Thunderstorm Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣷⣦⠀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣼⡿⢟⣛⣯⣭⣭⣽⣛⡱⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⣿⣿⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣭⣭⣭⣝⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢼⣿⣿⢟⣫⣽⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠻⠿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣮⡝⠛⠛⠛⠛⠛⠛⠛⠓
⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⠂⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⠂⠀⠀⠀⠀⢀⣶⣶⣶⣀⣀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣿⣿⣿⣷⣶⡶⠀⠀⢀⣼⣿⣿⣿⣿⣇⣀⣀⡀⠀⠀⠚⠛⢻⣿⡿⠋⠀⠀⣠⣦
⠀⠀⠀⣰⣿⡿⠀⠀⠀⠈⠉⠉⣽⣿⠏⠀⠀⠀⠾⠿⠿⠿⢿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣾⠟⠁⠀⠀⣼⣿⡟
⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢀⡀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢀⣴⣿⡇⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⣼⠟⠁⠀⠀⣠⣾⡀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣼⠋⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠙⠿⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠊⢸⣷⣶⣦⣤⣤⣄⡶⣞⣷⠸⣷⣜⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡠⣀⠄⠀⠀⠀⢿⣿⡿⣿⢿⠁⡅⡿⢻⡆⢿⣯⢿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⡅⡊⠊⠀⠀⠀⠀⠀⠛⠀⣡⣼⠀⠀⡆⣼⢣⡘⣿⡷⣀⢺⡏⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠂⣿⠁⠀⡄⣣⢇⣾⡧⢸⣿⠗⢸⡃⡄⠈⡛⠂⠀⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠁⠆⢋⠀⣿⡇⠘⡉⠒⠻⠿⢥⡆⠓⡤⠂⠐⢩⡷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠂⠀⣼⣯⣇⢰⣿⣗⣱⣶⡆⢠⠎⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⢸⢠⡿⢻⡞⢟⡧⡝⠛⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠀⣿⣻⡇⡆⢘⠰⠛⠁⠿⠃⠀⠁⣼⠖
⠀⠀⠗⢆⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⣿⠾⠁⠀⠨⡊⠁⠊⠰⠀⠀⠀⠈
⠀⠀⠀⠏⠀⠀⠀⠉⣿⣷⣶⣶⣻⣵⣦⡤⢤⡤⠀⠀⠀⡂⢉⡴⠋⠘⠉⠀⠄⡀⠀⠀⢠⠀⠠⠀⠀⢀
⠀⠀⠀⠐⠀⠀⠀⠀⣿⣿⣿⠫⣾⣿⢃⠀⢈⡄⢀⡠⠎⡔⠈⠀⠀⠀⠄⠎⣠⣜⠳⣷⢸⣄⡄⠀⠀⠂
⠀⠀⠀⢸⠀⠀⠀⠀⠹⠺⢛⡄⢿⢿⡠⠜⠋⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⣿⣽⣷⣌⢻⣇⣵⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣗⠋⠈⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠐⡫⠘⠁⢹⡄⢻⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⢠⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢿⡀⢿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⡁⠀⠁⢃⢘⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠈⠀⠀⣀⠔⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⢀⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠒⠀⠀⠀⠀⠀⡴⠃⠀⠀⡄⠀⠀⠀⢠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠁
⠀⠀⠀⠀⡠⠀⠀⠀⠀⣠⣷⣤⠤⣤⣤⣶⣧⣰⣄⣀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠒⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠔⡀
⠀⠀⢀⣾⡇⢠⠎⠀⠘⠛⠻⠁⠰⢿⣿⢿⣿⣯⠀⠀⠁⠠⠤⠀⣀⠀⠀⠀⠠⣤⡄⠀⠀⢀⠀⠀⠀⣀⠤⠂⠀⠀⠀⠁⠉⠛⡛⠉⠀⠀⠀⢀⠐⠄
⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⢟⣿⣶⣶⣶⡾⠋⠈⠉⠠⣄⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠠⠀⠀⠀⠀⠰⡍⠓⠲⠶⠚⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣵⡾⣿⣿⣿⣿⠿⣶⣤⣠⣠⠉⠙⠛⠁⠀⠀⡰⠀⢤⣤⡤⠤⠀⠳⣅⡀⠀⠀⢀⡈⠻⣶⣶⣾⡿⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠈⠁⠞⠔⠋⠁⠀⠑⢦⣌⣒⣭⣽⣇⠀⠀⢶⣄⣀⣀⣤⣮⣿⠛⣻⣟⣤⣿⣿⡿⠋⠀⣀⡄
⠀⠀⠀⠞⠀⢲⣄⣉⣂⠀⡀⠀⠈⠃⠄⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⣰⣶⢿⣿⠿⢛⣿⣿⣿⣷⡖⣹⣿⣟⡿⠋⠱⠁⠉⢀⣼⠟⢋⡀⠀⠀⠁
⠀⠀⠀⠀⠂⠁⣄⠀⠀⠀⠳⠆⠀⠐⠀⢠⡘⠉⠉⢉⠀⠀⡀⠀⠀⠈⢙⠁⠀⣴⡂⠊⢛⣿⡿⢋⣾⣯⣿⡟⠁⠀⠀⠀⠀⠈⢀⡔⠁⡰⠚⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢰⣆⠀⠰⣄⠀⠉⣓⡈⠁⢀⣌⢷⣤⢤⣶⡽⠂⡴⠛⡁⣰⣼⠟⣴⣿⡿⠟⠋⠀⠀⣴⡅⠀⠀⣰⣯⡀⠊⠀⠀⢀⠞⠁
⠀⠀⠘⣿⣷⣶⣆⣀⡴⠂⠀⠙⠛⠟⣹⣿⣿⣿⣿⣿⣿⣿⣷⣾⢫⡥⠀⠀⢀⢀⣰⣿⣧⡶⣠⢿⡃⠀⣠⣤⡾⢟⡖⠀⠀⢑⣏⠀⠀⠀⠀⠁⣠⠆
⠀⠀⠀⢋⡿⣯⠟⣵⠾⣠⡔⡡⢂⣼⣿⣿⠟⢻⣿⠿⣿⣿⣟⣽⣋⠠⢀⡴⢇⣭⣿⣿⢋⣴⡣⠈⣠⠊⠿⣩⣶⣿⡷⠀⢴⡿⠋⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠋⠀⢀⠞⢃⡿⠋⠀⢰⡾⣻⡿⠂⢠⡔⠂⠀⠀⠁⠭⠨⣻⡆⢈⣴⣿⣿⣿⣡⣿⣿⡿⡿⠛⠀⠘⠛⢻⠟⠂⠠⠍
⠀⠀⠀⠀⠀⣀⡰⠋⠐⠀⠀⠊⠀⠉⠀⠀⠀⢀⣊⠐⢛⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⣚⢞⣁⡀⠀⡐⠀⢠⡜⡁⠈⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠛⡀⠀⢠⡄⠀⠀⠀⠀⠀⣀⣾⣿⣷⣟⣴⣿⣿⣿⣯⣭⣿⣿⠟⣿⣿⣷⣖⣠⣥⣹⠞⡀⠈⠀⠀⠾⠻⠃⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⢀⠚⠿⠿⠁⠀⠀⠀⡠⠊⠁⠁⣰⣿⣿⣿⣿⣷⣽⣯⣼⣿⡿⠃⠾⢿⣿⢯⠿⢿⡿⣿⣿⣅⢠⠄⠀⠈⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠟⢻⣿⣷⣿⡿⣛⢿⣿⣿⣿⡟⣡⡞⣡⡯⠀⠈⠈⠉⠉⢱⣶⣴⣦⣿⣿⣿⣿⣷⣶⣶⣖⠃
⠀⠀⠀⢀⠂⠀⠀⢰⡇⠀⠀⠠⠀⠀⠀⠀⠠⡾⠿⢋⠜⠙⠁⣼⣿⣿⣏⣸⣿⡿⢈⣦⣾⣥⣤⣤⣴⡿⠟⣋⣩⣿⣿⣿⣿⣿⣿⣿⣯⣷⠀⠁
⠀⠀⠀⡤⠊⠀⠀⢼⣇⠀⠈⣆⢠⠀⠀⠀⠀⠀⠜⠃⠀⢀⣾⣾⣿⣿⣿⣿⣿⣧⣿⠛⡁⠾⢿⣿⣫⣵⣾⣿⣿⢟⣿⡛⣩⣷⣧⣍⣙⣷⣶⡄
⠀⠀⠀⠟⠀⠀⠀⣿⣟⠂⢼⡿⣿⡂⠀⠀⢀⡀⠀⠀⠀⢺⣿⣿⣭⣭⣽⣿⣿⡿⣡⠌⠑⠠⠘⠿⢻⣿⣿⣿⣁⣙⣻⣿⣿⣿⡏⠻⣿⣿⣟⣦
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⣸⣿⣿⡇⢰⡄⣾⡆⠀⠀⠰⠃⢾⣿⣿⣿⣿⣟⡿⠈⠁⣀⠄⠀⠀⠀⠀⠀⠸⢿⠿⣿⣿⣿⣦⣼⡰⣿⣿⣷⣿⠁
⠀⠀⠀⡀⠀⠠⢾⣿⣿⢿⣿⢫⣿⣇⣸⣟⣿⣟⠀⠀⣀⢀⣐⣫⣽⣭⣩⣭⣤⣉⠛⢁⣀⣀⡀⠀⠀⠀⢀⡘⡀⡟⠃⠉⠿⢁⠔⠙⠛⠋⣁⣀⣀
⠀⠀⠀⣿⣾⣿⣿⣿⣿⣾⣿⢟⣿⣻⡧⣿⣿⢾⣾⣿⣿⣿⠿⡟⣿⣿⠟⢻⠻⣻⣶⢃⣿⣧⠻⢶⣿⣧⣿⣿⣿⣿⣿⣧⣥⣧⣶⡷⠘⢿⣿⣿⣿⡆
⠀⠀⠀⣉⣍⣵⣾⡿⣧⣌⣵⢿⣿⣟⣡⣿⣿⣷⣙⣉⠁⣤⡤⠄⡀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠉⠉⠉⠉⠙⡇⠁⢹⡇⠉⠉⠀⠀⠈⠛⠛⠛⠁
⠀⠀⠀⡽⠿⠟⠿⡛⠒⠙⠛⠻⠉⠬⠽⠷⠼⠿⣦⣬⣅⡀⠀⠀⠐⠛⠛⠛⠛⠻⠿⠛⠓⠈⠀⠀⠒⠂⠀⠈⠀⣿⡀⢸⡇⠠⠀⢴⣤⡤⠤⢤⡤
⠀⠀⠀⣷⣴⣶⣶⣶⡶⣶⣦⣤⣴⣦⣤⢀⣀⠀⠀⠉⠉⠉⠋⠉⠒⠐⠒⢴⣤⣤⣤⣄⠀⠀⠀⠀⠀⠤⠤⠶⠖⠚⠷⠚⠻⢿⣿⣿⣿⡿⣿⣿⣷
⠀⠀⠀⡻⠓⠚⠉⠋⠀⠀⠈⠁⠐⠒⠛⠛⠛⠛⠻⠿⣶⣶⣶⣤⣶⣶⣾⣿⣿⣿⣿⣿⣷⣽⣿⣾⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠴⠿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠤⢀⠔⠉⠐⠄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⡠⠁⠈⠔⡠⢂⠂⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠨⠂⠄⠀⡠⠠⠂⠀⠀⠠⠊⠀⠀⠀⢀⠔⡡⡀⠁⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⣀⠀⠀⠀⠀⢀⢀⠁⠈⠈⠈⠉⡠⠁
⠀⠀⠀⠀⠀⠀⠃⢠⠀⡠⠀⠴⠀⠀⠖⠀⠰⠂⢀⠄⠀⠀⡐⠁⢀⠟⠉⠐⠁⠀⠤⠂⠀⠀⠀⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠠⠂⠠⠀⠰⠀⠀⠃⡔⢀⠊⠀⠀⠠⠮⠄⠀⢲⣂⠀⡰⠂⠀⢈⡀⠀⠚
⠀⠀⠀⠀⠀⠘⠀⠀⠀⡔⠀⢀⠀⢠⠀⠀⡰⠂⠀⠰⠇⠒⠒⠒⠒⠀⠀⠀⠄⠀⠊⠠⠆⠀⠈⡠⠀⠐
⠀⠀⠀⠀⠀⠀⡌⠀⠀⡄⠀⠉⠀⠃⡤⠀⢀⠀⠀⠀⠰⠂⠠⠊⠀⠀⢀⠔⠁⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠐⢀⠀⠰⠁⡄⠀⠀⠊⠀⠸⠁⢀⠔⠁⠀⢀⠔⠁⡰⠀⡀⠀⠊⠀⠀⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠘⠀⠈⠀⠠⠂⠀⠔⠁⠀⠀⠀⠀⢀⠀⠀⠈⠀⣀⠀⠋⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⢠⠇⠀⠀⠄⡠⠂⠀⡀⠈⠀⠀⠀⠋⢀⠀⠀⠈⠁⠀⠜
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⢠⣊⣀⡀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡨⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠊⠁⠀⡔⠀⠀⠀⠀⡠⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠉⠉⠉⠙⠶⠖⠋⠉⠉⠙⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣀⡀⣀⣀⡀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣐⡢⠀⠀⠀⠀⠀⠈⠏⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡖⠁⠀⠀⠀⠀⠀⠀⠀⠚⠓⠒⠭⠶⠖⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡯⠤⢀⣀⠀⢀⣀⣀⣠⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣔⣰⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠒⠋⠉⠀⠀⠉⠳⣞⣡⣖⣠⠖⠳⢴⣠⢖⣰⣦⠾⠛⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣰⠃⠈⠁⠀⢀⡤⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢰⠃⠀⠀⢀⡴⢋⠗⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠰⠿⠶⢞⡶⠂⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢠⡯⠴⠒⡾⠃⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣠⠖⠉⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⢞⡵⢋⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⣾⠗⠉⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀
⣟⣛⣻⣛⣟⣟⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣟⣻⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠿⠿⠿⠿⠷⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠿⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⠉⠉⠉⠉⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⣠⣾⠿⠷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠹⠟⠉⠉⠛⢿⣆
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⢷⡄⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠁⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠿⢟⣿⣶⠀⠀⠀⠀⢀⣤⣤⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⣀⣐⣿⡅⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⣸⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢻⣦⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣸⡏⠀⠀⠸⣿⣿⣦⣄⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⣷⣿⠃⣀⣤⣄⠀⢀⣿⣿⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠻⣿⡿⠿⠶⠶⠶⠾⠟⠋⠁⠀⠈⠿⠾⠛⢹⡿⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠛⠿⠷⠿⠿⠆⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⢀⣀⡀⢀⣿⠁⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣽⣿⣆⠀⠀⠀⢠⣿⠉⠻⣾⠏⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣸⡏⠀⡀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⢻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣱⣿⡿⣷⣄⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡆⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⢀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⢸⣿⠃⠀⠀⠙⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠀⠀⠀⠀⢿⠏⠀⠀⠀⠀⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠾⠿⠿⠿⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠿⠿⠿⠿⢷⣶⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⣀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠙⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠉⠙⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣹⣧⣤⣤⣀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⢀⣴⡿⠛⠋⠉⠉⠛⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠉⠉⠉⠛⠿⣶⣄⠀⣀⣤⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣄
⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠛⠉⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣴⣶⣶⣶⣤⣄⠀⣿⡆
⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠛⢿⣾⡇
⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⢿⣦
⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻
⠀⠈⠙⠿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣽⡟⠉⠉⠉⠉⣩⣿⠋⢿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⢀⣼⣿⣥⣤⣬⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣃⣀⣀⣀⣀⣈⠉⠉⠉⢉⣿⠏⠉⠛⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⠶⠶⠶⠶⢶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⢻⡿⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⢠⣾⣿⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣁⣀⣀⣀⣀⣉⠉⠉⠉⣩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⣿⡏⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⠉⠙⠓⢦⡀⠀⠀⠀⣠⠴⠶⠶⢤⡀⣠⠴⠚⠛⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠖⠚⠁⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠙⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠲⣄⡀
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⣸⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⡄
⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠒⠚⠛⠉
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⢀⠀⠀⢀⣡⠞⠛⠓⠶⣾⠟⠁⠈⠙⠓⠶⠦⣤⣤⣤⣴⣶⠟⠛⠶⢦⣤⣤⠶⠞⠋
⠀⠀⠀⠀⠀⠙⠳⠶⠤⣤⡤⠶⠾⠛⠁⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠛⠛⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⡠⠂⠀⠴⠗⢚⡽⠃⠀⠀⠀⠀⠀⠀⠀⠠⣠⡾⠃⠀⠀⣠⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⢀⠜⠁⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⠴⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠁⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣥⡤⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣀⠀⠀⠀⠀⠀⠀⠘⣩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠖⠚⣹⠏⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣰⠃⠀⠀⠀⣠⠾⠋⠀⠀⠀⢀⡔⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⣰⠃⠀⠀⣠⠾⠁⠀⠀⠀⠀⡠⠊⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣬⡁⣠⣤⠈⢷⣣⡀
⡆⠠⡀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢈⣿⡳⣜⢿⣎⢻⣷⣔⠽⢿⣄⢠
⠈⠢⣀⡐⠒⣋⠀⠀⠀⠀⠀⠀⢈⠐⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣶⣯⠛⡛⢎⢢⡹⣷⣝⢿⣯⣂⠻⣧
⠀⠀⠀⠈⠉⠀⠀⣤⡤⢤⠖⠀⠈⠑⠀⠀⠀⢀⠀⠀⠀⠀⣠⡀⢴⣶⣷⣝⡿⠋⠙⠳⠘⠀⠑⠙⠊⠉⠀⠙⠁
⠀⠀⠀⠐⠒⠒⠚⠁⠀⠀⠉⠓⠀⠀⢀⣀⣤⣼⣿⣶⠲⣿⣿⣿⣦⡹⠿⠻⠃⠀⠀⠀⣀⡀
⠀⣀⢀⠀⠀⢀⣀⡀⠀⠀⢀⣤⡰⣾⣷⡙⢿⣿⣿⣿⣷⡌⠣⠉⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⣿⣿⣍⢿⣿⣿⣿⣿⣧⡻⣿⣮⣿⣿⡦⠻⣿⣿⣇⡈⠒⠐⠠⠤⠔⠀⠀⠐⠒⠈⠀⢀⠀⠀⣤⣀⠰⣝⢷⣗⠄
⠀⠀⠈⢿⣿⣦⠻⠟⠹⣿⣿⣿⣮⣿⣿⣿⣆⢲⣾⣿⣿⣿⣿⣿⡻⣷⣿⣿⣶⣦⡺⣿⣿⣟⢦⡻⣿⣷⡳⠌⠢⡉⠳
⠀⠀⠈⢲⣝⢿⣷⣿⣧⡹⣿⣿⣿⣷⣝⢿⣿⣷⡙⢿⣿⣿⣿⣿⣿⣮⡿⣿⣿⣿⡛⣈⣿⣿⣷⣕⢮⢿⣿⣦⡢⠈
⠀⠠⡙⣝⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣷⣿⣿⣷⡼⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣮⡻⣛⣿⣿⣯⣷⣝⠿⡿⡄
⠀⡝⢮⠺⣿⣿⣿⣿⣿⣏⠉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄⢭⣝⣛⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣷⡝⠞
⠀⠈⣿⣿⣟⢿⣿⣿⣿⣿⣷⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣼⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⡿⣦
⠀⠠⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⢆⣻⣿⣿⣿⣿⣿⣿⣿⣄⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⠿⢯⡘⠌⠢
⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⡿⠟⡛⢮⢠⣧⠳⡀
⠀⠐⢠⡎⡙⡟⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⡛⢯⡻⢷⢰⡆⣿⠀⣸⣿⣀⡈
⠀⠀⢐⢯⣿⢰⣿⣸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣿⣿⣷⣿⣿⣿⡿⣿⣿⡏⣤⠁⡆⣼⣿⣿⢃⣼⣿⡇⠙
⠀⠀⠂⣼⣿⣵⣿⣏⠷⠈⠻⠿⠟⠍⠛⠟⠗⠙⠛⠉⠊⢛⠋⠛⣛⠛⠻⠿⠮⠙⠐⣿⢘⣷⣸⣿⣿⣿⣿⣿⣷⡄
⢠⣾⣷⣿⣿⣿⣿⣿⣶⣶⣦⣤⣾⣿⣶⣦⣶⣿⣷⣦⣔⢻⠿⣿⣿⣿⣷⣶⣶⣦⢸⣿⣿⣿⣿⣿⣾⣧⣿⣿⣿⣶⣦
⠀⠉⠋⠁⠀⢀⣈⣀⣉⣉⣋⣉⣉⣉⡙⠛⠋⠉⠉⠤⣤⠤⠀⠀⢤⣁⠤⠭⢙⣐⣿⣿⣿⣝⣿⣿⣿⣭⣽⢿⣿⣶⣍
⠀⠐⠀⠂⠢⠀⠐⠀⠀⠈⠓⠐⠒⠶⠒⠀⠀⠀⠃⠙⠛⠉⠛⠋⠁⠀⣀⣠⣬⣴⠼⠗⠒⠛⠛⠛⠛⠋⠛⠻⠿⠛⠫
⠀⠮⠙⠛⠻⠿⠿⠛⠛⠛⠛⠛⠓⠒⠒⠛⣀⡘⢛⣡⣛⠛⣻⡓⠁⠀⠀⠀⣀⣀⣠⣤⣤⡶⡶⠶⢶⠿⠛⢿⣶⠦⣜
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⣄⣒⣟⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣟⣛⣋⣉⣉⣉⣁⣀⣀⣀⣠⣀⣀⣀⣀⣈⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⣚⡆⠙⠉⢦⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠈⠠⣽⣿⠃⠀⢸⡀⠀⠀⠀⠀⢀⢴⣶⢖⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠐⠀⠀⡀⣾⠇⠀⡀⠃⠀⠀⠀⠀⣞⠋⠍⡷⠪⣳⣕⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⡀⡀⣴⡇⠈⠀⠐⡄⠈⢂⠀⠀⠀⠢⢵⡥⠃⠀⢑⢻⣱
⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⢰⣷⣽⠟⠀⠀⠀⢀⣷⠀⢠⡡⡀⠀⠀⠀⠀⠀⠀⠀⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣴⣿⣿⠏⠀⣀⠄⠀⣸⣿⣆⠀⠹⣮⣐⠀⠄⡀⠀⠀⠠⣗⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⢫⣾⣿⡿⢠⢾⡿⣛⡻⣄⠀⠻⣿⡧⣀⠨⠀⢠⢓⢚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⣿⡿⠋⡴⢅⣿⡋⠀⠸⣿⣭⣠⠈⢹⡽⢿⢶⠿⣡⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣊⡴⢩⠖⠃⣀⣤⣶⣿⣿⣷⢀⢿⠃⢙⠋⡱⣾⢽⣏⡤⣾⣷⣦⡤⡀
⠀⠀⠀⠀⠀⠀⠀⢼⣷⡝⢁⣴⢞⣿⣿⣵⠎⣼⣿⣾⡇⠢⠃⢀⢷⡃⡟⠉⠉⠛⠻⢿⡗⠘
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⠾⢟⣡⢯⡜⠃⢼⠏⣴⣯⣤⠷⣒⣻⢟⣽⡇⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣞⢯⡊⠿⣶⣤⠺⣩⣿⣿⡏⢹⡭⢸⣌⣵⣄⠀⠀⠀⠀⢉⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣈⢸⡇⣋⣸⠶⠛⣏⢸⡧⠃⠈⠉⠁⠈⠛⠷⣗⡂⢄⠀⡎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣷⡘⢹⢲⠔⠃⡼⣖⠵⢐⣂⠀⡒⠀⢤⡀⠙⠢⣽⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⡛⣻⣿⡿⠲⢄⣐⣥⢾⣳⠾⠟⠻⣍⠈⠉⠉⠉⠀⠀⢸⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣏⠥⣿⠙⣾⣹⢽⣿⣷⠧⠚⠿⣶⡸⣮⠖⡀⠄⢀⠀⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣭⡯⣛⡭⠟⠁⠀⠠⢊⢙⡇⡿⠷⣞⣷⣿⣖⣶⣅⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠢⠬⠞⠁⠀⠀⠀⠀⠀⠀⢀⣈⣷⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠫⣪⡟⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣘⣋⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡅⢿⣿⣄⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣫⣭⡁⠀⠀⠻⡆⠘⢷⣞⢤⡤⣶⣀⠀⠀⠀⣀⣤⠀⠀⠀⠀⢀⣤⡀
⠀⠀⢠⣤⣤⢾⣿⣿⠿⣿⣿⡿⠀⠐⠀⠀⠀⠀⠀⠙⠷⠄⠀⠈⠳⠶⠾⠿⠛⠻⢶⣄⣤⠚⠉⠻⢆⡀
⠀⠀⠀⠁⢈⣀⣉⠃⢴⣿⡿⠗⠀⠀⠀⣠⣶⣾⣿⡔⢤⡀⠀⠀⠲⠦⣤⢀⠠⠞⠛⢻⠈⠀⠀⠀⡀⠉⠓⠉⠉⠑⠢
⠀⠀⠀⠀⠈⠋⠉⠢⠈⠉⠣⣤⣄⣴⣾⣿⣿⠿⣿⣧⣠⣅⡀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⢻⣿⠙⠃⠎⠀⠟⣿⡿⢿⠀⠀⠀⠀⢀⣤⣾⡟⠀⠀⠀⠀⣾⣿⡆⠀⠀⠤
⠀⠀⢰⣤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⢀⢠⠞⠉⣤⣈⣤⡀⠀⠀⠀⠉⠻⣿⣤⣄⠀⢸⣿⣿⣷⠞⠁⠉
⠀⠀⠸⢿⠛⠓⣿⣶⣄⣤⡀⢀⢠⣿⡄⢀⠀⠈⠀⠒⢼⣿⣿⠿⠟⠆⠀⠀⠀⠀⠈⠙⠀⠀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⣿⣾⣿⣧⣿⣷⠀⠀⢀⣤⠈⠙⠓⠶⡀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣠⣄⠀⣀
⠀⠀⠈⠓⠻⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣼⣿⣿⣇⣀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⡿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠙⠛⠛⠛⠛⠻⠿⠿⢧⣼⣿⡀⠀⠀⣤⣠⣿⡿⠻⠿⢿⣿⣿⠟⠛

