Singly Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⢁⡠⠠⠀⣀⢀⣄⢀⢀⠉⡉⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠠⡈⣦⣴⣷⣿⣿⣿⣿⣿⣿⣷⣮⣵⣷⡶⣊⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣦⣼⣿⠟⣹⠋⠙⣿⣏⣻⣿⡻⡍⢻⡿⢿⣿⣶⣶⢖⠉
⠀⠀⠀⠀⠀⠀⠀⠠⣼⡞⠋⠀⢻⠀⡷⣷⣾⣾⣿⣿⣿⣷⡇⢈⡇⠀⠈⠛⢿⣷⣿⣤⠤
⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠸⡆⢣⡻⣿⣿⣿⣿⣿⣣⠃⣸⠃⠀⠀⠀⢀⡠⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠒⠦⣀⠀⠀⠀⠙⢦⡙⠺⠭⠯⠭⠚⣁⡴⠃⠀⢀⡤⡒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⢤⣭⣳⢶⣶⢶⡿⠯⠶⠒⠊⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⠉⠉⠉⠀⣀⣀⣀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⢀⣀⣤⠤⠖⠚⣿⢹⠉⠉⠉⠉⢳⠀⣻
⠀⠀⠀⠀⠀⠀⣠⠚⠁⢀⡤⠶⡏⡏⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⢸⠀⣽
⠀⠀⠀⠀⠀⢰⠃⠀⡴⠋⠀⠀⡇⡇⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⢸⡀⣿
⠀⠀⠀⠀⠀⣾⠀⢰⡇⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⣻⣸⣤⣤⠶⠾⠿⣍⠛⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⡇⣇⣀⣤⡤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠈⠳⣄⠉⠲⢤⡀
⠀⠀⠀⠀⠀⡇⠀⢸⣀⣤⠴⠞⣿⣿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡇⠀⠀⠙⣲⣤⡀
⠀⠀⠀⠀⠀⡇⠀⡿⣍⠀⠀⠀⠙⢾⡻⣝⢦⡀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣋⣩⡤⠖⠛⠋⠉⠉⠙⠷⣄⡀
⠀⠀⠀⠀⠀⡇⠀⡇⠈⠑⢄⡀⠀⠀⠙⠮⣷⣷⣀⣤⠴⠒⣋⡭⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⡇⠀⢸⠳⣄⠀⠀⠀⠈⠙⢦⣀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⣤⡤⠤⣤⡀
⠀⠀⠀⠀⠀⡇⠀⣸⠀⠈⠙⢦⣀⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉⠁⣀⡀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠷⣤⡿⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠛⠉⠁⣀⣠⡤⡶⢺⣭⣿⡟⣷⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⢀⣀⡤⣶⠚⣋⣭⣴⢾⡇⢸⡏⠁⣀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠉⠻⢷⣄⡀⠀⠀⢀⡟⠀⢠⡞⣋⣽⡆⣿⠟⠋⠉⢀⣀⡇⢸⡗⠋⠁⢿⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠉⠻⢶⣤⣼⡇⠀⣼⠛⠉⢹⡇⢻⠴⠖⠚⠉⠁⣿⢸⡇⢀⣀⣼⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠉⢺⡇⠀⣿⠒⠚⢹⡇⢸⠀⠀⢀⣀⣤⠿⠼⢛⣉⣥⣴⠏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⢸⠇⠀⣿⠀⠀⣸⣇⣼⠖⠛⣉⣥⠤⠖⠛⠉⠁⠀⢸⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣼⠀⠀⣿⠚⣉⣡⠭⠗⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠉⠓⢦⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣆⠀⠀⠀⠀⠀⠀⢸⣤⠴⠒⠚⠒⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢀⡤⠶⠖⠒⠒⠒⠶⠿⢷⣦⣄⣀⢀⣤⠖⠛⠉⠛⠻⣧⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠟⠁⠀⠀⠀⠀⠀⠈⣧⣀⣀⣀⣤⣴⣞⣁⡀
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠉⠉⠀⠀⠀⠈⠉⠳⣆
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⣀⣀⣤⡴⠖⠛⠉⠈⠛⢦⣤⣤⣤⣴⠋⠁⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠙⢿⣖⠒⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⣤⡤⠞⠉⢷⡀⠀⠀⠀⠀⠀⡿⠉⠛⠲⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣠⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⡴⠶⠶⠤⣔⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⣾⠿⢿⣿⡶⠿⠿⠶⡌⢦⠘⣿⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⣿⡁⠻⣏⠰⣿⣇⠲⠛⢛⣦⢀⣴⣠⡿⠀⠀⠀⣡⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⣿⢹⡀⠘⢷⣌⠉⢉⣉⣥⣴⣿⣿⠛⠁⠀⢀⣼⣿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣄⠻⣦⣤⣌⣉⣿⣯⣍⣉⣀⣀⣀⣠⣴⣾⣿⡿⠀⣘
⠀⠀⠀⠀⠠⡀⠀⠀⠘⢷⣶⣄⡙⠛⠛⣿⣿⡿⣿⣿⣿⣿⣿⡿⡛⢿⣿⡟⠁⣰⣿⡁⠀⠀⡀
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢻⣿⣿⣿⡛⠛⠻⣿⣷⣭⠳⡌⠀⠀⣠⡿⢏⣠⣾⣿⠏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣷⠩⣶⠾⣿⣾⣿⣿⣿⠇⠀⠀⠇
⠀⠀⠀⠀⠀⠀⡀⢹⣶⣤⡀⡀⠀⢀⣀⣉⣛⣿⣿⡟⡁⠀⠀⠀⠀⠟⣸⡿⠇⠀⠀⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠙⠿⣿⣿⣿⣿⡛⢟⣻⣿⣿⣷⡝⢂⡠⢀⣀⡴⠋⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠾⢿⡏⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠁⠉⠀⠀⢀⣀⣀⣠⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠤⠤⠖⣿⠋⣿⢿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣿⣿⠏⢙⣿⡇⠀⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠰⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢹⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⣾⣿⣿⣷⣶⣽⣽⡿⢷⡀⣇⠀⠀⢀⣤⣴⣶⣶⣾⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⡿⣫⣳⣏⢀⣴⣿⠿⣟⣻⣫⣔⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⣿⠎⣥⣶⣿⣿⣷⣯⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠁⠀⠀⠀⠀⠀⠀⠈⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡄⠀⠀⠘⢰⠀⠀⡆⠃⠀⠀⢠⠁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐⠀⠂⡀⢠⠀⢿⠀⠀⡿⠀⡀⢀⠔⠁⣂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡊⠐⣅⡈⠀⠸⠄⠨⠇⠀⢁⣨⠂⢑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⡈⠐⢼⣿⠀⠀⢓⡚⠀⠀⣿⡧⠂⢁⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠂⠀⠈⠳⢌⣄⠻⠀⠀⢸⡇⠀⠀⠟⣠⡣⠞⠁⠀⠐⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⠐⠀⠙⢆⠀⠀⠈⠁⠀⠀⡰⠋⠀⠀⠀⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⡈⠀⠈⠓⠀⠒⠀⠢⢄⠰⠆⡠⠔⠀⠒⠐⠚⠁⠀⢁⠠⠀⡄
⠀⠀⠀⠀⢔⠝⠻⠆⣿⡀⠀⠀⠄⡀⠀⠀⠈⠀⠁⢀⠀⠈⡀⠈⠀⠁⠀⠀⢀⠰⢀⠀⢀⣿⠰⠟⠫⡢
⠀⠀⠀⠀⣿⠀⢀⠀⠹⡿⣄⠉⠂⠐⠓⠁⠲⠄⡀⠀⠀⠀⠀⢀⠠⠖⠈⠚⠂⠀⠉⣰⣿⠏⠀⡀⠀⣿
⢀⣤⠀⠀⢭⣓⠄⣈⣠⢌⠠⡐⠀⡀⠀⢀⠂⠄⠀⠳⣀⣀⠎⠀⠀⠐⡀⠌⢀⠀⠂⠔⡠⣄⣁⡠⣚⠬⠀⠀⣤
⠁⠀⠀⠀⠁⠈⡉⠒⠂⡡⢘⡈⣀⢉⡀⠈⠓⠀⡀⠀⢹⡏⠀⢀⠄⠚⠁⢁⡉⣀⢀⡂⢈⠑⠒⢉⠁⠊⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⣫⡮⠥⢤⡨⡀⡁⠙⠢⡄⠈⠣⡈⢀⠜⠀⢠⠔⠋⠈⠀⣅⡤⠬⢵⣝⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠙⠀⣀⠀⠈⠓⢄⡸⢇⡠⠚⠁⠀⣀⠀⠋⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠚⢣⡜⠒⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠐⠋⠍⠉⠉⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠈⠉⠛⠋⠀⠀⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢦⣤⣤⡤⠀⠈⣁⣀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢽⡿⢠⠖⡚⠁⠠⠗⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⡱⣿⠏⠡⠔⠒⠛⢃⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣀⠀⠸⡄⠀⠀⠀⠀⠀⠘⡧⠀⠀⣰⠃⢀⣤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠹⣦⣀⣀⣠⣠⡼⠁⣸⡞⢡⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣭⢤⣦⣤⣴⡾⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠃⠀⠀⠀⠀⠀⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡴⠿⣻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠹⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡠⠎⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡅⠀⠀⢰⠁⠀⡴⣼⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠸⣦⡀⠀⠀⠀⠀⡇⠀⠀⣞⣠⣾⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣝⣷⡄⠀⠀⠀⣸⠀⠀⠻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠢⡀⠀⣸⠀⠀⠀⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠘⠄⣿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⡄⢀⠀⠀⠀⠘⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠙⠿⠿⣿⣷⣦⡀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣄⠀⠀⠈⠉⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣴⣄⣠⡀⣈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠛⠛⣳⢦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⡡⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡠⠀⠀⠀⠀⠀⠀⣀⠀⢀⡔⢁⡴⢦⠀⢠⠀⢨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠊⠀⠀⢀⠔⢁⣠⠞⣟⣴⣯⠞⠋⠀⠘⣧⡸⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣤⣾⠷⠚⠉⠀⠀⠋⠁⠀⠀⠀⠀⠀⠈⣿⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣭⠉⠁⠰⠦⣄⡀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠃⢸⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣴⣤⣍⠋⢠⠇⠀⠘⠛⣡⣤⡀⠀⢸⡇⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡻⠿⡋⠀⡾⠀⠀⠀⠀⢿⡿⢃⠀⢸⡇⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠉⠁⠸⣧⣀⠀⠀⠀⠑⠚⠁⠀⢸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡄⠤⠤⠤⠶⠶⠦⠤⠠⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣀⣀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢶⠞⠉⣏⠉⠉⠉⠈⢻⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠚⠛⠉⣹⠁⠈⣆⠀⠈⠀⠀⠀⠀⣼⠀⢹⡍⠙⠛⠲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢄⠀⠀⠀⠀⡇⠀⠀⠙⢆⠀⠀⠀⢀⡴⠃⠀⢸⡇⠀⠀⠀⡄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⢆⠀⠀⠀⢻⣀⣠⡄⠈⠳⣄⣠⠞⠁⣤⣄⣼⠁⠀⠀⠸⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠈⠂⠀⠀⠈⠉⠀⠳⢤⣀⣸⠁⣀⣠⠇⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢻⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⣧⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢯⣉⠓⠦⢤⣀⣰⡀⢸⡀⠀⠀⠀⢀⡇⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣠⠴⠚⠉⣨⡷
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠶⠶⣤⣽⡇⠀⠉⠓⠶⠚⠉⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⡤⠶⠞⢻⡏
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⢀⣀⣿⣤⡄⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⠛⠚⠷⢦⣤⣤⣤⣤⠤⣤⣤⣄⣸⣇⣇⣀⣠⣤⡴⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⣧
⠶⠶⠶⠶⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⢶⣶⣿⠛⠛⠛⠉⠙⠛⠛⠿⣿⡦⠀⠀⠀⠀⠀⢀⡿⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⣽⠀⠀⠀⠀⠀⠀⢀⡰⢬⣳⣄⣀⣠⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠛⠲⠶⠶⣤⣤⣀⡾⣭⡳⣝⣽⠏

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣴⣤⣤⣤⣄⡀
⠀⠀⠀⠀⢀⣤⣾⠿⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠻⠿⣶⣄
⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣿⣿⣶⣷⣦⣤⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⣶⣶⣿⣿⣿⣷⣶⣴⣤
⠀⠀⠀⠀⠤⢽⣿⣿⣿⠿⠛⣿⠉⢹⣿⠉⣻⠛⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⢻⡍⢩⡿⠏⢹⡏⠛⠿⣿⣿⡿⠥
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠤⣀⠘⠷⣤⣤⣴⠏⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠠⠈⠳⢦⠤⠴⠟⣀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣦⡄
⠀⠀⠀⠒⠛⠉⠉⠛⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠺⠿⠿⠟⠛⠉⠉⠉⠒⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠛⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠻⢿⣿⣄
⠀⠀⠀⠀⢀⣴⠶⣶⣶⠦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣶⡶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⣤⣤⡤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⢲⣾⣶⠄⣀⠀⠉
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠛⠓⠚⠃⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠁


⠀⠀⠀⠀⠠⠴⠶⢦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠶⠒⠦⠄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣉⠉⠛⠀⠀⠀⠀⠀⠈⠉⡁⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠟⠻⠿⠿⣿⣦⣄⠀⠀⠀⠀⠀⢀⣴⣾⠿⠟⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠠⣶⠋⣿⣿⡟⢦⡀⠀⠀⠀⠀⠀⠀⣠⠚⣿⣿⠏⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠋⢃⠃⢉⡍⠁⠀⠀⠀⠀⠉⢠⡀⠉⡘⠈⡙⠳⠆
⠀⠀⠀⠀⠀⠈⠓⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣌⣙⡳⣶⡄⠀⠀⠀⠀⣴⠞⣋⣁⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢋⣛⣿⣿⡿⠿⠶⠦⠀⠀⠀⠀⠀⠾⠛⢛⣭⣭⣍⡉⠉
⠀⠀⠀⠀⠀⢻⣿⡙⠛⠛⠻⠄⠀⠀⠀⠀⠀⠀⠜⠛⠛⠛⣹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠰⢴⡅⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠨⣦⠎⠄⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠐⠋⠉⠉⠙⠛⠿⣷⣶⠀⠀⠀⠀⠀⣰⣶⠿⠟⠛⠉⠉⠉⠳⠀⠀⠀⠀⠀⠀⠰⠾⠛⠛⠛⢛⣛⠛⠛⠆⠀⠀⠀⠀⠀⠀⠰⠛⢛⣛⣛⣛⠛⠛⠷⠆
⠀⠀⠀⠀⠀⣠⡤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢿⣽⡟⠱⡀⠀⠀⠀⠀⠀⠀⠀⡜⠹⣽⣿⠉⠳⠄
⠀⠀⠀⠀⠀⠙⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁


⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣤⣦⡀
⠀⠀⠀⠈⠉⠉⠙⠛⠷⠶⣦⣄⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⢋⣉⣠⣬⡽⢯⣝⠛⠀⠀⠀⠀⠉⣭⠿⢷⣶⣤⣌⡙⠿⢦
⠀⠀⠀⠀⢀⣄⣤⣤⣤⣄⡄⠈⠀⠀⠀⠀⠀⠉⠠⠤⣶⣶⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠁⠛⠛⢃⣚⣫⠃⠀⠀⠀⠀⠘⠙⢯⣌⠛⠛⠁⠛⠦
⠀⠀⠀⠀⠈⠛⠄⠠⠭⠁⠺⠀⠀⠀⠀⠀⠀⠀⠚⠀⠐⠐⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⢶⠶⠶⠂⠘⠛⠓⢶⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣁⡄⠙⠶⠀⠀⠀⠂⠀⢿⣏⢁⡽⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠋⢻⢿⣿⣍⣡⣔⡦⠀⠂⠹⠿⠦⣤⡟⢒⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⣴⠒⠶⠖⠶⣤⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢙⣿⣿⢾⡿⣖⡠⡀⠈⢻⠟⢻⡿⣛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡥⠿⡇⢨⣿⡥⢹⡟⣿⠛⠛⠒⠻⣶⠖⢲⡛⢏⢳⣿⣤⣀⠀⠀⠀⠀⠈⠹⣿⣿⣿⡟⣷⣇⢳⢺⡇⣿⣷⡨⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀⠀⡷⠞⣿⠀⠀⡇⣿⠀⠀⠀⠀⣸⡇⠀⠁⠈⠁⠋⢷⠘⣧⡛⢶⡄⠀⠀⢙⣷⡇⡾⣿⣿⣿⣸⣷⣿⢸⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⣸⠇⡇⡿⠃⣸⠃⣿⠀⡀⣆⢠⡿⠀⠀⠀⠀⠀⠀⣼⣷⡟⠙⠀⠀⠀⠀⣾⣿⣿⣿⠿⣿⣿⣫⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣴⣏⣴⣷⣤⣴⣋⣠⣏⣠⣿⠿⢾⡇⠀⠀⠀⢀⣤⣾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⣰⠆⣿⣿⣿⣿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡏⠉⠉⠉⣿⠁⠀⠀⡧⠀⠀⠸⣾⠁⣞⡁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣀⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣿⠀⠀⣸⠇⠀⠀⠀⢹⡄⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿⠀⢀⡟⠀⠀⠀⠀⠸⡅⣥⡀⠀⠀⠀⠀⢀⣀⣤⠆⠀⠀⢀⡩⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⣄⣾⠁⠀⠀⠀⠀⠀⡇⠈⠙⠛⢻⡟⠛⠉⠉⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣷⠴⠶⠟⢻⣁⠲⠒⠲⠖⠒⠒⠚⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⠀⠀⠀⢀⡿⢸⡇⠀⠀⠀⠀⠀⢸⡇⢀⣠⡴⠛⠉⠀⠀⡀⠀⠀⣠⠄⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠲⣤⡾⠁⣸⡇⠀⠀⠀⠀⢀⡾⢻⣿⠁⠀⠀⠀⠀⠀⠹⣦⠞⠋⠀⢰⡆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣿⠁⠰⠏⢻⣄⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⡷⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢹⣦⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣜⣿⠀⠙⣷⣤⡀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⣰⡟⠈⣿⡟⠀⠀⠀⠀⠀⣀⣠⡴⠊⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢻⣀⣴⠟⠀⣠⠶⠃⠀⠀⠉⠁⠀⢀⡤⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠙⠛⠻⠟⠛⠛⠛⠛⠛⠛⠻⠶⡼⠇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢠⡟⠉⠉⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠀⠀⠀⠀⠉⠛⠲⢦⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⣤⣤⡀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀⠳⠶⠶⢶⣶⣶⣶⠶⠶⠾⠿⢯⣤⣀⡀⠀⠈⠻⣦⡀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣷⣶⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠈⠻⣦⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⢳⣄⠀⡀⠀⠀⠀⠀⣀⣠⣴⢞⣫⠵⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢠⡾⢉⡶⢋⡝⢻⠿⠷⣾⡶⠒⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⣿⡇⣾⠀⢸⡀⢿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠙⢷⣼⣇⠈⢷⡸⣧⠀⢸⡄⠀⠀⠀⠀⠀⢀⣠⣄⣤⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⡶⢛⣩⣤⢂⣠⣆⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⠁⣀⣤⣾⠿⠶⠶⠶⠾⠛⠛⠛⠛⠛⠻⠿⢿⣏⡉⠉⣉⣩⡥⢛⣽⠛⢿⠿⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣯⣤⡴⠋⢁⡔⠁⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢾⣯⣄⣤⡾⠋

⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣦⣷⣼⣶⣷⣶⣦⣶⣿⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣷⣾⣿⣷⣶⣶⣶⣶⣿⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⡀⠠⠤⢒⠖⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⡀⣀⡀⣂⣄⣤⠆⠶⠒⠛⠁⠉⠉⠉⠉⠀⠉⠉⠁⠤⣄⡀
