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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⢉⠲⣄⣠⣀⠤⠒⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡜⢂⠰⡉⡁⠎⠀⣉⢰⠲⠀⠀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⠻⢶⣄⠢⠀⠤⡀⣀⣈⣒⡂⠀⠀⠠⠞⠳⣄
⠀⠀⠀⠀⣠⠔⠋⣁⠀⠤⠤⠤⠀⢀⡀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⢳⣀⣀⣀⣽⡿⢿⠾⢟⠓⠲⣬⡙⢲⡄⠡⣌⠇⡅
⠀⠀⢀⣾⡱⢊⣁⣀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠲⠤⠹⢶⣦⠀⠀⠀⡰⢏⡤⢰⢋⠀⠈⠀⡆⠑⢄⡘⢏⠲⠛⠓⢦⡴⠁
⠀⢀⠎⠈⣿⢯⣭⣉⣉⣉⠛⠛⠓⠲⠶⠶⠶⠶⠖⣺⠿⡅⠀⢀⠞⠁⠀⢀⡘⢦⡀⠸⣀⣯⣥⣤⠹⠋⠀⠀⠀⠘⣦
⠀⠸⠀⣴⠃⠀⠀⠀⠈⠉⠉⠛⠛⠲⠶⠶⠖⠛⠛⢿⠦⣷⠀⢈⠰⢐⠀⢸⣍⣠⠝⠛⠋⠓⠦⣤⡄⢴⣂⠀⠰⢢⡙⡄
⠀⣶⠊⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⠀⡜⠋⡡⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⢷⢬⡜⠀⠈⠒⠁⡿
⠀⡇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢰⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡀⠯⢱⠸⡁
⠀⢻⣹⠀⠀⠀⠀⠀⠀⠰⠆⠀⠠⡀⠀⠶⠀⠀⠀⠀⠀⢰⠖⢷⡎⠴⠢⠀⠀⠰⠆⠀⡠⠀⢀⡄⠀⠀⠈⠛⡒⢧⣀⡇
⢀⡼⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡗⠒⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠺⠭⠞⡟⠳⡀
⣞⠀⠘⡀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⡎⠀⠀⠀⠀⠘⡇⠀⠈⠓⠀⠀⡴⠄⠀⠀⠀⠀⢀⡇⠀⡇
⠈⠲⠴⢧⡀⠀⠀⠀⠀⠀⠰⣄⠀⠀⣠⠂⠀⠀⠀⠀⣠⠞⠀⠈⠦⡀⠀⠀⠀⠘⢆⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡼⠧⠛
⠀⠀⠀⠀⠉⠢⢤⣀⣀⣀⠀⠀⠉⠉⠁⢠⠤⠤⠒⠊⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠈⠁⠀⠀⠀⣀⣀⡀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠽⣷⡦⢄⣤⢴⡿⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣄⣀⣀⣠⡼⢧⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⡴⠙⠳⢶⠶⠋⢳⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⣰⠏⡸⣻⠟⢯⡝⢯⠹⣆⠙⢆
⠀⠀⠀⠀⣠⠊⠀⠀⣼⡳⠤⣤⠀⠠⠠⣚⣧⠀⠀⠀⠈⠓⢦⡀⠀⠀⡴⠃⣰⠃⠏⡀⠀⢀⣸⡇⠹⡄⠈⢧
⠀⠀⢀⠔⠁⠀⠀⣴⠀⠉⠓⠒⠐⠒⠃⠁⢸⡆⠀⠀⠀⠀⠀⠟⠀⡾⠁⢀⣧⡜⠀⡙⠷⠋⣽⢷⠀⢳⡀⠈⢧
⠀⣴⠯⠀⠀⠀⢸⠣⣍⡒⠠⠄⠠⠤⠄⣒⠥⢻⡦⡀⠀⢀⠜⠀⣰⠁⠀⡜⢩⠃⠀⢡⠀⢠⠇⢀⡀⠈⣧⠀⠘⡄
⢀⠟⠛⠲⠆⢴⡇⠀⠀⠈⠉⠒⠒⠈⠉⠁⠀⠈⣧⠈⠓⠿⣄⣰⠃⠀⢰⠁⢸⠀⠀⠀⢣⡞⠀⢸⡇⠀⠸⡆⠀⠰⡀
⠘⣆⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢠⡟⠀⠀⠆⠾⣿⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢿⠀⠀⢳⡀
⠀⠘⢦⡀⠀⠀⠀⠙⣦⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡾⠀⠀⡸⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇⠀⠀⢷
⠀⠀⠀⠙⣆⠀⠀⠀⢈⢇⡈⠈⠳⣄⠀⠀⠀⠀⠀⢸⣀⣴⣅⡁⢀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠟⠀⡀⢈⣇
⠀⠀⠀⠀⢸⠙⣦⠶⡋⠈⠀⠀⠀⠈⠢⣀⠀⠀⠀⢸⡅⠀⠙⠛⢾⠀⠀⡶⢂⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⠴⠓⠚⠉⠓⢄
⠀⠀⠀⠀⣾⠡⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⣇⣀⣠⡄⢸⠦⣦⣄⡽⠀⠀⠀⠀⠀⠀⠙⠚⠺⠶⢳⡏
⠀⠀⠀⠀⣿⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢻⠉⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠙⠲⠦⠤⠴
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢺⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠐⠉⣆⠀⠀⠀⠀⠀⠤⡆⠀⠀⣧⣁⡠⣠⠀⠀⠀⣀⣀⣀⠀⠀⣀⣀⢀⣀⡤⠃
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠛⠀⢸⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⣗⣒⡂⣀⣉⢹⡏⠈⢹⡙⢫⡍⠉⠀⣿
⠀⠀⠀⠀⠀⣿⣤⣤⡤⠶⢤⣤⣄⠀⣸⡤⠤⠤⠤⢤⣬⣇⡀⠀⠈⡷⠉⠀⠀⠉⢾⠁⠀⢸⡇⠀⠐⠀⠁⣿
⠀⠀⠀⠀⣼⠡⢀⣤⣤⣤⣤⣄⠘⡃⡏⢀⣤⡴⡦⢤⡀⠘⢱⡀⠀⣧⣀⣠⡤⣀⣸⠀⠀⢸⣇⣤⣤⣤⣀⡇
⠀⠀⠀⠀⣿⢠⠎⣀⡠⠭⠥⠞⢣⡅⣾⠋⠺⢟⡁⠀⠙⡆⠻⠀⡔⠉⢈⣼⡀⠯⠹⣇⢀⡞⠣⠔⠰⠒⢫⢷⡀
⠀⠀⠀⠀⠸⣼⠋⠀⠀⠀⠀⢀⢸⡇⣇⢀⣀⣀⣉⣒⣢⣼⠃⢠⠘⠂⠸⣶⠃⠀⠀⣿⢸⠈⠀⠀⠈⣶⠏⢨⡇
⠀⠀⠀⠀⠸⠭⠁⠀⠒⠒⠒⠒⠒⠋⠓⠒⠒⠒⠒⠒⠒⠚⠀⠘⠛⠛⠛⠛⠛⠋⠙⠛⠙⠛⠛⠛⠛⠛⠚⢛⣁⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣿⣷⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣿⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣷⣶⣦⣄⠀⠀⠀⢠⣴⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣷⣄⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢰⣾⣿⣿⡿⠋⢁⡀⠀⠀⢀⠈⠻⣿⣿⣿⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⢁⠀⠁⠀⠀⠀⠀⠈⠢⠘⣿⣿⣿⠀⠀⢿⣿⡟⠀⣀⡉⠉⠙⠉⣉⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⠹⠿⠟⠁⠘⠿⠿⠿⠉⣿⠃⢺⣿⣿⣧⠂⠐⠒⠀⠀⠀⠐⠲⠀⢡⣿⣿⣿⡀⠀⢸⣿⠇⣬⠀⠀⠀⠀⢀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⣼⣏⠀⠀⣴⡄⠀⠀⢩⣿⡆⡿⠻⠈⣿⡿⢿⡈⠁⣨⣦⠀⠀⠈⣼⣿⢸⠟⣿⣿⠏⢉⠣⠙⡆⠿⠷⠀⠀⠀⠿⠷⠀⢰⠋⡁
⠀⠀⠀⠀⠀⠀⢩⠀⠈⠋⠁⠀⠀⠈⠉⠀⠇⠀⠀⠹⢿⣬⡇⠀⠉⠁⠀⠀⠀⠉⠁⣼⣾⣿⠋⠀⡄⠀⠀⠃⠀⠀⡀⡀⢀⡀⠀⠀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠒⠀⠰⠀⠀⡘⠀⠀⠀⠀⢨⠉⠓⡀⠀⠳⠶⠆⠀⠀⠠⠋⡻⢄⡀⢀⠀⠀⠀⠈⠢⡀⠈⠒⠊⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⣀⢀⡀⠈⠀⡀⠠⡌⢀⠂⠀⠀⠈⠠⡤⠀⠠⡤⠊⢀⡀⠦⠞⢁⠆⠀⠀⠀⠀⣄⣀⣷⠀⠐⣞⣀
⠀⠀⠀⠀⠀⠀⠀⡰⠍⠒⠃⠀⠐⠓⣖⠋⠀⢀⠨⠁⠀⡀⠴⢓⠈⠉⠂⠐⠉⢫⠁⠀⡀⡄⢃⠀⠀⡠⠄⠚⢁⠁⠑⠉⠉⠈⠁⠉⠓⠠
⠀⠀⠀⠀⠀⡔⠁⠀⣨⢠⠀⠀⠀⠀⠈⠈⠏⠀⠀⡀⠬⠀⠀⢸⠀⠀⠀⠀⠀⠈⠄⠅⠀⠘⢄⠁⠈⣁⢀⠂⠄⠀⠀⠀⠀⠀⡀⠃⣀⠀⡜⢀
⠀⠀⠀⢠⠂⢴⣤⣖⡠⠼⠀⠀⠀⠀⠀⠀⡈⠀⠀⡀⠚⠒⠯⡜⠒⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⡐⠀⠀⠀⠀⠀⢰⠉⢑⣛⣋⣁⠠⠃
⠀⠀⠀⠈⠂⠀⠀⡆⠳⠄⠁⠀⠀⠀⠀⠀⢰⡀⠀⠀⠉⠉⣇⠕⠎⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠄⠁⠀⢀⣀⣀⡀⠻⡉⢽
⠀⠀⠀⠀⠀⠀⠀⣋⠔⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠅⠊⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡇⠁⠐⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⢰⣦⣄⣀⡀⠀⠀⣀⣀⣠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⠌⠀⢀⣀⡀⢠⠂⠀⠀⠀⠘⠭⠙⢛⠛⢛⣛⢛⡛⠫⠍⠃⠀⠀⠀⠀⠀⠈⠢⠀⢤⠀⠀⠀⠂⡄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡧⢴⠀⠀⠀⠀⠀⡦⠸⠀⠀⠀⠀⠀⢺⠰⠇⠀⠀⠀⠀⠳⠰⠇⠀⠀⠀⠀⠀⠀⠀⢸⠠⠇⠀⠀⠀⠀⠸⠆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⡄⠀⠀⠀⠀⡂⠀⠑⡀⠀⠀⡐⠁⢀⡆⠀⠀⠀⠀⣄⠀⠑⡀⠀⠀⠀⠀⠀⡐⠁⠀⡆⠀⠀⠀⠀⢸⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⡃⣤⠞⠁⠀⠀⠀⠀⠙⠲⠆⠁⠀⠀⠉⠒⠋⠀⠀⠀⠀⠀⠈⠓⠊⠁⠀⠀⠀⠀⠀⠩⠶⠋⠁⠀⠀⠀⠀⠈⠙⠶⠩⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⢠⠀⠀⠀⠀⣀⡀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠔⣶⠀⠀⠀⠛⠃⢀⠈⠄⠀⠀⠀⠀⠠⠀⠀⠾⠆⠀⠀⢰⡶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠙⠁⠂
⠀⠀⠀⠀⠀⠀⠀⣩⠀⠀⠀⠀⢠⡄⢀⡀⢀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⡠⠤⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠘⠀⠛⢏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢁⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⠥⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡘⠀⣀⡀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣀⠁⠠
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣶⠀⣶⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠸⠤⣿⠤⣿⢤⣧⠤⠅⠀⠤⠀⠠⣤⠤⠤⠤⠤⠤⣿⢤⡧⠄⠀⠀⠤⠄⠀⠀⠤⠴⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠀⠀⠀⠀⠙⠋⠈⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⢆⡔⠀⠣⠠⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠉⠀⠀⠀⠀⠀⠠⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣀⠀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠞⠃⠀⠀⠀⠀⠰⣦⣦⡀⠀⠀⣼⣿⣬⠀⡏⠁⠀⠒⣐⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢥⡀⠀⠻⠿⠇⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠲⢦⣀⠆⣲⠂⠐⢢⣄⠀⠀⠀⠀⢠⠀⠞⢦⠀⠀⠀⠀⠀⠀⡏⣙⣿⠄⠀⠀⠀⠀⠀⣿⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⢁⡈⡦⣀⠠⠔⠆⠀⠀⠀⠀⢠⠈⠀⠀⢠⣷⠀⠀⠀⠏⠉⣉⠇⠀⠀⠀⠀⠀⣿⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢉⠙⢄⣠⣂⣤⠀⠀⢀⡔⠁⠀⠀⣰⠿⠶⢧⠀⠀⠀⠉⠁⠀⠀⣀⣀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣐⣛⣤⣵⣭⠋⠵⠂⠁⡀⠀⡀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⡴⠋⠁⠈⠀⠀⣿⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⣀⢀⡛⠂⠖⠀⠉⠀⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣄⣤⡄⠀⣿⠐⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⠭⠄⠈⠁⠀⠤⠀⠐⠉⢀⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⢁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣍⡀⡠⠄⠈⠁⢠⠄⠀⠈⠁⡠⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠾⠶⠶⠶⠾⠳⠖⠋⠉⠁⠰⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇⢸
⠀⠀⠄⠀⠸⠀⢰⠂⢀⠀⠀⠈⠀⠀⠈⠐⠂⠀⠀⠀⠐⠐⠂⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢘⠁
⠀⠀⠀⠠⠀⠈⠀⠀⠘⠂⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⢀⡄⠀⠄⡀⠄⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⢀⠞⠉⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆
⠀⠈⠀⣤⠀⢠⡄⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠘⣿⠟⠉⠉⠉⠙⢿⣿
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⡀⢠⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠀⠶⠀⠰⠆⠘⠉⠀⢀⢀
⠀⠀⠠⣶⠂⢰⡦⠀⠀⠀⠀⠁⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡿⠀⠘⣇⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣿⠁⢸⡇
⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠁


⠀⠀⠾⠿⠃⠻⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠇⠻⠷⠄
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣦
⠀⠀⠀⠀⠀⢀⡄⠀⡀⠀⡀⠀⠦⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠀⡉⠉⡙⠧⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⠴⢮⣆⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠐⠀⠐⠂⣹⡁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠐⠀⠒⠈⠻
⠀⠀⠀⠀⠀⠠⠈⠲⠶⠂⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠱⠤⠤⠆⣍⠃⣄⠀⠀⠀⠀⠀⠀⡠⢿⡦⣄⡠⠄⠀⢀
⠀⠀⠀⠀⢰⠀⠀⣤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠈⢐⠀⠀⠀⠀⢠⠗⠀⠀⠀⠀⠐⠀⡠⠃⣀⠀⠀⠀⠰⡏
⠀⠀⠀⠀⢨⡄⠉⠜⢗⢦⡄⠌⣃⣀⡀⠀⠀⠀⠘⠀⠨⢈⣦⠐⠀⣀⣀⣃⣀⠀⠀⠀⠐⠦⡅⠀⢴⡤⢄⠠⡄⣹
⠀⠠⠀⠀⠀⠑⠐⠀⠚⠉⠄⠠⠤⠤⢤⡤⠴⠐⣦⠑⢤⣼⣸⣟⣻⠤⠶⠶⠒⣦⡧⠐⠰⣄⠳⠄⣸⣎⣏⣛⣳⡔⠲⠶⠶⣴⠄⢠⡀
⠀⠐⠤⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡘⠓⠤⠤⠀⢀⣠⣀⣀⣀⡀⠠⠠⠄⢲⠏⠑⠒⠀⠤⠤⠤⠠⠤⠤⠤⠤⠄⡔⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠈⠙⠃⠈⠋⠉⠉⠙⠉⠙⠃⠀⠘⠛⠑⠀⠀⠉⠙⠋⠁⠑⠒⠒⠒⠂⠀⠰⠛⠙⠀⠈⠁⠘⠉⠟⠓⠓⠒⠃⠀⠛⠛⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣤⡤⠀⠀⠀⠀⠀⠀⢤⣤⡄⠀⠄⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠈⠀⢀⡀⠄⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡐⢿⡷⠀⠀⡀⠀⠐⢿⡗⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠓⠐⠚
⠀⠀⢀⡠⠀⠀⠈⠉⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡠⠾⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈
⠁⠄⢊⠃⡐⠀⠀⠀⠀⠀⠀⢀⠀⠁⢄⠀⠀⠀⠐⠂⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠦⠆⠀⠐
⠀⠠⠈⠀⠀⠀⠂⣀⠀⠀⠄⠀⠐⠄⠀⠡⠀⠈⠁⠀⠄⠉⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⡀⠀⠀⢢⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠀⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠆⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠒⠂⠐⠂
⠀⠀⠀⠄⡀⠀⠀⠀⠀⢀⢀⠠⢄⣀⡼⠃⠀⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠈
⠀⠀⠀⠀⠀⠀⠀⢈⢁⣉⣀⠤⣀⠋⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠈⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⠁⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣀⠀⠀⠀⠀⣀⠴⠆⠀⠀⠀⠲
⠀⠀⠂⠁⠀⠀⠠⠀⠀⣀⣈⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠈⢃⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈

⠀⠀⠀⠛⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠑⠐⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠛⠛⠉⠉⠙⠛⠛⠋⠛⠉⠒⠒⠒⠒⠂⠒⠛⠛⠛⠿⠏⠀⠈
⠀⠀⠀⣤⢀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀
⠀⠀⠀⣿⣤⣴⣤⣤⣄⠀⠘⠛⠛⠛⠋⢩⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠨⠉⠭⠍⠩⠥⠬⠤⠤⠴⠄⠀⠀⡀⠙
⠀⠀⠀⠿⠉⠉⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠃
⠀⠀⠀⠶⠴⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣤⢠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣄⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⣉⠉⠉⠉⢙⠁
⠀⠀⠀⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣉⣁⡈⠀⠀⠀⠀⠀⠀⠀⡀⡀⠒⠒⠒⠲⠶⠶⣦⣤⡈⣉⣉⣩⣭⡄
⠀⠀⠀⣭⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠿⠁⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣟⣛⣿⡟⠻⠷⠶⠾⠷⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠒⠒⠒⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠙⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⣤⣾⠒⢃⠀⠀⠀⡘⠒⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢈⢋⠔⠁⠀⠀⠀⠙⢦⢉⡠⠁⠂
⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠲
⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⢻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠂⣀⠹⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⡏⠀⠉⠫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣀⣆⡀
⠀⠀⠀⠀⠀⣟⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠓⡂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠀⠀⠀⠸⠿⢧⠈⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠂⠉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⠲⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡜⠄⠀⠂⡐⡄⣀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠉⠀⠳⢠⠀⠀⠀⠀⠀⠀⢤⣇⠀⠀⠀⡀⠀⠈⠁⠀⠠⠶⠂
⠀⠀⢀⣀⡤⢂⠖⢹⠆⠀⠙⡑⠀⡑⡄⠀⠀⠀⠀⠀⠃⣸⠀⣀⠀⠀⠀⢀⡤⡀⠀⠠⢀⠀⠀⠀⢀⠀⠘⠓⠒⠲⠶⠟⠁
⠀⠀⠀⢿⠃⢨⡀⠀⠃⠀⠀⠙⠀⠀⡈⢄⠀⠀⢀⠀⠚⠛⢆⠈⢀⡀⡈⣁⠀⠱⠒⠋⡤⠄⣠⣦⠤⠶⠂⠀⢀⡆⠀⠀⠀⠀⠐
⠀⠀⠀⡜⢠⢼⣄⠀⠀⠀⢰⡄⢂⠀⠈⠈⠄⠀⠀⢉⡑⣶⡆⠀⠠⠄⡀⠳⠶⢶⢴⠯⠒⠉⢀⣀⠀⠤⠂⠀⠀⠀⡖⠀⠀⡀
⠀⠀⡐⠄⢠⣤⡀⠁⠠⠤⠀⠇⢘⠀⠀⣁⠊⠹⠒⠀⣠⡉⠉⠀⠀⠀⢀⠀⠀⠒⠂⠒⠠⡀⠠⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠰⣄⡀⠁⢀⠀⠀⠀⢤⣊⠀⠁⢁⢀⣶⢶⠯⠔⠛⠁⠀⠀⠀⠀⠒⠀⠀⢤⠀⠀⣧⠁⠀⠁⠁⠄⠀⠠⠦
⠀⠀⠀⠀⠀⠁⠈⢱⣶⣦⣄⣀⠀⠈⣉⡋⠉⠀⣀⡀⠤⠀⠲⢯⠉⠋⠓⠶⠆⠐⠃
⠀⠀⠀⠀⠀⠀⢼⡐⠀⠀⠀⠀⠋⠉⠞⠓⠘⠛⠻⡀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠺⠧⢤⣄⣀⡘⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠄⠁⠃⠂⠻
⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠌⣡⠨⠀⠚⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠔⠉⠉⠑⢄⠀⠀⠀⢀⠔⠚⠓⠢⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⠀⠈⠈⠑⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⢐⡮⠒⠚⠉⠉⠉⠉⠉⠛⠲⠤⢢⡀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢀⡠⠒⠑⡤⠒⠒⠒⠒⠺⠦⡀⠀⠀⠀⠀⠀⠀⣀⣧⠞⠁⠀⠀⠀⠀⠀⠀⢀⠔⠢⡀⠀⠣⠤⡀⠀⠀⢧
⠀⠀⠀⠀⠀⡜⠀⣞⠉⠉⢉⣡⣀⠀⣿⠊⠉⠉⠉⠉⠙⡛⢆⠀⠀⠀⠀⢨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠐⠠⠄⣀⠘⡄
⠀⠀⠀⠀⢠⡅⠀⠘⢆⢀⡎⣄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠚⠦⡄⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢂⡭⠷⠁⠀⢀⠇⢣⣀⠴
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠉⠀⡏⠑⠤⢴⢄⡀⠀⠀⠀⠀⠀⠀⣀⠧⠀⠀⠀⠈⠙⡖⠖⠲⠤⠤⠒⢊⡒⠈⠁⢰⠁⠢⠔⠉⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠈⠑⠒⠚⠒⠲⠦⠚⠀⠀⠀⠀⠀⠀⡃⣀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣰⠀⠀⢀⡀⠀⠀⠀⠀⠒⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⡀⠀⢀⠀⠀⠀⠀⠈⠢⠤⢚⠦⠊
⠀⠀⠠⡔⠁⠀⠀⠀⠀⣰⠃⠀⠀⠀⢠⠀⠀⠀⠈⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠈⠛⠉⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠈⠡⣀⣄⠀⡜⠫⠳⡢⢄⣀⣀⣉⡈⢠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢒⠶⡆⢀⡽⡲⠂⡀
⠀⠀⠀⠀⠀⠀⢨⡟⠁⠀⡇⠈⠉⢹⠗⠒⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠉⠀⠆⠀⠈
⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⢁⠀⠀⠸⠈⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡌⠀⠆⠀⠘⡀⠀⠀⡀⠈⠢⠄⡀⠀⠀⠀⡀⠀⠦⠀⠀⠀⠀⠀⠀⠄⠀⠈⢰⠀⠀⠀⠀⠀⢈
⠀⠀⢀⠂⠀⠈⠑⡄⠘⢄⡀⠈⠑⠢⢏⣢⣤⠀⠈⠑⠢⡤⣀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⡹⢀⠀⢀⠄⠊⠀⠀⣠⡂
⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⠉⠐⠢⠤⢄⠀⠈⣳⠜⠢⠤⠃⠈⠑⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠋⠀⠀⣀⠄⠒⠈
⠀⠀⠀⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⢁⠀⠠⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠠⠈⠃⠶⢒⡠⠔⠂
⠀⠀⠀⡇⠀⠀⢰⠃⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⡀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⣴
⠀⠀⠀⠣⠀⠄⠘⠀⠀⠀⠈⠂⢄⢀⣀⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠈⠉⠈⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⡊⠩⠀⠀⢁⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠂⠄⡐⠈⠀⣐⡨⠭⠒⠈⠁⠀⠀⠀⠀⠀⠀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡬⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠠⢤⠖⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠑⠆⡀
⠀⠀⠀⠀⡠⠤⠚⠁⠀⠀⠀⠀⠠⣃⣀⡀⠓⠤⣀⠀⠀⠀⠀⠔⠛⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠇⠀⢈⠇⠀⡖⠊⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢠⠐⠓⠆⠤⢄⡀⢀⣀⣀⡀⠄⠆⠀⠓⠒⠃⠀⠀⠱⣀⣀⣀⠀⢀⣀⢤⣀⠝⠢⣄⠀⠀⠀⡐
⠀⠰⡅⠀⠀⢓⠀⠀⠀⠀⠠⠀⠀⠸⠀⠀⠀⠀⡔⠠⡀⠀⠈⠍⠁⠉⠁⠀⠐⠂⠀⠈⠈⠳⡄⢠⡇
⠀⠐⠤⢤⣰⡀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⣄⡔⠀⠀⠘⠆⠈⡀⠀⡀⠀⡀⠈⠀⠀⠈⠤⠴⢻⠃⠈⢢
⠀⠀⠀⠈⠀⠑⢄⡀⠀⠁⠀⠀⠀⢀⣠⠔⠁⢠⠀⠀⢀⠞⠀⠈⠢⢈⡁⠀⠀⠀⢀⡀⠄⣺⠀⠱⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠠⠚⠳⡖⠒⢒⣛⢣⡤⠌⢚⣉⣤⠞⠊⠀⠀⢀⠂⠉⡟⠫⠝⠋⢸⠉⠙⠆⢱⡄⠈⠉
⠀⠀⠀⠀⠀⢠⠃⢀⡼⠁⢰⠋⠈⢃⠑⢢⡀⠙⡃⠀⠀⠀⠀⠠⠀⣰⠃⠀⠀⠀⠘⣄⡀⠼⠸⡉⠔⡷
⠀⠀⠀⠀⠀⠘⣸⠃⠀⠀⠸⡀⠀⠀⠂⠀⢡⡀⠘⡀⠀⠀⠀⠀⠢⡿⠀⠀⠀⠀⠀⠈⡑⠀⡅⠧⠾
⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠈⠑⠠⠤⠐⡋⠀⠀⡅⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⢀⠔⠀⠠⠀⠀⡠
⠀⠀⠀⠀⠀⠰⠄⣀⠀⠀⠀⠀⠀⠀⣀⢼⠡⣠⠐⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⣦⡤⠂⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠁⢰⠂⢋⠉⠕⠻⣄⠁⠀⠀⠀⠀⠀⠀⠀⢱⠂⠐⢄⡠⠊⡑⠢⢄
⠀⠀⠀⠀⠀⢀⡤⣞⠢⣀⢀⡾⣤⣃⠔⡄⠀⠈⢣⠠⠂⠡⢀⣤⣀⣔⢅⠀⢀⡏⢦⡊⠀⠀⠀⠑⡄
⠀⠀⠀⢀⠞⠉⠄⠀⠝⠀⢈⠀⠈⡧⠄⠃⠀⠀⡸⠀⠀⠰⠁⠀⠀⠀⠀⠈⡝⠀⢨⠀⠀⠀⠀⡰⠀⠠⡀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡌⠀⠀⠘⠤⠤⠤⠒⠁⠀⠀⠘⢄⡀⠀⠀⠀⢀⠆⠀⠈⠢⢀⡠⠞⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠈⠁⠉⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠂⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠚⠁⣠⠄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢹
⠀⠀⠀⠀⠜⠉⢳⡙⡀⠢⡀⣥⠀⠀⠀⠀⠀⠀⠈⣄⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠀⠀⠀⡀⠋⠀⠄
⠀⠀⠀⠀⠀⡀⠀⠋⠁⠚⠒⠀⡀⠀⠀⠀⠐⢈⠄⠈⠳⠀⠀⠀⡂⠀⠀⠀⠀⢈⠀⠈⠒⠿⠛⠉⣖⡀⠘
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠁⠀⠄⠀⠀⠀⠀⠆
⠀⠀⠀⠀⡜⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⣐⢐⡇⠀⠀⠀⠀⠀⠀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠃⠺⠗⠒⠒⠋⠋⠂⠀⠀⠈⡄⠀⠀⠀⠀⠀⢠⠘⠆⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠤⠄⢸⢲
⠀⠀⡀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⢛⠀⠀⠀⢀⡀⠀⠰⠖⠐⠀⠀⠀⢈⡴⠲⠒⠒⠂⠠⢤⡤⢦⠄⠀⠀⠼⢂⠀⢀⠀⠀⠀⠀⠠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠚⠻⠀⠀⢀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠈
⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂
⠀⠀⠀⠀⠀⢠⣾⠭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣧⣠
⠀⠀⠀⠀⠀⢸⠁⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⢹⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⠊⢣
⠀⠀⠀⠀⠀⠈⠀⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⣊⣉⡉⢠⡬


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

⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⢀⣰⣶⣾⣧⣴⣦⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⢰⣿⡿⠻⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⣾⡿⠿⣿⣿⣿⡇⢀⣠⣴⣿⣿⣿⣿⡆
⠀⠀⠀⢸⡏⢡⡄⠀⣭⠻⣿⡟⠀⠈⠉⡉⣿⠃⠼⡄⠀⣿⡁⠀⠈⢻⣿⡇⠸⣿⠟⠻⠿⠿⠻⣿
⠀⠀⠀⢠⣇⠈⠀⠀⠀⣾⣭⠀⠛⠀⠐⠋⠍⠀⠰⠃⠈⢥⠛⠀⠘⠃⢸⠁⠀⢿⢴⡁⠀⣤⠀⡿
⠀⠀⠀⠈⠛⠓⠆⠰⠺⠉⠀⠀⢈⠛⡁⠀⠀⠀⠀⠀⠀⢸⣦⣈⢋⣠⣾⠀⠀⠈⠀⠠⠀⠀⠀⠁
⠀⠀⠀⢂⣠⣆⠀⠀⣔⣄⡠⣾⠁⠀⠁⢧⠀⠀⠀⠀⠁⢠⠉⠁⠀⠙⠹⡀⠀⠀⣠⢲⠀⣦⣄
⠀⠀⠀⠀⠊⠂⠀⠀⠒⣇⢱⠏⠀⠀⠀⠘⡆⠀⠀⠀⠀⢨⠀⠀⠀⠐⢲⡀⠐⣀⠀⠀⠀⢀⣆⣀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⣭⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⢰⠀⠀⠀⠠⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠆⣶⣶⣶⣶⣶⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢶⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠷⢶⢶⢶⠶⠗⢸⣿⣿⠿⣿⣿⡆⠀⠀⠀⠀⣠⣀⣀⣀⣀⣠
⠀⠀⠀⠀⢠⡄⠀⠀⠤⠄⠀⠉⠉⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠋⠋⠋⠉
⠀⠀⠀⠀⠐⠂⠀⠀⡐⠀⠀⢉⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⡛⠀⠀⠘⡃⠀⠀⠀⣀⠁⠀⠈⣈
⠀⠀⠀⠈⠒⠁⠀⠀⠙⠊⠠⡠⠂⠀⠐⢄⠀⠀⠀⠀⠀⣄⡠⠀⠀⠠⣀⡄⠀⠀⠉⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠚⠛⠛⠓⠒⠤⣀
⠀⠀⠀⠀⢀⠠⠐⠀⠀⠈⠀⠈⠐⢢⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⡠⢂⠅⠀⢀⠄⠂⠁⠀⠀⠀⠈⣇⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⢀⠌⡔⠁⠀⠂⠁⠀⠀⠀⠀⠀⠀⢠⢿⡄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠤⠤⠤⠤⠤⠴⢲⠟⢁⣀⡄⠀⠀⠀⡠⠄⡀
⠈⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⠃⠀⢫⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠊⠐⠀⠄⠤⠭⠩⠑⠈⠃⠀⠂⠉⠀⠀⠣⡄
⡇⠀⠀⠀⠀⠀⠀⠀⡠⠐⡡⡾⠁⠀⠀⠀⠑⢄⠰⣄⠄⠀⠀⠀⠀⠀⠀⠀⢣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡉
⠁⠀⠀⠀⠀⠄⢒⠡⠐⠉⠀⠀⠀⠀⢀⠀⠀⠀⠑⢿⠺⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠘⡄⠔⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢸⢴⠀⢰⠂⢓⡄⠀⢇⢹⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⢸⢤
⠀⣘⣼⠀⠀⠀⠰⠆⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⢠⡇⠀⡦⠊⠉⠖⠁⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠎⠔⠁
⠀⠣⡟⢧⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⡔⣃⠧⠋⠀⠀⡜⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠉⢐⡷⡄⡀⠀⠀⠀⠀⠀⠀⣀⠤⡲⢻⠪⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠲⢄⣒⡲⢶⡂⠀⠉
⠀⠀⠀⢸⠀⠇⢋⠁⠀⠀⠀⣟⠁⠀⡲⠤⠻⠀⠘⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠈⠱⠠⠽⡴⠟⠉⠢⢄
⠀⠀⠀⠘⡀⠀⠈⠀⠀⢀⠚⡍⠛⠛⠤⠐⠁⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠑⢄
⠀⠀⠀⠀⠈⠂⠰⠀⠠⠃⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡐⠀⠢⢔⡯⢀⠀⠀⠀⠀⠀⠀⢀⣑⡤⠁⠀⡁
⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠐⠾⣤⡠⠤⠀⠒⠀⠀⠀⠈⠐⠶⣒⠒⠒⠂⠓⠠⠤⢠⣀⡸⠁⠉⠉⠀⠀⠀⢯⣀⣱⡤⠊⠁⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠈⠚⢷⡦⠠⠀⠤⢄⠠⣁⠀⠀⠠⠤⠤⠖⠀⠘⠑⢀⡀⠀⠀⠀⠀⠀⠠⠶⠒⠒⠚⡛⢋⣁⣀⣀⣀⣀⡀⠀⠤⡤⡤⠬⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⣤⣤⣤⣤⣤⡤⠶⠐⠋⠹⢏⠉⠛⠒⠒⠊⠉⠙⠉⠙⠓⠂⢠⣠⠀⠀⠐⠛⠠⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠉⠉⠉⠣⡀⠀⠀⠈⠋⠐⠠⡀⠀⠀⠈⠫⠐⠒⠠⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠲⢄⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⠀⣀⠀⠀⢸⠂⠀⠀⠸⠆⠒⢸⠃⠐⠀⠘⠓⠒⠘⡃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠸⠀⠓⡄⠀⠀⠤⠀⠒⠒⢺⣁⢀⡎⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡄⠀⠀⠀⠀⢠⡃⢄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠘⢄⠠⠼⠀⠀⠀⢸⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⠀⡇⠀⠈⠒⠔⠦⠈⠐⠌⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⣀⠀⠀⠀⠀⠳⣀⣣⡀⠀⠈⢶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⣀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡍⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣤⣤⣤⣀⠀⣤⡾⠿⠿⠿⠟⠛⠛⠛⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣤⣤⡤⠶⠶⠒⠾⠟⠛⠉⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠙⠟⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡄
⠀⠀⠀⠀⠀⠀⡄⢜⣯⠀⠀⠀⠀⠀⠀⠈⠒⠂⠤⠄⣀⡀⠈⠉⢙⣿⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⢸⠁⠈⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣶⣾⣿⣿⣿⣿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⢀⠀⠀⠙⠧⣤⣶⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣯⡀
⠀⠀⠀⢀⣠⣿⣥⡔⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣀⣤⣾⣿⣿⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⡶⠒⠛⠉⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⠟⠓
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠂⠀⠀⠀⠚⠛⡆⠀⠀⠀⡌⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⠶⡆⠀⠀⠀⠴⠓⠐⢶⠀⠀⢠⠗⠒⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⣶⡟⠁⠀⢸⠂⠀⠀⡇⠀⠀⠘⠀⠀⣸⠀⠇⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠈⠀⠀⢀⠁⢀⠀⠀⠀⠈⢐⢀⣀⠀⣄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡉⢄⠉⠊⠀⠀⠀⠀⢈⡄⢘⠀⠀⣾⣿⣧⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⠈⣄⡆⠀⠀⠀⣆⡆⣧⡆⠀⠀⠀⠉⠉⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣿⠀⣸⡁⠀⠀⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠤⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠈⠉⠉⠉⠁⢢⠀⠀⠀⠢⡤⠊⠉⠐⠤⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢀⠄⠠⢀⣀⣀⠔⠀⠀⠀⠀⡆⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠎⠀⠀⠀⠨⠀⠀⠀⠀⠀⡀⡵⠄⠐⠊⣹⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡲⡀⠀⠀⠀⡆⠀⠀⠀⡠⠊⠀⢇⣃⠀⡜⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠰⡅⠘⠠⠠⢌⡀⠀⠀⡰⠁⠀⠀⠈⠢⠉⠔⠊⠀⣐⠱⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢋⢢⡀⠠⣺⡁⠣⡄⠃⠀⣀⢀⡀⢈⠊⠀⠀⢠⠋⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⠁⠀⡞⠈⠈⠈⠁⠀⠀⠀⠀⠸⠀⠀⠀⡟⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⢀⠔⠉⣀⡀⠀⠀⠀⠀⢀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⠃⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠸⢀⠴⠶⣁⡀⠔⠤⠄⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⡸⣀⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢀⠸⡀⠀⠀⣀⠤⠀⠒⠂⢸⡇⣠⠤⠀⠀⠒⠂⠀⠉⠁⣰⡞⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠉⡉⠉⠀⠀⠀⠀⠀⠀⢨⢸⡀⠢⡀⠀⠀⠀⢀⡠⠐⠋⠀⢀⠔⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⡿⢬⣦⡀⠵⠂⠉⠉⢢⡀⠀⡰⠁⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠤⠄⡀⠀⠀⢀⡠⡥⡀⠀⠀⠈⠂⠐⠂⠁⠈⠊⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠉⢀⠤⠊⠁⠀⢃⠁⠀⠀⢠⠒⠤⣀⠄⠀⠀⢀⠤⠐⠒⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠐⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠉⠀⠈⠉⠉⠀⠀⠀⠀⠈⢷

⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢗⢀⠃⠁⢐⠸⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣴⡰⣲⡄⠀⣤⡟⣥⡏⠐⢀⠀⠄⠂⣺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢹⠐⣿⠁⣼⡟⠫⣴⠀⢃⠄⠐⠈⡌⠎⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⡜⢳⡆⠀⡖⣿⠀⣞⣀⠢⠉⢐⠒⠠⠁⢀⡂⠠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⢳⡄⢰⡀⡇⢸⠀⣿⠂⡎⠃⠐⠂⠠⠀⠠⢂⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠻⡄⢳⣼⢸⡄⣸⣆⢣⠁⠀⠀⠄⠀⠄⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠹⡄⠹⣶⠇⠘⣷⡾⡊⠲⠆⢀⠰⢠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡄⢿⣷⠀⠘⡿⣵⣬⢁⣂⡲⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡿⣎⠌⢦⠀⠈⢮⠉⢉⠽⠊
⠀⠀⠀⠀⠀⠀⡇⠘⣇⠈⢳⡀⢸⠀⡌
⠀⠀⠀⠀⠀⠀⡇⠀⠻⣆⠀⠙⣯⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠒⠒⠖⠒⠤⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠷⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡰⠋⠑⠀⠀⠀⡀⠂⠀⠀⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⡁⡄⣷⡀⠀⠀⠈⢦⠀⠀⠀⢀⡞⠀⠀⠀⠈⡶⠊⠀⠀⠀⠀⢀⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⣿⣧⠀⠀⠀⠀⢣⠀⠀⡸⠀⠀⠠⠐⠈⠀⠀⠀⠀⡠⠂⠁⠀⠀⣀⡤⠞⠓⠒⠒⠒⡆⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⣧⣿⣿⢧⡀⠀⠀⠀⢳⡖⠉⠚⣅⠀⠀⠀⠀⠀⢀⠚⢄⠀⠀⣠⠞⠁⢣⡀⠀⠀⠀⠈⠙⠢⡈⢢⡁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣏⠁⠘⢷⡄⠀⠀⠀⠱⡀⠀⠘⣦⠄⠀⠀⠀⠂⠀⠀⠑⠴⠁⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⢸⡄⢀⡔⠛⠃⠷⣄⠀⠻⣄⠀⠀⠀⠰⡄⠀⠘⡄⠀⠀⡆⠀⠀⠀⢀⣧⡠⠔⠈⠻⣿⠿⠗⢀⠀⠀⡀⠄⠛⠿
⠀⠀⣠⠒⠉⠁⠙⣇⠬⠷⢀⠀⠀⠈⠳⠤⠈⢂⠀⠀⠀⠘⢄⡰⠃⡏⠈⠁⠂⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⡇⠀⠀⠀⠈⢧⡀⠀⠀⠉⠒⠤⠄⠀⠀⠘⢧⡀⠀⠀⠈⢣⣀⠘⢆⠀⠀⠐⠀⠀⠷⢄⠀⢀⣀⣄⠀⠀⠀⠀⡀
⠀⠀⢣⡀⠀⠀⠀⠎⠙⡇⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠹⡑⡄⠉⠈⠹⠤⠤⠒⡀⠙⣿⣿⣿⣷⡄⠀⠈⠁⠀⢄⣀⡰
⠀⠀⠀⠉⠀⠀⠒⠛⢄⣘⣦⡽⠂⠀⠀⠀⠀⠀⠀⠀⣈⣦⠀⢀⠀⣧⣸⠀⠔⠃⠀⠀⠀⠑⢤⠘⠻⠿⢏⠀⠀⠀⠀⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⠊⠑⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠈⠧⠄⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤
⠀⠀⠀⠀⠀⠀⡔⠶⢖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠐⠈⠱⡤⢈⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡆⢠⣻⣓⢲⣹⡇⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠘⠙⢻⠶⠊⡁⠀⠙⠃⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⠄⣿⠡⣤⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠴⠆
⠀⠀⠀⠀⠀⡿⠿⣶⣷⣾⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠿⠜⣿⣧
⠀⠀⠀⠀⠀⠌⠂⠐⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠚⠻⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⢀⠀⠺
⠀⠀⠀⠀⠀⠞⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⣭⣧⣀⠀⣀
⠀⠀⠀⢰⣤⣀⡈⣯⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣖⣐⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⡽⢉
⠀⠀⠀⠀⠉⠙⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣾⣇⣹⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⢤⣶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠙⠋⠉⢉⣿⣷⣴⣿⣦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢴⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⠁⠉⠉⠢⠀⢀⡐⢀⠀⠀⠀⠀⠀⠀⢠⡟⠘⠛⣿
⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠁⠀⠀⠀⠀⠘⠂⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⠆
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⢤⣦⡀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⣴⣥⠞⠀⠀⡠⢴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠚⠶⡀⠀⠀⠀⠀⠠⢿⣀⠸⣄⣁
⠀⠀⠀⣀⠀⡷⠟⠁⠀⠀⠀⠹⠯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠁⠠⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠈⠳⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡬⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠀⠈⠹⠾⠇
⠀⠀⠀⠀⠀⢻⣿⣶⣤⣀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⣇⡀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣮⠁⠀⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡟⠛⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣽⠟⢋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⠉⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠈⡹⣿⣴⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⣤⣧⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⢰⣥⣤⠆⠈⠙⠂⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⠶⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠘⠀⠜⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⢀⡀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣀⡘⣁⡀⢀⣾⣿⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠾⠟⠀⠀⣠⣾⣿⣿⣿⠟⣿⠿⢿⣿⣿⣧⣌⣉⣩⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡎⠉⣩⣤⣔⠀⠈⠀⢸⣿⣿⡟⠉⠀⠀⠀⠀⠈⠿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⡇⣿⣿⣩⣽⣦⠑⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠦⠀⠀⣨⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣮⣿⣿⠿⠿⣲⣄⠸⣿⣿⣿⡖⣀⡀⠀⢠⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣩⡽⠛⣋⣵⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⢻⣿⣿⠿⣿⣿⣿⣿⠋⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣿⣿⡿⠋⣿⣿⠋⠵⠚⠉⠁⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠻⠿⠧⠈⠩⠀⠀⢀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣅⠀⡰⠀⠀⢠⣠⣾⣿⣄⣴⠾⢋⠁⣁⣀⠀⠤⠛⣁⢀⢰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⣸⣤⣾⣿⣿⣿⢏⣁⠴⠊⠀⢹⡄⠡⠠⢾⣟⢢⣸⣿⠀⠀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢘⣭⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡿⠉⠂⢀⣐⣢⡶⣾⡇⢸⣿⣨⣿⣿⠿⠛⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⣿⠏⠁⢸⡏⠙⠛⠟⠛⠋⢀⣉⣠⣽⣿⣷⣦⣍⡤⠿⢿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠁⠀⠀⢤⣷⣤⡀⠀⠀⠀⠈⡻⢿⣿⣿⣿⣿⣿⡇⢠⣿⣬⠀⠄⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣭⣶⣾⣾⡇⣤⠀⠈⠋⠁⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⠀⢾⣿⠇⣀⣴⢀⡀⠀⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡛⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⢿⣦⣉⣀⣿⣿⣿⣷⣤⣶⣶⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠒⠠⠘⠟⠀⠀⣸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠹⠿⠿⠿⠿⠒⠾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡛⠻⠿⢿⡿⠿⠷⠄⠜⠛⠛⣿⣿⣶⡞⠃⠀⠘⣿⠿⣿⣋⣉⠚⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣿⣿⣦⡜⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⠀⡀⢭⠔⠀⠀⠉⠁⠀⢹⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣿⣿⡀⠀⢀⠴⢤⠄⢀⣴⣋⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠿⢿⣿⣶⣄⣤⠀⢀⣀⣼⣿⣿⡿⠛⠁⣠⢄⠱⣦⣄⣴⡿⢟⣛⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣶⣆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣾⣿⣾⡿⢟⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠙⠿⣿⣧⠉⠙⠛⠋⠺⠷⠿⠛⠻⣽⣴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⡀⠤






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣴⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠉⠛⠲⣄⠀⠀⠀⢠⠟⠁⠀⠀⠀⣀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠰⣏⢉⣳⡄⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢾⠀⣠⠞⠛⠳⣌⣋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠙⣟⠉⢷⠀⠀⢸⠀⣠⠞⠉⠛⠲⣤⡀⣿⠀⠀⠸⣶⠇⠀⠀⠀⠀⠉⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢷⡀⢸⠘⣇⠀⠀⠀⠀⢸⠇⣿⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠻⣿⡀⠙⠷⢤⣤⠶⠋⠀⢹⡄⠀⠀⠀⠻⣤⣀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠙⠓⢶⣾⣳⣶⠤⠦⢼⣇⢀⡤⠶⠤⢾⠉⠉⣿⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠀⢰⣏⡼⢧⣙⡆⢀⠀⢹⡟⠀⢀⠀⠘⠳⠖⠋⠀⢀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠀⣀⣭⣥⣼⡆⢸⠇⠀⣿⣤⣤⣄⣀⣤⣤⣼⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⣇⣤⣄⣾⣥⣿⣤⣤⣤⣿⣤⣤⣤⣄⣼⣅⣿⣿⣤⣤⣤⣤⣼⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠛⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣀⠥⠂⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀⠀⠠⠀⢠⠰⢂⣀⡌⢱⠞⠉⠒⡀⢤⠌⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡋⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⣗⠀⠀⠁⠀⠀⠉⠀⡈⢩⠰⡴⠻⠁⢄⠀⣀⠄⠂⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⢀⡔⢀⡀⠀⠈⠂⠀⠃⠀⠀⠀⡇⠀⠀⠀⠀⠠⠛⠀⠀⠈⠒⠀⠀⠀⠀⠈⠡⠀⠐⠁⣠⠤⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠿⠀⢄⣙⡆⠀⠀⢀⡀⠈⠲⠆⡁⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣾⠀⠀⠀⠀⠀⠀⠤⠄⡀⠀⠀⢰⡄⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣇⡆⠀⣀⡂⠀⠀⢀⣴⢕⣁⣤⣦⣀⡀⠀⠀⠀⠀⡠⠐⠂⠉⠲⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣓⢷⣄⡈⢛⣡⣶⡿⡿⣿⣿⣿⣿⣿⣽⢧⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠠⢀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣧⢽⣻⣿⡫⢵⠶⣽⣿⣿⣿⣿⣿⣿⣷⢿⡀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣴⣬⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠐
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠙⠦⣄⢀⠤⠒⠁⠀⠀⠀⣠⠁
⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⠒⠒⠀⢺⣯⠇⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⢸⠉⠙⠿⢿⡿⠃⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠘⣿⣿⣿⣿⣿⡿⢏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠏⠀⠀⠀⡼⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣀⣸⠛⣟⠛⠉⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠸⡀⠀⠀⠀⠣⣄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢅⣀⣐⡂⠞⡠⠈⠆⠐⠀⠀⠀⢡⠀⠠⠀⠀⠀⠀⠀⠀⠀⢼⠁
⠀⠀⠀⠙⢆⠀⠀⠀⠈⠣⣠⡴⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣤⠀⠀⠀⠀⡀⠔⢐⣉⢢⠀⠈⠀⠀⠀⠀⠉⠀⠰⠁⠀⢼
⠀⣀⣀⣀⣈⣳⣄⠀⠀⠀⠘⠯⣍⣤⠟⢿⡿⠿⡿⣿⣟⣟⣏⣣⡐⠚⠘⠛⠒⠠⠊⠄⠀⠍⠂⠀⠡⡀⠀⣀⠀⠀⠀⠀⠑⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠈⢳⠦⢀⠀⠠⠋⡏⠠⡈⢷⠗⠉⣼⠻⠈⠁⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠙⠀⠉⠈⠉⠈⠀⣼⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠑⢄⠀⠀⠢⣠⠌⠒⢋⡟⠄⠀⠀⣺⠀⠀⠀⢙⡉⠀⢡⢀⣰⠟⢀⠀⡀⢰⡄⠀⠀⠀⠀⢰⡟⠀⠀⠈
⠀⠀⠂⠀⠀⢠⠂⠀⠀⠀⠀⠈⢉⡉⠉⠀⠴⠞⣺⣄⢈⣀⠁⢟⣒⡚⢐⣚⣖⣐⡰⡞⠏⣰⡟⢀⠇⣉⣎⠀⠀⠀⣰⣿⠀⠀⠀⡆
⠀⠀⠀⢀⣴⣃⣈⣠⣩⠤⣤⢴⣖⣒⣂⣩⣉⣠⢧⣤⣤⣄⣈⣉⣉⡁⠀⣀⠀⢀⠩⢧⡎⠉⠀⠉⠁⠀⠀⠀⠀⡐⠿⠈
