Armour Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣤⣤⣤⡶⠖⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⣀⡀⠀⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣧⣐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⡿⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⣿⣿⣷⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⡟⠓⠀⡀⣀⠀⣙⠛⠿⠿⠿⠿⠟⢣⠀⠁⠁⠒⢸⣿⣿⣿⣿⡟⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢰⣿⣿⠂⠀⡜⠁⠀⠀⠀⠙⢦⣀⣀⣀⡴⠀⠀⠀⠀⠀⠀⡟⢻⣀⢀⣿⣾⣿⠏⢻⡇
⠀⠀⠀⠀⠀⣿⣿⡿⠦⢼⣿⣶⣤⣦⡀⠀⠈⡟⢫⡙⣦⠀⠀⠀⠀⠀⠀⢿⡈⣦⣿⣿⣿⠏⠀⣘⣷
⠀⠀⠀⠀⢀⣿⣇⣀⡀⢸⣿⣿⣿⣿⡷⣶⣶⡁⠀⠈⢸⣧⠀⠀⠀⠀⠠⠄⢿⣽⣿⣿⣿⠀⠀⠈⠻
⠀⠀⠀⢀⣾⣿⡿⠛⠋⠌⠻⢻⣿⣿⣿⣿⣿⣷⣴⣷⠞⣟⠀⠀⠀⠀⠀⢃⠂⣻⣿⣿⣿⡀⠀⢴⣾⣦
⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⠐⠐⢐⣿⣿⣿⡯⣿⣿⣿⣷⡟⠹⣿⡀
⠀⢠⣾⣿⣿⠇⠀⠀⠀⣠⣿⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⡄⣿⣇
⠀⠘⣿⣿⣯⠀⠀⠀⠐⣿⡟⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢙⣿⣿⡏⠀⠀⢸⣿
⠀⠀⢸⣿⣙⡀⠀⠀⠠⠛⢡⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⠿⠛⠛⢻⣿⣿⣄⠘⢿⣿⣇⢀⠀⣿⡇
⠀⠀⢸⣿⠿⠿⢯⡅⠀⠀⣾⣿⣿⣿⣿⣿⣿⡉⠁⢀⣿⡖⠀⠀⡀⠀⠀⠈⢻⣿⣿⣦⢸⣿⣿⣷⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⣿⡆⠈⣿⣿⣿⣿⣿⣿⣟⡀⢀⣀⢫⣠⣶⡆⡁⠀⣶⣄⡈⢿⣿⣿⣿⣿⡏⠁⢈⣿⠁
⠀⠀⢸⣿⠂⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⡯⠤⠴⠯⠭⢥⣿⣿⣿⡍⢠⣿⣿⠷⢿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⢸⣿⠀⢀⣰⡿⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣾⠿⠉⠀⠀⠀⠈⢿⣷⠈⠻⣿⣿⣿⣷⠀⣸⣿
⠀⠀⢰⣿⣿⣧⣿⡇⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⢻⣿⡿⠿⢶⣿⠿
⠀⠀⠀⠉⠉⠉⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡟⢷⡀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⢀⣴⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠁⠀⢺⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣀⠀⠼⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠟⠋⠙⡉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣽⣿⣿⣷⣶⣤⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣤⣬⣉⣹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣟⠻⣿⣿⢯⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⢀⣤⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⣓⢵⣿⣄⣇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣻⡿⠟⠋⠀⠀⠻⢟⣿⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⢁⠀⡀⠀⠀⠀⠀⠈⣻⣆⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⡟⠙⣿⡀⠀⠀⠈⠻⣿⣛⣛⣾⣷⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⣾⣿⣷⣶⣠⣤⣀⠀⢿⣿⣷⣦⣴⣶⣿⣭⣿⣿⣿⣥⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⣾⣿⣿⠀⠀⠀⠙⠆⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣤⠀⠀⠀⠀⠀⡁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣾⣧⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⢸⡀⣶⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠸⣿⣽⣿⣿⣿⣾⣿⣿⣏⣴⣿⣿⢿⡀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣾⣇⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠘⢿⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣠⣿⣿⣿⣿⠻⣿⡟⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠛⠻⠿⣿⣿⣿⣿⣿⣆⠈⠻⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢹⡇⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣦⣀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠄⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣾⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣶⣄⣀⡈⠀⣟⡛⢿⣿⣿⣻⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣔⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠿⠉⠀⢀⣸⣿⣿⠛⠉⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⢺⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠁⠘⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠿⠿⢿⠟⠀⠀⠀⠀⠀⠀⠀⠐⠿⡿⠿⠟⠟⠁





⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢿⡇⠀⠒⠢⠼⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⠀⠀⠀⢠⠏⢑⠠⠤⠤⠠⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢦⣤⣸⠟⡆⠈⠄⡀⠀⠀⠸⢻⣦⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢱⠂⠀⣼⡏⣀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡧⢄⣰⣏⣸⣿⠂⠀⢀⠀⠀⠀⢠⢾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣾⠟⣾⣿⠃⠀⠀⠀⠀⠀⠀⢰⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠶⢶⣼⣿⡃⠀⠀⠀⠑⢄⡀⠀⣼⢫⡿⠰⠂⣤⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠿⣧⠄⢄⣀⡀⠀⠁⠶⣄⣾⡗⠀⠀⠸⡄⠀⠀⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⡿⠢⠀⢿⡟⠀⠀⠀⣾⠿⣧⠀⠀⠀⢃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢄⣰⡇⠀⠡⠘⣷⠀⠀⢰⣿⠀⠘⣷⠀⠀⠺⣄⠀⠀⠀⠀⢠⠆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢻⡅⠀⠀⠱⢘⡧⣀⣾⣇⠄⠀⠘⣆⣰⠂⠉⠀⠀⠀⠀⠃⠂⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡈⣿⠂⢀⡀⠀⢠⡬⣿⣿⣷⠀⠠⢸⣿⣤⠶⠀⠀⠀⠀⢸⣧⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢿⣻⣁⣀⣀⣡⠀⠈⠀⠻⡿⡿⢷⣤⣨⠟⠉⠀⠀⠀⠀⠀⠈⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠿⠍⠁⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⡇⠀⠀⣠⣶⣹⣁⣀⠀⠀⡆⠀⠀⠀⢀⣤⣤⡴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠣⠀⣼⡟⠁⠀⠀⠉⣥⣴⡄⠈⡦⡐⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡇⠀⠀⠀⠀⠀⠀⡀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣞⡃⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢻⠀⠀⠀⣠⠀⠛⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⣀⣀⢘⠀⠀⢄⡠⠗⠀⠀⢠⣼⠄⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠏⢻⠀⠀⢸⣴⣆⡀⠄⠟⠻⢆⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠒⠒⠲⠔⠻⠀⠀⠀⠉⠉⠉⠛⠀⠲⠶⠿⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠋⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠘⠗⠛⢿⣄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⠾⠿⣟⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣴⡄⠀⣼⣿⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠐⠈⠠⡦⠀⢰⠶⢼⡟⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠐⢠⣤⢀⡹⠃⠚⠁⢀⣴⠟⣿⠀⠀⠀⠀⠀⠀⠀⢛⠀⢸⣿⠀⠸⠀⠈⣻⡿⣇
⠀⠀⠀⠀⠀⠀⠆⢀⣾⠷⡚⠁⠀⢀⣸⣿⡿⠀⠸⣄⠀⠀⠀⠀⢀⠀⢼⡿⠟⠽⡁⠀⠱⣿⣿⠀⠛⣦
⠀⠀⠀⠀⠀⣤⢀⠼⣇⠘⣿⣀⣼⣿⣿⣿⠙⠃⠲⣹⡷⠀⠀⠰⠀⡤⣿⠟⢰⠀⠉⠀⢾⣿⣿⠀⠀⢙⡷
⠀⠀⠀⠀⠀⠈⠋⠓⣦⣾⣵⣏⡈⠉⠙⣿⣶⠢⠔⡿⠀⠀⠀⠀⠀⢹⣿⣆⣃⣀⠀⠀⣙⣿⣿⣧⡴⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⠖⠂⠀⢻⣿⠀⢰⠇⠀⠀⠀⠀⣄⢸⡆⡟⠉⠉⠉⠉⠙⣿⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣦⠀⠀⡀⣨⣿⠀⢾⠀⠀⠀⠀⠀⠀⣿⣷⡁⠀⠀⠀⠀⠀⢻⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣽⡗⠆⠀⢸⡿⠠⠞⠀⠀⠀⠀⠀⠘⠿⠿⠁⢦⣶⡶⢶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⠟⠀⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⣀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⣿⠀⠀⠧⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠾⣿⡇⠀⠸⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠇⠀⢡⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡀⠀⠀⣿⡇⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠘⢶⣇⢀⡀⠐⣷⡀⣀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣆⡄⢩⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⢰⣎⣭⣿⠟⠛⠘⠋⢯⣬⣽⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⠉⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣶⠀⠀⠸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⣛⠀⠉⠀⠘⠢⣰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⣿⣯⣽⣆⣀⣤⣀⠀⢈⣇⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠻⡍⠙⢿⡇⣸⣿⢤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠰⡦⠀⡾⠀⠸⠗⡾⠇⠆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⠀⠀⠀⠜⠣⣄⠀⠀⠀⣄⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣉⣠⣤⣀⡀⠘⠤⢶⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠈⠻⣿⣟⡿⠿⠟⠻⢾⣿⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠⣄⠀⠀⠀⢹⣿⣤⠲⠄⠀⠀⠁⠹⣏⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢣⠀⠆⠀⠀⠀⠀⠀⠌⠀⠈⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣍⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⠋⠁⠀⠚⠂
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁
⠐⠦⢤⡀⠀⠈⠂⣦⠀⠀⠀⡠⣄⠀⠀⠀⠀⢰⣇⠊⠀⠈⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⣢⡄⠀⠀⠂⠀⢀⣀⣠⡄
⠀⠀⠀⠀⠙⠒⢦⠈⠀⠀⠀⠁⠈⠀⠀⠄⡀⠀⠟⠛⠶⠶⣤⣤⣐⣂⣀⣀⠀⠀⠀⠀⡈⣈⣉⣤⡶⣶⠞⠉⠀⢱⣯
⠀⠹⡤⠰⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⢟⠚⠀⠀⠀⢀⠀⠉⠉⠉⠉⢉⠉⠙⠉⢉⠉⠻⡾⠉⡳⠆⡀⠀⠻⣧⡄
⠀⠀⠲⢇⣀⠀⠀⣀⠛⠓⠀⣰⠀⠀⠀⠀⣴⠲⡞⠟⠲⠶⠶⢤⠄⠄⠤⢀⠀⠡⠠⠶⠤⠀⠒⠀⢹⣾⣠⠛⢤⠀⠀⠈⢲⡀
⠀⠀⠀⠀⠀⠉⠛⠁⠀⡶⠚⠉⠙⠒⠀⣠⡝⠙⢳⣆⠰⠤⠀⠸⠆⠰⠀⠀⠀⠀⢤⠄⠀⠀⠐⠒⣸⡟⠑⠄⠀⢤⡦⠒⠉⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠠⠤⡇⠒⢤⣄⠀⠀⠈⠙⢆⠈⠉⢽⠇⠘⠷⠂⠒⠲⠲⠦⠀⠀⠒⠚⠒⢈⢩⠟⠀⠠⠤⠄⠀⢓⠈⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣉⠳⢤⡀⠀⠨⡄⢱⣽⡀⠐⠒⠉⠀⠀⠤⠄⠀⡀⠐⢂⣠⢠⠏⠀⠀⡅⠒⠁⡀⠿⣧⣄⣤⣤⣄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢅⡠⠄⠃⢸⠁⣝⣙⠃⣄⣉⢹⡟⣛⠈⠿⠀⠀⠴⠀⠀⠀⠀⢗⣛⠩⠐⠉⢻⡙⡟⣟⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠓⡬⠭⢤⡤⣤⣤⣤⡭⠐⢦⣶⠶⡄⠀⠀⠀⠔⠉⠁⠀⠠⢾⣎⡿⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⠐⠀⠀⠀⢻⠁⠸⢶⣶⣟⣤⡽⠒⠇⠀⠀⢀⠀⢀⣴⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⡅⠈⠁⠀⠀⡏⠀⠀⣶⠋⡁⠀⠀⠄⠀⠀⠀⠈⣦⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢠⠄⠀⠒⠀⠀⡓⠀⠚⣁⠀⣠⠀⠀⠀⢰⣤⣤⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠂⠘⠒⠀⢘⣧⠀⠰⣿⣶⣥⢀⡀⣤⣾⣿⣎⢧⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠓⠶⠀⠐⠄⣤⣼⣧⣠⣤⠠⣮⣾⡿⣿⡦⠀⠙⢿⠒⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢀⠀⠰⠀⠤⠀⠀⠀⢀⠀⠓⠀⠉⠰⣟⡃⠀⢀⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣷⣶⣶⣶⣦⣴⣤⣤⠶⣶⣴⣿⠿⠿⣦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠈⢧⠀⠀⠀⠈⠞⠁⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣀⣀⣠⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣤⣤⣤⣤⣤⣤⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠛⣿⡟⠛⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣿⡇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢤⣀⣀⡀⠀⣇⣀⠀⣿⣇⢀⣀⣿⠀⢀⣀⣀⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠈⢩⡟⢻⣿⠉⠉⠉⠉⠉⢉⣿⡟⢻⡀⠀⠀⠀⠈⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⣴⣿⣿⠉⠉⠉⠉⠉⠉⠉⠙⠛⡛⣿⣿⣄⡀⠀⠀⠈⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡓⠢⠔⣖⣿⠉⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡌⠉⣯⡒⠲⠔⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠬⣭⣭⠿⢃⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣝⢶⡘⠯⣭⣭⣽⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡉⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣼⠀⢊⠁⠀⠀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠦⣀⣀⡠⠵⠇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣿⠀⠸⠤⢀⣀⣢⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢷⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡏⠀⠀⠀⠀⡜⠁⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠈⡳⡆⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠃⠀⠀⢰⠊⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡇⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡏⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⢀⡯⢍⣉⣉⣉⣉⣉⣉⣉⣉⣉⡩⣽⢿⡀⠀⠀⢸⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⡇⠀⠀⣼⡓⠒⠶⠤⠤⠤⠤⠤⠤⠤⠶⠖⢚⣛⣧⠀⠀⢸⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⣨⡇⠀⢰⡏⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠊⠉⠉⢫⢹⡆⠀⢸⣀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣼⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠛⠛⠛⠛⠛⠷⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⠀⠀⠀⠀⠠⣀⣀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠿⠿⠿⠿⣶⣤⣠⣿⣿⣿⣀⣠⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠛⠋⠉⠉⠉⠀⢘⣿⣿⡿⠛⢉⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⢨⡿⠋⠀⢠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⠒⠉⠉⠁⠀⣀⣴⡿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⣶⠶⠶⠋⠉⠀⠈⠻⣿⣷⣤⣠⣤⣤⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢿⣿⠇⠀⠀⠀⠀⣀⣤⠞⠋⢛⠀⠀⣠⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠀⠀⣠⣦⡀⠌⠰⢷⣼⣿⠆⡀⠀⣠⣿⣿⡿⠀⠀⣾⣶⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡻⣷⠛⠁⠈⠻⣗⣶⠈⣿⣿⣿⢿⣿⣿⣿⣿⡿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠟⠷⠀⠀⠀⠀⠙⣻⣿⣿⠀⢸⣿⣿⣿⣿⣿⣦⣴⠿⠛⠋⠉⠉⠋⠉⠉⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢢⣤⣀⠀⠀⠀⠀⣿⣿⡟⠀⠈⢿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣿⣿⣶⣶⣶⣾⣿⡿⠀⠀⠀⢸⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠆⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠃⢀⣠⣤⣤⣼⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢀⠀⡀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣴⣿⣿⣿⣿⠉