Coffee Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡬⠥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⢀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⡀⣀⣴⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⣀⡸⠣⠴⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠛⠻⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠞⣛⠩⠀⠀⣀⣀⣄⣀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠭⠭⠅⠐⠒⠂⠉⠁⠀⠀⠀⢸⠁⠀⠀⠀⠙⡆⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣠⣴⠟⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠾⠟⠛⢉⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠳⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣀⣠⠤⣔⠊⠉
⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠙⢦⡙⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠂⢄⠀⠉⠓⢄
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⠤⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⢴⠖⠉⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢳
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⠤⠬⠭⠭⠭⠭⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠤⣀⣈⠁⠒⠂⠠⠤⠄⠀⠤⠠⠤⠀⠠⠤⠀⠐⠒⠈⢁⣀⣠⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⣾⣟⡯⠍⠓⠒⠒⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⣽⣿⣧
⠀⣿⠈⠉⠻⠓⠒⢶⣶⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⣶⣶⣶⡶⠟⠛⠉⢸⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⣠⣤⣤⣀⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠟⠋⣉⣉⣉⣙⠻⣷⣄
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⠟⠋⠉⠉⠛⣷⡌⢿⣆
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠃⠀⠀⠀⠀⠀⠘⣿⠸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠃⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿
⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡟⢸⡿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢀⣼⡟⢠⡿⠃
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣠⣾⠋⣰⡿⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣠⣴⡾⠋⣡⡾⠋
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠛⠋⣁⣴⡿⠋
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣴⡾⠛⠁
⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠁
⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡾⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣽⣭⣬⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠲⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣚⡭⠵⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠲⠤⣍⣓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⢀⣀⣠⣤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⢤⣄⡀⠀⠉⠛⢦⡙⢦
⠀⠀⠀⠀⠀⢠⡏⡏⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⠀⠉⠙⠲⣄⠀⣿⢸⡆
⠀⠀⠀⠀⠀⢸⣇⢣⡞⠉⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠋⠀⡤⠤⡄⢹⡀⠀⠀⠀⠀⠀⢀⡿⣡⢾⡇⣀⣤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⢸⡈⠳⣌⡳⢤⣀⡀⠀⠀⠸⡄⠀⠀⢰⠖⠒⠚⠁⠀⢹⢸⡇⠀⢀⣠⡴⢞⣡⠞⣡⠞⠋⠁⠀⣀⣀⣀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠓⠮⣍⣛⠒⠶⠿⠦⢤⣬⣷⣤⣤⣤⠤⠿⠾⠒⢋⣩⡤⠖⠉⠀⠀⣧⣀⡤⠖⠋⠁⠀⠉⠙⣦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⠶⠶⠶⠶⠶⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢧⣄⣀⣀⣀⣀⡤⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠈⠉⠉⠁⣀⡠⠔⠋
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠻⣍⠛⠋⠉⠉⠁
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⢸⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⢦⣌⡙⠲⠦⢤⣄⣀⣀⣀⣀⣀⣀⡤⠤⠖⠚⣋⡥⠞⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠤⠤⠤⠴⠶⠒⠋⠉⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠙⠪⣝⡲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⣚⡭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠺⠭⣍⣓⣒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠴⣖⣒⣻⠭⠷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢠⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⠁⠀⡾⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣰⠇⠀⣸⠃⠀⢠⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⡟⠀⢰⠇⠀⢠⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠖⣾⠀⢸⠃⠀⡿⠀⢠⡟⠀⢀⡏⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣭⠴⠒⠀⠀⡏⠀⣿⠀⢸⡇⠀⢸⠁⠀⣼⠀⠀⠠⣌⣙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢾⣿⣶⣞⣋⣥⣤⣿⣤⣿⢠⣼⡇⣤⣼⢠⣤⣿⣀⡀⠒⠦⣭⣛⠆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠈⢉⣠⠟⢑⣶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣍⠓⠂⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⣉⣀⣤⡴⠾⣛⣵⡆⣚⣛⠧⣄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⢋⣉⣉⣭⣴⡶⠿⣛⣭⠀⠉⠉⠙⣾⡇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠃⢸⣟⡳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣉⣉⣥⣴⡶⠿⣛⡅⠈⠑⣂⡴⣻⡣⢠⡏
⠀⠀⠀⠀⠀⢀⣤⠞⣋⡥⢒⡠⠄⠀⠀⢻⣟⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣉⣉⣉⣩⣭⣴⠶⢟⡋⢡⣞⣛⣽⣾⢟⣵⠟⢶⣄
⠀⠀⠀⠀⣰⠟⣡⡞⢡⡞⢉⡴⠊⡀⠀⠈⢿⣟⠒⠄⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣭⣭⣴⡶⢿⡛⢡⡿⢿⣛⣭⡶⠛⠁⣀⢢⡈⢷⡄
⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⣿⡀⢾⡀⠀⠀⠀⠻⣯⡒⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣭⣥⣤⠴⠶⣞⡛⣱⣿⣿⣿⣿⣿⢿⡌⢳⡈⣆⢳⠈⡇
⠀⠀⠀⠀⠻⣆⠹⣧⣌⠳⢮⣛⠶⣬⣁⠀⠀⠀⠈⠻⣥⣄⣀⠀⠀⠀⠀⠀⠤⠤⠶⢶⣶⣾⣿⣿⣿⣿⣿⢸⣿⡜⣿⢎⡇⢸⠇⡏⢀⡼⠃
⠀⠀⠀⠀⠀⠈⣳⣮⣙⠷⣦⣍⣙⠓⠦⠄⠀⠀⠀⠀⠈⠙⠻⢯⣿⣛⣛⣛⣛⣛⣯⣽⡿⠿⢛⣫⣵⡿⢋⣾⠟⣰⠟⠘⠁⢋⣠⣶⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠨⠽⠿⠶⢮⣭⣙⣓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠽⠷⠶⠶⠶⣚⣛⣋⣭⡵⠾⠛⢁⣈⣡⣤⣶⣾⣿⣿⠿⠭⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣉⣹⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣛⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣟⣛⣛⣛⣛⠛⠛⠛⠛⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⣩⣭⣭⣭⣭⣭⣭⣉⣉⡉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣡⠶⠋⡹⠋⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣠⠞⠂⠀⣼⠁⢠⠀⠀⠀⢈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠛⠀⠀⣴⢃⣠⡦⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠂⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⡤⠚⠀⣠⣦⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢾⠆⠀⠀⠀⠀⠀⠀⠌⠃⠀⠁⠈⠉⢒⡆⠀⠙⣿⠀⡆⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⣤⢀⡉⠻⠀⢇⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⠇⠀⢸⡿⠀⠀⢠⠗⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠱⣤⠦⠈⠢⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠏⣰⡿⠚⠁⠰⠟⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠉⠻⠆⠐⠶⠚⠾⠃⠀⠀⠶⠀⠀⠒⠛⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠠⡟⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⣦⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠀⠀⠈⠁⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⣧⠀⠀⠶⣦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠶⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠀⠀⠀⠀⠀⠀⠐⠒⠲⠚⠀⠀⠀⠀⠀⠀⢀⣤⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢤⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣶⣶⣶⣶⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘






⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣾⣶⣶⣶⣾⣶⣾⣶⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣻⡟⢿⡿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢁⠈⠀⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠂⠀⠈⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠶⢺⡀⠀⢁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⣹⠀⠀⣏⠉⢵⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢋⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠒⠡⠐⢧⠀⠱⣀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣀⡀⠀⡈⡗⢤⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡜⢀⢊⠐⠁⠀⠀⠀⠑⢤⡈⠑⠂⠷⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠴⠊⠉⠀⠀⠀⢀⠻⠲⡐⠄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⡼⠉⠑⠂⠠⠀⢤⡀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠡⠁⠀⠀⠀⠀⠀⣠⠞⠀⠃⠀⢨⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠙⢒⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀⣁⠤⠖⣒⠼⠶⠶⠦⠤⠶⠚⠁⠀⠀⠀⠠⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠠⠆⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⣠⠴⠶⠒⢉⣡⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠌⡁⠂⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⠀⠔⠊⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠴⠒⢈⠠⢐⡮⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⠀⠀⠀⠈⠉⠁⠒⠒⠀⠤⠤⠤⠤⠤⠤⠠⠤⠤⠶⠶⠾⠛⠛⠓⠊⠉⠉⠠⠀⠂⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠃⠐⠒⠒⠒⠒⠒⠒⠒⠂⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⡀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⡴⠀⠀⠰⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡎⠀⠀⠀⢀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠠⠀⡘⢡⡷⢤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⡄⢖⠀⣠⣴⣶⣶⣿⡿⢷⣼⣿⣿⣿⣿⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠷⢂⢹⣧⣤⣬⠍⠉⠉⠁⠉⠉⠉⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠑⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠠⠐⠞⠻⣦⡀⠀⠀⠀⠀⠀⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠷⣤⣍⣓⣶⣶⡤⠴⡋⠁⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣌⠻⣿⣿⣿⣿⣦⣌⣉⣉⣉⣁⡀⠔⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠀⢠⡴⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠋⠉⠉⠀⠀⠀⣀⣀⣤⡔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⠉⠁⠀⠈⠁


⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡈⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠄⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠐⢄⠊⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠦⣍⣘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⣸⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣠⠔⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⢀⢤⣶⠿⠛⠉⠀⠀⢀⢠⠃
⠀⠀⠀⠈⢇⠀⣈⣉⠐⠠⣀⢮⠢⡐⠄⠀⠀⠀⠀⠀⠙⠓⠲⢦⡴⢒⡵⠟⠁⠀⠀⠀⠀⢄⡴⠃
⠀⠀⠀⠀⠀⠑⠄⡀⠁⠲⣈⣧⠦⠬⠱⡾⣽⡴⡦⡄⣀⣰⣯⣥⣔⣉⡀⡀⠠⢀⡂⠬⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣉⣗⣾⡿⣿⠬⡯⢑⢔⡉⣀⠛⣧⣶⡖⡖⠒⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠤⠠⡿⠿⠹⣞⣛⢯⡉⣯⣽⣿⣘⢣⢻⣯⢏⣿⣾⣤⢀⡠⡀
⠀⠀⠀⠀⠀⠀⠜⠀⢘⢐⣝⡭⠨⣽⢛⠖⢚⣿⣓⠾⠏⠉⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠤⠂⠈⠑⠋⠉⠒⠃⠘⠯⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢦⠹⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⡄⣿⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢰⠃⣿⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢀⠏⣸⠇⠀⢰⢿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟⠀⠀⡞⢈⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣾⠁⠀⡼⠁⣼⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣰⠏⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⠶⠖⠚⢻⡀⠈⠁⣿⠀⠈⠀⠀⠀⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣋⣥⣤⠴⠶⠶⠒⠈⠻⠦⡘⠙⢧⡘⠛⠳⠶⠶⠤⢤⣀⠉⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣥⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠾⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⣀⣼⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡇⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢹⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠖⢋⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⠶⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⡁
