Heap Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠦⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠰⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢠⣂⣀⠌⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠢⢂⠀⣤⣤⡀⠀⣀⡀⢀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣑⡃⠀⢒⣆⣀⣀⠤⣼⡤⠠⣀⣱⣤⡠⣴⣏⡤⣀⠀⠀⠤⢄
⠀⠀⠀⠤⠤⡀⠀⠀⠀⣎⣁⠀⢡⡞⠉⠉⠓⠛⠛⣓⠸⣖⡓⠤⢬⠈⡳⠦⠨⢤⠔⠦⠌⠉⠁⠤⠄⢗⠀⠀⠤⠄
⠀⠀⠀⠀⠤⠼⠀⠓⠒⠒⠈⠉⠉⠉⠉⠃⠀⠀⣀⡀⠀⠒⢌⣉⡩⠭⠤⠒⠒⠤⠤⠗⠚⠀⢀⣀⠠⠤⠕⠀⣀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠝⠁⠀⠀⠠⠐⠉⠀⢀⣀⠄⠂⠉⠀⠀⠀⠀⠂⠀⠒⠢⢄⡈⠂⠀⠉⠒⠠⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣦⠀⠀⠀⠀⠈⠀⠀⢀⣈⡂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣒⡈⠉⠉⠉⠉⠉⠉⢁⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠉⠁⠉⠈⠉⠁⠉⠁⠉⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠤⣤⣁⣐⠒⠀⠀⠀⠀⠐⠒⣒⣈⣡⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠉⠉⠙⠉⠉⠉⠉⠉⠉⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⡐⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⢈⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠉⠑⠒⠒⠒⠖⠒⠒⠒⠒⠒⠚⠉⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⠬⢧⡀⢀⡀⠀⠀⡀⠀⢀⡀⢀⣀⡄⢠⡾⠤⠤⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠢⠤⠥⠤⠠⠌⠹⠶⠦⣤⠤⡤⠤⢬⡤⠶⠶⠾⠭⠤⠠⠄⠤⠤⠴⠚⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠠⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠤⠐⣟⠁⠀⠤⢀⣀⣀⡀⠀⠀⢀⣀⣀⠠⠄⠐⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⠑⠒⠤⠤⢀⣀⣀⣀⡀⡤⠤⠄⠖⢺⠁⠈⠐⠢⠤⠤⢀⣀⣀⣀⢠⡤⠤⠔⠒⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⠄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣷⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠠⣭⢀⣁⣀⣋⣀⣐⣈⣉⣉⢉⡥⠸⡆⠀⠨⡍⢀⣀⣈⡁⣐⣒⣙⣋⣉⢩⡥⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⡅⠀⠀⠈⠀⠀⠀⠀⢠⠿⡄⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⣰⠃
⠀⠀⠀⢠⣤⠀⠒⠒⠒⠛⣶⣐⠂⠄⠀⠇⠀⠀⠦⠤⣀⣒⡶⠟⠒⠻⣆⣒⠀⠀⠀⠀⠀⠠⠦⠤⣐⣃⡶⠛⠒⠒⠐⠒⠒⡂⣄
⠀⠀⣴⠃⠈⠙⠋⠋⠀⠘⠂⠉⠙⠛⠛⠛⡟⢛⡟⠉⠉⠉⠁⠁⠀⠛⠈⠉⠛⠛⠛⢻⡛⢛⠟⠉⠉⠉⠁⠃⠐⠃⠀⠉⠛⠉⠈⢧
⠀⣸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⢠⠇⢤⣁⡁⠀⠀⠠⠄⠀⠀⠠⠄⠀⠘⣀⡽⠢⣄⣀⠀⠀⠀⠤⠀⠀⠀⠤⠀⠀⢒⣈⡽⠤⢀⡀⠀⠀⠀⠄⠀⠀⠀⠦⠀⠀⣃⡉⢹
⡼⠀⠀⠈⠉⠉⠁⠒⠒⠒⠒⠒⠋⠉⠉⢁⡇⠀⠀⠉⠉⠉⠐⠒⠒⠒⠒⠒⠋⠉⠉⢰⠃⠀⠀⠉⠉⠋⠐⠖⠒⠒⠒⠚⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀
⢻⡐⠂⠤⠤⠀⣀⣀⣀⣀⣀⣠⡀⠤⠤⠼⣇⠒⠠⠤⠄⢀⣀⣠⣀⣀⣀⣄⠠⠤⠤⢸⣇⠒⠠⠤⠄⢀⣀⣅⣀⣀⣀⣀⠠⠤⠤⠐⢊
⠘⣆⠉⠐⠒⠀⠤⠠⠤⠤⠤⠤⠤⠀⠒⠂⢹⡈⠁⠐⠒⠠⠤⠤⠤⠤⠤⠤⠤⠐⠒⠈⢳⠈⠁⠒⠒⠠⠤⠤⠤⠤⠤⠤⠄⠒⠒⠉⣸
⠀⠘⣆⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⣀⡞⠳⣄⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⣠⠏⢳⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⠃
⠀⠀⠈⠓⠦⣭⣉⣨⣅⣤⣬⣭⣭⠤⠖⠋⠀⠀⠙⠶⢬⣭⣥⣍⣤⣤⣭⣭⡭⠴⠚⠁⠀⠀⠙⠦⢬⣭⣥⣍⣀⣤⣭⣭⡥⠔⠚⠁
⣿⣿⡿⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠈⠉⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⣀⠤⠒⠒⠒⠒⠚⡶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⣀⠤⠔⠺⡒⠒⠒⠢⣄⠀⡰⠁⠀⢀⣀⣀⡀⠀⢀⣀⠤⢸⠒⠉⠉⠉⠉⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡠⠒⠁⠀⠀⠀⠀⠱⣄⠀⠀⣨⣾⣒⡨⠥⠄⠐⣲⠖⠻⠽⣦⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠟⢋⠇⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⢹⡻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠁⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡜⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠃⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⢄
⠀⠀⠀⠀⠀⠀⡼⠀⠀⢱⠀⠀⠀⠀⠀⢠⢾⠋⠀⠀⡜⠁⠀⠀⠀⠀⠀⠣⣀⠀⠀⢀⠔⠻⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⢣⠀⠀⠀⠀⣾⠃⠀⣠⠾⠐⠒⠈⠉⠉⢉⡐⢦⣀⠉⠉⠀⣠⠗⠒⠒⠒⢦⠀⠀⠀⠀⠀⣀⡀⠀⢸
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠳⡀⠀⢸⣃⣔⠟⠁⠀⠀⠀⠀⢠⠖⠁⠀⠀⡇⠑⣴⠋⡜⠀⠀⠀⠀⡈⢣⣀⠴⠚⠉⠀⠈⠑⣧⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠢⣼⠋⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⠁⡜⢘⡞⢀⡠⠔⠒⢢⡇⣼⠁⠀⠀⠀⠀⠀⢠⠁⠱⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏⠑⠢⢄⡀⠀⠀⠀⠈⠀⠀⠀⢀⠇⢀⡥⣺⠖⠉⠀⠀⠀⣼⣧⡯⠤⠤⢄⡀⠀⠀⡜⠀⠀⢱
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡘⡄⠀⠀⠀⠀⠉⠙⠒⠺⡖⠒⢲⠞⢋⡡⡏⡝⠀⠀⠀⠀⡰⠋⡿⣇⠀⠀⠀⢱⠀⢠⠃⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠋⡠⠋⠀⣹⠁⠀⠀⢀⡾⠀⣴⠃⠈⠢⡀⠀⢸⢀⠇⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠁⠀⢦⠀⠀⠀⠀⠀⠀⢀⠔⢁⠟⠢⢄⣼⣁⣧⠤⠚⠁⣣⠎⠸⠀⠀⠀⠈⠑⣶⠮⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⢀⡴⠋⢠⠃⠀⢠⠎⠀⠀⠀⠣⣀⢼⠁⠀⡇⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⡆⠀⠀⠀⢙⣦⡖⠁⠀⢠⠃⣀⠔⠁⠀⠀⢀⡠⠊⠉⠺⣤⣰⣃⣀⣠⠴⢊⠇⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠤⠧⠄⠒⠊⣁⣀⠨⠷⠤⡞⣊⣁⣀⠠⠔⠚⠁⠀⠀⠀⠀⢧⠎⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠄⢺⡊⠉⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢆⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠑⠤⣠⠃⠀⠀⠀⣀⠤⠊
⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢠⠏⠈⠉⠉⠁
⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⢈⣢⣀⠠⠄⠚⠁⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⠠⠔⠁
⣿⢿⠿⠿⡿⠿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣦⣦⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⢤⠛⣷⡝⠄⠈⣯⣍⡹⠷⠦⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣉⠃⠈⠁⠀⠀⠒⠀⠀⠀⠀⠁⠐⠃⠛⠗⣶⢀⣀
⠀⠀⠀⠀⠀⠀⣤⡆⣲⡶⣽⠠⠰⡴⠂⣠⡀⣴⣄⣛⢋⢀⣀⢀⣿⡀⠺⠟⠁⠈⠙⠢⢦⣤⣀⡀⣀
⠀⠀⠀⠀⠤⣼⠠⠈⠉⠀⠠⠄⠀⠐⣃⠉⠁⣄⠀⣁⠀⠀⠀⠀⣨⣄⠘⠷⢇⡂⠛⢗⡤⠴⠩⠀⣝⣃⣠⢠⣄⡀
⠠⠐⠈⠒⢲⠿⣀⣤⣡⣤⠄⠈⠁⠂⠫⠞⠀⣬⣰⣿⡮⠰⣶⣤⠌⢿⡟⣃⣼⢁⠒⣘⠁⣶⡧⠴⠄⣿⡛⠻⠁⠦⠔⠠⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠟⠐⠚⠐⠂⠘⠋⠉⠁⠙⠃⠲⠀⠠⠀⠄⠠⠀⠀⠀⠘⠛⠀⠈⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠉⠀⠀⣇⠈⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠉⠒⠢⢄⣸⠀⠀⠀⢀⠎⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣤⠤⠶⡶⢤⣀⠈⢧⠀⢀⠎⠀⠀⠈⢳⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⠋⠉⠀⠀⠀⠀⡇⠀⠉⢳⡄⢣⠏⠀⠀⠀⠀⠀⢷⡠⠎⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠑⢤⡀⠀⣀⠼⣀⠀⠀⠀⣧⢸⢣⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⡇⠀⠀⠀⠙⡏⠀⠀⠈⠑⠒⠢⣽⡇⠀⢣⣀⣠⣤⣞⠉⠉⠙⢦⣀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⠋⠉⠉⠁⠈⢧⡀⠀⠀⠀⠘⢄⠀⠀⠀⠀⣠⡟⠛⠉⠉⣁⠴⠥⠬⠷⣄⠀⠀⠙⢦⡀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠈⠺⡄⣠⠴⠦⣤⡏⠉⠓⢶⣤⣤⣬⡷⠴⠒⠛⠁⠉⠛⢲⣏⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⠻⣇⣀⣷
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢠⡟⠁⠐⢾⣽⠀⠀⠀⠀⠈⠳⡄⢤⡤⠄⠒⠒⠒⠘⣅⢹⡄⠀⠀⢀⡴⠛⠉⠒⢤⣀⣠⣎⣧⢸⡆
⠀⠀⠀⣠⠖⢺⡷⢦⣄⣀⡤⠟⠯⣉⠉⠉⠙⠦⣄⠀⠀⠀⠙⣼⣠⣄⠀⠀⠀⠀⠈⣶⣇⣨⣴⣏⣁⠀⠀⠄⢠⠃⡸⢀⡿⠚⠧⣄
⠀⠀⢰⣟⣒⣻⣿⡽⠛⠁⠀⠀⠀⢈⠷⡒⠒⠒⠸⣦⠀⣠⣖⠋⠁⠈⠙⣆⠀⢀⡼⣏⠀⠀⠀⠀⠉⠉⠛⠲⣾⣠⣷⣋⢉⠖⠒⠚⣷
⠀⠀⠀⠉⠉⣼⠽⠒⠒⠢⠤⠤⡔⠁⠀⠱⡀⠀⠀⣸⣀⡏⠈⣱⠒⠒⠤⠬⢷⣾⡰⠉⢉⡖⠒⠒⠒⠂⠠⣠⡾⢰⣯⡭⣿⠛⢛⠛⠉
⠀⠀⠀⢀⣀⡸⣄⡀⠀⠀⠀⠀⠘⡄⠀⣀⣱⡤⢾⣛⣛⣷⣴⠁⠀⠀⠀⢀⡼⠋⠳⣤⣼⣀⣠⣤⡤⠴⣚⣁⠐⠀⠉⠉⠁
⠀⠀⠀⢿⣿⠟⠀⠉⠉⠉⠛⠛⠓⠛⣩⠿⣶⠀⠀⠀⠉⠉⠈⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠯⠾⠃
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠰⣯⡿⠆
⠀⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡇⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠉⠻⣆⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢻⡏⠀⠀⠀⠀⠀⣿⣛⣿⠟⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠸⣧⡴⠶⠓⠒⠶⢿⡉⠙⣦⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢻⡟⠁⠀⠀⠀⣠⣤⣤⣷⣠⠞⠛⠓⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⡀⣿⠀⠀⠀⠀⢸⠁⠀⠀⠈⣧⠀⣠⡤⢴⣟⠛⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡾⠋⠁⠈⠉⠻⣿⣠⠞⠓⣦⢸⡄⣠⡴⠶⠛⢻⡏⠀⠀⠈⠳⡄⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢹⡇⠀⠀⠀⢠⡴⠞⠛⢦⣀⣸⡿⠿⣅⡀⠀⠀⢸⣧⣤⣤⣄⠀⣿⠳⣆⠀⣸⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣸⠇⣀⣀⡀⢻⡀⠀⠀⣸⠏⠉⠀⣀⣈⢻⣶⣴⣏⡀⠀⠀⣸⠷⠛⠚⠛⠶⣏⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠟⠷⣼⡏⠉⣻⠶⠛⠛⢻⣇⣀⣤⣾⣉⣨⡟⠉⠀⠈⠙⣦⠶⠷⢦⣀⠀⠀⠀⢹⡏⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⡈⠻⣾⠃⠀⠀⣰⠋⡟⠁⢠⡟⠁⠈⠙⡆⠀⠀⠀⣸⡇⠀⠀⠙⣧⠀⠀⢰⡟⢦⣽
⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⣠⡴⠶⢻⡷⣄⣠⣿⣖⠿⣦⣼⣧⣤⣀⢰⠇⢀⡾⠋⠉⠳⣄⢀⠀⣿⢂⣴⣟⠀⠀⢹⡶⠚⠳⣦
⠀⠀⠀⠀⠀⠀⣾⠀⣷⣄⠀⠀⢰⡏⠀⠀⠈⣷⠋⠉⠉⠉⠙⣯⠀⠀⠀⠙⣿⠞⠋⠻⢦⣤⡴⠛⠉⠉⠙⠻⣄⢘⡇⠀⢀⡷⠖⠒⠿⣄
⠀⠀⠀⠀⠀⠀⢻⣼⠋⠉⠙⠓⢾⡿⠟⠛⠛⢳⡄⠀⠀⠀⠀⢸⣇⣠⡤⠤⢿⣤⡶⠒⠲⢦⣿⡄⠀⣠⡴⠚⠛⠛⢷⡴⠟⠁⢀⣀⣀⡿
⠀⠀⠀⠀⢠⡞⠉⣻⣄⠀⠀⠀⣿⠀⢀⣠⣤⡀⣷⣀⣀⣀⣠⠟⠉⠀⠀⠀⠀⠙⣧⠀⠀⠀⠘⣿⠛⠁⠀⠀⠀⠀⠀⢷⠶⢾⡉⠉⣧
⣰⠟⠙⢷⠸⣇⠀⣿⣨⠿⠿⠛⠻⣆⣸⣅⣀⡿⠋⠉⠉⠉⢳⡄⠀⠀⠀⠀⠀⣠⣏⣠⣄⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣠⣿⣀
⢿⣀⣀⡾⢀⣈⡉⢹⡃⠀⠀⠀⣠⠞⠋⠉⠙⣧⡀⠀⠀⣀⣸⡷⢦⣤⣶⠶⠛⠋⠉⠀⠈⠙⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠷⠶⣏⠀⢀⣹⠆⠀⠀⣀
⠀⠉⠉⢰⡋⢉⡷⠈⠙⠛⠓⠛⣿⠀⠀⠀⠀⢸⣇⡶⠋⠉⠉⢹⡆⡞⠁⠀⠀⠀⠀⠀⢀⡼⢿⠾⠷⢦⣤⣤⡴⠞⠋⠀⠀⠀⣸⠛⠉⠁⣰⠞⠋⠉⠙
⠀⠀⠀⠈⠓⠛⠁⠀⠀⠀⠀⠀⠹⣆⣀⣀⣠⠾⠹⣇⣀⣀⣤⠞⠁⠻⣦⣀⣀⣠⡴⣞⣋⠀⢻⡀⠀⠀⣷⠈⠛⠶⠶⠤⠶⠞⠁⠀⠀⢸⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣍⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⢁⡴⠛⠉⠉⠻⣄⠻⠦⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⠀⠀⢀⣸⠇⠀⠀⠀⠀⠀⣤⠶⠶⠶⢦⡀⠀⠀⠸⣇⡀⠀⠀⣠⡿⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠸⣇⡀⢀⣀⡼⠃⠀⠀⠀⠈⠙⢛⡿⣭⠀⠸⣇⣀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⣀⠠⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠎⠀⠈⠁⠀⠀⠀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⡔⠀⠀⠀⠀⢄⠀⠀⠁⢉⡑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⡎⣀⠴⡀⢠⠦⠼⠀⠀⠘⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⠘⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠠⠂⠁⠀⠀⠀⠁⠢⡀⠀⠀⢇⠀⠠⠀⠈⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠘⠐⠀⢀⠀⢄⠀⠐⠂⠄⠄⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣀⡤⠄⠀⠊⢀⠠⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠋⠀⢠⡠⠊⠁⠀⠈⡄⢀⠎⠀⠀⠀⠀⠂⠄⠀⠀⠀⠖⠤⢀⡡⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠅⠀⠀⠀⠄⡘⠀⠀⠀⠀⠊⠐⠠⢀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠠⠀⡀⠀⠀⠀⢀⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠈⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡀⠌⠁⠂⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠱⠀⠠⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡒⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⢄⠀⢀⠄⡀⣃⠈⠀⠀⠈⠂⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢠⠁⢀⡀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠈⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠈⠂⠁⠑⠤⠎⠑⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠀⠀⠀⠈⠈⠉⢓⠊
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠦⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠁⠤⠀⠀⠀⠄⠀⠤⠤⠀⠀⠀⠀⠓⠢⠴⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⢆⠠⠒⡄⠈⠄⠂⠁⠐⣀⠔⠀⠀⠀⠒⠀⠠⠢⡀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⡁⠄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⢀⠤⠂⠑⡄⠀⢀⠔⠁⠑⢄⣠⠃⠀⠈⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠐⠂⠄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⣀⡀⢠⠙⢗⠁
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⢣⠀⠀⠆⠉⠠⠈⠀⠈⠀⠘⡀⠀⠀⠀⠀⡄⠈⠁⠀⠊⠀⠈⠉⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠐⢤⠔⢦⠠⠠⠄⠀⠀⠀⠀⣇⡠⠚⠢⠔⠀⠁⠀⠀⠀⠀⡠⢄⠀⠈⠁⠒⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠑⠁⠀⠠⠄⠀⢀⠄⠀⠀⡄⠄⠀⠢⡀⠀⠠⠂⠠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢰⠁⠐⠐⠀⠁⣈⣀⣁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⣀⠀⠀⠀⣀⠤⠀⠤⣀⣀⣀⠤⠤⢄⠀⠈⠐⠀⠀⢠⠖⠒⠂⠁
⠀⠀⠀⠀⠀⢢⣀⠤⠐⠈⠁⠀⠀⠀⠈⠳⡄⣀⠔⠉⠢⣀⣀⠔⠁⠀⠀⠀⠀⠀⠉⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⣀⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⣄⡀⠀⠀⠀⠀⡀⢀⣀⣤⡴⠼⠖⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠠⠄⠀⠀⠐⠀⠀⠁⠁⠀⠀⠀⡴⠁⢘⢀⡠⡴⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡀⣣⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢕⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠤⣠⣄⣀⡀⣀⣀⣄⣠⡴⠴⠶⠛⠇⣾⡏⠁
⠀⠀⠀⠀⠀⠈⠀⠐⠀⢄⣩⡲⠒⠌⡫⢄⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢀⠈⠠⡔⠋⠀⠇⠀⠀⠈⠐⠝⠢⣄⠀⠀⠀⣇⡇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠈⠂⢝⠦⢀⣿⡁
⠀⠀⠀⠀⠀⠁⠂⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠐⡘⠁⠀⣤⠠⢊⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠂⠀⠀⡇⠄⢠⣆⠀⣫⣤⣾⠴⠷⢾⠉⡃⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢈⠃⠀⢻⡏⢀⣟⡀⠡⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠁⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⡀⢆⠇⠈⠇⣈⡹⠞⠋⠁
⠀⠀⠐⠂⠄⠀⣄⠰⣥⡌⠀⢰⠀⠈⠀⠄⠀⠡⠀⠀⠀⠀⠀⠀⠀⡠⠃⣤⠁⠀⡃⠲⢦⡾⠚⠁
⠀⠀⠀⠈⠀⣀⠈⠉⠙⢁⠘⠲⠿⢄⡀⠆⢠⠀⠂⠀⠀⠀⡄⠀⠸⠁⣀⠙⣇⠘⣧⡴⠊
⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠒⢀⠀⠀⠙⠿⣿⠂⠀⠀⢀⠀⠀⠐⡆⠀⢻⠀⣈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⡀⠁⣀⠀⢂⠉⠱⣄⠀⠸⠆⢀⡀⠈⠦⣼⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠅⠀⠤⠉⠀⠤⠈⢷⡦⠐⠀⢱⡆⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡠⠊⠈⢳⡀⠦⣸⡷⢃⠠⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠍⠩⠅⠀⡀⠀⢳⠀⢀⡟⠁⠀⠀⠀⠠⠳⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡾⠀⠀⠀⠀⢽⡿⣼⠗⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠉⠁⠀⠀⢀⡔⠀⠀⠀⣴⡇⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡿⠀⠀⠀⠀⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣷⣄⣀⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠈⠛⢿⣦⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣠⠤⠒⠒⠒⠒⢲⡴⣤⡄
⠀⠀⢠⡞⡡⠚⠁⠀⠀⠀⢀⡿⣶⠇
⠀⠀⣿⢸⡁⠀⠀⠀⢀⡰⠂⠀⣿
⠀⠀⠹⣾⣇⠀⣀⣤⠞⠁⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠲⠤⣀⢀
⠀⠀⠀⠈⢻⣿⠟⠁⠀⠀⢀⡜⠁⠀⠀⠀⠀⣀⠤⠖⠉⠰⡎⠈⠇⠀⠀⠀⠀⠀⠀⠀⢸⡥⠱⣤⠀⠀⠈⠓⢆
⠀⠀⠀⠀⠀⠙⠷⡶⠶⠖⠋⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠙⠛⣓⡄⠀⠀⠀⠀⠀⡴⠃⠀⠀⠛⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠰⠀⠀⠀⠉⣷⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠐⣄⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡴⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡇⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡞⠀⢀⡀⠀⢀⡀⡰⠁⠀⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⣠⠴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠿⣿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⠶⠶⠶⢶⣶⠶⠶⠶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡱⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢏⣳⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⠀⠀⠀⠀⢰⠁⢀⣤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠔⠒⠒⠢⡤⠊⠁⣇⡠⡴⠒⢦⠀⠀⡏⢇⠀⡴⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢣⡠⠒⠒⠋⠀⢀⣠⣎⣀⠀⠀⡇⠀⣇⠀⢈⠗⠒⠙⢆⠙⠚⠛⠹⡀⠀⠈⡢
⠀⠀⠀⠀⠀⢣⠀⡰⠁⠀⠙⠒⢄⡀⠰⠉⠀⠀⠈⠑⢄⣙⡯⠤⠴⠊⠔⠚⠉⠙⠢⡀⠀⢀⣇⣀⡘⠁⠀⠀⣔⢦⡀
⠀⠀⠀⠀⠀⡏⢠⠃⠀⠀⠀⠀⠀⢱⣀⣀⡀⠀⠀⠀⠘⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠓⠒⠚⠦⠀⢹⣦⠖⠚⢱⡏
⠀⠀⠀⠀⣀⠜⠉⠉⠉⠉⠉⠲⠖⠉⠁⠀⠨⠕⠒⠒⠒⢤⡜⠈⢾⢯⠀⠀⠀⠀⢀⣤⣤⣤⣄⢀⣼⣁⣀⡀⢀⡇
⠀⠀⢀⠞⠁⢀⠴⠒⠂⠀⠀⠀⣀⠤⢄⠐⠋⠳⠀⠀⠀⠀⠹⠖⠉⠋⠀⠀⠀⠀⢀⣧⣍⣀⡇⠉⠀⠀⠀⠉⠻⠓⠒⡄
⠀⠀⢎⣀⣔⡡⠤⢞⠉⠉⢑⣦⣀⣀⣈⣑⡒⠚⠀⠀⠀⣀⠤⠴⠒⠒⠒⠒⠤⢄⡀⠀⠀⠈⢃⡤⠊⠉⠓⠒⠒⠒⠊
