣄⠈⠁⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⢀⣳
⠀⠀⠀⠺⡀⠀⠁⣐⣸⡁⣠⠶⠃⠀⠀⠀⠀⠀⠀⠚⠃⠠⠀⠀⠀⠀⢰⠌⡇⠐⡄⠀⠀⠀⠠⣿⣿⣉⣉⣉⡉⣀⡶⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠁⠂⠴⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠣⠈⠞⠃⠀⠀⠀⠀⡇⠻⢫⣭⠅⠀⡈⠀⠀⠀⠀⠀⢠⠄⠀⠀⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⡀⠐⠀⠀⠀⠀⠀⣀⣼⡇⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠠⣀⡀⡙⠚⠶⠄⠠⠜⠹⠓⠂⠒⡀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣴⡷⢻⣤⣶⣾⣿⠻⠛⠿⢛⡁⠈⠁⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠑⠠⠄⣀⣀⣀⣴⠏⠁⠀⠀⠈⠀⠀⠀⠀⠂⠀⣉⣉⡉⠉⠉⠉⠀⡄⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠳⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⣀⡀⣀⠈⠃⠀⠀⠰⠀⠀⢠⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⡀⠀⠀⠈⠁⠈⢀⠀⠐⠆⠈⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠙⠄⠀⠀⠀⠀⠈⠃⠂⠀⡀⠀⠀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢵⡆⠀⣿⣿⣿⣿⠁
⠀⢀⣴⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡤⡄⠀⠀⠀⠀⢻⣿⡿⠁
⠀⠈⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠄⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠠⠄
⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂
⠀⠒⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢡⠀⠀⠐⠒⠀⠀⢀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⣤⣤⡆⣤⣤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠉⠛⠃⠙⠋⠊⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠓⠀⢠⠤⠤⠤⣤⠄⢠⣤⠀⠚⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠷⡄⠀⠀⢀⣶
⣿⣿⣿⣿⣿⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠀⠀⡾⠃
⢿⣿⣿⣿⠟⠣
⠒⠒⠒⠒⠒⠉⠀⠀⠰⠄⣶⣤⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠆
⢀⠆⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠌⠀⣠⡟⠁⠀⠀⢀⣠⣴⣶⣿⣿⡏⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣀
⠀⡴⠋⠀⢀⣴⣾⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⢿⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⠙⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⣄⢀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠈⠑⠂⡀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠄⠀⠣
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⢤⢀⢀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡠⠄⡗⠂⠈⠀⠀⠀⠀⠀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒
⠀⠀⠀⠄⠀⠀⠀⠄⠀⠉⠦⠜⠈⠁⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰
⠀⠀⡈⠀⠀⡨⢐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠁
⠀⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠒⢈⣉⠉⠭⠭⠭⠌⣽⣀⡐⠒⠂⢄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣠⠔⢋⡠⠔⠂⠉⠣⠀⠀⠀⠀⠀⠐⠁⠀⠀⡩⠖⠢⢄⡁⠢⢄⠀⠀⠀⠀⢀⠀⠀⡠
⠀⠀⠀⡠⢊⡤⠂⠁⠀⠀⠀⠈⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⠐⢄⠑⢄
⠀⢀⠊⢠⠋⠉⠉⠈⠉⠁⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠄⠠⠀⠀⢀⣀⠀⠀⡆⠈⡄⠀⠀⡄
⠀⠸⠀⠸⡀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⢁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈⠝⠁⢲⠀⠀⠀⠈
⠀⠀⢣⠀⠈⠢⢄⣀⠤⠊⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠒⣌⡠⠤⠒⠁⢀⠐⠁
⠀⠀⠀⠑⠠⢀⣤⠉⠁⠒⠢⠤⠤⠤⠤⠾⠤⠤⠤⠐⠒⠒⠉⠀⣁⣀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠤⠤⠤⠤⠀⠤⠤⠤⠀⠀⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⠿⠿⠿⠿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⠿⠛⣋⣉⣤⠴⠂⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⠟⠋⣉⣤⣶⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠛⠉⠁⣀⡤⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠿⠛⠉⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠟⢋⣩⣴⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣾⢟⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⢯⣭⣥⣬⣉⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣬⣭⣿⣷⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡾⠉⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣶⡾⠿⠛⠻⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠈⢩⠿⠿⠿⠿⢷⣶⣄⣨⣿⠃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⣤⣿⠟⠀⠒⠶⠶⢤⣤⣤⣄⣀⣀⡀
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⣤⣶⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⢷⣶⣶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣉⣙⡛⠛⠿⠿⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠿⠿⣷⣶⣦⣬⣍⣉⠛⠛⠻⠶⢦⣤⣀⡀
⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠀⠀⠀⠀⠀⠉⠉
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠛⣛⣛⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉⠁⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠁⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢰
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⣴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡆
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣹⣿⠉⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⡠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⡿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⡇
⠛⠻⠿⠶⢶⣶⣶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⢷⣶⣶⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⠿⠷⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⢀⠀⠀⠀⡈⠉⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠙⠲⠤⣀⣀⣠⡤⠤⠠⣄⠀⠘⠛⠻⢦⣜⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠻⣆⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⠀⠀⠈⠉⠛⠿⢶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣴⠆⠀⠀⣿⣀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠜⠀⠀⠈⣏⠉⠉⠉⠙⠓⠲⠤⢤⣀⡀⠈⠉⠉⠒
⠀⠀⠀⠴⠀⠀⠀⠀⣀⡀⠀⠀⠈⠛⢶⣌⣉⣹⡆⢠⣂⢐⡒⠲⣦⣤⡤⣤⣄⣀⣀⣀⣶⣖⡂⠀⣿⠶⠦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠓⠾⠶⠶⠾⠧⠤⠶⠶⠿⣿⠿⠋⠀⠀⠀⠀⠈⠉⠒⠲⠤⠀⣤⣰⠃
⠀⠀⠀⠀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⡀⣀⣀⣄⣀⣀⣀⣀⣀
⠀⢠⠚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣷⣶⣤
⠀⣼⡀⠀⠀⠀⢀⣀⡀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣴⢿⡄⠀⠀⠀⠀⠀⠀⠀⢈⡿⢋⢹⣝⡿⠿⡟⢻⣿⢩⣿⠿⣷⣄
⠀⢹⣿⣶⣶⣶⣭⣶⢄⣽⣿⡯⠉⢿⡀⠉⢩⣿⡇⠀⠲⣦⣞⡟⠀⠀⠈⠀⠁⣶⠀⠀⠀⢠⣿⣿⠇⣼⡎⠛⡟⡇⢻⡇⡘⡿⣠⡜⣿⣦
⠀⢸⣿⣿⠿⠻⣿⣿⣉⣽⣿⠶⠾⠿⠷⠶⠷⢶⣤⣀⣤⣹⣿⣥⡄⠀⠀⠀⣾⠏⠀⠀⠀⠘⣿⣿⣶⣿⣷⢣⢶⡶⣶⣶⣷⣷⣿⣷⣿⣿⡆
⠀⠀⠉⠛⠻⠷⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢾⡺⠐⢛⠻⢛⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⣿⣴⣶⣭⣴⢠⣾⣾⣿⡿⠃
⠀⠀⠀⠀⠀⣀⠔⡸⠋⠿⣖⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⠤⢤⡤⠤⢤⡀⣀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠁
⠀⠀⠀⠀⣴⢡⠂⠠⠃⠀⠀⠠⠀⠀⠉⠉⠉⠒⢄⣀⣀⣀⠀⠀⠀⠙⣿⣟⠀⠀⠀⠀⠀⠉⠐⠀⢄⡉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⡀⠁⠀⠀⠀⠠⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⢦⡄⢀⡘⠓⠲⢤⣠⣤⣀⠀⠀⠀⠀⠀⠀⠢⢀
⠀⠀⡎⠁⢠⠁⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢀⡔⡀⠀⠈⠀⠀⠠⠆⢹⣍⡟⠢⠄⡀⠀⠉⠉⠛⠒⠶⠦⠄⠤⣀⡱
⠀⠀⣗⣰⡏⠀⠀⡄⠀⢀⡾⠁⠊⠀⠀⢀⡽⠁⠀⢀⠀⠀⢼⠋⠈⠉⠀⠐⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀
⠀⢰⠫⢿⣧⣄⠈⠀⢠⣿⣧⡀⠁⠀⢠⡜⠀⠀⢠⠀⠀⠐⠃⠴⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⡤⠂
⠀⠘⡠⠓⠳⠶⠖⠒⠛⠲⢶⠉⠀⣀⣿⠀⠀⠀⠀⠀⠀⢈⡎⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠈⠉⢩⡀⠈⠉⠉⠙⠃
⠀⠀⣧⣄⢠⠦⡜⠋⠐⢀⠀⢁⣛⠛⠛⠠⡀⣀⣤⠀⢲⣾⡁⠀⠠⠀⠀⣰⠇⠀⠀⠀⣠⠀⠰⢄⣀⠄⢂⡀⠔⠒⠁⠀⠀⠀⠣⢀⣀⡠
⠀⠀⠀⠉⠁⠀⠳⢶⣤⠾⢴⠊⠀⠈⡢⠀⢠⡎⡛⠓⠛⠋⠧⡐⠀⠁⢠⡟⠀⠀⢄⠀⢀⡃⣀⣠⠴⠚⠉⠉⠐⠂⣈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣦⠔⠻⢴⡃⠀⠀⡔⠀⠠⠒⠍⡁⠓⠿⠀⠀⣀⠤⠔⠛⠈⠁⠀⣴⡄⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠓⠢⣧⣀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡴⠆⡀⣾⠀⢀⣤⠀⠠⣤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣹⣾⠋⢀⠴⢡⡟⠀⠘⡀⠳⣄⠀⠙⢤⡙⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⣴⠟⠁⢀⠂⣰⡿⠁⠀⠀⠡⡄⠘⣧⡀⠠⠷⣄⠹⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⡏⠀⡴⠁⣰⡿⠁⠀⠀⠀⠀⠸⠀⠸⣿⣄⠀⠸⣦⠙⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣾⣿⠃⠀⠀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠠⡀⠸⣿⣧⡈⠻⡄⠹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢰⡏⠑⢶⣀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⢿⣿⣷⡄⠳⠀⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⢳⡶⠎⠉⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠉⣲⠀⠈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠂⠀⢀⣶⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠃⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠘⠁⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠄⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⢠⣤⣴⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⣀⡴⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡿⠾⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⢀⣤⣸⡿⠀⣠⡄⠀⠀⣄⠀⠀⠀⠀⠀⣰⡄⠈⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⣿⣇⣤⠞⠀⠀⠀⠀⣿⣿⣿⣷⣾⣿⣧⠀⠀⠈⠻⣦⡦⠀⠀⢿⡇⠀⣻⡇⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⢀⣿⡯⠃⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠳⣦⡀⠀⢸⣇⣤⣿⡗⠀⢺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⢀⡾⠿⣧⣤⠀⠀⠀⣼⣿⣿⣧⡿⠉⠇⢿⣿⣿⣷⡀⠀⣶⣿⣛⠂⠸⣿⣿⡿⠁⠀⢸⣿⡇
⠀⠀⢀⡾⠀⢀⣄⠀⠀⠉⠁⠉⢀⣴⣿⣯⣴⠀⣼⣿⣿⡟⠀⢰⡆⠀⣿⣿⣿⣿⣿⣿⡟⣭⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⢃
⠀⠀⢸⡇⠀⠘⠉⠀⠀⠀⠉⠹⣿⣿⣿⣿⣿⣾⠻⣿⣿⠁⠀⠉⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣭⣿⠏⠀⠀⠀⣠⠜⣏⠘⣷
⠀⠀⢰⡇⠀⠀⠐⠁⠀⠀⠙⢷⣶⣾⣿⣿⣿⣏⣶⡿⠧⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣯⡿⠃⠀⠀⠀⡰⢋⣾⠇⠀⠸
⠀⠀⢨⡇⠀⠀⠀⢠⡔⢰⣄⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠖⠀⠀⠀⣀⠉⣳⠛⠁⠀⠀⢸⠁
⠀⠀⠀⠻⣄⠀⠀⠈⠀⠘⣿⡄⠀⠈⠻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠁⠀⠀⠀⣰⣧⠾⠉⠀⠀⠀⠀⢸⣀
⠀⡼⠀⠀⠈⢢⠀⠀⠀⠀⢌⢻⣆⠀⠀⠈⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣧⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⢨⡏⠀⢀
⠀⢿⠀⠀⠰⣦⡿⢦⡀⠀⠀⠀⠙⢧⡄⠀⣽⡿⠿⣿⣾⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡀⣠⣾⠟⠁⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠈
⠀⠛⡀⠀⠀⠈⠂⢤⠉⢠⣀⠀⠀⠈⢻⣶⣿⠁⠀⢹⣶⡿⣦⣿⣿⣿⡿⣿⠋⣼⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢰⠎⠁
⠀⠀⠳⡀⠀⠀⠀⠀⠀⠐⠟⠳⢦⣀⠈⠻⣿⡇⠀⠀⠙⣇⠀⠉⠉⠀⠀⠏⠸⣿⣿⣿⣿⠏⠁⠀⠀⣀⣴⢞⠀⠀⠀⠀⠀⠀⢀⣰⠆
⠀⠀⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣙⢿⣿⣄⠀⠀⠀⠛⠀⠀⠀⠀⢰⠀⠘⣿⣿⣿⣿⣀⣤⣤⡾⡟⠛⠁⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠄⠀⠰⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠏⠉⠈⠀⠀⠀⢠⣄⠀⣀⣤⡿⠁⣶⡀
⡜⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠉⠀⠀⢀⡀⢂⣠⣬⡛⠛⠉⠀⠀⠀⢹⡇
⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣀⣹⣦⣸⣿⣿⣄⣀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣶⣶⠖⢉⣷⠔⠛⠁⠀⠀⠀⠀⢀⣤⡤⠊
⠀⠀⠹⢾⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢿⣿⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠋⠁⠀⣀⠀⡄⠀⠀⠀⠴⣿⠟⠁
⠀⠀⠀⠀⠙⠴⠦⠀⢤⣤⣤⣠⣤⣤⣀⣤⣤⣴⣤⡄⣡⡀⠉⠉⠉⠉⠉⣿⣿⣿⣿⣶⣶⣴⣿⣿⣿⣤⣷⡿⠞⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠉⢁⠀⠀⠀⠀⢸⣿⣿⣿⡏⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⣤⡴⣾⡆⠀⠀⢀⣤⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡯⠤⠬⠤⠶⠶⠶⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣤⡶⣶⡄⠀⠀⠀⣨⣿⣯⣽⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠷⠬⠀⠀⠴⠦⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣶⣾⢶⡄⠀⠀⢀⣼⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣩⡾⠀⠀⠀⢈⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠿⠁⠀⠀⠀⣨⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣧⡤⠀⠀⠀⠀⢉⣿⣿⣿⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠚⢁⣤⣿⡞⣳⡄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⡟⣿⣿⡿⣿⡆⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤⣽⣿⣿⣦⣦⣤⣤⣼⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⠿⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡟⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣶⣾⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⡀⢠⠀⠀⢀⠃⠀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠁⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡞⠀⠬⠮⣤⠨⠅⠀⢀⠉⡜⠈⡠⡪⠃
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⣤⠤⡤⠷⣦⡀⡄⡠⠀⠀⠀⢀⠀⠀⠀⢀
