Doorstep Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣁⠀⠀⢀⠉⢂⠀⠀⠀⠀⠀⠀⠠⠓⣃⠀⠀⣈⠑⢄⠀⠀⠀⠀⠀⠀⡐⠀⡀⠀⠀⡀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠓⠃⠀⠃⠃⠀⡆⠀⠀⠀⠀⠀⡇⠘⠷⠃⠘⡚⠃⢸⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠘⠚⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢰⣶⠀⠀⢠⠃⠀⠀⠀⠀⠀⢇⠀⠀⢰⡆⠀⠀⡸⠀⠀⠀⠀⠀⠈⡄⠠⣔⡷⣕⡦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⡀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠈⠓⠤⢀⡀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠄⠊


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢐⠄⠀⣐⠄⠈⡀⠀⠀⠀⠀⠀⠊⢔⠒⢒⡒⠒⠢⠑⠀⠀⠀⠀⠀⢀⠂⠰⡒⠀⠰⡢⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠈⠒⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⠉⠉⠑⢀⠔⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠂⡀⠈⠀⠀⠀⠀⠀⠀⠣⡐⠁⠀⠀⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡀⡀⠀⡀⡈⢢⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⡔⠡⡄⠀⠀⣤⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠛⠂⠀⠛⠃⠀⠃⠀⠀⠀⠀⠀⠁⠐⠒⠀⠀⠒⢂⠈⠀⠀⠀⠀⠀⠘⠀⠘⠛⠀⠘⠛⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢠⠐⠒⢤⠀⢠⠁⠀⠀⠀⠀⠀⢆⠀⠀⢎⡱⠀⠀⡰⠀⠀⠀⠀⠀⠘⡄⠀⢶⣶⣶⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⢀⠀⠄⠊


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠀⠐⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢰⠒⡆⠐⡖⢲⠙⡄⠀⠀⠀⠀⠀⠂⢀⣊⡀⢀⣁⡂⠑⠀⠀⠀⠀⠀⢀⠆⡔⢤⠀⠠⠔⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠉⠁⠉⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄⠉⠚⠀⠙⠊⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⢄⠘⠉⠉⠃⡠⠊⠀⠀⠀⠀⠀⠀⠈⠢⡘⠦⠚⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉

⠀⠀⠀⠀⠀⢀⠀⣀⣠⣤⣤⣀⡀⢀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣀⣀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⡀⢀⣀⣤⢤⣄⣀⠀⡀
⠀⠀⠀⡠⣢⠔⠋⠁⠀⠀⠀⠀⠉⠓⢮⡢⡀⠀⠛⠃⠀⠠⢀⡵⠊⠁⠀⠀⠀⠀⠈⠑⢮⡐⢀⠘⠿⠟⠀⢀⢔⡥⠚⠉⠀⠀⠀⠀⠈⠙⠦⣔⢄
⠀⠀⢌⡼⠁⠀⣠⢶⠆⠀⠀⢶⢦⡀⠀⠙⣎⠆⠀⠀⡐⡴⠋⠀⡴⠛⣷⠀⠀⣾⠛⢦⠀⠙⢦⢂⠀⠀⠠⣱⠋⠀⢠⡴⣤⠀⠀⢠⠶⣄⠀⠈⢧⡡
⠀⠈⡾⠀⠀⠀⢿⣯⠗⠀⠀⢯⣿⠇⠀⠀⠘⣎⡀⠀⣼⠁⠀⠈⠁⠀⠉⠀⠀⠉⠀⠈⠁⠀⠈⣧⠂⢀⣱⠃⠀⠀⠸⢶⠿⠀⠀⠹⢷⠿⠀⠀⠀⢷⠁
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⢀⢧⠀⠀⠀⠴⠛⣦⠟⢧⡾⠳⠄⠀⠀⢠⣏⠀⠰⢷⡀⠀⠀⠀⠻⣤⣀⣀⣤⠟⠀⠀⠀⢀⡞⠂⠈⢹⡄⠀⠀⠀⠉⠻⠉⠉⠹⠋⠁⠀⠀⠀⡼⡀
⠀⠀⠈⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠁⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠈⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠁
⠀⠀⠀⠀⠨⢓⠦⢤⣤⣀⣠⣤⡤⢖⠫⠁⠀⠀⠀⠀⠀⠀⠨⢛⠶⢤⣤⣀⣠⣤⡤⠶⡊⠅⠀⠀⠀⠀⠀⠀⠈⠝⠲⢤⣤⣄⣠⣤⡤⠴⡚⠅
⠀⠀⠀⢰⣶⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠐⢿⠇⠀⣠⣶⣄⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⢀⣤⡄⠀⠀⢠⣦⡀⠀⠉⠀⠀⠀⠀⠀⠁⠀⣴⡆
⠀⠀⠀⠈⠉⠠⣀⡤⠤⠶⠦⠤⣄⡀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⢀⣤⠤⠶⠶⠤⣤⡀⢀⠀⠘⠿⠏⠀⠀⠈⠉⠀⢀⣠⠤⠴⠶⠤⢤⣀
⠀⠀⢀⢔⡥⠚⠁⠀⠀⠀⠀⠀⠀⠉⠲⣔⢄⠀⠀⠀⠀⠀⣨⠞⠉⠀⠀⠀⠀⠀⠀⠉⠳⣕⠀⠀⠀⠀⠀⡀⣪⠖⠉⠀⠀⠀⠀⠀⠀⠈⠓⢬⡢⡀
⠀⢀⢢⠞⠀⠀⣆⠀⣴⠀⠠⣄⢀⡦⠀⠈⢦⢂⠀⠀⢌⡞⠁⠀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠈⢳⡡⠀⠀⡐⡼⠁⠀⣴⢫⣷⠀⠀⣰⣏⢳⡀⠀⠳⡔⡀
⠀⠀⡟⠀⠀⠀⠈⠋⠁⠀⠀⠉⠉⠀⠀⠀⠈⣧⡄⠈⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠃⢠⣼⠁⠀⠀⠻⠾⠟⠀⠀⠸⠿⠾⠃⠀⠀⢻
⠀⠘⡇⠀⠀⠀⢤⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣿⡇⠀⡇⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠀⠀⠀⠀⠀⢀⡴⠶⣄⠀⠀⠀⠀⠀⢸⠇
⠀⠠⢳⡀⠀⠀⠈⠙⠛⠒⠒⠛⠉⠀⠀⠀⣰⠧⠀⠈⡹⡄⠀⠀⠀⠠⠞⠁⠈⠳⠄⠀⠀⠀⢠⢟⠁⠀⠼⣆⠀⠀⠀⠀⠘⢦⣤⠟⠀⠀⠀⠀⢀⡞⠄
⠀⠀⠁⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠍⢓⠶⠶⠤⠴⠶⢖⠻⠁⠀⠀⠀⠀⠀⣀⠀⠐⠩⠓⡲⠶⠤⠤⠶⣖⠛⠉⠂⠀⠀⠸⣿⠇⠀⠀⠨⠝⣲⠶⠦⠤⠶⢖⠚⠉⡀
⠀⠀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠂
⠀⠀⠀⠀⡠⣀⡴⠒⠋⠉⠉⠓⠲⢄⡠⠀⠈⠀⠀⠀⠀⠀⡀⢀⡤⠖⠚⠉⠉⠓⠲⢤⡀⢀⠀⠠⣶⠆⠀⠀⢀⢄⡠⠖⠚⠉⠉⠙⠒⠦⣀⢄
⠀⠀⠠⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡢⡀⠀⠀⢀⢊⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡑⡀⠀⠀⢀⢔⡴⠋⠀⣀⠀⠀⠀⠀⢀⡀⠈⠳⣕⢄
⠀⠰⣱⠃⠀⠀⣤⣤⡄⠀⠀⣤⣤⡄⠀⠀⢳⡴⠀⢀⢣⠏⠀⠀⠛⢦⣀⠀⠀⣀⡴⠋⠀⠀⠹⡜⠀⠀⢠⡞⠀⠀⠀⣉⣷⠆⠀⢾⣏⠁⠀⠀⠘⣎⠄
⠀⣀⡇⠀⠀⠀⠈⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⣷⡇⠀⡟⠀⠀⠀⠀⠘⠋⠀⠀⠙⠃⠀⠀⠀⠀⢻⠀⢰⣾⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡀
⠀⠀⡇⠀⠀⠀⠀⣆⠀⠀⠀⢀⡦⠀⠀⠀⠀⡿⠃⢀⣧⠀⠀⠀⢀⡴⢷⣰⣆⡾⢦⡀⠀⠀⠀⣼⡀⠘⢿⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠀⠀⠀⠀⢸⠁
⠀⠐⠹⣄⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⢀⡼⠁⠀⠀⠘⣆⠀⠀⠈⠀⠘⠁⠈⠃⠀⠁⠀⠀⣰⠃⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠂
⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠀⠰⣿⡇⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠾⠆⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⣀⣤⢞⠁
⠀⠀⠀⠀⠈⠐⠈⠉⠛⠛⠛⠋⠉⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠛⠛⠛⠛⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠉⠙⠛⠛⠛⠉⠉⠂⠁

⠀⠀⡠⠊⠁⠈⠢⡀⠀⢀⠔⠁⠀⠑⢄⠀⠀⡠⠊⠀⠈⠢⡀⠀⢀⠔⠁⠈⡑⠄⠀⠀⡠⢊⠀⢈⠢⡀
⠀⠀⡅⢈⠀⢈⠀⡀⠀⠨⠀⢁⢀⡁⢸⠀⠀⡇⠈⠀⢈⠀⠇⠀⢠⢀⠁⠀⣁⠀⠀⠀⡀⢉⠀⢈⠀⠇
⠀⠀⠈⠢⠅⠥⠊⠀⠀⠀⠑⠤⠠⠔⠁⠀⠀⠈⠪⠌⠅⠊⠀⠀⠀⠑⠅⠠⠑⠁⠀⠀⠈⠢⠄⠤⠊
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⡔⢡⠀⢀⡑⠄⠀⢠⠊⣄⠀⡌⢢⠀⠀⡔⢡⠀⢀⠑⡄⠀⢠⠊⡀⠀⡈⢂⠀⠀⠔⣥⠀⢠⡑⡀
⠀⠀⢂⠀⠄⠄⢀⠄⠀⠐⡀⠠⠠⠈⡱⠀⠀⢇⠀⠀⡀⢀⠇⠀⠰⡀⢄⢠⠄⡀⠀⠀⢄⢤⠀⠠⢀⠃
⠀⠀⠀⠑⠂⠒⠁⠀⠀⠀⠈⠒⠐⠊⠀⠀⠀⠀⠑⠂⠒⠁⠀⠀⠀⠈⠒⠐⠊⠀⠀⠀⠀⠑⠂⠒⠁
⠀⠀⠀⡠⠄⠤⡀⠀⠀⠀⢀⠤⠠⢄⠀⠀⠀⠀⡠⠄⠤⡀⠀⠀⠀⢀⠤⠠⢄⠀⠀⠀⠀⡠⠄⠤⡀
⠀⠀⡌⠠⠀⠠⠌⠂⠀⠐⠁⠄⠀⠄⢱⠀⠀⣎⠐⠀⠐⢈⡆⠀⠰⠁⠄⠀⠄⠁⠀⠀⠊⠠⠀⠤⠈⡄
⠀⠀⠣⡀⠰⠀⡠⠂⠀⠘⢔⠒⠒⢒⠜⠀⠀⠫⡐⠒⠒⡨⠃⠀⠘⢄⠂⠀⢀⠄⠀⠀⠂⠰⣉⡱⡠⠁
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢀⠄⠂⠒⢄⠀⠀⠀⡠⠐⠐⠢⡀⠀⠀⢀⠔⠂⠒⢄⠀⠀⠀⡠⠒⠐⠢⡀⠀⠀⢀⠔⠂⠂⢄
⠀⠀⡇⠒⠀⠐⠂⠁⠀⢀⡠⠁⠀⢡⣜⠀⠀⡇⠘⢀⡈⡀⡆⠀⢘⠀⠂⠀⠂⠐⠀⠀⠁⠚⠀⠘⠀⡆
⠀⠀⠑⢄⠂⢀⠔⠁⠀⠀⠢⡀⠁⡠⠊⠀⠀⠑⢄⠀⢈⠔⠁⠀⠈⠢⡙⠊⡡⠂⠀⠀⠑⢌⠂⢊⠔⠁

⠀⣠⠖⠋⠉⠉⠓⠦⡀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠉⠓⢤⡀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠙⠒⢄⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠙⠲⣄
⡼⠁⢀⣀⠀⢀⣀⡀⠙⡄⠀⠀⠀⡎⠀⢀⡀⠀⢀⡀⠀⢱⠀⠀⠀⢀⠏⠀⣀⡀⠀⢀⡀⠀⢳⠀⠀⠀⢠⠃⠀⣤⡀⠀⣠⡄⠈
⡇⠀⠁⠈⠁⠈⠀⠁⠀⡇⠀⠀⠸⡀⠀⠉⠁⠀⠈⠁⠀⢀⡇⠀⠀⢸⠀⠀⠉⢁⣀⠈⠁⠀⢸⠀⠀⠀⢸⠀⠀⣉⣀⣀⣈⡀
⠹⡄⠈⠑⠒⠒⠋⠀⡰⠃⠀⠀⠀⢣⡀⠈⠓⠒⠚⠁⢀⡜⠀⠀⠀⠈⢧⡀⠘⠉⠉⠉⠂⢀⠞⠀⠀⠀⠘⢆⠀⠳⣄⣀⠴⠁⢠
⠀⠈⠓⠤⠤⠤⠴⠊⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠚⠁

⠀⠀⣠⠤⠴⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠦⠤⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠦⠤⣄
⢠⠋⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⡴⠁⠠⡀⠀⢠⠄⠙⣆⠀⠀⠀⢀⡜⢁⠀⠀⠀⠀⠀⠑
⡏⠀⠰⠗⠀⠐⠿⠀⠀⡇⠀⠀⢰⠃⠀⠦⠖⠀⠲⠴⠂⠈⡆⠀⠀⢸⠁⣀⣉⣁⣀⣉⣁⡀⢸⡀⠀⠀⢸⠀⢈⠽⠂⠀⠺⠅
⢧⠀⠐⠒⠒⠒⠒⠀⢀⠇⠀⠀⠈⣆⠀⢠⠖⠒⠢⡄⠀⣰⠁⠀⠀⠘⡄⠳⡀⠀⠀⠀⣠⠇⡸⠀⠀⠀⠸⡀⠀⡠⠖⠒⠦⡀
⠈⠳⢄⡀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠈⠢⣄⡀⠀⢀⣠⠔⠁⠀⠀⠀⠀⠘⠦⣈⠓⠒⢋⣡⠜⠁⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⡠⠞
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⢀⡴⠊⣡⠀⢠⣉⠳⣄⠀⠀⠀⠀⢀⠔⠋⠀⠀⠈⠉⠢⡀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠈⠙⢢⡀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⠑⢦
⡞⠀⢠⣦⠀⠠⣦⠀⠈⡆⠀⠀⢀⠏⠀⢢⣄⠀⣠⡔⠀⠹⡀⠀⠀⢠⠃⠐⣦⡖⠀⢢⡔⠀⢱⠀⠀⠀⢰⠁⠀⢦⡄⠀⢠⡄
⣇⠀⠀⣀⣠⣀⡃⠀⠀⡇⠀⠀⠸⡄⠀⢉⠀⠀⠀⡉⠀⢀⠇⠀⠀⢸⡀⠈⠁⠉⠀⠁⡈⠀⢸⠁⠀⠀⢸⠀⠀⣉⠀⠀⢀⡁
⠘⢄⠈⠁⠀⠀⠉⢀⡜⠁⠀⠀⠀⠳⣄⠀⠉⠉⠉⠀⣠⠞⠀⠀⠀⠀⠳⡀⠉⠉⠉⠉⠁⣠⠏⠀⠀⠀⠈⢣⡀⠈⠉⠋⠉⠀⡠
⠀⠀⠙⠒⠲⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠒⠊⠁⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠒⠖⠒⠋

