Peanuts Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣤⠶⠒⠞⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣏⣩⡍⣭⢉⣒⣖⣯⠭⠆⡛⠂⠛⠘⠃⢼⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣨⠅⢶⠆⠠⠀⠮⠄⠰⠒⠈⠉⠀⠀⠁⠂⠰⢉⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⠇⠘⠁⠀⠀⠀⠀⠦⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⡠⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢆⡀⠠⠀⠀⣀⠔⠋⠁⠀⠈⠁⠀⠐⠃⠈⢁⠤⠒⠒⠒⠀⠒⠂⠤⠀⢀⠤⠔⠒⠉⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⠀⠀⠀⠀⠀⠀⢄⠀⣂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⣈⡉⢠⡀⠀⠀⠀⣀⠀⣀⠁⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡦⡤⡀⢀⡀⠀⣀⢠⣤⣦⢤⠁⣤⠀⣠⡄⣠⡌⠤⢼⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣥⣩⣥⣬⣭⣤⠶⢿⣿⣾⡞⣲⡄⢴⣦⣶⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠒⠒⠞⠛⠓⠒⢦⣀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⡠⡀⢠⣀⣦⣴⣀⣀⣟⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡻⠎⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠉⠛⠛⠳⣦⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣮⣄⠀⢀⣀⣀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣤⣶⣽⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⣽⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠚⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣤⣁⡀⠀⢀⢀⣠⣶⠟⠁⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠓⠤⣤⣤⣤⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⠄⠀⠾⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣪⠟⠉⠀⠀⠀⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣀⠒⠤⢤⣀⣀⢉⠉⠛⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠄⢀⠒⡢⠤⡀⣀⣀⢀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⡀⠁⠀⡀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠈⣀⡞⠉⠉⠉⠉⠀⠈⠙⠢⣐⠀⡀⠂⠀⠁⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⡴⢃⣀⠀⡤⠿⠿⠖⠂⠀⠠⣄⠀⠀⠀⢀⡬⠷⢶⡤⠤⠤⠴⠶⠚⠁
⠀⠀⠀⠀⠀⠘⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠛⣁⠤⠤⢒⠭⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠸⡘⢄⣀⣀⠀⠀⠀⠀⠀⢀⡤⢚⠔⠁⠀⠀⣔⡁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠑⠄⡀⠈⠑⠒⠶⠂⠉⣩⠔⠁⠀⠤⠤⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠉⠀⠰⠇⠀⠀⠀⠀⠈⠑⠨⡑⠂⠤⠤⠤⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠤⠤⠄⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠇⣰⢀⠀⠀⠀⡀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠈⠃⠇⠃⠃⠃⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢤⠆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣋⣭⡤⣍⣍⠐⠀⠤⠤⠤⡤⠶⢖⣛⡭⠷⠒⡛⢛⠛⠦
⠀⠀⠀⠀⠀⣾⠀⠴⢃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠈⢁⣠⢤⢦⡬⢀⠇⠀⠐⣂⣟⣻⡩⢥⠶⠖⠺⡏⠉⠙⠁
⠀⠀⠀⠀⣸⠃⠀⢰⡇⢊⡜⢡⡆⣼⠇⢹⢀⡞⣰⠃⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢣⠤⡤⣤⢤⣤⣤⠤⠤⠼⠤⢷⠓⢒⢒⠂⢨⠉⣦⠀⡀⠄⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠘⣛⡗⣿⠶⣶⣶⡶⠶⣖⣶⣳⣛⣋⠏⠉⡁⡉⢈⠀⣙⣀⣥⡤⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣁⠙⣿⡂⠀⠀⠀⠀⠉⠛⢿⣯⠓⠓⠓⠚⠤⠧⣍⣀⣉⣉⡉⣩⣀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⣐⡒⠒⠂⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⡠⠛⠋⠉⠀⠀⠀⠀⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢷⣦⣄⡉⡑⠒⠢⣄⣰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠉⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠫⠴⠶⢦⣍⡳⣤⣀⠀⠀⠀⠀⣀⣀⣠⡴⢟⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣌⡀⠀⠀⠀⠀⠀⢀⡀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣯⣍⣓⣛⣛⣋⣩⣅⡶⠋
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠉⠑⠢⢄⠀⠴⠃⠀⠀⠈⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠐⠲⢼⡄
⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠉⠙⠲⣤⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⠶⢶⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣠⡾⢉⣔⡉⠀⠂⠀⠄⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠂⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠄⠼⠒⡾⠀⠀⡄⠀⠀⡠⢁⢸⠽
⠀⠀⠀⠀⠀⠀⠁⠒⠢⠤⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠂⠐⠀⠀⠘⠀⠀⠰⠣⣀⡏⠰⡀⢀⡴⠚⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠐⠀⠀⠀⠀⠀⠀⠀⡀⠃⢘⡞⢠⠋⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⣣⣄⠈⠀⡀⠀⠀⢠⠀⢐⣠⠴⠚⠉⣠⡇⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣨⠋⢀⡜⠁⠀⣠⠋⠀⠀⢂⡤⡨⡿⢁⡴⠂⠁⠉⢠⡀⠀⠀⠀⢀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⡃⠀⣜⠀⠀⣼⠃⡀⠀⢀⡼⢠⣻⢁⡾⠀⠀⠀⠀⠀⢶⡀⠀⢀⡜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣽⠀⠀⡇⠀⣰⠇⡀⢈⡤⢎⣱⢿⠇⡼⠁⠀⠀⠀⠀⠀⢸⣧⡴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⣆⠀⢧⡀⡝⠠⠴⣩⡶⠟⠋⢰⠀⣇⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣢⡄⠣⢈⡴⠋⠁⠀⠀⠀⠀⠂⠻⣤⣤⣤⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉


⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⢤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠂⠉⡀⠤⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⢈⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⡀⠀⠀⢀⠀⠀⠀⢀⠔⠒⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⢁⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⢰⠀⠀⠀⠀⡈⠛⠋⠲⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢩⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠐⠀⢨⠀⠀⠀⠘⢧⣁⣤⣔⠂⠀⠀⠀⢴⡀⠀⢰⡆⠀⠐⠀⠀⠙⠳⠿⠿⠶
⠀⠀⠀⠀⠀⣠⠒⡁⠒⢄⠀⠀⠈⠓⠤⣁⡀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣴⣠⣴⣾⠁⠀⠈⢧⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠳⠤⠤⠄⠀⠀⠀⢀⣴⣄⠄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡍⠁⠀⠈⣽⡆⠀⠀⠀⠀⡠⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢹⠹⡄
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⡄⠀⡀⠀⠀⣴⠚⠉⠁⠀⢀⡄⣶⡼⠃⢀⡀⣠⠎⠀⠀⠀⠀⣻⠀⠀⠀⠀⣠⡾⣏⠀⣐⣿⠑⡇
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠡⡀⠀⠀⢸⠇⠀⢀⣤⠒⠋⠛⠻⠒⠃⠀⠀⠀⠐⢡⣴⣖⣯⠏⠀⠀⣠⡞⠁⠀⣿⣿⣿⣯⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢁⠀⠀⢸⣾⣴⣿⡧⣂⠀⠀⢀⠀⡄⠄⠀⡐⢞⣣⠿⠟⠃⠀⠀⠀⣿⠀⠀⡠⠏⣫⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠼⠀⠀⠀⠉⠉⠉⠹⣿⣤⣫⣾⠾⠿⠶⠷⠿⠟⠃⠀⠀⠀⠀⠀⠀⡙⠛⢋⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡤⠖⠋⠉⠉⠁⠀⠀⠈⠉⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠶⢤⣄⣀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⣠⠞⠁
⠀⠀⠘⠦⠤⣄⡀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠙⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢀⡴⠉⢀⠀⠂⠈⠁⠂⢄⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡞⠀⡠⠁⠀⠀⠀⠀⠀⠀⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢸⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⡏⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⢀⠌⢀⡞⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⡠⠞⠁
⠀⠀⠀⠀⣰⠃⠀⠰⡀⠀⠀⠀⢀⡠⢀⡴⠋⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⣰⠃⠀⢀⣀⣈⠂⠁⠈⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⠃⠠⠊⠀⠀⠀⠈⠑⢄⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣀
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣀⠀⠀⠀⠀⠀⣀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠦⣄⠀⠹⣄
⠀⠀⢻⡀⠠⠀⠀⠀⠀⠀⠀⡰⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠈⣇⠀⠈⠳⢤⣀⠀⠀⠀⣀⡴
⠀⠀⠀⠱⣄⠑⢄⠀⣀⠤⢊⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠈⠓⠦⣅⣀⡴⠋⠀⠀⠀⣠⠖⠛⠛⠛⠲⠦⢤⣤⣀⣀⣀⣀⣠⣤⠤⢤⣤⣄⣀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠸⡄⠀⠀⠀⣀⡤⠖⠉⢀⡤⠖⠋⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⠑⠚⠉⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⠤⣤⣀⣀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⣠⠴⡒⠈⢛⡋⢐⠒⠦⢄⡀
⠀⠀⠀⠀⠀⢀⣠⠤⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠙⢦⠀⠀⠘⠁⠶⠶⠤⢤⡉⠁⠢⠈⣢⠍⠢⡀⠀⠀⠀⠠⠤⠄⢄⣀
⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠠⠑⠦⣀⠒⠈⠲⣄⠠⠘⠀⠀⠁⠀⡠⠐⠈⢉⢓⡲⣤⣍⡢⡀
⠀⠀⠀⡼⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠸⡄⢤⢦⡀⠳⢤⡈⠀⠑⠂⠀⠀⠀⠀⣷⠒⠋⠉⠉⠉⠉⠉⠙⠛⠦⣄⡀
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠋⠉⠀⠀⠀⠀⠈⢄⠣⢜⠃⢤⠚⠒⢤⣀⠀⠀⠀⠂⢀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠒⠊⠉⠀⠀⢀⡀⠤⠤⠤⣄⡀⠀⠀⠀⠈⠣⡀⠳⢤⡉⠁⣠⠐⠀⠀⠀⣦⢌⠓⠶⠤⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠊⠉⠁⠀⠀⠁⠀⠐⠙⠆⠀⠀⠀⠀⠈⠢⣄⣉⠒⠦⠤⠌⣍⠢⣉⠒⢭⠒⢤⠬⠉⠳⣄⣀⣀⣀⡀⠤⠖⡫⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠶⠛⠋⠉⠓⢤⡈⠐⠤⠀⣀⡀⠀⠠⠀⠀⢀⠀⢒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⡀⠀⠙⠁⠀⣩⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣤⣤⡤⠤⠒⠋⣁⣤⠖⠒⠒⠒⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠄⡠⠤⠴⢋⢁⡠⠤⠐⠊⠉⠙⠁⠀⠆⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣠⠖⠉⢁⣀⣀⠠⠀⠀⠀⠂⣀⣬⠵⠖⢒⠛⠿⠅⡈⠳⠤⠾⠀⠀⢳⡄⠀⠀⠀⢀⡔⡉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⢠⠋⠔⠚⢋⣁⡀⠤⠀⣤⠀⠀⢀⣀⡀⠠⣄⠀⣀⠆⠀⠥⠤⠀⢀⠀⠀⡜⠀⠀⠀⠀⡞⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⡏⠠⠖⢰⣊⣁⠩⠤⠴⠚⠊⣉⠃⠁⠐⠚⠁⠤⠠⠄⠀⠠⡐⠂⠣⢀⠞⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞
⠀⠀⠀⠀⢻⡉⠋⠈⣈⣀⣠⠴⠔⠊⠉⠀⠀⠀⠠⠄⠀⠀⠀⠃⠀⠐⠀⣣⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⡀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠑⢄⠀⣀⣀⣀⠤⢀⠈⠀⣀⡤⠚⠁⠀⠁⠒⠒⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠼⠤⠖⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠃⣴⢃⡀
⠀⠀⠀⠀⠀⠁⡀⠘⣇⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠵⣄⠃⣪⢁
⠀⠀⠀⠀⠀⠀⣰⣦⡈⠃⠁⠠⠀⠂⠀⠀⠀⠀⠀⠀⢩⣛⣷⡄⣳⢂⠘⠀⠀⠀⢀⣠⡤⠀⠶⢤
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⢀⣀⢰⡄⠀⠈⠉⠻⠆⠛⠙⠳⠶⢷⣾⢶⡷⠠⠄⢈⠁⢁⠐⢚⣻⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⡀⣹⠗⣠⣤⡐⠛⠙⠟⠰⠛⠓⣦⠀⠤⢬⡀⠐⠒⠶⠉⡤⠶⡶⠉⣴⢿⣄⡷⠀⠀⠀⠀⡀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⢻⡏⣠⡥⠌⠒⠛⢻⡖⠚⠛⣷⡦⢠⣶⣶⣶⠰⠾⠛⠿⣂⣠⣶⢟⣀⢿⣵⣿⣶⣈⠨⢤⡍⣤⣄⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣧⣵⡶⢦⣍⠙⠻⠄⢢⣄⠻⠋⠭⠀⠲⠎⢽⣶⣿⣿⣿⣏⣵⣿⣫⣤⣿⣿⣃⣬⣤⣀⢤⣀⣩⣤⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⡛⠶⠦⠴⢿⠷⣿⣿⠀⣤⠶⠶⠈⣙⠛⢰⡝⢿⣿⣿⣟⣯⣶⣿⣿⣯⣉⣉⣬⣭⠰⢾⡍⣹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠋⠉⠉⠉⠉⠛⠶⣶⣶⣿⣿⡏⢰⣷⣾⣷⣿⣽⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣷⡶⠿⠿⠧⣿⣷⣮⡯⠟⠋
⠀⠀⠀⠀⠰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢟⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣨⡭⠽⠿⠛⠛⠫⠤⣀
⠀⠀⠀⠈⢧⣝⠢⢄⣀⣀⠀⠀⠀⢀⣀⣄⣀⣤⣶⣿⢿⣿⣿⣿⣽⣯⣽⣿⣿⣿⣿⣿⣅⣀⠀⠀⠀⠀⠀⠀⠈⣷⣤
⠀⠀⠀⠀⠘⡼⣿⣧⣭⣭⣷⣶⣾⣿⣿⣟⣛⣿⣿⣧⣿⣿⣭⣿⣿⡿⠟⠋⠉⠉⠉⠉⠙⠛⢿⣦⣄⢀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠐⢮⣿⡿⢿⣽⠯⣭⠔⠛⣛⡥⠛⠉⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠃⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⡸⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣆⣀⣠⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠞⡀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠀⡀⠀⢀⣀⡈⠂⠀⠂⢠⡃⠀⠀⠜⠁
⠀⠀⠀⠚⠀⠀⠀⢠⡾⠋⠉⠉⠙⢻⡁⠀⠀⠉⢲⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⢀⠀⠀⠀⠀⡈⣳⡀⠀⡄⣅⣀⡠⠖⠀⠐⠪⣠⡞⠁
⠀⠀⠀⠈⢁⠀⠀⠙⠛⠒⠠⠤⠤⣚⢙⣒⢒⣶⠞⠀⠀⠈⢄⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣰⡏⠉⠉⠉⠙⣿⣁⡀⠁⠀⠀⠀⠀⡅⠈⠂
⠀⠀⠀⠤⠄⠀⠐⠁⠀⠑⠠⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠒⠃⠀⠀⠀⠈⠃⠉⠉⠀⠒⣉⠍⠉⠀⠀⠀⠠⢤⡚⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⠤⣀⠀⣦⠀⠀⠀⢀⠖
⠀⠀⠀⠀⣀⣠⠤⠤⣤⡞⠉⠀⡀⠄⢹⠃⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⢀⠗⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⠉⠐⠄⢠⠈
⠀⠀⠀⣮⠉⠀⠀⡠⠒⠑⠒⠛⠃⠀⠋⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⠀⠀⡬⡶⠁⠀⠀⠀⠀⠀⠀⢀⠄⠀⠚⠁⠀⡀⠀⢀⠀⠀⡼⡯⠀⢀
⠀⠀⠸⠈⠉⢉⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⡀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠤⠂⠀⠀⠃⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠲
⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠊⠙⠀⠐⠂⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠠⠴⠁⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠂⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠀⠀⢨⠅⠀⢐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠴⢄
⠀⠀⢀⠔⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⡆⠀⡠⠁⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠐⠈⠁⠀⠀⠙⢤
⠀⠀⢈⠀⢀⣤⣤⣠⣴⠂⠀⣀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠑⠦⣄⣻⡀⡧⠴⠋⠀⢤⡄⠀⠀⠢⣄⠀⠀⠸⡄⠀⣀⣀⣄⣀⣀⣀⣤⡀⠐⠂⡄
⠀⠀⢰⡋⠁⠀⢀⡨⠷⠶⢾⠁⠸⠂⠐⠀⠀⠀⠀⢢⠀⠀⣠⠖⠚⠉⠙⢻⣁⣠⡔⠀⡨⠁⠀⠀⠀⡀⠑⠶⣤⣿⠋⠀⠈⠉⠛⡅⠀⠈⠉⢲⡄
⠀⠀⠧⠉⠉⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⠈⠀⢠⡻⠤⢤⠔⠒⠋⠤⠘⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠛⠂⠲⠶⣚⠉⣓⠒⣦⡾⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠠⠄⠊⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠢⠄⠠⠐⠁⠀⠉⠒⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⢦⣤⡠⢤⡤⣄⢠⢴⠀⠀⠀⠀⠀⢀⡤⠔⠛⠈⠢⢀⢀⣀⡀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠉⠀⠀⠀⠀⢀⡠⠾⠀⠀⣀⣤⠞⡩⠉⠀⠀⠀⢨
⠀⠀⠀⠀⠀⢄⣀⡴⠖⠁⠀⠀⠉⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢀⡼⠋⠀⠀⠀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⡤⠚⠉⠀⠀⠀⠀⣀⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣀⢀⣀⠀⠀⠀⠀⠀⠈⠢⢀⡀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠙⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠉⠒⠲⢤⣀⣤⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⢰⣀⡀⠀⠀⠀⠐⠀⠀⠀⢈⢣
⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠉⠓⠆⢭⡠⠆⠒⠐⠰⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠄⠀⠐⠀⠈⠉⠐⡉⠀⢀⡀⣈⠔⠁⠀⠀⠀⠀⠀⠙⠲⡅⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⣤⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⡤⠀⠀⠀⠀⠀⣀⠀⣄⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠠⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⠀⢄⠀⠉⠣⡀⠀⠈⡑⠖⠍⠌⠒⠉⠐⠀⠀⢘⡀⠀⠤⡀⠀⠀⠀⠠⠂⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡦⠀⠐⠊⠈⠁⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠈⠑⠂⠒⠄⡀⠀⠈⠢⣀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢆⡀⠀⠀⡰⡄⠀⠀⠀⠀⠀⠀⠀⠤⣤⡤⡔⠈⠦⡀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⢀⠴⠋⡄⠀⠹⡄⡔⠁⠈⠒⡠⣤⡠⠐⠃⠀⠀⠀⠀⠀⠉⠢⡠⠟⠉⠑⠂⢤⠀⠊⠁⠙⢆
⠀⠀⠀⣠⠖⠁⠀⠐⠀⠀⠀⠹⢀⠀⠐⠁⠀⠀⠈⠢⠀⡀⠀⠀⠀⠀⠀⠀⠳⣀⡀⠀⠀⠀⠱⡯⠀⡀⠀⢇
⠀⠀⡜⠁⠀⠀⠀⠇⠀⠀⡠⠎⠀⠀⠀⠀⠀⢀⠸⡇⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁⠪⡂⢸
⠀⡘⠀⠀⠀⠀⢰⣀⡠⠄⢀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠘⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⡜
⠀⢇⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠁⠀⠀⢔⠉⣄⠀⠀⠀⠀⠀⣇⢀⣇⠠⢀⣀⢀⣠⠴⢊⠗⠢⢄⡀⢀⡠⠁
⠀⠸⡀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⠄⠂⠠⠤⠒⠁⠀⠀⠀⠈⠐⠠⣠⠉⠉⠀⠀⠀⠀⢠⠃
⠀⠀⠑⢄⠀⠀⠈⠢⡀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠉⠀⠒⠠⠬⢄⡀⡰⠋⢘⣄⣄⠀⠀⠀⠀⠀⡠⠂⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠊⠁⠘⠃⠀⠄⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠒⠊⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠛⢛⡛⠻⠿⠟⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠁⣻⣷⣾⠿⠟⢀⣀⠀⠰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⣿⣦⣾⡿⠛⢿⣷⣾⣿⠛⠁⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠛⠉⢀⣸⣿⣿⣄⣴⣿⠟⠛⠟⠀⢠⣶⡄⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⣠⣴⣾⠿⠟⠋⢙⣿⣿⣷⣦⠀⠀⠀⣼⣿⠁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⣠⣾⡿⠁⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⣿⣿⡟⠁⠀⢀⣤⡀⠀⠀⢀⡀⠀⠀⠰⣿⠟⠋⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣆⠀⠀⠀⠹⣿⣦⣠⣿⡏⠀⢀⣀⠀⠀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠻⣿⣆⠀⠀⠀⠈⣹⣿⣿⣦⣤⣿⡟⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠈⠻⣷⣤⣠⣾⡿⠋⠈⢉⣿⡿⠟⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⢠⣾⣿⠿⣿⣶⣤⣴⣿⠟⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⢉⣀⣤⣴⣿⠿⠛⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣷⢸⣟⡛⠁⢠⣷⡀⢸⣧⡀⣿⢰⡇⠀⣿⠘⢻⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠁⢸⣏⣁⢀⡾⠼⣧⢸⡇⠹⣿⠸⣇⣀⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠁⠁⠀⠈⠀⠁⠀⠈⠀⠈⠉⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠒⡋⠀⠐⠀⠀⠀⠂⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠥⠙⠉⡠⠀⠱⠤⡀⢠⡴⠟⣳⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠐⠿⡣⠀⠘⠀⣀⢀⣠⠀⠀⢶⡿⢟⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠻⡿⢔⣥⡘⠃⠠⠀⡾⢁⡀⠾⠿⢀⡀⢴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠰⣶⠚⡍⢛⡻⠉⣡⠔⠀⠏⣾⠆⡀⠃⠆⠀⠀⣼⣄⣾⣿⣷⣦⠎⡻⠁
⠀⠀⠀⠀⠀⠀⣔⣈⡼⣯⢆⡣⡀⢸⡷⣆⠛⠹⠙⣿⠟⡒⢿⡿⢒⡐⢠⣶⣼⣿⣿⣷⣿⣿⣿⠟⠁⠀⣀⠤⠄⠀⠀⠀⠀⠀⢀
⠀⠀⢀⡠⣆⠀⠛⡉⠐⡀⠻⣠⠄⠀⢩⣥⣄⠾⠻⡎⢣⠶⡆⢠⣿⣿⣯⣷⣿⣿⣿⠿⠿⠛⠁⡠⢒⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡐⢠⡀
⠀⣠⡾⠈⢋⡀⠈⠓⢂⠠⢄⢴⠏⢻⡄⠃⣚⣄⠨⠙⠈⢢⠾⠆⢿⡿⠇⠈⠉⠀⠀⠀⠀⢠⣺⡀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣵
⢠⢻⣟⣦⡙⣊⠀⠐⠇⢀⠀⠄⠑⣭⣷⢆⠻⠟⣁⠺⣷⠘⣴⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣜⢿⡲⢦⣀⡀⠀⠠⠄⠀⢀⣀⣀⣠⣾
⣼⣶⣿⣯⢾⡧⠄⠀⡀⠤⡴⠺⡂⠿⡖⣁⣄⢻⣣⢰⣵⣆⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⣴⠒⠦⣌⣳⣒⣷⣶⣯⣿⣷⠟
⣿⣿⣟⣿⣽⣦⣾⣷⣤⣜⠛⡂⣡⡄⣿⣿⣿⣤⣾⣿⠿⠋⠀⢀⣠⠴⠒⣪⡅⣀⣀⠀⠒⠂⠀⠀⠄⠀⠀⠀⠈⠘⠓⠛⠛⠛⠋⠉⠉⠁
⠙⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠿⠛⠂⠈⠉⠉⠉⠉⠙⠒⠦⢴⣛⡤⠞⠛⠛⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⣱⠳⠞⠋⠀⠀⠁⠉⠀⠀⠀⠀⠀⠈⠑⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢀
⠀⠀⠀⠀⠈⠙⠛⠻⢿⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠒⠠⠂⠀⠀⠀⠀⠀⠄⠀⠀⣠⣞⣡⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣘⣵⠆⠚⡮⠂⢴⢗⠦⡠⢫⡠⠀⣴⣷⣤⣶⡦⢴⣶⣶⡶⠘⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡶⢶⣾⣦⣼⣷⠟⠤⠖⠀⠀⠀⠉⠀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠂⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⣄⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠾⠃⠠⠅⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢤⣟⡫⠉⠓⠀⠀⠀⣀⡀⠀⠀⠘⡷⣄⠀⠀⠀⢀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⢿⡿⠿⠏⠁⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠤⠀⠈⠀⢨⡥⠤
⠀⠀⠀⢀⠀⠠⠐⠧⠄⠉⠁⠀⣤⠀⠐⠆⠃⠀⠘⠤⠀⠀⠁⠀⠀⠐⠂⣀⡴⡜⣿⠀⠈⠋⠁⢀⡤
⠀⢀⡀⠈⠀⠀⣠⣤⣜⣧⠀⠐⠈⠦⠤⠌⠂⢒⣀⣀⠈⠀⡀⠀⢀⡤⠄⠘⡂⢈⡀⠤⠰⣾⠁⡀⠒⠁⠀⠀⠀⠐
⠀⣨⡄⠐⢯⠀⠈⠷⠆⠀⠈⣀⡄⢀⢶⡾⠒⣏⠈⠳⠀⠀⠀⣀⣵⡦⠠⣼⣇⠎⢷⡇⣀⣉⡯⡴⢰
⢀⢛⣯⡀⠀⠀⠀⠀⣴⠃⣀⠼⣇⠌⠈⠃⢀⡠⠀⢶⡂⠃⠉⣟⠀⠀⠒⠃⠣⠀⣈⣉⣭⣠⣾⠆⠉⠂
⢀⢺⢏⣡⠀⢇⠀⠀⠈⠂⠁⣀⠠⠀⡴⠂⠈⠧⠀⠘⣧⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠻⠛⠋⠉
⠀⣿⠿⣿⡴⢴⡖⠂⠠⡄⠘⡏⠀⠀⠑⠀⠀⠀⠂⡂⠀⢀
⠀⠈⠋⠹⡾⠉⠧⡴⢂⠦⡄⠁⠒⢀⣀⢀⡀⡶⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠐⠓⠊⢹⡭⠺⡿⠂⠻⣟⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣼⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⢀⠠⠟⠙⠋⠀⠒⠠⠤⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠒⠛⢻⢓⡲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢴⡶⠶⡶⠶⠶⠒⠶⠚⠿⢹⣁⣀⡬⡦⢲⠇⠐⠐⣶⠌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⠐⠀⣀⣘⡀⢀⢒⠂⠍⠩⠭⠹⠉⠱⢀⣀⣸⠤⠶⠒⠋⢽⠡⢤⡇
⠀⠀⠀⠀⠀⠀⢀⡜⢃⡠⢔⡊⠙⠁⠘⠀⢈⣃⠴⠖⡒⠛⡗⠉⢻⣙⢁⠠⢂⠁⢄⣈⠥⣠⠟
⠀⠀⠀⠀⠀⠀⣼⢁⠩⠂⢀⠀⠐⠀⠀⠋⣉⣀⣀⠬⠤⢒⠖⣪⡽⠶⠒⠲⠛⠺⢭⡴⠚⠁
⠀⠀⠀⠀⠀⠀⣟⠅⢇⢠⠓⠐⠒⠁⡘⠋⠉⠠⢃⣀⣐⡰⠋⠁⠀⠀⠀⠀⢀⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠹⣆⠠⠈⠐⠋⣠⣈⣦⠤⠖⢋⣩⡥⣾⠁⠀⠀⣀⠤⠔⠚⠉⠀⢀⣾⠀⢀⣠⡤⠴⠶⠶⠦⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠮⢥⣌⣻⣭⡭⠶⠛⠉⠀⣰⠃⠀⠐⠉⠀⠀⠀⠀⣀⡴⢫⣯⠞⠉⠀⠀⣀⣀⣀⡠⠤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣦⣤⣀⣤⡤⠶⢚⣩⠖⢡⡏⠀⠀⠊⠉⠁⠀⠀⠀⣠⢴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠀⠀⠀⠙⠷⣦⣤⡤⠤⠴⢚⣩⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⢀⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠁⠀⠁⣶⢨⣤⣌⠢⠙⢢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠘⠁⢰⠁⠀⢀⡟⠈⠙⡟⠃⠀⣶⡶⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⠆⢠⠎⢙⠃⣤⣬⡅⡤⣄⢀⣀⠀⠈⢂⣀⡉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃⠠⠀⣁⣂⡈⠇⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⢳⢀⣯⣅⠁⠒⠀⠱⠀⠄⠐⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠋⠀⢩⣍⣐⡃⠺⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢃⠘⠒⠂⡀⠀⠀⢈⣿⠉⠀⠀⠠⠂⠉⠋⣦⢸⠋⠉⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡘⡀⢻⠀⠀⢤⠀⠀⢁⠄⠀⠂⠂⠀⠀⢤⣤⡄⠀⣥⢿⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣶⠛⠂⣀⣀⡀⠘⠀⠀⢀⠀⢀⡤⣤⣄⠁⣰⠛⡿⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢷⣼⣿⡆⢹⠈⠋⢰⠉⠀⠀⠀⡀⠀⠈⠱⢀⢬⣇⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⢁⠀⠀⡷⠿⠀⡀⠀⠀⠀⡀⣀⡠⣤⠘⡤⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠰⡶⠀⠇⠀⠀⠁⠀⠀⠈⠁⠀⠀⠂⢸⡟⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠰⠀⠀⢀⡀⣤⢰⠒⠂⠒⠂⠘⣂⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠀⠈⠀⠀⠀⠀⠈⠀⢠⠈⠲⠄⠠⠀⠈⠟⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⡄⠀⠀⠀⠀⠔⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣤⣴⡄⢠⣴⢶⡄⢸⠛⠏⠃⠹⠞⣠⣤⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣻⡇⣼⡟⠹⠃⡼⠟⠚⡇⢀⣀⣀⡀⠐⠟⠉⠈⠦⢰⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⡄⣨⣴⣶⢀⣶⣤⣀⡃⠀⠉⠉⠁⢰⡾⠛⠻⠆⠀⢁⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣮⠀⡟⠀⠈⠸⠃⠋⠀⠁⣰⠆⠿⡇⠀⣁⣀⠀⠁⢀⠚⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢧⡈⠓⢶⣇⠀⣟⡛⠓⡃⢈⠀⢸⡇⠀⣁⣀⠀⢀⠀⠀⠤⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢹⡇⠈⢫⠠⣽⣄⢄⡁⠖⡆⠰⡆⠀⠉⠉⠀⡿⠋⠹⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⣳⠘⣏⣧⡟⠻⣼⠃⣠⡽⢬⠃⣶⣶⠀⣼⣻⡿⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡅⣀⡘⣿⢳⡄⣿⡷⡏⢀⣼⠐⠋⠟⣰⢦⣍⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣍⠺⣿⣿⣷⡄⣽⣶⠀⣴⡀⠘⠃⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⣛⠻⠀⣿⡏⡼⠛⠁⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠀⠤⠘⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠠⠠⠰⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠂⠔⠀⠂⠀⠐⠄

⠀⠀⡸⡀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⡤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠁⠀⢀⠧⢀
⠐⠁⠈⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠈⠂⠀⠀⢀⡈⠃⢀⠤
⠄⠐⠠⢀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠉⠀⠹⡀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠴⣄⠀⠀⠀⠀⢠⡈⠀⠀⠀⢤⡀
⢆⠀⠀⠀⠁⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠈⠀⠀⣶⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⣁
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⡔⣋⠄⠀⠀⠀⠙⢦⡆⠀⠀⠀⠀⠈⠂⠀⠀⠘⠃⠀⠀⠳⢄⡀⠀⠀⠀⠘
⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠻⠈⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⡀⡀⠀⠀⠀⠚⠒⠀⠀⠀⠀⠽⠥⠤⠀⠁
⠀⣀⠦⠠⢀⣀⡀⠀⠀⠀⠀⠻⠃⠀⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⢀
⡾⠋⠀⠀⠀⠈⣿⣧⠀⠀⣠⡴⠖⠋⠉⠀⠈⡟⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠰⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁
⣇⠀⠀⠀⠀⣰⡿⠉⢀⡜⠉⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠀⠀⠁⠀⡀⠈⠂⠀⠀⢀⠀⠀⠨⠀⠀⠀⠠⠀⠀⠀⢠
⠛⠿⠶⠶⠚⠙⠀⣀⣼⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⣠⣀⣀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠰⠀⠀⢱⠐⠄⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⡶⠟⠉⠁⠀⠀⠀⡀⠀⡴⠊⠉⠀⠀⠀⢀⠀⠀⢰⠟⠛⠉⢉⠉⠒⠈⠀⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠟⠀⠰⣀⠀⠀⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠐⠁⠐⠇⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠀⠃⠠⠀⠀⠐⠂⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⠀⠀⡠
⠀⠀⠀⢧⡀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⢠⣞⠇⠀⠀⠈⠂⢄⡀⠀⠀⡠⠊⠀⠀⠀⠀⠠⠀⠀⠀⡌⠒⠀⠀⠀⠀⠀⠀⠢⠀⠠⠁
⡀⠀⠀⠈⢉⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠃
⠉⠉⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠢⢀⠄⠀⠀⠀⠀⠀⠰⡄⠀⠀⢠⡞⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⡀⠀⠀⠀⠀⠈⠀⢀⡀⠀⢠⠶⢦⣀⠀⠀⠀⢠⠁
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠑⠈⢑⡂⠀⢈⣶⠟⠀⠀⠟⠀⠀⣀⣤⠞⠔⠁⠀⠀⠀⠈⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡏⠀⠀⠉⠀⠐⠄⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⡢⠖⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡀⢀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠻⣆
⠐⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠐⠠⠂⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⡆⠀⠀⢹⠏⠀⠂
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣔⣴⠿⠛⠻⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⣇
⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠸⣇⠀⠀⠀⠀⠈⣷⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⢷⣀
⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⣽⣇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠈⠳⢤
⢉⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⢀⡀⠼⠀⠠⠉⠁⠒⠒⠟⠻⠋⠀⠀⠀⠀⠈⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠠⠐⠈⠀⠀⠀⠢⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠂⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠅⠀⠰⠀⠀⢀⠴⠀⠊
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀
⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⣀⠄
⠀⠀⢱⡄⠐⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡫⠂⠀⠀⠀⠉⣦⣀⡀⠀⠀⣀⡤⠤⠤⡀⠀⠀⣠⠤⣄⣼⠀⠀⠀⢧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⠏⠀⠀⠀⢀⣠⡎⠁⠀⠉⠉⠉⢁⣀⣀⡀⢈⣀⡴⠃⠀⢈⣿⠀⠀⠀⠀⠉⠉⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⣿⠀⠀⠀⠀⢋⣤⠷⠒⠒⠒⣦⡾⠃⠀⠈⠙⠋⠉⠀⠀⠀⣼⠃⠱⣤⡀⢀⡠⠤⡄⢠⠗⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠒⠉⠙⠐⠲⢄⣴⠏⠁⠀⠀⢠⠞⠅⢻⡀⠀⠀⠀⠀⠀⠀⡠⢞⠡⣤⣀⣀⡰⠋⠀⠀⡟⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⣴⣶⣏⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡏⠀⠀⠀⠉⡳⠒⠒⠒⠲⣧⣠⡏⠁⠀⠉⠉⠀⠀⠀⣰⠇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⢹⡄⠈⠙⠻⢿⣷⣀⠀⠈⣇⠀⠀⠀⢸⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢻⢧⣀⠀⠀⠀⠀⣀⣴⡥⠖⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠈⠉⠓⠛⠶⠤⣴⣿⣧⣀⠀⠀⢀⣧⡤⠤⣀⢀⣀⣨⣄⣬⠽⠿⠛⠛⠉⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣟⣛⣲⠶⠶⢶⣶⣶⣶⣶⣶⣶⡶⠶⠶⡖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠶⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⣤⣤⣤⢤⣤⠴⡶⢺⠋⠁⣔⣄⡤⡞⠠⠝⢤⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠍⠀⠈⠦⠈⢨⠤⠴⡒⢒⣒⢲⠞⢫⠉⠁⢹⣈⣤⡤⠴⢛⠖⢂⣿
⠀⠀⠀⠀⠀⠀⠀⢠⢾⠁⣀⡥⢔⡒⠋⠹⠀⠲⠀⣄⣌⢠⠶⡬⠴⢶⡺⠋⠉⠅⠠⠀⠀⢜⡀⣱⠏
⠀⠀⠀⠀⠀⠀⢠⡟⠀⡫⠥⠊⠀⠀⠂⢠⠔⠺⡉⠁⡔⣁⣄⡡⠤⣾⣊⣤⣭⢴⣤⡜⣋⡭⠞⠁
⠀⠀⠀⠀⠀⠀⣾⠑⢫⠀⢔⡁⢤⠤⢀⣢⠶⢓⡋⠋⠀⠉⢸⡵⠊⠉⠀⠀⠈⠀⠀⠉⢻⡀
