Difficulty Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⣣⣀⣀⣀⠀⣀⠔⠊⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠉⠙⡂⠈⠉⠀⠑⠢⢔⡻⢶⣊⡉⡉⠐⢢⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⢒⠧⣀⠀⠀⡔⠙⠶⣤⠤⢤⠖⠋⠉⠀⠙⢀⡸⠀⠨⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⢊⡀⠈⠳⣀⠄⠀⡠⢸⡞⠑⠒⠤⢄⣠⠂⠉⠀⢁⡾⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠔⠡⡃⠈⡆⠀⢸⢇⠜⢠⠃⠘⡀⢀⡠⠔⠉⣱⠒⡲⠯⢤⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠙⡄⠀⡩⠖⠓⠤⣼⣸⠑⠶⢤⠀⣵⠁⠀⠐⢹⠝⠛⠄⠀⢨⠀⠈⣱
⠀⠀⠀⠀⢰⣶⡦⠤⠤⠤⠤⠤⢼⠤⠤⣤⠾⢯⠤⠤⣤⠮⠤⠟⡄⢠⠀⣿⢸⠀⠀⡸⢿⠤⠤⣼⠤⢬⢷⡭⠤⠤⠤⠤⠤⠤⠤⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠣⠀⠈⣦⠴⠥⠤⡀⣠⠜⣎⢻⡇⡾⠀⠀⡇⢸⣀⡴⠃⠀⡌⢀⠕⡖⠢⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠴⢒⢽⣀⣀⡠⢾⠁⠀⢈⡟⢿⣧⠤⠤⠷⡊⢇⠀⠀⠀⣷⠃⠀⢸⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢸⠀⠀⠀⢀⡉⠶⠮⠒⠁⣻⣄⠀⡖⠙⢝⡗⠄⠠⡇⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢄⠀⡠⠋⠀⠀⡠⠐⡾⠁⠈⠑⢧⡀⠀⡗⠒⠊⠀⢀⠞⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠀⠀⢠⢊⡠⠊⠀⡠⠔⠒⠙⢏⠉⢳⠲⡖⠊⠁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⢋⠁⠀⡠⠊⠀⠀⠀⠀⠈⠢⢄⣣⠼⠀⠀⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣤⠀⠠⠄⡠⡀⠤⠀⢠⡄⠀⠀⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣛⡱⠄⠀⣣⡀⢶⠖⠈⠰⠈⠐⢶⠆⣠⣃⠀⠴⡙⡳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⢸⠶⠀⣋⡑⠴⡴⠞⠒⠒⢶⡤⠄⣉⣃⠰⢾⡊⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠈⢾⠷⡠⠴⣂⠁⡀⣄⠸⠦⡹⡁⠼⣠⡄⡀⢉⡲⠠⡰⢾⠎⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣉⠀⠚⠃⡰⠊⢠⡐⠈⠀⢨⠿⠂⠀⢻⠃⠂⠲⠯⠀⠈⠀⣠⠈⠲⡐⠛⠂⢈⣁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠈⠀⠢⠀⠟⠋⢦⠋⡻⠞⠁⠀⡰⠀⠀⢈⠀⠀⠐⡀⠀⠙⠾⡋⢳⠎⠋⠃⠀⠂⠘⠡⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡆⢀⡆⠀⠓⣴⣄⠀⠈⠉⠍⡃⠤⣀⡼⣆⡠⠄⡋⠉⠉⠀⢀⣴⡔⠃⠀⣇⠀⣶⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⡄⡚⠉⣙⠁⡄⠈⠛⠠⢇⣠⡇⠄⡀⢡⡉⡀⠀⣦⣀⠧⠘⠋⠀⡄⢙⡩⠑⡂⣤⡄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⣉⠣⣆⠨⣟⣛⠀⢀⢀⢁⡨⠶⣐⢬⣄⣘⣀⣬⢴⡐⠮⣀⢁⢠⠀⢘⣛⠉⡄⠧⣋⡁⠀⡀⡆
⠀⠀⠀⠀⠀⠀⠐⠐⡗⡯⠙⠧⢘⡯⢚⡅⣸⠀⢂⢿⠨⡅⠫⠉⣤⡼⠿⡿⣀⡹⠥⢐⡤⢸⢋⡀⢸⠀⢑⡿⣟⠠⠟⠯⡖⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⣶⣛⡃⠰⡽⠾⡄⠋⠈⢂⣹⣉⡝⠙⠅⢰⠁⠼⠙⣍⣹⣀⠂⠈⠁⡼⠻⡥⠀⢋⣳⡆⣀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡥⢍⡤⢻⡤⠜⠂⡤⡠⠶⠀⠃⠀⠇⠀⡠⡎⡤⡀⠀⠇⠀⠂⠰⠶⡤⡄⠚⠦⣴⠃⣌⠅⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⢀⠣⠀⠛⢘⠑⢄⠀⠠⠴⠇⠀⠀⠬⠅⠀⠀⡷⠠⠄⢀⠔⢹⠘⠃⠠⢓⠀⠙⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢀⡉⠀⠓⣖⠏⢵⠾⠑⠂⢄⠈⠀⠀⢰⠄⠀⠈⢀⠀⠒⠙⢶⠍⢑⡖⠃⠀⣃⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠈⢻⠖⣨⠒⢄⢐⠰⠐⠀⣾⠃⢀⠞⢆⠀⢻⡇⠔⠰⢀⢀⡔⢨⡐⢾⠋⠐⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢠⠛⠋⢡⠘⠣⠄⢄⠁⣼⠁⠘⠂⢹⡌⡁⠀⠤⠛⢠⡉⠛⠣⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣚⠉⠠⢷⡆⢉⡉⠛⢐⠘⠋⣉⠁⣶⠧⠈⢙⡧⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠮⠁⠀⠃⣤⠉⠓⠠⣈⡠⠐⠉⢡⡄⠃⠀⠩⠞⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠘⣦⡀⢀⣠⣴⠾⠋⠉⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣳⣆⠀⠙⠻⣉⠀⠀⢰⡖⠛⠒⠲⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠙⠢⡀⠀⠈⠳⣄⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠀⢠⡇⠀⠀⠀⠀⡘⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡄⢺⡄⠀⠀⠁⠄⠀⠀⠀⠀⣤⢺⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⣽⣾⣧⣤⣠⣌⠀⠀⠀⠀⢸⠿⣀⡀⠀⠀⠀⠀⠠⠧⠼⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣯⣇⡽⢏⡡⢄⠉⢩⣇⠀⠀⢠⡟⠀⠀⠙⠂⠀⠰⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠉⠂⣀⠳⣋⠾⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠇⠀⠀⣿⠀⠀⣄⡉⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣄⡄⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⣀⡀⠰⢒⣰⡏⠀⢰⣿⠁⠀⠈⢦⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠉⠉⠙⠄⠀⠈⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏⠉⠁⠠⠿⠟⠀⢠⡄⠈⢧⡦⢀⣤⣶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠼⢉⣧⠁⠀⠀⢀
⠀⠀⠀⠀⠀⢠⡄⠀⠀⡰⡞⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⡅⢀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢹⣙⣤⣤⠴⠋
⠀⠀⠀⠀⣠⠼⠿⠀⠚⠇⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⢸⡙⣏⢌⣽⡀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⢸⣿⡗⠓⢺⠁
⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣤⣿⣿⣦⡇⠀⠀⠀⡰⠟⠲⠋⠀⠀⠀⠀⠀⠻⠀⡇⠀⢠⡆
⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀⢧⣠⠴⠚⢋⣩⣿⣿⣿⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠃⠀⣼⠁
⠀⠀⠀⠈⠛⠓⠒⠒⠛⠳⠤⠤⠤⠤⠤⠤⠜⢹⣻⡧⠤⠤⠀⠈⠛⢛⡿⡾⠋⠉⠓⠦⢄⡀⢀⣀⣠⡥⠾⠷⣦⢰⡦⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢤⣀⠀⠈⠉⠉⠉⠉⣉⡽⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠒⠒⠚⠛⠢⡀⢀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠠⠴⠖⢛⢙⡒⠒⢤⣀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⡼⢠⠖⢈⡀⠤⠄⡉⠲⣈⠧⠀⠀⠀⠀⠀⠡⠞⠀⠀⠀⠀⢠⡈⠁⡀⣔⣀⡠⡄⠀⠀⠠⡑⣄⠠⠁
⣸⢰⠁⡰⠥⢈⣩⡉⣾⡆⠘⢼⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⢀⠁⠀⠈⢡⠄⠐⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠠
⣷⢸⠀⢇⠡⡘⠻⣦⡉⠁⠾⣿⠀⠀⠀⠀⠀⡴⠟⢀⠀⡀⠀⢀⠠⡁⢾⣀⠄⣐⢨⣬⡀⠀⠊⠀⠀⠀⠀⣀⣀
⠘⣆⠣⣈⠒⠬⠥⠖⢋⡼⣲⠄⠀⠀⠀⠀⠂⠂⠀⠡⣴⠀⠀⠀⠈⠁⠀⠀⠘⠎⠂⠈⠁⠀⠠⠤⠄⠀⠀⠁⠒⠉⠁⠀⠑
⠰⠿⣵⣮⣉⣀⣂⠐⢝⣮⠃⠀⠀⠀⠀⠀⠙⠛⠤⡀⠈⢀⢀⠀⠀⠀⠠⢀⡀⠀⠀⠀⢠
⠀⠀⢸⡋⠿⠟⠻⠭⠍⠙⠧⠖⠀⠀⠀⠀⠀⠈⠂⠀⠠⣤⠀⠀⠀⠙⠇⠶⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠼⠂⠀⡇⠀⡰⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠑⠄⠀⠄⠂⡁⠔⠀⠀⠲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠒⠇⠀⠀⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠴⠶⠶⠶⠶⠦⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠖⠒⠒⠲⠦⣄⡀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠰⠯⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠘⠓⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⣧⠖⠛⠛⣆⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⣼⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠁⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢀⡇⠀⠀⠀⢀⡼⠂⠀⠀⠀⠀⠀⠀⣸⠃⠀⠚⠂⠈⠙⠂⢀⡀⠀⠤⠞⠁⠀⢴⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⢸⡇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠙⠦⠶⡦⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡒⠒⠆⠀⠀⠀⠸⡄⠀⠀⠀⠀⣻⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⡇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣤⣤⠀⠀⠀⣄⠀⠀⣰⠋⠈⠉⠓⠲⢤⣄⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢻⠀⠀⠀⠀⠈⠳⡆⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⡟
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠙⠶⠤⣄⠀⠀⢠⡏⠁⠀⢀⣤⣄⡀⢈⣧⣤⠤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣼⠃
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⢠⡞⠀⢨⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⣰⠃⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢸⡄⠐⠋⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⢸⡇⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣞⠁⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠚⠻⠶⠶⠶⠶⠖⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣞⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣴⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣶⣆⡀⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⣿⣷⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣯⡻⠇⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠸⣿⣾⣿⣻⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣥⣤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢀⣈⡿⠟⠀⠀⠉⠿⢿⡿⢿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢿⡿⢯⣛⣿⡟⠻⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⡟⣦⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢸⣿⣯⣵⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣽⣿⡃⠀⠉⠓⠒⠒⠒⡖⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⣠⣿⣿⠟⠛⢿⣿⣧⡀⠀⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣻⣿⣿⠿⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠛⢻⢇⡀⠻⣿⡇⠀⠀⠀⣽⣿⠃⣀⡿⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠰⡆⠀⢿⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠃⠀⠙⠟⠀⠀⠘⠟⠁⠸⠧⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⠀⠛⠛⠂⠀⠐⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣦⠀⠀⠀⠀⢀⣴⣞⣛⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣯⣿⡇⠀⠀⠀⢸⣿⣷⣿⣧⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⢾⣿⣿⠀⠉⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⡾⠟⠀⠀⠀⠀⠈⠻⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⡇⣠⣬⡛⠁⠠⠀⠈⣿⣿⣶⣂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡇⢬⡽⠃⠠⠀⠀⠀⢻⣿⣏⣻⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣿⣟⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣞⣷⣹⣖⣦⣄⠀⠀⠐⣿⣿⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⣾⣿⣿⣿⣿⣿⡷⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⢗⣽⣹⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡿⢿⣿⢿⢿⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠶⣿⣽⠷⠋⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡾⠿⠿⢿⡿⠿⢾⡵⠿⢿⢶⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠈⠁⠒⠤⡀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡐⠀⢃⠈⢄⠀⠀⢀⠀⠀⠀⠀⠀⠢⠈⡆⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢧⠀⢄⡙⠀⠁⠀⢊⡑⠀⠒⢄⠀⣀⠀⡇⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠇⠀⠀⠀⠈⠇⠀⠀⠨⠔⡁⠙⡄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⢀⡀⢶⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡠⠜⠁
⠀⠀⠀⠀⠀⣤⠤⠄⡀⠀⠤⠀⠒⠂⠤⣀⠀⢠⠁⡆⠈⠀⠀⠀⠀⠀⠑⠤⣄⣐⣒⢀⠀⠀⡀⠠⢎⠑⢄
⠀⠀⠀⠀⢰⣈⠀⠠⢍⡕⠀⠀⠀⣀⠀⠀⢧⠈⠀⠁⢀⠀⠀⠀⠀⡔⠁⠀⠠⡉⠉⢓⡦⠼⠒⠦⠞⠀⢀⣀⣄⠀⠀⠀⠀⠀⢀⡠⠤⠒⡄
⠀⠀⠀⠀⠀⠐⠚⠙⠊⢸⠀⡠⠂⠃⠈⢠⡎⢃⡲⠂⠈⡄⠀⠀⡜⠀⠀⠀⠤⠘⠒⠫⣄⠀⢀⠀⢀⡆⢀⠀⢈⡳⠤⠄⠒⠉⠁⠀⠀⠀⡧⠒⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⢆⠀⢨⠉⠀⠇⠀⠀⠀⠁⠀⠁⠀⠀⠠⡀⠀⡇⠀⠀⠀⠀⠀⢠⠎⠀⠀⣄⣇⡨⠷⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠬⢣⣤⠵⠄⠀⢀⠖⠢⣄⠞⢀⡀⠀⠀⠈⠙⠳⣄⡀⠀⠀⠀⢘⣤⠦⠛⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡉⠐⠒⡲⡂⢉⡁⠀⠀⠈⠁⠁⠆⡁⢧⠀⠉⠉⠉⡉⠩⠀⡂⠥⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠬⠅⢈⠠⣄⠀⠀⠰⠤⢀⣀⡀⠷⣬⠷⠄⠐⠀⠅⢀⡠⣅⢠⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡘⢄⡀⠀⠀⢀⡀⢀⡇⠀⠀⢀⠀⠄⠂⠀⠀⠀⠀⠈⠢⣘⠦⣈⡫⡶⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⢅⠀⠀⠉⠉⠀⣉⠥⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡳⣬⡷⣽⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠍⢁⠔⠒⢺⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣙⢹⠀⢀⠀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⢃⠀⠀⠀⠘⠐⠠⡀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠓⠒⠂⠛⠂⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣾⣿⣿⣿⣿⡿⠿⣿⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣿⣿⣿⢿⣿⡿⢻⣿⡇⠒⠈⣿⣿⠿⠿⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⡎⣭⢡⣿⣿⡇⠿⠗⣸⣿⣿⠇⣼⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣛⡛⢿⣿⣿⣀⣾⣿⣿⣿⣶⣿⣿⣿⡧⣾⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢣⣾⣿⣿⡿⠿⣿⣿⣿⠟⣋⣙⣻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠈⠙⢻⣿⣗⣿⣿⣿⣿⣷⠗⣸⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣷⣬⣥⣾⣿⣿⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣶⣿⣷⣦⡉⠉⠁⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠀⠀⠀⠻⠿⠋⠉⠛⠿⠏⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣿⣿⣿⣿⣿⡿⢆⡀⠀⠀⠀⠀⢠⣿⣿⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⣿⣯⡡⣿⣿⣦⣀⣀⣀⣛⡻⢃⡞
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⣿⣿⣿⣿⣿⠇⠈⠻⠿⠿⠿⠿⠿⠇⠼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡛⠿⠿⢿⡿⠿⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣶⣶⣶⣆⣀⣀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⣶⡄⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠘⣷⡀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⢰⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠶⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣛⣻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⢀⣴⠞⢛⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡙⢷⡀
⠀⠀⠀⠺⠁⠐⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠐⣧⣀⠀⢳⣄⣀⣀⣠⣆⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣰⠏⠉⠛⠋⠉⠉⠉⠉⠙⠳⠶⣦⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠻⣆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⡟⠀⠀⢠⡏⠀⠀⠀⠀⠈⠙⠛⠛⠃⠀⢹⡆⣠⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⢰⡇⠀⡴⠋⠀⠀⠀⠀⠀⠀⠶⠖⠀⠀⠀⠘⠿⢋⡤⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣇⠀⠀⠚⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣾⠦⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣧⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⣁⡴⠋⠻⢶⣄⣀⠀⠀⠀⣀⣠⣴⡟⠁
⠀⠀⠀⠀⠀⠀⢀⣼⠋⣠⠾⠋⠀⠀⠀⠀⠈⢹⡟⠛⠛⠋⠁⠘⣧⣀⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⣿⢀⣠⣀⣀⣠⣤⠶⠛⢷⣄⣀⣀⣠⡴⠏⠉⠙⠳⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡿⠋⣉⣽⡙⣯⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⢻⡄⠀⡴⠞⠋⣉⣇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣴⠞⠋⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣀⣤⡴⠿⠶⠶⠒⠶⠻⠶⠶⢤⣤⣀⡀⠀⠀⠀⠀⣠⣤⠶⠖⠛⠛⠛⠛⠛⠛⠳⠶⢶⣯⣀
⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤
⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⡇
⢀⣼⣷⣿⣿⣿⠿⠿⠿⠿⠿⣿⡿⠿⣿⣿⣿⣿⣶⣦⣄⣠⣶⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣶⣿⣄
⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⢿⡛⠶⣬⡛⠶⡍⠛⠿⣿⣿⡿⠟⠉⣁⣤⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣤⣤⠶⠤⣤⣤⣤⡴⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠀⠒⠒⠒⠒⠂⠀⠒⠒⠒⠒⠒⠠⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⡀⠈⠉⠐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢠⢾⣷⣤⣼⣿⣽⡯⣿⣤⡀⠀⠀⠀⢠⣿⠟⠉⠀⠀⠈⠙⢿⣷⣄⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢸⡾⡼⣅⡏⢻⣿⢻⣾⣩⠻⡄⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠋⣿⣿⢹⣿⣛⣟⠳⣿⡧⣴⠃⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⣠⣿⡿⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢆⠠⠉⠯⣒⣹⣤⡷⠛⡟⠓⠠⠤⠄⠌⠷⣿⣦⣤⣤⣤⣾⣿⡿⠁⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠉⠛⠒⠚⠉⠓⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⠉⣀⡀⠤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠤⠤⠤⡴⢀⡴⠤⠤⠤⠄⠀⠐⠒⠐⠂⠙⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠃⠀⠀⠀⠀⠉⠢⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⢛⡋⠉⠉⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠈⠁⡢⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠠⠶⡄⠀⠀⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⠒⢄⡀⠀⠛⠠⢄⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠸⣿⡿⠛⠻⣿⡇⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎⠀⠈⠀⡀⡱⠀⠀⠀⢀⠇⠀⠀⠀⠀⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣇⡘⢻⡋⠃⠀⠀⠀⠀⠀⠀⠙⠂⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡴⠲⡖⢄⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⠔⢊⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡫⠁⠀⡸⢀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⡄⠠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠑⢲⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠃⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⡟⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡩⠒⠢⠄⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⢀⣠⣤⠒⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⡀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠚⢀⣠⣴⣆⣀⣤⣤⡀⠀⠀⣶⡿⢦⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡽⠁⠀⠀⢠⡆⠀⠴⠂⢀⠀⡀⢀⡄⠀⠈⠓
⠀⠀⠀⢀⠀⠀⠀⠺⣿⣿⣿⡿⢋⣭⣶⣿⡏⠀⠉⠐⣦⣷⠠⠀⠈⠀⠘⠀⠀⠀⠀⠈⠁⠂⠀⠀⠈⠀⠀⢠⣄⠘⠀⢀⠠⠆
⠀⠀⠀⠈⣓⢲⣥⡄⣿⡇⠉⣧⠈⣉⣛⣥⣴⣶⣄⡀⠉⠀⣀⣶⣤⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠉⠨⠄⠀⣉⠀⠁⠈⠐⣟⣁⣿⣿⣿⣿⢀⣤⢰⣿⣡⣜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⠃
⠀⠀⠀⢠⡄⢀⣾⣿⠿⠀⠀⠀⠀⠙⠻⣿⡿⠟⢹⣾⣿⠿⠋⢠⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠈⠀⠀⠀⡌
⠀⠀⠀⠄⡁⠛⢿⣿⡄⠀⠀⢠⡤⡀⠀⢻⡷⠄⠀⠀⠀⢠⠾⠁⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⣤⣤⠄⢊⠁⢀⡁⢸⠂⠀⠀⠀⠐⠀⡄
⠀⠀⠀⠈⠋⠓⠀⢼⣷⢠⢾⣧⣼⠟⠀⢀⡀⠀⠀⠀⠠⠌⠀⠤⠀⠈⠀⠀⠀⠀⠀⠴⠄⠛⠀⠀⠀⠉⠂⠘⠃⠀⠀⠀⠂⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⡘⠟⠩⣇⠀⠀⢁⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠘⠁
⠀⠀⠀⠀⠀⢀⣴⣦⣶⣯⣅⡄⠀⠀⠠⠆⠀⠀⠀⠀⠀⢀⢈⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⠲⠂⠊⠀⠀⠂⠈⠁
⠀⡄⠀⠀⠀⠀⠹⡿⣿⣿⠿⢧⣄⠠⣤⣾⣾⡆⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠋⠀⣠⡄⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢹⣿⡟⣿⣯⣅⡀⠋⡰⠾⣷⣷⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠀⠀⣶⣶⣿⣿⣿⣿⠊⠧⢈⣄⠉⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠲⠄⠆⣁⡀⡀⠈⢀⡤
⠀⠀⠀⠀⠀⠀⣀⠰⡂⠀⢰⣦⠐⢺⣿⣿⣿⣿⣟⣿⣷⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⡂⠂⠀⠀⠙⠀⠚⠷⢮⡄⠘⠙⠧⠀⡀
⠀⠀⠀⠀⠐⣲⣿⣦⡔⠆⠈⠃⠀⠀⠘⢙⠻⠇⠸⠍⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠢⠀⠀⡀
⠀⠀⠀⠰⠲⠛⠉⣿⠆⠛⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠒⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠠⠍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣉⣤⠒⠒⢲⠒⠠⢌⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠋⠐⣎⢦⠀⠘⡀⠀⣠⠜⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⠃⣼⣁⠀⡀⠈⣆⠻⠀⡦⢊⠴⠊⠀⠱⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⠂⠠⢤⣄⡉⠀⡀⠀⡋⠒⠒⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣁⠀⠀⣠⣞⡵⠞⣳⡌⢫⠉⠁⠀⢐⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠻⠙⢫⠋⠁⠈⠀⡇⢿⠉⢦⡱⡀⢀⠎⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠉⠀⡀⠀⢀⡸⠀⢀⡱⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠈⠈⣿⡉⢤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⠀⠀⠀⢀⣀⡀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠁⠀⠉⠉⠉⠁⠀⡀⠄⠂⠙⡗⠲⣤⣀⣀⣀⣤⠔⠚⠉⠀⠀⠀⠀⠀⠙⢆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⢀⠔⠈⠁⠀⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⡈⠀⠠⠂⠁⠑⢄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⣇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡔⢃⠀⠀⣀⣀⣀⣀⠀⢦⠀⠀⠀⠀⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠈⢠⡡⣀⠤⠒⠉⠀⡀⠀⠀⡏⠉⠉⠉⠀⠀⢱⠀⠀⠀⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⢈⠀⠀⢀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⡠⠉⠀⠉⠓⠴⢦⡂⠀⠀⠀⠀⠀⡕⠸⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡄⠀⠀⠀⠀⡰⠁⠀⢠⠧⢄⡀⠀⢩⠁⠒⠒⠂⠐⠈⠁⠀⠀⡶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣁⠀⠰⠃⠀⠀⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡀⠀⠀⠀⠈⠢⢀⡀⠀⢀⠄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣄⠙⠶⢤⣀⣀⣀⣀⡀⢀⣀⣀⡠⠴⠚⠋⣠⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⠲⠤⣤⣀⣀⣈⣉⣀⣀⣤⠤⠖⠒⠉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠊⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠆⢰⣿⢧⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠈⠊⠐⣿⠇⠚⠹⠆⠀⢀⠠⡡⢝⡤⣰⣄⣰⠀⠀⠀⠀⠘⢳⣷⡂⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠁⠋⢘⢀⠀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠣⠩⠄⣿⣧⠓⠀⠒⠀⠀⠀⠀⣿⣧⠔⠀⠀⠀⠀⠀⢰⣿
⢰⡄⠀⠀⠸⡀⢳⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠐⠁⢼⣿⡟⡀⠀⠀⢀⡤⣀⣀⣿⣿⣷⠂⠀⠀⠀⠀⣼⣿
⡼⣷⣤⣀⠀⢀⢦⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⡄⢀⠐⠀⠀⠀⠀⠐⠀⠟⣿⣿⣵⡶⠟⢃⣽⣿⡿⠛⠿⠛⢀⣀⢄⣈⣹⣿⣿
⡘⢿⣻⣇⡀⢻⣦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⡀⢀⣠⣴⠶⠛⠛⢉⡴⠊⠋⠉⠁⠀⢀⠀⢠⡞⠋⢻⣿⣿⣿⣿
⡝⠊⠛⠟⠻⠉⢳⡀⠀⡄⢀⠀⠀⠀⠀⡀⠀⢀⢀⣠⠤⠶⠞⠃⠀⠉⠁⠤⡴⢾⣿⣿⣇⡀⠀⣠⣤⠞⠉⠋⠂⠀⠈⠻⠿⡿⣿
⠀⠀⠀⠀⠁⠠⣈⣿⣤⣷⣾⣂⣀⣤⣿⣷⣾⠿⠛⢁⣴⠃⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⠔⠡⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠙⣿⠛⠉⠉⠉⠈⢹⣿⡏⠠⠄⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⣗⢌⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸
⠀⠀⣤⣄⠀⠀⠀⠀⢐⣶⠖⠀⢀⣴⣶⣿⣿⣷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⡿⠉⠈⠀⠰⢧⠀⠀⠀⠀⠀⠁⠈⢀
⠀⠀⠈⠛⡄⢀⣬⣤⠞⠃⡴⠛⠛⠉⣯⠈⠈⢿⣿⣿⣿⣿⣿⢿⣿⠿⠟⠻⠛⠛⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⢀⠉⣠⣾
⢀⠀⠀⡠⠷⠛⠋⠁⡠⢶⣄⠀⠀⠀⢻⣶⠀⠀⢀⡼⠿⣿⡝⠻⠿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⡾⢰⣿⣿⣿⢿
⣳⡆⣰⡀⠊⢀⠖⠁⠀⢸⡉⠀⠀⠀⢸⡿⢺⣦⠀⠀⠀⠈⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣰⣿⠿⠟⠁⠘
⣿⣟⢿⣄⠀⡃⠀⠀⠀⠸⣆⠀⠀⠤⠋⠀⢸⠻⣧⣄⠀⠀⠀⠀⠀⠐⣬⡷⢂⣴⠂⠀⠀⠀⢀⠴⡤⠠⠑⢈⡿⠛⠁⢀⠀⠠
⢿⣿⣷⣿⣾⣿⢆⠀⠀⠀⠈⠒⠁⠀⠀⠀⢷⣄⠈⠛⢦⡀⠀⠀⠠⠄⠺⠛⠛⠁⠀⢀⣰⠞⠋⠀⠀⠀⢀⣤⣠⣀⣠⣴⡼⠋⣰
⣀⠽⣿⣾⣿⢯⣤⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡿⢿⡷⠚⠛⠛⢁⣀⣬⡤⢀⣐⣴⣾⡿⠁⠀⠀⠐⠀⣰⠏⣼⠿⠟⠋⠁⠀⢀⣿
⣿⣧⣼⠃⠘⠀⢨⡄⠀⠈⠙⠋⠀⠀⠀⠀⠀⠴⣊⣀⣼⣿⢚⣉⣿⡏⠉⠀⠙⢿⡿⠇⡞⠣⡆⢀⣼⢏⣾⠟⠁⠀⢀⡀⣠⡿⢳
⢽⣻⣷⣀⡀⠀⣾⣦⡀⠀⢉⢉⡱⢖⠚⠓⠚⢛⣋⣹⣟⣿⡿⣻⢿⡗⢀⠀⠀⠈⠀⠀⠀⠙⠁⢻⣻⡟⣿⠁⠀⠀⢨⡿⣯⡴⠏
⡫⠘⢟⠿⣷⣄⣻⣶⣤⣴⣭⢆⡐⡤⠄⢀⣀⣽⣿⣋⣹⡿⠛⠁⣼⡟⢸⣿⣧⡔⠀⠀⠀⠀⠀⠀⢻⣷⣿⠤⣄⣲⡟⠀⡅
⠻⣇⡈⢇⠙⠻⠿⣿⣶⣬⣿⣿⣿⣟⣝⠋⠉⠃⠈⢻⣏⠀⠀⢸⣿⣳⢾⠻⣿⣷⠀⠀⠀⠀⠀⠀⠞⢹⠇⢸⡿⠋⠀⠀⣠
⣦⣉⠻⣯⣟⠒⢤⠾⢶⡅⠀⠀⢀⠉⢫⢆⠀⠀⠀⢸⣏⠀⠀⣾⣿⣿⣇⠨⢜⣛⣒⣲⡲⢟⠀⢠⣼⠃⢀⡟⠀⠀⠀⣀⠃
⣿⣿⣿⡈⠛⠷⣤⡀⠀⣠⠆⢻⠎⠗⠰⢏⠀⠀⠀⣾⠣⠀⠰⢿⣿⣿⣟⡷⣺⠷⣶⠮⣤⣤⣞⠒⠀⡴⠋⠀⠀⠀⠐⠋
⣿⠟⣿⣿⣶⣶⣾⡇⣠⡼⡆⣩⠀⠀⠀⢸⠀⠀⢀⡿⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠊⠀⠀⠀⠀⠂
⢸⣼⣿⣼⣿⡿⣿⣽⠏⠵⠇⠘⠋⠀⠂⠀⢰⢀⣸⡆⢀⠀⠀⠀⠀⠀⠘⠿⠏⠉⠁
⣸⣙⣿⣿⢿⡇⡿⣿⢾⡀⠃⠀⠀⠀⠀⢠⣼⡿⢫⣄⠀⣴⣿⣇⡄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⠀⡄
⣻⢿⣿⣿⡾⣷⡇⣟⠂⠀⠁⠀⠀⠐⠀⣾⠟⠀⣋⠀⢀⣿⣿⡿⠛⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢁⠆
⣷⣿⣿⣻⢻⠿⣷⠁⠆⠀⠠⠀⠀⢠⣴⡟⠀⠀⣥⠄⣺⣭⣿⣶⣢⡼⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣸⠄⠰⢀⣾⡇⠊⢄
⣷⣯⣿⣿⣿⣾⣞⣇⣇⣀⡀⠀⣀⣸⡿⠁⢀⣴⠒⠀⣿⠶⢻⠿⠿⠁⠀⠀⠀⠀⡄⢀⠀⠀⠀⢂⣾⣷⣾⡿⣠⣾⢿⠏⠀⢰⣾
⠛⢿⣿⣯⣼⣰⣏⡟⡙⠙⠉⠘⣿⣿⣁⠀⣹⠖⠛⢐⣿⠀⠀⡴⡀⠀⠀⢀⣦⡁⠈⠀⠀⠀⠀⣸⠋⢉⠋⡻⣿⣿⣮⣤⣼⠿⠋
⠀⠘⣿⢻⢿⡿⢻⣿⣷⣶⣦⣾⡿⠛⠃⣰⣿⣷⡿⣾⣛⠄⠀⠀⢡⠠⢠⣼⣿⣿⣾⠄⠀⠀⡰⢧⣀⠈⣆⢻⣷⣿⣿⡇⠀⠈
⣿⣆⠿⣿⣿⣶⢻⣿⣿⣿⣿⢟⡀⠄⠘⣿⣿⣿⡷⢿⣽⣶⣤⢀⣈⢡⣌⠻⣿⣿⣿⠀⠀⣴⢧⣾⣿⣿⣿⣼⣿⣿⣏⡁⠀⠀⣀
⡿⡻⣤⣌⣝⣻⣿⣿⢿⣟⠥⠈⣡⣴⣾⣿⣿⣿⣽⡇⠝⣙⣻⣧⣤⡈⠀⠀⢠⣿⣿⣦⠾⠁⠀⣩⣭⣿⠂⠀⠉⠙⠛⠀⠐⠟⡍

⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⢛⣿⡛⠋⠛⣹⣿⣿⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠇⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣦⡿⣿⣿⣿⣾⣻⣿⣟⣿⠃⣠⣴⢾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⣹⣿⠁⣽⣿⣿⣿⣧⣾⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢻⣿⣿⠀⣿⣿⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⣿⣿⣿⣀⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡿⣿⠾⣿⣾⠀⣿⣿⡿⢟⣩⣿⣿⠟⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⢻⣿⣿⠀⣿⢟⣴⣿⠞⣱⣿⣿⣿⡿⣸⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣁⣼⣿⡿⠀⣴⣿⡼⢋⣾⣿⣿⣿⣿⣿⢹⡿⢻⣿⠟⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡉⣿⣿⣏⢠⣿⡿⢱⣿⣿⣿⣿⣿⣿⣿⢀⡔⣿⣿⡄⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⢈⣾⡇⠸⢻⠿⠋⠠⠜⠁⡟⢸⠃⣸⣿⠁⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⢿⠿⣿⠧⠞⡅⠋⠀⠀⠰⠶⡇⠠⠀⠛⣫⣤⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢹⡇⣿⣿⣿⠘⣿⠀⢸⠄⠀⠀⠀⢠⡀⠀⢀⡶⢿⡿⠙⣿⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⣰⣿⣧⢸⣿⠀⠉⠀⣴⣆⣀⣬⣵⠄⡶⠂⠸⣏⠈⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⡄⣿⣻⣿⢾⡿⠂⢰⡄⠻⣿⣧⣼⠟⠀⠁⠀⣾⠃⢸⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⣿⣿⣦⢸⣷⠀⢠⡀⠀⠀⠈⠁⠀⡄⠄⢰⣿⡇⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⣿⣿⡟⢸⣿⠀⣶⣥⣄⣀⣀⣐⣶⠁⡟⢸⣿⡇⣿⣿⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠸⠟⠀⣿⣿⣿⣿⣿⣿⣿⢠⣏⣴⣿⡇⣸⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠇⠈⠀⠀⠙⢿⣿⣿⣿⡿⣿⠠⠟⣿⣿⠓⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⣿⣿⢛⣾⠟⠀⠀⠘⠿⠀⣿⡃⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣤⣤⣤⣤⠀⢀⢀⣰⠀⣦⣾⣿⣷⠟⠁⠀⠀⠀⠀⠀⠀⠙⠁⣿⠟
⠀⠀⠀⠀⠀⢀⣴⡞⣿⣿⣿⣿⡇⢸⣧⣰⡇⢸⣿⠀⣿⣿⣿⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡃
⠀⠀⠀⠀⢰⣿⣿⡇⣿⣿⣿⣿⣇⢸⣿⣿⡇⢸⣿⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣰⣿⠁
⠀⠀⠀⠀⢸⣿⣿⡇⢻⣿⣿⣿⡏⢸⣿⣿⡇⢸⡏⠀⣿⣿⢇⡀⠀⠀⠠⢀⠀⣠⣶⠠⣿⠏⠁
⠀⠀⠀⠀⠸⣿⣿⣧⣼⣿⣿⣿⣇⠘⣫⣾⡇⢸⣧⢘⣿⣷⣿⣷⣴⣿⡄⢸⠂⣿⣧⡼⠉
⠀⠀⠀⠀⠀⠈⠃⠀⠉⠻⣿⣿⣿⣼⣿⣿⡇⢸⣿⢸⣿⠟⣿⣿⣿⣿⡇⢸⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣾⣿⣿⡇⢸⣿⢸⣷⡆⠻⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣇⣸⣿⠸⣿⣧⠀⠀⠀⠀⠀⢸⣿⣧⠀⣠⣶⣤⣤⣶⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣯⣿⣿⢀⣿⣿⣧⣴⣶⣶⠀⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡆⠉⠛⠛⠻⠿⢿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⡏⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠿⠇⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣍⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣽⠛⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⠛⢛⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣬⣉⠙⢀⡄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡧⠥⡬⠀⠥⠬⠼⢿⣄⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠊⠀⠰⠀⠀⠀⠈⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢰⠞⠉⠉⠉⢳⡀⠀⠀⢀⡞⣁⣀⣀⡠⠤⠤⠖⠒⠀⠒⠢⠐⠄⠳⢤⣙⣧
⠀⡴⠖⠛⠂⠀⠀⠀⣸⡃⠀⣴⠛⠏⠠⠀⠀⡆⠀⠈⠀⠀⢘⠀⢰⠀⢠⠀⠘⡀⠉⣧
⢸⡇⠀⠀⠀⠈⢢⠀⠈⢿⡀⡏⢠⠀⢈⠀⠀⠁⠀⠀⠂⠀⠈⠀⠀⡄⠀⠀⠀⡇⠘⣸
⠀⣿⠂⠀⠀⡀⠈⠀⠀⣸⡿⠛⠀⢤⣚⣩⣭⣩⣒⡉⠉⠀⠴⠖⠒⢆⣀⡐⣖⣳⠂⠉⢳⡄
⢀⣸⡇⠀⠀⠁⠀⣤⠞⢹⠃⠀⠀⢰⠀⢀⠊⣴⡿⢷⠀⠀⠀⡴⢷⣦⠑⡄⠈⡇⠀⠀⠈⣷
⡟⠀⠀⠀⣆⠀⠀⡹⠶⠛⠠⡀⢀⡸⢠⣬⣄⣛⣿⠞⣠⣤⣄⢻⣿⠟⡠⣁⢀⡃⠀⠀⠀⣿
⠻⢤⣠⣤⠁⣀⢀⣠⡶⠓⠲⣦⣠⣾⣻⠟⠟⠩⢋⠕⠛⣏⠙⢌⠫⠑⠟⢿⢙⣷⣤⡤⠞⠁
⠀⠀⠀⠈⠉⠉⠉⠁⢀⣴⠞⠋⠉⠙⡇⠀⠀⠀⠀⠀⠚⡍⠂⠀⠀⠀⠀⠀⡋⠉⠉⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⠤⠔⠊⠘⠀⠀⠀⠀⠀⠀⠜⠙⢆⠀⠀⠀⠀⠀⡈⠉⠑⠤⢤⡙⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣀⣀⡀⢳⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡸⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⠄⠊⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸⠒⠂⢲⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠤⢽⡇⠀⠐⠒⠀⠤⢼⠀⠀⠀⠀⠀⠀⡸⠄⠒⠊⣿⣤⡴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠁⠀⠐⠒⠂⢎⣰⣣⣆⣴⣰⣠⠗⠒⠈⠉⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣛⣦⣤⣀⣀⣀⣤⣐⣿⣾⣓⣠⣄⣀⣀⣠⣴⣞⣯⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠉⠀⠀⠀⠈⠀⣼⣿⡀⠈⠀⠀⠀⠈⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⢷⡳⢄⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⣩⣽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄
⠀⠀⠀⢀⡶⠛⣻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⢳⡄
⠀⠀⠀⠛⠁⠘⠿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣤⡤⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠀⠈⠉⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠐⣧⣄⣀⣻⣦⣀⣀⣰⣦⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣰⠇⠉⠉⠉⠀⠉⠉⠁⠉⠛⠶⢦⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠹⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⡟⠀⠀⢠⡏⠀⠀⠀⠀⠈⠉⠛⠛⠃⠀⢹⡇⣠⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⢰⡇⠀⠴⠋⣀⠀⠀⠀⠀⠀⠶⠖⠀⠀⠀⠘⠿⢃⡤⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⠀⠀⠛⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⣧⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⣡⡴⠋⠛⢶⣄⣀⠀⠀⠀⣀⣠⣴⡟⠁