⠁⠀⠀⠀⢰⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢇⠀⠀⠉⠰⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⣩⣾⣿⣿⣿⡿⣠⠶⠾⢯⣍⠉⠛⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠉⠴⢆⢠⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠻⡟⠁⠀⠀⠀⠈⢷⣄⣀⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⢘⣿⣿⡿⢻⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣾⣅⡀⠀⠀⠀⠀⠀⣿⣿⡿⠛⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠀⠀⠀⠀⣄⠀⣿⣿⡇⠈⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⢀⣉⣉⣠⣴⠼⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡦⠤⠀⠀⢀⣻⣿⣿⣿⡟⠀⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⣀⣽⣿⣿⡟⠛⠉⠉⣹⣿⣿⢤⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠈⠛⢻⣿⣿⡆⠀⢀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣥⣿⡿⠛⠀⠀⠀⠀⢴⣿⣇⣺⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢠⣾⣿⡟⠀⣰⣟⢻⣿⣿⣿⠿⠷⠄⠀⠀⠀⠉⠛⢻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣠⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠈⠛⣿⣿⠁⣠⡿⠓⣾⣿⣿⣿⠶⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡾⠟⣁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⣿⣿⠀⣿⢃⡐⢻⢿⣯⣉⠀⠀⠀⡀⠀⢠⣤⣤⣤⣽⣯⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠺⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠇⠀⠀⠀⠀⢠⣿⣟⣾⠟⢀⠐⢠⣾⣿⢿⣤⣀⣀⣀⠀⢤⣀⣀⣀⡀⠀⣠⣴⣿⣿⣿⣿⣇⠀⠀⠈⠀⠀⠀⢀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⢰⣿⣿⠃⠀⠆⢰⠇⠀⠀⠀⠙⢿⣿⠛⠆⠚⠋⠁⠀⠙⡄⢻⡿⣿⣿⠟⣿⣷⠀⠀⠀⠀⠀⠀⢉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠀⠀⣻⣿⠟⠀⠀⢰⡏⠀⠀⠀⠀⠀⠘⡿⣧⠀⠛⠀⠀⠀⠀⢠⡘⣿⣿⣧⣠⣿⣿⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⣿⣿⠆⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠈⣃⢻⣿⣿⡏⣿⡇⠀⠀⠀⢀⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⢸⣏⣿⣷⣿⡇⠀⠀⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⣼⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⣿⠿⠁⠀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣠⣿⣿⣿⣿⣏⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡅⣽⡟⠁⠀⠀⢰⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⠀⠸⡇⢹⣿⣷⠀⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣿⣿⡖⢀⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⢻⡟⢁⣿⣹⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣄⣀⣠⣾⣿⣿⣦⠀⢀⣠⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡙⠻⣿⣿⣿⣿⢹⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣤⣿⣿⣿⣿⡟⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣾⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠐⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣷⣦⡉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⠿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠛⣿⠁⠀⠀⠀⢻⣿⣿⣿⣿⣯⠻⣿⣯⣘⣿⣿⠟⠁⠀⠀⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣰⡇⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣼⡟⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡋⠩⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢿⣿⠋⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡗⢰⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⣿⢹⡆
⠀⠀⠀⠀⠀⢀⠀⢀⣼⣿⣿⣷⣿⠁⠀⠀⠀⣰⣿⣿⠟⣻⣿⣿⣿⣿⣁⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⣿⣿⣿⠀⣿⣼⠇
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⢠⣿⣿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⠀⠘⣿⣿⡁⣿⡟
⠀⠀⠀⢸⣛⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣽⣿⣽⣿⡇
⠀⠀⠀⠀⣿⣷⠈⠻⣿⣿⣿⠏⠀⠀⠈⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣰⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢿⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⣱⣿⣿⠏⠀⠀⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡙⣿⣟⣴⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⡳⢈⠟⢿⣿⣿⣿⡟⠛⠛
⠀⠀⠀⠀⢸⣻⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⣸⠀⠀⠈⠉
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡹⡀
⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢇
⠀⠀⢀⣴⣿⡿⢶⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢾
⠀⣴⣾⣿⢿⣦⡀⣿⠶⠖⠀⠀⠀⠀⣿⣿⣿⡿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠼⡇
⠀⠠⠘⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣇⢳
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢣⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣷
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠁⠀⣠⠞⢛⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠞⡡⠞⣿⠟⠋⠙⠿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣀⠀⠀⢀⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⣛⣀⣼⣿⣤⣤⣀⣉⣀⠴⣷⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣤⣤⡄⣶⣽⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋⠉⢻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣶⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⠏⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢶⣿⣿⣷⡶⠂⠈⣻⣿⣿⢟⣿⡿⢃⣼⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠋⠓⢚⣋⡉⡸⢻⡟⠀⣠⡾⠛⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⣤⣿⡾⠋⠙⠛⠉⠀⠈⢻⠟⣽⣾⣿⡆⠀⠀⠈⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠻⣉⡒⢶⣶⠿⣫⣿⠃⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣟⣃⣀⣀⣼⣿⣾⣉⣡⣄⣀⣴⣶⣿
⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⠟⠹⣿⣿⣿⣿⣿⣿⡟⡞⠁⠈⠙⠛⢿⣿⣿⡿
⠀⠀⠀⠀⠑⡿⠟⠏⠈⠉⣹⣿⣿⣧⣄⣀⣀⠄⢠⣾⠃⢀⣨⣿⣿⣿⣿⣿⣿⢇⡿⠦⠀⠀⠀⢘⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣯⣾⠁⠘⢿⣿⣿⣿⣿⠿⢯⣴⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣴⡇⢠⠤⠀⠀⢐⣿⣿⠁
⠀⠀⠀⠀⠀⠙⢿⣞⣆⠀⣠⡽⠻⡟⠑⣤⣶⣿⣯⣭⣿⣿⣶⣧⣘⣿⣿⣿⣏⡏⣇⠀⠀⣤⠐⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⠫⣈⢃⣌⣶⣿⣿⠿⢿⣿⣿⣿⣿⣿⣋⠙⣿⣿⣷⣷⣿⢀⠴⠁⢼⣿⣿⠃
⠀⠀⠀⠀⠰⢾⡇⠀⠈⢿⡟⣭⣿⣿⣿⣿⢟⣛⡛⠫⠾⡏⠈⠙⣷⡟⢛⣿⣿⣿⣿⡟⢓⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⣿⣿⣿⣿⢿⠁⠀⠈⠀⠐⣓⡧⠀⠀⣽⣿⣿⣿⣿⢿⡿⣷⣴⣿⣯⠋
⠀⠀⠀⠀⠀⠀⠀⢡⣭⣽⠻⢿⡿⢚⢸⠀⠀⠀⠀⢌⣭⣾⡟⠛⠋⠉⣽⣿⣿⠘⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⡀⢻⡿⠛⠀⠈⣿⣿⣼⣔⢦⣿⡤⠇⠼⣽⣇⣤⡴⠿⢾⣿⡿⠀⢿⠟⠉
⠀⠀⠀⠀⠀⢨⡤⠊⠀⠀⠀⠀⠈⠱⠖⠿⣷⣥⣶⡿⠃⠈⢻⣷⣶⣶⢾⣿⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠤⠌⣈⣵⣿⣿⣿⣧⣤⣀⡤⠟⠓⠠⣼⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣥⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁⢉⣀⣶⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠛⢿⣍⣙⣿⠿⠿⠿⡿⠿⠿⠿⢿⠿⠛⠋⠁

⠀⠀⢸
⠀⠀⢸
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣴⣶⠋⠀⠛⠃⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠚⠛⠳⠦⣀⡀⠂⢀⠀⠀⠈
⠀⠐⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠛⠦⢤⡄⢀⡀⠉⠃⠀⠙⠀⠀⢀⣀
⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣤⣤⣤⣠⣄⣉⣉⣉⠀⠀⠀⠀⠈⣧
⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⣭⠥⡤⢤⣄⣉⣀⣤⡀⠤⠂⠀⠠⣿
⠀⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣶⣶⣴⠟⠛⠋⠉⠀⠀⣀⡶⣷⡏
⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⡀⢉⣠⣤⣴⠗⠈⢀⣼⡏
⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣛⣫⣤⣴⣶⣾⡿⠿⠷⣀
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠟⠓⢏⠛⠫⠉⠉⠉⠀⠀⠀⠀⠈⠰⡤
⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⢾⡑⢿⡄⠀⣈⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⡾⢷⡌⠓⢄⠝⣶⣿⠀⣠⣶⠆⠈
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣴⡟⠳⣄⡈⠁⠀⣴⠛⠉⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈
⠀⠀⠀⣿⣿⠀⠀⠀⠀⢠⣏⠁⠒⠀⠁⠀⣴⠃⠀⠃⠀⢻⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⢿⣿⠀⠀⠀⠀⢸⣿⠷⢶⣦⣤⣤⣾⠆⠛⠀⠀⠈⢻⡦
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⣀⣀⣀⣀
⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⡤⣀⣡⠀⢰⡏⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡟⠋⠀⠉⠛⠂
⠀⠀⠀⢸⣿⡇⠀⠀⣠⠃⠀⠉⠛⣤⣾⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠘⣿⡄⠀⠀⠀⣀⠐⠄
⠀⠀⠀⠀⣿⡇⣴⣿⣿⣤⣤⣄⣀⢘⣿⣤⣀⣀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⢿⡀⠀⣀⠄
⠀⠀⠀⠀⣿⡇⠻⣿⡟⠋⡉⠉⠙⣿⠛⠟⠻⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠻⠇⠀⠀⡼⢿⡾⠋
⠀⠀⠀⠀⣿⣧⣤⠿⠀⢰⣿⣴⣾⠛⠳⣶⣤⣤⣤⣤⣤⣀⡀⠠⠤⠤⠤⠤⠤⠤⠤⠀⠴⠆⠀⠇⠈⠁
⠀⠀⠀⠀⣿⣿⣍⡄⠈⠀⠀⣰⣿⣤⣄⣀⡀⠀⠀⢉⡁⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣼⡄⠀⠀⢀
⠀⠀⠀⠀⣹⣿⣟⠛⠛⢳⣾⣇⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢻⣿⠦⠤⣤
⠀⠀⠀⢾⢻⣿⠈⠉⠱⡾⠋⠛⠿⠶⣶⣤⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣾⠇⢀⡀⠀⣾⣧⢤⣄⡀
⠀⠀⠀⠺⣺⣿⠀⠀⢀⣶⣄⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣼⠟⠀⡾⢿⣄⠈⠻⣿⠉⠻⣷
⠀⠀⠀⢀⣿⣿⣥⡄⡿⠀⠙⠻⠿⠶⢶⣶⣦⣤⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠐⠁⠀⠙⢷⣄⡀⠁⠀⠙
⠀⠀⢰⡟⢸⣿⣿⣥⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣤⣀
⠀⠀⢸⣇⠀⠈⠛⢿⠀⠙⠛⠿⢶⠶⣤⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣠⠞⠁⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣶⣦⣤⣀⠐⠶
⠀⠀⠀⠙⢷⣤⣀⢀⣵⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢁⣠⡼⠟⠁⠀⠀⠀⠀⠈⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇
⠀⠀⠀⠀⣼⣿⣿⡟⢿⣿⣶⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣾⡟⠋⠠⠄⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣾⣿⣿⣿⣿⣶⣧⣼⡿⠏⠉⠙⠛⠛⠛⣿⡟⠻⢿⣇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠠⣶⠿⠿⠻⣿⠛⠛⠋⢹⡿⠁⠀⠀⠀⠀⢀⣴⠟⠃⣶⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⣽⣀⣸⣠⣿⠇⠀⠀⠀⠀⢀⣾⠏⠀⠀⠘⠀⢸⡇⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠀⠀⢿⡇
⠀⠀⠐⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠙⢿⠆⠀⠀⠀⠀⢀⡀⠀⢸⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢻⠃
⠀⠀⠀⠈⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⣰⡇⠶⠂⠉⠉⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠸⣿⣿⣿⣧⡄⢀⣀⠀⣸⣿⣷⣦⡄⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢻⣿⠟⠀⠀⠉⣉⠰⣿⣀⣼⠄⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠸⣿⠆⠀⠀⠀⠉⢾⡟⠛⣿⠄⠀⠀⠀⠀⠀⠀⠀⠺⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢸⣆⢠⣤⣤⣤⣴⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠏⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⡿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣹⠗⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢻⠟⣇⠀⠆⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠹⣷⠞⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢹⣤⠀⢀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡘⠀⠘⠿⣆⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣥⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⠋⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡌⠀⠘⠀⠀⠹⣧⡀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢠⡿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⣠⠀⠀⠀⠙⣷⡀⠀⠀⠀⠸⠀⢠⠶⣄⠀⠀⣿⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣽⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⠀⠀⠀⠀⠈⢻⣄⠀⠀⢀⡾⠃⠀⠀⠀⢀⡻⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠳⣾⢠⣦⠀⠀⠀⠀⠙⢧⡀⠉⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⠁⢹⡆⠀⣿⣏⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⢧⣤⣀⡸⠏⣉⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢸⡟
⠀⠀⢀⡀⠠⠶⠶⠋⠉⠀⠀⠈⠿⠞⠋⢉⡟⠀⠀⠀⠀⠀⠀⣠⣜⠁⠀⠀⠙⢁⣾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⣼⠇
⢀⣶⠀⠀⠀⠀⠀⠀⣤⢤⣄⣀⣀⣀⣤⡿⠃⠀⠀⠀⠀⣀⠚⠉⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⢠