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⡎⣤⣤⣤⣬⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣬⣀⠹⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠘⣿⡗⠦⣹⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠹⣷⣄⣌⠉⣯⣽⡯⣿⣿⣿⠿⣦⡠⠄⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣶⣧⣶⣟⣁⠷⠿⣸⢹⣶⣶⣾⣿⣶⣦⣄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣽⣿⣯⡴⠀⠄⠀⠙⢿⣿⣿⣿⣿⣿⣷⣦⣁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠿⠟⠛⣿⣿⣿⣯⡀⡶⠉⠀⠈⢻⣿⣿⣿⣟⣻⣿⣿⣦⡀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⣠⠟⢡⣷⣿⣿⣷⣦⣿⣿⣿⣿⡤⠀⢀⣘⣋⣿⣿⣟⣽⣯⣿⢿⣿⣿⣶⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠐⣇⣀⣺⣭⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢿⣽⡿⣿⣿⣷⣄⠈⣣⡀
⠀⠀⠀⠀⠀⠀⣿⣙⣿⣿⣿⠇⠀⠀⠀⠈⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣧⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⡧⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣮⣾⠿⣟⣩⣽⣻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⡁⠀⠀⠁⠀⠀⠀⠀⠙⢏⢳⣷⡄⠀⣙⡾⣽⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣟⣿⠶⣄⣀⣄
⠀⠀⠀⠀⠀⠀⠸⣿⠟⠠⠰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑⢽⡛⣿⣯⣀⠀⠉⢷⣴⣟⠛⣳⡾⣉⢛⣭⣻⢿⣟⣿⠿⣮⣍⣿⠶⣤⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢽⣟⡷⣶⣦⡿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣷⡟⠿⣿⣯⡹⣿⣿⡷⣼⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠷⣍⣻⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣿⡿⣆⠀⠈⠉⠚⠻⠿⠿⣯⣭⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠛⡻⣿⣿⣿⣿⣷⠶⠿⣧⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⡿⠛⠁⣀⡱⠂⠉⠛⢿⣿⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣻⣮⠴⠒⠋⠀⠀⠀⠀⠀⠀⢿⣿⣯⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡼


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠈⠉⠙⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⣷⠀⠀⠀⠀⠀⠀⠘⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠋⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣟⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣹⣿⣸⣷⡆⣀⣄⡀⣿⠀⢀⣠⠄⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⢿⣳⣿⣿⣇⣨⣽⣿⣟⣤⡓⣻⣛⣤⣶⣿⡀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣽⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠓⠀⠀⠠
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣉⣉⣉⠉⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣤⠀⠒⠀⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣴⣮⡃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣄⣶⣤⣤⣤⣤⡤⠴⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣃⣢⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡤⢤⣤⣥⣤⡤⠶⠶⠶⠒⠛⠛⠛⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿
⢿⡋⠙⠏⠀⠀⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠌⠉⠀⢿⣿⣿⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠒⠒⠒⠒⠒⠀⠀⠀⠐⠒⠒⠒⠂⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠄⠤⠤⠀⠀⠀⠀⠄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣈⡉
⠀⠀⣀⠀⢀⡀⢀⠀⡀⠀⠀⠀⠀⣈⢡⢎⣼⣿⣾⣷⣤⡒⢤⡀⠀⠀⠰⠶⠒⠀⢀⣐⠂⣠⡀⠀⣀⠀⠄⢀⡀
⣀⣨⣤⣤⣀⠒⣀⣈⣉⣀⡈⢉⣿⡏⠐⣩⢟⣵⣿⢟⣿⣯⠀⠤⣄⡀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡷⢶⠛⡿⡲⠮
⠈⠙⠉⢉⣁⣈⡉⠛⠙⠛⠋⢉⡭⢒⡠⣸⡽⣛⣵⣿⣿⣿⣦⠀⠈⢝⣆⢠⣰⣠⣀⣀⣈⠉⠛⠛⠛⠻⠿⠾⠞⠛⠒
⠤⢆⣚⠫⠭⣛⣫⣭⠤⣒⡨⠔⣪⠵⣚⣽⣾⣿⡿⣿⣿⣿⡟⣄⡀⠐⠛⠍⡿⢤⡈⣍⣬⠍⣉⠈⠙⢃⡴⣧⡦⢴⡃
⡉⠄⣐⣚⣭⠵⠶⢚⣩⠵⣒⡭⣒⣯⣟⠽⣛⣥⢞⣿⡿⣿⣿⣯⣋⣐⢀⠀⡉⠐⠉⠁⠉⠩⣘⠉⠈⠉⠀⠀⠩⠟⠉
⣫⣭⣽⡥⢴⣒⣫⢽⣾⡿⣗⡿⢟⣫⠥⣐⢭⡾⢟⣽⡾⢟⣽⢿⣿⣿⣶⣤⣬⣥⣤⣬⣴⣮⣵⣍⣱⣀⣒⣵⡶⣾⢣
⠩⠽⠖⣛⠭⢑⣚⢭⣶⠏⣁⡡⢈⣠⡴⠞⣉⡤⠞⣩⠾⣫⣵⠟⣫⣴⠟⣋⢜⣹⠟⡯⢚⣿⣿⣻⣿⣿⠿⣫⠾⢃⣟
⣶⣶⠶⠶⠿⣷⣶⣿⣶⣿⣷⣿⣿⣷⣶⣿⣷⣶⣾⣶⣾⣿⣶⣾⣿⣷⣾⣷⣿⣶⣿⣶⣿⣿⣿⣿⣿⣷⣶⣷⣶⣿⣿

⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⣀⠀⠀⠀⠀⠀⠈⣿⡟
⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⠀⠀⢩⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⣸⡿⠁⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠞


⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠰⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣥⠂⠀⠀⠀⠀⠀⠀⢚⣿⡟⠁⠀⠀⠀⠀⠀⣰⣿⣀
⠀⠀⠀⠀⢠⣿⣏⠀⠀⠀⠀⠀⠈⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠺⣿⡖⠀⠀⠀⠀⠀⠀⠋⣿⠃
⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣲⣿⣿⣿⣿⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⣠⣫⡾⠋⠀⢫⡝⠀⠙⢷⣝⣄⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣶⡾⠛⠋⠙⠛⢷⡤⡪⢟⡻⢕⢤⡾⠛⠋⠙⠛⢷⣶⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢿⣷⡔⡈⢀⢔⡟⣠⣴⣦⣄⢻⡦⡀⢁⢠⣾⡿⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⡐⠩⣱⣲⢿⣾⢏⡥⣖⣲⢬⡹⣷⠿⣖⣎⠍⢐⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⡴⣫⡾⢿⣄⢀⢔⣻⣷⣟⠽⠚⢺⡕⢪⡓⠓⢯⣻⣼⣿⣢⡀⣠⡿⢷⣕⢦
⠀⠀⠀⠀⠀⠀⣼⣾⠋⠀⡠⡫⠛⣩⠏⡵⣺⣀⠠⣢⣇⣸⣴⠀⣠⣗⢮⠹⣍⠛⢝⢄⠀⠙⣷⢧
⠀⠀⠀⠶⠶⢾⣿⣿⣏⠶⣿⡱⠰⣿⢸⢹⡱⣉⣉⣹⣿⣿⣏⣁⣉⢎⡏⡇⣿⠆⢎⣿⠶⣹⣿⣿⠷⠶⠶
⠀⠀⠀⠀⠀⠀⢻⢿⣄⠀⠑⣕⣤⣙⣆⡳⢽⠊⠐⠛⡏⢹⠻⠀⠉⡫⢞⣰⣋⣤⣪⠎⠀⣠⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠳⣝⢷⣾⠋⠈⠫⣽⡟⣯⣣⢤⢼⡣⢜⡧⡠⣔⣽⢻⣯⠝⠁⠙⣷⡾⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠀⠅⣐⡹⢽⣶⣿⣎⡓⠿⠭⢚⣱⣿⣶⠯⢇⣂⠈⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣾⡿⢃⢁⠈⠪⣧⠙⠻⠟⠋⣼⠕⠁⡈⠘⢿⣷⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠿⢷⣤⣄⣀⣤⡾⠳⢕⣮⣵⡪⠞⢷⣤⣀⣠⣤⡾⠿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠙⢝⢷⣄⠀⣜⣣⠀⣠⡾⣻⠋⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠽⣻⣿⣿⣿⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⣀⠰⣀⣤⣀⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢦⣰⣤⣀⡆⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣄⢠⣀⣠⡄⢀
⠀⠀⣄⣹⣷⣿⣿⣿⣿⣶⣿⠄⠀⠀⠀⠀⠀⢠⣌⣿⣾⣿⣿⣿⣷⣾⡥⢀⠀⠀⠀⠀⢀⣘⣿⣾⣿⣿⣿⣷⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣶⣠⡄
⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣃⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡷⠂
⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐⣺⣿⣿⣿⡿⠛⠉⠉⠀⠈⢑⠀⠀⠀⠠⣿⣿⣿⣿⡿⠟⠉⠉⠁⠩⠳⠀⠀⠀⠀⠀⠀⠄⠀⠈⠙⠟⠁
⠐⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠈⢩⢿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠉⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠺⠿⣿⣿⣿⣿⠻⡍⠁⠀⠀⠀⠀⠀⠀⠐⠻⢻⡷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⡦⣤⣴⠂⡄
⠀⠀⠀⠀⠀⠉⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠄⠠⠡⠀⠀⠀⠀⠄