⠀⠀⠀⢀⣀⣤⣶⠾⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀
⠀⢀⣴⡿⠋⠉⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⡶⠶⠖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄
⠀⢸⣿⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢤⣄⠀⠀⠀⠀⠀⣹
⠀⠈⢿⣦⡀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⠿⣶⣄⣈⠙⠿⢷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠞⠋⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⣶⣤⣬⣉⣉⡙⠛⠛⠛⠻⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠉⠉⠁⣀⣀⣠⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠋⠉⠉⠉⠁

⣦⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣭⣭⣿⣵⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣶⣶⣾⣷⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠃⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠠⠆⠉⠈⡈⠉⣁⠉⠀⠀⠀⠀⣼⣿⣿⠿⣽⣶⣶⣶⣮⣉⠙⠁⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠴⠀⠀⠀⠀⢀⠀⢲⣂⣀⣤⣤⣶⠂⠀⡀⠀⠁⠘⠛⣩⠉⠿⣿⣿⡆⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠄⠀⣸⠃⠀⠀⠀⠀⠀⢈⡀⠁⣙⡂⠐⠛⢿⢷⣄⡀⠈⠃⣬⣿⡿⠁⢘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢌⣙⠀⠤⣄⣀⠀⠈⠉⠀⠩⠁⣠⡀⠀⢐⠛⢛⣛⡻⠞⠋⣤⡞⢾⣏⣀⣄⣈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠻⣶⣶⠄⠀⠀⠀⡀⠐⠓⠛⠓⠂⣙⡋⠩⣷⣴⡀⢠⡿⣧⣿⡟⠁⠀⠙⣯⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠁⠀⠀⠉⠄⠈⢸⠏⣿⠇⠀⠀⣠⡿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⠶⠶⠛⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠷⠟⠛⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡴⣿⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢀⣶⣿⡿⢟⣳⡄⠈⠉⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢘⣿⣉⣻⣷⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⠀⠀⣛⡾⣿⣟⣩⣿⣥⡀⠀⠱⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢁⣯⣽⡟⠉⣻⠏⣿⣾⣧⣤⣀⣀⠀⣀⢀⠀⣀⣁⣠⣶⣿⣫⢊⡽⣒⣿⣿⣷⠞⠀⠀⠀⢘⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⣥⡀⣌⠉⢙⠛⠻⠿⠿⠿⠿⠿⡿⠿⠿⣿⣟⢍⣯⣱⣝⠋⠁⠀⠀⠀⢀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠞⠁⠰⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠉⠉⢁⠀⢀⡮⢶⡋⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠰⠤⠦⠤⠤⠤⣄⣢⣤⣄⣤⡐⠿⠯⠁⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⠰⠶⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠖⣊⣉⠭⡤⠤⢤⠤⠬⢭⣙⡒⠦⣀⠀⠻⠷⣀⣀
⠀⠀⠀⠀⠀⡰⣡⣶⣋⣭⣥⣴⣶⣶⣶⣶⣧⣔⠀⠈⠑⠪⡻⣄⠀⠀⠈⠑⣄⣀⣠⠄⠀⠤⠄⠠⢄
⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣺⠗⠀⠀⠀⡹⣏⣹⠦⠤⠚⡁⠀⣀⡴⠂⠈⠀⠠⠊
⡴⢊⡭⠭⣙⣿⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⢿⡇⢀⣠⠤⣚⣹⡟⠋⢀⠄⣸⣷⣚⡉⠉⠀⠀⣠⣆
⣇⢸⡀⢀⡼⣿⣿⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⣍⣴⣿⣿⣿⢧⣤⡷⢞⣉⠁⠀⠀⠀⣴⣉⣠⣄⣙⡢⢄
⠘⢦⡙⠯⣄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣿⠀⣌⢧⡀⠭⠀⢠⢶⡉⠻⠀⠉⠈⠉⠂⢘
⠀⣼⠙⡷⠦⣽⣟⠣⠀⠀⠀⠀⠀⠀⠀⠀⣌⠙⣿⣻⣿⣿⠇⢀⡿⣧⢿⣯⣭⣓⢦⣉⣀⣠⠤⠤⠶⠋⠉
⠀⢿⠠⠂⠀⠀⠙⢦⡀⣠⣄⣀⣄⣠⣀⣦⣿⣼⣿⣿⣿⣿⣶⣄⣠⡿⣻⣧⣩⣿⠟⠁
⠀⠘⢆⠠⠀⠀⠀⠀⠙⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⠤⠆⣱⣿⣍⣧⣽⣀⡀
⠀⠀⠈⠓⢤⡀⠀⢀⡀⠀⠀⠈⠙⠛⠿⠟⠿⠿⠟⠛⠉⠉⢀⣠⣾⣿⠛⢽⣿⣭⣿⢿⡆
⠀⠀⠀⠀⠀⠈⠓⠲⠤⣍⣁⣀⣀⣀⢀⣀⣀⣀⣠⣤⣶⣞⣩⣿⣿⣿⢀⡘⠿⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠛⠉⠉⠉⠈⠿⣯⠭⠶⣿⢿⣝⢧⣀⣾⣯⣽⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣎⢿⣼⠏⠳⠬⡅⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠶⠶⣦⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⣟⡿⠋⡁⠄⠀⣠⡀⠈⠁⠒⠬⢝⡺⢷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠕⣿⡷⠉⢂⠕⠊⠀⠀⠈⣃⠐⠀⠀⠀⠐⢍⠲⣝⢷⣌⠷⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡴⣫⠞⠁⣾⠏⡄⡔⠁⠠⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠈⠎⢿⣦⣤⠖⢘
⠀⠀⠀⠀⠀⢀⡞⡴⠁⠀⢸⣼⠘⠨⠀⠀⠀⠁⡀⠀⠀⣴⠄⡀⢁⠀⠀⠀⠀⠀⠀⠘⣼⡇⣠⢾⣧⡀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣿⢿⢀⠀⠐⡀⡀⠀⠘⠦⠄⣀⣄⠀⠈⠀⠀⠸⠀⠀⠀⠀⡇⣿⣥⣀⡀⠀⠈⠀⠒⠄
⠀⠀⠀⠀⢰⠃⠁⠀⠀⠀⢻⠸⡈⠌⢆⢳⣕⢄⡀⠀⠀⢈⠃⡴⣋⢔⠵⠁⠀⠀⡸⢸⢹⣿⠟⠻⠉⢗⡶⠦⠤⠃
⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠈⣧⠱⡀⠀⠀⠙⠻⢟⣓⠶⣤⡁⣴⡿⠃⠁⠀⢀⠜⡱⢣⣿⠛⠃⠀⠀⠈⣧
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠘⣷⣙⢦⡈⠲⢄⡀⠀⠉⠢⢱⠇⠀⠀⠀⠔⣡⢞⣵⣿⠁⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⢍⡲⠤⣌⣁⣀⣀⣀⣀⣠⠤⢖⣫⠴⣻⠞⠁⠀⠀⠀⠂⠀⠀⡟
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢂⠈⠻⡷⣍⣙⡒⠲⠾⠭⠶⠶⠒⢛⣉⡴⠞⠁⠀⡀⠠⠀⠌⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠠⡣⡈⠮⡛⠻⠷⠶⠶⠶⠖⠛⠋⠉⠀⠀⠠⠊⠀⢀⠌⠄⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠓⠀⢀⠀⠀⠀⠀⠀⠀⡀⠄⡪⢀⠕⡡⠊⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠂⢅⡀⠀⢀⠀⠈⠀⠀⠀⠂⠉⢀⠁⠊⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠄⣢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣗⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣖⣫⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠯⠭⣿⣿⠿⠭⠭⠷⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⣠⠄⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣼⡏⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢻⣧⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠻⣷⡀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢹⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⡼⠃⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠀⠒⠒⠂⠒⠒⠒⠒⠐⠒⠒⠒⠒⠐⠒⠀⠠⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡖⢉⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡍⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⣇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡰⠁⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⠤⠊⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⣀⣀⠤⠐⠁
⠀⠀⠀⠀⡠⠄⠒⠒⢈⣉⣭⣤⣬⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣧⣤⣤⣤⣉⡁⠒⠒⠠⢄
⠀⠀⠀⠀⠢⢄⣀⠀⠛⠿⠿⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢀⣀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⢤⡀⠭⠭⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠭⠭⢀⡤⠒⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠂⠀⠤⠤⠤⠤⠤⠤⠀⠐⠒⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⣰⡁⢀⣀⣤⣴⣶⣾⣿⣿⠿⠿⠿⣿⣿⣿⢿⣿⣟⣛⣛⣛⠛⠻⣛⡿⠿⣿⣿⣶⣦⣄⡀⠀⠀⠀⡉
⠀⠀⠀⠀⣦⠻⣿⣿⣿⠿⣿⣿⣿⣿⣷⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣷⣬⣉⣙⢻⣿⣷⣆⣴⢁⡼⢀⣠⡔⠂⠉⠀⠲⣄
⠀⠀⠀⠀⢻⡷⢈⣛⡷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣥⣾⣿⣿⣣⠴⠀⠀⠈⠙⣿⡄
⠀⠀⠀⠀⢈⠁⠀⠀⠁⠀⠉⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⡛⠛⢋⢡⣴⣾⡯⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠈⠼⣿⣿⡯⢸⣿⡿⠟⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⣻⣿⣧⣾⠇⠀⠀⠀⠀⠀⠀⠊⡀⠁
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣷⡏⠀⠀⠀⣀⠤⠚⣡⠜
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣤⡴⠒⢊⣁⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡞⣿⣿⡿⣛⣉⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠙⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢀⡎⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢁⣠⡤⠞⢋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣶⠟⠁⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣿⠁⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠻⣦⡀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⣸⡷⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢀⣴⠟⢁⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡾⠃⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠖⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠓⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣫⡭⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⣈⡙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣡⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣭⣽⣟⣷⣦⣄⣙⣦⢸⡷⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⢸⣇⣀⡀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⣠⣾⠉⠉⠻⡆⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⡏⠀⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⡿⠀⢀⣴⠏⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⢀⣾⣡⡴⠛⠁⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⣹⡷⢛⡿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠶⠶⢶⣟⠉
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⣴⠋⢀⡾⢁⣿⡿⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢰⡏⢀⡾⣿⣯⡥⠶⠶⠚⣛⣿⣧⣀⣤⣤⠴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠈⢷⣾⣡⣽⣟⠒⠒⠛⠛⢉⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢮⣭⣉⣉⡓⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠓⠒⠶⠶⠶⠶⠖⠒⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠠⠄⠐⠒⠒⠒⠒⠒⠀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⡀⠀⠀⠠⠄⠀⠀⠀⠤⠄⠠⢀⡀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⠂⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⢁⣀⠀⣁⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡋⠈⡙⢿⠀⠀⠀⠀⠀⠉⠐⠒⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡢⡑⢯⣀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⡨⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⠻⢶⣬⣉⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⢠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈⠛⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡤⠋⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣥⡴⠖⠋⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠡⡀⠀⠀⠀⠀⠀⠀⡀⢙⢦⣄⠀⠀⠀⠀⠀⠀⠀⣀⠔⣺⣿⠟⠉⠁⠀⠀⠀⠀⢀⢊⡞
⠀⠀⠀⠀⠀⠻⡂⠤⢀⡉⠉⠑⠢⢄⡰⢦⠳⣌⠢⢄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠲⠶⣦⡶⠚⣡⡾⠋⠀⠀⠀⠀⠀⠀⡀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠘⠦⡈⠉⠉⠢⢤⡀⢳⣆⣇⠈⡑⢢⡭⠖⡲⢂⣆⠀⠀⠀⠀⣠⡖⠚⠁⡠⠞⠁⠁⠀⠀⢀⡀⠤⢐⡨⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⠄⣀⣙⣶⣿⣆⡉⣏⣠⡿⡉⠛⣯⡝⠙⣍⣚⢛⠛⠛⠛⠛⠒⠒⢒⣐⡂⠭⠕⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣯⣼⣻⡟⢻⢔⠶⡗⢎⡵⢩⡀⣈⡊⠁⣾⣾⣳⢏⢫⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢠⣠⠀⣼⡿⣿⠧⢷⠾⣙⠮⡋⠹⣏⣉⣽⣿⠌⠫⣜⢿⣫⡼⢩⣿⣾⣦⣀⢀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⡀⠴⡌⠒⢳⡀⡸⠗⠛⡓⠿⢄⠾⣏⣭⣽⣿⠶⢥⢬⣿⠳⠛⠳⠟⠃⠡⠈⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠲⠤⠞⠒⠋⠁⠩⡟⠰⠥⢤⣓⣛⠷⣆⣘⣣⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠞⠁⠀⠀⠈⠁⠀⠈⠉⠁⠀⠙⠶⠞


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⠴⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠉⣀⣤⡤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠤⠤⣄⣀⣈⣷⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠛⠛⠛⠉⠉⠉⠁⢸⠊⠀⣀⣀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⠱⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠊⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡼⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⣀⠤⠊⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠀⠀⠒⠂⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠳⠦⠤⠤⣀⣀
⠀⠀⠀⠀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⠉⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⠤⠤⠬⠍⠉⣛⡒⠒⠒⠒⠛⠛⠭⠍⠉⠤⠤⠀⠐⠂⠀⠁⠀⠀⠀⠀⢀⣠⡴⠟