⠀⠀⣴⣿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠙⠻⢿⠞⠛⠛⡻⠛⠛⠛⠛⠛⠛⠛⠓⠚⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠉⠉⠁⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠉⠉⠉⠉⠑⢄⠀⠀⠀⠈⠢⣀⠔⠉⠉⠉⠢⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢠⠃⠀⠈⠙⠑⠒⠊⠁⠀⠀⠀⠀⠀⢣⡀⠀⠀⣠⠴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠔⡺⢫⠁⠉⠀⡠⢏⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⡀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⡔⠁⠀⢣⡘⡄⠀⡴⠁⢸⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠑⢄⡀⣀⡔⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⠹⡌⠚⣡⠤⠚⠀⢠⢔⡦⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠁⠀⠸⡣⡀⠀⠀⢀⣽⠟⢢⠀⠀⡜⠀⠀⠀⠀⠀⠀⢣⡰⠃⠀⠀⢀⡷⠋⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢣⣘⡆⠀⠘⠼⠄⣈⣛⣃⣁⠠⠤⠤⠐⠒⠒⠆⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣾⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢔⣉⠇⠀⠀⣀⠀⠀⣠⠞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⣧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠊⠉⠉⠒⠒⠊⠀⠉⠉⠀⠀⣀⡕⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠇⠀⡸⣹⠶⠆⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣇⠀⣀⣀⡠⠤⠒⠊⠉⠉⠁⢸⡇⡠⡖⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⣰⡿⠃⠀⠀⢀⠴⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡅⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣇⠘⢦⡀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⢀⡴⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣌⢣⡀⠉⢲⣤⠤⠖⠚⠳⡀⠀⠀⠀⡠⠋⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡏⠈⠙⠛⠋⠑⣇⠀⢀⣀⡠⠞⠢⣀⠜⠁⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⣲⠄⠀⠀⢀⡤⠔⡗⢢⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⢀⡤⠚⠁⠀⠀⢹⢸⠀⠀⠀⠀⡔⠓⠦⣀⡰⠀⠀⠀⠀⣀⠤⠒⠓⠲⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⡤⠔⠊⠁⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠈⠒⠊⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢀⠤⡂⠀⠉⠙⡌⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠉⠙⡆
⢀⠔⠁⠀⠈⠓⣄⠀⠸⠀⠀⡜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠙⢗⠢⢌⣒⠒⠁⠀⣀⣀
⡄⠀⠀⠀⠀⠀⠈⣇⠀⡇⡼⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢠⣚⡉⣉⣥⣴⢾
⡧⢄⣀⠀⠀⠀⠀⢸⠀⣧⣃⡠⠞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡴⣩⠖⠊⠉
⢱⡀⠀⠈⠉⠐⠒⡞⢺⠛⡅⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⡴⠁
⠀⠑⢤⣀⠀⣠⠞⠀⡞⢀⡹⠖⠁⠀⠀⠀⠀⣠⠔⠐⠒⢔⠒⠦⢤⡀⠀⠘⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠟
⠀⠀⠀⠀⠉⠉⠒⠋⠉⠉⠀⠀⠀⠀⠀⢠⠎⢀⡉⢦⡀⠈⢂⠀⢡⠈⢣⡀⠘⠢⢝⡒⠶⠤⠤⠴⣒⡪⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡔⠁⢀⡠⢷⠀⢸⠀⠘⠀⢸⢳⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢀⣀⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⡠⠃⠀⣀⣄⢸⠀⢠⣀⣸⠼⡇⠀⠀⠀⠀⠀⠀⠀⡠⢚⣅⡢⠾⣿⠷⢦⡑⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠸⡇⠀⠁⢀⠾⢄⣀⠈⠉⠉⢀⣀⢴⠇⠀⠀⠀⠀⠀⢠⢊⣴⢷⡏⢨⠞⢄⡜⠃⢼⠸⡆
⠀⠀⠀⣠⠔⠛⢧⡔⠋⠀⠈⢟⠦⡀⠀⠱⣄⠀⢸⠤⣀⠈⠉⠉⢉⣁⣠⠞⠀⠀⠀⠀⠀⠠⢡⣳⠗⢅⡜⠑⠦⣎⠈⡰⢺⠀⡇
⠀⠀⡌⠀⠀⠀⢠⡀⠀⠀⠀⡘⠲⣝⣆⠀⠈⠑⠮⢷⣤⣬⣭⡭⠽⠛⠁⠀⠀⠀⠀⠀⢠⢣⡿⠋⡦⡂⢈⠖⢄⡠⠋⡲⡞⢰⡇
⠀⡜⣸⠦⠤⡖⠁⠙⠦⡠⠴⣄⠀⣸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣼⢉⠶⣀⡼⠓⣤⡊⢩⠾⡕⡡⢣⠁
⠀⣏⠀⠀⠀⠱⣄⣀⡜⠀⠀⢠⠋⠈⢦⡇⠀⠀⠀⠀⠀⢀⠤⠀⠲⠤⡀⠀⠀⠀⠀⠀⡇⡧⠉⣦⡎⠑⠴⣁⡜⡳⢊⠔⡱⠃
⠀⠘⣄⢀⣀⠊⠀⠀⠑⢢⠖⠣⣀⠀⡼⠁⠀⠀⠀⠀⢠⠊⠐⠢⡀⠀⢈⣆⠀⠀⠀⢐⢠⡘⠶⣁⣨⡳⠤⢛⡩⢔⡡⠊
⠀⠀⠈⠳⢄⣇⠀⢀⠠⠯⣄⣰⠧⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠆⠊⢸⠀⠀⠀⠌⣬⠶⢋⡠⠴⡒⠉⠁⠂⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣠⠔⠋⢉⡷⠃⠀⣀⡌⢈⡵⢊⠕⠊⠁⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠒⠒⠲⢤⠤⣀⡀⠀⠀⠈⠉⠉⠁⠀⢀⠾⠉⢍⢫⠈⠀⠀⠀⠀⢸⡀⣀⣸
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⣀⣠⢤⠴⠤⢦⠈⡍⡗⠀⠀⠀⠀⠀⡰⡉⢦⢱⢸⠎⠀⠀⠀⠀⢀⣼⣖⣗⣮⣄
⠀⠀⠀⠀⢀⠜⢄⠀⡠⠴⠎⠙⠉⠀⠀⠀⠀⠀⠇⢸⠃⠀⠀⠀⠀⡾⠤⡙⡼⡤⠃⠀⠀⠀⠀⠀⣾⢏⡏⢿⢙⢿⣄
⠀⠀⠀⠀⠊⠡⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠑⢄⣈⡷⠁⠀⠀⠀⠀⢀⣎⡟⣹⠓⢼⠉⣍⣷⢆
⠀⠀⠀⠀⠀⠀⢱⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠾⢁⣼⠑⣾⠁⣤⢹⠛⠧⡀
⠀⠀⠀⠈⠢⢄⣼⣠⠁⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠘⣅⠘⡹⠃⠫⣀⣨

⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⠀⠀⠀⠀⠀⣾⣶⣦⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⣀
⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠤⠾⠶⠀⠀⠀⠀⢰⡿⣿⣿⣿⠄⠀⠀⢠⣶⣿⣿⣏⠃⠀⠀⠀⠰⣿⣷⣾⠀⠀⠀⠀⠀⠀⠘⠹⠿⢿⣿⣷
⠀⠀⣰⣿⣿⣿⣷⣤⡀⠀⠀⠀⡀⠁⣴⣶⣦⣀⢀⣤⣾⣿⣿⣿⣯⡀⠀⢀⢿⣿⣿⣿⣿⣆⠀⠀⠀⣠⣿⣷⣬⣄⠀⠀⠀⠀⠀⠵⠶⠶⠖⣿⣿
⠀⣼⠿⣿⣿⣿⣿⣿⣷⠀⠀⣼⠿⣿⢿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣆⢿⣿⡿⣼⣿⣷⠀⣀⣿⣿⣷⣾⡟⢳⡀⠀⠀⢈⣤⣄⣰⣺⣿⣿
⣰⣿⠂⠉⡁⠈⢿⣿⣿⡇⠀⣽⡖⣽⣷⣿⡉⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣏⡿⢷⣿⣿⣿⡗⢿⣿⣿⣿⣿⣧⣠⣷⠀⠀⢽⡀⠈⣿⣿⣿⣿
⣿⣿⢦⣀⣀⡓⣻⣿⡿⠟⢰⣽⢃⣿⣿⣿⣧⣸⣿⣇⣿⣿⣿⣿⣿⡿⠛⣿⣿⣟⣉⣉⠘⣿⢿⡿⣿⣿⣿⣿⣿⣿⡷⠿⠂⠀⣸⡇⣼⣿⣿⣿⣿
⢸⣿⠇⠨⠠⣄⣹⣿⡇⠀⣴⠹⢖⣒⣻⣿⣴⡟⢻⣿⣿⣿⣿⣿⣿⠁⠀⠈⡿⠀⠁⠀⠀⣿⣦⠀⠿⣿⣿⣿⣿⣿⣧⠀⠀⢻⠋⢹⣿⣿⣷⣿⣿
⠀⠻⣷⣾⣿⣿⣿⣿⠁⠀⠛⠂⠈⠿⠛⠛⠉⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠠⣷⠀⠀⠀⠀⢻⣿⠀⠀⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠈⢿⣿⣿⡿⠟
⠀⠀⢿⣿⣿⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⣧⣤⣀⡀⡀⢸⣿⡄⠀⢸⣿⣿⢿⡿⠁⠀⠀⠀⠂⠀⠀⣘⢋⠑
⠀⠀⢸⣿⡛⠹⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠈⢹⣿⣿⣿⣿⠁⠀⠀⠀⢨⣿⣿⢋⣉⣼⠉⠀⠀⠀⣿⡏⣿⣷⠀⠀⠀⠰⣦⣤⡄⣤⣤⡤
⠀⠀⢸⣿⠃⢀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⢨⣿⣯⠸⢿⣿⠀⠀⠀⠀⣿⠀⣿⡏⠀⠀⠀⢠⣿⣿⠀⢻⣿⡇
⠀⠀⢸⣿⡆⠘⣿⡿⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⢻⠿⢿⡏⠀⠀⠀⠀⠀⣿⣿⠰⣿⠇⠀⠀⠀⠀⣿⡀⣻⡇⠀⠀⠀⠸⠋⠁⠀⢸⠛⠇
⠀⠀⢸⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⢀⣄⣿⠁⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⢹⣀⠀⣿⡄⠀⠀⠀⠀⣿⡁⣹⡇⠀⠀⠀⢠⡆⡄⠀⢸⣀
⠀⠀⠘⣿⠇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⣏⠐⣋⡀⠀⠀⠀⠀⠿⠃⣾⡇⠀⠀⢠⣬⣿⡀⠀⢈⣿⡃
⠀⠀⠊⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠛⠁⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠈⠉⠁⠀⠸⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⠫⣼⣷⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣌⠛⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⡉⠁⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠇⠀⠀⠀⠀⠀⠀⣤⠀⠔⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠠⠤⠚⢶⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠻⢆⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠊⠀⠤⠄⠙⠋⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⣈⠁⠀⠀⠀⠀⠀⠀⠐⠉⠈⠑⢠⠀⣨⡇⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠒⠞⠊⢂⠂⠟⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴⠇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠘⠁⠀⠀⢀⣸⠃⠀⠠⣇⠀⣦⣄⡀⠀⠀⣸⠛⠳⠶⠰⡀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠤⠏⡶⠈⠁⢰⠃⠀⢻⣤⢴⠆⢱⠀⠀⠁⠈⠉⠑⡂⡇⣠⠀⠀⠀⠈⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠃⠀⠀⠁⠀⠀⠸⠀⢠⠀⠀⠀⠀⢘⠀⠦⠀⠀⢀⣠⣟⠋⠁⡀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠰⠆⢸⠀⠀⠀⢰⠎⠠⢒⠖⡴⠄⠀⠈⠀⠀⠁⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠒⢖⠂⠁⠘⠈⣄⠧⢀⠀⠎⠐⠐⠛⢦⡅⠰⠂⠀⠁⠀⠀⠀⠀⠀⢀⠀⡂⢠⠤⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣵⡚⢲⣤⣭⡀⢐⣊⠉⠤⠬⠄⠀⠀⢠⣄⠀⠘⠲⣆⠀⠀⡇⠀⢀⠀⠀⢨⢤⣈⣻⡀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠙⢉⠀⠀⠀⠸⠀⠀⠀⠀⢰⠀⠀⠀⠈⡄⠀⠀⠀⡱⢄⠁⠆⡞⠒⠐⠀⠶⠂⠀⠛⠣⣤⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠈⠺⣦⠠⣀⠠⠔⠇⢀⠀⠜⠀⠀⠀⠐⢄⠸⠐⠈⠀⠀⢀⠀⠶⠁⠠⠀⠀⠇⠀⠀⠸⢦⡟⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢻⡄⠀⠈⠳⣘⢄⠀⠀⠄⣀⡤⠄⠀⠀⠀⠀⠛⢃⡠⠄⠒⠀⠀⠐⠠⣤⣀⣀⠀⠀⠀⠀⢨⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⠲⡆⠀⡽⢦⠳⡀⠈⠀⠀⠀⠀⡀⠄⠀⠈⠀⠀⠀⢀⠠⠄⢂⣡⠴⠃⠈⢱⠀⠀⢀⣈⣤⣶⣐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠑⠜⢦⠀⠀⠈⠀⠀⢀⡀⠤⠄⣒⣨⠴⠖⠛⠩⢀⠀⢠⢄⣀⠀⠀⢸⡼⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠤⠤⠄⡄⠀⣄⣀⣀⣈⣢⡑⡄⠤⠒⣈⣡⠴⠖⠋⠉⠀⠀⠀⠀⢸⠀⠀⡀⠀⠈⠓⠤⣇⡇⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⠀⠀⠀⠀⠀⠛⡷⡞⠉⠉⠉⠈⠉⠁⠀⠀⠀⠀⢺⠀⢰⠀⠋⡗⠂⠤⣀⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⣸⠶⡀⡇⠛⠒⢠⠸⠀⣀⣀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⠒⠁⣇⠀⠀⠸⡾⢼⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠀⠀⠉⠁⠀⢸⣇⡞⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣻⡀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢀⣄⣀⣀⣀⣀⣀⣀⣀⣨⢼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣾⣿⣶⣶⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣬⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⣿⣎⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⣀⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣄⠀⠀⠀⠀⠀⠀⣤⣦⣴⣾⣆⣠⣤⡄⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⢠⣶⡿⠿⣿⣿⣿⣿⣧⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⣠⣾⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⣿⡿⢀⠀⠀⠘⢿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⠟⢻⣿⣿⣿
⠀⠀⠀⠀⠀⣼⠡⠀⣤⠀⠀⠀⣻⡿⡷⠀⠀⣿⡿⠫⠄⠀⠉⠩⠍⢹⣿⡇⢸⡟⠿⠛⠁⠀⠀⠙⠿⣿
⠀⠀⠀⠀⠀⠨⡄⠈⠋⠀⠀⠈⠛⢰⡄⠀⠀⠿⡇⠀⣴⠀⠀⢀⣶⢈⠟⠀⠛⠇⢿⡄⠀⠀⣾⡄⠁⠟
⠀⠀⠀⠀⠀⠀⣡⡀⠀⠁⠀⠀⢀⣌⠀⠀⠀⠀⡃⠀⠁⠀⠀⡀⠁⠘⠀⢀⣼⣧⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⢠⣿⠇⡰⠀⠀⠆⡙⣿⡄⠀⠀⠀⠀⠀⠀⠑⠛⠁⠀⠀⠀⠈⠛⠿⠦⢄⡀⠀⣀⣴⡿⠿⠆
⠀⠀⠘⠛⠺⠛⠉⡁⠀⠀⠀⠀⠀⠉⠛⠟⠛⠋⣠⣤⠘⠀⠈⠃⣤⠀⠈⠠⠀⠀⠂⠀⠀⠀⠁⠐⠂
⠀⠀⠀⠀⠠⠤⢄⡡⠄⠀⠀⠀⠀⠐⠄⠂⠀⠐⢸⡏⠀⠀⠀⠀⠹⡆⠀⠐⡀⣀⢘⠀⠀⠀⠀⡐⢀⢀⠂
⠀⠀⠀⠀⠀⠀⡆⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⢁⡟⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⡍⠀⠀⠀⠀⠀⡀⢩⠁⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⣿⣷⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠩⠛⡛⠷⠾⠛⠛⠉⠃⠀⠀⠀⢿⣿⣿⠋⢹⣿⣿⡿⠀⠀⠀⢛⠻⠶⠶⠶⠶⠾⢛⠃
⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠂⠃⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠂⠤⠚⠀⠀⠀⠀⠑⠤⠐⠀⠀⠐⣠⠔⠀⠀⠀⠀⠦⡄⠂⠀⠐⠠⠔⠁⠀⠀⠀⠓⠤⠐

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣿⣿⣿

⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣄⠀⠀⠀⠀⢴⣷⣦⡀⠀⠀⠀⢀⣤⢀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀
⠀⠀⢠⣿⣿⣟⠀⠀⠀⠀⠀⢀⠛⠛⠂⠀⠀⠀⣟⣿⣿⣿⠂⠀⢀⣶⣿⣿⣯⠁⠀⠀⠀⢺⣿⢿⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⡇
⠀⣠⣾⣿⣿⣿⣶⡄⠀⠀⣤⣤⣼⣿⣷⣦⣴⣾⣿⣿⣿⣿⣆⠀⣜⢿⣿⣿⢿⣧⡄⠀⢠⣿⣿⣷⣶⣀⠀⠀⠠⠚⠛⠛⣻⣿⡇
⢠⣿⠩⠛⠻⣿⣿⣿⡀⢀⣏⢹⣿⣿⠿⢹⣿⣿⣿⣿⣿⣿⣟⢰⣿⣾⣿⣿⣿⣿⣿⣀⣾⣿⣿⣿⣇⠹⡆⠀⢠⡎⠳⣾⣾⣿⣇
⣿⣿⡄⠈⠤⢼⣿⣿⡇⣸⡿⢸⣿⣿⣇⣺⣿⢻⣿⣿⣿⣿⣿⢼⣿⣿⠷⠚⠹⣿⣿⣧⣿⣿⣿⣿⣿⣾⡧⠀⠈⣧⢠⣿⣿⣿⣿
⢻⣿⠋⠍⢅⢸⣿⡇⠀⣽⢯⡩⢽⣿⣟⡟⢿⣿⣿⣿⣿⣿⠁⠈⢙⡏⠙⠉⠐⣿⡍⠸⣿⣿⣿⣿⣿⣇⠀⠐⡾⢻⣿⣿⣿⣿⣿
⠈⢿⣷⣿⣿⣿⣿⠁⠀⠓⠀⠻⠟⠛⠋⠁⢸⣿⣿⣿⣿⣿⠀⠀⢸⡆⠀⠀⠀⢿⣿⠀⢸⣿⣿⣿⢿⡏⠀⠀⠁⠀⠻⣿⣿⡿⠟
⠀⠀⣿⣿⣿⢿⣿⠀⠀⠀⣀⣀⣀⣀⣀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠸⣦⣦⣄⣀⣼⣿⠀⠀⣿⣿⣿⣿⠁⠀⠀⠐⠀⠀⠬⠍⣄
⠀⠀⢿⣿⠉⣿⣿⠀⠀⠀⢹⣿⣿⣿⡏⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⣾⣿⠟⣷⡞⠀⠀⠀⣸⡗⣿⡿⠀⠀⠀⢻⣶⡖⣶⣶⡆
⠀⠀⣿⣇⠀⣿⣿⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢺⣿⠀⣿⡏⠀⠀⠀⢹⡇⢸⡇⠀⠀⠀⣿⠿⠀⢸⣿⡇
⠀⠀⣿⣿⠀⣻⡟⠀⠀⠀⠀⠛⠙⣿⠇⠀⠀⠀⠉⠉⡋⠀⠀⠀⠀⠘⠟⠁⣿⠁⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣡⢀⠀⢸⠀⠁
⠀⠀⢹⣿⠀⠙⠓⠀⠀⠀⠀⠐⠉⢉⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⣿⠃⣟⠁⠀⠀⠀⠸⠇⣼⡇⠀⠀⣀⣻⡆⠀⠘⣷⠂
⠀⠀⠚⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠙⠁⠛⠃⠀⠀⠀⠀⠀⠙⠃⠀⠀⠈⠙⠃⠀⠸⠿

⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠀⠀⣰⣾⠤⠀⠀⠀⠀⡠⠄
⠀⠈⢄⠀⢀⣀⢀⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⣀⠔⠉
⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⢀⠀⠀⠡⣀⠀⠀⠀⠔⠀⠀⠠⢂⠴⠊⠀⠀⠀⣠⠆
⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⡠⠈⠀⢀⠀⠀⠉⠈⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠰⠛⠂⠀⢴⣢⣄⠀⠀⠉⠁⠀⠈⠀⠀⠀⠙⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠁⠀⠀⠀⠈⢪⠘⣆⠀⠀⠀⠀⣦⠀⠀⠀⠠⠀⠀⢀⠔⠉⠀⠀⠀⠞⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣺⠟⠓⠲⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⡀
⠀⠀⠀⠐⠤⣤⠠⠔⠀⠀⠑⠤⠀⠀⢠⠏⠓⣾⠇⠀⢇⠀⠀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⡞⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢜⠁⠀⣠⠞⠀⠀⠀⠀⠀⡰⠀⠐⠂⠀⠀⠀⠀⠀⠃⠀⢠⡞⠀⠀⠀⠀⢐
⠀⠀⠀⡯⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠊⠀⠀⢠⣼⡦⠀⠀⢃⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⢤⡦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠈⠲⢄⡀⠈⠀⠀⠀⣀
⠀⢸⡀⠀⠀⠀⠀⣠⣾⡀⣀⣀⠀⠀⠀⠀⠈⠁⠘⠉⠉⠀⠀⠀⠀⠴⠠⢰⠄⠀⠀⠀⡠⠀⢀⡴⠶⠀⠀⠒⠁⠀⢠⠀⡄
⠀⠀⠳⢤⠀⠐⠛⠛⠻⠻⣛⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⢀⡀⠀⠀⡐⠂⠀⢀⣠⠤⢤⠀⢀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠣⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡈⠁⠀⢀⣿⠇⠰⣿⣿⡆⠃⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣶⣤⠄⡘⠁⠀⠀⠈⡉⣄⣀⠉⠩⣠⡤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠹⢿⣿⠂⡴⠿⠁⠀⠀⠀⡇
⠀⠀⠁⠀⣀⠀⢀⠀⠀⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⡀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠚⠁⠀⠀⠀⠀⠠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣿⣿⣷⣶⣶⣶⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢀⣴⡾⠛⣋⣙⣛⡻⢿⣦⡀