⠀⢀⣀⠀⢀⠠⠐⠒⠠⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⢀
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠠⠁⠀⠌⠄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢠⣶⣟⣿⢿⣶⡠⡀⠀⠀⠀⠀⠀⠄⠒⠒⠂⢀
⠀⠀⠀⡠⣢⣾⣻⡿⣶⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⣿⡻⣶⡀⡀⠀⠀⠀⠀⢀⣸⣶⡿⢱⣿⡟⢸⣿⡆⢿⣿⡜⡄⠀⠀⡐⠁⠀⠀⢀⠔⡴⠁⢄⣀
⠀⠀⡜⣼⣿⠃⣿⣧⢹⣿⡌⡆⠀⠀⠀⠀⠀⢀⣂⣾⣿⢰⣿⡇⢻⣿⣘⡄⠀⠀⠤⣿⣿⣿⣿⣿⣿⣇⣸⡿⠇⣸⠟⠛⠛⠀⠀⢡⠀⠐⠈⠡⠈⠀⠀⠈⠀⠀⡄
⠀⠀⠉⠙⠻⠔⠛⠛⠸⠟⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠻⠘⠛⠓⠼⠋⠉⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠁⠐⠂⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠙⠛⠆⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡗⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠠⠲⣤⣤⣀⢀⠀⠀⠀⣀⣠⣦⢠⣶⣀⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡗⡏⣯⣼⡆⢿⣿⡼⣿⣿⣿⠃⠀⠄
⠀⠀⠀⠠⠀⢸⠆⠀⠀⠀⠀⠀⣠⢔⡤⣶⠀⠀⠀⠠⠚⢻⡟⢹⠏⢿⠅⡇⠏⢸⡇⠘⣿⣿⢟⠙⣻⣇
⠀⠀⠀⠈⠀⠀⠀⠀⣀⠠⠄⠀⢸⢠⢈⢁⠀⠀⠁⠀⠀⠉⠁⢨⣶⡿⠀⠆⣧⣾⢣⣆⠹⣧⣯⠔⣹⡇⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢼⠀⠀⠀⠀⠀⠀⠈⠸⡇⠀⡀⡆⣵⡇⣾⣿⠀⣿⡿⠃⣼⡅⡤⡀⢙⠉⢀⠠⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠁⠂⢄⠀⠀⠃⠀⠀⠇⣋⠆⣿⣿⠂⢉⡙⠒⠿⠿⢥⡆⠸⢺⢼⠀⠋⣿⢿⣶
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠐⣰⣿⣽⠀⣸⣯⣟⢰⣶⡦⠈⣶⡃⠠⢀⢀⢨⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⣴⣯⡿⢹⠃⣹⡿⣷⣿⣿⣇⡸⠯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠸⣿⢿⡟⢨⠁⠋⣰⠿⠃⣿⡇⠈⠃⢸⣆⡤
⠀⠈⢦⣀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⡻⢋⡟⠈⠀⣀⠥⠀⠠⢙⠄⠀⠀⠸⠋
⠀⠀⠘⢸⡹⠁⠈⢀⢰⣤⣀⡀⣀⣀⣄⡀⠀⠀⠀⠀⠀⢀⠀⠄⣀⠤⢀⡴⢹⡉⠂⠀⠀⠏⠀⠀⠀⠈
⠀⠀⠀⢸⡇⠀⠀⠁⠈⣿⣿⣿⣿⡾⣿⣿⠟⡂⢶⡶⠂⠀⠀⣨⠈⡤⠋⠀⠈⠀⢀⡖⠠⣆⠁⡀⣔⠀⠐
⠀⠀⠀⢨⡀⠀⠀⠂⠀⣿⣿⣻⡩⢺⣿⡏⡆⠑⣈⡄⢀⠰⠊⠠⠊⠀⠀⠀⠈⠂⠍⣠⣦⡉⠻⣧⣸⡤⣶⠇⠠⠊
⠀⠀⠀⠘⡇⠀⠀⠀⠈⠰⠻⠟⣥⢸⡿⣧⠰⠚⡅⡏⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣧⣿⣷⡈⢯⣇⣽⡿⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠊⡿⡚⠁⢉⠀⠀⠄⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⢟⡇⠻⠉⣷⡄⢽⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⣰⠀⠀⠃⠀⠀⣀⠈⡄⠀⠀⠀⠀⠀⠁⠀⠁⠃⢀⠀⡝⣿⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡊⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠅⠘⡇⠸⠃⠁⠁⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡂⢸⠐⠂⠀⠀⠀⣠⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠰⣊⡠⠬⠓⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⡤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⢀⣴⣿⣿⠿⠛⠉
⠀⢀⣴⡿⠛⠉
⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡤⠶⠒⠚⠛⠛⠒⠒⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠈⢉⣀⡤⠤⠴⠤⣤⣄⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡾⠉⠀⣀⡤⠤⠤⢄⡠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠙⢶⡄⠀⠈⠉⠉⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠉⢠⠔⠋⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣄⣤⣄⣀⡀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⣀⡤⠖⠛⠋⠛⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠈⠉⠳⢦⡀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⠇⠀⡠⠊⠁⠀⠀⠀⠀⠀⠢⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡈⠉⠛⢦⡄
⠀⠀⠀⣠⠖⠋⢀⣼⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠈⠉⠳⣄⠀⠹⣆
⠀⢀⡞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⢰⡄⠀⠀⠀⠀⠈⣧⠀⢹⡄
⠀⡾⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⡆⠘⡇
⠰⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠾⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⡇
⠀⠻⣄⠀⠳⣄⡀⠀⠀⣀⡴⠞⠉⣭⠗⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢀⡞
⠀⠀⠈⠻⠦⣬⠉⠉⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠛⠉⣹⠶⠤⠤⠴⠚⠉⣠⠟
⠀⠀⠀⠀⠀⣿⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡋⠀⠀⣀⣠⠴⠋⠀⠀⠐⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠈⠛⠓⠶⠶⠤⣤⣤⠤⠶⠞⠛⠦⣤⣀⣀⣀⣀⣤⣴⠖⠋⠁⢀⡬⣭⣭⣅⣤⣤⣤⠴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢤⣀⡀⢀⡟⠀⠀⠀⠀⡾⠒⣒⣚⣭⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠚⠋⠁⠀⠀⠈⠉⡿⠁⠀⠀⠀⠘⠛⠋⠉⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣤⠴⠖⠛⠋⡇⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣾⣿⣷⣶⣾⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡏⠈⣿⣟⢻⡆⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⣿⣿⡿⠀⠀⠸⣿⡀⠹⡄⠀⠙⢿⣿⣿⡙⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠉⢻⣷⠀⠙⣿⣧⠘⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠻⣷⣤⣤⣼⣷⣴⠟⠀⠀⠀⠻⣿⣧⣤⣬⣿⣴⡟⢻⣿⣿⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⡀⠀⢸⢿⣿⣿⠿⢻⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⢿⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠘⣿⣿⣧⠀⢸⢸⡇⠹⠀⢸⠀⠀⠀⠀⠀⠿⣿⢸⣿⡇⢼⠀⢀⣿⣿⡿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⣿⣧⢀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⢻⡇⢸⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡀⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠸⠇⣸⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⠿⣿⣦⣄⠀⡶⠟⠛⠛⠛⠛⠳⡦⢀⣠⣾⣿⠟⠁⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣿⣿⣶⣦⣤⣤⣤⣤⣴⣾⣿⡿⠋⠁⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠷⠂





⣿⠿⠿⠿⠿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⡀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⣤⣸⣾⣿⣷⣏⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⢾⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⣶⣶⣤⣀
⠀⣠⣿⣿⣿⣿⣿⣠⡀⠀⠀⠀⢠⠐⠒⠁⠀⠀⠱⠿⡻⡍⠀⠀⠀⠀⠃⠀⣠⣿⡿⠤⡁⠀⠀⠀⠀⡔⠒⠋⠀⠀⠈⠮⠟⣿⣿
⠀⠀⠸⠛⣻⠛⢧⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠤⠃⠀⠀⠀⠀⠐⠲⠿⠟⠠⠤⠤⠬⠆⠀⠀⠀⠣⠤⠤⠤⠤⠤⠤⠜
⠀⠀⠀⠀⠈

⠀⠀⢀⣄⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⣀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣶⣤⣄
⠀⢠⢿⣿⣧⠀⠿⠟⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠼⢦⠿⡼⠗⠐⠻⠓⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠁
⠀⢺⡸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠶⢀⠀⠶⠀⠀⠀⠀⠀⠺⢋⣯⡹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⠀⠂
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠁

⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⡀
⢀⣴⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⡀⠀⠂⣤⠀⡀⠀⡀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠁⣶⢈⢀⠀⣶⢀
⠀⠙⠛⠛⣛⡻⠛⠋⠀⠀⠀⠀⠀⣀⠀⣼⣿⣷⡀⣀⠀⠀⠀⠀⠀⠉⠛⢛⣿⠛⠛⠋⠁⠀⠀⠀⠀⠀⢇⣿⢸⢸⢐⣿⢸
⠀⠀⢀⢸⣓⠯⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣻⠸⣻⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣬⣭⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⣀
⣴⣾⣿⣿⣷⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠜⢀⣝⡁⠘⠋
⠈⢩⠋⠉⠀⠀⠀⠓⣢⣀⡸⠂⠀⠀⠀⠀⠘⠿⣿⣿⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢉⢆⡼⠁⠉⠉⠁
⠀⠀⠐⠀⠀⠀⠒⠒⠋⡃⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁⠀⠀⠀⠀⢠⣴⣷⣶⣿⣷⣾⣿⣤⠀⠀⠀⠀⠀⢠⣤⣾⣧⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠒⠶⢤⣀⣀⣀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢸⠁⠀⢀⡴⠋⠉⠀⠉⠙⢾⡁⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢇⣀⠤⠐⡲⠊⠁⠀⢠⠖⠛⠛⠒⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣸⠁⠀⢠⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡄
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢇⠀⠀⠀⠀⡠⠤⡀⠀⠀⢀⠀⠈⢢⢀⠖⠀⠀⠀⠀⠀⡝
⠀⠀⠀⠀⠀⠀⠘⢦⣀⡔⠁⠀⠀⠀⢸⠁⢄⠀⠀⠀⠀⠹⣄⠀⠀⠀⣴⠁⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠂⡐⠀⠀⠀⣐⠠⣉⡇⢠⢋⡁⠀⠀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠳⢤⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⢹⡍⠁⠀⠫⡈⠀⠐⢤⠜⠁
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠘⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠚⠁⠀⠀⠀⡄
⠀⠀⠀⠀⢀⣀⠇⠀⠀⠀⡀⠈⠓⢧⡀⠀⠀⠀⠀⠀⣠⠞⠀⡴⣻⢟⡳⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠓⢤⣀⠈⠑⡏⠁⠀⠀⢇⠉⠆⠙⠲⣉
⠀⠀⠀⠀⠀⢀⢆⠀⢀⣠⠇⠀⠀⠀⠀⠀⠈⢉⠉⠙⢦⡀⠀⠀⢹⠈⡀⠀⠠⡉⠢⢄⡀⠀⠀⣀⠤⣄⡀
⠀⠀⠀⠀⠀⠁⣿⠀⠀⠁⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠁⡇⠠⠀⠀⠈⠲⣄⢈⣑⠊⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢄⣀⠇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠐⠚⠀⠀⡀⠀⠀⢈⠀⢳⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠃⠀⠀⢀⠀⠀⠀⠀⠀⠠⠜⢀⡼⠆⠀⣀⡤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠈⢆
⠀⠚⠁⠀⠘⠀⠀⠠⠁⡸⠀⠀⠀⠀⠀⢠⢪⠀⠀⠤⠆⠀⠀⢀⡏⠀⠀⠀⠁⠀⠀⠀⠀⠈⠳⠀⠀⠒⢄⠀⡜
⠀⠀⠀⠈⢠⠂⠀⢤⣠⠇⠀⠀⠀⠀⢀⠁⢸⠀⠀⠀⠀⠜⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁
⠀⠀⠀⠢⠞⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠲⠊⠀⠀⠀⠀⢸⠀⠀⠻⢦⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠲⠊⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⢀⡴⠓⠒⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣒⣛⣉⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣀⠤⣴⠒⠊⠉⠙⠩⢛⣱⣄
⠀⠀⠸⣽⠶⠤⠤⠤⠔⠒⠊⠉⠕⢿
⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⠠⣴⠄⢰⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠻⣤⡀⠀⠀⠀⠀⠠⡾⠀⠀⡀⠀⠀⠀⠀⠀⠀⢛⡁⠀⠀⠀⠉⠀⠰⡄⠀⠀⢰⠀⠈⠁⠀⡀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⡊⠁⠀⠀⠀⢀⡤⠊⠀⠀⢠⡏⣞⠤⠒⠐⠂⣀⠉⠁⠀⠀⣀⢔⣠⠤⠀⠀⠀⠀⠀⢀⠀⡎⣠⠒⠀⠀⠴⠂
⠀⠀⠀⢠⡊⠉⠁⠀⠀⣂⠟⠀⠀⠀⠀⠈⢳⣿⡼⠉⠀⠈⠀⠀⠒⠀⢠⣏⣞⡵⠞⠛⠙⠐⠆⠀⠐⡤⣤⡇⠄⠀⠶
⠀⠀⠀⢟⣂⠀⣠⠒⢋⡝⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⡟⠋⠛⠁⠀⠀⠀⠀⠀⠈⣿⠃⠐
⠀⠀⠀⠈⠿⣭⣑⡂⠉⠙⣆⠀⠀⠠⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⢠⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠉⠓⠿⢠⡰⠃⢀⠠⠾⠿⢿⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⣾⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠾⠛⠋⠉⠉⠉⠂⠀⠀⡄⠀⠀⠉⠁⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠠⡤⠤⠤⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢀⣠⠆⣤⡭⠷⠖⠐⠚⠛⠒⠒⠀⠐⠒⠚⠃⢂⡀
⠀⠀⠀⠀⠀⢀⠀⡀⠀⡠⣣⠀⢄⣠⣬⣵⣾⣵⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢤⢰⡀
⠀⠀⠀⠀⢰⣿⣿⣷⣾⣾⣿⣾⠿⠿⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢀⡀
⠀⠀⠀⠀⠀⠛⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠐⠄⡀⠀⠐⣡⠤⣤⣶⡾⡿⢿⠿⠿⠿⣿⣿⢯⣿⣿⣗⣒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠀⢀⣰⣶⢾⡟⣪⣇⣴⣰⣶⣷⣾⣿⣷⣾⣿⣿⣿⣿⣻⠿⣶⣿⡱⢔⡠
⠀⠀⠀⠀⠀⠀⡀⠁⢀⢀⢀⠔⣷⣿⠿⣫⣵⣿⣿⣿⣿⣿⠿⢿⣿⢛⣻⣿⡿⠿⣿⣿⣿⣫⣶⣯⣿⣿⣾⣟⠖⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣩⢯⠟⢯⣿⣿⡟⣿⣿⣿⠇⠰⣆⣻⣷⣿⣿⡟⠋⠑⢨⡝⠻⢿⣿⣿⣿⣟⡏⣻⣿⣷⡀
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣹⠵⢡⡾⠛⠁⢰⣿⣿⣾⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⣸⣷⠆⠈⣟⠻⣿⣿⣿⣿⣬⣍⡿⡎⠁
⠀⠀⠀⠀⠀⠂⠐⠃⣒⣴⢇⣴⣿⠀⠀⠀⢸⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⣿⡗⠲⣿⣿⡟⠀⡞⠀⠘⣛⣿⣿⣿⣿⣷⠋⣀⠄
⠀⠀⠀⠀⠀⠠⠄⠐⢩⣿⣾⣿⣿⡀⠀⡄⠘⣿⣿⣿⣴⣿⣿⢻⣿⣿⣿⣿⣿⣷⣿⣿⠁⣠⡇⠀⠠⣽⣿⣿⣿⣿⣟⢉⠁
⠀⠀⠀⠀⠀⠠⠀⠰⣾⣿⣿⣿⣿⣿⣤⢋⠀⢻⣿⣯⣅⢿⡿⣷⣾⣭⣿⣿⣿⣿⡿⠋⣰⡿⠁⣀⣴⣿⣯⡽⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠛⠿⣿⣧⣕⣄⠀⠙⢿⣿⣿⣷⣿⡛⢿⣿⣿⣿⣹⣵⣿⣋⣴⣾⣿⣿⠿⠍⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣂⣈⣹⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡟⢛⠹⠂⠀⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠉⠉⠹⡿⠖⠈⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⢁⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⢠⡝⠐⠞⠋⠄⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣋⠠⠄
⠀⠀⠀⠊⠀⠂⠐⠢⠀⠀⣿⡅⠀⠀⠠⡀⠀⠀⠩⠄⠁⠉⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡳⠤⠄
⠀⠈⣉⠁⢀⠶⠤⢠⡀⠀⢉⠁⡄⠀⠀⠍⠀⢠⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣙⠀⠀⡶⠈⠀⠄⠀⢠⡜⠁
⠀⠀⡄⡀⢀⠀⡶⠀⢠⡄⠀⢰⠀⠀⡀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡐⠀⢠⠁⠀⠀⠠⣽⣇⠲⢀⠁
⠀⠀⡿⠄⠈⠳⠀⠀⣿⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⡇⠀⠁⢸⠀⠀⠀⡰⠑⠌
⠀⠀⢰⠀⠈⠀⠀⢰⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⢰⠃
⠀⠀⠸⡄⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸⠄⢀⣏⣀⣠⣤⣤⡀
⣀⣀⡀⣧⢀⣀⡀⣾⡯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠃⠈⠛⠛⠉⢉⣉⠉⠉
⠀⣞⣆⣿⣗⡙⠉⣿⡅⠈⠁⠀⢀⠀⠉⠛⠋⠉⠀⠀⠈⠉⣤⠤⢍⣀⠀⠀⠀⠁⠀⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐
⣾⣼⣿⣿⣿⣷⣿⣿⠃⣶⣦⢀⠈⠀⠠⠄⠐⡖⣦⣴⣤⣀⣀⣴⣿⣿⣶⣆⡠⣴⣦⣴⣿⡷⠒
⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠉
⣿⠿⢿⣿⠟⠛⣻⡟⠀⠉⠉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠠⠿⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣥⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⢤⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⢽⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⠽⠋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⢝⢦⡀⠀⢀⡤⡺⠛⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢱⣽⣿⡵⠋⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⢔⡫⠚⠛⢮⡢⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣠⣾⠗⠉⠀⠀⠀⠀⠙⢮⣢⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣯⣄⡉⠁⠀⠀⠀⠀⠀⠀⠙⠪⡲⣄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠨⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠈⠪⡳⣄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣈⠪⡳⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⣇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠜⡿⡟⢙⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣤⣶⣵⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠚
⠀⠀⠀⠀⠀⠀⢰⡿⣻⣯⣵⣿⣿⡏⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣠⣃⣤
⠀⠀⠀⠀⠀⠀⠸⣿⠻⣿⡿⣿⣿⢃⣾⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠏⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡾⠿⠿⠿⠊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⣶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠤⠴⠃⠈⠓⢢⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢁⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡟⡗⠚⡇⠀⠀⠀⠀⣠⡶⠶⠂⠐⣧⢒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⢄⣏⠸⣇⣇⣀⣙⣲⡄⢸⣾⢟⣶⣄⣀⠚⢉⣀⣀⣀⣀⣀
⠀⣁⡀⠤⠒⠚⠃⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠀⠸⠖⠠⢀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠉⠉⠉⠉⠑⠢⣄⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⠉⠀⠀⠀⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠔⠒⠒⢤⡀
⠀⠀⠀⠀⠀⢀⡤⠖⠚⠒⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢹⠴⠶⠤⣄⡀
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣀⠼⠗⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⣄⡀
⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠴⠒⠲⢤⡀⠀⠀⠀⠀⠀⠙
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠱⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠙⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⠞⠁
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⣠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠓⠒⠚⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠋⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⠦⣀⣀⣀⣤⡀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠲⢤⣀⣀⣀⣠⢤⡀⠀⠀⢀⡴⠓⠒⠚⣉⠀⠀⠀⠀⠀⠀⢠⡆⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⢠⢿⠀⠀⠀⠀⠀⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⣏⡼⠀⠀⢸⢧⡀⠘⣇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣶⠀⠀⠀⠀⢠⡏⢣⠀⠀⠀⠈⠀⠀⠀⠸⣄⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡾⡀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡇⠀⠀⠀⠈⠳⠞⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡞⠀⡇⠀⠀⠀⠀⡀⠀⠀⠈⠶⠃⠀⠀⣄⠀⠀⠀⠀⠀⢰⠉⡆⠀⠀⠀⠀⠀⣰⢻⡀
⠀⠀⠀⠀⠀⠀⠙⠞⠁⠀⠀⢀⡴⢻⠀⠀⠀⠀⠀⠀⢠⠋⡄⠀⠀⠀⠀⠘⠚⠁⠀⠀⠀⠀⠀⢇⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡼⠁⠀⠀⠀⠀⠀⠘⠖⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠾⡄⠀⠀⠀⠀⠀⠀⢠⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⡞⢱⠀⠀⠀⠀⠀⠸⣤⠇⠀⠀⠀⣴⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⡜⠹⡄⠀⠀⠀⠀⠀⠈⢧⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⡄⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⣄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠙⠴⠃⠀⠀⠀⠀⠀⢸⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⢰⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⢤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣆⠀⠀⠀⠀⠀⠸⣀⡼⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡄⠀⠀⠀⠈⣇⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⡝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣸⠉⢹⠇⠀⠀⠀⠀⠀⣠⠛⢲⠆⠀⠀⠀⠀⠀⢀⡼⠉⣩⠟⠀⠀⠀⡤⠦⠤⣤⠀⠀⠀⠀⠀⡰⢡⠏⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⢠⡇
⠀⢀⡇⠀⣯⠀⠀⠀⠀⠀⡴⢁⡴⠃⠀⠀⠀⠀⠀⣰⠋⠀⣾⠆⠀⠀⠀⣰⠃⠀⣰⠃⠀⠀⠀⠀⡰⠁⢹⠏⠀⠀⠀⠀⠀⠀⡴⢻⣁⠀⠀⠀⠀⢸⢇⣼⠃
⠀⠘⡦⢠⡎⠀⠀⠀⠀⡾⠤⠬⢿⢯⠟⠁⠀⠀⣠⠋⢐⡿⠂⠀⠀⠀⢠⠇⠀⢘⡷⠀⠀⠀⠀⠐⡿⠱⡷⠀⠀⠀⠀⠀⠀⢾⢀⡴⠁⠀⠀⠀⠀⢠⣌⢹
⠀⠠⡇⡸⠃⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⣠⠟⢰⡿⠀⠀⠀⠀⠀⢈⠇⡴⠋⠀⠀⠀⠀⠀⠘⡟⣾⠁⠀⠀⠀⠀⠀⢠⣯⠞⠀⠀⠀⠀⠀⠀⣸⠙⡇
⠀⢠⡟⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣿⠶⠋⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠘⣷⠃⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣸⢰⠃⣀⠀⠀⠀⠀⢰⠃⠀⡴⠃⠀⠀⠀⠀⠀⣰⠃⠸⢥⡄⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢰⢒⠆
⠀⢀⡞⠹⢦⠀⠀⠀⠀⠀⡏⠘⠋⡟⠀⠀⠀⢀⣏⣀⠘⢛⡗⠀⠀⠀⠀⠘⣳⢖⣴⡋⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⣼⠁⡷⠀⠀⠀⠀⠀⢀⠇⠸⡄
⠀⠘⡗⢠⡏⠀⠀⠀⠀⢰⣃⢴⠀⢹⠃⠀⠀⠀⢁⡏⣀⠘⣲⠂⠀⠀⠀⠾⣥⠄⡽⠃⠀⠀⠀⠀⣴⡃⢩⠇⠀⠀⠀⠀⠀⡟⠐⣷⠀⠀⠀⠀⠀⠘⢳⢰⠁
⠀⠸⣥⣠⠋⠀⠀⠀⠀⠈⠁⠾⢶⡜⠀⠀⠀⠀⠈⠉⡝⡰⠁⠀⠀⠀⢀⣔⠁⢾⠄⠀⠀⠀⠀⠀⢹⡷⠃⠀⠀⠀⠀⠀⠀⡽⠰⡇⠀⠀⠀⠀⠀⠀⣧⡽⠂
⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⢉⣟⡴⠃⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠶⢤⡴⠞⠋⠉⠁⠀⠀⠉⠙⠳⣦⣠⡴⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⢴⣿⡁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⣤⣴⠒⠂⠀⢸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣧⠴⠤⠤⠄⠀⠀⠀⠀⢀⣠⡤⠶⢤⣶⡋⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠷⠀⠀⠉⠉⠉⠙⠳⣦⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⠴⠲⢯⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢉⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⣿⣿⢿⣏⠁⠈⠛⠀⠀⠀⠀⠒⠚⠛⣷
