Fun Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣥⣴⣶⣶⣶⣶⣤⣌⡛⢿⣷⡀
⠀⠀⠀⢰⣿⣿⡿⠛⠛⠻⢿⣿⣿⡿⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⢀⠿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢟⣣
⠀⠀⢘⠋⡎⠉⠀⠀⠀⣦⠀⠀⠀⠀⢰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠚⡉⠀⠀⡀⡀
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠆⠀⢀⠀⠀⠀⠘⣲⠺⠂⠀⠀⠀⠀⠀⣸⡇⠘⠂⠻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠈⠡⠀⠋⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠰⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠈⠑⢄⠙⠻⢿⣷⠀⢰⣿⠿⠛⢉⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠒⣠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠒⢄⢀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠸⠿⠗⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠮⣹⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠐⠀⠀⠁⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣀⠀⠀⠐⠲⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠴⠈⠄
⠀⠀⠀⠀⣠⠀⡠⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⢁⣠⠼⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⡄⠀⠀⠀⠀⠴⢿⣆⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡿⠤⠠⠀⠄⠀⠀⠈⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢈⢁⡤⢏⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠀⠀⠈⠳⢄⠠⠤⠄⠆⠀⢂⡈⠁⢢⡀⠈⢧⡀⠀⡀⠠⣀⠁⡀⠀⠡⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⡀⠁⢈⠀⠀⠀⣂⢀⠂⠀⠀⠀⠒⠦⡀⠀⠀⠀⠀⠑⡌⢀⠊⠀⠀⠈⠀⣀⣀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠠⢀⠂⠐⡰⠇⠄⠀⠀⠈⠱⡀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⢰⠀⠀⠀⡠⠤⠀⠠⣄⠑⢆⠀⠰
⠰⣄⡆⠀⠀⡅⠀⠀⠀⢀⡤⡄⠀⢤⡄⠀⠀⠀⠀⠀⡆⠀⠀⠚⡅⠀⠀⡄⠀⠀⠀⡄⠀⠀⠰⠘⠲⠠⠄⠉⠀⢸⠀⡘
⠚⠻⣿⠋⠀⠁⠀⠀⠀⠀⣤⡄⠀⣄⡌⠀⠀⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⢊⡀⠀⢰⢤⡀⠈⢠⣀⠀⠉⡡⠀⠀⠃⡐
⠀⠀⠈⠳⠦⠴⡀⠀⠀⠈⠘⠀⠐⠒⠀⠄⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠿⠀⢰⣦⣤⡼⠁⡀⢡⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠤⢒⣿⠀⠀⠀⠀⢠⢷⠀⠀⠡⢄⠀⡠⠄⢀⣼⣿⣾⢇⠀⠘⠒⠊⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣴⣵⣤⡄⠀⠈⠋⠑⢀⡤⢄⡠⢴⣋⣿⠀⠀⠀⠀⣀⠀⠀⠀⣠⠋⢀⣨⠦⢀⣀⣀⡠⢒⡏⣀
⠀⠀⠐⢦⣤⣀⣻⠀⠉⠉⠒⠦⢖⠒⠉⡇⠀⠀⠀⠞⠟⠓⠂⠰⡎⡼⠁⠒⠈⠀⠰⠯⠥⠤⣤⡄⠀⢳⠈⠳⠟⠃
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⡟⢧⣰⡇⠀⠀⠀⠀⠀⠀⠀⣠⠿⠴⠶⢲⡆⠀⠀⠀⣀⣴⠟⠁⠀⠈⠷⠤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠁⠀⠀⠀⠀⠀⢠⣾⣥⣤⡀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢂⢀⢤⠊⠀⠩⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⠀⢀⠀⠀⣠⠐⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠹⡨⠄⠉⠁⠀⠀⠀⠄⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡀
⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⠃⠌⠠⠀⠀⣀⣀⣀⢠⡀⠀⠤⠆⢒⠀⠀⠀⠀⠔⡀⠌⠀⠀⠁⠘
⠀⠀⠀⢀⡌⡄⠀⠀⠀⠀⠂⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⡠⠁⠐⠤⠀⠀⠀⠂⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠄⠀⠊⠐⠠⡀
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠘⠀⠀⢂⣠⡀⢠⣀⠂⠀⢸⠀⡐⠀⠀⠄⠀⠀⢀⠈⢂
⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠐⢩⣆⠀⠀⣶⡙⠀⡈⠀⠡⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢃⠀⠀⡀⠀⠀⠀⠰⡦⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠉⠁⢀⡃⠐⢤⠄⣀⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠆⠈⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠑⢤⣀⣀⣀⣀⠴⢋⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠔⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠩⢀⡤⠶⣾⣷⠶⣄⠀⠀⠀⠀⠀⢀⠀⠌⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡾⠁⠀⣰⠋⠀⠀⠈⠻⣷⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠉⣼⠃⠀⠀⠀⠀⠀⠘⣿⡌⢧⠀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⢸⡄⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⢰⣦⡀⠀⠀⣿⢸⡇⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠈⣧⠀⠀⠀⢸⣿⡝⢦⡀⠏⢸⠇⡁⠀⡁⠀⠀⠐⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⠀⣸⠘⣷⡀⢹⡇⢹⠀⠈⡊⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠹⣎⣡⠙⢦⡀⡟⣶⣿⣧⣼⠀⠸⡆⠀⠀⢆⡀⠀⠀⠀⠀⠈⠐⠢⢀
⠀⠀⠀⠐⡆⠀⡀⠀⠀⠀⠀⠀⠈⣻⣷⣤⣉⣓⣟⣉⣵⡇⠀⠀⢧⠀⡓⠒⠒⡄⠀⣀⡀⠀⠀⠀⠀⠙⢀
⠀⠀⣰⣇⠈⡏⠀⠀⠀⡐⠈⠀⣤⣇⠼⣟⣿⣯⣿⡋⡜⠀⠀⠀⠘⣆⠀⠀⠀⡇⠉⣄⠙⠋⠐⠒⠲⠄⠤
⠐⢠⠇⠸⡄⣇⠥⡀⠇⣘⠀⠐⠿⠿⣽⣧⣟⡗⣏⠟⠁⠀⠀⠀⠀⠘⣆⠀⠀⡇⠀⠀⠀⠀⠀⣌⡉
⣠⡏⠀⠀⢹⡇⠃⠐⠀⠄⠾⠂⢼⡆⣼⣙⡮⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡁⠀⠀⠉⠈⠒⠂
⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠃⠈⠋⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢂⡡
⠈⠓⠤⠔⠊⠀⠀⠀⠀⠀⠐⠆⠠⡶⠖⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⢶⣤⡼⢃⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣀⣀⣀⣡⣤⣔⣀⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢧⣼⣿⣿⣿⣿⣧⣼⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⣦⣀⣠⠴⠶⠶⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡀⢀⡤⠖⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣠⠟⠹⣶⠋⠀⠀⠀⠻⣄⠀⢀⠀⠀⢠⡴⢰⣾⣇⡼⢧
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢰⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡧⢠⣌⠳⠖⠋⠁⠀⠀⣦
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⠏⢀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⠟⠀⠀⠀⢹⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣧⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠸⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠘⠛⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢷⣿⡄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠙⠿⠁⠀⠀⢀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⢻⣆⠀⠀⠀⠀⠰⣤⡀⢀⣠⡄⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⠀⠀⢳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⢷⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⠀⢠⣀⠀⢉⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣙⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣾⣇⠀⠀⢿⣷⣤⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠉⠉⠛⢳⡶⠤⠶⠚⠛⠉⠀⠀⠘⣏⠀⠴⠤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣤⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⠤⠚⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⠃⠀⠀⣿⡉⣹⠏⠉⠉⠉⠉⢁⣀⠀⣼⠿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠏⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⢰⡏⢉⣿⣿⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣼⢋⡴⠲⡄⠀⠀⣠⠖⠲⡄⠀⠀⠀⠉⠊⢹⡟⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠃⠈⠳⠞⠃⠀⠀⠛⣤⠾⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⣄⠀⠘⢷⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣼⠋⢙⡆⠀⠀⠀⠀⠀⠀⠘⢧⡴⠇⠀⠀⠀⢳⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⡀⠀⠀⠙⠓⠋⠀⠀⠀⢠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠳⣆⡸⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢀⡷⠀⠀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⣤⠶⣄⠀⠉⠉⠀⠀⠀⠺⣅⣿
⠀⠀⠀⠀⠀⠀⢸⡿⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⠿⠀⠀⠀⠀⠀⠀⢀⣀⢼⡇
⠀⠀⠀⠀⠀⢠⡟⠀⠙⣿⡽⣟⠓⣂⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣶⠾⣿⠛⠊⢳
⠀⠀⠀⠀⢠⡎⠀⠀⢰⡟⠀⡏⠀⠀⠀⠀⠈⠘⡿⢻⡄⠈⠉⠉⠉⠛⠛⢻⣿⠹⠉⢻⠀⢹⠀⠀⠘⡆
⠀⠀⠀⠀⠘⢧⣄⡀⠸⡇⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⢃⢸⡇⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠹⠀⠇⠀⠀⠀⠀⠀⠀⠘⠁⠆⠀⠀⠀⣀⣠⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⠛⠿⠶⠶⠶⠶⠾⠿⣶⣶⣶⣶⣶⣶⣶⠶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⡀⣠⡀⢻⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠙⠁⠀⠈⣻⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣺⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⣀⣀⡜⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢀⣨⠟⠛⠷⣤⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣅⣀⣀⢀⣀⣈⣉⣉⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⣿⠿⠿⡿⠿⠿⢏⠹⠿⢿⡿⠿⠿⠿⣿⣿⣿⣿⡟⣻⣛⠛⠛⠛⣛⡛⣛⢻⣿⣿⠛⠛⠛⠿⠿⠟⢻⠛⠛⠻⣿
⣿⣷⣿⣶⣶⣷⣾⣶⣾⣶⣾⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠛⠛⠛⠛⠛⠿⣶⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⣰⣿⠏⠀⠀⠀⣴⣾⣿⣦⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⡏⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠸⠿⠿⠛⢀⠶⠄⠘⢿⣉⠀⠀⠀⠀⠀⢠⠃⠀⣿⡟⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀⣼⡿⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠈⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠀⠐⡡⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡟⠋⠀⢀⣤⠟⠀⠹⣿⣿⣿⣶⣶⣶⠆⠀⣰⠀⢀⡷⠀⢸⣶⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣶⠿⠋⠉⣿
⠀⠀⠀⠀⢀⣠⣿⡋⠉⠀⢠⣾⣿⣄⣀⣠⣀⡙⠻⠿⠿⣿⣿⣦⣤⣿⣤⣼⣿⣤⣾⣿⡋⠀⢀⣀⣠⡾⠿⣦⡀⠀⠀⠀⠀⠀⣴⡿⣿⠀⠀⠀⠀⠀⣰⡟
⠀⢀⣴⠾⠛⠉⠉⠉⠻⣆⣸⡟⠀⠉⠉⠙⠛⢿⣦⡀⢀⣼⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⡿⠿⠿⣧⠀⠘⢷⣦⣤⣤⣤⣰⣿⠀⠘⠷⠶⣶⡶⠟⠋
⠀⣾⡇⠀⢀⣤⣤⡀⣀⣿⠟⠁⠀⠀⠀⢠⡄⠀⠙⢷⣿⣟⠛⠛⠛⠻⣿⡟⠹⣿⣛⣉⣀⣀⣠⣤⣶⣿⣶⣦⡄⠉⠁⢨⡿⠟⠃⠀⣠⣶⡾⠟⠀⠴⠂⠄
⠀⠈⠛⠛⠋⠉⠈⠙⡛⠁⡀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⠿⠿⠷⠶⣾⣿⠃⠀⣿⡛⠉⠉⠉⢉⣉⣉⣿⠇⠹⣿⣀⣴⣿⣦⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⢠⠀⠻⢷⣦⣤⣄⣠⣤⣴⣿⠀⢸⣿⣟⡛⠛⠻⠿⣿⣿⡁⢀⠆⠙⠛⠋⠀⠹⣷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠉⠉⣉⣽⣿⡄⢸⣿⡛⠿⣿⣷⣶⣤⣽⣿⢠⡇⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠟⢋⣩⣿⣷⣿⣿⣿⣦⣄⡙⢿⣯⡉⠀⠈⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣯⣴⡾⠿⣿⣿⣁⣠⣿⣿⡿⠿⢿⣷⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠿⢿⣶⣤⣶⠟⠙⠷⣦⡤⡈⠙⣿⠟⢁⣤⣶⠟⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡷⣦⡀⠀⠀⠈⢀⣀⡀⣠⡿⠀⠀⣠⣿⠀⠈⢻⣇⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠁⢸⣿⣤⣤⣾⠿⠛⠛⢿⣤⣴⠾⠋⠙⠷⣦⣤⣿⣦⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠈⠻⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠈⢻⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠈⠙⢻⣦⣤⣄⡀⠀⠀⠀⠘⠟⠀⠀⠀⠀⣿⡷⠾⢿⣷⣾⡏⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠛⣿⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⢈⣩⣿⠆⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⠃⢰⣿⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣾⠟⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⢷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⣤⣄⡀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⢳⡄⠈⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣧⣀⠀⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣾⣿⣿⣿⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⣉⣉⣉⣉⣉⡉⠙⠻⢿⣷⣦⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣙⢿⣿⡄
⠀⠀⠀⠀⣾⣿⣿⡿⠋⠁⠉⠛⠻⠿⢿⣿⠟⠻⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠟⠉⠻⢿⣿⣿⠿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟
⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠿⠧
⠀⠀⠀⣷⢲⡝⠛⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠁⡞⠚⡅
⠀⠀⠀⠣⣀⠃⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠠⣄⣄⢀⠀⠀⠘⠀⠀⠈⠀⣀⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⣥⡎⠀⣄⣴⣰⣀
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠐⠂⠠⠤⠐⠂⠀⠀⠀⠀⢀⠀⠀⠀⣽⠏⡉⢙⡀⠀⠰⠀⠀⠀⠀⠐⠫⠙⠁⠀⠀⠀⠀⠀⠀⣠⣿⣇⠀⢀⡀⠽⠧
⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⢀⠄⠂⠉⠂⠄⣉⠂⣑⠠⢑⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⠿⠛⣡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠄⠀⠀⠤⠀⠰⠊⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁⠑⣄⠈⠛⡿⣿⣿⠀⠀⣾⣿⠿⠻⠋⣀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠢⠤⠔⡀⠠⠁⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠌⢶⡀⠀⠅⠁⠀⢰⠁⢀⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠐⢄⠐⢃⡃⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠰⠘⢿⡟⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠑⠤⢎⠂⢈⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢻⣿⣟⠿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⠠⡑⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⢐⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⠞⠀⠀⠀⢸⠂⠠⠤⠤⠀⠂⠀⠈⠘⠄⠠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠀⠤⠤⠤⠀⢺⠀⠀⠀⠙⠂⠄⣤⡄⠐⠁⢀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠀⠀⠀⠀⠀⠠⢬⠠⢥
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢛⠐⠖⠈⠁⠀⠈⠐⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠊⠀⠀⠈⢂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠒⠉⠈⢋⣀⠃
⠀⠀⠀⠀⢀⡾⡀⣠⠂⠀⠀⠀⠀⠀⠀⠡⡀⡀⢄⢀⣀
⠀⠀⠀⠀⠐⠀⢈⠷⡀⠀⠀⠀⠀⠀⠀⠀⢡⣀⡴⠇⠔⠃
⠀⠀⠀⠀⠀⠈⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠂⠁

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⣾⠏⣀⣤⠞⠋⠉⠉⠉⠉⠉⠛⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠞⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢿⣄⣀⣀⠀⢻⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣿⣀⣿⣤⡟⠿⠁⢼⡟⠛⠛⣧⣀⣿⣆⠀⠀⠀⠀⢵
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣃⣼⠻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠟⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⡂⢠⠟⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠄⠀⢸⠃
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⡄⢀⡎
⠀⠀⠀⠀⠀⢹⣸⣇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠆⣸⠃
⠀⠀⠀⠀⠀⠀⣿⢸⣇⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠋
⠀⠀⠀⠀⠀⠀⢹⣆⠹⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣠⡾⠃⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠼⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣦⣤⣄⣀⡀⠀⠀⢀⣀⣠⣤⠞⠛⣿⡀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠙⢿⣄⠈⠉⠙⠛⠚⠉⠉⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠛⠿⠶⢶⣤⣀⣀⣀⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣠⡛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⠶⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠶⣦⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⢾⡷⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣧⣄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⡀⢠⣴⠀⢶⣾⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣶⣶⣦⣾⡿⠿⠛⠉⠁⣿⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⢻⡃⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠄⢦⣤⠤⠴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠶⠶⠶⢶⣤⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣠⣤⠚⠛⠛⠛⠛⢳⣄⣼⠋⠁⠀⠀⠀⠀⠈⠙⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢾⣧⣄⠀⠀⠀⠀⠀⠀⣨⣿⣤⣤⣀⣤⣤⣤⡤⠶⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢿⣿⣶⣶⣾⡛⠾⠉⠈⠙⠛⠋⠶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⠲⠙⠓⠉⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠉⡻⠛⠲⢤⡀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠀⡎⠀⢀⣀⣀⣀⠤⣤⣤⠤⢂⣁⣀⡀⠀⠹⡄⢠⠦⠀⠀⠠⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⢉⣉⠀⣤⣤⡄⠀⠀⠀⣤⡀⠀⣉⡑⠂⡗⠀⠋⠀⡀⠀⠑⠨⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡚⢡⣿⣿⣿⣷⣶⣤⡄⢠⣤⣶⣾⣿⣿⣿⡆⠙⡄⠀⠞⣡⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠀⠻⣿⣿⣽⣿⡿⠀⠀⢿⣿⣯⣿⣿⠟⠠⢚⣷⠀⠀⠁⠀⠀⠀⠔
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⣚⠁⠈⠛⠛⠛⠑⠛⠚⠊⠛⠛⠋⠁⠀⣐⠃⡼⢂⡠⠀⠀⠀⠀⠀⠮⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⢧⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣠⠏⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⣀⠞⢉⠉⠒⠤⡙⢦⣄⣀⣀⠀⠀⠀⣀⣀⣤⠞⠁⠀⠀⠀⠊⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⡍⠀⡀⠀⠀⠀⠈⣦⠤⠽⢿⣉⣉⣹⠿⠷⠦⢤⠀⡀⠤⢄⠄⡀⠀⠀⠀⢀⠀⠤⠤⠒⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠤⠄⠀⠀⠀⠀⠐⠋⢹⠀⠀⠀⠉⠉⠁⠀⠀⠀⡞⠘⠀⠠⠀⢀⠈⠆⠀⠀⠀⠳⣤⡀⠀⠀⠐⠴⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠠⠀⠀⠀⠐⠁⠠⠒⢄⠀⠀⣸⡀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠋⠀⠀⠠⢹⣦⣌⠛⠿⠶⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⠀⠀⠰⠀⠀⠀⠁⠐⠙⡇⠀⠀⢀⠘⠄⠀⠀⢠⡇⢀⠀⠀⠀⠂⠀⠀⠀⠁⠌⢉⡙⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣄⠤⢤⣿⣄⣀⡴⠛⣄⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⠀⠀⢀⠢⣀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠘⠁⠀⠀⠀⠀⠉⠑⢄⠀⠘⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡞⣿⠃⠉⠁⠀⢬⠀⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠊⣾⠁⠈⠉⠉⠉⠉⠀⠀⠘⣿⣿⣉⠀⠀⠀⢀⡴⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡷⠀⠀⠀⢠⣴⠀⠀⠀⠀⢿⠈⠻⢃⠂⠤⠞⠁⠀⠀⠤⠀⠴
⠀⠀⠀⠀⠀⡴⠊⠉⠉⠙⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢸⢹⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡂⠀⠀⠄⡀⠀⠈
⠀⠀⠀⠀⢸⠀⢀⠀⠈⠐⠄⣸⡆⠀⠀⠀⠀⠀⢀⣠⠊⣿⠀⠀⠀⠀⣾⠘⡆⠀⠀⠀⠀⢷⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢄⣸⣀⣀⣨⠉⠑⠉⠉⠀⠀⠉⡇⠀⠀⠖⠋⠀⢧⣤⡀⠤⠄⡞⠅⠀⠀⠀⠁⠘⠁⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠈⠻⠤⣄⣠⠶⣹⠋⠉⣹⠉⡁⠀⠰⠀⠀⣠⣖⣠⠀⡄⠀⠀⠀⠀⠀⠀⠄⣴⡒⠀⠀⠀⠀⠀⠀⢀⡠⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠈⠠⠴⠃⠠⠁⠀⠀⢀⣴⢋⡽⠁⠀⠀⠀⠀⠀⠀⡲⠘⠻⠶⠻⢦⣄⠀⠀⠃⠉⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣷⡏⠀⠀⠀⠀⡀⠀⡜⠉⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠙⡆⠀⠀⠀⠐⠂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠣⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠡⠀⠀⠂⡠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠛⠛⠛⠛⠛⠿⣶⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⣰⣿⠏⠀⠀⠀⣴⣾⣿⣦⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⡏⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠸⠿⠿⠛⢀⠶⠄⠘⢿⣉⠀⠀⠀⠀⠀⢠⠃⠀⣿⡟⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀⣼⡿⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠈⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠀⠐⡡⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡟⠋⠀⢀⣤⠟⠀⠹⣿⣿⣿⣶⣶⣶⠆⠀⣰⠀⢀⡷⠀⢸⣶⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣶⠿⠋⠉⣿
⠀⠀⠀⠀⢀⣠⣿⡋⠉⠀⢠⣾⣿⣄⣀⣠⣀⡙⠻⠿⠿⣿⣿⣦⣤⣿⣤⣼⣿⣤⣾⣿⡋⠀⢀⣀⣠⡾⠿⣦⡀⠀⠀⠀⠀⠀⣴⡿⣿⠀⠀⠀⠀⠀⣰⡟
⠀⢀⣴⠾⠛⠉⠉⠉⠻⣆⣸⡟⠀⠉⠉⠙⠛⢿⣦⡀⢀⣼⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⡿⠿⠿⣧⠀⠘⢷⣦⣤⣤⣤⣰⣿⠀⠘⠷⠶⣶⡶⠟⠋
⠀⣾⡇⠀⢀⣤⣤⡀⣀⣿⠟⠁⠀⠀⠀⢠⡄⠀⠙⢷⣿⣟⠛⠛⠛⠻⣿⡟⠹⣿⣛⣉⣀⣀⣠⣤⣶⣿⣶⣦⡄⠉⠁⢨⡿⠟⠃⠀⣠⣶⡾⠟⠀⠴⠂⠄
⠀⠈⠛⠛⠋⠉⠈⠙⡛⠁⡀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⠿⠿⠷⠶⣾⣿⠃⠀⣿⡛⠉⠉⠉⢉⣉⣉⣿⠇⠹⣿⣀⣴⣿⣦⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⢠⠀⠻⢷⣦⣤⣄⣠⣤⣴⣿⠀⢸⣿⣟⡛⠛⠻⠿⣿⣿⡁⢀⠆⠙⠛⠋⠀⠹⣷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠉⠉⣉⣽⣿⡄⢸⣿⡛⠿⣿⣷⣶⣤⣽⣿⢠⡇⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠟⢋⣩⣿⣷⣿⣿⣿⣦⣄⡙⢿⣯⡉⠀⠈⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣯⣴⡾⠿⣿⣿⣁⣠⣿⣿⡿⠿⢿⣷⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠿⢿⣶⣤⣶⠟⠙⠷⣦⡤⡈⠙⣿⠟⢁⣤⣶⠟⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡷⣦⡀⠀⠀⠈⢀⣀⡀⣠⡿⠀⠀⣠⣿⠀⠈⢻⣇⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠁⢸⣿⣤⣤⣾⠿⠛⠛⢿⣤⣴⠾⠋⠙⠷⣦⣤⣿⣦⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠈⠻⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠈⢻⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠈⠙⢻⣦⣤⣄⡀⠀⠀⠀⠘⠟⠀⠀⠀⠀⣿⡷⠾⢿⣷⣾⡏⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠛⣿⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⢈⣩⣿⠆⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⠃⢰⣿⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣾⠟⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⢷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⣤⣄⡀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⢳⡄⠈⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣧⣀⠀⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⡡⠞⠁⠛
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠡⠀⠀⢸⣯⣿⡇
⠀⠀⠀⠿⠿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⡏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠐⠶⠷
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠲⠂⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠘⣿⠛⠛⠛⣿⠀⠄⠁⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⣿⡀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠠⠀⠀⠀⠛⠋⠀⠻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣨⣠⣤⣍⢥⠀⠐⠐⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡄⠀⠀⡄⠀⠈
⠀⠀⠀⠁⣠⣤⣤⣀⡀⠀⠀⠓⠊⠀⠐⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠇
⠀⠀⠀⠀⠋⠉⢙⣿⡗⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⢀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⡟⡛⣛⡛⣻⠿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣠⣤⢀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣉⣁⣤⣤⣤⣀⣉⡙⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠄⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣹⣿⣧
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣼⣿⡿⠿⠛⠉⠈⠛⠻⠿⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⢿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠉⠃⠈⠃⢹⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢋⠅⡀
⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠘⠈⠁⠀⢀
⠀⠀⠀⠀⡕⢢⠙⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⢱⠿⠠⠂⠀⢀⠀⠀⠀⠀⢒⡶⢶⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⡗⠚⢾⡔
⠀⠀⠀⠀⠠⢈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠃⠀⠀⠆⠀⠁⢄⢀⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠁⡙⠿⣷⣦⣤⣄⠀⢀⣴⣶⣾⡿⠟⢉⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠁⠂⣀⠉⠙⠋⠀⠜⠙⠋⠀⢠⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⡐⢂⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⢹⣶⡄⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠃⠀⠀⠘⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢤⠀⠥
⠀⠀⠀⠄⡀⠑⠂⠀⠌⠀⠀⠠⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠂⠂
⠀⠀⠈⠁⠜⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⠄⠀⠀⠀⠐⢺⣧⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡴⠛⠓⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐⠒⠊⢦⡀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢳⡀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠩⠂⠈⠀⢀⡀⠐⠀⠀⠂⠄⠹⢀⠔⠀⠐⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⡆⡀⠀⠁⠃⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡁⠀⠀⢀⣀⠀⢀⡓⠢⠀⠘
⠠⣰⠀⠀⠀⠀⠀⠀⣀⣤⠁⢠⣄⠀⠀⠀⠀⢠⠀⠀⢒⡁⠀⣈⠁⠀⠀⠄⠀⠀⠄⠦⠠⡀⠁⠀⠇⢀
⠒⠿⡟⠁⠠⠀⠀⠀⠀⣤⠀⢤⡌⠀⠀⠀⠀⠇⠀⠀⡈⠀⠀⠀⡁⠀⢰⢄⡀⠀⠄⡀⢈⡁⠀⠐
⠀⠀⠉⠓⠒⠂⠀⠀⠈⠁⠀⠉⢀⠂⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⢀⣼⢶⠓⠀⡷⢶⠞⠀⠈⡼
⠀⠀⠀⠀⠀⠈⢄⠀⠀⠘⣄⠭⡛⠀⠀⠀⣠⢾⡆⠀⠀⠂⠐⠂⠀⠺⡿⠟⢂⠀⠀⠁⠀⣀⡼⠁
⠀⠀⢀⡀⠀⢸⠛⠻⠦⣀⣀⣀⠴⡏⠑⠊⢓⣾⡦⢀⣀⠔⡦⠀⠠⠐⢀⣴⠚⢁⡀⠒⣎⠸⣤⡦
⠀⠀⠈⠉⠛⠚⠂⠀⠀⠀⣰⣦⢀⡇⠀⠀⠀⠈⠀⠀⢈⣾⣁⣤⡄⠀⠀⠉⢉⣹⠏⠀⠘⣆⣈⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠙⠃⠀⠀⠀⠀⢀⣴⣏⣀⠀⢀⣷⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡠⠄⠠⡀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠋⠍⠉⣉⣩⣴⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠐⠄
⠀⠀⠀⡀⠒⠒⠁⠀⠀⠘⡀⠀⠀⠀⠀⣰⠟⠀⠌⢡⡴⠟⠉⠙⢿⣷⣌⠛⢦⡀⠀⠀⠀⠀⠀⢀⠂⠁⠁⠀⠀⠐⠒⠐⠂⠠⡀
⠀⡀⠜⠀⠀⠀⠀⠀⠀⠀⠁⢢⠀⠀⣼⠃⠀⠀⣰⠟⠀⠀⠀⠀⠀⠙⣿⣷⡈⣷⡀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢀
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣺⠃⠀⠋⣰⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡈⣷⠀⠀⠀⠢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠜
⠀⠀⠁⠂⠀⠠⣀⠀⣠⠄⠀⠀⢠⡿⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⡇⠀⠀⠀⠀⠀⠁⠉⠦⠤⠴⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣇⠀⣀⢸⡇⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⢘⣿⠀⡇⠀⠤⠤⡀⠀⠀⠀⠀⣀⠀⣠
⠀⠀⠀⠀⠀⠤⠠⠄⠂⠈⠀⠀⠀⣿⠈⠁⠈⣧⠀⠀⠀⠀⠀⡿⣿⡙⠦⣄⠸⠇⢠⡇⢸⠀⠀⢸⠀⠀⠀⠀⠢⢀⣸⠀⠀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠹⣆⠀⠀⠀⢸⡇⢻⣧⠀⠈⣿⠄⢺⠇⠈⠓⠖⠁⠀⠀⣎⠄⠀⠀⡘⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠰⠆⠹⣆⠀⠀⢸⠁⡈⣿⣇⠀⡾⠀⠸⡇⠀⠀⡇⡀⠀⠀⡀⠄⠁⠁⠐⠠⡄⢀⡀
⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠂⠀⠀⠀⠀⠹⣦⣾⣄⠘⠷⣄⣼⢸⣿⣿⡿⢾⠇⠀⠀⢷⠀⠀⡆⠉⠢⠠⠄⣀⢀⠀⠀⠀⠀⠁⠊⠢⢀⢄
⠀⠀⠀⠀⠀⢇⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣷⣶⣄⣙⣺⣉⣡⡤⡿⠀⠀⠀⢸⡄⠀⡛⠋⠉⢓⡆⠠⢈⣀⡁⠐⠀⠄⠄⢀⢀⠉⣠
⠀⠀⠀⢰⡆⠀⢹⠋⠀⠀⢀⠀⡠⠚⠃⠄⣀⡟⣠⢾⡿⣝⣿⣹⡽⠋⣰⠃⠀⠀⠀⠀⢳⡀⡀⠀⠀⢈⡇⠉⢁⢈⠹⠏⠒⠶⠰⢦⡤⢀⣅
⠠⠀⢠⠇⢹⡄⢸⠐⣂⠀⢸⠀⡵⠀⣀⢸⣿⣿⡧⣾⠓⡾⣿⢹⠽⣶⠏⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⡇⠀⠈⠃⠀⠀⠁⠀⠃⠲⠀⠀⢢
⠀⢰⡏⠀⠀⢻⣸⢶⡉⡇⠈⠑⠅⢂⡀⠈⣡⠈⠙⣿⣟⢦⣿⣸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠆⠀⠀⠄⡀⠀⠀⢘⣇⢢⡀⣀⡀
⣤⡞⠀⠀⠀⠈⣿⠊⠁⠈⠓⠐⠃⠺⢙⠠⢿⣋⣰⣿⣨⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠳⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠁
⢸⡇⠀⠀⠀⠀⢸⠄⣀⣀⠀⠀⠀⠀⠙⠀⠈⠋⠩⠄⠠⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⢈⠀⠀⠀⠀⠂⢪⣟⠊⢉⡄⠀⠀⠀⠀⠈⠁⠤⠄
⠀⠣⣄⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠈⢤⠀⢀⣄⣠⡀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠈⠛⣀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠅⠀⠨⠐⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣂⠂⠀⣀⢀⡁⢀⡄⠀⠄⠀⠀⠀⠀⠀⠈⠐⠈
⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠹⢶⣶⠿⣡⠆⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⡌⠌⠀⠀⠀⠀⠀⠀⢈⠀⠈⠍⠌⠠⠔⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⠀⠀⠀⠐⠀⠋⣀⢀⢬⠀⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠿⠿⠿⠿⠿⠿⠷⠾⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⢠⡟⣠⠴⠚⠉⠉⠉⠉⠛⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣽⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠹⡆
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⣤⣴⢻⣦⣤⡄⠹⣄⠀⠀⠀⢻
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⢰⣷⡾⠿⠉⠁⠘⠃⠀⢹⠶⣿⡆⠀⠀⠘⡇
⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⢰⡆⢠⠇⡿⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⠀⠀⡇
⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⢸⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⢉⠀⣸
⠀⠀⠀⠀⢱⣿⡄⠀⠀⠀⢀⡞⠁⠀⠀⣶⠄⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⣼⣡⠏
⠀⠀⠀⠀⠘⣧⠻⡄⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠘⢧⡀⠘⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣌⣣⡀⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠚⠁⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣶⡤⣀⣀⠀⠀⣀⣠⡤⠞⢿⡄⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠹⣦⣀⡉⠉⠉⠉⠀⢀⣠⡿⠁⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠚⠋⠁⠀⠀⠀⠀⢠⡀⠀⢀⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⢸⣟⡒⠤⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⡛⢃⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣶⣿⠿⢿⣷⣤⡿
⠀⠀⠀⠀⠀⠀⢿⠀⠶⡇⢺⠃⠘⠛⠛⠛⠿⠿⠛⠻⠿⠿⠿⠟⠛⠛⠛⠛⠋⠉⠀⠀⠸⣟⣽⠃
⠀⠀⠀⠀⠀⠀⠈⠷⣄⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⡇⠀⠀⠀⠀⠀⠀⠀⠛⠓⠒⠰⡶⠒⠚⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⣤⡤⣤⣄⡀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣠⠴⠚⠛⠛⠛⢦⣼⠋⠀⠀⠀⠈⠙⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣤⡀⢀⣠⣰⡿⢶⣶⡶⣶⢖⣛⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠓⠉⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠛⠛⠛⠻⣷⣤⣤⡾⠛⠛⠛⠛⠻⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⠶⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⣤⠾⢿⣀⡼⠋⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣾⠋⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⢠⡞⠃⠀⠘⠛⠀⠀⠀⠀⠀⠈⠷⣤⡀⣴⡄⠀⠀⢸⣟⠉⣻⣿⠶⠛⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠸⣷⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠀⣴⡄⠙⠿⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⡿⢿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢀⣿⠁⠀⠉⠁⠸⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣶⣾⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣴⡄⢸⠁⠀⠀⠀⠀⠰⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠹⣧⠙⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣦⣤⠀⠀⠈⠁⢸⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣷⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠈⠙⠻⠋⠀⠀⠀⢀⡸⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣀⠈⢿⣄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣁⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠈⢿⣦⠀⠀⠀⠀⠀⠉⠻⠶⣤⠶⠾⠟⠀⠀⠀⠀⠀⠀⠛⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠹⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⣤⡀⠠⣤⣄⣀⣀⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⣙⣻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣶⣿⣧⠀⠀⠀⣿⣷⣦⣤⣼⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⠏⠉⠙⠶⠶⣤⣄⣠⣤⡴⠶⠶⠊⠉⠀⠀⢹⣟⠀⣀⣤⣈⣿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⢀⣿⡉⠈⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⢦⡌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠛⠳⢤⣴⣾⣿⣷⣦⣤⣤⣤⣤⣴⡦⠚⠋⠀⢠⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢻⣿⠇⠀⠀⠰⣿⡀⢀⣿⠃⠀⠀⠀⠀⠀⠀⣀⣀⠀⣼⡿⠋⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⣾⠏⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⢰⡞⠉⢹⣿⣿⡇⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⣾⠋⣠⣤⣄⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠈⠻⠦⠛⢻⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣾⠃⢸⡏⢀⣿⠀⠀⠀⢰⡏⠀⣨⡗⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣾⠁⠀⠈⠙⠋⠉⠀⠀⠀⠈⠹⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⣦⡀⠀⠈⢻⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⣴⠟⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣾⠁⠀⠀⠀⠹⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠿⠦⣴⠿⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠷⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡜⠋⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⠀⣹⡆⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠈⠛⠛⠉⠀⠀⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣤⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠞⠛⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡞⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠈⢻⣷⠶⣿⡛⣉⣠⣤⣤⣤⣤⣶⣴⣦⣤⣤⣤⣦⣤⣷⣶⣶⣶⣶⣶⣶⣦⣶⣾⡿⠿⣿⠏⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢠⣿⠟⠀⣿⠁⠀⠀⠀⠉⠉⠉⢸⣿⠻⣇⠀⠉⠉⠉⠉⠙⠛⠛⠿⣿⡿⠹⡏⠹⣇⠀⠹⡆⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⣿⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢳⠀⢿⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣀⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠸⠀⠸⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢶⣦⣤⣤⣀⣀⠀⠀⠀⠀⠘⠃⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠃⠀⠀⠀⣀⣠⣤⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⠿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⣶⣶⣶⣶⣶⣾⣿⣿⡿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢠⣄⣀⣤⠄⠸⣷⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠙⠋⠀⠀⠀⢹⡟⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠛⣿⣾⣿⣄⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣁⣀⣠⡶⠛⠁⠀⠈⠛⠻⠶⠶⠶⠶⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣤⣀⣀⣀⣀⣀⣀⣀⣤⠴⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠻⡿⠿⠟⠿⠛⠿⠿⢿⡄⠛⠿⠻⣿⠿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣛⣻⣛⣛⣉⠉⣉⣉⣉⣛⣋⣻⣿⣿⣟⠛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠛⠛⣻⣿
⣿⣷⣾⣷⣶⣶⣾⣶⣿⣶⣾⣷⣶⣿⣶⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠐⢎⣀⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣤⡄⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⢸⠇⢰⠇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢏⠀⡐⡅⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⢸⠀⢸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⣿⡇⠙⢿⣷⣦⣄⣘⠀⣂⣀⣴⣷⡀⠀⠀⠀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠻⣿⣿⣿⠿⢿⣿⡟⠀⠀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⣠⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠸⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⢿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠖⠛⠉⠉⠉⠓⠶⣤⣿⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⢸⣿⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠁⠹⣾⡣⠀⣠⣴⣶⣤⣀⣰⣟⣹⡆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣀⣀⣤⡾⠟⠓⠒⠶⣄⣹⣯⠝⢦⡀⠀⢀⡇⠀⠀⢻⡗⣾⠋⢰⡖⠛⠉⠉⠉⣿⣷⠀⠀⣧⣀⡀
⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⠀⣀⠀⠀⠀⠙⢿⡯⡈⠻⣄⢸⠃⠀⠀⠀⢻⣻⣄⠸⣇⣤⡄⠀⠀⡿⣿⠖⠒⢻⡀
⠀⠀⢈⡏⠀⡣⠀⠀⠀⠀⠀⢠⣿⣷⢅⣴⠿⠛⠻⢷⡄⠀⠈⣿⣔⠀⠈⠿⠀⠀⣆⠀⠈⢷⡉⠛⠛⠉⠀⢀⣼⢃⣸⣄⣀⣼⡿⠛⠓⠒⠂
⠀⠀⠈⠁⠙⠃⠀⠀⠀⠀⢀⣼⣿⣿⣺⡇⠀⠀⠀⠈⢿⠀⠀⢸⣿⣷⡄⠀⠀⢰⣿⡆⠀⠘⣷⣤⣤⣴⡶⠟⣱⠟⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⡞⠀⠀⣼⣷⣿⠻⣦⣀⣸⠀⢻⡶⠾⠿⠉⢉⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡆⠠⡄
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠉⠉⣹⠿⣿⣿⣾⠉⠛⠛⠁⠀⠀⣰⣿⣿⣿⠄⠘⠛⠋⠀⠀⠀⠀⠀⠐⣤⠷⣖⠒⠒⠛⠦⠀⠀⠀⠈⢠⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⡰⠛⠻⣷⣧⣤⣤⣤⣴⡾⢿⣯⡁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡎⠈⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⠞⠁⠀⠀⠜⠋⠉⠉⠉⠀⠀⠀⠙⣷⡀⠀⠀⣠⠞⠋⠉⠉⠉⠲⣇⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡼⠃⢀⡴⠞⠛⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠒⠈⠦⡄⠀⠀⠀⠀⠀⣿⡼⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣠⣀⡎⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣦⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢲⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠀⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⣰⠏⢷⡀⠀⠀⠀⠀⠀⠀⡘⠋⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⣄⡀⢠⡶⠛⠛⠻⠶⣦⣴⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⡀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠈⠙⡿⠁⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣀⠀⠀⠀⠀⠀⢀⣿⡁⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⣀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣴⠟⠁⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠀⣙⣉⣁⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢰⠞⠉⠈⠙⢷⡀⢀⣤⣤⣤⣄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣿⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⣟⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣧⠀⣰⣾⢲⣀⡿⠀⠀⣀⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⢦⣿⣿⣿⣿⡇⠀⣾⣿⣋⣥⢸⡇⠀⠀⠀⠀⠀⣦⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠉⠉⠉⠀⠹⣆⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣰⣿⣆⠀⠀⠘⠓⣲⡶
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⠀⢰⡏⠈⠻⠿⣶⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠚⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣗⡀⠀⠀⠀⣰⣏⣀⣀⣀⣀⣀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣦⠀⢰⡟⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⣿⡀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⣇⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣴⡿⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠘
⠀⠀⠀⠀⠀⠀⠀⣿⡋⠀⠀⠀⠀⠀⣰⠋⠛⠀⠀⢠⣏⠉⠉⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⢀⣤⠾⢷⣄⠀⠀⠀⠀⢻⡄⠀⢀⣴⠟⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⡾
⠀⠀⠀⢀⡴⠋⠁⠀⠈⠛⢷⣦⣤⣤⣤⣿⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⢀⣿⠁
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠃⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢇⠀⠀⠀⣾⠁
⢠⡾⢀⣤⣾⠇⠀⠀⠀⠀⠀⠀⢀⡿⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢻⡏⢸⡆⠀⠀⣿
⣮⣴⠟⢡⡟⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣿⠀⢸⡇⢸⡇⠀⠀⢹
⠛⠁⠀⢸⡇⢀⡄⠀⠀⣤⠀⠀⣿⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉⠀⡿⢀⡿⠀⠀⣿⠀⠀⣿
⠀⠀⠀⢸⡇⣼⢷⡀⠀⣿⡄⢠⡟⠀⠀⠀⠀⠀⠀⠙⣿⣦⣄⣀⡈⣃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠋⠀⠀⠀⢠⠇⣼⠃⠀⠀⣿⠀⢀⡟
⠀⠀⠀⠘⣷⡿⠘⣧⠀⡏⢳⣀⡇⠀⠀⠀⠀⠀⠀⢀⡿⠀⣿⠉⠛⢻⡄⠀⣿⡿⠶⠶⠶⠾⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣀⣠⡏⣼⠏⠀⠀⢰⡏⠀⣸⠇
⠀⠀⠀⠀⠙⠗⠀⠘⢷⣧⠀⠛⠓⠀⠀⠀⠀⠀⠀⣸⠇⢸⡏⠀⠀⠸⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠛⣧⠀⠀⠀⣸⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣀⣰⠏⢠⡿⠀⠀⠀⠀⣿⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠴⣶⣠⠏⢠⡶⠾⠷⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠛⠶⣿⡁⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠀⢸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣀⠀⢀⣾⠃⠀⠀⠀⢸⡏⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⣷⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠻⣧⡞⠁⠀⢀⣀⣀⣸⡃⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⡏⠉⠛⠻⠿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⣄⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⢻⣄⡾⠁
⣀⣀⣀⣄⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣈⣡⣀⣀⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⡿⠿⣿⣿⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣼⣧⣠⣧⣠⣶⣦⣥⣼⣿⣱⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠾⠟⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⣴⠦⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⣀⣀⣴⣇⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢄⠀⠀⠐⡀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠦⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠰⠂⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠟⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⣈⠘⠁⠀⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠒⠂
⠀⠀⠈⠂⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠠⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠅⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠷⠀⠀⠀⢠⣄
⠀⠀⠀⠤⢤⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⡀⠉
⠀⠀⡀⠀⠀⠈⡁⠀⠀⠙⡀⠀⠀⠀⠀⠛⠛⢶⣦⣤⣤⣤⡶⠚⠛⠋⠀⠀⠀⠀⠀⠀⢾⣟⡶⠈⠰⣿⣟⠑
⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⡀⠀⠀⠠⡄⠈⠀⠀⠀⠔⠉⠢⠀⠀⠔⠁⠀⠀⠀⠤⠀⠐⠛⠿⢋⠔⠜⠛⠋
⠐⡆⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠈⠂⠄⠀⡀⠀⠀⠠⠐⠁⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠈⠇⢲⠀⠀⠐⠂⠀⠁
⠀⠀⠘⢤⠄⠠⠀⠠⠀⢂⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠘⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢼⡄⣀
⠀⠀⠀⠀⠁⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠠⠴⢤⣤⣀⡀⡀⣀⣠⡴⠶⠆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠶⠲
⠀⠠⡀⠈⠉⡙⠛⡉⠃⠀⡠⠁⠀⠀⠀⠀⠀⠀⢀⡄⠈⢠⠀⠀⠀⢀⡀⠀⠀⠣⠀⠈⠋⠀⠶⠀⠉⠁⠀⠈
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠠⠀⠄
⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂
⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣆⣰⣿⣿⣦⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠄⢊⣉⣁⣤⣤⣄⣀⣉⠙⠻⣿⣶⣄
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣻⣿⡆
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠿⠟⠛⠁⠉⠛⠿⠟⠛⠻⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠋⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⣿⣿⠏⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣫⢁
⠀⠀⠀⠀⣘⢍⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠃⠈⠀⠀⢀
⠀⠀⠀⠀⠉⢘⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠸⠂⠀⢰⠀⠀⠀⠀⠀⠀⠰⢒⠶⠶⠀⠀⠀⠀⠀⠀⢀⣿⡆⠘⠂⠛⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠶⣖⠂⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠱⣶⣤⣤⡄⠀⢠⣴⣶⣾⠿⠛⢁⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠉⠛⠁⠄⠊⠛⠉⠀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⡄⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠊⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠈⠂⠁⠐⠂⠀⠀⠀⠀⠀⠂⠊⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⣄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠑⠂⠄⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⢀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⡄⠀⠸⠠⢠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡰⠋⠀⠀⠘⠒⠒⠐⠀⠅
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢡⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠄⠀⠀⠀⢀⠶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⡀⡠⣴⠃⠀⠐⡋⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢰⡃⠀⠀⠀⠀⠁⡀⠒⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡆⢇⠠⠚⠒⡀⠀⠁⠀⠀⠉⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⠀⠙⠺⠄⠀⠀⠀⠀⠀⠈⠉⠀⠉⠖⡂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠍⠀⠀⠀⠠⠄⠀⠬⠈⠀⠀⠀⠀⠀⠚⡄⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⢫⡑⠢⡀
⠀⠀⠀⠀⠀⠀⣀⠤⣺⢀⡀⠀⠀⠀⠀⢨⡑⠒⠂⠢⠬⠤⠤⠤⠤⠤⠤⠔⠠⠒⠚⠋⡅⠀⠀⠀⠀⢠⢊⠄⠒⠀⠈⠁⠀⠉
⠀⠀⠀⠀⣠⡎⡄⠊⠀⠀⢀⢀⣓⠀⠄⠈⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠁⠀⠀⠀⠀⠈⠨
⠀⠀⠀⡼⠀⠑⣀⡀⠀⠀⠀⠄⠀⠀⠀⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⡠⠄⠐⠂⠰⢄⡀
⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⡄⠀⡆⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⡰⠁⠀⠀⠀⠀⠀⠠⠉⢆
⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⢸⠀⠀⡇⠀⣠⠖⡋⠉⠉⠙⡳⢤⠀⢰⠁⠀⡇⠀⠀⠁⠀⠀⠀⠘⠀⠀⡆
⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⢸⡷⠀⠀⢸⣷⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠈⡆⠀⠀⣀⠀⠀⠀⠀⠈⡿⠀⠀⡇⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠑⡶⠢⠤⠤⡤⠤⢤⠤⠤⡤⠂
⠀⠀⠀⠘⡄⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⠳⢤⣀⣀⣀⣀⣠⠔⢋⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢁⠀⠀⠀⠈⠉⠀⠀⠀⢸⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢆⣀⣀⣀⣀⡀⠤⠔⠚⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸⠤⣄⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣱⠏⠁⠀⠀⠀⠀⠀⠀⠀⢠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⢀⡞⢷⡀⠀⠀⠀⠀⠀⢀⠋⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠏⠀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⣼⠁⠈⠳⣄⡀⣰⠞⠛⠛⠶⢦⣼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠉⠙⠲⢶⣤⣀⡀⠀⠀⠀⣰⡯⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⡤⠀⣠⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢶⣞⠋⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣠⠴⠶⣄⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠹⣧⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡞⠁⠀⠀⠈⣧⡴⠞⠛⠛⢦⡘⠃⠀⠀⠀⠀⠀⢀⡿⠛⠉
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡇⠀⣠⣤⡀⣿⠁⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠘⢳⡄
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢳⣀⣿⣷⣿⣿⠀⣴⣾⠗⡀⣸⠀⠀⠀⠀⠀⣄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠉⠛⠛⠁⠹⣄⣿⣿⣿⣿⡟⠀⠀⠀⠀⢠⣿⡄⠀⠈⠳⠦⣶
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⢠⡏⠘⠿⢶⣶⣶⠟⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠋⠉⠉⠀⠀⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠿⣯⡀⠀⠀⢠⣿⣀⣀⣠⣤⣤⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢹⣦⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⢸⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠈⣷⡶⠛⠀⠀⠀⠀⣠⢤⡀⠀⠀⡾⠷⠟⠃⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⣇⠀⠀⢀⡼⠷⣦⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰
⠀⠀⠀⢀⡴⠟⠉⠘⢷⣦⣀⣀⣀⣹⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢸⠟⠁
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠉⠙⡻⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⢰⠟
⢠⡞⢀⣴⡶⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢛⡏⢼⠀⠀⢸⡇
⣮⡴⠋⢸⠇⠀⠀⠀⠀⠀⠀⣾⠁⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢿⠁⢸⡇⢸⡄⠀⢸⡇
⠋⠀⠀⣿⠀⣤⠀⠀⣤⠀⠀⡟⠀⠀⠀⠘⠷⣦⡀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠀⣾⠀⡿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⢰⡟⡆⠀⣿⡆⢸⡇⠀⠀⠀⠀⠀⠈⣿⠷⣦⣤⣨⡄⠘⣧⣀⣀⠀⢀⣀⣀⣀⣤⣶⠾⠟⠋⠀⠀⠀⢠⠏⣼⠃⠀⢠⡇⠀⣼⠁
⠀⠀⠀⠸⣿⠀⠹⣄⡇⠹⣼⡇⠀⠀⠀⠀⠀⢠⡟⢠⡇⠈⠉⣧⠀⢻⡟⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠦⣮⣼⠃⠀⠀⣸⠁⢠⡏
⠀⠀⠀⠀⠉⠀⠀⠙⠿⠀⠈⠁⠀⠀⠀⠀⠀⡾⠁⣾⠀⠀⠀⢻⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⠀⣹⠀⣀⣠⣟⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⢾⣅⣼⠃⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⢷⠋⢠⡏⠉⠉⠙⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⣿⠇⠀⠀⠀⣿⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⣆⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠺⣇⡼⠋⠀⢀⣀⢀⡏⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⢻⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣸⠛⠛⠻⠶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠛⣧⣼⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠻⠛⠻⡿⠋⡻⠛⢛⠻⠛⣟⡿⣛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣷⣿⣾⣿⣶⣿⣷⣾⣿⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⢀⣴⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⣿⣿⣾⣿⣿⣿⣿⣦⣾⣦⠀⠀⠀⠀⠀⠀⡠⠶⠛⠛⠋⠉⠉⠛⠛⠛⠷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣐⣨⣴⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣈⠻⢿⣿⣆
⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣴⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⣿⣿⠿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡟⠛⠋⠁⠀⠀⠉⠙⠛⠋⠉⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠁⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⢟⣉⢂
⠀⠀⠀⠀⠀⢟⡛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢴⠆⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠸⠀⠻
⠀⠀⠀⠀⠀⠓⢹⡈⠁⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⢰⡄⠀⠀⡄⠀⠀⡆⠀⠀⠀⠀⢠⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⢠⡀⢼⣴⣀
⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⣇⠀⠀⠃⠀⠀⠀⠀⠀⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡆⠀⠀⠀⠹⠧
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠙⠅⠁⠀⠀⠀⠀⠧⡀⠂⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣿⠟⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠄⠈⠘⠿⣷⣶⣶⡆⠀⢠⣶⣿⣿⠿⠛⠉⢀⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⣤⠂⠈⠛⠑⠀⠊⠙⠋⠀⠀⣤⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠂⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⠃⠀⠀⠀⠘⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡌⠻⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⢀⠒⢐⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⠚⣀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠠⠤⠲⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠤⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠑⠂⠄⠐⠈⠀⠀⠑⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠐⠊⠉⡆⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠀⠀⡏⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠻⣣⠇⠀⠀⡇⠁⠘⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠝⠁⠀⠀⠀⠉⠉⠉⠈⠀⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡄⠀⠀⠀⠀⠷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⡀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠒⠿⠟⠋⠉⠁⠢⣀⠠⠚⠛⠛⢷⠆
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠸⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⡀⠀⣣⣄⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣠⠀⢹⣀⣤⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠛⠃⢠⡛⠃⠀⠀⡐⠀⠀⠀⠀⠀⠂⠀⠐⠄⠀⠁⠀⠫⡁⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠉⠀⣨⠈⠉⠀⠀⢀⠄
⠀⠀⢋⠀⠀⠀⠀⠀⠐⠁⠈⠂⠀⠊⠀⢀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⠀⠁⠠⠄⠠⠤⠚⠁⠙⠒⠀⠂
⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠌⠑⠄⡀⠀⢀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⠋⠀⠀⠘⠓⠀⠀⠀⠉⠞
⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠄⢃⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⣠⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠋⠀⠀⠀⠀⠰⢿⣆
⠀⠀⠀⠀⠤⠤⣤⣀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠤⡀⠀⡀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠃⡠⠄⠀⠉⠂⠀⠀⠀⠀⠀⠀⠲⠿⠶⣶⣤⣄⣀⣀⣀⣠⣤⠶⠟⠛⡋⠀⠀⠀⠀⠀⠀⠠⠁⢰⣟⡷⣶⠌⠏⣴⣛⡲⠋⠡
⠀⠀⢰⠀⠀⠀⠀⠀⠘⡀⣀⠀⠀⠈⡄⠀⠀⠀⠀⢀⠈⢄⠀⠈⠙⠉⢉⠟⢋⠉⠃⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠰⠀⣤⣽⣟⣶⠀⡆⣨⣽⣽⠇⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠋⠀⠀⡠⠁⠀⠀⠀⠚⡇⠀⠀⠑⠂⠀⠒⠁⠀⠈⠑⠒⠒⠊⠀⠀⣠⠀⠀⠀⠹⠁⠀⠑⠤⣉⣋⠥⠊⠉⠒⠤⠕⠊⠀⢀
⠘⢳⡂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⢀⠄⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃
⠀⠀⢱⠦⣀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠈⣉⠒⠀⠄⠀⠀⠀⠀⢠⠤⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⡗⠢⡤⠄⠀⠠⠤⠐⠂⠉
⠀⠀⠀⢢⠀⠈⢘⠀⠀⡉⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⡇
⠀⠀⠀⠀⠙⢯⠀⠈⠀⠀⠋⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡄⠀⠀⠂⢉⠴⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣀⡶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠿⠛⠀⠀⠀⠀⠀⠛⠷⠄⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⠃⠀⠀⠀⠀⢀⣠⣀⡀
⠐⠛⠛⠻⢷⣶⣤⣄⣤⣤⣴⡶⠟⠛⠉⡍⠁⠀⠀⠀⠀⠀⡠⠂⠈⠁⠐⢄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⣀⠀⢈⣉⡉⠁⠀⡀
⠀⠀⢆⠀⠀⠉⠀⡨⠋⢍⠈⠁⠀⢀⠔⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣠⡄⠈⡄⡄⠀⠀⠈⠀⢀⣀⠀⠀⠀⠙⡄⠀⠀⠉⠀⠀⠟⠄⠀⠈⠀⠀⡠⠃
⠀⠀⠀⠉⠂⠀⠈⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⢫⠀⠀⢀⠀⠀⠈⢀⡰⢆⡀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀
⠀⠀⠀⡀⣠⠴⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠁
⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⠀⠀⢀⡀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠽⠟⠈⠑⠈⢙⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡬⠂⠀⠀⠀⠀⠀⠀⢀⠀⢀⡼⠂⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠓⢄⠀⠀⠀⠀⠀⠐⠓⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢜⡠⠤⠤⠶⠒⠒⠒⠒⠋⠀⠴⠦⣤⡈⢧⠰⡡⠀⠀⠂⠀⢌⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣠⣶⣤⣖⣛⠂⠀⠀⢐⣒⣠⣴⣦⡄⠿⡀⠀⣠⠄⠀⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⠀⢻⣿⣿⣿⣿⣿⠃⢹⣿⣿⣿⣿⣿⠇⡠⢇⠀⠑⠊⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢩⠦⠙⠿⣿⡿⢧⠤⢤⠻⣿⡿⠟⠃⢠⣔⢹⠄⠀⠀⠀⠁⠈⢀⣀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡒⢧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡼⠀⠍⠈⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠖⠒⠢⢌⡳⢤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠁⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠱⢀⡀⠀⠀⢨⠇⠀⠀⠀⠀⠈⢢⡬⠭⣿⡛⠛⣻⠯⠭⢤⡄⢀⠤⢀⠀⡀⠀⠀⠀⠀⢤⡠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠤⠀⠀⠀⠀⠀⠚⢹⠀⠀⠀⠉⠉⠁⠀⠀⢸⠑⠁⠠⠀⢀⠐⠂⠀⠀⠘⢦⣀⠀⠀⠀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠐⠂⡀⠀⢸⡄⠀⠀⠠⠀⠀⠀⠀⣸⠀⠀⠀⢀⠈⠃⠀⠀⠘⢳⣤⡙⠻⠷⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠢⡀⠀⠀⠀⡈⠀⠀⢀⠐⠀⡇⠀⠀⢀⠙⠀⠀⠀⡏⠠⠀⠀⠀⠂⠀⠀⠀⠀⠉⡄⠁⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⢉⡈⢀⡠⠦⠒⠼⢷⣤⠴⠋⠱⡀⡇⠀⠀⠀⠀⠀⠀⠀⡇⢀⣴⡄⢀⡠⠓⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠊⠀⠀⠀⠀⠀⠀⠑⢌⢀⢽⡇⠀⠀⠀⠀⠀⣀⠀⣿⣬⣻⠁⠀⠀⠀⠂⠀⠀⠀⢠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⣼⠀⠀⠀⠀⠀⠀⠀⠀⢹⡹⣧⣄⡀⠀⣰⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠤⠴⢤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡏⠀⠀⠀⢸⣿⠀⠀⠀⠘⡇⠈⠑⠂⠉⠁⠀⠀⠈⠁⠈⠅
⠀⠀⠀⠀⡔⠁⠀⠠⢀⡈⢳⡀⠀⠀⠀⠀⠀⠀⠠⢺⠇⠀⠀⠀⣾⢸⠀⠀⠀⠀⣻⠀⠄⠀⠀⠂⠈⠈⠑⠢⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡏⠀⠐⠀⠀⠀⢸⠃⠀⠐⠠⠤⠤⠚⠃⠺⡦⠀⠄⡤⠏⢸⡂⠀⠀⣀⣸⢇⠀⠀⠀⠄⠠⠀⠀⠄⢀
⠀⠀⠀⠀⠈⠲⣄⣀⢀⣤⢫⠗⠒⠶⠐⠀⠀⡄⠀⢀⣅⢀⠀⡀⠀⠀⠉⠉⠀⠀⢘⡤⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠜⠉⠀⠉⢀⡠⠃⢠⠂⠀⠀⣠⡞⢹⠋⠀⠁⠀⠀⠀⠀⡄⠲⢯⠷⣦⡀⠀⠐⠆⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣪⡗⠀⠀⠀⢀⠀⢠⠞⠉⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠳⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⢞⣉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⢲⠆⠀⠀⢸⣷⣄⠀⠀⠀⢸⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢⠆⠂⠀⠀⢈⣿⠻⣷⣄⡀⠘⠀⠁⢀⣆⠀⠀⠀⠀⣀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠈⠛⠿⢿⣶⠿⠛⠿⣶⣶⡶⢿⣿⠃⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣇⠀⠸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⣠⠞⠉⢀⣀⠈⠉⡿⣧⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⢁⣠⡿⠧⠤⣄⣰⣞⢧⡀⠀⢠⠇⠀⢻⣦⡾⢫⡽⠿⠚⠛⣿⡄⠀⣇⡀
⠀⠀⣀⣠⠀⠀⠀⠈⠙⢿⡟⠁⠀⣀⡀⠀⠙⢿⡇⠙⢆⣸⠀⠀⠈⢿⣧⣘⣧⣤⠀⢀⡿⡷⠒⢳⣀
⠀⠀⠧⢀⠅⠀⠀⠀⠀⣿⣿⣴⠟⠉⠙⢷⡀⠈⣷⣅⠀⠋⠀⣴⡀⠘⣧⠉⠉⠀⣠⣾⣥⣧⣤⡾⠋⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡀⠀⠀⢨⠇⠀⣿⣿⣷⡄⠀⡏⢧⣠⣼⠿⠛⠛⠉⢾⠁⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠠⠖⠚⠛⠛⢛⠿⣿⣿⡟⠒⠒⠉⠀⣰⣿⣿⡇⠙⠛⠁⠈⠁⠀⠀⣤⢦⡤⠬⠷⠀⠀⠀⠂⡠⠇
⠀⠀⠀⠀⠀⠀⢀⡴⠋⡰⠋⠻⣷⣦⣤⣶⠾⠿⣯⠀⠀⠀⠀⣀⣤⣄⡀⠈⠈⢦⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣧⠀⣠⠞⢁⣠⣄⡈⠳⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠔⢀⠀⠀⠀⠀⢹⣰⠃⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⣄⡆⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠟⠛⠛⠛⠛⠛⠛⠻⢶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣠⣴⣾⣿⣥⣶⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣈⣤⣴⣶⣶⣶⣿⣶⣶⣶⣤⣄⡈⠛⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣧
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢸⣿⣿⡿⠟⠛⠉⠀⠈⠙⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠙⠛⠀⠙⠿⠈⠻⠏⢻⣿⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟⢃
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠰⠀⠀⣤⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠁⡏⠱⠆
⠀⠀⠀⠀⠘⡿⠻⢿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⠀⠀⠀⠀⠆⠀⠀⠉⠀⢀⠐⠂⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣡⡄⠀⣀⢀⡄⡠
⠀⠀⠀⠀⠰⠎⢹⡆⠉⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈⡗⠃⠺⠁⠀⠀⡄⠀⠀⠀⠀⠫⣹⠛⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⡟⠉⠉⣷⡋
⠀⠀⠀⠀⠀⠢⢀⡃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠀⠀⠈⠁⠀⠀⢲⠀⠀⡑⢤⡀⣰⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠟⢈⠔⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⠀⠀⠀⠀⠀⢨⠀⠀⠈⢺⠋⠉⢻⣿⣦⣤⣤⣤⡀⠀⠀⣠⣤⣴⣶⣿⠿⠛⠁⣠⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⡡⠂⠘⠣⠀⠈⠟⠻⢿⢇⣀⣸⠿⡿⠛⠉⢁⣠⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⢀⠀⠀⠀⢀⣀⣀⠀⠄⡂⠔⠈⢀⠔⠈⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⢀⠇⡀⠄⠀⠅⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠀⠀⠀⠀⠅⠀⡀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢡⣀⠔⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠈⡠⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⢀⡀⠠⠐⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠁⠂⠘⠋⠁⠐⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣄⣀⠀⠀⠀⠀⠀⠘⠲⠀⢚
⠀⠀⠀⡀⠀⡔⡐⠉⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣠⢊⠅⠓
⠀⠀⠀⢉⠁⡞⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⠈⠁⠀⢹⠉⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠐⠈⠀⠀⠠⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂⢉⢶⡰⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⠿⠿⢿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠈⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⢀⣴⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣷⡈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⢻⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⢠⣿⠏⠀⠀⠀⠀⣀⣤⣤⣴⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠿⠷⠶⣤⡀⠀⠹⣿⡄⢻⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⢸⠃⠀⠀⢀⣴⡿⠟⠋⠉⠀⠀⠉⠙⠻⣶⡄⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠈⠙⣦⡀⠙⣧⠀⠹⣿⣧
⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠁⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠈⠀⠀⢹⣿⣧
⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠈⣿⣿⡄
⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⣷⣄⡀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀
⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠻⣿⡟⠻⠶⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇
⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣤⡤⠶⠋⠁⡾⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡞⠁⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣦⣄⡀⠀⢀⣠⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠙⠛⠛⠛⠋⢩⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⢷⣀⣰⣏⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠐⠛⠛⠛⢛⣛⣛⠛⠋⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⡀⣸⣇⣀⣀⣤⣤⣶⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠠⠠⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠆⠰⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣀⠀⢀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⠐⠉⠀⠈⠋⡀
⠀⠀⠐⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠀⠀⠀⠁⠁⠒⠦⠴⢤⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠐⢿⣾⣿⣿⣿⣿⣿⣿⣾⣿⡶⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢰⣤⢬⣉⡅⡉⠽⠉⠿⠯⠶⠷⠳⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠘⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠤⠽⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠄⠀⠀⢤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢄⠀⠁⡀⢠
⠀⠀⠀⠐⢪⣷⣻⠯⡑⠀⠳⠟⣓⡃⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⠹⠧⠃⡁⠤⠚⡤
⠀⠀⠀⠀⢸⡋⠚⠧⢵⠒⢲⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡜⣒⠙⠉⣀⢐⠸⡆⣀⡚⡀
⠀⠀⠀⠀⠀⣷⠋⠠⢈⣵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡟⠻⡶⢄⠻⣯⠑⣾⡟⢘⡟⣾
⠀⠀⠀⠀⠀⠘⣆⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⢛⢯⠟⣾⣠⣿⣯⣽⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠁⠚⠾⠉⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⡈⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢈⣀⣀⡀⠀⠀⠀⠈⠷⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠁⠀⠀⡠⠤⣄⠀⠾⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⣠⠀⠀⠀⢀⠎⠀⠀⠀⠑⣄⠒⠺⣄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⢊⠏⠁⠀⠠⠊⠁⠀⠀⠀⠀⠑⠈⢢⠀⢈⣧⠀⠀⠀⢀⠜⢁⡠⡙⢆⡠⢔⠂⠉⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡠⠀⡇⠀⠀⣀⣀⡀⣠⠒⠐⠀⢣⢀⠊⠀⠀⡠⠃⠴⠁⠁⣸⡤⠒⠁⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢘⠁⠀⢃⢰⠋⠀⠀⠈⠉⠁⠀⠀⠃⢺⣆⠀⡔⢡⠊⠀⣢⠴⡇⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⢄⣈⠉⠑⢄⡄⠄⠀⠀⠠⡶⠄⣀⠠⠃⠀⠓⠊⡔⡁⠖⠊⠁⠀⠈⢳⣈⣁⠀⠀⢀⡄⠈⠄
⠀⠀⠀⠀⠀⣀⣀⡀⡇⠀⠠⠒⠲⠤⣭⣐⡚⠷⠀⠀⠉⡉⠀⠀⠀⠈⠐⢄⣠⠊⠁⠀⠀⠀⠀⠀⠀⢀⡴⠁⢠⠊⠁⢁⠀⠠
⠀⠀⠀⠀⠘⠁⠀⠐⠛⠒⠲⢤⣤⣀⣆⣀⣸⣑⡒⢤⣔⠀⠀⠀⠀⠀⠀⣸⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⢱⠔⠋⠀⠀⠈⡆⠀⠇
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⣀⠕⠀⠀⠀⠀⠀⠀⣿⣿⣄⣀⣀⣀⣠⣴⡿⠿⢿⡃⠀⠀⠀⠀⠀⠀⡘⢈⡠⠒⠁⠈⢣⠜⠀⡰
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣹⠉⢆⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠛⠛⠛⠁⠀⠀⠀⠈⡆⠀⠀⠀⠀⢠⠃⠉⠀⠊⠉⢁⠆⢀⠜⠁
⠀⠀⠀⠀⠀⠈⡆⠠⡤⠭⡀⠉⢳⡳⡀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢁⣀⡀⠀⣀⡔⢁⣀⣀⡀⣤⠎⣠⠣⢄⡠⠔⢋⡔⠁
⠀⠀⠀⠀⠀⠀⢡⠀⢇⠀⠘⠤⠜⠣⡈⠢⡀⠀⠀⠀⠑⣄⣀⣤⠚⠁⠀⠉⠙⠛⠡⠭⢀⡠⠖⠁⡰⣁⠔⠉⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠈⠆⠈⡆⢰⠚⢉⢑⣛⠂⠈⠲⢆⣛⣉⡱⠚⢀⡔⠀⠀⠀⠀⣀⠝⠓⠲⡤⠴⠚⢏⢀⡠⠚⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠙⢄⠘⠂⠀⠀⡝⠒⠤⠤⠥⠤⠒⠁⠈⠉⠉⢏⠁⠱⡤⠦⠴⠁⠀⠀⠀⢣⠀⠀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠅⢂⡀⠚⠢⠤⠤⣀⣀⣠⠒⠒⠢⠤⠖⠋⣄⠠⠋⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⡐⠁⠈⢢⣀⣀⡤⠒⠀⡖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠠⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⡰⠁⠀⠀⢠⡇⣇⢘⠒⢤⠆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢲⠃⠀⠀⠀⢸⡇⡏⠁⠀⡸⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁⠀⠀⠀⠀⢸⢽⠁⠀⢠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⢀⠤⡀⣸⣿⡿⠀⢠⠃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⡤⠤⠔⠒⠊⢫⠀⢆⠀⠀⢀⠔⠁⠀⢳⡜⢿⠃⡠⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢣⣈⡦⠒⠁⠀⠀⠀⠈⠧⣏⠔⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠑⢄⠀⠀⠀⠀⠀⠐⠚⠉⠱⡀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣠⢔⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠻⢿⣟⣛⠿⢿⡿⣿⣭⣭⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⢘⡏⠒⠀⠈⡉⣉⣉⠛⠺⠿⠿⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠄⠀⠠⠤⠤⠄⠒⠒⠀⠀⠀⠀⠉

