Grandad Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣠⣿⣿⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⡀⠀⠀⠀⢠⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠧⠀⠀⠀⡴⠒⠿⠀⠀⠘⠃⠃⠀⠀⡀⠚⠘⠀⠀⠠⠗⠒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⡄⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡯⠴⠂⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠀⢀⡝⢂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠑⠦⠤⠴⠊⠀⠀⠂⣠⠇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⢤⡀⠀⠀⡤⠴⠒⠋⠁⠀⠀⠀⠀⣼⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡬⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢰⡀⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡜⠀⢀⡝⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠘⠿⠾⢿⣦⣾⣿⣿⣿⣽⣿⣿⣿⣿⣿⣶⣄⠀⠀⢀⣴⣮⣛⠿⠷⣤⣤⣤⣴⠿⢛⣩⣶⣄⠀⢸⡇⠀⠘⠁⠀⠀⠀⢈⣿⠉⠉⠉⠁⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣿⣵⣾⣿⣿⣿⣿⣻⣿⠿⣿⣿⣿⣧⡀⢸⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⠀⠰⠂⠀⠈⠃⠀⠀⠀⠀⢸⣿⣿⣿⠋
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣻⣿⣿⣾⣿⡟⠁⠀⢠⡍⠁⠹⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢳⡀⠀⢣⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠿⣯⠉⠀⠀⠀⠈⠋⠀⠀⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⢦⣸⠤⠊⠀⠀⠀⠀⠀⠀⢀⡠⠒⠖⠋
⠀⠀⠀⠀⢹⣿⣿⣾⠋⠉⠀⠀⠸⠇⠀⠰⠆⠀⠀⠀⢠⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠑⠲⡄⢠⡤⠤⠴⠚⠉
⠀⠀⠀⠀⠘⣟⣿⠉⠀⠀⠀⠀⠀⠀⠀⢶⡖⡏⠀⣠⠾⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠊⠋⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⢡⡀⢀⣀
⠀⠀⠀⠀⠀⠈⠻⢷⣟⣷⣄⣀⠀⠀⠀⠀⣩⠵⠚⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠘⢿⠂⠀⠀⠀⠀⠀⠀⢳⢀⠁⠌⠀⣵⡽⠮⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⢉⣉⡉⠉⠀⠉⠀⢄⠀⠀⠀⠀⢸⡟⠟⢻⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠲⢄⠘⣿⡭⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠾⣧⣄⠀⠀⠀⠀⠙⢦⣤⠴⠚⠁⢀⣠⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⠈⠟⠐⢕⡰⣙⠿⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣽⡿⠀⠀⠐⢤⡀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⢁⡄⠒⣂⠖⡰⢭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣉⣉⣉⣉⣉⣈⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠉⠉⠉⠐⠒⣤⢀⢀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠤⣤⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡦⡇⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⣄⠰⠆⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠩⡒⠝⠜⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡜⠉⠑⠂⠬⠭⠥⡍⠛⠛⡀⠀⢸⠀⠀⠀⠃⠀⠀⡔⠒⠂⠠⠤⠤⠤⠐⠒⠂⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⡄⢸⠀⠀⠚⠁⠀⠀⠈⠙⠀⠀⡇⢠⠼⡀⠀⠀⣣⣀⢠⠁⠀⠰⠛⠀⠀⠀⠛⠂⠀⢰⣀⡸⡀
⢀⡴⣞⡶⠀⠀⠀⠀⢸⠀⢦⠆⠀⠀⠀⠒⢀⣀⠒⠂⠀⠀⠰⣔⠀⡇⠀⠀⡁⢨⡝⠁⠒⢲⠉⢱⠒⡎⠉⡖⠂⠉⢅⠃⠰⠀⠀⠀⠀⠀⣴⣲⢦
⢿⡾⢁⠔⠣⢄⣀⡤⢄⠑⡆⠀⠀⠠⠢⣀⣁⣀⣀⠔⠂⠀⠀⠰⠊⠀⠀⠀⠑⢼⠀⠀⠀⠄⢉⡉⠉⢉⡑⠤⠀⠀⢸⠔⠛⣄⡀⠀⣀⠼⠀⠙⢯
⠀⠈⠉⠲⢄⢀⠏⠀⠜⠑⠰⡄⠀⠀⠀⠈⠒⠂⠁⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⣜⡀⠐⠛⠈⠙⢄⣀⠔⠊⠉
⠀⠀⠀⠀⠀⢏⢀⠎⠀⠀⡠⠊⡳⠤⢤⢄⣈⣀⣠⣤⠤⡚⡍⠲⡒⠒⠒⣖⠴⠒⡞⠒⢄⣀⣀⠀⠀⢀⣀⣠⠤⠚⣅⠀⠀⠀⠀⡄⡸
⠀⠀⠀⠀⠀⠀⠉⠒⠄⣠⠁⠀⠇⡆⡆⠀⣿⣻⠆⢈⠀⠇⡁⠀⢱⡔⡟⠁⠀⡌⠀⠀⢰⠀⠣⡉⠉⢁⠜⠁⡇⠀⠈⡆⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⠀⠁⠈⠉⠡⠆⠉⠉⠀⠰⢠⠀⠀⡀⡀⠀⠰⠀⠀⠀⠀⠓⠀⠚⣿⠓⠤⠐⠁⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⢸⠀⠀⠀⠠⠆⠀⠀⠀⢀⢸⠀⠀⡟⠾⠃⢹⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⠀⠀⠠⠆⠀⠀⠀⠈⢸⠀⢰⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⠈⡀⠀⠀⠠⠆⠀⠀⠀⠰⠘⠀⡆⠀⠀⠀⠀⢇⠈⠉⠉⠀⠉⠉⠀⠈⠉⠀⠈⠉⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡇⣇⠀⠀⠀⠀⠀⠀⢀⡆⣥⠴⠀⠀⠀⠀⠀⠆⠈⠉⠈⠁⠐⠒⠒⠒⠂⠈⠁⠉⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠈⠉⠉⠉⠉⠉⠈⠁⠙⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⠤⢠⢧⠤⠤⢎⡀⠀⠀⠀⠀⠀⠀⠀⠸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⣤⠤⠾⠰⠦⠤⣤⡼⠄⠀⠀⠀⠀⠀⠀⠀⠈⢲⣳⣞⣶⣾⠴⣷⣶⣁⢶⡏⠉
⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⠀⠒⠒⠎⠉⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠗⠒⠂⠐⠺⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠞⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠶⣄⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡴⢋⢉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡠⠐⠠⡑⠀⠀⠀⠀⠔⢀⠀⠀⣸⠁⠀⠈⠃⢣⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣼⠁⠋⣿⣻⠄⠆⠀⠈⢹⠀⣀⠔⠁⠀⠀⠒⣁⠎⠀⠀⠀⠁⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠘⢦⣰⡬⠟⠁⠀⠀⠀⠚⠈⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠢⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⣆⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣄⡀⣀⣠⠄⠈⠀⠀⠀⠤⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⠇⠀⠀⠀⠀⠀⢹⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠇⠀⠀⠀⠀⣀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡞⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣤⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣀⣀⣀⣀⣀⡠⠤⠖⠉⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⠏⠁⠀⢀⠈⠁⠀⠀⠀⠈⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠤⠄⠐⠂⠉⠀⢠⠋⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⢀⣠⠔⠉⠀⠈⠉⡗⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⢠⠁⡄⠀⢰⡠⠂⠀⢻⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⡯⣄⣣⠀⠫⠤⣤⠶⡼⣦⢀⣤⣤⡴⠁⠀⠀⠀⠀⠀⡀⠐⢁⠞⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠘⡏⠁⠈⠑⠒⠳⠀⠸⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡙⢁⠄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡴⠋
⠀⠁⢸⠀⠀⠀⠀⠢⠴⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣣⠂⡠⠊⠀⢀⠀⠙⠦⢄⣀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡗⡴⠁⠀⡠⡛⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢹
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⠒⠁⠣⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠉⠀⠀⠀⠀⢸
⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠁⠒⠒⠒⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠐⠒⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⣇⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡤⠤⠤⠤⠤⠤⠤⠤⣤⡤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠚
⠀⠀⠀⠸⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⣀⡀⠤⠔⠒⠉⠀⠀⢨⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠤⠔⢒⡒⠚⠉
⠀⢤⣤⠴⠃⠀⠀⠀⠘⠃⠒⠒⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⡀⣤⠰⠒⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣶⣶⣦⣬⣭⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⠔⠒⠒⠒⠉⠉⠑⠂⠒⠄⣀
⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⠀⠀⠘⠀⠉⠈⠉⠐⠢⢖⠁⠈⠑⢦⡀
⠀⠀⠀⢰⠁⠀⠀⡀⠀⠀⠀⣱⢄⣀⣠⠚⠀⢀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢁
⠀⠀⠀⡸⠀⠀⠀⠈⡖⠀⠉⠀⠀⠀⠀⠑⠲⡁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠆
⠀⠀⠀⢇⠀⠀⠠⠊⠀⢠⠤⠀⠀⠀⠰⢦⠀⠙⢄⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠔⠁⠘⠢⢀⡀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⡎⠲⠆⡦⢄⡀⢀⠄⢤⡀⣀⠄⢄⠀⣀⠼⡆⠺⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡄⠀⡴⠒⠈⠋⠁⠀⠀⠀⠛⠂⠉⠐⡄⠀⣰⡀
⠀⠀⠀⢇⠀⠶⠁⠀⠀⠙⠄⢠⠥⢽⠄⠔⠁⠀⠀⠱⡇⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠸⣠⠁⠀⠀⢠⣠⠀⠀⢤⡄⠀⠀⠀⢡⣘⠀⠱
⠀⠀⠀⠀⠙⡂⠀⠀⠀⠚⠤⢀⣀⣀⠠⠜⠂⠀⠀⠀⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢸⠁⠀⠀⢀⠄⠀⠊⠑⠀⠀⢀⠀⠀⠀⢡⡤⠃
⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠉⠒⠒⠊⠀⠀⠀⠀⠀⠆
⠀⠀⠀⢀⠤⡶⠋⠠⠄⣀⣀⠀⠀⠀⢀⣀⠠⠄⠚⠳⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⢠⠃⠌⠀⠀⠸⡀⠀⢀⠝⣣⡉⠣⡄⠀⢀⠃⠀⠱⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⢀⠎⠀⢈⠓⢖⠒⠤⠤⠔⡲⠒⡉⠀⠈⢫⠐⢄
⠀⠀⠇⢸⠀⠀⠀⠀⠑⠔⠁⠀⢈⡀⠀⠈⠓⠃⠀⠀⠀⢡⠀⠈⢢⠀⠀⠀⠀⠀⣰⠁⠀⠀⡌⠀⠀⠸⡀⠈⠢⣄⣠⠔⠁⢀⠇⠀⠀⠀⠃⠀⠑
⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⢢⡀⠀⠡⡀⠀⠀⡄⢠⠀⠀⠀⠁⠀⠀⠀⠈⠉⠁⠰⠆⠈⠉⠁⠀⠀⠀⠀⢀⠀⠀⢰⠤
⠀⡆⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠸⠀⠑⣄⣀⠧⣠⠞⠙⣆⣑⣲⠞⡀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⠜⢠
⢀⠁⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠱⣄⠞⠁⡴⠋⠀⠀⠁⠀⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⢬⠣⢾⠁
⢸⣀⣀⡎⢡⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⢿⣿⣧⠃⠀⠀⠀⠀⠀⢣⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢀⠆⠰⠀⠈
⡤⠤⠤⡇⠈⢆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⢸⠉⠁⠒⠐⠒⠒⠒⠒⠒⠒⠒⠀⠈⠉⠉⣤⠒⠀⢨
⢇⣀⡀⠂⠀⠀⠇⠁⠉⠉⠀⠒⠒⠒⠀⠈⠉⠁⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠄⠐
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣸⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠧⠤⠤⠿⡤⠤⠴⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⣞⣤⣠⣀⡴⠇⢧⣄⣠⣤⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣖⣡⣑⣪⡷⠺⣕⣊⣑⣍⡿⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣾⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⢩⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠁⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⢹⡆⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠇⠀⠀⠀⣼⣿⣷⡟⠗⠀⠀⠀⠀⢸⣿⣿⠛⠷⠀⠀⠀⢿⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢿⣦⣴⣿⣤⣤⣤⣶⠏⣤⣤⣙⣦⡴⠶⠦⣴⢟⣡⣌⠛⣶⣤⣤⣤⣼⣷⣤⣼⡷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢹⣿⡼⠋⠈⠹⣿⣿⡟⠁⠀⢹⡿⠀⠀⠀⢿⡏⠁⠈⢣⣿⣿⠏⠉⠈⠙⢻⣿⡟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡙⠫⣿⡅⠀⠀⠀⠀⠀⠙⠛⠛⣿⠳⣦⡄⠀⠀⠙⠳⠞⠛⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠿⣇⠀⠀⠀⠀⠀⠀⣰⠆⠉⠓⠛⠁⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⣤⠴⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣤⣀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠃⢸⡏⠙⢷⣄⣀⣀⣀⣀⣀⣠⣶⠛⠛⣿⠀⠙⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⠋⠁⠀⠀⠀⠸⡇⠀⠀⠻⣧⡉⠉⠉⣡⡿⠁⠀⠀⡿⠀⠀⠀⠀⠈⠹⣶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⣾⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈⢻⣦⣼⡋⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣷⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⣤⠶⠛⠉⠁⠉⠙⠷⢦⣼⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠸⣧⣀⣤⣤⣴⠾⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⣧⣀⣀⣀⣠⣴⠏
⠀⠀⠀⠀⣿⠀⠀⢠⡟⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⢹⡏⠀⠀⢸⡆
⠀⠀⠀⢰⡇⠀⠀⣼⠃⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠈⢳⡀⠀⠀⣷
⠀⠀⠀⢸⠇⠀⢠⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠘⣇⠀⠀⣿
⠀⠀⢀⡟⠀⠀⡿⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢿⡀⠀⢸⡆
⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠘⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠀⠀⠀⠀⠸⣇⠀⢸⡇
⠀⠀⣾⠁⠀⢺⡀⠀⠀⠀⠀⠀⣿⠉⠉⠛⠶⠶⠶⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠶⠶⠶⠛⠉⠉⣽⠀⠀⠀⠀⠀⠀⣹⠀⠈⢷⡀
⠀⣼⠃⠀⠀⠈⢷⡀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⣰⠞⠁⠀⠀⠈⢷
⣸⣧⣤⢰⡄⣶⣮⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⠿⣤⠀⠀⠀⠀⠸
⢿⡇⣿⢸⡇⡿⠻⠃⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡿⠸⠿⢤⠠⣤⠴⠛
⠈⢿⣿⣼⣧⠃⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠶⠤⢾⣧⣀⣀⡀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣀⣀⣀⣴⣷⠤⠶⠶⠶⠦⣴⡏⣹
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢈⡗⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣤⣤⣤⣤⣤⣤⡤⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠲⠦⣤⣤⣤⣤⣤⣤⣴⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⡧
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣟⠉⠙⠋⠉⠙⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣴⣶⣶⣶⣶⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠙⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠤⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⣼⠃⠀⠀⠀⠀⠀⢀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡏⢻⣿⠀⠀⠀⠀⣠⠶⠀⠀⠠⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠀⠀⠀⠙⡺⠀⠀⠀⠁⢀⠀⠀⠀⠈⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣀⡀⠀⠀⠰⣧⠄⠀⣦⡶⢀⡀⢠⣬⠙⡦⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣮⣥⣴⠟⢿⠀⠀⠀⠀⠉⠉⠁⠘⠓⠶⠟⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠻⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠃⠀⠘⠷⣄⠀⠀⠀⠘⢋⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣄⠈⠓⢤⣀⡀⠉⠓⣶⣤⣄⡀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠼⣧⠀⠀⠀⢸⣿⣷⣄⠀⠈⣹⠂⢀⡿⠀⠈⠙⠛⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⢋⠀⠀⠀⠀⠹⣆⠀⠀⠈⡿⢿⣿⣿⣿⠏⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⠀⠸⠀⠀⠠⠂⠀⠹⣦⡀⠀⠙⢦⣤⠝⠃⢀⣾