⠀⢀⡤⠒⠒⠒⠢⢄⠀⠀⠀⠀⠀⠀⣠⠔⠒⠛⠒⠢⣄⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠦⣀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠒⢤⡀
⣰⠃⠀⠖⠀⠐⠦⠀⠱⡄⠀⠀⠀⡞⠁⠐⢒⣀⡒⠂⠈⢳⠀⠀⠀⠀⡞⠁⠰⠤⠀⠴⠂⠈⢦⠀⠀⠀⢠⠎⠀⢠⡄⠀⣤⠀⠘
⡇⠀⠳⠔⠀⠘⠴⠃⠀⡇⠀⠀⢸⠀⠀⣰⠉⠀⠉⣆⠀⠀⡇⠀⠀⢸⠀⠀⠺⠆⠀⠸⠇⠀⢸⠀⠀⠀⢸⠀⠀⠛⠀⠀⠛⠃
⢳⡀⠈⠲⠤⠤⠊⠀⣠⠃⠀⠀⠀⢧⠀⠹⣞⣉⡳⠏⠀⡼⠀⠀⠀⠈⢆⠀⠰⠊⠉⠓⠄⠀⡼⠀⠀⠀⠘⣄⠀⠉⠉⠉⠉⠁⢀
⠀⠙⠦⣄⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠑⠦⣄⣀⣤⠴⠊⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣠⠤⠊⠀⠀⠀⠀⠀⠈⠲⢤⣀⣀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣷⠟⠙⢮⣗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠞⠁⢠⠔⠢⡙⢷⣵⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⡾⠁⠀⠀⢣⣀⡠⢗⣮⠶⣮⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡽⢋⣠⡤⣀⡠⣔⣭⠞⠋⢀⣀⡈⢧⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⠋⢠⣿⡿⣛⡽⠞⠉⠀⠀⠠⡋⠀⢙⠌⢳⡵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣵⠟⢁⠤⣪⡵⠚⢉⣴⣶⡢⡄⠀⠀⠑⠒⠊⣀⡴⢟⠼
⠀⠀⠀⠀⠀⠀⠀⡖⠒⠛⠚⠓⠚⠓⠛⠓⠒⠒⠛⠛⠚⠓⠚⠒⠒⠒⠒⠛⠛⠚⠓⢲
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢽
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⠼⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡖⠙⠆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣑⣒⣁⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼
⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠖⠒⠚⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢉⣡⣤⠶⠖⢶⣄⣀⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢋⣠⡴⠞⠋⠁⠀⠀⠀⠀⠉⠉⢻⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢉⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣻⣄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢉⣴⠞⠁⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⢉⣉⣉⣡⣤⣤⣌⠉⠉⠉⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣁⡴⠋⠁⢀⣀⣤⡴⠞⠋⣉⣤⡴⠶⠛⠛⠉⠉⠉⠀⠀⠀⠹⢦⣄⡀⢸⣆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢋⣠⠞⠋⣀⣴⠾⠛⢉⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣿⣆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⣁⣴⣛⣥⠶⠛⣉⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢹⡿⣆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⣁⣤⡾⠿⠛⣉⣤⠶⠛⠉⢀⣠⣤⠤⠤⢤⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠘⣇⠀⣧⠹⣆⠘⣧
⣠⣄⣀⣀⣀⣀⣀⣀⣤⣴⣾⡿⠷⠞⠛⣉⣡⡤⠶⠛⠉⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⢀⡼⢋⣤⠶⢤⡉⢳⡄⠀⠀⠀⠀⢻⡀⢹⡶⠛⠀⠘⣧
⢿⡀⠈⠉⠉⣉⣉⣉⣉⣤⣤⡴⠶⠞⠛⠉⠁⠀⠀⠀⠀⣰⠟⠁⠀⣠⠶⠦⠤⣀⠀⠀⠀⠀⠘⣧⣸⠃⣾⠑⢿⡀⢹⡆⣿⠀⠀⠀⠀⠘⣇⠈⣿⠀⣀⣀⣈
⠸⡇⠀⣀⣼⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣴⠛⢀⡀⠀⢸⡆⠀⠀⠀⠀⢸⣿⣇⠹⢦⣭⣥⠞⢠⡟⠀⠀⠀⠀⠀⢿⡀⢹⡋⠉⠉⠉
⠀⢿⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢰⡇⠀⠈⠓⠺⠿⣤⠀⠀⠀⠀⠀⡇⠙⠳⢦⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠸⣇⠘⣇
⠀⠸⡇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣿⠀⠀⠀⠀⠻⣤⣀⣀⠀⠀⠘⣇⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢻⡀
⠀⠀⣿⠀⢻⡀⠀⠀⠀⠀⠀⣠⠞⢋⣉⣉⡙⠳⣄⣷⠀⠀⠀⠀⢠⡞⠋⠙⡇⠀⢠⡟⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠘⣇
⠀⠀⢸⡇⠸⡇⠀⠀⠀⠀⢰⡏⣰⠏⣵⡌⢹⡆⢹⣿⡄⠀⠀⠀⠀⠻⣄⣀⡀⢰⡟⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡿⠀⠀⠀⢿⡀
⠀⠀⠀⢿⠀⢿⠀⠀⠀⠀⠸⣇⠹⣆⡘⢛⣼⠃⣼⠁⢳⣄⠀⠀⠀⠀⠈⠉⠛⠋⠀⢀⣴⠟⠀⠀⠀⢀⣀⣤⠴⠚⠛⠉⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣸⡇
⠀⠀⠀⢸⡆⠸⡇⠀⠀⠀⠀⠘⠷⣬⣉⣉⣥⠾⠁⠀⠀⠙⠷⣤⣀⣀⣀⣀⣀⣤⠶⠋⠁⣀⣤⡴⠞⠋⣉⣤⠴⠖⠛⠋⠉⠁
⠀⠀⠀⠀⣷⠀⢿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣀⣤⠶⠛⢉⣠⡴⠞⠋⠉
⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢉⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠈⣷⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠛⣉⣤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⢹⡄⠀⠈⢻⣄⣀⣠⣤⣤⡤⠴⠶⠛⠋⣉⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠈⠷⢤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠛⠋⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠴⢤⡀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠈⠾⠁⢍⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢦⣾⣴⣋⡀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⣩⡞⠟⠫⠢⡑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠘⠑⠹⠀⢀⣠⡀⠀⠀⢻⡄⣀⡀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⠃⠀⠰⢛⠉⠉⠀⠀⠀⢣⣤⣉⣁⣀⢠⣶⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣬⣥⣥⠆⠀⠀⠀⠄⠀⠡⠤⢈⠉⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠁⠀⠀⡀⢖⠒⠀⠀⡀⠰⡟⠄⢰⡆⢠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⠀⢀⡀⠀⠀⠾⠫⢄⡤⠀⣀⣰⠀⠀⠉⢠⣿⠥⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠛⠛⠻⢶⣤⣀⡀⣉⣤⣾⢿⣿⣶⣤⣠⣿⠃⢀⣉⣠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠁⡶⣃⡀⢀⠀⠀⢈⣹⢿⡿⣿⠆⡀⢀⠈⢉⣿⠗⠒⠘⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⡀⠀⠾⢋⣠⣶⣿⣾⣇⡀⠉⠈⠠⠟⠀⣼⣿⠒⠒⠶⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡿⣶⣶⣿⣿⠿⠋⠙⠿⣿⡿⠶⢄⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠙⠓⠄⠙⣿⠏

⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀
⠀⢀⣴⣶⠞⠭⠰⡖⠨⠝⢶⣶⣄⠀⠀⠀⠀⢰⠶⠽⣯⡀⠶⣀⣿⡧⢶⡄⠀⠀⠀⠀⠀⢀⣾⠟⢫⣽⣿⣍⠻⣷⡀
⠀⠈⠋⢸⠟⠲⢦⣤⠶⢺⢯⠉⠋⠀⠀⠀⠀⠈⠋⢁⣼⡼⣶⠇⣧⣉⠙⠀⠀⠀⠀⠀⣴⡾⠟⢰⣯⣻⣿⣽⡆⠹⢷⣦⡀
⠀⢀⢀⡾⠇⠀⠀⠀⠀⠐⠿⣆⢀⠀⠀⠀⠀⣠⢊⠉⣁⣼⣟⣦⡈⢈⠳⡄⠀⠀⠀⢼⣏⣠⣼⣆⣸⣿⣿⣇⣠⣧⣄⣹⣧
⠀⠈⣮⡄⠀⡀⠀⠀⠀⡀⠀⣼⡞⠀⠀⠀⠀⠀⡋⢰⠏⣠⣿⡀⠻⡆⢹⢀⠀⠀⠀⠸⣯⣾⣿⠋⡋⣿⣿⢙⠉⣿⣷⢿⠇
⠀⠠⢹⣵⣿⣿⣧⢰⣿⣿⣶⣽⡁⠀⠀⠀⠀⠛⠓⢾⡾⠉⡀⠉⢶⠷⠛⠛⠀⠀⠀⠀⠈⠛⠛⠻⢿⣎⣱⡿⠟⠛⠋⠁
⠀⠀⠀⠀⠟⣯⠈⠽⢿⡟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣶⢿⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠃
⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠈⢳⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣽⠇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢸
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢼⣿⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⣷⢾
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣈⣉⣅⣰⡶⡦⠀⠀⠀⠀⠀⢐⣓⣿⣧⣿⣾
⠀⠀⠀⠀⠀⠀⣼⣿⡼⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠨⢍⣩⡹⠋⠻⠶⠖⠀⠀⠀⠀⠀⠸⠇⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠸⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣷⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠀⠀⠉⢿⣿⣿⣿⣿
⠀⠀⠠⣶⣴⣶⣶⣿⡟⣩⡤⠉⢻⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣟⠛⠛⠉⠉⠀⣰⡞⣻⣶⡄⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⢀⣹⣿⣿⣿⣿⡀⠙⣛⡷⢀⣿⣿⣿⣿⣷⡀⠀⠀⢸⣿⣿⡿⡆⠀⠀⠀⠀⣿⣦⠤⢹⡿⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⢳⢻⣿⣿⣿⣿⣦⣬⣵⠿⠛⣋⣉⣉⣉⠁⠀⠀⢸⣿⣿⣿⡜⡄⠀⠀⠀⠈⠛⠛⢋⣠⣤⡶⠶⠶⠶⣿⣿⣿
⠀⠀⠀⠈⢧⡛⠛⠛⠛⣉⣡⣴⣶⠿⠟⠛⠛⠛⠂⠀⠀⢸⣿⣿⣿⣷⡙⣆⣀⣤⡶⠶⠾⠛⢉⣀⣤⣤⣤⣤⣿⣿⣿
⠀⠀⠀⠀⠘⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣄⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⣿⣿⣿⡿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⢈⣩⣤⣶⣶⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⣴⡾⠟⠋⠉⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⣠⣴⠿⠛⠁⠀⠀⠀⢀⣀⣠⣤⣬⣤⣤⣭⣭⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⣩⣴⡿⠛⠁⢀⣀⣤⣶⣶⠿⠟⠛⠋⠉⠉⠉⠙⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢛⣉⣥⣴⣶⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⠿⠿⠟⠛⠉⠁⠀⣠⣤⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⠿⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⢡⡆⣷⣴⣿⣿⣿⣿⡄⠀⠀⠀⠐⣿⣿⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⠘⠃⠙⠛⢿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣶⣿⣶⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⠇⣸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠻⣿⣿⠇⠀⠀⠀⠀⠸⣿⣿⣿⣧⣈⡃⢨⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⣀⣀⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣀⣤⣴⣶⠿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠻⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠷⠾⠿⠿⠿⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠁⠀⠉⠲⣄
⠀⠀⠀⠀⢀⠔⠁⠀⠀⣀⣀⠀⠀⠀⠘⡆
⠀⠀⠀⢠⠋⠀⢀⠀⠀⠀⠀⠙⡄⠀⠀⡇
⠀⠀⢠⠇⠀⢠⠆⠀⠀⠀⠀⢀⡇⠀⢀⡇
⠀⠀⣼⠀⠀⢸⠀⠀⠀⠀⢀⡼⠁⠀⣼⣁
⠀⠀⣿⠀⠀⠘⢧⣄⣠⡴⠛⠀⢀⣼⡁⠈⠙⠢⣄⡀
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠙⠲⢤⣀⠀⠉⠓⠤⣀
⠀⠀⠀⠈⠳⠦⣤⡤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⡀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠌⡜⠀⠀⠀⠀⠀⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢌⠞⠠⢂⢄⠌⡱⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠎⡠⢁⢊⠎⡰⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢼⡱⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡟⠉⣉⣉⣉⡉⠻⣿⡿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣷⡀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡇⠸⠿⠿⠿⠇⢀⣿⡏⢀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣦⣤⣤⣤⣴⣾⣿⣿⣿⠟⠋⣉⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⣿⠿⢿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣦⡀⠙⢿⣿⡿⠋⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿⣦⡀⠉⣠⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠙⠛⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⢀⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠿⠋⣉⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡟⢁⣴⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠟⢀⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠏⣠⣾⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⣿⡿⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⠒⠒⠒⠒⠒⠲⠀⠰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢨⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢎⠀⠁⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠃⠀⡖⠂⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠸⡀⠀⢌⠕⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⠠⠤⠤⠤⠴⠁⠀⠀⠀⠀⠀⠑⠢⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⢒⣒⣒⣒⣐⣒⣒⣒⣒⡒⠒⡆⠀⠀⡠⠖⠐⠑⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠁⠀⠀⡎⢡⠎⢡⠀⠀⠈⡆⠀⠀⢸⠁⠀⢎⡃⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡀⠀⠀⢣⡐⢃⡜⠀⠀⣀⠇⠀⠀⠸⡄⠀⠢⠆⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⠥⠤⠂⠀⠀⠈⠓⠰⠂⠋⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣋⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⡀⡄⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⠿⣛⣛⡛⠛⠛⣛⣛⣛⣛⣛⣛⡫⡉⠱⠀⠉⢸⣿⣿⣿
⠀⠀⠀⠀⣰⠆⠂⠈⠀⠉⠉⣴⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠉⡴⠐⠀⠂⠈⠉⢉⡴⠁⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⣄⡿⠀⠀⠀⡀⠀⠀⢠⡰⣶⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣠⡀⠀⢀⢜⡇⠀⠀⠀⠀⠀⠀⢀⡴⣶⡆⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣝⠝⠀⢰⣘⡧⠀⠀⢀⡷⠀⠀⠘⠙⠋⢁⣀⡤⢞⡤⠀⠀⠀⢸⣿⣿⣿⠀⠀⠨⣍⠜⠀⢨⣸⡇⠀⠀⢠⡇⠀⠀⠈⠛⠉⢀⣠⡴⣆⡆⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣷⣄⡀⣿⡇⠀⢀⣀⣀⣠⣬⣴⠾⠛⠁⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢀⣈⣻⣷⣄⢐⣿⡃⡀⠀⣀⣀⣠⣭⡴⠟⣫⣴⣿⣿⣿
⠀⠀⠀⠀⠶⠔⠒⠂⠘⠉⠉⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠐⠶⠐⣒⣂⣉⣉⣉⣉⣈⣉⣙⣛⣛⣋⣭⣭⣭⣴⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣉⣭⣍⣁⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⡴⠛⠉⠁⠀⠈⠉⠙⠷⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠀⣀⡠⠤⢄⡀⠀⠀⠀⠀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣏⣁⣤⣤⣀⠙⠶⣾⠛⠛⠛⠛⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣤⠿⣿⢯⡀⠀⢙⣧⡀⠈⠳⣶⣲⠖⢲⡖⢦⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠸⣧⣀⣘⣷⣝⡲⠿⣿⠇⠀⠀⣼⣁⡴⣋⡤⠞⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠉⠉⠉⠉⠉⠋⠁⠀⠀⢠⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢠⠎⠁⣰⠳⡄⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⡆⠀⢸⡀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡾⠀⢠⡇⠀⠀⠀⣰⠏⠹⣄⠀⠀⢀⡀⠀⢳⡀⠀⢻⡀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠃⠀⡾⠀⠀⣾⠛⠁⠀⠀⠈⠓⠚⠉⠉⠛⢦⢳⡀⠀⢻⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡏⠀⢰⠇⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢳⡀⠀⢻⡄⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⠁⢀⡟⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢳⡀⠀⢻⡄⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⡏⠀⣼⢁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢱⡄⠀⢳⡄⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⠀⢠⡏⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢻⡄⠀⢻⡄⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⠇⠀⡾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⢻⡼⢧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡾⠀⣸⠇⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⢻⡄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⠃⢀⡟⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⣆⠀⢻⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟⠀⣼⢁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣆⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⣠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⣟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣛⣩⣭⣭⣝⣓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⠀⠀⣿
⠀⠀⠀⣿⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡶⠋⠁⠀⣀⡀⠀⠉⠛⢮⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣱⠋⠀⠀⣴⠛⣫⡙⢳⡄⠀⠀⢻⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡏⠀⠀⠀⢿⡀⠻⢿⣟⠃⠀⠀⠀⢿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡇⠀⠀⠀⣤⣽⣶⡄⠙⡇⠀⠀⠀⣼⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢿⡀⠀⠀⢻⣄⡻⢃⣰⠇⠀⠀⢠⢏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡻⣤⡀⠀⠈⠻⠏⠁⠀⣀⡴⣫⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠿⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣽⣓⡶⠶⠶⢶⣚⣭⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⢂⠔⣈⡔⠊⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⣪⢚⠥⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⡤⠔⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠰⠑⠁⢼⠀⠀⠀⠀⠀⠀⠂⢀⣶⠶⠞⠺⢿⡖⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠤⠄⢀⠃⠀⠀⢸⠀⠀⣤⡤⠴⢖⠛⠙⡇⠁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠁⠀⠀⠈⠰⠠⠆⠼⠶⢯⡇⠀⠀⢸⢸⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠤⠖⢈⣀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠉⠀⠀⡘⠀⠀⠐⠗⠀⠀⢸⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠘⢀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠸⠀⢸⡇⠀⠀⢸⠀⢠⣷⠦⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠠⠀⠠⠀⠀⠀⢠⠰⠀⠀⠃⠀⠀⠀⠀⠐⠀⣹⡇⠀⠀⣸⣿⣻⡿⢶⣾⣯⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠂⠐⠨⠀⠀⠠⢰⡈⠋⠉⠙⢹⠀⢹⡗⠀⢨⣿⣿⠿⣧⣾⣯⣽⣿⡇
⠀⠀⠀⠀⠀⠸⡇⠄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢰⣖⣊⡇⣁⣾⣷⣸⢶⣷⡿⢛⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡇⠀⠈⠉⠛⡟⡟⠛⠊⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠉⠐⠀⠄⡀⡗⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣒⣒⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⡀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠽⠷⠾⠍⠽⠿⠉⠀⠀⠀⢀⡤⠖⢒⠒⠦⡀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣠⠾⠦⠀⠘⡆⠀⠀⣿⢸
⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠉⣉⠉⣉⠉⡉⢹⡄⠀⠀⠀⠀⠀⣄⣇⠀⠨⠭⠽⠀⢠⡇⠀⠀⣿⠸
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠉⠀⠁⠀⠁⢸⡇⠀⠀⠀⠀⠀⣜⢮⡲⣄⣘⣠⠴⣫⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣷⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣼⡇⠀⢰⣶⠀⠀⠈⠳⢭⣗⣒⣒⡋⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡽⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠋⠠⠤⠤⠀⠠⠤⠄⠀⠤⠤⠄⠀⠤⠤⠀⠠⠤⠤⠀⠠⠤⠄⠈⢧⡓
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠒⠒⠀⠐⠒⠂⠀⠐⠒⠂⠀⠒⠒⠀⠐⠒⠒⠀⠐⠒⠂⠀⠒⠒⠂⠙⣆
⠀⠀⠀⠀⠀⠀⢀⣼⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣏⣉⣉⣉⣉⣉⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣷⡄
⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣃⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠠⠤⠄




⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠎⠀⠀⡃⠀⠀⠀⡆⠀⠀⢨⠀⠀⠀⢐⠀⠀⠱⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⣠⣮⣤⣤⣤⣮⣤⣤⣤⣧⣤⣤⣤⣥⣤⣤⣬⣤⣤⣤⣼⣤⣤⣤⣵⣤⣤⣤⣵⣄
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡸⢇⠀⠀⡨⢇⠀⠀⢨⢇⠀⠀⡸⢅⠀⠀⡸⢇⠀⠀⠨⢇⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠀⣠⣱⣪⣥⣤⣵⣪⣥⣤⣭⣨⣤⣬⣅⣬⣤⣤⣵⣊⣅⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⠀⠀⠀⢹⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⡟⠛⠛⠛⢻⡟⠛⠛⠛⢻⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣧⣠⣤⣤⣼⣧⣤⣤⣄⣼⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⡿⠿⠿⠿⢿⡿⠿⠿⠿⢿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⢀⡀⣰⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣆⢀⡀
⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠭⡽⠿⠿⣯⣭⣭⡭⠭⠽⠿⢭⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡧⠮⠤⠀⠘⣓⣒⣒⠃⠀⠠⠤⠭⠗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠠⠤⠤⢄⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠤⡄⢠⠠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠈⠓⠀⠤⠐⠒⠁⠂⠀⠀⠈⡇
⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇
⢰⡀⠁⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⢀⡠⠖⠂⣷⡶⠒⠢⠀⠀⠠⠤⡤⠤⠠⠠⠤⠦⠷⣠⣄⣀⡀
⠸⠶⢼⠦⠀⠂⠀⠀⠃⢀⣿⣿⣷⣶⣾⣶⣶⣶⣶⣶⣶⢖⡀⠀⠀⠀⠀⣩⣀⣂⣐⡒⡋⠉⠉⢉⠿⣿⣟⣛⢿⢿⣉⣩⣾
⠈⡷⡾⡼⠼⠤⠄⠀⠁⢀⣾⣷⣿⣶⣾⠿⢿⡯⢿⣛⣻⠁⠀⠀⠀⠀⣾⣟⣿⣿⣷⡶⡔⣒⣒⡖⡦⠄⠄⠂⣀⣥⠤⠖⠚
⠀⡷⠶⠤⣦⠐⡤⠐⠀⠈⠉⠉⠙⠛⡟⠟⠻⠣⠾⠿⠿⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣶⣶⣿⡷⢈⠁⠈⣄⢰⣊⢩⡄
⠀⣙⠒⡗⡓⢺⣿⣉⡈⣁⡀⠀⠀⠀⢼⣾⢴⡤⡀⠀⠀⠀⠂⢀⣀⢀⢿⣿⣯⣷⣪⣭⣙⠛⠿⠶⣦⡔⠺⡛⣡⠶⢓⡊⠡
⠀⠀⠁⠙⠙⠋⠻⢉⣁⠉⠉⠀⢀⡀⠌⢹⠚⠒⢓⠁⠀⠀⠀⠹⠀⠀⠀⠀⠀⠉⠉⠛⠿⢶⣦⣶⡾⠚⡫⠭⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠑⠲⠚⠖⠐⠐⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⣉⣉⠉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡴⠚⠋⠉⠉⠉⠉⠙⠲⢄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣉⣉⣉⣉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⠘⣷⣨⣍⣁⣒⠻⢭⠩⠭⢛⠲⣶⣶⣄⠻⣆⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⡴⣪⡶⠛⣧⡈⣷⡆⠀⠉⠻⡆⢻⣶⢄⠑⢌⢿⣾⠇⠙⡆⠀⠨⣳
⠀⠀⠀⠀⠀⢂⠼⠉⠀⠀⠈⠉⠡⠤⠀⠐⠀⡄⣼⡯⡀⠀⠀⠈⢫⠀⠈⠁⠀⠠⢸⡄
⠀⠀⠀⠀⢸⠸⣅⠀⠀⠀⣀⣈⣐⣈⠏⢀⠜⣰⣟⠥⡂⠄⠀⡆⣿⡴⠃⠀⠐⠀⠀⢱⣄
⠀⠀⠀⠀⠈⡆⠀⠉⢁⣲⡬⢵⣂⡠⢐⡡⣾⠞⡣⠑⠂⢀⡰⣱⠁⠐⠀⠀⠄⡠⣠⡳⢦⣍⡒⠤⡀
⠀⠀⠀⢠⡊⢀⣾⢣⢶⢌⣤⣴⣶⣾⢷⣿⣫⠵⠬⠒⠊⢩⡔⣇⠂⣁⣁⣁⣪⣜⠉⠛⠶⡬⣛⠷⣮⣑⠦⣀
⠀⠀⠀⢈⢏⢾⣮⢧⢣⣿⠃⠀⠈⠻⠿⣶⣖⣦⢶⡜⡿⡔⣱⡸⠛⠛⠋⠉⠉⠙⠳⢶⢗⡮⣙⣷⣦⣍⡛⢶⣭⣒⠤⡀
⠀⠀⠀⢃⣾⣿⢣⢣⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⡎⢷⠐⡹⡽⣷⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⣗⠉⣛⢷⣮⣙⠳⣮⣕⡢⣄
⠀⠀⢰⣿⣾⢣⢣⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⠱⢱⡹⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠛⠉⠿⡻⣦⣿⣿⣗⣷
⠀⣔⣽⢯⢣⣳⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣷⣧⢻⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⠋
⡴⣿⢯⡷⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⢿⡆⢷⡱⡀
⢻⣳⡿⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣎⣿⡘⣧⣧
⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡷⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠁
⠀⣀⡀⠀⠀⡀⢀⠀⡀⢀⠄⡀⠀⢀⠠⠀⢄⠀⡢⠐⠖⠠⠀⠔⠠⠂⢄⠠⢀⢄⠠⡀⢄⠄⡠⡀⠔⡠⢀⢄⡠⣀⠄⡀⡀⢀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢉⣻⣭⠙⠢⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢟⠸⠦⠬⣽⡬⠤⠞⠘⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠖⡛⣾⡓⠲⣄⠀⠀⠀⠨⣈⠛⠲⠶⠦⠀⣒⣘⡇
⠀⠀⠀⠀⠀⠀⠆⠣⣄⣙⣧⣁⡤⠞⢰⠀⠀⠸⢈⣉⡛⠛⠛⢛⢉⡠⡆
⠀⠀⠀⠀⠀⠀⢼⠲⠤⠤⣤⣤⠴⠖⢐⠂⣀⣀⣤⣌⣉⣉⣉⣉⣡⣴⠡
⠀⠀⠀⠀⠀⠀⢉⠘⠲⠶⠦⠤⠴⠒⡋⡎⠁⢦⡿⠆⠈⣫⢩⣉⣡⣤⣆⣀⣀
⠀⠀⠀⠀⠀⠀⢘⢉⡓⠒⠶⠶⠖⠛⢃⣈⠓⠒⠛⠒⠊⣁⣜⢽⠉⢡⣽⠶⠉⢙⡔
⠀⠀⠀⠀⠀⠀⢰⣌⣉⢓⣶⣶⣶⣯⣭⣨⣉⠛⠛⠛⠛⣉⡨⣈⠓⠒⠿⠗⠒⢋⣨⢣
⠀⠀⠀⠀⠀⠀⠀⠤⡅⣏⠀⢤⣟⠃⠀⡱⣩⣉⣉⣉⣉⣥⠺⡩⣉⠚⠓⠚⠛⠛⣁⡌⠀⠀⠀⠀⠀⠀⠀⡀⢠⠖⢲⠄
⠀⠀⠀⠀⠀⠀⠰⠸⣦⣌⣉⡓⠛⢚⣉⣠⣎⣭⣉⣉⣤⣤⡾⣦⣈⣉⡉⠉⣉⣉⢅⣇⣀⠠⡤⠤⠐⠂⠉⠀⣀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⣷⡿⠿⡿⠭⢽⡏⠉⠉⠉⠉⢩⣥⣤⠔⠺⠋⠛⠛⠛⠛⠉⠉⠍⡀⣀⠀⠐⠀⠀⠠⣞⣯⠵⠛⠀⠀⠆
⠀⠀⠀⠀⠀⢀⡔⣇⡀⢷⣏⠃⢀⡽⣵⡶⠦⠖⢊⣥⣴⣶⣶⠶⠀⠀⠐⠈⢀⢰⡄⠉⠁⠀⠐⠁⠀⠀⢀⣀⣠⡄⠾⠒⠚⠁
⠀⠀⠀⠀⠀⠸⠠⢤⣭⣽⣿⣛⣻⣏⣀⣀⣀⣀⡀⠉⠉⠀⣀⣀⣀⣀⣀⣀⣀⣡⠤⠤⠄⠶⠒⠒⠊⠻⠷⠯⠯⠟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠓⠤⣤⣤⣠⣤⠤⠪⣿⣦⣶⣯⣭⣭⣭⣭⣭⣥⣤⣤⣤⣭⣿⣿⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣀⣸⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠶⠚⢛⣡⠶⠞⣛⣉⣉⣛⡛⠶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⠋⠁⠀⣴⠟⣡⠾⠛⠉⢉⣩⣭⣝⣷⣯⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣼⠃⠀⠀⣼⢃⡾⠁⠀⠀⠀⣾⣁⣴⠟⣷⠈⢻⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣿⠀⠀⢸⡇⣼⠁⠀⠀⠀⠀⠻⣥⣤⣴⠏⠀⠀⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣌⢷⣄⠘⣧⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣦⣙⠳⠘⣧⡙⢧⣀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣵⠟⢁⣽⠗⠂⣼⠻⣦⣍⡛⠶⠶⠶⠶⠞⣋⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣵⠟⠁⠀⡟⠁⠀⣼⠃⢠⡟⣽⠛⢻⡖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣴⢟⣴⠟⠁⢰⡟⠛⠃⠀⣰⠏⢠⡟⣼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⢋⣵⠟⡁⠀⣾⠛⠃⠀⠀⣰⠏⢠⡟⣸⠃⣀⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⠁⠀⣿⠙⠉⠀⠀⠀⣰⠏⢠⡟⣸⠇⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⢰⡟⢀⡟⣰⠏⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⣱⠏⠐⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣏⣤⡴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⣿⢿⠿⡿⠿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣛⣻⣿⣛⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⠶⠿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠒⠶⠶⠿⣼⣧⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣽⣷⣤⣄⢀⢄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⠉⠉⠉⠉⠁⠀⠀⣷⠢⠎⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢷⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢬⡆⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢏⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⣾⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⡇⠀⢠⠂⠀⠀⠀⠀⠀⠀⡏⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢐⠄⡸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣇⠀⠀⠀⡔⠀⠀⠀⠀⠀⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢜⠄⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠊⡀⠀⠀⠀⠀⠀⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⢎⠎⣽
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠞⣸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⡐⢈⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣧⠀⠀⠀⠀⠀⠀⢀⡾⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠊⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣬⠦⣄⣀⣀⣠⡴⡟⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠴⢥⣾⠵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⢪⢺⡆
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⠆⢀⠔⠀⠀⢀⠔⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡱⢣⢾⡇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠐⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⡰⢣⠞⡇
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⡡⠃⣧
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡱⠃⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠱⢁⢺⡇
⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠳⠦⢼⠇
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠻⠤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⠀⠀⠀⣤⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣄
⣿⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣋⣩⣤⣶⠶⢶⣦⣬⣍⡛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠞⠋⠁⠀⠀⣀⡀⠀⠀⠉⠛⢷⣌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣰⡟⠁⠀⠀⢀⣤⣾⠟⠿⣦⣄⠀⠀⠀⠙⢷⡌⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣸⡏⠀⠀⠀⠀⣿⠁⢠⣾⣆⠀⣻⡆⠀⠀⠀⠈⢿⡌⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣿⠀⠀⠀⠀⠀⢿⡄⠈⠻⢿⣟⡛⠁⠀⠀⠀⠀⠸⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⠀⠀⠀⠀⠀⣈⣛⣷⣤⡀⠈⢻⡆⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⣿⡀⠀⠀⠀⠀⣿⠉⢹⣿⣇⠀⢸⡇⠀⠀⠀⠀⣸⡇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⣷⡀⠀⠀⠀⠹⣧⣌⡛⣁⣤⡾⠃⠀⠀⠀⣠⡿⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⢿⣄⡀⠀⠀⠀⠹⠿⠿⠁⠀⠀⠀⢀⣴⢟⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣍⡻⢶⣤⣄⣀⣀⣀⣀⣤⣴⠾⣋⣵⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿
⣿⠀⠀⠀⠉⠉⠙⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣭⣭⣭⣭⣭⣥⡶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠉⠉
⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣰⠛⢦⣄⡀
⠀⠀⠀⠀⠀⢰⠇⡼⠓⠺⢍⠓⠦⣤⣀⣀
⠀⠀⠀⠀⠀⣾⢰⠇⢸⣷⠀⣳⡀⠀⠂⠌⠩⣛⠓⠶⠶⠤⢤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⢠⡇⠈⢤⡛⠁⣠⠏⡡⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠒⠂⠊⠚⠉⠭⢝⣓⠲⢦⢤⣄⣀⡀