⠀⢀⡀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠤⠄⡀⠀⠀⠀⢀⡀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⠄
⠀⠘⠀⠀⠀⠀⠀⠈⢄⠀⠰⠁⢀⠀⠉⠆⠀⠀⠌⠠⠊⠠⠀⠟⠊⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠈⠀⠡⣈⢀⡄⠐⠰⣲
⠀⠀⠐⠀⠐⠀⠀⢉⠣⠀⠀⠣⡀⠈⠀⠈⠆⠈⢠⠴⡀⠊⠀⠀⠀⠂⠀⠠⠀⠄⠀⠁⠀⠀⡠⠁⠀⠒⠄⠀⠉⠉⠀⢐
⠀⢀⠀⣋⡀⠀⠀⠀⠈⣀⢀⢀⠈⠒⠈⠄⢀⡴⠀⡯⢚⠁⠀⠀⢠⠐⠂⡐⠠⢀⢀⠀⢀⠜⠀⢣⢄⣀⡀⠀⠀⠀⠤
⠀⠘⠀⠀⡆⠀⡠⠊⡁⠀⠃⠂⢌⡑⢄⠀⠸⠋⠀⢸⡵⠀⠀⢸⠃⠀⢀⡅⡀⠘⠄⠐⠁⢀⠀⠠⣀⡀⠀⡀⠠
⠀⢠⠖⢦⠀⢄⣠⠒⡂⢜⣊⠼⠓⢊⠁⢠⠒⠒⠤⡀⠃⠀⠀⠀⠁⠨⠉⠁⠀⠀⠀⠀⠠⠂⢤⠀⠠⠀⠀⠁⠀⠈
⠀⠀⡆⣈⢂⠀⠀⠀⠀⠒⠒⠒⠈⠁⠀⡌⠀⠀⢰⠀⠰⢠⠀⠀⠂⠆⠀⢀⣴⣶⢻⣟⣳⡆⠀⡀⠀⢀⠀⢀⡰⠉⡘
⠀⠀⡄⠂⢠⠀⠀⠀⢖⠒⠀⠠⠀⢲⠆⠑⠀⠲⠈⠀⠀⠁⠃⠀⠀⠀⠀⢀⢀⠽⠉⠉⠉⠉⠀⠀⡀⠐⠀⠀⠉⠉
⠀⠀⠈⠉⠁⠔⢄⠀⡄⠉⠀⠒⠀⠁⢀⠀⣴⡢⡄⠀⠠⠀⠀⠀⢀⠖⡆⠀⠀⠀⠀⠍⠀⠁⠀⡀⠃⠀⠀⡔⠉⢆⠁⢠
⠀⠦⠀⠀⠈⠀⠀⠀⠀⢀⠀⢈⠈⠡⠏⠓⢉⠠⠣⢶⠠⢅⢤⡤⠀⠀⠀⡀⠈⣁⠀⠈⠈⠀⠂⠀⠀⠀⠨⠀⡠⠊⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠀⠀⠂⠈⢀⣐⣃⡠⠤⠗⢉⠀⠀⠀⠀⠰⠀⠀⡀⠼⢑⠀⠉⡂⠲⢀⠑⢀⡱
⠀⠀⠐⠀⠘⢀⠀⠀⠀⠀⠀⠤⠀⠀⠘⠀⠈⠉⣤⠀⠀⢘⠭⠁⣀⠀⠐⣁⠀⠀⠀⠁⠀⠀⠀⠀⠀⢈⠀⠀⠀⠎
⠀⠐⠂⠀⠀⡠⠄⠀⠀⠊⢟⠀⠀⢠⠶⠛⠉⠐⢹⣶⠄⠈⡠⠀⠀⠀⠀⠈⠀⢠⠄⠀⠀⠀⠀⠀⠀⣡⠀⠀⠘⠀⠂⠄
⠀⠰⠠⠍⢀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⢈⠠⠒⠀⠀⠀⠀⠉⠉⠉⠈⠐⠡⠄⢀⣀⠀⠂
⠀⢠⠀⠀⡀⠄⠈⡆⡕⠀⠈⠀⠐⠀⠀⠠⠒⠃⠀⠀⠀⢰⠀⠀⠀⠌⠀⠀⠄⠠⠏⠀⡀⠀⢀⠠⠀⠀⠙⠇
⠀⠀⠀⡀⢀⡄⠀⢡⠁⢀⡀⠌⠀⡄⠂⠂⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡰⠒⠀⠀⠒⢅⡀⢀⠰
⠀⠠⠐⠊⠸⠈⠆⠈⢀⠡⠀⠀⠦⠠⢖⠀⠀⠀⠈⠁⠀⠈⠀⠐⠀⡄⠀⢠⠂⠉⠀⠀⠀⠈⠀⢀⠜⡀⠀⠀⢻⠧⠂⠁
⠀⠀⠠⠨⠄⠀⠂⠀⠀⠒⠈⠀⠀⠀⠀⠄⠀⠀⠀⠠⠒⠃⠀⠀⣀⠀⢌⢀⠢⠤⠬⠔⡀⠤⠔⢊⠈⠃⡀⠈
⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠈⠠⠂⠀⣀⡀⡔⠒⠒⡆⠀⢢⠀⠐⠀⠀⠁⠀⠈⠀⠠⡈⡀⢄⠀⠠⠀⠈⠀⠀⠀⠀⠐⠁
⠀⢀⡀⠐⠀⠀⠀⠀⠂⠄⠁⠀⠀⠘⠄⠂⢘⣀⣸⠀⠀⡀⠀⠐⠄⢤⠆⠀⠀⠈⠚⠁⠡⠊⠈⠠⠁⠀⠤⠀⠐⠀⠒⢤
⠀⠀⣠⡶⢀⡶⠀⢀⠀⠃⡀⠂⠈⠠⠀⠀⡀⠄⠠⣀⠀⠀⠘⠀⢀⠀⠀⠀⠃⠀⠠⣀⣠⣤⠄⠀⠀⠀⢄⠀⠀⠀⠀⠔
⠀⢸⣆⠅⠊⠀⡀⠁⠀⡂⠀⠀⡀⢊⠀⠌⡠⠊⢉⡉⠠⠀⠈⠠⠊⠀⠀⠀⠁⠀⢀⠜⠈⠡⡀⠀⠀⠀⡈⠀⠈⠁⠁⢠
⠀⠀⠁⠂⠀⠀⠃⠀⠀⠈⠀⠉⠀⠈⠀⠋⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⡋⣶⣳⣦⣴⢤⣶⣤⣴⡒⣀⣤⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣾⣉⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣤⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣽⣴⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⣶⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣆⣀
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠿⠛⠛⠛⠉⠉⠉⠉⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡷⠄
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡟⣶⡀
⠀⠀⠀⢀⣷⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⣀⣤⣴⣶⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣭
⠀⠀⢰⣿⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⣴⣾⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⣿⡷
⠀⢀⣾⣷⣿⣿⣿⣿⠟⠁⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠁⠀⠀⠉⠙⢿⣆⣠⣄⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣇
⠀⣿⣾⣿⣿⣿⣿⠃⠀⠀⠀⣰⣿⡿⠋⢀⣠⣤⣶⣦⣤⣄⠀⠀⠀⣾⡏⠀⠀⢀⣄⡀⠀⠀⠈⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡀
⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⢠⡿⠋⢀⣴⡿⠋⠉⠀⠀⠉⠻⣷⡄⠀⢻⣧⠀⠀⠘⠛⠃⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇
⢠⣿⣿⣿⣿⠇⠀⠀⠀⠀⠘⠁⠀⣾⡟⠀⠀⠀⣠⣀⠀⠀⠸⣿⠀⠀⠙⢄⡀⠀⠀⠀⢀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇
⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠛⠛⠀⠀⢀⡏⠀⠀⠀⠀⠙⢛⠶⣾⣿⣿⣷⣶⣄⣀⣤⠶⠖⢦⠀⠀⠀⠀⠀⠀⠸⣿⣿
⢈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢒⣽⣿⣦⡀⠀⠀⠀⠀⣠⡾⣡⠀⠀⠀⠀⠀⠘⢷⣿⡟⢧⡀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠈⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠙⠻⠷⠶⣶⣟⡋⢰⣿⠀⠀⠀⠀⠀⠀⠈⢿⣄⢸⡇⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⣿⣿
⠀⠰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡾⠋⣽⠟⢻⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⠇⠀⠀⠀⠀⠀⠚⢡⡌⠁⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠛⠋⠉⠀⠀⠻⣦⡘⣿⣇⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⡤⠶⣤⠀⠀⠀⠀⠈⠙⠻⣿⣦⣄⣀⣀⣠⣴⣿⣿⢿⠀⠀⠀⢀⣴⣶⣾⡿⠁⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⢿⡿⠿⠿⠿⠟⠋⠁⢸⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⡀⢀⣠⣤⣤⣄⠀⠀⠀⠳⡄⠀⣀⣠⣤⣤⣼⣄⣀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠈⠙⣷⣄⠀⣀⣽⡿⠟⠋⠉⠉⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⣠⠶⠶⢶⣤⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⢀⠀⠙⣿⡄⠀⠀⣀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠈⠀⠀⠦⠀⠀⢘⣺⣄⡠⢀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠄⣠⡁⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠀⠋⠀⠀⠐⠀⠀⠀⠀⠀⢀⠘⠛⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⢀⠀⠀⡆⠀⠠⠘⢥⠽⠀⠀⠀⡀
⠀⡀⠀⠐⠀⢠⠀⠒⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠠⠄⠀⠀⠀⠀⢀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠰⣷⢄⠀⠀⠂⠀⠁⠀⢠⡇
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣊⢄⠀⠀⠀⠐⢜⢃⡀⢀⣀⣐⣤⡤⢊
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠑⠈⠉⣓⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠈⠀⠀⠀⠈⠆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠄⠀⠀⢀⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⢊⠀⠀⠀⠀⡊
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢈⡠⠀⠀⠀⠉⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⠀⠀⠀⠀⠐⠊⠀⠀⠐⡀⠀⠀⡰⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠠⡀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡉⢉⡒⡄⠀⠀⠀⠐⠠⠀⠀⠀⠀⣄⡠⠉
⠀⠀⠀⠀⠀⠤⢤⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⡄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⠀⠀⠈⢶⠀⠀⠈⢱⠂⠐⠒⠒⠋⠁⠄⠀⡇
⠀⠀⠀⠀⠀⢠⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠢⣄⡀⠐⢆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⠃⠀⠘⠀⠀⡠⡧⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠉⠘⣆⡀⠀⢀⠀⠲⡄⠀⠐⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠄⠼⠀⠀⠀⠀⠀⠀⠀⢌⠢⣄⠐⠀⢀⡴⠤⢛⢽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠐⠒⠂⠈⡼⠀⠀⠀⠀⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣠⠀⠑⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠞⠁
⠀⠃⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠛⠫⠄⠀⠀⠁⠀⠐⠤⢀⣀⣀⡠⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠘⠀⠀⠘⠋⠀⠠⠀⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⣰⠀⠀⠀⠀⣠⠄
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⡒⠒⢄
⠀⠀⠀⢠⠒⡉⢀⠠⠈⠀⠄⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠄⠀⠂⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣲⣤
⠀⠀⠀⠰⠀⠂⠀⠀⠠⠀⡔⢤⠀⢸⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⢇⠀⠉⢀⡄
⠀⠀⠀⠀⢃⠀⠰⠁⠢⠔⠀⠀⠣⠴⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠠⠀⠀⠈⠉⢉⠃⠈⠢
⠀⠀⠀⠀⠀⢄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢀⡤⡆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⢔⠞⡟⠁⡧⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⠜⢡⠂⡜⠀⠀⢣⡈⠑⠲⣑⣄⡠⡀⢀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⢀⣠⠃⠀⡇⠀⡁⠀⠀⠀⠑⡄⠀⠈⣦⡱⠁⡯⠷⡔⠚⡢⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠂⢫⠀⠸⢆⠀⢇⠀⠱⡀⠀⠀⢀⣇⣀⡠⣗⣌⠉⠵⢄⠘⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠦⠁⡶⠗⡢⡀⡨⠬⡁⡲⢯⢁⠴⢅⡠⢄⢀⢥⡠⠵⢤⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢒⠖⢱⠢⠤⠩⡇⠀⢨⠀⠀⡅⠀⠠⠀⢠⢁⠜⡚⠈⠉⠉⠉⠁⠒⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⡄⠸⡀⠰⠀⠈⡄⠀⡇⠀⡈⢀⠜⠁⠀⠀⢀⠄⡔⠓⠄⡀⠁⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠰⠀⠀⣀⣰⣀⠗⠤⣇⠀⠇⠀⠁⠀⠃⡎⠀⠀⡀⢫⠛⠉⠀⠈⢑⠇⠂⢀⠀⢰
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠛⠁⠈⠇⠀⠁⠀⠀⢸⠀⠐⠀⠀⠘⠀⢧⢤⡠⢄⠀⠉⠒⡀⠈⠁⠀⣀⠇⠉⠎
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢀⣄⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠱⣀⠈⠢⠒⠐⠄⠔⠑⠊⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠤⠤⠤⠄⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⣧⠶⠛⠛⠉⠉⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⢀⣀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣀⣀⣀⣀⣠⣴⡾⠛⡅⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⢻⠛⡟⢻⢻⠛⠙⢹⡇⢀⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡶⡟⣿⢷⣤⡾⣿⠀⡇⢸⠘⡀⡇⢸⣈⣀⣠⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⣴⢿⠉⡏⢻⡇⣧⢸⢸⡈⡇⢸⣀⣷⡾⠟⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠾⠿⠟⠛⠁⠀⠀⠀⠘⠃⠀⠀⢠⣴⣿⣾⡄⢇⢸⣧⣼⣼⣤⣇⣧⡾⠛⠁
⠀⠀⢠⡶⠋⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⢸⣿⣧⣼⣼⣿⠁⠈⠙⠛⠋
⠀⢰⡟⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠙⢿⡜⢿⡏⠀⠈⠉
⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣠⢾⣷⠟⠋
⠀⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⣼⣷⢸⣿
⠀⢸⣷⢹⠛⡿⢿⡿⡷⢶⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⡿
⠀⠀⠻⣿⣄⣧⠘⡇⡇⢸⢼⣧⠀⠀⠀⠻⠇⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠈⠙⠛⠿⠿⠿⠾⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⡿⣶⣤⣤⣀⣤⣴⡿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡇⠀⡏⢹⠉⢻⠈⡇⢣⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣷⣬⣦⣼⣤⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣿⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⣀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⠉⢠⠄⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⠄⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠂⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠐⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣥⣾⣿⣿⣥⣴⣶⡤⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠈⠐⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡿⠻⢿⡿⢿⣿⡿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⢠⣶⣤⣀⣠⠀⠀⡀⠀⠀⣀⠀⠹⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠫⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠳⠛⠋⠁⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠠⠖⠀⣀⠀⠀⠀⢠⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠅⠀⠀⠀⠀⠘⢄⠀⠀⠀⡀⠀⠀⡖⠀⠀⠁⠀⠀⡀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠐⢄⠀⠀⠐⠒⠿⠛⠀⠀⡀⠀⠀⠀⡀⠀⠀⠂⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠐⠈⠑⠲⠀⠤⠤⠄⠐⠈⠀⠀⠀⢐⠀⠀⠀⠄⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠁⢀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠄⠁⠒⠂⠂⠗⠀⠰⠁⠀⠀⠈⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⡰⠊⠀⠀⠀⠀⣰⠃⠀⠉⠀⢀⠤⠀⠀⡀⠁⠄⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⢀⡒⠢⠁⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠠⠆⡴⠂⠒⠀⠈⠉⠉⠁⠀⠀⠀⠐⠀⠀⠀⠀⠂⠁
⠀⠀⠀⡈⠀⠀⠠⠀⠀⠀⠀⠄⠐⠀⣠⠦⠐⠂⠈⡠⡄⠀⠀⠀⠀⠀⠀⣀⠤⠄⣀⠠⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠌⡄⠷⢤⠤⡤⠒⠂⠹⡁⠶⠷⡖⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠀⢸⠰⡁⠠⣀⠔⠀⣰⠁⠀⠀⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⠀⠀⠀⠀⠀⠘⠤⠉⠒⠀⢤⡭⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣶⣶⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⠠⠤⠄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡉⠉⠀⢀⣀⣀⣤⡤⠽⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⠋⠡⣤⢶⣛⢉⣡⣴⣻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠉⠉⡟⠋⠑⠨⠴⡀
⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⢣⣀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⠙⠒⠲⠤⣤⣀
⠀⠀⠀⡠⠖⠋⠁⠠⢔⣮⡅⠻⢼⣽⣿⣿⡶⠦⣤⣠⣇⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣇⣭⣵⠦⣄⠀⠈⠙⠲⢄
⠀⠀⠈⠀⠀⢠⣴⣿⣛⣻⣥⡴⠶⠛⠛⢛⣻⣿⣷⡿⣯⣿⣓⣲⡦⢤⣼⣀⣀⠀⠀⡷⠟⠛⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠦⣀⠈⠉⠉⠙⠳⠀⠀⢠⣶⣿⣯⠭⠶⠶⢒⣻⠿⣋⣉⣽⠿⠛⠷⠿⠿⠽⠇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠈⠙⠒⠦⢤⣄⣀⣀⠀⠉⠁⠀⠀⠀⠀⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠒⠒⠲⠶⠶⠶⠤⠤⠶⠶⠶⠒⠒⠒⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣴⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠑⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠡⠀⠐⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⠟⠛⠛⠉⠉⠉⠉⣩⣭⣭⣭⡀⠄⠀⠀⠀⠀⠠⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣉⣤⠁⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣼⣿⣿⠿⠿⠟⠿⠟⠛⠿⠁⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀⠀⢀⠴⠁⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⠀⠀⡀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡆⠰⡄⣶⣶⣦⣳⣾⣾⣇⣷⣤⣿⣾⣴⣷⣼⣰⣀⣷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⢍⢿⣳⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣾⣀⡆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣼⣿⣮⣿⣿⣿⠿⠛⠛⠉⠉⠀⠀⢀⣀⣠⣤⠤⠤⣄⡀⠉⠉⠛⠻⢿⣿⣿⣿⣣⣠
⠀⠀⠀⠀⠀⠀⠐⢦⡐⢦⣾⣽⣿⣿⠟⠋⣡⣶⣾⣿⣿⣶⣶⣶⣶⣿⣿⠋⠀⠀⠀⠈⠙⣷⣄⠀⠀⠀⠈⠙⠻⣿⣿⣧⡄⢠
⠀⠀⠀⠀⠀⠐⢆⠐⣿⣷⣿⠟⠋⠀⣠⣾⠟⠋⠉⠉⠉⠛⠛⠿⠟⠛⠁⢀⡤⠴⠶⠤⣄⡈⠛⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣀
⠀⠀⠀⣀⡀⣙⣿⣿⣿⠟⠁⠀⠀⣰⠟⠁⠀⣠⠤⠖⠒⠦⢤⡀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠙⢦⡀⠀⣀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣆⣀
⠀⠀⢄⣤⣹⣿⣿⠟⠁⠀⠀⠀⠐⠁⠀⣠⠋⠀⠀⠀⠀⠀⠀⠈⠳⣼⠁⠀⠀⢀⣀⡀⠀⠀⠀⢿⣜⣵⡆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣣
⠀⢀⢠⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣠⣤⡀⠀⠀⠀⣹⠀⠀⠀⠸⢿⠟⠀⠀⠀⣼⡟⠉⠉⠀⠀⠀⠀⠀⠀⠐⠨⢿⣿
⠀⣀⣽⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣤⣿⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⡿⣇⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣿
⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣩⠽⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠉⠻⣶⣶⣴⣶⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠱⠀⠆⢻
⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣀⣀⣤⡴⠛⠁⠀⠀⠀⠈⠻⣦⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⠀⢀⠂⢸
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⡀⢀⠀⠀⠀⠀⠀⠹⣿⠀⢹⠀⠀⠀⠀⠀⢐⠀⠀⠄⠀⠰⠄⢈⢈⣸
⣾⣿⠠⠀⠠⠂⢀⠀⠀⠀⠀⡄⢠⡨⠃⣀⠀⠀⠀⠀⠀⢰⡏⠉⠺⡀⠀⠀⠀⠀⠀⢹⣇⣼⠀⠀⠀⢀⡄⠢⡨⠆⠒⠂⢒⠒⠀⠃⢸
⣿⡧⠀⠀⠀⠂⠀⠀⢁⠂⠀⠀⠐⣦⠂⠀⠀⠀⣀⠀⠀⠄⠹⣄⡀⢳⡀⠀⠀⠀⠀⢸⣿⠋⠀⠀⢀⡀⢐⠀⠃⢶⠀⢠⠨⠉⡀⠂⣼
⣿⡇⠐⠀⠀⠀⠀⠄⡀⠀⠀⡄⠠⠋⠐⠀⠁⢀⠈⠁⠀⠨⠀⡘⠉⢟⠻⣦⣀⣀⣀⣼⠟⠀⠆⠀⣀⢀⢀⡁⢂⠈⡆⠁⠌⠠⢀⣠⣿
⢻⣿⢂⠂⠀⠀⠀⠀⠄⠃⠁⡇⠂⠂⠀⠁⠠⠀⠄⠁⠐⢢⠀⠰⠀⢸⡐⢾⡏⢉⠛⢁⠂⠤⠀⠁⠉⡀⢈⢘⠐⣠⡃⠠⠈⠀⠀⣼⡿
⠸⣿⡄⡠⠀⠈⠀⡀⠀⡅⠀⠙⢶⣄⣀⣈⣀⠀⠁⣀⣐⣤⣤⠶⠶⠟⠛⠛⠛⠓⠶⢧⣤⣁⡝⠠⢐⣀⣤⠴⠚⠁⡙⠂⢀⠀⣰⣿⠁
⠀⠹⣷⡀⠂⡀⠀⠠⠈⠀⢀⠀⠀⢀⠉⠉⠉⠛⢛⡋⠉⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⢳⣊⠉⠛⠛⠋⠉⠀⠄⠀⡂⠈⡀⠈⣰⣿⠃
⠀⠀⠹⣿⡁⠀⠀⢌⠘⢂⢀⠀⠄⢀⠀⡐⢀⠐⠀⠀⠀⠁⠀⡈⠀⠀⠀⣶⢦⣤⣤⠾⠣⠀⡀⠔⠨⢡⠄⠂⠑⠀⠠⢈⣼⡿⠁
⠀⠀⠀⠘⢿⣄⠠⠂⠀⠄⠀⠀⠀⠀⢀⣂⠀⠄⡠⠁⠐⠀⠀⢂⠀⠕⠆⠙⢘⠉⠓⢷⣠⠂⠀⠀⢀⡐⠈⠀⢠⠁⣠⣿⠟
⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠐⡈⠁⠀⠠⠀⠀⠁⠀⠀⠀⠤⠤⡀⢨⠤⠈⠀⣤⠄⠂⢧⠄⢈⢈⡀⠀⠁⣋⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⣀⣀⣀⡘⠀⠀⡰⢀⠀⠁⠠⠐⠠⠂⠀⠂⠌⢩⠀⡨⠠⢸⣈⠘⠀⣀⣬⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣷⣶⣦⣤⣤⣬⣄⣠⣠⣀⣀⣤⣦⣥⣦⣾⣶⣿⠿⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠂⠀⠀⠉⠉⠀⢠⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠉⠉⠀⢠⡄⠀⠀⠉⠉
⠐⠄⠄⠀⠀⠀⠀⠀⠀⡤⠔⢇⠀⠀⠘⠀⠀⠀⠀⠀⠐⠄⠄⠀⠀⠀⠀⠀⠀⡤⠔⣇⠀⠀⠘
⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠎⠀⢀
⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠄⠈⠉⠁⠀⠀⠀⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠄⠈⠉⠁
⠀⠀⠀⠘⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠋⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠘⠄⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⠄⠀⣀⠆
⠂⠀⢐⠘⠙⢢⠄⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙⢢⠆⠀⢘⠘⠉⢢⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙
⠀⠀⠈⠓⠒⠉⠀⢠⡄⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠒⠉⠀⢠⡄⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠
⠀⠀⠀⠀⠢⠀⠀⠉⠉⠀⢠⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠉⠉⠀⢠⡀⠀⠀⠉⠉
⠐⠄⢄⠀⠀⠀⠀⠀⠀⡬⠔⣃⠀⠀⠘⠀⠀⠀⠀⠀⠐⠄⠄⠀⠀⠀⠀⠀⠈⡭⠔⣂⠀⠀⠘
⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠆⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠂⠀⠀⠓⠲⠊⠀⢀⠀⠄
⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠒⠀⠀⠀⠈⠙⠂⠀⠀⢀⠄⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠊⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠖⠋⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⠄⠀⢀⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠄⠀⢀⠆
⠀⠀⢈⠘⠙⢢⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙⢀⠄⠀⢐⠘⠉⢢⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⠰⠙
⠀⠀⠈⠑⠒⠉⠀⢠⡄⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠒⠉⠀⢠⡄⠀⠀⣿⡄
⠀⠀⠀⠀⠀⡀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣤⣴⠆⠀⠉⠀⠀⠀⢠