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢨⡀⣀⣇⠀⠀⠀⠀⠀⠇⠛⠿⣄⠒⠀⠀⠀⠾⡀⠳⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠠⠹⠀⠀⠀⠀⠀⡄⠀⠀⠙⣧⣀⠀⠀⢀⡹⠀⢻⣄
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠠⡀⢷⠀⠀⠀⠀⠃⠀⠀⡜⠁⠉⠛⠒⠛⠁⠀⠀⢣⠱⣄
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣇⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠈⢦
⠀⠀⠀⠀⠰⠇⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢀⡮⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠈⢧
⠀⠀⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢧
⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⢣
⠀⠀⠀⠀⠃⠀⢸⡀⠀⠀⠀⠀⠉⠁⠀⠀⢸⡇⠀⠀⠀⠉⠉⠑⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⢳⡀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⠦⢤⣤⡤⠶⠶⠄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠻⠀⠈⣧
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠋⠀⠀⠀⢸⡇⠀⠀⠀⡄⠃⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠘⡇
⠀⠀⠀⠸⡀⠀⣀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣰⣠⠆⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢀⠀⠀⢹
⠀⠀⠀⠀⠙⠋⠁⠀⠀⠐⠚⠛⠛⠛⠃⠀⠀⠀⠀⡟⠀⠀⠀⠀⠉⠁⠀⢀⣀⣀⣀⣀⡤⠴⠖⠂⠀⡇⠀⠸⠀⠀⣸⣄⠀⢀⣀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⡇⠴⠛⠛⠋⠀⠀⠀⠀⠠⢛⡄⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⢆⢙⠀⠠⡆⠀⡇⣦⠿⢁⡙
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠛⠃⠸⡤⠞⠛⠃⠀⠈⠛⠘
⠀⠀⠀⠀⠈⠋⠉⠉⠙⠛⠉⠉⠉⠉⠁⠒⠖⢶⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡼⠀⠀⠀⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡏⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠶⠶⠶⠞⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠢⣤⣀⣀⣤⣴⠞⠀⠀⢻⡄⠀⠀⠀⢸⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣇⠀⠀⠀⠤⠤⠤⠤⠤⠀⠀⢀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢦⣄⠀⣸⣄⢿⡀
⠀⣠⠴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⢷⡐⠆⠶⠶⠶⠶⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠧⣤⡄⠀⡠⠇
⠙⠓⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠶⠲⠖⠒⠋⠉⠉⠓⠊⠉⠙⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠒⠲⢦⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣖⣋⠉⠉⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⢀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠚⠧⣀⠙⣦⣤⣤⣤⣤⣴⠏⠁⠈⠉⠉⠉⠉⠲⣄⠙⣦⣀
⠀⠀⠀⠀⢰⢻⡹⠝⣦⢾⣩⢤⡏⠀⣰⠀⠀⠀⠀⢰⡄⠀⠈⡗⠛⣶⠶⠶⠶⣟⢀⣤⠀⠀⣤⡤⠀⠀⠈⣧⣸⠈⣧
⠀⠀⠀⠀⢫⡃⠀⠀⠀⠘⡇⢸⡇⠀⠉⠀⠰⡄⠀⠈⠀⠀⠀⠀⣴⣿⠟⡦⣄⣽⠘⠉⠀⠀⠉⠈⠃⠀⠀⠁⣸⡇⣿
⠀⠀⠀⠀⠀⠑⣄⡀⣀⠴⠳⢤⣧⠀⡠⠊⠀⠀⠀⡀⠙⢆⠀⠀⣯⡀⠀⠀⢸⣿⠀⡔⠁⢀⡀⠀⠑⢆⠀⢠⠏⡇⣿
⠀⠀⠀⠀⠀⡼⢡⠿⣅⠀⠀⠀⠈⠳⢤⣀⠈⠛⠋⠁⠀⣀⣤⡞⠁⣽⢦⠞⠁⠈⠳⣄⣀⡙⠋⣀⣀⣠⠴⣏⠀⡇⢻
⠀⠀⠀⢀⡞⢡⠏⠀⠈⠲⣄⠀⠀⠀⠀⡼⠉⠉⠉⠉⠉⠁⠀⠙⢦⣺⠈⢳⡀⠀⢰⠃⠈⠉⠉⠉⢹⠁⠀⠈⡇⡇⢸
⠀⠀⢀⡞⠀⠻⠶⠶⠶⠶⠶⠿⠶⠶⠶⠷⠶⠖⠴⠖⠶⠶⠶⠶⠸⠻⠤⠴⠝⠷⠿⠦⠴⠦⠤⠤⠾⠦⠴⠴⠟⠁⢸
⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢾
⢠⢏⡴⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻
⡏⢸⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢦⣀
⣇⣘⣦⣤⣇⣀⣠⣤⠤⠤⢤⣤⡤⠤⢤⣤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠖⠒⠒⠒⠒⢒⡶⠶⠒⠲⢶⣒⠒⠚⠂⢸⠀⠀⠀⠀⠲⠒⠒⠒⠂⠀⠀⣨
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣀⣄⣀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡤⣤⢤⡀⠈⢦⠀⠀⢸⠄⠀⠀⠀⠒⠒⠚⠛⠋⠠⣄⡸
⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡞⠎⠉⢪⣇⠀⢹⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡇⠀⢾⡎⠀⢙⢹⠀⢸⣆⣀⡼⠁
⠙⠒⠒⠒⠒⠒⠒⠛⠛⣇⠀⠳⣕⣐⡯⠃⠀⡾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⡀⠈⠪⠴⠵⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⣤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠂⠉⠉⠉⠙⠁⠒⠀⠄⡀⠀⠀⠀⠀⢀⣀⠀⠀⠰⠁⠘⠆⠀⡠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⢀⠀⠱⡀⣿⠀⢀⠀⡔⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢢⠀⠑⠏⠀⠸⠜⠀⢠⠊⡠⠖⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣀⣀⢣⡀⠀⠀⠀⠀⠀⠈⠉⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢠⠶⠛⠋⠀⠀⠀⠙⠛⠛⠂⠀⠀⠀⠀⠀⠀⢅⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠘⠒⠢⡀⠀⠀⡙⠢⣀⡀⣠⣀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⠌⢁⡀⠀⠀⠀⠀⠀⣠⡀⢢⠀⠀⠀⠀⠀⡰⠁⠈⡲⠁⠀⠀⠀⠠⠁
⠀⠀⠀⠀⢀⠔⠂⠁⢀⠀⠀⠀⠀⡂⡀⠿⠇⠀⠀⠀⠀⠠⠿⠃⠂⡀⠀⠀⠀⡅⠀⠀⢰⠀⠀⠀⠀⠀⡂
⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠄⠁⠀⠀⡁⢐⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠹⠀⢣⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡠⠁⢨⢁⡀⠀⠀⠀⠀⠀⡜
⠀⡜⠀⠀⠀⠀⠀⢀⣀⣀⡆⠠⡀⠁⠐⠀⠀⠀⠀⠠⠀⠀⠀⣀⡠⠔⠈⠀⢀⢦⠀⠀⠑⠢⢀⠤⠊
⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠷⡀⠈⠐⠀⠶⠒⠀⠄⠒⠂⠉⠁⠀⠀⠀⡠⠔⠁⠀⡆
⠀⢇⠀⠀⠀⠀⠸⡀⠀⠀⡀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⡇
⠀⠘⡄⠀⠀⠀⠀⢹⠀⠀⢡⠀⠀⠀⠀⠐⠓⠠⡀⠀⠀⡠⠛⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⡄⠀⠀⠀⠀⡆⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⢲⡗⢆⠀⠀⠀⠀⠤⠀⠀⠀⢅
⠀⠀⠀⡇⠀⠀⠀⠀⣇⡠⠄⠈⠀⠀⠀⢀⠂⠈⠀⠘⠤⠀⠀⠀⠀⠀⠀⢄⠀⠄⠀⠑⢄
⠀⠀⠀⡳⠐⠂⠀⠀⢁⡠⠊⡰⠀⠀⠠⠁⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠁⠠⠼⠀⠀⠱
⠀⠀⠊⠀⠀⠀⠀⠐⠁⠀⠉⠣⡀⠠⠀⠀⠁⠀⠀⠀⠀⠑⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⡑⠂⣀⠀⣀⠤⠋⢀⠂⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠢⠤⠤⠤⠤⠤⠤⡄⠂⠁
⠀⠀⠑⠒⠁⠀⠀⠀⠀⠘⠦⣅⠀⠧⢤⠤⠀⠄⠐⠀⡰⠂⠈⠢⢀⠀⡀⠀⠀⠂⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠀⠠⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡄⠀⠁⠀⠁⠀⡔⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⢀⣠⠄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠾⠀⠁⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⢄⠀⠀⠂⠀⠈⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⢈⠀⢀⡀⠃⠀⠀⢸⠤⣀⠀⡠⠊⠉⠐⠒⠂⠀⡖⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠳⡀⠀⠀⠀⠖⠃⠀⠀⢣⠐⣄⢀⡀⠀⠀⠁⠡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠚⠁⠈⠀⠀⡠⠊⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢱⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠚⠂⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠴⠞⠋⠙⠛⠛⠛⠋⠉⠁⠴⠶⣤⣀⣀⡀⠀⠀⣀⣀⣠⡤⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠈⠉⠉⠁⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠳⣄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣉⠱⠄⠀⠀⠀⠀⠀⠀⠶⠲⣦⣄⠀⠀⠀⠀⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠰⡇⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠁⠹⠀⡀⠀⠀⠀⠀⢀⡀⠀⠶⠲⠆⠀⠀⠀⠀⠀⢀⠰⠛⠃⠀⠀⣀⡀⠀⠰⠄⠀⡾⠋⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⢾⠟⠀⠀⠀⠀⠼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⠘⠻⠅⠀⠀⠘⣿⠆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠻⡀⠰⡄⠀⠀⠀⢀⣤⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠓⢤⣄⠀⠀⢠⡜⠛⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠙⢄⠀⠰⠏⠀⠀⠀⠀⠈⣉⠉⠛⠛⢿⡿⠷⠶⣄⠀⠈⢷⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠈⠰⣶⠀⠀⠀⠀⠀⠀⠈⠛⠒⢂⣀⠀⠀⠀⠈⢳⡄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠋⠀⣀⣀⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠙⠣⠄⠀⠀⠀⠀⠈⠛⠛⠂⠀⠒⠛⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠑⣤⣴⠟⠁⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢀⠀⠀⠀⠀⠀⢀⣤⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⡏⠀⠀⠀⠀⠀⢠⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⠀⠀⠀⠀⠀⠈⡄
⠀⠀⢸⠀⠀⠀⠀⠀⢀⡎⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣇⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⣼⠃⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢹⡆⠀⠀⠀⠀⠧
⠀⠀⠘⠀⠀⠀⠀⣰⠇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⠀⠀⠠⠄
⠀⠀⡆⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠘⣆⠀⠀⠀⢀⡇
⠀⢰⡇⠈⠳⠄⣤⡏⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⣿⠦⠒⠉⠹⣇
⠀⠸⣷⣤⣀⣀⣿⠁⠀⠀⠀⠀⠉⠛⠛⠒⠤⣤⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⠀⠒⠚⡏⠀⠀⠀⠀⠀⠉⠀⠀⠀⣴⡟
⠀⡠⠀⠈⠉⠛⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⡟⠉⠁
⣴⠃⠀⠀⠀⠀⢀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠟⣀
⠛⣤⡀⠀⠀⠀⣼⡇⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣴⠻⠀⠘⠀⠶
⠀⠈⠙⠛⠓⠈⠉⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠛⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠉⠛⠛⠓⠶⠶⢶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠦⠶⠀⠙⠛⠋⠉⠉⠉⠉⠉⠰⢆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠓⠶⠖⠒⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠚⠛⠙⠛⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠈⠁⠉⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢰⣾⠟⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢾⣿⣄⣀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣀⣀⣠⣼⣷⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣤⣀⠀⠀⣀⠙⠛⠛⠛⠙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢀⠈⠻⠛⠛⠻⣿⡀⠀⠀⠀⣠⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠁⠀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠈⠉⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⡴⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡄⠀⠀⠀⢰⣶⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⠀⠀⣀⣀⣀⣀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠛⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆
⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇
⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⣀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠘⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣿⣷⠀⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠒⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠄⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣠⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠃⠀⠹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠏⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⢀⣀⣴⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣥⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣠⣼⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⣿⣿⣿⠿⠿⢿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⣀⡀⠀⢀⣀⠀⢀⣀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠿⠻⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣁⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠀⠀⠉⠀⠀⠐⣷⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠛⠀⠀⢀⣶⣯⢞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠈⠛⠒⠢⢤⣤⣤⣤⣽⣟⣫⠭⣶⣾⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⣴⡋⠀⢬⣿⣷⠶⢶⣶⣦⣷⣶⠾⢿⠵⠖⣋⣩⠴⠖⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣿⠁⢹⠦⠄⣉⣁⣀⣀⡤⢦⠀⣧⠀⢸⣏⡭⠥⠖⢒⣋⡥⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠙⠶⠋⠀⠀⠀⠀⠀⠉⠀⣸⠀⡟⠀⢸⡗⠒⢚⣋⣉⡥⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢦⣤⡀⠀⠀⠀⠀⠐⠦⠏⣰⣿⣿⣿⣿⣿⣯⣥⣶⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣷⡛⠋⠀⠀⠀⠀⠀⢀⣠⡾⢻⠏⣹⣿⡉⠀⢠⠇⠀⢸⠈⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠠⠤⣤⣤⣶⠟⢋⡴⠋⣠⠏⢹⣧⡴⠋⠀⢀⡟⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢁⡴⠋⢀⡴⠋⠴⣿⠟⠃⠀⣠⠟⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠙⠠⣔⡋⢀⡴⠚⠁⢀⣠⠞⠁⣠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢻⣿⠃⠀⠀⠀⠉⠉⠐⠒⠛⠛⠓⠒⢻⣿⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣟⣿⣶⣤⣤⣴⣶⣖⣶⣶⣖⣒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠁⠈⠙⠛⠋⠉⠀⠈




⣀⣀⡀⢀⡀⣀⣀⣀⣀⣀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠁⠄⠒⢄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⢷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠃⠁⠃⠀⠓⠊⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠯⡟⢃⡠⠊⡰⢙⡗⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣀⡠⢴⠧⡠⠁⡀⠈⠢⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⢀⠀⠙⢆⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⡆⠂⠀⠈⡄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡠⠁⠀⠐⠀⠀⠇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢀⠀⠈⠀⠀⢀⡀⠀⢎⠀⠀⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠤⢀⠗⠀⠀⠀⠀⠀⠈⠂⠎⡍⠉⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠄⠌⠀⠀⠀⠀⠀⠀⢰⢀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠁⠀⠀⠀⠀⢎⠀⠀⠈⠛⠢⠤⠄⡎
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣒⣐⣒⣒⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠉⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡶⠲⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠈⢻⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠙⣷⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠶⣤⡿⠁⠀⠀⢀⣴⠶⠶⠶⣤⠀⠀⣤⣤⣤⣄⡀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣸⣷⠞⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣀⢀⡾⠃⠀⠀⣴⠋⣀⣤⡶⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢴⣯⣤⣀⡀⠈⠻⣦⠀⠀⠀⠀⠀⠀⣹⣧⠶⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⢻⡇⠀⠀⠠⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣄⢸⡇⠀⠀⠀⠀⠀⢙⣿⠤⠄⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⠶⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣤⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⣀⠀⣿⠀⠀⠰⠞⠓⠀⢿⣇⣽⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣻⣿⡟⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣀⡈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡽⢷⡟⠀⠀⠀⠰⠋⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠐⢦⡉⠙⠃⠀⠀⠀⠀⠀⠀⣿⣏⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⣥⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠘⠷⢶⣤⣶⣤⠶⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠐⠟⠉⣷⢀⣾⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠁⠀⠀⠀⠀⠠⣤⣤⡶⠞⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠘⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣿⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⡏⠀⢠⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠷⠤⠿⠷⠶⠿⢦⣤⣤⣷⣀⣴⣤⡿⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⣤⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠻⠷⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣀⣠⣴⠾⠛⠁⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⢨⡿⠷⢦⣄⣀⣀⣀⣀⣀⣠⣤⡶⠟⢻⡿⠛⠛⠉⠉⢠⣶⣀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠙⣯⣭⡿⠛⠉⠀⠀⢠⡿⠁⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣄⣤⢴⡟⣁⣭⡿⣦⡀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⢀⣤⠶⢛⣻⡿⠒⠒⠢⣤⣾⣧⣤⡤⢤⣀⣿⠀⠀⠀⠀⠀⠀⠉⠁⣼⠋⠙⣋⠀⣿⠻⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⢀⠀⠀⠀⠀⢻⡆
⢠⣴⡾⠁⣴⠋⢁⡤⠞⠂⠀⠀⠉⢣⡈⠛⣦⡉⢿⡄⠀⠀⠀⠀⠀⠀⠀⢹⡷⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣛⡛⡿⠆⠀⠀⣸⡇
⣯⢰⡇⢸⡇⢀⡿⠁⢀⣤⣤⣄⠀⠈⣿⠀⠘⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠉⠙⠋⠀⠀⠀⢠⣿⠃
⢻⡌⢳⡈⣷⢘⡇⠀⣼⠃⠀⢘⣧⠀⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣠⡿⠁
⠀⠙⠻⣿⠿⠿⣿⡶⠿⢶⡟⢋⣡⠾⠃⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⣰⡟⠁
⠀⠀⠀⣿⠀⠀⣿⠀⠀⠈⠛⠛⠳⢦⣤⣤⠴⠟⠁⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠸⣿
⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⢿⣇
⠀⠀⠀⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⡿⢻⡇
⠀⠀⠀⢻⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠁⠀⠀⠘⣡⡿
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠻⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠾⠛⠋⠉⠀⢹⡷⠤⢤⣤⠾⠋
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠞⠛⠛⠛⠖⠀⠀⠀⠈⡇⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠛⠛⠳⢶⣄⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢉⣿⡄
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⣿⠀⠀⠀⠀⠀⢀⡾⠛⠉⠉⠉⠉⠛⢶⣤⣤⡴⠛⠉⢷⡀
⠀⠀⠀⣾⢷⣦⡾⢷⡀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠁
⠀⠀⠀⣟⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣶⣤⣤⣤⠞⠁⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉
⠀⠀⠀⠛⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠀⠀⢀⣀⣤⡴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠛⠛⠛⠛⠛⠛⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣻⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⢹⠀⣴⣩⡤⠿⠂⠀⠀⠀⠀⠀⠐⠻⠤⣍⣳⠀⡟⠉⠉⢹⡀
⠀⠀⠀⠀⠀⢸⡷⢶⡀⢸⠀⠋⠁⢀⣤⠀⠀⠀⠀⠀⠀⠀⣠⡄⠈⠙⠀⣟⠀⡴⢶⡇
⠀⠀⠀⠀⠀⣿⠀⢺⣇⠸⡆⠀⠑⠢⠉⠀⠰⠔⠒⠢⠄⠀⠈⠕⠚⠀⢀⡇⢸⣷⠀⣿
⠀⠀⠀⠀⠀⠘⠷⣬⣏⠀⠙⠶⠶⠒⢶⣃⡀⠀⠀⠀⢀⣘⡶⠖⠶⠶⠋⠀⢸⣧⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡀⠉⠉⠓⠒⠒⠉⠉⠀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠙⠲⠶⠤⠤⠤⠶⠖⠋⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣀⣨⣷⡀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⣠⠶⠋⢡⡟⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⢻⠻⣦⡀
⠀⠀⣴⠃⠀⠀⣼⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠞⠋⠀⠀⠀⡆⠀⠙⠷⣤⡀
⠀⣸⠇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣶⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠙⠳⣄
⢀⡿⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⢹⡶⣦⣀⡀⠀⠈⢷
⣸⠃⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠇⣽⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣿⠀⢸⠇⣼⢀⡟⢳⠀⢸
⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄⢿⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡟⢦⢿⣠⣏⣼⣡⣿⡴⠏
⣿⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠳⠎⠙⠲⠶⣶⠏⣹⢹⠁
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸
⣿⣀⣀⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣿⢸
⠈⣿⠉⠉⠉⢻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠺⠤⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠁⠀⢿⢸
⠀⣿⠀⢠⣶⠸⣧⠉⠙⠲⠶⠶⣤⣤⣤⣤⣄⣀⣿⣤⣄⣤⣤⣤⣤⡶⠶⠞⠛⠉⣹⠏⠀⠀⣿⢸
⠀⠸⣧⠸⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⢸
⠀⠀⠈⠳⡿⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠶⠶⠶⠦⣤⣤⣤⣿⣦⣤⣤⠴⠶⠶⠶⢶⣾⡁⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇⠀⠀⠀⠀⢿⣼
⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠠⠆⠀⠤⠠⠄⠤⠀⠄⠤⠠⠠⠤⠠⠄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣠⠴⠦⣄⣀⣀⣀⣀⣀⣀⡤⠤⠦⠤⣀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⢠⡶⠿⢶⠀⠀⠀⠀⡴⠾⢶⣄⠀⠀⠸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠤⢧⠀⢠⠇⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢇⠀⡾⠤⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⣷⠋⠀⢒⡶⠈⠁⠀⢱⠀⠀⠀⠐⠁⠀⠑⢶⡒⠀⠀⢸⢾⠁⠀⢹
⠀⠀⠀⠀⠀⠀⢳⡀⠀⡟⡆⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢧⠹⡀⣠⠖⠉⢉⣉⣩⣭⣭⣭⣉⣉⠉⠑⢦⠀⠀⡼⠁⡎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠙⠇⢀⠞⠉⠀⠙⠢⠤⠖⠁⠈⠑⣄⠈⣧⠞⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠲⢤⣀⡀⠛⠟⠃⠀⢀⣠⠼⠀⠁⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⠉⠉⠉⠉⠉⠁⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠙⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠐⠒⠒⠒⠂⠸⠦⣀
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣰⢆⢀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢀⡼⠁⠈⠣⡑⢄⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⢨⡀⢀⢠⡶⠋⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠌⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡄⠀⡇
⠀⠀⠀⠀⠀⣲⡸⠀⠀⡀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠸⣀⡃
⠀⠀⠀⠀⡜⠹⡁⠀⠀⠀⠁⠀⠘⠁⠀⠀⠀⠁⠀⠀⠀⢹⠉⠂
⠀⠀⠀⠀⠘⠤⢧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠔⠃⠀⠀⠀⠀⠠⢖⣦⣠⠤⠤⠤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠁⠀⠀⠀⠀⠁⠀⣠⣔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡀⠀⠀⠀⠀⠈⠂⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠑⣖⠂⠒⠒⠐⠂⣫⠇⠀⠉⠲⢄⠀⠀⠀⠀⡜⡡⠒⠁⢣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢀⠔⠋⡇⠀⠀⢹⠑⠂⠀⠒⠊⠹⠀⠀⠀⠀⢀⠗⢄⡠⠲⠋⠀⠀⣀⠀⠑⢄⢸⡆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡴⠁⠀⠀⡇⠀⠀⠘⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡌⠀⠈⢣⡀⠀⠀⠸⠟⠀⠀⠀⠉⠙⢆⠀⠀⠀⠀⢸
⠀⠀⢀⠎⠀⠀⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢲⡇⠀⠀⠀⢧⠀⠀⠀⠀⠐⠀⠀⢀⣶⡆⠑⠤⣀⢀⠆
⠀⠀⡌⠀⠀⠀⣀⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⢃⠀⠀⠀⠀⢡⠀⠀⠀⠸⣆⠀⠀⢈⠀⠠⠀⠀⠁⠀⠀⠰⣥⠊
⠀⠀⡇⠀⠀⠀⠏⢸⠀⠀⠀⢰⠀⠀⠒⠒⠀⠘⡄⠀⠀⠀⠨⡆⠀⠀⢨⣄⣑⠤⣀⠁⠀⠀⠀⠀⠀⣀⡀⡸
⠀⠀⠘⣤⡶⠚⠧⣸⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⡁⠀⠀⡸⠈⢧⣀⠈⣟⠐⠒⠒⠊⠉
⠀⠀⠀⠘⢤⣄⡤⠻⠦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣀⠁⢀⡰⠁⠀⠀⠀⠉⠀⢳
⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠲⡇⠣⡀⠀⠀⠀⡠⠊⠉⡇
⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡶⠻⢆⡈⠁⢲⠉⠁⠀⡰⠁
⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠉⠓⠚⠦⠔⢺⠁
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣷⣠⡑⢤⣀⡀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠢⣄⠀⠀⠀⡀⠀⠀⠀⢀⣠⡇⠇⠀⠉⠉⠛⠛⠛⢻
⠀⠀⠀⠀⠀⣼⠀⠘⡄⣀⣀⣀⣀⣀⣉⡀⠶⠤⠘⠊⠉⣀⣴⠀⠃⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠧⢄⣀⣀⣀⣀⣀⣀⡠⠤⡔⠂⠉⠀⠸⠀⠀⠀⠀⠸⢳⠀⠀⢸
⠀⠀⠀⠀⠀⡏⠀⠀⢸⡀⠀⠀⠀⡇⠀⠀⠀⠀⢣⠀⠀⠀⡇⢈⠄⠀⠀⠼⠸⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠈⢧⠀⠀⠀⠳⠀⠀⠀⠀⠈⢆⡀⠀⢇⠈⠇⠀⢀⡇⠀⢦⣤⣼⡄
⠀⠀⠀⠀⢀⡏⠀⠀⠀⡞⠲⢄⠀⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠿⢉⠹⠇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠑⠢⠤⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠠⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠊⠀⠀⠀⡄⠀⠀⠀⠙⡔⠀⠀⠈⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⠀⡠⠀⡁⠀⠀⠀⠀⠀⠦⠀⢀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⠃⠀⠧⣀⠀⠘⠂⠃⠀⠀⠀⠀⢀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠆⠀⠀⠐⠠⢄⡀⢈⠀⠰⠂⣀⠀⠠⠆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠂⠒⠤⢤⠤⠂⠘⠃⡀⠀⠀⠀⠄⠀⠀⠀⢌
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠄⡀⠀⡈⡁⠔⠌⠤⠇
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⠒⠊⠀⠀⠀⠘⠢
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠄⣀⠀⠀⠀⠀⠀⠠⠚⡀⠱⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢡
⠀⠀⠀⠀⠀⠀⢸⠀⣀⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠁⠈⠂⠀⠀⠀⠀⠤⠐⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠊⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⡅⠀⠀⠀⡈⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢐⠀⠇⠀⠀⠀⠇⠀⠀⠀⠀⠈⠑⠂⠒⠒⠒⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠅⢰⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠀⠀⠐⠤⠀⠔⠀⠀⠀⠀⠀⠀⠀⠸⠰⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠚⠄⠀⠀⠌⠒⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⢒⣲⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠋⣩⣧⣴⣇⠀⣹⣯⣽⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⢸⣿⣸⡟⠁⠀⠈⠙⠛⠻⣏⣿⠏⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣷⠸⠃⣀⣄⣀⢀⣀⣘⣿⡇⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢰⣿⠛⢻⠿⢿⡿⢺⠿⣿⣿⣧⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠘⣏⠰⣌⠓⣚⣁⣸⡷⢚⣿⡿⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠸⠆⠬⠭⡤⢼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢧⡀⠉⢉⣠⣿⣣⣴⣶⠖⣶⣤⡀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⢦⡀⠻⣿⣿⡿⠿⠶⣩⡿⣦⣿⣿⣟⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣀⣠⡶⢾⣏⣿⡀⠀⠙⢦⡽⢻⣿⣿⠉⣁⠀⠈⠛⢿⣿⡇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⣴⡞⣯⣿⣿⣿⡇⢸⣿⣿⣿⡄⠀⣰⣷⣼⣿⡏⠘⠛⠀⡀⠛⠋⣿⣧⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡟⣁⢹⣎⣿⣿⣿⡇⢸⣿⣿⣿⣿⣶⣿⣼⣿⣿⣥⡀⠀⣐⠟⢀⣾⣿⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡿⢸⣿⣾⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢷⢦⣈⣤⣾⣿⣿⣿⡄⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢇⣾⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⠁⢈⣆⠙⣿⣋⠉⣿⠻⣧⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣯⣿⡿⠁⢀⢸⡞⠉⠙⠦⠋⠉⢹⣿⠀⣌⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⠁⡀⢈⣾⣇⣀⡀⠐⠀⡀⢸⡏⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢁⣏⣭⣬⣿⣿⡿⠃⣰⣶⡟⠛⠭⠍⠻⣿⣍⣉⣩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡿⢛⣿⣿⡟⢿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠾⢿⣿⡷⠿⠿⠿⠛⠋⠁

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

⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣿⣶⣦⣴⣶⣿⠀⠀⠀⠀⠀⢸⣷⣶⣦⣶⣾⡇⠀⠀⠀⠀⠀⣿⣶⣦⣴⣶⣾⠀⠀⠀⠀⠀⢸⣷⣶⣤⣶⣾⡇
⠀⠀⠀⣿⣿⣿⢹⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⡇
⠀⠀⠀⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⡇
⠀⠀⠀⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⡇
⠀⠀⠀⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⡇
⠀⠀⠀⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⡇
⠀⠀⠀⢿⣿⡿⢸⣿⡿⠀⠀⠀⠀⠀⢸⣿⡿⠀⢿⣿⡿⠀⠀⠀⠀⠀⢹⣿⡿⢸⣿⡿⠀⠀⠀⠀⠀⠐⣿⣿⠇⣿⣿⠇