⠀⠀⠀⠉⠀⠀⠀⠀⠑⠚⠉⠀⠀⠀⠀⠀⠈⠉⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣀⡠⠤⠒⠂⣉⣀⢱⣀⡀⣄⠀⠀⠀⠀⠀⢀⠄⠀⠀⠉⠁⠀⠀⣀⢕⣠⠔⠉⠒⠤⣀
⢀⣤⡒⠋⢉⣀⠤⣖⣪⠭⠑⠂⢉⣀⠐⠀⠀⠀⠀⡠⠀⠀⢀⣀⣠⠤⠖⠚⠉⠈⠲⡁⠀⠀⠀⠀⠀⠉⠢⢄⡀
⡏⠀⢹⣪⢵⡚⠉⣁⡠⠴⠒⠋⠹⢅⡀⠀⠀⡠⠊⢠⠞⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀
⡇⠀⢸⡇⠀⢸⠉⠁⠀⠀⣠⠔⡋⢄⣁⣠⠮⠀⠀⠌⣄⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡤
⡇⠀⣿⡇⠀⢸⠀⠀⠀⣞⡴⠊⢉⣻⣫⠗⠤⡐⠀⠑⠬⢣⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⣠⠴⠚⣹⣀
⡇⠅⡇⡇⠀⣼⠀⣠⠖⠹⠕⠊⠉⠙⠛⠓⠦⣄⠀⠀⡆⠁⠹⣄⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠖⠋⣁⢔⡫⠖⠳⢄⡀
⠉⠙⣧⡇⡆⣏⣠⣴⣟⠢⢄⡀⠀⠀⠀⠀⠀⠀⠙⣦⣸⣴⠚⠉⢦⡀⠙⢄⠀⠀⠀⣀⣀⣤⠤⠴⠖⣺⣋⠵⣪⠔⠋⠀⠀⠀⢀⣹⠗
⠀⠀⠀⠛⠓⣟⡦⣍⡺⣕⢄⡉⠓⠦⣀⠀⢀⠴⠊⢁⣯⠈⠳⢄⠀⠱⣄⢠⡗⠛⠉⠁⣀⠠⢔⣒⠭⠝⠛⠻⣄⠀⠀⣀⡴⠚⢉⡷⣀
⠀⠀⠀⠀⠀⠛⢕⠦⡉⠒⢵⣮⣒⠄⠈⢽⠁⡠⢖⣵⡋⠑⠦⡀⠙⠦⣌⠻⢌⣀⠭⠴⠒⠋⠉⠀⢀⣠⠔⢻⣥⠖⠋⠁⡠⢖⡩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠪⣑⢄⡈⢙⢷⠒⠚⣩⠔⢻⡀⠙⠢⣄⠈⠳⢄⠈⠑⢦⣀⠀⠀⠀⣀⡴⠚⢉⡠⢚⠵⢻⣤⠔⣩⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠺⠗⠋⠁⠀⠀⠉⠢⣀⠈⠓⢦⡀⠙⠦⣀⠈⢓⣔⡫⠁⡠⢔⡭⣚⡥⠞⣯⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠉⠳⢤⡈⠓⠼⣗⣀⣬⣶⠿⠋⢁⡤⢚⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠉⠲⢄⡤⢚⠉⢀⡠⢖⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣾⣊⠀⢐⡩⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⣤⡶⠂⠈⠉⠀⠉⠁⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⠿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣠⣤⣄
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣦⣤⡤⠲⢖⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠐
⠀⠀⠈⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⣋⢀⣠⣀⢈⣉⡉⢉⠛⠛⠷⠖⠐⠄⣩⠙⠀⠀⠀⠀⠀⠀⠘⠃⢘
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣼⣿⣿⣿⣾⣿⣟⣿⣧⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⡀⢀⡃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣭⣀⣘⣲⡄⠀⠈⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⣄⠘⡇⣸⣶⣠⢼
⠀⠀⠀⠀⠑⢤⡌⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⣶⣶⣬⡇⢶⡇⠀⠀⠁⠀⠀⠀⡀⡻⣿⣶⣿⣿⣿⣟⣾⣴⣦⣖
⠀⠀⢀⣀⣸⣾⣿⣦⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⠆⠀⠔⠀⠝⠳⣼⣿⣿⣿⣿⡿⠟⣛⠏
⠀⢠⡽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣰⣄⣤⣴⣿⠟⠀⢉⣿⡁⠀⠘⡇
⠀⠀⠚⠉⠛⠛⠛⠛⠛⠛⠛⠛⠿⠟⠛⠛⠻⠟⠟⠛⠻⣿⡟⠛⠛⠻⢿⠟⢿⠿⠿⢛⡿⠻⢿⡿⢿⠿⠿⡿⠿⢿⢿⣿⣶⣾⣿⣿⡿⣿⠶⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠚⠿⠙⠓⠆⠀⠀⠀⠀⡀⣀⣤⣄⣀⠀⠀⠈⠀⠁⠂⠉⠀⠉⠉⠉⠀⠀⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠉⠉⠉⠉⣱⠖⠊⠉⠀⠀⠀⠀⢀⣀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢀⣽⣯⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⣸⠿⠎⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⢀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠐⣿⡶⠷⠛⠙⠶⠄⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢀⣡⣤⡾⠁⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⣿⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠈⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⣿⡇⣽⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣟⠳⠃⠀⠀⠀⣇⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⣀⡠⠔⠋⠙⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠁⠀⠀⠀⠀⢀⡈⠳⣯⡌⠉⠲⡄
⠀⠀⠀⠀⠀⠀⣴⡟⢠⡎⠀⠀⠀⠀⠀⠀⠈⠻⠿⢶⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣹⣗⡦⠀⣻⣶⡄⠘⣤⡀
⠀⠀⠀⠀⠀⢰⠟⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⢋⣷⠛⠛⠛⠻⣿⣷⠀⠁⠈⠲⡀
⠀⠀⠀⠀⢀⡟⣀⡼⠓⠒⠦⢤⡀⠀⠀⠀⠀⠀⠀⣿⠁⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣀⣀⠀⢠⢴⡾⠃⠀⠀⠀⠀⠀⢉⣻⢦⡀⠄⣷
⠀⠀⠀⠀⢸⣹⠁⠀⠐⣶⣦⡄⣷⠀⠀⠀⠀⠀⢸⠉⠀⠀⢸⣦⡄⠀⠀⠀⠀⠀⠀⠀⢸⢏⠁⠈⢛⣿⠟⠀⠀⠀⠀⠀⠀⣴⡌⠀⠈⣿⣦⡟
⠀⠀⠀⠀⢠⡇⠀⠀⢀⡟⣋⠁⢻⣇⠀⠀⠀⢠⠎⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣩⢠⣿⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢻⠯⠇
⠀⠀⠀⠀⣾⠁⠀⠀⠈⣙⠀⠀⢸⣿⡄⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⢠⣿⣿⣿⣯⢹⣿⡀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⡞
⠀⠀⠀⠀⠘⠦⠄⢀⣀⣸⣟⣇⡼⠤⡀⠀⢀⣰⣿⡄⠀⠀⠈⣿⠿⠥⠤⠤⠒⠶⢤⣼⣿⣿⣿⠿⠛⠁⠉⠙⠓⠲⢄⣀⣀⢛⣡⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠻⣿⣧⣄⣀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⣒⢲⣶⣦⣀⠀⠀⠀⠀⠰⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⣉⣙⡛⣷⣝⡛⠀⠀⠀⢸⣿⣿⡿⠛⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡋⢀⠀⠀⠀⠀⠀⠀⠀⠠⡾⢛⣹⠿⠋⠉⣹⣿⣿⡏⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠇⠛⠊⠇⢀⠀⠀⢠⠀⡉⠁⠀⣈⠀⠀⢀⣀⣴⣾⠍⠀⠀⠀⠀⠀⠀⠀⠙⣍⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⣠⠀⡀⠀⠀⠀⠀⠀⣀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⣰⣿⣯⡈⠛⢧⣥⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠙⡀⠀⣿⣂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠠⣀⣠⣶⣶⣶⣶⣐⣆⣼⣿⡿⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⣷⠂⠙⠃⠀⠀⠈⠀⠀⠀⠀⢀⡄⠀⣀⠔⣋⠔⠀⠀⠀⠀⠀⠉⠀⠁⠀⣻⣿⣿⣿⡟⠻⣆⡀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣴⠃⣰⣿⣦⣀⣀⣠⣤⣤⣀⣀⡀⣠⡿⣿⣿⣿⣿⡄⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⢲⠖⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⡇⠀⠉⠻⣿⣿⣿⣿⠟⡛⠉⠋⠛⠛⠻⠟⠿⣿⣿⣧⡴⣼⠃⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⢀⡠⠀⠖⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢀⡀⠈⠉⠁⠈⠀⠀⠀⣤⣤⣤⣤⣀⣀⣸⣿⠟⢁⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⣿⡿⣤⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢧⡈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢿⣿⠗⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠈⠉⠱⠌⢰⡵⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠂⣈⣻⡿⢫⡤⡤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠁⠛⠷⢿⣿⣷⣿⣵⣄⣰⣤⣶⣦⣤⣤⣠⣤⣤⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣧⣞⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⠿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣦⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⢻⡿⢿⢻⡟⠛⣱⢿⣟⡷⣿⡏⠈⠻⢷⣆⣀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⠈⠘⠁⠙⠸⠁⢻⣇⠀⠀⠀⠟⠉⠛⠶⢠⠀⠀⣄⣸
⣿⡇⠀⠐⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠠⠐⡀⠣⠀⢸⣿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿
⣿⡇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⢄⠀⠀⠀⢸⣿⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿
⣿⡇⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⣿⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⡏⢑⡄⠀⠀⠀⠀⣀⠠⡤⠤⠀⢼
⣿⣧⣀⣀⠀⣀⣉⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣥⣘⣃⡀⠘⠋⠁⢀⣀⡀⠀⣀⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢠⣤⣤⡤⣶⣄⣤⣄⣤⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠁⠁⠁⠉⠈⠁⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⢿⣿⣿⡿⠿⠋⢀⣉⣝⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⢁⡆⠀⠀⠀⢀⡄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡄⠛⠀⣴⡀⠀⠈⠋⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣿⣿⣿⣇⠀⠀⠈⠀⠀⠀⡰⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⢽⣍⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⣭⠀⠀⠀⠀⠀⠀⣿⠅⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢀⣀⣠⠔⣪⣭⡽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠞⠁⠀⠐⠩⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⣠⠞⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⣀⠔⠂⠀⠀⠒⠠⣀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠘⠀⣠⣤⣤⠀⠀⣤⣤⣄⠀⠁⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠉⠉⣡⣌⠉⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⡀⢢⠀⠀⠀⠛⠛⠀⠀⠀⡔⢀⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⡌⠀⠈⡆⢅⢀⡇⢀⡀⢸⡀⡨⢰⠁⠀⢡⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠰⠀⠀⡠⢈⡑⠂⠤⢀⣀⠤⠘⢊⡁⢄⠀⢀⠆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢀⠜⠚⢅⡀⠄⠛⢒⠤⠌⣁⠛⠠⢀⡨⠓⠫⡀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠘⡄⠀⠀⡔⠂⠉⠀⠀⠀⠀⠉⠐⢢⠀⠀⢠⠃⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠑⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠊⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠡⠀⠐⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠛⠀⡄⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢇⠀⠤⠖⠒⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠆⡠⡅⣸⠀⠀⠀⠀⠀⠀⢈⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢹⣟⡄⠀⢀⡔⢹⣷⣿⣇⠢⠀⢦⠔⠀⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠆⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢲⣎⠀⢸⣿⠘⣰⢎⢀⣿⣿⣿⣿⠀⠀⡟⢠⢻⣾⠁⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠸⣿⣿⠦⢀⣾⣿⣿⣿⣿⣧⣀⠀⡿⣾⣏⣳⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⢰⣿⣿⡀⠈⣿⢿⣟⣉⡽⠿⠛⠚⠛⠛⠻⠖⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⣿⣿⣧⡀⠘⢿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⢁⠀⡀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣇⣿⣿⣿⣿⡖⠋⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠈⡴⠤⢤⣶⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠃⢘⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠐⠐⠲⢄⡀⠀⠀⠀⠀⠀⣼⣀⣠⠾⠟⠁⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠠⠴⠾⠋⠈⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠂⠠⠤⣀⠀⠀⠀⠀⢀⠴⠾⣷⠶⠒⢀⣀⢀⡂
⠀⠀⠀⠀⠀⠀⠒⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠁⠀⠀⢀⡤⠂⠡
⠀⠀⠀⠀⠀⢀⠀⡄⠘⠂⠤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⡁⡀⠌⠈⠀⠀⢀⣠⡾⠟⠖⠀⣠
⠀⠀⠀⠀⠀⠀⡁⠂⢄⡀⠀⠀⠀⠉⠐⠒⠢⠤⢤⢀⢠⣀⠤⢖⠍⠀⠠⠀⠀⠀⠀⣀⠄⢊⠕⠋⢀⡠⢖⠩⢪⡁
⠀⠀⠀⠀⠀⠀⠠⠐⠌⠉⠓⠲⠠⣀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠔⠁⠀⠀⢀⡤⠔⢉⠤⠚⣁⠄⣚⠕⠎⠑⣊⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠦⠠⢠⢀⡉⠑⠒⠠⢤⣀⣀⠀⠀⢀⡤⠔⠊⢁⠤⢊⡡⢔⡩⠐⠉⢀⣠⠖⠋⠉⢐
⠀⠀⠀⠀⠀⠀⢐⠦⠤⢀⣀⣀⡀⠀⠈⠉⠙⠒⠲⠠⠠⢭⣉⣁⣀⠴⢊⠥⣂⠕⢊⠁⢀⡤⠒⠉⠀⢀⣀⡦⣿
⠀⠀⠀⠀⠀⠀⠐⠉⠑⠒⠂⡒⠨⠍⢐⣒⡂⠤⠀⣀⡀⠀⠀⣈⠴⣊⠕⣊⠴⣊⡴⠚⠁⣀⡠⣲⡭⢛⣡⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⢂⠀⠀⠀⠒⠀⠩⠍⠐⠪⠭⠭⠕⠩⣔⣫⠓⠏⣡⡤⢞⣫⠕⢟⡵⢖⡯⠕⢋⡇
⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⣩⡟⠏⠁⣒⣭⠴⣚⠩⠔⢀⡵⠚⢙⠴⡛⢹⣇⣀⡀
⠀⠀⠀⠀⠀⠀⢈⠈⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣶⢞⡍⠖⡊⠐⠞⠊⣀⣴⡜⠕⢊⢡⢸⡇⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠐⠀⠠⡀⣿⣛⠵⢃⠠⠀⠀⣠⡴⠜⠉⠀⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⣀⠀⠀⠐⠀⡋⠀⠀⢀⡠⠒⠟⠉⠂⢀⡠⠖⠋⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢈⠉⠙⠒⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠤⡒⠁⢀⡠⣮⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠄⣀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠁⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠠⠀⠁⠀⠀⢰⢐⣬⠞⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠀⠀⠈⠉⠉⠙⠉⠒⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⢀⣀⣀⠀⠀⠀⠀⠀⣀⠐⢶⣶⣾⡏⢀⠄⢀
⠀⠀⠀⢺⣷⠀⢰⠂⢀⣶⡿⠟⠋⠛⢿⡿⠛⠛⠛⠃⣴⡆⠀⠀⠀⢈⣦⣦⣻⣥⡌⠁⢀⣀⣀⠀⣀⡀⠘⠿⠆⠀⠛⠃⠀⢏⠀⣸
⠀⠀⠀⢸⣯⣤⡾⠀⠈⠉⠀⠀⢠⠀⠀⣑⣄⠀⠀⠀⠘⢯⡀⠀⠀⠀⠁⠘⣿⣋⠉⠀⠀⠀⠉⠙⠁⠀⠀⠀⠠⣦⣤⡄⠀⠀⠚⠛⠃
⠀⠀⠀⠈⠉⠙⠓⠂⠀⣠⣴⣶⣧⣶⣿⣿⣿⣷⠦⠤⠃⠀⠁⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡘⡄⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠁⣶⣶⡆⠠⢤⣦
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠇⢻⣷⠘⠀⣸⠀⠹⠀⠀⠀⠀⠀⠀⠀⡆⢹⣿⡷⠀⠟⠟
⠀⠀⠈⠑⠒⢦⣤⣄⡄⠀⠀⣤⣤⠀⢰⣶⣶⣿⠏⠀⠀⠀⠀⠀⡟⠘⠙⣿⣷⣦⣍⠀⠀⠇⠀⠀⠀⠀⠀⠀⠷⠈⠁⠀⠀⠀⠹
⠀⠀⠀⠁⠐⠂⠻⠿⠶⠀⠠⣿⡇⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⣄⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠈⠛⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢹⣟⢿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡄
⠀⠀⠈⠁⠀⢰⠀⠀⣗⣾⣿⠀⠀⠀⠀⠀⢺⣿⢧⢻⣧⠀⠀⠀⠀⠀⠀⢸⢠⣴⣦⡐⠂⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⢰⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠄⡿⣿⠀⠀⠀⠀⣧⠘⢿⠀⢳⡿⠀⠀⠀⠀⠀⠀⢸⢸⡿⣿⣇⠀⣡⡄⠀⠀⠀⠀⢰⣤⣀⡀⠀⠀⠀⠙⣷⢰⣸⢸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⣿⣿⣌⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⡉⢻⠀⠃⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠸⠿⠿⠚
⠀⠀⠀⠀⠉⠛⠛⠈⠛⠁⠀⠀⠀⠀⠀⠈⠛⠻⠁⠻⠇⠤⠆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠠⠆⠀⠀⠀⠀⠸⢿⣿⡿⣿⣿⣤⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡄⠀⠀⠀⠀⠀⣠⣤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⡄⠀⠀⠀⠀⠀⠉⠀⡾⠀⠀⠀⠀⠀⠀⠋⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢀⡴⠶⣤⡀⠈⢧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠁⠀⠀⠀⠀⣠⡶⠶⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠸⠇⠀⣸⠇⠀⠀⠀⠀⢀⣠⣤⡶⠟⠛⢛⠛⣿⢻⣾⣆⠀⢴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⣰⣿⣷⡆⠴⠟⠛⠛⠛⠋⠁⠀⠹⡄⠀⠀⠀⠀⡤⠆⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣡⣄⠀⣰⢏⣼⡏⠀⠀⠀⡀⠠⠄⠀⠀⠄⠀⢹⡄⠀⠀⠀⠀⢀⣠⡴⠚⠉
⠀⢀⠀⠀⠈⠑⢦⡰⢄⠀⠈⠁⠀⣿⡇⠿⡆⠀⠀⠀⠀⠀⢀⠀⠄⣤⣤⢴⡇⠀⠀⠀⠘⠉
⠀⠈⠻⢦⣀⠀⠀⠙⢦⡀⠀⠀⠀⢸⣗⣻⣿⠄⢀⡾⣻⣿⡋⠀⢀⡠⡿⠗⢿⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠖⠒⠒⠋⠁
⠀⠀⠀⠀⠉⠓⢦⣀⠀⠙⠂⠀⠀⠘⣿⣿⣧⠀⠀⠀⠹⠥⠀⠁⠀⢳⡈⠁⢸⡄⠀⠀⠀⠊⠉⠁⠀⣤⣤⡤⣄⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠀⠀⠀⠀⡗⢲⠀⠀⠀⠀⠀⠀⠀⠨⠴⠼⠋⢄⠀⡇⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⠂⠀⠀⠀⠀⢀⡎⣀⠤⠤⢖⠨⠃⡇⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠈⠳⣇⡀⠀⢳⡈⠀⠀⣱⠋⢉⣑⢰⠇
⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠔⠀⠀⠀⠀⢀⣼⡧⡹⢶⣄⣉⠀⣼⠑⠊⣁⡠⠽⢄⠀⠀⠀⠈⠦⠀⠂⠠⠶⠶⠄⣀⣀⡤⠤
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣇⠈⠲⢭⡛⢻⡇⠀⠀⡤⢠⣤⠾⣄⠀⠀⠀⢲⣶⠦⢤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣧⢹⡄⠀⠀⠈⢻⠃⠀⠀⠀⣀⠄⡴⢶⣤⣤⡀
⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠻⠿⠿⠿⠿⠿⠛⠀⢷⡄⠀⢠⣿⠀⠀⠀⠀⠉⢰⠂⠚⠉⠙⠛⠛⠶⢶⣤⣄⡀
⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣿⠃⠀⠀⠀⠀⣠⠿⡇⠀⠀⢠⠀⠀⠀⠀⠀⠘⣿⢦
⠀⠀⣼⡟⠘⣦⠀⠀⠀⣄⠀⠀⠀⠺⣦⡀⠀⠀⢀⡼⠋⠀⠉⠢⡄⢀⣼⣇⠀⣿⠀⢀⡾⠀⠀⠀⠀⠀⠀⢹⡜⣧
⠀⢸⡟⠀⠀⠘⣧⠀⠀⡇⠀⠀⠀⠀⠈⢋⣠⣾⡿⠓⠢⡀⠀⠀⠙⣾⣿⣿⡄⣿⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇
⠀⣿⠃⠀⠀⠀⠸⣇⢠⡇⠀⠀⠀⣠⣴⣿⡿⠋⠀⠀⠀⠙⣆⢀⡤⢹⣿⣿⣷⣿⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢠⡿⠀⠀⠀⠀⠀⠀⠸⠀⣀⣴⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠘⢎⣵⣿⣿⣿⣿⣿⠀⠀⠀⣸⠃⠀⢺⠀⠀⠀⠀⠀⠈⣷
⣾⠇⠀⠀⠀⠀⠀⣀⡴⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⣃⣼⠼⠟⢻⣿⣿⣿⠀⠀⣰⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠖⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠄⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠳⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢷⣄⠀⢀⡀⢠⡟⠀⠀⠀⠀⠀⠹⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢺⣌⣶⣻⣀⡟⠀⠀⠀⠀⠀⠀⠆⢱⡈⠢⡀⠀⠀⠀⠀⠀⡞⠙⠋⠉⠉⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⠛⠛⠋⠛⡉⠍⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⡆⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⡀⠀⠀⠀⠀⢤⡿⠼⠳⡄⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⣠⢴⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⣰⠟⠁⠀⠀⠀⢧⠀⡤⠀⣠⠟⠀⠀⠠⠈⡆⢀⣀⣀⣀⠀⠀⢀⣇⠀⠀⢀⠀⣿⢶⠶⠶⠶⠶⠶⠦⣼⣷⠋⣴⣩⠄
