BUG Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⡇⠀⠀⢨⣌⣠⡄⠀⠀⢄⠀⠒⠀⠀⠀⡸⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣱⣄⠀⡠⢼⣁⠀⣀⣀⣈⣋⡛⠁⠀⢰⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡖⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢤⡄⣢⠼⠶⠦⢤⣀⣀⣀⣀⡀⣀⣀⣸⠁
⠀⠀⠀⠀⠀⣀⣸⠈⡡⠔⠊⠁⢙⠃⠀⠀⠀⠀⠣⣀⣀⠏⠀⠀⠀⠈⢳⣝⢗⡲⠤⣄
⠀⡠⢰⠶⠿⠛⠛⠋⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⣇⠀⠈⠲⣄⣹⢦
⠎⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢣⠈⠻⣢⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣹⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠎⠸⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⢿⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡸⠀⠀⠙⠄⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠋⠀⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢘⣦⣄⠇⠀⠀⠀⠀⠙⠿⣆⣺⣑⡒⡦⠶⠋⠀⠀⠀⢰⠀⢱
⠀⠀⠀⠀⠀⠀⢀⣰⠍⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢸⡆⢶⡃
⠀⠀⠀⠀⠀⢠⣲⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢐⣣
⠀⠀⠀⠀⣰⢁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢧
⠀⠀⠀⢠⠉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠀⠀⠀⠐⣦⠀⢀⢈⠙⠛⢉⢀⠀⢰⡞⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⣀⠀⠀⠀⠀⠀⠀⠈⠢⣘⡎⠀⠀⠈⣞⡠⠎⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢼⡟⣀⣄⡘⣿⠅⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⣠⣴⣉⣩⣿⢸⣏⣉⣳⣤⣀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⢀⡴⠫⢏⣾⢟⣿⡿⠸⣿⣟⣿⣝⢏⠳⣄⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⢟⣿⣿⡧⢰⣿⣿⡟⢿⡆⠀⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣇⢀⣏⠘⣿⣿⡇⠀⣿⣿⡟⢈⣅⢀⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⣋⣿⣮⣿⡀⢻⣿⣷⢸⣿⣿⠃⣸⣯⣾⣍⡉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⠟⠁⢠⢞⣉⡉⠙⠛⢿⣶⣍⣉⣈⣉⣵⣾⠟⠛⠉⣉⢙⢷⠈⠻⣿⣶⣤⣀⣀⡀
⠀⠀⠀⠀⠀⢠⠚⠋⠉⠉⠉⠈⠉⠁⠀⠀⢸⡸⣿⣿⡀⣀⢠⠈⣧⢳⢰⢯⡎⢀⡆⣠⣤⣿⡿⢻⠀⠀⠈⠉⠀⠀⠉⠉⠉⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢹⣿⢡⠿⣿⡋⢏⠏⢇⠋⡖⡻⣁⣗⢹⣿⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢨⣿⣾⣶⣎⡿⣦⡘⡬⢠⣸⡟⣡⣼⣿⢟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢘⠉⢯⡽⣟⣼⣷⣀⡗⢰⣛⣏⣿⢐⡟⠺⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣀⣚⣘⡯⣤⢾⣉⡷⢾⣱⡖⣟⣨⣤⡄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠉⠉⠣⡤⢭⣟⢀⡯⠨⡿⢧⡇⡫⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠼⢮⣛⢎⡏⠠⡿⠰⠷⡞⣷⠼⢄⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⣾⣧⣀⣀⣠⣷⣀⠀⣬⣿⣶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠉⠉⠁⠀⠁⠀⠉⠉⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⡿⠿⠿⠿⠿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⠀⠀⠀⢀⣀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡄
⠀⠀⠀⠀⠀⢀⢸⣷⢀⠀⠀⠀⠙⣧⠰⣟⠳⡄⠀⢀⠴⢻⡇⣰⠟⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠸⣧⠇
⠀⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠘⢧⣊⣭⢾⣶⡾⢮⣕⣰⠋⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⣦⣼⣴⠆
⠀⠀⠀⠀⠀⠁⣿⣿⡇⠁⠀⠀⠀⠀⠀⡄⠀⠀⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⢺⣿⣟
⠀⠀⠀⠀⠀⠀⡟⠹⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢴⣶⣶⣤⣤⣤⣤⢷⣤⣤⣤⣤⣶⣶⠆
⠀⠀⠺⣽⣷⢦⣄⣀⣤⣶⣿⡽⠂⠀⠀⠀⠰⣿⣿⠀⠀⠀⠀⠈⠉⣛⣿⣿⡿⣾⣿⣿⣟⣋⠉
⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⡘⠁⠀⠀⠀⠐⢸⣷⣿⠁⠀⠀⠀⠐⠟⠛⠛⠉⠀⡇⠈⠙⠛⠛⠷⠀⠀⢠⣴⣤
⠀⠀⠀⠀⠙⠿⠋⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣄⣤⣄⣀⠀⣀⠀⢀⣠⣤⣄⡄
⠀⠀⠀⠀⠀⠀⡸⡿⡀⠀⠀⠀⠀⠀⣠⣾⡿⠡⠭⠻⣿⣦⡀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠈⠛⢻⣿⣿⣧⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠈⠉⠁⠀⠰⠇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠈⠉⠉⣿⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠘⣿⣿⡟⠢⠀⠀⡰⠛⣿⣿⠏⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⣼⣿⣤⠀⠀⠀⠀⠹⢟⣇⣀⣠⣦⣀⣠⣟⠟⠀⠀⠀⠀⢀⣮⠍⣦⠀⠀⠀⠀⠀⠀⢀⣶⣆
⠀⠀⠀⠀⢀⣾⡿⠿⣿⡆⠀⠀⠀⢐⣿⠋⠀⠘⠇⠀⠙⢻⡆⠀⠀⠀⠀⢸⣿⣾⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⡿⢙⣚⠹⣿⠀⠀⠀⠀⢻⣷⡴⠀⠀⠢⣶⣿⠁⠀⠀⠀⠀⠀⢻⣿⠏⠂⠀⠀⠀⠸⠛⠉⠿⠋⠙⠿
⠀⠀⠀⠀⠈⠁⠈⠉⠀⠉⠀⠀⠀⠰⠃⠈⠀⠀⠀⠀⠉⠈⠦

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢰⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡎⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠘⢿⣦⣀⢰⣾⣿⣿⣶⢀⣠⣾⠟⠀⠀⠀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣄⠀⠀⠈⢿⣿⡿⣿⣿⣿⡿⣿⣿⠏⠀⠀⢀⣼⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣴⣾⣿⣯⣿⣿⣿⣿⣽⣿⣿⣿⣶⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠋⠁⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠇⠀⠀⠉⠳⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣸⡇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⡀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠉⠛⠋⠛⠋⠁⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠤⣤⣀⣀⠀⠀⠀⣠⣤⡀⠀⠀⣀⣀⡤⠤⠤⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⣴⣿⡄⠀⠀⢈⡁⠀⠀⠈⠀⠀⠂⠀⢈⣉⣷⣶⣿⣿⣷⣾⣉⡁⠀⠐⠀⠀⠁⠀⠀⢈⠁⠀⢀⢠⣿⣦⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣷⡠⠀⡁⠀⠀⢀⠀⠠⠒⢀⠔⢁⠴⣻⣿⣿⣿⣿⣟⠆⡈⠢⡀⠒⠄⠀⢀⠁⠠⢈⠐⢄⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠀⠀⠐⠀⣿⡾⠠⠀⠁⠐⡡⢪⢾⣿⣿⣿⣿⡷⡕⢌⠢⠈⠀⠄⢶⣿⠀⠂⠀⠀⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀⠀⠀⠀⠀⡀⢠⣤⢊⡴⠋⢜⣿⣿⣿⣿⡡⠙⢦⡑⣤⡄⢀⠁⠀⠀⠀⢀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣀⠊⠠⠂⠀⠚⠉⡜⠁⡈⡘⢼⣿⣿⡿⢃⢁⠈⢣⠉⠓⠀⠐⠄⠐⣀⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠔⠁⡠⡺⠀⣠⠀⡇⢺⣿⣾⡟⢸⠀⣆⠀⢗⢄⠈⠢⠀⢱⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⠋⢠⡧⠊⠁⠀⠀⠸⣿⣿⠇⠀⠀⠈⠑⢼⣄⠙⠶⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠠⠋⠀⠠⠀⠀⠀⠃⢻⡟⠘⠀⠀⠀⠄⠀⠉⠆⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣠⠁⠀⠀⠂⠀⠀⠀⠀⡌⢡⠄⠀⠀⠀⠀⠀⠀⠈⣄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠼⠀⣀⣄⠠⠜⠁⠈⠣⠄⣠⣀⡀⠥⠄⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠢⢤⡀⠀⠀⠀⢀⣠⠤⢄⡀
⠀⠀⠀⠀⠀⣠⠒⢛⣛⡒⠢⢄⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⡄⠀⡴⠋⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⣇⠀⡏⠀⠉⠑⠦⣈⠳⢄⠀⠀⠀⠀⠣⡀⢀⡆⠀⠀⢱⣸⠁⠀⢰⠀⠀⢠⠇⠀⠀⠀⡠⠖⣋⠥⠒⠒⢢⠀⢘⡄
⠀⠀⠀⠀⠀⠘⢦⡙⢄⡀⠀⠀⠈⠑⠦⡙⢦⡀⠀⠀⠈⠉⠀⠀⢀⣸⣇⠀⠀⠈⠓⠒⠋⠀⣀⠔⣫⠔⠋⠀⠀⠀⢀⠞⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⠒⠤⣀⠀⠀⠈⠲⣝⠦⡀⠀⠀⠀⡴⠋⠆⠰⠑⡄⠀⠀⠀⣠⢞⡥⠊⠁⠀⠀⣀⡤⠒⣁⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢬⡙⠢⢄⡀⠈⠑⣍⢦⡀⠀⣇⠀⠑⠃⢀⡇⠀⡠⢊⠔⠉⠀⢀⡠⠔⢋⡡⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣌⡓⠢⢌⣷⠙⢦⡈⡷⠠⢶⢋⡤⠊⡔⣁⣠⠴⢊⣡⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣤⠀⠀⢹⠓⠒⢾⠋⠀⢀⣩⠥⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣋⡭⠤⡄⢀⣧⣤⣀⣸⡀⠀⣀⣉⡙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⣁⠴⠚⠉⢀⡴⠊⢀⣼⠀⠀⠀⠈⡟⢆⠑⢄⡈⠑⠲⢤⡉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⡠⠔⠋⠀⠀⣠⠔⠉⢀⠔⠁⡏⠉⠉⠉⠙⡇⠀⠱⣄⠙⠦⡀⠀⠈⠑⠦⣈⠙⠢⣄
⠀⠀⠀⠀⠀⣠⠞⠁⣀⠔⠋⠀⠀⢀⡤⠊⠁⡠⠞⠁⠀⢸⠷⠤⠤⠤⢤⡇⠀⠀⠈⠑⢄⠈⠓⢄⡀⠀⠀⠑⠦⡀⠙⢦⡀
⠀⠀⠀⠀⡴⠁⢀⠞⠁⠀⣀⠤⠚⢁⡠⠔⠋⠀⠀⠀⠀⢸⣄⣀⣀⣀⣰⠁⠀⠀⠀⠀⠀⠙⠦⣀⠉⠲⢄⡀⠀⠈⠢⡀⠙⢦
⠀⠀⠀⢰⡁⠀⠘⠒⠒⣉⣡⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡈⠓⠢⠤⠜⠀⠈⡆
⠀⠀⠀⠀⠑⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠛⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⣀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠦⠤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠃

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠿⠛⠛⠛⠱⣼⠀⠀⠀⠀⠀⠈⠱⣦⡀⠀⣠⣤⣤⣄⠀⢀⣤⠏⠁⠀⠀⠀⠀⢠⣇⡎⠛⠛⠛⠿⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠟⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⡴⠫⠈⣩⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠱⣄⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣿⣥⣤⣾⠿⠅⠹⠇⠻⠏⣋⠁⡿⣘⣻⣿⣦⠶⣿⣿⣤⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢯⣶⣾⣧⣦⣿⣴⣷⣀⣴⣿⣶⣿⣿⣿⣳⣿⣷⣬⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡖⣉⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠪⠉⡉⠉⠉⠉⠉⠉⠛⣛⣿⣿⣟⡛⠛⠉⡉⢩⠁⣤⠀⠀⠢⢦⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⡄⠀⠀⠀⢂⠐⣰⣾⣿⣿⣿⣿⣿⣶⣄⠡⡜⠇⠀⠀⠀⠀⠀⠛⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠊⠀⠄⠐⠀⠀⠀⠀⠁⠘⣿⣿⣿⣿⣿⣿⣿⠉⠇⠁⠀⠀⠀⠀⡀⠀⠀⡀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣶⠁⠀⠈⠔⠀⠀⠂⠀⠀⠀⠈⠻⢿⡏⠛⡍⠥⠡⠈⠀⠀⢆⠀⠄⠠⠀⠀⡀⠄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣯⠀⠀⠈⠎⡀⠀⠀⠀⠀⠀⣀⡙⠾⠁⠤⢱⡚⡆⢣⠀⠀⠈⡄⠠⠀⠂⠀⠀⣠⡎⣿⡆
⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣿⣿⢸⣿⡇⠂⠈⠰⢀⠀⠀⠀⡠⠳⣲⢳⣻⠠⠘⡀⢣⢭⠀⠀⠀⠀⠱⠀⠀⠠⠀⢸⣿⡧⢹⣿⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠋⠉⠉⠙⠿⣦⣸⣿⣧⠜⠟⠁⡢⠈⠃⣌⠠⠀⠠⣻⢧⣟⢾⣿⠀⠠⣹⡌⡂⠂⠈⠀⢀⡀⠠⠆⠀⠄⠄⠛⠃⢸⣿⣇⣶⡿⠋⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡐⠁⠋⣠⣽⣿⣿⣿⣧⡺⡿⣿⣾⠀⠱⣼⣧⣀⣴⣶⣿⣿⣿⣷⣦⣇⠘⡌⠀⠁⢸⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢁⠂⣾⣿⣿⣿⣿⣿⣿⣿⣿⢷⡛⢸⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠌⡰⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣦⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠣⡈⡆⠁⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠁⠍⡙⠛⠿⠟⠻⣿⠟⠠⠷⣿⠸⣌⣟⢼⠋⡟⠛⠋⣩⢩⠉⡔⣆⢩⠐⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠈⠀⠀⠂⡖⡲⠆⠀⠴⣗⡳⢽⢪⠐⢩⣾⡿⡇⢩⠀⢣⠤⠄⢤⠤⡸⡀⠆⠰⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⡄⣄⠀⢀⠁⠄⠂⢠⡄⢨⢐⢨⠰⠀⢺⡇⠀⢁⠀⡀⡰⢦⢱⠐⡀⢀⣱⣘⢂⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠻⣿⣧⣌⠀⡃⠍⢉⡴⢸⠚⢰⢸⢩⣿⢁⠀⠈⣀⠳⢨⣏⡐⣦⣿⣿⣿⣯⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠙⣿⣿⣷⡀⠂⣡⠜⢹⢾⠨⣽⠓⡏⢸⠀⡇⢯⠸⠄⡄⢡⣿⣿⣿⡿⢃⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠈⠻⣋⠡⡄⡁⠈⢹⢘⠀⡇⡀⡇⡘⠄⠹⢨⠙⠆⣡⢨⠝⣣⠞⠁⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⠀⠀⠈⠳⢍⡁⠠⣿⢸⡀⣧⡅⠂⡋⣆⢸⠄⠲⣷⢪⡠⠞⠁⠀⠸⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠒⠿⣼⣃⡘⡇⢁⡁⠥⠘⣁⠤⠚⠉⠀⠀⠀⠀⠀⠈⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄

⠀⠀⠀⠀⠀⠰⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠞
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠙⢧⣀⣤⣀⠟⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢷⠀⠀⠀⠀⠀⠀⢀⠞⠉⠲⡀⠀⠀⠀⠀⠀⠀⡴⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⣇⠀⠀⡴⠂⢀⡙⠚⠓⠚⠉⣉⠉⠢⡀⠀⣠⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⡄⢸⡁⣾⣿⣿⣿⠀⢸⣿⣿⣿⠀⡇⢠⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣉⣇⠙⢿⣿⣿⠀⢸⣿⣿⠿⢠⢏⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⠘⣿⡄⢸⣿⠁⣴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠊⣩⠶⠿⣷⡿⠿⣭⡁⠉⠑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣾⢁⣴⣾⣿⣧⠀⠀⢿⡇⠀⣰⣿⣿⣷⡄⠈⣟⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⢚⠇⢸⣿⣿⣿⣿⠀⠀⢸⡇⠀⢻⣿⣿⣿⡏⠀⠸⡈⠹⢆
⠀⠀⠀⠀⠀⠀⠀⡐⡘⠀⡼⠀⠀⠙⠛⠛⠁⣠⣾⣿⣿⣷⣄⠉⠉⠉⠀⠀⠀⢧⠀⢆⢣
⠀⠀⠀⠀⠀⠀⣸⢰⠃⢠⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⡀⠈⡄⢣
⠀⠀⠀⠀⠀⢠⣳⠃⠀⡜⠀⠀⣀⣀⣀⠀⠈⢿⣿⣿⣿⣿⡟⠀⠀⣀⣀⣀⠀⠈⠇⠀⠈⢴⣇
⠀⠀⠀⠀⣠⠞⠁⠀⢀⡇⢠⣿⣿⣿⣿⣿⣷⣄⠉⢹⡟⠉⣠⣾⣿⣿⣿⣿⣧⠀⢸⠀⠀⠀⠘⢦⣀
⠀⠀⣠⡼⠁⠀⠀⠀⢸⠀⠸⣿⣿⣿⣿⣿⣿⡿⠀⢸⡇⠀⢿⣿⣿⣿⣿⣿⠟⠀⠈⡄⠀⠀⠀⠀⠹⠕
⠀⠈⠑⠁⠀⠀⠀⠀⢸⠀⠀⠈⠙⠿⢿⡿⠟⢁⣀⣼⣧⣀⠈⠻⠿⠿⠛⠁⠀⠀⠐⣇
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⣇⢣
⠀⠀⠀⠀⠀⠀⢰⡁⢾⣠⣴⣶⣆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⣤⣤⡀⡇⢉⠄
⠀⠀⠀⠀⠀⠀⣼⡅⠘⣿⣿⣿⣿⠀⠀⠀⠙⠿⠿⣿⣿⣿⠿⠟⠁⠀⠀⣿⣿⣿⣿⠁⠘⡘⡄
⠀⠀⠀⠀⠀⢀⢃⠃⠀⢻⣿⣿⡟⠀⠀⣠⣤⣄⠀⠀⡇⠀⣠⣤⣤⡀⠀⠹⣿⣿⡿⠀⠀⡇⢱
⠀⠀⠀⠀⠀⣼⢸⠀⠀⠀⢿⡟⠀⠀⣼⣿⣿⣿⡇⠀⡇⠀⣿⣿⣿⣿⠀⠀⠈⣿⠃⠀⠀⠸⡀⢃
⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⠱⣄⠀⣿⣿⣿⣿⠇⠀⡇⠀⣿⣿⣿⣿⠇⠀⡰⠃⠀⠀⠀⠀⠳⣿⡀
⠀⠀⠀⠀⣠⠟⠃⠀⠀⠀⠀⠀⠈⠳⢝⡻⠿⠋⠀⠀⡇⠀⠈⠛⠛⢋⡠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠚⠓⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⡑
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠀⠇⠰⣲
⠀⠀⠀⠈⢃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣿⣷⣆⠀⠔⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠉⠙⠥⢔⣀⠀⠀⠀⣀⣴⣠⣴⡾⠛⠉⢻⣿⣦⣀⡀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⢿⣿⡋⠀⠀⣠⣾⣿⣿⣿⣷⣀⣀⢴⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠿⣿⣙⣻⣿⠁⢈⣏⣰⣿⣻⡟⢁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⣠⠄⠀⠂⠈⢹⣿⡿⣋⠛⠀⣣⡾⢻⠶⢋⡟⠻⡉⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠓⠦⡀⠀⠈⠙⢿⣏⣦⣾⣫⣔⣧⣚⣇⠈⢶⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⣠⣼⠟⡿⢛⣡⠾⣻⣿⣿⡟⠁⠀⡌⠀⢤⣤⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⣟⣦⡼⠿⢿⣷⣎⠁⠀⢀⡨⠀⠹⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⢿⣷⣤⡶⠟⠻⣿⣿⡜⠊⠃⡀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠙⣿⣿⣶⠶⠛⠿⣿⣾⡌⠁⠀⠀⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠸⣇⠈⠛⠿⣶⣿⣯⣻⡧⠀⠀⠀⠀⠘⠄⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠳⣄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣠⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣄⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣇⠀⠀⣠⣿⣽⡆⢰⣯⣿⣄⡀⠀⣸⣽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠿⠋⣿⠁⠀⠀⠠⣾⠙⠿⢷⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣏⡀⢀⣠⣷⢀⣼⣧⠀⣿⣆⡀⢀⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠻⢿⣿⣿⡿⡞⢿⡿⢳⢿⣿⣿⡿⠟⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⣀⣀⣠⠄⢼⣿⣿⡿⢿⣿⣿⣇⣤⣄⣀⣀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⡛⠿⣿⣿⠿⠿⠟⠁⠘⠻⠿⣿⢿⣿⠿⢛⡇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⣇⠀⣧⣭⣄⣄⣀⡀⢀⣀⣠⣤⣴⣿⠀⢸⡇⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣿⣆⢿⡿⠛⠛⡛⠃⠙⠟⠻⠟⣿⡿⣰⣿⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣸⠇⢼⣷⣤⣤⣤⡇⢠⣴⣤⣴⣿⣏⠸⣇⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⣜⡿⠀⣸⠿⢿⡿⢿⡄⢰⡿⢿⣿⣿⣇⠀⢿⣣⡀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠸⠏⡀⣾⣿⣤⣠⠀⠀⠀⡀⠀⠀⣄⣦⣿⣶⣄⠩⠅⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣦⡼⠋⢹⣿⣿⣾⣿⣾⣷⣿⣿⣿⣿⡏⠙⢿⣶⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣧⠀⠀⠘⣇⠀⠀⠈⢸⡇⠁⠀⢀⣼⠃⠀⠀⣼⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⢷⡀⠀⢿⣿⣿⣷⢾⣷⣾⣿⣿⡿⠀⢀⡾⠁⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⣿⡄⠘⣧⣀⣉⣼⣇⣁⣀⣼⠃⢠⣿⠁⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠈⣷⠀⠘⢿⣿⣿⣿⠿⡿⠃⠀⣾⠁⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⠀⠀⠀⠰⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠂⠀⠀⠀⠠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⡆⢠⣠
⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣦⣿⡀⠀⠀⠀⠃⠀⠸⡇
⠀⠀⠀⠀⠈⢴⡤⢆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⠉⠊⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⣀⣀⣠⣾⣤⣶⡿⠛⠁⠈⣹⣿⣷⣤⡄⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣿⣏⣀⠀⣀⣴⣿⣿⣿⡿⣿⣦⣤⢴⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⢿⣍⣝⣿⣏⠀⢨⣧⣴⠿⣹⡿⢋⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⢠⠤⠀⠘⠀⠉⣿⣿⠿⣉⠋⠀⣣⣾⠟⡶⠚⣹⠛⢏⠉⠛⠶⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠽⠷⠦⡀⠀⠀⠉⠻⣿⣍⣢⣾⢟⡵⣪⡔⣫⠈⠡⡆⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⢀⣽⠟⣿⠟⣉⠾⡻⣿⣾⣿⠖⠀⠀⡔⠀⣀⣀⡀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠈⣷⣿⡟⣄⣰⣿⣿⣿⣦⠖⠈⠁⡀⢙⠉⢹⡀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⠹⣿⣯⣙⣄⣴⠿⣿⣧⣄⢴⠙⡆⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠻⣿⣿⣅⣂⣹⣿⣿⣆⣤⠸⠂⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⡿⠛⢿⣯⣄⣠⣽⢿⣿⡆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠳⢄⡀⠉⠛⠻⠿⠷⠞⠃⠀⠀⠀⠀⠀⠈⠂⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠨⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠄⠀⠀⠀⠀⠀⠠⠞⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡖⠀⠀⠀⠀⠀⠀⠀⠐⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢠⠄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⢻⣷⣤⣀⡀⡄⡀⢀⣠⣴⣿⠃⠀⠀⢀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌⠉⠀⠀⠈⣻⣿⣿⣿⣷⣾⣿⣿⣿⠋⠀⠀⠀⠀⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡴⢚⣿⣗⣻⣶⣶⣦⣤⣌⣹⣿⣿⠿⢿⣿⣿⣉⣠⣤⠶⠶⠻⢒⣶⡶⠖⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣬⣤⣄⣀⠀⠈⣨⣿⣿⡿⢻⣿⣿⣿⣰⣾⣿⣿⣿⣿⣿⣿⣿⣷⠿⠛⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠜⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣂⣀⣀⣤⣤⠶⣮⣍⣀⡀
⠀⠀⠀⢀⣴⣾⣿⣷⣦⣰⣞⣂⣀⣀⣈⣉⣛⠛⠻⢿⣿⣶⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⢿⣛⣛⣉⢉⣙⣿⣇⣤⣭⣭⣿⣷⣤⡀
⠀⠀⠰⠛⠉⠈⢉⠽⠛⢿⣿⣿⣿⣿⣿⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⢿⡻⣿⣛⠉⠙⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⡿⣿⣿⣿⣿⣿⢿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡟⠻⠿⣿⣻⣯⡻⠙⠢⠁⠀⠙⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⢠⣿⠋⠀⢻⣿⣻⣿⣿⠛⢻⣿⣿⣿⣧⡿⠀⠘⢿⡀⠀⠈⠙⠗⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡴⠟⢻⣼⣿⣿⡁⢹⣿⣿⣿⠉⠻⡆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⣾⣿⣿⡇⣸⣿⣿⠏⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡀⠀⠈⠻⣿⣷⣿⡿⠋⠀⠀⢸⠇⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠙⢦⠀⠀⢀⡴⠊⠉⠉⠑⣄
⠀⠀⠀⠀⠀⢰⠁⢠⠖⠒⠦⣌⡙⠢⣄⠀⠀⠀⠀⠀⢸⡀⠀⠀⡀⠀⠀⢧⢀⡞⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡤⠖⣚⣉⣉⠉⠓⢆
⠀⠀⠀⠀⠀⠸⣆⠸⡄⠀⠀⠀⠙⠲⣄⡑⢦⡀⠀⠀⠀⠑⠤⠔⠁⠀⠀⢸⡼⠀⠀⠀⢇⣀⣀⠞⠀⠀⢀⡤⢚⣡⠖⠋⠁⠀⢈⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠈⠢⣈⠲⣄⡀⠀⠀⠀⠙⠢⣙⠦⡀⠀⠀⠀⠀⠀⢀⡠⠼⠧⢄⠀⠀⠀⠉⠁⠀⣠⠔⣩⠔⠋⠀⠀⠀⠀⣠⠞⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣉⠓⢤⣀⠀⠀⠈⠳⣌⠢⡀⠀⠀⢀⡎⠈⡃⡘⠃⢳⠀⠀⠀⣠⢞⡥⠊⠁⠀⠀⣀⡤⠖⢋⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣌⡙⠲⢄⡀⠈⠳⣌⠢⡀⠈⢆⠀⠈⠁⢀⡞⠀⡠⢊⡵⠋⠀⢀⡠⠔⠊⣡⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⡓⠲⠬⠇⠈⢦⡀⡟⠒⢲⢉⡴⠋⣰⣋⡠⠴⢚⣉⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣲⣤⠀⠀⡹⠓⠒⠺⡏⠀⢀⣈⡥⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⢚⣉⠤⢤⠀⢠⣧⣄⣀⣈⡇⠀⢀⣈⣉⠓⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢉⡤⠖⠋⠁⢀⠴⠋⢀⡞⠀⠀⠀⠀⡟⢦⡘⢦⡈⠉⠒⠤⣌⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⣀⠤⠚⠁⠀⠀⣠⠞⠁⢀⠔⢁⡟⠛⠛⠛⠚⡇⠀⠑⢄⠉⠢⣄⠀⠀⠉⠲⢄⡈⠓⠦⣀
⠀⠀⠀⠀⠀⠀⣠⠖⠉⢀⡤⠊⠁⠀⠀⣀⠴⠊⠀⣠⠞⠁⠀⢸⣤⣀⣀⣀⣀⡇⠀⠀⠈⠑⢄⠈⠓⢤⡀⠀⠀⠉⠳⢄⡈⠑⢦⡀
⠀⠀⠀⠀⢀⠞⠁⢀⡴⠋⠀⠀⣀⡤⠚⠁⣠⠴⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠙⢦⣀⠉⠲⢄⡀⠀⠀⠙⢦⡀⠙⢦⡀
⠀⠀⠀⠀⡞⠀⠀⢏⣀⠤⠔⠋⣁⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⢸⠉⠉⠛⠛⢻⠇⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠉⠓⠤⣄⣀⢙⡆⠀⢳⡀
⠀⠀⠀⠀⠣⣄⣀⣠⡤⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⢤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣀⡉⠉⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡁⠀⠀⠀⠀⢤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⠀⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣤⣴⡶⠶⠶⢿⣻⡇⣼⣧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢺⣿⣿⣟⠉⠀⠀⠀⠀⠀⠉⢸⣿⣦⣼⣿⣿⢦⡀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠘⠿⣿⣿⣷⣦⣀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣷⣴⣿⣤⠀⠀⠀⠒⠂⠁⠀⠀⠀⠀⣠⣄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠀⠀⠀⠙⠶⠤⢤⣈⡉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⡿⣷⠀⠀⠀⠀⢠⣴⣶⠾⠿⠿⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣦⣄⡈⠛⢿⣿⣿⣿⣿⣿⡇⠀⠈⠙⢻⡄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡶⣿⠟⠛⠛⠲⢤⣈⠙⠿⣿⣿⣇⠀⠀⠀⠈⣿⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣫⣾⣿⣧⠀⠀⠀⠀⠈⠑⢦⡈⠻⣿⣦⡀⠀⠀⣿⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣫⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣶⡤⢸⣤⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠉⢀⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣶⠟⠹⠟⠁⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠙⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠂⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣄⡀
⠀⢀⣤⣴⣾⡿⠃⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛
⣠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠻⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⢹⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣤⣄⡀⠀⠀⠀⢀⣤⣤⡤⣤⡀⠀⠀⠀⢀⣠⣤⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡀⣀⣰⣿⡼⠧⠸⠧⣿⣦⣀⢀⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠞⠛⠁⣿⠂⢀⡀⠐⣿⠈⠛⠳⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣧⣤⣤⣴⣿⢤⣾⣷⡐⣿⣦⣄⣤⣼⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠁⠀⠉⠛⠻⠿⣿⣧⣹⣏⣼⣿⠿⠟⠛⠉⠀⠈⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣤⣤⣴⣮⣩⣯⣿⠏⢹⣿⣿⣯⣷⣦⣤⣤⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠸⡏⠉⢻⢿⠛⠋⠉⠀⠀⠈⠙⠛⠛⡟⠉⣸⠃⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣿⡀⢸⣶⣶⣶⣶⡆⢴⣶⣾⣿⣿⡇⢀⣿⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢸⡿⢸⣏⠁⠀⠐⡄⠀⢁⠈⢈⣿⡇⢾⡏⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⣶⠇⢸⣿⣿⣿⣿⡃⢘⣿⣿⣿⣿⡇⠸⣷⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⡾⠚⢀⣼⢋⠉⠉⠙⠋⠈⠋⠙⠋⣛⣧⣀⠛⢷⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣁⣈⡾⠿⣿⣾⣤⣤⣤⣦⣴⣤⣿⣿⡿⢿⣗⣌⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣾⠋⠀⠐⡟⠉⠉⠙⢹⡟⠛⠙⠛⣻⠃⠀⠙⣷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠘⢧⡀⠀⢿⣶⣶⣤⣸⣇⣴⣶⣶⡿⠀⢀⡼⠃⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⢷⡄⠘⣏⠙⠛⢻⡏⠋⠉⣻⠃⢀⡾⠁⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠘⣻⡀⠹⣿⣿⣿⣿⣾⣿⠏⢀⣿⠃⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⣺⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣗⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠒⠂⠀⠀⠀⠉⠉⠲⠦⢤⣀⣾⣷⣤⠤⠶⠊⠉⠁⠀⠀⠀⠒⠂⠄⠀⠀⣠⣀
⠀⠀⠀⠀⣠⣿⣿⣷⠀⠀⠂⠁⠀⠀⠀⣠⠄⠀⠖⢋⢽⣿⣿⣿⣿⢍⠑⢂⠀⢤⡀⠀⠀⠀⠁⠂⠀⢰⣿⣿⣧⡀
⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠁⠀⢰⣷⡆⠀⠀⡀⠂⢔⣽⣿⣿⣿⣿⣕⢄⠢⡀⠀⡀⣴⣷⠀⠀⠁⢸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⠀⠀⠀⠉⢀⠀⣤⡔⣥⠎⢼⣿⣿⣿⣟⠈⢦⡕⣤⡄⠀⠈⠁⠀⠀⢰⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⡠⠂⠔⠀⠐⠋⡹⠃⡀⡘⣿⣿⣿⡟⠀⡀⠻⡉⠓⠀⠐⠄⠂⣠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠠⠊⣠⢼⠁⣠⠀⡇⢿⣿⣿⠇⡇⢠⡀⢻⢤⡈⠢⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊⢁⡦⠊⠀⠀⠀⢸⣿⣿⠀⠀⠀⠈⠢⣦⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠊⠀⠠⠀⠀⠀⠃⠻⠃⠃⠀⠀⠠⠀⠈⢂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⠀⠀⠀⠀⢰⠁⢱⠀⠀⠀⠀⠀⠀⢀⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠂⠒⠒⠂⠁⠀⠀⠁⠒⠒⠒⠚⠁⠁