⠀⠀⣀⣀⠴⠃⠘⠢⣄⣀⡄⠀⠠⣄⣀⡴⠃⠀⠘⠢⣄⣀⠄⠀⠠⣄⣀⠴⠃⠘⠢⣄⣀⡄⠀⠀⢤⣀⣠⠞⠀⠓⢤⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠉⠉⠁⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⢀⣄⠀⠀⠠⠌⠉⠁⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠒⠀⠀⠀⣤⡀⠀⠀⠀⠀⡀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠠⢄⠀⠈⣁⣀⣀⠠⠤⡌⠠⠀⠆⠀⠀⣀⠄⠐⠇⠉⠀⢀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢃⠀⠛⠀⣀⢀⠤⠀⢁⡈⠀⠀⠀⠀⠀⢀⡌⠀⠆⠉⡀⣦⢰⣤⠄⢀⠀⠀⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠀⢠⡈⢻⢰⠋⡀⠄⠀⠈⣀⣀⡉⠀⣠⣐⡏⠁⠸⠷⣤⣌⠷⢤⡶⠶⢄⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⢀⠀⣀⣤⣠⣤⠮⡀⢀⠀⠀⠙⢦⣜⡏⠀⢩⡀⠀⠈⠉⠃⠀⠛⠙⢃⠀⠀⠶⣦⣽⡄⠀⠾⠷⠈⠣⢀⠀⠀⠰⣯⡀
⠀⡔⠊⠉⠉⣀⠘⠒⠉⡄⠀⠛⢆⠀⠀⢀⡀⠙⡅⢠⠀⡶⠠⡖⠢⡐⢀⡰⢖⣺⢇⡄⢀⣇⠈⠀⡀⠀⠀⠀⣾⠇⢀⣠⡷⠍⣀
⡈⠀⡆⢀⡴⠋⠀⠀⠠⠀⠀⠀⠀⠐⣤⡈⠀⠀⠀⠆⠀⢀⠐⠃⠀⠻⠀⠙⠋⡉⠈⠟⣡⡿⣶⣄⠉⠀⠀⢀⣹⣶⠛⠉⠀⢢⡜⠆
⡇⠀⠠⡆⠀⠀⠀⠉⠐⠁⠀⠄⠀⠀⢨⠁⠢⢤⡀⠀⠀⣤⠙⠁⠠⠄⠠⣀⡸⠟⣸⡖⠙⣦⡀⣁⠙⡖⠉⠀⠀⠀⠀⣤⠴⠀⠀⢰
⡇⠀⠘⡇⠀⠀⠀⠄⢀⠀⠀⠀⠀⢀⡆⠀⡀⠀⠙⢄⠉⠀⠈⠁⠠⠔⠛⢈⣷⣼⠋⠀⢠⣀⢴⠷⠶⣤⠶⡶⡞⠂⠳⠊⠀⠁⠀⢸⡁
⡇⠀⠀⣧⠀⠀⠀⠀⠀⠈⠠⢈⠂⠈⠑⠒⠂⠀⣡⡌⠛⣷⢶⣄⣤⠷⠶⢋⡿⠉⡄⠀⣹⡆⠃⡄⣢⠉⡆⠨⠑⠄⡀⠀⠀⠀⠀⡾
⡇⠀⠀⣯⠀⢀⠀⠀⠄⠈⠀⠀⠙⡆⠀⠀⠀⠀⣈⠁⠀⠉⢲⡀⠀⢀⠀⡘⠁⠀⠀⠀⢹⠀⠁⣶⠋⢰⠄⠒⠈⣰⠃⠀⢀⡀⠀⠇
⢁⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠤⠰⠭⠀⠀⡀⠀⠈⠓⢤⡶⠁⠀⡀⠀⢰⡋⠀⢰⡄⠤⣀⡀⠀⠀⣸⠀⠀⠀⠁⣾
⢸⡆⠀⢧⠀⠀⠀⡀⠀⠀⠀⠐⠀⣿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠚⢴⠋⠀⠀⠐⠀⠀⠋⠝⠂⡤⠀⢤⠬⠤⡄⢰⠃⠀⠀⠀⢠⡿⠠⣀
⠸⠇⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⡀⢀⠀⠀⡠⢋⡤⣶⣶⣄⠖⠛⠀⡀⢰⠟⣀⣀⠠⠤⠶⠛⠓⠘⠃⠠⠤⠤⢀⣀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠠⠚⠀⣈⡵⢉⣀⣀⠀⠀⠀⠀⠀⣯⣀⡁⠀⢀⡰⠟⠁⠀⠙⠛⠻⠶⠶⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄
⠀⣾⠘⣠⡇⠀⠈⠁⣀⡴⠒⠉⠁⠀⠀⠀⠀⠀⠈⠉⠒⠂⣼⣤⣤⣤⠋⠁⣀⠀⠀⠀⠀⣤⣀⣄⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⠊⠀⣀⠔⠃
⠀⣿⢀⡬⣷⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢉⡧⠂⠉⠀⢀⠠⠴⠡⠴⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠉⢹⣾⡁
⠀⠹⠠⠄⠈⠤⠀⠀⣤⡄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠠⣘⢧⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⡽⠋⢫⠘⠵⣀
⠀⢸⠢⢄⣀⡀⠐⠋⠠⠤⢀⡐⠀⠀⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣰⠟⠁⠀⠀⠀⠀⣀⣀⢀⣄⠀⠜⣤⣁⠞⠀⠀⠉⣤⠘⢣
⠀⢸⠀⠀⣽⠁⠈⢉⡁⠶⢦⣭⣄⣉⠓⠛⠤⠴⠛⠲⠶⢶⣈⣁⡀⡀⠀⠀⣴⠄⠤⠶⠦⢀⠶⢤⣌⣁⠀⠤⠄⠘⠋⠘⢁⡤⢛⡁⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈
⠀⢸⢠⠠⣿⠀⠀⠀⢀⠐⠉⠀⠀⡠⢛⣉⠛⠰⠶⡦⢠⣤⢷⣀⡼⠉⠀⠉⠀⢀⡴⠛⠻⡀⠀⠀⠈⡀⠀⢁⠀⠀⣠⠶⣧⡇⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢈⠀⣿⠀⢀⠀⠀⠀⠀⣠⡞⢰⡆⠀⢉⠗⠚⠿⢁⡤⠊⠻⡀⠀⠀⣠⠔⠛⠀⠀⠀⠙⠒⠖⠛⢤⡘⠁⡤⠊⠀⠀⠈⣷⠛
⠀⠀⡜⢐⡧⡆⠈⠀⠀⠀⠀⠀⡥⢬⡉⠐⠉⠓⠠⠀⠠⠤⣴⠤⡈⠑⠺⡇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠘⣧
⠀⠀⡇⢸⡇⠳⣀⠀⠀⠀⡀⠀⠣⡈⠃⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠀⠀⠀⠀⠀⠙⡂⠀⠀⣀⠠⠒⠀⠀⠀⠀⠁⠀⣿
⠀⠀⡧⠼⠷⢤⡈⠓⠦⣀⠁⠀⡆⠈⠃⠈⠿⠛⠂⢤⠤⢤⣀⠀⣠⠋⠀⠐⠒⠛⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⣿
⠀⠀⡇⠀⢰⡄⠉⠦⣄⠀⠻⣄⠐⠆⠀⢐⠆⠂⢀⣹⠲⢆⠱⡦⢸⠀⠀⠀⢠⠴⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⣿
⠀⠀⢇⠀⣾⣌⠑⢠⣀⠙⠂⢀⡙⠲⣤⠟⠒⣤⣤⣰⠀⢸⡆⠀⢾⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠐⠀⠀⢰⡆⢸
⠀⠀⢸⡀⣿⠈⠳⣄⠉⠓⣄⠈⠁⠓⣤⡀⠀⠀⠈⢻⠀⠘⡆⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡆⠀⠂⡠⠂⠀⠀⣠⡇⠀⣾
⠀⠀⢸⡇⢻⡄⢰⡏⠙⠧⡀⠉⠲⣄⠀⠈⠚⠤⣀⣸⠀⠀⢀⠀⢸⠄⠀⡀⠀⢀⠂⠀⠀⠃⠀⠀⢠⡀⠊⠀⠀⠐⢀⠿⡇⠰⡏
⠀⠀⠈⣷⠀⣇⢨⡇⠀⠀⠈⠲⢤⡀⠘⠦⣀⠀⠀⢹⠀⠆⠀⠀⢸⠀⠀⠉⠀⠌⠀⠀⠀⠀⠀⠀⢈⡤⠀⠀⢀⠀⡌⢱⡇⢀⡁
⠀⠀⠀⣿⠀⢻⠀⠇⠀⠀⠀⠀⠀⠙⠲⣆⠈⠓⢦⣼⠰⡶⢴⣦⣸⣷⠀⠠⠀⠀⠠⠂⠀⣀⠰⢀⠀⢱⠄⠰⡉⢸⡗⢀⡗⢸⡇
⠀⠀⠀⢿⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣽⠀⠀⠄⠀⣸⡏⣷⠁⠈⡰⠀⠀⢀⠄⠈⠀⠀⢈⠷⡎⠀⠈⡇⢠⣇⢸⡇
⠀⠀⠀⠸⣇⣸⡅⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⠀⡀⠀⣾⠁⠈⠂⡐⠁⠐⠈⠀⢀⠀⣤⡄⠸⣧⣤⡤⠀⡇⢘⡇⢸⡇
⠀⠀⠀⠀⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇⠀⣹⠀⠈⠺⢅⣠⡤⠒⠒⢋⡁⠀⠀⠀⠈⢮⣆⠀⣟⣻⡁⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⡇⠀⢸⠀⠀⠀⠀⠉⡇⠘⠀⠈⠁⠀⠀⠠⠀⠈⢣⡟⠧⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⣼⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠗⢴⣍⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡇⠀⣹⠀⠀⠀⠀⠀⠀⠳⣀⠀⠄⣀⠀⠀⠀⠉⠐⠀⠀⠛⠙⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⣈⠐⢀⠀⠀⠀⠀⠀⠐⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈⠀⠒⠒⠀⠐⠒⠐⠇
⣶⣾⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾
⣿⣿⣿⡏⠀⠈⠀⢉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡈⠭⠭⢁⠢⢒⠩⠭⠭⡐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⠂⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠡⠐
⠀⠀⠀⠀⢀⡀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇
⠀⢀⠄⠐⡡⠂⠉⠁⠢⡑⠄⠀⠀⠀⢀⡀⠐⡌⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡐
⡐⡡⠊⠀⠀⠀⢀⠠⠄⠓⠀⠀⠀⠀⠒⠠⠅⡚⢦⡑⢄⠀⠀⠀⠀⠀⢀⠔⣡⢊⣀⠄⣂⠭⠀⠀⠀⠈⠭⢐⠂⠄⡀
⡇⡅⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⢢⡑⠤⡀⡀⠔⡁⢀⠅⠐⠢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⡂⠄
⠐⡌⡲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠹⡑⠤⠔⡘⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⢢
⠰⢰⠀⠀⠀⠀⠀⣀⣀⠀⠤⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢡⠡⠀⠰⠰⠀⠀⠀⠀⠀⠄⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢃
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠒⠁⠐⢄⡀⠀⠀⠀⠀⠄⡀⡄⢀⣀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤⠣⣠⠀⠀⠀⠀⠀⠸⠈
⡰⠸⠄⣤⡀⠬⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⣄⠀⠔⣐⠢⡙⢰⠁⠘⠛⢤⠋⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠑⠇⠀⠀⠀⢀⢀
⡇⡁⠀⠀⠈⠇⠠⠋⠉⣑⠄⠀⠀⠀⢔⡉⠉⠳⠀⠛⠀⠀⠀⠐⠸⠀⠀⢀⠈⠀⠊⠀⣨⡁⠀⠀⠀⢀⠔⠒⢦⠀⢮⠌⠥⠈⢂
⠡⠰⡀⠀⠄⠀⠀⠀⠘⠛⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠢⠀⠠⢁⢆⠣⡀⡁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠷⠀⠈⠀⠀⠀⠀⠀⡰
⠀⠑⠀⠂⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢒⠡⠂⠀⠑⢢⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⢀⠔⡡
⠀⠀⠀⠈⢆⢂⠀⠀⠀⠀⠐⠂⠀⠐⠂⠀⠀⠀⢀⠎⠌⠀⠀⠀⠀⠀⠀⠆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡭⠅⠊
⠀⠀⠀⠀⠀⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠣⠊⠀⠀⠀⠀⠀⠀⠀⠈⢢⡣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠎
⠀⠀⠀⠀⠀⠀⠈⠂⢅⡂⠤⠤⠤⠤⠄⣂⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢑⠂⠤⠄⠀⠤⠄⢒⠡⠀⠁
⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠃⠉⠉⠉⠙⠀⠒⠀⠄⡀⠀⠀⠀⢀⡀⡀⠀⢰⠁⢸⠀⠀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠘⡀⠘⠄⠀⠀⠘⠀⠎⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠱⡀⠈⠂⠀⠠⠈⠀⢐⣁⠔⠈⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⣀⣀⠀⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠸⠄⠐⠀⠀⠀⠀⠈⢀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠞⠋⠉⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⡆⠑⠤⠤⠤⢀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡠⠄⠼⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠉⢀⠀⠀⠀⠀⠉⠈⡦⠄⠀⡈⠑⠤⠤⠤⡤⠔⠁
⠀⠀⠀⠀⠀⢀⣰⠀⠀⠀⠀⠀⢀⠈⣠⡄⠀⠀⠀⠀⠀⣶⡄⠂⠰⠀⠀⠀⠔⠀⠀⡘⠀⠀⠀⠀⠰
⠀⠀⠀⠀⡰⠁⠀⠀⠄⠀⠀⠀⠥⠀⠛⠃⠀⠀⠀⠀⠐⠉⠁⢡⠀⠀⢠⠴⠀⠀⠰⠀⠀⠀⠀⢠⠃
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠤⡄⢀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⠀⠆⠀⠀⠀⠀⠀⠀⠈
⠀⡠⠃⠀⠀⠀⠀⠀⠀⢀⢀⠀⠆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠊⢀⠔⠁⢀⠂⠒⠤⡀⠀⠀⣠⠃
⠀⠁⠀⠀⠀⠀⢀⠐⠒⢲⡀⠑⢄⡀⢀⡀⠀⠀⣀⠀⠠⠀⠖⠈⠀⠀⡠⠊⠆⠀⠀⠈⠐⠈⠁
⠀⡄⠀⠀⠀⠀⡆⠀⠀⠀⠁⢄⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢠
⠀⢣⠀⠀⠀⠀⠘⡄⠀⠸⠀⠀⠁⠂⠄⡤⡀⠀⠀⠀⢀⣦⠂⠀⠀⠀⠀⠀⢸
⠀⠀⢣⠀⠀⠀⠀⢐⠀⠀⢃⠀⠀⠀⠀⠀⠈⠐⢄⣤⠃⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⠀⠀⢰⠀⣀⠀⡆⠀⠀⠀⠄⠐⠀⠘⣁⠕⠀⠀⠀⠘⢀⠀⠀⠈⠢⡀
⠀⠀⠐⣧⠠⠀⠀⠘⠋⡠⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠑⠄⣁⠀⠈⠢
⠀⢀⠀⠀⠀⠀⠀⠐⠉⠀⠪⡀⠀⡀⠀⠂⠀⠀⠀⠀⠠⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠈⠦⠀⠀⠀⠀⡰⠂⠲⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⢀⣀⣀⣀⣀⣀⠄⠊
⠀⠀⠣⠔⠁⠀⠀⠀⠀⠣⣠⠀⢠⢀⣀⠀⠀⠤⠄⡰⠂⠐⢄⠀⠀⠀⠀⠠⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠄⠀⠄⠀⠀⠀⠀⡔⠀⠀⠀⠀⠉⢄⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢃⠀⡂⠀⠂⠀⡔⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⠀⢇⣀⠄⠀⣆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠰⡆⢨⠀⠀⠀⠀⠈⠐⡀⠀⠀⠀⠀⣅⠀⠀⠀⠈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠈⠀⢀⡄⠃⠀⠀⡰⠄⠀⢀⠄⠁⠁⠒⠂⠀⣶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠄⠀⢀⡠⠚⠁⠀⠀⠂⠢⣀⢀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠈⠶⠋⠁⠀⠀⠀⠔⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢈⡀⠀⠀⠀⣀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀
⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⢍⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠔⠂⣀⠒⢂⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢈⠀⡀⠀⠙⠂
⠀⠀⠀⠀⠰⠀⣔⠀⠒⠂⠀⠈⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠂⠠⡀⠀⠣⠚⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠂⠑⠀⠨⠄⡑
⠀⠀⠀⠀⠀⠀⠀⢈⣱⡀⢐⣥⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢼⢓⡠⠐⠰⢓⡆⠀⠄⡀
⠀⠀⠀⠀⢀⠔⠊⠁⠈⠉⠉⠀⠀⠈⠑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⡀⢴⠷⠬⢀⡀⠙⠀⠢
⠀⠀⠀⠀⡄⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⡄⠀⠰⡄
⠀⠀⠀⠀⠃⢂⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠆⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⡀⠈⢢⠀⠀⠀⠀⠀⠀⠀⢣⠘⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⡘⠀⠀⣤
⠀⠀⠀⠀⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢣⠃⠀⠀⠀⠀⠀⢀⠔⠂⢄⠄⠀⠀⠀⠐⠤⢀⡧⣀⣀⣆
⠀⠀⠀⠀⠑⠠⠗⠤⠤⢤⣀⣀⣀⡀⠠⢮⠇⠀⠀⠀⠀⠰⣁⡀⠑⠌⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⠇
⠀⠀⠀⠀⠐⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢢⠀⠰⠅
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢉⡒⠒⡖⠲⠒⠒⡋⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⢀⠊⠀⠈⣀⡽⠰⣇⠈⠀⠙⣆⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠈⣀⠀⠈⠛⠀⠀⠰
⠀⠀⠀⠀⠀⠈⠑⠊⠀⠉⠀⠀⠈⠓⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⠉⠉⠉⠉⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠃⠀⠀⠀⠰⠒⠀⠀⠒⢉⣀⡀⠈⢢⠀⣣⣄⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⠀⠀⠀⢰⠂⢠⡶⠀⠀⠀⠉⠁⠀⠀⠳⡇⠈⡆⠀⢰⢦⠋⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⢤⠀⡏⠀⠀⠀⣀⠀⣀⠰⠀⠀⠀⠀⢰⡤⠁⠀⢀⠆⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢳⠁⠀⠀⠀⡁⠈⠈⣀⡴⠂⠀⣀⠸⢤⡤⠴⣥⢠⡚⠁
⠀⠀⠀⠀⠀⢀⠶⠃⠴⢃⠄⠀⠉⠺⡀⠀⠀⠀⠈⠛⠛⣿⠰⠚⠛⠣⡀⠀⠱⠀⠈⡄⠈⠲⡄
⠀⠀⠀⠀⠀⠸⢇⣀⠀⠀⠓⠠⠤⡠⠜⣦⣀⢀⡠⠔⠉⠀⠐⠢⡄⠀⠃⠀⠀⠀⢀⠃⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠠⡜⠀⠀⡠⠉⠀⠀⠀⢈⠉⠢⡀⠈⡄⠠⠀⢸⠀⡘⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⡂⠀⠀⠀⠀⡠⠳⣄⢸⡀⡇⣀⢀⡂⣰⣥⡁⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⡘⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⢠⠁⠀⠀⠰⠟⠀⠀⠀⠀⠈⠛⠂⠀⠈⡄⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠈⡦⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠘⢲⡇⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠙⡀⠀⡇⠀⠀⠀⠀⠀⠈⠈⣀⠀⡀⠁⠁⠀⠀⠀⠀⠀⡇⠀⠎⠉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⠑⡶⠛⠲⡇⠀⠀⠀⠀⠖⢀⡀⠁⠀⠀⡀⠀⠢⠀⠀⠀⠀⡗⠚⢲⠞⠁⡨⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠉⠶⠁⠀⠀⢡⠀⠀⠀⠀⠀⠀⠉⠒⠒⠈⠀⠀⠀⠀⠀⠀⣠⠁⠀⠈⠶⠊⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠞⠒⠒⠠⠤⠤⢦⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⢀⠴⣦⠤⠄⠐⠒⠛⡄⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠈⠀⠀⠀⠀⠀⡰⠁⠀⠁⠒⢦⠠⢄⣀⣀⡀⠤⠀⡖⠊⠁⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠄⢀⡀⠀⡸⠁⠀⠀⠀⠀⠘⠀⠀⢀⣤⠀⠀⠀⠇⠀⠀⠀⠀⠀⢣⡀⣀⠠⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠇⠀⠀⠀⠀⠀⠀⢃⠀⠀⣀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠛⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣲⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢑⡒⠠⠤⠤⠀⢀⣀⣀⣀⡀⠀⠠⠤⠄⢒⣂⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠉⠀⠀⠐⠒⠒⠒⠂⠀⠀⠉⠁⢠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⣿⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢡⡄⠀⢠⠿⡤⠀⠠⣼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣉⣀⣀⣀⢰⠀⣆⣀⣀⣀⣈⣱⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣶⣶⣶⣮⣭⣤⣬⣵⣶⣶⣶⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⣤⠐⠊⠁⡆⠀⠀⠀⢌⠁⠉⠈⠓⢆⠀⡄⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⡰⠁⠀⣀⠤⠊⠀⠀⠀⠀⠀⠙⠶⣖⠠⠬⡆⡀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠌⠹⣄⠃⣠⠟⣋⣄⠀⠀⠀⠀⠀⠀⠤⡤⡤⣁⠂⢠⠇⢱⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠈⢧⣴⠦⠛⠓⠒⢢⣠⠤⣤⡏⠘⠛⠓⢛⠒⡟⠀⣈⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠈⠲⠤⢤⠠⠔⠀⠀⠀⠀⠠⡴⠶⠅⠐⠁⢀⡆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠀⠀⢀⠄⠒⠂⠀⠀⢀⠀⡀⠔⠐⠊⠒⠢⡀⠀⠌⠡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⠀⠨⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠐⠠⠀⠀⠠⠔⠛⠓⠀⠐⠊⠁⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠄⡀⠀⠀⠀⠀⠀⠐⠂⠘⠛⠀⠀⠀⠀⣀⠔⠊⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡉⢄⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⢀⠖⠈⠀⠠⠂⢸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⠹⡄⠢⡀⠀⠀⠈⠒⠂⠈⠁⠒⠒⠁⠀⡀⠀⠀⢀⠃⡘⠈⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢦⠀⠹⡢⠀⢑⠦⠄⡀⠀⠀⠀⠀⠀⡠⢊⠀⠀⠠⠃⡰⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠈⢦⠀⠰⠐⠇⠀⠀⠘⢄⠀⠀⡠⠊⠀⠀⠈⡔⢁⡐⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⢆⠀⠀⠀⠈⠱⠊⠀⠀⠀⡠⠊⢀⠌⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠑⠀⠀⠠⠁⠀⠀⡠⠎⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠂⣄⠈⡠⠎⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢇⡀⠉⢀⠔⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢙⡻⡏



⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢈⠃⢀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦
⠀⠀⠀⠀⠀⠀⣁⠀⠉⠩⠀⠤⣭⣱⣆⣉⣉⣩⠀⠛⠁⠀⠀⠀⠀⠀⢼⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⢸⡌⠃⠀⠀⢀⢻⣿⡏⡸⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⡯⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣶⣾⣿⣿⣧⣷⣦⣾⡇⠀⢀⠀⠀⠀⢠⠐⠀⠁⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⣿⣿⡿⣿⣻⢂⡦⣄⠀⠀⠈⡃⠀⠀⠈⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢾⣿⠸⠈⢁⠤⠀⠀⠁⠀⠁⠀⠀⠀⢰⠆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣷⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⣿⣷⡿⢣⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⣿⣹⡾⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠬⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡌⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣆⠨⢿⣿⠇⠠⠤⠀⠀⠀⠀⠀⠀⠈⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣌⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠟⠛⠉⠉⠉⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣀⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡄⢸⡏⠙⠿⢿⣿⣶⡶⠶⠤⣀⠀⠀⠀⠀⠘⢿⡷⠶⠦⢀⣀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠘⣇⡘⠁⠀⠀⠀⠉⠁⠶⠶⣄⣀⡀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠉⠉⠙⠶⣦⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠛⠓⠒⢠⣄⣼⣧⠀⠀⠀⠀⠀⠉⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡎⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠈⢋⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⢀⣀⣠⠀⠀⠀⠀⣀⠀⠐⠒⠒⠀⠀⢀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢻⣿⠀⠀⠸⠂⠀⠀⣤⣤⣤⠀⠘⠃⠀⠀⠀⠀⠀⠁⠀⠀⣤⣦⣤⡀⠀⠐⠀⠀⠐⠂⢺⡿⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠉⣿⣿⠄⠀⠀⠀⢰⠀⠀⠀⠀⠄⠀⢀⣿⣿⡄⠈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣧⣠⡤⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡄⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠤⢤⣤⠤⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣷⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣦⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠔⠺⣿⠛⣦⠁⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⣿⡏⠀⠀⠀⠀⠀⠀⣾⠊⠉⠀⠁⠀⠀⠀⢠⣾⠟⣶⠋⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⢿⢱⣦⠀⠀⠀⠠⠚⠁⠓⡀⠀⠀⠀⠠⠜⢻⡏⣷⠁⠀⠀⠀⠀⠀⠀⣴⠟⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢻⣿⠀⠀⠀⣾⢃⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡴⣧⡿⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢸⡇⠀⠀⠀⢫⡼⢫⡄⠀⠀⢸⠀⠀⠀⠀⢀⣴⠟⣱⠏⠁⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠃⠀⢸⠇⠀⠀⠀⠀⢳⣄⠙⢦⣤⣀⠀⢀⣤⡴⠛⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠟⠀⠀⡈⠀⠀⠀⠀⠀⠀⠙⢷⣤⠈⠙⠛⠋⣤⡼⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣴⠏⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠋⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣾⡏⠲⠶⠶⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠙⠿⠶⠖⠀⠸⠃
⠀⠀⠀⠀⠀⢀⣼⣥⡘⠷⠒⢶⣄⡈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢀⣼⠛⢁⠀⡀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠸⣷⣶⣿⣤⠛⢠⣿⢸⡿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠁⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⣿⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠙⠲⠶⣄⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠒⠲⠶⢦⣄⠀⢈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠘⡇⠀⠀⠀⠀⠀⠘⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⣦⣄⠀⣠⠋⠀⠀⠀⠀⠀⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠤⠤⠀⠀⠀⠀⠙⠛⠋⠉⠁⠙⣦⣀⠀⠀⠀⣰⡟⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠤⠤⠄⠛⠻⠶⠶⠿⠁⢀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡏⠐⠶⠶⠶⠶⠶⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠶⠶⠶⠶⠶⠀⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⡷⠘⡇⠈⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢰⠃⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⡇⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣠⣶⣴⣶⣶⣶⣶⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣧⡴⠶⠈⣉⣉⣁⣴⠶⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣉⣉⣁⣀⣉⣉⣙⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠶⠊⠉⠉⠉⠀⠚⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⣤⣀⣀⣤⣤⣤⣀⣠⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⡂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠠⢄⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠔⢃⠀⠀⠀⢠⢄⠀⠀⡰⡏⢦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⡎⠑⡲⠒⠁⠀⠉⢉⠀⢱⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⣰⠠⠌⠀⠀⠀⠀⠀⠀⡦⠌⡆⠀⡀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠁⠀⠀⠀⠈⠢⣀⠀⠀⡀⠔⠁⠀⠀⠀⠃⠀⠀⢲⠁
⠀⠀⠀⡄⠀⠀⠀⠀⠠⢌⠀⠀⠀⠀⠀⠆⠂⠀⠀⠂⠀⢀⠂⠀⠀⠂⠰⠈⠀⠀⠀⠀⡣⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠥⠀⠀⠁⠀⠈⠈⠢⡀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠔⠉⠁⠀⢀⠀⠀⠬
⠘⠀⠀⠀⠃⠀⠀⠀⠠⠀⠀⠙⢀⠀⠀⠀⠑⠤⢀⣀⣀⣀⣀⡀⠄⠊⠀⠀⠀⡀⠋⠀⠀⠄⠀⠀⠀⠈
⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠑⢦⡀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⢀⡴⠊⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⢠
⠣⡇⠀⠀⠂⠀⠀⠀⢀⠔⢁⠀⠀⠀⠀⡭⢀⡨⠜⠓⢤⡤⠔⠫⢁⡀⢽⠀⠀⠀⠀⡈⠢⡀⠀⠀⠀⠐⠀⠀⢸
⠀⠈⢢⠀⠀⠠⠀⠊⠁⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠈⠑⠠⠄⠀⠀⡔⠁
⠀⠀⠠⢓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡘⠄
⠀⠀⠀⢃⡔⠁⡅⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢸⠈⢢⡸⠃
⠀⠀⠀⠀⠈⠈⠁⠑⣦⣀⣀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⣀⣴⠊⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠉⠉⠀⠐⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠂⠀⠉⠉⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠉⠑⠐⠒⠒⠒⠒⠀⠊⠉⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣦⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⢀⠖⠋⠉⠳⡄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⣀⡤⠼⠶⠖⠒⠒⠛⠲⢤⡀
⠀⠀⡴⠋⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠳⡄⠀⣸⠀⠀⠀⢀⡄⠀⢀⢶⡀⠀⠀⣧⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⣇⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⡇⢀⣿⠀⣠⣶⣋⡤⠖⠉⠀⠳⢤⡀⣿⠀⠀⠀⠀⠀⣠⠶⣄⠀⠀⠀⠀⡇
⠀⠀⢻⠉⢿⠀⣠⣤⣠⠤⣄⣠⣄⢸⠏⢙⠇⢺⡈⢻⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣟⡇⡴⠒⠒⠋⠁⠀⠈⠓⠲⢦⣀⣇
⠀⠀⠈⢑⡏⠀⠙⠛⠁⠀⠀⠛⠛⠈⣷⠉⠀⠀⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠘⡧⠀⣴⣤⠤⠤⣤⣤⡄⢸⠏⣸
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢸⡘⢆⠀⠀⠀⠀⠀⠀⠀⢀⡞⡼⠙⡇⠀⠙⠋⠀⠀⠈⠛⠁⠈⡟⠁
⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣠⡤⢚⡁⠀⠀⠀⠀⠀⠣⣌⠓⠦⣄⣀⣀⡤⠖⢋⡰⠃⠀⠘⣄⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⢠⠴⡞⠉⠉⠹⠦⠏⠙⢯⣬⠧⡀⠀⠀⠀⠀⣀⡽⠖⠿⠦⠿⠒⢾⣍⠀⠀⠀⠀⢈⣑⢦⡤⢤⡤⠖⠋
⠀⠀⡴⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠙⢆⠀⣠⠞⠁⣹⠀⠀⠀⠀⠀⣇⠈⠳⡀⠀⣠⠯⣬⡟⠓⠚⢻⠛⠲⣄
⢀⡞⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢿⣧⣠⣼⠁⠀⠀⠀⠀⠀⠘⢦⢀⣽⣰⠃⠀⠀⡇⠀⠀⢸⠀⠀⠘⠆
⢾⡳⠤⣄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠴⢚⡧⠀⠹⡀⠀⠀⠀⠀⠀⠀⡾⠙⢾⠃⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠘⡄
⠀⣿⢳⢶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⠚⣯⠀⠀⠀⢱⡀⠀⠀⠀⠀⡼⠁⠀⡟⠀⡄⠀⠀⡇⠀⠀⢸⠀⠀⢠⠀⢱
⠀⢿⢸⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡇⢸⠀⠀⠀⠀⡇⣀⡀⣀⣠⣇⠀⢸⠁⣸⡇⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇⠈⡇
⠀⠘⡄⢇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⢸⡇⠀⢀⡞⠀⠀⠀⠀⠀⠸⣄⡏⢀⡏⡇⠀⠀⡷⠒⠖⢾⠀⠀⢸⢹⠀⢻
⠀⠀⠹⡼⢾⡗⠲⠦⠤⣤⠤⠴⠶⢺⠀⠀⢸⡜⡇⠀⡾⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⢰⠃⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇⡇⠘
⠀⠀⢰⡗⠚⡿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⠃⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⡏⠈⢳⢤⣀⡇⠀⠀⢸⣀⣤⡞⠁⢹
⠀⠀⢸⡇⠀⡇⠀⠀⠀⣶⠀⠀⠀⢸⠀⠀⠈⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⣷⢶⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡶
⠀⠀⢸⡇⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢧⣼⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠺⣄
⠀⠀⢸⡇⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣅⡀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡇⠀⣧⠀⠀⠀⣿⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠙⡖⢤⣤⡤⠴⡞⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡇⠀⢹⠀⠀⢠⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡧⡇⢰⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡇⠀⢸⠀⠀⢸⣿⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⢸⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢸⡇⠀⢸⠀⠀⢸⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡇⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢸⡇⠀⢸⠀⠀⢸⠏⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣷⡇⣼⠀⠀⠀⠀⠀⠀⠓⠒⡖⢶⣶⢲⡖⠚
⠀⠀⢸⡇⠀⣸⣄⣀⣸⠀⣇⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⣏⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡟⢸⠃
⠀⠀⢸⡇⣾⣡⣤⣤⡽⠀⢯⣤⣀⣉⣷⠀⠀⠀⠀⠀⢀⡴⣯⣾⠟⢷⣼⠷⡄⠀⠀⠀⠀⢀⣴⣣⣾⢷⣌⣷⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤⣬⣯⣭⣤⣤⣤⣬⣭⣧⣤⣤⣤⣤⣾⣾⣭⣥⣤⣭⣷⣾⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠈⠉⠉⠁⠐⢢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠀⢀⣤⠀⠀⣠⣄⠈⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢳⡗⢶⡏⢻⠖⢲⠏⢵⠶⠺⡾⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠢⢀⣠⣬⣤⣤⣭⠁⣀⡔⠰⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠡⢄⠳⢃⠤⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢚⠂⣤⠀⣀⣀⠀⠤⠒⡓⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⢸⠀⠘⠒⠊⠑⠒⠃⠀⡇⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢧⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠳⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⠃⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠀⣗⣼⣦⢀⠀⣀⣀⢀⡀⣤⣦⢰⠃⠀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⣼⠀⠀⡀⠀⠉⠁⢀⡴⠦⡀⠈⠉⠁⢰⠀⢰⣇⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⢠⠀⠀⠀⢰⠀⠀⢡⠀⠀⠀⠘⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡌⠀⠀⢘⠀⠀⠀⡈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠃⠀⠀⠈⡀⠀⠀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠃⠀⠀⠀⡇⠀⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠚⠠⢼⠀⠀⠀⠀⢠⠠⠚⠐⠒⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⢻

⠀⠀⠀⠀⠀⠀⠠⠂⠈⡉⠈⠈⢉⢢
⠀⠀⠀⠀⠀⡐⠀⢊⡤⠈⢀⠄⠄⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠁⠈⠲⠄
⠀⠀⠀⠀⠀⡄⠀⡠⠄⢠⡥⠀⠐⣇⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠈⠉⠐⣀⠀⠄
⠀⠀⠀⠀⠀⢀⠘⠃⠀⠠⣄⡤⠄⠘⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⡠⡀⠀⣛⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⡈⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣁⣨⡀⠀⣈⢁⣆⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠔⠾⣤⡤⠗⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠁⠊⢀⠌⠠⠏⠁⢀⡂⢄
⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠒⠛⡟⠚⠀⢄⠀⠀⠀⠃
⠀⠀⠀⠀⢸⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠰⠠⠀⠀⢳⠀⠀⢳
⠀⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡰⠃⠀⠀⠀⡇⠀⠀⠆
⠀⠀⠀⠀⢸⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠘⡄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢀⠌⠀⠀⠈⠀⢀⠃⠀⠀⡆
⠀⠀⠀⠀⠸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⠀⠀⠀⠀⠀⠀⠀⡠⠂⢀⡠⠀⠀⠀⠀⠰⢤⠀⢣⣀⣀⣰⢁
⠀⠀⠀⠀⢐⠄⣰⢇⣀⣀⣀⡀⠀⠀⠀⣀⣠⣣⠃⠀⠀⠀⠀⠀⠊⠀⠒⢄⠋⠀⠀⠀⠀⠀⠀⠉⠃⠆⣀⣽⠁
⠀⠀⠀⠀⠀⡀⡼⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⡷⠀⠀⠀⠀⠀⠀⠁⠛⠆⠂⠀⠀⠀⠀⠀⠀⠸⠀⠀⠰⠞⠉
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⡸
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢹
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠆⢀
⠀⠀⠀⠀⠘⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞⠁⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⠙⡓⠚⡋⠉⣉⡀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⣉⡽⠀⣏⡉⠀⠀⠑⡄⠀⠀⠀⠀⠀⠑⠘⠀⠀⠀⠀⠀⢆⣀⠂⠈⠙⠂⠠⠐⡎