⠀⠀⠀⠀⠀⠀⢀⣼⠋⣠⠞⠋⠀⠀⠀⠀⠈⢹⡟⠛⠛⠉⠁⠘⣧⣀⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⣿⢀⣠⣀⣀⣠⣴⠶⠛⢷⣄⣀⣀⣠⣴⠏⠉⠙⠳⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡿⠋⣁⣽⡙⣯⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⢻⡄⠀⡴⠞⠋⣉⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣴⠞⠋⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣀⣠⣤⠿⠶⠶⠶⠶⠿⠶⠶⣤⣤⣀⡀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠛⠛⠛⠛⠛⠳⠶⢶⣯⡀
⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⢸⡇⠀⠀⢀⣀⣀⣀⣀⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⢸⡇
⢀⣼⣷⣾⣿⣿⣿⠿⠿⠿⠿⣿⠿⠿⣿⣿⣿⣷⣶⣤⣀⣠⣶⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣿⣀
⣾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⢾⡛⠶⣬⡛⠾⡝⠻⢿⣿⣿⡿⠟⠋⢁⣠⡴⠟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣥⣤⣤⣤⢤⣤⣤⣤⣴⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⠶⢶⣄⣀⠀⠀⠀⠀⣀⡶⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠉⠙⠷⠲⢦⡼⢧⡀⠦⢄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⠔⠲⡤⢄⢀⠀⠀⢀⣈⢥⠀⢈⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠀⢀⠉⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⠒⠃⠀⠀⢠⠀⠀⠀⠀⢀⣄⣼⠃⠰⠚⠁⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⣀⡴⢿⣰⠀⢀⣘⣿⣄⣠⣄⠀⣀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⠗⠚⡅⠐⠂⢹⡆⣸⠋⠁⠀⠀⣬⡟⠉⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⠓⠋⠀⠀⢠⠄⠀⠀⠀⢻⣧⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⠀⠀⠀⠀⠫⠇⠀⠀⠀⠀⠀⠀⡀⢺⡧⠤⢤⡈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣸⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡆⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⣿⡗⠒⠚⢁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⠒⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡦⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣑⣑⣚⣘⣃⣄⣤⣤⣤⠤⠤⠬⠶⠮⢕⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⣄⣀⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⢿⡯⣥⠤⠤⠤⣤⣤⡀⠰⠿⠿⠟
⠀⠀⠀⠀⣦⣄⣩⣽⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠈⢳⣾⣶⣶⣶⣶⣶⣶⡶⠀⠀⠀⠀⣔⠀⠀⠀⠀⠐⠀⠀⣰
⠀⠀⠀⠀⣿⣟⡉⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣾⣿⡿⣿⣿⣿⣿⠁⠀⢀⣄⣤⡿⠀⠀⠀⠀⠀⠈⠁⢸
⠀⠀⢀⢠⣿⠿⠿⣿⡿⢖⡀⢀⣀⣠⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰
⠀⠀⣤⡞⡍⠀⠚⠋⠉⠉⠉⠉⠉⠉⢹⡄⠀⠀⠀⠀⢸⣿⠻⠟⠉⠉⠉⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⢀⣤⣤⣶⡏⠀⣾⣿
⠀⠀⠉⣀⣠⠀⢙⠄⠀⠀⡔⠋⢱⣄⣘⢻⠀⠀⠀⠠⠘⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠁⢀⣤⣿⣿⣿⣿⣿⣿⣷⡄⠈⠙
⠀⠀⠀⠀⠈⠀⠁⠀⠠⢺⡄⠀⠈⠀⠘⣹⠀⠀⠀⢰⠰⣿⠀⠀⢀⣀⣀⣀⣀⣀⣠⡀⡄⠸⣿⠛⣿⣿⠻⠿⣷
⠀⠀⣤⣄⣀⣀⣀⣀⣢⣀⣀⣀⣀⣻⣾⣿⠀⠀⠀⢨⠀⣿⠀⠂⠸⣿⣿⣿⣿⣿⣿⣧⣇⣼⣷⡄⠀⠀⠀⠀⣀
⠀⠀⠲⢼⣏⣽⠉⣭⣽⣯⠭⢭⡿⣾⠿⠋⢀⣾⠀⠘⣀⣿⠀⠀⠠⣿⣿⣿⡿⣿⡏⠀⠀⣿⡿⠃⠀⠀⠀⣴⣿⡆
⠀⠀⠀⠀⠨⡟⣻⣈⣁⣩⣴⣾⠞⠋⠀⠀⣼⣯⠀⠀⠀⠹⣧⣶⣶⣾⣿⣧⣴⣿⡇⠀⠀⣿⡇⠀⢠⣶⡀⣿⣿⡁
⠀⠀⠀⠀⠀⠉⠉⣏⣿⣿⣽⣾⣦⡀⠀⠀⢿⡇⠀⠀⠠⡶⢿⡿⠿⣿⣿⣿⣿⣿⡇⠀⠀⣿⡧⠀⠀⣿⣧⢿⣟⣷
⠀⠀⠀⠀⠀⠀⢀⣼⠰⣿⣿⠿⢿⣌⢦⠀⣿⣻⡇⠀⢘⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⠀⠀⣿⡗⠀⠀⣿⣿⡄⠙⠃
⠀⠀⠀⠀⢰⠊⠉⠙⣄⢹⣿⡶⢤⣿⣿⣿⣿⣿⣇⠀⢸⣿⡚⠛⣛⣻⣿⡛⣾⡻⣧⣀⣀⣿⣿⡄⠀⢀⣤⣶⣤⡄
⠀⠀⢀⣶⣞⣓⢶⣄⠈⢹⣧⣘⢻⣿⣿⣿⣿⡟⡷⢤⡆⡈⢃⠈⠯⢿⠧⠀⣿⣿⣿⢭⠶⠶⠶⠶⠷⣯⣬⣿⣯⡅
⠀⠀⠀⡿⢿⢿⣯⣽⡦⢸⣁⣻⣼⣿⣿⣿⣿⡟⢡⣾⣏⣥⣽⣤⣤⣤⣤⣠⣿⣿⡇⡴⢶⠶⠶⠶⠶⣿⣿⣿⣿⡇
⠀⠀⢘⣟⢻⢾⣿⣿⢹⣾⣿⣿⣷⣿⣿⣿⡟⢱⣷⢾⣬⣭⣤⣼⡿⣿⣿⣿⣿⠛⠃⡆⡂⣴⡞⢛⡆⣿⠿⣿⣿⡇
⠀⠀⠈⣷⣼⢸⣷⠖⣶⣶⠶⢶⣾⡟⢹⣿⣴⡏⣿⣿⣿⠀⣽⣿⣷⣟⠛⢳⣿⣷⡃⡇⢀⣾⠃⠀⣷⠉⠀⠤⢠⠁
⠀⠀⠀⣿⣻⢻⣿⣀⣿⣿⣄⣸⣿⣧⣸⠻⢿⣿⣿⡧⠘⠀⢻⣿⣿⡏⡿⢿⣿⣿⡇⣷⣸⣿⡇⠀⠻⡀⠀⠐⠉⡀
⠀⠀⠀⣿⣸⣺⣧⣤⣤⣤⣤⣶⣶⣦⣾⣿⡿⢨⠀⣳⠠⠢⠀⠟⢻⣿⣿⣿⣿⣿⢷⣟⣛⠛⣛⠃⠒⠀⢀⠀⣂⣲
⠀⠀⠀⣿⣟⠻⣿⣶⡜⢶⡿⣿⢳⣿⣿⣷⣶⣷⣿⡇⣠⣶⣆⢸⣿⠉⣶⣷⣿⣿⣦⣭⣽⠿⡟⡿⢿⣿⠉⣿⣿⣭
⠀⠀⠀⢘⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⡏⣻⣿⡿⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣤⣿⣦⣾⣷⣿⣿⣿⣿
⠀⠀⠀⢙⣻⣋⣉⠉⢉⢉⡉⢟⠻⠻⢿⣿⣿⣿⣿⣁⣿⣷⣾⢸⣿⢰⡿⠿⣿⣿⣿⣏⣹⣿⣟⢿⠿⣿⣃⣿⣿⣻
⠀⠀⠀⣶⠛⣋⣭⣠⣤⣞⠓⠚⠁⠛⢲⣿⣿⡿⣿⣿⠏⠻⣏⢸⣙⣧⣀⣼⣿⣏⣿⣿⢧⣿⠉⣸⣿⣽⣿⣿⠋⣙
⠀⠀⠀⣿⢿⣿⣛⣏⢱⣄⣤⡤⣾⣉⣿⣿⣿⡏⢀⢹⡄⣠⡿⠌⣟⢙⣿⣯⢿⠹⣿⣏⠹⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⣤⣼⣿⠉⠩⡿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣦⣴⣏⣻⣶⣶⣿⣟⡯⣿⣨⣶⣿⣤⣶⣭⡁⠀⠀⠘⠻⢿⣿⣿
⠀⠀⠀⣿⣴⣷⡶⣶⣿⠾⢿⡾⢿⣿⣿⣿⡯⠀⠁⢰⡿⣿⣿⣿⠿⣿⣿⣿⣿⣋⣽⣿⣿⣿⣿⣾⣤⣆⡴⠞⢶⡭⡀
⠀⠀⠀⣿⣿⣿⣇⣸⣇⣷⣶⣿⣃⣁⣈⣭⣀⡀⠀⢀⡷⠛⠛⣿⠰⠾⡽⢩⣿⣿⣿⣿⣧⣛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠖⠓⠀⠀⠀⠘⢥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⣿⣿⣿⠿⠿⠛⠛⠛⠛⠿⠿⣿⣿⣿⣷⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⡿⠿⠟⠛⠉⢉⣀⣠⣤⣴⣶⣾⣷⣶⣦⣤⣄⣀⡉⠙⠛⠻⠿⢿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⣁⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣈⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠋⠉⠉⢹⣿⣿⠁⠀⠀⠀⠀⠈⣿⣿⡏⠉⠉⠙⣿⣿⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⡿⠁⠀⣤⣀⣼⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⣧⣀⣤⠀⠙⢿⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⣿⣿⡟⠁⠀⢸⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⡇⠀⠈⢻⣿⣿⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢹⣿⠀⠀⠀⠈⠀⠈⠙⠛⠋⠉⠀⠀⠀⠀⠉⠙⠛⠋⠀⠀⠁⠀⠀⠀⣿⡏⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⢿⣿⣿⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣿⣿⡿⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠘⢿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⡿⠃⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠙⠻⠿⣿⣿⠿⠟⠋⢀⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣶⣤⣄⣠⣤⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠩⠤⠰⢒⣲⣲⣟⣉⣡⣆⣖⣶⡶⡤⢄⣉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⣿⡷⠴⠛⠻⠿⠿⢿⢟⢻⢛⡛⠛⠃⢠⣤⣾⠃⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠃⠂⠈⠑⢾⣿⢿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡤⠊⠀⡠⣿⡟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠁⠕⠶⣤⣀⡀⠙⠻⣿⣿⣿⣹⣿⣿⣿⣿⠿⠋⠀⣀⣬⣤⢟⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⡿⣿⣶⣤⣀⡉⠛⢷⣆⡈⠛⢿⣿⣽⡿⠛⢡⠴⠶⠛⢉⣩⣴⠞⠋⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⣴⣬⡿⠀⠀⠉⠛⠛⠿⢷⣬⣥⡶⢤⣽⢫⣴⢶⣀⣤⡴⠶⠛⠉⠀⠀⠈⠁⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣱⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⡷⠊⠋⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠠⠟⠻⢏⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢎⢽⡄
⠀⠀⠀⠀⠀⠀⣗⠰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠻⡀
⠀⠀⠀⠀⠀⢸⣶⠁⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡰⠀⠀⠈⠥⠀⣧
⠀⠀⠀⠀⠀⣿⠘⠄⠄⠈⠳⢘⠲⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⠋⢻⡤⠂⠐⠠⢰⣽
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠐⢮⢀⣦⢠⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⣀⢰⣿⣿⠞⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⡿⠃⠁⠀⠀⠀⠀⣴⣆⣿⡜⠇⢠⡐⡠⣠⠀⠘⢁⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢻⣾⡿⠄⡖⡶⠆⠀⠀⠐⣿
⠀⠀⠀⠀⠀⣯⡆⠀⠀⠀⠀⠚⠟⢻⡿⣟⣿⣒⣛⣛⣈⣀⢀⠏⢐⢄⡟⠁⡽⠋⠉⠱⣖⠶⠺⡗⡻⢿⡇⡇⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⢇⡗⠀⠐⠠⠶⢻⣿⡝⠫⠍⠀⢹⠟⠛⠇⡮⠀⠴⠮⠒⠀⠀⠀⠀⠈⠁⠀⠀⠫⠞⢹⡗⠦⠄⠀⠀⠀⠀⢰⡾
⠀⠀⠀⠀⠀⢸⠜⠉⠀⠀⠀⠀⠸⠋⠙⠀⠁⠀⠀⠀⠀⠀⢳⣦⣾⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣥⣽⢿⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢺
⠀⠀⠀⠀⠀⠀⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡑⢀⡻⣿⠛⣛⠩⣶⢤⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠆
⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⡄⡀⠀⠀⡀⠄⠀⢨⡴⣟⣫⣻⡿⡖⠱⠘⠈⠁⠹⠃⢽⡠⣢⣠⣔⠀⢀⡀⠀⠀⠀⠀⣧⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⡲⡶⣿⡳⠛⠟⡿⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢧⡙⣲⠤⢆⣠⣤⣶⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢩⠏⠁⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠉⠉⢡⢁⣞⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠋⠄⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠈⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠈⠀⠚⠛⠛⠒⠒⠒⠠⠒⠚⠟⠉⠉⠛⠊⠉⠁⠀⠀⠀⠀⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠶⠄⠄⠖⢦⠐⢶⡶⠶⣶⣠⢶⣶⣶⡶⠲⠹⠟⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⢤⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⢽⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⠽⠋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⢝⢦⡀⠀⢀⡤⡺⠛⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢱⣽⣿⡵⠋⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⢔⡫⠚⠛⢮⡢⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣠⣾⠗⠉⠀⠀⠀⠀⠙⢮⣢⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣯⣄⡉⠁⠀⠀⠀⠀⠀⠀⠙⠪⡲⣄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠨⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠈⠪⡳⣄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣈⠪⡳⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⣇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠜⡿⡟⢙⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣤⣶⣵⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠚
⠀⠀⠀⠀⠀⠀⢰⡿⣻⣯⣵⣿⣿⡏⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣠⣃⣤
⠀⠀⠀⠀⠀⠀⠸⣿⠻⣿⡿⣿⣿⢃⣾⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠏⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡾⠿⠿⠿⠊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⣶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠤⠴⠃⠈⠓⢢⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢁⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡟⡗⠚⡇⠀⠀⠀⠀⣠⡶⠶⠂⠐⣧⢒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⢄⣏⠸⣇⣇⣀⣙⣲⡄⢸⣾⢟⣶⣄⣀⠚⢉⣀⣀⣀⣀⣀
⠀⣁⡀⠤⠒⠚⠃⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠀⠸⠖⠠⢀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⢀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀
⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄
⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷
⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿
⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⡅⠀⠀⠀⠀⠀⠀⢨⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃
⠀⠀⠀⠀⠈⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣤⣽⡄⠀⠀⠀⠀⢠⣯⣤⣬⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢨⠃⠀⠀⠀⠀⠘⡅⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣄⠀⠀⠀⠣⣀⡤⠋⠀⠀⠀⠀⠀⠀⠙⢤⣀⠜⠀⠀⠀⣠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡤⢀⠴⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡾⠛⠉⠠⠤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠳⣤⣤⣤⡤⠴⠚⢁
⠀⠀⠀⢠⠁⠀⠈⠁⠀⠀⢀⣀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠟⠛⠀⠑⠒⠒⠒⠛⠋⠉⡠⣄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠒⠚⠉⠁⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠶⠶⠖⠒⠒⠒⠒⠶⠤⢬⣽⠓⠦⢤⣤⣀⣀⣀⣀⣀⣤⠴⠶⠛⡏
⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠀⣤⠀⠹⠓⠒⠶⠦⠤⠤⠤⠤⠶⠒⠚⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢠⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠶⠀⠀⠀⠀⠿⠀⠀⠀⠘⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠰⠆⠀⠠⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠛⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠷⠶⠿⠿⠿⠿⠦⠼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢟⣫⡭⠶⠖⠛⠛⠉⠉⠉⠉⠛⠛⠒⠶⠭⠙⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡾⢛⣥⠞⠋⠁⠀⠀⢀⠀⠠⡤⠀⠀⢠⢄⢀⠀⡀⠀⠀⠙⠳⣬⡛⢷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣉⡾⠋⠀⠀⢠⠄⠰⣈⠞⠤⠄⠀⠀⠀⠓⠘⠜⠜⠋⣰⡢⢀⠀⠀⠙⢷⣔⢾⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣾⢏⡾⠋⠀⠀⣄⠈⠮⠒⠃⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠁⠠⢎⠖⠦⠀⠀⠙⢷⡙⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣷⡿⣡⠟⠀⠀⢴⡅⠙⠀⠀⠀⢀⣠⠶⠚⠋⠉⠉⠉⠉⠙⠓⠦⣄⡀⠀⠀⠁⢜⣡⢆⠀⠀⠻⣌⢿⣿
⠀⠀⠀⠀⠀⢀⡿⣱⠏⠀⢠⠄⠑⠊⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠀⠸⡁⠀⡀⠀⠹⣎⢯⡀
⠀⠀⠀⠀⣾⢻⢣⡏⠀⠀⠈⠑⠂⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠐⠊⠁⠀⠀⢹⡜⣿⣷
⠀⠀⠀⠀⢿⡌⣼⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣧⢻⡿
⠀⠀⠀⢀⣼⡇⡟⠀⢼⡿⢶⣄⢰⣷⢰⡿⠿⠿⠂⣿⠿⠿⠇⢸⡆⢰⡾⠿⡦⢸⡇⠀⢸⡇⣾⠀⠀⠀⠾⢿⡿⠷⠀⢻⢸⣧⡀
⠀⠀⠀⢾⣿⠃⡇⠀⣠⠄⢈⣿⢸⣿⢸⣿⠿⠀⠀⣿⠿⠆⠀⢸⡇⣿⡁⠀⠀⢸⡇⠀⠸⠇⣿⠀⠀⠀⠀⢘⡃⠀⠀⢸⢨⣿⡷
⠀⠀⠀⠈⢻⠆⣧⠀⠿⣷⡾⠋⠰⡿⠸⡧⠀⠀⠀⡿⠀⠀⠀⢺⠇⠹⢷⣶⠗⠀⠿⣶⡦⠁⢻⣷⣶⡦⠀⢸⡇⠀⠀⣼⢸⡟⠁
⠀⠀⠀⠀⣾⣂⢻⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⣷
⠀⠀⠀⠀⢿⣿⡜⣇⠀⠀⢀⡠⠄⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣠⣄⡀⠀⠀⣼⢧⣿⡿
⠀⠀⠀⠀⠀⠈⣷⡹⣆⠀⠈⠀⢈⠆⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠉⠀⠀⣰⣝⠾⠃⠀⣸⢏⣿⠋