⠈⠛⠿⠶⣶⣶⣶⣶⠿⠞⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣶⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠁⠈⠋⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠚⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣋⣈⣋⣿⣆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣈⢡⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣯⣭⢻⣈⡿⣯⣿⡏⠀⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⡟⢻⣷⡉⠛⢛⣶⠟⣿⠃⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠿⠿⠿⣏⡞⠁⠀⠀⣀⣤⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠰⣾⡿⣿⣿⡟⠛⠛⠉⠒⣤⠉⠛⠻⢿⣿⣿⣿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢠⡟⠀⠀⢹⣿⣿⠀⠀⠀⠀⣿⡀⠀⠀⢸⣿⣟⡡⠂⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣼⣿⣧⣴⠃⢹⡟⠀⠀⢀⣼⡿⢧⡀⠀⠘⣿⠛⢧⣀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠏⠀⠈⠻⠋⠀⢸⠆⢀⣤⣞⠋⡉⠈⠛⣦⡀⠀⠀⢀⣹⣿⠿⢦⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣀⠀⣠⠞⠀⠀⠀⣸⣿⣟⠉⠀⢀⣀⠈⢿⣿⠃⣰⠋⠉⠁⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣾⣿⣧⠁⠀⠀⠀⢠⣼⣿⣖⠯⡀⢰⠍⠀⠘⣿⠒⠀⢠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣧⠀⢠⣦⡿⣦⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⢙⣻⠛⠛⠿⠇⢠⡟⣧⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣈⣀⣧⣤⣴⣷⣦⢸⣦⣼⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢘⡛⠛⠉⣴⣏⠈⢹⡆⢹⡆⠠⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣄⠙⠈⢀⣿⢨⡏⠈⣟⣹⣇⠙⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡌⠘⠓⢠⠏⣿⢸⡇⠀⣟⠸⢋⡁⠾⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠾⠙⠋⢁⠟⠀⡟⢸⡇⠀⣿⢀⠈⣷⠒⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠰⠒⠀⠀⠀⣷⠘⠋⠛⠷⢈⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⠀⠀⠀⢸⠧⣀⣀⣠⡄⢸⠀⠿⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⢸⠀⠉⠉⠁⠀⢸⠀⠀⡃⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡇⠀⠀⠀⢸⣀⡀⣀⣀⣀⣸⠀⠀⠍⠁⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠠⠍⠀⠀⠀⠀⠀⠉⢹⣿⠏⠉⠀⠀⠀⠘⠀⠀⠀⢹⣿⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⣦⣠⣴⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠟⠛⠛⠛⠿⢿⣿⣿⣿⣶⣦⣤⣤⣤⣄⣀⣸⠿⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡀⠀⠈⠋⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠩⣵⣖⣦⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢷⣧⣀⠀⠀⠀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣶⣬⡿⢿⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢦⣾⡴⠛⠿⠿⠀⠸⠿⣦⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠹⡄⢀⣿⠀⠀⠀⠀⠀⠀⠀⠠⣼⣧⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣠⡽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠳⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠟⢉⣶⣿⣿⣿⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⣱⣿⣿⠛⠉⢹⣿⣿⣗⣶⡖⠀⠀⠀⠀⣤⣾⣿⣿⣿⣀
⠀⠀⠀⠀⠀⢸⡏⠀⠐⠛⠛⠛⠉⠀⠀⠀⢻⣿⣿⢻⣷⣄⢀⣠⣾⣿⣿⣽⣿⠁⠹⢿⣧
⠀⠀⠀⠀⠀⠈⢻⣷⣶⣤⣤⣴⠞⠲⣦⡀⠀⠛⣿⣉⣿⣿⣻⣿⣿⡿⠋⠀⣿⣵⣿⢟⣱⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣶⣶⣄⡙⢦⡎⠁⠀⠉⢉⣉⠉⣿⢷⡄⠀⠈⠻⡿⣫⣵⡦⢭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢻⣿⡾⣇⠀⠀⠀⠀⠀⠀⢿⡚⣿⡀⠀⠀⠺⣟⢡⣄⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡶⠀⠀⠀⠀⠀⣦⢸⡇⢻⣷⡀⠀⠀⠈⠃⢌⠣⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠹⠸⡿⠙⣿⣇⠀⠀⠀⠀⠀⠑⣍⠓⢀⡈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢿⠀⠀⠀⠀⠀⠀⠀⠶⠆⣿⡖⢻⣿⣧⠀⠀⠀⠀⠀⠀⠁⠢⠌⠀⠀⣀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⣤⣤⣤⣤⣤⣴⣤⣶⣶⣿⣷⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢯⡛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⢻⣿⡿⠀⠙⢻⣿⣿⡏⠀⠻⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⠀⣸⣿⡇⠀⠀⠀⢻⣧⠀⠀⠀⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡀⢤⣼⣿⠀⠀⠀⠀⠀⢻⣇⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠉⢿⡇⠀⠀⠀⠀⠀⠀⢻⣶⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣴⣿⡇⠀⠀⠀⠀⠀⠀⠈⣟⣀⣀⣀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠻⣿⡾⠀⠀⠀⠀⠀⠀⠀⠿⠇⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠛⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⡀⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠃⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⡿⡴⠂⠀⠀⠀⠀⠀⠰⠀⠀⡠⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⣰⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⢠⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠎⠁⢠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠤⠦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠫⠙⠉⠉⠀⠀⠀⠈⠀⠀⠀⠈⠙⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠛⠳⠁⠀⠀⠀⠀⡀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣶⣃⣀⠀⠀⣀⣠⣎⠀⠀⠀⠀⡱⠄⠀⠤⡀⠀⠠⡀⠀⠉⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠟⠈⢛⠟⠣⡴⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠒⠂⡐⢀⣘⠈⠙⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣻⠀⢀⡎⠀⠀⢸⡇⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠢⡮⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⣉⣄⠘⣇⡀⠰⣾⣿⣦⣀⣳⣄⢈⣿⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠅⠀⢈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣶⣶⣅⣄⣀⣀⡀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣮⡻⣿⡿⠻⣯⣽⣟⣻⣿⣿⣿⣿⠁⠀⠀⠀⠀⣛⣿⣿⣿⣿⣯⣭⣭⣿⡿⣛⣴⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠹⣿⣿⣿⣮⣹⣿⠿⢿⣿⣿⣿⡖⢤⠴⠒⠢⣼⡟⣻⣿⠿⢿⣯⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠋⠙⡿⢿⣿⣿⣆⠀⠹⠉⣿⡇⠀⠀⠀⢠⠉⢠⡿⠁⣠⣾⣿⡿⣋⣵⠟⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⡿⢷⣝⠳⡈⠻⣦⠀⢀⣿⡇⠀⠀⠀⢸⡀⠘⠁⣴⣿⠟⣡⠞⡛⠛⠀⠀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⠿⣷⣬⡑⠮⡂⠈⠷⢸⣿⡇⠀⠀⠀⠈⡇⢀⡼⠋⠠⣪⠥⠬⣡⣤⣤⣤⣤⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠁⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⢀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣾⣭⣭⣁⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠀⠀⡀⢰⣿⣿⣿⣿⣄⠋⠁⡤⠄⠊⠃⠄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠠⣀⣀⡀⠀⠻⠟⠉⠉⠙⢿⣷⣿⣤⣄⢀⣀⣄⣡⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢾⣥⣤⣤⣤⣤⣤⡽⠿⠿⠧⣤⣤⣤⡄⠈⠉⠀⠉⡙⢻⣿⣤⣬⣭⣽⣿⣶⣶⣶⣾⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣟⣹⢛⣟⢋⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⢰⡤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⢐⠀⠖⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⡿⢶⡶⢶⣿⡧⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣗⠾⢾⣶⣉⣿⣿⠁⠀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠳⣼⣿⣶⣤⣾⡿⣸⠇⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣠⣦⣶⣶⣶⣶⣾⠯⣍⣩⠽⢿⣦⣤⣴⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⠉⠙⢻⣿⣷⠀⠀⠀⠀⣿⠀⠀⠀⣽⣿⡿⢋⠜⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⣷⣦⣀⡖⢻⣿⠀⠀⠀⣰⣿⡄⠀⠀⢻⣿⢿⡁⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡴⠋⠙⢻⡟⠁⢸⡅⠀⣀⡴⠿⠓⠹⠦⣀⠀⠃⠀⠙⣷⣿⣆⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⡴⠆⠀⠀⢷⣾⡿⢃⠄⠀⢀⠰⣿⡶⠁⡰⠋⠛⠁⢈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⠘⠀⠀⠀⢠⣽⣿⡿⣎⠈⢥⠛⠀⢹⣧⠔⠁⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣹⠋⠀⠀⠀⠀⢰⣿⣿⣿⣾⣷⡎⠀⢀⣈⠿⣀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠈⠙⢺⠟⠻⣿⠅⢠⡿⡏⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣐⢃⣇⣠⣤⣤⣦⢸⣄⣹⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢘⡻⠛⠋⣥⣏⡍⢻⡎⣿⡌⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣎⠙⠁⢀⣿⢸⠏⠘⣗⣿⡌⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⣼⢉⠛⢠⡏⣿⢸⡇⢈⡟⠺⢉⡀⠏⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠙⠋⠁⠞⠀⡏⢸⡇⢘⡿⠄⢸⡗⠺⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠡⠂⠀⠀⠀⣷⠈⠉⠙⠃⠀⠀⡈⠀⢢⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣠⠀⠐⠀⢰⠣⣌⣀⣠⠆⡐⠀⢷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡀⠀⠀⠀⢸⠀⠀⠈⠀⠀⢰⠀⠈⣁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠸⣤⣀⣠⣠⣄⡼⠀⠠⡅⠀⠀⢀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⠀⠆⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠐⠀⠀⠀⣿⣿⣿⣿⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢐⣷⣶⣾⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠉⠉⠙⠛⢿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣼⣓⣀⣀⣀⣀⣠⣼⣿⣿⣿⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣟⡿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⢻⣿⣿⣿⣸⣿⢿⣿⢣⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣥⣶⣾⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠈⢿⡿⠿⠇⡿⠋⠉⠙⠻⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⣛⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠈⣿⣶⣿⠿⠿⢛⣛⣭⣭⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠶⢦⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⠀⠀⠀⠀⢹⣆⣀⣀⣤⣄⣀⣴⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣦⣤⣤⣽⣿⣿⣶⣤⣤⣴⣿⣿⡿⠿⠛⣿⡟⠛⢉⣠⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣦⣤⣾⣿⢹⠉⢹⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⡁⠀⠀⠀⠀⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⠃⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⡿⠀⠀⠀⠀⢸⡇⣿⣿⣯⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡆⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⣀⣤⣄⣀⡀
⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠁⠀⠀⠸⠇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⢦⣄⠀⠀⣼⠀⠀⠀⠈⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢤⣠⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⣿⣵⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢂⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣀⠀⠀⠀⢀⠐⠢⣄⠀⠠⠈⠠⣄⠀⠀⠀⠊⠁⣾⡏⠀⠀⠀⠀⢠⡄⢀⣀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣈⡀⠀⠈⠙⠀⠀⠀⠙⠃⡀⠀⠀⣸⡿⠂⠀⠴⠃⠀⠹⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠂⠂⠉⠉⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠠⠶⣎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠉⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣼⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠉⠉⠁⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⢶⣯⡴⠇⠀⠀⠤⠤⣿⣟⣠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠒⢄⡈⠳⣌⢀⠠⣼⠓⠀⢀⣿⠃⠀⠀⠀⠘⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢳⡄⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠈⠓⠀⠀⢹⣿⡟⠛⠛⠛⠓⠒⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⡀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣷⣿⠿⠆⠀⠀⠀⣴⣿⡷⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣿⡟⠁⠀⠈⡄⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣾⣿⠇⠀⠀⠀⠹⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


⣿⠿⠿⠿⡿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣟⣻⣟⣿⣿⣿⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⢀⠠⠐⠀⢀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⠃⠀⡔⠁⠀⠀⠀⠀⠑⣄⠀⠻⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢁⣀⣼⣥⠀⠀⠀⠀⠀⢠⣾⣆⣀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⠟⣿⣿⡗⡀⠀⠀⠀⠀⡸⣿⣿⣟⠿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⠀⠀⠀⣿⣿⣇⣧⠀⠀⠀⢠⣇⣿⣿⡇⠀⠀⠈⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⡀⠀⠀⠀⣻⣿⣿⡟⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⡀⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠋⠷⣶⣶⣿⣿⠿⠟⠁⠀⠀⠀⠀⠙⠿⠿⣿⣿⣶⡶⠏⢻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢀⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣄⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠈⠿⣿⠿⢿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⠿⢿⣿⠏⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣤⡶⠶⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⠶⣦⢄⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠰⠠⠆⢰⣾⣍⠻⡇⠀⠀⠀⠀⡿⢫⣽⣶⠀⠦⠰⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠐⠀⠄⢀⣿⣿⡇⠃⠀⠀⠀⠀⡧⣿⣿⣇⠀⠆⠐⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⢈⣿⣿⡇⠀⠀⠀⠀⠀⡇⣿⣿⣏⠠⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣤⠈⣿⣿⣷⠀⠀⠀⠀⠀⢨⣿⣿⡏⢠⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡀⠘⢀⡄⠀⣿⣿⡟⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⡄⠘⠀⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠖⢀⣀⣼⣿⣿⡇⣄⠀⠀⠀⠀⡀⣿⣿⣿⣄⣀⠐⠆⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠂⢾⣿⣿⣿⠿⣡⡇⠀⠀⠀⠀⣧⡹⢿⣿⣿⢿⠆⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢷⡏⣴⣾⠋⠀⠀⠀⠀⠀⠈⣻⣶⡍⣷⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠦⣄⠀⣀⡤⠾⠟⠛⠁


⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡃⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⠀⠐⠃⡀
⠀⢀⣀⠀⠐⣬⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠐⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⣠⠏⠀⠀⠀⢀⠴⠀⠀⣁⠈⠀⠀⠀⢀⠀⠀⡴⠋
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⢀⣀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⢀⡈⢉⠀⠀⠉⡠⠆⠀⠃
⠀⠀⠀⣀⠀⠀⠀⠋⠀⠈⠛⠂⠴⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠃⠀⠀⡄
⠀⠀⠂⠌⠀⠀⣀⢀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⡒⠤⠀⠀⢸⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⡀⢀⡠⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠄⠄⠀⠀⠀⠀⡀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⠀⢀⠌⠀⠠⠒⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠱⠄⠀⠣⡀⠐⠁⠿⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠐⠀⣀⠀⡀⠀⠀⠀⠀⠋⠒⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⡴⠒⠀⠀⠀⠀⠀⠐⡌⠓⠀⠃⠀⠀⠀⠀⠀⠁⠂⠀⠀⠐⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⡄⠐⠀⣠⠞
