Hesitation Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠓⢦⡴⠛⠉⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠈⠁⠀⠀⠀⢸⠏⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⡼⠋⠙⠲⣤⡤⠞⠋⠀⠀⠀⠀⣀⣀⠀⠀⠘⢧
⠀⠀⠀⣰⠋⠀⠀⠀⠀⣠⠶⣄⠀⠀⠀⠀⠹⡆⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠋⣉⣉⠙⢦⠀⠈⠛⠛⠲⢦⣄
⠀⠀⠀⡏⠀⣀⣤⠴⠛⠁⠀⠈⠓⠦⣤⣀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⡞⠁⠈⡧⢘⡇⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⣇⣀⡇⠀⠀⡀⠀⠀⠀⢀⠀⠀⢹⣀⣿⠀⢿⠀⢰⠋⠉⠳⠟⠉⢳⡄⠈⠉⢠⡖⠚⢁⡼⠁⢀⣀⣀⠀⣀⣀⠀⢹
⠀⠀⢰⡏⠉⠁⠀⠈⠁⠀⠀⠀⠉⠁⠀⠈⠉⢹⡄⠸⡆⠘⢦⣀⠀⠀⣠⠞⠁⠀⠀⢸⡇⢸⠉⠀⠀⡟⠁⠉⠟⠁⠉⣧⠈
⠀⠀⠈⡷⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣾⠀⠀⠙⣆⠀⠈⠓⠋⠁⠀⠀⠀⠀⠘⠧⠞⠀⠀⠀⠙⢦⡀⠀⣠⠼⠃⢸
⠀⠀⠀⡇⠀⢧⠀⠀⠀⠉⠉⠉⠀⠀⢀⡾⠀⣼⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣰⠚⢲⡄⠀⠀⠀⠀⠈⠉⠁⠀⣠⠏
⠀⠀⠀⡇⠀⠈⠳⢤⣀⣀⣀⣀⣀⡤⠞⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⡄⠀⠀⠘⠶⠞⠁⠀⠀⠀⠀⣀⣠⡴⠞⠁
⠀⠀⠀⡇⠀⠀⠀⠀⢹⠁⠀⠈⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⡠⠒⠒⢦⡀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⢀⣧⡤⣤⡤⠤⠼⠀⠀⠀⠧⠤⢤⡤⢤⣿⡀⠀⠀⠸⡇⠀⠀⢀⡇⠀⠀⠙⠲⣤⣀⣀⣀⣠⡴⠚⠁
⠀⡴⠋⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠙⣦⠀⠀⠙⠓⠖⠋⠀⠀⠀⠀⠀⠀⠀⠉⠁
⢸⠃⠀⠀⠀⠀⠀⠙⠓⠲⠶⠖⠚⠉⠀⠀⠀⠀⠀⢸⡆⠀⣠⠶⢦⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢧⣀⣠⠇
⢸⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⢸⡇⠀⠀⠈
⢸⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠸⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠸⠇
⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠇⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⠀⢀⡀⢀⡀⢀⠀⡀⡀⠀⣀⠀⠀⠀⠄⠀⡀⢀⠀⡀⡀⢀⠀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠗⠂⠀⠀⡗⠀⠀⠂⡼⠰⠸⠒⡇⡇⠀⠗⠂⠀⠙⢢⠀⠇⢸⠲⡁⡅⢀⠇⡗⠄⠐⠂
⣿⣷⠾⠿⠾⢿⣷⣿⣿⣿⣿⣷⣿⣾⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷

⠀⠀⠀⢀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡶⣦⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⢠⣯⠀⠀⠀⠀⠀⠀⠀⣮⢸⣿⣿⠀⠀⠀⠀⠀⠰⣿⣷⣄⡀⠀⠀⢸⣿⣿⣿⡟⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠳⠀⠀⠀⠀⠀⠀⠠⠻⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⢸⣿⣿⣿⡅⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡀⢸⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⠙⠁⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢤⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣾⣿⢠⡟⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⣹⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠸⡿⠘⢿⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣴⡄
⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠿⠇
⠀⠀⠀⠀⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢀⡰⠆
⠀⠀⠀⠀⣿⠊⠀⠀⠀⠀⠀⠀⠀⠠⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠈⠉⠿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⠿⠿⠿⣶⣦⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣥⡶⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⣤⠀⠀⠀⣠⠞⠁⣠⣴⠟⠻⣷⠀⠀⢰⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢀⣴⠞⠁⠀⠀⠀⠀⠀⣠⡶⠋⠀⢀⣴⢿⡿⠀⣠⡾⣋⣴⡾⠟⠁⠀⠀⢻⣧⠀⢿⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠁⠀⠀⠀⠀⣀⣴⠞⠋⣀⣴⡾⠟⠁⣸⣷⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⢿⣧⣼⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⣤⣶⣯⣵⡶⠿⠿⠿⣤⡄⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢴⣶⠿⢿⡿⠛⡋⢁⣄⣀⣤⡀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀⣀⠀⠀⣿⡿⢦⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡟⠀⠀⠻⠛⠛⠛⠉⠉⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⢾⡏⢉⡉⢉⡉⢉⠙⢷⡄⢻⡇⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⡇⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⣰⡆⠀⠀⠀⠈⠿⠾⠿⠿⠿⠾⠿⠟⠀⢸⣧⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡆⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠸⣿⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠁⠀⠀⠀⢸⣿⢡⣤⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡄⠀⠀⠀⢀⣀⣤⣤⣴⣶⠶⠿⠿⠿⠷⢶⣶⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⣼⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⠀⣠⣾⡟⠛⠉⢁⣀⠤⠤⠶⠖⠚⠛⠿⠿⠿⠿⣷⣍⡛⠛⢿⣶⣤⣀⠀⣿⡇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⢿⡾⠋⢸⡇⠀⣰⡿⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⠈⠻⣦⠀⠹⣟⠻⢿⣿⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠈⠀⠀⣿⠁⠀⣶⠿⠿⠿⣷⣶⣶⣾⡏⠀⢻⣷⢀⣿⠀⠀⠈⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⢠⣤⣭⣭⣿⡇⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢠⡏⠀⠀⠀⠀⠘⠿⣦⣤⣄⡈⠀⠀⠀⣹⣷⠀⠀⢹⣧⠀⠀⢀⠀⠀⠀⣿⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⢰⡄⠀⠀⢠⠀⠈⠙⠛⠛⠛⠛⠛⣿⡇⠀⠈⣿⣆⠀⠀⣷⠀⠀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣼⡇⠀⠀⣾⠀⠀⠀⠀⣤⠀⠀⠀⣸⣿⣄⠀⠘⣿⣄⢀⣽⣷⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠶⣶⣿⣄⣀⣀⣠⣿⣀⣠⣾⠟⠙⢿⡆⠀⠈⢿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⡏⠉⢹⡟⠛⢛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⡶⢿⣿⠀⠘⣧⠀⢸⡇⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠷⣦⣤⣄
⠀⠀⠀⠀⠀⢀⣤⣶⠾⠟⠛⠛⠋⠉⠀⠀⠀⠀⢸⣟⠀⠀⣿⠀⠈⣿⡀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠈⠙⠛⢷⣦
⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣄⢸⡏⠀⠀⣿⡇⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠈⠻⠿⣿⣦⣤⣿⣇⣸⣿⣿⣷⣦⣤⣴⣶⣆⠀⠀⠀⢀⣴⡿⢷⣦⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⠀⠉⠉⠉⠉⠉⠉⠁⣀⣼⠟⠁⠀⠙⣷⣶⣿⣿⡛⢳⣄⣿⡇⢠⡉⠀⠀⠀⠹⣷⡀
⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢿⣼⣷⡈⣷⣼⡿⠛⢿⣿⡁⠀⠀⠀⠀⠹⣷⡀
⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠛⠛⠉⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⢻⣷
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⢿⣧
⠀⢀⣿⠃⠀⠀⠀⠐⣷⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⢿⣧
⠀⢸⣿⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⢻⣧
⠀⢸⡿⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠈⣿⣆
⠀⣾⡇⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠸⣿⡀
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠶⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠈⠙⠿⣶⣤⣤⣤⣀⣀⣼⡿
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠉⠉⠉⠙⠛⠋
⠙⢿⣷⣤⣀⡀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠈⠛⠛⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⡾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡈⣹⣀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢥⡍⢹⣡⠋⢠⠟⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠉⠁⢴⡃⠀⠀⠸⡆⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠤⢤⣄⡈⢦⠀⠀⢻⡀⣉⡥⠤⣄⡀⠀⠀⣠⠴⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⢦⠀⠀⠹⡇⠀⠀⠀⠈⠓⢾⣵⢒⠂⠉⠉⠓⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡬⡆⠀⠀⢳⣴⣿⡆⠀⠀⠈⢧⡤⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠙⠧⣄⡀⠀⠀⣠⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣸⣾⠀⠀⠀⠀⠀⠀⠉⠭⠝⢋⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡤⢶⡶⢦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⡤⠤⡆⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⠈⠙⠦⣤⣈⡐⠄⠠⠀⠄⣀⡤⠾⢄⡈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠏⠀⢸⠀⠀⠀⠀⢀⣀⣀⡤⠤⠬⠉⢻⠛⠛⠉⠉⠀⠀⡰⢃⡴
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣄⢤⢸⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡴⣄⠞⣡⠏
⣻⠀⢸⠀⠀⠀⠀⠀⠀⣐⡆⠀⠀⠀⠀⠠⡤⢶⡤⠤⡇⠀⠀⢀⣠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠎⠀⠈⠻⡅
⢸⢰⠟⠇⠀⠀⢀⡤⠒⢒⠲⡄⠀⠀⠀⠀⠈⢻⡻⠒⠓⠤⠤⠍⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⣀⠀⢀⣀⠼
⢸⡟⣠⡀⢀⠤⣼⠶⢮⡽⡸⠁⠀⠀⠀⠀⢠⡤⠭⠽⣖⠢⠤⢄⣀⡀⠀⠈⠉⠉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⡶⠛⢾⡿⠋
⢸⡇⠈⡇⠙⠾⣖⠉⢣⠀⠛⠒⠒⠒⠒⠒⠒⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⠤⢤⣀⣀⠀⠀⠀⠀⠀⡼⠉⠲⡼⢟⣆
⠸⠧⠶⠃⠀⠀⠈⠓⠦⠤⠏⠉⠉⠉⠉⠉⢏⢻⠀⢀⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠲⠞⠁⠀⠀⣷⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⣻⠀⠀⠀⠈⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢤⣴⣏⡀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⠀⠀⢸⢁⡏⠸⡍⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣼⡠⡴⡇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠛⢋⡟⢉⣉⣽⣅⣙⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⣁⣹⠽⠝⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⠛⠛⠓⠣⠤⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⢩⣃⡀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⣤⢡⣣⠏⢠⠟⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠀⠰⢇⠀⠀⠘⡆⠀⣀⡀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⠔⠲⡦⣄⢱⠀⠀⠹⣀⡤⠶⠦⠤⣀⠀⣠⠖⠒⢢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⠀⠀⠹⡄⠀⡀⠀⠈⠙⢷⠏⠍⠀⠀⠈⠲⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⡇⠀⠀⠹⣾⠿⠀⠀⠀⠘⢲⣈⣁⠀⠀⠠⠼⠤⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⠀⠀⠀⠈⠓⠦⠤⣄⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣒⣒⠚⠿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⣖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠙⠋⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡴⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⡇⠀⠀⠀⢀⡴⠊⣷⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢥⠬⠤⠥⠖⠋⠉⢱⠄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢾⠀⠀⣇⢀⣀⠤⠖⠒⠊⠉⠉⠉⠀⠘⡄⠀⠀⣀⠀⡴⢁⠞⠁
⣿⠓⢲⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡇⠀⠀⠀⢈⡣⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢀⡜⠉⠙⢴⠃
⢹⢠⠾⡄⠀⠀⠀⡀⠬⠵⡀⠀⠀⠀⠀⠑⢾⣧⠤⠧⣀⢀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⢈⡧
⢾⣾⢁⡀⢀⣀⣸⣤⣖⡇⡞⠀⠀⠀⠀⢀⣈⣓⣒⡤⢄⣀⡀⠈⠉⠁⠒⠒⠒⠒⠒⠒⠠⠤⠤⠤⠤⠤⣤⠗⣽⣶⠚⠉
⢸⡅⢹⡃⠫⢾⡕⠲⡌⠘⠢⠤⠤⠤⠤⠼⢧⠀⠀⠀⠀⠀⠈⠉⠁⠒⠢⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⣰⠓⢦⣸⡷⡄
⠸⡧⠴⠃⠀⠀⠑⠦⢄⡸⠉⠉⠉⠉⠉⢲⢻⠀⠀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠤⠴⠁⠀⠀⣇⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢒⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⣿⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢤⣴⣇⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠁⠀⠀⢸⢀⠏⢸⡙⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣇⣼⡤⡾⡇⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⣉⣏⣀⣤⡼⡤⠬⡋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⣄⣴⣿⣷⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⣣⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣦⣤⣄⠀⠀⠀⠀⢠⣮⣙⠿⢟⣩⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣫⡻⢿⣿⣿⣷⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⣩⠶⠶⢦⡙⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣫⣵⣾⣿⣿⣶⣭⣛⢻⡇⠀⢸⣿⡿⠿⢿⠿⠿⣿⣇⢛⣼⡿⢪⡷⣸⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣡⣿⣟⣻⣿⣿⣟⣿⣿⣈⡁⠀⠘⣿⡸⣿⣷⣿⡿⣸⣿⣿⡇⡾⢛⣵⡿⣫⣝⠛⣩⡙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⠻⠋⠀⠀⠹⣷⣮⣛⣫⣶⣿⣿⣿⡇⠇⣿⣿⣧⡹⢿⣿⡿⢋⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣭⣭⣿⣿⡟⣼⡇⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⢋⣍⢻⣿⣿⣿⣶⣥⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣮⣛⠻⠿⠿⢟⣫⣾⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣮⣭⣾⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⣿⡇⠀⠀⠀⢠⣶⣶⡄⠀⠀⠙⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⢰⣶⣾⣿⣿⣶⣶⢄⣶⣤⡀⠀⠀⠈⠻⠿⠃⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣭⣛⣛⣛⣛⣥⣾⣿⣿⣿⡆⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⡇

⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⣠⠊⠉⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⢸⠇⠀⡠⠋⠈⡌⠙⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⣶⠀⢱⡓⠀⠁⡇⠀⢀⠀⢵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⣿⠀⢸⡁⠀⠠⡇⠠⠀⠐⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⣯⡄⢀⠓⠒⠚⠉⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⣷⠀⠄⠠⠀⡀⠀⠟⠂⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⢆⣀
⠀⣿⠀⠀⠀⠀⠈⠰⠶⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠎⡠⠊⠳⠈⢳
⠀⢿⡀⠀⢄⡀⠠⣀⢠⣤⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠜⠁⠀⠀⡏⠈
⠀⢸⣇⠓⠂⣙⢡⠁⠀⠉⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⠀⡆⠀⠀⠀⡂
⠀⠀⢿⣴⢀⡉⢀⠀⢛⠰⡦⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠐⠋⠁⠹⠤⠔⠊
⠀⠀⠘⡇⠘⠑⠛⠀⢀⣄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣠⠐⠶⠀⠆⠀⢀⠠⡀⠠
⠀⠀⠀⢷⠀⠐⠓⠀⠀⠉⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠘⠃⠀⠀⠀⠂⠀⠀⠀⠀⣰
⠀⠀⠀⠸⣇⠀⢠⡄⠀⠀⠀⠙⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠀⠂⠛⢀⡄⠛⠆⠀⠁⠀⢀⡍
⠀⠀⠀⠀⢧⡀⢀⠂⢀⠀⢀⠦⠬⢽⣖⣢⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠷⠀⠉⠉⠀⠀⠐⠿⠀⡼
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠰⠏⠀⠀⠀⠈⠉⠛⠹⠿⣦⣤⣀⡀⠀⠀⠀⠀⢀⡼⠁⢀⡀⠀⠤⠀⣀⠃⠛⠀⢀⠜
⠀⠀⠀⠀⠀⠘⣦⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢭⠒⠒⠤⠞⠀⠀⠈⠉⣤⡄⠀⠀⠐⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡠⠀⠀⢠⡀⠀⢀⡀⠤⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢂⡀⠀⠈⠁⣀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠁⢀⡤⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣉⣙⢀⣀⣀⣀⣀⣀⣀⣀⣹⣦⣀⣠⠆⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠛⠛⠳⢦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣫⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⡖⢀⣴⠿⣆⠀⣤⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣏⡤⠚⠁⠀⠀⢀⣠⠞⠀⣠⡾⣿⢀⣴⣫⡶⠛⠁⠀⢹⣆⢻⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢺⡏⠀⠀⠀⣠⣴⣋⣤⠶⠛⠁⠰⡿⠟⠋⠁⠀⠀⠀⠀⠀⠻⣶⡄⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⢤⡶⠿⠟⣛⣭⣦⣄⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⣶⡄⠀⢿⡷⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⢀⣼⣿⣷⡀⠀⣸⠇⠀⠀⠀⠀⣤⣶⣄⠀⠀⢸⡇⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⣿⡿⠁⢀⡿⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⢸⡇⢸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⢸⣧⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢰⡟⠉⣿⠀⡟⠉⣷⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⢷⣴⠟⠈⣷⠀⢹⡇⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣦⣤⣤⣤⣤⣤⣼⣷⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⢀⣾⠋⠈⠛⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠞⠛⣿⠟⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⢻⡗⠶⣤⣀⡀
⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠉⠙⢷⡄
⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠹⢧⣄⡀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠈⢻⡄
⠀⠀⠀⣼⠇⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠳⣄⠙⢷⣤⣀⠀⢀⣩⡿⠳⣦⡀⠀⠀⠀⠙⣷⡀⠀⠀⠀⢻⡆
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠙⠛⠛⠉⣤⠶⠈⢷⣄⠀⠀⠀⠈⢿⣦⠀⠀⠀⢻⡄
⠀⣸⡟⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠙⢷⣄⠀⠈⢿⡄
⢠⡟⠀⠀⠀⠀⠈⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠻⣦⡀⠈⢿⡄
⣿⡁⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠘⢷⡄⠈⣿⡀
⠈⠻⣶⣤⣄⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠈⠻⣦⠸⣇
⠀⢠⣿⠈⠉⠙⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠙⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠈⢿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠈
⠀⢸⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⣷⡀⠀⠀⠀⠀⢀
⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⠤⠶⠖⠚⠛⠛⠛⠛⠲⠶⠶⠤⠤⠄⠀⠀⠀⠀⠀⣿⡇⠘⢿⣦⣄⣀⣀⣼
⠀⢾⣇⡀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠙⠻⠷⢶⣶⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⣶⡄⣠⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣼⣿⣷⣿⡿⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⣀⠀⢀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⣿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠉⠉⠙⠛⠛⠛⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡇⠀⢨⣿⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡾⠿⡿⠛⠁⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣾⣷⣦⣀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⣿⣿⣄⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣶⣦⡌⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣾⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣶⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣧⣀⣠⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠛⠓⠒⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠷⣦
⣠⡾⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⢀⣤⣤⡿⠁
⠛⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣤⡛⠁
⠀⠀⠀⢛⣀⠀⠀⠀⣀⡴⠚⠋⠉⠉⠀⠉⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠐⣦⠀⠀⠀⠀⠀⢻⡀⠀⠉⠁
⠀⠀⠀⠈⠉⠀⣠⠞⠁⢳⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠌⠛⠦⣄⣀⡀⠈⣇
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠙⢦⣀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠈⠁⠀⣿⣀
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣈⣳⣄⠀⠀⠀⠀⢻⠀⠀⠀⠀⡞⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⠙⠲⠦⠤⣼⡆⠀⠀⠀⠹⣄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⠜
⠀⠀⠀⣰⢺⡄⠀⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠓⡄⠀⠀⠀⠈⠉⠱⣄⠀⠀⠀⠀⠀⠰⣏⣉⡷⠀⠀⠀⠀⠀⢀⠞⠉⠁
⠀⠀⠀⢹⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⠇⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠁
⠀⠀⠀⠀⠉⠚⢳⡀⠀⠀⠀⠀⠀⠒⠋⠙⠂⠀⠀⠀⠀⣠⠞⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⣿⠻⣶⡶⠦⠤⢶⣶⠾⢻⡏⢹⣶⣄⠀⠀⠀⠀⠀⠀⣀⢀⠞⠹
⠀⠀⠀⠀⠀⢠⠏⠙⠦⣄⣀⣀⡀⢀⣠⠴⣶⠄⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠺⣧⣤⡼⡆⠀⠉⣳⣶⡉⠀⢀⡞⣤⣴⠿⠈⢷⠀⠀⠀⢀⡼⠽⠋⠀⢀
⠀⠀⠀⠀⠀⢾⠀⢀⡴⠚⣹⢆⠉⠋⠀⢺⠓⡿⠉⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣴⠟⠀⢻⡴⠛⣇⣸⡟⠢⣼⠁⠙⣧⡀⠀⠘⣇⠀⢠⡟⠛⢦⣄⡤⠊
⠀⠀⠀⠀⠀⠸⣤⠞⢀⠜⠁⠀⠳⣄⣤⡞⠋⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠻⣦⠀⠈⣷⠀⡏⢹⡇⣰⠇⠀⣠⡾⠁⠀⠀⢻⢀⡾⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⢀⡏⠐⠋⠀⠀⢀⡼⢹⣿⡇⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢰⡇⠀⣀⠀⠀⠙⢷⡄⠘⣿⡇⠘⣷⡟⢀⣼⠏⠀⠀⠀⣷⢸⣿⠁⠀⠀⢀⡟
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣠⠏⠀⢸⣿⡇⠀⠀⠀⠀⢠⠀⠀⢸⡄⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠻⣦⢹⣇⢠⡟⣠⡿⠁⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢠⡇⠀⠀⠀⡴⠃⠀⠀⢸⣿⡇⠀⠀⠀⠀⣼⠀⠀⢸⠃⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⠀⠙⢷⣿⣾⡱⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠣⣀⣀⡚⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢹⠀⣠⠏⠀⠀⠀⠈⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣄⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠈⡟⠁⠀⠀⠀⠀⠀⠘⣦⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⣀⡀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠘⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡇
⣶⣶⡶⠶⠶⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⢀⣶⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⢀⣶⣬⣙⠻⠿⢛⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠟⣩⣤⣦⣬⡙⢿⣿⣧⣤⣀⣀
⠀⠀⠀⣿⣿⣿⣿⠿⢋⣴⣿⣦⣙⠻⢿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⠟⣥⣤⡍⢿⡆⢻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⣿⡏⣥⣶⣾⣿⣿⣿⣿⣿⣿⣶⣬⡉⣿⡇⠀⠀⢸⣿⡿⢛⣛⠻⠟⣛⣛⢿⣿⣆⣛⣰⡿⠿⢃⣼⡇⣸⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣛⣁⣿⣿⡟⢻⣿⣿⣿⡟⢛⣿⣿⣇⣛⠃⠀⠀⢸⣿⡁⣿⣿⣷⣼⣿⣿⠆⣿⣿⣿⣿⢰⣾⠿⢋⣴⣿⠟⢛⣙⠻⢛⣛⡛⢿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⢻⣷⣌⡛⢿⡿⠟⣩⣾⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⠰⣿⣿⣧⣾⣿⣿⢸
⠀⠀⠀⣭⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⣭⡄⠀⠀⠀⠀⠻⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣌⣉⣼⣿⣿⣿⣷⣌⠻⢿⠿⢋⣥⣾
⠀⠀⠀⣿⣇⢻⣿⣿⣿⣶⣶⣶⣾⣿⣿⡿⢠⣿⡇⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡟⣩⣬⡙⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⠟
⠀⠀⠀⣿⣿⣦⡙⠿⣿⣿⣿⣿⣿⠿⢛⣡⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣿⣿⣿⣧⣙⣛⣠⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⣿⣿⣿⣿⣷⠀⣶⣶⣶⡄⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠛
⠀⢀⣤⣴⣶⠰⣶⣶⣶⣿⣿⣿⣷⣶⣶⡆⣴⣶⣤⣀⠀⠀⠀⠀⠘⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⣰⣿⣿⣿⣿⣧⣙⠻⢿⣿⣿⣿⡿⠿⢋⣴⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⢠⣴⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠿⠟
⣿⣿⠟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣿⣿
⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿
⣿⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣾⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡐⢂⢄⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠑⠊⠉⠢⠤⣀
⠀⠀⠉⠀⠀⠈⠣⡄⡀⠤⠤⢀⠀⠀⡠⢁⣀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠀⠀⢈⠂⢀⣀⠔⠁⠀⠀⠑⡄
⠀⠈⠣⣀⣂⡠⠋⢸⡠⠒⠓⢄⡃⠘⡄⠑⠘⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⡠⠒⠤⣀⠈⠆⡎⠀⠀⠀⠀⠐⡩⢍⢢⠈⠈⠒⢄
⠀⡠⢤⡌⠈⣤⠤⠼⡄⠀⠈⢠⣵⠜⠀⠀⡠⢄⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⠶⠁⠤⠀⠠⠀⠣⠆⠆⢘⠙⠊⡁⠈⢰⢪⠊⡠⢤⠤⠄
⠈⠀⠀⠈⠉⠀⠀⢰⠈⡖⢒⠁⡘⡢⠀⠀⠑⠉⠠⡊⠀⠀⠀⠀⠀⠀⠀⠈⠒⡀⠀⠤⠀⠀⡖⡃⠈⢆⡉⠉⠀⠀⢘⡂⠀⠑⠤⠔⢁
⢀⢰⠀⠀⠀⠀⢠⠁⠙⠠⠄⠋⠈⡇⡔⡆⠀⡔⢢⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⠤⢤⠚⠀⡇⠀⠀⣈⡐⢢⠀⠘⠊⠀⠀⡤⠐⠁
⠈⠈⠀⠀⠀⠀⢈⢐⠀⠀⠀⠀⡂⡡⠌⢀⠄⠨⡥⠐⡎⠀⠀⠀⠀⠀⠀⡠⠆⠤⠘⠀⠐⢠⠄⠇⠀⠸⣀⡸⠀⠑⠠⠤⠄⠊
⠀⢸⠀⠀⠀⠀⠸⢉⡀⣀⣀⣀⡏⡇⢠⠐⠒⢰⢳⠒⠒⢄⠀⠀⠀⠀⢰⠀⠀⠈⠑⠒⠒⠉⠀⠀⠁⢠⠒⡀
⠣⢬⠀⠀⠄⠀⠘⠀⠀⠀⠀⠀⡇⢇⠸⡀⠀⢸⢸⠀⠀⡀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠃



⠀⠀⠀⢀⣀⡀
⠀⠀⢰⣁⡀⠜⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠉⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢸⠪⠀⠀⡇⠀⠀⠊⠉⠒⠉⢱⠀⢸⠈⠁⢢⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠠⠔⡊⠁⠀⠠⠒⠤⠒⡄⠀⣏⠀⠀⠀⠈⢢
⠀⠀⠈⠠⠤⡤⠃⠀⠀⠑⠄⣀⠤⠊⠀⠸⡂⢀⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠎⠀⠚⡬⠂⠀⣈⡀⢤⣔⣁⠀⢇⢏⠐⢲⡀⢘
⠀⠀⡠⠚⠤⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⢁⠦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⡍⠞⠁⢠⠜⢸
⠀⠀⠀⡌⠉⠢⢰⠀⢸⠉⠉⠀⠉⠉⡃⡆⢰⠑⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢐⠂⠁⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⡆⠀⢸
⠀⠀⠀⣇⡀⡀⢸⠀⠈⠢⠜⠀⠣⡴⠁⡄⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠤⠴⠒⢤⠀⠀⠀⠀⠀⠀⠀⢀⡨⠀⢧⣀⢸
⠀⠀⠀⡇⠀⠉⠈⠀⠀⠘⠀⠀⠀⠗⠒⠓⠊⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠂⠒⠄⠀⡇⠀⠀⠀⠀⠀⡐⠁⣈⠤⡉⠀⢹
⠀⠀⠄⠈⠉⢹⠋⠉⡒⠊⠉⡉⠉⢑⠊⠹⡉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡇⠀⢐⠀⢸
⠀⠈⠀⠀⠀⡌⠀⠀⡅⠀⠸⡀⠀⠪⠀⠀⢡⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠄⠀⠇⠀⠸⠀⠸
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠿⣿⣷⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣠⣿⡿⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣝⠻⣿⣿⣿⣿⣿⣿⣿⠿⢋⣵⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣦⣬⣍⣉⣉⣭⣴⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⠿⠿⠿⢿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠟⢿⣯⢸⣿⣿⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⣀⣾⣿⣿⡿⠟⠉⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣀⣠⣿⣿⣿⣿⣷⣄⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠑⠊⠉⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⣀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⠿⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡐⠁⠀⠀⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠈⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⠈⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣿⣿⡟⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⣤⣴⣿⠟⠋⠙⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣥⡀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣟⠀⢀⣾⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣧⡘⠉
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣦⠴⠚⠛⠉⠉⠉⠙⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠳⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠞⠉⠙⢦⠀⠀⠀⢠⠞⠉⠙⢦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢠⡖⠒⠒⠒⠒⠒⠒⠒⠒⣦⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠈⠙⠒⠒⠖⠚⠉⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⣤⠤⠤⠶⠚⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄⣠⣤⣀⣴⠀⠀⠀⠀⠀⠀⠠⣤⣶⡖⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠙⠉⠉⠁⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⢰⡶⠄⠀⠀⠀⠾⠀⠀⠀⠀⠺⠛⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣀⣠⡴⠂⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠶⣤⣀⡀⠁⠀⠀⣀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣰⠞⢩⣿⡛⢿⡷⣦
⠀⠀⢠⣏⣰⣿⣿⣿⣶⣿⡟⣷⣄
⠀⠀⢀⣟⣹⣯⡼⢻⣿⢻⣧⣿⢹
⠀⠀⢸⡟⢿⣟⣿⢿⣿⣾⣿⣷⡟⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠈⢷⡈⠻⢭⣽⣿⡿⠿⠿⠃⠀⠀⠀⣀⡴⠞⠋⠁⢀⣀⣤⣤⠀⣀⠀⣉⡙⠳⣤
⠀⠀⠀⠀⠉⠛⠛⠹⣇⠀⠀⠀⠀⠀⣠⠞⠉⠀⣠⣶⡿⠿⠛⠉⠉⢠⢚⣼⡿⡑⠄⣤⠘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⡾⠁⣀⣰⣿⠟⡁⢅⠂⡥⠌⣠⣴⠿⠋⢂⢃⣼⣿⣦⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠠⢤⣾⢟⡠⡔⢁⣐⣬⣶⡿⠛⢁⠌⠒⣡⣾⡟⠀⡙⣿⣤⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣱⡿⠃⣪⣴⡾⢟⡛⢉⠑⠐⠎⣣⣬⡾⠋⠀⠻⣶⣔⠈⢿⣦⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠸⠟⣠⡿⢋⡁⢒⣠⣤⣶⠶⠶⠟⠋⠁⠀⠀⠀⠀⠈⢻⣮⠘⢿⣆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣌⢇⣾⢟⡔⣢⡷⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⢲⡀⢀⡀⠀⢻⣯⠸⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡇⣼⡏⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⢻⡇⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣹⣟⣼⠋⠀⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⠀⠀⠘⣿⣿⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠙⣿⠁⠀⠀⢿⡀⠀⣿⠀⠀⠀⣠⠄⠀⠀⠘⣇⠀⢠⡟⠀⠀⠀⣿⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣿⠀⠀⠀⠈⠛⠛⠁⠀⠠⣾⣁⠀⠀⠀⠀⡈⠛⠋⠀⠀⠀⠀⢿⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⠀⠀⠀⣴⠟⠃⠀⠀⠈⠁⠀⠀⠈⠙⢶⡀⠀⠀⠀⠀⣾⣿⣭⣙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⡇⠀⠀⢠⡾⠛⠙⠓⣦⠀⠀⠀⡗⠀⠀⠀⢠⣿⠐⡩⡻⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⡻⢷⠶⣄⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠁⠈⠛⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⠛⠛⠛⡛⣛⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⣍⠙⠛⠁⠀⠈⠋⢋⣽⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠦⣤⡀⠀⠀⠀⡾⠀⠊⠀⠙⠷⣦⣀⣀⣠⣾⠛⠁⠈⣠⠈⣷⠀⠀⠀⣠⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣬⠿⠛⠀⠀⢹⡆⠀⢀⡇⢨⣄⠀⠀⠀⣿⠉⠉⠈⣿⠀⠀⠀⣠⡀⢹⡄⠀⢰⡃⠀⠀⠛⠻⣯⣀
⠀⠀⠀⠀⢠⡾⠋⢀⣠⣤⢶⡞⢛⢿⡄⢸⡇⢸⡏⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢿⡇⢸⡇⢠⣿⡛⢲⡶⢦⣄⡀⠙⢶⡀
⠀⠀⠀⠀⠻⠴⠞⠋⠁⠀⠀⢿⡀⠊⣿⡄⡇⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⢸⣇⣾⠣⠀⡾⠀⠀⠈⠙⠛⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠜⣿⡇⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢺⡇⢸⣿⠫⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠃⠘⣷⣸⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣾⣧⣾⢃⠇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠐⠀⠽⠛⠿⠄⠀⠐⢿⠀⠀⠀⡿⠀⠀⠂⠿⠛⠋⠐⠀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⠿⠿⠿⠿⠿⠿⠿⢷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠀⠀⠀⠉⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠷⢶⠾⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣷⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⡏⠀⠀⠀⠿⠿⠿⠿⠿⠈⠿⠿⠿⠿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣶⣤⠄⠀⠀⠀⠙⠉⠉⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⣉⣙⠻⢿⠀⠀⠀⠐⠒⣶⠖⢄⠀⠀⠆⢀⠤⣶⡖⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠠⠄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⣄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⣿⣶⣤⣀⣀⠀⠀⠀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⣀⣀⣀⣀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠟⠛⠻⢶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣫⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡴⠁⣠⡾⣦⠀⢠⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⣸⡏⣠⠞⠁⠀⠀⠀⣠⠞⠀⣠⡼⣿⠀⣴⣋⣴⠞⠋⠀⠸⣧⢸⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⢀⡤⢞⣡⡴⠞⠋⢠⣿⠿⠛⠉⠀⠀⠀⠀⠀⠹⣧⡄⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⢸⡇⣤⣴⡾⠿⢛⣩⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣤⣄⠀⢸⣷⣄⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣷⢸⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠈⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⣴⣶⣶⡀⠀⢠⡇⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣻⠀⠀⠀⠸⣿⣿⣿⠃⠀⣾⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⢰⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⣿⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠹⢶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢀⡾⠛⣷⡀⣶⠛⣶⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠸⣧⣀⣾⠃⣿⠀⢸⣇⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠈⠉⠁⠀⢿⡄⠀⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⣀⣀⣀⡀⣀⣀⣸⣧⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⣹⡿⠛⠷⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡿⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢻⣦⣄⣀
⠀⠀⠀⠀⣀⣤⡤⠶⠖⠛⠛⠛⢻⣏⠀⠈⠻⣦⣄⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠉⠛⠶⢤⡀
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⣀⣼⠟⢿⣄⠀⠀⠀⠀⠀⠀⢀⣼⣷⡀⠀⠀⠈⢻⡄
⠀⠀⢀⣿⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠙⣷⠾⣿⠻⠶⠟⢳⣤⠾⠛⠁⠈⢿⡄⠠⠀⠀⢻⣆
⠀⠀⣸⡇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡆⠀⠀⠘⣿⠀⠀⣠⣴⠿⣷⡇⠀⠀⠀⢻⡄
⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣧⠀⠀⠀⠘⣷⠞⠋⠀⠀⠈⠻⣦⡀⠀⠀⢿⡀
⠀⢸⡏⠀⠀⢠⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠈⢻⣆⠀⠈⢷⡄
⠀⣿⠁⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡆⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠙⣷⡀⠈⣿⡀
⠀⣿⠀⠀⠀⠀⠀⣿⠀⢰⠖⠒⠒⠛⠛⠛⣷⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠈⢿⣄⠘⣷
⢰⡟⠀⠀⠀⠀⠀⣿⠀⠈⠓⠒⠒⠒⠚⠋⠉⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠻⣦⣹⡆
⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷
⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣿⡇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⢷⣄⠀⠀⠀⠀⠀⣼
⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠻⠶⠶⠶⠾⠏
⠻⣦⣄⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠙⠛⠛⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣄⣀⣀⣀⣸⣇⣀⣤⣤⣴⠶⠟⠛⠉

