Jeweler Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⠆⠀⠀⠀⠀⠀⠀⠊⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖
⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠴⢁⣾⡟⠀⠁⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠱
⠀⠀⠀⠀⠀⠀⠸⣁⢀⡗⠀⡄⢀⣾⠊⡅⠀⠀⠀⠀⠂⠀⠀⣀⣀⠀⡀⠀⠀⠄⠀⠁⢸⡿⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢿⠶⠀⠀⣂
⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⣷⡤⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠸⠀⠀⠀⠈⢀
⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠻⣷⡟⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁⠀⠀⢠⠇⠀⠀⠀⠀⢇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠴⢦⡄⠀⠀⡀⠀⠀⠀⠁⠀⠀⢦⣤⣴⠾⣇⡠⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⠓⠶⠇⠀⣈⣉⣙⠳⡄⠀⠀⠀⢸⣯⠤⠖⠋
⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡔⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠒⠚
⠀⠀⠀⢀⠤⣄⠀⠀⠀⠀⢀⣤⡾⠂⠈⠛⠂⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⢂⠀⠈⠁
⠀⠀⠀⢯⣀⣸⠂⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠛⠁⠈⢻⣆⠀⠀⠡⡀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣝⡳⠶⢂⠀⠀⠀⠀⠈⣯⢿⣀⠀⢀⡼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠂⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠙⠉⠀⠀⠀⠀⠀⠙⢧⣍⣛⣩⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡋⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠂⠓⠶⣄
⠀⠀⢾⣿⣿⣟⣿⣷⣿⡇⠀⠀⠀⠀⠀⣠⠖⠛⢦⡀⠀⠀⠀⠀⠠⡟⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⣀⣠⡚⠛⣀⣀
⠀⠀⠈⣿⣿⡟⢿⣿⡟⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠰⠿⠻⠚⠋⠚⠹⠧
⠀⠀⠀⠘⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠳⣄⣀⡤⠋⠀⠀⠀⠀⠀⠈⠳⢤⣤⢤⠶⠋

⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢰⡟⠛⠛⠛⢿⡆⠀⠀⠀⠀⠀⠀⣶⠀⠀⣤⠀⠀⠀⠀⠀⠀⣴⡾⣧⡀⣠⣶⣤⠀⠀⠀⠀⠀⣔⣉⣟⣙⣏⣑⣄
⠀⠀⠘⡇⡠⠒⢄⣸⠁⠀⠀⠀⠀⠀⠀⢠⡀⢀⡉⠀⠀⠀⠀⠀⠀⢹⣧⢹⣷⣿⢡⣿⠀⠀⠀⠀⠀⠳⡘⡄⠀⡜⡠⠋
⠀⠀⣾⣥⣤⣤⣤⣬⣧⠀⠀⠀⠀⠀⠀⢸⢱⡞⡇⠀⠀⠀⠀⠀⠀⠀⠻⣾⢿⣯⣾⠏⠀⠀⠀⠀⠀⠀⠈⢺⣼⠊
⠀⠀⠙⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠚⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⡠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⡖⢲⠦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣰⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠉⠉⢲⢃⠀⠀⠀⠀⠀⠐⠀⡙⠻⢋⣯⠽⠀⠀⠀⠀⢠⣴⣶⣶⢰⣶⣶⣤
⠀⠀⢧⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢘⠀⠀⢸⠨⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣙⠋⠀⠀⠀⠀⠸⣿⣶⡶⡶⣶⣾⡿
⠀⠀⠈⠳⢴⣿⠴⠚⠀⠀⠀⠀⠀⠀⠘⢍⡖⢰⣉⠋⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢹⠁⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠑⠠⡦⠔⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢼⣯⡛⢿⣿⣿⣄⠀⠀⠀⠀⠀⠳⣄⠀⠀⢀⡰⠃
⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣽⡟⠀⠀⠀⠀⠀⠀⠈⣡⣿⡍
⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠠⣾⡆⠀⠀⠀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠘⠻⠛⠀⠀⠀⠀⠀⠀⠀⢾⣄
⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⢠⣶⡇⠀⠀⠒⢫⡆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠆
⣀⡀⠤⠞⠉⠀⠀⠀⠀⠀⠸⣿⠧⠀⠀⣠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄
⢘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣿⣿⣿⡇
⠘⠣⣄⠀⠀⠀⠀⠀⢀⡠⣤⣄⣣⣤⡋⠤⣺⠟⠁⠀⠀⠀⠀⠀⢀⡥⣫⣥⡀⠀⠀⠀⠀⠀⠀⡎⠛⠈⣻⣿⠇
⠀⡀⠈⣇⠀⠀⢀⠾⠋⠉⣩⡍⠉⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⠏⣻⣷⠀⠀⠀⠀⠀⠀⠉⠂⠈⠉⠁⠀⠀⠀⠀⢰
⡀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠢⡀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀
⠀⡘⣿⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⡇⠻⠏
⡆⣷⠟⠀⠀⠀⠀⠀⠀⣀⣤⡼⣽⡷⡤⣤⠀⠀⠀⠀⠀⠀⣀⢓⠤⡄⠀⠀⣤⠶⣦⡀⠀⠀⠀⠀⠣⣷⢸⡄⣤⡄⣠⡀⣼
⠉⠀⠀⠀⠀⠀⠀⠀⣤⣿⣹⣶⣿⣷⣏⠼⢷⡄⠀⠀⠀⠀⠳⡄⡀⠁⠀⠀⠻⠶⠾⠃⠀⠀⠀⠀⠀⠈⠚⠳⠟⠧⠿⠶⠟
⠀⠀⠀⠀⠀⠀⠀⣠⣇⢡⣏⣿⣿⣷⣿⡿⠁⠃⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⣄⡀⠀⠀⠀⠀⠀⠙⡏⢳⡟⡟⣿⣿⣿⣇⣉⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⠀⠀⠀⠀⢀⢀⣤
⣿⣵⠀⠀⠀⠀⠀⠘⢧⡴⣻⣿⣮⣿⠏⢶⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢵⡆⡳⠄⠀⠀⡿⢾⣿
⠽⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠌⣳⡇⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠁⠀⠀⠀⠻⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠄
⠘⢷⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿
⠀⠀⠉⠉⢷⣄⠀⠀⠀⠀⠐⢄⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣅⠈⠀⢀⣿⣅⠈⠀⠀⠀⠀⠀⠀⢀⠰⠂
⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠀⠀⢸⣾⣿⠄⠀⢸⣾⣿⠀⠀⠀⠀⠀⠀⠀⣺⠟

⠀⠀⢠⣤⣤⣄⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾
⠀⠰⣿⣿⣉⣭⣭⣭⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠉
⠀⠀⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⢰⣿⣻⣟⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⢀⣀⠤⠂
⠀⠀⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣀⡤⠔⠚⠉
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠠⠾⠟⠒⠒⠚⠋⠉
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⣾⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀⣾⣿⣿⡏
⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⡔⣶⣢⡴⣶⣢⡄⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⡀⢀⣼⣿⣿⡟
⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⣿⡄⣿⣿⢠⣿⠏⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠉⠋⠁
⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠙⢷⢻⣏⡿⠋⠀⠀⠀⠀⠀⢰⡒⠿⠿⣶⣬⣟⣛⣛⠛⣛⣫⠤⠖⠶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣉⠣⢤⣈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠆
⢀⣀⣠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠺⠭⠭⠭⠤⠤⠤⢤⡀
⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⢸⣿⡅⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡆⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⡇⠀⠀⠀⣴⠋
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣯⡀⣸⡇⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⢀⣧⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⣦
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⠃⠀⠀⠀⠀⠀⢸⣾⣷⡄⠀⠀⣼⣿⡇⠀⢀⣿⣿⠀⠀⠀⣤⣾⣷⣄
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⣆⠀⠀⠀⠀⠀⢸⣿⣿⠄⠀⠀⣿⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣴⡿⠋⠀⠈⠻⣷⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿
⠀⠀⣾⣿⡇⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢸⠁⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⠀⠀⠘⣿⣿⣿⣿
⠀⠀⣿⣿⣿⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠹⡿⠃⠀⠀⢿⡿⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣿⣿⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣿⣿

⠀⠀⠀⢐⠐⡆⠀⠀⠀⠀⡄⠀⢄⠀⠀⠀⠀⢸⢜⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⡖⠉⢲⠀⠀⠀⢀⠇⠀⢸⠀⠀⠀⠀⡸⠯⡀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⢀⠋⢁⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⡴⠿⠶⠿⢶⠀⠀⢀⡿⠿⣧⠀⠀⠀⣠⣼⣯⣄⠀⠀⢰⡟⠛⠛⠛⣷⠀⠀⢬⣀⣨⡄⠀⠀⠀⣿⣿⠀⠀⠀⠀⡼⣛⣛⣛⢷⡀
⠀⠀⡇⠀⠀⠀⢸⠀⢠⡏⠀⠀⠀⢳⠀⢠⡏⣿⣿⡟⣇⠀⢸⡇⠀⠀⠀⣿⠂⢀⡞⠉⠉⣦⠀⢀⡖⠋⠙⢢⡀⠀⢸⢱⣿⣿⣿⣷⢳
⠀⠀⣇⣀⣀⣀⣸⠀⠀⢣⣄⣀⣠⠎⠀⠀⢿⣿⣿⣷⠇⠀⠘⢧⣦⣶⣦⠟⠀⠀⢳⣶⣶⡋⠀⠸⡄⠀⠀⢠⠇⠀⠸⣜⢿⣿⡿⢏⡏
⠀⠀⠀⣞⠉⣰⠀⠀⠀⢠⡛⠛⢓⠀⠀⠀⠀⡽⢫⠁⠀⠀⠀⠈⣿⣿⡟⠀⠀⠀⢸⠀⢨⠁⠀⠀⠉⣶⣾⠁⠀⠀⠀⠙⣿⣶⣾⡟
⠀⠀⠀⢰⠒⡆⠀⠀⠀⠀⡆⠀⠸⠀⠀⠀⠀⢸⣻⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠧⠆⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠐⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠛⠛⠛⠁
⠀⠀⢠⡴⠖⠐⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢢⠀⠀⡴⢢⠀⠀⠀⢀⢄⠀⠀⢀⢄⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣠⣄
⠀⢸⡃⠀⠀⠀⠀⠈⣇⠀⠀⣰⣯⣆⠀⠀⠀⣈⣏⠀⠀⣨⣏⠀⠀⠀⣈⣎⡀⠀⣈⣎⡀⠀⠀⠀⢠⣿⣿⣆⠀⠀⠀⢠⣿⣿⣇
⠀⠸⣆⠀⠀⠀⠀⢀⡿⠀⢰⠁⠀⠈⢧⠀⢰⣿⣫⣗⢰⣯⣫⣗⠀⢸⣟⢙⣿⢸⣿⢙⣿⠀⠀⢠⣻⣽⣿⣝⢆⠀⣰⠋⠁⠉⠙⣆
⠀⠀⠉⠷⠶⠴⠶⠏⠁⠀⠘⢦⣀⡤⠋⠀⠈⠻⠾⠃⠈⠻⠞⠁⠀⠈⠙⠛⠋⠀⠙⠛⠋⠀⠀⢠⣿⣿⣿⡿⡀⠀⢿⡀⠀⠀⢀⡿⠃
⠀⠀⣄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡠⠄⠤⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡾⠁⠀⠈⢻⣶⣶⡿⠁
⠀⠀⠈⢆⠀⠀⠀⠀⡼⠁⠀⢀⡌⠀⠀⠀⠈⢆⡀⠀⠀⢱⡀⠀⠀⢀⡼⠀⠀⣠⡴⠶⠶⠶⣤⡀⠀⠛⠛⠁⠀⠀⠀⠀⢿⡿⠃
⠀⠀⢴⠆⡑⠠⠐⣊⢴⡄⠀⢈⡅⠀⠀⠀⠀⢸⡅⠀⠀⠀⠋⠒⢖⠋⠁⠀⠀⠙⠛⠶⠶⠞⠛⠁
⠀⠀⠀⠈⠃⣙⡇⠙⠀⠀⠀⠀⠱⢀⡀⣀⠐⠆⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠈⠀⢀⡲⠃⠀⠈⣇⠀⠀⠀⠀⣾⡆⢠⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣸⡆⠀⢨⡇⠀⠀⠀⢸⠇⠀⠀⢠⣾⡆⢠⣿⡄⠀⠀⠀⡤⣀⣀⣀⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⢉⡧⠀⠈⢇⡀⠀⢀⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠎⠉
⠀⠀⠀⠀⠀⣼⣢⠀⠀⠀⢠⠀⠈⠉⠉⠀⠀⠀⠀⠳⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠤⣀⠀⠀⠀⠀⡠⠊⠁⠈⢢
⠀⢀⣤⠀⠀⠛⠛⠀⠀⠀⣬⠀⠀⣰⣾⣿⣷⡄⠀⠀⠉⠀⠀⠀⠀⠤⠀⠀⠀⠀⡎⠀⠀⠀⠀⠈⡆⠀⠀⢰⠁⠀⠀⠀⠀⢣
⠀⠀⢽⡆⠀⠀⠀⠀⠀⣴⠇⠀⠀⠳⣌⠙⣿⡗⠀⠀⠀⠀⠀⠀⣶⣿⣷⠀⠀⠀⣣⣤⢀⣀⢠⣤⠃⠀⠀⠘⠷⣦⣤⣴⡾⠛
⠀⠀⠀⠘⠟⠲⣷⠘⠿⠂⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠈⠉⠙⠟⠉⠋⠀⠀⠀⠀⠀⠈⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠐⠛⠙⠙⠛⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠄⠄⠀⣀
⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⢠⡛⠀⠀⠀⠀⠀⣤⠄⠀⠀⢤⣄⠀⠀⠀⠀⠀⡎⠀⠰⢤⠄⡠⡤⠆⠀⢰⠄
⠀⠀⠀⠻⢢⣄⢀⡀⣤⡼⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠉⠒⡤⠤⠤⠤⢤⣦⡖⠋
⠀⠀⠀⠀⠈⠁⠻⠟⠉⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⢀⠀⠀⠆⡀
⠀⠀⠀⢠⠉⣩⣤⣍⡉⢄⠀⠀⠀⠀⠀⢀⡴⢋⡟⠉⠉⢿⠑⢄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠁⢸⣿⣻⣿⣷⠈⡄⠀⠀⠀⠐⢿⠶⢾⠒⠲⠒⢒⠗⢶⡷⠀⠀⠀⠀⠀⠀⡅⠀⠀⢨⡄
⠀⠀⠀⢠⠘⠿⣿⡿⠋⡸⠀⠀⠀⠀⠀⠀⠙⢌⢧⠀⢀⠞⡴⠋⠀⠀⠀⠀⠀⠀⣾⢹⡀⢀⡞⢷
⠀⠀⠀⠀⠲⠂⠀⡐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠃⠘⠿⠿⠂