⠀⠀⠤⠀⠀⠓⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣴⢟⣉⡁⢀⠀⡀⠀⠀⠀⠀⣉⡡⠤⠔⠋
⠀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠷⠀⠀⠀⢀⡤⠀⡆⠀⠔⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠒⠋⠁⠐⢀⣡⣌⠢⣀⡤⠟
⠀⠉⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠋⠉⠉⠈⠉⠈⡧⣈⠂
⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⣠⡄⠀⠹⡀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⡰⠃⠀⠀⠀⠄⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⡀⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠢⢌⠙⢤⠀⠀⠀⠀⠀⠁⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⢇⠀⠀⠀⠀⢤⣀⠀⠀⠀⣠⠂⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠒⠒⠀⠀⠀⠀⠀⠐⢄⠀⠈⠀⠀⠀⠀⠀⢀⠀⡸⠁⠀⠀⠀⠀⠈⠗⢠⡎⠀⠀⠀⡀⠀⠀⠀⠀⠈⡦⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡤⠒⢀⠀⠄⢀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢀⠤⠀⠀⠀⠀⠃⠀⠀⠠⠀⠸
⠀⠀⠀⠀⠁⠊⠀⠀⠀⠈⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠐⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⢒⣂⣀⣀⣐⠒⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡦⢖⣉⠤⠒⠊⠉⠁⠀⠀⠀⠀⠈⠉⠑⠂⠬⣉⡲⣔⢤⣄
⠀⢰⡖⢀⠀⠀⠀⠀⡨⣋⣠⡍⡐⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⢹⣄⡙⢄⠀⠀⠀⠀⠄⣶⡄
⢠⣾⠻⡀⠑⢄⡠⠊⡠⠊⠀⠀⠀⠀⠀⠀⠀⣠⣶⠻⢉⠉⡛⣶⡀⠀⠀⠀⠀⠀⠀⢈⠑⢄⠑⢄⡠⠊⢠⠟⣷
⢠⡽⠄⠈⢆⠈⠣⡎⠀⠀⡇⠀⠰⡀⠀⠀⠀⡿⠛⠛⠾⠿⠛⠓⣿⠀⠀⠀⢀⠃⠀⢸⠀⠀⢣⠌⠀⡔⠁⡠⢯
⠘⢇⠈⠢⡀⠑⢄⠈⢂⠀⡀⠆⠀⢱⠀⠀⢰⡇⠐⠶⠀⠀⠶⠀⢸⠆⠀⢀⡇⠀⠀⢨⠀⠔⠁⡠⠊⢀⠔⠀⡼
⠀⢠⡳⣀⠈⢦⡈⠣⣄⣑⡵⠀⠀⠀⠑⢴⣾⣷⡦⠠⠀⠠⠄⣶⣾⣷⠤⠁⠀⠀⠀⣦⣊⣤⠞⢀⠔⠁⣠⢾⠄
⠀⠀⠙⠉⠳⠶⠛⠟⠋⠊⠉⡖⢤⡄⠀⠀⠀⠻⠧⡘⣟⣻⢃⠼⠏⠀⠀⠀⢠⠤⠐⠉⠃⠛⠻⠛⠶⠞⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠙⢦⡄⠀⠀⠀⠘⠋⠙⠡⠂⠀⠀⢠⡴⠃⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠘⠇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡸⢁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠉⠄⠀⠀⠀⣸⣇⠀⠀⠀⠐⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠐⠀⠁⠀⢉⠉⠀⠉⠀⠂⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⠃⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠁⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⢿⣶⡤⢤⡤⢤⣶⡿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠁⠢⢄⡘⢃⡠⠐⠙⠄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱⡏⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠆⠳⡀⠀⠀⠄⠀⠈

⠀⠀⠀⠀⢀⡀⠀⡀⠀⣴⠖⠀⢠⡄⠀⠀⢠⡆⠀⠀⣲⡆⠀⠀⠀⢰⣯⡀⣰⣿⠂⢀⣤⡀⠀⡏
⠀⠀⠐⠒⠉⠄⠱⢅⢀⠞⠁⠈⢻⠇⠀⠀⢺⡧⠾⢻⡯⠽⢶⣶⡦⠹⢿⣧⢼⡉⠐⠺⡇⢀⠀⣅
⠀⠀⠀⠸⠇⠀⡀⠀⠀⠀⣀⡤⠛⠒⠒⠒⡛⣡⠀⠘⢡⣄⡤⢤⡤⢤⠰⢃⣠⠄⠋⠘⣀⠚⢿⣿⠖
⠀⠀⠀⡀⠀⠸⠃⣠⠴⠚⠀⠀⢠⣠⣠⡘⠣⠉⣔⠉⠨⠿⢴⠃⠼⢭⢧⡌⣑⡒⢦⠀⠽⢠⢌⣴⣤⠗
⠀⠀⠀⠃⠀⣠⠜⠁⢠⡀⣳⡴⠟⡫⠀⠊⠈⠈⠇⠣⠀⠀⢊⣉⣀⣬⣤⡀⣀⣠⠀⢂⡀⣈⣿⠁
⠀⠀⠀⢀⡠⠀⠀⢲⣔⠷⠧⠀⠈⠂⡠⡼⠚⣳⣤⠋⡉⠹⡟⠒⡂⣦⢀⡌⠁⠬⠡⠜⠃⠀⠸⠆
⠀⠀⠀⠸⠁⠀⠒⡦⢣⡆⢨⢀⠔⠋⠇⠉⠀⠈⠓⠓⠀⠀⠏⡇⢡⣏⠸⣁⣶⣠⠆⠀⢡⣀⢰⠄
⠀⠀⠀⡄⠀⠈⠿⢄⠁⠐⡶⢦⡝⡦⠚⠡⠷⣀⠈⣀⠖⣾⢺⡅⣟⡫⢖⡵⡜⠹⢠⡘⣄⠡⢨⡀
⠀⠀⠀⠆⢘⣿⠷⡦⣡⣷⠖⠀⡖⢱⠀⠀⠀⢑⠖⣶⣈⣻⡖⠶⣹⣿⢏⢬⠄⠁⠈⡃⠉⢠⡘⡀
⠀⠀⠀⣧⡌⠁⢈⣄⠋⡆⠈⡓⣀⣠⠰⣦⢤⣞⢲⡆⢿⣿⣿⢺⡃⣿⢁⡤⣌⠐⠁⠐⠧⢀⣿⠅
⠀⠀⠀⠘⠀⠀⠿⣛⣷⡾⠶⣬⣥⣵⡿⢼⠗⠚⢷⡑⢶⠯⠽⠙⣼⢏⡿⢼⡛⣇⢀⠀⢾⣿⣿⡀
⠀⠀⠀⠀⠀⠶⠀⠛⠙⡁⠉⣉⠁⠀⠀⠀⠘⠶⠂⢉⠗⡳⢞⠋⠹⡌⠳⠞⢣⡿⠨⠀⢨⣷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠜⠀⠓⠁⠱⢤⣗⣠⠀⠂⢅⠀⡙⠀⠑⢢⡈⠉⡟⠡⠄⡂⢀⠟⢰⢸⡂
⠀⠀⠀⠀⣧⠀⠀⠀⠐⠲⡀⠂⣤⣶⣶⡌⢿⣿⣿⠈⠀⢄⡒⠠⢈⡘⡃⠽⣶⠶⠟⠈⠠⠃⢸⠃
⠀⠀⠀⠠⢿⠄⠀⠀⠀⠀⠈⣲⣽⣿⣿⠇⠈⣿⣧⠖⠦⠆⠒⣉⣠⡖⣒⢶⣂⣠⡶⠄⣉⡺⢬⡇
⠀⠀⠀⠀⠀⢀⣤⣤⣶⠖⠈⣰⡟⡋⢁⡺⣀⣌⠀⢁⣴⣿⣿⢿⡿⠭⣈⣩⣽⡷⠒⠚⠉⠉⢻
⠀⠀⠀⣀⢬⣷⠷⠋⣡⣴⠞⢁⠉⢸⡆⠙⠢⣤⠴⢉⣼⡟⢡⣌⡴⠞⠡⠐⠛⢿⡦⢴⠄⠀⢸
⠀⠀⠀⠀⠀⠉⠀⡺⠹⠏⠀⠈⠀⢏⠀⠂⡄⡪⢶⣻⢟⣡⠶⣯⡀⠀⠁⠀⠀⠾⣿⡎⢀⡅⢸
⠀⠀⠀⠀⠀⠀⠈⠀⢀⢶⠀⠀⠀⠸⠤⣾⡍⣀⡾⣡⢏⡅⠀⠈⠹⣦⠀⠀⠀⠴⢿⣧⣟⢁⣼
⠀⠀⠀⠀⠀⣄⠀⠈⡈⠉⠀⠀⠀⠀⠠⠘⡻⠀⢳⡏⠀⠀⠐⠀⠀⠈⠻⣤⣮⠶⠛⣿⣭⡄⢸
⠀⠀⠀⠀⣀⠚⠀⠰⠁⠈⠰⠀⠀⠀⠀⠂⢉⣾⠋⠙⠳⢦⣶⢶⣄⣂⣼⢟⣧⣀⢈⡙⣿⠄⢸⠄
⠀⠀⠀⠀⡇⠠⣌⠠⠀⠇⠀⠀⣜⠀⠀⣇⢩⣇⡐⠈⠀⠘⠻⢻⣿⣿⣇⡸⠛⢙⣇⠀⢻⣶⣾⡀
⠀⠀⠀⠀⠁⠐⡀⠀⠁⠀⠠⡄⠔⠀⠀⠰⢾⣿⣻⣧⡠⡀⢰⣾⡿⣡⠈⠙⠶⣏⡉⢻⡾⣿⢸
⠀⠀⠀⠀⠇⠀⠐⠁⠠⠀⡇⣿⠀⠀⠛⠂⣟⠉⠿⢿⠙⢿⡿⡟⣷⣶⠷⢦⣴⣷⣏⡓⢯⣿⣿⢀⡀
⠀⠀⠀⠤⣂⠀⠈⢁⠀⠀⡀⢉⠸⠼⠥⡬⢥⡠⠤⣤⠠⡬⢤⡩⠭⠭⣄⣞⠩⡤⠭⠭⠭⠭⣿⢿⠷⠖
⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⣨⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠟⠀⠈⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆⠀⠀⠀⠲⠀⣇⣀⣤⠤⠶⢶⡶⠚⠉⠉⣇⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠏⢡⠞⠃⢠⡀⠀⢤⡶⠋⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠎⠉⠣⠀⠀⠘⢦⠀⠀⠙⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢀⠇⠀⠀⠀⠀⢠⠂⠀⡄⠀⢀⣄⣀⣸⠇
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⢠⡏⠀⢸⣃⡴⠛⠉⠉
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡾⠁⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⡙⠉⠉⠙⠛⠓⠶⢒⡶⢄⡀⠀⠠⠏⠀⠀⠀⢀⡾⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣤⣤⣤⣤⡄⠀⠀⠢⢍⡂⣤⠀⠀⠀⣠⡾⠁⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⠔⠁⠐⡁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣮⣅⡀⠀⠀⠀⠀⢀⣴⠊⣠⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁⠀⠈⠛⠿⣿⣶⣶⣶⣿⣿⣿⣿⣯⣉⡙⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⠻⣷⠀⠀⠀⠀⠀⢸⣿⠛⠿⠛⠛⠛⠛⠿⢿⣿⣿⣮⣙⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠁⠀⠀⠘⣆⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠙⠛⠳⠎⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀⣾⣧⠀⠀⠀⠀⣠⣤⣤⣴⣧⣤⣤⣀⣀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋⠙⢻⣿⣤⣴⠶⠛⠉⠉⢿⣛⠒⠶⠶⠶⠦⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⢠⡾⠛⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢀⡿⠀⠀⠀⠀⢠⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣸⣇⡔⠀⢰⡆⢸⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⣿⡟⠀⠀⢸⣇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠓⠀⠀⠀⡆⠀⡇
⠀⠀⣼⠏⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢠⡇⢀⡇
⠀⠀⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⣄⣸⡇⢸⡧⢄
⠀⣼⣁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠒⠚⠃
⢼⣋⣉⣉⣓⣦⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡔⣍⣁⣀⣀⣈⣹⡵⣄⣤⣤⣤⣤⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢸⣿⣿⣿⡟⣳⣾⣤⣠⡈⢀⠀⠀⣀⣱⣧⣼⢻⣿⣿⣿⡆⠈⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⢠⡟⠀⣹⢟⣣⣹⣏⣘⣿⣯⡉⢹⡄⣿⣿⣿⣧⠀⠀⠀⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⢤⣿⣿⣿⣿⡶⣟⣽⣿⡿⠟⢛⡛⠿⢿⣿⣿⣿⣇⣿⣿⣿⣿⡄⠠⢀⣯⣉⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸⣿⣿⣿⡇⢠⣿⡟⠁⠐⠀⢸⠃⠀⠂⠉⢿⣷⡌⢸⣿⣿⣿⡇⠀⠀⠀⠸⡗⢻⣿⣷⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠀⠛⠀⠀⢸⡀⠀⠀⠚⠈⢿⣷⢸⣿⣿⣿⡇⠀⠀⠀⢀⠟⣫⡿⠿⢿⡃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠱⢤⠤⢼⣿⣿⣿⡿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣾⣿⣿⣿⡷⢤⡤⣤⣼⡼⡿⠁⠰⣟⣙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇⢻⣿⡄⠿⠀⡀⠀⠀⢀⡀⠚⣰⣿⠇⣿⣿⣿⣿⠀⠀⠀⢸⣩⠼⠃⠠⣖⠻⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⡄⠻⣿⣶⣄⣁⠐⠂⣀⣥⣾⣿⠋⣸⣿⣿⣿⠇⠀⠀⠀⡘⠁⠀⣴⠲⣎⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠹⠅⠛⠻⢿⡿⠿⠿⣛⡫⣹⠙⠛⠛⠛⠛⠉⠉⢉⡉⠁⣴⠶⡜⠓⢾⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣾⣽⢷⡞⣷⢠⡶⣦⢠⢶⡄⣿⢹⡆⠹⢦⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠼⣿⡁⢸⣧⠟⠸⣠⡇⢙⣿⣇⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠚⠋⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠂⠀⣀⠀⠀⡠⠴⠂⠀⠀⠀⠘⢳⠆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣠⠀⠠⠊⢃⣠⠀⣀⣀⢤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⡄⠉⠁⠀⠀⡄
⠘⣭⠁⠀⠀⢀⠤⠀⠀⠀⠀⡠⠂⠀⣀⠵⠛⢿⡁⢀⠄⠁⠸⠋⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠠⠁⢈⠁⠀⠤⡇
⠘⠫⠀⠀⠀⠀⠀⠀⠤⠤⢀⢰⠗⠂⠀⠀⠀⣀⠄⠁⠀⢀⠄⠐⢡⠀⠀⠀⢀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⢌⠀⠀⠠⠈⠀⠀⠄⠄⠀⠀⠀⠴⠖⠋⠀⠀⠀⣀⠉⡔⠃⠹⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠁⡐⠀⠄⢀⠀⠠⠉
⠀⠀⠈⠐⠒⠒⠓⠒⣚⠉⡇⠀⠐⢠⡈⠁⠀⠀⠀⠀⡤⠖⢀⡀⢠⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠠⢀⣉⡥⣦⠠⠔⢛⠀⡆⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣠⣾⣿⣽⣽⣯⣽⣿⣿⣿⡿⠋⠉⠉⠉⠉⡉⠉
⠀⠀⠀⠀⠀⠀⢉⣍⠀⠀⡞⡇⣠⠎⢀⡶⠶⠶⠶⠶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⡄⠀⠄⠠⢀⡤⠀⡁⠀⠓⠈⢀⡜⢸⠀⠄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⢀⢲⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠤
⠀⠙⠀⣠⠀⠈⠀⠀⠛⠂⠀⡄⠊⠀⡏⠀⠀⠐⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⢋⠘⡀⠀⠀⢀⠄⠠⠀⠀⠀⠀⠀⡤⠀⠂
⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⡆⠀⢸⠁⡆⠀⠀⠀⠀⠄⠀⢠⠀⠀⠀⠀⠀⠐⠃⠀⠀⠐⢋⠠⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠂⣠⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠈⠀⠇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⡰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣤⣮⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣝⣩⣭⣹⣍⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⢠⠆⠀⡆⢣⠀⠐⣄⠀⠀⠈⣷⠀⠀⠈⣦⠀⢠⠇⠀⠀⠀⠀⠀⡌⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠘⠄⡈⠀⠀⣧⠀⢀⢀⠈⡄⢠⠀⢱⠀⢠⡄⠈⢁⣼⠀⠀⠀⠄⡀⣸⠏⡄
⠀⠀⠀⠀⠐⠒⢾⣇⠀⠀⠹⣇⢿⣏⠀⠸⠛⠀⢸⡆⠘⠇⠀⢸⡇⠀⣆⣰⠏⣴⠋⠀⠇⢠⠴⠂
⠀⠀⠀⠀⠀⠀⠻⠉⠉⠲⣄⡛⢘⠃⢠⣤⣰⣦⣼⣧⣶⣤⢠⡟⣤⣀⣍⣩⣼⢇⠀⢒⠀⢀⡴⠁
⠀⠀⠀⠈⠲⠦⡄⣄⣿⠀⠈⣷⣿⠟⠊⠙⠈⢡⠈⠁⢠⠁⠀⠁⠉⢻⡿⣿⡃⢸⣟⠃⢠⡞⠀⠄
⠀⠀⠀⢀⡀⠀⠀⣿⣿⣷⣤⡉⠉⢁⠀⠀⢈⣤⣼⣲⣬⣄⣀⠀⠒⠈⡃⠈⢷⣬⢨⣴⡿⠉
⠀⠀⠀⠈⠻⡖⢤⣤⢻⠏⠸⡇⠷⢦⡆⠀⠈⠙⠛⠉⠋⠁⠀⢻⡓⢺⣷⣶⠛⡿⢳⡟⢀⡄
⠀⠀⠀⠀⠀⠈⣀⠉⣿⡔⠄⠺⠗⠋⠁⠐⢂⣴⣦⣰⣦⠀⠀⠈⠓⠺⠆⠻⠒⢻⡈⠇⠞⠀⣠⠆
⠀⠀⠀⠶⡖⠀⢼⣷⣜⡀⠈⣦⠉⠀⠀⠒⠀⠈⠉⠩⠄⠀⡤⠀⠀⠀⢸⠗⠀⢨⣇⣼⢶⡾⠛⠉
⠀⠀⠀⠠⠼⣄⣤⣿⡿⢃⠀⣀⣠⡤⢤⣤⡉⠰⢿⠾⠗⢀⣤⠔⢶⠶⢼⣀⠀⠈⢿⣷⠨⢄⡀
⠀⠀⠀⠀⠊⠁⠰⡌⡅⢠⡞⢋⡤⠶⠲⠦⣙⣾⠀⠀⣴⢊⠴⠶⠶⠶⠦⡍⣳⠀⢲⡟⢀⣦⢤⠂
⠀⠀⠀⠀⠉⠑⣄⠙⣦⣸⣡⡷⠶⠶⠖⢶⠟⡸⠃⠈⢫⡈⢟⡛⠓⠓⣳⣞⣦⣱⣿⡴⠋⠄
⠀⠀⠀⠀⠀⡀⠸⢿⡿⠓⠸⣿⣍⣛⣭⣱⠞⣁⣼⣷⣀⠙⢳⠶⢿⣯⠿⠟⠃⠀⠻⠟⢠⡀
⠀⠀⠀⠀⠀⠃⠀⠐⢄⠐⠀⠀⠉⠉⢈⠹⠉⣫⣿⣿⣯⡁⠀⠈⠀⠉⠀⠀⠀⠠⠊⠀⠀⡁
⠀⠀⠀⠀⠀⠰⡄⠀⠀⠱⣤⡴⢶⣄⢈⡰⢤⣿⣿⣿⣿⣿⢷⣦⡀⣾⡟⢳⡈⠀⠀⢀⡈⠁
⠀⠀⠀⠀⢤⣤⡰⣦⣤⡶⠄⣴⣎⣉⡽⠷⠁⣹⠏⠾⡛⣷⠀⠘⢫⣍⣷⠗⠙⠓⣶⡞⠛⣿⠟
⠀⠀⠀⠀⠀⠹⣖⡀⠋⠀⡀⣠⠿⢻⣿⠲⣤⣧⣤⣤⣤⢼⣤⣴⣿⣿⠙⢦⡁⡈⠀⠐⠶⡃
⠀⠀⠀⠀⠀⠀⡼⢂⣄⠀⡼⠃⢤⢸⣿⣦⣻⣅⣅⢹⡁⣸⣿⢿⣹⡧⢰⠈⢳⡄⣠⠌⠀⠁⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠙⣻⡁⠀⡴⠀⠟⢻⡿⣿⣠⣄⣇⠿⢴⢦⢱⠇⡜⠆⣀⡹⣇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⡄⠀⠀⡰⠇⠿⣆⠀⠐⣢⠛⠀⢈⡁⠻⡇⠈⠘⠘⣏⠛⠀⣠⠟⠉⠻⣦⠄⠀⠃
⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠈⠳⣄⢹⠇⠬⠅⢢⡶⢾⠐⠂⠀⣿⢋⡴⠁⠠⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡿⣀⠀⠀⠘⢦⠞⠀⠂⠀⣸⣧⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠁⠀⠀⠘⣆⠀⠀⠀⣠⠞⠀⠙⢦⡀⠀⢀⡀⠀⢀⡼⠃⠈⠳⣄⣀⣀⣰⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠈⢙⣛⠉⠁⠒⣀⠒⣚⠳⣦⣶⣷⣤⠞⣛⠈⣦⡙⢈⡉⠩⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠁⠚⠀⠕⠒⠞⠓⠈⢩⡾⠁⠘⠓⠒⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⠙⠳⡤⠶⠠⠀⡀⠀⠀⢀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠠⣼⣿⣓⠦⣄⡀⠀⢀⣀⡀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠳⣄⠉⠉⠀⠀⠉⠻⠋⠉⠩⠋⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠴⣚⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣾⠷⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⣥⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠳⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠁⠀⠀⠀⠀⠁⢐⡟⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠈⠛⠻⢯⣍⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⣻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⠀⢾⡛⠉⠋⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⢰⣴⣮⡀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡼⠃⢿⠹⣿⡏⠇⠙⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣠⢤⡖⠒⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠉⠁⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⠀⡤⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠲⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡏⢸⡄⠀⠀⠀⠀⢀⣄⠀⠀⠀⢘⡆⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠀⠀⠙⠶⣤⣤⠶⠋⠹⣄⣀⣠⡾⠁⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢨⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠊⠀⠀⠀⠀⠀⠀⠙⠙⠂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠋⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠀⠐⣂⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡖⠀⠀⠀⠀⠀⣼⡆⠀⠀⠰⡀⠁⠋⠀⢹⡵⠋⠀⠰⢀⠀⢀⡁⢀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡭⠁⠀⠃⠀⠈⠓⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠊⠰⠄⠀⠀⠀⠀⠊⣭⢤⡼⢃⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠸⣦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠛⠒⠒⢿⡣⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⡾⠛⠓⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡄⢸⣷⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⣀⡏⠉⠁⠐⠘⠀⠀⠀⣤⠠⠞⠃⠰⠆⢠⣤⣌⡶⡚⠀⡆⠀⢩⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣼⠃⠁⠀⢰⠆⡴⠃⠰⠦⠐⠤⠄⠀⠃⠀⣀⣀⣤⢅⡆⠇⢴⡀⢈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⠗⠀⢠⡖⠐⠺⢦⡀⡦⠤⢿⣷⣶⣶⣦⣭⣥⣴⣾⣿⣷⣆⣀⠘⣿⣯⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⣦⣒⠲⣽⣮⣻⣯⣙⣭⢙⠉⢻⣭⣿⣤⣯⣛⣛⡛⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠉⠀⣯⣽⠙⣈⣁⢉⡞⠂⠨⣻⣿⡃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢊⠉⠹⠻⡇⠳⢷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠉⡛⣻⡆⢀⠤⡆⢄⢟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⢸⠀⣘⡇⠸⡀⣴⣿⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣼⣿⡏⠗⢀⣀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠃⠿⠿⠁⠶⠌⠁⠀⠙⠋⢹⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠰⠀⣀⠀⠀⠉⠙⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⣲⣿⣠⣄⢲⣴⣤⣤⣦⣬⣿⣟⡾⣠⣤⣀⣰⣽⣿⣵⣆⣀⠀⠠⠤⠶⡤⣦⣤⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠤⠤⠬⠿⠿⠶⠌⠐⠂⠒⠚⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢦⡀⠀⠘⣶⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠤⢾⠋⠀⠀⣴⠃⠂⠠