⠸⣿⣷⣷⣀⡔⠙⢿⣡⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠛⠙⠋⠉⠳⣤⡼⠿⣿⣶⠀⠠⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⣾⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢷⣤⡿⢿⡞⠀⠀⠀⠀⠈⠙⢦⣤⠲⠒⠠⣼⡏⠃⢔⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣧⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠲⠛⠳⣍⠣⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠻⣷⣀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠉⠉⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠈⢻⣾⣤⣄⠀⠀⠀⣧⠀⠀⠀⠐⠂⠀⠀⠀⠀⣿⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠀⠀⣠⠋⠻⢾⡷⣶⣿⣦⣀⡀⣴⡀⠀⡄⠀⠀⢻⣆⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣤⠀⠀⠘⠠⣌⢻⠿⠿⠧⠿⣇⡁⠀⠀⠀⢿⠀⢴⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⣆⠠⠀⠡⡈⠲⡀⠀⠀⠀⠀⠹⡄⠀⠀⠸⡀⠈⠇⠀⣯⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⢰⡶⠞⠉⠒⠀⠀⠀⠐⣄⠀⠀⠙⡄⠐⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣼⣇⠀⠀⠀⠀⢒⠀⠀⠈⠇⠀⠀⠈⢦⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⡄⠀⠀⠀⠀⢣⠀⠀⠀⠙⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⢰⡀⠀⠀⠀⠀⠆
⡀⢰⢲⣠⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠱⡄⠠⣄⡠⠃
⠓⠉⠄⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠩⡀
⠀⢸⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠠⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢸⠀⡇⢀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⣸⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢈⠀⠣⠈⡃⠀⠀⠘⡄⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⡾⢠⡁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣜⠀⢼⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠢⣽⠀⠀⠀⢰⠀⠀⠀⠀⡆⠀⠀⠀⡀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⡀⠀⢠⠀⠇⠀⠀⠈⠂⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⡠⠀⡠⠂⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⠀⠀⠂⠀⠆⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠓⠁⠈⢄⠀⠀⠀⠀⠀⣇⠦⡴⠼⠀⠀⠀⠀⠈⡰⠀⠻⢶⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⢸⡀⠀⠀⠀⠀⡁⠁⢸⢛⠀⠀⠀⠀⠈⠁⠘⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣠⠞⠂⠀⠀⠀⠀⠛⠀⠁⠉⠁⠀⠀⠀⣨⣦⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⢠⠂⢸⠀⠀⠀⠀⢀⠀⠘⠁⢀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⡀⡀⠀⠀⣀⠀⠀⠈⠓⠄⡟⡀⠈⠀⠀⠀⠀⠘⣥⠠⠒⠁⠀⣀⢀⡀⢀⠀⢀⢀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠷⡀⠀⠀⠀⢰⢁⠁⠀⠀⠀⠀⢀⣀⣡⠀⠀⠀⠐⠉⠋⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠤⢀⡤⠤⠀⢠⣄⣐⡂⠠⣴⠦⠀⠈⠲⡂⣀⡈⡛⠀⠀⠀⠀⠀⠉⠈⠉⣠⣄⠔⠀⠀⠄⢰⠆⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠈⠉⠐⢅⡀⠀⠙⠇⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⠀⡠⠖⠀⠀⠀⠑⠀⠛⠦⠀⠄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠶⠷⡇⠀⠀⠀⠐⢸⠀⠀⠀⠀⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⠸⠖⠂⠀⠀⠻⠀⠰⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠆⠀⠀⠒⠂⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠊⠀⠀⠀⠀⠀⠀⠀⣤⠒⠰⢷⣄⣴⠂⠠⢤⣀⡀⠾⠀⢀⣄⠀⠀⠀⠀⠀⠙⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⠀⠀⢸⠄⠋⠀⠀⠀⠀⢠⠃⠄⠑⠀⠠⠀⠀⠀⠀⠈⠑⠀⣠⠀⠀⠀⠀⠀⠘⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢠⣀⠁⡀⠀⠀⠀⠀⠀⠈⠂⢆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢃⡀⠀⠐⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡠⠈⡁⠄⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢻⠡⢀⠁⠰⠀⠈⠈⠱⢤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠃⠀⣽⠠⠄⠠⠄⠀⢠⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠠⠀⠸⠄⣀⡀⠤⠤⠤⠘⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣴⢶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠤⠄⠄⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠾⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡑⠢⠍⠭⢽⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣶⣿⡇⠀⣼⣿⡆⠀⢀⣠⣬⣿⣷⣾⠻⠿⣿⣿⣾⣿⣿⣿⣷⣄⣀⣄
⠀⠀⠀⠀⠀⣾⢏⣿⣿⣾⣿⣿⠅⠀⢺⠋⠉⠙⢿⣴⣤⣄⣈⣿⣿⣿⣿⣿⣿⣿⣼⣿⣧⣦⣄
⠀⠀⠀⠀⠀⠋⠀⠃⣿⣿⣿⣿⡄⠀⠈⢇⢰⡿⣳⣿⡿⣟⣿⣍⣉⠍⠉⠙⠻⢿⣿⣿⣿⣿⡟⣧
⠀⠀⠀⠀⠸⠆⠀⢸⣿⣿⣿⣿⠁⠀⠀⣾⡟⢿⣷⣭⣼⣿⡿⣻⣿⡷⣦⣄⠀⠀⢹⣿⣿⣿⣿⣿⣆⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣉⠙⠁⢀⣤⣤⣙⣳⣴⠟⠙⣛⣿⣾⣿⠏⣼⣿⢿⣿⣷⣾⣿⣿⡟⢹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣤⣾⡟⣻⣭⢠⣾⣿⣿⣿⣿⣿⠿⢞⡾⠟⠛⢿⣿⣿⣏⣼⣿⣿⣿⣿⣿⣷⣿⣅⣻⣿⣷⡀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠙⠻⣧⣤⣷⣿⣿⣿⣭⣭⠇⡼⣦⣿⠁⠀⠀⢰⡏⠹⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⡛⠤⠀⠀⣾⠀⠀⢸⡇⢙⣉⠻⠟⣿⢻⣿⣿⣿⣶⣿⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠿⢂⣠⣾⠠⠄⡀⢀⡀⠠⠀⣿⠿⢿⣿⣿⣿⡟⢻⣷⣤⣤⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡟⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⣠⢾⠇⢀⡄⠀⢠⡀⠀⢀⡄⠀⢠⠀⢸⡷⣄⠀⠀⠀⠀⢀⣴⡆⠀⡀⠀⠠⡄⠀⠀⡤⠀⠀⡀⠀⣶⣄
⠀⠀⢰⠃⢸⢀⡼⣿⠀⠈⡇⠀⣼⠀⠀⣿⢧⢀⡇⠘⡆⠀⠀⠀⣾⣿⡀⢠⣿⠀⠀⢻⠀⢰⠇⠀⢸⣧⡀⢸⣿⡆
⠀⠀⢸⠀⠈⠋⢠⠇⠀⣠⣿⣀⡿⣄⠀⠹⡄⠛⠁⢀⡇⠀⠀⠀⣿⣿⣷⣿⠏⠀⢀⣼⡀⣸⣄⠀⠈⢿⣷⣿⣿⡇
⠀⠀⠈⠳⢤⡀⠻⣄⠀⡷⠈⠉⠀⢯⠀⣠⠟⢀⡤⠞⠁⠀⠀⠀⠙⠿⣿⣿⣄⠀⢨⣿⣿⣿⣿⠀⢀⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠙⠲⢬⣙⠁⢹⢠⡏⠈⣋⡥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⢻⢳⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠈⣇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣼⡜⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠤⠤⠴⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⣛⣛⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣓⣒⣚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣭⣭⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠬⣭⠭⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣛⣛⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⢒⡿⠉⡁⡀⠈⡆⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣮⣿⠽⣛⠿⢛⠀⠀⠀⠙⣄⠄⣶⢀⡇⠟⠾⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⣀⡀⢀⣞⡩⠽⣿⣷⡟⠁⠀⠀⡀⠁⣨⣥⣿⡤⠁⣼⣁⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢣⠋⠙⠉⠈⣹⣿⣷⠟⠁⠀⣀⡼⠟⣆⠀⣚⣴⣿⣟⢛⣩⣷⣾⣿⣏⣄⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⢻⣧⢀⣿⣤⣴⡲⠾⢿⣿⠇⠀⠠⠾⣩⣤⣿⣿⣿⡟⠟⣷⣴⣿⡟⠛⠒⠛⢺⡻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⣴⣿⠿⠸⠛⠉⠁⠀⢀⣘⠟⠀⡤⠀⣲⣿⣻⣿⣿⣧⣿⡾⠿⢿⡿⠓⠒⠲⡞⠀⡟⠃
⠀⠀⠀⠀⠀⠀⠀⡴⢿⢏⠁⢀⡴⠾⣿⡏⠀⠀⠀⠀⠀⠀⣬⠧⠀⢼⡆⠈⣟⡉⣭⣼⣿⡿⡛⢲⣤⠀⠀⠉⣿⡴⠧⣼⠁
⠀⠀⠀⠀⠀⠀⣼⣿⠇⣠⣿⣿⡇⢀⣿⠃⠀⠰⡀⠀⠀⣐⠛⡤⣥⣤⣶⠋⠉⠁⠈⣿⣿⡍⠀⠈⣻⡆⣠⣢⣟⣀⣴⠃
⠀⠀⠀⠀⠀⢠⣿⣿⣰⣿⠁⠀⣧⣾⠏⠀⠀⠀⠘⢓⣾⣯⣾⠏⠉⠉⠁⠀⠀⠀⠀⠀⠻⣿⣀⠨⣦⡟⠙⠁⠈⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⡤⠖⠁⠀⠀⠀⠀⡴⢋⡞⠁⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡼⢿⢿⣿⣇⡤⠔⠒⠠
⠀⠀⠀⠀⠀⠀⠹⣯⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢸⠀⠀⠈⠱⣤⣤⣖⡲⢛⡹⠶⠛⠛⠙⠁⢨⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣉⣉⣋⣙⣿⠛⠁⠀⠀⠀⣀⣴⡿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠛⠛⠉⢉⡽⠟⣀⠀⠀⣠⣶⣽⡟⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⢤⡀⠀⠀⠀⡀⠤⠐⠀⠈⠃⢀⣩⣿⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠵⣤⣴⣦⣤⣦⣤⣴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⢿⡟⢿⢿⡿⡿⢿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣷⣿⣶⣷⣷⣿⣷⣿⣼⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣶⣶⢿⡶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⢿⣷⡦⣄
⠀⠀⠀⠀⠀⠀⢠⡾⣁⣴⠏⣵⠋⢠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣇⠘⡏⣻⡌⢳
⠀⠀⠀⠀⠀⣰⣿⣴⡟⢠⡞⢀⡔⢃⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠨⣆⢸⡄⢻⡎⣧
⠀⠀⠀⠀⢠⣿⣯⠴⠄⢸⡖⣋⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣈⣾⡅⣘⢻⣿⡀
⠀⠀⠀⠀⣾⡿⠄⠀⠻⣮⠛⣛⣿⡋⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣯⡉⣽⠖⠀⢛⣿⡇
⠀⠀⠀⢸⣿⠿⠤⠋⠉⠈⢸⣿⣿⡇⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⣠⠖⠈⠀⠀⠀⣹⣿⣿⠁⠙⠉⣄⣹⣿⡇
⠀⠀⠀⠸⡹⣮⠀⡡⢰⣶⣿⣿⡯⠀⠀⠀⠀⠀⠘⡄⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠈⠻⣿⣷⠢⡈⠄⣠⢿⠇
⠀⠀⠀⠀⢿⣿⡟⠻⢿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠟⢳⣾⡷⠁
⠀⠀⠀⠀⠀⢻⣷⠈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠏⠁⣾⡏
⠀⠀⠀⠀⠀⠘⢿⡂⢼⣟⣿⣿⣶⣀⣴⠾⢻⣦⠀⠀⣿⣿⠀⠀⣴⡟⠷⣦⣀⣶⣿⢿⣓⡷⢐⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠺⣭⣉⡽⣻⣿⣧⣀⠀⢻⣷⡾⢿⡿⠿⢾⡟⠀⣀⣼⣿⣟⢫⣉⡭⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣷⣿⣿⣿⣶⣿⡅⣼⠀⢁⡻⣶⣿⣿⣿⣼⣿⡾⠋
⠀⠀⠀⢀⡀⠠⢤⣤⣤⡤⢄⣸⡃⠀⠉⠙⢿⣿⣟⣥⠁⠀⢠⣻⣿⡿⠋⠉⠀⢘⣇⡠⢄⣤⣤⡤⠄⢀⡀
⠀⢀⣴⣿⠟⠛⠋⠉⠙⠛⢿⣿⣵⣢⣀⠀⠀⣿⣿⣧⡟⢤⣼⣿⣿⠀⠀⣀⣀⣾⣿⡿⠛⠋⠉⠛⠛⠻⣿⣦⡀
⣰⣿⢋⣠⣴⣲⣶⣿⣯⣵⣾⡧⣄⡉⠛⢿⣶⠟⣫⠭⢭⡭⠭⢛⢿⣶⡿⠋⠉⣠⢼⣗⣪⣽⣷⣶⣖⣦⣄⡙⣿
⠉⣑⣭⡿⠋⠉⠀⠀⠉⠙⢻⡿⠶⢮⣷⣄⡽⣸⣿⡀⢸⠀⢀⠆⣄⣯⣰⣾⡽⠶⢻⡿⠋⠉⠀⠀⠉⠙⢿⣭⣊
⠞⠽⠋⠀⠀⠀⢀⢀⣠⣤⣿⣦⣤⣄⣈⣙⡇⣿⠃⢱⠠⠀⡆⠀⠿⢻⣋⣀⣠⣤⣴⣷⣤⣄⣀⡀⠀⠀⠀⠙⠯
⠀⠀⠀⠀⢀⣾⣷⠟⠋⠉⣿⠉⠉⠉⠉⠙⡇⠆⠀⠀⠀⠀⠀⠀⠐⣸⠋⠉⠉⠉⠙⣿⠉⠙⠻⢿⣷⡀
⠀⠀⠀⢠⣾⡟⠁⠀⠀⣠⣿⣶⣾⠶⠾⠻⣷⢨⣖⣀⣀⣀⣀⣀⠅⣿⠟⠶⠶⣿⣶⣿⣄⠀⠀⠀⢻⣷⡄
⠀⠀⠀⠋⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⠀⢸⡿⣿⣿⠟⠛⠻⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣷⡄⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⡿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣤⣾⣿⣿⠃⠀⠀⠀⠀⠀⢻⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡇⠀⠀⠀⠀⠀⠠⣿⣿⣟⣀⣀⣽⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣉⡉⣿⣿⣏⠀⠀⠀⠀⠀⠀⢸⡇
⠤⠤⠤⠤⠤⠤⠤⢄⠤⣾⠇⠠⠤⢠⠤⠀⠄⣿⣿⣟⠛⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣷
⠁⡀⠀⠀⠀⠀⠀⢀⢀⣇⡀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻
⣉⠀⣀⢀⡀⣀⣀⣀⢨⡇⣀⢀⣀⡀⣀⠁⣂⢻⣿⡷⠶⢶⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⡀⠀⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⡠⢶⣿⢿⣿⡉⠛⣿⡿⣿⣷⢄⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠞⢁⣿⠁⠈⣿⣧⢰⣿⠀⠈⣿⡀⠳⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⣧⣾⣿⣴⣶⡿⣿⣿⠿⣶⣦⣿⣷⣼⠀⠀⠀⠀⠀⠀⠀⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠒⠒⠒⠐⠂⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠉⠉⠉⠉⠉⠙⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢹⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⢀⡜⠀⠀⠀⠆⠀⠴⠀⠄⠀⠀⠇⠀⠀⠰⣄⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠉⢹⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡀⠂⠀⢀⠢⢀⠀⠀⡘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⣀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⠄⠀⠀⡄
⠀⠀⠀⠀⠀⢺⢁⠘⠓⠰⢰⣿⢷⡄⠀⠀⠀⠀⣾⢿⡇⠀⠈⠇⠌⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠗⠀⠀⣾⠿⡆⠀⠀⠀⠐⣰⢿⣷⠀⠀⡞⠀⠁
⠀⠀⠀⠀⠀⠘⠦⣀⢲⡀⠈⠉⠉⡀⠀⠀⠀⠀⠈⠉⠁⢀⡰⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠁⠘⣄⠀⠙⠋⠁⠀⠀⠀⠀⠈⠙⠁⠀⡠⠂⢀⠎
⠀⠀⠀⠀⠀⣀⣀⣀⡤⠚⠓⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠋⠠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠀⣠⠄⠓⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠂⠠⡀⠘⡄⡀
⠀⠀⠀⠀⠀⡇⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠂⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⡇
⠀⠀⠀⠀⠀⠇⡜⠀⠀⠊⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠠⠴⡥⢲⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠂⡽⠦
⢀⣀⡀⠀⢀⡀⡀⠀⠄⢀⡐⠀⠐⠀⠑⡀⠀⢀⡀⠠⠬⠁⠀⠀⠀⣇⠀⠀⢀⣀⣀⡀⠀⠀⣇⡀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣀⣀⠀⣸⠀⠀⠀⠀⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⡀⠆⠀⠀⢀⡀⡀⢠⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡅⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⢀⣀⣀⣰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⢀⠠⠂⠀⠀⠀⢜⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠐
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠄⠄⠒⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠂⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒
⠀⠀⠀⠀⣄⡀⢁⣀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠸⠀⢀⠔⠀⠘⡀⠀⢀⢆⠘⠉⠉⠉⠉⠉⠉⠁⠒⡀⡀⡀⠂
⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⡁⠈⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠈⠁⠾⣜⠂⠀⠀⠀⠀⠀⠀⠀⠐⠒⡃⠇⠉⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠈⠉⠁⠀⠠⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠀⠀⠀⠀⠒⠀⠀⠀⠐⠒⡒⠐⠒⠒⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠀⡆⠀⠀⠂⠐⠂⠀⠅⠀⠂⠠⣀⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠂⠀⠄⢰⠀⠄⠀⠀⠀⠰⠂⢀⡆⢀⠄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠄⠀⠌⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠋⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⡄⢀⠀⠁⠀⠰⠁⢀⠀⢠⣶⡢⢤⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣭⣿⣿⣿⣷⣿⡷⠀⠀⠄⢸⣿⡿⢿⣯⡭⢽⣧⡀⠀⠀⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⢀⣽⡏⠀⠀⠈⠻⣿⣯⡅⠁⠇⢸⣿⡷⠂⠀⠀⠀⢹⣧⠀⣠⢣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⡄⣸⢿⠁⠀⠀⠀⢠⣽⣯⢿⢟⠃⠜⢻⣤⠀⠀⠀⠀⠀⢿⣶⢛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⣿⡿⣆⠀⠀⠀⢤⣿⣻⣿⣷⢠⣢⣴⣿⣷⣇⠚⠀⠀⣠⡾⣿⠎⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⡀⠀⠀⠠⠓⣻⣿⡀⠀⠘⢟⣹⣿⠗⠋⢝⣿⣿⣿⡇⠀⢀⣄⣀⠛⢸⣆⡀⣀⣴⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠁⠀⠈⠻⢷⣄⣀⡊⢘⠻⠐⠃⢻⣿⣿⣿⣧⣠⣟⠝⠁⠀⠀⢈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠙⢿⢉⣶⢀⣡⣤⣿⣿⣿⣿⣿⠓⠁⠀⢀⡔⠃⠘⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠈⠑⢤⣀⡤⢪⡊⢄⢉⡭⣽⣽⣿⣿⣿⣀⣤⠔⠉⠀⠀⠀⠣⠀⡀⠀⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠑⠠⠒⠀⠀⠀⠀⠀⠀⠐⡀⠶⠀⣀⠀⠀⠙⣿⣿⠹⢾⣾⣿⣿⣿⣿⣿⣿⡟⢁⣀⣠⣤⡤⠶⢸⠉⠁⠀⠀⠀⠉⠙⠂⢤⣀