⠀⠀⠀⠀⠈⠻⢍⡑⠒⠢⠤⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡤⠤⠤⠖⠒⢚⣩⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⣄⣀⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣀⣠⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠶⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢁⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠖⠒⠛⠋⠉⠀⣾⡀⠀⠠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠑⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣾⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠶⠶⡶⠶⠶⠶⠶⠶⡶⣶⣶⡶⣶⣶⣶⡶⣶⣶⢶⣶⡶⢶⣶⣶⢶⣶⣶⠶⣶⣶⡶⢶⣶⡶⢶⣶⣶⣶⢶⣶⡶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⢉⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡤⠃⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⣀⡠⠤⠤⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠔⠋⠀⠀⠀⣠⠶⠦⡌⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢁⡾⠃⠀⠀⠀⠀⠀⡏⠀⣄⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣾⠃⢀⣴⠶⠶⣄⠀⠹⣄⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡀⢸⢣⡎⠳⠈⡇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⣧⠘⣎⠳⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠻⣆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢻⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⠠⠃⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣶⠋⠉⢀⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠂⠘⠋⠀⠀⣀⣀⣀⣀⡤⠤⠤⠴⠒⢋⣠⠾⢻⣷⣒⡓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠚⠉⠀⠀⣸⠋⠉⠙⡇⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡰⠃⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠐⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢀⡄⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢀⠀⠀⠀⠀⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣠⠴⠟⠁⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣤⣀⣀⡈⠉⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⣭⣉⣉⣁⣀⣀⣀⣀⡀

⠀⣹⣿⣿⣆⡀⠀⠀⠀⠀⠐⢆⠀⢀⣠⣿⣟⣛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣹⣿⣿⣂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠿⠐⠀⠀⠀⠀⠀⠀⠀⠈
⣾⣿⡿⠍⠉⠀⠀⠀⠀⠈⠀⠈⠉⠛⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠠⣤⣤⣴⣾⣿⡿⠍⠁⠀⠀⠀⠀⠀⠀⠤⢤⣴⣾⡿⠟⠃⠀⠀⠀⠀⠀⠀⠠⣤⣤
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠐⠒⠢⠭⠭⠭⠗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠨⠭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⢀
⣿⣷⣶⡴⠆⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠠⣼⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶
⠉⠉⢙⠷⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⣻⣧⣀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠈⠀⠀⢸⡷⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⣀⣬⡀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⣴⣿⣿⡅⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⢂⡀⠀⠀⢠⣼⣿⣯⡁⠀⠀⠀⠀⠀⠀⠁⣀
⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢈⢙⣺⠿⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠤⠴⠾⠿⠃⠠⠀⠀⠀⠀⠀⣉⢋⣬⠿⠉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⢠⠀⠀⠀⠀⠀⣾⣶⣿⣯⣥⣶⣬⡈⢻⡆⠀⠀⠀⠀⠀⢤⣶⣿⣿⣿⣷⣶⠶⢂⡀⠀⠀⠀⠠⣼⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⡀
⠀⡀⢰⣾⠇⠀⡀⠀⠀⠀⠉⠉⠛⠛⠛⠋⢿⣿⡁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢈⠆⠀⠀⠀⠀⠀⢀⢦⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠉⠡⠀⠀⣀
⠀⣸⣿⣿⣗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠋⠁⠀⠀⠀⠀⠀⢀⡀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠈⢱⡄⠀⠀⠀⢠⣿⡇
⣾⣿⡿⢏⠁⠀⠀⠀⠀⠀⠀⠰⠤⠶⠶⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠤⠴⠿⠟⠃⠀⠀⠀⠀⠀⠒⠢⣤⣶
⠏⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠪

⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠿⢿⣷⣄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣰⡤⠀⠀⠀⠀⠀⠀⣿⣾⣯⣭⣄⣶⣤⡈⢻⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿
⠀⢈⣿⡏⣉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣽⣧⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⢻⣿⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣿⣤⣀⠀⠀⠀⠀⠀⠐
⣤⣾⣟⣯⠁⠀⠀⠀⠀⠠⣀⠀⠀⢀⣼⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⠋⠀⠀⠀⠀⠢⡀⠀⠀⣠⣾⣿⣯⠅⠀⠀⠀⠀⠀⠀⠙⣄⣀
⡛⠋⡉⠀⠀⠀⠀⠀⠀⠀⢈⡉⠽⠿⠭⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠾⡿⠛⠁⠀⠀⠀⠀⠀⠀⣈⠩⠽⠿⠍⠥⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⣿⣿⣇⡀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠐⣿⣿⣯⣭⣶⣶⣤⠉⣻⡀⠀⠀⠀⣰⡀
⣀⣀⣴⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣁⣉⣀⣀⣀⣈⣑⣂⣀⣀⣄⣀⣠⣾⣿⣃⣀⣀⣀⣀⣀⣉⣉⣙⣙⣉⣉⣛⣋⣀⣀⣀⣀⣉⣑⣀⣀⣠
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣬⣉⠙⠿⢦⣤⣄⠙⠫⠉⠁⠀⠐⠦⡀⠀⠤⠤⠀⠤⠐⣦⣤⣤⣤⣴⠋⠁⠈⠁⠀⠀⠀⠠⡈⣿⠏⠀⢁
⠀⠈⠉⠙⠒⠦⠬⣑⡒⠤⢤⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡏⠉⣿⣿⣿⣇⣀⣀⠀⢀⣠⣿⣴⣵⠇⡐⣱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠪⢭⣖⣧⣤⡶⠀⠀⠤⠶⠿⠿⠿⠿⠿⠿⠓⠒⠒⠒⠚⠛⠻⠛⠛⠙⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⢁⠀⠀⠀⠀⠀⠀⠀⠸⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⡇
⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡏⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡂⢸⠇
⠀⠇⡀⢻⣷⡶⣶⣶⡖⣂⣉⣻⢰⡇⠀⠀⠠⠁⢀⣤⠉⣿⢺⣿⣿⡃⠀⠠⠀⠀⠀⡄⠀⠀⢸⣦⠘⠄⢴
⠄⢱⡄⠾⠁⢁⣩⣿⣇⣿⣬⣿⠛⠉⠙⠛⡳⠗⢶⣾⡦⣿⣤⣬⣿⣿⣇⣉⣀⠀⢀⣙⣫⣭⣿⣷⣒⣶⡿
⠀⢸⡏⣽⢻⣶⣾⣿⡟⠛⠉⠿⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠉⠛⠛⠙⠛⠉⠉⠉⣽⢻⣿⠿⡿⠿⣿⣿
⠠⢺⠿⠻⡿⠻⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⢿⡟⠓⠶⠶⠿
⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⢺⣽⣷⠄
⢀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⢸⣿⣧⡄
⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⢠⣀⣾⠛⠿⠷⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣶⣛⡿⠿⠷⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠶⠶⠤⣭⣉⡓⣶⣄
⠀⠀⠀⠀⠀⠀⠸⣅⠙⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢛⣹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠉⠓⠦⣬⣉⡉⠛⠓⠲⠶⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠴⠶⠒⠋⢉⣠⣾⠃⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣤⡀⠀⠉⠙⠓⠒⠶⠦⢤⣤⣄⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣠⣤⡤⠶⠒⠛⠉⠀⢸⣀⣸⠇⠀⠈⣳⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠰⣿⣿⣿⠂⠀⠰⡷⠀⠀⠀⠀⢀⣀⡀⠈⠉⠉⠉⠉⠉⠉⣡⣤⣄⠀⠀⠀⠀⢸⣿⠀⣿⠀⠈⢷⠀⠈⠋⢹
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡗⠀⠀⠀⠃⠀⠀⢸⣿⣿⣿⡇⠀⠤⠀⠈⠁⠀⡟⠀⠀⢸⠀⣀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⠀⠀⠈⠀⠀⣠⣤⣄⠀⠀⠁⠀⠀⣀⣄⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⡀⢰⡇⠀⢀⡾⠘⠋⠀⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⣿⡗⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠻⠟⠀⠀⢀⣤⡀⠀⠀⢰⣿⣿⠀⠀⠀⣸⠤⠖⠋⠁⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⠈⠛⠋⠀⠀⠀⡟⠰⠀⠀⣾⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⢠⣿⣿⣦⠀⠐⠂⠀⠈⠛⠉⠀⠀⠀⠀⢠⣶⡄⣸⠷⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠘⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣰⣶⠀⠀⠀⠉⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⠛⠛⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡆⠀⠀⠈⠁⠀⡄⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⣀⣠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠙⠓⠶⠦⠤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠚⠋⠉⣰⠋
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠒⠒⠶⠦⠤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠚⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⢠⡀⠤⣄⣀⣤⠤⠀⠀⢀⠀⠀⠀⢰⡀⠠⢤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣶⣶⣶⣶⠶⣤
⠀⠐⢾⣿⡉⠁⠈⠀⠀⠀⠁⠀⠀⢠⣀⣶⠈⠏⠖⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠈⠈⠒⠁⠀⠀⠀⠀⠀⠛⠒⠃⠚⠁⠀⠀⠀⠀⢠⠄⠠⠠⠀⠀⠀⠀⠀⠤⣀⣀⡠⠞⠉
⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⡿⣿⣶⣶⣄⡸⠁⠐⢠⡄⠀⠠⢤⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⠿⡃⠐⠀⠈⣇⠀⠀⠐⠒⠀⠀⠀⢀⣀⣤⠤⠤
⠀⠀⠀⢀⣀⣀⣀⣉⣀⠤⣤⣀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠂⠀⠀⠀⠋⠁⠀⠀⣐⠶⠚⠋⠋⠉⠉⠉
⠀⠐⠉⠀⠀⠀⠉⠙⠛⠉⠀⠀⠁⠀⠀⢠⣤⣤⣤⣀⠀⢲⡰⢄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⣤⣴⣴⠆
⠀⠀⠀⠀⢲⠟⣶⡆⡀⠖⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⣟⠂⠀⠀⠀⠠⠤⠤
⠀⠀⠀⠀⠼⠀⠹⣯⠁
⠀⠀⠀⣤⡶⠀⠀⣿⡆⡠⠀⠀⠀⢠⣶⣶⡀⠀⠀⠀⠣⢀⡠⠃⠀⠀⠀⣠⡾⢿⣶⣄
⠀⠠⣶⣷⠧⡤⠈⠛⢋⠁⠀⠀⠀⠈⠛⠙⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠹⣱⣞⣞⣿⣇⣰⣥⠤⠤⠄⠂
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠚⠙⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠈⢉⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠃⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⠀⠀⠀⠀⠀⢿⡀⠀⠔⠁⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢎⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⡇⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠤⠀⠒⠒⠒⠒⠂⠐⠒⢎⠀⡞⠓⠒⠒⠂⠠⠤⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⣀⣠⣤⣴⣶⣶⣶⣿⣿⣿⣿⡿⣿⠿⠦⠘⠻⣶⣶⣶⣶⣤⣤⣄⣀⡀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢸⠀⢴⣿⡿⠛⠛⠉⠉⠉⠀⠀⠀⠀⢀⡀⠙⠶⠶⠦⠴⠀⢈⣉⣭⠿⠛⠛⠿⢿⣷⣦⡀⠈⡆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠀⠘⠿⢷⣾⣿⣷⣶⣶⣶⣤⣝⠦⠤⠤⢬⣽⣒⣒⣒⣻⣙⢧⣀⣀⣀⣀⣀⣠⣼⣿⠇⣸⣉⣠⣤⣀⡈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣯⣭⣓⣒⣛⣻⣟⣛⣩⣭⣽⣷⠿⠿⠿⠛⠛⠉⠁⠀⠀⡟⠁⠀⠀⠀⠉⢢⡀⠹⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠴⠊⠀⠐⢄⠀⠃⠀⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⠀⠀⠀⠀⠀⠀⠐⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠇⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⠄⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⡔⠁⡘⢠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⡠⠊⢀⠜⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠒⠙⣄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠳⠄⠤⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠈⠁⠀⠀⠒⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠿⣤⠀⠠⠴⠚⠉⠒⠢⣀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠐⠛⠿⠿⢿⣲⠤⣄⣈⠑⠒⠠⠤⢀⣉⡁⠀⣀⣀⠲⣄⡀⠈⠅⢀⣀⠀⠀⠀⢐⣀⡩⠄⢒⣠⠆⠀⠈⠉⣉⡃⠤⠄⠒⠉⢀⣀⣤⣾⣿⣶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠬⡉⣓⠒⠦⠤⣤⣄⣉⣉⣉⣁⡈⠛⠛⠒⠒⠲⠶⠤⠴⠶⠒⠒⠚⠛⠓⣂⣉⣉⣡⣤⡤⠴⢶⣲⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⠀⠤⠭⠍⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠥⠭⠥⠀⠐⠒⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣕⣂⠲⠶⠶⠿⠶⠷⠶⠶⠶⣶⣖⣒⣲⣶⣿⣷⣾⣿⣿⣮⣤⣬⣭⣀⣀⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣿⣿⣿⡿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⣿⣿⡿⠀⣸⣷⠖⠁⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⠃⢠⣿⡛⠈⠁⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠀⣾⠏⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣽⣿⣿⣧⣲⣏⡀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣽⣿⣿⣿⣶⡿⠋⠁⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣄⣀⡂⢤⣤⣤⣄⣾⣿⣴⣀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣾⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠰⡓⠚⠦⠘⠙⡟⠛⡿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣦⣴⣦⣬⣤⣤⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⣳⣤⣃⠉⠀⠀⠉⠻⠿⠿⢿⣿⣯⣭⣽⣿⡿⠽⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣤⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠶⠤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⣘⣛⣿⠯⠭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠉⠐⠬⠢⡀⠀⠀⠀⠀⠈⠈⠀⠀⠀⡠⠊⡠⠐⢈⡉⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠀⡀⠀⠐⠷⡄⠀⠀⠀⠀⠀⠀⠀⡔⢡⠊⡠⠊⠁⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢌⡄⠀⠀⠀⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠀⠀⢀⠂⠀⠀⠀⢀⡀⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⢐⠒⠒⠒⠂⠄⣀⠀⠀⠀⠉⠀⠘⢀⡠⠐⠂⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠎⡠⢂⠥⠀⠀⠀⠀⠀⠉⢀⡑⢢⡀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠔⠁⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⣀⣀⠀⠤⠤⠤⠄⡀⠀⠀⢀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡠⠊⠀⠀⡠⠀⠀⠉⠀⣀⡀⠀⠀⠀⣀⠸⡀⠀⠈⠁⠀⠀⠀⠀⠀⢐⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠀⢸⡄⣠⣐⣀⠁⠀⠠⠄⠂⠈⠁⠀⠆⢀⠠⡄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡢⠀⠳⢁⠠⢀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢡⠀⢀⡄⠀⠀⠠⡄⠀⠀⠀⠀⢸⠁⠀⢈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⣠⡀⠁⢰⠆⠀⠀⠀⡄⢀⠎⢀⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣢⠀⠀⠀⡀⠀⠀⢃⠀⠀⢄⠠⠌⣿⣗⠂⠈⠁⠀⣤⠊⡰⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⡀⠈⡄⠀⠀⢀⡔⠀⡆⠑⠂⠀⠀⣡⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠖⠀⠀⠶⠀⡅⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠑⢄⠰⠂⠀⠀⠀⠀⢀⠔⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠄⠀⠀⠉⠒⠒⠒⠂⠉⠀⢀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠤⠀⠅⠀⠀⠤⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠐⠀⠀⡒⠀⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⣿⣿⡿⠛⡛⢿⣿⣿⠀⠀⠀⣥⣤⣤⣤⣤⣤⣤⡄⠀⠀⠃⠀⠀⠀⠀⠀⠘⠀⠀⠍⠉⠉⠉⠉⠩
