Trashcan Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠄⠠⠄⠉⠍⠉⠃⠧⠠⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣀⠁⠀⠀⠐⠀⠠⠊⠀⠀⠀⠀⠀⠈⠐⠀⠁⠀⢁⡐⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠐⠀⡀⠀⠀⠨⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠀⠀⠀⠀⠀⢰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠠⠤⠤⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠠⠤⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⣀⠀⠀⢘⠀⠀⠊⠁⡆⠀⠰⠀⠀⠀⠀⠂⠉⡆⠀⠈⠀⢀⢀⡀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠇⢃⠑⢨⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠐⢁⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢸⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠘⠀⢸⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⢸⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡂⠄⢀⠀⠐⠠⠂⠀⠀⠀⠠⠀⠀⠀⠔⠀⣀⠠⠐⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢁⠂⠀⠀⢀⣀⠒⠒⠀⠐⠒⢂⡀⠀⠡⠐⢂⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⠀⠤⠤⠠⠠⠄⠀⠐⠂⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠒⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⣀⠀⠒⠀⠀⣙⣓⣒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣷⣶⠾⣿⣿⠿⠻⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠿⠋⠨⢠⣾⣿⣥⣆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡯⣽⡇⠈⠁⠀⠀⢠⠞⠛⠋⠁⠈⠉⠶⢿⡫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣦⠀⡴⠖⠻⣦⠀⠀⠀⣀⡼⢔⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣮⣿⣿⣿⣿⣻⣗⠒⠒⠒⠊⢻⣷⠀⣴⠿⠋⠊⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣽⣿⣿⣏⣠⣄⢠⣄⢠⢿⣷⢿⡄⢶⣔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣙⣿⣿⣿⣷⣿⡄⢹⠀⢻⡀⠸⡇⢸⡇⣻⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣿⣿⣿⣿⣾⡇⢸⠀⢸⠀⢰⡇⢸⡇⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣇⢸⠀⢸⡆⢸⡇⣸⡇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⣿⢸⡆⢸⡇⢸⡇⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣿⣿⣿⣿⣿⣿⢸⡇⢸⡇⢸⡇⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⣛⣽⣶⣿⣿⣿⣿⣿⣿⣿⡼⡇⠸⡇⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠠⢚⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡧⣷⠀⡇⢸⡇⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⠀⠃⠘⡃⡿⠂⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣿⣿⣿⣿⣿⣿⣿⣯⣏⠈⠀⢂⡤⠴⣞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⡅⠀⠀⢸⣯⣀⣯⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠷⠶⠿⠟⠛⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠰⡯⠭⠭⠭⠭⣽⠀⠀⠀⠠⡤⠺⠤⠤⠼⠦⡤⠀⠀⠀⢲⡖⠛⠒⠒⠛⠒⡖⠀⠀⠀⠰⡆⠈⠀⠀⠁⠐⠆
⠀⠀⠀⡇⡆⢰⠀⡆⣼⠀⠀⠀⠀⡇⢰⠀⡆⢰⠀⡇⠀⠀⠀⠀⡇⢷⠐⡇⢸⢰⡇⠀⠀⠀⠀⠃
⠀⠀⠀⡇⣷⢸⠀⡇⡎⠀⠀⠀⠀⡇⢸⠀⡇⢸⠀⡇⠀⠀⠀⠀⡇⢸⠀⡇⣼⢸⠁⠀⠀⠀⠀⠁⢰⠀⠀⠀⠸
⠀⠀⠀⣇⠘⠸⠀⠇⡇⠀⠀⠀⠀⣇⠈⠀⠃⠈⢀⠇⠀⠀⠀⠀⣧⣈⣀⣁⣉⣸⠀⠀⠀⠀⠀⢃⣘⣀⣀⣁⡀
⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀
⠀⠀⢰⣶⣶⣿⣶⣶⡖⠀⠀⠀⠀⢶⣶⣿⣿⣷⣶⠆⠀⠀⠀⠠⣤⣼⣿⣿⣧⣤⠄⠀⠀⠀⠀⠀⠉⠻⢷⣽⡁
⠀⠀⠘⣏⣿⡏⣿⢹⡇⠀⠀⠀⠀⢸⡟⣿⢻⡟⣿⠀⠀⠀⠀⠀⣿⢿⣿⢻⡿⣿⠀⠀⠀⠀⠀⣤⣤⣤⣤⣬⡍
⠀⠀⠀⣿⢻⡇⣿⢸⡇⠀⠀⠀⠀⢸⡇⣿⢸⡇⣿⠀⠀⠀⠀⠀⣿⣿⣿⣸⡇⣿⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⡇
⠀⠀⠀⣿⢸⡇⣿⢸⡇⠀⠀⠀⠀⢸⣇⣿⢸⡇⣿⠀⠀⠀⠀⠀⣿⣽⣿⣹⣇⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠾⠿⠿⠿⠿⠿⠷⠀⠀⠀⠘⣛⣛⣛⣛⣛⣛⠃⠀⠀⠀⠀⣤⣾⣥⣬⣿⣤⠄⠀⠀⠀⠀⣻⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃
⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠀⣿⢸⡇⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⢼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣤⣿⣼⣧⣿⠀⠀⠀⠀⢠⣾⣽⣿⣿⡿
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠈⠉⠉
⠀⠀⠀⢰⡶⠶⢶⡄⠀⠀⠀⠀⠀⡠⠤⠶⠦⠤⡄⠀⠀⠀⠀⢀⣠⣴⣾⣷⣦⣤⡀⠀⠀⠀⠀⡤⠤⠶⠶⠤⣄
⠀⠈⣿⠉⠉⠉⠉⠉⡟⠁⠀⠀⠀⠓⠒⠂⢲⢰⡎⠀⠀⠀⠀⠘⣷⡒⠒⢒⢒⢺⠃⠀⠀⠀⠀⢳⠒⠒⠒⠒⡟
⠀⠀⣿⠀⡇⣗⣿⢀⡇⠀⠀⠀⠀⢰⡄⡀⠈⢸⠁⠀⠀⠀⠀⠀⣿⡇⠁⢸⢸⢸⠀⠀⠀⠀⠀⢸⢸⢺⢸⢸⡇
⠀⠀⢸⠀⠇⠗⠻⢸⡇⠀⠀⠀⠀⢸⡇⡇⠀⢸⠀⠀⠀⠀⠀⠀⣿⡇⠆⢸⢸⢸⠀⠀⠀⠀⠀⢸⢸⢻⢸⢸⠁
⠀⠀⠸⠶⠶⠶⠶⠾⠃⠀⠀⠀⠀⠰⠧⠧⠤⠾⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠘⠾⠾⠾⠾
⣿⣿⠿⠿⢻⠿⡿⢻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⢛⣛⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠞⠛⠛⣻⠟⣃⠟⠉⠉⠻⣥⡟⣯⠙⠛⠛⠲⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⠀⠙⠛⠓⠒⠒⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⣼⡟⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⣿
⠀⠀⠀⠀⠀⢸⣯⡑⢤⣀⡉⠙⠛⠒⠒⠒⠶⠶⠶⠶⠶⠖⠒⠒⠛⠛⠛⠉⠉⠀⠀⠤⣊⣿⡆
⠀⠀⠀⠀⠀⢸⡟⠹⠷⣦⣬⢉⣒⣒⡲⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⣒⡒⠈⠥⠴⣶⠛⠁⢉⡿
⠀⠀⠀⠀⠀⠈⠳⣶⡤⣤⡤⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣤⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡄⢰⡄⠀⢻⠀⠀⠀⣧⠀⠀⠀⣧⠀⠀⢺⠀⠀⢸⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⡇⠀⢸⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⢸⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⠀⢸⡆⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⠀⠀⣸⠀⠀⣿⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣿⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⡇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠸⡇⠀⠀⢸⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⡇⠀⠀⢸⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⢸⡇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⡇⠀⣷⠀⠀⢸⡄⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⢸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣿⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠀⡏⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⢰⡇⠀⣼⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⣿⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣦⣿⣀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⣸⣧⣤⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⠾⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣤⡶⠾⠷⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣦⣀
⠀⠀⠀⠀⢠⡾⠋⣹⡷⠛⠁⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠈⣻⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠄⠀⢀⣀⡀⠀⠀⠈⠙⠳⢶⡾⠛⠛⠻⣆
⠀⠀⠀⣠⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣾⡷⠾⣿⠀⣿
⠀⠀⠀⢸⡗⠛⢺⣿⢿⣿⠛⠛⣛⣫⣛⡙⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⣩⡽⠿⠶⠶⠟⠁
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⣴⠞⠋
⠀⠀⠀⠀⢿⣤⣤⣄⣠⣤⣤⣤⣀⣄⣠⣠⣀⣠⣀⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠶⢶⣶⠿⠋
⠀⠀⠀⠀⠀⠈⠉⣿⠉⠉⠉⠉⠉⠹⣏⠉⢩⡟⠉⠉⠉⠉⠀⠈⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⣾⠁⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠳⠄⢿⣄⡾⠁⠀⠀⠀⠀⠀⠀⠘⢷⣤⠟⠀⠀⠀⠀⠶⠆⠹⣦⡾⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⡀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢟⣋⣭⡽⠿⠿⢿⣿⣿⠇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⡾⠋⠉⠀⠀⠀⠀⠀⠈⠹⣧⡀
⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣴⠏⠀⠀⣠⡴⠶⡶⢦⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⢰⡏⣾⠛⢻⡇⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⡀⠀⠈⢻⣝⣛⣫⡽⠋⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⣛⣚⣓⣒⣒⣒⣒⣒⠒⠒⠒⠶⠤⠤⢴⣿⣌⣷⣄⠀⠀⠈⠉⠁⠀⢀⣠⠟⠁
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⠶⠶⠶⠞⠛⠁
⠀⢰
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⢿⣿⣶⣿⡶⣶⣦⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢲⣋⣩⣤⣶⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡸⣿⠿⡍⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣯⢻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⢟⣿⣿⣿⢿⣿⣾⣿⣿⣿⣿⣿⣧⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣟⣿⣦
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇⣼⣿⣭⢁⣿⣛⣛⡛⣾⣿⡿⢿⠻⣿⡟⠟⠛⢻⣿⣛⣛⡉⣿⣯⣭⣾⣷⣿⣾⡟⠉⣳⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣧⣿⣿⣿⣿⣹⣿⣛⣿⣿⣼⣿⣯⣯⣥⣿⣿⠿⣿⣟⣰⡯⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠻⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣭⣭⣭⣭⣭⣭⡍⠉⠉⠉⠠⡤⡶⣾⣿⣿⣿⣿⡵⣿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡖⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠒⠒⠒⠒⠒⠛⠛⠋⠹⣿⣥⣽⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⣿⣿⡏⣿⣿⣿⠻⣿⣿⣿⣿⢿⣇⠀⢀⠀⣠⡀⠀⠀⢠⣤⠀⠀⢸⡏⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⣿⣿⡇⣿⣿⣿⠀⣿⣿⣿⣿⠸⣿⠀⠸⠀⢸⡇⠀⠀⢸⣿⠀⠀⣿⡇⢸⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡀⣿⣿⡇⣿⣿⣿⠀⣿⣿⣿⣿⢈⣿⠀⠀⠀⢸⡇⠀⠀⣿⣿⠀⠀⣿⡇⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⢻⣿⣧⢹⣿⣿⡄⣿⣿⣿⣿⢨⣿⠀⠀⠀⢸⡇⠀⠀⣿⡏⠀⠀⣿⡇⣏⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⢰⣿⠀⠀⠀⢸⠁⠀⠀⣹⡇⠀⠀⣿⠁⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢸⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⠈⣿⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⠀⣿⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢸⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⢸⣿⢀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠈⣿⣿⣸⣿⣿⡇⣿⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⠸⣿⢸⣿⣧⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⣿⣿⡇⣿⣿⡇⢻⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⢰⡿⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⣿⣿⡇⣿⣿⡇⢸⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⢸⡇⢸⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⣿⡇⣿⣿⣿⢸⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⠰⣿⠁⠀⢸⡇⢠⣿⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⡇⣿⣿⣿⢸⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢼⡇⢟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⢸⣿⣿⢻⣿⣿⢸⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠘⣿⣿⢸⣿⣿⢸⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⢸⣿⠀⠀⣿⠁⢿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣿⣿⢸⣿⣿⠸⣿⣿⣿⠀⣿⠀⠀⠀⣾⠀⠀⢸⣿⠀⠀⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⢸⣿⣿⠀⣿⣿⣿⠀⣿⠀⠀⠀⣇⠀⠀⢸⣿⠀⠀⣿⠈⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⣿⣿⣼⣿⣿⡆⣿⣿⣿⡄⣿⠀⠀⠀⡿⠀⠀⢸⡇⠀⢀⣿⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⢸⣿⡏⣿⣿⡇⣿⣿⣿⠇⣿⠀⠀⠀⣷⠀⠀⢸⡇⠀⢸⡿⠘⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣻⣧⣿⣿⡇⣿⣿⣿⡇⣿⠀⠀⠀⡏⠀⠀⢸⡇⢀⣸⣷⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⠟⢇⠿⠶⠶⠀⡷⠶⡒⣻⣋⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠛⠒⠒⠛⠋⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠭⠭⠽⠯⠭⠽⠯⠭⠭⠽⢦⠀⠀⢺⡿⣽⠿⠿⠿⠭⠭⠭⠭⠭⠭⠭⠭⠽⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡶⠶⢶⡶⠶⠶⠶⢶⡶⠶⢶⡾⠀⠀⠀⠳⣾⡶⠶⠶⢶⠶⠶⠶⠶⢶⠶⠶⢶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⢸⡇⠀⢸⠇⠀⠀⠀⠀⠙⣿⠄⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⣿⠛⣿⠆⠀⠀⣾⠀⠀⠀⠀⠀⠀⢹⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠰⣿⠂⠀⠀⣿⡆⠀⣿⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠙⠿⠐⠿⠋⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⡇⠀⠀⠀⠀⣰⢫⡭⣝⢦⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣆⠸⡀⢣⡀⢦⠘⣾⣿⢸⡇⠀⠀⠀⠀⢿⡸⠦⢟⣼⡄⢰⡀⠰⡀⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠈⠉⠉⠉⠉⠉⠉⠉⠉⠘⠛⠁⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠈⠉⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠁⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⣀⠤⠀⠐⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠀⠄⣀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠪⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠕⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠈⠁⠐⠒⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠐⠒⠂⠈⠁⠀⠀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠉⢓⠂⠀⠀⡤⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢤⠄⠀⠐⠚⠉⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⢠⠀⠀⠀⠃⠀⠈⠈⡏⠉⠉⠉⠉⠁⠀⢸⠀⠀⠀⡀⠀⠀⠀⡃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣈⠑⠂⠀⠠⠄⢀⣀⣨⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣄⣀⡀⠠⠄⠀⠐⠈⣁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢉⠁⠂⠀⠄⠠⠤⢀⣈⣉⣉⣉⣉⣉⣉⣉⣉⠁⣀⠠⠄⠠⠄⠒⠀⠉⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠃⠀⠀⢠⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣉⠑⠂⠠⠸⠄⣀⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⣀⠠⠤⠀⠒⠊⣉⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠙⠀⠒⠠⠄⠤⠄⣀⢉⣉⣉⣉⣉⣉⣉⡉⢀⠠⠄⠠⠄⠐⠀⠏⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠠⢄⡀⠀⠁⠀⠀⠇⠀⠀⠠⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⢀⡠⠄⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠁⠙⠀⠀⠂⠠⠤⠼⠤⠄⠤⠤⠄⠼⠀⠀⠛⠈⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠒⠒⠒⠒⠀⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⣒⣉⣉⣉⡓⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡴⠛⠉⠉⠉⠉⠙⢦⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠀⠀⠀⠠⠰⠲⠀⠲⠒⠗⠲⠖⠶⠶⠰⠤⠾⠶⠞⠒⠒⠒⠒⠒⠒⠲⠖⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⡄⡄⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠀⠒⠒⠐⠀⠄⠐⠐⠀⠂⠀⠘⠐⠀⠀⠃⠂⠘⠀⠀⠂⠃⠀⠒⠂⠀⠒⠀⠃⠇⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡹⢋⠟⡹⢋⠟⢉⠏⡹⢋⠟⡹⢋⠟⢹⠛⠾⢋⡿⢋⠟⡹⠛⡹⠚⢘⠖⡰⠒⡶⢲⢖⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠉⠀⠁⠀⠁⠉⠀⠁⠈⠀⠁⠈⠠⠎⡴⢡⠞⢡⠞⢁⠔⣡⠞⡠⢊⠔⣡⢊⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⣦⠀⠀⢰⠶⡄⠀⠀⣴⢦⠀⠀⢠⡴⣄⠰⢁⣴⢥⡴⢁⢜⡥⡎⡰⢋⣼⣥⠊⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢺⢸⡀⠀⣿⠂⡇⠀⢠⡏⢸⠀⠀⢸⠅⠇⠀⣋⡿⢘⡖⣡⢾⠁⢟⡰⢛⡇⢰⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⡇⠀⣿⠀⠀⠀⢸⡇⢸⠀⠀⢸⡂⠀⠀⣡⣟⢀⠞⡡⢿⠀⣟⡴⣻⡇⣰⢞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⡇⠀⣟⠀⠀⠀⢸⡇⢸⠀⠀⢸⡃⠀⠀⡱⠿⠀⢟⡴⣿⠀⣏⠔⣹⡇⣴⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢹⡇⠀⣿⠀⠀⠀⢸⡃⢸⠀⠀⢸⠀⡁⠀⠔⣿⠀⣋⡔⣛⠀⣏⢞⣽⠂⣧⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠈⠃⠀⢹⡀⠀⠀⢸⡇⢸⠀⠀⢽⠁⡇⢠⠞⡯⠀⣋⠜⣟⠀⣧⠊⣸⠀⣧⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡁⠀⢸⠄⠀⠀⢸⡇⢸⠀⠀⢴⠄⡇⠠⢞⣟⠀⣥⢊⣿⠀⣧⠎⣹⠀⣧⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⢸⡁⠅⠀⢸⡇⢸⠀⠀⣾⠀⡇⠰⢋⣗⠀⣵⢋⡿⠀⣇⠞⣻⠀⣏⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⢸⡆⠀⠀⢸⡃⢸⠀⠀⢼⠀⡇⠐⢉⡧⠀⣷⠊⡯⠀⡏⣴⣻⠀⣯⠾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡀⠇⠀⢸⡂⢰⠀⢸⡇⢸⠀⠀⢺⠀⡇⠀⢁⡟⠀⣥⠊⣿⠀⣯⢊⣽⠀⣧⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⢸⡁⣤⠀⢸⡁⢸⠀⢸⠧⢸⠀⠀⢼⠤⡇⠘⠫⠿⠰⡿⢞⡿⠼⡿⠋⣿⠛⣿⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢸⡅⠘⠀⢸⡂⢸⠀⠘⡇⢸⠀⠀⢺⢈⡇⠐⣠⣦⢀⡴⣪⣷⢤⡤⢂⣷⢤⣿⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢸⠅⠠⠀⢸⡂⢸⠀⢈⡆⢸⠀⠀⢼⢼⡇⠀⣡⣤⢠⡷⣠⣤⣠⡤⣠⣤⢠⣤⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⢸⡆⢘⠀⠸⡇⢸⠀⢸⡇⢸⠀⠀⣻⣀⡇⢈⡵⣏⠀⡟⡡⣾⠀⢟⡥⣿⠀⣟⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⢸⠀⠀⡇⢸⠀⢸⡇⢸⠀⠀⣺⡀⡇⠀⠔⣏⠀⣯⢞⣽⠶⣯⣾⣿⠀⡟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢧⡴⠀⠀⣇⡸⠀⠀⣇⡸⠀⠀⢹⣀⡇⠀⢺⣇⣠⢗⡿⣻⣴⣫⢖⣿⣠⢏⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣠⣢⣀⡄⠄⠄⠀⠠⠀⠄⠠⠠⠠⠄⡞⠑⠋⡕⢫⠞⢡⡞⣱⢋⣞⣱⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠤⢖⠤⢒⡠⠔⣠⠄⣠⠴⢀⡠⢀⡠⢆⡠⠀⣀⢄⣠⠄⡠⢀⡠⢀⡠⢀⡠⢀⠄⣠⠄⡀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠁⠒⠉⠔⠋⠐⠊⠁⠊⠁⠒⠉⠔⠊⠖⠉⠔⠩⠖⠡⠖⠡⠖⠡⠚⠱⠋⠔⠫⠖⠑⠋⠘⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠛⠛⠿⢶⣦⣤⣄⣀⣰⡾⠛⠛⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠶⣦⣤⣄⣀⡈⠉⠙⠛⠿⠷⣶⣤⣤⣀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠷⣶⣤⣤⣀⣀⠉⠉⠛⠻⠿⢶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠶⣦⣤⣄⣀⡈⠉⠙⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⣿⣿⣷⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⠀⠀⢠⡄⠀⠀⣤⡀⠀⢠⣤⠀⠀⣠⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⡆⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⡟⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠈⣿⠀⠀⣿⡇⠀⢸⡇⠀⢠⣿⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢸⣷⠀⠀⣿⠀⠀⣿⡇⠀⢸⡇⠀⢸⡿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠈⣿⠀⠀⣿⠀⠀⣿⡇⠀⣼⡇⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⡀⠀⣿⡇⠀⣿⡇⠀⣾⡇⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿⡇⠀⣿⡇⠀⣿⡇⠀⣿⠇⠀⣿⠃⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢹⡇⠀⣿⡇⠀⣿⡇⠀⣿⠀⢀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⢻⡇⠀⣿⡇⠀⣿⠀⢸⣿⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠋⠀⠘⠃⠀⠙⠁⠀⠛⠀⠈⠋⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⠀⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⡆⣾⣿⣿⣿⣿⣿⣿⣿⣿⡘⡀⠀⠀⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⡸⠀⠀⠀⡘⣸⣿⣿⣿⣿⣿⣿⣿⣿⡆⡆
⠀⠀⠀⢞⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣳⠀⣞⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣳⠀⢰⣃⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠆
⠀⠀⠀⢸⢰⠒⠒⠂⠀⠉⠉⠉⠉⢹⠀⡇⠀⢸⢰⠒⠒⠂⠈⠉⠉⠉⠉⢹⠀⡇⠀⠈⡐⠒⠒⠂⠀⠉⠉⠉⠉⠉⡇⢸
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡗⠆⠀⠀⠀⠀⠀⠀⠀⢀⠁⡈
⠀⠀⠀⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡇⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠇⠃⠀⠀⠀⠀⠀⠀⠀⡄⠀⠃
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⢸⠀⢘⠀⠀⠀⠇⠃⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢠⠁⢸
⠀⠀⠀⢰⣺⣴⣶⣶⣶⣶⣶⣶⣮⣤⣞⣒⣒⣒⣲⣼⣶⣶⣶⣶⣶⣶⣮⣤⣖⣒⣒⣒⣚⣼⣶⣶⣶⣶⣶⣶⣾⣤⣼⣲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣀
⠀⠀⠀⠀⠀⣤⣾⣶⣿⣼⣦⣶⣤⡀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣬⣽⣶⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⢹⣿⢹⣿⢻⡏⠀⠀⠀⠀⠀⠀⠂⠀⠂⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⢸⡿⢸⡿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣌⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⢰⣧⢸⣧⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢉⢀⠀⡉⡈⠀⠀⠀⠀⠀⠀⠀⣿⡃⢻⡟⠂⣿⡇
⠀⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠄⠀⠀⠀⠀⠀⠀⠠⣤⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣧⣴⣿⣿⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣮⣿⣿⣳⣷⡿⣿⣯⡆⠀⠀⠀⠀⠀⢸⣿⠿⣴⣤⢿⣿⡇⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⡟⢹⡿⣿⣿
⠀⠀⠀⠀⠀⡇⣿⣷⣿⣯⣷⣿⠇⡇⠀⠀⠀⠀⠀⢸⣿⣄⣹⣍⣤⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣦⢻⣾⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣁⣻⠁⢹⡟
⠀⠀⠀⠀⠀⡇⢹⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠛⠋⠓⠋⠛⠚⠛⠚⠃⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠈⠛⠻⠿⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠁

⣶⣶⡶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢸⠀⠀⡇⡈⣍⣙⠓⢦⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠶⠤⠕⠀⠁⢹⣀⣀⣟⠠⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠐⢅⡛⠍⡒⠵⢢⡈⠉⠒⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠈⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣤⠚⠑⠢⠀⠒⠤⢄⣀⠀⠀⠀⠈⠉⠉⠒⠲⠤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⡇⠉⠉⠉⠉⠉⣉⣁⣒⠩⠄⢐⡂⠤⠀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠀⠠⠤⠤⠤⣤⣶⠏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⡔⠲⠧⠤⣬⡙⢦⠀⠀⠁⠀⠀⢲⠭⢍⣓⣂⡀⠀⠄⠀⠀⢀⣈⣉⡉⠉⠉⠛⣛⡆
⠀⠀⠀⠀⠀⠀⢸⡇⡔⡩⠒⠿⣿⣿⣶⣯⡙⣼⠀⠀⠀⣠⠒⠉⠀⣀⡀⢹⠁⠉⠉⠑⡖⢒⠲⣒⠒⠀⢨⠋⠁
⠀⠀⠀⠀⠀⠀⢸⢧⡁⣇⠀⠀⢸⠀⠀⣀⠇⣼⠀⠀⠀⡇⠀⠘⠊⡁⢱⣼⠉⡇⠀⢰⡇⠀⠀⠈⣧⢀⠋
⠀⠀⠀⠀⠀⠀⢸⠂⠳⢬⣙⣒⣛⣒⣚⠥⠚⢹⠀⠀⠀⡇⠀⠀⠀⡇⢸⠁⠀⠑⣦⢸⠀⠀⠀⢀⡟⡘
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⢠⠃⠀⠀⠀⠇⣸⠀⠀⢠⡟⠞⠀⠀⢠⠎⢰⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢀⠀⡇⠀⠀⡼⠀⠀⠀⢸⡏⠀⡞
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⡇⠀⠀⡿⠀⠀⠀⢸⢸⠁⠀⢠⡇⠀⠀⢀⢿⠃⢠⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⣼⢸⠀⠀⢸⠁⠀⠀⡎⣼⠀⣸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢀⠀⢸⠁⠀⠀⡇⠀⠀⠀⡇⣼⢀⢠⡏⠀⠀⡰⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⢸⠀⡼⠀⠀⢠⠃⠀⠀⢰⠃⣇⠐⢸⠀⠀⢠⠇⢸⠁⢸⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⡟⢀⡇⠀⠀⢸⠀⠀⠀⡞⢰⠇⠀⣾⠀⢀⡞⠀⡏⠀⡿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡇⠀⢸⠁⢸⠃⠀⠀⢸⠀⠀⣸⠁⡞⠀⠀⣧⡴⠋⠀⣸⠃⢰⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⢀⠏⠀⡾⠀⠠⠄⢻⣀⡴⠁⢰⠃⠀⠀⢧⠀⠀⢠⠏⠀⢸
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠤⠌⢷⣋⣀⣠⠇⠈⠁⠀⠸⣅⠀⢀⠏⠀⠀⠀⠈⠳⠖⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢿⠑⠲⠤⣀⣀⡉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⠤⢖⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⣍⣉⡉⠉⠒⠒⠒⠒⢒⣒⣒⣒⣉⣉⣉⣭⠥⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢒⡬⠤⠐⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠘⢄⡀⠀⠀⠀⠀⠀⠀⣠⠂⢠⠬⠭⠥⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠉⠢⡀⠀⠀⠀⠈⠐⠊⠭⢒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⢠⠄⡶⠓⠊⠀⠀⠀⠀⣀⠤⠀⢖⣊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣐⠑⠁⠀⠀⠀⠈⠀⠡⣒⢤⠀⠀⠀⠈⠢⢘⡥⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢉⠱⣈⠆⠀⠀⠀⠀⠀⢀⢀⡦⠃⠀⠀⠀⣀⠤⣘⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠣⠯⠀⠀⠀⠀⠀⠃⠏
⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⣤⣤⡀⢀⡤⠤⠤⠤⠤⡄⠀⠀⢀⡤⠤⡴⠦⠤⠤⠤⡤⠤⠤⠤⢤⣠⢤⡤⢤⠤⢤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⣀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠳⢤⡔⠋⣀⠔⠁⡤⡄⠀⠀⡇⢀⢔⠁⡠⢉⢍⡠⣪⠆⠘⡡⡢⢉⢸⠃
⠀⠀⠀⠀⠀⢸⣀⠀⠀⠘⠬⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⡀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⠀⡇⢐⠅⡨⢒⠕⢅⡞⣱⣟⡝⣾⠔⣡⣎⡳
⠀⠀⠀⠀⠀⠈⢻⡻⡄⠀⠀⠀⠀⠀⠫⠂⠀⣦⠶⠃⠁⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡲⠷⠦⣈⢔⠕⣡⣠⠆⢿⡉⣀⠼⢖⢕⠕⡇
⠀⠀⠀⠀⠀⠀⢀⠗⡇⠀⠀⢀⡀⠀⠀⠀⠀⠑⠤⢤⡴⠞⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠻⠇⣀⣠⡧⡢⢊⠝⡺⢊⠔⡡⢀⢔⢡⠗⢁⡇
⠀⠀⠀⠀⠀⣠⣮⠜⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⠀⠀⢠⢄⠀⠀⠀⡸⢊⢈⠔⡡⢊⡰⣡⢊⢔⠕⣱⣡⣾⠁
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠌⠊⠈⠈⢱⠀⠀⠁⠀⠀⠀⡇⠔⡡⢊⢰⡙⠝⠉⡇⡡⠪⣪⢜⢆⣳
⠀⠀⠀⠀⠘⢷⡄⠀⢀⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣢⠀⠀⢀⡠⡰⠀⠀⠀⢤⢶⢲⢷⣾⣴⡡⡠⠙⠒⡲⢋⢔⣟⢕⡑⡰⣹
⠀⠀⠀⠀⢠⠎⠀⠀⠻⣍⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠐⣉⠔⠁⠀⠀⢸⡦⠃⠁⠀⠀⠀⢹⡔⡡⠊⡠⣳⠛⡹⣏⡱⢈⢼
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠻⢇⣀⠐⠒⣊⢽⠖⡡⠪⣪⣪⡾⢊⢄⢔⡕⡏
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀⡴⡖⡶⢦⡀⠀⢾⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡹⢝⠍⠐⢡⡊⢎⠝⡠⣫⣖⣻⡼⠿⠗⣧⣤⡶⠶⠶⠶⠶⠄⣀⣀⠐
⠀⠀⠀⠀⢸⠋⠀⠀⠀⢰⢏⢜⡞⢋⣉⣱⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⡗⡧⡯⠪⣪⣪⣎⠮⠴⠯⣐⡛⠻⠿⢿⡷⠖⣲⣖⣛⡛⠓⠋
⠤⠐⢒⣒⠸⠦⠤⠶⠶⠟⣛⣛⠋⠹⠿⠽⢒⣒⠶⢿⣶⡿⠶⠖⣛⢒⠒⠶⢶⡶⣶⡶⠾⣷⢿⣿⣿⣽⣿⣟⠛⢩⣯⣬⣷⠶⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⢶⡶⠶⠶⣶⠶⠶⠶⣆⠀⠀⠀⣴⡶⢶⣶⣴⣦⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⠀⠀⠹⣏⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡍⠉⢹⡏⠉⠉⠉⠉⡏⠉⠉⣿⠁⠀⠀⠀⠘⢿⣟⠉⠉⢹⡏⠉⠉⠉⠉⣿⠉⠉⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⠇⠀⠀⠀⠀⠇⠀⠀⡟⠀⠀⠀⠀⠀⠈⣿⡀⠀⠸⠇⠀⠀⠀⠀⠿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣿⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⣐⠿⠋⠿⢗⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠸⣿⣁⠀⣀⣻⡷⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛⠟⠀⠿⠛⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢳⠀⠀⠀⠀⠀⣠⠟⣛⣛⢦⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡷⡀⢰⠀⢦⠀⢆⠀⢦⣿⡇⢸⠀⠀⠀⠀⠀⣿⢸⣅⣽⢨⡇⠀⡄⠀⣄⠀⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠏⠳⠷⠶⠷⠾⠷⠾⠷⠾⠛⠷⠞⠀⠀⠀⠀⠀⠈⠳⠶⠶⠿⠿⠶⠿⠶⠾⠶⠾⠃