⠀⠀⠀⠀⠀⠀⠙⢳⣦⣤⣤⠄⠀⠀⠀⠀⠀⠀⣿⠁⠈⢻⣿⣿⣷⣄⣤⡄⠀⠀⠀⣤⣄⣴⣿⣿⣿⠋⠀⢹⡆⠀⠀⠀⠀⠀⠀⠻⣷⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢺⣿⣿⣿⣿⡿⠁⠀⠀⠀⠹⣿⣿⣿⣽⣿⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⢿⣦⣀⣀⣈⣩⣴⠋⠀⠀⠀⠀⠀⠀⠈⢷⣭⣉⣀⣀⣠⣾⠇⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⣀⠀⠀⠀⠀⠀⡀⠈⠙⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠈⠉⠛⢻⠟⠻⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠛⠛⠛⠛⣿⠁
⠀⠀⠀⠀⣠⣴⣯⡁⠻⠶⢤⡀⠀⠀⣠⠏⠀⠀⠈⢻⡶⠶⣤⣤⣤⣤⣤⡾⢿⡷⠶⠶⠶⣦⡴⠶⠶⠶⠛⠉⠘⣆⠀⠀⠀⠻⠶⠛⢻⡇
⠀⠀⠀⠀⢀⡴⢋⣠⡤⠒⠁⠀⠀⣴⠋⠀⠀⠀⠀⠿⠦⢤⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⢀⣀⣀⡀⠀⠀⠀⠸⣆⠀⠀⣀⠀⠀⣸⡇
⠀⠀⢠⣾⡵⠚⠉⠀⠀⠀⠀⢀⣾⣁⣤⢤⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢸⡇⠀⠀⠰⠟⠋⠉⣹⠃⠀⠀⠀⠀⠹⡶⠟⢻⡀⠀⣿⠁
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡞⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⢠⡏⠀⠀⢰⡄⠀⠀⠀⠀⠘⣇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡴⠞⠁⠀⠀⠀⠀⡄⠀⠀⠀⢾⣥⠶⠚⣿⠀⠀⣼⠀⠀⠀⢸⠙⠛⣹⠃⠀⠀⢻⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣀⡴⠋⠀⠀⠀⠀⠀⢠⠞⠷⢤⡄⠀⠀⠀⠀⠀⢿⠀⢰⠇⠀⠀⠀⠾⢲⢰⠇⠀⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠞⠉⠀⠀⠀⠀⠀⠀⢴⠗⡶⢠⠞⠀⠀⠀⠀⠀⠀⢸⠀⡾⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠃⠀⠀⠀⠀⠀⠀⠀⢸⢠⠇⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣤⣤⣤⣀⣠⣤⣤⡀⠀⠀⠀⣿⣿⣿⣿⠟⠛⠛⠿⠟⠛⠛⢿⣿⣿⣿⡇⠀⠀⠀⡠⠤⠤⢄⡠⠤⠤⣄
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠀⣠⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠑⢄⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⣀⠀⠀⠠⡄⠀⠀⠀⠀⠀⡀⠰⡆⠀⠀⠀⣤⠀⢰⠄⠀⢀⠀⢰⡄⠀⠀⢀⠀⠀⠀⠀⠦⠀⠀⠀⢀⠀⠀⠀⣀
⠼⢀⡄⠈⠃⠀⠘⡇⠀⠀⣧⠀⠀⠀⡄⠀⠛⠀⠀⠀⠀⠿⠀⠈⠁⢀⠀⢽⠀⠀⠀⠀⠚⠀⢀⡀⠸⠆⠀⠀⣽
⠀⠘⠗⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠨⠆⠀⠀⠀⢶⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⢠⡄
⢀⠀⠀⠀⠀⠛⠀⠀⠀⡧⠀⠀⢀⠀⠀⣠⠀⠈⠆⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡀⠀⣆⠀⢘⡄⠀⠘⠖
⢸⠀⠀⢼⡄⠀⢀⠀⠀⢀⠀⠀⢨⡆⠀⠸⠀⠀⠀⠀⢸⡇⠀⠋⠀⠀⢸⡅⠀⠀⠀⢀⠀⠀⠸⣥⠀⠁⠀⠈⠁⠀⠀⣆
⠀⠀⠀⠈⠀⠀⢻⠆⠀⠘⠇⠀⠀⣀⣀⡀⠤⢀⠲⠒⠺⢧⠤⠄⣀⣀⣀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⢀⡄⠀⢠⠀⠉
⠀⠀⣷⠀⢾⠀⠀⠀⣀⠔⢂⡩⡾⢝⢉⡒⢕⣊⠅⠀⠀⢨⡆⢅⠂⢀⠈⠉⠛⠳⢖⡒⠠⠤⣀⡀⠀⠀⠸⣾⠀⢪⠄
⢀⠀⠙⠀⠀⣠⡐⠙⠀⡠⢺⠭⡢⢋⠥⡂⠅⠂⠀⠀⠀⠀⠘⡀⢇⠂⠈⢀⠀⠀⠀⠈⠳⣕⠆⠈⢉⠢⢄⡀⠀⠀⣤
⠸⠆⢀⡔⡫⠔⠁⠀⡰⠁⢑⠥⡪⢔⠨⢒⠁⠀⠀⠀⠀⠀⠀⠃⢑⠄⠀⠀⠢⠀⠀⠀⠀⠈⢧⡢⢀⠠⠀⠈⠢⡀⠙⠁
⠀⡠⢧⣀⡄⠀⠀⢰⣁⢔⢕⣪⡲⠑⠍⡀⠄⠀⠀⠀⠀⠀⠀⢸⠴⠊⡀⠠⠀⠁⠀⠀⠀⠀⠂⢳⣕⡠⢁⡀⠀⢜⢆
⣼⠖⠓⠓⠒⠧⢀⡨⢔⢅⣕⠵⠮⠖⠒⠖⠤⠦⣤⣠⣀⠀⠀⢐⠨⡐⠠⣘⣪⣥⣀⣂⠤⠀⣀⠄⢳⣊⣔⠤⠙⠒⠺⡆
⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠓⠦⣘⡵⠟⠋⠁⠀⠀⠀⠀⠀⠉⠈⠉⠺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⡇⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣇⠀⢀⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣉⣉⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠐⠒⠄⠢⠀⠄⡪⠐⡡⢂⠅⡪⢀⠔⡠⡂⢔⠔⡠⢂⢔⠔⡠⡢⢀⠀⡢⢀⢄⠀⡀⡀⡀⢀⠄⡀⡀⢀⢀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡠⠀⣀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⡶⠁⠉⠉⠀⠐⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠎⠅⠐
⠀⢀⠌⠀⠀⠀⠉⠀⠀⢸⠷⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠝⠀⠀⠀⠀⢀
⠀⠀⠆⠀⠀⢉⡁⠐⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠟⠀⠀⠀⣀⠔
⠀⠀⣆⠂⠠⣈⠁⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⠀⢰⠀⠀⠀⢸⠞⠄⠁
⠀⠀⠋⠀⠀⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠀⢸⠀⠀⢠⠇⠈
⠀⠀⢰⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⡎
⠀⠀⠘⡄⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠸⠤⠼⠳⠶⠚⠛⠛
⠀⡩⡉⣷⡏⠷⢶⣿⠀⠒⠀⠒⠚⠛⠶⠾⠓⠒⠛⠓⠒⠚⣒⠚⠐⠒⠒⠚⠒⠒⣛⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⡴⣱⣶⣿⣶⣤⣴⣿⢢⡤⡀⠐⠂⠀⡀⠀⡀⣄⣀⣀⠀⠉⣠⣾⣯⣠⡀⣀⣤⣀⣤⣦⠤⠂
⣗⣿⣿⣿⣿⣿⣿⡗⢰⣯⣴⣧⠄⠀⠀⠀⠁⠈⠉⠈⠙⠛⠛⠛⠛⠟⠛⠨⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒
⣿⠿⣿⣿⠟⠛⣿⠇⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠁⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠈⢳⣤⣤⣤⣤⣤⣄⣴⠾⠿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠀⠙⢿⣄⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠖⠂⠀⣠⣴⠶⠿⠿⠿⠿⠿⠿⢿⣷⣦⣄⣀⣼⠟⠋⠉⠀⠀⠈⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣧⣄⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⠀⠀⠀⠉⢻⡟⢁⣤⠶⣦⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⣾⣡⣦⠘⡧⠀⠀⢀⡾⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⡼⠿⣿⣇⠀⠀⣸⣵⣶⡀⣸⠃⣀⣀⣀⣀⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣠⡶⠶⠶⠶⠶⣤⣿⠿⠛⠳⠶⠤⠀⠉⠙⠻⢏⣰⠿⠋⠉⠉⠀⠈⠉⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡿⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⢀⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢋⡽⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡶⠋⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠶⠦⠤⠤⠤⣤⣤⡿⢱⣶⣶⡄⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⣆⠀⠀⠀⢀⣀⡤⠶⣻⡽⣿⣿⡇⣴⡶⠶⠿⠛⠋⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡬⠿⣷⣖⣋⣩⡴⠖⣋⣡⢟⡶⡗⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠒⠒⠒⠚⠛⠋⠉⠉⢿⣆⠀⣀⡬⠟⠛⠛⠟⣻⠟⢁⣞⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠿⣧⣤⣀⣀⢀⣼⣁⣀⡼⠛⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠙⡻⠋⠛⠉⢹⠇⠀⠀⠘⢿⢿⣷⣶⣤⣤⣤⣶⠟⠋⠉⠛⠿⢿⣷⣶⣦⣤⣤⣤⣤⣶⡾⠟⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣾⠀⠀⠀⠀⠈⢣⡈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢦⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠛⠳⢦⣀
⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡾⠛⠉⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⢀⣤⠿⠶⠶⢶⠏⠀⠀⠀⠀⠀⠀⠹⡷⠶⠶⠶⣤⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠋⠉⠉⠉⠙⢷⡄⠀⠀⠀⠀⠹⣆
⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠻⠞⠛⠛⠷⣦⣿⡀
⠀⣠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠉⠛⢛⡛⠛⠛⠛⠛⠛⠛⠛⠻⣟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⣠⣴⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡴⠋
⢀⡾⠋⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣍⠉⠁
⠘⣷⣀⣀⣼⠃⠀⠀⠀⠀⠀⣴⠟⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⣠⡴⠟⢻⡄
⠀⠈⠉⠉⠁⠀⠀⠀⣀⠀⠸⣏⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⣿⠀⠀⣾⠋⠀⠀⣾⠁
⠀⠀⠀⠀⠀⢀⣤⡾⢻⡇⠀⠙⠛⠛⠋⠀⠀⠀⢀⣴⣦⠀⢸⡏⠀⠀⣸⡏⠀⠀⠻⣤⣤⣼⠏
⠀⠀⠀⠀⣰⡟⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⣿⠀⠈⠻⠶⠾⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠹⣦⣀⣠⡿⠁⠀⠀⠀⠀⠀⠰⣟⠀⠀⢠⡿

⠀⠀⠀⠀⠀⣴⣿⣿⠇⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⠀⣾⣿⣿⡗
⠀⠀⠀⠀⢰⣿⣿⠏⠀⣀⡀⠀⢀⣾⣿⣁⣠⣤⣤⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⣾⣿⣿⣾⣿⠏⠀⠀⡼⠿⠟⠛⣻⡿⠁⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⢀⣿⣿⣿⣷⡞⠀⠀⠀⣸⣿⣿⠃⠀⣀⡀
⠀⠀⠀⠸⠛⠋⣹⣿⠏⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⢀⣼⣿⣿⣷⣶⠖⠀⠀⠘⢉⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣾⣿⠏
⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⢀⣾⣃⣤⡤⣤⠀⠀⠛⠋⢹⣿⣿⠋⠀⠀⠀⠀⣾⣿⣿⡤⠀⠀⠀⢸⡿⠟⢩⡿⠃
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡴⠁⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠘⠉⣿⠟⠁⠀⠀⠀⠈⠀⢠⡟⠁
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠳⣤⣤⣤⡤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⡴⠋⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⣴⠃⠀⠀⠻⣤⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢾⡷⠶⠖⠛⠛⣛⣃⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⣚⡛⠛⠓⠲⣾⠷⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠿⣶⠞⠛⢋⣩⡅⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣭⣉⡛⢛⣶⠿⢦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠷⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠛⠷⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠈⠙⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣦⣄⡀⠀⠀⠀⢀⣠⡶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠉⠙⠛⠋⠉⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣠⠔⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢠⠞⡱⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢀⡔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢸⡇⠀⣄⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢻⡄
⠀⢀⠎⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠈⢷⡀
⢀⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠘⣧
⡜⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⢸⠇⢸⡇⠀⠀⠀⠀⠀⠀⢹⡄
⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣿⠀⠀⠀⣾⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠱⡀⠳⢄⣀⣀⣀⣀⠤⠤⠤⠤⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠙⠢⠤⠤⠤⠤⠐⠒⠒⠒⢤⡈⢳⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠳⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⡏⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣈⠙⢿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢰⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣇⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣹⣆⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇⠀⡀⢸⡇⠀⠀⠀⢀⣼⣁⣀⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⠈⠉⠉⠉⠛⢦⣤⠶⠶⣧⠴⠒⠛⠲⣤⣸⠟⠋⠉⠙⢳⡞⠛⢶⠋⠉⠉⠉⠉⠁⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣀⣀⣀⣀⣀⣀⢰⡇⠀⢰⠃⠀⠀⠀⠀⠈⣯⠀⠀⠀⠀⠀⣿⣀⡾⠀⣀⣀⣀⣀⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠒⠚⢧⣀⠀⠀⣀⡴⠿⠦⣤⣠⡤⠞⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠁⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠐⠒⠁⠂⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠈⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠐⠀⠀⠀⠀⢈⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⢀⡠⠰⢀⠤⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⣠⠂⠁⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢌⠀⢠⠃⢜⡂⠀⠀⠠⠀⣠⠖⠊⠀⠂⠀⡀⠈⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠉⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⢀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐
⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⠷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠿⠿⠿⢷⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠛⠻⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠙⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣤⣄⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⣠⣾⠿⠛⠉⠉⠉⠙⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠉⠛⠿⣶⡄⢀⣤⣶⣶⠶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠀⠀⠀⠀⠀⠙⠻⣷⡀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⠶⠶⣶⣤⣄⠸⣿
⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠻⣿⣿
⠀⣠⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷
⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠙⠻⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠟⠉
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⢀⣠⣾⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠟
⠀⢠⣾⡿⠋⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁
⠀⣿⡏⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷
⠀⢿⣧⣀⣀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣷⡀⠀⠀⠀⠀⣀⣴⣾⠟⠉⣿⡆
⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⣿⡇⠀⠀⢀⣾⡿⠋⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡆⠀⠈⢿⣦⣄⣠⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢹⣿⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿⡆⠀⢸⣿⡁⠀⠀⠀⣸⣿⠁⠀⠀⠈⠻⣷⣤⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁⣿⡇⠀⠀⠻⣷⣦⣤⣾⠿⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣴⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⡾⠟⠁

