Cupcakes Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠂⠑⢄⠀⠀⠉⠭⢭⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠳⢤⣄⣀⣤⡴⠋⠀⠀⠀⢀⡴⠁⠀⢠⠀⠀⣹⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠤⠖⠚⠁⠀⢀⡴⠁⠀⢀⣿⠓⢮⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡶⠾⠿⠟⠛⠛⠛⠉⠉⠉⠀⠀⠀⣀⣠⠤⠚⠁⠀⠀⢀⣾⡏⠀⠈⢷⡘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⡴⠶⠶⠶⠖⠚⠛⠉⠁⠀⠀⠀⢀⣠⣴⡿⠋⣿⠀⠀⢸⣧⢸⡷⢦
⠀⠀⠀⠀⠀⠀⠀⠞⣋⣭⣥⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠿⠿⠛⠁⣠⣾⠃⠀⠀⣸⡏⢸⡇⠈⢧
⠀⠀⠀⠀⠀⢀⣴⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⠿⠛⢁⡰⠁⣰⡿⠁⣼⠃⠀⢸
⠀⠀⠀⠀⠀⣿⡏⠀⣀⣤⣶⣶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⢀⣠⠔⠋⣠⣾⠟⠀⣴⠏⠀⢀⡿
⠀⠀⠀⠀⠀⢿⣧⣼⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠖⠚⢉⣠⣴⡿⠛⠁⢀⡾⠃⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣶⣶⣶⣶⣶⠿⠿⠟⠛⠛⠉⠉⣀⣀⣤⣴⣾⣿⠋⣁⣀⣤⣶⣯⣤⣶⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣿⠁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⣿⠉⠛⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠟⠋⣹⡇⠀⢸⠇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠸⡇⠀⢸⡇⠀⠸⣷⠀⠀⣿⠀⠀⢸⡇⠀⠀⡿⠀⠀⣿⠀⠀⡟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢳⠀⠀⣷⠀⠀⣿⠀⠀⢿⠀⠀⢸⡇⠀⢸⡇⠀⢠⡇⠀⢸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⡆⠀⢸⠀⠀⢹⠀⠀⢸⠀⠀⢸⠁⠀⢸⠁⠀⠸⠀⢀⣦⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣦⣤⣤⣸⣀⣀⣸⣀⣀⣸⣀⣠⣼⣤⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢛⢦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡂⠅⡚⠟⢃⠇⠀⡆⡄⢣⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢊⡔⠈⢀⠤⠊⢀⠔⢠⠁⣰⠛⢍⢣⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡂⠭⢀⣒⠚⠙⠳⠴⠖⠷⠒⠓⢉⠔⡀⠊⠰⢀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠤⣒⣾⣮⠐⢈⠁⠠⠤⠀⠀⠀⠀⠐⠒⡈⠕⢊⣀⡔⣡⣪⠃⡜⢢
⠀⠀⠀⠀⠀⠀⢸⠭⠬⠉⢉⣙⠋⠥⠈⢛⣉⠨⠌⢚⡋⠭⠑⠋⣉⠠⠐⢉⠤⠊⡠⢊⡇
⠀⠀⠀⠀⠀⠀⠘⣈⢈⠡⠄⢀⣒⠀⠭⠀⠒⠈⣉⡀⠤⠐⢂⡉⠤⠒⣈⠤⠐⣈⣔⠅⠣⡀
⠀⠀⠀⠀⠀⢠⠊⠈⠂⠢⠭⠤⡄⠐⣒⣈⣩⡅⠀⣒⣈⢭⡀⠒⠬⠵⠒⠚⠉⠉⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠸⡖⢢⠄⢠⡀⢀⣀⠀⡀⡀⡀⠀⠀⠀⠀⢀⢀⣀⣀⡀⢀⡠⠠⡤⠀⠶⠈⣿
⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⡇⠀⠉⠀⢸⠀⠸⠁⠘⠁⠈⠁⠈⠁⠈⠀⠀⠀⠀⠁⠀⡆⢠⡇
⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠁⠀⢰⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢠⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⠘⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡄⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⡃⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢀⣇⠀⡄⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⡀⠤⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠁⠈⠁⠈⠁⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣶⠛⢿⣷⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⣸⡟⠀⠀⢀⡀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢸⡇⠘⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⠻⣄⣈⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⢦⠉⠛⢿⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣰⡿⠛⠛⠛⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠾⠛⠁⠀⠀⠀⠀⠀⠈⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⢿⣄⣀⡀⠀⠀⢀⣀⣀⣀⣀⠠⠄⠀⢀⣀⣠⣤⡴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣤⣭⣭⣭⣭⣭⣤⣤⡶⠶⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣼⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⢉⣠⠾⠋⠉⠉⡙⠻⣷⣦
⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢒⣉⣤⡴⠞⠋⠁⠀⠀⠀⠀⠙⣦⠈⣿⣇
⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⣿
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⠶⠶⠶⠞⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⡿
⣸⡏⣼⠁⠀⠀⠀⠀⠀⣠⠖⢉⣠⡴⠶⠞⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡿⣧⣤
⣿⡇⢿⠀⠀⠀⠀⠀⣸⢃⣼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣤⠞⠋⠀⣨⢻
⢻⣷⡸⣆⠀⠀⠀⠀⠟⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠞⠛⠋⠉⠀⠀⠀⢀⠞⣸
⠈⢿⣷⣝⠦⠤⠄⠀⣀⣾⣿⣄⠀⠀⠀⠀⢀⣠⣤⣍⡉⠓⠦⣤⣀⣀⣀⣀⣀⣀⣀⠤⠔⠊⢁⣀⣤⡴⠶⠛⠉⠉⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⠴⠞⠉
⠀⠀⠉⠛⢻⣿⡟⠛⠋⠀⠙⠻⠿⠶⠶⠛⠋⠁⠉⠙⠻⠷⣶⣤⣤⣭⣭⣥⣤⣤⣴⠶⠾⠛⠋⠙⠿⠷⠶⠶⠶⠶⠛⠛⠉⠉⠉⠉⠉⠀⢀⣠⡄
⠀⠀⠀⠀⠘⣿⣿⣷⣤⠶⠻⢿⣶⠾⠿⣷⣦⡴⠿⢿⣦⣤⡴⠿⣿⣬⣉⣥⣤⣄⣀⣠⠶⢶⣶⣄⣀⣤⠶⣶⣄⣀⡴⢾⣦⣴⠟⠿⠶⠾⣻⣿⠃
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⣤⠀⠀⠀⣷⠀⠀⠀⣷⠀⠉⠁⢸⡆⠈⠉⠉⣴⠈⠛⠛⠁⢰⡇⠉⠛⠋⠁⡖⠈⠙⠉⣴⠈⠉⠁⢸⠇⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⢹⡆⠀⠀⢿⡀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⡿⠀⠀⠀⡿⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠈⣷⠀⠀⢸⡇⠀⠀⢹⡄⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⢹⡆⠀⠘⣧⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢀⡟⠀⠀⠀⣼⠁⠀⢀⡿⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠈⣷⠀⠀⣿⠀⠀⢸⡇⠀⠀⠸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡟⠀⠀⢸⡇⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢻⡆⠀⢸⡇⠀⠈⣧⠀⠀⠀⣿⠀⠀⠀⡟⠀⠀⠀⠀⣿⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⢀⡿⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠘⣧⠀⠘⣧⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⢀⡟⠀⠀⠀⣼⠀⠀⢸⠇⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢻⡄⠀⣿⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢀⡟⠀⠀⡿⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠘⣧⠀⢸⡇⠀⢹⡄⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣼⠀⠀⠀⢸⡇⠀⢸⠇⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢻⡄⠘⣧⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⠇⠀⢀⡟⠀⠀⠀⣾⠀⠀⡿⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠘⣧⠀⣿⠀⢸⡇⠀⠀⢿⡄⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢀⡟⠀⢸⠇⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠛⠀⠸⠇⠀⡷⠀⠀⠸⠇⠀⠘⠃⠀⠀⠀⡿⠀⠀⠺⠀⠀⠀⠸⠇⠀⠿⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠾⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠚⠅⠈⢦⣀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠱⡀⠀⠈⠑⠲⣄⣑⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠸⠂⢀⠠⠀⢸⣿⠋⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⠀⣁⡤⢚⣇⡔⢂⣤⠤⠴⣂⣼⠻⣄⠀⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⣸⠀⣠⠞⠉⠀⠀⠈⣦⣀⣴⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⣰⠏⠀⣏⠀⠀⠀⠀⠀⣿⠻⡿⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢀⣠⠗⠀⠀⢀⣠⠴⠁⠀⢿⡁⢨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢀⣀⣀⣀⣀⣀⣀⠀⠠⠔⠊⢀⣩⠤⠶⠶⠚⠉⠀⠀⠀⣰⣿⣵⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠻⢄⠀⠉⠀⠀⠀⠀⣠⡤⠞⠉⠀⠀⠀⠀⣠⣴⣶⡾⡿⠿⡿⡿⢊⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠉⠳⠶⠶⠶⣾⣿⣦⣤⣤⣴⣾⣿⠟⠿⠋⠁⠊⣩⢄⣴⣼⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠲⣤⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠠⡠⠀⣀⣀⣧⣶⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⠸⡏⠘⡷⠒⢿⠶⣿⠾⢿⠿⣿⠿⣿⠿⢻⠛⢻⠁⣿⡿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⡆⢿⠀⣿⠀⢸⠀⣿⠀⣾⠀⣿⠀⣿⠀⣽⠀⣿⣿⣿⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠐⣸⡇⢸⡆⠸⡇⢸⢄⣻⡀⣹⠀⣿⢰⡿⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⣿⢸⣧⠀⣷⣼⣆⣿⣄⣿⢤⣿⣺⣧⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣟⣾⣿⡐⣿⣿⣇⣿⡽⣿⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⣿⣇⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣽⣿⣧⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣵⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢗⡚⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠛⠻⠛⠛⠟⠛⠋⠉⠉⠉⠁



⣿⠿⡿⠿⡿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣻⣿⣛⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠉⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠉⠙⠳⢦⣤⣀⣀⡀⠀⠀⢀⣀⣀⣠⣤⣴⠶⠞⠛⠛⠛⠛⠛⠉⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠳⢦⣄
⠀⠀⣀⣴⢿⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠉⠻⣦
⢀⣼⠋⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⣹⣷⡄⠀⠀⠀⠀⣴⣶⣄⠀⢀⣴⣦⡀⠀⢀⣴⣦⠀⠀⣠⣶⣄⠀⠀⣴⣶⡄⠀⢀⣴⣦⡀⠀⣠⣶⣦⠀⠀⣠⣶⣄⠀⠀⣠⣿⡅
⢸⡏⠀⠻⣦⡀⣠⡾⠃⠀⠹⣶⠟⠁⠈⠻⣶⠟⠁⠈⢷⡾⠋⠀⠙⢷⡾⠃⠈⠻⣶⠟⠁⠈⢻⡾⠋⠀⠙⢷⡾⠋⠀⠙⣷⠾⠋⠈⣿
⠀⢿⡄⠀⠈⠻⡏⠀⠀⠀⠀⠸⡆⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⡿⠀⠀⠀⢸⠁⠀⠀⠀⣼⠁⠀⠀⢠⡏⠀⠀⣼⠏
⠀⠈⣿⡀⠀⠀⢹⡄⠀⠀⠀⠀⢷⠀⠀⠀⠘⡇⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢀⡟⠀⠀⠀⢠⡇⠀⠀⠀⣼⠁⠀⣰⡏
⠀⠀⠘⣷⡀⠀⠀⢳⡀⠀⠀⠀⠸⡇⠀⠀⠀⢷⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⣼⠀⠀⠀⣸⠃⠀⠀⠀⡼⠀⠀⠀⢠⡇⠀⣰⡟
⠀⠀⠀⠘⣧⠀⠀⠈⣧⠀⠀⠀⠀⢻⠀⠀⠀⢸⡀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⡏⠀⠀⢀⡟⠀⠀⠀⢰⠇⠀⠀⠀⡾⠀⢰⡟
⠀⠀⠀⠀⠹⣇⠀⠀⠘⣇⠀⠀⠀⠈⣇⠀⠀⠈⡇⠀⠀⡇⠀⠀⠀⢸⡇⠀⢸⠃⠀⠀⢸⠃⠀⠀⠀⡾⠀⠀⠀⢰⠃⢠⡟
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠹⡄⠀⠀⠀⢹⡀⠀⠀⢿⠀⠀⡇⠀⠀⠀⢸⡇⠀⣾⠀⠀⠀⡟⠀⠀⠀⢸⠇⠀⠀⢀⡟⢠⡿⠁
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢹⡀⠀⠀⠀⣧⠀⠀⢸⡄⠀⡇⠀⠀⠀⢸⠇⢀⡇⠀⠀⢸⠃⠀⠀⠀⡟⠀⠀⠀⣸⢃⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⢷⠀⠀⠀⢸⡆⠀⠘⣇⠀⡇⠀⠀⠀⢸⠇⢸⠁⠀⠀⡟⠀⠀⠀⢸⠃⠀⠀⢠⣏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣬⣧⠀⠀⠀⢷⠀⠀⢿⠀⡇⠀⠀⠀⢸⠃⡿⠀⠀⢸⠃⠀⠀⢀⡟⢀⣠⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣼⣆⣀⣸⡆⡇⠀⠀⠀⢸⢠⡇⠀⠀⣟⣀⣤⣤⠾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠭⢪⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣸⠙⣏⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠈⠀⣠⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣄⠈⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⢂⡤⢈⣟⣿⡃⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠰⠀⠠⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢁⡀⠑⠀⠲⢀⣀⡀⠀⣀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠛⣕⠈⠘⠃⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠈⣠⠀⠾⠂⠀⢴⠗⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠘⠃⣠⡀⠂⠀⣴⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠛⠃⣨⡀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⠙⠁⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣍⠕
⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠻⠉⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠰⡿⠀⠀⠀⠈⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⢀⠀⠀⣶⣶⠄⠀⠀⠻⠿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠙⠋⠀⠈⠛⠀⠀⠙⠁
⠀⠀⠀⠈⠛⠛⠛⠶⠞⠳⡶⠟⠶⠖⠲⠞⠁⠀⠀⠀⠀⠔⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠄⠊⡁⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⠁⢀⣀⠤⠔⢋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠃⢉⡠⠄⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⣿⢿⠿⠿⢿⡟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⣷⠒⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢹⡀⠀⠀⢀⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠚⠛⠀⠀⠈⢱⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠙⠶⣄⣀⡀⢀⣠⡴⠚⠁⣠⠏⠀⠙⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⣤⠤⠶⠒⠛⠋⠉⠁⣀⣤⠞⠁⠀⠀⢠⡏⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⠀⠀⠀⠀⠀⠀⠠⠶⠖⠚⠛⠉⠁⠀⠀⠀⠀⢀⡞⠀⠀⠀⢸⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣰⠋⠁⣀⡴⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣼⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⠁⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⣴⠃⠀⠀⢸⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢃⡞⠁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡾⢿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⠁⢀⡴⠋⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⢀⡼⠁⣸⡟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⡞⣠⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⣠⣞⣠⡾⠿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⢿⣿⣷⣧⡇⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⣀⣀⣤⣾⠟⠛⣿⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡈⢿⠛⠉⢻⣭⠴⠒⣆⣀⡴⢦⡀⢀⣠⢿⣤⣤⡴⠶⣶⣿⡏⠉⠻⢿⠟⠀⠀⡟⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⣇⠀⠈⣇⠀⠀⢸⠁⠀⠈⣿⠋⠀⠀⢹⡏⠀⠀⠈⣿⠀⠀⠀⣸⠀⠀⢸⠃⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢹⡄⠀⢿⠀⠀⢸⡄⠀⠀⢿⠀⠀⠀⢸⡆⠀⠀⠀⡟⠀⠀⠀⣿⠀⠀⡟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢷⠀⢸⡆⠀⠸⡇⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⢸⠇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⣇⠀⣧⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢀⡇⠀⠀⢰⡇⠀⡾⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠀⢻⠀⠀⣿⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠁⠀⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠉⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⢈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉

⡤⠀⡆⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠤⣀⣄⡉⠀⣁⣀⡴⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠠⠇⠀⠀⠀⠀⠀⠀⠀⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠲⠆⢠⡄⠠⠄⠀⠙⠉⠀⠀⠀⠠⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⢴⡶⡀⠙⠀⠀⠀⢀⢀⣀⡠⠥⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⠀⠀⠊⠀⠀⠈⠂⣀⣠⣀⣀⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⣀⣀⣀⣤⡤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⢀⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⣤⣄⠀⠰⠶⠒⢀⠀⠀⠀⢀⠠⢴⠄⠀⠀⠀⠀⢈⠉⠀⢤⡴⠀⠀⠀⠀⠀⠛⠋⢀
⠀⠀⣀⡚⢃⠉⡄⠄⠀⠀⠀⠀⠀⠀⠀⠾⠃⠈⠉⠀⠀⢁⠔⠁⠀⠀⡠⢠⠀⠀⠘⠄⢁⡀⠆⠀⠀⣄⠈⠁⠀⠄⠀⠀⠄⡀⠀⠈⠓