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⢀⢀⠀⢀⢀⡤⠆⠀⠐⠒⠚⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣄⡅⠰⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠀⠀⠀⢱⠶⠒⠲⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⢀⣀⡴⠊⠉⠀⠀⠀⢠⠆⠀⣰⠃⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠐⢢⠀⠀⠀⠀⠀⠀⢠⠃⣀⣀⡇⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣀⠀⠀⠘⠐⣆⠀⠀⠀⠘⢡⠋⠉⠉⠳⡄⣀⠀⠀⠀⢘⣇⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠈⢦⡀⠀⠀⢸⡄⠀⠀⣴⣏⠉⠀⠀⣠⣾⡽⠋⠁⠀⠉⡆⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣑⣄⣱⠀⠀⠀⠙⠦⠤⣺⠏⠑⠒⠊⠉⠀⣄⣀⣀⠤⠚⠁⠀⠀⠀⠉⠐⢒⣶⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢉⣉⣉⡉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢰⣋⠉⣠⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠛⠚⠋⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣤⠼⠓⠙⠒⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⠤⠴⠚⠉⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⣸⠃⠀⠀⠀⠀⠘⢧⡀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢰⠚⠉⠉⠤⠛⠚⠙⠧⡄⠀⠀⠀⡸⢉⠂⠈⢲⡌⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⠇⠀⠀⠠⠀⠤⢤⡶⠃⢠⠒⠧⣍⡛⣄⠀⠘⠁⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⣯⠜⠉⠀⠉⠉⠀⠀⢸⠀⢦⣴⣿⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡹⡁⠀⠀⠀⠀⣀⡇⢸⣀⣀⣤⡀⠀⠀⢀⣀⠴⡠⣈⠀⡽⢏⠒⠄⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⡄⠀⣴⠛⢻⠀⡞⠁⠀⣶⠟⠒⣋⠭⠒⠉⠀⠈⡏⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢜⡦⠃⢠⡏⢠⡃⠀⠀⣟⡏⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢉⣶⢰⠃⡜⠉⠑⠒⠿⠀⠀⠀⠀⠀⠀⢠⠞⡇⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⢠⡿⠻⡜⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠇⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢿⠝⢦⡀⡸⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠔⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠒⠒⢻⠉⣩⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⡏⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠰⢾⠶⠇⢀⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⢻⡟⢶⠶⣲⡾⣿⣽⠿⠫⢿⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣷⣴⡶⠞⠋⠀⠈⠛⠷⠶⣴⣧⠾⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠳⠬⢭⢆⡀⠀⠀⠀⠀⢀⡀⢀⣀⠀⢰⠁⣸⠇⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠒⠒⢴⠃⠐⠛⡀⠀⠀⣠⢽⡉⠋⠘⡆⡌⢀⣿⡰⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠈⡄⠀⠀⡇⣀⣜⢣⠈⠱⢤⣀⡷⠦⢾⡿⡱⠣⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣅⡀⠳⣄⢀⣷⣖⡩⢿⡃⠈⠂⠁⢀⡾⠉⡇⠀⠸⠫⠛⠓⣬⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢁⣀⡀⠀⡈⠉⢯⠁⠀⠀⣇⠀⠀⠀⡿⠀⠀⠘⠒⠛⠒⢲⣎⣸⠀⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠉⠏⠀⠠⠋⠉⢉⣩⣴⣾⣛⣒⣚⣒⣒⣒⣐⣩⠭⠭⠖⠚⢲⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢄⣀⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⡀⣶⠶⢸⠳⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠧⣀⠀⠈⡙⠊⠀⠀⠰⠶⣶⠗⢻⣟⣻⢍⠆⣎⣦⣼⠹⣹⣿⣽⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠢⢄⣙⢋⣿⣯⠄⠀⠀⠀⣿⠀⠸⠋⠉⠓⠃⠋⠀⠀⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⠢⡄⠀⠀⠤⠤⣄⠀⠀⠀⠀⠤⠤⠄⡀⠀⠀⠠⠤⠤⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⢤⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠀⠰⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠦⣠⣇⣀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠀⢱⠒⡜⢾⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⣀⠤⣄⣼⠖⢠⡀⢰⠀⠇⠀⠉⠢⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣄⣀⣀⣠⣤⠤⢤⣀
⠀⠀⠀⠠⡎⠉⠉⣑⡀⠈⠉⠉⠑⠀⠀⢸⣄⣀⣀⡠⠵⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⣧⠀⠀⡏⠉⠉⠁⠀⢨⠛⠫⡉⠉⠁
⠀⠀⠀⠀⠳⣤⡼⠷⠈⠉⠒⠊⠉⠑⠢⣼⣼⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢃⡿⢓⠞⠓⠶⠬⣉⣉⣉⣦⣄⡈⠑⡄
⠀⠀⠀⣀⡜⢀⠀⣠⠀⠀⠸⢆⠀⠀⡠⠈⡟⢾⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠈⣧⣈⡒⠢⠤⡤⠋⠉⠽⠤⣇⡈⠉⠓⠢
⡴⠚⢁⡤⠴⠋⠙⣆⣡⡀⠀⠀⢱⠀⠀⠈⠀⣸⢤⣬⣉⠉⠀⠀⠀⠐⠒⠒⠚⠁⠀⠀⣉⣉⣤⠴⠋⠀⠉⠙⠒⠠⠤⠤⠴⠦⠤⢬⡷⠦⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠴⠠⠒⠒⠉⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄⣡⠀⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠋⠀⠀⠀⢙⡶⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢠⠏⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠁⢠⡀⠀⠀⠀⠀⠈⠀⠈⠑⣄⢀⠀⠀⠀⠀⢇⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⠀⢣⣄⠀⠀⠀⣄⠀⠀⠬⣿⠁⠀⠀⢀⠔⠀⠋⠀⠀⠈⡷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣦⣘⡆⠀⠀⠈⠒⠒⣒⠏⠀⠀⠀⠀⠀⣄⣀⢠⡴⠚⠁⠀⠀⠀⠈⠁⢙⡳⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⡤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠰⣏⡁⣠⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠋⠘⠉⢀⡠⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡤⠶⠛⠈⠓⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠚⠉⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⡰⠃⠀⠀⠀⠀⣈⠳⣄⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡂⠀⠀⠐⠛⠋⣉⠗⣤⠀⠀⠀⣘⢉⠇⠈⠳⡄⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡶⠃⢠⠞⠓⠬⡙⣦⠀⠈⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡠⠜⠁⠀⠀⠀⠀⠀⠸⠂⢰⡴⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⢠⣇⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠉⢹⠃⡜⠀⠀⣴⡜⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⣵⠂⠀⡞⢀⣇⠀⠀⢏⡏⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⢀⡵⢆⠃⣸⠀⠈⠐⠚⠁⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⣄⣾⠛⠾⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠹⠘⠓⠄⣠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢀⣀⣀⠠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠂⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠹⡇⢸⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⡇⡜⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠈⢛⣟⣋⣀⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⢹⠿⠽⢟⣭⠿⠿⣾⣿⠗⠻⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣿⡶⠾⠛⠁⠀⠀⠈⠙⠛⠶⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠀⠀⠠⠀⠀⠀⠄⠄⠄⠄⠀⠀⠀⠀⠀⠆⠆⠀⠄⠀⠐

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣾⠿⠂⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⡄⠂⠀⠀⠀⠀⣠⣼⡆⠀⠀⠀⠀⠀⢀⣴⣿⠿⠝⠿⣳⣴⡆
⠀⠀⠀⠀⠀⢀⣤⣷⣾⣷⣦⣤⡀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡟⠁⠀⠀⣌⣀⠀⠀⠀⢀⣼⣿⣿⠃⠀⠀⠀⠀⠀⡀⠉⠁⠀⠐⠿⠿⠟
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠿⣿⣿⣿⠟⣶⣧⡼⢿⣿⣿⡿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠻⣿⣿⡄⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣄⣛⣟⢁⣼⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣷⠉⠓⠀⠀⠀⠀⡄⠀⣿⣟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⠿⣿⣶⡆⠀⢠⣾⣷⣶⣿⠏
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠘⣿⣿⣿⣟⣿⡿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠻⡟⠉⠁
⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠂⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⣀⣤
⠀⠀⠀⣠⢴⣞⣉⣁⣀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⢻⣛⣛⣛⣛⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⠛⣙⣿⣿⣿⠇
⠀⠀⠀⠱⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⢹⣿⡿⣿⣿⣟⢹⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⡀⠀⠀⠉⠉⠙⠋⢀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⢼⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣈⡀⠀⢀⣴⣿⣴⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠝⠋⠛⢛⠟⠋⡇⣀⠀⠀⠀⠀⠀⢸⢰⣿⣿⣿⣿⡷⢸⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡿⠀⠺⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⣿⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡵⠋⠁⠀⠀⠀⠀⠀⣼⠂⠀⠉⠉⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿


⠀⠀⠀⣤⣶⣾⣿⣿⣶⣶⣄⠀⠀⠀⠀⢀⣠⣤⣤⣦⣵⣤⣤⣀⠀⠀⠀⠀⠀⣴⠶⠶⠚⠛⠛⠛⠛⠷⠶⢦⡄
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢈⢛⣟⣛⣛⣛⣛⣛⢛⡁⠀⠀⠀⠀⠸⠶⢶⣶⣤⣤⣴⣤⡶⠶⠾
⠀⠀⠀⢸⣿⣿⢛⠛⣿⣿⠀⠀⠀⠀⢏⣻⠐⡇⢸⡇⢸⠀⣿⢰⣏⡇⠀⠀⠀⠠⣴⣄⣄⣀⣀⣀⣀⣀⣤⡄⠀⠀⠀⠠⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦
⠀⠀⠀⢸⣿⠉⣿⠿⢙⣿⠀⠀⠀⠀⠀⢹⠈⡇⢸⡇⢸⠀⣿⢸⡇⠀⠀⠀⠀⠀⣀⡀⠈⠉⠉⠉⠨⠀⣀⠀⠀⠀⠀⠀⢹⣶⣾⢻⣶⣾⣻⣷⣾⡟⣷⣶⡏
⠀⠀⠀⢸⣿⣷⣾⣶⣿⣿⠀⠀⠀⠀⠀⢸⡀⣷⢸⡇⢸⠀⣿⢸⠇⠀⠀⠀⠀⠀⢉⡋⠛⠛⠓⠛⠛⠉⣉⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⣿⢹⣿⣿⡇⣿⣿⡇
⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⢸⣇⣿⢸⡇⣸⠀⣿⣸⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣶⣶⣾⣿⡏⠀⠀⠀⠀⠀⠶⠿⠿⠾⠿⠿⠾⠿⠿⠷⠿⠿⠶
⠀⠀⠸⠟⠿⠿⠿⠿⠿⠟⠿⠀⠀⠀⠀⠀⠺⠭⠽⡧⡿⠭⠽⠒⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⣿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⠶⠶⢶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⣭⠴⢶⣒⣲⣶⣶⣐⣂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⡶⠿⠛⠛⠋⠉⣉⣉⣉⣙⣛⣛⣻⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣭⣶⠶⠾⠟⠛⠛⠛⠛⠋⠉⣉⡉⠙⠛⠛⠛⠿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⢏⠉⠀⢀⣠⣀⣀⣠⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣄⣀⣂⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣡⣤⣶⠾⠿⣿⣿⣿⣿⢿⣿⣽⠿⣷⣟⡝⣡⣋⣽⢿⣟⢿⣿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣏⠘⣛⣿⣶⣿⣿⡟⣇⣁⢹⣏⣷⡿⠛⣟⠛⢻⣷⣴⣶⣶⣿⣶⣥⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢃⢎⣸⣶⣿⣭⣶⣀⣙⣿⣥⣹⣾⣏⠹⣯⣵⢩⠙⢁⣦⣿⡿⢛⣋⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⢿⣿⠄⠀⠈⣉⣉⡉⠛⠛⣛⣻⣧⣬⣭⣽⣿⣿⡿⠛⠛⠟⡟⢻⡯⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠛⠛⢁⣠⣼⣿⠁⠿⢸⡿⣌⣿⠀⣸⡏⠙⣏⠀⢠⡖⢶⡄⠂⣰⢶⣦⠀⢸⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣿⣯⣤⣤⢸⡇⠀⣿⠀⣿⠇⠀⣿⠀⢸⠀⢸⣷⠀⣿⢸⣿⣶⠾⠿⠟⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣶⠶⠿⢸⡇⠀⣿⠀⣿⠀⠀⣿⠀⢸⠀⢸⣿⠀⣿⠸⣿⠛⣷⡶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⢸⡇⠀⣿⠀⣿⠀⠀⡇⠀⢸⠀⢸⣿⠀⢹⠀⣿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡇⣺⡇⠀⣿⠀⣿⢠⡄⡇⠀⢸⠀⢸⣿⠀⢸⠀⣿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⡇⠀⣿⠀⣿⠀⠀⡇⠀⢸⠀⢸⣿⠀⢸⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⣿⡟⠀⡏⠀⣿⠀⠀⡇⠀⢸⠈⢸⣿⠀⣾⠄⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠢⠀⣿⡇⠀⡇⠀⣿⠀⠀⡇⠙⣿⠀⢸⣿⠀⣿⠀⣿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢰⡇⠀⣿⠠⢀⡇⠀⢿⠀⢘⣿⠀⣿⢰⡿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣄⡀⠈⠛⠋⠀⠀⢿⣀⣸⠃⠀⠸⣦⣼⠟⣀⡈⠋⠁⣸⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣏⣽⠲⢽⣷⡦⣤⣝⣒⣀⣉⠀⠀⣀⣀⣀⣀⣬⡭⠵⣖⣻⢵⢛⣿⠀⠀⠀⢀⣤⣤⡀
⠀⠄⠀⠀⠀⢰⣿⣿⢶⠾⣤⣙⢷⣄⡒⠈⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⢶⠶⠶⠚⠛⠉⣀⣬⣼⢏⣴⡾⢿⣾⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠻⣷⣮⡠⢎⠽⠿⣯⡛⠻⠿⢷⣦⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⡶⠶⢟⣛⠉⢿⣿⣛⣛⡹⣙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠜⢿⡔⢡⢖⣽⣫⣻⡆⠓⢀⣠⣤⣀⠈⠉⠉⣩⣿⣷⣶⣶⡆⢾⣿⣿⡗⠀⣍⢛⠛⡛⠛⠋
⠀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣾⣮⣵⠟⠀⠀⢿⣿⣻⣿⠆⠈⠉⠻⢷⣤⣤⡾⠇⠀⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡼⠮⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠙⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠔⡖⠀⠀⠀⠀⠀⠧⠀⡴⡶⠛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡴⠊⠉⠈⢉⡶⠊⠀⢰⡧⠔⡞⠉⠙⡆⠀⠸⡄⢣⡀⢾⡙⣿⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠹⣠⠚⠉⠙⠁⠀⢀⡤⠯⢄⡀⠀⠸⡀⢈⣧⠀⠀⡿⠚⠉⠉⣆⠈⠉⠉⠁⢧⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠈⢦⠀⣰⠁⠀⠈⠑⠢⣄⠀⠴⠁⠀⠀⠀⠈⠳⣄⣙⣧⠤⠴⠚⠡⠒⠉⠉⠙⠢⡀⠀⠀⣼⣀⣀⡘⠀⠀⠀⢰⡲⣄
⠀⠀⠀⠀⠀⢰⠃⢰⠃⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡶⣄⠀⠀⠀⠀⠀⠀⠀⠹⠒⠒⠣⠄⠀⢱⣠⠴⠶⠁⡟⠁
⠀⠀⠀⠀⠀⢈⡕⠛⠚⠋⠛⠓⠢⢤⠔⠊⠉⠉⠉⡣⠤⠖⠢⠤⣀⡼⠙⢎⢾⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⡼⠁⠀⠀⠀⢹
⠀⠀⠀⢠⠖⠉⠀⣀⡤⠤⠀⠀⠀⠀⠀⣀⠀⢀⠴⠦⡀⠀⠀⠀⠈⢣⡤⠬⠷⠇⠀⠀⠀⠀⠀⡟⠒⠚⢹⠡⠚⠉⠉⠉⠓⢦⡿⠤⣄
⠀⠀⢰⠃⠀⣠⠎⠁⡠⠤⠖⢦⣀⠐⠋⠉⠱⢄⣀⣀⠇⠀⠀⠀⠀⠀⢁⣀⣀⣀⣀⣀⠀⠀⠐⠋⠉⠙⢺⠀⢀⡤⠤⢄⣀⣀⣁⡤⠼
⠀⠀⠘⠶⠾⠓⠋⠉⠑⢄⢀⡤⠜⠓⠲⠒⠒⠒⠢⢤⣀⣀⣠⠴⠒⠉⠉⠀⠀⠀⠀⠈⠉⠒⠦⠤⣤⡤⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢤⡄⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⢠⠤⠴⠶⠦⠤⡄⠀⠀⠀⢠⣤⣴⣴⣦⣤⡄⠀⠀⠀⠰⡀⠕⠂⠈⠀⠀⠀⠀⠀⢀⡤⠾⠛⠋⠉
⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠆⢠⠀⠀⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠰⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠢⠠⠠⠌⠄⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠄⠄⠤⠤⠃⠀⠀⠀⠀⠘⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⡼⠷⡄
⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠎⠀⣠⣇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠞⢫⠟⠛⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡴⠃⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⡏⠀⢀⡿⠒⢲⠀⠀⠀⠀⢀⡟⠀⠀⠀⠠⠎⠙⠓⣲⠖⠲⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠾⠀⠐⠋⢀⠴⠚⣓⣦⣀⣴⣯⢷⣦⠀⠀⠀⠀⠐⠚⠁⣀⡴⠋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠛⠒⠒⠚⠒⠒⠒⠒⠂⠙⣋⣿⣠⣷⠖⠛⠂⠀⠀⣀⣤⣤⣭⡥⠴⠋⠁
⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠉⢻⣀⡼⠻⠄⠀⢀⡞⠁
⠀⠀⠀⢀⡞⠀⠀⣤⣄⣴⣶⠄⠀⠈⢷⣄⠀⢠⡀⠀⠈⠉⠛⠀⠀⠀⠀⣼
⠀⠀⠀⣼⠀⠀⠀⣸⣿⣿⣷⡄⠀⠀⠀⠻⣧⠈⢻⡄⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⡇⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠀⢻⠀⣸⠇⠀⢤⣄⣀⣤⣤⡟
⠀⠀⠀⣧⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣾⠀⠋⠀⠀⠀⠠⣤⣀⣤⣿
⠀⠀⠀⠹⣷⣞⣋⣀⡀⠈⢻⡀⠀⠀⢀⣼⠏⠀⠀⠀⠑⣦⡀⠠⢤⠤⣾
⠀⠀⠀⠀⠀⠉⠉⠉⠉⣧⣸⠇⠀⠀⠈⠀⠀⣀⣠⢤⡞⠁⠉⠛⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠖⠒⡟⢿⡁⠰⣄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡷⠶⠚⠋

⣿⣷⠿⠷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣖⣒⣶⣖⣒⣶⣖⣖⣲⣄⠀⠀⣤⣤⣤⣤⣤⣴⣖⣒⣒⣒⣒⣒⣒⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⠇⠀⠘⣯⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡇⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠈⢿⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠁⠀⠀⠀⠉⠀⢰⡇⠀⠀⠀⠀⠈⣇⠀⠀⠁⠀⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠠⣶⢲⣦⠀⠀⢸⠇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⣿⠀⠀⠰⣷⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠛⠸⠛⠁⠀⣸⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣄⡇⠀⠀⠀⠀⠀⠀⠀⣏⣤⡀⠀⠀⠀⠀⢸⣧⣄⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⣧⡀⢀⠀⡀⢀⠀⣀⣿⠀⡇⠀⠀⠀⠀⡏⡔⢲⡽⡆⢀⠀⢀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢧⡼⠻⣧⣼⣤⣷⣬⣦⡽⠻⣤⠇⠀⠀⠀⠀⠳⣽⣯⣼⣧⣬⣦⣼⣦⡽⠃