⠀⠀⢀⠰⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⣶⣄⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⢡⣦⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣧⡼⠻⠋⠉⠁⠀⠀⠀⠀⠀⠀⠑⢤⡤⠊⠁
⠀⠀⠀⠈⣿⠀⢀⢘⠛⠻⠃⡄⠀⠀⠀⠀⢨⠿⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣴⡶⡶⡀
⠀⠀⠀⠀⠙⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣟⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠒⠀⠀⠤⠀⠀⠀⡀⠀⠀⠀⣸⡿⠛⠛⠃⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠒⠂⠠⠄⠀⢀⠀⠀⠀⠀⠀⠚⠒⠀⠀⠀⠛⠁⠀⢠⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠚⠡⠆⠀⠀⠀⡦⣀⠀⠀⠀⠀⢀⠈⠧⡀⠀⠀⢰⣿⡟⣷⣹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⢂⠣⡀⡱⡆⠀⠙⠿⠚⢚⠉⠀⡄⠀⠤⠬⠄⠀⡘⣀⠀⠘⠿⠿⠽⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢹⠉⠅⠁⠁⠀⠀⠀⢀⡈⠀⠀⠤⠀⠨⠁⠀⠘⠂⡀⠁⣿⣷⣶⣦⣤⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠤⠄⠀⠘⠀⡀⠀⠃⠁⣿⣿⣿⣿⡇
⠀⠐⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⡒⢉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠃⠀⠀⠀⡇⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠏⠀⠀⠀⠀⠁⣬⣽⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠄⠄⠀⠁⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢠⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣤⡄⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⢰⣧⠀⠀⣸⣢⡟⠀⠀⠀⠀⠀⠀⢢⣿⠿⠃⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢈⡇⠀⠀⠉⣩⣄⠀⠀⠀⠀⢀⡤⣠⣼⠟⠋⠀⠀⢐⣦⢀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠆⠀⠀⢰⡇⠀⠀⠸⡇⣸⡇⠀⠀⢠⠟⠁⣹⣿⢀⡇⠀⠀⠀⠹⣆⣯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠈⡇⠀⠀⠀⠉⠉⠀⠀⠀⠈⠀⠀⣿⣿⡿⣇⡀⠀⠀⠀⡹⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠀⣿⣿⡗⢛⡟⠀⠀⢠⠧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠽⠟⠀⠒⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⢧⠈⠁⠄⢨⣤⡛⠀⢸⡆
⠀⠀⠀⠀⠀⢀⣴⠟⠋⣿⠁⠀⠀⠀⠀⢐⣧⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢨⣭⡄⡔⠁⠈⠀⠀⡩⠀⠀⡄⢀
⠀⠀⠀⢀⣴⠏⠀⠀⠀⣿⠀⠀⠀⠰⠶⠎⠉⠁⠀⠀⡀⠻⠿⠶⠶⠖⢀⠀⢸⣿⢧⠀⠀⠀⠀⠀⣧⣀⠀⣀⣼⡇
⠀⠀⢠⣿⠃⠀⠀⠀⠀⣿⠀⠀⠠⣿⣇⢸⡀⠀⠀⠀⠁⠀⠀⠀⢹⡆⢸⠀⢸⣿⣴⡇⠀⠀⠀⠀⠘⢻⣿⣿⠟⠁
⠀⠀⣿⣿⣇⠀⣆⡀⠀⣿⠀⠀⠀⠸⣿⣼⣿⣦⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠠⣿⣿⣿⣧⣿⡇⠀⣿⠀⠀⠀⡷⢿⣝⢿⣍⠠⠄⠀⠀⠀⠀⠀⣗⣸⡄⠘⣿⢹⡅⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢿⣿⣿⣿⣿⡇⠀⢿⠄⠀⠀⠙⠾⣿⢿⣷⣶⣶⣶⣶⠿⠟⠿⠛⠛⠁⠀⣿⣸⡇⢠⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⢿⣿⣿⣿⣿⣾⣾⣦⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢿⡇⠄⣸⡿⠗⢀⣠⣤⣾⡇
⠀⠀⠀⢸⣭⠿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠈⢻⣇⡲⡆⠀⠀⠀⠀⠀⠀⢸⡷⣮⣜⡰⠟⢉⣥⣼⣿⡄
⠀⠀⠀⠀⠱⣶⣾⡉⢭⢛⡷⢉⣀⣲⠀⣀⠀⠀⠠⡌⣠⣿⠃⠀⠀⠀⠀⠀⣀⣨⡷⠟⣁⣴⣪⠟⣾⣧⢾⣿⣷
⠀⠀⠀⠀⠀⠈⠑⢶⣶⣿⣧⣜⡩⠄⢚⣿⣀⣀⣼⠟⠋⠹⣧⡀⠀⠀⢠⣾⣟⣥⣶⣯⣿⣽⣷⣾⠧⢛⣿⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⣿⣍⠉⠁⠀⠀⠀⠙⢿⣇⠀⠛⣼⠿⣶⣿⣶⣟⣛⣿⣮⠟⣿⠏
⠀⠀⠀⠀⠀⢠⡤⠤⡀⢠⣤⡀⠀⠀⠀⠉⠻⣿⡀⠀⠀⠀⠀⠀⢻⣆⡄⠺⢿⣿⣮⠿⠀⠉⠉⠖⠛⠁
⠀⠀⠀⠀⠀⠘⢿⣾⡿⠘⢿⠇⠀⠀⠀⠀⠀⠈⠻⣶⣦⡀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠈⠉⠻⢦⡀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⢟⠛⠧⠸⠂⠀⠀⠀⠀⠀⠘⠻⢷⣄⠈⢿⡇
⠀⠀⠀⠀⠀⢴⣿⡛⠉⠶⡖⡠⠄⢰⡶⠾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠻⢷⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠹⢶⣶⣶⣶⣿⣿⣷⣾⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⠋⠀⠀⠀⠀⠀⠀⢀⣪⡏⠀⠀⠀⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡃⠐⣋⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡹⣇⠅⠈⠀⢃⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣮⠛⢒⣤⡾⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⠉⠀⠀⠙⠛⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠁⠀⣶⠀⠀⣶⣄⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣄⠀⠙⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⡀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠛⠻⢿⠀⠀⠿⠀⢀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⣄⣀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣄⣀⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢐⡆⠰⢆⢲⣖⠊⠴⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠐⠀⢨⡏⠂⠀⠐⠀⢹⣨⡷⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⡀⠐⢀⠀⢁⠀⡈⠀⢀⡱⠓⠁⠁⠪⢆
⠀⠀⠀⠀⠀⡠⠔⠒⠒⠂⠄⣘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡐⣥⣼⣷⢨⢬⡣⠸⣇⠀⠂⠀⢀⡕⢣
⠀⠀⠀⢠⠊⡠⠂⠉⡀⠀⠀⠈⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢏⠌⠸⠺⡼⠶⠚⢸⣦⡇⡠⡪⠊⠉⠈
⠀⠀⠀⡆⠐⠀⠀⢏⣈⠂⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠟⠛⠚⠛⠚⠢⣮⢏⢴⠎
⠀⠀⠀⢃⠘⡀⠀⠢⡤⠃⢀⠄⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠁⡠⠊⠉⠉⠉⠢⡈⢻⠁
⠀⠀⠀⠈⠢⡈⠒⠄⠀⠔⢉⢠⠁⠉⠉⠉⡩⠭⠭⠭⠭⡉⠉⢉⢟⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠑⢄
⠀⠀⢰⠉⠅⠭⠩⠩⠉⠉⠉⠉⠉⢢⠤⠤⠭⠬⠭⠬⠭⠥⠤⡎⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠊⠛⠻⠛⠛⠋⠉⠉⠉⠉⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢆
⠀⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠒⠒⠂⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠂⠛⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣽⣉⣉⣋⡦⢰⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠋⠑⠛⠀⢸⣀⡠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢀⡠⠄⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⡇⠀⠠⠔⠁⠀⠰⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣐⣀⣀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠥⠚⢧⠀⢀⠗⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠂⠈⠓⡴⠡⠤⠀⠓⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠠⠀⡀⠀⡀⡀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⣄⢋⣰⠈⢢⠠⠤⠤⠤⠼⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢀⠨⠈⠂⠀⠀⡀⡀⠀⠈⠁⠆⡈⢸⣀⡀⠀⠀⠀⠀⡎⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡘⠂⠀⠀⠀⠀⠈⠧⡟⢱⠒⡦⢄⡡⠂⠀⠀⠉⠉⠁⠐⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣧⡠⠊⠀⠀⢀
⠀⠀⠀⠀⠀⢀⢤⠖⡫⠉⠉⡐⠠⢍⡁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⡠⠊
⠀⠀⠀⡠⠚⡱⠃⠀⠀⠀⡠⠃⢠⢄⡌⢝⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠇⠀⢀⣠⠴⡁⠑⡄⢀⡠⠖⠉
⠀⠀⡜⠀⠘⠂⠤⢀⣠⠌⠀⣐⠣⠆⣠⠋⠲⡑⢨⠃⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⢇⠀⠘⣄⠜⠉
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠉⠒⠠⢉⡒⠦⢌⠔⢀⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡆⠀⠀⠀⠑⢄⡇⠀⠀⠃
⠀⠙⠒⠠⠤⠤⠤⠀⠀⠀⣀⣀⡀⠈⠑⠢⣤⠎⠀⠀⠰⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠢⠄⣀⢤⣀⠀⡀⠀⢀⡁⠤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠤⠤⠤⠄⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡤⠤⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠤⠴⠶⠖⠒⠛⠛⠉⠉⠉⠉⠀⠀⠈⢧⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠉⢹⡏⠁⠀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⠉⠓⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣠⠟⠀⠀⠀⠀⠀⣰⠏⠀⢀⣼⣷⣤⠀⠈⢳⡄⠀⠀⣠⣤⡀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁⠀⢀⣀⠀⠀⠀⡏⠀⠀⢿⣯⣤⣀⠀⠀⠀⣷⠀⠐⣇⣀⡽⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡇⠀⠠⡏⠉⣳⠀⠀⣧⠀⠀⠀⠉⠙⣿⡷⠀⠀⣿⠀⠀⠈⠉⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠀⠀⣇⠀⠀⠙⢛⡥⠖⠒⢻⣆⠀⠘⠛⢿⡟⠁⠀⣰⠃⠀⠀⠀⠀⠀⡴⠋⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢿⠤⣤⠞⠉⠀⠀⠀⣸⠋⠛⠦⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⢀⣸⣇⣀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⣀⣠⡼⠞⠁⠀⠀⢀⣠⠞⢁⣀⣀⣀⣤⡤⠤⠴⠶⠖⠒⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⢀⡴⠛⠋⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡴⢦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⣩⣴⡏
⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠳⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠘⢧⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢻⣷⡀
⠀⠀⠀⠀⠀⠀⣼⡟⠉⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠈⠙⢿⡄
⠀⠀⠀⠀⠀⢠⡟⠇⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠟⣧
⠀⠀⠀⠀⠀⢸⡇⠆⠀⠀⠀⠈⠳⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠛⠀⠀⠀⠀⠂⣿
⠀⠀⠀⠀⠀⠀⢿⣣⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⢠⣻⠇
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⣀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣜⣿⠷⣷⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣷⢿⣟⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠁





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

⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⠀⢸⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⢠⡜⢛⠛⠀⠀⠀⠀⠀⢸⡗⣿⢺⡇⠀⠀⠀⠀⠀⠀⢀⡞⠉⢻
⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣟⠀⠀⠀⠀⠀⠀⢀⣿⡧⠄⠀⠀⠀⠀⠀⠀⢘⡋⣿⣙⠃⠀⠀⠀⠀⠀⢀⣿⡗⠓⢺
⠀⠀⠀⣨⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⠀⠀⠀⠀⠀⠈⠱⣿⣿⡇⠀⠀⠀⠀⠀⣨⣿⣷⡶⠚
⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠅⠀⠀⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠄⠀⠀⠀⠀⠀⠙⠻⠟

⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠿⠿⣿⣯⣽⠿⠿⠃⠀⠀⠿⠿⣿⣿⣽⠿⠿⠃⠀⠀⠿⣿⣿⣿⣿⣿⠿⠃⠀⠘⠿⠟⡍⣉⡩⠻⠿⠃⠀⠀⠿⠟⠍⣉⡩⠻⠿⠃
⠀⠀⠀⠀⣿⣿⣿⠀⡶⠀⠀⠀⠀⠀⣿⣿⣿⠀⣦⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣏⠉⢱⠀⡶⠀⠀⠀⠀⠀⣊⠉⢱⢀⣆
⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢄
⠀⠀⠀⠀⢸⣿⡇⢜⢦⠀⠀⠀⢀⠞⠛⢷⣦⣄⠀⠀⠀⠀⠀⢸⣶⣶⣶⡇⠀⠀⠀⠀⠀⢸⠛⠛⠛⡏⠀⠀⠀⠀⠀⠀⣤⣴⣦⣾⣧
⠀⠀⠀⠰⢁⣀⠀⣇⠀⠀⠀⠀⡜⠠⠸⡋⠉⠈⡄⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠘⢤⣀⡠⠃⠀⠀⠀⠰⢶⣤⣠⢣⠀⠀⠸⡀⠀⠀⠀⠘⢻⣿⡿⠃⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⣶⣿⣿⣿⣯⣤⣤⡀
⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠷⠶⠛⠃⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠸⠶⠿⠷⠇⠀⠀⠀⠀⠿⠛⠛⠛⠻⠿⠿

⠀⠀⠀⠀⣤⣿⣤⣀⣤⡀⠀⠀⠀⠀⣤⣿⣦⡀⠀⠀⠀⠀⠀⠀⣤⣿⣤⡀⠀⠀⠀⠀⠀⠀⣤⣿⣤⣀⡀⠀⠀⠀⠀⠀⣤⣿⣤⣠⡀
⠀⠀⠀⠀⣇⣀⣸⠉⠄⢣⠀⠀⠀⠀⣇⣀⣸⣿⣿⡇⠀⠀⠀⠀⣇⣀⣨⣿⣷⠄⠀⠀⠀⠀⣇⣀⡰⠛⡑⢄⠀⠀⠀⠀⣇⣀⣹⣿⣷⡀
⠀⠀⠀⠀⣿⣿⣿⢗⢂⣀⡆⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠍⠉⠀⠀⠀⠀⠀⣿⣿⣿⠆⣤⡄⠀⠀⠀⠀⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠻⠿⠟⠈⠋⠉⠀⠀⠀⠀⠻⠿⠟⠈⠉⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠈⠁⠀⠀⠀⠀⠻⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣤⣤⠀⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠶⠶⠶⠆⠠⠶⠶⠄⠀⠀⠀⠀⠀⢀⡴⠖⠛⡛⠳⢦⣄⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⣈⣁⡄⠀⠙⣧⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠤⠤⠤⠴⠤⠦⠤⠤⠴⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⠦⠤⣤⠀⠀⢹⠆⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡄⠀⣤⠀⢠⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⢷⠹⣄⠀⠐⠒⡒⠋⠀⢀⡿⠀⠀⠀⢸⡇⠛
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣌⠳⣝⠶⣤⣈⣁⣤⠴⢋⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣇⣙⣛⣛⣋⣉⣉⣉⣉⣉⣙⣙⣛⣋⣸⡇⠀⠀⣤⣤⡄⠀⠀⠈⠳⣬⣛⡲⠶⠶⠶⠚⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⡴⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⢷⡘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣡⠟⣀⣀⣈⠉⠉⣁⣀⡈⠉⠉⣁⣀⡉⠉⠉⣀⣀⡉⠉⢉⣀⣀⡉⠉⢉⣀⣀⠉⠉⢉⣀⣀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⢁⣀⣈⣉⣉⣁⣀⣈⠉⠉⠁⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣏⣉⣉⣉⣉⣉⣉⣙⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠖⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠐⠛⠛





⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣵⣷⣮⡴⣢⡄⠀⠀⠀⠀⣠⡒⣮⣳⣽⣾⣶⣆
⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠉⠙⢿⣿⣷⠀⠀⠀⢼⣿⡿⠛⠉⠉⠉⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠿⣿⣿⣯⡾⣠⠀⢠⡸⣮⣻⣿⡿⠿⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⡀⠀⠀⠙⢿⣿⠟⠀⠈⠿⣿⠟⠋⠀⠀⣀⠈⣿⣿
⠀⠐⢦⣄⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠁⠀⠀⢀⡴⠒⠀⠀⠀⢸⣿⣇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣿⣿⠁
⠀⠀⠀⠙⢷⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣾⠟⠁⠀⠀⠀⠀⠀⠻⣿⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣶⣤⡀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣷⣷⣦⣀⠀⠀⠀⠀⠀⢀⡠⣶⣶⣿⣾⣦⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠿⢿⣿⣿⣿⣦⠀⢠⣿⣿⣿⣿⠿⠿⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠿⠿⣿⣿⣿⣶⡀⠀⠀⣀⣾⣿⣿⣿⠿⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣶⠛⣦⡀⠀⠉⠻⠿⠃⠀⠈⠻⠿⠋⠀⠀⣴⠞⣵⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⣿⡏⠀⡀⠀⠈⠘⣿⠿⠁⠀⠀⠙⢿⡿⠉⠀⠀⡀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⢻⣿⣇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⡇⠀⣢⡴⠀⠀⠀⠀⣽⡿⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⡇⠀⢠⠦⡀⠀⢠⡔⣾⠛⠁⠀⢶⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⡃⠀⠈⠚⠁⠀⠎⠀⣿⣿⡀⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⡀⡧⠀⠀⠀⠀⠀⠂⠀⣿⡇⠉⠀⣴⠙⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢻⡏⠁⠀⠀⣇⠀⢀⢀⠀⠀⠀⠀⣭⡅⠀⠀⠉⡆⠈⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢸⡇⠀⣨⣋⡀⠀⠀⠈⠉⢹⡁⡄⣿⣧⠄⠀⠀⢷⣦⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⡄⢸⡇⠀⢸⣧⣷⡄⠀⠀⠀⠀⡁⡇⣿⣿⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⡇⠀⠻⣽⣿⣤⣤⣤⡤⣤⠷⠃⢸⣿⠀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣧⠀⠀⠀⠈⠻⣿⠏⠀⠀⠀⠀⠸⣏⣸⣻⡦⠤⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣭⡿⣿⡿⢆⣠⢀⠀⠀⡙⣻⠇⠀⠀⠀⣀⣿⢟⣩⡴⣿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠷⣯⣴⣾⣥⡴⠋⠙⢧⡀⢠⣾⣿⣾⣻⣿⡾⣻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⣀⠀⠀⠙⢷⣆⠀⠀⠈⢻⣄⠹⣿⡿⠏⠛⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃⠙⠃⠀⠀⠀⠙⠷⢦⡀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠲⠼⢂⢀⠀⠀⠹⢦⡀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⣀⣻⣦⣼⣿⣿⣿⡇⠀⠀⠀⠙⢾⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠿⠟⠁⠀⠀⠀⢠⡆⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠘⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠖⠢⠼⠋