⠀⠀⠀⠀⠤⣄⡀⢷⣌⢿⡏⠑⠢⢤⣈⣀⣀⣱⣤⣴⣋⠀⠀⠀⣀⣴⠟⠉⠁⣨⢟⣷⣠⠋⢈⣢⣀⣠⠟⠁⠈⠳⡁⠀⠂⠀⠀⠀⠉⠻⣿⣡⡆
⠀⠀⠀⠀⠀⠈⠻⡲⣏⣾⡇⠀⠀⠀⠀⠀⣴⠯⢽⡉⠉⢙⣶⠞⠉⠀⠙⡞⢉⡾⠋⠉⠙⢶⠋⡟⣹⡏⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠈⢻⡅
⠀⠀⠀⠀⠐⠶⣲⡿⠋⢫⡉⠛⠓⠶⣤⣀⣿⡀⠀⢱⢀⠎⠘⢧⠀⠀⣀⣈⣾⣏⠉⠉⣉⣽⠧⣧⡟⠹⡀⠀⠀⠀⠀⡰⠋⠳⢄⡀⠠⠄⠀⠈⣧
⠀⠀⠀⠀⠀⣼⣋⣀⣀⣠⠵⢄⣀⣠⡟⢻⠈⠛⠛⢻⡏⠀⣴⠿⣛⡉⠉⠁⣈⣉⣻⣾⣏⠀⠀⠘⣧⠀⠙⣄⣤⢶⣶⣥⢶⠶⢦⣌⠙⢢⣤⣤⢼⡆
⠀⠀⠀⠀⠀⠙⣦⣀⡀⠀⠀⣠⣤⣼⣇⣀⠇⠀⠀⢘⣧⣼⠃⠀⠀⠬⢹⠏⢯⠀⠀⣽⣈⣦⣀⣀⡼⠷⣴⠏⠀⢸⡘⣳⣾⣠⠞⣉⣧⣸⣄⣠⡞
⠀⠀⠀⠀⠆⠀⠀⠈⠉⢻⡟⠃⠀⠀⢈⠹⣷⠒⡒⠛⠁⢯⣤⣤⣄⣀⣸⠀⢸⣀⡴⣿⠧⠤⠭⠟⢷⣼⡇⠀⠀⢠⠋⠀⢸⠷⣾⣩⠋⠈⣿⠡⢄⣀
⠀⠀⠀⠀⣾⣟⡛⢻⡆⠈⠳⠦⣤⣤⣤⡶⠋⠀⠉⠀⠀⠀⠂⠀⠀⠤⠉⠛⠛⣉⣀⣈⠛⠒⠒⠶⠚⠁⠙⠶⠶⠾⠚⠛⠋⠄⠘⠷⠶⠟⠁⠀⠀⠈
⠀⠀⠀⠀⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡍⢉⡩⠟⢳⡆⠉⠀⠀⠀⠀⠈⠛⠀⠀⣴⣚⣳⡄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠒⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⡀⠀⢠⣤⣀⠀⢠⣤⣤⣠⣤⣤⡄⠀⠀⢀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡤⣤⣦⣴⡆⠀⢠⣶⣶⣶⣦⡶⣶⣦⣤⣤⣤⣤⣤⣴⢶⣴⣤⡤⠀⢀⡀
⠀⠀⠀⢠⣶⣤⣍⠀⣿⣿⡿⢛⣵⡜⠿⠿⠟⠋⠀⢀⣠⣌⠐⠠⢠⣿⣿⣿⣿⣯⣿⣬⣿⣛⣿⡀⠀⠈⠻⠿⣛⣽⣿⣿⣿⠻⣿⣯⣿⣟⢿⣷⡽⠟⢀⣿⣿⣧
⠀⠀⠠⣿⣿⡿⢁⣾⣿⣿⣾⣿⣿⣿⡀⠠⠀⡄⠀⠸⢿⣿⡠⠀⣿⠟⣩⣶⡘⢿⣿⣿⣿⣿⣾⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⡁⠐⢿⣿⣿⡛
⠀⠀⢠⠹⠏⠀⣿⣿⡿⣻⣭⣽⠻⠟⠀⠘⠿⢀⣶⣿⣦⣶⠙⠀⠀⣾⣿⣿⣷⣶⣿⣿⣿⢛⡛⠟⣁⠰⠐⠀⠻⣿⣿⣿⣏⣠⣿⣿⣾⣿⣿⣿⡷⠀⠘⣿⣿⡗
⠀⠀⢸⡇⠀⠘⣿⣟⣄⢿⣿⣿⣷⡄⠀⠀⣤⣾⣿⣿⣿⣷⡟⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⡆⢲⣷⡀⠈⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠠⠘⢿⡇
⠀⠀⠘⣷⡄⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⣿⢻⣿⣿⣿⣿⡿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣎⣾⣿⣿⣟⡿⡎⠟⠃⠰⣷⡜⢿⣿⣿⣿⣏⣥⣮⣝⠻⠀⠀⣴⣮⠃
⠀⠀⠀⣿⣧⠀⠘⢛⣿⣿⣿⣿⣿⠂⠀⢘⣿⣿⣿⣿⣿⣿⡇⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢩⣻⣿⣿⣿⣿⣷⣷⣿⠛⠁⠀⣾⣿⡿⢂
⠀⠀⠸⢿⡇⠐⠀⣿⣿⣷⣿⣇⠀⠀⡆⠈⣿⣿⣿⣿⣿⣍⠀⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠀⡿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣿⣿⣡⠏
⠀⠀⢰⣶⡀⠀⠀⢻⣿⢿⣿⡿⠀⠀⠁⠀⣿⣿⣿⣿⣽⡟⠀⡆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠳⢿⣿⣿⣿⣿⣿⡷⠀⠀⠙⣿⣿⣿⠇
⠀⠀⢸⣿⣿⡄⠀⠐⣿⣿⣿⠇⠀⡀⠀⢈⣿⣿⣿⣽⣿⡧⠀⡇⠘⣿⣿⣿⡟⣭⣼⣭⢹⢛⣛⡿⢿⣻⣿⠁⠘⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠀⠀⠀⠘⣿⣿⡆
⠀⠀⣾⣿⣿⡇⠀⢀⠺⠿⢏⠀⢸⡇⠀⠈⣿⣿⣿⣿⣿⠃⢸⠀⠲⣿⣿⣿⣿⣿⣿⣿⡆⣄⠈⠁⠉⠱⢿⠀⠴⡄⢀⣼⢠⣦⣈⢹⣿⣿⣿⢀⣠⠀⠄⠸⣿⣦
⠀⠀⢹⣿⢋⣼⡇⠛⣴⢯⡿⢃⣼⣿⣄⠀⡙⢿⣯⣿⣏⠀⠹⠀⢀⢿⣿⣿⣿⣿⠿⢋⠀⠀⠀⠉⠀⣷⣶⣾⣆⢡⣿⣿⣾⣿⣼⣷⣽⣿⡟⢨⠁⠆⣈⣳⣝⣯
⠀⠀⠈⠉⠛⠛⠓⠺⠛⠳⠘⠟⠛⠛⠿⠄⠙⠒⡈⢉⣓⣀⠠⡐⠃⠄⠙⣻⠿⡿⢡⠅⠸⠐⢰⡄⣷⠌⠝⠿⠿⠧⠿⠛⠛⠛⠫⠿⠿⠫⠴⠺⠤⠼⠿⠛⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣮⣿⡷⠶⠤⣤⣤⣠⣴⣶⠧⠚⠛⠉⠀⠀⠈⡀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⢿⣉⡉⣉⠙⠟⣳⡮⣗⠒⠖⠻⠿⠳⣦⣤⣤⣄⡤⢀⣀⡟⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠉⠉⠁⣠⠏⠀⠀⢹⡆⣌⡀⠀⠿⠛⠉⠉⠉⡠⠜⠛⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠉⠀⠀⣰⣿⣗⣻⡿⠇⠀⠄⠒⠛⢁⠀⠀⠀⠒⢛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡖⠂⠄⠒⠀⠀⠄⠐⣦⣬⠟⠋⠁⠀⠀⠀⠀⠀⠖⢀⢴⡾⠃⢺⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠷⢿⠟⢁⣀⡀⢀⡀⣠⣤⠄⣽⠇⠴⠟⠲⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⡤⠶⠛⠂⠈⠴⣉⣀⠻⠜⠓⠒⠠⠄⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢀⣦⣶⣤⣤⠙⠀⠀⠤⠄⠀⠬⢁⣀⡐⠠⠀⠀⢄⠉⠀⠀⠀⠀⠀⢀⣼⡷⠋⠁
⠀⠀⠀⠀⠀⠀⡀⣬⡆⠀⠀⠀⠒⠀⠀⠀⢶⡶⠟⠁⣠⠆⠀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⢀⠀⠉⠉⠀⠦⠀⠐⣾⡆⣠⠿⣁⣤⠀⠈⠉⠁
⠀⠀⠀⠀⢀⡶⠈⠀⠀⠀⠀⠀⠠⣀⠀⡀⠈⠙⣄⠠⠬⠀⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⡷⠦⠀⠀⠀⠀⠀⠀⢀⡀⢀⠈⡀⠀⠀⠀⠀⢠⣄⣤⣰⡄
⠀⠀⠀⠈⠙⠁⠀⠀⠀⠠⣀⣀⠴⠟⠐⠚⠂⠚⠃⣀⠀⠀⠀⠀⠐⠒⠚⠃⠀⠀⢤⣶⢾⡁⣀⣀⠀⢀⣨⣧⠀⠀⢻⡥⠘⠃⠀⠀⢠⡀⠀⠾⠛⠛⠋⠁⠁
⠂⠀⠀⣀⢠⣴⣶⠀⠀⣰⣟⣀⠀⠤⡀⣤⣤⣤⣤⠽⠆⢄⠀⠀⠐⠦⣤⣤⣤⠄⢀⣄⡀⢀⣀⠈⠀⠘⠁⠀⠀⠀⠀⠀⠐⠀⠐⢨⣸⣿⡆⠀⠀⠀⠄⡀
⠀⠀⠀⠛⠛⠉⡁⠀⣴⡟⢈⡀⣠⠔⢛⣻⡏⠀⣀⣀⠠⠴⠃⠤⣀⢴⠄⠉⠁⠀⢸⡿⠀⠀⠉⠀⠀⣴⣿⣦⠀⡀⣀⣀⣀⣤⣤⣾⠋⠀⠀⠀⠀⣀⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠁⠚⠋⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠠⠈⠉⢻⠛⠂⠀⠀⢙⡟⠉⠙⠛⠀⣡⡈⣶⣶⣿⣿⣿⣟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣠⣿⣟⠃⢀⣴⣶⡆⠀⠀⢙⣶⣶⣽⣌⢩⡌⠛⠻⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⢾⡛⠉⠁⠀⠀⢠⡓⣚⣁⣠⣾⢾⠿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⣀⠀⡀⠀⢁⠍⠉⢉⣷⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡄⣀⣠⣾⡿⢀⠔⠁⠀⢠⣿⡿⠟⠉⠀⠀⠀⠀⠀⣆⢀⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣤⣶⣾⣿⣟⠟⠋⠚⠁⠀⢀⡴⠋⠁⠀⠀⠀⢀⣀⣀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⣡⠇⡀⠀⠀⣤⡴⠋⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⡿⢿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠀⢀⣼⠃⠀⠀⠀⠀⠀⠶⢤⣶⣿⣿⠿⠟⠟⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢴⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠃⠀⠰⠤⣤⣄⣰⣄⣤⣠⣴⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠟⠛⠛⠓⠶⢤⡄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⡠⠄⡠⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⢠⡰⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠈⢍⠀⡀⠈⠄⠈⠄⣀⣀⣀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢠⡔⣳⠬⠶⢯⢥⣴⡆⢠⢬⢄⣤⡐⢁⣀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⡎⠠⢆⡸⠅⠈⣤⠲⢾⠃⠼⠀⣍⠂⠈⡥⢋⡀⠰⣁⣈⡠⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠙⢆⢠⡀⠆⣷⡭⣞⣱⡛⡶⢮⡿⣗⣦⡴⣙⠛⣉⠂⠮⣁⡠⡍⣤⠡⠐⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⢜⠀⡭⡓⣄⠳⠜⢻⣧⣄⠘⠀⣌⢆⠐⣉⠢⣈⠁⠂⠠⠙⠦⢸⠡⠕⢈⣆⣠⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠰⢄⣀⣀⡞⢈⡠⠂⠶⡫⠤⢛⣀⣐⢦⠠⢤⡀⠀⢆⡀⠁⠊⠀⠰⢄⠑⢦⣴⢈⠙⣊⠵⠈⡴⡀⠀⠀⠉⡀⠰⠦⠈⠁
⠀⠀⣀⠀⠀⠀⠐⠒⠊⠫⣥⡄⠉⠚⠁⠀⣠⣠⢬⣹⣦⠤⣀⢬⡜⣭⠝⠀⣀⠄⡠⠑⠀⣰⠋⠐⠀⣄⣁⡄⡈⠕⠐⠀⠀⠌
⠀⠀⠀⠀⠀⠨⡭⠅⠀⠂⠁⠀⣀⠴⠬⠬⠉⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠵⠛⠁⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⡤⠄⠀⠹⠟⣻⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠠⠀⠠⠔⠂⠂⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣀⡴⠞⠀⠂⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⢟⣿⣿⣿⣿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠉⢀⣼⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢢⣐⠀⠀⠀⠀⠀⠀⣠⣦⣴⣶⣶⢐⣿⣿⣿⣿⠟⢜⡾⣳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⢈⣿⣦⡀⠀⠀⠀⣾⡿⠽⠛⣿⠅⣿⣿⣟⡽⠇⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠄⠻⠛⠋⠉⠳⣄⠀⢀⡀⠀⢤⡤⣾⣿⣽⣿⣷⢟⣵⡶⣿⣿⣿⣿⡄
⠀⠀⠀⠸⠟⠓⠛⠓⠒⠚⠛⠯⠞⠉⠉⠈⢻⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⢹⣶⢸⠙⢰⡆⢁⣣⣿⢟⠜⣵⡿⣫⣾⠏⣹⣿⣿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣆⡆⠀⠀⠀⠀⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⠀⣼⠙⠘⣏⡴⢫⡿⠋⡞⠱⠧⠮⠞⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠠⢀⣾⣿⣷⠀⣴⡶⣰⢯⠟⢾⡗⣽⠻⢩⣿⣿⣎⣤⣼⣶⣶⡷⠚⠋⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⡁⠀⢋⡛⡟⣿⣿⣿⠋⠐⣿⠏⠬⠀⠘⢡⢧⣿⣿⠿⣿⣽⣿⣿⠁⠀⠀⠀⠀⠀⠀⣰⠞⢫⡴⡛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣖⠋⣤⣴⣶⠿⣛⣤⣿⣿⣿⣀⣶⣬⡄⣠⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣧⣴⣀⠀⠖⣦⣤⣶⣽⣶⠋⠀⠎⢽⢿⠏
⠀⠀⠀⠀⠒⠒⠀⡀⢸⣬⣿⣿⡛⠛⠛⠿⠿⠛⠛⠉⠉⠭⣭⣀⣈⣙⡛⠒⠒⠚⠁⠀⣈⡉⢙⣿⣿⣿⣤⣼⠿⠟⠛⠛⠉⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠈⠁⠀⠀⠈⠃⠚⠃⠁⠀⣀⠐⠂⠀⡂⠉⠀⠚⠃⠈⠙⠛⠛⠛⠻⠿⠌⠉⣉⠀⠸⢧⣤⣤⣄⠀⠀⣉⣀⣠⠙⠳⠶⢦⣶⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣼⠋⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⡾⠉⢸⣿⣷⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⠀⢀⣽⣿⠃⠀⠀⠀⢰⠀⠀⠀⡇⠀⢈⣽⣿⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣿⣀⣸⣿⣿⡃⠀⢠⠖⠋⠀⠀⢀⣇⣀⣸⣿⣿⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠘⠷⠤⠄⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢸⡟⠋⠉⠉⠉⠉⠛⣿⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠀⠈⠉⠉⠉⠉⠉⠉⠉⠋⠀⢠⡖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⣾⣿⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⣽⣿⣾⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⣠⣤⣤
⠀⠀⠀⠀⣄⠀⠀⣀⠀⠀⣠⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⠃⣀⠀⠀⠀⠀⠠⡀⡀⢸⠐⠂⠐⠒⠒⠊⠀⠀⠀⠀⢠⣶⡀⠘⣿⡷⠋⢠⠊⠀⡇
⠀⠀⠀⢀⣿⠆⠀⡇⠀⠸⠟⠋⠁⠀⠀⠙⢏⠀⠀⠀⠀⠐⢿⠄⠀⠀⠀⠐⠟⢻⣾⡿⠷⠀⠀⠒⠢⠤⣤⠔⠀⠀⠉⠁⠀⠀⠀⠀⠉⠂⣴⣧⡀
⠀⠀⠀⠰⠿⢿⣿⢁⠀⠀⠀⠀⠀⠈⠀⣀⣦⣷⣄⡀⠀⢠⠈⠵⠆⠀⠀⠀⠀⠀⠿⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢲⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣶⣿⣿⣷⣿⠿⠿⠟⠙⠛⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠉⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄⢀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⡄⠆⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣏⠀⢖⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣸⠀⢻⣷⠈⠀⢨⡄⠈⡅⠀⠀⠀⠀⠀⠀⠀⡆⢘⣿⣿⠀⠹⠛⠃
⠀⠀⠀⠁⠂⠠⡄⣤⠀⠀⠀⠀⡤⡤⠀⢠⣶⣶⣾⡟⠁⠀⠀⠀⠀⠀⡏⠘⠙⣿⣷⣦⣌⡁⠀⠨⠀⠀⠀⠀⠀⠀⠀⠷⠀⠁⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠼⢧⡄⠀⢀⣿⠇⢀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠠⣿⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠙⠻⠂⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠆⠀⢀⡀⠀⢠⢄⣴⣾⠀⠀⠀⠀⠀⢠⣿⣮⢻⣗⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀⣀⠀⢠⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠇⠀⠈⠘⠟⣿⠀⠀⠀⠀⢀⠘⣿⡇⢣⡻⡇⠀⠀⠀⠀⠀⠀⢸⢸⣿⡿⣇⠀⠘⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠲⣤⡀⡄⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⡟⠀⠀⠀⠀⢸⣦⠈⣧⠀⢹⠁⠀⠀⠀⠀⠀⠀⠘⠈⢛⡵⣹⠀⢰⠛⠀⠀⠀⠀⠀⢸⣿⣲⣦⣤⣄⡀⠀⠘⣧⣡⣇⡴
⠀⠀⠀⠀⠠⣤⣄⣄⣠⣴⠞⠀⠀⠀⠀⠀⢸⣿⣷⠬⢠⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣻⡇⢀⠀⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠒⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠛⠛⠛⠓⠛⠓⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⢠⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣤⡄⠰⢶⣶⣆⡀⠀⠀⠀⠀⠀⢐⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠛⢿⣿⠶⠂⠀⠀⠀⠀⠀⠀⠔⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢻⡇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⠾⠸⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠈⠣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⣀⡀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣦⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣉⠁⠀⠀⡀⠀⠀⠀⢀⡀⠘⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠃⠀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠾⢿⠷⣆⠀⢀⠀⠀⣾⡟⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⠃⠀⠀⠀⠀⢀⡀⠐⠀⠀⠀⢂⡴⢄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠛⠢⠋⢰⣿⣤⣾⣿⠀⠀⠀⢰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣽⣦⣤⣤⣴⣾⣿⠃⠀⢀⠤⠚⠁⠀⠀⣠⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⣽⣯⠔⠚⠁⠀⠀⠀⠔⠋⠀⠑⠄⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣠⠆⣾⣻⠟⢋⠍⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣿⣿⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⣠⠀⠀⡼⠋⡔⢁⠌⢉⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⢮⣾⣿⣿⡟⣛⠛⠉⠉⠀⢀⡤
⠀⠀⠀⠀⠋⠁⠰⠂⢀⡼⢂⡜⣀⣾⡟⠁⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣢⣿⣿⣿⣷⣿⣿⣿⣿⣷⣤⠤⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣼⡶⢡⡟⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣫⣿⣿⣿⣿⣿⣿⡿⠁⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢣⣾⡏⢀⡿⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠁⠀⣾⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠁⠘⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⡿⠃⠰⣰⠏⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠰⠤⠤⠄⠀⠤⠄⠄⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⣄⠀⣰⣿⣿⣿⣿⣿⣿⡿⣦⠀⢀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣿⣿⣿⣯⣿⣿⣿⣿⣿⡇⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠀⠙⣿⡿⢿⣻⣿⠯⣯⡟⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣾⣿⣿⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⢸⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠉⠙⠻⡛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠐⠀⢳⡎⠀⠀⠀⠀⠙⣄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠫⢤⣀⣀⡤⠾⠷⠒⠶⣄⠀⠀⠈⢆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠤⠶⠷⢦⡔⣉⣀⠀⢀⡤⢒⣽⠶⠶⠶⠦⣦⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢹⡄⠈⡟⠛⡶⠋⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣀⣤⡤⠿⠴⠷⠶⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⣼⠉⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⢹⣠⡤⠴⠶⠚⠛⠛⠺⣧⡟⢄⡀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⢧⣀⣀⣀⢸⡀⠀⠀⠀⠀⣠⠴⠶⠶⠶⠶⣄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠳⣄⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣉⠀⠀⠀⡏⠉⠉⠉⠳⣄⠀⠀⢾⠁⠀⠀⣀⣀⣀⣸⡟⠛⣦⡴⠖⠒⠲⠶⢤⣀⢸⠇⠀⠈⢛⠓⢚⡿⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠓⢒⣺⣤⣄⣀⠀⢨⡟⠓⣺⠶⠚⠛⠉⠉⠁⠈⠙⢶⡏⠀⠀⠀⠀⠀⠀⠉⠻⣶⠉⠙⣶⠓⢾⠁⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⣠⠞⠉⠀⠀⠀⠉⠙⢻⡄⢸⡧⣄⡀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠛⠛⠛⢧⡀⠀⣸⣌⣳⡀
⠀⠀⠀⠀⠀⠀⠀⡋⠀⠈⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⡧⣼⠃⠀⠉⠓⣤⢤⡴⠚⠒⠶⠷⠶⣄⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⠀⠀⠹⣶⠁⠀⠈⡇
⠀⠀⣠⠞⢻⡓⠒⠓⠺⠭⡟⠀⠀⡠⠤⢤⡀⠀⠀⠀⠃⠘⢦⡀⠀⣾⠀⣾⠀⠀⠀⠀⠠⠀⠙⢧⠀⠀⢀⣀⣠⣼⠋⠀⠀⠀⠀⠀⣿⠈⠉⠉⠳⣄
⠀⣰⠋⠀⠀⠉⠢⣄⡀⠸⣇⣠⠤⠷⠶⠶⠓⠒⣳⠾⠿⠛⠿⣏⣉⠉⠉⠛⠦⣄⠀⠀⠀⠀⣠⡾⠛⠋⠉⠉⠉⠉⠙⣦⠀⠀⢀⡴⠃⠀⡠⠀⠀⢸⡆
⠀⡟⠀⠀⠀⠀⠐⠉⠙⣢⣬⣧⣤⣤⣀⣀⣀⣌⡀⠀⠀⠀⠀⠀⠉⢻⠖⠒⠒⠛⠻⣶⢾⡋⠉⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠛⢿⡀⠀⠀⠀⠀⠀⢸⣇
⢸⡇⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⣃⣀⣀⢀⣠⠴⢋⣈⡽
⢸⡇⠀⠀⠀⠀⢰⣯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠤⢤⣤⣤⣾⡀⠀⠀⠀⠀⡿⠀⠀⣷⠀⠀⢀⣀⣀⠀⠀⣿⢹⠟⠋⠉⠉⠉⠙⠻⣶⠊⠉
⠀⠻⠦⣄⣀⣀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠀⠈⢳⡄⠀⢀⣴⣇⣀⣼⣋⢒⣒⣧⡤⠾⠶⣴⢁⡏⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⡤⠶
⠀⠀⠀⠀⠈⠁⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⢸⡧⠚⠉⣠⠞⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⠾⣆⠀⠀⠀⠀⠀⣀⣴⡿⠶⡖⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⢤⣤⣤⣤⠶⠶⢯⣁⠀⠀⠀⠀⢀⣀⣠⠞⠀⠀⠀⣇⠀⠀⣀⣤⣤⣤⠤⠤⢤⣴⡛⠀⠈⣛⢷⡛⠛⠛⠉⠸⡀⠀⠈⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣬⣽⣟⣛⣛⠋⠉⣡⡤⠤⠤⣄⠈⠑⡞⠣⣀⡀⠀⠀⠀⠀⠀⠉⠳⣜⠧⠤⠟⠀⠀⠀⠀⠙⠒⠛⠋⠁
⢠⠞⠛⠛⠛⠛⠛⠲⣤⡀⠀⠀⠀⠀⢾⣍⡀⠀⠀⠀⠀⠈⢹⠀⢸⣅⣀⣀⣠⡾⠀⢸⡇⠀⠈⠉⠉⠛⠛⠛⠓⠛⠓⢹⡇
⡟⠀⠀⠀⠤⠤⠤⠀⢈⡇⠀⠀⠀⠀⠀⠣⣌⣉⣉⣩⣤⡴⠋⠀⠀⠉⠉⠉⠁⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠈⠓⠦⣤⣀⣤⣤⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠉⠙⠓⠦⣄
⠀⠀⠀⢀⠖⠉⢀⡤⠖⠛⠉⢙⡓⠦⣄⠀⠙⣄
⠀⠀⢠⠏⠀⡴⠁⠀⠀⢰⠚⠛⢦⠀⠈⢧⠀⠘⡆
⠀⠀⡟⠀⢸⠁⠀⠀⠀⠘⢧⡀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠴⠶⠶⠤⠤⢤⣀⡀
⠀⠀⢷⠀⠸⡄⠀⠠⣄⣀⣠⠇⠀⠀⢀⡞⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣡⠴⢒⡒⠛⢛⣻⣿⣖⣲⣤⡉⠓
⠀⠀⠘⣦⠀⠘⠦⣄⡘⠁⠀⣀⣠⠴⠋⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠣⣝⣛⣶⠖⠋⠀⢀⣘⣉⡤⠏⢀
⠀⠀⠀⠘⢷⣦⣀⡀⠉⠉⠉⠉⣀⣤⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣷⣶⣤⣤⣉⣉⣉⣉⣉⣁⣤⣤⣶⣿
⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⠒⠒⠒⠒⠶⠤⣼⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡤⠖⣋⡍⠉⢉⡭⠿⢭⣙⡶⢄⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠙⠮⢭⣛⣛⣉⣀⣀⣈⡥⠴⠋⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣾⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⠟⠋⠉⠉⠉⠉⠛⠿⣿⡿⣿⣯
⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⡀⠀⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⡴⠒⠋⠙⠓⠲⣄⠈⠻⣿⣿
⢠⠔⢋⣩⠤⠴⠶⠒⢒⣶⡦⠤⣍⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⣷
⣏⠀⢯⣐⠲⠧⡤⠖⠉⠀⠹⠛⣃⡽⠀⢸⣿⣿⣿⣿⣽⣿⣿⣯⣿⣿⣿⣿⣿⠀⢸⠠⢴⡇⠀⢀⡴⠛⢦⡀⢹⠀⢸
⣿⣷⣤⣌⣉⡙⠛⠓⠚⠛⠋⣉⣁⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠘⡆⠀⠳⠦⠞⠀⠀⡼⠉⢸⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣷⡀⠘⢦⡀⠀⠀⠀⠀⠀⢠⠞⠀⣸
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠈⢿⣿⣦⡀⠉⠓⠶⠤⠶⠚⠁⢀⡴⠃
⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣤⣤⣤⣤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⠉⠉⠳⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃⠀⣀⠜⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡬⠂⠀⠀⠀⠀⠀⠀⠀⡃⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠉⠁⠒⠒⡆⠀⠀⠀⠀⠀⡝⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡨⠀⠀⠀⠀⢀⣤⣄⡅⠀⠀⠀⠀⠀⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠹⠀⠀⠀⢀⠨⣀⠨⡄⠀⠀⠀⠀⠀⡀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠢⡀⠀⠀⠀⢀⣀⢀⡞⠉⡗⠀⠀⡁⠀⠀⠀⠀⠸⡶⠅⢪
⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠠⠉⠐⠐⠶⣚⠁⠀⠀⠀⠈⠉⠉⠀⠈⠛⢻⣻⠃⠀⠀⠀⠀⠀⣫⣁⡯⣀
⠀⠀⠀⠀⠀⠀⠀⠅⡠⡍⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠣⠄⠀⠠⠔⣊⢙⣯⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢰⠒⠲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⠉⠘⣼⣻⠀⠀⣷⠘
⠀⠀⠀⠀⠀⠀⠀⡘⠶⣤⡞⢉⡱⠈⠉⣽⡄⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠜⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠘⢄⢠⠄⠀⠈⠓⠒⠦⠏⠢⡜⠗⠊⠙⡀⠀⡔⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣀⣀⣀⠀⠁
⠀⠀⠀⠀⠀⡄⠀⠚⠈⠉⠀⠀⢁⠀⢠⠐⠉⠀⠀⠀⠀⠈⢱⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢉⠉⠉⣤⣹
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⠖⠛⠀⠈⢄⡤⠶⡄⢀⣀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡄⠀⡟⠋
⠀⠀⠀⠎⠉⢒⣁⠀⠀⣠⣬⡓⠈⢁⠀⠈⢡⠐⠁⢺⠀⣥⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⠀⡅
⠀⠀⠀⠣⠤⠏⠉⠀⠘⣇⡀⣹⣤⠼⠀⠀⠈⢄⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣘⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠒⠒⠒⠂⠐⠚⠛⠛⠓⠂⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠟⠛⠿⠿⣿⣿⣿⣿⠿⠿⠛⠻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⢈⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣦⣴⣶⣿⣿⣿⣿⣶⣦⣤⣶⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠋⠙⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⠋⠙⠻⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⠀⢠⣿⣿⣿⣧⣴⣦⣼⣿⣿⣿⡀⠀⠀⠀⠀⣠⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿⣯⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣿⣿⣿⣿⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣶⣤⡀⢀⣴⣾⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠟⠋⠛⢿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠛⠛⠶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡄⠀⠀⢹⣆
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢹⣆
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢻⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣠⣤⠶⠿⠖⠛⠛⠛⢶⣄
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⣤⡀
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⢠⠞⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢹
⠀⠹⣆⣤⣤⣤⠤⠤⠤⠤⠤⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⠤⠀⠀⢸
⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠔⠒⠛⠋⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⢳⡄⢄⣀⣀⣀⠀⠀⠀⠀⢀⡀⠀⢸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣧⠒⠒⠀⠀⠀⠀⠀⠀⠀⣿⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠸⣆⠸⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⣹⠟⠉⠉⠛⠛⠛⠒⢓⣒⣒⣶⣶⣿⣿⣍⠉⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁
⠀⠙⠲⣷⣀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠁⠀⢀⣤⠖⠛⠛⠉⠉⠉⠉⠁⠀⠀⡴⠉⢳⡄⠀⠙⢷⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⡟⠈⠓⠦⢤⣄⣀⣀⣀⣀⣀⣼⠃⠀⢸⡇⠀⠀⠀⠈⠉⠉⠉⠛⠲⠶⠶⠖⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠉⠉⠉⠁⡇⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⢀⣠⣠⣷⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣴⣶⡄⠀⠀⢀⣀⣸⣿⠠⠄⢈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣠⠈⢛⣿⣷⣾⣿⣿⣿⣿⣦⣴⣿⣿⣷⣶⣤⡀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣼⣿⠇⠀⠉⢹⡿⣿⣿⣿⣿⣿⡄⠉⢟⣉⠙⠛⠻⣿⣼⣿⣿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⡿⠿⠶⣦⠀⠙⠁⠘⢿⣿⣿⣿⣷⠀⣘⢛⡑⢿⠇⡋⢹⡟⢻⣘⣀⠁
⠀⠀⠀⠀⠀⢠⣶⣾⠿⣿⣷⣫⣤⠀⠶⠾⣷⣿⣿⠟⠈⠀⠀⠉⠉⠀⠻⣿⣽⣶⣾⣿⣿⣷⣿⠿⠋
⠀⠀⠀⠀⠀⠸⣏⣠⣴⣶⣿⢉⣡⣶⣦⡄⠋⠛⠛⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⣘⠟⣅⠈⡙⣿⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠹⣛⣿⡦⣸⣤⣿⣿⡟⠀⠀⠀⠀⠀⠈⠁⠈⢡⣾⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⡇⠀⠈⢹⠇
⠀⠀⠀⠀⠀⠀⣀⣬⣷⣾⡿⠟⠙⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⠁⢹⣿⣈⣥⣤⣀⣿⡇⠈⠀⠈⠁
⠀⠀⠀⠀⠀⣾⡿⠿⣿⣿⣶⠀⠀⠀⢿⣷⠀⠀⢀⣤⣤⣀⠀⠈⠄⠀⠃⠐⢿⠟⠻⠿⠿⣯⡁
⠀⠀⠀⠀⠀⠈⠀⠠⠙⢿⡇⢀⣴⣶⣿⠿⢇⣀⣾⠛⠛⠛⢶⣦⣠⣤⣤⣤⣸⠐⣀⠐⢸⠉⠳⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠁⣼⣿⣿⡇⠀⠐⠭⠇⠄⠀⠤⠸⣿⠛⠛⣿⣿⣿⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⣘⣛⣛⣾⣿⣶⢾⣿⡄⠀⠻⣿⠟⠃⠀⠘⠃⠀⠀⠀⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣀⣭⡭⠭⠍⣉⡉⠿⠿⢿⣶⠀⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⣹⡀⢀⣴⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠒⠛⠋⠉⠉⠉⠉⠉⠙⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠁⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠊⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠶⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠡⡤⠤⠄⠀⠈⣃⡀
⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⣠⠴⠃⠀⢀⠀⠆⠀⢀⠀⠈⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠤⣄
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡀⠀⠸⠂⡄⠀⠻⠂⢀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢈
⠔⠊⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⡐⠀⠀⠈⠀⠀⠐
⠈⠀⠄⠀⠀⠀⠀⠰⣧⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⢸⡀⠀⠀⢀⡦⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠞⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡾⠻⢦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⣀⣤⠤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠩⢽⠈⠢⠄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡰⠈⢧⡀⠄⠀⠍⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢰⠄⡤⢇⢀⡂⠁⠌⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⡀⠠⠃⡀⠄⠈⣇⡂⠀⣰⠄⢀⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⡏⠀⡂⠈⠉⠈⠟⢐⡃⠄⣀⠠⠍⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡕⠠⠄⢠⢄⡀⡎⡂⠅⠂⠀⡇⢀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢰⠣⠌⢂⠀⠀⢰⢑⠢⠁⠐⠨⠅⠅⡂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠰⡕⠂⠀⢐⣀⠀⢸⠃⠁⠀⡀⢸⢐⠠⢀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢄⡀⠀⠐⢏⡚⡎⠀⡴⠀⣜⠄⡊⠔⠠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠘⠁⠀⠀⠀⠀⠀⠀⠀⠐⠈⠸⡀⠄⢀⠘⠁⠣⢠⠏⠀⢗⠨⢂⠁⢂⠅⣂⢣
⠀⠀⠀⠀⠀⠀⢠⠜⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠉⡯⠂⠠⠀⠀⠐⠀⠸⡄⠀⠸⣂⠕⢊⠄⣊⠔⠀⣀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠹⡂⠄⢐⠀⠂⢀⠌⣮⠀⠼⡅⢊⠔⡨⠐⡈⠀⣆⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡎⡂⠡⠐⠒⠀⢈⠰⠓⠀⣄⢨⠔⡩⢐⠡⠀⠔⢉⠘⣄
⠀⠀⠀⠀⢀⠂⢀⡜⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠔⠇⡇⣒⡨⠅⠀⠀⠬⠂⠀⠰⠊⣀⣡⣂⣕⡈⡐⣀⣄⡠⠈⢢
⠀⠀⠀⠀⢠⠀⣸⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⢠⠭⠀⡂⠅⠀⢸⠍⡢⠅⠀⠈⠁⠠⣠⠅⠀⠠⢘⠵⣪⠅⢸⡆
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠈⣠⠆⠀⠀⠘⡎⠅⣂⠈⠀⣘⠡⢂⠁⢀⠇⠀⠀⠲⠒⠖⢢⢒⠠⢒⠤⠀⢿⡲⠄
⠀⠀⠀⠀⢰⡀⣷⡄⠀⠀⠂⠀⠀⠀⡔⠣⠀⠀⠀⢠⡃⠅⠐⠀⡄⠳⡌⠂⣠⠎⠀⠈⠀⠀⠀⠀⢸⠅⢂⠅⢂⠀⠿⡧⡀⠈
⠀⠀⠀⠀⠀⠙⠢⡁⠠⠀⠀⠀⠄⢰⠁⠠⠀⠀⢀⡖⠀⠁⠐⠀⠀⠀⢂⠎⢑⠀⢐⠨⠀⠈⡥⠤⠸⣊⠄⡈⠄⠀⣠⠇⢀⡠⠐
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠴⠢⠸⠤⠉⠀⣰⠃⠀⠄⠀⠀⠃⠈⢰⠑⡈⠅⠀⠁⡀⠕⡠⢀⠀⠀⢷⡠⢂⣤⠎⣁⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⡄⣀⣁⠒⠨⠀⠀⠲⠴⡀⢞⠄⠀⠅⠂⠡⠂⠅⠪⠀⠅⢀⡄⠒⠉⡀⠄⢀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠈⠉⠒⠒⠲⠤⠀⠲⠚⠒⠒⠒⠒⠀⠖⠃⠀⠀⠀⠀⠐⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⠀⠀⠀⢀⠀⠈⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⢄⠀⠀⠀⠑⠄⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢮⣻⣎⠐⣧⠀⠀⠀⠀⠀⢘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⡟⣿⣿⡆⢲⣣⠀⠀⠀⠀⠘⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠱⣿⣿⣷⠀⠱⡀⠀⠀⢸⡄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠙⣿⣿⣧⠀⠀⠀⠀⣽⡇⠀⠀⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠶⠖⢿⣿⣿⣦⣀⡀⠘⠟⠷⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡅⠀⠀⠀⢘⣿⣿⣿⣼⣿⣷⣦⣄⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢘⣿⣿⣿⣯⢿⣿⡻⣿⣦⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣠⣀⣹⣿⣿⣿⣾⣷⠍⠻⣮⡻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠉⠛⠿⠃⠀⠙⠻⡻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣶⣶⣰⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠛⠉⠉⠙⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣯⣶⣌⣁⡤⣀⠀⡂⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡵⠾⢿⣳⣍⢀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢿⣿⣿⣿⡆⢸⣷⣿⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣻⣹⣿⣿⣧⣿⣿⣶⡛⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠽⣿⣿⣿⣿⣿⣿⣿⣿⣏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⡟⠫⡐⠀⡄⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣾⣯⡒⠻⠄⠇⣠⣧⣜⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣶⣄⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⠄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡯⣉⣩⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣧⡀⠀⠀⠀⠀⠰⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠉⠙⠛⠲⠶⠄⠀⠀⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⡀⢀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠒⠚⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠈⢳⡸⠄⣿⣿⣿⣿⡋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⣸⣡⣎⠙⠿⢟⣻⠴⢚⡈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡷⠒⡛⠋⠁⢀⣤⠅⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠯⠥⣄⣀⣀⣀⣰⢈⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢛⡇⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⡷⢈⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠼⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣗⣻⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⣶⣿⣶⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠋⠉⠉⠙⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⣿⣿⣿⣿⣿⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣾⣿⣿⣏⣁⠀⠙⠿⠿⠿⠋⠀⢈⣹⣿⣿⣷⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⠀⠀⠠⠾⣿⣷⣄⣀⣀⣴⣿⣿⠦⠀⠀⠈⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⠓⠒⠚⠉⠉⠉⣿⠋⠉⠉⠓⠒⠚⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⣉⣉⣩⢲⣾⣶⡆⣶⣾⣷⣾⣽⣭⡝⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣠⣿⣯⣿⣿⣿⣿⣇⣻⣿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣶⣶⣒⣚⣉⣙⣒⣿⣒⣋⣉⣓⣒⣶⣶⣻⣆⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⣭⣯⣭⡭⢶⠶⢭⠁⢩⢵⠶⡶⣭⣭⣭⡅⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠛⠛⠻⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⣀⣤⣴⡿⠉⠉⢿⣦⣤⣀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⣿⣿⣇⢸⡇⣸⣿⣿⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣿⣄⢸⣿⣿⣿⣿⣿⣿⡇⣠⣿⡆⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡀⠀⠀⠀⢻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣦⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⢿⣿⣿⣷⣤⣀⠙⠻⢿⣿⣿⡿⠟⠋⣀⣤⣾⣿⣿⡿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⣄⠀⠀⣠⣴⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣾⣟⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣶⠟⠻⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⣿⡿⠟⠋⠀⠀⠀⠀⠉⠻⢿⣿⣿⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣖⠚⠋⠉⠉⠉⠁⠀⠀⠉⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢳
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢀⣀⡀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣼⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠈⢧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠚⠛⠒⠦⣄⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠬⡗⠶⢤⣄⣀⠀⠀⠀⠀⣀⣀⡀⢸⡇⠀⠈⠓⢦⡀⢀⣀⡀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠸⣅⣂⣀⠟⠀⠀⠈⠉⠉⠛⢻⠋⡁⠈⠹⡾⣇⠀⠀⠀⠀⡷⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠛⠛⠲⣜⢄⣈⢀⣀⠗⠙⢦⣄⣀⢀⡇⠀⠀⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠳⢄⠀⠈⢧⠈⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠠⠄⠄⠄⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠅⠮⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢆⠡⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⣀⣀⠀⠀⠀⢠⠶⠛⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡐⢆⠀⣤⡀⢑⡤⠴⢴⠀⡠⢄⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡒⡀⠾⣯⠿⠛⣉⡑⠻⠒⣯⡐⢻⠳⢽⠅⠠⣵⢢⡤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠿⠀⠝⠤⠓⣌⣰⣟⣩⠳⢥⣺⡖⢦⠍⣠⣖⣡⡬⢌⢲⠤⢴⠛⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄⠙⡔⢾⣘⡘⢷⡺⣥⢯⡧⠟⢲⠛⣿⡿⠞⣽⡉⢤⡉⠛⠼⢌⠲⠴⣆⢂⠑⣄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡉⡰⠍⠰⠞⡱⢌⣱⠊⢱⠿⠦⡎⠐⠴⡣⠈⡠⢐⡤⠀⠁⡘⠆⠹⠠⣏⡲⣁⠼⡥⢦⠀⠀⠀⣀⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⡄⠐⠒⣤⣀⡼⢁⡤⣞⠜⢻⠷⠖⣩⣆⡐⠷⠖⢶⢆⣀⣧⡄⠕⡋⠠⢘⠢⠌⣛⣶⣆⠄⢉⠔⠂⣠⠢⢄⠀⠂⣈⢈⠽⠓⠀⠈
⠀⠢⢤⡀⠄⢀⣀⣛⠏⠉⢶⣖⣀⠽⠇⣬⢑⣥⠼⠦⣽⠶⠶⣄⢶⠍⠧⠌⠀⣁⣖⡱⠊⠁⢸⡋⣐⠂⠰⡤⠤⠈⠄⠁⠀⠈⠡⠆⠁
⠀⠀⠀⠀⠑⠒⠤⠀⠒⠂⠤⠀⣀⠴⠒⠴⠉⠉⠀⠈⠀⠈⠈⠁⠀⠀⠉⠁⠉⠉⠉⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠁⠀⠀⠀⠁⠈⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠀⠁⠁⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣤⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⢿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢁⠀⠀⠈⣉⠹⡏⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠻⠿⡟⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⢈⠉⠉⠙⣛⢹⣿