⠀⠀⠀⠀⠀⢻⣿⠀⣾⡿⠀⣿⡟⠀⠀⠀⢹⣿⢁⣴⡦⢹⣿⠃⠀⠀⢸⣿⠟⣩⡻⣿⡇⠀⠀⢠⣤⡤⠤⣤⣤
⠀⠀⠀⠀⠀⢸⣿⣷⣤⣤⣾⣿⡇⠀⠀⠀⢸⣿⣦⣉⣡⣾⣿⠀⠀⠀⢸⣿⣜⠟⣡⣿⡇⠀⠀⢸⣯⠰⠗⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠈⠛⠛⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣰⡤
⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠂⠀⠀⠀⠀⠀⠒⠖⠻⠂⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡖⠒⠀⠀⠐⠒⠒⠶⢶⣷⣿⡴⠟⢻⣶⡄⠀⠀⠀⠀⣿⣤⣤⠀⠀⠀⠀⠠⠀⡤⣤⣤⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣿⡇⠂⠀⠀⠀⠀⠉⠈⠏⣿⡿⠀⠀⠀⢹⢹⠀⠀⠀⠀⣿⣿⢷⣦⣠⡶⠄⣴⣄⣷⣄⣰⣿⡟⠛⠿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⠁⠉⠀⠀⠃⠹⡟⠁⣿⣿⡇⠀⠐⡙
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⣿⡏⠀⠀⠀⠀⣿⣿⡍⠀⠀⠀⠀⠀⠀⡅⠀⢿⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⢀⣼⠟⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⡃⠀⣼⣿⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢾⣿⣧⣮⠏⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⢠⡇⢀⣭⣿⣇⣠⣾⠇
⠀⠀⠀⠀⠀⠀⢻⣧⢠⡆⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⡀⠀⠀⠀⣀⣾⡇⢀⣿⣿⣿⠝⠋
⠀⠀⠀⠀⠀⠀⠘⣿⣷⡷⠀⠀⠀⠀⠀⢸⣽⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⠀⠛⠁⠛⠛⠹⣷⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⡓⠐⠒⣀⣉⣙⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣍⣉⡙⠛⠙⢙⣟⣟⣿⣽⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣀⣀⣭⣿⠆⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣤⡀⠀⣏⠀⢹⠁⠀⣿⠀⠀⡇⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⠻⢦⣽⡶⠿⠶⠶⠿⠶⠶⣧⣤⣞⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⢀⣤⣤⣼⠇⢠⣤⣤⡶⠶⢶⡶⠶⣟⠛⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠳⠶⠛⠳⠟⠁⢠⡟⠀⠀⠀⢸⡇⠀⠀⡇⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢀⡼⠃⠀⠀⣧⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡛⠷⠶⣤⡾⣯⠀⠀⠀⠀⢿⡀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣠⡾⣳⣤⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⣤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣶⣶⡶⠶⠶⢶⣶⣶⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠈⠛⠻⢶⣦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⠾⠟⠋⠀⠙⠟⠛⠛⠻⠶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠻⣷⡀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠘⣷⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣸⡟⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣰⡟⠁⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⣾⠏⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣠⣴⠟⠁⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡶⠟⠋⠁⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁


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

⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠄⡀⠀⠀⠀⢰⣰⡀⠀⢠⠀⠀⠀⡠⢤⠀⠀⡔⠀⡀⢀⠄⠀⠴⠀⡄⠀⣀⢀⠀⠀⡠⠀⡐⡀
⠈⡇⠀⠐⠒⠂⠐⠀⠨⠀⢀⠄⢀⣴⠂⠀⠁⠀⢰⠀⠀⠃⠀⢇⡠⠀⠤⡈⠁⡀⠠⢀⠔⠈⠀⠁⠰⠥⠼⠀⡊⢀⠂⢠⠈⢒⠒
⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⢀⠹⠀⠀⠀⠀⠘⠀⠑⠀⠄⠀⠀⠀⠠⢂⢠⣀⡀⠀⡀⠀⢀⠀⠀⡀⡀⠆⠣⠀⠠⠄⠠⠀⡠
⠀⠀⠀⢂⡲⡴⠿⢀⢀⡀⠠⢀⠀⠀⡀⠀⠈⠀⣰⠀⠀⠀⠃⠤⠀⡀⠠⠀⠈⠁⣠⠀⠈⠐⠈⡀⠀⠐⠶⠉⠆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠔⠀⠀⠈⠐⢄⠀⠀⣿⣿⣾⣿⣿⠀⠈⠀⣀⠀⣀⠄⠀⡐⠀⠑⡈⠀⠀⣤⣷⠾⡯⣷⡆⣄⠀⠠⠃⠁⠒⠠⡀
⠠⠀⡄⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠀⠿⢿⠿⠿⠇⠀⢊⠩⡤⠯⡈⠆⠓⠄⠀⠠⢁⠀⢂⢿⣄⣀⣾⠟⠁⠃⠁⠐⠀⠀⠀⠀⢠
⠀⠀⠁⠀⠁⠄⠀⠀⠀⠀⠠⠀⢂⡁⠠⠂⠀⠀⢀⠀⢼⣤⠲⠒⡣⢲⠤⠐⠒⠄⡀⠀⣊⠑⠄⠀⠀⠀⠈⠀⡀⣀⠈⠠⡀⢀⡔
⠠⡶⢤⣈⠷⢡⠠⡖⡲⠄⠀⠠⠂⠀⠀⠂⢀⢁⠀⡀⠀⠈⠉⠓⠋⠈⠀⢀⢀⠂⢀⠑⢌⠀⠀⠀⠀⠀⠄⠊⠠⢃⢀⠁⢠
⠀⠐⡂⢸⠀⠸⠀⠁⠁⠀⠀⠆⠐⠀⠀⠂⠁⡀⠄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠬⡀⠀⠆⢠⠂⡐⠀⠀⠰⠁⠐⠒⠁⠀⠑⢄
⠀⠀⢃⢸⠀⡆⡘⢠⡅⢀⡤⠰⠂⠀⡄⠀⠸⣄⢀⠠⠀⠀⢀⡖⠛⠂⠀⠀⠀⠀⠀⠀⠐⠃⢀⠙⠁⡠⠀⠀⣀⡀⠀⠉⠀⠀⢸
⠀⠀⡜⠻⠒⡟⢣⠀⠀⠢⠒⠚⠀⠴⣀⠃⠀⠀⠀⢀⣀⡀⠀⠻⠶⣤⣄⡀⢰⠁⠀⠀⠀⠀⠐⠎⠀⠀⠀⠀⠙⠣⠇⠀⠀⠀⢸
⠀⠰⣀⣆⠀⢰⣠⠆⢠⠔⠀⠀⠀⢄⠀⠉⠀⣠⡞⡋⢀⡽⢀⣴⣠⡖⠀⠙⠎⡇⠀⠀⠘⠶⠶⢶⣤⣄⠀⢘⠀⠀⠀⠀⠀⠀⢸
⠠⠀⠈⠀⢉⠁⢁⠀⢣⠀⠂⠒⠂⠱⠃⠀⣰⡏⠀⢈⡯⠁⣸⠃⣿⢩⠧⢀⡤⠀⠀⠀⠀⠀⣀⣀⠀⠜⠀⠆⠀⠑⠀⠤⠀⠂⠈
⠀⠐⠚⣀⠣⠔⡀⠁⠈⡀⠀⢀⠀⡀⠀⠀⣿⡄⢀⠸⣧⠔⡿⢸⡇⠻⠤⠸⠦⠀⠀⠀⠀⠐⠅⠠⠁⡥⡨⢠⣈⠍⣭⢅⣠⡄
⠀⠐⠕⢆⠀⡠⡠⠀⠀⠇⠄⠹⠀⠂⠀⠀⠈⠛⠉⠀⠀⠚⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠘⠀⠀⠀⠀⡨⠁⠀⠀⠁
⠀⠀⠀⠳⡀⠏⠀⠀⠀⠢⣀⠀⡀⠄⣤⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠰⠂⠉⠤⠀⠉⠵⠂⠰⠁⠀⠁⢀⠐⠤⠀⠀⢰⡀
⠀⡦⠶⠶⠶⠶⠖⢰⠈⠀⠀⢐⡠⠀⢹⠀⠉⠁⠉⠈⠁⢘⠀⡐⠁⠀⡆⠀⡔⠀⠀⠀⠀⢸⡀⡠⠀⠒⡁⡂⠡⣄⣀⡤⠋
⠀⢃⠀⠀⠀⠀⢀⡆⠀⡀⠀⠒⠠⠁⠁⠂⠀⠀⠀⠀⣱⠀⠑⠳⠀⠊⠀⢀⣤⣤⣤⣤⣤⣤⡄⠡⠀⠈⠤⠃⠀⠀⠀⠀⠠⠦
⠰⡈⠑⠤⠤⠴⢊⡼⢀⠄⠃⢀⠐⠀⠀⠈⠴⠴⠛⠋⠹⠀⠠⠐⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠉⢀⣀⣀⣶⣆⣀⡀⠈⠡⢄
⠀⠀⠈⠀⠀⠉⠀⠀⠈⠀⡉⠁⠀⠀⡄⠠⠀⠀⠀⠀⠀⢠⢅⡀⠀⠀⢀⣁⠉⠉⠉⠙⠋⠉⠀⠀⠀⠘⣀⠀⠉⠀⢀⠁⠸⣀⠌
⠀⢠⢚⡁⠠⠄⠁⢀⠀⠊⡀⠀⣀⠀⠁⠚⠀⠀⠀⠀⠈⠈⠂⠈⠁⠀⠸⠋⠀⠀⠠⠠⠒⠄⠀⠀⠘⠯⢌⡐⠒⠒⠁⡈⡲
⠀⡀⠀⠀⠐⠀⠀⠈⠀⠀⣀⡀⣀⣀⣠⡀⠀⣀⣀⡀⠀⠀⠃⠆⠈⠠⠠⠀⢸⠀⠀⠠⠀⠀⠐⠀⠀⡄⠀⠁⠁
⠀⠐⢄⠉⠀⠐⢀⠘⠀⠀⣿⠀⢿⣿⣿⡟⠸⣿⣿⣿⠱⠀⠃⠳⠀⠐⠆⠢⢘⠂⠀⠀⠈⢀⠖⠄⠘⢧⣀⠀⡀⠠⠀⣀⠲⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠁⠀⠙⠋⠀⠘⠛⠛⠛⠀⠀⠐⠠⠀⠐⠀⠀⠀⠈⠀⠀⠀⠒⠂⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠤⠤⠤⠤⢤⣄⣈⡉⠙⠲⠤⣄⡀
⠀⠀⠀⠀⣀⡤⠖⢒⡒⠒⠦⢤⣀⠀⠀⢠⠎⠈⢇⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣄⡉⠓⠦⣄⡀
⠀⠀⢀⡞⣡⠖⠉⠉⠉⠁⠀⠀⠈⠙⠲⢧⡄⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⡉⠓⢤⣀
⠀⠀⡾⠰⠃⢠⠴⠚⠋⠉⠉⠙⠲⡄⠀⠀⡇⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡈⠓⢦⡀
⠀⢀⡇⠀⢰⠃⠀⠀⠀⠀⠀⠀⢠⠟⠐⠚⠁⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠉⠳⣄
⠀⠀⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠑⢄
⠀⠀⢧⠀⠘⣆⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠱⣄
⠀⠀⠘⣆⠀⠈⠳⣄⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⢧
⠀⠀⠀⠈⢧⡀⠀⠀⠉⠓⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⡆
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⣄⣀⣀⠷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣧⡤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠳⠤⠤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⢀⢺⣤⢏⡴⠲⡄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⣈⠉⠁⠈⠳⠚⠱⣅⡼⠁⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠤⠤⠤⠤⠤⠤⠶⠖⠚⠛⠉⠁⠀⠀⢀⣠⠤⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡏⠈⠳⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣠⠤⡤⠤⣄⡀⠈⠲⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⡇⠀⠀⠷⠊⠁⢀⠇⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠉⠳⣄⠘⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠹⡄⠀⠀⣀⠄⠀⢴⠟⠀⠴⢦⢠⡋⣳⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠘⡆⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠙⠛⠉⠀⠀⠀⢯⡤⠀⠳⠞⠀⠉⠁⠀⠀⠀⠀⠀⣀⣤⡎⠀⠀⠀⠀⠀⠀⣀⣰⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠉⠉⠛⠛⠛⠋⠉⡡⢋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠦⢤⣀⣀⣀⣀⣠⡤⠶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣖⣺⣿⠿⠿⠿⠿⠻⠖⡶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⢶⣿⣿⣆⣀⣄⣀⣀⠀⠀⠀⠀⠉⠺⢝⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣼⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⢿⣿⣌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⢻⡇⠏⢠⡴⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⡐⠀⠈⢻⠈⡇
⠀⠀⠀⠀⠀⠀⢠⡶⡫⢢⠀⢸⣿⣌⠂⠀⠄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⣀⢀⡜⣸⡟⠷⠶⠤⣄
⠀⠀⠀⠀⠀⣰⢫⡞⠉⠳⣦⡀⣏⠈⠳⢮⣂⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣨⠾⣿⢁⡤⢤⣀⠌⣧
⠀⠀⠀⠀⢰⡇⣿⣳⡀⠀⠠⠙⣿⡀⠀⠀⠈⠏⠓⠲⠶⠭⠤⠤⠤⠭⢶⠖⠚⠉⠁⢼⣯⡏⢀⣠⠼⢣⣿
⠀⠀⠀⠀⢸⠁⠟⣧⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⠈⠉⢉⣠⠴⠛⢻⡆
⠀⠀⠀⠀⠸⣇⢀⠙⢷⣤⣤⣤⠂⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⠖⠋⠉⠀⠂⢶⠂⡇
⠀⠀⠀⠀⠀⠻⣎⢣⡀⠛⢿⣿⡿⠛⢾⢦⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⣤⠟⠁⠀⠀⠀⠀⠀⠀⠩⢸⠇
⠀⠀⠀⠀⠀⠀⠙⢧⡙⢆⠀⠉⠀⠀⠈⣷⡙⣷⢧⣄⣀⣀⣀⣀⣠⡤⠞⢋⠵⠂⠄⠀⠀⠀⠀⠀⡠⢣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣅⠢⢀⠀⠀⠈⠻⣧⡀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣬⣙⠒⠾⢿⣿⣦⡀⠈⠳⢄⡀⠀⠀⠴⣒⠦⠤⢔⣂⡵⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⠿⢿⣷⢂⣴⠿⢦⡬⠤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠈⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⢤⣸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢿⣷⣦⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⢹⣷⡀⢰⣿⡄⠀⢸⣿⡆⢀⣀⣤⣤⣤⡌⢻⣿⣿⣿⡿⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⣸⣿⡇⣼⣿⡁⠀⢸⣿⡇⠈⠉⣿⣿⠀⠀⠀⠈⠻⠿⠁⢸⣿⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠙⠛⠀⣿⣿⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⢀⣴⡾⠛⣶⡄⣼⣿⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⢹⣿⡄⠀⣿⣿⡇⠀⢠⣿⣿⠀⣾⣿⣀⣠⣿⠇⣿⣿⣀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⣴⡦⠸⣿⣇⣼⠿⣿⡇⠀⢸⣿⣿⠀⣿⣿⠉⠉⠁⠀⣿⣿⠉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⡶⠟⠁⠀⠈⢉⣅⠀⠈⢠⣄⢸⣿⡏⠀⠘⢿⣆⣀⡾⠃⢹⣿⠀⠀⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠠⠶⣶⣶⠖⠂⣾⣿⠀⠀⣿⣿⠀⠉⣤⣶⣤⡀⠀⣰⣦⣠⠴⣦⣄⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⢠⣿⣿⠀⠀⣾⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠸⡿⠀⢹⣷⠀⣿⣿⠃⠀⣿⣿⠀⣼⣿⡆⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠚⠛⣿⣿⢀⣴⡶⢦⣿⣿⠀⣿⣿⠀⢠⣿⣿⠀⣿⣿⣿⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠀⠀⣿⣿⠀⠀⠻⠃⠀⠀⠻⠟⠸⣿⡇⢀⣿⣿⠀⢿⣿⠀⢸⣿⡟⠀⢹⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⣴⠶⣬⡁⢀⣾⣷⠀⠀⣾⣿⡄⠀⠈⠉⠉⠿⠇⢀⣶⡄⠀⣠⡶⠶⣦⡈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⣿⣷⢸⣿⣟⠀⠀⣿⣿⡇⢰⣿⠋⠛⣷⡄⠈⣛⠁⢠⣿⡇⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠿⠟⢸⣿⣿⠀⢠⣿⣿⡇⣸⣿⠀⠀⢹⣿⢰⣿⣇⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢿⣿⣤⡾⠻⣿⠇⣾⣿⣀⣤⣿⠏⢸⣿⣷⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⢀⣾⡆⣴⣦⡀⠀⢀⣤⡄⢻⣿⠉⠉⠀⠀⢸⣿⡟⠸⣿⡇⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠛⠋⠀⠸⣿⣿⣶⣿⣿⠃⠸⠿⠀⠀⠀⠀⠈⠛⠃⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠃
⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⢠⣜⣍⣬⣭⣿⣆
⣿⣿⣿⣿⠟⡉⢿⡞⢛⡛⠞⠛⣋⣷⡶⢶⡞⣙⢛⠛⢻⠋⣳⡶⢶⣾⢋⠛⠛⠛⠞⣛⣷⡶⢶⣶⠛⢋⠛⠞⣛⣶⠶⣶⡞⡙⠚⣙⠲⢋⣳⡶⢶⡞⢉⡟⡙⠒⢛
⣿⣿⣇⣬⣤⣾⣿⣟⣨⣧⣰⣆⣁⣽⣷⣾⣇⣉⣀⣠⣙⣠⣿⣷⣼⣿⣰⣾⣠⣄⣄⣁⣿⣷⣾⣟⣠⣠⣶⣌⣁⣿⣶⣿⣅⣶⣇⣋⣴⣰⣿⣷⣾⣇⣐⣃⣴⣟⣀

⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠶⠖⠒⠒⠒⠛⠛⠛⠛⠛⠛⢒⣒⣒⣶⣶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣎⠡⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡢⣀⣀⣉⠉⠀⣚⣹⣧
⠀⠀⠀⠀⠀⢸⠇⡟⠛⠛⠿⠯⢽⣿⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣭⡭⠿⠖⠛⠋⠉⠀⣿⡆⢀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢸⠭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢀⣀⣀⠀⠠⠄⠐⠂⣿⡷⠋⣩⠴⠖⠲⢦⣍⠻⣶⡀