⠀⠀⠀⠀⠀⢀⡴⢒⣶⠶⣶⢦⡀
⠀⠀⠀⠀⠀⣾⣠⣿⣿⣿⣼⣿⢳⣦
⠀⠀⠀⠀⠀⣿⣹⣯⣞⣿⡟⣿⣸⢻⠆
⠀⠀⠀⠀⠀⣏⢻⣯⣟⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠙⢦⣝⣳⣾⠟⠛⠛⠁⠀⠀⢀⣤⠴⠖⠛⠛⠛⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⢀⡴⠋⢀⣠⣴⠶⠿⠶⢶⣤⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⣴⠟⠉⢀⣀⣤⣄⣀⠈⠻⣦⡈⠑⠛⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⡀⣀⡴⠃⠀⠈⠁⣠⡾⠛⠉⠉⠉⠉⠻⣶⡘⠃⢀⣤⣤⣀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⡉⠉⠀⠀⠀⠀⣴⠟⠀⠀⣠⣶⣶⣶⣤⡈⠛⠀⠈⠉⠈⠙⣷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠸⢶⣤⣤⣤⡾⠋⠀⠀⣰⠿⢦⣍⠳⡌⠻⠄⠶⠶⠶⣄⠀⢹⡇⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⣀⠈⠉⠀⠀⠀⢀⣼⠋⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠘⣧⠈⢿⡀⣧⣀⡤
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠃⢨⡟⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠀⢸⣆⡈⠳⠦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠁⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠘⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡼⠛⣿⠀⠀⠈⣇⠀⣸⠁⠀⢀⡴⠀⠀⠀⠸⡀⢀⡾⠀⠀⠀⣿⠛⢷⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣤⣿⠀⠀⠀⠈⠉⢁⣀⠀⠛⠶⠀⠀⠀⣀⡉⠉⠀⠀⠀⠀⣿⣤⡼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢿⡀⠀⠀⠀⣰⠋⠁⠀⢀⣀⣀⡀⠀⠈⠙⢷⠀⠀⠀⢀⣿⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡄⠀⠀⠘⠀⠀⠰⠏⠁⠀⠙⠦⠀⠀⠘⠀⠀⢀⡾⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⣶⡶⢶⣶⡖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢶⠾⠏⠉⠉⠉⠽⠒⡶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⢀⡞⠁⢀⠹⣧⡀⠀⠀⠀⣠⡾⠁⠆⠙⣧⠀⠀⠀⣠⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣽⠷⠂⠀⠙⣆⠀⢸⡇⣠⡥⠠⠌⠛⠛⠾⠛⠋⠐⡄⢠⡁⢹⠀⠀⣾⠁⠀⠐⠿⣯⡀
⠀⠀⠀⠀⢠⠞⠉⣀⣤⠶⣾⠛⢻⣆⢸⡇⣿⠁⢣⡄⠀⠀⠀⠀⠀⠀⢈⢸⡇⢸⡆⣰⡟⠛⡶⠶⣤⣀⠙⢦⡀
⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠘⣆⠀⢻⣼⡇⣿⡄⡆⡁⠀⠀⠀⠀⠀⠀⠈⢸⡇⢸⣷⡟⠀⣸⠃⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢿⡇⢿⡇⢁⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟⡐⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⠌⢿⢼⡇⢈⠀⣀⠀⣂⡒⠂⠀⢂⣸⣷⡟⠐⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣿⡁⠀⠀⠀⠀⣀⣤⣶⣶⣤⣀⠀⠀⠀⠀⢀⣤⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⣰⡟⠁⠀⠀⠈⢻⣆⠀⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⣿⡀⠀⠀⠀⠀⢀⣿⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⢿⡆⢰⡾⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⣿⠁⠈⢿⡄⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⡟⠀⣾⢷⣤⣤⡾⣷⡀⢻⣿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢰⡿⠀⣼⠏⠀⠀⠀⠀⠹⣧⠀⢻⡆⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠿⠁⣰⡟⠀⠀⠀⠀⠀⠀⢻⣆⠈⠿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣡⣾⣏⢉⣿⣿⡇⠀⠀⠀⣰⠋⠀⠀⠀⠃⠀⠀⢠⣀⣋⡀⠿⢸⢟⡙⠛⢳⣿⣿⡟⢛⣉⣿⡟⢉⡿⣿⡿⣷⣿⣿⠷⠛⢠⣜⠻⡄⢹⣷⡿
⣿⠾⢿⣭⢠⠟⠀⠀⠄⣠⡇⣀⣠⢀⡄⢀⠀⣤⠀⣩⣥⣤⣄⣤⣤⣤⣷⢨⣷⡾⢏⣿⡟⢻⡇⠭⣵⣿⣱⢏⠙⠣⣴⡷⣾⡟⠉⢲⣿⣩⣿
⣤⣤⠾⡇⠀⠀⠀⠀⡼⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⢠⣥⠹⠄⢻⣄⠰⠀⠀⣨⠉⠸⢥⡤⢨⠉⣭⣥⡼⠟⢀⠠⣿⡷⣶⣋⠿⣾⡽
⣿⣿⠞⠁⠀⡠⠀⣀⠁⣄⣀⠀⣀⣀⡀⠀⢀⠀⢀⢠⣄⢡⣀⠤⢠⠀⣤⠼⠧⣠⣄⣠⣿⣶⡀⣤⠀⢸⣷⡏⢰⡾⣦⣤⣌⣅⣹⣻⣤⠙⣧
⣉⡉⠀⠀⡐⢀⡾⠁⠀⠀⠀⡘⠁⣼⡃⠀⠀⠀⠈⠀⠐⠀⠉⠀⠀⠀⠧⠀⠀⢠⠴⠿⠛⠈⣃⠀⢀⣬⡉⣼⡛⡿⣅⠁⢨⣿⣇⢹⣅⠑⣷
⠋⣁⡁⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣠⣀⣀⣠⣆⠰⣾⡂⢁⣼⢿⣤⢤⢉⣉⠛⣦⣕⣠⣝⢁⣼⣭⣿⣼
⠛⠛⠻⠟⠛⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣉⠙⣋⠉⢉⡉⠉⠋⠑⠉⢹⣿⠛⠛⢿⣿⢿⣿⡿⠋⢻⡦⢯⡙⠻⣿⣿⣹⣧⡙
⣀⣤⣦⣤⣀⣶⣀⣤⣴⣆⣀⣰⣆⣠⣤⣴⣰⠶⠶⠂⢉⣥⣤⣿⣤⣤⣍⣻⠿⠷⣶⣯⣍⡰⣦⢬⣍⣛⠿⣜⣿⣄⣤⠀⠿⠷⢦⡬⠿⠿⣿
⡿⠽⠿⣍⣉⣩⣏⣉⣁⠩⠿⣯⣤⣬⡟⠋⣉⣴⣶⡿⠿⠛⠛⠛⠉⠙⠛⠿⣶⣤⣉⠿⠻⣿⢛⡷⠾⣿⡶⣾⠤⢾⡟⠳⢄⢶⡀⠓⠲⢦⡝
⠄⠠⠴⠿⠄⠤⠼⠿⠥⠤⠤⢿⡿⠁⢀⣴⡿⠋⢁⣤⣾⣿⣿⣿⣿⣿⣿⣶⣦⡙⠻⣶⡄⠉⢹⣧⣶⣯⡟⠿⣶⡾⠷⢦⡤⣼⠿⡶⢤⣄⠻
⣦⣶⣤⣤⣴⣶⣤⣤⣴⣦⣴⠞⠀⢠⣾⠏⠀⣰⣿⣿⣿⡿⢿⠻⣿⠿⢿⣿⣿⣿⣦⡈⢻⣿⡀⠈⢻⣯⢶⣶⣾⣷⠤⢦⣴⡿⠶⢶⣤⡻⠶
⢶⡶⠾⠷⣶⡤⠷⢶⣶⠉⡟⠀⠀⣾⠏⠀⣼⣿⣿⡟⠋⠀⠀⠀⣿⠀⠀⡙⢿⣟⢿⣷⡀⠙⣿⡄⠈⠹⣷⣿⣿⣿⣿⣷⡿⠦⠠⢤⣿⠿⢶
⣾⣅⣀⣠⣿⣄⣀⣼⠃⠴⠃⠀⣾⡏⠀⣸⣿⣿⡿⠁⠈⠃⠀⠀⣷⢼⠁⠟⠈⢿⣾⣿⣷⡄⠘⣿⡄⠀⠻⣶⣤⣥⣈⢿⠦⠀⣀⣄⠻⠆⠈
⠁⠀⠀⣸⠁⠀⠀⡉⠀⠃⠀⢠⡿⠀⢠⣿⣿⢿⣇⣀⣀⣀⡀⠀⡟⢺⣀⣀⣠⣨⣿⣿⣿⣷⡀⠸⣷⡀⠀⠘⣧⣘⣧⣀⣀⠀⠨⣷⣤⣀⣤
⠐⠒⠒⠛⠖⠒⢲⠃⡀⠀⠀⣿⠃⠀⣾⣿⡿⠈⢨⡿⠀⢹⠀⠀⣿⠈⣿⡇⢸⣿⣏⣿⣿⣿⣧⠀⠹⣷⡆⠒⠻⣧⣨⣿⣿⡿⢦⣀⣀⣬⣴
⠠⠀⠀⢀⡀⠀⠈⠀⠗⠀⣰⡇⠀⢸⣿⣿⡟⠀⢰⡗⠀⢹⠀⠀⡿⠀⠘⡇⠈⣿⣆⢻⣿⣿⣿⣇⣠⡿⣷⡀⣸⣷⣾⣿⡅⢠⣌⣽⣁⡀⢀
⠀⠀⠀⠈⠁⠀⠁⡀⠀⠀⣿⠁⠀⣾⣿⣿⠃⠀⠀⣏⣀⣠⡀⢀⣷⣹⣴⣧⣤⣿⢀⣹⡏⣿⣿⣿⡟⢦⣽⡅⠈⠐⢾⣯⡄⢉⣿⣥⣼⣷⣀
⠀⠀⢀⡀⠀⢀⡀⠀⠀⢸⡏⠀⢠⣿⣿⢿⠀⠀⣴⡏⣹⠉⣩⣿⡿⠿⠿⣿⠏⣽⣟⠁⣿⣹⣿⣿⣷⠀⠙⣿⡗⠚⠻⣯⣸⣿⣿⣿⣿⣯⣿
⠀⠀⠀⠁⠀⠘⡃⠀⠀⣾⠁⠀⣸⣿⡏⠀⠆⡀⢸⡏⢻⣾⡏⢹⠀⠀⠀⠘⠀⠀⢻⡀⢯⡝⠛⣿⣿⣧⠀⠹⣧⠀⢀⣸⣿⣿⣿⣿⣷⣿⣿
⠀⠀⣄⠀⠀⣮⠀⠀⠈⡟⠀⢠⣿⣿⡇⠀⣄⠁⢸⣇⢹⡄⠀⠈⠀⠀⠀⠀⠀⠜⠀⠑⣺⣷⡄⣿⣿⣿⡆⠀⢻⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡧⠀⠀⢸⡇⠀⢸⡏⢿⠁⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢸⡇⠃⢿⣿⣿⣿⡀⠈⣷⠀⠋⠻⣿⣿⣿⣿⣿⣻
⠀⢶⠀⠀⣾⠀⠀⠀⢼⠀⠀⣸⡇⡘⠀⠀⠀⠀⢀⡏⣿⠀⠐⠀⠀⠀⠀⠀⠀⠘⠆⡀⢸⡇⠂⠂⣿⣿⣿⡇⠀⠸⣇⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠠⢀⡟⠀⠀⠠⣮⠀⠀⣿⢠⣇⠀⠀⡆⠀⢸⡇⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⡉⠸⣧⢀⠀⣽⠉⣿⣿⡀⠀⢻⡄⠀⠘⣿⣏⣽⣏⢻
⠀⠖⠀⠀⠁⠔⠂⠀⠁⠀⢀⣿⠾⠉⠃⢀⡇⠀⢰⣇⣸⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⣶⣾⣬⣆⢿⡇⢻⣿⡇⠀⠀⣿⡀⠙⠻⣿⣿⣿⣿
⠀⣀⣀⣤⣤⣴⣆⠀⢀⠀⢸⣿⠀⡇⠀⠘⡇⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣼⣷⠘⣿⣿⠀⠀⠘⣷⡀⠰⣿⡿⠋⢻
⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠺⠏⠘⡇⠀⠀⡁⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣹⣿⣿⣿⣿⣿⣆⢹⣿⣇⠀⠀⢿⣄⡀⢸⠁⠀⠸
⠀⠛⠻⠟⣿⣿⣛⣿⡦⠀⣾⠇⠀⡇⠀⠀⣥⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣸⣿⣿⡄⠀⠈⣿⠀⣶⣴⣾⣶
⠀⠀⠀⠈⠈⠋⢩⡯⠀⠈⣿⡆⠀⣇⠀⠀⢩⣸⣿⣿⣿⣿⠀⠀⠘⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⡇⠀⢀⣼⣿⣿⣿⣿⣿
⢀⡁⠀⠄⠀⠛⢠⣤⣤⣴⣿⣷⣤⣿⣤⣤⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣷⡠⠾⣿⣿⡿⣿⢿⠛
⣮⣿⣶⣤⡀⠀⠋⠉⠻⣏⠻⠁⢁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣠⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⢿⣿⣿⣿⠛⠉⢹⣿⡆⠈⠀⣤⣴⣶⣶
⣿⡿⣿⣿⣯⡀⣀⢤⡶⣿⠄⣀⣾⣄⣀⠀⣤⣠⣿⣿⣿⣿⣿⣿⡀⠀⠀⠰⡀⠰⣄⣴⢾⣷⣿⣿⣾⣿⣿⣧⣄⣀⣤⠿⢧⣴⣾⣿⣿⣿⣿
⢿⡿⡿⢟⢻⣉⣉⣀⣀⡀⠀⢉⣉⡉⣋⣙⣋⣹⣿⣿⣿⣿⣿⣿⣏⣀⠀⠀⢰⣦⣤⣴⣿⣿⣿⣿⠟⠛⠋⠉⢽⣯⣿⣀⣀⠿⢿⣿⣿⣿⣿
⠀⠀⠥⠀⣾⡟⠛⠙⣿⠛⠟⠛⠛⠛⠙⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣡⣼⣿⣿⣿⣿⣿⣿⣅⡠⠀⠀⠀⠺⠋⠈⠛⣿⠃⠚⠉⣀⠉⠁
⠷⡄⠀⢀⢋⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣖⣾⡁⠀⣾⠀⠛⡀⠀⠀⠉⠄⠒
⣆⠀⠑⢀⠘⣿⣄⣀⣀⣀⣀⣼⣦⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣟⢉⣛⣄⣻⣄⣈⣿⣿⣴⣤⣤⣤
⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠏⠀⢿⣿⢿⣿⣿⡞⣻⡿⠛⠿⣿⠿⣾⣿⣏⠉⠉⠉⠉⣉⣉⣈⣁⣀
⡿⠛⣿⠟⠿⢻⡶⢶⣶⣶⡶⢴⣶⡿⠿⠻⣿⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣽⡇⠀⠀⢻⡆⠁⠀⠙⣿⡛⠛⠛⢿⠉⠉⠉⠉
⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⠀⠈⠁⠀⠈⠉⠉⠀⠀⠘⠛⣿⠛⠋⠉⠈⡇⠈⠁⠘⠳⡀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢀⠁⠀⠀⠀⠘⣿⣤⠟⠉
⣧⣀⣀⣀⣀⣀⣤⣤⣿⣤⣤⣤⣴⣦⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠓⠒⠞⠳⢶⣿⣁⣴⣄⣀⠀⠀⠀⠀⢠⣶⢀
⡙⠛⠛⠛⠛⠋⠙⠋⠉⠒⠀⠙⢊⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠈⢏⠉⠉⢉⡹⠿⠿⠉⣹⡛⠻⠿
⠛⠀⠀⠀⠀⠠⣀⡤⠤⠐⠶⣾⣿⣅⡀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⣿⣥⠔⠂⠀⠀⠀⠀⠀⢀
⢠⠟⡀⠒⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⢷⡄⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣱⡄⣠⣴⠖⠒⠛⠉⠀⠀⠒⠀⣈⡉
⠂⠘⠃⠀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠂⠰⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⡙⣋⣀⠀⠀⠆⠀⠀⠀⠀⠀⠘⠛
⡭⠉⠀⠀⠈⠙⠚⢢⡀⢀⡀⠁⠀⠀⠀⠀⠒⠀⠒⣀⠀⠹⣦⣤⣀⣀⣀⣀⣀⠴⠊⠉⠀⠈⠉⠉⠉⣰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢦⠤⠽⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠿⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠢⣀⡛⣄⠀⣀⠸⡓⠀⠀⡠⠔⠒⠃⠀⠀⠀⠀⢠⡤⠒⠋⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠾⠱⣿⡄⠀⠧⢤⣈⠀⠀⠀⠀⠐⠺⣜⣠⠤⠖⠛⠒⠛⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠶⣖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡴⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⣀⣀⡤⠔⠊⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣒⠿⠛⠉⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠠⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠈⡙⢲⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⡙⠁⠘⡆⠈⢶⢤⣀⠀⠰⡶⢄⠀⡆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢦⠀⢸⡀⠀⠇⠀⠘⣆⠈⠉⠐⠛⠈⢺⠥⠤⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡜⠀⠀⠃⠀⠈⠀⠀⠘⠆⠀⠀⠀⣰⡃⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡧⠤⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢾⣿⡀⢿⠿⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⣖⡀⠛⢁⡀⠳⡄⠰⠒⣻⠁⠀⠀⠙⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⣇⣠⣅⠀⢠⠇⠀⠀⠀⠀⠈⠒⡛⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠦⣀⠀⠀⠈⢑⣿⠉⠬⠤⠞⠢⠄⠀⠀⠀⠀⠘⢅⣠⠇

