Abandomnetcore Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠶⣶⣒⠺⢯⣉⠻⣯⡉⢳⣖⠒⢶⣤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠉⠁⠢⢍⣓⣦⠤⠙⡗⠊⣿⠉⢹⡏⠉⣿⠉⢩⣿⠓⢺⡿⠹⢶⠤⣀
⠀⠀⠀⢀⣠⣤⠶⠞⡏⠑⠒⠒⠒⠒⠒⠈⠁⠘⡌⡇⠀⣷⠀⢸⠀⢸⡇⠀⣿⠀⢸⡏⠀⣿⠃⠀⡜⠀⠀⠉⠲
⣠⡶⠛⣉⡡⠤⠒⠋⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢹⠀⢸⠀⢸⠀⢸⠇⢠⡇⠀⣾⠁⢸⡏⠀⠰⠁⡠⢄⡀
⣷⠁⡞⠁⠀⣀⡤⠖⠚⡆⠀⠀⠀⠠⠀⠀⠀⠀⢸⠘⡆⠘⡆⢸⠀⢰⠀⢸⡇⢀⡟⠀⡿⠀⢀⠆⢠⠁⠀⠈⢱
⢹⡆⢷⠒⠉⠀⣠⠤⠖⢣⠀⠀⠀⠀⡇⠀⠀⠀⠈⡆⢡⠀⡇⠸⠀⢸⠀⢸⠀⢸⠃⢸⠃⠀⡜⠀⡎⢀⠥⠃⡎
⠈⣷⠸⡖⠊⠉⢀⣀⠤⠺⡀⠀⠀⠀⢸⠀⠀⠀⠀⢁⠀⠀⢀⠀⠀⢸⠀⢸⠀⡞⠀⡏⠀⢰⠁⡸⠀⠐⡈⡸
⠀⢸⡆⢣⠔⠊⠉⢀⡠⠤⠇⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠈⠀⠀⠸⠀⡆⢀⠇⢸⠁⢀⠇⢠⠁⠤⢂⢡⠃
⠀⠀⣷⠘⡤⠒⠈⠁⢀⡠⢼⠀⠀⠀⠀⢡⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⡜⢀⠇⠐⡨⢄⠏
⠀⠀⢸⡆⢣⠤⠒⠉⠁⣀⡤⡇⠀⠀⠀⠀⡄⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡜⠀⠀⠁⡜
⠀⠀⠀⣷⠘⡤⠔⠂⠉⠀⣀⢥⠀⠀⠀⠀⠀⠀⠀⠀⢨⡤⠤⠄⠒⠒⠒⠒⠒⠀⠤⣆⠀⠈⠑⠒⢺⠁
⠀⠀⠀⠸⣆⠙⠢⠤⠒⣉⠤⠚⠓⠢⠤⠀⣀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠐⠁
⣤⣤⣤⣤⣬⣷⣶⣶⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠛⢋⣩⣤⣾⡿⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠿⠟⠛⠉⢁⣠⣤⣶⠾⠟⠛⢉⣠⣴⣾⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⠿⠟⠋⠉⢀⣀⣤⣴⣶⡿⠟⠋⢉⣀⣤⣶⣾⡿⠿⠛⠋⠉⣁⣤⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⡿⠿⠛⠋⠉⠁⢀⣀⣤⣶⣾⠿⠟⠋⢉⣀⣤⣴⡾⠿⠛⠋⠉⢀⣀⣤⣶⣾⣿⣿⡿⠿⠟⠛⠉⠉⢹⣿
⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠛⠋⠉⠀⠀⣀⣠⣤⣶⣿⠿⠛⠋⢁⣀⣤⣶⠾⠟⠛⠉⢀⣀⣤⣴⣶⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢠⣶⡿⠿⠛⠉⠁⠀⢀⣀⣤⣴⣾⣿⠿⠛⠋⣁⣠⣤⣶⠿⠟⠋⢉⣀⣤⣴⣶⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⣾⣿⣧⣀⣀⣤⣴⣾⣿⠿⠛⢋⣁⣤⣴⡶⠿⠛⣉⣁⣤⣴⣶⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⣿⠛⠿⣿⣿⣿⣥⣴⣶⣿⣿⣿⣿⣿⣶⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⣿⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠉⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣿⣿⡿⢿⣿⣶⣦⣶⣤⣴⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⣿⣿⠁⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⡿
⣿⣿⡟⠉⠛⠻⠿⠿⢿⣿⠿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠁
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⡿⠟⠛⠉
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠟⠛⠉
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠟⠋⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⡿⠟⠋⠉
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣀⣤⣴⣾⣿⣿⠿⠟⠛⠉
⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⡿⠟⠛⠉
⠀⠈⠻⣿⣷⣶⣤⣤⣤⣴⣶⣾⣿⣿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠈⠙⠓⠂⠄⣀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠐⡒⢢⣴⡶⣻⡿⢿⣆⣀⠀⠀⠀⠀⠀⠀⠙⢒⠿⠗⣒⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠸⠚⠚⠙⣀⠼⢿⣯⢷⢄⡤⣄⠀⠀⠀⠀⢺⣶⣀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣉⠤⠀⠤⢆⠉⠀⣀⣾⣿⠾⢦⣬⡝⡆⠀⠀⠀⠈⣿⠈⠆⢳
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⣩⡴⠮⠷⢹⡟⣶⡝⡍⢽⣻⣷⠀⠀⠙⠷⠇⡀⠀⠠⠀⠻⣿⣆⠼⠦⣄
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⣀⠀⠀⠘⣯⣙⡀⢫⣿⣿⣇⠀⠀⠀⡀⠀⡁⠀⠀⠀⠳⢺⡾⢽⣾⣇
⠀⠀⠀⠀⠀⣸⠃⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣼⠿⠃⢐⣖⢶⣮⣿⡿⡻⢿⢿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠌⣿⣿⠀⣿⠏
⠀⠀⠀⠀⠀⣿⡀⠒⡦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠤⠀⠀⠹⠟⡇⠀⡅⠀⠋⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣄⠈⢆⠀⠀⠀⢀
⠀⠀⠀⡞⣻⡏⢱⣷⡙⢂⢀⠀⠀⠀⠀⠀⠀⣟⣀⡰⡴⠤⢀⣐⣧⠀⠠⠀⢀⣐⣌⢿⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣖⣭⣷⣄⠀⠘⡇⡐⠲
⠀⠀⣼⣉⣿⡇⠬⠈⢳⢠⠞⡃⠂⠀⠀⠀⣀⣿⣟⣒⣒⣶⣶⣬⠛⡀⠃⣃⣩⣹⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡆⠉⢻⡿⢀⠛⠏
⠀⠀⣯⡿⠙⠮⣦⠵⢫⣬⡦⠈⠀⠀⠀⠀⢹⠆⠀⢘⠀⠉⠻⣿⣤⣴⠾⣭⣿⣏⢝⢻⣟⡄⠀⠀⠀⣠⠀⡊⠁⠀⠀⠀⢻⣄⠀⢧⠚⣠
⠀⠀⢸⠷⢛⠷⣀⠞⠋⠀⠀⠀⠀⠀⠀⠀⣨⣤⣔⣚⣲⣾⠛⠿⢷⣄⠱⡀⠙⣿⡞⢾⡽⢦⢤⣀⣠⣿⣆⠀⠨⠀⠀⠀⠀⣽⡇⢈⣷⡷⠂
⠀⢀⡏⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠉⢩⢟⣩⣤⣬⣙⠢⡄⠙⣧⢱⣶⠛⣫⠊⠂⠀⢦⣿⣈⡻⣿⡆⠀⠀⡠⠀⠀⠘⢿⡯⢤⣼⣷⡀
⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢘⢘⣹⠸⣿⣏⣿⢈⡶⠿⠚⠁⠀⠀⠀⠀⠀⠀⠻⡉⠙⠛⠁⠀⠀⠀⠀⠐⠠⢸⣿⣭⣌⣿⠁
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡶⣿⢃⣾⢡⠇⡿⡿⡇⡏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⣄⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⡾⣯⡟
⠘⠳⠶⢷⣶⣶⣶⣶⣶⣶⣶⡶⣶⣼⣧⣶⡏⣾⠃⠊⠁⠀⡷⣷⡇⠀⠀⠀⠀⠀⠠⠀⠘⠂⠀⠀⠀⠈⢢⡿⣿⠿⣛⣿⣭⠾⠗⠒⠛⠁
⠀⠀⣰⠃⠙⠂⣭⣽⣿⣿⣿⣿⣿⣿⣿⡏⣼⢃⠀⢠⡆⢀⠲⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢢⡉
⠀⢠⣀⣓⣚⣯⣭⠽⠯⠭⠭⠴⠒⠒⢊⡼⠃⠀⠀⠀⠀⠀⠘⢺⡎⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢂⠀⠀⠒⠀⠑⢄
⠀⠀⠉⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇⢃⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣠⣀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⣿⠆⣠⠟⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠹⡌⠀⡀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⢀⢋⣛⣮⢳⡀⣹⠄⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣃⣾⠋⢀⠀⠀⠀⢀⣠⠃⠀⠀⡄⠀⠠⠱⣧⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠉⠣⢽⣧⣉⣼⣇⠀⠀⠀⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠰⠠⠶⠛⢣⣞⢀⢈⣤⣠⣼⡿⠀⠀⠀⠀⠀⠀⠀⢿⢆⠀⠘⠃⣀⣾⠷⠶⢦⡀⠀⠀⠀⠀⡠⠛⠉⠙⠋⠐⠛⠛⠛⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠑⢷⣬⡀⠈⠐⠄⠀⠈⠎⠆⠀⠀⢸⡏⣎⣉⣷⠝⡆⢀⡴⠛⠀⠀⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠘⣦⣄⣜⡄⠀⠈⠳⣛⣻⣭⠞⣳⡉⠀⠀⠀⠈⠋⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢞⣦⡂⣀⡨⠜⠷⠻⠿⢯⣉⣡⣶⣾⣶⠿⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠐⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠷⢎⣉⣀⣤⣴⠶⠟⠋⠉⠀⠀⠀⠀⠉⠙⠛⠶⣦⣤⣀⣈⡡⠶⠧⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠆⠂⠀⠀⠠⠤⠐⠤⠤⣄⡀
⠀⠀⠀⠒⢄⣀⣤⣀⡠⠤⠠⢀⠁⠀⠀⢀⠀⠄⠀⢀⣀⡀⠀⠰⠂⠀⠀⠀⠀⠢⠦⠬⠥⠒⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⠙⠛⠶⠷⠦⠤⠊
⠀⠀⠀⠠⠀⠈⠉⠉⠛⠂⠁⠀⠀⠀⠀⠀⠀⠁⠒⠋

⠀⠀⠀⡀⠀⠄⠠⣀⠀⠀⢀⣀⣀⢀⡀⣀⣀⡀⠀⢀⣠⣤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⡤⣄⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⣥⣀⡨⠁⠈⣁⣀⣽⠿⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⠛⠛⠦⢘⡀⠐⠀⢘⠠⠞⠓⠉⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⢀⡄⠀⠀⣀⠀⡀⠰⠀⣀⢀⠀⠀⠀⢠⡰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠔⠉⠄⠚⠋⠉⠉⠛⢇⡜⠋⠉⠉⠙⠓⠦⠷⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠬⠒⠂⠀⣉⠚⠃⠀⣀⠀⠀⠀⠀⠀⠀⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠤⣤⣶⣤⠠⠖⠀⠀⠈⠀⢀⡀⠤⠐⠸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⢀⠤⠒⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢌⡑⠦⣌⣄⣀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⠤⠀⠀⠀⠀⠠⠤⠤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡤⠒⣶⣤⣄⡀⣀⣠⠔⠒⠢⠄⠄⡀
⠀⠰⣤⡄⠴⠖⠉⠛⠄⠀⠈⠀⠰⠄⠀⠒⠀⠁⠀⠀⠉⠙⠾⠦⣀⣠⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣵⠿⠛⡿⢾⣥⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠈⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⠀⣉⠙⠲⢦⣀⠀⠬⡶⠀⡠⠀⠀⡀⠀⠈⠄⠀⣠⡴⠒⢋⡁⠂⠄⠀⠀⠀⠀⠠⠀⢐⠀⠀⣀⣀⡀⠘⠀⣀⣀⠀⠀⢣⠀⡀
⠀⠢⢀⠨⠃⠀⠀⠉⠑⠒⠚⠛⠀⠀⠀⠘⠛⠒⠒⠋⠁⠀⠀⠘⠀⠀⠂⠀⠀⠀⠀⠀⠙⠲⠖⠛⠉⠁⠀⠀⠀⠀⠉⠙⠓⠶⠛⠁⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠐⠀⠀⠒⠠⣤⡐⠂⣤⡄⠒⠂⠀⠢⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠉⠀⠉⠉⠀⠈⠒⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠐⠰⠀⠀⠐⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠘⠐⠀⠘⠈⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣤⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡾⠋⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢼⡇⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣄⠀⢀⡀⢀⣜⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣽⠿⠶⠾⠿⣯⣅⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣚⣫⠭⠒⠀⠀⠀⠀⠠⠦⠒⠠⠂⠭⢛⡳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣚⠭⠲⠟⠋⠁⠀⠀⢀⠀⠠⠀⠆⡀⡀⠀⣀⠀⠈⠙⠛⠷⣭⡻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⢞⣯⡾⠁⠁⠀⠀⠀⠀⠢⠀⠀⠀⠁⢁⠀⠁⠀⠀⠃⠀⠀⠀⢀⡤⠀⠈⢗⢍⡳⣄
⠀⠀⠀⠀⠀⣠⡞⣱⠟⠁⠠⡀⡄⠀⠀⠀⠀⠂⢀⠨⠄⠠⠬⠤⠀⠬⠀⢀⡐⠀⠀⠀⠀⢶⠂⠀⠈⠙⣎⠳⣄
⠀⠀⠀⢀⡼⣋⠊⠁⠀⢤⠁⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⠢⠁⠀⣀⠐⠡⢮⠄⠀⢵⣜⢷⡀
⠀⠀⢀⡾⣳⠟⠀⠐⠕⠀⠀⠈⠡⠀⠁⠀⠀⢀⣀⣤⣶⣶⣶⣶⣶⣶⣦⣤⣾⠃⠀⠀⠐⠅⠀⢄⠀⠀⠀⠀⢻⣮⢷⡄
⠀⠀⡾⣱⠋⠀⠀⠀⠀⠀⠁⠦⠠⠀⠀⣠⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠠⢆⠐⡉⠢⠀⠀⠀⠀⠹⣿⢿⡀
⠀⣸⢣⡏⠀⣐⠉⠀⡘⢠⠂⠈⠂⢀⣾⡿⢈⣙⣽⡙⣻⣛⣛⣏⣿⣏⠛⣻⣿⣏⣿⣷⣲⠞⠁⠀⠀⢃⠀⠰⠙⠀⢻⡟⣧
⢀⡟⣾⠁⠠⠄⠀⠰⢀⠌⠀⠀⢠⣿⡟⠃⢻⣿⣿⣿⣿⣿⡿⢿⣿⡟⢻⣿⢟⣽⣿⣿⣻⡄⠀⠀⠐⡀⠆⠀⠍⠁⠈⣷⢹
⢸⢧⡏⠀⠒⠂⠀⠀⠰⠀⠀⠀⣾⣿⣿⠇⢰⡿⢻⡿⣿⣿⣇⣸⣿⣿⣻⠟⡏⡞⣿⣿⣿⣿⡀⠀⠀⠀⢀⡀⠀⠒⠂⢸⡟
⢸⢸⡇⠀⠤⠀⠘⠃⠃⠀⠀⢰⡟⠋⣁⣠⡿⣳⡠⡳⣙⠟⢩⢭⠟⣛⢋⣜⢯⣿⣿⣿⣿⣿⡇⠀⠀⠈⠈⠃⠀⠒⠆⢸⡇
⢸⢸⡆⠀⠈⠀⠀⠄⠀⠀⠀⠘⢻⣿⣿⢻⣷⣕⢝⢝⣢⠵⠛⠈⠏⢥⣿⡣⣻⣿⣿⣿⣿⣿⡇⠀⠀⠠⠤⡄⠀⠴⠆⠀⡇
⢸⡞⡁⠀⠀⢠⠀⡀⢀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⣮⣩⠃⠀⢠⣷⣭⣹⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⠀⠀⣀⠀⢘⢧
⠈⣧⢃⠀⠠⢂⠀⠈⠀⢆⠀⠀⠘⢿⣸⣽⣿⣯⣿⡾⢿⡿⠁⠘⢻⣿⢺⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠁⠋⠀⠐⠛⠀⡼⣸
⠀⢹⡜⡆⠀⠁⠀⠀⠐⠈⠄⠀⠀⠈⠻⣝⢿⣅⢿⣷⢿⣷⣤⣤⣼⣿⣾⣿⣿⣿⣿⣿⠋⠀⠀⠀⠒⠄⠀⠀⠀⠀⢠⢧⡏
⠀⠀⢳⡹⡄⠀⠀⢀⠀⠀⡤⠂⠀⠀⠀⠈⠳⢬⣛⠿⣻⣾⣷⣶⣎⣿⡿⢟⣫⡾⠟⠁⠀⠀⡀⠢⠄⠀⠀⠶⠀⢀⢟⡾
⠀⠀⠀⢳⡽⡄⠀⠛⢀⡄⠀⠀⠐⠀⠀⠀⠀⡀⠉⠙⠻⠿⠥⠤⠶⠶⠛⠉⠁⠀⠀⠀⠀⢊⡀⠀⠀⠠⡈⠁⢠⢟⡾⠁
⠀⠀⠀⠀⠻⣞⢦⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⡕⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡀⠀⢢⠄⠀⠀⠠⡀⠀⠀⡰⣫⠞
⠀⠀⠀⠀⠀⠈⢷⣕⢄⠀⠁⠐⡔⢀⠀⠀⠚⠀⠀⡔⠀⢀⠀⠀⡂⠀⣰⡀⠀⠛⠀⠀⠀⠀⠪⠃⠀⣠⣞⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣕⠤⡀⠀⠈⠚⢠⠄⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠐⠤⠆⠁⢀⠠⣪⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣕⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣢⡵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⢷⣶⣒⣄⣤⠤⠤⠤⠤⣤⣀⣰⣦⡬⠷⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠁⠐⢄
⠀⠀⠀⣀⠤⠤⣀⡠⠒⠴⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠒⠲
⠀⢀⡤⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠐⡀⠀⠈⠓
⡠⠇⠀⠀⢀⠀⢀⠀⠀⢈⢫⡛⠮⣎⡔⡔⠀⣈⠱⡀
⠑⠄⠀⠈⠥⠡⠥⠥⠤⠅⠡⠵⢄⠀⠉⠚⠬⣤⣞⠠⠠⣄⣐⣒⣒⣒⣒⣒⡒⠤⠤⣀⡀
⠀⠀⠂⠂⠂⠂⠄⠀⠄⠈⠊⠐⠀⢣⡱⠀⠀⢶⣾⣿⡵⠗⠛⠛⠛⠛⠓⠲⠿⢿⣶⣦⣌⣉⠒⠤⣀
⠀⠀⠂⠂⠆⠄⡀⡀⠐⠐⠀⢠⠚⣷⣷⣼⠦⣀⠙⢶⡲⠿⠿⠿⠿⠽⠯⠭⢽⣲⣶⢭⣛⠓⠭⣒⠤⣑⠢⣄
⠀⠀⠈⠀⠀⠀⡀⠐⠔⠀⡆⢡⣾⠿⠁⡭⣿⠟⠳⣄⠈⠢⡀⠀⠀⠀⡀⠄⠀⠀⠈⠙⠒⠿⣶⡤⣉⠲⢕⢦⡑⢄
⠀⠀⠀⠀⠐⠀⠀⠰⠀⠐⠀⣏⡯⢁⣞⠞⠀⠀⠀⠀⠛⢥⡈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣕⣦⠑⢝⡄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣞⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠙⠦⢨⢩⡱⣢⠀⠀⠀⡀⢀⠀⠐⠀⠀⠈⠳⣷⡈⢿⡄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⣗⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣍⣈⠴⡑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢣⠈⣿⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⢸⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢦⠢⠀⠉⠒⠤⢄⠀⠀⠀⠀⠘⣿⠀⠸⠀⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢻⡧⠈⡞⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⢄⣤⠄⣠⠇⠀⠀⠀⢨⣽⠀⢸⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⢇⠈⢿⡄⠘⣬⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠙⢦⡥⢚⡃⠀⠀⠀⣀⣞⠎⢀⠂⢀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢢⠀⠙⢆⡈⠳⢍⡳⢄⡀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⡴⡻⠋⡠⠂⢈⡞⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠱⣄⡀⠉⠂⠄⡉⠒⠪⠷⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣔⡯⠕⠋⡠⠞⠀⠠⠎⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⠢⢄⡀⠀⠉⠒⠢⠄⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⣈⠖⢁⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠘⠑⠢⢤⣀⣀⠀⠀⠉⠉⠐⠒⠒⠒⠒⠒⠉⠁⠀⢀⣀⡅⠎⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⣀⠀⠉⠋⠛⠒⠒⠶⠼⢬⢬⢴⢴⠶⠖⠚⠩⣡⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠠⠤⠀⠀⣀⡀⠀⠤⠖⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⢐⠄⡈⠔⠠⢂⠅⡠⢐⠠⢀⠄⡂⠔⡠⢀⠀⡂⢀⠀⡂⢐⠠⢂⠄⡠⢀⠄⡠⢀⠄⡠⡂⢐⢔⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠦⠀⣀⣤⣤⣶⣶⢶⠶⣶⡶⢶⣤⣤⡄⠠⠤⠀⣀⡀
⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠭⠭⢄⠀⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⣎⣩⣉⣵⡆⠈⠲⠀⠃⠀⠀⠀⢠⡀⡄⡄⢀⡀⠀⣄⣄⢀⡀⠀⢠⠀⠒⠀⠤⠀⠀⢇
⠀⠀⠀⠀⠀⢹⣿⣽⣿⣗⢋⠀⠀⠀⡀⠀⠀⠉⣧⣻⡇⢸⣷⡇⢸⣿⣿⣷⡞⡞⠁⢠⠊⠉⠉⠀⡈
⠀⠀⠀⠀⠀⠈⣿⡿⣿⣿⣺⠀⠀⠀⠁⠀⠀⠀⢹⣸⣇⠀⣿⣇⢸⣿⣿⢋⢣⠁⠀⠂⠀⠂⠀⣴⠃
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣻⡄⠀⡀⠀⠀⠀⠀⠘⡇⣿⠘⣻⣿⣾⣿⣿⡜⡆⠀⠀⠀⠀⠀⠘⡸
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇⠀⡄⠀⠆⠀⠀⠀⢷⣻⠀⣿⣿⣿⣿⣇⣾⠀⠀⠀⠀⠀⠀⣄⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⢿⠀⢀⡀⢀⠀⠀⠀⢺⣾⡇⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢰⡸
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣽⡇⠀⠀⠀⠀⠀⠀⠈⣿⣇⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⡀⢴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⡿⣿⠀⠀⡀⠰⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠤⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠥⠀⣀⠀⠀⠡⠘⡀⠀⠀⢀⣿⣿⠿⡿⠿⠀⠀⠀⡀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠁⠤⠄⠒⠀⠀⠉⠀⠀⠐⠒⠢⠬⠭⠴⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣄⡄⠄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣿⣿⣿⣷⣿⢮⡢⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣯⠈⢽⣿⣿⣧⢱⣹⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣶⣿⣿⣿⡇⡇⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡇⡄⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢱⡸⣿⣿⣿⣿⣿⣿⣿⡿⣰⠃⠉⠀⠂⠀⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⠿⣿⣿⣿⣿⢫⣏⡁⢀⠄⠤⢀⡀⠀⢠⡀⠀⠀⠀⠠⢜⣦⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢤⠤⠁⠀⠀⠀⠐⠒⠒⠚⠋⠉⠉⠐⡯⠿⠤
⠀⠲⢶⣶⣶⣶⣶⣶⣶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⣀⠉⠉⠀⠈⠉⠀⠂⠀⣀⡀⠁
⠀⠀⠀⠀⠉⠉⠉⠛⠿⣿⣶⣤⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠦⢄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣦⣘⡀⠄⠐⠁⠀⠤⠀⠀⠀⢀⣀⣴⣿⣿⣿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣶⣿⣶⣿⣿⣶⣷⣾⣿⣷⣾⣿⡇
⠀⠀⠀⠀⢸⡆⠀⠀⠀⣷⠀⠀⢸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀
⠀⢠⣤⣤⣼⣧⣤⣤⣤⣿⣴⢦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀
⠀⢸⣿⣿⣯⣽⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⣾⠛⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⡿⣿⠿⠛⠉⠛⠛⣿⠟⠛⠛⠋⠛⠹⠋⢻⣿⣭⡉⢰⣯⠋⠙⢻⣿⠿⣿⣿⣿⣽⠛⠏⠙⠛⠛⠃
⠀⠸⠋⠉⠉⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠙⠀⠀⠀⠀⠀⢀⠀⠀⠩⣆⠰⣩⠆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣺⠿⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢀⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢻⡇⣤⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⣋⣉⣥⣤⡤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣋⣤⡶⢾⣛⠥⢷⡾⠖⠒⠓⠷⠚⠒⢢⠞⢭⣛⡛⠿⢿⣶⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢋⣴⠾⣛⠴⠛⠉⠁⢠⠀⠀⣀⢀⡀⢀⠀⣀⠀⠀⠀⠁⠈⠑⠲⢭⡛⢿⣽⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡶⢏⣱⠪⠀⠀⢆⠂⠈⢐⡈⠥⠰⠀⠂⣾⠀⠂⠨⠉⡒⠀⠀⡁⠀⡀⠈⢲⣉⠿⣿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡾⢁⣼⢟⣵⠎⠁⠀⡀⠒⡀⠀⠈⡀⡄⠀⣀⣀⢀⠉⡀⡀⠀⠀⠀⢉⠐⠡⡐⠠⠀⠀⠑⢑⣍⠿⣿⣷⣄
⠀⠀⠀⠀⠀⢠⠟⢠⡿⣹⡞⠀⢀⡐⢣⠔⠉⠀⠀⡀⠱⢂⣠⣅⣬⣬⣦⣭⣈⣇⣀⠰⣁⠚⠁⠈⠁⠌⠠⡀⠀⠲⡦⡹⣿⣿⣦
⠀⠀⠀⠀⢠⠏⣰⠟⣴⡗⠀⠠⠂⡜⢣⣄⠀⠈⣃⣥⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣦⣀⡛⠀⠀⣀⡡⠊⢂⠀⠐⡵⡘⣿⣿⣧
⠀⠀⠀⢀⡟⢰⡟⣶⠉⠀⢀⠙⢶⣄⠀⠀⢠⣾⣿⣿⣭⣿⣿⡿⢉⠄⠨⢩⡟⢿⣻⣿⢿⣿⣿⣆⢠⠉⠀⠈⡄⠂⠀⠐⣹⡜⣿⣿⡇
⠀⠀⠀⣼⠁⡿⢬⠇⠨⠁⠂⠎⠀⠀⡙⠲⠯⡉⠋⢉⡏⠛⠃⣑⠈⢠⣤⣦⡩⡎⠀⠄⡈⣸⣿⣿⣴⢪⡄⠀⠈⠄⢂⠀⠈⢇⢹⣿⣿⡀
⠀⠀⠀⡏⢸⣏⡞⠀⣖⢉⡐⢠⣋⠨⢅⣤⠤⣬⠑⠦⣄⣠⢀⣴⠲⡸⣿⣿⢣⣏⣐⡌⢻⣟⡈⡙⠙⢇⠚⠀⡤⢈⣁⢤⡀⠙⡜⣿⣿⡇
⠀⠀⠀⡇⣾⢧⣧⠀⠀⠀⡄⠀⠠⡄⣾⡆⣀⠈⣌⣧⠀⣭⣻⢿⣷⠆⠏⢦⡿⣿⣷⣳⣹⣮⣭⣶⣷⣹⠈⡅⠈⠈⠀⡀⠀⢠⡇⣿⣿⡇
⠀⠀⠈⠇⣿⡻⡇⠀⠀⠀⠆⠀⠸⣿⣿⣷⠈⠐⠻⣆⠴⠹⡷⠀⡄⢿⣿⠆⣇⠀⣿⡇⣽⣿⣽⡿⢻⣿⡇⠃⠀⠀⣇⡇⠀⢸⡇⣻⣿⡟
⠀⠀⠀⡇⢻⡇⣗⠀⠀⣀⣃⢠⡄⠈⣿⣿⣷⣔⢄⢟⢆⠰⣷⣀⣽⣦⠶⣾⣿⣛⠾⣤⣛⡛⡞⣳⣿⣿⠁⠀⠀⢀⣿⡇⠀⢌⡇⣿⣿⡇
⠀⠀⠀⢇⢸⣇⢯⠀⠀⠁⠠⠀⠀⠀⠘⣿⣿⣿⣧⣉⠀⠢⠈⢙⡋⢻⣿⠸⣟⡟⡋⡄⢙⡍⣶⠸⣿⡏⠀⠀⠀⡈⠉⠀⠀⢾⢓⣿⣿⠇
⠀⠀⠀⢸⡀⣿⠸⡆⠀⠀⠆⢣⠀⠀⠀⠹⣿⣿⣿⣿⣿⣦⣬⣘⡠⠶⡇⢰⣋⣰⣧⡆⣿⣷⣶⣿⡟⠀⠀⠀⢠⣤⠄⠀⢰⡆⣼⣿⡿
⠀⠀⠀⠀⢧⠘⣷⡹⣄⠀⢈⡀⠆⠰⠖⣄⠈⠻⣿⣿⣿⣿⣿⣿⡏⠀⠇⢻⣿⣿⣿⡇⣿⣿⡿⠋⠀⣤⡀⡠⠛⠎⠀⢰⡿⣱⣿⣿⠃
⠀⠀⠀⠀⠘⣧⠸⣧⠹⡄⠉⠂⠐⠁⢄⠀⠀⠀⠈⠻⢿⣿⣿⣿⠁⠀⠠⢸⣿⣿⣿⣀⠟⠋⠀⠀⠘⢎⠴⣀⢀⠒⢰⡝⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠈⢧⡙⢷⣙⢦⠀⠶⠀⢄⠓⣄⠀⠀⣤⠀⠈⠉⠛⠛⠛⠚⠛⠛⠋⠉⢀⡀⠀⠀⢀⠴⡉⠆⡑⢂⣴⢟⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠳⣌⠻⣦⡻⢗⣄⠀⠙⠢⠝⠰⠏⣀⠀⠀⠀⠀⡆⣶⠀⠀⠀⠀⢈⢡⠀⢎⠡⠒⠁⢀⣴⠞⣣⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣈⠻⣮⣝⢳⣦⣀⠀⠉⠒⠠⠭⢑⡒⠲⠰⠂⠆⢐⣒⡫⠽⠀⠋⠁⠀⣀⣼⢟⣱⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠛⠻⣾⣙⢻⠶⣤⢤⣀⡀⠀⠂⠎⠀⠸⠀⢀⣀⣠⢤⣤⡼⢟⣫⣼⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣙⠛⠶⣮⣭⣉⣒⡛⡿⢾⢿⠟⡗⣚⣛⣹⣽⣶⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠬⠅⢬⠤⠤⠤⠀⠀⠀⠈⠀⠀⠀⠉⠉⣛⠍⠀⠀⠀⠈⢁⡉⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡙⢀⠀⠀⢰⣰⣴⡖⠒⠒⠒⠒⠒⠒⠒⠀⠐⠒⠀⠀⠀⠀⠐⠂⠐⠒
⠐⠒⣒⢒⡓⣶⣶⣿⣿⡃
⠀⠀⠠⠀⠀⠼⠽⣧⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠄⠉⠀⠡⠭⢽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠉⣉⣉⣩⡭⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⣂⣀⣐⡒⢀⣀
⠒⠀⢐⣃⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⣒⠒⣐⠒⠒⢀⠀⠒
⣀⡐⠂⣁⣐⣛⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠖⠂⠀⠐⠒⠀⠐⡶
⠒⠂⠀⠂⠄⢲⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠈⠉⠀⠠⠅⠉⠅
⠀⠀⠠⢤⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠤⠥⠨⠤⠄⠩⠥⠬
⠂⠀⠀⢥⣯⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⠀⠀⠀⠤⠥⠄
⠠⠄⣤⠭⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠠⠤⠬⠭⠉⠉
⠀⠀⠉⠩⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠈⠉⠉⠀⠈⠉⠉
⡍⠁⠉⠉⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠩⠭⡤⠤⠤⠄
⠀⠀⠈⠀⠈⠉⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠋⠋⠉⠁⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⡀⢀⣀⢀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⠀⢀⣀⡤⢴⡶⠒⢺⡞⠛⣿⠛⠓⡗⠒⢲⡦⠤⣀⡀⠀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡟⠉⣧⠀⠀⡇⠀⢸⡇⠀⣿⠀⠀⡇⠀⢸⠁⠀⣸⠉⢻⠁⠀⢨⡇
⠀⠀⠀⠀⠀⢀⡤⠖⠚⢋⣉⡇⠀⠀⢷⠀⢸⡀⠀⢿⠀⠀⡇⠀⣿⠀⢰⡇⠀⣼⠀⠀⡏⠀⡏⡄⣀⣼
⠀⠀⠀⠀⠀⠸⡄⠐⠋⣉⣡⣽⠀⠀⠘⡆⠀⣇⠀⢸⡀⠀⡇⠀⣿⠀⢸⠃⠀⡏⠀⢸⠁⢸⢁⠁⢩⡇
⠀⠀⠀⠀⠀⠀⢧⠀⠉⠁⢀⣈⡇⠀⠀⢳⠀⢸⡀⠈⡇⠀⣷⠀⣿⠀⢸⠀⢠⡇⠀⡟⢀⡏⠈⠘⣽
⠀⠀⠀⠀⠀⠀⠸⡆⠘⢋⣉⣤⢿⠀⠀⠘⡆⠀⣇⠀⢷⠀⢻⠀⣿⠀⢸⠀⢸⠀⢰⠇⣸⠙⠁⢣⡇
⠀⠀⠀⠀⠀⠀⠀⢷⠀⢉⣠⠤⠾⡇⠀⠀⢳⠀⢸⡀⢸⡀⢸⠀⣿⠀⣼⠀⣾⠀⣼⢀⡏⠚⠴⣼
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣀⣀⡤⢿⠀⠀⠘⡆⠀⣇⠘⡇⢸⡆⣿⠀⡿⠀⡇⢠⡇⣸⠱⢠⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⣉⡤⠤⠞⣇⠀⠀⢳⣀⣽⠤⠷⠚⠓⠛⠒⠳⠶⠧⣼⣀⡏⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⠤⠶⠒⢻⡀⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢶⣍⣹⣿⣍⣥⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠋⠉⠉⠉⠉⠙⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢠⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣄⣠⣾⣿⠉⠉⠙⣦⣠⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣦⣀⣤⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠿⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡴⠖⠐⠋⠙⠀⠀⠀⠁⠈⠀⠁⠈⠈⠒⢰⢠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⢊⣉⣀⣀⠴⠤⠆⠒⠒⠒⠂⠐⠒⠒⠒⠢⠤⣄⣀⠀⠀⠘⠁⠧⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡛⢀⣰⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢤⣀⠀⠈⠘⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣁⣶⠟⠉⠀⣀⠀⢴⡋⠀⣀⠀⠀⠀⢤⠀⠀⠀⠀⡄⠠⠄⠀⠀⠀⠀⡀⢦⡤⠉⠛⢦⣄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠁⣠⡿⠋⠀⠀⠐⣦⡼⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⠀⠀⠀⠀⠀⠀⠐⠀⠀⠋⠡⠙⠀⠀⠉⠻⢦⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⢠⠞⡁⣢⡾⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡟⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠙⣦⡀⠀⠐⢤
⠀⠀⠀⠀⠀⣤⣷⣼⣾⠏⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠐⢦⠙⠀⠀⢲⣀⣧⣭⣜⣤⣼⣼⣑⣏⠀⠀⢚⠀⠲⢠⠎⠐⢀⡠⠂⢀⠀⠀⠀⠛⣦⠀⠀⢧⠄
⠀⠀⠀⠀⠶⢾⣿⡟⠁⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠄⢛⣠⣴⣾⣾⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣶⣄⡘⣧⡿⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⢣⡀⠈⠺⣄
⠀⠀⠀⠤⣐⣾⡟⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠈⣣⣴⣿⣿⣿⣿⢻⡟⢿⠺⠛⡟⢿⠿⣿⣿⣿⣿⢉⣿⣯⣤⡀⠀⠀⠀⣠⠔⠁⡀⠀⠀⠀⠀⠳⡀⠀⠹⠆
⠀⠀⡤⢠⣿⡟⠀⠀⠀⠀⠀⠀⠄⠀⠙⠓⠀⣠⣾⣿⣿⣿⣹⣯⣿⣿⡇⢀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢟⣙⠺⣿⣆⠐⢎⠁⠀⠘⠀⠀⠂⠀⠀⠀⢷⠀⠀⠛⡀
⠀⠰⠦⣾⣿⠀⢀⡀⠀⠀⠁⠂⠀⠀⡀⠀⣼⣿⠿⣿⡿⣿⢻⠋⠛⠛⡟⢻⣾⣿⣿⡄⣐⣼⡟⣿⣿⣿⣿⣿⠗⢈⣿⢷⡌⣱⠄⠀⠀⠀⠌⠀⠀⠀⠻⡇⠀⠘⣯
⢀⡆⢰⣿⡇⠀⠬⠁⠀⡀⠀⠀⠀⢠⡅⣼⣏⣙⡀⠁⡀⢀⣈⣀⣀⣀⠳⢾⣿⣿⠛⠓⢈⣭⣿⠿⣟⣻⣿⣿⢀⡈⣹⢈⣓⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠸
⡀⢁⣿⣿⠁⢀⡖⠀⠠⠤⠀⠴⢦⣬⢰⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣮⣤⣿⣿⣿⣿⣿⣿⣇⣿⣋⣸⣿⣿⣿⣿⣷⢘⡓⢤⡀⠀⠀⠐⠋⠀⠘⣆
⡷⢈⣿⡿⠀⠾⠶⡄⠀⠀⠀⠖⠚⠷⣼⣿⣽⣿⣿⣿⢩⣿⣿⠿⠻⣿⣿⠿⣡⣤⣶⣬⡿⢿⣿⣿⠿⠟⣿⣟⣿⡟⣿⡟⣼⡿⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣋⢀⣿⡿⣷⡜⣿⡻⣤⣿⠰⣿⣧⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⢰⣾⣶⣿⣿⢋⣿⠟⣼⣿⣼⡇⠀⠀⠀⠀⣄⣤⠀⠀⠀⢺
⢯⠔⣿⡇⠀⠀⠀⠀⠀⠠⠀⠀⠉⠉⣿⣿⣽⣿⣎⢳⣍⢻⣿⣿⣥⣤⣤⣈⢿⣿⣿⡿⢃⣠⣴⣬⣿⡷⣿⣿⠘⢃⣼⣏⣾⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⣙
⢐⡈⣿⣿⠀⠆⠀⠀⠁⠤⠀⠀⠀⠀⢿⣿⣿⣯⡹⣷⣿⢿⣿⣿⢿⣿⣿⣽⡿⣾⣿⡶⣿⣿⣿⣷⣿⡞⣿⣏⢰⡟⣨⣿⣿⣿⠀⠀⠀⠀⠄⢹⠃⠀⠀⢀⡿
⠈⣁⢿⣿⡄⠀⠀⠀⢀⠀⠖⠛⠋⠀⠘⣿⣿⣿⣿⣦⡹⢿⣬⣉⡛⢎⣿⢫⠜⣿⢿⡏⢻⣿⡿⣿⣙⠷⣿⣯⣠⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠇⠀⢠
⠀⠤⢸⣿⣇⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣾⣥⣼⣿⢫⡿⣟⢿⢺⡗⢻⡿⠟⢛⣭⣶⣿⣿⣽⣿⣿⣿⡟⠀⠀⠀⠀⡈⠀⠀⠀⠀⢀⡟⠀⠀⠾
⠀⠘⡄⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣟⣭⣭⡡⣿⣾⣷⣯⣾⣿⡷⣶⣾⣿⣿⣿⣿⣿⣯⣿⣿⣿⠟⠀⠀⠀⠀⠀⡻⠻⠀⠀⠀⣾⠃⠀⣸⡏
⠀⠀⢩⡼⣿⣿⡀⠀⠀⠀⠀⠈⠁⠀⠰⠒⢀⠀⠻⣿⣿⣟⢿⣿⠇⣿⣿⣿⡟⣿⠸⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⡌⠙⠀⠈⠤⠁⠀⠀⠀⣸⠃⠀⣰⡟
⠀⠀⠀⠳⠈⢿⣿⡄⠀⢠⠒⡀⠀⡄⠂⠀⠊⠁⠀⠀⠙⠿⣷⣶⣿⣿⣿⣿⠇⠉⠀⣿⢸⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠘⠀⠐⠂⠠⠐⠗⠀⣴⠋⠀⣰⠿
⠀⠀⠀⠀⠛⠎⣿⣿⣦⠀⠀⣿⣆⠀⠤⡁⢠⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣷⣿⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⢀⠀⢀⠀⠀⣠⡖⠀⢀⡾⠃⢀⣼⡟
⠀⠀⠀⠀⠀⠘⠳⠛⢿⣷⡄⠀⠠⠗⠀⠀⠀⠀⠃⢄⡀⠀⠟⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠿⡀⠠⠀⠁⠀⠀⠀⠀⠉⢄⣠⠟⠀⢠⡿⠏
⠀⠀⠀⠀⠀⠀⠈⠣⠈⠹⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⡅⠙⠀⠀⠀⢀⠀⢠⠈⢡⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⣤⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡌⠹⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠀⠀⣰⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢫⠙⠿⡿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠄⠚⠀⠀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠈⠋⡱⠶⢤⡤⣀⣀⡀⠀⠀⠀⠈⠃⠀⠈⠀⠀⢀⡀⠠⠄⠀⠀⠀⣀⢠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⡐⠀⡴⣌⠻⢀⡉⠛⠒⠀⠀⠀⠐⠛⠀⠀⠀⣀⠀⣤⣴⠖⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠚⣺⣇⣠⣤⣤⣦⣴⣰⣄⢛⠘⠀⠋⠉
⣿⣿⣿⡿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡈⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣴⣾⣿⣷⠌⠁⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠿⠏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣬⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⢷⣶⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠙⣍⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣧⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣴⡿⢿⡄⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣦⡾⠋⠉⠉⠻⣿
⠀⠀⠀⠀⠀⣾⠏⠀⠈⢿⣄⡀⠀⠀⢠⠖⠒⠦⣌⠹⣿⣿⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠻⣦⣄⠸⣄⠀⢀⡞⢀⣿⣿⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠈⠙⢷⣮⡳⠟⣀⣾⣿⣿⣿⡄⠀⠀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠙⣷⣤⡀⠀⠀⠀⣇⠂⠈⠉⠋⠉⠙⠻⣿⣿⣿⡀⠀⢸⡏⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⠉⢻⣦⣄⡈⣧⠀⠀⢠⡞⠉⠙⢓⡎⢿⣿⣧⢠⡾⢣⡟⠈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠙⡙⠻⣿⣷⣤⣀⠳⣄⠀⣾⠁⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠇⠀⢇⢛⢻⣿⣿⣶⣭⣧⣾⣿⣿⣿⣿⡶⣤⣄⡀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣦⣤⣤⣀⠀⠚⠸⡞⠀⠈⠉⠉⠉⠁⢿⣿⣿⣿⣷⠦⣭⢻⡄⠀⢠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠈⠙⠻⠶⣤⣤⣄⣀⣀⣀⣀⠀⠘⣿⣿⣿⣿⡆⣸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⢉⡏⠙⠻⠿⢿⣿⣷⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣄⣀⣀⠀⠀⢘⠃⠆⠀⠀⠀⠀⠉⠉⢿⠿⣿⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡟⠛⠛⢻⠟⠛⣻⠿⠟⠛⠛⠛⢋⣷⡉⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣐⣋⣴⣎⣁⣄⣀⣀⣀⣴⡟⠁⢳⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠛⠛⠛⠋⠉⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡟⢉⠙⠛⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠛⣁⠀⣠⢠⣿⣿⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣏⠀⣄⣀⠀⠉⢸⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠘⡂⠈⠁⢀⠔⠉⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠃⠀⢤⣿⠋⢀⣇⠀⢀⢰⠖⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⣰⣧⣬⣄⡉⠁⢘⣷⣄⢸⢀⣆⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⠀⣸⣿⣿⣿⣼⣿⣿⣿⣿⣿⡇⠉⠁⢠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢀⠆⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠎⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢠⠃
⠀⠀⠀⠀⣠⠤⠤⠤⠼⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡏⣹⣿⣿⣿⣦⣤⣎⡤⠦⣀
⠀⠀⣠⠌⢱⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⢻⠋⠉⠉⠀⠈⢿⣿⣿⣿⡟⠁⠀⠀⠈⠳⣎⠽⡀
⠀⠀⢹⡆⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣀⠜⠻⠿⠋⠀⠀⠀⠀⠀⠀⠈⠳⣇
⠀⠀⠈⣧⠀⢰⠀⠀⠀⠀⢀⣠⡔⠀⣀⠀⡀⠁⠀⢱⠀⠤⡚⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢳⡄
⠀⠀⠀⢹⡈⡀⡆⠀⣀⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠈⠢⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢹⡀
⠀⠀⠀⠘⣇⠀⢸⠀⠁⠀⠀⠐⠀⣀⣀⣀⣀⣀⠀⠀⢱⠀⠀⠀⠈⢢⡀⠀⠐⠀⠀⢀⡠⠖⠉⢈⣉⡠⠇
⠀⠀⠀⠀⢻⡰⢀⣦⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⣿⣿⣦⣀⣠⣶⣿⣦⣴⣾⣏
⠀⠀⠀⠀⠘⣷⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠉⠀⠀⠀⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣄⡀
⠀⠀⠀⠀⢀⣠⣤⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⡀
⠀⠀⠀⠀⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠋⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⢸⣿⡀
⠀⠀⠀⠀⢹⣿⣯⣿⣿⣷⣽⡻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⣠⣴⣾⣿⣿⠟
⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⡻⣿⣾⣝⡿⣦⣄⠀⢀⣤⣶⠿⠛⠁⢀⣤⣶⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⣿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣽⡾⣝⣿⠟⠉⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢿⣿⣿⣿⣮⢿⣿⣿⣿⣿⣿⣟⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⢀⣤⣶⠾⢿⣿⣯⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⠁⣸⣿⣄⡀
⠀⣿⠿⣦⡀⠀⠻⣿⣿⣿⣿⣿⡿⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣙⣿⡦
⠀⣿⡆⠙⢿⣦⡀⠙⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣿⢿⣯⡀
⠀⠈⣿⣶⡀⠙⢿⣦⡈⢻⣿⣿⡇⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⡿⠿⠿⢋⣿⣿⠆
⠀⠀⣿⣿⣿⣦⡀⠙⢿⣦⡹⣿⣿⣾⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⠟⠋⠁⠀⠀⣀⣴⣾⣿⡋⠁
⠀⠀⣿⣜⢿⣿⣿⣦⡀⠙⢿⣮⡛⠛⠉⠉⠁⣀⣤⣶⠿⠛⠉⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢻⣿⣎⢿⣿⣿⣿⣦⡀⠙⢷⣄⣠⣶⠿⠛⠉⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠈⢻⣟⢯⢻⣿⣿⣿⣿⣦⡀⣿⠋⠁⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⣤⣶⣿⣿⣮⢧⡻⣿⣿⣿⣿⣿⣿⣆⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⢉⣿⣷⣦⡀
⣼⣿⠿⣿⣿⣿⣧⢳⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦
⣿⣿⢷⡈⠻⣿⣿⣷⡹⣼⡟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿
⣿⣿⢦⡙⢦⡘⢿⣿⣷⣽⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣿⠇
⢻⣿⣦⡙⢦⡙⢦⡙⢿⣿⣧⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢰⣿
⠘⣿⣯⡳⣌⠻⣄⠻⣆⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠸⣿
⠀⠹⣿⣷⣌⠳⣌⠳⣌⠳⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷
⠀⠀⠈⠻⣿⣷⣌⠳⣈⠳⡌⠳⣌⠻⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠟
⠀⠀⠀⠀⠘⢿⣿⣧⡙⢧⡘⢧⡘⢧⡘⢿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡙⢦⡙⢦⡙⢶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡙⢦⡙⢾⠁⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣌⠳⣼⡂⠀⠀⠀⢀⣠⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⡇⢀⣤⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⢉⣉⣀⣀⣀⣉⡉⠙⠛⣿⠛⠋⠉⠉⣉⣉⣉⣉⣉⣉⡙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠡⠖⠊⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⡟⠀⠀⠊⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⢀⠤⠄⠒⠒⠉⠉⠉⠉⠉⠒⠀⢰⡇⠀⠤⠔⠒⠊⠉⠉⠉⠉⠉⠉⠂⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⢀⣀⡠⠤⠤⠤⠤⠤⣀⠀⠀⢸⠁⠀⠀⠀⣀⣀⠠⠤⠤⠤⠤⣀⡀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⣸⣿⣿⠀⠐⠊⠁⠀⠀⠀⣀⣀⡀⠀⠀⠁⠀⡟⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⢀⣿⢧⡇⢀⡤⠔⠒⠉⠉⠉⠀⠀⠈⠉⠓⠀⢠⡇⠀⢀⠤⠔⠒⠊⠉⠉⠉⠉⠒⠒⠦⠄⠘⣿⣿
⠀⠀⠀⠀⣸⡿⣾⠀⠀⢀⣀⡠⠤⠔⠒⠒⠒⠒⠤⠄⠀⢸⠃⠀⠀⠀⣀⣀⠤⠤⠤⠤⠤⠄⣀⡀⠀⠀⣿⣿⡇
⠀⠀⠀⢠⣿⢷⡇⠀⠚⠉⠀⠀⠀⣀⣀⣀⣀⡀⠀⠄⠀⣼⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⣿⣇
⠀⠀⠀⣸⡿⣼⠀⣀⠤⠒⠊⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⡏⠀⠤⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠐⠒⠢⠀⢸⣿⣿
⠀⠀⠀⣿⣿⡇⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⡦⢤⣄⡀⢠⡇⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄⣐⠂⠈⣿⢹⡇
⠀⠀⢰⣿⣿⢁⣤⣾⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣾⣧⣾⣿⣿⠿⠿⠿⠷⠶⠶⠾⠿⠿⠿⣿⣿⣿⣶⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣭⣥⡶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⣿⠇
⠀⠀⠘⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⣀⣘⡿⠟⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡛⠁⠀⠀⢰⡟⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⡀⠀⠀⡞⠀⡠⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠚⡢⠊⢀⣴⣿⣷⣦⡀⢀
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡿⠋⠁⠀⠈⠻⣾⣧
⠀⠀⠀⠠⣾⣿⣿⣿⣦⣷⣿⣿⣿⣄⠀⠀⠀⣀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣪⠄⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⣦⢸⣿⣿⡆
⠀⠀⠀⢸⣘⣛⣛⣛⣹⣿⣿⣿⣿⣿⣿⣿⣗⢡⢀⣆⠀⠈⢀⡾⡞⠓⣂⣀⣀⠘⣿⠀⠀⠀⠀⠀⣸⡗⣞⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⡃⠉⡏⠀⠈⠀⠠⣀⡉⢫⠀⡍⠀⠀⠀⠀⣰⠋⡼⣹⣿⣿⡇
⠀⠀⠤⠤⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠓⡄⠀⣀⠀⠀⠀⠀⠈⠙⢺⠣⡀⠀⠀⣰⣷⣞⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢹⠉⠁⠀⠈⠉⠢⣄⠀⠀⠀⠶⠱⠈⠀⣴⣿⣿⣿⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⣀⠜⣿⣿⣿⡟⠛⠟⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠃⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣀⠤⠄⠒⣈⡡⢤⡐⠧⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣚⠔⠁⠈⠑⠢⢀
⡔⠻⡤⣐⠾⠟⠊⣉⡠⠴⡖⠀⠀⠀⠀⠀⡠⡔⠒⠊⠉⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠉⠒⢄⡀
⡇⠀⡟⠉⢱⠂⠉⠀⢀⠴⢂⣁⣠⠀⠀⠀⣄⠑⣄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠈⠒⣤
⡇⢀⡇⠀⢸⠀⠀⢰⡵⢉⣙⣯⡓⠀⠀⠀⠌⢢⡈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⢀⡠⠖⡻⣅
⡧⣼⢱⠀⡏⠀⣎⡉⠋⠀⠀⠉⠉⠓⢤⡀⢧⢀⡱⣄⠑⣄⠀⠀⠀⠀⠀⠀⠀⣀⣙⡎⢁⢔⡮⠊⠉⠲⢄
⠀⠘⠻⢤⡷⡚⢿⡨⡑⠦⣀⠀⠀⠀⡠⠜⠾⠣⣀⠈⢦⡈⢣⠤⠴⠒⠒⢉⡩⢅⡿⢮⡒⠁⠀⢀⣠⠔⡏
⠀⠀⠀⠀⠻⡫⡑⠪⣮⣄⠀⠙⣲⢉⡠⣶⠟⠢⣈⠑⢤⡑⢿⣂⣀⠤⠝⠒⠉⢁⣠⢴⣋⠴⠚⢉⠤⣪⠝
⠀⠀⠀⠀⠀⠈⠑⢕⡄⠉⢻⡖⢚⡭⠚⢇⠙⠢⣀⠑⢤⡈⠲⢄⡀⠀⣀⡤⠒⢉⢔⡮⢻⡤⣒⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠓⠁⠀⠀⠀⠑⢤⡈⠓⢤⡈⠲⣄⢉⡎⠁⠀⣪⣶⠵⢊⡿⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠈⠳⢄⡉⢓⡶⠟⠉⡠⢔⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⠑⠀⢔⡭⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠄⠀⠀⠀⠀⠀⠁⠢⢀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠊⢀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢎⡎
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠤⠀⠀⠀⠐⠀⠀⠀⠀⢠⢢⣾⡗
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠠⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣣⣿⣿⡁
⠀⠀⠀⠐⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⡐⣿⢯⣾⣿⡿
⠀⠀⠀⣈⣋⣒⣒⣂⠬⢔⡠⢁⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣼⢿⣿⢏⡾⠁
⠀⠀⠀⠹⠠⠀⣈⠈⠉⠛⠺⠿⣿⣦⣄⠀⠀⠴⠀⢀⣀⣠⣤⡤⡤⠤⢤⡄⣀⡀⠀⠀⠀⢜⡾⣡⣿⢫⡞
⠀⠀⠀⠀⠥⡿⠞⠚⠈⠉⢙⣻⣯⣭⣿⣿⣾⣾⣿⣯⣶⣶⣶⣶⠶⠒⠒⠂⢀⡈⢙⠲⢼⡾⣱⡿⣣⡟⠁
⠀⠀⠀⠀⢷⢶⣏⣹⠟⠛⠋⠹⠯⠝⡶⣓⣒⣹⣽⣿⣷⣿⡭⣟⣻⠕⢒⠶⢄⣈⠉⢊⡹⣿⡟⣰⠟⠀⠀⠀⢀
⠀⠀⠀⠀⡄⠄⢿⢿⠀⢜⢾⠢⠵⠿⠷⠶⠆⠁⠑⠒⠦⠄⢾⣿⣿⡫⠗⠦⠄⡈⠍⠲⣕⣟⣼⠋⠀⠀⠀⠠
⠀⠀⠀⠀⡇⢸⠘⣿⣬⣖⣀⡀⠤⢀⠀⠀⠀⠀⠀⢀⡀⣧⡾⣫⣮⡁⠀⠀⡀⠬⠑⠢⢼⡾⠁⠀⠄⠐
⠀⠀⠀⠀⢸⠀⠀⠙⣇⢉⠉⣉⣉⣽⡽⣿⠶⠶⠭⡿⣐⣫⠾⠍⡝⠁⠀⠀⠀⢀⣀⢤⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢷⢤⣾⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⣧⡄⠀⠀⠀⡡⠤⠚⠊⠉⠉⠉
⠀⠀⠀⠀⠀⠘⢧⢿⡇⠐⠴⠈⠀⣀⢀⣀⢀⡀⠀⣀⡀⢀⣀⢉⣸⠛⠀⠈⠀⡀⠤⠐
⠀⠀⠀⠀⠀⠀⠈⠛⠭⠍⠩⠤⠀⠓⠒⠒⠒⠉⠉⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠛⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⡀⡀⠀⢀⣾⠛⠛⢤⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠙⠃⠀⠈⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠿⣇⠂⠀⠀⠀⢀⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠈⡅⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢱⡀⠀⠀⢀⡐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢀⠀⠀⢐⡏⠉⠲⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠙⢦⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⠀⠀⠀⠘⣃⣀⡀⠀⠀⠀⠀⠀⢀⣤⠶⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠛⠋⢹⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣶⡿⣆⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠶⠶⠄⠀⠀⠠⡔⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠂⢀⣭⣭⣭⣵⣾⣿⡿⣛⡛⢛⣉⡭⠿⣛⣫⡄⠀⠀⠀⠀⠀⠀⣗⡀
⠀⠀⠀⠀⠀⠀⡀⢀⣠⣀⠀⠀⠀⢱⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣴⠉⠁
⠀⠀⠀⠀⠔⠫⠒⠕⣫⡵⢛⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠏⣠⣤⣶⡿⡿⠟⠒⠂⠤⢤⣤⡤⢤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠈⠉⠉⠙⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⠿⣅⣀⠸⣤⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠆⠀⣀⣀⠀⠀⡠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠰⠇⠀⠐⠒⠋⠀⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠰⡀⠀⠀⢀⢓⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣮⢔⢆⠀⠘⢆⠢⣀⣀⠈⠁⢐⠀⢠⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠠⠀⠐⠂⠀⠀⠻⣾⣧⡢⡀⠠⠑⢄⠈⠉⠃⡴⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡔⢄⠀⠀⡁⢰⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠤⠚⢿⣿⣦⠀⠀⠉⠠⠀⠀⠐⠀⠀⠄⠀⡀⠀⠀⠀⠈⠐⠠⡀
⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠻⣏⠻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡤⠤⠄⢂⣔⣀⠨⡐⠠⡀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠑⢄⡀⠀⠙⢿⣽⣷⡜⠀⠀⠀⠀⠔⠀⠠⠅⠐⣠⣀⡁⠀⠀⠁⠪⠀⠡
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠂⢉⡢⠒⠃⠀⠀⢀⠀⠀⠀⡀⢤⢊⡄⡀⠀⠀⠀⠈⣆⢀⣀⡀⣀⠆
⠀⠀⠄⠀⠀⠀⠀⠀⠄⡁⠀⠀⠀⠀⠀⠀⠀⢀⠀⢉⡢⠵⢆⠈⠀⠚⠊⠳⠶⠦⠛⠂⠘⠋⠉⠀⠀⠀⠀⠀⡂
⠀⠀⢆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⡘⠒⠊⠉⠉⠁⠀⠀⠊⠳⠀⢀⣀⠤⠤⠂⠀⠀⢂⠠
⠀⠀⠀⠵⠄⠀⠠⠤⠀⠀⣰⣦⡀⠈⠁⠀⠀⠀⠀⠀⢀⡠⠤⠒⠈⠁⠀⣀⠠⠤⠒⠂⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⢿⣆⡀⠀⠀⢀⠀⠀⢀⠀⢀⠂⠄⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡔⠛⠋⠛⠒⠒⠒⠳⠄⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠶⠶⠶⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠍⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣴⣶⣶⠶⠶⢲⡶⠒⠀⠀⠀⢤⡤⢤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣀⢀⡈⣀⣀⣀⣀⣀⣘⣤⡀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢿⠟⠋⠻⠟⠿⠋⠑⠿⠃⣿⠉⠉⠁⠀⠀⠀⠉⠋⠙⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣶⣶⣾⣶⣶⣟⣤⣿⣦⣰⣾⡐⣯⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠛⠻⠿⠿⢿⣿⣿⣿⠿⣿⣿⣿⡀⠀⠀⠀⠀⣀⣀⣀⡉⠀⣀⠀⠀⢀⣀⣤⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢶⣾⠟⠛⠉⠉⠉⠉⢉⡴⠖⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠐⠂⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠒⠶⠦⣤⣤⣤⡴⠋⣩⠤⠂⠀⠀⠀⠀⠀⢀⣸⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠇⠀⠀⠀⠀⢀⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠒⠶⠤⠤⠤⢤⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⣠⠴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠲⠤⢤⣀⠖⠙⠉

⠀⠀⠀⠀⣼⣿⢸⣿⢀⣿⣿⣿⣿⣶⣿⠿⢿⣄⠀⢸⠈⣀⠜⢳⣆⠹⣧⢰⣿⣿⢸⣿⢿⣿⣿⣿⡇⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣿⡇⣿⣿⣼⣿⣿⣿⣿⣿⣟⠀⢀⣿⣿⠈⢸⣿⣶⢿⣿⠀⣿⣾⡇⠀⢴⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⡿⣻⣿⣿⣋⠀⠚⠿⣿⣿⣿⣇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣾⣿⣿⠀⣾⣿⣯⣾⡿⠀⣿⣿⡇⠀⣼⣿⣿⣿⠛⣫⣽⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⡀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣽⣿⣿⡀⣿⣿⡟⠀⢠⠈⣿⣿⡇⠀⣿⣿⣿⣻⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣹⣿⣿⣿⠃
⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠉⣿⣿⡏⠀⢸⣿⣿⣿⢰⣿⣿⣇⠀⣿⣠⣿⣿⡇⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇⣸⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣏⠀⣿⣿⡿⠀⣸⣿⣿⣿⣾⣿⣿⣧⠀⣿⣿⣿⣿⡇⠘⣿⣹⣿⣿⣿⡇⣿⣿⣿⠟⣀⣿⣿⡇⢺⣿⣿⣿⣿⡇⣭⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡗⢰⣿⣿⡇⠀⣿⣿⣿⣿⡿⣿⣿⠏⢠⣷⠌⣿⣿⡇⢰⣿⣿⣿⣿⣿⡇⣿⣿⣿⠀⠐⣿⣿⡇⢰⣿⣿⣿⣿⣆⣽⣿⣿⡏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⢸⣿⣿⡗⠀⣿⣿⣿⣿⠂⣿⣿⠀⢸⣯⣖⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⢻⣿⣿⠂⢨⣿⣿⡇⠀⣿⣿⣿⣿⠃⣿⣿⣿⣇
⠀⠀⠀⠀⣽⣿⣿⣿⣿⠇⢸⣿⣿⡇⢸⣿⣿⣿⣿⠀⢻⣿⠀⣸⣿⠃⠛⣿⡇⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⡀⢸⣿⣿⡇⠀⣿⣿⣿⣿⠘⢿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⡇⢸⣿⣿⣿⣏⠀⢸⡟⠀⣿⣿⠀⠀⣿⡇⢸⣿⠼⢿⠿⠫⢟⢫⢝⣻⡇⢸⣿⣿⡇⠀⣽⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⢸⣿⣿⡇⢸⣿⣿⣿⡏⠀⣿⡇⠀⣿⣿⡗⠀⢿⡇⢸⣿⣴⢀⣠⠈⠀⠀⠐⣼⡄⢸⣿⣿⡇⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⣾⣿⣿⡇⢼⣿⣿⣿⡇⠀⣿⡇⠀⣿⣿⡇⠀⢺⢣⢁⣿⠉⠀⠀⠄⣰⡇⠰⣛⡇⢸⣿⣿⡇⠀⣿⣿⣿⣿⠀⢺⣿⣿⣿⡅
⠀⠀⠀⠀⢾⣿⣿⣿⣏⠀⣼⣿⣿⣧⢸⡿⣿⣿⡇⠀⣿⡇⢨⣿⣿⡇⠀⢾⡧⠄⠸⢆⠀⠀⠈⠁⠀⠀⣨⡇⣸⣿⣿⡇⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣾⣿⣿⡗⠸⠗⣛⣸⡇⠀⣿⠇⢸⣿⣿⡄⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠿⣇⢺⣿⣿⡇⢠⣿⣿⣿⣿⠀⣼⣿⣿⣿⡎
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⢻⣿⡇⠀⠀⠀⠀⡇⠀⠋⠀⠈⡇⣿⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠿⣿⣿⡇⢸⣿⣿⣿⡿⠀⢿⣿⣿⣿⡏
⠀⠀⠀⠀⢸⣿⣿⣽⣇⠀⢿⢸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡌⣿⡇⢸⣿⣿⣿⣧⠀⢺⣿⣿⣿⡗
⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣸⣿⡇⢸⣿⣿⣿⣿⠀⢺⣿⣿⣿⡇
⠀⠀⠀⠀⣸⣿⣿⣿⣏⠀⣾⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⢸⣿⣿⣿⡿⠀⣻⣿⣿⣿⡧
⠀⠀⠀⠀⢸⣿⣿⣿⡷⠀⢿⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⡃
⠀⠀⠀⠀⣾⡇⢿⣿⡟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⣄⡤⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⢸⣿⣿⣿⠁⠀⢹⣿⣿⣿⡇
⠀⠀⠀⢸⣿⡇⢸⣿⡧⠀⢸⣿⠀⠄⠀⠀⠀⠀⠀⢠⣦⠀⠈⢣⢀⡀⠀⢀⣶⠁⢃⠀⠈⡀⠀⠀⠀⠀⠀⠀⣿⡟⠟⠀⢨⣿⣿⡇⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⣺⣿⡧⠀⣿⡇⠀⠘⣿⢀⡆⠀⢠⠀⠀⠀⠀⠻⠿⢿⡟⠁⢻⣿⣆⠼⠆⠙⠁⡤⠧⠀⠀⠀⢀⡀⠀⡿⠁⢠⠀⠈⢻⣿⠁⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣯⡀⠹⡅⠀⢠⡏⢹⠀⠀⠈⣧⣀⡀⠀⠀⠀⢸⡇⠀⠄⢿⠟⣰⡄⠀⠀⡇⠀⣠⠆⠠⠘⠁⠀⠰⡀⠀⠱⣤⣾⠋⠀⠀⠀⣧⢸⣿⣿⣧
⠀⠀⢀⣿⣿⡏⣷⣾⣧⣴⠿⣃⡜⠀⠀⠀⢻⣿⣿⣶⣶⣶⣾⣿⣶⣴⣶⡾⢋⡛⢠⡄⠉⠀⠁⠒⠒⠃⠀⠀⠀⢱⡦⠐⠊⠁⡂⠶⠀⠀⢻⣿⣿⡟⢢
⠀⠀⢸⣿⣿⡇⢿⢿⡿⢡⣾⠟⠄⠀⠀⠘⢿⡛⠛⠋⠉⠉⠉⣿⣿⣿⣿⡇⠉⣿⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠓
⠀⠀⢐⣿⡿⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⠈⢿⣧⣸⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⢉⡀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠈⢿⣿⣠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠰⠖⠂⠰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡜⢃⣀⡀⣀⣋⠐⣤⣤⡤⢤⡶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠉⠀⠀⠐⠁
⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠔⠀⠻⢷⣀⠀⠀⠈⠉⠛⢻⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠓⠒⠒⠒⠒⠶⠤⠤⠤⠤⢄⣀⣀⣀
⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡋
⠀⠀⢠⣴⣎⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠄⠀⢸⡄
⠀⢠⠇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠓⠒⠒⠶⠶⠶⠤⠤⣤⣤⣄⣄⣀⣀⣀⡠⠞⡁⠀⠀⢀⠔⡩⠃
⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠀⠈⠀⢀⠔⢁⠎⢀⠜
⠀⠀⡜⠉⠉⠓⠒⠒⠲⠦⠤⠤⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠔⠁⡰⠁⡠⠏⣀
⠀⠀⠘⡆⠤⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠒⠦⠤⠤⠵⣾⣅⣠⠎⠀⡔⠁⢀⠔⡇
⠀⠀⠀⠉⠒⢤⡀⠀⠉⠙⠻⢭⣁⠒⠒⠂⠤⠤⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⢎⡠⠚⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⠉⠙⠲⣤⡀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠀⠤⠤⣷⠗⠉⠀⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠈⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠈⠙⠢⢤⣀⡠⠒⠁⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠍⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣠⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠦⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⢀⡐⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⡛⠉⠀⠐⢢⡤⠀⢀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢆⠀⠀⠀⠀⠠⠘⠃⠀⠀⠉⠛⠁⠀⠙⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧⠑⡀⠀⡐⠀⢀⢟⠀⠀⠀⠀⠢⠄⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣎⣜⢤⠀⠀⠸⡌⠦⡀⠀⠈⠙⠁⢰⠀⠀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠄⠐⠈⢿⣬⣧⡣⡀⠀⠝⠦⡈⠉⠓⢒⠆⢈⠀⠀⠁⠁⠀⠂⠀⠑⢄
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠠⠀⠀⠈⢻⣿⣷⡜⢆⠈⠄⠈⠢⠀⡀⠴⠁⠀⠀⠈⡀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⢻⣿⣿⣆⠢⠈⢄⠠⡀⠃⠀⠠⠀⠀⢀⠀⡀⠀⢠⠀⠀⠀⠐⢄⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢸⢀⠰⠖⠋⠙⢿⣿⣷⡑⡀⠠⠀⠒⠀⠀⠀⠁⠂⠰⠄⠀⡄⡀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠈⢻⣧⠙⣽⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣤⠤⠔⠒⢢⣔⣀⠀⢌⠒⢤
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠙⢿⣌⣿⣦⠓⠀⠀⠀⠀⢀⠄⠀⠠⠍⠑⢢⡠⡚⠁⠀⠂⠉⠒⢅⠀⢣
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠣⠍⠒⢄⠤⠎⠀⠉⠉⠀⠀⠀⠠⠀⠀⡡⠊⡀⠀⠀⠈⠉⠉⠳⡄⠀⠄⠀⠆⠠⡀⠁⠈⠉
⠀⠀⢀⠁⠂⡠⠀⠀⡀⠀⠐⠂⠀⠒⠂⠀⢠⠊⠁⠐⠀⢀⠒⢤⡑⣄⡀⠠⠊⢡⣇⣸⣁⣂⡀⣠⠤⠀⠠⠷⠶⠒⠊⠉⠉⢁⣀⣀
⠀⠀⠘⠀⠀⠀⠄⠈⠀⠌⠐⠁⠀⠀⠀⠀⡆⣄⣀⣀⠴⠦⠬⠖⠋⢈⡳⣄⠀⠀⠠⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒
⠀⠀⠘⢄⡀⠀⠀⠀⡐⠠⠇⠒⠀⠒⠒⠀⣯⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⠤⠔⠒⠛⠉⠀⠀⠀⡕⢒⡂
⠀⠀⠠⠤⠛⠂⠒⠐⠒⠢⠍⠀⣜⣗⡄⠉⠀⠀⠀⠀⠀⠀⠀⢄⡠⠤⠒⠊⠉⠀⠀⢀⡀⠤⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⢿⣶⢄⠀⠂⠀⡀⠀⠀⠀⣀⠀⢀⡂⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⠟⠀⠀⠀⠀⠁⠢⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢘
⠀⠀⠀⠀⠀⠀⠔⠁⠀⠤⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⡠⠃⠀⠁⠐⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡋⠀⡔⠁⠀⠀⢀⢤⠢⠓⠃
⠀⠀⠀⠀⠀⠐⢶⣤⣤⣠⣄⡀⣤⣄⣀⠀⠠⠤⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠎⠀⠀⢄⠎⠊⠁⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠂⠒⠦⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡡⡰⠕⠈⠀⠀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠐⡷⢆⣀⡄⣀⠀⠀⠀⠀⠉⠐⠉⠲⢀⡠⢀⡀⠀⢀⢄⠢⠑⠁⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⣠⢊⣀⠀⠀⠀⠉⠒⠚⠴⢆⡠⢀⡀⠀⠀⠀⠈⠐⠺⠃⠁⠀⢀⡤⢖⠟⠁
⠀⠀⠀⠀⠈⡏⠈⠉⠉⠉⠉⠐⠒⠂⠙⠒⠚⠛⠛⠾⠥⠶⢄⣤⣀⣠⠴⠚⠉⡔⠁
⠀⠀⠀⠀⢠⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡠⠊
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠒⠂⠠⠤⠤⠤⠤⠤⠤⠊





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

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⣰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠁⠀⠀⠀⢀⡠⠖⠉⣠
⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠰⣿⣟⠳⠶⣤⡀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⡀⠔⠉⠀⠀⢀⣾⡦
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠉⠙⠲⠾⣿⣦⣠⣞⣣⣤⠤⠤⠤⠤⠴⠋⠀⠀⠀⠀⣠⠼⠋
⠀⠀⠀⢰⣿⠗⠶⠶⠦⢤⣀⣀⠀⠀⠃⠀⢀⣠⣴⣶⠾⢉⣉⡈⣿⡀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠻⠯⣍⡙⠒⠦⣄⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⣼⣿⣯⣿⣭⣭⣴⣶⣶⣿⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡉⠤⢀⠀⠀⠀⠀⠉⠓⠦⣤⣷⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠠⠀⠂⠁⠀⠀⠐⠃⠢⢄⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣌⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣉⡦
⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⠀⠙⠿⣦⡈⠛⢦⣀⠀⣀⠀⣀⠤⠒⢉⣠⠄⠀⣀⣀
⠀⠀⠀⠀⡺⠛⢦⡀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣽⠕⣁⠀⠀⠈⠙⢷⣦⣈⡿⢖⣉⣠⠴⠚⠉⢀⡴⠊⠀⠀⠀⠉⠉⠉⠒⠂⢠
⠀⠀⠀⠀⠈⢎⠢⠈⠢⡀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⣴⣷⣻⢔⢯⣵⡄⠀⠀⠀⠀⠈⠛⠷⠛⢫⡀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻
⠀⠀⠀⠻⣗⠀⠑⢄⠀⠈⠢⢀⣀⣀⣀⠈⠙⢷⣄⠀⣼⣿⣿⣷⣾⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡴⠋
⠀⠀⠀⠀⠘⢧⡀⠀⠳⣄⠀⡂⠀⢠⣤⣤⣤⣴⣿⣿⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠤⠤⣄⣀⡀⠀⠀⠀⠀⡠⠊⡰⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠒⣒⢫⣛⣩⠽⠟⠛⠉⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠛⠤⢤⣀⡉⠉⠑⠒⠚⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⠿⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠓⠒⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⣠⣖⣉⣩⣷⣖⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⣩⡷⣾⠛⠛⠉⠉⠻⡿⢿⣿⣿⣿⣿⠿⡷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠙⠛⣿⡏⠀⠘⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⣿⡿⣿⣶⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠻⣷⣘⠛⠁⣠⣾⣷⣤⡀⠀⠀⢠⢶⣻⣿⣶⢿⣿⣿⣛⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢨⡛⠛⠛⠋⠀⠀⠙⠿⣷⣄⣸⣾⣿⡶⣿⣿⣿⡾⣿⣿⣼⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⢶⣦⣴⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣿⣿⣿⣿⣿⣻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡅⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠋⣉⣉⠉⢻⡟⣿⣽⣿⣿⣿⡇
⠀⠀⠀⣀⣠⡤⠶⠛⢻⡄⠀⠀⠀⠀⣀⠤⢤⣤⣄⡀⠀⠀⠀⠠⣯⣀⣀⣀⣠⣤⠶⠛⠁⠀⠀⠀⠀⠀⠉⠀⢀⣼⣷⣿⣿⣹⣿⠋
⣴⡾⠛⢉⣁⣤⡴⠀⠀⠻⡄⠀⣠⢞⡵⠚⣉⣀⠀⠹⣆⢀⣀⢸⣏⠉⠉⢉⣥⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠞⠙⢿⣿⣻⡿⡿⠁
⠘⣽⣆⠙⢉⣡⣠⠶⠞⠀⠙⢶⣅⣯⣴⣯⣥⣾⣿⣿⣿⣯⣨⡇⠙⠛⠿⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠿⠟⠁
⠀⠹⣿⣆⠙⢉⣀⣀⣴⣛⣿⣿⣽⣿⣍⠙⢿⣛⣻⠋⢹⡅⠀⠀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠘⣯⣧⡈⠉⠀⠀⠈⠉⠉⠀⠀⠈⢷⡀⠈⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⣄⡀⢹⠷⠶⠶⠶⠶⢾⡇
⠀⣴⢾⡏⣷⣽⡞⢳⠀⠀⠀⠀⣀⣤⠞⠋⢹⣄⣀⣠⣬⡇⠀⠀⠀⠀⠀⣸⡿⢿⣿⣶⣤⡆⠀⣠⣾⡇
⣠⢇⢿⢰⡘⣌⣇⠸⣦⣠⠴⠛⠉⠀⠀⠀⣿⢉⣉⣀⣀⣿⠀⠀⠀⠀⠀⢈⡴⣿⠀⠟⠁⣠⠞⣽⠉⣇
⠹⣌⠻⡷⡝⠈⠙⠀⠙⠧⣄⣀⡀⠀⠀⠀⢿⠉⠉⠉⠀⢿⠀⠀⠀⠀⣰⠏⢰⠇⢀⣤⢞⣥⡼⠃⠀⣿
⠀⠙⠳⣾⣄⠀⠀⠀⠀⠀⠀⠉⢻⠛⣶⠶⠾⠦⣤⣤⣤⣼⣧⣤⣤⣴⣯⣤⣿⣴⣿⠤⠼⢧⣤⡀⠀⢹⡀
⠀⠀⠀⠀⠉⠛⠲⠶⠤⣤⣀⣀⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⣠⣇⣒⣰⠆⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢿⣛⠛⠛⠛⠛⠛⣺⠗⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⠋⢉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣸⠇⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠈⠉⠉⠉⠉⠉⠉⣹⠏⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣯⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡟⢸⡦⠶⠶⠶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠒⠒⠒⠚⡇⠸⠷⣶⣶⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠛⣛⣿⣿⣿⣥⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⢿⣿⡿⠿⠛⠋⠘⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⡠⢼⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢹⡤⡀
⠀⠀⠀⠀⠀⠀⠀⡆⡇⣾⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⢸⡇⡇⡆
⠀⠀⠀⠀⢀⣼⠉⡇⡇⡟⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠈⡇⡇⡏⢹⣄
⠀⠀⣠⣤⣿⣽⢸⢇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⢷⢸⣿⣧⣤⡀
⠀⠀⣿⣿⣿⡟⢸⢸⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⢸⠸⣿⣿⣿⡇
⠀⠀⣿⡿⡏⡇⢸⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⢸⠀⡏⣿⣿⡇
⠀⢸⣿⠰⠁⡇⣸⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⠀⡇⠡⢹⣿
⠀⢸⣿⠀⠀⡇⡏⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡈⡇⡇⠀⢸⣿
⠀⣸⡇⠀⠀⠀⡇⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡇⡇⠀⠀⠀⣿⡀
⠀⣿⡇⠀⢀⠀⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⢈⠀⠀⣿⡇
⠀⣿⠇⠀⠈⠀⠃⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠇⠈⠀⠀⢿⡇
⠀⣿⠀⠀⠀⠀⠀⠄⣐⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢐⡀⠄⠀⠀⠀⠀⢸⡇
⢸⣿⠀⠀⠀⠀⠀⠀⣀⡉⠑⠺⢽⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⢋⣁⣠⡤⠼⠘⡅⠀⢸⣿
⢸⣿⠀⢀⠄⠀⠀⠀⠠⠬⠭⣓⣲⠬⣝⡿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⣟⡭⢤⣚⡯⠭⠥⠶⠒⠒⠒⢧⠀⠸⣿
⣼⡇⢀⠂⠀⠀⢀⣐⡒⠤⠬⣑⡒⠯⢝⣛⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢟⣛⠭⠗⣒⡩⠥⠔⣂⣀⠀⠀⠈⢂⠀⣿
⣾⣇⣒⠢⠭⢍⣒⠲⠬⢍⣓⠢⠬⣙⠒⠬⣙⠲⢍⡛⢿⣆⠀⠀⠀⢠⣾⠟⣋⠵⢚⡩⠔⢚⡩⠤⢒⣋⠭⠴⢒⣊⡭⠥⢒⣂⣿
⠙⠛⠿⠿⣷⣶⣤⣭⣙⡒⠢⠭⢑⠂⠌⠑⠂⠉⠂⠈⠂⠙⠳⡀⡰⠛⠁⠊⠀⠊⠁⠒⠉⠅⢒⡩⠥⠒⣚⣩⣥⣴⣶⡿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣤⣽⣥⣄⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡾⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠷⣶⣤⣴⣿⡿⠿⣿⣿⣷⣤⣴⡶⠿⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠄⠒⠂⠈⠉⠉⠁⠒⠒⠒⠤⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⡠⠤⣖⡒⠂⠉⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠓⠒⠲⠶⠤⢤⣤⣀⣀
⠀⠀⠀⢸⢡⠀⠀⠀⠉⠉⠓⠒⠲⠤⠤⣄⣀⣀⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⡟⠁
⠀⠀⠀⠸⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠤⢤⣄⣀⣀⡀⠀⠀⣀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠸⡇
⠀⠀⠀⣀⣑⣲⣤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠷⠓⠤⢤⣄⡀
⠀⠀⠀⠙⡟⠲⠤⣄⣀⠀⠈⠉⠙⠒⠲⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠐⠚⢷⣦⣀⣀⣤⡶⠿⠿⠗⠒⠒⠊⠉⠉⠀⠸⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠉⠙⠒⠤⢄⣀⠀⠀⠀⠀⠈⠉⠉⠛⠓⣶⣦⣤⣄⣀⣈⣇⣠⣤⡶⢿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⡶⠶⠒⠛⠋⠉⠉⠀⠀⠀⠉⠉⠙⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣀⣈⡿⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⢶⣿⣶
⣰⡿⠛⠳⠷⣦⣭⣛⣿⣶⣤⣀⠀⠀⠀⠀⠀⣼⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠔⠒⢒⣋⣩⣭⡥⠶⠶⠛⠛⠋⢹⠃
⣿⠁⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣶⣤⣀⣠⠏⡄⢀⣀⣀⣤⣤⠶⠶⠒⣚⣋⣩⣭⡤⠶⠖⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣯⣭⣴⡶⠶⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣘⣦
⠀⠉⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠶⠶⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣖⣷⣀⣀⣤⣤⡶⠶⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡀⠀⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⡺⠃⠀⠤⣄⠀⡠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠄⠀⠀⠀⠐⡃⠀⠈⠉⠁⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⢂⠀⠀⢠⠨⡀⠀⠀⢬⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢿⡗⣕⠀⠀⠳⡁⠂⠤⢄⠀⠃⠀⠂⠀⠀⠀⠢
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢻⣿⣎⢂⠀⠈⠓⠄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⡑⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠉⠀⠙⢿⡟⢶⡄⠀⠈⠀⠀⠀⠀⠈⠀⢐⠀⠀⢀⡀⢀⠀⠂⢀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠻⣦⣻⣆⠠⠀⠀⠀⢀⡀⠀⡘⠫⠁⠠⠔⠁⠑⠈⠂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠠⠓⢄⡠⠏⠉⠉⠀⠀⠀⠀⠀⡠⢂⠀⠀⠈⠉⠓⡄⠀⠀⠄⢠⡸
⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⡢⢔⢤⠀⠊⠡⠧⠧⣤⣄⠴⠄⠀⠗⠚⠉⠀⠉
⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠴⠶⠒⠚⠀⠉⠀⠂⠑⠄⠀⠀⣀⣀⠤⠀⠀⠀⢀⡀
⠀⠀⣈⠦⠀⠄⠄⢀⠀⢀⣤⡀⠂⠃⠀⠀⠀⠀⠀⢀⡀⠤⠐⠊⠉⠀⣀⠠⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡿⣧⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠄⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠴⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⠢⢭⣙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣈⠒⠦⣌⡉⠒⠦⣍⡛⠿⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡐⠤⢍⣒⠦⢌⣙⠲⠤⣀⡀⠀⠈⠙⠻⣦⣄⠠⠐⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠒⢬⣝⡲⠬⣉⡒⠤⢌⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⣙⣳⣦⣍⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣛⢻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⠉⠉⠉⠉⠛⠷⢶⣤⣄⣀⠀⠀⢀⣀⣼⠇⢀⣀⣠⣤⣶⠶⠞⠛⠉⠉⠉⠉⠉⠉⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⢶⡶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡟⢧⡛⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⠾⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠸⠀⠈⢷⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇⠀⠀⢀⣀⡈⣻⣿⣤⣴⣶⣶⣶⠶⠶⠶⠾⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣀⣠⣴⠶⠞⠛⠋⢹⣿⣴⠾⠟⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠿⠶⠶⠶⠶⠶⠶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣆
⠀⢀⣴⠿⠛⠉⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⣛⣒⣒⣒⣒⣒⡒⠲⠶⠶⠤⠤⠤⠭⠭⠭⠭⠉⠉⠉⠉⢉⣿⡉⠉
⢠⡿⠁⠀⠀⠤⣄⡀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣂⣀⣀⡀⠀⠀⠀⠀⠈⠭⠭⠭⣭⣭⣭⣭⣅⣒⠒⠀⠀⠀⠀⣾⠀⠉⠓⠦⣄⡀
⣿⠁⠀⠀⠀⠀⠀⠈⠉⠂⠀⠘⣷⡀⠀⠀⠀⠐⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⢭⣍⣉⣉⣉⣉⣒⡒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠙⠳⣦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⣐⡒⠲⠶⠶⠶⠶⣤⣤⣤⣤⣄⣀⣀⣀⣒⡒⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢷⣀⣀⣤⣤⡴⠶⣶⠟⠛
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠷⠶⠶⠶⠾⠿⠿⠿⠿⣿⣿⣽⣿⣭⣭⣤⣀⣀⣀⣀⣀⣀⣐⣒⣀⣀⣀⣙⣏⠁⠀⠀⢰⠇
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣼
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠠⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠐⠂⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡀⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡴⠶⠶⠶⠚⣛⣋⣉⣥⣤⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣌⣛⣬⣑⠒⠀⠀⠀⠀⠒⠚⢉⣉⣉⣩⣭⣤⡴⠶⠶⠚⠛⠛⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡇⠀⠀⡀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⢰
⣿⣿⣿⣧⣬⣽⣿⣾⣿⣦⣀⣩⣿⣿⣶⣶⣿⣿⣿⡇⠀⠀⠛⠒⠋⠉⠀⠉⠻⡿⠒⠄⠉⠉⠉⠁⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢁⠈⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩⠋⠁⠐⠀⢈⠐⠠⢀⠉⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⠕⠊⠁⠐⠀⣠⠐⠀⠀⠀⠀⠉⠂⢄⡈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⣠⠶⢬⣐⠠⣀⠀⠄⠁⠀⠀⠁⠀⠈⠀⠀⡈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣀⡐⠯⣁⠀⠀⠈⠑⠦⣕⠠⡐⠀⠀⠀⠀⠀⠀⠂⠀⠑⠢⢀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⣠⡾⠋⠀⢀⠀⠉⠂⠀⠃⠳⠈⠙⠢⣄⠢⢀⠒⠀⠁⠂⠠⠈⠀⡠⢂⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⠁⢀⡴⠋⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⢀⠀⠀⠀⠑⠢⣁⠢⢄⠀⠀⢀⠔⣠⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠔⢁⡴⠋⠄⠀⡠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠱⣄⠈⠐⣡⣾⢋⢔⣷⡀
⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⡠⢟⡄⠻⠀⢴⡔⠺⣧⡮⠀⠗⠈⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣠⣾⢏⢕⣵⡿⢿⡿⠂
⠀⠀⠀⠀⢠⡊⠵⠃⢠⣊⠀⠈⠉⠠⠦⠰⢁⡠⠂⠀⠀⠄⢀⠀⠚⣤⠀⡃⣠⢀⢄⣴⠞⣡⣾⢟⢕⣵⣿⣧⣽⣿⠁
⠀⠀⠀⠀⠸⣿⠷⣤⡀⠈⡳⣤⡘⠦⢹⢧⡼⣠⣆⠜⠃⡾⣧⠀⣄⠁⠀⠄⣽⣿⠟⣡⠞⢛⢕⣵⣿⠫⣛⣿⡟⠙⣧⡀
⠀⠀⠀⠀⠀⣿⢒⠢⡙⠳⣬⡐⢝⡦⠀⢤⡷⢌⣞⠀⣈⡓⢘⠠⠀⠘⠠⣪⠟⣡⣾⡏⢀⣵⣿⣓⢍⣻⣾⠏⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠻⣶⣍⡢⢕⠠⡙⠳⣬⡒⠤⡀⠞⣃⠚⠚⠉⡢⠀⠀⢠⠞⢁⣾⡿⢋⣴⣿⣟⣾⣵⣿⡿⡫⣪⡾⠋
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣶⣍⠪⢕⠄⡉⠑⠮⣑⠀⠲⢦⡀⠀⢀⡴⢁⡴⠿⢋⣴⠿⢫⢎⢝⣿⡿⠋⣨⣾⠏
⠀⠀⠀⠀⠀⣶⢦⣍⡻⣿⠿⣿⣆⡉⠀⢄⡈⠘⠵⣦⡑⠩⠓⢈⣴⣿⠗⣴⡿⢯⣔⢕⣠⣿⢟⢕⣾⠟⠁
⠀⠀⠀⠀⠀⢸⡗⠪⣙⠻⢿⣷⣽⡻⢶⣤⣌⡐⠥⡢⣙⠳⡶⢟⠝⣡⡾⢫⢘⢕⣹⣿⡿⡡⣴⡿⠃