⠀⠀⠀⠀⣴⣴⣀⣔⣵⣢⣴⣶⠿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠧⣀
⠀⠀⠀⠀⠿⡿⠿⠿⠛⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⣢⢤⣤⣴⡿⢿⠿⠿⢿⣿⣷⣖⣶⣒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣠⣶⡾⣟⣾⣧⣤⣶⣷⣾⣾⣾⣿⣿⣿⣿⡟⣶⣿⡵⣂
⠀⠀⠀⠀⠀⠄⢀⠄⢤⣜⣿⡿⣯⣷⣿⣿⣿⠿⠛⣻⣟⣽⣿⣿⣛⠿⣿⣿⣾⣿⣿⣿⣿⣝⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣵⢍⣿⡿⠛⣿⣿⣻⢠⣷⣾⣿⣿⣿⠁⠀⢸⣇⡉⢿⣿⣿⣿⣧⡟⠿⣆⡠
⠀⠀⠀⠀⠄⠠⠆⢄⣼⣉⣾⠁⠀⢀⣿⣿⢟⣿⣿⣿⣿⣿⣿⡦⢤⣾⣿⠄⢸⠈⠹⢿⣿⣿⣿⣯⠖
⠀⠀⠀⠀⠀⡄⠐⣻⣵⣿⣿⠀⢀⠀⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿⣶⣿⡟⢀⡼⠀⠀⣻⣿⣿⣿⣟⢋⠁
⠀⠀⠀⠀⠠⠀⠲⣿⣿⣿⣿⣷⣜⠄⠹⣿⣯⣜⣿⣿⣿⣽⣿⣿⣿⡿⢃⣼⠏⣀⣴⣿⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⠛⢿⣶⣥⣀⠈⠻⢿⣿⣿⣍⣿⣿⣿⣭⣾⣿⣵⣿⡿⠿⠛⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⠶⠿⢿⣿⣿⠿⠿⣿⣿⠻⠍⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠇⣸⣽⣿⣶⣾⣿⣮⣟⣢⡦⠤
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣖⣀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢓⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣒⣦⣦⢤⣄⣀⠀⠀⠀⢀⣤⣿⣾⣿⣆
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⡍⣽⣿⡛⠿⡿⣿⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣯⣷⣶⣶⣶⣶⣶⣶⣶⣮⣽⣝⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣇⠀⠀⠀⠈⠙⠟⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢛⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠇⡝⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡿⠹⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣭⣯⣿⣌⡋⣙⡀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠃⠀⠀⠀⠀⠀⠀⠀⣠⣼⣇⣭⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣥⣀⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣀⣠⡴⠟⢫⣼⣶⣻⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣦⣖⠲⢾⡍⠏⡽⣛⣵⣿⣿⣫⠇
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣍⣁⠋⠙⠋⠙⠀⠍⠙⠛⠈⠋⠉⠁⠀⠀⠐⠋⢊⢩⣿⡿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢲⣰⠁⢺⣟⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⠿⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⢐⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣭⣥⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣂⠀⠀⢀⡬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⠀⢰⢖⣖⡺⠍⠁⠀⣀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⡶⣭⡥⣿⢧⣾⡍⠱⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠁

⠛⠛⠣⣤⣤⡜⠛⠛⣤⠀⠀⠀⠀⠀⠀⠀⠰⠂⡰⠃⠀⠀⠀⠀⠀⣀⠼
⠀⠀⠀⣿⣿⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡞⠒⡇⠀⣠⠄⢂⢀⠏
⣿⣿⣿⠀⠀⢾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡄⢀⠠⣀⡄⠀⠀⢠⠄
⠛⠛⠛⣤⣴⣾⢋⠻⢳⡦⠀⠀⠀⠀⠀⠀⠖⡁⡸⠃⠀⢀⡴⠟⠈⠀⠙⠀⠀⠀⠀⠀⣠⡴⠶⢦⣤⡄
⠀⠀⠀⠿⠿⣟⡂⠀⡸⠇⠀⠀⣤⡋⠀⠀⠀⢠⡧⠶⠂⠈⠑⠎⠀⠀⠀⠀⠀⠀⡀⣸⣃⣴⠶⠳⣎⣿
⣿⣿⣿⠀⠐⢺⣯⣾⡇⠀⠰⣘⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣹⣿⣇⠂⠀⠘⣻⣿⡇
⠛⠉⢽⣦⣶⣿⠭⡫⠀⣌⣂⣤⢰⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠀⠈⠉⠉⠀⠉⠀⠉
⣀⣀⣺⣿⣿⣿⣅⣚⡐⠬⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠛⠂⢁⣼⣿⣷⣶⡿⠿⠂
⣿⣿⣿⠃⢚⣿⣿⡷⣞⡍⢀⣀⣀⣤⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠈⠻⣿⣿⣿⣿⣿⣿⡇
⠀⢠⠿⣿⣿⣿⢔⡛⡓⠻⣿⣿⣿⡻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⣠⣿⣿⣿⣿⣃⣄⠈⠙⢿⣿⣿⣿⣿
⣤⣤⣴⠛⠉⢽⣿⠬⢄⣾⠿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣿⣿⣿⣿⣿⣷⣆⢸⣿⣿⣿⣿
⣿⣿⣻⣏⠀⢹⣯⡭⠘⠁⠀⢻⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠙⠃⢣⠛⡏⠙⢻⣿⣿⣿⣿⡞⢡⡈⠓⢠
⠀⠀⠨⣿⣿⡇⠒⠃⠀⠃⠀⠠⡆⠘⠻⠂⠁⢀⡀⠀⣶⡟⡋⡎⢢⣿⣿⣀⣀⠐⠀⠉⠉⠉⠀⠈⢡⣶⠆
⣀⣀⣤⣭⣭⣀⡴⠶⠀⠀⠀⠤⠁⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠙⣿⣿⣿⣿⣿⠃⠀⠀⠁⠀⠀⠀⠔⠶⢦⠦⠀⢤⠄⡴⠂⠀⠀⠀⠀⠀⠖⠄⠀⠀⠀⢸⡿⠂⠁⡒⣞⣲
⠀⠘⢻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡖⠉⠊⠐⠋⠁⠀⠀⠀⠀⣠⠆⢠⠀⠀⠀⠀⠈⣿⡃
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠉⠙⠛⠚⠁⠀⠄⠁⠀⠰⠶⠆⠉⠃⠀⢀⡀⡀
⠀⠀⢴⣄⠰⢿⣿⣿⣿⣿⣿⣧⣠⣀⣤⡼⢶⠲⣾⣿⡿⢀⣶⠄⣴⣶⣶⣶⣶⣶⣶⣶⣶⡘⣷⡈⣿⣿⣶
⠀⠀⠈⠛⢷⢄⡌⠽⣟⣻⣿⣿⣿⣿⣿⣧⣼⣶⣿⣿⣣⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣼⣿⣿
⠀⠀⠀⠀⣠⣽⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠴⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣒⣠⣴⡾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠤⣀⣶⡶⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣰⡷⣻⣿⣿⠟⠋⠀⠀⠀⣠⡤⠔⠒⠒⠃⠈⢉⢉⠉⠉⠀⠀⠀⠲⠦⢤⣦⣀
⠀⠀⠀⢠⣿⣿⠿⠋⠀⠀⠀⠀⠠⣌⣠⣦⣴⣾⣿⣷⣶⣷⣶⣿⣿⣶⣶⣦⣤⣴⣞⠉⠁
⠀⠀⠀⠈⠟⠀⠀⠀⠀⢤⣤⣤⣷⢿⡿⢟⣛⣯⣿⣿⡀⠀⠀⠀⢹⣿⠉⠁⠀⠉⠉⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⡟⠁⢸⢊⣹⣿⣿⣿⣿⣿⣶⢀⠀⣸⣿⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠏⠀⠀⠸⣿⠋⢹⣿⣛⣃⣽⣷⡀⠞⣥⠇⠀⠀⠀⢀⣤⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣸⡟⠀⠀⠀⠀⠻⣶⣿⠙⠋⢻⡟⠛⢧⠀⠃⠀⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⠀⠀⠀⠈⠈⠁⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠒⠂⠀⠀⠒⠄



⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⣿⣿⣿⣿⠋⣻⣿⣿⣴⠚⠻⡦⠀⢀⣉⠛⢿⣿⣿⡆⢰⣿⣿⣿⣿⣿⢿⠛⠻⠿⠿⣏⢻⣿⣿⣿⢻⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣄⣸⣿⣿⠀⠉⠃⠀⠀⣟⡉⢢⣿⣿⣿⡇⢸⣿⣿⣿⣿⡿⠶⠀⠀⠐⠒⣛⢹⣿⣿⣿⡜⢿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠇⢀⣤⡀⠀⠀⠁⢸⣿⣿⣿⣿⣾⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⠀⠒⢀⣴⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⠠⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠀⠙⠓⠒⠚⠛⠛⠛⠛⠛⠛⠁⠈⠛⠛⠛⠛⠀⠀⠀⠀⠛⠀⠘⠙⠛⠛⠛⠛⠛⠛⠛
⠀⠀⣶⣶⣶⡖⠀⠀⠀⠀⠐⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢰⣶⣶⣶⣶⡦⠴⠆⠀⠀⠲⠶⠶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⣿⣿⣿⣷⣤⠀⠐⠶⠻⡿⠿⣿⠻⣿⠿⢿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠴⣶⣄⠀⠀⠀⢴⣶⣾⣟⣿⣿⢻⣿⣿
⠀⠀⣿⣿⣿⣿⡍⡇⠀⠀⠀⠐⠊⠀⠁⠉⠀⣸⣿⣿⣿⡇⢸⣿⣿⣿⡟⠴⠁⠀⠆⠀⠀⠐⠀⠈⠻⣿⣿⣠⣿⣿
⠀⠀⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡇⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⣿
⠀⠀⠿⠿⠿⠿⠿⠶⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⠿⠿⠿⠃⠘⠿⠿⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠛⠛⠛⣉⣉⡦⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣂⣀⣂⣀⣨⣅⡤⣕⣀⢠⠄⠘⣀⣀⡀⠀⠒⠐⠒⠠⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣖⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠓⠒
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣦⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⠟⠋⠙⠙⠛⠻⠿⣷⣦⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣭⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⢿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠈⠙⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠟⠋⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⣄⠀⠻⣿⣿⣴⣀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠁⢙⠛⣱⣿⢇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣯⣉⣉⣿⠀⠙⠻⣦⡈⠻⣿⣿⣶⣶⣿⣷⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⣻⣟⣟⣿⣿⣟⡆⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠉⠙⠓⠦⠌⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠉⣠⣤⣽⡿⣶⣄⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣶⣍⠙⠓⠦⠬⠽⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠆⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠛⢟⡭⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣟⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢉⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣧⣼⣽⣿⣹⡽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢾⣿⣭⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣤⣴⣦⠀⠀⠀⠀⠀⠀⠘⠚⣿⣿⡾⢟⣯⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⣾⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢨⣽⢟⡯⣻⢿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⣭⣾⣽⢟⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢽⣾⠘⣋⢿⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⣾⡽⠀⢴⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇⠀⠘⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠄⠀⢠⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⣿⣦⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⡿⠿⠿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⣀⣤⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣶⣤⡀⠙⢿⣧
⠀⠀⠀⠀⠀⣠⡾⠋⠁⢀⣠⣶⠿⠛⠉⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⠻⣿⣦⡈⢻⣷⣄
⠀⠀⣀⡴⠟⢁⣠⣴⣾⠟⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⢻⣿⣦⣙⠻⠷
⠀⠈⠉⠒⠛⠋⠉⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⠙⣿⣿⣿⡿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⡟⠻⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⠀⠀⠀⠀⠀⠈⠻⢿⣾⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣀⣼⣿⣶⡶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣄⣀⡀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣀⣤⣴⠶⠛⣋⣩⣤⣤⣶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠛⣛⣛⣫⣭⣵⠶⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠇⠀⢀⣴⣿⣿⣿⣿⣷⡀⠀⠀⢰⣿⣿⣿⣿⣷⣆⠀⠀⢣⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⢧⠀⠀⠀⢸⠀⠀⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⠀⢸⡀⠀⠀⢠⠏⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢣⠀⠀⣿⠀⠀⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡟⠀⠈⡇⠀⢠⠃⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠱⢄⡇⠀⠀⠈⠙⠛⠛⠉⠀⠀⢀⣀⣀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⣇⠴⠁⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⣀⡤⠤⡀⠀⠀⠀⠀⣠⠤⣄⡀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠘⠦⠤⠤⠞⠁⠀⠀⠈⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠚⠟⠛⣛⡷⠟⠛⠉⠙⢓⠲⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⢚⣫⠞⠁⠀⠀⠀⡴⠉⠁⠠⡀⠀⠘⢶⡄⠀⠀⠠⣁⠘⢫
⠀⠀⠀⠀⠀⣠⣰⢟⡁⠰⠛⠉⠀⣀⠔⠃⢠⡾⠁⢈⡀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠙⡎⠁⠀⠀⠈⠒⠦⣄
⠀⠀⠀⢠⡿⢻⠷⠛⠁⣠⠀⠤⢾⣥⡤⠖⠚⢃⠀⠈⠓⢤⣀⠀⣤⠖⠁⠀⠀⠀⠠⠃⣄⠀⠠⠀⠁⠀⠘⣗⢤⡀
⠀⠀⣴⡿⠀⠀⠀⠀⠼⣧⡀⠀⠀⠋⠀⠀⠀⢸⡄⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠠⡇⠀⠈⠀⠀⠀⠀⠀⠀⠈⢀⠘⢷⡀
⠀⣼⣸⠁⢀⡿⠇⡤⡄⠈⠛⠦⣀⠀⢀⠆⢀⡾⠃⠀⠀⠀⣐⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⣿⣄
⢀⣿⢻⡅⣯⠀⢸⠀⠀⠀⠀⠀⢸⡤⠊⠀⢸⡇⠀⠀⠀⠘⠋⠁⠀⡀⠀⠀⢀⣧⠀⠀⢈⡇⠀⠀⠀⢠⠈⠀⠉⡗⠁⠉⣙⢷⡄
⣸⡃⠈⢀⣿⠀⠆⠀⠀⠒⠀⢠⣾⠁⠀⠀⢸⣧⣤⣠⣤⣴⣶⠤⠄⠠⠄⣠⠜⠁⠀⢰⡟⠀⡔⠀⢢⢸⠀⠀⠀⢳⣠⡄⠈⠼⣿⣄
⣿⡇⢠⣾⡇⠀⠀⠀⠀⠀⠀⢠⢿⣦⡴⠛⠉⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣸⡀⠐⠉⠀⢼⠈⠀⠀⠀⠘⣸⠇⠀⠀⠈⢻⣦
⢻⣷⠀⠁⢷⡀⠀⠐⠀⠀⠀⠀⢺⠏⣡⠀⠀⣀⠆⠁⠀⠀⠤⠤⠀⠒⢈⣉⣀⣈⣉⡉⠑⢦⡶⠦⠜⠃⠀⠀⢀⠾⠋⠀⠀⣠⣾⢻⢻
⠀⠟⢷⣀⠀⠙⠦⣤⠀⠐⢀⣤⡊⠀⠻⣵⠘⠁⡄⢠⡀⠄⠀⠀⠀⠀⠀⠐⡄⠈⢉⠁⠀⠀⠗⠀⠀⠐⠀⠀⠈⠁⠀⠁⠀⠙⠿⢆⣲
⠀⠸⣾⠤⣤⣤⣼⣧⣰⣷⡽⢿⠁⠀⠁⠀⠐⠀⠃⠀⠀⠀⠀⠀⣀⣶⠀⣴⣇⠀⠙⣓⠀⠀⠈⠀⠀⠄⠀⠀⠆⢀⡶⠂⢫⣴⣢⠾⠛
⠀⠀⠈⠲⢤⣽⣿⣿⣿⠿⠗⢻⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡴⣿⣿⣿⣴⡀⠐⠓⠛⠙⠗⠒⠛⣿⣦⣤⣤⣴⣿⣶⣶⣦⣿⣷⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣄⣀⣤⡴⣶⣶⠗⢿⣿⣿⣿⣿⣿⣟⣿⡷⡲⣤⡰⠄⢛⠲⣤⣤⣐⡛⣻⣿⣿⣿⡟⠻⣯⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡋⣉⡀⠝⠁⠀⠀⠙⠻⠿⣿⣻⣿⣿⣿⣷⡿⣽⣷⣝⢳⣯⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣌⢛⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠒⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣾⢗⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢕⡥⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣦⣈⢢⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⡊⢀⣤⣶⣴⣦⣤⣀⣀⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣥⡳⡠⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣀⣄⠀⠀⠘⢿⢿⣿⣿⣿⣿⣿⣷⣥⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡻⠋⠁⠈⠀⠈⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠈⠘⠿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣔⣄⣄⡠⣀⣠⣤⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠡⢻⣿⣿⣿⣿⣿⣿⣿⣶⣾⣦⣤⣤⣤⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠻⣿⣿⣿⡿⠉⢿⣿⣿⣿⠉⠿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢻⡟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀⡉⣁⣀⡠⢬⡿⢿⠧⠀⠀⡿⣿⣿⣗⠀⠀⠀⠀⠀⢿⡿⠛⠋⠻⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⡌⠀⠀⠀⠁⠉⠉⠋⠻⠿⠿⡿⠟⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⡙⡍⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠛⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣞⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢂⢊⢊⢦⢂⠀⠀⠀⠀⠀⠀⠀⠈⠋⠿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣯⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢕⣵⣵⣽⣷⡧⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣵⣿⣿⣿⣿⣿⣿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣮⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣽⣾⣟⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⡀⡀⠀⠀⠀⠀⠀⠀⢼⣽⣿⣞⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⢞⣞⣞⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣻⣿⣯⣿⣻⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⢐⣵⣵⡵⡿⡵⡷⡷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣪⣮⣮⣾⢿⢿⢟⢞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣓⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣷⣾⣿⣷⣦⠀⠀⠀⠀⠀⠀⢸⢟⢕⣵⣷⡿⡷⡿⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣤⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⢔⣵⣿⡿⡫⣪⣾⢿⢟⣽
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⢿⣽⣿⣿⣿⣿⣿⣵⡀⠀⢐⢝⣿⣫⣾⣿⢿⣵⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣼⣿⣿⣿⢿⢿⡪⢀⢕⣯⡪⣿⢕⣵⣿⣻⣾⣾⢞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣑⣽⣮⣗⣝⣯⣪⣗⣕⡿⡫⣮⣿⢕⣵
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣫⣿⢷⣿⡪⣾⣟⢵⡿⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠻⣿⣿⣿⣿⣿⣷⢿⣪⣿⢽⡿⣾⢿⣷⡿⣫⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⢿⡿⣿⢿⡻⣷⢽⡯⣻⢗⣿⡿⣿⢿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⠀⠀⠀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣠⣶⠟⡉⢀⣴⡶⢀⣀⣉⣛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣾⣿⡟⢰⢱⣿⣿⠿⠷⠦⠀⢍⡓⢮⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣜⢿⣿⣿⣇⢰⣿⣿⣿⣿⣿⣛⠿⣶⡌⢦⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡈⢿⣶⣝⠿⣿⡌⠻⣿⣇⠉⠛⢿⣷⣌⠁⠀⠹⣦⡙⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠣⢖⣤⠶⠟⠛⠋⠀⠁⠀⢰⣮⡁⠀⠀⠉⠻⣧⠀⣄⠘⣷⡌⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠡⠞⣡⣴⠖⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠈⠃⢸⣆⠘⣿⡌⢿⣆
⠀⠀⠀⠀⠀⠀⢠⣿⢣⣤⠞⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⢸⡿⠀⢻⣿⡌⢿⡄
⠀⠀⠀⠀⠀⠀⣾⠇⣩⡎⣼⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡀⣷⢠⣬⣿⣿⡜⣿⡀
⠀⠀⠀⠀⠀⣸⣿⢰⣿⣸⣿⠃⠀⡠⠶⠶⣶⣤⣄⡀⠀⠀⢸⡆⠀⢀⣠⣤⣴⣿⣼⡇⣿⣿⣿⣷⠹⣧
⠀⠀⠀⠀⠀⣿⣿⢸⣿⡿⡃⠀⠀⠀⠀⠀⠀⠈⠙⠻⠇⠀⠀⡇⠀⠞⠋⠉⢰⡏⠈⣷⠸⣿⣿⣿⣇⢿⡄
⠀⠀⠀⠀⢀⣿⣿⢸⡏⠘⡇⠀⠀⢠⣤⠼⣿⣥⢄⡀⠀⠀⠀⢸⠀⠀⣤⣬⣿⣃⣸⣿⠀⠹⣿⣿⣿⡘⣧
⠀⠀⠀⠀⠈⣿⣿⡄⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⡆⠀⠈⣻⠋⠉⢸⣿⢠⡆⠹⣿⣿⡇⢻
⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢰⠃⠀⠀⣾⡇⢸⣷⠀⠙⣿⣿⠘⡇
⠀⠀⠀⠀⠀⠘⣿⡝⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠈⠀⠀⠀⣿⢠⣿⣿⡄⣦⠈⢿
⠀⠀⠀⠀⠀⣷⡘⣷⠀⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⡇⣿⣧⠘⠃
⠀⠀⠀⠀⠀⣿⣿⠈⠣⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠈⡇⢸⣿⣿⡇⣿⣿⡀