⠀⠀⠀⠀⠀⠘⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣐⠸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠶⠆⠶⠤⠴⠦⠴⠶⠶⣿⣿⣭⠽⠾⠆⠾⢿⣿⣿⣿⣷⣠⣤⣀⣀⣀⣀⣀⢡⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⣻⠵⠶⠶⡿⣾⣿⣿⣿⣏⢀⠀⠀⠀⠈⠉⠉⠁⢾⠉⠉⠁⠸⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣀⢀⣠⣤⠤⠶⠾⠛⡿⠚⠚⠛⢛⡛⣿⣿⣿⡟⠛⠳⠶⢦⣤⣀⣀⡀⢸⠆⠀⠀⠀⠀⠙⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠛⠋⠉⠀⠀⠀⠀⠀⢻⡒⣂⢀⣁⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠉⠙⠂⢸⡇⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠉⢋⣟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⢴⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠉⠳⡄⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣀⣀⣄⢸⣧⣰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠙⢦⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠭⠽⣿⣿⣿⣶⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⢾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣀⣠⣤⢰⡿⢿⣿⣿⣿⣿⣿⣿⣤⣤⣄⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢾⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⠙⠷⠂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⡿⠿⠿⣿⣿⡏⠉⠛⠻⣄⣀⣤⠒⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⡆⣴⣾⡿⠇⠀⠀⠙⠛⠉⠁⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⡀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠘⢻⣿⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣾⣿⠟⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⢺⠉⠗⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡌⢿⣿⣆⠀⠀⠀⣸⣿⡀⠀⠀⠀⣼⣿⠏⣸
⠀⠀⠀⠀⠄⠢⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠰⠂⠄⠀⠀⠀⠀⠘⣆⠛⣦⡹⢿⣿⣦⣼⣿⣿⣿⣦⣶⣿⠿⢋⡼⢋⡼
⠀⠈⠀⠀⠁⠀⣀⠊⠀⠀⠀⠀⠀⠀⠀⠘⣀⡀⠀⠁⠀⠂⠀⠀⠰⣌⠳⣬⡙⠶⣬⣹⣿⣿⣿⣿⣿⣉⡥⠞⣋⡴⢛⣱⠂
⠀⠀⠀⠀⠁⠐⠤⢀⣀⠀⠀⠀⠀⠀⢐⡀⠤⠔⠀⠡⠀⠀⠀⠀⠀⠙⠳⢦⣍⣓⠶⢼⣿⣿⣿⣿⣿⠷⢖⣛⣩⠶⠛⠁
⠀⠁⠀⠀⠄⠈⠁⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠲⣤⣄⣀⣀⣉⣉⣽⣿⣿⣿⣿⣿⣍⣉⣉⣀⣀⣤⡤⠞
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣿⣿⣿⣿⡇⠀⠈⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣽⣿⣿⣿⣵⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠟⠻⠿⠛⠿⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢻⣿⠷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⣼⠏⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠑⣇⠛⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠈⠩⠀⠐⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢿⡀⠘⠃⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢈⡁⢀⠀⡜⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠁⠀⠸⡰⣦⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣀⠀⣻⣞⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⡷⣻⢛⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠃⠛⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢙⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣀⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡆⠀⡘⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⣀⣰⣿⢃⡀⠀⠾⠀⠀⠀⠀⢀⠀⣄⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠘⣣⣷⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⣿⠿⠟⣁⠈⠟⠊⠀⠀⠀⠀⠀⠘⣂⠘⠣⣔⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣭⣟⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣇⠈⠀⠉⠙⢀⡀⠀⠀⠀⠀⠀⢐⣏⠀⠀⣩⡀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠠⣤⣿⣟⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⠖⢤⡤⣀⠀⢨⡅⠀⠀⠨⢋⣶⡀⠀⣄⣀⣌⢿⣿⣿⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠘⢿⡷⠞⠛⠋⠀⠈⣷⠿⣦⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣧⣽⣿⠿⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⠀⠀⠀⢀⣧⣀⠀⠑⢤⣸⡏⠀⢹⡄⠀⠀⠀⠀⢨⣿⡟⣾⣿⡿⠟⠋⠉⠀⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⡷⠆⠀⠀⠃⡙⠳⣄⡀⠉⡁⠀⣸⠀⠀⠀⠀⠀⣸⣿⡇⣿⣧⣤⡄⣠⣶⡴⢛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠒⣄⠈⠑⠊⠉⣿⠃⠀⢸⡀⢀⡀⡄⣰⣿⣿⡟⣿⣿⠿⠟⠋⢁⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠬⣻⢦⣀⠀⠈⡆⠀⠸⣷⣶⣾⣧⣿⣿⡿⣱⣿⣿⣄⠀⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠉⠀⠭⠿⣿⡇⠀⠀⠙⠋⠙⢫⠉⠋⣼⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⠀⠀⠀⠀⣀⣤⣤⣤⡶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠢⠤⠀⢀⡀⣟⠀⠀⠀⠀⠀⠀⢈⣾⣿⣿⣿⡟⠀⢀⣰⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠿⠉⣿⣠⣤⣤⣤⣀⣤⣿⣿⣿⣿⣿⣷⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠘⣻⣿⠋⠉⢉⣙⣛⣉⣭⣿⣿⣿⣿⣅⣴⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠁⠉⠉⢉⡟⠁⠈⠉⠉⠉⠉⢿⣿⣿⣿⡟⠉⠉⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⢀⣰⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⠉⠉⠉⣹⣿⣿⣿⣿⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⢨⣦⡄⢠⣤⣀⣀⣤⣿⣿⣿⣿⣇⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠤⠤⣤⣠⣤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⢰⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠒⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀⠀⠀⣨⣿⣿⣿⣟⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣿⡿⢿⣶⣿⣿⣿⣿⡟⢧⣌⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⡀⢸⣷⣾⣧⣄⣸⣿⣟⣀⣼⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠚⠛⠉⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡀⠈⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡆⠂⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠠⢄⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣓⣦⣄⣀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⢀⠊⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠲⠎⠓⢃⡂⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠔⠀⠀⠀⠁⠈⠀⠀⢄⠄⠈⠁⡠⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣾⢇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⡀⠐⠋⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠁⠈⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠒⠂⣀⡰⠀⠰⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠈⠁
⠀⠀⠀⠀⠀⠃⠠⠄⠁⡀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⡀⠈⠁⠒⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣂⢄⠀⠀⠠⠤⠐⠂⠁⠠⠄⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⠰⢄⠀⠁⠈⠲⠀⢨⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡴⢂⠠⠀⡀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠄⠃⠉⠐⢀⠄⣀⣀⠇
⠀⠀⠀⠀⡰⠂⠀⠉⠠⢈⠈⠀⠀⠂⠀⠢⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠀⠀⠀⠁⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⠀⠀⠘⠀⠀⠀⡀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠸⡀⠀⡠⠃⠀⠀⠀⠐⡅⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⣘⣆⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠤⢠⠃⠀⠀⠀⠀⠰⣀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡖⢀⢀⠀⠈⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠀⣠⠽⠞⠀⠀⠀⠀⠌⠠⢄⡙⠛⡚⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠇⢀⡀⠈⠉⣃⠀⠉⠀⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠤⢠⡥⠀⠀⠀⠀⠀⡰⠮⡓⠊⠉
⠀⠀⠀⠀⠀⠀⢠⠈⢀⠂⡘⠁⠈⠀⠌⠂⠀⣰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⡀⠆⠓⠺⡇⠀⠀⢀⠀⣼⡁⠀⠐⠢⣄⠀⣀⡤⠂
⠀⠀⠀⠀⠀⠇⠀⠳⢄⠀⢠⠀⠀⠀⠈⡤⠄⠁⢀⠼⠁⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠂⡘⠃⢀⡘⠒⠤⣀
⠀⠀⠀⠀⠀⠈⠢⢀⡀⠀⣡⠀⠀⠀⠀⠰⠂⠀⠀⢀⠀⠔⠃⠀⠀⠀⠈⠀⠀⠀⡎⠀⠀⠀⠞⣇⠐⣠⡖⠀⠀⠀⠀⠠⡄⠈⢀
⠀⠀⠀⠀⠀⠢⠤⠤⢤⡠⠘⡆⠀⠀⢀⠚⠠⣈⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠊⠀⡇⠀⠀⠀⠀⠘⠋⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡃⠀⠈⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡡⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠹⠀⠀⢨⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⣶⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡞⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡡⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⣶⠶⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⠃⠀⠀⡏⢆⢀⣀⣤⠀⠀⠀⠀⢀⡞⢡⢃⣳⡤⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣄⠀⢀⣤⣖⡃⢀⡠⣞⠁⠀⢴⡆⠀⠚⠒⠋⠉⠈⢧⠀⠀⣀⡤⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⣿⠀⠀⠀⠀⣧⡏⠉⠛⠆⠀⠀⠀⣀⡤⠶⠶⢤⣄⡈⠂⠼⢉⣤⣶⡿⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⢀⡼⠃⠀⠀⠀⠀⣙⣀⠀⠀⢀⣤⣤⡀⠋⠀⠀⣠⣤⣬⡙⠳⣴⣿⡟⠁⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣠⠞⠁⠀⠀⠀⢠⣾⡟⣹⠇⣴⡿⠋⢩⣿⡇⠀⣾⡿⠉⢹⡇⢰⣿⡟⠛⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠁⢠⠟⠛⣷⣄⣼⣿⠟⠁⢰⣿⠁⠀⣼⣿⠁⢰⣿⡇⠛⠋⠀⣿⡿⢀⡴⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢹⣿⡆⣾⠀⢀⣼⣯⠘⣿⡀⣠⠿⣿⣠⡾⢻⡏⠀⣸⢿⣇⠀⢀⡞⣿⣇⡾⢁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠻⣿⣿⠀⣏⣸⠇⠀⠈⠙⠉⠀⠈⠁⠀⠈⠳⠶⠃⠈⠙⠛⠋⠀⠙⠿⠁⢸⡏⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡀⠈⠙⢀⣩⣵⡶⠿⠟⣛⣛⣛⡛⠓⠶⠶⣤⣄⣀⠐⠶⢤⣤⣤⣽⠇⠀⠘⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠟⠀⠠⡿⠋⠁⢠⡾⠛⠋⠉⠉⠙⠋⣠⣤⣤⡀⠉⠛⡳⠶⢦⣤⣤⡼⠇⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢠⡶⣌⠁⠀⢠⣤⡀⠀⣰⣿⠋⢈⡇⠀⠸⡿⠂⠀⠀⠀⠀⢀⣠⣬⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⣿⡆⠀⠀⠳⣼⣆⠀⢈⣋⠀⢀⣿⣿⣠⠞⠁⠀⠀⠀⣰⡶⠶⣶⢀⣿⡿⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⣿⠀⠀⠀⢸⡏⠉⣿⡿⠀⠸⣿⡏⢁⡴⢶⣄⢀⣼⣿⣋⣠⠟⢸⣿⡇⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡄⠀⠸⣿⣆⠀⠀⢸⡇⢰⣿⡇⠀⢠⣿⣧⡟⢀⣼⣿⠉⣿⣿⠉⠀⢀⡾⠻⣿⡄⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⢀⣼⠀⠀⢿⣧⣠⠟⠘⣿⠇⣿⣾⠇⠀⠘⢿⣤⣤⠞⠁⣠⠾⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡀⠀⢠⠀⠀⡐⠀⢀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢋⡴⠊⢿⣤⢸⡆⢀⡇⣤⡿⠑⢦⡙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢀⡞⠀⢀⡼⣿⢸⡇⢸⠀⣿⢦⡀⠀⢳⡀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⡼⠀⢀⡾⢠⡇⢸⠀⢸⠀⢹⡄⣷⡀⠀⢧⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢠⡇⢠⠏⠀⡞⠀⡏⠀⢸⠀⠀⢳⡀⠹⡄⢸⡄⠀⠀⠁⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢴⢁⠏⠀⢸⠁⢰⠃⠀⠘⡄⠀⠈⣇⠀⠹⡈⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢠⠀⠀⠸⣏⣀⣤⠇⠀⢸⠀⠀⠀⡇⠀⠀⠘⣦⣤⡹⠇⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣧⠀⠀⠀⢀⣽⠃⠀⠀⡏⠀⠀⠀⣽⠀⠀⠀⠘⣯⡀⠀⠀⠀⣼⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢿⣷⣦⣾⢟⡜⠀⠀⠀⣧⠀⠀⠀⢸⠀⠀⠀⠀⢣⡻⣷⣴⣾⣿⡽
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠾⠓⢏⠀⠀⠀⠀⢸⣆⢀⡀⣸⠀⠀⠀⠀⠀⡹⠚⠷⠒⣻⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⢀⡜⠀⠀⠀⠀⠐⣿⡾⢷⣿⠂⠀⠀⠀⠀⢣⡀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⡸⣏
⠀⠀⠀⠀⠀⠙⣾⣄⠀⠀⠀⠀⠀⠀⠘⣿⣶⣗⣧⠀⠀⠀⠠⣿⣿⠁⠈⣿⣿⠀⠀⠀⠀⣼⣺⣶⣿⠃⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠠⣿⣿⠋⠘⣇⠀⠀⢠⠿⠋⠀⠀⠙⠿⡄⠀⠀⣸⠀⠙⢻⡿⠀⠀⠀⠀⠀⢀⣴⡷⠃
⢻⣆⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠻⣿⣦⡀⠙⢦⠀⡏⠀⠀⠀⠀⠀⠀⢹⠀⡴⠁⢀⣴⣿⠟⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⣰
⠀⠻⣷⣤⡀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⠈⠻⣝⣦⣀⣿⣇⠀⠀⠀⠀⠀⠀⣸⣿⣁⣼⣫⠟⠁⠀⠀⠀⣀⣴⡿⠁⠀⠀⠀⠀⢀⣤⣾⠟
⠀⠀⠈⠙⠿⣶⣄⡀⠀⠀⠀⠙⠣⠿⣷⣶⣤⣀⡈⠙⣟⡾⢏⠷⣄⣀⣀⣠⠞⡽⢷⣻⠋⣁⣀⣤⣴⣾⠯⠟⠋⠀⠀⠀⢀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡤⣤⣄⣀⣀⣀⠉⠉⠛⠻⣽⡇⠀⠁⠈⠛⠛⠁⠈⠀⢹⣭⠟⠛⠉⠉⣁⣀⣀⣠⣤⢤⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⠿⠿⢷⣖⣺⠃⠀⠀⠀⢀⠀⠀⠀⠀⠸⣗⣲⡾⠿⠿⠿⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡁
⠀⠀⠀⠀⢀⣀⣠⣴⣤⣴⣴⣶⣶⣶⣭⣭⡭⠭⠽⠿⢿⣧⣄⠀⠀⠀⠀⠀⠀⣠⢾⡿⠿⠯⠭⢭⣭⣭⣶⣿⣶⣶⣦⣤⣦⣄⣀⡀
⠀⣠⣴⠾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣷⡙⢷⣶⣤⣤⣶⡿⢋⣾⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠷⣦⣄
⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠁⠀⡘⢻⡶⠦⠤⠤⢴⢶⣟⢇⠀⠈⠙⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠁⠘⠃⠀⠀⠀⠀⢈⣙⡼⠀⠀⠀⠀⠀⠈⠛⠻⢧⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣠⣄⡀⠉⠙⢛⣻⣯⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠙⠛⠦⠤⠤⠴⣾⡻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⠶⠀⠐⠂⠀⠲⣃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢦⡈⠙⠋⢉⣵⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⠀⠀⠉⠉⠉⣍⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠈⠉⠁⠈⠉⠁⠉⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⠀⠀⠀⠀⡌⠀⠘⡇⢸⠀⠀⢢⠀⠀⠀⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡘⠀⠀⠀⢹⡏⠀⠀⠀⢣⠀⢀⡴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣲⣤⣤⣞⣳⣤⣤⣖⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣠⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢸⡀⢸⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⠀⢸⠀⣸⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠳⠜⠀⠈⠃⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠋⠀⢘⠴⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⢀⡀⠀⠊⠀⠀⠀⠂⠄⠀⣀⠀⠀⢠⠎⠀⠀⠈⠃⠀⣀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠘⠐⠏⠀⠀⠀⠀⠀⠀⢀⣀⣔⣻⡄⠈⣜⠀⠀⠀⠀⠀⠀⠈⠗⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠠⠀⠀⢐⠔⠹⠃⠿⠘⠃⠀⠀⠀⠀⠀⠈⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠊⠁⠀⡀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠙⠂⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠃⠀⠀⠠⠤⡞⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠀⠀⠀⠀⢊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠈⣁⠀⠀⠀⠀⠀⠀⠀⢈⡁⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠊⠉⡄⠀⠀⠀⠀⠀⢀⡌⠉⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠠⠁⠀⠀⠀⠀⠁⠨⠂⠀⠠⠀⠠⠆⠈⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠲⠀⠸⠤⠄⠤⠄⠀⣄⠀⠰⠣⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⢄⡀⠀⠀⠀⠀⢀⠄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠓⠀⠤⣤⡤⠄⢊⠀⠀⠀⠀⠀⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠠⠠⠄⠀⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠀⠄⠀⠀⠀⠤⠽⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠐⠳⠄⠺⠤⠤⠤⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡤⠄⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣠⠶⠖⢿⠉⠉⢹⡏⠀⢸⡿⠉⠉⠉⠁⠀⠸⣁⠀⣀⠄⠀⠀⣲⠍⢁⣉⡁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⣠⣾⠋⢀⡤⠇⠀⠀⠉⠷⢴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⢤⡾⡯⠀⠓⠚⠛⠒⠀⠉⠉⠀⠀⠀⠁⠐⠂⠄⠀⠘
⠀⢸⢇⢨⡟⠯⡀⢑⣀⡀⠀⠀⠸⣟⠡⢀⠀⠀⠀⠀⠀⢀⣠⡀⣠⣿⢋⡇⣢⠴⠖⠒⠂⠀⣀⡀⠂⣶⣋⣽⣿⡯⠯⢽⢀⣠
⠀⢸⡈⠙⠒⣤⠾⠍⠡⡄⢸⠆⠀⠉⢿⠓⢾⠶⣾⠒⠲⣛⣪⢹⡛⠿⠽⠥⠾⢥⣤⠔⢲⣾⡟⠀⠀⠀⠉⠁⠁⠒⠒⢈⡡⠃
⠀⠘⣿⠃⠘⠃⠉⠚⠂⠙⠀⠀⠀⠀⠀⠂⠈⠦⠈⡛⡄⢠⡉⠉⣓⡐⠒⠒⠒⠚⠲⠾⠿⠓⠿⠯⠶⠶⠶⠶⠶⠞⠊⠇
⠀⠀⠈⠣⣄⠀⠐⠒⠠⢤⡀⢠⡀⠀⠀⠀⠸⣆⠀⠨⢷⡈⠐⢦⡉⠙⠁⣢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠘⢦⣀⠀⠀⣸⠃⠘⠀⠀⠀⠀⠀⠘⠀⠀⠈⡇⠀⠀⠙⣶⢄⠀⠐⢽⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⣀⠀⠀⠀⠀⠈⢣⣷⡀⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡠⠊⢀⣤⣶⣦⣄⠚⠻⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⡉⢹⠉⣵⣿⡛⣿⣿⣿⣷⠄⢱⠀⡠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⢎⠷⠿⠻⠿⣸⣔⡝⠛⢛⡿⢷⢏⢞⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠕⠉⢹⠫⣇⢉⣀⢀⣸⠀⠠⠤⠷⠉⢿⠉⠑⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠘⢟⣠⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⣇⠮⠄⠀⠤⠧⠨⠇⠀⠬⢦⡏⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠲⡄⠠⣤⣤⣤⣤⣤⡤⠀⡴⡂⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠈⠙⠛⠛⠉⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⣀⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⣠⣾⣿⠟⠁⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⢀⣿⣿⠋⠀⢀⣠⣴⠞⠙⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣼⣿⣷⣶⣿⠿⠛⡁⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣤⣤⣴⣾⣿⣿⣿⡏⡧⠔⠊⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡐⠉⢸⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣀⡠⠄⠐⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⣿⠀⢀⠎⠀⢀⣼⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⡤⠒⠉⠀⢀⡠⠞⣻⠏⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣿⠀⠸⣤⠤⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣠⠖⠁⠀⠀⠀⣠⠏⣠⣾⠇⠀⠀⣰⡟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢻⡄⠈⡏⠈⢳⣯⠃⠀⠀⠀⠀⠀⣀⣀⣠⣊⠀⠔⠁⠀⠀⠀⠀⢰⡏⢠⣿⡏⠀⢀⣴⡿⠁⢸
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠘⣆⠀⢳⣄⠐⡄⠀⢻⡄⠀⣀⣤⠴⢾⣿⡿⣿⣿⠆⠀⠀⠀⠀⠀⠀⣟⣓⣻⠿⠀⣠⣾⡟⠀⠀⡼
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠛⠦⣄⡈⠳⣄⠙⢦⡈⢶⣼⠷⠋⠉⠀⠀⠈⠡⣾⣿⡿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡾⠿⠋⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠙⠦⣌⠓⢄⣽⠟⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠁⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠉⠙⠳⠤⡈⣷⣿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠇⠀⠀⣀⣀⣀⠀⣈⣿⢿⡏⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠛⠿⢷⣝⣿⡿⠛⠉⠀⢩⢻⣦⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠈⠙⢷⣦⣤⣤⣴⣾⣿⣿⣿⠏⠑⠠⢀⣀⣈⣼⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠸⢷⣤⡀⠀⠀⢹⣿⣿⣿⣿⣿⡿⣃⠙⢦⣀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡀⠀⠈⠻⣶⣶⣿⣿⡟⣿⠿⡋⢢⡈⠳⣄⠉⠓⠶⢤⣤⣤⡤⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣇⠀⠈⠛⣦⣄⣠⣿⡿⠿⠟⠁⠀⠀⠙⣆⠙⢦⡈⠓⢤⣄⣀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⡉⠉⠻⣦⣀⣸⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠹⣄⠀⠈⠙⠳⣤⡀
⠀⠀⢰⣷⣦⣤⣤⣶⣴⡉⠻⢦⣀⣸⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠙⣧⠀⠀⠀⠀⠙⠆
⠀⠀⠀⢉⡟⣁⣠⡼⢿⣟⣦⣨⣿⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣿
⠀⠀⠀⠈⢿⡟⠁⣠⣾⣿⠳⡯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠻⠿⢟⡷⢠⣷⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⣼⣷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡑⠀⠘⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⠀⠀⡠⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⠤⢶⠊⣠⢼
⠀⠀⠀⠀⠀⠀⢠⠖⡉⠂⠄⡻⣀⠀⠀⠀⠀⠀⠀⢠⣮⡤⢤⡈⣙⣿⣯⡫⠠⣌⣷⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⡟⠘⣧⠀⠉⠀⢸⠄⠀⠀⠀⠀⠀⣿⣿⡇⢀⢼⠋⢩⡇⠳⣸⣃⢀⣼⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠙⣦⡙⡒⠤⠤⣤⠇⠀⠀⠀⣀⣼⣿⣿⢷⢃⡏⡄⠀⠙⡲⣿⡻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢀⠠⢤⣸⠙⠈⣦⣄⣚⣻⣷⣴⣿⣿⣿⡿⠋⢸⡾⠀⠀⠀⢠⠗⣻⣷⣿⣿⠟⠛⠋⠉⠁
⠀⠀⠀⢀⣏⠿⠼⠃⠀⠀⣿⣿⣟⣿⣿⣿⣿⣿⣿⡄⠀⢸⡇⠐⠀⠀⡞⣼⣿⣿⣿⠋
⠀⠀⠀⠘⢧⣤⠀⢀⣠⠞⠙⢿⣾⣿⣿⣿⣿⣿⣯⢀⣤⣿⣿⡀⠀⠀⣷⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣻⡿⠿⡿⢷⣤⡾⠿⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⣛⡿⡟⠁⠀⠀⠀⢸⡛⠂⠀⢀⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⣿⣿⡇⠀⠀⠀⠀⢸⣧⣿⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⣿⣿⣿⣿⡿⠀⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡛⠕⠉⠈⢳⡀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠈⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢿⣷⣄⠀⠀⠀⢳⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠀⢀⡟⠉⠙⢆⠀⠈⢧⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⢀⡤⠊⠀⠀⠀⠈⣇⠀⠈⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⢀⣤⠤⠴⠊⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄
⣤⣾⠏⠉⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠛⢆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠙⣶
⢸⣧⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⣏⠀⠀⠀⢣⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⣆⣿
⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣤⠞⠁⠀⠀⠀⢿⣧⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠁⠀⠀⠀⠀⠀⠀⠻⠳⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠂⠀⡝⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢫⠀⠐⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⢈⠀⡰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢂⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⡖⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢲⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠸⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⡀⠠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠤⢀⡀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢊⠁⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠈⠱⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠈⡄⠀⠀⠀⢆⠀⠀⠀⠀⢀⡴⠺⠚⠻⡖⢦⡀⠀⠀⠀⠀⡰⠀⠀⠀⢠⠁⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢃⠱⡈⢄⠀⠀⢀⡦⢄⠀⣠⠎⡠⠋⠉⠉⠛⢄⠱⣄⠀⡠⢶⡀⠀⠀⡰⢁⠎⡘⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠑⡈⢢⡬⠲⣗⡉⠉⠀⠀⡝⠃⠌⠀⠀⠀⠀⠀⠀⢣⠘⢫⠀⠀⠉⢉⣺⠔⢅⡔⢁⠊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠡⡈⢢⣕⢄⡈⠪⣕⢠⣎⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⢱⡄⣪⠕⢁⡠⣪⡐⢁⠜⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠢⣘⠯⡀⠑⠬⣺⣀⡍⠒⠬⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠒⢩⡀⣗⠥⠊⢀⠽⣃⠔⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡱⠦⢕⠤⣋⡙⠐⠠⢴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠄⠀⢋⣙⠤⡂⠱⢎⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠑⠠⠀⢈⡸⠋⠖⠲⢴⡅⠀⠀⠀⠀⠀⠀⠀⠀⢨⡦⠔⠲⠙⢇⡁⠀⠄⠊⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠂⠄⠀⣀⡀⠀⠈⠉⠐⠒⠒⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠒⠒⠀⠉⠀⠀⣀⣀⠀⠤⠐⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠼⣀⣈⣉⣉⣋⡁⠠⠄⠲⡀⣀⣀⣀⣀⣀⢀⣀⠖⠠⠄⢉⣹⣉⣉⣁⣀⠧⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠤⠤⠤⠤⠤⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠒⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⣀⣀⣀⡀⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⡦⠤⠄⠀⠀⠠⠤⣖⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⢀⠎⠱⠀⠀⠀⠀⠀⢀⠉⢢⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠼⡀⠀⣣⡀⠀⠀⣠⡎⠀⢀⠦⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢾⡷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⢸⡇⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⢰⡇⢸⡇⢸⡆⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣿⣀⠀⠀⠀⠀⠸⡇⢸⡇⢸⠇⠀⠀⠀⠀⣀⣿⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠙⠓⢶⣄⡀⠀⠀⢸⡇⠀⠀⢀⣤⡶⠚⠉⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠳⢦⣼⣧⡴⠞⠋⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⡾⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠋⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣧⠀⠀⢀⣠⡴⠞⠛⠉⢉⣉⣉⠛⠳⢦⣄⡀⠀⠀⣼⢷⣄
⠀⢸⠛⠲⢦⣤⣀⠀⠋⠀⠘⣧⣴⠟⠁⠀⠀⠀⠀⠈⠉⠉⠙⠳⢦⣈⠻⣦⣼⠃⠀⠙⠀⣀⣤⡴⠖⠛⡇
⠀⣾⠀⠀⠀⠀⠉⠛⠳⠶⣤⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡈⢻⣤⠶⠞⠛⠉⠀⠀⠀⠀⣷
⢸⡇⠀⢀⣄⣀⣀⣀⡀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢻⡄⢀⣀⣀⣀⣠⡀⠀⢸
⣾⠁⢀⡿⠁⠉⠉⢉⣙⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣋⡉⠉⠉⠈⢿⡀⠈
⠀⠀⣼⠃⠀⣴⢶⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻⡶⣦⠀⠘⣧
⠀⠰⠏⠀⣸⠇⠀⠉⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⡟⠉⠀⠸⣇⠀⠹⠄
⠀⠀⠀⢠⡏⠀⠀⠀⠘⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣾⠃⠀⠀⠀⢹⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠻⣦⡙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠈⠛⢦⣌⡙⠁⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀⠀⠀⠀⣼⠛⠛⠓⠲⠖⠚⠛⠛⣧⠀⠀⠀⢀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡾⠃⢠⡗⠀⠀⠀⠀⢲⡄⠘⢷⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⡾⠁⠀⠀⠀⠀⠈⢷⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠶⠛⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣤⣀⣈⣉⣉⣥⣤⡴⠶⠛⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢰⡐⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢲⣄⠀⠘⣬⡀⠀⠀⠀⠀⠈⠣⡀⠀⣠⣤⣤⠄⠶⠤⠠⠠⠀⠠⢄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠓⠛⠒⢢⣌⠙⢶⡀⢹⠧⠀⠀⠀⠀⣀⣤⣹⣾⠋⢴⢉⠑⠁⠀⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠘⢋⣉⣉⣉⣑⣄⣀⣈⣧⣀⣷⡀⢾⡀⠀⣠⣒⠞⢛⣡⠻⡖⢦⡥⣀⣐⡂⣨⣄⡉⠈⠉⠛⠀⠀⠀⠀⣿⠃
⠀⠀⣰⣶⣶⣢⣀⣠⣤⣴⣶⣗⣿⣿⣿⣿⣿⣯⣿⣿⢿⣿⣿⣿⣿⣿⣿⠷⣶⣾⣘⡶⠚⠁⠀⢱⠀⠈⢀⠌⠈⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⣸⠷⠆⣛⠱⠽⠟⢯⠿⠉⡟⠘⢏⠉⣸⠏⢙⡞⢹⡿⠿⢛⣿⠿⣿⠟⢿⣷⣾⣭⣿⣿⣶⣶⡿⣀⣴⠡⠐⡚⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿
⠀⢣⡤⠁⢤⣤⡶⡦⠈⣇⠀⣮⠀⢸⡀⢠⡇⠁⢧⣾⠟⠠⣀⠁⠈⠹⠈⠈⢿⣻⣿⣿⣿⣩⣿⣿⡭⠤⠒⠊⡠⣠⣞⡍⡡⢂⠔⠀⠀⠀⠀⣿
⠀⠘⢷⣀⡀⠊⢹⡝⠴⠻⠄⠙⠷⠈⠓⠚⠷⣄⣼⡟⠀⠐⠀⠂⠀⠀⠀⠀⠀⣿⡿⣛⡩⣿⣟⡟⡏⠑⠒⠨⣴⣯⠫⣜⠐⠁⠀⠠⠀⠀⠀⣿
