Chemist Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢄⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⠀⡀⡼⠿⠄⠀⠀⠀⣠⠤⡤
⠀⠀⠀⠀⠀⠘⢶⢿⣿⡽⠿⣿⣿⠀⢠⠀⠀⠀⠀⣠⡞⡇⠀⣠⣶⣴⣾⡟⢦⡀⠀⠀⠈⠻⡵⠃⠀⠀⠀⠈⣁⣨⣦⠤⠤⠤⠴⠶⠒⠊
⠀⠀⠀⠗⠀⠀⢸⢷⠀⠀⠀⢽⣿⠀⠀⠀⠀⠀⣰⡷⠛⡄⠀⣻⣿⢟⣽⡓⡆⢳⠘⠤⡀⠀⠀⠀⠀⠀⠀⡎⢉⠁⠈⠲⣄⠀⣎⠀⡶⢀
⠀⠀⠀⠀⠀⢀⠎⢌⠀⠀⠀⣼⢺⠀⠀⠀⠀⠀⡷⠒⠚⣃⣀⠈⠉⠛⠶⠞⣁⣸⠀⠀⠀⠀⠀⢰⠶⠰⡶⠙⢄⠀⠀⠀⠀⠳⡀⠀⠀⠉
⠀⠀⠀⠀⠀⡜⣀⣬⣅⣤⣴⣶⣼⡆⠀⠀⠀⠀⢱⡀⣾⢋⣩⣳⠀⠀⢀⣼⣿⡟⠀⠀⠀⠀⠀⠀⡰⠚⠋⠑⠒⢿⣄⡐⣈⢀⡧⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠐⠅⠑⢌⡻⡿⠟⠀⣾⣿⡿⠋⠀⠀⠀⠀⠀⢀⡀⣧⡀⠐⠊⠀⠁⠀⠻⡉⠉⠀⠈
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡠⣒⢦⡀⠀⠀⣈⠉⠉⠉⠉⢁⡤⣄⠀⠀⠀⠘⠛⣈⡶⠟⢛⡓⢶⡒⠒⠒⠚
⠀⠀⠀⠀⠘⢿⣿⣛⣿⣿⣿⣿⠿⠟⢱⢹⠧⢗⡇⠀⠸⣱⠗⠀⠀⠀⠘⠢⠚⠀⢰⣲⡀⠘⠉⠀⠐⠀⠀⠀⢱⠀⠀⠀⠖⠫⠰⠄⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠈⠉⠀⣀⠀⢀⣀⡀⢀⣤⡘⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡀⢀⡞⠉⠀⢸⡄⠴⡀⢸⠤⠄⡰⣆⠶⡀⢸⠀⠸⢄⠀⠀⣏⠀⣤⢴⡁⠀⡏⠁⠀⠈⠉⢿⣾⡟
⠀⠀⠀⠀⣦⢴⢠⡆⣶⢠⠀⡄⣄⡄⣀⠈⠦⠤⠄⠘⠀⠀⠃⠘⠒⠀⠁⠉⠀⠁⠘⠀⠐⠋⠀⠀⠁⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠃⠈⠈⠀⠁⠈⠀⠀⠋⠃⠛⠰⠀⠀⠀⠀⢀⡀⢀⣤⣤⠀⠀⢀⣤⡀⢀⠀⢀⣤⣤⡄⣀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠲⠲⠶⢤⣀⣀⣀⠀⠀⠀⠀⢼⠀⢸⠀⠀⢸⠉⣿⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠘⠁⢸⠿⡏⠉⠉⢩⠉⡇⠈
⠀⠀⠀⠀⢀⠠⠄⣰⠋⠀⠀⠀⠈⠁⣀⢹⣇⠜⠁⠀⠀⠀⢹⢶⣾⠀⠀⢸⠀⣿⠀⠀⠀⣿⣿⡇⢸⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠠⡇⡇
⠀⠀⠀⠀⠀⠁⢠⠃⠀⠀⢀⠀⠀⣪⡽⠟⡇⠀⠀⠰⢶⡶⢾⣾⣿⠶⠶⢾⣷⣿⠶⠶⠶⣿⣿⡷⢾⠖⠀⢻⡟⠀⠸⢉⣇⣀⣄⣼⣥⡇
⠀⠀⠀⠀⠀⡴⣻⣄⣐⠬⠔⠚⠉⠀⡴⢡⠇⠀⠀⠀⠀⡇⢸⣿⣿⠀⠀⠸⣿⡇⠀⢀⣀⣿⣿⣇⣸⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠛⠉⠙⠦⣀⣀⢀⣀⡠⠞⢁⣤⣤⠀⠠⢼⡷⠾⠿⠿⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁⠀⣦⠀⠀⠈⠻⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣭⣥⠀⠀⢸⠸⡮⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⣉⠇⠀⣀⣸⡋⠙⣳⠀⠀⢄⢀⠤⠒⢤⣌⣀
⠀⠀⠀⠀⠬⠁⠀⠴⢒⡚⠿⢍⣹⠋⢹⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⡷⠀⡀⠀⣡⠶⠛⠉⣂⠝⣀⡇⠀⠀⢰⠃⣠⣀⣀⠺⣗⠂
⠀⠀⠀⠀⠀⢠⠲⡛⠛⣻⠉⣇⠀⠐⠸⠀⠀⠀⢚⢠⠒⠁⢢⣴⠶⠺⣄⣀⠜⠀⠈⠉⡼⠋⡠⠞⠉⡝⢀⣽⠀⠀⠐⢸⠀⣧⡄⡞⣶⡟⠂
⠀⠀⠀⠀⠀⢸⠀⣉⠁⠀⠀⡇⠀⣀⢀⡇⢠⠀⠀⠉⠦⠤⠞⠀⠀⠀⠀⠀⠀⠀⠠⢸⠁⡼⠁⠀⣶⡇⢸⣧⣄⠀⠀⠁⠳⣼⡕⣷⢟⠑
⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⢇⠀⠀⠀⡇⠀⠿⠃⠀⣀⣤⠀⠀⣀⣀⠀⠀⠀⠀⠀⣿⠀⢇⡀⠀⠙⣤⡿⠜⠁⠀⠀⠀⠀⣸⢳⣇
⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⠀⢸⣀⣀⡡⠜⠀⠀⢀⣼⣿⣏⣀⣰⣿⣃⣴⣮⣱⠀⠀⢿⠀⣶⣯⠙⠓⢻⡖⢢⡀⠀⠀⠀⠀⣿⣿⣿⢀⢀
⠀⠀⠀⠀⠀⢸⣀⡠⠤⠤⠤⠼⠁⠀⠀⠀⠀⠚⢻⣷⣆⠀⠀⢹⣷⡄⠉⠉⠁⠀⠀⠈⢷⣬⣅⣀⣀⣀⣱⠞⠀⠀⠀⠐⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠂⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠒⠒⠒⠀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠠⠀⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠒⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠤⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣄⡀
⠀⠀⠀⠀⡄⠁⢀⠄⢄⠀⠉⠀⠀⠈⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⠉
⠀⠀⠀⡗⡁⠀⢈⠀⢈⠀⠀⡁⢈⠀⠀⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠉⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣁⡌
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠐⠐⡁⢸⢀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠐⣇⣂⠿⡀
⠀⠀⠀⡇⠐⠄⢠⠀⠀⠀⠀⡇⢰⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣈⠛⠀⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⣀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠃⠂⠘⠈
⠀⠀⠠⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠄⠘⠚⠀⠀⠒⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠠⠄⠈⠁⠠⠭⠬⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⡀⢀⣤⠀⣠
⠀⠀⠀⠀⢰⠋⠙⠦⠤⠤⠤⠤⠜⠋⢱⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⠖⠶⣧⠀⢀⡇⠠⡟⣋⡛
⠀⠀⠀⠀⢸⠀⣀⣀⣀⣀⣀⣀⣀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡎⠑⠒⠢⠤⠄⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠨⢐⡂⢸⠁⢙⡀⢠⢣⣛⣸⠆
⠀⠀⠀⠀⢸⢀⣒⣒⣒⣒⣒⣒⣒⣒⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡁⢸⠀⠐⢅⢨⠀⡆⠒⡀
⠀⠀⠀⠀⢸⠀⠒⠒⠒⠒⠒⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⢘⢨⡅⢺⠀⠸⡤⠰⠀⠊⠯⠇
⠀⠀⠀⠀⢸⠀⠨⠭⠭⠭⠭⠍⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠰⠄⢺⠀⠀⡇⢸⠀⢸⢻⡼
⠀⠀⠀⠀⢸⠀⠨⠭⠭⠭⠭⠍⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣴⣿⠀⢘⡇⢘⠀⢸⣶⣾⡆
⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠁⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⠉⠉⠿⠆⠾⠇⠠⠀⠀⠯⠤⠇