⠀⠀⠀⠀⠀⠀⠹⡎⠀⣀⠁⡠⠆⠀⢐⠀⣀⣠⠥⢖⣒⣰⠋⠀⠀⠀⠀⣀⣠⠴⠂⠀⢸⡇⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⢮⣀⠢⠀⠶⡷⠾⢏⣡⡤⠶⠛⢹⠏⠀⣀⠴⠚⠉⠁⠀⠀⠀⣠⣿⢁⡴⠒⠋⠉⠉⠉⠉⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠛⠛⠉⠁⠀⠀⠀⣟⣀⠀⠀⠀⠀⠀⢀⣠⡴⢛⡵⣻⠋⠀⢀⠤⠴⠖⠒⠒⠋⠉⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣞⣻⣭⠴⠞⠋⠐⢿⣄⡀⠀⠀⠀⠀⢀⣠⠴⢚⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣛⣫⣭⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⡉⢁⠄⡀⠀⣀⠉⣑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡗⠀⠠⠁⠂⠀⠀⠀⢢⠁⠈⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠓⠀⠂⢀⠀⠀⠀⠰⠀⣄⠄⠐⠄⠂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠉⢢⠀⠉⢉⠉⠁⢀⠄⢠⠀⠂⠐⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠁⠀⠀⠆⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⡆⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠐⠄⠀⠀⠀⠀⡀⠀⠀⠀⠘⠤⠔⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠤⠄⠀⡀⡉⠀⠀⠀⠀⠀⡀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠸⠀⠁⠘⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠄⠐⠀⢀⠀⢀⠠⠀⠀⢆⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠐⠢⡀⠠⠀⠠⠄⠀⡈⢄⠀⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠐⠂⠠⠂⠸⡀⠀⠀⡠⠀⠀⠀⢸⢆⠀⠄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠄⠈⠁⠢⠀⢀⠀⠀⠀⠀⠀⡀⠀⡀⢠⠐⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠰⠄⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢄⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⡔⠒⠀⠀⡀⠀⠤⠄⠈⣀⠀⠀⡁⠉⢅⠀⠰⠒⠂⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃⠀⠀⠀⠇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠠⠤⠀⠄⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠤⠀⠊⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⡄⠐⠂⠀⠤⠀⠠⡀⠀⠤⢄⠀⡐⢁⠅⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣑⠐⠃⢄⠀⠄⠀⠁⠀⠜⠀⢈⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠠⠤⠤⠤⠀⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⢀⡴⠚⠋⠉⢀⡼⠉⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⠴⠂⠀⠀⠀⠀⢳
⠀⢠⠏⠀⠀⢠⡴⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡤⠖⠋⠁⠀⠐⢒⠀⠀⢠⡟
⠀⠁⣀⡰⠋⠁⠀⠀⠀⠀⠀⠀⢀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣧⣀⠈⠀⠠⠄⠂⣁⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⣈⣱⣄
⠀⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠠⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⢹
⠀⠘⢌⠢⡀⠀⠀⠀⠀⠑⢄⡀⠀⠱⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠓⢤⣀⡀⠀⠀⠄⣀⣉⣶⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠀⠀⠉⠉⠉⠙⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⢰⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⢀⡀⠀⠀⠉⠉⠁⠀⠀⠀⠐⠄⠙⠷⣦⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡌⠰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⠶⠋⠁⠀⠀⡠⠤⠶⠄⠒⠒⠒⠦⠄⠀⠀⠘⣧⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠀⢹⡀⠀⠀⠀⠀⠀⠀⠑⠦⣄⡀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣁⣀⠀⠒⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠖⠂⣀⡴⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⠒⠶⢦⣤⣤⣌⣁⡀⠀⠀⠀⠀⠐⠛⢀⣁⡤⠴⠖⠛⠉⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠀⠠⠄⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠉⠛⠲⠤⢤⣑⠒⠒⠶⠤⠔⢂⣉⣉⣠⣤⠾⠋
⠀⠀⠀⠀⠀⠀⣠⡔⠢⣍⠛⠉⠙⢛⡉⠉⠗⢦⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠛⠉⠉⠉⠉⠀⠀⠀⢀⡓⢦⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠰⣁⠁⠀⢿⣤⠟⢦⣄⠙⢶⠄⠀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡆⢀⣀⣀⠠⡄⠀⠀⠀⢠⠂⠁
⠀⠀⠀⢀⠖⢦⡈⢦⡤⠦⡄⠀⠀⢹⡇⠘⠆⢀⠀⠸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⡤⠘⠲⣄
⠀⠀⠀⠀⠀⢀⡽⠇⢴⠉⠁⠀⠴⠋⣼⠟⠓⠞⢤⠳⡈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⣼⠃⠀⠀⢸⣆
⠀⠀⠀⢀⣤⠆⠀⡴⠋⠀⢈⠓⠦⢄⡈⠳⡀⠀⠀⠀⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣦⡆⣸⣴⠏⡀⠀⠀⣾⠻⣆
⠀⠀⠀⠙⣎⠁⠀⠣⣀⣠⠋⢙⡂⠀⠱⢤⣑⠂⠠⡠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⣷⢿⠋⠀⠑⣿⢸⣿⠀⠈⠁
⠀⠀⠀⠀⠫⡀⠤⣄⡀⢠⡒⠋⠈⠙⠳⡄⠈⠳⢤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⡇⠉⠀⢀⠀⠘⢿
⠀⠀⠀⠀⠀⠈⠃⢀⢀⠀⢈⠂⠀⣤⡴⠻⣶⡄⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⡀⠔⣉⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⣈⣀⣠⣄⣀⣤⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠦⢤⣤⣤⠤⠶⠚⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠒⠒⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠠⠗⡾⠋⣥⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠶⢦⣀⡀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡡⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠐⠢⢨⡷⠈⠴⢻⣦⣸⠆⣒⣿⣀⣠⣆⣀⡀⠀⠀⣇⣀⣀⣀⣠⠜⠋⠁⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⣈⣳⣦⡀⠀⠀⢈⢖⣀⡴⢻⡁
⠀⠀⠀⠀⠀⠘⣠⢡⡟⢑⣸⠖⣾⣽⠛⢠⠯⣍⠿⢷⣌⢻⣥⡀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⢀⣀⠔⠀⠟⠈⣻⣿⣷⣤⣴⣿⡿⠝⠉⠁
⠀⠀⠀⠀⠀⢀⢃⡟⣰⣿⣿⣿⣿⠟⢳⡄⢻⢡⡶⠶⣧⣤⣿⣷⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⠛⠃⠀⠀⠏⠀⠀⠀⠀⣷⡾⠃⠀⠈⠻⠷
⠀⠀⠀⠀⠀⢤⡿⠤⣼⡿⠿⠛⠉⢀⠺⣿⠾⠷⣶⣦⣿⣿⣿⣿⠇⠁⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡄⠀⠀⠀⠐⠀⠐⠋⠉⠀⠀⠀⢆⣀⡄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠛⢭⣯⠤⠿⠟⢃⣠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⡤⠀⠀⠀⠀⠓⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠁⠀⣆
⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⢻⣿⣿⣷⡄⠀⠀⡀⢀⠀⠀⠀⢀⠀⠀⠒⠐⠒⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⠀⣰⢟
⠀⠀⠀⠀⠀⠀⠨⠔⢉⣭⣔⠿⢻⠁⠂⠆⠀⠀⢻⣦⡇⣀⠤⡷⡀⠀⠀⠀⠀⢀⣽⡀⠀⠀⠀⠀⠐⣆⠀⠀⠐⠌⢉⣩⠭⠁⠀⠀⡔⠉⠐⠂⠀⠴
⠀⠀⠀⠀⢀⢁⣴⣶⣻⡟⠩⠄⠋⣠⠎⠀⠀⠀⠀⠙⠓⠒⠊⠉⡐⠠⠀⠀⠒⠀⠘⠓⠂⠀⢀⣠⣼⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣀⠀⡶⣾
⠀⠀⠀⠀⠈⠙⠿⠿⣧⠠⢀⠨⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠲⠤⠥⠾⠾⠞⠁⠀⢀⣤⡶⠶⠆⠀⠀⠀⠀⠻⣿⣿⣷⣿⣷⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠞⠩⢀⡈⠛⢃⣤⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡔⣒⣶⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⢧⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣶⣼⠷⠛⣍⠁⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⡿⣽⣤⣴⣧⣼⣿⡁⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠄⣘⡻⠶⠋⠵⠿⠷⠐⠉⠀⠐⠀⠀⢀⡄⠀⠢⢴⡶⣸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣦⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠢⠴⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠴⣲⣤⣦⠒⢈⡠⠂⠀⠀⢀⡾⠀⠀⠲⣧⡘⠁⠈⠳⣄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠍⠒⣢⣄⣤⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⣇⢀⡤⠀⠀⢿⣦⣤⣀⠜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠷⢤⣬⣉⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⣀⣴⣄⣫⢋⣋⡁⢠⣜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠘⠋⠀⠀⠄
⠀⠀⠀⢀⠈⠀⢀⣉⣺⣿⡭⣠⡝⠿⠟⠃⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣴⣦⡀
⠀⠀⠀⣌⣀⢠⢾⣿⣿⣿⣷⣿⣽⣟⣉⢶⣶⡟⠀⠀⣀⠤⠀⠀⠀⠈⠁⠀⠀⢀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⡓
⠀⠀⠀⠙⠛⣱⣿⡿⠛⠁⣨⣼⣿⣿⡋⣷⡾⠃⠀⢰⣏⡄⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣤⣦⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠛⣿⣵⣿⣿⣿⣽⠿⢊⠷⠞⠀⠀⠀⠘⢿⣇⣀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠀⠀⠱⣄⣤⣼⠆⠀⠀⠀⠀⠉⠛⠉⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠘⠻⠶⠿⠥⠟⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐

⠀⠀⣴⠞⢻⡟⠛⢻⡟⠛⢻⡟⠛⢻⡟⠛⠛⣿⠛⠛⢻⡟⠛⠛⣿⠛⠛⢻⡟⠛⢻⡟⠛⢻⡟⠛⢻⡟⠳⣦
⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⣿
⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠸⡇⠀⢸⡇⠀⣿
⠀⠀⣿⣀⣸⣇⣀⣸⣇⣀⣸⣇⣀⣸⣇⣀⣀⣿⣀⣀⣸⣇⣀⣀⣿⣀⣀⣸⣇⣀⣸⣇⣀⣰⣇⣀⣸⣇⣀⣿
⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣗
⠀⠘⣷⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣾⠋
⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠛⠶⠶⠶⠶⠶⠶⢶⣤⣤⣤⣤⣀⣀
⣿⠶⠛⠛⠛⠛⠓⠛⠻⠶⠶⠶⠶⠶⠶⠶⣶⣶⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷
⢻⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠛⠉⠉⠉⠉⠉⠉⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⡅⠀⠉⠛⣦⡀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠄⠰⠄⠀⠀⠘⣷⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⢀⣠⡴⠟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⣋⠉⢀⡀⠀⠀⠀⢀⣴⠾⠋⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣸⠏⢀⠉⢠⠌⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⢻⡆⠈⢁⡀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠻⣦⣈⠁⠀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⣀⣀⣀⣀⣀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾
⣿
⣿
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾
⣀⣀⣙⣳⣶⣶⣶⣶⣶⣶⣶⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣳⣶⣶⣶⣶⣶⣶⣶⣞⣋⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⣶⡶⠶⢶⣶⠶⠶⣶⠶⠶⢶⡶⠶⠶⣶⠶⠶⣶⡶⠶⢶⣶⠶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣼⣧⣤⣼⣧⣤⣤⣿⣤⣤⣼⣧⣤⣤⣿⣤⣤⣼⣧⣤⣼⣧⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣧⣴⠟⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠟⢳⣄⠈⠻⣦⡀⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⡀⠀⠙⣧⡀⢨⣷⣦⡀⠀⠀⠀⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢷⣄⠀⣨⡿⣯⡀⠈⢻⣄⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣽⣿⡋⠀⠈⢷⣄⠀⢹⣧⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠾⠋⠁⠈⢷⣄⠀⣠⣿⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢶⣄⠀⢀⣽⣿⡉⠀⠹⣧⠀⣿⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⠀⢹⣷⣏⠁⠈⢷⣄⠀⠻⢄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢷⣟⠁⠙⢷⣄⡀⠙⠃⢀⣾⠃⠀⠀⠚⠛⠛⠛⠛⠛⠛⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠛⠶⢦⣬⣥⣴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡶⠶⣶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣳⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣄⣀⣤⣶⠟⠃⠀⢿⣿⣿⣿⢻⠻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⣿⣿⣿⣇⣷⢹⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⢹⣿⣦⢠⣤⢀⡀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢠⠦⣠⣶⣦⣤⣤⣤⣵⣦⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣏⣾⣿⣿⣶⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⢠⣀⣴⣿⣿⣿⡿⢉⠀⠀⠀⠰⣦⣤⣄⣀⠀⠀⠉⢩⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣧⣶⣴⣸⣿⣿⣿⣿⣶⣶⣿⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠛⠇⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠
⡄⠀⠀⠀⠀⠀⠀⠀⠉⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣼⣿⣿⣿⣿⣿⣿⡇⣹⡁⢀⣴⡀
⣿⣶⣄⣠⣾⣿⣿⣿⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠛⠉⠿⣿⣏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣯⣷⣿⣷⣿⣿⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠰⣶⣆⠙⠶⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⢷⣤⠛⡋⢰⣠⡀⠀⠀⠀⠀⠔⠀⠁⠀⡶⠾⣁⣀
⠀⠀⠀⠀⢣⣄⡤⠛⠷⠤⣌⡙⠿⠃⠀⠀⠀⣠⡖⢀⠀⠐⠛⠡⠒⠒⠚⢻⣧⠀⠀⣠⠀⢀⡔⢨⣉⣴⠋⠀⢠⡦
⠀⠀⠀⠀⠈⠋⠃⣀⠠⣦⣀⡉⢛⠁⠀⠀⠘⢍⡣⢤⣄⡀⢀⣴⠄⠶⢛⣣⡞⠀⠀⢁⠀⠈⣡⣤⠞⠁⣀⣏⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠺⠏⠉⠁⠀⠀⠀⠀⠀⠛⢄⣰⠠⣴⣶⣶⣿⠿⠋⠀⠀⠀⠀⠒⢛⢉⠁⣰⠶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠉⣉⣵⣤⡀⠀⠀⠀⠀⣀⢀⣴⠒⠄⣀⡄⠀⠀⠀⠀⠀⡀⢿⠀⣹⡀
⠀⠀⠀⠀⢠⠋⠀⠀⣀⣤⠶⠏⡍⠩⣿⣿⡄⠀⢀⠞⣧⣮⡀⢿⣧⠙⠤⡀⢷⢄⠀⠀⠱⠀⠁⠈⡍⠀⠀⢀
⠀⠀⠀⠀⠸⣼⡶⠋⠁⠀⠀⠀⣀⣸⣿⣿⠇⠀⠘⠾⣦⠀⡏⠈⠙⠀⠀⠀⣘⡷⠀⠀⠀⠀⠀⠒⠁⠀⠐⠈⡷
⠀⠀⠀⠀⠀⢻⡁⠀⠀⠀⡀⢠⣴⣿⣿⡿⠀⠀⠀⠲⣤⣉⣛⡶⠦⠦⠤⠔⢋⡼⠃⠀⠀⠀⠀⢀⡀⠈⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠳⣶⣤⣴⣶⣷⣾⡿⠟⠁⠀⠀⠀⠀⠈⠻⢧⣜⣥⣝⣩⡍⠀⠁⠀⠀⠀⠀⠀⠀⠁⢊⡹⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣤⠶⠖⠛⠛⠛⠛⠛⠛⠓⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠚⠛⠛⠛⠛⠛⠛⠲⠦⣤⣀
⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⠙⠲⣄⡀⢀⣤⠖⠋⠁⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠳⣄
⠀⢀⡾⠁⠀⠀⢀⡤⠖⠋⠉⠉⠀⠀⠀⠈⠉⠉⠛⠲⢦⣄⡈⠙⠋⢀⣠⡴⠖⠛⠉⠉⠀⠀⠀⠀⠉⠉⠛⠲⣤⡀⠀⠀⠈⢷⡀
⢀⡾⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠙⢦⡴⠋⠁⠀⠀⠀⣀⣠⣤⡤⠤⠤⣤⣄⡀⠀⠀⠙⣦⠀⠀⠈⢷
⣸⠃⠀⠀⡾⠁⠀⠀⢀⣠⠶⠛⠉⠉⠉⠉⠉⠛⠲⢦⣄⡀⠀⠀⠀⣀⣠⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⢳⡄⠀⠈⢷⠀⠀⠘
⣿⠀⠀⢸⠃⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⡇
⣿⠀⠀⣿⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣿
⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿
⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿
⣿⠀⠀⣿⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⣿⠀⠀⣿⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⣿
⣿⠀⠀⣿⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡿
⣿⠀⠀⢻⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡇
⣿⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡇
⣿⠀⠀⢸⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⡇
⣿⠀⠀⢸⡄⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⢹⠀⠀⢸⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⡇
⢸⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢸⡇
⢸⠀⠀⢸⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡇
⢸⠀⠀⠸⡇⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⣸⠁⠀⢀
⢸⡆⠀⠀⠻⣄⠀⠀⠙⠷⣤⣄⣠⣤⡤⠶⠖⠛⠛⠛⠛⠛⠛⠒⠒⠶⠶⠶⠶⠤⠤⠤⠤⠴⠶⠶⠖⠛⠋⠁⠀⠀⣠⠟⠀⠀⢸
⠀⢻⡄⠀⠀⠙⠷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋⠀⠀⢠⡟
⠀⠀⠙⢦⣄⡀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⣠⡴⠋

⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀
⣤⣶⡿⠿⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠻⠿⣶⣦