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣻⣄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠝⢽⠭⡢⣱⠑⠀⢤⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠒⠒⢄⠴⠊⠁⢱⠊⠀⢈⣳⣄⠔⢭⠎⣰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠭⠽⢷⡾⠦⢤⣠⠎⠀⠀⠸⢋⣉⢳⢄⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠉⠉⠉⣡⠙⠶⠊⠀⢠⣶⠆⠀⣤⣴⠿⠋⠉⢉⠐⠢⡄⠀⠀⠀⠰⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⡀⠀⠀⠀⣀⠔⠁⠀⠀⢀⣔⡧⢁⣠⠴⠿⣀⠀⠀⢀⠆⠀⢀⠇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠈⠉⠉⠉⠀⠀⢀⣠⠶⣉⢎⡔⠁⠀⠀⠀⠈⠳⡄⠘⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢩⠼⠊⢁⡎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠒⢤⣈⡬⠄⠉⠁⢇⠉⠀⠀⠀⠀⢀⣀⡠⠔⠊⠁⠀⠸⣄⣂⠀⠀⠀⠀⠀⢸⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡜⠁⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⠿⠆⠀⠀⠀⠀⣇⢀⠎
⡠⢮⠿⣳⢊⡱⠀⠀⣃⣤⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠄⠐⠒⠚⠀⠀⠀⣸⠁⠈⡻
⢩⠁⠀⢠⢃⡠⠐⠊⠁⠀⠈⢉⣗⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠒⡶⠀⠒⠈⠁⡃⠀⢀⢃⠀⢰⡁
⢈⠃⠀⡜⠇⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠉⠙⠓⠶⠖⠠⠤⣄⣿⠀⠀⠀⡘⠀⠀⠀⠘⠅⠲⠀⢰⣺⠁⡜⠁
⠸⠀⡠⠤⣞⣀⠔⠊⠉⠚⣄⠀⠀⠉⢐⣒⣶⡖⠛⠛⠛⠛⠛⢋⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⣉⣀⣤⠇
⠈⠊⠁⠀⠉⠀⠀⠀⢠⢤⣌⡩⠍⠬⠷⠌⠙⠓⣶⠠⠤⣄⣀⠀⠀⠀⠨⠀⠀⣀⡠⠤⠐⠊⢯⣸⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣡⠤⠦⠴⠖⠒⠂⠈⠀⠀⠀⠀⠀⢉⠃⠒⠖⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣋⣁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠉⠉⠉⠉⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡏⠀⢸⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⢾⣿⠶⠶⢾⣷⠶⠶⣿⡷⠶⣾⡇⠀⠀⠀⠀⣿⣿⣿⣿⠉⢻⣿⣿⣿⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠃⠀⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⡿⠿⢿⡿⠿⢿⣿⠿⢿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣻⡇⠀⣸⣇⠀⢸⣿⠀⣸⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⢻⡟⠛⢻⡟⠛⢻⡟⠛⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣸⣷⣀⣸⣇⣀⣼⣇⣠⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠙⣿⠋⢻⡟⠙⣿⠟⢻⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣿⡀⢸⡇⢀⣿⠀⣾⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢶⡒⠒⠶⣶⡶⠞⠋⠉⠉⠉⠙⠓⠶⠦⣤⣤⠶⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⠀⠈⠷⠀⠀⠋⠀⠀⠰⡄⠀⠀⠀⠀⠀⠘⠋⠀⢀⡀⠀⠀⣿⣷⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣟⠳⢤⡈⠛⢦⣀⣴⢦⣤⡦⠶⣶⠿⣤⣄⡀⠀⠀⠠⠀⣠⠞⠀⢀⢠⡿⠉⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠉⢳⡦⠤⣤⣀⣿⠠⣞⠁⢤⡀⠸⣄⠀⢨⡟⠻⣟⠛⢾⡗⠶⢦⣾⠈⠁⠀⣠⡟⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⣴⠋⢤⠀⢀⡴⠋⠀⠀⠈⠉⢻⣤⡼⠷⢦⣍⣀⣈⣀⣼⣃⣀⢌⣉⣈⣉⣷⣀⣹⣟⣄⣠⣯⠀⠀⠄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢈⣤⠀⠀⠀⠀⣴⠟⠙⢧⢀⡀⠀⠉⠉⠉⠉⡿⠙⠻⠿⠿⢯⡍⠉⠉⠉⠀⠀⠻⡄⠀⠀⠀⠀⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣀⡴⠋⠀⠀⠀⠀⠚⠁⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠘⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠻⣆⠀⠀⠄⣤⠀⠀⡀⠀⣷⠀⠀⠣⠂⣤⠀⠀⠀⠀⠠⠀⠈⢧⠀⠀⣄⠀⠀⠀⠀⡆⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⡄⡇⠀⣈⣤⡤⣼⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⠞⣧⠀⣀⣤⠴⠦⣄⣿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠻⠟⠋⠉⠀⠀⠈⠛⠛⠛⠛⠓⠒⠒⠖⠖⠒⠋⠀⠀⠈⠉⠉⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⠀⠀⠰⡂⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⢿⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⣼⢃⠀⠘⠉⠓⠶⡄⠀⠀⠀⢸⣦⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⣟⠂⠀⠀⠀⠁⠀⢹⡄⠀⠀⠀⠻⣟⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡇⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠙⠲⢦⡈⠂⠀⣠⡟⠀⠀⠀⠀⠀⠈⠳⢮⣟⡲⠦⢤⣤⣤⣤⣀⣀⣀⣠⣤⠴⠖⣛⣭⠞⠋⠀⣠⢶⠛⢻⠉⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠙⠒⠚⠋⠀⣠⡴⠞⢦⡀⠀⠀⠀⠈⠉⠛⠲⠶⠶⠤⠤⠤⠤⠶⠶⠚⠋⠉⠀⠀⠀⠀⣏⠈⠆⡀⠀⣈⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⡈⢿⠛⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣤⣷⣋⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⡳⣤⣙⠀⠁⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣍⣛⣻⡶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⡰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠂
⠀⠀⣀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⠀⠀⠀⠀⠀⠀⠀⠓⠢⣄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣆⠀⠀⠀⠈⠀⠀⢂
⠀⠰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠠
⠀⠀⢸⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠘⡀⠀⢸⢰⢢⣶⡄⡖⠀⣶⢰⣴⣦⡖⠂⡇⠀⡇⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⡇⠀⢸⠸⡼⡟⡆⣧⠤⠟⣿⢻⠎⢧⠄⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠁⠀⠈⠂⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠐⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⣀⡴⠟⠳⣦⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢀⠀⠈⠋⠀⣀⠐
⠀⠀⢠⠴⠧⣤⠀⠀⢀⡷⠂⠀⠀⠓⣆⠀⠀⣠⠴⠧⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢁
⠀⠀⢸⠀⠀⣿⠀⠀⠘⣧⣀⣀⢄⣠⡿⠀⠀⣿⠀⠀⣿⠀⠈⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠩⠁⠀⠀⠀⠀⠈
⠀⠀⢸⠀⠀⣿⠤⢄⣀⠈⠉⠑⠚⢉⣀⡀⠤⣿⠀⠀⣿⠀⠀⠀⠀⠈⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⡠
⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣲⣯⠤⠶⠒⣒⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠊⠉⠀⠀⠘⠉⠉⠉⣉⣩⡭⠽⠶⠦
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠬⠭⠝⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠉⢷⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣞⣀⣀⠀⠀⢀⣀⣀⣀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣞⣉⣡⡤⠤⠤⠭⢽⡯⠤⠤⠤⠤⢵⠄
⠀⠀⠀⢀⡤⠒⠒⠢⣄⡇⠀⠀⠐⠓⡒⠒⠠⠤⠄⠒⠊⠙⠒⠒⠲⣖⡭⠖⠲⣄
⠀⠀⢀⡏⠰⠞⠓⠆⠈⡁⠀⠙⣆⢙⠦⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠀⠠⠾⣁
⠀⠀⠘⢄⠠⠤⠒⠈⠀⠈⠲⣄⠈⢿⡀⢰⠁⢠⠏⠉⡽⠉⠉⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠉⠳⠤⠮⠤⠴⠛⠇⠀⠀⠀⢀⣀⡤⠖⠋⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⡀⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠇⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡤⡾⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠲⢍⡑⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠳⠴⠋⠽⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠦⡀⣱⠚⠀⠰⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠄⠀⠀⠀⠀⡄⠉⠀⠀⡿⠧⠤⢄⣀⣀⣀⣀⣀⠀⠤⠇⠰⠋⣿⢄⠀⠀⡜⢰⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⡿⠂⠀⢸⠲⢄⣀⣿⠒⠒⠒⠒⠒⠂⠀⠀⠈⢉⠁⢄⡴⡇⠀⢑⡄⢙⠎⢆⠀⠀⠠⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⣳⠀⠀⠈⡇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠀⠀⡇⠀⢸⠀⡎⠀⠎⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠉⣿⣟⠁⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠪⡀⠀⡇⠀⡼⠀⡇⠀⠀⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡇⢻⣤⢄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣴⠇⠀⠇⠈⠁⠀⠀⠀⠀⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡨⠝⠧⠼⠍⠉⡗⡶⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠂⠀⠘⡄⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢰⢣⠃⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠎⡞⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠇⠀⠶⠶⢾⠾⡃⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠺⡄⠀⠀⠀⠀⠀⠑⢢
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡇⠀⠀⣮⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢠⣰⠇⠀⠀⠀⠀⠀⣴⢶⡢
⠀⠀⡖⠋⠉⠉⢹⣟⡟⢛⣛⡿⠿⢿⣿⣛⣛⣫⣋⠉⠉⣙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠛⠿⠶⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠉⠓⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠆⠀⠀⠀⠀⠀⠀⠤⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣉⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠠⡄⢻⡀
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠓⠘⣧
⠀⠀⠀⠀⠀⣼⠁⠀⡂⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣤⣤⣼⣧⡀
⠀⠀⠀⠀⣿⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⡇
⠀⠀⠀⠀⢹⣅⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⣶⠶⠶⣶⠶⠶⠶⢶⣶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡅⠀⢸⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡄⢠⡟⠀⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢀⣾⣁⣀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣤⣤⣼⡧⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣴⠶⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠐⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⢸⣿⡂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣄⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣥⣼⣿⣧⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣾⣬⣭⣭⣤⢤⣤⣘⣹⣷⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⡤⠖⣚⠋⢉⠡⠤⣒⠺⠛⡋⠉⠛⠉⠀⡠⠂⢸⠀⠱⣌⠂⢍⠛⠫⣄⡂⠄⣀⠀⠭⢉⡒⠦⠤⣤⡀
⠀⠀⠀⣼⣯⣩⠐⡀⠀⠂⢈⠠⠂⠉⠀⠔⠀⢀⠔⠁⠀⢘⠀⠀⠘⣆⠀⠈⠢⡀⠉⠒⠤⡉⠓⠢⠄⠉⢒⣼⣷⠄⡀
⠀⣴⣄⡹⠩⠙⠛⠻⠶⢾⣤⣤⣤⣈⣀⡀⣀⠊⠀⠀⠀⢸⠀⠀⠀⠈⢆⠀⠀⠘⣦⣀⣠⣬⣷⣶⠶⠿⢛⣛⠿⢀⣷
⠀⢹⡍⠉⠛⠲⠦⠤⣀⣀⡀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⢀⣀⣠⣬⡴⠶⠿⢛⣝⣷
⠀⠀⠙⢳⣶⣤⣤⣀⡀⠉⠉⠙⠓⠒⠶⠶⠦⠤⠤⣤⡤⠤⣤⡤⠤⠤⠤⠤⠠⠶⠖⠒⠛⠛⠉⠉⢁⣀⣤⣤⡾⠟⠟⠋
⠀⠀⠀⠈⣇⠀⠈⠉⠉⠛⠓⠲⢶⣾⣛⣻⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠶⠶⠗⠛⠛⢻⡏⣿
⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠁⠀⠀⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣿
⠀⠀⠀⠀⢹⠗⣤⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢤⣾⢻⡇
⠀⠀⠀⠀⠘⡇⢻⠀⢸⡋⢱⣤⠖⢢⠀⣤⠤⡀⠀⣀⣄⠀⠀⣠⣄⠀⢀⣤⢄⠀⣴⠒⣄⡞⢨⡇⢘⡇⣸⠃
⠀⠀⠀⠀⢠⡇⢸⡁⢸⡇⠀⣿⡀⠀⣼⡇⠀⢰⣼⠁⠀⣷⣸⠁⠀⢶⣿⠁⠈⣿⡇⠀⣿⡇⢸⡇⢸⡇⣿
⠀⠀⠀⠀⠀⣇⢸⡇⢸⡇⠀⣿⡇⠀⠿⡇⠀⢸⡇⠀⠀⣽⣿⠀⠀⢸⡟⠀⠀⣿⡇⠀⣿⠀⣸⠇⢸⠃⡏
⠀⠀⠀⠀⠀⢶⠸⡇⢸⡇⠀⣻⡇⠀⣶⡇⠀⢸⡏⠀⠀⢹⡇⠀⠀⣼⡇⠀⠀⣿⠁⠀⣿⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠌⠀⣷⠈⡇⠀⣿⡇⠀⣹⡇⠀⢸⣧⠀⠀⢸⡇⠀⠀⢿⡇⠀⠀⣿⠀⢰⡿⠀⢸⠀⣾⢸⠁
⠀⠀⠀⠀⠀⢸⡆⣿⠀⣿⠀⢻⡇⠀⢻⡇⠀⢸⣿⠀⠀⢼⣧⠀⠀⣸⡇⠀⢠⠟⠀⢈⡧⠀⣿⠀⡟⢸
⠀⠀⠀⠀⠀⠸⠇⢿⠀⣿⠀⢸⡇⠀⢸⣷⠀⠸⣿⠀⠀⢸⡿⠀⠀⣿⡇⠀⢸⣦⠀⢸⡇⠀⣿⠀⡇⡞
⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⠀⢻⡇⠀⢸⡿⠀⠀⣿⠀⠀⢻⡇⠀⠀⢹⡇⠀⢸⡟⠀⢼⡇⢀⡿⢰⡇⡇
⠀⠀⠀⠀⠀⠀⣀⢸⡇⢹⠀⢸⣷⠀⢈⡇⠀⢰⣿⠀⠀⢾⡇⠀⠀⣼⡄⠀⢸⡇⠀⣼⡃⢸⡇⢸⢡⡇
⠀⠀⠀⠀⠀⠀⢻⠸⡇⢸⡇⢸⣿⠀⠸⡇⠀⢠⣿⠀⠀⢼⡇⠀⢸⣿⡇⠀⢺⡇⠀⣿⠇⢸⡇⣸⢸⠁
⠀⠀⠀⠀⠀⠀⢸⡄⡇⢸⡇⠘⣿⠀⠰⣷⠀⠸⣿⠀⠀⢠⡇⠀⠸⣿⡆⠀⣿⡇⠀⣿⠀⢺⠃⡿⢸⠇
⠀⠀⠀⠀⠀⠀⢰⡆⣇⢸⡇⠀⣿⠀⢀⣿⠀⠀⣿⠀⠀⣽⣿⠀⢰⣿⡇⠀⣿⡇⠈⣿⠀⣿⢀⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠘⡇⣿⠀⡏⠈⣿⠀⠸⣿⠀⠀⣿⠀⠀⠀⣿⠀⢸⣿⠀⠀⣿⠀⢰⡿⠀⣿⢠⡇⣿
⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⣿⠀⣿⠆⢀⣿⠀⠀⣿⡆⠀⢾⡏⠀⠸⣿⠀⠀⣿⠀⢸⡇⠀⡟⢸⣇⡟
⠀⠀⠀⠀⠀⠀⠀⣇⢸⡅⣿⠀⣿⡇⠀⣿⠀⠀⣿⡇⠀⣼⣇⠀⢠⣿⠀⢰⣿⠀⣸⡇⢀⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⣿⡆⠀⣿⡀⠀⣿⡇⠀⣸⣿⠀⢸⣿⠀⢸⣏⠀⣿⡇⢸⡇⡙⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠈⡇⢸⠁⢻⡇⠀⣿⡄⠀⣿⡇⠀⢙⣿⠀⢸⣿⠀⢸⣿⠀⣿⠃⢸⠇⣇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠷⣸⡀⢸⡇⠀⣿⡇⠀⣿⠇⠀⢸⡇⠀⢸⡟⠀⢸⡇⠀⣿⠀⢸⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡁⠻⠧⠀⣽⡇⠀⣿⡆⠀⠻⡟⠀⢸⣷⠀⣼⡧⠤⠟⢂⣩⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⠀⣀⠀⠙⠁⠀⠛⠋⠀⠈⠉⠀⣀⠤⠖⠚⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠈⠉⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠤⠄⠀⢰⣶⣦⠈⢻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠇⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⢄⡀⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⠀⠀⠀⠀⠀⠉⠂⠄⠀⠀⠀⠤⠀⣬⢻⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠈⠀⢉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠇⠶⠟⠓⠘⠛⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⡟⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⡄⠄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠄⠠⠄⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿
⡇⠐⠒⠒⠛⠙⠃⠈⠀⠐⠛⠋⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠉⠛⠻⠿⡄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣥⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠗⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⣧⣄⣀⣀⣀⣀⣀⡀⡀⢀⣀⣀⣀⣀⣀⣀⣠⣼⣿⣿⣿⣿⣶⣶⣶⣤⣆⣀⣀⣀⣀⣤⣤⣾⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⢦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⠉⢿⡀⢸⠉⠉⠻⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⢠⠀⠀⠆⡀⢉⣭⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡷⢾⣤⣇⣞⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣍⣀⠠⠍⠛⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣄⣿⠿⢦⣬⣙⣺⣯⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣿⡿⢛⣉⡭⢟⣽⠛⡟⠛⣏⠙⡏⠻⢿⣭⡛⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⡴⠚⠉⢁⣴⠟⠡⢄⢳⡀⠀⠠⠈⠀⠀⠈⠙⢷⣄⠙⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠋⠀⠀⠀⠀⣰⠟⠁⠀⠀⠈⠄⠳⠀⠀⠀⠘⢦⡀⠀⠀⢀⡈⠂⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠈⠀⠀⠀⠙⠆⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⡧⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠄⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠨⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠂⠀⠀⠘⠷⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠟⠋⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠦⠴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⢀⣴⠟⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⢋⡁⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠛⠋⠉⠄⠀⠈⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠰⢄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⡆⠀⠀⠀⣴⠟⠹⣇
⠀⡀⠀⠀⠀⠀⠀⣴⠋⠁⢿⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠉⠉⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠻⣧
⠀⠀⠀⠀⠀⢾⣁⣀⣤⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠋⠀⠀⠀⢀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠷⠶⢦⣤⣭⣿⣶⣦⣤⣤⣀⡀⠀⣀⣈⡁⠀⣀⣀⣀⣈⣁⣀⣤⣤⠶⢾⣯⣥⣤⣴⣾⣤⣤⣤⣤⣤⣤⣦⣼⡷
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠛⠛⠋⠉⠉⠉⠉⠉⠁
⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠡⢄⠀⠀⡄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡀⠀⠱⠀⢡⠔⠉⠀⠀⠀⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠑⠢⡀⠈⠢⡀⢡⢸⠀⢀⠔⠊⢁⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⡀⠀⠀⠘⠶⠠⣈⠲⠤⢖⢁⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢒⣀⡴⠟⠴⠰⠠⢎⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡐⠂⠂⠁⢠⠊⠀⠀⠡⠀⠁⠂⢌⡐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠀⠀⠀⡰⠁⠀⠀⠀⠀⠣⡀⠀⠀⠈⠐⠄⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠂⠀⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠳⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠔⠁⠀⠀⠀⠀⠀⢰⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⣀⡀⠀⠈⠓⢄⠀⠀⠀⠀⠀⣀⡀⠠⠤⠄⠀⠐⠲⠶⠥⠤⠤⠤⠤⠤⠄⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠓⠂⠈⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠳⡐⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢭⣉⠛⠲⠦⣌⣢⡀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠉⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠾⠳⢦⣀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢚⣋⣉⣀⠀⠀⠀⠀⠀⢶⡬⣷⣄⠀⠀⠀⠀⠹⡄⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⡡⠖⠉⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠙⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠒⠒⠒⠶⠤⠈⠳⢄⡀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⡔⠒⢄⠈⢇⡀⠀⠀⠀⣿⠗⢃⠤⠒⠂⠈⠉⠁⠀⠀⠀⠀⢀⡽⠒⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠣⡀⠈⠃⠀⠉⠙⠛⠉⠁⠀⠉⠙⠒⠒⠤⠀⠀⠀⠀⠀⠴⣥⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⢤⠀⠀⠊⠁⢀⠔
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢀⡄⡘⠀⠀⠀⠀⡎⢃⣰⣁⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣠⠎⢻⢡⠀⠀⠀⠀⢈⠎⡠⢤⡧⢀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⢀⡖⠚⠛⠓⠲⢖⠒⠒⠒⢦⣀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠲⢞⠁⠀⢸⠉⠁⠀⠀⠈⠉⠐⠧⠒⡗⠠⡎
⣀⣐⣇⠱⣀⡠⠐⢈⣉⡇⠀⠀⠀⠀⢠⠙⠢⢄⡀⠀⠀⠘⠂⠀⠀⠀⠀⣀⡀⠈⠓⠹⠦⠀⢹⡀⢈⡉⠀⠀⠀⠀⠀⠑⡤⣚⣥⠵⢥
⠑⣤⡬⠀⢀⡐⠊⠥⡄⠙⣄⣀⡠⢤⠇⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠒⠉⠁⠈⢳⠶⠤⠤⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠣⣽⣡⣐⣚⣢
⠈⠃⠄⠊⠁⠈⠉⠉⠁⠀⠈⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠔⠣⡑⣤⠚⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⣢⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⡤⠠⣠⣀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣼⠀⠀⠀⠀⠲⠓⠒⠳⠶⠆⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡹⢮⡧⡽⠒⠤⢄⣀⠄⠀⢸⡆⠀⠀⢀⣀⠴⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⠭⢄⣐⣒⡒⠒⠀⢸⠄⠀⠀⢏⣉⠉⠉⠀⠀⠀⠈⢀⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣎⣰⣀⣀⣀⣐⣄⣡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⡨⠅⠈⣀⠄⠀⠀⡠⠀⠀⠀⠀⢄⠀⠀⠤⡀⠀⠩⢁⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢊⣁⠐⣈⣁⣀⣐⡊⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⢑⣀⣀⣈⠑⠀⢉⠰⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⠀⠠⠤⠤⠤⠤⠤⠀⠀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣀⡀⠀⠀⠀⠤⠤⠤⠤⠄⠀⡖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠁⠈⠀⠘⠓⠐⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠒⠂⠀⠀⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡩⠭⢉⠉⠉⠁⢀⣀⣒⠐⠒⠒⠒⢒⡒⠒⠒⠒⠒⣀⣀⡀⠉⠉⢉⡍⠭⢍⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢⢄⣸⠀⡇⠀⠀⡇⠀⠀⡇⠀⠈⡇⠀⠀⠊⠀⠈⡇⠀⠀⢸⠀⠀⢙⠀⠀⠈⠀⠀⠈⠀⣇⡠⠔⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠏⠑⠢⣸⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣇⠔⠊⠱⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⢆⡀⠈⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⣆⠀⠁⠀⡴⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡙⠠⡄⠀⠀⠀⡇⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⡇⢠⠔⢋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡜⢰⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢠⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⡇⠀⠀⢀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡇⠀⠀⠸⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⡇⠀⠀⢈⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡑⠒⠃⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠐⠒⢊⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠤⣈⠑⠢⠤⡀⣈⠓⠒⠁⠀⠀⠠⡀⡠⠇⠀⠀⠈⠒⠊⣀⣀⠤⠔⠊⣁⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡉⠑⠂⠤⢀⡈⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⣁⡀⠤⠐⠊⢁⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠠⠤⢀⣉⣉⡁⠀⠀⠀⠀⠀⣈⣉⣉⡁⠤⠤⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢰⣿⣯⣭⣭⣭⣭⠙⣿⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⠉⠉⠠⠾⠛⠀⠀⠀⠀⠀⠀⠛⠧⠀⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣶⣤⣤⣤⣀⣐⣒⣒⣒⣒⣀⣀⣤⣤⣤⣶⣾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠽⠟⣒⣲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣙⣛⠛⠿⠷⠶⠶⠶⠶⠶⠶⠾⠿⠛⢛⣉⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠉⢻⣿⠛⠻⢷⣶⠶⠶⢶⣾⠟⠛⣿⡟⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⡏⠀⠀⣿⡇⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢸⣿⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⠇⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿⠀⠀⢸⣿⠀⠀⣼⡇⠀⠀⣿⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣿⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣿⡄⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⡟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣧⣤⣼⣿⣀⣀⣿⣧⣤⣼⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠄⢀⡠⣤⠴⢁⣉⣑⡲⠶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣭⣍⣉⡀⡄⣀⣔⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣩⣿⡛⢻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠁⠈⠁⠀⠉⠑⢠⡀
⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢰⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠸⠉⠃⠀⢠⣿⣿⣧⣀⠀⠀⢀⣠⠤⠒⠈⠉⠉⢻⣇
⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⣿⣿⣿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠿⢿⡇⢀⠖⡒⠤⢀⡀
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣸⠗⠊⠀⢃⠀⠀⠈⠁⠀⠉⠉⠉⠉⢰⠄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡮⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⢰⡀⠈⠑⠀⠾⠤⠤⠤⠄⠀⠀⢤⣶⠃⢠⠃
⠀⠀⠀⠀⡇⠀⠀⠀⢀⣴⠞⠛⠛⠛⢛⣿⡟⠻⠿⢿⣷⣶⣶⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⣴⣿⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠘⡉⠙⠛⠛⠋⠉⢁⠃⠀⠀⠀⠀⠈⠉⠉⠉⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⢀⣼⠂
⠀⠀⠀⠀⡇⠀⠀⠀⢠⠃⢒⡊⠉⡑⠒⠀⡆⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣶⡿⢿
⠀⠀⠀⠀⡇⠀⠀⠀⠈⡄⠈⢉⣡⣤⠄⠀⣷⣦⣤⣤⣤⣤⣄⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⠀⢀⣠⣧
⠀⠀⠀⠀⠉⠐⠒⠒⢲⣁⠀⠀⠀⠀⠀⣀⣿⠿⠿⠿⠿⠿⠿⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠚⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠒⠒⠒⠒⠒⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣼⣥⣤⣤⣼⣥⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⢶⡆
⠀⠀⠀⠀⠀⠀⠀⣶⢶⣤⣸⡇⠀⢸⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⠁⠀⣹⣿⡇⠀⢸⣇⣤⡶⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣄⠈⢻⡇⠀⢸⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⣼⡟⠁⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣼⣧⠀⢸⣿⣧⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⣿⣧⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⣿⣿⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⠀⠘⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡇⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢸⣿⣿⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⣼⣿⣿⠀⢸⣿⣿⠀⠀⣿⣿⠀⢀⣿⣿⣧⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣉⡙⠛⠻⠿⠿⠿⠦⠴⠿⠿⠿⠟⠛⢋⣁⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⣶⣶⣶⣶⣶⣶⣶⠶⠾⠿⠛⠋⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⣀⡀⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⠀⠶⠘⠀⠀⠀⠘⠂⠇⠀⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢒⠩⠐⠀⢁⠄⠊⠁⡠⠊⠀⠀⠀⠈⠣⡄⠈⠐⠠⡉⠀⠊⠅⡒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣉⡀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠤⠤⠤⠤⠤⠤⠄⠀⠀⠐⠒⠒⠒⠐⠀⠀⢈⣙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠬⠭⠭⠥⠄⠉⠉⠉⣉⣉⣁⣀⣈⣉⣉⡉⠁⠈⠉⠠⠬⠭⠭⠥⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠐⣒⣒⠒⠒⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠒⠒⣒⡒⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⣀⠸⠀⠁⠀⠈⠀⠀⢰⠁⠈⢢⠀⠀⡔⠀⠀⠆⠀⠀⡞⠁⠈⡆⠀⢰⠁⠀⢸⠀⠁⣀⠠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢐⠐⠤⣙⠀⡄⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⣋⠤⠂⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢢⡀⠈⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠁⢀⡔⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠢⡄⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠃⠀⢸⠀⠀⠈⢨⠔⢉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡆⡃⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⡀⢰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⢠⠀⠀⡇⠀⠀⡇⠀⠸⠀⠀⢸⠄⢠⡇⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⡇⠀⠀⡇⠀⠠⠀⠀⢠⠄⠸⠇⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠂⢰⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⡇⠀⠀⡇⠀⠸⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠣⠤⠀⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠘⠤⠜⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢌⡑⠢⠤⣀⡀⠑⠠⠊⠀⠀⠸⡀⢀⠇⠀⠀⠓⠄⠂⢀⡀⠤⠐⢊⡡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢌⠑⠂⠠⠌⣉⠐⠂⠐⠂⠒⠀⠀⠀⠂⠂⠒⢂⣉⠡⠄⠒⢊⡡⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠠⠤⢀⣉⣉⣀⡀⠀⢀⣀⣈⣉⣉⠠⠤⠀⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡶⣦⠖⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⢿⡛⢯⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡟⢿⡉⢧⡈⣧⠈⣧⠹⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢀⣴⣾⡿⢿⠹⡆⢻⡄⢳⡈⣧⢘⣆⣼⣶⠿⠛⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⣿⡏⢳⡘⣇⠹⣆⣻⣤⣷⡾⠛⠋⠉⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡌⢷⣼⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⢀⣄⣠⡀