⠀⠀⠀⠀⠀⠸⣷⣅⡂⢕⠢⢉⠳⢯⣷⣮⢿⣮⣕⡤⡑⢸⠕⣡⡾⢛⠢⠢⢕⣶⠋⠩⣪⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣬⡒⠅⡢⢉⠋⠒⢾⣽⡻⢶⣤⡾⠻⣶⣷⣍⣠⡿⠋⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣌⠁⠁⢀⠈⠚⠿⣷⣦⣴⡻⡿⣿⣿⠟⣱⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣍⡒⠄⡢⢙⠻⢦⣶⡟⣡⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢯⣖⠬⡑⠃⣾⠉⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣥⣉⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠒⢿⠀⠀⠀⢀⣤⡴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢼⠀⢀⣤⠞⠛⢁⣠⣿⡍⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⢀⣿⡟⠛⠻⠿⢶⣋⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠀⢸⠘⡿⠆⢀⠛⠀⢉⡛⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⠀⢸⣋⣠⠦⣈⠢⢾⣨⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣷⡈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣧⡻⣿⡿⠿⠿⢿⣿⣿⣿⣿⠀⢸⡿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⡿⠻⡟⠶⠻⣇⡀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣧⠀⢹⠒⢤⣼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⢿⣿⣧⣸⠀⢸⡀⠉⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠻⣿⡿⠒⠚⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠘⢿⠉⢳⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⡟⠛⢻⣿⣿⠟⠋⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⣧⠀⢸⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⠘⣆⠘⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡼⠀⠘⣆⢧⠀⠀⠀⠀⠀⠀⢸⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⢸⡸⡀⠀⠀⠀⠀⠀⠈⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⣧⣷⡀⠀⠀⠀⠀⠀⢣⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣇⠀⠀⠀⠉⠙⠷⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠄⠐⠂⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠐⠀⠉⠀⡠⠒⠀⠁⠀⠂⠄⡀⠉⠐⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢆⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⣀⠀⠠⠘⠂⠀⠀⠀⣀⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠐⠐⠀⠈⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠈⠑⠠⡀⠀⠀⢀⡀⠤⠄⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡠⠂⠁⢀⣈⡢⢄⠠⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠎⠁⠒⢠⡀
⠀⠀⠀⠐⠁⠀⡔⠁⠀⠈⠢⠑⠢⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠠⠄⠒⠂⠈⠁⠁⠀⣀⠠⠐⠊⠁⠄
⠀⠀⠀⠐⡀⠀⢄⠀⠀⠀⢀⠇⠀⠀⠑⠠⠰⠤⠤⠴⢒⡀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠊⠉⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⢄⡈⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠠⢤⠀⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⣨⡕⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠖⠋
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠉⠘⠙⠒⠦⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⢁⠔⠁⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⠄⣀⠀⠀⠠⠀⠀⠀⠀⣀⠤⠐⠊⠁⠀⡠⠚⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠒⠓⠐⠚⠌⠁⠈⠀⡐⠈⠁⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⠀⠤⠄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢈⠀⠀⢀⠃⠀⠀⠀⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠒⠠⠤⠈⠒⡠⠄⠊⠀⠀⠈⡀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠠⠥⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⢷⣦⠀⠀⣰⠟⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⣾⠋⠀⣼⠏⠀⠀⠀⠀⠀⣠⡦⠀⠀⢀⣤⠄⠐⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⠿⠶⢶⡿⢀⣴⠟⣳⠂⢠⣿⠖⣿⠆⣾⡷⢺⡇⢘⣿⠃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣿⠃⢼⣃⢼⣏⣠⣿⣗⣾⠏⣸⣿⣲⡿⠁⣾⣃⢼⡟
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠘⠧⠀⠀⠻⠦⠈⠁⠀⠉⣼⠇⠈⠀⢠⡟⠈⠁⠀⣀⣈⣡⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠿⠀⠀⠀⠸⠃⠀⠀⠸⠯⠴⠟⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⣠⡄⠀⠠⣄⠀⢀⣄⠀⠀⠀⠀⠀⠸⢿⠆
⠀⠀⠀⠀⢀⣴⠶⣾⠟⠓⢦⠀⠉⣉⡉⠙⠛⠓⠶⢶⣿⣥⣾⣏⣀⣀⣠⡿⠀⣼⠏⠀⠀⠐⠂
⠀⠀⠀⠀⠘⠇⣼⣯⣤⣤⠞⣀⣤⡍⠡⣞⡆⠀⢠⡿⢁⡾⢁⣭⡭⢉⣡⡤⣴⠏⠀⢀⣴⠶⡤⠀⣠⡖⢀⡶
⠀⠀⠀⠀⠀⣸⡏⠈⠀⢈⡇⣸⠏⢀⡴⣫⡟⢀⣾⠁⣸⡷⢫⡿⢡⡟⠁⣰⡏⠀⣠⡿⢁⣾⠃⣰⡟⢠⣿⠃
⠀⠀⠀⠀⠰⢿⣤⣤⠶⠋⠀⠛⠚⠉⠀⠻⠞⣻⡇⠀⠟⠀⠘⠓⠚⠛⢛⡉⠛⠛⠉⠓⠁⠛⠂⠙⠓⣽⠇
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⣰⠏⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⠲⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣾⣏⣀⣠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢉⣩⣿⠟⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠾⠿⠛⠋⠁⡀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣯⣽⣿⣭⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⣀⠀⢀⠀⠀⢀⡠⠒⠊⠉⠀⠀⠀⠀⠀⠉⠐⠢⣀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⢰⡀⠀⠀⠀⠀⡇⠐⢘⡀⢂⠇⢠⠋⠀⡄⢠⢠⣤⠄⡀⠀⢀⠀⠀⠀⠀⠑⠄⠀⠢⠅
⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⡦⣸⣼⠛⢁⡄⠀⠀⠀⠅⠀⠀⠀⢩⠀⠸⡀⠘⠛⠏⠾⠭⢸⣥⣠⣏⡀⣴⠚⡇⡴⠀⢀
⠀⠀⠀⠀⠂⠀⢠⢰⣼⡞⠖⠃⠀⡔⠁⠀⠀⠀⠐⠦⠠⡢⠐⠁⠀⠘⢌⠢⢄⡀⠀⠀⠀⠀⠀⠁⠁⠉⠉⠑⠀⣀⠀⠑⡀
⠀⠀⠀⠁⠀⠀⠈⠛⠈⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠒⠬⠉⣐⠒⠀⠤⠤⠤⠄⠐⡂⠨⠠⡑⢄⠘⡀
⠀⠀⠀⠀⡀⢁⠀⠀⠀⡠⢊⠔⠁⠀⠀⠀⠂⠀⠒⠁⠀⠀⠁⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠢⢂
⠀⠀⠀⠀⠐⠀⡀⠀⡀⠐⠁⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠈⠂⢄⠀⠀⠐⠀⠀⠉⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠆⠀⠀⠁⠀⠀⠀⠄⠃⠀⢠⡖⡆⠲⣌⣳⡐⡯⡅⢸⠀⠀⡐⠀⠀⠀⠀⠱⠘⡆⣯⠈⠋⠀⡀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠤⡀⠀⠘⠢⠇⠘⠘⠃⠁⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠀⢀⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠈⡆⠀⢀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠁⡀⠤⠀⠀⠀⠀⠀⠀⠀⡉⢀⡀⡀⠀⠀⠀⠐⠀⠆
⠀⠀⢈⠀⠢⡀⠀⠄⠀⠀⠀⠐⠀⠀⠈⠁⠉⠂⠂⠐⠀⠂⠀⠀⠁⠀⠀⠀⠘⠈⠀⠀⠐⠋⠀⠀⠐⠀⡨⢁⠇⠖⡇⠀⠈⠈
⠀⠀⠤⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠒⠒⠐⢄⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠜
⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠠⠀⢰⢤⡦⠶⡶⠲⠀⠴⠴⠦⡆⠠⠆⠖⠴⠀⠀⠀⢀⠀⠀⠐⠡⠢
⠀⠀⠀⠀⠀⠠⠀⠀⠁⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡐⢄⠀⠀⣀
⠀⠀⠀⠠⢖⠔⠊⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡎⠀⢀⡖⡶⠛⢶⠂⠀⠠⡁⠀⠀⠀⢰⠰⢥⠪⠕⠯⢲⠕⠀⠂⠀⠀⠀⠀⠔⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠
⠀⠀⠀⢠⠁⠀⢀⣄⡀⡀⣁⣠⠀⠀⠈⡆⠀⠀⡄⠀⠀⡆⢶⢡⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠀⠂⠀⣀⠀⠀⠁
⠀⠀⠀⢂⠀⠀⠛⠘⠙⠛⠙⠓⠀⠀⢀⠃⠀⠀⠁⠈⠉⠋⠈⠈⠉⢀⢈⠀⡀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢠
⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⢀⣀⠤⠒⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⢀⡠⠌⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠈
⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠤⠤⠤⠄

⠲⣌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⠀⠀⠈⠉⠉⠉⣏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⣀⣀⠀⠀⢠⡀⠀⠈⠁⢰⠀⠀⠀⠠⡈⠁
⠀⣿⡸⠀⢀⣠⣄⠀⠀⠀⠀⠀⠘⠻⠛⠛⠛⠛⠛⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⢥⣤⣤⣤⣤⣤⣤
⣐⣻⠠⠀⡈⠈⠀⢡⡀⣠⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀⠈⠙⠁⠀⣀
⠈⠉⠀⠀⠀⠠⢀⣾⡆⠈⠙⣦⡀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠠⣾⡿⠟⠛⠿⣿⡆⠀⠀⠂⠀⠀⠀⠁⢀⣀⠤⠴⠚⠓⠀⠠⠈⡏⠈
⠐⠐⠆⠀⠠⠶⠿⠟⠀⠀⠈⠛⠿⢦⡀⠀⠀⠀⢀⠀⠤⠀⠠⠀⣺⠇⠀⠀⠀⣿⠃⣀⠀⠈⠈⠀⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠆⠀⠀⣤⣠⡀⠉⠉⠛⠶⣄⣠⣤⣤⣤⣤⣤⣀⠀⠉⠙⠒⠶⠤⠴⠚⠋⠉⠉⠁⠀⠀⠀⠀⢢⡁
⠛⠛⠛⢀⡀⠀⡀⢤⠀⠀⠀⠀⠠⢺⠟⠁⠙⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⣠⣆⣤⣤⣤⣠⣤⣴⠖⠀⠀⢰⣴⠶⠶
⣤⣤⠀⠀⠀⠏⠁⠀⠉⠀⠀⠀⡦⡍⠈⠀⠁⢸⢼⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠉⠟⠁⠀⠈⢻⠟⠃⠂⡄⠰⠈⠈⢠⣤
⠝⠉⠉⠋⠀⠀⠄⠠⠀⠚⠀⢀⡯⠦⡄⠀⠀⢾⡡⠕⠀⠀⠀⡴⠦⠀⠀⠠⠐⠂⠄⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠐⣠⠂⠀⠀⠠⠀⣄⠋⠀⠀⠀⠀⠀⠀⠀⣠⢀⡴⠋⠀⠀⠑⢆⠁⠀⠀⠀⠀⠀⡈⠙⠛⠒⠤⡤⠤⠒⠒⠒⠀⢀⠀⠀⠈
⠈⠀⠀⢠⡾⠛⠿⠷⠶⣦⣷⡄⠀⠀⠁⠀⢀⠐⠋⠘⠋⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐
⠶⠶⠀⠉⢤⣤⣄⣀⣀⣀⠰⠿⣤⣀⠀⣀⣬⠤⠴⠒⠂⠀⠀⠂⠀⠂⣤⣤⠀⠒⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣄⣂⢀⣀⣤
⠀⠀⠀⠀⠉⠉⠉⠈⠋⣙⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡤⠠⠤⣰⣶⣶⣶⣦⡄⠀⠀⠉⠉⠁
⠀⠀⢀⠀⠐⢋⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠴⡤⠀⠐⠀⣀⣀⣀⣠⣤⣤⣴⣄⠀⠀⠀⠨⠇⠉⠀⠁⣹⠛⢿⡅⠓⡍⠀⣀⣀
⠀⠀⠉⠉⠛⠒⠶⠤⡴⠖⠛⠁⠀⠀⢐⡀⢀⠀⠀⠀⠚⡿⠉⠉⠉⠀⠀⢀⣙⠃⠀⠀⠘⠀⠂⠀⠂⠂⠀⠠⠀⡼⠏⠉⠄⠨⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠼⠖⠺⠿⠿⠛⠻⠟⠀⠒⠓⠶⠦⠤⣄⣀⣠⠤⠶⠶⠂⠠
⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⣴⣄⡠⠖⠒⠒⠀⢀⡀⠀⠈⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠠⠀⠀⢀⡴⡶
⠉⠀⣀⣠⠤⠶⠚⢠⠀⢀⠀⠄⢀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠈⠁⠂⠀⠂⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠬⢤⣀⡀⠀⠘⠑⠁
⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡞⠛⢿⣿⣷⡀⠀⠀⠀⢲⡄⠀⠘⠉⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠉⠉⠊⠀⠈
⣠⡤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⢀⣸⣿⡏⠀⠀⠄⢻⣿⡟⠀⠀⠤⠭⣴⣁⡐⠀⠐⣀⣠⠄⠀⡘⠀⠀⠂⠀⠀⠀⠀⠀⣀⢀⣀
⠁⣤⠀⠀⠀⡠⢀⢀⡀⡀⠀⠐⠛⠦⣄⣡⠹⢇⡀⡀⢀⡟⠛⠀⠀⡀⠀⠀⠀⠀⠉⠁⠋⠉⠀⠀⠠⡇⠤⠤⡀⠀⠀⣀⣀⠘⠛⠉⠉
⠾⠷⢀⠀⡁⢠⡼⠃⠛⠻⡾⠄⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠀⠈⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⣤⠡⠌⢅⠑⡤⠼⠷
⠀⠀⠀⠀⠀⠘⠅⠄⠠⠀⡞⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠁⠀⠀⠒⠀⠀⠀⠀⠀⢱⠀⠉⠀⠀⡦⠄⠀⠄⠰
⠉⠉⠉⠓⠢⢤⠤⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⣀⣠⠴⠖⠒⠀⠈⠀⠀⠀⠛⠛⠛⠛⠾⠶⡤⡤⠴⠒⠒⠒
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣄⣀⣀⡒⠳⠤⣤⣤⠬⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠒⠀⠉⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠁⠐⠂⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⢦⡀
⠀⠀⠀⠀⠀⢠⡑⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢧
⠀⠀⠀⠀⠀⡦⠡⠁⠀⠀⠀⠀⢀⡀⠤⠀⠒⠀⠀⠀⠀⠀⠀⠐⠂⠄⡀⠀⠀⢰⠀⠀⠀⢀⠤⠒⢀⠈⠭⠥⠤⠤⠭⠥⠄⢀⣀⠂⠠⠄⡀⠀⠈⠌⣧
⠀⠀⠀⠀⣜⢠⢃⠀⠄⣒⠈⠠⠄⠒⠀⠈⠉⠀⠀⠀⠀⠈⠉⠀⠒⠤⠀⡑⢄⢸⠀⡠⡊⠔⠊⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠤⠤⠀⠉⢐⣂⠑⠺⠞⡄
⠀⠀⠀⢸⡃⣉⣀⡀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣀⣤⣄⠀⢺⠎⢀⣄⣀⣂⣈⡉⢉⡉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠒⠂⠀⠀⠨⠄⠀⣹⡄
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉






⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠚⠛⣷
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⣀⣤⣄⢠⣿⠤⣤⠀⣠⡤⠤⠶⠶⣦⣠⠖⠒⠲⢶⣤⠶⠒⠒⢲⡆⢀⣀⣀⣀⢀⣠⠖⠚⢳⡄⠀⣠⠞⢛⣛
⠀⢀⡾⠁⠀⠘⠋⠀⠀⠸⣶⢻⠀⠀⠶⣶⢿⠻⣄⡀⠘⠯⣏⠓⢢⠀⠛⣷⡿⢁⣄⠹⣾⡇⢠⣺⡿⠷⢾⠋⠀⠈⢙
⠀⣾⡇⠀⠀⡀⠀⣀⠀⠀⠹⣸⠀⠀⠴⠿⡾⣷⠒⠋⠀⢠⣿⠉⠉⠀⢠⢿⠇⢀⡁⠀⣿⢇⡀⣩⡀⢠⣾⠀⠀⠈⠛
⣸⠻⠤⠤⠼⠙⠶⠉⠲⠒⠒⡿⠶⠶⠦⣴⡇⠙⠶⢖⡶⣿⠘⠒⠒⣺⣯⠘⢓⣯⣈⣒⣿⣄⣭⡁⣳⡟⢸⣐⣲⣒⣶
⠻⠤⢤⣤⡴⠳⠤⣾⠳⠶⠚⠛⠓⠛⠛⠋⠙⠛⠛⠋⠀⠙⠛⠛⠋⠉⠉⠉⠉⠀⠉⠁⠀⠀⠀⠉⣿⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠲⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣶⠂⠀⠀⣠⡾⠋⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠖⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⢀⣀⣀⣀⣠⡀⢀⣤⣤⣤⣄⣀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⣤⣬⣧⡀⠀⢀⡤⠤⢤
⠀⠀⣼⠛⠉⠳⡞⠉⠉⢻⣠⢿⠉⠁⣠⣤⣷⣟⠀⠀⢤⣼⡿⠤⣄⠀⠾⣥⣴⠛⠛⢳⣴⡏⢀⣴⣼⣇⣰⠿⠀⠰⠿
⠀⣼⡇⠀⠀⠀⠀⠀⠀⠈⢿⢸⠀⠀⣨⣿⣹⣬⠭⠗⠀⢨⣷⠶⠞⠀⢀⣿⡇⠘⠃⠈⣿⡄⠘⠚⠁⢀⣹⠀⠀⠠⠾
⢰⣿⣀⣀⣰⢦⣠⠶⣄⣠⣬⣾⣄⣀⣈⣀⡇⢧⣀⣀⣤⡾⠻⣤⣤⢤⡎⠼⢤⡼⠳⠤⣿⠙⢒⠚⢦⡟⣿⠰⣄⣀⣀
⣿⣀⣁⡈⣡⣤⣁⣤⣤⣠⡴⠧⠤⠤⠤⠟⠷⠤⠤⠞⠋⠳⠤⠤⠶⠚⠳⠶⠞⠉⠛⠛⠉⠉⠙⠳⢺⡇⠈⠛⠒⠒⠋
⠀⠉⠉⠉⠁⠀⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠤⠶⠶⠶⠶⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠉⠙⠷⣄⡀⠀⠀⢻⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣻⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣤⣤⡒⠒⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡤⠚⠋⢦⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣰⠟⠻⠟⠉⢻⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢹⡅⠀⢀⢠⣠⣠⣷⣤⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠀⠀⠆⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⢷⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⡀⠀⣠⠎⣱⣶⡶⠖⠒⠍⣁⠴⠀⠀⠀⠀⠳⠀⠀⠀⠈⠻⢶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡜⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠜⣡⣾⠿⠃⠀⠀⠀⠀⠈⠄⠰⣄⡀⠀⠀⠠⠀⠀⠀⠀⠀⠙⠻⠦⠤⠄⠀⠀⠀⠁⠀⠀⠀⡇⠰⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⡰⢁⣾⡟⠁⠀⠠⠄⣄⣀⡀⡠⠂⠀⠈⠛⠀⢄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⣤⣴⠷⣔⠀⠺⣿⣿⢋⢩⠙⣿⠐⠄⢹⣿⣿⣿⡇
⠀⠀⢠⡴⠿⠋⠀⠀⠀⠀⠀⠀⠢⢄⣀⡀⣀⣀⠂⠀⢸⠆⣆⠀⠀⠀⠀⠀⢠⡀⠀⢀⠀⠀⠀⠐⠁⠀⠒⠀⠈⠹⡿⣿⣿⣄⣁⣤⡿⠿⠦⠞⣛⠋⠙⠁
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⢀⡀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣘⣧⡤⠶⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣠⣏⡦⢄
⠀⠀⣠⣄⢀⠀⠀⠠⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣷⣽⣿⣴⣏⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠄
⠐⢻⡟⢠⣏⡄⣰⣦⢰⣦⡆⣼⡃⠂⠀⠀⠀⠀⣿⡿⠋⣿⢻⣿⣿⡿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠰⣆⡸
⠀⢿⠀⣾⢸⡇⠇⣻⢘⡙⣇⠛⠹⠄⠀⠀⠀⠀⣿⡏⣼⢇⡆⢁⠸⢃⢹⠃⢿⣿⡇⠀⠀⠀⠀⣤⣾⢓⡖⠀⢐⠒⢈⠀⠀⡤
⠀⠀⠀⠀⣾⣰⢹⠋⢸⡇⡟⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡞⡷⡻⢛⡞⢺⢿⣾⣿⡇⠀⠀⠀⠀⢸⠃⢸⠙⡇⡏⡇⡾⢿⢸⣧⡁
⠀⠀⠀⠀⠀⣃⣬⣤⢤⡖⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⢣⣇⣿⣇⣮⣸⣿⣿⡇⠀⠀⠀⠀⠈⠀⠀⣶⢰⣠⠆⣴⢠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠩⠃⠙⠀⠙⠘


⣠⣤⡶⢀⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⣴⠖⣠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣴⠆⣠⠀⠈⡈⠀⡁⠀⠀⣀
⢠⡟⠀⡾⢴⢠⡞⡆⣼⢶⡆⣾⣔⠁⠀⠀⠀⠀⢸⠏⢰⠇⡶⣼⢱⢠⡷⡆⢸⣃⠂⠀⠀⠀⠀⢻⡏⢰⡗⣷⣾⢹⢰⡿⡇⢸⣧
⠈⠃⠀⢣⡜⣨⢡⡅⢩⡌⣅⠉⠙⠂⠀⠀⠀⠀⠘⠀⠘⢠⣃⡍⣼⠘⣅⢣⠘⠘⠃⠀⠀⠀⠀⠸⠀⠘⢠⣏⣌⣼⠈⣥⢫⠌⠘
⠀⠀⠀⠘⢣⠇⠿⠀⠺⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣸⢹⡇⠸⠇⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣸⠙⠇⠀⠟⠻⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢾⣆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠤⠤⣦
⠀⢻⣆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀
⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣠⠊⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣿⣿
⠀⠀⠀⠹⣧⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂⠈⠀⠀⠀⠀⢀⣿⣁⣤⠀⢀⣴⣶⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡷⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠘⣷⡄⠀⠁⠀⢀⠠⠐⠉⠀⠀⠀⢀⣠⣤⣶⣾⡿⠟⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠘⠀⠀⠀⠀⣀⣴⣾⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡇⠀⣠⣴⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠉⠒⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠐⠦⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣙⠯⠋⠀⠤⢤⣀⠀⠀⡤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⡀⠞⠕⠀⠀⠒⠚⠁⠀⡚⠂⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡕⢄⠀⠀⡐⠀⡈⡦⠀⠁⠀⠀⠦⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⢊⢢⡐⠀⠀⣧⠱⡀⠠⠀⠐⠶⠖⢠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣀⡀⠌⢿⣿⢳⡱⡀⠀⠨⠣⡈⠓⠒⠦⠤⡄⠘⡄⠀⠃⠂⠀⠤⠈⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠈⠠⡀⠀⠀⠻⣿⣿⣌⢢⠀⠡⠈⠲⢄⠀⠈⢀⠞⡇⠀⠈⡀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠠⠀⠀⢀⠻⣿⣿⣦⡱⡀⠡⡀⢨⠀⡇⠁⠀⠁⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡆⠀⣆⣤⠾⢿⣯⣷⣌⢄⠑⠌⠂⢅⠀⠈⠀⠦⢀⢀⣐⠀⠀⠁⠀⠀⠀⠀⠑⠢⣀