⠀⠀⠀⢀⣀⠤⠔⠊⢉⠉⠑⢢⡀
⢀⠔⠀⠀⡀⠠⠈⠁⠠⠀⠂⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢢
⡏⠀⠈⠀⠁⠀⠀⠀⢀⠀⠀⠄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⡁⠀⠈
⢃⠈⢀⠀⠁⠀⢁⠀⠀⣀⠀⠀⢠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠼⠁⠲⠏⠘⡆
⠈⡆⢀⠀⢀⣀⠀⠀⡘⠁⠱⡀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠂⠀⠹⢄⣀⣀⡴⠣⡀
⠀⠘⡈⠀⡏⠀⠉⠊⠁⠀⠀⠙⠒⠃⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠂⠀⠀⠂⠀⠀⠄⠀⠀⠈⣆
⠀⠀⢡⡀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣀⣤⣶⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠈⠘⡄
⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣀⢔⠞⡿⠁⡿⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡱⠋⡜⠀⠀⢃⠙⠲⠧⣠⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡸⠁⢰⠁⠀⠀⠈⠳⡄⠀⠀⠑⢟⠲⣉⠆⡔⢦⡀⢀⣠⡀⠀⡇
⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⣀⡠⠴⡾⠀⠀⡇⠀⠸⡀⠀⠀⠀⠀⠘⡄⠀⠀⢸⠖⣏⣀⡏⠉⠙⢍⢀⠝⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠉⠹⡀⢀⠟⢄⡀⢱⡀⠀⠱⡀⠀⠀⠀⣠⣇⣀⠤⢺⡵⣤⠤⠜⠶⡒⢚⠏
⠀⠀⠀⠀⠀⠀⠀⡠⣖⡊⠭⣶⣓⡚⢄⡀⣨⠭⡍⢒⡶⢍⠑⡲⢭⡁⡠⢄⠀⣨⣄⣀⡱⣤⠤⡸⠃
⠀⠀⠀⠀⠀⠀⠚⠠⠤⢤⠤⢳⢄⣈⣑⣙⡅⠀⠈⡋⠀⠀⡙⠀⠀⡍⠀⢈⠏⢀⡯⠗⠒⠚⡉⢑⠒⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⡀⠈⡄⠀⢣⠀⠀⠇⠀⠀⡇⠀⠀⡇⠀⣸⠖⢁⠀⠀⠀⠁⠀⡠⡀⠈⠀⠄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢱⠀⢱⠀⠈⡆⠀⢸⠀⠀⡇⠀⢰⠀⡰⠁⠠⠀⠀⡄⣰⡩⠮⠤⠬⣐⡆⠐⠀⠀⠈
⠀⠀⠀⠀⢀⠠⠄⣠⣇⣀⣀⡠⠤⣧⡔⠋⠒⢼⠀⠈⡄⠀⡇⠀⣘⢠⠃⠠⠐⠀⠀⢷⡁⠀⠀⠀⢀⡠⠃⠡⢀⣀⡀
⠀⠀⠀⢠⠃⠀⠀⠘⠃⠀⠈⢂⠀⠈⠀⠀⠀⢸⡇⠀⠃⠀⠂⠀⠃⠘⣤⢶⣀⠤⡀⠀⠈⠹⢉⠉⠅⠂⠈⣀⡜⠀⢈
⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⠜⠀⠉⠉⠉⠉⠉⠉⠁⠈⠣⡀⠀⠱⠤⠔⠲⣌⣀⠖⠒⠔⠁⠀⡠⠊
⠀⠀⠀⠀⠈⠒⠤⢀⡀⠤⠊⠉⠒⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⡀⠀⠀⠀⠀⣀⡠⠔⠊
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠔⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣿⣇⣀⣏⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢨⠇⠀⠀⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣏⣌⠥⠀⠈⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠈⠀⢸⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⣀⣠⡤⠀⠀⠀⢀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣀⣤⣤⣶⣾⣿⣿⡿⢃⣗⠀⠄⢼⡄⢃⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠀⢀⣄⣾⠀⠈⠁⣤⡀⣀⡭⠴⣙⡻⣿⣿⡿⡟⢿⠿⣿⡟⠋⠁⢸⣿⠀⠂⣿⡇⢠⡄⠈⠑
⠀⠀⠀⠀⠀⠀⢀⣿⡀⣼⠿⠓⠁⡠⠟⠪⠴⠀⠩⠈⢁⣄⢂⡡⠈⠉⠉⢠⠀⢀⡄⠁⠊⠲⠄⢄⣠⠈⠙⠛⠻⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠸⡟⠁⠡⠀⠀⢊⠑⢠⡆⡀⠠⣀⠀⠈⠁⠀⠀⡠⠐⠈⠀⡄⢸⣇⠀⠀⠀⠀⠀⠘⠩⢂⠀⠀⡉
⠀⠀⡀⠀⡀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠸⡧⠓⠀⡀⠀⠀⠀⣸⡆⣀⠄⠀⣰⣷⣾⠿⠄⠲⢂⠀⠂⠀⠄⢸⠀⠀⠇
⠀⠀⠃⠠⠀⠀⠤⠈⠐⠒⠀⠾⢿⣿⠿⠆⠀⠈⠉⠀⠚⠃⠾⢧⠈⣉⠈⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠁⠲⠄⠀⢰⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠒⢂⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠸⡇⠹⣆⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⢠⣞⣣⠶⣤⠀⠀⠀⠀⠀⠈
⠀⢀⣤⣤⡄⠐⡄⣤⢰⠀⠀⠁⠶⠀⠂⣈⠀⠀⠀⠀⣧⠀⣷⠀⠙⡆⠀⠐⠉⢃⠀⡀⠈⠀⠀⠀⣴⣻⢞⡽⣻⠎⠀⠀⠀⠀⠀⠀⣤⠚⣶
⠀⠀⠀⠀⠀⠀⠓⠃⠘⠀⠀⠈⠛⠋⠛⠉⠀⠀⠀⠀⢹⣁⠀⢂⡼⠃⠀⠀⠒⠊⠉⠀⡂⠀⠀⠈⠐⠫⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠿⠵⠋
⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⢀⣤⣤⣠⣦⣤⠀⠀⠀⣀⠐⠄⣈⠀⠀⠀⠀⠀⢠⣄⣠⣄⣤
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⡒⠦⠀⠀⠀⢠⣤⠙⠃⠀⡼⠂⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⣴⠆⠀⠑⢿⡄⠀⠀⠘⠛⢿⣿⠋⠁
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⣤⠐⠂⠛⠁⠠⠀⠀⠀⢀⠈⠀⠔⠊⠀⠀⠀⠀⠁⠀⠀⡀⠀⣦⠀⠠⠟⠃⠐⠛⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠈⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠛⠛⠛⠛⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠐⣶⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠚⠀⠁⠒⠀⠀⠀⢤⡀⠈⠹⠇⢤⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣀⡀⠀⠈⠿⠀⠀⠀⠀⠀⠠⠁⠀⠀⡙⣷
⠀⠀⠀⠀⠠⣶⣤⣤⠀⠀⢾⡀⠁⠀⠀⠀⠰⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⠀⠀⠀⠠⠞⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁
⠀⠀⠀⠀⠀⠿⣮⠗⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠘⠃
⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⢠⠦⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠉⠛⡂⠀⠀⠀⠀⢰⠀⢃⣠⠖⠀⡀⠀⠀⠄⠀⠰⠀⠀⠀⣴⣶⣶⣶⡶⢤⣀
⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⡣⠀⠠⢤⡄⣤⠖⠉⠀⠀⠀⣅⢠⠄⠀⣀⡀⠀⠀⠀⠳⢄⠀⠀⡴⠖⠋
⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠹⠁⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠡⡄⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢺⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⢾⡿⠁⠀⢀⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣰⠀⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠐⣄⠁⠠⣀⡀⠀⠀⠀⠀⠀⠀⠐⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠰⠀⠀⠐⣿⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠈⠁⠀⠀⠀⠀⢀⠈⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⣷⢄⡇
⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠄⠀⠉⠑⠀⠄⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠰⡇⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⣀
⠀⢆⣀⣀⠀⠀⣀⣠⠀⠀⠸⣿⣍⣶⣍⣻⡀⠀⠀⠀⠰⠒⠆⠀⠀⠀⢀⠉⠉⠀⠀⠉⠉⡀⠀⢀⡜⠁⣴⣴⣀⡶⢤⡀⢠⣾⣛⢛⣿
⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢶⣤⣄⣤⠶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠠⠀⠃⣜⢁⣸⡧⡴⠃⠀⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣶
⠀⠀⠀⠘⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢤⣤⠀⠀⠀⠀⠘⢿⡿⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡆⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⡄⠀⠀⠀⠀⠀⠂⢠⠁⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠄⣀⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠙⠀⠀⠠⠀⠀⠀⠘⠳⠶⠶⠟⠉
⠀⠀⠀⠀⠄⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⢀⠒⠀⢀⡇⠀⠀⠀⠀⠀⠤⢤⠐⠂⠀⠀⠀⣶⣦⠙⢦⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠾⠷⠂⠐⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠉⢠⡜⠛⣅⠀⠀⠉⠉⠉⠀⠀⠃⠀⠀⠀⣷⡾⡿⠻⣆⠀⠀⠀⠀⠀⣾⢀⡀⠟⠀⠀⠀⣤⣤⠀⠀⢀⠀⠀⠀⢀⠸⠋⠙⠉⠙⠁
⠀⠀⠀⠉⠀⠉⠂⠉⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠳⠊⠐⠓⠀⠛⠁⠀⠀⠠⣅⠈⢨⠅⠀⠀⠀⠀⠉⠠⣦⠆⠻⠃⠀⠀⠘⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠓⠂
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢃⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⢀⠤⠀⠀⣤⠀⠀⠀⠀⠀⣤⣤⡀⠀⡀⠀⠀⠀⠿⡇⠀⢀⣠⠤⢄
⠀⠀⠀⣠⣶⣶⠀⠀⠀⠀⠠⠀⠀⠘⠇⠀⠀⠀⣴⠾⠟⠉⠁⠀⠐⠀⠀⠁⠀⣘⣦⡄⠀⠀⠀⠀⠀⠀⢠⡀⣠⠡⠀⠀⠀⠁⠐⠀⠀⠿⠁⠟
⠀⠀⠀⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠟⠁⠀⠀⢈⣿⡷⠒⠾⠟⠂⠀⠀⠀⠀⠀⡞⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣦⣶⣴⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠻⠛⠛⣿⠋⢻⠛⢛⡛⠟⣛⣿⢛⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣾⣿⣶⣾⣿⣷⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣦⣀⣉⠞⢾⣋⠚⣴⠈⢵⡛⣿⡿⣿⣿⣿⣽⣯⣯⠉⢀⠸⡃⠶⠈⠷⠈⠃⠙⠻⣷⡦⠟⠀⡞⠁⢸
⣿⣿⣿⣿⢿⣶⠪⣁⡀⠉⠑⠀⠋⡛⣝⣿⣿⣟⣟⡶⡿⡳⠉⢨⣧⣤⣅⠂⠀⠀⠀⠀⢿⠀⣈⠂⡄⠔⢋
⣿⣿⡛⣛⡷⢻⡟⣛⡛⠋⠀⠀⠠⣟⠓⠈⠉⠈⠉⠉⠃⡀⠘⠏⢙⣟⠟⠄⡘⠠⠀⠑⠀⠀⠀⣤⡾⠖⠛
⣿⣿⣿⣿⣷⣿⣛⣫⣵⢾⠣⣤⠽⢁⡀⠀⠀⠀⠀⠀⠀⠁⠠⠎⠈⠁⠀⠲⢀⣄⢀⠒⡗⠀⠀⠘⠁⠀⠒
⣿⣿⣿⣿⣿⣿⣿⣿⠗⡞⠉⠁⠀⠈⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢸⣦⡇⠀⠀⠤
⣿⣿⣿⣿⣿⣿⣿⣧⣴⣷⣿⢃⡀⢀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⢠⠀⠀⢟⡹⡳⢿⢟⣯
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠲⠄⠀⠀⠀⠂⢀⠀⠮⠀⠀⢛⡠⢼⠀⠀⠀⠀⠀⢶⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠶⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠖⠣⡄⠀⠀⠀⠀⠀⠀⠚⣫
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣞⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠛⢠⡀⠰⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠂⠀⠀⠈⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⢠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠛⡷⠞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⠤⢀⠈⠀⠀⠀⣀⡋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⢫⣾⣤⣀⢳⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⣶⡟⢺
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⣠⣤⠀⠸⠇⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⢭⣁⣷⣏⣻⡧⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠟⠀⠀⠀⠀⠀⠀⡲⠶⡛⣭⡿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢃⡤⠍⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠠⡀⠀⠀⠀⠀⢀⣴⣮⣤⣷⣶⣄⠀⠀⠀⠀⠀⣠⠦⠀⢤⠌⣄⠀⠀⠀⠀⢀⢖⣦⣾⣿⢦⣿⣇⠢
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⢿⣿⡆⠀⠀⠀⢸⠀⠁⡌⠤⠡⠐⠀⠀⠀⠀⢤⣿⣿⣧⣀⣀⣸⣿⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠡⣿⣿⣿⣿⣿⣿⣿⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠠⠄⣀⠊⠀⠀⠀⠀⠀⠂⠦⠤⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⡀⠆⠀⠀⠀⠀⠀⠀⠐⠊⠛⠋⠀⠉

