Hurrah Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣶⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⣿⠀⠀⠉⠉⠛⢿⣿⣿⣧
⠀⠀⠀⠀⢀⣾⣿⣿⣿⡶⢾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢹⠀⠀⢀⡀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⠏⠀⠿⠟⠉⠞⢋⡅⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠘⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⢸⠠⠈⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⢼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⡘⢀⠇⠀⠀⠀⡀⠈⠓⠚⠁⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⠀⠀⠀⢠⠄⠀⢿⡀⠀⠀⠀⠀⠀⣠⡟⡛⠛⠖
⢀⠠⢄⡀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠐⠢⣄⡀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠘⣧⠀⠀⠀⠀⠀⣇⠀⠃
⢃⠀⠀⠀⠀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠙⣿⡄⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢈⡇⠀⡠⠔
⠈⠳⣤⣤⣴⣿⣿⡿⣃⣤⡀⠀⠀⠀⢠⡟⠻⢦⣨⠂⠀⠀⠀⠀⠸⣿⠿⢿⡿⠿⡇⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣼⠁⢠⡇
⠀⠀⠈⠉⠛⢉⡵⠏⠉⠄⠉⢷⠀⠀⠀⠻⢦⣤⣀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠔⠃⠀⢀⣴⠿⠤⠤⠤⣄⣀⣀⣰⠇⠀⡾
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠀⣿⠀⠀⠀⠀⠀⠈⠙⠳⣶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠛⢷⡀⠀⠀⣋⣈⣉⣁⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⣄⠈⠳⣄⠀⠀⠀⢀⡴⠋⠁⠀⠈⠉⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠈⢿⡖⠚⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⠷⠴⠞⠋⢱⣀⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⣤⠶⠚⠋⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⠶⠶⠶⠟⠛⠉⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠉⣻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣟⠀⠀⢰⡟⠀⠉⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣁⠂⠄⣠⡟⠛⠛⠛⠛⠛⠉⠁⠀⠙⠻⣶⡟⣩⠷⠂⢀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠉⣿⠶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⢁⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣤⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⠈⠛⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣄⣀⣽⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠁
⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣀⣀⣠⣀⣀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⠄
⠀⢰⣿⣿⣿⣏⣿⣿⣿⣺⣿⠿⣄⠀⠀⠀⠀⠀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣻⡇⡀⠀⠀⠀⡰⠛⢆⢀⡀⠀⠀⣀⢀
⠀⢸⣿⣿⣿⣿⣿⣿⣽⣿⣿⡿⣏⠑⠀⠀⠀⠁⠙⡿⢂⣀⣧⡀⠈⠈⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠘⣿⣿⣿⣿⣿⣿⣻⣿⣿⣶⢩⣀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡁⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⠀⠂⠀⠐⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠒⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣎
⠀⢠⣾⣾⣿⣿⣿⣿⣿⣿⣟⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠄⠀⠀⠀⠐⠀⠀⠀⢣⠄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠐
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠂
⠀⢻⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢰⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠠⠇
⠀⠘⣿⣿⣿⡟⠻⠂⠻⣿⣿⣿⣷⡄⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣯⣼⣟⠀⠀⠠
⠀⠀⠛⠿⠡⠄⠀⠀⠀⢿⣿⣿⣟⣻⡄⠀⠀⠀⣾⣿⣿⣦⡄⠂⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⣿⣿⣿⣧⠀⠀⠀⢀⠄⣨⣿⣻⡁⠈⢷⡀⠀⡀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧⡀⠀⢈⡿⣿⣿⣿⡆⠀⠹⢦⣄⠀⠀⠀⢢⣼⣏⣰⣿⣿⣿⣿⡆⠀⠀⠛⠀⣹⣿⣿⡗⠀⠀⠻⣤⠓
⠀⠀⠀⠀⣀⣠⡤⠶⠀⠀⠀⢹⣿⣿⣯⡁⠀⡸⠀⢿⣿⣿⣿⣦⠀⠀⠉⠑⢤⢀⣿⣽⣿⣿⢿⣿⣿⣿⡟⠲⣤⡀⣰⣿⣿⣿⠀⡀⠀⠀⠈⢷⣄
⠀⠀⣴⣿⣿⠗⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣾⡇⠀⠈⢿⣿⣿⣿⣿⣶⣦⣤⣤⣾⣿⣿⠋⠁⣼⣿⣿⣿⠃⠀⠀⣶⣿⣿⣿⣿⠐⣏⠁⠉⠖⡈⠻⡷⣄⠳
⠀⣺⣿⣿⢫⣤⣤⣾⣿⣶⣂⣴⡄⣿⣿⣿⡿⠗⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠇⠀⠈⠀⠳⠈⠁⠀⠘⠫⣟
⠀⣿⣿⣿⣇⠙⠛⠻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠖⠛⣴⣿⣿⣶⣎⣤⣋⡀⠈⠃
⠀⠚⢿⣿⣿⡆⠸⣆⢠⠬⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠘⠻⣿⢯⣤⣿⣿⡿⠋⠋⡙⠁
⠀⠀⣻⡟⢙⣧⠶⢻⡇⣄⢰⣾⣿⣿⣿⣿⣥⣾⡿⠋⠀⢰⣄⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⡀⢤⡀⠱⣄⣄⣠⣿⠂⢀⡾⠋⠀⠀⠜⡟
⠀⠐⣿⡶⢊⣿⣄⣲⣿⣿⣿⣿⣿⣯⣶⡾⠛⠁⠐⠂⠤⣹⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡠⠶⠄⠐⠊⠳⡄⠙⢿⣿⣿⣿⣿⣷⣀⠀⠀⢀⣫
⠀⠀⠍⠻⣩⣿⣿⣿⣿⣿⣿⡟⠛⣿⡟⢠⣴⣾⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠉⠛⢶⣶⣾⣿⣷⠀⣤⣴⡍⢻⣿⣉⣥⠿⣷⣶⠀⢉
⠀⠀⢠⣾⣿⢯⣿⣿⣿⢿⣿⡷⣼⢿⣇⣸⣿⣿⣿⣿⡽⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣤⣶⠀⢠⣾⣿⣿⣿⠟⠀⠉⠻⣿⣿⣿⣿⣿⣷⣼⣿⣃⢶
⠀⠀⢈⣿⡏⢷⣩⣟⣤⡈⢻⣵⣦⠻⣿⣿⣿⣿⣿⣿⣗⣿⣿⣿⣻⣿⣿⠛⠛⠛⠃⠚⣿⡿⢛⣡⣶⣿⣿⣿⣿⣿⣇⡀⢀⣷⣿⣿⣿⣻⣿⣿⣿⣿⣷⣶
⠀⠀⠄⠾⢿⣮⣙⣝⣿⣿⣿⣿⣿⣧⣾⣿⡿⣟⡿⢿⣿⣿⣿⣏⣿⣿⣿⣆⡠⠄⠀⣠⣾⣤⣾⣧⣬⣟⣿⣿⣿⣿⣿⣷⣄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⡿⠁⠀⢹⣿⣿⣧⣿⣿⠟⠁⣠⣤⣼⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣟⠋⢻⣿⣿⣿⣿⣆⢀⣾⣿⣿⣿⡿⠁⠀⠀⠻⣿⣿⠿⣿⣾⣿⣿⣿⣿⣿⡿⠋⠉⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠂⠀⢰⡟⠛⠁⠀⠀⠄⠙⠀⠘⣿⣿⣿⣿⣿⣟⣿⣿⣿⠟⠀⠀⠀⢠⣦⣿⣥⣀⣙⣹⣿⣿⣿⣿⣿⠃⠠⠂⣷⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢠⣿⣷⣶⡄⠀⣀⣆⣡⣶⣠⣈⣷⡘⣿⣿⣿⣿⣿⣿⣿⣧⣦⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡝⢳⣄⠀⠹⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠸⣿⣿⣿⢧⣾⣿⢋⣿⡿⠉⠈⣹⣿⣾⣿⣿⣿⣻⣿⣯⡷⣯⣐⣌⣉⣉⢿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠇⠈⠓⠒⠶⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⢿⣿⣿⣿⣷⣿⣿⣷⣶⣠⡽⣯⡛⠿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣟⠁⠑⠚⠛⠰⣿⣿⣿⣏⠡⠰⠠⠐⠶⣦⣤⡚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢰⣶⣾⣿⣿⣿⠋⣉⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣯⣾⣿⣿⣿⠿⣿⣿⣿⣿⡆⠴⢠⢴⣾⣿⡿⢿⣾⣧⡀⠀⠀⠀⠸⣿⣿⣷⣮⣿⣿⣿⣿⣿⣿⠛⣹⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⢦⠀⠀⠀⠀⠠⣾⣿⣿⣿⣤⣄⠀⠀⠈⠻⣿⣿⣿⣮⡍⣿⠿⣿⣌⣿⠆
⠀⢸⣿⣿⣿⣿⣿⣿⣉⠽⠿⣿⢿⣿⣏⣾⣿⣿⣿⣿⣻⣿⣿⣿⣟⢿⣯⣵⣿⣿⣈⣦⣤⣄⣀⣀⣈⡙⠻⣿⣿⠾⠷⠶⣶⣶⣿⣿⣿⣿⡿⠘⠖⡘⢿⠻
⠀⠀⠉⠻⣿⣿⣿⣯⢻⣶⡴⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡏⢼⣿⣆⠻⣿⣿⣿⣭⣹⣟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣡⣌⡻⣿⣯⣀⢀⣠⣶⣦⡤
⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣧⣼⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣆⢹⣿⡇⢙⣿⣿⣿⣿⣿⢷⣦⣤⣬⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⢠⠀⡀⠀⠈⠛⣁⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣷⣮⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢰⣤⣈⡻⠃⠉⠁
⠀⠈⠁⠀⠀⠀⠀⠙⠛⠛⢿⠟⠻⣿⣿⣙⠿⣯⡉⢹⠟⠛⣿⣦⣿⣿⣿⣿⠄⠀⢀⣠⣟⠿⢿⡟⠙⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠁⢀⣐⣆⡄
⠀⠀⠜⠂⣀⡀⡀⢀⡐⠲⢿⡇⠀⡙⣿⣿⣶⣾⡿⠿⣿⣧⣴⣾⣿⣿⣿⡿⠁⠀⠈⠁⠙⢷⣾⡷⡿⢶⣴⣿⡿⣿⣿⡏⠉⠹⠿⣿⣿⣿⣷⣰⡍⠩
⠀⠀⠀⠁⢠⡤⠠⠦⠈⠙⠲⠚⢳⣷⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣼⣿⣷⡄⠀⢐⠄⡘⢿⡇⠀⣨⣽⣷⡗⡤⢁⠈⠀⠰⣷⣦⣝⣿⣿⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠐⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⣿⣿⠿⠟⠀⠈⠀⠀⠀⠉⠀⣁⣬⠍⠛⠿⠄⠀⠀⠓⠀⠠⠍⠛⠿⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠰⠤⢀⠒⠂⠀⢠⣀⢤⠄⠀⠁⠀⠂⠀⠀⠀⠀⠀⠞⠛⠛⠂⠀⠀⠀⣴⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⠀⡠⢲⣤⡀⡀
⠀⠁⠈⠀⣀⠀⠀⠀⡀⠸⠿⠃⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠀⠀⢠⣀⠐⡂⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠂⠀⠀⠀⢴⣀⡀⠁⠈⠀⢠
⠀⠀⠘⠀⠀⣰⢀⣉⣀⠰⢆⠀⠀⠈⠠⢠⠀⠀⠠⠔⠀⠞⢠⣤⡤⠸⠂⠀⠸⠀⠀⠀⠤⢀⢆⠀⡂⠀⠈⠁⠀⠀⠀⡄⠈⡊⣀⢾⠀⠀⠀⠔
⠀⠀⠀⠠⠀⠼⠀⠛⠀⠀⢘⢃⠀⠀⡌⢡⠀⠒⢀⠌⠒⠀⢀⠀⢀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⣠⠐⠒⠀⡄⠀⠙⠏⠀⠀⠢⠀⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⢀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⠀⠀⠀⠀⠀⠐⠋⠤⠠⠁⠒
⠀⠀⠀⠈⠀⠈⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠒⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⣠⠏
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡠⠋⠀⠀⠀⢀⡞⠁
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠟⠿⣿⣿⡇⡴⠁⠀⠀⠀⣠⠏
⢸⣿⣿⣿⣿⣿⣿⣿⡟⠁⣰⠟⠁⣰⡿⠛⣿⠟⠻⡟⠋⢉⡉⠙⣿⠋⢉⠉⠙⣿⠟⠉⠈⢹⣿⠋⢀⣼⠋⠀⣴⣿⣿⠏⠀⠀⠀⢀⡼⠃
⢸⣿⣿⣿⣿⣿⣿⠏⠀⠰⠋⠀⣰⠟⢀⣼⠏⢀⡾⠁⠰⠋⠀⣰⠃⠀⠋⠀⡴⠃⢀⡄⠀⣾⠃⠀⠈⠁⠀⣼⣿⣿⠋⠀⠀⠀⢠⠞
⢸⣿⣿⣿⣿⣿⠏⠀⣀⠀⠀⣰⠏⠀⡾⠃⢀⡾⠁⣀⠀⢰⣿⠃⠀⠀⢰⡟⠁⠀⠀⠀⣸⠃⠀⣴⠂⠀⣼⣿⡿⠁⠀⠀⠀⡴⠃
⢸⣿⣿⣿⣿⠏⠀⣰⡏⠀⢰⣿⡄⠀⣠⠀⣼⠁⢠⣿⡄⢈⣇⣀⣾⡄⢈⣇⣠⣿⡇⢀⣏⠀⣸⡏⠀⣸⣿⡟⠀⠀⠀⢀⡞⠁
⢸⣿⣿⣿⣿⣶⣤⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣠⠏
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠭⣿⠋⣤⡌⢻⡏⠨⢙⣿⣿⣧⠀⣽⡇⠹⠏⢹⡇⠰⣿⣿⣿⣿⣿⠏⠀⠀⠀⡰⠃
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣦⣉⣡⣾⣇⣤⣘⣿⣿⣿⣠⣿⣇⣼⣇⣼⣇⣈⣿⣿⣿⣿⠏⠀⠀⠀⣼⠁
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣼⣿
⢸⣿⣿⣿⣿⠃⢠⡄⢹⣿⠀⢸⣿⣿⣿⠃⠀⢸⣿⡟⠀⣤⠈⣿⣿⠁⣤⡀⢻⣿⣿⣿⣿⣿⠏⠀⠀⢀⣾⣿⣿
⢸⣿⣿⣿⣿⠀⣸⣷⣾⣿⠀⢸⣿⣿⣿⠀⡇⠘⣿⣷⡀⠙⢿⣿⣿⣄⠈⠻⣿⣿⣿⣿⣿⠏⠀⠀⢠⣾⣿⣿⣿
⢸⣿⣿⣿⣿⠀⢸⡏⢹⣿⠀⢸⣿⣿⡏⠀⡁⠀⣿⡿⠻⣦⠀⢹⡿⠻⣷⠀⢹⣿⣿⣿⠋⠀⠀⢠⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣦⣈⣠⣾⣿⣄⣀⣁⣽⣇⣼⣿⣀⣿⣷⣄⣀⣠⣾⣿⣄⣈⣠⣾⣿⣿⠏⠀⠀⢠⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⢿⠟⠻⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⠿⢿⡿⣡⣾⣦⣀⢈⡉⠁⣠⣾⠏⣠⣿⠏⠀⠚⠋⠀⠋⠙⠛⢿⣿⣿⠏⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⠏⢄⠄⠐⣫⣿⣿⣿⠟⠁⣴⣿⠟⣺⣿⠃⣰⡶⢀⣾⠟⢙⣿⠟⢻⣿⠏⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣷⢏⡜⢠⣿⣋⣁⠀⢀⣤⣤⠁⣴⠟⠁⡼⠋⢠⠟⠁⣰⠟⠁⣰⣿⡏⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⡏⠘⣠⣿⣿⡿⠃⢠⣿⣿⣿⣀⡄⢀⡄⢰⣧⣠⣾⣄⡤⢀⣼⣿⡟⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣧⣴⣿⣿⣿⠁⢰⣿⣿⣿⣿⡟⢀⡞⢀⣿⣿⣿⣿⡟⢁⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡁⠈⢠⣾⣿⣿⣿⡿⠀⣼⣿⡿⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡃⠀⣰⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⣠⡾⠋⠁⠉⢻⣦⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠈⣿⡇⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠸⣿⡀⠀⣠⠃⢰⣿⡇⠀⣸⣿⠀⢠⣦⠀⢠⣤⢰⣟⡟⠀⠀⢿⣼⣀⡀⣠⣴⠶⡄⠀⠀⣿⢇⡴⣶⡄
⠀⠀⠀⠀⠀⠉⠛⠋⢁⣤⣼⣿⣤⣤⣿⣧⡤⣸⡿⠀⣸⡟⠀⡝⣹⡗⠀⡰⢱⣿⣷⡿⠁⢠⣝⠀⣸⣿⠏⢰⡿⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣿⡇⠀⣸⣿⠁⠀⣿⢃⣼⣿⢁⡜⠀⣿⠃⡴⢁⣿⣿⣿⠁⡰⣿⡏⡰⣿⡏⠀⣿⠇⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⣿⡇⠀⠀⠛⣋⣈⣿⣯⣤⣤⣿⣏⣀⣸⡿⠈⣛⡋⠁⠙⠛⠀⠙⠃⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⢸⣿⠀⣤⣾⣿⠿⠛⠋⠉⠉⠉⠉⠉⠉⣿⡷⡞⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠜⠃⠊⠁⠀⣤⠐⠀⠀⠈⠀⠀⠀⠀⠉⠉⠁⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⠂⠶⠤
⠀⠀⠀⠀⡀⣐⡀⣍⠀⠀⠀⡀⠀⢀⠀⢠⢀⣈⠱⠀⠠⠂⠀⢀⠠⠀⠄⠀⠂⠀⠀⣀⢐⠂⣼⠂⠀⡀
⠀⠀⠀⠠⠇⠉⠄⠈⠁⠀⠄⠁⠐⠠⠑⠀⠀⠀⠄⠐⠀⠀⠀⠀⠘⠉⠀⢀⠀⠂⡰⠀⠙⠁⠀⠆⢀
⠀⠀⠠⠀⠄⠀⠀⠀⠄⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠢⠀⠀⠀⠀⠀⠁
⢀⡠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣠⠀⠀⠀⠀⠀⠀⠁
⠃⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⢃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⠌⠁⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠄⠐⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠈⠁⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⢁⠀⡜⠦⠀⠀⠀⠤⠠⠔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠃⠀⠒⠌⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠀⠀⠀⠀⠀⠀⠠⠖⡀⠜⢦⠀⠀⠀⠈⠠⡄⠀⠊⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠂⠀⠀⠀⡂⡄⠀⠀⠀⠆⠄⠀⠀⠀⠀⠗⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠳⡔⠀⠁⠀⡀⠀⡀⠀⡄⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠐⠀⠀⠆⠆⠁⠀⢙⠂⠐⠀⠁⠄⢉⡀⣧⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⣁⣀⡀⠀⠀⢀⡀⣄⣀⠄⣁⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣼⠃⠀⠁⠀⢽⠷⠶⠒⠶⠉⢠⡟⣀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⠉⠄⢈⠀⠀⠀⡀⡰⢆⠀⠐⠢⣀⠸⠼⣷⣤⣂⡀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠤⠀⡈⢁⠀⠀⠂⠈⠀⠙⡠⡈⢁⡀⠘⠁⡠⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠐⠀⠀⡀⠰⠀⠀⠀⠀⠀⠘⠃⠀⠤⠀⢀⠋⠰⢤⡄⠚⡡⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢘⠀⠀⠀⠀⠀⢠⠀⠆⠀⠀⠀⠀⠐⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠂⠀⠀⢀⠀⠀⢨⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⢄⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠓⠀⠈⠠⠄⠰⠀⢀⣀⠴⠀⠳⠀⠂⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣀⣀⣪⣤⣤⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄⠀⠰⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢉⠕⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠅⠎⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠄⠀⠀⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠱⠔⠁⠏⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠁⢀⡀⠘⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠄⠩⠲⡤⢨⠇⣽⠀⠆⠠⠄⠸⠆⣀⣀⡀⠀⡀⠀⠀⠀⠀⢠⣄⠻⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣀⠠⢂⠁⠠⠰⡃⠀⠈⠀⢡⠀⡀⡈⠠⡄⠀⠉⠉⠄⡔⡄⣈⢆⢣⠙⡜⣆⢂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠈⡷⠼⠀⠛⠘⠘⠗⢠⠢⠆⠜⣠⠈⠀⠁⠐⢸⠀⠀⠀⠀⠡⠈⡌⠃⢉⡈⢀⡆⢈⣀⢀
⠀⠀⠀⢀⠀⠄⠀⠀⠀⠠⠖⠂⠲⠀⠀⠀⠐⠀⠀⠀⡼⡄⠓⠀⠀⠀⡀⡆⠀⠀⠀⠄⢴⠢⠉⡤⠾⠤⠤⢠⣀⠀⠀⠀⠀⠀⠀⠀⠠⢟⠎⠈⢅⠢⡀
⠀⠀⢪⠀⢂⠞⠁⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠍⠀⠠⠈⠣⠄⠀⠀⠁⢧⠀⢀⡀⡤⠂⠁⠈⠁⠀⠀⠀⠀⠀⠆⠘⢂⠢⢠⠀⠘⠀⡖⠀⠀⢀⠀⠐
⠀⠀⠚⠘⠚⠗⢂⠀⠀⠀⠘⠀⠀⠄⡺⠀⠄⡘⠀⠀⠒⠒⡬⠆⠀⠀⠀⠃⠀⠈⡆⢀⠈⠠⠀⣄⠀⠀⠀⠀⠀⠀⡀⡈⠀⣄⠀⠀⠀⠠⡌⠲⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡸⠀⢠⠚⠁⠀⠐⡄⡇⡆⠔⣠⠀⠀⠀⠁⡀⠃⠀⠀⡆⠀⡀⠆⠧⠮⡼⠀⠀⠀⠂⠀⠈⠀⠀⠀⠉⢁⠚⢛⠀⠀⠀⠀⠈⠂⡀⠀⠀⡄
⠀⠀⠀⠙⠀⠘⠀⠀⠃⠀⠀⠀⠠⠐⠉⠁⡇⠀⢀⠀⠀⠸⠀⠀⠀⠰⠀⠀⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⡀⠘⡀⠀⠀⠀⠀⠀⠀⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠐⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠀⠃

⠀⠀⠀⣄⡄⡠⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡤⠤⠄⠀⠈⠁⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⢀⡀
⠀⠂⠀⢀⠈⠀⠀⠀⢿⠏⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠂⠀⢈⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠁⠘⠋⠊⢀
⠀⠀⠀⠀⢠⣄⣉⡀⠶⠀⠀⠈⠀⡤⠀⠀⠄⠀⠆⠄⣤⠀⠂⠀⠈⠁⠀⠠⢀⢄⢐⠀⠀⠉⠀⠀⢠⡀⣑⢁⣼⠉⠀⠠⠂
⠀⠀⠀⠄⠼⠄⠋⠀⠀⠉⠀⠀⠈⠂⠐⠀⠎⠐⠀⠤⠀⠄⠀⠃⠀⠀⠀⠀⠈⠋⠀⠀⢠⠀⠈⢠⠆⠈⠛⠀⠀⢓
⠀⠀⠀⡀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⡀⠀⠀⠈⡐⠀⠀⠁⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⣤⣾⣿⣿⣦⣤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡟⢽⠆⠀⣿⣿⡿⣿⣿⣿⣿⣧⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠄⠌⣠⠥⠰⠂⣠⠊⣀⣉⠀⣀⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⣐⣀⠀⠀⠀⠀⠂⢀⣀⡀
⠀⠀⢦⣀⣀⡠⢀⡆⢻⣆⣀⣯⣷⢷⣿⣏⣿⣿⣿⣿⣿⠻⣽⣿⣟⣽⣧⣾⡋⣻⣧⡀⠀⠀⣾⠿⢟⣿⠂⠚⠘⠠⠐
⠀⠀⠈⢀⠈⠃⠺⠄⢿⣿⣿⣻⣷⡒⢿⣿⣿⣟⣿⣹⣛⣚⣟⣛⣟⣛⣻⣿⣿⣿⣿⣿⣾⡆⣷⠒⠻⠟⢓⣠
⠀⢠⣷⠌⠷⠶⠠⢶⣼⣿⣿⣼⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⢤⡆⠀⠢⣴⣆⠀⣴
⠀⠀⠀⠀⠀⣠⣤⡾⠿⣶⣫⣽⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣴⣦⣆⠀⠀⡋⢀⠃
⠀⠀⠀⢶⣿⣼⠥⢿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣼⣷⡔⠐⠟
⠀⠀⠀⡄⢹⣛⣿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣯⡼⣻⣴⣯⣅
⠀⢀⣴⣇⡊⣿⢫⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢠⣾⠧⠦⠙⠸
⢀⡄⢹⡿⢿⣿⣿⢽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⣠⠘⣷⣆⠄⠀⢀
⠈⠁⠈⡽⠘⢋⣿⣾⣿⠛⣻⡋⣿⣿⣿⣿⣧⣛⣓⣋⣻⣧⣿⣟⡓⠛⠊⡹⢿⠿⠟⣟⡿⣻⣿⠇⣇⠀⣿⡿⠈⠀⢩
⢸⣇⣱⡗⡀⣌⣿⣿⣇⣸⢉⣿⣯⡙⣩⠟⠹⠻⡿⢿⡿⢿⣻⣿⢛⢋⣿⡩⡽⣆⠀⢀⢰⣿⣓⣿⣿⣦⡿⣼⡄⠀⡟
⠘⠘⢈⣇⣇⣿⣹⠈⠏⠘⠃⠙⠿⠷⠟⠁⠴⠞⠙⠓⠒⠛⠁⠀⣹⣿⠿⠛⠛⣡⣶⣭⠈⣿⣿⡇⣟⢸⣧⣹⠇⣰⠈
⠀⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠼⣿⠆⠤⠇⣦⠀⠍⠀⠀⠉⠉⠀⡥⠸⠓⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠻⡛⢯⠐⠚⠒⢸⣦⠀⢇⠀⢀⡨⢦⠓⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢣⠀⠀⠼⠾⡛⠺⠒⠒⠂⢙⡉⣥⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠲⢧⠀⢣⠠⠒⡆⠹⡂⠀⠀⣀⠯⣥⣀⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠶⢆⢄⠃⠀⠉⡿⣄⡀⠱⡀⠀⣀⠀⠀⢹⠮⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣡⡄⢐⣂⡀⢹⡀⠜⠸⡶⠶⡉⠀⠀⡏⠥⠌⠒⢄⣠⢟⣉⢲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣆⠤⠄⢛⣃⠀⠘⡄⠱⡀⠀⡇⠂⣣⣭⣄⣾⡧⢷⠾⣟⠋⢁
⠀⠀⠀⠀⣀⠜⠄⠌⠡⠀⠤⠄⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠠⡄⣀⠀⢀⢈⠀⠀⡘⡄⠱⡄⠀⢀⣙⡄⠙⣦⠤⠶⢧⠈⢻⠉
⠀⠀⠀⠀⠫⢠⡄⣀⠀⠆⠀⠀⣀⡀⡘⡀⠀⠀⠀⠀⠀⠀⠀⡠⣉⠎⣇⢨⠿⠿⡿⢋⣠⡤⠼⡄⠘⠯⠽⠋⡙⡄⠘⢆⠐⠀⢳⡀⢣
⠀⠀⠀⠀⠀⠀⠀⢃⠂⠠⢁⣀⣨⡄⠴⠐⠂⠀⣀⠀⠀⢠⣆⠛⠉⢉⠟⡗⢤⡀⢄⠁⠑⠆⠀⠀⠈⠀⣀⣀⣤⡼⡄⠈⢿⣋⠭⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠈⠉⠉⠀⣀⣀⣠⣠⣴⣦⣽⣿⠛⣧⢹⣄⠀⣸⠀⣱⡄⢈⣬⣷⠶⠶⠶⡚⠛⠉⠉⠁⢀⠀⠸⡄⠈⢳⣈⣉⣀
⠀⠀⢴⣆⢀⣺⣆⣴⡀⢲⣞⣿⠉⢿⡅⠸⠯⠿⠓⢛⠀⢉⠀⠠⡄⠈⠀⠀⠁⠀⠀⠀⠀⠀⠈⠘⡶⢢⡁⢀⣤⠤⠶⠺⠄⠀⣛⣻⠫
⠀⠀⠘⣿⡏⠉⠿⠜⠷⠚⠛⠉⣁⠀⠄⠀⠐⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢳⣸⡄⠀⠰⡁⠀⠀⠀⠀⠀⢾⠊
⠀⠀⠀⠉⡄⠀⠆⠀⠛⠀⠋⠀⠀⠀⠠⠀⢀⣘⣨⣤⣴⣄⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠡⡀⠀⠉⠁⢀⣀⣀⡈⠑⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣽⢾⠿⠃⠸⡕⢏⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣛⣉⠩⠤⠔⠃⢀⣠⣧⡤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠔⡩⠋⠩⠙⠙⠒⠀⠀⠙⡎⣠⠗⡢⠀⢀⣀⡀⠤⠄⠂⠆⢉⣩⠒⠤⢤⡶⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢗⠲⠤⣄⣀⠀⠀⣴⢺⠂⠀⠀⠀⣀⣀⡼⡆⠀⠀⠘⣏⣁⣠⣤⣶⡶⠶⠛⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⢆⢀⣴⣒⡍⠶⠟⠓⠒⠂⠉⣁⣀⣠⣴⠃⠀⠀⠀⡘⡏⠹⡅
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠄⠈⢦⠳⣀⣠⣤⣤⠶⠾⠟⠛⠋⠙⢍⡉⠉⠀⠀⢣⠱⡙⣄⣧⣤⣤

⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⢸⡳⡜⣎⢳⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣰⢏⡞⡰⣻
⠀⠀⢻⡳⡝⠌⠂⢿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢸⠏⠚⠼⡵⢻⠇
⠀⠀⠈⢳⡀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠙⠦⡄⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢧⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠟⢹⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⣠⠏⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠳⡄⠀⠀⠀⠀⠸⣿⣿⡿⠛⠫⠭⠉⠉⠀⠀⠀⡤⠄⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⡴⠃⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠙⣦⠀⠀⠀⠀⢿⣿⠃⠀⣀⣤⣀⠀⠀⠀⢀⣤⣄⠀⠈⢿⣿⠃⠀⠀⠀⢀⡞⠁⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠈⢧⡀⠀⠀⣏⠀⡀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠈⠁⠀⡀⢸⡇⠀⠀⣠⠏⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢱⡤⣄⠈⠒⢷⡀⠀⠀⠀⡴⠶⠶⡄⠀⠀⠀⣸⠗⠋⢀⡤⣴⠃⠀⠀⢠⠟
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠈⢷⡤⠞⠉⠀⠘⢦⡀⠀⠱⢄⡀⠀⠉⠓⠋⠁⠀⣠⠞⠁⠀⣠⠞⠀⠈⠙⠦⣤⡏
⠀⠀⠀⣠⠟⢧⡀⠀⠀⠀⠀⠐⢯⠀⠀⠀⠀⠀⠀⠙⠦⣤⠶⣟⠓⠶⠤⠶⠖⢋⡷⢦⡤⠞⠁⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠙⢧⡟⠁⠀⢠⡀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠹⣄⠉⠓⢄⣀⣠⠴⠋⢁⡼⠁⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠈⠀⠀⣠⠞⢧⡀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠈⠢⠤⣶⣿⣻⡶⠤⠊⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡾⠉⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣀⣀⣀⣀⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⣿⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣀⣀⣸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣛⣟⣛⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⣛⣟⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢻⠆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡟⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠃⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡟⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠀⣾⣖⣒⣛⣻⣭⣭⣭⣭⣭⣽⣿⣓⣲⣾⣶⣖⣾⣿⣿⣿⣿⣿⣿⣿⣟⣒⣒⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⣠⣤⠀⠀⠀⡸⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠈⠀⠀⠀⠙⠋⠀⠀⠀⠃⠀⠀⣀⡀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⣠⣷⡞⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠒⢤⡀⠀⢰⣿⡿⠉⢈⡇⠀⣠⡤⢤⣄⠀⠀⢀⣀⣀⠀⣾⡏⣿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⠋⠁⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⢀⡼⠁⠐⣿⠀⠀⠘⡆⢰⣿⠁⠙⣧⠙⣿⣿⣤⡄⠀⣴⣿⠟⢻⣶⡄⢠⣦⠀⠀⣠⡄⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡧⠛⢀⡀⠀⢹⣿⡇⠀⣿⢀⣿⣿⢦⣿⠞⠁⣼⣿⠇⢸⣿⠃⢀⣾⣿⢀⣿⡿⠀⢠⣿⣿⠀⣾⠃
⠀⠀⠀⠀⠈⠉⠀⢀⣤⠾⣿⣇⡼⢻⣿⡄⣿⣿⠙⢺⡿⢾⣿⡇⢠⡿⠀⠀⣿⣿⢀⣿⣿⣤⡾⣿⡇⣼⣿⡇⠀⣾⣿⡇⢀⡃⠀⠐⠒⠋
⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⢿⣿⠃⢸⣿⠀⣹⣿⣤⡿⠁⠈⠿⣷⠿⠁⣀⡀⠙⠿⠟⠈⠛⠋⠀⠙⠟⠙⣿⣷⡾⢻⣿⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠘⣿⡄⠸⣿⣤⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠛⠓⢀⣀⣤⣤⣄⣀⣀⠀⠀⠀⠈⠉⢠⣿⠇⠐⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠈⠀⠀⠈⠉⣀⣤⣶⣾⣿⣿⣷⣶⣦⠄⠀⣴⣿⠛⠉⠉⠉⠉⠙⠛⠶⢦⣄⣠⣿⡟⠀⢀⣴⡿⠁⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢺⡾⠀⠀⠈⠉⢙⣿⣿⣿⣿⣿⣯⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠙⠻⠿⠟⠋⠁⠀⠀⠀⠘⠿⣦⣤⣀⣀⣀⣠⣤⣶⡿⠟⠁⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠈⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠄⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⣴⣿⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⢿⣆⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⣴⡿⠋⠀⠀⠈⢧⡻⡄⠀⠀⣤⡆⠀⠀⢀⣴⠟⠁⠀⠀⣠⣴⠿⡟⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⡾⠋⠀⠀⠀⠀⠀⠈⢧⢱⡀⠀⠛⠃⠀⣠⡾⠋⠀⢀⣴⡾⠋⠁⠀⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠞⣡⣤⣤⣤⣄⡀⠀⠀⠈⣧⢧⠀⢀⣠⣾⣿⣤⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢠⡾⠛⠋⠉⠉⠛⠻⣷⣄⠀⠈⢿⣷⢟⣛⣩⣭⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠇
⠀⠀⠀⢀⣀⣤⣤⣶⣶⠿⠟⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢸⠁⠿⠛⢛⣻⡷⠀⠀⣰⣶⣶⣤⣤⣴⣶⣶⣶⣶⣾⠿⣦⣄⣀
⠀⠀⠀⣿⠛⠻⠿⠿⠷⣶⣶⣾⣿⣷⣶⣶⣤⣤⣤⣤⣤⣴⣾⣿⣾⣿⣿⠟⣿⠛⠁⠀⠿⠿⠿⠛⠻⢿⣿⣍⣩⣭⣭⣥⣤⣶⣿⣿⣿⣿⣷
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⢉⣽⠿⠋⣡⡿⢫⡿⠋⠁⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⠛⠛⠛⠋⠉⠉⠉⠉⡏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⣼⠋⢠⡟⠀⠀⠀⠀⠀⣶⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡼⠃⢀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠘⠻⠿⣿⠿⣿⣿⣿⣶⣶⣶⠟⠁⠀⠀⢸⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣾⣿⣿⣿⣿⠇
⠀⠀⠀⣶⠀⢰⡇⠀⠀⠀⠀⠈⣹⠏⢀⣠⡆⠀⠀⠀⣸⢷⣶⣶⣦⣤⣄⣀⡀⢸⡇⢀⣠⣤⣾⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⢠⣇⣴⠿⠋⠁⠀⠀⢀⡏⠀⠀⠉⠉⠛⠛⠿⢿⣿⣿⠿⠟⠛⠉⢹⠀⠀⢠⣿⣦⠀⠀⡸⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡟⠁⠀⠀⢸⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡆⢀⡟⠀⠘⣇⣰⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢡⡾⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣶⣦⣤⣄⣀⣀⠀⠀⠀⣸⡇⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣶⣾⣧⣶⣶⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⢀⣐⡍⠑⣠⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⡠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢤⠛⠶⢤⠷⢸⡷⠞⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⣀⣾⢦⣷⠠⠂⢉⡳
⠀⠀⠀⠀⠀⠀⠓⡐⢲⡎⠋⣍⠉⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢸⡧⠓⢁⠋⢉⠉⠉⠉⠉
⠀⠀⠀⡀⠀⠀⠠⠀⢨⣇⢠⠈⡑⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⡂⡒⢄⢀⡠⠐⠂⣻⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢰⣁⡀⢸⢀⡀⢡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⡌⠈⠁⠀⠈⠃⠀⡁⠀⠀⠀⠀⠐⠀⠀⠊⢉⡹⢢⣍⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⠀⠀⠈⠀⠈⣡⣬⠈⠁⠉⠈⢃⣀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢁⡈⢈⠉⡇⠀⡦⠉⢀⡲⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠐⠀⠈⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠈⠀⠀⢡⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣀⠅⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠂⠚⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⡁⠀⠀⣀⡸⠀⢸⣀⣀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⣄⠀⠦⡀⠀⢀⠀⠟⡀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣀⠀⢀⣀⣀
⠀⠐⢄⠂⠈⠒⠁⠀⠀⣻⡆⠀⠞⠤⠊⠠⠖⠡⠂⠴⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠦⠶⣛⢉⡩⢒⡛⠛⠻⣟⡓⠁⡀⠀⠑
⠴⠂⠀⠒⠒⠰⠷⠾⠲⠿⠿⠿⠒⠒⠢⠔⢲⠄⠈⠰⠐⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⡇⠀⠈⡀⠀⠀⠀⢆⠀⠀⡄⢠⠊⠉⠀⠁⠈⢛⠂⠀⠀⢀⡀
⠐⠀⠂⠉⠀⡀⠀⠀⠀⠰⠀⡎⠐⠂⠀⠀⠀⠂⠲⠘⢳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠜⠈⠉⣷⡋⠀⠻⠈⠁⠉⠈⠾⡅⠈⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠀⣠⣄⡀⠀⠀⡀⠄⠀⠘⡆⠀⠀⠣⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡧⠸⣷⠟⠃⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠉⡻⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠩⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⣠⡆⠐⠃⠇⠤⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⡇⠀⢰⡶⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⢀⢤⠤⠤⢄
⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣑⣦⡋⣐⡠⠂
⠀⠀⠀⠀⡈⠉⠘⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡀⢀⠀⡀⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠉⠛⡿⠞⡋⠀⠀⠀⢀
⠀⠀⠀⠠⠇⠀⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠒⠒⠲⠶⠶⢧⡀⣅⡞⢁⠴⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠠⠖⠠⠄⣀⠀⢰⠂⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠴⠂⢤⠀⠀⠉⠁⠀⠀⢠⠄⠙⣀⠄⠀⠀⠈⠉⠈⠁⠓⠀⠤⠆⣀⠀⠇⠀⡨⠦⠛⠉⠁
⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⣀⠀⠈⠀⠐⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰
⠀⠀⢠⠹⣧⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢲⠖⠊⠀⡇⠀⣀⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⠀⠀⠈⠐⠁⢲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠸⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⢠⢀⠀⠴
⠀⠈⠀⠂⠀⠄⢻⣷⡀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠚⠗⢰⠁⠀⠀⣠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⣿⢰⡆⣸⡷⠖⣤⠀⠀⠀⠀⢸⡠⣴⡆⣭⢀⡄⠀⠀⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠀⣿⠸⡇⠸⣧⣠⡏⠀⠀⠀⠀⡾⣰⡿⢸⡏⣹⢉⣡⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟⠀⣿⠀⢿⠛⠀⠐⠛⠃⠀⠱⠿⠁⠿⠀⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠘⠁⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠈⣧⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠶⠀⢠⣤⣤⣄⣀⠀⠀⣀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢻⠀⠀⢠⣾⠟⠑⣆⠀⠤⠀⠀⠀⠀⣽⣇⣠⣿⠇⠀⣿⡆⠀⢶⡀⠀⣰⠇⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣾⡦⠀⣿⡏⠀⠀⢸⡆⣠⣾⠟⣤⠀⣿⣿⣯⠀⠀⠸⠀⢿⡀⠈⢧⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡟⠉⣇⠀⣿⡇⠀⢀⡞⢸⡿⠁⠀⣿⠇⣿⠸⣿⢀⣤⣧⡴⠾⣧⠀⢸⡏
⠀⠀⠀⠀⠀⠀⢰⡿⠃⢹⣷⠀⢹⠀⠘⠛⠒⠉⠀⢻⡇⠠⢄⡟⠀⣿⠀⢹⣆⠹⠁⠀⠀⠘⣧⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠘⡇⠀⠀⠀⣤⣄⠈⠛⠂⠉⠀⠀⠋⠀⠀⠻⣆⠀⠀⠀⠀⠀⠘⠀⠀⣸⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢻⡀⠺⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣄⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠟⠁⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠘⠻⣶⣦⣄⡀⠀⢻⣷⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄⢻⣷⡶⠿⠿⠿⠿⠿⢿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣤⣀⣀⣀⣀⣀⣀⣀⡙⣿⣧⣿⣷⣶⣶⣤⣤⡀⠀⢀⣨⣼⡿⠟⠻⠿⣶⣦⣤⣀⡀
⠀⠀⠀⣀⣀⣠⣤⣴⣶⣾⡿⠿⠟⠛⠛⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡿⠛⠉⠉⠙⠛⢿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⣿⣷⣦⣤⣀
⣶⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣿⣦⣬⣿⣿⡀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣶⣶⣶⣶⣶⣶⣶⣾⣿⣯⣁⡈⠉⠛⠿⣷⣦⣄⡀
⣿⣯⠙⠿⣷⣦⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡾⠿⠛⠛⠉⠉⠉⢛⣿⣷⣄⠀⠀⠀⠀⠀⢀⣿⣯⣀⣤⣤⣄⣀⠀⠀⠈⠉⠛⠿⣿⣦⡀⠀⠀⣉⣹⣿
⣿⣿⠀⠀⠈⠙⢿⣿⣶⣶⣶⠿⠻⣿⣉⡁⠀⠀⠀⠀⠀⠀⣀⣤⣾⠏⣹⣿⢿⣷⣶⣶⣾⣿⣿⡿⠛⠋⠉⠛⠻⣿⣦⡀⠀⠀⠀⠈⣿⣿⣾⠿⠛⠋⢹
⢿⣿⠀⠀⠀⠀⠀⢽⡿⣿⣶⣄⢸⣿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⢁⣴⠟⠁⠀⣠⣿⣏⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡷⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⢰⣿⠃⠈⠳⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠏⠀⠀⠀⣿⣿⣴⣶⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣦⣤⣶⣶⣿⠟⢋⣁⣼⣿⠀⠀⠀⢸⣿⡟⠛⠉⠀⢸⣿⠻⣿⣦⣄⡀⠀⠀⣀⣤⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡿⠟⠛⠋⠙⢿⣿⡿⠿⣿⣿⠀⠀⠀⠘⣿⣇⠀⠀⠀⢸⣿⠀⠉⠻⠿⣿⣿⢿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢿⣿⡀⠀⠀⠀⠹⣿⣆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⣧⡀⢀⣀⣀⣬⣿⣦⡄⠐⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⢿⣿⡿⠿⠛⠛⠉⠉⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⣿⡀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⢿⣧⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⢻⣷⡀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿
⠀⠀⠀⠀⠻⣿⣄⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢀⣠⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⢸⡇⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⢸⣿⡏⠀⠀⣀⣠⣴⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⣿⣷⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠟⠋⠁
⣿⣿⣿⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠤⠤⣄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠛⢄⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⡆⠈⢳⡀⠀⢹⡤⠴⠖⠒⠒⠒⠂⠂⢄⡜⠁⠀⠀⣀⣠⣦⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠱⡚⠉⠙⣄⠀⢇⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⣴⠋⠁⠀⠘⣆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠁⠀⠹⣦⡱⣄⠀⠘⣆⠘⡄⠀⠀⠀⠀⡴⠁⣠⠔⠃⣀⡽⢦⣀⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡳⣄⠘⡆⢹⣦⡀⢀⠞⣡⣞⡥⠖⠋⠁⠀⢀⣩⣷⣄⡇⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢻⡲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣿⡞⠉⢩⡿⠛⠉⢁⣀⡤⠴⠚⣋⣉⣀⣀⣼⠇⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣿⡀⠀⢸⠉⠒⠚⠛⠛⠛⠛⣉⣉⣩⡭⠥⠖⠋⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠉⠢⣄⣀⡤⠴⠖⠋⠉⠀⠀⢀⣨⠽⠻⡏⠁⠉⠉⠙⠯⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⢨⡇⠀⠀⠀⣀⣠⠴⠚⠋⠁⠀⠀⠈⢢⡀⠀⠀⠀⠈⠳⢦⡀⠀⢀⣀⠤⠒⠋⠉⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠢⣀⠀⢸⡇⠀⠀⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⣀⡤⠖⢻⡟⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠑⠼⣇⠀⠀⣿⠈⠑⢦⡀⠀⠀⠀⠀⠀⢀⠠⠄⠊⣯⠁⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠻⣤⡇⠀⠀⠀⠈⠢⢀⠠⠐⠈⠀⠀⠀⠀⢸⠄⠀⢀⣸⡧⠤⠶⠚⠉⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⢈⡗⢄⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⣀⡼⠓⠛⢹⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣸⠀⠀⣼⠁⠀⠈⠒⢦⡀⣿⠀⣀⣠⠴⠚⠉⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⠀⠀⡟⠀⢰⠇⠀⠀⠀⠀⠀⠉⡟⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣸⠇⠀⣾⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⣇⠀⢀⣀⣠⣤⠼⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⡤⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢸⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢸⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠚⠁
⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⢶⡄⠀⠀⠱⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⣀⣀⣀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⠤⣤⣙⣦⣄⠀⠙⣆⢣⠀⠀⠀⠀⠀⠀⡠⠃⢀⠔⠉⠀⠀⡘⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡈⠙⠻⠷⣤⣈⢙⣠⣤⣤⡀⡠⠜⡡⠔⢁⣀⣀⣀⣔⣁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠠⠀⡀⠀⠉⠛⠒⠒⠒⠊⢹⣀⠀⢡⡏⣐⠛⣉⣉⣁⣀⣀⣀⠀⠉⠳⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣄⣀⠠⢤⣤⡄⠁⠒⠒⠒⢒⠶⠾⣿⣆⣸⡇⠀⠙⣦⠨⢭⣉⣀⠉⠀⠁⣰⣧⠴⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠈⠁⠈⠒⢶⡎⣧⢀⣈⣉⣉⣁⠰⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⠤⢄⣀⣀⡤⠖⠇⢸⣦⡀⠀⠀⠉⢻⡇
⠀⠀⠀⠀⠀⢀⠁⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡀⡸⠀⠸⡇⠉⠛⠲⠒⠋
⠀⠀⠀⠀⠈⠓⠒⠛⡇⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⠀⠀⢸⡃⠀⢠⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡾⠉⠢⣸⠀⠀⢨⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠷⠒⠛⢆⢠⠇⠀⠀⢹⠀⠀⠸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⠀⠀⠀⢸⠀⠀⠀⠈⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠈⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡰⠀⡆⠀⠀⢸⠀⠀⠀⠘⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⠀⡇⠀⠀⢸⠀⠀⢀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣠⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⡀⠀⡇⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁
⣿⣏⢹⣟⠻⣿⣟⠟⡻⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⢃⢺⢳⣟⣶⢻⡗⡄⠀⠀⡏⢻⠀⢸⠉⡇⠀⠀⢀⡤⠿⢯⣉⡙⢳⡀⢀⣀⡀⢀⣀⣠⢤⣤⣤⣤⣄⣸
⢸⠈⣉⡈⣿⢹⢨⡥⠃⠀⠀⣿⢸⡀⢸⠀⣧⢔⣒⠺⣰⣿⡷⢹⣹⠆⣷⠛⣆⠙⠋⣰⡏⣸⠀⣿⢁⡾⢹
⠸⡧⠏⠧⢿⣿⣿⠹⣿⣟⣽⣿⣎⣛⣛⠀⣿⠻⡤⠇⡷⢮⡴⠋⢡⡾⠃⠆⢸⡇⢸⣥⣇⣿⢸⣇⡼⠁⢸
⠀⠀⠀⠀⠈⡇⣉⠀⣯⢿⢨⣼⡅⡏⢹⡇⣟⠓⠖⠋⠀⢸⡇⣸⣄⠳⣖⢲⣼⣧⣼⢧⢿⣟⣻⣛⡇⠀⢸
⠀⠀⠀⠀⠀⠷⠟⠓⠛⠛⠛⠛⣇⣷⢸⣧⣼⠀⠀⠀⠀⠸⣧⡿⠈⠧⠽⠆⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⡶⢶⣄⠀⠀⢠⣷⠀⢰⣯⠉⢳⣀⡀⣠⠔⠢⡄⠀⠀⣿⡞⠉⣧⠀⢠⠤⢤⠀⠀⣧⠀⠀⣸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆⠀⢸⡇⠀⢨⡿⠒⠒⡏⠀⢿⣀⠀⢸⡀⢠⡿⠀⢀⡿⠀⠁⠀⢸⡇⠀⢹⡄⠀⣿⠁
⠀⠀⠀⠀⠀⠻⣦⡀⢹⡇⠀⢸⡇⠀⢸⡇⠀⢀⡇⠀⢸⡿⢄⣈⡇⢸⣇⡠⠞⠁⢀⣤⠶⢾⡇⠀⠀⠻⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⣤⣸⣷⠀⢸⡇⠀⢸⠇⠀⢸⡇⠀⢸⠋⢸⡿⣧⠀⢀⣾⠁⢠⢿⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⠁⠈⢷⣄⠞⠀⠀⢸⡇⢀⡏⠀⠘⣿⠈⠃⠘⣿⡰⠃⢸⡀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⣿⡀⠀⠀⠀⠀⠀⣀⠀⠙⠋⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠈⣃⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠁⠀⠀⠀⡏⠞⢹⠀⠀⡤⣀⡀⠙⡏⢁⣰⣇⠀⡆⢠⡞⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢩⠷⡖⠤⣄⠀⠀⠹⣄⡞⠀⠀⡆⡱⠃⢰⡇⠀⣽⠉⠀⠁⠘⢆⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠀⠈⠳⡀⠀⠈⡤⣄⢄⠉⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⡀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠲⣷⡇⠀⠹⣠⠎⠀⠀⠀⠚⠓⠀⠘⠛⠀⠀⠀⠓⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⠀⠀⢀⣄⣈⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣤⡀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠛⢻⣿⡏⠉⠉⠀⣰⣳⡿⠛⣧⠀⠀⣿⣿⣿⠻⣶⡄⠀⠀⠀⣼⠟⣿⣟⠄⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⢰⣯⣿⠃⠀⢹⡆⢰⡇⣿⡿⠀⢸⣿⡆⠀⢸⡏⠀⣿⣿⠰⠈⢿⣿⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⣸⣿⣿⠀⠀⢸⡇⢸⣿⣿⡇⠀⢈⣿⣿⢀⣿⠁⠀⣿⣿⣸⠀⠈⢿⣷⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣽⣿⠀⠀⣸⡇⠸⡟⣿⡇⠀⢰⣿⡟⢸⣿⠛⠛⣿⣿⣿⠀⠀⠀⢉⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⢸⣿⣿⠀⠀⣿⠀⠀⣷⣿⣇⠀⣾⡿⠁⣸⣿⠀⠀⣿⣿⡇⠀⠀⢀⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⢻⣿⣧⡼⠃⠀⠀⢹⣿⣿⡾⠋⠀⠀⣿⣾⠀⠀⢿⡿⣧⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠉⠀⠉⠁