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⡺⠭⠭⠆⠀⣿⠱⣿⠃⠀⠀⠀⠀⠙⡆⠘⣿
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡫⠭⠄⢀⣿⣤⠟⠀⠀⠀⠀⠀⠀⡿⠀⡿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣒⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⠃⣼⠃
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣀⠒⢀⣿⠃⠀⠀⠀⣀⣠⠴⠚⢁⣼⠋
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠥⠴⠀⣼⣏⣤⠶⣚⣋⣩⣴⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡍⠁⣰⣿⡯⣲⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣒⠀⣰⣿⣨⣼⣋
⠀⠀⢀⣀⡤⠶⠒⠛⠛⠹⣇⠀⠀⠀⠀⢲⣶⠀⠀⠀⠄⠀⠀⠠⠤⢔⡒⢢⣿⠉⠉⢉⣽⣿⣿⣷⣶⣤⣄⡀
⡰⠚⠉⠀⠀⠀⣀⣤⡄⠀⢿⡆⠀⠀⠀⠈⠭⠭⠭⠍⠀⠨⣭⣥⣴⡷⠶⣺⡿⣶⣶⠾⠿⢯⣍⠀⠈⠁⠈⢻⡆⠀⠐⠀⠀⡀
⠡⣄⠀⠀⠀⠀⠁⠁⠁⠀⠈⠳⢦⣤⣌⣉⣉⣉⣉⣉⣉⣉⣭⣥⣴⣶⠿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠈⠙⠳⢦⣤⣤⣀⣀⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⣠⣤⣤⣤⣠⣤⣤⣴⣶⠿⠛⠁⠀⣀⣈⣉⣉⣉⡉⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠾⠭⣭⣍⣛⣛⣛⡛⠛⠛⠒⠒⠒⠒⠒⠛⠛⣛⣛⣉⣩⡭⠶⠟⠋⢁⣀⣀⣐⣒⣒⣒⣒⣒⣒⣒⣒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⠁⢀⣀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠂
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠶⠶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⢟⠿⠛⢁⡀⠀⣀⠐⠒⠒⠬⢍⡛⠽⡲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⢶⣟⠗⠡⣂⠅⠂⠀⠈⣺⢁⠀⠀⠀⠀⠪⡑⢮⡻⣷⣙⣢⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⡡⠊⢠⣿⢋⢀⠞⠀⣠⠆⠁⠀⠀⠃⠀⠀⠀⢀⠀⠈⠢⠙⡌⣿⣄⣠⡤⠘⡆
⠀⠀⠀⠀⠀⠀⣰⢋⠞⠀⢀⣏⠇⠎⠊⠀⠀⠐⠂⠀⠀⠀⣀⠀⠘⠀⠀⠐⢀⠀⠀⠀⠸⣹⡆⢀⣴⢣
⠀⠀⠀⠀⠀⢰⠃⠃⠀⠀⢸⢸⠀⠀⠀⠄⠀⠀⠀⢦⡀⢀⠁⠐⠁⠋⡈⠰⠸⠀⠀⠀⠀⡏⣧⡊⠉⠉⠁⠀⠀⠠⣄
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢾⡌⡇⢇⢂⢘⡔⢄⠀⠀⠀⠁⠻⠇⣠⠆⡠⡡⠃⠀⠀⢠⢠⢳⣿⡿⠿⡒⢦⣤⣀⡀⡨
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣧⠹⡈⠀⠀⠙⢷⣵⠶⣤⣄⡐⠐⣡⣯⠎⠂⠀⠀⣠⢃⠎⣾⠿⠇⠀⠀⠀⢵
⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠙⣷⡙⣄⠀⠤⡀⠈⠉⠒⠬⠻⢴⠟⠁⠀⠀⣠⠞⡡⢋⣾⠏⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⣿⣮⣓⢦⣈⠑⠢⠤⠄⠈⠘⠀⠀⢀⣠⠴⣋⡴⣿⠏⠀⠀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⠸⢿⣯⡓⠮⢭⣓⣒⣒⣒⣒⣒⣩⡭⠖⢛⣥⠞⢁⢄⠀⠀⢠⠂⠀⠀⣾
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢂⢠⠙⢝⠷⣦⣤⣀⣉⣉⣁⣬⣤⡴⠞⠋⠀⠀⡠⠂⠀⢠⢂⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠑⡑⢄⠑⠤⡉⠉⢋⠉⠉⠀⠀⠀⠀⠀⠀⡨⢀⢀⠔⡡⠃⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠠⠀⠀⠂⠀⠄⠤⠄⠀⠀⣀⠥⢊⠔⡁⠈⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠐⣂⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢾⡢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⢔⣪⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠯⣿⣶⡦⢤⣤⣄⣀⣀⣀⣤⠤⠤⣖⣺⡭⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠒⠛⠉⠉


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

⠀⠀⠘⢿⣶⡀⠀⠀⠘⢿⣶⡀⠀⠀⠘⢿⣦⡀⠀⠀⠘⢿⣶⡀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⢨⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢨⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣼⡿⠁⠀⠀⠀⣼⡿⠁⠀⠀⠀⣼⡿⠁⠀⠀⠀⣾⡿⠁⠀⠀⠀⣾⡿⠁
⠀⠀⢸⣿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⣸⣿⠃
⠀⠀⢿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠘⢿⣧⡀⠀⠀⠘⢿⣧⡀⠀⠀⠘⢿⣦⡀⠀⠀⠘⢿⣦⡀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⣀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⣿⣿⣿⡿⢋⣭⣭⣙⣿⣿⠟⠛⢿⣿⡟⣫⣍⣻⠟⣩⣙⣿⠟⠻⢿⣿⠿⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⣿⣿⣿⠁⣿⣿⣿⣿⣿⠃⣿⣿⡆⠻⠃⠿⢿⡿⠸⠿⣿⠁⠚⣛⣸⠇⠚⢛⣠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡄⢿⣿⣿⡿⢻⣧⣙⣛⣡⣿⠇⣿⣾⣿⢸⣷⣿⣌⡛⣋⣿⣆⡛⢋⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣶⣤⣤⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⣀⣤⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣀⣀⣀⣈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠆⠶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡫⡄
⠀⠀⠀⢀⣀⠀⠀⠀⡄⣦⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⠶⠶⣿⠇⠂⡂
⠀⡠⠀⢠⠁⠟⣆⣴⣿⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣅⣴⡟⣼⠂⢡⡄⡟⢻⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⡤⠶⠚⢳
⠀⠘⠿⢿⣏⢀⡜⣳⡆⢸⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⣠⣤⡾⠃
⠀⠀⠀⠀⠈⢦⣤⣭⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠀⠀⠀⠀⠀⠀⠂⠀⠸⣿⠂⠶⣦⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⣀⣀⣀⣀⣾⡃⢀⣿⣴⣾⣻⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣋⣥⣾⣿⣿⠯⠴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢤⠤⣤⣾⣷⣶⣶⣾⣿⣿⣿⠟⠏⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣿⣿⣿⣿⡿⢿⡟⠉⠀⣀⣴⣾⣦⡞
⠀⠀⠀⠀⢀⣠⡶⠚⠋⠀⠀⠸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣯⣋⠀⡼⢳⣿⣿⡿⠟⠛⣡⣿
⠀⠀⢀⡼⠿⡍⠀⣤⠆⢀⣀⣀⠸⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣾⠃⠈⣱⣿⣿⠁⠴⢶⣤⡾⣯⠰⢽⡆
⠀⠀⣾⡇⢀⠐⣦⠃⢰⣿⣿⣿⣿⣿⣶⣤⡀⣀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡗⢋⡾⣯⣥⣾⣶⣳⣜⣿⣷⣿⡿⢷⢿
⠀⠐⢿⣿⠀⢀⣿⠀⠘⣿⣿⣿⣿⣿⡎⠛⠁⠈⢳⣶⣤⣿⣿⣿⣿⣿⡿⢷⣶⣿⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣸
⠀⠀⠀⠻⠳⠆⠠⢁⡀⠘⢮⣿⣭⣝⣛⠀⠀⠀⠀⣷⣿⣾⣿⣿⣷⡶⠾⠿⠿⠟⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠐⢌⡻⣶⡀⠀⠀⠀⠙⠿⣷⣤⣤⣤⣴⣶⡏⠻⡷⢦⣤⣀⡀⠀⠀⠀⠀⢀⣀⠠⠴⠊⠀⠀⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠙⠲⣝⣓⠚⠀⠀⠀⠈⠉⠛⠛⠛⠛⠓⠶⠤⢄⠨⣍⠛⠿⣶⣶⣤⣄⡀⠀⠀⠀⠀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣉⠓⠛⠈⠀⠰⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠓⠚⠓⠒⠒⠛⠛⠛⠛⠛⠓⠒⠈⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣤⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣽⣿⡿⢿⡏⣯⢹⠿⢿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣽⣿⣾⣷⣧⣿⣽⣬⣿⣼⣿⣿⣿⣿⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣛⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣽⣽⣿⣿⣿⣿⡟⢿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⣿⣿⡟⠻⡟⠻⣿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣤⣽⡿⠿⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠁⠟⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⣽⣿⣿⣷⡀⠀⢀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣥⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⡇⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢈⣿⡿⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠉⠁⠈⠛⠳⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢠⣴⠀⠀⠶⠿⠃⠀⠀⠀⠀⠉⠉⠉⠰⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠤⠀⠀⠀⠀⢿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠛⠋⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⡀⠀⠸⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⠈⠀⣮⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠰⣿⣿⣤⢰⡄⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠋⠰⠤⡈⠉⠉⠀⠀⣙⣿⣿⣿⣿⣿⣿
⣿⣿⣦⠀⠀⠀⠀⠿⣿⣿⣿⠁⠀⠙⠛⠉⠁⠁⠸⠛⠏⠁⠈⠟⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢮⣍⠃⢉⣿⣿⣿⣿
⣿⣿⣿⣿⡆⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠈⣿⣿⣿
⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠑⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⡀⠸⠁⠀⣸⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣾⣆⠀⠀⢄⣠⠀⠀⠀⠀⣀⣀⣠⣴⠟⢀⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣠⣴⣿⣿⣿⣿⣿⣶⣤⣦⠀⠀⠀⠶⣌⠉⠉⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣇⢠⣶⣿⣠⣾⡷⣤⣀⣀⠀⠀⠀⡀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠒⢶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣶⡿⠋⠀⠀⠠⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⢀⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⠄⠈⠛⢻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠉⠛⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠙⣿⣼⡟⠋⠛⠻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣥⣼⣦⢸⣿⣿⣿⣿⣿⣿⠿⠋⣉⡉⢁⠈⠙⢷⣶⡾⠛⠉⠀⠀⠀⢀⣄⡉⠻⠿⠋⠀⠀⠀⠀⠀⣠⡿⣤⡻
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠘⠃⠀⠙⠛⠿⠿⠋⠀⠀⠈⠉⠉⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠿⠨⠟⠀⠀⠀⠀⠀⠀⠀⠈⠳⡾⠟⠘
⣿⣿⣿⣿⣿⣿⣿⣵⡄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠈⠀⠀⠖⣀⠀⠀⠀⠀⠀⠈⠁⠤⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠰⠾⢿⡷⠀⠀⠀⠀⠀⠐⠷⣾
⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠤⠤⣤⠀⠀⠀⠈
⣿⣿⣿⣿⣀⠀⠀⠀⣴⠻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣧⣄⠀⠈⠉⠉⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⢸⡏⠹⠗⠐⠀⠀⣀⣤⣶⠞⠃⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠠⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⡀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⣤⣶⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣤⡀⠁⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣶⣿⣿⣿⣿⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀
⣿⣿⣿⣿⣿⣿⡿⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠤⠤⠄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠂⠀⣁⡀⠀⠤⠀⠤⠤⠤⠤⠤⠄⠠⠤⠀⣀⡉⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠤⠤⠤⠤⠄⠀⠀⠀⠀⠠⠤⠤⠤⠄⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠜⡆⠀⢁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⣥⠀⢸⡩⠹⡄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠘⡕⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡠⢊⠤⠊⠄⠘⢆⠈⠢⣰⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣀⠤⠄⠀⢹⡲⢄⠑⠄
⠀⠀⠀⠀⠀⠀⠘⠠⠂⠀⠀⠀⠀⠀⠑⣤⣀⠁⠚⠒⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠖⠉⠀⠀⠀⠀⡘⠋⠢⠣⠈⡄
⠀⠀⠀⠀⠀⠀⢀⠹⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⠀⠀⠲⣄⡸⠄⣀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⣉⢀⣁⠀⠀⠀⢀⣠⠞⠁⠀⠀⠰⢁⠃
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢀⡀⠤⠚⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⣈⣉⣉⢉⣁⣀⠤⠔⣂⡩⠕⠊⠉⠉⠛⠋⠉⠀⠀⠀⠠⠔⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠉⢁⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⡠⠐⢈⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⠐⠀⠈⠁⡐⠂⠠⠤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠶⠶⠾⠛⠛⠉⠉⠀⠐⡀⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠂⠀⠀⠀⠐⠒⠒⠒⠀⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠁⠀⠀⣀⡀⣠⣤⢠⣤⠤⠤⠤⣤⢤⣤⣤⣌⣀⠀⠀⠀⠀⢰⡦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠂⠉⠀⠄⠁⢠⡤⠀⠀⠀⠀⠀⡶⠿⠿⠟⠿⣿⣿⣿⣶⣧⣄⡈⠁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠦⠀⠀⠀⠀⢠⠀⠰⣖⣀⣠⣤⣤⡶⠂⠀⢀⠀⠉⠉⠙⠃⣭⠉⠻⣿⣿⡇⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢦⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢁⠀⠁⢙⠂⠔⠚⢛⣻⣦⣀⠀⠙⢀⣀⣿⣿⠇⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡙⠒⠤⣀⣀⠀⠀⠀⠛⠁⠈⣉⠑⢀⡀⠀⠀⠶⠿⠿⡶⢶⡯⠟⣁⡴⢳⣾⠃⢀⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⢶⣦⣮⣍⠁⠀⠀⠀⠀⠠⠶⠿⠷⠤⠰⠿⠒⢛⣍⣡⠀⠀⣿⣿⣰⣿⡟⠉⠉⠻⣖⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠃⠀⠀⠀⢀⣰⢶⠗⠀⠈⠉⠛⡀⠘⠃⣿⢿⣿⠁⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣼⣏⣀⣠⣴⠿⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣰⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣿⣅⣉⣉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠷⠊⠙⠉⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣿⣟⠿⠿⢷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⢠⠠⠃⠀⠀⠀⠀⠀⠴⣻⣷⣵⣆⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⣰⠽⣿⡿⣋⡬⣿⠀⠈⠀⡈⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢴⡁⢹⣧⣠⣹⣿⣆⢀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠁⠀⢀⣩⠾⣿⡾⣥⣾⣷⠤⡀⠀⠐⠀⠘⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠉⣹⣭⣿⡋⠥⣺⠏⣻⣷⣷⣦⣄⣀⡀⠀⣀⠀⡂⢀⣉⣀⣴⣾⣿⣱⢊⣽⣓⣺⣿⣾⡷⠏⠀⠁⠀⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢛⣡⠀⢀⠉⠉⠛⠻⠿⠿⢿⣿⣿⠿⣿⡿⠿⠿⣿⣻⢍⣫⡙⣌⠛⠉⠀⠀⠀⠀⢀⠠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣥⠬⠋⠀⣴⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⠋⠛⠉⠀⠀⢀⣥⢴⢊⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⣀⣀⣀⣀⠠⣄⣀⠀⡀⠤⣶⢎⢃⠈⠿⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⠂⠀⠂⠁⠁⠀⠀⠀⠘⠓⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠾⠿⠷⠶⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢋⡕⣢⣽⣤⣾⣷⣶⣵⣶⣬⣍⡛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣡⣾⣿⣿⠟⠛⣋⣉⣉⣍⣉⡛⠛⣿⣿⣿⣶⣍⣻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⣿⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣯⢟⣿⣿⣿⣿⣿⣿⢫⣥⣿⣿⣤⣬⣿⣿⣿⣿⠿⠟⠛⠉⠉⠊⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣵⣪⡿⣧⣹⢿⣿⣿⣯⣸⣿⣿⣿⣿⣿⣿⡿⠿⠏⢀⣠⣴⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⣽⣞⣿⡿⢷⣯⣝⣛⣻⣛⣿⣻⣫⣭⣥⣴⣶⣿⣿⣿⣿⣿⣿⡿⠟⠉⣼⡄⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣯⣰⣿⢼⣹⡇⠀⠀⠉⠙⢛⠛⠛⠉⠉⠁⠀⠉⠉⠛⠛⢛⡉⠉⠀⠀⠀⠀⠸⣴⠀⠀⠀⠐⠊⠁