⢀⣀⣀⣀⠀⢀⣀⡤⠀⠀⢀⡀⠠⠶⣀⠈⠡⠄⠠⠤⢔⣀⣀⠀⠀⠀⠀⠀⠀⠒⢋⣠⠶⠋⠀⠀⠀⠀⡀
⢁⡝⠁⠀⠀⠀⠈⠲⠀⠀⠀⠀⡤⠊⠀⠀⠀⠀⠀⢠⣶⡉⣀⣀⡀⠀⠐⢐⣨⠖⠉⠀⠀⠀⠀⠀⠀⠀⢷⠀⣀
⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣞⠀⢀⠠⢐⡤⠖⠋⠀⠀⡀⠀⠀⠀⣶⠀⣀⠤⠚⠉
⢤⡀⣀⠀⠀⠀⠀⠀⠐⡖⠒⠁⠚⠁⠀⠠⠀⠀⠀⣤⣬⡠⡟⠘⠉⠀⢣⡀⠀⠀⢃⠀⣀⠤⠚⠉
⠒⠒⠒⠒⠶⠶⡒⠒⠂⠀⢈⡁⠀⠠⢮⣤⢤⣤⢤⡬⠉⡁⠀⢸⡆⠀⠈⣷⣀⠤⠚⠉
⠀⢠⡏⠀⠀⢰⠇⠀⠀⠀⢸⠁⠀⠀⠀⡟⠀⠀⢸⡇⠀⢻⡀⠈⣷⡤⠒⠉⠁
⡀⣸⡀⢀⠀⢸⡀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠸⡇⠀⠸⣗⠯⠁⠈⠀⠀⣀
⠒⠒⠒⠒⠒⠂⠒⠒⠒⠒⠐⠒⠀⠀⠀⠈⠀⠀⠀⡽⠒⠁⠀⠀⠀⠂⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠂
⠶⠲⠶⠴⢆⠲⠀⠀⠤⠤⠤⠤⠄⡀⠀⡄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠄⣿⠀⠀⠀⢀⣠⠴⠞⠋⠁⠀⢀⣠
⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣿⠀⠰⠋⠁⠀⣀⣠⣴⡶⠿⠛⠉
⠄⠀⠘⢤⠀⠀⠑⢄⠘⠀⠀⠀⢀⣁⡀⡇⠀⠀⠀⠄⠀⠀⢠⣷⠀⠀⠀⣾⣿⠀⠀⠶⠶⡟⠛⠋⠁⣀
⡀⠀⡁⠀⠀⠀⠀⠀⠉⠂⠐⢆⠈⠉⠁⡇⠀⠀⠀⠂⠀⠀⠸⣿⣿⠀⢀⠿⣿⠀⠀⠀⠀⠃⠀⠀⠀⢀⢈⣁⠐
⠷⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠘⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡙⠛⠩⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠙⢆⡇⠀⠀⠀⠀⠀⠁⠀⣀⣿⠀⠀⠀⢀⡆⠀⠀⢸⠁⠙⠦⠈
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠁⠀⠠⠬⠴⠖⠛⠋⠁⠀⠀⠀⠀⠸⠇⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⡀⣶⡆⠀⠀⠀⠀⠀⠀⠤⣠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⡧⠄⠀⠀⠀⠀⠀⠀⢀⠴⠶⣾
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⡀⠀⠀⠄⠀⠀⢀⢈⡆⠠⡀⠀⣦⠀⢤⣀⠀⠠⠁⢀⣀
⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢀⠀⡔⠉⠁⠻⠏⣿⡀⣹⡇⢀⠆⠘⡟⠀⣠⢳⠀⠈
⠏⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⢈⣦⠀⠴⠀⠚⠑⡿⠀⣀⠈⠻⡄⠰⠀⠀⠀⢚⠱⠉
⡄⠃⠈⢧⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠁⢀⣀⠰⣄⠀⠀⢀⠀⢁⠀⡈⢀⡄⠀⢀⠎⠀⢀⠘⢰⡦⠆
⠁⢀⣀⠤⠗⠒⠉⠉⠉⠉⠉⠓⠤⠀⠀⠀⠀⠀⠞⠏⠀⢀⣸⣿⡆⢀⣠⠈⠀⠈⠀⠙⠖⠋⠀⡠⢻⡀⠈⠳⣆
⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⠇⣀⣀⠀⣸⠃⠀⠚⣿⡰⡦⠀⠠⠤⠀⢀⠴⠶⢤⠇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⠄⡿⠀⠼⣿⢀⣽⠂⠉⢰⣿⡠⢁⣄⡄⢀⡄⠿⠀⠼⡇⠀⢀⡷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⢻⣄⠀⠀⠉⠁⠀⠀⡾⠈⢑⡛⣉⠁⡶⣀⠀⠀⢀⣤⣼⣤⣤⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣐⡿⢥⠄⠀⠒⠺⣿⣵⡄⠷⠶⠭⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠘⣿⣭⡇
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠙⠻⠦⠠⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠐⢀⠀⠈⠳⢄⣀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⠟⣉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠄⠓⢀⡴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⣀⣽⣿⣿⣭⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠌⠛⠻⠟⠿⣷⣶⣍⡢⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣢⣖⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠂⢁⣀⡼⠧⠐⠲⣿⡆⢿⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠸⠛⠁⠀⠀⠀⠀⢩⣶⠞⠓⠦⠄⡀⠀⠀⠀⠳⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⢠⠆⠀⠁⢀⣾⠀⠀⢀⠄⡍⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠁⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠣⠀⢠⠇⠀⠀⠀⠸⠃⠀⠀⠊⠘⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠁⠀⠀⢔⡾⠀⢰⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢇⠈⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠁⠀⠀⠀⠟⡀⠐⠁⠀⠀⠀⠀⠀⣠⣴⠶⠚⠁⠀⠸⡄⢻⣆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⠀⠀⣠⠞⣁⣀⣀⠀⠀⠀⠀⠈⡛⠀⡀⠤⢤⡔⠆⠉⢼⠻⡀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣰⠿⠛⠉⠉⠉⠀⢀⠀⠀⠀⠠⡼⠷⠀⠘⠁⠀⠀⠠⡀⠡⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢈⡆⠀⠀⠐⣿⠁⣠⣤⠀⠀⠠⡏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣧⠀⠈⠀⢨⠠⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠄⢠⡆⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠁⡄⡴⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠒⠚⠏⠁⠀⠀⢀⠀⠈⠀⠀⠀⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠴⠒⠒⠉⠀⠀⠀⠀⢰⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣀⠀⠀⠀⠀⠀⣇⠘⣧⠀⠁⠀⠀⠁⠀⠉⠀⠀⣠⡤⠶⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣼⠀⠀⠀⠀⡘⠸⡀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡿⢡⡇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣴⠏⠰⢠⠃⠀⡀⠀⠡⡀⠙⢿⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⣿⠾⠝⣡⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣤⣴⡴⠃⢀⡄⠑⠀⠀⠀⠀⠀⠹⠈⣱⠐⠠⠤⠤⠤⠤⠒⠂⠉⢀⠀⠀⢛⣐⠞⠕⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⠿⠟⣉⡀⠀⠭⣤⡔⢀⡠⠊⣿⢶⡀⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⣀⠄⣠⠊⠁⠀⠀⡀⠀⠀⠀⠀⠈⠁⠠⠊⡠⣾⠏⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⡀⢀
⠀⠀⠀⢀⠜⠁⠜⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠡⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⡦⠐
⠀⠀⠀⠀⠀⠀⠀⠊⠊⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠐⠷⠖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⡀⠃⠀⠀⠀⠀⢀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡄⢀⣀⣤⣤⣶⣶⣶⣯⣤⣤⣀⡀
⠀⠀⠀⠀⠸⢷⣦⡀⠀⠀⠀⠀⠀⠁⠀⣰⢋⣴⣿⣿⣥⣤⣈⣉⡙⠛⠛⢻⣿⣿⣵⡢⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⠀⠀⠀⣠⣤⣽⣿⣿⡿⠟⠟⠻⢻⣿⣯⣶⡶⠇⠈⠉⠽⣿⣷⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣿⣥⣴⣶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣒⠀⠙⠻⣿⣷⣕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⠋⣉⣠⣤⣤⠤⠤⣴⣤⣀⣟⣻⠻⣿⣿⣿⣿⣿⡦⠍⠺⢿⣿⣿⣧
⠀⠀⠀⠾⠿⠛⠛⠋⠉⠁⢠⢾⠃⠈⣁⣀⣠⣤⣤⣤⣤⣤⣏⣉⡗⢀⢸⡏⠉⣻⣿⣿⣷⡉⠳⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡟⠀⠉⢉⣁⣤⣤⣤⣭⣴⡿⢷⡄⠀⠛⠛⠁⢠⣾⣿⠟⠿⣶⣶⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢡⡶⠶⠛⠉⢩⡿⢯⠻⢿⣿⣦⣉⠀⠀⠀⢠⣿⣿⠟⠃⡀⠠⣽⣿⣿⡽
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣼⣧⡰⣤⣸⣆⠀⠈⠃⠀⠁⠈⠋⠁⠀⠀⠘⣿⣷⡀⠀⣙⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⡼⠀⠀⢀⣴⣿⡿⠥⡈⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣸⢯⣤⣽⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⢧⡀⠳⣿⣿⣿⣋⣉⣓⠢⡄⠀⣿⠀⠀⠀⠀⣀⣴⡿⠇⠘⠿⠿⢫⣵⣶⡌⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠿⣿⣭⣍⣉⡉⠙⢿⠃⢋⠀⠀⢀⡼⠋⠁⠀⠀⢀⣠⣶⢾⣿⢟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠠⠴⠖⠋⠀⠀⠊⠀⠀⠀⢀⣼⡿⢧⣌⡥⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣦⣀⣀⣀⠀⠀⠀⡀⣀⣀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⠩⠿⣁⣩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣎⠒⠶⢤⠘⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⡨⢠⠈⢀⡰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⠁⠀⠈⠑⠢⢤⣠⡏⠀⠀⠁⠃⠀⠨⠓⢄
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⢴⡾⡋⠢⠤⠄⠀⠀⢀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⢀⣠⠖⠋⠁⣀⠐⢱⠀⠀⠴⠆⠑⠂⢄⡀⠀⠈⡌⠀⠀⠀⠀⠀⠀⠜⢀⡀⠀⠑⡀
⠀⠈⢻⣄⠀⠘⠄⠸⠃⠈⠢⡀⡀⢀⠤⡪⠀⣈⡴⢻⠁⠀⠀⠀⠀⠀⠘⠀⠀⠑⠀⠀⠈⠠⡀
⠀⠀⢸⠈⠓⢤⡀⠀⡀⠈⢆⣀⠍⠓⣨⠴⠚⠓⠂⠣⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⢄⡠⠣⡀
⠀⠀⢸⠀⠀⠀⠙⠢⡉⠀⢀⣠⠖⠋⠁⠀⢀⠴⠋⠀⠀⠉⠓⠂⢀⡴⡀⠀⠀⠀⠀⠀⠀⢑⡢⡈⡆
⠀⠀⢸⠀⠀⠀⠀⠀⢹⠚⠉⡇⠀⠀⢺⠚⠁⠀⠀⠀⢀⣠⠔⠚⢁⣀⡜⠀⠀⠀⠀⠀⠀⠐⠠⠬⡀
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⠀⢖⠈⣿⠀⠉⠉⣏⢸⡇⠀⠀⠀⠀⠀⠈⠒⠤⠥⠃
⠀⠀⢸⡀⠀⠀⠀⠀⢸⠀⠀⣧⠀⠀⡜⢆⠀⠀⠈⡄⣿⠀⠀⠀⡿⠈⡇⠀⠀⠀⢠⣶⣻⣛⡏⣻⢙⣓⡄
⠀⠀⠈⠳⡄⠀⠀⠀⢸⠀⠀⣿⠀⠀⡇⠈⢆⠀⠀⠸⡇⠀⠀⠀⡇⠀⣷⠀⠀⠀⠈⣿⣟⣏⣹⣉⣯⣿
⠀⠀⠀⠀⠙⢦⠀⠀⢸⠤⠒⠋⠉⢹⠀⠀⠈⢧⠀⠀⢣⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⡀⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⣄⢸⠖⠒⠋⠉⠁⠀⠀⠀⠈⢧⣀⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⠺⠇⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠞⠁

⠀⠀⠀⠀⢠⠖⠀⢀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⡐⠆⠀⡀⢀⠀⠄⠄⡀⠀⣠⡦⠖⠉
⠀⠀⠀⠀⠆⠀⢸⠈⡼⠃⡏⡟⣌⢰⢘⢋⠀⠀⠃⠐⡇⠀⠀⣇⠄⡞⠙⡎⠐⡇⡞⢁⠞⠀⡇⡆⠏⢸⠋
⠀⠀⠀⠀⠘⠄⠈⠃⠓⠀⠃⠀⠛⠚⠘⠈⠀⠀⠀⠐⠃⠀⡀⠇⠀⠂⠀⠃⠀⠃⠓⠈⠂⠀⠃⠃⠀⠘⠈
⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⣷⡀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠴⡤⣤⣄⠀⠀⣠⣾⠃
⠀⠀⠘⡝⠦⣤⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢟⡱⠃
⠀⠀⠀⠈⢦⠇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠢⡀⠈⣇
⠀⠀⠀⠀⠧⠀⠀⠰⠶⠶⠶⠖⠀⠀⠣⡀⣀⡠⠃⠀⡿⠀⢀⣀⣶⣴⡀
⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢱⠀⠀⠹⡇⠉⡇
⠀⠀⠀⠀⡘⠀⠀⠀⠀⢀⣴⣿⣿⢾⣿⣷⡀⠀⠀⠀⢸⠀⣠⠞⠁⢠⠇
⠀⠀⠀⠀⢹⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠀⠀⢀⡎
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⢠⠃⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱
⠀⠀⢾⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢨⣦⣶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁
⠀⠀⠉⠻⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⢀⣀⡤⠷⠀⠀⢀⣀⣀⣀⣀⣀⣀⢤⡄⠀⢐⣟⡦
⠀⠀⠀⠘⠓⠓⠊⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉

⠢⠄⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠐⠒⠉
⠀⠀⠀⠀⠀⠀⡀⠀⣀⢀⠀⠀⠈⠄⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠂
⠀⣈⣀⠀⠀⠀⣀⠀⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⡸⠆⠀⠀⠀⣈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠁
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠁⠠⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢴⡇
⣷⣤⡀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠀⠘⠦⣄⡀⠀⠘⡆⠄
⣟⠻⢿⣶⢛⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠷⣼⣧⠰⠀⠀⠀⠀⣤⣀⣀⣀⡀
⠛⠻⢧⣽⣿⣷⠶⢿⣶⣤⣤⡀⠀⠀⠀⠀⠈⠲⣿⣿⠗⢦⣄⡀⠀⠈⠉⢻⠿⠟
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠿⣿⡷⣶⣤⡀⠀⠀⠀⠘⠏⠲⠀⠉⠀⠀⠀⠀⠈⠀⢯⠀⠀⡀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣽⣿⣿⠶⡄⣢⣀⣦⣠⣀⡔⠘⠀⠀⠁⠀⢸⡀⠀⢩⣤⣦⠀⢰⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣿⣿⣻⣿⣿⣿⢆⢆⠀⣤⣡⡶⠀⠀⠀⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣯⣰⣿⣿⡄⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡉⡛⣿⣿⣿⣁⡀⣧⡄⠘⠻⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣻⣿⣿⣿⣿⡿⠁⠀⠀⣾⣿⢦⠀⠀⠐⣾⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⣿⣿⣿⣿⣿⣇⠀⠀⣼⣿⣿⡗⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⣀⠀⠀⢠⣀⣠⣾⣿⣿⡏⢿⣿⣷⣿⣿⠀⣼⣿⣿⠿⢀
⠀⠀⠀⠀⠀⢀⣄⣠⣾⣟⡯⠤⢩⣶⣶⣿⣿⣿⣿⣿⣿⣷⢸⣿⣮⣿⠿⢠⣿⣿⣷⠆⠄
⠀⠀⠀⠀⠀⠐⣿⣿⣅⡠⠟⠒⠛⠛⠉⠙⠿⣿⣿⣿⣿⣿⣿⡿⢿⡇⠛⢸⣿⣿⣟⠁⠀⠀⡄
⠀⠀⠀⠀⢀⣠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣛⣿⣿⣿⣿⣶⣿⣶⡂⣸⣿⣿⣿⠀⠀⠀⢘
⠀⠀⢀⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⠿⠛⠃⣿⡇⠛⢻⣿⢿⡓⣧⠀⠈⢷⠤⠠⡀
⣤⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠉⠀⢀⣄⣠⡶⣟⡀⠀⠀⠻⣿⠇⡟⠡⡀⠘⠓⢦⣄
⣯⣁⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠈⠻⠿⠿⣿⠀⠀⢠⣶⡶⠂⠀⠀⣿⣄⠀⠀⠈
⢇⡘⠇⠀⠐⢿⣷⡆⢀⣀⣦⣤⣶⣤⣤⣴⣿⡿⢛⣇⠂⠀⠀⠀⠀⠈⠁⠛⠉⠀⠀⠀⠀⠀⢠⠀⠉⠑⠾
⠋⠁⣀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣭⣿⠟⠁⢸⣟⠃⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣤⠤
⠓⠆⠉⠹⢧⡄⠰⣿⣿⡿⣿⡟⣻⠿⠛⠁⢀⡠⣿⣿⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠙⢤⣀

⠀⠀⠀⠀⠀⠀⢼⠢⣄⠀⠰⡖⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠉⠒⠼⠆⠀⠀⠉⠒⠢⠤⠤⠤⠤⠶⠒⠒⠚⠛⠋⠉⠉⢳⡀⣀⣀
⠀⠀⠀⢠⣄⣀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢅⠠⠼
⠀⠀⠀⠈⢧⣈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠈⣛⠶⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⡴⠂⠀⠀⢀⣀⠤⡴⠂⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⡳⢛⡫⠿⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠸⣇⢶⠀⠀⢀⡤⠞⠒⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠖⠛⠓⢦⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠈⠈⠙⠚⢁⣤⣒⣲⠆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠏⠀⠀⠀⠀⣳⣀⠤⠤⣌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠛⠋⠉⠙⠿⡽⠉⠒⣤⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠣⡀⠀⠀⣀⢧⡀⠀⠉⢀⠇⠀⠀⠀⠀⠀⠀⠀⡠⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣨⠍⡉⠁⠀⠙⠣⠉⠁⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠣⠤⠆⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢠⠀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣂⠔⠋⠀⠀⠀⠀⢸⣉⡹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⣀⣀⣠⠤⠖⠋⡙⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣀⣀⡀⠀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⡏⡟⠉⠈⠉⠉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⡟⠗⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⡿⠗⢾⣦⣀⣀⡀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⢰⣦⡀⣿⣴⣷⣆⣤⣶⣶⣶⣤⣶⠶⢶⡾⠛⣷⣴⣷⣶⣦⣴⡾⠿⣿⣿⡿⣷⣶⡆
⠀⠀⠀⠀⠀⣿⣷⣽⣿⣿⣿⢤⣿⣷⠿⣿⡷⠶⣿⠤⢤⣿⣿⣷⡾⢿⣿⡶⠞⣿⡏⣻⠾⣿⣇
⠀⠀⠀⠀⠠⠟⠈⠿⣿⠿⠟⠘⠋⠙⠻⠿⠶⠞⠛⠶⢻⡇⠿⠛⠲⢾⢷⠶⠟⢿⡟⠻⢦⣤⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸




⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣾⣆⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣷⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠾⠇⠙⠃⠙⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠋⠘⠋⠸⠷⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⠀⠈⠳⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠞⠁⠀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠅⠀⢀⣀⣀⠈⠀⢀⣴⠶⠞⠳⠶⣦⡀⠀⠁⣀⣀⡀⠀⠨⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠉⠀⣴⠏⠀⠀⠀⠀⠀⠀⠹⣦⠀⠉⠻⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠆⢸⡇⠀⣀⣀⡀⢀⣀⣀⠀⢸⡇⠰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢻⡾⠋⠉⢻⡟⠉⠙⢷⡟⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣴⠟⠁⠀⢀⣼⣧⡀⠀⠈⠻⣦⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡾⠁⠀⠀⣠⡞⠁⠈⢳⣄⠀⠀⠈⢷⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠼⠋⠀⠀⢀⡾⠋⠀⠀⠀⠀⠙⢷⡀⠀⠀⠙⠧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣧⣤⣤⣤⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣼⣧⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠯⠧⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⢀⣀⡤⡈⠑⠤⡀⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⠉⠂⢘⠑⠄⡈⠀⠀⠀⢠⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⢀⠀⠀⠀⠉⠢⢄⡀⠀⠰⠀⠀⠀⠐⡨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢺⣈⠉⢙⠉⠙⠂⢰⠂⠀⠈⠐⠄⢀⠀⠀⠮⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠝⣖⣲⡔⠒⠄⠈⡧⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠉⠁⠂⠀⠒⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢡⠈⠀⠄⠀⢂⠀⠀⠈⠆⠀⠀⠀⠀⠈⠑⠂⠄⡀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡀⠀⠀⠀⠁⠀⠀⠀⢀⢀⣠⠀⠀⢀⠸⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢖⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠈⢄⢀⡠⠒⠉⠀⠠⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠑⠢⢀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣸⠀⠀⠈⠷⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢸⠀⠀⠀⠀⡀⢀⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡄⠓⠁⡖⢏⠁⠀⢓⣣⢷⠐⢂⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠉⢢⠀⠀⠀⠀⠀⠀⡸⣀⡀⠀⡰⠐⠤⢚⠆⠹⣁⡑⠮⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠑⠗⡀⠀⠀⠀⠀⠐⠉⠀⠉⠱⠀⡀⠙⡀⠂⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⡅⠀⠀⡈⠉⠀⠀⠀⠐⠐⠀⡉⢤⣚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠢⠈⢄⠀⠢⢀⠀⠀⠀⠈⠀⠉⠀⠤⢒⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢪⠦⠀⢢⡤⣤⠄⠂⠀⠀⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡢⠤⠧⠤⢥⣀⠒⠊⠉⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠲⠊⢩⡭⣕⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢄⠀⠀⠀⠠⣄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⠈⠉⠱⠀⠀⠀⠀⠀⡼⢁⡤⠚⠀⠦⡀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠈⠀⠁⠒⠢⠤⠤⢈⡶⣍⠉⣁⠒⠠⢅⡤⠊⢀⣮⡤⣄⠙⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠲⠖⠒⠛⠓⠈⠀⢠⣾⣏⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⣀⡤⠚⡍⠀⠀⠀⠀⠀⠀⠀⠚⠑⠒⣂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡊⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠒⣦⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡎⢹⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠆⠗⡗⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃
⠁⡖⠠⠄⣀⠀⣑⣌⡦⠗⢒⠆⢀⣀⣀⠀⠀⢀⠀⠀⠀⡇⠈⠲⣄⡇⢰⠀⠀⠀⣋⡰⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⢸⠐⠴⠑⠈⠳⠤⣴⠁⠘⠒⠒⠒⢒⣒⣒⣺⠁⠒⠒⣷⠀⠀⠈⢙⣁⠤⠔⠊⠉⡨⠇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢓⠄
⢀⣦⡏⠉⠉⠉⠩⡙⠭⢃⠏⠀⠈⠉⠂⠀⣯⣏⠠⢤⢾⢚⣳⢶⡉⠁⠀⠀⠠⠐⠙⠤⡀⠀⠀⠀⠋⡉⣆⣈⣉⠒⠔⡊⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⡴⣡⠜⡛⠀⠀⠈⠓⠖⠁⠀⠀⢀⣠⡼⠗⠀⢀⠀⠭⢥⣄⣈⠹⠘⠀⠀⠠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⡥⠚⠁⠘⡗⠀⠀⠀⠀⠀⠀⠀⠰⠗⠛⣟⠤⣒⠅⠊⠒⠂⠀⠀⠀⠉⠁⠉⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⣔⣐⣒⡷⠠⠤⠀⠀⠀⠠⠤⠄⠒⢺⠉⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣸⢬⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣤⣿⣧⣤⣿⡟⠶⠶⠶⠦⣤⡤⠀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠋⠛⠛⠾⠭⠍⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡄⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠎⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⢆⠀⠀⠀⠀⠏⠑⡄⠀⠀⠀⠀⠀⢈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡂⠀⠇⠀⠀⡘⠀⢠⠁⠀⠀⠀⠀⢀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠒⠂⠐⠂⠠⢎⢀⡈⠀⠀⣰⠷⠀⠊⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠡⠚⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡴⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠒⠀⠉⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠠⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⡄⠀⢌⠢⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠡⠌⠦⢕⠚⠂⠀⠀⠀⡌⠀⠀⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠀⠈⠢⠐⠀⠀⠀⣀⡼⠀⠀⠀⣀⠠⠄⠚⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⣨⢃⡹⠊⠁⠀⠀⣀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢘⡠⢶⠟⢹⠀⠀⠄⠒⠈⠁⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡖⠀⠀⠀⠀⠀⣸⠂⠁⠄⢜⠁⠀⠀⠀⣀⣀⣀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢡⡶⢀⠀⠤⠞⢙⣈⣤⣠⢭⠉⠁⠒⠤⢠⢡⠀⠀⠤⠠⡔⠠⠀⡠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡼
⠈⢆⠀⠀⠀⠀⠀⣐⡏⡀⠘⠀⠀⠀⠀⣈⡀⠀⠀⠀⢀⠟⠄⢊⣀⠐⡈⠑⣀⠌⠀⢠⠂⢄⠀⢀⠂⠀⠀⠀⠀⠀⡇
⠀⠘⣆⠀⠀⠀⡰⢹⡠⠉⠙⠂⠨⣉⢡⠀⠈⠦⠒⡖⠁⠀⠀⠀⢠⠀⠀⠉⠓⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⢄⠀⠀⠄⠃⢇⠀⠀⠀⠀⠀⡄⢰⡄⠳⠈⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠇
⠀⠀⠀⠀⠒⠰⠀⠐⣚⣧⣐⣀⣀⣀⣈⠁⠈⠊⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⡰⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠏⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⡴⠶⠋⠉
⠀⠀⠀⠀⠀⠀⢠⡶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠞⠛⠉⠀⠀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⠋⠉⠀⠀⠀⣀⣀⣤⡴⠾⠛⠋⠉⠁
⠀⠈⠉⠁⠉⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠟⠛⠋⠁⠀⠀⢀⣀⠤⠤
⡀⢈⠁⣀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣂
⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣠⢴⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢛⣡⢶⣍⠉⠙⠙⣿⡷
⠀⠀⢀⠀⠀⡀⡀⢸⡧⠤⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠈⠀⢸⠛⠶⣯⣄⣠⠾⠋
⠀⠀⣿⣿⣶⣶⡇⢸⡇⢠⣶⠆⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠈⠀⣜⠉⣙⣿⣳⠄
⠀⠀⣿⣿⣿⣿⡇⠀⡇⠀⣀⢠⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣤⡀⢰⣿⡝⠋⠙⠉
⠀⠀⣿⣿⢿⣿⡇⠀⡇⠀⠀⠀⠀⣿⡄⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣷⠤⢈⣠⡂
⠀⠀⣿⡿⢾⣿⡇⠀⠡⠴⠒⠚⠉⠉⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣷⡀
⠀⠀⠙⠓⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠲⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠂⠀⠈⣩⣤⣤⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡟⠀⠀⠀⠀⢀⣿⣿⡖⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠄
⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠈⡿⢋⣾⣿⣿⣿⣿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢤⣍⠛⣤⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⡀⠠⣄⡈⠀⣲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⠃⠲⠂⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠒⠶⣶⡶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⡀⢇⣾⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢼⣿⣿⣿⣿⣿⣿⣄⢸⣟⡴⢶⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡏⠛⢻⣿⣿⣿⡿⠻⣿⣿⣿⠮⠧⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠒⠢⠄⡐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⢀⠄⡤⠂⠚⠁⠀⢀⠀⠁⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡀⡠⢀⢄⠈⠀⠀⠋⡄⠀⠀⠈⢷⡀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡿⠀⠀⠀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠙⣦⡀⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠈⠉⠉⠀⠀⠀⠹⢧⡀⠀⠀⠈⠛⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠉⠳⢄⠀⠀⢄⠀⠀⠀⠀⡿⠀⠀⠀⠀⠘⡇⠘⠄
⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⢣⠀⠈⣇⡒⠶⠞⠁⠀⠀⠀⠀⠀⢷⡰⣦⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⣄⡐⠶⢄⡀⠀⠀⣄⣸⠃⢀⠤⠤⢾⡀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠻⣝⠛⠾⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡧⢀⡵⠀⠀⡏⠓⠢⡅⠀⣠⠚⣷⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣾⠃⡀⢠⣬⣷⣄⡇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢊⠥⠚⠉⣠⡞⠐⠠⣀⠀⠰⢿⠁⠀⠹⡜⠄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢿⡀⠈⠛⢷⡇⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠺⠷⣶⣶⣿⣄⣀⣀⠀⢱⣧⠙⣧⠀⠀⠈⠦⠠⠄⠀⠀⠀⠀⠀⣸⡃⣀⣀⣙⣧⠀⠀⣾⣷⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠉⠉⢙⣻⢷⣿⠀⣿⣧⠀⠀⠀⢨⡖⢀⠂⠤⣤⣤⡿⠋⣿⠉⠉⠛⠂⣼⠃⢻⡄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⣾⣋⣶⣿⣯⣄⣼⠘⢧⣄⠀⡾⠀⠘⣿⣿⣶⠟⠀⢀⡇⠀⠀⠀⣼⠁⠀⠈⣷⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠈⣽⠏⠙⣻⡟⠛⠛⣿⠿⢧⡀⠀⢸⣿⣿⡀⠀⢸⡇⠀⠀⣰⡏⠀⠀⠀⠈⠂⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⣸⡏⠀⣰⡟⠀⠀⢈⣙⢶⡿⣶⣄⠉⣉⣉⡷⠀⠸⢇⣆⢰⡟⠀⠀⢀⣀⣶⡶⠶⣆⠤⡀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⠟⠀⠀⠀⢸⡿⠻⡉⠛⠛⠉⠉⠉⡇⠤⠤⣠⣷⡶⠟⠛⠛⠉⠉⠀⠀⠀⠈⢳⡌⠳⣄⠀⠹⣆
⠀⠀⠀⣠⠋⠙⠛⢿⠀⠀⠀⠀⠀⠀⢀⡠⢁⣟⣴⠿⠶⠶⠶⠶⢾⠁⡴⠀⢠⠀⠀⠠⠤⠗⠂⠀⠈⠳⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠎⠿⠧⠀⢻⡄
⠀⠀⣴⠃⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠘⡄⠓⠀⠀⡃⠀⠠⠤⠄⠂⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠀⠀⢫⡓⠒⠒⠀⠀⠀⠀⠀⣰⡿⠀⢀⣀⣀⣠⣤⣤⠶⠖⠒⠚⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠓⠓⠲⠖⠒⠒⠊⠉⠙⠛⠛⠛⠋⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠒⠛⠉⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣎⣁⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠚⠧⠴⢋⡞⠀⠀⠀⠀⠀⢢⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⠃⢀⠠⠀⠀⠉⠓⠲⠤⣤⣀⡀⠃⠀⠈⢆⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢺⣶⣶⠦⠴⣦⣀⣤⠤⠒⠲⣼⠉⢹⠖⠀⠈⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⠁⠀⠀⠰⠿⢿⣭⠟⠁⠀⠀⠀⣴⠃⢀⡏⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⣠⠤⠀⣀⣠⣔⣊⣡⡴⠚⠛⠒⠦⣬⣀⣂⡶⠃⠀⢿⣶⡞⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⣙⠦⢄⡀⠀⠀⠀⠈⠉⠉⠉⢉⣭⠉⠁⠈⢷⠲⠚⠉⠉⠀⢀⡞⠘⠀⠀⠀⠀⠀⠀⠠⡀⠸⡄
⡴⠞⠽⣤⡔⠊⠉⠑⠒⣶⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡓⣒⡒⣦⠀⠀⣸⠀⠀⣧⠤⠃
⠈⠳⣄⠉⠳⣄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⣩⣍⣡⠯⠤⠴⠋⠋⠉⠁
⠀⠀⠈⠳⣄⠈⠳⣄⠀⠈⣯⠙⠲⣄⡀⠀⠀⡤⠖⢢⠀⠀⠀⡜⠉⠉⠓⠊
⠀⠀⠀⠀⠈⠢⡀⠈⠳⣄⠘⣆⠀⠀⠙⢦⡜⡁⠀⣠⠃⠀⡜
⠀⠀⠀⠀⠀⠀⠈⢢⡀⠈⠳⣜⣆⠀⠀⠀⠙⢶⡋⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠈⠛⢦⠀⠀⠀⠀⠱⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢳⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢲⠀⠀⠀⡇⢎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠱⡄⠀⠀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣋⣩⣀⠀⠀⣧⠼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠹⡄⣠⠟⠚⠓⠒⠒⠒⠒⢲⠶⡶⠒⠒⠒⠒⢲⡖⠒⠋⡧⠽⣎⣧⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠋⠀⠀⠀⡶⠂⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡤⠖⠒⠙⠃⠀⠙⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠁⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠤⠤⠤⠤⠤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣘⠀⢡⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢯⡉⠭⠘⣦⠀⣧⣉⠀⠒⢢⠤⣀

⣿⣿⣿⣿⡜⣿⡌⠻⣦⣿⣧⡹⣦⡙⠻⢷⣶⣤⣬⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⡿⢟⣫⣾⣯⣶⣿⡿⢃⣴⣟⣴⠟⠀⣴⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⠘⣿⣆⠈⠻⣿⣿⣾⣿⣶⣤⣀⠉⠙⢿⣯⠙⣿⣿⣿⣼⣿⣿⣯⡾⢛⣩⣽⣿⣿⣿⣿⣿⣿⠟⢁⣴⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡆⠘⣿⣧⠀⢹⣿⢉⡉⠛⠻⢿⣷⣄⢸⣿⣀⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⡿⠟⠋⢉⡉⣿⣏⣼⣿⡿⠋⣼⡿⣸⡿⣿⣿
⣿⣿⣿⣿⢿⣿⣄⠘⣿⣇⣿⡇⣿⣙⡓⢦⡀⠉⠻⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⠟⠁⣠⡴⡛⢍⣷⢹⣿⡿⠋⣠⣾⠟⢀⣿⢣⣿⣿
⣿⣿⣿⣿⣧⡻⣿⣦⠸⣿⣿⢸⡇⡟⣖⢶⡙⣦⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡼⠕⣶⡿⣦⢹⢸⣿⣣⣾⠟⠁⢀⣾⣣⣿⣿⣿
⣿⣿⡜⣷⡙⢷⣝⣿⣷⣿⣿⢸⡇⣧⣜⣛⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣕⡻⣾⣶⢸⢸⣿⡿⠋⣀⣴⣿⣿⡿⣻⣿⣿
⣿⡿⣿⡸⣿⡌⠻⣿⣿⣿⣿⣯⣧⢫⣮⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣽⡅⣾⣾⣿⣷⣿⣿⡿⢛⣥⣾⡿⣿⣿
⣿⢿⣾⣿⣿⣿⣦⣌⣻⣿⡛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⣛⣿⣿⣿⣾⣿⡿⣫⣾⣿⣿
⣿⠷⠄⠉⠛⢿⣿⣭⡉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣽⣿⣿⣿⡿⠿⠛⢻
⣿⣛⣛⣭⣽⣿⠿⠿⠿⢦⣄⠀⠀⠀⠀⠀⡞⠁⠀⣿⠻⣆⠀⠀⠀⠀⠀⡴⢛⣯⠀⠙⣇⠀⠀⠀⠀⠀⠠⠾⠿⣿⣿⣿⣿⣿⣿⡿⢿
⣿⣿⣿⢿⣿⡿⠟⣛⣛⠃⠈⠀⠀⠀⠀⠀⢳⡀⠰⣿⠀⢸⡆⠀⠀⠀⢸⠁⢸⣿⠀⢠⠇⠀⠀⠀⠀⠀⠀⢙⣛⣛⠿⣿⣟⡻⠿⣿⣿
⣿⢋⣴⡿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣿⣠⠞⠀⠀⠀⠀⠈⠧⣄⣿⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣗⠦⣿
⣿⣟⣩⣾⣿⠏⣀⣤⠴⣲⣶⣶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡯⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⢶⣶⣶⢦⣤⡈⢻⣿⣿⣿⣷⣾
⣿⣛⣿⣿⣏⡾⠋⣠⡾⠋⠁⠀⣴⣿⠏⠛⠶⣤⣀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⣀⣴⠾⠛⢹⣿⣆⠀⠈⠙⢷⡄⠙⣦⣿⣿⣹⣿⣿
⣿⣿⣿⢿⣿⡇⣰⠏⠀⠀⠠⠞⠋⣼⡄⠀⠀⠀⠉⠛⠓⠒⢦⣹⡿⣅⠶⠒⠚⠋⠉⠀⠀⠀⢨⣇⠙⠣⡀⠀⠀⠙⠄⢸⣿⣿⣻⡟⣸
⣿⣏⣇⠈⣿⡇⠋⠀⠀⠀⠀⠀⠘⣹⣥⠀⠀⠀⠀⠀⠀⠀⣰⠃⣇⢻⡄⠀⠀⠀⠀⠀⠀⢠⣼⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣷⣿
⣿⣿⣿⡄⠸⣷⠀⣄⠀⠀⣀⣠⡾⢻⣿⠀⠀⠀⠀⠀⠀⣠⠏⣰⣿⡄⢻⡄⠀⠀⠀⠀⠀⢸⣿⡻⢦⣄⡀⠀⠀⡄⢀⡿⢹⠃⣿⣿⣿
⣿⣿⣿⣿⣄⠘⠇⢹⡷⣤⣁⣠⣴⡿⢿⣴⠀⠀⠀⠀⢰⠏⣴⠃⡇⠹⣆⢻⡄⠀⠀⠀⢸⣾⡿⣿⣦⣀⣀⣤⢾⡇⠼⠁⠀⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣦⣤⣽⣦⣶⣟⣉⡀⠘⣿⣰⠂⠀⢠⢏⡼⠁⠀⡇⠀⠘⢦⢳⡀⠀⠰⣸⣿⠁⢀⣙⣷⣦⣶⣯⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣝⢿⣿⡌⠉⢙⣿⣿⣿⣿⣭⣉⠻⣦⠸⣿⠀⠀⢸⣿⠀⠀⠀⣧⠈⠀⢨⣿⡇⠀⠀⣿⠁⣴⠛⣩⣽⣿⣿⣿⣟⠋⠉⣰⣿⠟⣩⣿
⣿⣿⣧⠙⣿⣤⣿⣿⠻⣿⡙⣿⣿⣧⣹⡆⣿⢦⡀⠸⣿⡇⠐⠂⡀⠐⠀⢸⣿⠁⣠⠾⣿⢸⣇⣾⣿⣿⢫⣿⢛⣿⣷⣰⡿⠃⣾⣿⣿
⣿⣿⣿⡆⠘⣿⡿⣿⣆⣿⣷⣿⡏⢿⣿⣷⡿⠀⠙⣆⣿⣧⠀⠀⠛⠀⠀⣿⡏⡴⠃⠀⣿⣿⣿⣟⢻⣿⣾⣿⣼⡿⣿⣿⠁⢸⣿⣿⣿
⣿⣿⣿⣇⠀⢿⣧⡊⣿⣿⠛⣿⣧⣾⣿⣿⠁⠀⠀⠸⣿⣿⠀⠇⠀⠈⢰⣿⣿⠃⠀⠀⠻⣿⣿⣷⣾⣿⠙⣿⣿⢑⣿⡏⠀⣾⣿⡿⢸
⣿⠸⣿⣿⣄⠘⣿⣿⣿⣩⣈⣢⣬⣿⣿⣟⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣥⣅⣲⣙⣿⣿⡿⠁⣰⣿⡿⢃⣼
⣿⣿⣾⣿⣿⣷⣤⣭⣛⣛⠿⢿⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣼⣿⠇⠀⠀⠀⢰⣿⣿⣿⣿⠿⢟⣛⣩⣥⣴⣾⣿⣿⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠈⠁⠁⠀⢠⣖⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢰⠞⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⢛⠂⠼⣇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠁⠀⠀⠀⣠⡦⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠠⠀⠀⠀⢀⠴⠚⠋⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠘⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡈⣠⣠⠁⠌⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⢱⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠠⢀⣿⡇⠀⢀⣄⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢹⡀⡀⡆⠀⠀⢳⣘⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣏⡀⠀⢡⠀⠀⠀⣿⣯⠀⠀⢀⠀⢀⡈⠃⢹⡇⠀⢠⣸⡏⡇⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡄⢸⠻⠃⠀⠈⡧⠀⢸⠛⠿⠃⠀⠀⠛⠉⠁⠀⠈⠃⢠⢘⠾⠃⡁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠐⠀⠀⠃⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡄⣧⠀⠀⠀⠀⠑⠿⠓⠚⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠁⣿⡈⠀⠀⢠⡤⠤⠤⠤⠀⣀⡀⠀⠀⠘⠁⠀⠀⣰⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠈⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣿⡆⠀⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⣷⡀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠀⢿⣿⠀⢸⣷⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣮⣀⠀⠀⠀⢰⣂⣤⡴⠾⠋⣽⣿⡿⠗⠂⠀⠉⠐⢻⠿⣄⢸⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⣏⡿⢿⠟⠛⠛⠋⠉⠉⠠⠀⠀⢰⣿⠓⠊⠁⠀⠀⠙⠲⣾⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡁⢠⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢨⡇⢠⡀
⠀⠀⠀⠀⠀⢚⡶⣦⠄⠀⠀⣤⣶⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢸⠁⡜⠁
⠀⠀⠀⠤⣤⠏⠸⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⡼⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣿⠁⠀⠔⠂⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡎⣹⣀⠌⣿⣻⣿⡄⠀⣰⠆⣤⠔⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡰⢹⣧⣼⣿⣿⣿⣯⡷⢋⣼⠒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣷⣾⣿⣿⣿⣿⣏⢮⡏⢶⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⠃⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⠑⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣉⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⢄⣤⣄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠁⠀⠀⣀⣨⣿⣿⣷⣤⣤⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠹⠿⠛⠛⣿⣿⡏⠙⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣫⣭⣤⣤⣬⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣝⠛⠳⡄
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡀⢁⠈⠹⣏⣿⣿⣿⣿⣿⣿⡇⣹⠇
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠙⠻⣿⣿⣿⣶⣄⢻⣜⣿⣿⣿⣿⣿⣳⣏⣀⣤⠃
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣰⣥⣦⣴⣴⣾⣿⣻⣿⠿⠿⠿⠿⠿⣿⣿⣿⡿⠿⡿⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢧⡟⠀⠀⠠⠖⠛⠛⠶⢦⣤⠀⠀⠀⢀⡴⠟
⠀⠀⠀⠀⠉⠐⠀⠤⠀⣀⠀⠀⢀⣿⡿⢿⣿⣯⡾⠁⠀⠀⠀⠀⢠⣒⡀⠀⠀⠀⠀⠀⣼⡔⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠋⠱⠋⠋⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⡏⠁⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡍
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠓⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠛⠀⢉⣈⣿⣦⣈⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡌⠀⠀⠀⠀⠀⠀⠰⠓⣴⣋⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣤⣶⣶⣤⣶⠀⡘⠀⠀⠀⠀⠀⠀⠀⣶⡊⢠⡈⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣦⣴⣶⡇⡰⠀⠀⠀⠀⣀⡀⣀⡀⠈⢿⡧⠈⠶⣀⢀⡠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠢⡀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠾⢃⠁⠀⠀⠀⠀⠈⠳⠀⠙⠇⠀⡀⠀⠈⣤⠉⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠿⣿⣿⠿⢿⠿⡿⢿⣿⠐⠀⠀⠁⠂⠤⠀⡀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠄⠁⠀⢠⣡⠌⠘⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣄⠘⡀⢀⣠⣴⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠒⠠⢤⡑⢤⠞⣄⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠈⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡿⠿⠿⠿⡟⠉⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠂⠢⣀⣀
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⢄⠈⠁⠐⠒⠤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠤⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠔⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⠆⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰

⠀⢀⣤⣀⠀⠀⣠⡀⠀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀
⢰⣿⣿⣿⡆⢸⣿⣿⠀⣿⣿⣿⣿⣄⣿⣿⣿⢿⣿⡿⠿⢿⣿⡿⣿⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⢸⣿⠿⠿⢿⣿⠿⠿
⣿⣿⠀⠻⠟⣼⣿⣿⡄⣿⣿⢀⣿⣿⣿⣇⣀⠸⣿⣇⡀⢸⣿⡇⣹⣿⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⢸⣿⣀⠀⢸⣿⣀
⣿⣷⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡿⠁⣿⡿⠟⠀⣿⡿⠃⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⢸⣿⠛⠀⢸⣿⠛
⢿⣿⣄⣶⣾⣿⡟⣿⣿⣿⣿⣿⣿⡀⣿⣇⣀⣨⣿⣧⣄⣸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⡀⣿⡇⢸⣿⠀⠀⢸⣿⣀⣀
⠘⢿⣿⡿⣿⣿⠃⢻⣿⣿⣿⠹⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡀⠀⠀⠀⡆⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⡇⣸⣿⠀⠀⣾⣿⣿⣿
⠀⠀⠉⠀⠉⠉⠀⠉⠉⠩⠍⠤⠩⠭⠩⠉⠉⠍⠭⠩⠉⠉⠭⠁⠌⠭⠅⠀⠡⢠⠥⠠⠄⢤⡀⠤⠠⠄⠀⠀⠀⠠⠄⠨⠭⠉⠉⠩⠭⠭⠁⠭⠩⠤⠠⠭⠭⠍⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠟⠛⠛⠛⠛⠛⠛⠻⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣀⡀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢿⣍⣉⣉⣉⣽⡟⠁⠀⠀⠀⠀⠀⠻⣯⣉⣉⣉⣩⣿⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠛⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⡶⠶⠿⠿⠶⢶⣦⣽⣷⡀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⢶⣦⣤⣤⣤⣤⣴⡶⠾⠟⠛⠉



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

⠀⠀⠀⠀⠀⣀⣤⣤⠶⣤⣄⣀⣤⠾⠛⠛⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⣤⣄⣀⣴⠞⠛⠛⠳⢦⣄
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠙⠷⠶⠶⢦⣄⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⣤⣀
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⢻⣄⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⣸⠇⠀⠈⢷⡀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡴⠿⠛⠛⠛⠛⠛⠛⠛⠷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⢶⣄⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣴⢦⣀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢶⡾⠁⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠉⠛⠳⠶⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠟⠉⠻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡏⠀⠀⠀⠀⠉⠙⠛⠻⢶⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣨⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠜⠃⠙⠓⢦⠀⢠⡎⢣⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠻⢦⣄⣀⢀⣀⣠⡶⠋⠀⢻⡄⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣿⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣿⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⡄⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣠⠗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠺⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠍⠀⡇⠈⢃⢸⠈⡏⠁⡜⣃⠀⡯⣕⢸⠀⠀⣯⠁⠀⠐⢍⡀⠉⡏⢸⣅⡝⢰⠉⠙⡆⣧⡎⠀⡍
⣀⣀⣀⣀⣀⣀⣀⣀⣼⣤⣤⣧⣤⣃⣼⣀⣧⣰⣉⣹⣤⣦⣼⣨⣤⣤⣧⣤⣀⣠⣤⣇⣠⣇⣸⣄⣵⣘⣦⣴⣁⣇⣹⣤⣧⣤⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣆⡤⠔⠒⢈⣽⣋⡤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠡⠞⠁⠀⠀⠈⠉⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡼⠀⠀⠀⠀⠀⠀⠀⢛⣉⡠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠧⣀⢀⣴⠒⠒⠲⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠀⠀⢰⠃⠀⠀⠠⠂⡜⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣰⡁⠀⣇⠀⠀⡠⠔⠓⠒⢤⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠻⣿⠀⡎⠀⠀⠀⢀⣾⣄⣀⡠⠓⠒⠢⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢉⣉⠛⠟⠀⠑⠦⠘⠻⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠈⣀⡄⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⢀⠈⡧⡾⠉⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⣥⠀⠀⠀⡏⠉⠁⠛⠓⠒⠒⠒⠀⠀⠒⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠸⡆⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢻⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⡠⠤⡀⠰⠒⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢠⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⢇⠀⠑⠸⡀⢰⠀⡰⢁⣍⡓⠦⣀⣤⣀⠀⡵⠠⣙⠇⠀⢀⠜
⠀⠀⠀⢰⠋⠛⢶⣄⡐⠧⢸⡰⢁⠂⠀⠉⠓⡞⠀⠈⠉⠳⡀⠙⢦⣀⣈⣣⣄⣀⠀⠀⠀⢀
⠀⠀⠀⠈⠑⠢⠤⠬⠥⠤⢄⡀⠚⡤⠀⠀⠘⠳⣤⡀⠀⢀⠳⠔⢣⡙⢹⢻⠁⠀⣷⡨⠭⠄⡁⠀⠀⠀⡴⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⣅⠀⠀⣠⠛⠉⢧⣠⠾⠠⠄⠀⣙⠣⠤⡆⡰⠁⠀⠀⠀⠀⠀⠀⡰⠁⡰⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⢄⡨⢶⡊⠫⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡠⠄⠲⠊⣠⠴⠓⠉⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣽⠖⠦⣬⣉⣐⣀⠀⠀⠤⠄⠀⠐⢂⣹⠀⣀⡼⢃⠴⢤⡐⠶⠶⠦⠖⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣀⠀⠶⠀⢈⡁⢽⡏⢉⡉⠻⠟⢉⣉⡟⠸⣀⡠⠖⠀⠀⠈⠓⠤⣀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⠀⠛⠁⣤⡄⠘⠃⣠⡄⠘⠋⢠⣤⠘⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⢆⣉⡁⣼⡟⠛⣟⣻⣿⣉⠩⠴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣒⠒⢒⣒⡀⣴⠛⡿⠭⠭⠭⢥⣏⣿⠠⠤⠔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠉⠉⠉⠉⠉⢹⣂⠊⠉⠉⠩⢤⠖⠃
⠀⠀⣀⣀⣀⣀⣠⠄⠴⡒⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⡒⠶⠒⣲⣄⣀⣀⡀
⠴⡭⠥⠀⠀⠈⠭⠭⠙⠴⠤⣤⠤⠤⣤⣤⣀⣀⠀⡀⠀⠀⢀⣠⣤⠶⡛⣩⣩⣁⣀⣀⣀⣀⡽⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡉⠙⣁⣀⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⠟⠀⠉⠉⠉⠛⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠰⠆⠀⠷⣄⠀⣠⡀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣾⣿⡇
⠀⠀⠀⡇⠀⠀⠀⠛⠃⠀⠀⢀⡙⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠙⠛
⠀⠠⠴⠷⠀⠠⠘⠱⣶⠀⠀⠚⠳⣤⣄⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠄
⣿⠀⠀⠀⠀⢶⣄⣀⡀⠃⠀⠀⣀⣀⠛⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠶⠆⠈⠉⠉⠀⠀⠀⠀⠀⠀⣠
⠙⠶⢠⡄⠀⠀⠉⠉⠁⠀⠐⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⣀⡀⠲⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⢸⣇⠀⠀⢠⣿⣦⣤⢤⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠈⠙⠛⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡀⠶⠂⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠦⣀⣀⣀⣀⣀⡠⠶⠶⠊⠉
⠀⠀⣤⣶⠀⠀⠺⣿⣶⣦⣀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠈⠁⠚⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢠⠀⠈⠉⠈⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠟⢡
⠀⠀⠀⠀⠀⠘⠷⢤⣤⣤⡀⠺⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⣾⡆⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢠⣿⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⢸⣿⣇⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠁⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡗⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠺⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣇⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣀⣀⣀⡙⠛⠀⠛⣃⣀⣀⣠⣶⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣟⠉⠁⠙⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣠⣤⣤⣤⣤⣤⣄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠶⢟⠓⠄⡀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠉⠉⠑⠧⣼⠂⣀⠑⢤⠙⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠙⠦⣁⠰⠁⠀⢀⠕⠂⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⢤⠤⠤⢤⠠⣄⡀⡈⠑⠂⢄⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⣄⡠⠐⠀⢧⠀⠀⠀⠀⠐⠲⢌⠉⠒⠂⠤⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠑⣏⠂⢠⠀⢸⢀⠄⠀⠄⠀⠀⠀⢙⠂⠄⡀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠈⠂⠊⠀⠈⢁⡈⢃⠆⠀⠀⢠⠌⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠞⠨⠥⠤⠺⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢖⠠⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠀⠱⣤⡔⠈⠀⠀⠂⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⡀⢈⡘⠝⢧⡀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢝⢺⠀⠀⠀⠛⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣻⡘⢄⢤⣤⡴⠄⠢⠤⢢⠥⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⡀⠀⠀⠀⠀⠀⢸⠃⠀⠈⡃⢋⡩⠍⠹⡛⠣⣸⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢖⠺⣄⠀⢀⢀⠐⣡⠛⠉⠐⠧⢀⠐⠧⠤⢾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⠢⠻⢿⠈⠁⢨⠛⠂⠀⠀⠣⠜⢑⣃⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⢄⠣⠀⠱⢄⠀⠀⠀⠀⠀⢋⠠⠔⡻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢑⡵⠒⢺⢄⡶⠖⠀⠉⣀⠔⡪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠕⠒⠚⠒⠲⠦⠉⠙⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠃⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠙⠛⠿⠆⠀⠉⠉⠉⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡏⠀⢀⣀⡀⣭⣍⣁⠀⠀⡄⠀⠀⠀⣀⣤⡤⣀⣀⡀⠀⣸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠉⣥⠭⠤⠤⠤⠤⠿⠿⠬⢇⢰⠁⡠⠤⠮⠤⠿⠧⠤⠤⠤⢤⠍⠈⠙⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠐⠪⣦⡀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢃⠀⠀⠀⠀⠀⠀⠀⡴⠁⢠⠖⠂⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣀⡇⠉⠒⠒⠒⠒⠒⠒⢹⠀⠀⠈⠓⠒⠒⠒⠒⠒⠋⠀⠀⢸⣀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠴⢗⣚⣉⠝⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣄⣀⣀⣀⣀⣠⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠛⠻⠿⠿⠿⠿⠟⠛⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⣀⣀⣀⣀⣀⣀⡠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣴⣾⣿⣿⣿⣷⣦⣴⡀⠀⠀⠀⠀⠠⠤⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⢁⣈⡆
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⠏⠀⠀⠀⢹⢷
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⣠⡄⠀⠈⢸
⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢹⣿⡆⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠛⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠉⠁⠉⠉⠙⠛⠛⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣀
⣿⠿⠿⢿⠿⠿⠿⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⡿⢿⠛⣿⣿⡟⠿⠿⠿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢄⡞⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⣸⡋⠡⠀⠀⠀⠀⠀⠀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠛⠋⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠰⠶⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠁⠀⠑⠄
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠜⠋⠉⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠀⠀⡿⠁⡘⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⡀⠀⠐⠒⠀⠂
⠀⠀⠀⠀⠀⠈⠓⠦⠄⠄⠒⠚⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⣄⠀⠐⡀⠀⢲⢀⡀⠀⠀⣦⣠⣀⣰⢄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠈⢢⢼⣇⣠⣇⣤⡴⢠⠂⠀⡄⣝⠉⡋⢨⣸⠄⠀⠀⠀⡀⡤⢤⣠
⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠢⣡⡰⠋⠁⠀⠀⠈⢣⠊⠀⠀⠀⠌⠑⠃⠊⠀⠀⠀⠀⠰⠆⠀⠀⠈⠻⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢇⠀⠀⠁⠀⡄⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠻⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠲⠚⠄⠀⠀⠀⠀⠀⠀⠉⠐⠑⢂⠀⠀⠀⠀⠀⠐⠚⠒⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡰⠾⢿⣿⣷⣿⣿⣷⣾⣷⣶⣤⣤⣦⣄⠀⣀⢠⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⣤⣀⣀⣀⣀⣄⣀⡀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣄⠙⠻⣦⡳⣝⠄⠀⢄⠳⣽⣶⡐⢮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠉⢽⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣾⡛⢦⢀⠘⢿⣮⠂⡰⠀⠱⣼⡝⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣟⡿⣦⡁⠳⣄⠻⣿⡌⢦⡀⠱⠻⣶⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣽⣿⣿⣿⣿⣿⣶⣝⢿⡿⣷⢾⠟⣦⠘⠢⡘⢷⡜⢿⡦⡘⠙⢿⣎⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠣⣹⡿⣿⣿⣿⣿⣿⣧⡙⢬⣷⣵⣌⠳⣄⠈⠂⠻⣆⠹⣄⠀⠬⢻⣷⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣽⣿⣿⣿⣿⣿⣿⣧⣙⢿⣯⠳⡘⢧⡀⠁⠘⢦⡈⢷⡀⠑⢿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⡄⡀⠀⠀⢀⠀⠰⣿⣦⡭⡿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣜⠺⣿⣆⠀⠈⠻⡄⠳⠀⠀⠉⣹⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠠⡀⠛⢷⡄⠀⠙⣆⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠐⢶⣶⣶⣶⣶⣿⣿⣿⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⡄⠙⠷⡤⠈⢳⡀⠀⠀⠀⠙⡛⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡂⠈⠀⡀⠙⢦⡀⠀⡀⠈⠝⣧⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠻⣿⣿⣯⠽⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠈⠠⡈⠙⢆⠀⠀⠀⢻⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⣿⣧⡙⣿⣛⣿⣷⣿⣋⠻⠹⣿⡟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠂⠀⠀⠀⠀⠀⠹⣟⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣿⣿⣧⠈⢧⡙⠻⣿⣿⣦⠀⢼⣧⠘⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡐⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠑⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⢆⠀⠐⡀⠀⠀⠀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣸⣦⡀⠀⠀⠀⠀⠀⠀⠐⣤⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⠁⡀⢾⡀⠀⠀⠈⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢠⣸⣶⣦⣶⣾⣿⣿⣷⣦⡄⠀⢠⡀⣤⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠐⠌⣷⡀⠀⢧⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⢟⡿⣛⣛⣻⣻⢿⣿⣧⠹⠀⣦⢿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣣⠈⢸⣷⡀⠈⢄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠉⢩⡏⠉⡏⠈⠙⠛⠃⠀⠀⠀⠡⠿⣉⣷⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠠⡈⢷⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⣰⣿⣇⣚⣷⣶⢦⣤⠀⠀⠀⠀⠀⠀⢘⢻⡿⣾⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣆⠐⠘⣧⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣿⣿⣿⣿⣿⣿⠿⢿⣿⡷⣤⣶⣶⣤⣬⣯⡾⢿⣿⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠃⠹⣆⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣶⣿⣿⢿⣿⣼⣷⣾⣯⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡄⠀⢻⠀⠈⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠟⠻⣿⣄⣻⣻⣿⣯⣽⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢣⠀⠈⣇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣻⣾⣿⣥⣼⢿⣭⣿⣿⣿⣿⣿⣿⣿⣌⢿⣿⣿⣿⣿⣿⣿⣿⠘⣇⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⣿⣿⣿⣿⣿⣆⢿⡑⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⡜⣧⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⠈⠻⠿⣿⣿⣿⣿⣷⠸⣆⠀⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠉⠙⠙⠋⠋⠉⠀⠉⠉⠉⠉⠻⠿⠟⠛⠿⠿⠻⠿⠟⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠀⠈⠨⢷⠼⠅