⠀⠀⠀⠀⠀⢹⣿⠀⠀⣆⠘⣿⡀⠀⠀⠀⠀⠀⢰⣿⣿⣉⣉⣹⣿⡆⠀⠀⠀⠀⠀⣼⣿⡟⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠘⣿⢰⠀⣿⡀⠸⣷⡀⠀⠀⠀⠀⠀⠙⠿⢭⡭⠿⠛⠀⠀⠀⢀⡾⢠⡿⠋⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠙⢸⣧⣿⣧⢧⡘⢷⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⣈⣴⠀⠀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡞⣿⠆⠈⠀⠙⠦⣄⡀⠀⠀⠀⠀⣀⡴⣿⣿⠇⣾⣿⡏⢠⠃⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡜⢧⣀⡀⠀⢀⡀⠉⠛⠿⠿⠿⠋⠀⣼⠏⣸⣿⡿⠁⣾⣴⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣭⣽⠎⠘⠁⠀⠀⠀⠀⠀⠀⠐⢃⣼⣿⡿⠁⣼⣿⣿⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⡻⠿⠿⢛⣁⣀⡠⢀⡀⠀⠀⠀⠀⠀⠶⠿⠿⠋⣠⣾⣿⢟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠐⠤⣤⣴⣾⡿⠟⣡⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠿⠶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡑⢄⣀⣀⡀⠈⠉⠙⠛⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢈⢀⣀⣈⠉⠑⠒⠦⢤⣀⠀⠈⠙⠛⠶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠖⠛⠛⠓⢊⠔⠉⠀⠀⠀⠢⢄⡀⠀⠉⠑⠢⢄⡀⠘⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣟⠋⠀⠀⠈⠑⠒⠴⡃⡀⠀⠀⠀⠀⢀⠀⢀⠉⠡⢄⠀⠀⠀⠀⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠑⠒⠤⢀⠀⠀⠈⠑⠲⠤⣀⣀⡁⠐⠁⢀⣀⠀⡀⠦⢄⡀⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⢉⣁⠀⠀⠀⠈⠑⡀⣿
⠀⠀⠀⣠⣤⡶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢄⣀⠀⠀⠈⠑⠤⠐⠃⠀⡇⣿
⠀⠀⢸⡏⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠝⣦⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⢸⡟⡉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⣰⠁⣿
⠀⠀⢸⡇⢸⠑⠢⠄⡈⠐⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠊⠁⡸⠀⣿
⠀⠀⢸⡇⢠⣍⠀⠀⠀⠁⠂⠄⣀⠉⠒⠤⣄⡤⢔⠂⢀⡄⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⣿⠀⢰⡟
⠀⠀⠈⣧⠐⢤⡙⠢⢄⡈⠐⠠⠀⠉⠒⠤⣀⠃⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⠶⠟⠀⠈⠛⠛⠁
⠀⠀⠀⢻⣤⣾⠙⠳⢦⣌⡑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣈⠉⠒⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡇⠀⡄⠀⠀⠀⠀⢀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣇⣤⡴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣦⣶⡶⣒⣶⡶⣿⣿⣿⣷⣶⣦⣔⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⣻⡿⠿⢫⡽⠟⠛⠛⠛⠛⠛⠿⠿⣻⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣶⣶⢾⣿⣿⣻⣿⣿⣿⠿⠿⠻⠟⠱⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡫⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣿⣿⡿⣫⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡿⣿⠟⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⠀⠀⢀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣷⠶⢿⣿⣿⣿⡖⢶⣿⣤⣶⢁⡞⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢿⣏⢹⣆⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣵⣯⣋⣴⡿⣁⣰⠆⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢿⣇⢸⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣶⣶⣮⣭⣛⣿⣯⣴⣾⣿⣾⡿⣃⡤⠎⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⠶⣵⣬⣿⡿⢿⣿⠟⠛⢿⣿⣿⡉⠙⠛⢻⣿⡇⠉⠛⠷⣮⣝⣿⣿⣿⣯⣠⣴⠟⣣⡤⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠳⣶⣾⡿⠛⠀⠘⣿⣶⣶⣿⣿⣿⣿⣿⡿⣦⣙⣷⠀⠀⠀⠀⠉⠛⣼⣿⣿⣿⣿⣟⣩⡤⢚⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⠿⠁⠀⠀⠀⢻⣷⡞⣿⣿⣿⣿⡿⣿⣿⣿⠃⠀⠀⠀⠀⢀⠔⠀⢿⡙⠻⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈⠀⠀⣀⣤⣾⠏⠀⠀⠀⠀⠀⠀⠈⢿⣾⡿⣭⣿⣯⣽⣿⠟⠃⠀⠀⠀⡠⠔⠁⠀⣔⡛⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⡙⢆⠀⠀⢀⣀⡀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠀⠠⠐⢀⣤⣾⡿⢿⡝⢝⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⣹⡼⢋⡴⢬⣥⢑⠖⡒⠀⣀⠀⠀⠀⠀⢀⢠⣠⣤⣿⡿⠿⣧⡀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠏⠐⠹⡇⠋⡞⢡⣷⢟⡇⢿⡆⢷⡟⡆⠳⡙⣿⡇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠛⠀⠇⢻⠆⠸⠇⠃⠀⠁






⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⠛⠳⢶⣄⡀
⠀⠀⠀⠀⣤⣤⣄⣀⣀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⢰⡏⠀⠈⠉⠉⠛⠛⠻⠶⠿⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠇⠀⠀⠀⠀⣿
⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢀⣼⠏
⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⣾⠉⠛⠛⠛⡛⠛⠛⠛⢻⣷⡶⠶⠶⠶⣾⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠚⢛⣽⠇
⠀⢠⡟⠀⠀⣴⠟⠛⣷⠀⠀⠘⠛⠃⠀⠀⢠⡟⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠉⠀⠀⠀⢰⡟⠁
⠀⣼⠇⠀⠀⠙⠷⠶⠟⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀
⠀⣿⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣤⡀
⠀⢻⡆⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⢸⣿⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠈⢿⣄⠀⠀⠙⠛⠛⠁⢠⣶⣦⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠙⠷⣄⣀⠀⠀⠀⢈⣉⣥⠾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠛⠉⠀⣸⡇⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢠⣼⣷⣄⠀⠀⠀⠀⣿⠀⠀⢠⡟⠉⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠉⢻⡆⠀⠀⣿⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢻⣿⠃⠉⠻⢦⣀⠀⣿⠀⠀⠘⣧⣀⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣀⣼⠃⠀⠀⣿⠇⠀⣀⣴⠞⠋⢉
⠀⠀⠀⠀⠀⢈⣿⡄⠀⠀⠀⠙⠻⢿⠆⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣿⠶⠟⠋⠁⠀⢀⡾
⠀⠀⠀⠀⠀⣸⠏⢻⣄⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⢀⣀⣀⣤⣤⣿⣄⠀⠙⣦⡀⠀⠀⠀⠀⠀⢸⡿⠶⢦⣤⣄⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡶⠶⢿⡇⠀⠀⠀⠀⠀⢀⣼⠋
⠙⠛⠛⠿⠶⢦⣽⣿⣦⣌⠻⢦⣄⠀⠀⠀⠈⣷⡀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⢀⣿⠁⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⢻⡆⠀⠀⠈⢷⣄⠀⠀⠀⠀⣀⣤⣤⣀⣤⣤⣀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢶⣄⣀⣾⠋⠁⠀⠉⠀⠀⠙⢷⣀⣠⡶⠋⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⣤⣤⣤⣤⣤⠶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⡴⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣦⣤⣴⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⢿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠛⠾⣭⣟⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣭⡛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠟⠳⢦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠙⠳⢮⣙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠀⠀⣠⣾⠟⡋⠹⠙⠛⠻⠷⣦⣄⡀⠀⠀⠀⠀⠀⠙⠳⣬⡻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢿⡛⠙⠳⣤⣀⠀⠀⠀⣿⣥⠂⠀⠀⠀⠀⠀⠀⠈⠙⡻⣷⣦⡀⠀⠀⠀⠀⠈⢻⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠉⠲⢄⡀⠉⠳⢦⣀⠈⠛⢷⣦⣄⡀⠀⠀⢀⠐⣨⠷⢚⣿⣿⠀⠀⠀⠀⠀⢸⡇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠉⠛⢦⣄⡈⠙⠻⣶⣴⣠⣟⣥⣶⠟⠋⠉⠛⢦⣄⠀⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠈⠙⠶⣄⡀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠶⣼⡇⣿⡇
⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠛⡇⣿⡇
⠀⠀⠀⢸⡿⣥⡀⠀⠀⠀⠀⣴⡟⠙⢀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠈⠛⢦⣄⡀⠀⣠⡴⠛⠉⠀⠀⢀⡿⢹⡇
⠀⠀⠀⢸⡇⠈⠙⢦⣄⠀⠀⢿⡻⣷⢄⡉⠰⣀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⣨⣿⡟⠁⠀⠀⢀⣠⠶⠛⠁⢸⡇
⠀⠀⠀⢸⣧⠀⠀⠀⠉⠻⢦⣀⠙⢶⣕⣭⡲⣄⠘⢠⠀⠀⣀⣤⣶⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢻⠁⢸⠇⣀⣤⠾⠋⠁⠀⠀⣀⣼⡇
⠀⠀⠀⢈⣿⣦⣀⠀⠀⠀⠀⠉⠳⣦⡈⠛⢿⣮⣷⣤⣚⡯⣷⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⢸⣠⡼⠟⠋⠀⠀⠀⣠⣴⡿⠛⠉
⠀⠀⠀⠀⣿⠀⠉⠳⢦⣀⠀⠀⠀⠀⠙⠳⣤⡙⠻⠿⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⢀⣠⡴⠛⠁⠀⠀⢀⣠⣶⠿⠋⠁
⠀⠀⠀⠀⣿⣆⠀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⢀⣤⠾⠋⠁⠀⠀⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠈⠳⣦⡀⢀⣠⠶⠋⠁⠀⠀⣠⡴⠞⠋⠀⠀⠀⢀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠈⢻⠋⠁⠀⠀⣠⡴⠛⠉⠀⠀⠀⣀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠈⠛⢦⣄⠀⢸⠀⣠⡴⠛⠁⠀⠀⠀⣀⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠈⠛⢿⠟⠁⠀⠀⠀⣀⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⠀⠀⢸⠀⠀⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣼⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⢦⠀⠠⠁⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠀⠀⠀⠀⢐⡂⠀⠀⠀⡁⠔⠈⠀⠀⠀⡔⠁⠀⠱⡀⠀⠀⠁⠂⢄⠁⠀⠀⠀⠰⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠢⡀⠄⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠆⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠸⠅
⠀⠈⠀⠀⠀⠀⢐⢢⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠰⠄⠀⠀⢸⢀⡀⠤⠄⠤⠤⠤⢸⡤⠖⠒⢰⠂⠒⠠⠀⠧⠀⠀⠀⠀⠐⠂⠀⠼⠘
⠀⠀⠀⠈⠁⠀⠀⠀⠁⢀⣴⣄⢠⣆⢠⣦⢴⣶⣶⢨⣆⢠⡦⣠⣄⣴⡄⢠⣄⢠⣤⠀⠀⠀⠀⡀
⠀⠀⠠⠀⠀⠀⠠⠀⠀⣼⣯⣿⡼⣿⣾⠇⠀⣿⠀⠸⣿⣾⡇⣿⠛⢻⣿⣿⡿⣿⡿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠁⣉⠋⣴⠀⠉⠀⡖⠉⠉⣀⠋⣤⠀⠉⠘⠁⠈⠁⠀⠀⢴⠀⠀⠀⠀⠂
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠘⠧⢺⠀⣿⣠⠐⣇⠀⢈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⢐⡃⠀⠀⠮⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣟⣛⣟⣛⣻⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⣁⠀⠀⠀⢀⠀⠠⠈⠄⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄
⠀⠀⠀⠀⠀⢀⠁⠄⠀⠀⠀⠀⠈⠑⠊⠀⠀⠀⠀⠀⠄⠠⠀⠊⡀⠄⠀⡆⠄⢀⠀⠀⠀⢰⣿⢃⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠂⢠⡕⡀⠄⠃⠀⠤⠁⠀⠀⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠂⠀⠀⠠⠬⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢄⣄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⡀⠀⠀⠈⠙⠓⠒⠒⠜⠂⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣽⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠄⢀⡼⠟⠁⠢⠀⡀⣶⢀⣴⠀⠀⠀⠀⠀⠀⢀⠈⠄⠀⠀⠀⠈⠉⠀⠀⠀⠴⠆⠴⠦⠁
⠀⠀⠀⠀⠀⡀⣸⠀⠀⠀⠀⠀⢀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⠀⠐⠂⠀⠀⢤
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠂⢸⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠘⢠⡇⠀⣿⡂⢀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⢠⠗⠣⠊⠄⠩⠄⠠⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⠿⠿⠛⠻⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠋⠁⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⢸⡇
⠀⢀⣀⣀⣀⡀⠀⢀⣠⣶⣿⡿⠟⠋⠁⠀⠀⠀⠀⢠⣿⠋⠉⠙⣷⠀⠀⠀⢸⣿
⣼⣿⣿⡿⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣿⠇⠀⠀⢸⣿
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣼⠟⠀⠀⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠻⢶⣤⣸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⡶⠟⠋⠉⢀⣀⡀⠀⠈⠙⠻⢶⣄
⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣤⣴⠶⠟⠛⠿⢾⣿⣥⣀⠀⠀⠀⠈⠁⠀⣀⣠⣤⣶⣾⣿⣀⡀
⣿⣿⣿⡇⢀⣠⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣶⣤⣶⠾⠟⠛⠉⠁⠀⠀⠈⠉⣛⣷⣦⣀⡀
⣿⣿⣿⡗⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠟⠋⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠛⠋⠉⠈⠉⠛⠿⢶⣤⣄⡀
⣿⣿⣿⣿⡿⠟⠻⠷⣶⣤⣤⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⣀⣠⣴⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣤⣄⣀
⣿⣿⣿⣅⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣦⣄⣀
⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⣀⣤⣴⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠿⠛⢻
⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠟⠛⠉⠀⠀⠀⠀⢸
⠀⢸⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢸⣿⣿⡇⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣤⣴⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠛⠉⠁⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣤⣴⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⠶⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡜⢀⡆⠀⡴⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⠀⣼⢁⡾⣡⡞⣠⠞⣡⠞⣁⡴⠋⡠⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣸⢃⣿⣼⠷⠿⠿⠿⡿⠿⠾⢷⣾⣯⣶⣫⡶⢋⣀⠔
⠀⠀⠀⠀⠀⠀⠀⢠⢰⣧⢿⠋⠁⢄⣵⣾⣿⣷⡷⠮⡤⠀⠘⢏⠙⠻⢾⣯⡥⠖⣊⡥⠂⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡿⠋⢀⡇⠀⣑⣾⣿⣿⣿⣿⣄⣀⣼⣁⠀⠘⡀⠀⠀⠉⠻⣿⣭⠶⠞⣋⡠
⠀⠀⠀⠀⠀⣠⠎⠀⠀⠸⡇⠀⠼⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⡇⠀⠀⠀⠀⠈⢻⣿⣭⣭⠤⠖
⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⢧⠀⢈⠽⣿⣿⣿⣿⣿⣿⠿⠉⠀⡸⠀⠀⠀⠀⢀⣴⣟⡛⠦
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⠣⡀⠐⠁⠏⠿⠻⠙⠄⠁⢀⠔⠁⠀⠀⣠⣶⡛⠷⣄⠙⠂
⠀⠀⠀⠈⠉⠛⠓⠶⠤⣄⣀⡀⠈⠒⠤⢄⣀⣀⣠⠤⠒⢁⣀⣤⣶⢿⡟⢮⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠛⢶⠲⣶⢶⡖⢾⡟⢿⠙⢧⠙⠆⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠓⠈




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢐⠞⡇⠉⣶⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⡴⢿⡁⠀⠀⠀⠀⠔⠋⢳⣤⡴⢢⠀⠀⠀⢀⣠⣤⣦⣀⠀⠀⠀⠀⣄⣂⠀⠀⠀⢀⠤⣂⠻⣏⣵⡄
⠀⠀⣐⡯⢐⡇⠀⣩⡅⠀⠀⠀⡈⢳⣶⡇⠒⠀⠀⠀⡨⠀⣨⠈⠢⡄⠀⠀⢻⣴⣼⣟⣶⡟⠀⠀⠀⡼⠽⢿⣿⣿⡆⠀⠀⠐⠀⠠⠆⠀⠀⢎⠲⢯⣿⡯⠵⡮
⠀⠀⠂⢹⣶⣧⣠⡽⠀⠀⠀⠀⠁⣨⠉⠱⠄⠀⠀⠀⢠⡴⢫⡀⠀⠀⠀⠀⢈⡛⣠⠟⡅⠀⠀⠀⠀⡇⢰⢄⡈⠀⠀⠀⠀⠈⢠⠉⠀⠀⠀⠀⠉⢹⠋⠛⠓⠁
⠀⠀⠀⠀⠉⡛⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⣸⠡⠂⠀⠀⠀⠀⠉⠁⠀⠒⠀⠀⠀⠀⠘⠯⠩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠆⡀⠀⠀⠀⠀⠀⣠⣴⣦⡀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣇⠀⠀⠀⠀⠀⠘⢷⣤⡀⠀⠀⠀⠀⠀⢘⠣⠄⣠⠀⠀⠀⠀⠠⠊⠀⠀⢱⠀⠀⠀⠀⠀⡿⡛⢻⠏⠀⠀⠀⠀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⠈⣣⣬⢅⡠⠀⠀⠀⠀⠀⢀⢸⠃⣑⠀⠀⠀⠀⠒⠊⣹⣟⠂⠘⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠀⢿⠀⢘⠀⠀⠀⠀⠀⢹⡟⠙⠛⠋
⠀⠀⠀⢀⡴⣤⡚⠶⢼⠄⠀⠀⠀⠀⠀⠐⣎⣸⠗⠉⠀⠀⠀⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠘⠶⣦⣦⠞⠀⠀⠀⠀⠀⠸⣆⡜⠀⠀⠀⠀⠀⠈⢦⣠⡄
⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠆⠀⠰⢶⣿⠗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣣⠇⣀
⠀⠀⠉⢰⣶⡦⠚⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡼⣿⢀⡄⣤⢸⡏⠀⣠⢠⡄⢀⠀⠀⠀⢠⣄⣄⠀⠀⢀⡤⡶⢛⠿⠷⡛⠓⠂
⠀⠀⠀⠞⢉⠃⠲⠄⠀⠀⠱⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⡗⣿⢼⡷⢿⣿⡇⢀⣿⠼⣷⣿⣴⣶⡖⣸⡏⡟⠀⠀⢼⡏⢰⠃⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⢸⠃⣿⡆⠀⠀⢸⣇⡜⠀⠀⠸⣿⠃⠀⠈⠁⠀⠷⠂⠀⠘⠣⠸⢆⣀⣴⢁⣔⠡⠃⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⡀⢀⣄⠉⠀⣴⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⡧
⠀⠀⠀⠐⣤⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⢀⡄⠀⠄⠀⠀⠀⠀⠀⢻⡇⣼⣯⠀⣸⣿⣾⣤⢀⣾⠇⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣷⢻⣿⣀⡇
⠀⠀⣾⣁⣈⣦⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⣧⡏⢿⡠⢿⡟⣱⣿⠙⣿⣸⢿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢾⡿⠋
⠀⠀⢸⣶⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⢸⣟⠻⠏⠀⠀⠀⠛⠀⠀⠀⡄⠀⠐⠀⠔⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈
⠀⠀⠘⠃⠁⠀⠀⣀⢄⠀⠀⠀⠀⢀⣤⣤⢀⡀⣴⡶⢶⡶⢦⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⣀⠌⢄⣀⠀⠀⠀⠀⠈⠻⠓
⠀⠀⠀⠀⠀⠀⡴⠉⣿⣷⠀⠀⠀⢸⣿⠏⣿⣋⣿⡧⢺⣟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⡞⠫⠥⠤⢬⡙⣤⠀⡀
⠀⠀⠀⠀⠀⠀⡆⠀⣷⢏⠀⠀⠀⠸⣿⣆⡿⠇⠛⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠢⠠⠄⠤⠠⠜⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡽⡆
⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡞⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⠒⠋
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄
⠀⠀⠀⠞⢽⠟⣃⠀⠀⠀⠀⠀⠄⡀⡶⣆⠀⢠⠀⠀⠀⠀⠀⠤⠺⠰⠇⠻⠂⠀⠀⠀⠀⢀⠀⠈⢠⡤⠀⠀⠀⠀⠀⠀⠈⠃⢠⠇⠂⣀
⠀⠀⢘⡀⣲⣮⣽⠀⠀⠀⠂⢀⣷⣿⣶⣯⣀⣀⠀⠀⠀⠀⠀⠇⠀⢸⡁⢀⡇⠀⠀⠀⠀⠘⢇⠀⠁⠀⡀⠀⠀⠀⠀⠀⢰⣒⣹⡥⡀⠈⠀⠀⠀⢀⣀⡀⠋
⠀⠀⠀⢳⣿⣯⠃⠀⠀⠀⠀⠀⠈⢙⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠚⠀⠀⠀⠀⠀⠒⠈⢑⣀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠉⠢⠴