⠀⠀⠀⠂⢀⣀⣀⣀⣀⣀⣘⠃⣀⣀⣈⠀⣠⠄⣀⣄⣠⣤⣍⡀⠀⠀⢠⣀⣀⡄⢩⣤⣤⣤⣠⣤⠀⠉
⠀⠀⠀⠀⠈⢯⡉⢉⣩⣭⣥⣀⢁⣍⠬⠀⣍⡇⡭⠁⠉⠉⢳⣀⠜⠃⡁⠀⠀⢠⣷⣤⡤⠀⣼⠇⠀⠄
⠀⠀⠀⠄⠀⠈⢷⡈⣿⣟⣽⠃⢊⣨⠊⡂⠘⠲⣇⠇⡯⡄⢸⡿⠿⢿⣧⠘⢆⠘⣿⣿⣇⣼⠃⠀⠀⠐
⠀⠀⠀⠀⠴⢶⠌⢁⡜⢋⠵⠴⡆⣩⢸⡗⠆⢹⡏⠒⣤⠇⢜⢧⠀⢱⢻⣶⡎⡰⡄⢈⣄⠩⠶⠶⠓
⠀⠀⠀⠀⠆⠀⠀⠀⢳⡌⠠⣦⡅⠆⢸⠃⠃⣼⠁⠠⣜⢀⣘⡘⠆⢸⡿⠜⢈⣷⢊⡾⠃⠀⠀⠀⠠
⠀⠀⠀⠐⢶⣦⠄⠀⠀⠛⡎⠃⢳⢆⠦⣔⠋⡅⢰⡇⢠⣂⢉⢹⣟⣉⠷⢀⠄⢉⣾⠁⠀⠀⠠⢴⠖
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⢟⣱⢎⣭⡼⢫⠖⢣⠮⠸⡄⡈⡻⠇⢈⡈⢳⣌⢃⠿⠁⠀⠠
⠀⠀⠀⠀⡄⠀⠀⠀⠂⠀⡄⢡⡾⢨⠇⠋⠀⡸⠀⠒⡃⠃⠀⠀⠈⠧⠀⢼⣿⣀
⠀⠀⠀⠀⠄⢰⠀⠀⡃⠀⡴⠋⠀⢚⣠⡴⠶⠷⡉⡭⠉⢉⣻⡟⠷⣶⣤⡨⢡⢿⡄⣀⠈⠀⠷⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠆⠀⣠⠥⢚⢛⣩⡄⢒⡶⠶⣶⣶⣿⣿⡦⢭⣝⠻⣰⣎⠱⣾⣆⠠⢘
⠀⠀⠀⠀⠀⢰⠀⡀⢁⣟⣋⣩⠉⠙⣛⡽⣛⣡⣶⣿⣿⣿⣾⡦⢄⣙⠒⢶⣤⠬⢽⡃⠘⠀⠙⠀⣀
⠀⠀⢀⣄⡀⠈⠀⠈⠸⣭⢔⣂⣈⣭⠴⠞⣯⢹⣿⣿⣿⣿⣿⡍⡳⠲⠤⠂⢙⣳⣦⡇⠀⠤⠀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠂⠀⠀⣸⠋⠑⠀⠀⠖⣠⡅⠺⡿⣿⣿⢻⡿⠃⠘⢰⡀⠆⠀⠉⠹⡓⠀⡀
⠀⠀⠀⠀⠐⠐⢀⠤⡒⡿⢠⡔⣶⣦⠀⣁⢾⠛⡏⣧⠿⠬⢴⣇⣶⢦⡈⣴⡏⣱⡆⡧⢰⡟⡀⢀
⠀⠀⠀⠀⠀⠐⠁⡈⠋⣧⢸⣧⢸⣽⠟⣱⣺⣭⡤⠄⠴⠴⠶⠸⣌⠳⣿⣿⡇⣿⢧⣧⡯⠺⡅⠈
⠀⠀⠀⠀⠀⠂⡶⣞⣷⣾⣧⣛⠀⣿⣿⠉⠉⠀⠁⠀⠸⠀⠄⠀⠈⠉⠙⣿⠃⣶⣿⣿⠻⠚⢳⢄⡀
⠀⠀⠀⠀⡠⠞⠐⢧⡞⠹⢍⠇⠀⠨⡀⠄⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⢸⡀⣹⡴⡲⠉⢔⡎⠂
⠀⠀⠀⠸⣇⡈⡓⣀⣹⣧⣿⣶⠀⢀⠀⡄⠒⠠⠴⠊⠐⠀⠐⠒⠀⠠⠐⣁⠎⡼⡿⣿⣷⣿⣷⠒⣩⡛
⠀⠀⠀⠛⣍⠷⣀⣨⡛⣉⣁⣴⣆⢀⣀⡄⣀⣐⣸⡛⠋⡟⢙⡛⣳⢶⣤⣁⣴⣷⣦⠈⣿⠛⢙⠃⡙⢁⣀
⠀⠀⠀⠀⢸⣸⣭⣹⣭⠤⢝⡛⠿⡿⢟⠛⠛⠻⢿⠇⠰⠁⠸⡿⠟⠛⢛⡛⠏⠉⠣⠄⠂⠶⢛⠁⣽⠆⠹
⠀⠀⠀⠀⠸⠁⢿⣉⣥⣤⣴⣶⣶⣶⣷⣾⠳⢦⣾⡇⠀⠀⣺⣧⠆⠲⡶⢰⣿⣵⣾⣿⣿⣶⣦⠹⡌⣆
⠀⠀⠀⠀⠰⢦⣀⡙⠛⠿⠟⠙⠻⣿⣿⣯⠳⠄⣈⡉⠉⠉⢉⣀⡤⠞⠁⣿⣿⣋⡹⠟⡋⢩⣯⡲⣶⡣⠁
⠀⠀⠀⢀⠀⢸⣿⣧⣿⣥⣾⣿⣿⣿⡿⠿⡆⠀⠀⣿⠉⠉⣿⠀⠀⠸⡶⢿⣿⣿⣿⣿⣱⣿⣿⣿⣿⡷⡅
⠀⠀⠀⠀⢣⠐⠈⠻⠿⣿⣿⣥⡆⠈⠠⡀⠀⠀⣰⣿⠅⠀⣿⣀⠀⠀⠉⢀⠈⠉⠩⣿⣿⠟⠿⡋⢃⡀⠁
⠀⠀⠀⠀⠀⠀⠂⠰⢤⠀⢀⢈⡤⠖⠀⠀⠻⣶⠏⢿⢀⣠⣟⠳⠴⠆⠀⠀⠀⠋⠥⠄⠶⠎⠐⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠚⠪⠕⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⡄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡶⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡯⣄⣼⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣾⣗⣲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠉⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⢠⡄⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⡀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⢨⣿⣾⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠆⣶⣶⣾⣷⣄⠀⠓⢯⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠿⠞⠘⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⠀⠀⠀⠀⢸⡇⢿⣿⣽⣿⣿⣗⣦⡀⠙⠂⠀⠲⣶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⡄
⠀⠀⠀⠀⠀⡀⢤⣿⡿⡇⢠⡄⠀⠀⠈⣿⣅⣽⣯⡟⢉⣯⣿⣿⣢⣀⠈⠲⣿⣫⡴⠀⠀⠩⠀⠀⠀⠀⠀⠀⠠⣿⠀⡇⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⣿⣦⣾⣵⣿⣿⣗⠋⠘⣛⡽⣿⣿⣱⣋⣵⣄⡈⠙⢿⡷⠁⣀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠰⠇⢠⣤
⠀⠀⣠⣾⠿⢃⠀⠀⣾⠸⠲⠿⣿⣿⢿⣿⣿⣿⡟⠿⠛⣴⣿⣿⣿⡿⣻⡏⣹⡶⣞⣷⣾⣿⠃⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⢾⠟⠁⠀⠀⢷⠞⣹⡗⠀⢀⣈⡿⢨⣱⣷⣽⡿⢟⣚⣵⣿⣿⣿⣷⣿⣵⣿⣿⠾⠿⣿⡏⠀⠐⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠘⣯⣟⣳⣢⡟⠻⣶⣿⣿⣯⣟⣿⢺⣯⡟⢿⣁⣿⣿⣿⣿⣿⣿⣦⡀⢈⠃⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⢿⠿⣤⣿⡿⠿⠟⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢢⡀⠀⠀⠋⡀⠀⣠⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣀⠀⠘⠿⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣯⣿⣯⣽⣿⠇⠀⠐⠂⠀⠊⠛⢿⣿⣿⣷⣄⡀
⠀⠀⠀⢶⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣾⠻⣿⣿⣿⣿⣿⡷⣦⡀⠀⠀⠀⠀⠀⠈⠛⢿⣏⣽⡆
⠀⠀⠐⣲⣆⡚⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡏⢿⠛⣻⡗⢾⣟⠳⣄⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠉⠀⠁⠀⠀⠀⣠⠞⠋⠟⡿⠉⠉⠻⠛⠛⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⡳⠷⡘⢌⣽⣦⡙⠇
⠀⠀⠀⢠⣄⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢿⣚⣿⣿⣿⣿⣿⣿⣿⣿⣾⠉⠿⣮⣻⣽⣷⣄⣀
⠀⠀⠀⠺⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⠀⠁
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢀⡀⠀⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣿⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡼⣻⢿⣤⣾⣿⠀⢹⣿⣿⣿⣿⣿⡻⢱⣾⡿⣿⣿⣟⢻⣿⣿⣿⣷⣤
⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⠾⣻⣿⣟⢀⣶⣿⠿⣿⣿⣯⣿⣟⠾⠛⢌⡻⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠐⠟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡿⣿⣦⣼⠿⣻⣿⠈⠀⣀⠀⠑⢤⣙⣿⣻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠀⠈⠀⠉⠒⠿⠃⠀⠴⠿⠶⠀⠈⠻⣿⡿⣏⠙⠿⣿⣿⣇⡀
⠀⠀⢀⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢗⠄⠙⠏⠙⢆⢀
⠀⠀⠀⠙⠛⠉⠁⠀⣾⢹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠉⢷⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠇⠀⠙⢂⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⠀⠹⣧⣤⣤⣀⡿⠲⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⣿⠟⠛⠉⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⣠⡔⠀⠀⠀⠀⠠⠶⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⣀⣠⡤⠴⠤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠟⠋⣭⣿⠶⠶⠶⣮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢰⠀⠀⢀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢄⡀⠀⠀⠘⢀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⡉⠚⢲⣾⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠄⠙⠚⠋⠴⠉⠿⠔⠋⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⣤⣀⣀⣀⡤⠖⡁⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠛⠃⢀⠴⠒⠀⠈⣯⠁⠙⠀⠀⠁⠀⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⢁⡄⠀⢀⠚⢁⢆⣻⡄⠘⠂⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⢎⠠⡄⠱⠎⣻⣆⠀⣽⠇⠀⠷⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⡀⠀⠐⠎⠀⠀⣿⣿⡟⠋⠀⠠⡇⠈⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⣄⡀⠠⠖⠛⠁⢻⣆⠀⠀⢀⡅⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⡁⠀⢀⠀⠀⠠⠄⠞⣠⠿⢆⠚⠣⠴⢶⠶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠘⠶⠀⠀⠀⣴⢦⣘⡏⠀⡈⠀⠀⠀⠀⣦⠀⠈⠓⢄⡀⢰⡞⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢴⠦⠜⠻⢇⠀⣾⣯⣼⣤⣷⣤⣴⢶⡾⢫⣄⣀⣀⡀⠀⢺⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡾⠂⠀⠀⠈⠛⢋⡉⠉⢹⠀⠈⣥⡀⠛⠉⠉⠉⠉⢻⣾⣿⣿⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⡈⠀⠀⠀⠀⠀⠀⢨⡀⠀⠈⢡⡀⠗⢸⣷⡵⢾⣿⣿⢃⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⠈⠀⠀⠳⠸⠇⠝⣧⢰⣾⣩⡻⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢿⣁⣁⠈⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⢠⢛⣶⣮⣭⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⠀⠀⠀⢶⡅⠀⠀⠐⠒⠒⠀⠀⠐⣯⡈⠁⠀⠈⠃⠘⡿⠟
⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢻⡆⠈⠀⠀⠀⠀⠀⠀⢀⣿⡆⣄⠀⠦⣀⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠲⡶⠾⠇⠀⠐⠀⠀⠀⠀⠀⢨⠈⣷⣄⠉⠐⠈⠁⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⡉⠀⠈⠁⠀⠀⢀⠿⠀⠈⣿⣄⡁⣀⢠⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢁⡄⠚⠹⢏⠀⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠃⠈⠲⡄⠀⢨⣀⢤⠴⠤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⠀⠤⠤⣄⣨⠀⠀⠐⠂⡀⠀⣏⠀⡀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⢱⡴⣿⠤⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣐⣂⠠⢼⣴⡏⠀⠀⠀⠀⢸⠆⢸⣆⠘⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡀⡉⠛⣻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣨⠄⠀⠀⠀⠀⠀⠻⠦⠀⢀⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⢧⠀⠂⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⢀⣷⡄⠀⠀⠈⢡⣠⣶⣶⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡵⠀⢀⣀⣀⣀⣀⠘⠛⠇⣀⠓⠀⣘⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡠⠴⠦⣼⣶⣾⣿⡿⠟⠉⠀⠀⡀⣀⣠⡜⠙⠛⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣸⣿⠟⠁⠀⠀⠀⠀⠈⠀⠙⠛⢧⣄⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠶⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⢸⣄⣀⣰⡀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⡈⢃⣴⣾⠷⠿⠟⠓⠚⠻⢿⡿⣷⣍⣰⡀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣢⡿⠛⠩⠀⠐⠀⢴⠀⡁⠠⠀⠀⠀⠈⠙⠳⣮⣐⠀⡖
⠀⠀⠀⠀⠀⠀⠙⠒⣾⠿⣡⡐⢋⡍⡠⠏⢠⡇⠀⣭⠀⢙⣫⣤⡀⠹⡄⠈⠹⣾⣤⡆
⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠱⠅⣽⢸⠷⠖⣷⠁⢠⣿⣤⡼⡉⠀⢿⣞⡃⠀⠀⠈⢻⡿⣠⠂
⠀⠀⠀⠀⢀⣲⣿⠃⠲⠀⣿⡄⣆⣈⢳⡤⠗⣛⢾⣿⠶⠿⢧⣤⣿⣟⣣⣤⡤⠟⠀⢻⣧
⠀⠀⠀⠀⢩⠟⠩⠻⡿⣷⣮⣦⡼⢛⠻⡗⠀⠉⣸⡿⠀⠂⡈⢸⢿⣿⡿⢿⣀⣾⡧⠈⣷⡛
⠀⠀⠀⢠⣟⢀⠀⠂⣹⣽⣷⣿⠿⢟⣟⣻⣦⣶⣻⣯⣵⣤⣤⣾⣿⣿⣷⠞⠻⣇⣴⣿⢿⡟⠂
⠀⠀⠀⢸⣿⣾⣿⣾⣯⣫⠿⡟⠄⠐⠀⠀⢹⡉⣿⢹⠏⠏⠀⠀⢠⣿⣿⠦⣆⣿⣿⣉⣶⡏
⠀⠀⠀⠈⢉⡧⠐⢻⡧⠁⠀⣹⡆⠀⠈⠂⠀⢠⣴⣛⣿⡀⠀⠀⢸⡈⠻⣠⣶⣟⢩⣿⣿⡇
⠀⠀⠀⢰⣾⣿⡻⠛⠀⠀⠀⢻⡄⢀⣤⣴⣦⣬⡙⢿⢏⣥⣦⣤⣼⡃⠀⠀⠀⠹⣟⣿⠿⠳⠄
⠀⠀⠀⠀⢿⣿⡁⠔⠉⢡⣥⣬⠻⣿⣿⣿⣿⣿⢇⣀⠸⣿⣿⣿⣿⢇⣀⡁⠉⢲⣯⣧⣼⣤⣆
⠀⠀⠀⢠⣯⣨⢷⡠⠿⠻⠟⠊⣴⣈⠙⠛⣉⠰⢹⣿⡶⢨⣩⣽⠇⠉⠿⠿⠿⣦⣿⣟⣯⡿⣦⠞⠁
⠀⠀⠀⠀⢹⣿⣾⣷⡀⠀⠀⠀⠉⣇⠈⣥⢨⠆⢾⣿⣿⡤⢻⣇⠀⠀⠀⠀⠀⢸⡟⣻⣷⣿⠧⣄
⠀⠀⠀⠈⠹⠻⠻⣿⣿⣶⡄⠀⠀⠈⠻⢧⡀⡄⢾⠏⢳⠐⣸⠟⠀⠀⠀⠀⠀⢸⡇⣦⣽⡛⠀⢸⡆
⠀⠀⠀⢀⣿⣻⣶⢸⠜⠻⢾⠀⠀⢠⣶⣾⣷⣶⡼⢼⠎⣴⣾⣶⣦⡀⠀⠀⢠⡟⢿⣿⣏⣻⣄⡀⢀
⠀⠀⠀⠘⢛⣿⣿⣷⣿⣶⣮⡀⠀⠀⠉⠹⡟⢿⣗⣖⣺⣿⣿⠉⠋⠁⠀⣠⣾⣽⣥⡿⣏⠻⠋⠈
⠀⠀⠀⠀⢁⣷⡿⢛⣿⣿⣉⣽⢶⣤⡀⠀⣿⠓⣈⡉⡀⣠⡏⠀⢀⣴⣾⠟⢡⡿⣟⣢⣿⣾⣷⣤
⠀⠀⠀⠐⢋⢘⣿⣿⣿⡾⠟⣋⣿⣉⣹⠷⠾⣿⣿⠻⢿⣿⣷⣶⠟⢹⣤⡶⣿⢷⣿⢿⣿⣿⠇⠴⠎
⠀⠀⠀⠼⣿⡄⣩⢤⠸⣿⣽⡛⠻⢿⣋⢲⢤⣄⣙⣉⢛⣀⣠⠼⣿⣿⣿⣧⣴⣦⣴⣶⡿⣶⠀⠀⣄
⠀⠀⠀⣻⠛⠿⡟⢸⣶⣯⡿⠿⣧⣄⡙⠳⢷⣥⣹⣙⣏⣋⣭⡶⠿⢭⣯⣿⣯⣟⢿⣿⠓⣼⣶⣞⠁
⠀⠀⢰⣿⣶⣿⣉⣩⠟⡿⣿⣦⡌⠛⠿⣾⣦⣤⣬⣥⣠⣄⣠⣤⡶⠟⡛⣹⣏⣝⣿⠿⢤⣿⣭⣵⣦
⠀⠀⢸⣟⠁⣏⣀⣿⣀⢀⡁⠛⢛⣿⣶⣢⣠⠈⡍⠉⣉⡁⡈⣄⡀⣤⣬⣿⠿⠛⠀⠀⢠⡿⠿⠀⢹⡇
⠀⠀⢸⡿⠀⠁⠀⠈⣿⣉⣷⣀⣠⡈⠛⠋⠛⠙⣶⠟⠿⠟⠿⠙⠯⠋⠁⢁⣀⣀⣼⣍⣏⠀⠀⠊⠉⠇
⠀⠀⠀⢡⡀⢈⡆⠀⠉⠁⠈⢹⣧⠿⠶⣴⡟⣦⣀⣤⡴⣦⣀⣤⣟⢦⡤⠿⣯⡟⠁⠉⠉⠀⠦⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠟⠁⠀⠈⠻⠋⠉⠈⠻⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣥⡶⠟⠳⢦⡀⠀⠋
⠀⠀⠀⠀⣠⡞⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠉⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠂⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⣴⣾⡿⣭⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠘⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠆⠀⠈⢛⣿⣯⠾⠃⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⢺⣦⡀⠀⢸⣿⣧⡀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⠿⠟⠀⣤⡀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣼⣿⣅⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⡾⠁⠀⠀⠻⠿⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠈⠁⠀⠀⠀⠘⣿⡯⣿⣿⡿⠂⠀⣀⡴
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠀⠀⠀⠑⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣸
⣿⣤⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⣶⣿
⠻⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠰⣾⣿⣿⢲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⢈⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡄⢀⣠⣴⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⡿⠍⠙⠛⠛⠿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠻⣿⡿⢿⡟⠛⠋⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⢀⣀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⠀⠀⢀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⢸⣿⠿⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣟⠁⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢦⣀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣄⠀⠀⠀⠰⣦⡄⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⣧⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠛⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢿⣿⣦⡀⠀⠀⠀⢀⠀⠀⠀⠒⣀⡀⠀⠀⠀⠀⢸⡇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⢿⣷⡀⠀⠀⠈⢶⡄⠤⠽⠋⠀⠐⠓⢶⠂⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣙⡛⢀⠀⠀⠘⣧⣶⣤⣤⣶⣾⢟⣿⣷⣿⠋⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣰⠖⢙⣻⠿⣿⣟⡟⠂⠒⠀⠀⢹⣿⣿⣤⠈⣀⣼⣿⣿⣿⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⣍⣹⣿⣦⣻⣿⣿⣿⡟⠃⠀⠀⢠⣾⣿⣿⣯⡠⢾⣿⣿⣿⡿⠉⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠙⠉⠁⣸⣿⣿⡷⠀⢠⡀⣀⣬⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠈⢻⣯⣸⣿⣛⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⢴⣿⣏⢸⠃⠙⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⢠⣼⢿⣿⣿⣦⡀⠀⠙⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠀⠀⠁⠈⣿⣿⣿⣷⣄⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣆⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠙⠿⣿⣿⣿⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢠⠄⠀⠀⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧⠀⠀⢀⣴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣾⠁⢙⠴⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣀⢋⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⣀⣠⣤⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣠⡾⠛⠧⠞⠻⣦⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⣿⠀⠀⠀⠀⠀⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣜⠷⢤⡤⡤⠞⣃⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣴⣶⣶⣖⣛⣀⣀⡀⡀⠸⠛⠒⢒⣒⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢻⡍⠁⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢀⡽⠃
⠀⠀⠀⠀⠀⠰⣯⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠒⠲⠶⠤⠤⢤⣤⣄⣀⣀⣀⣀⣼⠗⠋⠁

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

⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⣤⣤⣤⣤⣀
⠀⠀⠀⢰⠂⠈⠀⠀⠀⠀⠀⠉⠋⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠛⠛⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠲⠌⡃⠀⢀⠠⠤⠲⡄⠀⠀⠀⠄⠂⠒
⠀⠀⠀⡠⠠⠀⠀⠀⠀⣠⣀⡄⠀⠀⠅⠠⠧⠂⠀⠓⠿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢂⠀⠀⠀⠀⢀⡀⢀⢀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⣿⡍⠀⠀⠀⢶⣶⣶⣶⣾⣷⣾⡇⠀⠀⠀⠀⠀⢀⣒⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠄⠤⢽⠀⠀⠀⠁⠻⢿⣷⠾⢾⠿⠟⠻⠦⠤⠚⠛⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢐⣀⠀⠸⠀⠐⠠⠀⠀⠀⠤⣄⣹⣶⣛⣳⡈⢛⣃⣛⣛⣛⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⢀⣀⣴⣶⣀⠀⠀⠀⢐⣿⣿⣽⣿⣿⣿⣯⡄⢘⠡⠀⠭⠥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠃⠸⠿⠿⠻⠍⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⣳⣤⠄⢀⣠⣴⡾⠛⠚⠛⢶⣶⣤⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢕⢶⣿⡿⠟⢻⠿⠒⠣⠀⠀⠀⢉⠙⢿⣤⡄⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⠋⠀⠃⠁⠀⠀⣀⣄⡀⠀⠀⠽⣿⠿⣾⣷⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡿⠖⠀⠀⠀⠔⠉⠀⠀⠈⠁⠀⠀⠈⠀⠈⢻⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⡿⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣫⣷⡁⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠉⠛⠦⣌⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢫⡻⢿⣿⣷⡷⣄⠙⠚⠳⢤⣤⠒⠅⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠛⠚⠛⠛⠋⠙⠿⠿⠿⠿⠛⠓⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠒⠒⠊⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡇⣸⣿⠷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣷⣿⣯⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣿⣋⣾⣿⡃⡙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣿⣭⣭⣍⣛⠛⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣬⣭⣽⣿⠆⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣶⣶⡶⠤⠖⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⡆⠀⠀⠀⠀⢻⣿⣿⣿⣷⣦⣤⣴⡟⠻⠶⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢷⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⡟⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣃⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠉⢱⣦⣀⣠⣿⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⠀⠀⠙⠘⢻⣾⣿⣿⡿⠿⠿⠿⣿⠿⡿⢿⣿⣿⣿⠏⠀⢀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠁⠁⢠⣤⠀⠀⠃⠀⣿⢰⡇⢠⣏⢠⣮⡶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⢲⣄⡀⠀⠀⡀⠃⣼⣿⣴⡞⠳⢶⣿⢾⣥⢌⠉⣥⣬⣥⠉⠘⠛⠚⠓⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢿⣿⣷⣄⣰⣏⣼⣿⣿⣿⡟⠛⠘⠃⠀⠁⠀⠀⣿⣿⣿⠈⠀⠀⠀⠀⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣷⡶⢹⣿⣯⣿⣿⣿⣿⣿⣿⠀⠟⠀⠀⠀⠀⠀⠀⣿⣿⣯⠀⠀⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⢰⣀⣿⣿⣿⣿⡏⢈⡿⠁⠀⣿⣿⣿⣿⣿⣿⠀⣦⡄⠀⠀⠀⠀⠀⣿⣿⣿⡀⢀⣀⣀⣀⣀⣈
⠀⠀⠀⠀⠀⠀⠀⢘⠻⣿⣿⣏⠉⣉⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⣷⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⡞⡀⢀⣉⣽⣿⣟⠁⠀⠀⠀⢾⣿⣿⣿⣿⣿⡏⠙⠙⠛⠛⠉⠉⠉⢹⣿⣿⣿⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢨⣿⣷⠿⣫⡍⠿⠛⠃⢀⣀⠀⢰⣟⣿⣿⣿⣿⣷⢢⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠸⣿⣫⣾⠉⠀⠤⣶⢉⣿⢋⣀⡿⢋⠉⠹⣿⣿⣿⡸⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣟⣂⣀⣤⣼⣭⣿⣿⣿⡅⣀⠀⠀⣿⣿⣿⡇⢉⠁⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠛⣿⣿⣷⣭⣶⡾⠿⠿⣿⣷⡟⠃⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣼⣯⣿⣿⣿⣿⣿⠿⣿⣿⣿⠛⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣿⠿⠿⠇⡀⢿⣿⣿⣿⣿⡿⢻⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣟⣀⣶⣿⡄⢸⣿⣷⣾⣿⣿⡞⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠇⠸⣿⣿⣿⣿⣿⣿⠓⡶⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⣿⡿⣿⣿⡟⢿⣾⢳⠇⠀⠀⢸⣿⣿⡆⠀⠀⠠⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⣿⣇⣿⣿⣧⠸⣿⣧⠺⠓⠀⢸⣿⣿⣷⠀⠠⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⣿⣿⣿⣿⣿⠀⢿⣿⣾⡱⠖⠈⣿⣿⣿⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⢻⣿⣿⣿⣿⠀⠸⣿⣿⣿⣧⡘⢻⣿⣿⠆⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⢻⣿⣿⣿⣿⣆⠹⠃⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⡇⠙⠷⢻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⢺⡇⢠⠈⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣧⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡏⢸⠀⣿⣿⣿⣷⣄⠀⠀⢸⣿⠉⣿⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⢀⠉⠀⠉⣡⠌⠀⠀⠈⢻⣦⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⣶⣶⣾⠁⠀⠀⠀⠀⠀⠻⣷⣤⣴⠿⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⢹⣿⡟⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⡿⣿⠃⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡀⠀⠀⠀⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣧⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣦⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠒⠉⢡⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣶⢖⣿⣷⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃⢾⡿⠿⠋⠉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣌⠳⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣶⣿⣭⣁⠀⠀⠀⠀⠀⡀⠘⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣶⣟⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣷⣝⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⢻⣿⠟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠁⠀⠀⠈⠢⣄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣶⣄⡀⠀⢠⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶⣌⣻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠉⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⣻⡿⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡻⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠙⠛⠛

⣿⠿⠿⠿⡿⢿⠿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣛⣛⣿⣿⣿⣟⣿⣿⣿⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣤⣤⠴⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣄⠀⠀⣠⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣆⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⣿⣿⢿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟⠛⠿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⡿⠿⠃⣸⣿⣿⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⠃⠀⡀⠀⠀⢍⠛⠛⠛⠛⠛⠛⡅⠀⠀⠀⠀⢿⣿⣿⠿⢿⡛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣘⠁⠀⠀⠀⠉⢲⣀⣠⡤⠖⠀⠀⠀⠀⠀⢸⡉⡦⢴⣿⣿⣿⠏⠘⣧
⠀⠀⠀⠀⠀⠀⣼⡿⠿⠂⢺⣿⣶⣶⣦⡀⠀⢸⡏⠡⠐⢆⠀⠀⠀⠀⠀⠈⢧⢨⣾⣿⣿⡏⠀⢠⣿⠆
⠀⠀⠀⠀⠀⠀⣿⢯⣤⣀⠈⣿⣿⣿⣿⣿⣶⣾⡄⢀⠀⢸⡦⠀⠀⠀⠀⠂⠊⢷⣿⣿⣿⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣴⣿⡖⠋⠑⠀⡙⣻⣿⣿⣿⣿⣿⣿⣾⣇⣉⣅⡀⠀⠀⠀⢈⡄⣨⣿⣿⣿⣧⣀⢰⣿⣿
⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣠⣭⣿⣿⣿⣸⣿⣿⣿⣷⡉⢿⡇
⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⢤⣹⡟⠸⣿⣿⣿⣿⣿⣿⣿⣟⣟⣋⣟⣿⣿⣿⣿⣿⠃⠸⣿⣿⣿⠿⠓⢸⣷⡄
⠀⠀⠀⢹⣿⣿⡇⠀⠀⠈⢻⣏⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣾⣿⣿⠀⠀⢸⣿⠁
⠀⠀⠀⠀⣿⣿⣅⣀⠀⠀⠉⢠⣿⣿⣿⣿⣿⣿⡟⠋⠉⣛⠛⠛⠛⠉⠉⠁⠹⣿⣿⡆⠈⣿⣿⣄⣤⣼⣿
⠀⠀⠀⠀⣿⡟⠛⠓⢶⡀⠀⣾⣿⣿⣿⣿⣿⣾⠀⠀⠰⡿⢃⣀⠠⠀⠀⡀⠀⣿⣿⣿⣦⣿⣿⠿⠛⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⡿⠓⠀⣄⡤⠆⣾⣿⡤⠆⢘⣿⣦⣾⣿⣿⣿⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⣾⠃⠀⣸⣿⣿⣿⣿⠋⠀⠁⠉⠉⣹⣿⠿⠚⠀⠐⢻⣿⡉⠻⣿⣿⠙⣋⠀⠀⣿⡇
⠀⠀⠀⠀⣿⣧⡄⢴⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⢻⣷⠀⠸⣿⣿⣿⣧⣼⣿⡁
⠀⠀⠀⠀⠸⠿⠿⠿⠛⠀⠀⣿⣿⣿⣿⠃⠀⠀⠠⣼⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⢻⣿⡌⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠙⠄⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠋⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠯⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉


⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⡏⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢹⣆⠹⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡁⢘⢦⠹⡀⣣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣳⠟⠉⠈⠉⠉⠙⠓⠒⠦⣤⣄⣀⣤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⢦⡀⠀⠀⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠳⡄⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠛⠿⡀
⠀⠀⠀⠀⢰⠲⣠⠎⣀⣀⣀⡀⠀⠀⠀⠘⠲⣇⡀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⠲⡤⢄⡀
⠀⠀⠀⠀⠈⠳⠔⠋⠁⠀⠀⠈⠑⠦⣄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠲⢦⣀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⡹⠶⠒⠶⠶⠦⠤⠤⣄⡰⠀⠀⠀⠀⠀⠀⠉⢷⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⠼⠖⠒⠒⠛⠓⣒⣯⡀⢀⣀⡤⢤⣔⠒⠒⣇⠀⠉⠓⢄⠀⠀⠀⠀⣀⡼⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⢀⡠⢄⡀⢀⣇⠀⠀⠀⠀⠀⠐⠯⣵⠟⠋⠁⠀⠙⠟⠀⠀⢸⡀⠀⣠⠼⠣⠀⢰⠺⣁⣀⠬⠧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⢺⡀⠀⠙⠇⠈⠓⠲⠤⠤⠤⠔⢺⣇⠀⠀⠀⠀⢀⣀⣤⣶⠉⠁⡼⠃⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠈⠓⠦⣀⠀⣀⡀⠀⠀⠀⠀⡼
⣏⡈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡙⠛⠛⠛⠉⠉⠁⠁⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⢀⣬⣿⢁⡨⠇⠀⠀⡸⠁
⠀⠉⠉⠉⠉⠒⠒⠲⡒⠀⢤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣉⡉⠲⡄⠀⠀⠀⠀⠈⢧⠀⠀⣰⢷⣯⢽⠟⠉⠂⠠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣙⠢⢎⣩⠖⢤⡀⠠⡀⠀⣠⠤⣄⣉⠈⢉⡡⠃⠀⢀⢷⠘⣄⣸⠶⣋⠴⠊⠈⢁⣀⠔⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣍⡐⠃⠬⠆⢻⡏⣇⡞⠁⠈⠉⢣⢄⣀⢀⡼⠺⢀⣯⣴⣊⣡⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣉⣒⣲⠚⢷⣬⡇⠀⠀⠀⣇⣀⣠⡭⠔⠚⠉⢠⣞⡥⡴⢶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠲⠉⢹⠀⠀⠀⠙⢯⠛⢿⢉⣡⠴⡉⣠⠀⠀⢠⠃⠀⠀⢸⡌⢇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠯⠵⠮⠤⠃⠀⣦⢍⡋⠀⢸⡧⣀⡰⠋⠁⠀⠀⢸⣄⡀⠀⢸⢇⣼⡜⡀⡁⢈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠹⠵⠉⠉⢀⣠⠁⠀⠀⠀⠀⠸⡄⠉⠛⢻⠻⢤⢻⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣩⠟⣽⢒⣒⣶⣶⣖⡿⠀⠀⠈⠒⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠛⠊⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢈⡏⠁⣠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠁⠃⣨⠙⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠶⠀⠸⠇⢀⡾⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⡱⣦⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠧⠀⠀⠋⠠⠋⠔⠁⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠃⠀⠀⠀⢠⣄⣂⣀⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⡄⠇⠀⠀⠁⠀⠀⠀⠉⠙⠛⠻⠷⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠒⠶⠲⢀⣤⠄⣠⠴⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⠟⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠰⠀⠐⠂⠘⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠙⢻⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠆⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠞⠢⠄⠀⠤⠊⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄⡀⣀⡀⠠⢀⠐⠃
⠀⠀⠀⠀⠀⠀⡬⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣷⡾⠷⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⢛⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠀⡄⠐⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⢤⡀⢤⠈⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟
⠀⣠⠀⠀⠀⠀⠈⠀⣀⠒⣀⠀⠂⡐⢂⣤⡄⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⣀⠀⡀⣀⡠⠆
⠘⢋⡙⠉⠉⠁⡀⠀⠻⠿⣃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠚⠃⠰⠂⠉⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠼⢇⡨⠍⠀⠀⢀⠄⠀⠛⠀⠀⠠⠀⠐⢢⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡃⠐⢀⠀⠀⡁⢠⡤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠛⠂⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡼⠃⠀⠚⠃⠀⠀⠀⠁⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤
⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠁⠀⠀⢠⡀⢃⡰⠐⠶⠀⠑⠲⢄⣠⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡰⠚⠎⠀⣠⣀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀
⠀⠀⠐⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠉⠀⠀⡰⠀⠀⠀⠀⠠⠤⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀
⣀⣀⣀⣀⣈⣉⣉⣁⣀⣘⣢⣄⣀⣀⣐⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡶⣻⣽⣏⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡤⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣆⢄⣀⡠⠈⠉⠉⠉⣛⣛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣩⣷⡶⠈⢻⣭⠤⣤⠒⠒⠒⠊⠉⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⡁⠉⠁⠚⣡⣾⠛⠉⡠⠖⠊⠉⠋⠈⠁⠀⠄⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠙⢻⡔⠚⠉⢈⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡿⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣠⣀⣀⡀⠤⡰⠁⠀⣾⠗⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡈⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣿⠀⣿⣴⣿⠉⣠⠃⠀⣸⡏⣠⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣆⣼⣿⣿⢀⠇⠀⢠⣿⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡲⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠈⠓⠚⠛⠿⣿⣿⣿⠏⠀⠠⡞⢋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡁⠀⢨