⠀⠀⠀⠀⠀⠘⢯⣻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠘⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣿⣇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡛⢦⡈⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠙⢦⣙⣷⠿⣿⣧⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡿⠋⠀⠘⣿⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠸⣿⡆⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠙⠂⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⠇⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠈⣿⣇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣿⢷⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢸⡷⠆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣀⠀⠀⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣼⣿⣿⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⡇⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣿⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣿⡟⠀⠀⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⣿⡟⢀⠀⠀⣿⣿⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠉⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣸⣿⣤⣴⡶⠾⠷⢶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢻⣿⠫⠄⠐⠋⠀⠓⠀⠿⢉⡻⣷⡄⠀⠟⠀⠀⠀⢀⣾⡿⠁⠀⠾⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠁⣌⢻⣆⠀⠀⠀⢠⣿⡿⡿⠿⡿⠷⣦⣀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⡄⣀⣴⣿⠋⠀⠁⠀⠁⠒⢊⡻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣄⣀⣀⣀⣀⣥⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⣿⣏⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⣶⣦⣤⣴⣶⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣀⠤⠀⢀⣀⣀⣭⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⠿⠿⣛⣿⠟⠋⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣿⠛⠹⠋⠁⠀⣸⣿⡿⠛⠦⢄⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣥⣄⠀⢠⣬⢥⡀⠀⠀⠀⠈⣙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⡿⢿⠿⠭⠽⠶⢿⡆⠀⣴⠟⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣭⣽⣿⣦⡀⣀⢀⣄⢠⣿⡺⣇⢰⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣍⣻⣿⣿⡾⣇⢸⡆⢹⠀⢸⡆⣿⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⢸⡇⢸⠀⢸⠁⣿⢸⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡷⣿⢸⡇⢸⡄⢸⠀⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡏⡇⢸⡇⢸⢨⡟⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣻⣷⣿⣿⣿⣿⣻⡇⣇⢸⡇⢸⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠡⢾⢿⣿⣿⣿⣿⣿⣿⣿⣇⣿⠀⠇⢸⢸⣿⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢪⣿⣿⣿⣿⣿⣿⣿⣿⢹⠀⠁⠘⢨⠆⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⢿⣿⣿⣿⣿⣽⣝⠀⠀⢀⡄⢶⡂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣾⣿⣿⣇⠀⠀⢸⣇⣨⡵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠈⠁⠀⠀⡤⠀⠒⠀⠒⠀⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡠⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢾⣉⣹
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢀⡞⣩⣉⣓⠲⠦⢯⣌⣇
⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⣠⠏⡴⣣⠴⢭⠉⠛⠒⠦⠬⣍⣛⡦
⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⠀⠀⡴⢣⠞⠁⢻⠶⠚⣇⠀⠀⠀⠀⠸⡌
⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⠀⠀⠀⢠⠞⣱⠋⠀⠀⠈⡧⠶⠚⠁⢀⣤⡀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠿⠼⣧⢰⡏⣉⣧⠀⠀⠀⠀⣯⡤⠿⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠴⠖⠒⠦⠼⠚⠛⠥⠼⠒⠲⠤⠴⠚⠲⠤⠿⠒⠺⠦⠴⠒⠲⠤⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠠⠖⠲⢤⡤⠖⠲⢤⡤⠖⠲⢤⡤⠖⠶⢤⡴⠖⠶⢤⠴⠖⠶⠄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠰⠦⣤⣤⠴⠦⣤⣤⠶⠦⣤⣤⠶⠦⣤⡤⠶⠦⣤⡤⠶⠄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠤⢤⣄⣠⠤⢤⣀⣠⠤⢤⣀⣠⠤⢤⡀⠀⠀⠠⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠦⢀⣀⠀⡀⠀⠀⠀⣀⠀⠤⠐⠃⠀⢀⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⢀⠀⠒



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢙⣢⣄⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⣴⣶⡶⠂⠁⠈⠳⣶⡾⠒⠲⣶⠞⠀⠁⠀⠀⠀⠀⠀⠈⠀⣩⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⣿⠉⢙⠧⣠⠞⠁⠄⠀⢀⣀⣈⠟⢓⡸⠛⢿⣀⢠⣄⡚⡆⠀⠀⡔⠒⢏⡹⡟⢻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣯⡗⢄⡄⠁⢾⡀⢻⣤⢀⣤⠾⡁⣀⡠⠐⠛⠁⠐⢾⢍⡉⠛⡻⢄⢄⡴⣋⠶⣏⣠⣏⣹⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢿⣷⣤⣌⣭⠛⣚⢛⡇⢀⡀⠉⠖⣩⡤⢤⣤⠼⠄⠉⠧⠄⠠⠿⠤⠾⠛⠃⠀⠉⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡾⠻⣿⣿⣏⠹⢿⣿⣷⠛⢛⣿⡿⠛⢻⣿⡛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⣿⠱⣟⣷⡟⢣⡘⣷⣛⡯⠙⣶⡋⠀⠙⣶⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⢽⣷⡏⢉⡟⣭⠿⠒⢽⣷⠊⠀⠙⣤⠞⠀⢗⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣸⠏⠱⣬⡜⠋⣳⣌⠾⡯⠳⣄⠞⠉⠳⣀⠾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⢿⣿⣶⠛⣄⠐⣼⠻⣇⣟⣵⠽⢦⢔⣶⠛⣶⢏⡁⠀⠀⠀⠀⠀⠀⣀⢤⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⠉⠐⣬⣟⠀⠢⣼⣿⡋⢀⣼⣿⡀⠀⣼⢿⣗⣂⠀⠀⣀⡴⢮⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣴⡇⠱⣴⣋⠘⣷⣟⡏⠀⣳⣿⠁⡒⣻⡞⠁⠈⣽⣷⣢⣾⣿⣇⣈⢳⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣧⡜⠙⣶⡴⢃⡹⢦⡟⢧⠹⣇⠾⠭⠙⢦⠞⣿⢹⣧⡟⠯⣷⣼⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣿⣷⣠⠾⢇⠀⡠⠛⣦⡀⡴⠋⢳⣶⣶⣟⢷⣆⡷⠋⣧⣳⣾⢿⣡⣿⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⡾⣏⢹⣹⣿⡵⠖⣤⠿⣞⣂⣩⠟⣴⡽⣷⡿⣖⠚⣿⠿⣿⣰⢿⣿⠀⠀⠀⣠⢾⠖⣾⣏⣹⠗⢚⣏⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⣿⣁⢸⣷⣛⠳⣿⣿⢉⡒⢪⣿⡛⠎⢙⣿⡿⠀⢈⢿⡷⢃⣿⡟⢿⣟⣀⡀⢼⠤⡶⠲⣅⠰⡆⣽⢁⡝⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡿⣧⣜⠤⠿⣴⠋⣈⢳⣶⠋⢈⠻⣴⠎⣱⠛⣴⢫⢘⣷⣞⢠⣿⡞⣻⠀⠘⠻⠶⣀⣦⠸⣄⣙⣛⡈⠒⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣷⣀⣼⡙⢶⣲⠞⠹⣧⣠⠟⣹⣟⣱⠊⠈⢧⣴⣏⠻⣿⢿⢷⡯⠀⠀⠀⢠⡀⠸⢃⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⣷⣫⢻⣄⢈⠝⠳⣷⢦⡜⠻⣌⣤⠼⠿⣤⣤⣿⡿⣇⣴⢛⣾⣾⡷⣊⣒⠤⢼⠏⢀⣥⠄⢠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠞⣦⡤⠟⠋⠉⣼⠘⡾⠿⡂⠁⡼⢯⡙⣁⡾⢧⣤⣄⡼⢧⡸⣷⡟⢳⣡⣾⣿⡏⠉⠉⠀⠀⣠⡿⢦⠊⠒⠁⣿⢽⠦
⠀⠀⠀⠒⠒⢈⠀⠀⠈⠁⠀⠀⠀⠀⠈⠙⠒⣶⣿⣿⡅⠀⡹⢎⠀⠈⣿⣯⠖⠀⣹⣏⠣⢄⡿⢯⣿⣿⢿⣧⡿⠁⠀⠀⠀⠀⠙⠓⠫⢳⡴⠶⠆⠚⠃
⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⢀⡀⠔⠛⠻⢿⣯⣻⣾⡧⢤⣷⡞⢇⡉⣳⣞⣁⣨⡷⣿⡖⣾⣿⡷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠳⠾⠿⠿⠷⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡹⠖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⠀⠀⠀⠫⣱⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠈⣀⢤⠀⠀⠀⠀⠰⡁⢠⣤⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡔⠊⠉⠙⡶⠊⠀⣾⠤⡶⠋⠳⡄⠀⣇⠣⡀⢖⣺⣆⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⢠⠃⢧⡎⠉⠉⠀⢀⠤⠾⠤⣀⠀⢣⣀⠧⠀⢀⠗⢉⣉⣣⠈⠉⠉⢸⠀⠀⢈⠇
⠀⠀⠀⠀⠀⡱⢀⠎⠀⠀⠈⠑⢦⡀⠃⠀⠀⠀⠀⠱⡤⠝⠒⠚⠁⠚⠁⠀⠈⠑⣄⣀⣸⠶⠦⣌⠀⠀⣰⢯⡲
⠀⠀⠀⠀⠘⢅⣼⡠⠤⢄⣀⠀⣀⡧⠤⠤⠄⢀⣀⡀⠀⢀⡿⣳⡠⠄⠀⠀⠀⠀⠈⠉⠉⠓⠀⢸⠞⠉⠁⢻⡀
⠀⠀⠀⣠⠔⠁⠀⢀⠀⠀⠀⠉⠁⠀⠀⢀⣊⠁⠀⠈⠙⢮⣀⣙⣵⠀⠀⠀⠀⠰⡶⠤⠶⡆⡤⠿⠶⠦⢄⣼⣁⡀
⠀⠀⡰⠁⢀⠔⠉⣠⡤⢤⣀⠠⠒⠑⢦⡁⢀⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠓⠒⠲⡇⠀⣀⣄⣀⠀⣁⣀⡹
⠀⠀⠳⠶⠗⠊⠉⠳⣀⡠⠴⠓⠄⠒⠒⠺⠥⣄⣀⠤⠔⠊⠉⠉⠀⠀⠉⠉⠒⠦⠤⠤⠤⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠂⡁⠀⡆⢀⠀⡀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⢂⠂⢠⡘⣗⣀⠀⡀⠀⠁⠄⠀⠈⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢑⠚⡷⠞⡀⠁⡀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡤⠶⢄⠀⢀⠴⠦⡄⠀⡠⠶⢤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⢸⡔⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⡏⢣⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⡇⠈⡆⠀⠀⠈⢻⠟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⡇⠀⠹⡀⠀⠀⠀⢣⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⡇⠀⠀⢧⠀⠀⠀⠈⡆⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣧⠀⢸⡄⢸⠀⠀⡇⢰⡇⠀⣾⠀⢸⡇⠀⠀⠘⡆⠀⠀⠀⢳⣴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢿⠀⠘⡇⢸⠀⠀⡇⢸⠃⠀⡿⠀⢸⠁⠀⠀⠀⢹⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣀⣠⠃⠸⣄⣀⠇⠸⣄⣀⠇⠀⢸⠀⠀⠀⠀⠀⢇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠜⠀⠀⠀⠀⠀⠘⣆⡼⠁

⠀⠀⠀⠀⠀⢀⠤⠤⠤⠀⠀⠒⠒⠀⠀⠀⠤⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡎⠐⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠁
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⢀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⡠⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠤⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡇⠀⠀⡈⢠⣄⡄⠀⢠⡄⣤⡀⡤⢠⢠⠄⢸⠀⢠⠀⢀⠀⠀⠒⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⢰⠀⠀⣁⢸⠟⡇⠀⣯⡇⡹⡀⡇⢸⠇⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⠤
⠀⠀⠀⠘⡀⠀⠹⣘⣀⣓⣂⣃⣁⣙⣁⣁⣘⣘⣃⡼⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠄⠀⠀⠀⠀⠇⠀⠀⣀⡄⠘⠉⠀⠀⡠
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠠⠀⠠⠊⠁⡤⠀⠀⠠⡞⣀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⢀⣣⡀⠀⠀⠀⠀⣎⡋⠀⠀⢟⣱⠀⠀⠀⠀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀⢀⡠⠜⠁⠀⠀⠀⠈⠳⠶⠮⠃
⠀⠀⢸⠇⠀⠈⡇⠀⠀⢰⡏⠀⠀⠀⠀⢸⡆⠀⠀⡏⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢸⡃⠀⠀⡇⠀⠀⠀⠻⡦⢴⢲⡶⠟⠁⠀⠀⡇⠀⠀⣿⠀⠀⠠⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠁⠈⠀⠈⠁⠀⠀⠁⠀⠠⠤
⠀⠀⠸⣅⣀⣠⠟⠒⠂⠤⠤⠠⠤⠍⠀⠤⠄⠒⠉⣇⣀⣀⡝⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠉⠀⢆⣀⠀⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣤⣤⣶⣶⣂⣀⣠⣤⣴⣾⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⠿⠿⢋⣽⡿⠛⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣿⡟⢻⠿⠛⠁⠀⢀⣿⣿⡿⠿⠦⣤⢀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣿⣇⣁⠄⠀⠀⡐⣫⡉⠁⠀⠀⠀⠐⡋⠭⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣾⣛⣋⣻⣆⠀⠀⢠⣴⣽⡻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⣿⢿⣿⣿⣿⣿⠉⠉⠉⠉⠉⢿⣇⣠⡿⠋⢃⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣷⣾⣿⣿⣿⡤⢤⠠⢶⡀⠾⣿⠽⣷⠼⣷⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢵⣽⣿⣿⣿⣿⢿⡇⢸⡇⢸⡇⠀⢻⡄⣿⠂⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣧⢸⡇⢸⡇⠀⣿⠀⣿⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⢸⡇⠈⡇⠀⣿⢀⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣭⣿⢼⡇⠀⡇⠀⣿⢸⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣖⣇⠀⣿⠀⣿⢸⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⠀⣿⠀⣿⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠠⣺⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⠀⢹⠀⣿⢸⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠋⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⠘⠀⠿⣸⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣟⣟⣿⠽⠁⢤⣀⣀⣭⡴⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠐⣿⡂⢸⣇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣤⣤⣤⣿⣷⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⢠⠀⠈⠀⠈⠆⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠂⠁⠀⠰⠠⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠄⠀⢠⠠⡀⠈⡆⠀⠀⢀⠀⠀⠀⢡⠀⠀⠈⠀⡠⠀⠀⠃⢄⠄⠀⠀⠰⠀⠠⠀⠀⠀⡔⠢⢀⠀⠀⠀⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠄⠀⠀⠐⢌⠄⢀⡠⠈⠂⠀⠀⠀⠀⡟⢀⢈⣉⢲⠀⠀⢀⠀⠐⢀⠀⠁⠐⠀⠀⢀⡇⠀⠈⠑⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⣀⠀⠠⢀⡍⠁⠀⠠⠊⡁⠀⠐⢒⢠⠀⠨⠐⠃⣀⣐⣐⣙⠦⠀⠀⡀⠁⠀⠠⣀⠀⠁⢉⢃⠈⠀⠀⡃⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⠂⠈⠀⠀⢅⠀⠠⠀⠀⠀⠀⠀⠠⠈⠅⠁⠁⠠⠄⠚⠁⠀⠀⠀⠈⠗⠀⠭⠓⡐⠒⠈⠀⠢⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠳⢶⣤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⠋⢰⠶⠂⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⠤⠭⠅⡳⢄⠈⡑⠛⠯⣒⠲⠤⠤⠤⠤⣤⣄⣠⠤⠤⠴⢔⡒⠭⠐⡋⢞⢤⡖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠒⠲⠤⢤⡈⠀⠄⠀⠀⠀⠄⠁⠈⠁⠒⠒⠒⠂⠒⠚⢦⣉⠁⠶⢠⠘⢂⣙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⡄⠀⠉⠀⠀⠀⠀⠀⣰⠀⠀⠠⠀⠀⠀⢠⡀⠀⠀⠹⡀⠞⠪⠐⠅⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⡨⢂⢵⠌⠂⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠦⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢕⠀⡊⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢅⠈⠂⢀⢐⢈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⢀⠄⣠⣾⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠀⠈⠀⠈⢄⠢⠀⠀⠉⠐⠂⠤⠠⣄⣀⣀⣀⢀⣤⡄⠵⠦⠖⠚⠉⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠍⢙⡒⠂⠀⠀⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⠀⠀⠠⠀⢠⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢆⠉⠁⠈⠉⠀⠀⠀⠈⠉⢩⡥⠤⢼⠀⠰⢦⠉⠻⣷
⠀⠀⠀⠀⠀⠀⠔⡯⠿⢍⢑⠒⠤⢄⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠒⠚⠷⠬⠑⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⣈⠦⠀⠀⠨⢉⡀⠀⠀⠁⠐⠒⠠⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⡏⠑⠒⠒⠒⢓⣂⡩⠖⠀⠠⠄⢀⠀⠀⠀⠈⠉⠉⠒⠒⠒⠀⠠⠄⠀⠀⣤⣤⡼
⠀⠀⠀⠀⠀⠀⡇⠀⢀⢠⠐⠫⠤⣭⡓⡄⠀⠉⠀⠀⠠⠤⠐⣒⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢛⡷
⠀⠀⠀⠀⠀⢠⡇⠎⠔⠹⠿⠿⠿⢯⡑⡇⠀⠀⡰⠊⠁⣀⢄⠘⣅⡀⠉⢹⠃⠐⢦⡀⠀⡽⠁
⠀⠀⠀⠀⠀⢠⡠⡄⢄⣀⣸⣀⣀⠴⣣⡇⠀⠀⡇⠀⠉⠀⠀⡶⠀⣅⡀⢸⠀⠀⠀⣱⠐⠁
⠀⠀⠀⠀⠀⢸⠇⠈⠓⠒⢶⠛⠒⠊⠀⡇⠀⠀⡇⠀⠀⠠⠀⡇⠀⠈⣇⡇⠀⠀⣠⠃⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡼⠀⠀⠀⢠⡇⠀⢠⠁⠀⠀⢀⢸⠁⠀⢸⠀⠀⠀⣴⠃⣸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡌⡞⠀⠀⡏⠀⠀⢰⡏⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⡇⠀⢸⠃⠀⢀⢣⠇⢸⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡇⠀⠀⠀⣸⠀⠀⡏⠀⠀⢠⢃⡇⠀⡞⠀⠀⡎⢸⠀⡞