⠀⠀⠀⠀⠀⠀⠙⠒⠛⠐⠉⠀⠀⠈⠛⠒⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠒⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠒⢤⡀
⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⡏⠀⡼⠀⢀⣠⣄⠀⠀⠀⣠⣄⠀⠘⡄⠀⢣
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⡴⠛⠒⠒⠒⠒⠒⠚⠙⠢⡀⠀⠀⠸⡄⠀⠀⠀⡠⢧⣴⣥⣤⣜⡩⣻⡤⠤⣴⣉⡫⣥⣤⣽⣤⡦⣄
⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⠏⠀⠴⠶⠦⠀⠀⠶⠲⠄⠀⢱⠀⠀⠀⢧⠀⠀⠀⡅⢸⢫⠀⠙⠓⠄⠾⠀⠀⠹⠤⠜⠛⠀⢸⠙⠀⢸
⠀⠀⠀⠀⠀⢸⠂⠀⡔⣼⠀⡤⠒⠁⢒⠀⠐⠊⠒⢦⠀⠘⣤⠲⡀⢾⠀⠀⠀⠙⢻⠀⠳⠤⣤⠶⣾⡶⢶⡶⢶⡦⠤⣤⠞⢸⠓⠁
⠀⠀⠀⠀⠀⠈⡇⠈⠧⢇⠀⠀⠀⠀⢁⣀⡀⠀⠀⠀⠀⠀⡸⠴⠀⡏⠀⠀⠀⠀⠘⡄⠀⠀⠸⣄⡈⢷⠾⣀⡠⠃⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠈⠢⣠⣈⣦⡀⠀⠀⠀⣤⡄⠀⠀⠀⠀⣴⣡⡠⠊⠀⠀⠀⠀⠀⠀⣈⣦⣀⠀⠀⠉⠉⠉⠁⠀⠀⢀⣤⠮⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠛⣶⠢⣀⣀⣀⡠⠖⣶⠛⠢⢄⡀⠀⠀⠀⠀⢀⠔⠉⠁⡏⠀⢹⡃⠒⢲⣶⠒⠚⡹⠁⠀⡇⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣿⠀⠈⢦⡴⠁⠀⣿⠀⠀⠀⠙⢦⡀⠀⡰⠁⠀⠀⠀⡇⠀⠀⠑⠊⠁⠀⠉⠒⠁⠀⠀⡇⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⡿⠖⠚⠉⠉⠓⠶⢿⠀⠀⠀⠀⠀⢳⡼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢇⠀⠀⢠⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢀⢟⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⢧⠀⠀⠀⢇
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⠈⣗⠋⠸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢸⠁⠳⡖⠒⡇
⠀⠀⠀⠀⢀⠁⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⡟⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢧⠀⢰
⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠘⡄⢸
⠀⠀⠀⠀⠈⡀⢠⠻⣄⠀⠀⠀⣃⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⣿⡆⠀⡇⠀⠀⣯⢓⣞⣦⠤⠤⠤⠤⠤⠤⠤⠤⢤⢾⣖⡪⢿⠀⠀⠀⡣⠘⡇
⠀⠀⠀⠀⠀⡏⣹⠀⢸⠉⠒⠢⠞⠀⠀⠀⠀⠀⠀⠷⠔⠒⠉⣷⣁⡷⠲⡇⠀⠀⠇⠀⠀⠉⠉⠐⠒⣒⣒⣒⠒⠈⠉⠉⠀⠀⡎⠀⢀⢮⠀⠀⢘
⠀⠀⠀⠀⠀⢧⡿⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣼⣿⣇⠃⠀⠀⢸⠀⠀⠀⠀⠀⢸⠁⠀⠈⡆⠀⠀⠀⠀⠀⡇⠀⢸⡞⠓⠒⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⠀⠀⢃⠀⠀⠀⠀⢠⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠈⡄⠀⠀⠀⡌⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⣏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡇⠀⠀⠀⠀⠀⢡⠀⠀⢰⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢒⣊⣀⢴⠀⠀⠀⡦⣀⣙⡒⣤⣀⠀⠀⠀⢀⡤⠐⠒⠒⠧⠤⢠⡇⠀⠀⠀⠀⠀⢸⡄⠤⠾⠐⠒⠂⢾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠤⠀⠒⠊⠀⠀⠀⠑⠒⠀⠤⠤⠜⠀⠀⠀⠚⠤⠤⠤⠤⠤⠒⠚⠁⠀⠀⠀⠀⠀⠈⠓⠒⠤⠤⠤⠤⢼⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠉⠉⠉⠉⠁⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠒⠤⣀
⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠣⣀
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡆⠀⠀⠀⣠⡤⠀⠀⠀⠀⠠⢤⡀⠀⠀⠸⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⡃⠀⠠⢤⠜⠱⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡆⠀⡤⢄⡀⠘
⠀⠀⠀⠀⢀⡱⠾⠂⠁⠀⠰⠃⠀⠀⠀⠀⢠⡀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠰⣀⠇⠀⠱⠊
⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢀⠔⠒⢄⠀⠀⠀⠀⠀⠀⢰⠉⢢⢠⠃
⡐⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⡒⠊⠉⠐⣢⣔⠂⠉⠑⠂⠀⠀⠒⠁⠀⠈⠃
⡇⠀⠰⠁⠈⠀⠒⠐⠀⠯⠀⠀⠑⣨⠧⡉⠉⠁⠀⠀⠉⠉⠉⡹⠁⠀⠀⠀⠀⠀⠃
⢱⡐⠃⠀⠀⠋⠀⠀⠀⣀⠀⠀⠀⠸⠀⡛⠢⠤⠤⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⢀⠂
⢈⠘⠆⠀⠀⠀⠓⠠⠖⠉⢀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠢⣀⠀⠀⠀⠀⠀⠖⠄⡠⢀
⠈⠠⠄⠀⠀⠀⠒⠀⠒⠊⠁⠀⠀⢀⢃⡰⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠈⣆⠈⢢⠀⠀⠀⠀⣇⣼⠀⠆
⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⣀⡤⠖⢻⣉⡤⢜⡁⠀⠀⢀⡀⡤⠞⠉⠉⠁⠀⠀⠀⠰⠀⠀⢆⠀⠀⠎⠀⠘⢧⠞
⠀⠀⠀⠀⢨⠉⡏⠉⠁⠙⣅⣀⡠⠔⠁⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠄⠀⠈⠆⠀⡤⠋
⠀⠀⠀⢀⠃⢀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸⠀⠀⠂
⠀⠀⠀⠈⡆⢼⠀⠀⠀⠀⠀⠀⣀⠼⠒⠈⠶⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⠤⠒⠤⠁⠀⡘
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠉⠉⠀⠀⡰⠆⠴⠤⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⡄⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⡆⠀⠈⠂⠤⠄⠐⠒⠀⠈⠁⠀⠀⡰⢲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⠒⠤⠤⠐⠁
⠀⠀⠀⠀⠈⠒⠒⠀⠰⡒⠒⠂⠈⠉⠐⠲⣎⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⠀⠀⠀⠀⠙⠷⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠉⠐⠒⠂⠠⠤⠤⠤⠄⠠⠀⠒⠒⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⣿⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠦⠠⠠⢤⠛⠤⠀⠠⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣁⣀⣀⣀⠀⢼⠀⡥⠀⣀⣀⣀⣈⢦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣶⣶⣶⣶⣬⣥⣤⣬⣵⣶⣶⣶⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠐⠖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠒⠒⠒⠒⠒⠸⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⡰⢆⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⡜⡀⠀⠀⢀⡞⠁⠈⠢⡑⢄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢰⣔⠀⢀⠴⠋⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢸⠀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡄⢘
⠀⠀⠀⠀⠀⣸⡼⠁⠀⠄⣤⠀⢀⡀⠀⠀⢠⡄⠀⠀⢡⣬⡀
⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠇⢘⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠑⠛⢄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠞⠒⠁⠀⠀⠀⠀⠀⡠⠞⡁⠄⠂⠀⠀⠀⠀⠉⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⡀⠀⣀⣀⠀⣀⣤⠖⠧⣀⠀⠀⠀⠀⠀⠀⡜⠁⢈⣀⡠⠤⢠⠂⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⣠⡔⠉⠀⣯⣉⠈⠉⢁⡠⠾⠀⠀⠀⠑⢦⡀⠀⡠⠶⠐⠋⠁⡀⠀⢠⠇⡠⡄⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡠⠊⠀⡇⠀⠀⢸⠀⠉⠉⠁⠀⡆⠀⠀⠀⠀⠎⠙⢆⢣⡄⠀⠀⠺⠇⠀⠰⠏⠀⢣⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢀⠎⠀⠀⠀⡇⠀⠀⠈⡀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠈⢇⢠⠀⠀⠀⠀⠀⠀⠀⢠⡞⢣⡘⠀⠀⡜
⠀⠀⠀⠎⠀⠀⠀⢀⠁⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⠘⣼⢦⠀⠀⠐⠂⠀⠀⠀⠀⠀⠑⢤⡜
⠀⠀⢸⠀⠀⠀⣵⢺⠀⠀⠀⢰⠃⠒⠠⠄⠀⢃⠀⠀⠀⠀⣇⠀⠀⠀⣯⣤⡵⠦⣄⡀⠀⠀⠀⢀⣠⢤⡠⠃
⠀⠀⠘⡄⣀⡤⣆⢸⠀⠀⠀⡌⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⡸⠂⠀⢀⠿⢘⠢⢀⣀⢏⠉⠉⠉⠠⠅⠀⡄
⠀⠀⠀⠘⣏⠁⢈⡼⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢤⢙⠀⡠⡜⢰⡌⡗⠀⠀⣀⣱⠀⠀⢸⢀⡴⠁
⠀⠀⠀⠀⠈⠉⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⢠⣇⠑⠤⣷⣘⡶⠋⠀⢨⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⡠⠟⠳⢄⡀⢹⢇⠀⣠⠊
⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡇⠀⠀⠀⠉⠙⢯⡙⢦⡀
⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠋⠭⠥⢒⣒⠀⠤⠈⣆⠱⡄
⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠐⢄⡀⠀⠀⡀⠀⠀⠀⠀⢀⡸⢸⣁⠒⠒⠠⠤⠤⣉⣉⣻⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠈⣄⣀⠀⠀⠀⣈⢁⡠⠤⠥⠞⠓⠋⣁⡇⠸⢤⠭⠍⣑⣒⣒⠒⠦⢼⠤⠤⠂
⠀⠀⠀⠀⠀⣸⠇⠀⢧⣤⣀⣀⣀⣀⣀⡀⣤⠤⠔⠒⠉⠀⠇⠀⢸⠀⠀⢈⠀⠀⢻⡀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠸⡀⠀⠀⠈⣇⠀⠀⠀⠀⡆⠀⠀⣸⠀⠀⢸⠄⠀⡍⠀⠀⢾⣡⠇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢷⠀⠀⠀⡇⠀⠀⠀⠀⠐⣄⡀⢸⡀⠀⢸⡤⠘⢳
⠀⠀⠀⠀⠀⣿⠀⠀⠀⡜⠓⠤⣀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠑⠒⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠋⠉⠀⠁⠉⠉⠙⢳⣄
⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⣠⡶⢶⣿⠋⣻⣿⣄⣤⣀
⠀⠀⠀⠀⢀⣾⣿⠓⠦⢤⣄⡤⠺⣥⠒⢦⡤⠒⠋⣿⣧⠇⠀⢀⡟⠽⢣⢿⣛⣵⡿⡷⠽⢻⣆⣀⡀
⠀⠀⠀⠀⠈⠳⢿⡄⠀⢣⡈⠣⠴⠋⠂⠘⡀⠃⢠⡿⠉⢀⣾⣻⡷⠒⠋⠀⠀⠀⠙⠛⠛⠛⢿⣻⡗⣀
⠀⠀⠀⠀⠀⠀⢈⣿⣦⣀⠑⠢⠤⠤⠀⠊⣀⣴⣟⠁⠀⣾⢽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡃
⠀⠀⠀⠀⢀⡴⠋⣹⠉⣽⠛⢲⡖⣶⠒⢻⡟⠙⣿⠳⣤⣟⣿⠛⠢⣄⡀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣸⣧⠟
⠀⠀⠀⣠⠞⠀⠀⡏⠀⠹⡶⣾⡋⠙⠓⣾⠁⠀⢹⡄⠈⠻⣿⡀⠀⢀⡍⠻⡋⢳⣶⣉⣱⢶⡋⠉⢸⣻⡇
⠀⠀⣴⠋⠀⠀⢸⡇⠀⠀⠀⠈⣧⠀⢸⠃⠀⠀⠈⣧⠀⠀⣸⣧⡀⠀⢢⡀⠈⠉⠀⠉⠁⡠⠁⢠⡟⠉
⠀⠸⣧⡀⠀⠀⣾⠁⠀⠀⠀⠀⠸⣇⡟⠀⠀⠀⠀⢹⣦⡞⢁⡾⣿⠶⣤⣌⣐⣒⣒⢒⣉⣠⣶⢿⡀
⠀⠀⢸⡿⠲⢶⡏⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠈⣿⣷⡾⠁⣿⠀⠀⠈⠻⣛⠿⠋⠉⠁⢹⡄⠻⣄
⠀⠀⢸⠇⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠹⣆
⠀⠀⣾⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⢈⣧⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢹⣆
⠀⠀⣿⠀⠀⣿⠧⣤⣀⣀⣀⡀⠀⣸⡆⣀⣀⣀⣤⠴⠛⢻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⠻⣇
⠀⠀⣿⠀⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢹⠟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢻⡄
⠀⠀⢿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇
⠀⠀⠸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⢰⡇
⠀⠀⠀⢻⡄⣿⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⢸⡇⣸⣦⠤⣤⣤⣤⣤⣤⣠⣤⠴⠶⠞⢋⡹⡇⣸⠃
⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢀⡴⣄⡀⠀⠀⡰⢢⡀⠀⠈⡤⠤⢿⠏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⡇⣿⠰⡍⠷⢠⠇⠀⢈⠱⡦⠰⠀⢀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠲⠚⠉⠀⠀⠀⠤⠂⠊⠔⡊⡐⠄⠀⢿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠆⢉⡊⢡⠀⠀⠀⠁⣑⡁⠅⠀⢸⡆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠠⣋⡑⠚⣀⠇⠀⠀⠀⠀⠀⠀⣀⡼⠇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⠙⠓⠦⣤⣤⣵⣶⣤⣤⡤⠴⠶⠶⣿⠋⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢹⡀⠀⠀⠀⣿⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠙⣷⣦⠤⠤⠤⣤⣤⣤⠾⣷⠤⠶⠖⠒⠛⠛⢯⡀⠀⢀⣼⡗⠓⢲⣄⣻⣀⣤⠖⢚⡳⢦⡀
⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠹⣶⠁⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟⠀⠈⠉⠁⠘⣿⠋⠀⠁⠁⠀⠈⣷
⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣀⣀⣠⡴⣿⠷⢤⣄⣀⣠⣀⡠⠴⣿⢸⡟⠛⠛⠛⣛⣛⣿⣛⠒⢒⢒⣚⣩⡿
⠀⠀⠀⠀⠀⠙⠷⠬⠭⠭⠭⠭⠤⠴⠟⠓⠒⠒⠒⠒⠒⠛⠛⠋⠀⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠹⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠈⠉⠉⠁⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠁⠀⠀⠀⢀⡈⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡄⣀⠀⠀⠉⢁⠀⠀⡀⡉⠀⠀⡀⠠⠰
⠀⡖⣄⠦⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠂⡀⡐⢈⡀⡘⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠴⣠⢂⡀
⠋⡞⢀⠔⠂⠄⢀⠄⢀⠁⠂⠀⠀⠀⠢⡄⢀⠀⣠⠔⠀⠀⠀⠐⠀⡀⠠⡀⠠⠀⠃⣀⢱⠝
⠀⠀⠈⠀⠄⢀⠈⠀⠈⠐⠰⡄⠀⠀⠀⠈⠂⠊⠀⠀⠀⠀⣠⠄⠂⠁⠀⠀⡀⠠⠀⠁
⠀⠀⠀⠀⠀⠆⣀⠀⠀⠀⡠⠈⠀⠀⠠⠀⠀⠀⠠⠄⠐⠈⠈⠢⠀⠀⠀⢀⡰
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠄⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠒⠀⠈⠓⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠬⠀⠀⢈⠐⢈⠀⠀⢨⠄⠀⡒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡰⠂⢦⠀⠀⠉⠁⠀⣠⠆⠲⡈⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠐⠀⠀⣀⠀⠀⠉⠀⠘⠀⠀⠡⡀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⣤⡄⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠠⠣⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠇⢵⡅⢅
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣄⣒⣘⣂⣠⣀⣘⣀⣀⣀⣀⣀⣶⣤⣤⣜⣢⣤⣤⣆⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⣯⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⠀⠘⠿⡶⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢛⣿⠛⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠅⣤⣤⣤⣤⣤⣤⣶⡖⢿⡏⢿⡄⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⡏⠈⣿⣿⣿⠀⠀⣈
⠿⠟⠛⠛⠋⠉⠉⠁⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣉⣿⣿⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡿⣿⡏⢳⣤⡀⠀⠀⠀⠀⢰
⡀⠀⠀⢀⣤⣤⣄⣾⣿⣿⣿⣷⡟⢻⣟⢿⣿⣿⣿⣿⣿⠻⣿⣿⡿⣿⣿⣧⣀⡛⠻⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡽⡎⢿⣷⡈⢻⣿⡆⠸⣿⣿⣿⣿⣬⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣾⣤⣼⣿⣧⠀⠻⣧⠀⣿⣿⣿⠟⡋⠉⠛⣯⣿⠿⠖⠃⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢷⣿⠋⡛⠁⣆⡛⠚⣍⣍⠀⣀⣍⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠁⢀⠐⣤⡉⠫⠄⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢻⡿⠟⠉⣛⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣸⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⠆⠀⠀⠀⠀⢸⡟⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠉⣀⣀⠠⣾⣿⠉⠃⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠿⣷⣆⠈⠃⢘⠒⠂⠀⢈⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠘⠛⠲⣼⣶⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠸⣶⣦⠈⠸⣿⣿⣿⣿⣿⣿⣦⣄⣀⣠⠴⣄⡀⠀⠹⣿⣷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢓⢿⠿⢧⣾⠿⠋⣅⡀⣀⣉⡿⠟⣉⣠⣦⢀⣹⣦⠀⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣀⠀⠀⠀⠌⠛⠙⠛⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣦⣌⣭⣿⣿⣦⣬⣽⣿⣿⣭⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠄⠐⠒⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡅⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠘⠁⠐⠒⠐⠒⠒⠒⠚⡆⠀⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠸⡀⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⣄⠀⠀⠀⡄⠀⠀⡇⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⢉⠉⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠒⠀⠀⠀⢀⠈⠀⠀⡄⠀⠀⡞⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢸⠺⠾⠒⠳⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⡈⠀⠀⠀⠀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢼⠀⠀⣀⠜⡏⠐⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⡏⣈⢶⠊⠁⡸⠁⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠉⠈⠃⠑⠈⠀⠀⠀⢀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡉⢀⡀⠤⢀⡀⠀⡜⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⡀⠐⡄⠀⠀⡀⠀⠨⠑⠒⣚⡁⢀⡇⠀⠀⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠈⠉⠁⠀⠀⠉⢩⠉⢰⠋⠉⣩⣙⣲⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣇⡬⠁⢲⠂⠀⠤⠤⠤⠐⠒⠚⠓⠊⠉⠁⠘⠋⢁⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠀⠀⠒⠒⠒⠀⠀⠒⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠆⠀⠉⠉⠉⠉⠉⠉⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢉⡤⠶⠂⠉⠉⠉⠉⠁⠲⣄⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⣈⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⠋⠀⠀⣀⣀⠠⠄⠤⠤⢤⠀⡀⠉⠀⠀⠱⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠠⠚⠉⢉⠀⠀⠀⠀⣠⣄⣉⠲⠀⠀⠀⠛⣠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⡀⠂⠉⠀⢈⡗⠀⢰⠊⠀⠈⠙⣗⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⡇⠀⠰⡄⠚⠛⣁⣼⠗⠀⠻⣘⠶⠀⢤⡬⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⠘⠁⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠈⠀⠀⠀⠠⢤⠤⣄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡟⠀⡜⠃⠀⣠⠞⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⢠⠄⠀⠀⠀⣇⡠⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢰⠀⡘⠃⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⠎⠀⠀⠀⠀⢛⠤⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡠⠞⠛⠓⠐⠒⠶⠀⠀⠈⠉⠀⠳⠎⠁⠀⠀⠀⠀⠀⣿⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⢀⡜⠳⢄⠀⠀⠀⠀⠀⠉⠲⢂⡄⠀⠀⣠⣁⣀⡀