⠀⠀⠘⣟⢃⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣥⣀⣀⣀⣀⣀⣀⣠⡶⠟⠛⠛⠉⠻⡏⢿⠇⠀⠰⠀⠀⠙⢷⣭⡆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⣶⡆⡟⢺⠯⡙⣄⢀⡨⠛⣤⠀⢩⣞⠁⠀⠂⠀⠀⠠⣻⢯⣥⡆⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠊⠁⢙⣾⣯⠝⣿⠙⢾⣧⣤⡂⣠⠏⠀⠀⠀⠀⠀⣠⣶⡡⠮⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣴⡶⠊⠁⢀⣤⣾⢽⡿⠁⣸⡇⠀⠀⠀⠘⢻⠯⠌⠻⠲⡠⠰⠂⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠖⠋⡽⣇⡞⠁⠀⣽⠇⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⠀⣼⣿⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡬⢼⠋⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠐⣁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⡴⠶⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢣⠀⢈⠀⠁⠀⣇⠀⠀⢠⡃⡇⠀⣸⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡇⢸⠀⠀⠀⡏⠀⠀⡞⠙⡇⣰⢻⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣺⠈⢦⠴⠀⠧⠄⠰⠃⢰⢀⠇⢰⠇⠀⠀⠀⠀⠀⡜⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣞⡴⠋⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⢀⡴⠋⣀⡔⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡤⠊⠉⡏⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⢶⡤⣤⣄⡀⠈⣽⠂⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉⠀⠀⣀⣤⣀⠀⠀⠀⠙⠒⢊⡠⠴⠚⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⣁⡡⣭⣷⠾⡤⠶⠻⠙⠋⠋⠛⠓⠀⠀⠀⠀⢀⡞⠁⠀⢄⠹⡄⠀⠀⠰⡅⠀⠀⠀⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠫⠙⠀⣧⡾⢶⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠺⠀⠀⠀⠈⠇⣹⣀⠀⠀⠀⡔⠃⢠⠏⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⣠⢊⡞⠉⠁⠀⠀⠈⠳⣄⠀⠀⠀⢠⠈⠁⠀⠤⠴⣄⣀⣠⠀⠉⠉⠙⡄⠀⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢧⡞⠀⠀⣀⣀⠀⠀⠀⠘⠆⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠁⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⡇⠈⡗⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠉⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⢽⡀⠀⠈⠓⢄⢀⡠⠤⢄⡀⠀⠀⠀⠀⠈⡇⠀⢀⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⠙⣄⠀⠀⠙⠦⣀⣀⣀⡴⠃⠀⠀⢳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⠉⠉⠓⠲⠶⠶⠚⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⠉⠒⠢⠤⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄
⠀⠀⠀⠀⠀⡰⢿⡿⠤⢿⡿⢄⠀⠀⠀⠀⠀⠀⠀⠖⠐⠀⠉⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⢿⣿⣿⣶
⠀⠀⠀⢸⠭⠤⠔⢀⣀⡌⠢⠤⠯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠁⠈⠉⠙⣿⡇
⠀⠀⠀⢸⠀⠀⢻⣿⣿⣿⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠛⠋⢸⣿⠠⠁⠀⠀⠀⠀⠀⠀⢸⣿⣤⣶⣶⣷⣶⣦⣿⡇
⠀⠀⠀⢸⢤⣠⣾⣿⡿⠿⠟⠛⠀⢃⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⣿⣿⣿⠟⣿⡇
⠀⠀⠀⢉⠀⠉⠉⠀⠐⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠾⠆⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠿⣄⢤⣥⣌⣤⣤⠿⡇
⠀⠀⠀⠀⠀⠣⡀⢠⠒⡀⠀⠊⠀⠀⠀⠀⢀⣀⣰⣶⡶⠄⣀⠄⢄⣤⣶⢄⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⠉⠁⠀⠀⠙⠛⠋⠉⠉⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣤⡤⠤⣾⣿⡧⠤⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢞⣷⣿⡳⣤⣄
⠀⠀⠀⠀⣿⣿⣿⡏⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⣿⣶⣬⠛⠛⣵⣶⣿
⠀⠀⠀⠀⡿⣿⡿⢀⣴⠛⢿⣿⠿⠀⠀⠀⠀⠀⠀⡠⡔⢶⢾⡣⡣⡦⢄⡇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⣿⡟⠁⣾⣯⣶⡈⣿⣿⠀⠀⠀⠀⢠⡶⣭⣵⣷⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣤⣿⣷⢠⣯⠈⠁⠀⣿⣿⣤⠀⠀⠀⠀⠁⠘⣿⠿⢿⣯⣿⠏⠀⠀⠀⠀⠀⠀⢹⣿⠻⣟⣻⣷⡿⠃⣿⡏
⠀⠀⠈⢿⣯⣥⣤⣶⡴⢶⣦⣭⣿⡟⠀⠀⠀⠀⠀⢀⣈⣻⣟⣉⣁⣀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣬⣥⣴⡾⠿⣿⠇
⠀⠀⠀⠀⢅⡐⠋⠛⠙⠀⠐⣀⠍⠀⠀⠀⠀⠨⠿⠛⡛⠛⠃⠑⠘⡛⠻⠯⠀⠀⠀⠀⠀⠙⠛⣏⠉⠀⣰⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀⣾⣿⣋⡁⠀⠀⠀⠀⠀⠀⠤⠶⠿⠶⠶⠄
⠀⠀⢀⣀⡀⠀⢠⠾⠿⣿⣿⠛⢿⣿⡛⠛⠆⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣄⡀
⠀⠀⠀⠈⠁⠀⠀⠀⣾⠟⠁⠀⠀⠻⣿⡞⢷⣾⡷⣀⣀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢠⣄⠀⠋⠀⠀⠀⠀⠀⠘⠛⠛⠉⢻⣿⣟⠛⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠰⠞⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠙⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣴⣦⣦⣦⣤⣤⣴⣶⣤⡀⢤⣤⣤⢠⣤⣶⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠈⠁⠉⠉⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⢸⢠⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡎⠀⠀⢸⢸⢸⠀⠀⠈⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⠤⣀⢸⢀⡠⠔⠊⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⠀⠀⠀⠀⠈⢹⠁⠀⠀⠀⠀⢸⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⣀⠤⠖⢚⣒⡢⢄⡀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠒⠦⢤⣸⠞⠁⠀⠀⠀⠀⠉⠳⣝⢾⣠⠤⠖⠊⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣠⠤⠤⢤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢧⠤⠤⢴⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠇⡰⢾⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⠶⡀⢧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠃⠀⢻⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠃⠀⢳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠈⠳⣝⠢⠀⠀⠀⠀⣀⡴⠋⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⢠⠟⡓⠒⠒⡙⢧⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣸⠃⠀⠀⢻⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⠀⠀⢀⣀⣀⡤⢆⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠊⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⢀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡡⠊⢑⡄⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⢐⢑⠔⠁⠈⢐⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠣⠔⠀⠀⠀⢠⣿⣿⣿⣿⠛⠛⠛⠛⠉⠁⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠅⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢅⠕⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠠⠄⠀⠀⠀⠀⢸⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁
⠠⢁⠓⢄
⠱⣆⠀⢎⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣤⣶⣶⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢨⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠻⠿⠿⠉⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡨⠖⠋⠑⢌⣳
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠠⠈⠒⢿⡷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣩⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡽⠚⠈⠀⠀⠀⠂⠀⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢄⠀⠈⠀⠠⣿⡄⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⢆⠀⠀⠀⠀⠀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠈⠙⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠛⠛⠿⡇⠀⠀⢸⣿⡿⠿⠛⠓⠄

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⡇⡟⡝⠉⣫⡏⡯⡏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⣷⢇⢇⣀⣷⣗⡷⣇⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⡴⠋⠁⠀⠀⠙
⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⡄⠅⠀⢼⡦⠚⡤⢄⣠⠞⣛⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⡀⠀⠄⠂⠀⠀⠄⠈⠀⢇⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣷⣿⢷⣶⣦⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠄⠀⠀⢸⠀⠀⠀⠁⠀⠀⠈⠀⢀⠤⢴⡾⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⢠⢂⠀⠰⠀⠀⠀⡀⡀⣄⡞⠈⠳⣤⣀⣀⣠⣤⡴⢾⠀⠀⠀⣀⣤⠤⠤⣮⣭⣉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠠⠺⢤⣹⣦⣾⡐⠊⠈⠉⠀⠀⣀⣾⣲⡶⠈⠻⣍⠀⠑⠀⢿⠟⠁⠀⠐⠐⠂⠉⠻⡆⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠄⠀⢸⣿⣿⣿⣿⡆⠀⣀⡴⠚⠍⠉⣿⠅⠀⠀⠊⠙⠳⠦⡈⠳⣄⠀⠀⠀⠰⠀⠈⠳⡶⠶⠒⠒⠒⠒⠒⠢⡄
⠀⠀⠀⠀⠀⢸⠃⢐⣾⣿⠿⠿⠛⠀⢐⠀⠀⠀⠀⢀⣟⠀⠀⠀⠀⠀⠀⠁⢿⠀⠈⠙⠶⣤⣀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠛⠖⠊⠀⠀⠀⠀⠀⠄⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠉⠙⠲⠶⠤⣤⣀⣀⣁⣀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⣤⣤⡄⠄⠀⠰⡆⠰⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⣶⣀⣄⡰⠋⠀⢦⠀⣀⣤⢤⠒⠶⠉⠈⠁⠁⠀⠰⡄⠀⠀⠀⣧⠀⢀⡋⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⢁⣠⣼⣿⣿⣿⣿⡦⣠⡾⠟⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠻⣾⣥⣄⢻⣿⡌⣷⡆⣏⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠁⢀⣰⣾⣿⣿⡿⢠⠛⣥⠞⡹⠁⠀⠀⠀⠀⠐⠈⢦⣄⡶⣦⣿⣿⣷⣝⣿⣯⣸⣿⣧⡿⣣⣟⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢠⣤⢿⣿⣿⡿⠟⠁⠐⢸⠟⡔⣡⣼⣔⠁⠂⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣻⣧⣶⣶⣭⣾⣿⠟⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢠⣶⠟⣡⣿⣿⠏⠀⠀⠀⠀⣁⣼⡁⠍⣥⣡⣬⢲⣆⠀⢠⣿⣿⣿⣿⡿⢿⡉⠋⢉⡽⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣾⡟⢡⣾⣿⡟⠁⠀⠀⠀⠰⣊⣿⣿⣟⣽⣿⣿⣿⡿⣿⡳⡾⢿⣿⡋⠉⢶⣄⠹⠆⠸⣧⡈⢙⡭⠋
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⣴⣿⡿⠋⠀⠐⠒⠒⠒⢺⡿⠿⠟⠛⠉⠛⠛⡛⠚⣛⣻⣧⠀⠙⣇⠀⢈⡿⢈⡇⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡟⢣⣾⡿⠋⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡆⢠⡏⠀⣸⠇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠴⠛⠁⠀⠀⠀⣀⣀⣀⣀⠀⠁⠀⣤⡀⣰⠄⣤⠀⣶⣶⣿⣿⣿⣿⠃⢸⠇⢠⡅⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⠙⢉⠉⠛⠻⣿⣿⡿⠀⠀⠀⠀⠀⠙⢿⣯⢻⣿⠟⢠⠟⠀⡸⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣩⣥⣶⣶⣶⣶⣷⣿⣿⣿⣿⠿⠛⠁⢠⣀⣠⡤⡲⠏⠘⡿⠟⠉⡴⠋⠀⠀⠁
⠀⠀⠀⣤⢀⠀⠟⣉⢉⡁⠠⠄⠻⠭⠬⠛⠛⠋⠁⢀⣤⣿⣿⣿⣿⡟⠷⠎⣸⠃⠀⠃
⠀⠄⠀⢿⣄⣁⠈⠁⣀⢀⢀⣀⣄⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⡿⢃⢁⣠⠞⠁
⠀⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⡽⠧⠴⠚⠉
⠀⠀⠀⠀⠈⠛⠟⣻⢿⣿⣿⣿⣿⣿⣻⡛⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠗⣠⡂⠶⠀⡏⢤⠸⠰⠇⠰⢲⣦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⡀⢘⣥⠬⠗⠚⠛⠛⠛⠛⠛⠛⠳⠦⣮⣸⡕⢃⢗⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢄⣬⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣾⣇⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⣹⠞⠁⠰⣆⣀⠀⠀⠀⠐⣿⣆⣀⣤⠰⣶⡀⠀⠀⠀⠀⠀⠉⠳⣤⡋⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠏⠀⠀⠀⢇⠉⠁⠀⠀⠀⠀⠉⣁⡀⠀⠹⣷⣄⠀⢀⠀⣀⠀⠀⠈⠻⣄⡴⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣛⡇⠀⠀⢀⣤⣾⡆⠀⠀⠀⠀⠀⠀⠙⠿⠦⠐⠋⠉⠀⢘⣧⡿⣁⠀⠀⠀⠈⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡥⢸⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⡀⠈⢳⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣠⣤⣀⠀⣀⣠⡄⢰⣾⣿⣿⣶⡶⠂⠀⠀⠻⠿⠃⠀⠈⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⣧⣶⣿⣿⣿⣿⣷⠌⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⡿⣆⠀⠘⣟⠛⠉⢻⠁⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢺⡟⠹⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⡄⠀⠒⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠈⢧⠀⠈⠳⠶⠟⢠⡟⠁⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⣨⡅⢴⠖⠀⠸
⠀⠀⠀⣠⡴⠶⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⡾⠀⠀⠀⠸⡇⠀⠀⢸⣦⡀⠀⣀⡀⡈⠉⠀⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠻⣤⡀⠀⣴⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⠞⠁⠀⠀⠀⠀⢷⠀⢀⠸⠆⠀⠀⠉⠻⠉⠀⠀⠀⠀⠀⠀⢰⠁
⠀⢴⡞⠛⠋⠀⠀⠈⣳⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣤⣶⣄⠀⠀⠸⣄⠿⠗⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⣠⠏
⠀⠀⠙⠳⣦⠀⠀⢼⡁⠀⠀⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⡴⠃
⠀⢸⣿⠉⠉⠀⠀⠀⠉⠉⠒⢦⡙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠙⢶⣄⣀⠀⠀⠀⢀⣀⣤⠶⠋
⠻⣏⣉⠀⠀⠀⠀⠀⢀⣔⠒⠉⠀⠀⠙⢿⣿⣷⣶⣤⣀⡀⠀⠀⠈⠉⢻⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉
⢠⡏⢍⠙⠛⠲⢲⡆⠀⠉⠛⠓⠒⠠⣤⡀⠈⠙⠻⢿⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⣶⣦⣤⣀
⡼⠃⠉⠀⠘⠆⣾⠁⠀⠀⢈⣿⡏⠉⠁⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⡀
⢧⠺⣤⣂⣠⣶⠛⠚⠛⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠉⠉⠛⠛⠿⢿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣀
⠘⢧⣈⣉⡴⢻⣶⠏⠀⠶⠆⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⢿⣷⣦⣄
⠀⠀⠉⠉⠀⠈⣇⠀⠧⠄⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠈
⠀⠀⠀⠀⠀⠀⠻⣌⡓⣋⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⡀⣀⣀⣤⣤⣴⣶⣶⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⣶⣶⣶⣶⣶⣶⣶⣿⣿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠟⠛⠉⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣠⣴⣶⣶⢤⠀⠀⠐⢶⣶⣶⣶⣤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡤⣶⡞⣿⣿⣿⣿⣿⣿⡟⠋⠉⠛⠷⣷⡀⠀⠀⠀⣸⡿⣛⣿⣿⠯⢤⣤⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢻⡇⠀⣷⡼⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⠀⠀⡸⣷⣤⣀⣠⣾⣴⠛⠉⠉⠉⠛⠛⠩⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢲⠀⢿⡀⠀⣿⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⠃
⠀⠀⠀⠀⠀⡀⢃⢸⡆⢾⡅⣸⣿⣼⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⡯⢻⣾⡦⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⡁⣿⣶⡿⢻⠏⠻⠏⠈⠿⠹⠟⣿⡀⠹⣿⣉⢿⣟⢿⣟⣿⣯⣬⣝⣛⣿⣛⡋⠡⡯⠓⠽⢶⢾⣿⣴⣤⣤⣀⠢