⠀⠀⠀⠀⠀⠀⣟⡆⠀⠀⠀⠀⠀⣸⣧⠀⢀⣿⡇⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⡇⠸⡆⠀⠀⣀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⣀⠀⠀⠀⠀⠀⠉
⠀⡠⡄⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠻⠟⠀⠀⠉⠀⠀⠀⢸⢦⠀⠀⠀⠀⠀⠀⠙⠤⠃⠀⢰⣿⡆⠀⠛⠀⠀⠀⠀⠀⠀⢿⡷⠀⢠⠛⡆⠀⠀⠀⢀⡆
⠰⣧⠇⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠳⠼⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⡜⡄⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⣆⠀⠀⠿⠇⠀⢀⡄⠀⢸⠁⡽⠀⠀⠈⠛⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠶⡄⠀⠀⠀⣿⡄⠀⠀⢀⠀⠀⠘⣿⡇⠀⠀⢰⣷⠀⠀⣰⡄
⠀⢾⡿⠀⠀⠀⠀⠀⣾⣷⠀⠘⠞⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⡄⠀⠈⠳⠃⠀⠀⠈⠿⠇⠀⢀⡞⡆⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⢿⡷
⡄⠀⠀⢀⡀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⣶⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣸⠀⡟⠀⠀⠀⣿⠀⠀⠀⢠⡀⠀⠀⠀⠀⣼
⡟⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⢰⣷⠀⠀⠀⢸⣿⡆⠀⠘⠚⠁⠀⢀⡀⠁⠀⠀⠀⢮⡳⠀⠀⠀⠀⠸
⠀⠀⠘⠿⠟⠀⠀⢠⡀⠀⠀⣸⣧⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⡞⢆⠀⠀⠀⠀⠘⠟⠀⠀⠀⠘⠻⠃⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⡏⣇⠀⠀⠛⠋⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⣇⢸⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⢠⡀⠀⠀⠀⠀⠸⠟
⣶⡀⠀⢀⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⡎⢣⠀⠀⠀⢠⠀⠀⠈⠛⠀⠀⠀⡞⢡⠀⠀⠀⣼⡀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠸⠇⠀⢠⡆
⠿⠇⠀⢸⣿⠀⠀⣄⠀⠀⠀⠀⣴⡄⠀⠙⠋⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠧⠜⠀⠀⠰⣿⠇⠀⠀⠈⠉⠀⠀⠀⣰⣄⠀⠀⠀⠀⠈⠁⠀⠀⡀
⠀⠀⠀⠈⠉⠀⢰⠉⡆⠀⠀⠀⣿⠇⠀⠀⠀⢠⡀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⣦⠀⠀⠸⣄⣹⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⡀⠀⠀⢼⢹⠀⠀⣼⡆⠀⠀⠀⢀⠜⡄⠀⠀⠀⡇⡳⠀⠀⠀⠀⢠⢶⠀⠀⠀⠀⠈⠋⠀⢀
⠃⠀⠀⠿⠇⠀⠀⠈⠁⠀⠀⠀⡔⣇⠀⠀⠀⡄⠀⠀⢰⣿⠀⠀⠈⠉⠀⠀⠻⠏⠀⠀⠀⣮⠀⡇⠀⠀⠀⠉⠁⠀⠀⠀⢠⡇⢸⠀⠀⠀⢀⠀⠀⠀⠸
⠀⠀⢀⠀⠀⠀⠀⣰⡆⠀⠀⣼⠀⡇⠀⠀⣰⣿⡀⠀⠈⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⢀⣦⠀⠀⠀⠀⡀⠀⠈⠧⠞⠀⠀⢰⢡⡄
⠀⢀⠟⡆⠀⠀⠀⠿⠇⠀⠀⠘⠚⠁⠀⠀⢻⣿⡇⠀⢀⡄⠀⠀⣸⣷⠀⠀⢰⡆⠀⢰⡀⠀⠀⠀⠸⠿⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢻⡴⠇⠀⣠
⠀⣾⢀⡇⠀⢠⡀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠉⠀⢀⠇⡇⠀⠀⠘⠛⠀⠀⠘⠁⠀⡎⢹⡀⠀⠀⠀⠀⠀⣀⠀⠈⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣿⣧
⠀⠈⠋⠀⠀⢿⡇⠀⠀⠀⢸⡇⠀⠀⣾⡆⠀⠀⠀⠘⠶⠋⠀⠀⢠⠀⠀⠀⠀⠀⠀⢳⣈⡇⠀⣰⡆⠀⠐⡿⠀⠀⠀⠀⠀⠀⡎⣇⠀⢀⣷⡀⠈⠻⠟
⠀⣤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠋⠁⠀⢠⡀⠀⠀⠀⠀⢀⣿⡆⠀⠀⢠⡀⠀⠀⠉⠀⢠⣿⣷⠀⠀⠀⠀⢀⠴⡀⠀⠀⠙⠃⠀⠘⠿⠇⠀⢀
⢸⣿⠆⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⣤⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠛⠁⠀⠀⢯⠇⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠸⣼⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⡞⡆
⠀⠁⠀⠀⢸⢦⠀⠀⠀⠀⠀⠀⢼⣸⠀⠀⠀⠈⠁⢠⣾⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⢠⣷⠀⠀⣰⣿⠀⠀⠀⠳⠃
⠀⣀⠀⠀⠻⠜⠀⠀⢀⡼⡀⠀⠈⠁⠀⢀⣤⠀⠀⠘⠟⠀⠀⣠⠀⠸⠏⠀⠀⠀⠀⠀⠸⣿⠆⠀⣤⠀⠘⠿⠁⠀⠀⠀⠘⠿⠁⠀⣿⣿⠇⠀⠀⣀
⢰⣿⡆⠀⠀⠀⠀⠀⡞⠀⡇⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⢠⡇⢣⠀⠀⠀⢀⣾⡆⠀⠀⠀⠀⢸⣸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢸⣿
⠈⠛⠁⠀⡀⠀⠀⠀⠹⠴⠁⠀⠀⢀⡀⠀⠁⠀⡜⡇⠀⠀⠈⢧⡸⠂⠀⠀⠈⠿⠃⠀⣠⡄⠀⠀⠉⠀⠀⣾⣿⠀⠀⢀⢼⡄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⢸⠁⡼⠀⠀⡄⠀⠀⠀⠀⣶⡄⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠈⠁⠀⠀⡎⢀⡇⠀⠀⡤⠀⠀⣾⣇
⠀⠀⢸⣿⡟⠀⠀⠀⠀⣿⡄⠀⠀⠻⠿⠃⠀⠈⠚⠁⠀⢸⣷⠀⠀⠀⠀⠻⠃⠀⠀⠀⠈⠀⠀⣰⡆⠀⠀⠀⣄⠀⠀⠹⠞⠁⠀⣸⠘⡇⠀⠙⠋⠀⡰
⠀⠀⠀⠉⠀⠀⢀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⢠⡄⠀⠀⠀⠸⢆⠀⠀⠀⠀⠷⠇⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⢰⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠦⣤⠴⠚⠉⠉⠉⠉⠉⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⢀⣤⠴⠒⠒⠛⠓⠶⢤⣀⠀⠀⠸⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠴⠋⢀⡤⠖⠚⠉⢓⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⣀⣠⠼⣥⣀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣷⣶⣶⡲⠦⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⢰⠃⠀⣠⠖⠋⠁⠀⣀⣀⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⠿⢿⣿⡷⠀⠈⠻⣆⠀⠻⠤⣄
⠀⠀⠀⢀⡤⠊⠀⣞⡀⠀⠀⢰⡿⠿⡿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠉⠙⢦⠀⠀⠙⠁⢀⡤⠒⠛⠒⠢⣄⠀⠹⣄
⠀⢀⡴⠁⢀⡴⠋⠁⠀⠀⠀⠀⠁⣼⠁⠀⢸⣟⣿⣦⣀⣴⢷⣄⣠⣾⡿⢻⣿⠄⠀⠈⡇⠀⠀⣰⠁⢠⡀⠀⠀⠀⠈⢧⠀⢹⡄
⠀⡞⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⡀⠈⠛⢛⣻⡏⠁⠈⠉⢻⣍⠛⠛⠋⠀⢀⣼⡗⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠈⡇⠈⡇
⠰⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠛⠻⠛⠉⠀⠀⠀⠀⠀⠙⢝⠓⠲⠞⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⡇
⠀⠳⣄⠈⢧⣀⡀⠀⣀⡤⠞⠋⠭⠇⠀⢀⣤⠶⠿⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡼⠁⡼⠁
⠀⠀⠈⠳⢶⡌⠉⢻⡁⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠙⠿⣷⣤⣠⣽⣦⠀⠀⢀⡆⠀⠀⠙⠉⣽⠲⠶⠶⠒⠋⣠⠞⠁
⠀⠀⠀⠀⠘⣇⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠐⠋⠲⣄⠀⠀⠀⠀⠈⠉⠉⠁⢹⣀⠴⢎⣀⣀⣠⠤⠞⠁⠀⠀⣹⠞⠛⠁
⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠈⠉⠛⠛⠒⠒⠒⠒⠛⠉⠙⠲⣤⠤⠤⠤⠶⣞⠉⢀⣠⠴⠲⠶⠶⠶⠶⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⡀⠀⠀⢀⣀⡴⠞⠛⠶⢤⣤⡏⠀⠀⠀⢰⣏⣭⠭⠶⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣀⣠⡤⠶⡆⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠀⠀⠀⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣠⣤⣾⠿⠿⢿⣿⣭⣤⣤⣄⣀⡀⠀⠰⠆
⢿⣠⣴⠶⣶⡟⠁⡈⠃⠀⠀⠀⣠⣤⣤⣤⣀⣉⡉⠛⠛⠶⠶⢦⣤⣄⣀⠐⠲⢤⣤⡀
⢸⡿⣤⡀⠈⠻⣤⡀⠀⠀⠀⠋⠀⠀⠀⠀⣤⣉⣉⣙⡛⠲⠶⣤⣤⣈⡉⠛⠲⠶⣤⣌⣉⠒⠂
⠸⣇⠈⠛⢦⣄⡈⠙⠳⢶⣤⣄⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠶⢤⣍⠙⠳⠶⣦⣤⣈⡉⠙⠶⢦⣤⣀⡀⠀⠀⢤⡀
⠀⠙⠳⣄⡐⠈⠙⠲⣦⣄⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠉⠙⠳⢶⣤⣄⣉⠙⠶⣤⣄⡈⠁
⢀⡀⢠⠜⠉⠒⠦⣄⣒⡩⠝⠛⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣈⡙⠲⣤⣀⠀⠀⠀⠀⢀⡀
⠈⠙⢷⣄⣀⠀⠀⠀⠉⠑⠚⠡⠄⢀⠉⠉⠛⠒⠤⠄⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⣀⠀⠀⠀⠤⢀⣀⡤⠀⠀⣀⣉⠙⠷⣤⡉⠓⠤⡀⢢⠈⠁
⠀⠀⠘⢷⡉⠛⠂⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠦⣤⣀⣉⡐⠛⠻⠥⢤⣤⣀⣈⣿⠆⠀⠀⠈⣷
⠀⠀⠀⣀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠶⠤⠤⠤⠤⠤⠤⠤⠤⠛⣦
⠀⠀⠠⠀⢀⡤⠈⠙⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⣰⠏
⠀⠀⠸⣦⡞⠁⠠⢀⡀⠀⠀⠉⠉⠛⠛⠳⠶⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠉⠛⠛⠓⠲⠶⠶⣶⠖⠛⠋⠁
⠀⠀⠀⠈⢿⡶⣤⣄⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠛⠛⠓⠶⠶⠶⠶⠶⣾⠶⠞⠃
⠀⠀⠀⠀⠀⠻⣦⡈⠙⠛⠲⠶⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠶⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠞⠁
⠀⠀⠀⢉⠀⠀⠀⠙⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣤
⠀⠀⠠⣿⠀⠠⣇⡠⠆⠀⠉⠙⠳⠶⢦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⢛⣻⡿⠋
⠀⠀⠀⠻⣦⡀⠈⢿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⠶⠶⠶⠶⠾⠛⢋⣹⡖
⠀⠀⠀⠀⠈⠛⠶⣼⣆⠈⠉⠛⠛⠶⠶⢤⡤⢀⠀⠈⠉⠀⠀⠐⠒⠂⠒⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠞⢋⣽⢧⣄⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣖⠲⠦⣤⣤⣀⣀⣀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠶⠶⠾⠛⠋⢉⣀⣤⠶⠋⠁⠀⣨⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⣀⡈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠒⠊⠉⠀⠀⢀⣤⠞⠋⣸⣇⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡙⠙⠛⠻⠶⠶⠶⠦⠤⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠋⢁⣠⠾⠋⠉⠛⣦⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⣤⣤⣄⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠀⠐⠂⠀⠀⠀⡀⠠⠄⠀⠠⠔⠋⠁⠀⠀⣠⡴⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣍⣉⣉⠉⠉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠴⠛⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡉⠙⠛⢛⠟⠛⠛⠛⣛⡉⠀⠀⠀⠀⣀⠢⠤⠚⠀⢀⠐⠂⢀⡠⠾⠿⣷⠶⢶⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⢤⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⣠⠶⠛⠁⢀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠷⣦⣤⣤⣤⠴⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠛⠉⠁⠀⣠⣾⣷⡶⣦⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢷⣤⣄⣀⡀⠐⠲⠆⠀⠈⠀⠀⠀⠀⠀⣠⡴⠞⠁⠀⣀⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⡿⠛⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⢀⣤⠶⠞⢋⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⢾⣤⣤⠤⠔⠒⠂⠀⠀⠀⢀⠀⠈⠁⢀⣤⣴⡟⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⡤⠄⠀⠀⠀⡶⠀⠀⠀⠈⠉⠀⠈⠻⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠐⠀⠒⠂⢀⣀⣠⣤⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣨⡿⠷⠖⠂⠀⠀⠀⠀⠀⢉⣉⣡⠤⠖⠋⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⠤⣄⣀⠛⠀⠀⠀⠀⠀⠉⠉⠙⣶⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⣤⠶⠿⠷⠦⠄⠁⠀⣀⠀⠀⣰⠒⠛⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣟⠳⢦⣀⡀⠀⠀⠀⠁⣤⡤⠍⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠁⠁⠀⠀⠉⠙⠛⠳⠶⠶⢼⣷⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣷⡄⠀⠀⠀⠀⠀⠀⢲⣦⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠴⠄⠀⠀⣀⣈⡓⣶⡿⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⠑⠀⠀⠀⣀⣀⢸⣇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠈⢲⣶⠖⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢘⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡀⠀⠀⣼⠀⠀⠀⣀⡠⠒⠉⠀⠉⢳⡀⡀⢀⡀⠤⢄
⠀⠀⣀⠠⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠠⠠⠄⠀⠀⠀⠉⠀⠀⢀⡨⠀⢠⠊⠀⡡⠤⠤⡀⠀⠈⠁⠈⡇⠀⠀⠀⠗⠢⡀
⠀⠉⠉⠁⠀⠀⠂⠀⠌⡑⠂⠤⢀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠒⠪⣀⠠⠔⠀⠀⢸⠀⠀⠓⠀⠀⠹⠀⠀⠀⠴⠃⢀⣀⠀⠀⠀⠚⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⢀⠈⠒⠠⠄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣣⣀⠠⠄⠐⠁⠀⠀⠀⢀⠀⠈⡸⠀⠀⠐⠀⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⣐⠀⠀⠄⢀⡀⠉⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠈⠉⠉⠀⣠⣤⡤⠶⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠒⠤⠤⢌⣁⡀⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣒⡒⠒⠬⣔⠄⡀⠀⠀⢠⠀⠀⠀⠈⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠢⡀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠒⡶⠶⠦⠤⣄⡀⢀⣸⡄⠀⠈⡆⠀⠉⠉⠁⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠐⢤⡀⢀⠤⠶⠦⣄⡀⠀⠀⢱⠄⠀⠖⠀⡹⠁⠈⠳⠤⠖⠁⠠⣄⠀⠀⠈⡧⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠤⠉⠒⠀⠀⠒⢿⡀⠀⡆⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⢀⡴⠄⠊⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠴⠦⠀⠤⠞⠉⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⡠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢋⠀⠀⠀⠀⠀⠠⠤⠀⠀⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⣽⠃⢴⠛⣃⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠖⢉⡀
⠀⠀⠀⠌⠉⠄⠠⠄⠄⠀⢲⣇⠀⠀⠠⢀⠀⠐⠉⠡⠘⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⠄⡍
⠀⠐⡋⠀⠀⠠⢀⡀⡄⠀⠘⡉⢀⠀⠀⠈⠃⠀⡀⠀⠀⠀⠉⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⢠⡈⠂⠀⠀⠀⣠⠚
⠀⠀⠁⠀⠀⠁⠤⠀⢀⡀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⡩⠀⠀⡄⠀⠀⠀⣐⡖⠡⡄
⠀⠀⣿⡀⠘⠂⠁⠀⣼⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⡇⠁⠀⢀⠃⠀⠀⢀⠞⢉⠀⠀⠁
⠀⠀⢸⠀⠀⠀⠂⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⠀⠀⠀⡞
⠀⠀⢠⡄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠘⡅⠀⣸⠀⠀⠀⣀⡀
⠀⠀⠀⣧⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠧⠶⠛⠛⠛⠛⠛⠋
⠉⣩⡍⣹⣾⠻⠓⣾⡗⠙⠒⠉⠙⠛⠛⠿⠿⠻⠋⠉⠉⠛⠛⡉⣛⠛⠉⠉⠛⠛⠋⠙⢋⡁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂
⣴⢱⣿⣿⣷⣇⣶⣾⡗⢠⡀⡀⠐⠒⠀⡀⢀⠀⢀⣀⣀⡀⠈⠉⣤⣾⣯⣠⡀⢀⣠⣄⣠⣴⡤⠤⠐
⣷⣿⣿⣿⣿⣿⣿⣿⡇⣸⣟⣴⣦⡀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠺⠟⠻⠻⠿⠿⢦⠝⠋⠉⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆
⣿⣿⣿⣿⡿⠿⢿⣿⠂⠿⠿⠿⠋
⠉⠀⠉⠉⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⢅⣀
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠊
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣦⣶⣴⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠲⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⣀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠇⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⢁⠠⠐⠊⠀⠁⠀⠐⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠂⣰⣤⡃⠀⠀⠀⠀⠀⢀⡠⠀⠄⠀⠀⠀⠀⠈⠂⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⡇⠙⠅⠀⠸⠁⢸⡄⠸⠀⣡⠀⢀⡛⠀⡀⠀⠀⠀⠀⢁⠀⠀⠒⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣤⡇⠀⢆⢢⡀⠱⣎⠘⠀⢸⣇⠡⣄⡀⠀⠁⠠⠀⠀⠁⠀⠦⢄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠺⢏⣇⠁⠀⠘⢦⣱⡀⠹⡿⢿⠾⣽⣷⣘⢙⣄⣀⠀⡀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⡠⡎⠻⣄⡀⠀⠀⠀⠙⢦⡹⡄⣀⢀⠈⠛⠓⠉⠉⠛⠓⢲⣶⡆⠀⠀⢘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣺⣆⡇⠀⢻⣟⠲⣄⡀⢄⡀⠉⠳⣿⡏⠁⠁⣀⠚⠐⠀⠛⠛⠋⠁⡠⢆⠈⠀⢠⡗⠀⠐⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣼⣿⡇⠀⠘⣿⢰⡆⠉⠉⢻⣤⣀⣀⠉⠙⠢⢄⡁⠀⠀⠐⣾⣿⡿⠛⠁⠀⠠⠛⠀⠀⠐⠈⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢠⣿⠁⡻⢷⠀⠀⠈⢿⣇⠀⢠⠆⠀⠉⠉⠛⠳⠒⠀⢬⢿⣓⣶⣿⠿⠃⠀⠀⠀⠀⠀⠀⢠⣤⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⠞⠃⠀⠀⣪⣷⣄⡀⠀⠉⠻⣟⢛⣛⠳⠶⠤⢤⣤⡤⠤⠩⠽⠌⠷⠒⠀⠀⠀⢈⡀⠀⠀⣮⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠙⢾⣤⣀⠉⠉⠟⢻⢻⣷⣶⣌⠙⠳⢬⡑⠀⠠⠀⣀⡀⠀⠀⠀⠀⠀⣀⢴⣶⡾⠉⠖⠒⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⡀⠀⠈⢻⣷⣦⣈⣤⣞⡼⢻⣿⣿⣿⣿⣶⢸⠀⠀⡆⣆⠌⣽⣻⣿⣿⣿⢶⡛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡝⣶⣦⡀⢻⠁⠠⠈⠻⣇⣼⢿⣿⠻⡿⣿⠠⡆⠀⢀⣯⣸⠿⠟⠋⢉⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢧⡟⣼⣿⣗⢷⡀⠀⠂⣿⣧⡿⢝⣼⣗⣿⡆⢹⡶⠊⣩⠰⠖⢛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣥⣿⣿⣿⣿⠆⠹⠀⠀⣿⣿⠏⢡⡹⠖⠚⠛⢀⣽⣤⡀⢀⣠⡿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⡀⣦⣷⢦⠻⣿⠟⠁⠀⠀⠀⣴⣿⣽⣧⡽⣿⡽⠉⠁
⠀⠀⠀⢀⣀⣀⣀⣀⣠⡤⢶⡿⣝⣿⡿⢻⣿⣿⣿⣷⣶⣾⣿⡄⠸⠀⢰⣄⢀⣸⡿⡣⣏⣽⣿⡟⠁
⠀⠀⠈⠉⠛⠂⠐⠂⠈⠛⠚⠉⠉⠁⣠⣾⣿⡿⣿⡿⣿⣿⣿⣟⡻⣶⣾⠋⢻⣟⣿⣷⣿⠟⠋
⠀⠀⠀⠀⠀⠀⣤⣶⣦⣤⣤⠴⠶⠞⠛⠋⠁⣰⠏⠀⡼⠁⠀⠉⠙⠛⠛⠳⣿⢻⡟⣿⣹
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠘⢦⢳⡸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⣽⣜⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡷⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠘⠒⠒⠲⠤⠠⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢤⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢄⠀⠀⠀⠀⠀⠐⢉⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠹⢛⠒⠀⠀⠀⢭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠅⠀⠸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⣛⣚⡀⢸⠂⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⡀⢀⠀⠈⡀⠁⠂⣼⡿⠆⢠⣜⡄⠈⠀⠀⠀⠀⠤⠄⠄⡀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠊⠀⠀⠀⢲⡀⠀⢒⣿⢇⠀⠐⠀⠱⠀⠀⠀⡀⠊⠀⠰⠃⠂⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠂⠠⡈⢄⠀⢀⣈⣿⡋⢘⣽⡆⢇⠂⠐⠁⠀⠀⠀⠢⠀⠠⣈⣈⡤⠃
⠀⠀⠀⠀⠀⠀⢀⠐⠀⠄⠁⠀⠀⠈⠀⠙⠲⣤⡝⢿⣞⣍⢀⠺⠄⠀⠀⠦⠀⠀⠈⡆⢘⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠈⠑⠀⠙⣿⡟⠒⡴⠀⠀⠀⠔⠔⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠐⠠⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄
⠀⠀⢀⡀⠀⠐⠀⠀⠀⠄⠀⡀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠄⠀⠀⠈⠁⠠⠀⠀⠐⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠆⠐⠂⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠈⠁⠀⠀⡄⠀⠐⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⡈⠀⠀⡀⠀⠀⠄⠀⠀⠁
⠀⠀⠂⠀⠀⠀⢀⡀⠀⠀⠀⡀⢈⣠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⣪⣴⣶⣾⣶⣿⡋⠁⠀⠀⠠
⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⢊⣼⣿⣿⣿⣿⡇⠀⠀⣀⣀⣤⣤⣼⣿⣿⣿⣿⣿⣿⠿⠗⠁⠀⠀⠀⠈
⠀⠀⣀⠀⠀⠀⠀⠈⠡⠡⢞⣿⣿⣿⣿⣿⡗⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢳⠶⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧⠀⢏⠻⣁⠀⠐⣨⢋⣠⣤⣤⡀⢂⠀⠀⠀⠈⠀⠀⠂⠈
⠀⠀⠀⠈⠀⠀⠀⠁⢈⣴⣿⣿⣿⣿⣿⣿⣿⣄⢀⣑⣘⣟⣤⣦⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠈⠀⠀⠀⠀⠐⢻⣿⣯⣭⣽⣾⣿⣿⣿⣿⠧⠤⠴⠿⠛⠛⠋⠁⢻⣿⣿⣿⣿⠈⢄⠀⠀⠀⠄⠀⠠⠄
⠀⠐⠀⠀⠀⠀⠀⠈⢿⡿⢿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣶⣄⡀⠀⡀⠄⠀⠀⢀
⠀⠀⠀⠄⠀⠀⠀⠀⠈⢿⣦⣙⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠚
⠀⠈⠀⠠⠀⠀⠀⠂⠀⠄⠙⠿⣳⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣟⢏⢱⢃
⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠈⠛⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠼⣧⠀⠉⠸⠀⠀⠈⠡
⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠁⠠⠀⠈⠃
⠀⠀⠈⠁⠀⠀⠀⠀⠠⠀⠀⠀⠆⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡔⠁⠀⠀⠀⠀⠀⠀⢲⣆⣮⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⢮⣤⡝⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣟⣛⣟⣛⣻⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠉⠀⠀⢀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠸⠇⠀⠀⣀⠀⠀⠀⠀⠈⠛⠀⠀⠈⠁⠀⠀⢸⡆⠀⠀⠘⠟
⣧⠀⠀⠀⠸⠂⠀⠀⠉⠀⠀⣸⡄⠀⠀⡀⠀⠀⠀⠰⠇⠀⠀⡄⠀⠀⣠⠀⠀⠀⠀⠉⠀⠀⠀⠶⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡄
⠉⠀⠀⢀⠀⠀⠀⠀⢠⡀⠀⠈⠀⠀⠠⠷⠀⠀⠀⡀⠀⠀⠸⠇⠀⠀⠉⠀⠀⠀⢸⡄⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠘⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⢀⠀⠀⠀⣷⠀⠈⠁⠀⠠⠷⠀⠀⠘⠁⠀⢀⣆⠀⠀⠀⠀⠀⣄⠀⠀⢠
⠀⣠⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠾⠀⠀⠀⢀⠀⠀⠀⠸⠆⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠁⠀⣴⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠸⠂⠀⡀⠀⠀⠀⠀⢰⡆⠀⠀⠛⠀⠀⠀⠙⠀⠀⢠⣷⠀⠀⠈⠃⠀⠀⠀⠀⠀⣼
⠀⠀⡀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣸⠄⠀⠀⠀⢠⣧⠀⠀⡀⠀⠘⠃⢰⡄⠀⠀⣰⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠸⠇
⠀⠘⠃⠀⠈⠋⠀⠀⠀⠀⠿⠀⠀⠀⣄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠃⠀⠀⠀⠘⠃⠀⠀⠙⠁⠀⠀⠛⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣰⠀⠀⠻⠂⠀⠀⢀⡄⠀⠀⣰⠀⠀⠀⣤⠀⠀⢀⡄⠀⠀⢠⡀⠀⠀⠀⠀⠀⠺⠀⠀⢀
⠀⠀⣿⠀⠀⠀⠉⠁⠀⠀⠀⠀⣾⠄⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠘⠃⠀⠀⣼⠀⠀⠀⠀⠀⠸⠇
⡆⠀⠀⠀⠀⣆⠀⠀⢰⡆⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢰⠄⠀⠰⠆⠀⠀⣾⡄⠀⠀⢾⠀⠀⣄⠀⠀⠷⠀⠀⡀⠀⠀⠉⠀⠀⢰⡆
⠁⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⢰⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠿⠀⠀⠀⡀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢸⠆⠀⠀⢠⣧⠀⠀⠀⠙⠃⠀⣠⡀⠀⢸⡆⠀⠀⠀⠿⠀⠀⢰⡆⠀⠀⠛⠀⠀⣦⠀⠀⠀⠀⠸⠇⠀⠀⠀⣴⠀⠀⠃⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⣆⠀⠀⠀⡄⠀⠉⠀⠀⠀⠀⣾⡀
⢰⡄⠀⠀⢰⠆⠀⠀⠀⠘⠛⠀⠀⣧⠀⠀⠀⣆⠀⠀⠿⠀⠀⠘⠃⠀⠀⠀⠁⠀⡀⠀⢠⡄⠀⠈⠋⠀⠀⠈⠉⠀⠀⠀⢰⡆⠀⠉⠀⠀⠘
⠀⠀⢀⠀⠀⠀⢰⡆⠀⠀⢠⡀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⡆⠀⠀⠀⢰⡆⠀⠀⣷⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⢰⡆⠀⠀⠀⠀⢀⣆
⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠘⠃⠀⠀⠀⠀⣰⠀⠀⢠⠀⠈⠛⠀⢰⠀⠈⠉⠀⠀⠀⠀⠀⠀⠼⠇⠀⠀⠁⠀⠀⠀⠀⠀⢸⡄⠀⠈⠉
⠰⠄⠀⠀⢠⡆⠀⠀⠀⢠⠀⠀⠀⣸⡆⠀⠈⠀⠀⠸⠇⠀⠀⠀⡀⠀⠀⠀⣴⠀⠀⣦⠀⠀⠀⠀⠀⣤⠀⠀⢸⣇⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠉⠀⠀⠉⠀⠀⣶⠀⠀⠛⠀⠀⠀⠁⠀⢀⡄⠀⠀⠉⠀⠀⣦
⠀⠰⠇⠀⠀⠀⠿⠀⢀⠀⠀⢰⠄⠀⢀⡄⠀⠉⠁⠀⠀⣾⠄⠀⠀⠀⢀⣇⠀⠀⠀⣠⠀⠀⠀⢀⠀⠀⠀⢴⠀⠀⠀⠀⠁⠀⠀⡄⠀⠀⠉
⠀⠀⠀⢰⡆⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⡿
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠸⠗⠀⠀⢀⣇⠀⠘⠁⠀⠀⢠⡧⠀⢠⡄⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⠋⠀⢀⠀⠀⠀⣄
⠀⠛⠀⠀⠀⢸⡆⠀⠀⢰⠀⠀⡀⠀⠀⢠⡆⠀⠀⢀⡀⠀⠉⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠈⠁⠀⢠⡆⠀⠀⢼⠆⠀⠀⠿
⠀⠀⢀⡄⠀⠈⠀⠀⡀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠀⠙⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠃⠀⢀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡄⠀⠀⠻⠃⠀⠀⢠⡇⠀⠀⠉⠁⠀⠠⠇⠀⠀⠈⠁⠀⠀⠀⠺⠀⠀⠀⠿⠀⠀⠸
⠀⣴⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⡀⠀⠀⠘⠗⠀⠀⢀⠀⠀⠀⢠⠁⠀⠀⠀⣶⠀⠀⠀⠀⣆⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⡄