⠀⠀⠀⠀⠀⠀⣿⣷⡙⣦⠀⠀⠑⢋⡕⢀⠀⠀⠈⠙⠶⢤⣄⣀⣀⣀⣀⣀⡤⠶⠋⠁⠀⠀⠠⣄⢠⠶⠀⠀⣼⢯⡾⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣌⢷⣄⠀⠀⠲⠴⡱⠂⢀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⢠⢤⡲⡀⠝⠀⠀⢠⡾⢡⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣍⢷⡄⠀⠀⠁⠺⠏⡀⡔⡄⠠⠤⠀⠀⠄⠐⠒⡼⡉⠀⠰⠃⠀⠀⣠⡾⣫⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠗⢠⡀⢦⣄⡀⠀⠀⠀⠁⠑⠃⠀⠀⠚⠀⠀⠁⠀⠀⢀⣠⡴⢛⡥⠾⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣳⣮⣍⡂⠰⠦⣤⣤⣀⣀⣀⣀⣤⣤⠴⠶⣚⣫⣵⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⡟⠶⠲⣶⣶⣶⣶⠶⠶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠈⢳⣤⣤⣤⣤⣤⣄⣴⠾⠿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠀⠙⢿⣄⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠖⠂⠀⣠⣴⠶⠿⠿⠿⠿⠿⠿⢿⣷⣦⣄⣀⣼⠟⠋⠉⠀⠀⠈⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣧⣄⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⠀⠀⠀⠉⢻⡟⢁⣤⠶⣦⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⣾⣡⣦⠘⡧⠀⠀⢀⡾⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⡼⠿⣿⣇⠀⠀⣸⣵⣶⡀⣸⠃⣀⣀⣀⣀⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣠⡶⠶⠶⠶⠶⣤⣿⠿⠛⠳⠶⠤⠀⠉⠙⠻⢏⣰⠿⠋⠉⠉⠀⠈⠉⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡿⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⢀⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢋⡽⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡶⠋⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠶⠦⠤⠤⠤⣤⣤⡿⢱⣶⣶⡄⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⣆⠀⠀⠀⢀⣀⡤⠶⣻⡽⣿⣿⡇⣴⡶⠶⠿⠛⠋⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡬⠿⣷⣖⣋⣩⡴⠖⣋⣡⢟⡶⡗⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠒⠒⠒⠚⠛⠋⠉⠉⢿⣆⠀⣀⡬⠟⠛⠛⠟⣻⠟⢁⣞⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠿⣧⣤⣀⣀⢀⣼⣁⣀⡼⠛⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠙⡻⠋⠛⠉⢹⠇⠀⠀⠘⢿⢿⣷⣶⣤⣤⣤⣶⠟⠋⠉⠛⠿⢿⣷⣶⣦⣤⣤⣤⣤⣶⡾⠟⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣾⠀⠀⠀⠀⠈⢣⡈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⣀⣀⣀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⢠⣾⣿⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣄⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠈⠻⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠙⠿⠿⠿⠟⠁⠀⠀⠀⠀⠙⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⡶⠶⠶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢁⣴⣶⣦⡌⢻⣿⡇⠀⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢸⣿⣿⣿⠇⣸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⣠⣾⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⣿⣿⣿⣿⡇⠀⠉⠉⠉⠉⠉⠉⠉⢹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣉⣹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⣦⣄⡀⢰⡟⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣭⡶⢷⣦⠉⠻⣾⡷⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⣦⣄⡀⢠⡿⠛⢻⡄⠀⠀⣴⠟⣁⣀⣀⡈⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣦⣉⠻⣾⡷⣦⠾⢃⡘⠷⠶⠾⠋⣉⣉⠻⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⡈⢻⡆⠀⠈⠻⠶⠶⠾⠛⠋⠙⠷⢶⣶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⣠⡾⢿⣄⠛⣛⣿⡇⠀⠀⠀⠀⣴⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⡇⠀⣿⠀⠀⠙⠛⢋⣉⣤⣤⣶⠶⣶⣿⣄⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣿⠆⠀⠀⢰⡟⠉⠁⣀⣀⣤⡙⣷⣭⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣾⠋⠀⠀⠀⠀⠈⠻⣦⣀⠙⢿⡝⢷⣮⣵⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢠⡶⠟⠛⠛⠷⣮⠙⣷⠈⣿⢀⣴⠾⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠹⡇⢿⠀⠿⢼⠇⠀⠀⠀⠀⠈⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠁⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠐⠒⠁⠂⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠈⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠐⠀⠀⠀⠀⢈⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⢀⡠⠰⢀⠤⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⣠⠂⠁⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢌⠀⢠⠃⢜⡂⠀⠀⠠⠀⣠⠖⠊⠀⠂⠀⡀⠈⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠉⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⢀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐
⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠂⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⡡⢣⠊⠠⠝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⢄⣲⣿⣿⣿⣿⣿⣿⣿⠀⠂⣰⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢇
⠀⠀⠀⠀⠀⠄⠈⠳⣌⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⡉⠛⠢⣤⣶⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡯⠿⣾⣶⣾⣦
⠀⠀⠀⠀⠐⣀⠈⠻⣿⡅⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢠⠀⠀⠉⠐⠛⣷⠀⠀⠀⣀⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠸⣆⠀⠐⠛⠛⡛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠨⣶⣄⠤⠜⠋⠉⢉⣉⠭⢭⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠈⢣⡀⠀⠀⠐⠊⠛⠃⢀⣀⣀⠈⠀⠀⢀⠠⠄⠺⠿⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⣠⡾⠟⠻⠿⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⠒⠙⠟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣭⡅⠈⠀⠀⠀⠀⠀⠀⠙⣻⣿⣶⣶⣦⡀⠀⠀⠀⠈⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⢛⣻⣶⣤⡀⠀⠀⠲⠶⢻⣿⣿⣿⣿⣿⣿⣆⠐⠀⢀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠤⠉⠋⠙⢛⣻⣤⣤⣀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡤⢒⣡⠾⠏⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠁⠀⠈⠗⡿⣿⣮⡿⢿⣿⣿⣿⣿⣿⡶⠋⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣷⣤⣬⠾⠛⠠⣿⠁⠀⣀⣴⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣶⣿⣿⡇⠀⢀⣰⠟⠋⢛⣷⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⠙⣃⣷⣼⣓⣁⣤⣶⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⢷⣶⣾⣿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣾⠙⠛⠟⠛⠿⢻⣯⢽⣟⠰⡄⠀⡤⠮⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠥⠝⠂⣰⠁⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⡀⠀⣀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⢔⡡⠋⠀⠀⠀⢀⡀⠰⣿⣿⢰⣤⡴⢦⠀⠰⠿⣿⡆⣿⣿⡟⠛⠉⠉⠁⣀⠀⠈⠙⢳⣈⣭⣙⠛⠓⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⠏⢀⠞⣹⠋⠀⠰⠷⠦⠀⠀⣼⣧⠾⢿⣿⠁⠀⠀⡴⠋⢁⡠⠖⠋⠉⠁⠈⠙⠛⠿⢷⣌⠻⣦
⠀⠀⠀⠀⢀⢴⠆⠀⠈⠁⠀⣾⢰⣁⣀⣠⡀⠀⠀⠀⠀⠈⢡⠏⣹⡇⣤⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⣧⠻⡽⣧⠙⣧
⠀⠀⠀⠠⣥⠎⢀⢼⠇⠀⢠⢿⡠⣬⠤⢥⡙⠄⠇⠀⠀⣠⣠⣴⠉⡇⡸⢀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠈⣠⡴⣿⢡⣳⣯⣸⢻⡄
⠀⠀⠀⠀⠀⠀⠧⠊⠀⢀⡏⢸⡅⠲⠌⠒⠛⠀⠃⢀⠀⠀⡽⠃⢀⡟⣃⣀⣉⠁⠀⠈⢁⠀⠀⠀⠀⣴⢋⡼⢣⣾⡿⣿⡏⣾
⠀⠀⠀⠀⠀⢀⣰⠂⠀⢸⡇⣄⠧⣄⣀⣀⣤⢴⡚⣁⡴⠋⠁⢠⢎⠾⠃⢀⣀⡀⠀⠀⠈⠀⠀⠀⠘⠁⣼⣱⣿⡟⡅⣾⣴⡇
⠀⠀⠀⠀⢠⢃⡎⠀⠀⢸⡇⠈⠲⣄⠨⠉⡇⣠⣟⣿⣄⠀⣰⣿⡏⠔⡩⣐⡩⠤⡄⠀⠘⣠⡴⠂⢀⣼⣿⣿⡿⢌⢞⣿⡿
⠀⠀⠀⠀⠘⠊⠀⠀⡴⠟⠶⠭⣍⠙⠲⡄⠿⠃⡟⣻⡝⢶⢋⡿⠓⠛⠉⠁⠀⠤⠀⣠⣾⢋⣠⣴⢟⡫⡿⣳⣾⡷⣫⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣸⢡⠴⠖⠒⢦⡀⠀⣿⠀⣸⡾⢋⣷⣤⣼⣇⣐⠲⠚⠛⠓⢦⡀⢻⣿⣿⣿⡴⡫⡾⣡⠗⣪⡾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢰⣇⣴⠋⠀⢿⣴⠟⣿⢻⡟⣁⠄⠂⠀⠈⠲⡄⢹⡿⢻⣿⣏⢞⣼⢿⢿⡾⡋⣴⠏
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢰⠟⠋⠁⠀⠀⢸⣃⣴⣿⣾⡟⠛⠋⠛⠛⢦⠀⢻⣾⠇⢸⣿⡱⣻⡳⣣⡼⠇⣼⡟
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡿⠃⣿⡟⠀⠀⠀⡀⢀⣾⣀⡿⠃⠀⣾⣿⣿⣿⣿⣿⣞⣼⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣿⣠⣾⠋⠀⠀⠀⠘⢁⣾⡏⠉⠁⠀⠀⢻⣿⣿⣿⣟⠿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⣼⢿⣇⠀⠀⠀⠀⠈⣿⣿⣿⣿⣞⡜⢿⣷⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⠇⡠⠀⡀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣼⠣⡾⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣎⢿⣧
⠀⠀⠀⠠⡀⠈⡇⠀⠀⡟⢰⠃⢠⠇⢠⠂⠀⢰⡇⠀⠀⠀⢀⡼⣿⣟⡾⣸⡇⠀⠀⠀⠀⠘⣿⣷⢻⣿⣿⣿⣞⣿
⠀⠀⠀⠀⠱⠀⡇⠀⠰⣇⠚⠀⡟⣰⠃⡰⠀⣼⠀⠀⠀⣰⣿⢿⣿⣿⣿⢷⡇⠀⠀⠀⠀⠀⢻⣿⣧⡎⡿⡿⢫⣿⡆
⠀⠀⠀⠀⢀⡼⠁⠀⠀⠈⣇⠀⠀⠁⣈⡤⠴⠋⠀⠀⣰⣯⣴⣾⠟⠛⠒⠛⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⣧⡾⣿⡇
⠀⠀⠀⠀⣾⠶⠒⠛⠛⢦⣬⣽⣳⠞⠁⠀⠀⠀⠀⢠⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣿⣾⣾⣿⡷
⠀⠀⠀⠀⣷⢤⣀⣀⣀⣀⣘⣣⣤⣾⡇⢀⡀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣍⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⢈⠛⠳⠤⠭⠿⠿⠛⠿⣟⡡⠾⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣤⣶⣛⣻⡿⠋
⠀⠀⠀⠐⠉⢀⡆⢠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡀⠀⠀⣼⠀⠀⠀⣀⡠⠒⠉⠀⠉⢳⡀⡀⢀⡀⠤⢄
⠀⠀⣀⠠⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠠⠠⠄⠀⠀⠀⠉⠀⠀⢀⡨⠀⢠⠊⠀⡡⠤⠤⡀⠀⠈⠁⠈⡇⠀⠀⠀⠗⠢⡀
⠀⠉⠉⠁⠀⠀⠂⠀⠌⡑⠂⠤⢀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠒⠪⣀⠠⠔⠀⠀⢸⠀⠀⠓⠀⠀⠹⠀⠀⠀⠴⠃⢀⣀⠀⠀⠀⠚⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⢀⠈⠒⠠⠄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣣⣀⠠⠄⠐⠁⠀⠀⠀⢀⠀⠈⡸⠀⠀⠐⠀⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⣐⠀⠀⠄⢀⡀⠉⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠈⠉⠉⠀⣠⣤⡤⠶⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠒⠤⠤⢌⣁⡀⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣒⡒⠒⠬⣔⠄⡀⠀⠀⢠⠀⠀⠀⠈⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠢⡀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠒⡶⠶⠦⠤⣄⡀⢀⣸⡄⠀⠈⡆⠀⠉⠉⠁⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠐⢤⡀⢀⠤⠶⠦⣄⡀⠀⠀⢱⠄⠀⠖⠀⡹⠁⠈⠳⠤⠖⠁⠠⣄⠀⠀⠈⡧⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠤⠉⠒⠀⠀⠒⢿⡀⠀⡆⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⢀⡴⠄⠊⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠴⠦⠀⠤⠞⠉⠉







⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠈⠻⠿⠏⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⡏⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⢀⣠⣴⣶⣶⣶⣦⡘⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣛⠿⢿⣿⣿⣿⠿⢿⡇⣿⡿⣋⣭⣬⣭⡻⠟⣩⣥⣭⣝⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⡇⣿⢱⣿⣿⣿⣿⣷⡞⢿⣿⣿⣿⣇⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠛⠛⠉⣴⣿⢸⣿⣿⣿⣿⣿⢣⣎⠿⠿⢿⡏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⣿⣆⢳⣶⣶⡙⡟⣸⣿⣿⣿⡿⣱⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⣭⡝⣿⣷⣌⠻⣿⣤⣿⣿⠟⣫⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⢸⣿⢠⣤⣤⣤⣤⣤⡄⣿⡇⣿⡇⣛⢻⣿⣿⣮⡙⢟⣵⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⢸⣿⠈⠛⠛⠛⠛⠛⠁⣿⡇⣿⡇⣭⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⣛⣡⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⢠⣾⡇⠀⠀⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣄⠀⠀⣾⠟⣆⢀⡾⠁⣷⢀⡼⠟⣿⠀⢀⣠⣴⡞
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢽⡏⢳⣀⡟⣰⡘⠿⣡⢿⣘⢟⣵⡇⣷⡶⢛⢉⡟⠀⠀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠈⣿⡶⣄⣸⡇⣷⣭⣥⣏⣹⣻⣯⣬⣿⣯⣼⣿⡶⢞⣏⢸⠷⠞⠛⣻⠟⠁
⠀⠀⠀⢠⣀⠀⠀⢹⣇⢢⣝⣥⣿⣬⣿⣿⣽⣿⡻⣾⣟⣷⣿⢿⣟⣿⣿⡗⣻⡏⣼⣡⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠻⣿⠓⠶⠿⣜⣧⣽⣿⣿⣟⣟⡾⡷⣽⣟⣼⣿⣹⣾⣿⣹⣯⡻⣿⠿⢥⣤⢌⣭⠟⠉
⠀⠀⠀⠀⠀⠙⢷⡙⣟⢛⣿⣿⣟⣿⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣧⡻⣝⣷⣿⡷⠿⢶⣤⣄⣀⡀
⠀⠀⠙⢿⣟⠛⣛⣋⣾⣿⣿⣟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣸⡿⣷⣭⡟⣂⡤⠿⠛⠋⠁
⠀⠀⠀⠀⠙⠷⣬⣻⣆⣿⣻⢿⣷⣾⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣵⣏⡷⣿⡻⣍⡛⢶⣄⡀
⠀⠠⢴⣶⡖⠛⣭⣼⠟⣿⣻⡿⣷⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⣿⣯⣿⢿⣽⣟⣿⣿⣾⣟⣩⣤⡬⠿⠷⠄
⠀⠀⠀⠉⠛⠳⢶⡍⣿⣾⣿⣾⣿⣯⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⣻⢿⣿⣙⣦⡹⣦⣀
⠀⠀⢀⣠⣴⡞⠋⠶⠯⣤⣿⣽⣷⡷⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⠿⣿⢿⡟⢿⢿⣮⠥⣬⣿⣦
⠀⠀⠈⠉⠉⠛⠛⢛⡿⢸⣟⣻⣮⣳⣿⡟⣷⣿⣏⣿⣟⢾⣯⣻⣟⣻⢟⣽⣿⣿⠾⢧⠹⣆⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⣴⣋⣀⣩⣭⡽⣿⢻⣷⣿⣟⣳⡼⠿⣬⡿⣶⣞⣭⡿⣿⣷⣼⡌⣿⠓⠦⣿⣇
⠀⠀⠀⠀⠐⠉⠉⠉⠉⢁⡞⠱⢛⣫⡻⣿⣩⣿⣿⢛⣿⣿⢋⡿⠾⣶⡏⣾⢮⡁⢿⡆⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡶⠟⠋⣹⠃⢋⣵⣧⢹⠟⣼⡍⠿⢱⡿⣎⠁⣿⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣿⡾⠋⠁⣿⣠⡾⠃⢻⣠⣿⠁⠘⢿⡟⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣿⠟⠀⠀⠈⣿⠃⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⢀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣤⣤⣤⣀⡀⠈⠉⠀⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢀⢾⣧⣤⣼⣿⣽⡧⣿⣤⡀⠀⠀⠀⢠⣿⠟⠉⠀⠀⠈⠙⢿⣷⣄⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢸⡾⡼⣅⡋⢻⣿⢻⣾⣩⠻⡄⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢻⣷⠂⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠋⣿⣿⢻⣿⣛⣟⠳⣿⡇⡴⠃⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢆⠠⠉⠫⣐⣹⣤⡷⠛⡟⠓⠀⠤⠀⠈⠷⣽⣦⣤⣤⣴⣾⣟⠿⠁⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠉⠛⠒⠚⠙⠃⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠉⣀⡀⠤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠤⠤⠤⡴⠀⡴⠤⠤⠤⠄⠀⠐⠒⠐⠀⠙⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠃⠀⠀⠀⠀⠉⠢⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⢛⣋⠉⠉⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠈⠁⡢⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠠⠶⡄⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠒⢄⡀⠀⠛⠠⢄⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠸⣿⡿⠛⠻⣿⡇⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠈⠀⡀⠱⠀⠀⠀⢠⠇⠀⠀⠀⠀⠑⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠘⢳⡋⠃⠀⠀⠀⠀⠀⠀⠙⠂⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⣴⠲⡔⢄⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⠔⢊⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠁⠀⡸⢀⣈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⠈⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠄⠀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠑⢲⠊⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠃⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⡟⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡩⠒⠢⠄⠠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⢀⣠⡤⠒⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⠲⠤⢄⡤⠴⠒⠒⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉⠉⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⡆⠀⠀⠀⠀⠀⣠⢋⡜⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠀⡴⠒⢄⢀⣠⠗⠉⢀⣀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡄⠑⠒⣺⢡⠃⠀⢀⡎⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠘⢄⣀⡤⠊⠑⠤⠖⠁⠘⠦⠴⠋⠳⠔⠃⠘⠦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠀⠀⠀⠀⠀⠀⣠⠎