⣠⡾⠛⢿⣷⠀⠀⠀⣠⣴⢦⡀
⣿⠃⠀⠛⠋⠀⠀⣼⣿⠃⣸⢃⣤⣀⠀⠀⠀⠀⣤⣀
⠙⢧⡀⠀⠀⠀⢀⣿⣿⠞⠁⠹⠿⠋⠀⠀⠀⠀⣿⣿⠂⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠉⠉⠒⠒⢺⣿⡏⠀⣴⣿⠇⢀⣾⣷⠘⣯⢉⣴⡶⣶⡄⠀⢠⣾⡟⠋⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⣿⡿⠀⣸⣿⡇⢠⡏⣿⣿⠕⠛⠀⠀⠘⠿⠧⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠼⠿⣤⣄⣀⠙⠓⠚⠁⠙⠓⠋⠀⠙⠻⠦⠞⠁⢀⣤⣶⠶⠶⢶⣤⣾⠃⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣌⠛⢿⣦⣠⣶⡄⠀⣠⣶⠛⠙⣷⡆⠀⢺⣿⣅⣀⣀⡤⠟⠙⠻⢶⣤⣶⠀⣙⡿
⠀⢀⣤⡄⢀⣀⢠⡿⠋⣘⡓⠙⣿⣿⣅⣼⣿⣇⣠⣾⠟⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢰⣿⡏⠉⠁
⠀⣼⣿⠁⣿⣿⠀⣿⢸⣿⡇⢠⣿⡟⠉⣿⣿⠋⣩⣤⠀⣠⡶⠒⢢⢶⣾⡆⢠⣾⡗⠀⣿⡿⠀⢀⡀
⠀⢹⣿⣠⠿⣿⣴⠏⠸⣿⣃⡼⣿⣧⢀⣿⣿⡞⣼⣿⢸⣿⡇⠀⣾⢿⣿⠇⣼⣿⠁⢸⣿⡏⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠉⠈⠉⠁⢿⣧⠎⠻⠷⠞⠁⠈⠛⠋⠈⠻⠖⠋⠙⠿⣤⡼⠋⠀⠀⣶⣶
⠀⠀⠀⢀⣤⡶⠶⠒⣶⣄⢠⠞⢹⣿⡄⠀⠀⣀⡀⠀⢀⣠⣤⡀⢀⣶⡄⣠⢶⣶⡀⠀⣀⣀⡰⠛⢻⣿⡟⢛⣿⣾⢻⣷
⠀⠀⢰⣿⡏⠀⠀⠀⣿⣿⠏⣠⣾⡿⢁⣶⣿⢋⣿⢰⣿⡏⣿⣿⢸⣿⣿⣣⣾⠟⢠⣿⡟⢩⡟⠀⣿⣿⠀⢸⣿⣇⠸⠋⢸
⠀⠀⠘⠿⡧⠄⠀⢸⣿⡿⢿⣯⡁⠀⢸⣿⣟⠋⢁⣽⣿⣤⣿⡇⣿⣿⢿⣯⡁⠀⣸⣿⡇⠁⠀⣰⣿⡇⠀⢀⡟⣿⣦⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠈⠻⣧⡤⠞⠉⠛⠛⠋⠀⠀⢸⣿⠃⠙⠛⠀⠻⠧⠞⠉⠻⠿⠶⠚⠹⣿⡇⢀⡞⠀⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠛⠛⢛⣿⡿⣤⣄⡀⠀⠀⠀⠀⠀⠀⢲⣦⡀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣠⡾⠋⠀⠀⠉⠛⠻⢶⣤⣄⣀⣠⣾⡿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣶⣶⣾⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠊⠉⠉⠉⢱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⣁⠀⠀⢀⡀⠀⣀⠀⣺⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠉⠘⢷⠿⠇⢀⡩⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡏⡘⠀⡨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠉⠳⣤⠖⠉⠉⠉⠉⣧⡜⠀⣇⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣀⣀⣶⡜⢀⠀⠀⠀⠀⢠⡏⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠽⢿⣀⣀⣠⠶⠋⠀⠀⣸⠗⣩⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠙⠃⠂⠀⠀⠀⠀⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠠⢀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⠤⠅⠠⢤⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠂⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢼⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠣⠀⠀⠈⢱⣦⡀⠀⠀⠀⠀⠀⠀⡠⢊⡠⠚⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⡠⠊⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠀⠀⠀⠑⣄⠀⠀⠀⢀⡴⠋⠐⠈⠀⠈⢷⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣼⠀⠀⠀⠑⢈⠓⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⣠⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢹⠀⠀⠀⠀⡠⢛⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣄⡸⠀⠀⡠⢊⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡧
⡤⠖⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀⡀⡸⢇⣀⠈⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣀⡨⠤⢴⣾⢿⣀⣀⡀
⠧⢤⣤⣤⣤⣀⣠⣤⣤⣤⣄⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣸⡀⠀⠀⠉⠉⠙⠛⢻⣍⡰⠒⣲⡖⠒⠒⠒⠂⣉⣉⣉⡁⠤⣴⠊⠀⠀⠀⠁⠀⠈
⡏⣠⣤⣤⣤⣤⣤⣤⣤⠀⡏⠈⠉⠉⢹⡏⠉⠉⠉⡿⠚⠛⠛⠛⠒⠒⠒⠒⠚⠛⠟⠓⠶⠶⢾⣼⠓⡿⠘⡁⠁⠀⠀⠀⠀⠇⠘⡄⠀⠈⠟⠇⠀⠀⠀⠀⢠
⡇⠀⠀⠉⠻⢿⣿⣿⣿⠀⡇⢠⠀⠀⢸⡇⠀⠀⠀⡇⢘⠛⠻⣿⣿⣿⣿⣿⣷⠈⠀⠀⠴⠀⠀⡇⢸⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⣄⠀⢀⡶⡴⠋
⡇⢠⠀⠀⠀⠀⠉⠻⢿⠀⡅⢀⠀⠀⢸⠇⠀⠀⠀⣇⢰⠀⠀⠀⠙⠻⣿⣿⣿⢀⠀⠒⠠⠀⠀⠃⠀⢸⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠙⠉⠀⣇
⡇⢸⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠁⢸⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠙⢿⢸⠀⠀⢀⡄⢸⠀⠇⢸⣁⠼⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠱⠴⠶⠖⠋
⠀⠘⠲⠶⠶⠂⠉⠁⠼⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⣀⡄⢸⠀⠀⠀⠀⢸⢠⡇⠸⢷⣶⣶⡒⠋⠙⠛⠉⠉⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠶⣶⣶⣶⣶⣶⣦⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸⠈⠉⠛⠛⠓⠒⠀⠀⠇⢸⠀⠠⠆⠀⢸⣉⡁⠀⣀⣰⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠆
⠀⢠⠀⠀⠙⢿⣿⣿⣿⠀⡇⠀⢀⠀⡘⠀⠀⠀⠀⢸⢀⠛⠿⣿⣿⣷⣶⣶⣆⢸⠀⠀⠀⠀⢸⠀⢈⡿⠿⣏⠙⠒⠂⠶⠶⠂⠲⣶⠶⣎⠉
⠀⢸⠀⠀⠀⠀⠈⠻⢿⠀⡇⠀⠈⠀⡇⠀⠀⠀⠀⠈⢸⠀⠀⠈⠛⢿⣿⣿⣿⢸⠀⠒⠀⠀⠘⠉⠀⠁⠀⠈⠑⣆⠀⠀⠀⣤⠖⠃⠀⠉⠀⠉⠐⠢⠤⠀⢀
⢀⢸⣀⣀⠀⠠⠄⠀⢺⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠻⣿⢸⠀⠀⠀⠀⡇⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⣀⠀⠉⣹
⠸⠤⠭⣄⣀⠒⠒⠂⠘⢐⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⡷⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⢄⣘⠶⢋⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢤⣉⣉⣐⡒⠒⠒⠒⠆⢸⠀⡀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢌⡉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠀⠤⠄⠤⠤⠤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠉⠒⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢪⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠁⠀⡴⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⢀⠎⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⠎⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⡆⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⢀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⡯⠯⠛⣛⣒⣒⣒⣋⣉⣟⡛⠒⠒⣒⣓⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⠁⠐⠒⠒⠀⠀⠤⠤⣷⠤⠭⠤⠤⠤⢼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠬⠍⢁⣀⡐⠒⠒⠒⠘⠛⢲⡶⢾⣋⣉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡑⠒⠤⠤⠤⠤⠄⠀⣀⣀⣀⣈⡇⢠⣄⠤⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠒⢲⢲⠒⠒⠒⠒⠒⠒⡇⠀⠙⣏⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸⠸⡀⠀⠀⠀⠀⢰⠃⠀⠀⢥⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⡇⡇⠀⠀⠀⠀⠈⠓⠒⠋⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣉⠓⠠⠍⠘⠛⣟⠒⠟⠁⠀⠀⢀⣠⠴⠚⠉⣆⡀
⠀⠀⠀⠀⠀⠀⢀⡔⡫⠛⠉⣩⠭⠿⣖⡦⠤⢀⣀⠈⢹⠒⠒⠒⠉⠉⠁⠀⣀⣠⢤⡶⠯⠭⢬⡑⢄
⠀⠀⠀⠀⠀⠀⢸⡚⠦⠾⠥⠧⠤⠤⠬⠅⠂⠨⠶⠾⢭⡯⠽⠯⠽⠿⠭⠭⠤⣈⣉⣸⣓⣀⣀⣙⣼
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠑⠂⠉⠉⠉⠁⠐⠒⠒⠤⠤⠤⠭⠥⠴⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣦⠄⣀⣴⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⡿⢃⣾⡌⢿⣿⣿⣿⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⠛⠛⠻⠿⢿⣥⣿⠛⣿⣦⣿⠿⠿⠛⠛⢻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⡆⢿⡿⢿⣿⣿⣿⠀⣿⣿⣿⣿⢿⣿⢃⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣌⣿⣆⠻⣿⣿⣄⣿⣿⡿⢣⣾⣏⣼⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⢡⣾⣿⣦⠹⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧⡘⢿⣿⠟⣰⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣎⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣌⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣶⣦⢹⣿⣿⣿⣿⣿⣶⣤⣍⣉⣛⣛⣋⣉⣥⣴⣬⣙⠛⠻⠿⠿⠛⠛⠉
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⣛⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠿⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⡄⣤⠤⢠⠶⢠⡄⢦⡤⢠⣤⠠⣤⠄⣤⠦⠀⢠⡄⢠⡄⣤⢠⡄⣤⢠⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢙⡇⣿⣉⢨⣹⢆⡇⢸⡇⣾⠺⡆⣿⠀⣿⣉⠀⣸⢻⠏⡇⣿⢸⠙⣿⢸⣄⡾
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⢀⠀⠄⠊⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠐⠁⠀⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠡⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣴⣿⣭⡁⠂⠀⠀⠀⢰⣾⣍⠃⢂⡀⠄
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠂⠀⠒⠀⠁⠀⢰⡿⠛⣿⡧⠀⠀⠀⠀⢸⣉⣽⠇⠀⠑
⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⠂⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠐⡀
⠀⠀⠀⠀⢀⠠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⣇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⣀⡀⠀⠀⠐⠄⠂⠐⢤⡊⠀⠀⠀⠣
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⡀⡄⢀⣰⠀⠀⠀⠀⠀⠁
⠀⢀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⢂
⠀⢸⣠⠁⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⠠⢆⠀⣀⡀
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠐⠠⠚⠄
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠁⠜⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠⠊
⠀⠀⠀⠀⢤⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠠⠀⠀⠐⠂⠀⠂⠐⠂⠙⠀⠊
⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠠⠠⡀⠥⡀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⠒⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠁⠀⢶⠿⡧⣤⠀⠀⢠⡀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠚⠦⡀⠙⣖⣿⣾⣇⡠⡜⣡⠐⡆⢀⠈⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡌⢠⣳⣼⣿⣿⣿⣿⢦⣟⠁⠴⣅⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣼⣿⣿⣿⣿⣿⣿⣿⡔⠁⣇⠰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⣦⣜⣿⣯⣿⣿⣿⣿⣿⢷⣼⠃⠔⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠞⠩⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠟⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠼⣾⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠀⠀⠀⠈⠿⠿⠟⠋⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡻⢛⡛⡛⣿⢻⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠻⣷⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠹⡇⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⣾⣯⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣶⠇⠀⢸⠀⠀⠀⠀⠀⠀⠙⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢉⠟⡀⠀⠸⡄⠀⠀⠀⠀⠀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠲⢦⣄⣀⣤⠞⠋⠉⠉⠁⠘⠦⣤⡤⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢀⡀⠀⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢆⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠞⠁⠋⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣄⣀⣀⣀⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢠⣤⡤⠴⠶⠲⡆
⠀⠀⢸⣇⣀⣷⠀⠀⠲⣦⡴⠲⣦⡴⢦⡀⠀⠘⢻⡇⠀⠀⠀⣠⢤⣀⣀⣄⣀⣀⣀⣠⣀⣀⣀⠀⢠⡤⣄⠀⢀⣠⣤⣠⠄⣀⣤⣄⣤⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⢻⡏⠉⠿⢀⡄⠀⣿⠀⠀⣿⠀⠀⢻⡄⠀⢸⡷⠶⢦⡀⣁⣤⣿⠈⣿⠁⠈⠋⢸⡟⠁⠙⠁⢈⣠⣼⡇⣿⢤⣤⣿⠘⡧⣤⣼⡇⣴⠋⠛⣦⣴⠋⠙⣿
⠀⠠⠼⠷⠤⢤⣼⠇⣠⣿⣄⢀⣿⣄⠀⣈⣧⣀⢸⣁⠀⣸⢿⣁⢀⣿⣀⣿⣀⠀⠀⣈⣇⡀⠀⠀⣏⡀⣸⡃⡿⠦⠤⠟⢸⡷⠤⠴⠋⢿⡋⠙⡛⣧⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠈⠉⠉⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠻⠶⣀⢀⣀⠈⠉⠉⠑⠊⢻⣷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⣷⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⣿⣘⡀⢠⢹⡄⠈⠻⢶⡀⢀⣀⣀⣠⣤⡤⢤⣹⡇
⠀⠀⠀⢶⣤⡀⠀⠀⠀⠀⢸⣏⠓⠾⠯⢿⣄⣀⡀⠓⡍⠉⠙⠃⠿⢿⣿⣿⣿
⠀⠀⠀⢠⣧⡹⣶⣄⠀⠀⢸⣿⣤⡀⢸⣼⣿⡦⢤⣾⣆⠀⠀⠀⠀⣤⣿⡿⠛⠋⠁
⠀⠀⠀⠀⣿⠶⣶⣻⣷⡀⠘⠿⠋⠻⣆⠁⠉⠈⠀⠉⠛⠃⣀⠀⢠⣿⣿⡶⠀⠘⢀
⠀⠀⠀⠀⠉⠼⠿⣟⠛⢿⣶⣤⣠⣤⣼⣦⣀⣀⡀⠀⠀⣼⠛⢷⠰⠉⠉⠉⠲⠶⢶⡶⣿⣿⣷⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠿⠿⢿⣿⣭⣽⣿⣿⢿⣿⣆⣀⣸⣿⠉⠓⠒⠒⠒⠛⠥⠤⢹⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠿⣿⣿⣿⣅⣀⠟⠛⠿⢿⣿⣶⠄⣀⣀⠀⠀⢀⣴⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠿⠄⢲⣦⣤⣶⣶⣾⡿⠿⢷⣶⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠛⠿⠻⠆⢿⡟⠏⠹⠛⠛



⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠶⠤⣶⣋⣱⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠁⠀⠀⠀⠀⠉⠉⢩⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⢏⣁⣿⡆⠀⠀⠒⠘⠳⣤⡝⠋⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⠘⣿⣿⠇⠀⣴⣤⣤⣴⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⡿⠃⠀⠀⢀⡀⢸⡟⠉⠛⠛⠓⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡻⠋⠀⠉⠁⠀⠀⠀⠈⠉⠁⠀⣀⠀⠀⠀⢀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⢈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡜⢩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⢿⡋⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠘⠋⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠁⠀⠸⣿⡀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⢀⡠⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⡀⠀⠀⣄⠈⣿⡀⠀⠀⠈⠛⢷⣄⡠⠶⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⣷⠀⠸⠛⣆⠘⣿⡀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡏⠉⠀⠀⠀⠻⣶⣼⣷⠀⠀⢰⣿⣥⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢠⡞⠁⠀⠀⠀⠀⠄⠈⠛⢿⣷⣶⣄⣻⣿⣏⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠸⠇⠀⠀⢈⠀⠀⠀⠀⠀⠀⠸⣿⣉⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⢸⣿⣇⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⠀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠘⣿⣷⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⠀⠀⠀⣿⡇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⣿⣇⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠸⠂⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⡄⣀⠀⠀⢻⣀⣾⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢀⡹⠀⠀⣿⡟⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣀⣀⣿⡟⠀⠀⣿⠄⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⠟⢻⡇⠀⠀⠛⣀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⢸⡇⠀⠸⣷⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢸⠁⠀⠀⠈⠀⣿⢂⣀⣀⣀⣀⣀⠀⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠸⣿⣧⣤⣤⣤⣴⣦⣤⣴⣿⠃⢀⡀⠀⢸⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠟⢶⣦⣤⣤⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⣘⠻⠏⠱⣤⣤⣤⣾⡿⠁⠀⠈⠃⠀⣹⠀⠀⠀⢸⣿⣿⣿⣿⠻⠟⠛⠛⠀⠸⠧⠤⣿⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢻⡇⠀⠀⢰⡆⠈⠉⠁⠶⠶⢶⠷⠈⠛⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⠆⠀⠸⣦⡀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⢀⣼⠇⠈⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡆⢰⣌⠛⠉⠉⠉⣿⡉⠉⠉⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠹⣿⡟⢳⡟⠃⠀⣴⠀⠀⠈⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡎⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠿⠿⠿⢶⣦⡤⣾⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⡃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⢀⠈⠌⠹⣿⡄⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣆⠀⠃⠀⠐⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠈⠃⠀⠆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠁⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⡄⠀⡀⠀⠀⠿⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠈⠀⠀⠀⣻⣦⠀⠀⠀⠀⠀⠠⠄⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣄⠀⠀⠀⠰⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⡙⠛⣦⡀⠠⠀⢈⣤⣾⠟⢠⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡘⣿⣿⣿⣶⣤⣤⣄⣀⡀⣰⣄⣀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡶⠁⠀⣹⣷⣤⣿⣿⣿⠏⡰⠊
⠀⠀⠀⠀⠀⠀⠀⣼⣄⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⠁⠀⠀⣀⣸⠀⠘⣀⣠⣾⣿⣿⣿⣿⢿⡧⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣤⣌⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣋⣬⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠙⢿⠿⢿⣿⠿⣿⣿⣿⣾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣤⣄⣀⠛⠋⠬⠟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⡏⠀⣿⣿⣿⠇⠀⠀⣶⡾⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⠁⠀⢠
⣿⡇⠀⢸⣿⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⢸⣶⣿⠾⠃⠀⠀⠀⣼
⣿⡇⠀⠘⣿⣇⠀⠀⠀⠊⡀⠀⠀⡀⢨⣶⣼⣮⣶⣅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣷⠾⠇⠀⢀⣴⣿
⣿⣿⠀⠀⠸⣿⠀⠀⠀⢀⢴⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠄⠀⠀⠀⠰⠿⠛⣋⣿⣷⣶⣿⠾⠿⠟⠁
⠹⣿⣷⡀⠀⠹⣄⠀⠀⠘⢺⣿⣿⣿⣿⣿⣿⡿⣿⠿⠗⠀⠀⠀⠀⠀⠀⢠⠀⠀⢲⡿⢟⠛⣿⡀⠊⠁⠀⠀⣠
⡌⠹⣿⣇⠀⠀⠈⠣⡀⠀⠀⠈⠉⡻⠛⠿⠛⠃⠁⠀⠀⠀⠀⠀⠁⠈⠄⡰⠁⠄⠀⠀⠀⠀⢻⡇⠀⢀⣠⡶⠿
⣿⡀⠀⠻⣇⠀⠀⠀⠴⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡇⠀⠀⠀⠠⠀⠀⢁⡤⠀⠀⠁
⣿⣧⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠈⠐⢀⠒⡑⠀⠀⠀⠀⠀⡔⠀⠐⠀⠨⠀⠀⠀⠀⠀⠀⠈⠱⢰⡶⠂
⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡃⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⣸
⡀⠙⠿⣿⣄⡀⠀⡀⠀⠀⠀⠀⠀⠁⣀⡐⠀⢊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⣢⣴⠾⠋
⣷⣄⠀⠉⠉⠛⠧⠷⠤⠀⠀⠡⡀⠀⠁⢂⠤⡡⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠡⠘⠓⠉⠄⣰
⣿⣷⣀⣄⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⣠⡄⣤⣾⣿
⣿⣿⣿⠻⣿⣦⡀⠘⠻⠶⡶⣀⠀⠀⠠⠀⠁⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣠⣾⡿⢛⣿⣿⣿
⣿⣿⣿⣷⣌⣹⣿⢿⣦⡀⢀⠁⠀⠀⠀⠀⠄⠑⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢋⣤⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣦⡟⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢟⣵⣾⣿⣿⣿⣿⣽
⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣶⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⢿⣿⣿⣿⡄⠀⠀⠉⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⢠⣶⡄⠀⠉⠀⢀⣴⣤⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣇⠀⣠⣴⣿⣿⡟⠀⠀⠀⠸⣿⣿⣷⣶⣴⣶⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠸⢿⣿⠿⢋⡀⠀⠀⠀⠀⢈⠛⠿⠟⠛⠃⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣧⣸⢻⡄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⠿⡇⠀⠀⠀⡀⠀⠻⣦⡤⠖⠀⠀⠀⣈⣷⣤⡀⣿⢣⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣷⢀⡤⠾⣧⣤⣤⣤⣤⡤⠶⠚⣋⣽⡏⠈⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢿⣶⣤⣤⣤⣴⣶⠿⣿⡟⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⢷⣄⣉⣉⣀⣀⣤⠿⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠉⠉⠉⠉⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠦⣤⣀⣀⣀⣀⣤⠴⠋⢹⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⢾⡆⠀⠀⠀⠁⠀⠀⠀⠀⢸⣅⣩⠍⠉⠛⠳⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠰⢾⣇⡀⠀⠀⠀⠀⣀⣤⣖⣋⣭⣤⠤⠶⠶⠚⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠦⢤⣤⣤⣤⣈⣙⣷⣤⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡂⡹⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢡⡍⢹⣡⠏⢀⠗⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⠁⢠⡃⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠠⢤⣀⠈⢦⠀⠀⢳⡀⢈⣥⣤⣀⡀⠀⠀⣠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠳⡜⢆⠀⠀⠹⡇⠀⠀⠀⠉⠓⢾⣵⠒⠒⠉⠉⠳⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡨⡄⠀⠀⢳⣰⣶⡆⠀⠀⠘⢯⡤⠄⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠙⠧⣄⡀⠀⠀⣠⠏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣸⣾⠀⠀⠀⠀⠀⠀⠉⠭⠝⢋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡤⢶⡶⠦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⣀⠤⠤⡄⠀⠀⠀⠀⠀⣀⢤⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀⠀⠄⠠⠀⠄⣀⡤⠮⢀⡉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠏⠀⢸⠀⠀⠀⠀⠀⣀⣀⡤⠤⠬⠉⢻⠛⠛⠋⠉⠀⠀⡰⢃⡴
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣄⢄⢸⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡴⣄⠞⣡⠋
⢿⠀⢸⠀⠀⠀⠀⠀⠀⣐⡄⠀⠀⠀⠀⠀⡤⢤⡤⠤⡇⠀⠀⢀⣠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠎⠀⠈⠻⡅
⢸⢠⠟⠃⠀⠀⢀⡤⠒⢒⠲⡄⠀⠀⠀⠀⠈⢻⡛⠒⠓⠠⠤⠍⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⣀⠀⢀⣀⠼
⢸⡿⣠⡀⢀⠤⣼⠷⢮⡝⡸⠁⠀⠀⠀⠀⢠⡤⠭⠭⣖⠢⠤⢄⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠒⠒⠒⠒⠒⠒⡶⠛⢾⡿⠋
⢸⡇⠈⡇⠙⠾⣖⠉⢣⠀⢙⡒⠒⠒⠒⠒⠒⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⢤⣀⣀⠀⠀⠀⠀⠀⡼⠉⠢⡼⢟⢆
⠸⠷⠖⠃⠀⠀⠈⠓⠦⠤⠏⠉⠉⠉⠉⠉⢏⢹⠀⢀⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠲⠞⠁⠀⠀⣣⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⣻⠀⠀⠀⠀⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⢤⢴⣇⡀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠉⠁⠀⠀⢸⢀⡏⠘⡉⢏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣇⣼⡤⡴⡇⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠛⢉⡏⢉⣁⣽⣥⣉⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⣀⣹⠿⠟⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⠉⠙⠛⠛⠓⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠒⠋⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣅⠄⢀⣤⡖⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢤⠀⠙⠒⠋⣎⣀⠀⠀⠀⠈⢦⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⢀⠒⠀⠀⠀⠀⠉⢑⡒⢦⠤⣀⠀⠑⢖⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠙⣛⣭⡭⣽⠶⠚⠉⠉⢩⠇⢀⠏⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣋⡵⠤⢄⣀⠈⠍⢙⡣⠀⣤⣀⣴⠋⠀⡜⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢫⣍⠁⠉⠉⠈⠛⠓⠒⠒⠒⡚⠛⠋⢻⠤⠤⠞⠛⠁⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⣠⠴⢺⣅⠤⠛⠒⠢⢤⡤⠤⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠶⠠⢤⡀⠀⢰⠆⠈⣆⡸⠆
⠀⠀⠀⠀⠀⠈⠳⣖⠘⢦⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⡉⣉⡼⠥⠤⠊⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠙⢦⡀⠀⢻⡉⠒⢄⡀⠀⢠⠖⠒⡄⠀⠀⡜⠉⠉⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⢦⡀⢧⠀⠀⠙⠦⣇⠀⣠⠆⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⠢⣧⡀⠀⠀⠙⢯⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠳⡀⠀⠀⠀⠁⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠰⢷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠘⡄⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⠤⠇⠀⠀⢃⢬⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡘⣆⠀⢸⠷⠦⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⢿⣒⣏⢲⠀⢸⠉⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠸⠚⠁⠀⢠⠤⠤⠤⠤⠿⠚⠒⠒⠒⠒⠚⠳⣄⣈⡭⠼⠿⠀⠘⠦⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⠒⠠⠤⠔⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⡤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡇⠹⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣄⠒⠊⢱⡀⢡⡱⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠒⠂⠉⠁⠈⠛⠒⠶⠊⠀⠭⠆