⣿⡇⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀⠀⠀⠀⣿
⣿⡇⠀⠀⠙⠛⠛⠿⠿⠿⠿⢷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠿⠿⠿⠟⠛⠛⠁⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿
⠉⠻⣿⠿⠿⣷⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⡿⠿⢿⣿⠋
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⣿
⠀⠀⣿⠀⠀⠀⣿⡿⢿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⠿⣿⣷⠀⠀⢸⣿
⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁⠀⠀⠀⠀⠈⠙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⣀⡀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⢠⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⡟⠀⠀⠀⢲⣤⣠⣤⡶⠶⠟⢿⣏⠁⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠈⠋⠉⠻⣦⣀⣀⣤⣤⣼⢷⣾⠛⠃⠀⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠛⠛⢿⣅⠀⠀⠀⠘⠃⠀⠀⠀⠀⠉⠛⠿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠲⣦⡀⠀⠀⠈⠙⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣤⡀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣶⣤⡀⠀⠀⠀⢀⣺⣦⣤⠶⠾⠛⠛⠻⣧⡀⠀⢀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⠀⠈⠉⠈⠛⣶⣄⣤⣤⠶⠾⠻⣿⡋⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠉⠙⢷⣄⡀⠀⠀⠈⠁⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⣀⣀⣠⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠉⠛⠿⢷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⡿⠟⠋⠁⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢻⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠙⠻⢷⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠿⠿⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡂⠀⠐⡀⢆⠛⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣤⣄⠒⠛⠉⠩⠀⠀⠀⠀⢙⠓⠒⠐⠀⠒⢶⡇
⠀⠀⠀⠀⢀⣄⠀⢘⡶⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠄⠀⠀⠇⠀⠀⠀⣊⠉⠉⠀⠒⠋⠹⡃
⠀⠀⠀⠀⢤⣼⣿⣏⡕⣋⡉⠂⠀⠀⡀⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⣶⡿⠟⠉⠁⠀⠀⠀⣿⠷⠤⠠⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠛⠠⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣦⣄⣀⣀⣀⠀⣠⠁
⠀⠀⠀⠀⠀⡠⠤⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠦⠬⠄⠚⠀⠀⠀⠀⠙⠳⠤⠀⠤⠀⠘
⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣜⠀⠀⠀⠀⠰⠄⠀⠔⢷⡀
⠀⠀⠀⠀⣶⢾⣿⡤⢀⣤⡠⠂⠀⠀⠈⣴⣯⣷⣦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠣⠳⠶⠤⠤⠴⢛⡴⡤⠴⣃
⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⢿⣿⡿⠆⠀⠀⠀⠀⠀⢀⡖⠒⢋⡭⣵⠞⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⡀⠀⠉⠙⠲⠤⣄⡀⠉⠑⠦⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢤⢼⣈⣙⠒⠤⠤⠆⡀⠀⠈⠉⠒⠤⠤⣀⣀⡐⠀⠀⠈⢉⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⡡⠔⠀⠐⠆⠀⢤⣈⠑⠖⠤⣄⠉⠐⠂⠠⢀⠀⠄⠀⢉⠛⠓⠲⠤⣀⠀⠘⢏⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣴⠷⠒⠒⠒⠒⠦⢄⣘⠀⠉⠐⠖⠌⠉⠉⢟⢓⡲⠬⣔⡠⢅⠀⠀⠀⢀⠈⠑⣆⢸⡄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⠉⠢⣄⠊⠀⠀⠀⠆⠀⠉⠁⢋⠐⠒⠒⠒⠢⢄⡈⠑⠠⡙⢦⡈⠑⠒⠤⠤⢄⢨⣾⣀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢧⡀⠌⠑⢦⡤⠞⠂⠉⠉⠅⠉⠈⠉⠒⠾⣄⠀⠈⣳⣄⣈⣤⣹⣄⢀⣀⣀⠤⠚⢹⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣕⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⡋⠭⠐⠒⠒⠂⠀⠉⠑⠦⡀⠂⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⢍⡳⢀⠀⣀⡀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠤⠤⠤⠿⠷⢤⣀⠀⢀⠀⣀⣀⣀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠤⠶⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠒⠒⣆⠐⡖⠒⠒⡆⠀⢐⣶⡄⠀⠀⢲⢦⡀⠐⡖⠀⣶⠂⠀⢲⠂⠴⠒⣶⠒⠆⢠⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠒⠚⠁⠀⡏⠙⢀⡄⢀⡾⠦⢿⡄⠀⢸⠀⠙⢦⡇⠀⢻⡀⠀⣸⠀⠀⠀⣿⠀⠀⢬⡉⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠉⠉⠁⠉⠁⠀⠈⠉⠀⠉⠁⠀⠀⠁⠀⠀⠉⠉⠁⠀⠀⠈⠉⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠎⠁⠁⠉⠀⠁⠁⠀⠀⠀⠀⠉⠈⠈⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣶⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣟⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⠀⠈⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⠗⣶⣹⠇⠀⢠⣤⡈⡆⠀⢠⡇⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣿⡁⣺⠗⠋⠀⠀⠀⠈⠙⣷⠀⠈⠀⠑⠀⠀⢀⠱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢹⣿⢿⢇⠈⠀⠀⠀⠀⠀⠀⢄⠉⠀⠀⣀⣀⡀⠀⠈⠀⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠀⠀⠹⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌⠍⢻⣖⠂⣭⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠑⠀⢸⡯⢳⣻⣿⡄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠣⠀⠀⠀⠀⠀⢀⣬⠁⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠔⠋⣀⠞⣡⢿⣇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠤⠐⢚⡡⠞⣵⣿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡩⢒⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⢶⣿⠻⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠔⠐⠋⠁⠃⠀⣏⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢀
⣿⣟⣟⣿⣟⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣛⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢴⡪⠙⡩⠒⠉⠀⠈⠀⠀⠀⠀⠀⠈⠉⣉⠶⢢
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢶⡶⣦⢤⠤⢤⠤⣤⠤⡤⢤⣶⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠈⠀⡇⢸⠀⢸⠀⢸⠀⠇⢹⠁⣟⣹⣼
⠀⠀⠀⠀⠀⠀⢀⡆⠜⠿⢾⡦⣧⠼⢄⣸⣀⣨⣤⣤⣼⠷⣿⡟⣧
⠀⠀⠀⠀⠀⠀⢨⣙⠢⢤⣀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣸⡿⠜⡇⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢸⠘⣯⠐⠂⠬⠉⢉⠉⠁⠀⠀⠈⢩⡭⠭⠐⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⣿⠁⠀⢀⡤⠂⠄⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠐⠀⠀⠔⠉⠉⠉⠂⠀⠀⠀⣿⣿⠀⠀⠸⣄⠀⠀⠈⢠⠀⠀⠀⢄⡒⠂⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢹⢸⠀⠀⠀⠉⠳⠦⠤⠶⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⣷⠀⠀⠀⢄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢰⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡙⡄⠀⡠⠀⠂⠀⠠⠀⠀⠠⢉⠀⠨⢄⣠⡴⠆⢤⡜
⠀⠀⠀⠀⠀⠀⠸⣼⣿⣏⡛⠷⠶⠶⠾⠶⠶⠶⠶⠒⠚⢻⣿⣿⡅⠀⠀⠠⣜⠓⠒⠒⠒⠓⢳⠻⠟⠉⠡⠔⢳⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⡭⠆⠀⠀⠀⠈⠳⠴⣷⡳⠾⠖⠐⠛⠛⠓⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠀⠘⠓⠒⠒⠒⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⣀⠤⠴⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠶⠤⢤⣀
⠀⢸⡷⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⢾⡄
⠀⢸⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⢀⡿⠒⠶⠤⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠴⠖⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠒⠦⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠶⠒⢯⡀
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⢸⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨
⣾⠈⠉⠙⠓⠒⠶⠤⠤⠤⠤⣤⣤⣤⣤⣤⠤⠤⠤⠴⠒⠒⠋⠉⠀⣿⠀⠀⠀⠀⠀⠀⣾⠈⠉⠛⠒⠲⠦⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠖⠒⠋⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣴⢒⠲⢄
⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠐⠁⡷⠲⢃⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡸⠡⢀⠈⠐⠨⡇
⢻⠀⠀⠀⠀⠀⢀⡤⠒⡋⠀⠈⡏⠉⢢⣌⠊⠈⢣⡠⠔⠢⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡗⠂⠄⡀⠁⢢⠇
⢸⠀⠀⠀⠀⢀⠼⠠⢎⠉⠑⡖⠉⠺⡉⠙⣖⠁⠈⠡⠀⠈⠑⡄⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡴⠪⠈⠐⠠⠀⣱⠃⢠⠔⠓⡓⣦⡀
⢸⠀⠀⠀⠀⢸⠀⢀⣨⠒⠒⢧⠔⠲⣅⠤⠛⡄⠀⠀⠂⠀⠦⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣎⠐⠠⠈⠀⠂⠆⣿⠀⡟⠀⡀⠠⠀⢺
⢸⠀⠀⠀⢰⠁⡏⠀⢈⣧⣀⠼⠤⣀⡼⠑⢴⣡⢆⠀⠀⢠⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣦⠁⠂⠀⡌⢀⢀⠇⠀⠱⣁⠄⠄⠂⡘⣆
⢸⠀⠀⠀⢠⠋⠉⠙⡅⠀⠈⡆⠀⣸⣀⣀⠎⠁⠀⠑⠢⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠳⢥⣂⣴⠤⠋⠀⠀⠀⠘⡖⢀⠈⠄⠊⠳⡀
⢸⠀⠀⠀⢈⠦⢤⠔⠓⠠⡎⠉⣯⠠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠄⠀⠀⠠⢷
⢸⠀⠀⠀⠘⢄⣈⡦⠤⠜⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣒⠘⡀⠡⣰⠇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠉
⢸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡇⠀⠀⠀⠀⠀⠀⢸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠘⢧⠈⠉⠙⠒⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠒⠋⠉⠁⣰⠃⠀⠀⠀⠀⠀⠀⠘⣆⠈⠉⠙⠒⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠒⠒⠛⠉⠁⣰
⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⢤⣤⣤⣤⣤⡤⠤⠤⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⢤⣤⣤⣤⣤⣤⣤⡤⠤⠴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠂⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠐⢁⠄⠀⡀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠐⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠐⠄
⠀⠀⢀⡀⠄⠄⠀⠀⠂⠀⠁⠀⠀⠀⠀⠁⠀⠀⠑⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⢀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣐⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠀⢲⡄⠀⠀⠀⠀⠀⣤⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢸⡟⠛⠛⠛⠋⢰⡏⢰⠇⣰⢂⠀⣠⡄
⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⢹⠀⠀⠀⢰⠇⡼⢣⡾⢱⡆⢀⣿⢀⡿⢸⢳⡟⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠈⠀⠸⡷⠊⠀⢸⡇⣼⢁⡟⠘⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠿⠀⠈⣷⣀⡀⠀⠀⠀⠁⠘⠇⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠀⠠⠟⣸⡤⠖⠚⠛⠛⠛⠶⢼⢃⡶⢰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣯⠁⡀⡆⢸⡆⢰⡆⢀⡿⠃⠛⠛⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⣴⠃⢸⡆⣿⠸⣇⠇⣿⠀⣾⠁⣾⠃⣼⢠⡆⢀⣤⢀⣀
⠀⠀⠀⠀⠀⠀⢠⡆⠁⠲⣄⠀⠘⣇⣿⠀⣼⠀⢿⠘⣇⠰⠏⢸⡇⣾⡃⣼⠃⣼⠋⣹⢻⡏⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⢷⡄⡏⢿⡄⠙⠂⠘⠇⠉⠀⠀⠀⠀⠈⠁⠙⠺⠃⣼⠃⣼⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⠇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠓
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⢿⣦⡀⠀⠀⠉⠒⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠀⠀⠈⠉⠻⣶⣄⠀⠀⠀⠀⠁⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢸⣦⣶⣤⠞⠉⠙⠃⠀⠀⠐⠀⠺⢿⣷⣄⠤⠒⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⣿⢻⣷⣤⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢐⣦⣄⠀⠀⠀⠀⠀⠀⢶⣶⣤⣿⣮⣿⣿⣿⣦⡀⠀⠀⡆⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠀⠀⠀⢀⡀⣀⠀⢉⠤⠈⠁⠈⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠘⠀⡀⠀⠀⠄⠨⣿⣨⠶⠆⠂⠀⠀⠀⠀⢸⣿⣦⣅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠘⢿⠀⡄⠀⢀⡄⠁⠰⣞⣏⡻⠀⠀⠀⡾⠻⠝⢿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠘⣷⠀⣤⠁⣀⠀⠀⢺⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠐⠆⠀⠀⠰⠏⠀⠀⠄⣸⣆⠀⠘⠏⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠰⣿⣏⠈⠙⠲⣦⢄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠷⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⢀⣀⡀⡿
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠐⢠⣤⡴⠂⠀⠀⣿⠁⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠃⠀⠋⠀⠀⠰⡾⠏
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⡗⢚⠉⢋⡡⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠰⠞⠂
⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣀⣠⣴⣶⣾⣿⣶⣶⡶⠄⠀⠀⠀⣠⠀⠀⠀⡠⠉
⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠲⣦⣄⡀⠀⠀⣠⠚⠉⠉⠉⠙⠲⡄⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣠⠞⠋⠉⠉⠉⠁⠺⠗⠛⠉⠉⠙⠒⢤⡀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣼⠁⠀⠀⠀⠀⡖⡖⡴⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢀⠔⠚⠉⠙⠛⠲⣄
⠀⠀⠀⠀⠀⣠⠾⠚⠛⠓⠲⢤⡇⠀⢀⣄⣠⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠉⠁⠃⠋⠛⠸⠅⠀⠀⠀⡠⠂⠁⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢀⡟⠀⠀⣀⠉⠊⢷⠠⢀⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠉⠀⠀⢀⣾⠃⠀⠀⠀⠀⢀⢀⡀⡄⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠑⠙⠳⢂⠄⢈⠛⢳⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣟⠁⠀⠀⠀⠀⠘⠈⠚⠈⠁⠀⠀⡾
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣄⡀⢀⣀⣀⣤⠴⠖⠛⢿⣿⠁⠀⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠈⢠⠇
⠀⠀⠀⢳⠈⠒⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠻⣿⣿⡿⠁⠀⠀⠀⠀⠙⠲⠤⣄⣀⡀⠀⠀⠀⢀⡠⠤⠶⠶⢿⣄
⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠭⠭⠭⠟⠉⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⢠⠀⡒⠢⠄⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠐⢤⠪⠂⠛⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠼⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠽⠧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠚⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⣠⠶⠛⣋⡙⠳⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⢠⡞⢁⡴⠟⢁⡴⣤⢹⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⠏⠀⣠⠿⣥⡟⣸⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣫⣾⣁⣴⠾⣥⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣩⠞⢉⡴⢋⡵⣞⣭⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣴⠏⣠⠟⢳⣴⠟⣷⣫⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇⡼⢁⣠⣞⣁⡼⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣇⠑⣋⣥⠾⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠷⠦⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