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠄⢸⡟
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠠⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠶⢶⠶⢂⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣤⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠟⠉⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡿⢀⠀⠉⠉⣀⢹⡟⠀⠀⠀⠀⠀⠀⠀⢸⡿⠹⠛⠿⡟⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⡟⠠⠀⠠⠀⠄⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⠀⠄⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⠀⠀⠤⠀⠀⣀⣿⣿⣄
⠀⠀⠀⠀⠀⠸⣿⣿⣷⣤⠀⢠⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠻⣿⠿⠃⠀⠘⠻⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣃⣀⣘⣻⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠴⠶⠤⠤⠤⠤⠤⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⢷⣴⣶⣿⣶⣶⣶⡿⣿⡆⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣿⣿⠋⠛⠉⠉⠙⣿⣷⡄⠀⠀⠀⠀⠀⠈⣿⣾⡿⠟⣿⠿⠛⣿⣿⣿⠁⠀⠀⠀⠀⢠⣿⣿⠋⠙⠿⠟⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢻⣿⢈⠀⠀⠀⡀⢸⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⡀⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⠈⢿⡿⢀⠀⠀⠀⡀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣍⠀⠀⠀⠀⠀⠀⣩⣿⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⢀⣠⣾⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣤
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⢊⢈⠀⠀⠀⠀⡁⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠌⢣⡀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠛⠛⠛⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡆⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢊⡠⠠⣓⠵⠟⢷⣆⡀⠀⠀⠀⢰⠋⠄⠊⠀⢰⡿⣿⣿⣿⣿⣿⢢⡀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣇⢸⡆⠀⠈
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠠⡀⢔⠬⠀⠀⠄⣾⣿⠃⠀⠀⢰⠋⠀⠀⠀⠀⡾⠀⠉⠛⠙⢿⡇⠱⣽⣷⡄⠀⠀⠀⠸⣿⣿⠃⣙⣿⠡⢁⠀⠞⢲⡀
⠀⠀⠀⠀⢀⢞⡠⢖⠠⠀⢀⠤⠐⢉⡠⡐⢋⣿⣟⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠄⠂⠠⢼⣿⣞⡚⢏⣿⠀⠀⠀⠀⣽⣿⡎⠀⠘⣧⠄⣼⡆⠀⠁⢂
⠀⠀⠀⡴⠟⣍⠤⠢⠑⠈⡐⢔⠪⠑⣁⣾⣿⣿⣍⠄⠀⠀⠙⣦⠀⠀⢀⣸⠇⠀⠄⠀⢀⡴⠏⠏⣉⣛⣯⠄⠀⠀⠀⠙⠿⣧⡀⠈⠉⣷⠸⣯⣆⠀⠀⢣
⠀⠀⢰⠗⠎⠠⠀⠜⠪⠑⡬⣒⡶⣾⠩⠯⠙⣉⠎⠀⠀⠀⠀⠈⢳⣤⣽⣧⣤⣕⣈⡵⢾⣷⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⠒⠠⣄⢀⠁⢈⣓⣴⡾⠁
⠀⠀⡜⠀⣠⠒⣼⣾⣷⣾⣯⠿⠷⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠯⠒⠋
⠀⠀⠑⠠⢁⠂⠾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠛⣷⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣼⣯⡟⠁⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠉⠙⢿⣿⣶⣶⠤⣀⡀⠀⠀⠀⠀⠘⣿⣿⡿⣾⣽⣟⡁⠽⠟⠒⢦⡀
⠀⠀⠀⠀⢴⣯⣟⡋⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢷⣦⡄⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⣿⣽⣶⣿⣿⣆
⠀⠀⠀⠀⠈⢿⡍⠙⢶⣄⠀⠀⠀⠀⣠⣿⣧⠀⠀⠀⠀⣰⠁⠀⢀⢀⡀⠰⡄⡀⠀⠀⠀⠀⠀⣀⡶⠀⠀⢰⡿⠟⠿⢬⣩⣛⡙⡻⠿⢿⣿⣿⣨⣷
⠀⠀⠀⠀⠀⠀⠘⠦⣴⢿⡀⠀⠀⣰⣿⣿⣿⠁⠀⠀⠀⠙⡄⣶⣦⣈⢻⣶⣄⠙⠷⢤⣀⣀⡰⠏⠀⠀⠀⠀⠳⣀⠐⠀⠈⠩⠉⠙⠳⠾⣵⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣁⣠⣾⡽⣿⡿⠋⠀⠀⠀⠀⠀⠘⢮⣿⣿⣷⣌⡬⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⣹⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠠⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣽⣯⣙⡓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⢘⢿⣧⡀⠀⠀⠀⠀⠀⢸⣇⠙⢦⠈⠙⠛⠭⢷⣦⡀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣬⣐⡩⣟⣲⣄⡀
⠀⠀⠀⠀⠀⣠⠔⠒⠃⠠⠐⠁⢘⣦⣡⡀⠀⠀⠀⠀⠈⣿⣿⣯⡑⣄⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠸⡌⣧⢾⡭⢿⣳⣯⣝⣛⡲⢦⣄⡀
⠀⠀⠀⠀⠐⣝⠲⣤⡀⠀⢴⠾⠾⢚⡿⣇⠀⠀⠀⠀⠀⣿⣾⣾⡶⠎⢳⠀⢀⣰⡾⠃⠀⠀⠀⠀⠀⠀⠱⡈⠓⢭⡤⣉⡻⠶⣭⣟⢷⣦⣯⣓⡶⣤⢤⡀
⠀⠀⠀⠀⠀⠈⢦⡀⢉⠛⣦⡀⠐⠋⠀⠈⡀⠀⠀⠀⠀⣿⣿⣿⣿⢋⡏⣠⣾⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢀⠑⢮⣙⣷⣮⣝⣷⠶⠟⠛⠛⠛⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠺⡇⡇⢹⣀⣀⡴⠦⠖⠀⠀⠀⠀⠘⣷⠿⠖⣿⣴⣟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡜⠀⠀⠙⠫⠟⠋⠁⠀⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠻⠭⠇⠀⠀⠀⠀⠀⠀⠀⠘⠻⠯⡿⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠠⠀⠀⢀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⡿⠛⢲⣶⡶⢖⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠗⣩⠆⠄⠙⠋⢠⣞⣿⣿⡆⠀⠀⢀⣾
⠀⠀⠀⠀⠀⢀⣴⢻⠇⠁⠘⠁⠀⠀⢠⣴⣿⣿⡿⠫⢻⣿⢏⡿⢸⡆⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⢰⣿⢟⡁⠀⠐⠠⠊⠂⣤⣿⣩⣿⡟⠀⠀⠀⠉⣴⢱⡎⡇⠀⠀⠀⣶⠀⠈⣟⢶⠲⢶⠐⠚⢏⣿⣆
⠀⠀⠀⠀⢸⡀⠈⠁⠒⠊⠉⠁⣨⣿⣿⡿⠾⠤⠤⠤⠶⠚⡀⣸⡷⠢⠤⠤⠴⠛⠀⠀⡏⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠈⡛⠳⡶⢴⠶⠿⡾⠋⣡⣻⡇⠠⠀⠀⠤⠤⢀⣁⣿⠁⠚⠀⠀⠀⠸⠀⠰⡷⣀⡀⠀⡀⢀⢀⣤⠼⢲⠄⠀⢀⣨⣿⣦⡀
⠀⠀⠀⠀⠀⣛⣦⡁⠨⠤⢊⡠⠬⡟⢿⡟⠓⠒⠚⠛⠋⠉⠹⡏⠀⠀⢄⣀⠤⠔⡆⠐⠗⠧⢉⣉⣉⣉⣀⣥⠾⠞⣷⠚⠛⠟⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠬⠙⠋⠀⣄⣉⠒⠼⠿⣾⡇⢠⠀⠀⠀⠀⠀⢶⢴⣶⣤⣤⡶⠶⠂⡇⠀⡄⠀⢀⣄⣉⣉⣥⣤⡀⠀⣿⡀⠀⠀⠀⢀⣽⠿
⠀⠀⠀⠀⢰⡆⠒⠺⠤⠀⠉⠛⢛⣰⣿⡇⠀⠀⠀⠀⠀⠀⢸⡰⣦⠉⠁⣄⠀⢰⡇⠀⠁⠀⠘⡾⢿⠧⣄⢸⡇⠀⣿⠷⠒⠐⠛⠋⠁⢰⡆
⠀⠀⠀⠀⠀⠛⠛⠺⠶⠾⠛⠋⠉⠁⣯⡇⠀⠀⠀⠀⠀⠀⢸⡉⢻⠤⠀⠈⠀⢸⡇⠈⠃⠀⢈⡀⢨⡏⠈⢸⡇⠀⣿⡇⢇⠀⠀⠀⣠⡌⢰
⠀⠀⠀⠀⠀⢤⠀⠟⠹⠛⠉⠙⠀⢠⣿⣿⠀⠀⠀⠀⢀⠀⢰⢡⣤⣤⣦⣶⣶⠮⣇⠀⠀⠀⢸⡄⢁⠅⠀⣘⡇⠀⠽⣇⠉⠱⠶⠟⠻⠃⡀
⠀⠀⠀⠀⠀⣞⡂⠘⠀⠀⠀⡌⠀⢤⣿⣿⠀⠂⠀⠀⠸⠀⢸⡄⢨⡍⡉⠉⢠⠀⣏⠀⡧⠀⠘⠻⠋⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⣆⡛⠀⠰⠆
⠀⠀⠀⠀⠀⢿⢙⡄⠀⠈⠐⢀⡌⠈⢽⣿⡀⡀⠀⣁⡀⠀⠨⡇⠬⠯⢤⠀⠀⠀⣿⠀⡖⠀⠀⢀⣀⣀⣀⠀⠀⢈⠀⢹⡄⠀⠀⣉⣠⢔⡄⢀
⠀⠀⠀⠀⠀⢨⡄⠱⠀⠀⠀⠈⠀⠐⢫⣿⠦⣤⡤⠤⠤⠶⠒⡇⢒⠛⠉⠀⠀⣀⠸⠀⠃⠀⢀⣺⢯⡿⠻⣦⡀⠈⠀⢸⡇⠀⠿⠿⠛⠿⠁⠸
⠀⠀⠀⠀⠀⠘⢲⣁⠁⡀⠀⠀⠀⠐⢨⣿⠀⠀⠀⠀⠀⠀⠀⡇⢺⠿⢿⣿⡿⠛⠓⠂⠄⠀⢨⣿⡾⣹⠀⣸⠇⠐⠀⠈⣿⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢰⡞⢁⡄⠈⠢⠀⢀⡀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢹⠀⣄⠀⠀⠄⢀⣸⡆⠃⠀⠈⠻⣻⠋⠃⠁⠀⢠⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠀⢂⠀⠦⠤⠐⢄⠙⢹⣿⠀⠀⠀⠀⠀⠀⠀⢼⠘⢋⠀⡀⠰⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢶⣤⣌⣁⣉⡉⢀⣈⣁⣼⣿⠀⠀⠀⠀⠀⠀⠀⡟⠐⠛⠀⠀⠀⠐⠸⣇⠀⠀⢰⣶⡴⠶⠶⢶⡆⠈⠀⠀⠈⣿⠀⢀⣀⣤⣶⡇⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠋⠉⠉⠉⢸⣿⡀⠀⠀⠀⠀⠀⠀⠁⢰⣶⣶⣶⣶⡶⠦⣿⡃⠀⢨⣿⣿⢳⠗⢸⡇⠀⠀⠀⠀⢿⠀⠙⠛⣡⣬⡁⢬⡇
⠀⠀⠀⠀⠀⠀⢆⠉⢄⡀⠄⠤⠝⠁⢼⣿⡇⠀⠀⠀⠀⠀⢀⣸⠀⡊⠉⠋⠁⣀⣀⣿⠁⠀⠸⣷⣽⡇⠀⢸⡇⢐⠂⠀⠀⠸⡇⠌⠀⢻⠜⢁⡄⢓
⠀⠀⠀⠀⠀⠀⠠⠖⣀⠘⠂⠀⠐⡁⢸⣿⣇⣀⣀⣐⡒⣀⣈⣼⢀⡟⣆⠀⠀⢉⢩⣟⠃⠀⠀⠻⢵⠥⣰⠀⠂⠈⠀⠀⠀⠀⢷⡆⠀⣤⣼⣿⡇⡸⡀
⠀⠀⠀⠀⠀⢀⠠⠓⣁⠔⢆⡈⠁⠁⢸⣿⣇⠉⠉⠉⠉⠉⢁⣹⠀⠴⠟⠀⠀⠸⠋⢿⠓⠀⠀⠛⠛⠀⡀⢀⣠⣼⡄⠀⠀⠀⢸⠄⣄⠉⠁⠀⠀⡅⡆
⠀⠀⠀⠀⠀⢸⣤⡘⠒⠒⠂⠀⠛⢂⣸⡟⠙⠻⠷⠿⠾⠿⠟⠛⡰⢿⣿⣿⡿⡿⠟⡼⠿⣿⣽⢰⣆⣽⠿⡿⠿⠛⠃⠀⠀⠀⠀⢇⡐⠀⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠘⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠦⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣦⡟⢠⡼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⡏⢁⡾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⡶⢿⡿⠀⠰⠃⠀⣠⠀⠀⠀⢰⡄⠀⣀⠀⠢⠀⠀⠀⠐⣿⡷⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠝⠇⠃⠀⠀⠀⠀⢂⠀⠟⠀⠀⠠⠈⠀⠀⠟⠀⠀⠀⠀⠛⠋⢤⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣠⢸⡇⠀⠀⠀⠀⡂⠀⠃⠀⣤⣤⡀⠀⠀⠛⠛⠙⠛⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣌⣳⠀⠀⠀⣤⠶⠋⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⢨⣍⡇⠙⢷⡏⠀⠐⠀⣴⠀⠀⣀⣀⠀⠀⠐⢀⣠⡀⠈⢶⣶⣾⡷⠆⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠅⠀⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⣿⣟⢾⣿⣿⣇⠀⢀⣄⣠⣤⣤⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⣀⣠⣤⡾⣻⣿⣿⣿⣿⠏⠉⠙⠻⣶⣾⣏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣿⠃⠀⠉⠉⠉⠉⠰⣶⣤⠄⠀⠀⠙⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣴⣿⡇⠀⠀⣠⣾⠇⠀⠀⣠⣴⣖⣷⠀⠀⣀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣀⣀⠀⠀⢶⠷⠆⠉⣻⠃⠄⠀⣴⢶⣶⣧⠀⠙⠿⠿⠟⠀⠰⠟⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠹⠉⠃⠀⠁⠀⠀⠰⣿⠅⠀⠀⠙⠺⠟⠁⠀⡀⠠⠀⣀⣀⠀⠀⠀⠠⠰⠟⣡⣦
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⠠⠈⢻⣿⠀⠀⠄⠀⠀⠀⠙⢡⣾⠄
⠀⠀⠀⠀⢴⣿⡾⠀⠀⠀⠀⡀⠠⠀⠠⠀⠀⠀⠘⠛⠁⢠⠄⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣭⢳⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡀⠀⣤⡦⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢀⠀⠀⠿⣠⣌⠸⡷⠃⠀⠀⠀⡋⠁⢸⠮⠁⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉⠉⠀⠀⠈⠲⣌⠳⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡍⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣴⠃⡀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⣆⣠⠴⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⢀⠟⠁⠀⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢰⠀⠀⠙⣄⠀⢀⣠⣄
⠀⠀⠀⠀⠀⣀⠴⠚⢏⠉⠛⠒⠲⢤⡑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⠀⠀⠀⠈⢳⡁⠀⡼
⠀⠀⢀⡠⢴⠁⠀⠀⠈⢣⡀⠀⠀⠀⢹⣇⣀⣤⣶⣶⠖⠒⠒⠒⠒⠲⢤⣀⠀⠀⠀⢸⠇⠀⠀⢀⣀⣬⣇⣀⡇
⠀⠀⣼⠀⠘⣆⠀⠀⠀⡠⠔⠲⢄⣠⢼⠏⠢⠀⠀⠈⠉⠙⠓⢦⡀⠀⣀⠬⠽⠛⠒⠚⠲⣄⣰⠋⢆⠀⠀⠈⢳⡀
⠀⠀⠈⠓⠒⠘⠓⠒⠘⠣⢤⣤⠼⠤⠺⠄⣀⣀⣀⣀⠀⠒⠒⣁⢗⠶⢧⣀⠀⠀⣀⣀⣀⣀⠟⠒⠚⠶⠤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⢧⣀⣠⠏⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣭⣯⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⡶⠶⠶⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠘⠙⠙⠛⠁⠈⠙⢲⡆⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠋⠁⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠹⣾⣶⣠⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣈⣱⡜⠋⣻⠀⢠⠀⠀⠒⠒⠒⠂⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠐⠂⠀⠀⠉⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⣶⣭⣝⠳⢤⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠠⣏⡀
⠀⠀⠀⣤⣤⣄⡀⠁⠀⠐⣿⡟⠫⣷⠖⠲⠖⢲⣶⣶⣶⣶⣿⠿⢓⠲⠿⠿⣿⣷⣿⣿⡏⠙⠟⠒⠚⠉⠀⢻⣿⡟⠀⠀⠀⠀⠁
⠀⠀⠐⠉⠛⡿⠛⠷⠤⣀⠞⢁⣿⡇⠀⠀⠘⡿⠋⠒⠠⠄⠀⢀⡠⠄⠀⠀⠀⠀⠘⣿⣿⠂⠀⠀⠀⠀⠀⠀⠹⡃⣀⣀⡤⠄⠀⢰
⠀⠀⠀⣶⣄⠁⠀⠀⠀⠈⠉⠁⠩⢀⠀⠀⠣⢇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠈⠙⠻⠦⣀⣀⠀⠀⠀⠈⠙⠒⠧⠄⢤⣀⠀⠀⠀⠈⠀⠀⢀⣀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣀⡀⠀⠀⠀⠈⠉⣿⢶⣦⡰⠊⠀⠁⠀⠀⠀⠀⠀⠀⡖⣀⠠⠶⠒⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢶⣤⣤⣽⣏⠀⠀⠀⢠⣀⣤⠤⠔⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠀⠂⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⣩⠦⠔⢒⣒⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡡⠞⠉⠀⠀⠀⣭⣽⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢮⣡⠋⠀⣀⣠⠴⣶⠺⡉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠚⠛⠛⠋⠟⠁⠀⠀⡷⠶⣫⡬⣿⠒⢋⡳⠃⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣆⠀⠀⠀⠸⠀⣀⣠⣤⠓⠂⠁⠀⡿⠒⠉⡀⣠⠊⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡶⠒⠚⠁⠀⠉⠙⣒⠾⠋⠁⠀⠀⣱⠀⠀⠀⣳⠄⠒⠈⣳⡀⠀⠀⡇
⠀⢀⡤⠖⠒⠋⠉⠉⠁⠀⠀⠀⢀⡠⢤⣚⣉⡄⠀⠀⠀⠀⠀⢾⠄⠀⣜⠁⠀⠀⣠⠾⠤⠤⠾⡁
⠀⠈⠓⠶⠤⠤⠤⠐⠒⠒⠈⠉⠁⠀⠀⠟⠛⠒⠈⠉⢉⡗⣺⠛⠀⠀⠈⠓⠛⠋⠀⠀⠀⠀⠐⠚⠋⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⣉⣉⠛⠧⠞⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠈⠃⠀⠀⠀⠀⠀⠠⠔⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⢀⠤⠎⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⡄⠀⠀⠘⠲⠤⠖⠒⠉⠓⢤⣀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠄⠀⠀⠀⣀⠔⠁⠀⢸⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣤⣤⣄⣀⣀⠀⠀⣀⣠⠴⠊⠁⠀⠀⠀⡆⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡏⠉⢷⠤⣄⠈⠉⠉⠉⠀⠀⠀⠀⠀⡤⢌⣳⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡀⢸⠄⠀⠀⠀⣠⠤⠤⠤⠞⠀⠉⠒⠉⠀⠉⠉⠉⠉⠉⢳⣀⣠⣤⣤⠇⡀⢙⡅⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢍⡿⠛⠘⣆⣠⠴⢻⡆⠀⠀⠀⠀⣀⠀⡀⢀⠤⡀⢄⠀⠀⠈⡇⠒⠒⢲⡄⣹⢯⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡧⠴⡅⢠⡔⠊⢯⡇⠀⡀⠀⠀⡛⠢⢠⡸⣀⣄⠈⢦⠤⣀⡇⠀⠀⠀⠉⠹⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡤⠚⢹⡖⠃⠀⠘⠷⡏⠀⢀⢄⠸⣐⠄⣎⡂⠸⠚⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢊⠉⠁⠀⠀⠀⠰⠀⠀⠘⠤⠃⠾⠀⠀⠀⠀⢀⣀⣘⡆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢰⡆⠀⢠⣇⣀⣀⣀⣀⣀⣠⣤⣴⣶⣿⣿⡗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣸⠛⠛⠋⠉⠉⡽⢩⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⣠⣤⣤⣧⣼⣤⠤⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣗⢲⠤⠤⠤⣤⣤⣇⣀⣠⢤⠴⢓⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⢻⣉⠤⠒⣫⠕⠋⠪⢶⣤⣘⣉⡇⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠭⠤⠖⠚⠉⠀⠀⠀⠀⠀⠈⠙⠛⠭⠭⠝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠈⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣶⣶⡶⣶⣶⣾⣿⣶⣶⡀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠹⣿⣿⠿⠿⢻⠇⠈⢻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⢛⣿⠀⠀⠈⠉⠀⠀⠀⠀⠀⡟⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⣾⣿⠆⠐⠂⠀⠀⠀⠀⠀⠘⡇⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣾⣏⣉⣀⡀⠀⠀⠀⠀⠀⠀⡸⠉⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⠯⡤⠤⠀⠀⠀⠀⠀⠠⢁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣧⠀⠀⠀⠀⠀⢀⣴⣿⠿⠓⠀⣠⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣶⣴⣶⣶⣾⡿⠟⠁⠀⠀⣴⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣫⣤⡄⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⢿⣿⠿⠿⣿⡿⢛⣿⣫⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣥⣤⣼⣿⣾⣿⣿⣿⣿⣿⣿⡿⠿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠛⠿⣿⣿⣿⣿⣾⣿⣿⣿⡽⣿⢻⣉⣍⣭⣿⣻⣿⣿⣿⣿⣿⣿⣴⣿⣿⣟⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡇
⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⢿⣿⣿⡿⠋⠛⠛⠛⠛⠋⣿⢿⣿⣿⡟⣟⠛⢿⠿⣿⣿⡂⣿⡇
⠀⠀⠿⠿⠟⠛⠛⠋⠛⠛⠋⣹⣿⣿⡟⠉⠀⠀⠀⠀⠀⣨⣿⣿⡷⠆⠀⠀⠀⠀⠀⢀⣄⠃⢻⡆⢸⢡⠆⣵⣻⣏⣸⣿⡇
⠀⠀⢀⡀⠀⠀⢀⠀⠴⠾⡿⢿⣿⣿⣿⣿⣶⣶⣶⡾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⢶⡀⢸⣤⢸⣿⣿⡏⣽⣿⡟
⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠰⢉⡀⣹⠎⢿⡏⣿⣿⠟
⠀⠀⠀⠀⣀⣀⣀⣴⣶⣾⣿⣿⣿⣿⠟⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣔⣼⣝⡸⣷⣦⣾⡿⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣫⣡⣿⣿⣿⠁
⠀⠀⠻⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠉⠛⠿⣿⣿⣷⣦⣄⣀⣀⣀⣀⡤⣠⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠋⠙

⠀⠀⠀⠀⠀⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠄
⠀⠀⠀⠀⠀⠇⠀⠂⡀⠀⠀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢩⡲⣶⠃
⠀⠀⠀⠀⢈⡄⠀⠀⣽⠀⡆⠀⠰⡄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠳⠅⠀⠀⡀⢈⠑⣄⢠⠂⡀⢀⣈⣴⠞⣿
⠀⠀⠀⡀⠼⣇⡀⡘⣇⣸⠃⠀⠀⢁⡥⠃⢠⠐⠞⠳⡀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣦⣺⣴⣿⣦⣀⣨⣾⠿⣛⠛⣧
⠀⠀⠒⠛⠓⠛⠓⠛⠛⠻⠄⠀⠐⠚⠛⠓⠛⠀⠀⠐⠛⠒⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠋⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣖⣀⣀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣶⠦⢵⠟⢻⡿⠟⣟⢐⣶⣧