⠀⠀⠀⠀⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄
⠀⠀⠀⠀⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⡟⠿⢻
⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⢸⡇⠀⢸⡀
⠀⠀⠀⠀⢸⡇⣶⠿⠯⠭⠭⠽⠿⣆
⠀⠀⠀⠀⢸⡇⢻⣤⣤⣤⣤⣤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⣠⣿⣿⠟⠻⠛⠛⠻⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⡴⠒⢺⢛⣡⢄⠀⠈⠀⠀⠀⠈⠁⠀⣠⠈⡛⣗⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠉⠙⠻⣿⣿⠟⠿⢿⣿⣿⡄
⢧⡀⢻⠀⠉⠉⠀⠀⠢⠄⠠⠔⠁⠀⠈⠉⠁⣽⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣇⠀⠰⠀⠀⠀⠰⠆⠀⠘⠷⢀⣀⣿⠿⣿⠃
⠀⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⡁⠀⠐⠤⠔⠀⠀⠘⠑⠀⠀⠁⠀⠀⢸⠆
⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣤⣾⣄
⠀⠀⠀⠀⠐⡿⢻⠛⡖⠒⠒⣲⠛⡟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣛⡖⠒⠒⠒⢛⡏⠛⣿⣿⡿⠟⠛
⠀⠀⠀⠀⢰⠃⡞⠀⠘⠒⠒⠁⠀⢻⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠈⢳⡙⣟⠿⢿⠋⠙⠦⡀⢸⡿⠿⠄
⠀⠀⠀⠀⡏⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠀⢘⡇⢸⠀⠀⠀⠀⠀⠸⡆⢧
⠀⠀⠀⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢀⡼⢁⠞⠀⠀⠀⠀⠀⠀⣇⠘⡄
⠀⠀⣠⠃⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢶⠿⡖⠉⠀⠀⠀⠀⠀⠀⠀⢹⠀⠳⡀
⠀⠀⠙⠒⢺⡗⡶⢶⡦⠤⠴⢶⡶⢶⡞⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⠀⣷⡶⣶⠤⠤⠴⢶⡶⢶⠛⠒⠚⠁
⠀⠀⠀⠀⢸⡇⢧⠀⠈⡗⢺⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⠀⢹⡒⢺⡅⠀⢸
⠀⠀⠀⠀⢸⡇⢸⡄⠀⡇⢸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢻⠀⢸⡇⠀⡇⠀⢸
⠀⠀⠀⠀⢸⡇⢈⣇⠀⣇⣸⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠘⡆⠀⡇⠀⣧⠀⣸
⠀⠀⠀⣴⠚⠉⠉⣽⣀⡿⢹⣀⣺⠉⠉⠛⣢⠀⠀⠀⠀⠐⠒⠀⠒⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⡗⠛⣇⣀⡏⠉⣿⣀⡸⠓⠒⠢⣄
⠀⠀⠀⡏⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡗⠒⠒⠲⠶⠶⠶⠒⠒⠒⠚⠉⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠂⠀⠀⠀⢒⠤⢹⡶⡖⠒⠒⠒⠀⠂⠀⠀⠐⠒⢒⡊⢳
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⡦⡸⢹⡄⣀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠇⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄⠀⠉⠁⠀⠘⣡⣿⡞⠃⠀⠀⠀⠀⠀⠈⠀⠈⢉⡀⠀⠀⠇⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⠀⠀⠀⠀⣽⡏⢡⡀⠀⠀⠀⠀⠀⡤⠀⠀⠻⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣀⠀⡈⠀⠀⠻⢣⣸⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣯⠘⠋⠁⠀⠀⠀⢀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⢘⡫⠀⠀⠀⠀⠐⠂⠎⠸⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣯⠀⡀⠈⢉⣭⣿⣶⠆⡖⠰⣦⠴⠆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⣿⣯⣿⣟⢚⠸⡱⠿⢿⣻⣖⠀⠀⠀⠀⡀⠀⠀⠐⠛⠆⣳⣾⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣯⣭⣭⣟⣻⣿⠏⠁⣀⡀⠙⠛⠳⠶⠶⣤⣤⣄⣀⣀⠚⠛⠉⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠤⠤⣈⣀⣐⠶⠯⢥⡀⠉⢟⣛⠿⢿⣶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠙⠛⠛⠙⠒⠚⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⢛⠙⢋⠍⢛⠻⠖⣲⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣖⠒⠟⡛⠩⡙⠫⡛⠓⠶⣤⡀
⠀⠀⠀⠀⢀⣤⠞⢩⢂⠄⡰⢁⠔⠁⠀⢁⠔⠋⠀⠚⢉⠝⢳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠏⠉⠑⠀⠙⠢⡈⠀⠈⠢⡈⢦⠠⡐⣝⠳⣤⡀
⠀⠀⠀⣠⣟⣰⡴⡷⢾⠼⣧⣏⣴⢁⢔⡡⠂⡀⠄⢀⠠⠒⡡⢚⠿⣷⡄⠀⠀⠀⠀⠀⢠⣾⠻⡓⢌⠒⠄⠀⠀⢀⠑⢌⡠⡈⣦⣽⣼⠷⡷⢾⢤⣆⣳⣄
⠀⠀⣼⡟⢯⡉⠆⠘⡄⠁⠁⠀⠙⡙⠿⣴⣊⣴⡪⢄⣂⠠⠒⢁⠔⣹⠃⠀⠀⠀⠀⠀⠘⣏⠢⡈⠒⠌⢐⡠⣕⣤⣑⣦⠿⠋⠋⠀⠈⠈⢠⠃⠰⢁⡽⢻⣧
⠀⣼⣇⠙⢆⠡⠈⠆⣈⡾⠛⠋⠉⠛⠓⠮⣿⣿⣊⡥⢒⣒⣭⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣭⣒⡒⠬⣑⣽⣿⠵⠚⠛⠋⠙⠛⢷⣁⠠⠁⠌⡰⠋⣨⣧
⣸⠧⢤⡑⠀⠀⠀⢐⡾⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢷⡂⠀⠀⠀⢊⡤⠼⣇
⡟⠒⠲⠌⣀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠡⠖⠒⣻
⣏⣩⠁⠀⠀⠀⠈⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠁⠀⠀⠀⢈⣍⣽
⢻⢒⣒⣀⣀⣤⠤⢽⣧⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣼⡯⠥⣤⣀⣀⣒⡒⡟
⠘⣧⠔⠋⠁⠀⠀⠀⠀⠀⠉⠑⢯⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡽⠊⠉⠀⠀⠀⠀⠀⠈⠙⠲⣼⠃
⢰⠏⢠⣴⠶⠾⠻⠻⠶⣧⣂⠀⠈⢹⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⡏⠁⠀⣀⣼⠶⠟⠟⠷⠶⣦⡄⠙⡆
⣿⡴⠻⡕⢂⠀⢀⠀⠀⠈⢻⡤⠀⠀⣗⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣺⠀⠀⢤⡟⠁⠀⠀⡀⠐⡀⢪⠟⢦⣿
⠀⠀⠀⠙⢷⣮⣎⣄⣆⣤⠟⠀⠀⢀⡧⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢼⡁⠀⠀⠻⣤⣠⣠⣳⣵⡾⠋
⠀⠀⠀⠀⠀⠙⠺⢬⣉⣈⡂⣀⣀⣾⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣷⣀⣀⢐⣁⣉⡭⠗⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⠤⠶⠶⠶⠶⠶⠦⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⣚⣻⡍⠉⠁⠀⠲⠖⠒⠛⠓⠀⠀⠀⠀⠀⠲⠶⠦⣭⡉⠙⠓⠦⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠻⢯⣍⠉⠉⠉⠛⠛⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣤⣤⣤⡤⠴⠶⠶⠖⠛⠛⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣧⡆⠀⠰⠖⠛⠛⢻⡆⠀⠀⠀⠀⠀⢀⡏⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣻⣏⣀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⣸⠃⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠐⣶⠀⢀⣄⣀⣤⣤⣼⡇⠀⠀⠀⠀⢀⡟⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣌⣽⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠐⠛⣻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣼⠁⠀⠀⠀⠀⣼⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡟⠀⠘⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠐⣿⠄⢠⣤⣤⣤⡤⣿⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠶⣿⠒⠀⠀⠀⠀⠀⠾⠧⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢘⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣿⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⣁⡀⠀⠀⠀⠀⠀⣼⡇⠀⠴⠶⠶⠚⠋⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⣤⣴⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠖⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⢀⡴⢿⠛⠟⠹⠹⢹⢻⣿⣿⣿⣦⣄⠀⠀⠀⢀⣴⣾⣿⣿⣿⢻⠹⠹⠙⠟⢻⠶⣄
⠀⠀⠀⣴⠟⠃⠀⠀⢀⠀⠀⠀⠈⠀⠎⠙⢿⣿⠋⠁⠈⢻⣿⠟⠉⠄⠈⠀⠀⠀⢀⠀⠀⠀⠛⢷⡄
⠀⠀⡸⠃⢀⡤⣾⣟⣿⣾⢾⣄⢄⣠⡂⠀⡠⠃⠀⠀⠀⠀⠣⡀⠀⣀⣐⢄⣼⢾⣾⣞⣾⡦⣄⠀⠻⡀
⠀⢠⠃⣰⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⠖⠊⠀⠀⠀⠀⠀⠀⠀⠈⠒⢾⣿⣿⣿⠿⠿⣿⣿⣿⣿⣧⡀⢣
⠀⢸⠀⣿⣿⣿⡿⠃⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠰⠀⠀⠈⠁⠀⣼⣿⣿⡇⠀⠀⠈⠻⣿⣿⣿⡇⢸
⠀⢸⢸⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⠟⠛⠀⠠⢀⣀⠀⢀⣀⡠⠀⠘⠛⢿⣿⡇⠀⠀⠀⠀⣻⣿⣿⣿⢸
⠀⠸⣸⠛⣩⣉⡙⠲⣦⡀⠀⣿⢡⣿⣿⣶⣶⣄⣩⣶⣮⣁⣴⣶⣾⣿⣧⢹⡇⠀⣠⡶⠛⣉⣩⡙⢯⡸
⠀⠰⢧⠹⣿⣿⡍⢦⠈⢿⡄⣹⡍⢻⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣿⠋⣿⡁⣾⠏⢠⠎⣽⣿⡿⢱⠧
⠀⠀⠀⠱⡜⢿⣿⡾⠀⣼⡟⢣⠠⠌⠉⠙⠋⠀⢋⣿⣏⠃⠈⠛⠉⠉⠤⢠⠛⣿⡄⠸⣾⣿⠟⡴⠁
⠀⠀⠀⠀⠙⠳⣤⣤⣶⡿⠃⠸⢦⣤⣠⣤⡑⠠⠺⡿⡿⠠⠐⣁⣤⣠⣤⠿⠀⠻⣷⣦⣤⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣧⣀⠀⠀⠀⢀⣠⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠹⡿⣍⡛⠳⠛⣋⡽⡿⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠙⠦⣍⣙⣉⡥⠞⠁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⠀⠬⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣄⣀⣄⣀⢀⣀⣀⣀⣠⡀⣼⠀⡓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢸⠀⡇⠀⡉⢉⠉⢉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣀⠀⠁⠀⠀⠀⢸⠀⠱⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠈⠑⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠳⠚⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⢤⡀⠀⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡷⠀⠄⠘⠦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⢒⡒⠒⠒⠒⠒⢺⠀⠀⠀⠀⠀⠀⠸⣆⠀⠘⠒⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⢒⠒⠒⣒⠒⣐⢒⠒⡒⡒⠒⡒⠒⢲⠀⠀⠀⡠⠶⠉⠋⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠋⠀⠀⢠⠟⣡⠌⠻⡄⠀⠀⠙⡆⢸⠀⠀⣜⠃⠀⢰⠛⠄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⢸⡀⢈⡦⢀⡇⠀⠀⠀⡇⢸⠀⠀⢏⡀⠀⢀⣙⠆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠳⠤⠤⠤⠷⠦⠴⠾⠤⠤⠤⠖⠁⢸⠀⠀⠈⠣⣤⠀⡁⢀⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠈⢿⠋⠉⠀⠈⠉⠙⠛⠋⠙⠛⠻⠟⠛⠏⠘⢿⣿⣦⣀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠀⠀⠀⠈⠁⠀⠄⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢻⣿⣿⣷⣄
⠀⠀⠀⠀⣰⣿⣿⣿⣟⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠃⣿⣿⣿⣦
⠀⠀⠀⣼⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⣿⣿⣿⣷⡀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣷
⠀⢰⣿⣿⣿⣿⠟⠛⡋⠀⠀⠀⠀⠄⠰⠶⠶⣤⣄⠀⠀⢠⣤⠶⠶⠦⣄⠀⠀⠀⠕⠀⠸⣿⣿⣿⣿⣿⡇
⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠐⠤⠀⠀⢤⡤⢴⢤⡀⠀⠀⠀⡠⢤⠤⡤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣦⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡋⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣦⣁⢀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣼⣿⣿⡛⣿⣿⣿⣿⣿⣿
⠀⠘⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢱⡇⣿⠸⡇⠻⡀

⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⣀⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣤⣀
⠀⠀⠀⠀⠀⠀⠠⠬⠭⠤⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡤⠀⠁⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠘⠣⣄⡀⠀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⢷⣄⢀⣼⠢⠀⠀⠀⠻⠦⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡀⢀⠁⠀⠀⠀⢼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠐⠁⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⣦⡄
⠀⠀⠀⠀⣠⣴⣾⣿⣷⣄⠀⠀⠀⠀⠀⢧⠈⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣤⣾⠿⠗⣀⡀
⠀⠀⠀⡾⢿⣿⣿⣿⣿⣿⣷⡶⣄⠀⠀⠈⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠐⡺⠋⢁⡤⠖⠋⠀⢀
⠀⠀⠀⠻⣮⣿⣿⣿⣿⣿⡿⣷⠹⠦⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠒⠈⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠐⣜⣿⣿⣿⣿⣿⣟⡵⠃⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠎⠓⠍⣩⠽⠛⠁⠀⠀⠀⡙⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠶⠗⠓⠋⠁
⠀⠀⢷⣤⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠉⠑⠒⠐⠀⠁⠀⠀⠀⠀⠀⠀⢰⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠚⠦⠀⠀⠀⠀⠀⠀⠀⣠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢿⣟⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠙⠳⠆⠀⠈⠀⠀⢉⣻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠷⠄⠀⠀⠀⣀⣴⠿⠛⣋⣉⣉⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⢀⣼⠟⣡⡾⠟⢛⣙⠛⠿⣶⡌⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠙⠀⠀⢀⣾⠃⣾⠏⣴⡾⠛⠛⢿⣦⠘⣿⡀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠲⠀⠀⢸⡏⢸⡏⢸⡏⠀⠀⠀⠀⢻⡇⣹⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⡶⠶⢶⣶⠶⠶⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠙⢷⣦⡀⢸⡇⢸⣇⠸⣷⣄⠀⠀⢀⣾⢃⣿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⢿⡄⠀⠉⠳⠀⠀⠙⠃⠀⠈⠙⠂⠈⠻⠟⠛⢿⠶⣶⣤⣤⣭⣿⣿⣿⡄⠻⣷⣬⣉⣠⣴⠟⣡⡾⢃⡾⠃
⠀⠀⠀⠀⠀⣀⣤⣶⡟⠋⠉⠙⠷⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⡙⠊⠉⠻⠍⠙⠿⣶⣬⣉⣉⣩⣥⡾⢋⣴⠟⠁
⠀⠀⠀⠀⠺⣿⡁⠘⠿⠆⠀⠀⠀⠀⣀⣤⣴⡶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢷⣶⣤⣤⣄⣀⣉⣉⣉⣩⣥⡶⠟⠁
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⢀⣠⣶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣌⣛⣿⣷⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⢻⣿⣿⣿⣿⡿⠛⢉⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠋⣠⠾⠋⣁⣠⣄⣀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣿⣿⣿⡟⢀⡾⠃⣰⠞⢋⣉⡉⠙⢻⣄⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⠁⣼⠃⣼⠃⣰⣿⣿⣿⣷⠀⢻⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⣿⠀⣿⠀⢻⣿⣿⣿⣿⠃⣼⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠐⢶⣶⣦⡐⣶⣶⣄⠠⣤⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢻⡄⠘⢷⣄⣸⣿⡿⠋⣰⠏⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠲⣿⣿⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣷⣾⣿⣦⣶⣦⣀⢤⣀⠀⣀⣀⠈⠻⣦⣄⣉⣉⣁⣤⠞⢁⡴⠋
⠀⠀⠀⠀⢀⣠⡀⠘⢿⣿⣿⣶⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢿⣾⣿⣷⣦⣀⣉⣉⣉⣉⣤⡶⠛⠁
⠀⠀⠀⠀⠈⢻⣿⣶⣾⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠙⣿⡿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⢂⠔⣈⡔⠊⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⣪⢚⠥⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⡤⠔⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠰⠑⠁⢼⠀⠀⠀⠀⠀⠀⠂⢀⣶⠶⠞⠺⢿⡖⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠤⠄⢀⠃⠀⠀⢸⠀⠀⣤⡤⠴⢖⠛⠙⡇⠁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠁⠀⠀⠈⠰⠠⠆⠼⠶⢯⡇⠀⠀⢸⢸⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠤⠖⢈⣀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠉⠀⠀⡘⠀⠀⠐⠗⠀⠀⢸⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠘⢀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠸⠀⢸⡇⠀⠀⢸⠀⢠⣷⠦⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠠⠀⠠⠀⠀⠀⢠⠰⠀⠀⠃⠀⠀⠀⠀⠐⠀⣹⡇⠀⠀⣸⣿⣻⡿⢶⣾⣯⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠂⠐⠨⠀⠀⠠⢰⡈⠋⠉⠙⢹⠀⢹⡗⠀⢨⣿⣿⠿⣧⣾⣯⣽⣿⡇
⠀⠀⠀⠀⠀⠸⡇⠄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢰⣖⣊⡇⣁⣾⣷⣸⢶⣷⡿⢛⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡇⠀⠈⠉⠛⡟⡟⠛⠊⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠉⠐⠀⠄⡀⡗⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⡠⢋⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡑⣄
⠀⠀⠀⠀⠀⡴⢡⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠈⢆
⠀⠀⠀⠀⢰⠁⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⡄
⠀⠀⠀⠀⡌⠀⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡇
⠀⠀⠀⠀⢠⠀⣿⣏⢦⣀⠀⠀⣀⣀⠠⠤⢤⡖⠶⡎⠀⠀⠉⠟⠁⠀⠀⠱⠆⢲⠤⠤⢄⣀⡀⠀⠀⣀⡜⣸⣿⠀⡇
⠀⠀⠀⠀⠈⢆⢹⣿⣄⠀⠀⠀⠀⠀⢀⣠⣴⣁⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⠀⣰⣦⣄⣀⣀⠀⠀⠀⢀⣴⣿⠏⡼
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠶⣿⣿⣿⣿⣿⡿⠓⠂⠀⠀⠀⠈⠟⠁⠀⠀⠀⠐⠺⢿⣿⣿⣿⣿⣷⢾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⡙⠛⠛⣿⢿⣒⠄⠀⠀⠀⢀⢀⠀⠀⠀⠀⠠⣒⠿⣿⠛⠛⢋⣉⣉⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⠈⠛⢻⣿⣿⡇⠛⡏⢳⣄⣰⠀⠠⠿⡀⠰⣄⣴⠛⡹⠓⢸⣿⣿⡟⠛⠁⠈⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⢠⣿⡀⡈⠛⢻⡇⠀⠀⠀⠀⠀⢹⠟⠋⢁⢠⣿⡈⠒⠀⠀⠀⠊⣡⣴⣆⢀⡋⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⣿⣷⠀⣼⠳⠀⠀⠀⠀⠀⠞⣦⠀⣿⣿⢿⣷⡀⠀⠀⠀⠈⣿⠻⡀⣉⡀⠙⠣⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢻⠞⣿⣿⣆⢹⣦⡀⠀⠀⠀⢀⣴⠏⣸⣿⡿⢸⡟⢳⣄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠘⠀⢸⣿⡟⠉⠙⢛⣀⣀⣐⡛⠉⠙⢿⣿⡇⠈⠁⠸⢦⣉⣁⡐⢢⠈⢎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡅⠈⣿⣦⣬⣿⣥⣴⡿⠁⢸⣿⣧⠀⠀⠀⢀⠀⠀⠈⣌⠣⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢣⣠⡆⢟⢿⡏⣻⣾⣯⣿⣿⣿⣽⣷⣛⢹⡿⡿⣸⣄⡆⠸⠀⠀⠀⠈⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⢿⣿⣎⢸⣿⣿⣿⠟⠉⠀⠉⢻⣿⣿⣿⠁⣵⣿⡟⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡙⢿⣇⠻⢿⣿⣶⣶⣶⣶⣾⣿⡿⠟⣼⡿⢋⠔⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠻⢿⣄⠈⣿⣿⣿⣿⣿⣿⣿⠀⣰⡿⠛⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⡀⠙⢦⢹⣿⣿⣿⣿⣿⢇⡾⠋⢀⣠⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠛⠦⠌⢳⣿⣿⣿⣿⣿⠟⠠⠔⠛⢉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢿⣿⣿⣿⡿⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠠⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⢀⣉⣍⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠨⢉⢉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢘⠆⠀⠀⠆⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠉⠀⣀⠒⠈⢌⡐⠂⠤⢀⣀⠠⣤⠼⠉⠠⠶⠏⣤⣀⡀⢐⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠉⠉⠅⢘⡂⠸⠁⠀⠈⠉⠁⠀⠀⠀⠀⢀⣀⣤⣠⠀⠐⢊⠓⠟⠇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠂⠀⠐⠢⠖⠚⡇⠀⠀⠀⠀⠀⠀⢀⠂⣅⡉⠉⢙⠃⠀⠀⠐⠛⠛⠉⠲⠦⠤⠀⡠⠀⠀⠀⢲⡀
⠀⠀⠀⠀⠀⠖⠒⠒⠠⠴⠀⡄⠸⠂⠀⠀⠀⠰⠤⠜⢃⣀⡀⠄⠤⣄⢰⢒⣋⡕⣒⠷⠶⣣⣅⣒⡦⠤⡁⠀⠉⠓⢈⡉⢄⠔⢒⠉⠉⡅
⠀⠀⠀⠀⠀⠃⠤⠀⠠⠼⠁⡅⠰⠦⣀⣀⣀⣀⣀⡠⢞⠈⠉⢈⠢⣿⠸⠀⠒⠨⠍⠛⠁⠀⠬⣁⡈⠉⠀⠍⠐⠽⢒⠉⢉⠩⢀⠀⠀⡇⡀⣠⢀
⠀⠀⠀⠀⠐⠣⠤⢤⡗⠟⠋⠭⣝⠄⠀⠀⠀⠀⠀⢀⡼⢄⠀⠀⡥⣴⠶⢄⠤⣠⣤⣀⣤⣄⣤⣄⡀⠀⠀⢀⣤⠤⠶⠎⣒⣩⡅⠖⠓⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣐⣀⣠⣆⣀⠀⠀⠀⣙⣰⠞⠓⠲⠒⢚⢭⢆⢲⠂⠀⠀⠀⠀⠈⠙⠙⠻⠽⠀⡠⠤⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠁⢪⡁⠒⠀⠐⡒⠂⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠐⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠎⠀⠀⡃⠀⠀⠀⡆⠀⠀⢨⠀⠀⠀⢐⠀⠀⠱⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⣠⣮⣤⣤⣤⣮⣤⣤⣤⣧⣤⣤⣤⣥⣤⣤⣬⣤⣤⣤⣼⣤⣤⣤⣵⣤⣤⣤⣵⣄
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡸⢇⠀⠀⡨⢇⠀⠀⢨⢇⠀⠀⡸⢅⠀⠀⡸⢇⠀⠀⠨⢇⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠀⣠⣱⣪⣥⣤⣵⣪⣥⣤⣭⣨⣤⣬⣅⣬⣤⣤⣵⣊⣅⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⠀⠀⠀⢹⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⡟⠛⠛⠛⢻⡟⠛⠛⠛⢻⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣧⣠⣤⣤⣼⣧⣤⣤⣄⣼⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⡿⠿⠿⠿⢿⡿⠿⠿⠿⢿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⢀⡀⣰⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣆⢀⡀
⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇

⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆
⠀⢿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠙⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠁


⠀⠀⠀⠀⢠⡖⠉⠀⠀⠀⠀⠀⠉⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦
⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⢻⣷⠄⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁

⠀⠀⠀⢀⡠⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⢤⡀
⠀⠀⢰⠃⡜⣁⣠⡤⠤⠶⠶⠤⠤⣤⣀⡀⣀⣀⣀⣀⣤⣴⣦⣤⣀⣀⣀⣀⢀⣀⣤⡤⠤⠴⠶⠤⢤⣤⣈⣣⠘⡄
⠀⠀⠘⣦⣘⣿⣄⣀⡀⠀⠀⠀⠀⠀⠈⠟⡏⠀⠀⠀⡞⠁⠈⢳⠀⠀⠀⢹⢻⠁⠀⠀⠀⠀⠀⢀⣀⣠⣿⣃⣴⠃
⠀⠀⠀⠀⠙⠛⠛⠒⠉⢁⣀⣄⣀⡀⠀⣰⡇⠀⠀⠀⢹⡀⢀⡏⠀⠀⠀⢸⣆⠀⢀⣀⣠⣀⡈⠉⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠻⣍⠀⠀⠐⠭⣽⣿⣝⡒⢤⣀⣠⠇⠸⣄⣀⡤⢖⣫⣿⣯⠭⠀⠀⠀⣩⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣸⣯⢠⣈⡂⠉⠀⣲⣖⠀⠉⢐⣁⡄⣽⣇⣀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠋⢻⣠⠞⠁⠈⠳⣄⡟⠙⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⢻⠀⠀⠀⠀⡟⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣼⠀⠀⠀⠀⣧⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣰⠾⡄⠀⠀⢨⠷⣆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠁⠀⠀⠈⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⣼⢫⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣧
⠀⠀⠀⠀⠀⣸⠏⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠹⣇
⠀⠀⠀⠀⢠⡟⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢻⡄
⠀⠀⠀⠀⢸⡇⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⣸⡇
⠀⠀⠀⠀⠘⣧⢆⠀⠙⠲⠤⠤⠀⠀⠀⠀⢠⣤⣤⣤⣤⣴⣿⣿⣦⣤⣤⣤⣤⡄⠀⠀⠀⠀⠤⠤⠔⠋⠀⡰⣼⠃
⠀⠀⠀⠀⠀⠘⢧⣑⠤⣀⠀⠀⠀⠀⠀⠀⢸⠇⠤⠤⠀⠀⡏⢹⠀⠀⠤⠤⠸⡇⠀⠀⠀⠀⠀⠀⣀⠤⣊⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠮⠥⣤⡀⠀⢠⠟⠈⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠁⠻⡄⠀⢀⣤⠬⠵⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣊⣉⡉⠀⠀⢠⣤⡲⡀⠀⠀⠀⠀⠀⠀⢀⢖⣤⡄⠀⠀⢉⣉⣑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠀⠀⠀⡄⠀⠘⠮⠻⢈⠖⠀⠀⠀⠀⢲⡁⠟⠵⠃⠀⢠⠀⠀⠀⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠚⠉⠉⠓⠲⠇⠈⣆⠀⠀⢁⡠⠂⠀⠀⠐⢄⡈⠀⠀⣰⠁⠸⠖⠊⠉⠉⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⠀⠀⠀⠀⠀⠀⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⣤⣤⣤⣤⣤⣤⡄⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣤⣼⣿⣿⣧⣤⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠭⠟⠛⠛⠻⠭⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠒⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠢
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠀⢰⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠀⠀⠀⠀⠐⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢾
⠀⠀⠀⠀⠀⠸⣿⡄⠣⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⡠⠎⢀⣴⠃
⠀⠀⠀⠀⠀⠀⣿⣿⠄⠑⣤⡀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣳⡤⠖⠊⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⡀⠙⠳⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣤⣶⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢫⡍⠐⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢀⣠⣭⣬⣶⣦⣄⠀⠀⢴⣫⠀⢀⣠⣶⣿⣋⣁⠀⢺⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠁⣼⣿⣿⣿⣿⣿⣿⣧⣤⠶⠿⣷⣾⣿⣿⣿⣿⣿⣧⠈⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠠⣿⣿⣿⣿⣿⣿⣿⣿⢏⣠⣄⠛⢿⣿⣿⣿⣿⣿⣿⡇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠻⣿⣿⠿⠿⠛⠁⢀⣾⣿⣿⣧⠈⠛⠿⠿⢿⣿⡿⠂⠀⠀⢗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠤⠔⢤⡠⢄⡀⣼⣿⣿⣿⣿⡄⠀⡴⢶⣦⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⣀⣤⡴⠶⢦⣌⠀⠛⢿⣿⣿⡿⠁⠀⣸⡧⠶⢦⣤⣀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⣼⣿⡟⡀⠀⠈⠀⠀⠀⢸⢿⣷⠀⣸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⡆⠻⢿⣿⣿⣷⣿⣷⣷⣿⣦⣴⠟⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⢿⡌⢲⠈⠃⠙⠈⠏⠘⠀⠃⣀⠆⣾⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⡎⠁⠀⠀⠀⠐⠀⠑⠈⠁⠐⠁⡜⢉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⢿⠎⠡⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡿⠻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡌⠳⣄⠰⠄⣀⡀⢀⠀⠀⠀⢻⡞⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠈⠷⣶⣻⣠⣼⣀⣠⣠⠾⠁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠁⠉⠛⠉⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣵⣿⣿⣿⣿⢿⣾⣙⣻⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣽⣿⣿⣿⢿⣵⣿⣿⣾⣿⣿⣿⣭⣭⣝⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⠀⠀⠀⠈⠿⠋⠈⠋⠁⠉⠉⠛⠛⠋⠉⠉⠉⠛⠛⠻⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣏⠀⠀⣀⣠⣤⣶⠿⠛⣉⣠⣤⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣶⡿⠟⢋⣁⣤⣴⡶⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣶⡿⠿⠛⠋⠁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⣏⣿⣿⣿⣿⣿⣏⣀⣤⣤⣶⡶⠶⠂
⠀⠀⠀⠀⠨⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⢻⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠴⠶⠒⠲⣖⠒⠒⠒⠘⢯⡉⠉⢀⡼⠀⠀⠀⠀⢷⡀⠉⢉⡽⠓⠒⠒⠒⣲⠖⠒⠶⠦⠤⣤⣄⣀⡀
⠀⠀⠀⠀⢀⡾⠋⠉⠉⠉⠛⠶⣤⡀⠀⠈⣧⠀⢀⣀⣸⠷⠞⠉⠀⠀⠀⠀⠀⠀⠉⠳⠾⣇⣀⡀⠀⣼⠁⠀⢀⣤⠶⠛⠉⠉⠉⠙⢷⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⡈⣻⡶⠚⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⢶⣟⢁⣀⣀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢿⣤⡴⠞⠋⠉⠉⠉⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠉⠉⠉⠙⠻⢦⣤⡿
⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⢀⣤⠖⠋⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠾⠛⠙⠲⣤⡀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⡿⠁⠀⠀⣠⣶⣟⠋⠙⢿⣛⠛⠛⠛⠛⠓⠚⠛⠛⠛⠛⣛⡿⠋⠙⣻⣶⣄⠀⠀⠈⢿⣤⣶⠟⠁
⠀⢠⡤⣤⣀⣀⠀⠀⠀⠀⠙⠻⣶⣤⣼⠏⠀⠈⠙⢦⡀⠈⢷⡄⠀⠀⠀⠀⠀⠀⢠⡾⠁⢀⡴⠛⠁⠀⠹⣧⣤⣶⠟⠋⠀⠀⠀⠀⣀⣀⣤⢤⡄
⠀⠀⢷⡀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠈⢷⡀⠀⢻⣆⠀⠀⠀⠀⣰⡟⠀⢀⡾⠁⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⢀⡞
⠀⠀⠀⠻⣦⡀⠀⠀⠀⢀⣠⣤⠶⠛⢿⣇⠀⠀⠀⠀⠘⣧⠀⠀⢿⡆⠀⠀⢰⡿⠀⠀⣼⠃⠀⠀⠀⠀⣸⡿⠓⠶⣤⣄⡀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠈⠛⠶⠶⠛⠋⠁⠀⠀⠀⢸⣿⢻⣶⣤⡀⠀⣿⠀⠀⢸⣷⠀⠀⣾⡇⠀⠀⣿⠀⢀⣤⣶⡟⣿⡇⠀⠀⠀⠉⠙⠛⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣏⠈⢻⣷⣿⡆⠀⢸⣿⠀⠀⣿⡁⠀⢰⣿⣾⡟⠁⣹⠇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠈⠛⠋⠙⣿⠃⠀⣸⡟⠀⠀⢻⣇⠀⠘⣿⠏⠙⠋⠁⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⣿⠀⢠⣿⠃⠀⠀⠘⣿⡄⠀⣿⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢰⣿⢠⣿⠏⠀⠀⠀⠀⠹⣿⡄⣿⡆⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⣿⣾⠋⠀⠀⠀⠀⠀⠀⠙⣷⣿⡇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⣆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣶⡄⠀⠀⠀⠀⠀⠀⢠⣶⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣄⣈⣙⣷⠶⠶⠶⠶⣾⣋⣁⣠⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣤⣀⣀⣀⣀⣀⣀⣤⠿⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⠶⠿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠒⠶⠶⠿⣼⣧⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣽⣷⣤⣄⢀⢄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⠉⠉⠉⠉⠁⠀⠀⣷⠢⠎⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢷⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢬⡆⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢏⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⣾⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⡇⠀⢠⠂⠀⠀⠀⠀⠀⠀⡏⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢐⠄⡸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣇⠀⠀⠀⡔⠀⠀⠀⠀⠀⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢜⠄⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠊⡀⠀⠀⠀⠀⠀⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⢎⠎⣽
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠞⣸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⡐⢈⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣧⠀⠀⠀⠀⠀⠀⢀⡾⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠊⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣬⠦⣄⣀⣀⣠⡴⡟⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠴⢥⣾⠵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⢪⢺⡆
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⠆⢀⠔⠀⠀⢀⠔⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡱⢣⢾⡇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠐⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⡰⢣⠞⡇
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⡡⠃⣧
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡱⠃⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠱⢁⢺⡇
⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠳⠦⢼⠇
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠻⠤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠴⠬⠓⠛⠚⠓⠧⠶⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣔⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠄⡀
⠀⠀⠀⣀⡴⠛⠋⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⣔⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠄
⠀⠀⢰⡏⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡁⡀⠀⠀⠀⠈⠁⢢⠀⠀⠀⡴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢷⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢇⠀⠀⣸⠃⠀⠘⠂⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶
⠀⠀⠀⠈⢧⡀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠢⣄⠰⠄⠀⠀⠀⠀⠠⠆⣠⠔⠀⠐⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠆
⠀⠀⠀⠀⠈⢳⡄⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚
⠀⠀⠀⠀⠀⠀⠹⡾⠃⢠⡤⡀⣹⠶⡀⢨⡦⡄⠀⣦⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬
⠀⠀⠀⠀⠀⣧⠴⠋⠒⠋⠀⠈⠁⠀⠈⠉⠀⠈⠛⠁⠐⠲⠛⠇⠀⣦⠀⠀⠀⠀⠀⠈⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃
⠀⠀⠀⢠⡀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢙⠂⢠⠀⠀⠀⠀⠘⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠷⠃
⠀⠀⠀⠀⢟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠈⠓⢻⠆⠀⠀⠀⠀⠀⠚⢢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠉
⠀⠀⠀⢤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠃⠀⠙⢿⡄⠀⠀⠀⠀⠸⢤⣼⠀⠀⠀⠀⠀⠀⠈⠉⠋⠷⠰⣖⡦⠶⠺⠊⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⢘⢻⠀⠀⠀⠀⠀⢠⡉⢀⠀⠀⠀⢀⡀⠀⠂⠀⡀
⠀⠀⢹⡞⠃⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⣠⣄⠰⣠⠀⠀⡀⠀⠀⠀⠀⠀⠙⠏⠀⠀⣰⠩⠈⠁⠊⢙⣿⠀⠀⠀⠀⡔⠉⡆⠀⠀⠀⣿
⠀⠀⣀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢣⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢧⣤⠆⠀⣿⡏⠉⠉⢹⡉⣯⡁⡀⠀⠈⠀⡠⠃⠀⠀⣼⣿⣄
⠀⠀⠈⠟⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠈⠃⠀⠀⣏⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⡇⠀⠀⠠⡀⢀⣼⠂
⠀⠀⠀⠠⢾⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡗⠚⠀⠀⠘⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⡴⠷⣄⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⣀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠃⠀⠀⠰⡀⢀⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⢱⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡊⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⢀⠀⠉⠉⠁⠀⠀⣸⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⡽⣤⠒⢦⣀⠠⢤⡀⡠⠦⣄⡎⠉⡿⠁⠀⠀⠀⢡⠀⠈⠄⠀⣠⢾⣩⡞⠀⠀⠨⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠸⠋⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠛⠓⠁⠀⠈⠂⠘⠋⠀⠀⠀⠀⠀⠈⠓⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⣤⠤⠶⠒⠒⡶⠀⠀⠀⠀⢾⠒⠒⠶⠤⣤⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠶⠺⠿⠯⣭⣉⡁⠀⠀⠉⢲⡀⠀⠀⠀⠘⣧⣀⣤⠞⠁⠀⠀⠀⠀⠈⠳⣤⣀⣼⠃⠀⠀⠀⢀⡖⠉⠀⠀⢈⣉⣭⠽⠿⠓⠦⣄
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠉⠻⣶⡀⢀⣨⣷⣤⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠦⣤⣤⣼⣅⡀⢠⣶⠟⠉⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢿⡆⠀⠀⣀⣤⣤⣤⡤⢬⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡥⠤⢤⣤⣤⣀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠘⢿⣴⠟⠋⠁⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠈⠙⠻⣦⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⢀⣤⠞⠋⠉⠛⠷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠛⠉⠙⠲⣤⡀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣿⠃⠀⠀⠀⣠⣶⣟⠋⠙⠻⣝⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣫⠟⠋⠙⣻⣶⣄⠀⠀⠀⠘⣷⣤⣾⠟⠉
⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⣾⠋⠀⠉⠳⣦⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣴⠞⠉⠀⠙⣷⣀⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⢻⡄⠉⠙⠛⠻⠷⠶⠶⠶⠶⠾⠿⢿⡏⠀⠀⠀⠀⠈⢷⡄⠀⠙⣷⡀⠀⠀⠀⠀⢀⣾⠋⠀⢠⡾⠁⠀⠀⠀⠀⣹⡿⠿⠷⠶⠶⠶⠶⠶⠟⠛⠋⠉⢀⡟
⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠀⠀⠀⠀⠀⠈⣷⠀⠀⠸⣷⠀⠀⠀⠀⣾⠇⠀⠀⣾⠁⠀⠀⠀⠀⠀⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠙⠷⣦⣄⣠⣤⡶⠞⠛⠉⠀⠀⣿⣷⣤⣄⡀⠀⠀⢻⡇⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⢰⡟⠀⠀⢀⣠⣤⣾⣿⠀⠀⠉⠛⠳⢶⣤⣄⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⡟⠿⣶⣄⢸⡇⠀⠀⢸⣷⠀⠀⣾⡇⠀⠀⢸⡇⣠⣶⠿⢿⡏⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⢷⣄⣨⢿⣿⣟⠀⠀⣼⡿⠀⠀⢿⣧⠀⠀⣾⣿⣿⣅⣠⡾⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⣿⡇⠀⢀⣿⡇⠀⠀⢸⣿⡀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⣿⡇⠀⣼⡿⠀⠀⠀⠀⢿⣧⠀⢸⣿⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣿⡇⣸⡿⠁⠀⠀⠀⠀⠈⢿⣇⢸⣿⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣿⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣶⣿⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⢿⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠙⢿⣇⣀⠀⠀⠀⠀⢀⣸⡿⠋⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣦⡶⠾⠛⠛⠛⠛⠛⠛⠷⢶⣴⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⡤⠤⠤⢤⣤⣴⠞⠋