⣶⣶⣶⣶⣶⡾⢶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠾⠾⣶⣾⡿⠷⠶⠶⠶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⡶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠉⠀⠀⠀⠀⢰⠀⢀⡦⠀⣴⠄⣰⠄⠀⣶⠀⠀⠀⠀⠀⠉⠉⠛⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡘⣸⠇⢨⣟⢠⡟⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⢷⣶⣤⣤⣀⣀⣀⠘⠁⠀⠛⠀⠘⠋⠘⠁⠃⠛⠀⠀⢀⣀⣀⣠⣤⣤⣶⠿⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣾⡏⠉⠉⠛⠛⣿⡟⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⣿⡟⠛⠛⠉⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣀⣻⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⢀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠟⠻⠿⢶⣶⣦⣤⣿⣧⣤⣤⣀⣀⣿⣇⣀⣀⣤⣤⣿⣧⣤⣶⣶⡿⠿⠟⠻⣿⣤⣀
⠀⠀⠀⠀⠀⠀⣼⡟⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⢻⣧
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⣿⡟⠛⠿⢷⣶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⠾⠿⠛⢹⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡤⠚⢙⡻⣍⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⢁⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡄⢰⣿⢰⠟⣀⣀⠀⢀⣀⣀⡀⢀⡀⢀⣀⠀⣀⠀⢀⡀⣠⣿⡁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⢀⣾⡡⠟⣴⡏⢀⣿⠃⣿⠗⣹⡟⣸⡏⠀⣿⠇⣿⡏⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠻⠧⠴⠋⠀⠈⠿⠦⢖⡿⠷⠚⠿⠶⠿⠀⠿⠷⠿⠷⠚⠿⠖⠿⠷⠊⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡁⣀⠀⡀⣰⣆⣶⢀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⢧⣏⢼⣇⣟⣸⣇⣿⣊⠝⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⣿⡏⠛⠻⠿⢶⣶⣦⣤⣤⣤⣌⣀⣀⣁⣀⣀⣈⣀⣁⣈⣁⣀⣁⣠⣤⣤⣤⣴⣶⡶⠿⠟⠛⢹⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⠀⢹⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠹⠿⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠃⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄⡀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣤⣀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⣿⣿⣿⣦⣀⣀⡀⠈⠉⣿⣿⣿⠟⠃⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣿⣿⣶⣾⣿⣿⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡿⠀⠀⠸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡆⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⡇⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⠤⣶⣾⣿⡿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⠿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠋⠙⠖⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠑⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠐⠢⣀⠀⠀⠀⠀⠀⠈⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠔⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄
⠀⠀⠀⠀⡀⠤⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢄⡤⠠⡄⠀⠀⠑⢄
⠀⠀⠀⠀⢇⠀⠈⠀⢠⠂⠀⠀⢀⡠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⣠⠂⠀⠀⠀⠀⠑⢤⠀⠀⠀⠀⡔⠂⣄⠄⢤
⠀⠀⠀⠀⠈⠢⡀⡰⢃⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠌⠀⠀⠀⠀⠀⠀⠀⠘⠦⠖⠁⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⢣⠀⠀⠀⡰
⠀⠀⠀⠀⠀⣀⠜⠊⠀⠀⠀⠀⠀⠀⠀⡠⠀⢤⠔⠂⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠳⣀⠜
⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠳⡀
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⢀⠇⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠆⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠈⠲⠉⠑⠄⠀⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⣘⡠⠖⠉⠀⠀⢀⡀⠀⢀⡀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠁⠀⠀⠀⠀⠰⡀⠈⠛⠀⢹⠀⠀⠀⠀⠙⠢⡀⠀⠠⠲⠖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⣀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠑⢄⠀⣠⠃⠀⠀⣀⠠⠂⠁⢉⣢⡈⠢⡞⣀⣀⠤⠀⠀⠀⢠⡄⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠁⢀⠜⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⢀⡠⠊⣡⠔⠚⠉⠈⠀⠙⠂⠉⠉⠀⠀⠀⠀⠀⡐⠀⠠⠍⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡐⠀⠀⠀⠙⢇⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠁⠀⠀⠀⠀⠀⢣⠈⢢⡀⠄⠒⠀⠈⠁⢀⠀⠀⠈⠁⠐⠒⠤⣀
⠀⠰⡂⠒⠐⡆⠀⡘⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⡄⠰⠂⠀⠚⠀⠀⠀⠀⠀⠀⠀⢣⠀⢂⠀⠀⠀⠀⠀⠉⠉⠒⠒⢢⠀⠀⠀⠀⠑⢄
⠀⠀⠑⡤⠊⠀⢠⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠠⠀⢀⢀⡏⠀⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⢣⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠱⡀⡘
⠀⠀⠀⠀⠀⠀⡆⢀⡃⠀⠀⠀⠀⠀⠀⢠⠊⠁⠈⠑⢧⡀⠀⠀⡀⣿⠀⠀⠃⣨⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠘⡀⠀⠀⠀⠀⠱⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⢀⡡⠊⠉⠉⠁⠐⠢⢄⡀⠈⠠⡀⢀⠔⠀⠈⠢⠎⢠⠁⠀⣰⢂⠡⠐⠂⠉⠉⠁⠀⠈⠉⠘⠣⡀⠃
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⡄⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡇⠘
⠀⢀⢀⣀⡀⢸⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⡇⢰
⠀⠘⢄⡠⠃⠈⡀⢰⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡈⠀⠑⠶⠠⠄⠀⠀⠀⠀⠤⠔⠊⠁⠀⠀⠀⠀⠇⢸⠀⠀⠋⠁⠒⠉⢰
⠀⠀⠀⠀⠀⠀⢃⠀⠆⠀⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡄⠀⠀⠱⡀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠈⡄⠘⡄⠀⠀⠀⠈⢓⡒⠂⠠⠄⠓⠒⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⡄⠀⠀⠀⠀⠉⠂⠤⣀⣀⣀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠉⠀⢀⠌
⠀⠀⠀⠰⡀⠒⠂⡰⠆⠀⠈⠢⣀⠀⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠉⠉⠉⣀⣀⠀⠤⠤⠔⠒⠁⠀⠀⠰⢍⠐⠂⡠⠆
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠲⡒⠒⠒⠒⠒⣒⠶⠒⠒⠒⠂⠈⠉⠙⠥⣀⣀⣀⣀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠛⢛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⠀⠰⣶⣶⣆⠈⢻⣿⣿⣿⣿
⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡋⠉⠁⠀⠀⠀⠀⠀⠈⠙⠀⡀⢻⣿⣿⣿
⣿⣿⠟⡡⠂⠑⠀⠀⠄⠀⠢⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⣄⣿⣿⣿
⣿⣏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢀⠔⠀⠙⠒⠤⢄⣀⠀⠠⣤⢠⠴⠓⢏⠻⠿
⡟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠈⣥⣤⣴
⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⠠⠘⠁⠀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⣀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠈⢻⠟⠿⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠠⠤⠤⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡏⠀⢀⢄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠛⠻⢿⣿⣿⣿⣿⡏⠀⠐⡀⠒⠠⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠁⠀⠐⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⠿⠿⠻⠀⠀⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠐⠀⠀⠀⠘⠠⠄⠐⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡘⠂⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣥⣤⣵⣢⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠠⠀⠀⠕⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣹⣶⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣶⠊⠀⠀⠀⠀⠀⠀⢸⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣌⣀⣠⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠠⠄⠀⠤⠀⠸⠀⠀⠀⠀⠠⠜⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠈⠀⢀⠄⠣⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣦⣄⣀⠀⠀⠀⣈⣁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⡀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢲
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⡇⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⣧⣼⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⢀⣀⣠⣤⣤⣀⣀⣀⣀⣀⣀⠀⢤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣤⣀⣿
⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⡿⠿⣿⣿⣿⣿⡁
⠀⠀⣾⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⢻⣆
⠀⢸⡟⠿⢿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿
⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠻⠟
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠀⢀⣤⣤⠀⠀⣤⣄⠀⣠⡀⣀⣀⠀⢀⡀
⠀⢸⡇⠀⠀⠀⠀⢾⣿⣛⣿⡇⢹⣿⠶⠦⠀⣼⣯⣿⣧⠀⣿⠻⣷⣿⠃⣿⡟⠀⣿⡇⠙⣿⡟⠓⠀⠀⠀⠀⡄
⠀⢸⡇⠀⠀⠀⠀⠸⣿⠍⠀⠀⠘⠟⠛⠛⠛⠛⠁⠈⠛⠀⠛⠀⠘⠛⠀⠙⠻⠶⠟⠁⠸⣿⠃⠀⠀⠀⠀⠀⡄
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⣠⣀⣤⣤⣤⣤⣤⣤⣄⣠⣀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠘⣿⣯⣿⣇⢸⣿⠀⢻⡇⠈⣿⡇⠀⣿⡏⠀⣿⡿⠬⠅⣼⣿⣙⣿⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⢿⠷⠾⠟⠈⠛⠓⠛⠁⠀⠛⠁⠀⠙⠃⠀⠛⠛⠲⠂⠻⠏⠻⡿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⡠⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⢸⠁⠀⢸⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⡇⠀⠘⠄⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠐⠄⠎⠀⠀⠀⡇⠀⠀⠐
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣠⡀⠄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⢩⡗⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⠉
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠲⠄⠊⠁⠘⠋⠀⠀⠀⠀⠀⠀⠈⠓⣾⣿⠒⠂⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⣀⠎⠁⡴
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢼⢁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢰⠧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠀⠐⠀⡇
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠛⠉⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⡿⠋⠁⠀⢀⠀⠂⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⡡⠊⠀⢀⠠⠂⠁⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣫⠔⠉⢀⠤⡚⠁⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⢀⡠⠊⠁⡔⠀⠀⠀⡠⠂⢠⡟
⠀⠀⠀⠀⠀⢸⠇⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⡰⠋⠀⢀⠎⠀⢀⠔⠊⢀⣰⡟
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢹⣇⡜⠀⠀⡴⠁⠀⡴⠁⢀⢔⣿⠏
⠀⠀⠀⠀⠀⢻⠀⠀⠈⠀⠀⠀⠄⠙⢷⡀⠀⠀⠀⠈⢻⣦⡠⠊⠀⢠⠎⠀⣰⣷⠟⠁
⠀⠀⠀⠀⠀⢸⣇⠀⠀⡄⠀⠀⠈⠆⠈⢻⣆⠀⠀⠀⠀⠙⠻⢶⣶⣧⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠸⣇⠄⠀⢰⢀⠀⠀⠘⡄⠀⡽⣧
⠀⠀⠀⠀⠀⠀⣿⠘⡀⠘⡀⠡⡀⠀⠰⡀⠘⢿⣧⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣇⢡⠀⢣⠀⠘⢄⠀⠱⡄⠈⢿⡆⠀⠀⠀⢠⣾⠟⠙⠢⢄⠀⠉⢻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢻⣌⡆⠈⢆⠀⠈⢆⠀⠘⢆⣼⡇⠀⠀⢠⣿⠁⠑⢄⡀⠀⠑⢄⠀⠱⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠈⢦⡀⠈⢆⢀⣼⠟⠀⠀⠀⢸⣟⠢⣄⠀⠙⠢⡀⠀⠱⡄⠈⢆⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣷⣶⠾⠟⠉⠀⠀⠀⠀⠈⢿⣄⠀⠑⢤⡀⠀⠑⠠⠈⢆⠀⠑⢝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡦⡀⠙⢆⠀⠀⠀⠈⢂⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣆⠀⠑⢄⠀⠀⠀⠡⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⡁⠠⠀⠀⠀⠄⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⣀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠈⠁⠢⡀⠀⠀⠀⡰⠊⠉⠉⠑⠢⠤⠀⠄⠀⠒⠠⡀⠀⠀⠀⢀⢴⡶⠟⠛⠉⠁⠉⠉⠙⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢐⣅⠠⠀⠀⠀⠀⠀⠀⡀⢡⠀⠀⠸⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠂⠀⠀⠘⢶⡤⠭⠀⢀⣀⠀⢀⡤⠴⠶⠟⢠
⠀⠀⢀⠔⡈⠤⠀⠔⠒⠤⢍⡛⡶⡤⠤⣀⣒⣚⣛⡻⠀⠀⠘⢮⣝⡀⠂⠀⢋⣠⠄⢙⣲⢲⣤⣿⡁⠀⠀⠀⠀⠙⠛⠛⣻⠖⢀⠀⠀⠀⠀⠐⢫
⠀⠸⡍⡒⠒⠒⠠⠦⠤⠀⠰⢒⢃⡎⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠉⢉⣉⠭⣭⠴⠋⢊⠟⠁⠉⠹⡾⡂⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠱⡀⠈⠉⠉⠉⠉⠉⠀⠀⡼⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄⠈⠀⠀⡠⢗⣒⡒⢢⠭⢒⡇⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⣀⡤⠞
⠀⠀⠀⠙⠢⢅⣠⡀⣀⡠⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡛⣀⣄⠒⠮⠔⠃⢠⡔⠴⠦⠞⠀⠀⠀⠀⠀⠀⠙⠶⠤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⣁⡊⢀⢂⠸⠟⠁
⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡤⢤⢄
⠀⠀⠀⠀⣀⡔⠁⠀⠑⣤⣀⠔⠛⠫⡉⠉⠉⠐⣄⠀⠀⠀⠀⠀⣠⠶⠒⠀⠀⠐⠂⠀⠀⠈⠙⠁⣀⠀⠀⠀⣠⠋⢠⡞⢠⠇⠈⠃
⠀⠀⠀⡼⠻⣿⣠⡖⠉⠉⠛⣆⠀⠀⠈⠆⠀⠀⠈⡡⠀⠀⠀⠀⣷⣀⠀⠀⠐⠀⢀⡀⠀⠠⠄⢴⡌⠀⠀⢸⠁⢠⡞⢀⡞⠀⢸⣂⡤⠔⢒⠢⢤⡀
⠀⠀⠀⣄⠀⠀⠁⠀⠀⠀⢠⣿⠃⠀⠀⡐⡀⠀⡰⠃⠀⠀⢠⢎⠁⠘⠉⠉⠁⠤⠒⠁⠀⠈⢉⣆⡃⠀⠀⠘⡄⢿⠀⡞⠁⡴⠋⣡⠔⣊⡴⠚⠉⣤
⠀⠀⠀⠹⣷⡀⠀⠀⢀⣠⡿⠧⡀⣀⣴⡋⠤⠚⠁⠀⠀⠀⠸⣦⣁⣄⠀⠀⠀⠠⠄⢀⣀⣀⣉⣴⠇⠀⠀⠀⠀⠀⠀⠀⡎⠠⢊⣴⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⠛⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠉⠁⠉⠁⠀⠈⠁⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⢥⣚⢀⡤⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣖⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⣄⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⣠⣾⠤⠤⠤⣄⢠⡾⠁⠀⠀⠀⠀⢾⠀⠑⢒⣀⣠⡏⠀⠀⠀⠀⠀⠘⡄⠀⠀⠠⠀⠀⠀⠓⠂⣠⠄⠀⠠⠵⢀⡀
⠀⠀⠀⢸⡀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠈⠢⢊⠁⠀⣠⠈⠑⣔⠚⠊⠀⠠⠂⠀⠀⠡⠀⠀⡠⠐⠉⠉⠉⠉⠐⣀⠀⢸
⠀⠀⠀⠀⠉⠁⢛⠀⠀⠀⢀⠠⣒⣀⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠈⠁⠀⠀⠓⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠡⣀⡀⠠⠄⠀⠂⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣶⡤⠕⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠒⠈

⠀⠀⢰⡖⠒⠂⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠒⠒⠒⠂⠤⡄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⠃⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡀⣿⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇
⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣽⣯⣭⣽⣿⣯⣴⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠃
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠉⠛⠛⠿⠿⠿⠿⢿⣿⠿⠿⠿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠛⠋⠛⠿⢿⣿⣿⣧⡀
⠀⠀⣰⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠈⣿⡄
⠀⠀⣿⠿⣽⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⡟