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡙⠗⠀⠀⠸⠁⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠷⠶⠾
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⢻⣄⢄⣀⣰⡦⠜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⢀⠀⢘
⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠈⠀⠈⠻⠏⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠀⠳⠄
⠀⠀⠀⠀⢀⣤⣊⡍⠀⣀⣀⠀⠀⢀⠀⠀⢀⣠⠤⠀⠁⢀⠭⠄⠀⠈⢾⣶⠆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⢠
⢀⡀⢀⡀⠉⢀⠈⠙⡆⢀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠒⠂⠡⠠⠀⠀⣀⣤⠁
⠈⢳⣾⠇⢾⣿⣿⡆⠘⣄⣃⠀⠀⠀⠀⠸⠀⠀⣠⣤⣀⠀⠀⠀⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰
⠀⠈⠁⠈⢀⠉⠛⢃⢀⣿⡟⠁⠀⠀⠀⠀⠉⠈⡻⣿⣿⣧⠀⠠⢄⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠐⠂
⠀⠀⠀⠀⠀⠛⠻⣿⣿⣶⡶⡆⠤⣀⡦⢀⣀⡀⣀⠈⠁⠀⣰⢬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠛⠂⠀⠸⠙⠚⠹⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠂⠀⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣤⣤⣤⣠⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⠻⠘⣿⢿⠉⣫⣟⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠷⣼⣡⣾⣼⠛⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢿⣽⣿⡏⠀⣀⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢀⣤⣴⠶⠶⢿⣿⣻⣿⣿⣿⣿⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠈⣿⣿⢒⣄⡤⠤⠥⣬⣭⣭⣥⣤⣴⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢀⣤⣖⣬⢃⡾⠃⠀⠈⠉⠉⢙⣿⡟⠉⠀⢻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠉⠛⣷⣼⠃⠀⠀⠀⠀⢠⣿⣿⡷⠀⣀⣼⣿⣿⣯⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⣼⡿⣿⠀⠀⠀⠀⠀⠈⠻⣿⠃⢈⢉⣿⣿⣿⣿⣿⡟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣄⣀⣤⣼⢿⣇⣀⢳⡄⠁⠀⢀⣤⠾⠿⣦⣤⢸⣿⡏⢸⣿⠟⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⣎⢹⡿⠃⠀⠟⢁⠈⣧⠀⢀⡾⠁⠀⠀⠈⣿⣿⣿⠇⠀⣿⡗⠚⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⡼⣿⣿⣿⠀⠀⢘⡗⠋⠀⢸⡿⠛⠃⠀⠀⠀⠀⠛⣿⣿⠀⠀⣿⣧⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢷⠃⡿⣿⠇⠀⠀⢌⣀⣂⠀⠘⠧⠤⠀⠀⡀⠤⢠⡼⣿⣇⠀⠀⢸⣿⣯⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⢀⡟⠀⠀⠀⣾⣿⣭⣷⣦⣰⣶⣾⣄⣁⣤⣶⣾⣿⣷⣤⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣤⡼⠁⠀⠀⠀⣿⣾⣿⣿⣿⣿⣏⣿⣇⣿⣿⣽⣿⣻⣿⡅⠀⠀⢹⡿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠘⠳⠀⢀⡀⢸⣿⢃⣟⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡛⣿⣿⠀⠀⠀⢻⠞⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣎⣠⠄⠐⢷⣀⠾⣿⣿⣿⣿⣷⣿⡿⢿⣿⣿⣧⣤⣭⡿⣿⣿⣿⡿⣶⡄⠀⠃⣠⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠤⠄⠤⠘⠉⣽⣏⣿⢯⣿⠟⠁⠀⠈⣿⢿⣿⣿⣯⡾⠋⠀⠘⣷⣿⣧⠈⢾⣟⣥⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠾⠛⡿⠋⠀⠀⠀⢠⣿⣼⣿⣷⡿⠀⠀⠀⠀⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡞⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢿⠇⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⣰⣿⡿⣿⣿⣿⣿⡇⣿⡄⠀⠀⠀⢸⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣁⣀⣀⡀⣰⣿⠋⣴⠇⠙⢁⢘⣿⣿⢃⡀⠀⣠⣼⠇⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⠞⠍⠉⠉⣿⣿⡏⢰⡟⠀⣰⠏⠘⢿⡛⢸⢛⡿⠁⠻⡆⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⣴⡗⠀⠀⠀⢸⡏⠀⠸⠃⠀⠛⠀⠒⠘⢳⣼⣟⠁⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣎⠟⠓⠒⠶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⢈⡷⠒⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⡀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢀⣠⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠛⠛⠻⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣠⠌⠿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣤⣤⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣯⣅⣀⣶⠉⠉⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠟⠛⠻⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣤⣶⣦⣄⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡾⠋⠀⠀⠈⠹⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠃⠀⠀⠀⠀⠀⢹⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢻⡄⠀⠀⠀⠀⠀⣸⠃⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⣼⣷⣤⣤⣤⣤⣤⣏⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣴⡿⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⢷⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠮⢭⣭⡴⠶⠛⠋⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠈⠃⠁⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠘⢁⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠈⠉⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣴⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡇⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠛⠛⠁⠀⠘⠋⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠆⠀⠀⠀⠀⢿⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠃⠀⠀⠀⠈⣷
⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣸
⣾⠁⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠖⠒⠒⠚⠛⠛⡇⢠⠖⠛⠛⠛⠳⣄⠀⢸⠛⠛⠲⣆⢰⠶⢦⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣰⡟
⢻⡄⠀⠀⠀⠀⠀⠀⠀⣇⠀⣀⠀⠀⠀⣀⣀⣀⡇⡏⠀⠀⣀⡀⠀⠘⡆⣸⠀⠀⠀⠸⡟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣰⡟
⠈⢿⡀⠀⠀⠀⠀⠀⠀⠉⠋⢹⡄⠀⠀⡏⠉⠉⠀⡇⠀⠘⠷⠗⠀⠀⡟⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⣰⡟
⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢰⠀⠀⠀⠹⣄⣀⠀⢀⣠⡾⠁⣇⠀⢸⢷⡀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⣾⠏
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠙⠋⠀⠛⠛⠒⠞⠃⠀⠀⠀⠀⠀⢀⡼⠃⣠⣴⠿⠋⠁
⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠟⠋⠁
⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡶⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡤⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⣀⡀⣠⢤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⢀⡀⠄⠀⠀⠀⠁⠀⠀⠐⢦⡤⠾⠋⠀⠘⣁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡄⠀⠂⠀⠀⣼⠫⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⣠⠌⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠶⠤⠀⢄⣴⠋⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠈⢡⡄⠀⠀⠀⠀⠛⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⠀⠄⣼⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠿⠀⠂⠀⢀⠁⠀⠀⠀⠀⠀⠐⠀⠈⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠲⣌⠁⡰⠛⠋⠉⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣦⣀⠀⠈⠀⠀⣼⠀⢄⡀⠀⠀⣀⠀⠐⢦⣶⡾⠋⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣾⣧⡧⢾⣦⣼⣶⠿⠷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡆⢾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡦⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣀⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠀⠀⠀⠀⠀⠀⢺⣿⠓⢶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠂⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⣿⠏⠁⣤⠆⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣛⣿⡿⠀⠀⣾⠠⣤⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣤⣀⠀⠒⠀⠀⠔⠀⠀⠀⠳⠒⠶⠟⠷⢄⡀⣹⠀⠉⠀⠰⠄⠀⢀⣀⠀⠀⠀⠀⠀⠃
⠀⠀⣽⣽⣿⣷⣿⣤⣽⣤⣴⣀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠀⠇⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠁⠙⠛⠖⠺⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠁
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠒⡀⠐⠒⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣌⡙⠛⠛⠛⠛⠛⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠉⠉⠉⠉⠁
⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣶⣶⣶⣦⣾⣦⣾⣧⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣾⣷⣦⣄⠀⢀⣤⣄⡀
⠙⠿⠿⣿⣷⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠘⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⢈⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣠⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⠛⠙⠛⠋⠉⠉⠀⠀⠀⠉⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠊⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⡤⠐⠐⢠⣿⣶⢞⣷⡀⠀⠀⠀⣰⠀⢤⢀⠀⠙⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠈⠻⢿⣾⡿⡇⠀⠀⠀⡟⢀⢀⠀⢠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠀⠉⢠⡀⠀⠋⠔⠁⠀⠀⢰⠀⠀⠸⡇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⠀⡘⠈⠀⠀⢱⣶⢶⠀⠀⠀⠀⠸⠀⠀⠀⠩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢚⡄⠀⠀⡠⠛⢁⣘⡄⠀⠀⠀⣀⡀⠀⠈⠓⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠉⡉⠔⠋⠀⣠⢷⣌⠁⠀⠀⠀⣹⡁⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣤⣤⣀⣠⡇⠘⠁⠀⠀⠀⠀⣼⣿⣶⣄⠀⠲⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⣄⠐⠛⠋⠉⠟⠛⣛⡻⢾⡷⡂⠀⠀⠈⢲⣅⠈⠉⢻⣧⠀⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣷⡿⠷⢦⣤⣀⣀⡨⠄⠡⠀⠀⢡⠀⠀⠀⠈⠘⠀⠀⠀⢹⡄⢋⠇
⠀⠀⠀⠀⠀⠀⠎⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠉⠻⣧⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢸⠿⠁
⠀⠀⠀⠀⠀⠀⠢⠤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢦⡀⠂⡴⠿⠶⣄⠀⠀⠀⠀⠠⠶⠖⠋
⠀⠀⠠⡠⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠤⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠠⣔⣮⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠴⢺⣽⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⣻⣿⡏⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣔⠁⣘⣷⠀⠀⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣸⡇⣼⣿⣿⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⣿⣿⣿⢠⣴⡶⢟⡳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢹⣯⣙⢛⣻⡏⠉⣉⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢿⣿⣿⣿⠇⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢰⣝⢿⣿⠸⠟⢋⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⢹⣿⣿⣿⠿⡦⠺⣿⣿⡾⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠘⣿⢠⣀⠌⠻⣿⣿⡟⠀⠀⠀⠿⣿⠟⠁⣀⡄⠊⢀⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⣈⣷⣿⢸⠁⠀⠀⠈⠛⠄⠀⠀⠀⠀⠀⠀⠀⠉⢠⣞⡋⢀⣀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⢸⡆⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣠⣤⣀⣿⣿⣿⣿⢸⡇⠀⢰⡀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⢻⣿⢸⣟⢀⣤⣷⣤⣤⣼⣿⣤⣤⣄⣀⣀⠀⢸⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠂⠼⠟⠿⠿⢿⡏⠸⣿⠈⢻⡙⠛⠛⠛⠛⣿⣿⠛⢻⡛⠛⠋⢀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣴⣾⣶⣦⡀⠘⠷⢶⣿⣤⠤⢶⠃⠀⠀⠀⣿⣿⠀⠀⢳⡄⠀⣼⣿⡟⠛⠿⣿⠇
⠀⠀⠀⠀⢳⣿⣿⡟⣵⣤⡄⡀⠀⠶⠶⢒⠀⠀⠀⠀⣿⣿⠻⣆⠀⠛⠀⠁⠀⠙⠦⢤⡟
⠀⠀⠀⠀⠀⠙⠛⠃⣿⣿⠏⡗⠀⣚⣈⣉⠀⠀⠀⠀⠛⠁⠲⡀⠀⣀⣠⣤⣶⡎⣿⡞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢧⡸⣧⠀⠤⠤⠚⠀⠀⠀⠀⣀⣤⣴⢹⡿⠟⣋⣙⣉⡵⠿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠼⣯⠀⠀⠀⠠⡶⡶⠖⠛⠉⠁⠉⣼⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢀⡵⠒⠃⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠞⠋

⠑⠀⠈⢠⡀⣧⠛⠾⡓⠀⠀⠀⠀⠀⠀⢀⠸⡃⠀⢀⠀⡀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠐⣠⠄⠀⣠⠊
⢀⣶⣕⠌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠠⠂⡄⠀⠀⠀⠁⠀⠐⠁⠘⢆
⡉⠀⠁⢠⠶⠈⠀⠀⠀⡀⠆⠘⠖⠒⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠈
⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⠓⠀⠀⠀⢀⠐⠑⠆⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡠⠐⠂⠁⢀⠀⣀⠀⠁⠀⠰⡀⠀⠀⠀⠀⠀⠘⠉⠁⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠋
⠀⢀⡄⠀⠂⢘⠀⠀⠀⠀⠏⠈⠀⠀⠢⠀⠀⠀⠁⠐⠀⠀⠀⠸⠂⢈⠄⠋⠈⠁⡆⠓⡀⠀⢄⠀⠀⠀⠐
⡈⠀⠀⠀⠀⠠⠛⠈⠁⢐⠀⠊⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠥⠀⠀⠰
⠉⠁⠀⠀⢀⠤⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠂⠠⡄⠀⠀⢨⠥
⠓⠠⠄⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰


⣉⣉⣀⠤
⣉⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⣉⣤⡴⠖⡊
⠉⣥⠶⢃⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠛⢿⣶
⣈⣴⣮⣅⣤⣴⣎⣠⣠⣄⣤⣄⣠⣇⣤⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣀⣤⣤⣈⣤⣌⣵⣦⣈

⠬⠒⠃⠀⠀⠀⠈⠀⠄⠀⠀⠂⠀⠀⢀⠀⠘⠦⠀⠀⡀⠀⠅⠀⠁⢧⡀⢹⠋⢻⣬⢺⣣⣼⣼⡿⠉⡽⠁⡨⢋⠁⠃⠼⢁⠞⠹⠃⢈⡼⠋
⣅⡀⠀⠀⠀⠀⠀⠈⠀⠈⢢⣠⣤⣄⣤⣀⡀⠀⠠⠀⠀⠀⠀⣠⠖⠾⣇⡀⢱⢋⡷⠻⠋⠀⠈⣷⡼⢁⣀⠀⡈⠀⠀⢀⡀⠀⣀⠐⠁⠀⠈⠀⠀⢀⡴
⢤⢉⠁⠤⣀⠀⠀⠀⠀⠀⣾⠉⠀⡀⢀⣉⣻⡖⡦⠥⡀⣀⣴⠧⠁⠀⠙⠉⢨⠮⣤⡀⠀⠀⠰⠟⠗⠛⠉⠙⠉⣹⠋⠁⠹⠏⢉⠷⡄⠀⡀⠔⠊⠉⠉
⣓⠀⠀⡀⠀⠐⠀⠀⠀⠀⢻⡍⠀⠈⠯⡉⠁⠀⠁⠀⠀⠉⠉⠗⠲⠄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠉⠀⠀⠸⠀⠀⣈⡗⠂⠁⠀⠀⠠⠐
⠦⣠⠀⠁⠉⠒⠀⠀⠀⠀⠻⣆⡄⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⣲⢤⠤⢤⠀⠀⠀⠀⢲⡋⠀⠀⠀⠀⠠⣄⠼
⠂⠀⠅⠀⠂⠀⠀⡀⠀⠀⠀⠈⣟⠁⠀⠀⠀⠀⢀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⡀⠀⠠⠆⠀⠈⠹⠀⢀⣸⠃⠀⠀⣀⣠⠇⠀⠀⠀⠀⠀⠀⢀
⠙⠃⠀⠀⠀⠀⢀⢀⣠⣔⡓⢺⡋⣷⠒⠀⠀⠀⠀⠀⠀⠈⠁⠉⠠⠄⠀⠀⠱⠀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡰⠞⠁⣴⣃⣠⡀⠀⠀⠀⠀⠄