⠉⠙⠒⠦⠤⣄⣀⡀⠀⢸⢨⣿⠻⡿⣮⢿⠁⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠀⠀⠀⠀⠿⠤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠋⠉⠉⠓⠋⠀⠀⠋⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣦⡀⠀⠀⠀⠀⢀⡀⠐⠀⠀⠀⠀⠉⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠀⠀⠀⠀⣠⠞⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⠁
⠀⠀⠀⢀⠤⠚⠉⠀⠀⠀⠀⠁⠐⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡆⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢠⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⢻⣿⡿⡄⠀⠀⠀⠀⠀⢠⢾⣫⡽⠯⣝⡳⣄⠀⠀⠀⠀⠀⣠⡾⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠈⢿⣷⣽⣦⣄⣀⣠⡴⣯⡟⠁⣄⣠⡄⠹⣿⡳⠶⠶⢶⣻⠿⣁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠈⢿⣿⡵⣾⣭⣿⡶⣿⠀⠀⠀⠀⠀⠀⢹⡿⣿⢉⣿⣇⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⣿⣿⣧⢻⡄⠀⠐⠟⠀⠀⣼⢽⣧⣻⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣝⣮⠿⣽⡦⣤⣠⠤⣺⣙⣾⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⠟⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠀⠀⠀⠀⠀⢀⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠘⠀⠀⠀⠀⠀⠈⠃⠈⠗⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠲⢤⡀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠄⠋⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⠾⠾⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠐⠒⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⠛⠒⠒⠶⠶⠶⠶⠶⠤⠶⠶⠶⠶⠶⠖⠒⠒⠒⠛⠛⠉⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢖⣋⣙⠲⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡸⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⡤⣊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡃
⠀⠀⠀⢀⡠⠤⠒⠚⠉⠉⠉⣈⣉⡷⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⢯⣀⣉⠉⠉⠉⠒⠲⠤⣄
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠲⠶⠤⠭⠭⠭⠯⠿⠷⠿⠿⠯⠭⠤⠤⠴⠒⠚⠉⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠈⠙⠒⠦⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠴⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠖⠛⠛⠉⠉⠁⠉⠉⠛⠓⠒⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⣠⠶⠿⠉⠀⠋⣼⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⠳⣤⡀⠀⠀⠀⠀⠀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡖⣠⣤⠙⣿⣟⡛⠀⢋⣽⣿⣿⣧⠄⡀⣭⣿⡄⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣼⣿⣿⣿⣿⣾⣷⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡙⢸⣿⣿⣿⣿⡿⠛⡿⣿⣿⣿⣛⠋⠸⣻⣿⣿⠀⠀⠀⠀⣷⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣟⡊⠉⣿⣷⠶⠀⣀⢤⣾⣿⣿⣿⣤⣲⣾⣿⣿⣯⣤⣴⣾⣿⣿⡇⠀⠀⠶⢿⠶⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣿⣫⣾⣷⣽⣿⣿⣷⣿⣿⣿⣿⡿⠛⠋⠉⠉⠛⠿⣿⣿⡻⠃⢻⣿⡇⠀⠀⠀⢸⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⣿⡿⢻⣿⣿⡿⣿⣿⣿⡟⠁⣤⡤⠒⠋⠀⣤⠀⠈⠙⢿⣶⣼⣿⠀⠀⠀⠀⢸⡆⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢺⣷⡆⡨⣽⣏⠀⠸⠿⠋⠀⠀⠀⠳⠐⠂⣰⡇⠀⠀⣄⠀⠻⣿⣿⡆⠀⠀⠀⢸⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⡾⠷⠀⣰⣿⡏⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠺⠩⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠀⢂⣼⣿⠟⠁⠀⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⢠⣾⣷⣦⣿⠟⠀⢠⡄⠀⠀⠀⠀⢀⣄⡰⣄⣴⠋⠀⢀⣴⣿⣿⠿⠁⠀⠀⠀⠀⣴⠋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠋⠀⠀⠈⣿⡀⠀⠀⠀⢼⣿⣷⠋⠀⣀⣴⣿⣿⠿⠁⠀⠀⠀⠀⣠⠞⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠋⠉⣸⠃⠀⠀⠀⠀⠘⢿⣦⡀⠀⢀⣙⣥⣴⣾⣿⡿⠟⠁⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠀⠀⢀⡘⠃⠠⣄⠀⡀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⣀⡠⠴⠚⠁⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣀⡆⣼⡟⠁⠀⢀⡟⢶⣁⠆⢠⠂⠈⣿⣿⣿⣿⣿⣿⠒⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⡆⣷⣿⣿⣿⠀⠀⠀⠘⢀⡇⡼⠳⣏⠀⠀⠘⣿⣧⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢀⢸⣷⣿⡿⠋⠘⣇⣀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠛⣋⡀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⣶⢸⣾⣿⣿⠁⠀⠀⠹⠉⠓⠦⡄⣀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢌⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢀⣾⠂⠀⠀⠀⠀⠀⣿⣸⣿⣿⠇⡄⠀⣆⡀⣦⠀⠀⠀⠉⠙⠲⠶⠦⠤⠤⠤⠿⣿⣷⡘⣆⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡞⢙⠟⢠⠋⠛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠴⠒⠉⠀⢠⠏⣠⠋⡳⠃⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⣤⠀⠀⠀⣰⠃⡴⢁⡞⠀⡴⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣾⠆⠀⠀⠀⠁⠀⠀⡴⢃⠞⣠⠋⢀⠞⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡖⠀⠀⠀⠀⠀⣞⡵⠀⢾⢡⠊⠰⡅⣠⠏⠀⠓⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢱⠀⠀⢀⡄⠀⠀⠀⡤⡀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⡆⠀⠀⠀⠀⡴⠚⠁⠹⡄⢀⠑⠃⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠙⣄⠀⠀⢀⠧⠄⠀⠀⢺⠈⠀⢀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣧⣄⣈⣑⠒⠉⠀⠀⠀⠀⠀⣑⣴⣻⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⠁⣻⠀⠀⠤⠭⠉⠉⠉⢩⠝⠍⠀⢰⡟⢀⡾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢹⡇⠀⠀⢀⠀⠤⠂⠀⠀⠀⠀⢼⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠈⡳⢤⠤⠖⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠉⠙⠛⠋⠒⠒⠛⠋⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣗⠄⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⣄⣀⠀⠀⢀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠙⠛⠿⣿⣇⠀⢸⣾⣶⡶⠶⠶⠶⠖⡟⠛⠁⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠊⠀⠀⠐⢀⠈⠁
⠀⠀⠀⠈⠟⠒⠚⠏⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣬⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄
⠤⠄⠀⠀⢸⠀⠈⡂⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⢾⣯⣿⠏⠈⠀⠀⠀⠀⢸⠤⠄⣀⣀⣠⣬⠤⠶⠖⠚⠛
⢀⠀⢻⣷⣿⠀⠀⣷⣶⣶⡆⠀⠸⡇⠀⠆⠀⠸⣼⣿⡇⡀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠁
⠀⠀⠀⡇⣿⠀⠀⣧⡆⡇⠀⣴⡲⣿⣶⠄⠀⠄⠛⠋⠁⠃⠀⠀⠀⠀⠀⠀⢸
⠀⠄⠀⡇⣿⠀⠀⣿⡇⡇⠀⠸⣷⠞⢷⡇⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⢸
⡉⠀⠀⠀⣿⠀⠀⣷⡃⡇⡀⠀⠓⡀⠈⣀⠀⠀⠀⣀⡀⠤⠼⠀⢘⠀⠀⠠⠚⠛⠛⠙⠛⠛⠉⠉⠉⠉⠉
⡛⣀⠀⠀⣿⠀⠀⣿⡀⡏⣁⣀⣧⣧⣾⣿⣇⠀⣼⣿⣇⣀⣠⣴⣿⣶⣒⣶⣒⣒⣒⣐⣶⠒⠒⠲⠖⢶⣶
⣿⡇⠀⠀⢸⡄⠀⣧⠄⡇⢀⣤⢾⣯⣿⣧⣭⣸⣿⣿⣿⢿⣿⣿⣻⡇⣿⣿⣽⣿⣿⣿⢻⣿⣿⣿⣿⣿⢺
⣿⡇⠀⠀⢸⡇⠀⠛⠀⠃⠻⠿⠿⠿⠟⣛⣉⢹⣿⣯⣿⣼⣿⣿⣀⡀⣉⠁⡉⣙⣿⣿⢘⠙⠛⢻⣿⢿⠺
⣿⡇⠀⣀⣸⡇⠀⠀⣭⣿⣟⢻⣿⣿⣿⣿⣿⢸⣿⣿⣷⣿⣿⣿⣾⡇⣿⡇⣿⣿⣿⣿⢰⣶⣶⣶⣤⣽
⣿⡟⠉⠿⢿⠗⠰⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠸⣿⣿⣿⣿⣿⡿⣿⣧⣿⠇⣿⣿⣿⣿⢸⡿⣿⣿⣿⡟
⠯⡇⢥⣿⣿⠉⠈⠀⣉⣩⣤⣤⣤⣶⣶⣶⣶⠆⣿⣶⣿⣿⣿⡟⣶⣍⣝⢸⣯⣿⣿⣿⢸⣿⣿⣿⣿⡗
⣿⣿⣛⣯⣿⣶⢶⠀⡯⣶⣿⣿⣻⣿⣻⣿⣯⡇⣿⣻⢿⣿⣿⣧⣿⣿⣿⢸⣿⣿⣿⣿⣶⣭⣭⣙⣛⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣉⣛⣛⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣂⠀⠁⠀⠀⠐⠂⠒⠀⠀⠉⠀⠀⠀⠀⡠⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣤⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⢀⡀⠀⠀⠁⠉⠙⠒⢦⣀⠊⣀⠀⠀⠀⠀⠀⢠⡄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⡏⠀⠀⣀⣤⡆⢖⣶⢶⣶⣶⣶⣤⠐⠉⠁⠖⠠⠙⣋⠀⠀⠆⠀⣠⡀⠉⢳⣶⡄⢠⣀⠔⠁⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣧⣠⣾⣿⣿⣄⡊⠵⣿⣿⣿⣿⡏⠘⡀⠠⢆⠀⢌⡛⢁⣐⣈⣥⣿⡁⢢⡀⣿⡏⠘⠁⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣌⠛⢿⣿⣿⣿⣷⣂⣠⠟⠫⠟⣧⠀⠀⠶⠀⣛⣠⣴⣯⣿⣿⣾⣿⣷⣾⡿⠛⠀⠀⢀⡤⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠻⣿⣦⣤⣉⡙⠛⠻⠿⠿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⡉⠁⠀⠀⠐⠒⠁⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⡼⡆⠨⡍⠙⠛⠛⠛⠷⠶⣶⣦⣤⣤⣭⣭⣥⡤⠤⠤⠄⠀⠐⠂⠉⠀⠀⠀⠀⢀⠀⠀⠀⣀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⡀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠉⠀⠅⡀⢄
⠀⠀⠀⠀⠀⠡⢞⣿⣿⠿⣛⣻⣿⣿⣷⡄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⢢⡀
⠀⠀⠀⡰⢀⠀⢨⣟⠁⠻⣿⠿⢚⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⣧⠑⡀
⠀⠀⢀⠀⠸⡄⠀⢸⣿⣿⠿⣿⣿⣏⣿⡟⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⣧⠐
⠀⠀⠀⠄⠀⠀⠀⠀⠉⢻⣶⣿⣿⣿⣿⠂⠤⠉⣿⣿⡷⣶⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠶⠟⠉⣂⣀⢰⣿⣿⣽⣈⣫⣯⡭⢍⠫⠍⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⢃⠞
⠀⠀⠀⠀⠀⠀⢄⠀⢄⡀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠛⢻⣿⣿⡁⣠⠜⠀⠀⠀⠀⠀⠀⠀⣤⣴⢦⡀⠀⠀⣴⣾⣷⣾⣛⠃⠀⠀⠠⢀⠟⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠒⠀⠀⠀⠙⠻⠿⠿⠟⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠛⠙⠛⢋⠀⢀⡨⢖⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠵⣚⠥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⠦⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⢛⡋⠩⠁⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⡷⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣪⢔⣠⣴⣿⣿⣶⣼⣿⣿⣿⠏⠀⠀⠀⠀⠠⠀⠆⠀⠀⠀⠷⠂
⠀⠀⠀⠀⠀⠀⣴⣤⣉⣉⠙⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐
⠀⠀⠀⠀⠀⠐⣿⢸⢸⢹⡘⡘⡇⡇⠱⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠊⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠆⠃⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠖⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⢰⡀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠘⠈⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣾⣿⣿⡷⠄


⠀⠀⠀⠀⠀⠀⢀⢄⢠⣖⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⢷⣿⢠⣿⡄⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡄⢀⠀⠀⠀⠀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⠈⠻⠃⠿⡧⢘⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢵⣄⠐⠛⠛⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢤⡀⠀⠀⠈⠙⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠈⠀⢀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣳⠾⠦⢤⣴⣤⣀⣠⣤⣤⣤⣤⣀⣤⣤⣤⣶⡶⠖⠋

⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠳⢤⣀⣙⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⠉⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢤⡴⢳⡡⡗
⠀⠀⠀⠀⠀⠀⠀⠙⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢤⡞⠀⠙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣭⣉⣩⣭⣭⣭⣭⣭⣭⣭⣟⣴⣾⣍⣐⣒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢑⣀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠤⡐⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢄⡀⡇⢀⡤⠀⠀⠀⠀⢸⠜⠀⡇⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠤⠤⢽⣿⡧⠤⠄⠀⠀⠀⢸⣀⡠⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠴⠁⡇⠙⠆⠀⠀⠀⠀⣮⣄⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢉⣉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⢀⡔⠉⠀⠸⣄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡰⠃⠉⠓⢦⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠧⣀⠀⠀⠀⠉⠉⠉⠉⠓⠒⠒⠒⠛⠉⠉⠉⠀⠀⠀⢁⡼⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣈⠉⠒⠒⠦⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⠤⠤⠴⠒⢉⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠦⠤⠤⠤⠤⠤⠤⠤⠤⠖⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠶⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠶⠶⠶⣤⣤⣄⣀⡀