⠀⠸⣿⡄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⢰⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠸⣿⠇⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠀⣴⣶⡶⠶⠶⠀⢀⣶⣿⣷⠀⠀⣾⣿⣦⡀⣶⡆⢰⣶⡶⠀⢠⣤⢠⣤⣤⣀⣀
⠀⠀⣿⠀⠀⠀⠀⠀⠈⣿⣿⣭⣽⡿⠃⢹⣿⡿⠿⠃⠀⣼⣿⣼⣿⣧⠀⣿⡏⠻⣿⣿⡇⢸⣿⡇⠀⣼⣿⠀⢩⣿⡟⠙⠁⠀⠀⠀⠀⢰
⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠸⠿⠿⠛⠛⠋⠛⠁⠀⠙⠛⠀⠛⠃⠀⠙⠛⠃⠈⠛⠿⠾⠿⠃⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠶⣶⣦⢰⣿⡇⠀⣿⡏⠛⢻⣿⠛⠛⢻⣿⡟⠋⣿⣿⡛⠛⠃⢰⣶⠶⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠾⢿⣧⠀⣿⣧⣀⣿⡇⠀⢸⣿⠀⠀⢸⣿⠇⠀⣿⣿⣛⣛⠀⣿⣿⢷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠋⠀⠈⠙⠛⠉⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠉⠉⠛⠛⠁⠛⠃⠈⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⠞⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠇⠘⠀⢴⠏⠘⠀⣼⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣦⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⢀⡀⠀⠀⡇⠀⠀⠈⠂⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠂⠀⠀⢆⡰⠃⠀⠀⠀⢸⠀⠀⠀⠀⠇
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⠀⠀⣀⠀⡄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠀⠒⢶⡄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠁⡀⠲⠄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⢠⣿⡀⣀⣀⠀⠀⡠⠂
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣹⠿⠷⠈⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠉⠀⠀⢊⡴⠁⢀⡴
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⡼⠋⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⡟⠲⠤⠤⠄⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⠤⠄⠀⠂⠀⠈⠀⢸
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠒⠒⠒⠒⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠂⠈⠉⠉⠉⠉⠉⠉⠀⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠙⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢏⠀⡁⠀⠀⠀⠀⠀⠀⠀⢸⣀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⣀⣀⣀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⢀⣀⡀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⠁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢀⠄⠐⠂⠀⠐⠂⠤⡀
⠀⠀⠀⠀⠀⠀⡀⠄⠖⠋⢢⡀⠔⠈⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰
⠀⠀⠀⢀⠔⠉⠀⡂⢀⡴⠛⠑⠂⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡠⠁⠀⠀⠀⢀⠋⢀⠄⠀⠀⠒⢌⡄
⠀⡰⡐⠉⠉⢁⡠⠁⠀⠂⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⢀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⢠⠁⠀⠀⠀⣼⠁⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⡋
⡄⠈⠣⠄⠰⡇⠀⠀⠀⠀⠈⠑⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣁⣀⡸
⡇⠀⠀⠀⢠⠉⠁⠀⠒⠒⠒⠀⠀⠤⠬⠶⢆⠀⠀⠀⠀⠀⠉⠉⠓⠄⠀⠀⠀⠀⠀⠀⡠⠔⢹⣀⠅
⡇⠀⡠⠐⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠈⢲⠖⠒⠁⠀⠀⠀⢀⠤⠊⠀⠀⢸
⢠⢸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠒⠊⠀⠀⠀⢀⡠⠂⠁⠀⠀⠀⠀⣈⣀⣀⡀
⠀⢫⢄⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢀⡤⠒⠁⠀⠀⠀⠀⡀⠠⢰⠁⠀⠀⢻
⠀⠈⢆⠈⠉⠀⠳⠒⠒⠀⠀⠤⠤⢤⡤⠀⠀⠀⠐⠂⠉⡏⠉⠁⠈⡆⠐⠂⠀⠉⠀⠀⢀⠎⠉⠀⠀⠙
⠀⠀⠀⠱⣄⠀⠀⠱⡀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚⠃⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠑⠤⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢈⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠉⠁⠐⠂⠀⠀⠰⠆⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠙⠀⠀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠁⠂⡸⢀⠊⠉⠆⠀⡗⠢⠘⠂⠪⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡎⠀⢀⠁⠀⠀⠀⠐⢄⠈⠀⠈⡀⡄⠀⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣉⡁⠀⠢⣀⣀⣀⠘⠀⠓⠒⠀⠉⢉⡀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡌⠀⠈⡆⠀⠀⠤⠤⠀⠀⠖⠀⡆⠘⠁⠈⠄⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠐⠀⡆⠀⠀⢰⠀⠁⠀⠃⠠⠀⢁⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠢⠀⠁⢃⠀⠀⡀⢠⡀⡐⠀⠣⠤⠊⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡀⢄⠀⠀⢈⡉⠁⠀⠀⠀⠀⠀⡀⡀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠁⠀⠉⠑⠀⠊⢀⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⣈⠀⠀⠈⠒⠀⠒⠀⠂⠀⠈⠃⠂⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠑⠀⠉⠀⠉⠑⠀⠂⠈⠆⠄⢢⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠃⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣈⠁⠒⠊⠀⢣⡀⢀⡀⠴⠀⡀⠀⢀⡝⠤⠐⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠂⠀⠈⠐⡄⠀⡤⠤⠤⢄⡀⠀⡀⠀⠀⠀⠔⠀⠀⢀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠦⠀⢀⣀⡀⠀⡁⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⢀⠀⠀⡀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⡤⠤⠀⢀⡀⠀⢈⣐⠀⠒⠀⠁⠈⠒⠒⢒⠊⠀⢠⠤⠄⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⡆⠀⠈⠁⠀⢱⠀⡄⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⠀⣠⠃⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠉⠁⠀⠈⠀⠑⠀⠀⠠⠤⠆⠀⠄⠠⠄⠂⢀⠭⡀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠒⡉⠉⢢⠐⠂⠁⠐⠄⡠⠂⠒⡄⠀⣂⠠⢂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⣑⠊⠐⠤⠤⠔⠀⠤⠀⠔⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠲⠶⠶⠦⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣠⡤⣤⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡴⢋⡤⠖⣂⡙⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠁⠀⠀⠀⢀⡞⡱⠋⣠⠾⣥⢇⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⡤⣾⠾⣥⠞⣧⠔⣡⠞⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠞⣥⡞⣡⣞⣵⡿⠗⠋⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⢣⠞⣡⡟⢡⢞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡉⢞⣡⠟⣩⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⣀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣟⠀⠀⢺⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠏⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣟⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠙⠛⢿⣿⡇
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠙⠷⢦⡤
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣼⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢿⣿⣷⡄
⣼⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⢻⣿⣿⡄
⣿⣿⣿⣷⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⣤⣾⣿⣿⠇
⠘⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠛⠋⠉
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⡿⠟
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠾⠿⠿⠿⠿⠟⠿⠿⢿⡿⠿⠿⠟⠛⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠶⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢠⠽⠷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣇⠃⠀⢀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⣘⠀⢼⠟⡡⠤⠤⣤⠀⠬⠻⣶⣶⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⣚⠛⠊⣩⠄⣰⠋⣠⢄⠀⠈⢧⠀⢀⠄⠉⠘⡋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⣡⠄⠀⠈⠉⠀⡴⠃⠀⡏⠸⣿⣾⠃⠀⢸⠀⠀⣉⠓⠀⠸⡀⠀⡀⠤⣤⠙⣦⠰⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠊⣡⠆⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠃⢀⣠⣤⣤⣀⠋⠀⠀⠈⠀⠀⠀⡇⠘⢧⡀⠈⣧⣀⣉⡉⠙⡆⣠⠤⢤⡀
⠀⠀⠀⠀⠀⣰⠁⡀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠀⠁⠐⠉⢀⣠⣄⠈⠁⠀⠀⠀⠀⠀⢀⡇⠀⠀⣡⠀⠶⠝⡋⠳⠀⣿⠃⣠⠞⠁
⠀⠀⠀⠀⢠⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠁⡀⠀⠈⠀⣄⡀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣧⠀⠐⠉⠈⠉⠀⣡⠞
⠀⠀⠀⠀⠈⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠂⠀⠀⠀⠀⠀⢉⢛⠓⠖⠀⠀⠀⢰⡏⠀⠀⠀⠈⢳⠀⣤⠤⠶⠞⠁
⠀⠀⠀⠀⠀⠘⣦⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠊⠀⠀⠀⠸⣆⠚⠦⠀⠀⠀⣿⠀⠀⠀⠀⠀⡝⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣉⡒⠀⠀⠐⢒⣁⣼⡇⠀⠀⠀⠀⠀⠠⣌⡛⣁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⢁⣸⠇⡈⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠹⡄⣒⠢⠤⣤⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⢖⣛⣯⠭⣶⠆⠀⠀⠋⠀⠈⠀⠀⠀⠀⠀⠀⠅⠀⠀⢠⠈⠉⠛⠲⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣌⠣⣄⣀⢰⡏⠘⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⡀⠀⠀⢀⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣍⡛⠦⢍⡀⠐⢰⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢍⡹⠂⠀⢹⠠⠈⠀⠀⠀⡀⠀⠀⠀⠐⠰⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠓⣾⠀⠀⢠⡇⠠⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠈⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠰⠴⠿⠀⠀⣿⠁⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⣤⠤⠻⣦⠠⠀⢄⠀⠀⠁⣈⠀⠁⠀⢤⣤⡿⠁⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⡦⢤⠀⠄⠠⠀⣒⣴⢿⡋⣼⠀⢚⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠤⠞⠀⡟⢒⡚⠛⠉⠁⠀⣵⣼⠤⠾⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⣤⣀⣠⡽⣦⠁⠀⠀⠀⢠⣯⣍⠻⡶⣦⣎⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠊⢢⣩⡦⣽⡕⠈⣉⢠⡀⠀⠀⠀⠑⠾⢳⣤⡍⠉⠑⣀⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠬⠈⣹⡭⡭⠴⠖⣋⡴⠇⠀⠀⠀⠀⠀⠀⠉⠛⠥⢬⡭⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠶⠶⠀⠐⠐⠂⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡌⣉⢛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⢛⡉⣩⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⢸⢰⣶⣶⣶⣶⡆⡇⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⡿⣿⣿⣿⢸⢸⣿⣿⣿⣿⡇⡇⣿⣿⣿⢿⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣶⣤⣬⣉⣉⣘⠘⠛⠛⠛⠛⠃⣃⣉⣉⣥⣤⣶⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡝⣿⣿⣿⡟⡑⠿⣛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢋⣴⣿⡿⣻⠀⢒⡨⢴⡿⠛⠛⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⠿⠯⣯⡴⠃⢤⣴⢺⣿⠀⢴⣿⣷⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣩⠊⣼⣷⣶⡆⠮⠭⢓⢸⣿⣿⣿⡿⠏⠈⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣔⢧⣾⣿⣿⣿⡇⢧⣭⢏⣬⣩⣍⣉⣀⣤⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣶⣤⣭⣍⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣦⣤⣬⣍⣉⣉⣉⣉⣉⣉⣩⣭⣴⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⣤⡤⣀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠠⠀⣛⡁⢄⣛⠂
⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⡚⣿⣷⠀⠀⢠⠀⠀⠀⠀⢀⣾⣃⣩⠿⠐⢊⠁⠭⠛⠐⢿⠀⠀⠀⣀⠴⠒⠒⠒⢦
⠀⠀⠀⢿⣿⣧⡀⣀⡈⠀⠀⠀⠀⠸⡞⣿⣷⡀⢐⡆⠀⠀⠀⠈⡿⢃⡤⠀⣀⡀⠀⠒⢠⡾⠂⠀⢀⠞⠁⠀⠀⢠
⠀⠀⠀⠈⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⠙⢿⣿⣷⡿⠃⠀⠀⠀⠀⠘⢿⣇⡀⣀⠠⠔⠂⠁⠀⠀⠀⡎⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠎⠠⠆⠐⠒⡲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⣾⡀⠑⢐⣄⣄⣠⣧⡿⠀⠀⠀⠀⠀⠀⠀⣠⣞⣛⡻⢷⣶⡀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⣾⠟
⠀⠀⠀⠀⠀⠀⢠⡈⠭⠷⠿⣅⣠⣄⣤⢅⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠙⣮⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⢻⡈⣁⣀⣈⣽⡉⢁⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⢀⣤⠤⠰⠖⠒⠐⠲⢤⡀
⠀⠀⣠⠒⠀⠀⣀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠀⣰⣍⠀⠀⠂⠀⠐⠒⠀⣺⣿⡀
⠀⢰⣿⡂⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡴⢶⣄⠀⠀⠀⠀⠀⣃⡒⡒⢬⣌⢡⠄⣤⣀⣤⣶⡇
⠀⠀⠹⣆⠀⠀⠀⢈⡇⠀⠀⡈⠀⠀⠀⠀⢀⣦⠀⠀⠀⢻⣺⡭⣟⡇⠀⠀⠀⠀⠸⢥⡴⣶⡀⠀⠈⣁⣂⡾⠟
⠀⠀⠀⢸⠀⠀⠀⣨⠀⠀⢰⣷⣄⣠⣤⣥⣾⣿⠀⠀⠀⠈⣿⠤⢹⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠂⠀⠀⡰⠃⠀⠀⠀⠙⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣡⠥⠆⢹⢧
⠀⠀⠲⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡦⢆⠘⣠⠾⡆⠀⠀⠀⠀⠀⢀⣤⢶⣫⣵⣿⣾⠄
⠀⠀⠀⠀⠀⠀⣠⣶⢶⣶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣞⣛⣿⣺⡇⠀⠀⠀⠀⠀⣾⣿⣿⠟⠛⠁⠀⠌
⠀⠀⠀⠀⠀⢠⣟⢯⣬⠓⠞⢉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠻⣯⣅⡀⡠
⠀⠀⠀⠀⠀⠈⠻⣯⣷⢜⡃⠼⠐⡷⣢⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⢲⠲⣄⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠿⣴⡥⣼⢡⢻⡇⠀⠀⠀⢀⠼⠁⢊⠀⢀⣊⠷⢿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠠⠂⠋⠽⣷⣄⠀⠀⠈⠛⠿⠿⠟⠁⠀⠀⠀⠸⣆⣉⠓⣿⠋⢀⣤⡾⡲⡆⠀⠀⠀⠀⢰⡗⠾⠶⣍⣻⣷
⠀⠀⡐⠀⢀⣀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠩⢑⡋⠉⠉⣏⣁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿
⠀⠀⢰⣶⣿⠿⣻⡽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣅⠠⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠠⠾⠛⠁
⠀⠀⠀⠀⠉⠉

⠀⢠⡄⠀⠀⠠⠶⣀⠀⠀⠠⠄⠀⠀⠀⠀⣤⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠰⠀⠶⠴⠀⡀⠀⠀⡀⠘⠁⠀⡄⠀⣀⠀⡄⠀⢠⡄⢠
⢀⣀⠀⡀⢠⡄⠀⠀⡀⠀⠠⠀⠰⠀⠀⠀⡀⠀⠀⢸⡿⣀⣰⠖⠀⠛⠀⠀⠲⢀⠀⠒⡀⢀⠈⠋⢠⠄⣁⠀⠴⠀⠀⠀⣠⠀⠀⠠⠆⡀
⠀⢡⡄⠁⠀⠀⠀⠀⠰⠜⠗⠀⡀⠀⡀⠷⠀⠀⠀⣠⣴⠄⠀⠀⣀⠀⠖⠀⠀⠈⠀⠀⠀⠈⠀⠠⠐⠀⠉⠀⠀⢠⡄⠀⠙⠃⠀⠀⠀⠀⠘
⠀⠀⠀⢤⡄⢠⠀⣤⠀⡴⠦⠈⠛⠶⠉⠀⠀⠀⢀⠃⠴⠀⠀⠋⣉⠢⠀⢀⠀⡀⠘⠃⣁⠀⠀⠀⠀⠛⢂⠀⠀⢀⠰⠀⠀⠘⠁⠀⠀⠀⠄
⠀⡀⠀⢰⠀⠁⠲⠂⠀⠀⠀⢀⠀⠀⠘⠛⠃⠀⠈⣥⠀⠠⠄⢠⣄⠁⠀⢀⠀⠸⠠⠀⠙⠙⠒⠈⣁⠆⠀⡤⠀⠀⠀⠀⠸⠇⠰⠖⠈⠛
⠀⠁⠀⠘⢀⡀⠚⠇⠀⢠⣴⠎⣤⠀⢀⡄⠘⠂⠀⠀⡀⢄⠏⠄⠀⠄⠀⠀⠤⠀⠠⡦⠀⠘⠃⢀⡙⠅⢠⡄⠰⠆⠺⠀⠀⠈⠁⠀⢤
⠀⠐⠷⠀⠀⠀⣀⡀⠠⡄⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠁⠀⠀⠰⠀⣀⠀⠐⠀⢀⡇⠀⠀⠘⠓⠈⠑⠂⠀⠀⠁⠀⠁⠘⢣⣀⡀⠀⠀⢀⡀
⠀⡀⠀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠃⠉⠤⠋⡤⠈⠁⢀⠠⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠃⣬
⠀⡛⠀⠆⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠛⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠁⠶⠀⠐
⢈⠁⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⣀
⠈⣉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠉⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠘⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣄
⠶⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣤⠀⡄
⢠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠐⡇⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠘⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠐⠆⣀⢀⣤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠾⠀⠐
⠸⠓⢀⠀⢤⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣉
⢙⡄⠰⠶⠀⣀⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠔⠊⢀⡀⠙⠁
⠀⣁⠀⠀⡀⠉⠀⢶⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠙⠀⠀⠳⢀⠰
⠀⠉⠀⠀⠐⠦⠀⠐⡃⠀⣀⡐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⣀⣀⡀⠀⠄⠒
⠀⠛⠂⢴⡄⠀⠀⠶⠾⠀⢈⡀⢀⡀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡀⠀⠀⠘⠋⠈⠀⠀⢠
⠀⣷⠶⠀⠀⠒⠆⠀⣤⡀⠈⠁⢈⠁⣤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣶⠄⠘⠃⢀⡀⠠⡦⠀⣾⠆⡤
⠀⡟⠀⠀⠀⣀⡀⠀⠈⠀⠀⠀⠿⠇⠀⠘⠂⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⠈⢩⠀⠀⣃⡀⠀⡤
⢠⣷⡄⠀⠀⠰⠇⠀⠰⣭⢂⡠⠜⠂⠀⠽⠿⠀⢀⡀⠀⣤⡀⠈⠢⠀⠠⠄⠘⠁⠀⠀⠀⠦⠰⡶⠀⠀⢰⣶⠀⠈⠁⠀⠀⠀⠘⠿⠂⠀⢠
⠀⣻⢀⡀⠐⠒⠀⠀⠀⠀⠘⢡⣄⠤⢀⠀⠀⠀⣈⠃⢤⡄⠀⠀⣠⡠⠄⠀⠀⠀⢀⣀⠀⠚⠃⠀⢠⡄⠀⠁⠖⠘⠃⠀⠀⠀⠀⢠⠄⢰⡆
⠀⣇⠀⠉⠀⠄⠀⠀⠀⠀⠀⢠⡀⢠⡄⠀⠀⢠⡯⠀⠀⠀⠀⠂⠉⠀⠀⣾⡀⠟⢿⣯⡀⢠⣄⢠⡄⠀⠀⠶⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢀
⠀⠽⠆⠀⠀⠰⠗⠀⢠⡆⢄⠘⠁⠀⣠⣤⠀⠦⠀⠀⢀⣀⠀⠀⠀⠐⠓⠙⠁⠀⠀⠈⠁⠀⠀⣴⡆⢠⡀⢠⡀⠠⣤⠄⠀⠀⠸⠿⠀⠀⣤
⠀⠸⠶⠇⠀⠠⣦⠀⣼⠟⠁⠀⠀⠀⠙⠛⠰⠞⠃⠠⣾⣼⠃⠀⠰⠗⠰⣶⠀⠀⠒⠀⠾⣤⡄⠈⠁⢠⡄⠈⠱⠆⠀⠰⠆⠲⠀⢀⡀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠉⠀⠒⠒⠦⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⣠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢸⠁⠀⢀⠤⠐⠒⠊⠁⠀⠀⠀⠀⡔⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠒⠁⠀⡰⠁⠀⠀⠀⢀⠀⠀⠀⠀⡴⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⢀⠎⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⡄⠀⠀⠜⠀⠀⠀⠀⠈⠀⢀⠖⠉⢁⠀⠈⢀⠎⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠂⠒⠁⢀⠞⠁⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠛⠤⠀⠀⠁⣠⠊⠐⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⣠⠤⠐⠒⠊⠀⠀⠀⡀⠀⠀⠀⠘⠁⠢⠀⠀⠀⠀⡠⠒⠈⠁⡀⠀⢀⡼⠀⠀⠀⠀⠐⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔⠁⠀⠀⠀⠀⡀⠀⠀⢈⡤⠀⠠⡄⠀⠀⡠⠄⠀⠈⠀⠀⠠⠀⠈⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡀
⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⣀⠤⠒⠁⢀⠀⠀⠀⣀⠜⡀⠀⠀⠦⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⣀⠀⠀⠀⠈⠢⡀
⠀⠀⢰⠁⠀⢀⡰⠃⠀⡠⠀⣠⠁⠀⠀⡔⠁⢀⡀⠀⠈⠁⠐⢊⠑⠄⠈⢀⣀⡠⠴⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⠀⠱⡀
⠀⠀⡆⠀⠰⠁⠀⡠⠎⡰⠋⠀⡠⠶⡊⠀⠀⠀⡠⠄⠀⠀⠀⢀⠔⠂⠉⠁⠀⠀⠀⢃⣀⣀⠤⠤⠤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠃
⠀⢠⠁⢀⠆⠀⠘⠀⢀⡀⠤⠊⠀⠀⠀⡠⠔⠊⠀⠂⡰⠀⢡⠃⠀⠀⢀⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡸⠁
⠀⠸⡀⡜⠀⠀⠁⠀⡏⡀⠀⠐⠀⢀⢎⡀⠈⠂⡠⠊⠉⢠⠂⣠⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⢤⣀⡀⠑⣄⠀⡀⠠⠄⢒⡡⠊
⠀⠀⢣⡱⡀⠀⡔⠋⠀⠈⠂⠀⣀⠜⠀⢀⣠⠎⠑⢀⡴⠁⡜⠁⠀⠀⠀⢀⣀⡤⠴⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢮⡆⠀⠐⠈⠁
⠀⠀⠀⠱⣁⠘⠀⢀⣀⡀⣠⠎⠀⣀⠴⠁⢀⡠⠖⠁⢀⡞⠀⢀⡀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸
⠀⠀⠀⠀⠈⠓⠦⠥⢀⣈⣀⣒⡊⠤⠤⠒⠉⠀⠀⠀⣎⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⡀⠀⠈⠛⠋⣉⣁⡤⠤⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠐⠤⠤⠃⠰⠂⠀⠂⢠⠀⠀⠈⠀⠈⠀⠀⠐⠒⠀⢰⠀⠀⠀⠀⠀⠀⠐⠤⠤⠃⠰⠁⠀⠂⢀⡀⠀⠈⠀⠈⠀⠀⠐⠂⠀⢰
⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⡠⠀⠀⠀⠀⠀⠈⢦⠀⡸⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠒⠂⠤⠁⡠⠄⠀⠀⠀⢄⠀⢤⠀⡸
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠤⢤⡀⠑⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡀⠈⠀⠀⠀⠀⠄⠀⠉⠂⠈
⠀⠁⠀⠀⠤⠀⠀⠀⠀⠀⠀⠘⠉⠐⠁⠀⠀⡀⠀⠐⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠀⠀⡀⡀⠆⠀⢁⠄
⠀⠀⠀⠀⠈⣋⣀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⣠⠰⠀⠀⠀⠀⠀⠀⡀⠤⠤⠄
⠂⠀⠆⠈⠢⠭⠤⡤⠀⠀⠀⠀⠀⠀⠱⣄⢀⡰⠂⠀⠠⠄⡀⠀⠀⠠⠂⠀⠆⠀⠢⢥⢤⠜⠀⠀⠁⠀⠀⠀⠱⡀⢀⠠⠂⠀⠠⠄⠀⠀⡀
⠘⠏⠀⠀⠀⣀⠀⠀⠠⡀⢠⠤⢀⠂⠀⡈⠁⠀⠀⠘⢄⣀⠔⠀⠃⠴⠘⠋⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⢀⠄⠀⠈⢁⡀⠀⠐⢄⣀⠐⠀⠃
⠀⠀⠀⠀⠀⣀⢀⠀⠀⠙⠢⠄⠁⠀⠀⣀⠠⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠘⠢⠄⠁⠀⠀⣀⠠
⠀⠀⠀⠘⡄⠉⠉⡡⠀⠀⠀⠀⠀⠀⠠⣀⠀⢀⠃⡃⠄⣀⠀⠰⠀⠀⠀⠀⠀⠘⣄⠀⢈⡴⠀⠀⠄⠀⠀⠀⢄⠀⠀⢠⠁⢊⠄⠈⠁⢠
⠢⡀⠂⠀⠈⠒⠊⠀⠀⠀⢀⠐⠄⡀⠀⠀⠉⠉⠀⠀⠀⠈⠈⠀⢰⠁⠑⢀⠀⠀⠈⠓⠁⠀⠀⢀⠀⠈⠆⡀⠀⠉⠒⠁⠀⠀⠀⠙⠐⠊⡜
⢀⡈⠀⠀⠀⠠⠀⠀⠀⠀⠠⣀⠠⠄⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⡀⠀⠀⠀⠀⠀⠉⠢⠈⠢⢄⠔⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠐⠀⠀⡠⠀⠀⢀⠄⠀⠀⠀⠀⢀⡀⠀⠃⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠀⠀⡠⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠁⡀⠀⠄
⠠⠒⠁⠃⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠐⢁⠄⢤⠂⠁⠀⠀⠒⠁⠀⠀⠀⠀⠐⠀⠀⠠⡄⠆⠀⠀⠀⠀⠀⠀⠀⠁⠤⠴⠀⠁
⠐⠤⠐⠃⠰⠁⠀⠀⢠⠀⠀⠘⠀⠐⠁⠀⠠⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⠄⠰⠃⠰⠈⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠈⠉⠀⢲
⠀⠀⠀⠀⠀⠀⠐⠂⠔⠀⡤⠀⠀⠀⠀⡀⠢⡀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠚⠂⠢⠁⡤⠀⠀⠀⠀⢀⠈⢦⠀⡸⠀⢀⡀⠀⢀
⣄⡀⠰⠀⠀⠀⠀⠀⠀⠈⠠⣀⡈⠀⡀⠀⡀⠈⠀⠃⠀⠀⠀⠉⠂⠀⠰⠦⠄⠈⠀⠀⠀⠀⠀⠀⠤⣀⡈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠐⠁⠀⠀⠀⠀⠀⠆⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠉⠐⠁⠀⠀⠀⠈⠂⠀⠠⠂
⠀⠀⠀⠀⠈⣉⣀⠠⠀⠀⠈⠀⠀⠀⠀⠠⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣠⠘⠀⠀⠈⠀⠀⠀⡄⠤⠤⠄⠀⠀⠈
⠀⠀⠄⠀⠲⠥⠀⠴⠀⠀⠀⠀⠀⠀⠱⣄⠀⡴⠃⠀⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠲⠥⠠⠔⠀⠀⠀⠀⠀⠀⠱⣄⠀⡠⠂⠀⠀⠄
⠚⠃⠀⠀⠀⡀⠀⠀⠠⡀⢠⠄⢠⠀⠀⠈⠁⠀⠀⠘⢄⢀⠔⠁⠀⠈⠑⠦⢀⠆⠀⡀⠀⠀⠀⡀⢠⠤⢠⠄⠀⠈⠁⠀⠀⠐⢄⢀⠔⠀⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠢⠔⠁⠀⠀⣀⡀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠈⠢⠔⠁⠀⠀⣀⡀⠄⠀⢠⠀⢀
⠀⠀⠀⢠⡈⠉⠉⡠⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⠃⠁⠂⣀⠀⠐⠀⠀⠀⠀⠀⢰⡈⠉⠉⡠⠀⠀⢀⠀⠀⠀⠠⣄⠀⢀⠃⠡⠂⣀⡀⠸
⠢⡀⠀⠀⠑⠔⠊⠀⠀⠀⠀⠘⠄⠆⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁⢰⠀⠲⠀⠀⠀⠑⠔⠊⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠃⡜
⠀⠀⠀⠀⠀⠠⠀⠈⠂⠀⠢⢄⣀⠄⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠸⣄⡀⠠⠄⠀⠀⠀⠀⠈⠢⠀⠠⢄⡀⠄⠀⠄⠀⠀⠠⡀⠀⠀⠀⠀⢄
⠠⠀⠀⠀⠐⠀⠈⡠⠀⠀⠀⠄⠀⠀⠀⠀⣀⡀⠀⠃⠖⠁⠒⠀⢀⠀⠤⠀⠀⠀⠐⠀⠈⡠⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠁⠆⠀⠒⠀⢀
⠐⠒⠀⡃⠀⢀⠀⠀⠀⠀⠀⡀⠂⢀⠀⠀⠀⠀⠀⠀⢀⠑⠚⢂⡀⠀⠀⠂⠁⣀⠀⢀⠀⠀⠀⠀⠀⡄⠆⢀⠀⠀⠀⠀⠀⠀⢀⠓⠚⢂⠈
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⣰⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⢛⡛⠁⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡿⠋⡽⠉⠉⠈⠇⠀⠰⢋⣾⡥⢹⠀⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡇⠀⡇⠀⠀⣠⠀⠀⠀⣿⢟⡥⢸⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣿⡇⠀⢣⣀⣴⣿⣷⣤⣶⠟⠋⠀⡞⠀⠀⠀⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⢠⡇⠀⠀⢠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣿⣿⣿⠿⣿⣿⣦⣄⣀⠀⢀⣀⣼⣀⣤⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣐⠅⡁⢀⢀⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡠⠺⠓⠻⠘⠁⠀⠂⠁⠀⠀⠀⠀⠀⠀⠂⠤
⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠔⠒⠒⠤⠤⣄⣀
⠀⠀⠀⠀⠀⣤⢀⠴⠚⠉⠀⢀⠠⠔⠒⠋⠉⠉⠉⠉⠀⠈⠙⢦⡄⢀⠀⢀
⠀⠀⣔⠻⠋⠸⠁⢀⠠⠐⠈⠀⠀⢀⡠⠤⠖⠒⠋⠉⠀⣠⡤⠖⣋⠵⡶⠃
⠀⠀⢈⠗⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠉⠁⠀⢉⣵⠞⣇⠀⠀⠀⠀⠀⠀⠀⣀⠤⢔⣲⠶⠖⠒⠒⠲⠤⣄⠀⠀⢀⣠⣤⣀⡀
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠀⠀⢀⣀⣠⡴⠞⠋⠀⠰⣸⡀⠀⠀⣠⠴⡶⢊⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠒⠻⢦⣏⣤⠂⠈⠙⣆
⠀⡞⢠⠁⢠⠃⠀⠀⠀⢠⠀⠀⠒⠒⠒⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣿⠇⢠⡞⡕⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠽⡝⣆⢉⡀⢸
⠀⡇⢸⠀⠈⢀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢸⢀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⢄⣀⠀⠀⠀⠀⠀⢀⡽⣼⡄⠀⣸
⠀⡇⠘⠆⡰⢁⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠁⠀⣸⢸⠸⢀⠧⣄⣀⣤⣀⡀⠈⠉⠒⠤⣀⠀⠉⠲⠦⣤⡴⢞⠅⣧⡤⠔⠁
⠀⢿⠀⢰⢣⡏⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠀⢀⡇⠈⢇⣸⠠⠀⢈⡄⠀⠉⠙⢦⡀⠀⠈⠳⡀⠀⠀⠀⠐⠋⡰⡇
⠀⠸⣇⠀⠈⠣⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⢇⡞⠀⠀⠀⢻⢀⠄⠀⠁⠀⠀⠀⠀⠳⢸⡍⢣⠁⠀⠀⢀⠀⡰⢣⡇
⠀⠀⠹⣦⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣀⠴⣫⠏⠀⠀⠀⠀⢸⡈⠂⠀⠀⠀⠀⠀⠀⠀⠈⠶⠞⠀⠀⠀⡡⠜⢁⣾⠁
⠀⠀⠀⠈⠳⢬⣓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣡⠞⠁⡀⠀⠀⠀⠀⠀⢳⡤⠤⠞⠁⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⢀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠦⢄⡀⠀⣀⣤⠤⠶⠒⠉⠀⠀⠙⠃⠀⠀⠀⢀⣄⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠘⡆⠀⢒⣋⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢽⡀⠾⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠉⠀⠀⠀⠀⠉⠓⡆⠀⡠⠖⠲⠿⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠁⠀⠉⠢⡀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⠀⠀⠀⠀⡠⠂⠙⠒⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢄⠀⠀⠀⠀⢀⠘⡄⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠈⡄⢇⡀⠀⠀⠀⠈⠛⠿⠃⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠒⠂⠀⠀⠀⠀⠀⢡⠀⠈⠑⠒⠤⢤⠖⢦⣤⠤⠖⠋⠹⠀⠀⠀⠀⠀⠸⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢣⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢏⣢⠀⢀⡠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⡩⡅⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡤⣆⠀⠀⢠⣴⢺⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⠊⠀⠰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣈⠉⠒⠛⠃⠛⠋⠁⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡴⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⠙⠛⠒⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣊⣑⠄⠑⠲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠭⠃⡔⢚⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⣄⣤⡄⠀⠠⠶⠎⢀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⣠⠀⠠⠛⠀⠁⣀⠀⠀⠀⠀⢄⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠰⠞⠉⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⡄⠀⠀⠉⠉⠀⠰⠒
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⡀⠀⠈⠀⠐⡄⠀⠀⠀⠀⠀⠀⠈⣆⢀⢀⡄⠤⠖⠁⢲⠐⢀⣄⠀⠀⠀⡐⠀⠰⠀⠀⡔
⠀⠀⠀⣀⠀⠀⠀⠀⢀⡤⠀⠀⣦⣄⡠⠂⠈⠁⠈⠀⡄⠀⠀⢈⠄⠀⠀⠈⠛⠶⡦⡀⠀⠀⡌⠀⠀⠘⠒⢶⡄
⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠨⠛⠈⠁⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠒⠱⣎⠀⠈⠋⠀⠀⠠⠄
⠀⠀⠀⠀⢀⣠⡄⠚⠁⠀⠀⠀⠀⠀⠈⠈⠀⠀⣂⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠂⠰⠀⢠⡅
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢃⡔⠀⠠⠶⠒⣤⡤⠄⠀⠁⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠖⢲⣦⣄⡀⠀⠀⠀⣀⣴⡾⠻⠋⠟⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⢿⠋⡰⢠⠇⡏⡏⡟⣦⢀⠞⠉⠈⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠲⣖⠒⠒⠒⠒⠲⠦⠤⣼⢏⡎⠏⣰⠁⠘⡘⣸⢃⢃⠙⣏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠠⡀⣸⢹⢸⢸⠰⠁⡀⢠⡗⡸⢀⡎⠘⢹⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢱⡇⠞⡀⣼⠇⢰⠇⡘⢠⠃⡞⣠⠆⣿⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣼⠀⠀⠀⢹⢀⢀⡔⢀⡇⠀⡘⠉⣼⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠚⠋⣿⡄⢀⡆⢋⠎⡜⠀⡜⣠⠎⡴⣱⠓⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⢸
⢀⣤⠴⠒⠋⠉⠀⠀⣀⠀⠐⠠⣄⣿⠉⠈⠀⡸⢠⠇⠐⡫⣇⢎⡴⠁⠐⠀⠹⣄⠢⠁⠀⠈⠀⠀⠀⠀⠀⠀⡀⣾
⠀⠈⠙⠲⢤⣀⠀⠀⠈⠓⡠⠄⠨⣿⢡⠃⡰⡱⠋⣠⠆⢠⣷⠟⢧⡀⠀⠈⠑⢌⠳⡑⢄⠀⠀⠀⠈⠀⢀⠀⢀⡏
⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡀⠀⠀⢿⠈⠀⡠⢀⠜⢁⣴⠟⠉⠉⣉⡩⠝⠉⠉⠉⣉⠽⠳⣅⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠾⣷⣴⣡⣮⠶⠋⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⠟⠲⢤⣄⣀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⡄⠃⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠃⠁⠀⠀⣧
⠀⠀⢀⣀⣠⣤⠤⡴⡶⠖⠒⠒⢲⣻⢛⠛⠛⠛⠓⠒⢺⠦⡄⠠⠀⢸
⠀⡞⠉⢁⠀⠀⠀⠇⠇⠀⠀⠀⠈⢸⠈⡀⠀⡇⠀⠀⢸⠀⢹⢀⠁⢸⡇
⠀⣇⠀⠘⠀⠀⠀⢠⢠⠀⠀⠀⠀⠘⠀⡇⠀⡇⠀⠀⢸⠀⢸⣀⡤⠔⠛⢦
⠀⢻⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⡇⠀⠇⠀⠇⠀⠀⢸⠀⢸⠀⠀⠀⠀⠈⣇
⢀⣾⠤⡤⠧⠴⠖⡾⠚⠒⠒⠒⠒⠛⠓⠓⠒⠛⠒⠒⠒⠒⠛⣦⠠⠀⠀⠀⢹⡄
⠘⡦⠀⡄⠀⠀⠀⡑⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⣧
⢸⠁⠀⡇⠀⠀⡤⠇⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⢸⡄
⡿⠀⠀⡆⡞⠼⠀⠀⢀⡠⠄⠀⠀⠉⢀⡈⠉⠉⠒⢄⠀⠀⠀⢸⡟⡇⢃⠀⠀⠀⣇
⡇⠀⠀⠳⠁⠀⢀⠔⠉⠀⢀⠔⠉⠉⠉⠳⡄⠀⠀⠀⢳⠀⠀⠀⡇⢹⡈⠀⠀⠀⢸⡄
⡇⠀⠀⠀⠀⡰⠃⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿⠀⠀⠀⠸⡇⠀⢰⡇⠈⣇⠀⠀⠀⠀⣇
⣷⠀⠀⠀⢠⠃⠀⠀⠀⢠⠋⠁⠀⠀⠤⣖⠁⠀⠀⠀⢠⡇⠀⢸⡇⠀⢸⡌⠀⠀⠀⢹⡀
⢹⡀⠀⠀⢸⠀⠀⠀⠀⢇⠀⠄⠀⠀⠀⣸⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⣧⠀⠀⠀⠸⡇
⠈⣇⠀⠀⠀⢧⡀⠀⠀⠈⣇⡠⠤⠤⠞⠁⠀⠀⢀⠔⠀⠀⠀⡿⠀⠀⠀⠸⡆⡔⢄⠀⣇
⠀⠸⡆⠀⠀⠀⠙⠢⠤⣤⣈⣀⠀⠀⣀⡀⠤⠚⠁⠀⠀⠀⢰⠇⠀⠀⠀⠀⢷⢱⡀⠉⠹⡇
⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⣳⠀⠀⠀⠀⢸⡏⠀⠀⢰⣷
⠀⠈⠻⣤⣈⣐⡒⠒⠒⠒⠂⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠴⠞⠃⠀⠀⠀⠀⡴⠧⠀⠀⢸⠘⠒⠦⢄
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠉⠉⠉⠉⠉⠀⣀⡤⠤⠤⠤⠤⠶⠶⠒⠃⠀⠀⠀⠃⠀⠀⠴⠾⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣟⣋⣉⣁⣀⣀⣀⣀⣀⣀⡬⠿⠉⠉
⣿⣷⠿⠷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠉⠉⠉⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠒⠒⠒⠒⠒⠒⠒⣒⡒⠒⠒⠒⠒⢒⡒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡞⡥⠼⢀⣀⣀⠀⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢾⡁⡽⠆⡏⢿⣃⡇⡇⡖⢌⢣⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠧⠇⠀⠧⣛⣻⠆⣇⣙⡵⠋⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡤⠒⡄⠀⠀⠀⠀⢀⠴⡄⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠎⠀⢀⡇⠀⠀⠀⡰⠃⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡴⠃⡀⠀⡼⠀⠀⢀⡜⢁⡀⢠⠃⠀⠀⠀⠀⠀⠀⡟⡇⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠸⠴⢺⠃⢰⠇⠀⢀⠎⣠⢋⡇⡼⠀⠀⠀⠀⠀⠀⢀⡇⡇⢿⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⡎⢀⡏⠀⢀⠎⠐⠧⠜⠀⢣⡀⠀⠀⠀⠀⠀⢸⢀⡇⢸⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⡼⠀⡼⠀⠀⠘⠤⠶⢦⡀⢀⡤⠜⠀⠀⠀⠀⠀⡟⢸⠃⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢰⠁⡰⠁⠀⠀⠀⠀⠀⢰⡇⢸⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠉⠀⡠⢤⣀⣠⢄⡀⠀⠓⠋⠀⠀⠀⠀⠀⠀⡸⠀⡞⠀⢸⠀⠀⣧
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠉⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⠇⠀⢸⡄⠀⢻