⠀⠀⠀⠀⠀⠀⢤⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠈⠻⣀⠀⣀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣦⠀⠀⠀⠀⠀⠀⠀⢈⡽⠙⢯⡀⠀⠀⠀⠀⠀⠀⠀⡰⢹⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡇⠀⠀⠀⡠⠄⠀⠾⠤⠤⠤⠗⠒⠒⠤⡀⠀⠀⢰⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢸⡄⠀⡞⢀⣴⣶⣶⣦⠀⢠⣴⣶⣶⡄⢰⠀⢀⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣵⡤⣇⠸⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇⢸⠦⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⣦⡈⠛⢿⣿⠀⢸⣿⡿⠛⣠⡟⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡆⢸⣿⡀⣸⣿⣀⡼⢅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⣡⠖⠛⠛⣿⡟⠛⠲⣄⣀⡀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣺⠁⣴⣿⣿⣿⣆⠀⠀⢿⡇⠀⢠⣿⣿⣿⣷⡀⠘⣯⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢚⡇⠐⣿⣿⣿⣿⡿⠀⠀⣸⡇⠀⠸⣿⣿⣿⣿⠇⠀⢸⡈⠹⢢
⠀⠀⠀⠀⠀⠀⠀⢀⡎⡏⠀⣼⠀⠀⠈⠛⠿⠛⠁⣴⣾⣿⣿⣷⣆⠈⠙⠛⠁⠀⠀⠀⣇⠀⢆⠣⡀
⠀⠀⠀⠀⠀⠀⠀⡜⢰⠁⢠⠇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢹⠀⠈⠄⠣⡀
⠀⠀⠀⠀⠀⠀⠰⣱⠃⠀⡸⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠐⡇⠀⠘⢄⣷
⠀⠀⠀⠀⠀⢀⡞⠃⠀⠀⡇⠀⣴⣿⣿⣿⣶⣦⣄⠙⠛⢿⡿⠟⠁⣠⣶⣾⣿⣿⣿⡆⠀⢸⠀⠀⠈⠙⢦⡀
⠀⠀⠀⢀⡴⠋⠀⠀⠀⢸⠃⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠸⡆⠀⠀⠀⠀⠹⣖⠄
⠀⠀⠔⡟⠀⠀⠀⠀⠀⣾⠀⠈⠻⢿⣿⣿⣿⣿⣿⠃⠀⢸⡇⠀⠘⣿⣿⣿⣿⡿⠛⠁⠀⠀⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠉⠙⠛⠋⣡⣤⣶⣿⣿⣶⣤⣌⡉⠉⠉⠀⠀⠀⠀⠀⡗⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⡿⣸⡆
⠀⠀⠀⠀⠀⠀⠀⢾⡔⢻⣤⣶⣿⣷⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣴⣶⣶⣄⡃⠨⠰⡀
⠀⠀⠀⠀⠀⠀⢀⣷⡆⠸⣿⣿⣿⣿⡇⠀⠀⠈⠙⠻⠿⢿⣿⡿⠿⠛⠁⠀⠀⠀⣿⣿⣿⣿⠇⠀⡆⢣
⠀⠀⠀⠀⠀⠀⡸⣸⠀⠀⢻⣿⣿⡿⠁⠀⢀⣤⣦⣄⠀⠀⡇⠀⢀⣤⣤⣄⠀⠀⠻⣿⣿⡿⠀⠀⢱⠈⡆
⠀⠀⠀⠀⠀⢀⡇⢹⠀⠀⠈⢿⡿⠃⠀⢀⣿⣿⣿⣿⡇⠀⡇⠀⣿⣿⣿⣿⣇⠀⠀⠙⣿⠃⠀⠀⠈⡆⠸⡀
⠀⠀⠀⠀⠀⢸⠁⡏⠀⠀⠀⠀⠳⣄⠀⢸⣿⣿⣿⣿⡇⠀⡇⠀⢿⣿⣿⣿⣿⠀⠀⣠⠋⠀⠀⠀⠀⠸⣶⡇
⠀⠀⠀⠀⠀⣸⡾⠁⠀⠀⠀⠀⠀⠈⠳⣜⠿⣿⣿⠟⠀⠀⡇⠀⠈⠻⠿⠿⠃⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣤⣴⣧⣤⣤⣄⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢦
⣠⢖⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⢹⡇⠀⠀⠀⠀⡴⣶⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠴⣫⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣧⠀⠀⠀⠀⣶⣥⣤⡦⠄⠤⠈⠐⣤⡀⠋⢙⠀⠀⠀⠀⡴⢡⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠸⣀⠀⠀⢀⣀⣞⠁⠀⠀⠀⠀⠀⠘⢛⠛⠛⠀⠀⠀⣼⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠉⢱⡶⠉⠀⠉⠉⠉⠉⠉⠉⠛⠛⠚⠓⢦⠀⣀⣰⡇⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢄⢀⣻⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠠⠶⠖⠛⣋⡭⠿⣅⡾⠋⠁⠀⠈⠉⠙⣷⠲⠶⠶⢶⠶⠂⠒⠒⠒⢿⣄⠠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢀⡤⠖⠋⠁⠀⠀⣘⠁⠀⠀⠀⠀⠀⠀⠘⢦⣀⣠⠞⠀⠀⠀⠀⠀⠀⠹⣷⣼⠯⣍⠓⠶⢦⡀
⠀⠀⡠⢤⣶⡲⡿⠟⠛⡻⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠀⠙⢦⡀⠈⢳⣤
⣠⢾⠖⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢾⡉⠘⣳⢤⢀⡀
⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠳⠀⠀⠁⠟⢟⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣸⠁⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣣⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢠⠇⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠻⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⡜⠀⠀⠀⠀⠈⠀⣄⣀⡀⠀⠀⣾⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⡄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⣤⣤⠇⠀⠀⠀⠀⠀⠀⠈⠛⢿⣌⣹⡏⣘⡛⣿⠶⠟⠉⠀⠀⠀⠀⠀⣷⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡙⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⢤⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠮⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠘⣻⠆
⠀⠀⠀⠀⠀⠀⠀⣠⢪⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣥⢷⡀
⠀⠀⠀⠀⠀⠀⣴⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢧⡀
⠀⠀⠀⠀⠀⢸⠛⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⢧
⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠈⢿⡄⢼⡟⢦⠀⠀⢀⡴⢻⡇⢰⡟⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣨⠇
⠀⠀⠀⠀⠀⠈⣹⣿⣯⡁⠀⠀⠀⠀⠳⡄⢙⣲⣷⣤⣾⣖⡋⣠⠏⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⢠⣤⣯⣤
⠀⠀⠀⠀⠀⠊⣼⣿⣿⠑⠀⠀⠀⠀⠀⢨⠉⠀⠈⣿⠁⠀⠉⡁⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢹⠟⡟⠀⠀⠀⠀⠀⠀⠈⠂⠀⠄⠛⠠⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡗
⠀⠀⠀⠀⠀⠀⠸⠲⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣤⣀⣀⣀⣀⡀⣰⡀⣀⣀⣀⣀⣠⣄
⠀⠀⢰⣲⣦⣄⡀⠀⢀⣠⣴⣖⣦⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣧⣿⣿⡿⠿⠿⠛⠉
⠀⠀⠈⠻⢿⣷⣿⣶⣾⣿⡟⠟⠁⠀⠀⠀⠀⢸⣯⣷⠀⠀⠀⠀⢀⣴⣾⡿⠿⠋⢹⠙⠻⢿⣷⣦⡄⠀⠀⠀⡀
⠀⠀⠀⠀⢾⣿⡿⠛⢿⣿⡷⠀⠀⠀⠀⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡦
⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⣤⣾⠿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⠸⣖⣲⣶⣤⣠⢦⣠⣴⣶⣷⡷⠃
⠀⠀⠀⠀⠀⠀⠈⣛⡡⠀⠀⠀⠀⠀⢰⡾⠟⠃⠩⠅⠙⠿⣷⡄⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠡⢿⣿⡿⣿⢿⣿⠯⠌
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⣀⠀⠀⠀⠀⣠⣴⣶⣶
⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⢹⣯⣿⠉⠣⠀⠀⠎⢉⣿⣽⠃⠀⠀⠀⠄⠀⣤⡀⠠
⠀⠀⠀⠀⠀⢀⣶⣿⣷⡀⠀⠀⠀⠀⢛⣽⣦⠤⢾⡧⠤⣼⣯⠋⠀⠀⠀⠀⠀⣾⡐⣼⡄⠀⠀⠀⠀⠀⢀⣼⣿⣤
⠀⠀⠀⠀⠀⣿⣟⠛⣻⣿⡀⠀⠀⠀⢸⡏⠁⠀⠘⠁⠀⠈⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⣿⠩⣭⡍⢿⡇⠀⠀⠀⠘⣿⣷⠎⠀⠀⠲⣾⣿⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⠀⠀⠀⠛⠉⠈⠏⠀⠙⠓
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠈⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠧

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢰⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡎⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠘⢿⣦⣀⢰⣾⣿⣿⣶⢀⣠⣾⠟⠀⠀⠀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣄⠀⠀⠈⢿⣿⡿⣿⣿⣿⡿⣿⣿⠏⠀⠀⢀⣼⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣴⣾⣿⣯⣿⣿⣿⣿⣽⣿⣿⣿⣶⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠋⠁⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠇⠀⠀⠉⠳⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣸⡇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⡀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠉⠛⠋⠛⠋⠁⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣏⡤⠁⠀⠀⠀⠀⠀⠀⠀⢤⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⡞⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠘⢄⢀⣀⡀⠈⣿⣿⣶⣤⣀⡦⢠⣀⣤⣶⣿⣿⠀⠀⠀⠀⠌⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠉⢠⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣆⣐⣒⣒⣒⡿⣷⣶⣦⣤⣤⡉⣹⣿⣿⡿⢿⣿⣿⣿⢁⣠⣤⣴⡶⠿⠶⢶⣶⡶⠶⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⣉⣁⣀⣈⠀⠈⠉⠉⣿⣿⣿⡿⢟⣿⣿⣿⢀⣤⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⡿⠟⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⢿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⡏⠁⢹⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣏⣿⣡⡤⠀⠀⠀⢀⣀⣠⣴⠛⠃
⠀⠀⠀⠀⠀⣠⣴⣞⣋⣉⠁⣐⡮⠍⠉⠛⠻⠿⠿⣿⣿⣿⣿⣇⣀⣼⣿⣿⣿⣯⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠿⣛⡯⠉⠛⠻⠶⣦⣄⡀
⠀⠀⠀⣠⡾⠿⠻⠿⢿⣿⡿⣿⣶⣶⣶⣶⣷⠖⠒⣶⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣶⣶⣶⣴⣶⣿⣷⣿⣿⣿⣿⣷⠿⢿⣦⡀
⠀⠀⠀⠁⠀⠀⠀⠘⠉⠀⠙⢛⣻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣮⡙⠷⠙⠷⣦⡀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠉⠩⢻⣿⣿⠿⠿⠟⢛⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡏⠙⢿⣯⠉⠙⠻⢿⣽⡳⣍⠀⠉⠆⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⢠⣿⠏⠀⢀⣿⡇⣿⣿⣿⠋⢙⣿⣿⣿⣿⣷⣟⠀⠀⠹⣧⠀⠀⠀⠉⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⢰⠾⠋⣧⣿⣿⣿⡅⠈⣿⣿⣿⣿⠉⠻⡇⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠹⣾⣿⣿⣏⢐⣿⣿⣿⠇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⣧⠀⠀⠘⢿⣿⣷⣺⣿⠿⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤


⣿⣷⡾⠿⠿⠷⣾⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣷⣿⣿
⣿⣿⣧⣤⣤⣦⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⢧
⠀⠀⠀⠀⠀⠀⠰⡆⠀⢀⣿⣾⣿⣧⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠂⢰⣷⢰⣾⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠙⢤⡛⠟⠀⠿⠿⡧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣼⠽⢿⣿⣅⠔⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡘⠛⢿⠿⢛⢠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠰⣾⣿⡇
⠀⠀⠀⠀⠀⢀⡞⠁⣸⣿⣇⣾⡤⢸⡘⡆⠙⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠀⣿⣿⠛⠆
⠀⠀⠀⠀⠒⠁⢀⡞⠻⣿⣿⣿⣿⣾⣿⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣄⢀⣿⣿⣄⠸
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠙⠿⠿⠿⠟⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢈⠹⣿⣿⡿⢉⠂
⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⠁⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢀⡄⡀⠀⣶⠀⠀⠐⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠘⣇⣱⣤⣿⣤⣦⣠⠇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠢⠄⢀⡀⠘⢦⡽⢟⠻⣿⣿⣿⣿⠴⠃⣀⡤⠤⠐⠈⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠒⠊⠉⠁⠀⠒⠛⠻⠯⣭⠉⣷⣦⣄⣀⣉⣲⣶⣟⣿⣿⣿⣿⣿⣿⣶⣯⣁⣀⣀⣤⣴⣒⣋⠭⠻⠛⠒⠀⠀⠈⠉⠐⠢⣄
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠤⠴⠀⠄⠰⠆⠛⠓⡾⠻⠿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠤⣤⠥⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⡤⡶⠁⠀⠀⣸⣇⢠⣿⣿⣿⡜⣿⠈⡟⢏⠈⣷⣄⡀⠀⠀⠀⠀⠈⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⢀⠔⠋⠁⠀⡇⣠⠀⠀⠁⣤⣿⣿⣿⣿⣿⣆⠀⠙⠈⡌⣿⠀⠉⠓⢄⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡂⠀⢀⡀⠴⣧⡀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣧⣀⢐⣿⡿⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠋⣷⠙⢿⣿⣿⣿⣿⣿⡿⠃⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⠿⠿⠿⠋⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣀⣤⣄⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣸⣿⣾⡀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⢸⣁⣿⠙⣽⣀⡏⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣵⣭⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⣉⣻⣿⣎⣉⡀⠀⠱
⠀⠀⠀⠀⠀⠀⢀⢼⡧⢀⡀⠈⠀⣀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢰⠛⠋⠙⣻⡋⠉⠛⡃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⣷⣀⢹⢀⣴⣆⢧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠢⣦⠀⠀⠀⣰⡦⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠃⡍⢸⡟⢨⠘⡏⢈⠈⠁⠈
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠉⠀⠐⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢹⠀⠳⣼⠞⠀⣾⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢤⣀⣤⣀⣀⠀⢸⣿⣿⣯⣿⣿⣿⠀⢀⣀⣤⣄⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠵⣠⣿⣷⣶⠿⢭⣅⢰⠟⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⠤⣄⢀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣰⣿⣿⣿⠁⠀⠀⢹⣇⣀⠀⠀⠀⠒⠂⠉⠉⠁⠠⠀⠀⠈⡡⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣻⣿⣿⣧⣤⣶⣿⣛⡣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⢤⣿⣿⣿⣿⣿⣿⠝⣩⣉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣠⣤⣬⣿⣿⣿⣿⣿⣥⣥⣄⣀⠀⠀⠘⣇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⢀⡖⠋⣽⣿⣿⣿⡿⠿⠏⠛⣻⡉⠉⣇⠀⠀⠘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣼⠇⠀⣿⣿⣿⣿⣿⣿⣿⠟⣿⡷⠀⣿⡆⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠆⠀⣿⣿⣿⣿⣿⣿⣿⣬⣿⡗⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠂⠀⢿⣿⣿⣿⣷⣶⣶⣶⣿⡇⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣽⡿⠀⠀⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠙⠀⠀⠀⠹⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡤⠄⣤⢤⣦⣿⣦⡥⣤⠄⢤⡄⢀⠀⠀⠆⠀⠀⠨⠔⠪⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠁⠀⢀⣶⠃⢳⣆⠀⠀⠀⢀⠠⠀⠀⠀⡠⠚⣤⠀⠀⠀⠀⠀⠀⠀⠠⠚⡿⠿⣥⠛⠿⠓⠀⠀⠀⠀⠀⠀⠐⠄⢴⡀⠀⠀⣈⡆⠰
⠀⠀⠀⠀⠐⠂⠂⣈⠻⣄⣾⢋⡀⠂⠚⠀⠀⠀⠀⠀⠅⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠉⢻⠟⠉⠡⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠘⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠡⣌⠳⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠈⠀⠀⡈⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⢤⡠⠄⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠆⠀⠙⢢⢠⠚⠁⡰⠆⠀⠀⠀⠀⠠⡀⠳⠈⠈⠠⠊⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢻⡇⣾⠃⠀⠀⠀⠀⠈⠑⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⡀⢀⡱⢼⡇⠀⣿⠴⣁⠀⡀⠀⠀⠀⠀⠈⠶⠈⡋⠀⠖⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣇⣹⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⢧⡈⠳⠀⠋⠘⠃⠴⠋⣠⠃⠀⠀⠀⠀⠠⠀⠋⠇⠋⠀⢄⠀⠀⠀⠀⠀⠀⢻⢭⣄⣀⠹⢃⣀⡽⣽⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠉⠂⠀⠀⡀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠂⠈⠁⠂⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠛⠛⠋⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠉⠒⠠⠐⣤⠦⢦⡔⠠⠒⠊⠁⠀⠀⠀⠀⠀⡄⠀⠠⡆⠀⢆⠀⠀⠀⠀⢈⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡁⠀⠀⠀⠰⣧⢡⢻
⠀⠀⠀⠀⠀⠀⠀⠻⡀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡠⠊⠙⠢⡨⠀⠀⠀⠀⠈⠄⠘⡆⠀⠀⢰⣶⡆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠠⣿⡀⣼
⠀⠀⠀⠀⠀⢀⢠⡎⠁⠙⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠶⠶⠰⢁⠀⠀⠀⠀⠀⠀⠀⢿⠠⢜⣻⣿⣟⡧⠠⡟⠂⠀⠀⠀⠀⠀⠠⡿⢦⡇
⠀⠀⠀⠀⠠⠂⠘⢃⠀⢠⠛⠀⠡⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠀⠀⠈⠑⡄⠀⠀⠀⠀⠃⠾⣄⠈⠙⠿⠋⠁⣰⠗⠡⠀⠀⠀⠀⠀⠀⣟⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠰⢀⠀⠀⠠⢓⠀⠈⠀⠀⠀⠀⠁⠀⠹⠆⠀⠀⠀⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠁⢂⡄⡀⣄⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠒⠚⡅⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠟⠤⠥⠔⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠛⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠐⠀⠀⠀⠀⠀⠘⢦⡀⠀⣁⡁⠀⡰⠂
⠀⠀⠀⠀⢠⡇⠀⠀⡀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠠⡄⠀⠁⠸⡶⠳⠀⠁⢠⠄⠀⠀⠀⠀⠀⠀⠲⠓⠒⠚⠦⠂⠀⠀⠀⡂⠐⠀⠋⠀⠉⠁⠀⠐
⠀⠀⠀⢀⠂⠃⠀⠀⡇⠀⠀⠈⢃⠀⠀⠀⠀⠀⠐⣃⠔⢁⠒⠄⠖⡈⠂⡀⠆⠀⠀⠀⠀⠀⠘⢀⠀⠠⠀⡄⠐⠀⠀⠀⠀⠠⡄⠀⢠⠁⢀⠄
⠀⠀⠀⠀⠀⠐⣄⠀⠀⢀⠔⠀⠀⠀⠀⢰⡀⠤⠀⠘⠀⠜⠋⠀⠘⠃⠀⠑⠀⠤⠀⣦⠀⠀⠀⢈⠂⠀⠀⡰⠀⠀⠀⠀⠀⠠⡐⠄⠀⠡⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠪⠙⠑⠉⠌⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠠⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠀⠒⠐⣖⣶⡟⣷⣶⡖⠐⠂⠀⠀⠘⠀⠀⠐⠁⠂⠀⠀⠂
⠀⠀⠀⢀⡄⠀⠼⡇⢸⡷⠀⠀⡄⠊⠀⠀⠰⠁⡞⠀⠀⠀⠀⠀⠀⠈⠈⠉⢺⠌⠉⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤⠴⠎⠈
⠀⠀⠀⠀⠐⠂⠃⠉⠉⠘⠓⠂⠀⠀⠀⠀⠐⣀⢣⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠉⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠄⠄⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⠀⢀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠂⠀⠴⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣤⠐⣤⠄⠀⠀⠄⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⢀⣰⠁⢧⣀⠊⠀⠀⠀⠀⠈⢄⡁⠶⢈⡄⠄⠀⠀⠀⠀⠀⠀⠀⢸⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⢸⡘⠧⠘⠂⠛⠴⠃⡸⠀⠀⠀⠀⡀⠐⠂⠂⠀⡀⠀⠀⠀⠀⠰⣞⣆⡈⠿⢁⣰⣳⠆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠟⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠠⢄⡀⢀⢀⣀⢀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡖⠠⠀⠀⠀⢠⡠⢤
⠀⠀⠀⠀⠀⠈⡿⠉⡏⠀⠀⠀⠀⠀⠀⠀⠘⢀⠠⠣⢀⡂⠀⠀⠀⠀⠈⡇⠀⠀⣤⣤⠀⠀⢸⠃⠀⠀⠀⠀⢸⡏⢸⠁
⠀⠀⠀⠀⠀⢠⡊⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠤⠼⡁⠀⠀⠀⠀⠀⢸⡀⣔⣿⣿⣢⠠⡿⠀⠀⠀⠀⠀⢸⡯⡟
⠀⠀⠀⠀⠂⠘⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠁⠢⠀⠀⠀⠈⠘⣇⠈⠛⠟⠁⣰⠇⠁⠀⠀⠀⠀⢸⣃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⠂⠀⡀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠐⠄⠤⡆
⠀⠀⠀⠀⢀⠧⠬⠤⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠒⣏⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠁⠀⠀⠀⠢⡀⢐⣒⠀⡠⠂
⠀⠀⠀⠀⠇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢐⠀⢀⠘⠒⠃⠀⠀⡂⠀⠀⠀⠀⠀⠘⠁⠊⠑⠂⠀⠀⠈⠀⠀⠃⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡔⠁⡈⠁⢡⠀⢀⠁⠀⠀⠀⠀⠀⢰⠀⠀⠰⠀⠀⠀⠀⠀⠂⠀⠂⠀⠃
⠀⠀⠀⠀⠀⠷⠤⠤⠪⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠈⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠛⠻⣦⣴⠟⠛⠃⠀⣠⠀⠀⠀⠀⠀⠀⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣆⠖⠀⠀⠲⣰⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢺⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣶⣶⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣤⠀⠀⣠⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣿⡥⣴⣦⣬⣿⡃⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠀⠀⠀⢀⣤⡾⣧⣴⣼⣿⣿⣧⣦⣼⣷⣤⡀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣀⡴⠋⢞⣼⡿⣱⣿⡏⢸⣿⣎⢿⣦⠳⠙⢦⣀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣧⠀⢸⡟⣹⣿⣿⡇⢸⣿⣿⣏⢻⡇⠀⣸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⡀⣬⡀⢻⣿⣿⡇⢸⣿⣿⡏⢀⣥⠀⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⢉⣹⣧⣽⣇⠘⣿⣿⣧⢸⣿⣿⠁⣸⣯⣾⣏⡉⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠟⠁⢀⡞⡋⡉⠉⠛⠿⣷⣮⡉⣁⣈⢉⣵⣾⠿⠛⠉⠉⠙⢻⡄⠙⢿⣷⣦⣀⡀⡀
⠀⠀⠀⠀⠀⢀⡴⠞⠊⠉⠉⠉⠉⠉⠁⠀⠀⢸⡰⣿⣿⡁⢀⡀⡈⠑⡌⣇⢸⢋⡞⠁⡤⣀⣀⣼⣿⠇⡇⠀⠀⠉⠉⠉⠉⠉⠉⠙⠳⣤
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣿⣟⣸⣿⡗⠹⡛⡹⢇⠟⠥⢿⢉⣿⢛⣿⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣻⢮⣧⣤⢽⣦⡁⢰⡜⢈⣷⡷⢏⣹⣾⣿⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠟⢷⠛⣿⢏⣻⡉⢸⡇⣸⣿⡗⣿⠉⣿⢥⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠂⠨⠯⢽⡋⣛⣗⢺⣇⡼⢖⣍⣿⠰⠃⠈⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠲⠶⡶⢚⠹⢷⡻⢺⡇⢻⣿⡇⡖⠺⠛⠋⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠠⣌⣽⣙⢻⠗⢸⡇⢘⣏⣹⣇⠺⡀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⠙⣾⠌⠴⠀⢸⡇⠈⠋⠯⣽⡚⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⠿⣧⡤⠤⠾⠷⠤⠤⣾⡿⠿⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡀⠀⠀⠀⠀⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡀⢀⣀⣀⠀⠀⣴⣿⣿⣟⣻⣿⣷⡄⠀⠀⣀⣀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢟⣆⢈⣿⣽⣭⣯⣟⡋⢀⡾⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⠀⠈⢻⡎⠁⠀⠀⠀⠤⠤⠐⠒⠚⠉⡉⠁⠀⠐⠀⠁⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠽⠿⢿⣿⣿⣆⣀⣀⣾⡿⢏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⣀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠘⣿⣿⣿⣿⣿⣿⡻⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡠⠼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⣛⢿⣿⣿⣿⣿⢄⠘⠋⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠀⠀⠀⣐⠿⢻⣿⣿⣿⣿⣿⡟⠿⣿⠿⠷⣆⠀⠀⠘⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⣼⠃⠀⣿⣿⣿⣿⣿⣤⣤⣤⣶⣧⠀⣿⡄⠀⠀⠀⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⠃⠀⣿⣿⣿⣿⣟⣛⣛⣛⣻⡟⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠘⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠶⣤⣴⣶⣦⠶⣂⡄⠀⠀⠀⠘⠳⣄⠀⠀⠀⢀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⢠⣒⠦⣴⣶⣦⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣷⣤⡾⢿⣿⣷⣤⣾⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣾⣿⣶⣶⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⡴⠲⢈⣽⢿⣿⣿⣿⣿⣿⣿⣿⣍⠀⠱⣄⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣿⠤⣤⡾⠻⡈⠟⠙⠟⣋⠡⢇⣻⣿⢷⡲⢿⣿⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣾⣶⣾⣧⣿⣄⣴⣿⣿⣿⣿⣿⣿⣿⠦⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠴⠶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠑⠀⠂⠀⠀⡀⡀⣀⣤⣿⣿⣷⣤⣀⢠⠰⡆⠰⠀⠀⠉⠻⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠂⠀⢂⠀⠀⠀⠠⠀⣿⣿⣿⣿⣿⣿⣿⢦⠙⠀⠀⠀⠀⠀⠀⡘⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡄⠀⠀⢀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠿⠛⡉⡀⠀⠀⡄⠀⡀⠀⠀⠈⢀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢫⡃⠀⠀⠄⠀⠀⠄⠀⠀⢀⠐⠾⠃⡄⣧⢶⠰⡀⠀⠘⡄⠀⠀⠀⠈⢀⣴⣿⣆
⠀⠀⠀⣀⣀⣀⠀⠀⢀⣿⡏⣾⣿⠀⠀⠠⢁⠀⡐⠀⡼⣦⣟⣷⠠⠰⡘⣏⡇⠠⠀⠄⠱⠀⠀⠀⠀⣾⣿⢹⣿⡀⠀⠀⣀⣀⣀
⠀⠀⠀⠉⠉⠙⠻⣷⣼⣿⡧⠛⠋⢀⠀⠁⣌⠠⠀⡸⡿⣼⣷⢿⠀⠀⣡⠉⡐⠀⢁⣀⣀⠠⢆⠠⠠⠘⠃⠨⣿⣧⣾⠟⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀⢃⠂⣴⣾⣿⣿⣿⣦⣿⣿⣿⠀⠱⡾⣧⣤⣾⣿⣿⣿⣷⣮⣆⠡⠀⠀⠐⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠅⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⢸⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠄⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠰⠩⢿⣿⣿⣿⣿⣿⡟⢿⣿⢸⠻⡿⢿⢿⣿⣿⢿⢿⠿⢛⠱⡜⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⡒⣀⣩⠁⠉⣫⠠⣹⣽⢈⠷⣧⣿⠘⠄⠄⠘⡘⠐⡂⠆⢁⠀⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠂⡃⡅⡄⡉⠒⢩⢨⠀⢼⡏⠉⠀⠁⢠⢇⢡⠩⠉⠈⡀⠤⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡼⣿⣦⣀⠀⠂⠂⣏⡥⣷⠚⢰⣒⡞⠁⠀⠀⠸⠌⡞⠨⣄⣷⣿⣿⡧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡘⢿⣿⣦⡀⠀⡅⠖⡷⡈⢼⠓⡇⠘⢐⠺⡁⠆⣅⢁⣿⣿⣿⡟⣱⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠻⣏⠠⣀⠁⠉⠆⠁⡎⠀⠇⡙⠄⠃⡍⠃⢸⠨⠝⡡⠊⢰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⠃⠀⠈⠑⠬⡀⢾⡇⣆⢲⡆⠀⠩⡆⣷⠐⠻⣈⠕⠊⠀⠀⠻⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠳⢧⣌⠓⠠⠄⢂⡩⠔⠊⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠀⠀⠀⠀⠠⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⢀⣀⣀⣀⣠⣀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡾⠟⠛⠋⠉⠉⠲⢫⣾⡟⢲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⣿⣿⣷⣄⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣏⣹⣄⠀⠀⡀⢠⠤⠐⠈⠁⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠳⣄⣈⡙⠛⠿⢿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠉⠀⠀⠀⠀⠀⠀⠉⠙⢲⣤⣀⠙⠻⣿⣿⣿⣿⣿⣿⡇⠁⠙⠾⣇⠀⠀⠀⣸⡟⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡼⡿⠿⠿⠦⢄⡉⠻⢿⣿⣿⡇⠀⠀⠀⢻⡄⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣫⣾⣿⡄⠀⠀⠀⠉⠳⣄⠙⠻⣷⣄⠀⠀⢸⡇⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣫⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⢄⠈⢿⣷⣦⠈⣇⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣉⣁⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠾⠟⠉⠛⠁⠀⢠⣶⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣴⣾⣷⣦⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀
⠀⠀⢀⣠⣴⣿⠃⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠶⠦
⢀⣾⣿⡟⠉⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠋⠃⠙⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⣼⣿⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⣸
⠀⠀⠀⠀⠀⠹⢄⠼⠟⢹⣿⣆⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⣸⣧⣿⣃⠰⠃
⠀⠀⠀⠀⠀⢀⣤⡰⠶⡶⢟⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢴⣿⡏
⠀⠀⠀⠀⠰⠃⣸⣿⣷⣷⣸⣼⡄⠳⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⢸⣿⡏⢂
⠀⠀⠀⠀⠀⡼⠈⢿⣿⣿⣿⠏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⢦⣼⣿⠷⠉
⠀⠀⠀⠀⠰⠃⠀⠀⠉⠈⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⠛⠋⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢠⠀⠀⢰⠀⠀⢆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡘⣦⣷⣿⣶⣶⠇⡼
⠀⠀⠀⠀⠀⠠⠤⠒⠲⠶⣶⣖⡒⢤⣈⠁⠒⢤⣙⣯⣦⣿⣿⣿⣯⣤⠚⠉⠀⣀⠤⠴⢶⡶⠒⠒⠠⠄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠈⠥⠬⠿⠯⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠍⣉⢁⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡎⠀⠀⡸⢇⣼⣿⣿⡹⠅⢝⢧⢹⡦⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⢀⡇⠂⠀⣠⣾⣿⣿⣿⣿⣆⠈⢰⣾⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢻⡶⢾⠻⣿⣿⣿⣿⣿⡟⢳⣼⡿⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠻⠿⠟⠋⠀⡧
⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⠆⣤⣦⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠾⣻⠦⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠂⠘⠶⣧⣜⠖⠁⠠⡀
⠀⠀⠀⠀⠀⡀⣦⠀⠈⠋⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠶⠛⢻⡟⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⣦⡈⣁⣴⣄⡆
⠀⠀⠀⠀⠀⢠⠓⠶⠀⠀⠰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠸⡁⢈⠇⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⠉⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡀⠹⣬⣦⣴⣿⣿⣿⣶⣾⣥⠆⢀⣠⠞⠁⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⢀⠠⠒⠚⠛⠛⠛⠿⣷⣶⣦⣄⣀⠀⠀⠹⠶⣼⣿⣿⣆⣤⣀⣿⣿⣷⡶⠟⢁⣀⣤⣴⣶⣿⣟⠛⠋⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠒⠋⠉⣩⠛⠛⠛⢷⣶⡾⠿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡞⢯⣉⣀⣉⣲⣄⣀⠈⠉⠂⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡤⠒⠋⠙⣻⣶⡾⠛⠉⠀⠀⣸⣿⣿⣯⣽⣿⣧⠀⠀⠈⠂⠙⣟⠛⢥⡀⠈⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠜⢡⡧⠀⠀⠀⢀⣰⣾⣿⣿⣾⣿⣿⣿⣷⣄⠀⠀⠀⣀⣧⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣠⠴⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠓⢦⣤⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⢠⠏⢿⣿⣿⣿⣿⣿⣿⠋⢷⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠉⠛⠛⠛⠋⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⣠⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣷⣖⣲⣉⣿⣩⣽⣿⣷⣤⣾⣿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⢀⣄⠀⣄⣀⠀⠀⠀⣀⣠⠼⣻⡍⠉⠁⣤⡀⠀⠀⠀⣤⣬⡝⢻⣦⣄⣀⠀⠀⠀⠀⣠⣤⣄⣤⣤⣤⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡇⣴⡟⠹⠏⠀⢉⣀⠀⠀⠻⣿⡀⠀⢸⣿⠇⠀⠀⠀⠘⠛⠳⣆⠀⢸⣿⣿⣿⠿⠟⠛⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⡿⣣⣴⣤⡂⠜⡉⢠⢀⣀⠈⡀⢀⠀⠙⠀⠀⠀⠙⠀⢀⣤⣌⠳⣌⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⣹⠁⢴⡏⣁⢉⣼⣴⣶⢾⡄⠁⡰⡴⠚⢃⡠⣄⠂⢲⣦⣤⣤⡙⠋⠀⠹⣦⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⣏⣀⣎⣴⣤⣍⣭⣬⣉⣉⣡⣯⡴⢛⠶⣮⣿⢿⢿⣷⣽⣥⣶⣿⣄⡖⠀⢹⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⣿⣿⠛⠿⠻⠛⠛⢛⡟⠻⢿⡿⠿⠟⠿⣾⣿⣿⣿⢻⣿⢿⡟⢛⣿⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢯⣽⣻⣿⠏⢰⣿⡇⠃⠀⣠⠀⠸⢿⣷⣤⣤⣤⣿⡏⠀⣰⣯⡵⢿⣿⣯⢻⣯⡽⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡿⠇⣼⣿⠀⢸⣿⡇⠀⠀⣿⣇⠀⠈⡻⣿⣿⣟⣽⠃⣐⣿⠏⠐⠀⣿⣿⡎⢹⣷⡼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣾⠃⢐⣿⡇⠀⣿⣿⡇⠀⠀⣿⣿⠀⠀⠁⢻⣿⣿⡌⠀⢿⡼⣇⠀⠀⣿⣿⠇⠈⣿⣳⢾⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⡟⠀⢼⣿⠀⢠⣿⣿⡃⠀⠀⠙⣿⡀⠀⠀⢸⣿⣿⠁⠀⢾⣷⡿⠀⠀⡞⣿⡆⠀⠰⣿⠈⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⢠⡃⠀⣿⡟⠀⠸⣿⣿⡇⠀⠀⠀⣿⣧⠀⠀⢸⣿⣿⠀⠀⡙⣿⡇⠀⠀⢿⣿⣷⠀⠀⣿⠀⢊⡞⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⡇⢰⡆⠀⣾⡁⠀⠀⣿⣹⠇⠀⠀⠀⢸⡋⠀⠀⠘⡏⠇⠀⠀⢈⡿⠀⠀⠀⢸⣽⡿⠀⠀⣿⠀⠐⡇⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢻⠀⠀⣷⡇⠀⠀⢻⡧⠀⠀⠀⠀⠸⢗⡀⠀⢴⡿⠀⠀⠀⣾⡇⠀⠀⠀⣽⣻⠇⠀⢀⣿⠁⠀⡇⢉⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢸⠀⠀⡑⠃⠀⠀⢸⣆⣠⠀⠀⠀⠐⣿⡄⠀⠸⣷⠀⠀⢀⣽⡇⠀⠀⠀⢿⠋⠆⠀⠀⣿⠀⠀⡇⢸⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡃⠀⣽⡃⠀⠀⢺⠟⠀⡂⠀⠀⠀⡾⠇⠀⠠⣿⠀⠀⠀⣇⠀⠀⠀⠀⠙⢷⠂⠀⢸⣿⠀⢸⡇⣸⣿⢺
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡇⢿⡇⠀⢯⠑⠀⠀⠀⠀⣈⠇⠀⠀⠀⠨⠂⠀⢀⣿⠀⠀⠠⣯⠀⠀⠀⠐⣿⣿⠁⠀⠀⣿⠀⠠⡇⣿⣿⡇
⠀⠀⣀⣤⣤⣶⣿⣿⣿⣷⣾⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⢀⣾⣷⠀⠀⢸⣿⠁⠘⢢⣿⣿⣇
⣴⡟⠋⠛⠛⠉⠉⢘⣏⢻⣿⣇⠈⣗⠀⢠⡇⠀⠀⠀⢠⠀⡃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢈⣿⡷⠀⠀⢸⡧⠀⢈⣿⣿⡟⢿⣷⣶⣄⣀
⠻⠀⠀⠀⠀⠀⠀⣿⣿⠛⠋⢹⠀⣿⠀⠀⣏⠀⠀⠀⠨⡾⣥⠀⠀⠀⠀⠐⠀⢀⣿⠀⠀⣾⠁⠀⠀⠀⢸⣿⡅⠀⠀⢸⡇⠀⠘⣿⣿⣟⣿⡯⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⠈⣇⣿⠀⠀⢸⣇⠀⠀⠀⣷⡿⠀⠀⠀⠀⠐⠀⠰⣿⠀⠀⡟⠀⠀⠀⠀⢿⣿⠀⠀⠀⣠⠄⠀⣴⡏⠻⠿⢿⣷⡄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠘⡎⢿⣆⠀⢹⣿⣦⠀⠘⣿⠃⠀⠀⠰⣅⠃⠀⠀⠀⢰⡆⢸⣿⠆⠀⣟⠀⠀⠀⢰⣿⡷⠀⠀⠰⡟⠀⢠⡟⠀⠀⠀⣾⣻⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⣿⣇⠀⢻⣼⡆⠀⠼⣣⠀⠀⠀⡏⠀⠀⠀⠀⢘⡇⢸⣿⡇⢰⡯⠀⠀⠀⣸⢿⠁⠀⢠⣿⠀⢠⡟⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡘⣿⣷⡀⢻⡟⣄⠀⣻⡆⠀⠀⢙⣛⠀⠀⠀⠘⡇⠸⣿⡇⢸⡇⠀⠀⠀⣻⡗⠀⠀⣸⠏⣰⠏⠀⠀⢀⣼⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⣿⣷⡀⠘⢿⣄⠘⢷⣄⠀⠈⠻⠂⠀⠀⠀⢿⢸⣿⡟⠸⠇⠀⠀⢰⣟⠁⠄⣼⣻⡾⠃⠀⠀⣰⣿⢧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⡇⠀⠀⠻⣧⣄⠙⢷⡄⣻⡄⠀⠀⠀⣮⢘⣿⣇⣾⡄⠀⠐⣼⣿⣼⣯⡾⠋⠀⠀⠀⣸⣿⠃⣾⠁
⠀⣀⣤⣤⣶⣤⣴⣾⡿⣿⡿⠋⠉⠉⠁⠀⠀⠀⠀⠙⠻⢾⣏⣿⣧⠀⠔⠰⡏⢸⣿⣿⣋⣶⡀⣞⣿⠿⠟⠋⠀⠀⠀⠀⠠⣿⣿⣰⣿⡀
⠀⡟⠋⠀⠛⠛⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣦⣾⣧⣼⣿⣟⣿⣡⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⢿⣧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣋
⣿⣿⣿⠟⠻⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣦⣶⣶⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⡀⡠⠂
⠀⠀⠀⠀⠀⠀⢘⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿
⠀⠀⠀⠀⠀⠀⠘⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢹⣇⡿⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⣶⡦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⡆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⢢⣾⡯⢷⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠂
⠀⠀⠀⢀⡴⠛⣽⡇⢳⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⠒⣟⣿⣿⠂⡀⠀⠀⠀⠀⠀⠀⠄⢹⣿⣿⣏⠡⡀
⠀⠀⠴⠋⠀⡼⢹⡇⡎⢷⠀⠉⠣⠀⠀⠀⠀⠀⠐⠈⠐⢿⣿⡿⠂⠁⠀⠀⠀⠀⠀⠀⠐⠙⣿⣿⠏⠁
⠀⠀⠀⠀⠐⠁⠈⣇⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠹⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡄⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣤⣾⣷⣾⣷⣤⣯⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⢣⡀
⠀⠀⠀⠀⢼⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡟⢡⡈⢿⣿⣿⣿⣿⡿⢁⡜⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⡇
⠀⠀⠀⢦⣤⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡋⠿⣿⢻⡾⢷⡟⣿⠿⢉⣾⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢳⣤⡴
⠀⠀⢀⣜⣵⠟⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⠀⠘⠿⣷⣶⣶⣹⣦⣴⣏⣶⣶⣿⠿⠃⢀⣀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣣⡀
⠀⠀⠀⠹⣿⠎⠀⠀⠀⠀⠀⢀⣴⠿⣧⣼⢿⣻⣿⣿⣷⡿⠿⢿⣿⣿⣿⣿⡿⠿⣿⣾⣿⠿⣿⡽⣧⣾⢿⣦⡀⠀⠀⠀⠀⠀⢱⣿⠏
⠀⠀⠀⠰⣿⡄⠀⠀⠀⠀⠀⣾⠛⣿⠕⠦⢤⣈⣍⣁⣠⠬⠍⣉⣻⣿⣿⣟⣉⠫⠥⣀⣉⣩⣁⡬⠗⢪⣿⠛⣷⠀⠀⠀⠀⠀⢰⣿⠂
⠀⠀⠀⠀⢹⡝⣦⠀⠀⠀⢠⣿⣿⣬⡝⠓⠒⠂⣀⣲⣖⣛⠙⠻⣿⣿⣿⣿⡟⠉⣛⣒⣋⣁⠶⠒⠚⢻⣵⣿⣿⡀⠀⠀⠀⣴⢻⡏
⠀⠀⠀⠀⠀⠳⣌⣷⣶⣦⣾⠛⢈⣽⠛⣟⣛⣛⣯⠉⣭⡄⣉⢸⣿⠿⠻⣟⡃⣍⢨⣭⡉⣭⣛⣛⣿⠛⣯⡉⠛⣷⣶⣶⣾⣡⠟
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠚⢠⣥⣤⣿⡻⠿⠿⢛⡻⠷⢾⣿⡧⠆⠰⣽⣿⡷⠾⢟⣛⠿⠿⢿⣷⣦⣬⡄⠹⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⠮⠟⠋⢁⣢⠟⠓⢉⣉⠩⠴⠬⠻⣷⣾⠟⠵⠦⠌⣉⡉⠚⠻⢔⡈⠙⠻⢿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣠⡔⠚⢁⡠⠖⢊⡡⠤⠒⢒⣒⣛⡜⢣⢛⣒⡒⠒⠤⣌⡑⠢⢄⡉⠓⢦⢤⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢣⡴⡞⣉⡴⠊⣁⣤⣤⣤⣍⣽⣤⣤⣤⣈⠐⢤⡉⡲⢦⡜⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣿⡿⣻⣿⣿⣾⡏⠀⠀⣿⣷⣿⣿⣟⢿⣿⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢲⣆⢀⣠⣿⣿⡷⢶⣴⣿⠟⣚⣻⡿⣙⣿⣠⣄⣿⢋⣿⣟⡛⠹⣿⣦⡶⠾⣿⣿⣄⡀⢰⡖⣄
⠀⠀⠀⠀⠀⠀⣤⠲⣯⣿⣿⣾⢿⢡⡟⠀⠀⠸⣿⡟⢊⠻⣷⣽⣾⣕⣪⣿⣯⡾⠟⡑⢻⣿⠇⠀⠀⢿⡎⢿⣷⣿⡿⣽⠖⣤
⠀⠀⠀⠀⠀⣾⡉⣱⡟⢺⡿⠹⣧⣿⣤⣤⣤⣼⣿⣋⠀⠐⣩⣿⣿⣗⣾⣿⣿⣍⠂⠀⣙⣿⣦⣤⣤⣤⣿⣼⠏⣿⡟⢿⣎⢩⣧
⠀⠀⠀⠀⣰⣧⣽⡿⢁⣿⠃⠀⢹⡏⠉⠉⠀⠉⢻⣷⣞⡡⠀⢛⣿⣿⣿⣿⡓⠀⢍⣲⣾⡟⠉⠀⠉⠉⢹⡏⠀⠸⣿⠈⢿⣯⣼⣆
⠀⠀⢀⣴⣹⣿⡟⠀⢸⣿⠂⠀⠰⡗⠀⠀⠀⠀⢀⣿⣋⣤⠖⣩⠴⣿⣿⠧⣉⠲⣤⣙⣿⡀⠀⠀⠀⠀⣾⠀⠤⢐⣿⡇⠀⢻⣿⣏⣦⡀
⠀⠀⣾⢠⣿⣻⠇⠀⠸⣷⠛⠛⠺⢷⣤⣴⠶⠟⠋⠙⢻⣿⣯⡤⢚⣿⡿⡓⢬⣽⣿⠟⠋⠙⠻⠶⣦⣤⡾⠗⠚⢋⣾⠇⠀⢸⡟⣿⡤⣷
⠀⠀⠿⣾⡟⠛⠀⠀⠀⠻⣧⠀⠄⠀⠙⣧⠀⠀⠀⠀⠀⢹⣿⣞⣵⣿⣿⣮⣳⣿⡏⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣼⠟⠀⠀⠀⠛⢻⣧⠟
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠙⣷⣄⣀⡡⣼⡧⣤⣄⣤⡾⠋⠙⠻⢿⡟⢻⡿⠟⠉⠛⢷⣄⣠⣤⣼⣧⣈⣀⣠⣼⠋⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⢨⣿⣯⣅⠀⠀⠂⠙⣧⡀⠀⠀⠀⣨⣿⣿⣅⠀⠀⠀⢀⡿⠉⠐⠀⠄⣨⣽⣿⡄⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⣿⣏⣠⣹⣷⣦⣀⣀⣸⣷⠶⠶⢴⡿⠁⠈⢿⡶⠶⠶⣾⣧⣀⣠⣵⣾⣏⣀⣹⣿⠀⠀⠀⢠⡟⠉
⠀⠀⠀⠀⠀⠀⠼⡇⠀⠀⠀⠸⣯⣿⠽⠊⠈⠙⠿⣭⡀⠀⠀⠐⢷⠀⠀⡾⠀⠀⠀⢀⣭⠿⠋⠉⠑⠯⣿⣽⠃⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣆⡀⠀⠀⠀⠈⠙⠳⢦⣤⣸⡆⢰⣇⣤⡴⠞⠋⠁⠀⠀⠀⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⠕⠄⠀⠀⠀⠀⠀⠀⠈⠙⠳⠟⠋⠁⠀⠀⠀⠀⠀⠀⠠⡣⢣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⡥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣖⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠰⡏⣍⠱⡄⠀⠀⠀⠙⢆⠀⠀⠀⠀⢀⠀⢠⠞⠀⠀⠀⠀⢠⠚⢉⠙⡆⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⠛⢠⠇⠀⠀⠀⠀⠈⢧⢸⠧⡤⢿⣴⠃⢀⠀⠀⠀⠀⢆⠀⠭⠴⠃⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⣰⣋⡡⣌⠙⣿⠋⢰⡆⠈⡿⠋⢉⠭⣦⡀⠀⠈⠓⠆⠤⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⡇⢸⠇⣹⠀⠈⡇⠰⡇⠀⣅⣰⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣙⣁⣠⢏⠀⣀⣄⣀⠙⣦⣄⣥⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢯⠉⣻⣟⠋⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣦⡀⠀⢀⣴⢞⡿⣿⣷⡒⡓⢶⣤⡀⠀⣠⣞⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠋⠀⠈⡹⢟⠐⣱⢏⡔⠋⣿⠓⠘⡆⠙⡙⢶⡋⠈⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠟⡰⢡⡎⠀⠀⠹⠀⢱⡈⢄⠘⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠠⠀⠂⡼⢀⠏⠀⢠⠀⣼⠀⡀⠁⠈⠃⢸⡄⢸⡄⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⡷⠻⡦⢀⡄⢠⢃⡄⢀⢂⡄⠀⠁⠀⠸⡄⢡⠀⡀⣇⢈⣇⠀⠳⢽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⠃⡞⣼⠀⡸⢸⠀⡇⠀⠀⠇⠃⠈⠀⣷⢸⢈⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⢀⡇⡇⠀⡇⢸⠸⠇⠀⠀⡅⢰⠀⡇⡟⠘⢸⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣦⣿⡀⡇⣷⠀⣇⢸⢰⠀⢠⠀⡇⠈⢠⠀⠇⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣼⠁⣿⣿⣇⢧⢸⡄⠸⡀⡘⡆⠀⠀⠃⡆⡎⠰⢀⠂⠄⡆⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢹⣿⡿⣞⣆⢻⡀⢳⢷⢳⡄⡘⣼⠘⡰⠁⠈⡐⡐⢰⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⣿⣷⡘⣜⢦⠹⡌⢿⢎⣿⣱⠏⡰⢁⠌⡜⡱⣣⠃⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣹⠃⠹⢿⣝⢦⡳⣼⣎⢣⣿⢃⡔⣡⢊⣾⣞⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⠙⠳⢽⣮⣙⣽⣿⣮⣾⣵⠿⠋⠁⠀⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⢿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡝⡎⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣻⣻⣻⣿⣿⣿⣿⣻⣻⣿⣿⣟⣟⣿⣿⣿⢩⠟⢿⣟⢻⠿⢿⣿⣿⢿⡟⡝⣿⣛⢻⠿⠟⢛⣿⠿⣿⣩⣿⡟⠛⣿⣿⣿⠟⡟⡿⢿
⣿⣿⣿⣯⣭⣭⣽⣿⣿⣿⣿⣭⣏⣿⣿⣿⣿⣰⣦⣤⣭⣾⣦⣶⣴⣦⣸⣇⣠⣤⣵⣴⣦⣧⣬⣤⣮⣿⣑⣤⣤⣴⣤⣴⣵⣶⣧⣷⣼
⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡶⢠⡀⣀⠀⠀⠀⠀⠀⢀
⠀⠘⠃⢈⡋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⠖⠙⣋⡀⠀⠀⠀⠀⠀⠈⠓⠐⠒⠿⢡⣋⣼⡦⣠⠀⠀⠊⠆⠘⣄⠚⢃⡄
⠀⠈⢷⣿⡇⣙⠀⢿⠆⣴⡄⠀⠀⠀⠀⠈⢿⡿⠁⣀⣦⣠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢠⡄⠴⠀⠀⠀⠀⠁⢈⣩⡤⠴
⠀⠀⠈⠀⠀⠙⠁⢶⣤⣿⣿⡄⠀⠀⠀⠀⠂⣴⡆⠁⢬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⢩⠁⠀⠀⠀⠀⠀⠀⠉⠄⢾⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠉⡁⠰⠄⠀⠀⠀⠉⠁⠲⡴⠂⠛⠀⠀⠀⠀⠀⠀⠠⣤⣟⡏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠏⣠⣄⠀⠀⠀⠀⠀⠀⠛⠃⡛⠀⠀⠀⡠⢔⢞⠰⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠴⠟⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⠀⣄⡀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢙⡲⡍⠀⠀⠻⠧⡟⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣴⢽⡁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢠⣤⠓
⠀⠀⠀⠀⠀⠀⠀⣀⠀⡖⢴⡌⠃⠀⠀⠀⠘⢷⡿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞⢛⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠚⠷
⠀⠀⠀⠀⠀⠀⡄⠿⠿⠡⣷⠀⠀⠀⠀⠀⠀⠘⣷⠿⠇⡟⣠⠀⠀⠀⠀⠀⠀⠀⢨⡣⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣧⠸⣷⡶
⠀⠀⠀⣀⢠⡄⡿⠈⠤⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠞⣬⢛⡳⣆⠀⠀⠀⠀⠀⠸⡜⢛⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⢀⡤
⠀⠀⣴⣥⢜⢻⣒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⣛⡋⣀⠀⠀⠀⠀⣀⠚⣫⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⢉⣄
⠀⠀⠻⠿⠎⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⢡⠅⠀⠀⠀⠀⠨⠱⡄⠀⠀⠀⠀⢀⣤⠀⣴⠆⠻⠇⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⣰⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⡼⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢟⠋⠉⠀⢠⡄⡰⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠖⠚⠀⠀⢠⡂⢈⡂⠀⠀⠀⠈⢆⠹⡄⠄⠄⣸⡄⠓⠂⣡⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⢀⠘⢎⠈⠒⠊⠁⡹⣄⠘⠷⠷⠦⡤⠤⠖⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⠀⠀⠀⠤⠄⢑⠄⠀⡀⠀⠀⠀⢀⣉⢬⢦⠀⠀⡠⠛⠉⠙⢾⠁
⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⢀⢤⠮⠥⠔⠋⠀⠀⡀⠀⡊⣠⠞⠀⢀⡇⣠⡤⢖⠋⠀⠈⢇⠑⢄⠑⡄⠤⢠⠇⢀⠄⢈⠴⢶⡀
⠀⠀⢠⣀⡤⠾⠁⠈⠑⠦⠖⢛⠀⡀⢀⠠⡀⠀⠀⠉⠉⠁⢀⣠⡴⠟⠯⡀⠈⠀⠀⡠⠀⠀⠀⠑⠾⢿⣷⣆⠥⠚⠁⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠒⢡⠁⠀⢀⡀⠔⡪⠗⠉⠀⠀⠀⠈⠁⠉⠉⠀⠀⠀⢈⢠⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢈⠛⢉⡤⠒⣩⠔⠉⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⠆⠐⣠⢈⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠊⢁⡤⠊⠄⠀⠀⠀⠀⠈⠀⠀⠀⠠⡇⠀⠈⠳⠚⠉⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠊⠀⢀⠠⠀⢁⡄⠀⠈⠲⠤⠔⠊⠀⠀⠀⠀⡀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠱⢄⣀⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠰⣿⠶⠴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⣰⢊⡉⣩⣲⣤⣔⠋⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡩⢽⣻⡵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠿⣿⡿⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠚⠿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠸⠿⣿⠿⣿⠿⣿⢿⡿⢿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠈⠉⠉⠉⠉⠛⠛

⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠂⠀⠉⠛⠿⠛⠻⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠟⠛⠉⠉⠀⠀⠀⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠐⠒⠂⠒⠒⠐⠂⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⡄⠀⣰⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣷⢤⠀⠀⠀⡀
⠀⠀⠀⡐⡿⢿⠿⠉⣠⣦⣄⠂⢐⣤⣇⡄⢠⣾⡿⣿⡿⢉⡘⠿⢆⡰⡟⣄
⠀⠀⣴⣷⣤⣀⣴⣿⣿⣿⡿⣷⣆⠙⠋⢣⡘⠘⣡⣁⣐⣿⣷⣦⣶⣧⣤⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠑
⢠⣶⠛⠛⠋⠉⠉⠉⣩⣵⣦⡈⠁⣤⣶⣿⠙⣆⠘⣿⣿⣿⣿⠉⠉⠛⠛⠻⣷⠀⠀⠄⠀⠈⠀⠀⠀⠠⠤⠄
⢸⣿⢾⣿⣶⣿⣿⣾⣿⣿⣿⣷⠀⣿⣿⣿⡆⢻⠀⣿⣿⣿⣿⢸⢠⠀⠀⢰⣿
⢸⣿⣼⣿⣿⣿⣹⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣾⡆⣿⣿⣿⣧⣼⣼⣧⠀⢸⣿
⠀⣿⣹⣿⣿⣿⣹⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⢰⣤⣀⣀⣴⣦⣠⣶⣦⢸⣿⣦⡀⠀⡆
⠀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠈⠿⠻⠿⠿⠛⠉⠻⢿⠿⠿⠿⣿⡿⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣻⣿⣿⣷⣿⣿⣿⠀⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣤⣀⣤⣤⣶⣦⣤⣄⣤⣄⣀⣀⣀⣼⣿⠀⢸⣿⣿⣿⣿⣿⣏⡛⢛⣉⡛⠛⠛⢀⡇
⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃
⠀⢠⣤⣤⣠⣠⢤⣠⣤⣤⣄⣄⢠⣠⣄⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠰⠲⠖⠒⠲⠰⠒⠀⠲⠒⠗⠖⠖⠶⠲⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠤⠄⠤⠶⠰⠆⠦⢤⡔⠒⠖⢖⡀⣀⣰⠀⠓⠚⠓⠛⠉⠀⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢹⠏⢻⡏⠹⡧⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠀⠀⣼⣶⡏⠀⠀⣦⣤⢰⡤⣤⣤⢂⠀⠀⠀⢡⣾⡄⠰⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠿⢷⠋⠀⣰⣧⣾⣸⡇⣿⡹⣿⡀⠀⠈⢛⠻⣷⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢾⠀⢀⣿⡾⠛⢻⣷⡟⠛⣿⣷⡀⠀⣟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠘⠛⠧⠤⢼⠿⠧⠤⠚⠋⠁⠀⢀⡀⠀⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⣿⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⣿⣿⠌⣡⣌⠙⢩⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠿⢃⡀⠁⠀⠘⢿⣷⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠸⠿⠋⠀⢱⡗⡆⠀⠹⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢚⡿⠛⠁⠀⢐⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣷⠈⠀⠀⠈⠦⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⡴⠋⠀⠀⠀⠀⢸⣇⣶⢀⣶⡄⠀⢀⣶⣷⡄⠀⠃⠀⢰⣶⣦⡀⠀⢠⣶⡀⣶⡇⣷⡀⠀⠀⠀⢧⡈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡘⠃⠀⠀⠀⠀⠀⠸⡏⡅⠹⣿⠃⠀⠀⢿⣿⡗⠀⠀⠀⢸⣿⣿⠇⠀⠈⠿⠀⠛⣅⣿⡇⠀⠀⠀⠀⠳⡜⠛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⠁⠀⠀⠀⠀⠀⠀⠀⢲⢁⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠈⣇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⣤⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢
⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢳⠰⡀⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⣺⣷⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠀⣣⢱⣹⣴⣦⠀⠀⠀⣀⡀⢸⠀⠀⣀⠀⠀⠀⣿⡇⣰⣿⣿⠁⢉⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠀⠀⠁⢸⣿⣿⡋⠠⠀⢀⣿⡇⢸⠀⠘⣿⠆⠀⡄⣹⡿⣿⢿⠁⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠿⡛⣦⣷⡀⣦⢱⣾⡄⣠⣧⣼⣽⣿⣿⣷⡇⠀⠀⠀⠀⠀⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠿⢟⣿⢳⡼⡟⢯⡷⣿⣿⣿⣿⣿⡫⠎⠀⠀⠀⠀⠀⢀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡓⠹⠾⡟⡟⠟⠻⠉⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡛⠢⡀⢠⠉⠳⡤⠋⠙⡿⢋⣉⣉⡉⠉⢻⢻⡟⡿⠋⠈⢹⢹⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠀⣉⣁⠀⠈⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⣄⣀⣇⠀⠈⠿⠀⠀⠁⠀⠀⠀⠈⠁⠀⡗⢀⣾⢈⠞⠀⠀⠀⠆⡌⠀⠀⠀⠀⢀⣀⠠⠤⠄⠀⠒
⠀⠀⠀⠀⣧⣄⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣸⡟⣴⠶⣦⠾⠞⢒⣀⣈⠉⠁
⠀⠀⠀⠀⡆⠈⢣⡀⠠⡄⠀⣀⣶⡀⣀⠀⠀⠀⠀⠀⠛⠑⠲⢧⡀⡇⠀⠀⠀⠀⠉⠀⠈⠉⠑⠀⠐⠆⠠⠄⠀⢠⣄⣀⣀⣀
⠀⠀⠀⠀⡇⢰⠋⠙⠠⣿⠋⠀⠀⠙⢫⡀⠀⠀⠀⣀⣤⣤⠒⡞⢁⡇⠀⠀⠀⠀⠀⠤⠤⣀⠂⠉⠉⠀⠀⠀⠀⣶⡤⢤⣀⣀⠀⠈⠉⠉⠉⠁⠐⠒
⠀⠀⠀⠀⢣⣾⢄⠀⠀⠈⢦⠀⠀⠀⠀⢻⡍⠀⠀⠀⠉⠛⢦⡟⠀⡇⠀⠀⣤⣤⣅⣀⣤⣿⣿⣷⠀⠀⠀⠀⢰⣸⠁⣿⢿⠿⢻⣿⣶⣦
⠀⠀⠀⠀⡿⡹⣻⠷⢴⠄⠈⠳⣄⣤⡴⠆⠀⠀⠀⠠⣄⡀⢀⡱⠊⡇⠀⠀⡆⠀⠀⡽⢇⣉⠷⣞⠛⢶⠀⠀⢸⢜⠀⠐⠂⠠⠈⠛⣿⣧
⠀⠀⠀⠀⡇⢱⠙⠀⡿⠄⠀⠀⢀⠈⠱⠄⠀⠀⠀⠀⠈⢻⡏⣇⢴⠇⠀⠀⡇⢠⠊⠀⠀⠀⠀⠈⠆⢸⠀⠀⠘⢾⣛⡛⠶⠴⣒⣀⡿⣿
⠀⠀⠀⠀⡇⠸⠀⢸⠓⡦⢅⣢⣾⡆⠀⢠⡄⠀⠘⡶⢠⡴⢇⡏⢰⡀⠀⠀⠁⣇⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⣠⡤⠲⣿⠟⣿⡶⢶⣿⡟
⠀⠀⠀⠀⡇⠀⡇⢼⠀⡇⠀⠀⠈⣷⣄⡜⠻⢦⣤⡅⠀⣅⢸⡧⢸⠀⠀⡸⠛⠛⢦⡀⠀⠀⢋⠚⠃⢸⣠⡞⣉⡵⠶⠛⣛⣩⠟⣡⠶⢤⣙⡆
⠀⠀⠀⠀⡇⠀⡇⢸⠀⡀⠀⠀⠀⠹⣆⠙⡄⢸⠀⡇⠀⢸⠈⡇⢸⠀⠀⢇⣀⣤⣾⠛⠻⣦⣿⣄⣠⣟⣩⠾⡉⣁⣤⡿⠻⠅⠀⣉⣀⢠⡇⡇
⠀⠀⠀⠀⡇⠀⠃⢸⡄⡇⠀⠀⠀⠀⣿⣆⠘⣾⠀⡇⠀⠀⣿⡏⢸⠃⠀⢸⢈⡭⠷⣠⡄⣀⢀⣤⢄⣹⣭⢓⡿⠋⠀⢘⡏⠉⠉⣩⠓⢺⡄⡇
⠀⠀⠀⠀⡇⠀⠃⠈⣇⢸⠀⠀⠀⡆⣛⢸⣄⠘⣼⠁⠀⠀⢻⣀⢾⡀⠀⠻⣯⡇⠀⠐⠟⠟⠛⢳⡞⢻⠟⠋⡇⠀⠀⡘⠧⢀⣠⣄⣀⣨⡇⡇
⠀⠀⠀⠀⢣⠀⡆⠀⣿⠈⡇⠀⠀⡿⠋⠉⠉⠙⠿⣄⠀⠀⢸⠃⢸⠃⠀⠘⠟⠙⣤⢄⣤⣀⣀⢀⠜⢻⠀⠀⡇⠀⠀⠐⡏⠉⠀⣶⠒⠚⣶⡇
⠀⠀⠀⠀⠀⢂⡀⠀⡼⢃⡸⠖⠾⠟⠙⠖⠲⢤⣄⠙⢦⡀⢸⡄⢹⠀⠀⠀⣳⠞⠁⣠⡿⠣⠼⠥⠤⠾⠀⠀⡇⠀⠀⠸⣧⡀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⡿⠄⠳⡀⢧⠚⠀⠀⠀⠀⠀⠘⡄⠀⢈⠐⢶⣿⢾⣁⢾⠀⣰⠋⢁⣤⠞⢹⡃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⡟⠙⠛⣿⠛⢛⣷⣇
⠀⠀⠀⠀⡇⠈⣦⠙⣿⡀⠀⠀⠀⠀⠀⢀⡥⠤⠴⠦⠤⣌⣋⣻⣾⣀⣹⣤⣎⣸⣦⡤⣼⣦⣤⣤⣤⣤⣤⣐⣓⣒⠒⠛⠓⠴⠮⠭⠭⠭⠭⠭⠍⠙
⠀⠀⠀⠀⡇⢠⠈⢣⡈⢷⣦⡀⢀⣀⣜⣡⣤⡖⢲⡋⠰⡏⢻⠇⢹⠏⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⡇⠸⠀⠘⠧⠘⡇⠉⡏⠉⠉⡇⢠⠇⠈⡇⠀⢧⠘⣀⢾⡄⢀⣀⣠⣀⣀⣀⡀⠀⠀⢀⣀⣤⣀⣄⡀⢀⣀⡀
⠀⠀⠀⠀⡇⠀⠀⢰⠀⢃⢁⠀⢀⠀⠰⠇⢸⠀⠀⡇⠀⠸⣈⠃⢸⠆⠚⡛⠀⠀⠀⢸⠿⢿⡿⢿⠁⠀⠀⠀⡙⠓⠀⠛
⠀⠀⠀⠀⠇⡀⠀⢸⣀⣸⣾⠀⠸⡦⠘⠓⠙⠙⠁⣀⠀⠀⣿⡠⢿⠀⠀⡄⠀⠀⠀⢹⠀⢸⡇⢸⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠁⠀⢨⠁⠀⢸⠀⠀⡆⠀⠀⠀⠉⠛⠋⠷⠶⢿⠃⣸⠀⢠⠁⠀⠀⠀⢈⠀⢸⡇⢸⡀⠀⠀⠈⠇
⠀⠀⠀⠀⠶⠶⠤⠚⠷⠤⢸⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢉⠀⢸⠀⠀⠀⢀⡏⠀⢸⡇⠘⣷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢘⡠⠟⠀⢨⡄⠀⠀⣯⣀⣠⣼⣧⣀⣹⣄⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⣸⠀⢸⡋⢳⡏⠉⠉⠉⠉⠉⠉⠉⠉⣿⣁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣥⠊⢫⠀⣼⠹⡟⠀⠀⠀⠀⣀⢄⠀⠀⠀⢸⡇⠀⣆⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣄⠀⠀⠀⢀⣀⠀⠀⢀⣍⠀⣾⠀⡿⠀⣿⠀⠀⠄⠛⣙⣻⣷⡴⠆⢸⢀⡴⢉⡿⠃
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠶⢤⡍⠉⠦⣀⠞⠉⠉⠟⠁⣉⠛⣣⡇⣘⠁⠀⠿⡀⠀⠀⠀⠉⡽⠋⠁⠛⠻⣎⣠⡿⠤⢤
⠀⠀⠀⠀⢐⠀⠰⣏⠉⠛⠦⠀⠀⢠⡀⠛⢀⡄⢀⠒⡾⢁⣏⡽⡿⠋⠒⣶⣾⣧⣤⣀⣀⣴⠃⠀⠀⠀⠀⠈⡟⠀⠀⡸⠃
⠀⠀⠀⠀⢨⡀⠒⠚⠓⠂⠀⠀⠀⣤⠛⡩⢢⣽⠃⢠⡧⠞⢻⡠⡿⠀⣀⣈⣀⣀⣀⣀⠀⠘⣆⠀⠂⠐⠁⣠⡛⠓⠘⣷⣄
⠀⠀⠀⠀⢸⠛⠒⡶⠖⠃⠉⠹⢧⣭⣬⣶⡯⠾⢒⠟⠀⠀⢸⠁⣞⢸⡟⠛⠛⠛⠛⢿⡇⠀⣨⣟⣶⠶⢻⡏⢹⠀⢠⡏⠀⢳
⠀⠀⠀⠀⢸⠀⣞⠣⠟⠛⠛⠓⠂⠛⠢⠤⢤⢰⠋⠀⠀⠀⣼⠊⡿⠛⣇⠀⠀⠀⠀⢸⡃⠘⡿⠟⣥⠀⢉⡛⣹⣤⠎⠀⢀⡾⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⠀⠀⣀⣤⣤⡤⠶⠶⣤⡀⠀⢸⠀⡇⠀⠀⠀⠻⢤⣷⡶⢿⣀⠀⣀⡠⠋⣀⢾⣿⡆⠒⣛⡿⢠⣧⣸⠀⠰⠛⡏⠉⠛⡉⠉⠉⠉⠁
⠀⠀⠀⠀⢸⡔⠋⠉⠀⢉⣴⠦⢤⣴⣭⣦⣸⠀⡇⠀⠀⠀⠀⠀⠀⠙⠛⠿⢯⣽⣶⡞⣹⢿⡷⢤⠾⠯⣴⣎⠸⠎⠙⠛⠛⠉⠓⠚⠁
⠀⠀⠀⠀⢰⣞⣩⠋⡉⠉⠀⡔⠁⣀⠠⠘⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠣⠽⣆⣒⠁⠠⠀⠯
⠀⠀⠀⠀⠈⠱⠄⠈⠻⠶⠾⠿⠓⠄⠠⠖⠁⠷⠤⠠⠀⠤⠤⠤⠴⠦⠶⠲⠶⠶⠶⠶⠶⠶⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶

⠀⠐⠿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰
⠀⠀⠀⠉⠙⠿⢦⣤⡀⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡄⠀⠀⠀⠀⠀⣠⣤⠾⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⣄⠀⠀⠀⢹⡈⡇⠀⠀⣀⠀⣀⠀⠀⠀⡈⡾⠁⠀⠀⠀⣴⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠦⣇⠀⠀⠻⠷⡶⠏⠉⠉⠈⠹⢶⡶⠷⠃⠀⠀⣄⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣌⠆⠀⡠⠘⠒⠛⠒⠒⠒⠛⠒⠓⢦⠀⠀⡜⢡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣔⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠒⠒⠒⠢⡤⠄⠀⢠⣤⠤⠴⠶⠒⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢻⠀⠀⠀⠀⠀⠘⢄⣀⠜⠁⠀⠀⠀⠀⠀⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⠋⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⡀⠀⠀⠀⢹⡘⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢢⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⡏⡅⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢸⣼⢻⡀
⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡄
⠀⢀⣤⡴⠞⠁⠀⠀⠀⢠⣸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠛⢧⣀
⠿⠇⠁⠀⠀⠀⠀⠀⠀⣬⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡟⡃⠀⠀⠀⠉⠳⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⢑⡃⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢾⠈⢆⠀⠀⠀⠀⠀⠀⣧⡄⠀⠀⠀⠀⠀⢀⡼⣧⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠈⠀⠈⢦⡀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣠⡞⠁⠈⢳⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⠀⠀⠙⢦⣄⣀⠀⣾⠀⠀⢀⣠⠔⠋⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⠄
⠀⠀⠀⣼⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠖⠒⠚⠋⠉⠉⠛⠛⠛⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠈⠉⠀⠈⡄⠀⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣷⡄⡈⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢸⡇⠑⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⡀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡀⠉⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠐⣼⡆⠀⠀⠀⢄⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⡟⣷⡄⠀⠠⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠹⡇⠐⢨⣶⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⣤⠄⢀⢹⣿⣿⠞⠀⠀⠈⠽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠐⢀⡶⠖⠒⠒⠻⡄⠀⠀⠉⠁⠀⠀⠀⠀⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠉⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣤⠀⢨⣿⣷⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⡯⠀⠀⠉⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣠⡀⠀⠀⠀⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣿⡿⠁⠀⠀⣠⣿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠿⠇⠀⠀⢀⡙⠛⢻⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⢀⣤⣤⠀⠀⠀⣤⣤⣴⣿⡇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠃⠀⠀⠀⠈⠻⠟⠛⠃⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠉⠻⠟⢀⡀⠀⠀⠀⢀⣠⣤⡀⠀⢶⣶⣤⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⣀⡀⠀⠀⠀⢻⣿⣿⣦⣤⣿⣿⡿⠀⠀⠀⠹⠟⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢿⡟⠁⠀⠀⠀⠀⢙⣿⣿⡿⣿⣿⡆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣴⡆⠀⠙⠋⠀⠈⠉⠀⠀⢿⣿⣦⣤⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣿⣿⠃⠀⠀⣶⡄⠀⠀⠀⠀⢺⡿⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⠛⠟⠀⠀⠠⣿⣿⣶⡿⠂⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠀⠀⢻⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⡆⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣄⢐⣶⣾⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣫⣶⣿⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⢡⣿⣿⣻⢿⣿⣿⣿⢯⣿⡿⡽⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⣿⣿⡿⣿⣿⣿⣿⣿⡿⣷⣶⣶⣶⣤⣄⢠⣶⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣿⣿⠀⣿⣿⣧⠟⠉⢀⢿⡟⡴⣼⠟⠟⢿⡿⠏⢹⣾⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣿⢻⡏⠀⠀⣠⣾⣽⡇⠁⠀⠀⠘⢿⡇⣸⡟⢿⣿⡿⠈⣿⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆⠻⢿⣾⡿⠉⠀⣿⠀⠀⢀⣠⣾⡇⣺⠂⠸⣿⣯⠀⢻⠉⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⡿⠻⠶⣿⢇⠰⣷⣿⣿⣿⣿⣿⡿⣸⠏⠀⠀⠉⢿⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⠋⠁⠀⠀⡴⣣⠖⠀⠈⢿⣿⡟⠀⣿⢁⠏⠀⠀⠀⠀⠀⠉⠂⠀⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡄⠀⠀⠀⢀⣾⠞⠁⠀⠀⠀⢸⢻⡇⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⣼⡿⠃⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠄⠀⣿⠀⠀⠀⠀⢀⣼⠋⠀⠀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣻⠀⠀⠀⠀⠾⠃⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡀⠂⠀⠀⠀⠀⠧⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⡇⠀⠀⠈⠃⠀⠘⠁⠀⠀⠀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣰⠇⠀⠀⢀⠦⠀⢤⡀⠀⠀⠀⣇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠈⢙⢃⢀⣘⡃⠁⣀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠛⣱⢫⣿⠸⣿⣻⡍⠙⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡄⠉⣿⣿⠐⣿⣿⠁⠀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⠛⠓⢤⣹⣿⠠⣿⢃⠴⠟⢛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣿⣿⣦⡀⠈⢦⠤⡖⠁⢰⣿⣿⣿⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠏⠀⢻⣿⣿⠃⠀⢸⣀⠁⠀⢰⣿⣿⡇⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⣿⡿⠀⠀⠀⣿⠀⠀⠈⢿⣿⡀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢼⣿⣿⡆⠀⠀⠀⡿⠀⠀⠀⣿⡿⣿⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢻⣿⣻⠀⠀⠀⡇⠀⠀⢀⣿⡏⢻⡇⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⠘⣿⣿⣷⣄⠀⣇⠀⣠⣿⣿⡇⣸⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢻⣿⣿⣿⡆⡧⣼⣿⣿⣿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠈⢿⣿⣿⣧⣿⢻⣿⣿⠏⠀⢿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠁⠀⠀⠈⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜

⠀⠀⠀⠀⠀⢠⡀⡀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠬⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠤⠤⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠒⠒⠓⠒⠒⠒⠒⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠡⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠐⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠐⡄
⠀⠀⠀⠄⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠
⠀⠀⡘⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣰⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡤⠐⠀⠚⠁⠓⠂⠄⡀⠀⠀⠀⢂⠀⠀⡀
⠀⢀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⠴⠃⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠢⠀⠈⢄
⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⢀⡀⡁⠀⠘⡀
⠀⠀⢄⠂⡀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠃
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢠⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⣀⣀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠁⠀⠀⡀
⠀⠀⢀⡀⠈⠂⡀⠀⠀⢠⠻⠀⠀⠀⠀⢠⠀⠁⠉⢀⡐⢀⠇⠀⠀⠀⢰⠈⠛⢫⣉⣛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢰⠁
⠀⠌⠀⠈⠉⡀⠈⠢⢀⢣⠁⠀⠀⠀⠐⠁⠀⠀⢀⡞⠀⠈⠛⢦⡀⠀⠀⡀⠠⡿⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠒⠢⠆⢘⡀⠠⠀⠌⠀⠀⠐⣤⠲⢣⠀⠈⢢⣀⡽⡴⢊⡕⢄⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⡘⣄⠀⣠⡎⠂⠴⠷⠖⠋⠈⠀⠱⣏⡰⠂⠁⠀⠠⠤⠖⠠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⣠⠇⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠂⠂⠰⠀⠦⠄⠀⠤⠀⠀⠀⠀⠈⠑⠀⠀⠀⡀⠴⠂⠀⠀⠀⠀⠀⣰⠆⠀⠀⡀⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠡⠖⠀⢠⡠⠚⢥⠛⠀⡔⠁⠳⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡀⢀⣀⠠⠤⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠉⠉⠁⠒⠠⠀⠀⠈⠉⠉
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢀⡀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣯⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⣳⣿⣿⣿⣿⣿⣿⣿⣵⣾⣿⣿⢃⣿⣿⣿⣿⣿⣿⡿⢟⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⢸⣿⡿⣿⣿⡿⠟⣑⣤⢞⣽⠏⢡⣾⣿⣿⣿⣿⣿⣩⣾⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣸⣿⣄⡿⠿⠃⠉⣉⣉⣿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠸⣿⣿⣿⡿⠋⠀⠀⢀⣀⣼⣿⣿⣿⠞⠛⢡⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⢀⡤⠒⠉⠀⣠⣴⣿⣿⠀⢸⣿⣿⠥⠒⠾⢛⣉⣿⣿⣿⣿⠋⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠁⠀⠀⢀⣾⡿⠛⢙⣯⠀⢀⣿⣧⣴⣶⡿⠻⣿⣻⡿⣫⢆⣼⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡠⢻⣏⣠⣴⣿⠿⢃⠉⡩⠥⠶⡶⠂⢠⣽⡇⣾⡟⣼⢣⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠛⣡⣾⣥⣤⣶⣶⣶⣿⠀⡸⢇⣵⢏⣼⣧⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⢠⡶⠋⠉⠁⣼⣿⣿⣧⣾⣿⣯⣭⣭⣭⣶⣾⣿⠸⣷⠟⣩⣾⣿⣿⣿⣿⣿⣿⣿⠟⡿⠁
⢠⠋⠀⠀⢀⣠⣿⣿⠿⠟⠉⢀⣤⣾⠿⣿⣿⣿⣿⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⡟⣡⠞
⠇⠀⠀⣠⣿⣿⣟⣵⣾⣯⣶⣿⣿⣿⣤⣿⣿⡿⠋⢀⣼⣿⣿⢿⣿⣿⣿⣿⣿⠞⠁
⠀⠀⣼⣟⣹⣿⣿⣿⣿⠻⠟⣿⡈⢿⣿⣿⣿⢂⣴⣿⣿⡿⠑⢸⣿⣿⣿⡿⣿⣷⣠⢄
⠀⠀⣿⠟⣵⣿⣿⣿⣿⡀⠀⣿⣧⠸⣿⣿⣵⣿⣿⣿⡿⠀⢠⣾⣿⣿⣿⣷⡈⢍⡻⣿⣆
⠀⠀⣿⡸⣿⣿⣿⣿⣿⣧⡀⠻⣟⢿⣿⣿⣿⣿⢿⣿⡇⢰⣼⣿⣿⡇⢻⣿⡿⣦⠀⠚⢿⡄
⠀⢰⡟⠀⢹⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣷⣿⡿⠀⣼⣷⣟⢸⣿⠘⣿⣷⠈⢷⡀⢘⣷
⠀⣼⠁⠀⣿⣿⣿⣿⣿⣿⡈⠙⢿⣿⣿⣿⣿⣿⡿⠁⣰⣿⣿⣿⣾⣟⠀⠙⠿⢧⡈⣧⣀⢻
⠀⢻⡀⠀⠘⢿⣾⣿⣿⣿⣷⡄⠈⠻⣿⣿⣿⡟⠀⡴⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⠛⠻⠿⣧⣄⡀
⠀⠈⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣶⣄⣄⣿⣟⣆⡨⠄⣹⣿⣿⣿⣿⣿⣿⡿⣦⡀⠀⠀⠀⠀⠸⣏⠙⠦⢤⡀
⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢹⣆
⠀⠀⠀⠈⢧⣼⣿⣻⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣷⣄⠀⠘⣿⡀
⠀⠀⠀⢠⣾⡏⢹⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣽⠿⣿⣿⡛⠿⢿⣿⣷⣾⣿⣷⣾⣿⣉⠙⠶⣤
⠀⠀⠀⣿⡿⠁⠘⣿⣿⣿⡿⠋⣆⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⣄⠀⠱⣄
⠀⠀⠀⣿⠁⠀⣾⡿⣿⣿⣡⠞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡐⢆⠘⠢
⠀⠀⠀⠀⠀⣰⡿⠀⢹⣿⣯⠀⠸⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⢿⣿⣿⣿⣶⣀⣀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣧⠀⠡
⠀⠀⠀⠀⢸⣿⠃⠀⢸⣿⣇⠀⠀⢿⣿⠈⠻⣿⣿⣿⣸⣿⣿⣿⣿⣷⣿⣿⣿⣟⣿⣿⡍⠁⠈⢳⡄⠀⠀⠀⠀⠹⣿⡇⠀⣤
⠀⠀⠀⠀⣸⡏⠀⠀⠸⣏⣿⡆⠀⠘⣿⡀⠀⠈⢿⣿⣿⣝⡛⢛⣿⣿⣯⣿⣿⣿⣿⣿⣿⣦⣄⠀⠹⣆⠀⠀⠀⠀⢻⡇⠀⠉
⠀⠀⠀⠀⢿⠁⠀⠀⠀⣽⢿⣿⡀⠀⠘⢧⡀⠀⠈⢿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠙⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠈⢆⠀⠀⢰⡏⠀⢹⣇⠀⠀⠀⠙⠂⠀⠘⣿⣏⠙⠳⣄⠈⠻⣧⠈⠻⣿⣿⣿⣿⡿⣿⣿⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠁⠀⢸⡀⠀⠀⢿⣦⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⡈⢆⠀⠸⢷⣤⣿⣿⡟⠋⠁⠙⣏⠛⠛⠻⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠙⠳⠤⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣵⡄⠈⠀⠙⢯⡓⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣷⣄⠀⠀⠰⠈⠒⠲⢤⣤⣤⣤⣄⣀⣀⡀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠠⡄⠀⠤⣤⣄⡙⠳⠦⣍⠙⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣤⣤⡤⠀⠈⠳⣤⠀⠀⠉⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⢤⣤⣀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠘⣿⣳⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣿⣯⣦⣀⠀⠀⠀⠀⣀⣠⣾⣿⠏⠈⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠀⢈⣿⠛⠉⠁⠉⢩⣉⠉⠉⠉⣶⣂⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⣿⡀⠀⢀⠘⢻⠋⢀⣀⢀⣼⣿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⣿⣿⣿⣿⣤⣤⣤⣿⣿⣿⣿⡏⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⣟⠛⠛⠋⠉⠀⠀⠉⠙⠛⠛⢳⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣤⣠⡀⡀⢰⣾⣶⣦⣤⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢁⣇⢸⠟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡘⢿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿⠁⢀⣴⣮⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⣿⠂⠉⠉⠁⠀⠀⣤⣠⠈⠁⠁⠀⣀⣻⡟⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⣿⣦⣤⣤⣤⣀⣀⣨⣧⣄⣤⣴⣶⣿⣿⣧⠀⢸⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠟⠋⠃⠿⣿⡇⠸⠇⠶⠖⠗⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠖⠛⠛⠳⠓⠚⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⡟⣏⡟⣏⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⣧⢹⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣿⠁⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⢹⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⢠⣄⣀⣾⡟⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣶⣷⣼⣿⣿⣿⠃⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠷⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢣⡀⠀⡀⢠⣠⠠⣤⠀⡄⠀⡰⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠑⢄⣠⣶⡟⠂⢾⣷⡡⠚⠀⠀⠀⠀⠀⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⢙⠀⠃⠀⠀⠸⢣⠀⠀⠀⠀⠀⠒⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⣠⡼⠄⣀⡀⠚⠃⠧⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢄⡀⢠⠏⠀⠀⠀⠀⠀⠀⢀⢸⡄⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⠀⡄⠀⠀⠑⠀⠐⠟⣼⡷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣃⠀⣀⣘⣦⣀⣈⣡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠁⢸⠈⠉⠹⠇⣽⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢻⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠻⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠁⠀⣿⣆⠀⠀⠀⠀⠀⡆⣿⣾⡇⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⣿⠀⠀⣄⢀⠸⡷⣿⣿⣿⠀⠀⠀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⢴⣿⣿⣼⣆⠹⢸⣷⣷⣿⣿⣿⣷⡀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠏⠀⠀⢠⠜⠹⣏⣿⡏⣸⢸⣿⣿⣿⢻⡏⠀⢧⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⢿⣿⣷⡀⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠻⢿⣧⣿⣿⣿⠟⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠐⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠉⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣯⡍⣆⠀⠀⠈⢢⠀⡀⠀⣀⢀⠞⠀⠀⠀⠐⢉⡙⡆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠌⠥⠃⢀⡤⠀⢤⣧⠋⡚⠻⣇⠤⢄⠀⠀⠢⡀⠂⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⠙⡆⡧⠀⡇⢸⠁⡌⢩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢭⣉⣤⢃⣠⣤⣌⣳⠬⠭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣿⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⣠⣼⣶⣯⢼⢤⣀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⢈⡷⡻⣻⣯⢞⣿⢢⢣⠹⡳⣞⠈⠙⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡜⡵⡱⢃⡾⢉⣂⢧⠳⡹⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣾⢤⢸⡻⢸⣱⢁⣎⢃⡟⢫⡆⢣⢡⢻⠹⡇⢻⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⣼⡇⣧⡇⠠⠈⢠⠀⢐⠸⠘⠈⡌⡬⣽⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡗⢻⡀⠆⡇⠆⢠⢸⠀⠀⠀⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡲⢻⣿⡆⡇⡇⢰⠁⠇⢸⠸⠀⠠⠈⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠋⠈⣿⡿⣵⡹⡄⣾⡟⣼⠀⡄⠀⠀⠀⠀⡑⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⣷⣽⡷⣝⣜⠆⣿⡜⡱⢠⢢⣣⣶⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇⠈⠻⢮⣪⡻⣤⣿⣔⣕⣵⠿⠋⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣎⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠈⠉⠛⠻⣯⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⢀⣠⣤⣽⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣴⠋⠉⠀⠀⠈⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣴⠾⠛⠛⠛⠿⢶⣶⣟⣍⡻⣦⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⢋⡭⣍⡻⠆⠀⠀⠀⠀⠀⠀⠈⠻⣦⡷⢸⡇⠀⠀⠀⢶⣽⣦⡀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡟⠁⠀⠀⠀⠀⠙⣮⡳⣄⣠⡶⠟⠛⠉⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠳⣭⡽⠆⠀⠀⠀⠀⠀⠀⠀⢰⣀⣐⣟⣻⣄⣀⣀⠀⠀⠀⠈⢻⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡀⠀⠀⠀⠀⠛⠃⠀⠀⠈⢯⣽⠟⠉⠉⠉⠉⠀⠀⠀⠀⠀⢿⡆⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⠶⠿⠿⣿⣿⣿⣿⡂⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣦⠀⠀⣀⣤⣤⣤⣀⢀⣴⠿⠇⠚⠛⠉⠉⠉⠛⠿⣿⣿⣿⡀
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠁⠀⠉⠻⢦⡀⠀⠀⠉⠛⠷⣶⣦⣄⠳⣄⣻⣿⣿⡏⣰⠞⠉⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢦⡐⣦⢻⣾⡇⠙⠿⣦⣭⣭⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠶⠛⠻⣦⣠⣀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⠾⠇⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⣤⣤⣤⣴⣶⣶⣶⣶⣶⢶⡶⢿⠿⠿⢿⠿⡿⠛⢻⠛⠛⡛⠛⡻⠛⡋⠉⡍⠉⢍⠉⢉⠍⠈⢣⡀⣠⣊
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠉⠉⠀⠹⡄⠀⠆⢡⠀⡰⠀⠈⣆⢀⣆⣀⣰⣀⣎⣀⣀⣧⣴⣥⣤⣴⣶⠷⠶⢾⣶⣿⡾⠛⠛⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠛⠋⠉⣹⣿⣿⣿⠟⠉⠉⠉⣠⣿⣿⣿⠟⠁⠀⢀⣴⣿⣿⠟⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⣴⣿⣿⡿⠃⠀⠀⢀⣾⣿⣿⡟⠁⠀⠀⣰⣿⣿⡿⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢫⣤⣤⠸⠿⠿⢿⣁⣀⣀⡰⠿⠿⠿⣯⣤⣤⣤⠘⠋⠉⣩⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣿⣿⡿⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠿⠿⣇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⣀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣼⣿⣿⡇⠈⠉⠉⣿⣿⣿⣿⠉⠉⠁⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢸⣿⣿⣧⠀⠀⠀⢻⣿⣿⣿⡆⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠈⠉⠉⣶⣶⣶⣾⡛⠛⠛⢳⣤⣤⣶⡟⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠘⣿⣿⣿⣷⡀⠀⠀⢻⣿⣿⣿⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡱⣶⣶⣶⣾⠛⠛⠛⠓⣤⣤⣤⣽⠿⠿⠷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⠀⠀⠀⠘⢿⣿⣿⣷⡀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠉⠛⠣⣤⣤⣤⣬⡟⠛⠿⠿⢀⣄⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠹⣿⣿⣿⣿⣆⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣶⡟⠛⠛⠛⠛⠣⣤⣤⣤⣼⡛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣶⣶⣶⠉⠉⠉⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⢁⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠑⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⢸⣿⣿⡇⠀⣿⣿⣿⠀⢰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠈⣿⣿⡇⠸⣿⣿⡇⠀⣿⣿⣿⠀⣿⣿⡟⠀⣼⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣧⠀⠘⣿⣷⡀⢹⣿⣿⠀⣿⣿⡇⠀⣿⣿⡇⢰⣿⣿⠁⢰⣿⡿⠁⢠⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⡀⠘⢿⣷⡀⠹⣿⣧⠀⢿⣿⡆⢹⣿⡇⠀⣿⣿⠃⣼⣿⠇⢠⣿⡿⠁⣠⣿⠟⠀⢠⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠹⣷⣄⠈⢻⣷⡄⠹⣿⣇⠘⣿⣧⠘⣿⣇⠀⣿⡿⢠⣿⡿⢀⣾⡿⠁⣰⣿⠋⢀⣴⡿⠃⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠹⢷⣄⠈⠻⣧⡀⠙⣿⣄⠘⣿⡆⠹⣿⡀⣿⣿⢠⣿⡇⣸⣿⠁⣼⡿⢁⣼⡟⠁⣠⡾⠋⢀⣠⠾⠋⠀⣠
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠳⣤⡀⠙⠷⣄⠈⠻⣦⡈⢻⣆⠘⣿⡄⢻⣇⢸⣿⢸⣿⠀⣿⠇⣼⡟⢀⣾⠋⢠⡾⠋⢀⣴⠟⠁⣀⡴⠛⠁⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠙⠲⢤⣀⠙⠲⣄⡈⠳⣦⡈⠳⣄⠙⢧⡈⢿⡈⢿⡈⣿⢸⡏⢸⡏⣰⠟⣠⡟⢁⡴⠋⣀⡴⠋⣀⡴⠚⠁⣠⠴⠚⠉⣀
⠀⠀⠀⠀⠀⠀⢄⡈⠑⠲⢄⡈⠛⠢⢄⠙⠲⢄⠙⠦⡈⠳⣌⠳⡌⢷⡘⣇⢻⢸⠇⡟⢰⠏⡴⠋⡴⠋⣠⠞⢁⡴⠚⣁⠤⠖⢋⣀⠤⠒⠋⣁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠒⠀⠀⠉⠒⠀⠈⠒⠀⠉⠂⠈⠓⢌⠣⡘⢆⠣⠹⠘⠿⠸⢡⢋⠜⡡⢊⠔⠋⠀⠊⠡⠐⠊⠀⠀⠂⠁⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⢠⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣞⣁⣀⣀⣀⣙⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⡄
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣤⣤⣴⡶⣶⣶⣶⠶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠁⠀⠀⢀⣾⣿⣿⣿⣼⣿⣿⡇⠉⠙⠻⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢻⠀⠀⠀⠀⢸⣇⣸⣿⣿⣿⣿⣿⣿⣿⣧⣾⡀⣦⠀⣠⡍⠛⠳⢦⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠞⢉⣀⡎⠀⣠⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡄⠀⠀⠈⠉⢻⣶⡤
⠀⠀⠀⠀⠀⣴⢿⡿⡆⡼⠉⢀⣼⡿⡿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢀⡼⠃⠉⠁⣰⠟⠁⠀⠀⠀⠀⢀⣤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⠀⠀⠀⠀⢿
⠀⠀⠀⣠⣴⠋⠀⠀⢀⣼⠿⠶⠶⣤⠀⠀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢀⡄⠀⠀⢸⡇
⠀⠀⠾⠋⣿⢀⡀⢀⡾⠃⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣼⣿⠀⠀⠈⣧
⠀⠀⠀⢀⡿⠘⣷⡟⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢰⡀⠀⢿
⠀⠀⠀⣸⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡿⢦⣄⣀⣤⣤⢦⣤⣄⡀⠀⠀⢸⣿⣿⣿⣾⣧⠀⢸⡇
⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣟⡄⠀⠀⠀⣼⢣⢎⣽⠟⠉⠀⠀⠀⠀⠉⣻⠶⣒⠻⣿⣿⣿⣿⣿⣠⠄⣧
⠀⠀⢸⠇⠀⢠⡿⠶⠶⣤⣀⢀⣴⣿⠟⠃⠀⠀⠀⠹⣦⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱⣸⣿⣿⣿⡿⠋⠀⣿⣶
⠀⠀⣿⠀⠀⢸⠇⠀⠀⠀⠙⣿⡿⠋⠀⠀⠀⠀⣀⣰⣿⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⣤⣾⡽⠻⣿⣿⡿⠁⠀⠀⣸⠇
⠀⢰⡇⠀⠀⣾⣁⣀⡀⠀⠠⣿⠁⠀⠀⠀⠀⠛⠛⠛⠛⢿⣽⠃⠀⠀⠐⠒⠀⠀⠀⠀⣼⠇⠀⠈⣿⠀⠀⠀⢠⡟
⠀⣾⢁⣠⣤⣿⣿⠿⢿⣿⣦⣿⣦⣀⣀⣤⣤⣤⡀⠀⢠⣟⣁⣀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢐⣿⣄⣀⡴⠾⠷⢦
⢠⡿⢋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣍⠀⠀⠀⠙⢳⣈⣿⣿⣟⡡⢎⣤⣴⡶⡛⠉⠀⠀⢀⡴⠚⠉⠀⠀⡀⠀⠀⢈
⢘⣿⣿⣆⣀⣶⣄⠀⢰⣦⡀⢠⡀⢀⡈⣧⣀⡀⢀⡀⠈⠛⢧⣴⣶⠛⣷⣸⣷⢣⣞⡴⠖⠋⠀⢀⣤⣤⣾⣿⣴⣦⣼
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣾⣿⣾⣿⣿⣿⣿⣶⣿⣦⣀⠀⣳⣼⣶⣿⣿⣿⣿⣟⣴⣠⣎⣴⣿⣿⣿⠿⠟⠛⠛⢻
⣿⠁⠀⠀⠀⠀⠀⠉⡽⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⠿⢿⣿⠿⠋⠁⠀⠀⠀⠀⠀⢸
⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣀⣀⣀⣀⣈⣀⣀⣀⣚⣋⣁⣀⣀⣘⣁⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⣄⣀⡀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠩⠭⠩⣉⡉⠑⠒⠦⣄⣀⡀⠀⠀⢀⢀⢠⣈⡋⢗⠦⣄⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠈⠉⠑⠒⠤⣌⣉⠓⠶⣦⣿⣿⡪⠇⠀⠀⠈⠻⡶⣤⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⣾⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⣍⠋⠀⠀⠀⠀⠀⡀⢋⣤⡶⠶⢞⡁⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣦⣤⣤⡴⠖⠛⢁⡧⢀⣾⣿⣾⣽⣿⡾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣟⠂⠄⣀⠘⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡌⢻⡿⠁⠀⠀⠙⠙⠧⣁⡹⣿⡻⣿⢿⣶⡍⢿⣷⡕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣶⣼⣕⣷⣭⡉⠳⠦⢄⡀⠀⢀⠀⣠⣿⠇⡾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣷⣄⣲⣿⣆⠙⣿⣦⣆⢀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣿⣞⣔⢅⣌⢲⣤⣿⣿⢋⣾⢱⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⣿⣿⣿⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣾⣿⡿⢣⣾⡏⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠟⡵⢿⣻⣦⣤⣀⣀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣿⣶⣦⣤⣤⣬⣭⣭⡯⠵⣿⣿⣧⠰⠄⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣏⡞⣠⣤⣿⣿⣧⡉⠙⠳⠤⠈⠃
⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠁⠈⠻⠿⣿⣯⣽⣿⣿⣿⠿⠛⠀⠀⢈⣉⣽⣧⠀⠀⠀⠀⠡⣴⣾⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠐⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⡾⡿⢿⡿⠃⠀⠀⠀⠀⠤⠀⠈⣽⣿⣟⠛⠻⣿⣼⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣿⡗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣳⡄⠘⢿⣿⣿⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣻⡄⠀⠀⠈⢻⣿⡛⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢡⡀⠀⠀⠘⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡁⠀⠀⠀⠀⢤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⠀⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣤⣴⡶⠶⠶⢾⣻⣇⣼⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢺⣿⣿⣟⠉⠀⠀⠀⠀⠀⠉⢸⣿⣦⣼⣿⣿⢦⡀⠀⠀⠀⠀⠀⠀⢀⠠⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠘⠿⣿⣿⣷⣦⣀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣷⣴⣿⣤⠀⠀⠀⠒⠃⠁⠀⠀⠀⠀⣠⣄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠀⠀⠀⠙⠶⠤⢤⣈⡉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⡿⣷⠀⠀⠀⠀⢠⣴⣶⠾⠿⠿⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣦⣄⡈⠛⢿⣿⣿⣿⣿⣿⡇⠀⠀⠙⢻⡄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡶⣿⠟⠛⠛⠲⢤⣈⠙⠿⣿⣿⣇⠀⠀⠀⠘⣿⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣫⣶⣿⣧⠀⠀⠀⠀⠈⠑⢦⡈⠻⣿⣦⡀⠀⠀⣿⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣫⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣶⡤⢸⣤⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠉⢀⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣶⠟⠹⠟⠁⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠙⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠂⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣄⡀
⠀⢀⣤⣴⣾⡿⠃⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛
⣠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⢹⣦

⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡛⠒⠦⣀⠀⠀⡠⠖⠋⠉⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡰⠂
⠀⢠⠞⠑⠢⡀⠈⠓⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⣲⣬⣧⣼⣥⡤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠚⡅⠀⠀⢀⣀⣀⣀⡀
⠀⢿⠀⠀⠀⠈⢢⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡯⠉⢣⠋⠁⢀⣽⢷⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣱⠀⡴⢋⠤⠒⢤⡉⢦
⠀⡞⢦⡀⠀⠀⠀⡱⢤⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⣧⣆⣜⢦⣀⣻⣼⡀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⣰⣅⡇⢰⡉⠀⡷⠈
⠀⠀⠐⠋⠓⡖⠾⣕⢈⣹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣹⠁⠀⠋⣹⠀⠙⠏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠘⠧⣈⣓⣒⠭⢍⣀⡰
⠀⠀⠀⠀⠀⠁⠀⠘⣯⡀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⢤⣉⣉⡑⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓



⣠⠴⢤⡀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠙⢄⡀⠙⢦⠀⠀⣳⡤⠚⢗⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡖⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⣇
⡴⠋⠉⣙⣾⠟⠉⠙⢷⣄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡀⣣⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠉⠀⢹⣄⡀
⠙⠒⠋⠁⡇⣀⣀⡈⣾⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢵⣊⠀⠈⠑⣄⣀
⠀⠀⠀⠀⣏⠥⠀⢛⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠻⠗⣞⠁⠈⠑⣄
⠀⠀⠀⠀⠘⢶⣉⠔⢊⢟⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡏⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⣐⡄⠘⠾⣁⠀⣸
⠀⠀⠀⠀⠀⠀⠉⠒⠧⠼⠖⠛⠂⠀⠀⠀⠀⠀⠀⡞⢻⠛⠟⢻⠉⣅⡽⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠛⠋⠋⠃⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣲⡀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢄⣙⢦⣀⡴⣋⣁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠖⢺⠐⡖⠒⣤⡀
⠀⠀⠀⢠⠖⠉⠉⠉⠙⠲⡴⢻⡀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢛⣛⡿⠯⢍⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⣇⣴⠼⢴⣁⠔⠁⢱
⠀⠀⢸⢿⠄⠀⠀⠀⢠⣴⣿⣆⢣⠀⠀⠀⠀⠀⠀⠀⠙⡞⠋⠘⢨⠳⣯⡑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠀⠀⢠⠧⠤⠤⣾
⠀⠀⡇⠈⢶⡋⡰⠦⢻⣅⣠⠾⠆⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠲⢤⡜⠁
⠀⢠⠁⠀⢼⠈⢹⠛⠉⠀⠙⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠓⢶⠋
⠀⠀⠀⠀⡎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠓⠒⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠹⣏⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡁⠀⠀⠀⠀⢹⡆⠀⠀⠀⠉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⢠⡟⠀⠙⢦⡀⢏⡵⠀⣷⠰⣏⡇⢀⡟⠘⢧⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣀⡾⢁⡾⠀⠀⠀⠀⠀⠀⣾⠀⠀⡠⣄⠹⣆⠀⠀⢸⡆⠀⠀⣼⠁⠀⠈⢷
⠀⠀⠀⠀⢠⡞⢳⣾⣁⣠⠋⠳⣾⠛⢿⢛⣦⠀⠀⣸⠋⠛⢦⡓⠊⠀⠘⣧⠶⠚⠓⢦⣼⠃⢞⣱⢀⡿⡇
⠀⠀⠀⠀⠸⡏⡿⠦⣴⣋⡉⠉⡿⠉⠙⡏⢘⡇⢠⣏⠀⡤⢤⠹⣆⢠⡞⠁⠀⠀⠀⠀⠹⡆⠀⣠⡾⠁⢿
⠀⠀⠀⠀⠀⢙⡷⢤⣾⠬⢭⣭⡏⠉⢹⣏⣽⡷⠋⠈⢷⡝⠋⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣷⠞⢫⠒⡄⣸⢧⣀
⠀⠀⠀⠀⠠⡇⠠⣶⢸⣴⢋⡉⠻⣟⡿⠯⡄⢻⡀⡤⢄⠹⡄⢀⡞⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢈⣩⠟⠁⠀⣹⠛⢦⡀
⠀⠀⠀⠀⠀⢹⡶⢖⡋⣹⣌⣁⣠⠏⠀⠀⠁⠈⣷⠑⠋⠀⢻⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷⠞⠋⠑⢏⠇⣰⣯⢤⢸⡇
⠀⠀⠀⠀⠀⠀⢳⡈⠛⠋⠈⠉⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⣠⡾⠁⠙⣺⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⡀⠀⠀⠀⢸⣃⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠻⠧⢴⡷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠓⢾⡷⠶⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣹⢒⣋⣙⣒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠈⠤⣭⠟⠿⣭⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢠⡀⢀⣄⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠑⣟⣃⣝⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠲⣾⣷⠖⠒⠉⠉⡏⠀⠀⠀⠈⠃⢠⠈⠁⠄⠸⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠔⠧⠤⠤⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐⠈⢹⣧⣼⡏⠁⠂⠈⢻⣧⠀⠀⠀⠀⠽⣠⣄⠿⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠠⠸⣷⣻⢣⠂
⠀⠀⠀⠀⢀⢠⣤⣄⡀⠀⠀⢀⠠⣺⣿⣟⣿⠠⡀⠀⠀⠻⣧⣀⠔⣿⣶⡿⢿⣾⣿⠲⣀⣼⠟
⠀⠀⠀⠑⠃⢸⣿⡇⠚⠊⠀⠈⠀⠸⣿⣿⡍⠀⠀⠀⠀⠀⢈⣻⡀⡋⣿⡇⢸⣿⠙⠀⣟⡁⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠒⡏⠊⢻⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⠭⠿⣧⣹⠷⠾⢏⣼⠿⠯⡝⢦⣀⠀⠂⠀⠀⡇⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⣀⡇⠀⢸⣰⠀⠀⠀⠀⠀⢸⠀⢺⠇⠈⠉⠉⠀⢸⢿⡆⣤⣙⣯⣸⣍⣤⣤⣶⠏⠀⠉⠉⠙⡀⠢⢳⣄⣠⣷⡀⣠⠏⠀⡠
⠀⠀⠀⠀⠁⠱⣴⠇⠈⠀⠀⠀⠀⠀⠈⣧⡾⠀⠀⠀⠀⠀⢨⢸⣧⣝⣦⠀⠀⣼⣞⣷⠇⡀⠀⠀⠀⠀⠈⢢⣀⣿⣿⣿⣿⣏⣀⠜⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⡎⠹⢾⢚⢶⣧⢛⡫⣉⢀⠇⠀⠀⠀⠀⠀⡀⣭⣽⣿⣿⣿⣯⡅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⢻⡇⠀⠀⠀⠀⠀⣿⢉⣱⢸⢩⡇⣟⡆⡉⣼⠀⠀⠀⠀⠀⠉⠀⣼⣿⣿⣿⣿⣿⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠈⡀⠀⠀⠀⠀⠀⠈⠧⣴⣀⣨⣇⢉⣾⠼⠃⠀⠀⠀⠀⠀⠀⠀⠏⣿⣿⣿⣿⡿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⣄⡁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉

⠀⣤⠶⠿⢿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⡤⠶⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠦⡀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⠿⠷⢦⣄
⡼⣁⣀⣴⣾⣟⠁⣨⡿⠛⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣀⡴⢾⣿⡟⠻⣯⡀⢙⣿⣶⣄⣀⡽
⣷⣿⢋⣡⠾⠛⣶⣿⣄⢀⣼⠏⠻⣾⣝⢶⣄⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⣴⢞⣭⡾⠋⠿⣄⢀⣼⣷⡞⠻⢦⣉⢻⣷
⠸⡼⢿⣁⣠⠾⠋⢉⣿⣿⣧⣀⣸⠁⠉⢻⣮⡻⣦⡀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡾⠀⠀⠀⠀⣠⡶⣫⣿⠋⠁⢹⣀⣠⣿⣿⣏⠉⠻⣦⣀⣹⠿⡼
⠀⠙⢾⡻⣅⣠⡴⠟⠋⢁⣼⡿⠿⣤⣤⣾⠏⠙⢮⣻⣦⠀⠀⠀⠹⡄⠀⠀⡼⠁⠀⠀⢠⡾⣫⠞⠉⢿⣦⣤⡼⠿⣿⣄⠉⠛⠷⣤⣀⡽⣻⠞⠁
⠀⠀⠀⠙⢿⢻⣀⣴⠞⠋⠁⢀⣴⡿⢿⣷⡄⠀⠀⠙⣿⣷⡄⠀⠀⢳⣤⣴⠃⠀⠀⣴⣿⡟⠁⠀⠀⣴⣿⠿⣷⣄⠀⠉⠛⢶⣄⣸⢻⠟⠁
⠀⠀⠀⠀⠀⢻⢻⣀⣠⠴⠚⠉⠀⠀⠀⣼⣿⣄⠀⠀⠈⢻⣿⡄⠀⠹⣿⡿⠁⠀⣼⣿⠋⠀⠀⢀⣼⣿⡄⠀⠀⠈⠙⠲⢤⣀⣸⢻⠃
⠀⠀⠀⠀⠀⠀⣟⢫⠁⠀⠀⢀⣠⡴⠟⠋⠉⠛⣻⣶⣶⣾⣿⣿⠀⣾⠟⢿⡆⢸⣿⣿⣶⣶⣾⡛⠋⠉⠛⠷⣤⣀⠀⠀⠀⢩⢻⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣴⠶⠟⠋⠁⠀⣀⡤⣖⣿⡿⠿⠟⠛⣻⡇⢸⡟⠀⠘⣿⠀⣿⡟⠛⠿⠿⣿⣗⡦⣄⡀⠀⠉⠛⠷⢶⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠶⠖⣋⣵⡾⠟⠁⠀⠀⠀⢠⣿⡇⢸⡇⠀⠀⣿⠀⣿⣧⠀⠀⠀⠀⠙⠿⣶⣍⡓⠶⠦⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠀⠀⠀⠀⠀⠀⢸⣿⡇⣸⣇⠀⠀⣿⡀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣄⣼⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄
⠀⠀⠀⠀⠀⢠⣾⣿⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢸⡏⠉⠁⣿⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠈⣿⣤⣿⣧⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⡀⣸⡿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣧⣼⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⢻⣿⣿⠃⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣾⡛⣿⡇⠀⢀⣾⣿⣶⡶⣦⡀⠀⠀⠀⠀⠀⣿⡇⠘⣿⡟⠀⣿⡇⠀⠀⠀⠀⠀⣠⡶⣶⣾⣿⣆⡀⠀⣿⡟⣻⣦
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠁⠀⢸⣿⣀⣤⣤⣄⢠⣿⠁⠀⠈⠀⠀⢹⣧⢀⣤⣤⣄⣸⣿⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⣿⣿⣿⡁⠀⢀⣾⣿⣿⠋⠉⣿⣿⠏⠀⠀⠀⠀⠀⠈⢿⣿⡏⠉⢻⣿⣿⣄⠀⠀⢹⣿⣿⡟⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠾⢿⣶⡿⠿⢿⣷⣶⣿⣿⡿⠿⠗⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠿⠿⣿⣿⣷⣶⣿⠿⠿⣷⣾⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡇⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣧⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⡿⠿⣿⣶⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠃⠀⠒⠒⢲⣤⣶⣿⣿⣧⣄⣀⠀⠀⠈⠀⠈⠉⠙⢛⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠞⠉⠀⣀⣴⣾⣿⡶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣶⣿⣿⡟⠻⠿⠋⠛⠻⠿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢲⠋⣿⣿⡿⠃⣠⣴⡶⢖⣴⣾⣿⣿⣿⣿⠀⣾⠃⠀⣀⡀⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣟⠛⠢⠀⣄⡀⠉⠙⠿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡿⢁⡞⠉⠀⠀⢈⣿⣿⣿⣿⡿⠁⣰⣿⣷⣿⣿⣿⣿⣷⣦⣄⠀⠈⠙⢿⣿⣿⣿⣿⠀⠀⠀⣴⣭⣄⣤⡀⠈⢧⠉⠻⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠀⣼⣿⡇⢨⡅⠀⠀⠀⠈⠛⢿⣿⣏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠙⢿⣿⣿⣄⣀⣤⣿⣿⣿⣿⣿⣦⣈⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⢾⣴⣿⣿⡇⢸⣿⣿⠀⣶⣶⣶⣿⣿⠏⢠⣿⣿⠋⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⣭⣽⣿⣿⣿⠇⠈⣻⣿⣿⣯⢧⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠷⠈⠻⣯⣁⣻⣿⣿⣿⣿⡄⢸⣿⣿⣷⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢠⣾⣿⣿⣿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣧⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡳⠀⠻⣿⣿⣿⣿⣸⡿⠉⠻⢿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠉⠉⠙⠻⣿⣿⣿⣿⡃⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠒⠤⣈⠙⠛⢻⣿⣿⣦⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣾⣿⣿⣿⣇⠘⣿⣿⣿⣽⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠋⠀⠀⠀⠀⠉⠒⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠙⣿⣿⣏⣤⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣶⣿⡿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠙⠛⠻⠛⠿⠟⠻⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣀⣀⣠⡤⠶⠚⠉⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠿⠓⠒⠶⠿⠿⠿⠿⠿⠛⠛⠉⠁
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣦⣤⣤⣤⣴⣯⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠉⠉⠛⠛⠳⠿⠖⠺⠟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡔⠊⠀⠀⠀⠀⠀⠀⠇⢀⡀⠀⠀⠀⠀⠀⠀⢂⠀⢀⠀⠂
⠀⠀⠀⠀⢐⢀⠂⢐⠀⠀⠀⠀⠀⢉⡜⠐⣗⠀⠂⠀⠀⠀⠀⠐⡗⠀⠀⠀⠀⠀⠀⠀⠖⠁⢠⡄
⠀⠀⠀⠀⢀⡙⠀⠠⠂⠀⠀⠀⠀⠊⡇⢰⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡁
⠀⠀⠀⠀⠃⠀⠀⢀⠓⠀⠀⠀⠈⠀⢇⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠘⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⣀⠀⡔
⠀⠀⠀⠀⠌⠀⠐⢁⠁⠀⠀⠀⠀⠀⠀⢀⠨⠦⠐⠶⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⣬⡂⣡⣤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣀⣀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⡠⢊⣠⠞⢈⢡⠂
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠐⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠳⠿⠥⠒⢡⠻⡄
⠀⠀⠀⠠⠖⣋⣽⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠿⠛⠋⠙⠳⠴⠤⠄⠀⠀⠢⡀⠀⠄⣢⡮⠈⠤⠀⠤⠂⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠘⠋⠋⠀⠘⠀⠀⠀⠀⠀⠀⠄⠔⡒⣂⣀⠂⣠⠤⠤⠈⠁⠀⠀⠀⢀⢈⡔⡀⢲⠟⠁⢈⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣂⣀⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠁⠁⠀⠡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⡀⠀⢀⠀⠀⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⡔⢻⠉⡟⢹⣿⢿⣿⠏⠉⠉⠀⠈⡏⢸⠛⡏⢹⣿⣿⠁⠉⣡⠼⠛⢉⡽⠟⢹⡟⠉⠉⢿⠉⢹⡅⠉⡏⡿⢘⣿⠉⠁⢹⠙⣏⠈⠉⠁
⠀⠀⠀⠸⡇⠘⡄⡟⢸⡇⡎⢻⠀⠀⠀⠀⢰⠃⢸⠀⡇⢸⣿⢹⣳⣾⣇⣠⣔⢋⣠⡴⠛⡆⠀⠀⠘⠀⢈⡇⠀⡇⢣⡿⡹⠀⠀⢸⡇⢻
⠀⠀⠀⢸⣇⠀⢣⡏⠸⡇⡇⢸⠀⠀⠀⠀⡼⠀⡇⠀⡇⠈⣿⢸⡿⠃⠀⠉⠉⡏⠀⣿⠈⢇⠀⠀⢸⠀⠀⡇⠀⡇⢹⠁⠇⠀⠀⡎⣧⠸⡇
⠀⠀⠀⠘⣿⠀⠘⣿⠀⣧⠇⢸⠀⠀⠀⢠⡇⢰⠁⠀⣿⠀⡿⢰⠃⠀⠀⠀⢰⠁⢰⢿⡆⠸⡀⠀⢺⠀⠀⡇⠀⡇⠀⣾⠀⠀⢰⢠⣿⠀⣷
⠀⠀⠀⢰⡇⢇⠀⢹⡄⣿⢸⢸⡀⠀⠀⣸⠀⣸⠀⠀⣿⠀⠁⡞⠀⠀⠀⠀⢸⠀⣸⠀⣷⠀⢳⠀⣾⠀⠀⣇⠀⡁⠀⡇⠀⠀⢸⢸⣿⠆⢻
⠀⠀⠀⠀⡇⠸⡀⠀⢷⢸⣸⠈⡇⠀⠀⡟⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⡏⠀⡟⠀⠸⡆⠈⡆⣿⠀⠀⣿⠀⡇⢸⡇⠀⠀⢸⢸⠸⣆⣸⣤⠤⠄
⠀⠀⠀⠀⡇⠀⢧⢀⡾⠛⠛⠉⠙⢷⣸⡇⠀⡇⠀⠀⢸⠀⠀⡇⠀⢠⣤⡴⢧⣤⣧⠀⠀⢳⠀⠸⣿⠀⠀⡏⠀⡇⢸⡇⠀⠀⢸⢸⣴⠋
⠀⠀⠀⢀⡇⠀⢸⡞⠁⠀⠀⠀⠀⠀⠹⣇⠀⡇⠀⠀⢸⠀⠀⣇⣠⠟⠀⠀⠀⠀⠘⠳⣄⠀⡇⠀⢻⡀⠀⣷⠀⡇⠈⡇⠀⠀⢸⡾⠁
⠀⠀⠀⠸⡇⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢸⡅⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⡀⠘⣇⠀⢹⠀⡇⢠⣶⣤⠀⢸⠁
⠀⠀⠀⢰⣧⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣧⣤⣄⣸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⠀⢻⡀⣸⠀⡇⢈⡇⣧⠀⢸
⠀⠀⠀⠘⡏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠹⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠛⠓⠛⠉⢻⡀⢻⣸⠇⢻⡀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠻⣼⡇⠀⠀⢳⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⣰⡇⠀⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡆⠀⢰⢃⣵⣀⡷⠀⡿⠉⠉⠉⠉⠉⠉⠒⠒⠒⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⣤⡤⠤⠤⠤⠤⣼⣧⡀⠘⠛⠁⠀⠀⡼⠻⡇⠀⠐⠂⠒⠒⠒⠒⠒⠀⠒⢇⠀⠀⠀⠀⠀⠀⠀⢼⣼⠀⠑⣄⠒⠆
⠀⠀⠀⠀⡇⠀⠀⢀⣿⠥⠶⠖⠲⠖⠒⠒⠒⠺⣧⠀⠀⠀⢀⠜⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠀⠀⠀⣞⢿⢿⣶
⠀⠀⠀⠀⡇⠀⠀⣨⠟⡉⠀⠀⠀⠀⠀⠀⠀⠈⢯⡀⠀⢀⠎⠀⠀⠀⠀⠘⡄⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠵⠦⣴⠤⢤⠘⠁⠀⠀⠈⢣
⠀⠀⠀⠘⡇⡰⠋⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠘⣆⠀⢸⣥⣤⣠⣤⣤⡤⠤⡤⠤⠄⠘⡖⠲⡄⠀⠹⠶⠂⢀
⠀⠀⠀⢰⡟⠁⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠘⢳⡀⠀⠀⠀⠀⠀⠈⢦⣼⠧⠤⠤⠤⠤⠴⠶⠔⠀⠀⠛⠛⠒⠳⢦⣤⡤⠴⠉
⠀⠀⠀⠈⡇⢠⠁⠀⠀⣀⡀⠀⡠⢈⠄⣀⣠⣄⠠⡄⣀⢀⠀⣇⠀⠀⠀⢄⡠⡄⢮⠉⠛⠶⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣦⣶⣿⣎⡁⣙
⠀⠀⠀⠠⡇⢸⠀⠀⠀⠀⠀⣤⣀⠀⣀⣀⣠⡀⣀⠀⠤⠀⠀⢸⠀⠀⠀⢚⣷⣿⣶⠀⠀⠀⠀⣻⡹⠯⠋⠛⡅⠁⠀⠀⠈⣯⡄⠣⡜⠣⡀
⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⠉⠁⣄⣀⣄⢀⡤⠀⠀⠀⠀⣀⢼⠀⠀⢀⡏⢻⢄⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⡗⠓⡆⢀⣃⢉⡖⠠
⠀⠀⠀⠀⡇⠀⠁⠐⠒⠒⠠⠤⠤⠤⠤⠤⠤⠔⠒⠒⠋⠉⠀⢸⣄⠀⢀⠋⠁⡀⠉⢢⡀⠀⠀⠃⠄⣉⢀⡀⣹⠀⠀⠀⠀⣟⣀⣹⣀⠽⠤⢴
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⡀⠀⢀⡋⣱⢟⣠⠀⡁⠀⢰⢀⣀⡕⠀⠀⠀⢀⣛⣛⠟⣨⡿⢢⣦⣠⡏⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⡹⠂⠉⠀⠈⡮⠁⡔⠀⠀⠸⡀⢉⡴⢤⡖⣿⠏⠋⠙⢻⣿⠿⠿⠛⠛⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⡜⠀⠀⡀⠀⠀⢸⡄⠳⡀⢀⡠⢟⠙⠮⣠⠟⢓⣢⣤⡿⣤⣤⡀⠀⠀⠀⣷⠤⠤⠔⠒⠒⠋
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠂⠀⠙⣄⠀⢈⣀⣠⣎⠀⠀⠀⠉⠇⠀⠑⢢⡤⠟⢹⣿⣿⣷⠾⠋⠁⠀⠀⠀⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠄⣠⢼⠋⣹⣽⣿⣿⣿⣇⠀⠀⢀⡛⠐⠒⢚⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡁⠘⢇⠻⢿⣾⣿⣾⠿⠤⣀⠀⡹⢒⠒⡏⠀⠀⢀⣀⠠⠤⠐⠒⠈⠉
⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡁⠁⢘⣷⣤⡟⠛⣿⠂⠀⠀⠈⡇⢸⠀⢷⡂⠉
⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡈⠙⢋⠁⠀⡿⠢⢄⡀⠀⠇⠈⡱⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠂
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠃⢠⠛⠀⠀⣀⡰⠃⠀⠀⠈⠉⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⣚⣩⠥⠖⠒⠉⠁
⠀⠀⠀⠀⡏⠲⢤⣀⠀⠀⠀⠀⠀⠀⡇⢸⠓⠶⠆⠀⠶⠚⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⣀⡤⠶⢚⣩⠴⠚⠋⠉
⠀⠀⠀⠀⡇⠀⠀⠉⠙⠛⠒⠒⠒⠐⠃⠈⠙⠒⠂⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠛⣋⡥⠖⠛⠉
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⣋⠥⠖⠚⠉
⠀⠀⠀⠀⠈⠙⠓⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠛⠛⠛⠑⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠒⠒⠒⠲⠦⠤⣄⣀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠁⠀⠠⠤⠤⠤⠤⢤⣄⣀⠀⠈⠙⠒⢦⡴⠋⠉⠀⣀⣀⣀⣀⣀⣀⡀⠈⠉⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠀⢸⡇⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠙⠀⠀⠀⣯⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡟⠀⠀⠐⠒⠚⠛⠓⠒⠒⠶⠤⣄⡀⠀⠀⢸⡇⠀⢀⣀⣤⠤⠤⠤⠤⠤⢤⣄⣀⠀⠀⢸⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣧⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢰⢃⡏⠀⠀⠀⠒⠒⠛⠛⠛⠒⠲⠦⢤⣀⠀⠀⠀⢸⡇⠀⠠⠴⠒⠒⠛⠛⠛⠓⠒⠲⠤⠀⠀⠀⢸⡀⣇
⠀⠀⠀⠀⠀⠀⠀⡞⣸⠁⣀⣠⠤⠤⠴⠶⠶⠶⠦⠤⢤⣀⡈⠀⠀⠀⢸⠁⠀⠀⢀⣀⣠⠤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⣧⢸⡄
⠀⠀⠀⠀⠀⠀⢰⠃⠛⣉⣁⣒⣛⣉⣉⣉⣉⣉⣙⣓⡒⠶⠭⠓⠦⢄⣸⡤⠖⠯⠭⠷⣶⠐⠂⠀⣰⠶⠿⠿⢛⠛⠶⠦⠿⠌⣧
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠓⢶⣶⡶⠚⠋⠉⠉⠉⠈⠀⠀⠁⠀⠀⠀⠈⠉⠉⠙⠛⠒⠻⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⠾⠛⠛⢛⣿⣿⣿⠉⣿⣿⣿⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⢀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢀⡀⠈⠙⠛⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⡹⠀⠀⢀⠀⢸⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇⣼⡄⣰⣷⡀⠈⠙⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⣶⡤⠃⢀⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⢹⡟⠂
⠀⠀⠀⠀⠀⠀⠟⣹⠟⢦⠏⠀⣠⠟⠛⠋⠉⠉⠙⠛⢻⣿⠿⠟⠛⠛⠙⠛⠛⠿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⢀⣾⣃⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⢀⣴⡏⠁⠀⠀⢠⡟⠉⠉⠙⢶⡄⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⢰⡀⠀⠀⢻⡄
⠀⠀⠈⠋⢸⡇⢠⡆⣰⠏⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣿⡇⠀⠀⠸⡇
⠀⠀⠀⠀⢠⡇⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⢦⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣷⠀⠀⣷
⠀⠀⠀⠀⣾⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⢞⣓⣦⣤⠶⠖⠛⠲⠦⣄⡀⢀⣀⣿⣿⣿⣿⣿⡆⠀⢹
⠀⠀⠀⢰⡇⠀⢸⣟⠀⠀⠀⠀⠀⠀⣰⢿⡆⠀⠀⠀⢼⠁⣯⡾⠋⠀⠀⠀⠀⠀⠀⠸⢟⡩⣍⢻⣿⣿⣿⣿⣷⡞⢸⡇
⠀⠀⠀⣾⠁⠀⢸⡏⠛⠛⠶⣤⣠⣾⣿⠋⠁⠀⠀⠀⠘⢳⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣧⣿⣿⣿⣿⠏⠀⠀⣿⡗
⠀⠀⢰⡏⠀⠀⢺⠁⠀⠀⠀⠈⢿⡟⠁⠀⠀⠀⣀⣀⣴⣟⣧⢀⣴⠀⠀⠀⠀⠀⠀⠀⠰⠶⣶⠋⠛⢿⡿⠋⠀⠀⢀⡿⠁
⠀⠀⣸⠁⠀⠀⣿⣤⣠⣀⡀⠀⣿⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⣶⠋⠀⠀⠈⠉⠁⠀⠀⠀⣰⡏⠀⠀⢸⡇⠀⠀⠀⣼⠃
⠀⢀⣿⣤⡴⠾⢷⠶⠿⠿⠿⣗⣛⣷⣤⣤⠶⠶⢶⣄⡀⠰⣿⣥⣤⡄⢀⠀⠀⠀⢀⣀⡴⠏⠀⠀⠀⣹⡷⠤⠴⠚⠛⠛⢦
⠀⣸⣏⣥⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣄⠀⠀⠀⠈⠻⣤⣿⡿⣿⠥⢪⣴⠶⡟⣫⠉⡀⠀⢀⡤⠚⠁⠀⠀⢀⡀⠀⠀⢸
⠀⣸⠋⠹⣦⣠⠷⢦⡀⢸⣶⡄⢠⣄⢠⣀⣷⣤⣀⠀⣀⠀⠉⠳⣶⢺⡏⢹⣦⣿⣁⣜⡵⠛⢉⠀⢀⡴⠦⠞⠋⠲⠛⠒⣿
⠸⢿⣤⠤⠴⠶⢦⣤⣍⣛⠀⠙⠛⠉⠻⠏⠁⠀⠉⠙⠛⠳⣄⡀⣼⣿⣷⠞⠋⠀⠉⣿⢵⣦⢾⡥⢏⣠⣤⠴⠚⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠛⠛⠛⠛⠲⠶⣤⣄⣀⣤⠤⠴⠯⢥⡤⠿⠛⠛⣳⣶⠷⠖⢒⣶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠄⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣾⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠉
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣋⠉⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠋⠁⠘⣿⣿⣧⣀⠀⠀⢀⡀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠉⠙⠋
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⠻⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠗⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣴⣀⠀⣞⠋⣿⣦⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣜⣿⣭⣷⣉⠛⣛⣷⣾⣾⣤⣴⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⣸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣽⣿⡭⣿⣿⣯⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣾⠛⠉⠉⠉⢰⣷⠋⠈⠙⠽⠿⣏⡻⠿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠠⡾⠟⢿⣷⣶⣾⣿⣷⡄⢤⣶⢻⣿⣧⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠠⣺⢿⣶⢤⣬⣬⣍⣸⣿⣿⣇⣠⣬⣶⣿⣿⡞⢯⣂⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠈⠁⣬⡥⣄⣀⣉⣠⣼⣿⣿⣏⣉⣉⣡⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢽⣥⣤⣬⣴⣾⣿⣿⣿⣿⣭⣥⣥⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⢯⣤⡤⡿⣿⣿⣿⣿⣿⣧⣶⣽⣿⣿⠃⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⢿⣤⠤⠼⣿⣿⣿⠿⠶⢷⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⣛⣿⣟⣝⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡄⣿⣿⣿⡏⣰⣿⣿⠟⠛⠛⠛⠛⠛⢛⣛⣛⣛⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠹⣿⣿⣿⡇⣿⣿⡿⢰⣿⡿⢃⣾⣿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⡄⢹⣿⣿⡇⣿⣿⠇⣾⡟⣡⣿⠟⣡⣶⣶⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡙⢿⣿⣿⡄⢿⣿⡇⣿⡿⢸⡟⣰⡿⢃⣴⣿⠿⠋⣉⣥⣤⣤⣤⣶⣶⣶⣶⣿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣉⠛⠻⣿⣦⡙⢿⣿⡘⣿⡇⣿⡇⡿⣰⡟⣰⡿⠟⣡⣴⣿⡿⠿⢛⣛⣉⣭⣭⣤⣶⣶⣶⣾⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣛⠻⠿⢶⣤⡉⠻⣦⡙⣷⢹⡇⣿⢸⢳⢏⣾⢋⣴⡾⠟⣋⣥⣶⣿⡿⠿⠿⠟⢛⣋⣉⣭⣤⣴⣶⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⢷⣶⣬⣝⡳⢌⡳⡜⡯⡇⣟⢆⣯⡞⣵⠟⣩⡴⢟⣫⣭⣶⣶⣶⠿⠿⠿⠿⠿⠟⢛⣉⣩⣤⣴⣶⠾⠇
⠀⠀⠀⠀⠀⠀⢐⣛⣛⣛⣛⣛⣓⠲⠶⠦⣭⣙⠲⣝⠎⠆⠡⠀⡸⣯⢞⡵⣫⠵⣞⣋⣭⣥⠴⠶⠶⠶⠶⣶⣶⣶⠿⠿⠛⣛⣩⣥⣴⡶⠾⠁
⠀⠀⠀⠀⠀⢀⣭⣭⣭⣭⣭⣭⣭⣍⣛⣓⡲⠮⣝⡦⡱⠄⠀⠀⠐⠱⢫⡪⢗⣫⠭⠶⠖⠒⠒⠛⠛⠛⠲⠶⠶⣶⣶⡾⠿⢛⣋⣩⣤⣶⠾⠿
⠀⠀⠀⠀⠀⣤⢶⡶⠶⠶⠶⠶⠶⠶⠮⢭⣭⣝⣒⠍⠊⠀⠀⠀⠀⠀⠠⠊⠥⠶⠉⠩⠭⠭⠭⢭⣉⣛⡛⠳⠶⢶⣶⣶⡿⠿⢛⣋⣭⣴⠶⠟⣃⡄
⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠷⠶⠾⠧⠀⢴⣶⣶⡦⠀⢴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣬⣍⣛⠲⠶⣤⣤⣶⠾⢟⣋⣭⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠻⠿⠿⣷⣶⣥⣶⡾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⢙⡄⢸⡿⡿⣿⡄⠀⠀⠀⠀⠀⠈⠙⠁⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡾⠑⠾⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⡶⣤⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠈⠙⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⣠⣾⣿⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠞⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢰⢿⡟⠉⠉⠀⡀⣦⡀⣀⣠⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⣿⣾⠁⠀⢠⣤⣧⣯⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⢀⣼⣿⣿⣿⣀⡀⠘⢿⠟⠻⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣦⣂⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡦⠀⠀⣰⣿⣿⣿⡿⠋⠐⠋⠉⠻⠛⢻⣿⠁⠀⠠⠀⠀⡇⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣦⣌⠙⠋⠁⠀⠀⢀⣤⣄⣴⣾⣿⣿⣿⣿⡧⢸⣧⣿⣿⣿⣿⡶⠒⢶⣤⣤⢀⡠⠀⠀⠀⢀⣠⣤⡌⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⠛⣇⣀⢠⣴⣶⣦⣾⣿⢿⣿⣿⣿⣿⣿⣿⣃⣾⣿⣿⣿⣿⣿⣧⠀⢠⣷⣭⠿⠇⣀⠀⠀⠘⠚⠃⠁⢠⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡛⣿⡉⠋⣻⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠄⠫⠆⠲⠁⢀⠀⠂⠀⠀⡜⠛⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⣀⣠⡰⣈⠀⣀⡤⠈⠀⣀⢐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠋⠙⠀⠀⠀⠀⢸⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡴⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢆
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢶⣢⣴⠦⣶⢦⠄⡀⠀⠀⠀⠀⠀⠀⣤⠐⠈⠉⡩⠛⣳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢠⠊⢸⢣⠀⢈⣄⠑⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠁⣶⡿⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⠟⠲⣿⡟⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠙⣧⠘⠋
⠀⠀⠀⠀⠗⠢⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⢀⣠⠤⠚⠛⠰⢞⠻⣨⠃⠀⠀⠀⠀⢠⠔⠉⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⣋⠰⣼⡫⠜⢻⡄⠀⣀⠀⣠⣌⠓⠘⡻⠀⢀⠜⠁⡄⠀⠀⠀⠀⠀⢫⠀⠀⣀⣀⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⣯⠐⠈⡷⠐⠋⠀⠀⡩⠑⠄⢄⠀⠀⠇⠀⢨⠀⠀⠘⠢⢄⡀⠀⠀⠈⢣⠞⠁⠀⢹⡇⠀⠘⠳⣄⣀⡀⠀⠀⢀⡅
⠀⠀⠀⠀⢈⠒⠉⠀⠀⠀⣠⠞⠀⠀⠐⡄⠂⠀⡆⣀⠸⡄⠀⠀⠀⠀⠈⠀⠈⠈⠇⠀⠀⠀⣨⣷⣄⠀⠀⠀⠙⠛⠃⠺⡃
⠀⠀⠀⠀⡷⠄⠀⠀⠀⠈⠀⠀⠀⠀⢠⠇⢠⠐⠉⠉⠀⣸⣦⡀⠀⠀⠀⠀⣀⠴⢇⡠⠒⠉⠀⠀⠹⣑⠢⣠⣄⠠⠔⠊
⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣴⢃⡀⠀⣠⡞⠁⠈⢫⠹⣶⣿⢻⢇⠀⡎⠀⣤⣖⠋⠀⠀⠈⠁⡩⢗
⠀⠀⠀⠀⠀⠈⡦⢄⣀⣀⣀⣤⣶⡿⣳⣦⡽⠷⡀⣙⠉⢱⣾⣾⣤⣬⣀⣸⡍⠓⡴⢄⡐⠋⠓⢤⡀⣀⣰⡇⠸⡇⠀⠀⣀⡠⠴⢤
⠀⠀⠀⠀⠀⣠⠦⠐⠂⢠⣿⢄⢻⣎⡠⢉⠖⡲⠆⠉⣦⣀⠀⣀⣀⣀⣀⡀⠈⣻⠃⢴⢻⠀⣀⢀⡱⣄⠱⡰⠄⢠⡷⢾⣏⠔⢀⠞
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠉⢹⡷⣇⠈⠑⠣⠎⠠⠚⢓⡿⢹⡀⡸⠀⠓⠰⠧⡞⠻⡀⠈⠉⠑⠙⠛⣒⠿⣄⡈⠚⢻⢦⣌⣡⣴⠃
⠀⠀⠀⠀⠿⡀⠀⠀⠀⢀⠼⣃⡩⠗⠒⠤⢤⠐⣢⠾⢳⡀⠻⢤⣀⠀⠀⠲⣃⣰⣷⡀⠀⠀⠀⠻⣴⣶⢾⣿⠉⣙⢰⠯⠛⠁
⠀⠀⠀⠀⠀⠈⠑⠒⠊⠁⠀⠘⣄⠀⠀⠀⠈⠉⢁⡔⠊⠻⡖⣤⣭⣗⣶⡀⢹⠞⠁⣿⣦⣀⠀⠀⢹⡷⠮⠽⠧⡏⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⡤⢐⣪⡿⣷⣄⠀⠛⣯⠿⢿⠿⠇⠐⣯⠉⠉⠁⣹⠄⠀⠀⡇⠀⠀⠰⠂⠆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡀⠀⠀⠷⠩⣭⡧⠞⠻⡆⠀⡇⠀⢳⣤⠒⠸⡏⠀⠀⡔⠁⠀⠀⣸⠁⠀⠀⢀⢀⣠⡾⢧⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠉⠉⠲⢍⢢⡤⠊⠁⠐⢆⡼⠀⡰⠀⠀⣀⣽⡀⠀⠈⢶⡾⢿⠏⠁⠽⠛⠛⣻⠟⣟⠡⣼⡤⠾⠷⢔⡂⢄⣀
⠀⠀⠀⠀⢀⠊⠀⢀⡴⠒⠒⢤⡀⠈⡿⠋⢻⡄⢴⡆⠀⠀⢧⡴⢾⣯⣽⣷⣤⣀⣸⠀⣸⣀⣠⣤⣶⢾⣿⡄⢻⡔⠁⢀⣀⣀⠀⠈⠢⡙
⠀⠀⠀⠀⡾⠀⢠⡿⠀⠀⠀⠀⢹⠋⢀⠴⣿⢱⣿⣷⡀⠀⠀⡆⡎⡄⡜⠟⣼⠀⣿⠀⢃⠻⡷⢿⣿⠟⣟⣧⠈⣻⣰⠟⠉⠀⠹⡄⠀⢻
⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⡰⢃⡴⡏⠀⢙⡘⠉⢏⣿⠆⣿⠀⢣⠠⠁⡈⠌⣰⠏⠀⢆⠢⠈⠡⢊⣼⡟⠈⡆⢹⡇⠀⠀⠀⠀⢱⠀⠈
⠀⠀⠀⢀⢷⠀⠘⢳⠀⠀⡼⢁⡾⠀⢻⠀⠈⠷⢍⣐⣡⠾⠃⠀⠀⠑⠒⠲⠚⠉⠀⠀⠀⠉⠀⠐⠻⢡⠁⠀⢿⡀⢆⠀⠀⠀⠀⢸
⠀⠀⠀⢸⡈⡀⠀⢾⡄⢸⡷⠊⠋⠀⠈⠃⠀⢹⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⡮⠳⣼⡆⢠⡄⠀⡸⠀⢰
⠀⠀⠀⢸⡇⢡⠀⠈⣧⡀⠁⠀⠀⠀⢠⠀⠀⢸⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠘⠃⠀⠀⠀⡇⠀⣸
⠀⠀⠀⢸⠛⠀⢢⠀⠈⢷⡄⠀⠀⠀⡼⠀⠀⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⣀⠀⣀⢰⡀⠀⣿⠀⠀⠀⠀⠀⡼⠀⢀⠟
⠀⠀⠀⢸⠀⠈⣠⢳⡄⠀⠙⠶⠤⠔⠁⠀⣰⣣⠃⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⣠⡴⢀⡠⢤⡁⠀⠀⣿⣄⠀⠻⣆⠀⠀⢀⡜⠁⢠⠎⣰
⠀⠀⠀⢸⣤⠚⢡⡘⢫⡢⣄⡀⠀⠀⢀⣴⠗⠁⠀⠀⠁⠈⠁⠉⠉⠈⠉⠉⠙⣆⣀⠁⣀⢴⣋⡀⠚⠁⢹⡄⠀⠙⠓⠚⠉⢀⡔⣣⠜
⠀⠀⠀⢸⣅⠤⣀⣰⠦⠃⠛⠉⣭⢿⣃⣀⠀⠒⠋⠀⠀⢠⡄⠀⠀⠉⠐⠃⠀⠉⠙⠉⠉⠀⠉⠑⠛⠐⠋⠈⠓⠶⠶⠶⣦⣵⢾⣥⡤⠰
⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑
⠀⠀⠀⢸
⠀⠀⠀⢸
⠀⠀⠀⠀⠆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠋⠉⠉⠉⠉⠛⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠂⠤⠄⣀⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⠴⠂⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⣼⠀⣴⠋⠉⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠓⢄⠀⠀⠀⠀⠀⠀⠀⣾⢡⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⣿⠀⢿⣀⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡼⠇⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⡼⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢠⡇⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢸⠁⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠙⠦⠀⠄⠀⠀⠀⠒⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⡾
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⡀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉
⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣰⣿⣀⣠⣾⠟⠉⠉⠉⠉⠉⠛⠂⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⡴⠛
⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠟⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣀⣠⡤⠖⠛⠁
⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⣀⡠⠔⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣄⣀⣀⣀⣠⡤⠴⠖⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡠⣤⡤⢤⡤⢐⠖⠲⠶⠤⡠⣤⣤⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣶⣤⣤⣴⣶⢟⣥⣴⣶⣿⣧⣍⡛⢿⣿⣿⣿⣿⡟⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⠿⣳⣿⣿⣿⢳⣿⣿⣿⣷⣼⣷⠊⠛⣿⣷⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣮⡻⠿⡿⣿⣿⣿⡀⢁⠉⠛⠟⣿
⠀⠀⢀⣠⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣤⣼⣿⣻⣤⣾⣷⣄⣀⣾⠆
⢀⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⢿⣿⣿⣿⣯⣭⣝⣛⣋⡉⠉⠀⠀⠉⡛⠋
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣥⣷⣶⠛⠂⠀⠀⠈⢑⣤⡉⠙⠿⣿⢇⠰⠆⠀⠀⣾⣿⢢⣄
⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣾⣿⣟⣛⣠⣤⠀⠀⠀⠀⠀⠀⠉⢣⣾⣠⠀⠀⠀⠀⠀⣼⣿⣟⠟⠛⡇
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣶⣷⠶⠄⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⣹⣿⣿⡇⢤⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⣿⡑⠀⠀⠀⠉⠀⠀⠀⣄⣀⡀⠀⠀⠀⠀⢠⣿⣿⣿⠇⡿
⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠛⣆⠀⠀⠒⠒⠚⠛⠉⠁⠀⢤⣤⣤⣤⣼⣿⣿⢇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣯⡙⠛⠿⣿⣿⣿⣎⣳⢦⣴⣿⣄⣤⠄⠀⠲⠶⠮⠍⢭⡭⢛⣁⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠙⠻⣿⣿⣯⣮⣯⣶⠦⣤⣀⠀⠀⢠⡾⠕⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠷⠄⠀⠀⠀⠀⠀⠈
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⢶⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠛⠛⠀⡀⠀⣦⣶⣿⣿⣶⣤⣀⠀⠀⠀⣉⣉⣙⣿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣠⣤⣤⡄⠀⣀⣊⠀⣠⣶⣿⣿⠛⣿⠟⠛⠻⠿⢿⣿⣿⣾⣿⣿⣯⣉⣫⣅⠉⠙⠛⠿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠋⣼⣿⠋⡴⠛⠋⠙⣿⣿⣿⣿⠟⢰⣿⣤⣶⣶⣤⣄⡀⠉⠛⢿⣿⣿⣿⣧⠀⠀⢘⣢⠀⠈⠙⢟⠳⣦⣄⡀
⠀⠀⠀⠀⠀⢀⢀⣿⡏⢸⠁⠀⠀⠐⠿⣿⣿⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠙⢿⣿⣿⡀⣀⣼⣿⣿⣿⣦⡘⠀⠀⠈⠉⠓⢤⡀
⠀⠀⠀⠀⠀⢮⣼⣿⣇⢸⣿⡇⣰⣶⣶⣿⡿⠀⣿⣿⠋⠉⠉⠛⠿⣿⣿⣿⣿⣷⣤⣀⣈⣭⣿⣿⣿⠆⢙⣿⣿⣿⣦⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠻⢧⣼⣿⣿⣿⣧⠈⣿⣿⣷⣦⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣠⣿⣿⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠙⢿⣿⣿⣿⠄⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣤⣀⠙⠿⣿⣾⣿⣀⠉⠛⠛⠛⠛⠉⠉⠀⢀⣀⣤⣤⣹⣿⣿⣷⠨⣿⣿⣟⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⠀⠀⠉⠒⠤⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⡟⠉⠻⣿⣿⣀⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣀⣾⣿⠿⠏⠀⠀⠀⠀⠀⠀⠘⣿⡏⠛⠛⠛⠿⠟⠻⢿⣿⣿⣿⣿⡟⠁⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁
⢤⡤⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠛⠛⠉⠚⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠈⠻⠿⣶⣦⣤⣀⡀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠳⣤⣀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⠟⠶⠶⢦⣤⡤⣤⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣄⡀

⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢂⢀⠖⠢⣱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠢⡄⠰⣡⢀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢀⡀⠀⢀
⠀⠘⠚⠀⠀⠉⠰⡂⠀⠀⠀⠀⠀⠀⠀⣶⣛⣶⡁⠀⠀⠀⠀⠀⠀⠀⠀⣒⣪⡙⢙⡗⠒⠊⠁⠀⠀⣰⠂⠉⣽⠿⢧⠠⠔
⠀⠠⢚⡄⠀⠀⠀⠉⢳⠀⠀⠀⠀⠄⠘⣩⢿⢏⠃⠀⠀⠈⠀⠀⠀⠔⠉⠀⣴⡟⡙⡞⠦⠄⠀⠀⣀⠀⠛⠀⣁⠀⣈⢁
⠀⠀⠸⣅⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠠⠋⣿⢈⠂⠀⠀⠀⠀⠀⠔⠃⠀⠲⠟⠔⠙⠀⠀⠀⠀⡠⠋⠘⠀⠀⠉⠀⣡
⠀⠀⠀⠈⠀⠈⠑⠘⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⣀⣰⠀⠐⠀⡆⠀⠤⠜⠀⠀⠀⠀⠄⠀⠀⠤⠈⠇⠐⠅
⠀⠀⠀⠀⠒⠲⠈⡧⠆⣀⠀⠤⠀⠀⠀⠀⣶⠀⡀⠀⠀⠑⢃⣱⡤⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⡠⠐⠻⠀⠜⢳⠄⡒⠒⠓⠄⠀⠀⠀⣇⠀⠀⠂⠂⠀⣉⠙⡌⠢⡑⠄⠀⠀⠀⠀⠀⠀⠀⢈⠀⡆⠲⠀⣂⠐⠁⠈⡐
⠀⢠⠁⠀⠀⠀⣉⣡⠎⡟⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠈⡆⠀⠀⢁⣀⣠⣀⡀⠈⠁⠈⠔⠄⠍⢆⠀⠕⠗⠂⠀⠉
⠀⠀⠀⠀⠀⠉⢀⡜⠀⠁⠀⠀⠈⣃⠀⣄⣠⢤⠀⡴⠀⠀⣤⢦⠀⡠⠂⠀⡏⠀⡤⡄⠐⠠⠀⠠⣈⣶⠶⢞⣻⡢⣀
⠀⠀⠀⠀⠀⢈⠁⠀⠐⠂⠀⠀⢰⡇⠀⣿⠁⣼⢀⠉⢢⣺⡗⠊⣰⡇⠀⣸⡇⠀⠑⢤⠀⠀⠈⢠⢋⠛⢛⣋⣭⠘⡌
⠀⠀⠀⠀⠀⠈⠀⠈⡀⠀⠀⡘⠈⠓⠁⠛⢠⠙⠊⠓⠛⠉⠙⠉⣄⢙⡋⠉⠋⠁⠒⠋⠀⠀⠀⠀⢸⠀⠉⠵⠇⠀⠃
⠀⠀⠠⡀⠀⢀⣀⠄⣑⠂⣀⠁⠄⠀⠀⡔⢸⢠⡆⢢⡔⢰⣤⠂⣿⠸⠀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⡄
⠀⢄⡀⠠⠭⡜⢠⠁⢘⡇⣀⣀⠀⠀⠄⠈⠈⠁⠁⠁⠈⠈⠀⠁⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠣⣼⣻⠞
⠀⠀⠀⢀⡀⠀⠀⠀⠈⠣⡀⠈⠀⠀⠀⠖⠒⢹⣋⠀⠀⣢⡀⠀⡀⣰⠀⠀⡽⠀⠀⠀⠀⠀⠀⠒⠀⢀⠀⡿⢿⡠⠀⠐
⠀⠔⠊⠁⠀⢀⠀⠐⡆⠀⠈⠀⠀⠀⠀⠀⣠⠉⠰⡲⠶⣶⠃⠀⠀⠉⠹⠋⠛⠲⠴⡒⠒⠤⠄⢀⠀⢀⠔⠭⠽⡛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠐⢀⠀⠚⠤⠖⠀⠈⠆⠀⠁⠀⠀⠈⠀⠀⡀⠀⠀⠀⠑⠀⠀⠀⡀⠊⠀⢯⠭⠇⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠀⠀⠀⢀⡴⡆⠀⠀⠀⠀⠀⠀⠀⠙⢆⣠⣾⣷⣤⠋⠀⠀⠀⠀⠀⢷⢿⡑⠤⣤⠋⢄⡀
⠀⢀⡀⠀⠀⠀⠀⠀⠄⠡⠄⢶⠂⢸⠐⠁⠀⠀⢀⣤⠆⠀⢀⠀⠀⢭⠀⠀⢅⣀⠠⠀⠂⠨⠤⠼⣆⠘⢷⣌⣤⣌⣄⠀⠒
⠀⠨⠁⠀⠀⠀⡀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠘⣢⣌⡗⠂⠀⠀⠀⠸⠺⠂⠀⢾⠁⠀⠀⠀⠀⠀⠀⠟⠀⠚⠀⠀⠑⠈⠁
⠀⠀⠈⠒⠤⣄⣀⢀⡈⠑⣆⠀⠀⠀⠀⢀⠈⢿⡿⡅⠐⠀⠀⠀⠂⢪⠀⠀⠈⠢⠀⢀⠀⠀⠀⠀⠒⠰⠆⡼⡤
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠑⢄⡡⠂⠀⠆⠀⠠⠾⠋⠀⠀⠀⠀⢀⡀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠐⠐⠉⢀⡀⢪⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⣄⠠⢊⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣗⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⣤⣄⣀⡀⠀⠀⠀⢀⣭⣿⣁⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣿⣩⠀⠀⣠⠖⠋⢉⣠⣤⣶⣶⣶⣶⣶⣶⣤⣤⣉⡒⠦⣀⠶⢿⣧⡴⢾⡷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢻⣯⣤⣴⠟⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡁⠺⣷⠤⠿⣟⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣆⡀⠀⢠⣯⣽⣷⡞⢁⣴⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣄⠚⢻⣉⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣯⠴⠀⢚⣷⣾⠏⢠⣾⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣧⠘⡟⠿⣟⡀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠘⣻⣦⠴⠋⣫⣿⠏⢀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠈⢿⣿⣿⣇⢹⡾⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣻⣷⣶⣉⣹⡟⠀⣸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠠⢤⡀⠎⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⣿⢿⣿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢹⢉⣛⣉⣽⡇⢠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠠⠖⡄⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⡇⢻⠈⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠉⠛⢻⠁⢸⣿⣿⣿⡄⠀⣀⡀⠀⠀⠀⠀⠁⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⡿⣿⡇⠸⡇⢻
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣼⣧⣤⣤⡾⠀⢸⡿⠏⠀⢻⡘⠧⠁⠀⠀⠀⡀⠀⣰⠧⠤⠤⠄⣀⣀⣀⣠⣴⠋⠀⠀⣠⠟⠘⠓⠚⠒⠲⠒⠒⠖
⠀⠀⠘⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⢒⠀⠙⣦⣀⠀⠀⠠⠤⠴⠃⠄⠔⡀⠅⠈⠅⠂⠂⢀⣠⠴⠛⠁⡀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣉⣋⣓⡲⠈⣩⢐⢂⢠⣐⣠⣤⡴⠶⠛⠉⠁⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⢠⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠀⠀⠁⠀⠀⠀⠏⠀⠀⠀⢠⡄⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⢆⡀
⠀⠀⠀⠐⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠐⠂


⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⡻⡿⠛⠿⠿⠿⢻⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢄⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⡈⠁⡀⡀⠀⢀⠀⠀⠐⠂⠀⠀⠠⡄⠀⢀⣈⠀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⣲⠀⢁⢀⡱⢶⠀⠈⣽⠀⠀⠄⠲⠀⣠⠱⠆⠈⠁⠀⡀⠀⢇⡄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢢⡤⠈⡆⠈⢱⡘⠆⣴⢀⡇⢸⠇⠀⠻⠀⠐⢀⠀⢀⠁⠀⠈⡇⠀⠀⠖⠀⢈⠋⢀⡤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⣞⠁⠀⠀⠀⠀⠀⠻⣌⢳⡶⠋⢤⡀⠈⣿⠉⢾⡀⢠⡄⠀⠀⠀⠃⠺⠄⠀⠀⢧⣀⡄⠀⠀⠘⣶⠋⠀⠄
⠀⠀⠀⠀⠐⠀⣀⠹⣠⠉⠀⠀⠀⠀⣤⠤⣄⢾⡀⡗⠀⠬⣅⠀⣼⠀⠈⣧⣸⠀⠢⣀⣀⡀⢐⡆⠀⠀⠀⠈⠙⢦⡀⣠⡿⠴⠶⢤⡀⣀⠖
⠀⠀⠀⠀⠀⠀⠙⠲⣧⠀⠀⠀⠀⡀⢁⡄⠫⠀⣽⡧⠀⠀⠻⣴⡏⠐⠀⠈⢿⡀⠊⠀⠀⢧⣼⠃⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠰⠤⠤⣾⡀⠀⠀⠀⠁⢺⠀⠛⣷⠞⣧⣀⠀⠀⢸⡇⠀⢂⠀⣸⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣅⣶⣄
⠀⠀⠀⠲⠀⠀⠀⠀⠈⢳⣄⢀⣀⠀⠈⠣⣰⡇⠀⠈⠛⠳⠶⣬⣧⠀⢫⡀⣿⠀⠁⠀⠀⠀⠸⣧⡀⠀⠀⣠⣶⠿⠣⠾⠿⠃⠀⢁
⠀⠀⠀⠀⠀⠤⢠⣀⣀⣀⡼⠛⠛⠻⣶⣄⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⡿⠀⠀⠀⡀⠀⠀⠘⣿⣶⡿⠋⠁⠀⠠⠄⠀⠀⠀⡘⠀⠀⠄
⠀⠀⠐⠁⢒⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣻⣆⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⢰⡀⠄⠀⠀⣰⣿⠃⠀⠀⠀⠀⠰⢂⡴⠈⢉⣁⠔⠒⠢⡀
⠀⠀⠀⠐⠎⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠻⠷⣦⣤⣴⣦⠀⠀⠀⣿⣀⣀⡸⠃⠀⠀⠀⣿⡇⠀⠀⠀⠀⢣⡴⠟⠃⠔⠛⣹⠄⠀⠀⠘
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠤⡄⠀⢙⡀⠀⠀⠀⠀⠀⠈⠹⣿⣄⠀⠀⣿⡏⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⢠⣿⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠄⠂⠀⠀⠈⠛⠘⢧⠀⠀⠀⠀⢀⡀⠀⠹⣿⣆⠀⣿⣧⠀⠀⢀⣤⣾⡿⠃⠀⠀⠀⣀⡼⠋⠀⢀⣀⡾⠁⣠⠆⠀⡄⠆⠉
⠀⠀⠀⠀⠀⠐⠒⠒⢧⣄⣀⣀⣀⠀⠀⠈⢧⡀⠀⣴⣿⠏⠀⠀⠻⣿⣆⣻⣿⢀⣼⣿⣟⣉⣤⣄⣀⣠⣾⣋⣠⣤⠾⠋⠉⠉⠚⠣⠤⠾⣡
⠀⠀⠀⠀⡀⠀⣤⠖⠋⠉⠉⠉⠉⠛⠓⠶⢾⣷⣼⣿⣯⣤⣄⣀⠀⢹⣿⣿⣿⣾⣿⡿⠛⠉⠉⠉⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⢠⠄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣦⣴⣤⠴⠀⠊⠉⠛⠛⠛⠛⠛⠛⠛⠛⠂⠠⢤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠦⠄
⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣤⣭⣻⣟⠿⠿⠿⠛⠛⣛⣁⣴⣿⡇⠀⣾⣿⡀⣦⠀⣧⡀⠀⠳⢶⣤⢠⣭⣬⣉⡛⠛⢩⣥⣤⣌⣭⣭⣿⣿⣿⣿⠾⠂
⠀⠠⢤⣤⣾⣿⣿⣿⣿⣿⣿⣛⡛⠛⣁⡴⢀⣶⣿⣿⣿⣿⣿⡟⠈⣿⣿⣿⣿⠄⠹⢿⣿⣆⠘⣿⣦⡹⣿⣿⣿⣷⣦⣽⣟⣻⣿⣿⣿⣿⡿⠟⠛⠓
⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⢟⣱⣿⠟⣡⣿⣿⣿⣿⣿⡿⠋⣠⡆⢸⣿⣿⣿⡄⢶⣦⡙⢿⣆⡙⠿⣿⣿⣿⢿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉
⠀⠶⠶⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⢏⣴⣿⣿⣿⣿⣿⣿⢇⠀⢻⣇⠸⠿⠿⣿⣿⣷⡙⣿⣦⣬⣙⣦⠈⠻⢿⣾⡷⣶⡿⠟⡛⢿⣿⣿⣿⣿⠟⠻⠄
⠀⠠⠾⣿⣿⣿⣿⣿⡟⣼⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⡿⣸⣧⢸⣿⣷⣿⣦⠨⣭⣽⣿⣾⣿⣿⣿⣿⣤⡜⢶⣴⣶⣤⣈⡛⠿⠿⠷⢿⣿⣿⣤⣤⣤⠄
⠀⠀⣀⣹⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⣿⣿⣿⣿⣧⣌⡛⠿⢿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣾⣿⣾⣿⣏⣀⣀⡀
⠀⠀⠀⣤⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣴⣌⢻⣿⣿⣿⣿⣿⣷⣶⣮⣝⠻⣿⣿⣿⣿⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⡦⠄
⠀⠀⣠⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⡤
⠀⠀⠀⠰⠿⣿⣿⡿⡿⠿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠿⠿⣿⣿⡿⢿⣿⣿⣿⣿⣷⣦⡄

⠀⠀⢀⡀⠀⢢⡀⢀⡀⠀⡔⠀⠀⣀⡀
⠀⠀⣻⣇⠀⠀⠱⣼⣯⠞⠀⠀⠀⣿⡃⠀⠀⢀⣄⠀⠀⠀⣸⣷⣶⣤⣴⣶⡄⠀⢀⣾⣶⣦⣤⣶⢦
⠀⢸⣿⡟⡄⠀⠀⣹⣏⠀⠀⠀⢸⣿⡇⠀⠀⢈⣿⡀⠀⠀⠀⠀⣈⠛⠉⢳⡝⠂⠀⠀⠀⠙⠋⠙⢮⠓
⠀⠆⣿⣿⠱⠀⢠⡟⢻⡄⠀⠀⠀⣿⡟⠀⠀⢸⣿⡅⠀⠀⢠⣼⣿⡖⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⡏⣿⠀⠀⡞⠀⠀⢳⠀⠀⠀⣿⡇⠀⢀⣘⣿⢻⠂⠀⠀⠀⢹⣿⣿⡳⡀⠀⠘⠛⠿⣷⣾⣟⢦
⠀⠀⡥⡇⠀⠰⠁⠀⠀⠈⠇⠀⠀⢿⠓⠀⠈⠈⠉⠸⡄⠀⠀⠀⠀⠀⠀⠳⠉⠀⠀⠀⠀⠈⠀⠘⠆⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢠⣤⠀⠀⣠⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⣿⢿⠾⠿⠟⠳
⠀⢀⣼⣯⠀⢠⣼⣯⠀⠀⠀⢘⣿⡀⠀⠀⠀⠘⡿⢿⠶⠶⠤⣄⠀⠀⢰⣿⡶⠶⠶⠀⠐⢓⡊⠀⠂⠒⠂
⠀⢛⣿⣇⠀⢛⣿⠛⠀⠀⠰⢏⣿⠇⠀⠀⠀⠠⠃⣸⠀⠀⠀⠉⠀⠀⢠⣿⠀⠀⠀⠀⢀⣿⣇⣀⣀⡀
⠀⢸⢿⡷⠠⢮⣿⣧⠀⠀⠀⠐⠿⢷⡦⠀⠀⢀⣀⣁⣀⣀⣠⣤⠀⠀⠘⣿⠁⠀⠀⣠⣾⣟⡁
⠀⢀⡏⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⡷⠀⠀⠘⠛⣛⠿⠉⠁⠈⠀⠀⠸⣿⡀⠀⠀⠉⢹⣿⠁
⠀⠈⢁⠀⠀⡀⢈⡁⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⢀⢀⡀⢀⡀⠀⠀⠀⠀⠛⠁⠀⣀⡀⡘⠊
⣶⡖⡶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⢀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠄
⠀⠀⠀⠀⠀⢀⣾⣇⡀⠀⠀⠈⢻⡄⢾⡛⢆⠀⢀⠔⣻⡇⣼⠋⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢧⡾
⠀⠀⠀⠀⠀⡴⣿⣿⢄⠀⠀⠀⠀⠹⢤⠭⠿⣿⡿⠯⢥⠴⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣾⣶⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢆⠀⠀⢻⠇⠀⢀⠆⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⢘⣿⣿⡁
⠀⠀⠀⠀⠀⠀⣇⣹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⣀⣀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⣰⣶⠀⠀⠀⠛⠿⣿⣿⣶⣶⣞⣳⣶⣶⣿⣿⠿⠛
⠀⠀⠙⢿⣿⣷⣤⣴⣿⣿⡿⠃⠀⠀⠀⠀⣿⣻⡅⠀⠀⠀⢀⣠⣴⣾⡿⢻⡟⢿⣷⣦⣄⡀
⠀⠀⠀⠀⣿⣿⡟⢿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠈⠉⠉⠀⠀⢸⡇⠀⠀⠉⠉⠁⠀⣰⣿⣷⡄
⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠘⢿⣿⣷⣦⣔⣦⣴⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠬⠕⠀⠀⠀⠀⠀⠾⠟⠋⢀⣉⠈⠙⠻⠆⠀⠀⠀⠀⠀⢿⡷⠀⠀⠀⠀⠐⠿⠿⢻⡟⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⣷⢆⠀⠀⢀⠴⣿⣻⡟
⠀⠀⠀⠀⠀⢀⣾⣇⡀⠀⠀⠀⠸⣿⣇⠀⣁⣤⣁⢀⣿⡿⠀⠀⠀⠀⢀⡜⢣⡄⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢀⣾⢿⢿⣷⠀⠀⠀⠀⣼⡟⠋⠩⠟⠉⠛⢿⡆⠀⠀⠀⠀⢸⣷⣾⡇⠀⠀⠀⠀⢠⣾⣿⣿⣧⡀
⠀⠀⠀⠀⣼⡟⡲⡚⢿⡇⠀⠀⠀⢹⣷⣤⠄⠀⢠⣴⣾⠁⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠠⠿⠛⢻⠟⠛⠷
⠀⠀⠀⠀⠙⠁⠉⠁⠈⠃⠀⠀⠀⡜⠉⠁⠀⠀⠀⠉⠉⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⠀⢀⡞⠁⠈⠁⠈⢣⡀⢀⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠙⠷⣄⠀⠴⣀⡀⠦⢀⡴⠛⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⢑⣿⡿⢿⣿⠉⠀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢀⣨⣷⣶⣤⣿⣀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡿⣿⡁⠀⡏⢹⡇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣶⣿⡆⣼⣿⣿⡇⢸⣷⣦⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣠⣼⣿⢻⡿⡿⠓⣿⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣯⡇⠘⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠛⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⡀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⠿⢷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⠦⡀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡏⠀⣿⣿⡎⠻⣿⣿⣿⣿⠟⠁⠀⢸⠀⢹⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⣷⠀⠻⣿⣿⣷⣮⣿⣏⠁⠀⡀⣠⠏⠀⣿⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡜⠁⠀⠀⠈⠻⣿⣿⣿⣿⣲⣦⠟⠁⠀⠀⠈⢦⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠙⠦⣀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⣀⠴⢫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⢾⡷⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠠⠄⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⣠⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄
⠀⢣⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠁⠀⠀⠀⠀⡀
⠀⠀⢱⡂⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠪⠀⠀⠀⠀⠀⢐⡊
⠀⠀⠈⢷⡀⠀⠈⠀⠘⠢⣠⣑⢄⠀⠀⠀⠀⠀⠀⠐⢿⣿⡷⠆⠀⠀⠀⠀⠸⢾⣿⡿⠂⠀⠀⠀⠀⠀⠀⣠⣊⣀⠔⠃⠀⠉⠀⢀⡾⠁
⠀⠀⠀⠈⠿⠂⠈⠛⠶⢤⠟⠉⠾⢣⡀⠀⠀⠀⠟⠀⠀⠉⠉⣲⣿⣿⣿⣿⣖⠉⠉⠀⠀⠻⡀⠀⠀⢀⡞⠷⠉⠫⡤⠶⠛⠁⠐⢿⠁
⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⢐⣶⠄⠁⠙⢦⠀⠈⢣⣄⠀⢀⣤⢿⣿⣿⢿⣿⡻⣤⡀⠀⣠⡜⠁⠀⡴⠣⠈⠀⣗⡂⠀⠀⠀⠠⢤⠁
⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠉⠈⠀⠀⠠⡀⠑⢄⠈⣿⣀⣾⠛⣿⣿⣿⣿⣿⣿⠓⣷⣀⣿⢁⡠⠊⢀⠄⠀⠀⠀⠉⠀⠀⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⣳⣌⢹⣮⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣵⡎⣀⣞⡁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠉⠛⠉⣙⣿⣿⣿⣯⠝⠋⢠⡿⣿⣿⢿⡅⠙⠻⣽⣿⣿⣿⣋⠉⠛⠉⠀⡀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠊⠀⢀⠼⣿⣿⠀⠀⢠⣿⣿⣿⣿⡆⠀⠀⣿⣿⠧⠀⠈⢐⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠜⡿⡇⠄⠀⣼⢻⢹⡏⡟⣥⠀⠠⢸⢿⠣⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠔⢀⣾⣧⣇⠀⠀⢠⣼⣿⣷⣇⡄⠀⠀⣸⣼⣷⡀⠢⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⣏⣿⡏⠀⠀⠈⠘⣾⣷⠃⠁⠀⠀⢹⣿⣹⣇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⢻⣿⣤⡀⡄⣧⣿⣿⣴⢠⢀⣤⣿⡟⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⢿⣿⣷⣦⣼⣿⣿⣷⣴⣾⣿⡿⢃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠈⠻⢿⣽⣯⡿⠟⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠁⠀⠀⢀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠦⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤
⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠐⠀⡠⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠈⠳⠂⠀⠀⠀⠀⠀⠀⢀⡞⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢇⠀⡃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠇
⠀⢻⠀⠀⠀⠀⠀⠀⠈⠣⣄⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⡼⠛⠋
⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣰⣶⣀⣴⡟⠛⠋⠉⠙⠓⠂⠀⠀⠀⣸⠀⠀⠀⠀⣀⠰⠛⠁
⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡀⠠⠖⠛⠉
⠀⠀⠀⠀⠙⠂⣀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠳⣄⡀⠀⠀⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⡀⠀⣿⣿⣿⡀⠀⢸⣿⣿⠉⠀⢲⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⠟⠀⠻⠟⠛⢳⣶⣾⠉⠉⢲⣶⡞⠛⠃⣀⣘⣿⣿⠛⠛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⢻⠘⠋
⠀⠀⠀⠀⠀⠀⠀⡄⣀⣨⠛⠛⢶⣶⣎⠉⠉⣶⣤⡟⠿⢆⡉⢰⣤⡿⢷⡀⠂⣄⣀⡀⠀⠀⠀⠀⢀⣀⣠⣴⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢆⣤⣼⠋⠉⣶⣶⡏⠉⢱⣦⣼⠿⠿⡉⠁⣤⣸⢶⣾⠻⢧⣅⣹⣿⣿⣀⣀⡿⠛⢳⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠸⠿⠟⣤⣶⡎⠉⠑⣶⣶⠉⠙⣤⣤⡿⢿⡏⠙⢠⣄⣿⣶⡟⠻⣦⣤⡿⠿⢧⣤⡎⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣿⣄⣠⣜⠋⠉⣶⣿⡁⠈⢳⣶⡎⠛⢣⣄⣸⣷⣞⠛⢧⣌⣹⣷⣟⠛⢷⣤⡞⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠸⡿⠟⣴⣶⡍⠁⢹⣿⡇⠉⢱⣶⡜⠻⢇⡀⢰⣶⣾⠿⣿⣉⣹⣶⡟⠛⣷⡶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣴⣾⡀⣀⡝⠋⢱⣶⡏⠀⢹⣶⡎⠛⣣⣄⣸⣿⡏⠙⣷⣤⣾⣿⡏⠉⣷⣶⠉⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⢸⣿⣧⠤⠜⠛⠓⠒⠚⠿⠧⣤⣿⣿⠀⢀⣷⣶⠿⢿⣃⢴⣿⣿⠉⣹⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠾⢿⣇⣀⣾⣿⡟⡉⣷⣶⡟⠉⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⢿⣿⣥⣤⡿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⣄⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣢⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⢓⡾⢷⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣷⠀⠀⠀⠀⠀⠀⢀⣠⡞⣀⠀⢹⣆⣀⣀⡀⠀⠀⠀⠀⢠⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⡆⠀⠀⣠⠊⠁⣀⣉⡉⠉⠉⠉⠁⣀⣀⠈⠳⡀⠀⢀⡎⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠻⡀⠀⡇⢠⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣷⠀⡇⠀⣌⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣵⠒⢧⠈⢿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡟⢀⡟⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⣧⣀⠉⠛⣿⣿⠀⣿⣿⡟⠋⣠⡾⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣧⠀⣿⣿⡀⣼⣿⣁⣰⢧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠊⠉⢁⡴⠖⠚⢿⣿⡿⠛⠶⣄⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡾⠁⣴⣶⣿⣿⣦⠀⠀⠐⣿⠀⠀⢀⣿⣿⣿⣷⣆⠀⢸⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠤⢶⠃⢸⣿⣿⣿⣿⣿⠆⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢻⠘⢛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢱⠀⠀⡞⠀⠈⢿⣿⣿⣿⠟⠀⣠⣴⣿⣦⣄⡈⠻⠿⣿⠿⠋⠀⠀⠈⡆⠀⢆⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡎⠀⣸⠃⠀⠀⠀⠀⠉⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⡆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡜⡼⠁⠀⢸⠃⠀⠀⢀⣀⣀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⣀⣀⡀⠀⠀⢷⠀⠀⠑⣴⣇
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⡯⠀⣠⣾⣿⣿⣿⣿⣷⣦⣄⠙⠛⢻⠿⠛⠁⣠⣶⣿⣿⣿⣿⣿⡆⠀⢸⡀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⡇⠀⠀⠀⠀⠈⠻⡗⠤
⠀⠀⣠⢾⠏⠀⠀⠀⠀⠀⢸⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⡀⠀⠘⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⠉⠛⠻⠿⠿⠋⣀⣠⣴⣿⣷⣤⣀⡈⠛⠛⠛⠉⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⡈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⣽⠀⣀⣤⣤⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⢀⣀⣀⠀⢸⠘⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⡏⠸⣾⣿⣿⣿⣷⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢠⣿⣿⣿⣷⡞⠀⢡⢰⡀
⠀⠀⠀⠀⠀⠀⠀⢸⢻⠇⠀⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠙⠛⢻⡟⠛⠋⠉⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⢸⡄⢧
⠀⠀⠀⠀⠀⠀⠀⡎⣼⠀⠀⠘⣿⣿⣿⠟⠀⠀⣠⣴⣷⣶⣄⠀⢸⡇⠀⢠⣾⣶⣶⣄⠀⠀⠻⣿⣿⣿⠀⠀⠀⣇⠘⡆
⠀⠀⠀⠀⠀⠀⢰⠇⣿⠀⠀⠀⠹⣿⠟⠀⠀⢠⣿⣿⣿⣿⣿⠀⢸⡇⠀⣿⣿⣿⣿⣿⡄⠀⠀⠈⣿⠇⠀⠀⠀⠸⡀⠸⡀
⠀⠀⠀⠀⠀⠀⡸⢀⡏⠀⠀⠀⠀⠈⢦⡀⠀⣼⣿⣿⣿⣿⣿⠀⢸⡇⠀⢿⣿⣿⣿⣿⡇⠀⠀⣠⠋⠀⠀⠀⠀⠀⢳⣴⡇
⠀⠀⠀⠀⠀⠀⣫⡾⠁⠀⠀⠀⠀⠀⠀⠙⢦⡘⢿⣿⣿⡿⠃⠀⢸⡇⠀⠈⠿⣿⣿⠿⠁⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠿⣄
⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣉⣁⡀⠀⣸⣇⣀⡀⡀⢀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀
⠀⣀⣴⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣠⠔⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⠭⠙⠛⢛⡩⣉⣉⣉⠉⠭⠍⠛⠛⠛⣉⢉⣉⠝⠋⠀⢀⠋⠉⠉
⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠄⠀⢰⡀⠥⠄⠘⢋⡅⠀⠀⠀⠠⠴⠞⠛⠉⠁⠀⠀⠐⡧
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠓⠀⢀⣤⡞⢠⠄⠶⡈⠁⠀⠉⠁⠒⠀⠤⠤⣤⠤⠤⠤⠒⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢊⡡⢺⠏⠁⠀⠀⡇⢠⠀⠀⠘⠲⠤⢤⡤⠤⠤⠐⠂⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠐⠛⠂⠤⠚⢋⠐⠁⠀⠀⠀⠀⡇⠀⡶⠂
⠀⠀⠀⢸⠀⠀⠀⢀⠄⠂⠐⠀⠈⢁⠐⠊⠀⠀⢀⣴⠋⠀⠀⡠⠶⠠⢤⠁⢸⠃
⠀⠀⠀⢸⠀⠀⠀⠑⠀⠀⡤⠆⠀⠁⠀⢀⡤⠒⠈⠃⠐⠀⠈⠀⠀⠀⠸⠀⡶⠀⠆⠀⠀⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⢠⣶⡖⠟⣯⠙⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠈⡳⠊⠉⠈⠢
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣖⣀⡀⠀⠀⠀⠀⠀⡆⢰⠃⢀⠜⠁⠀⠀⠀⠀⠈⢂
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢛⡹⣯⠁⠀⠀⠀⠀⢰⠀⠸⠀⠸⠀⠀⢀⣀⡄⠀⠀⢀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⡌⠀⡆⠀⠀⠑⢀⠈⠙⠀⣀⡀⠊⠐⡀⠀⠀⢀⡀⣠
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡇⢈⠁⠀⣠⡶⡀⢈⠿⠋⡽⠁⠢⣤⠛⡆⠰⡧⢰⡟⢈
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢃⠀⠀⠐⣦⢀⣀⠀⢠⠀⠀⠀⠀⢰⠀⢸⠀⣜⠿⠧⣆⣍⡀⢁⡠⠂⢠⡾⠓⣣⠀⡐⠚⠀⠰⡃
⠀⠀⠀⢸⠀⠀⢀⠔⠂⠐⠈⡶⠋⠃⠈⠋⠁⡠⢀⣀⣀⡀⠀⢸⠀⣠⠈⠁⣠⣄⣦⣢⠈⢣⣀⠄⠛⡄⠀⡽⠀⠀⠀⠀⠀⠱
⠀⠀⠀⢸⠀⠀⠈⠀⠀⢀⠋⠀⢀⠀⢶⠦⢽⡀⣉⠀⢈⡇⠀⢸⠀⢱⠀⠊⠉⢰⠋⠙⠀⠁⠀⠀⠀⡇⠨⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢸⠀⠀⠈⢀⠖⠀⠀⠀⢸⠁⠉⣄⡤⠌⡁⠀⢀⡧⠤⢼⠀⢸⠈⠶⠶⠿⠀⠀⣠⣈⡁⢀⡋⡇⢀⠀⠲⠒⠃⠀⠀⠀⡘
⠀⠀⠀⢸⠀⠀⠐⠁⠀⠀⠀⠈⠂⣄⠤⡋⢀⠸⠋⠉⠁⠀⠀⠀⠆⠨⠆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠳⣇⣠⢒⡄⣀⣀⡀⠤⠈⠀⣀
⠀⠀⠀⢸⠀⠀⢘⠷⡖⠂⣀⣴⠲⣏⠐⠨⠤⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⣀⠙⠢⢴⢯⠡⢜⠁⠀⢸⠊⡹⠄
⠀⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⠀⠀⣈⠙⠓⣶⢤⣤⡤⠀⢈⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣀⣄⣨⠓⠢⣄⣀⡤⠈⠤⣠⡿⢢⡏
⠀⠀⠀⠈⠀⠌⠑⠀⠚⠒⠚⠛⠒⠈⢁⠀⡛⠀⠀⠀⠈⠉⢻⠉⠙⣿⠟⠛⡷⠛⢃⠋⡩⠟⠉⠙⠯⣄⠀⠀⠀⠀⢠⣤⠟⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠉⣱⡖⠰⠂⠒⢶⡿⠒⠒⢧⣀⣐⡆⠀⠀⢸⡀⣀⡀⣄⣸⠃⠀⢂⠀⡥⠓⢄⠀⠀⠈⠒⠠⠀⡀⣀⣏⣼⠟
⠀⠀⠀⠀⠀⠀⠴⢿⠀⠀⠀⠀⢘⠀⡀⠀⡞⠁⠀⠀⢃⠀⢾⣿⣍⣅⣏⣿⣧⣠⣄⡹⢄⢀⠼⠣⢤⣤⣀⣀⣀⣠⣯⠝⠁
⠀⠀⠀⠀⠠⠄⠁⠌⠀⠀⠀⢀⠄⠋⠀⠀⠀⠀⠀⡀⠌⢀⡏⠀⣹⣻⣟⢛⣿⣿⣏⠀⠀⠙⢢⢤⣠⡀⣉⣈⣁⣠⡄⠆
⠀⠀⠀⠀⠀⠐⠈⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠉⠁⠀⠀⠀⠁⠀⠀⠁⠈⠁⠉⠙⠓⠚⠂⠀⠉⠈⡯⢹
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠙⠣⠄⡀
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⡠⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠀⣡⣄⠈⠀⠈⠓⠀⢺⠂⣦⠒⡆⢹⣂⠀⠠⠀⠀⢚⡇⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠙⣿⠁⠀⣀⣾⠒⠋⠁⠀⠀⠰⡈⣹⡂⠐⠂⠀⣈⠁⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠸⡄⠑⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⡿⣿⣿⡷⡤⠪⠵⣄⣠⠾⠁⠀⠀⠛⣟⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢠⠈⠢⣇⢀⣀⡈⢀⡢⢄⣀⡠⠴⠎⣀⠁⠂⠀⠻⠴⠶⢆⡀⠀⠀⠈⠁⠓⣦⠀⢠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⢀⡀⠂⠁⠺⠀⢀⠤⠐⠂⠠⠈⠉⠂⡁⠤⠙⠠⢀⡠⠾⣳⣶⠯⣈⣐⣶⣶⠶⠟⠁⠀⠌⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠣⣀⠀⠀⣶⠁⠀⠀⠀⠀⢈⡄⠀⢘⡀⠀⠀⢀⠃⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣠⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠂⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