⠀⠀⠀⠀⠀⠀⡐⠋⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠁⢂⠀⠀⠀⢀⠄⠠⢀⠀⡠⠐⠂⠄⠀⠀⠀⠀⠀⠀⢀⣴⣶⡖⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢤⣀⣀⣀⣀⣀⡤⡎⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⢀⠀⠀⠀⠀⠀⣛⡀⠀⠙⠉⣀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⢠⠀⠀⠀⠀⠀⠀⠏⡛⠶⠶⠷⠶⠖⢚⠑⠀⠀⠀⠀⠀⠢⢀⡘⢢⠐⠂⠀⠀⠀⠀⠀⣯⣭⣙⣛⡒⠶⠦⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠒⠋⠀⠀⠀⠀⠀⠁⠐⠀⠉⠍⠉⠛⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠄⡀⠀⠀⠀⠀⢀⣶⣿⠟⢻⣿⣿⣆⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣠⠤⣄⠙⣦
⠀⠀⠀⠀⠐⢁⣀⠀⠀⡀⠀⠀⣈⠄⠀⠀⠀⢸⠉⠛⠀⠀⠋⠀⣿⠀⠀⠀⠀⠀⠻⢿⣿⣶⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⡇⠸⡇⠄⠸⢀⡿
⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠠⡜⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠘⢦⣙⠳⠶⠛⠁
⠀⠀⠀⠀⠠⡀⠀⠆⠀⠀⠀⠰⡠⠁⠀⠀⠀⠀⠀⢄⠈⠉⢑⣆⠀⠀⠀⠆⠀⢃⡀⠀⠀⠀⠀⠀⣀⡸⠀⠐⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠈⠂⠤⠠⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠅⠀⠀⠀⠀⠒⠤⢈⠉⠉⠉⠉⠉⠁⠠⠀⠁⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌
⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢨⡙⢿⣿⣿⣿⣿⡧⠸⠅⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣬⠀⠀⠀⠀⠀⠀⠀⠄⠢⠐⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠂⠙⢿⣿⢋⠂⠈⠀⠀⠀⠀⣠⢶⠾⡿⣿⢿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣭⡌⠀⠀⠀⢀⠴⠚⠒⠒⠶⣄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⣯⣯⣽⣿⣯⣟⣻⣻⣿⣿⠟⠀⠀⠀⢀⡀⠀⠀⣬⣤⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠙⠋⠙⠉⠋⠈⠀⠀⠀⠀⢂⣼⣀⣠⣤⡌⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢐⠇⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠘⠣⠹⠧⠐⠀⠀⠀⠀⠀⠘⠤⠠⠀⠀⠀⠼⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⡉⠈⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣣⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢽⡓⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡖⠒⢲⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠉⢻
⠀⠀⠀⠀⠀⠀⡠⠟⢤⡀⢹⠀⡜⠀⡸⠙⢆⠀⠀⠀⠀⢠⣄⠤⠖⠢⡀⠀⠀⠀⠀⠀⠀⣧⠈⠓⢾
⠀⠀⠀⠀⠀⢠⠃⢀⣀⠱⣜⣴⣁⠔⠁⣠⠎⢧⠀⠀⠀⠈⢦⢀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡟⠓⢦⣸
⠀⠀⠀⠀⠀⠸⡎⠁⢀⡽⠋⣿⠯⣐⣉⡠⠤⢺⠀⠀⠀⠀⠀⣳⣾⣎⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠙
⠀⠀⠀⠀⠀⠀⠳⣀⠎⠀⠀⡇⢆⠀⠀⠀⣠⠃⠀⠀⠀⢠⡋⠙⠀⢨⢷⠀⠀⠀⠀⠀⠀⢠⠀⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣧⣀⣱⠶⠚⠁⠀⠀⠀⠀⡟⠋⠠⡦⠀⣀⡇⠀⠀⠀⠀⠀⢸⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢹⠀⠀⠀⠀⠀⠀⠀⠀⡷⠆⢀⣄⠀⠉⡇⠀⠀⠀⠀⠀⢸⣤⡀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⣀⠐⠶⡇⠀⠀⠀⠀⠀⠸⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⢀⣀⡀⠀⠀⠀⠀⢻⣁⠀⡉⢀⣼⠁⠀⠀⠀⠀⠀⠸⠶⣄⡀⠱
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⢾⣶⣖⣉⣠⡼⠀⠀⠀⠀⠈⠫⢤⣫⡴⠃⠀⠀⠀⠀⠀⠀⠀⣆⡀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⣹⠛⡷⢄⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠷⡄⠀⠀⠀⠀⠀⠀⠀⡇⠙⠢⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢹⠀⡄⠀⠙⠂⠀⠀⠀⠀⠀⢀⡇⣀⠤⠼⠀⠀⠀⠀⠀⠀⠀⢸⠦⣄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣤⠤⠼⠆