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠙⠀⠐⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠈⠀⠀⠀⠐⠀⠂⠘⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠠⠈⠐⠁⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⣠⣶⠿⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⡀⠀⠀⢰⣿⠋⠀⠀⠙⣿⡆⠀⣸⡿⠁⠀⠀⠈⣿⡇⠀⢠⣾⣿⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠈⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⣽⡟⠀⣿⡇⠀⠀⠀⢀⣿⡇⢀⣿⣿⢃⣾⣿⠏⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣾⠿⣷⣶⣄⠸⣿⣄⠀⠀⣠⣿⠃⠀⢿⣿⣄⣀⣤⣾⡟⠁⢸⡿⠁⣾⡿⢁⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⣽⣿⠀⠘⠻⠿⠿⠟⠁⠀⠀⠀⠙⠻⠻⠟⠋⠀⣠⡄⠀⠀⠉⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣴⣾⠟⠁⠀⠀⢀⣀⣠⣤⣤⠤⠴⠦⢤⣤⣄⣀⡀⠀⠀⠀⠚⠃⣀⡈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣠⣤⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣤⣀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦⠀⠀⠀⢀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⢰⠞⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⣠⠟
⠀⣠⡶⠋⠻⣦⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⣩⠟⠀⠀⠀⠀⠈⠻⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡶⠞⠁
⢀⡏⠀⠀⠀⣿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⣤⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢸⡇⠀⠀⠀⠘⢷⣄⣀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣅⣰⣦⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠘⣧⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⢀⣀⣀⣤⣤⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⡟
⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠀⠀⠀⣠⠟
⠈⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⠀⠀⠀⣰⠏
⠀⠀⠙⠿⢏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠠⠁⠀⠀⣰⠏
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⣀⣤⠾⠁
⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠶⢶⣤⣤⣤⣤⣤⣴⠶⠶⠶⣤⣤⣀⡀⠀⠀⠀⠀⣠⡴⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣷⣶⣶⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠻⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣶⡆
⠀⠀⠀⠀⠸⣿⠁⠀⠉⡙⣿⠁⠀⠀⠀⠀⢰⣿⠟⠿⠿⢿⣿⠆⠀⠀⠀⠀⣾⡏⡙⠛⢛⠛⣿
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠸⡇⢀⣀⡀⢰⠏
⠀⠀⠀⠀⠀⠘⢦⣠⣄⠜⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢻⣬⣽⣭⡾
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠉⢥⣀
⠀⠀⠀⠀⠐⠛⠛⣒⠚⠛⠛⠀⠀⠀⠀⠀⠐⠛⠟⠛⠛⠛⠷⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⣀⣾⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⢰⣿⡿⠻⠋⠻⣿⠀⠀⠀⠀⠀⢸⡟⡀⠉⠉⡙⡿⠀⠀⠀⠀⠀⢸⡟⠟⠿⠻⢿⣷⡀
⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠀⣿⡇
⠀⠀⠀⠀⣾⣷⣄⡀⢀⣴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣀⠀⣀⣶⣿⡿
⠀⠀⠀⠀⠘⠿⠟⠀⠘⠻⠟⠁⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣩⣏⣀⣹⣟⠛
⠀⠀⠀⠀⠐⠛⠒⠚⠒⠚⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⣀⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⣴⣦⣠⣤⣤⣤⣴⣤⠀⠀⠀⠀⢠⣾⣿⣿⣿⣶⡄
⠀⠀⠀⠀⢸⣿⠛⠋⠉⠹⣿⡄⠀⠀⠀⠀⢻⣿⠟⠻⠟⠻⣿⡏⠀⠀⠀⠀⣿⡟⠉⠻⠛⢻⣿
⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠻⠇⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣄⡀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⢀⣠⣄⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣠⣄
⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣦⣤⣿⡀⠀⡴⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢀⣴⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⡖⠃⡮⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣸⣿⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢒⣧⠀⠀⠠⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⢿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠧⠦⠔⠂⠉⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⢸⣿⣿⡿⠿⠟⠛⠛⠉⠉⠀⠀⠀⣀⣠⡤⠀⠀⠀⠀⢀⠀⠀⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢀⣀⣠⣤⣴⣶⣿⣿⣿⡿⢃⣷⢠⡠⠀⣿⠀⠃⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⣰⠀⠀⡀⢰⡇⠀⠘⠂⢠⡀⠀⣐⣒⣔⠒⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⡛⠀⣸⣿⡐⠠⢰⣿⡄⣈⡩⠉⡉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣆⣿⡤⠎⠉⢻⠥⡤⢀⠀⢞⠑⠚⢉⢡⠘⠛⣓⠿⠛⠫⠁⠡⠈⡀⠪⠩⣖⢄⢀⠈⠉⡃⠶⠸⠿⡧⠘⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠰⠋⢉⠀⣠⢶⡁⢀⠈⠁⠀⠈⠐⠀⠾⠖⠒⠉⠀⢀⡠⠄⠳⠀⢰⣇⠀⢄⠀⠉⠑⠙⢸⡧⣄⠀⠐⠈⡶⠚⠁
⠀⠀⠀⠀⠀⢀⠀⠘⠫⣀⠔⠂⠀⠀⠠⠀⡀⣿⢠⡆⠈⠒⠂⠀⠀⠀⢠⠀⠂⠁⢀⠀⢠⡇⣸⣿⠀⠀⠁⠀⠀⠒⠠⠒⠉⡆⠀⠀⡇
⠀⠀⢄⠀⡠⠀⠀⢀⠀⠀⠀⠀⠁⠈⣀⣀⣀⡻⠊⠁⢀⣄⠄⠀⠀⠀⣼⡆⠤⣎⣀⡠⠼⠷⠿⠛⠓⠉⢋⠄⠂⠁⠀⠀⠁⡇⠀⠀⠃
⠀⠀⠋⠠⠦⠁⡂⠠⠀⠈⠁⠂⠀⠻⠿⠿⠽⠿⠂⠀⠀⠀⠀⠈⠙⠀⠿⠷⠈⣀⡀⠪⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠶⠄⠘⠀⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠃⠀⠠⠦⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠋⠭⢜⡀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⣴⣶⣤⣤⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡔⣿⣿⣽⣿⠿⠟⢿⢻⣻⣻⣿⢛⡿⣷⣾⣤⣄⠀⡴⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣙⣿⣿⡿⢿⣻⡏⡆⢻⢸⣼⣿⣿⣿⡏⣸⣡⣿⢟⡿⣹⣿⢷⣼⡁⣰⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣾⣿⢿⡋⠳⣸⣯⣿⣿⣬⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⣵⡿⣯⣶⡿⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣰⣿⣿⣿⣿⡛⢦⣽⣶⣾⡿⠟⠿⠛⠍⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣾⣟⣽⣶⣫⣴⡶⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣼⣽⣿⡿⣿⣌⣻⣿⠿⠋⠁⠀⢀⣬⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣥⠶⣫⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢼⣿⡿⠋⠳⣾⣿⢟⣥⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠿⢗⣩⣀
⠀⠀⠀⠀⠀⠀⠀⠭⡭⠟⠋⢀⣤⣽⡿⠁⠀⠊⠈⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠏⠀⠀⠀⠀⠀⠁⠀⢹⣿⣿⣶⣿⣩⣵⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⡾⢟⡻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠠⠂⡄⠉⢸⠛⣿⡿⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠿⡇⠈⠀⠈⠀⠤⠀⠈⠄⢀⣀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴⢃⣼⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⢴⡟⢻⡷⡶⢶⣤⣭⣁⡒⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠚⢁⣤⣾⣿⢿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠸⠀⠁⠁⠻⠫⢻⡿⢿⢶⣶⣦⣤⣌⣭⣉⣉⣉⣁⣁⣤⣤⣶⣾⡿⡟⢯⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠚⠊⠸⠻⡿⠇⢿⢽⢹⡿⡿⠿⣿⠛⡇⠑⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠹⣿⣿⣷⡄⣿⠀⠘⢮⣿⣄⠀⡜⢿⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣴⣶⠋⣰⣾⣿⡿
⠀⠀⠀⠙⣿⣿⣿⣿⠀⠀⠀⠈⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠿⠿⠟⠻⣿⣿⣿⣃⣴⣾⣿⣿⣿⢏⣰⣿⡿⠏
⣷⡄⠀⠀⠈⢿⣿⣿⡄⢤⢤⣾⣿⣿⣿⢿⣿⣿⣿⡿⠿⢿⣿⠷⣠⣥⣀⣀⢀⢶⡇⣀⡀⠉⠉⢿⣿⣿⡿⢛⣩⣥⣤⣔⣚
⢿⣿⣦⠀⠀⠀⠃⠈⣿⣴⣾⣿⣿⣿⣤⣾⣿⣿⣿⠿⢨⣠⣿⡇⣿⣿⡿⠟⣷⢾⣟⣩⣽⠟⠁⣸⣿⣿⣿⣿⡿⠛⢯⣭⣽
⡈⠻⣿⣷⡄⠀⠀⠺⣿⠿⠿⠿⢟⣿⣿⣿⣿⣿⠅⠀⠈⢙⣛⣷⣿⣿⣷⡀⣿⠀⠙⠋⠀⠀⠀⠀⠀⣹⡟⠹⡆⠰⣄⠺⣿
⣿⣦⡈⠛⠀⠀⠀⠀⠀⢀⣤⡄⣼⣿⣿⣿⣿⣶⣿⣟⣋⣭⣭⣴⡶⣿⣿⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢀⣾⣭⠁
⣿⠛⢻⣦⢀⣠⣥⣄⣀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢈⣿⠀⠀⠈⠛⠗⠀⠀⢀⠀⠀⠀⠀⡀⠂⠀⠀⣴⣿⡿⠋⠈
⠛⠀⠸⣿⣷⣿⣷⣨⣿⣠⣴⣿⣿⣿⣿⣿⣿⣤⡭⣻⡷⣮⢹⣤⣄⠀⠀⠀⠀⠀⠈⠀⠀⣤⠈⠁⠀⢂⣾⡿⠋⠀⠀⣠⣴
⠀⣘⣿⣿⣉⡟⠻⣿⣿⣶⣿⡟⢹⣿⠉⢻⡟⠉⠀⠀⠝⠒⠀⠉⠓⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣠⣾⠟⠉
⣤⡘⣿⣿⠏⣵⠚⢁⢼⠟⢿⣿⣿⣿⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠿⠋⠀⣠⣶
⣿⣿⣿⣿⡄⢉⣰⠆⠠⠶⣾⡿⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁⠠⠐⠄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠽⠚⣩
⠉⠛⠿⣿⣦⠺⣝⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣀⠀⠀⡠⠔⠋⠉
⡀⠀⠈⢿⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⣀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠛⠋⠀⠀⠀⠈⠙⠢⠀⠀⠀⠀⠐⠿
⠀⠀⠀⣼⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠛⠋⢠⣄⣠⣤⣤⢄⡀
⣤⣤⣠⣧⠄⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠛⠉⠉⠉⠁
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⣀⣀⠀⠀⢀⣄⡀
⠈⠉⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⢀⣙⠂
⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡈⠻⢿⡗
⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⠀⠠⠀⠀⢸⣷⠀⠀⠀⠠⡄
⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣿⣄⡀⠀⠄⠀⣰⣿⣿⠀⠀⠀⠀⠛⢶⡀
⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⢫⣹⣧⠰⣶⣾⣿⣿⣿⡶⠀⠀⠀⠀⠈⢧
⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣶⣿⣿⣷⣌⣻⣿⠟⠥⣦⡀⢶⣶⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠸⣿⣧⣀⣠⣤⣴⣾⣿⠐⢀⡤⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡶⠂⠠⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⢿⣿⣿⣷⡄⣴⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣶⣿⣃⡀⠀⢸⠟⢹⣿⣿⣿⣿⡆⠀⣵⣦⢀⣤⢹⠿⣿⣿⡏⡅⢩
⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣸⠛⠉⢉⣹⣆⣤⣤⣤⠙⠛⠛⠿⠿⠈⣦⣶⣎⣛⣵⣤⣿⣠⣷⣶⠂
⠀⠀⠀⠀⠀⢻⡟⢋⣉⣻⡟⠛⢃⣌⠉⠹⢿⣿⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⡿⠙⠿⣿⣏⣤⣤⣀
⠀⠀⠀⠀⠀⠈⣿⢿⣟⡛⣅⣺⣿⣾⠏⠀⠀⠀⠀⠐⠂⠠⢄⣤⣼⣶⣴⣶⣦⠈⠁⠐⣾⣿⠋⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠈⣺⣏⣾⣿⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⠻⠟⠛⣿⣿⠛⢀⠀⢸⣟⠃⠂⠀⠆⠁
⠀⠀⠀⠀⠀⣴⣾⣿⣿⣯⡅⠀⠈⢹⣷⡀⠀⠀⢀⣀⠀⠀⠙⡀⠀⠄⢸⣷⠿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠋⠁⣘⢿⣿⠁⢀⣤⣼⣿⡦⠀⢰⡿⠿⠿⣤⣄⢀⣀⣀⡀⡟⠄⠀⠄⠚⠳⣤⣤⠖
⠀⠀⠀⠀⠀⠰⠀⡀⠀⠙⣀⣿⣿⣿⢁⠠⣟⡛⡀⠀⣀⢸⣿⠻⠿⣿⣿⣿⣶⣷⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⠶⠶⣴⣶⣤⣶⣿⡀⠀⢻⣿⠿⠉⠉⠻⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣈⣉⢉⠉⠛⠉⢾⢿⣷⡦⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⠀⠀⢹⡁⢀⣴⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢛⣷⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⠀⠀⠀⠉⠉⠛⠛⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡉⠉⠉⠓⠒⠦⠤⣄⡀⠀⠈⠉⠛⠷⢦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠧⣄⠗⡢⠒⠒⠒⠠⢀⠀⠈⠉⠒⠢⢄⡀⠀⠈⢹⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠛⠛⠛⠥⢀⡀⣰⠀⠊⠀⠀⠀⠀⠀⠀⠉⠲⠦⠀⠀⠀⠈⠉⠒⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠞⠋⠙⠣⠤⢄⡀⠀⠀⠀⠈⠑⠼⣀⡀⠀⠀⠀⡔⢁⠄⠂⠈⠙⠂⢀⠀⠀⠀⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⠀⡀⠀⠀⠉⠑⠒⠦⠧⣨⣤⣤⣤⠤⠤⠈⠉⠒⠤⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠠⢄⠀⠀⠀⡀⠀⠀⢸⡇
⠀⠀⠀⣠⣶⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⣀⠀⠀⠀⠑⠆⠈⠀⠀⡁⢸⠃
⠀⠀⠐⡏⣉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⢹⡄⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⣿⢉⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣠⠁⣸
⠀⠀⠀⣿⠈⡍⠓⠢⢄⠉⠐⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠊⠁⣸⠀⣿
⠀⠀⠀⣿⠀⣍⠐⠀⠀⠈⠐⠠⢄⡈⠉⠒⠤⣄⣀⡤⣒⠀⢀⣀⠠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣷⢶⣟⠁⢀⣿
⠀⠀⠀⣿⠐⢬⡙⠲⢄⡀⠑⠢⢄⠈⠑⠢⢄⡀⠙⠀⢸⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⡴⠏⠀⠙⠿⠟⠃
⠀⠀⠀⢸⣇⢰⡟⠷⣦⣈⠑⠢⢄⡀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠋⠉
⠀⠀⠀⠀⠉⠋⠁⠀⠀⠉⠛⠶⣤⣈⠙⠲⠤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⢹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⢸⠀⠀⠀⣀⣠⣴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⣷⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠋

⣿⣷⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣶⣶⣶⣿⣶⣶⣶⣄⣦⡠⢖⣀⡰
⣿⣿⣷⣆⠀⠀⠀⠀⠠⢿⡿⡆⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢠⢀⡄⠀⣤⡀⠀⠀⢀⣜⢿⠇⠉⠁⣠⣽⣿⣿⣝⣿⣿⣰⣿⣿⣿⡆
⣿⣿⣿⣿⣇⣴⣿⠀⠀⢼⢿⣿⣦⡄⠀⢢⣃⠀⠀⠀⠐⠀⠠⢿⠟⠻⣿⣿⡿⠃⡰⠁⠀⠀⣰⣿⡏⠈⡇⣰⣾⢿⣿⣿⣿⣿⡜⢃⣿⣿⣿⣿⡃
⣿⣿⣿⣿⣿⣿⣿⠀⣠⠀⠀⢿⡟⣿⣄⠈⠛⢣⡘⠃⠀⡇⠀⢸⢀⣴⠻⣿⡛⠿⡇⢠⢟⣼⣿⣿⡿⣿⣶⡿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣧⣿⣷⣤⠰⣠⣷⣾⣧⣄⣀⢳⡀⠀⢱⠀⢸⠘⠻⠤⡤⢥⠽⠚⢙⡾⠋⠁⠀⣀⠀⠀⠙⢿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠖⠿⢿⣿⡏⡿⣿⣍⢷⡀⢸⠄⠸⡀⠀⠀⡇⢀⠀⣠⣿⡇⠀⣧⠀⠹⠗⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠐⣶⢳⡄⢻⡇⠇⢻⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣅⠈⢻⣶⡄⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢰⣤⠀⣰⣿⡇⢀⡻⣿⣿⣿⣿⣿⣟⠿⠿⠯⢿⣿⣿⣿⣿⣿⣿⣿⠷⣄⠉⠁⠀⠀⢀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠂⠰⣿⣿⣿⣾⣿⣿⣿⡯⠑⢺⣿⠀⠀⠀⠀⣿⢿⣿⣿⡿⠁⠀⠀⠈⠦⠄⠀⠀⠀⠉⠲⢤⡀⠀⠙⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠋⠉⠣⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⢿⣿⠈⡻⣿⠀⠀⠀⠀⡏⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⢠⡄⠹⣿⣿⣿
⣿⣿⣿⡟⠁⢼⣇⠀⠀⠀⠀⠀⠀⠸⣷⡌⢿⣷⢰⣶⣶⣾⣿⣿⠄⠀⠀⣀⣇⣾⠟⠁⠀⣠⣾⡇⠀⠀⠀⠀⢫⢻⡄⠀⠀⠀⠀⣴⠏⠀⣿⣿⣿
⣿⣿⣿⡀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⣿⣿⡜⣿⡆⣿⣿⣿⡓⢺⠠⠥⠐⠛⠉⠀⠀⢀⡼⢻⣿⠃⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠞⠀⣀⠀⣿⣿⣿
⣿⣿⣿⢻⠀⣦⢀⣄⠀⠐⠀⠀⠀⠀⢸⣿⣿⣿⣧⣿⠍⢉⣯⣴⡒⠀⢀⡀⣴⣷⣦⠘⣇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣿⣿⣿
⣿⣿⣿⡾⠀⣿⠈⣿⠘⢦⠀⠀⠀⠀⠀⢿⣿⣿⣿⣽⣿⠿⠿⠿⠰⠉⣸⣼⣧⣿⣿⡀⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⣿⣸⣿⣿
⣿⣿⣿⡇⢸⡿⣸⣿⠀⠈⠏⠑⠀⠀⢠⣄⠻⣿⣿⣿⣿⡓⠒⣚⡘⢂⡹⣿⣼⣿⣿⣿⢻⣸⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠘⣿⣿⣿⣿
⣿⣿⣿⠃⢸⣷⣿⡏⠀⠀⠀⠀⠀⢀⠀⠻⣷⣽⣿⣿⡯⣉⣉⣉⣉⣉⣁⣾⣿⣿⣿⣿⠆⣽⡿⠁⣸⠀⠀⠀⠘⠀⠀⠀⠘⢦⣿⣿⡆⣿⣿⣿⣿
⣿⣿⣿⠀⢻⣿⣿⠀⠀⢻⢰⡄⠀⠰⡑⢄⠈⢻⠿⢿⢛⣛⣟⠛⠛⠛⠛⠛⠛⢿⣿⣿⣴⣿⢃⣴⠋⠀⠀⠀⠁⢠⠚⠑⢦⡈⣿⣿⡇⣿⣿⣿⣿
⣿⣿⣿⣆⠸⣿⣿⡆⠀⢸⠀⣇⠀⠀⢻⡀⢻⠛⣟⢧⠉⠋⠋⠉⠉⠉⠉⣋⣐⣀⣹⡿⠋⣱⠋⣀⠤⠲⢦⠄⡀⠃⠀⠀⠀⢷⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣆⢹⣿⡇⠀⣿⠀⠹⣆⠀⠈⣧⠘⠀⠹⣿⠟⠋⠉⠉⠋⠛⠛⠛⠛⣫⣽⡇⠠⠁⠀⢁⡠⠒⠁⡜⢰⡁⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⠀⠀⠀⠀⠀⠙⢦⡀⠀⠻⢿⣿⣿⣿⣿⣯⣼⣤⠀⠉⠁⠀⠀⠀⠰⠃⠀⠀⠀⢡⠀⠧⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⠀⠀⠀⠀⡀⠀⠈⠙⢦⣄⡀⠉⠉⠙⠛⠛⠛⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⢿⣿⣿⢸⠀⠀⠸⡄⠀⠀⠀⠑⠀⠀⠀⠀⠈⠡⠀⠐⠂⣀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠋⣠⣞⣿⣿⣷⡇⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⣽⠹⣟⢿⣿⣿⠏⠀⠀⢻⣿⣿⣿⣿⣿
⢀⡀⠉⠁⠙⠿⠿⣿⠖⢄⡀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⠀⠀⠀⠀⠀⢠⡞⠍⠁⠀⠀⠀⢸⡀⠹⣆⡙⠿⣿⡃⠀⠀⠻⣿⣿⠿⢿
⠈⠀⠀⣀⣠⠴⠚⠁⢀⡆⠙⣷⣄⠈⠢⡙⠳⣦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠂⠀⢀⣿⠀⠀⠀⠀⠀⢰⠀⠳⡄⠈⠻⣦⡈⠙⠷⣵⣄⠙⠃
⠀⠀⠋⠉⠀⠀⠀⢀⠋⠁⠀⠀⠙⠀⠀⠈⠳⣄⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠄⠀⠀⠱⠄⠈⠓⠀⠈⠙⠢⠦⣀⡉⠛⠆
⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠚⠲⠑⠤⠀⠖⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣾⣷⣶⣤⣌⣁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢲⠀⠀⠀⠀⣠⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣗⡣⠀⠀⠀⠈⠀⠉⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠈⢹⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠂⠀⠀⢀⠐⠳⠦⣤⣤⠠⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣠⢠⡛⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠐⢄⣀⣠⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠜⠃⢀⠀⢀⢠⠊⠚⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠐⠄⢈⠄⢀⠂⡰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠉⠁⣃⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠁⠀⠁⠀⠀⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⢀⣀⠀⠀⢦⣦⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⠄⠐⠊⢉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⡀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡶⠒⠿⠃⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⢓⢤⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢣⠀⣠⢌⡀⠌⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠖⡻⠛⠉⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠙⢄⡨⢀⣠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠒⠂
⠀⢈⠈⡉⠀⠐⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄⡙⡺⢈⢉⠀⠀⠀⠀⠀⠀⢀⠂⢀⠨⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠆⠀⢸⢠
⠀⠎⢰⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢈⠀⠀⠀⠀⠀⠀⠀⠀⡈⠢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠄⠀⢸⣿