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠈⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢀⣀⣀⣴⣤⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠛⠛⠻⢿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⢸⣿⠏⠀⠀⠀⠀⣠⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣿⣿⠀⠲⣦⣠⡾⠁⠀⢨⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠈⠋⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡍⢿⣿⣶⣶⣶⣾⣿⠏⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢡⣦⣤⡌⠙⠋⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢿⡟⠀⠀⠀⠈⢿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠦⠉⠉⠟⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⠶⠆⠀⠀⠀⠀⠀⠀⢀⣖⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠓⠲⠶⠄⠀⢂⣬⢆⣀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⠰⣝⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⣤⣶⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠛⠛⠷⠶⠤⣤⣤⣀⠀⠀⠀⣀⣤⡶⣉⡙⠒⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⡀⠀⠀⠀⠂⠄⢀⠈⠉⠙⠓⠶⠻⠭⣬⣉⣛⢳⣦⣄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠿⣶⣦⣤⣝⣒⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠉⡉⠙⠛⠻⢷⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠛⠻⠿⣿⣿⣷⣶⣦⣤⣂⣄⠀⠐⠂⠤⣄⣀⣀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⣹⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⣟⢲⡀⢦⡍⣉⠋⠉⠛⠛⠻⢿⡷⣶⣶⣦⣤⣅⣈⠁⠐⠂⠀⠀⠀⠀⢀⡾⠃⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⡲⠿⠌⣹⣞⢷⣿⡀⣞⠳⡄⢠⣄⡀⠙⠓⠿⢿⡿⢿⣶⣶⣤⢤⣴⣿⠃⠀⢤⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⢤⣤⣙⡃⠘⠧⢿⠎⢷⠻⣄⢟⢳⡄⣤⢤⡀⠉⠉⠃⠙⢻⣧⠐⠈⢆⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⢬⣅⣉⣀⠀⠹⠈⠉⢿⡌⣯⢳⡀⠈⢻⣆⠐⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⣅⣘⡋⠳⠀⠀⢻⣆⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠲⠿⠟


⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠄⠀⣀⡄⢀⡉⠉⣉⣉⡁⠉⢐⣒⡒⠒⠒⠒⠒⠂⠤⠤⠄⠠⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣤⠶⢛⣿⡟⢁⣤⠾⠧⠖⠋⣠⡾⠯⠴⠛⣰⣿⣍⣹⠟⢀⣴⡟⠙⣦⠀⢰⣦⠀⣴⣤⢀⣤⡤⠄⠉⢩⣭⣭⣍⡈
⠀⠀⣀⣀⣉⣀⣀⣈⣉⣀⣉⣁⣀⡀⠀⠚⠁⠀⠀⠀⠰⠟⠀⠠⡿⠀⢻⣿⣀⣴⡟⠀⠘⣿⣶⣿⠃⢸⣿⡷⠒⠀⠘⣿⣆⠈⢿⣦⡀
⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠐⠒⠈⠩⠍⠉⠛⠛⠓⠛⠛⠿⠿⠷⠷⢶⣶⣶⣯⣥⣤⣤⣤⣉⣙⣁⣀⣈⣙⣛⣛⣒⡀⠛⠻⠶⠟⠛⠃
⢀⣀⣀⣀⢀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠉⢉⣉⡙⢛⠛⠛⠛⠻⠛⢛⠓⠒⠲⠶⠦⡤
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠢⡀⠀⠈⠪⡉⠐⠂⠔⠁⠀⠀⡠⠊⡹
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⡲⠏⢉⡒⠈⢐⣄⣀⡈⣦⠎⣀⣀⡔⠪⠔⣊⠉⠲⢄⣀
⠀⠀⠀⡤⣒⠊⠉⠀⠀⠀⠀⠄⢀⣀⡀⠁⣂⣬⠭⡿⣤⠼⡯⣥⣄⠈⢁⣀⡀⠀⠜⠀⠉⠉⠒⡢⡄
⠀⠀⠀⡆⠀⠉⠑⠢⠤⣠⡄⣒⡩⠥⠒⠉⠁⠂⠉⡠⠋⠢⣀⠁⠀⠁⠐⠢⠍⡒⢤⡤⠔⠊⠁⠀⡆
⠀⠀⠀⡇⠀⠀⠀⠀⢀⠗⠋⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠈⠣⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠢⣄⣀⢀⡴⠁⠀⠀⠀⠀⢀⠔⠉⠉⠒⠢⢄⣀⣀⡤⠐⠊⠉⠢⡀⠀⠀⠀⠈⠂⠀⣀⡤⠂
⠀⠀⠀⠀⡇⠀⢋⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⡗⢄⠀⠀⢠⠛⠉⠀⡇
⠀⠀⠀⠀⡇⠀⠈⢆⠀⠀⢀⠜⠓⡧⢄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⡷⠊⠢⡴⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠈⠣⡤⠃⠀⠀⡇⠀⠈⠉⠒⠤⢄⣀⣃⡠⠔⠊⠁⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠧⢄⡀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠈⠑⠢⠧⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⢄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡠⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣀⢸⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠓⢤⠀⠀⠀⠀⢀⠴⠒⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠔⢲⣄⢳⠀⠀⣰⠃⢀⠴⠒⠦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠙⢎⢧⣠⠇⢰⣋⠴⠊⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠛⠒⠚⠛⠒⢺⠆⠀⠀⢯⠤⠤⠶⠥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠛⠛⢦⠀⠀⠉⠀⠀⡴⠉⠉⠳⣆⠀⢸⠁⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⡀⠀⠀⠀⡼⢁⡔⡂⠀⡧⠀⠄⠀⠀⡇⠠⠙⣦⡘⣆⡼⠀⡴⠋⠀⣀⣀⠙⡆
⠀⠀⠀⠀⠀⡜⢁⣠⣄⡀⠉⢦⡀⠀⡇⣾⠃⠀⡰⠃⣀⣄⣀⣀⢱⢄⡀⠘⢧⢳⣇⡜⢀⡴⢋⡠⠊⢀⡇
⠀⠀⠀⠀⠀⢣⡈⠑⠤⣉⣳⡀⠱⠶⢣⣡⠴⠞⠓⠉⠁⠈⣇⠈⠉⠉⠙⠓⠲⠆⠍⠁⣜⣉⣁⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠙⠢⢤⣤⠤⢼⠂⠈⠈⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠉⠉⠙⠓⢢
⠀⠀⠀⠀⠀⢰⠒⠉⠁⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀⣇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢡⠀⠀⠀⠆⠂⠈⠀⠀⠀⢩⠀⠀⠀⠙⠀⠀⠀⠗⠀⠠⠄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠤⠀⠀⠈⠈⠄⠀⠀⠰⡀⠀⠀⠀⠈⠙⡆⠀⠀⡄⠀⠀⢀⣀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⠠⠀⠚⠀⠀⠀⢃⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⡈⠀⠀⠁⠀⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⣆⣀⣀⠴⠴⠒⠉⠉⠒⠒⠲⠤⠤⠴⣄⣀⣀⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⠴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⣰⡟⣟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣻⢻⣆
⠀⠀⠀⠀⡿⡽⣜⠎⢷⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡞⠱⣣⢫⢿
⠀⠀⠀⠀⠱⡜⠀⠀⠈⢸⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⡄⠁⠀⠀⢡⠏
⠀⠀⠀⠀⠀⠙⢦⣀⡤⠞⠳⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⠞⠳⢤⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠛⠿⠛⠉⠛⠋⠁⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢹⣿⡏⠀⠈⠉⠁⠀⠀⠀⠈⠉⠀⠈⣿⣿⠇⠀⠀⠀⢀⡜⠁⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠱⣄⠀⠀⢸⠋⠀⠀⠚⠉⠁⠀⡀⠀⠉⠉⠃⠀⠈⢹⡄⠀⠀⡰⠋⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠳⣄⠈⠳⢷⡀⠀⠀⢀⡤⠤⢤⠀⠀⠀⢠⠷⠞⠀⣠⠞⠁⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠳⣄⠀⠱⢄⡀⠀⠙⠒⠋⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠈⠳⣤⡤⡟⠒⠶⠶⠖⠚⣿⢤⡤⠚⠁⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⢠⡏⠀⢹⠓⢦⣀⡴⠊⡇⠀⢳⡀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠸⡄⠀⠸⡖⣿⣿⣿⢺⠁⠀⣰⠃⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢠⡏⠀⠀⣇⣻⣿⡏⡾⠀⠀⢱⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⢠⡞⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠈⢧⡀⠀⢹⣿⣿⣿⠇⠀⢀⣾⣤⠀⣴⠋⠀⠀⠀⠀⡄⠀⠀⢳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢳⡀⠈⣿⣿⡿⠀⣠⠏⠀⠀⠀⣿⠀⠀⠀⢠⡞⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⣄⢻⣾⢃⡼⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠻⡿⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⢠⣾⡿⠋⢀⣠⣶⣿⡿⠟⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆⠀⢻⠀⠀⠀⠀⣰⡿⠋⢀⣴⡿⠟⠋⣁⣤⣶⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⡀⠀⠹⣇⠘⠀⠀⠀⣰⠏⢀⣴⠟⠉⢀⣤⣾⣿⡿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⢻⠀⠀⠀⠰⠃⢠⡞⠁⢀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⢹⡀⠸⡇⠀⠀⠀⣰⠏⠀⣰⡿⠋⠀⠀⢀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢠⡄⠀⠀⠃⠀⠃⠀⠀⠰⠃⢀⣼⠏⠀⢀⣤⣾⣿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⢀⡾⠁⢀⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡏⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣀⣼⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⢠⣴⡾⢿⣿⠉⢁⡟⢠⣾⠟⣄⠀⢀⣴⣿⠆⠀⢠⣿⡷⠶⠀⣠⣄⠀⠀⢀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠉⠀⣿⠃⠀⣸⠁⣾⠏⢀⣿⢀⣾⡟⠁⣿⠀⣾⠁⠀⣰⡿⢋⣴⡄⢠⣿⠃⣰⡟⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⡟⢸⡟⠀⣸⡇⢸⡿⠀⣸⠏⢰⡇⠀⣸⡟⢁⣾⣿⢀⣿⠏⣰⡿⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢰⠃⢸⡁⣰⡟⠀⠘⣧⣴⠏⠀⠘⠀⠀⠻⡷⠟⢸⡏⢸⣏⡜⣼⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⣰⠇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⢠⡏⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠟⠀⠀⠠⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠚⠁


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

⠀⠀⠀⠀⢰⡄⠀⣤⠀⣰⣿⣀⡀
⠀⠀⠀⠀⠈⠳⠤⠝⢲⡿⠉⣀⠿⢂⡀
⠀⠀⠀⠀⠠⣾⡿⢠⣿⠃⠐⠋⢀⢽⡇⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠿⠄⠻⠧⠄⠀⠀⠣⠾⠃⠀⠸⣇⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣾⣿⠧⠀⠒⠒⠒⠋⠉⠓⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⣀⠀⠀⣈⣑⡒⠀⠘⠏⡇⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢣⠞⢁⣼⢶⣾⠟⢩⣿⣿⠂⣰⣫⣿⠆⣼⣿⠃⠀⣰⣿⠂⢀⣾⡆⠀⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢋⡞⣽⡟⣿⢏⡴⣿⡿⢃⡴⢳⣿⠃⣸⣿⠃⡠⣾⡿⠁⣠⣾⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠈⠛⠋⠀⠙⠋⠀⠙⠷⠋⠀⠈⠓⠊⠉⠛⠊⠀⠘⠃⢊⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣀⠤⢒⣾⠏⠒⢄⡀⠀⠀⢀⣤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣰⡿⠀⠀⣿⣠⣄⡀⡞⠁⣠⣾⠃⠀⠀⠀⠳⣄⠀⠸⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣠⣿⠓⢀⣿⠆⠀⠘⣃⡙⠛⠋⣵⡞⢓⠀⠀⠀⠘⢦⣀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡠⢾⣏⣴⣿⢋⣿⠗⣿⠰⣿⠒⡡⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠻⠃⠈⠉⣸⣏⠜⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⣀⣀⣀⣀⠀⠀⣠⣾⣟⣽⠆⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣶⡀⠸⣿⣶⣶⣶⣾⣿⠿⠟⠿⢿⣦⡀
⠀⢠⣿⣿⡿⠿⠿⠿⣿⣿⡿⠛⣡⣾⡟⠁⠀⠀⠀⠀⢀⣀⣀⢸⣿⢡⡿⠀⠀⣈⣉⠉⠉⢱⣶⡄⣶⣆⠀⢹⡇
⠀⠘⣿⠏⠀⠀⠀⢰⣿⣿⠁⢰⣿⡟⢀⣾⡟⠙⣷⢰⣿⡁⢸⡇⣻⣿⡁⢀⣾⣿⢋⣶⡆⣾⡿⢀⣿⣿⣠⡿⠁
⠀⠀⠀⢀⣠⣤⣤⣿⣿⣇⣀⣿⣿⣡⣼⣿⡟⠀⣿⣼⣿⣥⣼⡿⠋⣿⣿⢾⣿⣵⣿⣿⡴⠿⠷⢋⣼⣿⠋
⠀⠀⠀⣾⣿⡿⢻⣿⣿⠉⢹⣿⣿⠁⢸⣿⠓⣺⠏⠹⣷⡴⠟⠀⣸⣿⣿⣈⠛⠋⠈⠉⣀⣴⠾⢋⣿⡟
⠀⠀⠀⠹⢿⠇⢸⣿⣿⠀⠘⣿⣿⠀⠀⠙⣛⠉⠀⠀⢀⣠⣴⣾⣿⣿⠋⠉⣿⠀⠀⢰⡟⠁⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⡀⠀⠹⣿⣦⡤⠈⢿⣿⣿⣿⣿⠟⠋⠀⢻⣿⣀⣠⡟⠀⠀⠘⠧⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣤⠤⡀⢿⣧⡟⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⢰⣟⠁⠈⢣⢀⣴⣷⠤⣷⠔⠋⢩⣿⡟⠀⣰⣿⠟⢉⣽⡿⠀⢠⣶⡆⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣧⢀⡖⣶⣆⠀⠈⢻⣶⠒⢚⡇⢿⣇⢀⠇⠀⠀⣿⡟⠀⣰⣿⣏⡴⢻⣿⠃⣠⣿⡟⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣸⡅⠸⣿⣇⠀⣾⣿⠀⣸⠁⠀⠉⠉⠀⠀⠀⠻⠧⠜⠁⠈⠉⠀⠈⠛⠚⠉⠿⡧⠖⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠹⢿⡴⠃⠙⠓⠁⠀⢀⣀⠄⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣤⣤⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⠀⠀⠀⠀⠀⠀⣴⣾⠟⠁⣴⣿⠀⢸⣷⣄⡀⠀⢠⡞⠉⠀⠀⠀⠀⢀⣾⡿⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⣼⣿⡟⠀⠀⠻⣿⡿⠀⢸⡀⠀⠀⠀⣀⣴⣿⠟⠁⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠿⠟⠋⠁⠀⠀⣴⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⢠⡄⠀⣠⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠘⡿⠷⠀⠀⠀⠈⠁⠀⠉⠀⣶⠀⡟⠀⠆⡀⠀⠸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠈⠃⠀⠀⠀⠀⢀⡀⠒⢀⠀⠀⠀⡄⠀⠛⠀⠀⠀⠁⠁⠀⡤⠀⠀⠒⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⠆⠀⠀⠀⠀⠀⠀⢰⡆⠀⠈⢁⡒⠈⠀⠀⠀⠀⠀⠀⠀⣤⣼⣦⡄⠀⠀⡴⣠⠀⠰⠟⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠃⢴⣶⡶⠶⠂⠀⠀⠁⠀⠀⠈⠁⢰⣷⣰⣦⢠⡞⠀⠀⡼⠻⡏⠀⠀⠘⠱⠃⠀⠀⠀⠀⢀⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠟⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠘⣬⠃⠀⣠⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠘⠃⠀⠾⢧⡤⠀⠀⣸⣿⠟⠳⣆⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠲⠌⠀⠀⡀⠀⠀⣰⠟⠁⠀⢠⣿⡿⠀⣠⡟⠀⣾⠋⢻⠀⠿⠟⠛⢹⣿⠋⣩⡍⠉⠉⠀⣀⡀⠀⠀⠘⠃⠀⠀⢠⣾⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠁⠀⠀⠀⣼⣿⡿⠿⠋⠀⠀⠈⢹⠟⢛⣿⠆⢠⣿⡏⢠⣿⠃⠀⣠⢾⣿⠁⠀⣀⡀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣷⡤⠄⠀⠘⠃⠀⠀⣿⣿⠁⣴⢏⣿⠏⣰⠋⢠⣿⠏⢠⣿⣿⠀⢸⣯⠴⠋⣀⣿⣿⠶⠛⠋⠀⠀⠰⠀⠀⠉⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠟⠹⠏⠀⢀⣀⠀⠀⠀⢸⣿⣿⠀⠿⠋⠻⠞⠁⢠⣿⠏⠀⣾⣿⡇⠀⠀⢀⣴⠟⣻⣿⠃⠀⠀⢠⢀⠀⠀⠀⠀⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⡀⠰⠆⠀⢶⣿⡛⠉⠀⠀⠀⢸⣿⡏⠀⢚⣛⣛⡀⠀⣸⣿⠀⠀⣿⣿⠁⠀⢰⣟⠁⢠⣿⠏⠀⠀⠐⠂⡌⠀⠀⢹⣾⣷⠶
⠀⠀⠀⠀⠀⠈⡋⠰⠆⠀⣠⡾⠟⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠈⠁⠀⠀⠘⠿⠶⠿⠋⠀⠀⠀⢀⡀⠀⠀⠀⠘⠙⠇
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠉⠀⢀⠀⠀⠀⠙⠀⢀⣀⣤⣴⣶⡶⠶⠶⠶⢶⣦⠀⠀⠙⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣾⡤⢶⡇⠀⠀⠀⠐⠖⠒
⠀⠀⠀⠀⠀⠠⠶⠶⠓⠀⠀⠀⠈⡄⠀⠀⠀⡾⠟⠛⢉⣿⠇⢀⡄⠀⠀⢠⣶⠀⠀⣴⢻⣿⠇⢀⣤⣶⠶⡄⠀⠀⠀⠀⣾⣤⠞⠃⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⣈⠁⠀⣠⣿⣿⢂⡾⢁⣿⡟⣰⣿⡟⢁⡴⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠿⠀⠀⠀⠀⠒⠀⠂⠀⢠⣿⡇⢠⣿⢀⡞⣿⡿⢻⡿⠁⣸⣿⠃⣿⡿⠛⠋⠀⢀⡤⠀⠀⠀⠀⠂⣈
⠀⠀⠀⠀⠀⠀⠀⢤⠄⠴⠀⠀⣀⣴⠆⠀⠀⠀⠀⢸⣿⠀⠘⣿⠟⢰⣿⡇⠀⠀⢀⣿⣿⠀⣿⣧⡀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠘⠛⠓⣶⠆⠀⠀⠀⠸⠿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠈⠛⠛⠉⠀⠀⠀⠐⣾⣷⠂⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠸⠿⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠉⠁⠀⠀⠀⢠⡄⠀⠀⢀⠀⠁⠈⠀⣩⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⠆⠀⠀⡀⣀⠀⠀⠀⠖⠀⠀⣠⡄⠀⣀⣀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣴⣄⢀⡴⠀⠈⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⠞⠁⠷⠚⠁⣿⠀⢀⠀⠰⣶⣿⡤⠀⠸⠃⠹⠞⠀⠀⠰⣌⠳⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠆⠀⠀⠀⡰⠀⠤⠀⠘⠀⠀⠀⠀⠋⠻⠀⠀⡤⡐⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠉⣠⣦⡀⠀⠀⠀⢀⠀⡀⠀⢀⡀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠀⠀⠀⠀⠘⠀⠁⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡄⠀⣠⡶⠛⠛⠷⣤⠀⠀⣿⠀⣿
⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢸⡇⠘⣇⢰⡏⠀⠀⠀⠀⢸⡇⢸⡇⢰⡏⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⢠⡟⠁⠀⠀⠙⣷⠀⠀⠀⣿⠀⢿⡈⢷⡀⠀⠀⢀⣼⠃⣾⠀⣾⠁⠀⠀⣾⠋⠀⠀⠈⢻⡄
⠀⠀⠀⠸⣧⠀⠀⠀⢀⡿⠀⠀⠀⢹⡆⢸⣇⣀⣙⣛⣛⣋⣁⣰⡏⢠⡟⠀⠀⠀⢿⡀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠈⠻⠶⠶⠛⠁⠀⠀⠀⠈⣷⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⣸⠇⠀⠀⠀⠈⠛⠶⠶⠟⠁
⠀⢀⣀⣤⣴⠆⠀⠀⠶⣤⣤⣀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣤⠶⠀⠀⠰⣦⣤⣀⡀
⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⣿⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⠀⢈⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡁⠀⡄⠀⠀⠀⠀⠀⠀⢠⡄
⢻⡆⢸⡇⠀⠀⠀⠀⠀⠀⣾⠀⣼⠇⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠸⣧⠀⣷⠀⠀⠀⠀⠀⠀⢸⡇⢰
⠀⠻⣾⡇⠀⠀⠀⠀⠀⠀⣿⡾⠋⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠙⢷⣿⠀⠀⠀⠀⠀⠀⢸⣷⠟
⠀⠀⢸⡇⠀⠀⢠⡀⠀⠀⣿⠀⠀⠀⠸⡇⠀⠀⢠⡶⢶⡆⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢀⡄⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣷⢸⡇⠀⠀⣸⠇⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢀⡗⠀⠀⠀⠀⢻⡄⠀⠀⣿⣼⠁⠀⠀⣿⠀⠀⠀⠀⢸⡀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⠀⠀⢰⡏⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣷⠀⠀⢸⡿⠀⠀⣸⠇⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⢸⡆⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠸⠇⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢰⡇
⠀⠀⠀⢸⣇⣀⣸⣇⣀⣼⠇⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠸⣧⣀⣸⣇⣀⣸⡇
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⠓⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⣦⡄⠀⠀⠀⢀⣤⣄⠀⣀⣤⣿⣆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣙⠗⠛⣠⣄⣠⣶⡄⠀⢰⡟⠉⠀⢶⣄⣼⠇⣶⡄⠀⣿⡁⣿⠿⢿⡍⠀⠙⠛⣷
⠀⠀⠀⠀⠀⠀⣤⣶⠿⢻⡟⢋⣩⡿⢿⡇⠀⠈⠃⠀⠀⢿⡿⢋⣠⣿⣿⡶⣿⣿⣟⣀⣈⣿⡄⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⢛⣿⣩⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠈⢩⣿⠈⠿⠘⠿⠛⠋⣩⣌⠁⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⢿⣻⣹⡿⠀⠀⠀⠀⠀⠀⣀⣉⡀⠈⡋⠀⠀⢠⣄⠀⣾⠏⢁⣀⣤⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡙⣻⡇⠀⠀⣠⣴⡶⢿⣟⠉⣥⡀⢹⣆⣀⣼⣿⠀⣿⡾⠟⠛⢉⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠿⠃⠀⠀⠈⠀⠀⠈⣿⡀⢸⣧⠈⣿⠛⠉⠹⡧⠈⠻⠶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣿⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣶⢶⣄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠓⠀⠻⠋⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣶⡾⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⣿⠿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣶⡶⠶⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡗⢻⣟⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⠆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠄⠀⢀⣀⢀⠘⠀⠀⠀⠀⠀⢠⣿⣿⠍⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣶⣿⠟⠃⠀⠀⠀⠀⠀⢸⣟⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠁⠀⠀⠀⠀⡆⠀⠀⠀⢸⣿⣿⡅⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⣿⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣥⠀⠀⠀⠀⠈⠀⠀⠀⢸⣇⣾⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣬⣷⠀⠀⢀⣀⠐⢦⣤⣼⣿⣿⣿⣿⠷⡤⢾⣇⠀⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⣶⣿⣶⣿⠿⠈⠀⠀⠀⠀⣰⣬⣭⣴⣶⡞⠁⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢉⡀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⡏⣀⡀⠀⠀⠀⠀⣤⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⡗⠙⠁⠀⠀⠉⠉⠙⠋⠉⠉⠀⠀⠙⠀⠀⠀⢰⣿⣿⠿⠀⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⡀⠄⠀⣰⣦⢀⣴⣿⠟⠁⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣧⠼⠋⠀⠀⠀⠀⠀⠀⠀⡠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠻⣦⣶⣦⣴⣶⣶⣾⣿⣿⣟⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⣋⡉⠉⠉⣾⠏⠀⠀⠀⠀⣸⣦⣶⣾⣿⡋⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠃⠈⠉⢳⢀⣼⠋⠀⠰⠀⠀⣠⣿⣿⣿⣿⣿⡿⠓⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡇⠀⢀⡴⠋⣸⣇⣠⠾⠅⢀⣴⣿⣿⣿⠻⠿⠏⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡄⠀⠐⠒⠀⠀⠀⣠⣼⣿⣿⣿⣿⣦⣭⣤⣾⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠛⠛⢻⣶⣿⣿⣿⣿⠟⠹⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡁⢸⡿⠿⠟⠋⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠉⠙⠛⠛⠛⠋⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠋⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠘⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢞⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⣻⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢽⣿⣿⣿⣽⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢷⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡪⠽⣿⣿⣿⣯⡔⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣸⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡐⠻⡿⣿⣽⠾⣋⣴⠆⡀⠀⠀⠀⢀⣤⣿⢡⡄⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⢻⣷⣾⣯⣵⡾⣋⣠⡾⢀⣼⡣⢁⣾⠃⠀⠀⠀⣈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⡿⣿⣿⣷⣿⣿⣯⣶⣿⣿⣥⡿⣯⣴⣶⣾⠷⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣏⣹⣿⣿⣿⣿⣿⣿⠿⠋⠙⢻⣿⣿⡯⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠙⠋⠿⠿⣟⣋⠁⠀⠀⠀⠈⢱⣬⣁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣶⣶⡾⢿⡏⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣻⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⢬⣿⣿⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⣭⣽⡯⠅⠀⠀⠀⠀⠀⠈⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠛⠃

⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⠈⡭⠠⠠⠆⠀⠀⠀⢠⣤⣔⣤⣶⠀⠀⠀⢀⡠⠤⢠⠀⢄⡠⠀⠀⠀⠀⠀⣸⣦⣜⣇⣲⣶⡆⠀⠀⠀⠀⣠⠀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠶⠀⢀⡀⠀⠤⠄⠻⠀⠀⠈⣵⠠⣬⣦⠀⠀⠀⢠⠀⡀⢀⢀⠀⠀⠀⢠⠀⠀⠀⣦⣦⡆⢰⢶⠖⡂⠀⠀⠀⣀⣀⣈⣃⣂⣈⣉⡀
⠀⠀⠀⠀⢠⠀⠈⡀⠤⠶⠄⠂⠀⠀⠀⣍⡠⣭⡀⠀⠀⠀⠀⢠⢀⡄⠠⡤⢠⠀⠀⠀⠀⠀⣤⣤⡄⣎⣰⣦⣵⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠈⣁⠀⡤⠴⢤⠀⠖⠀⠀⠀⠉⠙⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠉⠀⠀⣤⠘⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣠⠢⠰⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠁⠀⠙⠂⠀⠀⠀⠻⠛⠻⠻⠟⠬⠟⠀⠀⠀⠀⠈⠉⠀⢀⡀⠀⠀⠀⠀⠀⣄⠀⠰⠠⢀
⠀⠀⠀⠀⠀⡴⣻⢽⣿⣷⣄⠀⠀⠈⠐⠈⡀⠃⠃⠂⠀⠀⠀⠠⠷⠷⠇⠃⠎⠧⠟⠇⠀⠀⠠⢀⠀⢀⠤⢤⠄⠀⠀⠀⠀⢬⠰⡰⠀⠈⡅
⠀⠀⠀⠀⢸⣿⡿⠉⠹⣿⣿⡀⠀⠀⠀⠀⠀⠉⠹⠀⠀⠀⠀⠀⠀⠸⠿⠿⠳⠇⠀⠀⠀⠀⠀⠄⠀⠠⠄⠠⠐⠀⠀⠀⠀⢦⠰⠀⠂⠃⠈
⠀⠀⠀⠀⠈⠉⠁⠀⣀⡁⠉⠀⠀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⡄⣠⢀⠤⠤⠄⠀⠀⠀⣸⣿⣶⣏⣯⣜⣯⣹⣾⠀⠀⠀⠀⠀⠄⠄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡆⠀⡀⢀⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣬⣬⣭⣭⣭⣭⣬⣌⠀⠀⠀⠄⠄⠐⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠸⠄⠠⠠⠠⠄⠀⠀⠀⠤⠤⡄
⠀⠀⠀⠀⠀⠀⢠⣦⣤⣶⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⣨⡭⡍⣬⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⢠⠀⠀⠀⠀⠀⢀⣶⠆⠀⢰⣶⠀⠀⠀⠀⢠⣶⣤⠒⠔⡦⠶
⠀⠀⠀⢠⣠⣨⢫⠋⣧⣯⣝⣅⡤⠀⠀⠠⡤⠀⡀⠀⠀⠀⢁⠀⠈⢈⠀⠀⠀⠀⠀⠀⠉⢀⡀⠈⠋⠀⠀⠀⣴⠶⠦⣦⠶⠦⠶⠶⠲⠼⠄
⠀⠀⠀⠘⠛⠙⠈⠛⠙⠛⠓⠋⠒⠀⠀⠀⠐⠀⠀⠁⠀⠀⠈⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠠⠷⠷⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣄⣠⣾⣿⣿⣷⣤⡀⢤⣶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠄⣾⠿⢛⣛⣩⣭⣭⣤⡭⠭⣭⣭⣭⣍⣛⡛⠿⣷⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⣋⡤⠀⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠄⣨⣙⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡖⠼⣋⡴⠟⠉⠀⠀⠀⠀⡀⡄⠀⠴⢲⠀⠀⢀⣴⡀⣤⣄⠀⠀⠀⠀⠉⠻⢦⣝⠻⡶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⣁⠆⠀⠀⠀⠀⢠⠒⡌⢲⡇⠻⠔⠘⠊⠀⠀⠊⠉⠓⠋⠉⠾⠽⢢⢤⡀⠀⠀⠈⠳⣄⠺⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⣣⡾⠋⠀⠀⣀⠠⡽⠆⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠟⡵⢒⠄⠀⠀⠈⢳⣌⢿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⣴⠏⠀⠀⣠⣜⢾⠁⠁⠀⠀⠀⢀⣠⠴⠖⠛⠛⠛⠛⠛⠛⠲⢦⣄⡀⠀⠀⠀⠉⠉⣼⡡⢄⠀⠀⠙⣦⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠏⣼⠃⠀⠀⡀⡓⡇⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⢖⠋⠃⠀⠀⠈⣧⡹⣟
⠀⠀⠀⠀⠀⣠⣾⡟⣼⠃⠀⠀⠀⠻⡏⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠈⢲⣉⡇⠀⠀⠘⣧⢹⣷⣄
⠀⠀⠀⠀⢸⡏⢸⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆⢿⣿⡇
⠀⠀⠀⠀⠀⢿⡇⣾⠁⠀⠀⣀⡀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⡀⠀⣀⠀⠀⣀⡀⣀⣀⣀⣀⢀⣀⣀⡀⠀⠈⢿⢸⣿⠁
⠀⠀⠀⠀⣠⣿⡇⣿⠀⠀⢠⣿⣷⠀⠀⣿⠛⠛⣷⠀⣿⠛⠛⣿⠀⣿⠛⠻⣿⠀⣾⠟⠛⢿⡄⢿⡆⢀⣿⠃⣿⠟⠛⠋⢸⣿⠛⢿⣧⠀⢻⠈⣿⣆
⠀⠀⠀⠰⣿⣿⠀⣿⠀⠀⣾⣏⣿⣇⠀⣿⡷⠾⠟⠀⣿⡷⠾⠟⠀⣿⡷⣾⡟⢰⣿⠀⠀⣾⡇⠸⣿⣼⡟⠀⣿⡿⠿⠀⢸⣿⠀⢘⣿⠀⢀⡀⣿⣿⡇
⠀⠀⠀⠀⠙⣿⠄⣿⠀⢸⡿⠉⠙⣿⡀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢹⣧⠀⢿⣦⣠⣿⠃⠀⢿⣿⠃⠀⣿⣇⣤⣤⢸⣿⣤⣼⡏⠀⣼⢡⣿⠏
⠀⠀⠀⠀⠀⣾⠀⢿⡀⠉⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠙⠋⠁⠀⠀⠈⠉⠀⠀⠉⠉⠉⠉⠈⠉⠉⠉⠀⠀⣿⢸⣿
⠀⠀⠀⠀⢸⣿⣶⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⡇
⠀⠀⠀⠀⠀⠛⢿⣇⢻⡄⠀⠀⢰⣋⡷⡀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⢐⣆⠀⠀⠀⢠⡟⣸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣆⢻⡄⠀⠀⢡⣠⡕⠀⠀⠀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⢐⣍⡉⠁⠀⢀⡈⣱⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠻⣄⠀⠀⠁⠒⡟⣀⣀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⣀⣠⣤⠤⠶⠛⠉⠀⠀⠀⠀⠀⡳⡜⠊⠀⠀⣠⡟⣱⡛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣧⡙⢧⡀⠀⠀⠀⠧⠜⣰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡰⣜⡦⠛⠁⠀⢀⣴⠋⣼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡙⢦⣀⠀⠀⠈⠳⠋⣖⡶⢀⣠⠄⣀⣀⠀⠀⣠⣄⠤⢦⢰⡦⡘⠤⠇⠈⠀⠀⢀⡴⢟⠱⠞⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣦⠁⠀⢠⣀⠀⠀⠈⠁⠙⠚⠀⠿⠁⠀⠀⠧⠼⠠⠝⠈⠁⠀⠀⠀⠀⣤⠾⣋⣴⠿⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣦⣝⡳⠶⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⢠⣴⣾⣥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠛⢷⣆⣠⣍⣛⣛⣛⠲⠶⠶⠖⣛⣛⣛⣩⣭⣶⡾⠛⠓⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠛⠛⢿⣿⣿⡿⠟⠛⠛⠽⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡷⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠰⠛⠻⠉⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡷⠀⣠⣦⠀⠀⠀⣤⣴⣿⣾⡇⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⣰⣿⠏⠀⠀⠀⠀⠚⠉⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣿⣿⠟⠁⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⢠⣿⠏⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣣⣤⣾⠿⠖⠀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⢓⣆⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⢀⣤⡾⢿⡏⠀⣸⠏⠀⣰⣿⠟⣋⠀⣠⣶⡦⠀⠀⣴⣿⣷⡾⠟⠋⣠⡾⢿⣇⠀⢠⣿⠇⢀⣾⡆⠀⣸⡏
⠀⠀⠀⠀⠀⣴⣿⠋⠀⣾⠀⠀⣿⠀⣸⡿⠁⣰⠏⣼⣿⠋⢰⡗⠀⣿⣿⠁⠀⠀⣼⠏⢠⣿⡟⢀⣿⠏⡠⢸⡿⠀⢀⡟
⠀⠀⠀⠀⠀⠉⠁⠀⢠⡏⠀⢸⡇⠀⣿⢁⡼⠋⢰⣿⠁⢠⡿⠀⢸⠀⢿⡆⠀⢸⡏⣰⢿⣿⠁⢸⣟⡼⠁⣾⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡄⠀⠘⠋⠀⠀⢸⡇⣠⡟⠁⠀⠇⠀⠈⢿⠀⠸⠾⠃⠸⣿⠀⠻⠟⠀⢰⡿⠀⠀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠘⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⣠⣤⠀⢀⡿⠁⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⣠⠀⠀⠀⠀⢀⣼⠋⠀⣠⠟⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⠲⣶⣾⣿⡖⠀⠀⠀⠘⠁⠐⠋⠁⠀⠀⠠⢼⣷⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣀⣦⣀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠟⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡴⣖⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣲⢦⡀⠀⠀⠀⢸⣿⣿⣿⡿⢋⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢍⡛⢿⣿⣿⣿
⠀⠀⠀⡿⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⢻⣿⠀⠀⠀⢸⣿⣿⣿⢠⢱⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣷⡽⠈⣿⣿⣿
⠀⠀⠀⡇⡇⠀⠀⣠⣶⣄⠀⢀⣴⣿⣿⠟⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⢸⢸⣿⣿⠟⠉⠻⣿⡿⠋⠀⢀⣼⣿⡇⠀⣿⣿⣿
⠀⠀⠀⡇⡇⠀⠀⠙⢿⣿⣷⣿⣿⠟⠁⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⢸⢸⣿⣿⣦⡀⠀⠈⠀⢀⣴⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⡇⡇⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⢸⢸⣿⣿⣿⣿⣦⣀⣠⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⡇⡇⠀⡄⣠⣠⣄⣄⣠⣀⣀⡄⣠⡀⢸⣿⠀⠀⠀⢸⣿⣿⣿⢸⣸⣿⢿⠿⠿⠿⠿⠿⣿⢿⣿⠿⣿⣧⡄⣿⣿⣿
⠀⠀⠀⣇⡇⠠⠿⡟⢹⠹⠯⢿⠻⠇⡇⣿⠆⢸⣿⠀⠀⠀⢸⣿⣿⣿⢸⢴⣖⣐⣰⣦⣶⣸⣋⣧⣿⣺⣁⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠹⣾⣢⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣖⣵⠏⠀⠀⠀⢸⣿⣿⣿⣆⠣⠝⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠯⢊⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠲⡄
⠀⠀⠀⠀⠐⠀⣁⠃⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠉⠡⡀⡄⠠⠐⢀⠀⣀⠀⠀⠠⣄⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿
⠀⠀⠨⠀⠁⠀⠀⠄⠈⠃⠋⠀⠀⠀⠀⠐⠀⠀⢒⠀⢢⣦⣄⠱⣤⣠⠊⢁⣨⠴⠠⠀⠂⠀⠀⠀⠈⠉⠉⠉⠉⠋⠉⠙⠋⠁⠉⢉
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂⠀⠙⡿⠛⠓⢾⣃⣒⣚⣃⣀⣀⣀⠀⠀⠌⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠀⠈⠉⠉⠀⠀⠀⠀⠸⠁⠀⠀⠄⠀⠀⠿⠠⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠈⠀⠀⠐⠢⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠊⠐⡄⠀⠀⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠰⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠠⠃⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠀⢹⠂⡀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠜⠐⡇⠂⠀⠀⢰⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠒⠓⠀⠘⠒⠂⠀⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠠⢀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠘⢾⢀⡠⠞⠀⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡶⠀⠀⠀⠀⠀⠀⠀⢀⡁⠂⠀⣳⡤⠨⡭⠖⠐⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⠅⠀⠄
⠀⠀⠀⢀⡠⠄⢲⠁⠀⣀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠤⠖⠁⣀⠀⢄⣀⠠⣀⠀⠀⢠⣠
⠀⠀⠀⠀⠤⠠⡰⡀⡠⢊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠲⡈⠳⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡁⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠙⠀⠘
⠀⠀⠀⢀⣀⠀⢀⣿⢋⡀
⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡀⠈⠠⡀⣜⠄⢸⣀⣄⢬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢨⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠽⡶⢃⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸
⡄⠀⠤⠄⠀⠢⢼⠄⠸⠴⠤⠴⠶⠦⠤⠤⠄⠀⠀⠀⠀⡀⠐⠰⠁⠀⠀⣀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠠⠤⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠃⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠄⠰⣆⠀⠀⠀⠂⠀⠀⠠⠁⠀⠠⠁
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠃⠀⠀⠀⣀⡀⠘⠀⠀⠀⠀⠀⡀⠀⠀⡘
⠀⡴⠞⠁⠒⢄⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⠀⠀⢀⣉⣡⡔⠂⠀⠀⢀⠐
⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠻⠏⠙⠃⠀⠀⠀⠁⠀⡤⠂
⠀⠀⠀⠈⠀⠢⠀⠑⠂⡀⠀⠀⠀⠀⠈⠐⠀⡀⠀⠀⠐⠊⠶⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠐⢀⡀⠉⠐⠀⠀⠀⠀⠀⠈⠀⠀⡔⢳⠆⠀⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⡁⠀⠀⠀⠀⠐⠉⠃⠀⢠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⣀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⢁⡀⠀⠀⠀⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠁⢀⠠⠔⠧⠦⠤⠀⠀⠒⠶⠤⠤⠚⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠐⡆⠀⠀⠞⠉⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠉⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⠰⠆⢹⡄⠀⠀⠀⠀⠈⠀⢢⡄⠐⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠖⠰⠀⢀⡈⠀⠀⣁⠔⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢾⡄⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⣀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⣀⣀⠆⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠶⡛⢃⡄⠀⠀⠘⠁⠀⠀⢄⠁⢸⡆⠀⠀⠀⠁⡀⢀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⡀⠀⡀⠀⠀⣼⠋⢠⠀⠄⣈⠛⠀⢁⡀⠀⠀⠀⠈⡁⠰⣆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⢠⠁⠀⠀⠁⡒⠀⢀⠘⠁⠠⠀⠈⠀⠀⠀⠀⠀⠋⠀⠀⢀⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⡗⠀⠀⠘⠛⠖⣿⣶⣀⣀⣀⣲⠦⠘⡓⣾⠃⠘⠖⠂⠈⠁⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠚⠀⣀⣀⣲⡤⢻⡿⡄⠀⠀⠀⠀⠐⠉⠛⠫⠉⠙⡋⠀⠀⣵⢟⣤⡄⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠆⠸⢆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠻⠤⠤⠂⣠⠀⠀⠀⢀⣀⠰⠎⢳⡀⠐⡊⢧⡀⠠⢤⣾⣿⣷⣤⣂⣠⠀⠀⠐⠀⠀⠀⠁⠀⠀⠜⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠢⠀⢀⠀⠀⠌⠛⡁⠀⠘⠃⠘⠁⠀⠉⢤⣠⠋⠀⠋⠉⠛⠃⠀⠀⠂⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠀⠀⠀⢀⠃⠀⠐⠖⠒⡁⠀⠀⠀⠀⠀⠀⢠⡶⠉⠀⢘⠃⠀⢀⡴⠂⠁⠀⠀⢀⡾⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠆⣀⡉⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠙⠀⠀⠸⡀⠈⠉⣼⠖⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠂⠀⠁⠀⢈⠀⠀⠀⠀⠀⠀⠀⣠⠀⠰⠃⠀⠈⠀⠀⠀⠈⠀⠀⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠤⠄⠤⠄⠀⠠⠄⠀⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠜⠁⠆⠀⠐⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⣀⣀⣀⠀⣀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠃⠀⠘⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⡀⠀⠀⠒⠦⠀⠀⠤⠠⠀⠰⠂⠀⠀⢀⣀⠠⠤⡦⠀⠐⠃⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠁⠀⣰⡀⠀⡀⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠿⠿⠿⢿⣿⣿⣿⠿⢿⣿⣿⠿⢿⣿
⣿⣧⣴⣦⣴⣧⣴⣶⣶⣦⣶⣬⣧⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣿⣿⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⣀⠀⠀⠀⣶⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⣛⣠⣿⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⡀⠀⠀⣀⢀⣴⣶⣿⡿⠄⠀⠀⠀⠀⠈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠈⠿⠋⠁⠀⠀⠀⠲⣆⠁⠀⠀⠀⠀⠀⢀⣀⣸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⢰⠄⠄⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠉⢹⡏⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣷⣦⡀⠀⠀⢠⣿⠃⠀⠀⡀⠀⠀⢀⡠⠎⠀⠀⠑⠄⠀⠁⠀⠀⠀⠀⠘⠁⠀⢸⡇⠠⠰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠤⠄⠼⢤⠦⠤⠤⣤⠄⢠⡀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⣿⠀⠀⣀⠠⠄⠉⢀⣠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠷⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣵⣿⡶⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣷⡀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⠀⠘⣷⡄⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⠿⠿⠿⠿⣿⣿⣿⣿⣦⡈⢻⣦⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣿⣦⣀⠀⠀⠀⠀⢻⣿⠀⠀⠀⢠⣾⠁⠀⠀⣀⣤⣶⣦⡴⠄⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣷⣶⣦⣤⣼⣿⡀⠀⠀⢸⡟⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠉⠀⠀⠀⠈⠉⠙⠛⠻⣶⣦⣼⣷⡀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣿⣿⣷⣾⣿⣆⠀⠀⠀⠀⠀⠀⡀⠀⠈⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠁⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣦⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣿⣿⣦⣄⣠⡛⠳⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠓⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡆⢸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠰⠊⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡅⠀⠀⠀⢠⣿⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠀⠀⢀⣠⣤⣄⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢸⠇⠲⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣀⣼⡁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡹⠃⠀⢠⣿⣿⣿⣯⡙⠀⠙⢿⣿⣿⣿⣿⠃⠠⠀⠀⠀⠀⢀⡦⠰⣿⠆⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣟⣥⡴⠋⠁⠀⢀⣿⣿⣿⣿⣿⡇⠀⠐⠀⠉⠻⣿⠛⠓⠀⠀⠂⠀⣤⠎⠁⠂⣽⣶⡆
⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡏⠀⠂⠀⠀⠀⠀⠈⠉⠉⠒⠉⠁⠀⠀⠀⠀⠈⢹⠇
⠀⠀⠀⠀⠈⢻⣦⡀⠠⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠌⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠹⣷⣶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⢰⠗⢀⠀⢠⡄⠀⠀⠀⠀⠄⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠈⣧⣔⣀⡀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⠈⣻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡄⢀⣴⣾⣿⣏⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡃⠈⠿⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡽⠛⢛⣿⡷⣿⣿⡟
⠀⠠⠤⣴⣤⣄⡀⠀⠀⠀⠀⣾⡟⠀⠇⠀⢰⡄⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⣎⣹⣷⣿⠋
⠀⠀⠀⠀⠉⠻⣿⣦⠀⠀⣸⣿⠁⠐⠀⠀⠸⡇⠀⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣼⠆⢀⣻⡅⣼⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣇⣠⣿⡇⠀⠀⠀⠀⠀⠻⣄⢀⣄⠙⠀⠋⠉⠁⠉⠈⠉⠑⠆⠉⠙⠂⠉⠉⠈⠉⠁⢠⣤⡄⠘⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⣀⣴⡆⠀⠀⠀⠀⠛⠈⠿⠓⣶⢴⡆⢠⡀⠀⠀⣀⠀⠀⢘⡀⠠⡧⠴⠆⠁⡸⠏⠃⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡾⠟⠛⠃⠀⠀⠀⠀⠀⠀⠙⠀⡤⢠⠀⠈⠁⡀⠉⠉⠉⠈⢉⠁⠀⠀⢠⡄⠞⠃⠀⠀⠀⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠃⠀⠂⠀⠃⠘⠀⠘⠃⠀⠀⠀⠀⠀⠠⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠹⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠠⠈⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣦⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠰⠂⣶⡄⢴⠆⢠⡆⢰⡄⢰⡆⠲⠀⢰⡆⢲⣦⢰⡆⢰⣦⠰⠶⠀⠐⠆⢠⡦⢴⡆⠲⠀⠰⡆⢰⡦⠰⡆⢴⠀⡆⠀⢶⡶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠸⠇⠸⠀⠈⠀⠟⠀⠽⠂⠙⠀⠇⠈⠟⠀⠀⠸⠂⠀⠟⠀⠿⠀⠿⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡀⢀⣄⠀⣠⣀⣀⡀⢀⡀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠻⣏⠙⠛⣿⠋⠈⣿⠛⠛⣿⠙⡯⢉⣿⡿⢻⠏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣾⣷⣶⣷⣶⣶⣿⣶⣷⣿⣷⣶⣾⣿⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⢤⠀⠀⢠⠏⠀⠀⠀⠀⢸⠀⠀⠀⣠⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⢀⡟⠀⠀⠀⠀⢠⡎⠀⢀⡞⠁⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣰⠃⣼⠁⠀⠀⠀⠀⣾⠀⢀⡾⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⡟⢠⡏⠀⠀⠀⠀⢰⡇⠀⣼⠁⠀⠀⠀⠀⣼⠃⢀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⡧⠞⠷⣄⠀⠀⠀⣾⠀⣸⠁⠀⠀⢀⡴⠞⢧⣀⡞⠁⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⡁⠀⠀⢸⠂⠀⢰⠃⣰⠇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⣠⣤⣤⣽⡇⠀⠀⢼⣤⣤⣾⣤⡟⢀⣤⣤⣤⢸⠆⠀⠠⣧⣤⣤⣤⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠷⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⢤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠴⠲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢀⡴⠶⠶⢤⣀⠀⠀⠀⠈⠳⡶⠖⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠖⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠲⠶⠞
⡟⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⡇⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠱⣄⡀⠀⠀⠀⠀⠀⠹⣆⠀⣇⠀⣰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠘⣧⣣⡾⠁⡼⠛⠛⠛⢻⡇⠀⠀⢾⠛⠛⠛⠛⣷⠟⠋⠋⠉⣻⠇⠀⠸⡏⠛⠋⠙⠻⡄
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠸⡿⠁⠈⡇⠀⠀⠀⢸⡁⠀⠀⢸⠃⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⢀⡿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⣧⣄⣀⡇⠀⠀⠀⠀⠙⠲⠒⠋⠀⠀⠀⠀⢸⣀⣠⣀⡀⠈⠓⠒⠛⠁⣀⣠⣀⣀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⡏⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⢹⡄⠀⠀⠀⣼⢻⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⢧⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣠⡼⠀⠀⠀⠀⠙⣾⣅⣠⣤⡀
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⢹⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⣠⠖⠒⢦⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⢶⡄⠀⠀⣠⠶⠶⠶⠶⠛⠳⠶⠶⠶⠿⠀⠀⠘⠷⠶⢦⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠌⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠂⠙⢄
⠀⠰⠀⠀⠀⠀⠉⠐⠒⠀⠒⠂⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠤⠤⠤⢄⡀⠀⠒⠠⠐⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⡠⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡶⠁
⠀⠀⠈⠿⡁⠲⢤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⣀⡤⠶⡎⠁⢰⡄
⠀⠀⠀⢸⠃⠀⢠⠂⠀⠈⠁⠐⠂⠤⠄⣀⠠⠀⠀⠀⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠤⡐⠂⠁⠀⠀⠀⡁
⠀⠀⠀⠸⣣⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠱⠄⠀⠀⢇⠀⠀⠀⠀⠀⡠⠂⠀⠂⠀⠀⠀⠀⠀⢹⡟⣼⡂
⠀⠀⠀⠛⠁⠀⠿⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠃⠀⠀⠀⠀⠈⢦⠀⠈⢦⠀⢀⡠⠞⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠉⠑⠠⠀⠀⠃⣀⠄⠲⠒⠲⠀⠷⣀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠡⠠⠏⢁⠔⠚⠂⠀⠒⠠⣍⠃⡀⢀⠄⠄⠀⠦⢄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡇⠰⠁⠀⠀⠀⠀⠠⠀⠀⢠⡗⠁⠀⠀⠀⠀⠀⢣⠀⢀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠓⠄⠀⠀⠀⠀⢠⢁⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠘⠂⠀⢶⡶⠶⠶⠶⠾⠁⣠⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠠⠀⣄⡐⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡑⠤⡀⠀⠀⠀⢀⠔⠉⠀⠀⠀⢀⣠⣞⠁⠀⠀⠀⠞⢋⡵⡋⠖⣇
⠀⠀⠀⠠⢀⠀⠀⠀⡄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠣⣀⠀⠋⢘⡣⢑⢦⠀⣀⠽⠢⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠲⠄⠁⠀⠀⠀⠘⢃⠠⡀⠀⠀⠴⠀⠁⠀⡀⠀⠀⢠⠉⠑⢄⢃⠉⠃⠈⠁⠀⠀⣸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢺⠀⡜⠉⠋⠉⠋⠓⠙⢹⠇⠐⠀⠠⠂⠀⠒⠀⠂⠐⠂⠈⢋⣠⣀⡀⣷⠒⢋⠠⡀⠠⠂⠉⢱⡠⣄
⠀⠀⠠⠀⠀⠀⠀⢸⡐⠅⠠⣂⠀⡠⠀⠀⠨⠜⠃⠀⠀⠑⠀⠀⠐⠃⠀⠰⠀⠀⠀⠉⠛⠁⢽⣉⠋⢝⣺⠛⠛⢦⠂⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡣⠈⠇⠀⡀⠉⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠈⢱⠀⠸⣆⠀⠮⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⡀⠒⢺⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠱⠆⠊⢩⠺⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠤⠀⠉⠂⠤⢀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡆⢀⠸⡀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠓⠦⣄⡀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠐⠠⢀⡉⠐⠂⠠⠤⠤⠤⠄⠒⠈⢀⣠⠔⠊⠀⠀⠀⠀⠉⠓⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇⠀⠀⠀⠀⢸⠉⠙⠓⠒⠒⠒⠒⠂⢩⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⠠⠄⠐⠒⠃⠃⠈⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠤⠄⠒⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⢠⠄⠀⠀⠼⠀⠀⠀⠐⢀⠀⠀⠄⠀⠀⠀⠑⠖⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢧⠀⠀⠀⠈⢂⠀⠀⠀⠀⠄
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠱⠤⠀⠤⠂⠀⠀⠙⠦⣤⡄⠞⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣾⣷⣿⣶⡷⢾⣾⣿⣿⣿⣿⣿⣷⣶⣿⣾⣾⣿⣿⣿⣿⣶⣶⣶⣿⣷⣿⣾⣷⣶⣶⣶⣾⢿⡶⠾⠿⢶⣶⡶⢿⣾⠿⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⣀
⠀⠀⠀⡐⠒⣲⡔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣧⣧⢜⡆
⠀⠀⠀⣘⣬⡼⢦⠾⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⣀⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠉⡝⢉⠟⡇⠀⠀⠀⣀⣴⢶⠚⢒⣊⣳
⠀⠀⢀⠀⠀⡄⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠚⡀⡎⠀⡇⠀⠀⠈⡷⢧⡬⢶⣶⣮⣅⣄
⠀⠀⢸⡄⣠⠇⠸⠤⠤⠤⠤⠤⢼⠀⠀⠀⢸⠉⠉⠉⠋⠉⠉⣉⠉⠉⠍⡇⠀⠀⠀⡄⠀⣠⠃⡇⡴⡇⠀⠀⠀⣠⠊⢠⡎⢇⡱⣘⣾
⠀⠀⢸⠤⠤⡀⠀⡤⠤⠤⠤⠤⢼⠀⠀⠀⢸⠀⠁⠀⠈⢀⠀⠀⠀⠄⠀⢁⠀⠀⠀⡗⠊⠁⡠⢻⠀⡇⠀⠀⢸⣁⠜⠁⡳⠚⢀⡴⢻
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠐⠂⠀⡄⠀⠐⠀⠀⠆⣸⠀⠀⠀⡗⠒⠉⠀⡼⢀⡇⠀⠀⠀⡁⡠⠊⠀⡴⠋⡐⡻
⠀⠀⠈⠉⠀⠓⠚⠓⠒⠒⠒⠈⠋⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⢀⣀
⠀⠀⣀⣫⣄⣦⣠⣯⣖⡀⠀⠀⠀⠀⠠⡀⠐⡆⠀⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠤⢤⣤⣿⣧⣤⣤⠜⠀⠀⠀⠰⣒⢞⣛⣛⣯⡭⢀⣁⣱⠀⠀⠀⠀⠀⠀⠠⣈⣇⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⢄
⠀⠀⠀⠘⡀⢸⡇⢀⠇⠀⠀⠀⠀⢠⣡⠁⠀⠀⡀⠀⠗⠾⠂⠀⠀⠸⡁⠩⠍⠤⠤⠭⠯⠯⠭⠭⠭⠭⢽⠀⠀⠀⡏⠉⠉⠉⠉⠉⠷⡀⠀⠡⡀
⠀⠀⠀⠀⣇⢸⡇⣸⠀⠀⠀⢠⣔⠟⢹⢠⠀⠀⠇⠀⡄⠀⡅⠀⠀⠀⡄⢒⢐⡒⡂⢒⢖⠒⡒⣒⣒⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⣤⠊
⠀⠀⠀⠀⠘⠾⠷⠃⠀⠀⠀⠀⠈⠀⠸⠤⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠃⠠⠥⠬⠬⠤⠥⠤⠤⠤⠥⠤⠞⠀⠀⠀⠣⠠⠤⠤⠤⠤⠤⠦⠼