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣻⣿⡿⢿⢿⣿⣛⠻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇⠉⠈⣱⡟⣯⣾⡇⡞⣼⢡⡾⢶⡤⠀⠀⠁⡠⠹⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠿⠛⣻⣿⡿⠛⠀⠀⠀⠀⣿⣇⣿⣹⣹⣻⢃⡾⢁⡾⣷⡄⠀⠀⠀⠀⠀⠘⠻⣿⣿⡛⠻⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡶⣶⣾⣏⠁⠀⠀⠠⠀⠁⠀⣿⣿⣷⣿⡷⣣⢏⡴⡛⣼⣿⡇⠀⠀⠀⠑⠀⣀⠀⠀⣉⣿⣶⢶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠇⠀⠀⠛⠀⠀⠀⠈⠢⢤⣀⣼⣿⣿⣿⣯⣾⣷⢞⣵⠿⣿⣥⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡄⠀⢿⣿⣿⣿⣿
⣿⣿⣟⠋⢀⣔⠋⠉⠈⠉⠉⠉⠙⠻⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣾⣿⣿⣭⣿⡀⣤⣶⣄⡀⠀⠀⠑⠀⠀⠈⠉⢳⣄⠈⢛⣿⣿
⣿⣿⠗⢃⠛⠋⠀⢠⠂⠀⠀⠀⠀⠀⠛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⠃⢀⣿⠹⣿⣷⣦⡀⠀⠀⠀⢢⠀⠈⠛⢅⠛⢿⣿
⠿⣧⣴⣇⣀⣀⠊⠀⠰⣤⣦⡀⢫⠆⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⠁⠀⣼⡟⢸⣿⣮⣝⠿⣦⡀⠀⠀⠈⢢⣀⣀⣷⣼⡿
⠀⠁⠀⠉⠙⠁⠀⠀⠀⡘⠛⠁⠀⠀⣠⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢤⣾⣿⣇⣾⣿⣽⣷⠷⣾⣿⣷⢄⡀⠀⠙⠉⠁
⠄⠀⠀⠀⠀⠀⠀⣠⣶⢛⣠⣴⣾⣿⣿⣷⠻⠛⠛⠻⣿⣿⣿⣿⣿⡏⠀⢸⣿⠏⢾⣿⡿⠟⠉⠀⣀⡀⣻⣾⣿⣦⡀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⣴⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠸⣿⣿⣿⣿⣿⣿⣤⡾⢃⣴⣿⣿⣧⣴⣾⣾⣿⣿⣿⣿⣴⠄⢿⡆
⠀⣀⣠⡾⠟⠛⠛⠉⠉⠉⠉⡿⠛⠛⠻⠿⡟⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⡟⣰⣶⠾⣷
⠀⢇⠑⠛⣒⣀⣠⣁⡀⠀⠘⣿⣿⣶⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣷⣾⣿⡀
⠀⢸⣦⣋⣯⣽⣶⣿⣶⣖⣀⣤⠉⠛⠹⣷⠀⠀⠀⢀⣸⣿⣿⡇⢼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣟⣴⣿⣿⣿⣅⣄⡀
⠀⣸⢿⣿⣏⣀⣀⣼⣟⣋⣿⣿⠀⠀⠀⢿⢧⠀⢼⣿⣽⣿⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣴⣼⣷⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠻⠿⠾⠿⠛⠛⠛⠻⠿⠿⠷⠆⠀⠀⠈⠁⠁⠀⠈⠉⢩⣿⣯⣯⣿⣿⡇⠀⠀⠈⠹⣿⣿⣿⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠁⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡏⠘⣯⡁⠂⠀⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠣⡀⠈⠙⠲⢤⣀⠐⡸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠀⢱⣤⠀⢀⠀⠹⣿⣾⠇⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⢀⠤⠐⠀⣠⠃⢈⠤⠀⣀⣈⡀⠐⢿⡇⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠒⠒⠛⠂⠙⠱⢂⠋⠁⣡⠶⠋⠀⠈⢢⠀⠀⣩⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⠃⠀⠀⠀⠀⢨⣷⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠤⠴⠤⠞⠃⠀⣼⠀⠀⠀⠀⠀⣸⠁⢸⣄⠈⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠚⣁⣀⣠⡤⠖⠋⠁⠀⢠⡧⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⢦⣶⠖⠒⠶⠒⠒⠉⠓⠀⠀⢀⡴⠞⠋⠉⠉⠁⠀⢀⣀⣤⣴⣿⣿⣛⡳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠉⠢⢤⡀⠀⠀⠀⣠⣶⡛⠁⠀⠀⣀⣀⣤⣴⣾⠿⢟⠭⠋⡙⡉⡍⢐⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠀⠀⠀⠀⠉⠋⠛⠛⠛⠛⠟⡛⠚⡛⠟⠻⡏⢍⠁⠀⠈⠀⣾⣶⣾⣷⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠹⡶⣤⣄⣠⡄⠀⣀⠀⣀⢀⣄⣀⢡⣈⣠⣇⣴⣥⣼⣿⢷⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢧⢸⡆⠸⡏⠉⣻⠉⢿⠉⢻⡋⢻⠛⢻⠍⢉⡇⢠⣯⢸⣿⣭⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⡄⣷⠀⣷⠀⢹⡀⢿⠀⣼⠀⢸⠁⣸⠀⣸⡇⣸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠰⣸⡆⢻⠂⠸⣇⢼⠣⣸⡅⢸⠆⣿⢈⣿⣹⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⣿⣟⣹⣧⠀⣏⣾⡇⣿⣧⣿⢐⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡿⣿⣻⣲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡼⣿⣿⣷⣿⡟⣿⢿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣤⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣷⣿⣿⣷⣿⣿⣷⣿⣏⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡷⠿⡏⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣏⠿⠿⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠻⠟⠻⠿⠻⠛⠛⠛⠛⠛⠋⠛⠉⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠈⠉⠉⠀⠀⠁⠀⠒⠂⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠃⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⣀⣀⣠⡶⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣉⣀⣀⣀⣠⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠉⠉⠝⢻⡇⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⠙⠛⢶⣄⠀⠀⠈⠑⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⡇⠀⠀⠠⠙⠿⠷⢶⣄⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⠶⠃⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠾⠶⠶⣶⣶⡶⠶⠶⠟⠛⠉⠉⠀⠀⠀⠀⢀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠁⠀⠁⢰⣿
⠀⠀⠀⠀⠀⠀⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⠀⠀⠀⠀⠤⡦
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣖⣤⣤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠦⠀⠀⠀⡄⠀⠀⢀⡎⠀⠁⢀⡖⠀⠀⠀⠀⢸⠃
⠀⠀⢸⣶⣤⡀⠀⠀⠀⠀⠀⠉⠛⠒⠳⠶⢢⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠄⠀⠉⠀⠀⠀⣀⡤⠖⠉⠀⢀⣤⢸⡂⠀⠀⡘⠋⠀⠀⠀⠐⣿
⠀⢀⠉⠉⠛⠛⠽⠆⣶⣶⠀⢸⡗⠒⠛⡿⠟⠁⠀⠀⠀⢀⣀⣀⠈⠉⠑⠂⠀⠀⠀⢀⣀⠦⠚⠉⠀⠀⠀⠀⢰⠃⠉⠀⡻⢣⡞⠁⠀⠀⠀⠀⠜⠁
⠀⠈⠻⢿⣶⣤⡄⢀⣀⠀⠀⣾⠂⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠈⠉⠀⠀⣀⡠⠴⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⢁⡌⠁⠀⠀⠀⢶⣾
⠀⠀⠀⠀⠀⠉⠁⠉⠛⠛⠀⠁⠀⠀⠀⠀⠒⠲⠀⠀⠀⠀⢀⠠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⣇⡄⣴⠉⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⣀⣤⡤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⣿⣤⠍⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⠄⠂⠈⠀⠉⠉⠓⠶⠤⣀⠀⠀⠀⠀⠀⠀⡾⣿⡁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠛⠒⠒⠚⠛⠋
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⡄⡀⠀⠀⠀⠸⣿⣿⣿⢷⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠤⠦⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠉⠛⠿⣿⣮⣭⡙⠳⢦⣤⠔⠒⠀⠀⢠⣼⠃⠀⠀⠀⠀⠀⠠⡶⠋⠉⠋⠉⠓⣆
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠉⠙⠟⢿⣷⣀⣠⡦⠔⠋⠉⠀⠀⠀⠀⠠⠎⠈⠐⢦⣄⣀⣀⠀⠀⣸⠷⣦⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠰⠛⠙⢸⡟⢛⣏⠀⠀⠘⡇
⠀⠀⠀⠑⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡄⣠⠀⠀⠀⠀⠈⡿⠁⠀⠀⢰⠇
⠀⠀⠀⠀⠈⠙⠿⠿⢿⣏⣀⣚⢛⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣤⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⡀⠑⢤⣴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠲⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣴⡦⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠟⠀⠀⠀⠀⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡈⠁⠀⠀⢄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣌⠉⠁⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢀⣆⠈⠁⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠛⠻⡀⠀⠀⠀⠐
⠀⠐⠺⡖⠀⠑⠀⠤⠐⠊⠀⠁⠐⠂⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠐⠀⠀⠁⠒⠊⠁⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠈⠦⢀⡀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⠁⠈⠈⠁⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⢀⢀⠤⠊⠁
⠀⠀⠀⠀⠠⡄⠁⠀⡃⠀⠀⡃⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠘⠪⡁⠀⠃⠀⠉⠁⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠐⠀⠀⠂⠉⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⠒⠒⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⢀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠁⠈⠉⠈⠉⠉⠙


⠀⠀⠀⠀⠀⠀⣴⠶⠖⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠂⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠄⠀⠀⠀⠀⢴⣦⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠐⠀⠠⡀⠀⣀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠈⠁
⠀⠀⢀⡀⠀⠀⠉⠙⠀⠀⠀⠀⠐⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⣠⣤⠑⣊⣤⣤⣌
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡈⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠻⣿⣿⡿⢃⠛⢿⡿⠟⠀⠀⠀⠤
⠀⠀⠀⠀⠑⠂⠄⠀⠀⠀⠀⠀⣀⠠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠠⠊⠉⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠠⠈⠉⠄⠊⠁⠂⠀⠐⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠀⡀⠂⠐⠀⡱⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠤⣠⠖⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⣀⠄⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠄⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⢃⡠⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠭⠤⢀⠀⠀⡀⠉⡓
⠀⠀⠀⠀⣉⠉⠙⠿⡷⠶⣶⡶⢶⢿⠛⠉⠁⠰⠀⠀⠀⠀⠀⠐⣊⠁⠠⣤⠄⠀⠇⠤⡤⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠱⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠘⠦⡀⠀⠀⡠⠂⠘⠄⡀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⣠⠧⢀⡀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⢀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣀⡀⠠⠊⠓⠀⠤⠐
⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠁⠀⠀⡇⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢶⣦⣀
⠀⠀⠀⠀⢠⡞⠐⠒⢲⠐⠀⠐⡎⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠈⣁⠤⠄⠚⠒⠒⠀⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠧
⠀⠀⠀⠀⠑
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣥⣬⣧⣴⣠⣽⣮⣼⣯⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣙⣛⣉⣉⣩⣍⣤⣭⣍⣉⣉⣉⣭⣍⣙⠃⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⢿⠁⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠄⠀⠀⠀⢀⠔⠊⠉⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣊⣀⣀⣀⣀⣀⣀⣀⡀⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢃⠀⢀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠘⡋⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠛⠛⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣖⡲⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢳⡏⢀⣀⣀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠸⡉⡝⣛⠛⠉⡆⠻⠀⢸⡿⠿⠛⢣⣴⠿⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢈⣀⢸⣶⠀⠆⣡⡀⣀⠄⠀⣥⠀⠀⠀⠀⠆⠀⠀⠐⢞⠛⠛⢛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡤⠀⠀⣈⣬⣤⣭⣌⠉⠉⠁⠀⢸⡇⠉⣩⡽⣿⠛⠛⢦⡘⢂⡀⢉⣥⠘⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⣿⡇⠀⣜⣿⣿⣿⡝⣿⣠⣴⣾⣶⣿⢛⠏⡘⠀⠘⣷⠀⡸⣠⢴⠀⢸⣿⠘⢲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⠓⠋⠻⢿⣿⣿⢿⠉⠉⠉⣽⣿⡿⠘⢄⣇⠀⠀⣸⣿⣏⡗⠐⠀⣸⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⣀⣀⣑⣒⡚⠛⠃⠀⠀⠀⢈⣉⣉⡉⣀⠬⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⠶⠻⠿⢏⣸⠛⠻⣯⣹⣿⠿⡛⠫⣿⠿⠿⢿⣟⣷⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡉⠇⣬⢄⡀⢸⡜⠦⠀⠛⠚⡟⠒⠂⠥⠿⠀⣀⣸⣾⣿⡿⡛⠛⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠸⠶⠆⠉⣀⡀⢀⠀⠫⢷⣴⣶⣿⣷⠶⣦⡴⠶⣶⣶⡾⠟⣏⣭⣇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢀⠀⠀⣸⡦⢄⣀⠀⠀⢀⣈⣴⢸⣽⣿⣿⣿⣷⢸⣿⠀⠀⠁⠀⣠⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠩⠭⣉⣑⣀⣄⡀⠀⠠⠤⠟⠉⠻⢿⣿⠿⠿⣞⣅⣀⠥⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⡿⢿⡉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⣉⣉⣉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⠶⣯⣤⣀⣈⣁⡈⠐⠽⢛⣿⣭⣍⡉⠉⠉⡉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠋⠉⠉⠂⠒⠛⠛⠛⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠃⠀⠀⠀⠠⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠈⠀⠐⠂⢄⡀⠄⠀⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠃
⠀⠀⠀⠀⠂⠀⠀⣀⡀⠈⢀⣄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣄⠈⠁⣠⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⣘⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⡰⢄⠁⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡰⢄⠈⠀⢀⠌⠀⠀⠀⠀⠀⢀⡀⠐⡀⠀⠀⠙⠻⠀⠀⠀⠠
⠀⠈⠙⣍⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠈⡗⢤⠀⢀⠀⡀⣀⠀⢠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠘⣆⢀⠨⢀⡀⠀⡀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠺⠄⢀⡀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⢀⠈⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣆⣀⣀⣀⡀⠆⠀⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⠀⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠁⠀⠀⠀⠀⠙⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠁⠀⠀⠀⠀⠛⠻⠄
⠀⠀⠀⠀⠀⠠⠶⠄⠀⠀⠀⠀⣤⡤⠄⠀⠀⠀⢀⠀⠀⠀⠀⠔⠋⠀⠀⠉⠢⡤⠂⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⢄⠀⠀⠀⠠⡀
⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⠆⡷⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⢾⣿⣿⣿⠄⣷⣿⣿⡇⠀⠀⢀⡀
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢀⠔⠣⢀⣀⡀⠔⠀⠀⠀⠀⠀⠚⡇⠀⠡⠈⠛⠛⢋⠔⠢⢉⡉⠠⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠓⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⠀⠤⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⠶⠛⢀⡰⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠀⡀⠀⠀⢈⣛⠃
⠀⠀⠀⠉⡏⠙⠻⢿⡶⢦⣶⣴⣶⡟⠋⠉⢩⠆⠀⠀⠀⠀⠀⡎⠉⣀⣠⠄⠸⠅⢤⡄⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠈⠀⠀⠈⠂
⠀⠀⠀⠀⠑⢄⠀⠀⢀⠐⠀⠢⢀⡀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⡰⠆⢀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠟⠀⢀⠛⠃⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠠⠊⠂⠤⠠⠔⠁
⠀⠀⠀⠀⠀⢀⢤⠂⠀⢉⠀⠀⠈⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢶⣤⣀
⠀⠀⠀⠀⢠⠛⠚⠒⡆⠉⠀⢸⠉⠉⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⢀⠸⠵⠄⠂⠋⠈⠀⠉⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠁

⠀⠀⠀⣀⠤⣄⠀⠀⠀⢀⠔⠒⢄⠀⠀⠀⠀⠀⡞⠙⢦⣀⡖⠉⢳⠀⠀⠀⠀⡠⠖⠒⠒⠒⠢⣄
⠀⠀⠀⡇⣀⠼⠖⠒⠒⠚⠦⣄⡸⠀⠀⠀⠀⢀⡝⠀⠀⠀⠀⠐⡎⠂⠀⢀⢎⠤⣀⡰⠉⢁⠀⠘⡆
⠀⠀⠀⡜⠁⠀⠀⠀⢀⣀⠀⠈⢇⠀⠀⠀⠀⡞⠀⠘⠀⠀⠒⠀⢸⠀⠀⢸⠘⢤⠀⠀⠄⠸⠀⠀⡇
⠀⠀⢸⠀⠀⡩⠀⣖⠂⠀⠀⠀⡸⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡀⠀⡄⠀⠂⢠⠀⠀⡇
⠀⠀⠀⠣⣼⣀⠀⠁⢀⣀⡠⠔⡡⠁⠀⠀⠈⢦⣀⡀⠀⠀⠀⢀⡜⡀⠀⠀⢣⡀⠈⠒⠐⠃⠀⡠⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠀⠀⠀⠀⠉⠒⠲⠖⠒⠉
⠀⠀⠀⢠⠒⠄⠀⠀⢀⠤⡄⠀⠀⠀⠀⡠⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⢀⠗⠉⠉⠉⠓⢴⢣⠃⠀⠀⠀⣇⠀⠀⠧⠤⢄⡀⣠⠤⡀⠀⠀⠀⠘⣤⠴⡧⡤⣴⠋⢲
⠀⠀⠀⠘⠀⠀⠀⠀⢀⠈⡆⠀⠀⠀⠀⡜⠀⠀⡀⠀⠀⠉⠀⠀⡜⠀⠀⠀⢰⢁⡀⠀⢀⠀⢻⡫⠂
⠀⠀⠀⠸⡀⠁⡺⠗⡆⠀⠇⠀⠀⠀⠀⡇⠀⠈⠡⡄⠒⠀⠀⢠⠃⠀⠀⠀⡿⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠣⢄⡉⠈⣁⠔⡰⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠈⠀⠍⡒⠤⠤⠐⠁⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⢢⠀⠀⠀⢀⡀⡀⠀⠀⠀⢀⡠⣄⡀⠀⠀⠀⣠⣄⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢸⣀⡀⡰⠃⠀⡜⠀⠀⠀⢸⢀⣠⣷⣀⣀⣞⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠈⠁⠠⢰⠡⠀⠀⠀⠀⡏⢀⡘⠃⢃⡀⠹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⢀⠇⠂⠀⠀⠀⣰⠀⠈⠁⣤⠈⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⠀⠀⣠⢎⠄⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠩⠕⠁⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣥⣼⣿⣷⣾⣷⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠋⠉⠉⠁⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⡷⠾⠶⠾⢆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⡗⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢸⡟⠀⢀⣴⢦⣤⣀⡀⠀⠀⠀⠀⠰⠀⢸⣯⠀⠀⢐⣤⣽⡯⢼⣿⢿⣰⣦⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠸⡇⠀⠈⠉⠈⠁⠙⠣⠑⠀⠀⠈⠘⠀⢸⡏⠀⠀⠀⠀⠹⠉⠀⠁⡀⠘⠿⢣⠏⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⡄⣿⣦⡄⠆⢀⣹⣷⠀⠀⠀⠀⠀⠄⣆⠀⣿⣄⠀⠀⠀⠀⠀⠀⣥⣦⣄⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠹⣿⣿⣿⣿⠻⣬⠀⠀⠀⠀⠀⠈⡕⣻⡄⢻⣿⣿⣿⡅⡟⢀⣀⣤⣶⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣝⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⣳⡀⢿⣿⣿⣿⣿⣿⣷⡿⢿⣍⡉⠁⠶
⠀⠀⠀⠀⠀⠀⠒⠲⣤⣀⠈⢻⣎⢿⣿⢻⠁⠀⠀⠀⠀⠀⠀⠉⢷⡈⢿⣿⣿⣿⣿⣿⣧⣄⣹⣿⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠙⠯⢿⠈⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⣆⠻⣿⣿⣿⣿⣿⣿⣿⣿⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣧⡝⢿⣿⣿⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣞⡆⠀⠀⠀⠀⢰⣦⡀⠀⠈⣿⣷⡌⢿⣿⡟⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⣆⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠻⢷⠀⠀⠨⠿⢿⡌⣿⡇⠀⣧