⠀⢄⢠⢏⡀⡹⢃⣳⣀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠌⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⣿
⠀⠂⠀⢡⠥⠐⠉⢡⡉⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣦⣤⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⣄⠀⢼⣀⠀⠔⢉⡽⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⣿⣿⡇
⠀⢿⡀⠀⡀⠄⠂⠀⢀⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿
⠀⠸⠈⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣿⣿
⠀⠀⠡⣦⣀⠀⠀⠀⣿⣅⡹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡟
⠀⠀⠀⠙⠻⠿⣷⣾⣿⣿⣿⣿⡿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡟⠉⠣⣀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡄⠋⠠⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢻⠀⠀⢢⢄⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠒⠃⠀⠀⠀⠀⢶⠌⠙⠦⠄⠀⠀⠀⠀⠀⢩⠽⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡓⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢻⠇⠀⢀⡠⠤⢀⣰⠋⠇⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠥⢄⣀⡿⠠⣞⣭⠤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢶⣦⡄⢰⣉⣤⡶⠚⣻⠷⠈⠣⡉⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠈⣿⠸⣿⡄⠀⠀⠉⠀⠀⠀⢣⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠐⠐⠛⠋⠁⠀⠘⢧⡀⢨⡀⠀⠀⠀⣰⡀⠀⠈⢾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠈⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠠⠸⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣜⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣵⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠔⠠⠄⠀⠐⠶⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠄⠀⠀⠓⠂⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠⠤⠤⠄⠠⠠⠦⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣷⣶⣶⣷⣾⣷⣤⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋⠿⠍⠁⠀⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠤⠀⠀⠀⠀⠶⢼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣀⡔⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣶⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠉⠀⣴⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⠟⠀⠀⠲⠶⠶⢶⡶⣾⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠉⠛⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠄⠀⠂⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⣿⠆⠐⠒⠀⣀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⣿⠁
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠘⣅⠀⠀⠀⠀⠀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿
⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠈⠱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠈⠻⣦⡀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣶⣶⣿⣿⣦⡀⠀⠀⠀⠈⠛⠻⢿⣷⣶⣦⣤⣤⣀⣀⣤⣤⣶⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣠⣶⠃⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⣿⠇⠀⢠⠉⠻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢾⣏⠀⠀⠀⠻⣄⣹⣿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⡟⠀⠀⠀⠀⢀⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⣿⡇⠀⠀⠀⠀⢈⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⡇⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠟⠛⠛⠛⢻⡋⠁
⡟⢁⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢸⣧
⠁⣼⢇⡀⠀⠀⣀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡂⠀⣤⠛⠀⠀⢴⣦
⠀⠀⠻⣷⣦⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠸⠆⠀⠀⠀⠀⠠⠻⣷⣀
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠁⠠⣶⣤⣴⡦⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠿⠿⠿⠿⢷⣦⣤⡀
⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣭⣴⣶⣶⣶⣶⣶⣶⣶⣦⣽⣿⣷⣦⣄⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠛⠋⠉⠉⠉⠉⣩⡿⠛⠋⠉⠉⢉⣿⡿⢟⣛⣛⠿⣿⣿⡿⠛⠛⢛⣿⡿⠟⣛⣛⠻⢿⣷⣄
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣿⠏⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⣰⡿⢋⣴⣿⣿⣿⣿⣾⢿⣷⡀⢠⣿⠏⣴⣿⣿⣿⣿⣷⡿⣿⣧
⠀⠀⠀⠀⣿⣿⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⢰⣿⠇⣾⣿⣿⣿⣿⣿⣿⣯⢿⣇⣾⡏⣼⣿⣿⣿⣿⣿⣿⣷⢻⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣧⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⢸⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣰⣿⣿⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⢿⣷⡌⢿⣿⣿⣿⣿⢟⣿⣿⠁⠘⣿⣆⠻⣿⣿⣿⣿⣿⣷⣿⡟
⠀⣼⣿⡿⠿⣿⣿⡟⠁⠀⠈⢻⣿⣿⠟⠀⠀⠈⢿⣷⣦⣤⣤⣤⣤⣤⣼⣿⣷⣤⣤⣤⣤⣬⣿⣿⣶⣭⣭⣭⣶⣿⣿⣧⣤⣤⣼⣿⣿⣶⣭⣭⣶⣿⡿⠋
⠀⠉⠁⠀⠀⣿⣿⡅⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡿⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⣿⣿⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠙⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣶⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⠟⠃⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠋⠉⠀⠀⠀⠉⠻⣿⣶⣦⣤⣴⣶⡿⠟⠉⠀⢿⣿⡟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠋
⠀⠀⠀⠀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣀⣀⣀⣀⣠⣤⣴⣾⣿⡿⠛⠁
⠀⠀⢠⣾⣿⠏⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠛⠋⠁
⠀⣰⣿⡟⠁⠀⠀⠀⣠⣶⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⢰⣿⡟⠀⠀⠀⢀⣾⣿⠟⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⣾⣿⠁⠀⠀⠀⣾⣿⠃⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⣿⣿⠀⠀⠀⠸⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢻⣿⡆⠀⠀⠀⠙⢿⣧⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠈⢿⣿⣄⠀⠀⠀⠈⠙⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠈⠉⠻⢿⣿⣆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠈⠛⢿⣿⣶⣄⡀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠉⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣤⣶⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢡⣿⡟⠀⠀⠈⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⣾⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⣸⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⢣⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣧⡀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⣻⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣦⣌⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠤⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠈⣇⢀⠈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⡉⠁⢀⡇⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢹⠀⠀⠰⠀⠂⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠀⠀⠀⠀⠰⠀⡄⠀⠀⠸⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡄⠀⠐⡆⢸⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⠀⡆⢠⠀⠀⠀⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⡇⠀⣰⠀⠀⠀⠃⡆⠀⠀⠀⠀⡇⢠⠀⠀⠀⡆⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⡃⠀⠀⠀⣇⠀⢸⠀⠀⢀⡀⣇⠀⠀⠀⢀⡇⢘⠀⠀⠀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠛⠛⠋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣉⣈⣉⣈⣉⣉⣉⡉⠉⠉⠙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡰⠁⣀⡠⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠂⠠⠤⠤⢄⡀⠱⡀
⠀⠀⠀⠀⠀⠀⣡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡇
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣀⢀⣀⢀⣀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⡀⢀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣘⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⣿⡿⢟⡿⠿⢟⠛⠟⡛⠫⢙⠛⢛⡛⠿⢿⢿⢿⣿⣿⣶⣤⡀
⠀⠀⢀⣴⣾⣿⣿⠿⠟⠋⠁⠂⠉⡂⠨⡒⠪⢙⢐⠡⡃⡂⢌⢀⠁⠑⡪⢬⢐⠬⢛⠢⢍⠿⣿⣿⣷⡀
⣠⣴⡿⠟⠻⠋⠁⠈⠑⠐⠄⠀⠠⣀⣀⢐⢅⣰⡪⣮⣾⣷⣵⣶⣿⣿⡮⣑⣴⠭⡒⢄⠀⠑⠪⣻⢿⣿⣦⡀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡨⣺⣟⣵⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣮⣕⠬⡂⢔⠀⠑⢽⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣟⠋⣍⣄⣀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣈⠙⠿⣿⣮⡐⠁⢈⠐⢘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⡥⢏⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣈⠻⣿⣦⢑⠈⡂⢻⣿
⡀⣀⣠⣤⣤⣶⡾⠿⡛⣑⣭⣾⣿⣿⣿⣿⣿⡟⣻⠟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⣿⣷⡥⠊⠀⠈
⠛⢿⡿⡟⡛⣁⣨⣶⣾⣿⣿⣿⡿⣿⠙⢛⢹⣧⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣯⡙⣿⣆⠄
⠀⠀⠈⠉⠙⠻⢛⣿⣿⣿⣿⣜⢷⣷⠞⡜⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠃⠐⢫⣹⡿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠈⠊⣙⠽⢟⣯⣍⠷⣧⠁⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⡂⣙⣩⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠂⠑⢟⢷⣄⡙⠦⣄⣀⠈⠿⣿⣿⣿⣿⣿⣿⡿⠛⠁⡀⡠⠴⠺⣻⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡨⠛⠷⢦⣍⡙⠒⠦⠬⠭⠭⠭⠔⠒⢚⣉⣠⢤⠲⠿⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠕⠀⠐⢋⠽⠛⢻⢶⠶⠶⠿⠿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢀⠤⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣐⣒⣆⣄⣇⣄⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡀⡀⠀⠀⠀⡴⡄⠀⢀⠔⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡧⠊⠹⠝⠀⠀⠃⠀⠀⡌⡄⡌⢰⣡⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠖⠋⠈⣱⠀⠠⡪⠕⠒⢒⡲⠶⡤⡷⢖⡂⠐⠒⢤⡀⠀⠀⠱⠋⠀⢘⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠒⠊⢩⡆⡜⠀⠀⡔⡁⠀⡌⠀⠘⡄⠈⢢⠀⠙⠙⡄⠀⠀⢠⡊⠅⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠑⢤⠚⢱⠀⠀⢠⠁⠀⢀⠁⠀⠀⢁⠀⠀⡇⠀⠀⡇⠀⠀⠉⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠊⠈⠒⠤⢤⠈⡄⠀⠈⡆⠀⠘⠀⠀⠀⠸⠀⢀⠇⠀⢠⠱⠀⢀⡜⠉⠱⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⡐⣄⠘⢸⠀⠘⢄⡀⠘⢆⠀⠇⠀⠀⡆⢀⠎⣀⠔⠉⠁⢠⡟⠈⠢⣀⡸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠁⡛⡲⢦⣀⡀⠈⠓⠂⠍⠸⠄⠐⠒⠃⢈⣀⣀⣀⡀⠀⢱⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣈⠀⠀⠀⠀⢄⡀⠀⣠⣀⢴⣤⠒⠈⠁⠀⠀⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⢁⡢⠘⢄⣊⡣⠀⠀⡠⣒⠄⠉⠁⠀⡼⢃⢀⠑⣀⡸⢻⠃⠀⠎⠉⡊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠑⠀⠈⠁⠀⠸⣔⡰⣂⠾⠚⢰⢁⠞⢉⢈⠁⠀⢆⡉⠒⠒⠎⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⡊⠁⣬⠖⠂⠐⢉⡠⢄⠀⠚⢘⡿⠠⡫⠠⠘⠈⡀⠨⡃⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⠗⠉⠀⡆⣤⡍⡑⢙⢋⠀⣔⠝⡸⢩⢯⠀⠪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣠⠁⢹⠀⡇⠸⣽⠋⡙⠊⣀⣸⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣠⠃⠀⣕⠠⢒⠜⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣶⣶⡶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢣⡰⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⢄⣤⠥⠤⣑⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠁⢀⣅⠀⠀⣼⠀⢈⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢀⣙⢃⠲⣤⠀⢤⣀⠄⠈⠉⣀⣤⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢌⢏⡤⠹⡀⠀⠀⠀⠁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠩⠑⠛⠶⣤⣤⣤⣤⢴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠤⠤⢤⣀⣀⡀⣠⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⣎⠻⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠉⠈⠉⠐⢬⡹⡄⠀⠀⡀⠀⡜⠉⣷⣀⣀⣠⡗⠼⠆⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢮⣀⠀⠀⠀⠀⠀⠐⢌⢦⡀⢉⡸⠇⠂⢀⣨⣵⡞⠁⣀⠀⢰⡝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢱⡟⠉⢂⣶⣶⢿⡟⠿⡀⣸⡫⣷⣌⠛⠒⠮⠑⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠒⠦⢰⠋⣴⡶⡿⠹⠃⡀⢁⡤⠖⠒⠓⢯⡛⢷⡶⢩⣟⡏
⠀⠀⠀⠀⠈⡉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣯⣴⡏⠙⠃⢀⠁⡔⡉⠀⠀⠀⠀⠀⠉⠉⠛⠽⠋
⠀⠀⠀⠀⠀⢧⠦⡻⣶⣇⣤⠀⢠⡄⣰⣄⣮⣿⢘⣷⡇⠰⠆⢠⣞⣾⣟⠶⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢬⣙⡛⠛⡛⢋⠃⣬⡟⢿⣿⠁⣀⠀⡞⠆⡄⠏⣱⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠓⠚⣿⣾⡟⠃⢈⠒⡏⠙⠒⠦⡀⠀⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣞⣹⡀⢉⠁⢇⡀⠀⠴⢻⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣗⠯⠉⠣⠀⠿⣿⣿⡽⠷⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⠀⠀⠠⢄⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⣭⣉⣉⠉⠀⠀⠉⠉⠉⠉⠙⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⠉⠈⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣿⠉⠛⠛⣿⣿⠿⠟⠒⠤⠀⠀⠀⠀⠈⢻⡟⠛⠓⢤⣤⣄⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠹⣄⠁⠀⠀⠀⠀⠀⠙⠛⠦⣤⣄⢀⣀⡀⠀⠀⠹⣆⠀⠀⠀⠀⠙⠻⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠶⠿⠷⠀⠀⠀⠀⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢸⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣄⠀⠀⡀⠀⠀⠀⠈⠀⠀⠰⡶⠒⠀⠀⠀⠀⡶⠀⠈⠉⠀⠀⠀⠀⡀⠀⠀⣀⢈⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠉⢻⡇⠀⠀⠃⠀⠀⠺⣿⣿⠆⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠻⣿⡷⠀⠀⠀⠀⠀⠀⠈⣿⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⠾⠿⠉⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠰⢿⠟⠂⠀⠀⠀⠀⠀⠀⡟⠀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠘⠛⠛⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣿⢿⡘⠃⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⢹⡿⠉⠉⠉⠉⠀⠂⣤⠔⠒⠀⠋⠉⠉⠉⢀⣼⢿⣼⠃⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⢸⠇⣤⠀⠀⠀⢀⡴⠋⠶⠀⠀⠀⠀⢀⠰⣿⠻⣾⠃⠀⠀⠀⠀⠀⢀⣼⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⣿⡆⠀⠈⣰⡿⡀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⣰⣉⡿⠁⠀⠀⠀⠀⠀⠀⣾⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⣿⠀⠀⠀⠙⣠⠿⡀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠟⣧⠟⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠀⠀⡿⠀⠀⠀⠀⠛⣄⠛⢦⣀⠀⠀⢀⣀⡰⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢀⠁⠀⠀⠀⠀⠀⠙⢷⣄⠉⠛⠛⠋⣡⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠀⠉⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⣿⡅⠚⠛⠛⢦⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠈⠛⠿⠟⠀⠘⠃
⠀⠀⠀⠀⠀⢠⣿⡤⠘⠛⠛⣳⣄⣙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⢠⡞⠉⣤⢀⡄⢰⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠘⠿⠿⢿⣞⠁⢾⡇⢿⠋⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⣀⡀⠀⠀⠀⠀⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⠇⣾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠉⠛⠷⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢰⡏⠉⠉⠙⠛⢶⣄⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠹⡀⠀⠀⠀⠀⠀⠉⠲⠆⣀⣀⠀⠀⠀⠀⠀⢀⣈⣹⣷⣦⢠⣆⠀⠀⠀⠀⠀⢀⡽⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣔⢀⣠⣴⠟⠁⠀⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠀⣷⠀⣀⡀⢀⣀⠀⠀⠈⠙⠛⠛⠛⠂⠈⠉⠉⠛⠛⠛⠛⠋⠀⠉⠙⠛⠛⡃⠠⣤⡤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⣿⠀⠉⠉⠉⠉⠉⠉⠑⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠂⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠾⣟⢻⡏⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢨⠇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠶⠶⠶⠖⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣠⣄⣴⣤⣤⣤⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣠⡤⠄⠙⠛⣂⣠⣤⡄⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠉⠉⠉⠉⠉⢛⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠴⠖⠉⠉⠉⠉⠲⠿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣀⣀⣀⣀⣀⣀⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡶⠶⠶⣦⣄
⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠟⠋⠁⠀⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢷⣦⣶⣷⠿⠟⣿⣿⡟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣿⣿⡟⠿⢷⣷⣦⣶⠆
⠀⠀⠀⠀⠀⠘⠛⠻⢷⣄⣀⣌⣩⣤⡄⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠮⢁⣤⣬⣉⣄⣀⣼⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠠⡄⠀⢀⣸⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣰⣧⡷⡶⡚⢛⣛⡟⠉⢛⡗⢲⡿⢤⣀⡗⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣠⠖⠋⢹⢇⢻⠃⠇⣼⣋⡃⣠⣿⣵⡟⡰⢢⢎⠙⠳⣦⣗⢦⣀⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠴⠋⢑⢠⣠⣾⣾⣟⣞⣼⣿⣿⣿⣿⣿⣿⣫⡴⣫⣦⣴⡿⡯⠉⣳⣯⣷⣧⣀⠎⠀⡐⢀⡴⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⣀⣾⣼⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣾⡿⢟⣩⣾⣿⣛⣾⠶⣩⠔⢁⣐⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣰⣶⣿⣿⣿⣿⣿⠿⠿⣛⣿⣿⣿⣿⣭⣍⡛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠟⢓⣀⣤⢶⡋⣠⡴⢋⠌⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣼⣞⣽⣿⣿⡿⢋⡵⢆⣴⣿⣿⣿⠿⠿⠿⠿⣿⣿⣷⣄⠀⠝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣭⣶⣿⠟⢁⣴⡥⢞⠅⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡶⣿⣷⣿⣿⣿⡟⠁⠁⣰⣿⣿⡿⠋⣀⣤⣶⣶⣦⣄⠙⢿⣿⣧⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⢏⣸⣭⣶⡧
⠀⠀⠀⠀⠀⠀⢠⡞⣰⠾⣿⣿⣿⣿⣿⣿⠁⠀⢰⣿⣿⡟⠀⣼⡿⢩⡴⢶⢦⡙⣷⡀⢻⣿⡇⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣽⣯⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠡⢞⣡⣾⣿⣿⢻⣿⣿⡟⠀⠀⣾⣿⣿⡁⢸⣿⠁⣿⢈⠀⢀⡇⣼⠇⢸⣿⡏⠀⢸⣿⣿⣿⣿⠠⣻⣿⡿⣉⣛⠿⠭⠥⠄
⠀⠀⠀⠀⠀⢀⣤⢾⢷⡌⢿⡁⠘⣿⣿⣇⠀⠀⣿⣿⣿⡇⠈⢿⣧⡘⠷⣭⣥⠾⠋⣠⣿⣿⠃⠀⣼⣿⣿⣿⢗⣴⡿⢋⣊⣉⠉⠛⢲⠄
⠀⠀⠀⠀⠀⢿⣛⣛⣒⣤⣽⢿⣦⣿⣿⣿⡀⠀⠘⣿⣿⣿⣆⠈⠻⢿⣶⣶⣴⣶⣾⡿⠟⠁⢀⣼⣿⣿⣿⡿⠛⠽⠓⠻⢤⣈⠙⢦⡀
⠀⠀⠘⢦⡀⠉⣉⡉⠤⠈⠛⠿⣿⣿⣷⢤⣀⡉⠐⠛⠿⢿⣿⣷⣤⣀⠈⠉⠉⠉⠁⣀⣤⣶⣿⣿⠿⠟⣫⣴⣟⡓⢾⡝⢦⡙⣧⠆⠑
⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠐⠠⠜⠉⡠⠡⠶⠀⡄⡄⡀⡈⢉⡉⠙⠛⠓⠛⡛⣋⡉⣉⣉⣠⣴⠿⣿⠛⠿⡻⢆⠙⠆⠉⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡁⡘⢱⡅⡙⢰⢖⠻⢶⠛⠻⣌⢻⡘⡍⠙⣿⢷⣄⠙⣦⠙⠈⠳⡘⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠁⢈⠀⠇⠨⠘⠀⠈⠳⡄⠙⡀⡀⠑⠀⠸⡄⠙⠃⠈⢆⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣴⣭⣤⣦⣤⣷⣥⣾⣥⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣤⣶⣶⠄⠀⠀⠀⠀⠀⠀⢀⣤⡴⣶⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⢠⣾⠿⠿⠿⡇⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⣛⡛⠓⠿⠀⠀⠀⢀⡀⠀⠁⠐⠂⠀⠀⠀⢿⡿⠿⠃⠀⠻⠿⠿⠃
⠀⠀⣀⠂⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠁⠀⠀⢈⠁⠀⠀⠀⠀⠁
⠀⠀⠁⠀⠒⢠⣤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠶⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣿⣿⣿⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠂⠀⠀⠀⠀⢠⠙⢿⠟⣁
⠀⠀⠀⢴⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⢻⣿⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠛⠀⠰⠆
⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠰⠆⠀⠀⠀⢴⠈⠉⠀⠀⠠⠃⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠘⡟⠀⠀⠀⠀⠘⡛⠀⠀⠀⢀⡀⠠⡰⠄⢀⣤⣄⣴⣾⣷⣆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠂⡄⡁⠀⠉⡈⠔⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣻⡿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠿⢛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠁⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡄⠀⠠⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⡷⢆⣤⣥⣀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣤⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠣⠁
⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⡀⠁⢀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣿⡟⣟⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠉
⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣧⠀⠉⠛⠿⠿⠷⠿⠿⠿⠿⠛⠋⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠
⠀⢰⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⡟⣆⠀⠀⠀⠀⣠⣶⣶⣶⣦⡄⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠐⠀⠀⠉
⠀⣼⣿⣿⣿⣿⣿⣿⠀⠁⠐⠀⠀⠀⢸⡇⢻⣆⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠻⣿⣿⣟⣿⣿⡟⠀⠀⠀⠀⠀⠀⣾⡇⠸⣿⠆⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠁
⠀⠀⠈⠉⠉⠋⠉⠀⢀⣂⠀⠀⠀⠀⠉⠉⢈⠀⠀⠀⠀⠀⠉⣡⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠐⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣿⣿⡶⠂⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠁⠈⠉⢉⣽⣿⣿
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⠀⠀⢀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⣾⣿⣿⢏⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⣰⠋⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⢃⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⣼⠃⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠇⣼⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡴⠇⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⠋⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⠗⣴⣶⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠠⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣋⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⡾⠿⠿⠛⠛⠛⠛⠛⠛⠷⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⡞⠟⠛⠛⠛⠛⠻⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣋⣡⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀⠀⠉⠛⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡴⠟⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣶⣤⣄⠄⠉⠙⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⠞⠉⠀⠀⣾⣿⣿⡟⠉⢹⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⢿⣷⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠁⠀⠀⠀⢰⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢣⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⠁⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣈⣦⣀⣀⣀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠙⠻⢷⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠑⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢹⠏⠉⠉⠀⠀⠀⠀⢠⡎⠃⠀⡼⠁⠀⠀⠄⠀⠀⢀⢀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠻⠶⠀⣾⠀⠀⣠⠤⢄⡀⠀⣾⠂⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠁⠈⠶⡁⠐⢤⡀
⠀⠀⠀⠀⠀⠐⠄⠁⢄⠰⢀⢀⠃⠀⠀⠀⠀⠀⠀⠚⢻⠟⠚⣼⠁⠀⣼⠃⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⡐⢱⠤⠊⣁⠘⡆⠁⠐⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠁⢘⡢⡁⢠⠀⠏⠀⠀⠀⠰⢤⡴⠏⠀⠀⢿⡴⠊⢿⡤⠸⣥⠤⠂⢧⡤⠔⠀⠀⠐⢠⠀⠄⠀⠙⠦⠅⠀⠀⢀
⠀⠀⠀⠀⠉⠀⠀⠀⠈⣁⠉⢲⠀⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠀⣀⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣼⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠈⠄⠀⠀⠀⢹⠄⠀⢰⠚⣰⠀⢀⣴⡿⠗⢦⠀⣰⠞⠛⡆⢠⣞⠛⠃⠀⠀⠠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⡾⠀⢀⠆⢰⡏⠀⢠⣟⠀⠀⡼⢸⡟⠒⠊⢁⡀⠉⢻⠄⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠁⠀⠑⠄⠀⠀⠀⠀⠀⠐⠀⠀⡀⠳⠖⠋⠀⠘⠶⠚⠛⠙⠶⠚⠁⠈⠳⠤⠒⠈⠳⠶⠋⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠆⣒⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡄⢀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⣠⠀⢀⠀⢚⠡⡬⠒⠆⣀⡑⠀⢬⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡴⠄⠐⡲⠁⠄⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⡀⠊⠁⠤⠂⢐⠖⠀⣈⠐⠚⠠⠤⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠉⠁⠀⠀⠈⠡⡀⠈⠦⠀⠷⢀⣈⡡⠤⠔⠐⠂⠀⠀⠓⠀⠢⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠈⠁⢢⡅⢚⡰⠖⠈⢠⠄⠘⠢⠀⠀⠀⠀⢀⠀⠉⠡⢤⠄⣠⡤⡡⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡔⠩⠀⠔⠀⠈⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠂⠒⠁⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠐⠀⠀⠄⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠸⠀⠔⠀⠀⠢⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠲⠀⠰⠀⠈⡄⠀⠀⠀⠀⠀⠀⢸⠁⠤⠂⠀⠢⠀⢡
⠀⠀⠀⠀⠀⡇⠀⠀⣀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⢀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠠⠄⠀⠀⠘
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⣐⣁⡄⠀⠀⠀⢀⠀⠀⡀⠈⠢⠀⠀⠀⢠⣊⣠⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⣐⣁⡄⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠈⠉⠉⠲⠊⠁⠀⠉⠑⠋⠀⠀⠀⠀⠀⠀⠉⠉⠑⠖⠉⠁⠈⠉⠂⠁⠀⠀⠀⠀⠈⠉⠉⠰⠊⠁⠀⠉⠑⠊
⠀⠀⠀⠀⠀⠀⠂⠠⠀⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠂⠐⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠂⠕
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⢀⠂⠠⠀⠀⠤⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢀⠎⣠⠀⠀⡤⠈⢆
⠀⠀⠀⠀⠀⡌⠀⠐⠀⠀⠂⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⠇⠀⠀⠐⠇⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠲⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠒⠣⠤⠠⡀⡠⠄⠀⠤⢄⡠⠀⠀⠀⠐⠐⠒⠤⠄⢀⢀⠤⠀⠠⠤⣀⠅⠀⠀⠀⠒⠢⠤⠄⡀⡠⠄⠀⠤⢄⡨
⠀⠀⠀⠀⠀⠀⢠⡤⠈⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠌⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⡸⠈⠠⠆⠰⠆⠁⢡⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠁⠀⠊⠀⠈⡆⠀⠀⠀⠀⠀⠀⢸⠀⠐⠂⠀⠒⠀⢹
⠀⠀⠀⠀⠀⠇⠀⠀⠠⠤⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠄⡄⠀⢃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠤⠤⠄⠀⠘⡀
⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⢄⡡⢄⠀⠀⠀⢀⡀⠀⣀⠀⢂⠀⠀⠀⠠⣈⠤⡀⠀⠀⠀⣀⠀⢀⡀⠐⡄⠀⠀⢄⡡⢄⠀⠀⠀⢀⡀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠈⠉⠀⠑⠃⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠚⠁⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠑⠋⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠘⠊⠓⠀⠋⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠚⠐⠀⠁⠫⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠿⢿⣿⣦⡀⠀⠀⠀⣴⣿⡿⠿⠿⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣠⣴⠶⢶⣤⡀⠀⠀⠀⠀⠀⣠⡴⠶⢦⣄⡀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠈⠻⣆⠀⠀⢠⡞⠁⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⣠⣀⢹⣆⢠⡟⠀⠀⠀⠀⣀⣤⣀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⣿⣿⡿⣷⣿⣾⠃⠀⠀⠀⣰⣿⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⣿⣿⣷⣾⣿⣿⠀⠀⠀⠀⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠘⢿⣿⣿⠟⣿⣿⡀⠀⠀⠀⠘⣿⣿⣿⢟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠉⠁⣸⡟⢸⣧⠀⠀⠀⠀⠀⠉⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⢀⣴⡿⠁⠀⢻⣦⡀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣶⣶⡿⠛⠀⠀⠀⠀⠙⢿⣶⣶⣶⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠀⠀⠒⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⠁⠀⠉⠀⠒⠒⠒⠒⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠖⠒⠒⢲⣍⠤⠦⠤⠄⠀⠴⣆⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⡇⢐⣰⠀⢀⠕⠀⡀⠄⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡠⠊⡇
⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⣀⣈⣑⣐⣈⡡⠂⠁⢀⠤⠂⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡡⠋⢀⠎
⠀⠀⠀⠀⡏⡉⠭⠭⠭⠄⠀⠒⠒⠒⠒⡆⣀⠔⣊⠥⠤⠄⠀⠉⠁⠒⠢⡉⠑⠄⠀⠋⢀⡐⠒⢒⠞⠁⠀⡴⡇
⠀⠀⠀⠀⢧⢃⠀⠀⠀⣎⣑⠀⢀⡠⢐⡉⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣠⠊⠙⠁
⠀⠀⠀⠀⢸⠸⠤⠤⠤⣤⣴⠾⠗⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠃⢀⠔⠁
⠀⠀⠀⠀⠘⣆⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠤⠤⠤⠤⠤⠤⡀⣠⢎⠌⠀⡠⠋
⠀⠀⢀⡠⠒⠉⢸⠀⠀⠀⠈⠉⠩⠀⠈⠩⠭⠥⠄⠒⠒⠀⠀⠐⠒⠒⣒⣒⠒⠚⠁⡘⢀⠞
⢠⠞⠥⠤⠀⠀⠀⠛⠒⠒⠒⠒⠒⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⠦⠔⡱⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠊
⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⢤⠤⠞⠁
⠀⠀⠈⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⡇⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⡼
⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣤⣴⣄⣠⣠⡀
⠀⠀⠀⠀⢀⣼⠿⡿⠟⠿⣿⣿⣿⡟⠻⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⡦
⠀⠀⠀⢠⠋⠁⠀⣇⠀⡸⣿⢿⣿⡿⠀⠀⠘⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⢸⠛⢻⣿⡧⠀⡛⠀⠉⠛⢿⣿⠗
⠀⠀⠀⠹⠦⣄⡀⠘⢄⡀⣈⣀⡽⠁⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣄⠀⠀⠳⢤⣌⣠⠾⠁⢀⣠⣴⠞⠛
⠀⠀⠀⠀⠀⠀⠉⠉⠒⠺⠏⠗⠲⠛⠋⠁⠀⠀⠀⠀⠀⢀⢀⣤⣰⣾⣼⣼⣾⣰⣶⣠⣀⠀⠀⠀⠀⠈⠉⠓⠒⠂⠄⠒⠖⠛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠿⠋⣹⠟⢹⣿⣿⣿⣿⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⡇⠀⠀⢻⡄⠺⢿⡿⢋⣿⠀⠈⢻⣮⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠲⢶⢤⣽⣶⣤⣴⣾⣣⣴⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣵⣯⣾⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣴⣿⣿⣿⣿⡷⣶⣤⡀
⠀⠀⠀⠀⣠⡿⠟⢫⣿⣿⣿⣿⣿⡏⠻⣿⣭⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⣾⣿⣿⠋⢀⡾⠿⠛⣼⣿⣿⣿⣧⡀
⠀⠀⠀⣴⠛⠀⠀⠘⡿⠻⠿⣿⠸⡟⠀⢈⣟⠿⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠘⣷⢠⠘⢿⣿⣿⣟⣿⣿⣄
⠀⠀⠘⠛⠒⠲⠶⠤⠼⠦⠤⡤⠾⠴⠶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣙⣷⣵⣶⣿⣷⣼⣿⣟⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⢶⣼⣶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣾⣧⣀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢻⡟⢯⣿⣿⢿⡏⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠁⠘⣇⠈⠛⠋⣸⠇⠀⠀⠈⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⢷⣤⣌⣲⣾⣾⠧⠴⠀⠂⠐⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣾⣿⣿⣷⣶⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⡏⠹⢿⡿⢿⣿⣿⣿⡿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⠿⠀⢿⣿⢀⠇⡿⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⢠⡆⠐⠋⠇⣈⡅⣉⣤⠀⣽⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⣶⠰⠋⡝⢸⡆⣇⢈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣸⡇⣿⣤⡆⣿⣄⡿⣇⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣷⣿⡼⠧⢾⣶⢻⡏⣿⠝⡼⢛⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⡿⢹⠋⣿⢋⣭⡿⢿⢹⡇⣿⡙⡇⢻⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣟⢪⡇⣿⡇⣾⠋⠘⢸⡇⣿⡿⢲⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣷⣺⡜⣿⢨⣯⡀⢨⠸⡇⢻⡗⣷⣼⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣝⡗⣽⡇⣿⣿⣾⣈⣧⣽⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣼⣿⣿⣿⣤⣷⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⡿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⢠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣾⣿⣿⣷⣾⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⢛⣿⡿⡿⢿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠟⢛⣿⣿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⢺⠿⢹⠃⡇⣼⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⡏⣿⠇⣻⢿⠏⡇⣾⢻⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⢿⣶⣤⡆⡇⢹⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⡿⠷⢯⣿⢸⣧⣀⢸⢠⡟⡬⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡍⢿⢿⢿⡇⣷⡿⢚⠁⢟⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠁⢶⠾⠟⣻⢡⣟⡿⣿⣷⣿⢩⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠘⢸⠀⡇⢀⣦⣼⣃⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⢸⣧⣼⡇⡻⢸⢈⠁⣿⡅⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣆⣸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⣷⣼⣿⣿⣷⣿⣾⣿⣷⣿⣿⣧⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣤⣄⠀⠀⠀⠀⠀⢀⣤⠀⢀⣄⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠛⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠿⠀⠻⠏⠀⠀⠀⠀⠀⠀⡁⠀⠀⡁⠀⠀⠀⠀⠀⠈⠉⣀⠋⠃