⠀⠀⠀⠀⢀⢆⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡠⠉⢢⡀
⠀⠀⠀⢠⢸⠴⣟⢰⣦⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠂⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣋⣄⣊⣁⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣤⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠄⠰⠛⠛⠂⠀⠤⠂⠀⠀⠀⡆⠀⠙⠛⠛⠛⠛⠋⠀⡂⠀⠀⠀⠀⠈⠀⠠⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠄⠀⡃
⠀⠀⠀⠸⣀⢀⢘⠀⡸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣃⠀⠀⠀⣀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⡀⡀⣈⣈⣀⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡟⠀⢀⡠⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣾⣓⣮⠉⣼⢳⣄⣾⢳⣈⢿⡆⢠⡶⢻⣿⠁⣿⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢿⡼⣿⣹⠛⣿⣹⠃⠘⠷⠻⠷⠋⢿⠜⠛⠊⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀⢀⠠⠆⢸⣿⢄⢃⡀⠀⣠⠖⠊⣿⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢼⣿⢿⣿⠀⢸⣿⢸⣿⡂⢸⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠁⠞⠈⠀⠘⠀⠈⠈⠀⠈⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⣄⣠⡾⠛⢳⡆⢠⡾⠛⠻⣦⢘⡛⢘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣿⢻⣧⡀⠺⠃⠾⢀⣠⣤⣿⢸⡇⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⠀⣉⡻⣷⡄⣴⠟⠋⠉⣿⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⣸⠇⢰⡏⠀⣼⠟⣏⣀⣠⣾⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠛⠛⠁⠀⠈⠉⠉⠀⠛⠈⠋⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠛⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣄⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠘⣿⣿⡄⠀⠀⠀⠀⠀⣿⣧⢀⣤⠤⢤⡈⠛⠛⠻⠷⠶⠶⢶⣿⣿⣦⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⣹⣿⠈⠛⠒⠒⢻⡖⠒⢶⣄⠀⠀⢸⣿⡟⠀⠀⠀⠉⠉⠉⠙⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢈⡇⠀⢸⣿⡇⠀⣾⣿⠁⠀⢀⣤⡄⠀⠀⠀⣀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⢸⠃⠀⢸⣿⠃⢠⣿⡟⠀⠀⣾⡟⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣠⡿⢃⣴⠗⣶⡿⠀⢸⠀⠀⣿⡿⠀⢸⣿⠇⠀⣸⡿⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⢀⡴⠟⢁⣾⠏⢠⣿⠇⠀⠺⠀⢰⣿⠇⠀⣿⣿⠀⠀⣿⠇⠀⢀⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡷⠋⠀⠀⣼⡏⠀⣼⣿⠀⠀⡇⠀⣼⣿⠀⠀⣿⣿⠀⠀⢿⡀⣠⢫⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⣿⠀⢰⠻⣿⠀⣸⠁⠀⣿⡇⠀⠀⠹⣿⠀⠀⠈⣉⣡⣿⡿⠤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⢿⣀⠆⠀⠙⠶⠃⠀⠀⠻⡇⠀⠀⢀⡠⠔⠊⠉⣠⣿⠏⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⢀⣀⣥⣴⣶⣤⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣾⡿⠟⠛⠉⠁⠀⠁⢸⣿⠀⠀⡎⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠁⠀⠀⢰⣶⠄⠀⠀⣿⣿⠀⠀⠙⠛⠉⠁⠀⠀⢠⣶⣿⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠉⣿⣿⣿⡆⠀⢀⣘⠃⠀⠀⢸⣿⡏⠀⣠⡀⢠⢺⣿⠂⣰⣿⠟⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠁⠀⠀⠀⢹⣿⣿⡇⠀⣾⣿⠇⠀⠀⣼⣿⠇⡞⣼⡵⢁⣿⠃⢰⣿⡯⠔⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⡟⠀⠀⢀⣿⣿⡾⠀⠿⠁⣼⡏⠀⣼⡿⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⣿⡇⠀⠀⡜⣿⣿⠃⠀⠀⠀⣿⠁⠀⠸⣷⡀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠸⣇⠀⡰⠀⠙⠿⠀⠀⠀⠀⠙⠁⠀⠀⠈⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠉⠛⠁⢀⣀⣠⣤⠴⠶⠒⠒⠛⠋⠋⠉⠉⠉⠉⠛⠛⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢧⣄
⠀⠀⠀⠀⢠⡞⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡭⣭⣭⣭⣽⡇
⠀⠀⠀⠀⢿⣄⣀⡿⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡼⠀⠀⠀⡠⠓⠀⠀⠀⠀⠀⠀⠀⢀⡼⡡⠀⠀⠀⡀⢀⠳⣄
⠀⠀⠀⠀⢀⡭⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣛⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣛⡓⠛⠖⠲⡄⣀⣳⣞⣀⡀⠀⠀⠀⠀⠀⠈⢳⡑⠀⠀⠀⠃⠌⡴⠋
⢠⡶⠶⠚⠋⠀⠀⠙⠓⠶⢶⡄⠀⠀⠀⠀⠘⣏⠙⠒⠲⠶⠶⠶⠾⠛⢺⡷⠄⠀⠀⠀⠘⡏⣿⣤⠏⢻⠛⣯⢯⢩⠷⡟⠀⠀⠀⠀⠀⠀⠀⣿⣓⣚⣚⣛⣻⡇
⠈⢷⣐⡒⠰⠆⠲⠂⣒⣢⡞⠁⠀⠀⠀⠀⠀⠹⣆⡻⠒⠗⠐⠆⠲⢂⡿⠁⠀⠀⠀⠀⢰⡇⡇⠀⠀⢸⠀⡗⠈⠉⢠⡇⠀⠀⠀⠀⠀⠀⠀⠏⠉⡹⢩⡙⠹
⠀⠀⠉⠛⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠶⠶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠓⠶⠶⠶⠚⠀⠛⠲⠖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠂⠆⣁⣎⢰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠺⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⣰⣳⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠚⠋⠙⠑⣦⡀
⠀⣠⠞⠓⢦⣈⣁⡴⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⢠⠔⠗⢤⡴⠾⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⣿⣭⣭⣽⣋⣿⠀⠀⠀⠀⠀⠀⠀⠤⠈⡟⡏⠉⠉⢹⢻⠃⠤
⣞⠡⠈⡴⠋⠁⠉⠳⣄⠡⢜⡆⠀⠀⠀⠀⠀⡤⣿⢤⣠⢼⠧⣄⡤⡷⢄⠀⠀⠀⠀⠀⠀⠀⠻⠂⠈⢳⡷⠁⠐⠾⠁⠀⠀⠀⠀⠀⠤⠤⠀⡇⠀⡜⢣⠀⢸⠆⠠
⠈⢩⠟⠁⠀⠀⢀⡀⠈⢳⡉⠀⠀⠀⠀⠀⢸⣅⣀⣠⣧⣀⣀⣼⣀⣀⣠⡃⠀⠀⠀⠀⠀⠀⣰⣛⡀⠈⡇⢀⣜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡔⠻⣄⢸⡀⠦
⠀⢯⡀⠀⠀⠀⠨⠅⠛⣀⡗⠀⠀⠀⠀⠀⠘⠯⣭⢭⣭⡭⠍⠉⢭⣭⠽⠃⠀⠀⠀⠀⢀⡽⢭⣭⣽⣧⢧⣯⣭⣭⢿⡀⠀⠀⠀⠀⠀⠀⠘⢿⡉⡀⡉⢋⡽⠃
⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠘⠗⠀⠘⠚⠀⠀⠙⠂⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠿⠭⠭⠥⠭⠽⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⡠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠧⢸⡇⠸⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢬⡑⠘⠁⣋⠴⢆⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⣄⣄⡤⢄⣀⡀
⠀⠀⣏⠀⣠⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠌⠙⢦⠘⠃⡰⠛⠡⠦⠀⠀⠀⠀⠀⠀⠉⡥⠶⠌⠀⠀⠡⢒⠲⠆⠀⠀⠀⠀⠀⠀⣯⡧⠄⢒⣿⢾⡗⠲⠶⣭
⠀⠀⠈⠓⠿⠿⠀⡏⠉⠉⢣⠀⠀⠀⠀⠀⠀⠲⠶⠶⠀⢮⡵⠀⠶⠶⠶⠀⠀⠀⠀⠀⠘⠃⣰⠊⣱⣄⠀⣠⣈⠙⠈⠃⠀⠀⠀⠀⠀⡷⣗⣋⣩⣿⣻⣏⣙⣛⠶
⠀⣠⠤⠴⢶⡄⢰⣶⡮⢀⣠⠃⠀⠀⠀⠀⠀⠘⢋⣴⠎⢠⡄⠙⣦⡘⠒⠀⠀⠀⠀⠀⠙⠀⢹⠟⣛⡏⢺⠧⣼⡇⠙⡁⠀⠀⠀⠀⠀⣏⡯⠤⠴⣿⣽⡷⠤⢭⣛
⠠⠋⠀⢀⣀⡇⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⢘⡃⠰⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⢴⡇⢸⣓⣺⡇⠀⠁⠀⠀⠀⠀⠀⠧⠗⠒⠪⠉⠏⠙⠒⠲⠾
⠘⠦⣄⣘⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢹⠀⠈⡛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠡⠂⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡠⢺⣏⡽⡗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡄⢀⢀
⠀⡔⠲⢅⣿⣅⣸⡷⡬⠒⢢⠀⠀⠀⠀⠀⠀⠀⢰⢤⠈⡵⢶⡄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡦⡴⠠⠀⠓⡤⢢⡔⠀⠀⠀⠀⠀⠀⠀⢀⣲⣡⠖⣧⣬⠳⣄⣗⡀
⠀⢇⢈⡏⡀⢠⠄⣁⢻⠀⡘⠀⠀⠀⠀⠀⢰⠲⢎⢹⣲⢰⢁⢤⢉⡃⡄⠀⠀⠀⠀⠀⠈⠒⠰⢤⡴⠁⠈⠣⡤⠥⡐⠁⠀⠀⠀⠀⠀⡭⣧⣾⠚⠧⠸⠓⢻⣤⠯
⠀⠘⠀⠇⠈⢈⡀⠁⡸⢠⠁⠀⠀⠀⠀⠀⠀⣙⡙⢶⣿⢸⠫⢭⣴⠉⣗⢺⠀⠀⠀⠀⠘⠹⢃⡠⣃⠀⠀⢀⢇⣈⠃⠃⠀⠀⠀⠀⠀⣄⡷⣼⠋⣀⢠⡈⣻⡼⣁
⠀⠀⠷⢚⣿⠀⠀⣿⡓⠜⠀⠀⠀⠀⠀⠀⠑⢬⠿⢢⡻⣌⠓⠄⣱⣚⠍⠁⠀⠀⠀⠀⠘⠉⡈⣄⢠⠃⠐⣇⢠⣄⠈⠃⠀⠀⠀⠀⠀⠐⠣⠾⣍⣳⣴⣋⡽⠴⡗
⠀⠀⠀⠀⠘⢆⠀⠛⣩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠄⢢⡋⠠⡥⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠳⠞⠣⠀⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠙⣿⡏⣻⣿⣭⣿⣿⣿⡏⢿⣿⡟⢟⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⠟⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠸⠟⢉⣿⣿⡇⣿⢈⣿⢿⣿⠉⠟⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣷⣾⢸⣿⣿⣧⣋⣼⣿⣿⣧⣷⣾⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠹⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣴⠟⠋⠉⠉⠙⠻⣦⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⠏⠀⠀⠀⠀⢀⣤⠜⣷⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢿⡎⠻⣦⣠⣴⠟⠁⢀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢿⣄⡈⠉⠀⢀⣠⡾⠃⠀⠀⢸⣿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠘⠟⢠⡾⠋⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠶⠶⠶⠶⠶⢀⣤⡾⠛⠛⢶⣤⡶⠛⢀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⣀⣀⣀⡀⠀⣿⠉⠛⢶⣄⠀⠙⢷⣴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠉⠉⠉⠁⠀⠹⣦⠀⠀⠙⢷⡀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⠘⢷⣄⠀⠈⣷⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠙⠳⠶⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠫⠉⠁⠀⠂⠐⠀⠈⠉⠝⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠂⠀⠀⢠⣤⡀⣶⣦⢠⣶⡄⣤⣀⠀⠀⠐⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⠀⠀⢀⣄⢺⣿⠌⣿⠃⣿⠷⢸⣿⢃⣿⠏⣼⡓⣠⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠘⢿⡷⠟⣀⠬⠔⠒⠒⠒⠒⠢⠍⣘⠃⣼⣿⠏⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣏⡄⣀⣀⣀⣀⣀⢈⣴⠊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⠑⣤⣁⣀⣀⣀⣀⣀⢠⣹⣄⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⣤⡀⢠⣤⣤⡀⣤⣤⣄⢠⣤⣤⡀⢀⣤⣤⡀⣤⡀⣠⣄⣤⣤⡀⣤⣤⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⣿⣇⢸⣿⣹⡇⣿⣏⣿⢸⣿⣽⡗⢸⡏⣿⡇⣿⣇⣿⡇⣿⣏⠀⣿⢽⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⣿⣾⣿⢸⣿⠉⠁⣿⡏⠉⢸⣿⢹⣇⢸⣇⣿⡇⢸⣿⣿⠀⣿⣏⠀⣿⣾⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠛⠈⠛⠙⠛⠀⠀⠛⠃⠀⠘⠛⠘⠋⠈⠛⠛⠁⠈⠛⠋⠀⠛⠛⠁⠛⠛⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠠⡄⠄⠂⠀⠀⠀⠀⠀⢤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡤⠀⠂⠀⠀⠀⠐⠠⢠⠄⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡔⠀⠀⠀⠀⢀⡴⣦⣉⠢⠤⣀⣀⣀⣀⣀⣀⠤⠔⣉⣴⣦⠀⠀⠀⠀⠠⣢⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠻⠿⢁⣾⢣⣶⠆⣤⡄⣤⣤⠀⣶⢀⣿⡌⠻⠇⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠉⠁⠛⠿⠸⣿⡇⢽⣿⠘⠿⠇⠙⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠲⠶⠶⠖⠒⠚⠋⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⢀⡔⣂⠀⠀⠀⠀⣶⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠙⠟⠀⠀⠀⣠⠛⠦⠀⠀⠀⠀⠉⠻⠏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⢀⠼⠑⠆⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠁⠀⠀⠀⠀⠀⠛⠙⠂⠀⠀⠀⠄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠁⠀⠀⠀⠀⠀⢠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢴⣦⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⣰⢚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣤⠀⠀⠐⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠗⡀⠀⠁⠀⡀⠀⠀⡠⠀⢓⠴⠊⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠈⣠⠀⠈⠑⢤⢠⣩⠚⠋⠀⠀⠀⠀⠀⠀⠀⣠⡄⣠⡖⠰⠿⠁⠛⠁
⠀⠀⠀⠀⠀⢠⣿⠋⠀⢀⣾⣿⣷⡤⠀⠶⠚⠀⠀⠀⠀⠀⡀⠀⠀⠀⠲⠏⠀⠉
⠀⠀⠀⠀⢀⣾⠃⢀⣴⣿⣿⠿⠋⠀⠈⣠⣄⡈
⠀⠀⠀⠀⣼⠃⣠⣾⡿⠛⠁⣀⣤⣶⣿⠿⠛⠋⠁
⠀⠀⠀⣸⢃⣾⠿⢋⣠⣴⠾⠟⠋⠁
⠀⠀⢰⣷⣿⠵⠞⠋⠉
⠀⠀⠛⠉