⠀⠀⠀⠀⣼⠃⢰⠀⠙⠛⠛⠁⠀⠠⠀⡄⠀⠀⢀⠉⠀⣀⡀⠀⠐⠆⠀⠰⠄⠀⠀⠉⠉⠒⠢⣬⣍⣻⣶⣤⣀
⠀⠀⠀⢰⠇⠀⣼⠀⠀⠀⠀⠀⠨⢱⠀⠀⠀⠀⢈⡴⢚⠋⠉⠻⡻⢦⡒⠠⠀⠀⠀⠀⠀⢠⣼⠋⠀⣤⡌⢧⠉⡷
⠀⠀⠀⣾⠁⡖⠛⠲⣄⠀⠠⢠⠁⠀⠀⠀⠀⢠⠋⣄⠀⠀⠄⠀⡈⠘⡅⠀⠀⠖⠲⠀⠄⠀⣺⡀⠰⠾⠀⢸⢠⠇⠀⠀⠀⢀⣤⠞⠻⡆
⠀⠀⠀⣟⠀⠇⠰⡇⣸⠀⠀⠀⠀⠀⠀⠀⣴⠃⢀⢀⠁⢀⠀⠔⠃⢰⠁⠀⠀⠀⠀⠀⠑⠀⠈⠛⠲⠶⠾⠋⡞⠀⣀⡴⢚⡽⠛⠛⢦⡸⡄
⠀⠀⠀⠈⠙⠾⣶⣖⡫⠀⢀⡀⠀⡠⠮⠽⠸⣒⣉⣠⡀⠀⠷⡀⢠⢋⣦⠀⠀⠀⠁⠀⠊⠀⠀⠀⠀⠀⠎⣰⠷⢛⠥⢶⡟⠰⢿⠀⠀⡇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⢥⣥⣠⣀⣀⡀⣤⠀⠀⣉⣉⠁⠛⠚⠷⠢⠀⠒⠂⠀⠀⠀⠀⠀⣄⣼⢠⡟⠉⠀⠀⢠⣧⣀⠉⢁⡼⠃⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠓⠓⠶⠶⢯⣭⣥⣌⣒⠆⠠⢄⡀⢠⠎⣩⢻⡇⢸⠇⠀⠀⠀⢀⠈⠻⠛⠁⠀⠀⢵⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠶⠒⡋⠩⠀⠈⠙⠙⠓⠶⣮⣝⣆⣛⡼⢂⡿⠀⢠⡄⠂⠈⠀⢢⠀⠀⠀⠀⢾⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⢫⡩⠇⠒⠦⣌⡈⠀⠀⠀⠀⠀⠀⠈⡁⠀⣠⢞⠛⠷⣤⣾⢇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⢫⡉⢳⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡄⡆⠲⣾⠄⢸⡇⠆⠀⠀⢀⠀⠀⠀⠁⢸⠃⠆⠀⠠⠀⢻⡀⢣⠀⠀⠀⠀⠀⡀⠀⠀⠀⢘⣌⣩⣮⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠖⠶⠦⠼⣧⡙⠦⣚⣃⣼⠿⠀⠀⠀⠠⡀⠀⠀⠀⢸⠐⠦⠠⠀⢀⣸⠀⠐⣏⢴⣂⣠⡈⠀⡀⠄⣂⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢴⡶⣏⠸⠉⣉⡉⠓⣌⣷⠀⠠⣀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⢘⡄⠀⡄⠢⡀⠈⣓⠦⠛⠘⣉⠤⣈⣨⠶⡟⠉
⠀⠀⠀⠀⠀⠀⡠⠻⢸⡁⣿⠸⡄⢹⣟⠀⣽⠹⣧⠀⠱⡆⠀⠀⠀⠒⠈⠀⠀⠀⠀⡲⣍⡮⠀⠐⠉⣁⡠⣄⣲⡭⠶⠛⠉⢱⠀⣧
⠀⠀⠀⠀⠀⢸⢡⠀⢸⡅⢿⡇⠈⠾⠥⡴⠿⠁⠸⣇⠀⡍⠉⣬⠙⣆⠀⠀⠀⠀⣲⠉⠁⣀⣐⣦⡽⠶⠛⠉⣁⠀⠀⠀⠀⣸⡆⢹
⠀⠀⠀⠀⠀⡌⠸⠂⠘⡇⠈⣷⠀⢢⠄⠀⠀⠀⠀⠹⣆⠘⠦⢘⢣⡜⣀⣀⣬⡥⠶⢶⠟⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢇⠸⡇
⠀⠀⠀⠀⠀⡇⢸⡇⠀⣷⠀⣻⡄⠘⣇⠀⠀⠀⠠⠠⠜⠶⠒⠛⢛⡛⠉⠁⢉⠀⢤⡆⠀⢸⢂⡤⠀⠀⡠⠄⠀⠀⠀⢸⠸⠧⡸⠀⣇
⠀⠀⠀⠀⠀⡇⠸⠇⠀⣿⠀⢸⡇⠠⡃⢡⡌⢢⠀⠀⠀⠀⠀⢀⠪⠙⡖⢁⣈⣄⠀⠩⠶⠼⠔⠷⠓⢲⡀⡠⠄⢀⣒⣩⣷⣾⠷⠚⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⣾⢿⠀⠳⢀⣛⣼⢃⠀⠀⢄⠠⠍⠉⠉⠀⠀⠤⠄⢀⣠⣤⣀⣖⣦⣭⠶⠶⠗⠛⡟⣩⠙⡇⢿
⠀⠀⠀⠀⠀⡇⢰⠀⠀⢹⡄⠙⢮⢛⡷⢛⠻⠗⠻⠖⠖⠾⡶⠶⠿⠿⠿⠛⢛⡛⡩⡩⠉⠥⠀⠂⡶⠠⠀⠴⢔⢧⣝⡵⠃⣾
⠀⠀⠀⠀⠀⢇⠸⠀⣰⠎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉⣉⣉⣉⣉⣛⣛⣛⣛⣏
⠀⠀⠀⠀⠀⠸⢠⡾⠫⠍⠤⠤⢶⠶⠶⠀⠀⠀⠀⠐⠀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⡉⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠙⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⠃⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠉⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⣠⣌⣿⠁⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠁⢀⣀⠁⠐⢲⡝⣚⡅⠀⠀⠀⠀⡀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⠃⣿⠁⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠐⢶⡶⠶⠶⢠⣉⢄⠰⣄⢠⡆⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢄⠀⢸⢀⠄⢁⠄⠀⢀⠈⢛⡀⠀⠈⠋⠀⠀⠂⠀⠁⠀⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣀⠧⠐⠁⠀⠀⠀⠀⠀⡀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⡂⠀⠁⠀⠄⠀⠠⠀⠀⢀⠈⡰⢡⠠⠄⠄⣰⠀⠀⠉⠀⢰⡄⠀⢹⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡄⠱⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠁⠀⠨⠁⠰⠐⠀⢐⣀⡀⠸⠇⠀⢸⡇
⠀⠀⠀⠀⢀⠀⠀⠀⠁⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠈⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⢤⢤⠀⣀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠐⠀⠀⢀⠄⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠹⠂⡆⠀⠈⠇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣀⢀⣤⣄⡀⠀⠀⡀⣠⣥⠴⠼⡅⡠⠀⡈⢀⢁⡀⢀⠸⠀⠠⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠉⠈⠀⠀⠙⡁⠀⠀⠐⡉⠁⠡⢁⠌⡴⡽⠡⣻⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠢⠐⠄⠶⠦⠄⠶⠷⠴⢶⣤⣤⣵⣶⣤⣾⡾⡿⣿⣿⢁⡈⡉⡛⡦
⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⢒⠖⡒⠲⡒⠀⠋⠀⠀⠀⠀⠀⠈⠉⠙⠺⠿⣿⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠴⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠎⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣉⣀⣀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⠭⢥⢄⡀⠀⠀⠉⠉⠙⠶⣄⠀⠠⡀
⠀⠀⣠⢈⣀⡴⠉⢀⡯⠶⠒⠾⣶⢾⣆⠀⠀⠀⠠⠄⠈⠳⣄⠈⢄
⠀⣬⠇⠀⡟⠀⠀⠈⢄⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠄
⢨⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰
⠀⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠃⢀⠛
⠀⠸⣏⠦⡀⢁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠁⠀⠀⠀⢆⠂
⠀⠀⠜⢦⣙⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣄⡀⠀⢶⢄⠀⠑⠄
⠀⠀⠀⠈⠛⠛⢷⡦⣭⣩⣒⣒⣒⣒⣦⠀⠻⣷⡀⠑⢿⢢⡈⠳⡕⢄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢋⢿⣳⣤⣿⣷⣄⠀⠑⢌⢢⡈⠣⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⠩⠻⣾⣷⣄⠂⠙⢽⣦⣌⢮⣧⣄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣶⣶⠁⠤⠤⡄⠿⠷⠦⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣄⠀⠙⢟⡷⡘⣾⡢⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣓⣴⣶⣝⢜⣮⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠑⢝⢦⡑⢟⢄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡀⠀⠳⣽⣌⠳⣽⢦⡀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣤⣼⣿⣷⡌⠷⢽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⣉⡭⠷⡄⠀⠀⠀⠀⢠⡏⠗⡦⣄⣀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣉⡡⠴⢚⡉⠀⠀⠀⢹⠀⠀⠀⠀⡼⠛⠲⢤⣈⠈⠟⢲⢤⣀⡀⠈⠙⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣉⡤⠖⢋⡥⠖⣚⣨⠵⠚⠛⣆⢸⠀⠀⠀⢠⠇⢀⠀⠀⠈⠙⠓⠦⣌⡁⠿⢳⠦⣄⣀⠀⠉⢣
⠀⠀⠀⠀⠀⠀⠀⡞⢀⡜⠁⢀⣠⠤⠒⠋⠁⠀⠀⠀⠀⣿⢸⠀⠀⠀⢾⡀⠈⠁⠲⠄⣀⠀⠀⠀⠉⠛⠲⢤⣁⠛⠹⠖⢾
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠉⠓⠦⢄⣀⠈⠁⠲⠄⣀⠀⠀⠀⠈⠉⠓⠦⣼
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⡇⠀⠀⣠⠖⢋⣉⣉⠙⠲⠄⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣨⠀⡀⢠⠞⣡⠚⣉⣤⣍⠙⠂⠀⢺⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⡏⡼⠁⡴⠁⡠⠄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡋
⠀⠀⠀⠀⠀⠀⠀⣧⣽⠘⡇⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⣚⣩⠷⠒⣿⡉⣿
⠀⠀⠀⠀⠀⠀⠀⡿⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⣀⣠⠤⠖⠚⣉⣡⠤⠖⠋⠁⠀⢀⣀⣻⡧⠿⢤
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⢀⣠⠴⢺⡇⢸⢸⠀⠀⣴⣋⣁⣠⣤⣖⣋⣭⡤⠤⠴⠖⠚⠋⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡆⡇⠀⢀⣠⢴⢚⣭⠰⣼⠀⡇⢸⢸⠀⠀⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⢚⣹
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⣿⣾⢉⣿⣯⡇⢹⢸⡇⢸⢸⠀⠀⡗⠶⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⢹
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⣿⣿⠉⠸⠇⢃⣨⠼⠃⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⢺
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⡏⠋⣀⠴⠚⠉⠀⠀⠀⣼⢸⠀⠀⡷⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⢾
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⠛⠉⠁⠀⠀⠀⠀⣠⠞⠁⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃⡇⠀⠀⠀⢀⣠⠤⠖⠋⠀⢀⡴⠋⠀⠀⣟⠒⠒⠚⠉⠉⠀⠀⠀⠀⣀⣀⣠⠤⠴⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠙⠒⠚⠋⠉⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠙⠶⠶⠒⠒⠒⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠾⣇⡀⣀⣀⡤⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠋⠉⠙⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⢤⣀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⠀⠀⠀⠀⠈⠉⠲⠤⣀
⠀⠀⠀⠀⠄⠂⠈⠀⠀⢎⡀⠤⣈⠓⢦⣄⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠙⠲⢬⣝⠲⢼⡟⢲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠛⢦
⠀⠀⠀⠀⠈⠓⢤⣀⠀⡄⠈⠀⠀⣨⡙⠲⣤⣀⠻⢿⠷⣦⣀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠉⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠈⡜⠀⢰⠀⠀⠀⠀⠀⠀⠈⠙⠲⣶⣿⡟⣿⣦⣤⡴⠟⢋⡠⠔⠚⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠐⣒⣒⡒⠒⠉⠉⠉⠁⠒⢂⡹⠋⢿⣷⣋⠡⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⢼⡁⠀⠀⠀⢨⣬⣭⣭⠥⠴⣶⣖⣲⣿⠿⣋⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠓⠄⡈⠓⠦⣜⡛⠛⠛⠉⣉⠉⠝⠓⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⣉⡙⠲⡤⠤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡴⠶⢶⣶⣾⡧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣩⣿⣿⣥⣶⠶⠶⠾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⡶⢧⡀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣯⣶⣶⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⡾⣷⣦⣞⣱⠃⠈⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣿⣟⡿⣿⣟⣿⣿⣶⣇⣀⡟
⠀⠀⠀⠀⠀⠀⢠⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⢻⣿⣿⣿⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡛⣾⣿⣳⢿⣿⡄
⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⣸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣯⣿⣿⣿⣿⣿⡪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⡟⢃⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣻⢡⣿⣗⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣇
⠀⠀⠀⠀⠀⠉⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣛⣛⣉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢿⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠙⠻⣿⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠟⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠁


⠀⠠⠂

⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠳⢶⣤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⡟⠁⢀⣠⣒⡒⠢⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠳⣦⣀
⠀⢀⣿⠄⠀⣾⠛⠉⠙⣧⢸⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠷⣄
⠀⢸⣿⢠⠀⢻⣄⠀⢀⣿⠟⠀⠀⢠⡟⠉⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠙⢷⡀
⠀⠸⣿⣤⠃⠀⠉⠙⠉⠁⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢸⣷
⠀⠀⣿⡟⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡄⣿
⠀⢠⡿⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠘⣰⡟
⢀⣾⠁⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢀⣰⡿
⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠀⠀⠀⠀⢸⣿⠁
⣿⡀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⡁⣮⣷⠄⠀⠀⠀⠸⣿⣆
⢻⣧⠀⠀⠀⠀⠙⠤⢄⣀⡀⠀⠀⠐⠈⠀⠀⠀⠐⠘⣏⠢⡱⡄⠀⠀⠈⠛⠶⠶⣦⡀
⠈⢷⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢣⡑⢌⢢⠀⠀⠀⠀⠀⢾⣷
⠀⠈⠛⠷⣵⣦⣤⡤⠀⢀⣀⣈⣡⣌⣉⡲⣄⠀⠰⡉⠀⠑⡙⢄⠢⠑⠀⠀⠀⣇⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠿⣿⣧⠀⠙⢄⠀⠘⢎⠳⣄⢌⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠈⠢⡀⠀⠣⡙⢦⡡⡑⠄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣦⡑⢄⠀⠀⢄⠳⣌⢈⠢⡀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣎⣀⠤⠤⡑⣜⢦⡑⠑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣷⠈⠣⡱⣌⠨⠢⡀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠤⢀⠈⢮⠳⡄⢄⢄⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣦⡑⡄⠑⢝⢦⡣⡠⡀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠠⡱⣌⢎⠠⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣄⡀⣀⠈⢌⢣⡑⡑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⣬⣅⠀⠱⡝⢮⡢⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣬⣉⡋⢪⡳⣔⢄⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣇⣀⣱⣌⢷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣶⣾⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠀⠈⠑⠒⠒⠤⠭⠭⠭⣉⣛⣋⣉⣉⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡜⠀⠐⣶⠾⠤⢤⣀⣠⣤⣤⡀⠀⠀⣤⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠼⠏⢀⠀⠙⠃⠹⠛⠛⠒⠺⣿⠯⠁⢠⢫⠇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣶⣛⣡⠥⠤⠤⠤⣤⣤⣀⡀⠀⠀⠀⠈⠁⢀⢤⢮⠎⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⠭⠔⣒⠀⠀⠀⠰⣄⢠⣀⠈⠉⠉⠉⠙⠓⠒⠦⠌⠉⠓⢦⠀⠄⠔⣵⡵⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠟⠳⠶⠦⢤⣤⣤⣤⡤⠶⠖⠋⣐⣒⣭⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⠏⠀⠀⣐⣶⠾⣅⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣈⠥⠒⢋⣡⠴⠶⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⠏⠀⠀⠀⠈⢿⣭⣶⡶⠿⣙⡛⢿⡦⠀⢀⡴⠋⢀⡴⠚⣉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⠒⠦⠤⢤⣄⣀⠀⠀⠀⠀⠼⠋⠓⢠⠋⢀⣴⢋⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⢭⣙⡓⠦⢤⡀⠀⢠⠃⢀⡞⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⠀⡾⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠤⠧⠿⠽⠤⠤⣿⢸⠷⠶⠶⠶⠶⠶⠶⠶⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⠀⠀⠀⣿⢸⠩⣭⣭⣭⡍⠉⠉⠉⠉⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⢾⣉⠓⠀⠀⣿⠸⣬⣤⣧⣥⣤⣤⣤⣤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢤⣈⡷⠀⠀⣿⠀⠀⠀⠀⣠⡶⢚⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⡯⠀⢿⣗⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣛⣛⣻⣿⣟⣛⣛⣿⠀⠀⠀⠀⠻⣤⣙⣋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⡄⠀⠀⣨⣭⣭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡞⠉⢉⡿⣯⡿⡏⠉⠙⣦⡇⠀⡾⠉⣴⠦⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣇⣀⣈⣧⣻⣣⣏⣀⣠⡿⡇⠀⣧⠀⢮⠗⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠾⠿⠿⠿⠿⠿⠿⠿⠿⠶⠇⠀⠈⠛⠺⠞⠋⠁





⠀⠀⠀⠀⠀⠀⠀⠀⣏⡷⣰⣷⠘⣶⠃⣷⣴⡇⣿⠍⢸⢦⡆⢹⡏⠀⡷⣤⣷⣺⠭⠈⡟⢸⡦⣼⣴⠋⣦⣾⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠈⠁⠉⠀⠁⠁⠁⠉⠉⠈⠀⠁⠈⠁⠀⠁⠁⠉⠈⠉⠀⠁⠈⠀⠈⠈⠉⠁⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠻⣄⣀⠟⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⣠⠴⠖⠚⠛⠋⠙⠛⠓⠲⠦⣄⠀⠈⠓⠒⠋
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠻⠭⣉⠙⠒⠒⠶⠤⣤⣄⡀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢀⣠⣤⠤⠴⠒⠒⠋⣉⠭⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄⡀⠀⠀⣹⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⣏⠀⠀⢀⣠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠁⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⡀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⡸⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢇⠀⠀⠈⠙⠲⠦⣄⡀
⠀⠀⠀⠀⢀⣠⣴⣚⣉⠤⠤⠴⠒⠚⠋⠉⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠉⠙⠒⠒⠢⠤⠤⣉⣓⣦⣄⡀
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⡀⠀⠉⠓⠲⠤⢤⡤⢤⡤⠤⠖⠚⠉⠀⢀⡴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡴⠃⠀⠀⠀⠀⠀⠀⣼⠉⠉⣧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⣋⠬⠭⠬⣍⡛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⠕⠉⠀⠀⠀⠀⠀⠙⢷⣬⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢧⣤⣤⣤⣄⣀⡠⠤⢒⣚⣻⡶⢯⣅⡐⠊⠩⣒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⠍⠚⡇⢸⣥⠈⣿⠉⠛⢝⡻⣯⣐⠲⢦⣌⡐⠬⡓⢤⡀⠁⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠔⡁⢄⠳⡹⣮⣿⣤⣿⡤⢄⡀⠉⣪⢲⣙⢳⣬⡙⢦⡈⢪⢳⣄⠀⢣⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠤⣈⠳⠁⠀⠘⢿⣽⣿⣿⣦⠱⠀⠤⠧⢹⡧⡹⣍⢌⢳⡄⠑⢻⡄⠀⡌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⢐⠒⠠⠀⠀⠀⠠⡸⣇⣿⣿⣿⢀⢄⡒⠂⡀⡇⠠⢹⡄⢆⢿⠀⡇⣷⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠄⠀⠀⠀⠀⠀⠑⠬⠽⠟⠕⠁⡲⢌⠀⡇⡿⠘⢸⡇⠈⢸⡇⠀⣿⢀⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢬⡑⠀⢡⡇⠀⢸⡇⢠⢹⣥⢃⣟⡋⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠀⠀⢈⣢⡀⠀⠀⠀⠀⢄⠣⡱⠌⠂⢠⢃⡞⢀⡌⡾⠀⣌⡾⢣⡞⠣⡙⠸⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡞⠃⣠⢤⡖⢁⡜⠈⠢⡀⢠⢌⠢⡣⠑⢀⠔⣡⠊⡠⢊⡾⢁⢜⡞⣱⣿⡄⠀⣽⡄⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡴⠡⠊⣰⣿⠂⠀⠀⠧⢤⠤⣄⣐⣪⠔⣋⠔⣊⣴⡋⢔⣥⡞⣹⣧⡹⣧⡀⠘⢿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢊⠜⣡⠟⡉⣻⣷⣾⢛⠟⠫⣿⠋⢡⣈⣥⠲⣿⠸⠠⣾⣿⣿⠇⠸⡈⢿⣌⢳⣄⠀⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⡵⢃⡾⠃⢐⡼⢃⣿⠇⠋⠀⠸⠇⠂⣸⣿⣟⢸⡿⠸⠀⣿⣿⠿⠀⠀⢇⠀⠹⣿⡮⣧⡀⠘⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡡⢊⡴⠋⢀⡼⠋⠀⣌⡗⡘⠁⢰⣟⢘⣶⣾⢿⠀⢼⡏⢰⠚⢿⡱⠆⡆⠀⠘⡄⠀⠈⢿⣕⢅⠀⠀⠻⠶⣄
⠀⠀⠀⠀⠀⠀⣀⠼⣡⠞⣠⠏⠀⣰⠏⠀⠀⣸⡿⢡⠁⠀⡾⠋⠁⣼⡏⣽⠀⣸⡏⡃⠀⠀⣩⢀⠰⠀⢀⢱⠀⠀⠀⠹⣾⣺⣦⠀⠑⣜⣇
⠀⠀⠀⠀⠀⠸⢃⡜⣡⡞⠁⢀⠞⠁⠀⠀⢠⣟⠇⠄⠀⣸⣷⠀⠀⣿⡇⡟⠀⢿⡷⠁⠀⠀⠻⣟⠀⠆⠀⣺⡆⠀⠀⠀⠘⢿⣷⡱⡄⠈⢆⢻
⠀⠀⢀⣴⠖⡡⢀⢴⠉⢀⡴⠁⠀⠀⠀⠀⣾⠏⡐⠀⢰⡟⠁⠀⠀⣿⡇⡇⠀⣿⠁⠀⠀⠀⠀⣿⡸⠸⠀⠉⢥⠀⠀⠀⠀⠈⠻⣶⠿⣆⠀⢻⡶⢤⡀
⠀⠀⣷⢣⠊⡠⡿⠀⡠⠋⠀⠀⠀⠀⠀⢸⡯⠀⠀⠀⣿⡇⠀⠀⠀⣿⠃⡇⠀⣻⡇⠀⠀⠀⠀⠲⡆⠇⠃⠀⠪⡆⠀⠀⠀⠀⠀⠙⢷⡝⣧⡈⠿⣈⢻⡆
⠀⠀⢸⠁⣴⣗⣱⠎⠀⠀⠀⠀⠀⠀⢀⣿⢃⠁⠀⣼⣟⠀⠀⠀⠀⣿⠇⡇⠀⡿⠀⠀⠀⠀⠀⠰⣿⡀⠸⠀⠁⢷⠀⠀⠀⠀⠀⠀⠀⠻⣎⠟⢦⡙⣶⠁
⠀⠀⠘⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣮⡇⠀⠀⣰⠟⠃⠀⠀⠀⢀⣿⠢⠇⡞⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠃⢹⡘⡇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡜⠠⠀⢠⣿⡀⠀⠀⠀⠀⠘⣿⣴⠀⢱⡄⠀⠀⠀⠀⠀⠀⢸⡁⠀⠘⠘⡏⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣾⣣⠃⠀⠀⠀⠀⠀⠈⢮⢀⡼⠃⠀⠀⠀⠀⠀⠀⠈⢷⣧⣀⠀⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠀⢀⣤⣤⣀⠀⠀⢀⣾⣿⠛⡁⢈⣙⣿⣷⡀⠀⠀⣀⣤⣤⡀⠀⠈⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢰⣿⣿⣿⣿⡆⠀⣼⣿⣿⣄⡁⠘⠻⣿⣿⣧⠀⢰⣿⣿⣿⣿⡆⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⠻⢿⡿⠟⠁⠀⠸⣿⣿⠛⠃⠘⣀⣿⣿⠇⠀⠈⠻⢿⡿⠟⠁⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣇⣸⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣼⣧⣀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⠁⠀⠀⠈⢿⣿⡇⠀⠀⣿⣸⣇⠉⠀⠀⢸⣿⡿⠁⠀⠀⠈⢿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣷⠀⠀⠀⠀⣼⣿⡇⠀⠀⡀⢹⡏⣿⠀⠀⢸⣿⣧⠀⠀⠀⢀⣾⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠻⢿⣿⣿⣶⣶⣿⣿⣿⣿⣄⠀⠙⢻⡟⠁⠀⣠⣿⣿⣿⣿⣶⣶⣿⣿⡿⠟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠷⠦⠬⠥⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠳⠾⠋⠉⠩⡔⠲⠀⣩⠍⣱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⣭⣥⠤⣀⠀⠀⠀⠈⠙⢀⣮⣥⣾⣻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠂⠉⠁⠀⠀⠉⠙⠛⠶⠤⣀⣴⣿⣯⣷⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠲⠋⡍⠛⠲⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡟⠂⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢔⣪⠑⠊⠀⠀⠀⠱⢤⠀⢈⠑⣅⡨⢯⢟⣦⠠⣣⢡⣿⣿⣿⡛⢪⣦⣾⣿⣇
⠀⠀⠀⠀⠀⠀⣄⠒⢉⡴⠞⠅⢉⡁⠀⠀⠀⠀⠀⠀⠉⠉⠉⡠⢒⣽⣿⣿⡀⣱⣿⣾⣟⣀⣴⣟⣽⣯⢏⡾
⠀⠀⠀⠀⠀⢰⡿⣛⠲⠤⣌⣓⠠⣌⡀⠀⡠⠤⣤⡄⡠⣰⡭⣺⣿⣿⣿⡿⢀⣽⣟⠂⠀⢀⢹⡷⡿⣱⣏⡀
⠀⠀⠀⠀⠀⢘⡇⠈⢛⣒⢦⢭⣙⣶⣍⣉⠤⢞⣩⣔⣮⣷⣿⣿⣿⣿⣿⡝⠋⠁⠀⠀⣠⡼⢪⣿⣷⣯⣒⡚⠑⢶⠤⢄⣀
⠀⠀⠀⠀⠀⠘⠣⢄⡀⠠⢍⣋⠒⠮⢙⠿⠯⣗⣿⣟⣯⡿⣿⣿⠋⠓⠂⢀⣩⡿⣿⣪⣳⡼⢾⣿⡥⠴⠚⣇⣀⡂⡰⠔⣨⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣦⣭⣛⠲⠬⣴⠒⣾⣿⡿⠟⠷⠂⠀⠀⠉⢁⢎⣾⣿⠧⡳⣡⡎⠁⠀⢠⠂⠉⠚⠵⢖⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡺⠗⠂⢉⣉⠒⠤⣼⠏⠽⠀⠉⠁⠀⠀⢀⢔⣵⣟⣽⣿⢀⡴⠃⢀⡠⠕⠁⠀⠀⠀⠐⠒⠈⠙⡏⢙⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⢰⡎⠤⠸⠥⠀⢓⣀⠠⠬⢗⣒⠀⠀⡠⠔⣒⡖⢒⣵⡷⣳⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⣁⡀⣀⠘⡀⠈⢐⣡⠔⣫⡀
⠀⠀⠀⠀⠀⠀⢸⡯⠭⠍⢀⣒⣒⠒⠀⠉⠩⠍⠋⢉⡀⠐⠫⢒⠽⣣⣮⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢐⠯⢒⡩⢔⣫⣼⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠭⠔⢒⣒⣒⣀⠭⠍⠭⠤⠶⠒⠀⠭⠽⠤⣾⣿⡿⡿⠿⣶⣖⠒⢲⣖⠢⢄⣀⡠⠄⢀⡢⢭⣒⣬⣴⣾⢿⣿⡯⢛⠯
⠀⠀⠀⠀⠀⠀⠀⠈⣟⢆⡳⣖⠶⡴⠤⢭⠭⡭⢩⠏⡭⡩⢭⠍⡽⣿⡫⡂⠀⠈⠙⣯⣒⠦⡭⣤⣖⠲⢽⣗⣿⣿⣿⡿⠿⢛⡉⢔⣚⣽
⠀⠀⠀⠀⠀⠀⠀⠨⡎⠁⠉⠊⠑⠦⣀⢁⠊⠐⠑⠠⠊⠔⠡⠢⢂⡜⣿⣾⡀⠀⠀⠀⠱⠍⢨⣿⣿⣾⡿⠿⢛⡩⢕⡂⢬⡰⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠣⡀⠀⠀⠀⠀⠈⠙⡳⣖⠈⣉⡽⠖⢲⣶⣿⣿⣿⣿⡿⡶⠄⡀⠀⠀⢸⡿⠽⣔⣶⡯⠿⠛⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠘⡟⡙⠢⣴⢾⣿⣽⣿⣿⡷⠝⠋⠀⠀⠈⠐⠠⠜⠛⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⠀⠈⠮⠀⣼⣿⣯⠿⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠤⠟⠉
⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠐⠀⠀⠄⡠⢀⠄⡠⢀⠀⢀⠀⡠⠀⠀⡀⢀⠄⡠⢀⠄⠀⠀⡠⢀⠄⠀⢀⠀⡠⢀⠀⡠⢀⢄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠊⢔⠄⡀⠄⡀⢀⠄⡠⠔⡁⢀⠅⠠⢊⠄⡨⢀⠀⣀⠄⠀⡠⡠⢊⠔⠡⢀⠔⡡⢊⠔⡡⢊⠔⡁⠨⠪⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠐⠁⠊⠐⠁⠐⠀⠊⠐⠡⠂⠄⠡⢂⠀⠡⠪⠈⠄⠁⠊⠐⠁⠈⠀⠁⠈⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠋⢳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣉⣉⠳⣿⡷⠛⣉⣙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢠⠋⠁⢈⡇⢸⠀⢮⠁⠈⢳⠈⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡴⢇⠘⣄⠀⠻⠴⠋⠳⠼⠃⠀⡼⢀⡷⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡞⠀⠈⢦⡘⠦⣄⡀⠀⠀⢀⣤⠞⣡⠞⠀⠈⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣧⠀⠀⠀⠙⠢⢌⡉⠒⠚⣉⡤⠞⠁⠀⠀⢀⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠣⣄⣀⢀⣀⡤⠛⡹⣏⠣⣄⣀⠀⣀⣠⠞⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣭⣭⣭⡴⢋⣠⣈⠓⣦⣭⣍⡩⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡉⢩⠤⢤⡉⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠻⣀⣠⠇⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠤⠴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠛⠛⠛⠛⠛⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡭⠗⠀⢠⡖⠒⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢲⣆⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠀⠸⣧⣀⣽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣀⣀⣀⣀⣯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠶⠶⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠞


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠒⠒⠀⠲⣖⡶⠒⢲⠖⠒⠒⠒⠒⠒⠒⢒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⠤⡄⠀⠀⠎⠂⠉⢦⠈⠀⢠⠛⣄⠠⠒⠦⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⢸⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠦⡀⠀⠀⠀⢸
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡀⠐⠀⠀⠘⡄⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢰⠀⠀⣠⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠒⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠣⣀⠤⠆⠒⠊⠉⠀⠈⠉⠉⠉⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠢⣄⡀⠀⠀⢀⣀⡇⠀⠀⣇⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⡀⢸
⠀⣂⣀⣀⠀⠌⠀⡄⢨⣶⣀⡉⠦⢀⠀⣀⣀⡀⠀⠀⠉⠒⠉⠀⠀⡇⠀⠀⡏⠉⠁⠀⠈⠉⠁⠐⠒⠁⠀⠈⢣⣀⣀⠔⠒⠢⠖⠊⠁⠀⠀⠈⢾
⠀⠀⠀⠀⢈⠀⡘⢝⢸⣇⡜⢧⡤⠀⢩⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⡇⠀⠀⡏⠉⠉⠁⠉⠉⠀⠀⠐⠒⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠒⢸
⠀⠀⠀⠀⠈⡶⢣⡀⡱⡊⠈⢉⠇⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠙⠠⣈⣚⣒⣚⠩⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡀⢀⢆⠀⣾⠅⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⣾⣷⣶⠶⠶⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣀⡐⣌⣾⣈⠀⡯⠀⣜⣰⠕⢲⠆⣀⣤⣴⣾⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⡋⢰⡥⣟⡹⢒⡷⢻⣹⣏⣿⣝⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠓⡄⠀⠀⢀⠤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠈⠀⢁⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠙⢀⠔⠁⠀⠘⢢⠀⠈⣉⣒⡶⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣄⠀⠀⢀⡼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠥⠴⠃⠀⠉⠉⡁⠀⠰⣌⠣⡀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠦⣤⠴⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢸⡆⠀⢧⡆⠀⠈⠉⠉⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⠀⠀⠀⠀⢀⡤⠀⠉⢷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⡰⠉⠀⠈⠉⠉⠙⠉⠉⠀⠀⣠⠞
⠀⠀⠀⢀⠔⠋⠙⠲⡴⠋⠉⠉⠙⢦⡤⠶⠶⠶⠤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢦⠀⠀⠀⠀⠀⢰⡇⠀⠀⡀⠀⠀⠀⢀⡟⠛⠉
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠉⠛⠋⠛⠦⢤⠴⠚
⠀⠙⠒⠒⠻⣄⡀⢀⣀⡼⠲⠶⠶⠚⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢧⣀⡀⢀⣤⣀⠀⢀⣠⠏⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⣄⣀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⢿⠿⠿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⢀⣀⣤⡶⠞⣋⣠⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣿⣿⣯⣵⡶⠟⣋⣡⣴⠶
⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠛⠋⣉⣀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠿⠛⠋⠉
⠀⠀⠀⠈⢿⠿⠛⠛⠛⠛⠛⠛⠛⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢸⢆⢠⢣⢀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⠒⠦⢈⣃⣀⣁⣈⡋⠸⠔⡹⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣏⠉⡠⠊⠁⠀⠀⠀⠀⠈⠑⠢⡉⢡⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢠⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠉⠉⠛⢶⡄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣔⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠱⠓⠒⠉⠉⠉⠑⠒⠢⢄⡀⠀⠀⠀⢸⠠⣔⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⢀⡌⢠⡐⠓
⠀⠀⠀⠀⢠⠔⠒⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠⠊⠰⡒⠳
⡠⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⣓
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳
⣬⣵⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣯
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣦⣤⣀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢟⣛⢛⢿⣿⣿⣿⣦⣤⣤⣤⣴⣤⡄⢹⣿⣿⣿⣷⣶⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡸⢟⠸⣣⣿⣷⣾⣶⣷⣿⡾⢓⣉⣠⣼⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⢉⡉⠻⣿⣿⣿⣿⣇⠸⣿⠿⠿⠿⠛⠛⠛⠛⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢁⡴⠋⠙⢦⡀⠰⡴⢴⣶⣶⣶⢶⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡴⣫⣾⣷⡿⢷⣝⢦⡀⠀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⢀⡴⣫⣾⣿⣿⣟⡵⣫⠿⣷⡝⢦⡈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⢠⣤⣶⣾⣿⣿⣿⢋⡴⣫⢞⠍⠛⠫⢝⢿⣾⣥⣾⣿⡄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠀⠾⠇⠃⠾⣷⢀⠀⡏⣿⣿⡿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠓⠛⣠⢣⡿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠀⠀⠠⣄⠈⣩⡉⢙⡛⢛⣡⣤⣶⣮⡷⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢷⣯⡞⣡⣿⣿⡿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠿⠟⠿⢋⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠖⠁





⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⡀⠀⠀⢀⣀⠀⡀⣀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⣯⡽⢢⣿⣆⠹⡞⠀⡿⠾⡇⣸⠭⠀⡿⣼⡇⢹⡏⠀⢸⡿⠾⡇⣿⠭⠀⢹⠉⢸⠦⣇⢸⡉⣹⠀⡏⣹⠆⠀⡏⢾⡉⠁⣾⠉⣳⢸⠷⣼
⠀⠀⠀⠀⠁⠀⠉⠀⠉⠀⠁⠀⠁⠀⠁⠈⠉⠁⠁⠈⠁⠈⠁⠀⠈⠁⠀⠁⠉⠉⠁⠈⠀⠈⠀⠉⠈⠉⠉⠀⠉⠉⠀⠀⠁⠈⠉⠁⠈⠉⠁⠈⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠘⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢋⣠⡄⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠃⠘⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠀⠀⠰⣾⣿⣿⡆⢾⣿⣿⣷⠰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠚⢉⣤⡶⠀⠀⠀⠀⣠⣤⣤⣄⢀⣤⣤⣤⡀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢸⣿⡇⠀⠀⠀⠀⠙⠛⠛⠛⠈⠛⠛⠛⠃⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢸⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⠆⢿⣿⣿⡷⠸⣿⣿⣿⠆