⡿⢿⠿⠿⡿⢟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣻⣿⣿⣿⣻⣿⣛⣿

⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡠⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠒⠒
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣞⣀⣈⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⢠⡞⠙⠉⠉⠉⠉⠀⠈⠉⠉⠉⢏⠃⠀⠀⠀⠀⠀⠀⠀⠠⠴⠒⠒⠢⠴⠦⢄⣀⣀
⠀⠀⠠⡟⡀⠀⠤⠀⠀⠀⠀⠠⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠑⠒⡄
⠀⠀⠐⠑⡀⠀⠀⠀⠀⠀⠀⠉⠉⢁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⣠⡇
⠀⠀⠀⠈⠊⠢⠄⢀⡀⢀⣀⠀⠀⠁⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠉⢇⠀⢀⠠⠂⢈⠢⠀⢃⠀⢀⠀⠀⠰⣗⠀⠀⡠⠀⠀⠀⢪⡆⠀⠀⢀⡄⠀⢠⡄
⠀⠀⠇⡀⠁⢀⠀⠀⠀⠈⠀⠀⠀⠙⠀⠑⠐⢤⡚⡤⠀⠀⠀⠂⣧⢀⠄⢠⠄⠁⠰⡈⠠
⠀⠀⠀⠀⠀⠈⠈⠁⡇⠈⠀⠀⠁⠐⠈⠐⠀⣀⣉⣃⡄⡀⡞⠀⠈⢇⡀⢸⠈⠀⠸⠁⠸
⠀⠁⠀⡀⠀⠀⠠⠤⠅⠤⢄⠀⡄⠂⠢⠀⠀⠐⠒⠄⠋⠇⠑⠀⠀⠁⠀⠈⠀⠂
⠚⠓⠚⠓⠚⠒⠒⠒⠒⠒⠚⠒⠛⠒⠚⠒⠚⠒⠒⠒⠒⠚⠒⠂⠀⠀⠄⠀⠆⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠢⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠀⠀⠀⠀⠘⠋⠑⠢⣤⣄⣀⠀⢀⡀⡀⠀⣀⣀⡀⢀⣤⢀⢠⠀⢀⠀⡄⣤⢀
⠀⠀⡠⣺⡀⠀⠠⢰⠉⠉⠰⡿⡀⠀⠇⠀⠀⠀⢀⣄⠔⠊⠈⠀⠀⠀⠈⠁⠉⠀⣀⣀⣀⠀⠀⣈⣀⣀⠀⠀⠉⠁⠈
⠀⠀⠉⢹⢷⠀⠁⠀⣦⡤⠼⠒⠚⠲⣛⠒⠦⡀⠀⠁⠀⠀⠀⢀⣀⡀⠀⠀⢠⠀⠯⠟⠟⠀⠘⡟⠩⡟⢠⡀
⠀⠀⢀⢾⠀⠀⢠⠊⠀⠱⣇⠀⠾⠀⠸⢆⢀⠜⠀⠀⠀⠄⠀⣷⣖⣺⠇⠀⠀⠀⢰⠀⠀⠀⠀⡅⢐⠇⠀⠀⠀⣘⠋⠁
⠀⠀⠀⠉⣤⡀⠈⠢⡦⢀⣏⣱⢤⣤⠖⠚⡇⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⢸⠀⠀⠀⢠⣄
⠀⠀⠀⠀⢙⠒⠀⠀⠀⠀⢣⠀⠀⣸⠑⠒⠁⠀⠀⠀⠀⠀⢀⠟⠀⠈⢄⠀⠀⠀⢸⣿⣾⠀⠀⣿⣿⡆⠀⠀⢸⣿
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠸⠈⠢⠚⠁⠀⠀⣀⡀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢣⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠲⠪⣿
⠀⠀⠀⠀⠈⠘⣦⠀⠀⠀⢀⣤⠴⠞⠒⠉⠁⢱⠀⠀⠀⠸⢄⣀⣀⡀⠤⠔⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⡆⣀⣀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠻⠟⠁⠀⠃⢹⣾
⠀⠀⠀⠀⢀⠄⠀⣠⡤⠶⠞⠋⠉⠀⢡⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⡀⢀⣀⠀⠀⣄⠀⠀⢸⣯⡀
⠀⠀⠀⠀⠀⠈⠀⢱⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢇⠀⠰⠴⠺⠠⠀⠁⠛⠲⠀⠀⠀⠉⠈⠉⠈⠀⠁⠀⠀⠈⠋
⠀⠀⠀⣴⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠼⢠⠀⠀⠀⠀⡿⠺⠄⢘
⠀⠀⠀⢸⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠈⠆⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢄⠀⠀⠀⠀⡠⠧⡀⠀⠈
⠀⠀⠀⢸⠀⠀⡀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⢀⠔⠋⠀⠀⠑⢤⡠⠊⠀⠀⠈⠢
⠀⠀⠂⢻⠉⠀⢣⠀⠀⠈⡄⠀⣀⠤⠄⠒⠒⠃⠀⠀⠀⡐⠀⡀⠀⠈⠉⢣⠀⠀⠀⠀⠀⢀⢣
⠀⠀⠀⢸⠀⠀⠀⠑⢄⠀⠘⠈⠀⠀⠀⠀⠀⠀⠘⡀⠴⠘⢂⠋⢠⠀⠀⠀⢆⠀⠀⠀⠀⡎⠀⠄⠀⠀⡀⡀
⠀⠀⠔⢸⠒⠂⠀⠀⠀⠉⠒⠒⠆⢴⠀⠀⠀⠀⠀⠐⡆⠙⡰⠆⠁⠀⠀⠀⣨⠒⢒⣒⠒⠀⢀⠞⠉⠁⣴⢶⣄
⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⢰⡆⠙⠂⠍⠡⠁⠀⠀⠀⠀⠐⠁⠀⣷⣦⠇⠀⠈⢀⡄⠀⠻⠿⠃
⠀⠀⠠⢼⠤⠤⠤⠤⠼⠤⠤⠤⠴⠤⢇⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⡄⠀⠁⠀⣀
⠀⠀⠐⠏⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠀⣄⡴⠆⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣽⣤⣴⣊⡤⠄⢀⣀⣀⠀⠀⠀⠀⣀⡤⣴⠶⠿⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠸⣿⡟⠉⠀⠀⠀⠈⠁⠁⠘⠓⠲⢶⡖⣿⣿⡗⠁⠀⠀⠀⠀⠀⢸⡁⣹⠄
⠀⠀⠀⠀⠀⣠⣿⣄⠀⠀⠀⠙⢦⡀⠀⠀⡴⠻⣆⠀⠀⠀⠀⢨⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⡼⠋⠁
⠀⠀⠀⠀⣠⣿⣿⡿⣇⡀⠀⠀⠀⠉⠛⠛⠁⠀⠻⢦⣀⣀⣠⠾⠋⠀⠀⡠⠒⣤⣀⣠⠂⠹⡇
⠀⠀⣴⣾⣁⣁⣘⡄⣈⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠐⠤⠊⠀⢀⡷⠚⠓⡴⠙⣦
⠀⠀⣿⣳⣷⣾⡿⡥⠄⢸⣿⠆⠀⠀⠀⠀⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣷⠀⠀⠘⠒⠋
⠀⠀⠈⠛⠻⠿⠿⠭⠶⠚⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠣⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢶⡶⠶⡶
⠀⠀⠀⠀⠿⠿⡟⠘⠟⠿⡇⢻⢿⣿⠘⠟⠙⡇⢸⣿⣿⠀⠀⠀⠀⠀⢀⣡⣤⣄⡀⠀⢸⠀⠀⡇
⠀⠀⣀⣆⣿⣶⣇⢨⡆⠠⡇⣸⣿⢁⠀⡧⣠⡇⢸⠉⠁⣺⣤⠀⠀⠀⠈⡟⠚⢻⠁⠀⢸⠀⠀⠆
⠀⠀⠺⣿⢻⣟⣃⠼⡷⢞⣡⡾⠿⡞⠀⠓⡞⡃⠘⣶⣞⣸⡇⠀⠀⠀⠀⠇⠀⢸⠀⢀⠆
⠀⠀⢚⡏⣿⠟⡏⢸⠟⠛⡁⢹⡿⠉⠈⡟⠉⡇⢹⠉⠙⢹⡇⠀⠀⠀⠀⡄⠀⢸⠀⡈⠀⡀
⠀⠀⠀⡇⠸⠀⡇⢸⡄⠀⡇⢸⡇⠀⠀⡇⠀⡅⢸⠀⠀⢸⡇⠀⠀⠀⢀⡇⠀⠸⣤⡟⠛⠃⠔⠺⠛⡆
⠀⠀⠀⡇⡀⠀⡇⢸⡇⠀⠃⢸⡇⠀⠀⡇⠀⡇⢸⠀⢸⢸⡇⠀⢀⣎⣁⢀⠀⠀⠀⢉⡀⠀⠀⠀⠀⣿
⠀⠀⠥⣇⣄⢠⡇⢸⣇⣰⡄⣸⠃⣀⢀⡇⢠⡇⣸⠀⣨⣸⡇⠀⢸⣷⠖⠒⠀⠂⠒⣶⣿⣤⣤⣤⣺⣿⠂
⠀⠀⠴⣏⣹⣿⣧⣀⣛⣛⣓⣈⣻⣇⣀⡛⣟⣃⣘⣿⣿⣼⣧⠀⠘⣿⣆⣀⠀⠀⣤⣼⠋⠉⠉⠉⠁
⠀⠀⠰⠋⠉⠉⠈⠉⠁⠉⠁⠈⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠯⠀⠀⠈⠙⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠃⠀⢀⣄⠠⢾⠋⠀⠀⠀⠀⠀⠀⢨⡀⠤⢆⡵⣠⠀⠀⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠁⠀⠀⠀⠀⠀⠀⡅⠀⠀⢘⣄⠀⠀⢀⠀⠠⢀⡴⠇⠚⢌⠅⠀⠀⠀⡖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠀⠀⠀⠀⠀⠘⠂⠁⢀⠀⣐⠀⠀⠀⠀⠀⠀⢈⠐⠀⠀⢸⠀⠀⠀⠀⠔⠱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣻⠀⡐⠀⢀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢻⠲⠄⠀⠀⡀⠠⠂⠀⠀⣀⣼⣖⠈⠀⠘⠼⠱
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣿⠘⠏⡋⢘⠀⡀⠀⠈⢀⡀⡀⣀⡀⠸⠀⠃⢠⣠⠇⠀⠀⠔⠊⠀⢸⠀⠀⠠⠤⡎⠁
⠀⠀⠀⠀⠀⠀⢠⢾⡿⠁⢁⠀⠐⠓⠀⠀⠳⠷⢤⠀⠀⠉⠉⢐⡀⠨⠀⠀⠁⠀⠀⢀⣄⠀⠀⠈⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⢂⡀⠊⢐⣠⠄⠀⠀⠘⠀⠀⠀⠀⠀⠐⠀⠀⠂⠈⠠⠀⡸⣇⣠⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢀⠠⠀⠀⠀⢴⡤⠁⠩⢬⠥⣾⡦⠄
⠀⠀⠀⢀⣀⡠⠀⠀⡀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢭⠀⠀⠉⠁⠈⠃⠀⠀⠀⠀⢸⠁⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠑⠀⡠⠊⠀⠀⠀⠀⣩⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⡀⢘⡆
⠀⠀⠀⣠⠤⠼⠷⠀⢠⠀⢀⠲⠷⡽⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠠⠤⠤⠾⠷⠿⠗⠳⣷⣶⡲⠶⣶⠒⠲⢰⠀⢀⡀⢰⣾⣠⣤⣤⠴⣤⣄⢀⣰⣶⣚⣶⣶⣶⣤⣤⣤⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠘⠋⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠀⠀⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣯⣿⣿⣽⣯⣭⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⣠⠤⣄⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣒⢦⡀⠀⠀⠀⠀⠀⠀⣠⣷⣉⣀⣀⡈⠢⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣧⢏⡭⢽⡇⠀⠀⠀⠀⠀⠀⣿⠁⣀⠀⠀⠉⠑⢮⡢⣀⠤⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡉⠘⢩⣿⡄⠀⠀⠀⠀⠀⠙⡷⠭⣼⢄⡀⠗⢠⠋⠉⠀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⢿⣇⠀⠀⠀⠀⠀⠀⢷⡀⢈⡀⣉⠉⠁⢀⡖⠚⠁⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⣿⣆⠀⠀⠀⠀⠀⣼⠑⠤⣉⣁⡀⠴⠊⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠓⢒⣺⡏⠛⠀⠀⠀⠀⣠⢞⡿⢫⠟⢦⠀⣠⠔⢫⠙⣖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠁⠀⡇⢠⣖⣦⡰⠋⠠⢯⡀⢸⣄⣾⢯⠀⣠⠃⠀⢯⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡧⢾⠀⢹⣀⡀⠀⡞⠁⢸⠈⡟⠈⣻⠃⣀⡤⠼⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇⢸⡾⠏⠀⠈⠉⠛⠤⣸⠀⡇⡴⠁⠀⢙⡆⠀⠀⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢾⣺⠭⠄⠀⠀⠀⠀⠘⢦⡟⠀⢀⠔⠉⠀⣠⠊⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣤⠤⠴⢒⠏⠈⠙⢯⡙⡏⠙⢆⠀⣠⠟⡦⣅⠀⠀⡴⠁⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢻⡇⠀⠀⢹⣅⡴⠁⠀⠙⢶⡁⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣰⣒⣒⣒⣲⣠⡇⠀⠀⡞⡜⠁⠀⠀⠀⠀⠑⢄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⡇⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⠀⣠⠋⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠁⠀⠀⠈

⠀⠀⠀⢸⠋⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⠋⢻
⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣄⡀⠀⠀⠀⠀⠀⣠⣀⣴⡶⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⢀⠜⢻⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⡠⠊⢀⡌⠈⠢⡄⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⢀⣔⡁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠘⡄
⠀⣜⣀⣀⠈⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⡼⢄⣀⣠⣄⡀⢘⣆
⠀⢿⣿⣏⣹⣶⣦⣶⣾⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⡄⠘⠂⢠⣬⡁⠘
⠀⠈⠻⣿⣯⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣈⣁⣀⣀⣈⣛⣁⣀
⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⢸⣿⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠬⠅⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠘⠿⠃⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠭⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣦⣤⡇
⠀⠀⠤⣤⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⣿⡇
⠀⢀⣤⣿⣯⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡶⡇
⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⢢⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⢤⡀
⠣⡀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣹⣢⣖⣏⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⢪⠀⢰⡗
⠀⠈⢢⠀⠖⢱⣶⣶⡶⠀⠀⠀⠀⠀⠀⢔⠉⢱⣏⠠⠄⣳⡏⠉⡶⠀⠀⠀⠀⠀⠀⡈⣆⢀⡟⠉⢉⡏⠁
⠀⠀⠘⣸⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⢛⡲⢞⡗⠙⡏⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⢸⠆⠘⠁
⠀⠀⠀⠛⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠉⢆⡰⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣤⣤⣤⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣨⠽⠛⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢶⣶⣾⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡄⢐⣶⣀⠀⠀⠀⠀⠀⠀⠘⠳⡌⣿⣿⣿⣟⣛⣥⡀⠀⠀⠀⠀⢶⠒⣶⣶⣶⣶⣶⣶⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⢀⣀⣀⣠⢮⢿⡟⠁⠀⠀⠀⠀⠀⠈⠑⠀⠈⠛⠛⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢻⣶⣶⣶⣦⡆⢸⣧⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⣿⣿⣷⣾⡇⣻⣿⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣿⣿⣿⠇⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⣀⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⡿⠛⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣆⣿⣻⣿⣿⠁⠀⠀⠀⣿⣿⣿⣶⣶⣦⣶⣶⣶⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⡀⢀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣥⣄⣽⣿⣟⣴⠟⣛⣟⣟⣛⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⣿⠃
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡏
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠟


⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⡷⣥⣦⣵⡞⠁⠀⠀⠀⠀⠀⠀⢠⡉⠟⠀⠀⠀⢰⣿⣿⣿⡿⣿⣆⠀⠀⠀⠀⠀⠘⠆⣠⣃
⢀⠎⢀⢹⣿⢇⠘⢆⠀⠀⠀⠀⠈⣷⣿⠉⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⣿⡀⠀⠀⠀⠀⠀⠰⠿⠻⠃⠄⠒
⠳⠼⠧⠤⠤⠤⠽⣮⠇⠀⠀⠐⢾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣷⡄⠀⠀⠀⢀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣀⡀⠀⠀⠀⠀⠐⠛⠂
⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠄⠀⡄⠓⢄⡀⠀⠶⠀⠀⠀⠎⠀⢹⣿⠃⠉⢄⠀⠀⠀⠀⠀⣤⡀
⠀⠻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠤⡀⠥⢀⠘⠁⠀⠦⠀⠀⠘⢰⣶⠖⠓⣶⣶⡎⠀⠀⠀⠀⠀⣻⡃
⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠭⣀⠀⢈⠻⠁⠀⠀⠀⠀⣶⣿⣿⣦
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⣀⠀⠇⢀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣠⣠
⠀⠀⡏⠈⡇⠀⣼⠀⠀⠀⠐⠿⡂⠉⢠⡀⢠⠈⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣤⣀
⠀⠠⠿⠄⡇⠀⣤⠀⠀⠀⠀⢀⣈⢶⠎⠀⢶⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁
⢀⡀⠀⢀⡇⠀⠋⠀⠀⠀⠀⠙⠁⠀⣦⠀⠀⠓⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⢻⡆
⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⢶⣤⣴⣧⡀⠀⠀⠀⠀⠀⠀⠘⠛⣀⠛⠃⠘⠓⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀⠀⠀⠀⡰⡺⠉⠉⢓⠢
⠀⢰⣋⣰⣏⣀⡀⠀⠀⠀⠐⠋⠈⠉⠀⠉⠑⠚⠃⠀⠀⠀⠠⣶⣾⡄⠀⣿⡇⠀⠀⢀⡴⠻⠦⠠⠞⠳⡆
⠀⠀⢩⣟⠉⡉⣿⠀⠀⠀⠀⣀⢀⠀⡀⢀⡀⢀⡃⠀⠀⠀⠀⠠⣿⣿⡿⠟⠁⠀⠀⠀⠄⢀⢀⣇⣠⠀⠄
⠀⠀⠀⠉⠛⣻⠋⠁⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⣴⣿⣾⣿⣷⣶⣆⠀⠀⠀⠀⠉⠋⠈⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠖⠒⠒⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⢲⠂⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⣷⣶⣤⣶⣶⠆⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠄⢄
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠑
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⣸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢣⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣏⣿⣿⠿⠁
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⣿⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠛⠛⠛⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⣿⣿⡟⣻⣿⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣯⣿⣽⣿⣿⣿⡿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠯⠭⠯⠭⠽⠽⠽⠭⠊⠀⠀⠀⠀⠀⠀⠀⠑⠽⠿⠿⠿⠿⠿⠿⠿⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣼⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠠⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⡇⡇⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠨⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢺⢰⠀⠀⠀⢀⣈⡃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠨⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣬⣬⣤⣤⣤⣤⣴⣧⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠶⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠇⢣⠀⠀⢀⣴⡆⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣴⣦⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠘⡀⠀⠀⠠⠄⣿⡇⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣤⡄⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⣇⠀⠉⠩⠅⣿⡇⠀⠀⠀⠀⠀⢸⢀⣀⣀⣀⣈⣀⣀⣀⣀⡈⡇⠀⠀⠀⠀⠀⠀⢸⣿⠠⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⢱⢱⠀⠈⠉⡅⣿⡇⠀⠀⠀⠀⠀⠸⡽⡀⠀⠸⡻⢀⠰⠆⣸⣳⠃⠀⠀⠀⠀⠀⠀⢸⣿⠠⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⢇⣀⣘⣾⣀⣂⣂⣠⢟⡇⠀⠀⠀⠀⠀⠀⠹⢟⣦⣄⣀⣻⣣⣔⡵⠋⠀⠀⠀⠀⠀⠀⢀⣼⣿⣈⣀⣸⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠛⠛⠛⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠓⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣏⣉⣙⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⡴⠶⠶⠶⢶⣦⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⢸⣵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠉⠉⠉⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠒⠒⠒⢮⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠇⠀⠀⠀⠀⠀⠘⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣀⣸⡞⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠠⠼⢿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠠⠬⣧⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠤⠽⡎⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣤⠤⠤⢤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠠⠬⣹⡸⣷⡀⠀⠀⠀⠀⠀⠀⢸⣼⡀⠀⢯⡽⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⢀⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠣⢿⣇⠀⠀⠀⠀⠀⠀⠀⢧⢳⡀⠀⠀⠰⡷⠀⣾⠆⠀⠀⠀⠀⢀⡜⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡳⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠜⡼⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠢⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⢀⡀⢀⠀⠈⡀⠀⠀⠀⠀⢠⣄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠿⠿⠿
⠀⠀⢾⣶⣶⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠛⠑⡒⠂⠘⢢⡄⡔⠁⠀⠑⠊⠂⠘
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⡄⢀⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢼⡿⡷⡶
⠀⠀⠀⠰⠰⠂⠴⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⠃⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠃⠘⠂⠛⠀⠓⢘⡃⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠀⢀
⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠁⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠄⢀⢠⠄⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘⠘⠃⠁⠉⠈⠀⠀⠀⠀⡀⡀⠀⠘⠶⠤⠤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠈⠀⠀⡀⢄
⠀⠀⠰⣶⣶⣶⣶⡆⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⡏⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⢐⠆⣀⢷⡄
⠀⠀⠐⠻⣿⣿⡿⠛⡀⠀⠠⠀⠀⠐⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢳⡜⢆⠏⠛⡾⣠⠈⠀⡜⠀⢸⣼⡇⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢠⣴⠶⠶⣶⠄⠴⣦⢴⣦⡀⢴⡆⠀⠀⠀⠁⠀⢸⠀⢀⣇⣸⡄⢸⠁⡇⠐⡇⡜⠀⡯⢻⠃⠀⣶⡆
⠀⠀⢀⠄⠹⡇⠸⠏⠘⠀⠿⢀⠉⣀⢸⠇⠀⠀⠀⠀⠀⠈⠁⠚⠚⠁⠧⠞⠀⠈⠜⠁⠁⠒⠃⠘⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠀⠄⠀⠀⠀⠀⠀⠀⢀⣻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣤⡀⠀⠀⠀⢠⢀⡒⡁⠀⠀⢠
⠀⠀⢀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣻⣻⣦⡀⠀⠀⠠⠖⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠂⠀⠀⠀⢀⠀⢀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⢀⣶⡄⠀⢀⠀⢀⣀⠀⠀⢰
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠠⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢼⣿⣿⠀⠘⠒⠂⠂⠇⠀⠸
⠀⠐⠛⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠐⠂⠀⠀⠀⠐⠐⠀⠐⠀⠀⠃⠀⠀⠃⠀⠐⠀⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡠⠶⠙⠿⠃⠶⢄⣤⡀⠴⠊⠻⠛⠲⠄⣠⣄⠠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠘⠿⠏⠀⠀⠀⠀⠀⠈⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣆⠀⠀⠀⠀⠀⢸⣿⡷⡀⠀⠀⠀⠀⢀⡀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢼⣿⡆⠋⠁⠀⠀⠙⢲⣿⣷⠚⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠙⠛⠳⢦⢀⣤⣄⠀⣀⣀⣀⣀⣀⣀⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠈⣭⣭⣭⣭⣭⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣛⠿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢘⣻⣭⣭⣭⣝⣛⠿⠿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡟⡛⣿⣿⣷⣶⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⡟⣭⢻⣿⣾⣿⣿⠿⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣷⣶⣿⣿⣿⣿⣇⡻⢃⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠈⠉⠙⠛⠛⠋⠉