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠃⠀⢸⠀⡇⠀⠀⡇⠀⠀⡘⢸⠇⢠⠃⠀⣸⠁⡇⢀⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⢀⡇⢸⠁⠀⢸⠇⠀⢠⠇⡞⠀⢸⠀⡰⠃⢸⠁⣸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⠀⡸⠀⣼⠀⠀⢸⠀⢀⠎⢰⠃⠀⢸⠊⠀⢀⡏⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⡴⠁⢀⡇⠐⠃⠈⡗⠁⢀⡇⠀⠀⠘⢦⡤⠎⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠒⠤⣀⣀⠉⠛⠉⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⣀⣀⡠⢴⠃
⠀⠀⠀⠀⠀⠀⠑⠒⠲⠤⢭⣉⣐⡒⠒⠒⢒⣒⣒⣒⣒⣂⣉⣩⠭⠥⠖⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠉⠀⢄⡀⠀⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣤⣤⣶⣶⣂⣀⣠⣤⣤⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⠿⠿⢋⣽⡿⠛⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣿⡟⢛⠿⠛⠁⠀⢁⣿⣿⡿⠿⠦⣤⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣿⣃⣀⠀⠀⠀⣐⣫⡉⠁⠀⠀⠀⠀⡋⠭⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣚⣿⣋⣻⣆⠀⠀⢠⣤⣽⡛⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⣿⢿⣿⣿⡿⢿⡉⠉⠉⠉⠉⢿⣆⣠⠟⠛⠂⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣾⣿⣿⣿⡤⢤⠠⢶⡄⠾⣿⠽⣷⠘⣷⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢥⣭⣿⣿⣿⢷⢻⡇⢸⡇⢸⡇⠀⢿⠀⣿⠂⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⣿⣿⣿⣿⣿⣧⢸⡇⢸⡇⠀⣿⠀⣿⢸⣯⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣭⣿⢸⡇⠈⡇⠀⣿⢀⣿⢸⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣭⣿⠼⡇⠀⡇⠀⣿⢰⣿⣼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣿⣿⣿⣿⣿⣿⡶⣇⠀⣿⠀⣿⢸⡿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⣿⠀⣿⢸⣷⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠤⢚⡿⡿⣿⣿⣿⣿⣿⣿⣿⣽⣧⣿⠀⢹⠀⣿⢸⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡜⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⢾⡀⠈⠀⠿⢸⡧⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⣟⣟⣿⠝⠁⠠⣀⣀⣭⡶⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣿⣿⡛⠀⠀⠐⣿⡂⢸⣇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⣿⣿⣿⣿⣿⣤⣤⣤⣿⣷⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢂⣀⣤⡤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠡⠞⠋⠉⠀⠴⠶⠿⠷⠶⠶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⡛⢭⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠒⠒⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣸⣀⣒⠶⠒⠀⠀⠐⢰⣛⠋⠉⠉⠙⡸⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡇⠀⢀⡀⢨⡥⠍⠉⠓⠊⠉⠀⠀⠈⠉⠉⠙⠣⠒⠈⠁⡲
⠀⠀⠀⢰⠁⠀⠁⠉⠀⠀⢄⡀⠀⢳⡀⡟⠲⢦⠤⢤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢄⣀⠇⠀⡜⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠙⠢⠴⠧⠔⠚⠉⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⣀⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢼⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠯⣉⠻⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠛⠦⠞⠩⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⡀⢙⡖⠃⠀⢄⠐⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢀⠀⠉⠉⠉⠿⠄⠀⢀⣀⠀⣀⣀⡀⠀⠀⠤⠀⠚⢻⣦⡀⠀⢀⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⠀⠀⠀⠦⢀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡄⠀⠈⢂⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⢶⣶⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢰⡀⠀⢠⠀⠀⡀⢸⡀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⢹⣀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠃⢀⣸⠀⢸⠁⠘⠁⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡆⢺⢘⡏⠈⢧⣄⣀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⠁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠈⠉⠁⠀⡾⡸⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⢇⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠀⠀⠀⣀⣂⣼⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢀⠀⣧⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠉⠙⢹⠋⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡸⡀
⠀⠀⠀⢀⣀⠤⢤⣾⣤⣄⣀⣀⣀⣀⣏⣀⣀⣬⣦⢤⡤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣧⠟⠀⠀⠀⠀⠀⠀⠾⠻⠟
⠀⠀⢰⣇⣀⣀⣀⣸⣦⣟⡦⠬⠟⠋⠛⠿⢶⣮⠥⠾⣄⣀⣠⡼⠃
⠀⠀⠀⠉⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠺⠂⠀⠀⠀⠀⠀⠀⠒⠓⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠀⠄⠠⠠⠀⠠⠐⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠂⠠⠀⠆⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠛⠛⠛⠛⠛⠛⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⢰⣶⡄⠀⠀⢠⣶⡆⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠭⠌⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡉⠟⠋⣀⣸⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⣿⡇⠀⠀⢹⡿⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣀⠀⠀⣠⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡯⠑⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃⠼⠃⠀⠠⠜⠳⢹⡿⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣅⣸⡟⠠⠀⠀⠀⠀⠀⢀⢸⣿⡄⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢡⡿⣿⣷⣤⣀⡀⠸⠆⣀⣾⠊⣿⣗⢻⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢰⣱⡏⠀⠈⠙⢻⠞⠳⣿⠃⠀⠈⢻⣆⡄⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠈⣿⠁⠀⢉⠀⣾⣄⣁⣏⠠⠶⢒⣬⡿⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠹⢷⣤⡴⠎⠿⠉⠁⠟⠀⠰⢼⡟⠁⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⡀⠀⣸⣿⣀⡀⠀⠀⠀⢠⣤⡤⠾⣇⠀⢸⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⢿⡀⣿⣆⣉⣉⣀⡉⠀⠀⠀⠀⠐⣏⣄⠀⣼⣿⣏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣿⠁⠘⣷⠏⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠹⣾⠀⠋⣿⣿⡸⣧
⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠏⢰⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠈⡇⢸⣿⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢹⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢳⠈⣿⢀⡟⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠗⢹⡄⢸⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢶⡁⣸⠓⡄⠀⠈⠷⣤⣶⣾⣷⣶⣦⣤⡴⠟⠉⠀⠀⢿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇⠀⠈⠛⠶⣤⣈⠙⠛⠿⠟⠛⠉⠀⠀⠀⠀⠀⣨⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡿⠷⠶⠶⠶⠶⠿⠿⣷⣶⠶⠶⠶⠶⠶⠶⠶⢛⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⣿⠲⠒⠒⢶⣖⣚⠛⣿⠛⢓⡖⠒⠒⠒⠚⢟⣽⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠙⢻⣿⠇⢸⡿⠀⢀⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡼⠀⢸⡿⠀⠈⡇⠀⢀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣇⠀⣾⠃⠀⠀⡇⠀⣸⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠟⠀⡟⠀⠀⠀⡇⠀⠃⠀⠀⡿⠀⠀⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠇⠀⠀⠀⠇⠀⠀⠀⠸⠇⠰⣶⣅⣠⣄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡉⠉⠛⠛⠷⢶⣦⣴⣟⣉⠁⠈⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⢶⣦⣤⣀⣈⠉⠙⠛⠻⠶⣶⣽⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⢶⣤⣤⣀⡉⠉⠛⠛⠷⠶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⢶⣦⣤⣀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⡀⠀⠀⣀⠀⠀⣀⡀⠀⢀⡀⠀⠀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⠀⠀⣿⠀⠀⣿⡇⠀⢸⡇⠀⢸⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿⠀⠀⣿⠀⠀⣿⡇⠀⣸⡇⠀⢸⡇⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀⠀⣿⡀⠀⣿⡇⠀⣿⡇⠀⣼⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡇⠀⣿⡇⠀⣿⡇⠀⣿⠃⠀⣿⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢸⡇⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⣿⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣇⠀⢸⡇⠀⣿⡇⠀⣿⠀⢸⣭⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⠀⢸⡇⠀⣿⡇⠀⣿⠀⢸⡗⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣿⠀⢸⡇⠀⣿⡇⢀⣿⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠿⠀⠸⠗⠀⠿⠇⠸⠿⠀⠾⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣗⡠⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⠋⠈⠩⢛⡶⣻⣇⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣾⡷⠁⠀⣀⡸⢟⣯⡿⠏⣿⣼⡟⢦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣭⡀⠀⣀⣴⣷⣾⡿⠫⠀⣰⡿⢻⣷⣿⣧⣥⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢢⡼⢿⣿⣯⣿⡵⡟⢿⠏⣴⣶⣿⣷⡾⠏⠱⣿⣿⣭⣭⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡴⣟⣳⣟⣻⡏⢱⢼⣿⢳⡿⣷⣷⣿⣿⣿⣿⢃⣼⣤⣼⣿⣿⣿⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢴⡟⢾⡿⠃⣿⣙⠷⣼⢻⣿⣿⣱⣷⣾⣭⣚⣿⣿⣷⣿⣿⣦⡤⢸⡿⠿⢿⣿⠯⡀⢠⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠏⠙⢿⣤⣇⡄⠠⣶⣿⣭⣟⣛⣿⣏⣭⣴⣨⣝⣻⡿⢝⣿⣿⢁⣀⣽⣇⠀⣾⠇⠀⠀⠀⣧⡼⢿⣿⣿⣦
⠀⠀⠀⠀⢠⣾⣛⡀⠀⣠⠞⢹⣿⡇⣿⣿⡿⢷⣶⣭⣿⡿⡌⠉⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣶⣤⣣⣿⠿⣴⣤⣭⠃
⠀⠀⠀⢠⣤⣿⣿⣿⣯⡋⢀⣨⡟⢳⣹⣿⣿⢿⣿⣷⣿⣿⡇⠀⠛⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣇⠂⠈⠿⣦⡀
⠀⠿⠻⠟⠿⠶⠶⠿⠟⠛⠛⠿⠗⠛⠛⠛⠁⠘⠙⠉⠉⠙⠈⠛⠐⠋⠛⠉⠻⠷⠾⠟⠻⠗⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠿⠿⠿⣿⣶⣦
⣿⣿⠿⠿⡿⠻⠿⢿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠸⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⠀⣀⡀⠀⠀⠀⠀⠑⠉⠁⡝⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠒⠲⣀⠀⣀⠀⠀⠉⠁⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠂⡍⠀⡱⠒⠀⠀⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣾⣚⣛⡛⠓⠦⣄⠀⣠⠔⠋⠁⠀⢹⠀⢺⠀⢀⠖⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⠈⠉⠒⣬⣿⣧⠀⠀⠀⢀⡿⠀⡤⠃⠈⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠒⠒⠒⠒⠒⠛⠛⠂⣀⠉⠉⢇⣀⣀⡠⠔⠛⠀⠸⠒⣋⣉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠐⢲⠤⠤⡤⠔⠒⠉⣀⠀⠀⠈⠑⠀⠀⠀⠀⠀⣤⣈⣉⠡⠔⣇⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠇⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠘⠋⠀⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢖⣂⣉⡦⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣓⠭⡎⠑⠢⠄⡈⠢⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠉⢡⡀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠸⡖⠒⠁⠀⠀⠀⠈⠒⢄⠀⠀⠠⣴⡒⣉⠀⠀⠀⠀⠀⠉⠐⠊⠁⢠⣤⣶⡏⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡈⠀⠀⠀⠀⠀⣠⣳⠀⣰⠕⢱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣦⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⣂⠴⡖⠚⢂⠖⠙⠋⣠⠔⠉⠉⠉⠉⢉⠿⠧⠄⠀⠀⠀⢰⠠⡼⠔⣣⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣗⣈⠗⠀⠀⠀⢀⡏⠙⠒⢦⣀⣮⠗⠀⠀⠈⠉⠁⠑⠤⣭⣭⡇⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠃⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⡀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠉⠓⠒⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣣⢣⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⣀⠖⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⢨⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⢀⠁⠈⢉⠉⠉⠉⠩⡉⠉⠍⡀⠀⢀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠁⠀⠀⠀⠀⠊⠀⠀⠐⡁⢀⣀⣀⣀⡈⠂⠀⠀⠐⠀⠀⠀⠀⠉⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠐⢒⣒⣒⡀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣒⡒⠂⢰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠤⠤⠤⠄⠄⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡇⠈⠉⠁⠀⠀⠐⠀⠐⠀⠀⢠⠒⠒⡄⠀⠀⠂⠀⠂⠀⢰⠈⠉⠁⠸⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢐⠂⢌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⢸⡡⠐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⠀⠀⡆⠈⠀⠐⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠒⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⠀⠀⡇⠔⡁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢸⠀⠀⠀⠀⢨⠀⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⡇⠀⢸⠀⠀⠇⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠤⢂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢈⡐⠠⠀⢀⡑⠒⠀⠀⠀⠄⠰⠀⠀⠀⠀⢊⡀⠄⠄⢊⡁⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢌⡁⠂⠀⠤⠉⣁⣀⣀⡀⢀⣀⣀⡈⠡⠄⠀⠐⢈⠡⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠂⠀⠠⠤⠤⠤⠀⠀⠐⠒⠂⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠍⡉⠉⠙⠛⠛⠛⠛⠛⠛⠓⠒⠲⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣶⡴⢶⡶⠶⢶⢤⣤⢤⣀⠀⠉⠳⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡟⠁⠁⠀⠀⠀⠀⠙⠀⢸⡇⠀⠀⠈⢇⠀⠹⣭⣉⣉⣉⠉⠉⠉⠙⢳⡄⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠇⠀⠀⠈⠀⠀⠀⢸⡇⠀⢸⣦⠘⣇⠀⢻⡤⠀⠤⠤⠤⠤⠤⠤⢷⡀⠸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⡇⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠘⣧⠘⡆⠀⢳⣶⣴⣤⣤⣤⣤⣤⣤⣷⡀⠺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠛⠚⠛⠐⠓⠒⠒⠒⠒⣺⡇⠀⣿⠀⢸⡄⠹⡀⠈⢿⢿⠉⠛⠛⠿⢿⣿⣿⣧⠀⠸⣧⠀⠀⠀⢀⡶⠶⠖⠲⠶⢶⡶⠲⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡇⢰⡿⠙⠇⠖⠖⠖⢶⡶⠶⣾⡇⠀⣿⠀⠘⡧⠀⠁⠀⠘⡞⣇⣶⢾⠦⣾⣿⣿⣿⡄⠐⢿⡆⠀⣀⣸⢨⡭⠉⣉⣙⣺⡼⡄⣻