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃⠀⠀⠀⠀⠀⢀⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠸⠇⠘⠗⠀⠀⠀⠀⠀⠀⠸⠀⠀⠿⠀⠀⠀⠀⠀⠀⠘⠃⠻⠿⠃⠀⠀⠀⠀⠀⣲⡂⠰⡷
⠀⠀⠀⠀⢴⡿⠛⠖⠀⠀⠀⠀⠀⠀⢰⡿⢿⡆⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⡿⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⣀⡤⠖⠋⣥⣞⣀⣀⣠⠤⠖⠚⣩⠟⠁⢀⣀⠴⢻⠃⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠛⠋⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢰⠃⠀⠠⠔⠒⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢸⡆
⠀⢀⢀⡟⣦⡾⣧⢀⡀⠀⠀⠀⠀⢿⣄⡀⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆⠀⢱⣿⡀⠀⠀⠀⠀⢸⡇⢀⡀⣼⢳⣴⢻⡆⡀
⢀⢻⡿⣧⢸⠇⣿⠏⡿⠀⠀⠀⠀⣸⢋⡹⡏⠀⠀⠀⠀⢰⣟⣷⠀⠀⠀⡄⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠙⣧⠀⣀⡀⠀⣾⠁⢿⠹⣿⠸⠇⣾⢛⡟
⠻⣯⣛⠀⠀⢠⡟⠀⡇⠀⠀⠀⠀⢹⡀⢹⡇⠀⠀⠀⠀⠘⠿⠟⠀⢠⠴⠇⠀⠀⠀⠀⢾⣿⡿⠀⠀⠀⠀⠀⠹⣶⢃⢼⡾⠃⠀⢸⠀⢻⡄⠀⠀⣻⣽
⠀⠀⠹⣇⠀⠈⠀⢰⡇⠀⠀⠀⠀⠘⣧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠏⣸⠃⠀⠀⢸⡆⠀⠉⠀⣸⠏
⠀⠀⠀⠈⢻⡄⠀⠈⢷⡀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡏⠀⠀⢀⡾⠁⠀⢀⡟⠁
⠀⠀⠀⠀⠀⢷⡀⠀⠈⢳⡄⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣯⠈⠉⠙⠛⠛⠛⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⣼⠷⠋⠀⠀⢠⡞⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡴⠋⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣠⡶⠋⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣰⣿⣿⣷⣶⣤⣤⣙⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣥⣶⣾⣿⣿⣆⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⡓⠶⠤⣤⣤⣤⡤⠴⢶⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠳⣄⠀⠀⠀⠀⠀⣠⠞⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠉⠒⠶⠒⠉⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠿⠿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠛⠓⠶⣄⡈⠻⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠋⣁⡤⠶⠛⠋⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠉⠳⢤⡙⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣉⡴⠋⠁⠀⠀⠀⢀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠙⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⠁⠀⠀⢀⣠⡴⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠤⣉⡑⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠓⠒⠲⠤⢌⡉⠒⠢⢬⣉⡒⠢⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢴⡯⣄⡀⠀⠀⠀⠀⢀⣨⡷⠤⠤⠤⣉⡉⠒⠤⢍⣙⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠋⠁⠀⠀⠀⠈⠙⠺⠷⣦⣤⣠⣿⠋⠀⠀⠀⠀⠀⠈⢓⣤⣀⠀⠙⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠭⣖⣤⣄⣀⣴⣊⠥⢖⣫⠽⠲⡇⢸
⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⣥⡔⠊⠁⠀⢀⣠⠇⢸
⠀⠀⠀⢰⠉⠓⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⣇⡠⠔⠚⠉⠀⠀⢸
⠀⠀⠀⠈⡦⢄⡀⠀⠉⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡅⠀⠈⠑⠢⢄⡀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠈⠉⠒⠶⠚⠉⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄⣀⣸⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⡀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠭⢥⣤⡉⢳⣦⣤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠵⢶⡶⣦⢸⣆
⠀⠀⠀⠀⠀⠀⢠⠖⣁⠔⢁⡤⢂⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠘⣇⣀⣇⣇⣠⣿⣀
⠀⠀⠀⠀⠀⢀⡟⢠⠇⡴⠋⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠦⢤⣄⡀⠀⠸⡀⠈⠳⣄⠀⠀⣼⠁⢀⡤⠚⠉⣉⣈⡇
⠀⠀⠀⠀⠀⣮⠀⣏⡸⠥⣼⠀⠀⠀⠀⢳⡀⠀⠀⢀⣴⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠙⠦⣄⠀⠙⠋⠁⠠⣿⡵⠚⠉⢉⣾⡃
⠀⠀⠀⠀⠀⣿⡶⢋⣴⡎⢿⠀⠀⡄⠀⢸⠃⢀⣴⠟⠋⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢣⠀⠀⠀⠀⠀⠀⣠⡶⢉⣤⢀⡆
⠀⠀⠀⠀⢠⡏⢰⡃⣋⠀⠘⣇⣰⡆⠀⡎⢠⠏⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⡳⡀⠀⠀⠀⠈⠛⠞⠋⠀⡾⠁
⠀⠀⠀⠀⢸⠇⠈⣠⠃⠀⠈⠉⠉⠀⢸⠁⠃⠀⠀⠀⠀⠀⠀⠀⢁⡈⠃⣦⢀⣎⡀⠀⠀⠀⠀⠀⠀⠀⢳⡻⣄⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⡇⢺⠁⢸⢁⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠐⢊⡴⠛⣁⣠⣤⣤⣈⠻⣷⣆⠀⠀⠀⠀⠀⠈⣧⠈⢻⡄⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠈⣇⠀⠀⢸⡸⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣠⣞⣵⢿⣿⢿⣿⠛⢻⡿⣮⡻⣦⠌⠀⠀⠀⠀⢻⡄⠀⢷⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⢻⡀⠀⠈⠀⠀⠀⠀⠀⡼⡆⡄⡠⠀⠀⠀⠈⢹⣷⣌⠻⣦⣭⣴⠟⢀⣾⣿⠋⠁⠀⠀⠀⠀⢸⠃⠀⠸⡆⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⣀⢀⡾⠁⢿⠁⠀⠀⠀⠀⠀⠀⢙⠛⠉⠁⠀⢠⠾⠿⠟⠋⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢳⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢸⠇⠐⠶⠚⢡⡞⠁⠀⠘⣷⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⡞⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣤⣀⣄⣒⣲⣤⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣭⣍⣉⣉⣉⣉⣭⡤⢶⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣓⡲⠾⠶⠧⠿⠶⠗⣛⣫⣽⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣆⡉⠉⠉⠉⠛⠛⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢯⣿⣆⣀⣐⣶⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢠⣶⠄⠀⠀⣀⢿⡇⠀⠀⣸⠃⠀⠀⣴⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⢀⡾⠋⠉⠙⠷⣄⣠⣿⡟⠀⠴⢦⣿⣤⣥⠴⠞⠁⢀⠀⡘⠛⠉⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠘⠁⢀⣶⠟⣦⢨⢻⣿⢧⣆⡔⣼⣱⣰⢰⣶⡷⣟⣿⢺⡏⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⡇⠀⢡⠂⢸⣿⠘⠋⠗⠛⠉⠈⢉⣀⡀⠉⠁⢻⡇⠀⠀⠀⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠧⠔⣁⣄⡘⢃⣤⠀⠀⣼⠇⢀⣾⢿⠇⢰⡷⡄⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣴⠏⢸⡇⢠⡁⠀⢰⡿⠀⣸⡿⠋⠀⠘⢿⣧⡀⢠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⡀⣼⢠⣿⠃⠀⣿⢻⡉⣿⡇⠀⠀⢀⡞⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⣹⠇⣼⡟⢀⢸⣿⢸⠃⠸⣧⣀⡴⠋⣠⣾⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⢠⡏⠀⢿⣧⠃⠈⠿⠋⠀⠀⠀⠁⠀⣼⠟⠁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⡟⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣴⠏







⡿⠿⠿⠿⡿⢻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣴⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⣿⣿⣿⡿⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⠿⢿⣿⠿⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠺⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⣀⡤⠞⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⠤⣤⣤⡤⠤⠤⠖⠚⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣥⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡀⠀⠀⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠙⠛⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠋⠀⠀⠀⠀⠙⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⢀⣤⠶⠛⠁⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠛⠶⣤⡀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⣀⡴⠞⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠉⠳⢦⣀⠈⠙⠲⣄⡀
⠀⢀⣤⡤⠤⡤⠞⢋⣀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣈⡙⠳⢤⠤⢤⣤⡀
⣠⡼⠛⠁⣀⡴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢦⣀⠈⠛⢧
⠙⠲⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡟⢻⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢰⡇⢸⡇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⢸⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡇⢸⡇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢸⡇⠸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣸⠀⠀⣧⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣿⠀⠀⣿⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣟⠀⠀⢻⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⡟⠀⠀⢻⡀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⢀⡼⠁⠀⠀⠈⢧⡀⣹⡆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢢⡀⠀⠎⡰⡏⢲⣦⠀⠀⡀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠌⠸⠁⠀⢿⠃⢤⡄⢋⡀⠰⠛⠓⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡖⣀⢌⡻⠣⠤⠛⠀⠠⡉⠌⠭⣦⢁⡀⠁⠀⠀⠀⠀⠀⠀⢀⠏⠐⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠓⠒⠄⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⢛⠁⠒⠀⠀⠀⠀⠀⠀⠈⢆⣠⠁⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠉⡏⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣼⠆⡸⠐⣄⠀⡄⠀⠀⠀⠀⢀⠐⣒⣤⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠤⠉⡀⡠⠁⠀⠴⠀⠀⠀⠀⠀⠀⣆⠁⠀⠁⠐⠤⠈⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡘⠀⠑⠴⠢⡀⠀⠀⠀⢠⣀⡘⠀⠀⠰⡇⠁⡆⢀⠀⠊⠉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡰
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠱⣄⠐⢷⣌⢀⠍⠂⡀⠀⠑⢤⣄⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠁⠈⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠌⠀⠀⠉⠁⠀⠀⠐⡠⠊⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⢄⠀⠰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣰⡟
⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⣠⣴⡿⠃⠀⠀⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠙⠻⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠁⠀⠀⣀⣴⣾⠟⠉⠀⠀⣀⣴⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠒⠋⠉⠀⠀⢀⣤⣴⠾⠟⠉⠀⠀⣀⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⢀⣀⣠⣴⣶⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⣤⣤⣤⣤⣤⣴⣶⠶⠾⠿⠿⠛⠋⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⣸⣧⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣶⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⣀⣠⣤⣶⠟⠛⠛⠛⠛⠛⠂⠤⠄⠀⣀⡴⠋⣠⠜⢃⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣾⣭⣤⣌⠑⠶⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢀⣠⣴⣿⣿⣿⣿⣿⡿⣿⡁⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠛⣿⣿⠃⢻⣿⣿⣤⣽⣿⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠉⠀⠀⠀⢻⣿⢢⠚⠛⠁⠀⣹⠏⠀⠀⣀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⠿⢆⣰⣿⣯⣥⣴⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠂⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠰⠿⢶⡄⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠰⠶⠶⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠶⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⢠⣤⣤⣤⣤⣤⡴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠈⠻⠷⣶⣶⣶⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠟⢻⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠐⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠐⠂⠀⢀⡄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡠⠂⢉⠉⠀⠂⠄⡀⠃⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡤⠀⠀⠶⡈⠀⢐⠰⠀⠰⠓⢂⠐⠇⠈⠇⠰⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠈⠐⠂⠀⠉⠁⠉⠀⠠⠐⡨⠁⠀⠆⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠄⠢⢀⠀⠀⠀⠀⠈⠐⠢⠀⠂⣈⡀⢀⡠⠀⠀⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠢⡀⠈⠢⡀⠀⢀⡦⠀⠈⠤⠤⠀⠀⢀⠀⠆⠀⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠄⠃⠉⠀⠀⠀⠀⠀⡀⠀⠤⠀⠀⡃⠄⠆⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢈⣀⣀⣀⡀⠀⣀⠔⠈⠀⠀⠀⠀⡠⢊⠀⠊⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⣀⠀⠀⠞⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡼⠈⡨⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠀⠙⠢⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡆⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠒⠂⣀⠀⠔⣲⠥⠐⠂⡀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡂⠍⠐⠂⠈⠁⠘⠀⠀⠀⠀⠀⣀⡵⠄⠒⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢊⠐⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡄⡆⠀⠸⢊
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡇⠁⠀⠀⠇⠀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⠔⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠰⢀⣊⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⢣⡀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠀⠊⠠⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠁⢄⠀⠈⠂⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⢠⠠⢤⡤⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡢⡀⠀⡠⠀⡀⠀⠀⠀⠀⣄⣀⣄⡀⡀
⠀⠀⠀⠀⠀⢀⡬⡏⠽⢾⠦⢅⣀⠀⠀⠀⠀⠈⢄⣄⠮⠎⠆⠀⠀⠀⠀⠔⢸⣁⢀⡑⢲⠋⡼⢃⠀⠀⠀⠐⠀⠋⣉⣈⣁⡀
⠀⠀⠀⠀⠀⢸⣰⡦⡆⣔⣢⣨⣼⠀⠀⠀⠀⠃⠚⠐⠒⠓⠚⠀⠀⠀⠀⢠⡈⠑⢚⣔⠛⢊⠗⡛⢓⠀⠀⢀⣬⣤⣭⣁⣉⣡
⠀⠀⠀⠀⠀⢠⣁⠀⡀⣈⡁⣈⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣻⣟⡛⣠⢬⣇⢸⠀⡇⠉⠀⠀⠀⠚⠛⡛⠿⠿⠟⠂
⠀⠀⠀⠀⠀⠸⠆⢧⠇⣼⠁⡮⠘⠁⠀⠀⠀⠀⠂⠤⠄⠁⠠⠀⠀⠀⠈⢹⠇⠘⠧⠖⢿⢸⣤⣭⠃⠀⠀⠀⠛⡀⠐⠒⠒⡟⡃
⠀⠀⠀⠀⠀⠸⡃⣔⣤⡆⡦⡆⠯⡀⠀⠀⠠⡀⠄⠄⠄⠂⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠘⠊⠈⠀⠁⠁⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⣤⢨⣤⢤⠤⡄⠀⠀⢠⢶⣐⣶⢂⣤⡄⢀⢀
⠀⠀⠀⠀⠀⢀⠤⣤⠄⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡁⣀⢠⠁⢀⣀⠀⣠⠀⠀⢰⠊⢻⠹⠞⡓⠓⢛⠚⡆
⠀⠀⠀⠀⠀⠨⣆⡛⠰⢿⡀⡊⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠉⠀⠈⠁⠁⠹⠈⠀⠀⠈⠓⠋⠀⠂⣎⣓⢯⡼⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣀⡀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠁⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢡⢼⠤⢸⠠⠠⠦⠜⠮⠀⠀⠀⡀⠀⠀⠐⣠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣈⡁⠀⠀⢈⡮⡶⣤⠶⡷⢶⣆⣏⣾⡢⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢒⡙⠓⠉⢚⡙⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠆⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠙⠉⠂⠛⠑⠊⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡤⣄⣤⣤⣠⠤⠀⠀⠀⢀⣀⡀⠀⠅⠤⠤⠢⠂⠆⣄⢀
⠀⠀⠀⠀⢸⡭⡴⣠⣲⣄⢦⢴⣦⢤⢴⠄⠀⠘⠦⠛⠘⠙⠳⠞⠀⠀⠐⢹⣴⣻⠲⠴⡴⠶⠖⣆⣴⣲⢺⣷⣤⡤⡶⠹
⠀⠀⠀⠀⠧⠜⠈⠀⠁⠁⠥⠀⠁⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⠀⠀⠐⡠⠤⠤⠠⠨⠤⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⢛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⢠⡜⢆⠰⡀⢰⡜⣠⡵⡡⣠⠂⡤⠀⣠⠄⡉⡙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⡀⠈⢦⣄⣰⣷⣾⣷⣿⣿⣿⣿⣿⣿⣷⣿⣶⣞⣥⣾⣪⣔⠕⠁⡙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⣄⠀⠒⣬⣷⣾⡿⢿⣿⠿⣽⣿⣿⠿⣶⣭⡻⣿⠟⠿⣿⣿⣿⣿⣷⣖⣋⠠⣀⣈⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢄⣈⣿⡿⠿⣿⡏⢀⡟⠁⠀⢸⡿⣿⣄⣸⣿⣿⡌⢯⠀⠘⣯⠙⠻⣿⣿⣿⣯⣟⡡⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣈⣶⠟⠋⠀⠀⣿⠀⢸⢡⣷⣶⣿⣾⣿⣿⣿⣿⣿⣿⢸⡀⠀⣿⠀⠀⠐⠈⢛⠿⣿⣿⣭⣒⣁⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡶⠏⠑⠀⠀⠀⠀⢿⡄⠸⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⠀⢀⡿⠀⠀⠀⠀⠀⠀⠁⠉⣻⠿⠷⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠙⣌⠻⣿⣿⣿⣿⣿⡿⢟⡴⠃⠀⣼⠃⠀⠀⠀⠀⠀⢀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⢦⣀⠀⠀⠀⠀⠀⠈⢳⣄⠈⠙⠲⠭⣭⢭⠥⠖⠋⠀⣠⡾⠁⠀⠀⠀⣀⣤⠚⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢄⣀⠀⠀⠈⠻⢶⣤⣄⣀⣀⣠⣤⣶⡟⠭⢀⡂⢥⠒⢏⠁⢀⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠐⠒⠲⠞⠛⢿⠟⠛⠿⠂⠙⠉⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣶⠿⠿⠿⢿⣿⣶⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣿⡿⠿⠿⠿⣷⣤⡀
⠀⠀⠀⠰⠟⠋⠀⠀⠀⠀⠀⠈⠙⠻⠿⡿⣷⣷⠀⠀⠀⠀⠀⣴⣾⢿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⡒⠂⠄⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⢠⠔⠲⣲⡤⣤⡀