⠀⠀⢀⣈⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⠓⠳⠯⣿⣷⠾⠽⠯⣽⣾⣿⣟⡻⡶⠦⠷⣦⡈⣱⡏⠉⠉⠙⠻⣾⣟⠲⠾⠤⣤⣤⡀
⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠈⠣⣈⡙⢷⡄⠀⠈⠳⠿⠿⣿⣶⣿⣶⣾⣽⣿⣶⣦⣤⣀⡀
⠀⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠙⢷⡈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠺⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠈⣷⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠀⠀⣠⣾⠟⠁⠉⠉⠙⢿⣦⡀⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡷⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣦⡐⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣾⠟⠁⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠙⢿⣦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⡀⠀⠀⠈⠁⠀⠃⠛⢀⡉⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⡴⠻⣧⣀⣠⣤⢤⣦⠆⣴⠻⣿⣤⣠⣤⢤⣦⠆⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠂⢸⣇⣐⣿⠵⠷⠿⢻⡟⢸⣇⣀⣿⡵⠷⠿⢻⡿⠀⠀⢺⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⢉⡁⣀⢉⠉⠉⠉⢀⠀⡉⠁⢀⡉⣉⢉⡉⢀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡅⠉⠈⠀⠁⠀⠈⠀⠁⠀⠈⠁⠉⠈⠁⣨⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⡀⠀⠀⠀⠑⠀⠁⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⠟⠡⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⠋⠀⠀⠀⠈⠻⣷⣄⣀⡀⣠⣾⠟⠁⠀⠀⠈⠻⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⣠⢋⠼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠰⡄⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⡣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢆⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢀⡿⢁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⣰⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢼⣥⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⡄⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠺⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠀⠀⢀⠀⢾⡇⠀⡀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⣇⣠⠒⡎⠁⠀⠀⣔⡁⠀⡔⠛⣦⡄⢠⡶⠛⡄⠀⠀⢶⡄⠀⠑⠦⠤⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠈⠂⢠⡞⡟⢀⠎⠀⠸⠿⠀⠸⠟⡄⠈⠆⠀⠈⢻⡽⠀⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⢀⣿⣤⠊⠀⢰⠀⠀⠀⠀⠀⠘⡀⠈⣦⠴⣏⣀⢀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣋⣠⠃⠑⢤⡇⠀⠀⠀⠀⠀⠀⣷⠂⠉⢆⣈⣹⠏⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣹⠟⠦⠤⢼⠣⠀⠀⠀⠀⠀⢀⢻⠦⠀⠞⣯⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠄⣾⠀⠙⠇⠀⠀⠛⠁⢸⡶⠀⠀⢀⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⡄⠀⠠⠴⣾⠀⠀⠀⠀⠀⠀⠀⢸⠷⠠⢒⠜⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠆⣀⠀⠀⠠⢾⡄⠀⠀⠀⠀⠀⠀⣸⠦⠐⡁⠚⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠘⠀⢈⡒⠀⠠⠶⢷⣶⠒⠒⠒⢾⣿⠿⠒⠀⠀⠀⠉⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠁⠀⠈⠁⠀⠀⠀⢰⡿⠦⠌⠉⠿⣿⠀⠀⠀⠀⢀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡀⠀⠀⠀⢸⣄⣀⡀⠀⣠⣼⠃⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⢸⡉⠉⠀⠀⠀⣱⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠂⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠄⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⣤⣳⣆⡀⢀⣴⣧⡄⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣠⠂⠀⢹⡈⠉⠁⣘⠁⠀⢢⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣄⣀⣀⣧⣄⣠⣇⣀⣠⠼⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⡇⢸⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⡇⢸⢸⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡛⠿⣿⣿⣧⡇⢸⣼⣿⣿⠿⢛⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣝⡻⠇⠸⢟⣫⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⢻⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡟⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣿⣿⣿⣿⠿⠟⢃⣘⠛⠿⣿⣿⣿⣿⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢤⣄⡀⠀⠞⠁⠹⠟⣩⣴⣿⣿⣯⣭⣍⡛⠶⣍⠻⠏⠈⠳⠀⢀⣠⡤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠉⠛⠲⠦⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢳⡄⠴⠖⠛⠉⠁⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢠⠶⠤⢤⡤⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡄⢤⡤⠤⠶⡄⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡟⠀⠀⠀⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⠀⡀⠀⢻⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⡾⠃⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠈⣧⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡻⣮⡛⠿⣿⣿⣿⣿⣿⣿⡿⢟⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣦⡍⠓⠿⠿⠿⠿⠛⢩⣴⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⣡⡟⣻⣿⣿⣟⢻⣌⠻⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢡⣿⣿⣿⣿⡌⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣾⣿⣿⣿⣿⣷⡾⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠯⠭⠥⠴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣠⢤⡀⣠⡀⣄⣠⠀⢠⠤⢠⢀⡤⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⢾⠻⢤⠟⢷⢸⠇⠀⢸⠋⢸⢈⣽⠆⡏⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⢯⠦⠤⣀⣀⠀⠀⠀⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⢀⡠⠖⡉⠄⠐⠂⠈⠉⠉⠑⠒⠨⢍⠒⢄⡀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⣴⢫⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢭⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⣿⣷⣾⣥⡀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⣦⣁⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⡿⠛⣿⠛⣦⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢛⣷⣿⣿⡷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠀⠀⠀⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢋⣿⢻⠈⠿⠇
⠀⠀⠀⠀⠀⠀⣄⡀⢀⡀⣧⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣘⠀⣠⣸⣄⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⢸⡿⢿⣿⡿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣻⡏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⣯⣶⡟⠺⣋⣿⠽⠯⠭⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠧⣼⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⣦⣵⣠⣭⣻⣍⣍⣈⣉⣉⣀⣀⣤⣀⣠⣌⣹⣿⣿⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⢻⡷⡖⠈⣇⠀⠀⠀⠀⠀⣀⡀⣄⣠⣼⣿⠿⣥⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⢬⡛⠶⣿⣛⣆⠀⠀⠀⠀⣁⡤⠴⢋⡡⠚⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠈⠑⠒⠬⠭⠉⣿⣩⠭⠤⠖⠊⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡤⢛⣽⡿⢻⡏⠈⢹⠁⠈⢈⣿⣷⢶⠤⢄⣴⡵⡗⣠⠀⢀⣴⣾⠟⣳⠤⢤⣀
⠀⠀⠀⠀⠀⣠⣺⠟⢿⠋⢡⠀⣡⡀⣼⠀⠀⣾⠟⠁⠊⠁⠀⠈⠀⢿⣟⡷⢉⢙⣈⡴⢯⡄⠀⠈⠑⠢⡀
⠀⠀⠀⠀⢀⡿⠋⡤⣒⣀⠿⣿⣿⢻⡛⣦⣾⠇⠁⠀⠀⠀⠀⠀⢠⠄⠉⠳⢾⠿⡋⠀⠈⢻⣦⣄⠀⢀⡈⠢⡀
⠀⠀⠀⠀⢺⠖⠊⣠⠏⠀⠀⠋⠘⠈⠃⢸⣿⡀⠀⠀⠀⠀⠀⢰⡃⢆⠀⠀⠀⠙⠪⡀⠀⠀⠈⠛⢤⣮⠻⣦⡘⣦
⠀⠀⠀⠀⠀⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⣳⡀⠀⠀⠀⠙⢧⣄⡁⠀⢰⢆⣠⣿⣤⣤⠀⠀⠀⠉⠳⣝⣝⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣪⠏⣹⠖⢲⡤⠀⠠⢤⣍⡲⠤⠥⣈⢿⠿⢿⣷⣄⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠤⣴⣷⣊⢤⢴⣿⠤⠾⡿⡚⠛⠛⢿⡛⠻⡓⠛⢋⣉⣀⣀⣉⣤⣤⡤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡇⠀⠀⢸⢿⠀⠀⢀⣧⠀⠀⠸⣷⠀⢈⣙⣃⠉⠀⠀⠀⠳⠶⠖⠐⠉⠃⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⣾⠀⠀⠈⢘⣄⠀⠀⠙⢇⣯⣁⣀⡄⣄⣒⡁⣀⡀⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙⠒⠆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠇⠂⡷⢷⡄⠀⠀⠀⠀⠈⢻⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⠉⠀⠀⠀⣀⠚⠛⠃⡠
⠆⢀⠎⠘⠁⠀⠀⠀⠀⠀⠀⢿⡗⠀⠋⠋⡀⠉⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠐⣴⣾⢄⡀⠀⢸⣿⡇⠂⠈⠀⠀⠀⠀⠠⠂
⣄⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣗⠐⠂⠠⠀⠈⠂⢇⠒⠂⠀⢸⣿⡇⠀⠀⠂⠀⠀⠀⠂
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣤⠤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣤⣤⣤⣤⣤⣤⣤⡄⠀⢀⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⣿⣷⣦⣀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡟⢿⢿⡟⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⢀⣤⣾
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠷⢿⣸⣇⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⡿⠿⠟⠛⢿⣷⣾⣿⣿⣿⣿
⣿⠟⠉⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠛⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢠⣉⠀⠘⣻⠷⠀⡸⠎⡍⠀⠀⠹⣿⣿
⣋⣀⣤⡀⠀⠀⠀⣰⣴⣆⡾⣿⡟⠁⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⢻⣧⡄⠀⡀⠀⠀⠀⠀⠀⠀⠹
⠄⢈⠙⢳⠂⠀⠉⣈⠙⡙⠜⣿⡇⠀⠀⠸⣿⣿⡟⣿⣟⣟⣿⣿⡟⣿⣹⡟⡟⣿⣿⣿⣻⡟⢀⠀⠸⣿⣿⠀⠀⠁⠀⠈⠀⠀⠄
⡀⠀⠀⠈⠀⠀⠀⠀⠀⠅⢠⣿⣇⡀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢠⠁⢠⣿⣿⠀⠛⠓
⣷⣶⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣧⡜⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⢀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⣿⣿⣶⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣷⣤⣀⣀⢀⣀⣠⣴⣿
⣿⣿⣿⠏⠩⡛⠛⠅⣿⣿⣿⣿⣿⠿⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢼⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠿⣿⣿
⡿⠋⠀⠐⠂⠽⠳⠀⠀⠙⣿⣿⣋⠨⠻⣶⡀⠀⠀⡠⠶⠂⠐⣿⣿⣿⠃⠀⠀⢀⠀⠀⢀⠀⠈⠻⣿⣿⣿⠟⠁⠀⠀⠀⠄⠀⠀⠈⠻
⠁⢀⡀⠀⠈⠀⠀⢀⠀⠀⠸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢿⣿⡏⠐
⠃⠀⠀⠀⠀⠀⠀⢤⠰⠀⢀⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣸⣿⡇⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠗⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠋⢕⡢⢄⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡉⠒⢑⣤⠀⠈⠻⢶⡄
⠀⠀⣶⠀⢲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⣿⣶⣄⠀⠙⣦⡀
⠀⢸⣿⠀⢸⣿⡀⠀⠀⠠⡀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣌⣻⡄
⠀⣿⡗⠀⢸⡿⡇⠀⠀⠀⠈⠢⢄⠙⢆⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⣿⠋⠑⢦
⢸⡿⢇⠀⣾⠃⡇⠀⠀⠀⠀⠀⠈⠑⢌⢳⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠈⡇
⢸⡇⠘⠀⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠢⣟⢤⣄⣠⣄⣀⠀⠀⠀⠀⠀⣿⣿⣄⣀⡠⢓⠰⣶
⣼⡇⠈⢸⣷⣠⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢿⡆⠹⣯⠖⠻⠿⣦⢄⡀⢠⣿⠿⠉⠉⢡⠟⡄⢸⢧
⣿⡇⠀⢾⣿⡏⠀⠀⠀⠀⠀⠀⠀⣴⡡⠞⢲⣿⣆⠘⢿⡦⣄⡀⠉⠉⣻⣿⡇⠀⢀⣿⠀⢷⠀⣸⡀
⣿⡇⠀⠀⠘⡽⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢸⡸⣿⠀⠈⠻⣮⠙⠦⣠⣿⡿⠛⠷⣮⣴⠀⠺⡀⣿⡅
⣿⣧⠀⠀⠀⣷⠀⠀⠀⠀⠀⢰⡳⠁⠀⠀⣿⣿⢿⡇⠀⠀⠘⠷⡀⢾⢿⣧⡀⡐⠈⣿⠲⣦⡇⣸⡇
⢹⣿⡀⠀⢰⢳⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⡿⣿⠈⢿⡄⠀⠀⠀⡇⢸⡎⣯⠛⢅⣨⡏⠀⣹⠛⣷⣇⡀
⠈⣿⣿⣷⣮⣦⣠⣤⠀⢀⣮⠃⠀⠀⠀⠀⢻⡹⣧⡌⠑⠶⠶⠚⣰⠟⣰⠋⢳⠀⠹⣄⣼⠏⡘⡾⠀⣙⠳⢤⣄⣤⠔⠛⠶⠂
⠀⠈⠻⣿⠋⠈⠉⠈⢳⣦⣥⣤⣤⣶⣤⣤⣤⣵⣝⠻⢶⣶⡶⢟⣫⡞⠁⠀⢸⡄⠀⠘⠏⡰⣡⠇⣼⠛⠉⠉⠈⠉⠓⠒
⠀⠀⠀⠙⠻⢶⣤⣤⣾⡿⠀⠀⠀⠀⣠⡤⣸⣿⣿⣿⣷⣶⣿⣿⣿⣧⠀⠀⠈⣧⣀⠀⢘⣿⢋⣾⠏
⠀⠀⠀⠀⠀⠀⢀⣽⠿⠿⣿⡿⠟⣿⣗⣛⠿⣿⣽⣿⡄⠈⠛⢿⣮⡻⣧⣄⡀⢘⣿⣷⣾⡧⣿⠃
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠊⢁⣴⠶⣿⠊⠙⢿⡶⢭⣽⣿⣄⠀⠈⢻⣷⣹⣿⣿⣿⣷⣿⣿⢿⠁
⠀⠀⠀⠀⢠⠞⠁⠀⣠⠶⠋⠀⣠⠇⠀⠀⠀⠈⠁⠊⠛⢻⣷⣄⣀⣿⣿⣿⣿⡿⠿⠟⠛⠛⡄
⠀⠀⠀⣴⠋⠀⠀⠀⠋⠀⠀⠀⠋⢀⣤⣀⣠⣶⣶⣾⣿⣿⣿⠿⣿⣿⢟⣽⠋⣠⡔⠂⢉⣴⣿⡆
⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣰⡽⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣫⣿⡩⠔⠚⠁⠠⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣩⣤⠖⠊⠀⢀⣽⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⡋⠁⠤⠂⠀⠈⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡦⣀⡀⠴⠊⠛⣻⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣯⡠⠔⠊⠙⢙⣷⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣀⣠⣴⡏⠙⢿⣿⣭⡋⠋⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⠹⠀⠀⢻⠋⠳⣄⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⣿⡟⣿⣿⠁⠀⠀⢸⡶⠾⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠸⡇⠘⡿⣤⣤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣴⣾⠿⠛⠃⠈⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣻⣏⣌⣛⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣿⣭⣽⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠠⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⢒⣒⣶⣤⡶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⢠⣾⣿⠷⠚⠃⣨⣿⣋⣭⠤⠤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢀⣴⡾⠉⢹⣃⠀⢀⡀⠘⣇⣩⠄⠂⠁⢀⣤⠆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡾⣅⠀⠀⣹⣯⣥⣦⠷⠋⠀⠀⢀⣴⣿⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣘⠾⠛⠫⣅⡀⠀⡠⢀⠀⠀⣰⠿⣿⢻⡿⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠼⠷⠾⠷⠶⣶⡦⠼⢷⣤⡖⠉⣴⠟⣉⠗⠛⠛⠳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⣿⠟⠁⠀⠀⠀⠁⠀⢀⠀⣠⣄⣿⠼⢧⡰⠿⣾⡟⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠏⠀⠙⠧⣄⠀⠀⠀⠠⢠⣁⡀⢘⣿⣥⡠⣼⠙⣌⠉⣿⣶⣶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣯⡤⣻⠴⠒⠛⠓⠙⢶⡄⠀⠀⠉⣹⣿⡿⢛⣡⠁⢦⣈⣳⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣯⡻⣿⣛⢻⣿⡏⠀⢀⠀⠀⢀⠠⡝⠾⠓⠿⠛⣿⣄⡚⡶⢿⣾⡿⠶⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠻⣿⣇⠘⠿⡆⢿⣷⣄⣠⣤⣾⢁⣴⣫⣧⠈⣿⣷⣠⣝⠻⣿⣶⢦⣤⡀⣀⣀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣿⡟⢿⡄⠀⢨⡿⠄⠈⣧⣼⢿⡙⣯⣬⡁⢉⢻⣿⣿⣿⣄⠙⠿⠿⠷⣤⡉⠋⡁⠉⠉⠂⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⢠⡞⣿⡿⠘⢯⠀⠓⡇⢰⣆⢀⣎⣵⣿⣿⡿⠛⠛⢿⣿⢀⠟⠷⣬⣈⠑⢦⡴⠶⢤⣙⢂⠳⡄⠀⠀⠀⠀⠀⠘⡀