⠚⠛⠓⠲⠆⠀⢰⣏⡀⠉⠀⠀⠃⠈⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠹⠻⡟⢲⡄⣀
⣛⠖⠃⠀⠀⢀⣿⠉⠀⣀⠴⠚⠁⠀⠙⠂⠀⠀⠀⠒⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠀⠀⠀⠀⠁⠀⠀⢀⡼⢋⣑
⠻⠍⣠⣶⠋⠙⠇⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶
⠉⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣀⠀⠀⠀⢠⣤⣄
⠤⠼⢿⣓⣤⣰⠯⠀⠸⣧⣀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠂⠐⠀⠀⠀⠀⠀⠐⠂⢠⠀⢀⡀⠀⠀⠀⠀⠁⠀⠉⠉
⠃⠉⠀⠀⠀⠉⢳⣀⠀⠀⠈⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠀⠀⠀⠀⠘⠆⠀⠀⠁⠢⠄⠀⠀⠀⠀⠀⠀⠀⠈⣠⣸⠇⠀⣠⡼⠟⠂⠒⠠⠤
⣴⡖⠊⠀⠀⠀⠀⠈⠛⠷⠖⢡⡀⠀⠀⠀⠀⠺⢤⠴⠦⠄⠀⠀⠀⠀⠘⡄⠀⡀⠀⠀⠀⠠⠀⠀⠠⠨⠆⠀⢀⠀⠀⠀⣹⡝⠀⢙⠀⠀⢄⣀⠀⠘⠑
⡩⠀⠀⠀⠀⠀⣀⠄⠈⠁⠀⣼⡅⠀⠀⠀⠀⠀⠀⠀⢆⣀⡔⠀⠀⠰⣤⠤⡀⠙⠆⠀⠀⢀⣄⠀⠀⡀⠘⠒⠋⢹⣷⣛⠁⢠⣴⡏⠀⠀⠀⠀⠉⠀⠈
⠂⠀⢀⡀⠄⠊⠀⠀⣀⡤⡾⠊⠳⠴⠤⠴⠻⣄⣀⣰⣯⠉⣅⡀⡐⢀⣽⡆⠁⠘⠲⢤⢶⡞⠉⠓⠚⠻⣤⣤⡽⠛⠀⠉⠙⠋⠁⠀⠀⠀⠀⠑⠢⢈⡐
⣪⠕⠉⠀⠀⣀⠔⠛⡉⠈⠁⠀⠀⠀⠀⢀⠀⠀⠁⡰⠃⢠⠻⣤⠶⣞⠈⡻⣆⠀⡀⣀⡾⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠁⠀⢂⡠⠋⣡⡴⡋⢀⣄⣠⠊⡰⢂⠔⣉⡴⠀⣰⠃⣰⣦⣢⠘⣶⡔⣾⣧⣨⣃⢹⡉⡀⠀⡀⠀⠪⠀⠰⣄⠀⠐⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠐⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠒⠲⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⡄⠀⣀⡄⠸⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣏⠀⣠⠀⠀⠀⠀⢠⠎⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣿⣿⣿⣿⡀⠁⠂⠀⠀⡐⠃⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⢿⣿⡏⠀⠀⠀⣄⠀⠸⠆⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⢀⡤⠀⠀⠀⠀⣶⣾⠟⣠⣶⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢰⠋⠐⠂⠀⢄⣠⡿⠷⠚⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠟⠀⣠⠖⠒⠚⠧⡀⢀⠀⠀⠀⠀⠀⠀⠀⠶⠛⢉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠸⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡤⠊⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠃⢀⡀⠀⠀⠀⠀⠏⢠⡴⠋⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣉⣽⣦⣀⡀⠀⠘⢀⠤⠒⠲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⣹⣿⣿⣿⣶⠆⠁⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠋⠉⢈⣿⣿⢀⡼⠁⣀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⣈⣭⠷⠞⠛⣿⠟⠀⣸⣿⣿⣿⣴⣾⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢛⣭⡶⢾⣿⣿⠏⠀⣼⣿⣿⣿⣿⣿⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⣿⣾⣿⣿⠏⠀⢸⣿⣿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣴⣿⣿⣿⠿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣹⣿⢛⣇⣶⣤⣼⣿⣿⠟⣦⣤⣴⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⢻⣿⣿⣿⣿⣿⣿⢋⡠⠏⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⠹⣮⠍⠛⢻⣿⣿⢿⣧⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⡇⠀⠀⠀⠀⣼⣿⡇⠘⢿⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢩⣿⣿⠁⠀⠈⣷⣄⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⡇⠈⠀⠀⢻⣹⡟⠀⠀⠀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢸⡇⢸⡇⠀⡆⣿⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠸⠿⢘⠃⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠘⢁⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣾⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠸⡇⠸⢿⣧
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡀⠀⠄⠛⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣶⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠘⢷⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⣀⢻⡄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠛⠛⠛⠛⠓⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⢀⡀⠀⠀⣀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⢀⣴⠿⠛⣧⣀⣼⠛⢿⣦⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣿⠃⠀⠀⠀⠁⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠹⢷⣄⣀⠀⠀⢀⣠⡾⠋⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠈⠉⠉⠉⠉⠁⢀⣴⡞⠋
⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣖⣒⡛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠸⠋⠉⠉⠙⠛⠛⣛⣻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⣶⡞⠋⠉⠉⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⢸⣿⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⣿⡟⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⢸⣿⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⢠⣿⠇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⣸⡿⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⢿⣧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠶⠶⠦⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢆⣤⣄⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢶⣅⡉⠉⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡼⣿⣶⡽⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠋⢻⡟⠉⠛⠃⠈⠛⠳⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀⣷⣾⡇⠀⠀⠀⠀⠀⠜⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣠⡞⢡⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⢀⡾⠋⣀⣼⣿⡟⠉⢹⣿⣿⣶⡆⠀⠀⢀⣴⣿⣿⡿⣄⣀
⠀⠀⠀⠀⢸⣄⣈⠉⠉⢁⣀⡀⠀⢻⣿⡿⣿⣦⣴⣿⣿⡷⢿⢇⣨⣿⡆
⠀⠀⠀⠀⠀⠙⠻⠿⣿⣯⣤⡝⠦⣀⠙⠓⠻⠿⠿⣿⣏⠀⠘⢿⣿⣗⣙⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣷⣾⠀⠀⠀⠉⠁⢹⢻⣇⠀⠀⢽⡛⣅⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡷⠂⠀⠀⠀⢼⢸⣏⣿⣆⠀⠀⠈⠪⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠁⠀⠀⠀⠀⢈⡸⣏⣸⣿⡄⠀⠀⠀⠈⠧⠁⠠⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠙⣀⣀⣀⣀⣀⣬⣅⣿⣥⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⢒⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⡿⠙⠻⣿⣿⠏⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⢸⣿⠇⠀⠀⠹⣧⠀⠀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠬⣾⡿⠀⠀⠀⠀⠹⣇⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣼⡇⠀⠀⠀⠀⠀⢻⠗⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢿⣿⠀⠀⠀⠀⠀⠸⠿⠙⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⣿⡇⠀⠀⠀⠀⠀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⠇⠀⠀⠀⠀⠀⠹⡄⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⡿⡔⠀⠀⠀⠀⠀⠂⠀⡄⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢸⣿⠇⠀⠀⠀⠀⠀⠀⠈⣗⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡶
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠋⠙⢿⣿⣿⣷⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣀⣀⡠⢻⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠋⠉⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡆⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣶⣷⣾⣿⣿⣿⠿⠃⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠛
⣠⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⢻⣿⣿⣿⣿⣿⣿⣿⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⣀⠀⠀⠀⢀⣀⠀⠀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠎⡀⠀⠈⢋⢲⠞⠋⠈⠉⠋⠉⠛⠁⠈⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⡞⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣢⠖⢢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡉⠡⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠢⠀⢀⠆⠀⠀⠀⠀⠀⠀⠘⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣁⠀⠀⣨⠀⠀⠀⠀⠀⠠⠀⢤⡠⠞⡀⠀⡀⢀⠖⠓⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⣗⠀⠐⠠⠀⠀⠀⠀⢠⣾⡶⣿⣆⠀⠀⠀⠀⠀⠀⠈⢹⣿⣧⣦⣤⡄⠀⠀⠀⠀⠀⠩⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠃⠀⠠⠀⣀⡀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠠⠘⠋⠻⠟⠁⡀⣀⠀⣄⣀⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢇⠆⢀⣤⠂⠀⠀⢀⡇⠂⠀⠀⠀⠀⠀⡄⠀⢀⣀⡵⣅⣩⠴⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠓⠋⠛⠉⢷⠀⠀⠀⠀⠀⢸⠛⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⣄⡴⠊⣿⡁⠀⠀⠀⠀⣿⠦⢤⡤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠀⠀⠀⠐⡂⢹⡤⢄⡀⠤⣀⣿⢐⠤⠀⠀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠆⢀⠀⣀⣄⠁⠐⠀⠀⣀⣀⣀⡿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠙⡏⠈⠛⠦⢾⡶⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⢧⡀⠀⢀⣠⠊⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠐⠒⢴⡉⠉⠋⡇⠀⣀⢀⣿⡁⠈⠉⠡⠀⠀⠀⠳⢦⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠀⢀⠐⠋⠁⢀⠀⠊⠙⠞⢄⠁⠀⠁⠀⠁⠀⠀⠀⠈⠁⢀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡃⠆⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠰⢎
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠛⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠐⠀⠀⠉⠑⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠛⠃⠉⠀⠀⠀⠀⠁⠀⠐⠐⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠉⠉⠉⠁⠈⠉⠙⠓⠊⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡖⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⣷⣧⠀⠀⢸⡄⠀⠀⠀⢀⣤⡾⣿⡋⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⠝⣷⡇⠀⣴⡍⠁⠀⣰⣽⢻⣿⡿⡛⠻⡆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢆⣠⣿⣺⠿⠅⢺⠋⣜⣻⣿⠧⠀⣀⣄⡹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡿⡶⡧⠬⣿⣷⣏⣓⣻⣶⣿⠚⠂⠒⠈⢻⡄⠠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⢿⣿⣿⢯⠤⠲⠛⣻⣿⣿⡿⠃⠀⠈⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢿⡷⠿⠁⠀⠀⡿⣿⡿⢶⣿⣿⡓⣸⣿⣷⣠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢿⣯⣾⠁⠀⠀⣠⣶⠁⠇⠄⠀⠀⣿⡇⣿⣿⣦⡍⠉⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣟⣷⣾⡻⠟⣿⢀⣬⠢⠤⠀⢸⡇⣿⣿⠆⠈⠑⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⢀⣼⡿⢃⢲⢒⢲⢲⢸⠏⢿⡷⢠⠀⠀⠾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠁⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⢑⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠁⠙⢆⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢇⠀⠀⠐⢂⡀⠘⠷⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠈⣈⢉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⡀⠀⠀⠀⢠⡟⠉⠉⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠿⠟⠁⢄⣰⠿⢆⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢸⠃⠁⠀⠙⢶⡤⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣆⠈⢣⣴⡿⣷⣦⣤⣶⠞⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠯⠃⢸⡟⢀⣈⣙⠻⠻⠃⢈⣧⠄⡀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠳⣼⣿⡙⠉⠛⠀⠀⠁⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠖⠀⣻⡛⢶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠲⣤⣈⢙⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠀⠀⣧⠀⠈⠈⢱⡿⡉⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⡌⡀⠀⠀⠀⡶⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠡⢸⠀⠀⠀⠀⠀⠀⠀⢡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡰⢀⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠑⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡠⠤⠀⠈⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠤⣴⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠀⠈⠀⠀⠀⠀⠀⢀⣈⣁⣀⣀⠜⠀⠀⠈⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣙⣛⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠁⠀⠀⢀⣼⣿⣿⣿⣷⡶⠦
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣏⣠⣀⣶⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠋⠉⠉⠉⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣀⣚⣷⣶⣶⣺⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿
⠀⠀⣠⣴⣿⣷⣶⣦⣤⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⠛⠛⠋⠉⠁⠀⠛⢿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣦⣭⣥⣤⣤⣤⣴⣤⣤⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠓⠚⠋⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠈⠁⢈⣿⣿⣿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣄⠀⠀⠀⣾⣶⣿⣿⣿
⠀⠀⠀⢠⣾⣿⣿⣥⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠓⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠉⠉⠁⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣷⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⡀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⣴⡿⠿⠋⠁⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⢰⣿⣿⣿⣿⣷⣤⣄⡈⠛⢿⣿⣿⣿
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⢤⣴⡟⠁⢀⣀⣤⣤⣶⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠙⠿⠿⠿⣿⣷⠈⠛⢛⣻⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣶⣾⣶⣶⣿⠏⠛⢿⣷⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠆⠀⠀⠀⠉⠀⣀⣴⣆⠈⠉⠉⠉⠉⣠⣿⣿
⠀⠀⠘⠟⠛⠻⣿⡏⠀⠀⠀⠹⣿⡏⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣾⣷⡆⢀⣾⣿⣿⣿⠀⠀⣤⣤⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⠿⠃⠀⠀⠀⠀⠉⠧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣱⣾⣿⣿⣿⣿⣧⣘⣿⣿⣿⣿⣿⣿