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⡟⠀⡠⡄⢠⣤⠀⠀⣠⡉⠀⠀⣠⡦⢦⠀⠈⣿⡔⠒⠆⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠁⠈⠀⠈⡏⠀⠀⢀⡷⠁⠀⠀⡸⠀⡼⠀⢀⠇⠘⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⠤⠚⠀⠣⠔⠂⠀⠣⠤⠴⠊⠳⠴⠚⠐⠃⠀⠘⠒⠊⠀⠀⠉⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠊⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠦⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠔⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣀⣀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡄⠀⠀⠀⠀⢨⣯⣿⣿⣿⠀⠀⠀⠀⠀⢠⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣝⢿⣿⣷⡀⣠⡖⢲⣾⣿⣿⣿⣿⣷⡖⢲⣄⣰⡟⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠹⣎⢿⣿⡷⢻⠛⠛⠉⠱⠎⠀⠱⡉⠛⠛⠟⢉⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠛⢿⡟⢺⡷⢠⡥⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⡛⣽⡷⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡌⣁⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⣀⣀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⣆⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣽⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⢰⣰⣤⡖⢦⣤⣢⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠰⣏⣩⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠶⢶⣶⣶⣾⣻⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠰⣄⣰⠈⢃⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⢤⡾⠟⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡁⠁⠀⠀⠀⠀⠀⠀⠀⢾⡁⢀⣺⡇⣿⣿⢘⢘⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠈⢘⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢫⣿⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣄⠡⠉⠩⠈⡭⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣪⣿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣍⣽⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠩⠛⠷⠞⠛⠉⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⢐⣏⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠸⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠊⠉⠉⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢠⡌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠲⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠶⠿⡟⢘⡓⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⣥⡻⠷⢴⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣾⣿⡷⣼⡴⢦⣤⡰⢂⣀⡐⣥⡴⢴⣦⣈⢿⣿⣿⡄⠈⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⣾⣿⡿⠁⠙⠷⠾⢻⣿⣿⣿⣿⠟⠷⠾⠃⠙⣧⢻⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⠂⠀⠀⠀⠀⢘⣛⣿⣿⣿⠀⠀⠀⠀⠀⠘⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠉⠉⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠀⠀⠀⠠⢤⡤⣖⠂⠀⠀⠠⠄⠴⠶⠦⠤⠀⠀⣠⠀⠀⠀⠠⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⣀⣤⠄⠀⠑⠀⠃⢀⠁⢬⠀⠀⠉⠀⠀⣿⣿⣿⣿⡆⠀⠀⢀⣴⣶⣾⣃⡌⠀⠀⣤⣄⣾⣾⣷⡖⠐⢀⣀⢀⠀⠀⡀
⠀⠀⠀⠄⠀⠀⠈⠀⢠⡽⣷⣿⣷⡄⣄⠀⠀⠀⡄⡀⠀⠀⠐⢣⣐⣒⢛⡁⠈⠛⠫⡄⢀⣩⢿⡭⣿⣷⣄⡘⢷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠿⠛⠁
⠀⠀⡀⣀⡀⣂⡀⠀⠈⠉⠙⢻⣿⣷⣿⡑⠄⠀⠈⠐⣀⢒⡊⠳⢦⣬⣽⣲⣦⠀⠀⠀⠀⠙⠷⣾⠛⣿⣿⢥⣿⣿⣿⣿⣽⣿⣿⡿⡟⠛⠋⠁⠀⡀
⠀⠀⠁⠈⠙⢿⡗⢀⣀⠠⠺⣿⣿⡿⣿⣆⣐⡀⠚⣤⣞⢎⣻⠟⠋⠉⠉⠁⠀⠀⠀⣀⣀⠀⢈⠊⢻⣿⣿⣟⣛⣿⡿⢿⣏⢿⣿⢾⣿⣿⣵⣾⣶⡀⠀⠂
⠀⠀⠀⡀⣀⣠⢷⣈⣵⣾⣷⣿⣻⣿⣿⣿⣿⣽⡿⠿⣆⣠⣼⣶⣤⣤⣤⣤⣶⣶⣿⣿⣿⡆⠀⠀⡀⣿⣿⣾⣿⣿⣿⣿⣷⣿⣧⣿⣿⣿⣯⣿⣿⣷⣶⣶⣦⣄
⠀⠀⢁⠈⠵⡄⢘⣛⢘⣿⣿⡿⠯⢿⣿⡍⢉⢽⣆⣐⣋⣽⣮⢻⣮⣭⣶⡈⠉⠉⠉⠉⢿⣿⣶⡷⠛⠙⢿⣿⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢀⣀⣁⠉⠾⢿⣯⣭⣭⣯⣭⣍⢿⣆⣔⣀⣉⣁⣚⣳⣾⣷⡙⢿⡟⠳⠎⠉⠉⠀⠈⠛⢿⡷⣄⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⡻⣿⣛⣾⠿⢿⠟⠉
⠀⠐⠛⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠙⠛⠛⠛⠋⠉⢡⡿⢆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓
⠀⢀⣀⣀⣀⣘⣻⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣀⢑⠒⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠉⠓⠟⠛⠓⠐⠛⠾⠶⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣭⠹⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠒⠂⠂⠐⠂⠐⠀⠂⠀⠀⠀⠀⠐⠒⠀⠀⠂⠀⠒⠀⠀⠀⡂⡀⠓⢂⡐⠂⣀⣀⣂⡀⠂⠒⠐⠀⠂⠐⠒⠐⠒⠂⠐⠒⠀⠀⠒⠀⠒⠒⠒⠒⠒⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⠉⠉⠉⠛⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠋⠉⠉⠉⠛⠓⢲⢤⣄⡀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣀⠀⠀⠀⠀⢠⣦⠰⣿⠇⠀⣭⡓⠦⣄⡀⠀⠀⠀⠉⢳⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠥⠼⠃⠀⢸⣻⣮⠟⠛⠙⠲⢼⡟⠁⠀⡀⠉⠓⢦⡀⢀⡟⣦⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⡄⢀⣤⢀⣸⠃⠀⠀⠀⠀⠀⠓⢤⡿⠋⠀⠀⠀⣙⡾⠀⣇⣀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠖⠃⠀⢹⡏⠀⠽⠂⠀⠀⠠⢶⡆⠈⢣⡴⠶⠂⠈⠉⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣯⡿⠁⣠⣄⣠⣄⣀⣠⡀⢀⡀⢸⠀⠀⠀⣄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠳⡀⠀⠀⠁⠀⠉⠀⠈⠉⠀⢈⡷⠤⠼⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠤⢤⣤⣄⣠⡤⣶⡿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠋⠙⠲⢶⣶⠶⢾⣋⣉⠀⢸⣧⠳⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⣨⣧⣄⣠⣼⣧⣀⣀⣈⣉⣠⣼⡇⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⣿⢻⠀⢠⡄⠀⣄⠀⠀⠸⠀⠀⠀⣴⢃⡜⠉⠛⠒⣶⠶⢭⡉⠁⠀⢸⡇⠀⢳⡀⢘⡆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⡇⠘⡆⣸⢧⢠⢿⡀⠀⠀⠀⣠⠂⣿⢿⡁⠀⠀⢸⠁⠀⠀⠙⠒⠦⣼⡇⠀⠀⠙⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣳⠈⢠⡄⠙⠃⠘⠋⣸⠁⠀⠀⠐⠇⠀⢁⣸⠧⠤⠤⣼⡀⠀⠀⢰⣧⡖⠃⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⠀⡼⠙⣆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠙⠒⣾⣲⣶⣏⣙⣛⣻⣛⣲⠭⣟⡷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠁⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠰⣯⣥⣤⠤⣾⠿⠭⢞⡿⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣧⣀⣀⣠⣧⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢡⢾⠉⡏⢇⠙⣍⠪⣝⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠳⣴⠃⣰⠖⢾⡘⡶⠋⣳⠸⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠉⠒⠋⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠄⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣀⠐⠘⢹⢼⣉⠏⠇⣿⣘⠺⠲⢦⣆⣀⢤⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠐⢒⣂⠀⣎⡇⡸⢺⣴⣤⡏⡟⣷⠎⡼⢀⠞⡰⢃⡖
⠀⠀⠀⠀⠀⠀⠀⠈⣾⡶⣤⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠏⠿⣿⠘⡸⢁⡜⣠⢃⡜⢡⠏⡴⢡⠞⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣹⣿⣻⣷⣰⣄⠈⠭⠀⠀⠀⠐⠒⠀⠀⠈⠀⠏⡴⢁⠞⡰⢃⡞⣠⠋⡼⢡⠎⡴⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⢧⣿⢧⣿⣗⢕⡤⠀⠐⣒⡂⠀⠀⠨⠔⣁⠄⠀⠉⠐⢁⠞⣰⢃⡞⣠⠋⡼⢡⠏⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣾⡏⣿⣿⣿⡫⣪⢧⢄⡀⠀⠀⠤⠄⠈⣡⠞⣡⠀⠀⠀⠀⠁⠞⣰⢃⡜⢠⠏⡼⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣽⣿⣿⣿⣮⢖⢕⡣⡲⣀⠀⠈⠉⠍⣡⠞⣡⠞⣠⠀⠀⠀⠀⠀⠈⠰⠃⡞⢡⠋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣗⡵⡫⡪⣚⢕⢎⡥⠀⠀⡠⠊⡠⠚⡠⠞⡀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡪⣪⢞⢕⡷⡫⡪⣤⠄⡰⢋⡴⢊⡤⢊⡠⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢞⢕⡕⣫⢪⠞⡞⢵⣫⣶⡋⡴⢋⡴⢋⡴⢋⡴⢂⡀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣾⡺⢻⢉⡇⠟⠿⢹⢸⣫⣄⠈⠐⠋⠴⢋⡴⢋⡴⢂⠀⠀⠀⠀⠀⣼⠏⣰⠆⠀⠀⠀⢪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⢷⣏⣾⣼⢸⣡⣧⡎⢟⣵⣦⠀⠀⠀⠀⠀⠋⠰⠉⡴⢀⠀⢀⣾⠃⣼⠏⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⣿⢿⣶⡏⣎⣤⢵⢳⠿⣫⣾⡿⣀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡾⢁⣼⠃⣼⠋⣰⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣿⣷⡿⣫⣾⢿⣭⡼⡘⣾⢟⣽⣿⢟⣥⡀⠀⠀⠀⠀⠀⢠⡿⢁⡾⢁⣾⠃⣼⠏⣠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣿⣾⣧⠫⣱⡻⠿⣷⡇⣷⡿⣻⣿⡿⣫⣷⣄⠀⠀⠀⣠⡟⢠⡾⢁⡾⢁⣾⠃⣼⠋⣤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⣾⣿⣧⢣⢳⣻⣺⡜⢽⣿⣿⣯⣾⣿⣽⣿⢇⡄⣰⠟⣠⡟⢠⡿⢁⡾⢁⣾⠃⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⣿⣿⣯⠮⢷⢹⠱⢶⣬⣿⣿⣿⣿⣿⡟⠟⠟⣫⢰⠏⣠⡟⢠⡿⢁⡾⢁⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⢦⢋⠷⠶⣤⠍⠛⠉⠉⠈⠈⠘⠛⠿⢿⣿⣹⣰⣟⣠⡿⢁⣾⣿⣿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢯⡞⠳⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠛⠛⠷⠿⣿⢯⣾⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠈⠉⠉⠉⠉⠉⠰⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢁⡄⢶⠋⠉⠉⣙⢻⠋⠑⠶⣤⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡾⠋⠹⣄⢳⡀⠀⣿⢸⠀⠀⠀⣠⠟⣦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢰⠏⠀⠀⠀⠙⣆⠙⣦⡁⠸⠀⡴⠚⣡⠞⠁⢻⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡤⠄⠀⠲⠶⠒⠒⠒⠒⠒⠒⠒⠲⠂⠀⠀⠄⢠⡏⠀⠷⠠⠤⢤⣤⣤⣬⣧⠘⠁⠘⢁⣀⠊⠁⠀⠀⠀⢳
⢀⣠⠔⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠗⠒⠲⠢⠤⣤⣤⡀⠀⡀⠀⠈⣉⠉⠐⠒⠒⠂⣾
⣿⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢀⣴⣋⣤⡾⢻⣆⠙⣏⠉⠉⠉⠉⠀⢶
⣿⠀⠀⠀⠉⠙⠓⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣵⣤⣵⡦⢲⡶⠛⡩⠛⠉⡄⢸⡟⢶⡈⢷⡀⠀⠀⢰⠇⢰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠘⢂⡀⠈⠀⠀⠀⡇⢸⡇⠀⠙⣎⡱⣄⣰⠋⢠⠟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠑⠤⣀⡀⠀⣧⢸⡇⠀⣀⣨⡷⠉⢁⡴⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢈⠉⠁⠘⢻⡟⠈⣉⣁⣤⠖⠉
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢺⠟⠀⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿
⠀⣷⢠⡞⠙⠛⠦⠴⠦⠤⠖⠛⠛⠉⠉⢉⣛⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠛⠋⠉⠙⠛⠶⣄⠀⢠⡏
⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠁⠀⠈⢣⠉⠛⠳⠶⠶⠶⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇
⠀⢻⠀⣧⠀⠀⠀⠀⠀⠀⡞⠋⠀⠀⠀⣠⠂⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠇
⠀⢸⡆⢿⠀⠀⠀⠀⠀⣾⠀⢀⣀⠴⠚⠉⢣⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿
⠀⠀⡇⢸⡄⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇
⠀⠀⣷⠈⡇⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡹⠶⠖⠒⠐⠀⠒⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⢹⠀⣧⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡧⠀⠀⢀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⡇⣼
⠀⠀⢸⡇⢻⠀⠀⠀⠀⠀⠈⢷⠘⢦⡀⠀⣀⠴⠞⠁⠀⡧⠀⠀⠀⡟⠉⠉⠉⠁⠀⠀⠈⡇⠀⠀⠀⠀⣼⠁⣿
⠀⠀⠸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠑⠶⠞⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⢠⡏
⠀⠀⠀⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⢟⠀⡀⠀⠁⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢀⡏⢸⡇
⠀⠀⠀⢻⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠉⠳⢮⣵⣏⠀⠀⠀⠀⠀⠀⠀⢸⠅⡇⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠘⡆⢿⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢐⣇⡀⠀⠀⠀⢹⡗⠶⠠⠤⠤⠀⠀⠀⠶⠁⠀⠀⠀⣼⢠⡇
⠀⠀⠀⠀⣇⠸⡇⠀⠀⠀⠀⠀⢰⠋⠀⠀⢠⠏⠀⠉⠳⢦⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠇
⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⢃⣦⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠀⠘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟
⠀⠀⠀⠀⠀⣧⢸⡇⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⡄⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠇
⠀⠀⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣄⣀⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡇
⠀⠀⠀⠀⠀⠀⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠁
⠀⠀⠀⠀⠀⠀⢻⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢙⡆⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠞⠋⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⢦⣀⠈⠙⠛⠛⠲⠶⠶⠶⠶⠶⠶⠖⠛⠛⠉⠁⣀⡤⠾⠋⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠉⠛⠳⠶⠦⢤⣤⣀⣀⣀⣀⣤⠤⠴⠶⠛⠛⠉⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠦⢀⣀⣀⣀⣀⣀⣀⣀⡤⠶⠶⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠛⠓⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠴⠶⠒⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⢳⢏⣠⡄⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠞⣉⠁⠀⢀⣤⣉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⢀⡎⠈⡇⢠⡏⠀⠘⡆⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣠⠖⠒⠚⢿⣆⡇⢸⣦⣄⡠⠟⠋⠉⠁⠀⠈⠉⠂⠀⠀⠀⠐⠋⠉⠀⠀⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠉⠀⣇⠀⠀⠀⠀⠩⠤⢬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠘⢦⣀⡀⠠⢤⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠒⠀⠲⠤⠤⠀⠄⠀⡀⣾⣀⡀⠀⠀⣰⠛⠓⠚⠛⠃⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠨⣉⣡⡽⢀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡚⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠤⠤⢶⡀⠈⠁⠈⠳⠶⠛⠦⣤⣀⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠁⠁⠀⠀⠀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⢤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣸⠷⣄⠀⠀⠀⠀⠀⣀⡤⠟⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠶⠛⠁⠀⠈⠙⠲⠶⠒⠛⠉⠀⠀⠈⠓⠶⠤⠤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄⡀⠀⠠⡀⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⡄⣤⣬⣽⣶⣯⣶⣿⣷⣮⣷⣾⣭⣤⡀⣄⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣲⣿⣾⣿⣿⣿⣿⠻⣿⣿⣿⡿⢻⣿⣿⣿⣿⣾⣷⣒⡂
⠀⠀⠀⠀⠀⠀⠰⣤⢀⣆⣾⣿⣿⣿⡟⠻⣿⣿⣟⣔⣛⠿⢿⣃⣜⣿⣿⡿⠟⣿⣿⣿⣿⣆⣇⢠⡴
⠀⠀⠀⠀⠀⠀⠀⠵⣾⣿⣿⣿⣿⣿⣗⣮⣜⠁⠀⢿⡷⢀⡰⣿⠇⠀⢙⣬⣒⣿⣿⣿⣿⣿⣿⡶⠅
⠀⠀⠀⠀⠀⢀⢰⣿⣿⣿⣧⣈⢏⡏⠀⢘⠻⠅⣮⡎⠉⢷⠏⠉⣮⡆⠽⢛⠀⢈⣏⢏⣠⣿⣿⣿⣷⢀
⠀⠀⠀⠲⡶⣸⣼⣿⣿⣿⣿⠗⠋⠿⠧⠳⢷⣠⣼⣷⡿⢿⠿⣷⣿⣤⣰⠷⠣⠿⠏⠓⢿⣿⣿⣿⣿⣼⡰⡶⠂
⠀⠀⠀⠈⠔⣻⣿⣿⣍⣹⢺⣤⡄⡮⣶⣤⣾⠿⢃⣮⣾⣶⣾⣬⣆⠻⢿⣦⣴⡮⡆⣤⣼⢚⣉⣽⣿⣿⡑⠄
⠀⠀⠀⢠⡸⣿⣿⣿⣿⣷⠃⠙⡩⣄⢰⣿⢛⣼⣿⠟⠉⠉⠉⠛⢿⣿⣜⢻⣷⢀⡬⡝⠁⢳⣿⣿⣿⣿⡿⣠
⠀⠀⢠⣤⢳⣿⣿⡟⠛⡘⣴⣦⡉⠋⣽⡟⢠⣿⡏⠀⠀⠀⠀⠀⠈⣿⣧⠘⣿⡍⠋⣥⣶⡜⡘⠛⣿⣿⣷⢣⣤⠄
⠀⠀⠀⢠⢏⣿⣿⣿⣿⣼⠈⠁⢖⡷⢺⣿⠠⣿⣷⡀⠀⠀⠀⠀⣰⣿⠧⢸⣿⠲⣗⠂⠉⢸⣾⣿⣿⣿⣏⢧
⠀⠀⠀⠀⡘⢿⣿⣿⡿⢯⢣⣾⡖⣤⣬⢿⣿⣘⠻⣿⣷⣶⣶⣾⡿⢛⣸⣿⢯⣄⡔⣾⣦⢫⠻⣿⣿⣿⠟⡀
⠀⠀⠀⢀⣌⢿⣿⣿⣶⣾⣿⡀⠀⣑⣉⢀⣹⢿⣧⣵⣜⣩⣙⣵⣥⣿⢿⣁⢈⣑⡁⠀⣸⣽⣶⣾⣿⣿⢋⣔
⠀⠀⠀⠉⠉⠸⢸⣿⣿⣿⣿⠯⡳⡟⠃⠋⢋⡀⣼⡏⠛⣻⡛⠋⣿⡄⣈⠋⠃⠛⡷⡫⢿⣿⣿⣿⣿⠸⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠈⣛⣿⣿⣧⣴⣷⣷⡤⢼⡿⠂⠑⣡⣖⠫⢓⣦⡑⠁⠺⣿⠤⣴⣷⣷⣤⣿⣿⣟⡋
⠀⠀⠀⠀⠀⠀⢀⣮⠹⡿⣿⣿⣿⣿⡏⣑⣼⣦⡤⡿⢯⣀⡨⠿⡷⣤⣾⣔⡉⣿⣿⣿⣿⡿⡿⢩⣆
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠙⠿⣿⣿⣿⣿⣿⣿⣿⢈⣿⣿⣿⣏⢸⣿⣿⣿⣿⣿⣿⡿⠟⠁⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣭⡻⡝⢿⣟⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠟⡽⢫⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠚⣭⡓⠍⠛⠋⠕⣫⡝⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⣟⣶⣶⣶⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⣿⣿⣄⠀⣠⢤⣠⣿⣿⣿⣿⣠⡤⣄⠀⣼⢫⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡈⢻⡝⣿⣿⡞⡿⠶⠛⢫⡝⠉⢩⠛⠶⢿⠟⣣⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⢾⣧⢨⡅⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠲⠉⠻⢻⡶⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡦⢥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢤⣶⣤⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⠶⠏⠳⢦⡄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⣞⠙⠉