⠀⠀⢀⣤⠤⣤⣼⣿⡎⠳⣆⢠⣣⡹⣽⣦⠿⠋⠀⠉⠿⠿⣤⣆⣀⠀⢸⣯⠀⠀⠀⠈⠉⠒⠽⡆⠀⠈⠉⠉⠈⠀⠀⠀⠀⠀⠀⠰
⠀⠀⢨⠙⢈⣯⠉⠙⢳⡀⡘⣷⠽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⣼⣥⡟⠋⡤⠜⣴⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠧
⠀⠀⠈⢿⣷⣶⣶⠚⠛⢠⣭⡇
⠀⠀⠀⠀⠀⠉⠛⠶⠀⠿⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣾⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣴⣿⣫⣿⣿⣿⣿⡿⠟⠛⠋⢩⣤⡄⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⡘⠿⣇⠀⣠⣾⣿⣿⡿⠻⢿⣿⣿⡇⠀⠀⠀⢸⠟⠱⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡤⠀⠰⢿⠿⠛⠁⠀⠀⠀⠻⣿⡇⠀⣀⣀⡀⠀⠚⠛⠋⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⡄⠀⠀⣤⣤⠴⠶⠄⠀⣴⣿⡆⠀⢀⣾⠏⠈⣿⠀⣾⣇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡿⠀⠀⣿⡇⠀⠀⠀⠀⣿⣏⣿⠀⣼⣿⠀⠀⠀⠀⢸⣿⠶⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣤⡀⠀⣿⣿⠀⠀⠀⠀⣿⡏⢻⡇⢻⣿⠀⠀⠀⠀⢸⣿⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠙⣿⡆⢸⣿⡷⠛⠀⢸⣿⠃⠸⣿⠸⣿⡇⠀⠀⣴⠀⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡿⠘⣿⣧⡴⠶⠊⣻⡶⠚⠻⡅⠙⠿⣦⠴⠟⠀⠙⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡦⠾⠟⠁⠀⠉⠁⠀⠀⣼⡿⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡄⠀⠀⣼⡿⠁⢀⡴⠁⠀⠀⠀⠀⢀⣀⠀⠀⢰⣿⡗⠀⠀⣀⡤⠖⠋⠉⠁⠀⠀⠉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣶⠀⠙⠷⠃⠀⢰⣿⢃⡴⣋⣤⣶⡄⠀⠀⣴⡿⢋⡗⠀⢸⣿⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣿⡄⣿⡿⢀⣾⡟⠀⢀⣾⡿⢩⠞⠁⠀⣹⣿⣠⠼⣿⡗⠋⣠⠜⣡⣿⡇
⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠸⣷⡹⠛⠉⢿⣧⡠⢻⣿⣷⠃⠀⣠⢶⣿⠏⠀⠀⠈⠉⠉⢁⡞⢁⣿⠃
⠀⠀⠀⠈⠻⣄⣀⣠⠤⠞⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠀⠈⠻⠃⠀⣾⣤⣿⠏⠀⠀⠀⠀⠀⠀⠈⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠠⠄⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡀⢴⡠⢖⠒⢦⣄⡆⡄⠉⠐⠠⠄⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠃⠒⢸⠀⠀⠉⠀⠀⠙⠀⢀⠀⡀⠀⠀⠀⠀⠀⢸
⠤⠄⠒⠒⠂⠀⠀⠒⠒⠬⡢⣀⠀⡼⠐⠒⠠⡀⠀⠜⠀⠓⠒⢲⠄⠴⠭⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠹⣀⠤⠀⠤⠜⡼⠤⠤⣀⠠⠊⠀⠀⠀⠀⠀⠋⠒⢄⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⡔⠀⠡⡀⠇⠑⠂⠤⠄⡲⠀⢀⠆⠀⠀⠈⠑⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢡⠈⠢⣀⡠⠔⢡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⣀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⡆⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠶⠶⠚⠛⠛⠳⠾⢷⡤⣄⣾⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡸⣗⣤⢤⡾⠷⠞⠛⠛⠓⠲⠶⠦⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠙⡇⠀⠈⠉⠛⢳⣀⣠⡞⠛⠉⠁⠀⣸⠋⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡀⠀⠀⢀⡚⠋⠙⢳⡀⠀⠀⢀⡇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣼⣇⠀⢀⠞⠃⠀⠀⠙⠳⡀⠀⣸⢦⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣿⠀⠈⢣⡜⠀⠀⠀⠀⠀⠀⢣⡜⠁⠀⡿⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⢀⡆⠘⢧⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⡼⠃⢠⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢸⡇⠀⠈⢧⣎⠀⠀⠀⠀⠀⠀⠀⠀⣧⡾⠁⠀⢸⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣾⡁⠀⠀⡜⢻⣄⠀⠀⠀⠀⠀⠀⣠⡏⢣⠀⠀⢘⣧⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠙⡷⡀⢰⠃⠀⠙⠳⢶⣤⣤⡶⠞⠋⠀⠘⡆⢀⢾⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠘⣷⣆⠀⠀⠸⡵⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢪⠃⠀⠀⣰⡿⠃⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠻⣝⢦⣀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣀⡴⣫⠟⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠈⠓⢮⡓⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢚⡵⠚⠁⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣦⣄⡀⠉⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⢀⣠⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⡀⠀⠀⠀⠀⠙⠻⢵⡢⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⣐⡮⠟⠋⠀⠀⠀⠀⢀⣀⣤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣤⣤⣤⣤⠤⠌⠙⣧⣠⠤⠶⠶⠶⠶⢤⣄⣼⠋⠠⠤⣤⣤⣤⣤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣥⠴⠶⠶⠶⠶⠦⣬⡋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡤⠤⠤⠴⠤⣤⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣤⣄⣤⣤⣤⣤⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⢀⣀⣀⣀⣀⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⡇⠀⠈⡆⠀⠀⢰⠀⠀⢹⣄⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠤⠴⠻⣤⣤⠟⠤⠤⠞⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠲⢶⣄⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣷⡾⣆⠀⠀⠀⠀⠀⠀⠀⣟⣸⡇⠀⠀⠀⠀⠀⠀⣴⢷⣿⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠛⠀⢹⡆⠀⠀⠀⠀⣴⠟⠉⠉⠳⣆⠀⠀⠀⠀⢸⡇⠈⠛⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠁⣀⡀⢀⣀⠘⣧⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠈⣧⠀⠀⣼⢃⡞⢹⡇⠸⡏⢳⡜⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠘⣧⢠⡏⠀⠙⠛⠀⠀⠙⠛⠁⢻⡀⣼⠃⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠘⣿⢧⣤⣄⣀⣀⣀⣀⣠⣤⢼⡿⠁⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠈⠙⠳⢦⣄⣀⠀⣿⢤⣤⣄⣈⣉⣉⣁⣠⣤⡴⣿⠀⣀⣀⣤⠶⠛⠁⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⢀⠈⠻⣤⡀⠀⠀⠀⠀⠈⠉⠙⣿⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⣿⠋⠉⠁⠀⠀⠀⠀⢀⣠⠟⠁⡀
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠛⠶⣤⣄⣀⣀⣀⣀⡇⠀⢰⡇⠀⠀⠀⠀⠀⡀⠀⣿⣀⣀⣀⣀⣠⣤⠶⠛⠁⢀⡼⠃
⠀⠀⠀⠀⠀⠀⣤⠈⠻⢦⣄⠀⠀⠀⠉⠉⠉⠉⣷⠀⢸⠁⠀⠀⠀⠀⢸⡇⠀⣿⠉⠉⠉⠉⠀⠀⢀⣠⡴⠟⠁⣤
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠈⠙⠛⠶⠶⠶⠶⠶⣿⠀⢸⡀⠀⠀⠀⠀⢀⡇⠀⣿⠶⠶⠶⠶⠖⠛⠋⠁⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⣀⣀⠀⠀⠀⢀⣻⡀⢸⡇⠀⠀⠀⠀⢸⡇⢠⣏⠀⠀⠀⠀⠀⣀⣀⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠉⠉⠹⣇⠈⣷⠀⠀⠀⠀⣼⠁⣸⠋⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⣧⠀⠀⢰⠇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠘⠧⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠈⠛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣀⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠶⠶⠛⠉⠙⠳⠦⣤⡴⠃


⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣶⠀⠀⢸⠳⡄⠀⠀⢀⡀⠀⠀⠀⠀⢀⡼⢫⠏⣧⢉⣳⠤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠈⠀⠀⣀⣠⠇⠙⢊⡭⠛⠁⢀⣀⠀⠀⣾⠤⠞⠒⠻⡏⠀⠀⠀⢀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠙⢻⡆⠀⠙⠓⢲⡆⡖⠮⣷⣄⠀⠈⠋⠀⠀⣀⣀⣀⠀⠀⠹⡄⢠⡶⠋⣉⣤⣬⣭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⡼⠁⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢰⠟⠉⠉⠉⠙⠳⢦⣄⠉⣠⣾⡿⠛⠉⠉⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⣠⡞⠁⠀⠀⠀⢀⣴⣶⣶⡄⢀⣴⣶⠿⣦⣤⠀⠀⣰⣾⡿⢷⡄⠈⣿⣿⣏⣀⣀⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠾⠋⢀⣤⣄⠀⠀⣼⣿⣃⡾⠁⣾⡟⠁⢀⣿⡿⠀⢰⣿⠏⣀⡼⠃⣸⣿⡟⠀⣉⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⢰⡏⠀⢻⣷⡴⣿⡿⠉⠀⣸⣿⠁⠀⣼⣿⠇⠀⢸⣿⠀⠉⠀⢀⣿⣿⢀⡾⢋⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠹⣿⣧⢸⠀⢀⡴⣿⡇⠹⣿⣀⣰⠟⢿⣤⠞⠹⣿⠀⢀⡟⢿⣄⢀⣠⠟⢿⣇⡾⠁⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠙⢿⣿⡄⣾⣠⡿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⠓⠋⠀⠀⠉⠋⠁⠀⢘⡛⠁⢰⣿⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡀⠀⠉⠁⣨⣽⣶⠾⠟⢛⣛⣛⣛⣛⠛⠳⠶⢦⣤⣄⣀⠒⠶⣤⣤⣤⣽⠇⠀⠈⠻⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠟⠀⠀⣾⠟⠉⠀⣰⡾⠛⠛⠉⠉⠙⠛⢀⣤⣤⣄⠈⠉⠛⠷⠶⣤⣤⣤⣴⠟⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣤⣤⡙⠀⠀⣠⣤⡀⠀⢠⣿⡟⠁⣹⠀⠀⢾⡷⠀⠀⠀⠀⠀⠀⠀⣈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠙⣿⣆⠀⠀⠹⣌⣷⠀⠀⠙⠛⠁⠀⣿⣿⢀⣰⠏⠀⠀⠀⠀⣠⣴⣶⣦⠀⣴⣿⠟⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣿⣿⠀⠀⠀⠈⢻⠓⢲⣾⡷⠀⠀⣿⣿⠛⣡⣤⣤⠀⠀⣼⣿⠏⢀⡿⠠⣿⣏⢀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢸⣿⡇⠀⠀⠀⣼⠀⢸⣿⡇⠀⠀⣿⣿⣰⠏⠀⣿⡷⠺⣿⡿⠛⠉⠀⣰⢿⣿⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢻⣿⡄⠀⢀⡿⠀⢸⣿⡇⠀⣼⢻⣿⡟⢠⢞⣿⠃⠀⢿⣧⠀⢀⣴⠏⠀⣙⣿⣤⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⡾⠃⠀⠀⠙⠷⠾⠁⠀⠙⠃⠘⠟⠋⠀⠀⠀⠙⠛⠋⠁⠀⢾⣉⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⢶⣶⡌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⡏⡈⠀⠀⠀⠈⢛⡛⠡⠀⠀⠀⣴⠋⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⣿⡡⣤⢀⡴⠄⣠⠤⣤⠄⢀⣤⠀⣿⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⠛⠀⠿⠀⣿⠹⡇⠠⠿⠴⡿⡠⢾⣁⠼⠟⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⠀⠀⠀⠈⠁⠀⣩⠤⡀⠀⠍⠀⠀⠁⠀⠀⠸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠄⠀⢀⠂⢼⠇⠐⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠸⣐⣀⣶⠀⣿⠆⣻⡤⡾⠁⡸⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⠀⠹⡤⠊⠀⢯⠜⠋⠠⡿⠀⠧⠐⢀⠚⣷⠀⢠⣴⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄⢀⠀⠀⢀⠀⠀⣠⠖⢲⠀⠀⠑⠚⠁⠰⣾⠟⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⣀⠡⠀⢀⠀⢤⠀⣿⠀⢁⡀⣀⠀⠀⠀⠀⢸⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠰⠂⢿⠇⣼⠀⣿⢀⣾⡠⣿⠀⠀⠐⠶⠾⠿⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠠⠀⠀⢿⠄⠿⠂⠌⢰⡿⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⡗⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⢸⡿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣗⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿⠀⢹⣷⠀⠀⣄⠈⢻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣿⣤⢼⣿⠀⠀⠻⠲⢀⠉⠘⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠈⢛⢻⣷⣾⣿⡀⠀⠀⠀⠀⠳⣤⣀⠉⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢈⣿⠁⣽⡇⠀⠀⠀⠁⡀⠈⠹⣶⣄⠈⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣼⣷⠆⣿⠃⠀⠀⠀⠀⠘⢦⣴⣿⣿⣶⡆⠙⢻⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣠⣿⣿⣿⣾⡟⠀⠀⠀⠀⡀⡆⠀⠙⣿⣿⣿⣿⣦⣄⠘⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⡇⡇⠀⠀⠸⣭⠀⢨⣿⣿⣦⣀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠏⠋⣿⣦⠀⠀⠠⣶⣀⠀⣇⡇⠀⣴⡦⠀⢂⡀⠐⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣄⠀⠀⢐⣹⣀⠆⠀⢨⣿⣷⣿⣿⣿⣿⠀⠙⢢⠈⠙⠻⠿⠉⠀⢠⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣓⣾⢽⡅⠀⠀⢸⣿⣿⣹⠙⣿⣿⡇⠀⠀⠀⠀⣸⠀⠀⢸⡞⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣻⠟⡚⣳⡀⢀⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⣀⣤⡈⠑⣶⣯⡼⠋
⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢳⣿⣧⡘⠁⣿⠏⢀⡇⠸⣿⣿⣧⣼⣿⣿⣧⣘⣻⡟
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡰⠁⠐⣶⣾⣿⣿⠀⠀⣯⡴⠿⢿⣄⣸⣿⣽⣿⣿⣿⣿⣿⠓⢄⠀⠀⢹⡆⠀⠀⠀⣄
⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠀⠀⣯⠀⠀⢈⣿⣿⣯⣿⣿⣿⣿⡿⠋⠀⠀⠂⠀⠈⠰⠀⠀⠀⢙⣆
⠀⠀⠉⠁⠘⠉⠁⡠⠀⠀⠈⠀⠀⠀⠀⠀⠈⢹⣟⣦⡀⢰⣀⠀⣈⠿⣻⣿⣿⣻⡉⠀⠀⠀⠀⠀⠀⠑⠀⢈⡇⠀⠀⠀⣹
⠀⠀⠀⢸⡷⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⣼⡿⣿⣷⣦⣅⣤⣿⣿⣿⣿⣽⣿⡇⠀⠀⠀⠀⠀⢀⡄⣦⠾⠀⠀⠀⠰⢿
⠀⠰⡄⠀⠑⠢⣠⠀⢁⠀⠁⠄⠀⠀⠀⠀⠀⣿⡇⠉⢻⢻⣿⡟⣛⣹⣿⠟⢹⣿⣿⠀⠀⠀⢀⣺⣿⠟⠁⠀⠘⢄⠀⢀⣾
⠀⠀⠀⣄⠀⡠⠁⠘⠮⠁⠀⠀⠀⠘⠅⠀⢀⣿⠀⠀⠀⢷⣽⢹⠩⡿⠇⠀⢸⣿⣿⣶⠾⠟⠛⠁⠀⣀⣠⠤⡤⣄⣡⡨⠏⠀⣤
⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣆⠀⠀⣼⣿⢼⡇⠿⡤⠀⣼⣿⣿⣀⣠⣤⣴⣒⡉⠬⠖⠒⠋⠉⠁⠀⠀⠰⠃
⠀⠀⢀⠎⠀⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣿⣷⣾⣏⣁⣸⣇⣤⣷⣶⣿⣿⡿⠛⠛⠉⠉⠀⠀⢀⣀⣀⠀⠀⠀⢠⠞
⠀⠠⠀⠀⠀⠈⠉⠉⠐⠃⠀⠤⠀⢀⣠⣤⢠⣼⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣧⣴⣶⣶⣶⠯⠴⠿⠶⠿⠓⠚⠒⠃⠀⠈⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⢈⣿⣿⣥⣤⣀⣤⣾⣿⣿⣿⣉⣉⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘
⠀⠀⠀⣦⡀⢤⣄⣀⣀⢄⣰⣶⠤⠍⠀⠚⠛⠛⠚⣿⠶⣶⢶⣿⣷⣶⣿⣿⣿⣿⠛⠛⠛⠛⠓⠶⠿⠷⣶⣶⠶⠾⠿⠟
⠀⠀⠀⠈⠋⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡤⢶⡂⠀⠀⣈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠓⠾⠟⠛⠛⠿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣦⣤⣀⣠⣤⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⣽⠉⠉⠉⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣿⣿⣶⣤⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⣮⡀⠀⢈⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⣿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⢷⣾⣄⣀⣠⣿⣿⣿⣯⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣿⣿⣿⣿⣿⣿⡿⣿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣿⣿⣿⢿⣿⣿⡟⢿⣦⣴⣿⣿⡿⢻⣿⣿⣿⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⡿⡏⢻⣿⣿⣿⣉⠉⢿⣿⣿⠓⠀⣿⣿⡿⠯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠛⠉⠙⠣⠾⠛⠁⠙⠚⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠔⠀⠀⠀⠀⠀⣤⡀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠄⠀⠀⠀⣀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠚⠣⠬⠁⣀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠒⠋⠉⠉⠉⠉⠀⠀⠈⠁⠛⢀⡂⠒⠁⠀⠐⠊⠁⠨⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡙⠀⠀⠀⠀⡀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠤⡀⠀⠀⠈⢣⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⣠⠀⠠⠔⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢅⠀⠀⠈⠆⠀⠀⠀⣿⢁⠁⠀⠀⠀⢀⡠⠔⠉⠁⠤⠤⢤⠀⠀⢠⠊⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⣀⠀⠈⢢⠀⠐⢄⠀⠀⠻⣘⣄⣠⡴⠚⡃⠀⠀⠀⠐⠠⣀⠀⠀⠀⢹⡄⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢃⠀⠸⡀⠀⠈⠃⠀⠀⠀⠉⠘⠜⠐⠀⡀⠀⠀⠤⠀⠈⠁⠠⠤⠊⠤⠐⠚⠉