⠀⠀⠀⠀⣠⡤⠶⠶⠞⠛⠲⠄⣠⣤⣤⡴⠊⠁⠐⠂⠀⠛⠒⠆⢀⣤⡠⠖⠊⠀⢠⠖⣩⠛⠉⠉⠉⠛⠲⣦
⠀⠀⢠⡾⠙⡆⠀⠀⠀⠀⠀⠀⠀⠈⢣⡘⢷⠂⢶⡀⠀⠀⢀⡀⠀⠀⠀⠀⣤⡞⢁⡼⠁⠀⠀⠀⠀⠀⠀⡏⢹⡄
⠀⢠⡟⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢣⣠⡟⠒⠲⠊⠁⠀⠈⢏⡡⠂⣠⠎⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⢳⡀
⢀⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠛⡀⠀⢸⠀⠀⠀⣠⠎⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠸⠁
⢸⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠘⡄⢸⠀⣠⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡇
⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠻⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠃
⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡷⠴⠾⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠛⢁⣀⣀⡤⢤⡤⢄⡙⢦⡄⣀⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢸⠃
⡇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡠⠎⠻⡀⣀⠞⠁⢠⡟⠀⣷⠀⢈⡛⢦⡙⠱⢌⡿⠀⠀⠀⠀⠀⡘⠀⠀⠀⢀⡎
⣇⠀⠀⠀⠀⠀⣭⡤⠖⢤⣶⡎⠀⠀⢠⡎⢹⠶⠊⢻⠂⢀⣿⠀⢸⡇⢸⡏⠲⢦⠛⠆⣀⠀⠀⠘⠃⠀⠀⠀⢸⠇
⣿⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠘⢦⡀⠈⢱⡟⠀⢠⠞⠓⠙⠻⡴⠛⠳⠾⢧⠀⠀⢇⠀⠀⠉⢶⣷⣴⠆⠀⠀⡿
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠉⠈⣅⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⠀⠀⠀⠀⠃⠘⠀⠀⣸⠇
⢻⣄⣀⣀⣀⣤⣤⠶⠖⠛⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠧⣄⠀⠀⠀⠀⠀⣰⠏
⠈⠉⠉⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠈⠳⣄⣀⣀⡴⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠇⠀⠀⠈⠉⠉⠀⠟
⠀⠀⠀⠀⡇⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡷⠚⠛⠛⠓⠒⠴⣦⡀
⠀⠀⠀⠀⣇⠀⢀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠉⠀⠑⠛⠓⢶⡇⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡗
⠀⠀⠀⠀⢹⡇⢨⠋⢀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⡾⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⣾⠁⠀⠀⡾⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠸⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⢸⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡆⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠘⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠸⢧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢀⣠⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠦⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠓⠻⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡿⠲⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡇⠐⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣾⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣿⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣿⢠⡿⠀⢀⣧⣀⣼⡇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⠖⠿⠀⠀⠀⢻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠿⣤⣭⣍⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣩⣭⠿⢧⣤⣤⣤⣼⠟⢩⡵⠲⠖⠒⠒⠒⠒⠒⠒⠲⠶⢬⡽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠛⠉⠉⠉⠉⠉⠳⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢀⣠⣤⣴⡻⠛⣿⣷⣀⣀
⠀⠀⠀⠀⠀⣰⣿⠶⢤⣀⣀⢀⡠⣆⢢⠤⣄⣀⡤⠶⢾⡏⡷⠀⠀⣾⢛⡶⣹⡷⠾⣹⢿⣏⣿⢿⡄
⠀⠀⠀⠀⠀⢿⣽⡇⠀⠰⡉⠻⣀⣸⠷⠄⠸⠁⠆⢀⣿⠟⠁⣠⢶⣿⡬⠴⠋⠉⠋⠉⠺⠷⠶⠾⣿⢽⣧
⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠙⢄⣀⠀⠀⢀⠐⠁⢀⡼⠋⠀⣠⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⡆
⠀⠀⠀⠀⠀⢀⣴⠟⡿⠛⣶⣤⣤⣌⣩⣤⣴⡾⠻⣿⣦⡀⣸⢾⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣷
⠀⠀⠀⠀⣠⠞⠁⢸⠁⠘⣇⣀⣻⠶⢿⣄⣼⠃⠀⢿⡌⠛⢿⣾⡇⠀⠉⠓⠦⣤⠤⣍⣁⠴⣄⣠⡤⠖⢺⡿⡏
⠀⠀⢀⡼⠃⠀⠀⣾⠀⠀⠈⠉⢿⡀⠀⢸⠏⠀⠀⠘⣧⠀⠀⢹⣧⠀⠀⢚⠀⠑⠤⠜⠙⠶⠋⠱⠖⠀⣼⠷⠃
⠀⢀⣾⠁⠀⠀⢠⡟⠀⠀⠀⠀⠈⢷⠀⡾⠀⠀⠀⠀⢻⡆⢀⡼⢻⣷⣄⡀⠳⠄⣀⡀⣀⣀⠴⠃⢀⣼⠏
⠀⠈⢻⣦⣀⡀⣸⠃⠀⠀⠀⠀⠀⠘⣷⠇⠀⠀⠀⠀⠘⣿⣏⢠⡿⢹⡏⠙⠳⢶⣦⣤⣤⡤⠴⠶⣿⠙⣧
⠀⠀⢸⡟⠉⠉⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⢸⡇⠀⠀⠀⠙⠚⠋⠀⠀⠀⠸⣇⠈⢷⡀
⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⢷⡀
⠀⠀⢸⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⣿⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣸⡿
⠀⠀⣿⠀⠀⢸⠟⠲⠦⢤⣤⣤⣀⣀⣿⣄⣤⣤⡤⠶⠚⠉⢹⣿⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋⢷⡀
⠀⠀⣿⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠸⣧
⠀⠀⢸⡆⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿
⠀⠀⠘⣧⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⣿
⠀⠀⠀⠹⣆⢸⡃⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⡇⢸⣷⠶⠦⣤⣤⣤⣤⣤⣤⡤⠶⠞⢟⢫⡙⣷⢰⡇
⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⠀⣠⢦⣀⠀⠀⠀⠔⢦⡀⠀⠈⢄⠦⢿⡟
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣷⢸⡇⣎⠹⠎⣼⠀⠀⣉⢗⡆⢸⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢿⣼⡇⠑⠶⠓⠁⠀⠀⠠⡤⠂⢂⠤⢎⡑⢄⠀⢻⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⠊⣓⠊⡂⠀⠀⠀⠃⢑⠃⡊⠀⠸⡇
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣔⠚⠤⠇⢈⠄⠀⠀⠀⠀⠈⠀⠀⣠⡗
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡟⠿⢤⣄⣈⣉⣢⣤⣁⣀⣠⣤⣤⣤⣶⠞⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣏⠉⠉⠉⠉⣿⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠸⣷⣄⣀⣀⣀⣀⣀⣀⣀⣼⣄⣀⣤⣤⡤⠴⠶⢾⡃⠀⠀⢀⣿⠤⢤⣄⠀⣿⠀⣀⣤⡴⢯⣄
⠀⠀⠀⠀⠀⢀⡾⠋⠉⠈⠉⠉⠉⠛⢿⣭⠋⠁⠀⠀⠀⠀⠀⠈⢻⡄⣰⠟⠙⠂⠐⠊⢳⣿⠞⠛⠒⠒⠁⠈⣧
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⠀⠀⠀⠀⠀⠀⢀⣸⡇⣿⢤⣤⣤⣤⣤⠼⣿⠤⣤⠤⡤⠤⠴⢿⡇
⠀⠀⠀⠀⠀⢸⣟⡓⠒⠒⠒⠒⠒⠛⣉⣿⣍⣉⣉⣙⣿⣭⣭⣭⣽⠇⠻⠤⠤⠶⠶⠖⠒⠛⠒⠒⠲⣲⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣷⣶⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠈⠉⠉⠉⠁⠀⢹⣿⣿
⠀⡀⠄⡀⠀⠀⠀⠀⠈⣿⠃⠀⢤⠂⠠⠄⠀⠠⠀⠸⡿⠈
⠎⡔⠁⠠⠀⠀⣀⠈⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠁⠀⠄⠀⣰⣿⠟⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢁⣦⠠⢀⡀⢀⡠⡄⢴⡎⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢸⡏⠀⠀⠺⠇⠀⡀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠐⠂⠀⠀⠈⣿⠀⠀⣤⣶⣿⠆⠀⠀⠀⢸⣿⡟⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠃⠀⠀⠀⣿⠀⠀⠛⠉⠐⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠉⢻⠃⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠈⡁⠀⠀⠀⣿⠀⠀⠀⠀⡀⡠⠔⡀⠀⠐⠀⠀⠀⠀⠀⠀⠠⢀⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠈⠁⠀⠀⢸⣿⢀⠀⠀⠀⠁⠭⣒⠤⢀⠀⠙⠀⠄⠀⠄⠐⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⢳⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⣽⣬⣄⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣭⣭⣬⣭⣽⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣯⣭⣽⣿⣯⣤⣤⣤⣤
⣿⣿⣤⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣻⡿⢷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣴⣷⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄
⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣢⡾⣿⣆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠏⠃⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⠋
⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⡇⠀⡀⠀⠀⠀⠈⠀⠀⠐⡇⠀⠀⢀⠠
⠀⠀⡣⠃⢠⣤⡀⠄⢀⣀⣄⢀⣀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡀⠀⠀⠀⠀⠴⠋⣿⡅⠐⠀⠀⠁⠀⠀⠀⢈⠀⣀⠀⠀⡘⢣
⠀⠀⣥⠀⡈⠐⠒⠲⢤⣦⣿⣬⣭⣭⢭⡋⠉⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⢰⢳⢿
⠀⠀⠹⣦⠁⠀⠠⠀⣞⣹⣿⢁⡟⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡿⢿⡧⠤⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⡵⢮⣶⢾
⠀⠀⠀⠀⠀⢀⣻⣶⣿⣿⣿⣼⣷⣯⣿⡇⠀⣀⡀⠀⠀⠀⡆⠀⠈⠀⠀⢠⠀⠢⠀⠀⠀⢐⣠⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠺
⠀⠀⠀⠀⠀⠈⠻⠿⣿⢻⣿⣿⣿⣿⣿⠁⡢⣆⠠⠀⠀⣹⠈⠠⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡷⣾⣿⢹⠈⠃⣀⠄⠂⠙⠈⠐⠀⠀⠀⠀⢨⠀⠙⠃⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣷⣿⡿⡀⠀⠈⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⢣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠞⠛⠁⢽⡿⣏⡼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠒⢶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⡿⢇⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⢐⣦⠄⢻⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠞⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠺⠩⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⠺⠿⠿⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡷⠀⠀⠀⣒⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⡿
⠀⠀⠀⢀⣤⣀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⡆⠀⠀⢰⡆⠀⢸⣀⠀⠀⠰⠟⠛⠉
⠀⠀⠀⠘⠿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡁⠀⠀⠀⡠⠀⠀⠀⠀⢈⣈⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠆⠂⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡀⠀⠀⠀⠁⠀⠦⠀⠁⠀⠀⣀⣀⠆⠀⣇⠀⠀⠣⢀⡀⠔⠁⠀⡸⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣄⠀⠀⠁⡀⠀⠔⠀⠀⡰⠃⠀⠀⠀⠈⠑⠒⢠⠄⠀⠤⠒⠊⠁⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣷⣦
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⡄⢠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⡿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠘⠲⢲⣴⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢰⣷⣾⡆⠀⠀⠀⠀⠀⠀⠀⣠⣮⣗⡶⠦⠤⢶⣲⣷⣦⠀⡈⠀⠘⠁⠀⠉⣭⠉⠉⢹⣿⣿⣷
⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠉⢛⠛⢧⠀⢠⣦⡘⠿⠿⢃⣶⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⡀⠠⡈⢁⠀⠁⠀⠀⠀⠻⢿⡁
⠀⠀⣼⣿⣿⣿⠿⢻⡉⠀⠀⠙⠀⢸⢠⣿⣿⣿⣷⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠑⢤⣇⠋⠀⠀⠀⢀⡠⠒⠚
⠀⠀⢹⣿⠇⠀⠀⠈⠁⢀⣁⡄⢀⣼⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠙⠀⠂⠉⠉
⠀⠀⠀⠛⠶⣶⣄⣀⣀⡀⣡⢔⣫⣿⡟⠀⠛⢻⡿⣿⣿⣿⠈⢻⣿⣆⠀⠀⠀⣿⣿⣿⡀⠀⠛⠿⣿⡇⠀⠀⠀⠀⠐⡍⣆⢡⣼⠏⢂
⠀⠀⠀⠀⠀⠊⠁⠤⣆⠀⠀⠉⠙⠋⠀⢰⡶⠎⠉⠈⠛⣿⣶⣶⣿⣿⡆⠀⠀⣿⣿⣿⣿⣦⣄⡀⠀⠁⠀⠀⠀⠀⠀⠇⡬⠸⣿⡀⠌
⠀⠀⠀⠀⠀⠀⠘⣀⣼⠇⠀⠀⠂⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⡇⠀⠀⡜⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠂⠘⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴
⠀⠀⠀⠀⠀⠀⢈⣛⡶⠶⠶⠶⠶⠶⠾⠿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠈⠁⣀⣀⡀⠉
⠀⠀⠀⠀⠘⠁⠋⠀⠀⠀⠀⠈⠉⠀⠈⠀⠈⠉⠛⠃⠀⢠⠾⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠌⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⢲⣀⠀⢀⣶⡁⠀⠀⠀⠀⠀⡀⠀⠀⡈⠀⠀⠀⠀⣀⢀⠠
⠀⠀⠀⠀⠀⠀⠐⠀⣴⣚⣺⣚⣒⣗⣓⢓⣠⡤⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⢗⢶⢲⠖⡇⡗⡶⢺⢺⡒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠽⠭⠭⠯⠋⠋⠙⠋⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⠤⠤⠀⣀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠊⠒⠒⠒⠀⠒⠒⠒⠒⠂⠘⢆⠀⠀⡇⠀⠎⠀⡠⠒⠒⠀⠀⠐⠒⠢⢄⡀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠁⠀⠀⠞⠙⠂⠀⠀⠚⠙⠆⠀⠈⡆⢠⠃⠘⢀⠎⠀⠀⠀⢀⠜⢆⠀⠀⠀⠈⢢⠀⢠
⠀⠀⠀⠀⠀⠀⡎⠈⡇⠀⠠⠄⠊⠑⡄⠀⠀⠊⠒⢤⠀⠀⢣⠋⠙⢢⡄⠀⠀⢀⠴⣧⠀⠈⡷⢄⡀⠀⠀⡆⡘
⠀⠀⠀⠀⠀⠀⠱⠄⡇⠀⢀⠤⠠⠬⠄⠠⠤⠤⢤⡀⠀⠀⢨⢄⠜⠰⢳⠔⠺⢏⣈⣹⠁⠹⡉⢉⠿⠓⠒⡗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠥⠀⠤⠤⠴⠤⠤⠄⠠⠽⠆⢀⠎⠀⠀⠘⠲⡄⠀⠀⠀⣀⣀⣀⠈⠁⠀⠀⢀⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠦⣄⡀⠀⠀⠀⢀⣠⠤⢖⡁⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠷⠆⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⢀⠤⠖⠁⠀⠰⣜⠬⠍⠀⠭⢋⠜⠀⠀⠈⠑⠤⣀⠀⢀⠔⠊⢩⠉⠢⡤⢠⠖⠉⡉⠑⠢⣀
⠀⠀⠀⠀⠀⡔⠁⠀⠃⡠⠲⡀⠀⠉⡱⠲⣍⠁⠀⠀⢀⠤⡀⡇⠀⠳⡇⠀⠀⠈⡀⣀⠵⠮⣀⢀⠀⠀⠀⢨⠑⡄
⠀⠀⠀⠀⠎⠀⠀⢰⢊⡔⠓⢌⠢⢊⡔⠑⠢⡑⠤⢊⠥⠒⢌⡃⠀⠀⠘⡄⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⢸⠀⠈⡄