⠠⠬⠛⠅⡭⠀⣰⡏⠀⠀⠀⠀⠀⢸⡇⠚⠂⠀⠀⠑⠿⢆⠀⠂⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠸⠠⠚⠁⠀⠀⠀⠀⠀⠀⢸⠄⠀⢤⠴⠃⠒⠒⠚⠉⠙⠂⣠⡔⠛⠀⠀⠀⠀⠀⠀⠀⡤⣄⠠⠤⠀⠀⠀⣀⣠⠤⠒⢠⠀⠀⠀⡰
⠈⠳⠦⣀⢀⣀⠤⠐⢳⡀⠀⠀⠀⠀⠢⣤⠄⠠⠈⡀⠀⠐⠀⢲⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠒⠐⠀⠒⠒⠐⠐⠀⠀⠀⠀⠀⢡
⠲⠤⠤⣀⠠⣀⣠⣔⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠐⠂⠠⠆⠐⠒⠀⠉⠁⠀⠠⡀⠐⠂⢂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠃⠀⠀⠀⠀⠀⠀⠀⡎⠆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⢤⣰⣀⣤⠷⠜⠂⠀⠀⠀⠀⠀⠀⠀⠲⠷⡄⠈⠀⠀⠀⣀⣀⡠⠤⠖⠋
⠂⠤⣤⣤⡤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣀⠀⠠⠄⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠲⠐
⠀⠀⠀⠀⢀⠠⡲⠂⠀⠀⠀⠀⢀⠁⠀⠤⠖⠀⠀⠀⠀⢀⠁⠁⠠⠀⠀⠀⠀⢀⡀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣀⡤⠤⡄⠀⢒⠟
⠔⢠⠀⠀⠈⠂⠠⠄⠃⠀⠀⢦⠀⠁⠐⠂⠀⠀⠂⠄⠀⠀⠁⠀⠀⠀⠀⢀⣠⡋⣁⠀⠄⣆⠀⠀⠀⠀⠀⠀⣠⢟⠁⠈⢁⡀⡊⠤⠂⠁
⠠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⡂⠔⢀⣰⠏⠀⠀⠀⠀⠀⢸⠇⠁⠀⠀⠀⠈⠛⢅⡀⠉
⡞⠙⠀⠀⠀⠀⠀⠀⠀⣴⠤⠄⠒⠀⠀⠀⠠⠤⠖⠊⢠⠀⠀⠀⡰⣄⢀⡀⠀⠘⠐⠋⠀⠀⡀⠀⠀⠀⠀⢸⠂⠀⠤⠚⠉⠙⠛⠉⠁⠉⣁
⣥⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣬⣭⣥⣤⣭⣤⣭⣨⣥⣠⣤⣤⣤⣥⣤⣌⣿⣦⣤⣤⣤⣴⣤⣭⣤⣤⣤⣤⣄⣴⣶⣤⣤⣤⣤⣤⣴⣤⣯⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⡰⠆
⠀⠀⠀⢘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⢪⠂⠀⠀⠀⠀⠀⢠⠔⣺⠃⠤⡀⠀⠀⠀⠀⠀⣀⣠⣷⠆⠀⠀⠀⠀⠀⠀⡤⢒⣒⠰⠤⠄
⣠⡖⠢⢿⡿⠖⢀⣄⠀⠀⠀⣤⣶⣿⣿⣿⣦⡀⠀⠀⠀⠠⠀⠉⠤⠀⠈⠂⡀⠀⠀⡸⢅⡁⠠⠴⠚⠉⢦⠀⠀⠠⡎⠉⠘⠛⠃⠙⢄⠀⠀⠀⡞⠉⠒⠙⠒⠂⣛
⠿⢿⣶⣶⣶⣶⡿⠿⠀⠀⢜⠉⢀⠀⠀⡀⠀⢸⡄⠀⠀⡁⠀⠀⠀⣁⣠⡤⡍⠀⠘⣆⢲⣄⣀⣀⣠⣤⢤⠃⠀⢸⡀⠀⣀⣀⣀⣤⢤⠇⠀⢾⣉⠉⠑⠒⠒⠊⠉
⢸⡼⡇⣿⢹⢰⣇⡇⠀⠀⠈⡟⣿⣿⢻⡏⡞⡿⠀⠀⠀⢸⢹⡿⡇⣿⢸⢳⡇⠀⠀⠘⡏⣇⣿⢸⣇⡇⡏⠀⠀⠀⣯⣿⢸⣿⡇⡿⣸⠀⠀⠀⢻⢳⢲⡖⣿⣿⢹
⠀⢇⣇⣿⣿⢸⣻⠃⠀⠀⠀⢻⣿⣸⢸⣷⣇⠇⠀⠀⠀⠘⣟⡇⡇⣿⣾⡸⠀⠀⠀⠀⢇⣿⣿⢸⣿⣷⠇⠀⠀⠀⠸⣹⢸⣿⡇⣇⡏⠀⠀⠀⠸⣸⣿⡇⣏⣿⡾
⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢠⣶⣴⣶⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⣴⣤⣷
⠀⢀⡀⣤⣧⣄⠀⠀⠀⠀⠀⠀⢘⣿⣿⣏⠀⠀⠀⠀⠀⠀⠰⡶⠀⠐⣠⡄⠀⠀⠀⢀⠴⣙⠛⠁⣓⢄⠀⠀⠀⠀⢀⡴⢿⣿⣷⣀⠀⠀⠀⠀⢀⣎⡛⠋⠛⠋⣢
⣴⣇⣈⠛⠛⣁⣨⣧⠀⠀⢀⠾⠟⠛⠒⠓⠽⢆⠀⠀⠀⡄⠀⠀⠀⠀⠙⠁⢄⠀⢀⠀⠒⠄⠀⠉⠁⠀⠱⡀⠀⢈⠇⠀⠀⠀⠀⠈⠱⡀⠀⢀⣯⣀⣈⣉⡁⣀⣬
⢹⢻⡿⣿⣿⠿⡟⡏⠀⠀⢲⣀⣠⣄⣠⣄⣀⣸⠃⠀⠀⢦⣀⣠⡤⣤⢲⡏⡇⠀⠈⢳⣬⡦⣤⢴⡖⡿⣿⠀⠀⠘⣦⣄⢴⣤⡴⣶⢹⠃⠀⠻⣶⣀⣀⣉⣉⣁⣠
⠘⡟⡇⣿⣸⢸⣧⡇⠀⠀⠀⣇⣷⣻⢸⡇⡏⡏⠀⠀⠀⢸⣼⡟⡇⣿⢸⢹⠃⠀⠀⠀⡇⣧⣿⢸⣷⡇⡇⠀⠀⠀⢷⣿⢸⣿⡇⡟⡾⠀⠀⠀⢹⢹⣹⡏⡿⣿⣸
⠀⠳⢧⣿⣽⡼⠞⠀⠀⠀⠀⠘⠿⣼⣼⡿⠷⠃⠀⠀⠀⠀⠣⢧⣧⣿⠿⠞⠀⠀⠀⠀⠱⠿⣿⣼⡧⠟⠃⠀⠀⠀⠘⠾⣼⣿⣧⠷⠃⠀⠀⠀⠈⠾⢿⣧⣯⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣔⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄⠀⠀⠀⠀⠀⠈⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠤⢀⡀
⢀⡠⢀⣀⣀⡤⣀⡀⠀⠀⠀⢀⣿⣿⣿⡿⣅⠀⠀⠀⠀⢀⣀⠁⢠⡈⠤⣄⠀⠀⠀⢠⠚⠝⠛⣛⡭⠢⡀⠀⠀⠀⡀⠂⠀⠁⠀⠤⡀⠀⠀⠀⣠⠿⠅⣤⠈⢤⣵
⣾⣧⣤⣉⣉⣥⣴⣷⠀⠀⠠⠛⠋⠉⠉⠙⠛⢣⡀⠀⠀⠆⣠⠀⡄⠖⠀⣑⡦⠀⢠⡃⢉⠂⠀⠀⠀⣀⣘⠄⠀⢨⠃⠀⠀⠀⠀⠀⣘⡆⠀⣠⡷⠦⠤⠴⣀⡤⠶
⢸⣹⡏⣿⢿⢛⡏⡇⠀⠀⠹⡤⡴⣤⢴⡦⣤⣾⠁⠀⠀⢳⢤⣴⡖⣶⢹⢧⡇⠀⠀⠹⡖⡟⣶⢻⡏⡏⡿⠀⠀⠈⡷⣶⢺⣶⡞⣿⢹⠁⠀⠉⣿⢤⣤⣴⣦⢤⢴
⠈⡇⡇⣿⣼⢸⢷⠇⠀⠀⠀⢧⣿⣻⢸⡇⡇⡇⠀⠀⠀⠸⡾⡎⡇⣿⣸⢸⠁⠀⠀⠀⡇⣿⣿⢸⣧⣇⡇⠀⠀⠀⢻⢿⢸⣿⡇⡏⡿⠀⠀⠀⢸⢸⣼⡇⡿⣿⣼
⣀⣙⣳⣿⣿⣾⣋⣀⣀⣀⣀⣈⣻⣾⣼⣿⣋⣁⣀⣀⣀⣀⣙⣷⣷⣿⣟⣉⣀⣀⣀⣀⣈⣻⣿⣼⣿⣋⣁⣀⣀⣀⣈⣻⣾⣿⣷⣟⣁⣀⣀⣀⣀⣙⣿⣧⣷⣟⣁
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠙⡏⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⢀⡠⢽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡡⠐⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡠⠒⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠂⢁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠀⠈⡧⢂⠡⢺⠁⠀⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠳⠥⠄⠐⠀⠀⠀⠀⠀⠑⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠔⠂⠀⠀⠀⠀⠀⠈⠏⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠢⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠠⠔⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡀⠀⠀⡏⠁⠒⠆⠀⠀⢤⠀⠄⣤⠠⠀⠖⠒⠈⢹⠀⠀⢠⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠰⠀⠀⢰⠀⠀⢰⠀⠀⢸⠀⠀⡇⠀⠀⡆⠀⠀⡆⠀⠀⠆⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠇⠀⠀⡄⠀⠈⠀⠀⠈⠀⠀⠁⠀⠀⠁⠀⢠⠀⠀⠸⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⡀⠀⢠⠀⠀⡆⠀⢠⠀⠀⠀⠀⠰⠀⠀⡄⠀⢀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠅⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠈⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠐⡀⠀⡄⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⢰⠀⢀⠂⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠧⠄⢠⡀⢰⠀⢠⠀⠀⠆⠀⡆⢀⣄⠀⠼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⣀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠹⠟⠿⣆⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠔⠁⠀⠙⠂
⠀⠀⢸⣁⣀⢀⢠⡇⠀⠀⣤⣶⣾⣷⣦⡀⠀⠀⠛⠲⠞⠚⠋⠀⠀⠰⣀⠉⠉⢀⠇
⠀⠀⢸⣧⠈⣀⣿⠀⠀⠀⠓⠀⠉⠉⠙⠃⠀⠀⢰⣶⣶⣴⠀⠀⠀⠀⣀⣀⡀⣠
⠀⠀⠈⠛⠛⠟⠛⠀⠀⠀⠐⠲⠶⠄⠒⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣄⡀⠀⠀⠀⢀⣤⠐⠒⠤⣄⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣚⢘⣿⠟⡀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢧⣤⢀⣄⣩⡇⠀⠀⠀⠰⣁⣖⡐⣠⣽⠀⠀⣠⠾⣯⣿⣦⡀⠀⠘⠛⠯⠍⠉⠈⠕⠛⠃
⠀⠀⠀⢀⠀⡀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠍⡀⠀⠉⠉⡁⠀⠀⢀⣄⢀⣠⢀⣄⡄⢀
⠀⠀⠀⡀⠁⠀⡀⠀⠉⠛⠉⠉⠀⠀⠀⠰⠆⠉⠒⠒⠚⠁⠀⠀⠉⠛⠂⠚⠀⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠙⠁
⠀⠀⣼⡉⠀⠀⡀⠀⠀⠀⢀⡠⠤⠶⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⣤⡉⠀⠒⢦
⠀⠀⠉⠀⠉⢁⡉⠀⠀⠀⠚⠶⠤⠤⠤⠖⡃⠀⠀⠀⢀⡪⠾⠷⠶⢵⠀⠀⠀⠀⠘⠦⠴⠤⠤⠿⠾⠇
⠀⠀⢸⡈⠂⠰⠇⠀⠀⠀⠸⣄⣀⣤⣶⣾⠁⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣦⡀⠠⢀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠂⠀⠐⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠢⠤⡤⠀⠀⠀⠀⠈⠁⠶⠶⠒⠂⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⣸⠀⠀⠀⠀⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⢦⡒⠄⠀⠀⠀⠀⢀⠀⢈⡂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢺⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢁⠈⠁⠰⣄⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢸⣯⠸
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠴⠟⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠀⠀⠀⠀⠀⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⡶⠁⠀⡹⠃⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢒⡀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠋⠠⣠⣾⣵⣿⣟
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣦⣤⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢀⣨⣀⣀⣾⡄⣾⡟⣿⣛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⢳⣾⣷⣄⣰⣿⣴⡿⣶⣦⡀⠆⣀⣀⣀⣦⣦⣄⡀⢀⣻⣿⠟⠟⠻⠿⠿⣿⣿⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠁⠈⠈⠻⣿⣷⣿⣾⡿⠋⠙⠿⣿⣿⠟⠁⣼⠀⠀⠀⢉⡷⠋⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠀⣼⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⡟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⢺⡇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣧⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢾⠇⠀⢰⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⢐⡿⠀⠀⠀⠀⢀⣿⡧⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⢸⣿⣧⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⢰⣿⣷⠀⠀⠀⠀⣄⣽⡇⡀⠀⠀⢰⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠡⠀⠀⠬⠿⠿⠧⠁⠠⠀⠘⠉



⣿⣿⠿⠿⡿⠻⠿⢿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⢰⡄⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢿⠀⠀⠀⠀⣾⣿⣧⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣴⣾⣿⡟⠃⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠀⣻⠾⢻⣿⣿⣥⣾⢹⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠄⢺⣿⠿⢇⣼⣿⣽⣿⣷⣿⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠌⠰⣶⠻⣿⢛⡿⠟⢿⣿⣧⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠈⠋⠉⢈⣀⡴⠦⡷⠟⠟⣋⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠄⠀⠀⢀⣸⣆⠠⡀⠀⠢⣄⡰⠒⢈⡿⣿⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⣻⣯⣿⠿⡿⢛⢁⣴⣾⢡⣿⢁⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠖⠂⢩⣽⢡⠬⣈⣴⣾⠷⠶⣛⣙⢳⣾⡿⡉⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⢒⡩⠔⠚⠋⠁⠀⠰⠄⢈⡿⠏⠉⠀⣀⣼⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⣀⣠⠤⠾⠿⣄⡀⣀⣀⣀⡤⢶⣾⣿⣥⣴⠶⠟⣟⣊⡵⣾⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠈⠉⠉⠁⠀⠃⢿⣿⣿⢥⠾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠗⠨⠆⠐⡄⠀⣤⠀⠠⣀⠀⠀⡀⠀⢂⡰⣦⡔⠐⣾⠉⣿⣸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡀⠃⠀⡄⣀⠃⡄⠀⢣⠀⠀⡇⠀⢸⡀⢹⣷⣤⣯⣴⣧⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⢿⡤⣻⡏⢙⠐⢾⣹⣄⠀⣇⡴⢸⣇⣨⣄⢹⡅⢿⣿⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡈⢧⠈⣧⣜⣤⣄⢸⣯⢈⡿⠀⣼⠈⢻⠻⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢳⢻⡾⢿⠉⣮⠑⢸⣷⢤⣷⣿⣿⡆⣼⣷⣿⣼⣿⣿⣷⣺⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣧⢸⣷⢿⣇⡈⣿⠙⣿⢃⣿⡻⣿⠹⣿⣿⣿⣿⡼⢤⣔⠂⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣻⣏⣯⢸⡟⠳⣿⣷⣿⡞⣿⣿⣾⣹⣿⣾⣟⠷⣬⠷⣵⣠⠄⢀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣶⣿⣿⣿⣷⣿⣿⠿⣿⣷⢫⢟⢙⡣⡭⢉⠥⠞⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠾⣟⠽⠻⠴⠇⡭⠍⠁⠓⠊

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⢠⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢷⠀⠀⠀⠀⣾⣿⣇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣴⣾⣿⡟⠃⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠀⣻⡶⢻⣿⣿⣥⡾⢹⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠀⢚⣿⠿⢇⣼⣻⣽⣿⣷⣿⣽⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠈⠰⣾⠛⣿⡛⡿⠟⢿⣿⣧⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠋⠉⠈⣡⡴⢶⠶⠟⠟⣋⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣸⣧⠠⡀⠀⠢⣄⡨⠶⢈⡿⣿⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠁⠀⠀⠉⣻⣿⣿⠿⣿⢛⠀⣴⡾⢃⣿⢀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠖⠒⢪⣽⢁⠤⣀⣴⣾⠷⠷⣛⣹⢳⣾⡿⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⣒⡩⠔⠛⠋⠉⠀⠰⠄⢉⡽⠋⠁⠀⣀⣼⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⣀⣠⠤⠾⠿⣄⡀⢀⣀⣀⡤⢴⣶⣿⣥⣴⡶⡟⣛⣋⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡁⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠈⠉⠉⠉⠀⠃⣿⣿⣿⢿⠶⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠗⠠⠆⠐⡔⠀⣤⠀⠠⣀⠀⠀⡀⠀⢀⡰⣦⡴⠳⣾⠉⣿⣸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡀⠃⠀⣅⣀⡗⢀⠀⢣⠀⠀⡇⠀⣸⡀⢻⣷⣤⣿⣰⣧⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣷⡀⣿⡇⢹⠛⢾⣾⣦⠀⣧⡴⢹⡧⣨⣄⢹⡅⢻⣿⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡈⢷⠈⣦⣸⣦⣀⢸⡿⢸⡿⠁⣾⠈⢻⠻⣿⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢳⢽⣦⣿⠁⣏⠙⢺⣷⢠⣧⣾⣿⡦⣸⣷⣿⣽⣿⣿⣿⣾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣧⣸⣶⢿⣇⡀⣿⠛⣿⢃⣿⡗⣿⡻⣿⣿⣿⣿⡜⢠⣔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣏⣿⢸⣟⠃⣿⣶⣿⡾⣿⣿⣿⣽⣿⣿⣿⠳⣬⠿⣕⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣿⣷⣿⣷⣿⣿⡿⢿⣷⣯⢟⢙⡂⠭⢉⠍⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣟⠙⠋⠴⠅⠮⠈⠁⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣚⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⢼⡷⠆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣺⣄⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢁⠞⠁⡜⠑⠒⠲⡊⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⡰⠃⠀⡜⠀⠀⠀⠀⢣⡀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠰⠁⠀⢠⠃⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠙⢦⡀⠀⠘⡄⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⡰⢅⣀⡠⠤⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠉⠓⠤⠤⠔⠛⠒⠦⠤⠒⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠦⣀⣠⠊⠁⠱⡄⡠⠊⠉⠳⡄⡠⠊⠉⠢⣠⠊⠉⢢⣀⠤⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠅⠀⠀⠀⢨⠁⠀⠀⠀⢈⠁⠀⠀⠠⠄⠀⠀⢀⠅⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘⡀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⠸⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢡⠀⠀⠀⢡⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⢀⠇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠘⡆⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⡘⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢱⠀⠀⠀⡇⠀⠀⢸⠀⠀⢰⠁⠀⢠⠇⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢇⠀⠀⢣⠀⠀⢸⠀⠀⠸⠀⠀⠸⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢄⣈⣀⣀⣘⣀⣀⣈⣀⣀⣁⣀⣀⣃⠤⠞



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢕⣠⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣚⣊⡤⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣉⠭⠉⠀⠀⠐⡷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⣀⡠⠄⠀⠀⠀⢀⡗⢆
⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠐⠒⠒⣊⠩⠀⠼⠓⡄⠀⠀⠀⠀⠀⠤⠐⠒⠂⠐⠒⠂⠀⣉⠥⠄⠤⠋⡆
⠀⠀⠀⠀⠀⠀⠤⠒⠒⠀⠀⠤⠔⠊⢁⠤⠄⠀⢠⣁⠀⠀⠀⠀⠀⠔⠒⠒⠂⠤⠤⠒⠉⢀⠤⠄⠀⢀⠇
⠀⠀⠀⠀⠀⠾⢦⡄⣀⣀⣤⣴⣂⣀⣀⡤⠤⡤⡸⠻⠀⠀⠀⠀⠀⠚⠒⠒⠀⠔⠒⠒⠊⠓⠉⢊⠉⡇
⠀⠀⠀⠀⠀⢘⢤⢨⠀⠙⠉⠉⢉⠁⢀⠉⡈⢀⠀⡆⠀⠀⠀⠀⠀⠈⠄⢆⢰⠀⡄⣄⢰⠀⡇⡘⡘
⠀⠀⠀⠀⠀⠈⡘⠘⠀⠰⠀⠀⢸⠀⠈⢀⠃⠘⢰⠀⠀⠀⠀⠀⠀⠀⠘⠘⠘⠀⠇⠻⠈⠠⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠑⠤⠤⠤⠤⠤⠀⠀⠀⠀⠠⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠄⠀⠀⠀⠤⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢖⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠈⠢⣩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⡡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢓⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠀⠀⣈⠂⠀⠀⠊⣢⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠲⠀⠀⠁⠉⠉⠀⠥⡀
⠀⠀⠀⠀⠀⠀⡇⠸⡉⠒⡒⠤⢤⣉⠒⠒⢉⣀⣰⠀⠀⠀⠀⠀⠀⡄⢘⠦⣀⡀⠀⠐⠤⢀⠠⠔⠁⢄
⠀⠀⠀⠀⠀⢠⡯⣁⣈⣓⢌⣓⣦⣠⢕⡢⢤⡠⠠⡅⠀⠀⠀⠀⢀⠣⣈⠢⡀⠤⡉⠑⠢⢄⠀⠐⠐⠂⣀
⠀⠀⠀⠀⠀⠈⡄⢠⠀⡄⢁⠀⠀⠀⡄⢀⠀⡰⢀⠃⠀⠀⠀⠀⠟⠶⣤⣤⣬⣀⣈⢭⣤⣤⡬⣤⠶⠞⡙
⠀⠀⠀⠀⠀⠀⠘⡌⠇⢳⠈⡇⢸⠀⡇⣸⠀⠇⠎⠀⠀⠀⠀⠀⢨⢀⢨⠀⡉⢈⠀⢩⠀⠀⠀⡄⢰⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢇⠈⠘⠀⠇⠘⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⡘⠘⠀⠡⠈⠀⠘⠀⠇⢰⠁⠎⡘