⠀⠀⠀⢀⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⡷⣧⣤⣝⣶⠃⠀⠀⠀⠀⠀⠀⠀⣀⠛⠇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠰⠀⣰⣁
⢀⡜⠁⢰⣿⣞⠈⢦⡀⠀⠀⠀⠀⢐⣶⡿⠁⠀⠀⠀⠀⠈⠉⠙⢿⣷⣼⣿⡆⠀⠀⠀⠀⠀⠀⡼⡿⢿⠃⠀⠆
⠮⢴⣎⣀⣉⣀⣱⣆⣳⠀⠀⠀⢦⣾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢴⠦⠄
⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⡀⠀⠀⢆⠀⠀⠀⣄⠀⠀⠀⡠⠋⢹⣿⡟⠑⢄⠀⠀⠀⠀⠀⢀
⠘⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠁⠀⠀⡇⠀⡱⠀⠀⣀⠀⠀⠀⢁⣤⣄⣽⣡⣤⣌⡀⠀⠀⠀⠀⠸⣷⠆
⠀⠈⠻⣿⣿⣿⣿⡏⠀⠀⠀⠈⠑⠁⠈⠲⠀⠀⠁⠉⠀⠀⠀⠺⡿⠋⠀⠙⠿⡟⠀⠀⠀⠀⢀⣤⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠐⠊⠁⠀⠀⠀⠀⠸⠿⣿⡿⠗
⠀⠀⠀⠀⡄
⠀⠀⣶⠀⡄⠀⠔⡀⠀⠀⠀⢀⣘⠁⣤⠀⠀⠀⠴⠀⠀⠀⠀⠀⠐⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣷⠀⡇⠀⠰⠇⠀⠀⠀⠘⠛⢄⣀⠶⠀⡶⠀⡀⠀⠀⠀⠀⠰⠿⠿⢆⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣶⠦
⠀⠐⠛⠂⡃⠀⢰⡆⠀⠀⠀⢀⣤⠜⢻⠂⠘⠛⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣰⣄⣀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠉⢀⡘⠗⠀⠀⠁⠀⠀⠀⠀⢄⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠉⠉⠉⠉⠉⢹⡆
⠉⠉⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢶⣤⣴⣧⣄⠀⠀⠀⠀⠀⠀⢀⠉⢁⡘⠙⣀⠛⢃⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣆⠀⠀⠀⢀⢞⡟⠉⠉⢙⠲
⠀⢰⣏⣠⣟⣀⡀⠀⠀⠀⠀⠸⠉⠉⠉⠈⠉⠁⠘⡃⠀⠀⠀⠠⣾⣷⡇⠀⣸⣿⠀⠀⠀⣤⠞⠳⠄⠰⠚⠳⡇
⠀⠀⢉⣿⡋⢉⢹⠆⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠂⠂⢿⣿⣿⠿⠃⠀⠀⠀⠠⡀⡀⢸⡀⣠⠀⠄
⠀⠀⠀⠙⠛⢻⡟⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⣠⣶⣾⣿⣿⣶⣶⠀⠀⠀⠀⠈⠛⠋⠉⠑⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣒⣚⣉⣉⣉⣍⣉⣉⣙⣒⣒⣶⡦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠰⢞⣉⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣷⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠓⠶⠤⠭⠭⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠤⠴⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⡶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⣴⡶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣀⡀⢠⠀⠀⠠⠄⠠⠀⠄⠠⠄⠠⠀⣉⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡾⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠎⠑⠠⢤⣤⣤⣤⣀⣀⣀⣀⣤⣤⡤⠤⠄⠀⠀⠈⢧⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⠋⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡱⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡜⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡼⠁⠀⠀⠀⠀⢯⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡜⢦
⠀⠀⠀⠀⠀⠀⣠⢏⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⢧⡀
⠀⠀⠀⠀⠀⡰⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢳⡀
⠀⠀⠀⠀⡼⣱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡱⡄
⠀⠀⢀⠞⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡙⣄
⠀⢠⢎⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣆
⠠⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠼⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠃
⣦⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠
⢯⠳⢭⣓⠲⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⣋⡵⢫
⠀⠑⠦⣄⣉⠙⠒⠲⠮⠭⢭⣥⣒⣒⣒⣒⣒⣢⣤⠤⠤⠤⠤⠤⠤⠤⠒⠒⣒⣒⣒⣒⣒⣋⣩⣭⡭⠤⠶⠒⠚⣉⣠⠴⠋
⠀⠀⠀⠀⠉⠉⠛⠒⠲⠦⠤⠤⣤⣀⣀⣈⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣀⣠⡤⠤⠤⠶⠒⠚⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠘⠁⠀⢈⠂⠀⠀⠀⠀⠺⣿⣿⣿⡿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⠿⠿⠇⡄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠜⠀⠀⠀⠀⠣⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣷⣷⡄
⠀⠀⠀⣀⣀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⣧⣦⣤⣤⡄⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⣿⣿⣿⣿⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣿⣿⣿⠀⠀⠈⣿⣿⣼⣿⣾⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⡟⣾⣿⡇⠀⠠⣶⣶⣶⣶⣶⡆⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣠⣿⣿⣿⣄⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠿⣿⣿⣧⣿⡿⠃⠀⠀⠙⠿⢿⠿⠟⠁⠀⠸⢿⣿⣿⣿⣿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⢤⣤⣤⠀⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⠀⠈⠉⢱⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣀⣸⣄⣼⣀⣸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣾⠀⢰⣤⣼⠀⢰⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢙⣉⣻⠋⢻⣉⣻⠋⡟⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⣿⠿⢿⣿⣿⠿⢿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⢸⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⠀⢸⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⢸⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⡂⢸⣿⣿⠀⢸⣿⡿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⣿⣶⣾⣿⣿⣶⣷⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣷⣶⣶⣿⣷⣶⣾⣿⣶⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⡀⠀⠀⠀⠀⠠⡆⠀⠚⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠁
⠀⠀⠀⠀⠐⠁⣠⠀⠠⡌⠁⠀⠀⠀⠀⢀⣦⠈⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠠⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡂⠉⢂⠤⢄⡀⠀⠀⠀⠀⠀⠀⠐⡐⢀⣀⡀
⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠸⠅⠀⢨⡀⠀⢠⠀⠀⠀⠀⠈⠉⡅⠒⢦
⠀⠀⢀⡄⠴⠥⠤⠤⠤⠤⠴⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠙⠶⠃⠀⠀⠀⠀⠀⠘⠀⠀⠸⢧⡆
⠀⠀⠂⢵⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢈⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰
⠀⠀⠀⢸⡀⠀⠀⠀⣀⡴⠁⠀⣯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠈⠉⠀⠀⠀⠂⢀⡀
⠀⠀⠀⠨⠟⠀⠀⠈⠋⠀⠀⠀⠀⠴⠀⠀⠀⠀⠰⠀⢡⠀⠀⠀⠀⠀⠈⠁⠃⠙⠿⠶⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠘⠀⠀⠀⠀⠀⠀⠀⠰⢸⠀⠁⠀⠀⠰⢁
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣤⣤⡤⠀⠀⠀⠀⢰⠁⣇⠀⠄⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢧⠀⠀⠀⠀⠀⠁⠁⣆⡆⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠀⠀⠀⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠲⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣴⣾⣿⣷⣤⣤⣤⣤⣤⣴⣤⣦⣶⣶⠄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠂⠠⠀⠀⡂⠀⠀⠤⠀⡂⠀⡇⠀⠂⠀⡆⠀⠂⢠⠀⠀⠀⠀⠀⠀⢰⠀⠰
⠀⠻⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣯⡀⠀⠐⠃⠐⠐⠀⠂⠂⠂⠐⠀⠒⠀⠐⠂⠂⠒⠃⠐⠃⠘⠐⠂⠂⠒⠀⠂⠐⠂⠚
⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢂⡀⠀⢀⡐⢄⠀⠀⠠⢀⡁⠀⠀⣁⠢⠀⠀⢀⠔⠈⢠⣄⠈⠂⡀⠀⠀⡄⠊⠀⠀⢈⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠠⡀
⠀⠈⢀⣼⣶⣒⣾⣧⡀⠄⠠⢡⣿⣷⣀⣰⣿⣧⠀⠀⠀⠀⣠⣼⣿⣆⡀⠈⠀⡐⠀⠀⣴⠟⣦⡀⠀⠂⢀⣀⡤⠀⠤⠄⠀⢄⣡
⠀⢀⠈⠈⠧⣶⠼⠀⠁⠀⠐⠀⠀⠀⣽⣅⠀⠀⠀⠀⡀⢸⣟⠁⠋⢿⡇⠀⠀⢡⠀⢠⠿⠾⠋⠁⠀⡄⠈⠹⣀⣀⠞⢆⣀⠼⠘
⠀⠀⠁⢐⠚⠛⠓⡂⠐⠀⠀⠐⠄⡘⠛⠛⠀⠔⠁⠀⠈⠨⡉⠉⠉⠉⠡⠂⠀⠀⠲⠼⠀⠀⠀⡀⠈⠀⠀⠑⠄⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⢀⠀⠁⠀⠀⠀⠈⢠⠀⠀⡐⠁⢰⣆⢠⠎⠐⠀⠀⠠⠊⣼⣷⣠⣶⣶⡄⠀⢀⠀⠀⠀⣦⠀⠀⠢⠀⠀⠔⠁⠐⣺⡀⠀⠀⡀
⠀⠀⢠⣶⣤⣶⣤⣤⣄⠇⠰⢰⣧⠀⡁⠀⣤⡄⠐⠀⣤⣶⣿⣿⣿⣿⡿⠃⠀⠂⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠠⠀⠀⠐⠀⠁
⠀⠠⡴⣿⣿⡿⢿⣿⠟⠀⠈⠀⠉⠤⠀⣠⣀⠀⡐⠀⠟⢿⠿⠟⠛⠉⠀⠀⠀⠃⠀⠀⠀⣇⠀⠀⢀⠂⠈⡀⠀⣶⣷⣦⡄⠀⡀
⠀⠀⠐⠠⠀⠀⠀⠄⠀⠀⠀⠈⠢⠇⠈⠉⠩⠀⠀⠀⠀⠀⠄⠀⠀⠠⠐⠁⠀⠀⠑⠠⠴⠿⠦⠄⠁⠀⠀⠈⠠⠿⣿⡿⠇⠊