⠀⠀⠀⢦⣁⡀⠠⣤⣴⠶⣿⣦⠠⠔⠐⠂⠦⣄⣤⡴⠖⠚⠚⠻⢿⣿⣿⣻⣿⣴⣆⣀⣰⣿⢸⣿⣿⣦⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠉⠛⠙⠲⠟⠙⠿⠲⠶⠶⠶⡈⡘⠿⠾⡵⠤⠶⠾⠿⠻⢿⣿⣯⡬⠻⠿⠿⡿⠿⠿⠿⠿⡻⠓⠂⠒⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣦⣥⣿⣵⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠟⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⢸⣿⣷⣄⣀⣀⣠⣾⠟⢻⣷⣄⣀⣀⣠⣿⣿⡇⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠙⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣷⣌⠙⠛⠻⣿⣿⣿⣾⣷⣿⣿⣿⠟⠛⠋⣡⣾⣿⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣟⠛⠛⠓⠀⠀⣿⢻⡟⣿⣿⢻⡟⣿⠀⠀⠚⠛⠛⣻⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠛⠷⣦⣀⠀⠀⠀⠈⠁⠉⠉⠈⠁⠀⠀⠀⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⣉⠳⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢉⣀⡄⢰⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣿⡆⠀⣿⣷⠀⢰⣶⡀⣶⡆⢠⣶⣾⣦⢸⣿⠛⠃⢸⣟⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣼⡇⣸⣟⣿⡄⢸⡿⣷⣿⡇⣿⡏⣶⣿⢸⣿⠿⠇⢸⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠁⠿⠋⠻⠷⠸⠇⠘⠿⠃⠙⠿⠿⠿⠸⠿⠿⠆⠸⠟⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢿⣷⣔⣦⣀⡀⠀⠀⠀⠀⣠⣶⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡛⢷⣀⠘⡿⠷⠋⠉⠙⢣⣠⡀⠛⠛⠛⢛⣿⠀⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣄⣈⡀⠀⠙⠣⢬⡗⢒⡒⠐⣶⠛⣦⣄⡀⠀⢀⠸⠒⣿⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠓⣦⣤⣀⣀⣀⣁⣶⣾⣿⣿⣿⣿⣿⣿⣿⣾⣻⠁
⠀⠀⠀⠀⡀⠀⠀⠑⠒⠚⠛⠛⠛⠛⠻⢛⣟⣛⢛⠛⠛⠛⠛⠛⡟⠛⠛⠛⠿⠷⠶⠶⠤⠤⠆⠀⢠
⠀⠀⠀⠈⠁⠀⠀⠀⠊⠀⠀⠉⢀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠴⠘⢀⡌⠉⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢁⠁⣀⠃⠲⠀⢃⠀⠈⣠⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠱⠆⢀⠀⠶⠄⠀⢙⢀⡉⠀⠀⡙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⣉⢴⡀⠚⢀⠀⣤⠒⢘⢃⡶⣇⢀⡀⢠⠘⠡⠘⠉⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠞⠡⠆⠀⠈⠂⠀⠐⠀⠀⣠⠄⠀⠉⠀⠻⠄⠀⠠⠀⠀⠠⢇⡃⠠⡈⢧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣊⠉⠰⠒⠀⠀⠄⠀⠀⠀⡀⢀⡸⠆⡀⠛⠠⠁⠆⡀⢐⡀⠈⠁⠲⠀⠀⠀⠒⡈⢀⠉⣇⡀
⠀⠀⠀⠀⠀⠀⣀⡐⠚⠀⠽⠇⠀⠠⠀⠀⡀⠆⠀⡀⠀⠉⠪⠀⠀⠀⠀⠀⠘⠁⣬⠨⢁⡀⢃⠀⠀⠀⠶⠦⠀⠀⠛⠁⠈⠒⡀
⠀⠀⠀⠀⠄⠜⠠⣅⠀⠀⠁⠂⠀⣘⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠲⠒⠈⠀⢀⠈⠱⠆⠂⠀⠀⠐⢋⡀⠀⠊⠁⠀⠠⠜⠒⠄⡄
⠰⡱⢤⣒⠓⠖⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢩⡀⠀⠀⠀⠀⠀⢐⠣⡀⠈⢠⠐⠈⡀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠈⣑⢲⢀⢀
⠀⠀⠀⠀⠓⠢⠼⣄⠀⢂⡠⠤⠀⠄⠀⢀⠀⠡⢔⡈⠀⠒⠀⠀⢀⠘⠲⢆⡀⠈⠂⡌⠁⠛⢢⠀⠘⡂⠀⠀⠀⠐⠲⠀⡐⢀⠮⠠⠋⠙⠒⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠓⠒⠒⠚⠣⠤⠤⠣⠤⠤⢄⡠⠬⠴⡘⢀⠴⡍⡇⢅⠤⠸⠔⠣⠄⠀⠂⠈⠀⠐⠊⠉
⣀⣀⣀⣄⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣄⣤⣤⣤⣤⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⢐⠥⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣠⡄⣀⡀⠀⠀⠈⠁⠐⡆⠃⠀⠀⠀⠉⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢤⢁⣩⠀⠀⠀⠛⠒⡋⠀⠇⢇⣀⢄⢀⢀⠀⠀⣠⠊
⠀⠀⡠⡂⠤⢀⣀⠀⠀⠀⢀⠂⠠⠖⢢⠄⠀⠀⣈⢁⡀⠀⠠⠀⠀⠈⠉⠓⠥⣅⣠⡧⠟⠒⠄⡀
⠀⠀⠀⠙⠮⣄⣀⠌⠄⢘⠓⠰⢞⠋⠀⠀⠤⢬⡍⠅⠀⡠⠁⠐⠀⠀⠀⠀⠀⡠⠛⠷⢬⢭⠵⠮⠷⣄
⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⠤⠥⣌⣑⣤⣘⣛⡀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠐⠐⠀⢌⣸⢂
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢒⡒⣶⢤⡤⠀⣀⣀⣀⡀⠠⠊⣠⣍⠐⠂⠤⢀⠀⠀⠀⠀⠀⠁⠂⢤⠔⠁⠠
⠀⠀⠀⠀⢀⠰⠍⣀⠀⠀⠐⠈⠀⢀⣀⡒⠷⠒⠀⠀⠚⠛⠋⢀⣀⣼⡶⠤⠄⠉⡆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣈⠁⠈⢀⠔⠊⠉⠁⢈⣉⣈⠉⠒⠂⠤⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⡀⠀⠈⠒⠁⠀⠀⢀⠔⢁⠶⠤⠈⢁⣐⠒⠶⢅⠴⠍⠁⠀⣂⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣈⠉⠂⠠⢤⣐⣊⡡⠎⠙⠈⠠⠔⡩⠁⠈⢀⣾⠦⠄⠀⠈⢀⠴⠒⠉⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠳⢥⣀⡙⠲⣖⡡⠴⠊⠍⠀⢀⣠⠶⡹⠊⡠⠷⠂⢀⠊⢛⠶⠡⠤⣴⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠠⢌⠉⠘⠚⠹⠵⠲⠾⠟⣛⡒⠀⠠⠒⠂⠅⡠⠃⠀⠆⢐⠴⠋⠀⠐⣂⠍⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢄⡢⢕⠨⠒⢈⡐⠤⢄⠀⠀⠀⠀⠀⠈⠉⡒⠢⠤⢊⡀⠀⢀⠀⢒⠢⠀⣀⡊⠁⢀⠯⣮⣷⠡⠱
⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⠐⠠⠄⠀⢀⠠⢉⠲⠖⣒⣭⠴⠦⣄⡀⠀⠀⢇⠁⠒⠒⠠⠤⠎⣘⡚⠂⢀⣤⢐⠂⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠈⠀⠈⢠⠖⠁⠈⠙⠣⣤⣘⠙⡆⠀⠈⠳⣕⡀⡀⡠⡠⠀⢸⢀⢒⠤⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠔⠁⠀⠀⠀⠀⠀⠀⠉⠓⠃⢂⠀⠀⠀⠉⠙⠒⠲⠵⣭⣿⣅⣀⡤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠑⠡⠂⠋⠐⠕⠅⠠⠂⠄⠀⠨⠂⠀⠢⠊⠔⠀⠊⠔⠐⠁⠪⠂⠐⠅⠀⠠⠠⠠⠀⠀⠀⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢋⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠰⠟⣻⡆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠉⣿⣧⠀⣀⣠⠀⠀⣈⣯⠴⠖⠚⢉⣉⣩⠤⠴⠒⠒⠚⠻⠟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⡿⠟⠉⣁⠤⠒⠋⣁⣀⣤⣖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢛⣛⣠⠔⢋⣡⣴⣾⣿⣿⣿⣿⣿⡟⣴⣶⣦⣄⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠜⠛⣡⣶⣿⣿⣿⣿⣿⣿⣿⠿⠋⠾⣿⣿⣿⣿⣷⣄⠀⣀⣀
⠀⠀⠀⠀⠀⠰⠋⢁⡴⠊⢁⣤⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁⢀⣀⡄⠈⢿⣍⠻⣿⣿⡆⠩⡽⠃
⠀⠀⠀⠀⠀⢀⡴⠋⣠⠖⠉⠸⣿⡿⠋⣿⡿⠋⢀⣀⣨⣉⣋⡥⠖⠂⠀⠁⠀⣼⠁⠀⠈⠃
⠀⠀⠀⠀⢀⡞⣠⠞⣡⣤⣄⠀⣿⡇⢸⡿⠁⠀⢀⣠⣤⣄⠉⢀⣤⣶⣷⠀⠀⣿
⠀⠀⠀⠀⢸⡟⠁⠀⠈⠿⠟⠀⢸⣇⠈⠃⠀⣴⡿⠟⠉⣿⠀⠸⠋⠀⠙⠀⠀⣿⡴⣦⡀
⠀⠀⠀⠀⣼⠓⠦⣄⠀⠿⠀⢀⡀⣿⠀⠀⠀⠉⠀⢀⡀⠀⠰⣆⠀⣴⡄⠀⢰⣿⡖⢺⡇
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣴⠋⣩⣿⡆⠀⠀⠀⠀⠻⠟⡀⠀⢻⡄⢙⠁⠀⢸⣿⡇⣼⠇
⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⢻⡄⠀⠸⣿⡀⠀⠀⠀⢀⡼⠃⠀⠈⣷⠈⠛⠂⠘⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢿⡧⠀⠀⠀⠀⠀⠀⢿⡦⠟⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣇⠀⠀⠀⠀⠀⠀⠈⢃⣀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠸⢿⡟⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠠⠾⢶⣄⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡟⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣤⣼⠷⠛⠁⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣤⣴⣶⣶⣶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠟⠛⠉⠙⠛⠉⠉⠙⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠃⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠁⠀⠀⠹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠟⠀⢀⣶⣿⣷⣆⠀⠘⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠘⣿⣿⣿⡟⠀⠀⠈⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠋⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠈⢻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣆
⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣆
⠀⢠⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧
⣠⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣧
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣄⣀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢦⡀⠀⠀⠀⠀⢠⣠⡿⢻⡿⣿⣿⡿⣻⣟⣭⠿⣛⡛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡴⠋⠒⠤⢀⣴⣿⣟⡃⢿⣽⢡⠟⣷⣿⡟⠑⠉⣀⣀⠀⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⡙⠲⢤⣀⠻⣟⠻⣵⣸⣿⣿⢀⣿⣿⣇⣰⣿⣭⢿⡋⢻⢹⡏⢿⡉⢶⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⠻⣷⣦⣍⠳⣌⠻⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣅⠸⣇⠀⠀⣧⠘⣷⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠳⠦⡉⠛⣧⠙⢆⠈⠻⣿⣿⣿⣿⣿⡿⣇⠛⢻⣷⣌⡛⠶⣌⠓⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⣄⠈⠣⡈⠷⣤⡞⠻⣯⡉⣿⣧⣼⣿⣦⠰⣍⡻⢦⡙⣦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢆⣽⡿⢦⢀⣶⣾⣷⣦⡈⣿⣇⣨⡁⢙⣿⣷⠮⢉⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢉⡴⠞⠉⣹⢳⡟⣳⣿⣿⣼⣿⣿⣿⣏⣹⡟⣹⠎⡉⢹⣟⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⡄⠻⠒⣨⣽⢿⢋⡝⠙⣿⢋⣿⢹⣿⢻⡿⣿⡿⡿⣿⣽⠛⣾⡿⢃⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⠟⢁⣴⣿⣿⢳⣾⣿⠁⡾⢨⢸⡇⣸⣿⡜⡷⢹⢻⡇⠹⣿⣿⢿⣡⠞⠁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡴⢻⣿⣿⡏⣸⣿⣿⣤⣷⢨⣿⣇⣿⣿⣧⢧⢸⠈⣿⠀⣿⢿⣧⠙⣤⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⣿⣸⣿⣿⡗⢸⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣞⣴⣿⣇⣿⣾⣿⡄⣿⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣷⡸⣿⡿⢿⣿⠿⠿⢿⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠉⠉⠛⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣻⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣄⣀⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡏⢹⣷⡄⠀⣀⣭⣿⣧⣈⣉⣼⡟⠻⣿⣦⡝⢊⣿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢁⡴⠟⢁⠎⡾⣡⣽⣟⣿⠋⠙⣿⢡⡾⢩⠅⡀⣩⠞⢰⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⢀⣰⠟⠰⠐⡁⡟⣿⠏⠀⠀⠹⡟⢡⠏⠎⡴⢃⣴⠟⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠁⣴⠃⡼⢀⡼⠋⠀⠀⠀⠀⠙⢏⣠⡾⠑⣼⠃⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠻⠓⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⠀⢀⠀⠀⢀⠀⠀⣀⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠶⠀⣻⡀⡇⡌⠀⠀⢿⠀⡇⡜⠀⠀⢘⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⡇⠀⡇⠛⡇⢧⠀⠀⠈⠻⠀⡇⠀⠀⢸⠀⡗⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠉⠁


⣤⣤⣤⣄⣤⣀⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣤⣀⣤⣀⣤⣄⣠⣄⣀⣀⣀⣀
⣿⣛⣟⢻⣛⣿⣋⣙⡛⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣛⣙⣟⣛⣻⣿⣟⣻⣿⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠉⡰⠃⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠛⠲⢤⡀
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠜⢁⡀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠱⡄
⠀⠀⠀⠀⡞⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⡇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⣠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢻⠀⠀⠀⠀⢤⡀⠀⢦⡀⠀⢀⣼⠃⠀⠹⣦⡀⢀⣼⠏⡴⠏⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢸⠰⢦⣄⠠⢤⡛⡷⡄⠻⣤⠻⠹⠀⠀⠀⠘⢝⡎⡞⡜⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢸⠀⣤⣙⢿⢦⣿⣶⡼⠖⠓⠓⠚⡱⠻⣷⣦⡸⢰⡿⢀⠏⣾⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠘⡆⠀⣙⣷⣱⡹⡟⠀⠀⠀⠀⠐⠁⠀⠈⢿⡇⢸⡇⡞⣸⣿⣘⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠉⡏⠁⠀⢘⡷⢧⡀⠀⠀⢀⣀⣀⣀⡠⠾⣇⣸⣷⡉⠉⠁⢹⠇⠀⣇⣀⣸⣀⣀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠚⠋⠉⠈⠉⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠁⠄⠁⠀⠐⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣿⣷⡆⠀⠀⢀⢀⠁⠁⡶⠳⠢⠶⡆⣠⠤⢶⠴⠤⠴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡟⠘⡇⠀⠀⠃⣚⢃⠀⡁⣴⣟⣈⣹⡇⢦⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⣰⢤⣾⣋⡴⢤⡗⣸⡁⣩⡟⠉⣻⡇⢸⣆⢠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡶⢿⠒⢤⡙⣷⣿⠿⠁⠉⠀⢻⣿⣿⠁⠀⠹⣿⣷⢸⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⢟⢸⠉⠢⢬⣿⠉⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⣸⡗⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⡚⠘⠆⣽⣿⣿⡇⠀⢺⡆⠀⠀⠰⠀⠀⠀⢠⣯⣵⠇
⠀⠀⠀⠀⠀⠀⠠⠒⢻⣷⡟⢦⠄⡾⠙⠛⡿⠶⣞⠛⠛⠟⢛⣷⣶⣶⣿⡇⠘
⠀⠀⠀⠀⠀⠀⢠⢳⠚⢿⠀⠈⠀⢀⡴⢆⠟⠤⠂⠠⠌⠡⠽⣿⢿⣿⠙⢌⡒
⠀⠀⠀⠀⠀⠀⢘⣋⠅⠈⠂⠀⠀⢸⡏⠓⠲⠁⢠⠀⠀⠀⡆⠰⠖⢿⣓⠄⠙⢺⡄⣤
⠀⠀⠀⠀⠀⠀⣼⠁⠀⢠⠀⠀⠀⢰⠇⠀⠀⠄⢸⡄⠀⢠⣿⡙⠀⠈⣿⠀⠀⣸⢻⠟⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⢀⣤⣬⠂⠀⠀⢸⣿⠿⢠⣿⣿⡁⠀⠀⢻⣧⣴⣿⢻⢺⠁
⠀⠀⠀⠀⠀⠀⡛⢀⡴⢣⣶⣿⣿⣿⠀⠄⠀⢸⡟⠀⠀⠹⣿⣇⠀⠀⠀⢻⣿⣯⡰⠟
⠀⠀⠀⠀⠀⠀⠙⢾⣵⠟⢻⡟⠁⢿⠀⠀⠀⢾⡇⠀⠀⠀⣸⣹⡆⠁⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣸⠀⠀⢸⠓⠶⣶⢿⣿⠀⠀⡼⡿⢛⢻⡟⠛⠫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠸⡏⣱⣿⣾⡏⠰⣶⣗⣡⠊⢠⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣶⣇⠙⢹⣾⣿⣶⣧⠭⢥⣠⣾⣞⡛⠲⣆
⠀⠀⠀⠀⠀⠀⢀⣸⣿⣴⣶⣿⣿⣿⡿⣤⠯⠞⣿⣿⣿⠀⠀⠈⠈⢿⣿⣿⣷⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠶⣯⣿⣿⣿⣿⣿⣿⡿⠀⠁⠀⠀⠸⣿⣿⣦⣀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⠂
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣦⣤⣼⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣤⣾⣿⣿⠿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⡀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠉⠙⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠛⠛⠟⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠒⠒⠒⠶⠶⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠛⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠍⣀⡀⠀⠀⠁⠛⠟⡛⠷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠻⠿⠶⢶⣤⣤⣀⣀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠉⡉⠛⢿⠻⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠒⢢⠖⠯⠤⣄⣀⣀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠙⡖⠤⣈⠉⠓⠲⠤⠤⣤⣉⣉⣙⣾⣡⣀⣠⣤⣤⡤⢤⠤⠤⠤⠤⠤⠤⠴⠛⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠤⠄⠐⠒⠂⠀⠉⠉⠙⡄
⠀⠀⠀⠀⢀⠐⠠⠏⠀⠀⠀⠈⢀⠂⠀⠑⢦⠀⠒⠒⠀⠀⠁⠀⠀⠀⠐⡀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠀⠉⠒⠤⣀⡁⠂⠀⠐⠨⢅⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠌⡛⠓⠒⠬⠴⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢁⣀⡀⠴⠤⣤⡀⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⢸⠵⠒⠁⠀⠀⠀⠀⠀⠀⠙⠊⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠀⠀⡀⠌⢳
⠀⠀⠀⠀⢠⠒⠾⣈⠉⠒⠤⢄⣉⠉⠒⠒⠦⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢉⣀⠤⠊
⠀⠀⠀⠀⠈⠒⠤⣄⡉⠑⠒⠦⢄⣉⠑⠒⡤⠤⢀⣀⡈⠉⠉⠁⠒⠲⠤⣄⡀⠀⠀⠀⢀⡠⠖⣉⠴⠊⠉
⠀⠀⠀⠀⠀⠀⠀⢢⠈⠉⡒⠢⠤⢌⣉⠒⢃⣀⡠⠤⠛⠋⠑⠲⠤⠤⣀⡀⠈⠉⠒⡎⣁⠔⠉⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠨⢐⡂⠈⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠋⠀⠀⠀⠂⣸
⠀⠀⠀⠀⠀⢀⠠⢼⠀⠀⠈⠀⢂⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠃
⠀⠀⠀⠀⠀⠈⠁⠒⠩⠍⣒⠠⠤⠤⠴⠊⠀⣀⣀⣀⠠⠤⠤⠤⠔⠒⠈⠉⠉⠉
⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠐⠒⠚⡁⢀
⣿⡫⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠈⢷⡯⡄⠂⣴⠂⡂⢰⣾⠐⣫⡯⠬⡷⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣇⡼⣤⠈⠃⡷⠟⣻⣬⡏⣤⡟⠀⡀⡶⢰⣁⣠⣄⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠍⣆⠁⠰⣰⣿⢺⢚⣠⠋⢀⡿⣉⠽⠻⡟⡟⢷⣾⠍⢩⣔⠃
⠀⠀⠀⠀⠀⠀⢰⠃⣰⡋⠒⢻⣽⢹⣼⠟⣟⣼⣇⠱⡽⣟⢦⣤⣹⡴⣟⣳⠟⣿⠁⡁
⠀⠀⠀⠀⠀⠀⣘⣶⠏⣿⣿⣞⡹⢾⣽⡿⠟⣱⡟⣺⣿⢘⢮⣟⣿⣄⡅⣀⡶⠄⠀⡗
⠀⠀⠀⠀⡜⢊⡭⠿⠺⡟⢳⡞⠿⢸⣼⣿⣟⣭⡿⣿⣿⡀⢞⣤⡿⣿⢳⣿⡻⣆⡿⢁⠆
⠀⠀⠀⡀⠀⠘⠒⠉⣎⣤⣾⣿⠘⢻⣟⢿⣯⣿⡟⣹⠸⠙⢻⣏⡰⡿⡀⣹⣏⣹⠉⠋
⠀⠀⠀⠀⢸⣧⣾⣀⠞⢻⣯⣭⠙⣗⢰⣞⣿⣷⡿⣷⢠⠳⣷⣟⣷⣿⠷⣿⣿⣃⡏⢠
⠀⠀⠀⠀⠈⡀⢫⡟⢏⣰⣞⠧⣕⣏⡹⢻⣷⣼⠀⠎⠈⠧⠹⣹⣿⣉⠦⣿⣿⣿⡿⡸
⠀⠀⠀⠀⠘⠉⠈⢹⣻⡄⣽⣶⡿⣮⢻⣯⣿⣁⣷⣠⣾⠚⣾⣏⠟⠑⢮⣼⣸⣿⣰⢧⠐
⠀⠀⠀⠀⠀⢹⣅⣡⣷⣿⢹⡿⣇⣧⣟⠃⡟⣁⡟⢉⣿⣭⣿⢿⡆⠒⣬⣿⢹⣇⣀⣟⠠⠈
⠀⠀⠀⠀⠀⠀⢾⠈⠙⢈⣿⣿⣿⠳⣿⣷⣶⠏⢹⡏⣽⡿⢙⢯⡠⣶⣻⣏⠀⣿⡊⠻⡖⠒
⠀⠀⠀⠀⠀⠀⠈⠒⢱⡜⠉⣾⣷⠖⣿⣿⣇⣸⢌⡑⢫⢹⡋⡑⠀⡽⠀⢿⣽⡿⡇⢀⡁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠣⣿⣭⣉⣿⣞⣡⣽⡰⢷⠈⢾⠹⠅⠒⢣⣠⣿⣅⣳⣴⡍⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡧⠁⠀⠀⠈⠈⠱⠎⠑⠀⢧⠀⢀⠀⠉⠟⣯⠙⢿⣷⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠒⣿⣋⡧⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣟⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⢧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠰⡄

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

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠆⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠃⠀⠐⢀⡀⠀⢀⢤⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣌⠿⣯⠃
⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⡷⠀⡄⠀⠈⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢼⠀⠀⠄⠀⠀⠉⠢⡄⣀⡶⣂⠂⠀⢉⣠⣾⢿⡇
⠀⠀⠀⠀⠀⣸⡧⠠⠀⢾⡇⢰⡇⠀⠀⡸⠁⣃⠲⠀⣀⡠⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣌⣬⢇⣦⣷⣤⠁⠈⢳⣼⣿⡿⡶⢿⡇
⠀⠀⢀⣰⣧⣦⣾⣧⣔⣶⡷⣿⡀⠀⠄⣄⣼⣯⣆⣰⣇⣀⢀⣠⣷⣄⡐⠀⠀⠀⠀⠀⠐⠿⠟⠛⠶⠿⠿⠿⠿⠿⠿⣿⣿⣿⣅⡶⠦⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡄⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣖⣴⣿⣿⣷⡶⣿⢥⡀⣸
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⢀⠀⡀⠀⠄⡀⠀⠀⠀⢀⣀⣀⣤⣷⣤⣿⣟⣹⣻⠃⠈⠛⣋⣷⣿⣰⣾⣿⡇