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠠⠤⠤⠤⠤⠐⠒⠃⠀⠀⠀⠀⠀⠀⠀⠑⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠐⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⢦⣤⣤⣄⣤⣤⠶⠞⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠂⠀⠀⠀⠀⢀⣀⠀⠀⠐⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢷⣤⣀⣀⣀⠀⢀⣀⣀⣠⣠⡴⠗⠋⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠉⠉⠉⠉⡉⠉⠉⠀⢀⡀⠄⠂⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠓⠦⢦⣠⣠⣀⣀⣀⣀⣀⣀⣠⣠⣠⣤⠤⠖⡋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣀⣀⣀⡀⠀⠈⡈⠉⠉⠉⠀⠀⢀⣂⣈⢀⣀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠉⠉⢹⠉⠛⠋⡟⠛⠛⢹⡏⠛⠋⡟⠉⠉⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⠀⠸⡟⠀⠀⣇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⠀⠀⡧⠀⠀⢿⠀⠀⢸⠀⠀⢸⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠁⠀⡇⠀⠀⢸⠀⠀⢸⠐⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢻⠀⠀⢸⠀⠀⢸⠀⠀⡞⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠦⠤⠼⠤⠤⠿⠤⠴⠓⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢩⢃⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢊⠔⠁⠀⠀⠀⡜⠀⢠⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⢀⡴⠁⠀⠀⠀⠀⡰⠀⠀⢸⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⡠⠋⠀⠀⠀⠀⢀⡔⠁⠀⠀⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡔⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⡜⠀⢠⠇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣸⠀⠀⠀⡠⠚⠁⠀⠀⠀⡠⠊⠀⢠⠮⠤⣈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠐⢢⠜⠊⠉⢀⣉⠉⠥⠞⠒⠒⢒⡂⡰⠿⠀⠠⠘⠓⠢⡀⠀⠑⡄⢱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⠉⡆⠑⢎⡠⠔⠊⠁⠀⠀⢀⠠⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢡⠀⠃⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢻⡀⠉⠳⢤⣊⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⢀⠆⠀⠸⢄⠀⠙⡂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠉⠑⠒⠤⢄⣉⡱⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⠀⠀⠀⠀⢠⠊⣀⡤⠃⠈⡀⠀⢰⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⣚⣆⡀⠀⠀⠀⠀⠀⠉⠁⠒⠐⠠⠤⠄⣠⡀⠒⠉⠀⣀⣀⠀⠤⠤⠄⠚⣋⡭⠚⠁⠀⡰⠀⠀⢀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡦⣀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣀⣀⡀⠤⠤⠒⠂⠉⠀⠀⠀⢀⡴⠁⠀⠀⡸⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⣀⣤⣉⡚⠒⠒⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⢒⠖⠋⠀⠀⠀⡰⠁⢀⠾⠁
⠀⠀⠀⠀⠀⠀⣀⠤⢲⠞⠉⠉⡁⠀⠀⠉⠑⠂⠤⠄⣀⣀⣀⡀⠀⠤⠤⠤⠀⠐⠒⠒⠈⠉⢁⡀⠤⠐⠊⠁⠀⠀⢀⡠⠊⡠⠖⡡⢁⣀
⠀⠀⠀⠀⠀⠎⠁⢠⠁⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠒⠐⠉⠀⠉⠀⠀⠀⠀⢀⡠⠄⠒⣉⠔⢊⠤⠊⠑⢆⠀⠉⠑⠢⡀
⠀⠀⠀⠀⠀⢫⠷⣾⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠒⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠒⠈⢁⣀⠴⠞⠑⠈⠁⠀⠀⠀⠠⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠱⡀⠻⡹⡖⢢⡀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠁⠙⠠⠤⠤⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⢀⣀⡠⢤⡞⡾⢉⢃⠊
⠀⠀⠀⠀⠀⠀⠀⠐⡄⠑⡜⢆⠡⡈⢇⠈⢳⠉⠑⡖⠐⢦⠔⠢⣀⠴⠠⣀⠠⢄⣀⠤⠠⢄⡤⠤⢠⠔⠢⠔⠉⢰⠋⢉⠏⢠⢃⠌⡐⡱⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢆⢢⠱⡀⢣⠀⢃⠀⢡⠀⠀⡄⠀⢰⠀⠀⢸⠀⠀⢹⠀⠀⢰⠀⠀⡘⠀⡘⠀⢠⠃⢀⠎⢠⢃⠎⡐⢐⡕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢆⢣⠑⡀⢃⠈⡆⠘⠂⠀⢡⠀⠀⡄⠀⢸⠀⠀⢸⠀⠀⡌⠀⢠⠁⢰⠃⢀⠆⢀⠆⢠⠃⠎⡰⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢆⢣⠱⡈⡆⠸⡀⢃⠀⠈⡀⠀⠃⠀⢸⠀⠀⠘⠀⠀⠃⠀⡌⠀⡎⠀⡌⠀⡘⢀⠆⡜⠰⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⡄⢃⢢⠘⡀⠃⠘⠀⠀⡃⠀⢸⠀⢸⠀⠀⡄⠀⢸⠀⢠⠁⢰⠀⢰⠁⢠⠁⡌⠰⢁⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⡈⠌⡄⢡⢘⠀⠇⠀⢡⠀⠈⡀⢸⠀⠀⡇⠀⡌⠀⠨⠀⡆⠀⡆⠀⡆⢠⠀⠇⡌⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣡⢠⢰⠘⡀⡄⢸⠀⢰⠀⠀⡇⢸⠀⢀⠁⠀⡇⠀⡆⢀⠁⠀⠆⢰⠀⠚⠘⢰⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠦⣆⣇⣃⠀⡆⠈⡀⠀⠃⢸⠀⢸⠀⢀⠁⢀⠁⢸⠀⣐⢀⣸⡠⠧⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁⠀⠉⠈⠁⠉⠀⠈⠁⠉⠀⠈⠉⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⡶⠶⠶⢶⣶⡶⠶⢶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠴⠶⠀⠐⣛⣇⢢⣴⠄⠀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠈⠀⢀⢈⣈⠋⠉⠁⠉⠀⣤⣭⣅⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣙⠋⠀⠁⠀⠀⠀⠝⠐⠛⠀⠉⠉⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡢⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠸⡄⡆⢫⠘⡇⠵⡀⠚⡁⠘⡇⠀⣇⠸⡇⢰⡯⠀⢹⠃⢸⢩⠏⡏⡿⢀⡏⠬
⠀⠀⠀⠀⠀⠀⣧⢁⢰⠀⢇⠰⡇⠀⡇⠈⡇⠀⠊⠸⠇⢸⡇⠀⡿⠀⡸⢸⢠⢇⠇⢸⠀⡎
⠀⠀⠀⠀⠀⠀⠸⡞⡄⡆⢸⠀⡇⠀⡇⢀⡅⠀⡆⢨⠀⠘⡇⠀⣇⠀⡇⡼⢸⢨⠀⡜⢰⠁
⠀⠀⠀⠀⠀⠀⠀⢧⢇⢱⠈⡀⢸⠀⡅⠀⡇⠈⡇⢸⠁⢩⡇⠀⣯⢀⠇⡇⡇⣼⢀⠇⢘
⠀⠀⠀⠀⠀⠀⠀⢹⢸⢸⠀⡆⢸⠀⡇⠀⡇⠠⡇⢸⠃⢼⠇⠀⡇⢸⠁⠇⡇⡇⢸⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠘⡆⡎⠀⠲⢸⡀⡇⠀⡇⠀⡂⢸⠀⣸⡅⢀⡇⢸⢸⢰⠃⡇⡨⠀⠃⡨
⠀⠀⠀⠀⠀⠀⠀⠀⢇⢡⠇⢸⠸⠇⣷⠀⣯⠀⡃⢸⠀⣾⠁⢸⠁⡞⢸⢸⢠⠀⠧⢀⠜⠁⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠂⠬⢀⣁⡂⠐⠋⠀⠅⠸⠀⡏⠀⣸⡆⠧⡌⠼⠠⠀⠈⠀⠀⠄⠠⠀⠈
⠀⠀⠀⠀⠀⠀⡈⠐⡀⠐⠀⠀⡁⠀⠀⠀⠈⠁⠀⠈⠀⡁⢈⠔⠀⠠⢂⠈⠀⡁⠀⠀⠂⠐⠡
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⢊⠔⡠⠀⠀⢀⠄⡀⠀⠀⠐⠁⠈⠀⡀⢁⠈⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠁⠊⡠⢀⠔⠡⠂⠔⠀⠈⠔⠠⠀⠈⠀⠁⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣔⠿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠷⠬⠝⠖⠃⠀⢀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠉⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⢀⣀⡀⡀⠀⠀⠀⠀⢰⡆⢰⡆⢲⡆⢺⡇⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡰
⠀⠀⠀⠀⠀⠀⢹⡿⣷⣶⡿⢻⣿⡟⣿⠁⠀⠀⠀⠀⠀⠸⣷⢸⡇⢸⡇⣾⠁⣿⠀⠀⠀⠀⠘⠛⠿⠛⠻⡿⠟⠻⠿⠃
⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⡇⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠘⡇⢸⡇⣿⢠⡿⠀⠀⠀⠀⢰⣦⣀⣠⣤⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢹⣄⣇⣸⡇⣿⣸⠇⠀⠀⠀⠀⠀⠛⠻⠛⠻⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⢹⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠉⠁⣠⣬⣍⠀⠃⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⣹⡿⠿⣿⣯⣀⣿⣿⠆⠀⠀⠀⠀⠀⣿⣿⣿⣷⡿⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⣿⣿⣿⠉⣿⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠥⠿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⡧⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡀⠙⢆⠀⠀⠀⠀⠀⡄⠀⠀⠰⠀⠈⢀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠒⠂⠀⠀⠈⠀⠀⠀⡐⠀⠀⠠⠄⠀⠀⠀⠈⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣁⣀⣀⣀⣀⣀⠁⠀⠀⠀⠀⢀⡀⠄⢀⣀⠤⢀⣀⠤⣀⠀⠀⠀⠐⢄⠄⠐⠄⠠⡀⠲⠄⢁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⡏⣹⣿⢻⣿⠀⠀⠀⠀⠀⠛⠃⠀⠘⠋⠀⠙⠛⠀⠛⡃⠀⠀⠀⠀⠐⣿⠀⢸⣿⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣷⣿⣧⣬⡟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠻⠀⠸⠿⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐

⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠈⠛⠀⠀⠀⠄⠀⠐⠁⠈⠁⠀⠀⠀⠀⠀⠂⠐⠀⠁⠐⠀⡀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠀⠄⠁
⠀⢀⠀⠀⢀⠤⠀⠈⢁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⡄⠠⠀⢀
⠠⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠐⡄⠀⠀⠀⠀⠀⠐⠀⣤⢦⣀⣀⢠⣤⣖⠀⣰⠆⠀⣀⡀⡀⠀⣰⠂⣀⡈⠂⠀⠀⠀⠐
⠀⢣⠀⠀⠀⠀⠀⠀⠐⣧⡿⠿⠟⠿⠿⠟⠂⣧⠼⠿⡿⠻⠵⠿⠿⣿⢿⠇⠀⠐⡢⢤
⠀⠈⠁⠀⠀⠀⡁⠀⠀⠀⠈⠉⠁⠉⠉⠉⠀⠀⠈⠁⠁⠒⠀⠀⠀⠀⠀⠀⠠⠈⠤⠔
⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠

⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠠⢀⠀⠤⠀⠀⡤⢄⢠⢠⠄⠄⢠⠠⡄⡄⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⣄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠊⠸⠃⡈⢱⢄
⠀⠀⠀⠀⠀⠀⢑⠠⢀⡙⠓⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠎⠀⠀⠄⠀⠀⢀⡠⠀⡰⢷
⠀⠀⠀⠘⢄⣀⡄⠐⢂⠁⠈⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⠁⠈⠓⣠⣄⣒⣈⣥⣴⠞⢊⠼⡱⢄
⠀⠀⠀⠀⠀⢸⠂⠐⠈⠁⠀⠠⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠂⡀⠀⠒⠀⠀⠀⠀⢀⣠⡄⡠⠊⠘⠘⢧
⠀⠀⠀⠀⢀⢚⡧⡀⠀⠐⠀⠄⢀⢡⢧⡑⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢾⡁⢉⣀⣀⠤⢔⣋⠭⢀⠤⠒⠀⡠⠃⣮
⠀⠀⠀⣠⠰⡘⣗⠤⠅⡒⠁⡀⠘⠁⣠⢷⣌⣸⠄⠀⠀⠀⠀⠀⠀⣾⡷⡀⠣⣄⠀⠀⠒⣋⡠⠀⠊⢀⡠⠔⢊⡠⠞⣴⢿
⠀⠀⠠⠜⣄⠈⠈⠁⠀⣨⣭⣄⣀⣀⣴⡳⢋⡏⠰⠀⠀⠀⠀⠀⠀⠘⣷⠉⢦⢮⣥⣛⡃⢒⠂⠐⠚⡒⢒⣀⡩⠔⠊⡛⡎
⢀⣼⣀⠄⠀⠀⢂⠤⠤⠴⠒⠀⠀⠀⠑⢊⠝⡀⢀⠾⣄⠀⠀⠀⠀⠀⢹⣣⢈⠘⠀⠁⠹⠈⡿⠉⠋⠉⠿⠉⠀⠀⠀⡼
⣼⣄⠈⠢⠤⠔⠶⢤⣀⠀⣈⡭⠤⠴⠠⠒⠛⠀⠁⣠⡿⡆⠀⠀⠀⠀⠀⠁⡄⡆⡅⠤⠀⠀⠁⠁⡄⢀⠀⢠⠀⠀⣼⠃
⢣⠘⠋⣷⠐⢦⡤⢦⣤⢭⣀⡠⠤⢤⡴⢢⠶⠲⠇⢹⢠⠃⠀⠀⠀⠀⠀⠀⣽⡔⢰⢰⠀⠀⢰⠀⡇⢸⠀⠈⠀⢠⡟
⠈⢧⢃⢸⠀⠀⠀⡄⡁⢠⢸⠀⠀⠈⠀⠈⢀⢸⠀⣧⠏⠀⠀⠀⠀⠀⠀⠀⠘⢧⣸⠸⠀⡂⠸⠀⡃⡸⠀⣇⣤⠾⠁
⠀⠘⣆⠀⢆⠀⡄⠁⢡⠸⠸⠀⠀⢰⠈⠀⠈⣸⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠷⠶⠶⠼⠟⠚⠉
⠀⠀⠈⠳⢼⣀⡄⠀⠸⠀⠀⠀⠀⠸⠀⣠⣆⡧⠞⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠈⠉⠓⠚⠶⠴⠶⠗⠚⠓⠉⠁⠀⠀⠀⠀⠀⢀⡔⡁⠀⠀⠀⠀⠀⠀⠐⠂⠆
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⢿⣷⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⡤⢄⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⣡⣉⣩⢿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠘⠀⠂⠰⡄
⠀⠀⠀⠀⠀⠀⠐⢄⣁⠎⠀⠀⠀⠀⠀⠀⣰⢟⠅⠀⠀⢠⡞⢸⡜⡌⠳⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠂⠁⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣿⠼⢖⣆⡤⠊⠀⠨⠃⠁⡀⢸⡀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠄⠈⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⣉⠁⠘⢓⣯⣤⣴⡑⠊⣇⢀⣃⡥⣺⠽⣷⢄⠀⠀⠀⠀⠀⠈⠓⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⡁⠤⠄⠠⠤⠤⠤⠤⣬⣭⡭⠍⠛⣻⣄⣴⠏⡆⢳
⠀⢠⠤⠊⡇⠀⠀⠀⠀⠀⠀⠀⣎⣡⠒⠂⠉⠉⢉⣻⡉⢉⡧⠤⠮⠣⠌⠐⣋⠞⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡠⣴⠛⣋⣉⠉⠉⠁⠀⡀⣔⣪⣵⣚⣁⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠙⢾⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠰⣻⢿⡷⠋⠰⡾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣧⣇⣀⣴⣾⣇⢤⡤⣦⡾⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⣿⠟⢳⣤⠤⠘⠉⠁⣪⣽⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠉⠀⢀⣤⣶⣿⣿⣿⣿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠿⠛⠛⠋⢉⡁⠀⢀⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠙⠒⠒⠶⠄⠸⠿⠛⠋⠁⠀⠀⠀⠀⠀⠈⠟⠄⠾⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⡻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡃⠀⠀⠉⠀⠒⠒⠶⠴⠦⠦⢤⠶⢶⡶⠒⠛⢋⡉⠹⣤⣧⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣝⠢⣤⢀⣤⣀⢀⣀⠀⠠⠐⠀⢲⣜⣿⣶⣜⣤⣠⠯⣾⣷⣎⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠉⠙⠴⠋⣩⣳⠞⣟⡰⠋⠻⣽⡀⠙⠿⣿⡄⠈⢹⣿⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇⠀⣀⠀⠀⠁⠀⢠⡇⢀⠀⠀⢠⣇⠀⠀⣿⣇⠀⣾⢇⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣿⡀⠀⣤⠀⢰⡇⣾⣰⠀⢸⣿⡄⢰⣿⡏⠀⣿⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇⢀⣸⠀⢨⣿⣿⡿⠀⣾⣿⡃⣸⣿⡇⢹⡿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣷⣸⣿⣧⢸⡷⣿⣷⢀⣿⣿⠀⣿⣿⡧⣿⣻⣿⣿⢔⣲⣾⠅⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣿⣿⠵⣿⣿⠌⣿⣯⣿⣾⣿⣿⡀⣿⣿⣅⣿⣿⣿⣯⣷⣶⡲⠾⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠿⠶⢶⣿⣯⣬⣿⣿⣶⣿⣟⣿⣿⣿⣿⣿⣿⣻⢟⣋⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠋⠉⠛⠋⠉⠁⠉⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⠃⢀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠁⠀⢀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠋⠀⠀⡀⠈⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠘⠂⠀⣤⡀⠈⠁⠀⢀⣀⠀⡗⢄
⠀⠀⠀⠀⠀⢠⠎⠀⠑⠄⣀⣀⣀⣀⡀⠀⠄⣀⡀⣉⠰⣡⢀⠦⡀
⠀⠀⠀⠀⠰⠁⢄⠐⠃⠀⠀⠀⠒⠀⠀⠠⠤⠀⠀⠀⢀⠤⢊⣀⠈⢆
⠀⠀⠀⠀⡆⠀⢀⠉⠒⠀⠤⠀⠀⠀⠀⠤⠄⠀⠒⠈⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠈⠁⠀⠀⠋⠀⠀⠀⠐⠒⠀⠀⠘⠁⠀⠉⢁⡀⡴⠁
⠀⠀⠀⠀⠀⠸⠈⠉⠀⠀⠉⠉⠀⠉⠉⠀⠉⠁⠀⠉⠁⠉⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠩⠶⠦⠀⠀⠀⠀⠤⠶⠭⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠢⠴⠃⠀⠀⣀⢄⢶⠋⢉⠔⠖⢄⣀⡀⠤⠐⠈⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡤⠠⠤⠄⠀⢀⠔⢠⢂⡆⢀⠊⠘⠀⠱⠱⡤⠄⠒⠁⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣀⠧⠀⠀⠀⠤⡜⠀⠂⢻⠃⣬⡀⠿⠆⠀⠀⡇⠠⠤⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⢄⢰⠀⠸⠈⣻⠏⢀⠀⡜⡰⠁⠒⠠⢄⣀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠈⢀⡠⠘⢢⡀⠀⠄⠰⡀⠈⣎⠔⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠒⠁⠀⠀⠀⠈⠑⠚⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣴⣶⡄⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣰⣾⣷⡄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⠟⠁⠀⠀⡄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢂⢘⡣⠀⢿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠠⠇⠀⠀⢻⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠈⠉⢀⠀⠉⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠀⠈⠀⠃⠀⠀⠹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⠂⠀⠘⠀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⠂⠀⠀⠀⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⡀⣀⢀⡀⢠⠀⣀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄⠤⠀⡄⢈⠀⡄⠠⠈⠋⠃⠙⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠒⠂⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠰⠖⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⠄⠄⠀⠃⠀⠘⡉⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠖⠀⠰⠆⢈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠒⠒⠚⠋⠱⠋⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠐⠀⠂⠀⠀⠀⡀⠀⠀⠉⠀⠏⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠢⢤⠴⠒⠲⡖⣒⡲⠿⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣅⡡⢄⠘⠀⠀⠀⠀⠀⠦⠤⠀⢐⡀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠈⠀⠢⠀⠀⠄⠀⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢈⡀⠀⠁⠀⠈⠀⠀⠒⢀⡀⡀⠁⠀⠀⠀⠀⠀⠀⢀⢠⡄⡁⠰⠀⡀⢂⠀⠠⢀⡀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢀⠀⡀⡀⣀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⢀⠀⠀⠀⣀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⣷⠒⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢹⡀⠀⠀⢀⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠐⠚⠛⠀⠀⠈⢱⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠙⠶⣄⣀⡀⢀⣠⡴⠚⠁⣠⠏⠀⠙⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⣤⠤⠶⠒⠛⠋⠉⠁⣀⣤⠞⠁⠀⠀⢠⡏⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⠀⠀⠀⠀⠀⠀⠠⠶⠖⠚⠛⠉⠁⠀⠀⠀⠀⢀⡞⠀⠀⠀⢸⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣰⠋⠁⣀⡴⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣼⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⠁⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⣴⠃⠀⠀⢸⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢃⡞⠁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡿⢿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⠀⢀⡴⠋⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⢀⡼⠁⣸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⡞⣠⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⣠⣞⣠⡾⠿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⢿⣿⣷⣧⡇⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⣀⣀⣤⣾⠟⠛⣿⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡈⢿⠛⠉⢻⣭⠴⠒⣆⣀⡴⢦⡀⢀⣠⢿⣤⣤⡴⠶⣶⣿⡏⠉⠻⢿⠟⠀⠀⡟⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⣇⠀⠈⣇⠀⠀⢸⠁⠀⠈⣿⠋⠀⠀⢹⡏⠀⠀⠈⣿⠀⠀⠀⣸⠀⠀⢸⠃⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢹⡄⠀⢿⠀⠀⢸⡄⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⠀⡟⠀⠀⠀⣿⠀⠀⡟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢷⠀⢸⡆⠀⠸⡇⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⢸⠇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⣇⠀⣧⠀⠀⣇⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢀⡇⠀⠀⢰⡇⠀⡾⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠀⢹⠀⠀⣿⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠁⠀⣣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠉⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⢈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠙⠛⠛⠁⠀⢀⠀⠀⠀⠀⠘⠑⠛⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠘⠙⠛