⠀⠀⡠⢢⣶⣿⣷⡄⠢⠀⠀⢠⣾⣿⠀⠀⠈⠂⡀⠀⢀⢀⡵⠒⠒⠢⣔⠄⠀⠀⣤⠀⠀⠀⠀⠑⢀⠀⠀⡠⠂⠀⠀⠤⠀⠢⡀
⠀⠀⢀⣼⣏⣩⣹⣣⡄⠁⢀⠁⠀⠠⠀⣀⠀⠀⠀⠀⢠⡏⢰⣿⣿⡟⠛⡆⠀⡈⠹⣿⣦⣄⠀⠀⠀⠀⠀⠂⢸⣷⣦⠀⠀⠀⢠
⠀⠠⢀⣠⠽⣿⡿⣄⡀⡀⠈⡀⢠⣶⣶⣿⣿⣶⡶⠀⠘⣧⡴⣿⣿⡇⢠⢇⠀⢁⠀⠈⠙⠛⠳⠀⠀⠀⠀⠀⣼⣿⣿⣶⠀⠉
⠀⠀⠙⠅⢀⠀⢀⠨⠋⠀⠀⠐⠀⢙⠛⢛⠛⠉⠁⠀⠈⠪⠓⠦⠤⠒⠅⠀⠀⠀⠀⠄⡀⠀⢀⠲⠊⠀⠀⠑⢾⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠐⠂⢩⣷⣶⣄⠀⠀⠀⠐⠊⣇⠀⠗⠢⡀⠀⠀⠄⠂⠈⣩⠁⠢⡀⠀⠀⠀⠊⢀⠀⠈⠐⢄⠀⡀⠀⠀⠀⢀⠀⢀⢀
⠀⠀⠀⠀⠀⠾⠿⠿⠋⠃⠀⠛⠲⣀⡼⠦⣄⣀⣀⠀⠈⣰⣆⠀⣿⠀⣿⠀⡀⠀⠀⠀⣤⠷⣏⠀⠀⠂
⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡄⢀⣼⣁⡃⣸⡋⢉⠀⢸⣿⣿⣿⣿⣧⣿⠀⠃⢀⠀⢸⡄⠒⠨⠅⠀⠀⠀⠀⠀⠄
⠀⠀⠻⠛⠛⠒⠛⢃⠐⠁⠀⠘⠣⡎⠁⠉⢻⠄⠃⠀⠈⠻⣿⣿⣿⣿⡿⠊⠀⠀⠢⢀⢷⣤⡶⠡⠊⠄⠀⠀⠀⠀⠄⠠⠄⠠
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠠⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⢈⣉⡱⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⡇⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⢃⠀⠀⠀⡠⠐⠉⠀⡀⠀⠀⢤⠀⠉⠐⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢆⡔⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⢀⠀⠠⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⢈⠀⠀⠘⡄⠁⠀⠒⠀⠀⠀⠀⠀⠆⠈⡂⡽⠒⢄⠐⠕⡠⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡸⢃⡀⠅⠹⣄⡔⣉⣂⣄⣢⣠⣠⣄⣱⣤⣇⣁⣀⢑⢿⣋⣡⣴
⠀⠀⠀⠀⠀⠀⡐⠠⠊⠬⠾⠤⠲⢵⢬⣔⠷⢦⢹⡉⠑⢁⡬⠆⠄⠀⠀⠸⡬⠐⠒⢔⠀⠀⣊⡴⠇⡆
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠀⢀⡴⢻⠜⠣⡪⡧⢯⡁⢕⣙⠒⠲⣶⠀⢐⣌⡪⣐⣗⡡⠪⠂⢕⢇⠁
⠀⠀⠀⠀⡰⠀⠀⠀⠈⠂⠊⡤⣨⡂⢕⣕⣄⡪⢐⣎⡗⠻⣕⠀⠀⣀⠀⡋⠀⠉⠖⡡⢈⢔⢕⡽⠸
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠈⢍⠪⣺⣇⢠⡿⡊⣽⡿⣤⣷⡑⠉⢊⠤⡱⢂⢖⠭⡪⡱⢗⠵⡡⠁
⠀⠀⢠⠁⠀⠀⠀⠀⢅⠌⠀⠸⣋⣌⠞⡊⣕⠄⡪⢐⢡⣱⣹⡶⡋⢕⠅⡪⢚⢕⠈⣊⡔⢁⠔⠁
⠀⠀⠈⢔⠄⡀⠀⠀⠀⠀⠀⢨⠢⠐⠍⡪⠐⢅⢨⣲⡋⢇⡗⣬⣪⣕⡥⢪⠄⠵⠊⣉⠔⠁
⠀⠀⠀⠀⠈⠉⠀⠘⠁⠰⠶⠷⠾⠶⠿⠾⠿⠥⠉⠐⠊⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠠⡀⢀⠄⡀⢄⢠⠠⢀⠀⡀⡀⠄⠠⡀⢀⢀⠀⠀⠄⡀⠄⠀⡀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⣀
⠀⠀⠀⠀⠁⠈⠀⠀⠁⠈⠐⠁⠊⠀⠑⠁⠊⠂⠅⠊⠐⠀⠐⠑⠨⠀⠅⠊⠔⡩⠂⠕⠨⢂⠕⠪⠐⠡⡪⠠⠃⠔⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠉
⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⣤⣀⡀⠀⠀⢀⡀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡀
⠀⢀⡀⠀⠀⢁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⠙⠉⠑⠲⡀⠀⠀⠠⠀⠄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠐⢻⡁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠠⡆⠀⠠⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠅⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⡦⣀⠄⠀⠘⠛⡀⠀⠀⡤⠤⠶⠶⠶⢷⡤⠴⡏
⠀⠈⠀⠀⠢⡀⠀⠀⠀⠀⠀⠈⠃⠱⠠⠀⠐⠃⠨⠃⠀⠀⠀⣀⣈⣉⣀⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⠡⠋
⠀⠀⠀⠀⣰⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢘⢠⢁⣀⡄⠉⠩⠋⠊⠑⠀⠀⠀⠃⠀⣸⠀⠀⠀⠀⠀⣿
⠀⡤⠄⢠⣃⠀⠒⠂⠀⠀⠴⠶⠄⠀⠀⠀⠠⠤⠌⣀⠀⣀⡀⢀⠀⠀⢀⣆⠤⠬⠤⢬⠂⠉⠒⣐⣀⣀⣰⡿⠤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠖⠒⠚⢒⡶⠖⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡤⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⠋⠀⠀⢀⡴⠋⠀⢀⣀⣀⡀⠀⠙⢦
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡾⠀⢀⣞⣉⠀⠀⠉⢳⡀⠈⢧
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢰⠇⠀⡞⠈⠉⠉⠙⠓⠦⢷⣀⠸⡆
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⡀
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠈⣇⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠹⣄⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⢤⣀⠀⠈⠳⣄⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⢀⣀⠴⠒⠛⠛⠒⠦⣄⡀⠀⠀⢀⣠⠴⠒⠛⠛⠒⠦⣄⡀⠀⠀⣀⡠⠖⠒⠉⠒⠲⢤⣀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠖⠒⠋⠁⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠚⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣁⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠉⠉⠉⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠛⠛⠛⢻⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⢿⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣍⠛⢦⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟⠛⢳⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣉⡛⢶⣌⡛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠷⣤⠾⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣿⠉⣻⡆⠈⢛⣶⣬⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠙⠛⠋⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣴⠏⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠛⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢻⡆⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⣠⣤⣄⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠃⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠻⢦⠟⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠶⢤⣤⣼⣧
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⡀⠀⠀⢻⣄⠀⠀⠀⠀⢀⣀⣠⣤⠶⠞⠛⠉⠁⠀⢰⡟⢷⡄⠀⠀⣿
⠀⠀⠀⠀⠀⣠⡿⠶⠶⠚⠛⠛⠓⠶⠶⠶⢦⣤⣤⣤⠶⠶⠛⠋⠉⢷⡀⠀⢸⣏⠛⠛⣻⣿⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⢸⡇
⠀⠀⠀⠀⣰⡟⠀⠀⣴⠶⣦⠀⠀⣶⠛⣶⠀⠀⣠⣤⣤⡀⠀⠀⠀⠈⢿⡄⠀⢻⡄⠀⢿⣼⠟⠀⠀⢀⡴⠶⢶⡄⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⣰⠏⠀⠀⠀⠙⠷⠋⠀⠀⠘⠛⠃⠀⢰⣏⠀⠀⣿⠀⠀⠀⠀⠈⢻⡄⠀⠻⣦⠀⠀⠀⠀⠀⠸⣇⣀⣠⡟⠀⠀⠀⠀⢀⣰⠟
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⣦⣀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠈⠿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠞⠛⠉


⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦
⠀⣀⡀⠀⣠⣴⠎⠿⣿⡿⠃⢶⣄⡀⢀⣀⠀⢀⣤⡦⠹⢿⣿⠟⠰⣦⣀⠀⣀⡀⠀⣀⣤⡶⠂
⣼⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠉⢰⣿⣿⣿⡌⠁⠀⠀⠀⠀⠀⠀⠈⠁⣾⣿⣿⣧⠉⠁
⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣨⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠘⠛⠛⠁⠶⣦⡀⣠⣤⣤⡀⣤⡶⠎⠛⠛⠋⠰⢶⣄⢀⣤⣤⡄⢀⣴⠾⠛⠀⠀⠀⠀⣴⣦⣴⣦⣴⣤⣦⣤⣤⡆
⠀⠀⠀⠀⠀⠈⠰⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠁⢾⣿⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⢀⡘⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⢰⠾⠛⠁⠀⠉⠁⠈⠛⠷⢠⣾⣿⣶⡀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠁⠀⠀⢩⣭⣭⣭⣭⣭⣥⣭⣭⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣙⠛⠿⢿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⠀⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣶⣾⣿⣿⣿⣶⣶⣦⣤⣍⣙⠛⠛⠻⠿⠟⠛⠛⢛⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠿⣿⣿⣿⣄⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠠⣶⡄⣿⣿⣿⣿⣿⣿⣿⠟⢛⠛⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⣥⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠉⠛⠿⠿⢿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⠉⠉⠉⠉⠐⠤⡀
⠀⠀⠀⠀⠀⡠⠊⠁⡠⠄⠒⠈⠁⠀⠈⠁⠒⢄⠈⠳⡀
⠀⠀⠀⠀⡜⢀⠔⠁⠀⡀⠤⠀⠒⠂⠠⢄⠀⠀⠑⡄⠘⡄
⠀⠀⠀⡜⠀⠏⠀⢠⠊⢀⠔⠈⠉⠉⠢⡀⠱⡄⠀⠘⡀⠰
⠀⠀⠀⡇⠀⠀⠀⡎⠀⠎⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⢇⠀⢁⠀⠱⡀⠰⡄⠀⠀⠀⡠⠃⢠⠃⠀⠀⢀⠀⠃
⠀⠀⠀⠈⢆⠀⠀⢄⡑⣄⠈⢦⠒⠉⣀⠔⠁⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠑⠢⠄⣀⠘⡄⠀⡗⠉⠀⠀⠀⠀⠀⠀⠘⡀⠈⠄⠀⠀⢠⠤⠤⠤⣤⠀⠀⠀⠀⠀⡤⠤⠤⢤⡀⠀⠀⠀⠀⢠⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠒⠲⠻⠀⠀⠀⠘⡓⠐⠒⠐⠚⠃⠀⠀⠀⡟⠒⠐⠒⠚⠃⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⢀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⢤⣤⣤⣀⣰⠤⠀⠀⠀⠤⣧⣤⣄⣠⡧⠄⠀⠀⠤⢼⣤⣄⣀⡮⠄
⠀⠀⠊⠀⡄⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⡄⠀⢀
⠀⠀⠀⠀⠇⠀⠀⠀⠲⡉⠠⠆⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠐⠺⡄
⠀⠀⠤⠠⠄⠀⠀⠀⠀⠸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠠⡇⠀⠀⠀⠀⠀⠸⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⡇⠀⠀⢈⠁⠀⠤⢘⣋⠒⠂⣙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢣⠀⠈⠀⠀⠀⠀⠀⠀⠢⡀⠠
⠘⠉⡀⠂⡜⠇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠴⠃⡀⠂⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡎⠀⠠⠄⠀⠀⠐⠀⠊⠢
⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠂⠀⠤⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠂⠐⡄⠀⠀⠀⠀⡀⠀⠀⠐⡠⠀⠀⠀⠀⢀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠐⢧⡀⠠⠀⠀⠀⠤⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠈⡀⠀⠀⠀⠀⢰⠄⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠰⠄⠬⠄⠀⠩⠀⠤⠬⠥⠤⠤⠤⠤⠬⠤⠤⢤⡥⠤⠤⠤⠤⠤⡥⠤⠤⣬⠤⠤⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠀⠀⠇⠀⠀⠀⠀⠀⣿⡄⠀⣾⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠀⠀⠆⠀⠀⠀⠀⠀⠿⠃⠀⠸⠀⠀⠸⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⢸⠀⠀⠀⡅⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢸⠂⠀⠠⠃⠀⠀⠀⠀⠀⡇⠀⠂⡇⠀⠀⠀⠀⠀⢰⠂⠀⠆
⠀⠀⠀⠀⠀⠀⢀⡤⢒⣪⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣸⠆⠀⠀⠀⠀⠀⣥⠀⣀⠇⠀⠀⠀⠀⠀⢸⡀⠀⣄
⠀⠀⠀⠀⠀⣰⠞⡼⠋⠀⠙⠀⠐⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⡀⠀⣀⣿⣼⡁⠀⡠⠈⠉⠉⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄

⠀⠀⠀⠀⢠⠖⠉⠀⠉⠢⠀⠀⠀⡴⠂⠁⠐⠢
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠈⠢⣀⣀⡠⠞⠀⠀⠈⠦⡀⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠁⠀⠀⠀⠀⠀⠀⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⡈⠀⠀⡀⠀⠀⠀⠀⡠⠊⠀⡁⠂⠀⠀⠀⠀⠀⠀⣀⠥⢤⡁⠀⠀⠀⠊⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣀⠀⠀⠈⢂⠀⠀⢄⠀⠀⢀⠄
⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⡠⠒⠉⠉⠉⠲⡀⠀⠀⠀⠀⠊⠀⠈⢂⠀⠀⠀⠀⠀⠈⠑⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠
⡁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⠠
⠀⡠⠠⢀⠀⢇⠀⠀⠀⠡⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠈⠀⠀⠀⠑⠂⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠁⡸
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠑⠢⡀⠀⠁⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⡀⡄⢠⡄⢹⠀⠑⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⠀⡠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠁⠁⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠒⢄⠀⠀⠀⢀⠔⠋⠉⠉⠑⣄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠄⠀⠂⢈⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠔⠈⠁⠢⡀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠳⢄⣀⡤⠊⠀⠀⠀⠘⢄⠀⠁⠀⢀⠜
⡀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠈⠀⠀⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⠀⠀⡀⠀⡀
⣟⣿⣻⣻⣛⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣻⣻⣛⣿⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢰⢶⠷⢶⡶⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡿⠟⠹⠀⠀⠀⠀⠀⣸⡀⠀⠀⢠⠀⠀⠀⠸⠀⢀⣷⡀⠀⢠⣿⣬⣠⣀⡴⡟
⠀⠀⠀⠀⢀⡄⠀⠀⢰⣷⣍⠁⢀⠘⠀⠀⠀⠀⢀⡟⢃⠀⠀⠈⠀⠀⠠⠌⢅⣼⡿⠷⠆⠘⢏⣈⣉⣉⣡⡗
⠀⠓⠃⠀⠀⠁⠀⠀⢸⡟⠀⠿⣿⣆⠀⠀⠀⠀⡘⢠⣞⠞⠀⣀⡀⠀⢀⠀⠀⠙⠁⠀⠀⠀⠀⠉⠙⠃⠈
⠀⣤⡆⠀⠀⠆⠀⠀⢸⡇⢀⠀⠈⠉⣁⠀⠀⣠⣤⠞⠉⡷⠞⢣⡙⠈⠁⠑⠰⠾⠁
⠀⢠⣤⣤⡤⣦⣤⢶⡾⣷⡈⢻⣷⣾⡎⠳⠞⠿⠋⢀⡀⢱⣶⡞⣷⢰⡇⣤⡀⠛⠂⠀⠀⡀
⠀⠘⠋⠹⠇⠀⠹⠀⠈⠿⢁⠀⣿⣻⡧⠐⠃⠘⠘⠃⢠⢨⣿⠇⢸⣿⠀⡆⠿⠳⢦⣄⡘⠃
⠀⠀⠀⢀⣀⣀⣀⣀⠀⣰⠫⢏⡿⣿⣧⣤⣤⣤⣴⠀⠀⠰⠀⠀⢠⡟⢂⡀⠖⠀⠀⠈⠹⠓⢤⣀⠙⠉
⠀⠀⠀⠀⠠⠀⢾⠋⠀⣿⠀⠈⠃⣿⣷⣿⣟⣿⣿⠀⡀⠀⠀⠀⠀⠖⠛⠛⠀⠐⠓⠀⠄⢀⠀⠀⠗
⠀⠀⠀⠀⢀⡌⠆⠀⠀⣿⠂⠀⠀⣿⣿⣜⡟⠋⢸⢠⣼⣄⠀⠄⠀⠀⠀⠀⠀⢆⣤⡀⠀⠈⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠀⠀⡇⢀⣿⣷⠒⣿⣼⡟⡧⠀⠀⠀⠀⠀⠀⠀⢤⡉⠁⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠇⣤⠛⠩⠋⡃⢸⣟⠃⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠸⢄⣧⠀⠀⠘⠁
⠀⠀⠠⠠⠄⠀⠄⠚⠀⠿⠄⠀⢹⢠⠇⠀⠒⠒⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡼⠀⠀⠀⣀⡀⠘⠛⠁⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣿⣾⠟⣻⣏⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠻⠛⠓⠲⠷⡶⠂⠀⢸⠦⢿⠿⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢁⣁⢀⠀⠀⠁⣁⠀⣤⠶⠀⠀⠀⠄⠠⠐⠒
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⡖⠳⡀⠈⣿⡇⠀⠀⡐⠿⠀⠀⠀⣀⠰⠷⢽⡄⠐⢀⢈⣔⣁
⠀⠷⠖⠂⣀⡈⡅⠤⠀⠀⡿⠂⣄⠠⣟⡂⠀⠚⠋⠏⢹⢯⠀⢈⣾⠃⠒⠂⢀⡀⠀⠋⠁⠀⠀⠀⠀⠂
⠀⣤⣠⡸⣰⡇⠀⢠⢠⡤⢷⣁⡋⠀⣿⡗⠃⠀⠀⠁⠘⢆⢙⡎⣻⠇⠀⠀⣶⠀⠀⠀⡤⠀⡄
⠀⠛⠇⠙⠁⠘⣇⣈⣥⣴⢼⡏⡜⠀⠁⠉⠈⠈⡄⠑⣖⣀⡄⣠⣿⠂⠀⠀⣼⠄⠀⠀⡘⠆⠀⠘⠅⠴⠲⠆⢠
⠀⠂⢰⣶⣾⣿⡟⡟⠉⠻⠉⠁⠀⠘⠀⠀⠀⠀⠓⢀⠉⡜⡿⠽⠿⠧⠀⠀⠘⠁⠀⡀⠉⠈⠙⠂⠐⠂⠐⠲⠅
⠀⣿⠸⠟⠋⠛⠃⠁⠀⢸⠛⠋⠛⠛⠋⠉⠉⠉⠉⠹⡏⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⡀⠀⠀⣿⣶⣄
⠀⠁⠀⠀⠀⠀⠀⠒⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠂⡇⠀⠀⠛⠙⠋
⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⢲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠤⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡄⠀⠶⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠃⠀⠀⢠⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠈⠁⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡇⠀⠀⠀⢰⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣛⣢⠀⠀⠀⠀⢀⣠⠖⢁⣤⠖⠃⠀⠀⠀⠀⠀⠈⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀⠈⠉⠉⠒⠤⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡟⡩⠳⠀⣀⣴⣯⠵⠚⠉⡗⠒⠒⠒⠒⢲⠒⢲⠐⠏⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠉⠀⠒⠤⠀⣐⣿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡵⢀⣴⡾⠟⠋⠁⠀⠀⠐⢟⠓⠒⠒⠒⢲⡓⠒⢚⠟⢠
⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠷⢛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⠀⢀⡀⠤⠂⣠⡞
⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠳⡀⠀⠀⠀⢀⠼⠞⠒⢿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢴⠶⠞⠉
⠀⠀⠀⣀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠐⠆⠘⡄⠀⠀⡎⠀⠀⡀⣼⠱⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠣⣸⢄
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠘⠏⠁⠀⠀⠈⠦⣐⠃⠀⢰⠋⡟⢀⡾⠁⣀⣀⣿⣿⣿⠀⠀⠀⠀⠉⠙⠉⢹⠈⠁
⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⢿⢀⢤⠙⠀⡤⠃⠉⠉⣼⣅⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣸⣿⡇⣿⣇⡀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⣆⠠⠋⠀⠀⠇⠀⡀⠀⠀⢰⣿⢿⣾⣿⣻⣿⣿⣿⠀⠀⠀⢀⣼⡇⠀⢸⡇
⢠⣿⣟⣷⣿⣿⡿⡄⠀⠀⠀⠀⠀⠀⠣⡀⠀⣀⠠⠚⠁⠀⠀⠀⠀⠈⣈⠖⠦⠀⣿⣿⡀⣿⣿⣿⣿⣿⣿⠀⠀⢀⣾⠟⡀⠀⢸⡇
⢸⣿⣯⣿⣿⣿⣿⡜⠆⠀⠀⠀⠀⠀⢺⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⣀⣗⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣾⣏⡼⡇⠀⣸⣧
⣘⣛⣛⣛⣽⣿⣿⣇⠀⠳⣄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⠴⠾⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⢲⣿⠟⠒⠒⠾⠒⢻⣿⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠈⠳⣄⠀⠀⠀⠀⢀⣀⠠⢰⡋⠀⡄⠀⠀⢀⠀⠀⣠⣤⡤⠤⠤⠠⠀⠀⠄⠀⠾⠃⠀⠀⣰⠰⠄⢾⣿
⠀⠠⠤⠤⠴⡌⡃⠀⢻⠀⠀⠀⠀⠑⠀⠄⠒⠉⠀⠀⠈⠁⠀⠇⠀⠐⠘⣀⡀⣡⣎⢰⣶⠌⢻⣏⣠⣤⠀⣠⣀⣠⠆⠈⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⡇⡃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣐⣈⣻⣡⣭⡷⠶⠻⠋⠋⣿⣽⠃⠈⣩⡹⣇⠀⣹⡇⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠇⡇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣯⡿⢩⠀⣄⡄⠀⠀⡀⠀⣿⣿⡆⠒⡇⢠⠙⠛⣻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣿⣇⣼⡄⣽⡷⠘⣤⡃⡀⢸⣛⣇⡞⠁⠈⢢⣴⣿⣿⣟⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⢚⣿⡏⣿⡇⣼⡷⢶⣿⣿⣿⠿⠿⢿⡗⢲⣶⣼⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⣀⣠⠤⠄⠒⠊⠉⠀⠀⠀⠀⣠⣼⣿⣿⣿⣷⡿⠟⠛⠉⠁⠀⠀⠀⠘⣟⠀⣿⠃⠠⠦⠄⢳⠈⠋
⠀⠀⠀⠀⣀⣀⠾⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣻⣤⣤⣤⣤⡇
⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟

⠀⠀⡀⠀⠀⠀⠀⠑⠒⡎⠀⠀⠀⠀⠀⠀⠘⡆⠀⠴⠁⠀⠀⠠⠀⠀⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢘⠀⠀⠈⠁⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⡰⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠦⠒⠊⠀⠀⠀⠀⢠⠃⠀⠀⠀⢣⠀⠀⠀⠠⠊⠉⠒⠤⡤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⢄⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠄⡀⠀⠀⠪⠀⠀⣀⡌⠀⠀⠀⢀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀⠁⠀⠀⡀⠄⠀⠉⠀⠀⠀⠀⠀⠑⠂⠀⠐⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶
⡀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠥⢀⠀⠀⠀⠖⠁⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠈⠁⠢⠀⠀⠀⡎⠉
⠑⠊⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡡⠀⠀⠢⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⣤⠀⠀⠀⠈⠉
⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠈⠄⠀⡤⠒
⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠵⠒⠉⠉⠉⠲⣄⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀⠄⠨⡀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠤⠤⠤⠊⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠙⡄⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠒⠐⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠚⠓⠒⢤⡀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠘⢤⡀⠠⠄⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠔⠉⠁⠉⢢
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢆⠀⢀⠃⠀⠀⠀⡂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⡀⠀⠀⠀⢬
⠀⢠⠔⠒⠢⠀⠀⠀⡠⠒⠒⢧⠀⠀⠀⠀⠀⢀⠜⠉⠉⠑⡄⠀⠈⠀⠀⠀⠀⠀⠱⢄⣀⣀⠴⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⠫⠂⡔⠒⡄⠀⠀⠀⠀⠐⠠⠀⠤⠓
⠀⠀⠄⠀⢠⠃⠀⠀⡄⠀⠀⠀⠱⡄⢀⡀⢠⣜⠀⠀⠀⠀⡇⠀⠀⢀⣄⠢⠀⠀⠀⠀⠀⠀⣀⠴⠖⠲⠦⡴⠁⠀⠀⠑⡄⠀⠀⢘⠒⠁⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⢀⡠⠂⠀⠀⠀⠈⠲⠤⠤⠊⠀⠀⢀⠊⠈⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⡇⠀⠀⠪⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤
⠀⠀⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢀⡰⠁⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠂⠒⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂
⠀⠀⢀⠀⠀⣸⠀⠀⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡔⠀⠐⢄⠀⠀⠀⠈⠀⠁⢀⠄⠀⠀⠑⢄⣀⠀⠀⢀⠔⠁⠀⢀⠄
⠀⠀⠀⠉⠉⠀⠀⠀⠠⠒⠀⠀⠀⠈⠀⠀⠈⠀⠀⢄⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⠉⢢⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡱⠀⢀⠤⠤⢀⡤⠀⠀⠀⠀⠀⠀⡔⡄⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢂⠴⠒⢲
⠀⠀⠙⠂⡰⠁⠀⠀⠠⣐⣄⣀⠰⡌⠀⠀⠀⡃⠀⠈⡀⠀⠀⢀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠤⣄⠀⠀⠀⠈⢡⠇⠀⠂⠁⠀⢰⡉⢹
⡀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠑⠤⠊⠀⠀⠀⠈⠒⠀⠀⠀⠀⠇⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠲⡁⠀⠀⠀⠀⠈⠣⠜⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠙
⠁⠀⠀⠀⠈⠁⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠂⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⡳⠔⠦⠆⠀⠀⠀⠀⠀⠀⠖⠁⠀⠉⠢
⢄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠙⣄⠤⠄⣀⡰⠉⠀⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⠉⠁⠀⠀⡠⠀⠠⡀⠀⠀⠈⡵⠀⠀⠀⢀⠇⠀⢠
⠀⠉⠁⠀⠀⠀⣃⠀⣀⠀⡐⠁⠀⠀⠀⠙⡄⠀⠀⠀⣁⠀⠀⣀⠒⢂⡀⠀⠀⠀⣠⢞⡉⠀⠀⠈⠓⣤⢊⠉⠐⣄⠀⠀⣇⢀⠀⡈⠀⠀⠀⠈⠂⠠⢴⢊⠀⠀⠈
⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣿⡿⣿⣿⣏⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⢿⣿⣧⠈⢿⣿⣆⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣿⣄⠹⣟⣷⡜⢿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣮⣿⣁⠳⣿⣷⡝⢿⣿⣌⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⠻⣿⣯⠙⣿⣿⣿⣿⠛⠛⠿⠿⠿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠙⣿⣦⡹⣿⣷⡙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣷⡜⣿⣷⡈⢿⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣤⡤⢄⣀⣀⣀⣾⣿⣿⡌⢿⣿⡈⢻⣷⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⣿⢿⣿⣼⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢻⣷⡝⢿⣷⡝⢻⣿⣿⠉⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⢻⣿⣎⠹⣿⣾⣿⠋
⠀⢀⣤⣦⣤⣀⣀⠀⠀⠀⢀⣼⣿⣿⠹⣿⣆⢹⡿⣶⣿⣿⠏
⠀⣾⣿⣿⣿⣿⣿⣿⣟⣛⢿⣿⣿⣿⣤⣿⣿⣿⣿⣿⠏⠁
⠀⣾⣿⣿⣿⣿⡻⢿⣿⣟⣿⣿⠿⠿⠿⠿⠛⠋⠁
⢀⣿⣿⣿⡏⢿⣿⣄⣹⣿⣿⠃
⣿⣿⡏⢻⣿⣄⣿⣿⣿⠟⠁
⣿⣿⣷⣿⣿⣿⠿⠛⠁
⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠂⣨⡤⠖⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣭⣧⣤⣚⣡⠤⢀⣀⣀⠀⠀⠀⠀⢀⣠⢴⡶⠾⠗
⠀⠀⠀⠀⠀⠀⣶⡇⠀⠀⠘⣿⡟⠋⠀⠀⠀⠈⠉⠉⠈⠛⠒⢶⣖⣾⣿⡿⠊⠀⠀⠀⠀⠀⠀⣏⢈⡧
⠀⠀⠀⠀⠀⣠⣾⣤⠀⠀⠀⠙⢧⡀⠀⠀⣰⠟⣦⠀⠀⠀⠀⠀⣿⣻⠟⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉
⠀⠀⠀⠀⣠⣿⣿⡿⣧⡀⠀⠀⠀⠉⠙⠛⠁⠀⠘⢧⣄⣀⣀⠼⠛⠁⠀⢠⠔⢦⣀⣀⡔⠈⣯
⠀⠀⣴⣾⣡⣁⣘⡆⢈⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠦⠜⠁⠀⣸⠖⠛⢢⠎⢳⡄
⠀⠀⣿⣳⣷⣾⡿⣥⠄⢸⣿⠇⠀⠀⠀⠀⠙⠿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣹⡆⠀⠈⠓⠚⠁
⠀⠀⠈⠛⠻⠿⠿⠿⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⣶⠶⣶⠆
⠀⠀⠀⠀⢿⠿⠟⠘⡿⠿⡇⢻⢿⣿⠀⡟⠋⡗⢘⣿⣿⠃⠀⠀⠀⠀⠀⣨⣤⣤⣀⠀⠀⡇⠀⢸
⠀⠀⣠⣆⣺⣶⣇⢀⡇⠀⡇⢸⣿⢉⠀⡧⣤⡇⢸⡏⠉⢰⣦⠀⠀⠀⠀⠹⠓⠛⡏⠀⠀⡇⠀⠰
⠀⠀⠺⣿⢻⣟⣃⡴⣷⢶⣣⡾⠿⡞⠀⠓⡶⣃⠘⣷⣾⣋⡇⠀⠀⠀⠀⠸⠀⠀⡇⠀⡰⠀⠀⠀⢀
⠀⠀⢚⡏⣿⠟⡏⠈⡟⠛⡇⢸⣿⠉⠀⡟⠉⡏⢸⠋⠙⢹⡇⠀⠀⠀⠀⢠⠀⠀⡇⢠⠁⢀⠀⠀⠈⠄
⠀⠀⠀⡗⢸⠀⡇⠀⡇⠀⡇⢸⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢀⡼⠀⠀⢧⣾⠛⠛⠠⠒⠟⢳
⠀⠀⡄⡇⣀⠀⡇⢀⡇⠀⡇⢸⡿⢠⠀⡇⠀⡆⢸⠀⢰⢸⣇⠀⠀⣔⣁⣀⡀⠀⠀⡈⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠥⣧⣤⢀⣇⢀⣇⣠⡇⢸⠇⢘⠀⣧⢀⡇⢸⡇⢸⣸⡇⠀⢸⣿⡖⠒⠂⠐⠒⢲⣿⣧⣤⣤⣜⣿⡗
⠀⠀⠴⣏⣹⣿⣷⣀⣙⣛⣛⣈⣻⣏⣀⣙⣿⣓⣈⣻⣿⣎⣧⠀⠈⢿⣧⣠⠀⠀⢰⣴⡟⠉⠉⠉⠁
⠀⠀⠰⠋⠉⠈⠉⠉⠁⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠻⠄⠀⠀⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣷⣆
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣦⣤⣄⠀⠙⠃⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠁⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡀⠈⢿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠘⢿⣦⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣿⣿⣿⣷⡆
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⣀⣠⡼⠿⣟⠋⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⣿⡿⠋⠉⠀⠀⠈⠉⠻⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣾⣿⠿⠋⣀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣿⣤⣀⣀⣀⣀⣠⡴⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⡿⠾⠿⠿⠿⠿⠷⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠴⠀⠀⠉⠒⠂⠀⠀⠛⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢨⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⠀⠀⠀⣸⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢛⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⡸⠀⠀⠀⠀⣤⣴⣶⣶⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠿⠂⠀⠀⠀⠀⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠃⠀⠸⡿⠿⠿⠿⠛⠁