⣶⣶⣶⣶⣾⡾⢷⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⣶⠿⢶⠷⣶⣾⡿⠷⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠰⢆⡐⡀⢀⡜⠃⠀⢰⡀⠢⢀⠀⠀⠀⠀⠀⠐⡀⠁⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡼⠁⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣀⣰⣠⠼⡤⣷⣸⣆⠀⢘⢨⠇⣄⣀⣀⣀⡠⠠⢠⠐⠀⠠⡄⠀⠈⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠥⡀⡀⢀⠠⠤⠀⠀⣐⡁⠀⠁⠀⣠⣿⠟⠩⢠⣤⡶⢶⡎⠍⢷⣾⡇⠀⡸⠽⡛⠉⠀⠀⠀⠑⢠⠀⠄⠈⠲⣄⠀⡌
⠀⠀⠀⣴⠈⢢⣈⣁⡃⣄⣀⠀⠀⣀⠀⠀⢴⠀⠒⠈⠉⠊⠝⡿⠃⠀⣠⣟⠁⢈⡥⠈⠑⢷⡇⠻⠓⠀⠱⠂⣀⠤⣄⡃⠀⠠⠋⢠⡆⠀⠀⡙⢂⠀⣀
⠀⠀⠀⣉⣛⣯⠉⢅⡀⠀⠉⡆⠀⠨⠄⠁⠸⢆⡀⠀⠁⢀⢾⠁⠃⢰⠁⠈⠻⡅⠀⡜⠂⣈⢿⠎⢀⢊⠀⠥⠁⠀⣸⣟⢿⡖⣂⠰⣃⢀⠠⠀⠠⠌⡍
⠀⠀⠐⢮⣭⡧⠀⠄⣠⣾⠙⣆⠀⡴⠐⠐⠀⠀⠉⠂⢀⣊⠀⠀⠀⡌⠀⠀⠘⠏⢀⠁⣤⢹⠀⠐⠂⠀⠴⣤⣤⠈⠀⢸⡀⢻⠀⠈⠀⠡⣶⠤⠀⡴⠙
⠀⠀⠄⢤⣬⠷⠋⠌⠀⠻⣷⣾⡒⠀⠀⢷⠀⠀⠆⠀⠀⣿⠀⢈⠲⡆⠀⠀⠱⠆⠈⠀⠈⠀⢀⠰⠄⠀⠈⢻⡙⢿⠶⡾⡡⣸⠆⣇⠀⠀⢈⣶⠛⠉
⠀⠰⠦⣤⣤⣶⡠⠤⠒⠀⠀⡉⢿⠀⡔⠀⠐⢠⣀⡰⠁⡈⠀⠀⠺⣗⡀⠀⢠⡄⠚⡀⠤⣠⠀⠀⠂⠀⠀⠀⠝⠉⢨⠧⢒⠍⣆⢘⣳⠀⡆⠈⠀⠀⣄
⠀⠀⠀⠻⡶⠟⢉⠀⠐⢰⡠⡅⡰⠂⠀⠶⢇⠖⠉⠉⠒⡾⠀⠀⠀⢈⣳⣄⣀⠈⣀⡠⣤⣬⣱⠖⢠⣌⡥⠀⣈⣷⣛⡉⠁⠀⡇⠀⠀⠠⠂⠠⠈⠁
⠀⠀⠀⣼⣅⡵⠙⠐⠈⠀⢒⡢⠂⢴⣦⢰⢧⡀⠀⡀⠀⠸⣦⠀⠀⠀⠈⠉⣿⢛⣻⠾⠒⠢⣿⠆⠐⠾⣟⣻⡉⠀⠀⠂⢀⣼⢇⠍⠀⠀⠀⠘⠃⣀
⠀⣠⣀⣀⣉⣶⠂⠴⣶⠿⢽⠀⠀⣼⢽⡄⢠⠡⢀⢼⡄⠀⢻⣿⡖⠒⢒⣴⢿⡋⠀⡴⢢⡾⠛⢷⡾⢦⠀⡹⠷⡶⠒⣲⣿⡏⠀⠀⢄⠐⠾⡄⠀⠈⡧
⠀⠹⠌⠋⠙⠿⠀⠀⠀⡀⠈⢡⡊⡘⠀⡐⡮⢦⡇⣀⠀⡴⠸⣿⣍⡉⠉⣀⠤⡿⡟⠒⠸⠗⠤⠸⠇⠚⢛⢷⠤⣀⡉⣩⡿⠣⢠⡴⣼⢃⠀⣁⢰⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢘⠇⠁⢀⡴⠇⠈⠁⢝⢸⠄⠀⠈⠹⢓⠣⠴⠮⣴⡷⢀⡀⠀⠛⠀⢀⠀⠼⣦⠵⠆⢚⠋⠀⠀⠈⠃⠸⢆⡀⠈⠰⠀⠐
⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠈⣠⢰⠘⠆⡠⠀⢁⠈⠋⠀⠀⠂⠀⠀⠳⢄⠈⡸⢦⣀⣙⠂⠀⠘⣁⣠⡄⢉⠀⠰⠂⠀⠀⠀⠀⠀⢁⠨⠁⡄⣆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢸⠇⠋⢀⠀⢷⡄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂⠦⠠⠟⠀⡀⠃⠽⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢦⢀⣸⢀⠆⡡⠀⢀⠀⠢⠁⠀⠀⠀⠀⠀⠀⢈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠔⠀⡀⠀⡄⠲⠀⢇⡀⡴⠃⠁
⠀⠀⠀⠀⠀⠀⣰⠤⣄⠀⠀⠀⠤⠘⡿⠀⣀⠶⢀⠠⢄⠀⠠⠀⠀⠀⡤⠀⠀⢸⠀⠀⠀⠂⠀⠐⠀⠀⡄⠄⠀⠀⠄⠀⡠⠄⠀⠾⡄⠐⢿⠃⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠤⠀⠀⠈⠀⠀⠈⠔⢼⡠⡈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠠⣄⡷⠂⠃⠀⠀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠺⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠖⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢘⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠃
⠀⠀⠀⠀⠀⠀⡑⡄⠀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡀⠀⠀⠀⠀⠀⠀⠐⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠓⠒⠒⠒⠒⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠒⠀⠀⠀⠀⢸
⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣧⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠦⠾⣽⣷⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠃⠠⠄⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣤⠁⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⣺⣿⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠒⠋⢋⠉⠄⠰⠿⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⣉⢠⡤⠀⠆⣒⣂⣩⡤⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡫⣶⢀⣃⡴⠴⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣾⣟⣷⣶⣶⣶⣶⡶⡶⡖⣖⣒⣚⣛⣟⣋⣭⡩⡴⠦⠶⠒⡛⢋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⠟⠛⢛⡛⢉⡉⢉⡉⢩⡅⢡⡅⠠⣄⠀⡆⠰⠇⠘⣃⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⡄⠰⠆⠘⣁⣘⣃⣀⣥⣬⣥⣤⠥⠬⠥⠶⠶⠖⠒⠚⠋⠉⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣶⣾⣿⣿⣿⣯⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⠖⡒⠋⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠿⠛⠛⢛⡋⠉⡭⠉⢩⠉⠁⠄⠀⢰⠀⠀⠇⠀⣓⣀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢩⠄⠀⡆⠀⠩⢀⢀⡃⣀⣁⣀⣨⣡⣤⣤⠤⠴⠴⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⣂⣠⠤⠶⠒⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣷⣤⣤⣤⣤⣤⣤⡤⡤⢤⠤⠤⠤⠴⢶⢶⣶⣶⡶⢶⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣷⡿⣶⣺⢹⣦⣞⣼⣣⣷⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠓⠛⠛⠛⠚⠓⠓⠚⠛⠛⠛⣛⣛⣉⣩⡭⠿⠿⣛⠻⢴⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠶⢖⠒⢋⢉⣩⠉⠤⡀⡶⠀⠿⢀⣚⡤⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢐⠹⢂⣚⣡⣭⠴⠴⠴⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢻⡛⡟⣻⠟⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⢟⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣮⠻⣿⡧⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣟⢩⣛⠿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠀⠈⣿⣿⣿⣿⣿⣿⣿⣏⣰⣷⣞⣛⣭⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣯⣽⡆⢿⣷⣯⣟⣛⣻⣿⣿⣿⣿⣛⣻⣍⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡟⣿⣛⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣷⢧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣸⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⡏⣿⡜⣿⣿⣿⣿⣿⣿⣿⡿⣼⡇⣿⣿⣿⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⢻⣿⣎⢿⣿⣿⣿⣿⡿⣳⣿⢷⣿⣿⡟⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⡏⣿⣿⣷⣝⠿⠿⣛⣵⣿⡿⣾⣿⡟⣽⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⡸⣿⣿⣏⠀⢸⣿⣿⣿⢳⡿⣫⣾⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⠹⣿⣿⣿⣿⣿⡿⢋⣭⣾⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼⣿⣿⣿⣶⣟⣾⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⢿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣼⡻⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡿⣣⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⢤⢤⠤⠤⣄⢀⡤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠚⠚⠚⠒⠒⣿⢸⣛⣛⣛⣛⣛⣛⣛⣛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣄⠀⠀⣿⢸⢐⠶⡶⡶⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠘⠧⣌⠠⠄⣹⠘⠒⠖⠒⠶⢒⣲⣒⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠶⠟⠀⠀⣿⠀⠀⠀⢀⡼⠋⣭⡙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣠⣀⣄⣿⠀⠀⠀⠸⣇⠀⣽⡆⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠿⠧⠤⠤⠟⠀⠀⠀⠀⠙⠳⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡿⠿⣿⣿⣿⡿⠿⢿⣛⡇⠀⣰⠞⣛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣏⢺⡇⡷⠀⠀⣿⡇⢸⡇⠀⣻⡅⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣙⣶⣶⣾⣿⣿⣷⣶⣾⣉⡇⠀⠻⣦⣭⣥⠾⠁





⠀⠀⠀⠀⠀⠀⠀⢰⣖⡆⣴⡄⢦⡴⢰⣄⣶⢰⣒⢰⣆⣴⢲⡖⠀⣦⣠⡆⣶⡒⢲⡖⢰⣀⣆⡔⢲⣰⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠘⠉⠙⠘⠁⠘⠉⠘⠘⠒⠘⠈⠋⠈⠁⠀⠃⠁⠃⠛⠒⠈⠃⠘⠀⠃⠑⠚⠘⠓⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡀⢂⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣐⠂
⠀⠀⢠⣈⣾⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣄⡄
⠀⠀⠈⢿⡟⡛⢁⣶⣷⣽⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⣀⣤⣶⣾⣷⠀⠀⣿⣿⣿⠟⠋⠀⠺⣿⣷⣶⣀⣀⣀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡿⣩⣶⣦⡙⠛⢿⣿
⠀⠀⠀⢸⣿⣿⣌⠿⣿⢟⣽⠟⠁⠀⠀⢀⣀⣀⠀⠀⠉⠉⢻⣿⡿⠀⠀⠘⣿⣿⠀⠀⠀⠀⣿⣿⣿⠋⢉⣿⣷⡄⠀⣀⡀⠀⠀⠙⢿⣧⠻⣿⣿⢿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣾⠟⠁⠀⠀⣠⣾⡟⠻⢿⣿⣦⡀⠀⢠⣿⡇⠀⠀⠀⢸⣿⡆⠀⠀⢀⣿⣿⡏⠀⠸⣿⣿⠇⠀⢹⣿⣶⡄⠀⠀⠉⢷⣮⣴⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⡿⠁⣰⡆⠀⠀⠻⡿⠃⢠⡞⠻⣿⣇⠀⠸⣿⣧⡀⠀⣀⣼⣿⡇⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⣴⣿⠟⠋⠀⢀⣤⣀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⢀⢸⣿⣿⠏⠀⣰⣿⣷⡄⠀⠀⠀⢠⡿⠀⠀⠘⣿⣆⠀⠻⣿⣿⣿⣿⣿⣋⣀⣠⣴⣿⣿⣿⡀⠀⠀⠀⠀⣾⣿⠃⠀⠀⢀⣿⡿⠿⣷⠀⠈⢻⣿⣿⣿
⠀⠀⠀⢸⣿⠏⢠⣾⣿⠋⠉⠉⠀⠀⢠⣿⣷⣦⣤⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡈⣿⣿⣄⣀⣴⣿⡟⠀⢀⣤⣄⡀⠀⢻⣿⣿
⠀⠀⠀⢸⡏⢰⡿⢩⣿⣦⣤⠁⠀⠀⠀⠙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⣤⣄⢻⣿⡿⢿⣦⡀⢹⣿
⠀⠀⠀⠏⠀⣿⠀⠉⠙⣿⣿⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣿⣿⠈⢿⣆⠀⣹⣧⠀⢿
⠀⠀⢠⠀⠀⠻⢶⠶⠟⠁⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣆⠛⠛⠀⠀⡆
⠀⠀⠸⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⡄
⠀⠀⢸⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠁
⠀⠀⢸⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠈⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢾⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠶⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⠘⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢀⠹⣿⠟⢿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠹⣿⠻⢂⠀⠛⠛⠀⠸⡿⠃⠀⠙⠛⠀⠉⠁⠉⠈⠹⣿⣿⠏⡱⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⢘⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⠀⠀⠀⠂⢈⠀⠀⡀⠀⠀⡠⠆⠀⠆⣀⠠⠀⡀⠀⢀⠒⡳⣠⠎⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠀⠀⠀⢀⣡⣬⣶⣄⣠⡄⠰⠶⠶⠻⠇⢤⣤⣤⡉⢀⡑⠋⠀⢠⠜⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⢀⠀⠀⠀⢹⣿⣿⣿⡿⣿⣿⣿⣿⡿⠋⠑⣤⡀⣀⣶⣽⡇⣩⣀⣱⣖⠒⠉⠙⠒⣻⣤⣉⢧⣝⢶⠛⠋⢀⠰⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠃
⠀⠀⠈⠀⠀⠀⠀⠻⢿⣿⣷⣿⣿⣿⣿⣿⣾⣷⣀⠻⢿⣿⣤⣀⠙⢿⣿⠉⢳⣤⣴⠋⢸⢻⡿⠆⣈⣿⠵⢎⣤⣶⣾⣿⣿⣿⣿⣿⡿⠛⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠈⠈⠛⣧⡄⢸⣄⢺⣏⣹⣂⣾⣡⡶⠛⠉⠀⢠⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⠿⣿⣿⡀⠀⢀⣰⣾⠻⡄⠻⠷⣿⡿⠟⠋⠹⣅⡤⡔⠳⣤⣿⣾⣿⣿⣿⣟⠙⣻⣿⢛⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣤⣿⣿⣷⣾⣿⡿⠁⣤⠙⠿⣆⠀⠀⣤⡶⠒⣿⣷⣿⣦⣾⣿⣿⣿⣿⣿⣿⣯⣙⡵⠋
⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣈⠹⠟⠉⣿⠁⠘⠁⠀⠿⠀⠀⢹⠆⢠⡇⠀⠀⠿⠉⠀⣿⠉⣩⣹⣏⠻⣟⡿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⢸⣧⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠚⢻⡌⠓⠀⠐⢃⡂⠀⡀⠈⣴⣄⠠⠀⠘⠀⣤⠜⣃⠘⣩⡥⠬⠛⠣⠤⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⣼⡇
⠀⠀⠀⠀⣿⣦⠀⠀⠃⢡⠀⠀⢀⣄⠤⠀⣠⡴⠛⠃⡀⠀⠘⠣⠎⢁⠘⡥⢀⣥⣌⡁⠀⡃⠤⠐⠚⢉⠛⢡⠀⢓⠀⠱⠶⠁⠄⢲⣤⣀⡌⠃⠀⠀⣴⣿⡷
⠀⠀⠀⠘⣿⣿⣧⡀⠀⠘⠃⣈⡡⠄⢀⠠⣀⠀⠚⢠⠘⢀⠶⠀⠘⣹⠇⠠⠛⠄⡀⠀⠷⡸⠇⠘⢃⡈⠙⡄⠀⠘⣉⠤⡊⠰⠆⠀⡄⡸⠏⠀⠀⣼⣿⣿⡷
⠀⠀⠀⢀⣿⣿⣿⣿⣄⠀⠀⠙⠁⢀⡈⠑⠈⢀⠛⢠⡄⠈⠀⠀⢘⡧⠄⠳⠞⠀⠀⢰⠀⠀⠐⢎⠉⠙⠄⢹⢠⡶⡌⠂⠈⢀⣠⡤⠈⣀⠀⢠⣾⣿⣿⣿⡇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣧⡀⠀⡀⠀⠠⡼⢀⡌⠀⠂⠃⠀⠙⠀⡘⠖⠀⡉⠀⢢⡔⢀⡴⡄⠤⢹⠈⠃⠀⠛⠀⠀⠀⡰⠀⠈⠁⢠⠆⢁⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⡿⣫⣶⣮⡹⣷⣄⠲⠆⠀⡘⠀⡝⢠⣤⠀⠀⢀⠣⠀⠐⠃⠀⠁⠈⠀⠀⠀⠀⠀⠀⠰⠂⡠⠢⠔⠞⠀⢀⠰⠞⢃⣴⣿⣡⣶⣶⣍⣿⣿⡇
⠀⠀⠀⠐⣿⣿⣅⡻⣿⠟⣣⣿⣿⣷⣤⣀⠃⢶⡀⠀⠀⡠⠐⢠⠄⠲⠃⡀⢀⡐⣤⠄⡠⠙⠠⠄⠠⠀⠊⠀⠀⢤⠀⠀⢈⣤⣶⣿⣿⣿⣝⣿⡿⢋⣙⣿⣷⡀
⠀⠀⠀⡀⢻⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣈⡈⢡⡄⠀⠀⠠⠀⣠⡈⢠⠀⣀⠁⠀⠀⠀⢀⡀⡀⠠⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⠃⠁
⠀⠀⠀⠈⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠀⠀⠀⠐⠃⠘⠀⠘⠀⠉⠀⠐⠀⠀⠈⠀⠀⠒⠛⠛⠛⠛⠛⢛⣛⠛⠛⠛⡛⠛⠻⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣯⠶⢷⣻⣷⠂⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠋⠉⠀⠿

⠀⠀⠀⠀⣴⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⡄
⠀⠀⠀⣸⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⡀
⠀⠀⠀⣿⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⢸⡇
⠀⠀⢸⣿⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⢸⣿
⠀⠀⢸⣿⠘⠐⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠐⠘⢸⣿
⠀⠀⠀⣿⡀⠇⢀⢿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⢇⠀⠆⣸⡇
⠀⠀⠀⢻⣇⠸⡘⣌⢿⣟⣿⣿⣿⣿⣶⣶⣖⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⢒⣶⣶⣾⣿⣿⣿⣿⣿⢏⡞⡸⢀⣿⠃
⠀⠀⠀⠘⣿⣴⠣⡘⣆⢿⣮⢿⣿⣿⣿⣿⣿⠃⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢻⣿⣿⣿⣿⣿⢯⣿⢇⡞⡠⢳⣼⡏
⠀⠀⠀⠀⠘⣿⣦⠓⡌⠳⡙⢷⣝⢿⣿⣿⠋⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⢻⣿⣿⣟⣵⢟⡵⠋⡴⢣⣾⡿
⠀⠀⠀⠀⠀⠙⣿⣧⡹⣔⢌⢢⣽⣿⠛⠁⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠙⢻⣿⣥⢎⢔⡼⣡⣿⡟⠁
⠀⠀⠀⠀⠀⢠⡏⠛⠟⠋⠛⠛⠉⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠈⠙⠛⠋⠛⠟⠋⣧
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⣀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⡀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⡰⠫⠍⡒⠤⣀
⠀⠀⠀⠀⡰⠑⠀⢠⣀⠁⠀⠉⠒⠤⣀
⠀⠀⠀⡰⠁⠀⢠⡏⠻⠿⣶⣤⣀⠀⠀⠉⠒⠤⣀
⠀⠀⡰⠑⠀⢠⣿⡃⠀⠀⠈⠉⢿⣓⡄⠀⠀⠁⠂⠍⡒⠤⣀
⠀⠠⠀⠀⢠⡿⠷⠹⠦⣄⡀⠀⣼⡟⠃⠀⠀⠀⢀⣤⣀⠀⠀⠁⠀⠄
⠰⠀⠀⢀⡻⣏⠀⠀⠀⠈⠐⠂⠚⠃⠀⠀⣀⣄⢻⢿⣿⡇⠀⠀⠀⠁⠂⠅⡢⢄⡀
⠡⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠾⡿⠛⠁⢸⣿⡇⠀⠀⠀⠀⠀⣄⣀⠁⠀⠕⡢⢄⡀
⠀⠀⠁⠒⠤⡀⠀⠀⠀⠀⠀⢀⣴⣶⡍⠿⢳⣦⣀⣨⣿⡇⠀⠀⠀⠀⣼⡿⠏⠀⠀⣤⣀⠁⠊⠕⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⢻⣿⡇⠀⠀⠀⢸⣾⠏⠀⠀⠰⣿⡿⢿⣶⣤⡀⠁⠊⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⣀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠤⣄⠉⠀⠀⠐⣿⡟⠀⠀⠈⠙⣿⣷⡀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⣄⡟⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⠂⠀⠀⠈⠀⠀⢀⠂⣌⡀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠳⢗⣦⣠⣶⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠈⠉⠉⠀⠀⠀⠐⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠂

⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡆⢻⣇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣾⠇⡆
⠀⠀⢷⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⢠⡇
⠀⠀⠘⣆⠀⠈⠹⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⠟⠁⠀⡼
⠀⠀⠀⠘⣆⠀⠀⠈⠻⣶⣤⣉⠙⠛⠷⣦⡀⠀⠀⠀⠀⠔⠒⠛⠉⠉⠉⠉⠉⠙⠓⠒⠄⠀⠀⠀⠀⢀⣠⣴⠾⠛⢋⣩⣽⠟⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠉⠛⠿⢶⣶⣬⣽⣦⣀⣀⠀⠀⠙⠲⣦⡀⠀⠀⠀⣤⠖⠋⠀⠀⠀⢀⣴⣟⣩⣤⣴⡶⠟⠋⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡉⠛⠳⣄⠀⠘⢿⡄⢀⣾⠋⠀⢀⡴⠞⢛⡿⠋⠉⠉⠉⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢳⠶⠶⣷⣦⣤⣈⠃⠀⠘⣿⣼⡏⠀⠰⠋⣀⣠⣼⣥⣶⡖⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠸⣧⣀⠀⠀⣉⠛⢷⡀⠀⢻⣿⠁⠀⣰⠿⡋⠉⠉⢀⣼⠇⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠹⣦⡀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠈⠉⠀⠀⠁⠀⠙⠶⠾⠿⠋⠀⠀⠀⣼⠏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠉⠛⠻⠷⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠟⠋⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠁⠀⠀⣀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡟⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⢠⡿⠉⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢹⡀⠀⠀⠀⠀⠀⡿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⡈⠃⠀⠀⠀⠀⠀⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠀⠀⠀⠀⠀⠀⠘⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣀⣀⣀⠀⠀⠜⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣽⣯⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠀⠀⠓⠈⠅⠀⠀⢰⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠍⣾⠈⠙⠒⠒⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠠⡀⠀⠀⠈⠁⠀⠀⡀⠀⢀⡄⠀⡥⠀⠀⠀⣰⠊⠁⠈⠻⣆
⠀⠀⠀⠀⠤⠤⠤⣤⡤⠴⣦⣤⡤⠤⢆⠄⡀⠀⠰⣄⣀⣀⡈⡟⠉⣍⠂⡺⢀⡀⠀⠀⢠⣴⠇⠀⠀⠀⠀⠈⠉⠳⢦
⠀⠀⠀⣀⣀⢀⣌⠁⠀⠘⢻⣿⡿⠆⠈⠃⣄⣀⡤⠻⣿⣿⡐⡟⠀⡈⣠⠇⠀⢉⡆⠀⠈⢿⣆⠀⠀⡀
⠀⠀⠀⠀⠀⢺⣤⡀⣶⣿⣿⠆⠠⢀⠀⠀⠸⠀⠀⠇⢸⢸⣽⣷⣰⡷⢺⠀⠀⢸⢁⠀⠀⠀⠉⠳⠮⠤⠖⢂⣢⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠟⠸⠉⠀⠀⡀⠀⠀⠘⠢⣂⡀⠀⢘⢻⣇⣻⡷⠞⠅⠀⠈⠈⠑⠠⡀⢀⣀⣣⣼⡿⠿⣯⣙⠓
⠀⠀⠀⠀⠀⠀⠘⢯⠀⠀⢀⡀⠀⡠⠀⠀⠀⠀⠀⠉⠓⠺⠞⠋⠘⣏⠔⠁⠀⢂⡀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠈⠛⠕
⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠚⣮⡉⠉⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠘⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡹⣧⠀⠀⠶⠀⢺⣦⠤⢤⠤⠶⠶⠛⠛
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄⠀⠉⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⢧⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢰⠀⠀⢀⠀⡅⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣆⢡⠀⠀⠀⠀⠀⠠⣠⡴⠞
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡙⠳⣥⣀⣀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢷⡙⢏⠛⠛⠯⣉⠁
⠀⠀⠀⠀⠀⠀⢁⠀⡀⠀⠀⠁⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣽⣌⣳⣄⢀⠄⠉
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢤⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠟⣯⠁⠠
⠀⠀⠀⠀⠀⠀⢀⡯⡁⠀⠈⠈⠉⠉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡻⡇⣿⡀⠁
⠀⠀⠀⠀⠀⠀⣟⠁⠚⡇⠀⠀⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⢄⠨⠫⢸⠄
⠀⠀⠀⠀⠀⠰⣧⠀⠨⠇⠀⠀⢈⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⡄⠈
⣶⣶⠶⠶⠶⣶⣿⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣾⣶⣷⣶⣿

⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡝⠿⢋⣴⣦⡀
⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠠⠂⠠⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠀⠋⣉⠀⠉⠀⢉⠙⠀⠲⠢⡀
⠘⢫⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡝⠃⠀⠀⠀⠀⠀⠀⠀⣠⣦⠖⠀⠀⠀⠲⢦⣦⠀⠀⠀⠀⠀⠀⠀⢰⡇⠊⠁⠘⠧⣤⣀⣠⣀⣤⠼⠃⠀⠁⢸
⠀⠈⢱⣶⠄⠀⠀⠀⠀⠀⠀⠀⣶⡎⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢮⣔⡀⠄⠀⠀⠀⠀⠀⠠⢀⣢⡵⠞
⠀⠀⠀⠁⢾⣿⢆⣴⣦⡐⣿⡷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠋⠛⠓⠛⠚⠛⢻⢻⡅
⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠂⠒⠒⠒⠐⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡴⣆⡡⠤⠬⣀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠊⠉
⠀⠀⠀⡸⠀⢁⣠⣄⣄⡀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⠏⠈⠀⠉⠘⣏⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⡄⠀⠀⠉
⠀⠀⣸⠁⣰⣿⠿⣿⡿⣻⣦⠀⢣⠀⠀⠀⠀⠀⠀⢀⣴⣫⣤⣠⣏⣀⣠⣄⣄⣀⣘⣇⣠⣌⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢙⠁⠀⠀⠀⠀⡛
⠀⠀⣯⠀⣽⣿⣷⣬⣿⣿⣿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠙⢦⡑⠈⢣⠀⠀⠀⠀⠀⣰⠃⢘⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣆⠀⠀⠀⢠⢿⡀
⠀⠀⠘⡄⠈⠿⣿⣿⣿⠿⠃⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢷⠀⠀⠀⣴⠃⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡇⣿⡂⠀⠐⣾⠸⣗
⠀⠀⠀⠘⡄⡤⣀⣀⣀⣠⣄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣣⠀⣰⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣦⣾⡧⠀⠸⣿⣴⣿⠄
⠀⠀⠀⠀⠉⠁⠀⠀⠰⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠉⠋⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⡄⢲⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⡾⠻⣶⡢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠃⠒⠀⠥⣑⡂⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⡃⢙⣿⣦⡾⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⢀⡤⠞⠁
⠀⠀⠀⣿⣿⠆⠈⠉⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠙⢿⣿⡟⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠋
⠀⠀⠀⢸⣿⠀⠀⢀⠤⠭⠿⠻⠯⠥⢄⠀⠀⠀⠀⠀⠀⠀⣠⡿⢿⣦⡛⠉⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣻⡖⠾⠆⣟⣇⠆
⠀⠀⠀⠀⠿⠆⠀⠈⠉⠉⢬⣿⠍⠉⠉⠀⠀⠀⠀⠀⠀⠀⢹⣷⣴⡟⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣋⢾⠿⣿⡷⡷⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⠃⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠷⣟⠾⠷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢐⣛⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⡗⠀⠀⢺⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⢸⣿⡋⠉⠉⠀⠀⠀⠀⠈⠉⢙⣿⡇⠀⠀⠀⠀⠀⠈⣿⣷⣄⡀⠀⠀⠀⠀⠀⣀⣴⣿⡏
⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⢀⡾⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡷⠀⠀⠰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠙⠛⠁






⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠄⠀⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠸⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠇⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⡆⠰⣿⣿⠿⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠛⠓⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠛⠛⠳⢶⣄⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠱⡌⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⣿⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡀⠀⢠⡿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⣄⣤⠟⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⣩⠶⠾⠿⠿⢶⣄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠁⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣠⡖⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠐⢋⣽⠇⠀⠀⠀⠀⠀⡠⠶⢦⡀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⣠⠀⣠⡟⠀⠀⠀⠀⢀⣤⠖⠒⠂⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣱⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⢁⣴⠏⣰⡿⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⢠⡾⣯⣤⠟⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⣛⣉⣤⣤⡶⠟⠁⢠⣿⠇⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣧⣸⡏⠀⠀⠀⠀⠀⢀⠀⠠⠴⠚⠛⠋⣉⠉⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣄⡀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣄⡀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣶⣶⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠈⢿⣿⣧⠀⠀⠀⣠⠞⠛⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠘⢿⣿⣧⡀⠀⠉⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⢀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣀⠀⠀⠀⠀⠉⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣤⣤⠀⣤⡀⠀⠀⣤⠀⠀⠀⢠⣤⡄⠀⠀⡤⠀⢠⣤⣤⠀⣤⣀⠀⠀⣤⣤⠀⠀⢠⠄⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣿⣿⠀⠘⠇⠀⢸⣿⣇⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⢰⣿⣦⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠠⠀⣿⣿⡄⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠸⣿⣄⠀⠀⣿⣿⠀⠀⠀⠀⢀⡙⠿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠠⡄⠀⢸⣿⣷⠀⠈⠻⣧⠀⠀⠀⠀⢨⣿⣿⠀⢿⣿⠆⠀⠹⢿⣄⠀⠀⠀⠘⠃⠀⠀⠽⠃



⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠀⠔
⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠳⠀⠀⠀⠀⠀⠠⠧⠴⠀⠀⡄⣀⠆⠀⠀⠀⠀⠀⢈⠀⣤⡄⠈⠀⠀⠀⠀⢠⠘⠈⠀⡠⠻⡇⠐
⠀⠀⠀⠀⠀⠀⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠙⠬⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠆⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⢦⡀⠀⠀⠠⢆⠀⠀⠀⠀⠀⠽⠩⡇⠀⠀⢏⠹⡇⠀⠀⠀⢀⠈⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⢀⠎⠂⠰⠀⢰⠁⠀⠁⠀⠀⠀⠰⠚⠀⢻⠀⠔⠛⠀⢧⠀⠀⠀⠘⢴⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠈⠁⠋⠁
⠀⠀⠀⠘⠄⠀⠐⡇⠈⠀⠀⢰⠄⠀⠀⢸⠛⢿⠞⠀⡘⠳⡷⠞⠀⠀⠀⠐⠀⠀⠀⠠⠈⠀⠈⠁⠀⠀⠀⠀⠀⠈⣠
⠀⠀⠀⠀⠀⠋⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠠⡎⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡠⠤⠀⠁⠀⠀⠀⠀⠀⠀⡠⠼⠮⡀⠴⡷⠅⠀⠀⠀⠀⠀⢄⠔⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠶⠀⠀⠀⠀⢐⠀⠂⠘⣀⠁⠀⠀⡀⠀⠀⠀⠀⠒⡀⡤⢈⠂⢄⠄⠀⠀⠀⠀⠐⠁⣄⢸⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⠠⡞⡳⡖⣔⠏⢱⢶⠺⡹⠄⠀⠀⠀⠀⠒⠲⣧⡀⠀⢐⠀⠀⠀⢀⡤⠆⠙⠛⠰⠴⣤
⠀⠀⠀⠀⠀⠀⠘⠶⠽⠃⠀⠀⠀⠀⠀⠁⠀⠄⠁⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢼⠀⠉⠀⠀⠀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢐⣀⢀⠀⠀⠀⠀⠀⢀⠀⠐⡡
⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠈⢿⠤⠚⠉⠁⠈⡷⢌⠖⠀⠀⠄⠠⠛⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⢀⣀⡴⠃⢠
⠀⠀⠀⠀⠸⠀⠂⢰⠀⠺⠀⠠⠀⠀⠀⢨⠁⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⢢⠀⡀⠀⠀⠀⠀⠈⠛⠂⢀⣷⡁
⠀⠀⠀⠀⠀⠈⠠⠁⠵⠀⠀⠀⠀⠀⠢⠇⠙⠃⠘⠉⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠁⠑⠐⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⠀⠀⠀⠀⣆⠀⡾⠀⠀⢶⠀⣠⠀⠀⠀⠀⠀⠀⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⡰⢀⣿⣾⠁⠀⠀⠈⣷⢿⡀⢦⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠁⠀⠀⠀⣠⣾⣷⣻⢿⣷⣶⣀⣀⣶⣾⣿⣿⢾⣿⣤⡆⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⡘⣦⣶⣭⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣭⣶⣴⣇⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠈⢻⣟⣿⣿⣻⡟⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⠆⠈⢿⣿⣿⡿⠁⠰⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣉⠻⣿⣿⣿⣿⣮⠄⠀⠈⠃⠘⠁⠀⠠⣶⣿⣿⣿⣿⠟⣁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⣿⣿⠿⠃⠁⠀⠀⠀⠀⠀⠔⠀⠘⠿⢿⣿⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣾⣿⣶⣄⣀⣀⣄⣀⣠⣶⣿⣿⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢻⣿⡏⠿⠿⣿⣿⣿⠏⢸⣿⣿⣿⠿⠿⢹⣿⠋⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⢦⠀⠛⢿⣬⣆⣐⣂⢙⣿⣿⡁⣒⣀⣐⣥⡿⠛⢀⡔⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠈⣿⣿⣿⠏⠘⠿⠿⠗⠹⣿⣿⣿⠁⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢸⣿⡉⣿⣦⣤⣄⣀⣤⣴⣿⢿⣿⠇⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡻⠶⣤⣤⠶⢟⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⠀⠀⢰⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠘⠻⣿⣿⠟⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠂⡔⢲⢰⣆⣴⢰⣲⢠⣦⢰⣆⡆⢢⡔⠀⣶⣰⢀⣶⡀⣦⣰⡆⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁⠉⠋⠈⠈⠉⠈⠀⠈⠈⠈⠀⠁⠈⠀⠀⠁⠈⠈⠉⠁⠉⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠

⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠐⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠋⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⠘⠛⠛⠿⢻⣦⣤⣒⣆⣠⣤⠶⣤⣿⣿⠿⣿⣃⣠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣦⣠⣠⣞⣋⡈⢁⢸⣿⡇⢸⣿⣯⣁⣀⡁⢉⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠢⠌⠐⢢⣹⣿⠟⠃⢹⣾⠿⠡⡪⡬⢯⣿⣿⡛⠩⠽⠭⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⣤⣿⣿⠀⠘⠿⢿⣷⣾⣭⠟⣪⣿⣿⡁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡷⢾⡄⠀⠙⠉⠀⣰⣿⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⣅⠈⡇⠀⠀⠀⠀⡿⠁⣿⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠿⣦⣿⠀⠀⠀⢸⣷⣾⡿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡀⣿⣛⡒⠀⠐⣨⢿⣿⢑⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣿⡍⣿⣀⣚⣉⣸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⢛⣋⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡴⠶⢶⣶⣾⡧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣩⣿⣿⣥⣶⠶⠶⠾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⡶⢧⡀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣯⣶⣶⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⡾⣷⣦⣞⣱⠃⠈⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣿⣟⡿⣿⣟⣿⣿⣶⣇⣀⡟
⠀⠀⠀⠀⠀⠀⢠⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⢻⣿⣿⣿⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡛⣾⣿⣳⢿⣿⡄
⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⣸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣯⣿⣿⣿⣿⣿⡪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⡟⢃⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣻⢡⣿⣗⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣇
⠀⠀⠀⠀⠀⠉⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣛⣛⣉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢿⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠙⠻⣿⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠟⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠁


⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣦⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣛⡯⢞⣿⣿⣧⣼⣿⣿⡷⢽⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠽⣴⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣾⣻⡇⢸⡻⢿⡟⣿⣟⣻⡿⢻⡿⢝⡇⢸⣟⣧⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠱⡀⠀⣱⣯⣯⣻⣿⣇⠘⠳⣿⣿⣿⣿⣿⣿⣿⣿⠞⠃⢸⣿⣟⣽⣽⣆⠀⢀⡎
⠀⠀⠀⠀⢀⠂⠇⢸⣷⣮⣷⣿⡿⢻⣦⣄⣀⣹⣿⣿⣿⣿⣏⣀⣠⣴⡟⢿⣿⣿⣥⣾⡇⠸⠀⡀
⠀⠀⠀⠀⡂⢰⡤⠼⣿⣿⣿⣿⡏⣰⠿⠯⢥⠶⣷⣯⣽⣾⠷⡬⠽⠟⣆⢹⣿⣿⣿⣿⠧⢤⡆⢀
⠀⠀⠀⠀⠱⣀⣑⣤⣾⣿⣿⢿⣷⡿⡀⠖⠪⣀⠘⣏⣹⠃⣠⠗⠲⢀⢿⣾⡿⣿⣿⣷⣤⣊⡠⠎
⠀⠀⠀⠀⠀⠀⠀⣰⣺⣕⢎⣱⣮⣿⣓⠀⠈⠏⠀⠉⠉⠀⠻⠁⠀⣺⣿⣵⣎⡱⣪⣗⣆
⠀⠀⠀⠀⠀⠀⠠⣽⢞⣯⡿⠟⢿⣿⣿⡀⠀⠀⡆⠀⠀⢰⠀⠀⢀⣿⣿⡿⠛⢿⣽⡣⡯⠄
⠀⠀⠀⠀⠀⠀⠀⢈⣦⣉⠉⢉⣼⢟⣿⣿⠀⡇⠀⠀⠀⠀⢸⠀⣿⡿⡟⣧⡉⠉⣉⣴⡁
⠀⠀⠀⠀⠀⠀⡶⠯⠀⣊⣽⣿⢻⣤⣾⠿⣷⠘⡦⠀⠀⢔⠃⣾⠿⣷⣤⡟⣿⣯⣁⠀⠽⢶
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⠨⢫⢾⣿⣿⣿⣹⣾⡿⣀⣀⢿⣷⣏⣻⣿⣿⡷⢽⠅⠈⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣯⣞⣿⣯⡛⣿⡻⣷⣦⣴⡾⢟⣿⢛⣽⣿⣷⣭⣦⠞⠁
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠾⣿⣿⣿⣌⡳⢶⣏⣹⡶⢞⣡⣿⣿⣿⠷⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢏⣛⣿⡿⣿⣿⣛⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠙⠿⢿⣼⣇⣸⣧⡿⠿⠋⣿⠇
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⣉⡭⠷⡄⠀⠀⠀⠀⢠⡏⠗⡦⣄⣀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣉⡡⠴⢚⡉⠀⠀⠀⢹⠀⠀⠀⠀⡼⠛⠲⢤⣈⠈⠟⢲⢤⣀⡀⠈⠙⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣉⡤⠖⢋⡥⠖⣚⣨⠵⠚⠛⣆⢸⠀⠀⠀⢠⠇⢀⠀⠀⠈⠙⠓⠦⣌⡁⠿⢳⠦⣄⣀⠀⠉⢣
⠀⠀⠀⠀⠀⠀⠀⡞⢀⡜⠁⢀⣠⠤⠒⠋⠁⠀⠀⠀⠀⣿⢸⠀⠀⠀⢾⡀⠈⠁⠲⠄⣀⠀⠀⠀⠉⠛⠲⢤⣁⠛⠹⠖⢾
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠉⠓⠦⢄⣀⠈⠁⠲⠄⣀⠀⠀⠀⠈⠉⠓⠦⣼
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⡇⠀⠀⣠⠖⢋⣉⣉⠙⠲⠄⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣨⠀⡀⢠⠞⣡⠚⣉⣤⣍⠙⠂⠀⢺⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⡏⡼⠁⡴⠁⡠⠄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡋
⠀⠀⠀⠀⠀⠀⠀⣧⣽⠘⡇⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⣚⣩⠷⠒⣿⡉⣿
⠀⠀⠀⠀⠀⠀⠀⡿⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⣀⣠⠤⠖⠚⣉⣡⠤⠖⠋⠁⠀⢀⣀⣻⡧⠿⢤
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⢀⣠⠴⢺⡇⢸⢸⠀⠀⣴⣋⣁⣠⣤⣖⣋⣭⡤⠤⠴⠖⠚⠋⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡆⡇⠀⢀⣠⢴⢚⣭⠰⣼⠀⡇⢸⢸⠀⠀⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⢚⣹
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⣿⣾⢉⣿⣯⡇⢹⢸⡇⢸⢸⠀⠀⡗⠶⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⢹
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⣿⣿⠉⠸⠇⢃⣨⠼⠃⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⢺
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⡏⠋⣀⠴⠚⠉⠀⠀⠀⣼⢸⠀⠀⡷⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⢾
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⠛⠉⠁⠀⠀⠀⠀⣠⠞⠁⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃⡇⠀⠀⠀⢀⣠⠤⠖⠋⠀⢀⡴⠋⠀⠀⣟⠒⠒⠚⠉⠉⠀⠀⠀⠀⣀⣀⣠⠤⠴⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠙⠒⠚⠋⠉⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠙⠶⠶⠒⠒⠒⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠾⣇⡀⣀⣀⡤⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣄⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣿⣿⣿⣷⣄⡀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣀⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⢻⣿⣿⣿⣷⡀⢹⣿⡆⠀⠀⠀⠀⣤⣀⡀⠀⣰⣿⡿⣼⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢰⡄⠹⣿⣿⣿⣷⣨⣿⣿⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠹⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢤⣘⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⢰⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⣿⡆⠰⢦⣄⣀⣤⠆⠐⢮⣿⠈⠻⣿⣿⣿⣿⣿⣿⠿⡛⣽⣿⣿⣿⢟⣿⣭⡟⠁⠀⠀⠀⢀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⣿⣷⣤⠴⠿⠋⠉⠀⠀⠈⢿⣜⣆⠈⢻⠟⠋⢁⣴⣋⣤⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢰⡾⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣶⣦⣶⣿⣿⣽⣿⢟⣿⣿⣯⣽⣿⣦⣦⠆⠀⣠⡿⡄⢀⡴⠋⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠟⠋⢻⣿⣧⡀⠀⠀⢀⡄⠀⠀⢢⣽⡿⢸⣿⣿⣿⡟⣹⣷⣾⣿⣿⣿⣿⣿⣿⣿⣾⣷⡿⠀⠙⢾⣷⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⢱⣾⣿⣷⣼⣧⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠈⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⢺⠈⠻⢿⣿⣷⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣦⡉⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠛⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⣿⢿⣿⣿⣿⡟⠋⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢹⡇⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠻⣿⡏⠁⠀⢸⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⡁⠀⠀⠘⠀⡏⠛⢿⣿⣿⣿⣿⠟⠉⠛⠿⣾⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⠀⠀⡇⠀⠀⠀⠀⡇⡁⠀⠻⣿⡟⠃⡀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠇⡄⠀⠀⠙⠀⠀⠉⠀⠀⠀⢸⠀⠘