⠀⡀⣀⢂⠀⠀⠀⠁⠀⠀⠀⠀⢠⣠⠄⢙⡇⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢂⠀⠀⠈⠁⠀⠀⠀⠀⢠⣀⠄⢙⡇⠀⠀⠀⠐⣶
⠀⠱⠂⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡄⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠈⠓⠇⠀⠀⠀⠻⠲⡰⠦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠓⠇⠀⠀⠀⠻⠲⡰⠶⠀⠀⠀⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃⠢⠤⠴⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠦⠤⠴
⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠂⠈⠁⠀⢀⡀⠠⠀⠀⠀⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠂⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠂⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢤⠤⠤⣄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠁⡁⠀⠀⠀⠀⠀⠀⢐⠀⠀⡀⠀⢀⢤⠤⠤⣄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠋⠁⠀⡋⢠⠂⠂⠀⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠁⠀⡋⢠⠒⠂⠀⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡠⠃
⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢠⢚⠶⢀⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢰⠾⠶
⠀⠀⠀⠸⣦⣀⢀⣡⠟⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠠⠀⠐⠀⠈⠁⠀⢸⠁⠀⠀⠸⣦⣀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠠⠀⠂⠀⠈⠁
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠘⣿⣧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠀⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠉⠁⠀⠐⠀⠀⠸⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⠤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠘⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⣤⢤⣼⠁
⠀⠀⠀⡀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠡⠆⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⣀
⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣡⠰⠇⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣡⡰⠆⠀⠀⠀⠀⠟
⠀⠀⠀⠑⠀⠀⠀⠠⣄⡄⠀⠀⠀⡯⣑⠀⣘⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⢀⣄⡄⠀⠀⠀⣯⣁⠀⣘⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠐⢦⣀⡀⠁⠀⠀⠀⠈⠈⠘⠁⠀⠀⠤⡄⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡀⠁⠀⠀⠀⠈⠈⠊⠉⠀⠀⠤⡄⠀⠃⢀
⠶⠀⠄⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⢀⠄⠀⠀⠃⡀⠠⠍⠉⠁⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠠⠉⠉⠁
⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈
⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⠤⠆⠀⡴⠉⣉⡉⠃⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣀⠤⠇⠀⡴⠉⢉⡉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠄⠀⡅
⡆⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⢶⡽⠒⠀⠀⠀⠀⠘⠀⠤⠰⠊⠀⣠⣀⠀⠂⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⠶⠽⠖⠀⠀⠀⠀⠈⠀⠤⠰⠊⠀⣠⣀
⠃⠀⠀⢸⡀⠃⠀⠃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⣀⠀⠸⡄⠉⢰⠇⠀⠀⢸⡀⠃⠀⢃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠸⡔⠉
⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⢰⣿⡇⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⠀⣿⡇⠀⠀⠀⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢿⠟
⠀⠀⠀⠤⠄⠀⢀⣀⠀⢸⣶⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠂⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⢰⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⣀⠀⠀⡔⠋⠱⠤⢶⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢠⠒⣄⠀⠀⢀⠚⠠⠂⢀⠤⣄⠓⠛⢢⡑⢄⣆⠀⡀⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠑⣄⠀⡴⢶⠲⡒⠒⡖⡚⢲⠲⣆⣠⠙⠚⡗
⠀⠱⡤⡑⠼⠟⢍⡑⣄⡀⠘⠂⠊⠀⢤⡠⣵⠋⠁⢌⣀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢠⣍⣘⢃⠀⢣⢸⠀⢰⢀⠀⡇⣿⣁⠀⠘⢆
⢀⠞⠀⠀⠀⠀⣠⠞⠳⠥⠀⠀⠀⢇⣠⠇⢸⡧⣵⠀⢠⠒⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⢀⡀⢹⣤⠂⢸⢀⠀⢸⡄⢠⢇⡿⢹⣱⠒⠋
⢇⠀⢀⡴⢾⡉⠙⠢⣄⣔⡇⠈⡂⢀⠤⣤⠋⠧⣈⣁⡜⠀⠀⠀⠀⢀⡀⠤⡤⠤⠤⢤⣤⣤⣀⡀⠀⠈⠓⢻⠃⣧⠜⠋⠈⠒⠞⠚⠒⠛⠛⠉⠉⠀⠀⠉
⠀⠉⠉⠀⠀⠑⠤⠴⠛⠾⣌⣉⣠⡧⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⡎⠑⠠⠀⣌⡶⡷⠷⢶⡀⠤⣍⣣⠀⠀⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⡒⠲⠤⠤⡶⠤⠤⠼⠖⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⠀⠀⠀⡇⠀⠀⢀⡄⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠒⡩⠤⢕⢤
⠀⠀⠀⠀⠀⡰⠲⡋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠠⠤⠤⠧⠤⠤⠚⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢡⠀⡠⢀⠑⠤⠚⠀⣣
⠀⠀⠀⠀⠀⣧⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠒⠁⠀⠢⠴⠃⢀⣀⠸⣄⡆
⠀⠀⠀⠀⠀⢏⠳⣄⠈⠓⠤⣀⡀⠀⠀⢀⣰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⡄⠀⠐⢄⠸⣄⡨⠃⡠⠁
⠀⠀⠀⠀⠀⠘⢦⠀⠑⠢⠤⣀⣉⡉⢉⢉⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⣀⡴⠂⠈⠉⠑⠢⠤⣀⠀⠀⠈⠣⣆⠀⠁⡁⠄⣀⣰⠻⠥⣀
⠀⠀⠀⠀⠀⠀⠀⢹⠦⣀⡀⠀⠈⣉⡡⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠂⠁⠀⠒⠀⠤⣠⠀⠈⠁⠀⠀⠀⠀⣀⠄⠊⠉⡄⠀⠀⠀⠀⠈⠈⡑⢟⡉⢠⠻⡖⠊
⠀⠀⠀⠀⠀⠀⠀⡼⠀⡰⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣎⣁⡘⠊⢇⡀⠀⠀⠀⠂⠀⠀⠀⠉⠓⢂⠔⠊⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠉⠁⠈⠙⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣰⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠈⠉⢀⠿⣓⣦⣀⠀⠀⢀⣤⣀⠂⠀⢸⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠱⡀
⠀⠀⠀⠀⠀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⣞⡋⠤⡀⠀⠀⠀⠀⠀⠔⠅⠈⠀⠀⠀⢉⡹⠓⡄⠉⡆⠀⣸⠗⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠱⣄⢑
⠀⠀⠀⠀⢰⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⢬⠑⠒⠒⢮⠒⠉⠓⢦⠒⠒⠲⣤⠐⠀⢠⠀⡾⠊⠀⠀⠀⠀⠀⣠⠔⠛⠉⡡⢈⠩⠛⢦⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣆⠠⣀⣔⠠⠤⢄⣍⣀⠀⣀⣸⠀⡠⠞⠉⠁⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⣀⣀⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⢏⠠⡀⢀⡴⢋⢀⢔⢹⡏⠩⢑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠚⡟⠀⠰⡉⢁⣼⢅⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠤⣼⠗⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠈⢷⡤⢀⣷⣀⠴⡻⠉⠈⠎⣆⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠚⠉⠙⠢⡈⢆⠀⠈⣷⡖⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣄⣙⣺⣤⡤⠴⡟⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠐⠚⠉⠁⢄⠙⢆⠀⠀⠀⢹⠈⣄⡴⠙⠛⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⢆
⠀⠀⠀⠀⡀⠠⣀⡀⠀⠀⠔⠉⢍⡉⠲⣌⠢⠀⠀⠀⠑⠘⣀⣠⠤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠳⡄
⠀⠀⠀⠐⡀⠀⠸⡈⠑⠚⠒⢀⡀⠱⠀⠈⢆⣷⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠘⢆
⠀⠀⠀⠀⡝⠉⠀⡨⠛⢳⠔⠣⠤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠘
⠀⠀⠀⠀⢇⠀⣸⠽⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⣤⣤⣶⡿⠋⢁⡾⢃⠀⢸⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢶⡶⢚⣿⣇⢀⡿⣾⠏⠀⠀⣼⠃⠸⣄⠀⣷⠶⢶⡏⠙⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣁⣰⠏⣴⠏⡽⠉⠻⠃⣿⠀⠀⣶⠿⢷⡀⠙⠳⢾⣷⣿⣷⣴⡟⠛⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣰⠈⢹⣤⡟⠀⡇⠀⠀⠀⢻⣄⠀⣿⣤⣴⠾⠦⠤⣤⣀⡉⠉⠉⠙⠓⠀⠰⢬⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡙⢷⡟⠉⣷⠀⢳⡄⠀⠀⠀⠉⠛⠶⠶⠥⠄⠀⣴⠶⢻⡛⠛⠛⠛⠙⢻⣇⢀⣻⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢷⣄⣿⡀⠈⢧⡀⠙⣷⡄⢢⡀⠀⠀⠀⠀⠀⠐⣿⣤⣴⠿⣤⣄⣀⣀⠀⠙⠛⣁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢰⡏⠛⢻⣄⠀⠙⠢⣿⣄⣼⠛⠛⠳⠶⠶⠶⠶⣶⣬⣯⣤⣤⣬⣭⣿⣿⠟⡛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠃⠀⠀⠙⠷⣤⣀⠀⠈⠛⠓⠒⠶⠶⠶⠶⡴⣤⣤⣤⣤⣈⣡⣴⣿⣧⠞⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⠶⠶⠒⠛⠉⠀⠤⠶⠶⠾⠿⠟⠛⠋⠀⢠⡤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⢶⣆⢀⣤⡀⠀⣀⡀⠀⢀⡀⠀⠀⣀⡀⠀⣠⣄⡀⣠⣦⣄⣴⠟⣷⠟⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⣿⠋⠈⢷⠞⠋⢳⣶⠟⢻⣦⡾⠋⣿⠾⠋⣿⠿⠋⢸⠟⠁⢰⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢹⡄⠀⢸⡆⠀⠸⡇⠀⠈⡇⠀⠀⡇⠀⠀⣿⠀⠀⣿⠀⠀⣸⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣇⠀⠀⣇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡏⠀⠀⡟⠀⠀⡟⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢹⠀⠀⢻⠀⠀⢿⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠀⠀⠸⠀⠀⢸⠀⠀⢱⠀⠀⡇⠀⠀⡇⠀⠀⠃⠀⠀⢁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⣄⣀⣈⣀⣀⣈⣀⣀⣁⣀⣀⣁⣀⣠⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣿⠛⠓⠉⠉⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠙⠁⠀⢀⣀⡀⠀⠀⠀⠀⢈⢭⡝⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠰⣯⣭⠇⠀⠀⠀⠀⠙⠚⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⣾⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⢻⣦⣤⣤⣤⡴⢶⣦⣀⠀⠀⠀⢀⡠⣴⣀⠀⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⡇⠸⡇⢸⡇⢸⡆⢹⡏⢻⠏⣽⢀⡏⢉⡏⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⡇⢸⡇⢸⡇⢸⠀⢸⠀⡿⢸⡇⣸⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⡇⢸⠃⢸⠀⣼⠀⣿⠀⡇⢸⡇⣏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢧⣼⣀⣼⣀⣿⣀⣿⣀⣧⠼⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢞⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⠌⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠘⠁⠀⠀⡜⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⡠⠋⠀⠀⠀⡠⠀⠀⡇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢀⠎⠀⠀⠀⠀⠔⠁⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠠⠁⠀⠀⢀⠔⠃⠀⠀⡠⠁⢠⠓⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢀⡦⠀⠒⠒⢂⢤⡎⠀⠀⢄⣤⠊⠀⠤⠋⡈⠐⢄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡐⠋⠢⣌⡠⠔⠊⠁⠀⣀⠤⠒⠂⠁⠀⠀⠀⠀⠀⠄⠀⠀⡄⠹⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣣⡈⠛⠤⢋⠀⠀⣀⠔⠈⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠀⠀⠠⠁⢐⢦⠀⠱⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠈⠁⠒⠀⠤⢏⡀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⢀⣴⠡⡴⠊⠀⠆⠀⠃⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠛⡒⠓⠤⠄⠤⠀⠀⠀⠀⠀⠈⠁⠲⠾⠤⠤⠀⠒⠂⢈⠉⠤⠐⠊⠀⠀⡘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⣑⡠⠤⠀⠀⠀⠂⠀⠀⠀⠒⠒⠒⠂⠀⠀⠉⠀⠀⠀⢀⡀⠤⡴⠊⠀⠀⡰⠀⣰⠃
⠀⠀⠀⠀⠀⡀⠄⠖⠋⠙⡁⠀⠉⠐⠠⠄⠀⠀⠀⠀⠀⠠⠤⠄⠀⠒⠒⢀⡉⠁⠄⠂⠁⠀⠀⡠⢚⡠⢊⠆⢀⡀
⠀⠀⠀⠀⡊⡀⡌⠀⠀⠀⠈⠢⠄⡀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠐⠂⠉⠁⠀⠀⠀⠀⡀⠄⠒⡉⢔⡡⠐⠉⠢⠀⠈⠑⢂
⠀⠀⠀⠀⠑⡙⢳⣄⡀⡀⠀⠀⠀⠀⠀⠈⠁⠉⠄⠀⠀⠀⠀⠀⠠⡤⠀⠒⠂⢉⡁⠠⠐⠋⠁⠀⠀⠀⠀⢀⣂⢀⡔⠟
⠀⠀⠀⠀⠀⠘⢄⠡⡡⡘⡎⢶⠒⢤⠀⢄⠤⢄⡀⡀⢀⣀⠀⢀⠀⢈⣉⠉⢉⡀⠀⢄⡠⠢⡤⠂⡔⢰⢁⢃⢏⠨⠂
⠀⠀⠀⠀⠀⠀⠈⢂⠐⠜⠈⠄⢂⠀⠄⠘⡀⠈⠀⠈⠃⠀⢹⠀⠈⠁⠀⠸⠀⢨⠁⠰⠁⠀⠀⡜⠠⢡⢃⢂⡦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠌⠌⠄⢣⠘⠀⠁⠀⢃⠀⠀⠀⠈⠀⠀⠀⠀⠇⠀⡀⠀⠁⠠⠁⡘⢀⢃⢂⠆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⡌⠜⡀⠆⢁⠘⠀⠘⠀⠸⠀⠀⠀⢠⠀⠀⠀⢠⠁⡌⠀⠆⢰⠀⠀⠄⡌⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠐⡐⠐⠘⠈⡀⠃⠀⡁⠀⡄⠀⠀⢸⠀⢰⠀⠈⠰⠀⠰⠀⠆⠰⡘⠘⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠠⠀⠃⠸⢀⠃⢁⡣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠬⣰⢠⠀⡀⢰⠀⢀⢀⠀⡂⠀⡇⢰⠀⡀⢀⠀⣀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡈⠀⠉⠈⠈⠁⠀⠁⠈⠀⠉⠉
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠖⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⣰⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀⣞⡅⠀⠀⠀⠀⠀⠀⠀⡴⠂⠒⢦
⠀⠀⠀⠀⠀⠀⢮⣀⣄⣄⡼⠀⠀⠀⠀⠀⠀⡰⠣⠭⢽⡀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠦⡀⠀⠀⠀⠀⢀⠷⠔⠒⢺⡄
⠀⠀⠀⠀⠀⠀⠈⢟⠢⡱⠁⠀⠀⠀⠀⠀⠀⣗⢒⠒⢒⡇⠀⠀⠀⠀⠪⣀⢤⣀⡠⣀⡼⠀⠀⠀⠀⠸⡀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢸⠁⠀⠀⠀⠀⠀⠸⣴⣤⣦⠇⠀⠀⠀⠀⠀⠀⠈⢹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣱⠦⡀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠢⡀⠀⠀⠀⠀⢀⡠⠚⣧⣼⣦⠀⠀⠀⠀⠀⡰⠁⠉⠉⡖⠁⠀⠀⠀⠀⠀⡴⠷⠞⢄
⠀⠀⠀⠀⢰⣫⢤⣀⠤⣄⡠⢌⣲⠀⠀⠀⣏⣩⣭⣤⣤⣤⡇⠀⠀⠀⠀⠳⡒⠒⢲⠞⠀⠀⠀⠀⢀⣠⣏⣙⣉⣹⣄⡀
⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠯⠭⠭⠭⠤⠴⠃⠀⠀⠀⠀⠀⢹⢻⠏⠀⠀⠀⠀⠀⠈⠑⠒⣢⣖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠽⠦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠤⠶⠭⡉⠑⣄⠀⠀⠀⠀⢰⠣⢢⢔⢮⢣⠀⠀⠀⠀⠀⡔⠉⠀⠈⠱⡄⠀⠀⠀⠀⢸⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⡏⡞⡉⠳⡈⡆⢸⠀⠀⠀⠀⠸⡰⣐⣾⣮⡾⠀⠀⠀⠀⢰⠳⡖⢶⢲⡖⣷⠀⠀⠀⠀⢸⣿⣿⡛⢻⡇
⠀⠀⠀⠀⠀⠱⣱⢌⣒⣁⠷⠊⠀⠀⠀⠀⠀⠉⢲⣿⠃⠀⠀⠀⠀⠀⠈⣇⣧⢸⢸⢧⡏⠀⠀⠀⠀⢸⠋⠛⠻⠿⣯
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡔⠋⠀⢙⠲⡀⠀⠀⠀⠀⠀⠀⢀⣠⡓⢤⠀⠀⠀⠀⠀⠀⡞⡁⡙⡆⠀⠀⠀⠀⠀⠀⣀⣄⣤⣀