⠀⠀⠀⣼⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⡆⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣼⣿⣴⣿⣿⡿⣿⣿⣦⣻⣷
⠀⠀⠀⣿⣿⣿⡏⠀⠀⠉⠻⢿⡿⣿⡇⠀⠀⣿⣿⢟⣁⠉⠈⠙⢋⡉⠹⣿⣿⡄⠀⢟⣿⣿⣿⠋⠀⠘⢿⣿⣿⣿
⠀⠀⠀⡜⡏⠆⢀⣦⠀⠀⠀⣰⣶⢘⠷⠀⢀⣿⡇⠀⢀⡄⠀⠀⠀⣠⡀⣿⣯⠀⢠⠘⢿⣯⠀⠀⠀⢀⠉⠹⣿⣇⠀⠐⣶⣦
⠀⠀⠀⠀⣡⠀⠘⠛⠀⠀⠀⠛⠃⣨⡔⠀⠀⠀⡃⠀⠚⠋⠀⠀⠘⠛⠁⢇⠆⠀⠈⢰⡀⠛⠃⠀⠀⠛⠇⠀⣘⠉⠀⠙⠄⠁
⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⣷⡤⠀⠀⠐⠀⠀⢠⣀⣠⠄⠀⡀⠀⠀⣀⠀⣿⣇⠀⠀⢤⣤⠆⠀⢀⣿
⠀⠀⠀⠛⠿⠿⣶⢤⠀⡀⡤⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⣀⠈⢁⠀⠀⠀⠀⡤⠈⡠⢙⠛⠳⢤⡀⢀⣠⡴⠿⠿⢗
⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠁⠃⠒⠀⠀⠀⠀⠀⢠⣴⠆⠑⠀⠀⠁⣶⡁⠀⠀⠸⠁⠀⠸⠂⠁⠀⠂⠑⠀⠂⠀⠀⠄
⠀⠀⠐⠁⠀⡠⠂⠀⠀⠀⠀⠀⢂⡀⠀⠦⡀⡀⢠⡟⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠓⠛⢀⠉⠀⠀⠀⠀⠉⠰⠃⠚⠂⡰⠁⣾⠃⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢙⠌⠉⠀⠀⠀⠀⠩⠁⠍⠀⠀⠀⡘⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢄⠨
⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⣀⣀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣤⣀⠀⠀⠀⠀⢀⣀⣸
⠀⠀⠀⠀⠿⠷⣶⣦⣤⣤⣴⣶⠾⢿⠀⠀⠀⠀⢿⣿⣿⡿⠁⢿⣿⣿⣿⠇⠀⠀⠜⠈⠙⠛⠛⠛⠛⡛⠛⠍
⠀⠀⠀⠀⠈⠁⢐⠂⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠨⠀⠉⠀⠀⠀⠉⠉⠠⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⢰⠠⡄⠀⠀⠀⠀⡄⠠⠀⠀⠀⠐⠂⠠⡁⠈⢆⠀⡄⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠤⠉⠀⡆⠄
⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠘⠠⠔⠋⠀⠀⠀⠀⠙⠦⢔⠃⠀⠀⠋⠰⠚⠁⠀⠀⠀⠀⠙⠲⠨⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡠⠖⠋⠉⠈⠉⠒⢄⠀⠀⠀⣠⠔⠋⠉⠉⠉⠒⢤⠀⠀⠀⢀⠔⠚⠉⠁⠉⠓⢤⡀⠀⠀⢠⠖⠉⠒⠢⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⡰⠁⠘⢡⡄⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠑⢦
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡇⠴⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢣⡀⢿⠧⠀⠀⠀⠀⠀⡼⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠑⢄⠀⠀⠂⠀⠀⣀⠇
⠀⠀⠀⠑⠤⣀⣀⣀⣀⡤⠊⠀⠀⠀⠙⠦⣀⣀⣀⣀⡠⠚⠁⠀⠀⠈⠢⣀⣀⣀⣀⡠⠞⠁⠀⠀⠀⠀⠀⠙⠢⠤⠠⠔⠁
⠀⠀⢀⡀⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡤⠄⣀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣀
⠀⠀⡇⠀⠀⣠⠄⠉⠲⢄⠀⠀⠀⢀⣴⠁⢀⣴⣶⣄⠀⢱⣄⠀⢀⡔⠋⠀⠀⠀⠀⠀⠙⢆⠀⠀⡴⠋⠀⠀⠀⠀⡈⠙⢆
⠀⠀⢧⠀⠀⠀⠑⢕⠄⠀⠱⡀⠀⣾⠃⠀⠈⠿⠿⠏⠀⠀⠹⡄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠁⠀⠄⠀⢠⠀⡀⠂⠈⡆
⠀⠀⠈⢆⠀⠀⠀⠀⠈⡦⠀⢱⠀⠇⢠⣤⣄⠀⠀⠀⣠⣤⢀⠇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠐⠑⠁⠀⠇⠈⠌⠀⢂⡇
⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⢘⠀⠈⢾⣿⣿⠆⠀⠐⢿⣿⠏⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠳⡀⠀⠀⠀⡐⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠀⠀⠀⠉⠒⠢⠤⠔⠊⠁⠀⠀⠀⠀⠉⠒⠶⠤⠔⠚⠁⠀⠀⠀⠀⠉⠓⠲⠦⠖⠚⠁
⠀⠀⠀⢀⠴⠂⠉⠉⠑⠢⣄⠀⠀⠀⢀⡤⢆⣉⠉⠓⠢⣄⠀⠀⠀⢀⡴⠒⠻⠉⠓⠲⢄⠀⠀⠀⠀⠀⠀⡐⠳⢤
⠀⠀⡰⣡⡶⡻⣶⡄⠀⠀⠈⢣⠀⠰⠃⠀⠀⠀⠀⠀⠀⠈⢣⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢰⠄⠀⠀⠙⢄⡀
⠀⠀⡃⣿⠰⢡⣾⡇⠀⠀⠀⠈⠃⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⣿⣦⡀⠀⠀⠀⠑⣄
⠀⠀⢣⠈⠛⠟⠋⠀⠀⠀⠀⣰⠁⢣⠀⠀⢀⠀⠁⠀⠀⠀⣰⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⢴⣿⣿⣿⣿⣦⣄⠤⠔⠊
⠀⠀⠀⠳⢄⡀⠀⠀⠀⣀⠔⠁⠀⠀⠑⢄⡁⠀⠀⣀⣀⠔⠁⠀⠀⠑⢤⣀⠀⠀⠀⣠⠔⠁⠀⢀⠀⠙⡿⠛⠉
⠀⠀⠀⠀⠀⣉⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠁⠈⢀⣀⣀⣀⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠢⡀⠀⠀⡴⠋⠀⠘⣄⠀⠈⠲⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠓⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠑⣄
⠀⠀⡞⢰⡷⠀⠀⠀⠐⣷⠆⢹⠀⡜⠀⠀⠀⠀⠈⠱⢄⡀⢘⡄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡗⠒⠠⣀⠀⠀⠀⠀⠈⠉⠇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⣄⠀⠀⣠⣀⠀⠀⢀⠎⠀⠘⣄⠀⠈⠳⡄⠀⠀⢀⠜⠀⠘⢄⠀⠀⠀⠀⠀⠀⢀⡜⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠈⠑⠶⠬⠥⠤⠚⠁⠀⠀⠀⠈⠑⠢⠄⠼⠤⠒⠁⠀⠀⠀⠀⠑⠢⠤⠤⠤⠒⠉⠀⠀⠀⠀⠑⠢⠤⠀⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢐⢃⠈⠉⠁⠙⠀⠀⠀⠀⠀⠼⡶
⠀⠀⠀⠀⠀⠔⣿⣷⣾⣿⡟⠃⠀⠀⠀⠀⠀⠀⢿⣿⡿⠖
⠀⠀⠀⠀⢄⣼⣿⡽⡥⡄⢿⣆⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠤⠀⠀⠀⠀⠀⣿⣿⣤⡆
⠀⠀⠀⠀⢸⣻⣿⣿⣛⣷⡿⣿⠀⢳⠀⠀⢨⡤⣀⣀⣀⣾⠃⠀⠀⠀⠀⢸⣿⣿⣧⡗⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⣴⠳⢦⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣏⣿⣱⣿⣇⣼⢢⠆⢀⠛⡟⡯⠝⢁⠀⠀⡄⣠⢄⠙⠻⠋⠉⠀⠀⣠⡰⣿⣿⣿⣿⣷⡄⠀⠀⠀⢿⡖⣼⡄
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣾⣿⡿⣿⣠⢀⣿⣰⢺⠱⡖⣼⡅⢸⠁⢸⠀⢩⣶⣴⣶⣧⣸⣿⣿⣌⢻⣿⣿⣷⣧⡆⠀⢸⣞⠟⠛⠳⢄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⢻⡇⣸⣷⣿⠯⠍⠁⠡⣺⣼⡀⢸⣤⣿⣋⠀⠻⠛⣿⣿⣿⡿⠀⠀⠙⣿⣿⣇⠐⣳⢀⢠⣴⠃⠀⣀
⠀⠀⠀⠀⢸⡿⣿⣿⠿⠿⠿⠁⠀⢸⡇⣿⣿⢒⠓⠇⡤⢧⠼⡇⢠⠻⣿⡿⠦⠄⠀⠀⢹⣿⣿⠋⠀⠀⠀⢹⣿⣿⠀⣷⢾⠛⠁⠀⢀⣩⠁
⠀⠀⠀⠀⠸⣧⣤⣴⣶⣶⣶⣦⣤⣬⠓⣿⣿⢹⢘⣃⡋⢹⢼⠀⠈⠘⣿⡇⠀⠂⠀⠀⠘⢻⠋⣠⣄⠀⢰⣈⣿⣿⡄⣡⡿⠀⠀⠀⢰⡃
⠀⠀⠀⠀⠀⠸⣿⢽⡿⣿⣯⣿⢿⣿⠀⢹⣿⣶⣾⣤⢵⣾⡂⠀⠀⠀⢻⣧⣤⣤⣤⡴⢠⣿⠀⢀⠀⠀⠠⠀⣿⣿⡷⠩⠄⠀⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣽⣿⣽⣿⣿⢻⠀⠘⢿⣿⣿⣟⡛⠉⠀⠀⠀⢠⣤⣬⣉⣉⣀⣤⣼⣿⣶⣍⠁⠀⢀⣠⣿⣿⣃⠠⣤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⢺⣥⣿⢻⢿⣿⠟⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⠀⢸⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⠿⣾⣧⣤⣦⣀⣀⡀⠁
⠀⠀⠀⠀⠀⠀⢸⡿⠅⢸⠈⣿⣆⡻⠀⠀⠘⣿⣿⣿⣿⣿⣿⡛⠀⢸⣿⣿⡏⣿⣿⡟⠸⢿⣿⣿⣿⣿⣿⠿⠇⠀⠉⠋⠅⠝⢣⠄
⠀⠀⠀⠀⠀⠀⠀⣿⣶⣺⠀⣻⣝⠃⠁⠀⠀⣿⣿⣿⠹⣿⣿⠇⠀⠈⣿⣿⣧⣿⣿⡇⠀⠀⣻⣿⣽⣛⠛⠀⠀⠀⠀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢿⠸⠀⡆⠸⠆⠐⠀⠀⠀⣺⣿⡟⠀⣿⣿⠄⠀⠀⣿⣿⠃⣿⣿⠁⠀⠀⠸⠟⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠄⠀⠀⢧⠀⠀⠀⠀⢺⣿⣿⠀⣿⣿⡯⠀⠀⢿⣿⠀⣿⣿⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣸⠰⣿⠃⠀⠀⠀⢸⣿⣯⠀⣿⣿⡁⠀⠀⢸⣿⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠈⠀⡏⠃⠀⠀⠀⢸⣿⣿⡅⣿⣿⡏⠀⠀⠘⡟⢃⣿⡇⠀⠀⠀⢀⡾⠆⠀⠀⠀⠀⠀⠀⢀⡄⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⢲⠾⠄⠀⠀⠐⢿⠿⠇⢸⣿⡕⠀⠀⢰⣷⠸⡿⠁⠀⠀⠀⢸⣷⠆⣾⠄⠀⠀⠀⠀⠠⠁⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠄⠘⠿⠂⠀⠀⣰⣿⡇⠀⠉⠉⠀⠀⠀⠨⠀⠈⠧⠀⠀⠀⠀⠹⠿⠘⠧⠀⠀⠀⠀⠀⠀⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣾⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⢀⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⣿⣿⣿⣿⠙⢿⣿⣿⣿⢿⣿⡿⠀⣸⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢀⡀⠀⠀⠀⣹⡏⠀⠛⡛⠛⠋⠀⠈⠛⠛⠏⢸⣿⠇⠀⣿⣿⡇⠀⠀⠀⠉⠉⠉⠁⠀⠀⠹⣿⣿
⠀⠀⡅⢈⠀⠀⠘⠻⡇⢸⣿⡌⠂⠀⠀⢠⣦⠀⠀⠀⢿⣶⠀⣹⣿⠇⠐⠛⠃⠀⠀⠀⠐⠒⠆⢠⣿⠏⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠘⠤⠅⠈⠛⠃⠀⠀⠀⠘⠻⠃⠀⢀⡃⡘⠀⠛⣿⢰⣿⣄⠀⠀⠀⢠⣦⠉⠀⠘⣿⣠⠀⢀⡘⠿⢿⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⡀⡀⢀⣀⠀⠀⠀⠘⡉⠀⠀⠣⠼⠀⠙⠋⠀⠀⠀⠘⠻⠃⠀⢘⠁⡈⠀⠀⠑⠠
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠡⢀⠀⠈⠀⠀⠀⠀⢀⠀⠇⠠⠀⠀⠀⠀⠄⠀⠀⣄⡀⢀⣀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⡨⠂⠐⠄⡠⠀⠀⡦⠀⠄⠤⠀⢁⠀⢀⠀⠂⠀⠀⠀⠘⠄⡀⠈⠛⠊⠁⠀⡀⠄⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠌⣽⠉⠑⠠⠐⠈⠐⢠⢤⡄⠢⣀⠔⠀⠢⣤⢂⣈⡖⠀⠀⠲⠈⠀⠀⢀⢀⡠⠈⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠁⡚⠠⠁⠀⠀⠀⠀⠀⡁⢨⠀⠈⠑⡷⠄⡀⠀⠨⢡⠉⠀⠀⠒⠝⣋⠓⠉⠀⠀⢡⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠠⢁⠂⠀⠀⠀⠀⠀⠀⡀⡈⠢⣀⢀⠉⠀⡀⠹⠀⠁⠀⠀⠀⠀⠀⠱⡄⢠
⠀⠀⠀⠀⠀⠀⠀⡄⠌⠀⠀⠀⠀⠀⠀⢰⠀⢐⡂⠋⢉⣁⡀⢤⠀⠄⣀⠀⠀⠀⠀⠀⠀⢀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡙⢄⠃⠀⠀⠀⡀⠬⠤⠀⢀⣀⣀⣀⣀⣀⣈⡀⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡇⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠠⠤⠄⠀⠠⠤⠤⠀⠲⠋⠀⠀⠀⠥⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡐⢆⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⢢⠄⠀⡀⠠⠀⠀⠆⣀⣀⠀⡠⠃⠀⠠⠁⠢⣀⢀⠔⠁⠀⢂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠁⠀⡠⠁⠀⠀⠀⠀⠀⠑⠂⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠰⠀⣰⠀⠀⠀⠀⠀⠀⣄⢠⡄⠀⠈⠀⣐⡙⠀⠤⠤⠠⡄⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢨⠉⠁⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠈⠈⠈⠣⡀⠀⢻⠀⠈⠅⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠠⡀⠀⠀⠀⠀⠀⠀⡁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠨⠀⢸⢀⠄
⠀⠀⠀⠀⠀⢀⠊⠀⠀⣀⠀⠀⠀⠀⠀⢐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢂
⠀⠀⠀⠀⠀⠎⣑⠴⠚⠁⠀⠀⠀⠀⠀⠈⠳⣤⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊
⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣯⣭⣽⣽⣦⣭⣯⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠿⠛⠛⠛⠛⠿⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⢶⣦⣄⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⢸⡟⠀⠀⠀⠀⣠⣾⣱⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⢸⡇⠀⢀⣠⣾⣿⣿⠟⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠘
⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⢸⣷⢸⣧⣶⠟⠋⣸⡿⠋⠀⠀⠀⠉⠛⠿⢶⣄⡀⠀⠀⠀⠀⣿⣷⡀
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠉⠀⠈⠙⣷⣸⡟⠈⠉⣿⡀⠀⢫⣶⠾⠛⠃⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⢠⡿⣿⣿⡀⠀⠀⢠
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⢀⣀⡀⠀⠀⢸⣿⠁⠀⠀⢸⣇⠀⢾⡷⢶⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣴⠟⣡⡿⢹⡇⠀⠀⣼
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠙⠛⣛⠃⠀⣼⡇⠀⠀⠀⢸⡿⠀⠈⠿⠾⠛⠁⠀⠀⠀⠀⠀⠀⡾⢟⡋⢁⣾⠋⠀⢸⡇⢀⣼⠟
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢴⣿⣻⡏⠀⣿⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⣼⣷⠟⠃
⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠉⠉⠀⠀⢿⣄⠀⠀⠻⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⠄⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡍⠁⠀⠀⠀⠀⠈⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠈⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⢀⣠⣴⡶⠛⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠙⠻⣦⣤⣀⣀⠀⣀⣠⣾⠿⣦⣄⡀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣤⣤⣴⡶⠿⠛⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⣸⡏⠉⠛⠛⠛⠉⠀⠀⣿⠉⠛⠿⠶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⠐⡶⢆⣀⡀⠀⠀⠠⠜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠩⠭⡉⠉⠛⠿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿
⣿⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⠟⠛⠋⠓⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⣿⣿⣿⡇⣿⡾⣆⡟⡏⣽⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠐⠒⠀⠐⣾⣿⣿⣿⣿
⣿⠇⠤⠄⠀⠀⠀⠀⠀⠀⠀⠒⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠏⠉⠉⠉⠉⠁⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠔⠒⠂⠀⢸⣿⣿⣿⣿
⣿⢤⡄⠀⠀⠀⠠⠄⠀⠀⠀⠒⠋⠀⠀⠂⠀⠀⠀⠀⢠⡤⠀⠀⠐⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢀⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⡼⠤⠤⠤⠤⠤⠦⠤⠤⠤⠶⠦⠤⠀⠀⠀⠀⠠⢴⠾⠤⠤⠤⠤⠤⠡⠤⠠⠄⠑⠤⠤⠤⠤⠴⠉⠍⠍⠉⣿⣿⣿⡆⠀⠈⡁⠀⠀⠀⠀⣤⣹⣿⣿⣿
⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⡀⠀⠀⡄⠀⠀⠸⠉⠀⠈⠉⣹⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠀⠀⠀⠀⢰⣀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠀⠙⠢⢴⠁⠀⠀⡀⠀⠀⠀⢠⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⠻⠼⠇⠘⠘⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⣀⣉⠒⡄⠀⠀⠀⠀⣧⡀⠀⠀⣾⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣴⣿⠀⠏⢧⡄⡄⣄⣀⣘⡦⣸⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠈⠈⠀⠺⠉⣴⠃⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣷⣶⠂⠀⠀⠀⠂⢰⣶⣶⣶⣶⡂⠀⠀⠀⠀⢰⣶⠂⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⡆⠐⠀⠀⠀⠀⣶⣿⠀⠀⠀⠀⠈⠐⠈⠀⠀⠐⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣧⣤⣄⣠⣠⣼⣿⣦⣠⣄⣠⣠⣾⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⣿⣿⣶⣤⣤⣤⣤⣴⠀⠀⠀⠀⣀⣀⣤⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣴⣾⣷⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⢀⣤⣴⣶⣾⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⡟⠛⠿⠿⠿⠿⠟⠋⠁⠹⣿⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⡉⠉⠉⠙⠛⠋⠋⢹⣿⣿⣧⠈⣿⡇⣠⡤⠀⠀⠀⠀⣤⣀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⠁⠉⠉⠀⠀⠀⠈⠉⣁⠀⣿⣿⠋⡄⡿⠀⠀⢠⡀⠀⠀⠀⠀⣤⡀⢻⣤
⠀⠀⠀⠀⠀⢸⠛⡟⠀⠀⣼⡧⠀⠀⠀⣴⣿⠇⡟⢈⠀⠉⢲⠀⠀⠿⠟⠀⠀⠀⠸⠿⠃⠞⠠
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠒⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠲⠤⣤⠖⠀⠀⠀⠀⠀⠀⠈⢀⠣⡀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⡀⠀⡀⢀⠄⠀⠌⠀⡀⠀⠤⠠⠄⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⠄⠃⠀⠠⠒⢠⣴⠉⠀⠀⠐⠊⠀⠎⠆⠘⠁⠠⠄⠈⠂⠰⠇