⠀⠀⠀⠀⠘⠐⠀⠀⠀⠀⠀⠀⣗⠊⠁⠀⠀⠀⠻⣯⠟⢦⢦⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⠐⠀⠁⣀⡀⢀⡀⢀⢑⠤⣀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⠀⠈⠢⢀⠀⠀⠀⠹⢷⣤⠿⣷⡈⣤⠂⠀⠀⠀⠀⢀⠀⠀⠸⠮⣉⠁⢀⠤⠛⢚⠒⠌⡒⠄⠱⡀
⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢠⠑⣤⡀⠀⢈⡻⠳⠿⠿⡀⠀⠀⠀⢀⠐⠧⢀⠔⠉⠐⡀⠷⠵⢦⡀⠀⠀⠁⢈⠢⡀⢱⠀⡠⠤
⠀⠀⠀⠎⢀⠀⠀⠠⠀⠠⠀⣃⠀⢀⡀⢀⡉⢀⣨⠖⢉⠁⠄⡐⠠⢄⠀⠀⠀⣡⠤⣶⢁⡞⢠⠀⠀⠀⠀⠀⢹⣀⣀⣁⠬⠤⠼⠎⠀⠂
⠀⠀⢠⠄⠂⠔⢁⠂⡄⠀⠄⡡⠂⠀⠀⠀⠀⠎⠀⠀⠀⠀⠂⢨⣑⣬⠳⣆⠈⠀⠐⠛⠮⠷⠦⡶⠴⠛⠒⠀⠛⠉⠉⠀⠀⠀⠤⠀⠒⠒⠂
⠀⠀⢠⠀⠀⠀⠂⠀⠀⡈⠠⠁⠀⠀⠀⠀⢠⡴⡶⠶⠖⠛⠋⠊⠁⠐⠒⠬⠳⡄⠐⠂⠀⢀⣀⣠⠄⠀⠀⠠⡀⡀⠀⠀⠀⠀⠂⠉
⠀⠀⠨⡓⣄⠀⠀⡀⡐⢀⣉⠀⠀⣤⡤⢤⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠉⠉⠀⠀⢀⣀⡠⠴⠡⠤⠁
⠀⠀⠒⠊⠉⠉⠀⠀⠉⠙⠂⠈⢱⣿⡿⡀⠄⠀⠀⠀⠀⠀⠀⠁⠢⠔⠚⢉⡀⠀⢀⣀⠠⠔⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⣟⣻⣎⣄⠀⠀⠀⠄⡀⠀⠀⠤⣀⠨⠅⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠠⣻⡇⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠾⠀⠀⢣⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢠⢹⠁⠀⢀⡙⠀⠀⠀⠚⠋⠉⡉⠀⠀⠀⠀⠠⠰⣄⠰⣄⡻⣎⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣼⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠡⡈⢳⣜⢿⣦⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠙⡿⠛⡞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢓⣆⣰⣇⣀⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠓⠁⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢴⣶⣿⣿⣿⢿⣿⣿⣿⣯⣟⠛⣶⣶⣤⣄⡀⠀⠀⠀⠀⢀⠞⠓⠒⠤⣀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣟⢻⣿⣻⢿⡞⠟⠛⢻⣏⣷⡿⣾⠀⠀⠙⠻⠿⣿⣶⣤⣴⣿⡾⠋⠀⣠⡈⠌⠳⠲⣤
⠀⠀⠀⠀⠀⠀⠁⢽⠻⡿⠊⠈⠁⠀⠀⠀⠀⠀⢏⢸⣵⠃⢟⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠀⢠⠎⠁⠁⣀⠒⢄⠀⠙
⠀⠀⢀⠀⠀⠀⠀⠩⠀⡃⠀⠀⠀⠀⠀⠀⠀⢠⠏⣺⣼⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠔⠀⠐⠆⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠄⠀⠀⠀⢠⡑⠀⠀⠀⠀⠀⠀⠀⠀⢘⠗⣿⡖⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡒⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠒⠀⠀⠀⠀⣍⠃⠀⠀⠀⠀⠀⠀⠀⣾⡇⣷⡏⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆
⠀⠀⠀⠈⠁⠀⠀⣀⣽⢳⠀⠀⠀⠀⠀⠀⢸⠙⢶⣶⠅⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢁⠢⣀⠀⠀⠐⠀⣆⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠿⠶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠟⠋⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣻⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⠿⢛⣫⣿⣇
⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⡿⠟⣛⣭⣶⣿⢿⠛⡇⢹
⠀⠀⠀⠀⠲⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⠿⢛⣫⣵⣶⢿⢿⠏⡇⢹⠸⠁⠃⢸
⠀⠀⠀⠀⠀⢸⣏⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠟⣛⣭⣶⣿⢿⡟⡏⠏⢸⠘⠀⣃⣸⣤⠶⠛⠉
⠀⠀⠀⠀⠀⣸⡗⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⠀⠀⣀⣤⣴⣾⣿⣿⣿⠿⢛⣫⣵⣶⣿⡟⣿⢹⢸⠈⠁⡇⣀⣬⠴⠛⠙⠻⣶⣄
⠀⠀⠀⠀⠻⠿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⠿⣛⣭⣴⣾⡿⣿⢻⠙⠇⠃⠀⢀⣠⡴⠖⠛⠉⠀⠀⠀⠀⢀⣀⣉⣿⣦⣤
⠀⠀⠀⠀⠀⠀⢀⣉⣿⠿⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⢹⡿⣿⠻⢹⠁⠇⠟⠀⣀⣤⡶⣛⣉⣥⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠘⢿⣯⠀⠀⠹⣏⠛⠻⢶⣤⣄⡀⠀⠀⠀⣇⠘⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⡿⣿⠿⡟⣿⢻⡏⣿⢹⠉⡇⢸⠁⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠈⢳⣶⣶⣿⣿⣿⣿⣦⣼⣷⡾⣿⢿⠛⡏⣿⢹⡇⣿⢸⡇⡿⢸⠀⡇⢹⠈⠇⠏⠸⠀⢃⣚⣀⣇⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⢸⡇⡟⢸⡏⡇⡇⣿⣼⣧⣇⣼⣸⠀⡇⠘⠘⠃⣟⣸⣁⣥⣯⣴⣦⠶⠶⠶⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣸⠇⡁⢸⣃⣷⠿⠋⠀⢉⣉⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⡿⠛⠁⠀⢄⣐⣂⠠⠬⠭⠭⣍⣁⣒⣒⠶⠤⢤⣀⠉⠛⢿⣦⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠉⢀⠠⠈⠁⢒⡒⠤⠭⢭⣉⣒⣒⠲⠤⢬⣍⡑⠒⠀⠀⣠⡼⠛⢉⣉⣉⠙⠛⠛⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠁⡀⠂⠤⠄⣁⣐⡒⠠⠬⢉⣀⣒⠂⠤⢬⣉⠒⠂⠁⢀⣤⠞⠉⣀⡒⠠⠬⠭⢉⣉⡒⠢⠤⣈⡙⠛⠷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⢀⡐⠂⠠⠄⠈⡀⠂⠀⠭⠉⣛⠒⠢⠬⣉⠓⠒⠈⠀⣀⡴⠛⢁⣀⠀⠭⠭⠉⣛⠒⠲⠤⢍⣙⠲⠤⣍⡓⠦⣄⡉⠛⠻⢶⣤⡀
⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⣀⠒⠨⠭⢍⣓⣒⠢⠬⣀⣀⠒⠠⢉⡑⠀⠀⢀⣴⠞⠉⠀⢀⡐⠈⠈⠍⢉⡐⠒⠠⢍⣙⡒⠮⢍⣑⠢⢭⡑⠂⠁⢀⣤⠞⠉⢷⡀
⠀⠀⢰⡿⣏⣁⡐⠂⠤⠄⣀⣀⠀⠡⠉⢃⠒⠨⠭⣙⡒⠠⠍⠓⠀⠀⣠⠶⠋⠀⠀⠈⠀⠀⡐⠈⠩⠉⡐⠂⠬⢅⣐⠢⠍⣙⠒⢬⠙⠂⠀⣠⠴⠋⠀⠀⠀⣨⣿
⣀⣴⡿⠀⠀⠍⠉⠙⠛⠲⠶⠤⣭⣉⡓⠒⠤⢍⡐⠢⠌⠁⠀⢀⡴⠛⠁⢀⠒⠂⠠⠄⣀⡐⠒⠠⢅⣒⠢⠬⣁⠒⠬⣉⠒⠤⠉⠀⢀⡴⠞⠁⠀⠀⢀⣠⣾⠿⠋
⠙⠛⠶⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠒⢦⣌⠁⢀⣤⠞⠉⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠩⢁⡒⠦⢭⣐⠂⠍⡐⠂⠁⠀⣠⠖⠋⠀⠀⠀⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⢶⣤⣌⣀⠀⠀⠀⡠⠬⢿⣏⣀⡀⠂⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠄⡒⠠⠅⡂⠠⠉⠂⠀⣀⡴⠋⠁⠀⠀⢀⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⣆⠰⣿⣶⡍⠀⠀⠉⠁⠀⠀⠄⠤⣀⠀⠀⠨⣁⠂⠀⠉⠒⠄⠁⢀⣠⠞⠉⠀⠀⠀⣠⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⡴⠷⢦⣤⣀⡀⠀⠀⠀⠀⠉⠓⠶⣤⣉⡂⠀⢀⣠⠶⡫⠐⠁⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⣤⣄⡀⠀⠀⠀⠉⠛⢷⣟⡵⡊⠀⠀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣄⡀⠀⢳⡊⢀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣿⡿⠋

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢸⡏⠀⠀⠀⠀⠀⠉⠉⠙⠛⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠛⠉⠉⠁⠀⠀⠀⠀⠈⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⠓⡶⠛⠛⠓⠒⠒⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠒⠒⠛⠛⠓⡖⠋
⠀⠀⠀⠷⠤⡶⠶⠶⠶⠤⠤⢤⣤⣉⣛⠲⢤⣀⠀⠀⢸⠀⠀⢀⣠⠴⢚⣋⣡⣤⠤⠤⠴⠶⠶⠶⡶⠴⠇
⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠛⠓⠒⠚⠛⠿⠷⢿⣷⣦⣼⣤⣶⣿⠶⠿⠿⠛⠒⠚⠛⠛⠛⠛⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉




⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠒⠉⠉⠉⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⢀⠄⠐⠒⢠⣤⠀⢆⣠⢀⣼⣀⣀⣀⣀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠔⠁⠀⠀⠀⠀⠉⢇⣀⠀⠀⠀⣠⣀⣀⠠⡍⠉⢡⢄⣙⡻⢯⠈⠉⠉⢳⠉⣼⡊⠉⠉⠒⢤⡀
⠀⠘⠀⠀⠀⠀⠀⢠⠀⡇⠈⠀⠀⠘⠀⣿⣯⡆⠀⠀⠈⢸⣿⣼⠀⠆⠀⠀⢸⢹⠀⢹⠀⠀⠀⠀⠙⢆
⠀⡇⠀⠀⠀⠀⢠⠇⠀⠈⠘⠁⠀⠀⠳⠬⠩⠔⠱⠲⠌⠂⠭⠥⠚⠀⠀⠀⡶⠊⠀⠀⡇⠀⠀⠀⠀⠘⣆
⢐⠀⠀⣀⠠⠒⠁⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠹⣄
⠸⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⢑⡂⠚⠁⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⢀⣨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⡬⢥⠤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡇⢠⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢉⣥⠀⠋⠉⣲⡴⢎⡁⠙⠫⢄⠁⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠋⠀⠀⠀⢰⠁⠀⠀⠰⡄⠀⠀⠇⠀⠀⠀⠀⡏⠢
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠀⠛⠀⠀⢀⠁⠘⢀⡇⠦⡄⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠈⠁⠠⡀⠀⠀⠀⠀⠈⠀⠀⡔⠀⠀⠰⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠾⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡸⠀⠀⠀⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⢑⠶⡂⠀⠀⠀⠀⠀⠀⠀⠀⣢⠛⢏⠀⡆⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠒⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠐⠃⠀⠀⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡠⠊⠓⠒⠂⠠⠤⠔⠖⠚⠙⡀⠀⠀⠀⠀⢰⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡶⠶⠮⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠋⠑⠲⡶⠦⠤⢤⠴⠦⢲⠖⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠇⠀⠀⣸⠃⠀⠀⡸⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢁⡰⠃⠀⠀⠀⠇⠈⠓⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢠⠞⠁⠀⠀⠀⢸⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠔⠁⠀⠀⠀⠀⠀⠸⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡏
⠀⠀⠀⢀⡤⠚⠁⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰
⢀⡴⠚⠁⠀⣀⢆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠈⠢⡀
⡎⠀⠂⢀⠖⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠉⠢⣄
⠱⢤⡤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡙⠀⠚⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠉⠉⠉⠲⠄⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠠⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⣄⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⢀⠈⠛⠂⠀⠙⠳⠖⠄⠀⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣀⠀⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣴⣲⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⣍⢃⠁⠀⣈⠉⢳⡶⠀⠀⣠⢛⣉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⡀⢽⠷⠉⠀⠀⠉⠙⠛⠁⠀⢠⠙⠋⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡙⢦⠀⠀⠀⢀⠌⠀⠀⣷⡄⠀⠑⠀⠀⠀⣄⡀⠈⡀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣝⢄⠣⡀⠀⠎⠀⠀⣗⠸⡄⠀⠐⠀⠀⢀⣈⣍⡈⠀⡜⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣮⢷⡹⢞⠀⠀⠀⠙⣆⠙⢦⡀⠂⠀⠀⠉⠉⠀⣼⠀⠀⢢⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡠⢄⡀⣀⣠⠤⠂⠻⣿⣇⣻⣌⢧⡀⠀⠐⡌⢦⡀⠉⠓⠺⠶⠤⢔⠂⠍⡄⠀⠈⡍⠉⠒⠲⠄⠀⠈⠳⣄
⠀⠀⠀⢠⠎⠀⠀⠈⠁⡀⠀⠀⠀⠹⣿⣿⣻⣦⠙⣄⠀⠘⡄⠈⠲⣄⠀⠠⠘⠁⢠⠞⡇⠀⠀⠰⡀⠀⠀⠀⢣⠀⠀⠀⠓⣄
⠀⠀⡰⠁⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢨⢿⣿⣿⣧⡌⢦⠀⠘⡄⠀⠈⡓⠀⣆⠰⠃⠀⡇⠀⠀⠀⢱⡀⠀⠀⠈⡆⠀⠀⠀⠈⢆
⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡎⠈⢻⣿⡽⠿⣆⠳⡄⠘⢆⠠⣇⠀⢋⠀⠁⢰⠀⠀⣀⣀⠀⢣⡀⠀⠀⠁⠀⠸⠀⠀⠈⠦⣀
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⣜⣠⡶⠟⠻⣿⣿⣝⢧⡘⣆⠈⠢⡈⠁⢆⡔⠀⡆⠀⠚⠤⢌⣓⣄⡃⠀⠀⡄⢀⠃⢀⠀⠀⠀⠈⠑⠦⣀
⢸⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈⡧⠒⠋⠁⠀⠀⠀⠘⢿⣯⡛⠳⡮⢧⡀⠱⡀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠈⠉⢉⢝⡃⠃⡠⠃⠀⠀⢀⠀⠀⡀⠈⠓⢄⣀
⢸⡰⠁⠀⠀⠀⠀⠀⠀⠀⢰⠉⠲⢄⠀⠀⠀⠀⠀⠈⢻⣷⣆⠙⣦⢷⡀⠱⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⡴⠁⢸⡣⠴⠶⠒⠛⠉⢁⣴⣫⣤⣋⠑⢄⡈⠙⢦
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠙⢻⣟⡊⢷⣻⣄⠳⠁⠀⠀⠀⠀⠀⠀⣤⢋⠀⠀⣈⠈⠓⠦⡀⡠⠞⠓⠒⠂⠤⠀⡙⠢⢌⠁
⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⠃⠀⠀⠀⠀⢰⠈⣳⢄⡀⠀⠀⣀⠟⠛⠛⠳⠿⡀⠀⠀⠀⠀⠀⡴⠊⠣⠜⡠⠊⠀⠈⠁⡒⠚⠣⠛⢦⡀⠀⠀⠀⠀⠲⡈⠳⡀
⠀⠀⠀⠀⠀⡀⠀⢀⠀⠸⠀⠀⠀⠀⠀⠈⠉⠁⠀⢉⡧⠚⠉⠀⡀⠑⠂⢤⡀⠢⡀⠀⠐⠃⣀⡠⢤⠊⣠⡞⠀⡆⠀⠀⠄⠀⠀⠀⠹⡄⠒⠈⡉⣉⣉⣀⣘
⢨⠁⡠⠒⡁⠀⡔⠂⢀⠝⢃⠔⠚⠛⠋⠉⠉⢡⠎⠉⠀⠀⠃⠀⣀⠙⠲⣤⡘⢦⣱⠀⠠⠎⢉⣀⣽⣄⢻⣅⣘⢀⡃⢠⣦⡤⠠⠤⣼⠿⠛⠛⠉⠁
⠃⠈⠀⠰⠁⠘⠀⢀⠏⢠⠃⠀⠀⠀⠀⠀⠀⡜⢀⠀⠀⠀⢀⣠⣄⣁⣪⠄⡟⠃⠙⢧⡀⠀⠀⠀⠀⣉⣉⠉⠛⠛⠋⠩⠄⣀⡀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢉
⠀⠀⡰⠁⠀⠀⠀⡎⢠⠃⠀⠀⠀⠀⠀⣀⠀⣧⠽⡗⠒⠛⠉⠉⠈⠁⠈⠀⠉⠉⠓⠮⠛⢦⠐⠈⠉⠁⠀⣀⣀⣤⠤⠔⠂⠂⠸⠥⡀⣀⣊⠀⠀⡀⠀⠂⠉
⣄⠀⠀⠀⠀⠀⡜⠀⠚⠁⣀⣉⣉⣉⣀⣠⡄⣷⠞⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠉⠉⠁⠐⠀⠀⠤⢀⣂⣰⠉⣨⠌
⠿⠗⠒⠒⠂⠸⠔⢲⡉⠥⠬⢤⡲⢷⣲⠉⠁⠀⠀⠀⠂⠀⠀⠈⠐⠢⢤⠀⣀⣠⠤⠖⠋⠉⠁⠀⠀⠀⠀⣀⣠⠤⠔⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣕⣌⠑⠢⢄⡀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⣀⠤⠔⢀⣈⡤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣾⢘⣿⣮⢣⠀⠀⠈⠑⠢⢄⡀⠀⠀⠠⠖⣂⡤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣭⣭⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠓⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠠⢤⠖⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠙⠂⣀
⠀⠀⠀⠀⡠⠤⠊⠁⠀⠀⠀⠀⠠⣃⣀⡀⠓⠤⣀⠀⠀⠀⢀⠔⠛⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢈⠇⠀⡖⠈⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠇
⠀⠀⢠⠐⠓⠆⠤⢄⣀⣀⡀⢀⡠⠔⡆⠀⠓⠊⠁⠀⠀⠱⣀⣀⣀⠀⢀⡀⢤⡀⠝⠢⣄⠀⠀⠀⡐
⠀⠰⡅⠀⠀⢁⠀⠀⠀⠀⠠⠀⠀⠸⠀⠀⠀⠀⡜⠢⡀⠀⠈⠍⠀⠀⠀⠀⠐⠂⠀⠈⠀⠳⡄⢠⡇
⠀⠐⠤⢴⡺⡀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣤⣄⡔⠀⠀⠈⠆⠈⡀⠀⠀⠀⡀⠈⠀⠀⠈⠢⠴⡻⡃⠈⢢
⠀⠀⠀⠈⠀⠑⠤⡀⠈⠁⠀⠀⠀⢀⣠⡒⢁⣠⠀⢀⢀⠚⠀⠈⠢⢈⡁⠀⠀⠀⣀⡀⢄⣺⠀⠱⡀⠀⡇
⠀⠀⠀⠀⠀⠀⡠⠊⢛⡖⠒⢛⣙⠣⡤⠬⢙⣉⣤⠊⠋⠀⠀⢀⠊⠉⡏⠣⠜⠉⢸⠉⠙⢆⠱⣄⠈⠉
⠀⠀⠀⠀⠀⢠⠃⣀⠜⠁⢰⠃⠈⢁⠈⠢⡄⠙⡁⠀⠀⠀⠀⠠⠀⣰⠃⠀⠀⠀⠘⣄⡠⠼⠸⡉⠔⡗
⠀⠀⠀⠀⠀⠘⣼⠃⠀⠀⠘⢄⠀⠀⠁⠀⣱⣀⠘⡀⠀⠀⠀⠀⠣⡿⠀⠀⠀⠀⠀⠈⡑⠀⡅⠓⠾
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠑⠂⠀⠊⡇⠀⠀⡅⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⣀⠄⠀⠰⠀⠀⡰
⠀⠀⠀⠀⠀⠐⠤⢀⣀⠀⠀⠀⠀⠀⣠⣾⡢⡤⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠑⣇⡴⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠉⢰⠃⣩⠉⠕⠚⢔⠉⠀⠀⡀⠀⠀⠀⠀⡱⠃⠁⣆⡠⠋⡙⠒⢄⡀
⠀⠀⠀⠀⠀⣀⡴⢎⠢⢄⣀⡾⣔⡡⠜⡄⠀⠀⢣⠀⠂⠁⢠⠤⠤⢔⢅⡀⢀⡏⢶⡊⠀⠀⠀⠘
⠀⠀⠀⢠⠎⠈⠢⠠⠍⠀⢈⠀⠈⡗⠒⠁⠀⠀⡸⠀⠀⠰⠁⠀⠀⠀⠀⠀⠝⠀⢠⠀⠀⠀⠀⡜⠠⠠⣄
⠀⠀⠀⠀⢄⠀⠀⡀⠀⢀⠆⠀⠀⠈⠢⠤⠤⠊⠀⠀⠀⠘⠦⣀⠀⠀⠀⢀⠃⠀⠀⠢⠤⠤⠊⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⡫⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠭⣬⠉⠉⠛⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣏⡴⠒⣆⠀⠀⠀⠻⣿⣿⣿⣧⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠂⡿⡿⠀⣿⠛⢶⣶⣶⣾⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡉⠀⠨⠿⢷⡤⣌⠀⠸⠇⡘⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣀⡠⢾⠇⠀⠀⠉⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⣀⣀⡟⢲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣦⣄⣸⣿⣿⢀⡾⢁⣿⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣏⣹⡿⠯⠉⢸⡇⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣟⣉⣈⣁⣀⣀⣀⣘⣷⡀⡀⢀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡗⠀⠐⠂⠐⠐⠒⠒⠘⢷⣝⣧⡀⠀⠀⠀⢀⣠⡴⠟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠽⠿⡯⠭⠭⠭⠭⢯⣉⣙⡉⠭⢭⣽⣌⠢⢤⠒⢩⢎⠄⢊⡴⠻⣆
⠀⠀⢀⣀⠠⠤⠀⢞⠁⠐⠒⠂⠤⠤⢀⣀⣀⡀⠀⠀⠀⡰⠓⠮⠷⣶⣗⣐⣒⣒⣒⣒⣉⡡⠋⠀⠀⠀⠀⠉⠈⠛⠓⠠⠾⣉⣄⠀⠈⢷⡀
⠀⠀⠓⠒⡩⣩⠆⡤⡔⠲⢤⣀⠀⠀⠀⠀⠀⠈⠉⠉⠚⠓⠶⠆⠀⠀⡗⠒⠒⠒⢂⣐⣪⠔⢀⢠⠋⠓⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠢⠞⠡⢠⢇⠇⠀⠀⠀⠉⠒⠦⢄⣀⠀⠀⠀⢀⣶⡀⠀⠀⠀⡏⠀⠀⠀⠯⢛⠡⡔⣡⠯⠀⠀⠀⠀⠉⢷⠲⠤⣀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢿⣿⣟⣀⣠⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⢸⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⢁⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⠭⢵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠲⠤⠤⢄⣀⣀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠚⠉⠁⠀⠀⡇
⠀⠀⠀⠀⠀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⠞⠋⠉⠀⠀⠀⠀⠀⠀⣀⣤⡷⠆⠀⠀⡀
⠀⠀⠀⠈⡟⠀⠤⣀⡀⠉⠉⠙⠒⠲⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠚⠳⣦⣀⣠⣴⠶⠿⠯⠑⠒⠂⠀⠀⠀⢡
⠀⠀⠀⠀⡇⠀⠀⠀⠉⠙⠲⠦⣄⣀⠀⠀⠀⠀⠉⠉⣓⣒⣶⠤⢤⣄⣹⣄⣤⠶⢿⡏⠁
⠀⠀⠀⣀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⡞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⣠⣤⣟⡛⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠄⠐⢀⣠⣯⣿⠗
⢰⡿⠁⠉⠙⠛⠶⠬⣝⣳⣦⣤⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤⠔⠒⣒⣈⣩⠥⠤⠔⠒⠒⠉⠉⠉⠀⠀⡟
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣬⣤⣶⠶⣞⣛⣛⣭⣭⡵⠶⠖⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠙⠷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠛⠋
⠀⠀⠀⠀⠀⠉⠛⠳⠶⣦⣠⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡤⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠦⣤⣄⣴⠄⣀⣀⣀⣤⠤⠶⠗⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠂⠉⠁⡀⠄⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠠⠔⠂⢀⠂⠁⡂⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⢤⡴⠊⠀⠀⠠⠀⠥⢂⡀⠐⠈⠄⠂⠤⠐⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⡀⠀⠀⡀⡐⠂⠀⠠⠄⠀⠀⠠⠀⢱⡀⠀⢀⠡⠀⠀⠀⠂⠠⠒⠁⠀⠀⢀⠀⠤⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⢳⣳⡀⠀⠒⠂⠀⢀⣁⡈⠀⣀⡀⠀⠠⠀⠀⠀⠳⡀⠀⠀⠁⢂⠄⢀⠤⠐⠀⠔⠈⠁⡂⠁⠂⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢣⠀⠀⣀⠀⠠⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠒⡈⠐⡈⢐⠈⠀⠂⠁⠀⠐⡈⠄⠢⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡕⢧⠀⠐⠂⠀⠀⠈⠀⠀⠀⢀⡀⠀⠤⠉⠀⠀⠡⡀⠀⠀⣂⠄⠀⠁⠀⢀⡠⠐⠀⠁⠀⠀⠀⠡⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢇⠀⡤⠀⠀⠐⠂⠀⠄⠀⠀⠀⣐⠀⠀⠀⠀⠱⡀⠀⠐⣈⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢆⠱⡀⠀⠀⠀⠀⠘⡀⠉⣁⡀⠠⠀⠒⠀⠀⠱⡀⠫⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠈⣀⣤⠖⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⡆⠑⠀⠀⠀⠀⠀⠰⡀⠀⡒⠀⠠⠈⠠⠀⠀⠱⡄⠣⠀⠀⡀⠤⠒⢉⣀⣤⣶⣿⣫⣩⡤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⢄⠱⠤⠤⠀⠀⠀⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⣄⡉⣁⣤⣴⣾⡿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣈⣒⣒⡶⠶⣶⣶⣶⣴⣿⣿⣿⠿⠿⠿⠟⠛⠛⠣⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⢮⡒⢄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢰⠂⢈⠱⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⢄⢀⠴⠋⢱⡼⠠⢎⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⣿⡃⠀⠀⡼⠃⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠛⠛⠓⠒⠲⠤⠤⠭⠭⠥⢿⡁⠤⠜⠀⢀⡶⠄⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⡀⠌⠃⠀⠀⠀⡲⠂⠀⢈⡈⠛⣉⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢲⠋⠁⠀⠀⠀⠀⢀⡼⠁⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰⣀⡀⢀⠀⠀⢠⠀⡞⠀⠀⠀⠀⣠⢔⡳⠃⠀⠀⠀⡅
⠀⠀⠀⠀⠀⢀⡀⠀⢠⣀⣀⡤⠀⠀⠀⠀⠀⢀⠱⣶⠚⢬⠓⠂⠀⠀⢇⠀⠀⠀⠉⠱⠊⠀⠀⠀⠀⢠⡅⣀⡀
⠀⠀⠀⡉⣭⣤⡂⣾⣉⣭⡀⠀⠂⠀⠀⢸⠁⠉⠈⠠⡀⠂⢀⡀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠜⠊⣖⡓⠠⡀⠐⢄
⠀⠀⠀⢛⣶⡌⢁⣟⡃⣀⣀⠴⠦⠒⠒⢺⠃⠀⠀⠀⠈⠀⠄⡜⠁⡸⢻⢧⡠⢄⠀⠀⠐⠂⢉⡠⡤⡐⠎⠀⠘⠀⡇⢈⢦⣆⢂
⠀⠀⢠⠷⣍⢛⠂⡿⠛⢉⡁⢀⣀⠤⠤⣼⠆⠀⠀⠀⠀⠀⠀⠈⠰⠁⠨⠤⠒⠉⠑⠤⠒⢉⠔⠀⠀⠙⢢⣄⡌⠠⠀⠔⠠⠏⢊⣙⣉
⠀⠀⠐⠩⠛⠟⢨⡿⠿⠛⠁⡀⠀⢀⣀⣦⠀⠈⠁⠂⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢠⢤⡖⠂⠠⠤⠄⠀⠈⠑⠊⠨⠄
⠀⠀⠀⠀⠀⠀⠘⠁⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⣀⣠⠰⠄⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢰⣿⢛⣺⡞⢺⠃⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠉⠛⠙⠏⠍⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡓⠀⠀⠀⠀⠀⠀⠀⠀⡜⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠤⢄⣀⡰⠁⠰⡳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⠊⢱
⠀⠀⠀⠀⠀⠀⠘⢶⠦⣤⣀⣀⣀⠤⠤⠒⠒⠈⠉⠀⠀⠀⠱⢄⠀⠀⠀⣀⠤⠒⠊⠉⣀⣤⡄⡔⠂⠀⢸⡆
⠀⠀⠀⠀⠀⣲⡦⠬⢆⠀⠐⠂⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠙⡂⡄⣴⡦⢀⢠⣔⠼⠗⠈⠉⠀⠙⠀⠀⢸⣷
⠀⠀⠀⣀⣾⡃⢷⡀⠈⡆⠀⢹⣿⣿⡆⠀⠀⠀⣀⠀⠈⠁⠀⠀⣿⣿⣿⣭⠍⢉⠀⠐⠁⠉⠩⠄⠀⠀⢸⣏⠆
⠀⠀⠀⣧⢹⡵⣇⢃⠀⢻⡀⠀⢿⣿⣷⠄⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⡿⡄⠁⠊⠀⠀⠃⠀⠀⠀⠀⠘⣿⡜⡄
⠀⠀⠀⢹⡐⣷⣿⣾⡆⠈⣷⡄⠀⡀⠐⠢⠀⠰⠄⠐⠈⠘⠂⠀⠸⣿⡏⠔⣿⠀⠄⡂⠀⠈⠀⠉⠀⠒⠆⠻⣷⡘⡄
⠀⠀⠀⠘⣧⠘⣻⣿⡼⡄⢸⣷⡀⠠⡆⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢸⡇⠀⠀⠀⠀⢀⣀⡀⠐⠀⠀⢻⣇⢧
⠀⠀⠀⠀⢹⡄⢿⣿⢷⣧⠀⢯⣧⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⠀⠔⠂⣀⣀⣀⣉⠙⠂⠀⠀⣯⠸⡄
⠀⠀⠀⠀⠈⣿⠈⣿⡎⡿⡄⠀⠻⣦⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣋⣤⣶⣻⣿⣭⣟⣛⣛⠻⠿⣽⣿⣟⣟⣄
⠀⠀⠀⠀⠀⠸⡄⠘⣷⣷⢳⠀⢸⣋⠐⢄⠀⣹⣄⡀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⠿⣿⡾⢾⣷⣼⣮⢢⡀
⠀⠀⠀⠀⠀⠀⢿⠀⢸⣾⠸⡀⠀⠀⠀⠀⢳⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣟⣿⠿⠿⣿⣟⣿⡿⠿⣿⣶⣯⣿⣶⣾⡭⢥⠓
⠀⠀⠀⠀⠀⠀⠘⢦⠈⣿⣇⠧⠠⣤⣶⣾⣿⣿⣿⣿⣿⣿⠟⠋⠙⢯⣟⣛⣿⣟⡿⠀⠀⠀⠀⠈⠉⠛⠻⠾⢽⣭⣛⣻⠿⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⣻⣶⣾⣿⡿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢷⣮⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣠⣤⡭⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠠⠤⠤⠤⠤⠤⠄⠀⠒⠒⠒⠒⠒⠒⠒⠀⠉⠉⠙⡀
⠀⠀⠀⠀⠀⢰⠋⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡠⠤⠤⠤⠶⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠐⠈⠉⠉⠉⠉⠁⠀⢀⣀⣀⣀⠤⠄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠠⠤⠤⠴⠒⠒⠒⠊⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⡆⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠤⠤⠔⠒⠒⠒⠈⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠒⠉⠉⡭⠭⠥⠄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠴⠶⠤⠒⢒⣓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣧⠤⠤⠴⡇⠀⠀⢸⢠⠭⠭⠭⠭⠥⠐⠒⠒⠒⠊⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⣃⠔⢆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠐⠒⠒⠂⠐⠒⠒⠒⠢⠤⠤⠤⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⣉⢉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣉⣉⣉⡿⠶⣶
⠀⠀⠀⠀⠀⠀⢀⢎⡾⠟⠋⠛⠒⠨⠍⠴⠶⠦⠤⣬⣍⣉⣤⣒⠒⣠⣄⠤⣤⣤⣤⣶⣶⣼⡽⠦⢶⠖⢾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠸⣸⡏⠀⠀⠀⠀⠀⢀⣀⡐⠀⢀⣠⠄⠀⠀⠉⠉⠉⢉⡿⡙⢕⢅⢕⠀⡢⡪⢀⢕⢕⣐⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠷⣶⣴⣤⣤⣀⣀⣀⢀⠀⠉⠉⠉⠉⠉⠀⠀⠀⢸⡮⡪⢑⢕⣅⣪⣪⣶⢴⣻⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠈⠀⠉⠙⠉⠛⠛⠳⠦⠦⠭⠭⣤⢲⣤⣿⣶⣿⡿⠿⠗⠛⣻⣛⢛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡜⠑⢕⡄⠂⠀⠀⠀⢀⢀⣀⡀⣠⣤⣠⣄⣤⢴⢶⡾⣧⡿⡿⡻⢻⢛⢟⠟⠟⡛⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠡⡡⡀⢘⡺⢿⡻⡿⠺⠻⡻⢋⢟⠝⠝⡑⡡⡣⢣⢮⢪⢊⢎⠔⡑⡱⡡⢣⢊⢊⢊⠪⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣜⠄⠻⠀⣻⢔⠕⠑⡑⡡⡠⢪⢊⢌⠔⡔⠕⡕⡱⡡⣪⢊⢌⠔⡑⡡⡣⡣⣠⡵⣿⡗⣂⣤
⠀⠀⠀⠀⢀⡠⠔⠊⠁⠙⢮⣄⢀⣧⣠⣢⣎⣎⣼⣴⣥⣧⣧⣾⣮⣾⣾⣶⣶⣷⣷⣿⣾⡻⠿⣿⠿⠿⢿⣷⣾⡿⠋
⠀⠀⢀⣴⣃⣘⣥⣀⣠⡤⣠⠝⣛⡛⠺⠛⠿⠉⠹⠏⠍⣉⡩⠭⠭⣍⡡⠒⠀⠨⠄⠔⠂⠠⠥⠐⣲⣿⡿⠻⠜⡇
⠀⠠⣾⡏⠉⠀⢈⡉⡙⠋⠛⠒⠒⠷⠮⣭⣵⣶⣖⣒⠢⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⣟⠃⢂⡃⢰⣱⣄⡀
⠀⠀⣿⣇⠀⠀⣦⣴⠃⠀⣀⠀⠀⠐⠒⢂⡤⠤⠈⠉⠉⠙⠓⠻⠴⢶⡯⣽⣖⣒⣠⣥⣾⢿⢡⠙⠐⠌⣦⣼⠿⠻⠁⠙⠉⠑⢠⡀
⠀⠀⠈⠙⠛⠒⠒⣿⣦⣤⣤⣤⣀⣉⠀⠒⠀⠎⠥⠴⠒⠐⠆⠀⠀⠀⠀⠀⠀⢨⠋⡟⠘⠇⢂⢲⣼⡯⠟⠉⠙⠀⠂⣠⣴⣾⠟⠁
⠀⠀⠀⠀⢀⢴⣿⣽⣿⢯⣉⣉⠑⠛⠿⢓⡶⠦⣴⣂⣤⣀⣀⡀⠀⠀⠀⠀⠀⢸⢢⠐⡈⣰⣾⡟⠋⠀⠑⣂⢤⣶⢿⢋⡣⡏
⠀⠀⠀⠀⡇⡟⠊⠉⠉⠙⠳⠮⣭⠐⡂⠤⢀⡀⠀⠉⠉⠉⠉⠛⠛⠿⢶⣶⣦⣬⣷⣷⣿⡏⢁⣀⡶⣾⢿⢛⠧⠡⠣⠠⡱⣿⠄⣐⡀
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠠⢉⠑⠓⠲⢬⣅⡒⠤⢀⡀⠀⠀⠀⠀⠀⠁⠀⠉⣉⣜⣛⣯⠷⠋⢡⠢⣠⠡⠡⢁⣡⣵⡾⠗⠂⠒⠒⠂
⠀⠀⠀⠀⠜⣿⣒⡄⠀⠐⠠⠄⢀⠀⠙⠒⠆⡀⠈⠁⠋⠴⣮⣍⣐⠠⢅⣮⣴⡾⡿⡻⠩⡣⡱⢡⢡⠡⣣⣱⡥⣗⣿⠛⣉⠉⠁⠭⠍
⠀⠀⠀⠈⠀⠂⠌⠛⡵⣆⡤⣀⠐⠊⠲⢄⡀⠀⠉⠐⠀⠀⠀⠈⠉⡿⠛⠝⠕⡕⡑⡐⡱⡱⡡⣡⣵⢳⣯⠷⠛⠀⢀⣐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠆⡤⣀⡬⢉⠙⠒⠶⢀⠀⠀⠀⠠⣇⢂⢎⢌⠜⡐⣕⣴⢶⡹⠽⠛⠉⠉⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠒⠮⢔⡂⠤⢀⡀⠀⣧⢂⢢⣮⣾⣿⠷⠿⠍⣁⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠬⣅⡚⣿⣽⣚⢋⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⢀⣶⡦⠔⠒⠒⠒⠛⠉⠙⠍⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠊⠉⠀⠀⠀⠀⠈⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⢀⣠⣤⡒⡲⠦⢄⣄⠀⢠⡪⠅⢂⡳⠶⠦⢀⣀⠒⠒⠒⠂⠀⢿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⠐⠛⠠⠬⠁⢐⣉⣉⣤⣤⣬⣭⣶⣿⣶⣿⣴⣾⣿⣿⣿⣷⣬⣿⣝⡭⠉⠉⠉⠙⣿⣷
⠀⠀⠀⠀⡾⣿⣥⣤⣤⣷⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣏⣃⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣷⣿⣭⣽⡟⡆
⠀⠀⠀⠐⠷⠚⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⠠⠀⠠⢀⣀⣠⣤⣀⡀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⠖⠒⠀⢒⠈⠿⠿⣿⣷⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡉⠀⠀⠀⠀⠁⠀⠂⠌⡑⠢⢀⠉⠣⢀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⡎⢣⡑⠠⢀⠀⠀⠀⠀⠀⠀⠁⠂⠡⢢⣄⡁⠢⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠐⠀⠀⠝⢦⣀⠈⠒⠠⢀⡀⠀⠀⠀⠀⠀⠀⢉⡻⢛⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⠔⠁⡠⠊⠀⠀⠈⠉⠒⠖⠢⠀⡥⠀⠐⢺⡉⠤⠚⠉⠴⡅⠙⠛⠛⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠬⠄⠒⠈⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢩⣀⣀⣠⣼⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣷⣶⣀⣀⣐⢒⣂⣀⣰⣾⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡯⠭⠍⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡍⠁⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠹⠂⠡⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⠂⠘⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢁⣠⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⢔⡈⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⢅⠀⠀⠀⠰⢤⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢣⠤⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿
⡏⠉⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⢲⠤⠸⠉⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠉⠧⢴⣿⣿⣿⣿⣿⣿⣿⣿
⡷⠶⠀⠀⠀⠀⠀⠢⠤⠤⠤⠤⠤⢾⡶⠒⠆⠠⢠⠀⠀⣀⡇⠀⣇⡀⠀⠀⠠⠄⠀⠒⠾⣿⣿⠿⠛⠛⠛⠚⠻
⣇⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⡰⠉⠐⢪⢷⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⡏
⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠌⡏⡇⠀⢁⠀⠀⠀⠀⠀⠀⣀⠠⠄⡄⡞⠅⠀⢰⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣇⢄⠠⠇⢷⠀⢈⣤⣶⣶⣾⣿⣷⣴⡀⢀⣛⠑⡀⢀⣾⣿⣧⣀⣀⣀⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠟⠛⣹⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣤⣤⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠉⠛⠋⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠤⡀⠀⣀⡀⠀⢿⠟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡏⠓⠃⠀⠥⠄⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⣀⡀⠀⠀⢀⡼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣧⠉⢁⣠⠴⠋⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠈⠁⠸⠿⠙⢿⠋⠁⠀⠀⠀⠀⠉⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⡏⠀⠈⢀⡴⠁⢀⠀⠺⠆⠀⡔⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠁⠌⢠⠆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⡆⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣍⣛⡲⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠞⠀⠀⠖⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠅⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣽⡇⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⡴⠀⠐⠂⠀⠀⠛⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠙⠂⠼⠇⠀⠀⢀⡁⠀⢻⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⡀⠈⣧⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣀⠄⠤⠀⠤⠤⢤⣤⡸⣧
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣾⠷⠶⠶⠿⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⣿⣤⣦⣤⠤⠶⠶⠚⠛⠛⠛⠉⠉⠉⠁


⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢿⣯⣭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠈⠭⢙⡛⢷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⡛⠲⠤⢄⣀⡀⠀⠀⠉⠒⠍⠐⠛⠿⠷⢶⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠖⠒⠒⠒⠒⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢳⠛⠻⠿⠷⣶⣶⠤⢤⣀⡀⠠⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⢈⡉⠛⠻⡿⠿⠿⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡤⠤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⠄⠁⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢄⣠⣼⠀⠀⠀⠀⠀⣐⡂⠀⠀⠀⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡀⠀⠸⠤⣬⣉⠉⠓⠒⠲⠶⠤⢭⣁⣰⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠑⠢⢄⡀⠈⠉⠙⠓⠒⠶⠦⠤⠾⠧⠶⠶⠶⠖⠒⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⣉⣉⣉⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⢀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⡤⠤⠐⠒⢖⠂⠉⠉⠁⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⢠⠒⠉⠞⡁⠀⠀⠀⠠⠌⡐⠀⠀⠙⢦⠀⠒⠒⠀⠉⠉⠁⠀⠀⠀⠀⠰⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠈⠑⠲⢤⣀⠉⠒⠠⠁⠂⠠⢅⡂⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡙⠶⡦⠬⣐⡒⠬⡀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⡇⢀⣀⡀⠠⠤⠤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢠⠖⣉⡡⠒⠉⠀⠀⠀⠈⠉⠛⠧⣠⠴⠤⠤⠤⠀⠶⠶⠶⠶⠒⠚⠒⠒⠈⠉⠉⠉⠙⠒⠲⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣏⡁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣌⠉⠑⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡦⣈⠓⠲⠦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⣀⠤⠒⠉⡇
⠀⠀⠀⠀⠀⡴⠊⠻⢄⣀⠈⠓⠲⠤⣄⡀⠉⠉⠓⠒⠶⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢁⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠈⠲⠤⣄⡈⠉⠒⠲⠤⣄⣈⠙⠒⠶⣤⢤⣀⡀⠀⠀⠉⠉⠙⠒⠲⠦⢤⣀⠀⠀⠀⠀⠀⠀⣠⠴⢊⣡⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠙⠒⠦⠤⣀⡈⠙⠲⠶⠃⠀⠀⣈⡩⠿⠒⠲⠤⣄⣀⡀⠀⠀⠉⠑⠢⢤⠒⠉⡠⠖⠋⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⣒⠂⠤⢈⡉⠒⢶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⢄⡼⠖⠋⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠤⠈⣁⠒⠤⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⢤⣒⠚⠇⣀⡀⠀⠀⠠⢌⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠤⠤⠤⠔⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠮⠭⣒⣒⠒⠒⠚⠁⠀⣠⢠⠤⠤⠤⠶⠖⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠊⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠀⠒⢤⡈⠙⣦⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠈⠓⢌⣷⡖⠉⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢄⣀⣀⣀⣻⣠⣞⠥⠤⠶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠉⠛⠿⠆⢀⡴⠆⠋⠑⢮⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⢘⡌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠛⠛⠛⠛⢷⣄⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⣴⠟⠓⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠖⠚⠋⠈⠛⠳⠶⠞⠛⠁⠀⠀⠀⠀⠀⠈⠛⠶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⡙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢛⠛⠙⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⠀⢀⣤⡀⢠⣤⡤⣄⡀⢀⡀⠀⠀⠀⠀⢸⢰⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠸⡇⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡿⠙⡇⣼⠇⢧⠀⣎⠀⡾⠓⠻⠆⠀⠀⠀⠀⢸⢈⡇⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠟⠻⠿⣤⣶⣼⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣸⢃⡼⠋⣉⣳⡶⠂