⠀⠀⠀⠀⠀⡜⠡⠆⠀⠈⠀⢳⠀⠀⠀⠀⠀⡴⡏⢣⠈⡞⠁⠀⠀⠀⠀⡖⠚⠈⠘⠒⡆⠀⠀⠀⣰⡎⢱⡀⢀⠎⢑⡆
⠀⠀⠀⠀⠀⢸⣀⠀⠀⠙⢀⠎⠀⠀⠀⠐⣮⠣⣸⡤⠋⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢏⠀⠀⠀⠀⠙⢌⡏⠙⠋⢹⣡⠃
⠀⠀⠀⠀⠀⠀⠈⠙⣾⠉⠁⠀⠀⠀⠀⠀⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠋⠳⠬⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⢀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⢠⠎⠀⣠⠀⠈⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⢀⡴⠋⠀⢀⣏⠀⢀⠞⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢀⣴⣏⣤⡤⠶⠋⠁⠀⣀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠟⠛⠋⠉⠉⠀⢀⣀⣤⠴⠚⠁⠀⠀⠀⢠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⣀⣠⣤⣤⡶⠶⠞⠛⠋⠉⠀⠀⢀⡠⠀⠀⢀⡼⠃⢹⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠛⠉⠉⠁⠀⠀⠀⠀⣀⣠⡤⠖⠊⠁⠀⣠⡴⠋⠀⠀⠜⠀⠈⢻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠘⢯⣀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠀⠀⠀⠀⠒⠉⣀⣀⣠⠤⠀⠀⢀⠀⠀⣿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⠲⣄⠸⣯⣤⡤⠴⠦⠶⠶⠶⠶⠖⠚⠛⠛⠉⠉⢁⣀⣠⠤⠒⠁⠀⢠⡟⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠘⣧⣀⡉⠳⢦⣤⣷⣄⣤⣴⣶⠶⠛⠛⠛⠛⠛⠛⠉⣉⣁⣀⣀⣤⡤⠾⣫⡼⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣀⣉⠙⠂⠀⠠⢌⣉⠀⠀⠀⠀⠀⠈⠐⠊⠉⠉⠉⢉⣉⠉⠀⠀⠀⠞⠁⢰⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡉⢹⡷⠶⣶⠶⢦⣌⣉⣉⠙⠳⠶⠖⠛⠛⠛⠒⠚⠉⣀⣀⣀⣠⡴⠛⣿⢋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢳⡀⢹⡄⠘⡏⠁⢻⠓⢲⡆⠐⡖⠂⣿⠙⣳⠋⢁⡟⠀⣼⠁⣸⠃⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣧⠀⣇⠀⢻⠀⢸⡇⠸⡇⠀⡇⠀⡏⠀⡿⠀⢸⠃⢠⡇⢠⡏⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠸⡄⢸⠀⠸⡄⠀⡇⠀⡇⠀⡇⠀⡇⠀⡇⠀⡾⠀⢸⠀⡼⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢡⠀⠇⠀⡇⠀⢳⠀⡇⠀⡇⠀⠇⢠⠃⠀⡇⠀⡆⢠⠃⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠒⠒⠒⠲⠶⠶⠶⠶⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⡥⢋⡇⣿⢯⡗⢍⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠵⠀⢀⡴⠛⠁⣸⢈⡇⢸⡇⢸⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⠛⣛⣘⣻⢩⡴⠞⠃⠚⢁⣾⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠴⠒⠉⠉⣁⣠⣤⣥⣤⣀⣀⣀⣭⡾⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⣠⡴⠾⠛⠻⢋⣉⠠⠤⢶⢚⣡⣾⠇⣿⣿⡜⡏⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣴⠟⠁⡠⠔⠂⢉⣠⠤⠐⠋⠉⠁⣀⣁⡼⣻⡿⢃⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⣼⡇⣰⠋⠀⢠⠞⠉⢀⡠⠔⠒⠉⣁⣤⡤⠞⣩⠴⠟⢣⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠙⢧⢳⡀⢠⠇⢀⡴⠋⢀⣤⠖⠉⠁⣠⣶⣋⣀⣠⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⠓⠄⠀⢌⠙⠺⠧⠼⡧⣾⢿⣥⣤⢖⣻⣛⣧⣴⣯⣿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠞⢓⠛⠒⢻⠛⠛⢻⢹⣯⢨⡇⡏⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⠈⠀⡅⢸⢸⠀⡞⢸⢻⢸⢱⢳⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⢈⠀⡇⡏⢄⡏⡟⣼⣯⣯⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢧⢠⢀⢆⠀⠀⠐⢸⢈⣸⢸⣃⣷⣺⢿⣷⣿⣿⣿⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⢮⣞⣿⡄⡄⡄⣼⣿⣿⣼⣾⣾⣿⣸⣿⣿⣿⣿⢿⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⠆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣷⣾⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠿⡟⠟⠛⣛⣁⡂⠀⠂⠀⠀⠀⠀⠉⠉⠛⠿⠿⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⣩⣤⣼⠷⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⢄⣈⠉⠛⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⣡⣴⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⣈⡙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣇⠈⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⠟⢁⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣾⡿⠟⠋⠉⠉⠉⠁⠈⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⣻⣿⣄
⠀⠀⠀⠀⢀⣾⡿⠃⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠹⣿⣧⡀
⠀⠀⠀⢠⣿⡟⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠰⣧⣐⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠈⠉⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠙⣦⠹⢿⣷⡀
⠀⠀⢠⣿⠏⢄⣼⠃⠀⠀⠀⠀⠀⠀⠀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠃⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢿⣷⡀
⠀⢀⣿⡟⡄⣼⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠈⣧⡨⢿⣷
⠀⣼⡿⠡⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣀⠀⠀⠀⠀⠀⠘⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⣿⣇
⢠⣿⠇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠷⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⡿⠿⣿⣿⣷⣶⣶⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⢹⣧
⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣆⣩⣭⣛⣛⣿⡿⠿⠿⠻⣏⣽⠛⢦⣀⡿⠙⠻⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠾⣿
⣾⡇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠁⠀⠀⡉⠀⠙⠂⠀⠀⠈⠀⠀⠈⠉⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠁⣿
⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠆⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣿
⢿⡇⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⢸⠀⡆⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⣿
⢸⣷⠄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟
⠘⣿⡇⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣤⣄⣀⣀⣀⣠⣄⣤⣤⡤⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣸⡇
⠀⢻⡾⡄⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢀⡿
⠀⠈⣿⡈⠚⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣾⡃
⠀⠀⠘⣷⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⢶⡤⢶⠀⠀⠀⠀⠀⣴⠦⠀⡔⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⣾⡷⠁
⠀⠀⠀⠘⣧⡰⡈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣼⠁⡞⢠⡆⢰⠆⢠⡟⠁⣼⡟⢠⡆⢠⡦⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⣾⡿
⠀⠀⠀⠀⠈⢷⡅⠀⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠏⠰⠃⠸⠷⠚⠖⠻⠖⡿⢿⠚⡟⡷⠺⠶⠻⠀⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⣾⠕
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠎⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⣴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠈⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⣉⡴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣎⡙⠿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡿⠋⣡⣴⡯⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣌⡛⠻⢿⣷⣶⣤⣤⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣤⣤⣶⣾⠿⢟⣉⣥⣴⡮⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣤⣍⣉⣙⣛⠛⠻⠻⠿⠿⠿⠛⠛⡛⢋⣉⣡⠴⣾⡿⠽⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⣷⠷⠿⠿⠿⠽⠿⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⡦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠆⠀⠑⢦⣀⠀⠉⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢠⡀⠀⠀⠙⢶⣤⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠌⠀⣀⠀⣀⡉⠠⠀⠀⢸⣿⠃⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠖⠋⠳⢮⠄⠔⠛⣛⡥⠞⠉⠣⡀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣸⠏⠀⠀⠀⠀⣱⣴⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠴⠋⠀⣿⠀⠀⠀⠀⢀⡟⠁⣯⠅⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⢃⣀⣠⡴⠞⠋⠀⢀⣼⢄⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣖⠒⠒⠖⠒⠚⠃⠀⠈⣀⠤⠀⠉⠉⠉⠀⢀⣀⣤⣤⣾⣿⣿⡿⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠈⠲⢦⣀⣀⣀⣠⣴⣟⡁⠀⢀⣀⣤⣴⣾⡿⠟⣯⠛⡉⠉⣩⣀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⣀⠀⠀⠀⠈⠉⠉⠉⠋⠋⠙⠙⡝⠛⠙⠉⠈⠀⢀⣠⢼⣧⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡗⣶⠢⣶⣄⢠⣤⣠⣤⣶⣤⣴⣤⣽⣴⣶⡶⢿⡟⣻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢱⢸⡄⢸⡄⠈⡇⢸⡅⢸⡇⢸⡏⢹⠃⣸⠃⣸⣧⣿⣷⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢷⠈⣧⠀⣷⢈⡇⢸⡇⢸⡄⣾⢀⣽⢄⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⣾⡆⣿⡀⢽⣬⣧⢻⣇⣸⢄⣿⢪⡿⣾⣿⣿⣿⣿⣿⣿⣤⣄⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⢷⣻⡖⣼⣿⣿⣹⡟⣿⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣼⣿⣿⣽⣗⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⡁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠿⠃⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠟⠻⠛⠛⠋⠋⠁⠁⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⡿⢶⣤⠀⠀⠀⢀⣀⣀⣀⣀⠀⣀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣙⣛⣋⣿⣿⣦⣄⡿⣿⠶⠿⠛⠋⠉⠉⠛⠛⠛⠻⢯⡉⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣿⣀⣠⣴⠶⠿⠶⢶⡄⠂⠀⠀⠉⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠷⡆⠀⠒⣤⠀⠀⠲⣦⣄⠀⠀⠀⠀⢈⠻⣶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡝⠁⡀⠀⠀⡀⠀⠀⢸⡇⠀⠀⠈⢿⣷⡄⠀⠀⠛⠀⠘⡄⠈⠙⢶⡄
⠀⠀⠀⠀⠀⢠⣀⣴⠏⠀⠻⣝⠳⣤⣈⠙⠛⠛⠛⠛⠛⠹⢚⡡⢀⡼⠀⠀⣰⠃⠀⠀⣾⠁⢀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢹⠀⠰⠦⠙⣦
⠀⠀⠀⠀⠀⢨⡉⠙⣄⠀⠀⠈⠳⠦⣬⣉⣁⡀⡀⠠⠔⠚⢁⣴⠟⠁⣠⡾⠃⠀⠀⣰⠏⠀⠈⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠈⡇⠀⢠⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢳⡆⠈⠓⠢⠤⠀⠀⠀⠀⠀⢀⣀⣤⡴⠾⠋⢡⣴⠟⠋⠀⠀⠀⣰⠏⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⡀⠀⡇⠀⠘⡆⠀⠈⣆
⠀⠀⠀⠀⠀⣰⠋⢀⣍⠛⠻⠿⠾⠿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⡇⠀⠀⢹⡄
⠀⠀⠀⠀⢰⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠁⠀⠀⠀⢀⣴⡿⢋⡀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠁⠀⠀⡷⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠻⣮⣙⠓⠒⠶⠖⠒⠒⢚⣋⣉⣡⣤⣤⣶⣶⣿⡿⠟⠋⠀⠀⠀⣀⣤⡾⠟⠉⠀⠈⠁⣠⠂⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡇⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢈⡽⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠤⠴⠖⠛⠉⠁⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⣾⠁⠀⠀⣀⣸⠃⠀⢀⣿⠷⣄
⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⣰⣿⠏⠀⠀⣼⣿⣧⠘⢷⡀
⠀⠀⠀⣠⠟⢸⣀⣤⣤⣤⣶⣶⣶⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣤⣶⠶⠾⠟⠛⠉⠁⠶⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⣰⡿⠃⠀⠀⢰⣿⣿⠁⢠⡌⢷
⠀⠀⠀⡏⠘⠈⠙⢿⣿⡟⠁⢠⡄⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠉⠀⠀⣠⣾⠟⠀⠀⠀⣠⣿⡿⠁⠄⠸⠃⢸⡇
⠀⠀⠸⡆⠉⢠⡆⠀⠘⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣴⣶⣾⠿⠟⠋⠉⠀⢀⣠⡴⠟⠋⠀⠀⢀⣠⣼⡿⠟⠀⠶⠀⢀⣤⣿⠁
⠀⠀⠀⣧⠀⢺⡃⠀⢠⡌⠻⢷⣶⣶⣤⣤⣶⣶⠾⢿⣉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⣀⣤⣀⠐⠒⠛⠉⠁⠀⠀⣀⣤⣶⠿⠛⠁⠀⣀⣠⣤⣶⣿⡿⠃
⠀⠀⠀⢹⣧⡈⠻⣆⠀⠀⣀⡀⠀⠀⢠⡀⠀⠠⠐⠂⠹⣷⣦⣤⣀⣀⣀⣀⣀⣤⣤⣶⡾⠟⠛⠻⠿⢿⣷⣶⣶⣶⡿⠟⠋⠙⠀⣠⣴⣿⡿⢿⣿⠋⠁
⠀⠀⠀⠀⠻⣿⣶⣤⣀⠀⠉⠁⠀⠀⠈⠣⠀⡀⠀⠀⠆⠀⠙⠛⠻⠿⠟⠛⠛⠋⠉⠖⠀⢀⡄⠛⠀⠀⠁⣤⠀⠀⠀⠀⣀⣴⡞⢉⣿⣿⠀⣾⡏
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣾⣷⣤⣀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⣠⣠⣴⣶⣚⣁⣀⣀⣤⣴⣶⡿⢿⣿⡃⢸⣿⡇⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠘⣿⡇⢹⣿⣿⡀⠘⣿⣿⡿⣿⣶⣶⣾⣷⣶⣶⣶⣶⣿⣿⣿⣿⠛⠛⣿⣿⣿⠿⢿⣿⡏⠀⣾⣿⠀⣾⣿⠁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢻⣿⠀⣿⣿⡇⠀⣿⣿⡇⠈⠙⣿⣿⡏⠉⢹⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⣸⣿⡇⠀⣿⣿⢀⣿⡏⢸⠏
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⡘⣿⡆⢹⣿⣇⠀⢸⣿⣷⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⣿⣿⠁⢰⣿⡇⢸⣿⢃⣿⣤⣀
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣧⢹⣷⠘⣿⣿⠀⠸⣿⣿⠀⠀⢻⣿⡇⠀⠸⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⣿⣿⠀⢸⣿⠇⣼⡟⣼⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⡄⢻⣿⡇⠀⣿⣿⡆⠀⢸⣿⡇⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⣿⣿⠀⣾⣿⠀⣿⢧⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣿⣇⠀⢻⣿⡇⠀⢸⣿⣷⠀⠀⣿⣷⠀⢰⣿⣿⠀⠀⣿⣿⡇⠀⣿⡇⠀⣿⡿⢸⣿⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣤⣿⣿⠀⢸⣿⣧⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⣿⠀⠀⣿⣿⠁⢰⣿⡇⢰⣿⣇⣾⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣠⣤⣿⣿⣀⣀⣿⣟⠀⢸⣿⣿⠀⢀⣿⣿⢀⣸⣿⣷⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠻⠿⠿⠿⠛⠋⠛⠛⠛⠿⠛⠛⠻⣿⠿⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⢘⡦⠤⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠂⣸⡶⡏⠉⢀⠈⠰⠐⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣙⣿⣿⣤⠙⣄⠀⠆⠀⢃⡈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣽⣿⢿⡖⠾⣶⣆⣜⣛⣿⡷⠇⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣼⣇⣤⠾⠶⢟⣿⡹⣯⡤⣠⠆⠀⢉⡿⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢩⡿⠐⣿⣧⣌⡷⢛⣙⣃⣿⡾⣻⣟⡶⢈⡱⣧⡈⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢚⣿⣿⡌⢺⣧⡀⠻⢿⣿⣿⣿⣿⣧⣄⣷⣔⣫⣀⣤⣽⡿⠁⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢊⠙⢹⣿⠀⠈⠉⠓⠠⢍⣿⠟⠛⠛⠋⣹⣿⠟⠋⣁⣠⣴⠇⠀⣸⠀⠀⣿⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠈⠻⠿⣷⣄⣀⡀⠀⠀⠈⠙⠲⢦⡉⢉⠡⠒⠉⠀⠀⠀⡀⠐⠁⠀⣰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢿⣿⣷⡀⠀⢈⠻⡍⠁⠒⠒⠶⣶⣶⡶⠛⠁⠀⠀⠀⢀⣴⣾⠶⣲⡶⠚⡏⠈⢤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠹⠿⣿⣇⠀⠀⠻⣿⣿⣿⣛⣛⡉⣤⡤⠤⠴⠶⠖⠛⢩⠞⠀⠐⠋⠀⠀⡇⠀⠸⣦⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠈⠛⠟⣷⣿⡿⢟⡫⠉⠁⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⢠⠁⠀⣰⠃⠀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣦⣴⠶⠿⠶⠾⠧⠤⠤⠤⠀⠀⠀⣠⡶⠚⠉⢀⡠⠖⠃⠀⠀⠀⢀⣠⡴⠃⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠛⣿⣦⣤⣤⣀⣀⠀⢀⣀⣤⣶⠾⠷⣶⣶⣭⣁⣀⣀⣀⡤⠶⠛⠉⠀⠀⣀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣄⠉⠉⠉⠉⠋⠛⠋⠙⠐⠂⠁⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⢀⣀⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠻⠒⠰⠶⣤⡀⠀⣤⡀⣀⣀⣀⣀⣀⣠⣤⡤⠤⣶⠞⠛⠯⡙⢸⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⣆⠑⠀⠀⢀⢀⠀⠈⠟⠉⡁⠀⠊⠁⢨⡀⣤⢁⡄⣤⠀⡇⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣾⠀⢧⢰⡄⠈⠂⠀⡄⠀⢹⠀⠀⠀⢰⠀⡟⠀⠀⡇⢸⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢻⡟⣿⠀⣧⠀⠀⠀⡇⠀⢸⠀⠀⠀⠘⠀⡆⠀⠀⠃⡿