⠀⠀⠀⠈⢷⡤⣄⣐⣒⡶⠿⣻⣿⣷⡤⠆⠋⠈⠉⠝⠲⣭⣾⠿⡋⠉⠁⠈⠙⣻⣿⣿⣧⣿⣿⣾⣿⣬⣽⣽⣿⣈⣏⣿⣿⣷⣶⡀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠉⠁⠙⠉⠋⠺⠯⠭⠿⠗⠶⢦⣿⠿⣌⣁⠙⠷⠿⣽⢧⠴⠿⠿⠿⠛⢿⣿⣿⣱⣶⠭⡿⠿⢿⡿⠿⠯⠿⠿⠿⠿⠛⠓⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⣳⢟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢖⡀⡙⠉⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠈⠁⠹⠏⠒⡦⠔⣼⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠤⣹⡧⠐⠀⠀⠐⠓⡖⠂⣵⠂⣟⠁⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡆⠀⢸⡧⢍⡧⣼⠗⠂⠀⣼⠋⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⣨⠀⠀⠀⠙⠀⠀⡀⠏⡀⢤⣰⣿⣿⣽⡔⢷⣶⡄⠈⢁⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡇⢌⣛⢾⣷⣿⣿⣿⣿⣿⢧⣄⡬⠳⢟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠗⠀⠀⠀⠀⠀⠀⠄⢸⠀⠀⠐⠰⣁⡠⣬⡿⠋⣿⣿⣯⠿⣷⣿⣽⣷⣢⠞⣂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡴⠢⠆⢠⡟⣦⢼⡇⠀⠈⣿⣳⣦⣿⡿⣿⣿⡧⡿⣭⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⢀⣦⣍⡁⡦⢠⡛⣽⣻⣿⡇⠀⠈⢻⣿⢾⣿⣷⣿⣿⣛⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⠌⠁⠈⠃⠰⠀⢀⠘⠻⣏⢰⣲⣾⣷⣿⣾⠋⠈⠀⠀⡀⠹⢿⣿⣿⣿⣿⣿⠿⠁⠠⠆⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠈⡆⠀⢀⢀⡈⠄⠛⡿⠕⢿⣾⣿⣿⡿⠟⠋⠀⠀⡠⠎⠰⢠⠘⣿⣷⣿⣿⣿⣾⣼⡿⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠂⠀⠰⢱⡆⣠⣚⡟⣷⡄⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠂⠀⠂⠰⡄⢸⣿⣿⣿⣿⣩⢷⡀⠙⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⣠⢫⣿⠪⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠐⠀⠀⠤⣷⣂⠀⣿⣿⣿⣿⣿⣿⣹⡡
⠀⠀⠀⠀⠀⠀⠀⢠⣄⡌⡉⣿⣿⠿⢗⣮⣹⣳⠏⠀⠀⠀⠰⡀⠤⠤⠄⠀⠐⠘⢂⡘⠀⠀⠐⢼⣏⣀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⣼⣴⣼⣿⣿⣿⡿⢿⣿⡟⠀⠀⠄⠀⠈⡉⠀⠌⡀⠄⠀⠀⠂⠁⠀⣀⠀⢻⡿⢾⠿⢿⡿⠿⣿⣭⡷
⠀⠀⠀⠀⠀⠀⡀⠐⣾⣿⣿⣿⣧⣽⣟⣶⠟⠀⠀⣠⣦⡀⢃⡕⢀⠀⠠⠈⠐⡀⢀⢠⠈⠅⢀⢹⣷⣿⣷⣿⠷⣦⣈⢿⣿⠄
⠀⠀⠀⠀⠀⠀⠙⢀⣻⣿⣿⣿⣿⣿⡏⠁⠀⣠⣾⣿⣯⣽⡀⠚⢠⠌⠀⠁⠀⢂⠉⠀⠀⠖⣻⡶⢁⣿⣿⣶⣼⣿⣟⣧⣟
⠀⠀⠀⠀⠀⠀⢤⡏⡿⢻⣿⣏⣻⣁⣤⣤⣤⣿⣿⣿⣯⣽⣷⣤⣄⢈⠱⠐⢠⠀⢠⠑⣦⣷⣿⣧⣾⣿⣿⣿⢿⣿⡟⣿⣻⣗⢠⣀
⠀⠀⠀⠀⠀⠂⠈⠁⠙⠛⠛⠙⠻⣿⣿⡟⠻⣿⠙⣿⣓⣾⡷⠿⣿⣦⣤⣤⣶⣦⣶⣾⣿⣿⣿⣿⣿⣿⠿⣿⠄⢹⡖⡟⠉⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠰⠲⡟⠘⠋⣿⣿⣿⡛⢿⣿⣿⣿⠛⠻⠿⠿⠛⠉⠀⠙⠀⠈⠙⣾⡷⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⢹⡋⠓⢸⣿⡧⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣶⣶⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠈⠁

⢀⢀⡀⠀⠀⠀⠀⢁⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠄⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⢈⠂⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠠⠀⠀⠈⠀⠈⠀⠀⠀⠀⠄⠰
⠀⡀⠀⠀⠁⠁⠀⠁⠀⠀⢂⠀⠈⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⡄⠀⠈⠀⠄⠂⠀⠀⠈⠀⠀⡀⠀⠀⠠
⠀⠁⠁⠀⠐⠠⠀⠀⠄⠤⠈⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠠⠀⠀⢀⡀⠀⠀⠐⠀⠀⠈
⠀⠀⠒⠀⠄⠀⠀⠀⠀⠀⢀⡀⠀⠖⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠰⠂⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠁⠀⠀⠄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄
⠁⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠁⠀⠂⠀⢀
⡀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠇
⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠄
⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⢠⠀⠀⠈⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠄⠀⠀⠈⠀⢁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠄⠀⠀⠠⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠤⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠂⢀
⡄⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀
⠀⣀⠈⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆
⠀⠀⢀⠔⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⡀⠀⠒⠀⠀⠀⠠⠀⠀⠄⢀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠠⠀⠀⠄⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠠⠂⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠄⠀⠀⠠⠄⠄⠀⠠
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠂⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⠀⠀⠀⠀⠀⠁⠁
⠀⣠⠀⠄⠀⠐⠈⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⢀⠀⠐⠀⠀⠂⠀⠨⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠁⠀⠉⠁⠀⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⡿⠿⠿⠟⠛⠛⠛⠿⠿⢿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⠋⠉⢀⣀⣤⣤⣴⣶⣶⣶⣤⣤⡄⠀⠈⠙⠻⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣄⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⢀⣀⣄⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⢹⣟⣷⣠⣤⣤⣤⣀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⢟⣫⠽⠚⠉⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣛⣛⡿⠿⠿⢟⣛⣻⣭⡷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠹⣿⣝⣻⣿⣿⣿⣿⣛⣽⡿⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡈⠉⠛⣻⠃⣿⠛⠛⣋⣀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠞⠢⣀⣡⠟⠎⢿⠿⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⢠⠆⠤⢽⡯⣤⣴⢆⡔⢹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⡿⠨⣪⡀⠶⠛⠛⠿⢋⡾⡰⠁⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⢀⣠⣾⣿⣿⠃⣇⠀⠙⢿⣦⣤⣶⣶⣿⠟⠁⢠⣿⢻⣿⣿⣿⣿⢿⣶⡒⢢⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣤⣶⡾⢫⣿⣿⣿⣿⡄⣿⠄⠀⢀⣽⣿⣿⣋⠁⠀⣠⣿⡿⣼⣿⣿⣿⣿⣥⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣾⣿⣿⣿⣿⣇⣿⡈⠶⣿⣿⣿⣿⣿⠷⡼⢡⣿⢷⣿⣿⣿⣿⠟⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣿⣿⣿⣿⣼⣇⠀⠹⣟⣿⣿⠃⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣆⣴⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠒⢤⣄⢀⣀⡀⠀⠀⠀⠀⠈⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠊⠀⠀⠀⠀⠀⠀⠁⠠⣷⡄⢀⡀⠀⠈
⠀⠀⠀⣠⣾⠙⡉⠉⠉⠉⠑⠀⠃⠠⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠛⠁
⠀⠀⠀⠹⢿⣦⡦⡀⠀⠀⠀⠀⠒⠗⠀⠈⠐⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠁
⠀⠀⠀⠀⠀⠉⢣⠰⣄⡰⣄⣀⢠⠀⠀⡀⠬⣤⣘⠀⢤⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠪⣍⠳⢷⡀⢁⡑⠰⠡⡥⠚⢰⡊⢳⣶⡆⠠⢀⣠⠞⠀⣀⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠥⣤⣿⣥⣏⣽⣿⣟⣿⣧⣥⡼⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡳⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠰⣟⣢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⠁⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡀⠀⠀⠀⠀⢰⠃⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠤⠖⠤⣄⡠⠚⢩⠃⣀⡤⠤⣄⠈⠀⡎⡏⠀⡨⢵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⢀⡤⠤⠴⠁⠀⠀⣠⠊⠀⠀⢸⠉⢸⠁⠀⢸⡤⠤⠳⡌⠲⠽⠶⠻⡄⠀⠀⢦
⠀⠀⠀⠀⠀⢦⠀⢀⠏⠀⠛⠤⣀⠀⠀⡔⠋⠉⠉⠓⢤⠈⠶⣚⣁⡠⠞⡠⠴⠒⠺⢄⠀⠀⠀⡇⠀⠀⡎⠀⠀⢠⢄⡀
⠀⠀⠀⠀⠀⡼⠁⡞⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠛⠉⢡⣄⠀⠀⠀⠀⠀⠀⠀⢣⠤⠴⢍⠉⠑⣄⣠⠤⠊⡷⠋
⠀⠀⠀⠀⠀⢱⠾⠗⠒⠒⠲⠤⣀⠴⠓⠒⠒⢒⡤⠤⠤⢄⣀⡼⠳⡳⡒⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢠⠏⠀⠀⠀⢳
⠀⠀⠀⡠⠚⠁⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⡠⠤⡄⠀⠀⠀⠙⢦⠤⠵⠿⠀⠀⠀⠀⠀⡟⠒⠛⡿⠴⠛⠋⠓⠒⢤⡿⠤⡀
⠀⠀⡸⠁⢀⠔⠁⣠⠤⠖⠦⢄⠐⠉⠉⠢⣀⣀⠜⠀⠀⠀⠀⠈⣀⣀⣀⣀⣀⡀⠀⠐⠉⠉⠑⠇⠀⡠⠤⠤⣀⣀⣠⠤⠞
⠀⠀⠙⠚⠓⠊⠉⠉⠢⣀⠔⠚⠓⠒⠒⠒⠒⠢⠤⣤⠤⠖⠊⠉⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⠤⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣀⠀⠀⠀⡙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠉⠁⠙⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⢦⠀⠀⠲⣶⣶⡆⢀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣗⣉⣀⣙⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣿⣇⠀⠟⠛⠛⣿⡏⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡟⠀⣿⣿⣿⡀⠰⢤⣬⡟⢰⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢻⣧⠀⣿⡿⣿⣷⣤⣀⣀⣴⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡏⠁⠀⠀⠀⠀⢿⣿⡀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣄
⠀⠀⠀⠀⣴⣀⠀⠀⠛⢳⣤⠀⠀⠀⠀⢹⣿⣧⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠿⠿⢗⡦⠄
⠀⠀⠀⠸⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢻⠆⠀⣨⢻⢏⠝⡿⢻⢏⣽⣿⢿⣗⣀⣶⣾⡒⣦
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠁⠈⠀⠁⠘⠉⠁⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠛⠛⠓⢦⣙⣿⣿⣿⣿⣋⡴⠚⠛⠛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⡿⢿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡟⢩⣝⣦⡀⠀⠀⠀⠀⢀⣿⣷⣄⣀⣀⣀⡴⢋⣼⣿⠁⠈⣿⣧⡙⢦⣀⣀⣀⣠⣾⣿⡀⠀⠀⠀⠀⢀⣴⣫⡍⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⣬⣭⣴⣾⣿⣿⡇⠀⠀⢸⣿⣿⣷⣦⣭⣥⣾⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣿⣿⣿⣧⡀⠀⠀⠘⢿⣿⣿⣿⢿⣿⣿⣿⣿⣤⣴⣦⣤⣿⣿⣿⣿⡿⢿⣿⣿⣿⠃⠀⠀⢀⣼⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠉⠉⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠉⠉⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠸⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠇⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣶⣄⡈⠉⠀⠈⠉⠁⠈⠉⠁⠀⠉⢁⣠⣶⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣷⣦⣀⠾⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⢿⣿⣿⣿⣿⣿⣶⣍⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣿⣿⣷⣬⣙⠿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⣴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣵⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⡋⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠰⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⢀⡇⠀⠀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⢤⣄⡀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠉⠳⡀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢀⣙⡲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠐⢤⡀⠀⠀⠀⠀⠀⣷⠋⠁
⠀⠀⠀⠀⠀⠀⢞⠩⠄⢈⠆⠈⠙⠓⠒⠶⠶⡶⠚⠙⢲⣼⡃⠀⠀⠀⠙⡆⠈⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⣠⠶⠖⠲⠤⣜⢀⠀⠀⢀⡏⠳⣄⠀⠀⢀⠃⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠒⢄⠀⠈⢧⠉⠛⠉⠀⠀⠈⠉⠙⠛⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠁⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⢤⣒⠛⠓⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠉⠂⠀⠀⠘⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠖⠒⠲⠶⣤⣔⠁⠀⠀⠀⢀⡠⠤⢤⡤⠶⠤⣄⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⡤⠒⠉⠉⠃⠀⠀⠈⠀⠀⠈⠃⠀⠀⣷
⠀⠀⠀⢀⡤⠒⠚⠓⠀⠀⠀⠀⠀⢀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠱⣄
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠒⠲⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⣀
⠀⢀⡧⠤⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣧⡀
⠠⣏⣀⣀⣀⣀⣠⠟⠑⢢⡤⠤⠤⣄⣠⠖⠋⠉⠀⠀⠀⠈⠉⢳⡄⠀⢀⣀⣀⣀⠀⣠⠤⠤⢴⡋⠁⢀⡽
⠀⠀⠀⠉⠀⠀⠀⠙⠓⠚⠧⠤⠤⠼⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⡉⠁⠀⣉⣹⣆⣀⣀⡤⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠲⠶⠒⠚⠋⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠗⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠳⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢷⣄⠀⢀⡀⢠⡟⠀⠀⠀⠀⠀⠸⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢺⣎⣶⣻⣀⡟⠀⠀⠀⠀⠀⠀⠆⢱⡀⠢⡀⠀⠀⠀⠀⠀⡎⠙⠋⠉⠁⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⠛⠛⠉⠛⡉⠍⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⠁⢀⠀⠀⠀⠀⢸⡆⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⡀⠀⠀⠀⠀⢠⡿⠬⠳⡄⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣎⠀⠀⠐⠀⠀⠀⠀⣿⡽⢁⣠⢴⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⣰⠟⠁⠀⠀⠀⣧⠀⡤⠀⣠⠟⠀⠀⠠⠈⡆⢀⣀⣀⣀⠀⠀⢀⣇⠀⠀⢀⠀⣿⢶⠶⠶⠶⠶⠶⠦⣼⣷⠋⣴⣩⠄
⠀⠀⠀⠀⠤⣄⡀⢷⣌⢿⡏⠑⠢⢤⣀⣀⣀⣱⣤⣴⣃⠀⠀⠀⣀⣴⠟⠉⠁⣨⢟⣷⣠⠋⢈⣣⣀⣠⠟⠁⠈⠳⡁⠀⠂⠀⠀⠀⠉⠻⣿⣡⡆
⠀⠀⠀⠀⠀⠈⠻⡲⣏⣾⡇⠀⠀⠀⠀⠀⣴⠯⢽⡈⠉⢙⣶⠞⠉⠀⠙⡞⢉⡾⠋⠉⠙⢶⠋⣟⣹⣏⠀⠀⠀⠀⠈⢢⣀⠀⠀⠀⠀⠀⠈⢻⡅
⠀⠀⠀⠀⠐⠶⣲⡿⠋⢫⡉⠛⠓⠶⣤⣀⣿⡀⠀⢱⢀⠎⠘⢧⠀⠀⣀⣈⣾⣏⠉⠉⣉⣽⢧⣇⡌⢉⡀⠀⠰⠆⠀⣴⠋⠳⣤⣀⠠⢤⡀⠘⣿
⠀⠀⠀⠀⠀⣾⣋⣀⣀⣠⠵⢄⣀⣠⡟⢻⠈⠛⠛⢻⡏⠀⣴⠿⣋⡉⠉⠁⣈⣉⣻⣾⣏⠀⠀⠈⣧⠀⠠⣄⣤⢤⣤⣥⢤⠦⢤⣍⠙⢾⣥⣤⣽⡇
⠀⠀⠀⠀⠀⠹⣦⣀⡀⠀⠀⣠⣤⣼⣇⣠⠇⠀⠀⢘⣧⣼⠃⠀⠀⠬⢹⠏⢯⠀⠀⣽⣪⣦⣀⣀⡼⠷⣴⠏⠀⢸⡙⣳⣾⣀⠞⣉⣧⣸⣄⣠⡞
⠀⠀⠀⠠⠆⠀⠀⠈⠉⢻⡟⠃⠀⠀⢈⠹⣷⠒⡒⠛⠁⢯⣤⣤⣄⣀⣸⠀⢸⣀⡴⣿⠧⠤⠭⠟⢷⣼⡇⠀⠀⢠⠏⠀⢸⠷⣾⣩⠋⠈⣿⠡⢄⣀
⠀⠀⠀⠀⣾⣟⡛⢻⡆⠈⠳⠦⣤⣤⣤⡶⠋⠀⠉⠀⠀⠈⠂⠀⠀⠤⠉⠛⠛⣉⣀⣈⠛⠒⠒⠶⠚⠁⠛⠷⠶⠾⠚⠛⠋⠀⠘⠷⠶⠞⠁⠀⠀⠈
⠀⠀⠀⠀⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡍⢉⡩⠟⢳⡆⠉⠀⠀⠀⠀⠈⠛⠀⠀⣴⣚⣳⡄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣠⣦⡾⢻⡿⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣅⡠⠾⣿⣿⡵⠟⠁⠀⠀⠀⠹⣶⠀⠀⠀⣴⡎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠊⠉⠁⣠⣶⣴⣤⠀⠀⠀⠀⠀⢀⡀⡀⣠⠶⠋⠑⠀⢦⠀⠤⠄⣠⡀⠀⠀⠀⠀⠈⠉⢿⠾⠁⢀⣶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠄⠛⠁⠀⠀⠀⢠⡾⠛⠉⠀⠙⠷⣤⡀⢠⡀⠛⢩⠟⢁⠀⠠⠃⠄⢀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢀⣶⡥⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠙⠳⣄⡈⠁⠘⠊⠉⠀⠀⠀⠀⣿⣷⠀⣸⣿⣿⠀⢀⡴⢊⠿⠋⠠⠀⢠⠆⡴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣠⠀⡀⠀⠀⠀⠀⠀⢀⡈⢻⣦⡼⠋⠀⡠⠞⠀⢀⣿⣿⣤⣿⣿⣿⣗⠁⠀⠁⠀⠀⣠⠐⢋⣠⣾⡿⠟
⠀⠀⠀⠀⠀⣼⣆⡀⠀⠀⠚⢁⠔⠁⠈⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⠀⠀⠈⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⢊⣠⣴⠟⠋⠉
⠀⠀⠀⠀⠀⠙⠛⠿⣿⣧⣰⠋⣶⣄⢀⠀⠀⠀⠀⠀⠀⢠⣟⣁⠀⠀⠀⠸⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣇⣠⣦⠀⢡⣉⠁⢰⣷⣾⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⠿⣿⠟⠛⠛⢳⣦⡀⢀⢠⣼⣿⠟⠁⠀⣠⣦⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡄⣿⣿⣷⣾⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣄⠀⠀⠀⠀⠘⠿⢾⣿⣿⢁⡀⠀⣄⣿⣿⣷⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⠉⢀⡀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠛⠳⣤⡄⠀⠀⠀⠉⠉⠉⠙⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⢀⠠⠀⠊⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⣴⣾⡟⣀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣠⣶⣶⠗⠁⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠛⠓⠴⢄⡄⠀⠀⠈⠿⣆⣀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣰⣎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠙⠿⣧⣶⣦⣨⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠿⠛⠋⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢀⣴⠟⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠈⠙⠿⠛⣿⣿⣿⣿⣿⣿⣯⠉⠀⠁⠀⠀⢀⠀⠀⢀⣤⣾⣿⡶⠛⠁⠀⠀⠔⠋⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠿⠿⠟⠋⠉⠈⠀⠀⠀⠀⠀⠀⠁⠀⠐⠛⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠀⠉⠙⠻⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠿⠛⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣰⣿⠃⠀⠀⠀⠀⡀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠏⠀⠀⢸⡇⢸⡇⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡿⠿⠿⠿⢿⣷⣶⣄⠀⠉⠁⠀⠀⠀⣿⡇⢸⣷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡄⠀⠀⠀⠀⠻⠇⢸⣿⠟⠁⠀⢀⣠⣤⣶⣶⣿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠛⣿⣆⠀⠀⠀⠀⠀⠀⠉⠀⣠⣶⠟⠋⠉⠀⠀⠀⣠⣤⣽⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠋⣿⡄⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣴⠌⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⣿⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢰⡆⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⢶⣿⣦⣄⡀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠋⠀⠀⠀⠈⠙⠻⢶⣄⠀⠈⠳⠦⣄⣀⣀⣀⣀⡠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⡀⠀⠀⠀⣀⣀⣤⠞⣻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⣉⣹⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⢠⠴⠚⠉⠉⠁⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⡠⠚⠁⠀⠀⣀⣀⣀⣤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣿⡿⠿⠿⠛⠛⠛⠉⠉⠁


⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⢠⡟⠛⠛⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠉⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣦
⣿⠃⠀⠀⠀⠉⡀⠀⣀⣤⣤⣀⠀⠘⢿⣧⣤⣴⣶⣄⠀⠀⢴⣾⣿⣶⠟⢿⣿⠂⠀⠀⢸⡇⠀⠀⠀⢀⣤⡶⠿⢷⣾⡿⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⣰⣿
⣿⠀⠀⠀⠀⢸⣿⣾⠏⠁⠈⠙⣷⡀⢸⣿⠉⠀⠈⣿⡆⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⣸⣿⠀⠀⠀⠸⣧⣤⣤⣄⣿⠇⣰⡟⠉⠉⠻⣦⣠⡿⠋⠉⠙⣿⡀
⢻⣆⠀⠀⢀⣼⠟⣿⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⣷⠀⠀⠈⣿⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⣧⡀⠈⠉⣻⡷⣿⣷⠶⠶⠶⢿⣿⠃⠀⠀⠀⣿⡇
⠀⠙⠛⠛⠛⠁⠀⠙⢷⣦⣤⡾⠟⠐⠿⠿⠿⠀⢀⣼⠿⠷⢶⣴⣿⣦⣦⠀⠻⣧⣀⣴⠿⣿⣦⡄⠀⠾⠏⠙⠛⠛⠋⠀⠹⣧⣀⣀⣠⡄⢻⣦⣀⣀⣠⣾⣇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣭⣤⣤⣬⣽⣭⣥⣼⣿
⣿⣿⣿⣿⠛⠙⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡦⠀⢰⣿⡿⠿⠀⢀⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣾⣄⠀⣿⡇⠶⣿⡇⢸⣿⣛⣛⠀⣼⣿⣦⣿⠇⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⣿⣆⠙⠻⠿⠛⠁⠈⠉⠉⠛⠀⠛⠃⠿⡇⠀⣿⡏⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⡀⠹⣷⠄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⠿⠋⣠⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠁⢀⣴⡦
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡈⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣿⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣾⣏⡀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⣿⡟⣷
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⠃⠀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣀⡀⠀⠘⢷⣿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡆⠀⠀⣰⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⠈⣿⣿⠁⠀⠀⠈⣿⣿⠁⠀⠀⠈⣿⣿⠁⣠⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⢿⣿⡗⣸⣿⣦⣤⣤⣼⠟⠻⣧⣤⣠⣴⣿⣇⢸⣿⡿⠿⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⢿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⡿⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⣿⣿⣿⣿⣿⣿⣩⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⢽⣉⣉⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠆⠻⣿⣾⣶⣶⣷⣿⠟⠰⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⣶⣿⣿⠿⠋⠁⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠙⠿⣿⣿⣶⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣠⣾⣷⡆⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⢰⣾⣷⣄⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⢿⣧⣿⠃⢀⣿⡟⣿⣧⢸⣿⣟⣻⡀⠘⣿⣼⡿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⢸⣿⢻⣿⠋⠀⣿⣯⣭⡅⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⢀⢴⠖⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣾⢸⠀⠀⠀⠈⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢸⣿⠎⠠⠀⠀⠐⠒⠺⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⡆⠀⢸
⠀⠀⠀⠀⠀⢀⣼⣿⣠⣤⣤⣴⣾⣶⣾⣧⣌⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣧⣷⣀⣤⠶⠚⢤⡀
⠀⠀⠀⠀⠀⠚⠛⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠧⠄⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠷⠶⠽⠷⢶⠤⠷


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⣤⠤⠶⠒⠒⠲⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣦⣾⣷⣶⣆⣄⠀⠀⠀⡀⢹⡧⣭⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠿⠶⠟⠛⠋⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⡖⠨⡛⢳⣄⡀⠈⠁⢦⠉⡭⠢⠀⠉⣉⣹⡍⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠂⠉⠑⢴⢈⣂⠀⠀⠀⠀⠀⠐⠒⣶⠊⠁⠉⠁⠈⠙⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠠⡀⡰⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⢐⡜⠀⠀⠀⠀⠀⢀⠤⢀⣀⣠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢂⢢⣎⣀⡤⢲⣊⠭⠁⠈⠂⠄⠈⢋⢁⡔⢦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⢻⠛⠿⠹⣡⣀⠈⠉⢳⡒⠊⡛⠁⢨⠄⠀⢁⡿⡐⣦⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢄⡪⢝⡢⢁⠀⣼⡂⢠⠘⣽⢋⡽⠶⠶⠮⢜⣋⣅⢀⡲⠶⠄⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠀⠈⣑⠨⡑⠮⠙⢛⠦⠴⡌⣄⠖⣺⢃⠘⣼⡆⢹⠉⠅⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⣾⡾⡃⢛⣃⠘⢸⠀⠀⡇⣀⢀⡟⢈⣤⣠⢱⣼⢉⡅⢸⡄⣧⣀
⠀⠀⠀⠀⠀⢀⣀⠀⡴⠆⣄⢉⠒⠁⠀⠠⠋⠀⢃⠺⣄⣀⡇⡁⡟⠧⡯⣟⣷⣋⣾⠉⢧⡼⠇⡟⠉⡛⣶⢦⣀
⠀⠀⠀⢀⠚⠁⠜⢊⠀⢠⣌⠀⠑⠚⠢⢤⣴⠖⠪⠘⢻⢬⢿⡁⣧⢤⣷⣽⠵⠩⣀⣀⠾⢀⠀⣸⡚⠛⠉⠉⣀⣸⠂
⠀⢀⠀⠉⣠⡆⠀⡌⠁⠉⡔⢤⠒⣳⣦⠺⢁⠀⢀⡴⠞⢋⢈⣫⢉⣇⠀⢈⡶⢯⠧⡄⢀⣼⣀⡹⢃⡈⢀⣠⡿⢅⣀⣸⠆
⠀⠀⠉⠉⠉⢄⣠⠶⡄⠀⠀⣴⢟⡡⡌⠀⣠⡁⣙⢇⡀⠀⢄⣀⣀⣍⣉⡥⣠⣤⣧⣀⣀⣸⢯⣀⠀⣨⡏⠁
⠀⠀⠀⠀⠀⢄⠀⠀⣹⠗⢎⣁⣀⣤⡀⠀⢩⡿⡉⠀⢉⡿⠉⠁⠁⠉⠁⠀⠢⣀⣀⠼⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠨⣁⢀⣨⠏⠉⠀⠉⠛⠋⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠴⣀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠐⠲⣇⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡅⠀⠝⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠄⠀⠤⠀⠀⠀⠀⠀⠠⠾⠥⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣈⠲
⠀⠀⠀⠀⠀⠀⠀⢄⠬⢞⣿⠀⠀⠀⠀⠀⠀⠐⠐⠀⠛⠒⠲⠤⠀⠀⠀⠀⠄⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢴⠛
⠀⠀⠀⢀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢒⢀⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡄⣤⡀⢠⠆⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠘⠈⠁⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⡀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⡀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡠⠀⠁⠒⠄
⠀⠀⠀⠀⠰⠀⠀⠁⣁⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠁⠀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠤⠄⠉⠁⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠠⠄⠠⠀⠠⠀⠀⠀⠀⠀⡀⠘⠒⠒⠀⢤⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡀⠀⠀⠀⠀⢀⣐⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⠸⢳⣆⠀⠘⠈⠙⠁⠂⣶
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡉⠀⠀⠀⠀⠈⠉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠘⠛⢦⣌⠁⠀⣁⠲⠖⠛⠚⠛
⠀⠀⠀⠀⠈⢁⠠⠉⠈⠉⠋⠭⠀⠀⠀⠀⠁⢠⠀⠁⠅⠀⠀⢀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠴⠍⠋⠁⠀⡀
⠀⠀⠠⠤⠴⠀⠠⢀⠔⠒⠉⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠂⠀⠂⠀⠀⠂⠀⠀⡀⠑⢤⣤⡑⣀⡍
⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣤⠀⠐⣶⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠉⠁
⠀⣰⠖⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⠛⠃⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠨
⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡄⠀⠀⠠⠒⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⠀⠀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣠⡟⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⢀⠀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠦⠀⠀⡚⠙⡁⠀⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠴⠋⠁⠀⠠⠐⠠⠈⠋
⠀⠀⠀⠀⠀⠄⢆⠉⠀⠁⢈⣀⡀⠀⠀⠀⠐⠦⠚⠀⢀⡀⠦⣄⠐⠂⠀⠀⠀⠀⠀⠂⠀⠈⠁
⠀⠀⠀⠀⠀⡀⠀⠁⣴⡆⠈⠉⠁⠀⠀⠀⠀⠀⠀⠊⠈⠁⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠒⢀⠀⠀⠀⠀⠀⢀⢀⡰⡔
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⣐⢂⠌⠠⡑⠮⠕⢉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢜⠓⠛⠐⠠⠝⠑⠦⠓⠆⠠⠢

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡀
⠀⠀⠀⠀⢀⣼⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣸⣿⣿⣿⡤⠀⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡧⣿⠐⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣿⡀⢻⣿⣷⣿⣶⣶⣾⣛⣓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⢿⣿⣿⣿⣿⣭⣉⣙⡲⠦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣭⣿⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠷⣦⣦
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠛⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢁⠈⠀⢸⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠸⣦⡀⠠⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠀⠐⢈⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣀⣀⣤⣶⣤⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠁⠀⡂
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠈⠉⠉⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣷⡀⢀⣴⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡷⢒⠛⠂⠀⢀⣀⣀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢿⣿⣇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠶⠿⣿⣷⡀⠘⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠃⠀⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣻⣧⣼⡅⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠃⠀⠐⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⡿⠁⠀⣤⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠃⢀⣰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣄⡀⠀⣠⣀⣤⣴⣿⣿⣿⣿⣿⣿⠟⠁⠠⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⣀⡀⢀⣴⣿⠿⠋⠉⢞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠴⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠃
⢠⣭⣾⣿⣿⣿⣿⣧⣤⣤⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⣿⣧⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠏
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠄⠠⣻⣿⣿⣿⣿⣿⣿⡇⠀⠄⠂⠂⠙⣿⡆⠀⠀⣠⣾⣿⣿⣿⣿⠋⠉⠁
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠜⠊⢰⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣴⣿⣿⣠⣾⣿⣿⠿⠻⠻⠛
⠀⠈⠛⠻⠿⢿⡿⠿⠿⠿⢿⡇⠀⠀⠈⠹⠿⠿⠿⠃⠀⠀⠀⠀⠈⠼⠿⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠞⠛⠏⠉⠍⠉⣉⠋⠉⠫⠽⡶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠏⠀⣁⣠⠀⠀⠘⠳⢤⠄⠀⠀⠈⢈⡒⠟⣻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡟⣿⣿⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠹⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⢏⣿⣏⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⣦⣷⣹⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⣸⣿⣾⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⠁⠀⠐⠚⢀⠀⠀⠀⠀⠀⠼⠀⠀⢰⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⣱⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣴⣿⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡏⢱⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡞⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡷⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠿⠛⠛⣹⣛⣉⣴⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣰⢏⢴⡿⢿⠉⠀⢠⡆⢺⣿⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣦⡀⠀⣾⣧⣤⣾⠃⠏⠀⠀⠸⠑⣿⣿⢇⣿⣿⣿⣿⣿⣿⡧⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⡟⢻⣿⡟⠈⠀⠀⠀⠀⢰⣿⣿⣻⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣾⣿⣟⢹⢿⣿⡧⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠋⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣮⡻⣿⣿⣿⣿⡿⠿⣿⣛⡻⣿⣿⣤⡤⡄⠀⠀⠀⠀⠠⣿⣟⣫⠽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣶⡟⣿⣿⣿⣿⣶⣾⣿⡿⠶⠿⠉⡁⣀⣀⠀⠀⠀⣴⣦⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠟⠰⢶⠆⢁⣛⠁⢟⠁⢴⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⠦⠀⠤⠥⠄⠸⢶⣀⡀⢘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣶⣶⣶⣦⣀⣉⠁⢠⢩⡽⣿⣿⣿⣿⣿⢻⣿⣷⣿⢹⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⠻⠿⠿⠿⠻⠇⠤⠄⠚⢳⢿⣿⣿⣿⡭⢿⡿⣟⣶⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠈⠉⠙⠛⠛⠛⠛⠉⠀⢠⣤⣤⣴⣶⣶⣶⣮⣿⣻⣿⠛⠿⠻⠗⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⠿⢿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⡾⡟⢹⣾⣿⢿⡿⠿⠿⠿⠛⠛⠃⠉⠉⠉⢠⣾⣽⣿⣿⡟⣡⣴⣾⣿⣿⢪⣴⣮⣿⡎⢪⣾⣿⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⢿⡟⢉⣿⣿⣿⣿⣿⣋⣷⣿⣿⣧⡿⡎⣿⡿⠿⣿⠿⠛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⡶⢿⠟⠋⠁⠸⣴⣿⡿⠛⠹⠛⠁⠛⠛⢻⣿⣿⣿⣠⠸⣿⣶⣦⣤⣄⠉⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂⡀⢀⣴⣿⠛⠁⢰⡇⠀⠀⠀⣿⣮⣙⢛⠟⣿⣤⣶⣯⠉⢻⣿⣷⣆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡏⠛⣷⠀⢻⡄⠀⠀⢸⣿⠋⣠⠊⠀⢉⠀⠀⠀⢀⠈⠘⢉⣈⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⣤⣿⣷⡰⣿⡀⠒⢦⡎⣠⣿⠇⢾⠃⠐⢴⣿⣷⡄⠀⣸⣿⣷⣎⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣛⠾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠛⠟⠿⣿⣧⣸⣷⢿⣿⠇⢸⡀⠀⢘⣿⣿⡇⣾⡿⣿⢩⣍⡉⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿
⣽⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⣶⣦⣌⠙⠷⢠⡟⠀⠺⣿⠀⣸⠋⢛⠁⡈⠃⢸⣀⣈⡻⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠤⣤⡀⠙⢿⣷⣶⡿⣿⣿⣿⣿⣷⣄⡌⢠⣵⠈⣤⣁⣠⡾⠃⣿⠐⣿⣿⣿⡉⠓⠢⢼⣿⣿⣿⣿⣷⣿⣿
⣽⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣡⣄⡀⠈⢿⣿⡹⣿⣿⡿⢉⣿⣄⠈⣿⢃⡈⠛⠋⠀⣾⠁⠀⡽⠈⠙⢿⣷⠆⠜⢿⣿⣿⣿⣿⣿⣿
⣿⢿⣟⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠈⢿⣿⣦⣴⣶⣿⣿⣿⠛⣥⡜⣿⠆⠀⠀⠉⠀⠀⠃⣰⣦⠀⠙⠧⠀⢸⣝⣿⣿⣿⣿⣿
⣿⣻⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠙⢧⣠⣽⢻⡿⢛⠻⣿⣿⣿⠛⠋⠀⠀⠀⠀⡀⠀⠀⠈⣃⠈⠁⠀⠀⣰⣺⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣯⣟⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡄⠀⠙⠳⣾⣧⣸⣧⡘⣿⣿⡆⠀⠐⠀⠀⣰⠇⠀⠀⠀⠉⢻⡇⢀⢺⣿⢟⣿⣿⣿⣟⣿⣿
⣽⣟⣿⣿⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣶⣦⣀⠈⠛⢻⣷⣸⣟⢿⡄⣴⠀⢠⡟⠀⠀⠀⠀⠀⠈⢷⢘⢿⡯⠃⢸⣿⣿⣿⣷⣿
⣿⣿⣿⡿⣿⣟⣧⠀⠀⠀⠀⠀⠀⠐⠀⣤⣀⠀⠈⠈⠀⠀⠉⠐⢀⣿⡟⠉⣿⣿⡟⠀⣿⡇⠀⠀⠀⠀⢀⠀⠈⠀⠉⢀⣾⣷⣿⣿⣿⣽⣿
⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⡈⠁⠲⣬⣀⣀⣀⣀⣾⡇⢀⣼⠟⠙⠁⠀⢻⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡁⢻⣿⣿⣿⣿
⣏⣿⣿⣽⣿⣷⣾⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠀⠉⢿⡟⣰⡿⠁⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⡼⠃⠘⣿⣿⣿⣿
⢷⣿⡿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣦⣄⢀⡈⢳⣿⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⣤⣾⣿⣿⣿
⣿⣽⣻⣿⣏⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢿⣿⣿⣿⣿⣿⣦⡀⠀⣠⠆⠀⠁⠙⢐⡄⠀⡀⠒⠒⠲⠶⠶⠚⡩⠛⢻⣿⣿⣿
⢽⣿⣿⣷⣿⣿⣿⡻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠙⢻⣿⣿⠿⣆⡀⠀⠀⠘⣷⣜⠁⠀⠻⠷⣮⣤⠀⢰⣿⠟⣿⣿⣻
⡿⣾⣸⡷⣼⣍⣳⡳⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡀⢧⡈⠙⢄⣀⠀⠙⢻⡟⠛⡟⢷⣹⣿⣷⣿⠇⠀⣿⣿⣿
⣿⡿⣿⣿⣿⡿⣟⣿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠈⠀⠘⢢⠹⣦⡄⠈⠁⠀⠀⠀⠀⠀⠀⠙⠻⡏⢡⢄⠀⠹⣿⣿
⣻⣛⣿⣿⠛⢂⣙⣻⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⣀⡉⠛⠲⠖⠒⠊⠀⠀⠀⣠⣶⠿⢶⢶⣶⣶⣿⣻
⣿⠷⢲⡟⢙⣛⣿⣭⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡷⠲⠺⠷⠤⠴⠶⠛⠛⠁⠀⠘⠿⠀⣷⢸⣿
⣷⡾⣽⣿⡼⣿⣿⣯⠛⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠘⠓⠀⠀⠈⣿
⢿⣶⡿⣿⣶⣲⣴⢟⣿⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