⠀⠀⠀⠀⠀⢰⣼⣿⣿⠿⣿⣿⣟⣷⢮⡛⡆⠀⠀⠀⠀⠀⠀⠀⢠⢟⡥⣾⣿⣿⣿⠿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠙⢻⣿⣦⣽⣮⢯⡋⠀⠟⠐⠀⠀⠀⠀⠀⠀⠀⠂⠺⠀⠘⡮⣷⣮⣤⣾⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⣿⠀⣠⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⣿⠞⠁⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠘⢿⣄⠀⠀⢀⡀⠸⠋⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡞⣠⣿⣁⣀⣀⣉⣀⣰⣿⢃⣀⣀⡤⠖⢋⠁⠀⠀⡠⢤⠀⠀⠀⠀⠠⢤⣄
⠀⠀⠀⠀⠀⣰⣿⠁⡷⠐⣸⠁⣼⡏⣰⣿⢀⣤⡈⣿⡏⢹⣿⠀⣠⡀⣿⣇⣤⡜⢠⣿⢰⣿⠤⡄⠀⣸⠟
⠀⠀⠀⠀⠀⢿⣿⡀⢀⣴⠃⠀⠻⠧⠻⠷⠟⠸⠇⠿⡇⠹⠿⠞⠙⠇⢙⢋⠀⡀⠘⠛⠈⠁⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⣠⠤⣄⡀⠠⢤⣀⡠⠔⠊⣡⡖⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣿⡇⠀⠀⡇⣀⡋⣰⣿⣡⣄⢀⣼⢶⡄⢰⡏⢀⠤⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢸⣿⡇⠀⣼⠡⣿⠁⣿⣿⣡⡿⢸⡗⠊⡰⢋⣿⣧⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠞⠁⠀⠉⠀⠉⠁⠀⠀⠀⠉⠁⠀⠸⠿⠟


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣭⣭⣬⣭⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠒⠒⠒⠒⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢠⣴⣶⡄⠀⠀⣴⣶⣤⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣶⣿⣿⣿⣿⠃⠀⠀⢻⣿⣿⣿⣷⣦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⢿⣿⣿⣿⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀⠀⠀⠀⣀⣠⡤⠤⠤⢤⣄⣀⠀⠀⢀⣀⡤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


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

⠤⠤⢀⡠⠀⠀⠁⠀⢀⡠⢤⣄⣠⡉⠉⠁⢀⣄⡀⠀⠀⠀⠀⣠⢄⡀⠀⢄⠀⠀⠁⢀⡀⠀⠀⠉⡀⠀⡀
⠀⠀⢨⠁⠒⢰⠀⠀⠃⠀⠍⠹⠋⠛⠀⠀⠎⠀⠈⠂⠀⠀⠠⠃⠀⢰⠀⠀⠀⢀⠤⠈⢤⠤⠠⢴⠀⠀⠱
⠀⠀⠀⠀⠂⠀⠀⠠⠀⠠⠐⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠀⠂⠘⠅⠤⠬⠄
⠀⠀⡀⠘⠒⠂⠀⠘⠤⠈⠀⠈⣆⡀⠂⠀⠀⠤⠤⠀⠀⠀⠠⠀⠀⠀⢀⣀⡀⠀⣀⢀⣀⡀⢀
⠀⠘⠀⡀⠀⠀⠈⠋⠛⡆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠘⠁⠶⠀⠀⠀⣎⠑⢰⡀⠀⠈⢐⠋⠓⢲
⠀⠀⠘⠀⠀⠀⠁⠩⠉⠁⠀⠀⠹⠀⠀⠐⠀⠀⠀⠀⠀⠄⠘⠀⠀⠀⡇⠀⠀⠇⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄⠀⠀⠀⡼⠀⠸⡄⠀⠀⠀⠀⠀⠁
⠲⠖⠻⠚⠂⢠⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠐⠖⢦⡀⠀⠀⠠⠇⡀⢁⡀⡀⡀⠀⢀⠂⠀⠀⠀⢀⠶
⠀⠀⠘⠉⡈⠘⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠉⠈⠀⡀⠀⠀⠰⠀⠀⠸⠀⡀⠠
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠈⠉⠄
⣤⣄⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⠀⢀⠀⢀⠀⢀
⠀⠈⠅⠀⠀⣦⠁⠀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣶⡀⣱⡀⠀⠀⢸⡇⠈⣷⠀⠤⠄⠠⠀⠈⢡⠀⠀⢹
⠀⠈⠀⠀⠀⠩⡀⢀⡌⠀⠀⠀⠣⠀⠠⠂⠀⠀⢈⠁⠀⡠⡁⠀⠀⢨⠉⡉⢉⡀⠀⠀⠘⠀⠆⠈
⠀⠀⠀⠀⠀⠀⠓⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠅⠀⠸⠀⢀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⢀⣀⣀
⠄⠀⠀⠐⠉⠐⠀⠀⣀⠌⠓⠠⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⠘⡆⡄⠀⠀⡈⠐⠦⠀⠀⣄⠫⠸⠤
⠀⠀⠀⠇⢀⠀⡇⠀⠈⠂⢀⡀⠐⠀⠀⠀⠰⠠⠄⠊⠀⠀⠀⠠⠠⢠⠁⠀⠀⢸⠓⠐⢳⠀⠀⠙⢇
⠀⠀⠀⠈⠂⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⡄⠀⡮⠀⠀⠀⡀
⠀⠀⠄⠀⢄⡀⠂⠀⠀⡀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠈⠶⡄⠐⠐⠀⢠
⠀⠀⢀⠆⠉⠉⡆⢀⠀⠄⠀⠀⠀⠀⠀⢸⠇⠀⠀⠘⠀⠀⢰⠀⠀⠐⠂⠀⠀⠗⠀⠀⢠⠈⠀⢠⠆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠈⠂⠀⠊⠀⠀⠀⢀⠀⢀⠀⠄⠀⠀⠄⠂⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣟⣛⣿⣻⣿⣿⣶⣶⡆⢀⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⡯⠡⠏⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⢛⣵⣿⣿⣿⡟⠿⡧⠴⠶⠿⠛⠻⠟⠟⠋⠁⠀⠀⠘⠻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠿⠟⠿⣿⠀⢿⣾⣿⣿⠉⠀⠠⠤⠀⠠⠄⠀⠀⢀⣶⣿⣷⣤⣶⡄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣴⣟⣿⣶⡘⣦⣌⠛⢃⣀⢀⣠⣀⣄⣀⣀⠀⠀⠀⠈⠻⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢹⣿⡿⠟⠁⣨⣿⠁⠰⠿⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡏⠀⠀⠀⣠⣶⣿⢠⡇⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣰⣾⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⣿⡗⣾⣿⢟⣀⣈⣠⣾⣿⣛⣟⡃⠿⣿⠃⠀⠀⠀⠀⠹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠆⠈⢻⣿⡟⠋⠉⠀⠍⠻⠉⠙⠻⢿⣷⣦⣄⡀⠀⠀⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡁⢰⢃⣼⡟⠀⠐⠲⠦⠄⠀⠀⠀⢊⠄⣩⡙⠟⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⡷⢸⠿⡃⠀⠘⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⣡⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⣷⡖⠛⢀⠀⠀⠀⠄⡀⠀⠀⠀⠄⠀⠀⠀⠒⠀⠀⠀⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢛⣿⠀⢸⣿⡦⡀⠀⠀⡐⢦⣄⡀⠀⠀⢀⣠⣄⠀⠁⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠀⢸⣿⡅⣄⠀⠀⠀⠀⠉⠙⠛⠉⣈⣻⡏⠧⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠀⢸⣿⣗⡙⣺⣶⠶⠦⣀⡀⢀⣈⡿⢻⣿⣀⠀⠋⡄⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢿⣿⢿⣿⣿⣿⣿⣿⢇⡠⠟⠋⣿⣿⣿⣾⣿⢏⠠⡈⡇⡔⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⡟⠾⢿⡧⠿⠟⢽⡿⣿⡿⠛⠙⠙⣷⣾⣿⣿⡷⣾⠀⠫⠐⣢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣦⠤⣤⠀⣌⠉⢮⡉⠁⠀⠈⠙⠛⠛⠻⠇⢿⡄⢆⣀⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⠛⠉⡰⠛⠂⠛⠀⠀⠘⠛⢶⠀⠀⠀⠀⠀⢺⡏⠉⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣇⣀⣄⣀⠀⠀⠀⠀⢠⡀⣷⣶⣤⣶⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠉⠛⠻⠏⠙⡟⠣⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠿⠟⣛⣿⡿⠿⠿⠿⠭⣭⣉⡉⠉⠛⠛⠻⠿⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠋⠉⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢶⣾⣶⣦⣄⡀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⠀⢀⣼⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⠀⠀⠉⠛⠶⠦⣤⣤⣤⡤⠶⢞⣋⣡⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠒⠒⠒⠚⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⣰⣿⣿⡟⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠳⣿⣿⠞⠀⠀⠀⠀⠀⠀⡏⣿⣿⢹⠀⠀⠀⠀⠀⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠘⣿⣿⠁⠃⠀⠀⠀⠀⠚⠘⣿⣿⠀⠃⠀⠀⠀⠀⠀⢠⡿⢿⡆⠀⠀⠀⠀⠀⠐⠃⣿⣿⠐⠀⠀⠀⠀⠀⠰⢻⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣆⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠀⠀⠀⣠⣼⣧⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⢠⡏⣿⣿⢻⡀⠀⠀⠀⠀⢀⡟⣿⣿⢧⠀⠀⠀⠀⠀⠀⡞⣿⣿⢻⠀⠀⠀⠀⠀⢀⡟⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠾⢰⣿⣿⡀⠷⠀⠀⠀⠀⠾⢸⣿⣿⠈⠧⠀⠀⠀⠀⢀⡃⣿⣿⠈⡇⠀⠀⠀⠀⢘⠃⣿⣿⡜⡇⠀⠀⠀⠀⠀⠀⠇⣿⣿⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⢹⡏⢸⡇⠁⠀⠀⠀⠀⠈⢹⡏⢹⡇⠁⠀⠀⠀⠀⠀⠀⠀⡿⢻
⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⡇⠀⠀⠀⠀⠀⠀⠸⠃⢹⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⡇⠀⠀⠀⠀⠀⠀⣸⠁⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣯⡄⠀⠀⠀⠀⠀⠀⣀⣿⣇⡀⠀⠀⠀⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⢿⠀⠀⠀⠀⠀⠀⡿⣿⣿⣇⠀⠀⠀⠀⠀⠀⡞⣿⣿⢳⡀⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⡃⣿⣿⢸⡀⠀⠀⠀⠀⢀⣃⣿⣿⢸⠀⠀⠀⠀⠀⠸⠀⣿⣿⡀⠃⠀⠀⠀⠀⠀⣹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠻⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⣸⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡆⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇⠀⠀⠀⠀⠀⠀⡇⣿⣿⠸⠀⠀⠀⠀⠀⠀⠀⣹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠃⠀⠀⠀⠀⠀⠀⢁⡿⢻⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠝⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠴⠏⠀⠀⠯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠉⠙⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣆⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠈⢻⣿⡿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣋⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠻⣟⡿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⣀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠤⠒⣡⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠐⠤⡉⠒⢄⡀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⡠⠊⠁⠀⠀⠀⠀⠀⠀⠐⠲⠖⠋⠙⠓⠒⠂⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣷⣿⣿⣶⣶⣶⣤⣲⣞⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⡾⡻⣿⡁⢿⣿⣿⣿⠿⡌⠙⠉⣠⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣥⣤⣠⣏⣳⣾⣇⠈⠁⠱⢷⣶⠃⠀⢠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣾⣿⣿⡇⠻⢿⣿⣿⣷⣵⣤⣀⣀⠀⣴⣿⣿⣿⣿⣿⡿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡅⠀⠻⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠢⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠶⣄⣠⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣿⣿⣿⠛⡿⣿⣿⣷⡄⠀⠀⠈⠛⠻⢿⠿⠿⣿⠘⣿⣷⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⠘⠀⠀⢸⣿⣿⣿⣶⣷⣶⣿⡆⣠⣶⣾⣿⠰⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠉⠁⠀⠛⠈⠉⣴⡿⢻⣿⣿⣿⣿⣿⣿⣿⣳⣷⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣷⠟⠷⣿⣿⣯⣽⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠀⢸⡟⡿⠿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢀⠀⢘⡀⢠⣴⠟⠋⠀⠑⠆⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠹⣿⣷⣄⣈⣛⣿⣏⠆⠀⠀⠀⠀⠀⠀⠀⠒⠿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠐⠂⡀⠰⠀⠀⠀⠀⢰⠁⣴⣿⣿⣯⡙⠿⠿⣻⣤⣙⣲⣦⣄⡀⠀⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠙⠂⠀⠀⢀⠀⠀⠀⠀⠀⡈⠀⢿⣿⣿⣶⣿⣶⣾⣿⣿⣿⣿⣿⡿⢛⣦⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⢷⢽⣿⢇⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠷⠋⠀⠀⠹⠠⠄⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢘⢉⡉⣻⣟⠉⠁⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣼⢿⡷⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⣟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣐⠛⠿⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠂⠀⢀⣈⠁⠀⠀⠀⠂⠀⠀⠀⠄⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡷⠶⠿⠟⠁⠳⠶⢶⣀⡤⠀⠀⠛⢷⣦⣭⣼⣷⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠴⢧⠠⣤⠀⢸⣷⣽⣿⣿⣿⣷⡀⢃⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠻⠿⠷⠢⠀⠀⠠⠿⠶⠼⠿⠿⢿⣿⣿⣿⣷⡀⢷⡄⠀⠀⠀⠀⠀⠈⣥⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⣀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣉⡀⠀⠀⠀⢠⣼⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠂⠈⠙⠋⠉⠉⠉⠉⠉⠉⠓⠋⠉⠻⠻⠽⢽⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣠⣤⣤⣤⣤⣽⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠉⠉⠉⠀⠀⠈⠉⠉⠉⠛⠿⣿⣿⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠉⣠⠤⣒⣂⣭⣥⣤⣤⣬⣭⣙⣛⠶⣦⣄⣹⣿⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⣠⣿⣿⣧⡴⣋⡴⠊⠿⠿⢿⣿⣿⣿⣿⠿⠿⡛⠻⣦⣝⠻⣿⣿⣿⡯⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣿⣿⣿⣿⣫⡾⠏⠀⠀⠀⠀⠘⣿⣿⡿⠃⠀⡉⠀⠈⠈⠻⢷⣌⠛⠋⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠫⣞⠻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠀⠀⠀⢀⠴⠯⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡇⠀⠀⢀⣖⡯⠀⡼⠁⠀⣠⠤⠖⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢰⢡⠇⣸⢃⣀⣠⣾⣥⣶⣯⣄⣀⣼⣶⠭⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠸⡇⠀⢀⣿⡾⠿⣍⠁⠀⢯⠳⠝⢆⠀⢸⡿⡝⡗⠶⣤⣴⣊⣍⣉⣩⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡼⢰⣧⠞⠉⠙⢯⣗⠀⠱⡄⠀⠑⢦⡘⡆⠀⢧⠁⢹⠀⡸⢻⢻⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡿⣿⣗⠲⢄⠀⠀⠉⠙⠁⠀⠀⠀⠙⠃⠀⠀⠑⠞⠀⢱⡈⠀⡇⢸⠻⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠙⠲⢭⣤⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣙⠲⠃⠈⢣⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⢿⡄⠀⠀⠀⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⣴⠶⢶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢠⣿⡟⠁⠀⣸⠁⠀⢀⣴⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⡅⠀⢸⣿⣷⠀⠉⠛⠶⠤⣤⡤⠔⠃⠀⠀⠀⢀⣿⣿⠀⠀⣰⠃⠀⢀⣾⡟⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⢀⡼⠁⠀⠀⣾⡿⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⠏⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡵⠋⠀⠀⠀⢸⣿⡇⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣠⣶⡶⠷⢦⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡠⢤⣾⣿⠏⢀⣤⡀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⢸⣿⡏⢀⣾⣿⡇⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢸⣿⡇⠀⠈⠿⣧⠞⣿⣿⡇⠀⠀⢨⣿⣿⣇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢹⣿⡇⠀⣠⠋⢿⣿⣿⠀⠀⢀⠇⢸⣿⣷⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠘⣿⣿⡆⢀⡞⠀⠈⢿⣿⡄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠞⠀⠀⠀⠀⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⣼⡿⠁⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢸⣿⠇⢀⡞⣰⣿⢰⣷⢶⠀⢻⡏⠉⠀⢿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⡟⠀⣿⡇⣼⣷⡿⡄⢸⣿⠖⠀⠀⡙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⣿⠃⣿⣯⡴⠋⠘⣿⣠⠖⠀⠳⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠀⠈⠁



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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠄⠦⠹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠇⠈⠄⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⣿
⣿⣿⣏⠉⠉⠉⠀⠀⠀⠀⠀⠙⠛⠋⠉⣙⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿
⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⡤⢀⡀⢻⣿⣿⣷⣦⣄⠀⠀⠀⠀⢀⣤⣤⣼⣿⣿
⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠁⠀⠀⣴⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⡄⢠⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠙⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃⠐⢋⡙⠂⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠘⠃⠀⢹⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣀⣀⣤⣄⣀⣠⣶⣶⣾⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣿⣿⣿⠟⠻⣿⣿⣿⢿⣿⣿⡿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣿⣿⣿⡿⠀⠀⠈⠿⠃⡄⠹⠟⠁⠀⠘⣿⣿⣿⣿⡿⠋⠁⠀⠉⢿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿
⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠁⢘⣀⡛⠀⠈⣿⣿⣿⣧⣀⣤⣤⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⠀⠈⠉⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣴⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣀⠀⠀⠀⠀⢀⡀⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣷⡄⠀⠀⢀⣀⡟⠛⢻⣄⡀⠀⠀⠀⣴⡿⠿⣆⠀⠀⠀⣰⣾⠿⢷⣄⠀⠀⠀⠠⢂⢦
⠀⠀⠀⠀⠀⢻⠺⠟⠿⠇⠀⠀⠈⠙⠀⣀⡀⠛⠉⠀⠀⠀⠛⠀⠀⠻⠀⠀⠀⣿⡇⠀⠒⣿⠀⠀⠀⢼⠋⠉⠹⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⣤⣤⣶⡄⠀⠀⠹⢷⠀⢰⠟⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠀⡑⠀⠀⠀⠐⠀⠀⢘⠠⠀⠰⠀⠀⠀⠋⠙⡝⡏⠀⠀⠀⡠⡄⠀⢀⠄⠀⠀⡀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠠⠿⡝⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⠀⢀⣴⠶⣦⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⣰⠿⠿⣿⡄⠀⠀⠀⢠⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠈⠛⠋⠉⠃⠀⠀⠀⠸⠃⠀⠘⠃⠀⠀⠀⢜⠀⠀⢹⠀⠀⠀⠀⠨⠀⠀⠨⠁⠀⠀⠀⠸⠉⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠈⠾⠷⠏⠀⠀⠀⠀⠀⠐⠀⢂⠀⠀⠀⠀⠀⠠⠀⠄⡀
⠀⠀⠀⠀⢀⣠⣤⣠⣤⣦⡀⠀⠀⠄⣆⠀⢘⠻⡤⠀⠀⢴⡀⠀⢰⠆⠀⠀⠀⠈⡁⠒⢨⠁⠀⠀⠀⠘⠀⠀⠈⠙
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠀⣀⡀⠀⢀⣀⠀⠀⠀⠄⠁⠀⠁⡆⠀⠀⠀⢀⠁⠀⠈
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⡿⡿⢿⡆⠀⠀⠀⣰⣿⡿⢿⣆⠀⠀⠀⠠⠾⠛⠱⠄⠀⠀⠀⢠⣶⣿⣶⡄⠀⠀⠀⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠈⠐⠒⠒⠃⠀⠀⠀⣟⠙⠀⠊⣿⠀⠀⠀⠰⠏⠉⠙⠏⠀⠀⠀⢸⡿⠿⢿⡇⠀⠀⠀⣿⡿⠉⠟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿⣧⠀⢰⣿⠀⠀⠀⠀⡠⠀⣠⡀⠀⠀⠀⠈⢷⣾⣾⠁⠀⠀⠀⢿⣇⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠊⠰⠈⠂⠀⠀⠀⠛⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡀⠨⠉⠙⢀⡄⠀⠀⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣾⣿⣷⣄⣶⣿⣿⠀⠂⠀⠐⢲⠓⠀⠐
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠛⠛⠀⠙⠛⠛