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⠀⠀⡆⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⠿⠿⣿⣯⣭⣉⣉⣉⠉⠛⠛⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⣉⡠⠵⠛⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠓⠲⠶⠤⣄⣀⣀⡠⠶⠚⠋⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⡈⣄⠀⢇⠀⠀⠀⠀⠀⠀⠀⢠⢍⡀⠀⡇⣈⠋⠙⠉⠁⠀⠀⠀⠀⠀⢰⠃⡜⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⣳⠶
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠙⠻⠶⠾⢦⣤⣄⣀⣀⡀⠐⠁⠀⠇⠤⠀⡆⠀⠄⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇⠀⡇⠀⠀⠀⠀⠀⣀⣤⣶⡭⢒⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠦⠤⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢿⠀⡇⠀⣧⠀⢀⣠⣴⣾⠽⣚⣭⠖⢋⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⠶⣤⣤⣼⣇⣡⡀⢸⣗⡿⠟⣋⡴⠞⠉⠀⠀⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡘⠀⠀⠀⠆⠸⠇⢠⡤⠀⢀⡀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣭⠶⠛⠉⠀⠀⠀⢀⣤⠞⣩⣝⠁
⠀⠀⠀⠀⢠⣶⡞⢻⣤⣀⣉⠛⠁⠈⠀⠓⠀⠻⠆⠀⠎⠀⢠⠰⡎⢸⠃⢀⠃⣴⠀⡶⠀⢀⡶⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣤⠶⣫⡴⠋⡀⣩⡽⠿⢻⡄
⠀⠀⠀⠀⠀⠈⠉⠛⠶⢮⣭⣉⡛⠲⠶⠶⢤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠀⠙⠂⠀⠀⠀⠀⢸⡇⠀⣀⡴⢟⣵⡾⠷⠖⠋⠉⠀⡄⠘⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠛⠒⠦⣤⣄⣀⠉⠉⠙⠓⠲⠦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠞⣩⡴⠊⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⣿
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⣤⣀⡀⠀⠀⠀⠉⠉⠙⢓⣲⣶⣤⠤⢄⣀⠀⠸⠅⢚⠉⢀⣀⢀⡴⠘⢦⡇⠀⠀⡇⢀⢹⡇
⠀⠀⠀⠰⢷⣁⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢶⡶⠚⠛⠛⠉⠀⠀⣀⢤⠡⠄⢠⡄⠸⡇⢺⣽⠘⢇⠀⠈⠃⠀⢸⣀⡿⠞
⠀⠀⠀⠀⠀⢩⡟⠳⣦⣀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡆⠀⡇⠀⠀⢹⡾⡆⢾⠘⣷⣄⠳⠆⠉⠀⠀⣀⣤⠴⠶⣟⡁
⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣿⡳⢦⣭⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡇⠀⠁⠀⠀⠈⣷⠿⠞⠋⠁⢀⣀⣤⠶⠞⣋⣥⣤⡤⠠⠼⠿⠓⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣦⡀⠉⠓⠶⣤⣀⠐⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⡇⠀⠀⠀⣈⣤⡴⠒⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠈⠻⣆⣀⣤⣤⣬⠽⠶⠤⣄⠐⠂⠀⠀⠘⠀⠘⠀⠀⠁⠀⠒⠋⢁⡀⠀⠀⠀⠠⠀⠀⠀⢸⡄⠀⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢫⡙⢦⡀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⠈⠳⣄⠀⠀⢄⠀⠀⠀⠀⢻⠀⠉⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠙⢮⡳⣄⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠐⠁⠀⠀⢰⠁⠀⠀⠀⠈⡄⠀⢻⠀⠛⠀⠀⠀⠈⠳⣦⠘⢆⠀⠀⠀⠀⣦⡀⠀⠆⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢷⡀⠀⢀⡆⣿⠀⠀⠀⠀⠘⠷⠛⠛⡈⣆⠀⠀⠀⠀⠓⠀⠘⣅⠀⣠⡄⠠⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣤⡤⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡙⢦⣈⣇⡿⠀⠀⠀⠀⠀⡀⠀⡠⠁⠈⠂⠀⠂⠀⠈⠁⠀⣉⣀⣀⣤⣤⣤⡴⠶⠶⠚⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣿⠇⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣴⠶⠶⠶⠒⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠓⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠏⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣄⡀⠀⠈⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠉⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠐⠂⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⣦⣄⡀⠀⠀⠈⣒⡀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠚⠋⠉⠀⠀⠀⠉⠛⠷⣦⣄⣃⣀⣁⣠⣤⣤⣤⣤⣶⣶⣶⣶⡾⠿⠿⣟⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣉⣉⣀⠐⠒⠂⠒⠠⠤⠤⠄⣀⣀⡀⠀⠉⠛⠛⠋⠉⠉⠉⣉⠠⠔⠒⠉⠀⠀⠀⠀⢹⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠤⠭⠉⠁⠐⠒⠒⠂⠠⠤⠤⠍⢉⣙⡒⠢⣔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢨⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏⠉⠭⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠈⡆⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⡠⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣉⣉⡐⠒⠒⠢⠤⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠇⡇⠀⠀⢀⣠⣤⡶⠟⠋⠁⠀⠀⡰⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠟⠛⠷⠶⠶⣦⣤⣤⣌⣉⣉⠛⠒⠒⠒⠋⣰⣥⣶⠾⠛⠉⠁⠀⠀⠀⠀⠀⡜⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠋⠐⠒⠒⠀⠀⠀⠤⠤⠬⠭⠩⠛⠿⠿⠿⠿⠿⠏⠉⠀⠀⢀⣀⣀⣀⣀⣀⢀⠜⠀⠀⣠⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⡀⢀⣀⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠀⠀⠀⠀⠠⠤⠤⠤⠠⣄⠙⠀⠀⣰⣟⡠⠜⣛⡓⢦
⠀⠀⠀⠀⠀⠀⢀⣴⣞⠋⣤⣿⣃⣂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣆⣼⢟⡥⠚⠁⠀⠈⡎⡇
⠀⠀⠀⠀⠀⠀⠘⣶⠦⣽⣧⣤⣤⣤⣤⣤⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣒⣒⡒⠒⠒⠒⠒⠒⠊⣠⡿⠛⠁⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢒⠠⠌⠉⠻⣿⡭⣍⣛⠻⠭⢍⣉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡻⢟⡻⠟⠋⠀⠀⠀⠀⠀⠀⢀⡰⢡⠇
⠀⠀⠀⠀⠀⠀⠸⣙⠲⠬⣑⡀⠀⠘⠻⠓⠊⠉⠓⠲⠦⢭⣑⡒⠢⠤⢀⡀⠀⠀⠀⠀⣀⠔⡪⠔⠋⠀⠀⠀⠀⠀⠀⠀⡠⠔⣩⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣬⣙⡒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⢍⣒⣒⣩⡔⠋⡠⠔⠁⠀⣠⠄⠀⣠⠔⣋⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣍⣑⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⢉⠓⢢⣼⡧⢒⢕⠶⠊⣀⠔⢊⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣌⡙⠒⠦⠤⣀⡀⠉⠓⠪⢿⠞⢁⠤⢊⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⣤⣈⡙⠒⠦⠴⠚⣡⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⠶⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠒⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢷⡀⠀⠀⠀⠀⡤⠴⠒⠋⠉⠁⠀⣀⣠⡤⠤⠖⠒⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⢿⣟⣻⡇⠀⠀⣀⡤⠴⠒⠋⠉⠀⢀⣀⣠⡤⠤⠆⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢹⡜⣇⠉⠀⡀⠀⠁⠀⣀⡤⠴⠒⠋⠉⠁⠀⣀⣀⣤⠄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣷⡿⡏⣟⣿⠀⠈⠉⠀⣀⣠⡤⠴⠒⠋⠉⠁⠀⠀⢀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⢧⢻⣀⡀⠀⠀⠒⠋⠉⠀⠀⢀⣀⠤⠴⠒⠚⠉⠉⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⡟⢯⢹⣿⡷⠀⠤⠖⠚⠉⠁⠀⢀⣀⣠⠤⠶⠒⠚⠉⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣹⡘⣇⣉⡀⠀⠀⢀⣠⠴⠒⠋⠉⢁⣀⣠⡤⠤⠖⠒⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⠹⡍⢿⡿⠀⠉⣀⣠⠴⠖⠋⠉⠀⠀⣀⣀⣠⠤⠴⠆⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣌⣧⣻⡤⠦⣤⣀⠉⢀⣀⡤⠔⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣏⢷⠘⠯⠟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⢤⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣽⡜⣇⠀⠀⠀⣀⣠⡤⠤⣶⣒⣾⣿⣯⠭⠿⠓⠒⠒⣃⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⡶⠚⠯⠿⢶⠯⠭⢷⣒⣺⡭⠭⠶⠒⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠒⠒⠛⠛⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⡒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⣭⣭⡉⠙⢦⣀⠀⠀⠀⠀⠀⡴⠁⣰⠎⢹⣿⡿⠀⠀⣀⣀⠀⡏⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⢰⣿⡿⢋⡹⡄⠀⢻⣦⠀⠀⠀⣰⠁⢰⣿⠀⣸⡿⠧⣄⣰⣇⣈⣋⠁⢀⣉⣳⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣏⠀⢻⣿⢁⠏⣽⡇⠀⢸⣿⣆⡀⢀⠇⠀⣿⡇⠀⢋⡖⠀⣿⡏⢈⣿⡏⠀⣾⣿⡟⠀⢠⣶⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢄⣈⢛⣫⣾⣿⡇⠀⣸⡟⠀⢹⡾⠀⣸⣿⠀⢠⣿⠀⣸⡟⠀⣸⡿⠁⢰⣿⡟⠀⢀⣋⣵⢏⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⢻⠁⠀⡟⣰⡆⠸⠃⢠⣿⣿⠀⣿⣇⠀⢋⣴⡀⢛⣴⡀⠸⢋⣶⡀⠸⠿⣣⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣰⣿⡇⠀⠀⣼⣿⠏⠙⠿⠿⠛⠿⠿⠛⠿⠿⠟⠻⢿⠿⠛⠳⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣼⣿⡟⢣⣀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠀⠀⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⡄⠀⢠⠖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠃⠀⡟⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠏⠀⢰⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠋⠀⠀⣾⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⠚⢹⡆⠀⢠⠞⠉⣳⠀⢀⣼⠁⠀⠀⣠⡏⠀⠀⡴⠋⣩⣷
⠀⠀⠀⠀⠀⠀⢀⡴⢻⡿⠁⠀⣼⠀⠀⣏⣠⠴⠃⣠⢞⡇⠀⢀⡴⣻⠀⢀⣼⠃⠀⠙⣾⠶⠖⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠿⠁⠀⠀⢧⣀⠴⢧⣀⡠⠞⠁⠸⣧⠴⠋⠀⢯⡤⠞⠘⢆⣀⠴⠁




⣿⣿⣿⣿⣻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠟⣛⡛⠷⠀⠾⢛⣛⠻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢱⣿⣿⣿⣷⠀⣾⣿⣿⣿⡆⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣸⣿⣿⣿⡟⣀⢻⣿⣿⣿⣇⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⠿⢟⣩⣴⣿⣷⣍⡻⠿⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣾⣿⣿⣿⣷⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠻⠿⠟⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠰⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣠⣤⣶⣶⣶⣬⡛⠿⠿⠿⢟⣡⣶⣶⣶⣤⣄⡀⠀⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠘⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⡇⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡧⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣦⢻⣿⣿⠿⢛⣛⠻⠿⠿⢛⣛⡻⠿⢰⣶⣶⣶⡆⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣘⣩⣥⣜⠻⢿⣿⣿⣷⣦⡙⠟⣀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⢉⣛⠿⣿⣿⠟⢫⣭⣝⣛⠻⠃⠀⢀⣠⣶⣿⣯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠛⠉⠥⠶⠿⠿⠿⠷⠶⠀⠶⠶⠶⠶⠦⠭⠀⠀⠨⠭⠶⠶⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⡇⢸⠈⡟⢸⠤⡇⡞⢱⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠉⠀⠁⠈⠀⠁⠉⠋⠈⠀⠁


⡿⠿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠼⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠼⠄⠁
⠀⠀⣠⡴⠊⠁⢀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⡴⠚⠁⢀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄
⢠⡴⠋⠀⠀⠊⠁⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡴⠋⠀⠀⠊⠁⣠⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡳⢍⡗⠇⠀⠀⠀⣠⡤⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⡳⢭⡓⠇⠀⠀⠀⣠⠤⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⢭⣓⠀⠀⠄⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢭⣓⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⡈⠁⠀⠀⠀⠀⠀⠀⠖⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⡈⠁
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡙⢄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⠠⢶⡏⠁⠀⠑⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⡠⢶⡏⠁⠀⠑
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠼⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠼⠄⠁
⠀⠀⣠⡤⠚⠁⢀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⣠⡤⠚⠁⢀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄
⢠⡴⠋⠀⠀⠊⠁⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠊⠁⣠⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡳⢽⡓⠇⠀⠀⢀⡠⠤⠀⠀⠈⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⡳⢝⡓⠇⠀⠀⢀⡠⡤⠀⠀⠀⣤⠁
⠠⠀⠀⠀⠀⠀⠀⠉⠒⢭⣓⠀⠀⠄⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢭⣓⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⡈⠁⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⡈⠁
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢆⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢄
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⡠⢶⡏⠁⠀⠑⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⡠⢶⡏⠁⠀⠑
⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣉⡛⠒⡒⠒⠲⠦⠤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠴⠂⢄⣀⠴⠍⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠈⡝⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⠒⢲⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⣹⠀⠀⠀⠀⠀⠀⢀⡜⠁⠉⠁⠐⠒⠒⠒⠊⠀⠀⠀⠀⠀⢠⡟⢸
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⡟⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣠⡟
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⡟⣽⡟⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣰⡟
⠀⣠⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⡟⣼⠟⠀⠀⠀⠀⣰⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣰⠟
⠀⣯⡟⠛⠛⠿⠿⠿⠶⣿⣿⣭⣽⣿⣟⣛⣼⡟⣼⠏⠀⠀⠀⠀⠀⡏⠏⠛⠛⠻⠖⠶⠶⢦⡤⣤⣄⣀⣀⣀⣼⠏⣰⠟
⠀⣿⣅⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠋⣼⠏⠀⠀⠀⠀⠀⠀⡟⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠋⣰⠏
⠀⠉⠉⠛⠛⠛⠻⠿⠶⠶⣶⣤⣤⣤⣄⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠲⠿⠴⠶⣤⣤⣀⣠⠄⠀⢴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣶⣶⣶⢶⢶⢶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⢶⣶⡶⠶⣶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠁⠀⠀⠀⠉⠙⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢀⣠⠴⠶⠛⠓⠓⠶⢤⣀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡟⠀⢀
⠀⠀⠀⠀⠀⣰⠋⠀⢀⣼⣏⠁⠀⠀⠀⠀⠀⠀⠀⢈⣳⣤⠀⠀⢳⡀⣠⣶⡜⢿⣿⣿⠃⠀⠀⠀⣀⣀⣤⣭⣿⣿⣿⣶⡄⠉⠓⣄
⠀⠀⠀⠀⢰⠃⠀⢠⢿⡅⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⢧⣠⣾⣿⣿⣿⣷⣾⣿⣿⣴⣿⣿⣶⣿⣿⣿⣿⣿⣷⣿⣿⡆⠀⠘⠋⡄
⠀⠀⠀⠀⡏⠀⣂⣭⣴⣧⣤⣦⣦⣶⣶⡄⠀⠀⢀⣀⣠⣤⣼⣻⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣧⠀⠀⠀⠃
⠀⠀⠀⠀⣿⠰⣿⣿⣿⣿⣿⣿⣩⣿⣿⣽⣻⣏⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣧⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧⣐⣿⣿⠿⢿⡿⠃
⠀⠀⠀⠀⠈⠑⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣿⣡⣶⣿⣿⣿⣿⣿⣿⠛⢛⢛⣥⣶⣶⣶⣦⣕⢢⠘⠃
⠀⠀⠀⠀⠌⠄⠀⠀⢸⣿⡇⣾⣿⣿⣿⣷⣿⣿⣷⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⢷⢡⣿⣿⣿⣿⣿⣿⣿⣷⢡⠀⠄
⠀⢸⡀⠀⠐⠀⠀⠀⢸⣿⡇⠈⢻⣿⡿⢿⣿⣿⣿⣽⠿⠿⠛⠓⠛⠿⣿⣿⣿⠟⠓⠏⠁⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⢊
⠀⠀⠱⡄⡾⠂⠀⠀⢸⣿⡧⢤⣶⣿⣿⡛⠫⠭⣅⡀⠤⠀⠄⠤⣄⡀⠤⠤⠄⠤⣀⡤⡤⣽⡜⢿⣿⣿⣿⣿⣿⣿⠟⡜
⠀⠀⠀⠈⠙⠢⠄⠀⡻⠿⠟⠂⠉⠋⠉⠁⠀⠀⢉⣃⡤⠀⠀⠀⢹⠃⠐⠀⢢⣤⣋⠁⠀⠀⠙⠦⣝⣛⡻⠛⣛⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⢐⢺⢒⠨⠉⠔⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⣿⡧⠙