⠀⠀⠠⠰⠂⢀⡎⠀⠑⢄⡀⠈⠦⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⢀⠔⠁⢘⠖⠆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⡇⢀⠔⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠥⣀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠉⠙⠛⠛⠻⠿⢷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠉⠛⠳⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⠶⣦⣤⣀⣀⠀⣿⣧⣀⠀⠀⠀⠀⣨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⣿⣿⠿⢿⣿⢟⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠊⢏⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢈⣀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⠀⠈⢿⡿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠘⠲⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣤⣤⣄⣀⠀⣀⣠⠀⠀⠀⠀⣤⡲⢆⠀⣄⡀⠉⢻⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠾⠛⠋⢉⠙⠿⣿⢿⣿⣿⣿⣧⠠⡤⠀⣠⣿⣿⣿⣾⣿⣿⢿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠘⢿⠋⢀⣠⡴⠖⠛⢛⣚⣛⠳⣎⣿⣿⣿⣦⠾⢆⣿⣿⣷⠿⠗⠀⠝⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠚⠓⢁⣈⣥⣴⣶⣿⣿⣿⣿⣿⣾⣿⣷⡅⠉⠓⠑⣿⣿⣿⣾⣿⣿⣷⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⢻⠃⠀⣿⣿⣿⣿⣿⣿⠟⢡⠄⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣭⡉⠳⡀⠀⠀⢳⠀⠀⠀⠀⠀⡠⠐⠁⢀⠜⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣥⣴⣶⣾⡇
⠀⠀⠀⠀⠀⢸⣿⡀⠁⠀⠁⠀⢡⠀⠀⢸⠀⠀⠀⠒⠉⠀⣀⡴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣗⡨⠚⢿⣿
⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⢠⠀⠸⠀⢀⠜⠋⠀⢀⡠⠐⠚⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣂⢽⣿⡄
⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⢸⣄⣸⠖⢁⣠⡔⠈⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣀⢀⣠⣤⣤⣀⠀⢀⣿⣯⣹⢿⣿⠃
⠀⠀⠀⠀⣠⣯⡈⠙⠣⡀⠸⣿⠁⡴⠋⢠⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠙⠿⠿⠛⠿⢿⣿⣿⣼⣿⣿⣯⣻⣿
⠀⠀⠀⢠⣿⢻⠏⠳⠀⠘⢆⡏⢸⣇⣀⣼⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠿⣿⣿⢻⣿⣿⠇
⠀⠀⠀⢸⣎⣼⡄⠀⠀⠐⢮⣿⢟⣿⣿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⢶⢈⣙⢶⡶⢾⣿⣀⡍⢻⣦⣿⠏
⠀⠀⠀⠀⣿⣏⡇⠀⠀⠀⠀⣷⣬⣿⣤⡀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠶⣿⠿⠿⠛⠛⠿⣷⣶⠿⢿⣿⡿⢶⡿⠁
⠀⠀⠀⠀⠸⡿⢷⡀⠀⠀⠸⠀⠙⡿⣿⡿⡄⠱⡄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣿⣀⣴⣿⣇⡿
⠀⠀⠀⠀⠀⢿⡿⢷⠀⠀⠀⡇⠀⢹⢸⣿⡟⠀⠘⢦⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣶⣾⣿⣿⣿⣷⡟⠁
⠀⠀⠀⠀⠀⠸⣿⣪⣓⠀⣴⠄⢠⡟⠈⣿⣇⠘⢄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠙⠛⠏⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠞⠉⠀⠀⠹⣯⠻⣦⣷⡦⠀⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣶⣤⣄⣀⣀⣀⣀⣤⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⢀⠇⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡵⢀⡠⠐⠁⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⡔⠊⠁⠀⣠⢊⠁⣤⠀⠀⠀⠀⠀⠀⠀⠄⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠌⠁⣰⠂⠁⠀⢀⢠⣄⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠽⠟⠋⢈⠀⠀⠀⠀⣀⠀⠀⢈⡧⠀⠀⠀⠀⠀⠒⠒⠉⠉⠉⠉⠙⠓⠒⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⢀⡔⠚⠀⠀⠀⠀⠀⡀⢴⠀⠀⠀⠀⠀⣀⠀⡆⠀⠏⠀⡀⠀⠊⡠⠊⢁⡠⠊⣵
⠀⠀⠢⡀⡔⠡⢄⠀⠀⠀⠈⠂⡀⢈⡇⡼⠤⠄⠖⠛⣩⠃⠸⡄⠂⠉⠉⠀⠀⠛⠈⠁⠀⠀⡠⠸
⡀⠀⡙⠛⠀⠀⠀⠡⠀⠀⡇⠀⠂⠊⠀⠀⡀⢀⡀⡄⠴⠀⠀⠈⠚⠳⠤⠤⠀⠀⠁
⠣⡈⠀⠀⠰⠒⢴⠀⠁⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⢒⠤⢢⡀
⠀⠠⡀⠀⡀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠐⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⣙⡛⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣤⣄⡙⢷⡀⠀⠀⠀⠀⣀⣤⣤⡤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡙⣾⣧⠀⣠⠶⠋⠉⠀⠀⢀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⢀⣈⣙⣿⣿⡾⢉⣀⣴⡖⠲⠶⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡴⠞⠛⢻⣦⠶⠛⠉⠉⠉⣹⣿⣿⠿⢯⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⡟⣏⢷⡀⠀⠙⠷⣄⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡏⣾⡀⣿⠘⠷⠛⢧⡀⠘⢧⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⡾⠀⠈⣧⡿⠀⠀⠀⠘⣧⡾⠺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢸⡇⠀⠀⠛⠁⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣾⠀⠀⠀⠀⢰⣷⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢛⡏⠀⠀⠀⠀⢸⡏⠘⢷⡄⣠⣶⠀⣠⠶⡆⠀⣴⠖⠂⣿⣮⢖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇⣤⢸⡇⣿⠁⣰⡏⡼⡇⢠⣿⠀⡠⣿⡏⣼⠇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣤⣼⡇⠀⠀⠀⠀⠈⠿⠙⠋⠀⠈⠉⠈⠉⠁⠙⠁⠈⠉⠀⠈⠁⠿⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣸⡇⠀⢠⣄⠀⠀⢸⠁⢻⠀⠀⠀⣰⠏⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⢹⡇⠀⠀⢻⣇⠀⠸⣆⣼⢠⡄⢀⣟⡠⠞⠁⢀⣴⢲⡄⢀⡤⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇⠀⠀⠘⣿⠀⠀⡸⠧⣼⠆⢸⡧⠒⣦⠠⣾⠧⠞⢁⠿⣤⡛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠶⠶⠚⣧⠀⠀⠀⢿⣆⡜⠁⠀⣿⡠⠋⡕⣨⡟⠀⢿⣄⠔⠉⠀⣼⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢻⡄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠦⣤⣤⠼⠗⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡴⠞⠛⠷⣦⣤⣤⡴⠞⠋⠉⠻⠶⣤⣤⣤⡶⠟⠛⢶⣤⣤⣤⡶⠟⠉⠙⠳⢶⣤⣤⣤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⣀⣤⣤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡶⠟⠛⠉⠁⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⡿⣿⣯⡻⢿⠿⠿⠿⣿⡏⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣎
⠀⠀⣠⣴⣾⠿⠟⠛⠛⠻⣿⣿⣿⠿⢟⣱⣿⣿⣿⣶⠶⢿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣷⣄
⠀⣾⡟⠉⠀⠀⠀⠀⠀⠀⢹⣿⣶⠛⠉⠛⣿⣿⣿⡁⣀⣀⡈⢻⣧⠀⢠⡶⠛⠉⠀⢈⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣇⡀⠀⢠⣿⣿⣷⣮⡝⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠙⠻⠶⡶⠶⠶⠶⠖⠶⠦⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢛⣳⣾⣟⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣴⣿⣿⣿⣿⣷⣦⡸⣿⣿⣿⣿⣿⣷⣶⣝⣿⣿⣿⣿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣽⣵⣾⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⣿⣿⣿⡟⠛⠉⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠟⠛⠛⠋⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⢸⣿⠟⢁⣀⣠⣤⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠉⠻⣿⣇⠀⢀⣤⣴⣶⣦⣤⣿⣿⣿⡏⢹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣠⣾⠟⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⢀⣠⣤⣤⣜⣿⣶⣿⣯⣶⣶⣿⣿⣿⢸⣿⡇⠀⠻⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠉⠀⠀⠉⣿⠻⣿⣿⣿⣿⣿⣿⠏⢸⣿⡇⠀⠀⠈⠛⠿⠿⠿⠟⢛⣿⣿⡿⠋⠀⠻⣧⣀⣀⣠⡴⠶⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠈⠛⠛⠛⠛⠁⠀⣼⣿⣿⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣤⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠀⠀⠀⠀⢀⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⠀⠀⢱⠀⢠⡞⠀⢀⣤⠴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢙⡀⠁⠀⠙⡆⢠⣄⣌⠁⡜⠀⠀⣐⢉⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠛⡆⠀⠀⡷⠞⠀⠈⣿⣇⠀⣠⠛⠉⠻⣿⡄⠀⢀⠀⠀⠰⠆⠀⠉⠒⠆
⠀⠀⠀⠀⠀⠀⠐⠋⠁⠉⠀⠀⠀⠿⡓⠦⣀⣿⠃⠀⠀⠸⠰⠞⢸⠧⠀⠴⢿⠈⢉⣛⠀⠀⠀⢹⣿⡄⠊⠀⠠⠄⠀⠀⠀⠴⠦⠄⡀
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣀⣲⣀⠀⢀⡐⣿⡟⠀⠀⠀⡘⠳⠏⠀⠶⠀⠒⠀⠛⠋⠉⠀⠀⠐⠀⣿⣟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠤⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⢠⣿⠅⠀⠀⣀⣇⡀⠀⡀⠄⠀⠡⣀⣠⡤⢿⣀⠀⠀⠐⢹⣯⠀⠖⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠀⢤⡐⠀⠀⢹⠉⠂⣼⠏⠢⡀⠄⣿⠉⠋⡁⠀⠀⠀⠀⢀⠀⢀⠘⡿⠁⠂⠀⠰⣿⠐⠀⠀⠐⠊⠉⠉⠀⠒⣠⡄
⠀⢀⣤⣖⡈⠉⠀⠀⠀⠉⠀⠈⠈⠁⢰⡟⠢⠀⠈⢴⡏⠀⠀⠈⠳⢀⣤⡜⠋⠀⠈⠀⢿⡗⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘
⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠑⢌⢠⣼⡁⠈⠀⠀⠀⠆⠀⠠⠀⠐⠀⠀⣯⡖⠀⠀⠀⢿⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡛⢻⠗⠠⢀⠀⠘⢻⡇⠀⠀⠀⠀⢀⠀⣄⠀⠀⢀⢴⡴⠋⣀⠀⠀⢻⡇⣀⡹⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⣠⠛⡿⠀⠂⠤⢉⠒⣾⡇⠀⠃⠠⠀⢉⠀⠨⡴⠀⠈⢸⣷⡫⠔⠂⠁⢸⡇⠈⠱⣄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠟⠁⢰⠇⠀⠀⠀⠀⠈⠙⢷⠈⠀⠀⣀⡉⢀⡠⠀⣀⠀⡾⠋⠀⠀⠀⠀⠈⡇⠀⠀⠀⠻⡑⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠊⠀⠀⣸⠀⠀⠀⠀⠀⣠⠚⣇⣤⠚⠀⠀⠒⠂⠀⠬⠛⣾⣠⡟⢦⠀⠀⠀⠀⣇⠀⠀⠀⠀⠙⠈⣆
⠀⠀⠀⠀⠀⠀⢠⡇⡌⠀⠀⠀⡏⠀⠀⠀⡠⠊⡠⠊⠁⠈⢧⠀⠤⠤⠤⠤⠤⣴⠇⠀⠈⢦⠳⣄⠀⠀⢿⠀⠀⠀⠀⠀⠐⢿
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⢰⡇⠀⢀⠞⠡⠊⠀⠀⠀⠀⣵⠀⠀⠂⠀⠀⠀⢲⠀⠀⠀⠀⠣⣈⣆⠀⢸⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⢸⠇⠀⠚⠞⠁⠀⠀⠀⠀⠐⣬⢓⡚⠃⠀⣛⡫⢥⡛⠀⠀⠀⠀⠈⢦⣧⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡷⡎⠀⠀⠀⠀⠀⠀⠐⢩⡓⠤⣤⠀⡤⠞⣥⠓⠀⠀⠀⠀⠀⠘⡀⠇⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠃⠃⠀⠀⠀⠀⠀⠀⢠⡓⢮⡓⠦⠤⠒⢉⡄⣃⠀⠀⠀⠀⠀⠀⢱⡾⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⢾⠀⠀⠀⠀⠀⠀⠀⢀⠿⣕⢬⣁⡀⡉⣵⢹⡌⠀⠀⠀⠀⠀⠀⠸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠦⠀⠀⠀⠀⠀⠀⠀⠈⡡⠮⡓⠀⠀⢚⡵⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡳⠌⠁⠉⣩⣴⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⡈⢒⠀⠀⠛⡉⠲⠦⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⠴⠋⠀⢀⠜⠁⠈⠀⠀⠘⠇⠈⢢⡀⠑⠢⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠋⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⣴⠀⠀⠀⢢⠀⢐⣄⠀⠀⠀⣤⠀⠀⠄⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣶⣶⣶⣤⣶⣧⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣋⣙⣋⣸⣋⣹⣝⣨⣟⣻⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⡟⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⢀⣀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡏⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠈⣿⣼⠙⣷⡀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢰⢏⡠⠶⠶⣤⡆⠀⣠⡀⣴⢻⣆⠀⢸⣷⣀⡇⠀⠀⠘⣷⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠋⠀⠀⡴⠋⢷⣰⣿⠏⣿⣸⠿⣆⢸⠉⠉⠀⠀⢠⣤⣌⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⡞⠁⠀⢸⣟⢿⣠⠏⠉⠀⠉⠉⠄⠀⣠⣴⣶⣼⣿⣦⣤⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡞⠀⠀⠀⢸⣿⠀⠁⡀⠀⠀⠈⠀⠀⠈⢁⣴⣾⣿⣿⣿⣿⡛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣸⠇⣠⣄⠀⠀⠀⠀⢠⠀⠀⠟⣉⣥⡈⣿⡟⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⡴⢋⡾⢁⡞⠀⠀⠀⢠⠏⠀⢀⣾⠋⠀⡗⢸⡇⠀⣿⣿⣶⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣼⡧⠊⠀⠀⠀⢠⠏⠀⠀⣼⠇⢀⡼⠁⢸⡇⠘⢋⡟⠛⢿⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠖⣿⠀⠀⠀⠀⢠⠏⠀⠀⢰⡟⣠⠞⠀⠀⠘⣷⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢀⠎⢀⠁⠀⣼⡷⠃⠀⠀⣠⣤⢛⣵⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡞⠀⣾⠃⢠⣿⠞⠛⢳⣴⡿⠋⡼⠧⣄⡀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠎⠀⠐⣟⡰⢻⡇⢀⣴⣿⢿⣠⠞⠁⠀⢀⡽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣠⠎⠀⠀⠀⠀⠀⢈⡟⣫⡾⠃⠀⠀⠀⢀⡴⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⡞⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡾⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⣗⠀⡼⠁⠀⣠⠞⠁⠤⠌⠉⣁⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢀⣡⠤⠤⢤⣁⠠⠞⠁⢀⣠⡤⠖⠋⠉⠀⢀⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢠⡎⠀⠀⠀⠀⠈⢧⠐⠚⠉⠀⠀⠀⠀⠀⠙⠋⠀⠀⢠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⢸⡀⠀⠀⠀⠀⠀⢸⠀⠒⠒⣒⣒⠒⠒⠒⠒⠒⠂⢠⣿⠉⠀⢀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⠴⠖⠈⠳⣄⡀⠀⢀⡠⢋⠘⠒⠆⠛⢩⠟⢛⠷⢦⣄⠀⣾⣣⡴⠾⠟⠳⠍⠳⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⣠⠞⠀⣬⠉⣉⠉⣄⠈⠳⣄⠀⠀⠈⢉⣀⣤⡽⡿⢿⣿⣿⡶⠖⠞⢶⢲⡶⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⣰⠃⠀⣯⠀⠘⣆⠀⠈⠁⣠⣶⡏⡟⠸⢠⣾⢻⢻⡙⣟⢻⢶⣤⣀⠃⠙⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⢠⠏⠀⠀⣿⠀⠀⠙⠀⢀⡞⣿⠸⠀⠁⢰⡿⡜⠊⠀⢻⣙⣆⠈⠘⢹⢳⣄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠰⠖⠀⠀⠟⠀⢀⣠⣥⣤⣄⠂⠀⠟⠃⠋⠀⠀⢀⣿⡳⠉⠀⠀⠈⣏⢹⡄⠀⠈⠘⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⠴⠞⠉⢠⣤⣀⣌⠳⣄⠀⠰⠦⠞⠁⠸⡧⠓⠀⠀⠀⠀⢹⣉⣳⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠞⠁⠀⣀⣺⣧⣼⢩⣯⠞⠛⠀⠂⠀⠀⠀⠀⠁⠀⢠⣀⣀⠀⠘⡏⢹⡆⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣠⠤⡿⢩⡉⠰⡜⢆⡀⠐⠀⣀⡀⠀⠀⡄⠀⠀⠀⠀⠉⠀⠀⠀⡟⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡜⠂⠳⠀⣀⡭⢤⡙⠛⠛⠵⢭⣓⡦⠤⣤⣀⣀⣀⣀⣀⣀⣀⡟⠛⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣿⣾⣿⠉⠑⠚⠳⢭⣓⠦⣤⣀⣈⣉⣉⣀⣀⣀⣈⣉⣉⣉⣸⡷⠒⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡈⠉⣀⣀⡉⠙⢯⡀⠐⠐⠶⠒⠒⠂⢸⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠭⠭⠭⠅⣀⣩⡭⠥⠶⠶⠃⡄⢤⣥⣬⣭⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠐⠀⠒⠒⠒⠒