⠀⠀⠀⠀⠀⠉⣽⣯⣭⣥⡍⠞⠀⠀⠀⠀⠀⠀⢀⡠⠀⢄⡄⣷⣾⠀⡥⠀⢠⠀⠀⠀⡄⠀⠀⢠⠶⢩⣭⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠿⠿⠿⣿⣇⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠃⠿⠿⠀⠆⠀⠘⠀⠀⠀⠃⠀⠀⠘⣠⣸⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⡚⢦⣤⣼⠴⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢧
⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⠿⠗⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠘⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠿⡟⢘⡃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⣬⡿⠷⢴⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⣿⢧⣷⠶⣤⣜⣃⣀⣘⣤⠶⣶⣦⡹⣿⣿⣆⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣵⣿⣿⠏⠀⠙⠚⠋⣿⣿⣿⣿⠙⠓⠋⠀⢻⡜⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⣯⢿⠿⠿⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⡀⠀⢰⣦⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠃⣶⣿⣿⠿⠷⢶⡟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣾⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⣿⣶⠶⢿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⠀⠀⠀⠀⠀⢈⣿⣛⠿⣿⣯⡀⠀⠀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣿⣿⣻⣦⣼⣿⣿⠿⣿⣿⡟⠰⣿⣶⡀⠀⣿⡉⠉⣦⡈⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣆⠉⣹⣿⣿⣿⣿⣷⣶⣿⠿⠆⠀⠛⠛⠋⠀⠘⠟⠀⠿⣷⣽⣿⣿⣿⣏⠙⠻⢿⠃
⠀⠀⠀⠀⠀⠀⠀⢾⣶⣤⣀⠀⢹⣿⣿⣿⠿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠤⣤⣤⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡹⣿⣿⣷⣾⣿⣿⣧⣀⠙⠋⢙⡏⠉⠛⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠛⠐⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡟⢿⣿⣀⣄⡀⠀⣶⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⢿⣿⠉⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠻⣿⣷⣿⣿⣶⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⢿⣿⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠈⠻⣿⡏⠛⠿⠿⠿⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⠀⠀⠀⠀⣦⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠂⣤⡛⠁⠀⠀⣼⣿⣶⣾⣿⣷⣦⣄⠀⠀⠀⠀⢀⣀⣴⠿⣿⡿⠻⡿⠓⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⠀⠹⣿⣀⡀⠀⠀⠟⠉⣹⠿⣿⣿⣿⣿⣦⠀⠰⠟⠉⠛⠒⣾⣷⣿⣿⠶⠶⠀⠀⠀⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠈⠋⠀⠀⢀⣠⣾⡿⡄⠈⣽⣿⠿⠂⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣼⣇⣰⣷⠀⠀⡀⠈⢿⣷⠀⢈⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⡷⠺⣿⡶⠈⠛⠋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⡛⢿⣿⣿⣿⠃⣴⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠲⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⠗⠿⠷⠿⠿⠿⠿⠃⠸⠟⠁⠀⠶⠂




⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣷⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⢠⣤⠄⣤⣄⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣷⣶⣆⢠⣴⣤⣄⠀⠀⠀⡀
⣿⣿⣿⣿⡏⠀⠀⣿⣿⡿⠿⣿⣿⣿⡿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣤⣿⣿⣶⡀⠀⠁
⣿⣿⣿⣿⣷⡄⠀⠛⠋⠀⣀⠈⠉⠀⠀⠈⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷
⣿⣿⣧⣾⣿⣿⣶⠆⢠⣤⣤⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⡿⠿⠇⢀⠀⠀⠀⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⡟
⣿⣿⡇⢰⠆⠀⢀⠘⠃⠈⠙⠃⠠⣾⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣯⣀⣶⣿⣄⡀⢠⣶⡶⣤⣄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠿⣿⣿⠿⣿⠏
⠙⣿⣿⣿⣿⣿⣿⣿⣿⡝⠛⠁⠈⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣿⣷⣄⠘⠛⣠⡏
⣿⣯⣿⣿⣿⣿⡿⢙⣿⣷⣄⣀⣀⣀⣀⣠⣤⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⠟⠀⠀⠘⣿
⡄⠙⢻⣿⣿⣿⣧⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠇⠀⠀⠘⢿⣿⣿⣿⣤⠇⠀⣴⠏
⣀⠀⠀⠘⠿⣿⣿⡟⠛⢿⣿⣿⣤⣾⣿⣿⣿⡟⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⣸⡿
⠛⢧⡀⠀⠀⠈⢻⣿⣶⡶⣿⣿⠿⣿⣿⣿⣿⡀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠉⠠⠀⠀⠛⣿⡟⠀⢀⣀⠀⠀⠁⠀⠀⠉⢿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠇
⢧⣄⠀⠀⠀⠀⠀⠀⠀⢠⠿⠛⢧⣤⡄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⣠⣤⡀
⣾⣿⣷⣆⠀⠀⢸⡆⠀⠠⠿⢷⠈⠉⠀⠀⠓⢀⡀⢀⣦⣤⣶⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣤⣤⣴⡿⢿⣿⣶⡖
⣿⣿⣿⣿⣦⠀⠀⠀⢰⣤⡄⠉⢀⠘⠃⠀⠀⣾⡿⠟⠛⣉⣩⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠉⣻⠋⠉⠛⣿⡿⠿⣳⣾⣿⠛⠋
⣿⣿⣿⣿⣧⣤⡄⠀⠈⢿⣷⡄⠀⠀⠀⠀⡀⠉⠀⢠⡾⠿⠿⠿⠿⣿⣏⠛⣿⣿⣿⣿⣿⣷⣤⣄⠀⢀⣤⡀⢹⣿⣿⠿⠿⠿⢿⡿⢿⣷⣤⠿⣿⣷⣴⠺⠗
⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⢿⣿⣶⡀⢠⡞⠃⠀⠀⡾⠃⠀⢀⣰⣶⡌⢻⣿⠿⣿⣿⣿⣿⡿⠋⣉⠀⠈⡉⠛⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣯⠐⣿⣿⣿⡿⠟
⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠘⠿⠇⠀⠀⠀⠠⠆⠀⠰⣿⣿⣿⣿⠋⣄⠉⠀⠙⣿⣏⣹⣷⡾⠟⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣿⣿⣁⣿⣿⣴⡟⢻⣧⣤⣤
⣿⣿⣿⣿⣿⡿⢿⠏⣸⡆⠀⠀⣀⡀⠀⠀⠀⠀⣀⡀⢼⣿⣿⡿⠏⢀⡟⠀⢸⣿⣿⣿⡟⠉⠁⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⡾⣿⣿⠟
⣿⣿⣿⣿⣿⣷⠖⢲⣿⣷⠀⠘⣿⣷⡀⠀⠀⠀⠁⠀⠀⣹⣯⣠⣴⡏⠀⠀⢀⣿⣿⣥⣴⣶⣶⣶⣶⣾⣿⣿⣿⣏⣹⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⠓⠛⠉
⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣤⣴⣿⣿⣧⠀⠀⠀⣴⣶⡀⠘⢿⣿⡟⠁⠠⠀⢈⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣷
⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠇⠐⠀⠿⣿⠇⠀⠼⠿⠿⠿⠿⠇⠈⠛⠛⠆⠈⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣤⡉⠙⠛⠶⠤⠄⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢛⣾⣿⣿⣿⣶⣦⣄⠀⠀⠈⠛⠉⢁⣀⣤⡄⠀⠀⠀⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣾⣿⣏⠁⡘⠻⣦⣀⣿⣿⣿⣿⣿⣿⣿⠛⣤⡀⠻⢃⣬⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣽⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⣿⣿⣧⡍⠉⢩⡻⠿⠿⣿⡆⠺⣿⠏⠙
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⢿⣟⠻⢿⣁⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠉⣀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣠⣖⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣷⣾⠿⣿⣷⣾⣿⣿⡿⠿⣿⠟⣙⡿⣁⣤⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠛⠻⠟⠛⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⢉⣽⣿⣿⣿⣿⣿⣿⣷⣶⡈⢿⣿⣿⣿⣿⣿⣷⣄⠀⠘⣿⠟⠁⠀⠘⠻⠿⣆⠀⠀⠀⠀⠀⢹⣿⣿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣤⡿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⣽⣿⣿⣼⠿⢁⣤⣴⢷⣾⣿⣷⣿⣟⣥⣿⣿⣿⣿⣿⣿⣿⡶⣶⣶⣶⣦⣤⡄⠀⠀⡄⢿⣿⡟⠉⠻⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠰⡿⢁⣼⣿⣏⣠⣼⡋⣁⡾⢿⣿⣿⡿⠿⠛⠁⢈⣹⣿⣿⠛⠉⠀⢸⣿⣿⣿⣿⣇⠀⠀⣠⣼⣿⣧⢀⣀⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⠀⠀⠚⠿⡟⣻⣿⣿⣽⣿⣷⣿⣟⣿⣧⣄⣤⣾⣿⣿⣿⣿⣄⣀⠀⠘⠛⠛⠿⠟⢉⡀⠀⠀⠈⣿⣿⡀⠙⠻⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶⣄⣠⣾⣿⣿⣿⣿⣿⣿⣏⣟⠛⠻⣿⣿⣯⣹⣿⠿⠟⠛⠛⣁⡀⠠⠆⠀⠀⠛⠃⠀⠀⠀⠀⢹⡗⠀⠀⠙
⠀⠀⠀⠀⣶⣦⣀⣀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠹⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠈⠛⠛⠲⣿⣿⠄⢠⣶
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣄⣀⣿⣿⣿⣿⣿⣿⡁⠀⠀⢸⣿⡿⠛⠻⣿⣿⡟⠙⢿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁
⠀⠀⠀⠀⢠⣀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⢿⣿⣄⡀⠈⠉⠀⠀⢀⣀⣉⣀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠘⢿⣿⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⠟⠛⠻⢶⣶⡾⠿⠛⠿⠟
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠄⠀⢸⣿⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣘⣋⣬⣤⣽⣭⣥⣿⠘⠟⣉⣿⡶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⣮⣿⣿⠿⢿⢿⡿⠿⠻⠿⠛⠛⢻⣧⣉⣼⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢟⣷⡞⠛⡞⣳⢤⣶⠝⣷⣾⣇⠀⠶⠆⢼⣷⣿⣇⢹⣿⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⡶⢶⠿⠿⣻⢯⡶⠷⠛⠀⣠⣄⣛⣭⡝⣿⣶⣦⣾⣿⠌⢿⣧⠾⣿⣍⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠉⠉⠀⠀⠈⠻⣿⣣⢿⣿⡐⠿⣁⣹⣿⡿⠏⢀⣠⣼⣿⢻⣷⢺⡏⣷⣦⠈⣽⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠃⠸⣾⣷⣦⠀⠀⢠⠈⢿⢿⣾⣿⠞⠋⠑⠚⣿⣶⣾⣿⡿⣿⠘⢟⣸⣷⣍⠛⣵⢿⣷⠋⢋⠽⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠋⠀⠀⠀⢿⣿⡿⠀⠀⠀⠘⠄⢀⠅⠈⠙⢦⣤⣜⣿⢿⣟⣩⣥⡼⣷⣿⣻⣿⣿⣷⢿⡆⣿⣶⣿⡔⠸⡆⠀⢠⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⣤⣀⣀⠀⠀⠁⠀⢠⣴⣦⠀⢀⠎⠀⠀⠀⠀⠙⣆⣁⢛⣿⣤⣷⡆⢹⣿⣯⣿⣼⢛⣾⣷⣾⣿⣿⣽⣧⣿⠀⠀⠁
⠀⠀⠀⠀⠀⣸⣿⣯⣿⣿⣿⣿⣿⡻⣷⣦⡀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠙⣿⣯⡶⠻⢿⣿⣏⣻⢋⣿⠿⣏⡻⢿⣿⣿⣿⣇⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠰⠛⠁⣀⣠⣤⣤⣄⠀⠀⠀⠈⢿⢻⣛⠌⢿⡿⠂⡉⢿⣶⠛⢡⣤⠈⠛⢿⡇⠱⣾⣿⡀⠀⡀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⢶⣿⠹⠿⣈⢍⣁⠸⢛⡦⡀⠀⠘⣿⠿⢿⣦⣿⣤⣶⣼⢿⡇⡘⣿⡿⠀⢈⣷⣌⡻⢼⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⢿⣿⠀⣥⠼⠒⠅⠠⠼⣽⣢⡾⢿⣶⣄⡘⣖⡀⢼⣿⣿⣿⣇⣼⣷⣤⣄⣄⣤⣾⣿⣿⣝⢾⠇⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣮⣿⣼⣷⣿⣿⠁⣿⣰⠎⠁⠀⠀⠀⠀⠀⠈⣿⣦⣤⣿⣿⣿⣿⣿⣿⣇⡿⣿⣯⣿⣭⣭⣬⣱⣻⡿⣿⣿⣮⣻⡇⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⣿⣿⣿⣿⡀⠱⢧⠀⠐⡄⠀⠠⡀⢀⢰⡟⣫⠟⠻⢯⣠⣿⢻⣿⣿⣿⡿⣿⡏⠉⣩⡍⠹⣿⣷⣿⡸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣧⠀⠈⡆⠀⡧⠠⠐⢋⣿⣿⣿⡅⠁⠀⢸⣿⣿⣿⣿⣾⣿⣽⢻⡗⠈⢹⣿⠀⣿⣿⣮⣛⡖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⣿⣿⣿⣾⣿⣿⠀⠀⢰⡀⡇⠀⠁⠘⢸⣿⣿⣭⡶⣖⣫⣼⣿⣾⣿⣿⣿⣿⣿⡃⠀⠚⡁⢄⣿⣿⢿⣯⣟⣿⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⡇⠁⠀⠀⠀⠠⣿⣿⢿⣿⣷⣿⣿⣿⣏⠈⣹⣿⢏⣟⡳⠶⢶⣤⣾⣿⢿⣭⢿⣹⠏⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⣿⠀⠙⢾⣿⣿⣿⣿⣿⣿⣿⢾⣉⣿⡖⣎⣟⣻⠯⣞⣳⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣬⣻⣿⣿⣿⣿⠁⡀⠀⠀⢸⡇⠀⠀⠔⢲⣿⣿⡀⠀⠀⠈⠻⢷⣽⡻⢿⣿⣚⢿⣿⣹⣿⣿⣿⣿⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣿⣿⣿⣿⠀⠃⠀⠀⠘⣷⠀⠀⠀⣸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣟⣿⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠻⣿⢿⣆⠘⠀⠀⠀⢻⣂⠀⠀⣿⣿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⣿⠀⠀⠀⠀⠸⣟⠀⠈⣿⣧⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠛⡀⠀⢿⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡏⠀⠀⠀⠀⠀⠘⣴⠀⢸⣿⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠡⣟⠘⠟⣷⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣾⢈⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⣿⢿⡚⣾⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⢿⡇⠀⠀⠀⠀⢠⡟⢰⣌⡏⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⣼⣃⡿⣿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⢸⣟⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣷⡄⠀⠀⠀⠀⠀⠀⠐⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢰⣿⣿⣿⣶⣦⡄⠀⠀⢠⣴⣴⣿⣿⣿⡆⣀⣀⡀
⠀⠀⠀⠀⠻⣦⡀⠀⣠⣦⡀⠀⠀⠀⠰⣿⣿⣿⣄⣙⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣋⣠⣿⣿⣿⠆⠀⠀⠀⢀⣴⣄⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠈⠻⠞⠁⠀⠙⢠⣶⣿⣆⠻⢿⣿⣿⣿⣿⣟⣉⣩⡀⠀⠀⢀⣍⣉⣻⣿⣿⣿⣿⡿⠟⣰⣿⣶⡄⠋⠁⠈⠳⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣦⡙⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡿⢋⣴⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⢿⣿⣿⣧⣙⠛⠃⠀⠀⠘⠛⣋⣼⣿⣿⡿⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⡁⠺⣿⣿⢿⣿⣿⣿⠆⠀⠀⠰⣿⣿⣿⡿⣿⣿⠗⢈⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣴⣾⣿⣷⣼⣿⡘⠿⠿⠛⠀⠀⠀⠀⠻⠿⠿⢃⣿⣧⣾⣿⣷⣦⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠈⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠤⣄⡀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣰⠉⠉⠓⠲⢤⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⡀⠀⠀⠀⢢⠀⠘⢶⡖⠢⢄⣀⠀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⣰⢿⣦⣠⠤⠬⣃⣀⡜⠀⣔⣉⡴⠷⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣳⠦⠤⠬⣿⡆⠀⠀⠀⣀⠝⠀⢰⣷⣏⣀⣀
⠀⠀⠀⠀⡜⣱⠋⣠⠶⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⠠⡤⠈⠓⠒⠒⠉⠁⠀⠀⡷⢋⣡⡤⡌⢳⠀⡇⢸
⠀⠀⠀⠀⠀⡇⢸⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⡴⠋⠠⢁⠃
⠀⠀⠀⠀⠀⠉⠀⠑⠦⣥⣈⣁⣈⣀⣤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡆⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡆⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠁⠇⠀⣀⠤⠒⠊⣿⠫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⢹⠛⢲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠖⣉⠀⠄⢀⣀⣸⡯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⡰⠧⠤⢄⣀⣙⣢⣾⣠⣏⣙⣆⣀⣀⣠⣤⠤⠤⠤⠞⣆⠿⢋⡀⠒⠒⠢⢄⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⢙⡇⠠⡄⡈⠑⣄⢀⣴⠿⣦
⠀⠀⠀⠀⡰⢃⠖⣠⠞⣁⣸⠧⠤⠿⠭⠭⠭⠭⢉⣉⡉⢉⡩⠟⠉⠀⠀⠀⣠⠞⡇⠰⠁⠇⠀⠘⢟⠗⠕⠀⠁