⠀⠀⠀⠀⠀⠀⢀⠄⣽⣿⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⡠⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡦⡀
⠀⠀⠀⠀⠀⠀⢇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⠊⢀⣂⣴⣂⡠⢤⠀⠀⠀⠀⠠⠠⣰⡆⡴⠤⠈
⠀⠀⠀⠀⠀⠈⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⢤⠬⠇⠠⠀⠀⠀⠀⠠⠤⠃⡤⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⡌⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡸⠊⠀⠀⠀⠀⠀⠀⠀⠢⠐⠆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠰⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣶⠇⢸⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⡆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠉⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡟⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠠⡀⣀⣀⡀⠀⠰⣀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠠⠄⠴⠀⠀⠀⠀⠀⡆⢠⠀⠀⠀⠀⠀⠀⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢂⠀⠀⠀⠀⠀⢀⠂⠢⡀⠀⠀⠀⢀⠂⢂⠀⠀⠀⠀⠀⠀⠂⠐⡀
⠀⠀⠀⠀⠀⠀⢠⡁⠀⣠⠆⠀⠀⠀⠀⢠⡀⠀⢐⡀⠀⢀⠁⠀⣠⠄⠀⠀⠀⠀⠠⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠃⠌⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⠤⠃⠀⠀⠌⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⣀⣟⣉⣉⣏⣉⣙⣫⣉⣋⣙⣙⣛⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣭⣭⣽⣿⣭⣿⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⠛⠉⠉⠉⠉⠉⠙⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣴⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⣡⠿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⢽⣒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣾⣣⡼⠖⠋⠉⢑⡜⠂⠀⠀⠀⠉⠉⡿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢙⣲⣤⠤⠤⠤⣿⠄⠀⣀⣀⠞⠁⠀⠀⠀⢀⡀⠀⢧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠘⠄⠀⠁⠀⠀⠘⠉⠉⠁⠉⠁⠀⠀⠀⣠⣏⠏⠁⠀⣁⡤⣤⣄⠹⡄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⢛⡶⠋⠀⠀⣀⣀⣉⣁⣠⠼⠁⠀⣠⠀⠀⣦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠉⠒⠦⠤⠤⠤⠤⠴⠒⠋⠉⠀⠘⢉⡴⠚⠑⠺⣄⠀⠀⠈⠑⠢⠤⠴⢻⣀⣠⠿⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⣀⣀⣀⣀⣠⠤⠤⠒⠚⠙⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡖⠒⢜⣯⣍⣠⣟⠔⠊⣉⠆⠀⠀⠀⠀⠀⠀⠀⣠⣞
⠀⠀⠀⠀⢀⣴⣾⠏⠁⠉⠹⣿⣦⣄⠀⠀⠀⠀⠀⠀⠸⣉⣱⢇⡠⠤⣽⠏⠇⠠⢐⠄⠁⠀⠀⠀⠈⣦⡠⢾⡿⢿
⢠⠞⡑⠠⡟⠋⠑⣄⣀⣀⡠⠋⠙⠻⣆⠀⠀⣀⠤⡄⠀⢉⣒⣊⣀⣴⠃⠀⠢⢈⡁⠔⠁⠀⠀⠀⠀⣧⢠⠼⠀⠘⠓⡄
⡄⠀⠇⣶⡅⠀⢠⣿⣿⣿⣇⠀⠀⠀⣻⡀⢈⡟⢠⢞⣆⣸⡄⠀⠀⠈⠙⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢻⠈⢆⣠⠤⠚⠁
⠃⠀⠀⢻⡟⠀⠙⢿⣿⠿⠛⠒⢤⣴⣿⠁⠈⠻⡈⢧⠚⢇⣻⣄⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠚⠛⠦⡄⢡⣊⠙⢶⡄⠀⠀⢄
⠀⠀⠀⠈⢅⠀⠀⠀⡇⠀⠀⠀⢰⣿⠃⠀⠀⠸⢵⡌⠁⠛⠁⠈⠳⡀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⢺⠉⣇⠀⢣⠀⠈⡆⠱⠀⠀⠑
⠀⠀⠀⠀⠀⠑⠲⠾⢿⣶⡶⠖⠋⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠒⠊⠁⠀⠱⣄⠀⠀⠀⠀⢀⣴⣿⣁⡼⢾⡽⣇⢈⡿⠀⠁⠀⠄⠘⠃⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠤⠖⠉⠀⠀⠀⠀⠈⠫⣈⡼⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢠⠀⡄⢀⠀⡄⢀⠀⡄⠀⡀⠀⡀⠀⡀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣤⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣿⣿⡋⠀⠀⠀⣠⣿⣿⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⡇⢻⣿⢰⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠇⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢻⣿⣿⣿⡇⠀⠁⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⣿⡇⠀⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠈⢻⡿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⡆⣻⡃⡀⠀⢀⣸⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⠿⠿⠛⠀⠈⠹⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⡀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀⠀⢠⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡀⠀⠸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⣿⡀⠀⠀⠀⠀⠀⢈⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⢿⡿⣿⣿⣿⣿⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠶⠾⠟⣿⣿⣿⣿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠀⠀⠀⠀⠈⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣿⣿⠈⣿⡟⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡿⠀⢹⣿⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣄⠀⠀⠀⠀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠛⠃⠀⠈⠛⠚⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⣶⣦⣼⣿⣿⣤⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢥⣿⣿⣿⣿⣿⣿⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡯⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⣀⣤
⠀⠀⠀⠀⠀⠀⢀⣠⠴⣶⠶⢤⣄⠀⠀⠀⠀⠀⠀⢠⡏⢸⠟⣽⣵⠄⠀⠀⠀⠀⠀⠀⣠⠴⠶⡶⠶⣤⡀
⠀⠀⠀⠀⠀⢠⠎⠀⢠⡟⣆⠀⠈⢷⡀⠀⠀⠀⠀⣸⠷⠶⠶⠷⣿⡀⠀⠀⠀⠀⢀⡞⠁⠀⣰⠻⣄⠀⠹⡆⠀⠀⢀⡀⢶⠀⣇⢸⠆⡶
⠀⠀⣀⡀⠀⣿⠀⢀⡞⣀⢈⠳⣄⠘⡇⠀⠀⢀⡞⠁⠀⢠⢀⡀⠀⠹⣆⠀⠀⠀⢸⠷⠶⠞⣁⠀⠈⠳⠶⣿⠀⠀⢮⣹⡾⠛⠋⠙⠛⢷⣞⣉
⠀⠸⣧⣿⣀⢻⣤⠏⢠⣄⣈⣠⠈⣻⡇⠀⠀⢸⡁⠀⡎⠀⠀⠀⣠⠄⣿⠀⠀⠀⢸⡆⢰⣆⠁⠈⣁⡄⢀⡟⠀⠀⢳⡏⠀⠀⠄⠀⠄⠀⢹⡍⠀⠀⣴⠒⣦
⠀⠀⠀⠀⠉⢻⢿⣦⣄⣈⣉⣤⡖⠻⣿⠆⠀⠘⣇⠀⠙⠶⠶⠖⠁⢀⡟⠀⠀⣠⣾⠿⢦⣍⣉⣉⣁⡤⣾⣳⡄⠀⠘⣇⠀⠀⢤⣤⠄⠀⢸⣇⡤⠞⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠈⣧⣄⠀⠀⠀⠀⢘⣷⣤⣄⣀⣤⢴⣯⣀⠀⢻⠏⠁⢀⡴⡟⠉⠉⢹⣦⣈⠿⢿⣦⠀⠙⢦⣄⣀⣀⣀⣠⡞⠉
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠸⡏⠛⢶⣖⣶⠋⣹⠀⠀⠀⠀⠀⣿⠉⠛⣶⠛⡶⠋⢰⠇⠀⠀⠀⢻⡉⠳⣮⣃⣀⡴⠋⡏⠉⠉⠉⠈⢷
⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣀⣀⣀⣿⠀⠀⠉⠁⠀⡟⠀⠀⠀⠀⢀⣻⠀⠀⠙⠛⠁⠀⡼⠀⠀⠀⠀⠘⣧⠀⠀⢧⡼⠁⠀⣇⣀⣀⣀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢛⣚⣿⠛⠉⣿⣁⣀⣀⠀⠀⠀⠀⠷⢦⠶⠶⢶⡖⠛⠀⠀⠀⠀⠀⠀⠉⢉⡏⠁⠀⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠸⣧⣹⠇⠀⣯⡷⠀⠀⠀⠀⠀⢰⣋⣉⣹⠇⠀⢯⣥⣀⣽⠇⠀⠀⢠⡴⢾⡄⠀⢸⣗⣲⡄⠀⠀⠀⠀⠀⢾⣽⡇⠀⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢀⣀⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⣰⡛⠋⠈⠋⠙⠃⠘⠛⣧⠀⠀⠀⠀⣠⡴⠶⡶⠶⣤⡀⠀⠀⠀⠀⠀⣰⢶⣄⣶⡆⣠⡄⠀⠀⠀⠀⠀⣏⠙⣧⢼⡁⣹
⠀⠀⠀⠀⠀⢸⡉⢀⡴⠒⠛⠛⠛⠶⣌⠻⡦⠀⠀⡼⠁⠀⣰⢿⡀⠀⠙⣦⠀⠀⠰⣷⣾⠞⠛⠛⠿⣷⣷⡆⠀⠀⠀⢀⡽⠶⠟⠛⠳⣋⣷⣄
⠀⠀⢀⡀⠀⢰⣏⣏⣀⠀⠒⠰⠂⢀⡘⣿⡗⠀⢸⡧⠶⠊⡁⠀⠻⢦⣤⣼⡇⠀⢀⡟⠃⠀⠀⠀⠀⠀⠐⣦⠀⠀⢀⡾⠶⠶⠶⠾⠛⢿⣤⣹⡆⠀⠀⣴⠛⣷
⠀⢰⣏⣹⡀⠀⢸⡇⠉⣿⢲⡖⢺⠋⠀⣻⠀⠀⣸⣇⠀⢦⣀⣈⣁⠀⠀⣸⣁⡀⢸⡁⠀⠐⠂⠀⠃⠀⠀⢸⠇⣤⡴⣦⠀⠀⠟⠀⠛⠀⠈⢹⡇⣠⠞⠻⠶⠟
⠀⠀⠉⠉⠙⠳⢾⣧⡀⠙⠛⠛⠋⠀⣠⠏⠀⠾⠥⠟⠷⣤⣈⣉⣁⣠⠞⠿⠽⠟⠘⣧⠈⠓⠶⠶⠒⠛⢁⡞⠀⠈⠛⠛⣮⡙⠛⠒⠚⢋⣠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢨⡟⠶⠦⠤⠶⡾⢧⣄⣠⣀⣠⠴⣺⠋⠉⠉⠉⣿⢦⡀⠀⠀⠀⣨⣷⢦⣄⣀⣠⣴⣋⠀⠀⠀⢀⡤⢾⡟⠛⠛⠛⢿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢿⡀⠘⠷⠋⠀⢠⠇⠀⠀⠀⠀⢿⠀⠙⣶⢲⠞⠁⣿⠀⠀⠀⠀⢻⠙⠳⣞⣿⠉⢀⡞⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠳⡶⠶⠶⢶⡶⠼⠇⠀⠀⠀⢀⣏⣀⣀⣀⣀⣀⢸⡆⠀⠉⠉⠀⠀⠻⣶⠶⠶⣤⡼⠇⠀⠈⠁⠀⣼⣁⣀⣀⣀⣀⣀⣸⠆
⠀⠀⠀⠀⠀⣠⠖⠾⣇⠀⠀⢸⠗⠒⠶⡄⠀⢀⣠⣤⣿⠉⠉⢹⣏⡉⠀⠀⠀⠀⠀⢀⣠⣽⡀⠀⣸⣦⣀⠀⠀⠀⠀⠉⠉⣿⠉⠉⢹⡁
⠀⠀⠀⠀⠀⠻⠶⠶⠋⠀⠀⠘⠷⠤⠶⠃⠀⠘⠷⠶⠛⠀⠀⠺⣯⣽⠂⠀⠀⠀⠀⢿⡤⠶⠋⠈⠷⣤⡽⠇⠀⠀⠀⢰⣏⣹⠇⠰⣏⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⣠⣴⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣦⡀
⠀⠀⠀⠀⢰⣿⢹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠘⣿⣧⣿⣿⣿⡞⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⢻⢿⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠟⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣄
⠀⠀⠀⠀⣀⣤⣶⣷⣶⣤⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣬⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡅
⠀⢀⣄⡀⠘⢻⣿⣿⣿⠁⣠⣶⣅⡠⣶⡦⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡆
⠀⣾⣿⣿⡆⠉⠉⠁⢀⠉⣿⣿⣧⣮⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠈⢄⣺⣟⣛⣿⡁
⢸⣿⣿⣿⠒⠀⠐⠒⢈⠃⠸⣿⣿⣿⡟⢃⠀⢠⡀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⣠⣤⣞⡛⠿⠟⢻⢷⣤⣄⡀
⠛⠿⠟⢹⡁⠀⠀⠀⢀⣀⣀⠛⣿⣿⢹⡎⢦⡀⠀⠁⠀⢂⠀⠀⣜⣿⠁⠀⢸⣿⣿⣿⣿⣿⣷⣌⣙⡛⠋⣿⡄
⠀⠀⠴⣮⠁⢀⠀⠀⠀⠀⣀⢈⣿⣿⣿⠶⣎⠀⠰⢄⠀⠀⠀⠀⢹⣿⠀⠀⢸⣿⣿⣿⣿⣿⡴⠾⠏⠍⠄⢼⣿
⠆⠀⢀⣿⡍⠉⠉⠉⠁⠀⠀⣀⣿⣿⣎⣻⣾⡇⠀⢨⡁⠀⠠⠀⢸⣟⠀⠀⣼⣿⣿⣿⣏⡌⠀⣀⠀⡀⠀⣼⣿⡇
⠀⠀⢸⣽⠧⣄⠀⠀⠀⢀⣀⢈⣽⣿⣇⠙⠻⣿⡄⠈⢉⠀⠀⠄⠈⠏⣀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⣸⣯⣷⣼⣦⣤⣤⡬⡴⣿⡟⢻⣿⣆⠰⢟⠃⠀⠸⣦⠀⠀⠀⢹⣿⣿⡿⢻⣿⣿⣄⣀⣀⣀⡀⠀⣌⢻⣿⣿⡆
⠀⠀⣹⣿⣿⣿⣿⣿⣿⠞⣻⢿⡛⣻⣮⢿⠂⡲⠞⣇⠀⠘⠓⠀⡀⠈⠛⠋⠀⣿⣿⣿⣟⢍⡉⣉⠉⢠⣿⠈⣿⣿⣇
⠀⢈⡀⢿⣿⣿⣾⡿⣿⣞⣽⣿⣃⡛⢱⣴⣦⣽⡶⣿⣶⣦⣤⡶⠖⠁⠀⠀⠀⣿⣿⣿⣿⣷⣾⣿⣧⣾⣿⠀⢿⣿⣿
⠀⠈⠀⠸⣿⣿⣿⣿⣾⣿⣯⢽⣿⠋⠀⠺⣭⠚⠶⢀⠀⢿⠞⠂⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿
⠀⠀⠀⠀⠙⣿⣿⡟⢿⣿⣿⣟⣽⠀⠀⠀⢀⣹⡇⠀⠁⠈⢠⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣷⢻⣿⣾⣯⣶⠀⠀⠐⢪⣽⡆⠀⠄⠠⠓⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣿⣿⣻⡃⠀⠀⠀⠁⡶
⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⠿⣿⠃⠀⠀⠀⠢⣾⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⡇⢻⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣆⣿⣧⡞⠂⠀⠀⠀⢐⣾⡃⠀⡀⢈⡀⠀⠀⠀⠀⠀⠀⠀⢣⣿⣿⡇⢾⣿⣯
⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⡷⠀⠀⠀⠀⠈⠙⠀⠀⠁⠭⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠷⢾⣿⣧⡇⠀⠀⠀⠀⠀⠈⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣯⡅⣿⢿
⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⡃⢻⣯
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣩⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢨⡐⠂⠘⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣌⡆⠀⠀⠀⠀⠀⠀⣶⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡑⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣈⣻⣿⠿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠫⠉⢍⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⠍⠊⠀⠀⠀⠀⠈⠒⢍⡻⢿⣿⣿⣿⡿⠟⢋⠡⠐⠀⠀⠀⠀⠉⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄
⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦
⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠟⡁⠤⠈⢐⡀⠀⢀⡀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀⠄⠀⠒⡈⠈⠀⠙⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠟⡡⠊⠀⠀⠀⠀⠀⠁⠀⠒⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢩⠄⠒⠀⠈⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⢸⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⠁
⠀⠀⠀⠸⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠉⠉⠉⠉⠉⠉⠩⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠇
⠀⠀⠀⠀⢳⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡞
⠀⠀⠀⠀⠀⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠞
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⣤⣤⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⣤⣤⣶⣶⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⠬⠥⠤⠤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⢠⡞⡉⠛⠃⠈
⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣤⣶⣶⢶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⣠⠤⢤⣤⣴⣶⢾⣶⣶⣶⣶⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠸⠏⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠁⠀⠀⠀⠀⠉⠙⠆⠀⠀⠀⢈⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠗⣆⠀⠀⠀⠀⡞⠑⠊⢶⠀⠀⢀⡜⠒⠛⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣶⡿⠃⠀⠀⡗⠠⠀⠂⠁⠀⠀⠸⠀⠀⢈⡇⠰⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠚⠐⠘⠰⠂⠀⠀⠈⠀⢀⡀⠰⠀⠀⣼⣿⣿⣴⣸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⡅⠀⠀⠀⠀⢀⢀⠈⢀⡜⠀⠀⠈⠛⢻⠛⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⢿⣿⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⣿⠃⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠄⠀⠀⢨⡟⠀⠿⠀⠀⠀⠀⠀⠉⠶




⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⢸⣿⠘⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣶⢸⣿⡇
⠀⠀⠀⠀⠀⠘⣿⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣇⢸⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣼⡟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢨⡗⣿⣿
⠀⠀⠀⢀⡀⠀⠠⢻⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢸⡇⣿⣿⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠈⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢀⣀⢀⣀⣀⡀⠀⠀⡀⣠⣤⠿⠀⢻⡧⣿⣿⠤⠄⠐⠲⠶⠖⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠿⠋⠀⠀⠀⠊⠋⠀⠀⠀⠀⠀⠐⡶⣆⣠⣀⣀⣀⣸⣷⣿⣿⣠⣦⣴⢶⠶⡤⠿⠿⠿
⠀⠀⠀⠀⣰⣶⣶⣄⡈⢿⣿⠟⢀⡀⠀⠀⠀⠂⢘⠛⢚⣛⣛⣿⣏⣭⣭⣭⣭⣬⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⠶⣿⣾