⠀⠀⠀⣜⠀⠀⠀⣼⢋⡔⡙⣌⠲⢊⡔⣑⢦⡈⠒⢁⣠⠒⣄⢹⡀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠐⠖⠲⠶⠶⠴⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⣷⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⡔⠒⢢⡀⠀⡎⠀⠀⠛⠀⢀⠞⠁⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⢳⡀⠀⢡⠀⡇⠀⠀⡆⢀⠎⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠱⡀⠀⠓⠇⠀⠘⠥⠊⠀⢀⠎⣀⠴⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⢀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⣤⡶⠿⠀⠀⠀⠀⠾⠿⠷⠶⢄⠀⠀⠀⠀⠀⠀⡇⢐⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣁⣈⠓⠒⠒⠒⠠⣄⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⣼⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⡔⠁⠠⠀⠀⠀⠀⠀⠈⠀⠀⢩⡤⢄⠀⢠⠁⠑⠤⠤⠴⢦⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⣈⡄⠀⠀⠀⠀⠀⠀⠀⣤⡀⢡⠄⠀⠀⠀⠀⠀⣀⠂⠀⠀⢸⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⡠⠒⠚⠃⡀⠀⠀⠀⠀⠀⠄⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⣈⠿⠇⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠐⠂⠐⠂⠀⠀⠀⠀⡠⠛⠀⠀⠀⠖⠉⡄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠞
⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠲⢦⠀⣠⠀⠀⠀⠠⠀⠐⠐⠀⠀⠂⠐⠂⠐⠊⠁⠘⠀⠀⠀⡆⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠞⠀⢰⠆⠠⠤⡀⠀⠀⠀⠀⢀⡜⠁
⠀⡜⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⠀⠡⡀⠈⠑⠒⠒⠒⠂⠁⠠⠤⠄⠉⠈⢀⣁⡀⠔⠊⠁⠀⢀⠾⡄⠀⠀⠈⠓⠤⣀⠴⠋
⠀⡇⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠸⣇⠀⠈⠑⠂⠠⣶⠒⠂⠠⠄⠢⠔⠉⠋⠀⠁⠀⠀⠀⣀⠔⠁⠀⢰
⠀⢇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠁⠀⠀⠀⢸
⠀⠘⡄⠀⠀⠀⠀⠀⠘⣄⠀⠀⢸⠀⠀⠀⠑⠂⢄⣀⣄⠀⠀⠀⠀⠀⠀⢀⣤⠔⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠸⡄⠀⠀⢣⠀⠀⠀⠀⠀⠈⠁⠈⠢⣀⠀⡀⠠⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣧⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡀⠀⠈⢲⡟⠳⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠸⣄
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡿⢀⣤⡀⠘⠀⠀⠀⠀⢀⠆⢀⠇⠀⠂⠀⠜⠁⡀⠀⠀⠀⠀⠢⡀⠠⠄⠀⠈⠓⢄
⠀⠀⠀⠸⣆⣀⣀⠀⠤⠔⠛⠉⢀⠄⢠⠀⠀⠀⠀⠂⠀⠌⠀⠀⠀⠀⠀⢀⠘⡀⠀⠀⠀⠀⠈⠳⢄⡉⡂⠀⠀⠳⡀
⠀⠀⢀⡔⠁⠀⠀⠀⠀⢀⠴⠚⠡⢤⡂⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠨⣀⡐⢡⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢱
⠀⠀⢈⠀⠀⠀⠀⠀⠀⠨⣀⠀⡀⠀⠘⡄⠀⠓⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⢀⠛⠢⣠⠄⢀⡀⠄⠉⠀⡙⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠦⠰⠦⠤⠠⠤⠠⠴⠆⠈⠁
⠀⠀⠘⠧⠔⠁⠀⠀⠀⠀⠀⠀⠳⣀⡇⠀⢃⣀⣀⣀⡀⢀⠠⠤⠄⢰⠖⠂⠙⢦⡀⠀⠂⡀⢀⠀⠤⠀⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⠀⠠⡀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠈⠣⡀⠁⠀⠀⠀⠀⡀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢸⡀⠀⣥⠀⠀⡄⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠰⠀⠀⠐⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⠀⠀⡀⣰⠄⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢼⠂⠘⠃⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠉⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠃⠀⠀⠀⡰⠆⠀⠀⠈⣆⠀⠀⠀⠀⡠⠛⠒⠠⣤⡀⠀⢀⣤⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⢤⣀⠰⠊⠀⠀⠀⠀⣴⠃⠉⠓⣄⢨⡀⠀⠀⠀⠀⠀⠀⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠘⢄⠀⠀⣀⡔⠋⠄⠀⠀⠀⢨⠀⠑⢄⠠⢄⠀⠀⠀⠀⠘⠢⠀⠀⢀⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⡄⠌⠡⠞⠉⠀⠉⠀⠀⢀⠤⠃⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠂⠈⢣⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⡀⢀⡼⠤⠤⠤⠄⠀⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⢀
⠀⠀⣠⣶⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣦⡀
⢠⣾⣿⡿⠋⠉⣽⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡍⠉⠻⣿⣿⣆
⣾⣿⡟⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠘⣿⣿
⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢀⣿⣿
⠸⣿⣿⡄⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣼⣿⡿
⠀⠹⣿⣿⡄⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣼⣿⡿⠁
⠀⠀⢻⣿⣷⢸⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⢰⣿⣿⠃
⠀⠀⢈⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⣏
⠀⣰⣿⣿⠟⢻⣿⣿⠀⠀⠀⣿⣿⣏⠀⢈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⠛⢿⣿⣧
⠀⣿⣿⡏⠀⢸⣿⣿⠀⠀⠀⠹⣿⣿⣷⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠃⠀⠀⢸⣿⣿⠀⠈⣿⣿⡇
⠀⣿⣿⣧⠀⢸⣿⣿⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⢸⣿⣿⠀⢠⣿⣿⠇
⠀⠘⢿⣿⣷⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⠏
⠀⠀⠀⠉⠛⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣦⣤⣄⣀⣀⣴⣿⣿⠿⠛⠛⠛⠻⠛⠛⠛⠻⣿⣿⣷⣄⣀⣀⣤⣤⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⡝⠛⠿⠿⢿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⠿⠿⠟⠋⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠘⢿⣿⣷⣦⣄⣀⣀⣤⣾⣿⣿⠟⠀⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣆⠀⠀⠀⠀⢀⣴⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣶⣶⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠑⢄
⠀⠀⠀⠀⣀⠤⠔⠉⠁⠉⠀⠉⠉⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⣀⣸⡀
⠀⠀⠀⡔⠁⠀⠀⠀⠀⡄⠀⢀⡰⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⡇⠀⠀⠀⠉⠓⢄
⠀⠀⢠⠃⠀⠐⢢⠄⠒⠉⠀⠙⠤⡜⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠆
⠀⠀⢸⠀⠀⡔⠁⢠⠤⠀⠀⠰⠦⠈⢆⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠚⠉⠢⣀⣀
⠀⠀⠈⠙⣤⠃⠀⠀⣤⠀⠀⣤⠀⠀⠈⣄⠇⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠦⢴⣋⢉⡡⢄⢀⡠⢍⢀⣈⡦⠤⢦
⠀⠀⠈⢢⡁⠀⠀⠠⠀⠊⠙⢀⡤⠀⠀⢘⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⠂⠀⠈⡢⠼⠉⠆⠘⠁⠀⠑⢇⡸
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠈⠀⠒⠀⠒⠀⠀⠀⢀⠏
⠀⠀⢀⢴⠞⠱⠦⠤⢄⣀⢀⡠⠤⠔⢚⠻⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡵⢄⣀⠀⠀⠀⠀⠀⣀⡠⢮⣀
⠀⢀⠃⠇⠀⠀⢣⣀⠔⠙⠆⠙⠦⡠⠃⠀⠑⡈⠳⡠⣔⠊⠉⣩⡶⠊⠁⠉⠁⠉⠲⢪⡉⠉⢒⡤⠀⠀⣠⠊⠀⡜⠀⠀⠋⠣⡉⠀⣩⠏⢱⠀⠀⢣⠙⢆
⠀⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⡌⠉⠈⢩⠁⠀⠀⡰⡄⠀⠀⢸⠉⠉⠉⠀⠀⡏⡇⠀⢀⡁⠀⠀⠑⠐⠚⠛⠓⠒⠃⠀⠀⠈⡄⠀⢣
⠀⠃⠘⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠠⠆⠑⡤⠼⢄⠀⡜⡶⠒⢛⡀⢈⠓⠲⡜⡆⠀⠀⢀⠔⢑⠬⣂⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⢎
⢸⠀⢠⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠀⠀⢱⣈⣧⠀⠘⠇⠀⠀⠬⠭⠀⠀⡜⠁⠀⣴⣁⡔⠁⠀⠀⠀⡧⠤⠤⠀⢀⣀⣀⡀⠀⠠⠤⠤⢴⠓⠒⠋
⣘⠀⣺⠸⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠃⠀⠀⠈⠿⢏⣑⠖⢻⠒⢤⠄⢤⠔⠺⠓⠒⣆⡹⠿⠃⠀⠀⠀⠀⢱⠠⠤⠀⣀⣀⣀⣀⡀⠠⠤⠔⢾⡠⠅⣆
⣇⣀⡼⠀⠘⡉⠐⠒⠂⠀⠀⠐⠒⠂⢉⠁⠀⠀⠀⠀⠀⠀⠈⠑⡇⠀⠀⢹⠃⠀⠀⠷⠚⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣀⣼
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠨⠄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠑⠒⠒⠒⠒⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢡⠀⠀⢠⡄⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠠⣊⠀⠀⠀⠀⠀⠀⠀⢈⡦⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡠⠦⠀⠼⡧⠀⠼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⢶⢶⡶⡶⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣓⡒⢒⡒⣒⠒⡂⢐⢒⡒⣲⠇⠀⠀⢸
⣀⣀⣀⣀⣀⣸⣷⣶⣶⣏⣹⣶⣶⣾⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣤⣭⣜⣧⣭⣤⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣷⣿⣿⣛⣿⣿⣿⣟⣀⣀⣀⣀⣈
⣿⣿⡏⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠆⠀⠁⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣁⡠⠀⠀⠊⠐⠀⡄⠁⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠰⠆⠈⠒⠒⠒⠒⠒⠒⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡀⠀⣀⣸⡿⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⢿⠾⠟⠻⠁⠀⠀⡀⡄⠀⢀⠂⠀⠀⠀⠀⠀⠸⠂⠞⠀⠀⠈⢆⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠒⠀⠀⠈⠘⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠓⠄⠀⠀⠀⣄⠀⠐⠀⠀⠀⠀⠁⢀⠀⠀⠠⡄⠀⠄⠀⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠂⠀⠀⠀⠁⡀⡀⠀⠈⠀⠀⠀⠀⠠⢀⡔⠉⠀⠐⠀⠔⠚⠃⠀⢠
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠸⠂⠀⢘⠆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠀⠀⠀⠀⠈⢢⠉⠀⠀⠀⠀⠀⠀⢁
⢀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣆⠀⢈⠄⠀⠀⠀⠀⠀⠀⠇⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠐⠀⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀⠈⠑⠀⠄⠀⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠈⠃
⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠂⠀⠀⠒⠈⠉⠁⠢⠀⢄
⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠴⢄⣀⡀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠠⢤⣬⠀⠀⠀⢀⣑⠀⠤⣀⡀⠀⠀⢸
⠀⠀⠀⢀⠔⠦⢀⣖⠊⠁⠈⡀⡀⠀⠀⡀⡈⠁⠀⠀⢰⡀⡤⠼⡀
⠀⠀⠀⠸⠀⠰⡠⠁⠉⠐⢊⠐⢨⠒⢺⠐⢀⠖⠂⠉⠈⣀⠃⠀⠇
⠀⠀⠀⠀⠁⢠⠀⠀⠀⢀⠠⠉⠁⠉⠁⠈⠁⢄⠀⠀⠀⠀⡧⠊
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠑⠂⠐⠈⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃
⠀⠀⠀⢀⠄⠂⠉⠃⠈⡒⢦⠀⠄⣀⣀⠠⢠⠴⠐⠉⠀⠈⠁⠢⣀
⠀⠀⡰⠁⠀⠀⠀⡀⠀⠁⠀⠣⣄⠣⢀⡤⠂⠀⡇⠀⠐⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠤⣀
⠀⢠⠁⠀⠀⠀⠀⡇⠀⠑⠠⠀⠀⣉⡁⠀⠀⠒⠀⠀⠠⠀⠀⠀⠀⠀⢡⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠑⡀
⡎⠹⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢈⠉⡆⡇⠀⢀⠖⠄⡀⣄⠀⠀⠀⠰
⠑⣤⠑⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠑⡦⠀⠐⣡⠔⢁⢇⠐⠁⠐⠂⠈⠁⠛⠂⠄⡰⢄
⠀⠄⠁⠉⠪⠧⣤⣄⣀⣀⠖⠲⠤⡀⠀⡠⠤⠲⢄⣀⣀⣤⣴⠭⠂⠉⠩⠀⠸⣸⠁⠀⠀⠒⠠⠐⠀⠀⠀⢃⡀
⠀⠰⡀⠀⠀⠀⠀⣀⣀⢀⡀⢀⠰⠁⠀⠡⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⡘⠢⢄⡈⡄⠀⠀⠳⠶⠖⠁⠀⢀⠏
⠀⠀⠀⠈⠉⡙⠁⠀⠉⠉⠉⠑⠒⠒⠒⠒⠈⠉⠉⠉⠀⠀⢙⠉⠉⠁⠈⢲⠃⠀⡙⠂⢠⡄⣀⢠⡄⠐⠫⡀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠜⠀⠀⠀⠑⠀⠊⠀⠀⠀⢩⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⠑⠄⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠄⡀⢁
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠣⡄⠀⣀⣀⣀⡀⠠⡔⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢒⡀⡒⣖⣰⠂
⠀⠀⠀⠀⠀⠀⠣⠲⣀⠠⠲⠀⠴⣀⣠⢄⠠⠄⠠⢤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠦⢼⠤⠼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠒⠂⠉⠁⠈⠉⠐⠒⠂⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣀⣠⣤⠾⢦⣤⣀⣼⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠒⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣜⡐⠂⣣⡈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡏⠐⠀⡀⠒⠈⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢚⠛⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠃⢙⡫⠭⣻⠐⡖⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡼⠀⠈⠀⠀⠀⠀⢣⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢘⠓⠢⠤⠤⠤⠤⠤⠤⠔⠒⡃⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢰⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢺⠁⣡⠸⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣼⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⠊⠙⡗⢳⠀⡀⠀⠀⠀⢸⠹⠀⠀⠀⠀⢸⡘⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⡃⢸⠀⡇⠀⠀⠀⢸⠀⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⢸⠀⠈⡄⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢡⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠀⠀⢸⠀⢠⠁⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠀⠀⠘⠀⢸⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⡀⠀⠀⡇⠀⢸⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣀⣠⣧⣤⣤⡇⠀⣦⣤⣤⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡧⠀⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠈⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡡⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢌⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣠⣶⣶⣖⠀⠀⢰⣶⣶⣄⡀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡧⢾⠁⢰⣶⠈⡟⠑⡇⢴⡦⠀⠱⢴⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡏⡜⠲⠶⠴⢾⡉⠉⣹⠶⠶⠴⠋⡜⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠓⠰⠧⢀⣀⣀⠅⣀⠀⠨⠗⠒⠁⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢹⣦⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⡴⡎⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢸⠈⠦⢿⡒⢤⡤⠤⡤⠖⡻⠤⠁⡇⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡸⠀⠀⠀⠑⢄⡙⣞⣁⠜⠁⠀⠀⢃⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢹⣸⠁⠀⠀⠀⠀⠈⢆⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀⡠⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⠚⠧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠓⠒⠒⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢠⣀⣹⠉⠉⠛⠒⠒⠒⠒⡶⠒⠒⠒⠛⠋⠉⣭⣀⡄⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠞⠉⢹⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡟⠉⠓⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣸⠉⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⢀⡏⠀⢻⡀⠀⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠉⢹⠃⠀⠸⡏⠉⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⠭⢙⠻⠉⣿⠀⠀⠀⡟⠙⠟⡩⠭⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⢄⡠⠤⢗⣢⠟⠀⠀⠀⠿⣖⡾⠤⢤⡤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