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣯⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢯⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⡿⣿⡷⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣷⠾⣷⣿⣷⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⣶⣿⣿⣿⣿⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡏⢸⣿⣿⣿⣿⡧⡯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡼⠿⡟⢻⡿⠉⢿⠉⢉⣻⣟⠛⠿⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣭⡀⣰⡆⢸⡇⠀⠃⠀⠸⡆⠸⣿⣿⣷⠀⡆⢀⡍⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣻⡇⢹⣿⠸⡇⢸⡀⢣⠀⡇⢀⣤⡟⢻⠀⣇⢸⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⢸⡿⠀⢻⡀⣷⣼⡄⢷⠈⠛⠳⣾⡀⣿⣄⠿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⢛⡛⠿⣿⠛⠛⢿⡿⠾⠷⠖⣛⣥⣽⢿⣿⣷⠾⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠿⠃⢀⡟⠀⠀⠈⣇⠀⣰⡆⢠⡄⢀⣦⡄⠛⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⢲⡾⢿⡇⠀⠛⠀⢿⠀⠛⢀⣼⡇⢸⣿⣿⡆⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⡋⣰⠿⠧⣜⠅⢤⠸⠀⣄⢺⣿⡇⠈⣿⣿⣧⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠶⠞⠛⠷⠟⠳⠼⠷⢶⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠐⢠⠂⢀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠒⠬⢘⢭⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉
⠀⠄⠐⠂⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠄⠀⠀⠀⠀⠁⢀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘
⠠⠀⠌⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡤⠓⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠰⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠄⣖⡶⢔⡠⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠌⢘⣦⡀⠄⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠔⠚⠀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠀⠀⠀⠈⠀⠈⠄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⣰⣿⠟⠁⠀⣿⣿⡆⠀⣠⣴⡶⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣴⡿⠃⠀⠀⢀⣿⣿⠁⢰⣿⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⢀⣼⣿⠇⠀⢺⣿⡇⠀⢰⣿⢀⣤⠄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢇⣀⠀⣠⣾⡿⠃⠀⠀⠘⣿⣷⡀⣾⠇⠸⢿⣧⣤⣿⣷⠾⠛⠋⠉⠁⠈⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠉⠛⣩⣵⡾⠛⠛⠇⠀⠈⠛⣿⡿⣤⣤⡄⠀⣿⣿⠇⠀⢀⣠⡄⠀⢀⡀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣿⣿⠀⠀⢰⣿⡟⠀⢀⣾⣿⠀⠀⣰⡟⣰⣿⠏⠀⢸⣿⡿⠀⠀⣾⣿⠃⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⢿⣿⠀⣿⣿⠀⠀⢸⣿⣇⣀⡼⣿⣿⣀⣴⠟⢰⣿⡟⠀⠀⣿⣿⡇⠀⢠⣿⡟⠀⢀⣿⣿⠀⠈⣿⣿⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠿⠋⠀⠻⠿⠃⠀⠀⠉⠉⠉⠀⠀⠉⠉⠁⠀⢸⣿⡇⠀⣸⣿⣿⠀⠀⢸⠿⠷⠤⠚⣿⣿⢀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⡀⠀⠀⠀⢀⣀⣀⣤⣤⡶⠶⠶⠶⠶⠶⢶⣦⣄⠘⢿⣷⠾⠛⣿⣿⠀⠀⡾⢀⡴⠖⢺⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⣿⣿⣿⠋⠁⠀⠀⠀⣀⣀⡀⠀⠈⣿⣿⡇⠀⠀⠀⠀⢿⣿⣄⡼⠁⣿⣁⣴⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠸⠿⠿⠃⠀⠘⠛⠋⣠⠞⣿⣷⡄⠈⠙⠋⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠀⠀⠀⣤⣶⡆⠀⠀⣀⣀⠀⣼⠋⠀⣿⣿⡇⠀⣀⣀⠀⠀⣴⣾⣿⠟⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⣼⣿⡿⠀⠀⣰⣿⣿⢸⠇⠀⢸⣿⣿⣠⠟⣽⣿⡇⣼⣿⣿⠃⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⠇⠀⠀⣿⣿⣯⡟⠀⠀⣾⣿⡿⠁⢰⣿⣿⠀⣿⣿⡧⠤⠚⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⡄⣀⡴⠛⠿⠟⠀⠀⠀⠻⠟⠃⠀⣾⣿⡇⠀⠹⣿⣿⣄⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⢀⡼⠃⠉⠉⠉⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⣿⣿⣧⠤⣤⡀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠤⠴⠋⠀⠀⠀⠀⠀⢸⣿⠃⣼⡿⠀⠀⠀⢀⣠⠴⠻⣿⣷⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣤⣤⡤⠴⠞⠋⠁⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠋⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⣤⣄⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠔⠛⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡾⠀⣶⣄⣀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣧⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠲⣴⣿⣆⣀⣾⠃⠀⠀⠀⠀⠀⢸⡿⠀⠀⣠⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠰⠶⠛⠛⠛⠉⠙⠛⠻⠛⠛⠛⠓⠶⠀⣾⡇⠠⣾⠋⠀⠈⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣿⣧⣴⠇⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣠⣤⣤⣭⣿⣦⣤⣤⣤⣤⣤⡿⠀⠀⠀⠀⠀⢰⡿⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⡾⠛⠁⠀⠀⠈⠉⠉⠉⠉⠙⠋⠉⠉⢳⠀⠀⠀⢠⡿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⣠⡿⠀⠀⣠⡿⠁⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠿⣏⠀⢀⣴⠋⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣷⣾⣿⣿⣿⡏⠁⠀⠈⢠⣾⣇⡀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣝⠻⣏⡻⣧⡀⠀⢰⠏⠀⠈⠁⢀⣴⣿⠏⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⣷⣬⡙⠾⠛⠿⣏⠀⠀⢀⣶⣿⣯⡁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣿⠀⠻⣷⡈⠙⢦⣄⡀⠙⠷⠶⠿⠁⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⣿⠀⠀⢹⣷⠀⠀⠉⠙⠛⠳⠚⠒⠛⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡄⠀⠀⠀⠀⠀⣠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⣀⣀⠀⠀⠐⠻⠞⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢤⣴⣿⣿⡿⠿⠞⠛⠁⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠛⠛⠣⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⢰⡏⠀⠀⠀⠀⣍⠀⠀⢠⠖⠊⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠙⣧⠀⠘⠁⠀⠀⠀⠀⡿⠀⢀⠟⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡿⢰⡇⠀⠀⠀⠀⣴⠀⠀⡾⠀⠀⠀⠀⢀⡆⠀⢀⣤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⢠⠁⡀⠁⠀⠀⠀⠀⣿⠀⢸⠃⠀⠀⠀⠀⡌⠁⣀⠟⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢠⠃⢀⡎⠀⠀⠀⠀⠀⠏⢀⡙⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⢸⠀⢸⠃⠀⠀⠀⠀⣼⠀⡘⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⠀⣷⠀⠀⠀⠀⠀⠏⠐⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡀⠃⠀⠀⠀⠀⣼⡄⣇⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⣿⢀⡴⠾⣿⢆⡀⠈⠉⠳⣆⡀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢼⣿⡞⠁⠀⡟⠀⢻⣦⠀⠀⠈⠳⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⣿⠀⠀⢠⠁⢰⣿⠉⣦⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⡧⠀⠀⠀⢀⡀⣿⡇⠀⠀⠀⣀⢸⣿⠀⠘⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠋⠀⢠⣤⡉⠛
⢀⣤⠐⠆⢠⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⡿⠖⠒⠈⠉⠉⠉⣇⠀⠀⠀⠿⡉⠉⠒⢺⡇⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠉⠱⣆
⣷⠀⠀⠀⠀⠀⠛⢦⣄⠀⠀⡇⠀⢀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠈⠳⣄⠀⢸⡇⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⣀⡀⠤⠶⠤⠤⠾
⠻⣄⠀⠀⠀⠀⠀⠀⠉⢳⡀⠇⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⠳⣶⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⣠⡤⠒⠋⠉
⠀⠀⠙⠳⣆⡀⠀⠀⠀⠀⠙⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣆⠀⠀⠀⣴⠁⠀⠀⠀⡾⠷⠶⠤⠴⠶⠶⠚⢉⡠⠶⠋⠁
⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡼⠁⠀⠀⠀⣤⠃⠀⠀⠀⠀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⢠⠿⠶⠶⠶⠋⠉
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣾⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢩⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⣿⢦⣀⣷⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢡⡀⠉⠉⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢼⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠓⠛⠛⠒⠀⣤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢦⡀⠀⣠⣤⣿⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠚⡣⣈⠳⣦⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠙⣤⣼⡇⢸⣄⣤⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢶⡶⠄⢀⣀⣴⠋⠁⠀⠉⠉⣷⠌⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠶⠚⠛⠉⠱⣆⠀⠀⠀⠀⠘⠓⣷⣤⣠⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠉⣉⡿⠀⠀⠘⣿⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣠⣤⡀⠘⣧⡄⢠⡤⠚⣉⣉⠻⠀⠰⡟⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣀⣷⡞⢿⠖⠋⠳⠖⣇⣠⣿⠉⢱⡀⠖⠉⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠙⠃⠘⠛⠛⣢⡔⠁⠀⠈⠛⠛⠛⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⠄⠀⠀⠀⣠⣾⠟⠃⠀⠀⠀⠀⠀⠀⠿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢠⣄⠀⠀⠀⠙⠻⢿⡗⠀⠀⠀⠀⠀⠀⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠀⢀⣤⣼⠛⢀⣤⡀⣤⡞⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢻⣄⣸⡏⠉⣶⡏⠁⠿⡿⠇⠰⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠾⠀⠀⠙⠋⠁⠀⠛⠻⡟⢛⡄⠀⣓⠐⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠀⠀⠀⠀⠀⠀⡶⠖⣿⣏⣡⡎⠉⣤⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢤⡀⢀⡤⠘⢷⡈⣧⠀⠸⠿⢤⡜⠳⣄⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢐⠿⢧⣼⡧⠀⠈⠉⠀⠀⠀⢠⣴⠿⠀⠉⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢣⡜⠶⣾⡄⣴⡄⠀⠀⠀⠀⣀⣸⣅⡠⢄⡀⠀⠀⠀⠻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⣰⡈⠛⠋⠛⠀⠒⠒⡿⠙⢻⠉⣠⡼⠃⠀⢀⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⣾⠉⢛⣹⠇⠀⠀⠀⠀⣀⣿⠀⠸⠶⠟⠶⣶⠎⠉⢠⠄⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠚⠛⠀⠾⡧⣄⠀⠀⠐⠃⠈⠀⠀⠀⠀⠀⠀⣸⣦⣤⣾⠆⢰⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⢨⡇⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠈⠉⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡤⠤⢴⣟⠃⠋⠀⠀⠀⠀⠀⠀⠀⠘⠇⢰⣦⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣤⣀⣀⣀⣼⣅⠀⠀⠙⠒⠖⢻⣄⣀⣀⣀⣀⣀⣤⣄⣠⣿⣄⣀⣀⣀⣠⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠀⠉⠉⠉⠉⠉⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣄⠀⠀⠀⣠⣶
⠀⠀⣿⡇⠀⠀⢀⠀⠀⢀⣀⡀⠀⣀⣀⡀⠀⣀⣀⡀⠀⠀⣀⡀⠀⡀⠀⠀⡀⣀⣀⣀⠀⣀⣀⠀⠀⢸⣿⠀⠀⢠⣿⡇
⠀⠀⣿⡇⠀⢠⡟⣧⠀⣿⣉⣹⠆⣿⣉⣿⢈⣯⣉⡿⢠⡟⠉⠹⡆⢻⡄⣼⠃⣿⣭⣍⠀⡏⠉⢻⡄⠸⢿⠀⢀⣿⣿⠁
⠀⠀⣿⡇⠀⡾⠛⠻⡇⢿⠁⠀⠀⠿⠀⠀⠨⡇⠈⢷⠈⠳⣤⡼⠃⠀⢿⠏⠀⢿⣤⣤⠀⣷⡤⠾⢡⣶⣦⣀⣼⣿⡇
⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠿⣿⣿⡿
⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠈⠻⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⡤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠢⡀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣽⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⢟⢦⣹⠀⠀⠀⠀⠀⠀⡀⠀⢂⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣟⣾⣿⣥⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣑⡵⠞⠁⠀⠀⠀⠀⠀⠀⠃⡐⠄⠀⠀⠀⠀⠀⠀⠐⣫⣽⣟⢻⡿⡿⢿⡙⠃⠀⠀⠀⢀⠀⣠⣄⢀
⠀⠀⠀⠀⠀⠀⢠⠏⣠⣾⠏⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠁⠇⢀⣀⠀⠀⠀⠀⠀⠉⠸⢫⣾⡇⡿⠀⠃⠀⠀⠀⢚⣻⣿⣿⣻⣿⣷⠄
⠀⠀⠀⢀⣀⠠⠯⠚⠁⣠⡆⢴⣾⠿⠟⢚⡡⠃⠀⠀⠀⠠⢾⣉⣽⠀⠀⠀⠀⠀⠀⠀⣾⡁⠁⠀⠀⠀⠀⠀⠀⢉⡯⢿⣿⣿⣿⣟⠷
⠀⠀⠀⣏⠈⠱⠀⢀⣼⢯⡄⠘⡇⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⣨⠿⣤⠀⠀⠀⠀⠀⠀⠉⣤⠀⢀⡀⠤⡀⠀⠀⠩⢶⡟⠋⣿⣑⡝⠂
⠀⠀⠀⠹⣷⣴⢇⡾⠃⢹⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⠦⠿⠄⢀⡀⠀⠀⠀⣾⢿⢠⡏⠀⠀⣿⠀⠀⣴⠟⠁⠀⠘⠁⠁
⠙⠿⣍⠒⠦⣄⡼⠁⠀⠘⡇⠀⢠⣃⠀⠀⡎⠉⢆⠀⠀⠀⢨⣧⠀⢠⠊⠀⢘⠆⠀⣸⠏⢸⡌⠻⠿⠟⣁⣀⣤⢤⡤
⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⢷⠀⠀⢻⡟⢦⣛⠶⠜⣠⢾⠁⢸⠈⢷⡘⠷⠾⢋⣠⡄⠛⠀⠘⢣⢶⣿⠿⠋⢁⡴⠋
⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠸⡄⠀⠈⣷⠀⠈⠳⠞⠁⡼⠀⢸⠀⠈⠷⠖⠚⢩⠏⣸⠀⠀⠀⠀⠉⠁⢀⡴⠋
⠀⠀⠀⠀⠈⢷⠀⠀⠀⡀⠀⢧⠀⠀⢸⡄⠀⠀⠀⢠⡇⠀⣿⠀⠀⠀⠀⢰⠇⠀⡄⠀⡀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠈⣇⣠⡟⠙⢦⡘⣇⠀⠀⡇⠀⠀⠀⣼⠀⢠⡏⠀⠀⠀⢠⡏⠀⢰⠀⡼⡇⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⢹⣽⠃⠀⠀⠙⢿⣆⠀⢻⢀⠐⣄⡟⠀⣼⡥⠚⢳⠀⣾⠀⠀⣸⡜⠀⡇⢀⡞
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣆⠼⠃⠀⠘⠇⠰⠋⠀⠀⠀⢷⡇⠀⠀⠿⠀⠀⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⣟⣿⣭⡭⢯⣭⣿⣻⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣯⣓⣿⣿⣿⡿⢷⣾⣿⣿⣿⣿⣿⣯⣽⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣿⣶⣿⣿⡷⣟⣵⣿⣾⣯⣾⣿⢿⡿⣳⢿⣿⣿⡽⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⢄⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⢠⣿⢟⣿⡿⢿⡾⣿⠟⠷⠋⣽⢿⣿⡷⠻⠟⣿⣻⣾⣿⣶⢻⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⡀⠀⠀⠀⠀⣸⠀⠀⠀⣷⠀⠀⠾⠏⢈⣁⡀⠋⠀⠁⠀⠀⠘⠁⠀⠀⠀⠀⣾⣯⣿⣷⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⣠⡼⣧⠀⠀⠀⠀⣿⠀⠀⠀⡟⠀⠀⢰⢠⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣦⣹⡟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣦⠈⠁⠀⢹⡀⠀⠀⢠⡟⠂⠀⣸⠁⠀⠀⠸⡆⠠⢤⣤⣤⢤⠀⠀⠀⢀⣀⣀⣠⣀⠈⢻⡟⣛⡻⣮⢿⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⢀⣼⣇⠀⠀⣾⠀⠀⢀⡏⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠸⠾⣏⠁⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠉⠀⢻⡀⢀⡟⠦⠀⣸⠇⠀⠀⠀⠀⣧⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⢧⠾⠁⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⢠⣄⠈⠳⣶⠶⠂⠀⣠⠤⡀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⣤⠔⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡇⢲⠌⡑⠒⠶⠶⠒⠋⠁⠀⣹⡄⠀⠀⣰⠏
⠀⠀⠀⢀⠼⣇⠀⠀⠀⠠⡄⠀⠀⢀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠹⣌⠢⢄⠀⠈⠀⠈⠀⠈⠀⡼⠁⢀⡼⠋
⠀⠀⠀⣮⠀⠙⣧⠀⢰⡏⠋⣻⠟⠧⣴⠒⠒⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢷⣄⠈⠡⠴⠶⠒⠂⠉⣀⣴⡿
⠀⠀⠀⠘⢦⡀⠘⢧⠸⡗⠾⠁⠀⠀⡈⢷⣤⣤⡀⠈⡇⠀⠀⠀⣀⣀⣀⣤⠟⠙⠷⣦⣤⣤⣤⠖⠋⠁⠘⡇
⠀⠀⠀⠀⠈⣷⡀⠈⠳⣝⣶⣤⡀⠋⠀⠀⠀⠀⠀⠈⣧⡶⠿⣏⣿⣿⣿⣿⣿⣿⣶⣦⣬⣤⣤⣤⣤⣶⣶⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠹⣟⠢⠤⠞⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣄
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⣸⡷⣄⠀⠳⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⣠⣿⣷⡄⠀⠀⠀⠈⠙⠛⠉⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⣤⣀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⡿⣿⣿⡟⢡⣣⢽⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢛⣟⣻⢛⣿⣋⡛⢻⡟⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣻⣻⣿⣛⣿⣟⣛⣛⣻⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣾⣿⣿⡿⠟⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣶⡿⠿⠟⠿⠿⠛⠋⠉⠁⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣦⣾⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣼⣿⣿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠀⢻⣏⠉⠙⣿⡄⠀⢽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⣦⡀⠀⣿⡇⢸⣧⣤⣴⠆⠈⣿⡀⠀⣸⡇⠀⠘⣿⣿
⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⠿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⠀⢰⡟⠋⠙⢿⣆⠙⣷⡄⣿⠀⠈⣿⣀⣀⣤⡄⠹⠟⠛⠋⠀⠀⠀⢻⣿⡇
⠀⠀⠀⣴⣿⡿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⢶⣄⠸⣿⠉⣈⡽⠇⢹⣇⠀⠀⢸⣿⠀⠈⢿⡿⠀⠀⠙⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⠃
⠀⠀⠰⣿⣿⠀⠀⠀⠀⢀⡀⠀⠀⢶⡾⠛⢻⣦⠸⣷⣀⣤⠿⠀⢿⡟⠙⢷⣦⡀⠛⠷⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⣿⣿⡿⠿⠛⠁
⠀⠀⠀⢻⣿⡆⠀⠀⠀⣿⢿⣆⠀⠘⣷⡶⠾⠃⠀⢻⡏⠀⠀⠀⠘⠟⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠐⣿⣷⠀⠀⢀⣿⣤⠿⣷⡀⢻⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⡿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⢿⣿⡆⠀⠸⠏⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⣿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠻⣿⣷⣦⣔⣶⣿⣿⣿⡿⠿⠛⠛⠉⠁
⠀⠂⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣾⡟⠋⠉⢿⣴⠶⢤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢣⠀⠀⡼⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣳⡦⣤⡷⢶⣷⠲⢿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣊⡴⠚⢉⣴⠞⢹⡇⢸⠈⠳⣄⠈⠻⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣁⠀⣰⠟⠁⠀⢸⡇⢸⠀⠀⠘⢧⡀⣽⡇⠃
⠀⠀⠀⠀⠀⢀⣀⣐⠒⣠⠤⠤⠶⠶⣮⡭⠿⣞⣹⣄⠀⠀⢸⡇⢸⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⣠⠟⠡⡍⢻⡇⠀⠀⠀⠀⡿⣸⠃⠈⠛⣷⡖⠒⢾⠀⢸⠲⠶⠶⢶⡟⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠐⣇⠀⠀⠙⣾⣇⠀⠀⠂⢸⣿⠏⠀⠀⣰⣿⠀⠀⢸⠀⢸⠀⢀⠀⢸⠇⠀⣤⠶⠶⠛⠛⠛⢷
⠀⠀⠀⠀⠈⢧⣤⠖⠚⠿⢤⣴⠤⠟⠛⢶⣡⣾⣋⣿⠀⠂⢸⡇⣿⠠⠀⠀⣿⣤⣼⡇⠀⠀⠀⠀⠀⢸⣷⠞⡛⠳⣦⡀
⠀⠀⣠⡤⠶⣾⠃⠀⠀⠀⣴⠛⣆⠀⠀⠈⢿⡉⠛⠛⢻⡆⢸⠃⢻⣠⡴⠾⠋⢰⡂⣇⠀⠀⠀⠛⠀⢸⡇⢠⠇⠀⠈⠙⣷
⠀⠀⡇⠀⢰⣏⣴⣦⠀⣼⠃⠀⢸⡆⠀⠂⠈⣧⠀⠀⢸⡗⢺⠀⢺⣹⡇⠀⠀⠀⢳⡸⡄⠀⠀⠀⠀⢸⣧⠏⠀⠀⠀⢀⡿⢰
⠀⡆⢿⣠⣼⣯⣤⣽⣴⡇⠀⠀⠀⢿⠀⠀⠀⢸⡶⢶⡾⠛⣽⠀⢸⠀⢳⡀⠀⠀⠀⠹⣷⠀⠀⠀⠀⣼⡋⠀⠀⠀⣠⡾⠁⠁
⠀⠘⡄⣿⠀⠀⠁⠀⠉⣧⠀⠀⠀⢸⡇⢰⠷⣬⡇⢸⣇⡀⢸⠀⣸⣀⣀⣿⣦⠔⠒⠒⠚⣳⡤⡖⠋⠉⠉⠳⣶⠾⠿⠶⠶⠶⠶⣦⡈
⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣷⡏⠀⠘⠁⢸⡇⠁⢸⣼⡏⠉⠁⢠⠏⠀⠀⠀⡼⠁⠀⠈⢦⡀⠀⠀⠘⢷⣀⣀⣀⣀⣀⣼⠃
⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠁⢸⡏⠀⠀⠀⠀⣸⠄⠀⢸⡝⣷⡤⢤⣿⡴⢶⡄⢸⠗⠖⠛⠛⠻⢷⡀⢠⣄⡈⢿⠉⠉⢉⡏⠁
⠀⠀⠀⢸⡇⠀⠈⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠠⠀⣾⠠⠀⢸⡇⣿⢹⡄⠁⠀⠀⠙⠟⠀⠀⠃⠀⠀⠈⣧⠟⠈⠙⠻⠆⠀⣾⠁
⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⡋⠀⢀⣀⣿⣤⣤⣼⣇⣿⣄⣱⣤⣤⣤⣤⣤⡤⣤⣤⣤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠋
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠶⠶⠶⠾⠋⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠈⡆⡀⠈⡆⠀⢸⠆⠀⣤⡀⠰⡀⠀⠀⠀⠁⠈⡆⠀⠈⡇⠀⢸⠆⠀⣤⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠂⢰⣇⠇⢠⡇⠀⢸⠀⠀⡇⠀⢀⡇⠀⠀⠀⠀⢀⣇⠀⢰⡇⠀⢸⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⣠⠶⡆⢀⣀⡀⢸⠃⠀⢸⡀⠀⠋⢀⡞⠀⠀⠀⠐⠀⢀⣀⡀⢸⠇⠀⢸⡄⠀⠋⢀⡾⠁
⠀⠀⠀⢰⡏⠀⡇⢸⠁⡀⢸⡀⠀⢸⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡄⠀⢸⠁⠀⡞⠉
⠀⠀⠀⣿⡇⢸⠃⠼⠀⠐⠼⠃⠢⠾⠀⠴⠇⠀⠀⢀⣤⣀⠀⠡⠼⠀⠐⠼⠃⠢⠾⠀⠴⠇
⠀⠀⠀⣿⣧⠏⢀⣀⠀⠀⠀⣰⠟⣆⠀⠀⠀⡴⢦⣾⣯⣽⠀⠀⠀⠀⣀⡀⠀⠀⢠⣿⠀⢠⣶
⠀⠀⠀⢹⣿⣰⠻⣿⡀⢠⣿⣿⡀⣽⠀⣼⣿⣇⢸⠊⢉⣿⡟⠀⢠⣾⡏⣹⣷⠀⢸⣿⡀⣸⣿⠀⡴
⠀⠀⠀⠸⣿⡇⠀⢿⣧⠟⣿⡘⣷⡿⠞⠹⣧⣹⡾⠖⢻⡿⢀⣴⠋⠻⠿⠛⢷⠴⠁⠙⠛⢻⣿⡟⠁
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠉⠉⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡞⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⣷⣾⠀⣰⡶⣶⡄⢀⣶⡆⠰⣦⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠏⠻⠀⣿⡇⢸⡇⣸⣏⣿⡀⠙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠿⠡⠿⠛⠻⠧⠀⠿

⠀⠀⠀⠀⠀⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠔⠢
⠀⠀⠀⠀⣰⣀⡐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠩⠦⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡃⢐⡃⢘⡀⣓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠈⣠⠤⢀
⠀⠀⠀⠐⠀⠀⠀⠚⠘⠁⡇⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣠⡜⠀⠀⠀⠀⠀⠀⠀⠎⠁⣈⠄⠁⠄⠈
⠀⠀⠀⢸⢠⡥⢹⠀⣀⠖⠆⠀⠀⠀⠀⠀⠀⠀⠁⠰⠟⢀⠂⠀⠀⠀⠀⠀⠀⠀⡧⠶⠶⠶⠀⢠⢤⣾⠀⠀⠀⠀⠀⠀⠀⢀⣾⠡
⠀⠀⠀⢸⡀⠀⡘⠠⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠚⠓⠓⠚⠚⠓⠁⠉⠈⠃⠀⠀⠀⠀⠀⠒⠁⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠐⠆⠻⠉