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣰⠶⠶⠛⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠟⠉⠉⠙⢶⡀
⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀⠀⠈⠉⠉⠉⠉⣉⣁⡀⠀⣀⢀⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⠀⢽⣿⡇⠀⠀⠀⠀⢰⣹⣿⣸⠀⠻⣿⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⣽⣿⡟
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡟
⠀⠀⠀⠀⠀⠀⠸⡄⡞⠉⣭⣭⣭⣄⢸⠀⡏⠀⠐⠀⠐⢸⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⡇⡇⣸⣿⣿⣿⣿⠘⡀⡇⠢⠀⢴⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠻⠿⢿⣿⠿⠆⡇⢇⠀⠀⠀⠀⠄⠿⡿⢻⡟⠛⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⢈⣒⣒⣲⠶⠎⢍⣁⣨⠭⠭⠽⠿⠟⠛⢇⠸⡇⠀⠀⠈⠹⣿⣿⣿⣷⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⢠⣀⠤⡀⢸⡇⠀⠐⣄⠐⠆⠀⢸⠀⣷⠀⠀⠰⣶⣿⣿⣿⣿⣤⣤⠄⠀⠀⣼⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⡇⡇⢀⡠⠂⠈⠐⠀⢸⣷⠀⠀⠀⠗⠦⠄⠸⠄⠛⠀⠀⠀⠈⠻⣿⣿⣿⠟⠁⠀⠀⢠⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⡇⢸⢻⡇⠀⠶⠇⠀⠈⣿⠸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⣸⣿⣿⣿⡿⢃⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⣷⠘⠦⣤⣤⣤⣶⣖⣚⢉⣉⣉⣩⣭⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⣿⣿⠟⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⢸⢀⡤⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧⠐⠂⠀⠐⠂⢸⢸⠈⠁⢸⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠻⠃⠀⢸⠈⡠⠦⠆⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣅⡘⣒⣒⣒⣒⣊⣉⣀⣓⣀⣭⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣁⣀⣀⣀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣮⡻⢿⠿⢟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⡟⠉⠛⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠒⠀⠀⠀⠐⠚⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠸⡄⢸⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⡀⠁⠀⢁⣀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠴⠋⠁⠀⠉⠲⡀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⡄⠀⠀⠀⠀⠀⡇⢀⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠢⡀⢀⠡⣄⡈⠋⣀⡴⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠁⠀⠀⠀⠀⠁⢀⡉⠉⢁⡠⠤⢄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣀⣀⠀⠀⠐⠋⠀⠀⠁⠐⠀⠀⠙⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⢠⠔⠀⠀⣀⣠⣤⣄⣀⠀⢀⡤⠶⠒⠒⠦⢤⡀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠋⢀⡴⠊⠁⠀⠀⠀⠀⡴⠋⢀⠀⠀⠀⠀⠀⠙⠀⠉⠁⠉⠙⠢⠤⠔⠀⠈⠉⠁
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠚⠀⢀⣀⡤⠤⠤⠤⢤⣀⡀⠀⠀⠀⢀⡠⠴⠒⠒⠲⢦⣄
⠀⠀⠀⣰⠛⠁⠀⠀⠀⣠⠞⢩⠂⡀⢐⠆⠀⠀⠀⡭⠗⣦⠰⠋⠀⠀⠀⢀⣀⠀⠈⠳⣄
⠀⠀⢰⠃⠀⠈⠀⠀⡰⠁⠀⠀⠀⢀⢨⠆⠀⠀⠀⠦⣀⠉⢳⠀⠀⠠⣆⢠⠚⣢⠀⠀⠘⣇
⠀⠀⢸⠀⠀⠀⡀⢠⠃⠀⠀⠀⠉⠁⠀⢀⠠⠀⠀⠀⠀⠀⠸⡆⠀⠀⠈⠢⠜⠁⠀⠀⠀⣿
⠀⠀⠘⣧⠀⠀⠀⢸⡀⠀⠠⠄⢠⠀⠀⢙⠀⠆⢐⠄⠀⠀⢰⡇⠀⠄⠀⠀⡠⠀⠀⠀⣰⠃
⠀⠀⠀⠈⠓⠦⢤⠀⢷⡀⢸⢀⡠⠃⠀⠈⠑⠁⠀⠀⠀⢀⡞⠀⠰⠦⢤⣤⣥⡥⠾⠛⠁
⠀⠀⠀⠀⠀⠀⢲⡀⢠⠙⢦⣉⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⣄⠸⡄⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⣷⠈⢰⠂⠸⡍⠛⠓⡚⠛⢁⠄⠀⠀⠀⠀⠸⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⡟⠀⣾⠀⠀⠙⡄⢠⠇⣰⠋⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⢠⠇⠀⡇⠀⠀⠀⢸⠀⣸⠁⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⣠⠶⠒⠒⠲⠦⡀⠀⠀⡏⠀⣿⢠⣀⣀⠀⢀⡤⠤⣸⠁⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢫⡇⠀⣿⠀⠀⠈⠙⠲⠄⠀⠀⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠘⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢸⣇⠀⣿⠀⢸⡇⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣄⠀⢻⡦⠘⣋⣤⠥⠴⠶⠦⠬⣥⣘⠃⢰⡟⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠷⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠾⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡈⠻⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡰⠟⠃⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢃⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡰⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣃⣸⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠲⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡴⠚⠉⠉⠉⠙⢦⡙⠦⠤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠈⠙⠁⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠛⠮⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣤⣤⡀
⠀⠀⠀⠀⢀⣤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⠀⠀⠀⠀⠀⠀⠀⣼⢡⣄⡉
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡆⠀⠀⠀⠀⢀⡼⢡⣄⠉⠛⠷⠂
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠀⠀⠀⣀⡴⠋⣴⣄⠙⠃
⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡕⠧⠴⠖⣋⡁⢲⣆⠈⠻⣷
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢻⡆⠸⣧⠀⠉
⠙⢦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠏⠀⠈⠀⠀⠹⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠴⠶⠒⠒⠛⣏⠉⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⢲⠛⠉⠉⠉⠁⣀⡤⠖⠲⠦⣄⠀⠀⠀⠀⠈⠓⠦⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠋⠀⠀⠀⠀⡼⠁⢀⣼⣦⡄⠈⠳⡄⠀⢀⡤⣄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⢀⣠⣄⠀⢸⡇⠀⠸⢷⣦⣄⠀⠀⣧⠀⠻⣤⠞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠒⣇⠀⠸⣤⡼⣀⣈⣇⠀⢀⣤⣬⡿⠀⢀⡏⠀⠀⠀⠀⣠⠴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢻⣀⣀⡴⠋⠁⠀⣹⠳⣄⣀⣙⣀⡤⠊⠀⠀⠀⠀⢰⠃⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢀⣀⡼⠞⠋⠀⠀⣠⠔⢁⣀⣀⣉⣭⡥⠤⠴⠖⠒⠒⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠋⠉⠁⠀⠀⠀⡰⠛⢉⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⠴⡻
⠀⠀⠀⠀⠀⢠⠞⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣓⣒⡿⠃
⠀⠀⠀⠀⠀⠈⠳⡄⠀⠙⢦⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠳⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣆⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠞⠁⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⣟⣲⡲⣄⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣗⠒⠒⠒⠒⠒⠒⢲⢎⡷⠿⠧⠤⢬⣽⣷⣄⣀⢈⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣋⣀⣀⣀⣀⣀⣠⣿⢟⣄⣄⠀⡄⠀⠀⢘⣿⠉⠉⠉⠉⠉⠛⠓⠒⣒⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢽⡿⣿⡏⠩⠉⢹⠿⣿⣿⠟⠻⠿⠻⠷⠻⠿⠾⢿⡦⣿⣷⣧⣧⣤⣶⣸⣏⠩⡪⢂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠚⠋⢠⡶⣤⠀⠉⡸⠇⠂⠀⠀⡞⣋⡟⠀⠀⢸⣷⢿⡆⢀⠀⢸⣏⢙⣿⡆⢘⢕⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡠⣤⠀⠀⠒⣫⣤⢄⣈⠁⠀⢀⡂⠑⠔⠊⠀⠀⠓⠃⠀⠀⠀⠠⠈⠓⠃⣾⣇⠀⡩⠚⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠧⡾⠀⢀⣾⢋⣥⡈⡙⢷⡰⣷⡄⡆⠀⠀⢀⣀⠀⠀⠠⢊⠀⡐⠄⠀⠁⣿⣿⡀⠪⢀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠓⣲⣾⣷⣾⡷⠾⠷⠾⠿⠶⠾⠥⣤⣤⣯⣴⣆⠀⢀⠠⡑⣣⠃⠀⠺⣿⣘⣇⠀⢕⡹⣇
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⡋⠀⠀⠀⣀⣾⠋⢴⡄⠈⢻⣆⠀⣤⠀⠀⠀⠀⠀⢀⣠⠔⠻⡇⠀⠑⠒⠉⠀⠀⣀⣸⠀⢿⡀⠩⢆⢿⡄
⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠺⠶⢿⣯⣤⣤⣤⣧⣧⣿⣤⣀⣦⣄⣤⣴⣸⣏⠁⠠⡈⣧⠀⠀⠀⠈⠀⢸⢿⣿⠄⠸⣇⠐⠕⠸⣧
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⢹⠀⠈⡀⢹⣀⣤⠄⢤⡀⠌⠒⢻⠅⠀⢿⡀⠠⠂⢿⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⣔⢍⣿⠻⠲⢻⠇⠀⠀⢸⠇⡀⠸⡇⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢸⡷⠒⢲⣤⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠔⡡⣷⠨⠉⠡⠀⠀⠀⢸⡇⠂⠀⣷⠀⢠⠶⠸⣇
⠀⠀⠀⠀⠀⣾⡇⠗⡅⠀⣰⡖⠀⠑⣤⠊⠙⢄⡠⠋⠡⣈⡩⠋⠫⣉⡝⠛⣍⣹⣷⠀⠪⠀⢸⣠⣄⡀⠀⡔⠒⢌⣇⠁⠀⢸⡀⠐⠩⣂⢹⡄
⠀⠀⠀⠀⠀⣿⣅⠀⢈⠾⡁⠈⡢⡊⠀⠑⣴⠋⠑⢄⡠⠊⠢⣀⠴⠋⢄⡠⠚⢻⣿⡆⢀⠀⢸⡟⠙⠁⠀⠷⣷⠞⣿⠀⠀⣼⢟⠻⣻⣿⠿⠃
⠀⠀⠀⠀⠀⣿⢈⠶⡁⠀⡸⢎⠀⠈⣦⠋⠈⠣⣠⠊⠣⣀⡰⠋⢢⡠⠊⠑⢄⢼⣿⣷⠀⠄⡨⣇⣀⠀⠀⠀⠀⠀⣿⠀⣼⣏⣵⡾⠛⠁
⠀⠀⠀⠀⢠⡟⢄⠀⡸⢎⠀⠀⡱⡎⠀⠑⣤⠊⠈⢢⡔⠉⢢⡠⠊⠑⢤⠔⠉⢺⣿⢿⡄⠘⠄⣿⣧⡷⠀⠀⣰⣏⣻⣼⡿⠟⠁
⠀⠀⠀⠀⢸⣇⡠⠛⣄⠀⡱⢎⠀⠈⡦⡊⠀⢱⡔⠁⠑⣤⠊⠑⢤⠔⠁⠑⣤⢺⡿⣽⣧⣴⣇⣼⠛⠙⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⢸⡋⠢⣀⡠⠫⡀⠈⡷⡊⠀⢙⢶⠁⠈⡲⡊⠀⢳⡔⠁⠑⣤⠎⠀⢹⣿⣿⠣⢈⣽⠟
⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠿⠶⠾⠶⢧⣤⣵⣎⣀⡈⣤⠃⠈⢢⡔⠉⠣⡤⢺⣿⢯⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠳⠷⠤⣮⣌⣾⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣤⡄⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣤⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡄⠀⠀⠀⠀⢀⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⡶⠶⠶⠶⠶⠶⠶⢶⡄⠀⢸⡇⠀⠀⢀⡴⠋⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠁⢀⡴⠋⠀⣠⠞⠁⠀⠀⠺⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠳⠶⠶⠶⠶⠶⠶⠾⠃⠀⢀⡴⠋⠀⣀⠘⠁⠀⠀⢀⣤⣄⡀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⡀⣀⣀⠀⠀⠀⢀⡴⠋⠀⡀⠘⠁⠀⠀⢀⡄⣿⠀⠈⣷⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠁⠉⠉⠀⢀⡴⠋⠀⣠⠞⠃⠀⠀⠀⠀⠸⣁⠘⣛⠛⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠁⠙⠛⠁⠘⢧⡀⠘⠃⠀⠀⠀⢀⡄⠀⠀⠉⠉⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⠶⠂⠲⠶⠂⠀⠀⠙⢦⡀⠀⢀⡴⠋⣀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠞⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠙⢦⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠟⠷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣴⣾⣶⣄⠀⠀⢠⣿⣿⡰⠇⣷⣾⣿⣧⠀⠀⢀⣴⣾⣶⣄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⣿⣿⡿⠀⠀⢸⣿⣿⢿⡆⣦⢹⣿⣿⠀⠀⠸⣿⣿⣿⡿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣄⡀⠀⠈⠙⠉⠀⠀⠀⠈⠻⣿⣤⡁⣩⣼⡿⠃⠀⠀⠀⠈⠙⠉⠀⠀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠚⠛⠛⠛⠛⠛⠒⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⣀⣀⣤⣤⣴⣶⣶⣶⣦⣤⣤⣀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⢀⣤⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠦⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣋⣉⣉⣛⣶⣌⠳⣄⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠛⠉⠉⠉⠉⠉⠻⡇⠈⢳⡄⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢻⡄⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣸⠇⠀⠀⣀⡤⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⡼⠃⠀⠀⠀⣧⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢰⢃⡞⠋⠉⠳⢿⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⢸⡇⠀⠀⠀⠈⠳⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⠘⣎⣇⠀⠀⠀⠀⠀⠈⢷⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠈⣇⠀⠀⠙⢮⠣⡀⠀⠀⠀⠀⠀⠹⣞⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠳⣝⢦⡀⠀⠀⠀⠀⠸⡎⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢰⢯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣆⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠲⣝⡦⣤⣀⣀⡼⢣⡏⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⣠⢏⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿⠀⠀⠀⢀⣠⠖⣳⠀⠀⠙⢦⣀⠀⠀⠀⠙⠓⠒⠶⠚⠋⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⢀⡴⢃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⠟⠁⠀⣀⡴⠋⢀⡼⠃⠀⠀⠀⠀⠈⠛⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⠀⢀⡴⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⠟⠀⠀⠀⢀⡤⠞⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⠴⠋⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⣠⠖⠋⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⡶⠦⣄⣀⣀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠋⣁⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠟⠁⠀⢀⡴⠚⠁⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣠⡷⢦⣤⣈⣉⣉⠉⠉⠉⠉⣉⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣠⠞⠋⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⠞⠋⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⣀⡀⠀⣀⣼⠋⠀⢀⣤⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣡⡾⠟⠁
⢀⣤⠞⠋⠙⠛⠉⠀⢀⡴⠋⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣴⠞⠉
⣿⠁⠀⠀⠀⠀⣠⠴⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣡⡾⠛⠁
⣿⠀⠀⠀⣠⠞⠁⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠟⠁
⣿⠀⣠⠞⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣉⣴⠟⠁
⣿⡟⠁⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣠⡾⠋⠁
⢿⣷⡚⠧⣤⣀⡀⠀⠀⠀⢀⣠⠞⣉⣴⠟⠋
⠀⠈⠙⠻⢶⣬⣭⣙⡛⠲⣋⣠⡾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠯⣷⡄⠀⠀⠀⠀⠀⠀⠀⡞⡾⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠘⡇⠀⠀⢀⣶⣶⣶⣖⠛⡗⣒⣛⣿⣫⡽⠿⣤⠀⣀⣤⡀
⠀⠀⢸⣿⣏⣉⡉⡏⠉⠉⠉⠉⡏⠉⣹⣟⣿⡿⠛⠻⣿⠋⠉⠁⠉⠀⣰⠦⣿⣺⣧⠀⢹
⠀⠀⢸⠉⠉⠉⠉⠋⠉⠉⠉⢩⣯⣭⣭⣽⣿⣤⣤⡤⢼⠤⠤⠤⢿⣯⣽⣰⡇⠈⢹⡗⠾⣄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⡇⠀⠀⠀⣸⣀⣀⣀⣸⣾⡀⠙⠷⢤⣸⠁⠀⢀⣩⣷⣦
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠏⠉⠉⠉⠛⠉⠉⠉⠉⠉⡇⠀⠀⠀⠙⠓⢤⡼⡇⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⠸⠦⠤⠤⠤⠤⠤⠤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠢⣄⠀⠀⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠙⠦⣄⣿⠗⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⠿⠿⠿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠙⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡾⣷⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠉⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⢠⣾⢿⣶⣤⣀⣈⣛⣿⣶⣄⠀⢀⣀⣀⣀⣀⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠘⠿⣾⣿⣿⠟⠛⠉⠉⢙⣿⡿⠟⠋⠉⠉⠛⠛⢷⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠀⣴⣿⠋⠀⠀⣠⣤⣴⣿⠏⠀⠀⢀⣠⣄⣀⣠⣾⡿⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⡀⠀⠀⠀⣼⣿⠁⠀⠀⢰⣿⣿⣿⠃⠀⠀⣰⣿⣿⣿⣿⡿⠋⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⠠⣿⡇⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠹⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢫⡟⠐⣿⡇⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣳⠏⠀⣠⣿⣷⡀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢚⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⣰⡿⠁⠹⣿⣶⣤⣤⣤⣤⣬⣿⣿⣦⣀⣀⣀⡤⠔⢊⣡⣴⡾⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠏⠀⣼⡟⠀⠀⠀⠀⠙⣿⣇⣀⣀⣠⣀⣬⣿⡿⠿⣿⣴⣾⠟⠋⢻⣇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⠆⣼⠟⠀⠀⠀⠀⠀⠀⠻⣿⡏⠉⢻⠉⢻⡏⠀⠀⠛⠻⣿⣶⣤⠀⢻⣆⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⢋⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⢠⣿⡃⠀⠀⠀⠀⢸⣿⡅⠀⠈⢿⡄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢸⠀⣿⣇⠀⠀⠀⠀⠀⠻⢿⣷⣀⠈⢷⡀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠉⢡⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠘⣷⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣁⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⣾⣏⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠘⣧⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢸⡇⢿⣏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠹⡆⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⢸⡇⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠒⠒⠷⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⣲⠆⠀⠠⢨⡀⣀⠤⠄⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡺⠟⠿⠟⠁⠀⠀⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠁⠀⠀⠀⠀⠀⠐⠰⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⠠⠤⡖⠁⠀⠀⣀⠂⠀⡀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣉⠁⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⣀⣀⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡠⠂⢻⡄⠀⡸⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠒⠒⠛⢦⡎⠀⠠⠤⠓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠑⠢⠄⣠⠐⠢⡀⠀⠀⠀⠀⠀⢰⠁⢠⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⠀⠰⢄⠤⠞⢰⠱⠘⡖⠙⠂⠒⠃⠀⡸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢆⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠄⠀⠠⠀⠀⠀⡠⢤⣀⠀⠀⠀⠆⡎⢠⣤⣇⣀⣀⠀⠀⢀⡃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠑⠂⠀⠀⠀⠀⠀⠓⢾⢉⡷⠱⡶⣤⡠⠃⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠔⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⢤⠖⠋⠉⠉⢐⠲⢬⣙⠂⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⢀⡴⠋