⣿⣿⣦⣀⣈⡳⠿⣜⡓⠺⡴⠈⢱⡟⢿⣼⢿⣿⢿⣯⣽⣯⣯⠙⠀⠘⠟⠠⠆⠹⡇⠙⠃⠙⠹⣷⣦⠼⠃⢑⠟⠀⣹
⣿⣿⣿⣿⣿⣶⡤⢭⣁⠀⠉⠳⠀⡟⠾⢋⣾⣻⣿⠟⡟⣷⣿⢶⠍⠁⣿⣠⣀⡰⠀⠀⠀⠀⠀⢹⠃⠀⠓⢠⡡⠴⢛
⣿⣿⠿⠿⢷⡼⣿⠎⠿⠲⠖⠀⠐⠀⣾⠾⡍⠙⠛⠙⠙⠛⠇⠈⠀⣗⠛⢷⢟⢇⠀⠰⢀⠀⠰⠄⡀⠈⢁⣠⣧⠴⠟
⣿⣿⣷⣶⣿⣰⣿⠾⠟⣣⣤⣆⠀⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡴⠂⠛⠚⠀⣤⠁⡀⠈⠤⣤⠀⠀⠈⡿⠀⠀⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣕⠖⠋⠉⠁⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣓⢒⡆⡽
⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⡇⣤⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢾⣟⣼⣿⣷⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣰⡆⠀⠀⠀⠐⠚⠙⠀⠀⠀⡀⠀⠃⠀⠨⠷⠊⣮⠹⢓⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠗⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢀⣄⠈⠃⠀⢀⡩⠔⢸⠁⠀⠁⠀⠀⠘⢿⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⣫⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠱⣆⡀⢀⠄⠀⠀⠀⠈⠉⣵
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢞⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠉⠀⡄⠐⠲
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡏⠀⠀⠀⢸⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠆⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠿⠻⡶⠚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠺
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠂⠀⠀⠀⠰⠖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠀⠡⡶⣂⣂⠀⠐⡟⢦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣡⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢋⣀⣏⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣿⠃⠼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠦⣎⣴⣷⣗⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣿⣻⣼⡿⠗⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠀⠀⠀⠀⠀⠀⢠⡕⠛⣯⣰⡿⠳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢡⡦⠦⠙⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣉⡉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⠙⢛⣛⣛⣿⣿⣿⣿⣟⣛⠛⠛⠻⠿⠿⣷⣶⡌⠉⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⣫⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⠽⣒⣀⠠⣤⣄⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣵⣿⣿⣽⣿⣿⣿⣛⣭⣽⣿⣿⣿⣿⣟⣓⡛⠿⠿⠛⡿⢿⣟⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⣾⣿⣿⣿⣿⣿⣟⡋⠙⢛⡉⢉⠁⠀⠻⣿⣛⣫⠉⢒⡀⠉⠛⠾⣍⡂⠉⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⣿⣿⣿⠿⠻⣿⣷⣶⣶⣶⣧⣄⠓⢈⡙⢂⡀⠈⡙⠊⠙⢶⣌⠳⣌⡙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣿⣿⣿⣤⡾⢸⣿⡷⠿⣿⣿⣝⢿⣷⡄⡅⢠⠙⠆⢈⠳⠀⠀⠈⠻⣮⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⢿⣿⣿⡟⠀⢹⣽⣟⣛⢸⣿⣿⡏⣿⣿⢹⠈⠇⢢⡈⢣⠀⢠⠀⠀⠈⠳⡌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⡈⣿⣧⣳⡓⠈⠻⠤⠻⢛⣿⣿⢣⣿⡿⠈⠀⢠⡀⠧⠠⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡹⣎⠻⣿⣧⣝⣈⡃⠤⣿⣛⣵⣿⡿⠁⡄⢠⡀⢳⡀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡳⠌⠛⠿⠿⣿⣿⡿⠿⠟⠉⣠⠀⡠⠸⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢉⣉⠀⠴⠀⠐⣚⠁⠈⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠀⢐⣒⣋⣁⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣘⣛⣛⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠐⢤⣄⣀⣂⠰⢬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠁
⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⢰⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠙⢆⠀⠀⠀⠀⠀⠸⣤⠇
⠀⠀⠀⠀⠀⠀⠀⢰⠃⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣆⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠸⣄⡼⠀⠀⡰⢶⡀⠀⠀⠀⠀⠳⢄⣠⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆⠀⠀⣸⠁⢳⡀⠀⠀⠰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠘⢦⠴⠃⠀⠀⡇⠀⠀⢣⠀⠀⠀⠑
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣸
⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠃
⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠹⡀⠀⠀⠀⠀⠀⠉⠒⠒⠁⠀⠀⠀⡸⠉⠳⣄⠀⠀⠀⠀⢸⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠖⠃⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣼⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠀⠀⠀⠐⣀⠒⠒⠒⢲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⠋⣡⣤⣁⠉⣿⠏⠀⠶⡌⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠉⠉⠁⠘⢥⢄⡿⠀⢿⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠐⠊⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠆⠛⢧⣘⣆⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢰⡄⢠⡶⠛⡑⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡼⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⠆⢛⣧⡼⣧⣋⡽⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⠀⡀⠀⠀⢀⡞⠛⢶⡀⠀⠀⠀⠀⠠⡌⢿⡀⣿⠁⠀⠀⠉⢸⢫⠄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠚⢷⡀⠀⠈⠓⠒⣼⡇⠀⠀⠀⠀⠚⢣⣹⠧⢿⣄⣀⣄⡴⠻⣌⠛⢋⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠓⠓⠒⣻⠁⠉⠠⠄⡀⢀⣀⣼⡇⠀⠀⡍⠉⠉⢸⠀⡈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⡴⢻⢀⣇⠀⠀⠀⠀⠀⠀⡅⠀⣻⠶⠀⠂⢠⠄⠈⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠻⣤⠸⠀⠀⠀⠀⠀⠀⠀⡬⠁⠀⠀⠀⢀⠀⠸⠆⠀⡄⠀⠈⠀⠀⠉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢱⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⠆⠀⠀⠀⣴⠔⠀⠘⠃⣠⠄⠀⢷⡀⠀⢯⡉⣷⣀⠀⠀⣰⡆⠀⠘⠷⠾⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠛⠲⠈⠳⢦⠀⣸⠋⠀⣀⠀⠀⠏⠀⡀⠈⠳⠧⣦⣾⠿⣛⠻⠟⢛⣿⠶⣦⠰⠾⠛⠦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠁⣰⠟⠁⠀⠀⠀⠐⠁⠀⠀⠀⠘⠁⠀⣠⠞⠩⠐⠩⠐⠀⠉⡾⠁⢞⣀⡠⢄⡀⠉⠐
⠀⠀⠀⠀⠀⣠⢊⡴⠊⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⣀⠀⠁⡂⢀⡤⣪⣵⠐⠿⢿⠆⢠⣿⣿⣿⣦⢀⣒
⠀⠀⠀⠀⡐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠀⢈⠄⣠⡾⠋⠀⠀⠐⡫⠆⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⡼⠡⢣⠣⢋⣾⡟⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠡⠃⠣⢁⣿⣿⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣀⣠⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠿⠛⡿⢻⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⣚⣻⣭⣽⠿⠿⠿⢛⣛⣛⣉⣩⡭⠭⠭⠭⠤⠶⠶⠶⠶⠶⠶⠦⢭⣭⣭⣽⣛⣿⣿⣷
⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⣩⠥⠖⠚⠋⠩⠅⠐⠒⠂⠉⠉⠉⠀⠀⠀⠀⠒⠒⠒⠂⠀⠉⠉⠉⣉⣉⣛⣛⣓⣒⣿⣿⣿⣿⣿⣿
⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠒⠒⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣒⣐⣒⣒⣀⣀⣭⣿⡿⠛⠁
⣴⡿⠋⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠋⠉⠀⢀⣀⡠⠠⠀⢀⣀⣀⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⢿⣧⣀⠀⠀⣠⣴⣶⡿⠟⠋⢁⣀⣤⡴⠶⣚⣫⣭⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁
⠀⠙⡛⠿⢿⣿⣿⣧⣤⣶⣾⣿⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠉⠉⠁
⠀⠀⠙⢿⣶⣶⣶⣦⣭⣭⠭⠭⠭⠭⠭⠭⠍⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠙⠻⢿⣿⣟⠯⠭⢭⣛⣛⣛⣛⣛⡛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣶⣶⡶⠦⠤⠤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡯⠭⣍⣛⣛⣛⡒⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡶⢶⣮⣭⣭⣭⣭⣭⣉⣉⣉⣩⣥⣤⣤⣤⡤⠤⠶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⡶⠶⠦⠤⣤⣤⣤⣤⡤⠤⠤⠴⠶⠒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣛⣓⣒⣒⣒⣛⣛⣋⣩⡥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣓⣚⠫⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣶⣶⣖⣒⠒⠒⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⣒⣒⣒⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⠌⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢟⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣟⠮⠽⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⡤⠤⠦⢤⣀⠀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠈⢹⠃⠀⠀⢠⠶⠛⠉⠉⠙⠓⢶⡞⠉⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⠔⠒⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢳⡴⠋⠉⢠⠋⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠘⣆⠀⠀⠀⠀⣠⠖⠤⠀⠀⠀⠀⣀⠀⠈⠣⢰⠋⠁⢀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡰⠋⠀⠀⠀⠀⢰⠃⠀⣀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣰⠁⠀⠀⠀⠚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠦⠢⠄⠀⠀⠀⠀⠠⢈⠛⠀⢰⠛⢁⡀⠀⠀⠅⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡈⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠈⠳⣞⠛⠀⠈⢞⠑⠒⠀⢄⣀⡴⠁
⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⢱⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⢤⣀⡰⠓⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⠈⠳⣄⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⣄⣠⣄⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠡⠴⠃⠀⠀⠀⠰⠀⠀⠈⠙⣆⠀⠀⠀⣀⣀⣀⡴⠋⠀⢰⠃⢾⠓⢎⡀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢸⠀⠀⠀⠀⠈⠓⠦⣄⣀⣈⣇⠀⠀⠀⠘⢆⡃⢃⠀⠉⠣⣄⡀
⠀⠀⠀⠀⠀⠀⡠⠃⠀⢨⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠣⣄⡀⠀⠀⢹⠈⠆⠀⠐⢄⡉⠒⠤⡀⠀⠀⣀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠰⠁⡆⠀⠀⠀⠀⠀⠀⠠⢫⠀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠉⠉⠈⡇⠘⠀⠀⠀⠙⠦⣀⠀⠁⠒⠁⠀⠀⠀⠘⣦
⠀⠀⢀⠀⠀⡁⢀⡇⠀⠀⠀⠀⠀⠠⠁⢸⠀⠀⠀⠠⠜⠀⠀⢀⠀⠀⠀⢈⠀⢷⠀⠀⠀⠀⠀⠀⠚⠉⠉⠙⠂⠀⠀⠀⠀⡼
⠀⠀⢁⠀⠀⠉⠉⠀⠀⠀⣠⡄⠀⠠⣠⠞⠀⠀⠀⠀⠀⠀⠠⢸⠀⠀⡀⡸⠀⢸⣀⡀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠊⠑⢆
⠀⢀⠌⠀⠀⣠⠀⠀⠀⡰⠁⡇⠀⠀⠀⠀⠀⠀⢠⢖⠀⠀⠀⣸⠀⠀⠈⢀⡜⠁⠀⠀⠔⠋⠉⠉⠉⠙⠓⢦⡀⠀⠀⣀⠀⠀⢈
⠀⠀⠀⠀⡐⢠⠀⠀⠐⠀⢸⠃⠀⠀⣀⠀⠀⠠⠁⣸⠀⠀⠉⠁⢀⡄⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⣄⠞
⠀⠀⠀⢀⠀⢸⠀⠀⠘⠦⠞⠀⠀⠀⢠⠀⠀⡁⠀⡟⠀⠀⠀⢀⠊⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠠⣀⡼⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⡇⠀⠀⠙⠷⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⢀⣠⠞⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠚⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠁⠀⠀⠀⠉⠳⣶⠟⠉⠛⢶⡶⠛⠉⠉⠉⠉⠙⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡶⠾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠛⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡟⠀⠀⠀⠀⢀⣱⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠠⣴⠿⠛⠀⠀⠀⠀⣀⢀⡝⠀⠀⠀⠀⠀⠀⠛⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⣀⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠬⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢦⢠⣀⣔⣬⡺⠛⠁⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠈⢻⡄
⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠢⡄⠮⠓⠋⠁⠀⠀⠀⠀⣠⡌⠁⠀⠀⠀⠀⠀⠀⢈⣀⢀⣠⡴⠎⠀⠀⠀⠀⣀⠠⣴⡍⠀⠀⠀⠀⠀⢀⣿⣀
⠀⢸⡇⠀⠀⠀⠠⡀⢀⡠⢀⠎⠀⠀⠀⠀⠀⠀⡠⠀⠈⠒⠙⠞⠓⢉⠀⠀⢠⣠⡀⣄⣦⣴⣿⣿⡿⠋⠀⠀⠀⠘⠎⠳⠿⠿⠊⠀⠀⠀⠀⠀⢸⠀⠉⠉⠛⢷⡄
⠀⠀⠻⢦⣤⣤⡄⠉⠀⠉⠀⢄⠀⠀⡀⢄⠠⡔⢊⡄⡀⢀⡦⣄⡴⠄⢠⡐⢤⣙⢻⣿⠿⣫⣾⠛⡱⣄⢦⡠⣰⣵⣦⢂⡄⠀⠀⠀⠀⠀⠠⡰⡅⠀⠀⠀⣠⠼⣿
⠀⠀⠀⠀⠀⢸⣇⠀⡀⠀⠀⠀⠀⠁⢈⣤⣤⢾⢯⡳⣝⡲⣝⡲⠭⢃⣼⢯⣷⡿⢋⣡⣾⡿⣷⣷⠙⠮⠻⠟⠛⢉⣥⣿⢭⡇⡄⢐⢦⡪⣕⠿⠃⢀⡠⠖⢉⣼⠏
⠀⠀⠀⠀⠀⠀⠻⣦⣍⡂⠓⠌⣓⣩⡀⣀⡀⣀⠓⠝⠳⠝⢒⣉⣴⣿⣿⠟⠉⠀⠋⠉⣠⡾⣟⢮⣿⡿⢲⣖⠿⡛⣝⣾⡟⣰⣏⣪⣕⠎⢚⢁⣬⣤⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠙⣷⣀⣑⣈⡙⡿⢿⢿⢿⡻⣽⣿⣥⣶⠖⠀⣠⡾⣏⣽⣮⡿⢏⣴⣟⢮⣓⣽⡾⢟⣴⡿⣤⣥⣶⡾⢋⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⢷⣌⠈⢯⣷⣻⣾⡿⠋⠀⣠⣾⢻⣶⣤⣤⣴⣾⣿⣏⣈⣓⣉⣥⡶⣿⢯⡷⠾⠛⣩⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⡽⠋⠀⣠⣾⣿⣿⣿⡿⠿⠟⢛⣿⣿⠿⠟⣽⠿⠷⠶⠶⠶⠶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣤⢦⣄⢀⡰⠋⠀⠀⠔⠛⠛⠉⠁⠀⠀⢀⣴⣟⣋⣤⠴⠋⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠛⠋⠀⠀⣹⠟⠀⢀⣀⣤⣴⣶⣾⠏⠀⢀⣴⠿⠋⠥⠅⠂⠀⢀⣤⣾⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠿⠃⠀⠀⠀⠀⣠⣼⣧⣶⡿⢟⣿⠁⣲⠞⠁⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣠⡐⢶⡄⢄⡐⠻⡿⠯⢠⡿⠋⢁⡼⠋⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢶⣤⣥⣴⠞⠛⠛⠋⠀⣠⠟⢁⣴⡿⡫⠂⠀⠀⠀⠀⠀⠀⠀⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡾⢋⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣼⠟⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠉⢤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡾⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⡟⠁
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠟
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⠛⠛⠶⢦⣄⡀⠀⠀⣀⣠⣤⠴⠶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⣤⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⣀⣤⡴⠶⠶⠶⠦⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠀⠈⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⢦⣀
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠈⠻⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠀⠀⣀⣤⣤⢤⣤⣄⡀
⠀⠀⠀⠉⠙⠛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣶⣦⣤⠞⠉⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⣀⣠⣤⣤⣴⡟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⡾⠋⠁⠀⠀⠈⠀⠀⠀⣀⣤⠶⣷⡶⠂⠀⠀⠀⠀⠿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⢀⣠⡴⢞⣩⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠶⠶⠶⠶⠶⠶⠖⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣷⠾⠋⠁⠀⠛⠷⠶⠦⣤⣄⠀⠀⠀⢀⣠⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣻⡶⠀⠀⣀⣤⠾⠿⠷⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⢀⣴⣟⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠧⣤⣤⣀⣀⡈⠉⠉⠙⠛⠛⠳⠶⢶⣦⠄⠀⠀⠀⣀⣴⣾⠂⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣻⡷⠂⠀⠀⣀⡴⠟⠁⠀⠀⣠⠾⠋⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⢸⠏⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴⠞⠋⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠘⠷⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠈⣿⠛⠳⠾⠋⠁⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠘⠛⣳⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣠⣤⡴⠶⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⠀⠙⢿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠙⣿⣿⣧⡀⠀⠀⠀⠀⠉⠛⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⡇⠀⠀⠘⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢠⡞⠁⠀⠀⠀⠈⣿⣿⣿⣄⠀⠀⠀⠀⠀⠙⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠈⢳⡄⠀⠀⠀⠀⣿⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣴⠟⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢀⡰⠋⠀⠀⠀⣠⣾⠟⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⢸⡄⠀⠙⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢿⣿⠀⡠⠋⠀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⢠⠶⠋⠀⠀⠀⠉⠉⠉⣷⣄
⠀⠀⠀⣠⣤⣤⣤⣤⣶⠞⠋⠁⠀⢸⡇⠀⠀⠔⠊⠀⠀⢸⣷⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⠀⠀⠽⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⣇⣀⣀
⠀⠀⣼⠋⠀⠀⣠⡾⠃⠀⠀⠀⠰⣏⠀⠀⠀⠀⢀⣠⣴⡿⠋⠀⠀⠀⠀⣰⣿⡿⠁⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠈⠉⠙⣇
⠀⢀⡏⠀⠀⢸⡏⠀⠀⠀⠀⣠⠶⠾⡷⠶⠶⠟⠛⠋⠁⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⢹⣿⠶⠶⠶⠿⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠔⠋⠀⠀⠀⢈⡇⠀⠀⢠⠊⠀⢠⠞⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣾⣿⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠀⡠⠃⠀⢀⡤⠶⢶⡟⠛⠋⣿⠏⠁⠀⠘⣿⣿⡄⠀⠀⠀⠈⠙⠛⢶⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣰⠏⠀⠀⢠⣿⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠘⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠸⣿⣿⠛⠿⢷⣶⣶⡒⠚⠛⢫⡉⠉⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠉⠛⣷⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⢹⣧⣤⣤⣀⣀⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⡚⠛⠋⢩⡞⠁⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⢿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡀⢻⣆⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⢋⣽⣿⠟⠁⠀⣿⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⢀⣴⣿⢿⡅⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⣴⣿⠟⠁⠘⣿⡀⠀⠀⠀⠹⣦⡴⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠞⠁⠀⢠⣾⡟⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠲⠁⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠈⢧⡀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⠁⠀⠀⠀⠙⡆⠀⠀⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠀⢸⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡌⣧⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠚⠛⠛⠛⠒⠒⠒⠶⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⣠⠤⠤⠶⠶⠶⠶⠶⠤⠤⢤⣀⣀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⠦⣄⣠⡤⠤⠤⠤⠤⠤⠤⣤⣀⡈⠉⠲⢤⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠈⠉⠓⠒⠲⠦⠤⠤⠤⠤⠥⠶⠒⢚⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⣀⡉⠙⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠖⠚⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡉⠙⠒⠲⠦⠤⠤⠤⠤⠤⠴⠖⠒⢋⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣍⡙⠒⠒⠶⠤⠤⠤⠤⠴⠶⠒⠛⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣉⠙⠒⠒⠒⠶⠶⠖⠒⠒⠛⣉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣉⠓⠒⠒⠒⠒⠒⠒⠚⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢍⣙⡓⠒⠒⢒⣛⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢭⣭⣭⡭⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠖⢚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠛⠓⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠴⠶⠒⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⢳⢏⣠⡄⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠞⣉⠁⠀⢀⣤⣉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⢀⡎⠈⡇⢠⡏⠀⠘⡆⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣠⠖⠒⠚⢿⣆⡇⢸⣦⣄⡠⠟⠋⠉⠁⠀⠈⠉⠂⠀⠀⠀⠐⠋⠉⠀⠀⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠉⠀⣇⠀⠀⠀⠀⠩⠤⢬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠘⢦⣀⡀⠠⢤⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠒⠀⠲⠤⠤⠀⠄⠀⡀⣾⣀⡀⠀⠀⣰⠛⠓⠚⠛⠃⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠨⣉⣡⡽⢀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡚⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠤⠤⢶⡀⠈⠁⠈⠳⠶⠛⠦⣤⣀⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠁⠁⠀⠀⠀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⢤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣸⠷⣄⠀⠀⠀⠀⠀⣀⡤⠟⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠶⠛⠁⠀⠈⠙⠲⠶⠒⠛⠉⠀⠀⠈⠓⠶⠤⠤⠶⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⣲⣿⣿⠇⠈⠈⠙⠀⠀⠂⠀⠀⠐⠁
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⣿⡟⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠾⡿⠁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢀⠄⠒⠀⠂⡀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣿
⡋⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠒⠉⠉⠐⠄⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⡀⠀⣀⣀⣄⣄⢀⣤⣾⣿⡋
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⢦⠀⠊⠙⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⢿⣿⣿⣿⣯⣤
⠦⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⡠⠄⠀⠀⣻⡿⠜⢻⠻
⠀⠀⠀⠀⠈⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠠⣼⣦⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠱⡦⠀⡀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠑⠀⠀⠀⠐⠈⠻⠆⠀⠀⠀⠀⠤⢤⣾⣿⣷⠀⣿⣀⣠⣀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠤⠾⠿⣿⣾⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⠀⢀⠃⠀⣀
⠀⡄⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡅⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠉
⠀⠀⠼⢆⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⡂⡬⠞⡉⠀⠉⢧⡀⠀⠀⠀⠐⠂⠀⠄⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠎⠀⠀⠀⠀⠀⠀⠳⢀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠷⠶⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠢⠄⠤⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⣤⠀⠀⢀⣴
⠀⠀⠀⡃⠒⠂⠴⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢆⣠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢻⣤⣿⣶⣿⠿⠁
⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⣿⣧⡀⢠⠋⠀⠀⣠⣶⡥⠀⠀⠀⠀⠀⠀⣸⡿⣿⡁
⡀⠀⠤⠀⠄⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠒⠤⡀⠀⠸⣿⣿⣤⡆⢀⣾⣿⠋⠠⠀⠀⠀⠀⠐⢀⠂⠀⠀⠙⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⣀⣈⣠⣤⣤⣙⠛⠠⡘⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⠒⠒⠉⠭⠛⠛⢛⡿⡿⢉⣤⠚⣽⣯⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠾⠿⠿⠿⠿⠿⢷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⣀⣠⣤⣤⣤⣀⣀⣀⣀⣉⠉⠛⠯⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⡹⠃⣀⡤⠔⣚⣉⣭⣥⣴⣾⣟⣋⣉⠉⣛⠻⢿⡓⠢⢦⣿⣷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠏⠀⡴⣂⣭⡶⠞⠋⠉⡁⠠⠀⠂⣀⣈⠉⠉⠁⠂⠀⠄⣚⠲⣄⡈⠙⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠚⣀⡵⣿⢛⠁⢀⣤⣶⣟⡾⠿⠛⠋⠉⠙⠛⠛⠛⠷⠦⣄⡈⠱⣎⣿⢦⣹⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢠⠊⣿⢿⣇⣤⡾⢋⡴⣞⠵⠖⠒⣒⡒⠒⠲⣶⡒⠦⠤⡀⠀⠙⢦⣨⣹⡿⣻⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡙⢦⣝⡿⣿⣧⣆⣿⣫⠴⠚⢋⡐⠒⢲⣦⣤⣀⠈⠣⡀⣀⣱⣴⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠻⣮⣽⣿⣿⣿⣿⣶⣾⣭⣭⣭⣭⣭⣼⣿⣯⣿⣿⣾⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣿⡟⣿⣿⣿⣿⣿⠯⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⣟⡿⢚⣻⣿⠿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⠿⢛⣛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠼⣥⣽⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⠿⠿⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⠿⠿⢷⣦⣴⡿⠉⢀⣠⣤⡀⠀⠀⠙⢿⣶⣶⡶⢶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⣀⣤⡀⠀⠈⠉⠀⢠⡿⠋⠉⠀⠀⠀⠀⠈⠁⠀⣀⣀⠀⠉⠻⣧⡀
⠀⠀⠀⢀⣠⣤⣼⡟⠀⢠⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣆⠀⠹⣷⣤⣄⡀
⠀⣠⣾⠟⠋⠉⠉⠁⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠙⠻⣷⡀
⣰⡿⠁⢀⣴⡾⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⣦⠀⠘⣿
⣿⡇⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢹
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠈⢿⣦⡀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠄⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠙⠻⢷⣶⣶⣶⡾⠟⢻⣧⡀⠈⠛⠿⠷⠀⠀⠀⣠⣤⡀⠀⠀⠰⠾⠿⠋⠀⣰⡿⢿⣶⣶⣶⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣀⣀⣠⣤⣾⠟⠛⢿⣦⣄⣀⣀⣠⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠹⣷⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣼⡟⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⣹⡇⠀⢀⣼⡿⣷⡀⠀⠀⢸⣿⡀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠻⠷⠾⠟⠁⠀⣾⠏⠀⠘⣷⠀⠀⠀⠙⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣴⡿⠀⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢸⣿⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⠉⠁⠀⠀⢀⣾⠟⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⡾⠋⠀⠀⠀⠀⣴⡿⢻⣦⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⢹⡇⠀⠀⠀⠀⠀⠈⠻⠷⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⣶⠿⠃

⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⣜⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡻⣿⡧
⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⢠⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄
⠀⠀⠀⠀⠀⢾⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡝⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠹⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⢿⡷
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠻⣘⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣻⠏

⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⣼⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣿⣧
⠀⠀⠀⠀⠀⠙⠬⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠬⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠽⠋
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠠⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠏⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡶⣶⣷⣾⡏⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⠒⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠢⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠠⠀⠀⠀⠀⠀⣠⡠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢄⢱⡆⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠂⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣛⡍⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⡠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⣀⡀
⠀⠀⠀⢰⠁⣧⣶⣦⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⣴⣾⣶⡡⡾⠂⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠈⢧⣿⣿⣿⡷⣶⣆⠀⠀⡀⠀⣱⣤⣤⠶⠀⠀⠀⠀⠀⠀⢀⡠⠂⣠⣾⣷⢃⣶⣤⢹⣿⣿⣇⡀⠠⠃⠀⠀⠀⠀⠀⢠⢠⣿⣤⣶⣤⣀⣤⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋⣙⠦⣭⣿⣿⣿⣿⡶⢷⡀⣄⣀⣤⣶⣤⣶⣿⣿⣿⣿⣿⣿⡏⢸⣿⠿⢿⣦⣴⣷⣄⡀⠀⠀⢀⣨⣿⣿⣿⣿⣿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⠍⠉⠉⣉⣠⢮⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣷⣬⡛⢿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢹⡟⠿⠿⠿⠿⠛⠁⠈⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⡿⠿⠛⣁⠑⠄⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠈⠢⣾⣧⣴⣦⡀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠿⠿⠆⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠒⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⠉⠁⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⢠⣆⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣽⣿⣿⣿⣿⢀⣼⣿⣿⣿⡾⠿⢿⣿⣿⣿⣿⣿⣑⠃⠠⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⢋⡵⣿⣿⣿⠯⠛⠓⠚⠛⠋⠁⠋⠉⣁⣀⠀⠀⠀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⢀⣹⣍⡛⣧⠈⢿⣿⠛⠃⠀⠀⠉⠀⠐⠂⠀⠀⢾⣿⣿⣷⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⣿⣿⣿⠈⢿⡆⠀⣶⣴⣶⣶⣶⡶⣶⣤⡀⠀⠀⠉⠻⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠈⡟⠉⠀⠐⢛⣥⡆⡍⢸⣿⣿⣿⣯⣿⣿⡇⠀⠀⠁⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⣶⣦⣶⣿⠟⠓⠃⣸⣿⡿⡿⠟⣿⣿⠀⠀⠀⠀⠸⣿⣿⣷⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠙⣿⣿⠟⠛⠉⡙⣿⠛⠳⢾⣿⣤⣀⠀⠀⠀⠙⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠉⢰⢀⣽⠇⠈⠠⡤⠀⠀⠀⠀⢢⠌⣙⠿⠟⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⡇⢸⠿⠁⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠋⡙⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣷⡎⠋⣀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠐⠂⠀⠀⢠⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢹⣇⠀⣿⣿⠄⠀⠀⠀⠲⣦⣀⣀⡀⣠⣶⣄⠈⠈⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⠀⢸⣿⣠⡀⡀⠀⠀⠀⠈⠩⢁⣤⣿⡜⠆⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⣠⣼⣿⣥⣽⣿⡟⠑⣤⣤⣠⣼⣏⣽⣧⡄⠀⣄⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣹⣽⠹⣿⣿⣿⣿⣭⣬⣭⣡⣟⣻⣿⣿⣿⡆⢣⠟⡊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠉⡛⠋⠉⠘⠿⢹⠿⠅⠀⠠⠿⠿⣿⣿⢻⡇⠀⠁⢙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⠿⠽⠇⡸⣆⠮⠗⠁⠤⠤⣀⠀⠀⠈⠈⠣⣼⠲⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣤⡄⠘⠀⠀⠀⠀⠀⠀⠀⣯⣄⣀⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠚⠷⢿⠟⢿⠿⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⣚⣉⣉⣉⣭⣥⣀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣖⣺⣯⣵⡶⡟⠟⠛⢓⣒⣶⣶⣶⣶⣶⣮⣟⡚⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣪⠷⡻⠿⢛⡩⠤⠒⣂⣭⣿⣭⡥⣤⣤⣤⣤⣤⣉⡙⠫⣝⣯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡰⠟⠉⣠⡪⠔⢈⣡⣤⠶⠿⠛⣛⠿⠟⠛⠛⠛⠛⠲⠦⢬⣉⠙⣾⢿⣿⢣
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⡴⠋⢁⡤⢾⠟⢉⡀⠔⠂⠉⣀⣀⣠⣤⣄⡀⣀⣠⣤⣤⣌⣽⣿⣯⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⡟⢀⠶⠏⣠⡾⡩⣮⠥⣂⣥⠤⠶⠾⠛⠛⠉⠛⠛⢢⣔⣫⠽⣒⣯⣤⠴⠛⣿⣿⣿⢽
⠀⠀⠀⠀⠀⠀⡿⣷⠗⢷⣾⣿⡾⢞⣭⣶⢿⣥⡤⢤⣤⡶⣶⣞⡿⢿⣿⣶⣿⣿⣶⣭⣬⡷⣿⣥⠏⣿⡟
⠀⠀⠀⠀⠀⠀⠉⠘⠛⡿⣿⣿⣿⣿⣿⣿⣶⣶⣿⠿⣶⣿⣿⠿⢿⣿⣿⣿⣿⣿⣷⣶⡶⣛⣿⠶⡿⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣙⡿⠿⣿⡧⣭⣿⡿⣿⣿⣟⣿⣷⣺⣿⣿⣿⣿⣿⡿⣿⣟⣻⣿⣮⡽⠾⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⡿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠷⣶⣶⠶⠛⢻⣹⣹⡿⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣛⣛⣶⣶⣶⣶⣾⣧⣶⣶⣶⣶⣞⡫⠥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢽⣉⣛⡻⠿⠿⣿⡿⣿⣯⣭⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣟⣻⣭⣿⣛⣻⣷⡾⠿⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⣿⣷⢿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠛⠻⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⣠⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠉⠉⠛⠛⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠙⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢀⣤⣶⠾⠿⠿⠷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⡀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠛⠛⠛⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠛⠛⠛⠿⣶⣄⠀⠀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡿⠋⠉⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣶⣶⣶⣦⣄⡀⢿⡆
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠙⢿⣿⡇
⠀⠀⠀⢠⣶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄
⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠈⠻⣧
⠀⠀⠀⠘⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⢛⣿⠟⠛⠛⠛⢛⣿⠿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⣰⣿⣃⣀⣻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠈⠛⠛⠛⢻⣿⠋⠛⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⢻⣿⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⢀⣾⣿⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣀⣀⣀⣀⣈⠉⠉⠉⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⣻⡏⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢇⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⢦⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠛⠻⢦⣀
⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡾⠛⠉⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⢀⣤⠿⠶⠶⢾⠏⠀⠀⠀⠀⠀⠀⠹⡷⠶⠶⠶⣤⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠋⠉⠉⠉⠙⢷⡄⠀⠀⠀⠀⠹⣆
⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠻⠞⠛⠛⠷⣦⣿⡄
⠀⣠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡅
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠉⠛⢻⡛⠛⠛⠛⠛⠛⠛⠛⠻⣟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⣠⣴⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡴⠋
⢠⡾⠋⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣏⠉⠁
⠘⣧⣀⣀⣼⠃⠀⠀⠀⠀⠀⣴⠟⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣆⠀⠀⠀⣠⡴⠟⢻⡄
⠀⠈⠉⠉⠁⠀⠀⠀⣤⠀⠸⣏⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⣿⠀⠀⣾⠋⠀⠀⣾⠁
⠀⠀⠀⠀⠀⢀⣤⡾⢻⡇⠀⠙⠛⠛⠋⠀⠀⠀⢀⣼⣧⠀⢸⡏⠀⠀⣸⡏⠀⠀⢻⣤⣤⣼⠏
⠀⠀⠀⠀⣰⡟⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⣿⠀⠈⠻⠶⠾⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠹⣦⣀⣠⡿⠁⠀⠀⠀⠀⠀⠰⣟⠀⠀⢠⡿

⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣤⡀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠋⠁⢸⣇⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⢾⣿⣷⣤⣈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣤⡤⣿⡈⠉⡉⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣈⣈⣻⡀⢺⡇⠀⠡
⠀⠀⠀⠀⠀⠀⢠⣀⣈⣷⠉⠉⠉⠉⠙⠿⠛⠚⠓⠦⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠈⠋⠉⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⣠⣤⣶⣶⣶⣿⣿⣷⣲⡀⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⢿⠿⢿⣭⣽⣿⣟⣿⣿⣛⣿⣀⣀⡀⠈⠁⠀⠀⢀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣱⡗⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠓⠲⠶⢶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⣼⣁⣀⠀⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣀⣠⣤⣤⣤⠤⢀⣀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢀⡤⣒⣹⣁⣽⣿⣤⡤⠴⠶⣾⡿⠓⠛⠒⠂⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⠉⠉⢈⣿⣇⣀⣤⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣾⣿⣿⣾⣿⣿⣏⣧⡙⢆⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⢻⣿⣿⣿⣿⣿⣿⣧⠻⣿⠆⢉⢶⡿⢟⣟⣥⣄⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢃⣚⣭⣽⣶⣶⣶⣿⣿⠳⣦⣰⣿⣧⣿⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⡛⠿⣿⣿⣉⠻⢿⣿⣿⣿⡇⠀⣼⣿⡇⣿⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣗⣴⠂⠸⠿⠯⡉⠁⠀⠹⢿⣤⣴⣿⢿⣴⣿⣧⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣸⣏⡀⠙⠉⠛⣿⠿⠿⠷⣄⡀⠘⠛⠐⠆⢀⠀⠀⠉⠉⠀⠀⢻⣿⣷⣝⡳⠶⠤⠀⠀⣀⡀
⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠉⠀⠘⠙⠂⠀⠠⣍⠛⠛⢷⣦⣀⠐⠾⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠛⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣬⣥⣄⣀⡉⣛⡛⠂⠀⠀⠠⣤⣶⣦⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢿⡿⠿⠿⠛⠋⠉⠒⠀⠈⠉⠛⠛⠛⠃

⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⡇⠀⠀⠀⠀⠀⠀⠀⣖⡆⡆⢠⡀⣠⢰⣀⠀⢰⡶⠀⠀⡄⣦⣰⢰⣄⢐⢤⢀⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠓⠋⠃⠛⠋⠓⠘⠛⠀⠙⠛⠃⠀⠙⠘⠃⠘⠘⠙⠘⠘⠛
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠞⠿⠓⠏⠙⠧⠇⢯⠗⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⡤⢄
⡇⠀⠀⠀⠀⡀⣰⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠄⠐⠁
⡇⠀⠀⠀⠀⠠⠁⠱⠀⠒⠂⠀⠀⠀⠀⠀⡀⢠⠾⢍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⠀⢀⠰⠂⠐⠖⠃⢤⡉⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠓⠚⠓
⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣀⠡⡄⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⢈⠀⠈⢀⡀
⡇⠀⠀⠀⢀⣀⣀⣀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⠠⠈⠈⠐⠤⠄⠀⠀⠀⠀⠉⠉⠀⠉⠉⠁
⡇⠀⠀⢀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠠⠀⠆⠀⠀⠴⠄⡀⠀⠀⠀⠀⠠⠀⢠⠀⠀⢠⠀⠀⠀⡠⠄⡆⠀⢀
⡇⠀⠀⠈⡷⡤⠠⣾⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⢸⠔⠠⠄⠠⠀⢶
⡇⠀⠀⠐⠛⠁⠀⠛⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠀⠀⠐⠀⠈⢹⣦⢤⠤⠎⠋
⡇⠀⠀⠀⠙⠀⠀⠁⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⢢⠄⠆
⡇⠀⠀⠀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠠⠤⠁
⡇⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢠⣄
⡇⠀⠀⠐⠒⠉⠡⠈⠉⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣹⢷⡄⡀
⡇⠀⠀⠀⣄⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋
⡇⠀⠀⠀⠷⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀
⡇⠀⠀⠀⠀⠁⠀⢠⣤⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠄⠀⠈⠂⠀⠀⠀⠀⢠⣄⣀⣀⡀⣀⣀
⡇⠀⠀⠠⣖⢚⣫⠄⠠⠨⠽⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠔⠁⠀⠀⠀⠀⠀⠸⠉⠹⠻⠹⠟⠿
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠈⠻⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⢀⣴⡿⠋⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠻⣷⡄⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣾⣿⣄⠠⣾⠉⠀⣤⣤⣶⣶⣤⣤⠀⠀⣿⣆⣴⣿⣿⣿⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡻⠿⠷⠶⣬⣭⣭⣭⣭⣥⡶⠿⠿⠟⣛⣫⣭⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢶⣴⣟⣒⣒⠒⠒⠒⠒⠚⠛⠛⢋⣉⣡⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣭⣍⣛⡛⠛⠿⠿⠿⠿⠛⠛⢛⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣍⡛⠛⠛⠻⠷⠶⠶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣍⣛⠿⠿⠶⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣍⣛⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣉⣛⠛⠋⠁
⠀⠋⠃⠀⠂⠒⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⡠⠘⠃⡴⡀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⢱⠯⠀⠐⢿⠲⠀⠀⠀⠀⠀⠀⠀⢠⣼⠷⠇⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢰⠲⠤⠀⠀⣾⠀⠀⠀⠀⠘⠰⣤⠄⠀⠀⠀⠀⠀⢠⠀⠀⠋⢉⠈⣏⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣆⠀⠀⠀⠀⠀⠟⠮⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠉⠀⠈⠳⣤⣏⣠⡀
⠀⠀⠀⠀⠀⠀⣡⡞⠀⡀⢠⣄⡤⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠀⠀⠀⠀⠀⠀⠀⠿⣿⢿
⠀⠀⠀⠀⠀⠴⠏⠙⠀⠻⣾⣿⡇⠈⠇⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⣂⣄⣠⣿⡀
⠀⠀⠀⠀⠃⠀⠀⠚⠦⠀⠈⠋⡇⠀⣀⠀⠁⠈⠀⠀⠀⠀⠒⢷⡀⠀⠃⠤⠀⠀⠀⠀⠀⢉⡉⡀⠀⠀⢀⡷⠉⠉⠸⣟⡄
⠀⠀⠀⠀⠆⠀⠀⠀⠠⠀⡀⣀⢷⡀⠸⣶⠠⡀⠀⠤⠀⠀⠘⣿⣧⣠⡤⣤⣤⠄⠀⠀⠀⣈⠻⣿⣶⣿⠉⠀⠀⠈⠀⠼⠃
⠀⠀⠀⠈⠲⣷⠗⠀⠀⠣⠁⠀⢸⣿⣦⡈⠋⣑⣀⣀⡀⣀⣠⡜⠋⠙⠿⣋⣥⠦⠀⠀⠀⠘⠛⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠈⢶⣔⣲⡄⠀⡠⢿⣿⡞⢷⡶⣿⠿⣬⣙⠛⢯⡁⠀⠠⠀⣤⢤⠀⠀⢀⠀⠀⡆⠠⣄⣄⢀⡃⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠈⢷⡀⢛⣆⡾⠃⢻⡀⠹⢿⡿⠛⢠⡀⢠⡐⠘⠈⠇⠀⠈⠂⠀⠸⠀⠙⠀⠈⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠛⣆⠀⠀⠈⠇⠀⠀⢆⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠄⠈⠁⠀⠄⠘⠀⠀⠘⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⠀⠀⢀⣀⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠈⠁⠀⠁⠀⠀⠈⠉⠀⣀⣿⣿⣿⣿⡿⠋⠁⠈⢮⠩⠁⠀⠃⠉⠙⠗⠒⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠀⠀⣀⠁⢀⣁⣀⣤⣤⠤⠤
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣯⣽⣠⣖⣀⠀⠀⠀⠀⠀⡄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠳⢄⢠⣄⡀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠉⠙⢷⡄⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢰⢦⣄⢒⠟⠁⠀⡐⠋⢤⣭⡓⣦⡄⠀⠐⠀⠂⢀⣀⠀⢀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⣸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⢯⠀⣤⣊⠀⢀⡴⠛⠛⢿⡆⡀⠀⠀⠀⠚⣿⠾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢃⣿⣿
⠀⠀⠀⠀⠀⠀⠠⠲⠻⠁⠀⠙⢵⣿⣧⣎⡀⠀⠠⠊⠀⠀⠀⠀⡀⠌⢱⡄⠀⠉⢓⠤⣀⠀⠀⠀⢀⣼⣿⣷⣿⡿⠁⣠⡆
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢻⡼⣽⠛⠂⠀⠀⠀⠀⠀⠈⠂⣀⢹⣶⣦⣤⣽⣮⣽⣶⣦⡟⠛⣿⣿⠟⡑⣨⣻⠟
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠐⣀⣠⣴⡿⣿⣿⣛⡏⡁⠘⠁⠁⠉⠁⠁⠈⠐⠒⣠⣴⣿⡯⠊⣬
⠀⠠⡀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣿⣶⣿⣿⣿⣿⠿⡿⣶⠤⠂⠒⠒⠺⠟⠛⠛⢯⠵⠛⠠⡚⠁
⠀⣴⣄⣄⠢⠀⠀⠀⠈⠁⠀⠀⠉⠉⠸⠿⣿⣿⡿⠿⠊⢉⠀⠉⠹⡦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡦⠖⠁⡀⢀⡔⢠
⠀⢈⠻⣿⣶⠤⠀⠤⣀⠀⠀⠀⠀⠀⠐⠈⠉⠈⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠁⣠⠆⣪⡀⡊⠄
⠀⠀⠁⠉⣭⣷⣶⣄⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠛⠉⠀⢀⣠⣶⣦⠀⢿⡛⠀⠙⠒
⠀⠀⠀⠀⠨⠙⠛⠿⢿⣇⣄⣴⣏⣿⡇⠀⢈⣦⠀⣀⠀⠀⠀⠀⢒⡂⢠⣴⣾⣷⠶⣿⣷⡤⣿⣿⣿⣿⠗⠀⣝⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠥⠀⠀⠛⠁⢲⣿⡝⠻⢿⡖⠿⡿⠿⢿⣿⠿⠻⢿⣿⠿⠉⠉⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