⠀⠀⠀⠀⠀⠀⡁⢁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠁⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠁⠀⠁⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠖⢤⠔⠂⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠂⢤⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠚⢁⠱⠦⠰⠏⠈⠁⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠄⠀⡆⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠺⡏⠶⠒⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠃⠀⣤⣄⡀
⠀⠀⠀⠐⡨⠁⢀⡀⠈⠭⠀⠀⠀⠀⠀⢀⢠⠄⠊⠀⠀⠈⠣⢆⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠐⠉⠀⠀⠀⠀⠀⠀⠀⠚⢠⠤⡔⠤⣌⣈⢭⡈⠆
⠀⠀⠀⠘⢀⡠⠃⠘⠄⠀⠁⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠂⠒⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⢄
⠀⠀⠀⠀⢀⠂⠡⠄⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⢎⠃⠀⠈⠆
⠀⠀⠀⠀⠤⠄⠤⢠⠠⡤⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠓⣆⠃⠀⠀⠀⠀⠀⠐⢀⢰⠂⠐⠆⠒⠓⡀⡂⠀⠀⠀⠀⠀⠀⠆⡈⡁⠯⠘⠡⠄
⠀⠀⠀⠀⠠⠄⢀⡰⠠⠇⠀⠀⠀⠀⠀⢀⣠⣄⣀⣄⣀⣄⣡⣄⠀⠀⠀⠀⠀⠀⠘⢈⠘⠁⣆⠨⢳⡟⠁⠀⠀⠀⠀⡀⡡⠊

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡀⠈⢰⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡣⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡐⠛⠘⠂⢁⠒
⠀⠀⠀⠉⠘⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠰⡁⠀⢼⡱⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠘⢁⠐⠃⡚
⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠓⠖⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠡⠄⠁⠄⠌

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠶⠶⠶⠦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢞⡫⠽⠒⢋⣭⡉⠉⣭⡍⠉⣝⠒⠯⢝⡳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡽⣊⡁⠐⣿⣆⠈⣿⠇⠀⠿⠇⢰⣟⠀⣰⣷⠈⠑⢯⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠕⣁⡀⠹⣇⠀⣹⠭⠒⠚⠉⠉⠉⠉⠓⠒⠤⣝⠃⠀⠀⣀⡈⠪⡻⣆
⠀⠀⠀⠀⠀⠀⢀⡾⡳⠋⠈⠻⣇⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠙⠢⡀⠛⠃⠀⠙⢞⢷⡀
⠀⠀⠀⠀⠀⢀⡾⡱⠁⠻⣷⠆⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠈⢢⠀⠀⣀⠈⢎⢷⡀
⠀⠀⠀⠀⠀⣼⢳⠃⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⢳⠘⠛⠃⠘⡞⣧
⠀⠀⠀⠀⢀⡟⡏⠰⠿⠆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢹⢻⡀
⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣶⡆⢸⢸⡇
⠀⠀⠀⠀⢸⡇⡇⢠⣤⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠈⣇⣇⠈⠉⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⣤⠀⣸⣼⠁
⠀⠀⠀⠀⠀⢻⡼⡄⠀⢀⠀⢧⠀⠀⠀⠀⢿⣷⣶⣤⣄⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⠋⢠⢧⡟
⠀⠀⠀⠀⠀⠈⢷⡱⡀⠻⠇⠀⠳⡀⠀⠀⠈⠛⠛⠻⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣴⣤⡀⢀⢎⡾⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡵⣄⠀⠀⣀⡈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢡⣄⠈⠛⢣⢮⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⡢⡀⠛⠃⠀⢀⣝⠒⠤⢤⣀⣀⣀⣀⡤⠤⢒⠋⢤⣷⡀⠙⢒⢔⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣳⢄⡠⢾⡏⢀⣿⡆⢰⣶⠀⢶⣦⠀⢿⣧⠀⢙⡣⣞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢮⣕⣲⠼⣅⣀⣸⣋⣀⣸⣛⣠⠬⣗⣪⡵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⠶⠶⠶⠖⠛⠛⠉⠁

⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⢸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸⠛
⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠲⣾⡿⠛⣤⡀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣅⡀⠚⢹⣷⡄⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⣿⣿⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡉⢹⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠑⠿⣽⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠙⣿⣿⡆⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⣿⠁⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⠀⠀⠀⠐⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣴⣿⣆⠀⠀⠀⠀⠀⣿⣄⣠⣤⣄⠀⣤⣄⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⠀⠀⠈⢻⣿⠉⠿⠁⠙⠁⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢿⣄⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣴⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀⠈⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠏⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣼⡇⠀⢸⣾⡟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⢻⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠀⠀⠀⣸⢿⣤⠛⠀⠀⠀⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⠀⠀⣿⡀⠿⠀⢠⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⡀⠀⣀⢸⣿⠀⣶⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⣏⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⢹⣿⠟⠿⢿⣷⣤⠄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠉⠀⠀⠀⠉⣥⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⣤⣶⣾⠛⠁⠀⠀⠀⠀⠈⠛⠛⠓⢲⣤⠀⠀⠈⠉⠛⠿⢷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⣤⣀⣠⠀⠘⠙⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⣼⠟⠁⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠀⣤⡀⠀⠀⠀⠀⢀⣠⣤⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣤⠋⠀⠀⠀⠈⢰⡄⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠈⠛⢀⣀⣶⠇⠀⠛⠛⠁⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣤⣆⠀⠚⠀⠀⠀⠀⠉⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⣶⣤⣿⣿⣿⣶⣦⣤⣶⣤⣄⢀⣤⣾⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠉⣿⡟⠋⢹⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⢹⡧⠀⣿⣿⠆⠘⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⣤⠀⢾⠏⣀⠀⢸⣿⠋⢿⣿⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣿⣄⠀⣀⡨⠆⠸⣿⠄⠘⠁⠀⠀⠀⠀⠀⢀⣿⣦⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢈⣿⣏⠻⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⣧⣄⣹⣿⣷⠀⠃⠰⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⡟⠻⣿⠁⠀⠀⢣⣤⣿⠀⢀⣤⣤⣶⣾⣿⡏⠙⢿⣷⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠻⣦⡀⠀⢀⣽⣿⠀⠈⠻⣿⠉⢸⣿⠇⣶⠀⠿⠀⠀⡤⠐⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢀⠈⠁⠀⢼⣿⣧⣄⠀⣿⠁⠀⢸⣿⠀⠉⠀⠀⠀⢀⡀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅⢰⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⣿⠀⠀⢿⣿⠀⠀⠀⠀⠀⣾⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠓⣦⠀⠀⠀⠀⢠⣿⣿⣿⠀⠘⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⢸⣿⠀⠀⠐⠀⠀⣼⠛⠋⠀⠀⣀⠀⠀⠀⠤⠀⠠⠀⠀⣀⣿⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣿⡀⠀⠀⠀⠀⠀⢶⣤⡀⠀⠻⣷⣄⠀⠀⢰⡄⠀⠀⠙⠋⠀⣶⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠐⠂⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣶⡆⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠠⠀⣀⠀⠀⠀⠈⠉⠛⢿⣿⣿⡿⢿⣷⡄⢠⠀⠀⢰⣾⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠻⠄⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠉⣿⢇⣀⡈⠀⠀⠙⠁⠀⠹⣿⡿⠋⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣤⣾⣧⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣧⡄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⠙⢛⣛⣀⣄⡀⠀⠀⠀⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣄⠀⠀⠀⠀⠀⣴⣆⠐⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠺⠿⠿⢿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠈⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣴⣤⣤⣤⣶⣶⣶⣄⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⢀⣴⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⡦⢙⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡋⣴⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠎⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡀
⢠⣿⡟⠁⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠈⢻⣿
⣾⣿⠁⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⣿⡷⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⠈⣿
⣿⣿⡄⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢼⡿⠃⠀⠀⠀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠀⠀⢠⣿
⢹⣿⣷⡄⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠸⠿⢛⣤⣤⣴⣶⣶⣶⡿⠟⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢠⣾⣿
⠀⣿⣿⣿⣶⣦⣤⣀⣀⣀⣀⣀⣠⣴⣾⣿⠏⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣦⣄⣀⣀⣀⣀⣀⣤⣴⣶⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⠿⠿⣿⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⣼⡟⠁⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠘⠛⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠈⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃
⠀⠀⣙⣛⣻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⣛⣋
⠀⢰⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡆
⠀⣿⣿⣿⠏⠙⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⠹⣿⣿⣿
⢸⣿⣿⣿⠀⠀⠀⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠀⠀⠀⣿⣿⣿
⠸⣿⣿⣇⠀⠀⠀⠀⠈⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⠀⠀⠀⠀⣸⣿⣿
⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⠙⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⢾⣿⡿⠋⠀⠀⠀⠀⠀⠀⢿⣿⡟
⡀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⠟⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣿⣿⣦⠀⠀⠀⣾⣿⡆⠀⠀⠀⢀⣤⣤⡀⠘⠁⣠⣿⡿⠉⢿⡆⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⡟⠀⠀⢰⣿⣿⡟⠀⢠⣾⣿⣿⠙⣷⠀⣼⣿⣿⣧⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣸⣿⣿⡇⢠⣿⣿⣿⣿⠀⣿⢰⣿⣿⡿⢿⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣴⣾⣿⣿⣿⠀⣸⣿⣿⠙⢿⣦⣿⣾⣿⣿⡁⠀⠀⣸⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠋⣿⣿⣿⠀⢸⣿⣿⠀⠀⢸⡏⢹⣿⣿⠃⠀⢀⣿⣇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢻⣿⣿⢀⣿⣿⣿⡀⢠⡟⡄⠸⣿⣿⣆⢀⣾⠟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠻⢿⠾⠃⠈⠹⠿⠋⣼⠃⠀⠈⠛⠿⠟⠋⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠙⢛⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠾⠁⢠⣧⣄⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣤⣶⣿⣿⡿⠿⠿⢿⣿⣿⠇⠀⣀⣀⣠⣤⡄⢸⣿⣿⡆⠀⠀⣴⣿⣿⡏⠀⢰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣴⣿⣿⣿⣿⠃⢀⣠⣴⣿⠟⣡⣶⣿⠟⣻⣿⣿⠃⣸⣿⣿⠇⠀⣾⣿⣿⣿⡇⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣾⣿⣿⣶⠿⠟⠛⠁⣼⣿⡿⠃⣴⣿⣿⣿⢀⣿⣿⣿⣤⡾⠋⣾⣿⣿⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⢀⠀⢸⣿⣿⣣⣾⡿⢹⣿⣿⣼⠇⠀⠉⣉⣀⣤⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣷⣅⠘⠿⡿⠟⠋⠀⠈⠛⠛⠋⣠⣶⢿⡿⠛⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠈⠻⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⢰⣿⣷⡏⠀⢰⣿⣿⠇⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⣉⠛⠿⣿⣿⣿⣿⣷⣶⡶⠊⢿⣿⣇⢠⣿⣿⠏⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠤⣬⣭⠍⠀⠀⠀⠀⠙⠛⠛⠛⠁⠸⠋






⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⠶⣶⠶⢶⣶⡶⣶⠶⡶⢶

⢀⠀⠀⠤⠊⠔⡑⠀⠠⣀⠀⢴⢒⢧⡟⠿⢟⠟⢶⡾⡿⠛⠻⣧⡏⡀⣶⢶⢾⣪⣸⠀⠀⠁⣈⡁⡂⠄⠄⠀⠠⢄
⠀⢈⠬⠍⠄⠀⠀⠰⠔⠢⢌⢢⣀⣂⡄⠀⣸⢳⠽⡷⡾⣠⣄⣿⠃⡝⣕⠀⢀⣻⡽⣲⠀⠀⡕⠈⢑⠥⠶⠖⡌
⡁⠀⡀⠀⠀⠀⠀⠀⠀⢂⠄⠀⠟⠆⡛⣙⠉⢸⠛⡻⣽⠻⣿⠿⢔⢒⣺⡟⢵⢦⠜⡁⡁⠂⠄⠀⠀⠀⠀⠀⠘
⡅⠀⠀⠀⠀⡀⠀⠀⠔⡡⡎⣄⣀⡀⠀⠀⠀⠁⢸⣇⠈⣿⠁⠀⠀⣼⢿⡷⣋⣡⠀⢪⡐⠀⢀⠀⠀⠀⠀⠠⣄
⡅⠀⠤⠀⢖⠈⡀⢚⣠⣀⣠⣼⡵⣧⡀⠀⠀⠀⢀⣁⣠⣤⣄⣀⡀⠈⢛⣷⣵⡽⠑⠤⡀⠄⠌⠀⡀⠨⠄⠈⠀⠄
⢴⣦⠶⢖⢳⠦⣀⠬⢛⣿⣗⣶⡿⠋⢀⡠⠖⠋⠉⠀⠀⠀⠀⠀⠉⠳⢌⠋⠀⠀⠀⠀⣠⢴⡔⠈⠀⠀⠁⠠⠖⠒⡧⣔⠻
⣑⡜⠊⠱⣷⠯⠂⠈⢸⡩⣹⣿⣅⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣞⢩⢠⣧⠏⠉⠀⠀⠀⠔⣱⠌⠃⡂⡀
⠋⣡⣤⣰⠛⡟⠻⠂⢶⣯⡅⠿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⣳⣿⣔⣶⠄⠀⣠⣀⢀⡌⡦⠀⢚
⣐⣒⣭⣑⣢⣦⣤⣄⣰⢲⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢩⣿⣿⣄⡶⠵⣿⣦⡻⡜⠄⢁⠜⠒
⣀⠰⣻⠟⠯⣍⣍⠛⠛⠚⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⢈⡿⡀⠈⣼⢫⣿⣁⣤⠄
⣁⢬⣯⠄⣀⠝⡯⣆⣀⣀⡀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠖⠒⠛⠛⠛⠦⣄⠑⡛⠙⡸⣿⠽⠗⢽
⠀⠀⠋⣪⣿⠗⡉⠉⡀⣀⠀⠙⠢⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡹⠿⠷⢶⣏⠆⡀⢠
⡤⢒⡋⠭⠐⢊⠀⠀⠀⠀⠀⠑⠄⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠐⠰⠁⠀⠀⠀⠀⡑⠀⠀⡀⠀⠉⢿⣝⠓
⡔⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠸⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂⠀⠋
⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢡⠐⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡌
⡄⢂⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠂⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣆⠁⠐⡈⠀⠀⠀⠀⠀⠀⠀⠠⢈⠄⣰⠃
⡀⠀⠑⠀⠄⢀⠀⠀⠀⣀⠀⠀⠈⠈⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣷⠐⠀⠸⣆⠀⠑⣈⠀⠀⠀⠀⠀⠀⣀⣤⠞⠑⠀⢄
⠈⠲⠤⣤⢀⠀⠀⠀⠀⠀⠂⡀⢠⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢸⡧⡟⠉⠲⣺⣆⠡⠐⣀⣀⣬⡴⢖⠋⣨⠀⠀⠀⠀⠈
⠀⠀⠀⠒⠊⠏⡉⠁⠀⠒⠦⠴⡏⡀⢀⡂⢴⡛⠒⡒⠒⠒⠒⠒⣚⣷⠃⠀⠀⠉⢻⡟⡋⠁⠀⠸⣿⣽⡧⢣⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⣸⣼⣤⡴⢦⡄⠀⠀⠀⠋⠑⢻⣍⣩⣭⠭⢭⣭⣭⠭⣿⠶⢶⡤⠶⠶⠥⡠⠀⠀⢸⡿⣿⡷⠋⠛⣁⣸⣏⣽
⠁⠠⠁⠀⠀⠛⣻⢗⡵⢋⠀⠀⡀⠀⠀⠀⢸⡦⠶⠶⠶⠶⠶⠒⠒⣿⠀⠀⢂⣀⠀⡂⠀⠀⢐⣿⣿⣿⠁⠂⠈⠃⠀⠀⠣
⠈⠚⠂⠤⢄⡊⠛⣫⠖⣿⣷⣈⣔⠂⠀⣀⡸⠷⢶⣏⣛⣉⣉⣽⡟⢩⠀⠀⠁⠀⠀⠀⠀⣰⣯⣯⣻⡏⠠⡄
⠂⢀⠤⢄⠀⠀⠀⠻⠟⣫⣿⢽⡟⣢⠌⠁⠀⠤⠀⠀⠠⠄⠀⠳⢘⠀⠀⠀⠀⠀⠀⢀⣺⣯⣿⣿⠋⠐⢄⡀⠔⡀⠤⡕
⠈⡏⠀⠀⠉⠑⣧⠀⢀⠉⢁⣴⣿⣿⣶⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠠⢐⠄⣠⣴⣿⣿⠛⠒⠃⠀⠀⠀⣉⢖⡶⡥⢈⡀
⠀⠀⣀⣀⡠⠔⠁⠀⡀⠀⠀⠉⠿⣿⣿⣿⣧⣆⠁⠀⠀⡀⢰⡾⡆⢠⣌⣶⢷⣏⢌⣿⠟⠁⠀⠀⢀⠤⠖⠋⢣⢱⢀
⠄⠐⠀⠀⡀⣀⠤⡈⠀⠀⠀⠀⠀⠉⢷⣿⣿⢗⢈⡴⡚⠰⣴⢻⣽⡌⢯⠗⣾⢈⠉⠁⠀⢸⠀⠠⣇⠀⡀⠀⠀⣇⣇⠆
⠰⠎⠐⠈⠡⢔⡀⢰⠀⢀⠴⠚⠷⠉⠁⢉⠃⢡⢈⠄⡨⠁⢘⠾⠛⣉⠀⠀⠀⠡⠈⠉⠉⠂⢢⡀⠉⢙⠽⠦⠔⠋⡘⡅
⠁⠀⠀⠀⠀⠀⡇⡉⠀⢈⠀⠀⠀⠀⠀⠀⠀⠘⠀⠎⠀⠀⠀⠐⠌⠎⠀⠀⠀⠈⠀⠀⠀⠘⠀⢈⡰⣻⣴⣤⣁⣀⠴
⠄⠈⢂⢢⠀⠀⠄⠔⠦⡀⠐⠒⠠⠄⠀⠀⡠⡆⠠⠀⠀⢄⣀⠀⠀⢄⠠⠀⠐⠀⠀⠀⠀⡀⠀⠸⠈⠈⠀⠀⠛⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⢰⣾⡆⢀⣴⣾⣿⣷⣦⡀⢀⣴⣶⣿⣷⣦⡀⠀⣾⣷⣶⣦⡀⠀⢀⣶⣶⠀⢶⣷⡀⣰⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⣿⡇⣿⣿⠁⠀⢹⣿⣧⣼⣿⡏⠀⠈⣿⣿⢰⣿⣇⣨⣿⡇⠀⣾⡟⣿⣇⠀⠹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⢹⣿⠁⢿⣿⣄⣀⣼⣻⡏⢹⣿⣧⣀⣠⣿⡿⢸⣿⣿⣿⣍⠀⣼⣿⣷⣾⣿⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠘⠛⠀⠀⠙⠻⠿⠟⠋⠀⢀⢙⠻⠿⠟⠋⠀⠘⠛⠃⠉⠋⠘⠛⠋⠀⠀⠙⠃⠀⠛⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣷⣀⣀⣀⣠⣾⠇⠀⠀⠀⠀⣄⠛⠃⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣂⣀⣀⣠⡿⢃⡴⠒⠄
⠀⠀⠀⠀⠀⠞⠉⠀⠀⣸⡿⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠹⠘⠀⠀⠀⠀⢠⠊⠁⠀⠀⢸⣿⠉⠉⠉⠁⣰⡟⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⠀⠀⣴⡓⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⡁⡠⠊
⠀⠀⠀⠀⠀⠀⢀⡀⢠⣿⡇⣀⣠⠄⢀⣮⠀⠑⠀⠈⢉⠋⢹⣷⠀⠀⠀⠤⢢⡡⠆⠀⠀⢸⡏⠀⠀⠀⢸⣿⠄⢲⣆⠀⢀⣴⠖⠂
⠀⠀⠀⠀⠀⠀⠁⠀⣸⡟⠀⠀⠀⠀⣿⡧⡀⠀⠃⣠⠊⠀⣾⡏⢀⠎⠀⠸⠇⠳⢡⡄⠀⣾⡇⠀⠀⠀⣾⡟⠀⣿⡏⢀⣾⡿⠒⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠻⢧⠨⠟⠉⠀⠀⠀⠘⠓⠉⠀⠀⠈⠙⠀⠁⠀⠀⣿⠀⠀⢀⣀⠙⠁⠀⠙⠓⠋⠈⠳⠄⠊
⠀⠀⠀⠀⠀⣀⡴⠋⠀⢰⣶⣿⠀⠀⠀⣶⣶⡆⢠⡴⣶⣿⣶⣦⡀⠀⠀⠀⣰⣶⣦⠀⢰⡟⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣄⠀⢸⣿⣿⠀⠀⠀⣿⣻⠇⢿⣿⣇⡀⠙⠟⠋⠀⠀⢠⣿⣿⡿⡇⠈⠁⠴⠟⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠛⠦⢸⣿⣿⠀⠀⠀⣿⣽⠀⠈⠛⠿⢿⣿⣷⣆⠀⢠⣿⣿⠈⣿⣻⡀⠀⠀⠶⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠈⠛⠛⣃⠈⢿⣿⣦⣤⡾⣧⡟⠀⢿⣿⣦⣀⣨⣿⡿⢀⣿⣿⡿⠿⠿⣿⣧⠀⠲⢶⣤⣀
⠀⠀⠀⠀⠀⠀⢴⠟⠁⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠙⠛⠛⠛⠋⠁⠘⠛⠋⠀⠀⠀⠘⠛⠀⠀⠀⠹⢿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠗
⠀⠀⠘⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡟⠙⡆⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⣼⠿⡄
⠀⠀⢠⠿⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠟⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠟⠀⠐⠀⠀⠀⠀⠀⠀⠃⣰⡏⣰⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⡾
⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⣴⣿⣿⠟⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠙⢿⣿⡆
⠀⠀⠀⠀⠀⣿⡇⠘⣿⡇⠀⠀⠀⠀⠀⠙⠛⠿⠿⠆⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢻
⠀⠀⠀⠀⠀⣿⡇⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣼⡏⠀⠰⠟
⠀⠀⠀⠀⠀⣿⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⣉⣥⡄⠰⣶⣸⡏⠀⠀⠀⠛⣿⠿⠂⣠⡆⠈⣙⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢋⣤⣀⠘⠛⣿⡄⠀⠙⣿⡅⠀⠀⠀⢰⣷⠀⢠⣿⠁⣠⣿⡟⢠⣬⡙⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠁⠘⣿⣍⣿⣄⠀⠸⣧⡀⠀⠻⠇⣀⣀⣀⡸⠟⠀⢼⡏⣰⡟⢿⣶⣿⡿⠁⢠⣌⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠠⣶⣦⣄⠈⢿⣏⠹⠷⣀⣭⣶⣾⡿⠿⠿⠟⠿⠿⠿⣿⣷⣶⣭⡐⠗⣡⡟⢁⣴⣿⡛⠷⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢁⣤⣀⠹⣧⣽⠿⡶⣋⣴⣿⠟⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⡐⢿⣏⠙⠃⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢠⣿⣉⣿⠇⠘⠿⢠⣾⡿⠋⠀⠀⠀⣀⡀⠀⢴⣿⣿⠂⢰⣾⡦⠀⠀⠀⠀⠀⠙⢿⣷⡍⠀⠀⠀⠀⠀⠀⠈⣻⣷⣄⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠈⠙⠿⣦⠄⣴⣿⠋⠀⠀⠀⠀⠘⠿⠓⠀⠘⠉⠙⠀⠈⠉⠁⢀⣀⣀⣀⣠⣤⣤⣽⣿⣶⣶⣶⣾⣿⠿⠿⠿⠟⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣿⣿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⢉⣁⣀⠀⠀⣤⣤⣄⠀⣶⣶⣿⢿⡇
⠀⠀⠀⠀⢀⣠⣿⢇⣤⣄⣤⣤⣶⣶⣾⣿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠉⣀⣀⠀⠀⠀⣤⣶⣶⣶⣶⡄⣿⣿⠀⢹⣿⣿⣆⢠⣿⣿⣿⠀⢻⣿⣇⣀⣀
⠀⠀⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠁⣀⣀⣤⣤⣄⠀⢶⣶⣶⣾⣿⡟⣿⣧⢸⣿⡏⠀⠀⠀⠉⠙⣿⣿⠀⠀⢻⣿⡆⢸⣿⣿⣿⣸⣿⢻⣿⡀⢸⣿⡿⠿⠟
⠀⠀⢠⣤⣴⣶⣦⡄⠀⣾⣿⣷⡀⠀⢻⣿⡟⢻⣿⡇⠈⠉⣿⣿⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⣿⣇⢻⣿⣿⢸⣿⡇⠈⣿⣷⣀⣤⡄
⠀⠀⢸⣿⡏⠉⣿⣿⠀⣿⡟⣿⣇⠀⢸⣿⣧⣼⣿⡃⠀⠀⣿⣿⡀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠈⣿⣷⠀⣿⣿⠈⣿⣿⠈⣿⡧⠀⠿⠿⠿⠛⠃
⠀⠀⠀⣿⣿⣶⣿⠟⠀⣿⣇⣸⣿⡆⠈⣿⣿⠉⣻⣿⠀⠀⢸⣿⡇⠀⠀⠀⣿⡿⡀⠀⠀⠀⠀⠀⠸⠷⠇⠀⠀⠛⠛⠀⠉⠉⣀⠈⣀⣀⣤⣤⣤⣴⣶⣶⣶⣷
⠀⠀⠀⣿⣿⡍⠀⠀⢀⣿⡿⠟⢿⣿⡀⣿⣿⠀⢹⣯⡇⠀⠸⠿⠇⠀⠀⠀⠙⠉⠁⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⡿⠿⠟⠛⠛⠛⠋⢹⣿⠉
⠀⠀⠀⢸⣿⡇⠀⠀⠘⠿⠃⠀⠘⠋⠁⢉⣉⣀⣀⣤⣤⣤⣤⣶⣶⣶⣾⣿⡿⠿⠿⠿⠋⠛⠛⠛⠉⠉⠉⠉⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⣤⣤⣤⣤⣶⣶⣶⣶⣿⡿⠿⠿⠿⣿⣟⠛⠛⠉⠉⠉⠉⠀⢀⣀⡀⠀⣄⣠⡄⠀⢴⣶⠄⠀⠀⠀⠀⣴⣿⠏⣰⡗⢠⡀⠀⠀⠀⣾⡿
⠀⠀⠀⠐⠛⠋⠉⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠺⠿⠇⠠⢿⣿⠶⠀⠉⠉⠀⠀⢀⣤⣾⡟⢁⣌⠛⢷⣿⣁⣴⢆⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⢀⣴⡿⣦⡈⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣀⣤⣶⡿⠟⣁⣄⡹⣿⣷⣄⠈⠛⢫⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⣴⡿⠻⣶⠿⢡⣶⣦⠉⠛⠿⠿⣿⣶⣶⣶⣶⣶⣶⣿⠿⠟⢛⣉⢰⣦⠙⢿⣿⢾⣷⡹⢗⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡀⠀⢀⣴⣟⣿⠃⢰⡿⢳⣦⢠⣤⣬⣭⣭⢀⣤⠀⠀⠿⢿⣏⠀⠻⣧⡀⠻⣷⢉⣵⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣝⠉⢽⡏⢀⣿⠿⣿⠋⠀⣾⡏⠀⢻⣿⡏⠀⠀⠀⠈⣿⡆⠀⠹⢗⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣶⣤⣜⡛⠰⠿⠀⠀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⢘⡋⣴⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣷⣶⣶⣭⣤⣤⣤⣬⣵⣶⣶⣾⡿⠏⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢧⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄⣓⣤⣀⢳⣀⡤⠚⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⡤⠠⡄⠀⠀⣿⠀⠀⠈⠉⠙⡟⠺⡷⢮⣭⣉⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡀⣈⠳⣄⣠⣼⣖⣙⣤⣯⣤⠀⠀⠀⢰⠀⠀⠇⠀⠀⠀⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣭⡿⡥⢞⠻⢍⡀⠀⠘⣧⣤⣄⡜⠀⢸⠀⠀⠀⠀⢀⡿⣀
⠀⠀⠀⠀⠀⠀⠀⣏⠉⠰⢯⡴⠇⠀⢳⡄⣾⠷⠶⣏⠀⠀⡇⠀⡼⢍⠙⢯⠁⣹⡇⠀⡱⠂
⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣸⡥⠾⡀⠀⢏⠃⠀⠀⠹⣆⠠⠇⣸⣀⣀⣙⠦⣧⣼⣴⣋⣠⣤⡀
⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠙⠓⠦⣧⣀⠈⡆⠀⠀⠀⠹⣼⠀⢸⣁⣨⡭⠿⠿⣿⢿⠍⠀⠈⣷⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢾⣄⠀⢠⡶⠛⠛⠉⠁⠀⠀⣀⣀⣽⡎⣶⠚⠋⢻⡀
⠀⠀⠀⢰⣧⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⢧⣠⣤⠴⠖⢛⡏⠉⠀⠼⠛⢇⠐⣂⣈⣷⣠⣄
⠀⠀⠀⣰⠃⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⢹⡄
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⡾⠀⠀⠀⡐⡗⣀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⠃⠀⠀⠿⡪⡻⣗⠓⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠚⠋⣧
⠀⣾⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠛⠛⡛⠛⢻⠛⠛⠉⠉⠁⠀⢠⡄⠀⠀⢹
⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⣀⠀⠀⠀⠤⣶⣶⠄⠀⠀⠀⢀⣇⠀⠈⠀⠀⠀⠀⠠⡮⣶⢕⠵⠆⢸
⠀⣇⠀⠀⠀⠀⠀⢀⢄⣹⣳⣿⣼⣏⡠⡀⠀⠀⠘⠀⠀⠀⠀⢰⠁⠀⣱⠔⠢⢄⠀⠀⠀⠈⠡⡇⠁⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠈⠉⣽⠟⣿⡻⣯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠴⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⣸⠤⠤⠤⢀⣀⠀⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠀⠀⠀⠑⠤⠤⠚⠁⠀⠀⢀⡀⠤⠒⠉⢧
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠉⠙⠛⠓⠒⠒⠶⠶⠦⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠊⣉⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠸⠀⠈⢪⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠐⢄⠀⢣⠀⠀⢿⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡀⠀⠐⢤⡑⠄⠱⣄⠈⣧⡠⠤⡒⠁⠀⠀⣀⡀⠤⠊⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠖⣌⣉⣁⡀⣉⣉⣐⡊⣑⣦⣬⡗⢻⢡⠀⠘⣍⠉⠁⠀⠀⠀⠀⣀⣨
⠀⠀⠀⠀⠀⢸⠉⠉⠁⠐⠒⢲⠐⠂⢉⠭⠽⠟⠉⣉⢁⡉⣲⠰⠀⡴⠮⢍⠉⠉⠛⠛⠂⠈⠉⠉⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠂⠀⢸⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣤⠄⢀⡀⢀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⢸⠀⠀⢇⡀⢀⡀⠠⡤⠆
⠀⠀⠀⠀⢀⠜⢸⠀⠀⠀⠀⢸⠀⠀⢸⠉⠉⠉⠁⠐⠚⠒⡇⠀⠀⢸⠉⢹⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠓⠂⢹⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢣⠔⠂⠤⠄⠘⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠂⠐⠧⣀⡠⠄⢀⡀⢀⣀⣀⣀⣀⡀⢀⡀⠠⠴⠀⠠⠔⠐⠒⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢠⣤⡀⠉⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠙⢦⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡾⠁⠈⠱⡄⠀⠀⠀⠀⠀⠀⡇⣤⠀⠈⠆⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⣇⠰⠆⠀⡀⠀⠀⠀⠀⠀⠀⠱⣤⣀⡼⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢠⡶⢶⠶⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⣀⣀⣀⣀⡤⠞⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣴⠛⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿⡿⢿⡇⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠖⠋
⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠈⠉⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⣀⣀⡴⠇⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢀⡿⠀⢠⡿
⣠⣴⡟⠻⡀⠛⠳⣆⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣤⣤⣄⠀⠀⠀⠀⣾⠃⠀⣼⠃
⢿⡀⠐⡀⠘⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⢾⣇⠀⠀⠀⢸⠛⠛⠛⠛⠃⠀⣰⡏
⠈⠻⣆⡀⣀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠟⠓⠶⣤⣤⠤⠴⠶⠶⠚⠛⠛⠉⠉⠀⠀⠀⢻⣦⠀⣀⣰⠷⠶⠶⠶⠶⠶⠋
⠀⠀⠈⠉⠙⢷⡀⠀⠙⣦⡀⠀⠀⢀⣠⣞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠉
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠈⠻⠶⠟⠋⠁⢘⣤⢀⣀⣠⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣠⣤⠶⠟⠋⠉⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⣴⡶⣶⠿⠻⠛⠋⠉⢹⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣸⡇⠈⠙⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣠⡾⠟⠀⠀⢀⣾⠋⠙⠳⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠄⣀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣾⠾⠟⠛⠷⣦⣤⠀⣾⠷⣤⣀⠀⠀⠈⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡿⣶⣤⣈⠀⣼⡟⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⣯⡴⠟⠁⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣹⠟⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠉⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣤⣼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇⠀⠀⠘⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⡀⠉⠉⢻⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣁⣀⣀⣀⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣁⠠⠤⠤⠖⠒⠒⠒⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⡄⠀⠀⣞⢳⡀⠀⠀⠀⠂⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆⠀⣠⡶⢲⡄⠀⢠⣶⣶⣦⣄⣤⡀⠈⢸⠧⣀⡀⠀⣰⣿⣿⠃⠀⢀⣤⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡼⠋⣀⣼⡿⣻⣿⠟⢀⣴⣿⣿⠃⢠⡟⣴⣿⠇⢰⣿⡿⠁⠀⣠⣾⡿⠃⠀⣠⣿⡟⠀⠀⠀⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⣠⠞⣹⣿⢳⣿⠃⡴⢫⣿⡿⠃⣠⠟⣼⣿⠋⢀⣿⣿⠁⢀⢼⣿⡟⠁⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠹⠾⠿⠃⠘⠿⠚⠁⠘⢿⣥⠾⠃⠀⠹⠧⠴⠋⠻⠧⠔⠁⠘⢿⣄⠴⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⢏⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⢀⡴⠚⠉⣴⣿⠃⠀⠈⠳⣄⠀⠀⠀⢠⡾⠛⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢘⣻⡥⠴⢤⣄⠀⠠⡏⠀⢀⣾⡿⠁⠀⠀⠀⠀⠘⢧⡀⠀⠙⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⠀⢀⣿⣿⣡⠤⠒⣿⣥⠀⠀⠀⠠⣿⠇⠀⠙⠛⠛⠉⢀⣴⣶⠖⠲⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠖⣾⣿⠃⠀⠀⣼⣿⡟⠀⣠⣤⣀⡴⣶⣦⢀⣀⡀⢀⣾⣿⠃⢀⠜⠁⠀⠀⠀⠈⠙⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⣠⠞⣿⡟⣀⢼⣿⣿⠋⣸⣿⡿⢫⣿⡟⠸⣿⡿⠊⢁⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡤⠞⠁⠀⠈⠉⠁⣾⣿⠃⠀⠻⠟⠁⣼⣿⠁⡰⠻⠧⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢰⣿⣯⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⠀⠀⡀⠤⠐⠂⠤⣀⠀⠀⢀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⡏⠉⠀⠈⠉⡉⢉⢾⣀⣀⠀⢀⣀⢀⢷⠉⡏⠉⣆⢀⠏⢙⡓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣋⠀⠀⠀⢈⢀⡰⡁⠀⡌⠈⠅⠱⡀⢀⡎⢩⠀⢆⠈⢉⠆⠁⠀⠀⣀⡱⠙⢄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⣁⣀⣇⣰⠁⠈⣉⣹⠀⠀⢇⣀⣰⣎⣀⠆⠀⠸⣇⣀⠁⢹⣀⣀⣀⣀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⡇⠀⠀⠀⡀⢽⠀⠀⠸⠇⠸⠄⡟⠀⠀⢰⡇⠀⠀⠀⠀⠨⠤⡸⠄⠔⠃
⠀⠀⠀⠀⠀⠀⢰⢀⣀⡢⣀⡀⠀⠀⠘⢄⣘⠀⠀⠀⢡⢀⡔⠀⠀⠀⣸⡀⠀⣀⣀⠀⠀⣀⠜⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡧⠀⠥⠠⡧⢠⠀⠀⡄⣤⢇⠀⠀⠀⢾⠁⠀⠀⢠⠥⡇⠠⠅⠨⠤⢤⠄⠠⢄⡜
⠀⠀⠀⠀⠀⠀⠀⠈⢄⢠⠐⠃⠈⠂⢰⠆⠂⠈⡆⠀⠀⠈⢆⠀⢀⠆⠀⠳⢤⠀⠀⠀⠀⠄⢄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡒⡆⠀⡔⠺⡤⠄⠀⠘⡄⠀⠀⠈⢦⠎⠀⠀⡤⠐⠔⠢⡀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⠒⠃⠀⠓⠓⠒⡞⠙⡄⠀⠀⠈⢆⠒⠒⡗⠒⠀⠀⢳⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⢘⠎⠀⠀⠘⡄⠀⠀⠈⢣⠀⠉⠑⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠊⠀⠀⠀⢠⢻⡄⠀⠀⠀⠱⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢠⠏⢹⠉⢆⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠋⠢⡀⢉⠞⢢⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⡠⠁⠀⠀⠈⠀⠀⠀⠣⡀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢠⣞⠐⠊⠀⠀⠀⠀⠀⠀⠑⠂⣳⡄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣾⣁⡛⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢛⣈⢷⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣼⣃⣘⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣁⣈⣧⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣠⢠⣟⢘⣛⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣛⣃⢿⡆⣄
⠀⠀⠀⠀⠀⡀⣰⡏⡿⢟⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡻⠿⢹⣆
⠀⠀⠀⠀⢰⡇⣿⣾⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣷⣿⢸⡆
⠀⠀⠀⡄⣿⣇⣏⣵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣝⣼⣿⢠
⠀⠀⢸⣿⢿⣿⢟⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡻⣿⡿⣿⡇
⠀⢠⢸⣿⣼⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣧⣿⡇⡄
⠀⣿⡟⣿⡿⣫⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣝⢿⣿⢻⣿
⢀⢹⣿⣼⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣧⣿⡏
⢸⣯⢿⣿⡿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣮⢿⣿⡿⣽
⠘⣿⣎⣿⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣱⣿
⢷⡹⣿⣿⣿⣏⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣽⣿⣿⣿⢏
⠸⣿⣎⢻⡟⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢻⡟⣳⣿
⠀⠹⣿⣦⣷⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢿⣿⣼⣰⣿⠏
⠈⢷⣮⡿⣿⣿⠇⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠸⣿⣿⠿⣣⡾
⠀⠈⢿⣷⣌⢻⡄⣿⣿⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣿⣿⢠⡟⣡⣾⡿⠁
⠀⠀⢀⡙⢿⣷⣿⣿⣿⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⣿⣿⣾⡿⠋⡀
⠀⠀⠈⠻⣶⣝⡻⠿⣿⠘⣿⣷⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣾⣿⠇⣽⠿⢟⣫⣶⡟⠁
⠀⠀⠀⠀⠙⢿⣿⣦⣜⢷⣽⣿⡇⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢰⣿⣯⡾⣣⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣉⠻⠿⣷⣿⣿⣇⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⣸⣿⣿⣾⡿⠟⢉
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣭⣉⠙⢷⣬⡻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢟⣥⡾⠋⣉⣥⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣬⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣭⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠙⠻⢷⣶⣤⣄⣀⣀⣠⣤⣶⡾⠟⠋⠉⠛⠛⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣾⣿⣟⣛⣛⣻⣿⣷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠂⠀⣀⡀⠀⠐⠀⢀⣀⠀⠀⠘⣷⣤⡴⡿⠀⠀⢀⠀⠄⡀⠘⠂⠀⣀⡀⠀⠐⠀⠁⣀⠀⠀⠘⣷⣤⡴⡿⠀⠀⢀⠀⠄⡀
⠀⢸⠉⠀⠘⠀⠀⡴⠪⣥⠆⠀⠀⠉⡁⠀⠀⢰⠦⣴⡟⢡⠂⠀⢸⠋⠀⠘⠀⠀⡴⠪⣅⠆⠄⠀⠈⡁⠀⠀⢰⠦⣴⡞⢡
⠀⠀⠀⣀⠀⠀⡏⠁⠃⠀⠀⠀⠀⠀⠠⠺⠀⡹⢟⣿⢹⠆⠀⠀⠘⠀⢀⠀⠀⡎⠀⠂⠐⠀⠀⠀⠀⠠⠺⠂⡹⢟⣿⢹⠆
⠀⠰⠆⠈⠋⠀⠈⠠⠀⠀⠀⡴⠈⠀⡠⠾⠁⣤⡶⠤⠀⠀⢠⠀⠰⠆⠈⠋⠀⠈⠠⠀⠀⠀⡴⠈⠀⡠⠾⠁⣤⡶⢤⠀⡀
⡶⠂⠀⠀⠀⠺⠃⠀⠀⠂⠀⡀⠀⢄⠀⠀⠈⢟⠓⠚⢀⣿⣿⡶⠂⠀⠀⠀⠺⠃⠀⠀⠂⠀⡀⠀⢄⠀⠀⠈⢟⠓⠚⢀⣿
⠙⠀⠀⠀⠀⠀⡠⠀⣀⡀⠀⠉⠀⠀⡀⠄⠀⠘⠃⠀⣀⡘⠋⠙⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠉⠀⠀⠀⠄⠀⠘⠃⠀⣀⡘
⠀⠀⠀⠂⢡⣂⠀⠀⠈⠁⠀⠀⡄⠀⢀⠀⠐⠦⠀⠈⠛⠃⠀⠀⠀⠂⠢⢤⣂⠀⠀⠈⠁⠀⠀⡄⠀⢀⠀⠰⠤⠀⠀⠛⠃
⠈⠀⠐⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⡛⠑⣀⠀⠀⠀⠀⠈⠀⠐⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⡟⠃⣀
⣢⣀⡀⠀⠀⠀⠀⣅⠀⠀⠀⠀⣈⣀⠱⣶⣇⡀⠉⠀⠀⠀⠰⣢⣀⡀⠀⠀⠀⠀⣍⠀⠀⠀⠀⣘⣀⠱⣴⣆⡀⠁
⢀⠁⠀⢸⣧⠀⠀⢻⣷⡶⠾⠛⠁⠀⠀⠈⠉⠀⣠⣶⣶⣄⠀⢀⠁⠀⢸⣧⠀⠀⢻⣶⡶⠾⠛⠁⠀⠀⠉⠉⠁⣠⣶⣶⣤
⠘⣄⣦⣼⠟⠀⠀⠘⠉⠀⣤⠀⣀⠤⠴⣿⠀⢠⠘⠋⠑⡿⠀⠘⣄⣦⡼⠟⠀⠀⠘⠁⠀⣤⠀⣀⠤⠴⡟⠀⢠⠙⠋⠑⡿
⡀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⢸⣅⠀⢀⣬⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⢸⣄⠀⣀⣼
⠀⢠⡖⠒⢠⠀⠀⢐⡲⠀⡀⠀⠛⠿⠯⠉⠀⢀⡀⢀⣡⠆⡀⠀⢠⡶⠂⢠⠀⠀⢀⡲⠀⡀⠀⠛⠿⠏⠉⠀⢀⡀⢀⣡⠆
⠀⢘⠀⠀⠀⠀⡠⠋⠈⢛⠁⠁⠀⠀⠂⣠⡀⢼⣭⣿⢧⡘⠀⠀⢘⠀⠀⠀⠀⡠⠋⠈⢛⠁⠀⠀⠀⠂⣠⡀⢼⣭⣿⢧⡘
⠀⢀⡀⠰⡤⠀⠣⠀⠀⠀⠐⢀⠀⠀⠉⣨⠄⠊⣑⠛⠘⠁⠀⠀⢀⡀⠰⡤⠀⠡⠀⠀⠀⠐⢀⠠⠀⠈⣨⠄⠊⣑⠛⠘⠁
⣀⡈⠁⠀⠀⣠⡄⠈⠀⡀⠀⠋⠀⠀⠊⠉⠠⣿⣏⣩⠀⣦⣸⣀⡈⠁⠀⠀⣠⡄⠈⠀⡀⠀⠊⠀⠀⠊⠉⠠⣿⣏⣩⠀⣦
⢯⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠦⠀⠐⠈⠀⠀⢱⡄⠀⠐⢻⡿⢯⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠦⠀⠑⠀⠀⠀⢱⡤⠀⠐⢻
⠀⠀⠀⡀⠀⠀⠈⠀⠲⠆⠀⠀⠀⠀⠂⠁⢀⣀⠀⠠⣶⡆⠀⠀⠀⠀⡀⠀⠀⠈⠀⠲⠆⠀⠀⠀⠀⠂⠁⢀⣀⠀⠠⣶⡆
⠠⠀⢀⠄⠉⠒⠀⠀⠀⠀⠀⠀⠃⠀⢰⢤⣤⢍⠀⠀⠀⠀⠀⠠⠀⢀⠄⠉⠒⠀⠀⠀⠀⠀⠀⠃⠀⢰⢤⣤⢁
⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⢆⣀⡆⠀⠦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⢆⣀⡃⠀⠶
⠚⠖⠂⢠⡄⠀⠀⣷⣀⣀⣤⣤⠖⠒⠈⠿⠷⠂⠀⣀⣀⠀⠈⠚⠒⠂⢠⡄⠀⠀⣷⣄⣀⣤⣤⠖⠒⠈⠿⠿⠂⠀⣀⣀
⢰⠀⡀⢸⡿⠀⠀⢸⠿⠋⠉⠀⠀⠀⢀⣤⠀⠀⢺⡿⢻⣿⠀⢰⠀⡀⢸⡿⠀⠀⢸⠟⠋⠁⠀⠀⠀⢀⣤⠀⠀⢺⡿⠻⣿
⠀⠓⠻⠛⠁⠀⠀⠀⠀⠀⠛⢠⠖⠁⠉⠫⠀⠘⠀⠀⠀⠋⠀⠀⠓⠻⠛⠁⠀⠀⠀⠀⠀⠛⢠⠖⠉⠉⠻⠀⠘⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⢀⡀⡀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⢠⢤⡀⢠⢤⡀⣴⠚⠉⣹⠀⠀⠀⣀⣼⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢀⣻⡀⣷⣻⡀⣷⠈⢹⡄⡟⢀⠞⢩⣿⢹⣴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⠚⣿⠉⣿⢀⡏⣿⠀⡟⠀⢰⣷⠇⡟⢀⡞⣿⠀⡴⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣷⠀⠀⣿⠀⠘⠚⠁⠘⠚⣡⠀⠸⣹⠀⠙⠋⠀⠇⢸⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢰⣦⠰⡀⠀⣗⣧⣤⠄⣿⠀⡀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠛⠈⠉⠀⣁⠀⠈⣁⣉⡀⠘⠛⠁⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⢠⣿⠀⣸⡟⠁⠈⡆⠀⣸⠃⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢸⣿⠀⢻⡇⠀⠀⠁⢀⡟⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣸⣿⠀⠈⢿⡄⠀⠀⣸⣇⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠏⣿⠀⢀⠀⢻⡆⠈⣿⡇⠀⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠜⠀⣿⠀⠸⣀⣸⡿⠀⣿⠇⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⣿⣶⣄⠀⠀⠀⠀⣀⣿⣷⣰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⠻⣿⣦⣤⣶⣿⣿⣿⣿⣷⣙⠋⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠛⠉⠙⠻⢿⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⣠⡄⣠⡄⠀⠀⠀⢠⡄⢠⡄⠀⠀⠀⢠⣄⢠⣄⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣀⡀⣿⡇⣿⡇⢀⣀⡀⢸⡇⢸⡇⢀⣀⡀⢸⣿⢸⣿⢀⣠⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⣿⠇⣿⡇⣿⡇⠻⣿⠗⢸⡇⢸⡇⠻⣿⠟⢸⣿⢸⣿⠸⣿⡿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣷⣶⣿⡇⢿⣷⣶⣿⣶⣿⡇⢸⣿⣶⣾⣶⣾⡟⢸⣿⣶⣾⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⠉⠀⠀⠙⣿⣿⣿⠋⠀⠀⠙⣿⣿⣿⠋⠀⠀⠉⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠻⠿⠀⠀⠀⠀⠿⠛⠿⠀⠀⠀⠀⠿⠟⠿⠀⠀⠀⠀⠿⠟⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣿⠀⢸⡇⠀⠀⢸⡇⠀⣿⠉⠉⢻⡄⢸⡏⠉⠙⣧⠀⠀⢰⡿⣇⠀⠙⣧⠀⢠⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠶⠶⠶⣿⠀⢸⡇⠀⠀⢸⡇⠀⣿⣤⣤⡞⠁⢸⣧⣤⣴⠋⠀⢀⣟⠀⢻⡄⠀⠘⣧⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⣿⠀⠸⣇⠀⠀⣸⠇⠀⣿⠀⠀⢻⡀⢸⡇⠀⠘⣧⠀⣼⠋⠉⠉⣷⡀⠀⣿⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉⠀⠀⠈⠛⠛⠁⠀⠀⠉⠀⠀⠈⠃⠈⠁⠀⠀⠙⠀⠉⠀⠀⠀⠘⠁⠀⠉⠀⠀⠀⠙