⠀⠀⠀⠀⠃⣏⢰⡁⢊⡔⠀⣀⣀⣠⣤⠤⠶⠒⠒⣲⡞⠋⠀⠀⠀⠀⣠⠞⠁⠀⢿⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡏⠉⢹⠀⠈⠓⠦⣀⣰⠁⡇⠀⢀⣠⠴⠋⠁⠀⠀⠀⢸⡀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠆⡼⣸⠀⠀⢸⠀⠀⠀⠀⠈⠙⠲⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⣜⠀⢇⠻⣄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠇⢘⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡿⠿⢷⡶⠒⠒⠒⠒⠲⣶⡶⠶⣶⠟⠋⠉⠉⠁⠀⠈⠉⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠤⢾⠀⠀⠀⠀⠀⠀⣧⡤⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⠭⠿⠀⢯⠀⠀⠀⢀⣔⠾⣕⡶⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠁⠀⠀⠛⠒⠩⠥⠤⠬⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠪⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠭⣂⠀⠀⠀⠀⠀⠀⠀⠐⢂⠤⣂⠤⣀⣾⣽⣵⣣⠀⠀⠀⣀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢞⣖⣄⣀⡀⠀⠠⠭⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢔⣫⠵⣫⢿⣿⣿⣧⠸⢹⡼⣼⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢞⣞⣞⡼⡴⢮⣩⡐⣆⣠⣤⣤⣦⣶⣴⡴⡶⣴⡐⠀⠛⠘⠱⠛⣿⣎⣧⠥⠚⠁⠋⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⢾⣞⡞⠁⠲⣝⢿⣾⣿⣿⢿⡿⠁⡌⢿⣽⣵⠃⠀⠀⠀⠀⠀⠀⠙⣿⢛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠒⠒⣒⡺⣿⣟⢯⡫⡗⠀⠀⡜⢜⠎⠁⠀⠀⠀⠀⠀⠀⠀⢐⣚⣯⣟⣶⡄⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡻⣕⠏⠀⠀⠀⢎⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣲⣿⣿⣿⣓⡀⠀⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠯⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣻⣿⢿⣟⣒⣧⣴⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⢤⣶⣲⣬⡑⣔⢄⡢⡐⠀⠀⠀⠀⠀⠀⠀⠐⠰⡱⡱⣣⣶⢽⣲⠍⢹⣺⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠱⢦⠀⠀⠠⠭⣽⣿⣺⣿⢬⡢⡑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡱⢱⣿⣿⡺⠀⠐⣺⣿⡇⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠈⠭⡽⢓⣿⡯⣑⠍⠀⠀⠀⠀⢀⢀⣤⢀⠀⠀⠀⠀⠀⢁⣿⣿⣿⠀⠀⠀⢸⣿⣿⣝⠞⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠵⣿⣏⢀⢀⠠⠰⡐⡜⡌⢼⡳⣟⠄⡂⡉⣇⣧⣾⣿⡟⠁⠀⠀⠀⢸⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣦⣇⣇⣇⡣⣗⢿⠪⢳⣐⣟⢿⢿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⠰⠰⠡⢹⣿⣿⣿⡏⠉⠈⠉⠀⠀⠘⠘⠸⠸⢼⣮⣿⣿⣿⢸⣾⣴⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣟⣯⢥⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⡿⣿⣾⣾⡞⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣳⣿⣿⠀⠀⠀⠀⠀⡠⡴⣽⣾⠿⣷⣳⣯⢽⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠋⠂⠀⢀⡰⠒⠉⠈⠉⠛⠿⣷⣺⡯⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠌⠒⠒⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣌⢸⡇⢀⣀⠠⠤⠐⠒⠒⠒⠒⠲⠤⢤⣀⠀⣿⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣃⡼⠛⠉⠀⠀⢀⣠⠀⠁⠀⠀⠀⢀⣀⠈⠙⢻⣅⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⠿⣄⡀⠀⠀⠀⢋⠄⠒⠒⢄⡀⠒⠒⠠⡹⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠦⢤⡀⠀⠀⠀⡞⠀⠀⠀⠈⠷⠤⠘⠐⠁⠀⣤⡀⠀⠀⣠⣄⠀⠐⠰⡦⠏⠻⡀
⣦⣄⡀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠉⠢⡀⢀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠐⢀⠉⢀⠀⠃⠚⢅⡠⠂⠀⠀⠄⠀⡇
⢹⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠙⠦⣀⣠⡴⠧⠙⠢⠄⠀⢀⣀⡀⠀⠀⠀⠀⣇⣠⡴⢦⡀⢀⠇
⢸⡆⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⢛⠛⠛⢿⡀⠀⣀⠤⢾⠁⠀⠀⠐⠠⠀⡈⠙⠻⠶⠶⠖⠊⠀⡀⠠⠀⣻⠁
⠀⣇⠀⣀⡠⣶⡿⠛⠁⡠⠀⠀⠀⠀⠀⣠⠝⠋⠀⠀⣸⣇⠠⠀⡀⠀⠀⠀⠈⠉⠁⠈⠉⠁⢀⠠⠐⠈⣸⠳⢤⡀
⠀⠉⠁⠀⠀⠘⡆⡠⠊⠀⠀⠀⢀⡤⠊⠁⠀⠀⢀⡾⠁⠈⠲⣄⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢧⡀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣠⠞⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⡖⠒⠲⠒⠚⠉⠉⠀⠀⠈⣷⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠶⣿⣷⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⡾⠓⠤⣀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠙⠻⠿⠿⢷⡚⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⣤⣶⠿⠋⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢠⡇⠀⠀⠀⢀⣠⠶⣦⠀⠀⠀⣿⣷⡈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠋⣷⡞⠉⠀⠀⡶⢤⣀
⠀⠀⠀⠀⠓⠶⠶⠖⠋⠀⠀⢻⡀⢀⣠⠇⠈⢳⣦⣀⡀⠀⠀⠉⠙⠛⠲⠲⠶⠶⠶⠶⠶⠶⠒⠛⠉⠉⠀⠀⣰⠟⢧⠀⠀⢀⡇⠀⠈⠓⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⡇⠈⠙⠲⠦⣤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠚⠋⡇⠀⠈⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠙⡓⠒⠈⠉⠉⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣰⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠓⠄⠀⠀⠀⠀⠀⠀⠀⠠⢳⠆⠀⠀⠀⠀⠠⠞⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠠⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡘⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠒⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣁⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠃⠂⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠨⠈⠏⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢠⠟⠢⠄⣙⣦⣴⢮⠥⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⣇⣤⠞⠛⠫⣟⣷⡀⠒⠠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⢶⣿⠻⠛⢳⣶⠿⠽⣤⣿⣦⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡠⠊⠡⡿⠉⢲⣾⣡⢈⣦⢾⡟⢉⡿⣍⠳⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠢⢤⣰⣧⣞⡹⣿⢟⣻⡒⠿⣥⡎⢀⡽⡄⠘⢎⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⠒⡟⢿⡷⣰⡧⠤⢍⣧⣾⣧⠚⢉⣄⣿⠖⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢠⣇⠈⠻⣟⣛⣻⣿⣽⡿⣟⡿⡛⠻⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⣨⠭⠛⠷⢶⠚⢳⡀⠀⠈⠁⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠓⠦⠤⠖⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⢀⣀⣴⣤⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡚⠂⣤⣸⣿⣿⣩⣔⣁⠋⢩⠙⢆⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣼⠖⠶⣿⣿⣿⣧⣮⣿⣭⠽⣧⣤⢾⣧⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣔⣹⠟⢛⣺⡿⣿⣻⣿⣽⣯⣧⣚⣉⣽⣾⣇⡉⢌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢁⣮⣷⣞⡧⣶⣩⣿⣿⣿⣯⢝⣯⣩⣿⣿⡞⣷⠺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣘⡔⢹⣿⡟⢯⡿⣿⣿⣿⣿⣾⣿⣿⣯⣷⡿⣧⣜⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⢀⣼⠉⣷⣾⢻⣿⣧⣿⡿⣿⡿⢏⣿⢿⠛⡟⠁⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⣏⡷⣾⣿⣿⣽⣋⢿⣿⣿⣶⡏⠘⣾⣷⠇⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⢛⣛⣿⣟⢭⣿⢿⡿⢷⣿⣦⣰⣟⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢾⣱⣆⣹⣺⠃⣭⣽⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠉⠉



⡿⠿⠿⠿⠿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣟⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⠅⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠞⡞⣆⢦⢾⢶⠆⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠳⡘⠂⠃⢲⣤⣤⣶⣶⣶⣿⣿⠿⣛⡻⣆⠀⠈⠽⡼⡼⡜⠀⠘⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠿⠋⠁⣜⢴⣯⣪⣢⡀⠀⢡⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⢦⣶⣤⣤⣦⣀⡔⢿⣿⣿⠟⠋⠀⠀⠀⠀⣿⢿⣾⣮⣪⡿⡄⠀⠰⢃⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣦⡄⠂⠀⠀⠀⠠⢗⣝⣽⣟⢯⣾⣾⣦⡀⠚⣡⡟⣰⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣳⣿⣯⣾⡪⡫⡀⠋⣴⢋⡼⢃⡶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣟⣿⢿⣿⣿⡮⢡⣾⣡⣟⣰⣏⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣻⣽⣿⣿⡿⠯⡑⣿⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⢀⡀⣤⠰⣆⡻⠾⢳⡿⢟⡓⢌⡓⣲⣿⣿⣿⣿⠻⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⣷⣢⢄⠀⠀⠻⣈⢷⣘⣧⣬⣮⣆⠻⢍⡲⢌⡲⣟⣽⣿⣿⣏⣙⣻⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣄⡗⡸⣿⣿⣿⣿⣿⣍⠢⢍⣲⡿⣩⡿⠿⠦⠤⣬⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣿⣿⣻⢋⠋⡟⡻⣻⠿⣿⣷⣝⢲⣟⣮⣺⣛⣛⠒⠶⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⣿⣽⣟⣿⣾⣴⣣⡃⠎⣜⡹⠷⣭⣚⢵⣿⣿⣍⣉⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣻⣿⣿⣿⣿⣿⣯⡿⢿⣧⠊⠝⠿⡿⣿⣿⣯⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⣿⠟⠋⠀⠀⠉⠒⠄⢻⠖⠳⢯⣽⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⢠⠶⠾⠉⠉⠉⠷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠿⣿⠖⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⣠⠴⠿⢤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠏⠀⠀⢘⣇⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠘⠁⠀⠀⡞⠁⠀⣷⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⠞⢋⡍⠛⢦⡀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠺⣇⠀⠀⠛⠒⠂⣸
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⠃⠀⢨⣇⣀⡀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣧⡀⠀⠉⠉⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣶⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢛⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠘⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠛⣿⣿⡿⠟⣛⣉⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣼⣿⡟⣴⣿⣿⠛⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡿⢋⣥⣶⣶⣦⣄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡀⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢱⣿⣿⣿⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⢸⣿⣿⣿⣬⣭⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⠻⠛⡟⣻⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡘⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⠀⠀⠀⠀⠀⠀⠘⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢧
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠱⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇
⠠⠴⠐⠐⠚⠁⠉⠉⠉⠁⠉⠉⠁⠀⠐⠒⠀⠀⠤⠤⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠠⠠
⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⠁⠈⠐⠀⠒⠒⠐⠂⠒⠒⠋⠉⢉⠅⠤
⠀⠀⠀⠀⠀⠀⢀⣀⠐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠉⠉⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢀⠞⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⡀⠀⣰⠇⣰⡇⠀⣰⣆⠀⠀⠀⣠⡶⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢺⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠰⡏⠀⢠⡇⡜⢱⣣⠎⠿⡟⠀⡠⠊⢰⠇⡴⠚⡇⡴⣿⠀⣠⡆⠀⠀⠀⠀⢳⡀⠘⡦⢘⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⢠⠇⠀⢸⠏⠀⠸⠃⠀⠀⣷⠊⠀⠀⢸⠏⠀⠀⠟⠀⣿⠞⢹⢃⣤⠀⠀⠀⠈⡇⠘⣇⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⣸⠀⠀⣾⠀⠀⠀⢀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠏⢹⢀⢼⡇⠀⣿⠀⢇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡼⠁⠀⢀⣿⣤⣴⠾⠟⠋⠉⠛⣆⠀⠀⠀⠀⣀⣽⣦⣄⡀⠀⠀⠀⠸⠋⢸⠃⠀⣿⠀⠘⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⠟⢸⠁⠀⠀⠀⠀⢀⢸⡆⠀⠀⡜⠁⠀⠈⠳⣝⢦⡀⠀⠀⠀⢸⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢻⠀⠀⠀⠀⠰⠿⣸⠁⠀⣸⠀⠀⠀⠀⠀⣈⠙⡿⠒⠀⠀⢸⣠⠼⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢦⡼⠁⠀⠀⠀⠈⠳⣤⣄⣀⡤⠞⢁⡀⠀⢿⡀⠀⠀⠀⠰⠟⣠⠇⠀⠀⠀⣿⣣⢚⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡤⠋⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠋⣭⠀⠀⠀⠉⠓⢦⣤⡤⠞⠁⠀⠀⠀⢸⠛⢋⡽⠋⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣗⠺⠭⣄⠤⠼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢳⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠦⣄⡀⠀⠸⣆⠀⣠⣄⣤⡴⠶⢻⠖⢦⠞⣯⠤⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢯⡀⠀⠀⠀⠀⣀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠶⠶⠯⢤⡶⠀⠈⡻⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⣦⣤⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢽⣭⢭⡭⠭⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠠⠮⠤⠖⠊⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣃⣠⠴⠋⠁⠀⣀⠀⢦⣤⣉⣓⣦⣤⠀⣀
⠀⠀⠀⠀⠰⡇⠀⠀⣻⠄⠀⠀⠀⠀⠀⣀⣀⣀⡈⠳⣄⠀⠉⠉⠛⠓⠒⠒⠒⠒⠒⢒⣒⡿⠿⣍⠹⡄⢠⣄⠀⢹⠓⢬⣦⡀⠉⣩⠗⢴⠋⢠⡄
⠀⠀⢀⠀⠀⡇⠀⠀⠙⣦⠀⠀⠀⠀⠚⠯⢤⣉⡉⠙⠛⠒⢶⣶⡴⢲⣄⡤⣄⠀⣜⠉⠀⠀⠀⠈⠳⡝⣼⠀⠳⣼⡀⠀⠈⠉⠀⣳⢀⡼⠁⣾
⠀⠀⠈⢣⡀⠈⣢⠀⠀⠷⢤⡄⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠈⡇⢧⠀⠀⠀⠙⠉⠙⠖⠚⠳⠒⠲⠖⠿⠈⠀⠀⠈⠁⠀⠀⠀⢼⠁⣨⠇⢶⠁
⠀⠀⠀⠀⡇⠀⠙⢦⠀⠀⠈⠳⢦⣤⡤⠶⡄⠀⢀⡀⠈⢦⣤⣇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠈⣧⠀⠼
⠀⠀⠀⠀⠘⡆⠀⢎⠀⠀⠀⠀⠘⣧⠀⠀⠙⠋⠉⠓⠒⠋⠉⠉⠓⢦⢤⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠋⠀⡴⠁
⠀⠀⠀⠀⢀⠃⠀⠈⣷⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠙⠚⠙⠲⠛⠶⠶⡤⢦⠖⢶⡿⠙⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠛⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠉⠛⢻⣿⣷⣿⢿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡿⣛⣵⣴⣦⣀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⠿⠟⣷⣿⣿⣿⣿⣦⣀⡀⠀⠀⢽⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⠅⠒⠄⣶⣍⡻⢿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⣿⣿⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢼⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⠀⠀⠐⡟⢡⡿⣻⣿⣿⣿⣿⣿⠶⣄⣀⡲⢿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠆⠀⢘⢆⡾⣱⣿⣿⣿⠻⣿⢿⠑⠈⠹⣙⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣏⡿⠃⡭⡸⣿⣿⡿⠋⠙⣿⡝⠧⡄⣰⡀⣻⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣧⡞⣹⡇⢠⡶⠐⠻⣿⣿⣧⣶⢹⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣹⣿⣿⣿⠟⠁⢹⣿⠜⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡟⠉⠀⠘⣸⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⠯⠀⠀⠀⠀⠿⠼⠻⠿⠿⠿⡿⠿⠿⠿⡿⡿⠿⠇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣉⣁⠂⢀⣠⡤⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠈⡧⢸⠁⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⠀⠀⠓⠚⢁⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢁⡎⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⣊⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠃⡘⠠⠊⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⠁⠀⠉⠉⢹⡆⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⢠⠄⠀⠀⠀⠀⠀⠀⢼⠁⠀⠲⠤⠀⠀⠀⢸⠷⠛⠛⠛⣦⠀⠈⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⠙⡆⠀⠀⢀⡖⠻⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠸⠶⣄
⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠈⠓⠋⢁⠆⠀⢸⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠠⠾⢤⡀⠂⠒⠒⠢⣄
⠀⠀⠀⠀⠀⠘⣆⣰⢠⠀⠀⠀⠀⠀⠀⡰⠋⢀⡀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣀⢀⣴⣾⠶⠿⠓⠒⠦⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠉⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⢷⡄⠀⠀⠀⣸
⠀⠀⠀⠤⠤⠒⠚⠉⣀⠤⠒⠀⠀⠀⠈⠉⠑⠒⠲⠶⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠤⠐⠁⠀⠓⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⢀⡖⠀⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠐⠋⠒⠒⠶⠦⠤⠶⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⣀⠈⠀⣠⡟