⠀⠀⠀⢀⣻⣿⣿⣿⢿⣿⣷⣾⣿⣿⠟⠃⠘⠛⢻⣿⠛⠛⠛⠙⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⡇⠈⠁⠀⠀⠀⠀⠀⠀⢹
⠀⠀⣀⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣠⡆⠀⠀⢻⠄⠀⢰⣦⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣿⠃⠓⠟⠁⠀⠀⠀⠀⠀⢀
⠀⠀⣻⡀⢛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡇⠀⣄⠧⠡⠄⠀⠙⠀⠈⣁⣀⣀⣀⣀⣀⣀⣀⣛⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⣽⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠅⠸⠾⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠍⣭⠉⠁
⠀⠀⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⡤⠤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠚⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⠛⠿⢛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠈⢡⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⡀⠄⠐⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠠⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡐⠃⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠀⡀⢀⣠⣤⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠈⡇⠀⠀⡀⠀⢀⡄⠀⣠⠔⠀⠀⠀⠀⠐⡘⠀⠰⡄⠀⠀⡏⢀⠀⠀⠈⣦⠀⠱
⠀⠂⠀⠀⠀⢀⡠⠤⣤⡀⠀⠀⠀⠀⠀⣀⡤⠤⣦⡀⠀⠀⠂⡀⠠⠂⠀⠀⠁⠀⠀⠉⠀⠉⠀⠘⠁⠀⠀⠀⠀⠀⠀⠙⠀⠈⠃⠀⠀⠇⠀⠑⢠⠀⠑⢀⣖
⡄⠉⠷⢀⠂⠈⠀⢀⣀⠙⠆⠀⠀⠀⡸⢋⣀⠀⠀⠑⠺⠂⢰⠉⢀⠀⠀⠀⠉⠶⠀⢀⡶⠞⠚⠲⢦⠀⠀⠀⢀⡴⠞⠛⠶⣆⡀⠰⠆⠁⠀⠀⠀⠀⠈⠉⠉
⠃⠀⠘⠂⠀⠀⠀⣈⠉⠀⠁⠀⠀⠀⠀⠀⢉⣠⠀⠀⠀⢢⠀⠀⠀⠀⠀⠁⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠘⠀⠀⠴⠀⠀⠁
⠀⠑⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠀⠐
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠈⠢⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠯⠿⠯⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠂⠀⠀⠲⠐⠈⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⡠⠂⠀⠃⠤⠤⡄⠠⣤⠤⠃⠸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢂⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄⠢⠤⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⠅⠤⠼⠓⠒⠒⠊⠀⠀⠀⠁⠀⠀⠀⠀⠂⠐⠀⠀⠂⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢠⠁⠀⠀⠀⠀⡀⠤⠒⠤⠤⠤⠃⠀⠀⠀⠀⢠⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠒⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠹⠁⠀⠤⠘
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠁⠤⠠⠄⠀⠄⡀⢀⡀⠒⠀⢤⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣞⣂⣐⢂⡀⣐⠂⣀⡂⣛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠠⢀⣀⣀⣀⡀⠼⢀⣀⣀⣀⣀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⠄⠾
⠀⠀⠀⠀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠘⠂⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⡀⢀⡀⠀⠀⠀⡀⠔⢀⠀⠀⠀⠠⠄⠀⠀⠁⠀⠐⠂⠀⠀⠄⢠⠠⣄⣀⠀⠠⠦
⠀⠀⠀⠀⢸⣅⣤⡅⢀⡘⢠⣼⢡⣦⣇⣶⡀⣔⢲⣶⠒⡷⠢⡶⢶⣾⣘⣦⣦⡈⣾⣧⣼⣇⣀⣤⣶⣤
⠀⠀⠀⢰⣿⣿⣿⣃⣿⣿⣾⢿⣿⢻⣿⡟⣸⣿⣿⡿⢺⣿⣵⣦⣸⣿⣿⡿⠏⣿⣿⣿⣿⠿⢿⡟⠿⣿⣧
⠀⠀⠀⣼⣿⣿⢸⣿⣯⣿⠋⡘⢻⠓⠉⢱⠿⡇⡿⢳⣾⣷⣼⣧⡿⢻⠽⣿⠘⠛⢣⣯⣿⠀⣸⣷⣤⡟⣿⠂
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣫⢾⣶⡇⣾⣤⠁⣷⣾⠏⠈⣵⡏⠸⠛⠃⡌⠀⠀⠘⢿⣿⡤⣿⣇⣀⣿⠿
⠀⠀⠀⠀⠀⠀⣥⢈⣿⡿⠛⠃⠐⠿⠳⠋⠉⠀⠹⢧⡄⣷⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣾⠉⠹⣿
⠀⠀⠀⠀⠐⣰⣾⣿⣿⠀⠀⠀⠀⠰⠆⠀⠀⠀⠰⠀⠉⠁⢠⠂⢺⠆⠀⠀⣀⠀⠀⠀⡚⠿⣿⣷⣆⠻⣿⡄
⢀⡲⠀⠀⠀⠁⠀⠀⣶⣶⡶⢢⣴⡇⠀⡀⡀⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠙⠆⠀⣾⢃⡘⣷⣾⣿⣿⣦⣄
⠀⠀⠀⠀⠀⣠⠀⠈⠉⠙⠛⠈⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠻⠆⠀⠀⠀⠇⢀⡊⠻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⡶⠈⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠙⠀⠀⠈⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠔⠉⠑⢢⡄⡦⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢰⠶⠂⠉⠉⠀⠀⠁⠁⠒⢶⠀⠀⠀⠃⠀⢸⡎⠀⠀⠀⠀⠘⠁⠈⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠊⠏⢀⠀⡀⠀⠀⠀⠀⠀⠄⠈⠏⠂⠀⠀⢸⣆⠁⠀⠈⠀⠐⠂⠀⢀⣇⠀⠀⠃
⠀⠀⠢⡐⠀⠀⡀⠲⠄⠅⠀⠀⠀⡀⠠⠼⠀⠘⠀⢐⡠⢺⡇⠀⠈⠠⣄⣀⠆⠀⣸⡧⠁
⠀⠀⠀⠀⠀⠀⡷⠃⠀⠀⠐⠀⠐⠀⠀⠀⠷⡅⠀⣼⠀⡸⠛⠦⣄⠀⠍⠁⠀⣀⡏⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠑⠳⡄⠀⠀⢀⠀⠀⠀⣰⠓⠁⠰⢽⢈⠟⠑⢦⡸⡑⠒⠉⣯⢾⡁⣼⢠⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣋⠑⠶⠤⠄⢲⢋⡠⢗⠀⠀⠰⢏⣰⠀⠀⠙⠻⢄⡼⠁⠘⣇⢿⢚⠜⠱⡄
⠀⠀⠀⠀⠀⢀⡤⠊⠀⠙⠲⡀⢀⡦⠊⠀⢀⢳⠄⢱⡀⠹⠀⠀⠀⠀⠀⠉⠀⠀⠁⠈⡅⢀⣼⠁
⠀⠀⠀⠀⢰⠁⠢⢤⠀⠀⠀⠑⡉⠀⠀⠀⠸⢁⡄⠆⠜⢹⠀⠀⠀⠀⠈⠼⠀⠀⠀⠀⠛⠙⠈⢄
⠀⠀⠀⠀⢀⠑⠤⣼⠀⠀⠀⢰⡀⠀⠀⠀⢸⡋⠁⣄⡄⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⣠⠀⡂
⠀⠀⠀⠀⢀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢀⡏⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠼
⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠻⢄⣀⣀⣀⣀⣀⣀⣀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠒⡤⢤⡤⠤⣤⠤⠖⠊⠁⠀⠀⠀⠀⠀⠈⡁⢹⠉⠉⡏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣉⠦⢠⢏⢹⢄⠀⠀⠀⠀⠀⠀⠀⣠⢏⡹⠄⠠⠏⢹⢄
⠀⠀⠀⠀⠀⢠⠀⠈⠁⠑⢀⣻⣜⣂⠐⠉⠉⠀⡆⠀⢐⠁⠉⠄⠀⢰⡃⢘⡆⠐⠉⠉⠀⠂
⠀⠀⠀⠀⠀⠈⠙⠛⠒⠛⠒⠊⠀⠒⠛⠒⠖⠚⠋⠀⠘⠛⠒⠐⠛⠒⠀⠈⠒⠓⠒⠒⠚⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⠀⠀⠀⣀⢀⣤⣄⠀⣀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⢀⣤⣾⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⣀⣴⣴⣾⣿⣶⣦⣄⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⡿⠋⠿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢰⣿⡟⠛⠻⣿⣿⣿⣿⡆⠀⣤⣤⣶⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢀⣿⠋⢁⡀⠀⠈⢩⡹⣷⣿⡿⠥⠉⠉⠋⠍⢹⣿⡇⢀⣶⠀⠀⢸⡿⠁⠀⠀⠙⢿⣿⣿⠇⠀⣿⣿⠿⢿⣿⣿⣿⠿⣿⣷
⠀⠀⠀⠀⠘⣹⠀⠘⠃⠀⠀⠛⢸⣽⢿⠃⢠⡆⠀⠀⣴⢸⠿⠀⠈⣹⠀⠀⠾⡷⣶⠀⠀⢰⣎⠻⡿⠆⠀⢻⡟⠠⠆⠀⠀⠐⠀⢿⡟
⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⢀⣾⣿⠞⡀⠀⣀⠀⡀⠀⠘⠁⠀⠄⠏⠀⠀⠀⣇⠈⠀⡀⣀⠁⢀⡗⠁⠀⠙⡇⢿⠆⠀⠀⢷⠀⠸⠻
⠀⠀⠀⠀⠈⠉⣹⢻⠖⠐⠾⢿⡁⠀⠀⠐⢠⣈⠉⣁⠐⠁⢀⠀⠀⠀⠄⢢⡀⣿⣦⣀⠉⢁⣠⣾⡇⠀⠀⠀⠡⠀⠀⡀⣠⠀⠀⠊
⠀⠀⠀⠀⢰⠁⢠⠎⠀⠀⠀⠠⡀⠑⢄⢶⡇⠑⠀⠉⢾⡁⡀⠜⠁⠀⠀⠺⢀⠈⠅⠋⠀⠈⠓⣏⡇⠀⠀⠀⠀⠁⢠⡄⠀⡄⠊
⠀⠀⠀⠠⠘⢛⣳⣉⠀⠀⢰⣁⣟⣚⣇⣿⠁⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠟⠄⠀⠀⠀⠀⡀⡙⡀⠀⡀⠐⢣⠉⠓⠒⠛⠚⠂⢀
⠀⠀⠀⠀⠀⢠⡢⠀⠀⠀⠀⠡⢸⡄⢸⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢈⡷⢤⠀⠅⡄⠀⠀⠀⠀⠠⢑⣤⣼⠠
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠅⡀⠀⠀⣞⠀⠀⠀⠀⢡⡼⡶⠐⠊
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⢸⣐⠸⣤⣤⣀⣀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠠⣄⣄⠀⠀⡅⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⠀⢀⣠⡋⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠧⠠⠀⠀⠀⠀⠄⢰
⠀⠀⠀⠀⠀⠘⡟⢿⠻⠿⢛⠟⠟⠁⢸⣿⣿⡿⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣦⣤⣀⣀⣀⣠⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣇⡄⠀⠀⠀⣄⡇⠀⠀⠉⠙⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⠉⠏⠛⠉⠟⠙⠁⠀⠀⠀⡀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⣁⠁⠀⠀⠀⢁⡁⠀⠀⠖⠇⠀⠀⠀⡖⠆⠀⠀⠀⠀⠀⠀⠀⠰⠴⠀⠀⠀⢰⠴⠀⠀⠀⠀⢸⠀⠒⠁⠀⠒⠀⠃
⠀⠀⠀⠀⠀⣌⢀⡄⠀⠀⠀⣅⣈⡄⢀⠖⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⢀⠴⡀⠀⠀⠀⠸⠴⠆⠀⠀⠀⡦⠆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠆⠤⠂⠀⠀⠀⠳⠤⠣⠀⠀⠀⠀⠀⢰⣠⠴⠀⠀⠀⠐⢤⡘⠄⠀⠀⢀⠤⠀⠀⠀⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢅⡤⠂⠀⠀⠀⢦⡀⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⢠⣿⡿⢛⣛⠻⣍⣻⣷
⠀⠀⠀⠀⠀⠀⠈⢃⣁⠀⠀⠀⠀⣿⡿⢋⣛⡛⠿⣿⣷⠀⠀⠀⠘⠏⣴⣿⣿⣷⣬⡉⠿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⡏⢰⣿⣿⣿⣿⡆⣿⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⢠⣿⣌⠻⣿⡿⠟⣡⣿⡀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⣝⡛⠷⢒⡲⠿⠿⠿⠇⠀⠀⠀⠀⠀⢠⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⣡⠌⢰⡔⣿⣿⣶⡀⢠⣶⣿⣿⣌⠻⢛⣴⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣥⣾⣷⣍⣼⡝⣿⡇⣾⡿⢹⣿⣿⣿⣿⣿⣿⡙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠋⣤⢰⣶⣶⣿⠡⣿⣷⣶⣶⠀⣤⠀⠰⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⣭⣭⣭⣭⣭⣭⣭⡍⣿⣿⣿⢩⣭⣭⣭⣭⣭⣭⣭⡍⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠃⠙⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠿⠛⠛⠛⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⣿⣿⡃⠀⠀⠀⠀⢻⣿⣿⠃⠤⠒⠂⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡞⢿⡁⠶⠈⠉⢁⣤⠀⠀⠀⠀⠀⠈⢹⣶⡶⠹⡟⣳⠀⠀⠀⢠⡼⢿⠣⣤⣶⡁⠢⡀⠀⠀⠀⢀⣴⠈⠛⢿⣿⣿⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠡⡆⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⠃⢸⠋⢠⠀⠀⠀⢀⡈⢹⡀⠹⣿⣿⡆⠀⠀⠀⠀⢾⣿⣦⠀⠀⠉⡵⠊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠡⣤⣷⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⣷⣤⠛⠀⠀⠀⠀⠁⢤⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀⣤⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣇⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣀⠀⠀⣤⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠴⢛⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢇⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠙⠻⠟⠋⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠚⠿⢿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⣀⠤⡾⠿⠿⠟⠛⠩⠀⠈⠢⣄⢀⣸⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢉⣉⢹⠉⠁⠈⢙⣏⣁⣈⠑⡄⠀⠀⠀⠑⢄⣠⠁⠀⠈⠙⠻⣋⣉⣉⣛⠉⠀⠈⢹⣏⣉⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠃⢹⠀⢠⠀⠈⠀⠀⠁⠀⡇⠀⠋⠓⠤⢀⡀⠀⠀⠡⢄⡀⠀⠀⠀⠌⠠⠀⠀⠁⠒⠊⠁⠀⠃⠘⠂⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⡇⢰⡆⠀⠀⠀⠈⣆⠀⠀⠀⠀⢙⣶⡖⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠠⠊⠁⡄⠀⢀⡔⠉⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠉⢦⣀⠀⢀⠉⠑⢄⠀⠀⠸⡿⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣼
⠀⠀⠀⠀⠀⡁⠀⠲⠵⠾⠽⠆⡜⠓⠺⠀⠀⠀⠀⠀⠀⡗⠚⢢⠰⠯⠰⠮⠖⠀⠈⠀⠀⢀⠃⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡏⠠⣠⠊⠀⢣
⠀⠀⠀⠀⠀⠙⠀⠠⠀⠤⡤⠴⢇⣀⡔⠀⠀⠀⠀⠀⠀⢧⣀⡴⠦⣤⠤⠤⠤⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⢸⣧⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄⠘⠤⣏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠣⣜⠁⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣉⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢈⡇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠢⠤⠄⢀⣀⣀⣀⣀⠀⠠⠤⠐⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠂⠠⠤⠄⠀⠀⠀⠀⠤⠤⠀⠒⠈⠀⣠⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣤⣄⣀⣀⣀⠀⢀⣀⣀⣀⣠⣤⣶⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣷⢶⣤⣤⣀⣀⣀⣀⣀⣠⣤⣴⢶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢍⣛⠓⠾⠯⠭⣭⣭⣭⣭⠭⠿⠶⠛⢋⠩⠗⠀⠀⠀⠀⠀⠀⠃⢬⡉⡛⠛⠲⠾⠿⠿⠿⠿⠶⠒⢛⠉⣥⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠀⠀⠤⠀⠀⠀⠔⠒⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠒⠒⠒⠒⠒⠐⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠀⣖⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠒⡖⠀⠀⠀⠀⠀⠀⠀⠸⡒⠂⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠙⢄⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⢣⠀⠀⠀⠀⠀⢠⣃⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⢀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⢊⡉⢢⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡚⣉⡇⠀⠀⠀⠀⠠⡤⡽⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⡥⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⠿⠿⠿⠿⠿⢯⠻⠿⠿⠿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣛⣛⣛⣛⣛⣻⣿⣟⣛⣛⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⡿⠋⠻⣿⣿⣿⣿⢿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⠴⠫⣄⣠⠦⠦⡭⠛⢋⡼⠁
⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠈⠉⠐⠠⡀⠘⡖⠚⠁⠈⠒⠚⠁⢀⢸⡁
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⠦⣀⠀⠀⢀⠴⠁⠘⡷⠡
⠀⠀⠀⠀⠀⠈⠐⠠⠤⠀⠀⢀⡀⠔⠊⠀⢀⡤⠊⠉⠐⢲⡍⠠⢆⠁⠀⡰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠣⣀⢀⣀⠴⣚⠉⡆⢸⠠⠊⠀⠀⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⢉⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⠁⠁⠙⠆⠀⠀⠀⡇⠀⠀⠀⠀⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠅⠛⠀⠀⠀⢼⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡀⡤⠄⣌⢀⡀⠀⠀⣀⣘⣀⠀⢀⣸⡤⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡑⠘⠂⠈⠂⠘⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠃⠀⠒⠒⠲⠒⠠⠤⠤⠄⠠⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠉⠲⢤⣀⣀⣀⡤⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢁⠴⡄⢠⠋⠓⢨⠏⠀⠟⢠⠤⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠠⣮⡁⠘⣄⠑⢈⡁⠀⠀⠳⢬⠵⠁⠀⠀⢉⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠶⠛⠋⠉⠉⠉⠉⠉⠉⠙⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⠛⢷⣬⣦⣎⣽⣔⣛⣛⠲⠶⣄⠀⠀⡎⠠⣄⢻