⠀⣁⣈

⠀⠀⠀⠀⢀⠠⠐⠀⢈⠉⢉⠀⠀⢄
⠀⠀⠀⠔⠁⠐⠀⠀⠀⠄⢀⠀⠐⠀⠡
⠀⠀⠈⠀⠀⠠⠀⠔⠀⠀⠢⠐⠄⠀⠀⠁
⠀⠀⡄⠀⠀⠇⠘⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠐⠀⠀⠈⢄⠑⡤⠒⣁⠔⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠆⠘⠈⠀⠀⠀⠀⠀⠀⠠⡀⢀⡶⠶⢲⡀⠀⠀⢠⠶⠴⢦⡀⠀⠀⡰⠶⠴⣄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢁⣀⡀⢀⡁⠀⠀⣀⣀⡀⢸⡁⠀⢀⣀⣀⠀⣌
⠀⠀⠁⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢨⠈⢀⠁
⠀⠀⠀⠀⠀⠀⠡⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⠁⠈⡄
⠠⠀⠀⠐⠀⠀⡔⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⢄
⠈⠀⠒⠚⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠦
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠑⠤⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠠⠀⠀⠄⠀⠀⠀⠆⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠉⠀⠉⣍⠉⢉⠉⠉⠉⠉⡍⠉⠉⠉⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⠀⠈⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠅⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⡠⣤⡊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⠄⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⠇⠀⡄
⠀⠀⠀⢀⣎⠞⠁⠘⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉
⡄⠀⠀⢚⣛⡂⠀⠀⢀⠀⢰⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠀⡤
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠀⠀⣀⣀⡀⢀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠘⣤⣤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⠇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠐⠂⠣⢄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠄⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⡇⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⣀⣀⣀⣀⣀⣀⡁
⠀⠀⠀⠀⠀⠐⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢁⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡷⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣼⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢿⣷⣿⡿⠫⠊
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⣿⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢻⣿⣿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠉⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠀⠀⠀⠀⠀⠘⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣟⠉⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢿⣿⣿⣿⣏⣿⣏⣿⣿⣿⣿⣿⣿⢯⠇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⡣⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠉⠉⠉⠁⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⡀⠀⠁⠂⠀⠀⢄⠀⠀⠀⠀⠀⠈⢀⡠⠀⠺⠠⠤⠀⠀⠀⠀⠂⠂⠉⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠘⠆⠔⠀⠂⠀⠀⠀⠠⠂⠀⠲⠀⠘⣲⠎⢥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠲⣆⠀⣶⡄⠀⣀⡀⠀⣶⡄⢀⠠⠊⠀⠀⡘⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠐⠈⠐⢨⠁⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠁⠈⠉⠁⠀⠉⠁⠈⠉⠉⡃⠀⠀⠀⡄⠈⠢⡀⠀⠈⠀⠀⠀⠀⢀⢔⠕⠄⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⡆⠀⠀⠄⠂⠁⢢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣀⠁⡀⢁⠀⣁⠈⢀⠀⡀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢋⡃⠀⠀⠀⡠⠁⠈⠀⠀⠀⠀⡈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢈⠁⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⣉⡀⠀⠀⠀⠐⡀⠀⠀⠈⠈⠢⣀⡌⠀⠀⠁⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠍⣉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⠀⠈⡀⠀⡐⠀⠀⠀⢀⠁⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢰⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠡⠀⠡⡀⠀⠀⠀⢀⠒⠂⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠛⢶⠇⡇⠀⠀⠀⡖⠈⠀⠉⠐⠀⠀⠀⠸⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⢀⠘⠀⠄
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⣀⣀⣀⣀⠀⢢⠀⠀⠢⡀
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠈⠂
⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠧⠄⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⡆⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢲⢰⠖⠆⠴⠲⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠲
⠀⠀⠀⡏⠀⡙⢙⠀⠀⣀⡀⠀⠀⠀⠀⢠⡦⡧⢼⢼⠤⠦⠤⠴⢴⡆⠀⠀⠀⠀⠀⠀⡏⠀⡙⢙⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⠀⡸⠀⢨⡀
⠀⠀⠀⡧⠰⣧⣤⠇⠀⠀⠂⠀⠀⠀⠀⢸⡇⡇⢸⢸⠀⡂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡧⠰⣧⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⣀⡀⢀⣈⡆
⠀⠀⢀⣇⣀⣈⣁⣀⣰⣶⣾⡀⠀⠀⠀⠸⡇⠓⠚⠘⠒⠃⠒⠚⢸⡇⠀⠀⠀⠀⠀⢀⣇⣀⣈⣁⣀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⢀⡴⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠒⡆⠸⠛⣷⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣋⡓⠢⣄
⠀⠀⠀⠀⠀⠸⣇⠀⠀⣸⠇⠀⠀⠀⠀⢠⡞⠉⠉⢉⣉⠉⠉⠑⣦⠀⠀⠀⠀⠀⠀⢧⣤⡇⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⡞⣤⣄⠛⠁⣶⡜⡆
⠀⠀⠀⢀⣠⡾⠋⠛⠛⠁⠀⠀⠀⠀⠀⠘⠧⣀⡠⠋⠈⠆⣀⡠⠛⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣷⡇⠀⠀⠀⠀⠀⠀⠀⢧⠀⣦⠀⣴⣬⢁⠇
⠀⠀⠰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠇⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⣄⡠⠴⠋



⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢃⡀
⠀⠀⠀⠀⢀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢴⣷⣮⡤⠀⡀
⠀⠀⠀⢐⣯⡫⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣬⣥⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⡿⣿⠑⠈⠁
⠀⠀⠐⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⠿⠿⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠳⠀⠀⠁


⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣬⡀⠛⠛⢩⣥⠀⠀⠀⠀⠀⠀⢀⢀⡠⠦⠀⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⡀⠀⠀⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⣀⠰⠆⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢠⣤⣀⣈⡄⢀⣦⣤⣆⠀⠀⠀⠀⠀⠀⠀⠃⠀⢨⡄⣀⠰⠦⡄⠀⠀⠀⠀⠀⠀⢀⣷⣶⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⡟⠁
⠀⠀⠈⠛⠿⠛⠁⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠿⠷⠄

⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠟⠛⠿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏
⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣄⡀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠋
⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠉⠉⠀⣹⣿⡇⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⠆







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡄⠀⠞⠒⠤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⣾⣿⣿⣶⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢁⡴⠛⠁⠀⣆⣇⡴⠋⠉⠉⠉⠁⠀⠀⠀⠀⡎⠉⠉⠉⠉⠉⠁⠀⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠃⢀⠴⠊⢁⡤⠛⠁⠀⠀⠀⠀⠈⠁⠀⢴⢿⠆⠀⠀⣠⠀⡀⠀⠱⡀⠀⠀⠀⠀⢀⠜⠀⠀⠠⣳
⠀⠀⠀⠐⠈⠀⢿⣅⣠⠖⠉⠀⠀⣤⡄⠀⢠⠆⠀⡇⠀⠈⠉⠉⠀⠀⠁⠀⠇⠀⠀⠈⠒⣤⣴⠶⠁⠀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢠⡄⠈⠀⠀⢺⣀⣞⣀⣀⠀⠀⡄⢀⡀⠀⢠⠀⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣀
⠀⠀⠀⠀⠀⠘⠁⢰⣖⠤⣀⠀⢀⢀⠀⠀⠀⡟⢉⣁⣹⠃⠀⠈⠀⠀⢠⠃⠀⠀⠀⠀⠈⡕⠂⠉⠰⠦⠀⣀⢠⡯⠚
⠀⠀⠀⠀⠀⠀⢀⡮⠉⣻⠚⢁⠀⠂⠂⠀⠀⠙⠋⡼⢉⣹⠂⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣹⡎
⠀⠀⠀⢀⡤⠒⠊⠁⢠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⡹⠋⠉⢱⠄⠀⠀⠑⢄⠀⠀⠀⣀⠇⠀⠀⠀⢠⠀⣽⢒⡲
⠀⠀⠀⣿⣿⣶⣦⣄⠘⡆⠀⢰⡠⡇⡀⢰⠆⠀⠀⠀⢧⠄⠀⠉⠀⠀⠀⠀⠀⡄⠈⠁⠀⠀⣀⡀⠀⠸⡟⠉⠁
⠀⠀⠀⠙⢿⣿⣿⣿⣷⠇⠀⠘⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠘⣄⡜⠀⠊⠀⢀⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⢀⠤⣶⣶⣶⣐⡦⠀⠟⠳⠄⠇⠀⠀⠀⢹⡀⠀⢀⣴⠷⣴⠁
⠀⠀⠀⢀⠠⡞⢶⠀⠀⠀⠀⢾⣁⣀⣀⡤⠆⠀⠀⠈⢺⢻⠟⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⣴⠛⠁⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣴⠚⠋⠿⣅⠀⠀⠀⠀⠀⠀⠀⡞⣷⡏⢹⡄⠀⠀⠀⠀⠀⣤⠀⠀⠘⠂⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠠⠚⢧⠀⠀⠈⢛⣦⣄⣼⠆⠀⡀⠀⠀⠀⠀⢹⢻⣷⠘⣇⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⢾⣉⣁⣹⠀⠀⠘⠉⠈⠀⠀⠠⠈⡼⣿⡄⢿⠀⠋⠉⠹⡇⢱⠆⠀⠀⠀⠀⠀⠀⠘⠬⠜
⠀⠀⠀⢠⣄⠀⠀⢠⠖⠊⠛⢿⡉⠀⠀⠀⡤⠄⢤⠀⡀⠀⢣⢿⣷⣸⡇⠀⠀⠀⡸⢧⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠈⣁⠀⠀⠙⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠯⠚⠀⠀⠀⠘⠃⠀⢱⡄⠀⠰⣮⣭⣭⣵⠂
⠀⠀⠀⠘⠖⢃⣘⡛⠀⠀⠈⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠇⡇⣀⡤⠒⠁⣠⣀⠀⢸⠀⠉⡇
⠀⠀⠀⠀⢐⠛⠉⠉⢹⠛⠉⡇⠀⠀⠀⠀⠀⠀⣀⡀⠤⠶⠶⠤⠤⣀⣀⣀⣀⣩⣤⡄⠀⠑⠋⣠⠜⠀⠀⠡⡀
⠀⠀⠀⠀⠈⡄⠀⠀⢸⠇⠀⢰⠀⠀⠛⠋⠉⠉⠙⠓⠤⠤⠤⠤⠜⠋⠉⠁⠀⠀⠀⠠⣶⠀⠀⣷⣶⣿⣿⣿⡟⢄
⠀⠀⠀⠌⠀⠉⠉⠹⠥⠤⠔⠚⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