⠀⠀⠀⢀⡰⢊⡵⠁⠀⠀⠀⢀⠎⠀⣠⢤⡉⣒⠶⣐⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢴⡋⠱⡀⠀⢀⡠⠖⠁
⠀⠀⢠⠋⠀⠮⢄⣃⠀⢀⡰⠁⢀⠬⣀⠉⡰⠊⣴⢢⠙⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡄⠀⠀⠹⡁⠀⠑⡄⣸⠚⠇
⠀⢀⠇⠈⠀⠀⠀⠀⠉⠑⠢⠄⣈⠒⠦⣼⣁⠖⡤⠁⡜⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠇⠀⠀⠀⠘⢄⠀⡘
⠀⠼⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢉⠙⠀⡰⠀⠀⠀⡰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠸⠀⠀⠀⠀⠀⠑⠆⠀⢀⠆
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠒⠒⠒⠒⠒⠒⠀⠙⠛⠁⠀⠀⠐⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⣌⡉⠐⠚⠓⠂⠂⠀⠒⠒⠒⣈⡝
⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⡇
⠀⠁⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠁⠈⠋⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠦⠤⢤⡤⠤⠖⠒⢲⠖⠒⠒⠒⠒⡶⠒⠒⠒⠒⡶⠶⠶⢴⡦⠦⠴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡼⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢳⠀⠀⠀⢧⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡰⠁⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⢣⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠘⡆⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠤⠤⠤⠤⠾⠤⠤⠤⠤⠤⠶⠺⠒⠒⠒⠒⠒⠒⠺⠒⠲⠶⠶⠶⢿⠶⠶⠒⢠⠛⠒⠒⡒⠶⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⣼⡇⠀⢀⣶⠀⠀⣸⡀⠀⠀⣸⡀⠀⠀⣸⡀⠀⠀⣸⣀⠀⠀⡸⢇⣀⣀⡼⣄⣀⣀⣷⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠋⣿⠋⠉⠀⠉⠉⣹⣉⣉⣉⣀⣉⣉⣉⣀⣈⣒⣚⣁⣈⣉⣉⣀⣸⡁⠀⠀⠀⠈⣷⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣠⡿⠤⣀⠀⠀⢸⠉⠏⠉⠈⠀⠀⠀⠀⠀⠀⡀⣀⡀⣠⢤⠉⣉⠙⠉⠉⡇⠀⠀⡏⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠉⣉⣄⢀⣄⣸⡄⠀⢸⠐⡮⠀⣇⣀⣎⡳⢳⠞⠦⠃⢿⡦⠸⠻⠌⢓⣦⠀⢠⠇⣰⠚⠋⠛⢿⢤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣵⢰⣾⠁⠈⠉⢈⠿⠼⠀⢸⣄⡥⣤⡭⠤⠤⠴⠶⠲⠶⠶⠖⠒⠒⠒⠒⠚⠒⠒⠾⢘⣏⢠⣤⡴⣄⣤⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡶⠸⣄⣀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢰⠛⡄⠀⠀⣸⡇⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡟⢁⣀⠀⢠⢦⡀⠀⠀⠀⢀⣠⣤⣀⡤⢤⣄⠀⠀⠀⠀⠀⠀⢀⢶⡀⠸⣼⠀⢻⢲⡖⠋⢇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠛⢊⣙⠒⠓⣤⠿⡴⢲⣀⡀⡧⠴⣏⢀⡤⣀⠹⣄⣀⣀⡴⠶⢋⣦⢷⠴⣄⣴⢺⠜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠑⡦⢣⡠⠇⠐⢇⡤⢤⠈⢁⣽⣅⢖⡌⣻⢶⡚⢀⡏⢀⡟⢥⠀⣬⣹⣄⠀⠁⠸⡄⠀⠡⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠙⠣⠤⣶⠦⠴⠋⣛⠉⣆⢈⣿⠮⣉⡴⣇⣠⠟⡟⣉⠙⠧⣏⠛⠁⠀⣉⢧⣤⡖⠳⠼⠶⣏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⡸⠽⢀⠈⢳⠀⠸⠭⢇⠈⣻⠣⢴⠡⡏⣝⠹⣆⠙⠋⢀⡶⣏⣰⣀⣶⡼⠀⠀⡳⢀⠀⡘⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠚⣇⠀⣸⡟⢛⣶⠞⣆⡸⠉⢳⡠⠾⣆⡑⢚⣡⠋⡵⢚⡛⣿⠀⠀⢠⠿⢶⠀⡀⣱⠊⠒⠉⠑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⡁⣽⠋⠁⠻⢤⠃⢸⡃⢠⣙⣇⢯⡀⣤⠉⢻⣆⠑⠔⠋⡸⠛⣤⣾⣋⡈⠋⠉⣯⣤⠀⡀⢀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⣸⢯⠿⣤⠜⠳⠔⠓⠶⠋⠈⠒⠬⠤⠠⠜⠀⠉⠉⠉⠀⠀⠀⠀⠙⠦⠚⠓⠁⠘⠖⢹⠇⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⢀⢠⡤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⢤⡤⠤⠤⠤⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢰⡀⢠⠀⠀⠸⡆⠀⠀⢡⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⢠⡇⠀⠀⡄⢀
⠀⠀⠀⠒⠒⠉⠁⠉⠙⠁⠓⠒⠚⠓⠦⠤⠼⠆⠒⠒⠚⠓⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠛⠒⠒⠒⠚⠓⠂⠉⠙⠹⠦⠤⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡵⠚⠋⠉⠉⠉⠓⠮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣄⣀⠀⢰⠃⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢀⡤⣤⣈⡉⢹⠀⡧⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠘⠳⠶⣤⣭⣙⡀⠱⠤⣤⣉⠓⢦⠀⠀⠀⠀⠀⠀⢀⡸⢰⡃
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣀⣠⣤⣌⣉⠛⠛⠷⢦⣬⠇⢸⡇⠀⣠⠔⢊⣩⡜⣡⡏⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⠋⠁⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⣼⣡⠞⠁⠀⢿⣥⣴⠷⠃⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⢿⠏⠀⠀⠀⠀⣠⡴⠶⠛⠳⣄⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠙⣄⠀⠀⠀⠀⣀⣼⠃⠀⠀⣰⠏⣾⠀⠀⠀⠀⠰⡏⠀⠀⣀⣤⠞⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⡀⠉⠛⠛⠛⠉⠀⠀⣠⡾⠋⠀⡏⠀⠀⠀⠀⠀⠙⠛⣋⣁⣤⡴⠶⢄⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡉⠙⠛⠶⣦⣤⣤⡶⠟⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠋⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢷⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⣄⠀⢀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣟⡞⠁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⣀⣠⠤⠖⠒⢦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣹⡆⡞⡼⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣄⠀⣾⠉⠀⠀⠀⠀⠸⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠰⡏⡼⡹⣱⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢠⡶⠞⠋⠉⠀⠀⢈⠄⠙⠷⠶⢦⣤⣤⣤⡟⠀⣷
⠀⠀⠀⠀⠀⢠⡞⡹⣱⢳⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⣇⠀⠀⠀⢀⡴⠋⣠⠖⠛⠒⠶⠤⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡾⢠⢣⢣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠹⣦⣠⡴⠋⢀⡾⠃⠀⠀⠀⠀⣀⣠⠇⠀⠀⣾⠁
⠀⠀⠀⠀⣼⢁⠃⠇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣾⣁⣀⣤⠶⠞⠋⠉⠀⠀⣠⡾⠃
⠀⠀⠀⣸⠃⠄⠌⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠈⠉⠉⠀⢀⣀⣤⣤⠶⠞⠋
⠀⠀⢠⡏⠌⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⡴⠶⠛⠛⠉⠁
⠀⠀⠀⠱⠤⠚⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⣉⠉⠁⠀⠀⢀⡏⢝⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢒⣩⠘⡯⢫⢟⠱⡲⢶⢤⡤⣀⡀⠀⠀⠀⡇⠉⠁⠀⠀⠀⠀⢸⡮⢓⠭⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠘⠂⡯⢔⠥⡪⢖⢕⡠⠪⡪⢙⢝⢗⢲⡇⠀⠀⠀⠀⠀⠀⢸⡪⢐⠩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢗⠕⣪⣆⠵⠵⠒⠚⠚⠉⢹⢟⡿⢦⣀⠀⠀⠀⠀⠀⣿⡪⠕⣺⠁⠀⠀⣀⠤⠤⠀⠀⠒⣲⡤⢄
⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⣀⡇⠀⠀⠀⠀⡖⡉⠉⠄⠀⠐⠀⠀⠀⠀⠀⣾⠕⡊⢔⠮⣷⢄⠀⠀⠀⡭⣒⠅⡞⠀⠀⡜⢁⣠⡤⢴⣴⣼⣧⠾⠏
⠀⠀⠀⠀⠀⢠⠔⠱⠌⣗⢔⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢍⡪⢔⢺⣇⢕⠀⠀⠀⡯⢒⢭⠇⠀⡔⢁⡞
⠀⠀⠀⠀⠀⠀⢣⠀⠀⢻⣂⣺⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠿⠥⠮⠷⠽⠷⠵⠤⠤⠼⠯⠶⠼⠤⠔⢁⡎
⠀⠀⠀⠀⢠⠊⣉⣍⢭⣍⣩⡉⢉⡥⠤⠤⡤⠤⡄⢀⡴⠖⠒⠲⠀⡖⠒⡶⢖⣶⠀⣶⠚⢛⢛⠀⣰⠞⡺⣶⡲⢀⡞
⠀⠀⠀⠀⠈⠆⠳⣡⠀⠀⢻⢻⡈⣇⠁⠀⡇⠁⠁⢸⣥⠄⠀⢸⠀⡇⢰⡯⣐⡏⢸⣿⢌⠢⡸⠀⣿⠅⣪⡿⢀⡜
⠀⠀⠀⠀⠀⠘⡄⢹⣆⢠⡘⣟⣷⠸⡆⠀⡇⠀⢸⠘⡾⠀⠀⢸⠀⡏⢸⣏⣾⠀⣼⣿⣵⣭⠇⠰⣇⣅⡾⠁⡜
⠀⠀⠀⠀⠀⠀⠘⡄⠙⠛⠓⠛⠛⠁⠙⠒⠓⣒⡁⠀⢉⣉⣉⡁⠀⣉⣭⣭⣁⢀⣤⡤⢤⢦⢀⣤⡶⢦⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⣯⣟⣿⢣⢸⡍⡏⠀⢸⠀⡏⠀⠀⢸⢸⡇⢸⡳⡟⢸⢕⡏⡢⡝⢸⢻⢢⡏⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠹⡄⢙⢘⡆⣷⠇⠀⠌⠀⡇⠀⠄⡈⢈⠆⣿⣲⠇⣸⣹⣗⣼⠁⣏⣾⡞⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡙⠒⠂⠃⠘⠓⠒⠚⠀⢈⣀⣉⣁⣈⣙⣉⣉⣄⠬⠭⠥⠤⠤⢤⣤⠠⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣔⣂⣀⣀⢀⣀⣀⣀⣀⣉⣉⣉⡉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⡉⠉⢉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡞⣛⣏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠹⠛⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣥⣕⡼⣷⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣁⡽⣦⡆
⠀⠀⠀⠀⠀⢤⡠⢀⠱⣿⣿⣋⢴⠠⢄⠠⣀⠠⢀⠠⢄⠠⢄⠠⠀⣀⠀⣀⠀⡀⢄⡠⢄⠠⡀⠄⡀⢄⡠⣏⡻⠿⢗⡵⡠⣀⠤⣄⣤⡠⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠁⠈⠀⠈⠀⠈⠀⠈⠀⠈⠀⠈⠐⠀⠁⠊⠁⠂⠁⠊⠁⠊⠑⠉⠐⠁⠊⠁⠈⠀⠉⠉⠉⠁⠉⠀⠉⠈⠁⠉⠑⠉⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠁⠀⢠⣶⡆⠀⠀⣦⣄⣀⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠘⠿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⣄⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⡇⠀⠀⣿⠂⠀⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠉⠁⠀⠀⠁⠀⢀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠶⢒⠋⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠒⠉⣁⣠⠤⣤⠀⢿⣧⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠒⠒⠉⣉⣀⡠⠤⠒⠒⠉⠉⠀⠀⠀⠹⣄⠘⠛⣁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⠛⠉⣉⡠⠤⠶⠶⠞⠯⢥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡆⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠒⠈⣉⣀⡤⠤⠖⢒⡽⠋⠁⠀⣠⣄⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⣷
⠀⠀⠀⠀⠀⢠⠔⠒⠈⠉⢁⡠⠤⠔⠒⠉⠉⠁⠀⠀⠀⡰⠋⠀⠀⢀⣴⣿⣿⡿⣷⡦⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡀
⠀⠀⠀⠀⠀⠈⡄⠀⠻⢄⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⣿⡟⠀⣿⣧⠈⠁⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡇
⠀⠀⠀⠀⠀⠀⢱⠀⠀⣃⡠⠃⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠻⣿⣦⣼⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⠿⣿⣶⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠀⣷⢲
⠀⠀⠀⠀⠀⠀⠀⢻⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⢻⣿⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⡄⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢠⣤⣀⣀⣻⣷⣴⣿⠟⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢩⠆⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠈⠛⠿⠿⠿⣿⡟⠁⠀⠀⣠⠎⠀⠀⢀⣀⣠⠤⠔⠒⠉⢉⣁⡠⠤⢴⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠉⠀⣀⣤⠾⠵⠒⠚⠉⣉⡠⠤⠴⠒⠚⠉⠉⠀⢀⣀⢼⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣧⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡭⠷⠞⠛⠋⢉⣀⣤⡤⠖⠒⠉⡹⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⢠⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡆⠀⢱⠀⢀⣀⣀⠤⠔⠒⠊⠉⣉⣠⠤⠴⠶⣾⣿⣿⠟⠋⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢀⢸⡄⠀⣹⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠽⠀⠈⠋⣉⣀⠤⠴⠒⠚⠉⠙⠦⣀⠀⠀⠀⠀⠈⠋⠀⢀⣠⣔⣋⣀⠤⠤⠴⠒⠒⠚⠋⢉⣉⣀⠤⠤⠴⡶⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢧⠤⡔⠒⢯⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣩⡵⠶⠶⠒⠒⠊⢉⣉⣁⣠⡤⢤⡤⠖⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⡄⠸⡇⠤⠤⠔⠒⠒⠉⡉⣉⣁⡠⠤⠤⠔⠒⢲⣿⠛⠋⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⡉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⢀⣉⣀⣤⠤⠔⠖⠒⠚⠋⠉⠉⠀⠈⠑⠢⣄⣀⡀⠀⠁⢀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣉⣉⣁⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣈⣉⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⣦⣄
⠀⠀⠀⢀⡴⠋⣡⡴⠶⠶⢦⣄⠉⢷⣄
⠀⠀⢠⡟⢁⡾⠁⠀⠀⠀⠀⠘⢷⡀⢻⡀
⠀⢰⡟⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⢀⡿⠀⣠⡴⠿⠂⠀⠀⠀⠀⢀⣼⠃⣾⠁
⢸⡇⢰⠏⠀⠀⠀⠀⠀⠹⣟⠉⠀⣰⣿⣦⣄⡀
⢸⡇⢿⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⣿⡁⠀⠈⠙⠻⠶⣤⣀⡀
⠈⣧⠘⢧⣀⠀⠀⠀⣀⡼⠃⣰⠏⠈⠙⠳⢦⣄⡀⠀⠀⠉⠛⠳⢶⣤⣀
⠀⠈⠷⣄⡉⠛⠛⠛⣉⣤⠞⠁⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣄⡀⠀⠈⠉⠛⠶⣤⣄⡀
⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⣀⠀⠀⠉⠙⠷⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣄⡀⠀⠈⠙⠛⠶⣤⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣀⠀⠀⢉⡟⠁⣾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣾⠁⣼⠁⠈⠙⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠻⣶⣤⣀⠀⠀⠉⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢦⠉⠛⠷⢦⣄⣀⠀⠈⠙⠻⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠋⠀⠀⠀⣤⢦⣭⡷⢶⣤⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢠⡶⢤⣀⠀⠙⢶⡄⠀⠀⠈⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣤⣠⡟⠀⣰⠏⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢿⣄⣀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣈⣁⣀⣈⠀⠀⠠⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣀⡤⠖⠋⠉⡠⠀⢠⠉⠉⠓⡦⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠞⠉⠀⠀⠀⠀⣀⣎⣰⠇⡀⢀⠙⢏⠉⡁⠄⠢⢀⡀⠿⣽⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⢀⠜⢿⢰⢄⡳⢮⣴⣾⣅⢠⢭⠀⠅⡶⠄⢠⡄
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠂⠀⠀⠀⠀⣦⡀⠀⠀⠿⢶⣤⣵⣾⡧⢙⣿⠀⡆⠸⣾⣴⣣⢆⣞⣻⣷
⠀⠀⠀⠀⠸⠀⠀⠈⠏⠙⠉⠙⢳⢂⣿⡂⠀⠀⠀⣤⣤⣘⠒⢺⣿⡤⢠⡧⣴⣿⣴⣿⣿⣿⣿⢿
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢠⠏⣸⣿⠀⠀⠀⠀⠙⠟⠻⣿⣿⣾⣧⡾⠿⠿⠿⠿⠿⠿⢿⣽⣷
⠀⠀⢀⡎⠀⠀⠀⠀⢀⠄⢀⠎⢠⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡜⠀⠀⠀⠀⠀⢠⡴⡞⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⠀⠀⠀⠀⠁⠠
⠀⠐⠁⠀⠀⠀⠀⠀⠀⡿⠁⣼⡟⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⢌⡓⢄⠀⡀⠄⠐⠈⠑
⢆⡀⠂⠂⠂⠂⠠⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠳⠚⠥⠄⠀⠀⠀⠀⣀⣤⣶⠀⠀⠀⠀⠀⠀⣠⠤
⠈⠛⠿⠷⠶⢶⣶⣖⣶⡾⠟⠁⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠈⠓⣒⡒⣄⣖⡚⣛⡛⢻⣓⡀⠀⠀⠀⢠⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⣀⣠⠴⢺⣏⣚⡏⠙⠩⠉⢩⢿⡏⠙⠂⠀⠠⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢸⡏⠀⡠⠻⡛⣡⡟⡠⢀⣾⠾⠏⠉⠙⡃⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡇⡀⣱⠓⣸⣧⠷⣟⣻⠶⠔⠀⠀⠂⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣯⣴⡗⠚⠉⠙⠀⢘⠁⠀⣁⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡍⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⢶⣾⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣉⠉⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣀⣤⣤⡶⠶⢛⣻⣭⣉⠀⠀⠀⠙⠻⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠛⢛⣉⣡⠤⠒⠲⢿⣿⣿⣷⣦⡀⢿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⣿⡁⠶⢿⣧⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⢀⣾⣿⣿⡇⢰⣯⣃⠈⢿⣿⣿⣿⠇⠸⣿⣿⣿
⠀⠀⠀⠀⠈⣭⣤⣤⣤⣤⣭⣭⣥⣬⣭⣤⣤⣤⣬⣭⡅⠀⠀⠀⠀⣿⣿⣿⣿⣿⡖⠛⠛⠛⠛⠓⠒⠒⠛⠓⠛⠛⠛⠛⠒⠒⣿⣿⣿
⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⢀⣴⠖⢶⣄⠀⠀⠈⠻⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢀⣠⣿⣿⣿⠟⣉⣍⡛⣿⣿⣿⣧⣀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣧⣌⠻⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⢛⣭⣛⣿⣟⠀⠛⢮⡁⢸⣿⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣶⡄⠀⠀⠘⠷⠦⠾⠋⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡁⠈⢙⣿⣛⣛⣂⣙⣋⣥⣿⣿⣿⡟⠉⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣦⣤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡁⠄⠀⠨⣙⠳⣁⠀⠀⠈⠉⠙⢫⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡌⠀⠀⠀⣼⢸⠑⠄⠑⡀⠀⠀⠀⠀⢳⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⠀⢒⡀⠛⢧⠀⠈⠂⢑⠀⠀⠀⠀⠈⡎⠆
⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠀⠀⠢⡤⡀⠄⠀⣰⣄⢀⠘⠀⠂⠠⠄⠠⠀⣘⡘
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⡀⢿⣧⢠⠿⢤⡉⢅⢀⡀⢰⡄⠀⠀⠀⢰⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢐⡄⠁⠿⣟⣲⡾⠦⣗⣠⠄⣴⣴⣀⣀⢀⣯⡟⠉⠉⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡟⠠⣆⡏⢠⠀⠀⠀⣨⠀⠓⣾⠃⠀⠐⠢⡈⢳⣤⠋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢖⡏⠒⠤⢉⢀⠲⡦⣤⡘⢀⣣⣘⣦⠀⡤⠤⠴⠂⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠷⣀⡏⠀⠠⠠⠤⠀⣀⠀⠊⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠊⠀⠀⠠⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠁⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠲⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢐⠀⠀⠄⡠⠀⠁⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠐⠀⠄⠀⡀⠠⠀⠀⠐⠡⠀⠀⠀⠐⠀⡀⠀⠄⠂⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⡂⠈⠄⠂⠀⠀⠀⠁⠀⠀⠈⠀⣀⣈⠀⠐⠒⠊⠉⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠒⠒⠒⠐⠚⠉⠉⠁