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢻⣿⣿⠏⠀⢿⡀⠀⠀⣯⠀⠸⠀⠀⠀⣀⠠⠅⠀⠰⠺⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣷⣶⣿⣤⣤⣼⣧⣀⣀⣠⣮⣥⣤⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡫⢶⣿⣷⣾⣻⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠽⠿⠗⣿⠿⠷⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⣻⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⣴⣶⢾⣿⡄⣿⣲⡝⣛⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⠖⠒⠃⠀⢀⣠⠏⠉⠀⣼⣿⣦⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣀⣴⡖⠉⢀⣠⡴⠾⣟⣛⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣄⣠⣤⡴⠋⠀⡐⣦⣿⣿⣿⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡤⠖⠛⢿⣿⠗⠋⠀⣀⡀⢬⡿⢋⣵⠟⣿⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢠⣶⣉⣿⡯⢾⠋⢁⡴⢫⣶⣿⠷⠚⢋⣹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣼⣿⣿⠟⠁⠀⣰⠟⣴⠛⣻⣿⣿⣿⣿⣿⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠌⢛⠻⣶⣶⡾⠓⢚⡛⣛⠻⣿⢿⢿⣿⣿⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⣿⣿⣴⣆⡙⠈⠣⣀⠠⣾⣿⣛⣻⣷⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠚⣿⣄⣿⣿⣇⡀⢤⣬⣼⣿⣿⢿⣿⣿⣿⡃⣾⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠸⣿⠋⣿⡇⢤⡾⣿⡿⣿⣿⡄⠻⣿⡟⢁⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣧⢸⣿⠀⠀⣿⡇⠀⣿⠇⢰⣿⡇⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣿⡘⣿⣿⢸⣿⠁⢸⣿⠀⣾⡿⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢻⠹⢸⣿⡀⢸⡿⢸⣿⣿⣿⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⢼⣏⣸⣿⣃⣻⣿⣾⣿⣿⣿⣿⣿⣦⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⢿⣿⣿⡿⠍⠉⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣰⠂⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢠⡏⠀⠀⠀⠀⠀⠻⣦⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣻⡇⢸⡇⠀⡀⠀⠀⠀⠀⠈⠙⠛⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠛⠛⠛⠻⠇⠈⠀⣸⠃⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠴⠚⠉⢀⣠⠴⠖⠚⢻⡆⠀⢈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠠⠴⠚⠋⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⢀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡾⢷⠄⠀⠀⠀⠀⠀⠈⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⢀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣀⣀⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠠⠞⠋⠁⠀⠀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣦⣤⣤⣤⣤⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⡟⠀⡰⠀⠀⢻⡇⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⣦⡀⠀⠀⠁⢠⡇⠀⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠀⢰⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡀⠀⠻⣤⠏⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡾⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣦⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣼⣿⠏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⢻⣿⣯⡎⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣋⡈⢛⣩⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠛⠁⠈⠼⠴⠾⢛⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠰⢧⢤⣤⣴⡄⢉⡞⣿⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⡖⠈⣋⣵⠧⢶⡛⣷⡿⠂⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠓⠉⠁⠀⢐⣠⠟⢁⣠⣴⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠈⠉⠉⠒⠖⠒⠉⠉⠛⠛⠋⠁⣷⣿⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐⠀⠂⠰⠀⡆⠀⡄⠀⡄⠀⠐⣦⢐⡆⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣄⢰⡔⠠⢠⣆⠀⣇⠼⡆⣋⢻⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡈⡏⣧⣴⡄⢹⠡⡏⣰⠈⡟⣾⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣻⢻⣸⡆⢹⠷⣿⢻⣷⡿⣿⣿⣿⣟⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣏⣏⣿⢺⣷⣿⢿⣷⣷⣿⣿⣿⣵⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⡿⣻⢨⠶⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡇⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡖⠂⠀⠀⠀⡇⠀⡆⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢋⠟⢱⠁⠀⠀⠀⠀⠇⠀⠘⡀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⢡⠚⠀⠀⡘⠀⠀⠀⠀⠀⢀⠀⠀⠘⢦⡀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠆⠀⠀⢀⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠑⠒⠠⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠑⠒⢀⡀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡇⠀⠀⠀⠀⠇⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⢰⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⣠⠒⡛⠀⠀⠀⢀⣤⡀⠀⠘⡄⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⡠⢳⠊⠁⠀⠁⠀⠀⠀⠈⠀⠙⢆⠀⠹⢦⣀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠴⠂⠉⡝
⠀⠀⠀⣠⠃⢸⠀⠀⢀⠀⠀⠀⠇⡇⠀⠀⠀⠑⢄⠈⢆⠁⠢⠌⣢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⢀⣀⣀⠀⠀⠀⠤⠽⠃⠀⢰⠉⠑⡀
⠀⠀⠀⡈⠀⢸⠀⠀⠈⡆⠀⠀⠈⢷⠀⠀⠀⠀⠀⠑⢦⣄⠀⠀⠀⠉⠉⠁⠐⠒⠠⡤⣠⣒⠁⠀⢉⣉⣁⣀⡀⡉⠀⠀⠀⠀⣀⣀⠀⠤⠞⠂⠉⠙⢢⡀
⠀⠀⢀⡄⠀⠘⡀⠀⠀⢰⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠋⠤⠈⣉⠁⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⢀⠜⠁
⠀⢀⣨⠅⠀⠀⢃⠀⠀⠀⠱⡀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠤⢀⣀⠀⠀⠀⠀⢰⢩⣤⡄
⠰⢿⣿⡆⠀⠀⠀⠳⡀⠀⠀⠈⠢⣀⠀⠀⠈⢶⡄⠤⠀⣀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠶⣿⣷⣶
⣰⣾⣿⣿⣦⠀⠀⠀⠘⢄⠀⠀⠀⠈⠳⣄⠀⠀⠑⢦⡀⠈⠑⠒⠠⠄⣀⡀⠀⠀⠈⠁⠒⠤⠤⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣆
⢙⣽⣿⣿⣿⣷⡀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠑⠤⡀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠉⠐⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⠤⡀⠀⠀⣴⣿⣿⣿⣿⣿⣛
⠀⢡⣼⣿⣿⣿⣿⣦⣀⠀⠀⠈⠓⣄⡀⠀⠀⠀⠈⠓⠆⢄⡑⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠶⠆⠐⠶⠶⠶⠀⣄⠀⠀⠀⠈⠑⠦⣽⣿⣿⣿⣿⣟⠛
⠀⠸⡋⢉⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠂⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣤⣤⣤⣄⣀⣈⣽⣿⣿⠿⠿⡟
⠀⠀⢃⠘⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⢀⣠⣤⣤⣀⣠⣾⣿⣿⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⠄⠰
⠀⠀⠘⣇⠀⠻⠛⠛⠻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⡧⠀⠀⣰⠃
⠀⠀⠀⠈⡄⠀⡄⠀⠐⠿⠋⠀⠨⣿⣿⠿⢻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⡇⠉⠻⠿⠇⠀⢀⠀⠀⢠⠇
⠀⠀⠀⠀⢱⠀⢱⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠿⣿⠟⠁⠀⠹⣿⣿⡿⠛⠻⣿⣿⡟⠉⠻⣿⣿⡟⠁⠙⢿⣿⠇⠀⠈⠉⠀⠀⠀⡀⠀⠀⡜⠀⠀⠆
⠀⠀⠀⠀⠈⡆⠀⡄⠀⠀⢃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠋⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠁⠀⢠⠁⠀⡜
⠀⠀⠀⠀⠀⢰⠀⢰⠀⠀⠘⠀⠀⠘⣁⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡆⠀⠀⠠⠇⠀⠀⢀⠁⠀⠀⠀⡌⠀⠠⠆⠀⠌
⠀⠀⠀⠀⠀⠈⡄⠘⣅⠀⠀⡇⠀⠀⠹⡄⠀⠀⠸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇⠀⠀⢰⠀⠀⠀⡸⠀⠀⠀⢠⠁⠀⡎⠀⡜
⠀⠀⠀⠀⠀⠀⢡⠀⠘⠆⠀⢰⠀⠀⠀⢱⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠂⠀⠀⠀⡇⠀⠀⠀⠆⠀⡘⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⠀⠀⡆⠀⠀⠀⡆⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠆⠀⠀⢸⠀⠀⠀⡘⠀⢰⠁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⢰⠀⠀⠀⠰⠀⠀⠈⡀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⣜⠀⠀⠀⡆⠀⠀⣠⠁⠀⡎⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⡇⠀⠀⡄⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡏⠀⠀⢸⠀⠀⢀⡉⠀⢠⢁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠃⠀⠀⠀⢡⠀⠀⠀⠃⠀⠀⢸⠃⠀⠀⠀⡇⠀⠀⢰⠀⠀⠀⠃⠀⠀⡆⠀⠀⡌⠁⡠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⣀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⢸⠀⠀⢐⠀⣠⠰⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠤⠤⠼⣀⣀⣀⣦⣀⣀⣰⣀⣀⣀⣄⣀⠤⠼⠓⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⡿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠁⠑⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⢟⣴⣜⡿⠿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⣤⡀⠀⠀⠀⠀⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠇⠾⠿⢿⣿⣿⣗⣘⠿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣬⣄⡀⠀⠀⠀⠀⠀⠀⠤⠼⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡏⠺⣿⣿⣿⣿⣿⣿⡿⢟⡂⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣎⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠋⢾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣦⢹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⡐⢁⠉⠛⠛⠛⠋⣉⠑⠒⠒⠶⠶⠒⠊⠁⠙⡀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣵⢷⣤⣤⣤⣤⠶⣮⣭⣭⣉⣩⣭⣴⣷⡹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⣶⣤⣤⡦⣤⡤⢤⣤⣤⢦⣤⠞⣶⡶⢶⡞⠁⠀⠀⠀⠀⠀⣿⣿⣿⣷⡌⠉⠉⢉⡉⢙⣛⠛⢛⡙⠉⡌⠉⣉⠉⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⡇⢿⡇⢸⡇⠀⣿⡇⢰⣿⠀⣿⠃⣿⠃⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⠀⡇⠘⡇⠈⣿⠀⣼⡇⢠⡇⢀⡇⢰⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⠸⣇⢸⡇⠀⣿⠀⢸⡟⢰⡟⢰⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀⢹⠀⣷⠀⣿⠀⣿⡇⢸⠇⣸⠁⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢿⡄⣿⠀⣷⠀⢹⠀⢸⡇⢸⠇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣇⠸⡆⢻⠀⣿⡄⣿⡇⣼⠀⡟⢀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠣⠼⢀⣿⣀⣸⣀⣸⡁⠼⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣥⣼⣥⣿⣥⣿⣥⣭⣬⣵⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢶⣭⢣⢜⡿⣛⢯⡙⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣄⠀⢉⡤⠟⠀⣷⢹⡇⢸⡆⢻⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡼⠶⠿⠴⡶⣚⣡⣾⠧⢾⠃⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡠⠤⠒⠚⠋⠉⣛⣃⠀⠀⠀⠊⢋⣴⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⢀⣀⣴⣶⣾⠟⠛⠛⣛⣛⡽⢋⣽⡏⣿⣿⡻⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣠⠴⠋⢉⣠⠀⠓⠀⣈⠠⠤⠕⠐⠈⠇⢱⣿⣿⡇⣿⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⣾⠁⣠⠔⠁⢀⡤⠖⠋⠀⣀⡠⠄⠐⠒⢛⣩⠞⢋⣴⡟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠹⣿⢰⡃⠀⢠⠋⠀⣠⠖⠋⢁⣠⠤⠖⠛⣩⡵⠊⠉⢀⣼⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣄⡀⠈⠃⠷⣀⣏⠀⣼⠃⣠⣾⠋⠀⣀⣴⣿⡟⢒⡶⣾⣿⣿⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠂⠉⠑⠘⠠⠤⠍⢉⣙⣛⣟⣻⣟⣿⣽⣯⣶⢶⠿⡟⡿⣹⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠅⠀⢨⠉⠑⢩⠀⡿⢸⣻⢸⢃⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢹⢸⠀⠇⡆⠏⣼⣼⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⡀⡇⢸⠈⢨⠃⡇⢲⡇⣏⣿⣿⣿⣷⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⡆⡆⣌⡄⠄⠀⠁⢻⣼⣶⣾⢼⢿⣹⢸⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠼⣽⣷⣰⡴⠰⢿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⠿⠿⢚⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⡿⣿⣿⣿⣿⣿⢿⡿⠿⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀⠀⠀⢀⣀⣀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣝⣟⣽⣿⣦⣼⢿⠶⠛⠛⠉⠉⠛⠛⠛⢯⡉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣟⡛⠛⡓⠀⠀⣀⠀⠀⠀⠀⠙⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠙⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠈⠁⠀⠈⡆⠀⠈⢻⣦⠀⠀⠰⠀⠉⠙⠳⣄
⠀⠀⠀⠀⢠⣄⡴⠋⠙⣝⠢⢈⠙⠛⠛⠛⠛⠉⠀⠀⡼⠀⢠⠆⠀⢸⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠌⠣⡀
⠀⠀⠀⠀⠀⡍⠀⢀⠀⠈⠑⠀⠀⠠⠄⠀⠀⣀⡴⠚⣀⡴⠋⠀⠀⡞⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡼⠂⠠⢄⣠⣤⣤⣤⣤⠶⠖⠛⠉⠀⠚⠉⠀⠀⢠⠎⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⡄⢸⠀⠀⡀⠀⠠
⠀⠀⠀⠀⢨⡀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⣀⡴⢋⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠘⢮⣁⠒⠒⢒⣒⣒⣋⣩⣤⣤⣶⣶⠿⠟⠉⠀⠀⣀⠴⠚⠉⠀⠉⣀⠀⠀⠀⠀⢸⠇⠀⠀⢀⠇⠀⢰⡇
⠀⠀⠀⠀⠀⣠⠊⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⢀⡿⠀⠀⣤⡾⠀⢀⣾⠑⢄
⠀⠀⠀⢠⠞⠁⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣠⡤⠴⠒⠋⡁⠀⠀⠀⠀⢀⣴⡟⠁⠀⣰⡟⠁⠀⣼⣿⠃⠈⢣
⠀⠀⢠⠇⠀⠙⢿⣿⠋⢩⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠉⠀⣠⡾⠋⠀⠀⣰⡿⠃⠀⠘⢀⡆
⠀⠀⢸⡀⢀⡆⠀⠙⣆⡀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⣀⣀⣠⣤⣴⠶⠖⠛⠉⠉⢀⣠⠴⠞⠉⠀⢀⣠⡾⠟⠁⠐⠀⣠⣾⠁
⠀⠀⠈⣧⡘⣧⡀⠀⠈⠙⠛⠛⠛⠛⠋⠉⢦⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⣤⣄⣀⣀⣀⣠⣤⠶⠏⠁⣀⣤⣴⣶⡿⠟⠁
⠀⠀⠀⠘⢿⣮⣁⠀⠘⠃⠀⠀⠂⠀⠀⠀⠀⠙⠛⠷⠶⠾⠿⠟⠛⠍⠀⡀⠆⠈⠉⡉⠉⠁⠀⢀⣠⠚⣿⡏⢰⡟
⠀⠀⠀⠀⠈⠛⢿⣿⣷⣶⣶⣦⣤⣴⣷⣦⣄⡀⠀⢀⠀⠈⠀⠀⠀⣀⣤⣴⣶⣃⣀⣤⣤⣶⡾⣿⡟⢸⣿⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡆⣿⡄⢿⣿⡀⢸⣿⡟⠿⣿⣿⡿⠿⣿⣿⠿⢻⣿⠉⢹⣿⣿⠛⣿⡿⠀⣿⡇⣸⡏⢸⠇
⠀⠀⠀⠀⠀⠀⠀⣈⣿⡸⣇⢸⣿⡇⠘⣿⣇⠀⢸⣿⡇⠀⣿⣏⠀⢸⣿⠀⢈⣿⡏⠀⣿⡇⠀⣿⠀⣿⢇⣟
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣧⢻⡄⣿⣷⠀⢿⣿⠀⠘⣿⡇⠀⣽⣯⠀⢸⣿⠀⢸⣿⡇⠀⣿⡇⢸⡿⢰⣿⣸⣿⣿⣶⣄
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣼⣧⢸⣿⡀⢸⣿⡀⠀⣿⡇⠀⢹⣯⠀⢸⣿⠀⢸⣿⡇⢀⣿⠁⣼⡇⣸⣇⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡀⣿⡇⠀⣿⡇⠀⣿⣧⠀⢸⣿⠀⢸⣿⠀⢸⣿⡇⢸⣿⠀⣿⡇⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⣿⣿⣿⣷⣴⣿⣇⣀⣽⣿⠀⢸⣿⠀⢸⣿⠀⢸⣿⡇⣸⣿⣴⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣿⣿⣿⣿⠿⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠙⠛⠛⠉⠉⠉⠉⠛⠉⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠆⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠉⠉⠄⠉⠑⢄
⠀⠀⠀⠀⢠⠖⠉⠀⣀⠠⠂⠁⠀⠄⠐⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂
⠀⠀⠀⠀⠇⠀⠈⠀⠈⠀⠀⠂⠀⡀⠀⠰⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣬⡀
⠀⠀⠀⠀⢇⠈⢁⠀⠉⠀⠋⡀⠀⠀⠀⠀⠀⡠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⡎⠐⠾⠓⢹
⠀⠀⠀⠀⠈⡆⠀⠀⠀⣀⠀⠁⢠⠋⠉⢆⠀⠰⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠠⠀⠀⡧⣀⣀⣀⠜⢦⡀
⠀⠀⠀⠀⠀⠸⡌⠀⣾⠉⠙⠒⠋⠀⠀⠈⠂⠐⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠐⠀⡁⠐⠀⠬⠀⠈⠀⠠⠙⡄