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣤⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⡤⡄
⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠁⠀⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠂⠀⠀⠀⠀⠀⠈⠈⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⢽⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⢿⠟
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠻⠋⠀⠀⠀⠀⢀⡠⠐⢊⣱⡀⠀⠀⠀⠀⣀⡾⠤⣄⠀⠀⠻⠋⠀⠀⠀⠀⠀⣠⠖⠈⣱⡀⠀⠀⠀⢀⣠⡾⠤⣄
⠀⠀⠀⠀⠀⢀⡛⠀⠤⠊⠉⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠠⠂⠉⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣤⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⡤⡄
⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠈⠳⠄⣀⡀⣀⡤⠂⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠸⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡿⠇⠀⠀⠘⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠻⠋⠀⠀⠀⠀⠀⡠⠴⣚⣵⡀⠀⠀⠀⢀⣠⡾⠤⣄⠀⠀⠳⠏⠀⠀⠀⠀⠀⣠⠔⢘⣱⡀⠀⠀⠀⢀⣠⡾⠤⣄
⠀⠀⠀⠀⠀⢀⠛⡁⠐⠂⠉⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠁⠤⠊⠉⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢰⡄⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⠗⠘⠓⠒⢦⡒⠒⠓⠀⠀⠀⠀⠀⣀⣤⢄⣀⠬⠷⠶⠶⠻⢄⣀⠞⠀⠀⠀⠀⠀⠒⠲⣄⣈⣇⣼⡀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⣠⠃⡋⣉⡧⢀⣤⡟⠛⠒⠂⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⢠⣮⠉⣄⠀⠀⠉⢉⡽⣷⣮⢭⡉⠁
⠀⠀⠀⠀⠀⢳⠀⠙⠋⠁⢿⡤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠶⢤⡀⠀⡠⣄⠀⠸⡀⠀⠀⠀⠀⠘⣔⠤⠞⠰⠤⠆⠰⠬⢜⡷
⠀⠀⠀⠀⠀⠈⢣⡀⠚⠓⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠧⡘⠴⠃⠀⠵⠬⠃⢠⠇⠀⠀⠀⠀⠀⢦⡈⢩⠆⠀⠲⡂⢈⡡
⠀⠀⠀⠀⠀⣤⢾⠇⠀⠀⠀⠘⢯⡙⢢⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⠀⠰⠦⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⢉⡏⠀⠀⠀⣟⠉
⠀⠀⠀⠀⠸⠁⢸⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠳⠋⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠈⢢⡤⠀⠠⠤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠁⠒⣲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢱⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠂
⠀⠀⠀⠀⠀⡀⢀⡈⢱⣀⡠⠔⠛⠛⠛⢦⡀⠀⠀⠀⠀⠀⠠⣀⠐⣆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⢀⣠⠤⠴⠯⠤⣄⡤⠤⠄
⠀⠀⠀⠀⠠⠤⠤⣬⠞⣡⣖⠲⠐⣏⣿⠆⡇⠀⠀⠀⠀⠀⠀⣼⢷⡋⠈⣓⣲⣄⠀⠀⠀⠀⠀⢀⣞⡤⣄⢾⡅⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠳⠭⠞⣀⠴⢀⡞⠁⠀⠀⠀⠀⢰⠱⣟⡠⠇⠸⣛⣨⠻⡀⠀⠀⠀⠀⢸⡙⠓⢚⡴⣦⢰⠦⢄⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠓⣄⠀⠀⠀⠀⠘⠦⣄⠀⠠⠄⠀⠀⠀⣇⠀⠀⠀⠀⠀⡟⢲⡬⣿⠷⠋⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢦⠀⠀⢀⡎⠀⢸⠄⠀⠀⠀⠀⠀⢸⠀⢦⡀⠸⠀⠀⢸⡆⠀⠀⠀⠀⡄⢸⠀⠇⠀⠀⣰⠃⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠉⠃⠈⠀⢀⡜⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠃⠘⠂⠘⠀⢠⡇⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠹⢄⡀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠖⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣀⡠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠉⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⡤⠴⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⣉⣵⡤⠤⡧⠤⣖⠉⠉⠀⠀⠀⠀⠀⢀⡤⢴⢡⠤⡀⠀⣀⣙⢞⠓
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠓⠛⠓⠲⣍⣁⡀⠀⠀⠀⠀⠀⠀⢀⣠⠏⣬⣸⡇⢞⢩⡆⠀⠙⣏⠉⠃⠀⠀⠀⠀⡔⣛⢷⠄⣻⢰⡁⠌⣮⡙
⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠞⢩⡇⠰⠤⠴⠶⠶⠷⠶⠖⢀⡼⠀⠀⠀⠀⠀⠀⠀⢻⠀⠙⠁⠀⠉⠋⢸⡇⣠⠇
⠀⠀⠀⠀⠀⢠⡇⢀⡔⣻⢲⡀⢠⣖⢮⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡴⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⢼⠀⠀⠒⠒⠀⠀⠀⡏⠁
⠀⠄⠀⠀⢠⡟⡄⠀⠓⠒⣚⠀⠈⠛⠋⡜⢧⠀⠀⠀⠀⠀⠇⠱⣄⠀⠀⠀⠸⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⡄⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⢲⡤⠤⣬⡭⠭⢴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠤⠤⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣀⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠙⠲⢤⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠁⠀⠈⣡⠋⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠉⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢰⠇⠀⣠⡀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢃⠀⠀⡑⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⡔⣦⠠⡀
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠱⠀⡃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⣋⣸⢁⣤⡌⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠰⠞⠂⢀⡌⠳⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠐⠁⠀⢸⠆⠀⠀⠀⠀⠀⢀⡜⢣⡈⠉⠟⢻⡞⡁⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡦⢉⣉⢣⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠳⣄⣹⣾⣣⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠛⠃⠀⠙⠀⠀⠀⠀⢀⢒⡼⠀⠀⠠⠴⠖⠋⠀⠀⠀⠠⡀⣠⠏⢳⡄⠀⠀⠀⢀⠏⢰⢟⡥⠲⠌⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡔⠀⠀⠢⡀⠀⠀⠀⠀⢨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⣠⠋⠀⠀⠀⠀⢸⡀⠀⢸⣼⢧⢞⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⣀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢁⡼⠁⠀⠀⠀⠀⠀⠀⠉⠙⢦⠐⣨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢉⣉⣁⡀⠈⠑⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠱⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⢤⣤⠌⠀⢀⡏⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⢋⡾⣡⢆⡀⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠤⢀⡀⠀⣧⠀⠙⠒⠤⠖⠚⠉⠀⡇⢸⠀⠀⠀⠀⠀⠀⡠⢡⠞⣴⢣⡞⠡⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡈⣏⠀⠀⣀⠀⠀⢀⡴⠀⢸⣀⠴⠚⠳⡄⠠⡁⠈⠘⢡⠏⡐⠉⡂⠆
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡘⣎⠓⠋⣙⠓⠊⠁⠀⢀⠜⠁⠀⠀⠀⠷⠀⠙⣦⣀⠀⡸⢧⡞⠉⡆⢄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢻⣦⠞⠉⠉⠙⢦⡔⡁⠀⠀⠀⠀⠀⡎⢈⢦⠷⣈⠉⠀⢰⢀⡼⢁⠎
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣠⠶⢿⡀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⡇⠀⠒⢂⢮⡉⣇⠈⠉⠓⠋⠛⢳⡆
⠀⠀⠀⠀⠀⠀⠈⠳⣄⣤⡶⢫⠁⠀⠈⠓⠦⡴⠾⠋⠲⡄⠀⠀⠀⠀⠀⡇⠀⢀⠃⠀⡗⠹⡄⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡂⠀⠀⢀⠹⣆⠀⠀⠀⢸⡇⠀⠘⡆⠀⡇⠀⠙⢦⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠘⠀⡈⠳⢦⣴⠟⠀⠀⠀⢱⠀⢣⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠁⠀⠀⠰⠠⡀⠀⠀⢸⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠰⡀⠀⡟⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢧⢀⡀⠀⠀⢧⠑⢤⡴⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠦⡇⠀⠀⠀⠀⢸⡇⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠸⣦⣄⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣇⢀⣀⣀⣀⣀⢸⠀⠆⠀⠀⠀⢧⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⠤⠤⠤⠤⠴⠒⠃⠀⠀⠀⠀⠛⠛⠛⠛⢛⠓⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⡀⠀⠀⠀⠀⠀⣀⡤⣞⠁
⣠⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡾⠉⠛⠓⠒⠛⠋⠉⡇⠘⡆⠀⠀⠀⠀⠀⣠⠴⠒⠒⠲⢦⡀
⡇⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⣾⢰⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⢸⡇
⢹⡀⠀⠀⠀⠹⣆⡤⠶⠂⠀⠀⣛⠈⢃⡀⠀⠀⠀⠀⡴⠓⢧⠤⠷⠀⠑⢶⠁⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠳⣄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⡉⠉⠀⢧⠀⠀⢸⠀⠀⢀⣀⣀⡀⠀⠈⠀⠀⠢⣀⣀⡴⠛
⠀⠀⠙⠳⢶⣤⠾⠋⠉⠛⠲⠶⠾⣇⣀⣀⣼⠀⠀⠘⠦⠤⠤⠃⠈⠉⠙⠛⠛⠛⠛⠋⠉
⣿⣿⣿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠠⠄⠒⠉⠈⠢⡀
⠀⠀⠀⡐⠉⠀⠄⠂⡀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⡄⠄⠀⠀⠠⠀⠐⠀⠈⠀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⡗⠄
⠀⠀⠀⠰⠀⠀⢀⡀⠁⡐⠐⢄⠈⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠘⢄⣀⣀⠦⡀
⠀⠀⠀⠀⢃⠀⠇⠈⠉⠁⠀⠈⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠈⢢⠂⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣀⣴⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡩⢣⠂⠀⡑⠤⣆⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡰⠁⠐⠀⡆⠀⠀⠐⢄⠀⠉⠺⠲⡰⢠⢦⣀⣀⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⡀⠤⠔⠺⡇⠀⠀⠀⢃⠀⠀⠀⠀⡆⠀⢀⠏⠣⠼⣁⠉⢠⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢄⡧⢰⡈⠒⣘⠢⢀⡵⠤⢠⠐⢁⠀⠉⡋⠙⣄⠈⢢⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠭⢠⠼⡄⣁⠒⣎⠀⢑⠁⠈⡒⠁⢙⠁⢱⠊⡽⠧⠈⠓⠓⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠇⠈⡀⠰⠀⠀⠀⠀⡁⠀⡄⠀⠖⠉⠀⠀⢀⣀⠤⢄⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠘⡄⡡⢀⣁⠀⠇⠀⠀⠀⠁⡌⠀⠀⠁⡲⠓⠈⠁⠐⡴⠀⠐⠀⠱
⠀⠀⠀⠀⠀⠀⠊⠁⠙⠇⠀⢪⠀⠙⠀⠀⢹⠄⠰⠀⠀⠰⠀⣄⣄⣀⡀⠈⠒⠄⠒⠊⠀⠀⡘⠑⡖
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⣰⠀⠀⠀⢀⠌⠁⠀⠀⠀⠀⠀⠉⠢⡀⠐⠤⠒⠀⡠⠒⠀⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⠤⠄⠠⠔⠂⠁


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠛⡛⣁
⠀⠀⠀⠀⠀⠠⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⡐⠦⣄⣀⣀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⠈⠑⠒⠲⠔⠒⠒⢒⠇
⠀⠀⠀⠈⠑⠒⠒⠢⠤⠔⠊⡅⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⣽⡆⠀⠀⠀⠀⠈⣷⣶⣶⣶⣤⣴⣾⡟
⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⢁⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠀⡀⠀⠀⣀⣀⣀⣁⣀⣀⣀⠀⠀⠀⢀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⣠⣴⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣦⡀⢄⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠼⡏⣴⣿⣿⣿⣏⢻⣿⣿⠱⣜⡍⣿⣿⣿⣿⡆⢿⡇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⢐⣿⣿⣿⣿⡿⣿⣅⠈⢻⣿⣷⣿⣿⣿⣿⣿⠘⠣⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡆⠘⣿⣿⣿⣿⣿⣿⡏⢸⣿⣛⣵⣿⣿⣿⣿⡿⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⣃⣠⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣄⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣿⠿⣧⡻⠟⣉⠩⠄⠀⣀⣛⠀⠉⣉⣻⠿⢋⣾⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⣿⣼⣄⠀⠀⠀⠈⠀⠀⠀⠉⠛⠃⠉⠉⠀⠀⠁⠈⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣧⡈⠻⣿⡝⠓⠧⣬⣀⣀⣀⠀⠀⢀⣀⣅⣤⠶⠶⣿⡿⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⢠⣽⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⠁⠀⠨⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠤⠍⡻⠟⠭⠐⣻⣿⣿⣿⣿⡧⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠸⠟⠙⠏⠁⠀⠀⠀⠀⠀⠀⠀⠘⠉⠿⠹⢻⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠰⠷⡾⠷⠧⠾⠄⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠁⠁⠀⠀⠀⠀⠘⠋⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠁⠀⠈⠃

⠐⠀⠐⠀⠐⠀⠐⠀⠐⠀⡄⠀⠀⠀⢀⠀⡀⠂⢀⠀⠀⠀⠂⠀⣀⣀⢠⡄⢀⠄⠀⠀⠀⠀⠐⠀⠠⠄
⢈⠀⠈⠀⢈⠀⢈⠀⢘⠀⠁⠀⠀⠀⠰⠃⠀⠀⣈⡀⠠⠄⠀⠀⠉⠉⠀⠁⠉⠀⠄⠀⠄⠀⠀⠤⠀⣠⣤⣤⣄
⠈⠀⢈⣠⣬⣤⣌⠀⠀⠐⣾⣿⣷⣶⡀⢰⣿⣿⣿⣿⣿⣷⡆⠔⢀⣀⠦⠀⡐⠘⠀⢀⠀⢀⠐⠀⠀⣿⣿⣿⣿
⠀⠀⠿⣿⣿⣿⣿⢇⠀⢸⣿⣿⣿⣿⠀⡸⠹⠏⠏⣿⠙⠛⣇⠀⠈⠁⠀⠖⡆⢈⠄⠈⠂⠠⠤⠂⠀⣿⣿⣿⡟
⠀⠂⠓⠌⠉⠉⣡⠈⡄⠀⠙⠛⠛⠃⡀⠈⠀⡀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⣀⡠⡤⣀⠋⠉⠩⠑
⠀⠀⠠⠀⠀⠀⠀⠄⡠⠀⠀⠒⠄⡤⠁⠀⠀⠀⡨⡀⠀⠀⣔⠳⠠⢘⠍⠉⠩⠇⠠⠊⡩⠄⠀⠁⠀⠀⡤⠀⠄
⠰⠀⠠⠴⠀⠀⠠⠀⠀⢈⠠⢠⠴⠁⠈⠀⢀⠐⠀⣀⣤⣤⣤⣄⠀⡄⠀⠈⠀⠀⠀⠀⠇⠀⠀⢀⢀⡐⠆⠀⠘
⠀⠐⠀⠀⠀⢀⣃⠀⠀⠠⠊⠑⡸⠀⠀⠀⠀⠄⢸⣿⣿⣯⣿⡿⠂⠂⠠⠥⠈⠀⠀⠴⠋⠑⡀⠆⠪⠅⠦⠠⢐
⠀⠘⠄⠀⠀⠟⠛⢓⠀⠀⠀⠀⠖⢆⠁⠄⠀⠂⠈⠉⠉⠅⠀⠀⠀⠈⠉⠀⠀⠴⠀⢀⠄⠑⠃⠈⠢⠿⠁⠑⠬
⠀⣠⡂⠀⠀⢀⣤⡀⠐⡀⠀⣀⢸⠸⡀⠀⢚⡤⣶⣾⡿⣿⢿⡿⣿⣿⢿⠶⡦⣄⠨⠀⢀⠀⠀⠀⡠⠆⠀⠀⢀
⢀⠀⠀⢀⠀⠹⠛⠻⠀⠐⠊⠀⠊⠃⡓⠀⢾⣾⣷⣦⣿⣶⣿⣽⣦⣯⣾⣿⣧⣿⡆⠀⠀⡜⢀⠀⠉⠀⠀⠘⢨
⠈⠀⠀⠈⣤⡤⠤⠤⠤⠄⢠⠄⠀⠀⠀⡀⡘⠿⣿⣿⣯⣽⣽⣭⣭⣽⣿⣿⡿⠟⠠⠠⠼⣷⠚⢠⡁⠠⡄⢀
⠀⠀⠉⣶⠛⠒⠂⠀⠀⠀⠈⢈⠶⠒⠁⢀⠁⡀⢀⢤⣉⡉⠉⠉⠉⣉⡁⠀⠀⠢⢄⡀⠀⢠⠀⢀⠀⠈⠀⠂
⢀⡔⠁⠀⠃⡀⠤⡀⠀⠤⠔⢊⠀⠄⠐⠁⠈⠋⠓⠞⠛⢋⠀⠀⣾⣿⣿⡆⠀⢰⣄⣦⣶⡾⠀⠀⢁⣀⢈⡀⡀
⠀⡀⢀⡀⢀⡀⣀⠀⡀⢀⡀⠐⠢⠀⡆⠀⠈⠀⠁⠀⡀⢈⠀⠀⢹⠟⢻⡇⠀⠸⠛⢋⣡⡜⠀⢻⠀⠄⠀⠂⠂
⠀⡀⠀⠀⢀⠀⢀⠀⣀⠐⢄⣀⣱⢦⡅⠀⡘⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢻⡿⠿⠁⠀⠊⠀⠀⣀⣤⣴
⠀⢀⠈⠁⠈⠙⢌⠀⢀⡀⠀⡙⠚⡐⢡⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠚⠤⠵⠀⠀⠀⢸⣿⣿⡷⠾
⠀⢈⠀⡀⠀⠡⡑⠄⠀⠀⠀⡄⠀⠉⠈⠁⠀⠉⠛⠛⠛⠹⠀⠀⢀⣠⣣⣤⡀⠐⠀⡀⠢⠀⠀⡀⠘⠋⠁
⠀⢨⠀⠈⠂⠀⢓⠀⢀⡀⠀⠄⠰⠶⠶⠶⠀⠀⠀⠂⠀⡣⠀⢰⡿⢿⡟⠏⠣⢈⠀⠁⢰⠂⠑⠀⠁
⠀⣀⡀⠠⣀⠀⢀⣀⠀⢀⢀⠀⠘⠛⠂⢃⠀⠘⠂⠀⠒⠀⠀⠀⠐⠀⠀⠨⠀⠀⠀⡀⠉⠡⠄⠐⣂⡀⠩⡤⠐
⠀⠓⠪⠁⠒⠀⠐⠂⠄⠜⠚⠀⠸⠿⠿⠿⠀⠺⠿⠹⠏⠿⠽⠧⠻⠼⠆⠀⠀⠤⠇⠃⠀⠀⢰⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡅⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⣴⢠⠀⠀⠀⡄⠀⢀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⢀⡘⠀⠀⢀⠋⠀⠀⠀⣀⠀⠉⠿⠀⠀⠀⡇⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⡃⠀⠀⡧⣀⠀⠀⠀⠈⠐⠀⠀⢠⢿⡀⠀⠰⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⢣⠁⠀⠀⡀⠘⠀⠀⠠⠀⠀⠀⢀⠳⠋⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⡄⠀⠀⠀⠁⠂⠀⠀⡄⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⡀⢹⡄
⠀⠀⡀⠄⠀⠀⢸⡀⠄⠀⠀⠐⣆⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⢠⠄⠁⣠⣣⣀⣀⡄⠀⠀⣸⠀⠀⠀⡠⠐⣗⠀⠀⠀⢀⡀
⠀⠀⠀⠸⡀⠀⢁⠀⢀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠉⠀⠁⢀⠀⠀⠀⠉⠀⠀⠀⠼⢈⢿⡇⠀⠀⢸⠛⠃
⠀⠀⠀⢀⢾⣿⠧⡤⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠰⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⣤⢾⣷⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢀⠳⣌⣺⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⢸⠀⠀⢠⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠀⠀⠠⡄⠀⠂⠀⡤⠄
⠀⠀⠀⠀⠀⠀⢴⡿⣿⠀⠀⠀⠀⠀⠀⢠⠐⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠐⢀
⠀⠀⠀⠀⠀⠀⢋⠶⠸⡀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠛⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⣆⠀⠂⠁⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⢺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠣⠍⠁⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢋⠑⡄⠀⠀⠀⠀⠀⠀⢐⣞⣲⣧⡀⠋⠀⣐⠚⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⡿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⡀⢰⠀⠀⠀⠀⠀⠀⠈⠉⣏⠉⡁⠠⠆⠁⣇⠋⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠉⠠⠀⠀⠘⠀⠠⢿⡆⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠀⢁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠋⠈⠀⠀⣐⠀⡄⠀⠀⠀⠀⠀⠀⠀⡏⠀⠄⢠⡄⠉⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠸⣷⠑⡀⢀⢀⡁⢀⠁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡅⠀⠀⠀⠀⠀⠀⠀⢻⠆⢘⠨⠀⠀⠒⡧⡀⠰⡄⢧⠀⠀⠀⠀⠀⢴⣀⡀⢨⣎⠈⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠘⡀⠁⠀⢀⠀⠀⠈⠛⢦⣹⣄⣧⠀⠀⠀⠀⠀⠉⠀⠀⠈⢷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡠⠀⢸⣷⣤⠀⠈⠙⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⠰⠐⠱⠂⠈⠿⠓⠆⠀⠐⠂⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⢶⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣤⣶⣢⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⢀⢀⣨⣥⣤⣤⣤⣤⣤⣄⡉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣮⣿⣷
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠄⠀⠀⢸⣿⠟⠉⠁⠀⠀⠉⠉⠉⠉⠹⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⠟⠋⠈⠛⠿⠙⢿⡟⢿⡿⣿⠇⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⢱⠑⠄
⠀⠀⠀⢘⠛⡛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡄⡀⠀⠀⠀⠀⠀⠀⢄⣀⣄⠀⠀⠀⠀⠀⠀⢠⡆⠀⣄⢴⣔⡀
⠀⠀⠀⠈⠀⠓⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠸⠂⠀⠩⢀⠠⠀⠀⡄⡀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⣠⣾⣿⠀⠠⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠸⠛⠻⣷⣦⣄⣀⡀⠀⢀⣤⣤⣶⡾⠟⢉⣤
⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣈⠉⠻⠃⠀⠛⠟⠉⢁⣤⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡄⠀⠀⠀⠀⢴⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠈⠈⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠈⢿⣿⡻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠠
⠀⠀⠀⠀⢀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⣀⠀⠀⠀⠀⠲⠀⠂
⠀⠀⠠⠀⣀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡄⢨
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠑⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣶⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⠁⣀⡀⡱⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢹⡧⣟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⣷⠋⠀⣀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡚⢛⡛⠻⠟⠛⢓⠔⢻⠀⢸⣿⣿⠿⢟⢃⣤⣾⣿⣿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠰⣇⠎⡠⡀⢃⡄⠀⠈⠀⠈⣃⠈⠅⠀⠀⠠⡏⠁⠀⠀⣤⣴⣶⡶⠷⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠂⠀⡀⠐⢻⠀⠏⠦⠬⠤⠿⠧⠶⠉⠁⣦⡏⠦⠤⠦⣠⣤⣥⣤⣄⠐⣨⠒⠠⠦⢠⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⠑⠀⣠⣶⣿⣿⣿⣷⣦⠀⠀⢀⣀⣀⡷⣧⣀⠔⡿⠚⣿⢄⠀⠊⡗⠐⠒⠀⠠⣼⠐⣅⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣿⣿⠀⣠⣵⣿⣿⣿⣷⡜⣿⣴⣾⣿⣿⣿⣿⢩⠏⢰⠁⠀⠈⣿⡂⣤⢣⠶⠿⠄⠀⣿⣏⢉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡛⠛⠉⠻⢿⣿⣿⡿⣿⠀⠀⠁⢸⣿⣿⠿⠘⢄⣿⡀⠀⠀⣼⣇⣏⣨⠃⠒⠒⢰⡿⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣀⣀⠀⠘⠒⠒⠒⠛⠛⠀⠀⠀⠀⠈⠉⠉⠉⢉⣀⣈⠩⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣺⣿⣿⡏⢻⠿⣶⣶⡖⣶⣖⣶⡾⢿⢻⣿⣷⣿⣶⣶⠶⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⡟⢹⠀⠈⠉⢓⢺⡀⠀⢹⣯⣿⣿⣃⡒⠠⢸⡇⠀⠉⠙⢏⣾⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠚⠓⢺⡒⠦⠠⠾⠎⠉⠁⠈⠀⣌⡇⠀⠉⣉⣉⣑⠒⠠⠸⠟⠛⠿⠋⢩⣉⣉⣭⣽⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠇⠀⠉⢘⠛⢛⣀⠀⠀⠄⢀⣀⡈⠓⢛⡿⣿⣿⣿⣿⠛⢷⣶⠛⠻⠿⣿⠿⠟⢻⣷⣾⠧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠠⡀⠀⠀⢨⡾⠥⣀⣀⠀⠀⢀⣀⣨⡆⣸⣼⣿⣿⣿⣿⣷⠈⣿⡃⠀⠀⠠⠀⣀⣇⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠭⢍⣛⣉⠓⠤⢀⣀⠈⢉⣁⣀⡽⠿⠛⢿⣿⣿⣿⣿⣿⣾⣋⡈⣁⣂⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠲⠶⠶⠶⣦⣤⠦⠤⠉⠑⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡾⠵⠥⠤⠴⢲⡎⢳⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡛⠿⣤⣤⣤⣤⣤⣀⠢⠭⢛⣩⣿⣷⣦⣄⣀⡀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠀⠐⠒⠒⠛⠛⠛⠋⠉

⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣶⣶⣶⣤⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹
⠀⣰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣸⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀
⢠⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣲⣶⣾⣿⣿⠿⣿⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⡿⠁⡼⡀⠀⠀⠀⠶⣶⣶⣶⣿⣟⠉⢿⣿⣏⠁⡀⠉⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣦⡀
⠁⢸⣿⡇⠀⠀⠀⠀⠀⠀⠉⡁⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠿⡿⠿⠟⠉⠙⢹⣷⣴⣄⣤⣄⡀
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠟⠉⠀⠀⠉⢻⣿⣿⣶⣤⣶⣾⣦⣶⣶⣿⣿⣿⠛⠉
⣤⣾⡞⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣽⠟⣷⣶⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⠋⠘⣿⡏⠙⢻⣿⣿⣿⣄⣠⢾⣿⣦⣴⡿⣿⣿⣿⣿⡿⠛⠿⣿⡻⣿⣿⡇⠀⠀⠀⠀⠰⣜⠆
⡏⡀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠛⢳⣿⣷⣦⣬⣿⢤⣞⢻⣟⣹⣿⡷⣴⣿⣴⠿⣰⣽⣿⣋⣿⣤⡰⣴⣾⣹⠼⣟⠁⠀⠀⠀⠀⠘⣿⣆
⣂⠀⠐⡅⠀⠀⠀⠀⠀⠀⠈⣀⣤⣦⣿⡿⣟⡯⢾⠛⠿⢾⣿⣾⣄⠠⢝⣻⣯⣿⡿⠟⢫⣯⣻⡿⠟⣽⡿⠶⣿⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀
⡏⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢙⣿⡿⡿⠽⣿⠳⣦⣤⣀⣘⣻⣿⣿⣿⣿⡿⣿⣷⣰⣾⣿⠿⠩⣦⡴⢿⡾⡚⢻⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠄⢗⡉⢡⣂⣯⣴⣾⣽⡿⠿⠟⣻⣿⣿⣭⣍⣛⠟⠿⣷⣿⣖⣛⣿⣉⢹⣗⡏⢹⡷⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣾⣿⡇⠀⣤⠀⠀⠀⠐⢿⣷⡿⠷⠟⠫⣭⡟⠉⢤⣶⠖⠀⠻⠿⣿⣿⣿⣯⣤⡀⠀⠉⠉⣿⣿⣻⣿⣿⣵⣾⣷⡤⠀⠀⠀⠀⠀⠀⠀⠈
⢰⣿⣿⢁⣾⣿⠀⠀⠀⠀⠘⠿⢿⣷⢶⣿⣿⣥⣤⣜⠋⣴⣶⣾⣿⣿⣿⣿⣯⣿⡿⠈⢻⠷⠛⠚⠈⢻⣼⣿⠿⠟⠂
⣾⣿⡿⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠓⢿⣿⣯⡿⣧⣶⣿⣿⣿⣿⣿⠏⠉⠉⢀⣀⣀⣈⣁⣶⣶⣷⠿⢟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⣿⢿⣟⣿⣿⣿⣿⣿⣿⣷⣤⣤⣴⣾⠯⢽⡿⢫⣷⣿⠟⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠀⠶⠀⠀⣿⠀⠛⢿⡿⣻⣿⠃⡘⣿⣿⠉⣿⣧⣶⣞⢋⣿⣿⣿⠀⠚⠋⠰⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠾⠆⠀⠠⠄⣿⠀⣠⣼⣿⣿⣿⣷⣇⢸⠈⢀⣿⣿⣿⣿⡟⠋⢸⣏⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⣙⠴⠿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠓⠀⢸⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⢰⠁⠀⣾⣿⡇⠀⢸⣿⣿⣿⠻⣿⣿⡀⣿⣿⠟⠁⠈⠀⠀⢸⡇⠈⠉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣧⣿
⣿⣿⣿⣿⢠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠛⣀⠀⢸⣿⣿⣿⠀⢽⣿⣿⡿⣧⣠⣀⡤⠀⠀⣸⠇⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⣿⣿⣿⠀⢸⡏⠋⠀⣿⣿⣿⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠈⠛⠛⠻⠀⠘⠇⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⢀⠠⠄⠀⢠⢄⡀⡚⠑⠀⠂⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠠⣀⡀
⠀⣂⠳⣠⡴⢯⡁⠀⣈⠆⠀⢙⡤⠞⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠃⠴⠙⡌⠄⢉⠁⠉⢳⠆⠀⡆
⡠⠊⠀⠀⢀⠝⠻⠀⠀⠠⣀⠇⣷⢴⠈⡡⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠊⢀⡈⣣⠀⢸⠀⢸⡄⣈⡿⢗⡔⠊
⠣⠀⠖⠣⡈⢑⣔⡌⡈⢂⢐⡶⠙⠢⠤⠃⠀⠀⣠⠄⠰⠒⣒⡲⠂⠤⢄⠀⠉⣟⡴⠈⠁⠁⠋⠁⠉⠉⠉
⠀⠀⠀⠀⠈⠁⠀⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⠘⠂⠬⠄⣋⣉⣉⣑⡍⠴⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠉⠢⡀⠀⠀⠀⠀⠘⠀⣐⠀⠀⣀⠀⢀⣇⠜⠁⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⢠⠒⡋⢂
⠀⠀⠀⢀⡀⠣⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠔⠀⠀⠆⠈⠀⢎⡀
⠀⠀⠀⠀⡓⢄⡈⠢⠄⣀⢀⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠂⠠⠄⣀⠀⠀⠀⠀⠀⢆⠠⠀⠀⠀⠤⠜⡀⠁
⠀⠀⠀⠀⠘⢄⡈⠀⠂⠠⠀⡴⠊⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⡀⠀⠠⠖⠉⠀⠀⠉⠰⠆⠀⠀⠙⠠⠠⣠⠶⠊⣑⡢
⠀⠀⠀⠀⠀⡸⠈⠗⠒⠒⠉⠀⠀⠀⠀⠀⣀⣈⠓⢆⠀⠀⠀⠀⠁⠀⠲⠄⠀⠂⠀⠀⡸⠀⠀⠀⠀⠐⠂⠑⠂⠐⡄
⠀⠀⠀⠀⢠⠃⡌⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⡀⠛⠶⠄⣀⣀⠤⣄⠀⢸⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⢠⠃⡘⠀⠀⠀⠀⠀⠀⠣⢄⡂⣄⣀⣀⠬⠄⣀⡀⣀⠐⠉⠂⢈⡄⠈⠁⠀⠀⢀⡠⠴⠲⠶⢤⡀⠀⠀⠀⠂
⠀⠀⠀⠁⠴⠁⠀⠀⠀⠀⠀⠀⠑⠤⣈⡀⢀⡅⠀⠀⣇⡀⢀⡁⢀⠜⠚⠀⠀⠀⢀⠞⠁⠀⢀⣀⡀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⡀⠉⠀⠀⠀⠀⠀⠀⡟⢀⣀⠔⢁⣄⢹⠃⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣦⠀⠢⡀⠀⠀⠀⠀⠀⢧⡀⢸⡀⡨⠶⠚⠠⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠐⠂⠛⢍⢂⠀⢱⡄⠀⡃⠀⠀⠀⠀⠀⠀⠑⠯⣙⣦⡤⠴⠞⢁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⠈⠁⠀⠀⠑⡀⠀⢀⣦⠴⠋⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠱⡀
⠀⠀⠀⠀⠑⡒⠠⠤⠀⠑⠄⠑⠠⣀⡠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⢆
⠀⠀⠀⡏⠀⣈⣒⡖⠊⠤⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠁

⠀⠀⠀⠀⠀⠠
⠀⠀⠦⢄⣀⠚⢀⠀⠀⠁⢀⠭⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⣄⣐⢦⡄⠀⠀⠀⠂⠀⠄
⣀⡀⠀⢀⡭⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠐⠀⠀⠀⠩⠥⠶⠾⠇⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠤⡄
⠈⠉⠲⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠁⠀⠀⡅⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣑⠂⠤⣀⡞⠀⠀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠀⠀⠀⠀⢁⠆⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⣁⠀⠀⠑⠒⠠⠤⠃
⠀⠀⢀⠀⠀⠤⠤⠤⢄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣃⣠⣄⡄⠀⡀
⠀⠠⠀⠀⢠⣥⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣿⣿⠽⠟⠇⠀⠸
⠀⠀⠀⠀⠀⠀⡀⠂⠈⡇⠀⠠⠒⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡼
⠀⠀⠐⡏⠉⠛⠓⠺⢼⡀⠀⢄⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡄⠀⢀⡀⠀⢀
⠀⠁⠀⡇⠀⠀⠀⠠⠀⡏⠀⠀⠀⠀⠁⠀⢀⠀⠀⠄⡘⠐⠃⠴⢄⢀⡀⠀⠀⠀⠐⢤⡇⠀⠀⠀⠀⠉
⠀⢀⠀⠋⡉⠛⠒⠒⢼⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢂⠉⠒⠲⠤⢬⣀⣉⣂⠀⠀⠀⠑⣄⣀⣠⣤⡴⠞⠂
⠀⠀⠉⠓⠲⠭⠭⠭⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠶⢄⣠⣘⠐⠢⢠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⣀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢆⣼⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣧⡘⠀⠄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⠉⣴⣿⣿⡿⠗⠉⢋⠉⠙⠁⠀⠉⣡⣤⣄⢀⡀⠈⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠿⢇⣬⣥⣭⡹⣄⠹⠿⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⣿⡿⠃⣹⡷⠀⣿⢾⣿⣿⣿⣟⣷⣶⡄⠀⠀⡈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⣸⠁⡀⠀⣩⣷⡏⣰⠘⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣴⣶⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⢸⡟⢾⣿⣏⣠⣥⣼⣿⣛⣟⣛⣿⡟⠀⠀⠀⠀⠹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠇⡀⢹⣿⠋⠁⠀⠂⠙⠀⠉⡛⢿⣷⣦⡄⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣧⠐⣥⡾⠃⠄⠈⠛⠀⠀⠀⠀⠐⠰⡦⡉⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣙⢤⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡐⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡿⣿⠀⢰⣦⡀⠀⠈⠀⣀⡀⠁⠀⠀⠀⣀⠀⠆⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠀⢸⣿⣃⠀⠀⠀⠈⠙⠲⠖⠂⢹⣿⢷⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠁⢸⣿⣋⡓⣶⡶⠦⣀⡀⢀⣸⡿⣿⣆⠀⢊⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⣿⡿⣿⣿⣿⣿⣗⣀⣛⢋⡿⣿⣿⣿⢿⠀⣘⠧⢒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠛⠻⠷⠛⠋⣿⡻⣿⡋⠉⢉⣿⣿⣿⣿⢿⡀⠑⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢻⣷⡷⢠⣇⢨⡦⠂⣠⣄⡀⠀⠉⠉⠘⢣⡱⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⣀⡄⠀⠁⠀⠀⠀⠀⠀⠀⣟⡀⢀⣀⣀⡴⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⡿⠾⠿⠶⠀⠀⠀⠸⢦⣿⠿⢿⢟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠪⢈⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡿⠗⠀⠀⠀⠴⣭⠪⡵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢶⡴⠓⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠀⠀⠀⠀⠀⢘⢸⡰⠭⣵⢉⠄
⠀⠀⠀⢀⣀⣤⣤⣴⣶⣤⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⣡⣤⣬⣄⠀⠀⠀⠀⠀⢰⣿⠁⣉⣿⠏⠀⠀⠀⠀⠀⠈⠦⣙⣙⡥⠜
⠀⠀⣴⣿⣿⣿⠿⡿⣛⣹⡄⠀⠀⠀⢸⠀⠀⠀⠀⢀⢼⠿⠿⠿⡿⠻⠤⠀⠀⠀⠀⣺⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⣶⣷⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⢂⠂⢠⠀⠁⢇⠁⠀⢀⣴⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠢⠤⠼⠼⠌⠀⠀⠀⠛⠿⣿⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⢀⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⣴⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠫⠬⠯⠥⡞⠉⢃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢈⠖⠀⠒⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⡈
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢢⠀⠀⠀⡼⠉⡇⠀⠀⠘⠢⠤⠤⠤⠤⠗⠋
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠑⠚
⠀⠀⠀⠀⠀⢹⡢⣞⣢⢯⠇⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡚⢳⣤⡀
⠀⠀⠀⠀⠀⠀⣷⢾⣉⡾⠀⠀⠀⠀⢠⠞⠡⢀⠀⠈⠀⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡞⠙⣦⡴⠛⢏⣠⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠸⣞⣭⠇⠀⠀⠀⠀⠇⠨⢠⢰⣿⠿⣦⠀⠠⠀⠃⠀⠀⠀⠀⠀⠀⣠⣴⠚⢣⣠⠽⢏⢉⡷⣖⢋⣱⣶⠋⣱
⠀⠀⠀⠀⠀⠀⠀⢫⡟⠀⠀⠀⠀⠀⡇⢁⠀⢘⠦⠤⠊⠈⠀⠀⡇⠀⠀⠀⠀⠀⣾⢀⡸⢶⠋⣹⣴⠚⢻⣤⠞⢻⣀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢳⣆⣀⠂⠄⠂⠐⢀⣀⣾⠃⠀⠀⠀⠀⠀⠙⣯⣠⠼⢻⣁⡼⢿⡉⣹⣶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⠷⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⠚⢹⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁

⠀⠀⠀⢀⠐⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⢀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⢠⡟
⠀⡦⠶⠾⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡘⠁⠀⠀⢠⣌⠆⠀⠁⠀⠀⠀⠖⠈⠙⠉⠉⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⢠⡈⠀⣀⠁⠀⠀⠀⠀⠈⠂⠀⠈⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢃⠀⠀⠀⠀⠀⠀⠄⠃

⠀⠀⠀⠀⠠
⠀⠀⠀⠈⢀⠀⠀⠀⠀⢰⠉⠁⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠤⠖⣆⠀⠀⠀⠑⠀⢂⠐⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⠄⠀⠀⠀⠀⠈⠄⠀⠠⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠠⠆⠈⣁⣈⡉⠂⠀⠀⠐⣶⡀⢱⣮⠄⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠐⠈⠠⢻⠍⠀⠀⠀⠀⠂⠀⠨⠙⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣊⣄⡤⠄
⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣸⠀⠀⠀⠀⠀⠀⠠⢀⢀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⠀⢀⡴⠆⠄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⠀⢀⣀⣤⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠊⢁⠀⡀⠁⡄
⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠉⠰

⠀⠀⢀⢀⡀
⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡶
⠀⢐⠀⢫⠀⢀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠛⠁⠀⠀⢠⠀⠀⠀⢠⡀
⠀⠸⠀⠸⠀⠸⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠀⣀⠀⠀⠶⠞
⠀⠀⠄⠘⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⣠⣶⣷⣤⣶⣿⣷⣶⣾⣿⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠋⠉⠙⠉⠉⠋⠉⠛⠙⠛⢿⡇⢰⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣾⣿⣿⣿⡿⠛⠁⠀⠉⠛⠈⠻⠏⠻⠟⢻⡟
⠀⠀⠀⠀⠀⠀⠀⢀⢛⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢷⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠠⢸⡫⡙⠿
⠀⠀⠀⢄⣢⠗⢰⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢁⠓⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠡⠀⠀⠀⠀⠀⠀⠰⡀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠑⠈⢻⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠂⠄⠀⠤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⠋⠂⢀⠤⠀⠰⢄⠀⠒⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡐⠂⠀⡀⠀⢀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠐⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠂⠉⠀⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣄⠄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠑⠎⠱⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⡁⠀⠀⡀⠀⢈⠛⠀⠄⠉⠐⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠡⠐⠋⠠⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⢈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠧⠀⠈⠀⠐⢾⣷⡹⣿⢽⣷⣿⣿⣿⣿⣿⡟⠉⣽⡟⠁⠀⢀⠎⢩⣶⠆⠁
⠀⠀⠀⠀⠀⠀⠄⠁⠀⠠⢦⠑⠀⣰⢄⡀⡀⠘⠻⢾⣿⣿⣿⣿⣿⡇⠀⣿⣿⠀⠀⠉⠀⠈⠓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢤⡀⣀⠰⡎⠊⠙⠏⠁⠉⢳⣿⢇⣾⣽⣻⢿⣿⡇⠀⣿⡅⠀⣀⣀⣤⣴⣶⡿⡇
⠀⠀⠀⠀⠀⠀⢶⠘⠅⣌⠀⢁⠀⠘⣦⣀⣠⢏⣁⣾⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣯⣷⣿⡯⠇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢛⣉⠀⠛⠀⢀⢙⠩⢀⣾⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣟⣿⡋⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠈⢠⡾⠝⠩⣛⣷⣦⠀⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣷⣄⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠂⣻⣿⣇⣈⣤⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿⠿⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡇⠀⣀⡠⣟⣣⣴⣾⡇
⠀⠀⠀⠀⠀⠀⣿⠿⠛⠛⠛⠉⠀⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣷⣅⣀⣴⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠁⣽⣿⣿⣷⣧⣤⣥⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣥⣬⠿⡃
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠹⣿⣛⠛⠻⢿⣿⣏⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠀⠀⠐⡏⠠⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⡛⣿⣿⠛⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠧⠀⠼⠿⢇⡀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⡗⠀⠀⠀⠀⠘⣿⡿⠫⠀⠈⠉⠀⢀⣼⠁⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢸⡇⣷⠀⠀⠀⠀⠐⠻⠀⠒⠒⠠⠀⡠⢾⣿⠀⠛⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⢸⡆⠀⠀⠀⠁⠀⢸⠂⠐⡀⠈⣠⠈⣿⠐⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣼⣿⣦⣤⣀⣀⣼⡇⠀⣸⠇⠰⢻⡀⠀⠀⠀⠀⣦⡄⠠⠀⡇

⠀⠀⠀⠀⠀⠠⣒⣺⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠁⠘⡦⠀⠦⣤⣔⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡆⠛⠫⣿⢹⣼⣿⠅⠀⠀⠀⠀⠀⡓⠀⠘⢦⣀⡄⢫⠙⠒⠶⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⣐⣀⣦⣤⣈⣡⢿⣬⡀⠀⠀⠀⠀⠃⢐⠛⠘⣿⠋⠙⣳⣤⣿⡥⠐⠚⡆⠀⠀⠀⢤⣀⣀⠔⢞⠀⠁⠀⢙⡆
⠀⠀⠀⠀⠀⠩⡀⠁⠀⠁⡡⠄⠀⠁⠀⠀⠀⠈⠓⠤⢆⣴⣛⠶⠖⠃⠉⠀⠀⠀⠀⡇⠀⠀⢠⢲⠊⡁⠐⠀⣠⡀⢢⣼⡷⣀
⠀⠀⠀⠀⢺⡀⣀⡍⣠⣄⣁⣽⡀⠁⠀⠀⠸⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣨⣛⡁⠀⠈⠋⠀⢁⣈⣤⡴⠿⠂
⠀⠀⠀⠀⠀⣏⠉⠁⠉⠀⠉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠋⠀⠀⠀⢠⠀⠀⠉⠉⠁⠀⠈⠀⠀⢀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣣⣤⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠉⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢰⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣶⣄⣀⣀⣀⣀⣤⣼
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢛⣛⡛⠻⢟⣛⠕
⠀⠀⠀⠀⠀⠀⠀⣥⠀⠁⠀⠀⠀⠀⠀⢀⠆⠀⠀⢀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠢⣌⢣⣀⢸⡀⢠⣴⠁⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢙⠀⠀⠉⠉⠁⠉⠉⠙⠧⣀⡠⠊⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢀⡀⠀⠤⠒⠂⠐⠀⠠⠤⢄⠀⠈⠢⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⢒⢶⣧⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠠⡄⠀⢀⡄⢠⡄⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠄⣥⣮⣽⣿⠿⢀⠞⠋⠔⠡
⠀⠀⠀⡆⠠⠀⠚⠄⣸⣦⣾⢤⠖⢀⠆⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠂⠀⣄⡈⠛⠛⠋⠉⢉⡉⠙⣁⣔⣀⣀⣀⡀
⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠀⠉⠙⠋⠁⠉⠉
⠀⠀⠀⠂⠉⠢⠖⠐⠂⠄⠒⠀⠤⣄⠏⡴⠀⠀⠀⠐⠒⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣀⣀⠀⢀⣠⡄
⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⢀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⠀⠀⠀⠈⠛⠿⠛⠺⠽⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠒⠐⠃⠐⠒⠒⠂⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⡀⠑⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠸⠀⠘⠀⠐⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢈⢩⡙⠘⠒⠁⠀⠀⠆⠈⠀⠀⠁⠁⠢⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠘⣄⠢⡑⠀⠀⢠⠀⠀⠀⠂⠈⠀⠀⡆⠀⠀⠈⢆⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠈⠑⠀⠩⡥⠀⠀⠀⠀⠁⢈⠀⠀⠸⡀⠀⠀⠀⢣⠀⠡⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣁⣨⠀⠀⢄⢡⠀⠀⢕⢄⣇⠀⣁⠀⠄⠑⣇⠠⠀⠀⠀⢀⠂⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⢀⠔⠼⠢⠠⠀⠊⠀⢣⠀⠀⠦⢻⢀⢀⠠⠌⠦⠧⢀⠀⠀⠀⠀⠀⡠⠜⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⢲⠒⣦⣀⠀⠒⠐⣬⠡⡆⢤⡾⢠⣌⣒⣐⣒⠶⡊⠭⠩⠉⣁⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠑⠒⠒⠳⠤⠤⠜⠋⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠠⠀⢀⡀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢂⠀⠁⠀⠀⠐⠀⡰⠂⠀⢀⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠤⢀⡀⡑⢄⠀⠀⠈⠹⠀⠀⠀⠀⡏⠁⠀⠀⢀
⠀⡎⠀⢴⡁⠉⠉⠉⢇⣀⣀⠀⠀⡁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠁⠈⠀⠈⠀⠀⡀⠉⡀⠀⢀⡀⠁⢠⠀⠀⣠
⠀⠈⠒⠤⠤⣀⡴⠀⠀⠀⠀⠉⠉⠉⠒⠀⠀⠰⠤⠄⠀⠀⠠⡚⠀⠀⠠⢀⠁⡀⠉⠁⢠⠀⠀⠈⠐⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠧⢄⣀⠴⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠀⣀⣀