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⠴⡖⠲⠒⠒⠒⠠⢤⡤⣤⠤⣄⢴⣶⠦⣄⣤⣒⣲⡤⢤⡴⢶⣦⢤⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠭⣍⣉⣀⣀⣀⣀⣇⢀⣀⣀⡀⠀⠀⠹⡾⣠⡼⠦⢥⡵⣸⣂⣉⠈⢱⡄⢹⣄⣯⢻⢷⣏⠙⢻⢦⡉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡭⢭⠁⣍⣋⠿⠀⠀⣀⣤⣴⣫⡋⠁⠀⠶⠞⠉⠁⠉⠙⠯⠥⣠⠏⠀⡼⠀⣏⠧⠰⢇⡇⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⣯⠞⠉⠁⠀⠀⣀⠆⠀⠛⢯⣭⣉⣒⠒⠶⠶⢆⠁⠀⠀⣀⣒⣵⣚⣁⡼⠙⠋⢣⠀⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠷⠶⢷⣤⡠⠴⠒⠋⠡⠜⠀⠀⣻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⣻⣍⠲⠤⠴⢋⡦⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠴⢄⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠲⢄⠉⠻⣷⣶⣶⣫⡴⠒⡩⢿⡇
⠀⠀⢶⣒⡶⠶⠶⠒⠚⢫⠀⡅⣠⣾⠀⢠⠀⣤⢚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠱⣤⣿⠋⢻⡋⠁⠀⢠⣿⠁
⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⢲⠋⠀⡏⠀⠈⠀⢸⠛⠂⠀⠀⠀⠀⠣⣔⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢸⡃⠀⢸⣷⣤⡼⠛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢧⠰⣼⠷⠃⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣄⠘⣿⡷⡁⠐⣁⠀⠑⣼⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠳⣼⠀⠉⠉⢻⣾⠋⠁⢤⠈⣷⣄⡀⠹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠂⠀⠀⠀⢦⡼⢋⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢰⣄⡧⡀⠀⠄⢄⠡⢼⡞⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢿⡄⣀⠤⢾⢻⣇⠞⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⠟⢿⡌⠐⡁⣰⣴⡂⢁⢶⣶⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣷⣼⣋⡴⠖⡿⢋⠃⠈⠀⠀⠀⠀⢀⠀⣀⣀⣠⠊⠫⠍⡃⠀⠰⡇⢁⠑⠰⠇⠂⡈⠉⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣩⠖⢏⢠⡈⠀⡤⠈⢿⠷⠖⠛⠺⠾⣧⢱⠖⠛⠓⠛⡤⡘⢳⠀⠳⠀⠢⣀⣡⠖⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡤⢄⣴⠏⢀⣻⠃⠃⠤⠂⢠⣀⣸⡆⣀⠄⠀⠀⢈⣛⢯⡀⠀⠀⢁⡆⠨⡄⢱⠤⠄⠂⠈⠐⡌⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣏⠀⢈⠉⣷⡖⢩⣍⠋⠳⠄⠒⠋⠁⠀⠀⠠⠄⠀⠀⠍⠀⡀⠈⢦⣾⡷⠖⠉⠠⠒⠒⠠⠒⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⠁⠈⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠑⠙⣶⠈⢀⠒⠀⣀⠄⠀⠠⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⡉⢋⡄⠀⠀⠂⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠀⠁⠀⡪⠕⠒⣍⠁⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡠⠀⠀⡶⠁⠀⠀⢠⠏⠀⡀⡠⠁⢠⢃⠂⢠⠖⠈⡰⡢⠀⠀⡀⠀⠑⡆⠀⠀⣀⠴⠀⠀⠀⠠⢀⡔
⠀⠀⠀⠀⠀⣤⣦⡜⠙⠛⠢⣄⠈⠀⠁⠱⡀⡀⠀⠘⠀⠀⠁⠁⠀⠀⠈⠀⠀⠀⠀⠀⠁⠐⠀⠁⢀⢠⠫⠀⠀⠁⢤⣤⠦⠈⢣⠈
⠀⠀⠀⠀⠀⠿⣏⣀⠀⠀⢉⡄⠐⠦⡆⢀⢹⠀⠀⣀⡄⠀⠄⣠⢖⡒⢀⣀⣒⣢⡤⠤⣀⣤⡀⠀⢘⡇⠀⢠⡀⠆⣈⡙⢂⠔⠊
⠀⠀⠀⠀⠀⠐⠚⠛⠒⠒⠉⠀⠀⠀⠈⠉⠀⠲⠮⢩⠀⠉⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⠍⠵⠊⠀⠀⠀⠈⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡶⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣶⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢛⣋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣍⣋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣶⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢛⣋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣬⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣶⣶⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⣭⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣬⣍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠿⠟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⠿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣩⣤⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣦⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⣋⣿⡀⠀⠀⠀⠀⢀⣿⣭⡛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⠿⢻⣆⡀⣀⣰⣟⡹⠿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣾⣿⡏⣽⣯⡹⣿⣿⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣤⣿⡿⣧⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣅⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣇⣇⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠋






⡟⠉⢻⡟⠻⠿⠿⠻⢟⠻⠟⠿⢻⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⢿⣿⡿⠿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣪⣵⡾⠷⢶⡭⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣹⡟⠁⠀⠀⠀⠙⡌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣏⣓⢦⡀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⡷⠿⣻⣿⣸⠛⣇⠀⢀⣟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣽⠞⠉⠀⠀⢈⠝⣡⣾⣿⡤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⠖⠋⠀⠀⢀⡤⣚⡵⠞⣽⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣭⠞⠁⠀⢀⡠⢖⣽⠞⠉⢀⢾⠏⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠵⠋⠀⠀⣠⣖⣭⠞⣻⠏⠀⢀⣾⡟⠀⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⣠⣔⡿⠞⠉⠀⣸⠏⠀⢀⣮⡟⡇⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⣠⣴⡿⠞⠉⠀⠀⠀⣴⠏⠀⢀⢮⠟⠘⡇⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣾⠟⠁⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⣸⠏⠀⢠⢫⠏⠀⡆⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⣀⣴⣾⡯⠵⠒⠲⣼⡿⣅⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣰⢣⠏⠀⠀⢱⡇⠀⣧⡇
⠀⠀⠀⠀⣠⣞⡵⠛⠁⠀⠀⠀⠀⡸⢠⢟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣰⣣⠏⠀⠀⠀⢸⡇⢠⢻⠃
⠀⠀⢀⢞⡵⠋⠀⠀⠀⠀⠀⠀⢰⠃⡞⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣰⣳⠏⠀⠀⠀⠀⢸⡇⢸⢸
⠀⢀⣏⡞⠀⠀⠀⠀⠀⠀⠀⣰⢃⡜⢁⣿⡟⠀⠀⠀⠀⠀⠀⢀⣠⠷⢷⣴⣿⠏⠀⠀⠀⠀⠀⢸⡇⢸⢸
⠀⢰⡿⠀⠀⠀⠀⠀⠀⢀⠔⡡⠋⡠⣿⡟⠀⠀⠀⠀⠀⢀⡴⣫⠴⠚⢯⢹⣿⣆⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠈⢿⣄⠀⠀⣀⡠⢖⣵⣪⠴⣫⡾⠋⠀⠀⠀⠀⠀⢀⣞⡼⠁⠀⠀⢸⢀⡇⢻⡆⠀⠀⠀⠀⢸⡇⠸⣿
⠀⠀⠀⠙⠛⠿⠿⠿⠭⠥⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⡼⣸⠁⠘⡇⠀⠀⠀⢠⣾⠽⢿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⣰⣷⠇⠀⣸⠃⠀⠀⢠⣿⠃⠀⠈⣿⡜⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡆⠀⢀⡴⡿⠋⢀⣴⠃⠀⠀⠀⢸⣿⠀⠀⠀⢹⡇⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣽⠮⢴⣺⠟⠁⠀⠀⠀⠀⢸⣿⡀⠀⠀⣸⠃⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢧⣣⣀⣴⣟⣴⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣉⣁⣀⣀⣀⣀⣀
⣿⣿⠋⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⡤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⡏⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠾⢷⣺⠻⣅⠈⠲⠶⠃⣫⣽⡄⠀⠀⠀⠀⠀⠀⡰⠁⣿⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠇⠀⠁⢀⠀⠀⠀⠀⠀⣼⣽⣟⣱⡇⣿⣾⠗⠒⠆⡾⣽⡛⠫⢷⣋⡇⢀⡀⠀⠀⡠⠊⠀⠀⠋⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠢⢍⠒⠠⠤⠤⢭⢈⣤⡷⣲⢩⡏⠴⠒⡚⣡⣾⣟⠇⣲⡆⢐⠒⠒⠈⠡⠖⠁⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠠⡀⠀⠀⢠⢰⠠⠋⣎⡍⣑⡊⣇⢠⠎⢼⠋⣄⠇⠿⠌⡙⢥⡀⡀⠀⠀⠔⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⡀⢀⠌⢀⠆⠘⠿⠘⠀⠼⠁⠉⣃⢆⠙⣶⢀⠝⠐⢧⣋⡴⠲⡥⠘⣄⠐⣀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣙⣒⣣⠴⠆⢀⣀⠀⠀⠡⠞⡄⠀⠃⢹⠆⢀⠠⡀⠈⣡⡄⠒⠲⠾⠭⠥⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣩⣭⣷⠄⠀⠀⠀⠈⠹⢿⢷⡀⠀⢀⣇⠠⣦⣤⢀⡞⡀⢀⡾⢟⠋⠈⠁⠀⠀⢀⣿⣿⣶⡤⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⠻⣿⣿⣿⣤⣴⢞⣤⠂⢸⡆⠁⠀⠀⢁⠇⠹⠋⢸⠁⠀⠀⠸⠧⠈⢲⣝⠳⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠙⢛⣩⢴⠟⠃⠀⡀⠀⠀⡆⢀⡠⠂⢠⡀⠡⡁⠀⡅⠀⠀⡆⠀⠉⠿⠦⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠀⠀⠀⢇⠀⠀⡇⠀⠀⠐⠐⠐⠀⠀⠁⡇⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⡀⠤⠀⠀⠀⠠⠤⢀⡀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠷⠆⠀⠀⠀⠀⠀⠀⠰⠏⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡈⠉⠛⠛⠛⠛⠉⢁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠃⣤⣴⣠⡄⣶⢠⡴⣦⢠⡄⣦⢠⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢀⡿⣿⢿⣡⡏⣸⣷⡏⢿⣸⡇⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠉⠁⠈⠀⠁⠀⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠠⠖⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⠴⠚⠤⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠙⠋⠲⠶⣦⣄⣀
⠀⡠⠠⣭⡀⠀⠀⠀⠀⠄⠠⣀⠀⠈⠀⠛⠁⠠⠠⣦⣤⡀⠀⠉⣡⣶⡀
⠀⢀⣤⣴⣶⣶⣶⣤⣤⣤⣀⣲⡤⢌⣉⡐⠒⠢⢄⠀⡌⣫⡔⠛⠋⠹⣿⡄
⠀⢸⣿⣿⠟⠛⠻⠿⠿⠿⢿⡿⠿⣿⣷⣾⣭⣓⢶⣴⡜⠇⠀⠀⠆⠀⠹⣿
⠂⣾⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⠿⣿⣧⡹⣿⣎⠀⠀⠀⢀⣀⠀⠠
⡀⡇⠀⠀⠀⢀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⢹⡟⠀⠀⣴⢻⡿⠀⠀⠄
⡀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢿⣦⡀⢹⣆⣧⢀⢠⡄
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⡿⠇⢀⡿⠈⠰⡌⡇
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢁⢻⠀⢸⡗⡀⠀⢹⡟
⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠘⠛⡄⡜⡇⠇⠀⣼⠁
⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣠⠙⢱⡇⣆⠔⠁
⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⡅⠈⣸⣿⣿⣆
⠀⠠⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣠⡶⠟⢝⡯
⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣦⠔⠁⠀⠈⠁⠒⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⢙⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢄⣀⠀⠀⠸⣿⣿⠗⠲⠤⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠋⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠶⠀⠀⠀⢁⣒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣘⢿⣿⣿⣶⣤⣤⣀⣠⣄⠀⢀⣠⣄⣀⣤⣤⣶⣿⣿⡿⣃⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⠉⢿⣿⣿⣿⣿⣿⣿⡿⠁⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⣟⢻⣿⣿⡟⣻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⡿⢿⣷⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⡇⠀⠀⠁⠂⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣬⣭⣭⣥⣥⣀⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⢬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠏⢺⣿⡿⠟⠛⠛⠻⠿⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿⡆⢻⣷⣦⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⠀⠈⠛⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠤⠈⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠋⠀⠀⣀⡀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠹⠿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⠀⠀⠙⢸⣿⣿⣿⣷⡐
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠂⢀⡀⡺⣿⣿⣿⣿⣷⡄
⠀⠠⢀⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢠⠀⠈⣿⣿⣿⣿⣿⣿⣷
⠀⢠⣿⣿⣿⣿⡿⠉⠀⠐⠀⠈⠀⠀⠀⠊⠆⠈⠉⠉⠛⠷⠆⠀⠀⠰⠾⠛⠋⠉⠉⠓⠂⠀⠀⠁⠈⠀⠀⠙⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠲⢖⠚⠗⠦⠀⠀⠀⠀⠠⠖⠿⠒⠶⠀⠀⠀⠀⠄⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣧
⡆⣾⣿⣿⣿⣿⣧⣀⣤⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⠇⢿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣿⡀⠃⠀⠀⣀⢠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠘⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣰⣦⣤⣴⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢹⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠂
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠡
⠀⠀⠀⠀⠀⠠⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠛⢿⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡿⠟⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠛⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣤⢠⡄⣴⢄⣤⢠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠼⣇⢻⡎⢿⡀⢹⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣦⡀⠀⠻⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠿⠟⢻⡿⣟⣭⣤⠈⣿⣿⣿⣇⢻⣿⣿⣿⣿⣦⣄⠀⠈⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣀⣀⠘⠿⣧⣶⣿⢇⣴⣿⣿⣿⠀⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⡀⠀⣾⣿⣿⣿⡄⠛⣁⣤⡌⢿⣿⣿⠃⢰⣿⣿⣿⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡃⢿⣿⠿⢿⣿⡿⣿⣿⡇⢸⣿⢃⣠⣿⣿⣿⣿⣿⡇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠀⠰⢾⣿⣿⣷⣿⣿⠃⣌⠛⣿⠏⣽⣿⣿⣿⣿⢃⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⡿⢀⠹⣆⣿⠀⣿⣿⣿⣿⠏⣼⣿⣿⠿⣻⡁⢿⣿⣿⣿⠟⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⢀⣾⣦⡈⢋⣼⣿⣿⠟⠁⠾⠿⠟⠁⣾⣿⡇⠸⣿⡿⢋⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢾⣿⡄⠀⢿⡇⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⣀⡻⣧⣄⡀⣀⣀⠀⠀⠀⢀⣴⠟⠁⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠿⠇⠈⠛⠃⠻⠏⠀⣠⣴⣮⠉⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⢋⣽⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⣠⣶⣶⠁⠀⠀⠀⠀⠸⣄⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡦⣸⡟⢀⣴⠶⣦⡆⢰⡶⠀⣴⡆⢰⣾⢶⣦⢀⣶⠀⢸⡷⠉⢡⣶⣆⠀⠀⠀⢠⠞⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⠃⢿⣇⣠⣿⣁⣿⣇⣴⣿⣀⡾⠁⣾⣇⣼⣯⣠⣿⣇⡴⢋⣼⣿⣠⠄⠀⠸⣄⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠀⠉⠁⠈⠁⠀⠉⠁⠀⠀⠈⠉⠀⠉⠀⠈⠉⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠒⡒⢒⣲⠶⢚⡥⢚⡹⡧
⠀⠀⠀⢠⣀⣀⡠⠔⠲⢄⠀⡸⢆⠀⠀⠀⠀⢀⣠⠖⡋⠅⠀⠀⠀⠀⣀⣤⣖⡫⢵⣎⠅⠀⢁⠠⣿⣄
⠀⠀⠀⠈⠂⠤⠖⠒⣄⡠⠗⠿⣄⠳⣖⠒⠒⠻⢤⡁⠀⠀⠀⢀⠔⡫⠶⡶⠋⢀⣰⡏⠀⠀⠀⠠⡟⠢⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡝⣺⣶⢄⠈⠓⠃⠀⠀⠀⠀⠉⠳⣤⣔⣑⣪⠭⢋⡿⠠⣰⣞⡁⠀⠀⢨⡝⠁⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠃⠀⠀⠰⠀⠀⠀⠀⢀⣤⠾⢉⠕⠚⠁⡰⠈⣰⠎⠁⢠⡄⠴⠚⠁⠀⠀⠀⠀⠀⠈⢈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⣰⠂⠀⢀⣴⢝⡤⠊⣀⡴⠛⢁⡴⠖⢀⣴⣊⣁⣠⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣠⣬⡶⠚⢹⠀⠀⣼⣿⠊⢠⠜⢁⣤⠐⢁⡠⠖⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠙⠢⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⠀⠹⣿⠤⡅⢠⡇⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡗⢆⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡅⠀⢰⣿⢭⡷⣏⣴⣏⣀⣀⣀⡤⠤⠤⢤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣹⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢧⠀⠀⠻⠽⠁⠀⠀⢿⠀⠀⣀⣀⠀⠀⠀⠀⠀⠉⢣⡀⠀⠀⠀⠀⠀⣰⠃⠸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⢦⣤⣈⣙⣶⡄⠈⢻⡍⠉⠉⠉⠛⠛⠒⢺⠋⠉⠙⣦⣄⡀⠀⠀⢯⡀⠀⢿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡠⢀⡤⠖⠃⠀⢁⡞⠀⠘⣆⠀⠹⡄⠀⠀⠀⠀⠀⠀⢇⠀⢼⠁⠈⠉⠓⢦⡀⠉⢓⣿⡍⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠋⢀⡴⠛⠉⠉⠀⠀⠀⠈⠳⡄⠙⢄⠀⠀⠀⠀⠀⣸⠀⣹⠂⠀⠀⠀⠀⠈⢳⡈⢿⢣⡈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡳⠀⠀⠀⢠⠃⡾⠁⠀⠀⠀⠀⠀⠀⢢⢿⠈⡆⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢀⡾⠙⢮⡀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠂⢧⠀⡠⠔⠃⡸⠂⠀⠀⠀⠀⠀⢀⡴⢃⣼⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠾⠤⠤⠟⠀⠀⠀⠙⢮⡀⠀⠀⠀⠀⠀⠷⠦⡤⠞⠰⠥⠤⠴⠃⠀⠀⢀⠀⣊⠴⠋⠰⢿⣯⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣄⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣨⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠒⠒⠋⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢀⣶⡀⡆⢰⡆⣿⠆⡇⢸⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠁⠘⠚⠁⠋⠃⠙⠚⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢷⣶⠦⠰⣶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣠⣶⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⠀⠀⠀⠀⢸⣿⡿⢷⣶⣶⣿⣶⣶⣿⠋⠉⢿⡿⠋⠁⢸⣷⣠⣴⣶⡿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⡟⠻⣿⣧⠀⠀⠀⣿⡿⠁⢠⣿⠋⠻⣿⣾⣿⣧⠀⣠⡦⠀⠀⠈⢿⣿⠋⠀⠀⠈⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠋⠛⢿⣿⣿⣦⠈⢿⣷⣀⣀⠉⠀⠀⡸⠁⠀⣼⣿⣿⣿⣿⠟⠋⣤⣶⣶⣤⣾⣿⣿⣶⠾⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣶⣿⡇⠀⠀⠀⠀⣼⣿⣧⡀⠻⣿⣿⣧⣠⡾⠃⠀⣴⠟⠉⠉⢿⣿⠀⠀⠿⠟⣉⣉⣙⠛⢛⠁⠀⠀⠈⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡈⢿⣿⡇⠀⠀⠀⣼⠟⠉⠉⢿⣿⣿⣿⣿⡿⠿⢿⣿⡏⠲⣦⡀⠘⣿⡄⠀⣠⣾⣿⣿⢋⣾⣿⣿⣿⣿⣶⡄⢸⣿⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⢷⣀⣉⡁⠀⣀⣼⣧⣀⠀⠀⠀⢉⣿⣏⠀⣀⣀⣀⣸⣿⣷⣿⣷⡀⠸⣿⠿⢿⣿⣿⣿⣿⠟⣩⣿⢿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠛⠉⣿⣷⣾⣿⣿⣿⠛⠃⠀⢠⣿⣿⣿⣿⣿⣿⠟⠉⣻⣿⣿⣿⠷⣴⡏⠀⡸⠇⢿⣿⠋⣠⡟⠁⠀⣠⣿⣿
⠀⠀⠀⢀⣶⣦⣼⣿⠀⠀⠀⣀⠀⢀⢙⣋⠛⠻⠷⣦⠀⣿⣿⠁⠀⠈⠉⠉⠀⣴⣿⠛⠛⠃⠀⠹⠇⢠⣷⠀⠀⠑⠲⠿⢷⠿⣿⣿⣿⡋
⠀⠀⢠⣿⠋⣹⣿⡇⢠⡄⣸⡟⢀⣼⡿⠛⠀⢀⣴⣿⣾⣿⡿⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⠄⢠⣆⠀⣾⣧⡀⠀⠀⠀⠀⣠⣶⣄⠙⣿⣧
⠀⢠⣿⠃⢰⣿⣿⡇⢸⣷⣿⣷⣾⣏⠀⣀⣴⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣿⣿⣿⣿⠛⢿⣷⣄⠀⣼⣿⡟⠛⠿⠿⠟
⠀⣿⣏⣀⠀⠀⠙⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠋⠉⠛⠿⠃⠀⠀⠘⣿⣿⠿⠛⣿⣶⣦⣄
⠀⠙⢿⣿⣷⡀⠀⠸⣿⣿⣿⣏⠉⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠿⢿⣷⣄⠀⢠⣤⣶⣤⡾⠋⠁⠀⣶⣌⣹⣿⣿⣆
⠀⠀⠀⣈⣽⣿⡷⢿⣿⣿⣿⠛⢿⣦⣤⣄⣀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠙⠻⣿⣶⣿⡿⠛⠃⣴⣶⣾⣿⣿⠟⠻⠿⠟
⠀⠐⣿⣿⣿⠋⢠⡾⠋⣹⡏⠀⠀⢹⣿⣭⣿⡿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣶⣶⡄⢀⡈⠛⢿⣤⠀⠀⢹⣿⣿⣿⣿⣦
⠀⠀⠙⣿⣷⣄⠀⠀⠀⠈⠀⠀⢀⣾⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣿⣿⣷⣄⠁⢠⣾⣿⣿⣿⠟⢿⣿⣧⣼⣶⣤⡀
⠀⠀⠀⣿⣿⣿⣷⣦⣤⣶⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠋⢁⣨⣿⣿⣿⣿⡿⠛⠉⠀⠀⢀⣽⣿⣿⠿⢿⣿⣄⣤⣄
⠀⠀⢀⣿⣿⡟⠉⢻⣿⣟⣀⠀⠀⠀⢈⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠉⠻⣷⡄⠀⠀⣄⠘⣿⡇⠀⠀⣼⣿⣿⠙⣿⣇
⠀⣾⡟⠻⠟⠁⠀⠘⣿⣿⣯⣀⣴⡾⠻⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇⡀⠸⣦⡈⠁⠀⠀⠹⣴⡿⠁⠀⠘⠟⢋⣥⣄⠸⣿⡄
⠀⢻⣧⣴⣦⣀⣀⡀⢻⣿⣿⡛⠋⠀⠀⠀⢀⣈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⠈⢻⣦⣠⡀⢠⣿⣇⠀⢠⣷⣶⣿⣿⣿⣶⠿⠃
⠀⢾⡟⠛⠙⠻⢿⣷⣼⣿⣿⣷⡄⢠⣾⢷⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣽⣿⣿⣿⣿⣿⣄⠀⢿⡿⠋⣹⣿⣷⡀
⠀⣿⡇⠀⠀⣀⣴⣿⠋⠉⠙⣿⣷⣾⣷⣶⣶⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠉⠛⠻⣿⣿⠛⢻⣿⣶⢾⡇⠸⠟⠛⢻⣷⡀
⠀⠸⣿⣶⣿⣿⡏⠁⣦⣀⠀⣹⣿⣏⢀⣀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣴⣶⠀⠀⠘⣿⡆⠀⢿⠟⠀⢸⣿⣆⣀⣀⣿⣷
⠀⠀⠈⠁⠀⣿⣷⣶⠿⠿⣿⣿⣿⠿⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣦⣠⣤⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿
⠀⠀⢀⣤⣾⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠉⠛⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⣤⡀⠀⠀⠀⢠⣿⠟⠋⢁⣀
⠀⢀⣿⡟⠀⠈⠉⠻⠧⠈⠛⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠉⠻⢿⣿⣿⣦⣤⣾⣷⣶⣶⣿⣿⣷⣄
⣴⣿⡏⠀⠀⠀⠐⣦⣀⣿⣇⣀⣠⡖⢀⣾⣿⣽⣛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣀⠀⢀⣠⠿⠿⠋⠁⠉⠛⢿⡿⠟⠛⠛⢿⣿⡆
⢻⣿⣷⣦⣀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡏⠉⠉⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣧⡈⢿⣆⣹⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠙⠋⠙⣿⣦⣘⣿⣿⣿⣿⡟⠉⡈⠙⢿⣶⣤⣠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡘⢿⣿⣦⡀
⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⡀⠁⢀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢿⣿⣦⣀⡀⣠⣷⡀⠺⠋⠉⠁⠀⠹⣿⣷
⠀⣴⡿⠟⠛⠁⠀⠀⠀⠙⠿⣿⣇⠀⠈⠛⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣅⠀⠀⠙⣿⡿⠛⠻⣿⡷⠀⠀⠀⠀⣀⣠⣼⣿
⢸⣿⠀⣾⣦⠀⠀⠀⠀⣤⡀⠈⢻⡆⠀⢀⣼⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠀⠁⠀⠻⣆⠹⣷⣴⣾⣿⣿⣿⣿⠛⠛
⠸⠿⠿⠿⢿⣧⡀⠀⠀⣹⣷⠀⢸⣧⣴⡿⣿⣿⠿⢷⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⡄⠀⣾⣧⠀⠻⠀⠀⠀⠀⠈⠛⢿⣿⣇
⠀⠀⠀⠀⠀⣹⣧⣴⣾⣿⣿⣦⣾⣿⠋⣼⠟⠁⠀⣤⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣄⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠄
⠀⢠⣶⡶⠾⠿⠿⠿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣤⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠁⠀⠀⠈⢿⣆⠈⢧⣄⣀⣴⣶⣾⣿⣿⣿⠋⠁
⢰⣿⠟⠀⠀⢠⣴⣾⠿⠟⠛⢿⣿⠿⢷⣤⣀⣤⣿⣿⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠐⣦⣿⣷⣿⣿⣿⣄⠙⠻⠟⢹⣿
⠀⢿⣷⣄⠀⠀⣀⣤⣶⣦⣄⠈⠻⠷⢶⣿⣿⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⣠⣿⡿⠿⠟⠻⡈⠛⢿⣿⣷⣿⠏
⠀⠈⠙⠿⣷⣿⣿⣿⣿⡿⣿⣧⠠⣄⡀⣈⢉⣿⣿⣤⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠋⠉⠉⠉⢻⣿⣏⠀⠀⠀⠀⠹⡦⠀⠈⢻⣷⡄
⠀⢀⣀⣴⣿⠛⠁⠀⠀⠀⠀⢻⣧⣼⣧⠙⠿⣫⣿⠟⠛⠛⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⠛⡀⠀⠀⠀⡀⢠⡞⠀⠀⠀⠀⢀⣶⣤⣤⣾⣿⣿
⠀⠘⢿⣿⣿⣄⠀⠀⢠⣄⠀⠀⠹⣿⣿⣷⣾⡟⠁⠀⠀⠀⠘⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡋⠀⣴⣿⣇⣼⣧⣤⡄⢀⣤⣿⣿⣿⣯⣀⡀
⠀⠀⠈⠻⣿⣿⣦⡀⠀⠙⢿⣷⣆⠘⣿⣿⠏⠀⠀⠀⠀⠀⢀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣻⣿⣦⡿⠿⠏⠙⠻⢿⣷⣏⣿⣿⡿⠿⣿⣿⣧⡀
⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠙⢿⣆⣽⣿⣶⡇⠀⣀⣤⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣧⣾⣿⡿⠋⣤⡀⠀⠀⠀⢀⣿⠟⠋⠁⠀⠠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⣹⣿⣿⣿⣷⣾⣿⣿⣏⠀⠀⣀⡀⠀⠀⣠⣶⣄⣰⣿⣿⠿⠿⢿⣿⡀⠀⢿⠁⠀⠀⢀⠈⠁⠀⠀⠀⠀⢀⣠⣼⣿⡿
⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠛⠛⠻⠿⢿⣿⣿⣿⣟⠋⠛⠛⠿⢿⠿⣿⣿⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⠻⠆⠈⣧⠀⠀⠿⠀⠀⣀⣴⣶⣿⣿⡿⠿⠟⠁
⠀⠀⠀⢀⣾⣿⠉⠀⠀⠀⠀⠀⢀⣤⣾⣿⡉⠙⠛⢿⣶⣄⡀⠀⠀⢹⣿⣿⡟⠉⠉⠙⣿⣦⠀⣰⣶⣦⡀⠀⠀⣿⣄⣴⡀⣠⣤⣭⣭⣿⣿⣿⡄
⠀⠀⠀⠘⣿⣿⣷⣶⠀⠀⢠⣾⣿⣿⣿⠻⣷⠀⠀⠀⠀⠙⠛⠂⢀⣿⢿⣿⣧⣀⡴⠁⢀⣽⡿⠛⠋⠁⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠘⣿⣿⢀⣠⣤⣿⣿⣿⠉⢿⠀⢻⣇⠘⣷⢰⣶⣴⣿⡿⠃⠀⠙⢿⣿⡄⢰⡿⠋⠀⠀⠀⠀⢀⣤⡾⠻⣿⣧⡀⠀⠀⣤⣤⣤⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠻⣿⣿⠃⠀⠀⠀⣿⣶⣿⠿⣿⡿⢇⠀⠀⣀⣴⣿⠟⠁⣿⣇⠀⠀⠀⢀⣴⣿⠋⠀⢠⣎⠙⠛⣷⣶⣿⣿⡿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⢸⣿⡀⠀⠀⣠⣿⣿⡟⠀⠈⠀⠘⢃⣼⣿⣿⣿⣶⣾⣿⣿⣷⣤⠀⣿⣿⡇⠀⠀⢸⣿⡄⠀⠙⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⣿⣿⠇⠀⠀⠀⠀⢾⡟⠉⢹⣿⣿⣿⣿⡟⠻⣿⣿⡿⣿⣿⠀⠀⣿⠿⣿⣦⣀⣴⡾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣄⣀⣤⣦⣬⣿⡄⠀⠛⠛⠛⠛⠛⠀⠀⠉⠀⠈⢿⣷⣾⡇⠀⠈⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣿⣿⣿⣿⣿⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠙⠃⠉⠙⣟⠁⠀⡝⠉⠉⣿⠙⠇⢨⣽⠟⣹⡅⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣾⣿⣷⣾⣷⣿⣷⣾⣿⣿⣿⣷⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⣌⠻⢟⣵⣦⡀
⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠀⠀⠀⠀⠀⠀⠀⢄⡠⢄⡂⢄⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⠂⠈⢃⡅⠠⠁⠀⠅⠋⠀⢒⢤
⠈⢩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡉⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠂⠀⠀⠐⢶⣶⡀⠀⠀⠀⠀⠀⠀⢰⡇⢊⠀⠸⠦⣤⢤⣤⢤⡤⠞⠀⢈⠂
⠀⠈⢱⣶⢀⡀⠀⠀⠀⠀⣀⢰⣶⠍⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡄⡀⠀⠀⠀⡀⢀⣀⡤⠞
⠀⠀⠀⠀⢿⡿⢠⣶⣦⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⡍⠉⠉⠉⠉⠉⣟⣳
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀⢀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⡠⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢠⠛⠦⠒⠒⠢⠼⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⠤⠤⠤⢤⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠰⠀⠀⠰⠀⠀⠐⠄
⠀⠀⢀⠎⢀⣤⣶⣶⣦⣄⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢠⠏⠀⠀⠀⠈⢻⡄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢸⡄
⠀⠀⡎⠀⣾⣿⡙⠿⣫⣾⡧⠀⣧⠀⠀⠀⠀⠀⠀⠺⡷⢶⠶⣿⠶⠶⠶⠶⠶⠶⡷⠶⠶⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢠⡄
⠀⠀⢇⠀⢿⣿⣿⣾⣿⣿⠏⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠈⣆⠀⠀⠀⠀⣼⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣆⠀⠀⢀⡾⢷⡀
⠀⠀⠈⢦⠀⠙⠻⠻⠟⠋⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡘⢇⠀⠀⡼⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⣿⠄⠀⢼⡇⢸⡧
⠀⠀⠀⠈⢶⠓⠲⠒⠖⢺⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣆⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⢿⠃⠀⢻⢿⡿⡟
⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠜⣁⠒⠉⠭⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⠾⠻⣿⣄⣹⣮⡷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⣾⣭⠐⠒⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡿⣿⣷⣴⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡤⠞⠁
⠀⠀⠀⢻⣿⠀⠀⠀⣀⠶⣿⣿⡶⢀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣄⣾⠛⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣠⣥⣠⢤⢀
⠀⠀⠀⠘⣿⠀⠀⠾⠰⠎⣴⣬⠱⠎⠗⠀⠀⠀⠀⠀⠀⠘⣿⡈⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⣽⣴⣶⣼⡏⠶
⠀⠀⠀⠀⢄⠌⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⡏⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣼⣬⢯⣼⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⠧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡧⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣦⣤⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢉⣿⠁⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⣼⣿⡄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣴⡏⠈⣿⣤⡾


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣿⣟⣙⣟⣉⣋⣟⣛⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣭⣽⣯⣽⣿⣭⣭⣭⣭⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣀⣀⣀⣀⡐⠂
⠀⠀⠀⠀⡠⣤⡤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣬⣈⠬⠄⠁⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠉⠉⠉⠉⠈⡇
⠀⠀⠀⠀⡇⣿⡇⠀⣠⣄⡋⠀⢂⠀⠀⠀⠀⠀⠀⠀⢰⣀⣂⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⢨⢐⢒⢒⢒⢒⢊⠠⠀⠀⠀⠀⢀⣈⣀⡠⠀⠀⠀⣠⠀⠀⠇
⠀⠀⠀⠀⡇⣿⣿⡈⠷⠊⠑⠄⠀⠣⠀⠀⠀⠀⠀⠀⠰⠒⠒⠒⠂⠐⠲⢰⡆⠀⠀⠀⠀⢸⠨⢎⡨⢌⡨⢌⠀⠀⠀⠀⠀⡆⠀⢾⣵⠔⣁⣈⡀⠇
⠀⠀⠀⠀⡇⣿⣿⡇⡄⢑⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⢸⡇⠀⠀⠀⠀⢸⢨⣣⢠⡣⢘⡣⢐⠀⠀⠀⢀⣥⣤⣤⣶⣾⣿⣿⣶⣧⣤⡀
⠀⠀⠀⠀⡇⣿⣿⠇⠃⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣐⠆⣂⠆⢆⡒⢸⡇⠀⠀⠀⠀⢸⠐⡱⠔⡁⠄⠡⠈⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠤⠒⡠⡃⠢⠔⢸⠇⠀⠀⠀⠀⠘⢴⢲⣶⠒⠒⡦⠄⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠊⡒⠉⠒⠉⠑⡀⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠭⠤⠂⠀⠀⠀⠀⠀⠀⢸⣭⣭⣭⣭⣭⣷
⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⢁⣀⣸⣀⡀
⠀⠀⠀⠀⠀⣾⣒⣛⠃⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠓⠠⣤⠄⠃
⠀⠀⠀⠀⢨⣝⠩⣫⢡⡢⢙⣥⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣄⣰⣖⣲⢰⡦⣑⡀⠀⠀⠀⠀⠀⠀⠠⡏⠉⠉⡹⠋⣹⡂
⠀⠀⠀⠀⠀⣆⣀⣀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠄⠀⢽⢹⣿⡆⠉⠀⣯⠀⠠⠀⠀⠀⠁⠀⢀⣠⡀⡠⢊⣼⡶⠀⠀⠀⠀⠀⠀⣇⠠⠊⡠⠊⢸⡁
⠀⠀⠀⠀⠀⠥⠤⠤⠤⠥⠤⠄⡠⣪⣾⠗⢄⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡹⢊⣴⣿⠋⠀⠀⠠⠀⠀⠀⠀⡇⡠⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⡴⠀⡠⣪⣾⣿⠁⠀⡠⠃⠀⠀⠀⠀⢸⣿⠿⠇⠀⡇⠀⠀⠀⠀⠀⠖⠖⣪⣴⡿⣫⠇⠀⢀⠔⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠰⢀⣀⣀⣁⣀⣘⠿⡻⠁⡠⠊⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠐⠀⠛⢌⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠇⠀⢀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠃⠀⠐⠁