⠀⠀⠀⠀⠀⠀⢣⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠈⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡠⡶⡻⠃⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⣩⠊⡴⠁⠀⣇⠑⠤⣆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡰⠁⢰⠁⠀⠀⠈⢦⡀⠀⠉⠳⣽⢦⡔⢢⢀⣤⡀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣀⣠⡎⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⣦⣹⠃⢸⠤⠻⢖⠚⠉⡦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠔⠒⡏⠀⢸⣇⠀⠀⡇⠀⠈⢆⠀⠀⠀⠀⠀⡇⠀⠀⣠⠣⣈⠒⢛⣦⣀⠠⢵⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣱⢄⣾⠀⠑⠒⠚⠢⠄⣈⠷⠤⠠⡤⠚⠉⠉⠉⠀⠏⠉⠣⡀⠀⠑⢦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠓⠂⠉⢸⡁⠒⠪⠕⣤⠊⠉⠢⠔⠉⠑⠤⠊⠉⠲⠊⠙⢦⠖⠻⢶⣀⣉⠯⠽⠥⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠏⠀⡏⠒⢶⠐⠸⡆⠀⠐⡄⠀⠀⡆⠀⠀⡛⠀⠀⣇⡤⠚⡉⠀⠠⠀⠀⠂⠀⠠⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠰⠀⠘⡄⠀⢣⠀⠀⠇⠀⠀⡇⠀⠀⡇⠀⡼⠃⠀⠂⠀⠀⣠⢄⠴⠛⢤⣀⠀⢁⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢣⠀⢱⣀⡈⡆⠀⢸⠀⠀⠁⠀⢰⠁⡜⠀⠈⡁⠉⢔⡲⠗⠂⠉⠉⠐⢦⠇⠄⠐⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠲⣧⠄⠠⢔⠂⠘⠿⠁⠀⠈⢷⡀⠀⡆⠀⠀⠀⡸⢠⠅⠈⠀⠀⠂⡈⠣⢄⣀⣀⡀⠤⠊⡀⠀⢠⠲⢄⡨
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠘⡂⠀⠀⠀⠀⠀⢸⠥⠄⠁⠀⠀⠤⠅⠤⠵⡞⠦⠚⢢⠀⠀⠀⠁⠄⠐⢀⠁⢀⠤⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡸⣁⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠓⠒⠉⠑⠤⠔⠉⠁⠉⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⠶⠖⠊⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣀⣀⣀⣀⡀⠤⠔⠊⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⣧⣾⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠋⠀⢸⣿⣿⣿⠷⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠶⠀⠉⠛⠉⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠿⠷⣤⣤⣀⣀⣀⠀⠀⡀⠀⠒⠋⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⣦⣄⣀⡀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠔⠒⠚⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⢀⡀⠈⠁⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠠⠆⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⣀⠁⠀⠀⣀⣀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⢿⠿⠿⣿⡟⠛⢹⣿⡏⠛⢲⣶⣦⢤⣤⣤⣤⡴⣶⡞⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠸⡇⠀⢸⣇⠀⢸⣿⡇⠀⢸⣿⡇⠀⢸⣿⠀⢀⣿⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣷⠀⢸⣿⠀⢸⣿⣧⠀⢸⣿⡇⠀⣼⡏⠀⣸⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡆⠘⣿⡄⢸⣿⣿⠀⢸⣿⡇⠀⣿⡇⠀⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣧⠀⣿⡇⠀⣿⣿⠀⣼⣿⡇⢀⣿⠃⢸⡿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣿⡄⢻⣷⠀⣿⣿⠀⣿⣿⡇⢸⣿⠀⣿⣇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠛⠶⠿⠿⠦⠿⠿⠧⠞⠛⠚⠋⠁


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠑⠀⠀⠑⠦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣄⡀⠀⠀⠀⠀⠈⠉⠉⠍⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⣿⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠘⠄⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠓⠺⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠷⠃⠀⠀⠀⣹⠶⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡤⠊⠁⠀⣀⡀⠑⠌⢦
⠀⠀⠀⢀⡤⠴⠞⠧⣤⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠉⢉⡖⠊⠁⠀⠀⠀⠘⠤⠃⢀⡠⣏
⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠋⠁⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠠⠤⠤⠖⠚⠉⠀⠈⡇
⠀⠀⠘⢮⣂⠤⠀⠀⣏⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠴⠒⠒⠉⠉⠁⠀⣀⣀⣀⣀⣠⠼⠁
⠀⠀⠀⠀⠈⢹⠟⠒⠚⠛⠒⠚⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀⠀⠀⣠⠞⠁⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠘⠦⡴⠒⠛⠒⠤⣤⠤⠖⠋⠉⠙⠒⠦⠴⠚⠁⠀⠀⠀⠈⠉⡉⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠁⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠦⠤⠤⠤⠴⠶⠶⠶⠒⠒⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠴⣶⡒⠒⠒⠒⠲⠖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠑⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠒⠒⠈⣄⣙⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠔⠀⠀⠀⣀⠀⠀⠒⠂⠂⠀⠀⢀⣀⠀⠈⠃⠀⠠⠟⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡁⠀⠀⠀⠀⢀⡀⠁⢀⡀⠀⠀⠄⠀⡄⠀⠀⠠⠤⠀⠁⠀⠀⠐⠂⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⠆⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠂⠀⠈⠂⠀⢤⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢀⠘⠈⠈⠙⠠⠀⠀⠀⢀⡀⠀⠀⠤⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠸⡄⢂⠀⠈⠀⢀⠀⡀⢐⠀⣠⠀⠀⠤⠄⢀⠀⠀⠀⠠⠄⠐⠀⠲⠀⠀⠀⠀⠀⠀⠀⡀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢱⢠⠀⠀⢠⣀⡀⠑⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⡑⠐⡀⠀⠀⠀⠈⡀⢏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⡀⢟⠔⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠰⠉⡀⡀⠀⠐⠀⣀⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⡿⠶⢶⣀⣬⣤⡀⢌⠅⠀⣤⠒⠄⠀⣠⠰⡁⢀⠀⢠⣠⡤⡤⠚⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢠⠘⠀⠇⠘⠉⠉⠋⠓⡞⠛⢛⠛⠚⠛⠛⠑⠉⢹⠉⢘⠀⡇⠸⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⡄⠀⡇⠀⡆⠀⠁⠀⢸⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠁⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡈⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⠤⡄⠄⣄⣤⠤⠴⠒⠋⠁

⢠⡄⠀⠀⠀⠀⠀⠀⠘⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⡀⠄⠠⠄⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣀⡀⠔⡆⠀⢀⣤⡀⠀⠀⠀⢀⡤⠒⠒⢦⠀⠈⠁⠀⠀⠀⠀⠀⡼⢤
⠀⠀⠀⠀⠶⠀⠀⢀⡠⠔⡶⠔⠉⡆⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣗⢚⣄⣠⡾⠉⠁⠀⠀⠀⠰⢁⡀⢀⠀⠇⠀⠀⠀⠀⠀⢀⠔⠋⠉⠁⠒⡔⠃
⠀⠀⠀⢀⣤⣐⣈⣁⣀⣀⡁⠈⠭⠤⠄⠐⠒⢈⡆⢀⠀⠀⠀⠀⠀⢀⠤⠾⢽⣯⠗⢳⢄⠀⠀⢀⠀⠀⡶⠆⡆⡏⠦⣼⠀⠀⠀⠀⠀⠾⠠⠤⣠⠦⠜⠣
⠀⠀⠀⢸⣄⡂⠄⠀⠒⣒⣒⣖⣒⠂⠤⠘⠛⢀⠀⠀⢱⠀⠀⠀⢀⡟⣶⠚⠓⣿⣙⡏⠭⡄⠀⠈⠁⠀⡇⠀⣇⠃⠀⣿⠀⠀⠀⠃⠀⢂⠱⠀⠇⠀⡇
⠀⠀⡤⠚⠦⠄⣀⣀⡁⠀⠠⠤⠤⠤⠀⠒⠒⠉⢀⡠⠊⠀⠀⠀⠈⡄⡓⠄⢰⠢⠀⠔⢳⠃⠀⠀⠀⠀⢇⠀⠙⠀⠀⣹⠀⠀⠀⠀⢀⠔⢆⠁⠐⠀⢀⠁⢈
⠀⢨⡦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠒⠉⠀⠀⠀⠛⠀⠀⠘⠪⠬⢈⠀⠈⠔⠋⠀⠀⠀⠀⠀⠈⠈⠩⠈⠃⠀⠀⠀⠀⠀⠘⠤⢀⣁⡐⠠⠥⠒⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠀⠀⠀⢠⡀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠤⠤⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⢠⣄⠤⠐⠒⠀⣐⠢⡀⠀⠀⠀⠀⢛⡢⠌⢰⡈⠁⠀⠀⠈⠉⠀⠀⢨⠑⠞⠄⢀⠌⠀⠀⠀⠀⠀⠈⠃
⠀⠰⠆⠀⢀⠎⢰⠊⠓⡊⢯⠱⣈⢢⠀⠀⠀⠀⡤⠊⣁⡠⠊⠒⠈⢀⠂⢰⠀⠀⠛⢄⡁⠈⠀⠀⠥⡀⠀⠀⠀⠀⢠⠒⣏⠢⢄⠀⠓⢤⡀⠀⠀⠈⠃
⠀⠀⠀⠀⡆⡾⠅⠠⠋⠉⠘⣷⠌⡙⠇⠀⢠⠊⢀⡌⠀⠀⠀⠀⠀⠸⡀⡞⠀⠀⠀⠀⠈⠁⡏⠀⠀⠈⡆⠀⠀⠀⠈⠢⢌⠙⠲⠏⠉⢈⣀⠀⠀⣀
⠀⠀⠀⠀⡗⠢⡸⠣⡀⠀⣀⠟⣗⡇⡄⠀⡆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠠⡐⢄⠀⠀⣀⠀⠀⠱⠄⢠⠖⠀⡉⢑⣲⠄⠀⠀⠉⠊⢠⠆⡵⠲⡤⠒⠒⡍⠢⡀
⠀⠀⠀⠀⠘⢄⠣⠥⢎⡉⡉⡮⢚⠜⠀⠀⠘⣦⣈⠑⢄⠀⠀⠀⠀⠀⠀⠘⡌⠢⡀⠉⠀⠀⠀⠀⠈⢦⢠⠌⣉⣢⠀⠀⠀⠀⢀⢡⠒⠁⡠⠄⠦⣄⠉⠒⣷
⠀⠀⣀⠀⠀⠀⠑⠢⠤⠥⠯⠒⠩⠄⠀⠀⠀⠀⠉⠳⣄⢣⠀⠀⠀⠀⡠⠔⢁⡴⠃⠀⠀⠀⠀⠀⠀⠨⡼⠀⠉⠁⠀⠀⠀⠀⢸⢎⡀⠰⡀⠀⠀⢨⠀⢰⢸
⠀⢀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡉⠐⠒⢉⣠⡾⠋⠀⠀⣀⠤⡐⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢸⠀⠈⠒⠂⠁⣀⢜⠌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢔⠒⢲⡆⠒⢄⠀⠀⠓⠀⠀⠈⠻⣶⡢⠛⠉⠀⣤⢀⡊⠀⠑⡻⠖⠚⠀⠀⠀⢀⡀⠀⠀⠀⠋⠀⠀⠀⠑⠭⣈⡓⣤⡊⠔⠋
⠀⠀⠀⠀⠀⡠⢖⡛⠄⠈⢲⠀⠀⠃⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠘⠋⠑⣶⠠⣤⠀⢤⣬⠥⠤⣤⡤⣤⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⢎⣣⠀⠐⡆⠀⠈⠀⢀⠄⡠⠂⠀⠀⠀⠀⠀⡠⣔⠚⠑⠒⠦⡀⠀⠀⠈⠑⠠⠄⠿⠤⠽⠤⠤⠽⠤⠀⠤⠥⠨⠧⠤⠃⠀⠀⠀⠀⣠⠄⠀⠀⠉
⠀⠀⠐⠀⠀⠘⠇⠀⠀⢀⡠⢔⡡⠚⠁⢀⡀⠀⠀⠀⢌⡴⠋⢠⡇⠘⢷⡌⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⢀⡀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⠻⡀
⠀⠀⠠⣀⠀⠀⢀⢒⡪⠅⠊⠁⠀⠀⠀⠈⠁⠀⡠⢄⠀⠀⠀⠀⠁⠀⠀⠀⡈⢢⠶⠀⠀⠀⢀⠔⠁⢀⣀⡀⣀⡈⠳⡄⠀⠀⠀⠁⠀⠀⡔⠁⠀⠘⡀
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠑⡆⠉⠓⠒⠊⠒⠒⠉⡹⠁⠀⠀⠀⠀⡎⠀⡔⠁⢀⠤⢄⡀⠀⠘⡄⠰⠆⠀⠀⡰⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⡤⠒⠒⠠⡊⠁⠀⠑⢎⠁⠈⡆⠀⠀⠈⠉⠀⠉⠁⠘⠈⠉⠀⠀⠀⠀⠀⠀⡇⠈⡀⠀⢇⠄⠀⡱⠀⠀⡇⠀⠀⠀⠸⡀⠀⠉⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⡀⠀⠀⠘⡤⠚⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠱⡀⠑⢄⣀⣀⠴⠁⠀⡜⠀⣀⠀⠀⠀⢩⠶⣔⢶⣲⠞⠁
⠀⠶⠀⠀⢈⠦⢀⠀⠀⠀⠀⠰⣀⡠⠞⠀⠀⢀⠔⡩⠰⠒⡐⡶⣕⢆⠀⠀⠀⠀⢀⡀⠀⠀⢠⢞⠦⠤⠀⠀⠤⠔⠊⠀⠀⠉⠀⠀⠀⢸⢲⡋⣹⢥⠆
⠀⠀⠀⠀⡆⠀⠀⠑⠢⢄⣀⠔⠁⠀⠀⠀⡐⡅⢺⠣⡀⢣⠱⡐⡘⡆⠆⠀⡤⠀⠈⠀⢀⠔⡱⠃⠀⢀⡀⠀⠀⢠⠔⣠⡄⠀⠀⠀⠀⠈⠉⣯⡝⢰⠀⢀⡀
⠀⠀⠀⠀⡱⢄⠀⠀⡠⠂⠀⠠⡄⠀⠀⢀⠀⠦⡀⢣⠱⡀⢣⠱⣁⡀⠐⠀⠀⠀⢀⡔⡡⠊⠀⠀⠰⡁⠈⡃⠀⢏⣩⠕⢂⠀⠀⠀⠀⠀⡧⢦⢼⡎⠀⠈⠁
⠀⠀⠀⠀⢇⣀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⢆⡘⢄⢣⠑⡌⠆⣸⢃⠃⠀⠀⠠⢊⠜⠀⠀⠀⠀⠀⠉⠉⠉⠒⢄⡀⠀⠉⠀⠀⠀⠀⠀⢰⡵⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠈⠢⢌⠒⠷⠧⠜⢋⡥⠋⠀⠀⠀⠓⠁⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠁⠰⠆
⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋⠃⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣤⠀⠀⠀⠀⠀⠀⢀⠶
⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⢈⡟⠋⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⢀⢀⣸⣷⡀⠀⠀⠀⠀⢀⣠⣶⣄⡀⠀⠀⠀⠀⠀⠶⠀⠈⢀⠀⠀⠀⢠⠆⠘⠁⣰⢄⠀⠀⠀⣀⠴⢺⣿⠠⠀⠀⠀⠀⡰⣍⠾⣈⣀⣆
⠀⢰⣿⣤⣍⣭⣥⣿⡆⠀⠰⠛⠋⠉⠉⠛⢢⠀⠀⠀⠀⠀⠀⠀⣀⣰⠀⢠⡉⡐⠀⠉⢀⣀⣑⠀⢀⡇⠀⠀⠀⠀⣈⡄⠀⡼⠦⠤⠤⣤⠴⠾⡀
⠀⠀⣇⣏⣿⣿⣹⣼⠀⠀⠹⣶⣶⡶⡶⣶⡞⠀⠀⠸⣶⣶⡖⡏⡿⡏⠀⠀⢹⢻⢹⣿⣯⣇⡇⠀⠀⢿⣶⣿⢺⢹⢿⠁⠀⠙⡷⡤⡤⣶⣶⡾⠁
⠀⠀⠹⣿⣿⣿⣿⠏⠀⠀⠀⢯⣿⣇⣯⣿⠇⠀⠀⠀⢿⣟⣇⣷⣷⠇⠀⠀⠈⣿⣿⣿⣿⣿⠁⠀⠀⠸⣿⣿⣾⣾⡾⠀⠀⠀⢿⣗⣗⣿⣿⠇
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⢉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣴⣦⣶⠀⠀⠀⠀⠀⠀⣀⠀⠲⠀⠀⠀⠀⠀⢺⣷⡿⣀⠀⠀⠀⠀⠀⢰⣶⣶⡀⠀⠀⠀⠀⢀⣶⣾⣾⣷
⠀⢀⡤⠒⠾⠿⠆⢀⠀⠀⢀⣤⣾⣟⣷⣤⡀⠀⠀⠀⠎⠋⠀⠐⠶⡀⠀⠀⠴⡁⠀⠠⠜⠑⡄⠀⠀⡔⠉⠛⠛⠓⢄⠀⠀⢠⡞⠂⠤⠀⠐⣣
⠀⠸⡿⡷⣶⣶⣾⢿⠃⠀⢌⠁⣀⠀⣀⠈⣱⠀⠀⢈⡀⢀⣀⣠⣴⣾⠀⠘⣄⣢⣀⣀⡤⡤⡆⠀⠐⣇⣀⣀⣀⣠⣤⡆⠀⢮⣍⠉⠐⠒⠊⣙⡆
⠀⠀⣧⣧⣿⣿⣾⣾⠀⠀⠈⣿⣿⡏⣯⣿⡇⠀⠀⠈⣿⣿⡇⣇⣏⡇⠀⠀⠸⣼⣸⣿⣷⣷⠇⠀⠀⢻⣿⣿⣸⢸⣹⠀⠀⠀⣟⣟⡟⣿⣿⡟
⠀⠀⠘⠿⠿⠿⠿⠃⠀⠀⠀⠙⠿⠧⠷⠟⠃⠀⠀⠀⠙⠿⠯⠿⠟⠁⠀⠀⠀⠻⠿⠿⠿⠟⠀⠀⠀⠈⠻⠿⠿⠿⠋⠀⠀⠀⠙⠷⠯⠿⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⡆⠀⠀⠀⠀⠀⠄⠐⠙⠀⠀⠀⠀⠀⡼⣿⣿⢦⡀⠀⠀⠀⠀⣠⠤⠄⢀⠀⠀⠀⠀⢠⡾⠑⠲⠤⡀
⠀⢠⣾⡉⠛⠛⣃⣶⡄⠀⢠⠾⠟⠛⠻⠷⢄⠀⠀⠀⠓⠀⣂⡌⠙⣤⠀⢀⠚⠤⠀⠒⠉⠈⢆⠀⠀⡎⠀⠀⠀⠈⠑⠀⠀⣰⣏⣉⣉⡁⣉⣽
⠀⠈⡟⡟⣿⢿⢿⢻⠁⠀⢢⣤⣤⣤⣤⣀⡼⠀⠀⢠⣭⣤⡤⡴⣿⡿⠀⠈⢣⢵⢢⣴⣖⡟⡇⠀⠈⣧⣤⣶⢴⢲⣾⠃⠀⠻⣦⣀⣈⣭⣥⣤⠃
⠀⠀⢳⣷⣿⣿⣿⡿⠀⠀⠀⣿⣿⡇⣧⣿⡇⠀⠀⠀⣿⣿⡇⣧⣧⡇⠀⠀⠘⣼⣼⣿⣿⣿⠃⠀⠀⢹⣿⣿⣾⣸⣼⠀⠀⠀⣧⣧⣇⣿⣿⡏
⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠈⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣤⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠀⠀⠉⠀⢏⠓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⡞⠱⠁⠀⠀⠀⢸⠀⠈⡄⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⡔⠁⠀⡆⠀⠀⠀⠀⢸⠀⠀⠘⢆⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠋⠀⠀⡜⠀⠀⢰⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠑⠦⡀⠀⠀⠉⠒⠲⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠈⠑⠂⠤⢀⡀⠀⠀⠀⠀⠣⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠐⠇⠀⠀⠀⠱⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⡀⠀⠀⢃⠀⠀⢣
⠀⠀⠀⠀⢠⣀⡴⢉⡎⠀⠀⠀⡰⠢⡀⠀⢣⡀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠈⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡸⠀⣀⢠⢧
⠀⠀⠀⢠⠃⡃⠀⠈⠃⠀⠠⡄⠇⠀⠈⢆⡀⠹⡒⢄⡑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢑⡼⠁⠀⡞⢄
⠀⠀⠀⠇⠀⡇⠀⠰⠇⠀⠀⢲⡄⠀⠀⠀⠈⠢⣱⠀⠈⠑⠙⠲⠦⠤⣀⣀⠀⢀⡠⠂⠲⠀⠤⠬⠥⠖⠒⠀⠀⠈⠉⠀⠀⣰⠤⠀⠦⡀
⠀⠀⢸⠀⠀⡇⠀⠀⢃⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⠶⣆⣀⡒⠒⠒⠒⠀⠐⠒⠒⠈⠁⠀⠀⠀⢀⡜⠁
⠀⠀⣸⠀⠀⠱⡀⠀⠈⢢⡀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⠤⢀⣀⠀⠀⠀⠀⠀⠠⢻⣀⡀
⠰⣿⣿⡀⠀⠀⠱⡄⠀⠀⠙⠢⡀⠀⠈⠳⡄⠤⢀⡀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠰⠶⣾⣷⣆
⣴⣿⣿⣷⡄⠀⠀⠈⢦⡀⠀⠀⠈⠢⡀⠀⠈⠢⡀⠈⠉⠒⠠⠤⣀⠀⠀⠀⠉⠒⠠⠤⠤⠄⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣿⣿
⠨⡾⣿⣿⣿⣦⡀⠀⠀⠙⢄⠀⠀⠀⠈⠒⢄⡀⠈⠢⡀⠀⠀⠀⠀⠈⠙⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣀⠀⢸⣿⣿⣿⣿⣿
⠀⣴⣿⣿⣿⣿⣿⣦⣀⣀⣀⣉⣦⣀⠀⠀⠀⠉⠐⠢⠌⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠒⠒⡆⠀⠀⠀⠈⠑⠾⣿⣿⣿⣿⣯
⠀⠘⡀⣼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢉⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣶⣶⣶⣶⣾⣿⣏⠉⣹
⠀⠀⢣⠀⢠⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠛⢠⠃
⠀⠀⠈⢆⠀⡀⠀⠀⣿⠟⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣷⠈⠙⠋⠀⢀⠎
⠀⠀⠀⠈⡄⠸⡀⠀⢀⠀⠀⠘⠛⠋⠀⢻⣿⡿⠏⠉⢻⣿⣿⡿⠿⣿⣿⣿⠛⢿⣿⣿⡟⠙⢿⣿⡏⠀⠙⠛⠀⠀⠀⠁⠀⠰⠀⠀⡌
⠀⠀⠀⠀⢳⠀⢃⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠋⠀⠀⠈⠛⠁⠀⠀⠙⠋⠀⠀⢀⠉⠀⠀⡀⠀⠀⠀⡘⠀⢠⠃⠀⡘
⠀⠀⠀⠀⠘⡆⠈⡄⠀⠸⠀⠀⠸⠀⠀⠀⠠⠀⠀⠀⠀⠤⠀⠀⠀⠀⢠⠀⠀⠀⢰⡂⠀⠀⡎⠀⠀⢀⠃⠀⠀⢠⠁⠀⡎⠀⡜⠁
⠀⠀⠀⠀⠀⢃⠀⢣⠀⠀⡆⠀⠀⢣⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⢠⠇⠀⠀⣸⠀⠀⠀⡌⠀⣸⠁⡜
⠀⠀⠀⠀⠀⠘⡄⠈⣇⠀⠰⠀⠀⠀⠆⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⣸⠀⠀⢸⠀⠀⢀⡇⠀⠀⢰⠀⢠⠃⡰
⠀⠀⠀⠀⠀⠀⠃⠀⢉⠀⠀⡆⠀⠀⢸⠀⠀⠸⠂⠀⠀⠸⠀⠀⠀⠀⢸⠀⠀⠀⡈⠀⠀⡌⠀⠀⢸⠁⠀⢀⠂⠀⡎⢠⠇
⠀⠀⠀⠀⠀⠀⠰⡀⠸⠀⠀⢡⠀⠀⠀⠇⠀⠀⠇⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⡇⠀⢀⠃⠀⠀⠆⠀⠀⡌⠀⠰⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢆⠀⠈⡄⠀⠀⢰⠀⠀⠘⡀⠀⠀⢃⠀⠀⠀⠸⠀⠀⢰⠀⠀⢸⠀⠀⠸⠀⠀⢰⠁⢀⡧⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢧⡀⠀⠘⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⡄⠀⠀⡄⠀⠀⡘⠀⠀⠇⠀⢀⡧⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠦⢄⣀⣸⠀⠀⠘⡀⠀⠀⡇⠀⠀⡇⢀⣀⠧⠤⠖⠓⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⣧⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠄⢳⣴⣎⠑⢀⠉⢠⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣈⡛⣺⣿⣄⠷⡀⠈⠄⠀⠆⢩⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣉⣽⣿⣿⡥⣶⣷⣦⡴⢾⣿⣥⢍⢼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣹⡇⣊⣶⣦⢿⡿⢻⣟⣉⢁⣄⠀⠛⣿⡎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⣻⡟⢸⣿⣍⡰⣶⠾⡻⠇⣻⣅⢿⡷⣠⠄⢻⣧⠈⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⣾⣿⠻⣽⣇⠈⢯⣿⣿⣶⣾⣿⣋⡈⣧⡼⢾⠃⣀⣻⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢠⠛⡟⣿⣆⠈⠛⠳⠄⣉⠛⣿⡿⢿⠿⠿⣻⣿⣿⠟⠛⠉⣁⡴⠀⠀⠀⠀⢘⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠁⠶⣿⣿⣄⠀⠀⠀⠀⠉⠛⠲⢤⡄⠒⠛⢋⡥⠖⠛⠉⠉⠀⢀⠜⠁⠀⣼⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡄⠀⠈⠻⣿⠳⠦⠤⣤⣀⣀⣀⣽⡷⠒⠉⠀⠀⠀⢀⣤⣎⣠⣤⣤⢾⠁⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠠⣿⣿⣿⠀⡀⢰⣼⣦⣤⣤⣤⡾⠟⢋⡁⣀⣀⣀⣤⣤⠖⣻⠛⠉⡼⠋⠀⠸⠀⠈⣧⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣀⠁⠙⢿⢷⣦⣤⣮⣿⣿⠿⠶⠛⠉⠉⠉⠉⠉⠁⠀⢀⡼⠁⠀⠠⠀⠀⠀⡆⠀⠀⡼⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⠀⠤⣚⣿⣫⣽⡎⠁⠀⠀⠀⠀⠀⢀⣀⡠⠾⠋⠀⢀⡴⠀⠀⠀⠀⠀⢀⣼⠃⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠀⣀⣼⣷⣋⣀⡤⠔⠊⠁⠀⠀⣀⣴⡦⠖⠋⠀⢀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⣐⠛⠿⠿⠿⠶⢶⣦⡶⢾⢛⡋⠉⠉⠛⠛⠻⠿⠶⠶⠖⠛⠉⠁⠀⠀⣀⣴⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣤⣀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⢶⠟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠁⠀⡍⠙⠏⠁⠀⠲⢦⣶⠶⡖⠒⡲⠞⢻⠛⠉⠹⠏⠀⠂⢳⢠⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢨⡇⠘⡀⢰⠄⢠⠀⢠⠀⠀⠧⠀⠀⠀⠘⠃⢸⠃⠇⢸⠁⡄⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣧⣄⣇⢸⡄⠀⠀⠈⡄⠀⢰⠀⠀⠀⢸⠀⢾⠀⠀⡘⢠⡧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢿⣧⣿⣏⡿⢸⣧⠐⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⡕⠀⠀⣇⣘⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⣦⣿⣿⣿⣅⣀⡻⠆⠈⠀⢷⣀⠘⠀⣀⢀⡾⠀⠃⣐⣀⣩⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢿⡿⢽⣿⡿⣿⣿⣶⣶⣶⣷⣶⣶⣶⣿⣶⣶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡚⢿⣿⢿⣿⣽⣟⣛⣻⣿⡿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠛⠍⠛⠷⠛⠛⠉⠉



⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⡀⠉⠒⢤⣲⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢰⠀⠚⠃⠘⠀⣿⡏⠀⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠐⠒⠉⠰⢐⠉⣭⠒⠁⠑⡀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣀⣠⠞⢘⡁⠀⠀⢀⡟⣛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⠀⠀⠄⢐⣊⣠⡤⠖⠛⠁⣼⣠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠙⠫⣉⠉⠉⠀⣀⡤⠚⠉⠀⣀⣤⣴⣶⡿⡿⠿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠷⢀⠀⠈⠙⠛⠚⠛⠻⠒⡛⠛⠋⢉⢁⣁⣾⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⡟⡗⢲⠄⠶⢴⠶⡶⢶⢾⡷⢶⢻⣏⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠸⢹⠘⡇⢸⢸⡀⡇⢸⢸⣅⣾⣸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣿⡧⣧⢸⣾⣷⣿⣼⣼⣿⣧⣿⣿⣿⣿⣿⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠶⠶⠒⠒⠚⠛⠉⠉⠉⠉⠛⠛⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠁⠀⠀⣀⢤⣤⣰⡄⠀⠀⠀⣴⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⢸⠁⠀⠀⣼⠇⠀⠀⠀⠈⠳⠶⣶⣾⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⢻⡄
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣼⠇
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⢠⠖⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⢦⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠘⢦⣠⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣀⣀⣹⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⡤⠾⠋⠀⠀⠀⠀⠀⢀⣠⣴⠶⠻⣶⠶⠶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⠳⠶⢦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⠾⠿⠛⠛⠛⠛⠻⠿⠶⠶⠶⠶⠶⠿⠛⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢠⡆⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⢸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡀⢸⡆⠀⡄⠀⠀⠀⡇⠀⡄⠀⡇⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⢸⠀⢸⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢰⠀⣇⠀⠃⠀⠀⢀⡇⠀⡇⠀⡄⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⣸⠀⣸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡆⣿⠀⢸⠀⠀⢸⡇⠀⡇⠀⡇⠀⠀⠀⠀⢸⡄⠀⢸⠀⠀⣻⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃⢸⡄⢸⠀⠀⢸⡇⠀⡇⠀⡇⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⡟⠀⣿⠀⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⡇⠀⠀⠀⢸⡇⠀⡇⠀⡇⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⡇⠀⡟⠀⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⣀⠀⠧⠀⠀⠀⢸⠃⠀⠇⠀⡇⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⡇⢀⣇⣠⣷⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣶⣦⣤⣤⣤⣤⣤⣀⣤⣀⣀⣤⣤⣤⣤⣤⣶⣶⠶⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣤⣤⡴⠶⠞⠉⣰⣶⣦⣄⠙⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⠗⠀⠈⠉⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣈⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣈⣁⣤⣤⡶⠶⠞⠛⠛⠋⠉⠉⠉⠀⠘⠛⢶⣄⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠷⠾⠛⠛⠉⠉⠀⠀⠀⠰⣶⠆⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠙⠿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⣾⣾⣷⠄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠘⢿⡾⡿⣃⣤⠶⠞⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠷⠶⠶⢦⣤⣤⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠉⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⠗⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣟⣉⠈⣛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠶⠆⠀⠶⠶⠞⠛⠛⠛⠛⠛⠉⠉⠉⠉⠛⠛⠛⠛⡛⢋⣈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡾⢻⣦⡾⣷⣀⣾⣧⢠⣾⡆⣰⣷⠀⣾⡆⢠⣷⡀⢰⣷⡀⣼⣧⣠⣿⣆⣾⢷⣼⠿⣾⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡇⠘⣯⠀⢸⡟⠀⢻⡟⠈⣿⠏⠸⣿⠏⢻⡿⠈⢷⡿⠘⣿⠏⠘⣿⠁⢹⡏⢈⡿⢀⡿⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⠀⣿⠀⠸⣇⠀⢸⡇⠀⣿⡄⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⠀⣿⠀⢸⡇⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹⡆⢹⡆⠀⣿⠀⢘⡃⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⠀⣿⠀⢸⡇⢸⡇⣸⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢸⡇⠀⢿⡀⢸⡇⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⠀⣿⠀⣾⠃⢼⠃⣿⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣿⠀⣿⠀⢸⡇⠈⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⢰⡏⠀⣿⠀⣤⠀⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡄⢿⡀⢸⡇⠀⣿⠀⢸⡇⠀⢹⡆⢸⡇⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⣿⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢨⡄⠈⣿⠀⣿⠀⠸⣇⠀⢸⡇⠘⡇⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⢸⡇⢸⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠘⣧⠀⣿⠀⢿⡀⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⢰⡏⢸⡇⣿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⡄⣿⠀⢹⡆⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⠛⠀⢸⡇⢸⡇⣸⠇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸⡇⢻⡄⢸⡇⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⢸⡇⣿⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣸⡇⠘⣧⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⢸⠇⣿⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⢸⢻⣿⢿⣳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠛⢵⡞⢣⣾⠸⡏⢷⡏⠳⣌⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡱⢮⢀⢠⡖⠙⠁⢀⡿⢸⣧⠐⣿⠀⣿⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡾⠟⠛⠒⣻⣛⣩⡤⠿⠧⠺⠏⢠⣟⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⡠⠴⠒⠛⠛⠉⠁⢛⣃⡀⠤⠐⠊⠉⣋⣴⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⢀⣠⣶⣶⣿⠿⠿⠛⠛⣛⣻⡿⢛⣽⠏⣿⣿⡝⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⣀⡴⠟⠋⠉⣉⢹⠭⠤⠒⢊⣉⣻⣅⠴⢛⡯⢃⣿⣿⣿⢸⣇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⡞⠋⠀⣠⠖⠋⠁⢀⣤⠖⠂⠉⠁⠀⢀⣀⣨⡤⠞⣹⣿⠏⣸⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⣧⣿⠀⣰⠋⠀⠀⣰⠞⠉⠀⣀⠤⠖⠊⠉⠀⣀⣀⡴⠞⣋⠤⠞⠟⢳⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠘⢿⡄⣧⠀⠀⣴⠁⠀⣰⠞⠁⢀⣠⡴⠚⠋⠉⣠⡴⠋⠀⠀⣀⣤⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡳⢤⡀⠀⠉⠚⠧⣄⣿⡀⣼⡏⢀⣴⣿⠁⢀⣀⣤⢾⣿⡏⢛⡿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢆⠉⠉⠊⠓⠰⠤⣍⣉⣙⣛⣻⣛⣻⣟⣿⣵⣾⣴⣾⢿⠿⡟⢻⠏⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠈⡍⢩⠉⣏⠉⡯⢹⠇⢹⡇⣾⣼⢸⡇⣿⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡗⢸⠀⣸⢁⡏⡇⣸⢧⡏⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⡄⠀⠀⠀⠄⠀⢸⡆⡿⢘⠀⡾⢸⢃⣉⡏⣿⢸⣿⣿⣿⣆⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⡄⢡⠰⢠⠀⡆⢀⠰⠈⡇⡻⣸⢰⡇⣼⣴⢼⢳⣿⣾⣿⣿⣿⣿⣿⣶⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⡺⡎⣦⣇⠡⠀⡇⡆⣷⢾⣾⢸⣷⡟⣿⣿⣸⣿⣿⣿⣿⣿⣿⣯⡿⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠷⢿⣿⣄⣧⣓⣱⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⣿⡿⡏⠻⢿⠷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠐⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠀⠀⠀⠀⠀⠉⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣐⠮⠓⠉⠙⢦⠀⠀⠂⢀⠠⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠒⠉⠀⠀⡀⠀⣀⠀⠀⡀⠀⢀⠂⣧⡤⢠⡷⠴⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠆⠀⢀⠔⢁⣶⠋⠀⠈⠐⠳⠶⠷⠿⠛⠈⠁⠀⠀⢀⣰⣿⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠐⣰⠀⠈⡐⠁⢈⠀⠐⠄⡀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠁⠀⠀⠒⠀⠀⠧⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣚⠁⠀⢠⣠⠀⠙⠳⣆⣃⠀⠀⠓⠠⠀⣀⣉⣉⣉⣉⣀⠀⠄⢤⣠⠀⣀⣀⡤⠐⠛⠛⠓⢄⠐⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⢉⠟⠈⢳⠀⡀⠘⠦⣀⡀⠹⠷⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⠠⠤⢤⣄⡀⢠⡟⠁⠀⠀⠀⠀⠀⠅⠀⠀⣀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢘⢨⠐⡠⢐⠎⠢⠀⠀⠀⠀⠈⠫⡙⠗⠤⠀⢄⣀⡀⠂⠄⠄⢱⠀⡇⢀⣠⠄⠀⢀⡴⠷⠚⠓⢋⠂⠈⢢
⠀⠀⠀⠀⠀⣠⡄⣒⣂⠀⡠⠤⠤⡄⠈⡄⠳⡁⢀⠑⢌⡒⠦⢤⣀⡀⠀⠈⠂⢄⣦⣀⠀⠉⠙⠛⠶⠾⠞⠘⠉⠀⣀⡾⠋⠀⣀⠠⡊⡜⡠⠀⢨
⠀⠀⠀⠀⣼⡛⠈⠉⣩⠛⠀⡀⠀⠀⠀⠈⠀⡈⠀⠀⠈⠉⠁⠀⠀⠙⠛⢤⣤⣀⠈⠉⠉⠀⠀⠀⠀⠀⠄⠉⠉⠓⠉⠀⠉⠛⠀⣩⠠⠀⢀⡰⡻
⠀⠀⠀⠀⠇⢴⠀⠀⠋⡀⠀⠹⠦⢀⠑⠲⢦⣘⣃⡀⠂⢄⡉⠀⠒⠀⠤⠤⠬⠭⠿⠷⠖⠀⠤⠤⠤⠤⠤⠤⠤⠤⣤⠴⠶⠶⠾⠄⠚⠉⡉⠉⢂
⠀⠀⠀⠀⠀⢈⢧⡀⠄⠁⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠈⠉⠻⢕⡒⠢⠤⢄⣀⣀⠀⠤⠀⣤⣤⣄⣀⠠⠤⠶⠞⠉⠀⠐⠠⠄⢀⠀⠀⠐⢀⡣⢎⠑⡄
⠀⠀⠀⠀⢠⡀⠀⣉⠲⠌⠂⠀⠀⠀⠀⠐⠉⠓⠀⠀⠀⠳⣀⠀⠈⠉⠐⠒⠒⠒⠀⠉⠉⠐⠀⠀⢀⠠⠄⠐⠢⢤⣄⡂⠄⢀⣀⣄⡠⠖⠉⢀⠀⢀⠦⣀
⠀⠀⠀⠀⠘⢷⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡓⠀⠉⡉⡉⠑⠒⠂⠤⠤⢀⣀⣀⡠⠴⠖⠛⠒⠒⠒⠒⠒⠒⠒⠚⠛⠓⠊⠁⠀⠀⠀⣀⠔⢪⣸⠏
⠀⠀⠀⠀⠈⣦⠐⠀⢦⣀⠀⠲⣄⠢⡑⠢⢤⣤⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⠄⢀⡀⣐⠶⠾⢛⠋⠉⠨⠁⠀⠔⠂⠐⠒⠒⠀⢉⣀⣎⠉⠃
⠀⠀⠀⠀⠀⠿⣧⢉⠒⠛⠓⠤⠀⠀⣽⡄⠀⠀⠈⠉⠛⠒⠢⠤⠞⠃⠈⠛⠉⠉⠁⡀⠞⠁⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⢠⠶⠋⡶⠐
⠀⠀⠀⠀⠀⠀⠻⠩⠖⠲⣀⢀⠀⠀⠀⠈⠉⡀⠒⠂⠠⠤⠤⠤⠀⠀⠤⠤⢄⠒⠉⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⢐⠒⠤⠀⠘⠉⢻⡏⢠⡟⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠐⡐⢇⠀⠞⢓⡲⣄⠠⠄⠁⣠⢤⡄⠀⢀⣀⠀⠀⣠⢀⠀⠀⢀⡤⡀⠀⡴⠓⡄⠀⣛⠂⢀⡟⠀⢀⡆⠀⠠⡾⢠⠏⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠌⠢⡁⠀⠀⠃⠈⠁⢸⠀⠃⠹⢠⠁⠚⢻⠀⠑⠢⠛⡇⠈⠚⠁⠄⢠⠈⠀⠀⠸⠁⠀⠈⠀⠀⠐⠁⠎⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⣆⠠⡀⠀⠐⠀⠀⠀⠀⢰⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡼⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠀⠡⠈⠀⠂⠀⠀⠀⠀⡀⠀⡆⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠐⠀⠃⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠃⠀⠀⠀⠃⠀⢠⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⢸⠀⢸⠃⠀⣧⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣸⡀⢠⢸⠀⠘⠀⠀⠀⢀⠀⡄⢸⠀⡇⠀⠀⠀⣷⠀⠀⡇⡀⢸⠀⠀⢈⠀⣈⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠶⢠⣄⡀⠀⠀⠀⡇⠇⢸⡀⢧⠀⠀⠀⢹⡄⠀⣇⣴⠬⠤⠄⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠚⠓⠚⠓⠒⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⣾⡛⢩⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢎⣿⠁⠀⣜⣿⣍⣘⢹⣿⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢃⣾⣯⠀⠐⣻⡏⢸⡉⠩⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠰⠦⣀⡀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠗⢸⣿⣏⡀⠀⢼⣿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢈⠄⠀⢻⣿⣤⠀⠨⢿⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡰⠈⠀⠀⠀⢹⣿⣂⠀⢸⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⣇⡀⠀⠀⠸⣿⣿⣶⠘⣧⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⡄⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣅⠀⠄⡀⠀⠐⠀⠉⠻⢷⡌⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣿⠂⠉⠁⡂⠀⠀⠀⠀⠀⣴⡇⣹⣿⠂⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣿⡯⡜⣷⣿⢿⡀⠀⠀⠀⠀⢻⣇⣿⠟⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⡿⢧⣅⣿⠍⢦⡙⠁⠀⢀⣶⢼⠿⣧⣦⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣷⣜⡋⣋⠻⠂⠅⠀⢠⡟⠉⠂⠀⠈⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡒⠀⠟⢠⢀⠀⠀⠀⠀⠉⠇⠀⠀⠀⡐⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣤⡶⢶⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡻⣮⣿⡿⣶⣿⣿⣷⣦⣄⢶⡦⣬⣀⣤⣴⣦⣤⣬⣤⣝⣠⣤⣤⠴⠻⡿⠋⠉⠉⠁⠸⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡙⣟⢿⣄⣽⣧⠉⠻⣷⢾⠙⠻⢿⠿⠛⠟⠻⣻⡋⠉⠻⣿⠇⠀⠀⠄⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠈⢷⡈⠛⠄⠀⠀⠈⡀⠀⠀⠀⡇⠀⠀⣿⠁⠀⠀⡟⠀⠀⠐⠀⠀⡌⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠘⣷⠀⠘⡄⠀⠀⣇⠀⠀⢀⡇⠀⠀⢻⠀⠀⠀⠁⠀⠀⠀⠀⡰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢻⡄⠀⢵⠀⠀⢹⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠃⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⣷⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⢠⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣻⡄⠀⠀⠀⠀⢭⠀⠀⠀⠀⠀⡈⠀⠀⡆⠀⡌⢀⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⣀⣀⠀⡘⠀⠀⢀⠀⡀⠀⠀⢀⣇⣰⣀⡼⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⣿⣷⣵⡿⠤⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⣒⠺⣄⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⣈⣶⣽⣁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡥⡦⡙⡦⡍⢆
⠀⠀⠀⠀⠀⢀⣐⡙⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡈⠈⠀⢇⢣⠀⠀⠀⠀⠀⠀⠀⠈⡄⣦⢧⡑⢧⡆⡌
⠀⠀⠀⠀⠀⠹⣀⣹⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⠧⠤⠤⠼⠤⣆⠀⠀⠀⠀⠀⠀⠀⢎⡴⢋⣋⠷⣜⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢰⢲⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⠄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⡀⠉⠢⡀
⠀⠀⠀⠀⠀⣼⡓⢓⣲⣒⠛⣲⡄⠀⠀⠀⠀⠀⠀⠀⢏⠉⠀⠀⠉⡹⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⣏⢈⡧⠤⡇
⠀⠀⠀⠀⠀⣿⢿⣻⠇⠿⣻⢿⡇⠀⠀⠀⠀⠀⠀⠀⡠⠃⢀⡀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠁⣀⠔⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⢠⢖⠫⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⢎⠢⡑⢌⡞⠀⠀⠀⠀⠀⠀⠀⠀⣸⡒⣊⣑⢒⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡭⡣
⠀⠀⠀⠀⠀⡠⠚⡵⠑⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠚⠒⠒⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⣨⠟
⠀⠀⠀⠀⠈⠢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠒⠒⠒⠓⠀⠀⠀⠀⠀⠀⠈⠃⠍⠍⠩⠍⠋