⠀⣠⡴⠶⠚⢋⣉⣭⡤⠴⠶⠒⠒⠚⠉⠉⠙⠋⠉⠉⠉⠉⠉⠐⠐⠐⠂⠀⠀⠀⠀⠀⠀⠉⠁
⣾⠁⠀⣴⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⣿⠀⠈⢧⣄⡀⠀⠀⢤⣤⡰⣖⣶⣖⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⢤⣄⣀⣠⠖⠀⠀⠀⢸⡏⠉⠉⠉⠙⠛⠦
⣿⠀⠀⠀⠈⠙⠒⠿⠿⣿⣯⣭⣭⣍⣀⣈⣉⣙⣛⣛⣛⣛⣉⣻⣛⣙⣽⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⣸⣧⣤⣤⣄⡀
⢿⠀⠙⠛⠓⠶⢤⣤⣄⣀⣀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣀⣤⡤⠶⠒⠀⠀⠀⡉⠀⠀⠀⠈⠻⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠃⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠂
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⠎
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣐⣀⣴⣞⣉⣀⡀⡠⠋
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⠿⠿⠛⠻⠟⠿⠛⠛⠫⠭⠍⠉⠙⠛⠓⠒
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣱⠚⠉⠁⠀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⠶⢦⣶⣿⣿⣿⣿⣟⠻⢿⣿⣿⣿⢿⣿⣿⣿⣿⣷⠶⠞⠁⠀⣤⡶⠖⠛⠛⠒
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠒⠒⠶⢦⣬⣍⣉⣛⣛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣥⣤⠴⠖⠀⠀⣿⠶⠒⠲⣆
⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣸⠂
⠀⠀⢀⡾⠋⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀
⠀⠀⠸⣧⣄⣀⠀⠈⠙⠛⠻⠶⠶⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⡞⠁⡠⠊
⠀⠀⠀⠀⠉⠛⠿⣶⣶⣦⣤⣀⣀⣀⣀⣀⠉⠉⠉⠉⠉⠀⣀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠋⣁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠻⠿⠿⠿⠿⠟⠓⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠉⠀⠀⠀⠀⠀⠙⢷⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣤⣤⣤⣤⣄⣠⣠⣄⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠼⠟⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣛⣓⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣒⣛⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⣤⣤⣤⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⠀⡀⠀⠀⠀⠀⠘⠁⡄⢠⠀⠁⠆
⠀⠀⠀⠁⠀⠀⠀⠠⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡔⠀⠞⢸⠀⠖⣰⠂⠀⠘⠃⠁⠘⠊⠀⠀⠀⠀⠀⠀⡄⢳⠀⠀⣠⠖⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠤⠠⡄⠀⠀⠀⠀⠀⠀⠓⠀⠁⢠⢇⡞⢈⣁⣈⡁⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢢⠃
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠀⣄⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠄⠤⠤⠤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠛⠀⢀⣀⣀⠀⠀⠰⠁⠾⠀⠓⠲⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠃⠄⠀⠔⠀⠀⠀⠙⠛⠛⠁⠀⠀⢻⠉⠁⠀⠁⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣶⠤
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢈⣉⣉⣋⣋⣉⡀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠑⠂⠀⠤⠴⠂
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠙⠉⠉⠀⠀⠀⠙⠋⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⣄⡀⠀⠀⠀⠀⢰⢶⣤⡀
⠀⠀⠀⠀⢹⠿⠏⠀⠀⠀⠂⠚⠘⠛⠀⠀⠀⠛⠚⠳⠶⠖⠲⠒⠛⠛⠀⠀⠀⠀⠐⠆⠉⠀⠀⠀⠀⠘⣟⣀⣩⠗⠀⠀⠀⠰⠄⠣⠅⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣼⣧⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡷⠀⠀⠀⢴⣦⡜⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠐⠊⣷⠀⠀⠀⢸⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⢁⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⣿⣬⣭⣥⣭⣭⣇⠀⠀⠀⠀⠀⢠⡀⠀⢀⠆⠀⠀⠀⠘⠀⠀⠀⠀⠀⢐⣠⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠑⠂⠒⠒⠈⠀⠀⠀⠀⣦⣤⣤⣤⣤⠏⠁⠀⠀⠀⡀⠀⠀⠀⠐⠛⠁⠀⠀⠡⠏⠈⠀⠀⠈⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠰⣷⣴⣿⣧⣼⠆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀⠠⣤⡶⠤⠤⠤⠴⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠤⠤⠤⠶⠚
⠀⠀⠀⣰⠟⠁⠃⠈⠉⠉⠉⢁⠀⠀⡜⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠚⣿⠀⠀⠀⠀⠀⣠⡴⠤⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆⣿⡆⠀⠀⠀⣼⠋⠧⠀⠤⠄⠀⠀⡃⠀⠀⠀⢠⠀⠤⠀⠈⠁⠉⠀⠠⡀
⠀⠀⠀⠸⡧⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠃⢹⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⢸⡇⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⡄⠌
⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢰⠓⠖⠶⠶⣤⠶⢶⢸⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠄⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⡄⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠑⠂⠧⠶⠷⠜⠚⠚⠁⠀⠀⠀⠀⠘⠉⠩⠿⠿⠛⠒⠚⠃⠀⠀⠀⠀⠀⠉⠛⠓⠒⠒⠒⠛⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠳⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡇⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠂⠉⠀⠈⠐⠆⠤⠴⠋⠞⠀⡀⠠⠊⠀⠠⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠠⠀⢸⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠿⢏⣀⣀⠀⠀⠀⠁⠀⠉⠀⠤⠌⠁⠀⠀⠸⠄⠀⠘⠁⠀⠀⢔⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⣤⣤⣬⣀⣀⣒⣀⠢⠀⠈⠤⠭⠭⠭⠍⠉⠠⠟⢀⣆⣀⣀⣠⣈⠿⣞⡡⠇⠀⠀⣀⡤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣽⠙⠛⠻⠏⠹⠛⠛⠛⠛⠛⡟⠛⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⡟⢻⡟⠀⡤⠖⠉⠀⠀⣀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣒⢀⣈⣽⣿⠿⠿⣟⡃⠈⣊⠀⡟⢁⡤⠞⠋⠉⠙⢳⡄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠄⠠⠤⠶⣖⣲⣿⣯⠭⠂⠈⠵⠂⣷⠋⠀⠀⠀⠀⠀⠀⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣉⠁⢠⣭⠥⠶⢖⣛⡫⠀⢚⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠄⠀⠒⢚⣛⣉⡥⠄⢈⡁⢸⡇⠀⠀⠀⠀⠀⢀⡼⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠒⠀⢐⠛⣛⣋⣭⣭⠴⠖⠀⡭⠀⢸⡇⠀⠀⣀⡤⠖⠋⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡍⠠⠬⠥⠶⢖⣛⡋⠤⠒⠀⢸⣃⠉⠄⠀⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⣒⣛⣋⣭⡥⠤⠀⡩⠀⡟⣀⡤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡥⠨⠭⠵⠶⢖⣚⡋⢠⠆⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢒⣀⣲⣿⣿⡿⢃⠬⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⡤⠤⠤⠶⠶⠒⠛⢷⣄⡂⢀⠀⠀⠀⠠⡄⢰⣶⠀⣴⣶⣶⣾⣿⣏⠙⣠⣴⠿⠶⣶⣶⡶⢦⣤⣤⣤⣀⣀⡀
⠀⠀⢀⡤⠖⠛⠋⠉⠁⠀⣉⠁⠀⠀⠒⠒⠒⠛⠩⠯⡽⠒⠀⠀⠀⠀⠀⣂⣳⣤⡾⠿⠿⠟⠓⠒⠿⣿⠿⡛⠛⠛⠛⠶⠶⠦⣤⣈⠉⢉⠙⠻⣷⢦⣄
⠀⠀⠸⣧⡢⣄⣦⡀⠀⠘⠄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠙⠒⠶⠤⠤⢤⣤⣤⣤⣦⣼⣿⡿⠿⠷⠗⠚⠁⠤⠴⠂⠀⠠⠐⠀⢀⠠⠼⠃⣀⣴⣠⠼⣣⡟
⠀⠀⠀⠀⠙⠳⢯⣍⣛⠓⠒⠶⠶⠶⠤⠤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣤⣤⣤⣤⣤⣿⣿⣿⣿⠶⠶⠶⠶⠒⠛⣋⣩⣭⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⢤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠴⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢠⣾⠋⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⢹⣿⡄⠀⠀⠀⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠹⣿⣦⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣁⣀⣀⣀⣴⣟⣀⣀⣀⣀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠚⠛⣋⣉⣉⣉⣉⣩⣤⣤⣤⣤⣤⣤⣤⣬⣉⣉⣉⣉⣉⡛⠛⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠻⣿⣷⣦⣀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠾⠿⠟⠛⠋⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠚⠓⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡤⣄⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠙⡆⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠏⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣀⡴⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠚⠁⣀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡥⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣬⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣀⣀⣀⡀
⠀⠀⠀⠀⢀⡤⠴⠚⠛⠉⣉⣡⣤⣤⣶⣶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣶⣤⣤⣌⣉⠉⠛⠓⠦⢤⡀
⠀⠀⠀⠀⠘⢦⣀⡀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⢀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠤⢤⣭⣉⣉⣛⣛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⣛⣛⣉⣉⣥⡤⠴⠶⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢭⣍⣉⣉⡉⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⢉⣉⣉⣩⡭⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠒⠒⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠠⠠⡄⢀⠀⠀⢒⠒⠒⠯⣷⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣬⣜⠋⠀⣤⣉⡻⣿⢦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡙⠋⢘⣿⣿⠄⡀
⠀⠀⠀⠀⢸⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣭⣿⡿⢠⣇⡴⠞⠉⠙⣦
⠀⠀⠀⠀⠘⣿⣮⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⣼⡟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣭⣟⣛⠻⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣻⡍⡋⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏⡙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣽⣇⠁⢤⣀⢀⠀⠀⠀⠀⠀⢀⣀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠿⠿⠦⠧⠤⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠦⠤⠴⠶⠶⠶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠙⢿⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠛⠶⣄⠀⠀⠀⠀⢸⡇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠤⠒⠂⢀⡾⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢧⠀⠀⠀⠀⠀⠀⣠⢋⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⣨⣥⡶⠀⠀⢣⠀⠀⠀⢀⡜⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠕⢺⣿⣿⣿⠇⠀⠀⠀⠃⠀⠀⠞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠁⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠚⠛⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠀⠀⠀⠀⠀⠀⢀⣠⠤⢂⡠⠖⠁⡐⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠁⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠚⠯⠑⠒⠉⠀⠀⠀⢀⠁⠈⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠞⠁
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢖⠵
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢘⡄
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠙⠻⠶⣶⣤⣤⣤⣤⣤⣤⣤⣴⡶⠾⠛⠉⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢠⡾⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠒⠉⠀⠀⠀⠀⠀⠀⣠⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠖⣋⣡⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⡤⠤⠤⠤⠶⠶⢒⣚⣋⣩⠥⠔⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠂⠐⠒⠚⠛⠋⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠐⠒⠈⢉⣉⣉⣁⣀⣀⣀⣀⣉⣉⣉⡉⠀⠒⠂⠤⢤⣀⡀
⠀⠀⠀⠀⢠⠊⠉⣠⠄⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠤⡀⠈⠑⡄
⠀⠀⠀⠀⢸⠀⠀⢥⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⡶⣿⣶⣶⣶⢶⣶⡦⡤⣤⣀⣀⠜⠀⠀⡏⠉⠉⠉⠒⢄
⠀⠀⠀⠀⢸⠀⢄⡀⠈⠉⠙⠛⠀⠦⠤⠤⠤⠤⠬⠭⠭⠭⠽⠛⠟⠛⠛⠋⠉⠀⢀⠀⠀⡧⠤⠤⢄⠀⠀⢳
⠀⠀⠀⠀⢸⠀⠀⠈⠉⠒⠒⠠⠤⠤⠄⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⠔⠒⠒⠉⠁⠀⠀⡇⠀⠀⠀⢣⠀⠀⡇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡌⠀⢀⠇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠊⠀⢀⠎
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⢄⡀⠒⠒⠒⣋⡠⠤⠴⣁⣀
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡩⠐⠂⠉⠉⠈⠁⣀⣀⣀⠀⠉⠈⠉⠐⠒⢬⠑⡄
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠲⠶⢿⣛⡛⠛⡛⢿⠿⠿⣿⣿⣿⠶⠖⠚⠀⠤⠐⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠒⠒⠤⠤⠤⠀⠈⠉⠉⠀⠀⠀⠤⠐⠒⠉⠀⡷⠊⠙⣆⠈⡇
⠀⠀⠀⠀⠀⠀⢀⠜⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠏⢀⠇
⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠈⠑⠒⠢⠤⠄⣀⣀⣀⣀⣀⣀⠠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡤⠋⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⢒⣠⠤⠤⠤⠀⢀⣀⣀⠠⠤⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣁⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⣉⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠩⠜⢂⣒⣒⡒⡒⠒⠂⠐⠒⠒⡒⢒⣒⡐⠣⠭⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢾⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣬⣉⡩⠶⠶⠾⠿⠿⠶⠶⠶⢍⣉⣵⣿⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⢛⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⢀⡼⠀⢸⣿⣿⣿⣿⡇⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⢀⠞⠁⠀⣼⣿⣿⣿⣿⡇⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣌⣀⣠⣾⣿⣿⣿⣿⣿⡇⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣷⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣷⣶⣤⡈⠉⠑⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣷⣄⠠⣄⠈⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⣻⡀⣆⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢣⡿⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⣛⣵⠟⠁⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⢛⣫⣵⡾⠟⠁⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⢛⣩⣥⣶⡾⠿⠛⠉⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣡⡶⠾⠛⠉⠉⠀⠀⢀⣠⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣀⣤⣴⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢀⣠⣶⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠤⠄⠀⠀⠠⢤⣤⣤⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣶⣶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣼⣿⡿⢷⠈⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⢻⣿⡀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠹⣷⣄⣠⡈⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡌
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⠘⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠰⠁
⣠⣤⠶⠒⠂⠀⠀⠀⠀⢩⣤⡘⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⢣⡏⠀⠀⠀⠙⠐⠒⠒⠶⣦
⠙⢿⣷⣄⡀⠀⠀⣠⣴⣿⣿⣿⣌⠻⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⣱⣿⣷⣄⠀⠀⠀⣀⢀⡤⠞⠛
⠀⠀⠈⠙⠛⠿⢾⣟⣛⠻⠿⠿⠿⠷⠯⠿⣿⣿⣿⣿⣿⣿⣷⣶⡾⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠁⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⠉⠛⠛⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⢀⢀⣀⠀⡀⠀⡀⢀⡀⡀⡀⡀⢠⣀⡀⢀⢀⣠⠀⠀⣠⠀⠀⠀⣤⣀
⠀⠐⠠⠀⠄⠀⠀⠀⠀⢀⣨⣭⣍⠈⠉⠈⠀⢀⣐⣶⠀⠀⠀⠀⡆⠀⡐⠀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠠⠐⠂⠲⠒⣐⣰⠖⠒⢦⣄
⠀⠀⠀⠀⡀⠀⠀⣠⣴⣿⣿⠿⣿⢶⣶⣦⣴⣿⡀⢀⠀⠀⠀⠨⡇⠀⠂⠀⣰⠂⣤⣶⣼⣿⣿⡿⠃⠈⠈⣨⣶⠊⢩⡿⠶⣴⡎⠁
⠀⠀⠊⣾⡑⢀⢰⡎⠛⣿⣯⣶⡇⢼⣟⣹⣿⡟⣿⣿⣖⠠⠠⠄⡇⠀⢀⣾⣅⡀⢏⠙⠿⣯⣾⣷⡂⠀⠀⢹⣷⣦⠛⠁⠐⣷⣆⣞
⠀⢠⢸⣿⠟⠋⠀⠐⠚⣽⣿⡿⡿⢸⣿⣿⣿⣀⣼⣿⣷⢜⠀⢸⡇⠀⣿⣿⡟⠙⣤⠀⠙⣾⣿⠟⠇⠀⣤⣿⣿⠏⠀⠇⢺⣿⡿⣿
⠀⢀⢸⣿⡄⠀⠀⠤⡼⣿⡇⡁⠃⢸⣿⣿⡇⠈⠿⣳⣄⡀⠀⠀⡇⠀⠉⠁⠀⠸⣷⣀⢠⡸⣷⠾⡆⠠⢿⠿⢁⣿⡆⢀⣼⣾⣿⣿
⠀⢀⣾⣿⡇⠀⠀⠴⢿⠿⢉⣿⡭⢸⣿⣿⣧⠀⠀⠿⠛⠁⠀⠀⠅⠀⠶⠄⠒⠂⠀⢺⠬⠱⠟⠊⠁⠀⠀⢠⣨⠉⢿⣜⠛⠻⠧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⠃⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⠀⠀⠀⠀⣤⣤⡄⣤⣤⣤⣤⡤⠤⠤⠠⠄⠀⢀⠀⣤⣤⡄⢠⠀⠀⠀⠠⣤⡄⠠
⠀⢀⠁⠀⠈⢀⠀⠀⠄⠠⣦⢰⠆⢸⢿⣟⣏⣥⣤⣀⡀⡀⠈⢻⡇⠈⠙⠿⠁⠂⠀⣤⠈⠀⠀⠤⠄⡿⣧⣷⣷⣶⠆⠀⠘⠛
⠀⠀⠀⣠⣴⣏⣠⠄⠂⠐⠻⠀⢀⠘⠀⠘⢫⠿⠒⢻⠀⢨⡀⡀⠀⠀⢀⢴⡀⡄⢀⡷⣷⣾⣦⡟⠀⣇⣭⣤⣼⣧⡄⠴⠂⢀⣼⣦
⠀⠀⢿⣿⣿⠟⠃⢀⡀⠐⠀⠀⢹⠐⠀⡄⠀⠐⢶⠺⣮⣿⣜⠃⠀⠀⠸⣿⡇⠁⠘⣧⣾⠇⠉⠓⠁⣿⡏⣽⣿⣇⠀⠀⢀⡨⣿⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠁⣷⣠⠠⡠⠀⠀⢞⡷⢄⣸⡃⣿⣭⣯⢴⡆⠀⠀⣿⡟⠀⠀⢹⣿⠀⠀⡀⠀⣿⣷⣼⡝⡇⠀⠀⠊⣤⣿⢿
⠀⠀⠀⠴⠇⠀⠀⠀⠾⠟⠻⠶⣦⠀⢠⡙⠿⣾⣿⣶⣷⣾⣿⠻⠅⠀⠀⣿⡇⠀⠀⠼⠷⣌⣘⣥⠀⠁⣠⡼⣿⠂⠀⠀⠐⣻⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠟⠉⠙⠽⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡷⢄⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⣈⣙⣆⠀⠀⠸⣶⡊⣢⠀⠀⠀⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠃⠀⠀⠙⢢⣿⡇⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠸⠏⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⣤⣤⣤⠀⠀⠀⣀⣀⡀⢤⠤⠠⠄⠀⠀⣀⡀⣰⣿⣶⣦⠙
⠀⠀⠀⠀⠀⠀⠀⢰⣤⢷⡀⠀⠀⠀⠀⠀⡔⣿⣿⣿⣿⣿⡃⠀⠀⠀⠠⣤⣠⣀⣴⣶⣶⣶⣶⣤⡀⢛⣉⣽⣿⠿⢡⡇⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠹⢷⡄⠻⣄⠀⠀⠀⠀⠉⠠⣉⠙⠛⠋⠾⠿⠿⠤⣦⣤⡄⢉⡿⢶⣦⡌⠻⠿⠟⠚⠛⢋⣡⣴⣿⣷⡿⠵⠾⠍⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠲⢶⣶⣶⣿⣿⣿⣿⣿⡿⠛⠓⢤⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⠳⣄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢠⣼⣿⣿⣟⣿⣡⡾⠀⠀⠀⢠⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢣⡀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠙⢫⠡⢿⠃⠀⠀⣠⠆⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠱⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⣤⣴⠏⣾⣿⣶⠟⠁⠀⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣛⠋⢸⣏⠋⢀⣒⣨⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠑⣀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣟⡛⠛⠿⠛⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠜⢻⣿⣌⠳⣜⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠄⠐⣰⣿⣿⣿⣿⣷⠶⠦⢄⡀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡴⠋⠍⠀⠀⠀⢀⠉⠓⢼⣧⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠌⣤⣾⣿⣿⣿⣿⣿⣿⢺⡤⠂⠙⢢⠀⢽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡄⡐⠀⢺⠃⠂⠀⠀⠀⣠⠎⣴⠄⠀⠙⢿⣿⢦⠉⠐⢶⣶⡶⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠛⠈⠀⠀⠀⡠⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⢀⡀⠀⠘⠓⢾⠗⣒⣱⠀⠀⠙⣇⠉⠈⠀⠂⠢⣌⠉⠉⠀⠉⠉⠋⠈⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⣀⡞⢻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠑⠀⠀⠀⠉⡉⠁⠀⠀⠀⠙⠐⣀⠀⠀⠀⠀⢧⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠁⠀⢀⠯⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⣤⣤⣄⣀⣉⣀⣀⣐⣺⣿⣷⣶⣤⣤⢘⣋⣉⣉⠉⠀⣀⣀⠀⠀⠀⠠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠊⡟⠿⢛⠷⠛⠛⠋⠉⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⡿⠿⣛⣿⣿⣆⠀⢀⠀⠀⠀⠀⠀⣀⣀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡌⠀⢀⣌⠙⣿⣿⣿⣷⣦⢤⣁⣀⠲⢤⠙⡿⠷⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⡆⠙⠓⡀⠀⠀⢹⣿⣿⠿⣧⣀⣀⠒⠉⠛⠃⠑⢂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣤⡄⠭⣤⣾⣿⠿⣿⠿⠋⠀⠀⠀⢀⠴⠐⠂⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠈⠀⠀⠀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡆⠈⠋⠡⢄⠑⢄⠀⢀⣠⣄⣤⠁⠀⣀⣀⣀⣀⣠⣤⣶⣿⣿⠀⢲⢯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠈⠣⠇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢤⠀⠈⠉⢩⣁⣭⣿⣿⣿⣿⣿⢀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢶⣶⣭⣻⣶⢯⣿⣿⣿⢯⣿⣇⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠓⢉⣭⠟⣩⠟⠠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠓⠦⣤⡄⠀⠀⠀⠀⣀⡠⢴⢶⣿⠿⣄⣀⡀⣈⣱⣵⣾⣯⣴⡟⠁
⠀⠀⠀⠀⠀⠀⣰⡤⣀⣀⣀⣀⣀⢀⣌⠈⠁⠒⠋⠋⠉⣁⣀⣤⠷⢄⣿⣿⠿⣟⣛⣻⠶⠟⠋
⠀⠀⠀⠲⠶⠊⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⠤⠴⢶⣶⣶⠦⠶⠟⠷⠲⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⣶⣶⠾⠋⠁⠀⠀⠀⢀⡿⠛⠁⠀⠀⠀⠀⠀⠀⠈⢹⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⣠⡴⠖⠋⠁⠀⠀⠀⠈⡿⠃⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⣴⣿⠿⠉⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⣤⣼⣿⣯⠶⠟⠛⠛⠛⠿⣿⣿⣷⣄
⠀⣠⣾⠟⠛⠓⢦⣝⢦⡀⠀⠀⢀⣿⡄⠀⠀⠀⢀⣰⡏⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣦⡀
⢰⡿⠛⠀⣠⡀⠀⠙⢷⣻⡤⠠⢞⣿⣧⠀⠀⢀⣼⣿⡇⠀⠀⠀⣴⣿⣿⠏⠁⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿⣷⣄
⣾⠁⠀⠀⠈⢻⣶⣴⣼⣿⣿⣦⣾⣿⣷⣄⣿⣻⣿⣿⡿⣠⣾⣿⣿⣿⠋⠀⠀⢀⣠⢀⣤⣶⡖⠉⣩⣩⣽⣿⣿⣿⣿⣿⣿⣆
⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡆⣠⣴⣿⣿⣿⠿⠟⠛⠛⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣧
⠸⡄⠀⠀⢬⣿⣿⣿⡿⠟⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⢳⣦⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣶⡖⠰⠦⢤⣤⣤⣤⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠷⢂⠀⣀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠈⠂⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⢠⣴⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣤⣠⣀⣴⣆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠻⠿⠟⠛⠋⠁

⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠂⢀⠀⡠⠂⡠⣩⠃⠀⠀⠀⠀⢣⠡⡀⠀⡀⠀⠠⠚⠉⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠭⢙
⣤⣴⢆⠂⠀⠀⠀⣠⠄⠐⢐⣤⣤⡀⠀⠀⠑⠈⢣⣔⣪⠖⠁⠀⠀⠀⠀⠀⠀⠑⢔⡂⣤⠋⡐⠁⠀⠀⣀⣠⣄⠤⠀⠀⠀⠀⡀⠠⡤⣆⣀
⠁⠀⢂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠄⣸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⢙⠀⠀⠀⠀⠀⠉⠉⠨⠀⠀⠀⢠⠈⠀⠀⠈⠉⠙⠃
⠀⢀⣴⠁⠀⠈⣶⠁⠀⠀⠤⡀⠀⠀⠀⠀⠀⡴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡈⠂⠀⠀⠀⠀⣠⠄⠀⠀⣧⡓⠀⠄⣬⠠⠀⢀⣀
⠉⠀⡅⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠈⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡎⠈⠃⠀⠐⠀⠀⡁⠀⠀⠀⠻⠀⠀⠀⠀⠃⠐⠒⠒
⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⡘
⠀⠀⠀⠳⣀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠆⠀⠀⠀⠀⠀⠀⠀⠧⣀⠀⠀⠄⠀⠰⡁
⠀⢀⠠⣂⠜⠉⠉⡉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡚⢻⠳⠌⠀⠀⠀⠈
⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠠⠄⠉⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⣶⣴⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣶⣷⣶⣶⣾⣷⠀⠀⠐⡇⠀⠈⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣿⡇⠀⠀⢰⠃⢂
⠀⠀⠀⠀⠀⡂⠀⢒⡂⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⢴⣶⣶⣥⣶⠄
⣤⣴⣗⡄⠀⠀⢤⠠⠅⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣿⣿⡏⢴⡎⣿⣿⣧⠀⠀⢸⢸⣿⣿⠟⠁⢠⣾⣶⣶
⣿⣿⡿⣿⠦⡀⠀⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣶⣿⣿⣿⣧⡀⣼⣸⠟⠁⣠⣶⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣦⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⡇⢀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣵⣷⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣿⠯⠗⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⠇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⣀⣤⣀⣼⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⣿⣷⠈⠀⠀⠀⢘⣻⣋⡿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣦⣄⡤⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠤⠀⢐⢒⣒⣒⣠⣀⡀⠀⠀⠀⠀⣄⠐⡂⠒⠒⠒⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣤⣴⣶⠖⠃⠁⠀⠉⠘⠉⠈⠁⠀⠀⠉⠁⠁⠀⠀⠀⠁⠀⠀⠀⠁⠹⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⣛⠿⢥⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⣐⣤⡵⠆⠀⣀⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⣙⠛⠛⠿⠶⢶⢶⣷⣤⣤⢤⣥⣬⠦⠤⠵⠶⣶⣶⠶⠳⠛⠋⢉⡁⢰⡞⢣⢋⣠⡤⠄⠀⣀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⢹⢇⠀⠢⢤⣐⢂⣤⠞⣽⣾⣵⠒⠛⠛⠛⠲⣄⠻⣌⢲⣦⡉⢡⡀⠘⠟⠘⠀⣏⠂⠀⠈⠲⣠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠁⠀⠀⠀⠐⡛⠛⠎⠀⢉⣁⠀⡀⠳⡆⣀⡀⠀⡛⠟⢓⠋⠶⠱⠂⢑⠾⡀⡘⠀⠀⠀⠀⢠⢳⡀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢇⣘⠒⠰⠖⣴⣱⣤⠆⠁⠉⠔⡉⠹⢏⣛⠋⠀⠉⢸⣿⠙⢻⠀⡟⠈⢡⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠐⣧⠀⢛⡇⠀⡔⠻⣦⣉⢻⣄⠹⣤⡀⠀⢨⡅⠀⢸⠀⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⢸⠀⠈⡇⠈⢗⣺⢟⢏⣼⠅⢻⠿⠟⠀⠈⠁⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢘⠀⢸⠀⠀⡇⠀⠀⠃⢼⡹⠤⠖⣯⣄⡘⣿⢿⣿⠆⠘⠀⠇⠀⠸⠀⠀⠀⠀⢀⠌⠜⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠀⠀⠀⢨⠀⢸⣶⣄⣷⣶⢶⣀⠀⠀⠀⣴⣶⣻⣏⡁⡇⢰⣯⡞⡿⠃⠀⠰⠃⠠⠀⠐⠁⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⡂⠀⠀⠀⢴⣶⡾⢃⡀⠹⣿⢸⡟⡎⢻⣶⣾⣏⣿⣿⢰⠖⠛⠛⣇⠿⠃⠀⢸⣦⠄⠀⣀⣠⣔⡁
⠀⠀⠠⢄⣰⣦⡭⠶⢖⣶⡤⢛⡳⠀⠀⠀⢧⣿⣔⣿⢷⠄⢘⣿⣵⠁⠀⣭⣟⣈⣹⡌⠿⣏⠋⣸⡿⣇⠀⠀⠸⣓⡿⣛⣉⣙⠲⣮⢓⣲⠤⢤⡀
⠀⣵⣿⡟⣍⡂⠉⢪⡧⠟⡙⠚⠁⢦⡀⠙⢩⢚⣥⣌⢯⠺⠟⢘⣮⡷⠀⠾⠟⠻⡟⣧⣶⣶⡿⠿⠃⠀⠀⡴⠁⠵⠒⠋⠉⠉⠇⡴⢫⣭⣭⣳⡌
⠁⣿⢱⠳⠇⠀⠀⠘⠃⠔⠉⠁⠀⠀⠑⢦⣉⠛⠻⣿⢿⣦⣴⣯⡾⢿⣦⠀⣠⣶⢻⠯⣿⣋⠇⠀⢀⡤⠊⠀⠀⠀⠀⠀⠠⣄⡈⣴⠠⠀⢁⡽⢃
⠀⠪⣅⣓⠶⠄⠀⣀⠀⠀⠀⠀⢀⠁⢄⠀⠉⠓⢄⡻⢿⣿⡿⢿⣧⠀⠀⢦⣤⣽⣾⡿⣿⣿⣴⣸⠋⠀⢃⠆⠟⠁⢤⣀⠀⡀⠀⠀⡐⢊⡥⠖⠋
⠀⠀⠀⠈⠉⡛⠿⣶⣥⣲⣞⣂⣈⣁⣀⣀⣀⣀⣀⣛⡒⠇⠀⢄⢀⣀⡀⠈⠳⠯⠉⠙⣋⣉⣁⣀⣐⣒⣂⣐⣺⣥⣴⡶⠞⢛⣉⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠁⠂⠠⠉⢉⠈⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠛⠛⠒⠒⠖⠛⠛⠛⠋⠉⠙⠛⠿⢿⣿⠿⠿⠟⠋⣉⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢦⣒⣂⠀⠀⠤⠄⢀⢀⣀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠠⠤⠤⠀⠀⢒⣒⡨⠋⠁