Fair Emojis & Symbols

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡧⢖⡾⠟⢩⠞⠁⣼⠀⠹⡀⠓⢬⡛⢿⣳⣶⣤
⠟⠛⠛⠻⡛⡟⠿⣿⣿⠿⠛⠛⠓⠻⣤⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢀⡴⠖⢿⠻⢡⠶⢫⡄⡖⡇⠔⡴⢻⢰⠰⠳⠖⠆⡟⠒⢺⠛⠚
⢔⣶⢴⣄⡸⠿⠷⣿⣷⣲⣖⣶⣲⣐⣾⣷⠀⠒⠒⠒⠹⠉⠁⠁⠀⠈⠑⠑⠉⠈⠉⠎⡄⠋⠹⣃⡆⠛⠈⠊⠈⠀⠙⡏⢿⠆⠀⠀⢀
⡂⠂⠀⠀⠀⠀⠀⣿⣷⣷⠢⠿⠆⡿⠇⢻⠀⠀⠀⡀⢳⢤⠀⣀⠀⠀⠀⠀⠀⠀⠄⠴⡆⠰⢄⠥⠀⣀⣾⢿⢶⠹⠏⣤⠶⡶⡶⠆⢰
⠄⠆⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⡤⠀⢸⠀⠀⠀⢸⠀⠀⢀⣿⣤⢤⣿⡖⠒⡐⠂⣢⡿⣇⣾⡖⢩⣉⡀⢠⡀⡀⠀⠀⢰⠇⢸⣇⣖
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡦⠤⠤⢤⠦⠈⢹⣯⡄⠀⠐⢒⣲⣿⠉⠁⠈⡿⣇⣨⣍⣡⡮⡁⢡⢺⣷⡓⢀⣠⣄⡄⣹⢯⣴⣄⣂⡰⣤⡕
⠀⠀⠀⠀⠀⠀⢀⡯⣱⢁⠖⠛⠉⢯⠑⢺⣿⡁⠆⢀⣼⣩⡙⠇⠈⠈⠻⠇⠀⠀⠀⠀⠀⠀⠉⠈⠊⡅⠋⠘⠒⠘⠚⠛⠃⠀⠀⠁
⢣⠀⠀⠀⠀⠀⠀⣏⡣⢤⣹⣶⣦⠒⢲⡞⠏⠁⠀⢸⣿⡓⠚⠀⠠⠀⠀⠀⠀⠀⠴⠀⠴⠦⢠⠀⠠⠰⠀⠀⠀⠠⠖⠐⠚⠁⠀⠀⠄
⠈⢉⣤⣄⣤⣄⠈⠉⠛⠚⠞⠂⠐⢰⡀⠄⠁⠀⠀⠸⣶⠀⠀⠀⠀⢀⡉⢁⣤⣶⣦⢠⠾⠂⡶⡄⠶⠄⠤⠀⠀⠃⣤⡶⠆⠠⡶⠄⢼
⠀⣸⣿⡦⠘⠣⢤⠀⠄⢠⡄⣠⢀⣤⠶⠄⠀⠀⠠⠀⢳⣶⠀⠀⠀⢬⢀⠾⡿⣏⣿⠈⠌⠀⠉⠐⠀⠂⠀⠀⠐⠀⠈⠀⠀⢡⣶⣿⠖
⢈⠿⣭⢑⠶⢶⣾⣾⣿⣯⡴⠛⡈⠹⢼⣆⡳⢗⣦⣈⠉⠀⠀⠀⠀⠀⠀⠀⠓⡉⠃⠀⠀⠀⠀⠀⠐⠚⠁⡀⠠⠆⡠⡀⠀⠀⢻⣟⠒
⠃⠀⠸⠷⠾⣿⣿⣟⣹⣏⠀⣰⡶⠿⠮⣷⣆⠻⠅⠉⠀⠀⠀⠀⠈⠀⠀⠀⢺⡆⠁⠈⠀⠀⠘⠂⠀⠀⠀⠥⡠⠀⠀⠈⠀⠀⡤⠿⠷
⠇⡶⣤⠂⢦⢽⡿⣾⣤⡄⢐⠿⢤⠀⠀⡇⣿⡈⢠⡌⡤⠆⠀⠀⠀⢀⢠⠀⠸⠷⠀⣠⠀⢠⠀⠀⠀⠀⠀⠀⢀⣤⣧⠴⠮⠿⠷⠀⠐
⡻⠇⢫⢷⠠⣴⠤⠤⡝⣧⣄⠤⠄⠷⠶⠷⢹⡷⠀⢵⠁⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘
⣷⣶⠾⢿⠷⣿⣿⣷⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⡐⠢⠔⠀⠰⢄⡀⠀⠠⠄⠀⢀⠤⣀⠀⢀⠀⠀⠀⢀⣀⡀⠐⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⢉⠉⠄⠈⢁⠀⠀⠀⠀⠉⠀⡀⠙⠀⠀⠀⠈⠀⠈⠈⠉⠁⠉⠀⠀⠀⠉⠉⠉⠈⠉⠙⠋⡉
⠀⠀⠀⠀⠀⠀⠈⠧⠦⠤⠄⠤⠶⠀⢄⣄⠀⠠⠤⣤⠠⡀⢀⣠⡀⠤⠁⠀⠀⠀⣀⣀⣀⣠⣄⣀⣠⣀⣤⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡇⢰⠀⣾⠀⠀⠀⠀⡍⠀⠀⠀⠁⢶⠀⢩⢡⠀⠀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡇⠸⠀⣏⠀⡄⠀⠰⠇⢠⠀⠀⠀⢿⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢴⡗⠄⠘⡷⠀⠉⡆⠀⡧⠀⠀⠀⠀⣸⠀⠒⣸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠈⠆⠁⠾⠓⠈⠃⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠩⣌⠄⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠄⡄⠀⠘⢦⡁⠀⠐⢀⠀⠀⠀⢀⡀⠐⠒⠀⡆⢀⢀⠠⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠁⠀⡄⠒⠢⣼⠀⠀⠀⠀⡎⠀⠀⠂⠂⠨⠏⠙⠘⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢻⠛⡟⣻⠟⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⡉⢉⡉⣉⣉⢉⢉⡉⢉⡉⡉⢉⣉⣉⢉⠉⣉⣉⡉⢉⡉⢉⠉⣉⠉⠉⣉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠿⠼⠿⠿⠹⠼⠽⠿⠿⠥⠿⠿⠯⠯⠼⠿⠏⠿⠁⠸⠋⠿⠧⠿⠧⠿⠧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡤⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣮⠄⠀⠀⠀⠀⠀⠀⢬⣾⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠁⠈⠉⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠸⣿⡄⠀⠀⠀⠀⣠⡆⣦⢄
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⠀⠀⠀⢴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠸⢸⡇⣿⢸
⠀⠀⠀⠀⠐⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⠀⠀⠀⠀⠀⠀⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⣷⠆⠀⠀⠀⠘⢇⡿⢊⣀
⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠠⠀⢡⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠿⠇⠘⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⡾⠀⠸⡆⠀⠀⠀⠀⢴⣠⠀⠀⡜⠙⢦
⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⢰⢱⢠⡀⠀⠀⠀⢀⠀⠀⠀⠈⠉⡈⠑⠢⣄⣰⡇⠀⠀⣷⣀⡤⠒⠁⠀⠉⠀⠸⠀⠁⠀⠳⣄⠀⢀⣀⡠⢄⣀⣀
⠀⠀⠀⢰⢾⡧⢶⠠⣾⣱⣿⣸⣷⣹⣶⠀⠶⢿⠷⠆⠠⠶⠿⡿⠷⠦⠀⡟⠉⠁⠉⠙⡇⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠙⡷⣤⡤⠤⠤⢜⣃⣀⣀⡀
⢀⣤⡄⢸⣀⣀⣸⢸⣿⣥⣶⣶⣦⣽⣿⠖⡀⢸⠀⡰⢶⡇⢠⣿⠃⣷⢰⢃⣄⣠⣀⣄⢣⠀⢀⣀⡀⣠⠎⣠⣠⣄⣠⠤⢤⣀⣃⣈⡽⠶⠾⠭⣥⣀⣀⡬⠇⣠⣠
⠀⠰⠀⠸⠷⠷⠿⢺⠿⣿⠹⡿⠽⢿⡿⠆⠰⠾⠶⠀⠾⡿⠿⠿⠿⠿⠶⠶⠿⠾⠿⠷⠶⠶⠾⠿⠛⠛⠛⠛⠉⠫⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠂⠀⠚⠂⠁⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠁⠈⠀⠁⠀⠀⠈⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡖⠊⠋⡭⡐⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠅⠀⠀⠒⠊⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢢⠀⠀⠀⣦⢟⠥⠢⡕⡊⠘⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠠⣏⢈⣼⣾⠹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣼⣯⠡⡸⢏⡝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⡟⠉⢃⣠⣫⢜⣿⡦⢵⣬⣥⣾⣻⣧⠀⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠎⠃⡂⡂⡇⡤⣀⣧⣀⢘⣿⣿⣾⣿⢐⢐⢸⣿⡇⢳⣻⣧⠊⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠰⠣⠃⠀⠀⠀⣀⣷⣿⣿⣿⣿⣷⣯⣳⣿⠳⢤⣀⢀⢻⣧⣞⡷⠻⣧⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠞⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣾⡽⠻⢺⣿⣿⡇⠀⣽⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠨⠨⠀⠉⠉⠍⠀⠀⠉⠉⠉⣩⡁⠀⠈⠉⠩⠩⠠⠈⢸⣯⢭⢁⣼⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠭⢨⢈⣠⣤⣧⡾⠂⠰⣾⠿⣯⠁⣀⣤⣾⣿⣬⣬⣬⠟⢱⠀⠻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⡏⠙⠀⠀⠈⠀⠀⠀⠙⠌⣿⣿⣿⣿⡯⢰⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣶⣾⣶⣷⣶⣾⣿⣿⣿⣿⣧⣼⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠉⠙⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣠⠶⢺⡩⠭⢼⠥⠬⢭⡟⠢⣄
⠀⠀⠀⠀⣰⢞⣵⠚⠁⢧⠀⢸⠀⠀⡞⠈⠓⣤⠛⣄
⠀⠀⠀⣼⣣⠏⠀⠑⢄⠈⢆⢸⠀⡜⠀⡠⠊⠀⢳⣈⣦
⠀⠀⢸⠃⡏⠉⠒⠤⣄⣑⣼⣾⣾⣵⣋⡠⠴⠒⠉⢱⠸⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠶⠖⠒⠒⠒⠒⣺⣿⡁⣹⣿⡗⠒⠒⠒⠒⢺⠶⡇⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⢸⡆⣇⣠⠤⠒⠋⡱⢻⣿⣿⡛⢍⠑⠲⠤⣀⣸⢸⠃⠀⠀⠀⠀⣠⣺⢃⡈⣗⢄⡀⠀⠀⠀⠀⠀⡠⢯⠚⠠
⠀⠀⠀⢿⡙⣄⠀⡠⠊⢀⠎⣿⣿⢿⠛⠳⣄⠀⣰⢋⡏⢀⡶⠋⠻⠿⣦⢤⡤⠤⡤⢤⢤⠕⠂⠉⣩⡉⡯⣟⢖⠂⣶⣤⣄
⠀⠺⠿⠿⠿⣬⠻⣿⠻⡟⢠⣿⣿⡶⢷⢶⣿⠟⡵⠏⠀⣸⣠⣶⠿⢺⣷⡼⠃⠀⠃⠀⢾⣦⣉⡁⣅⡅⢛⣩⣿⠀⣰⠋⢏⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠠⠌⠿⢦⣽⣓⡾⢾⡟⣶⣞⣯⣴⣊⣡⡴⠿⠛⢧⣇⣴⣞⣴⣷⢲⡖⢴⠀⢸⡻⢽⣿⣿⣿⣿⠷⠋⠀⠁⠀⠈⠂⠀⡸⠞⣤⠽⠲
⣤⣤⣤⣤⡤⠤⢤⣤⣤⣽⣭⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⢶⣿⣯⣽⣭⣬⣯⣬⣥⣽⣥⣬⣥⣬⣭⣬⣥⣤⣤⣴⣶⣤⣤⣤⣶⣶⠧⡼⠽⡤⢬⣤

⠀⠀⠀⠀⠀⠀⠀⡀⠀⡤⣤⠀⡀
⠀⠀⠀⠀⢠⢤⡀⠰⠄⠰⠂⠴⠀⢠⠤⡀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠠⢁⡀⠉⠀⢀⠀⠄⠀⠐⠉⣀⠅⠀⠀⠀⢀⣠⣾⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠋
⠀⠀⢀⣀⣀⡉⠀⠤⢐⠜⢔⠨⠄⠀⣀⣀⣀⣠⣴⣿⡿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧
⠀⠀⠈⠀⢁⡤⠄⠀⠀⡹⡕⡀⠁⠠⢄⣸⢞⣵⢋⡾⢱⡇⣿⡘⣯⠳⣝⢶⣄⠀⠀⠀⠀⠀⠀⠀⡼⠁⠘⣆
⠀⠀⠀⠀⠀⢀⠶⠀⡰⠀⠙⣀⠒⢶⠛⠳⠟⠑⠛⣷⣿⣷⣿⣷⡟⠓⠙⠓⠟⠓⠒⠂⠀⠀⠀⣸⠃⠀⠀⠹⡄
⠀⠀⠀⠀⠘⠹⠆⠘⠉⢸⠈⠉⠄⠸⠙⠂⣤⣦⠀⣿⣿⣿⡿⠛⡇⠀⠀⠀⣠⣴⡄⠀⠀⠀⣰⠃⠀⠀⠀⠀⢹⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⢀⣶⣶⣶⣿⣿⡆⢋⠝⠛⠁⢐⡃⣰⣶⣶⣿⣿⡷⠀⠀⢠⠟⠀⠒⠚⠒⠂⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⢯⠁⠏⠀⣼⣷⣴⣿⣷⡇⠊⠉⠻⡅⠸⠁⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣇⣀⣀⣀⣃⣀⣀⣀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠴⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡟⠀⠀⠀⠀⣲⣾⣿⠃⠀⠀⠀⠀⠄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠢⠈⠁⠘⠟⠭⠩⡇⠠⢤⣀⣠⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦⠀⠘⠉⠀⠑⣄⠰⢟⣄⣀⣀⣰⣂⣿⠀⠀⠀⢼⣿⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠀⢀⠀⠀⢀⠜⠺⡍⠙⢢⡀⠸⠀⢹⠀⠐⠤⣞⡏⠀⠈⠢⡀⠀⣠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠤⣀⠀⢨⡟⣅⣀⠀⢱⡀⠀⠹⡗⣄⣈⡷⠀⠄⡌⠑⠀⠀⠀⠈⣻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢹⠟⢦⡀⠉⠉⠒⢿⣦⠞⠉⢧⡇⢇⡱⡾⠀⡆⢀⡱⣖⠋⠚⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡄⠀⠀⠀⡞⠀⠀⠈⢃⡢⢤⡞⠛⢿⡦⣬⡳⣼⡰⢁⡤⠷⠍⣀⠉⠀⠀⠀⢣⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡗⠒⠒⢺⣶⠶⠒⠃⠀⠐⢦⡘⠒⠶⠷⣾⣿⡿⣿⡗⠐⠖⡶⠄⠈⠳⠦⢤⡼⣦⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣿⠁⠀⠀⢸⠚⠏⠁⠒⠂⢀⡚⠀⢀⣠⠶⣿⣷⣾⣿⣿⣃⡚⢑⣀⠀⠒⠋⠁⠀⣏⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⡇⠠⠀⠀⡠⠁⣀⠼⡟⠁⢠⣿⣟⢿⣿⣗⠿⣝⡊⠉⠑⢄⡄⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢁⣌⠔⠊⣁⢀⡧⢠⢮⣿⣿⣸⣿⣿⠖⠘⢼⠄⠤⣈⣃⡑⡶⢰⠀⣤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⣀⠼⢟⠙⢌⠁⠀⠐⣇⠃⢸⣿⣿⣿⣿⡟⣇⠀⠀⢃⠀⡌⠉⡲⠠⣾⣬⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠀⠀⠈⠳⣄⠑⠄⢸⠏⣼⣾⣿⣯⣿⣿⡇⢙⠙⡀⣨⡿⡠⠊⠀⢠⣚⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠿⠿⠍⠀⠀⠀⠀⠈⠑⢦⣼⡾⠁⣿⣿⣿⣿⣿⡏⣆⣤⠼⠄⠋⠀⣄⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⢄⠀⠀⡔⠠⠉⠁⣿⣿⡟⡿⣿⡏⠁⠱⡀⠀⢠⣤⣿⣶⣤⠁
⠀⠀⠀⠀⣴⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣷⣾⣷⣧⢠⡀⡧⢺⣧⣿⢻⣧⡀⠀⣍⢦⡈⢻⣿⣿⡟⠀⠀⢤
⠀⠀⠀⣤⠘⢻⡁⠀⠀⠀⣠⠀⠀⠀⣴⣰⡆⣤⡀⠀⠀⢳⠀⠀⣤⣤⣶⠀⠀⣿⣿⡿⢿⣿⢺⢧⣾⡿⣿⣿⣿⣿⣶⠛⠈⠳⣞⠋⢹⡀⣀⣀⣼⡇⡀
⠀⠀⠀⣿⡇⣿⡷⠀⠀⡀⣿⡆⣼⡀⢻⣿⡿⢼⣷⡄⣴⡎⠀⠀⣿⣿⣷⣬⠉⠉⠁⣠⢼⣿⣼⠅⢼⣇⣿⣿⡿⠿⣬⣷⣤⣤⡽⣧⣼⣿⣿⣿⣿⣿⣿
⠀⣴⡄⣿⠷⣿⡇⠀⠸⡇⢻⣷⣾⠀⢸⣿⠇⢸⡟⣤⣿⣷⣶⣾⣿⣿⣿⣿⣤⣤⣶⣥⣼⣿⣿⣤⣾⣯⣭⣿⣿⣾⣷⣿⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⢻⠇⡙⠲⣿⢳⡄⠀⠀⠈⣿⡿⡂⢸⣿⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠉⠉⠘⠛⠀⠈⠁⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠙
⠀⠀⠀⠄⠄⢀⣀⣀⡀⢤⠀⠀⠀⡀⢀⡄⠀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⢠⢤⡤⠀⠀⠀⠸⣆⠾⠿⠿⠛⢿⠞⠛⢿⠿⣿⣿⣿⣿⠿⢿⣧⢤⣶⣾⣿⣿⣿⣿⣾⣯⠀⠀⠀⡄
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠘⢤⣶⡶⠶⠒⢀⡐⠃⠀⠀⠅⠀⠈⠀⠀⠀⠀⠻⠯⠁⠈⢁⢀⡴⠻⡀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠚⠲⠘⡅⠀⠀⠀⣰⡄⠀⠀⠀⡀⠀⠰⠀⡄⢀⡀⠈⠈⠀⠒⠊⠁⠀⠀⠉⠀⢩⣿⣥⠚⠂
⠀⠂⢻⡆⠀⠀⠀⠃⠀⠀⠁⠀⠁⠀⠀⠘⠿⠷⠆⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠻⣧
⠀⠀⠘⡇⠀⠀⠀⠀⠛⣐⣃⠛⢀⠀⠀⠀⠠⠶⠶⢰⠤⠤⠂⣠⣾⣷⣄⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠰⣦⢀⣿⣄
⠀⠄⠀⠀⣀⣀⡀⢀⣀⣘⣓⣀⣀⠀⢀⣀⡂⠈⣉⠘⠛⣠⣍⡟⣍⢫⣗⠀⠿⠛⠀⠐⠀⠀⠒⠆⠾⠟⣻⣭⣭⣿⢧
⠚⣤⠨⢈⡁⠀⠀⠀⠿⠾⣿⢿⠏⠀⠰⠈⠁⠀⠫⠼⠁⠸⠹⠿⠟⠂⠬⠠⠈⠀⠴⠆⠶⠤⠿⠜⠷⠂⠯⠿⣽⡷⣿
⠀⣠⡇⠈⣡⠀⡀⠠⢤⠀⠙⢈⣀
⠈⠉⠁⠞⠘⠉⠀⠀⠾⠷⣀⣿⣿⡇
⠀⠀⠠⠤⠜⠰⠀⠠⡀⣴⠛⠛⢛⣛⡃
⠀⠴⠀⠠⠀⠀⠀⡄⡀⠀⠀⠀⢸⠈⠁⢠⣤⠐⠓⠚⠒⠒⠒⠒⠒⣶⣞⠓⠀⠀⠀⠀⠀⠀⠓⣶⣗⠶⠛⠛⠛⠛
⠀⠐⠀⠀⢀⡀⣀⠘⢘⡃⠀⠜⠉⣀⣶⣾⡿⠷⠶⣴⣰⡶⠶⢤⣤⣽⣿⠖⣶⣶⡶⢶⣶⣶⣶⡯⢿⣶⣶⠶⠶⠄
⠀⠐⠿⠿⠎⠑⠙⠀⣼⣯⡏⠲⠲⠿⠿⠿⣶⣄⠀⣾⣤⠿⠆⣀⠀⣿⡟⠀⣷⣾⣷⣶⣶⠟⢻⣧⣶⣤⡿⢄⡴⠶⢀
⠀⠀⠀⠀⠀⠀⡀⠈⠉⠹⣇⠀⠀⠀⣀⣾⣿⣿⣆⣿⡆⠒⢶⣼⢼⡏⡗⢠⣷⢸⣿⢿⣿⣺⣼⡏⢀⣀⣾⡾⠇⢁⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢿⡏⠉⠑⠈⢹⣿⡿⠷⠿⠃⠀⠐⠛⠘⠃⠀⠈⠉⠜⡋⠁⠈⢻⣿⡗⠟⠛⠋⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠈⠘⠁⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣉⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠒⠒⠛⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣿⣿⣟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢡⠏⣼⢿⡿⡟⣆⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣰⠃⣰⠃⣿⡇⢳⠹⣄⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡴⠃⢠⠏⢀⡇⣧⠈⢧⠹⣆⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡞⠁⢀⡏⠀⢸⠁⢿⠀⠘⣇⠘⣦⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠏⠀⢀⡞⠀⠀⡾⠀⢸⡀⠀⠘⣆⠈⢧⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡼⠁⠀⢠⡟⠀⠀⢠⡇⠀⠘⡇⠀⠀⠹⡄⠀⠹⣄⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⢀⡴⠋⠀⠀⢠⠏⠀⠀⠀⣼⠀⠀⠀⣇⠀⠀⠀⠹⡄⠀⠈⠳⣄⠀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⠶⠶⠶⣶⡟⠶⠶⠶⢲⡿⠶⠶⠶⢲⡷⠶⠶⠶⢾⡶⠶⠶⠶⣿⡖⠶⠶⠞⢻⣶⠲⠶⠶⠻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠚⠁⠙⣲⠶⠒⠉⠈⠓⠶⠒⣿⠙⢒⣶⣾⣭⣟⠲⠶⠚⠁⠙⠒⠶⠒⠋⢻⠓⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢠⣿⠉⠉⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢀⡇⣿⣠⣤⠴⠶⠒⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⣸⠁⣿⣀⣀⣠⡤⠴⡇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⠉⠀⠀⠀⠀⡇⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠁⠀⠀⠀⠀⠀⠀⢀⣸⠃⠀⣿⠀⠀⠀⠀⠀⡇⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢹⠃⠀⠀⠀⠀⠀⢀⡏⠉⡷⠀⣿⠤⠤⠖⠒⠛⡇⠀⣾⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡲⠦⣄⣸⣻⠋⠀⠀⠀⣤⢤⣀⣀⣿⣿⠁⠀⣿⣀⣤⠤⠶⠶⡇⠀⠘⢶⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠮⠟⠀⠀⠀⠀⠛⠒⠶⠬⣭⡗⠀⠀⣿⠀⠀⠀⠀⢀⡷⠶⢖⣚⣻⡇⠀⠀⠀⠀⠀⠀⣤⠶⣚⣫⡭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠒⠛⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠤⠴⠶⠶⠾⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾

⠀⠀⠡⠀⠀⠀⠀⠀⢀⠠⠔⠒⣒⣒⣛⣒⣒⠒⠦⢤⣀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⢤
⠀⠀⠁⠀⠀⢀⡔⣊⠄⠂⠉⠁⠀⠀⠀⠜⠁⠨⠉⠒⢬⡑⢦⡀⠀⠈⢣⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠁⠀⠀⠀⠀⠀⠑⢄⣀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⡴⣫⠞⠛⠷⡀⠀⠀⠀⠀⠠⣀⠀⠐⠀⠀⠀⠈⠢⡙⢆⠀⠀⠀⠀⠀⠀⠡⠀⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⢠⠎⡔⠃⠀⠀⠔⠈⢆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠙⣆⢣⠀⠀⠀⢡⠀⠀⣀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡎⡜⠀⠀⠀⠀⠀⠀⢨⠓⠀⠀⠀⠀⠀⠀⣠⡀⠝⠈⠀⠀⢙⡄⡇⠀⠀⠈⠀⠀⠀⠀⢠⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⢸⢰⠧⣤⡀⠀⣀⣀⣐⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢁⢰⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠆⢸⡀⠀⠀⠀⠀⠉⠀⠀⠀⠙⣇⠈⡍⠀⠀⠀⢄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢢⢸⠴⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁⠀⠁⠀⠀⠀⠁⠁⠀⠘⠀⠂⡏⡸⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠘⡄⡆⠀⠀⠀⠠⠒⠉⠀⠀⠀⠀⠀⡀⢀⡠⠀⠀⠀⠀⠀⠀⣰⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⡐⢋⢰⡄⠀⠀⠀⠀⢀⣀⠔⠁⠀⠀⠀⠂⡀
⠀⠀⢱⡘⣄⠴⠆⠈⠀⠀⣰⠁⠸⡂⠐⠐⠈⢇⠁⡀⠀⠀⠀⣠⢃⠞⠀⠂⠀⠀⠀⠀⠀⠀⠐⠁⡀⠋⠈⠑⠒⠊⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄
⠀⠀⠀⠱⡌⠢⡀⠀⠀⡰⠃⠄⡀⡇⠀⠀⠠⠈⢆⠣⡀⢀⠔⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠼⢁⠞
⠀⠀⠀⠀⠈⢢⣌⠢⢴⣁⡘⠀⣶⠇⠀⠀⠀⢁⣈⡦⢚⣡⠞⠁⠀⠀⠀⢀⢑⠂⠀⠀⡼⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠀⡹⠒⢬⣉⣑⣛⡒⣒⣒⣈⡩⠥⠖⠋⠁⠀⠀⠀⡀⠂⡀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡰⠁⢠⠃⠀⠀⢹⡅⡌⠁⠀⠀⠈⠄⠀⢤⠘⠀⠀⠐⠀⠃⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠂⠀⢄
⠀⠀⠀⠀⠀⡰⠁⢀⠆⠀⠀⠀⠀⠠⠁⡃⠀⠀⠀⠘⡄⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠔
⠀⠀⠀⠀⡴⠁⢀⡞⠀⠀⠀⠀⢸⠀⠀⠓⠀⠀⠀⠀⠱⡀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢠
⠀⠀⠀⠈⠃⠐⠚⠂⠀⠀⠀⠀⠘⠀⠀⠀⠄⠀⠀⠀⠀⠓⠒⠒⠃⠀⠀⠀⠀⢀⠤⠀⣠⡀⠀⠀⠀⢄⡀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠈⠑⠠⣀⠄⠈
⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⡈⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⡀⠄⠊⠁⠀⠀⠀⠀⠀⠀⢀⡀⢶⠀⠀⢀⡀⡀⠰⡀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠈⠁⠈⠁⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠇⢸⠀⠀⠀⠂
⠀⠀⠀⠡⡀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⢱⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⡆⡌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠂⠀⠀⠀⠀⠐⡆⢦⢸⠀⣧⠸⡅⢸⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣰⡟⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠄⠀⠀⠀⠃⠀⠀⠀⠀⢃⡆⠀⠀⠀⠀⠀⢳⢸⡸⣄⠩⠃⠧⠈⠇⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣀⠁⠄
⠀⠀⠀⠀⠀⠁⠄⠀⠀⠰⠀⠀⠠⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠂⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡂⠀⠀⠇

⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠂⠃⠀⠀⡠⢒⠩⠐⠂⠀⠀⢰⠆⠐⠪⢝⡢⣄⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠁
⠀⠀⠀⠀⡠⢫⠞⠿⣄⠀⠀⠀⠀⢤⠠⠄⠀⠀⠉⠪⡳⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⡜⡱⠃⠀⠀⠛⢦⡀⠀⢀⢖⠀⠀⠀⢀⣀⡶⠳⡘⡄⠀⠀⡠⠀⢠⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⡘⣐⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠀⠶⠊⠀⠀⠀⢣⢠⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⡇⡟⠛⠘⠒⠒⠖⠀⠀⢤⡞⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠠⡇⣧⠀⠀⠀⠀⠐⡀⠀⠘⠃⠓⠀⠀⠘⠓⠒⠲⠦⢴⢰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⢡⢡⠀⠀⢀⠴⠚⠂⡀⠀⢀⠀⢠⣄⠀⠀⠀⠀⢀⡆⡾⠀⠀⠀⠀⠀⠀⢀⣚⣐⣆⠀⠀⣀⣀⡠⠞⠁⠀⠀⢂
⠀⠀⠀⢇⠳⡞⠋⠀⠀⡰⠁⢻⠁⠀⠌⢃⢢⠀⠀⢀⠌⡜⠁⠂⠀⠀⠀⠀⢀⢊⡡⠋⠀⠁⠈⠁⠀⠀⠀⠀⠀⠂⠰
⠀⠀⠀⠀⠳⡑⠄⡀⡰⢁⠀⣽⠀⡀⠀⡈⢆⢱⠤⡫⠊⠀⠀⠀⢄⠀⠀⢠⣋⠌
⠀⠀⠀⠂⠀⠘⠑⡬⣑⡂⠴⠿⠤⠧⠤⢒⣪⠵⠚⠀⠀⢀⠀⡀⠀⠀⢠⡎⠁⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡰⠁⡜⠉⠉⣿⠉⠋⠉⠀⢡⠂⢤⠐⠀⠘⠂⠃⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⡰⠁⡘⠀⠀⠀⠠⠂⢀⠀⠀⠀⢃⠈⢦⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡰⢁⢰⠁⠀⠀⠀⠀⠀⢘⠀⠀⠀⠈⣇⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⡈
⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⣀⠤⠒⠁⠒⠀⠀⠀⠁⠀⢦⠀⠀⠀⠀⢀⠤⡀⠀⢀⠀⠁
⠀⠰⠰⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠓⠠⣀⡄⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠁⠀⢀⡀⠔⠊⠁⠀⠀⠀⠀⢀⡰⣆⠀⣀⣀⣀⢄⠀⠀⠀⠘⡀⢠
⠀⠀⠀⢀⠀⠀⠀⠄⠠⠄⠀⠀⠀⠀⡜⠀⠀⠠⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠀⠀⣀⢀⢀⢀⡀⡤⢲⠀⠀⠀⠀⠂⠀⠀⠀⢺⢣⡇⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠢⠠⠀⠀⠀⠱⢀⠀⠀⠀⠀⢸⠸⣼⠈⣧⢉⠈⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠂⡀⠀⠈⠄⠀⠅⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠃⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠐⠂
⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠀⠀⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠿⣿⠁⣶⡎⢻⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣄⡀⣾⠉⢿⡄⠲⢤⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⣺⣕⡦⠴⠛⠻⣿⣶⣦⡝⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠮⢽⠯⢟⣫⣤⣔⠊⠙⠛⢦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣽⡞⠀⢾⣷⠄⣸⠿⣟⡠⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⢩⣿⡥⠌⣿⣇⣀⡤⣿⡻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣖⣠⣤⣖⣮⣟⣳⣽⣳⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢈⡹⠿⡛⠯⢟⠮⠿⡷⠽
⣿⣿⣿⣿⣿⣿⣿⡿⢁⣤⣾⣿⠞⢠⣤⠤⠘⠳⣾⣧⣄⡹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⠉⣭⡿⢛⣻⣿⣍⠉⠙⠻⣧
⣿⣿⣿⣿⣿⣿⣿⠃⠛⢿⣿⣿⣄⠐⠿⠟⠀⣠⣿⣿⠟⠃⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣥⡀⠀⠈⢿⣯⣕⣤⣿⠟⠀⠀⣠⣼⡆
⣿⣿⣿⣿⣿⣿⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢚⣿⣟⣛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣋⣭⠤⢶⡺⠉⢠⣤⠈⠳⣌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣛⡩⠔⠚⠉⠀⠀⠀⣈⣇⠀⠈⣽⡄⠀⢿⠜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⠕⠊⠁⣀⠀⠄⠀⠀⢀⠀⠈⠘⣧⣦⣄⡐⠼⠋⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡥⠔⠉⠀⣀⠀⠀⠉⠀⠢⡄⠁⠰⠀⠄⠀⠀⠉⠁⠀⠀⠀⢃⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⠰⠂⠁⠀⠀⠀⠀⢠⠀⠰⣠⠀⠎⠸⡀⠊⡀⠈⠀⠀⠀⠀⠀⠀⠀⣘⣧⠘⣧
⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠶⠖⠛⡩⠄⠒⠈⠀⠀⠠⠐⠀⢀⠀⠀⠀⡇⠂⠀⢣⠐⡀⢣⠈⠘⡄⢀⠀⠀⠀⠀⠀⣠⠽⢿⡄⠹⣆
⠀⠀⢸⡇⢡⡥⠼⠒⠶⠦⣄⡈⠁⠀⢀⠀⠀⠀⢀⠀⢀⡁⠘⡘⡆⠄⠹⠘⡀⠈⣇⢡⠘⠶⠃⢉⡡⠀⠀⠀⠀⠀⡇⢸⣦⠹⡆⢹⡄
⠀⠀⠀⣷⠀⡇⠀⠲⣶⡆⠘⡇⠠⠀⠠⠀⠀⡄⠰⡀⣋⡀⠀⡇⢡⡌⠄⢃⣉⠕⠘⠋⢠⣀⣠⣌⠀⣀⣀⠤⠔⢒⣙⣲⣴⣞⣁⣈⣧⡀
⠀⠀⠀⠘⣇⠙⢄⠀⠼⠧⣠⣷⠂⠀⠘⡄⠠⠃⠐⣷⠀⠄⢀⡇⠀⠘⠶⠀⠀⠤⣀⠤⠛⢉⡠⢈⣈⣥⣴⠶⠿⣻⣛⣛⠛⠛⣯⠉⠁
⠀⠀⠀⢀⣘⣦⠀⠙⠒⠚⠟⠉⠀⠀⠀⢳⠀⠶⡀⢹⣇⢉⡠⠷⠊⠀⢀⡠⠔⠉⣀⠤⣊⣥⠶⠿⠿⠁⢲⣲⠟⢉⣭⡉⠳⡄⢹
⠀⠀⣠⢺⡇⢹⡎⠠⢱⠄⠀⠀⠀⠀⠀⠀⣇⡀⠱⠖⠁⠀⠀⢀⡠⠒⠁⡀⢔⣪⣷⡟⠫⠀⠀⠀⠀⠀⣈⡁⠀⢘⣻⡀⠀⣿⢸⡀
⢀⡔⠻⢸⡇⢨⢿⡀⠀⢾⡄⣀⣀⡀⠈⠀⠈⠠⠀⠀⢰⠦⠂⠁⠠⢐⣯⡶⠛⡡⠀⠐⠂⡤⠀⠀⠖⠄⠐⢷⣄⡀⡉⣄⡴⠋⠘⡇
⣾⢐⠀⠈⣧⠸⣌⣧⠀⡼⠉⠠⣤⠙⣦⠀⠀⠀⠀⣀⠓⢀⣰⣴⢾⠋⠀⢃⠰⠐⢲⠐⠊⢱⠀⣰⠃⠁⠀⠈⠙⠉⠉⠁⠀⢇⠀⣧
⢻⠘⡇⠀⢹⡀⠈⠛⣶⠀⠳⣄⠙⢂⡾⢠⠀⣂⣭⣶⠞⠋⠁⣼⠘⠂⡄⢸⠀⠀⢸⡆⡀⢸⠀⡴⡄⢰⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿
⢸⠀⡇⠀⠘⣇⠀⢘⡜⠶⠶⠶⠿⠿⠖⡟⠛⣏⠈⣷⠘⣁⣀⡇⠀⠃⠐⢸⠀⠂⡀⡇⠃⠸⠆⢃⣃⠈⠀⢀⣀⠀⠀⠀⢒⣿⡇⢺⡇
⢸⠀⢹⠀⠀⣿⠀⠀⣶⠀⠀⠀⠀⠆⠀⡧⡤⢸⢠⠟⠠⠤⠼⠧⠆⠈⠓⠈⠉⠉⠁⠙⢉⡴⠋⠁⠀⠀⠀⠀⠀⠉⠒⢎⣭⠙⢧⠘⡇
⢸⠀⢸⠀⡀⣿⡀⢀⡝⢉⣉⠑⣄⠀⠀⠀⠀⠀⠀⣄⣠⣀⣄⠀⡀⠠⠤⠤⠤⠤⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡬⠭⠥⣝⣰⠷
⢸⡀⠀⡀⠁⢸⣇⠈⣇⠀⢿⠀⣿⠀⠀⠀⢀⣀⠫⠉⠁⠀⠀⠀⠤⠤⠤⠤⢤⡄⢰⠁⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠸⡇⢣⠃⠸⠀⣷⣀⣈⣱⣰⣾⣥⣯⣤⠤⠴⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠉⠃⡆⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣧⠘⠈⠀⡶⠭⠭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠸⡄⣣⠿⢥⣄⡤⠤⠤⠶⠖⠒⠂⠀⠉⠉⠉⠉⠀⢀⣒⣒⣒⣀⣀⣀⣈⣥⠥⠽⣦⠀⠀⠀⢸⠁⠀⠀⠇⡄⠀⠀⠴⠀⠀⠘⠂⠘⠀⠄⠀⠸
⠀⠀⠱⠏⠠⠤⠤⠴⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⢸⠀⠄⠘⠀⡇⠀⣐⠀⠀⡀⠀⢰⠀⠀⡂⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡆⠀⠁⠈⠈⠁⠙⠀⠉⠈⠉⠈⠀⠁⠙⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣶⣶⣯⣥⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⣀⠀⠀⣀⣀⣀⣀⡤⣤⣠⢤⣄⣀⣀⣠⣤⣤⣤⠀⡔⡖⣿⣱⣽⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡼⠿⠘⠁⠀⠓⠈⠐⢛⠛⣫⠏⡇⢟⡕⠙⠟⠛⠉⠉⠉⠉⠉⠁⠉⠩⠤⠀⠉⣼⣯⣿⠅
⠀⠀⠀⠀⠀⠀⢀⢔⡕⠁⠀⢀⢤⡠⡀⠀⠀⡰⣡⣾⡟⡼⢱⢿⡇⠀⡆⠀⠀⢠⣠⣤⣤⡄⠀⠀⠀⠀⡇⢩⣏⣿⡄
⠀⠀⠀⠀⠀⡠⡣⠋⠀⠂⠀⠀⠂⠑⠁⠀⠀⣿⣿⣧⢊⡆⣿⣿⡇⢠⠃⠀⠀⠊⠛⠛⠛⠀⠀⢀⠀⡜⡀⣾⣿⣷⠃
⠀⠀⠀⠀⡔⣵⣅⣤⣤⢠⢂⡀⡴⠴⢂⣀⣀⣰⡿⢃⢪⡇⣿⣫⣄⣁⣀⣀⠃⣀⡀⢃⣐⣛⣫⠭⡜⣸⡞⣼⣿⠃
⠀⠀⠀⢬⣬⣭⣿⠿⣶⠶⣶⣮⣭⣥⣭⣍⣙⣻⣣⣇⣬⣇⣿⣭⣿⣿⣿⣿⣿⣿⣿⣯⡭⠼⠶⠶⡾⣻⣿⣿⠁
⠀⠀⠀⢸⢻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣗⣒⣶⠿⣿⣄⣀⣈⣭⣙⣶⣖⣶⣿⣿⠇
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⢯⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⢀⣤⣄⣀⣀⡴⢦⣀⣠⣿⡿⠛⠉⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⣠⠞⠁⠈⠉⢁⣠⣶⡿⠟⠉⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⡾⠁⠀⠀⣠⣴⣿⠟⠉⢀⣠⠶⣤⣀⣀⣠⡄⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⡴⠋⠀⣀⣴⣿⠟⠋⢀⣠⠶⠋⠁⠀⠀⠉⠉⠁⠹⣦⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⣀⣴⣾⠿⠋⠀⢠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣤⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⣤⣾⡿⠛⠁⣀⠀⠀⢻⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣠⣶⡿⠛⠁⢀⣴⠞⠉⠀⠀⣼⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡿⠟⠉⢀⣤⠞⠋⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⢠⣾⣿⣿⠟⠉⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢀⣠⠶⠋⠁⣠⣴⡿⠟⠉
⠀⠀⠀⣰⣿⠋⢸⣿⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣤⣾⡟⠀⠀⠀⠛⠁⣠⣴⣿⠟⠋
⠀⠀⠀⢿⣧⡀⠀⠹⣿⣄⠀⠠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠈⠻⢿⣦⣄⠘⢿⣧⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠀⢀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⠻⣷⣄⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠙⠛⠉⠙⠳⣦⠞⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣆⣀⣠⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⢈⣿⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀
⠀⠀⢀⣿⡹⣯⡀⠀⠀⠀⠀⠀⣤⣤⣼⣷⣿⣂⠀⠀⠀⠀⠀⣿⣴⣂⣀⡦⡄⠀⠀⠀⣴⣿⣿⠿⢻⣿⠶⠀⠀⠀⠀⢰⣿⣿⣿⢿⡀
⠀⠀⠘⠿⠶⠿⠃⠀⠀⠀⠀⠀⠛⠛⠷⠶⠋⠁⠀⠀⠀⠀⠤⠿⠿⠽⠿⠧⠤⠀⠀⠀⠀⠙⠿⠾⠿⠋⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠁

⠀⠀⢠⣶⣶⣶⡀⠀⠀⠀⠀⠀⢀⣠⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡄⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖
⠀⠀⢸⣿⣿⣷⢲⡄⠀⠀⠀⠀⠾⢀⣞⠹⣿⠓⠀⠀⠀⠰⣟⣿⠀⠀⢸⣏⡷⠀⠀⠀⠀⣶⣒⣚⣛⣓⡂⠀⠀⠀⠀⠀⣰⣟⢽⣦
⠀⠀⠈⠙⠛⠙⠚⠁⠀⠀⠀⠀⠦⠌⠏⠠⠟⠁⠀⠀⠀⠀⠈⠁⠳⠶⠂⠉⠀⠀⠀⠀⠀⠀⠻⠿⠿⠯⠇⠀⠀⠀⠀⠀⠙⠿⠾⠋

⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠀⠠⡀
⠀⠀⢾⣿⣾⣿⡧⠀⠀⠀⠀⠀⢰⣿⣿⣤⣤⠀⠀⠀⠀⠀⢀⡴⣤⡐⢀⠇⠀⠀⠀⠀⠀⣿⣿⣽⣭⣿⡛⠋⠀⠀⠀⢰⠀⣺⠀⢨
⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⠘⠯⠿⠃⠁⠀⠀⠀⠀⠘⠛⠛⠒⠚⠓⠛⠁⠀⠀⠀⠀⠀⠂⠤⠄⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠿⣷⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣄⣀⣤⣤⣴⠶⠛⠋⣀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣁⣠⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣾⠿⠟⠛⠻⠿⣶⣤⡐⢶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡉⠙⠛⠻⠿⠿⠿⠏⣰⡿⢋⣤⣶⡿⠿⢿⣶⣄⠙⢿⡆⠹⠿⠿⠟⠛⠋⢁⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣤⡄⢸⡿⢡⣾⣿⡏⢰⣶⣦⣽⣿⣷⡀⣴⡄⢠⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⣿⡇⢸⣿⣿⣷⣬⣈⡙⢿⣿⣿⡇⠘⠃⢸⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣤⣄⢸⣷⡈⢿⣿⣏⠻⠿⠟⣸⣿⡿⢁⣿⠇⣀⣀⣤⣤⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣷⣌⠛⠿⣷⣤⣾⠿⠋⣠⣾⠏⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣧⡌⠛⠿⣶⣦⣤⣴⣶⠿⠛⠡⠾⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣐⣳⣧⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⢉⡱⠀⠀⠀⠀⡀⠀⠦⠄⠠⢒⣼⠒⢐⢚⣺⠀⣻⡃
⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠈⢉⠥⠤⠄⠒⣒⢲⡂⢎⢫⣻⡦⠭⢯⠟⡖⠋⠉⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⡐⠀⡄⠀⠂⠀⣀⠀⠀⠀⠀⠓⠂⠋⠉⠉⠉⠁⠹⠀⠚⠚⠁⠀⠀⠖⠀⠀⠐⠨⢍⣀⠀⠀⢄
⠀⠀⠀⠤⢶⡤⠀⠀⠀⣠⠐⣲⣾⡞⠨⢵⣶⠅⠐⣖⣈⢥⡀⠶⢁⠁⠀⠀⢀⠀⠀⠀⢈⡐⢁⡢⣀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣈⣤⠴⠧
⠀⠀⠑⠤⡐⠀⠐⠀⢄⠙⠒⢋⣜⣿⣀⢘⣻⠤⢀⣰⣏⡿⣥⡀⠀⠁⢂⡐⠀⠌⣃⢠⠐⠀⠦⢄⠠⠑⠤⣏⣀⣈⣟⣶⡞⠉
⠀⠀⠀⠀⠈⠁⠠⡀⠀⢢⠄⠀⣠⠤⢉⣉⠩⢅⠀⠒⠾⣟⠉⠤⠴⠂⢀⡌⠀⢠⠈⠃⠄⢀⣀⡀⣰⡄⠴⠄⠵⢤⠶⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⢡⣒⡶⠃⠀⠐⠁⠄⢬⡼⠿⣾⠷⣀⠀⠀⢉⡁⠀⡀⢀⣀⡀⣀⣠⣠⣜⣠⠒⠡⢶⡄⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡛⠉⠁⠀⠈⣿⣦⠟⠋⠛⠛⠋⠉⠀⠉⠁⠈⠀⠁⠀⠀⠌⠠⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣢⣀⡰⠆⠴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠴⠖⠚⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣻⣿⡿⠟⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⡏⠠⢙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡏⡇⡇⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢮⡷⣸⣢⣜⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣷⣷⣷⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣾⣷⣝⣫⣾⠿⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠐⠛⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡀⠐⠛⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⢀⣈⣻⣟⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣟⣁⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢹⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⠇


⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⢠⢦⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⣠⡀⣿⣿⣿⣄⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣉⣅⠀⠀⢨⣳⠀⠀⠀⣈⣉⣁
⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⢋⣿⣿⣾⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣤⣄⠉⣤⣤⣴⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣇⣿⣿⣿⣶⡆⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠉⠀⠀⠀⠉⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢻⠟⠛⢻⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣀⡼⢦⡀⢸⣿⡇⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⣠⢴⢖⣒⣾⣒⡒⣤⣄⡀
⠀⠀⠀⢀⡴⣏⠔⠚⡇⠀⡇⠀⠸⠓⠪⣹⢦⡀
⠀⠀⢀⣞⡴⠁⠓⢄⠘⡄⡇⢠⠃⡠⠊⠈⢢⣱⡄
⠀⠀⣼⢸⠉⠐⠢⢬⣳⣽⣷⣯⣎⠤⠔⠊⠉⡇⢷⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡟⢛⠒⠒⠒⣒⣾⣿⣀⣿⢗⡒⠒⠒⠒⡛⣻⠀⠀⠀⠀⠀⢠⠿⣆⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⢻⡸⡄⠒⠉⡠⠊⡽⣿⣯⣳⣌⠉⠒⢠⢧⡏⠀⣀⣤⣤⠦⠇⠤⠸⠵⠤⢀⠀⠀⠀⢰⢑⡃⠀⡀
⠀⣠⣌⣷⡙⣤⣎⣤⡼⢸⣿⣏⣆⣈⣳⣤⢋⠞⢀⡏⣁⣸⣹⡏⠹⠉⢻⠘⢳⡁⠤⠰⠲⢸⠼⣱⠀⢺⢽⠂
⠀⠀⠀⠁⣙⣦⣝⣺⠅⣼⣿⣯⡸⡟⣫⠴⢋⣠⡾⣟⡩⢁⡼⢹⣧⣀⣌⠀⢸⢿⢲⣾⣾⡶⣖⡿⠀⠎⠀⠃⠀⠀⡤⢀⣢⡄
⠀⠀⠀⣀⣤⣤⡈⢉⣞⣓⣷⣾⣿⣿⣿⣾⣯⣥⣤⡼⡟⢻⠛⠙⣇⠚⣸⠃⠐⠂⠙⢋⣛⠉⠁⣠⣤⣤⣤⣤⣤⣅⣐⣻⣀⣂



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢀⡤⣒⡯⠍⠾⠰⢩⢓⠤⡀⠀⠀⠀⠀⢀⡤⢪⡀
⠀⠀⠀⣠⢫⠛⣅⠘⡄⠀⢀⠂⢀⠝⢍⢆⠀⠀⢠⠞⢼⠋⠑⢄
⠀⠀⢐⢫⠓⠠⢀⡳⣼⣼⣦⢔⡁⠄⠚⡏⡇⠀⢸⢹⡗⠊⠴⢂⣦⣤⡄⠀⠀⠀⠀⠀⢰⢄⠀⡠⠶⡈⢑⠖⢄⠀⡠⡆
⠀⠀⢸⢺⠒⠂⠒⡺⣿⣅⣿⣇⡒⠒⠂⡒⣛⠀⠸⠄⠗⠂⠀⠉⢩⠟⣟⠃⠀⠀⠀⠀⠈⠤⠤⣤⢤⡤⢤⢤⡤⠬⠴
⠀⠀⠘⡭⠖⠈⢁⠔⡙⣿⣏⠑⢄⠈⢲⢧⠇⠀⠀⠀⠀⠀⠀⢠⡏⠀⠸⣆⠀⠀⠀⠀⠀⠀⣀⣷⠞⣵⣀⣸⣷⢾⡥
⠀⠀⠀⠘⢎⠶⡁⠠⢠⡿⡇⠂⢀⡱⣫⠎⠀⠀⠀⠀⠀⠀⠠⢱⠀⠀⠀⢊⢂⠀⠀⠀⠀⢸⡤⠀⢰⡶⢾⡴⠀⢚⡇
⠀⠀⠀⠀⠀⠙⠲⢇⣞⣲⣳⣨⠷⠊⠁⠀⠀⠀⠀⠀⠀⢴⣧⣦⣤⣤⣤⣼⣦⣦⠀⠀⠀⢜⠀⠀⠐⠃⠛⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢧⡀⠀⠀⠀⢀⠠⡖⣶⢲⢲⡤⡀⠀⢠⢰⠖⡆⣔⢲⢤⠀⠀⢀⠤⠐⠒⠀⠀⠐⠒⠤⢄
⠀⠀⠀⠀⢠⠀⠒⠑⠂⠒⠘⠊⠒⠀⠄⠀⠉⠙⠛⠻⢏⠉⠀⠀⠀⠉⡩⠛⠋⠉⠉⠀⠀⡇⣀⣤⠄⠚⢹⠂⢤⣄⡀⡇
⠀⠀⠀⠀⠰⠉⡉⠉⡏⡏⢹⠉⢋⠉⡄⠀⠀⠀⠀⠀⠀⢑⣤⠤⡴⡈⠀⠀⠀⠀⠀⠀⠀⣿⣶⢤⣉⣷⣼⣈⣤⣴⣾⡇
⠀⠀⠀⢠⠃⢠⠁⢠⢀⡇⡈⡄⢐⡀⠐⡀⠀⠀⠀⠀⠀⡇⡂⠀⢐⠈⠀⠀⠀⠀⠀⠀⠀⠙⠾⣼⣸⣸⣸⣸⣸⡼⠟⠁
⠀⠀⠠⠇⠠⡎⠀⢼⠉⡯⢹⡧⠀⢷⠀⢱⠄⠀⠀⠀⢠⠴⠶⠶⠶⠮⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣴⡿⢻⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣤⣶⣦⣄⣸⣿⠀⠀⠀⠀⠀⠀⢰⣿⢀⣸⣤⣶⡶⠶⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⠁⢸⠈⠉⠉⡇⠀⠀⠀⠀⢀⣠⣾⣿⡟⢻⠁⠀⡇⣰⡿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠹⢷⣼⣀⡀⠀⡇⢀⣠⣴⣾⡿⠿⠋⠙⢿⣾⣄⣠⣿⣟⣡⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⣤⣽⣙⣻⣿⣿⣿⣿⣭⣥⣤⣼⣤⣴⣶⢿⠟⠛⡟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠙⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⠀⠀⣀⣠⠴⢚⡽⣽⢆⠠⣀⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢀⣦⣄⣠⠀⠀⠀⣴⣄⠐⠟⠋⠁⢋⣸⣷⣿⣜⡷⡊⢹⠿⡶⢄⣤⡀⢀⡀⣠⣠⣤⡄
⠀⠀⠀⢶⣤⡄⣉⣱⣿⣾⣷⡱⣄⢿⣿⣆⠀⣆⢀⢯⣭⣭⣍⣭⡽⡿⡆⡬⠁⣼⣿⠇⡌⠡⣾⣿⣞⢛⠠⡀⣀⠔
⠀⠀⠶⢾⣷⡘⣮⣙⣟⣋⣵⣿⡿⠎⢿⣿⣽⠯⠠⢷⣏⣻⣿⣏⣷⣾⠽⣛⣴⡿⣿⣼⢇⣻⣿⣴⣿⢉⣤⣾⣿⣿⣦⡀
⠀⠀⣇⣚⣻⣷⡟⣻⣿⣿⣿⢡⣶⣶⠤⢛⣴⣿⣿⣷⣣⣟⠿⣣⣿⣼⣿⡶⣍⢛⡝⠂⢪⡛⣿⡷⣾⠺⢿⣿⣽⣿⣿⠁
⠀⠀⠙⢿⠿⢿⠿⣿⣿⣿⣿⣛⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣚⣮⣿⣷⣞⣯⢿⠿⠉
⠀⠀⠀⠀⠘⠛⠿⣿⣾⣿⣿⣽⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠛⠛⢛⠛⠟⡿⠛⣿⣷⣿⣿⣙⣿⣛⣹⣿⣿⣿⣿⣿⠿⠿⠿⠟⠻⠋⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢴⣿⣿⣿⣧⣤⣼⣿⣿⣿⣍⣿⣿⣿⣿⣯⣽⣿⢿⣷⣴⣰⣤⣦⡀⣄⣀⣀
⠀⠀⠀⢠⡄⠀⣿⡿⢿⣿⢼⣿⡿⣿⣿⣿⣿⣿⣏⣿⣿⣻⣿⣿⡟⣿⣿⣿⣌⣿⣿⣿⢿⣿⣿⣿⣿⣟
⠀⠀⣤⣤⣷⣤⡇⣳⣜⡁⢸⢏⣴⣼⣫⣽⡟⠳⠣⡿⠿⣿⡿⠏⣍⣙⣿⣟⠙⠯⠅⠸⠟⣾⠿⣾⣿⣿⣷⡇
⠀⠀⣿⣿⠹⣿⠟⠹⣿⠀⠺⣽⣿⣿⣿⣽⣷⠷⡘⠛⠏⠉⠠⢪⣽⣿⡿⠿⡷⡶⡁⠀⠈⢉⣿⣿⣿⣿⣟⡿
⠀⠀⠃⢸⣿⢻⣿⣀⣁⠀⠀⠸⣶⣿⣿⣿⣟⡢⠀⠀⠀⠀⠀⢸⣶⣾⣿⣄⣻⡙⠄⢄⠤⣾⠿⣿⢟⡏
⠀⠀⠀⡟⠛⠀⠋⠈⠀⠀⠀⠈⠟⡿⠿⠿⠃⠑⠆⠀⠀⠀⠀⠞⠻⠿⡏⠀⠀⠀⠀⠀⢶⠃⠠⠷⠋
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣾⣿⣁⡀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣾⣶⡿⠿⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⡆⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠙⠻⣟⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⠀⢀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⢩⣍⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣰⣿⣏⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⣿⣯⣿⣿⡄⠤⠀⠀⣀⡤⠤⣄⠀⠠⠤⠤⠄⣾⣿⣿⣿⠛⠿⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠛⣿⣿⡅⠀⣶⣦⠀⠀⠀⣠⣴⣾⣷⣦⡀⣿⡟⠙⢯⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⠒⣿⣿⠧⣼⣿⣧⣤⣤⣾⣿⣯⣍⣩⣿⣿⣿⣷⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠇⠈⢿⣿⣿⣿⠿⠁⠉⣉⣽⣿⠏⢽⣿⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣀⣀⣀⣉⣉⣡⣤⣤⣀⠈⣛⣁⣀⣿⣿⡄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠉⠁⠀⠀⠀⢸⣇⠀⠉⠛⢤⣀⠀⣹⢹⣧⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⡀⠀⣰⠞⠶⣀⢿⣄⠀⠀⠀⠈⠉⠻⢀⣽⣇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠴⠶⢾⣿⠇⠀⣿⣦⠀⠈⠃⠙⣷⣦⣀⠀⠀⠀⠀⠙⣿⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⣄⠀⠀⠈⠆⠘⠏⠳⢤⣀⣀⡄⠘⣷⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢷⣤⣤⣤⣿⣶⠀⠀⠀⠐⢶⣿⣾⣿⣻⣿⣦⠈⠇⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠁⠙⢏⡀⠠⣧⡀⠀⠀⠀⠙⢿⣿⣿⡙⢧⣄⠈⠀⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠁⠺⣿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣄⣽⣧⠀⢠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⢀⡀⢀⠈⠉⠛⢿⣷⡄⠐⠀⠀⠀⠀⠋⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢠⣾⣿⣧⣄⠀⠀⠀⠉⠑⠲⣆⡀⠀⠀⠀⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣾⠿⣷⠻⣿⢿⡄⠀⠀⠀⠀⢻⣅⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠀⢠⢜⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢲⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠿⢿⣿⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠠⠴⢬⣭⠷⢶⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠳⢦⣶⣦⣀⠂⠠⠍⢸⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠙⠓⠚⠂⠿⠿⠿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣄⣀⣀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣛⡿⢿⣿⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⢛⠹⠁⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡕⣃⢹⢰⣿⠷⠲⣦⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⣿⣅⠀⢿⣟⠀⢶⡏⣸⢛⡿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠌⣻⣶⠋⠉⢛⣾⡷⣷⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⢳⣦⡘⠦⢸⣇⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢋⣤⣠⣾⣟⠛⢿⣿⠉⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣴⣶⡤⣤⣞⡃⠀⠀⠀⠙⣿⡧⠾⡛⠛⠛⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣻⣿⣿⣿⣿⣿⡇⢿⣿⡃⠀⠀⢀⣼⠟⠳⣶⣿⣶⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⡇⢿⡋⢡⣄⡀⣀⠏⠀⣧⣉⣰⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠈⠉⣿⡃⢀⣾⢿⣿⡛⢻⠉⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣿⣿⣿⣿⣿⣿⣿⣇⢸⣇⣀⣴⣿⠂⠀⣼⡿⠈⣷⠈⢲⣶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣵⠃⠀⢰⣿⠃⠀⠉⠄⢡⢻⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⢃⠎⠀⡼⠂⠀⠀⠀⠀⠈⢆⢻⡍⢌⢷⣀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⡿⢡⠏⠀⣼⣤⡶⢾⡷⠶⣤⣄⠈⠊⢿⡌⢣⡻⣷⣄⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠛⢰⠋⠀⣴⣿⣿⡀⢸⡇⠀⠈⢻⠀⠀⠈⢻⡄⠀⢹⣾⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢻⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣹⡇⢀⣤⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡌⢿⣕⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣿⠀⣞⠇⡐⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣷⢀⠀⡄⠀⠀⠈⣿⣿⢞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢸⣿⣿⡃⠀⢠⠀⣄⠀⠀⠸⣿⠀⢻⣿⡟⠈⢀⣡⢀⠁⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣾⣿⣿⣧⠀⣞⡀⢿⣷⣤⣠⣿⣤⡿⠋⠀⠀⢿⡟⢸⣼⣈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢠⣿⡇⠀⠀⠀⠉⠛⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⣿⣿⢳⣼⣇⢠⢰⢠⣰⢠⣧⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣧⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢉⡉⣉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⢉⣉⢉⡉⣉⡉⣉⡉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠚⠓⠛⠛⠛⠗⠛⠛⠚⠒⠛⠛⠚⠓⠛⠛⠘⠃⠘⠋⠛⠓⠛⠓⠛⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠚⠉⠁⠉⠙⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠙⠿⠁⠀⠀⠀⠀⠀⠈⠿⠋⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⣤⣶⣤⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⣼⣹⠈⡏⡆
⠀⠀⠀⠀⢰⣶⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠄⠀⠀⠀⠀⠀⠻⣿⠿⠀⠀⠀⠀⠀⠠⣦⡄⠀⠀⠀⢹⢠⡧⠃
⠀⠀⠀⠀⠘⠟⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⢰⠀⣆⠀⠀⠄⠀⠀⢰⠋⠀⠀⠀⠀⠐⣿⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠠⣴⣤⠀⠀⠀⠀⡟⠀⢻⠀⠀⠀⠀⣴⣦⠄⠀⡜⠑⣄
⠀⠀⠀⠀⢀⠀⠀⠀⢀⢴⢷⡠⡀⠀⠀⠀⡀⠀⠀⠀⠉⣀⠙⠢⣄⣸⡃⠀⠈⣆⣠⠔⠋⠀⠉⠀⠰⠀⠀⠈⠣⣀⠀⣀⠠⠤⠄⣀
⠀⠀⠀⠲⢿⠷⡆⣾⣥⡿⣤⢷⣼⡖⠰⠾⡿⠶⠀⠶⠿⣿⡿⠶⠀⡏⠉⠉⠉⢹⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠈⢹⠦⡤⠤⠤⢘⣇⠀⢀⣀
⠠⠤⠄⣢⣤⣤⡃⣿⣷⡶⣶⣶⣾⣿⠃⠄⡇⠠⠛⣿⣠⣿⣃⣿⠰⢡⡦⣴⣴⡌⠇⠀⣤⣄⣠⣎⣴⣤⡦⠤⠒⠤⣤⠤⠤⠛⠛⠓⠧⠤⠤⠔⠃⣦⣴
⠀⠀⠀⠛⠛⠛⠛⢟⣻⡛⠟⢘⢿⡟⠀⠚⠛⠃⠘⠻⠛⠻⠛⠛⠛⠛⡛⠛⢛⠛⠛⠛⠛⠉⠉⠉⢉⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣬⢢⡢⣀
⠀⠀⠀⠀⠀⣄⣴⣳⢹⣦⣄⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢲⣤⣖⣶⣪⣠⣋⣄⣟⣌⣑⣨⣁⣴⣖⣦⣴⠂
⠀⠀⠀⠀⠀⢇⡟⢸⡜⡏⡇⠀⠀⠀⡒⣾⠹⡛⡟⡆⠂⠀⠀⠀⠀⢸⢿⢿⢿⢿⢿⣿⣿⣿⣿⣿⣿⢿⢿⡿⢿⠿
⠀⠀⠀⠀⡘⡼⢁⣾⣿⣄⠹⡄⠀⠠⠱⠏⣴⣿⣌⠙⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⠁⣿⣿⣿⡿⠿⣇⣀⢀⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⢀⣀⣯⣄⡀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠘⢱⡏⠉⡟⠁⠛⠿⠋⠀⣴⡟⠏⡿⠛⡿⠆
⠀⠀⠀⠀⠀⠀⠐⣶⠿⣏⠉⠉⣏⠉⠉⢳⠿⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⣶⡆⣼⡇⠀⠁
⠀⠀⠀⠐⠒⢤⣞⠁⢀⠜⣏⠉⣯⠉⣱⠯⡀⢈⣧⡴⠒⠀⠀⠀⠀⢰⢶⡶⡶⢶⢶⡿⣿⢿⠿⡿⣷⢶⡶⣶⢶⠶
⠀⠀⠀⠙⠛⡟⠈⢹⠳⢄⡘⢇⣻⡰⣃⡤⠞⣏⠉⣿⠛⠃⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠠⠤⠴⡧⠤⡷⠴⠴⠍⣏⠀⡱⢹⠶⠶⢾⠶⢾⠶⠦
⠀⠀⠶⠶⠂⢧⠀⢸⡤⢶⣭⡎⣍⢰⣟⡳⢤⣇⠀⡞⠰⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣷
⠀⠀⠀⠀⣀⡼⢿⡉⠘⣖⣏⡀⣇⢀⣳⣱⠃⢈⡟⠣⠤⠀⠀⠀⠀⠀⣀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡙⡇
⠀⠀⠀⢠⣤⡄⠀⠙⢶⣺⣀⠈⣏⠁⣀⡯⡶⠋⠀⢰⣶⠆⠀⠰⠶⠾⡷⣷⠷⡷⠶⠆⠀⠀⠀⣠⠊⠀⠀⠁⢷
⠀⠀⠀⠀⠀⠀⠀⠴⣯⠇⠉⠉⣟⠉⠉⢸⣽⠄⠀⠀⠀⠀⢠⣄⡄⢠⠀⣿⡀⢰⡀⢠⡀⢤⣜⣁⡀⠀⠀⠀⡞⡄
⠀⠀⠀⠀⠀⠀⠐⢾⡽⠀⠀⢈⣉⡀⠀⠀⢗⡷⠀⠀⠀⠀⠈⠩⠧⢹⣲⣿⢻⣀⠻⠌⠉⠀⠙⢿⣶⣭⣼⠀⢦⣧
⠀⠀⠀⠀⠀⠀⢀⣓⠃⠀⠀⠈⠉⠁⠀⠀⠸⣺⠀⠀⠀⠀⠀⢠⣤⣤⣼⣿⣦⣤⣤⡄⠀⠀⠀⠀⠀⠉⢹⠀⢨
⠀⠀⠀⠀⢀⣤⣾⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣶⣾⣶⣦⣀

⠀⠀⠀⡀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⢀⡀⠠⡆⠀⠀⠀⠈⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⣿⣸⣧⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠡
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡙⣿⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣆⢻⣷⡘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠹⡷⠈⠉⢀⢐⠂⠀⠀⠀⠀⢀⠀⠺⠇⠀⠀⠂⠀⠀⠀⠈⠉⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⢈⠛⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠒
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡔⠀⠀⠀⠀⠆⠀⠀⠀⠺⢠⣠⠔⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣐⠂⠁⠐⠚⡁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⠂⠂⠐⠒⠁⠀⠀⢀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠂⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⢀⣠⣴⣶⡿⠟⠋⠁⠀⠀⠀⠀⠒⠠⢀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠳⠜⠜⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈
⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠲⠒⠂⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠄⠀⠠⠶⠀⠀⠀⠀⠀⡀⠀⣠
⡿⠋⢁⡄⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⢠⠜⠀⠀⠀⠐⣠⣴⣿⣿
⠁⠀⠺⡤⠀⠀⢸⣿⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⡄⠀⠀⠤⠄⠀⣼⣿⡿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣶⣤⣤⣴⣾⡿⠋⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠠⠄⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⠟⠁⠈⠀⠁⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⡐⠠⠈⠀⠀⠀⠀⠀⠸⣿⣿⣿
⣿⡿⢋⠀⠀⠀⠑⠀⡀⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⠀⠀⢀⠄⠀⠀⠀⠀⠀⢿⣿⣿
⠁⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠈⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠲⠄⠀⠀⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢄⣀
⡠⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢨⠖⠐⠀⠁⢀⠤⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲
⠀⠀⠀⠁⠀⠀⠀⡀⠀⣄⠀⠈⠀⠀⠀⢀⠞⠋⠀⠀⠀⠀⠀⠀⢓⡆⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡖⠀⠘⡅⠀⠀⠀⠀⠀⢉⠂⠐⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠋⠀⠉⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠒⠢⡁
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⢇⠀⠁⠿⢤⡀⣀⠠⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠫⠀⠀⡀⣀⡴⡃⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⣠⡾⣿⢻⡿⢷⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠛
⣇⠀⠀⠐⠀⠀⠀⠀⣀⣤⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣀⣿⢸⣷⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘
⠀⠄⠀⠀⠀⠀⢠⣿⣯⢿⣿⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣾⣿⣶⣄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠔⠂⠂⡀⠀⠀⠀⠀⢀⡴⢟⡍⢈⣠⣤⣾⣶⢦⣀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠁⠐⡀
⠀⠀⡐⠀⠀⠀⠀⠀⠠⠀⠀⠀⣰⠏⠀⠀⡼⠋⠀⠀⠙⢿⣷⡌⢧⡀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠲⢠⡾⠁⠀⠀⠀⠀⠀⠹⣿⡞⢷⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠤⡀⠀⣀⠀⠉⢀⡟⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⠘⡇⠀⠀⠀⠀⠀⠒⢄⣀⠔⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠤⣧⠀⠀⠀⠀⢸⣷⣄⠀⠀⢹⠇⣿⢠⠐⠂⡀⠀⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡄⠀⠀⠀⣼⢿⡆⠙⢦⡉⢰⡇⠸⣀⡠⠃⠀⠀⠀⠁⠒⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡀⢷⡄⠀⠀⡏⠘⣿⡄⢈⡇⠸⡇⠀⠈⠀⠀⠀⠀⠁⠀⠐⠂⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⠘⢧⣠⣄⠹⣄⣀⡇⣿⣿⣷⣼⠁⠀⢷⠀⠀⠘⠦⢀⠀⠁⠀⠀⠀⠁⠂⠦⣄⡀
⠀⠀⠀⠀⢋⠀⢈⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣮⣉⣓⣏⣩⣴⡏⠀⠀⠸⡄⢠⠓⠋⢒⡄⠀⢉⣀⠐⠀⠀⠀⠀⡉⡠
⠀⠀⢀⢷⠀⢹⠁⠀⠀⠀⡐⠈⠀⢀⣼⡠⢾⡿⣿⣯⣷⠉⡼⠁⠀⠀⠀⢳⡀⠀⠀⠀⡇⠉⢀⠘⠋⠉⠳⠒⠦⠄⠥
⠐⢀⡞⠈⣧⢸⢈⣆⠈⢇⠜⠀⠄⠿⠿⣗⣿⣩⢻⡷⣟⡿⠁⠀⠀⠀⠀⠀⢻⡀⠀⠠⡇⠀⠈⠀⠀⠀⢠⡉⠁⠀⠑
⢀⡼⠀⠀⠘⣿⠹⠘⠤⠀⡁⣴⠯⢨⡧⢈⡟⢪⣻⠶⠋⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⡀⠁⠀⠀⠐⠀⠀⠘⠃⠓⠦⠂
⢹⠃⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡆⠈⡷⢚⡛⠉⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⢽⡦⠄⢒⠀⠀⠀⠀⠐⢀⡀
⠘⢦⣀⣀⣠⠎⠀⠉⠀⠀⠀⠐⡄⠀⣀⣠⡀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⠀⢠⡄⠠⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠟⠛⠔⢁⠀⠀⠂⠄
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⠄⠀⠠⠀⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⠤⣤⣴⣤⣶⣶⡿⢶⣶⣶⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠴⠿⠿⠿⠿⠿⠷⠾⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠤⡀⠀⠀⠀⠀⠀⠀⣠⣶⢛⠉⠉⣉⣭⣵⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠀⠒⠄
⠀⠀⠀⠄⠂⠚⠀⠀⠀⢨⠀⠀⠀⠀⢠⡾⠁⠀⠉⣴⠟⠉⠉⠻⣿⣦⡙⢷⡀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠐⠀⠐⠂⠠
⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢰⡟⣀⠀⢠⡞⠁⠀⠀⠀⠀⠈⠿⣿⣆⢻⡄⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀
⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠈⢡⡿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⢻⡀⠀⠀⠑⠀⠤⡀⣀⠀⠀⠀⡀⣀⠀⣀⠀⠠⠃
⠀⠀⠀⠁⠀⠒⠄⠀⠴⠀⠀⠀⢸⠇⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠘⡇⠀⠀⠀⠀⠀⠀⠈⠒⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⡆⣠⠄⣿⠀⠀⠀⠀⠀⢸⣿⢦⡀⠀⠀⡿⠀⡿⢀⠒⠒⢀⠀⠀⠀⢀⠂⠀⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⣇⠀⠀⢸⡆⠀⠀⠀⠀⣾⢿⣇⠙⢦⣀⠃⢰⡇⠸⡀⢀⡘⠀⠀⣀⠀⠁⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣻⡄⠀⠀⠀⡇⠘⣿⡄⠀⢹⡁⢸⡇⠀⠈⡋⠀⠀⠀⠆⠢⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠻⣆⠉⡀⠹⣆⠀⢠⡇⣶⣽⣿⣀⡿⠀⠀⣇⠀⠀⠃⢄⠀⠄⠀⠂⠁⠁⠀⠐⠠⠴⣀
⠀⠀⠀⠀⢠⡔⠁⠰⠀⠈⠀⠀⠀⠀⠀⠙⢿⣿⣦⣈⡳⠾⣇⡿⠿⠛⣹⠇⠀⠀⢹⡀⠀⣇⣤⢍⡁⠐⠂⠠⠀⠀⡀⠀⠀⠀⠉⠪⠄
⠀⠀⠀⢀⡀⠳⣤⠔⠀⠀⠀⠀⢀⣠⠀⠀⢠⡟⢻⣿⣷⣶⠶⡷⡶⠋⡿⠀⠀⠀⠈⣇⠀⠂⠀⠀⠸⡇⠬⠐⠶⣤⠤⣈⣈⣀⠂⠀⠠
⠀⠀⢀⡞⢧⠀⢸⢀⠀⠀⣄⠰⡁⠀⠈⣰⣾⣴⢋⣿⣹⣿⢾⡿⣄⡼⠁⠀⠀⠀⠀⠸⣆⠂⠀⠀⠘⠃⠀⠲⠐⠉⠀⠀⢁⡉⡙⠁⢚
⠈⢠⡼⠁⠘⣆⢸⢠⢧⡀⠻⣐⣃⠐⢂⠛⡛⠻⣽⣧⣞⢹⢹⣯⠟⠁⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢸⡁⠀⠀⠀⠀⠐⠀⢤⣀
⣀⣼⠁⠀⠀⠹⣾⠙⠒⠠⠄⣈⠀⡶⠆⢠⣷⢂⣼⠛⢮⣻⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠈⠙⠀⠀⠀⠘⠃⠓⠲⠚
⢱⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡆⠀⠷⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠨⣙⠦⠤⠐⠊⠀⠀⠀⠀⠀⠀⣀
⠘⣇⠀⠀⠀⣠⠇⠈⠀⠁⠀⠀⠀⢂⡀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠀⠀⠀⠁⠀⠈⠅⠐⠎⠁⠀⠀⠀⠀⠠⠀⠀⠐
⠀⠈⠉⠻⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⡃⠄⠻⠛⠋⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⡀⠀⡁⠀⠀⠀⠀⠈⠀⠤⠐
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⢳⣭⣤⡾⢃⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡐⠐⠀⠀⠀⠐⠈⠀⠀⠅⢐⠂⠂⢠⠸
⠀⠀⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⠠⠀⠒⠀⠀⣈⠀⠈⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡶⠶⣶⣶⣶⣾⣿⣿⡖⢶⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠿⠿⠿⠿⠿⠿⠷⠾⠿⠛⠋

⠀⠀⠀⠀⣠⠒⠛⢐⣤⣀⠀⠀⠀⢠⣴⣾⣿⣿⣶⡤⠀⠀⠀⠀⢀⣤⠤⢤⠠⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢻⣾⡿⣷⣿⠃
⠀⠀⠀⢾⣹⡇⠀⣿⣧⣿⠆⠀⠀⠀⢹⠀⢸⠀⢸⠀⠀⠀⠀⢰⣿⣷⣬⣿⣠⡾⠀⠀⠀⠀⢰⠾⣧⡿⢷⠀⠀⠀⠀⠀⢸⣯⡿⣯⣿
⠀⠀⠀⠀⠙⠄⣄⠬⠋⠉⠀⠀⠀⢀⣾⣄⣾⣄⣼⣄⠀⠀⠀⠘⠟⠛⠻⠻⠋⠀⠀⠀⠀⠘⠛⠒⠛⠓⠚⠛⠂⠀⠀⠀⠺⣿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣴⠹⡍⢩⠏⢦⡀⠀⠀⠀⢾⢉⢹⡄⢤⡄⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠈⣃⣜⢉⢟⡖⢤⠀⠀⠀⠀⢠⣤⠯⢧⣀
⠀⠀⢺⡇⣻⡇⣿⣿⠀⢸⡧⠀⠀⠀⢨⣧⣯⣵⣼⡄⠀⠀⠀⠑⣺⣿⣿⠀⣿⡒⠁⠀⠀⢺⠋⠀⢹⠺⠤⡎⠀⠀⠀⢶⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠉⠻⣰⣣⣞⣆⠞⠀⠀⠀⠀⢸⣿⣿⣿⣽⡇⠀⠀⠀⠀⠛⠛⠛⠀⠙⠃⠀⠀⠀⠈⠒⠐⠻⣄⡛⣤⠀⠀⠀⠀⠙⣿⠿⢿⡿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⣤⣤⣤⣶⣶⣄⠀⠀⠀⠀⠀⣸⣷⡆⠀⠀⠀⠀⠀⠀⢠⡿⠛⠿⣦⠀⠀⠀⠀⢠⠻⡈⢹⠐⢒⡆⠀⠀⠀⠀⣴⣶⣶⣶⣶⣤
⠀⠀⠀⣯⣽⣿⢺⢺⠾⡷⡇⠀⠀⠀⠂⠙⠛⡉⠛⠶⠄⠀⠀⠀⢘⠗⠀⠐⢟⠀⠀⠀⠀⠙⠒⠃⢸⠀⣎⣘⡄⠀⠀⠀⠘⢿⣿⣿⡿⠁
⠀⠀⠀⠛⠛⠛⠛⠿⠿⠋⠀⠀⠀⣠⣶⣶⡄⠂⠀⠀⠀⠀⠀⠠⣿⣿⣶⣾⣿⡧⠀⠀⠀⠀⠀⣀⣸⣀⡈⠁⠀⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠻⠛⠻⠂⠀⠀⠀⠀⡇⢴⡦⠀⣿⣻⣿⡇⠀⠀⢠⢝⢿⣿⡿⣆⡀⠀⠀⠀⢀⣀⣴⣒⣲⣆⣀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣷
⠀⠀⠀⠀⠀⠘⠂⠊⠀⠀⠀⠀⠀⣥⣤⣤⡄⠛⠛⠛⡁⠀⠀⠇⠀⠑⠀⠛⢩⡁⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣦⢸⡇⢿⣼⡿
⠀⠀⠀⠠⣾⣿⣾⣼⣿⡆⠀⠀⠀⣿⣯⣸⡇⠐⠟⠀⡇⠀⠀⠈⠢⢤⡄⢴⣿⣿⠂⠀⠀⠀⠿⠿⠷⠿⠿⠇⠀⠀⠀⠀⠉⠈⠁⢿⣾⠟
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠲⢄⡀⡴⠉⠀⠀⠀⠀⠈⢢⡀⣠⠴⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠈⠑⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠗⠋⠉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢀⣀⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⢠⠞⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⣄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⣀⢀⣠⠾⣄⠀⠀⢀⡠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⡴⠋⠀⠀⠁⠀⠀⠀⠉⠉⠉⠀⠈⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⣀
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠠⡤⠔⠋⠀⣰⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄⢀⡼⠉⠀⣠⣙⢆⠀⠀⠀⠀⠀⠀⠀⡴⠚⢉⠛⢷⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠉⢇⠀⠸⢿⠟⣸⡠⠒⠉⠉⠑⢦⣸⠀⢸⣿⡷⠀⡟⠂⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠈⠓⠤⠤⠚⢹⠁⠀⠀⠀⠀⠀⡏⠣⣄⣀⣠⠜⠁⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⠶⡏⠀⠀⠀⢀⡴⠒⠒⠒⠺⢆⠀⠀⠀⠀⣠⣃⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣄⣀⣀⣀⣀⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢼⠀⠀⠀⠀⠀⠀⢉⠓⢒⠋⠁⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠘⣆⠀⠀⠀⠀⠀⠈⠓⠊⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠈⠉⠒⠲⠤⠤⠤⠤⠤⠖⠚⠉⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢄⣀⣀⣀⣀⣀⣀⠤⠤⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡀⠀⠀⣀⣤⡴⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠿⠋⠁⣠⡾⠋⠀⠀⢀⡄⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣴⠟⠛⠷⠄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣤⣀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⢻⡆⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠻⢶⡶⠟⠁⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠈⠳⣤⣀⠀⠈⠁⠀⢀⣠⡾⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⣠⡾⠋⠁⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣴⠞⠛⠛⠻⠦⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢰⡏⠀⣤⣼⣧⣤⣤⣤⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠲⢿⡶⠶⠶⠶⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠳⣤⣀⣀⣠⡤⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠈⠉⠉⠁⠀⣠⡾⠋⢀⣠⣶⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠞⠛⠉⠀⠀⠈⠛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠟⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⡤⠔⢺⠒⠤⣴⣃
⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⣀⢀⡠⡟⠲⣀⡒⠒⣢⣤⡔⠒⠒⡖⠠⣄⠀⠀⠀⠀⠀⠀⠠⢤⡜⠁⡽⡆⢻⠃⡲⢅⠙⣤⠄
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢰⠊⠀⡇⠀⠀⢱⠮⣀⠈⢣⠀⠀⠁⠀⢨⠑⠀⠀⠀⠀⠀⠛⡟⠈⡟⠢⣼⣾⣶⡥⠚⠋⠘⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣚⣆⡀⣇⢀⠀⣺⣀⣀⣹⡟⠂⠤⡦⢀⣸⣰⣀⣀⡀⠀⣈⡉⢏⠉⡏⣩⣿⣿⣿⣯⣉⡙⢉⣏⣭
⠀⠀⣠⠀⠒⠂⠉⢹⠉⠁⠉⠀⠑⡧⣀⠀⣈⣉⡉⢨⠤⠀⠛⠃⢀⢼⠁⡘⠆⡇⠀⠉⢉⡽⣖⠛⣾⣿⣼⠈⣿⡞⢙⡾⣄
⠀⠀⠘⠒⠸⢠⡷⢾⠀⠒⡟⠋⠉⡏⠉⠉⠻⢠⣀⢈⡁⣀⢠⣿⣯⣘⠀⠙⠃⡇⠠⠀⢼⡇⠈⢱⣿⣬⣽⣇⡼⣷⠋⠀⠶⠆
⠀⠀⠀⠀⣰⣿⣧⣴⣀⣴⣷⣄⣀⣷⣦⠤⡤⢾⡟⠺⠻⠿⠿⣿⣿⠿⠿⠿⠒⣷⣿⣆⠘⠋⣩⣿⣿⣦⣼⡇⠀⣿⣧⢀⣄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠷⢤⣅⣸⣇⢀⠠⡴⡇⣽⣿⣠⣴⡆⣸⣿⣿⣿⣿⣠⣴⣿⣸⣄⣟⣷⢿⣼⣿⣿⣿⣷⡆
⠀⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠶⠶⣿⣿⣿⣶⢶⣶⣶⣿⣿⣾⣶⣶⣿⣿⣿⣿⣿⡶⣾⣿⣿⣷⣾⡷⠿⣶⣿⣿⣿⣿⣷
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠯⠵⠐⠒⠒⠃⠼⠭⠒⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⡽⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢮⡲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡄⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⡀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡇⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣼⠛⠛⣧⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢸⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡃⠀⠀⠀⠀⠀⠀⠀⢳⠀⣰⣿⠷⠾⣿⣦⠀⡞⠀⠀⠀⠀⠀⠀⠀⢘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡃⠀⠀⢘⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢣⠀⠀⠀⠀⠀⢠⣾⢯⠃⠈⠉⠉⠁⠘⡽⣷⡄⠀⠀⠀⠀⠀⡜⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠳⡀⠀⠀⣠⣿⢯⠏⠀⠀⠀⠀⠀⠀⠹⡽⣿⣄⠀⠀⢀⠞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠜⢢⣴⣿⢏⡎⠀⠀⠀⠀⠀⠀⠀⠀⢱⡹⣿⣦⡔⠣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⡾⢄⣀⠀⠀⠀⠀⠀⠀⣀⣠⢷⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⡾⠉⠐⠂⠯⠽⠉⠉⠯⠽⠘⠂⠉⢷⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⡼⠁⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢧⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶
⠀⠀⠀⠀⣰⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣆
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠈⠻⢿⣿⡿⠟⠛⠛⠛⠿⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⡿⠟⠁⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠉⠻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠁⠀⠀⠀⠀⢻⣿⣿⣿⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣷⣦⣤⣄⣀⣦⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠟⠀⠀⠀⠀⣨⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⢿⣿⠏
⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣴⣾⡿⠿⣿⣷⣄⠀⠀⢀⣤⣶⣿⣿⠿⠛⠉⠀⠈⠃
⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣦⣄⣠⡾⠟⠋⠁⠀⠀⠈⠻⣿⣷⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣶⣤⣀⣰⣿⡿⠋⡹⠿⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⡿⠋⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣤⣰⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠆
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢐⣆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⠑⣄⣀⢀⡀⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⣍⢩⡏⢉⡾⢶
⠀⠀⠀⠀⠀⠀⠐⠂⠀⢒⠐⠋⠃⠘⠲⣠⠚⠙⡇⠀⠀⠃⠈⠲⠀⠀⠀⠀⠀⠬⡟⠣⢔⡙⣤⣇⡎⡱⠔⢻⠥
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠸⡀⠀⠀⠀⠠⡅⠁⠒⡦⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠴⡧⢼⠆⢸⣿⣿⣿⠐⠶⠰⡖⢀
⠀⠀⡀⠤⠄⠐⢺⡂⠀⠛⠉⠪⡅⠀⠈⠙⠚⢉⡃⠀⠖⠀⢈⡟⠉⠧⠒⠀⠙⢓⣳⡴⢺⣿⢋⡏⢿⡝⠓⣼⡈⠉
⠀⠀⠓⠠⠦⣴⠼⠧⠀⡾⠒⠒⠋⠉⠻⠍⣀⢀⡂⠀⢠⣶⣅⡀⠘⠓⠀⠀⠀⢬⠅⠙⣾⣏⣩⣟⣈⣷⠊⠀⠤
⠀⠀⠀⢀⣼⣷⣤⣄⣴⣷⣄⣠⣶⡤⢤⠴⣿⠚⠿⠿⠿⣿⡿⠿⠷⠒⣲⣾⣆⢸⠂⣭⣿⣷⣤⣯⢀⣿⣧⢀⣄
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⢮⣄⣿⠀⠤⠼⢇⣿⡧⣤⠶⢸⣿⣿⣿⣿⠤⣼⡇⣧⢸⣿⢻⠼⣿⣿⣿⣷
⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⠾⣿⣿⣾⢷⣶⣾⣿⣿⣿⣶⣾⣿⣿⣿⣿⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⣶⣿⣷⡶⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡖⠫⠁⠂⠃⠀⠁⠸⣼⡼⠀⠁⠀⠃⠂⠭⠓⣦⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⣋⣙⣇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⢀⣟⣉⣻⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢄⠀⠙⠋⠁⢢⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠊⠈⠛⠁⢀⠜⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠑⠁⠀⠑⢀⠀⠀⠀⢣⠀⠀⠀⡀⠀⢸⠀⡀⡀⠀⠀⢠⠃⠀⠀⢀⠐⠁⠀⠐⠙⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡦⢄⡀⠀⠀⠀⠑⢄⠀⢀⠱⡈⠀⠀⠀⢸⠀⠀⠁⠈⡰⢅⠀⢀⠄⠁⠀⠀⠀⣀⠤⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⠆⠀⠈⠒⠤⡀⠀⢀⠙⢄⠀⠑⡄⠀⠀⢸⠀⠀⠀⡔⠁⢀⠜⢁⠀⠀⡠⠔⠊⠀⠀⢶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠴⠀⠀⠀⠀⠀⠀⠈⡑⠢⢄⠀⠑⠄⠘⡄⠀⢸⠀⠀⡜⠀⠔⠁⢀⠤⠒⡉⠀⠀⠀⠀⠀⠀⠰⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠉⠒⢤⡀⠘⣦⣼⣴⡞⠐⣠⠔⠊⠁⠀⠀⠤⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣈⣻⣿⠟⢿⣿⣋⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⡶⠿⢷⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⠿⣿⣷⣿⡿⢧⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢰⠿⠷⡆
⠀⠀⠀⠀⠀⠀⠲⠶⡞⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⡠⠔⠉⠀⣼⠛⢹⠙⢻⡄⠈⠑⠠⣀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠘⡶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⣀⠴⢖⠁⠀⠄⢀⡾⠁⠀⢸⠀⠀⠹⣆⠐⠀⠀⢑⠶⢄⡀⠀⠀⠀⠀⠀⠘⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠂⠀⢀⡠⠔⠊⠀⠀⠈⠰⣀⢀⡾⠁⠠⠀⢸⠀⠠⠀⠹⣆⢀⡰⠌⠀⠀⠈⠓⠤⣀⠀⢀⢢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⠒⠉⠀⠀⠀⠀⠀⠀⠀⣠⡟⠠⠀⡀⠀⢸⠀⠀⣀⠠⠘⣧⡀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠻⠷⡐⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠐⡰⠿⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠙⢦⡃⢀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⣣⠞⠁⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣅⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡒⠬⣐⣠⠆⠀⠀⢀⣸⣀⠀⠀⠀⢦⣐⡨⠴⣺⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣭⣭⡅⠀⠀⠈⠉⠙⠓⠒⣾⡖⠒⠛⠋⠉⠀⠀⠀⣭⣭⡝⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢳⡄⠀⢠⣲⣲⣲⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢹⣄⠀⣿⣩⣹⡇
⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠥⠤⠤⠤⠴⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧⠤⠤⠤⠤⠴⠽⠦⠿⠿⠿⠧⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡯⣀⠀⠀⠀⠀⠀⠀⡼⠷⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⢷⣦⡀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⣿⣿⣿⣾⣿⣇⠘⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠴⣻⢿⢿⠿⡟⡏⠁⠉⠛⠛⠛⠛⠻⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⣯⡟⡼⠈⡀⠐⠂⠀⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠠⠤⠐⠈⠀⠀⠀⠀⠀⠀⢘⡟⠠⠁⢸⣧⣀⡰⠼⠟⠓⠓⠓⠛⠃
⠀⠀⠀⣠⣦⣾⣷⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠀⠀⢀⡉⠉⠀⠀⣸⠀⠀⠀⠀⣠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⢀⣤⣿⣷⣤⣥⠶⠛⠛⠛⠛⠛⢣
⠀⠀⠸⠿⠁⠋⠛⡉⣿⢹⠀⢸⠛⠁⠀⠀⢠⣴⣿⠏⠀⠀⠀⢼⣿⢹⡥⡍⠀⠀⣹⣀⣀⣀⣀⣄⡇
⠀⠀⣤⠀⣦⣄⣦⣷⣿⢸⠀⠈⣿⡿⣷⣚⣻⣿⡏⠀⠀⠀⠀⠘⢿⣾⣇⣿⠷⠾⠟⠛⠛⠛⠛⠛⢇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢸⠀⠈⣟⠀⠀⢸⠇⠀⠀⠀⠀⠀⣤⣤⢸⡇⢹⠉⣿⡆⠀⠀⠀⠀⢀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡆⣇⡀⠘⠀⠀⢸⣳⣄⠀⠀⠀⠀⠘⢿⣾⣧⣼⣶⣿⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⣀⣿⣿⣷⣦⡄⠀⠀⠀⠹⠏⠻⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠙⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡛⣃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀⡠⠔⣤⠞⢡⡇⠸⢦⠙⠦⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢠⣾⢼⠧⡧⣷⢰⠀⠀⠀⠀⠀⠀⣠⢸⢦⢀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⣿⣽⣯⣴⣴⣍⣴⣴⣥⣶⣴⣥⣦⣴⣿⣯⣽⣶⣿⠁
⠀⠀⠀⠀⠀⠀⡜⢸⡇⢊⣆⢿⢸⡄⡄⠀⠀⠀⡗⣾⠇⢟⢸⡟⣗⠚⠀⠀⠀⠀⠀⠀⢿⢿⡿⡿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣻⡿⣿⢿⣿⡿⡿⢿
⠀⠀⠀⠀⠀⢠⢃⡿⢀⣼⣿⣌⠁⢧⠠⠀⠀⡰⢡⡟⢀⣾⣧⡑⠹⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⣿⣿⣿⢿⣿⣿⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠈⠀⠚⠛⠛⠛⠂⠈⠓⠁⠘⠁⠁⠀⠛⠛⠛⠛⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⣷⣿⡿⠟⣿⣿⣿⣸⣿⣿⣿⠀⠀⠀⢀⣴⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⣿⣿⣿⣧⡀⣿⣿⣿⡿⠋⠉⣛⣴⡦⣤⣾⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢶⡀⢀⣀⣤⣾⣤⣀⣀⠀⢀⡶⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠯⠀⠘⠏⠀⠋⡙⠛⠁⠀⢰⣿⠸⢃⣿⠛⠹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡾⢿⠉⠁⠀⢹⠁⠀⠉⢛⡾⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡟⢠⣶⡆⣦⣿⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠠⠤⣄⣀⡼⠋⠀⢈⣷⡒⠋⢻⠋⠓⢲⣞⠀⠈⠳⣄⣀⡤⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣿⣷⣾⣿⣷⣿⣿⣀⣀⣄⣀⣀⣀⣠⡀
⠀⠀⠀⠀⣤⣤⣬⡟⠳⢤⣴⠁⠈⢳⡀⣸⠀⣠⠏⠀⢑⣤⠶⠛⣧⣤⣤⠀⠀⠀⠀⠀⢲⣼⣦⣾⣦⣿⣴⣷⣼⣧⣾⣦⣿⣴⣷⣼⣧⣾⣦⣿⣴⠆
⠀⠀⠀⠀⠈⠉⣼⠁⠀⡸⠉⠙⠦⢄⣱⠼⢔⢉⡤⠞⠉⠸⡄⠀⢹⡀
⠀⠀⠀⠶⠶⠶⣿⠶⠒⡷⠒⠒⢒⠒⣇⠀⢀⢗⢒⠒⠒⠒⡗⠒⢺⡗⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠸⣶⣶⠆⢹⡄⠀⠸⣀⡤⡮⣬⡖⢩⠑⣮⠾⡶⢤⣰⠁⠀⣼⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⡧
⠀⠀⠀⠀⠀⠀⠀⣳⡴⠚⠙⢤⣍⡟⠀⢺⠀⠘⣯⢥⡴⠉⠓⣶⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢻⢻⠁
⠀⠀⠀⠀⠀⠒⠋⠁⠹⣦⡀⣜⡺⠓⠦⣾⠤⠖⠻⣫⡄⣠⠞⠁⠈⠉⠁⠀⠀⠀⢀⡠⣠⢎⡏⣧⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⢸⡆
⠀⠀⠀⠀⠲⠶⠾⠀⠀⠈⣹⣢⣧⣄⣀⣸⣀⣀⣤⣗⣻⠁⠀⠀⠘⠿⠟⠀⠀⠘⠛⠛⠛⠛⣿⡟⠛⡟⠛⠋⠀⠀⠀⠀⡠⠊⠀⠀⠀⡀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡯⡝⠀⠀⠀⢹⠉⠀⠀⠸⣖⡷⠄⠀⠀⠀⠀⠀⣠⡄⣴⠀⣤⠀⣿⣇⠀⢳⡀⢀⣄⠀⢤⣾⣁⡀⠀⠀⠀⡇⠈⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢽⠃⠀⠀⠀⠉⠁⠀⠀⠀⢯⢵⡖⠀⠀⠀⠀⠀⠙⠃⣿⡀⡷⢀⣿⡟⡶⠈⢿⣄⠛⠁⠈⠻⣿⣮⣭⣍⣒⠃⠀⣄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡟⠀⠀⠀⠘⠿⠟⠀⠀⠀⠘⣏⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢚⣿⣇⠙⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢺⡀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠿⠿⠿⠿⠿⠿⠿⠦⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⡻⢿⠛⡻⠿⡿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣿⡏⠉⢹⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠀⠀⠀⠀⣤⣤⣤⣶⣤⣤⣤⠀⠀⠀⠀⠀⣀⣴⠺⣳⣤⡀
⠀⢻⡇⠀⢸⡿⠀⠀⠀⠀⠀⢀⣶⢶⡶⣶⡶⣿⠿⣷⠄⠀⠀⠀⣹⣷⢻⢺⡟⡞⡇⠀⠀⠀⠀⠈⣿⢹⡍⢹⡏⣿⠁
⡠⢼⣷⡖⣾⡧⢄⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢧⣿⢸⢸⡇⣇⡇⠀⠀⠀⠀⠀⣿⢸⠇⢸⡇⣿
⠑⠲⠤⠭⠥⠖⠊⠀⠀⠀⠀⠀⠿⠇⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠙⠺⠼⠓⠋⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠄


⠀⠀⢠⡄
⠀⣰⣿⣀⣙⣤⠄⠀⠀⠀⠀⠀⠀⡎⡉⠉⠉⢱⣀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣟⠂
⠀⣿⡏⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⡇⠄⠀⠀⢸⡿⠀⠀⠀⠀⢐⣖⢀⣼⣧⡀⣲⡀⠀⠀⠀⠀⠀⠀⠀⠠⣽⠇
⠀⣿⡗⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⠠⠀⠤⠄⣯⡇⠀⠀⠀⢠⣿⠙⣾⡧⣻⣷⠉⣿⡀⠀⠀⠀⠂⠉⠁⢐⣀⣰⠋⠉
⠰⠿⠷⠶⠾⠿⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠞⠀⠀⠀⠀⠈⠻⠟⠙⠿⠟⠋⠿⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠈⢁⡀
⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣄⣤⠀⠀⠀⠀⠀⠀⠾⠎⠀⠙⠛⠋
⠀⠀⢀⣰⡿⠁⠀⠀⠀⠀⠀⢀⡴⣞⣵⠿⣾⣟⢦⣀⠀⠀⠀⠈⢶⡾⢷⡾⣷⡶⠁⠀⠀⠀⠀⠀⢀⣾⡀⣰⣇
⣿⣷⣿⣿⣷⣿⣿⡇⠀⠀⠀⠈⢸⣿⣧⡐⢺⣿⣷⠉⠀⠀⠀⠀⢸⡇⢸⡇⣾⣧⠀⠀⠀⠀⠀⣄⠜⣿⠻⣿⣿⢧⣤
⣿⣿⣿⣿⣿⣿⣟⠂⠀⠀⠀⠀⢸⣿⣧⣥⣾⣿⣿⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠸⡿⠀⣿⠀⢸⡏⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⢰⣴⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⡹⠹⠻⡄
⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠳⠖⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡖⠂⠀⡆⠀⢆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⣜⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠲
⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⢰⡆⠀⠸⠀⢀⣇⠀⠀⡁⠀⣿⠃⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠉⠻⠃⠀⡀⠀⠙⢆⢠⠗⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏⠀⠀⠌⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⠀⠀⢀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡅⠀⠐⠄⠀⠀⠀⠀⠀⠀⠒⠀⠂⠄⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡇⠀⠁⠀⠀⡄⠀⢀⠎⠁⡠⠈⡕⢀⣁⠉⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠤⠈⠂⠀⠁⠀⠀⠑⠉⡀⠀⠀⠀⠀⠘⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠈⡈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠒⠀⠀⠐⠈
⠀⠀⠐⠀⠒⠀⠀⠂⠀⠸⠅⠀⠀⠀⠂⢀⣀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠠⠄⠄⣀⣀⡀⠤⠐⠊⠀⠀⠉⠒⠀⠒⠐⠊⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣉⡄⠀⠀⠀⠈⠐⠂⠐⠂⠚⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⢄⡻⡧⠄⣚⠶⡄⠈⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠱⢇⠀⢀⢀⠊⠑⡌⠳
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠂⡇⠀⠾⣻⣷⠀⠀⢰⠰⣂
⠀⠀⠀⠀⠀⠀⢀⣤⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠠⡄⠰⡀⣰⠋⡍⠓⢄⡌⢀⠃
⠀⠀⠀⠠⠀⠚⣓⢒⠂⠉⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢰⠨⠀⢰⢡⠆⣃⢔⢴⡀⠀⠀⠀⠈⢢⣜⡻⠂⠷⢻⣈⡴⢣
⠀⠀⠀⠀⢀⣵⠀⣶⡇⢺⣮⠹⢦⣄⣀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠿⠀⠀⠀⠀⢘⠚⠠⡀⡸⠘⢰⠊⢷⢶⡍⢄⠀⠀⠀⠀⢈⠋⠐⠛⠈⢃
⠀⠀⢤⣶⠟⠁⠴⡿⠁⠘⠿⢗⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠡⠴⠲⢒⠲⠶⢦⡡⠈⠢⠕⠊⠀⠀⠀⠀⠸⠤⠤⠤⠴⠾
⠀⠀⠀⠘⠀⠀⠀⢷⡀⠀⢸⡎⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠅⠉⣷⣏⢈⣈⠈⢈⢁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⡀⠀⡄⡀⢸⣆⠀⠀⣧⠀⣼⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠚⠉⠁⠈⠈⠑⠂⡘⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⣔⣓⣚⣳⡄
⠀⠀⠀⠑⠀⠻⠅⠚⠓⠀⠀⠚⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠐⠒⠒⠒⠀⠐⠂⠐⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡠⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡔⠱⢹⡟⡝⢄
⠀⠀⠀⠀⠀⡀⢀⢀⣿⡇⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⣠⠊⢀⡇⢸⠇⠸⡈⠂
⠀⠀⠀⠀⢠⠟⠁⠈⠉⢁⡙⠛⠻⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⠀⠻⠖⢀⣔⡁⠀⣀⡁⢸⢀⣀⣠⡔
⠀⠀⠀⡠⡋⠀⠀⠀⠀⠢⠀⠰⠀⠑⡈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠁⠀⠀⠀⠉⠀⠀⠛⠁⢸⠙⠃⠈
⠀⠀⠴⢡⠛⠀⢠⠢⠀⠀⠊⠁⢢⡀⠐⠠⢑⣂⣀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠤⠄
⠀⠀⠀⠀⠈⠛⠋⠈⠛⠛⠀⠀⠈⠛⠒⠂⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓

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

⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠓⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠄⢠⢠⠄⢤⠆⢠⠀⢠⠠⡀⡄⠀⡤⠀⠀⢠⠆⠀⡤⠀⠄⢄⢠⠐⢄⢠⢠⠄⡤
⠀⠀⠀⠀⠀⠀⠀⢸⣁⢸⣀⡰⠰⠀⢸⢀⠃⢣⢸⣉⠆⣇⡀⣏⣁⠀⢈⡱⠀⡇⠀⠃⢣⠣⣀⡸⢸⠱⡀⣏⡀

⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⣸⣿⣷⡆⣤⡀⠀⢠⡄⢀⠀⣀⡄⣤⣤⣠⠄⡀⢀⣠⢤⡄⣤⢤⡄⠀⢠⢄⢠⡄⣤⣄⣄⡄
⠀⠀⠀⠀⠀⠹⣿⠿⠻⠏⣿⣷⠄⣸⣧⢸⣾⢿⡇⣏⢳⣸⠂⢻⣾⣿⠂⡿⣿⢸⡇⠀⢸⠞⣸⣧⣿⣏⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠟⠀⠀⠋⠙⠂⠃⠊⠳⠋⠛⠛⠂⠚⠛⠛⠒⠃⠛⠘⠃⠀⠘⠀⠋⠙⠟⠟⠁⠉


⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇
⠀⠀⠀⠀⠀⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇
⠀⠀⠠⢨⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡜⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠘⣼⠀⠀⡇
⠀⠀⠀⢠⠁⠀⠠⠖⠀⠀⠀⣠⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡞⠀⠀⠀⡼⢸⠀⠀⠀⠀⠀⠀⢤⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠠⡚⠀⠁⠂⢂⡀⠀⠀⠀⠀⡰⠁⠀⠀⢠⠎⠀⠀⠀⣰⠁⢰⠀⠀⠀⠀⠀⢠⡍⠵⠀⠀⣇
⠀⢀⠀⠀⠠⠡⠉⠀⠚⠀⠀⠀⠠⢨⠇⠊⠀⠀⠀⠔⡁⠀⠀⠀⢠⠇⠀⡈⠀⠀⠀⠀⠀⢸⣿⠜⡄⠀⡟
⠀⡟⠀⠐⠀⠆⠀⠀⠼⠀⡀⠀⠀⠀⠂⡤⠄⠤⣢⠎⠀⠀⠀⢠⠏⠀⠀⣿⠀⠀⠀⠀⠠⠳⠶⠶⠿⠀⣇
⢠⠃⢀⡑⢒⡚⣇⢂⠐⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⡠⣤⢀⡄⠤⠿⠤⠤⡤⠤⣐⡀⠀⠀⠀⠀⣿⢀⣀
⠈⠀⢸⡏⠈⠑⡀⠀⠀⠠⠄⠀⢀⡀⠠⠄⠔⠠⠂⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠠⣀⡄⠠⠄
⠀⡤⠾⠂⠀⠀⠙⠀⠠⠰⠋⠀⠈⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠺⠦⠺⠶⠮⠲⠴⢶⠓
⠠⠄⠀⠀⡀⠀⢰⠀⠀⢢⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⣀⡖⠀⠀⠈⠃⡁⠀⠐⠂⠈
⢰⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠈⠀⠀⠀⠀⡆⠀⢓⡀
⠀⠀⠀⠀⠐⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁⠚⠀⠀⠀⠠⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠠⠅⠘⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⢈⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⠀⢾⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠑⠾⠀⠀⠀⣇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠘⠆⠠⠄⠀⠈⠀⠀⠀⢠
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣮⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡠⡀⣄⡀⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋⠓⠋⠋⠃⠛⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠿⠰⣉⠸⠺⠸⠧⠟⠾⢮⡥⢿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⠦⡀⠀⠀⠀⡖⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⠶⣣⠄⠀⢤⡘⢢⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⣹⢁⠉⠈⠃⠹⢎⡜⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠁⢸⠸⠘⠶⣓⣒⡳⢚⠥⡀⠀⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⢦⣉⡆⠀⠀⠀⢇⣱⡸⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠆⠀⠀⠀⣀⠀⠀⠀⡀⠀⣠⣾⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡷⠂⡐⣞⢂⠀⠀⠋⠨⠀⠁⣁⣥⣿⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠌⠩⠾⠈⡑⠄⠁⢀⡀⢸⣿⣸⣿⣸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⠀⢆⡠⢧⠀⡧⠤⠥⠬⠭⠭⠭⠭⠥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠉⠀⠀⢰⠀⢸⡀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⠦⢤⢄⡀
⠀⠀⠀⠀⢀⡴⣫⠽⡖⠋⢹⠉⠓⡾⢭⡛⣤⡀
⠀⠀⠀⣴⢫⠞⠣⡀⠘⡄⢸⠀⡸⠀⣠⠞⢣⠱⡄
⠀⠀⢰⢓⠗⠤⣀⠘⢦⣹⣸⣰⣣⠞⢁⡠⠔⢳⢹⡄
⠀⠀⣿⣼⠀⠀⠀⠩⢵⣿⠋⢻⣷⠯⠁⠀⠀⢸⡤⡇⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢹⢸⡀⣀⠤⠒⣪⢿⣿⣿⢯⡓⠢⢄⣀⢰⢒⠇⠀⠀⠀⠀⡠⡞⠈⢗⢀⠀⠀⠀⠀⠀⠀⣔⡕⠂
⠀⠀⠘⣎⢫⠀⢠⠞⠁⡎⣿⣿⢗⠛⢦⡀⢠⢏⡟⢀⡴⠚⠿⢯⡬⣥⠭⡬⢭⢥⠕⠀⢀⣀⢸⢴⢂⠀⣰⣤⡀
⠀⠲⠦⠿⢧⡟⢿⡞⡟⢀⢿⣿⠾⡶⣼⠟⣣⠞⠀⣸⣤⢶⠗⣾⢦⠅⠀⠳⠀⢸⣦⣈⢨⣨⢘⣋⣿⠀⢰⠋⢏
⠀⠀⠀⠀⠐⠘⠳⢼⣑⣾⣺⣛⣶⣽⣴⣏⣥⠶⠟⠙⣷⣇⣶⣣⣾⡖⣖⢲⡀⢸⡛⢿⣯⣿⠿⠿⠋⠀⠁⠀⠘⠀⢠⠙⣣⡜⢚
⠤⠤⠤⠴⠶⠶⠦⠤⠾⠤⠽⠽⠿⠷⠿⠿⠷⠶⠾⠿⠿⠧⠬⠤⠼⠧⠥⠾⠥⠤⠥⠤⠤⠴⠤⠤⠴⠷⠶⠶⠶⠾⠾⠶⠷⠷⠾⠤⠤

⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈
⡄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣀⠀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡐⠈⠁⠀⠁⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠕⠀⠐⠃
⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⡐⠀⡐⠠
⡇⠀⠀⠀⡰⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⠜⢸⣧⠒⢄⠀⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⠀⠠⠀⢁⠔⢢
⡇⠀⠀⠂⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢅⡴⠋⠳⢕⠉⡠⠢⡄⠀⠀⠀⠀⠀⠈⣆⠢⠀⢀⡌⠀⡀⡎
⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣁⠉⠢⡀⢸⠀⠃⢀⠇⠀⠀⡆⡇⠀⠀⡧⠙⡒⠋⠀⢠⠁⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠀⠀⠀⠀⢀⢣⠀⠰⠀⠆⢰⠁⠢⠤⢣⠋⠀⠀⢀⠁⡆⠀⠀⢁⢀⡈⠀⠀⢸⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⣣⠁⠀⠀⠀⠀⠀⠀⢸⠀⠢⣀⠾⡀⢀⠑⢄⠀⣇⡀⠀⠀⠌⢰⠃⠀⠀⡉⠘⠀⠀⠀⠸⡇⢰
⡇⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢀⠀⠀⠀⠃⠸⠀⠀⠀⡔⠢⣀⡁⡸⠀⠀⣜⢀⠌⢠⠃⠀⠀⠀⢱⠆⠀⠀⠀⠐⠸⡀⢆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠐⠀⠸⠀⠀⠀⠈⢢⣀⠌⠀⠀⠀⡔⠁⡠⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠱⡀⢢⠄
⡇⠀⠠⠄⣤⠀⠀⠀⠀⠀⠀⠀⡤⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⣸⣊⣁⣠⡀⠁⠀⠀⠀⠐⠀⠀⠀⢸⠀⠀⠈⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢡⠰⢈⠀⠀⣀⠀⠀⣄⡀⠤⠖⠶⠂⡤⠤⠼⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⡇
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⡆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⢢⠔⠂⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⣀⡄⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡔⣴⢲⡢⣄
⠀⠀⠀⠀⢠⢴⣯⠯⡷⡦⠄⠀⠀⠀⢀⣴⣄⡀⠀⠀⠀⠀⠀⣷⣶⣿⣯⣤⣭⣤⣭⣤⣥⣤⣴⣾⣷⣴⡇
⠀⠀⠀⠀⡘⣼⢀⣧⠣⢧⠀⠀⢀⢣⡏⢫⡹⢻⠁⠀⠀⠀⠀⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⠟⠛⢻⠛⠛⠃
⠀⠀⠀⠐⠑⠁⠾⠿⠧⠈⠂⠀⠊⠊⠰⠿⠷⠀⠓⠀⠀⠀⠀⠀⠀⢠⣶⡶⣿⣿⣿⣿⣿⠀⠀⢸⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢳⠿⢿⡆⣿⣿⡿⠋⢙⢴⢦⣾⣿⣏
⠀⠀⠀⠀⠀⠀⣈⣹⡤⠔⠚⡿⠓⠦⣤⣏⡁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⡀⡍⣁⡀⣻⠈⠸⠁⠘⠁
⠀⠀⠀⠠⣄⣀⠞⠁⣹⡤⠒⡗⠒⢤⡏⠙⢧⣀⡤⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣷⣿⣷⣿⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠶⢶⠏⠓⡾⣁⠘⡄⡇⡰⠃⣨⢶⠊⢻⠶⠆⠀⠀⠀⠲⠿⠾⠾⠷⠷⠾⠾⠶⠷⠷⠾⠾⠶⠷⠇
⠀⠀⠠⢤⣼⠤⢼⠤⠬⠍⡎⠉⡦⠭⠤⢬⡤⢼⡦⠤
⠀⠀⠶⠶⠸⡄⠈⣄⢴⢭⡌⡎⢦⣷⠦⣼⠀⣸⠱⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢹⡆
⠀⠀⠀⢀⣠⠷⣏⠉⣖⣟⣀⣇⣈⣷⡞⠁⡹⠳⠤⠀⠀⠀⠀⢀⣰⣆⡀⠀⠀⠀⠀⠀⠀⢀⠔⢉⢻
⠀⠀⠀⢤⡤⠀⠈⢳⢿⣀⣀⣏⣀⣸⣵⠊⠁⠰⠶⠆⠀⠰⠶⠿⢾⡾⠾⠷⠆⠀⠀⠀⡰⠃⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠐⣿⠇⠀⢈⣏⠁⠀⢳⠗⠀⠀⠀⠀⢰⣦⡆⡄⢸⣧⡀⣦⠠⣄⠰⣮⡤⢀⡀⡀⢰⠃
⠀⠀⠀⠀⠀⠀⢻⠽⠀⠀⢠⣤⠄⠀⠘⡽⠁⠀⠀⠀⠀⠘⠃⠯⢼⡏⠧⠘⠃⠀⠀⠈⠻⠿⣶⡇⢸⡾⠂
⠀⠀⠀⠀⠀⢀⣞⡇⠀⠀⠀⠀⠀⠀⠀⣩⣇⠀⠀⠀⠀⠰⠶⠶⠿⠿⠶⠶⠆⠀⠀⠀⠀⠀⠀⡇⢀⡀
⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠒

⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠈⡀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠅⠀⠀⣠⠂⠀⠀⠠⠎⠀⠈⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⡠⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠄⠀⠀⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢄⡤⠀⡀⠈⠀⠀⠀⠀⠐⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠒⠀⠀⡀
⠠⡀⠀⠀⠀⠐⠈⠁⠀⠄⢠⠀⠀⠀⡄⢰⣉⣃⣘⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⡄⣶⣹⣁⣛
⠀⠀⠀⢀⠤⢠⡄⠀⠀⠀⠈⠃⠀⠀⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡃⠀⢠⠤⢠⠄⠀⠀⠀⠈⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠂
⠉⠀⠀⠌⠀⠼⠀⢠⡄⠀⠂⠀⠀⢀⠀⠀⠀⡀⠂⠀⠀⠀⠰⠃⠉⠁⠀⠀⠌⠠⠼⠀⢠⠄⠐⠂⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⡀⠀⠀⠀⠀⠄⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⣀⠀⢀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢢⡀⠀⢠⠄
⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠂⠰⠀⠊⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⡀⠄⠀⠀⠀⠀⢀⡐⠒⠢⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠠⢂⠐⠀⠂⠀⠀⠀⠀⠀⢀⠐⠒⠂⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠄⢂
⠀⠠⠠⠞⠀⠀⠀⠈⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠉⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠈
⡀⠀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟
⠀⠀⠀⠀⠉⠁⠐⠂⠀⢀⡀⠀⠀⠀⡀⢠⢰⡄⢲⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠔⠀⠀⢀⡀⠀⠀⠀⠀⣠⢰⠄⡦
⠈⢢⠀⠀⡀⠀⡀⠀⠀⠀⠘⠆⠀⠀⠃⠜⠃⠒⠒⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠚⠄⠀⠀⠃⠟⠐⠒⠂
⠒⠊⠀⠠⠉⢩⠁⠠⠔⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⢀⡂⠒⠒⠀⠀⠠⠉⢨⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⣠
⠀⠀⠀⠉⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⢂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠁⠀⠀⠰⠐⠈⣄⠃⠀⠘⠁⠀⡀⢀⠀⡄⠀⠀⠀⠀⠁⢁⠀⠀⠈⠁⠀⠀⠔⠀⠀⡌⠁⠀⠈⠀⠄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣴⠶⠟⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⣉⣭⣉⢉⣩⣯⡉⢉⣭⣍⡉⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣼⣷⣿⣿⣿⣾⣿⣿⣿⣾⣿⣿⣷⣿⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢿⣿⣿⡿⠿⠿⣿⣿⣿⠿⠿⠿⣿⣿⣿⠿⠿⢿⣿⣿⡿⠿⠿⢿⣿⣿⡿⠿⠿⣿⣿⣿⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠈⠿⠁⠀⠀⠀⠈⠟⠁⠀⠀⠀⠙⠏⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⣶⠰⡖⢶⠰⡶⢶⠶⡶⢲⠆⡶⢰⡆⣶⢰⡆⣶⣷⡆⣶⢰⡖⣶⢰⡖⣶⠰⡆⢶⠰⡶⢶⠶⡶⢲⠖⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠈⡇⠀⠀⠀⠁⠀⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢤⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡀⠀⡄
⠀⠀⠀⠀⢴⣷⣦⠀⠀⠀⠀⠀⠀⡰⠋⠀⢈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡆⠀⠀⠀⠀⠀⠀⢠⠋⠁⠈⠧
⠀⠀⠀⠀⠈⠉⠙⡇⠀⣀⣄⢀⣒⡁⣀⡠⠔⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⡴⣻⣿⡛⠀⠴⠀⠈⠻⢷⡀⠀⢀⡂⠀⡀⠃⠀⣀⠤
⠀⠀⠀⠀⠠⠤⠄⠈⠉⠙⢿⡿⠟⠁⠀⢾⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⢨⣿⣿⠿⣷⡄⠀⠀⡀⣀⡀⠁⠒⠹⣿⣾⡿⠁⠉⢴
⠀⢐⠒⠀⠀⠀⠀⠀⠀⠀⠘⢁⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⣰⢶⢦⢀⡤⠏⢸⣿⣻⠚⠈⠀⠆⠈⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠁⠈⢀⡰⢦⣤⣿⠿⢿⣦⢀⣴⣿⣿⣧⣀⣠⣌⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠷⠟⠀⠀⠉⠁⠀⠀⠩⠘⡟⠛⠙⠿⠋⣠⡦⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⡖⢲⠒⡖⢲⠒⡖⢲⠒⡶⢲⠒⡶⢲⠖⡶⢳⡖⣖⢲⡖⣖⢲⡖⣶⢲⡖⣶⢲⡖⣶⢲⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⢛⣞⠓⠛⠚⠓⠛⠚⠓⠛⠚⠓⠛⠚⠓⠛⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠛⠟⠛⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣲⡂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡇⠉⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⡖⠋⢹⠉⠛⢲⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⢀⡇⣀⡤⡟⠙⢯⣉⡋⢑⣦⢴⡖⠒⠈⢰⠒⠢⣤⡀⠀⠀⠀⠀⠀⠀⠐⢦⣴⡁⣠⠿⡎⢹⠉⣱⠧⡀⢙⣤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣧⠁⠀⡇⠀⠀⠈⡷⢏⡀⠀⠓⠀⠀⠸⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠛⢻⠁⢹⠓⢤⣽⣾⣶⣧⠴⠛⡅⢸⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠠⡿⡄⠀⡇⠀⠀⠠⡇⠀⠀⠀⠶⠠⠠⠤⢀⠀⠀⢀⠀⠀⠀⠀⠀⣒⡒⢻⠒⢛⠒⢺⣿⣿⣿⣿⡒⠚⡗⢘⣗⣃⡀
⠀⠀⠀⣀⡠⢤⠤⠒⢺⠓⠒⠚⡏⠙⠫⣏⠉⠉⠉⠛⠛⠃⠉⠈⠀⠀⠀⠀⠀⠀⠋⠘⢃⠀⠐⠒⠛⠛⡾⣦⠼⢾⣿⣟⢹⠙⢿⣏⠟⢦⣼⠈⠉⠁
⠀⠀⠸⣅⡀⣰⠠⣄⣸⡇⣀⣠⣧⠤⠤⡿⠷⠶⢶⡗⠋⠁⠈⠇⠀⠀⢰⣤⠞⠰⠀⠶⡄⢀⠀⠀⠀⣒⡏⠙⠦⣴⡿⠓⢻⡖⠚⣿⣠⠞⢀⣉⡀
⠀⠀⠀⠀⠀⢩⣼⡏⠙⠁⠀⣰⡇⠀⠀⣇⡀⠀⠀⠁⣽⣖⣲⣦⣴⣮⣿⣿⣷⣶⣤⣀⡀⢸⢀⣍⣀⣉⡏⢠⢤⣿⣿⠚⢻⡗⠚⢻⣧⠀⠀⠉⠁
⠀⠀⠀⠀⣠⣾⣿⣿⣾⣧⣾⣿⣿⣦⣤⡿⣷⠒⢛⡟⢹⡏⠈⠉⠛⣛⡛⣿⣿⠛⠛⠛⢉⣹⣿⣿⣧⣄⡇⠰⣾⡿⣿⢿⣿⣇⢠⡼⣿⣆⣰⣷⣀
⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⠿⠶⢶⣤⣼⣧⠠⡤⠶⠿⢤⣿⣿⢴⡾⡯⢠⣿⣿⣿⣿⣿⡧⢦⣿⡇⣿⡤⢯⣿⢻⠷⣿⣿⣿⣿⣿⣟
⠀⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡶⠶⢶⣿⣿⣿⣶⢷⣶⣶⣾⣿⣿⣾⣶⣶⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡷⠾⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠈⠉⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠈⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠿⣿⣿⠁⠀⠀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⢀⣀⠀⠀⠀⢸⣿⣆⣀⣀⡴⠲⢤⣀⠀⠉⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠉⠁⢀⣀⣀⣤⡤⠤⠶⠖⠚⠛⠋⠉⠙⢦⣀⡀⠀⣿⣟⠉⠁⠀⠀⠀⠈⠙⠓⠦⣄⣀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⡀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢲⠋⠀⠀⠀⠀⠀⢠⡞⠛⢷⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡆⠀⠀⠀⢠⡞⠉⢳⡄⠀⠀⠀⠀⠸⣇⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡆⠀⠀⠀⠀⠀⠘⠧⡤⠞⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣇⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠀⠀⠀⢹⡇⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⠀⢻⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣄⡀⠀⠀⢸⡄⠘⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡞⠀⣸⣿⣧⣀
⠀⠀⠀⢀⣠⣶⡿⠟⠉⢹⣿⡀⠸⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⠀⠈⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡾⠁⣰⣿⠋⠙⠿⣷⣦⣀
⠀⣠⣶⡿⠟⠉⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⣾⣿⡿⠋⣿⣿⠁⢀⣀⠈⢿⣿⣇⠀⢿⠀⢹⣿⠋⣀⡀⠙⣿⣿⡇⠀⠀⣼⠁⢠⣿⠏⠀⠀⠀⠈⠙⠿⣷⣦⡀
⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⠀⢸⣿⡿⠁⠰⢟⡋⠀⣭⣭⠀⢰⣿⣿⠀⢸⡄⠘⣿⣷⣭⡃⠀⣿⣿⡇⠀⣸⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡇⠈⣿⣿⣆⠈⠉⠁⢠⣿⣇⣠⣾⣿⡟⠀⠈⣇⠀⣿⣿⣀⣠⣾⣾⡿⠀⣰⠃⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿
⣿⡏⠙⠻⢿⣶⣄⡀⠀⠀⠈⣿⡇⠀⣧⠀⠘⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⢻⠀⢸⣿⣿⣿⡿⠋⠀⣰⠏⢀⣾⡟⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⣿
⣿⡇⠀⠀⠀⠉⠻⢿⣶⣄⡀⢻⣿⠀⢹⡀⠀⠀⠉⠻⠿⢿⣿⠿⠿⠛⠁⠀⠀⠀⠀⢸⡆⠘⣿⡏⠁⠀⠀⢠⠏⠀⣾⡿⠁⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣷⠀⠀⢠⡏⠀⣼⡿⠁⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⢠⡴⠲⣤⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿⡀⢀⡞⠀⣼⣿⣷⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⡀⠀⠀⠀⠀⠀⠸⣄⣀⡼⠀⠀⠀⠀⠀⠀⣸⡆⠈⣿⣇⡾⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢿⣿⠁⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠛⢦⡀⣀⣀⣠⣤⠤⠴⠶⠒⠚⠛⠉⠁⠀⢀⣀⣾⡿⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠉⠉⢀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⡿⠛⠋⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣶⣿⠿⠿⠿⠛⠛⠛⠉⠙⠿⠟⠉⠙⠻⢿⣷⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠙⢿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⡞⠡⡼⣹⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⡛⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⣽⢿⡍⣹⣿⣷⣸⣆
⠀⠀⠀⠀⠀⡠⢚⠙⢿⣿⡇⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⡻⣿⡄⠀⠀⠀⠀⠀⠀⢸⣇⠉⠈⢿⣪⠟⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠃⣚⡠⣼⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡲⢍⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣔⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠠⠤⠞⢡⣤⠀⠀⠀⠀⠀⠀⢰⡶⠒⡲⣒⣒⡲⡒⠲⣶⠀⠀⠀⠀⠀⢀⣾⠃⢸⣿⡦⠄⠛⠉
⠀⠀⠀⠀⠀⢠⣅⠐⣿⡇⣶⡆⢸⣿⠀⠀⠀⠀⠀⠀⢸⠀⠶⡹⣯⢜⡷⡱⠆⢸⠀⠀⠀⠀⠀⠘⠛⠃⠉⠈⡆
⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠸⠷⠤⣱⡯⠭⠷⠥⠴⠿⠀⠀⠀⠀⠀⠀⠀⡔⢪⠖⣩⡒⣴⢢
⠀⠀⠀⠀⠀⠈⠩⠤⠉⠡⠉⠡⠌⠩⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡺⣄⡞⣠⣟⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣄⡤⠭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠽⣋⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣶⣄⠁⣠⠟⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⢶⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠦⡝⢻⣿⣿⠃⠀⠀⠀⠀⣠⡞⠁⡘⠷⡧⣯⠞⠁⠻⣧⡄⠀⠀⠀⠀⣿⣿⣿⣿⡀⣫⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣿⣿⠃⠀⠀⠀⠀⠀⠈⠳⣄⣁⣠⠞⡅⣿⣦⣴⡿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠱⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠋

⠀⠀⠠⡀⠀⠀⢤⡀⠀⠂⢦⠀⠀⠚⠀⠀⠀⠀⢹⡆⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠄
⠀⠀⠀⠈⠢⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇⠀⠒⠰⠻⡧⢀⠀⠰⠖⠓⠀⢠⣀⠁⠈⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠒⠄⠀⠈⠁⢈⠴⠚⠁⠀⠀⣀⠁⠀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠰⠁⠀⠀⠀⠉⠠⠐⠊⢀⠄⠈⠩⠵⠂⢀⣄⠄
⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⢸⣶⣆⠀⢆⠀⠀⠈⠀⠉⢁⣀⡤⠀⠒⢉⠜⡠⢀⠃⠐⠂
⠀⠐⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⢸⣿⡏⠀⠀⠁⠀⠀⠀⠀⠠⡶⠿⠷⠗⠚⢊⣴⣧⡀
⠀⠀⠀⠀⡁⠀⠀⡁⢘⠀⠀⠀⠀⣿⠀⠀⠀⠀⢺⣿⡇⠐⠀⠀⠀⢠⡆⠀⠀⠈⠓⠶⠠⠶⠻⠀⠙⠛⣠⠄
⠀⠀⠀⠀⡄⠀⠀⠁⢈⠀⠀⠀⠀⠸⠀⠀⣀⠀⠼⠛⠛⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠹⣿⣄⣀
⠀⠀⠀⠀⡀⠀⣀⠆⠀⢀⠤⠀⠄⠉⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⢸⡇⢈⢀⡤⠐⠂⠀⠀⠁⠀⠈⢘⣿⠟⠋
⠀⠂⠄⠲⠀⠀⠀⠀⠐⠭⠿⠿⡖⢂⡀⠀⠀⠀⢠⣄⡀⠅⠀⠀⢨⣿⡏⢠⡠⠧⠖⠀⠀⠰⠏⡀⠀⠼⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠄⠩⢈⣀⣋⣉⣁⡄⡀⠀⠀⠈⣿⠧⠴⠓⣺⣉⠲⣄⠉⢳⠀⠀⠀⠤⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠟⣻⣥⢬⣴⣿⣽⣷⠂⠀⠀⢘⣿⣿⣅⠀⠀⠀⣨⠟⢀⣀⣤⡴⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠡⠖⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣅⠀⡀⠀⣰⠍⠩⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣦⣶⣶⣴⣦⣴⣿⣴⣷⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣯⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠛⢿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⠟⠉⠁⣀⣀⠈⠉⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⢴⣾⣶⣤⣿⣿⡏⣸⣿⣿⣿⣿⣿⡿⠁⠠⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡈⠙⠻⢿⣿⣿⠁⣿⣿⣿⣿⣿⣤⡄⠀⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢹⣿⡀⣿⣿⣿⣿⣿⠛⠀⠀⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠿⣿⠿⠀⣸⣿⣇⢹⣿⣿⣿⣿⣿⣷⡀⠘⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣤⣾⣿⣿⣿⡌⢿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠉⠉⠀⣀⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉

⠀⠀⠀⠀⠀⠀⡀⡀⢀⢀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡄⢠⡀
⠀⠀⠀⢸⢸⡇⡇⣿⢸⢸⢸⡇⠈⡟⢻⠋⠀⠀⠀⠀⠀⠀⢀⢴⣄⠀⠀⠀⠀⠀⠀⠠⢦⠈⢉⢈⠁⡉⢡⠶⢀
⠀⠀⠀⠈⣦⣤⠁⠁⣈⣈⣰⠁⠀⠇⢸⠀⠀⠀⠀⠛⠛⣛⠛⣿⢛⡛⠛⠛⠀⠀⢘⠓⠤⡐⡬⠂⠪⡔⡡⠄⠚⡂
⠀⠀⠤⢤⣿⣿⣼⣿⣿⣿⡷⠀⢸⡇⢸⡇⠀⠀⠀⠠⣄⠈⠁⣿⡀⠉⢠⡀⠀⠀⠛⠓⠐⡂⢅⠿⢷⡸⣐⠀⠚⠃
⠀⠀⠀⣠⢼⣇⣿⣿⣿⣸⡧⣄⠀⡇⢸⠀⠀⠀⠀⣰⣶⣶⣶⣿⣷⣶⣶⣶⠀⠀⠚⢃⡁⠔⢩⢁⠹⡕⠂⢈⠚⠃
⠀⠀⠀⢇⣉⡹⠉⠉⠈⢏⣉⡸⠀⡇⢸⠀⠀⠀⠀⠉⢉⠉⠉⠉⠉⠉⣌⡉⠀⠀⠀⠙⠃⣰⡇⢠⡀⢱⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠷⡾⠆⠀⠀⠀⠀⡏⠐⠀⠀⡄⠄⠈⡅⠀⠀⠀⠀⠀⠴⠤⠭⠥⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⢀⣤⣴⡀⠀⠀⠐⣀⡄⣰⡈⠀⠶⡀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠈⠉⠉⠉⠀⠀⠰⢶⡟⠃⡇⠀⠒⠧⡶⠴⠆⠃⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣟⡋⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠧⠤⠤⠤⠧⠤⠤⠄⠀⠀⢀⡠⢀⡖⡔⡕⠦⣀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⡿⣿⣿⢿⣿⡿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⣀⣀⣘⠉⠉⠁⠁⢠⣶⠏⠀⠀⠺⢻⣷⣤⣤⡀⠙⣿⣿⣶⡄⠀⠀⠀⡇⡇⢸⢸⠀⡇⡇⢸⢸⢸⡇
⠀⠀⠀⠀⠀⠙⢿⣿⣶⣦⣤⣴⣾⠇⠀⠀⠀⠀⣸⣿⣿⢿⡥⠀⠻⣿⠟⠃⠀⠀⠀⡇⡇⢸⢸⣾⣷⣇⢸⢸⢸⡇
⠀⠀⠀⢀⣄⣤⣤⣉⣉⣭⣤⣤⣄⠀⠀⠀⠀⠀⣉⣿⣿⣋⡀⢀⣠⣿⣄⡀⠀⠀⠀⡇⡇⢸⢸⣿⣿⣿⣿⢸⢸⡇
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠁⠁⠉⠈⠉⠉⠉⠉⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⡟⠀⠀⠹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢰⣶⣷⣶⡀⠈⠀⠀⠀⠀⠉⠉⠉⠀⢠⣧⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣏⢹⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡌⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠿⢷⣶⣶⣶⣶⡖⠀⢠⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⢻⣿⣿⣿⣿⡿⠁⠀⣷⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠰⡿



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⠢⡀⣀⠠⡀⢄⠀⠀⢀⠀⠠⠚⠁⠐⠀⡶⢀⣞⡐⡄⣴⢀⡠⠎⠀⠄
⠀⠀⠀⠀⠀⢠⡠⢂⣰⣦⠀⠠⠀⠃⠘⣍⠉⣀⡇⠀⠘⠃⠂⠀⠀⠀⠉⠁⣂⠀⠀⢠⠁⠉⠁⠀⠐⠤⢠⣮⣀⡔
⠀⠀⠀⠀⠀⢨⠟⠉⠉⠀⠀⠀⡀⠀⢀⡇⠈⠁⣯⣀⠀⢠⣤⣤⡤⡀⡀⢀⣄⠀⡀⢨⡄⣀⢀⣠⡀⠀⠐⠃⡟⠳
⠀⠀⠀⠀⠀⠀⢂⣤⡄⠀⠚⠋⠉⢍⣛⡉⠉⠋⠉⠡⠁⠈⠉⠈⠁⡀⡷⠀⠙⠏⠉⠉⠉⠁⠘⠛⠁⠘⠞⢢⣦⣤
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⠷⢧⣶⣦⣤⣤⣽⠀⣀⣥⣠⡶⡤⠤⠄⣧⡧⢠⠀⢇⠠⣤⡀⢒⠢⣿⡷⡶⠗⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠄⢀⢰⠀⢹⠹⠃⡟⡷⠦⣿⠿⣶⠶⡯⢻⡶⡤⢪⠚⠻⢰⡏⠀⢠⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣤⣌⠘⢘⣬⢱⡂⣂⡓⠤⠳⠀⢡⣀⡇⣈⡃⡁⠀⠠⠀⠀⢁⣄⢈⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠛⠛⢯⣼⣿⣇⣧⠏⠀⠀⢠⠶⠟⠹⠃⢄⣀⠀⠤⣧⢠⣤⢦⣿⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠨⠿⣇⡠⡄⠤⢋⣼⠥⠭⢗⠀⢴⡋⠑⠸⣿⣷⡿⠝⠀⠠⢽⣼⣿⡼⣿⠄⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠷⠶⢦⣤⣤⣤⡼⡿⣾⣀⡄⠤⠄⠀⠀⠀⢀⣛⣃⣀⠂⠀⠀⠀⣬⠷⣠⣴⣶⣷⡶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣀⠀⠀⢀⣤⡴⠶⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠿⠛⠁⣠⡾⠋⠀⠀⣀⣀⡀⠀⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⣾⠋⠉⠻⠇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣿⣄⣀⣀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣹⡏⠉⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣠⣿⠶⣦⣤⣴⠄⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⣀⣠⠾⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⣠⡶⠋⠉⠀⢀⡀⠀⠉⠻⢦⡀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣴⠿⠷⣦⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⠉⠛⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠘⢷⣤⣴⠟⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠘⠁⠀⢀⣠⡾⠋⢀⣤⣶⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠚⠋⠁⠀⠀⠈⠛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣄⣀⠀⢀⣠⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣾⠀⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠙⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡴⣛⣭⣭⣮⣭⣭⡓⢦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⢫⣾⣿⡿⠛⠈⠻⣿⣿⣦⡹⡎⢻⣿⡿⠿⣿⣿⠿⣿⣟⣯⣵⣿⣿⣿⠁⠙⣿⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⡅⠺⢸⣧⣼⣿⣿⣷⢹⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣧⣀⠈⠀⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡿⣷⢠⡉⢻⣿⣿⣿⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⢷⡹⣿⣿⣧⣈⠈⣁⣼⣿⣿⢣⡟⣴⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠳⣬⡻⠿⣿⣾⡿⠿⣛⡵⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠛⠉⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡿⡿⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣞⣯⣴⣿⣬⣧⣽⣷⣬⣗⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⡶⣾⡟⠻⠿⠬⠟⠃⢘⣙⣛⡒⢚⣛⣉⡂⡒⠲⠬⠾⠿⠻⣿⠶⣦⣄⡀
⠀⠀⠀⠀⠀⢰⣿⣻⣶⣷⣶⣾⣿⣷⣦⣬⣙⣿⣿⡇⣾⣿⣟⣫⣥⣶⣿⣿⣿⣶⣶⣿⣭⣙⣷
⠀⠀⠀⠀⠀⠀⠉⠠⠭⣭⣽⣽⣿⣿⣿⣿⣿⢿⣷⣾⣶⣿⡿⣿⣿⣿⣿⣭⣿⣭⣭⣭⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠘⠋⠛⢻⡟⠛⠛⠃⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢷⠀⠀⠀⠀⠀⢸⡇⠀⣴⣦⡀⠀⠀⢠⣄⣀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⣶⣾⣿⡀⠈⣿⣆⣄⣤⣿⣷⠀⢁⣿⣥⣤⣠⡌⢻⣿⡿⠿⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠊⡿⡋⠁⠈⠛⠻⠛⠻⠿⢹⣿⣿⡀⠿⠉⠙⢻⠛⠀⠈⠱⠉
⠀⢀⣀⣀⣀⣀⣠⣶⣶⣿⣿⣶⣶⣶⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣀⣀⣀⣀
⠀⠻⠯⠿⠶⠾⠟⠾⠿⣿⣛⣻⣿⣿⣏⢛⣿⣿⣿⣍⣉⣹⣿⣿⣿⠙⢻⣿⣿⣟⣻⣿⣶⣞⣷⣾⣿⡿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡾⠲⣇⢐⢐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠁⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡔⣤⣐⢦⣄⡰⣥⣄⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠠⠤⠄⠐⠊⠀⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣣⠶⣿⡦⠾⣷⠶⣿⣤⡽⠞⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠁⠀⠆⠀⠀⠀⣁⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⢠⡿⠀⣼⠃⣰⡟⠀⠀⢀⣀⠀⠈⢙⣟⣂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⠀⠀⠀⠂⠀⠀⠀⠉⠀⠈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⣼⠇⣼⠇⣰⡟⠀⠀⠀⢟⣿⡇⠀⠀⢳⣍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠆⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣿⢴⣿⣴⡿⣶⡿⠐⢀⣀⠓⠀⡀⠀⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠡⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣿⣵⣿⣶⣿⣴⣎⡍⠉⠙⢦⠈⠄⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠀⣤⠤⡤⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣽⢿⠿⣿⡇⠻⠿⠻⣿⣸⡇⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢀⠄⢠⣠⣨⡜⠂⠑⠀⠀⠀⠀⠀⠀⢠⣻⣟⡯⣿⣿⣿⠏⠀⢻⣿⣦⣀⣘⣩⠟⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠈⡄⠈⠀⠇⢀⠐⠀⡄⠀⣀⠀⠀⠀⠒⢿⡿⠿⠿⣿⠃⠀⠀⣈⠉⠃⡄⠄⠀⣸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠊⠀⠀⠀⠀⡇⢠⠈⠀⢠⡆⠀⠀⠘⢉⠍⠉⠈⠱⠴⠅⠉⠉⢈⣭⠍⣭⡍⠉⠈⠉⠉⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠙⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⢠⢃⠀⠠⠀⡀⠀⠀⠀⠀⢠⣎⢊⣄⠈⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⡄⠆⠀⠀⠨⠀⠆⠐⢠⡄⠀⢠⢠⠀⠀⡀⠀⠀⠀⡄⣄⠀⢠⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠰⡄⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⡄⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠂⠡⡂⠀⠀⠀⢘⠀⠀⢂⠀⣰⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⡇⠀⠀⠀⢈⡀⠀⠀⠀⡆⠀⠀⠀⠤⠀⠀⠤⠀⠄⠀⠀⠀⠠⠠⠀⠤⠀⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠁⠡⠀⠀⠀⠁⠀⠨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠀⠒⠒⠂⠒⠀⠂⠒⠐⠂⠐⠒⠒⠒⠀⠓⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠁⣉⢀⡉⠀⠈⡀⢈⠈⠉⢈⠀⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⠀⠀⠀⠀⢀⣆⣀⡀⠔⢠⣀⠀⠀⣀⠀⡀⠀⢀⡀⢀⢀⠀⣀⠀⡇⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠔⠀⠀⠀⠀⠘⠂⠀⠤⠤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠


⣾⣿⣿⣶⣶⡷⢿⣾⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⢷⡿⠿⡿⢶⣶⣿⡿⠶⠶⢶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⡆
⠀⠀⠀⠀⠀⠀⠚⠂⠚⠛⠓⠀⠀⠀⠀⠐⠋⠉⣿⠀⠀⢸⠀⡇⠀⠀⠀⠉⠁⠀⣀⣀⣀⠀⣀
⠀⠀⢀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⢸⡇⠀⣻⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉
⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣿⢸⡇⢸⣋⣉⣻⢰⡟⢛⣛⣛⡛⢻⣇⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂
⠀⠀⠀⠐⠒⠂⢀⣾⠀⠀⢀⣀⣿⣸⣇⣸⣉⣉⣹⣖⡇⢈⣉⣉⣁⢸⡏⠙⠲⠦⣄⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣠⡴⣯⣽⡆⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⠈⠉⠉⠁⢸⡇⠀⠀⠀⠀⠉⣷⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠘⡇⠙⢢⣼⠀⣟⣉⣉⣉⣉⣉⣿⢸⡇⢀⣀⣀⡀⢸⣇⡀⠀⠀⠀⠀⣿⡛⢻
⠀⠀⠀⠀⠀⠀⢰⣧⠞⠋⢸⠀⣉⣉⣉⣉⣉⣉⡁⢸⡇⢈⣉⣉⡁⢸⣏⣹⣇⡀⠀⠀⣷⣼⣿
⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢸⠀⣯⣉⣉⣉⣉⣉⡿⠘⡇⠈⠉⠉⠁⣼⠋⠁⠈⡙⣦⠀⣿⣳⣾
⠀⠀⠀⠀⠀⠃⢰⡖⢒⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡇⠀⠀⠀⡄⢸⠀⣿⢯⣹
⠀⠀⠀⠀⠀⡇⠈⠉⠉⠁⢸⠀⢸⡏⠉⠉⠉⢹⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⢸⠀⣿⠛⢻
⠀⠀⠀⠀⠚⣟⠒⢾⡙⠳⣟⠓⠚⠓⢶⠒⠒⠛⠓⢺⡟⠒⠒⠒⠚⣿⠒⠒⠚⠛⢻⡗⠛⠒⢻⠗
⠀⠀⠀⠀⠀⢹⡄⠀⠙⢦⡌⠳⢦⣀⡼⠛⠙⣦⠞⣋⣟⡳⣦⣤⣀⣿⠀⠀⠀⠀⣸⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢻⡖⠒⠛⣟⠳⣄⣉⡴⡞⠛⣏⠀⡳⢮⡅⢸⣦⣌⡙⠻⣿⣿⠛⣿⠒⢲⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠁⣤⠀⠘⣦⠈⠉⠀⢻⣀⣙⣦⣙⣏⣥⣟⣀⣼⡙⢲⣤⡉⠁⠈⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡛⢿⡉⠉⠉⣯⠉⠉⢹⡏⠉⠉⣹⠋⠉⢻⡤⣤⣤⣤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢽⣷⣄⡘⢧⡀⠘⡇⠀⣴⢋⣠⣴⣯⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠾⠷⠛⠛⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠉⢻⣿⣧
⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠻⣿⡧⠀⠀⠀⠀⠀⠀⣿⣿
⢀⣾⣿⣿⣿⣿⣿⣿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠹⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣄⣀⡀⠀⢀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⠟⠛⠃⠀⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢿⣿⣿⣿⣿⣶⣶⡆⠀⢰⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⣀⣠⣤⣤⣶⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠈⠁
⠘⣿⣿⣿⣿⣿⣿⣷⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣦⣀⠈⠉⠉⠉⠙⣿⣿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⠋⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠃⠀⠘⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣤⣤⣤⣤⣤⣄⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠉⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠻⠿⠿⠿⠿⠿⠟⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀⠀⢀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠹⣿⣷⣤⣀⣀⣀⣀⣀⣈⣻⣿⣷⣄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡷⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠄
⠀⠀⠀⠀⠀⠀⠀⡄⢤⡐⣮⢱⣆⡲⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣘⠹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠰⡠⠀⠀⠈⠱⢾⠿⠉⠯⡉⠁⠀⠸⡄⠀⠀⠀⠄⠀⠀⠋⠀⢉⡇⠈⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⣀⠘⣷⠀⠀⠀⠠⠆⠁⠀⢀⠀⠆⠀⠀⠀⠀⣼⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠄⠟⠁
⠀⠀⠀⠀⠄⠐⠀⢿⡀⢂⠉⠀⠀⠐⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⣠⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡍⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠐⠂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢡⠏
⠀⠀⡸⠀⠀⢹⠀⡁⢱⠀⣿⠁⢸⣿⡆⠀⠀⠀⠀⢀⢊⣾⣷⢢⣤⡤⠔⠀⠄⠄⠔⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠈⠀⡠⠋
⠀⠀⢻⡀⠀⠘⠀⠐⠋⠀⠷⠖⠘⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢵⣢⡀⢀⠜⠁
⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣠⠋
⠀⠀⠀⠀⠈⠙⠛⠶⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣠⢄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠅⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠂⡐⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠠⡀⠀⣖⠀⠹⠆⠀⠀⠀⠀⠀⠁⠀⠀⠈⣏⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠁⢠⡖⠆⢱⣀⡉⣡⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢿⣶⣦
⠀⠀⠀⠀⠀⠈⠣⣌⠁⣀⠞⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⢀⣤⠀⠀⠺⠂⠀⢠⡄
⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠒⠒⠤⡀
⠀⠀⠘⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⡤⠊⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⢄
⠀⢠⡆⠀⠀⠀⠀⢰⣿⣗⣀⣠⡤⠔⠋⡦⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣀⣀⣴⣁⣀⣀⡀⠤⠤⠒⠒⠉⠁
⠀⠀⠁⠀⠀⠀⢀⠌⢸⡋⠄⠀⠀⠀⠀⠁⠀⠀⠀⢠⣶⠾⣷⣶⣶⣶⣦⠀⠀⠀⠀⢀⣀⠤⠒⠁⣀⡤⠤⠴⠤⠤⢄⣀⣀⣀⠀⠀⠀⠀⣠
⠀⠀⠸⠃⠀⢀⠊⠀⠀⠓⢬⣆⠀⣠⠿⢒⡒⠛⡟⠉⡙⠙⢻⣿⠳⠬⣍⠙⠿⠛⣋⠁⠀⠀⠀⡎⠁⠀⠀⠀⣠⢔⡼⣻⢹⡜⣟⢦⣢⡔⠁
⠀⠀⠀⠀⠠⡟⠀⠀⢠⠄⠀⠻⡿⡳⣦⡬⠥⠼⡇⣀⣶⡄⣿⣿⡧⠴⠇⢀⣛⣖⣛⣤⣤⠀⠀⠙⠢⢄⣀⣈⣁⡨⠔⢋⣸⣅⠽⠚⠁⠉
⠀⠀⠀⡀⠀⣠⣤⣤⣼⣷⣤⡦⠀⠥⢼⡏⠉⠉⡕⠓⢲⣶⣿⣿⣷⣶⠒⠺⠛⠛⠟⠛⠻⠻⠒⠒⠛⠋⠉⢩⣿⣯⣹⣿⣿⣿⣿⣠⣽⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⠟⡿⠛⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣠⣤⣴⣶⣍⣹⣶⣶⣄⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢀⣙⣋⣿⣿⡿⠷⠒⠛⣿⣃⢿⣇⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠐⠛⣶⣴⠞⠁⠀⠀⠀⠀⣻⢟⣫⣿⣿⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣷⡖⣾⡟⠁⠀⠀⠀⡄⣠⢞⣵⣫⣄⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⡏⣻⢣⣾⢳⣼⣤⣯⠊⠙⡏⢸⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⠏⠙⠚⠛⣿⣜⣾⡏⢳⡴⣻⣽⠟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠁⡠⢀⢀⡞⣩⣿⣿⠏⠀⠀⠀⠀⣿⠀⣿⠀⢸⠻⠅⣇⢰⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡛⣗⢤⣄⣀⣀⣀⣐⠁⢈⡼⡿⣯⣿⣤⣤⣀⣤⣤⠷⠶⠿⡆⠀⠙⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢊⣻⣿⣟⣒⡫⠷⣶⣶⣶⡶⠾⣗⣭⣾⣯⣿⣷⣶⡒⣶⣶⣒⢲⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⡤⣉⡿⠿⢶⣶⡞⣱⣿⡜⢜⣿⡿⣿⣟⠿⣿⣿⣿⣻⣿⣿⣿⡿⢡⣿⣾⣿⣧⡏⣻⣿⡹⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡟⡴⣡⣾⣿⣮⠞⣹⢫⣾⡟⠻⣿⣮⣦⡱⣿⠿⠟⠒⠈⠭⠭⠭⠭⠉⢒⢿⣽⢿⠟⢻⣯⣾⣷⣇⢾⣾⠃
⠀⢀⠀⣀⣀⡀⠀⣸⠘⡵⣵⣻⣿⠣⠀⣇⠻⣿⣷⣴⡿⢃⡼⠛⠶⠊⢐⣐⡄⢿⣿⣿⣿⡗⣜⣿⣶⠟⡀⢯⣿⡞⢁⡿⣿⣧
⢠⣿⣹⣿⣍⠁⢠⡡⣾⢞⡵⣹⠃⣰⠆⣿⠃⢝⠻⠟⣰⠋⠀⠀⡄⠀⠀⠉⢹⡌⣿⣷⣿⢃⡿⠫⣮⡄⠀⡀⠈⠁⠈⢻⠿⣿⣶⣶⣲⡆
⢀⠙⣿⣿⣿⣶⠯⣚⣵⢟⡵⢃⡼⣡⣦⠸⡌⠀⠙⠾⠁⠀⠀⣸⣃⠀⠀⠀⢀⣧⣙⣛⣛⣸⣥⣷⣿⠿⢿⡁⠀⠐⠒⠉⠀⠀⠉⠛⠿⣇
⠿⣶⡿⠿⠿⣖⣛⣭⠶⠋⠀⣫⡾⠛⠁⢠⡷⠀⢠⠀⠀⠀⣴⠟⠙⠛⠛⠛⠉⠋⢰⣶⣦⠋⢸⡇⢸⠀⠀⠙⠒⠀⠤⠤⠤⠥⣥⣀⣄⠈⣷
⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠈⠁⠀⢀⡴⢋⣤⢸⠸⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠸⠷⢿⠆⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣴⣿⢿⣸⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⢛⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣝⡃⠻⣿⣯⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⢨⣬⣈⣀⠀⢸⡇⣾⠀⠀⠀⠀⠀⠀⢀⢴⣯⠻⠏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣝⢍⢻⣿⣿⡄⠰⣴⡄⠀⠀⠀⠀⠛⠛⠛⠀⢸⡇⣿⠀⠀⠀⠀⠀⣠⣿⣿⣿⣆⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⢻⢿⣿⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠈⠻⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣇⣀⠾⢃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠠⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣶⣾⣿⣿⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠭⠴⠛⠉⠉⡩⢲⠎⢩⡏⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⣉⣀⡀⠀⠀⠀⠀⢛⠈⣢⣉⣴⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠺⠑⠒⠀⠈⠉⠒⠚⠳⠦⣀⡀⣠⣾⣿⣫⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⢟⠒⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡿⠧⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣒⠭⠒⠁⠀⠈⢄⡀⠀⠐⠭⠩⢛⣶⡦⣄⢀⢆⢆⣿⣿⣿⡿⠏⣏⣠⣷⣿
⠀⠀⠀⠀⠀⠀⣀⠠⠔⣊⣥⢖⠙⠓⡀⠀⠀⠀⠀⠀⠉⠒⠒⠚⢈⡡⣪⣾⣿⠀⢫⣾⣿⡿⠟⢁⣶⢿⣿⢻⢫⡇
⠀⠀⠀⠀⠀⢀⣿⢴⣢⣉⡁⢦⣀⠑⠒⡀⠀⠀⠀⠀⠀⢀⡠⣒⢵⣾⣾⣿⣿⠁⣿⣿⡟⠟⠚⠻⣿⣕⡕⣵⠋
⠀⠀⠀⠀⠀⠀⡏⠑⢶⡬⣉⡓⠶⣭⣕⣲⠊⣩⡽⠟⡩⢵⣺⣵⣿⣿⣿⣿⣯⣖⠛⠈⠀⠀⣀⡄⣿⣮⣾⠋⠱⠶⢤⣀
⠀⠀⠀⠀⠀⠠⡏⠀⠀⠙⡑⠪⢝⣒⢿⣾⣽⣓⣞⣿⣾⢿⣿⣿⣿⠿⠿⠏⠓⠒⣦⣤⣄⢦⡛⣄⣿⡿⣙⣑⣦⠀⠩⠃⣩⡕⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢭⣛⡲⠬⣅⡂⠉⢘⠛⣿⣶⣿⣧⣟⠛⠈⠤⠤⠔⡪⣻⣯⡟⢺⠁⣹⠟⠋⠉⡀⠺⢕⡶⢄⣠⣶⣷⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠳⠦⢌⣙⡋⣼⡿⠟⠋⠬⠍⠀⠀⠀⢀⢞⣵⢿⣿⡏⢃⡽⠓⠀⢀⠔⠁⠀⠀⠈⣉⡚⠻⢿⠷⢇⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⢉⠴⠉⢰⡁⠒⠂⢬⣙⣋⠉⠁⠀⠀⣀⣀⣀⢔⣕⣟⡷⣿⣿⣦⠯⠄⠀⠒⠁⠀⠀⠀⠀⠤⠀⠀⠀⡜⠀⢄⢨⠉⣱⡆
⠀⠀⠀⠀⠀⠀⢐⡯⠬⠥⠉⣀⣀⡒⠊⠭⠭⠶⣢⡤⠤⠊⢁⠖⣃⢴⡿⡯⣾⣾⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⠂⣸⢄⣠⠴⢊⠥⢻⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠤⢂⣐⡒⠒⠀⠭⠭⠤⠐⢀⣀⣀⡨⡭⠍⠐⣫⣺⣿⣿⣏⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⡚⠭⢒⡩⣴⣫⣵⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢅⣒⣒⣒⡰⠯⠭⠭⠥⠔⢛⣒⣐⡤⠭⡍⡭⣿⣿⠿⡟⠙⠿⣿⢤⣘⠓⢬⠑⠲⣔⡢⠭⣐⣻⣵⣶⣿⢿⣿⠿⠛⡋⢬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢕⠬⡛⢌⡙⠩⢋⢟⡹⡋⢞⠑⡋⠛⠝⡒⢛⢷⣕⡢⠀⠀⠀⠹⢦⢽⣒⣺⣽⣕⣛⣭⣶⣿⡯⠟⢛⠩⢔⣊⣥⠔⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠉⠢⢤⣄⡀⢀⣁⣈⣀⡁⠈⣈⣰⣥⡿⣻⣄⠀⠀⠀⠈⠃⢰⣿⣿⠿⢟⢚⣉⣥⡒⠥⢒⠮⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠨⣛⢥⡀⠴⠚⣋⣽⢿⣿⣿⡿⣿⣟⠝⠒⢄⡀⠀⠐⣗⣖⣯⣽⠿⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠸⡩⡉⠹⢿⣻⣷⣟⣯⡿⠟⠊⠀⠀⠀⠀⠈⠁⠊⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⡀⠈⡆⢐⣿⡿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢐⠀⠂⠐⡠⠂⠔⡠⠂⡀⠀⠄⡠⢀⠀⠀⠀⠠⢀⠄⠀⠀⡠⢂⠐⠀⠀⠠⢀⠄⠀⢀⠔⡠⡠⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠊⠔⠥⢂⠔⢈⢔⡠⢊⠔⢊⠔⡠⢂⠑⡨⠐⠡⢊⠐⡡⢂⠀⡠⡠⢊⠄⠁⢒⡑⡠⢂⡔⢃⠔⡡⢪⠊⠔⠑⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠘⠁⠊⠁⠊⠐⠁⠊⠐⠡⠂⡠⢊⠔⠡⠪⠊⠀⠁⠊⠀⠋⠀⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣶⣾⣿⣿⣿⡗⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⡄⣿⢟⡻⣉⣿⢋⡿⣷⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡰⣻⡟⣿⣿⡟⡏⣷⠙⣧⣶⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⣼⢳⣿⣿⣿⢧⣿⣸⡀⡇⣼⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⡆⣿⠇⠈⢷⣿⠏⡜⠙⣿⣿⣿⣿⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⣿⣍⠛⢾⣦⡻⣾⡇⢠⣛⡝⣿⣟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⣆⢠⣄⣀⣀⣀⣀⣀⣀⣠⣤⣄⢨⣼⢻⡉⣽⣷⣮⣺⣿⣸⠃⠀⠙⠁⠘⠋⠁
⠀⠀⣀⣤⣤⣄⣀⣠⣶⣿⣿⠿⣿⠀⠈⢳⣮⣝⣻⠿⠿⢿⣿⣿⣿⣿⢸⠁⠀⠻⣿⡈⠉⠉⢿⣿⡆
⢠⣾⣟⣯⠿⠛⡟⠛⠛⠉⠀⢸⣏⣠⣴⣾⡧⠀⠉⠉⠙⢦⡻⣿⣿⢏⡞⠁⢀⣠⣴⣤⣀⣀⣀⣙⣷⡀
⣿⠘⡿⣿⠀⠀⠀⠀⠀⠀⠀⢈⣩⣿⣯⣼⣿⣀⣀⣀⣀⣀⣙⣺⣖⣋⣀⣰⣾⣿⣿⡟⢉⣽⣿⣿⣿⠁
⢿⡰⣧⠹⣧⡀⣦⣶⣶⣶⣼⣿⣿⣿⣿⠙⠻⣿⣿⣿⣿⡟⣿⡏⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠃⠀⠀⠀⡀
⠈⢧⡹⣧⠈⠻⣟⢿⣿⣿⣿⣿⣿⡟⠃⠀⢠⡿⠿⣿⣿⣷⢹⣧⢹⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣷⣶⣾⣿⣿⣿⣷
⠀⠈⠳⡌⢳⡤⡙⣦⢻⣿⣿⣿⡿⠃⠀⣠⡿⠁⠀⠀⠀⠉⠀⠛⢲⡟⣶⠚⠋⠉⠀⠛⠿⢿⣿⣿⣿⣿⣿⣏⠿⠁
⠀⠀⠀⠹⡄⢹⣰⣸⣾⣽⣿⣿⣣⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⣹⣿⣿⠆
⠀⠀⠀⠀⠸⣼⣻⣟⣻⣿⣿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠀⠀⠀⠀⠀⠀⠀⢰⡞⣴⣿⡿⠁
⠀⠀⠀⠐⢢⠎⠝⠻⣿⣯⣛⢿⣿⣿⣿⣿⣷⣦⣠⠀⠀⠀⠀⠀⠀⡷⡇⠀⠀⠀⠀⠀⢀⠀⢸⣟⣿⠟
⠀⠀⠀⠉⠈⠀⠀⠀⠈⠛⠿⣷⣮⣳⡄⡀⠈⣿⠋⣆⠀⠀⠀⠀⢀⣷⠃⠀⠀⠀⢀⣀⣉⣉⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡷⠇⠀⠻⠆⠘⠀⠀⠀⠰⠿⡏⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠿⠿⠟⠻⠿⠿⠿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⢁⣀⣤⠤⠶⠶⠶⠶⠤⣤⣀⡈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿
⠀⠀⠀⠀⠀⠀⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡈⠉⠛⠒⠶⠶⠶⠶⠒⠛⠉⢁⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣶⣦⣤⣤⣶⣶⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠉⠀⠀⠀⠀⠀⠙⠻⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠉⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢛⣿⡟⠛⠛⠛⠛⣿⡟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣾⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠸⣿⣷⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠛⠂⢤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣯⡗⣿⣾⡯⡇
⠀⠀⠀⠴⣿⢿⠦⠀⠀⠀⣠⡾⣿⣷⣄⠀⠀⠀⠸⠷⠂⠀⠀⠀⠀⠀⠀⠂⠿⠀⠀⠰⠶⠆⣤⣤⠀⠶⠖⠀⠀⠳⣷⣿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡇⣿⢸⡟⡇⠀⠠⡦⠰⠀⠀⠀⡾⡆⠀⠀⠐⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣾⢷⣿⡾⣇⢻⡀⠀⢀⡄⠢⠀⢸⠃⢳⠀⠠⠂⣄⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠰⢤⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣌⡟⠐⢸⡏⣄⠀⠛⠀⠀⠀⠀⠀⠴⠁⠠⡀⠀⡠⠀⠙⠄⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣷⣾⣶⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⡀⠀⠀⠀⠀⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣤⠀⠙⣦⡊⢀⣧⠈⣢⡾⠁⠀⠘⠃⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⠙⠛⠀⠀⣠⣴⡟⢹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⠟⠀⠀⠀⠙⢸⣿⠘⠁⠀⠀⣠⣄
⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠰⠶⠿⠿⣾⣧⣴⣷⠿⠿⠶⠦⠀⢀⠀⠀⠀⠈⠉⠉⠀⠀⣀⠀⠀⠀⠈⠉⠀⢠⣴⣿⣻⣟⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣿⣿⡿⣿⠀⣀⡤⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⠀⠀⠀⠀⢀⠀⠀⢀⡀⡀⡀⢀
⠀⠀⠀⠠⠶⠶⠆⠆⡆⠀⠀⠀⠘⠉⠉⣿⣩⣤⣿⠈⠋⠉⠀⠀⠀⢰⠲⢲⠰⡾⢇⡀⢘⣛⣌⣯⣄⠀⠀⠀⠛⠶⠣⣿⡧⠧⠷⠛
⠀⠀⠀⠀⠤⠤⠴⠤⠀⠀⠀⠘⠘⠙⠃⠋⠛⠙⠙⠙⠃⠃⠓⠀⠀⠸⠓⠿⠀⠿⠫⡅⠌⠹⠷⠬⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⢀⢀⡀⣀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⡇⠀⠀⡜⣱⡏⣼⠃⣿⠀⣷⢸⣇⢻⡎⣦
⠀⠀⠀⠀⠸⠁⠉⠁⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠈⠀⠀⠁⠀⠀⠈⠁⠀⠀⠈⠀⠁
⠀⠀⠀⣀⣀⡀⠤⢀⠀⡄⠀⠀⠀⠈⠀⠐⠘⠙⠋⠀⠀⠈⠀⠀⠀⠐⠀⠁⠀⠀⠸⠷⢶⠆⠀⠀⠀⢀⣠⡃⠶⠦⢘⣄⡀⣤⣼
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⣠⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⣀⣜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣟⡀⠀⠉⠀⠀⣾⡷⠀⠀⠀⠀⠀⠀⡀⣀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⡄
⠀⠀⡼⢫⣿⠟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠶⠾⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⠛⣃⣀⣸⣇⠀⠀⠀⠀⠀⠀⣠⣤⡀⣾⣿⣤⣼⡷⣖⢦⡦
⠀⠀⠀⣿⣧⠀⠈⠷⢤⣼⡋⠀⠀⠀⠀⠀⠀⠾⢸⣿⡇⠀⠀⠀⣯⣠⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⣿⡿⣿⠿⠆⠀⠀⠀⠀⠀⠥⢿⣿⣿⣼⣷⣾⠷⠟⠈
⠀⠀⣸⠏⠹⣇⢀⣀⡼⣯⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣥⣤⣦⣭⣅⡉⠉⠀⠀⠀⠀⠀⠐⠛⠋⢀⣀⣹⣿⣀⣀
⠀⠐⠋⠀⠀⠿⠄⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇

⠀⠀⢀⡔⠚⠁⢙⢈⠓⢢⡀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⠿⠿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⢠⠍⣁⠂⠁⠀⠈⠐⢊⠉⣆⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⡀⠀⠈⠙⡆⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣦⡀⡀⠀⠀⠀⠀⠟⠉⠉⠱⠄⢀⣀⡀
⠀⣬⠀⠄⠀⢰⣿⡷⠀⠠⠄⣼⠀⠀⠀⠀⣤⠀⠷⠐⢾⣿⣿⣤⣀⡄⠁⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⣻⡿⣿⡿⡿⠂⠀⠀⠐⠀⠀⠀⢠⠜⡏⠀⠘⢆⣠⠎⠉⢳
⠀⢸⡆⠀⢄⡞⠉⢱⡠⠈⢠⡆⠀⠀⠀⠀⠀⠈⠁⠚⢺⣿⣿⡤⠌⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⢀⠀⠀⠀⠀⠀⠀⠁⠰⠈⠁⠀⠘⢆⠀⠀⠀⠀⢀⡼
⠀⠀⠸⢦⣞⠁⢀⠈⣳⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠃⢻⣿⣏⠙⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⠒⠉
⠀⠀⢠⡎⠉⠀⠙⠁⠉⢹⡄⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⠷⠆⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⢠⡄⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡒⢤⣶⡜⠁
⠀⠀⢸⣾⣆⡀⠀⠸⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣌⢿⣿⣁⣿⣥⣤⣤
⠀⠀⢸⡟⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⢰⢼⠀⠀⠀⠀⠀⠀⠸⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣴⣾⠗⣨⠭⢗⡉
⠀⠀⢸⣀⣀⣘⣀⣀⣀⣀⡁⠀⠀⠀⠀⠀⠀⢀⣾⠷⣤⡀⠀⠀⠀⢸⣸⠀⠀⠀⠀⠀⠀⢸⣇⣿⣷⣤⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡻⣫⠗⣉⠔⠊⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⡾⢯⡀⠀⠉⠐⠂⣴⢸⣾⠀⠀⠀⠀⠀⠀⡿⢿⣿⠛⠃⠀⠛⠛⠀⠀⠀⠀⠀⠀⠘⡿⠏⡥⠋
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢴⠛⠁⢸⡇⠀⠀⢠⣤⣿⣾⣿⣤⠀⠀⠀⠀⣿⣿⣿⣿⣶⣛⣛⣓⣿⣿⣿⠆⠀⠀⠀⠈⠁⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠾⠶⣶⠶⠶⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⢴⡀⢺⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡺⠗⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⣶⣄⣤⣤⣴⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠙⣹⣿⣉⣉⣀⣀⣀⣀⣈⣁⠀⠀⠀⠉⢠⣿⢿⡄⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠘⣿⣿⠡⢒⢑⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣠⣄⣠⣤
⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢷⡀⠙⠛⠀⣀⣀⣀⣀⣼⣿⣿⡇⠀⠀⠀⠀⠈⠛⠷⢶⡶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣴⠏⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠁⢸⡁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡠⢄⡠⢼⡧⢄⡠⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠁

⠀⡀⡂⠀⠀⠀⢀⡀⠤⢒⣒⠀⢩⣍⣐⣒⠢⢤⡀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⢀⠔⣩⡖⠈⠀⠀⠀⠀⠸⠀⠀⠉⠓⠬⡓⢄⠀⠀⠇⠐⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠠⠤⠄⡀⠈⠈
⠀⠀⠀⡴⣡⠊⠉⢛⢄⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⢢⠱⡀⠀⠀⠠⠀⠀⠂⠂⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡞⡜⠁⠀⠀⠈⠈⢢⡀⠀⠀⠃⠀⠀⠀⠀⡔⠔⠁⢳⠱⠀⠀⠀⠇⠀⠰⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣸⢸⣀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠚⠈⠀⠀⠀⠐⠃⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⡅⢌⠉⠉⠈⠉⠒⡂⠀⠐⢦⠗⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⡇⢪⠄⠀⠀⠀⠀⠀⠀⠀⠚⠁⠓⠀⠀⡐⠁⠂⠐⠖⠠⡆⡜⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⢰⠰⠀⠀⠀⡠⠐⠓⠀⡀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⣰⢡⠇⠀⠀⠀⠀⠀⠀⠀⡜⡃⣄⠀⠀⠀⡀⢀⡠⠚⠀⠀⠀⢂⡀
⠀⠀⢃⠣⡴⠞⠁⠀⠀⡜⠀⢿⠈⠀⠈⢇⢡⡀⠀⠀⢠⢣⠞⠐⠂⠀⠀⠀⠀⢀⠘⣁⠌⠁⠁⠒⠊⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⢣⡑⢄⠀⠀⡜⠠⠀⣼⠀⠀⠐⡈⢆⠢⣀⠔⡡⠃⠀⠀⢀⠀⠀⠀⢀⠞⡰⠁
⠀⠀⠀⠀⠈⠂⢅⡚⠤⣆⣸⣏⠀⣃⣀⡡⠜⣊⠵⠊⠀⠀⠀⠀⠀⠍⠀⢀⠚⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡜⠉⠗⠒⠲⡤⠤⠶⠒⠂⢋⠉⢀⠀⠄⠀⠂⢀⠂⠀⢠⠞⠁⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⡜⠀⡜⠀⠀⠀⠛⡀⠀⠀⠀⠈⢆⠈⢇⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠜⠀⡸⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠘⡄⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠘⠠⠴⠇⠀⠀⠀⠀⠃⠀⠐⠀⠀⠀⠀⠼⢾⠾⠇⠄⠀⠀⠀⣠⢄⣀⡀⠀⠙⢦⣀⢸⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡰
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⣀⠀⠀⢄⣳⡀⠀⠀⠈⠙⣧⡀⠀⠀⢀⡴⢙⠦⣤⡴⠊
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡠⠴⠂⠈⠀⠀⠀⠀⠀⠀⠀⢀⠁⡀⠀⠀⠀⠸⡍⠳⣶⠏⡅⡐⢰⠃
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⢀⣀⣴⠙⣇⠀⠀⠀⠤⠫⣄⠀⠀⠀⠹⠀⠉⠀⠁⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢞⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢃⢰
⠀⠀⠀⠀⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠀⠀⢀⡀⡀⡤⢠⠀⡆⢲⠀⠀⠀⠀⢀⠀⠀⠀⠈⡼⢸⠄⠀⠀⠀⠀⢀⠢
⠀⠀⠀⠀⠂⠀⠀⠂⡀⠀⠀⠤⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⣇⢳⢃⠘⢧⣃⠘⡆⠀⠀⠀⠐⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠘⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡇⠀⠀⠀⠉⠄⠂
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠤⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠢⠤⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣫⣭⣭⣭⣝⣛⠿⣿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣘⣆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⠷⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠶⠤⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⡟⣭⣭⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠠⣽⣤⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣍⣉⣁⠀⠀⠀⠀⠀⠀⠀⣸⣁⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠏⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠛⡻⣿⣛⣿⣿⣿⣿⣿⣿⣾⣿⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢧⠛⠤⣀⡀⠀⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣙⣥⣛⠿⢿⣿⣿⣿⡿⠿⣛⣵⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢮⡭⡄⢄
⠀⠀⠀⠀⠀⣀⢠⣆⡟⡗⣄⣀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⢰⣄⣴⢦⣖⣉⡺⣁⣘⢇⣙⡂⡉⣠⣤⢶⣀⣴
⠀⠀⠀⠀⠀⡘⣼⠙⣅⢧⢿⢁⠀⠀⠀⡄⣶⢼⢹⡶⡦⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⢣⠏⣰⣿⣌⠘⣆⠀⠀⡰⣸⠇⣸⣧⡃⢳⡀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⣿⣿⣿⣿⣿⣿⠀⠀⠈⠁
⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠀⠈⠀⠀⠀⠁⠈⠉⠉⠁⠀⠁⠀⠀⠀⠀⢀⣀⢀⣶⣿⠏⣿⣿⣿⣿⣿⣿⠀⠀⠀⣴⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⡆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢏⡿⠛⢿⠆⣿⣿⣿⠋⢀⣬⢾⢳⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣄⣹⣦⠶⠚⢻⡟⠓⠶⢤⣟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⡀⡆⢀⣀⠈⣿⠈⠸⠃⠀⠋
⠀⠀⠀⢀⣀⡀⢀⠞⠉⢙⣦⠤⢺⡗⠠⢤⡏⠉⢫⡀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣿⣧⣿⣿⣼⣿⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⣤⣽⠟⠦⣴⠋⠘⣆⢸⠇⣠⠋⠈⣦⠴⠻⣧⣤⡄⠀⠀⠀⠸⣼⣧⣿⣽⣧⣿⣼⣦⣷⣼⣤⣧⣽⣼⣧⣿⡼
⠀⠀⠀⠀⠀⣾⠀⢰⠃⠙⠲⢬⠖⠠⡡⠖⠋⠈⣆⠀⣷
⠀⠀⣈⣉⡉⣿⠉⢹⡉⠉⡩⠝⠤⠤⠫⣍⡉⠉⡏⠉⡿⣉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄
⠀⠀⠙⠛⠁⠸⣄⣠⠷⣋⢯⡟⢰⡆⢻⣝⢉⡿⠦⣠⠇⠉⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣤⠏
⠀⠀⠀⠀⠤⠞⠙⢦⡀⡹⡽⠤⢼⡧⠤⢷⣋⢀⡴⠋⠙⠒⠀⠀⠀⢀⢀⡔⡝⣆⢄⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢸⡆
⠀⠀⠀⠰⠶⠖⠀⠀⢹⣻⢧⣄⣸⣄⣠⡼⣞⡏⠀⠀⠛⠿⠃⠀⠘⠛⠛⠟⣿⡟⢻⠛⠛⠀⠀⠀⣠⠎⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠐⣟⡎⠀⠀⢸⡅⠀⠀⢱⢽⠂⠀⠀⠀⠀⣰⣦⡖⢠⠀⣿⣧⠀⣦⠀⣦⡐⢶⣥⡤⣀⡀⠸⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣳⠁⠀⠀⣤⣤⠀⠀⠘⡮⡏⠀⠀⠀⠀⠀⠀⠟⠸⣁⣿⡏⣇⠘⠗⠀⠀⠈⠛⠿⣶⣶⡇⠀⣷⠧
⠀⠀⠀⠀⠀⠀⠀⡮⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣱⠀⠀⠀⠀⠀⢠⣤⣤⣴⣿⣧⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⡅⠀⠇
⠀⠀⠀⠀⠀⣤⣾⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣾⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣷⣶⣷⣶⣶⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣧⣹⣏⣛⣟⣙⣫⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡎⡞⡕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣡⡞⢰⣇⠹⣆⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡏⠁⣾⡟⠀⢸⣿⠀⢹⣷⡄⠙⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠃⠐⡿⠓⠠⣾⣿⡆⠘⠛⠓⠀⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡂⢀⠀⣿⣺⡇⣠⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠿⢿⠚⠂⣿⣺⡇⠁⡿⠿⡇⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣀⣆⣀⣴⣿⣾⣷⣄⣀⣦⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⡄⣦⢰⡄⣦⢠⡆⣴⢠⡄⠠⣤⣾⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣿⠤⠿⠆⣼⡷⠀⡀
⠀⠀⠀⠀⢸⡇⣿⢸⡇⣿⢸⡇⣿⢸⡇⠀⠈⡏⢹⠀⠀⠀⠀⠀⠀⣀⣀⠄⣠⢞⡗⣄⡠⢄⣀⠀⠀⠀⠀⣀⠒⠲⠀⠘⡄⢰⢀⡄⢀⠖⠒⢀⡀
⠀⠀⠀⠀⠀⣧⣤⡄⠀⠀⢀⣀⣀⡌⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠘⠛⠛⣛⠛⢻⡿⣛⠛⠛⠛⠋⠀⠀⠘⠒⠢⢤⣈⢂⠴⠒⠪⣄⢋⡤⠔⠒⠃
⠀⠀⢀⣀⣀⣿⣿⣇⣰⣿⣾⣿⣿⣧⠀⠀⣿⡄⢸⣷⠀⠀⠀⠀⠀⣀⠀⠉⠁⣸⡇⠈⠉⢀⡀⠀⠀⠀⠪⠽⠀⠠⠄⣍⢰⣿⣧⢨⠄⠤⠀⠾⠽
⠀⠀⠀⠀⠈⢹⢹⢸⢹⣿⡏⡇⡇⡇⠀⠀⠉⡇⢸⠉⠀⠀⠀⠀⢀⣭⣥⣤⣤⣼⣿⣤⣤⣬⣭⡀⠀⠀⢰⣒⡄⠐⢋⠌⡷⠀⢼⡧⣈⠓⠂⣔⣦
⠀⠀⠀⢀⠞⣹⢻⣾⠸⠿⠇⣷⡟⣏⠳⡀⠀⡇⢸⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⢀⣴⡀⠃⢰⠃⠸⠀⢣⠈⠂⣰⡀
⠀⠀⠀⠈⠧⣈⡤⠃⠀⠀⠀⠘⢤⣁⠼⠁⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠶⣄⡀⠀⠀⠀⠀⠉⠁⢬⡷⠀⠼⡀⢸⡧⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠦⢾⣦⠀⠀⠀⠀⠀⢸⡇⠲⠀⠀⢾⠀⡄⠀⣇⠘⠀⠀⠀⠀⠀⠀⠴⠤⠤⠭⠥⠤⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⢱⡀⣼⠈⠀⠀⠀⢴⠀⢸⠃⡏⠈⣦⣸⣤⠀⠀⡿⣀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷⠀⠀⠀⢀⣸⣸⣦⡇⡇⠙⢺⢸⠀⣗⡇⣸⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⢰⡃⡇⡇⠀⢸⢸⠉⡇⡏⢹⢘⡅⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣭⣭⡁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠼⠼⠦⠧⠷⠶⠾⠼⠦⠧⠷⠼⠴⠇⠀⠀⠀⢀⡠⠖⣱⢣⡞⣎⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠛⠛⠀⣠⣤⡖⠈⠀⠀⣼⢿⣷⣄⣀⡀⠀⠸⣿⣿⣦⣤⡀⠀⠀⠀⠀⣿⢸⡇⢸⠀⣿⢸⡇⣿⢸⡇⢹⠁⣿
⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣀⡀⠀⠀⠀⢀⣿⠟⠁⠀⠀⠀⠉⢸⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⣿⢸⡇⢸⠀⣿⢸⡇⣿⢸⡇⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣾⣾⡿⠿⠿⠀⠀⠀⠀⠀⠰⣼⡟⣿⡏⣻⡾⠀⠈⢹⣿⡏⠉⠀⠀⠀⠀⣿⢸⡇⢸⢀⣿⣿⣿⣿⣸⡇⢸⠀⣿
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣭⣍⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣈⣹⣿⣿⣉⡀⠀⣀⣼⣿⣇⣀⠀⠀⠀⠀⣿⢸⡇⢸⢸⣿⣿⣿⣿⣿⡇⢸⠀⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠛⠈⠃⠘⠈⠛⠛⠛⠛⠛⠃⠘⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠒⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⢻⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣼⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⢻⡟⡟⣿⢹⡟⡟⣿⢻⡟⡟⣿⢻⡟
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⠷
⠀⠀⢸⢹⡏⣿⢹⢹⡏⣿⣽⡿⢟⣛⣭⣭⣭⣭⣛⡻⢶⣼⣏⡏⣿⢹⡏⡏⣿⢹⡏⣿⣿⢹⡏⣿⣿⢹⡇
⠀⣀⣸⣾⣷⣿⣾⣾⣿⢟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡻⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣅
⠀⣿⢹⣿⡏⣿⢹⡿⣡⣿⣿⣿⣿⠿⠛⠀⠀⠛⠻⢿⣿⣿⣷⡌⢿⣿⢹⡿⡏⣿⢸⣿⡏⣿⢹⡏⡏⣿⢸
⠀⠙⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⡏⠀⠀⢰⣦⣤⣄⣾⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛
⠀⠀⣿⣸⢸⣿⡏⣾⣿⣿⣿⣿⣷⣄⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣧⢹⡇⣿⣸⢸⡇⣿⣸⢸⡇⣿⢸
⣤⣤⢿⣿⣿⣿⣇⢿⣿⣿⣿⣿⡿⢿⣿⣶⣦⠀⠀⠘⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⢿⣿⣿⣿⢿⡿⣿⣤
⣇⣿⣸⣇⣇⣿⣿⡜⣿⣿⣿⣿⣅⡀⠀⠉⠁⠀⣀⣼⣿⣿⣿⣿⢣⣿⣸⣇⣇⣿⣸⣇⣇⣿⣸⣇⣿⣿
⠉⡿⣿⢿⣿⡿⣿⢷⡜⢿⣿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⡿⣣⣿⣿⢿⡿⣿⣿⢿⡿⣿⢿⢿⡿⡏⠉
⠀⣇⣿⣸⣿⣇⣿⣸⣿⣦⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣾⣟⣿⣿⣸⣇⣿⣿⣸⣇⣿⣼⣸⣇⡇
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⠶⣮⣭⣭⣭⣭⣵⠶⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⢀⠀⠀⠀⠀⠀⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠆⠁⠀⠉⠁⠀⠘⠀⠀⠁⠒⠀⢈⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡀⠀⠞⠀⠉⠡⠀⠀⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⠆⠀⠳⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠄⢠⠀⡀
⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠠⠆⢉⠁⠀⠈⠀⠀⠀⡤⢄⠀⠆⠀⢀⠀⠈⠅⠀⠐⠉⡀⠄⠀⣀⡴⠂⠀⠞⠀⠃⡌⠀⢀
⠀⠀⠘⠀⠅⢀⡊⠉⠀⠃⠚⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠈⠀⠚⠀⠀⠚⡦⠅⠈⠉⠀⣠⡀⠠⠄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠐⠂⠀⠠⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣤⠀⠠⡄⠀⠀⢀⡀⠀⠔⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠇⢰
⠀⠀⠀⠀⠀⠘⠇⠐⠀⠀⠀⠋⠀⠀⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠄⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠁⠀⠀⠤⠠⣆
⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠞⠂⠂⠀⠀⠀⠀⠀⡀⠀⠀⠁⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⢶⣤⣀⠘⠄⠀⠀⣀⠀⠀⠙⠀⠀⠆
⠀⠀⠀⠀⠀⢁⠀⠁⠀⠀⠀⣰⠀⠀⢄⠀⠀⠀⣌⠉⠉⠉⠁⠈⠀⠀⠀⠀⡀⠠⠀⠀⠀⠈⠉⡤⠉⢹⡀⠁⠰⣞⠁⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠘⠻⠿⠷⠶⣦⣤⣤⣀⣀⣈⣀⣀⠀⠈⠃⠀⠀⠀⠂⠀⠐⠀⠀⠁⠀⠐⠀⠀⠀⠀⠓⠀⠀⢉⣁⣀⣀⣠⣤⣤⣴⣶⣾⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣟⡋⠉⠉⠉⠁⠀⠀⠈⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⠻⠿⠿⠿⠟⢻⣿⠿⠶⠖⠀⠠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⢸⠃⠲⠒⣒⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠘⠐⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉⠉⡵⠀⠈⣿⣦⡀⠀⠀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠲⣤⣷⡶⠟⠋⠉⠈⠉⠉⠉⠉⠁⠀⢄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢁⣼⠃⠀⠀⡤⢠⣶⡿⡿⢿⣖⣀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢀⣾⠃⠀⠀⠈⠁⣿⡇⠀⣷⠀⠈⠉⠀⠀⠀⠀⣸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣸⡟⠀⠀⠀⠀⠀⠙⣷⡀⣿⡆⠀⠀⠀⠀⠀⠀⠉⡼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢦⡀⠀⠀⠀⢸⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠁⣿⣧⣄⠀⢤⡀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣴⠁⠀⠀⠠⡀⠀⠘⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⣷⠈⣧⠀⠀⠀⠀⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡼⢁⣤⣤⡀⠑⢌⡓⠀⡀⠀⠀⠇⠀⠀⠀⢴⣿⣦⣤⣀⣙⠠⠔⣻⣶⡇⢰⠀⠀⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡞⠀⣿⣿⣿⡗⠀⢀⡉⣲⣾⣀⠠⣄⡀⠀⠀⠀⠙⠛⠿⠿⠿⠶⠟⠛⠉⠀⠁⠀⠀⢋⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⣿⣦⡈⠉⠁⠀⣜⣵⣿⣥⣼⡿⢭⡻⣯⣝⠒⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣝⢿⣷⣄⡀⠸⣿⣿⣿⣿⡿⣀⠀⣻⡎⢻⡄⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠙⢶⣝⠿⣿⣆⣿⣿⣏⡁⠀⣤⢋⣿⣿⣻⡿⠀⣴⣶⡶⢼⡇⠀⣤⠤⠆⠀⢈⣠⡶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣈⠓⢦⣝⡺⢷⣬⣟⠻⣿⣿⣿⣋⡟⠀⠀⠈⠉⠀⢸⠃⠀⠐⠲⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⠢⠽⣲⡮⢽⣳⣮⣝⣛⠿⠿⠯⠥⠀⠀⠰⠿⢿⠧⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠦⢤⣅⣈⣛⣛⣿⣿⡷⠦⣶⣤⣤⣤⣴⡶⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⡀⠀⠖⠀⠀⢠⣼⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⠐⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡤⠅⠀⠀⢿⡣⠃⠈⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠠⡀⢂⠆⠀⠀⡊⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢀⠊⠀⡠⠊⠁⠢⡅⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠁⠀⠀⠀⡀⠀⠀⠀⢈⠅⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⡠⠊⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠔⠉⣀⠌⡠⠊⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠆⠀⠠⠔⠠⠊⠀⡠⠊⠄⠀⠔⢁⠔⠆⠡⠞⠀⠀⠀⡄⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢀⠀⠀⠘⠤⠔⠁⠀⢸⣌⠔⠁⠐⣊⣐⣁⣀⠈⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠘⣁⣀⠀⠋⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠇⠀⡀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⡀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢀⠀⢄⠀⠙⠄⠀⠀⠄⠀⠀⢄⠣⠀⠀⢸⣀⣀⣀⠇⠀⢇⣀⣀⣀⠇⠀⠀⣇⣀⣀⣀⡸⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠈⠂⠈⠀⠀⠀⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠨⠠⡄⢀⠀⠀⠀⠀⠀⠀⡀⢦⣀⣀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠃⠀⡀⠆⠀⠀⠀⠐⠜⠀⠈⠠⠀⠆⠀⠦⠀⠀⠐⠦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⣄⢀⠘⠠⠀⠀⣀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠁⠘⠉⠁⠀⠀⣀⡀⠀⠄⠀⠀⠀⠧⣶⡖⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠿⠇⡈⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠄⠀⠀⠉⠉⠁⠊⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡌⠀⢀⡀⠀⠀⠀⡠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠈⠀⠀⠀⠡


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠤⣤⣁⠀⡀⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠄⢁⣀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