⠀⠀⠀⠀⠀⡠⠔⠊⠁⢀⣀⠠⢤⣀⠀⠀⠀⠀⠒⠢⢄⣠⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠺⡝⢠⣄⠈⢿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⡙⢦⠀⠀⠀⠀⠀⠈⠑⠄⠀⠐⠶⠖⠂⠀⠀⠀⠀⠀⢹⡆⢸⣧⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡔⠀⢹⠄⠃⠀⠀⠀⢸⡋⠉⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣘⠒⠊⠀⠀⠀⠀⠀⠀⠻⣈⡠⢆⠀⠀⣶⡄⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠃⠹⣄⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⣾⠁⢠⡘⢤⠄⢊⡑⠠⣤⢋⡄⠈⣷⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠈⢳⠦⢸⡛⠤⡴⠒⠒⢦⠴⢛⡏⠴⡾⢁⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣡⡘⡟⠖⠺⠆⠠⣾⡛⢨⡥⠛⣷⠄⠰⠗⠲⣻⢋⣌⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢂⠠⠴⢅⠰⣦⠤⠘⢉⠀⠸⠇⠀⠉⠁⠤⣴⢆⡨⠧⡄⡁⠲
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣦⠰⣎⠰⢬⣄⠁⠀⠘⠇⠀⣱⣆⠀⠸⠃⠀⠈⣠⡥⠆⣰⠆⣴⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠨⢴⣦⠛⠹⣏⡌⠈⠃⠬⡤⡄⠀⢡⡌⠀⢠⢤⠥⠘⠁⢡⣹⠍⠛⢴⡦⠅
⠀⠀⠀⠀⠀⠀⣶⢦⣠⠌⣧⡀⠉⠇⢰⣄⠐⠉⠘⠓⣤⣥⠚⡃⠉⠂⣠⡆⠸⠉⢀⣼⠡⣤⠴⣶⢀
⠀⠀⠀⠀⠀⠀⠋⠉⢨⡥⡂⠀⣦⡇⠈⠀⠁⡶⡰⢛⢨⡍⡛⢂⢶⡀⠀⠁⢸⣬⠀⢐⢮⡅⠉⠙
⠀⠀⠀⠀⠀⠀⠀⢨⠍⣭⠲⡼⠢⢤⡤⠆⠈⠈⠅⣂⢜⡡⠀⠠⠉⠉⠰⢤⡤⠴⢥⡖⣭⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⣈⢡⡘⣩⢁⣤⡐⠒⢇⠀⠘⠃⠀⡸⠒⢂⣤⡈⣍⣃⡈⣁⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢃⠠⣬⡙⠇⠙⠁⡁⢐⣤⠄⢸⡇⠀⣤⡂⢉⠈⠋⠸⠋⣥⠔⡘⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⡼⠗⢊⠚⢃⣑⠹⣗⠡⠎⣲⡏⢂⡌⠓⡙⠺⢇⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢾⠉⢸⡖⣩⡙⠋⡘⢋⣍⢲⡧⠈⡳⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠓⠀⠈⠰⠀⠩⠠⠄⠍⠈⠆⠁⠀⠚⠡


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠶⢶⡶⠶⠀⣶⠶⢶⣦⠀⢰⣶⠀⠀⢠⣶⣆⠀⠰⠶⣶⡶⠆⢰⡆⠀⠀⢰⡆⠀⣶⡆⠀⠀⣠⣶⠾⠷⣶⣄⠀⢰⣶⣄⠀⠀⣶⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣤⣼⠟⠀⢸⣿⠀⢀⣾⠏⢿⡆⠀⠀⣿⡆⠀⢸⣿⣶⣶⣾⡇⠀⣿⡇⠀⢠⣿⠁⠀⠀⠈⣿⡆⢸⣿⠻⣧⡀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠉⢿⣆⠀⢸⣿⠀⣼⡿⠓⠻⣿⡄⠀⣿⡇⠀⢸⡇⠀⠀⣸⡇⠀⣿⣇⣀⡀⢻⣧⣄⣠⣴⡿⠀⢸⣿⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠉⠁⠈⠉⠈⠉⠀⠀⠀⠈⠁⠀⠉⠀⠀⠈⠁⠀⠀⠈⠁⠀⠉⠉⠉⠁⠀⠈⠉⠉⠁⠀⠀⠈⠉⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠉⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠉⠉⠉⠉⠛⠲⣤⡀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠉⠉⠙⠛⠦⣄
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⢀⣀⠀⠈⠻⣆⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢳⡄
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠈⣷⠀⢰⠇⠀⠀⠀⠀⠀⣀⣤⣜⠛⠀⠀⠀⠸⡆⠀⣾⠁⠀⠀⠀⣠⡤⢤⣄⡛⠁⠀⡀⠀⢻⡀
⠀⠀⠀⢸⡇⠀⠈⠀⠁⠀⣁⣈⣤⣿⠀⠿⠂⠁⢸⡄⣿⠀⠀⠀⠀⠀⢾⡟⣥⠉⠓⠀⠀⠀⠀⣿⠠⡇⠀⠀⠀⠀⠃⣠⣿⠟⠙⠒⠋⠀⠀⢸⡇
⠀⠀⠀⠸⣇⠀⠀⠀⠈⠉⠁⠈⠉⠉⠀⠀⠀⠀⣼⠀⢻⡀⠀⡰⣿⣾⡂⢻⠉⢴⣿⣮⡂⠀⢀⡟⠀⣧⠀⠀⠀⢀⡴⠛⣻⠆⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠈⢷⡀⠀⠙⠟⠃⠀⠀⠐⠙⠟⠁⢀⡾⠁⠀⠸⣆⠀⣀⠊⠀⣈⣁⣀⣀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠻⢦⣀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠶⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠐⣀⠤⠒⠒⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠀⢀⡱⢺⡅⣊⠙⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠔⠁⠀⠀⢀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠐⠃⢈⡤⠂⠉⠀⢀⡏⠁⡠⠛⣘⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⣞⣤⡶⠋⠁⢸⢥⡴⢿⣕⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠲⠤⠂⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣲⣿⡾⠏⢀⣀⡃⣐⠎⣤⣼⢏⠀⠀⠀⢀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠠⡀⠀⡀⢿⣁⡲⢎⣱⣀⣼⢤⠶⡽⢾⣀⠀⠠⠒⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠶⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠢⡼⢏⠉⠉⢉⠍⣳⠾⢶⢿⡟⡤⠁⠀⠀⠠⣀⡀⢤⣼⡤⠒⠀⠀⠤⠀⠀⠀⠀⡸⠤⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⣀⡡⣂⡱⠂⠈⢾⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠒⠖⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠻⣷⣶⣤⣄⡀⠀⠀⠀⣀⣠⣴⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠻⡏⠙⢿⣆⢀⡴⠋⠁⠀⠈⠉⠛⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⡇⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠛⠛⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣤⡴⠾⠋⠉⠉⠻⢦⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⣄
⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠙⠛⠿⠶⣤⡀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⣀⣀⡀⠀⣀⣴⣿⠃⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⠦⣤⣄
⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢻⣿⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢠⣿⣧⣄
⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠁⠀⡹
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣤⣤⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⠀⢹⣿⡏⠁⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⡿⣿⣷⣿⣿⣿⣄⡀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⡿⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⢸
⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⢰⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⣿
⢹⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⣧⡈⠙⢿⣿⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣴⠿⠻⠿⠁⠀⠀⣿
⠈⢿⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣍⣷⠶⠀⠀⠀⠀⢠⣿
⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣿⣿⣿⡌⠻⣿⡙⠛⠛⠿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢿⡟⣿⡉⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠒⠒⠓⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠓⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠏⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡉⠃⠀⠉⠿⠁⢀⡞⠉⠉⣹⡅⠛⢐⣧⡿⢋⡿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠟⠿⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⣤⡶⣶⣦⡾⠟⢿⣦⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⣾⠟⠿⣦⣴⠶⣦⣄⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⢠⣿⠁⠀⠀⣉⣤⣶⠾⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢷⣦⣌⡁⠀⠈⢻⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⠿⠛⢋⣠⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡈⠛⠻⢶⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⣄⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠘⢷⣄⠀⠀⣿⠁⠀⠀⠀⠀⢀⡀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠙⠃⢀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣴⠟
⠀⠀⣸⡟⠀⢠⣿⠀⠀⠀⠀⠀⣾⠃⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⢻⡆⠀⠀⠀⠀⢸⡗⠀⠘⣿⡀
⢠⣿⠋⠀⠀⢨⣿⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡗⠀⠀⠈⢻⣧
⠈⠻⣦⡀⠀⠈⣿⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⢹⡄⠀⢸⡇⠀⢸⡇⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢸⡇⠀⠀⣠⡾⠋
⠀⢀⣾⠃⠀⠀⣻⣧⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⣤⣤⡴⠾⠋⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠈⠻⢶⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⣠⣠⣿⡃⠀⠀⢻⣆
⣴⡟⠁⠀⠀⣼⠟⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠙⢿⡄⠀⠀⠙⣷
⠙⢷⣄⠀⠀⣿⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣸⡇⠀⠀⣴⠟
⠀⣸⡟⠀⠀⢻⣆⠀⠀⠀⢻⡇⠀⣠⠾⠛⢛⣿⠇⠰⣶⣶⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣶⣶⡶⠀⢿⣟⠛⠻⢦⡄⠀⣾⠇⠀⠀⢀⣿⠃⠀⠘⣿⡀
⠀⣿⡁⠀⠀⠀⠻⣦⡀⠀⢸⣧⢸⣇⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⣿⠀⣿⠀⠀⣠⡿⠁⠀⠀⠀⣹⡇
⠀⠈⠻⢷⣦⠀⠀⠈⠻⠷⢾⣧⠀⠙⠷⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠁⢈⣿⠾⠟⠋⠁⠀⢠⣶⠟⠛
⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠈⣿⠀⠀⠀⢠⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⡁
⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⢻⣧⡀⠀⣿⣀⣿⠀⠀⣠⡶⢦⣄⣀⣤⣄⡀⠀⠀⣀⣤⣄⣀⣤⠶⣦⡀⠀⢸⣇⣸⡇⠀⣀⣿⠁⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣦⣌⡉⠁⠀⠀⠉⠀⠀⠙⠋⠁⠈⠛⠛⠛⠋⠀⠉⠙⠁⠀⠈⠁⠀⠀⠉⣉⣤⡾⢿⣥⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠉⠛⣷⣶⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⡿⠋⠁⠀⠀⢹⡇⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠛⢿⣦⠀⠀⢀⣿⠀⠀⠀⢀⣾⣿⡛⠿⠟⠉⢻⡿⠛⠿⠛⢷⣶⣶⠿⠿⠛⠛⢛⣿⠋⠉⠀⠛⣿⣦⠀⠀⠀⢿⣇⠀⠀⢠⡾⠛⠋
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢸⣏⠀⠀⠀⣿⣯⠉⠁⠀⢀⣴⡾⠿⠿⠶⠾⣿⣯⣄⣀⠀⠀⠀⣼⡟⠀⠀⠀⠸⣿⡀⠀⠀⠀⢸⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠛⣷⡀⠘⣿⣉⠀⢠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢶⣿⣧⣄⣀⠀⣾⡟⠁⠀⠀⣰⡟⠁⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⠀⠀⣿⠁⠀⢼⣯⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠃⠀⠀⠀⣿⠇⠀⠀⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠹⣧⠀⠀⠉⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠉⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶⣦⡀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⣴⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠀⠈⠙⠻⠷⢶⣦⣦⣴⣶⠾⠛⠉⠙⠻⠿⠶⣶⣤⣤⣄⣀⣀⣀⣠⡾⠋⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠾⠿⠿⠿⠷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣀⡤⠶⠶⠶⡶⠛⠛⠛⠻⠶⠛⠛⠛⠶⣌⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠙⢳⣄⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣾⡏⠀⡤⠖⠛⠛⠓⠀⠀⠀⠀⠈⠉⠉⠉⠙⠲⠄⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢠⡿⠀⠘⠶⠶⠶⠚⠁⠀⠀⠀⠀⠘⠓⠲⠒⠛⠀⠘⣧⠀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⢸⡇⠀⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⣼⡇⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣠⡤⠀⠀⠀⠀⠀⢠⣿⢁⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣬⣿⣄⣀⣀⣀⡀⠀⠀⠀⠉⣉⣁⣀⣀⣀⣠⣤⣴⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠛⠉⠉⢙⣷⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠟⠁⠀⠀⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⠀⣠⡴⠟⠋⠁⠀⣦⣤⡾⠿⢿⡟⠷⢶⣶⣶⣶⣿⣿⡟⢛⣛⣩⣥⣤⣴⠶⠶⣤⣄⡀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⢀⣾⠏⠀⢀⡴⠟⢛⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢻⣆⠀⠈⠉⠛⠛⠉⠁⠀⠸⣇⠀⠀⠀⠈⠙⠳⣦⡀⠹⣿⡄
⠀⠀⠀⠀⣾⣣⣴⠿⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠹⣆⢻⣷⡀
⠀⠀⢀⣼⡿⠋⠁⠀⠸⣇⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣄⣀⣀⣼⡃⣿⣇
⠀⣠⡾⠋⠀⠀⠀⠀⠀⠹⣆⠀⠈⢷⣄⡀⠀⣀⣀⣀⣀⣀⣀⣀⣼⣧⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢷⣼⣿
⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⢈⣿⡏⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐
⠀⢀⠴⠚⠉⠍⠍⣂⠤⣄⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⡀
⢠⢇⠞⡀⠐⠶⢀⢄⠑⠜⡂⠀⠀⠀⠀⠘⠁⠀⠀⠀⠰⡄⠀⠰⠠⡖⠂⠀⠀⠢⠱
⣞⠇⢰⠹⠀⣶⡂⠿⢀⡸⡇⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⢂⡄⠀⠘⠁⠀⠀⠀⠄
⠹⡰⡈⠢⣑⣀⡻⠆⣠⣉⠃⠀⠀⠀⠀⠈⠀⣂⠀⠀⠐⠰⠁⠋⠠⡔⠈⠙⠀⠀⡀⠀⠀⠀⠒⠬⠐
⠰⣷⣬⣒⠂⠄⣀⣺⣽⡁⠀⠀⠀⠀⠲⠂⡀⠙⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⢝⠹⠿⠻⠦⠉⠻⠴⠀⠀⠀⠀⠀⠐⠈⠀⣨⠀⠀⠀⠒⠰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⡄⠀⠀⠔⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠀⠀⠐⠚⢂⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⢀⣴⡇⠀⠀⠀⢀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⣾⣿⡆⠀⢀⣾⠏⣷⠀⢀⣤⡿⣿⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣄⠀⢠⣿⢁⢻⣄⣾⢋⡄⢻⣰⡾⢋⠀⡟⢀⣤⣶⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢐⣤⡀⠀⠀⢸⡇⠈⢷⣼⡏⣸⣇⣙⢃⡾⢿⣬⣉⡴⢿⡘⠷⠟⣉⡄⣼⠃⠀⣀⣠⣠⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢽⣟⠶⣄⣸⣇⢸⣧⣭⣥⡟⣈⣿⣻⣵⣬⣿⣿⣥⣾⣿⡷⠛⣿⡐⡿⠾⠟⠛⢫⡿⠁
⠀⠀⠀⠀⣀⠀⠀⠀⠈⣿⡀⣬⣙⢛⣼⡇⣽⣿⣷⣛⣽⠼⠶⢶⡚⠶⠴⢭⣟⡻⢾⣿⠿⠖⣻⠟⣴⣋⣀⣀⣀⡀⣀⢀⡀
⠀⠀⠀⠀⠘⢿⣷⠶⣤⣘⣧⠸⡏⢛⣿⣿⣯⡷⣻⣭⣶⣝⢾⢿⢹⡿⡝⣧⡭⣽⢳⣝⠻⣾⣿⣇⣋⣋⣋⠛⣛⡿⠟⠉
⠀⠀⠀⠀⠀⠈⢻⣆⠰⣬⣥⡾⣿⣾⣫⡿⡵⠶⣿⡢⣽⢎⣯⢧⡟⣇⡗⣏⣇⡿⡲⣮⢳⡜⢿⣏⣽⣟⢡⣞⡉
⠀⠠⢤⣀⣀⣀⣀⣙⣷⠜⣧⡰⡿⢹⣧⡶⠿⣯⣛⣭⣺⣷⣷⣿⣧⣯⣿⡿⣽⡹⡷⢽⢶⡝⣎⢿⡾⣯⣭⣉⠛⠿⢷⣶⣤⣀
⠀⠀⠀⠙⢿⣯⡉⢭⣭⣼⢿⣿⢷⣟⣸⡿⣭⠿⣿⣯⣿⣿⢟⣟⣛⡻⣿⣿⣿⢿⢝⡧⣟⢦⢻⡌⣿⣴⡾⢋⣥⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠙⠻⣦⡙⢷⣜⣿⣸⠇⣷⣿⣭⣽⡷⣾⣿⢷⣿⠛⠛⣿⡎⣿⣿⣿⢿⣺⢥⣾⢡⡇⣿⡛⢧⣉⠛⢶⣤⡀
⠀⠀⣤⣤⣶⠶⠒⣛⣡⡿⠛⣷⣺⡼⣭⢖⡯⣝⣻⣿⣿⣽⣿⣤⣤⣿⣿⣿⢿⣍⣟⣺⣏⡽⠏⡇⣾⣷⣶⣛⣓⣀⣉⣿⣷⣆⡀
⠀⠀⠈⠉⠻⠷⣦⣌⣙⠳⣮⢿⡼⣏⢾⣣⣟⣕⣿⣾⣿⣿⣮⣽⣭⣿⣿⣾⣿⣾⣳⣗⣛⣿⣸⢁⣿⠸⣏⠺⢿⣅
⠀⠀⠀⠀⠀⠀⣀⡼⠟⣰⡿⢛⣷⣹⡎⣹⢫⣟⣾⡿⣿⣿⡿⣾⣾⣿⣿⡿⣿⢽⣗⡚⡿⣰⢟⣾⢿⡷⡚⠳⠄⠻⣷⣄
⠀⠀⠀⠤⣶⣿⣥⣄⣉⣛⣛⢳⣾⢷⡙⢮⡝⣿⣸⣽⢿⢹⣸⣻⣻⣳⣷⣝⣿⣑⣻⢫⣵⢏⣾⣇⢻⡍⢾⣟⠛⠓⠾⠿⣆⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⠟⣸⣟⣘⣻⣮⡟⠮⣍⠹⣼⠿⡆⣗⣧⣳⡽⢮⢟⣭⡵⢋⣴⡟⣿⡿⢛⠻⠄⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣴⣟⣁⣈⣉⣭⣕⢿⡟⠙⣳⣭⣝⣛⠧⠽⠵⡮⡽⠞⣖⣋⣥⡾⣻⣿⣦⣿⠈⣿⠛⠶⣬⣿⣆
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠋⠉⠉⡼⢃⠾⠞⣫⠻⣿⢙⣽⣟⡟⠿⣻⣿⡟⢻⣿⡿⣄⡿⢱⣷⣝⠂⢻⡇⠀⠀⠉⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣥⣴⡶⠟⣿⢀⡿⢋⣼⡙⣧⡾⢛⡛⢷⡿⢰⣴⡘⠇⢸⡇⠉⠳⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠋⠋⠁⠀⢠⡟⣠⣴⠟⢹⡇⠋⣠⡟⢻⡘⢁⣾⠃⠻⣄⣿⡇⠀⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠁⠀⢸⣇⣾⠏⠀⠘⣧⣾⠇⠀⠀⠹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢸⠿⠁⠀⠀⠀⢹⠏⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠒⠒⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠾⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠢⠀⠀⠀⠀⠐⠒⠒⠐⠒⠀⠤⣀
⠀⠀⠀⣴⠶⠛⠻⠶⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡵⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣀⣀⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀
⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣠⣤⡤⠤⠍⠀⠀⠀⠀⠒⠒⠒⠒⠒⠚⠛⠓⠒⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠘⢧⣄⡀⠀⠀⢀⠀⠤⠤⠶⠒⠒⠒⠂⠀⠀⠀⠀⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣄⣀⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⠃
⠀⠀⠀⠙⠲⠦⢤⣤⠤⠀⠀⠁⠀⠀⠀⠀⠤⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠉⠉⠓⠒⠶⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢯⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣉⠓⢦⡀⠀⠀⠢⢤⣀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠀⠀⠁⠀⠀⢠⠞⣁⠀⠈⠳⡄⠙⡄⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠱⣀⣀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡒⠒⢉⡠