⠀⠀⠀⠀⠀⢀⣶⣋⠤⠼⡇⢸⡇⠀⠀⡠⠀⠀⠈⠇⠆⢿⠁⠀⣿⠀⢰⡇⣤⠘⠀⠀⢳⠙⣧⠘⢱⡜⣿⣿⣿⡇⠀⣼⣧⣴⣿⢽⢺⡿⠭⠿⠶⠒⠛⠛⣿
⠀⠀⠀⠠⣶⣸⣿⠀⣶⣶⡇⠰⡇⠀⠀⠃⠇⠀⠀⠀⠀⢸⠀⠀⣿⠀⣿⢠⡟⡄⢤⠀⢸⢇⣿⢤⣤⣉⣻⡅⣸⠛⢾⣿⣤⠟⠋⠘⢿⢯⡀⠀⣰⢿⣧⠀⣿
⠀⠀⠀⠀⣿⢾⣿⣠⣷⣿⡇⠀⣧⣤⣤⣄⣀⣀⣀⡀⣀⣿⠀⠀⡟⢰⠇⣼⠀⡇⢸⠀⠀⣿⡀⢉⣤⠿⠈⢻⣿⣿⡟⠃⢻⡄⠀⠀⢸⣏⠹⢦⣿⣿⣥⡴⠛
⠀⠀⠀⣀⣴⣻⣿⣵⡿⠋⠇⠀⣀⡀⢤⠄⠠⡤⡤⡄⣤⣽⠀⠀⣇⡸⢀⡏⠀⣧⠘⠀⠀⢸⠛⡋⠠⢠⣴⡿⣷⣿⡇⠀⣼⡇⠀⠀⠀⠈⢷⠀⠀⠀⠀⠈⡿
⠀⠀⣴⠋⢀⡀⠀⣹⠀⠀⠀⠀⣧⠸⢠⠀⠀⠀⠁⠁⠙⢸⠀⠀⣿⡇⢸⣡⡴⠟⠀⠀⠀⠘⡿⠷⠺⠿⠿⠶⠾⢿⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣴⡇
⢀⣴⡇⠉⠁⠀⠀⢸⠀⢸⠀⠀⠘⠀⠈⠀⠀⠀⠀⢀⠀⢸⠀⠀⠋⠀⢉⣁⠄⢲⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⢠⣼⡇⠀⢺⡇⠀⠀⠀⠀⣸⠟⠃⠛⠻⢿⡇
⠈⢻⠁⠈⣴⣶⣆⢺⠀⢸⡆⠀⠀⠀⠀⢠⠖⠉⠁⢨⡤⢿⡀⠀⠀⠘⠉⠙⠓⠞⢰⠂⠀⠀⠧⠷⠾⠴⢦⠶⠿⣿⠇⠀⣻⡿⠀⠀⠀⠀⣿⢀⣠⣤⣤⣰⡇
⣤⣾⡀⣼⣿⣿⣿⣼⣦⣾⠷⢶⣶⣶⣶⣾⣔⣶⣄⣼⣳⣾⣷⣶⣶⣶⣶⣶⣶⣶⣾⠓⠒⠒⠒⠒⠒⠒⠶⠶⠶⠤⠤⠤⢼⡇⠀⠀⠀⠀⣟⠀⣻⠙⣯⢹⡇
⠻⠶⢿⣿⠰⢾⣿⠟⢻⣯⢼⢺⡏⣿⢶⢹⢻⡿⠿⠿⠟⢻⣿⠱⡎⣿⠃⡆⣹⣏⡿⠛⢻⣿⣿⣿⣿⣟⣿⣿⣿⣶⡾⠛⠛⠃⠀⠀⠀⠀⣿⠀⣿⠀⣿⠈⢷
⠀⠀⠀⠻⢷⣿⠟⠀⠘⢿⣶⣿⣷⢿⣶⣾⡿⠃⠀⠀⠀⠀⠿⣷⡾⢿⣷⣾⣿⠿⠁⠀⠀⠹⢿⣿⡿⠿⣿⣿⣿⡽⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⠀⢹⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⡟⠀⣸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⢿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣏⢠⡄⠀⢁⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣟⠉⠙⣿⣿⣿⣟⠉⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠐⠿⣿⠋⠉⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣶⣿⣦⣰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆
⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣶⣶⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⢶⣛⣩⣿⣶⣶⣶⣶⣶⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡗⠒⣶⡖⢲⣶⠖⢶⣶⠒⢶⡖⠒⣷⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠙⡇⠀⣿⡇⢸⣿⠀⢸⣿⠀⣼⡇⠀⣿⢉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⣿⡇⢸⣿⠀⢸⣿⠀⣿⡇⠀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⢸⣿⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⢸⣿⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⢸⣿⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣿⡇⢸⣿⠀⢸⣿⠀⣿⣇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣟⣷⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢬⣍⣛⡛⠻⠷⠾⠟⣚⣋⣭⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠋⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣻⠏⠉⠟⠂⠀⠀⠀⠀⠀⠀⣴⣷⣿⣴⢎⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡟⢹⣿⣿⣿⣿⡀⠀⠰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⢻⣿⣿⣿⡇⠀⣰⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⡴⠀⠀⢀⠄⠠⠀⢰⣾⣦⠿⣿⣿⡿⠀⠀⢸⣿⣿⣿⣇⠀⣨⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡃⣴⣿⡁⢠⢶⣾⣿⣿⠟⣸⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⠀⢰⣶⠀⣿⠇⣰⣿⣿⠃⠀⠀⠀⠈⣿⣿⣿⣿⡄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣗⣿⡻⢀⠀⢿⣿⠃⣰⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠈⣿⡿⢁⣾⣿⣿⡟⠀⠀⠀⣤⠀⣀⣿⣿⣿⡿⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠈⠡⠟⣿⣿⡿⠀⠀⠗⣰⣿⣿⣿⣿⣿⣿⡇⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠁⠀⢠⠀⠀⠀⢰⣾⣿⡿⠁⠄⡀⠀⠀⠈⠉⠉⠛⡟⠙⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⣇⠀⣦⣴⣿⡟⠉⠀⠀⠀⠸⡄⠀⢀⣼⠁⣰⣷⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣶⢸⣿⣾⣿⣿⣆⢳⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠛⣿⣿⣿⣿⣿⣿⡆⢣⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣉⡇⣼⣿⣷⢻⣆⠁⣄⠀⠀⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⢻⠀⣿⣿⣿⣧⢹⣦⠈⣳⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠈⠰⣿⣿⣿⣿⡆⢿⣿⣿⣿⣤⠷⡿⢻⣿⣿⠛⠟⢻⡟⠟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣦⣿⣟⢻⡟⠀⠀⢸⣿⢿⡀⠀⠀⣇⣀⣻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⠏⢁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠈⠻⣿⣿⣿⠟⠆⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠤⢄⡀⢸⠵⠲
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡤⠄⠼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣣⠀⠀⡜⠉⠈⢦
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⡠⠒⡞⠁⠀⠚⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠷⡀⠑⣴⠋⠙⠣⡴⣠⣆⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣜⣄⠘⣏⠉⠉⡏⠥⡄⣀⣀⣀⠤⠴⠊⠁⠀⠀⠔⠁⠀⠉⢢⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣾⡄⠈⠒⠰⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣓⣢⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⢳⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣼⡢⢄⠠⡂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⡎⠢⡀⢳⡝⡳⠂⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡮⣹⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡶⢦⡇⠀⣸⠀⢀⠤⢦⣤⣤⡴⠶⠞⠛⠳⣋⣽⢆⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⠒⠢⣤⠭⠎⣀⠀⢸⠁⠀⠀⠀⠀⠀⢐⠉⠇⣤⣇⣉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠊⠈⠀⢣⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⢮⡉⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣬⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡤⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡇⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡔⠒⡶⢼⠒⠙⣰⣖⡖⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠠⡈⢶⡊⠑⠂⠋⢹⠤⠤⡘⠐⠯⢝⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠪⠽⢽⣹⡞⠀⠀⠰⢲⠋⡇⠀⠉⠊⠕⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠽⠕⠒⠒⠒⠚⠪⠽⠬⠃⠀⠀⠀⠁⢎⡑⠤⠤⠀⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣁⡠⠤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠝⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣀⣀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⣈⡉⠙⠛⠻⠶⢶⣿⣅⣀⡀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⢶⣤⣤⣀⣉⠉⠛⠛⠷⠿⣧⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⣶⣤⣄⣈⡉⠙⠛⠻⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣿⣷⣶⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⠀⠀⣶⠀⠀⣴⡄⠀⢰⡄⠀⢰⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀⠀⣿⡀⠀⣿⡇⠀⣼⡇⠀⣸⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢹⡇⠀⢻⡇⠀⣿⡇⠀⣿⠁⠀⣿⠁⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣇⠀⢸⡇⠀⣿⡇⠀⣿⠀⢰⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⣿⠀⢸⡇⠀⣿⡇⠀⣿⠀⢸⡛⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⣿⡇⢠⣿⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡄⢸⣇⠀⣿⡇⢸⣿⠀⣾⠇⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢹⡇⠘⣿⠀⢿⠇⢸⡟⠀⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣿⠀⠀⠀⡄⠀⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡀⠘⠀⠀⠀⢠⡄⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⠀⠇⠀⡀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⠀⠀⠀⣼⠇⠀⠀⡟⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢠⡆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡁⠀⠀⠀⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠈⠅⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣤⣄⡀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣠⣤⣀⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⠛⠛⠛⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠛⢙⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⣱⣾⣿⣿⣿⣿⣶⣶⡮⢻⣷⡄⠀⡀⠀⣠⣾⡿⢋⢀⣴⣿⣯⢻⡆⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣋⣴⣿⣿⣿⣿⣿⣿⡿⠋⣴⣾⣿⣷⣆⣤⣾⣿⢿⣷⣴⣿⣿⣿⣿⣿⠁⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣶⣤⣤⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠾⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣈⠙⠛⠋⠄⡀⠀⠈⣿⣿⣿⣿⠿⣻⣧⡀⠀⠀⠈⠉⢉⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣅⡀⠀⠀⠀⣠⠄⡠⢜⣿⣿⣿⢏⡾⢿⣿⡿⣦⣄⣀⠀⠀⠀⠀⠀⢤⣙⢻⡿⣋⣥⠶⠞⢛⠟⠛⠻⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢉⠁⠘⢿⣶⣿⣿⣧⢊⣴⠟⠋⠀⠀⢈⣠⣽⣿⣷⣶⣍⠛⠿⣶⣤⣀⣀⣠⣤⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣷⡘⠿⣧⠐⣶⣤⣤⣿⡿⠃⠀⢀⣠⣶⣿⠟⢛⣿⣷⣄⡉⠻⢦⡀⠈⢻⣿⣙⣿⣿⡿⠏⠀⠀⠀⠈⠀⢠⣿⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⣿⣦⡀⢉⣙⠂⠈⠷⣿⣿⣿⠏⠀⣠⣴⣿⠟⣩⠴⣟⡛⠶⣌⡙⠻⣷⣦⡈⢷⣤⣿⣿⠿⠏⢰⣧⢀⡂⠀⠀⢀⣿⡟⣸⣿⣿
⠀⠀⠀⠀⠀⢀⣤⡾⢿⣭⣁⠈⢉⣙⠀⣿⣿⣿⡶⠛⠋⣀⣴⣿⡿⢋⣵⣿⡋⠳⢬⡙⠷⣶⣤⣤⡘⢿⣿⣾⠿⠋⠀⡀⠲⣆⠛⢸⠁⠀⠀⣼⣿⣱⣿⣿⣿
⠀⠀⠀⢀⣴⣿⠉⠻⠆⠙⠻⠷⠈⣻⣿⣿⡿⠋⠀⣠⣾⣿⠟⣫⠾⣿⣿⣌⠻⣶⣤⣈⠢⣌⠻⣿⣿⠿⠋⠁⠀⢀⠀⢿⣆⠙⢷⡈⠂⡠⠀⣿⣿⡿⠛⢹⣿
⠀⠀⢠⣾⣿⣿⣷⡄⢶⣶⣦⣀⣼⣿⣿⠟⠀⢀⣴⣿⣋⣡⣾⠻⠶⣦⣤⣈⣀⠈⠛⠻⢷⣾⡿⠋⠁⠀⠀⠀⣄⠘⢷⡄⠹⣷⡀⠻⠆⠱⠘⠋⢁⣀⣴⣶⣿
⢀⣴⣿⣿⣤⣤⣤⣤⣀⣘⣿⣿⡿⠋⠠⢂⣴⣿⡟⠉⠉⠛⠻⢷⣦⣀⠉⡉⢛⣿⣮⡿⠟⠉⠀⠀⠀⠀⠘⢦⡌⠳⡄⠙⠆⠈⡀⣀⣤⣴⣶⣿⣿⣿⠏⠉⠛
⣼⣿⡇⠀⠉⠁⠺⢿⣿⣿⣿⣿⣄⣀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠌⢛⣿⣷⣽⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣒⣠⣴⣾⣿⠋⠻⠿⠋⠀⠉⠁
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⣿⣿⡟⠀⠀⠀⠀⠀⠀⠐⠦⢭⣛⣿⣧⣀⡀⡀⠤⠀⠀⠀⣀⣤⣶⣿⣿⣿⡟⢿⠏⠉⠁⠀⠤⠂
⢻⣿⣿⣀⣀⣀⣀⡠⠄⢀⣀⣤⣴⣿⣿⣿⣿⠀⠀⢠⠐⠂⢀⣀⡈⠲⣮⣽⣿⣏⣠⣤⣤⣴⣶⣿⣿⢿⣿⠋⠈⠉
⠀⢹⣿⣯⣭⣭⣴⣶⣾⣿⣿⣿⣿⡿⠋⡉⢻⣦⣤⣶⣦⣬⣴⣭⣝⣻⣾⣿⣿⣿⣿⣿⡿⢿⡿⠛⠋
⠀⠀⠙⠛⠿⠿⢿⣿⠛⠛⠋⠀⠉⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⣿⡿⠿⠟⠙⠛⠀⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣋⣙⣏⣈⣉⣹⣝⣫⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠹⠆⠩⠙⢳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣰⣀⢠⣠⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣍⣉⠝⠳⢿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⣷⣼⣟⣶⢯⣽⣦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⠭⠒⢉⡴⢃⢣⠈⠀⠃⠉⠛⢦⣙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠘⠀⠀⢀⡼⠉⠀⠀⠀⠃⠀⠀⠲⢄⠀⠀⡀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢀⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠣⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠂⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠈⠻⠦⠤⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠉⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢀⡏⠈⠀⢠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⣠⠾⡇
⠀⠀⠀⠀⢀⡼⠉⢧⠀⠀⠀⠙⠦⣄⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠁⠀⢹
⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠐⠋⠉⢧
⠀⠀⠀⠈⠛⠛⠳⠦⢤⣬⣷⣦⣤⣤⣀⠀⢀⣐⠀⢀⣀⣐⣀⣀⣤⠴⣞⣩⣤⣄⣀⣤⣤⣤⣬⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⢦⣤⣤⣤⣾⣟⣛⡉⠉⠻⢿⣶⠟⠋⠙⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠛⣻⡿⠀⠀⠈⠻⠤⡀⢀⡀⠀⠈⣳⠌⠀⣇⠀⠂⣄⣈⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠺⠿⠶⠶⠶⠶⠴⠶⠷⠶⠶⠶⠶⠿⠶⠛⠛⠛⠛⠉⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣲⠖⠒⠠⢤⡤⠤⣤⠤⠤⣤⠤⠀⠠⠤⢤⡤⠤⢤⡄⠖⠒⡶⢋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠓⢄⢀⡴⠋⠀⠀⠈⣱⠾⣁⠀⠀⠀⡴⠋⠀⠀⠀⠈⢲⠾⣁⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⡠⠛⠦⣀⠀⢀⠜⠁⠀⠀⠑⡶⢎⡀⠀⠀⠀⢀⡴⠁⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠿⢄⠀⠀⢈⠗⠥⡀⠀⠀⢠⠞⠀⠀⠈⠓⢤⣤⡊⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠑⣶⡋⠀⠀⠈⠓⣶⣃⠀⠀⠀⠀⢀⠞⠁⠉⠓⠤⣀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⡠⠊⠀⠙⢦⡀⢀⠞⠀⠈⠑⠦⣀⡴⠃⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⡸⠻⣄⠀⠀⠀⢀⠞⠙⠢⢄⡀⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠈⠳⣄⣠⠎⠀⠀⠀⠑⢦⣠⠋⠀⠀⠀⠀⠉⠑⢲⠾⢥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢀⠞⠓⢄⠀⠀⠀⢀⠜⠉⠓⢤⡀⠀⠀⠀⡰⠋⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣅⠀⠀⠀⠑⢦⡔⠁⠀⠀⠀⠀⠈⠓⢤⣞⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠈⠳⣄⣀⡴⠋⠙⠢⣀⠀⠀⠀⢀⡴⠁⠀⠉⣲⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠴⣿⣶⣤⣤⣤⣬⣱⣦⣴⣯⣤⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⢦⡄
⠀⠀⠀⠀⠀⢰⢶⡶⡾⢷⠶⠶⠶⠿⠶⠶⣦⠀⠀⠀⠀⠀⣾⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣷⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢺⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡿
⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⢸⣿⡟⢸⣿⡇⠀⠀⠀⠀⠀⢸⡀⢸⢷⠀⢸⡇⠀⣸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⣿⢸⣿⡇⣸⣿⠁⠀⠀⠀⠀⠀⠘⡇⢸⣾⠀⢸⡇⠀⣯⡇⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⡀⢸⡇⠀⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠘⣿⡇⢹⣿⢸⣿⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢿⠀⣟⡇⢸⡇⢸⢿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣷⢸⣿⢸⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⡇⢸⡇⢸⡾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⣈⣁⣈⣁⣈⣁⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