⢰⣶⣶⣶⣶⣶⣶⣶⡶⠶⣶⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠄⠠⠤⢄⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⢸⣿⣿⣿⣿⣿⣿⣿⡾⣛⣈⣿⣿⣿⣿⣿⣄⠀⢀⣠⠀⠉⠀⠀⢀⡇⠀⠀⠈⠑⢀⡀⠀⠀⠈⢻⠿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣼⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡇⢈⣛⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠐⣄⢂⡇⠀⠀⠀⠀⠈⠱⡀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣀⣰⢻⣿⣿
⢸⣿⣿⣿⣿⠻⠟⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠠⠆⠀⡄⠋⠉⢱⡤⠆⠠⠀⠰⠠⠑⠂⠐⠂⡀⠀⠈⢻⣿⣿⡿⢻⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⡄⠈⠲⢺⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⣄⡃⠈⠀⢾⡱⠄⠀⠀⠘⡀⠀⣢⡄⠀⠀⠐⢀⠀⠈⢿⣷⣼⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣟⣦⣤⠞⢻⢋⠀⠀⠀⠀⣌⢻⠃⠃⡼⠅⠻⡗⠢⠶⢏⠈⠏⣉⠟⠀⠁⠀⠉⠀⠀⠀⠀⠈⠆⣀⡄⣈⣿⣿⣿⣿⣿⣿
⠀⠛⢿⣿⣿⣿⠀⠁⠀⡇⠘⢦⡀⠰⠒⡋⠀⠙⠚⠛⠒⠚⠀⠂⠀⡀⠂⠐⠛⠀⠑⢻⡟⠀⣠⡀⠀⠀⣠⡄⠀⢠⠀⠀⠈⠙⣿⣿⣿
⠀⠈⠈⠻⢿⣿⣦⣄⠀⡖⠀⠸⣧⣀⡠⠁⠀⢄⠀⠀⠀⠘⠀⠠⡀⠁⡃⠀⠀⠀⣴⢀⠉⠈⠀⠙⠤⠤⣇⡁⠀⢀⣃⣀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠹⢿⡈⠁⢱⣄⠀⠈⠋⠀⠀⠀⠀⠐⠀⢠⣴⣤⢆⣠⡤⣇⣡⣤⣮⠊⢣⣄⣀⠀⠀⠀⠀⢸⢃⠈⠀⠀⠈⠁⠲⣿⡦⠈
⠀⠀⠀⠀⣠⣴⠾⠃⡿⢸⡿⠟⢒⣾⣀⡀⠀⠀⣴⣦⣉⠹⠑⠊⠁⠈⡬⠉⠴⣶⠶⠿⠟⠻⠂⠀⠀⢀⠀⢣⡹⢄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠉⠘⠲⠆⠉⠁⠉⠒⠐⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠐⠢⠀⠒⠚⠛⠚⠛⠁⠀⠀⠭⠄⠱⠎⠐⠞⠋








⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣤⣤⣄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣶⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡷⠟⠟⠛⠳⢦⣠⣤⣠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⢿⠿⣿⡿⢻⡁⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⠿⠟⢛⡩⠤⢖⠀⡀⢀⣀⡀⠉⠿⠿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣶⡸⣷⣶⣶⡄⢼⣷⣦⡄⢐⣊⣭⡴⠦⠓⠋⠓⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠛⠉⠃⢸⣿⣶⣧⣜⡿⠞⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠵⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⣤⣜⡇⢀⣒⡄⢀⣄⡀⠐⠉⢩⡅⠀⠂⣌⠀⣀⠈⠛⣤⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠦⠀⠀⠀⢠
⠀⠀⠀⣠⡿⣿⢧⣿⣵⣼⣿⡿⠛⠋⡆⠀⠀⢀⣽⡋⠑⠀⠠⠁⠁⠀⠛⠁⠀⠀⣠⠊⠩⠀⠀⠀⡴⢋⣋⠀⠀⢀⡟⠀⠀⡰⣇
⠀⠀⠀⣸⢷⡉⢀⡏⠠⣸⣿⡇⠀⠀⠹⠠⠿⠇⡙⠅⠂⡀⠀⠀⡀⠀⠀⠲⡄⠖⠁⠀⠀⠸⣦⠎⠘⠻⠿⠄⠀⠾⠅⠀⠰⠀⠸⠆
⠀⠀⠀⠀⠀⠁⠀⠃⣠⣿⣿⣿⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠶⠶⠒⠒⠒⠒⠲⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣓⡲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠯⣍⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⣤⣤⣀⣀⠀⠀⠀⢀⡀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⣀⠀⠀⠰⠖⠒⠒⠒⠒⠚⠓⠀⠀⠀⡠⠉⠈⠉⠁⠒⠚⠏⠁⠒⠲
⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠱⢄⡈⢳⡀⠀⣄⢰⡓⠀⠀⢀⡤⠒⠛⠉⠉⠉⠉⠁⠚⣤⠴⠚⠉⠉⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⠦⢿⠳⡽⡄⠀⠻⣄⣀⣉⡀⢀⣀⣀⠠⠴⣇⣂⣀⣠⠴⠾⠖⠒⠛⠋⠉⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡿⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢋⣩⣭⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠋⠉⠁⣈⡠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠒⠛⠛⣻⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠖⠒⠋⠁⠀⠀⠉⠒⠤⠀⡀⠀⠀⠀⠀⠀⠘⢉⡵⠚⢁⣀⣠⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢉⠀⠸⠉⢙⣐⠒⠒⠒⠒⠘⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠑⠰⣄⣸⠉⣁⡦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠢⣄⠀⡉⢳⡇⢈⡑⠒⠀⠄⠐⠒⡖⠂⠀⠀⣀⠀⢠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡀⠈⡙⠁⠘⢦⠀⢳⡦⣄⡀⠀⠐⣷⢦⠘⢻⠁⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠰⣄⠀⢳⠀⠀⢸⠀⠀⢳⡀⠉⠙⠒⠛⠀⠓⣏⣀⠤⠼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠁⠀⡇⠀⠀⣇⠀⠈⠀⠀⠀⢣⡀⠀⠀⠀⠀⡈⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⠀⠀⣈⠄⠀⠀⠀⠀⠀⢳⠀⠀⣠⣾⣿⣿⣿⣿⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢜⣩⣌⡽⠍⠀⣴⠀⠀⣀⣠⣴⠃⠀⠘⢿⠿⠛⠉⠁⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠛⠚⠒⠒⠒⠒⢺⠉⡛⠉⠁⢠⡇⠀⠀⠀⠀⠑⣄⣀⡴⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠤⢶⡴⣷⣦⠀⣾⠀⠀⠀⠀⠀⠀⠈⣡⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⡑⠦⢤⣀⣀⣀⡭⠽⡄⠴⠒⠒⠉⣲⠄⠀⠀⠀⠀⠀⠙⢤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠞⠉⠀⠀⠀⠀⠀⠉⠙⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠢⠐⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠐⠐⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⢒⣫⣿⡯⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠃⠀⠀⠀⠀⠀⠀⠠⠴⠶⠮⠥⢤⣀⣀
⠀⠀⣀⡤⢖⡉⠉⠉⠉⠉⠓⢒⡒⠒⢲⣆⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠤⢤⣖⣚⠉
⠰⢏⡁⠀⠸⡟⣼⠃⣤⣬⣦⡿⡇⢀⠞⠁⢠⠾⠭⢭⣑⠲⢄⣀⠤⠒⠋⠉⠉⢹⠈⠀⠀⠀⠀⢀⣀⣈⣉⡦
⠀⠀⠈⠳⢄⠚⠑⠒⠉⠉⠈⠁⠓⢿⠀⡔⠒⢕⠦⢤⣸⣯⡞⢳⣦⠤⣀⣀⡴⠋⠀⠀⠀⠀⠀⣀⣉⣒⣦⠄
⠀⠀⠀⠀⠀⠓⠛⠋⠉⣿⣿⠉⠛⠛⢄⠈⠂⠈⠁⠘⣠⠀⠳⢾⣗⡒⠋⠁⠀⠀⠀⠀⣀⡤⠤⢽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠀⠀⠀⠈⠓⢤⣠⠤⣤⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣚⣛⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠬⡤⣤⠤⠤⡴⠶⠋⠀⡝⣄⠈⠁⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⡆⠀⢇⢸⠀⠀⣱⡈⢆⢀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣷⠀⠈⠻⠯⢿⢻⡀⠾⠿⣧⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⢦⡤⣤⣤⣼⠃⠙⣹⠎⠹⡲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠓⢣⡀⠑⣝⣶⣗⠚⠀⠰⢡⣷⠛⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣻⡿⡤⠬⣙⡶⠠⠲⢍⣉⡹⠃⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡀⢀⣀⡄⢀⣠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⡿⣿⣶⣦⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠈⠑⠦⠀⠀⠀⠀⠀⠀⣰⣾⣟⠠⠈⣿⣿⣿⡏⠠⢸⣿⣿⣷⣄⠀⠀⠀⠀⢀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠦⡼⠿⡿⠿⢿⠶⠾⡿⠿⡿⠿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣼⣧⣼⣷⣴⣷⣴⣿⣶⣾⣿⣾⣿⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠃⠀⠁⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡀⠀⠀⠀⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣿⣿⡇⠀⠀⠈⠁⠀⠒⠲⠦⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⣾⣥⣬⣅⣨⣥⣬⣯⡤⢽⣤⣿⣤⣽⣷⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣿⡏⠻⣿⣯⠋⣿⣿⡇⣶⣿⣿⣇⣿⡙⠿⣿⡇⠀⠀⠀⢴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⢿⣇⣶⣽⣿⣿⠏⢸⣿⡏⠏⣹⣿⣿⣷⡄⣿⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠿⢿⣿⢏⣾⡆⣿⣿⡘⣿⣿⣿⠩⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢰⣿⠏⣘⣯⣭⣸⣿⣿⡿⣿⣿⣇⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⢏⣼⣿⡿⣿⣯⣿⣿⣦⢲⠌⣉⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣋⡁⣿⣿⡿⢑⡹⣿⣿⢁⣿⣧⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤



⣶⣶⣶⣶⣾⡿⢿⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣿⠿⠿⢾⠿⣶⣷⠶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣤⡄⠀⣤⠀⣤⡄⠀⣠⡄⢠⣄⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢠⣿⣇⢸⣿⠀⣿⡇⠀⣿⡇⠘⣿⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣸⡟⣿⣸⡏⠀⣿⡿⠿⣿⡇⠀⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⢿⣿⠃⠀⣿⡇⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⠉⠁⠀⠉⠁⠀⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⣴⣶⠀⣶⣶⠶⠆⢠⣶⠶⡦⠀⣶⡆⢶⣶⣶⣶⠆⢰⣶⣆⠰⣶⣶⣶⡶⢰⣶⡶⠶⠀⠀⠰⠶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣽⣿⠀⣿⣇⣀⡀⢻⣯⣀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣼⡟⣿⠀⠀⣿⡇⠀⢸⣿⣀⣀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣻⣿⠀⣿⡟⠛⠁⠀⠙⢿⣷⠀⣿⡇⠀⢸⣿⠀⢠⣿⣤⣿⡆⠀⣿⡇⠀⢸⣿⠛⠋⠀⠀⠀⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢽⣿⠀⣿⣷⣴⡄⣴⣦⣾⠟⠀⣿⡇⠀⢸⣿⠀⣼⡿⠉⢹⣧⠀⣿⡇⠀⢸⣿⣦⣶⠀⠀⠀⣶⡆





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣿⣿⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢯⣿⣿⡿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣳⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣠⣶⣶⣷⣦⣶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣦⣾⣶⣶⣆⡀
⠀⠀⠀⣠⠾⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣾⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⣄⠀⠈⠉⠉⠛⠛⠛⠇⠀⠀⠀⠀⠀⠻⠛⠛⠋⠉⠁⡀⢠⣀⣀⣠⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⣦⣴⣿⣿⣿⣿⣿⣾⣿⣯⣿⣶⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢶⣿⣽⣿⣿⣿⣿⣿⣟⣿⣂⣴⠂
⠀⠀⠀⠀⠀⠐⠲⠿⢿⣿⣿⣿⠛⢻⢿⣿⣿⡿⢿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡿⢿⣿⣿⣿⡿⠛⣿⣿⣿⡿⠿⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠽⣷⣌⠛⠛⠋⡀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠙⠻⠛⣡⢾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣶⣇⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠾⠯⣅⡕⠠⡈⠐⠤⡀⠑⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠘⠀⠓⠄⡈⢳⠈⠑⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠊⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢀⠄⠉⠐⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀⢰⡀⢄⠂⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠦⠄⠠⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠳⣟⠓⠀⡅⠈⡇⢀⠀⠀⠀⠀⠀⠀⠘⡖⠤⣀⡀⠀⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠘⠂⠐⠁⠀⠑⠁⠀⠀⠀⠀⠀⢠⣀⠇⠀⠀⠈⠁⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠰⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⢎⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢆⡀⢀⡌⠎⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⡟⢻⠀⠀⠀⢸⠢⡀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣲⠘⣄⢄⣀⣤⢆⣐⣄⠀⢄⠱⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠀⠀⠀⠀⠀⢩⠀⠈⠀⠃⠙⠀⡀⡀⠀⠈⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠀⣀⠀⠀⠀⠀⠀⢉⡤⠀⠠⡆⠀⠉⣴⣁⠀⣶⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡑⠀⢬⣀⠀⠲⠃⢀⠭⡀⠀⠀⢠⢀⠦⠤⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢄⠉⢃⠀⠀⡆⠀⠀⠀⠀⠀⠉⢠⠖⠛⢓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢄⠣⠀⠙⡢⠀⠀⠀⠀⡀⠤⠔⠊⡡⠫⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣵⠋⠉⠙⠣⠟⠋⢀⣀⠠⢔⠩⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⠄⠭⠌⠭⠛⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣵⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⣾⠿⠟⢿⣻⣿⣴⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣭⣿⣯⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠟⠛⠿⠛⠋⠋⠙⠙⣿⡿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠟⠋⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⣿⠬⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣧⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠤⠖⠛⠉⠉⠉⠀⠀⢹⡗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⢤⣄⢸⠟⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⣤⣴⣶⠄⠀⠀⠀⠀⠈⠹⣯⣿⣿⠀⠀⢸⡏⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⠀⠀⢷⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢸⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⢀⣠⢶⠕⠉⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡔⠈⠱⠒⠤⠠⠴⢒⣯⠄⠛⠁⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠉⠑⠳⠔⠈⠉⠁⢶⠀⠀⢀⡀⠠⠐⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠁⠉⠉⠙⡍⠁⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠰⠃⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢠⠀⠀⡀⠀⢀⠀⠀⠀⠀⣀⠀⢱⠀⠀⢠⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠛⠋⠻⠊⠹⠤⠼⠓⠤⠚⠿⠇⠀⢸⠀⠀⠃⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠐⠀⠚⠖⠒⠂⠀⠀⠀⠀⠀⠀⣸⠀⠀⢧⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⢿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠙⣿⣿⣿⣿⡿⢻⡿⢏⠈⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢿⠻⠯⠂⠀⠀⠀⠀⠀⠤⠸⠟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⠻⣿⠋⠀⠀⠀⠈⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⢿⣿⢿⡿⡿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣉⣉⡉⠀⠒⠈⠉⠉⠁⠀⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣚⣉⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠤⠾⣖⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠒⠒⠂⠀⢴⠒⠒⠒⢜⣋⣡⠦⠤⠒⢴⠀⠀⠀⠀⠀⠉⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡩⡟⠀⠀⢀⠞⢀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡄⠀⠀⠀⠈⢳⣌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⢄⣠⡊⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⣈⢆⠀⡀⠠⠼⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⡁⠀⠓⠚⠀⠀⠀⠀⠀⠀⠐⡒⠋⠀⡱⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠰⠁⣬⣑⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠔⣩⠌⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢀⡠⣄⡤⢀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⣶⣶⡖⠒⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢒⢘⣉⡉⠏⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⢊⣩⠀⠈⠉⠉⠉⠉⠀⢠⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢈⣽⣿⣿⣿⣿⣿⣶⣾⣿⣿⣶⠉⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠺⠁⠌⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⢋⠑⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⢲⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠛⠉⠁⠈⠉⠉⠙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⠶⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠆