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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⢀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠋⠉⠉⠉⠉⠒⢤⠄
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⢠⠋⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢰⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠒⠶⠲⡀⠉⠓⢤⠀⠀⢀⡠⠃⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠞⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⠎⠉⢹⠒⠒⠦⠄⠀⠤⠤⠤⠔⢚⠁⠀⠈⠉⠈⠀⠀⠀⠹⢀⣀⣠⣀⣀⡠⠔⠦⠤⠊⠙⡦⡀⠄⠀⠀⡇
⠀⠐⡧⠀⠀⠸⡁⠀⠀⠀⠀⠠⠄⠀⠀⡘⠀⠀⠀⠀⢸⠓⢄⠀⠀⢀⠤⠀⠀⠀⠀⠀⠐⠂⠀⠀⡇⠘⢦⡀⣴⡀
⠀⠀⠥⠤⢔⣦⠀⠀⠀⠀⠀⠐⠀⠀⠀⠣⣀⣠⣄⡠⠃⠀⠀⠑⡄⠈⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠣⠤⠞⡟⠁⠈⢆
⠀⠀⠀⠈⠃⠈⠢⡀⠁⠀⠁⠀⠀⠀⠀⠀⣀⠤⠋⣇⡀⠀⠀⢠⠴⠀⠙⠄⡈⠛⠉⠀⠀⠀⠀⠀⣀⢴⠊⠈⢆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢈⠵⢦⡤⡤⠤⠶⠶⣉⣉⡑⠲⠚⢇⣠⠶⠎⠀⠀⠀⡤⠒⢿⢶⣒⡲⠖⢸⠛⠲⣍⢑⣄⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⡇⠀⣠⠒⠳⡌⠢⡀⠁⠳⡟⠁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢸⡀⠀⠈⡆⡿⠆⢠⣄
⠀⠀⠀⠀⠀⠀⢸⢠⡔⠋⠀⠐⡇⠀⠀⠰⡀⠈⢢⠀⠑⡄⠀⠀⠀⠀⠀⡆⣞⡇⠀⠀⠀⠀⢀⠳⢆⠉⢡⢹⠈⣤⠃
⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⠑⢄⡀⠀⠀⢀⡼⢆⠀⢡⠀⠀⠀⠀⠀⠈⢺⠀⠀⠀⠀⠀⠀⠁⠎⠀⠸⠈⠈⠹
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠉⠐⠂⠉⡔⠀⠀⡰⠀⠀⠀⠀⠀⠀⠮⣤⣀⣀⣀⣀⡠⠊⠀⢀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠘⠤⢀⣀⡀⠀⠀⠀⠀⢀⣠⢾⡓⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠁⣗⡴⠊⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠉⠉⡿⠉⡏⠉⠕⠺⢥⠊⠁⠂⠀⠐⠀⠀⠀⠀⢈⡞⠉⠑⣄⡠⠊⢸⠒⠤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡞⢅⡀⠀⣰⣇⡴⢁⠴⡄⠀⠀⠱⡄⠠⠊⠃⢀⣀⣀⣠⢎⠀⠀⢠⠛⢤⠔⠁⠀⠀⠈⢳
⠀⠀⠀⠀⢠⠖⠹⡀⠈⡳⠀⠉⠋⠀⢹⣅⡠⠇⠀⠀⠀⡇⠀⠀⢠⠋⠀⠀⠀⠁⠉⠒⡞⠈⢈⠃⠰⠀⠀⠀⡼⠀⢀⠀⢀
⠀⠀⠀⠀⢃⠀⠀⠈⠉⠀⠀⢠⠂⠀⠀⢣⡀⠀⠀⣀⠜⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠃⠀⠘⢄⠀⠀⢀⠜⠀⠀⠈⠉⢄
⠀⠀⠀⠀⠈⠢⠄⠄⢂⡠⠤⠊⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠒⠤⠄⣀⡠⠎⠀⠀⠀⠀⠁⠊⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⢈⡀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⠀⠈⠀⠅⠀⢸⡇⠀⠀⠀⠀⠀⠘⠃⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⠀⠐⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠆⠘⡇⠀⣤⢠⠀⡀⠀⠤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠦⠤⠀⠖⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠙⠁⠀⠀⢀⠈⠃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠟⠘⠃⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⣹⠄⠀⢽⣷⠦⠤⠡⠀⠄⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠶⣠⣀⢸⡟⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⣽⠟⠃⡘⠇⠀⠘⢁⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣰⠏⣵⡨⣿⡉⠀⠀⠀⠀⠀⠀⣶⠀⠀⣄⣰⣭⠀⣬⣷⡒⣬⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣲⠤⠀⣼⣷⣾⣯⣽⣧⣉⠷⡂⢲⡄⢺⡒⣶⣿⣏⡴⣮⣥⡶⠶⣿⠷⠦⣛⣗⢲⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁⣽⠍⣈⢹⡶⠒⣛⣧⠤⢿⣭⠤⣴⣷⣡⣽⣅⣺⡞⣁⠲⢯⣼⠿⠭⣽⡟⠨⣭⠦⠤⠤⣴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠚⣧⡴⠤⠤⢿⣭⢍⣠⣏⡈⣡⣦⣽⣿⣋⣁⣿⣿⣃⢚⣓⣲⣚⣛⣻⣍⣉⡽⠧⠤⠤⢤⡜⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣶⠒⠒⠛⡶⠶⠶⣧⠴⠞⢻⡿⠶⠷⣦⠘⠳⡖⠐⠛⢶⡄⠛⠲⣦⠀⠛⠀⠐⢶⡒⠛⠃⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣉⡴⠉⠉⢙⡗⠊⠛⡶⠂⠈⠙⡿⠁⠛⠒⢖⠛⠛⢻⠚⠛⠻⡏⠛⠓⠛⠦⠈⠉⠉⠻⢯⣁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠲⠖⠒⠛⡲⠒⠚⠓⠲⠒⠚⠓⠒⠒⠒⠊⠲⡖⠚⠓⠶⣖⠐⠒⠒⠲⣖⠛⠛⠋⠙⠋⢍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣠⡆⠀⠀⣀⣰⡃⣀⣀⣀⣄⠀⣀⣀⣀⣸⡄⠠⣤⡭⠧⠤⠀⠤⣌⠛⠄⠰⠶⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⢀⡴⠃⠀⠀⣠⣃⡀⢀⣀⣠⠇⠀⢀⣀⡀⣹⣁⣀⣀⣈⣑⣤⡀⢀⣀⠀⠱⣄⡀⢀⢀⣀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠝⠁⠈⠀⢀⠟⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢳⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡙⣯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⢀⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠐⠋⠉⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⣀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⣼⣿⡧⠀⠀⠀⠀⢸⣿⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠘⠛⢁⡄⠀⠀⠀⠸⠿⠏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠲⠦⠶⠂⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⣀⢀⣀⣀⣠⣽⡿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣟⣉⣠⡿⠛⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠻⣎⣿⢋⣥⠾⠛⠉⠉⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣿⠀⠀⣼⠁⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠁⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠙⣧⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⢀⣴⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⣇⠀⠘⣷⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⢿⣄⣠⡴⠞⠋⠁⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣧⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⣤⠟⠉⠁⠀⠀⠀⠀⣿⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢷⡶⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣀⣀⣀⣀⣿⠀⠀⠀⠀⣼⡃⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠙⠛⠛⠛⠋⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢀⡀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡾⠁⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣠⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⣴⠋⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣸⢧⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢰⡏⠀⠻⣄⠀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⢀⡟⠀⠀⠀⠘⢷⣴⡟⠉⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠉⡿⠛⠁⠀⠀⠀⠀⠀⠀⢹⣦⣤⡟⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡏⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠛⠁⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠁⣽⣥⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⠀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⠉⢉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠄⣠⠀⠀⠀⠀⠀⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡴⠃⠀⠘⠃⡠⠃⠀⠈⠗⠀⠈⡇⠀⠀⠀⠀⠀⢀⡀⢠⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣽⢁⠀⠀⠀⠠⣀⣀⠀⠀⠀⠀⠀⣿⡦⣄⠀⠀⠀⡄⢱⠁⠚⣡⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠈⢲⠤⢄⣀⡈⠉⠀⠀⢀⠠⡀⣿⠁⡿⠀⠀⠰⡁⠀⠇⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣶⣦⣤⣤⣤⡤⠊⠀⠀⣿⠚⠁⠀⠐⠊⠣⣀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠭⣉⠭⠄⠀⠀⢀⣼⣣⠔⠋⠀⠐⡄⣠⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⠀⠀⠀⣀⣠⡴⠛⠉⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⢩⠟⠁⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡇⢈⡶⠴⢦⣸⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠈⠁⠀⠀⠈⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣠⣾⠀⠀⠀⢀⠁⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⢀⡾⠋⣿⠀⠀⠀⢸⠀⠀⢱⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠨⡈⢁⠎⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡀⠈⡇⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⣸⣼⠁
⠀⠀⠀⠀⠀⢀⡤⠖⡛⠛⠲⢦⣄⠀⠀⢙⡶⣞⢸⠀⠀⠀⠀⢰⡏⠙⠛⠛⠓⢦⠜⠉⠓⢦⣄
⠀⠀⠀⢀⡔⠋⠔⠀⠀⠀⠀⠀⠈⠳⣴⡿⠁⠈⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⡠⣄⡀⠀⠀⠙⢷⣄
⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢰⠀⠙⢦⡀⠀⠀⠹⣆
⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣸⠀⠀⠀⠹⣆⠀⠀⢻⡄
⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⢀⡟⠀⠀⠀⠀⢻⠀⠀⠸⡇
⢠⡀⠀⠀⠀⠀⠀⠀⠐⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⠚⠋⠁⠀⠀⣠⡞⠁⠀⠀⠀⠀⢸⠇⠀⢀⡟⠀⡰⢹⠆
⠀⠙⠦⣄⡤⠤⣄⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠈⠳⢤⣀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠘⡤⠒⠋⠉⠉⣠⠟
⠀⠀⠀⢿⢸⡱⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣀⡀⠙⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠻⢄⡠⠎⠉⠁
⣀⣀⣀⣈⣳⣶⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣳⣶⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣯⣿⣽⣽⣽⣽⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣭⣽⣿⣯⣭⣭⣭⣭

⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⣠⣼⣿⣁⠀⠀⠀⠀⠀⢀⣤⣿⣇⡀⠀⠀⠀⠀⣹⣿⣤⠀⠀⠀⠀⠀⠀⣠⣿⣷⡄⠀⠀⠀⠀⣸⣿⣦⡀
⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⠰⣟⣿⣿⣿⢷⡄⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣻⣿⣿⣿⠙⠆⠀⠀⠀⠀⣿⣿⣿⠞⠁⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⠀⠀⣿⠟⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⢿⡿⠀⠀⠀⠀⠀⢿⡏⢿⡆
⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⢠⣿⠁⠹⠗⠀⠀⠀⠀⠀⢿⠀⣿⠀⠀⠀⠀⠀⠀⣿⢻⠟⠃⠀⠀⠀⠀⠀⣿⡸⡇⠀⠀⠀⠀⠀⢸⡇⠘⣿
⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠀⠀⠀⠀⠀⠀⠛⠘⠃⠀⠀⠀⠀⠀⠀⠋⠁⠓⠀⠀⠀⠀⠀⠘⠃⠀⠘

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡃
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⡤⠴⢻⣿⣷
⠀⠀⠀⠀⠘⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉
⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⢴⠋⢸⡃⠀⠀⠀⠀⠀⠀⠀⢸⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⣦
⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈


⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⣸⣿⣧⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⣹⣿⣿⡿
⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣇
⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠆⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⢸⣿⢿⣿⠛
⠀⠀⠀⣿⠃⠸⣧⠀⠀⠀⠀⠀⠘⡟⣿⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⢹⡜⣇⠀⠀⠀⠀⢈⡇⢀⡟
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠛⠓⠂⠀⠀⠀⠀⠀⠀⠘⠃⠛⠀⠀⠀⠀⠈⠃⠈⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⠶⠖⠒⠒⠒⠶⠶⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠙⠛⠛⠓⠒⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡅⣴⠻⣶⣶⣤⣤⣄⠀⠀⠀⠀⢶⡦⣤⣀⣀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢧⠉⠀⠀⠈⠉⠒⠦⣤⡀⣀⡀⠀⠱⣄⠀⠉⠉⠲⡆⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠛⠂⠀⠀⠀⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡴⠁⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠘⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⡀
⠀⠀⠀⠀⠀⠀⣀⡤⣼⡄⠀⠄⠀⢀⣀⣀⠀⠖⠘⠀⠀⠐⠂⠀⣀⣀⠀⠂⠀⠀⠄⢼⡧⣄
⠀⠀⠀⠀⠀⢰⠃⠀⢸⠃⠀⠀⠀⠈⣻⣿⠀⠀⠀⡄⠀⠀⠠⠀⢸⣿⠄⠀⠀⠀⠀⢸⠀⠀⠓
⠀⠀⠀⠀⠀⠈⡄⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡎⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠘⢦⡀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠘⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⡾⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣷⠶⠶⠦⠀⢀⡀⡀⠀⠴⠶⠒⣹⣟⡷⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⡏⣄⠀⠀⢀⠞⠣⠀⠀⠀⢀⢴⡿⡜⠁⠀⠀⠀⢀⡼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢻⠆⠀⢸⢣⡄⠀⠀⡄⠀⠁⠀⠀⣠⢮⡟⠀⠀⠀⠀⠀⡾⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⣾⠀⠀⠀⢎⠱⣄⠀⠃⠀⠀⣠⢞⡵⠃⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⠀⠋⠀⠀⠀⠈⠳⣌⠙⠓⠚⣩⠶⠋⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣤⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡼⠁⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢀⡥⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⣴⠫⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠲⢦⠤⠀⠏
⠀⠀⠀⠀⣼⡧⠙⠒⢲⣌⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⢸
⠀⠀⠀⣸⣋⣴⢰⢀⡟⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠈⠉⠙⠧⢼⣧⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠒⠶⢤⣄⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠸⠃⠀⠀⠀⢄⡀⠀⠀⠀⣀⠀⠀⠀⣀⣀⠀⠀⣾⠛⠛⠛⢷⣤⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⡄⣤⠀⠀⠀⠀⠙⠻⠶⣤⣈⡙⠛⠶⠾⠿⠿⢾⣧⣤⠀⡀⣴⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠛⠿⣿⢿⣇⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡇⢿⠛⠛⠛⠛⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠒⠛⠛⠛⠛⠛⠋⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠰⠖⡟⢿⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡌⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣶⠀⠠⠤⠤⠤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⠐⠒⠂⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠘⣄⣤⣤⣤⣤⣤⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢷⣖⡃⠬⠥⠴⣒⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠤⠤⠤⠬⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣏⡁⠀⣀⣀⠀⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄⢀⣠⡀⢤⣤⣤⣤⣤⣤⣤⣼⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠁⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣀⣬⡿⠀⣀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣉⣤⣀⠀⢠⡟⠉⠙⣧⠀⢀⡀⢻⣄⣤⠶⢿⡏⠁⣴⠋⠉⢻⡄⠀⣀⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠙⠷⡈⢷⣄⣴⠏⡰⠟⠁⠀⣉⣀⠀⠀⢻⡄⠹⣦⣠⡾⢁⠾⠋⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠈⠀⠀⣠⡴⠃⣾⠋⠙⣷⠀⠀⣻⣄⡀⠁⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠛⣀⡀⢿⣄⣠⡿⢠⠾⠋⣩⡷⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣠⡾⠋⠁⠀⠉⠉⠀⢀⣤⠞⠋⢻⡄⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣧⣤⠀⣼⠋⣠⣾⡇⠀⠀⠀⠀⢸⡇⢶⣤⡄⠻⠄⣼⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢀⠙⠛⠋⢸⡇⠀⠀⠀⠀⢸⡇⢸⡇⠀⡆⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⠀⢸⠀⢸⡇⠘⢷⡶⢶⡶⢶⡾⠃⢸⡇⠀⡇⠀⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣼⣄⣼⠇⠀⢸⡇⢸⡇⢸⡇⠀⠸⣧⣠⣧⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠸⣧⣼⣧⣼⠇⠀⠀⠈⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⢿⣿⣿⣦⠀⠛⠁⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠈⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣴⣿⠟⢱⣶⡄⠀⠀⠀⠀⠀⠀⠈⠛⠇⢀⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠻⣿⣿⣿⣿⠗⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣶⠒⠒⠒⠒⠒⠚⠃⠀⢸⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣋⣀⣀⣀⣀⣿⠀⠀⡤⠤⠤⠤⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⠉⠉⠉⠉⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠟⠀⢸⡟⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣿⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠈⠉⠉⠉⠉⠉⠁⠠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡖⠐⠒⠢⠤⠄⠀⣀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢠⠖⠲⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠙⠢⢄⣀⣀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⠀⠀⠀⠀⠀⠀⠻⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠐⡟⠉⠋⠀⠈⠑⠢⣄⠀⠀⠀⣠⠜⢻⠰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠉⠁⢸⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠇⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⡀⠠⡄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢠⡀⢠⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡌⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸⠀⢸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⡇⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⡎⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⠀⠀⠀⡀⢾⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢐⣇⢀⡤⠀⠒⠂⢉⡧⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠉⡶⠶⠁⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠆⠀⢀⡤⠚⠉⠀⠀⠀⣀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢀⡸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⣄⠀⢀⣀⠠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⡜⠁⠀⠀⠀⢸⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢰⠁⢀⠾⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⡆⣀⡜⠀⠀⠀⠀⠀⠀⠸⡆⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣤⠋⠀⣉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠓⠠⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠸⠂⢀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠸⣄⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠎⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⣠⠜⢧⡀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀
⠀⠠⠔⠒⠈⠣⣄⡠⠷⣀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠫⡄⠀⠀⠀⠠⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀
⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢺
⠀⠀⠈⢦⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣀⠠⠟
⠀⠀⠀⠀⠱⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠂⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣀⡀⠤⠤⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣒⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⢀⡀⢀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠶⠒⠒⣇⢳⠶⠲⣤⣤⣴⠐⢰⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣻⠈⠉⣉⣠⣶⣄⢨⣧⣤⣀⡤⠀⡙⠀⣤⡉⢉⣼⠿⠒⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣲⣿⣶⣾⣿⣿⣿⡿⢿⣿⣿⣿⡷⣼⣿⣿⣿⣿⣿⣦⣴⡾⢷⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠀⢙⣿⣿⡿⣿⣿⣿⣥⣼⣧⡀⣀⣿⣬⡁⣹⣿⣿⢿⣿⣿⣄⡀⣀⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠉⠀⠉⣿⣿⣿⡇⣨⣿⣿⣿⡏⢀⠈⣿⠛⣿⣿⡍⣀⣹⣿⡁⠟⢿⣿⣷⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣤⣶⢠⣾⣿⣿⢏⣷⣿⣯⣩⡵⢻⣦⣼⠻⣦⡈⡾⢷⣿⣿⣿⡿⠗⢀⡀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠈⣶⣿⣿⣯⣦⣼⣿⣬⠷⠮⣄⣿⡅⠛⠾⡀⣄⡾⢷⣌⢻⣧⣶⡌⠿⣶⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠰⣿⠛⣿⣿⠃⠸⣟⢿⠀⠘⣿⢿⡏⠀⢸⡟⣿⠃⠀⢸⠇⣩⣿⡟⢠⢌⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⢀⣽⣷⡫⢙⣦⣤⠷⡿⣶⣾⣿⣸⣿⣶⣻⣧⣽⢧⡴⣿⣆⣠⢨⣿⢏⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣻⣟⣾⣿⣿⡞⢻⣿⣿⡿⠛⠉⠉⠉⠛⠿⢷⣿⣿⠻⣾⣯⣧⣼⣿⣶⣨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣧⣾⣿⣿⣶⣶⠾⠟⠻⣟⣻⣿⣿⣿⣽⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠛⡿⠀⣿⡇⠀⠀⠀⠀⢸⡇⠈⣏⠻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠀⢰⡇⠀⣿⡹⣄⡀⢀⡠⢃⡿⠀⣿⡀⠘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⢀⣼⡷⠠⢿⡜⣎⠻⠋⠀⢀⡧⠰⢿⡷⢄⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⢠⡆⠀⠠⠀⠀⣬⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣦⢴⣾⠃⠂⡁⠀⡀⣸⣷⡶⣶⣿⣯⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠿⣣⡾⣄⣀⣀⣷⠀⣴⣷⣿⣦⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠻⠁⠈⣹⢻⢻⠟⢿⣏⠻⡜⢷⣾⡙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢡⣿⢀⡆⢀⠀⡅⠀⢸⠇⡼⢸⠀⠀⢈⡾⣿⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣋⣠⠟⠁⠛⢦⣤⣤⣄⡀⠀⢀⣁⣀⢶⣻⣏⣤⣼⣿⣽⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⢠⡟⠀⠀⠀⠀⠛⠛⠋⣥⣶⣷⡭⠉⠍⢰⣿⣿⣯⣿⣿⣷⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠐⢢⡞⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⣾⣿⣿⡿⠛⠛⢷⡀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠎⠁⠀⠀⠀⠀⠀⣤⣠⣤⡼⠛⠛⠀⠀⠀⣨⣿⠁⠀⠀⠀⠈⢳⡏⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠈⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁⠃⠉⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠻⠆





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⠀⠀⣶⣿⣶⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⣴⣾⣷⠀⠀⠀⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠾⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⠃⠉⠀⠀⠀⠸⢸⣿⣿⠗
⠀⠀⠀⠀⠈⣯⡇⠀⠀⠀⠀⠀⠀⢹⣸⠃⠀⠀⠀⠀⢸⡏⣿⡁⠀⠀⠀⠀⣿⢿⡏⠀⠀⠀⠀⠀⠀⢸⡏⣧
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⢯⠀⠀⠀⠀⠀⠸⡇⢹⠇⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⠸⠇⣿

⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⠀⢀⣿⣿⠲⢤⡀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢘⣿⣿⡏⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⣷
⠀⠀⠀⠚⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⢻⣿⡿⠛⠆
⠀⠀⠀⣴⠟⠈⢻⡄⠀⠀⠀⣼⠏⠈⢻⣆⠀⠀⠀⠀⠾⣿⣿⡿⠆⠀⠀⠀⠠⣾⠟⢻⠀⠀⠀⠀⠀⣿⠿⣷⡀
⠀⠀⢠⣏⠀⠀⠀⠹⠶⠂⢰⣇⠀⠀⠀⠹⡦⠆⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⣼⡁⠀⠸⣦⠀⠀⠀⣰⡟⠀⢸⠧
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠉⠉⠐⠒⠢⠄
⠀⠀⠀⠀⠀⠀⣀⢀⡀⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠐⠀⠀⠠⢀⠀⠀⡀⠈⠀⡠⠤⠤⠒⠒⠢⢄⠀⠈⠤⡀
⠀⠀⠀⡠⠂⢁⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠈
⠀⠀⠈⢀⠔⠈⠉⢢⠀⠀⠀⠀⠀⠀⠀⢣⠀⡐⢁⠀⣀⣀⣠⣀⣀⡀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠰
⠀⠠⠔⠁⠀⠀⠀⠀⠣⠠⠤⢀⡀⠀⠀⢸⢐⠀⡈⠉⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡠
⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠄⠀⠈⠑⢤⡜⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢰⢓⠅
⠀⠀⠈⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠠⠀⠘⡀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢪⠦⣀⠀⠀⠄⢐⠈⠀⣀⠔⠁
⠘⠣⢄⣀⡑⠄⣈⡐⠒⠒⣁⡠⠔⠁⡠⠶⠆⠢⠠⠼⠤⡀⠀⠁⠘⢢⣤⣀⣀⣀⡠⠊⠀⠈⠙⢒⠶⠒⠶⣋⠁
⠀⠀⠀⠠⢠⠜⠁⠉⠉⠘⠉⢦⡈⡉⠀⢀⡦⠤⠤⡄⠀⠋⠉⠉⠋⠟⠐⢤⠤⠄⠒⡣⠤⠎⠀⠁⠐⠄⠀⠈⢆⡑⠒⠢⢤
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⡒⠁⠀⠘⠃⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⢀⡠⠬⢁⡀⠂⠠⣤⠤⠤⠄⠤⠆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢇⡀⠀⠀⠄⠠⠤⠤⠤⢼⣀⡀
⠀⢀⣳⡀⠀⠀⠉⠈⠈⢂⡔⠐⠒⠂⠀⠀⠀⠚⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⢀⡢⠀⠀⠀⢰⣀⣀⣀⣀⣦⠤⠔⠊⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡖⠒⠊⠉⠈⠳⣄⡴⠀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡀⠀⠀⠀⠀⠈⠁⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠈⢉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠛⠫⡉⢢⡉⠢⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡄⢄⠀⡄⠲⡈⠣⠑⢲⠤⡄⠀⣼⠖⠀⠀⠀⡠⢺⠋⢍⠒⠀⠀⠁⠀⠁⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣸⢸⡀⠣⠘⠀⠁⠀⠄⠈⣏⣥⠾⠃⠀⠀⠀⠀⡔⡏⢀⡀⠀⠀⠀⠀⡀⠀⠀⢸⡱
⠀⠀⠀⠀⠀⠀⠀⢠⠿⡀⠀⠂⠀⠀⠀⢀⡄⠀⡏⢀⣀⠀⠀⠀⠀⠀⠈⢇⠀⠁⡀⠀⠀⢀⡁⠀⠀⡬
⠀⠀⠀⠀⠘⣷⣀⣈⡽⠳⡀⠀⠒⠒⠒⠉⢀⡴⠡⡋⠈⠃⢀⠐⠠⡆⠀⠈⠢⡀⠙⠂⠐⠊⠀⢀⠔⠁⢠⡤⢤
⠀⠀⠀⠀⠀⣈⠉⠁⠀⠀⠈⢓⢶⡴⠦⣚⣉⣠⠔⠉⠁⠀⠈⠓⠚⠳⢄⣠⢤⡌⠓⡴⢤⣤⠖⠥⣤⠴⠚⢦⡼
⠀⠀⠀⠀⠸⣉⡹⠒⠒⠾⡉⠉⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⢤⠀⠀⠀⠀⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣀⣀⣀⠠⢧
⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⠀⠀⠀⠀⠀⢀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⡀⠀⢀⡱⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⠒⠒⠒⠲⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠒⠃⠑⠘⡀⣠⣻⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⣠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠦⠠⠃
⠀⠀⠀⠀⠀⠀⠰⢏⣈⡹⠃⠀⠰⣛⣃⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠁





⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⡀⢀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠚⠛⠋⢉⣉⠉⠁⠲⠦⢤⣀⠀⠀⠀⠀⣀⣤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠁⠀⠀⠀⠈⠉⢲⡤⠒⠉⠉⠈⠈⠉⠉⠒⠢⢄⡀⠈⠙⠒⠤⣀⠀⠉⠒⠶⣉⠁⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢀⣀⣀⣠⠖⢻⣷⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠑⠦⣄⠀⠈⠓⠤⣤⠤⣀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⠊⠁⢠⠎⠀⠀⠸⠘⣧⡀⠈⠑⠢⣤⣀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠉⠲⣤⣀⠈⠑⢤⡱⠄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠞⠀⣰⠃⠀⠀⠀⡇⠀⠈⢻⣦⡀⠀⠀⠉⠑⠂⠤⣄⡀⠀⠀⠀⠀⠈⠉⠒⠠⠤⣀⣈⣑⢒⣤⣽⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠆⡰⠁⠀⠀⢀⠞⠀⠀⢀⠞⠉⠻⣳⠦⢀⡀⠀⠀⠀⠀⠙⠐⠦⠤⠀⣀⣀⣀⣀⣤⡴⠎⠛⢁⡤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠁⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠉⠒⠤⢉⡁⠒⠒⠂⠀⠀⠀⠒⠒⠒⠈⠉⢁⣀⣴⡿⢥⣴⣏
⠀⠀⠀⠀⠀⠀⢰⠀⠠⣴⣉⣀⠠⠐⠉⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠙⠛⠒⠒⠒⠀⠀⠀⠐⠒⠲⠦⣉⣀⣀⣀⣀⣀⣽⠃⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠋⢹⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠘⡀
⠀⠀⠀⠀⢀⡜⠀⠀⢀⠆⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸⡀⠀⠀⢣⡀
⣀⠤⠤⠔⠋⠀⠀⢠⡌⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢱⡄⠀⠀⠙⠦⡄⣀
⠣⣀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠙⠄⣀⠀⠀⠀⢈
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠶⠤⢀⣀⣀⣀⣀⣀⣀⠤⠶⠐⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⡨⠥⠶⠶⣖⠒⠒⠒⠈⠁⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣰⠋⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢸⠑⣀⣀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢱⣤⣴⣶⡶⠿⠿⠛⠛⠋⠉⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠛⠛⠋⠉⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢧⠠⢤⣤⠤⠤⠤⠤⠀⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡤⣤⣤⠃⠠⣤⣄⣐⡲⣦⠼⣿⣟⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⢠⡄⢰⡆⢹⡿⠉⠙⠃⠐⡏⠈⠃⠛⠦⠞⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⢶⣾⠛⠛⠒⠛⠒⢋⣈⣷⢀⡛⢻⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⡎⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠀⠀⠀⠀⠀⠀⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠈⠙⣧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡇⠀⠀⠀⠀⢀⣀⠠⠆⠚⠁⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠄⠀⠒⠒⠲⠶⠒⠀⢤⣤⣤⡼⢧⣀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣀⣀⣀⠤⢤⣉⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣤⣤⡀⢀⣀⡀⣀⣰⡟⠒⠛⠛⠛⠛⠉⠉⠉⠁

⠃⠀⣠⠞⣁⢖⡡⠀⠀⠉⠱⡆⠀⠀⠀⡠⠞⠁⠀⣰⠧⠀⡰⠁⠀⠀⢠⡇⠀⠀⢀⡿⢃⡴⠟⠋⠁⠀⢀⣀⣤
⣠⠞⢁⡶⠟⠉⠀⠀⠀⠀⠀⡀⢈⢀⠚⠁⢌⢀⡾⠁⠈⡸⠁⠀⠀⠀⡾⠀⠀⢠⣿⡶⠋⢂⣤⡤⠖⠛⡉⣂⣤
⢁⡔⡹⠁⠀⠀⠀⠀⠀⠀⣰⣡⡞⢩⣼⡱⠶⠟⠓⠒⠒⠳⠤⢄⡀⠀⠁⢀⣴⣿⠿⠞⠋⠉⢀⣀⡤⠖⠋⠉
⠎⢰⠁⠀⠀⠀⠀⠀⢀⡼⣏⡾⠗⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⣋⣀⣄⠤⠤⠖⠚⠉⠁⠀⠀⣠⠴⠛
⠀⡇⠀⠀⠈⠀⠀⣰⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⡀
⠀⡇⠀⠀⠀⠀⢸⡟⠡⠊⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢀⡠⠴⠟⢁⢀⣴⠚⠃
⠀⣇⠀⠀⢀⣠⠏⠀⢀⡠⠂⠁⠀⠀⠀⠀⢀⣀⣀⠀⠤⣤⠴⠊⠀⠀⠀⠀⢀⡤⠞⡃⠀⠀⠀⠈⠓⠆⠍⢹⡀
⠀⢸⡇⠀⣾⡟⠀⠔⠁⡠⢄⣂⢂⣤⡩⢅⡶⠦⠄⠐⠈⠀⠀⠀⠀⣠⠴⠺⠉⠀⢀⣷⡀⠀⠀⠀⠀⠀⠑⢌⣷
⠀⠀⣿⣔⣿⣧⢀⣴⣽⣾⣿⠿⠿⠽⠷⣮⡤⣄⠀⠀⠀⢀⣤⢖⢋⠀⠀⣠⣴⠚⠋⠀⠲⡀⠀⠀⠀⠀⠀⠈⠛
⠀⣼⠀⣟⣿⡇⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠚⠉⠀⠈⠀⠀⠚⠁⠀⠈⠀⠀⠁
⢠⢻⡥⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⣤⣀⣀
⡆⢸⢰⣯⠏⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉
⣃⣾⠗⣁⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣯⠴⠚⣩⣼⡃⠀⠀⠐⠀⠀⠄⣀⠀⡀⠀⠀⢀⠂⠀⠀⠀⢀⠀⠠⠤⠄⢀⣠⡄⠀⠀⢀⡀⠀⠀⠀⠁⢐⣶⠛
⢗⠈⣴⠞⢍⡀⠤⡼⠇⠤⢀⠘⠁⠚⠉⠠⠤⡾⠀⠄⢠⠀⢡⠀⠀⠀⠀⠀⠁⡄⠀⠐⢁⡇⠀⠀⠔⠀⠈⠓⡀
⡰⣞⣠⣴⠞⠁⢠⠃⠀⣀⡾⠀⠀⠄⢸⣠⣾⠁⠀⣠⠇⣐⣝⠰⣂⣀⡀⡄⠀⢇⣠⢃⣯⡀⠆⠀⠀⠀⢀⠜⠁
⠘⠛⠳⠶⠚⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⠊⠀⡄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⡄⠄⠀⢠⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⢸⠀⡇⢠⠘⠀⡇⢠⣘⠚⡂⣈⠑⠁⠀⠀⠀⠀⠀⠀⢠⡾⢁⡼
⠀⠒⠶⢆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣧⢊⡴
⠀⠤⠀⢠⣝⣧⣤⣤⠄⠀⡄⠀⠀⠀⠀⠀⠤⣀⢠⣄⠠⠀⠀⠀⡄⠀⠠⡄⢤⣠⠀⠀⢀⠄⣠⠔⣝⣷⠟⡩⠊
⠀⠀⣀⠔⠚⠛⠋⠀⠀⠀⡇⠀⠈⢀⠀⠀⠀⠘⢷⣿⣆⠡⠀⠀⢰⠀⠀⡅⣜⣿⠀⠀⡆⣴⢋⠈⠞⠁⠈⠀⣀
⠼⡀⠤⠄⢤⠤⣄⣀⣀⠀⢰⡀⠀⠀⠂⠀⠀⠀⠀⠈⠻⣦⠱⡀⠀⢣⠀⢣⣿⣽⡇⡰⣰⢁⣨⣅⢀⠠⡂⠭⣒
⠢⠀⢄⠀⢀⠠⠴⠒⠘⠋⠛⣷⠀⠀⠀⠑⠄⠀⠀⠀⠁⢜⢷⣌⠢⡀⠣⡈⢎⣿⣷⣟⣧⣞⣡⡾⠷⠛⠛⢋⠙
⠂⢦⢘⡖⠁⠀⠀⠀⠀⠀⠁⠈⢦⡀⠀⠀⠀⠀⠢⣀⠀⠀⠀⢝⢷⣬⣂⣙⢌⠻⣿⣾⣿⠟⠿⢂⣈⢩⢩⠌
⠈⣖⡏⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠻⣦⡁⠀⠀⠀⠀⠉⠂⠤⣀⠀⠛⠿⣿⣿⣿⣾⣿⣧⡴⠯⣤⠒⢂⠋⠀⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠠⣂⠭⣄⣀⠘⠈⠙⠶⣄⣀⢀⠐⠀⠠⠤⢬⣭⣥⣴⠾⠻⣻⢻⣿⠙⢳⡌⠻⡍⠀⠔⠁
⠀⢹⠀⠀⠀⢀⠔⣥⣮⣍⡂⠈⡉⠑⠛⠲⢤⣠⡈⠉⠙⠛⠛⡛⠛⠯⡁⠀⠀⠀⡹⣆⡿⢊⠥⢛⡆⢱⠂⠀⢠
⠀⠸⡇⠀⠠⣵⡷⠒⠉⠉⠀⠀⠁⠉⠐⢀⠠⠀⠈⠙⠲⣤⣤⡀⠀⠀⠙⢲⣄⠀⢋⢸⡔⢁⢠⡎⢸⡌⣇⡸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⠁⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠀⠚⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⡿⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣷⠀⠀⢀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢀⠀⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⡄⠰⠿⠿⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⢸⣿⣏⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⠀⢠⣿⣟⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⢼⣿⣿⣤⡀⠀⠀⠀⢀⡾⣿⣿⠶⠂⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⢠⢿⣿⡇
⠀⠀⠀⠀⡸⣿⣿⡆⠉⠃⠀⢀⠞⠀⣿⣿⠃⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⣼⣿⣏⠙⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⢠⠃⢼⣿⡛⣄
⠀⠀⠀⠀⢀⣿⠻⣷⠀⠀⠀⠀⠀⢀⣿⢿⡆⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⢸⡿⣿⡀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⣾⠿⣧
⠀⠀⠀⢠⡞⠁⠀⣿⠀⠀⠀⠀⢀⡾⠁⠀⣷⠀⠀⠀⢀⣴⠟⠀⣿⠀⠀⠀⢠⡟⠁⢸⣇⠀⠀⠀⢀⣤⠾⣿⠀⠀⠀⢠⡾⠃⠀⢻⡄
⠀⠀⢠⣟⠀⠀⠀⠸⣧⣀⠀⠠⣾⠁⠀⠀⠸⣦⡀⠠⣿⡁⠀⠀⢹⣆⡀⠀⣾⠀⠀⠀⢹⣄⠀⠘⡟⠁⠀⣯⡀⠀⢠⣞⠀⠀⠀⠈⢳⣄⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠳⢦⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠲⢦⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⠷⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⣉⠙⠲⣤⠏⠀⢀⣀⣤⣼⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠼⣉⡛⠳⣴⣶⠀⣴⠿⠿⠟⢻⢹⠁⡏⢹⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣾⣀⡏⢙⣿⡏⢸⠀⢸⡆⠀⢰⢸⢸⠀⡇⢸⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠦⣤⣇⣀⠀⣼⣿⣽⣉⣛⣿⠏⠰⠀⡀⢸⣧⣦⡾⢾⠿⢿⣇⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⢙⣽⠟⡇⢸⣛⣿⠿⠂⠚⠈⢁⣈⣧⣤⢴⣞⠷⣦⣇⡉⠓⠳⢤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣄⣠⢤⣼⠛⠊⠉⣡⣿⣧⡤⠶⠞⠛⡛⠛⢿⣶⣶⣿⣿⡿⣿⣿⣷⣶⣤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢁⣀⣼⣧⣶⣾⣿⣿⣻⣷⣄⣴⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠾⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡫⣻⣿⣿⣿⣿⠿⣿⣿⣿⣿⣽⢿⣿⣿⣿⡏⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡿⠿⠿⠛⢛⠋⣩⠿⢦⣤⣄⣩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢻⢿⠿⣿⣿⣿⣿⣿⣿⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⢸⠉⠛⠙⠛⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣾⣾⣤⣼⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢻⢻⢿⣿⡂⡀⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠈⠀⢹⣗⣷⣶⡇⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⠀⠀⢸⡊⣿⣿⡷⡷⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⢠⣸⣕⠟⠁⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡿⢾⠼⣽⡃⢠⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢹⣦⣼⢠⡄⡗⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⢸⡀⢻⣿⣧⣿⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⡇⠀⢸⡄⣻⣮⡟⣏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠃⠀⢸⠮⡿⠁⠂⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠁⠐⢻⠏⠀⠀⠀⠚⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈
⣿⣟⣛⣛⣛⣻⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠁⠀⣼⡟⢻⡆⣿⣷⢸⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣇⣸⡇⣿⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠛⠋⠁⠙⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡀⠄⠊⠑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⠠⠐⢁⠀⠀⠀⠀⠂⠄⡈⠂⠄⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢱⡇⠀⡔⠁⢀⠄⠒⠒⢤⡀⠈⢂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⡇⢸⠀⠀⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⢸⠀⠘⣀⣤⠂⠐⠄⢰⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣸⡇⣈⣷⣾⣿⡟⠀⠀⠘⡇⢀⠌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⡁⠄⠂⢡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⢹⡇⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠀⣿⣿⡀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢸⣿⡀⢻⣮⣿⠦⣀⠀⠀⣴⡇⠀⢀
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠙⠛⠶⠦⠤⠤⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⣬⣉⡉⠉⠉⠉⠉⠉⠉⠉⠙⠲⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠋⠉⠈⠉⠉⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢠⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣷⠀⣿⠉⠛⠛⢿⣿⠿⠟⠒⠀⠀⠀⠀⠀⠈⢻⡞⠛⠂⢤⣤⣀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢹⣄⠁⠀⠀⠀⠀⠀⠙⠛⠶⣤⣄⠀⣀⣀⠀⠀⠹⣆⠀⠀⠀⠀⠙⠻⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠶⠿⠷⠀⠀⠀⠀⠀⢀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡿⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡄⠀⠀⡀⠂⠀⠁⠉⠀⠀⠰⡶⠲⠆⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠉⢻⡇⠀⠀⠃⠀⠀⠺⣿⣿⠂⠀⠃⠀⠀⠀⠀⠀⠄⠀⠀⠻⣿⡷⠀⡄⠀⠀⠀⠀⠀⣿⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⢾⠿⠋⠀⠀⠀⢸⠀⠀⠀⠀⠃⠤⠘⢿⠟⠃⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⠛⠛⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⢀⡀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣿⢿⡈⠁⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⢹⡿⠉⠉⠉⠉⠐⠒⣤⠶⠶⠀⠈⠉⠉⠉⢀⣼⢿⡼⠃⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⢸⠇⣤⠀⠀⠀⢀⠴⠋⠲⠀⠀⠀⠀⢀⠔⣭⢻⣼⠃⠀⠀⠀⠀⠀⢀⣼⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⣿⡆⠀⠈⣰⠿⡀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⢀⣰⣉⡿⠁⠀⠀⠀⠀⠀⠀⣾⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⣿⠀⠀⠀⠙⣠⠿⡀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠞⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣿⠀⠀⠀⠀⠛⣄⠛⢦⣀⡈⠀⢀⣀⡰⠾⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠃⠀⠀⠀⠀⠀⠙⢷⣄⠉⠛⠛⠋⣡⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢀⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡞⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠘⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣰⣿⡝⠓⠚⠛⢢⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠌⠛⠿⠓⠀⠘⠃
⠀⠀⠀⠀⠀⢠⣿⡤⠙⠛⠛⢳⣄⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⡞⠉⣤⢀⡁⢰⡿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠘⠷⠾⢷⣤⠁⢺⡇⢿⠋⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⣀⡀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠙⠟⢿⣇⣼⣀⣿⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣀⣀⡀⠛⢷⣤⣾⠅
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠋⠙⠀⠻⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠀⢸⡟⠉⠛⠛⠻⣷⣆⣤⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠰⣶⣤⣀⡀⠙⠲⠶⣦⣄⣩⣿⣿⣷⣴⣿⣃⡀⠀⠀⠀⣠⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠈⠛⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠙⠻⢶⠒⣀⣉⣉⠉⠉⠉⠉⢻⣿⣿⣦⣾⣾⠿⠁⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣴⣦⣦⠀⣶⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠐⠚⠿⠟⠛⠛⠛⠛⠛⠛⢛⣛⣉⣛⣻⣿⣿⣤⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣿⠀⣿⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠾⡛⢻⡏⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢨⠇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠒⠒⠶⠖⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⣠⣄⣶⣤⣤⣤⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣀⠤⠄⠈⢛⣀⢠⣤⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠉⠉⠉⠉⠉⠛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠴⠆⠉⠉⠉⠁⠶⠿⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⣀⣀⣀⣀⡀⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⣠⡄⣶⢦⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣦⣀⢄⡄
⠀⠀⠀⠀⢈⡟⠓⠙⠛⢷⣷⠁⠀⡰⠋⡹⠟⠟⠛⠳⢽⠯⡁⢱⠀⢀⡤⠞⣫⢶⣤⡘⠙⣶⢶⣶⢿⢿⣿⢿⣧⣤⣀
⠀⠀⠀⢀⡎⠀⠀⠈⠁⠀⢸⡀⠀⡇⣼⠁⡆⠀⠀⠀⡀⣇⢇⠇⠘⡏⢠⠐⠡⠀⠉⢻⢺⣿⡗⢁⠀⠀⠋⡀⢿⣿⣿
⠀⠀⠀⠙⢇⠘⢄⣀⡤⠂⡼⠋⢠⡷⠉⢆⠈⠒⠒⠊⣠⢿⠘⡄⠀⢣⡘⢤⠤⠂⢀⡼⠂⠛⣇⠘⠤⣀⠴⠁⡎⠹⡍
⠀⠀⠀⠀⢀⣱⠦⠤⢴⠊⠀⠉⠉⠀⠀⣀⣵⠒⠒⢎⠁⠀⠣⠝⢀⣠⡼⠒⢲⣒⡋⠀⢰⠷⠈⣲⠤⠤⢔⠊
⠀⠀⣀⢴⠉⡜⢦⡔⢺⢱⡄⠀⠰⣶⠊⢡⠟⢼⡀⢿⠣⡠⡦⠐⣇⡴⠁⠀⢸⣀⡟⠊⣁⣀⠼⡧⢖⠦⡜⢄⡀
⠘⠛⠀⢘⡟⠀⠀⠀⠈⡏⠈⢢⡄⠀⢠⠋⠠⠀⠀⠸⡀⠹⠖⠠⡎⠀⠀⠀⠸⠀⠁⠀⠟⣠⠊⠀⠀⠀⢰⡀⠸⠓
⠀⠀⠀⠀⣱⠲⠄⠤⢤⠃⠀⠈⠀⠀⠀⡽⠒⡶⡄⢴⠓⠀⠀⠀⡜⠉⠀⠀⢸⠁⠀⠀⠀⠙⠿⢖⠶⣲⡿⠵
⠀⠀⠀⣠⠃⠀⠔⡄⠘⡄⠀⠀⠀⠀⢠⠃⡸⠀⢧⠘⡄⠀⠀⠘⢦⠴⠓⠤⡜⠂⠀⠀⠀⠀⡤⣇⠀⠀⣣⢄
⠀⣀⠼⠤⣀⡞⠀⣟⣀⣸⡀⠀⠠⣊⣉⡻⠁⠀⠸⡟⠓⣄⣠⣒⣳⠀⠀⠀⣱⠒⣢⠀⢤⣚⣙⠇⠀⠀⠙⣉⣲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⠛⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⢸⡃⠈⡷⠴⡏⠀⢹⠀⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⠀⠙⠛⠁⠀⠉⠛⠉⢀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⡀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠈⡟⠒⠶⠶⠶⠚⡏⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣇⠀⠀⠀⠀⢀⡇⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣻⠇⠀⢻⠉⣻⠿⡏⢹⠃⠀⢿⡳⠤⣄⣀⡤⠖⠚⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢴⡃⠀⠀⠈⣇⢳⣶⢋⡏⠀⠀⠀⣹⠄⣰⠋⠳⡄⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢦⣀⠀⢹⡞⠘⣿⠀⠀⣠⠞⠁⢠⡇⣾⠁⢿⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢷⢰⢇⡴⠋⠁⠀⠀⠘⡇⠉⠀⢸⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠇⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣤⠴⠞⠙⡶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⣶⣰⣶⣶⣄⣶⣶⣦⣶⣶⡄⣶⣶⣆⢀⣶⡆⠀⣶⣶⡄⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣶⣈⣿⣶⣏⣿⠀⡉⢸⣿⣿⠀⣿⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⢿⣿⣿⣾⣿⣿⡋⠉⠉⠉⠛⢿⣿⣷⡛⠿⢻⡆⢿⠉⠁⢿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⢻⣿⣷⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢻⣿⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣀⣼⣿⣿⢹⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⢻⣿⠸⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⢈⣧⣆⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣾⣿⠟⣸⠿⠿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢫⠁⣰⣇⣿⠀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⠿⣿⣿⣿⣿⣯⣥⣀⣭⣴⠟⣹⢿⡿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣿⢿⣯⣿⠞⠛⣿⣿⣿⣦⣿⣾⠃⠀⠈⡿⢿⣿⡉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⣿⣿⠠⠤⣿⣿⣿⣿⣿⠏⠀⠈⣴⣧⡼⣫⣷⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢐⣀⢻⣿⣿⣄⣀⣿⣿⣿⣿⠋⢀⠀⠰⢻⣿⣿⢵⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⢻⣿⣿⣿⣿⣿⣿⣿⠌⠁⢀⣠⣦⠀⠸⣿⠝⡘⣟⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣘⢿⣿⠿⠿⠯⣉⣁⣤⣶⣿⠟⢻⣀⣶⣿⢷⣌⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⢿⣿⣿⣿⣿⣿⣿⡿⡟⠋⢩⣴⣻⢿⣿⣧⢸⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⣈⣉⣉⣉⣉⠀⠀⣿⣿⣿⣝⢻⡾⣿⣿⣼⠟⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢿⣿⣿⡏⠀⠀⢿⣿⣿⣿⡻⢇⡝⣿⠁⠚⠳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣸⣿⣿⣷⠀⣤⣾⣿⣟⡿⢟⣯⠞⠑⠄⠑⠒⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣴⣾⣿⣿⣿⡿⠼⠟⠛⠋⠉⠁⠀⣿⣿⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⠀⠁⠈⠁⠘⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⣀⣀⢀⠀⡀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠰⡄⠇⡰⢄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡿⠁⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠊⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠠⡀⡆⢠⠄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠀⠘⡶⠁⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠇⡠⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⡀⡇⢠⡀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠈⢰⠂⠐⠂⠁⠬⠐⠀⠁⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠂⠀⠱⣶⠃⠈⠂⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠂⠀⠀⡨⠠⠞⠀⠆⠇⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠦⠀⠅⠀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠈⡌⡔⠀⡤⠤⠤⢤⣀⠀⠀⠈⠀⢳⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣃⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⠭⠭⣙⣒⠀⠸⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⠭⢝⣒⣂⠀⠰⠿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣊⣩⣍⣙⣓⣒⠦⠥⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣛⣛⣒⣒⣒⠭⠭⣓⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣒⣒⣒⣒⣒⡦⠭⢭⣒⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠉⠙⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣁⣉⣀⣉⣉⣉⣉⣓⣒⣤⣼⣤⣤⣐⣀⣀⣀⣀⣀⣀⣀⣈⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣀⡈⠛⢶⣤⣀⣻⣿⣿⣏⣀⣠⣤⠾⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⣠⣤
⠀⠀⠀⣀⠀⠸⣿⣿⣿⣿⣷⠀⠀⠀⢠⣿⣿⣿⣆⠀⠈⠛⣿⣿⣿⠛⠋⠉⠀⠀⠀⠸⠀⣀⣀⡀⠀⠀⠀⢶⣀⣀⠀⠸⠿⣿⣿⡧⢀⣴⠞⠉
⠀⠀⠀⠙⠳⢦⣬⣿⣿⣟⣁⠀⠀⠀⠈⣻⣿⣟⠁⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣷⠀⢀⣾⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣤⣴⡶⠀⠀⠀⠀⠀⣨⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀⠀⣾⣿⣿⣿⡏⢿⣷⠀⢺⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⡁⢠⣿⣿⣿⣿⡇⠀⠛⠀⣼⣿⠉⢿⣿⠀⠀⠀⠀⠀⣿⢟⣼⣿⣿⣷⣄⠀⠀⠀⠀⠀⠛⢿⣿⢿⣿⠟⠗
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⣿⡇⠻⣿⠀⢹⣿⡇⠀⠀⠀⣿⣿⡄⣿⣿⠀⠀⠀⠀⠀⠀⣾⡿⠛⠛⠿⣿⡆⠀⠀⠀⠀⠀⢸⣿⠈⣿
⠀⠀⠀⠀⠀⢼⣿⣿⠀⢰⣿⠃⠀⠀⠀⠈⣿⠁⠀⠀⠀⣿⣿⡅⣿⣿⡆⠀⠀⠀⠀⠀⠙⠃⠀⠀⠐⠟⠁⠀⠀⠀⠀⠀⠈⣿⠀⣿⠂
⠀⠀⠀⠀⠀⠈⣛⣟⠀⢘⣿⠃⠀⠀⠀⢘⣻⠆⠀⠀⠀⠀⠀⠀⣻⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣻
⠀⠀⠀⠀⠀⠸⢿⣿⠀⠈⡿⠀⠀⣀⠀⢀⣯⠀⠀⠀⠀⢘⣶⡂⣿⣿⠃⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⡼⠀⣸
⠀⠀⠀⠀⠀⠀⠈⢿⣇⣰⣿⠁⠸⣧⣤⣼⣾⡃⠀⠀⠀⢻⣟⠀⣹⣿⠀⠀⠀⠀⠀⢤⡉⢴⡿⣿⡿⠗⠀⠀⠀⠀⠀⣀⢰⣷⣀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣼⡀⠀⠻⣿⣿⣿⣇⣴⠏⠀⠨⣻⢿⣿⣿⠀⠀⠀⠀⠀⠈⠿⣾⡿⣍⡁⣀⡀⠀⠀⠀⠀⠈⢿⣵⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠠⠶⣠⣿⣟⠛⠋⠀⠀⠉⣪⣿⣟⠁⠀⠀⠈⣵⣿⣟⠇⠀⠀⠀⠀⠀⠀⢴⣾⣿⣧⠀⠉⠉⠀⠀⠀⢀⣀⣰⣿⣿⣯⡀
⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⡗⠀⠀⠀⠘⠿⠿⡿⠁⢀⣀⣤⢽⣿⣟⣤⣄⣀⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠐⠈⠉⠈⠠⣬⣿⣯⣅⠉⠓⠦⢀
⠀⠀⠀⠀⠀⠈⠙⠋⠛⠋⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⢠⣶⣿⢟⢧⡀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣠⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⢩⣭⣭⠉⡭⢿⠍⢩⣭⣍⠈⢁⣵⡌⠉⣭⢭⡍⣩⣭⡍⣿⣿⣷⣿⣶⣂⡤⢄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⢸⣧⡄⠘⠷⣶⣤⣿⠀⠋⠀⣸⣏⣷⠀⢿⣤⠟⣿⣀⡀⣿⣿⣿⣿⣿⣿⣿⡽⢖⣀⡀
⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⢨⣷⣶⠄⣤⣴⠟⠙⠷⠶⠢⡿⠋⣽⡆⢼⡇⠀⣿⣤⡄⢿⣿⣿⣿⣿⢟⣿⣿⣻⡵⠖⣂⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣠⣍⡻⠿⣿⣿⣿⣿⣭⣿⣯⣥⠤
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡟⠻⣷⣮⣝⡻⢿⣿⣿⣷⣶⣋
⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠇⠈⡆⢹⣿⣿⣷⣬⡛⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⢾⣽⣿⣿⣿⣿⡇⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⢠⡇⣾⠁⠃⣿⠛⢿⣳⣬⢻⣿⣿⡍
⠀⠀⠀⠀⢽⠿⣯⣿⣿⣿⡇⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⠀⠘⡇⣿⠘⠀⠉⠈⢸⣿⣿⣼⣿⣿⣿⡆
⠀⠀⠀⠀⢰⡿⡿⣿⣿⣿⣧⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⡇⡇⠀⠀⠀⠀⡜⣿⣿⡏⣿⣿⣿⣷
⠀⠀⠀⠀⠐⣺⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⡄⠀⡇⠁⠀⠀⠀⠀⡇⣿⣿⡇⣿⣿⣿⡟⠆
⠀⠀⠀⠀⢚⣫⣾⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡇⠀⡇⢰⣦⠀⠀⠀⡇⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⡇⠀⡇⢸⠈⠀⠀⠀⡇⣿⣿⡇⣿⣿⣛⣿⠄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⢧⠇⢸⠀⠘⠀⠀⡇⣿⣿⡇⣿⣿⣟⣧⡄
⠀⠀⠀⠀⣛⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣩⣷⣶⣦⡀⠀⠘⣿⣿⣿⡼⠀⢸⠀⠀⠀⣶⡧⣿⣿⡇⣿⣿⣿⣟⡀
⠀⠀⠀⠘⣩⢾⣾⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣧⠀⠀⣿⡇⡿⣧⡄⠀⡄⢀⣤⣿⡇⣿⣿⡇⣿⣿⣿⣷⠄
⠀⠀⠀⠈⠵⣫⣽⣿⣿⣿⣿⢸⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠤⣽⠃⠀⠀⣿⡇⣷⣿⡇⠀⡇⠙⢿⡟⠁⣿⣿⡏⣿⣿⣿⣭⡅
⠀⠀⠀⠠⢞⡿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⡿⠀⣧⢳⡀⠀⣿⡇⣿⣿⡇⠀⡇⠀⣿⣷⠀⣿⡿⣧⣿⣿⣯⣿⣒
⠀⠀⠀⠐⣫⣾⣿⣿⣿⣿⣿⢸⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠋⡁⠀⣿⠿⡇⠀⣿⡇⣿⣿⡇⠀⠁⣤⣿⢸⢸⣿⣟⣿⣿⣿⣿⣭⣅
⠀⠀⠀⢈⡿⣿⣿⣿⣿⣿⣿⠘⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠋⠀⠀⢀⣿⡇⣿⣿⡇⡀⠀⣸⣿⢸⢸⣿⣯⣿⣿⣿⣿⡧⠄
⠀⠀⠀⢋⣾⣿⣿⣿⣿⣿⣿⡿⠋⠉⠘⠉⠁⠀⠀⠀⠀⠐⠂⠀⠀⢠⠃⠀⠀⢀⠈⠉⢿⣿⣿⢇⣷⣆⣿⣿⣸⣾⣿⣅⣿⣿⣿⣷⠶⠶⠂
⠀⠀⠀⢰⢫⣿⣿⣿⣿⣿⡿⠁⠀⢠⠇⠀⠀⠀⠰⠀⠀⠀⠠⠀⣶⣼⢶⣦⠀⠀⠆⠀⠈⢿⣿⣿⣟⣿⣿⣿⢸⣿⣿⡅⣿⣿⣿⠿⣛⠥
⠀⠀⠀⠀⣻⣿⣿⣿⣿⡿⠁⠀⢠⡏⠀⠀⠀⠀⡀⣀⠀⠀⡀⠀⢸⣿⣸⣿⣆⠀⠸⡀⠀⠈⢻⣿⣯⣻⣿⣿⣾⣿⣿⣋⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠐⠿⣿⣿⣿⡟⠀⠀⠀⠼⠀⠀⠀⠀⢀⢁⡏⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⢣⠈⢆⠀⠹⣿⣿⣿⣿⣿⡿⣛⣭⣾⣿⣿⣿⡭⠖
⠀⠀⠀⠀⢘⣿⣿⠋⠀⠀⠀⠎⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡀⠈⠇⠈⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⣿⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⣵⣾⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⠉⠉⣩⣭⣭⣭⣭⡭⠭⠽⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠉⠉⢛⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠄⠀⢠⠀⠀⠀⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡏⡇⠀⠀⠀⠸⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡆⠀⠀⣿⠀⢰⡄⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣿⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⡄⡼⠀⠀⡄⠀⡄⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠁⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠊⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠻⣗⠠⣤⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠰⠒⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠲⣄⣝⠙⡏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡧⠠⡴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⡟⡤⠃⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠰⠀⣀⣀⠀⡇⣧⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠛⠀⠘⠇⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠃⠘⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣶⣶⣶⡆⢰⠒⠀⢸⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡆⠀⣠⢂⡴⠠⠄⠀⠠⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡇⠸⠤⣄⣸⠀⠀⠀⠀⠀⠀⠈⡇⢀⡠⠀⢻⡔⢡⠞⠀⠀⠀⠀⠃⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣭⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⠈⠧⠃⠀⠠⢶⣖⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠱⢄⠁⠀⢀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡴⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⢠⠀⡄⢠⠀⡄⢠⠀⡀⢀⡀⠀⠀⡄⢠⠀⡄⢠⠀⡄⢠⠀⡄⢠⠀⡀⢀
⠀⠀⠀⠀⠀⢀⣼⣤⣧⣼⣄⣧⢸⡄⣧⢸⡅⣤⢠⡇⣼⢠⡇⣼⣤⣧⣼⣄⣧⢸⡄⣧⢨⡄⡄
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⠀⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⡇⢰
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⠀⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⡆⢰⠁⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⠃⡁⠀⠈⡇⢸⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⠸⡀⠀⠀⢣⠀⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⡞⠀⠀⠀⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸⠀⠘⡄⠀⠀⡴⠁⢸⠀⡇⢸⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠀⠁⠀⠀⠁⠀⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣠⣶⣿⣿⣿⣷⣦⣴⣾⣶⣴⣶⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⠀⠀⣀⠤⢄⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⠃⠀⠀⢳⠀⠀⠀⢰⠇⠀⠈⢣⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠸⣌⠁⠀⡾⠀⠀⠀⠸⡎⠁⢀⡜⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠑⠈⠀⠀⠈⠉⢦⡈⠒⠋⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣶⠄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣶⣾⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠉⠉⢈⡻⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠈⠛⠿⢶⣶⣤⣤⣤⣠⣤⣤⣤⡶⠞⢿⣇⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠋⠉⢹⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢿⣄⠀⣹⠆
⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀⡴⠚⠋⠛⠓⣶⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣞⠉⢳⡀
⠀⠀⠀⠀⠀⣰⠟⠀⣴⠏⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢳⡀⠙⠳⢤⣀
⠀⠀⠀⣠⠴⠋⢠⡞⠁⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⠳⠦⣄⠉⠛⠶⣄⣀⣀
⠀⣰⠟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠉⠳⠦⡀⠀⠈⠙⢦⣄
⣶⡋⢀⣴⣄⢰⡏⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣤⣄⣈
⠈⠛⠋⠁⠉⠋⠁⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠃⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⠶⣦⣤⣄⠀⠀⠀⠀⠀⠹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠁⠠⠀⡈⢿⣆⠀⠀⡀⠄⢀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠻⠷⣦⣄⣸⣿⣶⣀⣠⡶⠟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠀⠀⣠⡾⠉⠛⠁⠻⠟⠻⣦⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠛⠉⠁⣀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣯⣀⣀⣤⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠶⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣶⣾⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⡿⠿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣆⢻⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⢀⣈⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠉⠁⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⣠⣄⠀⠀⠀⠀⠀⣉⡙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠰⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠉⢁⡶⠀⠀⠀⠈⠛⠋⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢈⠃⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠈⠛⠛⠁⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣤⣤⡴⠶⢾⣿⣿⡻⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣿⣦⠴⢋⣡⣬⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠈⠳⣼⠓⠞⠋⠁⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡏⠀⢠⡇⠀⠀⠀⠀⠀⢠⣄⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡟⢷⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⣸⡇⠀⢸⠇⠀⠀⠀⠀⠀⠸⣇⠀⠻⣄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣏⠁⢀⣀⣴⠞⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣗⠄⣠⡿⠞⠛⠉⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣄⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣹⠇⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠶⠶⠷⠶⠶⠶⠶⠶⠛⣿⠀⠀⠀⠀⠙⠶⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠏⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢠⡞⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⣿⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡾⠈⠳⣄⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⣸⠃⠀⠀⠙⢧⣠⡞⠛⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠙⢳⠞⠃⠀⠀⠀⠀⠀⠀⠹⣦⣀⡿⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠾⠏⢩⣧⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⡀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠐⠁⠔⠊⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡑⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⢁⠤⠀⠀⠈⠉⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠦⠀⡀⣀⠀⠀⠀⠀⠀⡀⠋⠀⣠⠕⠁⢀⠠⠐⣀⠄⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠔⠀⡆⠁⠒⢄⣀⠨⠃⠀⠈⢀⣀⠊⠀⠀⠀⠀⠸⠀⠀⢐⣧⡞⠉⠐⣀⠥⠐⠊⠁⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⡀⢁⢁⠀⠀⠈⠁⠀⠀⠀⠀⣠⠀⠀⡀⠀⠀⢠⠀⠀⡠⠂⠊⠃⠀⠀⠡⠄⠀⠀⠀⣠⡁⠈⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢐⠀⠢⠀⠀⠀⢿⡆⠀⠀⡀⠻⠃⠀⠂⠀⠀⠀⠀⡜⠀⡀⠀⠂⠈⠀⢸⣷⠀⠀⠀⠹⠃⠀⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⠀⠀⠩⠂⠀⡀⠀⠀⠀⠀⠀⢇⠀⠋⡄⠀⠀⠀⠀⠀⠀⠀⠨⠂⠀⢀⣣⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠐⢄⠀⠀⠀⠀⠁⠤⠚⠀⠀⠜⠀⠀⠀⠀⠀⠘⡜⣄⣴⠖⡀⠀⠀⠀⠀⠈⠒⠀⠀⡠⢋⠏⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⢀⣀⡠⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⣷⠤⠀⠐⡦⠀⢠⡤⠄⠒⠉⠅⢨⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⢃⢱⢰⡯⢛⡒⠚⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠘⠀⠀⠀⠄⠀⠡⠰⣄⢆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢍⡳⡜⠞⠚⠉⠉⠁⠀⠐⠒⠓⠀⠐⠀⠒⠀⠀⠀⠀⠌⠉⠠⠅⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⠢⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⡄⠀⠀⠀⠀⠀⡒⠬⠝
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠐⡖⠶⠤⠤⠄⠤⠤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠀⠀⢰⠄⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠠⡶⠀⡘⠋⢠⡄⠐⠶⡒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡙⠁⣶⠄⠈⠋⢀⣄⠀⠰⠇⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⢠⣄⠀⠿⠀⣈⠁⢰⣦⠀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠠⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡀⠂⠀⠀⠀⠀⡆⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡨⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠦⢤⣀⡃⠀⠀⠀⠀⠀⢀⣀⡤⠔⠚⠀⡀⠀⠀⠀⠀⠀⠀⠠⠥⠒⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣤⣀⣑⣈⡆⠀⠀⠀⠀⠨⣠⣀⠤⠔⠚⠁⠀⠀⠀⠀⠀⠀⠐⠤⢌⣈⡀⠀⠀⠀⠀⠀⠐⠄⠆⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠙⠻⠿⠿⠿⠟⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠿⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡀⠀⣿⡇⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠆⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠻⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⡀⢀⣀⣀⠠⣶⡿⠉⠉⠀⢠⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣴⣾⠛⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⠿⠿⠂⢀⣀⣀⣀⡀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⠛⠋⠈⠙⠻⣦⣤⠀⠀⠀⣤⣤⡴⠛⠋⠀⠀⠀⠀⠈⢻⣿⠿⢿⣶⣦⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠓⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋⠉⠉⠉⠉⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⣿⠇⠀⣿⣤⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢴⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⣿⡇⠀⣿⠟⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠉⠀⠀⠈⡇⠀⢠⣿
⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠛⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿
⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠸⣿
⠀⠀⠠⣾⣿⣿⣇⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⣷⡆
⠀⠀⠀⠉⠁⠙⡿⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⠀⠀⠀⠀⠙⢇
⠀⠀⠀⠀⠀⢰⣿⠀⠀⣠⣤⡾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⢀⣤⣤⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡖⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠘⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠙⣥⡀⠀⠀⠀⢀⣶⡆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠘⠟⠁⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣀⣸⡿⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⠀⠙⠿⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠛⣄⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢈⡿⠀⠀⠀⠀⠀⠘⠿⣆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀⠀⠀⠀⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡿⠋⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠿⠋⠀⠀⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⠉⠁
⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠛⠋
⣀⣀⣉⣉⣉⣈⣹⣿⣿⣿⣿⣋⣁⣠⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⢈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠛⢄⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⢴⡁⠀⠀⠀⠈⢺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠑⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠤⠄⠠⠄⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠰⠁⡠⠐⠢⡀⠙⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢃⠀⢠⡀⣀⠄⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠢⢀⣈⣁⡠⠊⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠐⠠⣤⠀⠀⠤⠔⠚⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣙⠳⣤⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⠤⠄⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣭⠆⠀⠈⢩⠋⠝⣋⡛⢍⢫⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢁⢠⠸⡘⢅⠸⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⢀⠔⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡜⢠⣤⡄⠊⠀⡈⢆⠢⢍⣒⣒⡡⠊⠳⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢄⠁
⠀⠀⠀⠀⠀⠀⡜⠠⠁⠀⠀⢀⠀⠘⡌⢆⠀⠀⠉⠢⡀⠀⠈⠢⡀⠀⢀⠊⠀⠀⠀⢠⠊⠉⠑⡄⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡼⠠⠁⠀⠀⠀⠑⠃⠀⠘⡌⢆⠀⠀⠀⠈⠦⡀⠀⠈⢦⡌⠀⠀⠀⠀⠐⢄⣀⡠⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡰⢡⠃⠀⢠⡆⠀⠀⠀⠀⠀⠘⡜⡆⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣦⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⠀⣀⡀⠀⢤⣄⣀⡠⠤⠿⠐⠒⠺⠄
⠀⠀⡰⢁⠂⠀⠀⠀⠀⠀⠀⡔⠉⡆⠀⠀⠀⠱⡘⣄⠀⣀⣀⡀⠤⢤⣗⠒⠦⠭⠙⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢰⢃⠂⠀⢠⢢⠀⠀⠀⠀⠈⠛⠁⠀⡠⠠⡀⠱⡘⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢠⢃⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠁⠀⠡⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣣⣤⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡯⣯⣿⣿⣿⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⢿⣟⣿⣿⣷⣯⡻⠿⠿⠿⣟⣿⣿⣟⣷⣿⠿⠿⣿⣻⣿⣟⣽⣝⠿⠿⠿⢿⣻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢰⣿⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣷⣿⣿⣿⣿⣿⡿⡏⣾⣿⣿⣿⣿⣿⣿⣿⡾⣯⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠸⣿⡿⠿⠿⡿⢟⣾⠻⣿⣿⣿⣿⣿⣿⣿⣯⣽⣽⣿⣿⣿⣿⣿⣥⣥⣿⣽⣿⣿⣿⣿⣿⠹⣷⣿⣿⣿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⣵⣶⣿⣿⣷⣿⠀⠀⠀⣷⠀⠀⣰⡇⠀⠀⢺⡆⠀⠀⢀⣾⠃⢰⣷⠀⠀⠀⣿⣾⣿⣿⣶⣮⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣠⡾⠋⠀⠀⠀⠈⢷⣦⣴⠟⠁⠀⠸⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢠⣤⣤⣽⣧⣤⣔⡿⡶⣿⣶⠶⢾⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⣿⣤⣻⣟⣼⣯⣥⣤⣤⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⡿⣿⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣿⣿⣇⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⠉⠉⢹⣏⣭⣚⣻⣄⣿⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿⣒⣿⣿⣭⡹⡯⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⣀⣰⣛⣛⣯⣭⠍⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣿⡩⣭⣽⣛⣷⣂⣀⣀⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⣻⣟⣿⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣿⢿⣞⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠙⠛⠛⣛⡛⣛⡽⣯⣛⣿⡀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣿⢩⣿⠟⠛⢿⡛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⣿⠛⠛⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣿⣿⣿⣿⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠿⠿⣿⣿⡟⣿⠀⠀⠀⠸⣧⠉⠻⣦⠀⠀⠀⠀⢠⣴⠾⠛⠻⡟⠛⠷⣦⡀⣿⣻⣿⡿⢿⣫⡄
⠀⠀⠀⠀⣴⣶⣶⣾⣿⣿⣿⢯⣖⣻⣿⣿⣿⣿⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⠛⣻⣿⣿⣿⣿⣞⣿⡝⣢⢻⣵⣿⣶⣶⣿⣿⡆
⠀⠀⠀⠀⢿⣿⡿⠿⢿⡿⣻⡿⢿⣿⣿⡿⠿⢿⣿⢿⣳⣿⢿⣿⡿⠿⣿⢿⣴⡻⣿⣿⣿⠿⠿⡯⣿⣟⣼⣿⣿⣿⠿⢿⠿⠇
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣾⣿⣻⣿⣿⣿⣿⣿⣿⣶⣮⢿⣻⣿⣿⣿⣷⣶⣾⣟⣽⣿⣿⣶⣾⣿⣷⣶⡝⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⢯⡾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣼⠁⠀⠀⠀⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠹⣁⣄⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⡿⣠⣭⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⢳⡉⣩⣿⣿⣿⣿⡛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⣾⣿⠿⣿⣿⣿⣿⣦⡻⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣴⣶⣶⣶⣾⣿⠟⠁⠀⠈⡻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⢱⠆⠀⠀⠀⢟⣼⣿⡿⠛⣿⣿⣦⡀⣀⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣼⠆⠀⢀⣴⣿⡿⠋⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⢹⢆⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⠒⠒⠒⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠓⠋⠉⠀⠈⠙⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡜⢦⣀⣐⣠⣤⡼⣃⡤⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡸⠁⠀⠀⠀⠀⠀⠀⠉⢳⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠃⠀⢿⠇⠀⠀⠰⡿⠀⠈⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠵⠛⣶⢤⢄⢀⡤⡔⠾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡷⠋⢠⢀⠋⠑⢶⡴⠒⢣⢸⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⠗⢄⡼⣘⠀⠀⠈⠀⠆⠻⣼⡦⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢀⢾⣧⠇⠀⠀⠀⠀⠂⠚⢿⣇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣾⢄⡿⠀⠀⠀⠀⠀⠀⠀⢸⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡨⢠⣧⣀⠀⠀⠀⠀⠀⣀⣤⡞⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⠈⠉⠉⣉⡋⠉⠉⠊⠀⠩⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢰⠿⣇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⠀⡿⣚⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠉⢑⣤⣷⠋⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠒⠒⠒⠂⠸⠿⣍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠄⠀⣠⠦⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣔⠤⢊⣉⠤⠚⠁⠀⠑⢦⣘⣳⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡍⠒⠉⠉⠂⠀⠀⠀⠐⣋⡉⠙⠀⠍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⣰⠀⠈⣴⣶⣷⠀⠀⠀⢸⣷⣦⡉⠀⢰⡶⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢐⡇⠀⠀⠻⠿⠏⢀⠀⠀⠘⠿⠿⠃⠀⠀⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⡤⠈⣁⠀⠀⠀⠀⣤⡠⠤⠤⢠⡆⠀⠀⠀⢀⡍⠠⠴⠥⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡄⠀⠀⠀⠙⢾⡓⣛⠮⠁⠀⠀⠀⡌⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⡤⠄⠤⢤⠐⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡖⠒⠒⠒⠒⠂⠈⠉⠁⠢⠔⠈⠉⠁⠐⠒⠒⠒⠒⢲⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠀⠀⠈⠉⠀⠀⠀⠀⠚⠛⠚⠛⢻⠒⠂⠀⠀⠈⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡰⢀⠀⠀⠀⠀⠀⠀⠀⠠⣛⡓⠛⣻⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⡯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠻⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⠃
⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠀⠀⠠⠤⠤⠴⣈⢃⣊⡸⠤⠤⠄⠀⠀⠐⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀
⡖⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓
⡇
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠤⢤⡤⣤⠤⢤⡤⣤⢠⣤⣀⣀⣠⡤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠤⠤⠤⠤⠤⢤⣄⣀⣀⡤⢠⡤⣤⠤⢠⡤⣤⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢰⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠠⠤⢤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢢⡤⠤⠄⢲⠊⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣦⠾⣇⠀⠀⠀⠀⠀⢸⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⣶⡌⣌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢇⢱⣶⡈⠳⣄⠀⠀⠀⢸⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠓⠀⠀⢀⠞⠉⠐⢤⠟⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠹⡄⠂⠉⠳⠀⠀⠘⠉⢱⠀⠸⠀⠂
⠀⠀⠀⠀⢀⠦⢦⠀⠰⠀⠄⠀⠀⠸⠃⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⢀⠇⠀⠠⠄⠼⠀⠰⠦⢆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⡆⠀⢘
⣤⣤⣤⣤⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⡤⠤⠤⠤⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠻⠷⠶⠦⣤⡄⠀⠀⠀⠀⠀⠉⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣀⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⡏⠙⠛⢿⣿⠶⠶⠦⠀⠀⠀⠀⠙⣷⠒⠶⢠⣤⣀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣄⠁⠀⠀⠀⠀⠙⠛⠶⠤⢀⣀⣀⠀⠈⢳⡀⠀⠀⠈⠙⢷⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠚⠛⠀⠀⠀⠀⢀⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠘⠇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡄⠀⠀⡠⠀⠐⠒⠀⠀⢠⡄⠆⠀⠀⢠⡄⠀⠐⠒⠂⠀⠀⠀⠀⠀⠈⣿⣀
⠀⠀⠀⠀⠀⠀⢠⡞⠉⢹⡏⠀⠈⠃⠀⠰⢶⣶⠄⠈⠀⠀⠀⠀⠀⠀⠀⠒⣾⡶⠀⠀⠀⠀⠀⠈⣿⠉⠓⣄
⠀⠀⠀⠀⠀⠀⢾⠀⠀⠸⡇⠀⠀⠀⠀⠀⠴⠿⠁⠀⠀⠀⠂⠀⠀⠈⠁⠔⠻⠟⠁⠀⠀⠀⠀⠀⡇⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠈⠇⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⠀⠀⠀⠀⠀⠀⠀⠔⠒⠂⠀⠀⠀⠀⠀⠀⢀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢶⡿⣧⠉⠘⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⡿⠁⠉⠉⠁⠈⣰⡆⠋⠀⠀⠉⠁⣠⣾⢳⠏⠀⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠈⢃⣾⠀⠀⠀⠚⠁⠙⠄⠀⠀⠀⠊⣿⣹⠇⠀⠀⠀⠀⠀⣰⠏⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⡟⠀⠀⠸⠃⣦⠀⠀⠀⡆⠀⠀⠀⠀⣠⢞⡷⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢰⡇⠀⠀⠀⠻⡈⢧⣀⡀⠁⠀⣀⣠⡞⣡⠞⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠘⠁⠀⠀⠀⠀⠙⢦⡉⠛⠒⠚⣉⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣄⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⣸⠉⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠠⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⠑⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠠⠙⠷⠶⠀⠸⠁
⠀⠀⠀⠀⢠⣿⠦⠛⠋⢻⣦⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⢠⣞⣁⡶⢸⠀⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠉⠙⠿⣧⣼⣯⣸⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⢦⣤⣤⡀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⢿⠃⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠷⠶⣶⣄⡈⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣘⣻⣷⣦⣴⣟⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠿⠛⠇⠀⠀⠀⠀⠀⠀⠉⠛⠒⠷⠒⢊⣉⣉⡉⠉⢉⠙⣿⣿⣇⣶⣼⠟⠁⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⢠⣦⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠛⠛⠛⠛⠛⣛⣛⣉⣙⣿⣻⣿⣴⣶⢾⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠘⡏⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡾⢿⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⢸⠁⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠤⠤⠤⠤⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⠀⠀⠐⠒⠒⠂⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣄⣠⣤⣤⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣧⡴⠆⣈⣋⣀⠴⣤⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⣉⣉⠉⣉⣉⣻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠗⠊⠉⠉⠀⠐⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⢴⣶⣶⣶⡤⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠻⢷⣶⣤⣴⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣤⣦⣠⣤⣤⣤⣬⣭⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠑⠂⠀⠀⣸
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠒⠒⠲⠶⣤⣤⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⢻⣿⣿⣿⣿⡟⠩⠐⠆⢻⣿⣿⠇⠀⠀⠀⠐⠛⠻⢷⣤⣄⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⣆⣴⠀⠸⣿⣿⠀⠀⠀⠀⠀⠲⠆⠀⢸⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⢸⣿⣿⣿⣿⡆⠸⠇⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠙⠻⢿⣿⣿⣿⡿⠠⡄⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠆⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡟⠲⠄⠀⠀⠀⠀⠀⠀⣈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣄⠀⠉⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠓

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠹⠄⠀⣀⣠⣤⣤⣀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⢤⣄⡀⠀⠀⠀⣰⢞⣡⠄⠀⠀⠈⠙⢦⠀⢀⣠⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠉⠓⠀⢸⢣⡟⠁⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠸⡘⠃⠀⠀⠀⠀⠀⠀⠀⡟⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠤⠶⠖⠒⠀⠀⢳⡄⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢿⠙⢹⠇⠀⡴⠃
⠀⠀⠀⠀⣠⠄⠀⢳⠀⠀⠀⠀⠀⡾⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⡇⡏⠀⣼⠁
⠀⠀⢠⠞⠁⠀⠀⠸⣇⠀⠀⠀⣠⡇⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡇⣷⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢬⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡜⠦⠼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢱⣄⣀⣀⣀⣀⡀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠀⣒⣒⠒⡒⢒⣒⠒⣒⣂⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠈⠉⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢏⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⢂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡⠈⡎⡀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠾⢶⣦⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡠⠭⡀⢣⠀⠀⠀⠀⢠⠞⠁⡀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢀⠀⠀⠀⣠⠐⢀⠁⠀⢤⣽⣒⠶⣶⡄⠀⡦⠀⠀⢘⢸⠁⠀⢃⢸⠀⠀⠀⠀⠃⠀⠀⠠⠀⠀⠄⠀⠀⠀⠈⠇
⠀⠀⢠⡁⠀⢾⡟⠐⠊⠁⠀⠀⠀⢀⠀⠀⠘⢄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣄⠀⠀⠀⡄⢂⠀⠀⠀⡀⠀⠀⠀⢀⡀⡐⠠
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢇⠀⠀⠑⢌⠑⠒⠉⠀⠀⠀⠀⠈⡡⠃
⠀⠀⠀⠀⠀⠛⢆⠀⠀⠀⡀⠀⠀⠰⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣀⠀⠀⠀⠁⣢⠄⣀⣀⡤⠔⠊⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠈⠐⠊⠀⡋⢄⠁⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠙⣦⣤⠴⢊⠛⠜⠁⢠⡃⠑⢆⠀⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⠘⠧⠛⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⢸⡁⠀⠀⣀⣁⡠⠞⠐
⠀⠀⠀⠀⠀⠀⢀⠞⠁⡧⡸⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡉⠁⢸⠁⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠘⠢⠀⡷⠃⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢣⡰⠀⡀⡄⠀⠀⠀⠘⠤⠛⢤⡼⠀⠀⠀⠀⠀⠠⢀⡏⢀⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢬⡀⡀⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠪⡨⢛⢾⠇⠸⠃⡇⠀⠰⠒⠼⠀⣠⢤⡧⠀⢀⠀⠀⠀⠀⠀⢑⢀⡀⠎⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⠁⠀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠹⠤⢅⠼⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠵⠄⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠜

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⣶⣄⣴⣦⣤⣀⣠⣶⣾⣿⣿⣿⣿⣿⣾⣶⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠀⠒⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠠⠤⠄⠀⠀⠠⠤⠄⠀⢹⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣟⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡟⠀⠈⠁⡆⠀⢀⠈⠓⠉⠀⠀⠉⢰⢉⢬⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠠⣀⣈⣉⣁⣀⡀⠀⠀⠀⢠⠤⠵⣾⣿⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⣀⣠⡴⠚⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠟⡏⠉⡀⢙⡯⢹⡟⠳⣲⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡇⢠⠉⣁⡰⠁⡔⠁⠈⢢⠙⡏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⣠⡏⡜⠉⠉⢹⠛⠣⠜⠓⠄⠀⠘⣌⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡘⠋⠙⠿⣿⡜⠀⠀⠀⠀⢹⠀⠀⠀⠠⣄⠀⣿⠇⣮
⠀⠀⠀⠀⠀⢠⢷⡏⠐⠒⠠⢼⡇⠀⠀⠀⠀⠘⠤⢤⢤⣤⠛⡀⡿⢰⠁⡇
⠀⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⠸⠀⢿⡄⠈⢱⠼⡇⠸
⠀⠀⠀⠀⠀⣈⢀⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢣⠘⡇⠀⠈⢲⡧⠃
⠀⠀⠀⠀⠀⠙⢾⡶⠶⠹⢺⠁⠀⠀⠀⠀⢀⡀⠀⠀⡷⠹⠤⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢟⠲⢤⣀⣀⣀⣀⣀⣤⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠲⠯⠀⣤⢠⡤⠎⠱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⡔⠠⠤⠀⠉⠑⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢠⠁⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡌⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡜⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠃⢰⡇⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⡁⠹⡀⠀⠱
⠀⠀⠀⠀⠀⠀⡀⠀⢠⠊⠀⠀⡞⠁⠀⢁⠀⠀⠳
⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣀⢰⠀⠀⠀⠈⣇⣀⣼⣷
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⢠⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠃⠀⠀⠀⠀⠈⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⢔⣫⣝⢓⣀⠀⣀⠀⠀⠀⠀⠀⢀⠰⠶⠤⢀⠀⠀⣾⣀⡀⠀⢀⠀⣤⣀⠀⣠⠼⠇⢀⣠⠶⠒⠒⠲⢦⡀⠀⢄⠀⠆⣠⠤⠤⠤⡀⣷⠎
⠀⠉⡱⡣⡠⠀⠢⡂⠕⡆⠀⠀⠄⠀⠀⠀⠀⠀⠈⠆⠁⠀⠙⠁⠈⠉⠉⠀⠈⡆⠀⠀⠟⠀⠀⠀⠀⠀⠀⠓⠐⢈⣺⠊⠁⠀⠀⠀⠀⢻⡓
⠀⠠⡉⡉⠀⠀⠀⠀⡀⡰⠀⠺⠀⠄⠀⠀⠀⢀⠆⡘⠄⠐⡁⠀⠀⠀⠀⠀⡄⡘⠀⠚⡀⠂⠀⠀⠀⠀⡐⡠⠁⠘⢋⢀⠀⠀⠀⠀⡔⢀⠁
⠀⠀⠑⢄⡀⠠⠠⡐⠕⠀⠀⠀⠑⠠⣀⣒⣂⡡⠊⠀⠀⠀⠈⠢⠐⣀⣐⠪⠐⠀⠀⠀⠈⠢⢤⣐⣂⠬⠊⠀⠀⠀⠀⠃⢅⡂⢐⡊⠔⠁
⠀⠀⠀⠠⠎⠉⠉⠆⠠⢀⣀⠠⠀⠲⠈⢀⠁⠋⠀⠂⠆⠂⠦⠐⠂⠈⠁⠂⠂⠀⠤⢤⠔⠊⠅⠈⠁⠈⡈⠂⠴⠠⠐⠊⠨⠉⠉⠐⡦⡀
⠀⠀⠈⠀⠁⠀⠀⠈⡄⠀⠀⠀⠀⢸⠂⣒⠐⠃⠀⠀⠀⠀⠀⡰⠀⠀⠀⠈⡄⠀⠀⠀⠀⠘⡤⠠⡤⢤⠄⠀⠀⠀⠀⢀⠃⠀⠀⠀⠃⠘
⠀⠀⠀⠀⠀⡆⢰⠒⠁⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⡄⠔⠒⠀⠀⠀⠀⠀⠀⠈⡎⢁⠈⠀⠀⠀⠀⠀⠸⠐⡄⠄⠀⠊
⠀⠀⠀⠀⠒⠃⠈⠓⠀⠀⠀⠀⠀⠒⠃⠀⠘⠒⠀⠀⠀⠀⠀⠀⠒⠃⠚⠂⠀⠀⠀⠀⠀⠐⠚⠁⠈⠓⠃⠀⠀⠀⠀⠀⠒⠃⠙

⣿⠿⣿⡿⠋⠀⠀⡠⠁⠀⣼⡿⠁⠀⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣤⠘⠀⠀⠀⠀⠀⢠⢸⠟⠀⠀⠀⣿⣿⡏⣰⣿⣿⡏⣽⣿⣿⣿⣿⣿⣿⠀⣠⡀⠀⡀⠘⠂⢀⠀⠀⠠⠀⠤
⠛⠁⠐⠀⠀⠀⠀⣰⡿⠀⠀⠀⣠⣷⣿⣿⢼⠟⢻⣿⠟⢁⣾⣿⣿⣿⣿⣿⢰⣾⢪⣆⣧⣤⣄⣟⣀⣤⣤⣤⣤
⣀⣀⠀⣾⡄⢀⣄⡭⠁⠀⢠⣾⣿⣿⣿⣿⢸⣿⣛⠁⣤⣿⣿⣿⣿⣿⢿⣿⠀⠀⠋⠉⠉⠋⠟⠸⠛⠛⠛⠟⠛
⣿⣿⣿⡟⠻⠋⠉⠁⠀⠈⢿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢤⣠⣾⣼⡷⠀⣄⣰⣀⣼⣶
⣿⠿⠿⣾⣷⣿⣤⠀⣴⡄⠶⢿⣝⡋⠿⠿⠝⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢐⣿⡄⠈⠹⠿⠿⠿⠿⢿⣿⠿⠿
⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠒⠆⠀⣿⣿⣿⠻⠿⠿⣟⠃⠀⠉⠉
⣿⣿⣿⣿⣿⣏⣁⣭⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⢹⣿⣿⡿⢿⠟⢿⢒⣖⣂⡀⠐⠻
⠿⣿⣿⣿⣿⣉⣤⣡⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⢀⣀⡿⣾⣷⠉⠀⠈⠉⠀⠠⢻⡿⡅⠲⠦
⣿⣷⣍⣉⢉⢻⡿⢿⣿⣿⣿⣿⣿⡿⣾⣿⠛⢻⡟⢛⢛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣇⣀⠀⠀⠀⢸⡇⢸⣿⣴⡿
⣿⣿⣿⣿⣿⣿⣧⠀⣼⣿⣿⣭⣽⣧⣸⣿⡿⣿⣶⣶⣶⣿⣥⣀⡀⠀⠀⠀⣿⣿⣿⣿⣄⣄⢀⡡⠤⣞⡽⢛⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠄⠀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⠤⠞⠉⠈⠉⠉⠀⠙⠙⣮⡓⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣙⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦
⠀⠀⠀⠀⠀⠀⢰⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⣃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢀⠔⠉⢉⡁⠀⠀⠀⠀⠀⠀⠀⠰⠃⠰⠃⠾⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠏⡆
⠀⠀⠀⠀⠀⢀⠠⡀⢣⡆⠀⠀⠄⠀⠁⢀⡐⣀⣐⣄⣀⠀⠉⠛⠿⠄⠀⢀⣀⣀⣀⡆
⠀⠀⠀⠀⠀⠘⢷⣔⣈⡁⠀⡀⠀⠰⠂⡌⠙⠛⢿⣿⠯⢵⠀⠀⠀⠀⠀⠉⠉⠉⢉⠃
⠀⠀⠀⠀⠀⠀⠋⠙⣿⠃⡾⢡⣴⠀⠀⠀⠙⠳⡏⠀⠀⠀⠋⠉⢡⠀⠶⣾⣝⢳⣦⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢁⣯⠃⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⠞⠒⢾⠁⠀⠘⣙⡆⠀⠁
⠀⠀⠀⠀⢠⡀⡀⠀⢀⡀⢿⠙⠀⠀⠀⠀⠀⠀⠹⡄⢀⡠⠖⠣⠀⠀⠀⠀⠀⠀⡰⢃
⠀⠀⠀⠀⢀⠙⠦⢤⣌⣿⠾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠐⡤⠴⠊⠀⡎
⠀⠀⠀⠀⡄⠀⠀⠀⠈⢻⡄⢢⡀⠀⠀⠀⠀⠀⠀⣤⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢷⣼⠙⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⣼⠸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⢙⠓⠦⡟⢀⠀⡀⠀⠀⡌⠀⠀⠀⠀⠀⢠⣌⠄⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣼⡟⠁⠙⢄⠀⠀⠀⠀⠨⡁⠸⣇⠔⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠘⡄⠀⠀⠀⢠⡛⢳⣼⡇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠲⣿⡄⢸⠇⠀⠀⠀⠀⢷⣾⢁⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣤⡾⡀⠀⠀⠀⠀⡠⢉⣉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢩⡄⠡⡄⠀⠀⡌⡀⡆⢰⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠓⡀⠸⡆⢸⡟⣸⠇⡸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⢠⠀⠉⠈⢳⡟⢠⢣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢪⣿⡀⢸⠀⠰⣇⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣾⡟⠉⠉⠉⠉⠛⠛⠒⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⠆⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⢀⡤⠖⠋⠁⠾⠥⠤⠤⠔⠚⠉⢡⣞⣁⣠⠤⠖⢫⡇⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠋⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢀⣤⡀⣠⡀⠀⠀⠀⠀⠀⢻⠀⠀⢸⠃⠀⠈⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⣤⡀⠀⢷⣀⡀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡀⣠⣄
⠀⣤⣼⡏⣿⠉⣇⣤⡄⠀⠀⠀⠈⡷⢦⣼⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠿⡇⠀⠀⠀⠀⣸⠇⣤⣤⡏⢻⡏⣿⣴
⣴⡽⣮⠓⠘⠀⡟⢸⡇⠀⠀⠀⢸⡇⢢⣟⠀⠀⠀⠀⢸⣷⣿⠀⠀⣠⠃⠀⠀⠀⠀⣾⣽⡆⠀⠀⠀⠀⠹⣀⡴⠢⣠⡟⠀⣿⠘⡇⠈⠐⢫⡾
⠈⠛⢶⡀⠀⠜⠁⣸⠇⠀⠀⠀⠀⣧⠀⡟⠀⠀⠀⠀⠀⠉⠁⠀⢞⠉⠁⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠛⡴⢫⡏⠀⠀⢾⡀⠹⠆⠀⣠⠞
⠀⠀⠈⠳⣦⠀⠀⢻⡄⠀⠀⠀⠀⠘⢦⣭⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣼⠃⠀⢠⡴⠋
⠀⠀⠀⠀⠹⡆⠀⠀⢻⡄⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⡟⠒⠂⠤⠤⠤⠤⢤⡶⠀⠀⠀⠀⠀⠀⠀⢀⣧⡼⠃⠀⠀⡼⠃⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⣠⡞⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠙⢦⣀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠈⠓⠶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⢀⣠⠞⠃⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢀⣽⣷⣶⣤⣄⣈⣳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣡⣴⣶⣿⣇⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠲⠤⣀⣀⣀⣀⡀⠤⣴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⢧⡀⠀⠀⠀⠀⢀⡰⠋⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠓⠴⠶⠒⠋⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠫⠭⣉⡉⠛⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠛⢉⣉⠭⠝⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠉⠓⢦⣙⠻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⣋⡴⠚⠉⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠞⠁⠀⠀⠀⣀⣤⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣓⣶⣤⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣦⣤⣶⣒⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⠉⠭⠽⠯⣛⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⢀⣠⡄⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⣀⠤⢘⡉⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⡨⠟⠀⠘⣄⡥⠠⢄⡀⡠⠚⠉⠑⡄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡝⠁⠀⠉⠢⣄⠀⠀⢸⡀⠀⠀⠙⣶⠂⠀⢀⡇⣀⠴⠚⠉⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⢳⡴⠋⠓⠢⣕⣄⠀⢣⡖⠀⠀⢹⠄⠒⠋⠁⠀⠀⠀⠀⣐⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠑⢒⣒⠃⠀⠀⠈⠒⠂⠐⢶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠴⠶⠆⠀⠉⢇⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠤⠒⠳⡀⠈⠣⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠙⠄⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠴⢧⣀⠀⠾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢣⡜⢹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣠⠿⡄⠀⠀⠀⠀⠀⠉⠈⠹⡂⠀⠠⠤⣀⠀⠀⠀⢀⡇⠓⠳⣃⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⢀⠔⠉⣠⠤⢷⣀⡠⠶⠆⠀⠀⠀⠐⠷⠀⠀⢰⠛⠦⠔⠒⣉⠠⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠤⠔⢡⣔⠉⠀⠀⠈⠉⠀⠀⡄⠀⢀⠐⡀⠠⢳⡤⣸⡟⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣬⠇⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠃⠐⠀⠳⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡂⢈⠇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⠐⠒⠒⠋⠙⢍⠣⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠳⠻⣇⠀⠀⣀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣄⠀⠀⠀⠀⠀⠑⢤⣯⡗⣊⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠻⢿⣷⠦⣀⠀⠀⠀⠀⢿⣟⠓⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠛⣤⣀⠀⢠⠏⠀⠈⠑⠤⣀⠀⡾⣿⣺⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡢⠛⡿⠛⠉⠢⡀⠀⠀⠀⠀⡽⠺⢼⣷⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠡⠤⢴⡾⠃⠀⠀⠀⠀⠁⠀⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠘⠌⠁⠀⠀⠀⠀⠈⠃⠐⠂⠁⠀⠀⠤⠄⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠦⡀⠀⠀⠀⠀⣆⡙⢄⠐⠀⡀
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠈⠡⠚⣁⠀⠀⠀⠀⠄⣢⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⣶⠢⢹⡆⡴⢋⡩⠉⠻⣆⠴⢋⣽⡄⠠
⠀⠀⠀⠀⡅⠀⠈⡀⠀⠠⠀⠀⢠⣿⡄⢀⡀⠀⢻⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢧⡸⢟⣀⡼⢻⡇⣾⡿⠀⠀⣿⣤⣉⠈⠀⠀⠂
⠀⠀⠀⠀⠠⠀⠑⠚⣢⠅⠀⠀⠨⠟⠁⠀⠉⡷⠀⠀⢀⠀⠀⠀⢐⡀⠀⠀⠀⠐⠀⠀⠀⠙⠙⠃⠀⠀⠳⢬⣀⣀⠴⠟⠀⢹⠆⣀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢅⠈⠥⡀⠀⠀⠀⠀⠦⢠⡦⠀⠀⠀⠌⠀⠀⠀⠘⣭⡀⠀⠀⠐⠠⠒⠃⠤⠀⠀⠘⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠘⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣜⡤⢢⠴⠀⠀⠀⠀⠈⠂⠀⠀⡀⠀⠀⠀⠤⡀⣀⣀⡀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⢯⣤⣬⠦⠄⠠⠠⠀⠰⠀⠀⠀⠘⠭⠅⠀⠀⠀⢠⣀⡀⠀⢀⣤⢠⣅⠀⠡⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢰⢀⠋⡍⢸⠈⣿⡄⠀⠠⠆⣼⢃⠕⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠘⣷⡋⠀⠤⠀⠀⠛⢧⢊⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠀⠴⠀⠀⠋⢀⢨⡇⡀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠢⠤⠤⠤⠤⠤⠤⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡭⠥⠤⠤⠤⡯⢤⠌⠰⠇⠐⠆
⠀⠀⠀⠀⠀⠀⢀⠈⢀⢄⡰⢀⣀⣻⣤⣤⡀⠀⠀⠶⢒⣲⣤⠴⢶⠒⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⢥⡐⠅⠀⠁⠀⠉⠛⠛⠛⠁⠀⠀⠀⣀⣀⣀⠐⠨⠋⠉⠉⠙⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠩⠴⡮⢄⢀⡀⣂
⠀⠀⠀⢀⡄⢉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠈⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠓⠐⠒⠀⠀⠀⠒⠀⠂⠀⡁⠈⠑⠈⡋⣂
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸
⠀⠀⠀⢸
⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠠
⠀⢀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣷⣦⣴⣾⣿⣿⣿⣶⣴⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣤⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠻⣿⣿⣿⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⢸⣿⣿⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣴⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢷⡀⠀⣿⡏
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣆⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣼⡇⠀⢛⡛⣷⡀
⠀⠀⠀⢀⣼⠿⠿⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠏⠀⠀⠈⠁⣸⡇
⠀⠀⢀⣿⠁⠶⠶⠆⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛
⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠘⠻⢦⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠃⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡿⠷⢶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣴⣶⠟⠋⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠉⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠁⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣠⡾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁⠀⠘⠻⣦⣀⠀⠀⣀⣀⣀⣤⣶⠶⢶
⠀⣀⣀⣀⣀⣀⣠⣴⠟⠁⠀⠀⠀⣠⣶⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠁⠀⠀⢈
⣾⠏⠉⠉⠉⠉⠉⠀⠀⠀⠀⣠⡾⠋⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢷⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⢿
⢻⣦⣀⣀⣀⣀⣀⡀⠀⠀⠰⣿⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠛⣿⠆⠀⠀⣾⣿⣯⣥⣤⡿
⠀⠸⣿⡉⢉⣩⡿⣿⡄⠀⠀⢻⡆⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣄⠀⢀⣿⠁⠀⠉⠁
⠀⠀⠈⠛⠛⠉⠀⠈⠻⣶⡶⠟⠁⠀⠀⢻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠇⠀⠀⠀⠀⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠻⢶⣄⡈⠻⣦⡀⠀⠀⠀⣠⣴⠿⢷⡄⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠋⠁⠀⠀⠀⠉⠛⢷⣾⠿⣦⣴⠞⠋⠁⠀⠈⢿⣆⢀⣴⡞⣫⣝⣿⡿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠉⢻⣦⠀⠀⠀⠀⢀⣠⣶⠟⠁⠀⠈⢿⣆⠀⠀⠀⠀⠀⣿⣟⡵⣫⡿⠛⠁⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⢳⠀⠀⢹⠟⣹⠟⣺⣿⣉⠀⠀⠀⠀⠀⠀⠙⢷⡄⢀⣴⣾⡿⣫⡾⠋⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢣⡀⠈⠛⠛⢶⡋⠈⠻⣧⠀⠀⠀⠀⠀⠀⠈⢻⡟⣡⢮⣾⠋⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣵⡟⠁⠀⢀⣤⡾⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⠉⣉⠉⡉⢉⡉⢉⣉⣉⣉⠀⠀⡀⣀⠀⣀⣀⣀⠈⣻⣷⣶⡿⠛⣁⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠉⠙⠉⠉⠠⠦⠄⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠈⠉⠙⠉⠛⡀
⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠰⠧⠤⠄⣀
⠀⠀⢀⣠⡄⠒⠛⠓⠒⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠑⢢
⣀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠛⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣌⠳⠆⠈⠉⠉⠉⠉⠠⠾⢋⡴⢆⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠖⠉⠁
⠀⠀⠀⠀⣿⠛⡟⠀⠠⠤⠤⠤⠤⢀⡴⣛⡾⠁⣀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠠⠴⠶⢌⡷⡄⠚⠛⠋⠁⠈⠻⢄⣘⡷
⠀⠀⠀⡌⢿⠏⠀⠀⠀⠀⠀⠀⢀⢮⡙⠁⠀⠀⠀⠀⠀⢰⠄⠀⢴⠄⠀⠀⠀⠀⠀⠈⠳⣡⡄⠀⠀⠀⠀⠀⠀⠹⠇⣴
⠀⠀⣴⠃⠘⠀⠀⠀⠀⠀⠀⢠⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋⡸⠂⠀⠀⠀⠀⠀⠀⢸⠉⡆
⠀⠀⣅⢠⡆⠀⠀⠀⠀⠀⠀⡈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣴⠀⠀⠀⠀⠀⠀⢸⠀⡘
⠀⠀⠈⠲⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠟⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠻⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⢰⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⢴⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡝⠃⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⢾⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⠓⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣀⠀⠀⠀⠀⠙⠿⣦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⣴⠷⠋⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⡀⠀⠀⠀⠈⠛⠛⢷⡾⠷⠾⠟⠛⠉⠀⠀⠀⠀⢀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠂⢤⣤⡄⠀⢸⡂⠀⠀⣿⠂⢠⣤⡤⠰⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢹⠃⠀⠀⣟⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣼⡁⠀⠀⠉⠉⠉⢹⡇⠀⠀⣿⣉⡉⠉⠉⠀⠙⣧⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢸⠉⠳⣄⠀⠀⠀⠘⣇⠀⠀⠟⠈⢡⠀⢀⡴⠛⠋⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⡌⠀⠀⠈⢳⠄⢀⡜⠛⠳⠤⠀⠀⠈⠾⠯⠤⢀⣸⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⣧⣤⡤⠰⠛⠲⠞⠙⠲⠀⠀⠀⠀⠸⠀⠀⠀⠈⠉⠑⠒⢠⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠷⠶⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠶⠆⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡦⠤⠤⠀⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠈⢤⠤⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡀⠀⡀⠀⠀⣇⣀⣀⣀⡈⠉⠿⢂⠒⠒⠀⢤⣤⣄⣀⡀⡀⢀⡀⢀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠉⣟⠉⠉⢻⠋⠹⠟⠛⢩⣽⠛⠏⣭⡅⠴⡉⢹⠖⠒⠐⡟⠈⢿⡁⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠒⠋⢉⠉⠹⢇⠿⠲⠶⢶⡄⠀⠀⣰⠴⠶⢧⣸⠇⠀⠈⠙⠆⠛⢃⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⣄⣠⣾⡦⢾⣿⣦⡀⠀⠀⠀⣠⡟⢹⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠠⣿⡟⠛⠿⠀⠙⠁⣠⡛⠀⠱⠀⠀⠀⡛⠁⡾⢷⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣥⠄⠒⠒⠈⢀⣍⠉⠀⢿⠀⠀⣠⠃⢸⠀⠀⠳⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠲⠂⠈⠉⣹⠏⠉⢹⣿⠀⢠⡋⠀⠿⢢⡄⠀⢣⣀⡬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠛⠉⠠⠷⠤⠰⠾⠛⠳⡄⠙⠾⠆⠾⠇⠀⠈⠁⠖⠁⠶⠶⠄⠀⠀⠀⠀⠠⠶⣷
⠀⠀⠀⠀⠀⠀⢰⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠
⠀⠀⠀⠀⠀⢀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡄
⠀⠀⠀⠀⠀⡌⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢡⡀
⠀⠀⠀⠀⠀⠏⠀⠀⠱⢆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠸⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠈⢃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠧⠤⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⡀⠐⠀⣠⣄⣠⣀⣀⣀⣀⣀⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⢞⣛⡿⢿⠿⣛⣛⣫⣿⣬⡁⣿⡏⣨⣿⣯⣛⣛⣛⠿⠿⢿⣛⠢⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⠶⢂⣀⣀⣀⣀⡰⠶⠬⠓⠉⠘⠫⡴⠖⣀⣀⣀⣀⣀⠶⢦⠝⠁

⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣾⠉⠉⠉⠙⠛⠛⠛⣠⣤⣄⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣸⠿⠿⠟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠉⣿⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⣤⣤⣤⣤⡜⠛⠻⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣤⣤⣄⡿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⠉⠉⣤⣤⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣀⣤⣤
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠉⠉⠉⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣶⣦⡜⠛⠻⠿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠉⠋⣤⣀⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⡟⠻⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠿
⣤⣤⣶⣶⣶⡎⠉⠉⠉⠙⠛⠛⠛⠿⣀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⣱⣦⣤⣤⣤⣶⡎⠁
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣤⣀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠈⠙⠛⠿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁
⣿⡿⠿⠿⠟⢧⣤⣤⣤⣤⣤⣤⣀⡸⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣷⣦⣄⣀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠉⠛⠻⠇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠉⠉⠛⠛⠛⠛⣶⣶⣾⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣶⣄⡀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡞⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣴⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣶⣤⣰⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠙⠛⣁⣀⡀⠀⠀⠀⢸⣿⣿⡿⠿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠉⠛⠿⣿⡿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⢀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢡⣤⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⡿⠟⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠉⠛⢻⡀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣄⠾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣄⡀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠉⠉⢳⣶⣶⣶⣶⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠛⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⣀⣠⠛⠉⠉⠀⠀⠀⠀⣼⣿⣿⣿⣶⣦⡤⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠛⣃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⠻⢿⡏⠀⠀⠀⠀⠀⢀⣿⣿⣷⣶⣦⣤⡼⠛⠛⠛⠛⠛⠛
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣴⣦⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣇⣀⣤⣤⣶⣶⣶⡏⠉⠉⠉⠉⠛⢟⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⠼⢿⣿⣿⣿⣿⣿⡇
⠟⠋⠉⣸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⣀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠛⠛⠻⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠉⠻⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣀⣀⣀⣀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠙⢣⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⠟⠛⠉⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⠉⠙⠛⠿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣦⣤⣄⡸⠿⣿⣿⣿⣿⣿⣿
⣶⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⣤⣄⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇
⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣄⠿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⡇⠀⠀⠀⠀⠀⣀⠸⠿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠉⠙⠛⠿⢿⡇
⣿⣿⣇⣤⣴⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣤⣀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣦⣤
⠉⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠉⠙⠻⠿⣿⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣶⣤⣄⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿

⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⣿⣿⣿⣿⣿⢉⠍⠁⢨⣭⡍⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⡖⣶⣿⠗⠒⢶
⠀⣿⣿⣿⣿⣿⠀⠀⠀⣴⣿⡧⠸⠿⢀⣹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡋⠀⣸⣀⣰⡿
⠀⣿⣿⣿⣿⣿⠀⠀⠀⢾⣏⣴⠃⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡁⠸⢋⣿⣿⣇⣀
⠀⣿⣿⣿⣿⣿⠀⠀⠠⠞⣽⣿⢀⡈⠳⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣤⠀⢸⡿⣧⣍⠉
⠀⣿⣿⣿⣿⣿⠀⣠⣴⣾⣿⠃⣼⢁⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢐⣶⠞⠋⠀⢠⣾⢣⡿⠃
⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠉
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠂⠀⠀⠀⣀⡀⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠤⠤⠤⠀⠀⢀⠀⠀⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠩⠉⢉⣁⣀⠀⠠⢼⡯⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠉⠐⢄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣒⣒⡒⠒⠆⠀⠀⡜⠧⡀⠰⠀⠀⠀⠀⠀⠌⠀⠀⠁⠀⠀⠄⠀⠠⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠤⠀⠤⠭⠭⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠁⠀⠀⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠉⠉⢁⣐⡒⠢⠠⠭⣁⣺⢀⣀⠀⠀⠀⡂⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣒⡒⠒⠢⠬⠍⣉⣒⠒⠼⣿⠉⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠤⠤⠤⠭⠍⣁⣒⠂⠠⠬⢹⠿⠀⠀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠐⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠃⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠙⠒⠂⠤⢤⡨⣄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣩⠁⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⡇⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠙⠖⠀⠸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⡠⣴⠊⠐⠂⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠐⠆⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠄⡇⣿⠀⠀⢶⠃⠀⠀⠀⠀⠀⢀⡤⠞⢛⡛⢻⢻⣌⡃⣀⣜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⡛⣶⠛⢯⣁⠾⣄⠉⠀⠀⠀⠀⢀⣴⠟⠁⠀⣨⠃⠀⠳⠶⠆⠐⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠁⠀⣰⠀⠘⠷⠶⣿⠶⠾⢷⠀⠀⠀⣠⠞⠉⠙⣷⡔⠃⠀⠀⠀⠀⠀⠀⡼⢸
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠈⢹⡞⠁⠀⠀⠀⢈⡿⢀⡾⡽⠀⠀⢰⠏⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⢀⡜⣁⣾
⠀⠀⠀⠀⠀⠀⡾⠁⢠⠔⣹⣇⠀⠀⠀⠐⣺⢇⡜⢳⠁⠀⠀⢸⡀⠀⢸⣾⠄⠀⠀⠀⠀⠀⠠⣿⡏⠍⣿
⠀⠀⠀⠀⠀⢸⡇⠀⡜⢀⣽⠢⣄⣀⠀⢂⠟⣺⠀⠳⡄⠀⠀⠀⣧⠀⢸⠟⢦⣤⣀⣀⣀⣀⡼⢹⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⣰⣡⡟⠉⠙⠒⠺⠭⠭⢿⡉⢦⠀⠘⡆⠀⠀⢹⠆⠘⣦⠜⠁⠀⡁⠀⣽⠁⠘⡇⠀⡇
⠀⠀⠀⠀⢀⡏⠀⣹⠟⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⠙⢦⠈⠒⣄⢹⣆⣠⡾⠆⠀⠀⠹⣦⡏⠀⢀⡇⠀⢳
⠀⠀⠀⠀⠻⢧⡾⠃⠀⠀⠀⡆⠀⠀⡄⠀⠀⠀⠈⢳⡈⠺⠋⠙⠀⠈⣿⠃⠐⠂⠀⠀⢻⡄⠀⠈⡇⢶⡾⠁
⠀⠀⠀⠀⠀⢾⣄⣀⣀⣀⣠⡇⠀⠀⢈⠀⢀⣀⣤⣴⡷⠀⠀⠀⠀⢀⡿⣧⡀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠉⣹⠇⠈⠛⡿⠋⠉⢯⡉⠁⠘⣇⠀⠀⠀⠀⠀⣾⠀⠈⢷⡀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⡸⠁⠀⠀⠀⢳⡀⠀⢸⠀⠀⠀⢀⣾⠁⠀⠀⠈⢧⡀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣾⠃⠀⢠⡞⠁⠀⠀⠀⠀⠘⡇⠀⠈⡇⠀⢠⣾⡇⠀⠀⢀⡴⠋⣷⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⢰⡧⡀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢹⢠⡟⠀⠉⠳⢶⡟⠀⠀⢸⣆⢇⠀⢀⣹⣄
⠀⠀⠀⠀⢀⡞⠓⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡔⠹⢿⢦⡉⠁⢠⠏⠀⠀⠀⠀⢻⣦⠞⠉⠀⠹⣆
⠀⠀⠀⢠⡟⣇⠠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢈⠧⣽⢟⣷⡀⠀⠀⠀⠀⠀⠻⣟⣠⠞⡻⢟⡾⣷
⠀⠀⠀⠈⠻⡦⠸⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠃⣴⡗⢯⣤⠗⠀⠀⠀⠀⠀⠀⠈⢻⡴⢣⣮⡼⠁
⠀⠀⠀⠀⠀⠳⠤⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠽⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢰⢶⡄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡛⡓⠀⠀⠀⠀⠀⠀⠀⠀⡐⡐⠆
⠀⠀⠀⠀⠠⠐⠐⢃⠀⠀⠀⠀⠀⠀⠀⢤⡟⠐⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠑⠆⡀⠀⠀⠀⠀⠀⠐⢈⣤⡆⠀⢀⡆
⠀⠀⠀⠀⠀⢠⢨⡟⠆⠀⠀⠀⠀⠒⣲⠁⣹⠁⠀⠀⠀⠀⠀⠀⠙⡉⣇⣀⠀⠀⠀⠀⠀⠀⢀⢾⠀⠈⠁⢌⠂⠀⠀⠀⠀⢈⠋⡆⢨⣐⠕
⠀⠀⠀⠀⠐⢲⠁⠘⠁⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⢦⣀⣋⠑⠑⠀⠀⠀⠠⠦⠕⢺⠀⣠⠀⠀⠚⠃⠀⠀⠀⠘⢾⠀⢠
⠀⠀⠀⠀⠀⢘⠐⠚⠀⠀⠀⠀⠀⠀⢀⠂⠸⠀⠀⠀⠀⠀⠀⠀⠘⢯⠝⠲⠄⠀⠀⠀⠀⠀⠀⠈⠇⣰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠙⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢤⠀⠀⠀⠀⠀⠀⠀⢰⢀⣇⣐⠂⠷⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⡌⠑⡈⢄⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⠀⠈⠫⡂⠀⠀⠀⡰⢁⠴⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠑⠃⠃⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠰⠤⠀⠀⠀⠀⠀⠀⠐⠞⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠅⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣿⠀⠀⠀⠀⠀⠀⢸⡁⡍⠀⠀⠀⠀⠀⠀⡎⢻⠟⠀⠀⠀⠀⠀⠀⠀⠞⢻
⠀⠀⢀⡀⠀⠑⠜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣿⣧⠀⠀⠀⠀⠀⠚⢻⠓⠄⠀⠀⠀⠀⠀⢉⠿⡀⠀⠀⠀⠀⠀⠀⠀⡑⠙⠄
⠀⠀⠀⠻⡂⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣻⠂⠟⠃⠀⠀⠀⠀⠐⡆⣛⢤⠂⠀⠀⠀⠀⠐⡿⣠⢧⠀⠀⠀⠀⢉⣅⠘⠇⠐⠤
⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠉⢂⠀⠀⠀⠀⠀⢠⡃⠘⢑⠂⠀⠀⠀⠀⠰⡅⡀⡀⠆⠀⠀⠀⠀⠁⢚⠀⡰
⠀⠀⠀⠀⠀⣹⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢂⠀⠀⠀⠀⠐⠇⢰⠈⠀⠀⠀⠀⠀⠀⡗⢀⠂⠀⠀⠀⠀⠀⠀⠈⠘⢳⠁
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣠⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⡌⢂⠀⠀⠀⠀⠀⠀⠸⡀⠘
⠀⠀⠀⠀⠘⠀⢆⠐⡀⠀⠀⠯⡤⠮⠤⡴⠀⠀⠀⠉⠈⠫⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠙⡌⢆⠀⠀⠀⠀⠀⠀⢡⠀⢃
⠀⠀⠀⠀⡀⠐⠀⠣⣐⠄⠀⠠⣶⠔⢆⠧⠄⠀⠀⠀⣆⠀⢡⡄⠀⠀⠀⠀⡄⢸⠀⡀⠀⠀⠀⠀⠈⡠⠀⠈⡼⠂⠀⠀⠀⣠⢀⠂⠠⠀⡄
⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠛⠂⠂⠄⡀
⠀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠁⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠔⠕⠀⠠⠊⠀⠀⠀⠀⠀⣴⡀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢀⣀⣀⣠⡤⠀⢀⠀⠀⠀⣀
⠈⠈⠀⠀⠀⠀⠀⠀⢀⠀⡴⠀⢓⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠘⠂⠁⠃⠄⠀⠈⠀⠡⡀
⡆⠀⠀⠀⠀⠀⡠⢀⠡⠚⠀⠀⠀⠁⡀⢦⠆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⢃⠀⣀⠀⠠⠁⠂⠀⠀⠀⠀⢠⠀⠀⠀⢹⢉⠀⣀⢀⠀⠀⢴⡃⠀⠀⡤⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠂
⠀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⣀⠓⢄⠇⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡘⠑
⠀⢽⢣⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⢀⢴⣡⠖⠁⢀⠂⠀⠀⠀⠀⠁⢀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠔
⠀⠀⠀⡗⠤⡀⠀⠀⠀⠀⠀⠄⡒⡟⢃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⡖⠂⣀⠲⣆
⠀⠀⠀⠃⠀⠁⠀⠀⢠⢧⣀⠘⠂⠇⠀⠓⠁⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠈⠂⠛⠊⠀⠁⠢⡀
⠀⠀⠀⠁⠀⡀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⢁⡀⡆⠀⠀⠀⠀⠀⠀⠧⡠⠀⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠠⢆⣀⣀⠀⠄⠀⠀⠘⠠⢤⠀⠀⠐⠢⠀⢈⡁⡗⠐⠲⡀⠠⢰⣇⣣⡤⠒⠠⢤⣀⡀
⠀⠀⠀⠀⠀⠉⠻⢦⠤⠠⠄⠀⡁⠀⠀⠤⠤⢖⣀⣓⣀⠀⠀⠀⢀⣁⡒⣒⡒⣒⣚⣋⣭⣭⣥⣤⣤⢤⣶⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠖⠶⠶⠶⠒⠒⠀⠉⠙⢌⠉⠉⠉⠀⠨⡁⠉⠉⠹⣔⠈⠉⠉⠀⢤⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢖⡀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣤⠀⠀⢠⣀⣐⡄⠀⠀⡤⠄⠸⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠠⠆⠶⡄⠀⢀⠠⠄⠀⡖⠀⡙⠀⠀⠉⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢄⠄⡆⠀⠀⡁⠨⠁⠀⠀⠀⠀⠀⠀⠂⣀⠠⠀⠈⠒⠐⠇⠐⠀⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠪⡂⠄⠀⠀⠈⠢⢴⠄⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠀⠖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⠛⠛⣿⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⡿⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⣾⣿⣿⠻⣿⠆⠀⠀⠀⠀⠀⣿⡟⠋⠉⠉⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣴⣿⣿⣶⢲⣿⠄⠀⠀⠀⠀⠀⣿⡇⠀⢀⣀⣠⣤⠾⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠦⣄⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⣿⡷⠚⣋⡩⠅⠒⢀⣡⢬⣍⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⣴⣿⣿⡟⣿⡏⠀⠀⠀⣀⣠⠴⣿⡏⠀⢨⠀⠀⠀⠉⠁⠀⠉⠑⢶⢈⡳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣟⣛⣿⣧⡶⠆⣿⡁⣼⣦⣿⡇⠀⢀⡀⠀⡀⢰⠀⢀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣏⣉⣿⣿⡦⠀⢉⠀⢸⣽⣿⣷⠀⣾⡇⠀⢿⣈⣁⣾⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣨⠜⢻⡟⣿⣿⠿⠁⠀⣿⠀⢸⣸⣿⣋⣷⣾⣿⣿⣶⣛⣧⣹⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢛⡇⢸⡃⣿⡇⢸⣤⣤⣾⡷⣿⣿⣿⣿⡾⣿⣿⣧⡈⠻⣦⡙⢧⣉⣤⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢆⣽⣶⣿⣽⣿⣿⣥⣌⠳⣮⣼⠇⢉⣿⣿⣍⣹⣿⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣯⣷⣽⣯⡛⣯⣜⣠⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢿⣿⣿⣏⣰⣿⣿⠿⣿⣿⣿⣿⠟⠉⠙⡿⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢈⣿⣿⣟⠛⢿⣿⣍⣹⣿⣿⣿⣶⣿⣿⡿⠃⠒⠈⠋⣰⣒⣠⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⣻⣿⣿⢋⣁⣼⣧⣼⣿⣿⣷⣭⣻⣿⡏⢀⣀⡤⢶⣚⣯⣽⡿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣄⡉⣿⣿⢿⣟⣻⢿⢙⣿⣿⣿⠛⡿⣯⡉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⣹⣿⣿⣿⣿⣿⡏⠚⠛⢙⣿⣿⣟⠀⡇⡌⠙⠓⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⢛⣿⣿⡇⠀⠀⠀⣿⣿⣿⣀⡁⡆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⡍⣸⣿⡏⢩⣿⣿⡇⠰⣀⣤⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣼⣿⣈⣻⣿⣿⣏⣀⣠⣽⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⢎⡙⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣹⠛⣿⣿⣯⣿⣿⣿⣿⣉⣶⣾⣿⣿⣯⣽⣿⣿⣷⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠳⢛⣉⢝⣻⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⠟⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡿⠛⣁⣾⣿⣧⣸⣿⣿⣿⡙⢿⣿⣿⣿⢿⣧⠀⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⣿⣿⣿⣯⣿⣿⢿⣶⣾⣿⣿⠿⣟⢿⣷⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣟⣡⣿⣯⢫⣿⣇⡀⣸⣿⠟⠉⠀⠁⢸⣿⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⣻⣏⣸⠿⣿⣧⡿⠉⠻⣿⣿⠋⠀⠀⠀⠀⢸⣿⠀⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢩⡅⠉⡰⣾⢿⣧⡄⣼⣿⠃⠀⠀⠀⠀⠀⢸⣿⠂⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⢙⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⠃⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢈⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣋⣲⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣿⣿⣿⡿⣿⣿⣿⣿⠏⣉⣛⣿⠏⠉⠋⠋⣿⠉⠉⠙⢏⠉⠻⣯⡉⠻⠿⣿⡛⠛⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠠⡾⢿⡿⣿⣿⡷⣿⢿⣿⣿⣷⡿⣿⣿⣿⣴⣾⣾⣿⣴⣶⣶⣯⣿⣧⣶⣤⣬⣤⣤⣬⣵⣄⣤⣬⣿⣆⣠⣬⣭⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠢⡈⠑⢦⢀⠌⠁⢿⡀⠀⠀⠹⡏⠀⠃⠸⠀⠀⡏⢸⠯⡯⠁⡈⢉⠟⠀⠈⣹⠿⣋⠴⢋⡽⠋⠉⠉⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠢⡈⠣⡑⠢⣶⣷⡄⠀⠀⠹⣄⠀⢰⠀⠀⡇⠘⢸⠃⠀⡠⣪⢄⠠⣾⠵⣊⠵⠚⠉⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⡈⢄⠀⠈⠻⣦⣄⠀⢆⠀⢀⡆⠀⣧⠀⢨⠀⢀⣼⢇⣵⡾⠗⠋⠁⠀⠀⣀⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠂⠑⠀⠀⠀⠈⠓⢮⡄⠀⣆⠀⣿⢀⣾⣶⣿⠿⠊⡙⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠉⠐⢿⣾⣿⡿⠟⠉⠀⠀⠌⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠂⠀⠀⢰⠊⡟⣧⠀⠀⢀⠌⠀⢀⠴⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡀⢢⢸⡏⢻⠻⢿⣇⢀⠎⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣻⣷⣿⣹⣦⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣛⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣭⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⡉⠭⠍⣁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠊⠁⠀⠀⠀⠀⠉⢢⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡄⣂⠉⠉⠉⠉⠉⠉⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⢌⠢⢄⣀⣀⠤⢊⠜⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣁⠂⠀⠀⢂⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠉⠉⠉⠢⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⠉⡉⠉⢹⡁⠀⣀⠀⠀⠀⠀⢸⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠉⠀⠘⠋⠀⠀⠈⠁⠢⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠂⢀⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠱⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⣀⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡎⢐⠄⢀⠠⡀⠀⠀⠀⠘⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠁⢀⢀⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠃⠠⡄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠉⠰⠒⠉⠑⠂⠀⠁⢠⡍⠉⠒⠊⠉⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⢎⠈⡳⠸⠏⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠋⡀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠋⠑⢌⣉⡥⠚⠀⠤⢄⡀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢄⠀⠀⠀⠀⠁⠀⡀⠀⠀⠘⠤⢂⢌⠔⠀⢸⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢀⠌⠀⠀⠀⠀⠀⠈⠑⢠⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠈⠢⣑⠤⣀⠀⠀⢎⡰⠀⢀⡠⢐⠥⠊⠀⠀⢸⠀⠀⠀⠀⢠⣎⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠙⡀
⠀⠀⠀⠀⠀⠘⡤⠤⠤⠤⣬⠃⠀⠀⠀⠁⠒⠨⠭⠬⠭⠭⠐⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠪⣓⢦⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠱
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠇⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠈⢁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀
⠀⠀⠀⠀⠀⡰⠡⢤⠬⠥⡌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠂⡶⡰⣒⠂⠲⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⡜
⠀⠀⠀⠀⡰⠱⠁⠀⠠⡄⠀⠌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⠃⡣⡘⣢⡀⠀⠀⠀⠀⢀⠤⠐⠒⠂⠤⣀⠀⢀⠔⠁⢀⠜
⠀⠀⠀⠰⠡⠃⢀⡀⠀⠀⠀⠐⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠛⢊⡁⠈⠓⠤⣀⠜⠁⠀⠀⠀⠀⠀⠀⠑⡁⠀⡠⠊
⠀⠀⢠⢁⠂⠀⠀⠀⠀⠀⠀⠀⠐⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠲⢄⡀⠸⠀⠀⠀⠔⠁⢑⠀⠀⠀⢰⠈
⠀⢀⢂⠂⠀⠀⠀⠀⠰⣉⡁⠀⠀⠑⡐⡤⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠸⠤⠤⠀⠤⠄⢤⠀⠀⠀⢙⠀⠀⠀⠈⠐⠈⠀⠀⠀⢸
⢀⢂⠂⠀⠘⠃⠀⠀⠀⠀⠀⠰⠉⠆⠱⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣆⣘⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣁⣵⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