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

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⢟⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠁⠁⠀⠈⠈⠩⢿⣿⣿⣿⣿⠀⣾⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣻⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣿⣿⠀⣿⣿⣿⠿⣿⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠇⠀⣿⡟⣷⣿⣿⠿⠿⠯⠀⠘⢷⣦⣭⣙⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠘⠛⠀⣥⣤⠄⢴⣿⣷⣿⣿⢿⣿⣿⣿
⣿⣿⣿⡿⣡⠆⢀⣿⣿⣿⣿⠷⠁⢦⠀⠰⠀⢀⣿⣿⣿⡛⠛⣻⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⡄⢰⣿⢀⣿⣿⣐⠱⣿⣿⣿⣿⡿⣹⣿⣿
⣿⣿⢣⣾⡟⠀⣾⣿⣿⣿⣿⡇⠀⡳⣶⢓⠶⣟⣻⣿⣿⡇⠠⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⠃⢀⣉⣾⣿⣯⣧⡴⣿⣿⣿⣿⣇⣿⣿⣿
⣿⣡⣿⣿⠇⢰⣿⣿⣿⣿⣿⡇⠀⣿⠿⠞⠓⠛⢋⣁⣤⡄⠐⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠌⠿⢿⣿⣿⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠏⡾⣾⣿⣿⡿⠛⣛⡁⠀⣤⢶⣶⣿⣿⢹⡟⢷⡧⠠⢽⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠶⣾⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣟⣿⣿
⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⢆⠀⣆⣄⢌⡍⡛⢻⢾⡆⡇⠀⠈⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠰⣈⣃⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⣹⡎⠀⢻⣿⢿⢾⣆⠄⠀⠈⠀⠀⠀⠀⢠⣮⣧⡤⠯⠏⠙⠛⠀⢈⣄⠀⠈⢛⢻⠏⠫⠿⠉⠙⣿⣿⣿⣿⣿⣿⣧⣻
⣿⣟⣵⣿⣿⣿⣿⣿⣿⣹⣿⡆⠀⣿⣿⣿⣿⣿⣧⠒⠄⠀⠀⠀⠀⡟⠛⠀⣄⢤⡄⢀⠀⠀⠀⡿⢦⠀⠀⢰⣦⢰⡶⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟
⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠇⠀⣿⣿⡏⠙⣿⣏⡅⠀⠀⠀⡐⣼⣅⣴⣀⠁⠉⡁⠀⢤⣆⡀⠹⣿⣦⡀⢰⡄⠸⠕⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⡃⠀⣿⣿⡷⠀⢛⠛⠂⠀⢀⢌⡜⣡⣌⠃⡩⣷⡶⠂⠀⡽⣿⣁⡀⠛⡼⢷⡄⠀⠐⠈⠌⠩⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⣥⣠⠏⠟⠃⢀⡉⠁⠀⢠⣿⣿⠁⣿⡿⢸⣆⣿⣿⣦⡀⣿⣋⣙⣷⠠⣞⣾⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⡗⠀⠀⢸⡉⠙⣧⣉⣡⡾⣽⣿⣿⣿⣿⡿⢿⣿⡶⡦⠠⢾⣿⡿⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿
⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣏⣯⣿⣿⣶⡛⣷⣶⣶⣾⠿⠃⠀⠀⣀⣘⣿⣿⣿⣿⣿⡟⠛⠛
⠀⠀⠀⠉⠹⠟⠟⣻⡟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣌⠴⣏⣽⣿⣿⣟⣿⣿⣾⣾⣿⠏⠀⠀⠀⠀⠃⠘⠿⠍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣝⣻⣿⣽⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠲⡤⠤⠄⠀⠰⠮⣁⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠁⠀⠀⠘⡀⠀⠀⠀⠀⠀⠉⠢⣄⠐⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠔⠃⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⢸⠦⣀⣰⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣸⠀⠀⠀⠀⢣⠈⠑⠒⠓⠒⠠⠤⠤⠖⡐⠒⠖⠈
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢋⠀⠀⠀⠀⠘⡄⠀⢀⣐⣂⣀⣀⣠⠴⠒⠒⣷⠆
⠀⠀⠀⠀⠀⠀⠘⠶⡀⡰⢣⡄⢆⡴⠤⣄⢀⠶⠺⡏⠀⣠⠀⡏⠹⡀⠘⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠣⣗⠀⠉⠀⠀⠑⠤⠤⠚⠠⡤⠈⠉⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⣀⣀⠀⢀⣀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠒⠽⡅⢠⣏⣹⢨⠀⠀⡔⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈⠳⢅⢀⡯⢵⡤⢚⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⢄⠀⠀⠀⠈⠁⠀⠀⡄⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠙⠒⠤⣀⣀⣀⣀⡿⠋⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠄⡆⠀⠀⠀⠀⠘⡄⠀⠀⠀⡮⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⡆⠀⠀⠀⠀⡜⠀⠀⠀⠂⠇⠀⠀⠀⠀⠀⠱⠤⡀⠴⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡇⠀⠀⠀⠰⢁⠀⠀⠀⢀⣒⣀⣀⠠⡤⠄⠚⠁⠀⢢⠄⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃⠀⠀⠀⠀⢀⠜⢫⡉⠁⢀⣌⠀⢂⡇⠀⠀⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⡀⠀⢠⠚⠑⢢⣀⡉⢂⡺⠀⣈⠡⠶⣄⠀⠀⠀⡸⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢇⠉⢉⢇⠀⠐⠂⠒⠈⢣⠀⠂⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠆⢸⡀⠀⠀⠀⠉⠁⠈⡆⠀⠀⠀⣀⠎⠀⡄⠀⠀⠀⠀⡸⠀⡄⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠁⠀⠑⠤⣀⣀⣀⣀⠔⠉⠉⠉⠉⠀⠀⠀⠈⠂⠤⠤⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠃⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⣠⠖⢉⠗⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡇⠀⠀⠀⣠⠞⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣇⣸⠓⠒⢦⠞⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠉⠉⢀⡴⢁⡴⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⢳
⠀⠀⠀⡴⠒⠲⢄⣀⣀⣀⣀⣀⣀⣀⣞⡀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢶⠋
⠀⠀⢸⠀⠀⠀⣸⣏⣉⣉⣉⣉⣉⣉⣉⣉⣹⠆⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠑⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠖⢺⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡉⠀⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⣻⠆⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣦⠤⠤⣄⡀⢀⡴⢋⡴⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢳⣼⡆⠀⠀⠙⣯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⡟⣷⣤⠴⠋
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣾⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣛⣛⣛⣛⣛⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣴⣿⣿⣦⠹⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⢿⣭⣭⣿⢣⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢱⣶⣶⣶⣶⣶⡶⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⣵⠶⣮⢡⡌⣱⡶⣦⡙⢷⡀⢸⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⣾⣿⣧⣹⡆⢰⣟⣴⣿⣷⣼⣷⢸⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣛⣛⣋⣴⣦⣙⣛⣛⣿⣿⣿⢸⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢸⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣟⠑⠛⠛⠛⠛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⢏⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣐⣛⣛⣛⢛⣿⣿⣿⡿⣡⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⣠⣤⣄⡀⠀⠀⢀⣄⢸⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⣾⣿⣿⠸⠿⠿⠿⠿⠿⠿⠿⢡⣿⡿⠃⠾⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠇⣾⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣋⣍⡛⢿⣿⡿⢛⣉⣙⠻⣿⣿⠟⣋⢸⣿⣿⣿⣿⣿⡟⣸⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣷⣶⣶⣿⣿⢸⣿⣿⣿⣿⣿⠁⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠻⣿⣿⣿⠟⠛⠛⢿⣿⣿⠿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣴
⠀⢠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠈⠀⠴⠂⠉⠒⢊⡉⠕⠀⠢⠢⠰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⣿⠀⠀⠀⡀
⠀⠀⠀⠈⠈⠀⠀⠀⠄⠤⠠⠀⠄⡄⠠⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠠⠀⠀⢀⡀⡀⣀⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣍⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠠⡐
⠀⠀⢿⣿⣿⣿⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠥⠎⠃
⠀⠀⠀⠉⠉⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡄⢦⠦⡖⢲⣆⠔⠀⡖⠐⠖⠂⠠⢰⠀⠐⢦⠀⠒⠀⠀⠀⠐⠠
⠀⠀⠠⣶⠒⣲⠆⢲⡶⢶⠄⣶⠀⠀⠀⠀⠸⡀⣸⡄⡆⡞⠸⠀⡧⡏⣬⢩⠁⢼⡬⠀⠀⠸⠄⠘⠃⡀⠀⠀⠠
⠀⠀⠂⠘⠃⠋⠀⠈⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠈⠀⠈⠉⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⢀⠀⠄⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣾⣻⣷⣄⠀⠀⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⡉⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠠⠀⠀⡄⠀⡇
⠀⠴⠾⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⣶⢦⡀
⠀⠀⠀⢀⠀⣄⠀⠀⠀⠀⢆⡆⠀⠀⠀⠀⣹⣄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⣠⠞⢵⠀⠈⠢⠈⠓⠤⣄
⠀⠀⢀⠀⠀⢘⣆⠀⠀⣘⣙⣻⡄⠀⠀⠀⡇⢘⣿⠀⠀⠀⠀⢸⣏⠀⠀⠀⠈⢰⠞⠁⠀⠀⠀⠐⢆⠀⠈
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⣈⠉⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠹⡑⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⣀⢀⡀
⠀⠀⠀⢰⡶⣒⢿⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⡇⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⢸⣿⢿⠿⣿⠀⠀⠉⠉⠘⡇⠀⠀⠀⢐⠶⠖⠀⠀⢠⠄⠀⢤⠀⠀⠀⡇⢸⠈⡇⠀⢀⠔⠋
⠀⠀⠀⠚⠋⠉⠉⠛⠀⠀⠀⢐⠊⠀⠀⠐⠒⠒⠒⠒⠀⠀⠙⠒⠒⠚⠃⠀⠀⠁⠀⠚⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢜⡆⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⡀⢀⣨⡇⠀⠀⣀⠠⢚⡃⠀⠀⠲⠲⢔⠶⡲⢖⠶⡲⢲⠆⠀⠀⠀⢈⠀⠀⣹⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠀⠁⠈⠀⠁⠉⠈⠀⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉
⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⡤⢤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⡄⠀⠠⡄
⠀⠀⠀⢜⠓⠂⠀⢠⡄⠀⠃⢸⠀⢱⠀⠀⠸⠛⠛⢩⣧⠀⠀⠖⠀⠀⢸⠄⠀⠀⢠⡾⠛⠁⠀⣴⡇⠀⢸⡇
⠀⠀⠀⠘⠦⠀⠠⠞⠁⠀⠀⠸⠀⠀⠄⠐⠶⠒⠒⠚⠲⠆⠀⠰⠤⠤⠞⠀⠠⠊⠈⠀⠀⠀⠈⠀⠁⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠰⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡂⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⡠⠁⠀⠀⠀⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠴⣄⡀⠀⠐⠂⠀⠈⠁⠡⠐⠤⡀⠀⠀⠀⠀⠠⢀⣀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⣀⡠⠤⠐⠊⠁⢊⠀⠠⣂⠉⢐⠄⠀⠉⠑⠦⣄⡀⠀⠁⠂⠀⠀⠉⠑⠢⠀⠀⠈⠀⡀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠂⠂
⠀⠀⠘⢧⠀⠀⠀⠀⠀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀⠀⡀⠀⠀⢀⠇⠀⠀⠀⠐⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⢀⣀
⠀⠀⠀⠀⠙⠆⠀⠀⠈⠀⠀⠄⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠊⠀⠀⠀⠀⡀⠀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⡤⢀⡄⠁⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠑⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⡇⢠⠇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠄⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠌⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣒⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠄⠒⠀⠀⠀⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⠆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠈⠤⠤⠤⠬⠥⠁⠠⠤⠤⠤⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⢀⠈⠐⢀⠀⠀⡄⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠤⠄⠈⠀⠐⠂⠄⠆⠐⠀⠀⠀⠀⠀⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠀⠀⠰⠀⠀⠀⠀⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣄⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⢀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢈⡔⠘⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠜⠀⠁⠉⠉⠉⠜⠀⠉⠉⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⢤⠀⠀⢠⠤⠄⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⢀⠀⠀⢀⡀⠀⢀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠈⡁⠁⠀⠀⡆⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢘⠀⠀⢐⡃⠀⢘⡀⠀⢘⡃⠀⣚⠀⠀⢘⡂⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⠀⠠⠀⡀⠀⠁⠀⠀⠀⠁⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠈⠇⠀⢸⠀⠀⠀⡇⠀⢹⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠁⠙⠀⠀⠀⢤⠀⠀⠙⠀⠀⠀⠀⡇⢸⠀⠀⢀⠀⠀⢈⢈⠁⠀⡆⠀⢸⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⢀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⡀⠂⠁⠀⠀⡇⢸⠀⠒⠐⡄⠀⢸⠀⠀⠀⡇⠀⢸⠀⠒⢂⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠈⠉⠉⠉⠀⠐⠀⠀⠀⠀⡄⠀⠚⠀⠀⠀⡇⠈⠧⠤⠤⠁⠀⠀⠦⠤⠔⠁⠀⠀⠦⠤⠜⠀⢸
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢂⡀⠀⠀⠀⠀⢀⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠤⠁⠤⠤⠠⠄⠤⠤⠤⠤⠤⠠⠄⠄⠀⠤⠀⠀⠀⠈⠤

⣾⣿⣷⣾⣶⡷⢿⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣷⣶⢷⡶⠶⣶⠶⠶⣶⡿⠶⠷⢾⠷⠶⠶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣴⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠿⠟⣿⠋⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡿⣦⡀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠃⢿⡀⣀⣤⣤⣤⣤⣼⣧⠀⠀⠈⠻⣦⡀⠀⠀⠈⠻⣦⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣤⡄⠸⣇⠈⠙⢦⡀⠀⠀⠙⢷⣤⣀⣀⡈⣻⣦⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠛⠁⠀⠹⡆⠀⠀⠙⢦⡀⠀⠀⢿⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣤⣤⣤⣤⠀⢻⡄⠀⠀⠀⠙⢦⡀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⡿⠿⢿⣿⣿⣧⠀⢻⡄⠀⠀⠀⠀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⣾⣷⣶⣾⣿⣿⣿⣇⠈⢷⡀⢤⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣼⣿⣤⣤⣴⣿⣿⣿⣿⡆⠘⣧⠀⢹⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⣿⣧⣤⣤⣼⣿⣿⣿⣿⣿⠀⢸⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⣀⣄⠀⢀⣀⡀⣀⠀⡀⣀⣀⠀⡀⢀⡀⣀⢀⣀⡀⣀⣀⢀⣀⡀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢺⢸⡗⣇⢀⡇⢏⢀⡁⣿⠒⡇⣿⠒⢸⣿⣼⡇⣿⢘⢲⡄⢸⡇⢸⡷⣏⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠠⠤⠤⠤⠤⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠒⠒⠒⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠐⡆⠀⠲⡆⠀⢰⡀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠘⢓⠀⠛⠇⠈⠙⡇⠛⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠴⠀⢀⣀⠀⡀⣀⣓⡇⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⡇⠀⠀⠋⠀⠀⠨⢽⠛⠛⠩⠉⠍⠃⠒
⠀⠀⠀⠀⠀⠀⠀⢤⠈⢭⠀⣨⢽⣈⡳⢗⣐⠆⣎⡋⣧⡇⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⢫⣈⣉⠉⠈⠉⠉⠸⠭⢆⡋⣱⣬⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠈⠁⠈⠀⠁⠈⠉⠀⠀⠀⠉⠄⢀⣃⠀⠀⠀⠀⠀⠄⠀⠐⠀⠸⢄⠉⠉⠉⠉⢩⣀⣀⣀⣀⣨⣀⡀⠀⣀⣀⢠⣄⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣠⣤⣤⣶⣒⣒⡒⠒⠒⢀⣐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠸⣇⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠛⠿⠿⠻⠛⠟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠃⠋⠉⠉⠉⠉⠉⠉⣉⣉⡉⣁⠤⣧⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣥⡀
⠀⠀⠀⠀⠘⠀⠀⠄⠤⠠⠤⠤⠇⠰⠒⠛⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠢⢄⣄⣀⢀⡀⢀⢀⠀⠀⠀⠀⠀⢀⣠⣠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉⠊⠐⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠠⠉⠉⠄⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠠⠂⠤⠤⠐⠕⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠍⠉⠉⠅⠀⠀⠀⠀⠀⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠆⠀⠀⠀⠀⠀⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠉⣀⠀⠁⠂⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⢠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠊⠀⡐⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢑⠂⠉⠀⠀⠀⠤⠉⠒⠉⢁⡀⡇⠀⠀⠀⠀⠀⢀⡅⢀⡰⢘⠁⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠰⠀⠀⠀⢠⠒⠀⠀⠀⡇⠀⠀⠀⠀⡠⠁⠈⠐⠖⠁⠈⠀⠂⠄⠈⠂⡀
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠑⢕⠄⡀⠀⢀⡠⠀⡀⢐⠅⠀⠀⡇⠀⠀⠀⣔⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑⠄⠈⠄
⠀⠀⠀⠀⠡⠀⠀⠠⠃⠀⠀⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠗⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇⠤⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠂⠶⠚⠠⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡐⢒⠓⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡐
⠀⠀⠀⢀⠀⠀⠘⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠠⣤⠙⠠⡀⠀⡠⠊⠁⠀⠈⠀⡤⠊⢀⠔
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⠄⡀⠙⠀⠀⠀⠀⡄⠀⠈⡔⠁
⠀⠐⠂⣀⠀⠀⠈⠕⠀⡀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠀⠀⠠⠀⠀⠉⠀⠀⠈⠀⠁⠀⠀⡇
⡀⠄⡀⠈⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠇
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡞⠛⠛⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠀⠀⠈⠀⠀⠀⠈⠉⠙⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣴⠟⠉⠙⢳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⠻⣤⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠛⠁
⠀⠀⠀⠀⠀⣀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠛⠛⠳⣄⠀⠀⠀⠀⠉⠙⢷⠟⠉⠁⠀⠀⠀⠀⠉⠈⠙⠻⣦⣀⣠⡾⠷⠶⠞⠁
⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡴⠋
⠀⠀⠀⠀⠈⠛⠛⠛⠶⢤⣤⠀⠀⠀⢀⣄⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⠟⠉