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡼⠀⠀⢸⣇⣠⠼
⠀⠀⢀⣜⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣤⡤⠋⠉⠉⠉⠉
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠓⠒⠒⠒⠒⠒⠶⠶⠶⠤⠤⠤⠤⠖⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠒⠀⠈⠉⠁⠉⠉⠀⠐⠒⣤⠀⡤⢀⡀⠀⠀⠀⠀⠉⠀⠀⠉⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⡜⠀⡣⠤⠀⢀⣀⠀⣀⣀⣀⡀⠠⠠⡃
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠠⠒⠀⠀⢀⠀⠐⠢⠀⠀⢻⠀⠀⠀⠊⠁⠀⢀⠀⠀⠩⡂⠀⡈⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⡱⠀⠀⢈⠀⢠⡄⡇⠀⣤⠀⠀⠀⢸⠀⠀⠀⠠⡆⠀⢸⠴⠀⠀⡅⠀⠂⠀⠸⠀⡀⠈⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠓⠀⣆⣀⣀⡀⠀⠠⠃⠼
⠀⠀⠀⠀⠀⣠⣶⢿⠀⠀⠀⣀⣀⠀⡀⠀⡀⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠋
⠀⠀⠀⠀⣴⠏⠁⢸⠀⠀⠀⠟⠿⠼⠧⠿⠿⠏⠳⠀⠀⢸⠀⠀⠀⢠⣤⢤⢠⣠⣠⠀⠀⠀⠀⡇
⠀⠀⠀⠘⠃⡀⠀⢸⠀⠀⠠⡿⣿⡗⡏⢻⢻⣽⠧⠀⠀⢸⠀⠀⠀⠛⠻⠺⠞⠗⠛⠀⠀⠀⠀⡇
⠂⡴⠆⠀⠐⠀⠀⢸⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠡⣀⠔
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣤⣤⡀⠀⠀⠀⠀⠀⠁⠀⣀⣀⡀⣀⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠚⢀⡔⠉⠈⠁⠀⠁⠀⠀⠀⠀⠉⠳⠒⠀⠒⢤⣀
⠀⠀⠀⠀⠀⢄⠉⢀⠃⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⡇⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⣠⣶⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢰⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⣼⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢆⢸⢹⡁⠀⠀⣠⠄⠀⣀⠀⣀⠼⠤⠤⢆⣀⣜⡐⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⠈⣿⠉⠁⢒⣩⠭⢍⡉⠉⠉⠁⠀⣠⠤⠤⣄⠀⠉⠛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⣰⠚⢉⣩⣳⣤⠴⠲⣾⣷⣶⡄⠈⢻⢿⣤⣤⠎⠀⣴⣶⣿⡦⠖⢦⣾⣶⣶⡄⠈⢷⣶⣶⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣿⣿⡇⠀⠸⣿⣿⣿⠀⠀⢻⡻⠿⠃⢀⡞⠀⡟⠛⢆⠀⠙⠿⢫⠇⠠⠀⢯⡛⠟⠁⢀⠟⠉⠉⡧⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠊⠂⠀⠀⠀⠙⠦⣄⣠⠴⠃⠢⡴⠀⠉⠓⠚⠉⠀⢀⠇⠀⠀⠙⠒⠊⠁⠀⠓⠁⠀⠉⠒⠚⠁⠀⠀⢀⣇⡼⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣦⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⡞⠁⠰⠈⣳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡎⠈⠉⠉⠉⢛⠛⢉⣉⡉⠁⠀⢳⢱⡀⠀⢰⠃⡜⢸⠉⠉⠙⠛⠒⠛⠋⠉⢿⠸⡆⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣁⣸⠀⠀⠀⠀⣎⠁⠙⠊⠀⢸⠀⠀⠈⣥⠇⠀⠧⣤⡧⠁⠂⠠⠀⠄⠀⠀⠀⠀⠀⢣⣯⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⠢⣀⢀⡰⠃⠀⠀⠀⡇⠀⣀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣶⠤⢤⣤⣤⡤⣭⣤⠤⠤⡶⠊⠀⠀⠈⠀⠀⠑⡖⠦⠤⢤⡤⠤⡤⠤⠤⢴⡞⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⠥⠃⠀⠘⠧⣤⠴⠓⠒⠉⠁⠉⠉⠑⠒⠻⠄⣠⠜⠀⠀⠳⢄⣀⡼⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⡞⢛⡛⢶⡴⠶⣤⡀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢴⠞⣛⡛⢶⠒⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠞⣛⠓⠴⡿⠆⣈⠻⠛⡄⢳⣶⡈⡷⠶⢦⡀⣠⠴⠾⠯⣴⣶⠦⡙⠻⢓⡀⠻⠷⠀⣛⡛⢲⡄
⠀⠀⠀⠀⢀⡴⠞⠃⠺⠟⢁⡶⠶⠞⠉⠛⠛⠛⠦⢤⡇⢹⣷⠈⡿⢏⣴⣿⠂⣦⠴⠞⠛⠛⠋⠙⠲⠶⣇⠙⠿⠀⡟⠶⣄
⠀⠀⠀⠀⣼⣁⣾⠟⢸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣼⠀⣶⡖⢢⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⠀⢿⡆⢹⡆
⠀⠀⢠⡞⢡⣤⢤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣬⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⢀⣮⠙⣦
⠀⠀⣸⡧⠝⢋⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣈⡻⢁⣿⡀
⠀⢰⡏⢰⣶⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢰⣷⠈⣧
⠀⢈⡷⠚⡡⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⢋⠴⣏
⠀⢸⡇⢾⡷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⣿⠆⣿
⠀⠀⣿⠊⣠⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢋⡤⢾⡇
⠀⠀⢿⡈⢿⠃⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠿⠏⣸⠇
⠀⠀⠀⢹⠃⣾⣄⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢉⣴⡖⢺⠁
⠀⠀⠀⠘⢧⣉⠃⡘⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⣚⣋⣡⠟
⠀⠀⠀⠀⠀⢻⡀⠿⡗⢸⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠴⠿⠇⣸⠃
⠀⠀⠀⠀⠀⠀⠙⠲⡆⢸⣷⠈⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠏⣴⣿⠒⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣼⡀⣷⣦⠹⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣠⣶⡄⢤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⣽⠀⣶⣄⢹⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣁⣶⡦⢤⣭⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣬⣥⠰⣶⡌⢻⡤⣤⡀⠀⠀⣠⡤⢼⣃⣴⡦⢤⣭⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣬⣅⠰⣶⡄⢻⠶⠾⢁⣴⡶⢤⣭⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣬⣥⠸⣶⡶⢢⣬⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠦⠶⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⢀⠑⠐⠂⡐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢢⠀⢠⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄⠀⠀⠀⢁⠀⠤⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠂⡀⢠⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠆⠠⡜⠑⠂⠀⠈⠀⠀⠀⠘⠁⠀⠀⠀⢀⡐⠀⢀⢀⠀⢸⠆⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⡜⠂⠀⠀⣄⣠⠤⢄⣉⠀⠀⢰⠀⠶⢀⡀⠀⠀⠀⢈⡇⠀⠬
⠀⠀⠀⠀⠀⠀⠂⠀⡠⠀⠀⠀⣿⠀⠀⠀⢀⡞⠀⠀⠈⢀⠀⠦⠠⡄⠀⠁⠈⣏⠀⠀⠈⢣⡔⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠀⠄⠀⢀⡀⠀⠀⣿⡀⠀⠀⢸⡀⢠⠀⠀⠂⢉⠈⠁⢈⠀⠀⠈⢻⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠂⣈⠀⠀⠠⢸⡄⠀⠉⠘⣧⠈⠐⠀⠀⠀⠀⠀⢠⠀⠀⠀⡟⠐⠀⠀⠀⣏⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢈⡈⠸⣇⠀⠀⠘⠙⣇⢀⠀⢀⠀⠀⠀⠀⢀⠀⣽⠁⠀⠀⠀⢇⡟⠁⠐⡄⢈
⠀⠀⠀⠀⠀⠀⡀⠘⡖⠀⠀⠀⠤⣿⠀⠀⠀⣤⠈⠳⣄⡼⠀⠀⠀⢀⣡⠞⠁⠀⠀⠀⠀⢤⡷⠆⣀⡟⠀⠠⡤
⠀⠀⠀⠀⢀⠀⢀⡀⠉⠲⠲⢤⡀⣿⠀⢀⠀⠀⠀⠰⠈⠙⠶⣤⣵⠋⡅⠀⠀⢠⠡⠘⠀⣸⢇⡴⠟⢰⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢻⡏⠀⠀⠄⠀⢀⡀⣰⡶⠞⠋⠉⠛⠳⣦⣤⡉⠀⡀⠐⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⠀⠁⠖⢀⣤⠿⢯⡄⠌⢡⠠⠂⠀⢈⠈⣙⢿⣦⣤⣿⡇⠀⠀⠐
⠀⠀⠀⠀⠁⢀⠀⠄⠀⠠⠀⡂⠀⢹⣆⠠⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠐⠰⠉⠘⠀⠈⢻⣿⣷⠀⢊⠇⡄
⠀⠀⠀⠀⠠⠀⠀⣀⠀⠀⠀⡀⠀⠈⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠠⠀⢸⣿⣿⠀⠀⠀⣵⠒
⠀⠀⠀⠀⠀⠁⠑⣮⠀⠀⢠⡍⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⣿⡏⠁⣦⣴⠃
⠀⠀⠀⠀⠀⠀⠈⠙⢦⣸⣇⣠⣀⠀⣹⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠟⠿⠄⠀⣨⡄
⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠉⠉⠙⣷⣾⣿⡇⠀⠀⠀⠀⣴⣦⣴⣶⡄⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⢠⣿⡟⠻⣿⣆⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠸⣿⣿⠀⣿⣿⠆⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢀⣿⣿⢰⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣸⣿⣿⠿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢹⣿⡟⠁⢻⣿⠀⠀⠀⠀⠀⢸⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⢸⣿⡇⠀⠈⣿⣆⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣼⣿⣿⣿⣄⣀⣀⣾⣼⣇⠀⢀⣿⣿⠀⠀⠀⢀⣿⣿⣿⣷⣀⣀⣀⣀⡀
⣿⣿⣿⣿⣷⣶⣶⣶⠿⠿⠿⠟⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠒⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣽⡍⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣿⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣉⣉⣛⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⡿⣿⣿⣿⣻⣷⣿⣿⣿⣿⣿⣿⣿⣶⣾⣟⣟⣿⣟⣻⣭⣽⣿⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⠋⢉⡉⠉⠉⠁⢀⠀⢀⣠⠀⢀⠤⡄⠀⠀⡈⠉⠉⠉⠉⠉⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢣⠲⡂⢸⣰⡜⣆⠏⠀⠀⠠⠛⣴⣸⠉⠃⠋⠖⢐⡈⠂⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠸⠀⠗⠋⠀⠀⠀⠀⠀⢀⣘⣉⡀⠀⠀⠀⠀⠋⠉⠑⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⡶⢯⣍⢳⡄⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣭⣟⣱⢮⡜⣳⡷⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢀⣠⣴⣿⣣⣞⡱⣮⣟⡽⠃⣀⣀⣀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢀⣴⣾⣿⠛⣹⣞⡱⣾⡿⠟⢋⡴⢯⡵⢶⣉⠳⡄⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⡏⣿⢻⣴⠿⠶⠾⠿⡯⢤⡴⣮⢾⣥⠷⣏⢷⡷⡷⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠻⣵⢏⡦⢼⠒⢻⣛⣟⡩⢿⡚⣧⢚⣯⠽⣋⡿⠁⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠸⣆⣷⣈⣏⡩⢧⠜⣮⠝⣻⡭⠷⠒⠋⠁⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠀⠀⠀⠀⠙⢿⣷⣚⣶⣊⣯⠼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣶⣶⣤⣤⣤⣤⣬⣍⣉⣉⣀⣀⣀⣤⣤⣤⣤⣤⣤⣶⣶⣶⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣿⣿⣒⠚⠛⠛⠛⠛⠛⣛⣓⣚⣋⣿⣭⣽⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⡝⣯⢿⣿⣿⣿⣿⡟⣿⢿⣻⡿⣿⡿⣫⡿⣿⡟⣿⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣽⣏⣿⣥⣿⢾⣷⣿⢸⣿⡷⣿⡗⣿⢥⣿⣼⣳⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠷⠿⠾⠿⠿⠾⠿⠷⠿⠿⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⠶⢤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢞⡿⠙⠛⠉⠁⢒⡀⠬⠅⠀⠤⠀⠉⠉⠛⠿⠟⠛⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⢐⡩⠔⡠⠔⢊⣭⠅⠀⠀⠀⠀⠈⠐⠂⠀⠀⠥⢐⠒⢌⠠⡙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⣁⠴⠟⣋⠉⠁⠤⠾⠛⢉⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⢀⠀⠑⠬⣒⠺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⠄⠊⢁⠔⠉⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠉⠒⠶⠄⠈⠐⠀⢌⠂⠀⠈⠳⠌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡷⠋⢒⡴⠋⠀⠀⣠⡤⠖⠒⢂⡉⠁⠀⠀⠀⢀⣀⣀⣄⣀⣀⣀⠀⠀⠄⡀⠁⠂⠑⠀⠈⠘⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⣫⠀⢡⠋⠐⢁⢔⡟⠡⠀⠀⠀⠀⣀⠀⠀⠈⠉⠑⡂⠀⠒⠀⠤⢀⠀⠤⢀⠀⠉⠂⠄⠢⢄⠀⢁⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠏⡆⠁⢈⠔⡁⠁⠠⠀⠀⠀⠉⠀⠀⠀⠐⠂⠨⠤⠤⠤⠤⠌⠀⠀⠈⠀⠂⠈⠁⠒⠐⠲⠀⠳⣄⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⡅⠀⠸⠈⠀⡀⠄⠐⠒⠀⠀⠀⠀⢀⠲⠟⠛⠒⠲⠀⠐⣀⡀⠀⠀⠀⢀⣤⣬⣤⡀⠀⠀⠀⠌⠑⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⣧⢀⠃⠀⣈⠠⠀⠀⠀⠀⠀⠉⠁⠀⠂⠠⠀⠀⠄⠠⠀⣀⠀⠒⠒⠈⠉⠐⠂⠈⠉⢉⣉⡐⣒⣚⣶⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢰⡗⢬⣉⣛⡚⠒⠚⠀⠀⠉⠉⠉⢉⣐⣀⣀⡠⠄⢒⡀⠁⠀⠀⢀⣤⠤⠤⠒⠦⣤⣤⣬⣭⠴⠖⠚⠛⠉⣁⣄⡠⠨⢽⡦
⠀⠀⠀⠀⠀⠀⢸⡟⣶⣤⣤⣉⣛⠒⠲⠶⠶⠶⢶⣦⣤⣌⣉⣉⡛⠛⠓⠒⣀⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⠤⠤⠔⠒⠒⢉⣉⣤⠶⡶⠋
⠀⠀⠀⠀⠀⠀⠘⣧⡁⡟⠓⢝⠫⡙⠻⢗⡒⠢⠤⢀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠤⠤⠀⢀⠰⠞⠛⠍⢉⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⢡⠀⠀⠑⢌⡢⠄⠠⠁⢀⡀⠠⠀⠀⠀⠒⠒⠲⢚⡉⠉⠉⠉⠀⠀⠀⠁⠀⠀⠬⠁⠀⣀⠠⢐⡴⣫⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠠⡳⣷⡄⠀⢈⠒⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠁⠀⣰⠟⢨⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡐⢌⡻⢦⡄⢉⠒⠄⠤⠤⠀⠉⠁⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠒⠒⠀⠀⢉⠀⠀⣀⠼⢁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣬⣒⠤⣀⠠⠀⠀⠀⢒⠒⠲⠦⠤⠤⠤⠤⠀⠠⠤⠤⠄⠀⡂⠔⣨⠤⠒⠈⢁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⢅⠢⢤⣀⣀⣀⣉⡐⠒⣀⡂⠀⠀⠀⠀⣒⣈⡥⠖⠛⣉⣭⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣄⣀⣀⣀⠀⣀⣠⣤⡴⠞⠀⠀⠀⠀⣀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⡇⠀⡇⠀⠀⠀⠐⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡃⢧⠀⢹⡰⡄⠀⠀⠀⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢊⠋⠉⣹⣿⣿⠿⠿⠟⢛⣛⣲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⠘⣷⡹⡀⠀⠰⡀⠢⠂⣈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢉⣴⠿⠀⠉⡠⠄⠂⣀⡀⣠⣶⣶⡛⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡆⢻⡀⠱⣷⡑⠀⢐⠄⠡⡀⢸⣇⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠋⡳⣄⡿⣿⣿⣟⡆⠀⠀⠀⠀⠀⣴⣆⣻⣿⣇⠀⢧⠀⠹⣟⠀⠀⠀⣀⣀⡀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠠⠞⢋⣾⣽⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣌⢣⡀⠙⠧⢦⣦⡙⣿⣧⣈⣿⡄
⠀⠀⠀⠀⠀⠀⡔⠀⠀⡰⠁⠀⠀⠀⠀⠀⢀⠄⠂⣡⣿⣿⣿⣿⣿⡏⠀⠀⠁⠀⠀⠀⠉⢹⣿⡻⣿⣿⣷⣌⠳⣄⠀⠻⣟⣿⣿⣿⡏
⠀⠀⠀⠀⠠⢘⣴⠂⡔⠁⠀⠀⠀⢀⠀⢐⠀⠀⢸⣿⣿⣿⣿⠛⠁⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣌⡑⠦⢌⡙⠻⣿⡅
⠀⠀⠀⠀⠀⠞⠡⠊⠀⠀⠀⠀⢀⠁⢀⣾⣥⣶⣿⡿⠋⠀⠁⡀⠀⢀⠈⢛⣋⣿⣯⣭⣾⣍⠀⠁⣽⣾⠁⣿⣿⣿⣿⣷⣶⣭⡓⢮⡇
⠀⠀⠀⠀⠀⠀⣧⠀⢀⡤⢔⣲⠍⠐⣫⣾⣿⣿⠋⠀⠀⣰⠋⠀⠀⠀⢐⡼⠋⠀⠀⠀⠙⢻⣶⢾⠟⢁⣼⣿⠟⠛⠛⠋⠉⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠲⢽⣖⡻⢶⣂⣴⣽⣿⣿⣿⡦⣶⠞⠁⣀⣀⣠⣴⡿⠃⠀⠀⢀⣤⣴⠟⠁⣀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠛⠋⠁⠀⣯⠀⠊⠉⠉⠉⢋⣁⣀⣀⡠⠟⠉⣀⣴⣾⣿⡿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠛⠳⠤⣀⡤⠞⠉⠀⢉⣁⣤⠶⠛⣹⠵⣿⣟⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣶⣶⡶⠾⢿⣿⠟⠀⠰⠂⠀⣩⢵⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠿⣿⠥⠾⠁⠀⠀⠀⣠⠴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢯⣧⣴⣶⣤⡤⠶⠛⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠢⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢏⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⢁⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⣀⣀⣀⣀⣀⡴⠁⠀⠀⠀⠀⠈⠀⠀⠈⠑⣄⠀⠀⠀⠙⠒⠋
⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡘⡀
⠀⠀⡠⠉⠀⠀⠁⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠜⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⢤⣤⡤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡔⠉⠀⠀⠈⠓⣄
⣸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢣
⡇⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠃
⠃⠀⠀⠀⠀⠀⠈⠒⠦⣄⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⡀⢀⡀⣀⣀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⡄
⠈⢧⠀⠀⠢⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠑⠤⣤⣀⣤⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣤⠄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠉⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄⠀⠀⠀⠀⢀⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⠶⠖⠋⠁⠀⠀⠈⠳⣄⠀⠀⠻⠁⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠶⠶⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡦⠄⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢀⠁⣄⠀⠳⢤⢸⠃⡇⣶⠘⠡⣀⣆⡰⠀⠐⡎⢹⠸⡀⣸⠀⠠⠀⣙⡆