⣶⠛⠶⣄⣀⠀⢀⣠⡴⠞⠛⠛⠳⢦⣀⡀⠀⣀⣤⠶⠛⠛⠛⠶⣄⣀⠀⢀⣠⡴⠞⠛⠛⠳⢦⣄⡀⠀⣀⣤⠶⠛⠛⠛⠶⣤⣀⠀⢀⣀⣴⠞⢻
⣿⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣀⣀⣠⣤⣤⣤⣤⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⢛⣷⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠛⠛⠛⢻
⣿⠀⠀⠀⠀⠀⠚⠛⠛⠳⠿⠷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⣤⡤⠦⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠈⠻⣦⣠⡾⠀⠀⠘⠳⠖⠛⠛⠛⠛⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⠀⣀⣀⡀⠀⠀⠀⠀⠉⠛⢶⡶⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠐⠛⠛⠓⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⢸
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠤⠤⢤⡀⠀⠀⢀⣀⡤⣴⣂⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠤⢴⣆⡈⠉⠉⠁⠀⢤⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⠀⠀⠀⠙⣶⢤⡴⠖⠲⠤⢤⣀⡀⢸⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⠀⠀⢉⡴⠃⠸⡇⠀⠀⣀⠀⢹⠟⡷⠄⠀⠀⣠⣴⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠉⢄⠀⢀⣳⣤⣀⣁⡴⠋⠀⢇⣠⠴⠋⣨⢼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡄⠀⠀⠀⠀⠙⠢⣌⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢊⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⡻⣄⡀⠀⠀⠀⠀⠀⡴⠛⢳⡶⢶⠶⢶⠖⢻⠯⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣯⠹⡄⢙⠒⠦⣀⣠⠔⢣⠀⢰⠈⠁⢈⠋⣴⠋⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠗⠠⣄⡜⠀⠀⠀⠁⠀⢸⠀⡌⠀⠀⠈⢰⠹⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⣣⡀⠀⠀⠀⠀⠏⣤⠃⠀⠀⢠⡇⠀⡆⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⡏⠈⠁⠒⠒⢸⢠⠃⠀⠀⠀⠀⠇⠀⣥⣀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠀⠀⠀⠆⠎⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⢰⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⠀⠀⢠⠴⠚⠒⢄⠀⠀⠀⢠⠎⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡹⠢⠄⣨⠖⠀⠀⠀⡱⠤⠚⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠃⠀⠀⠈⡿⠤⠶⢊⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⠏⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⡠⠀⠀⠀⡀⠀⠀⠀⢔⠒⠒⣒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠁⠂⠄⢀⢀⠃⠀⠀⠈⠉⠉⠀⠉⠉⢫⠀⠀⠀⢘⠉⠁⡛⠀⠀⠀⠀⢨⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠑⠲⠤⠤⣀⡸⠂⠠⠀⡀⠀⠀⠀⠀⠀⢘⡄⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠈⠁⠀⠂⠀⠀⡇⠀⠀⡸⠀⠀⠠⡀⠀⠀⠀⠰⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠙⠦⠤⣄⣀⣀⢸⠂⠈⠉⠑⠒⠠⠤⢤⣀⣀⣀⡇⢠⠊⠀⠀⠀⠀⠈⢢⠀⢠⠃⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠇⠀⡆⠁⠀⠈⠉⠀⡆⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠀⢰⠂⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⡃⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠣⣀⠀⠀⢀⡠⠃⠠⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⢦⣤⡄⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⡦⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣎⣁⢠⠘⠳⠲⠙⠚⠉⠁⠀⠀⠀⢰⠀⠀⢈⠆⠀⠄
⠀⠀⢀⡰⠀⠘⣀⣀⡜⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠑⢦⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠑⠐⠣⠀⠀⡘
⠀⠀⠀⠑⠤⢴⠁⠀⠱⠀⢠⢠⡄⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠈⢸⡇⠈⠁⣏⠀⠀⠁⣿⠀⠈⢸⠁⠀⠀⠐⠉⠉⠣
⠀⠀⠀⠀⠀⠀⠑⠐⠃⠀⢸⢸⡇⠀⠀⠨⠁⠉⠒⠊⠀⠈⠠⠀⠀⢸⡇⠀⠀⣿⠀⠀⡇⣿⠀⠀⢸⠀⠀⠀⠘⠄⠠⠂
⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⠇⠀⠀⡿⠀⠀⠇⡿⠀⠀⢸
⠀⠀⠀⠀⠑⡃⣠⠴⡄⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⢀⡄⠀⠀⡄⠀⠀⡄⣤⠀⠀⢠⠀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠘⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢰⢸⡇⠀⠀⣧⠀⠀⡇⣿⠀⠀⢸⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⣶⢦⣀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣸⡇⠀⣀⣿⠀⠀⣇⣿⠀⢀⣸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠠⠀⣴⣿⣿⣿⣿⣿⣧⠈⢦⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢹⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡂⠰⣿⣿⣿⠿⣿⣿⣿⣆⠈⡇⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢺⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⡖⣿⡆⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⣿⢀⣀⣀⣀⣿⣇⣀⣽⣏⣀⣀⣸⣿⣀⣨⣿⣀⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣄⠀⠀⢀⠆⣾⠟⠀⠀⠀⠹⡿⠿⡿⠿⠭⣿⣿⣭⣿⣿⣿⣯⣿⣿⡯⠽⠿⡿⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠭⠽⠭⠅⠚⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡜⣿⣿⠟⣤⡀⠀⠀⠀⢀⣠⡸⣿⣿⠇⣤⡀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡌⠁⠀⠀⠀⠀⠀⠉⣼⣿⣷⠈⠁⠀⠀⠀⠀⠈⠉⣾⣿⣷⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠰⢦⡄⣤⣤⡀⣤⠶⠙⠛⠋⠰⣦⢀⣤⣤⡀⣤⡶⠃⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣿⣿⡗⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠁⠀⠀⠀⠙⠓⣼⣿⣷⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⣶⣶⣶⣶⣶⣶⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⠛⠿⢿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⡄⢸⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢰⣾⣿⣿⣿⣿⣿⣶⣦⣭⣉⣛⣛⣛⣛⣉⣭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠩⢙⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⡿⢩⣍⢻⣿⣷⣿⣿⣿⡿⠿⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣬⣥⣾⣿⣿⣿⣿⣏⢰⣿⠆⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠉⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢎⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⡘⠶⠶⠦⡄⠀⠈⠉⠁⠀⠀⠀⠀⠀⡠⠊⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠
⠁⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠠⢀⡇⠀⠀⢀⠄⢲⣿⡤⢄⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠏⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡇⠀⠀⠀⠀⠱⡀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⢌⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠘⠛⠒⢦⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⡠⠀⠀⠈⠢⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠀⠀⠀⢠⣤⡄⠤⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⡏
⠀⡇⠀⠀⡜⠀⠀⠀⠀⠀⢘⠢⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡖⠁⠀⠀⠀⠀⠱⠦⠤⠖⢏⣢⣀⠀⠀⠀⠀⡎⠀⠀⡵⠀⠀⠀⠀⢰⠁
⡰⠁⠀⠀⠐⠂⠤⠤⠀⠀⣤⠀⠤⠑⠒⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⡰⠚⠓⣄⠀⡎⠀⠉⢳⠀⠀⠀⠙⠢⠾⠁⠀⠀⠀⠀⠘⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠳⠄⡤⠃⠀⠙⠒⠒⠁⠀⠀⠀⠀⠀⡼⠋⠉⢢
⢀⡞⠉⢳⡀⠀⠀⠀⠀⠈⠻⡁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⡜
⠈⠧⡤⠞⢣⡶⠒⠒⢄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠤⠤⣄⠀⡏⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠠⠄⠀⠀⠄⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⡼⠌⢧⡀⠀⣠⠞⠀⠀⠀⠀⡞⠉⢲⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢠⣶⡄⣠⠀⠀⠀⠀⠐⠀⠀⠀⢈⡀
⠙⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠀⠈⠀⡀⠛⠀⠄⡀⠀⠀⠻⠃
⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠁⡰⠄⡀
⠶⠋⠀⠺⡀⠀⠀⢰⠉⠙⣦⣀⠄⡴⠋⠉⠳⡄⠀⠀⠀⠈⠁⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠌⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢠⠀⠀⠀⠑⠄⠀⠈⠑⠚⠁⠀⠈⢧⡀⠀⢀⡧⠀⣀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣠⣤⡤⢤
⠀⠡⡀⠀⠀⠈⠂⣀⡀⡀⠀⠀⠀⠀⠉⠉⠉⠀⡟⠁⠀⠙⣦⠀⢐⡎⠀⢹⠀⠀⠀⠣⡄⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠳⢄⣀⣠⠏⠀⠀⠈⠐⠁⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷
⠀⠀⢀⠃⠀⠀⠀⠀⠀⡀⠀⠐⠈⡆⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⢸⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⠗⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⡇
⠀⠀⠘⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠱⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢡⠀⠀⠀⡠⠂⠉⠉⠙⢢⡀⠀⠀⠀⠀⡠⢤⡜⠓⠉
⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢧⡀⠀⠀⠀⣰⠃⠀⠀⠀⠸⡀⢀⡸
⠀⠀⠀⠀⠀⠉⠒⠒⠐⠒⠈⠀⠀⠀⠀⠀⠀⢀⡖⠋⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢠⠀⠀⠀⠀⣟⠒⠒⠛⣥⢦⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⢤⡜⠷⢄⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⡎⠉⠙⡄⠰⡂⠀⢀⡇
⠀⠀⠀⠀⠀⢠⠖⢠⡀⠀⠀⠀⠀⢺⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠚⠁⠀⠉⠛⠉
⠀⠀⠀⠀⠀⢃⠀⣀⠇⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡃⢠⣤⣰⣶⣶⣶⣦⣾⣿⣿⣶⡆⠀⠄
⠀⠀⠀⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠔⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢠⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣉⣛⣉⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠂⠓⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⢀⢠⡤⠤⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠏⠉⢙⠇
⠀⠀⠀⡇⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢆⠀⠀⠀⠀⠀⠀⡰⠀⠀⠳⡀
⠀⠀⠀⠁⠀⠀⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠁⠈⠀⠁⠀⠉⠁⠀⠀⠀⣱
⠀⠤⠀⠀⠀⠀⠀⠀⡡⠀⠀⠀⠠⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⡀⣰⣾⠤⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠀⠀⠀⠀⠀⠀⢨⠂
⠀⠀⢣⠁⠀⠀⠀⡼⠁⠀⠛⠙⣄⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⠄⠀⣀⢀⣀⠠⠒⠁
⠀⠀⠀⠀⠀⢀⣀⡤⢐⣒⠚⠷⠊⠧⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲⠒⠾⠼⠟⣲⠀⠀⠀⠀⠀⠠⠶
⠀⠀⣀⠀⠀⠘⠀⡀⢰⢛⡾⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠻⠥⠦⣀⣩⡗⠒⠄
⠀⠠⠀⠀⠀⠰⢄⠀⠀⠘⡇⢉⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠰⣷⠀⠀⠐⠂⠀⠀⠀⡏⠉⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣍⣉⡗
⠀⠀⠛⠀⠀⠀⠀⠀⠢⠤⠇⠀⠀⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠠
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⣀⡀⠀⠄⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠓⠒⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⢀⣀⡤⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣓⣲⠖⠂
⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢈⠉⠁⠈⠢⠀⠀⡀
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⠈⠉⠁⡀⠀⠉⣠⠘⢂
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣧⠋⠀⢀⡠⠔⢚⠡⠐⠈⠑⢄⢢⡘⡆
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡡⠔⢊⡡⠔⡾⠁⠀⠀⠀⠀⠘⡆⠹⣶
⠀⢸⠀⠀⠀⠀⠀⠀⣀⣠⠴⢋⠴⢚⡡⠔⢊⡽⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⡸⢆
⡴⠸⡂⠀⠀⠀⠀⠀⢉⣢⢄⢾⡴⣇⠀⠐⠉⡜⠉⠉⠓⠄⠀⣀⠀⠀⢠⠃⢠⠇⠈⢧
⠛⠁⢱⠀⠀⡀⠀⠠⣀⢉⡎⡰⢣⠘⣆⠀⠈⡇⠀⠀⠀⣴⠀⠎⢀⣴⠣⢔⡗⠛⢦⢸⠆
⠀⠀⠈⠓⠉⠀⠀⠀⠈⠟⣼⠙⢎⢣⡘⣆⠀⠈⠂⠀⠄⠊⢀⠂⠉⠁⠀⠊⠀⡠⠋⡼
⠀⠀⠀⠀⠀⠀⠀⢀⠬⠔⠀⠑⠈⠆⠙⠈⡀⠀⠀⠀⡠⠐⠁⠀⠀⠉⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⣴⠁⠰⣂⡁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠹⢤⣀⣀⣀⣠⠤⠂⢱⠤⣄⡀⠀⠀⠀⠀⠀⠀⣴⠧⠆⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢘⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠓⠲⠤⣤⢤⡤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⡸⠀⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡒⠤⠠⠊⠉⣙⣏⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠌⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠏⠉⠀⠉⠁⢀⣘⣚⣀⠀⠤⠤⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⣃⣀⣠⠀⠽⠀⠀⠒⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠁⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢸⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠀⠀⠤⠤⠤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠄
⠀⡇⡁⠀⡇⢀⣀⠀⠀⣤⡦⣦⢴⠴⠤⠆⠦⡤⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠈⠄⠄⠀⠀⠀⠄
⠀⠇⠸⠂⡕⠺⠐⠀⠀⠂⠆⠀⠀⠀⠀⠀⠀⡇⠘⠀⠁⠀⠈⠁⢀⠀⠐⠀⠀⠀⠁⠐⡢⠄⠀⠀⠠
⠀⠁⠀⠀⠀⠁⠀⠁⠀⠉⠈⠀⠀⠁⠀⠈⠀⠁⠀⠀⠘⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉⠈⠙⠂⠀⠠⠀⠀⠂⠀⡀⠠⠀⠠⢀
⠀⠀⠀⠠⠔⠀⠀⢀⡀⠀⡅⠀⠀⠀⢉⠀⠀⠀⠀⠐⠐⡐⠬⠀⠀⠀⢀⠠⠌⡄⠠⠀⠀⠀⠀⠀⠀⠪⡀⠀⠁
⠀⠀⠈⢀⠐⠀⠂⠁⠠⠀⠀⠀⠀⠀⠀⠈⠡⠀⡀⠐⠀⢂⠔⠀⡀⠀⣋⠀⠀⠀⡚⡀⠀⢤⣈⠀⠰⠒⠦
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢈⠊⠀⠈⠁⠀⠀⠸⡀⠈⠦⢠⠄⠋⠀⡇⠐⠣⠑⡕⠀⠀⠀⠀⡀⢆⠆
⠀⠄⠀⠀⠠⠀⠀⡀⡀⠀⠠⠤⠁⠀⠀⠀⢀⡁⠀⡀⢀⡀⠀⠀⠀⠈⠲⠤⠤⠤⠒⠉⠁⠀⢀⠄⠄⠀⠐⠀⠀⠄
⠀⠀⠀⠀⢀⠃⠀⠀⢆⠀⢈⠈⠀⠀⠀⠀⠁⠀⠐⠀⠀⠈⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠁⣀⠀⠀⡠⠀⠈⠢⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠐⠀⠀⠁⠀⠀⠡⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠠⠀⠈⠃⠊⠐⠄⠀⢀⠀⠀⠠⡠
⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠡⠀⠀⠀⠀⣤⠀⠀⣴⣶⣶⡄⣴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠠⢲⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⠀⣿⠀⠀⣿⠀⢸⡇⣿⣠⡷⠀⠀⠁⠀⠀⠈⠀⠀⠀⠐⠈⠀⠀⠀⠀⢀
⠀⠀⣆⢬⢹⠀⠀⢸⠀⠀⠀⠠⡀⢀⠄⠀⣿⠀⠀⣿⠛⢻⡇⣿⢻⣧⠀⠀⡗⠂⠙⠻⡀
⠀⠀⠁⠀⠀⠀⡄⠬⠀⠀⠀⠀⠁⠀⠀⠀⠿⠟⠀⠟⠀⠘⠃⠛⠛⠃⠀⠀⠁⠀⠀⠀⡇⠀⠈⠀⠀⠀⠁⠀⠄
⠠⣀⠁⠀⢀⠠⠀⠈⠀⠀⠀⠦⠄⠀⡄⠀⢸⣟⣯⣟⢻⡛⣻⡟⣿⠀⠀⠀⠀⠐⠐⠀⡃⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⡀⠀⠄⠀⣂⠀⡀⠀⠀⠀⠀⣁⠀⡀
⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠈⠔⠀⠀⠀⠀⠀⠈⠀⠀⠣⠀⠀⠀⠈⡅⠀⠀⠀⠀⠊⠀⠀⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠂⠀⠀⠀⠀⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠈⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⡀⠀⠈⠅
⠀⠀⣿⢸⡇⣀⡀⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠁⠐⢈⠤⠀⠐⠀⠀⠀⠀⠀⠄
⠀⠀⣿⢻⡏⢹⡇⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡐⡀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠁⠀⠀⠀⠈
⠀⠀⠙⠘⠃⠛⠃⠙⠛⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⣿⣿⣿⣻⣛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣛⣻⣿⣟⣻