⠀⠀⠀⠀⣠⠞⣁⡤⠐⠒⠉⠉⠉⠉⠀⠒⠒⠤⣄⣀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠘⣧⡴⠒⢒⡾⠋⢹⠛⡛⢦⡀⠉⠶⡦⡈⢿⣀⡑⠒⠋⡼
⠀⠀⠀⡼⣧⠚⣁⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂⠟⣻⡆⠀⠀⠀⣼⠛⠴⠆⣞⢡⠀⢸⠀⢹⣀⠙⠦⣀⢱⠘⠋⠉⠙⢷⠞⠁
⠀⠀⡴⠁⢈⣿⠧⣤⣄⣀⣀⣉⠉⠙⠛⠓⠲⠶⠶⠶⠶⠖⣚⡽⢯⠀⠀⢠⠞⠁⠀⠀⣀⡙⢦⣁⠸⣄⣸⣬⣤⡤⠜⠋⠀⠀⠀⠀⠘⣶
⠀⢸⠃⢀⣼⠁⠀⠀⠀⠀⠉⠉⠙⠛⠛⠶⠶⠶⠶⠶⠛⠛⢿⡦⣸⡇⠀⠘⠠⠒⡄⠀⣧⣉⣀⡽⠛⠛⠙⠶⣄⣀⡀⢠⠦⠀⠀⠰⣂⡙⣆
⠀⢨⡤⠋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⣷⠀⡸⠖⢊⣡⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⣧⣈⣩⠃⠀⠱⠤⠃⣸
⠀⣿⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡏⢸⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⠄⠀⡖⢦⢰⡏
⠀⢫⡼⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠸⣃⣘⣄⣰⠃⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⣉⡜⠈⣧
⠀⠸⣧⡇⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢢⠀⠈⠁⠀⠀⠀⠀⠀⠀⣏⠉⠹⣇⠘⣡⠂⠀⠀⠘⠋⠀⠔⠂⠀⠰⠇⠀⠀⠀⠀⣩⣍⢳⣤⡟
⢀⡴⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠏⠉⠀⠀⠀⢀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠉⣿⠙⢢
⣿⠀⠀⢇⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⡏⠀⠀⠀⠀⠀⠻⡅⠀⠀⠙⠂⠀⠀⡶⠄⠀⠀⠀⠀⠀⢠⡏⠀⢸
⠈⠳⣤⡼⣄⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⣠⠋⠀⠘⢦⡀⠀⠀⠀⠀⠙⢆⡀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⣼⣯⠟⠁
⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠉⠒⠒⠋⠀⣀⣀⣀⠤⠖⠋⠁⠀⠀⠀⠀⠉⠓⠦⣀⡀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⣻⣄⣀⠀⠀⣀⣠⣯⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣆⠀⠀⠀⠀⠀⣠⡞⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡹⢟⣍⣉⠟⣹⢿⡈⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢻⣿⣿⣶⠶⢶⡿⢯⡉⠳⡄
⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⣰⠃⠀⠙⠛⠛⠁⠈⣯⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⣠⠟⢠⡍⣼⠃⠙⠒⠊⢳⠈⢧⠁⠙⣆
⠀⠀⠀⠀⢀⡔⠁⠀⠀⣴⣍⡛⠢⠤⠤⠤⠄⣊⡹⣇⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⣰⠋⠀⡾⢁⠇⣀⡀⠀⣠⣸⡆⠀⣇⠀⠘⣧
⠀⠀⠀⡠⠊⠀⠀⠀⣰⡀⠀⠉⠁⠒⠒⠊⠉⠀⠀⣸⡄⠀⠀⠀⠀⠀⢠⠏⠉⣹⠇⠀⣸⣦⡟⠀⡍⠻⠞⢀⡏⣷⠀⠸⡄⠀⠙⡆
⠀⣠⡾⡁⠀⠀⠀⢰⠷⣌⠂⠤⢀⣀⣀⣀⡀⠤⢊⡱⢻⣤⡀⠀⠀⢠⡟⠀⢰⠋⠀⢀⠏⠹⠃⠀⢣⠀⠀⢸⠁⢈⡀⠀⢳⡀⠀⠘⡀
⠀⣽⠷⢦⣤⣀⣠⡎⠂⠀⠉⠒⠲⠤⠤⠐⠒⠛⠁⠐⠘⣧⠉⠲⣴⣏⠀⣠⠏⠀⠀⡜⠀⣦⠀⠀⠀⢧⣀⠇⠀⢸⡇⠀⠈⣧⠀⠀⢱
⠸⡇⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠉⢛⡿⠀⠀⢀⢡⢤⡏⠀⠀⠀⠈⠋⠀⠀⢸⡇⠀⠀⢹⡆⠀⠀⢷
⠀⠻⣄⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣾⠃⠀⢠⡆⠈⢹⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠘⣧
⠀⠀⠙⢧⡀⠀⠀⠀⠀⠹⣗⠒⡶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢰⠏⡀⠀⣼⠁⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡄⠀⠀⠹⡆
⠀⠀⠀⠀⠙⣦⡀⠀⠀⢀⡜⣠⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣧⠴⣿⣤⡄⢀⡏⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⡇⠀⡀⠀⣿⡀
⠀⠀⠀⠀⠀⣯⠙⣶⠶⣋⠀⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⡇⠀⠀⠙⠛⢾⠃⠀⢰⠖⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⣷⠾⠛⠛⠉⠉⠳⣄
⠀⠀⠀⠀⠀⣿⢉⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠈⢻⡀⠀⣀⡀⢸⣧⣤⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⢷⠾⣶⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣿⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡟⠋⠉⠙⢺⠃⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡞⠳⢦⣀⣀⣀⣠
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡄⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡇⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⣠⠁⢸⡀⠀⠀⠀⠀⠀⠘⢺⠀⠀⠀⢻⣖⠒⠒⠆⠀⣀⣤⣤⡤⣤⣤⣤⡤⣤⣤⡄⢖⣶⠚⠁
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡧⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠸⣯⣭⣙⣁⣀⣀⣠⡏⠀⠀⣧⡘⠳⠆⠀⠀⠨⣿
⠀⠀⠀⠀⠀⠈⣷⣤⣤⣶⡶⠶⢤⣤⣤⠀⢀⣷⠬⠤⠤⠤⢤⣤⣽⣄⠀⠀⠀⣷⠒⠒⠒⠒⠒⢾⡇⠀⠀⢻⠁⠁⠐⠂⠈⢹⡇
⠀⠀⠀⠀⠀⣾⠡⠀⣀⣤⣤⣤⣄⡀⠈⠇⢸⠁⢀⣠⢤⣤⣀⡀⠀⠝⣷⠀⠀⢿⠀⢀⣀⣀⡀⢸⠇⠀⠀⢸⠀⣀⣠⣄⡀⢸⠇
⠀⠀⠀⠀⠰⣿⢀⡔⠉⠀⣙⣚⣀⡽⢆⢸⢸⡶⠫⣼⡷⠋⠀⠙⢦⡀⣿⠀⣠⠞⠉⢉⣟⠁⣏⠙⣇⠀⢀⡾⣉⡨⢉⡥⢯⢻⡄
⠀⠀⠀⠀⠀⣧⢚⡴⠒⠋⠉⠉⠀⠀⠸⡇⢸⠁⠀⠀⠙⠢⢄⣀⠈⣶⠆⢀⠡⣄⠀⣯⣹⠆⠈⠉⢹⠂⡟⠄⠈⠀⠸⣄⡤⠇⢹⡀
⠀⠀⠀⠀⠀⣸⠷⠄⠀⢀⡀⣀⣀⣐⣘⣷⣞⣒⣒⣒⣒⣒⣒⣒⣛⡿⠀⣼⣀⣤⣀⣵⣇⣠⣤⣤⣿⣠⣧⣠⣤⣤⣤⣼⣃⣠⣾⠁
⠀⠀⠀⠀⠀⠉⠐⠀⠀⠈⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠁⠉⠉⠉⠉⠉⠀⠀⠉⠀⠉⠁⠀⠀⠉⠉⠁⢈⡤⡴⢤⠶⠶

⠀⠀⠀⡖⠒⠒⠀⠀⠀⠀⠤⠤⠤⠤⠤⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⡇⢸⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣍⣉⣉⣉⣁⣀⣀⣐⣒⣒⣒⠒⠒⠂⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⢀⡄
⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠀⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠
⠀⠀⢰⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡄
⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⡟⠿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃
⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋⠀⠀⡠⠛⠁⠀⠀⠀⣤⠴⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⡇⢸⣿⣿⣿⠃⠀⠀⠀⢊⠄⠀⠀⡔⠀⠀⡠⠁⠱⣄⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸
⠀⠀⡇⢸⣿⣿⡏⠀⡀⢀⣰⠋⣰⠄⢊⣾⠠⠊⠀⣠⣄⠙⢿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⡇⢸⣿⣿⣿⠿⠐⣉⡬⠛⢁⣀⡈⠀⠀⠀⠘⠃⠀⠀⠀⠻⡍⢿⣿⣿⣿⣿⡷⠤⠤⠀⢈⣉⣉⣉⠹⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠇
⠀⠀⠁⢸⣿⣿⣿⢆⣶⠟⠀⠠⠟⠉⠁⠀⠀⠀⠀⠤⠔⠋⠀⠀⢣⣾⣿⣿⣿⣿⣧⠀⠩⠻⡍⠉⠁⠀⢀⣠⣼⡇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⠟⠪⠁⠀⠀⠀⠀⠤⠒⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⣿⣿⣿⣿⢟⡔⢢⡀⠀⠀⠀⠀⣸⣿⣿⣥⠀⡄⠘⣿⣿⣿⣿⣿⣿⡏
⠀⢸⠀⣿⣿⣿⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⠟⠁⠀⢀⣾⣿⠟⠉⠀⠈⠳⠿⠃⠀⠀⠀⢀⠈⣿⢹⠏⣆⣿⠀⢹⣿⣿⣿⣿⣿⡇⠐
⠀⢸⠀⣿⣿⣿⣷⣤⣴⣆⠀⠀⠀⠀⠤⣤⣄⠀⠐⣁⠉⠀⢀⣴⣿⣿⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⢤⣖⠛⠸⣇⡴⠹⣿⡆⢸⣿⣿⣿⣿⣿⡇
⠀⢸⠀⣿⣽⣿⣿⣿⣿⣿⣷⣤⣤⣀⡀⠀⠀⠀⠈⠁⠀⣠⣿⣿⣿⣿⣿⡇⠀⠉⢧⣤⣄⡀⠀⠀⠀⠉⠀⠘⣏⢣⠀⠹⣇⣸⣿⣿⣿⣿⣿⠀⡆
⠀⠸⠤⠤⠄⠀⠀⠀⢀⣀⣈⣉⣉⣉⠉⠑⢲⡤⠄⠤⠼⠟⠻⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⣇⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⢸⡇⠒⠒⠒⠒⠒⠀⠰⢆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⡇⠘⢦⡾⠥⠭⠉⠉⠉⠉⠤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⡤⠺⣻⠛⢿⣿⣀⡾⠁
⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢄⠀⢀⣎⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠋⠀⠈⣿⣿⠇
⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⢁⡞⡸⠁⠀⠀⠀⠀⠀⠀⠄⠚⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡞⣰⠁⠀⠀⠀⠀⠀⡀⠆⠀⢁⠀⠀⢸⠢⠀⠀⠀⠀⠸⠓⢦⣼⣟⠠⠚⠋⠁
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇⠀⠀⠀⡠⠂⠀⠀⠀⢀⡈⢦⡀⠀⠆⠀⠀⢀⠀⠀⠁⠒⡝⣯⢀⠀⠈⠃
⠀⠀⠀⠀⠠⢀⡀⠀⠀⡈⡀⣀⣀⣀⠀⠀⠠⠿⠟⠃⠂⠀⢠⡅⠀⠀⠀⠀⠀⢳⡀⠐⠀⣤⠃⠙⠀⠀⠀⠸⠊⠁⢸⡳⣤⠚⠀⠀⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⠛⠀⠀⠀⠀⠀⠀⠀⠳⡀⠳⡏⠀⠀⠀⠀⠀⠀⠈⠀⠚⠳⣬⣷⡤⠤⠐⢀⡈
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠐⠠⡀⠀⠀⠀⠀⠀⠀⢿⡿⠀⡤⢀⣀⠈⢀⠀⠀⠀⠀⠀⠈⢻⣷⠑⠒⠈⠁⠁
⠀⠀⠀⠀⢀⡄⣓⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠁⠀⠀⠀⠀⠀⢀⣿⣷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠈⠑⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠌⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠐⢀⠃⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡤⢄⠀⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉


⣿⣶⣶⡾⣷⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣾⣷⣶⣶⣶⣿⣶⣶
⣿⣿⣿⣀⣀⣠⣤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⡠⠊⢱⠤⠒⠒⠈⠙⢆⠀⠀⠀⣰⣿⡘⣯⣵⣶⣶⣶⡄⠀⠀⢀⢾⡁⠀⢀⡤⠤⣌⢢⡀⠀⢠⡸⣿⣿⠿⣛⣿⡛⢦
⠀⢠⣇⠀⠀⠣⣀⡠⠤⣄⣈⡆⠀⠐⢿⣿⣷⣝⠿⣟⣛⢛⡿⠀⠀⡎⠀⠉⠉⠁⠀⠀⠀⢣⠇⠀⣿⣿⣶⣶⣿⣿⣿⣿⡎⡇
⠀⠘⡏⠣⣄⢀⡎⠀⠀⠇⢀⠇⠀⠸⣮⡻⢿⡿⣸⣿⣿⣸⣿⠀⠀⢇⠀⣀⣀⡀⠀⠀⠀⡜⡆⠀⣿⣿⠿⠿⣿⣿⣿⣿⢇⡇
⠀⠀⠙⢄⣨⠏⠀⠀⣼⡠⠊⠀⠀⠀⠙⢿⢗⣵⣿⣿⢇⡿⠃⠀⠀⠈⠻⣀⠀⠈⠙⠒⣫⠜⠀⠀⠈⢵⣿⣿⣶⣭⣯⣥⠞
⠀⠀⠀⠀⠉⠑⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉
⠀⠀⠀⣠⡶⠓⠀⠺⣶⣤⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡙⠻⣦
⠀⠀⠈⠁⢀⣤⣤⡆⠀⠉⠀⠀⠀⢀⣾⣿⡟⠛⠛⣿⣿⣾⣧⠀⠀⠀⠀⠀⠀⠀⣠⡦⠙⢷⡄⠀⠀⢀⣾⣿⣿⠏⢙⣷⣌
⠀⢰⣦⠀⠸⢿⡿⠇⠀⣠⣶⡆⠀⠈⠻⣿⣅⡀⢀⣽⣿⠟⠙⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣾⣿⡿⠁⣰⣿⣿⠏
⠀⠀⠿⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⢦⣀⠙⠁⠀⠀⠀⠀⠀⠀⢈⠻⣿⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠐⢾⣿⡶⠁⠀⠀⠀⠀⠀⠀⠙⠏⠁⠈⠹⠟⠉⠀⠀⠀⠀⠀⠉⠳⠦⠀⠀⠀⠀⠀⠀⠘⢷⣦⡉⠻⠛⠁

⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣴⣶⣷⣶⣦⡀⠀⠀⠀⠀⢀⡤⠂⠈⠀⠒⢀⡀⠀⠀⠀⢀⢰⣶⣿⣷⣶⠄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠠⢁⠇⠀⠀⡴⠚⠉⠑⡀⠀⣰⡇⣾⣿⡿⢫⣴⣾⣷⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣿⣿⣿⣿⣿⣿⠀⠀⣧⡞⠀⠀⢸⠁⠀⠀⠀⡁⠀⠛⣼⣿⣿⠅⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣿⣿⣿⣿⠟⠀⠀⠸⡀⠀⠀⠸⡄⠀⠀⢠⠃⠀⠸⣿⣿⣿⡅⢿⣿⣿⡿⠁
⠀⠀⠀⠁⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠈⠂⠄⠀⠉⠶⠒⠁⠀⠀⠀⠈⠻⠿⢿⡦⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⣀⡠⠄⢀⡀⠀⠀⠀⢀⣿⣧⢀⣀⣀⣀
⠀⠀⠈⢿⢆⠀⠀⠀⠀⡀⠀⠀⠀⠙⣤⡀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠈⢀⠞⠁⠦⡀⠀⠈⢢⠀⠀⣾⣟⣵⣿⣹⢿⣿⣷⣄
⠀⠀⠀⠀⢂⠁⠒⠈⠁⣔⠀⠉⢐⡄⠘⣿⣶⣶⣿⡟⠽⣷⣶⣶⠦⠀⠀⢸⠀⠘⠂⠉⠀⠀⠀⡇⠀⢿⢹⣿⣧⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠢⠀⢀⣀⡁⠤⠀⠀⠀⠀⠘⠿⠿⠿⠷⠽⠟⠛⠛⠉⠀⠘⢌⣄⠀⠀⠀⠀⢀⡔⠀⠀⠘⠯⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁
⣟⣿⣻⣻⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣿⣿⣛⣛⣟⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠀⠀⠀⠙⠛⠃⠤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣤⣤⠔⠛⠁⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⢴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠇
⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠶⠶⠤⠀⠀⠀⠀⠀⠶⠶⠦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⣠⠶⠆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠶⣀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠳⣦⡀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⠋⠀⠀⣀⣶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠷⣦⡀⠈⠻⣄⠀⠀⠀⠙
⠀⠀⠀⠀⣿⠀⠀⠀⡞⠁⠀⠀⣀⣁⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⣡⣀⠀⠉⣷⠀⠀⠀⣾
⠀⠀⠀⣠⣿⠀⠀⢀⣧⣰⠶⠟⠉⠉⠉⡙⠋⠙⢿⡉⠷⢶⣄⣀⣀⣀⣴⠶⠎⢻⡟⠉⠻⠏⠉⠉⠙⠷⢦⣾⣤⠀⠀⣿⣄⡀
⠀⢠⠶⠉⡿⠀⠀⢸⣿⠃⠀⠀⠀⠀⠘⠇⠀⠀⠸⡇⠀⠈⢻⣿⣿⣿⡇⠀⢀⡘⠁⠀⠀⣸⡆⠀⠀⠀⠀⣿⣿⡃⠀⣿⠉⠙⣦
⠀⣿⠀⠀⠀⡄⠀⢸⣿⠆⠀⠀⠀⠀⠀⢠⣶⣶⣤⠆⠀⠀⣾⡏⠉⢿⡇⠀⠘⠁⣰⣶⣦⣼⠃⠀⠀⠀⠀⣼⣿⠇⠀⣿⠀⠀⠈
⠀⣿⠀⠀⠀⣷⠀⢀⣿⣷⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⢰⡿⠀⠀⠸⣇⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣿⡏⠀⢀⠁
⠀⢙⠀⠀⠀⠙⠓⠈⡙⠻⣶⡀⠀⠀⠘⣿⣿⣿⡿⢃⣴⡟⠁⠀⠀⠀⠻⣧⡄⢹⣿⣿⣿⡟⠀⠀⠀⣠⡿⠟⠁⠀⡿⠀⠀⠀⣤
⠀⠀⠿⡀⠀⠀⠀⠀⠷⠀⠈⠹⠿⠲⠆⠸⠿⠻⠿⠟⠉⠀⢀⡀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠇⠠⠾⠿⠋⠀⢰⣀⣾⠀⠀⠀⣠⠟
⠀⠀⠀⠻⣤⡀⠀⢀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠐⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠠⣤⣤⣤⣤⠠⢤⣤⣤⣤⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡅⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⢤⣤⣶⡖⠙⠛⢻⡇⠀⠀⠀⠀⠀⢸⡟⠙⠛⠻⣶⣦⡠⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁⠀⢀⣤⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠿⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠳⣦⣀⣀⡀⠴⠇⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠈⠷⣄
⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⠴⠶⣶⣾⠀⠀⠀⠀⠀⠀⠙⣶
⠀⠀⢀⣴⠉⠀⠀⠀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡆⠀⠉⠉⣉⣀⣀⣤⣤⡀⢿⡇⠀⠀⠀⠀⠀⠀⢀⣷
⠀⣠⠞⠉⠱⢦⣀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⡄⢰⣿⣿⣿⣿⣿⣿⣷⡄⣠⣀⠀⠀⠀⣀⡶⠋⠉⣶
⣄⠋⠀⠀⠀⠀⠉⣶⣄⠴⠶⠙⡏⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⡄⣿⣿⣿⣿⡿⠿⠿⠃⠙⠋⠁⣴⣶⠋
⣿⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠉⠻⣿⣿⣧⡈⠉⠉⠀⠀⠀⠀⣀⡀⠿⠇⠉⠁
⡿⠀⠀⠀⠀⠀⠿⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠉⠉⠻⢦⣀⣀⡀⠶⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠷⣀⠀⠀⠀⠀⠛⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠟⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠶⠋
⠀⠀⠻⣄⠀⠀⠀⠀⠻⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠰⠶⠾⣏⠉
⠀⠀⠀⠈⠷⣄⠀⠀⠀⠁⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢏⡀⠀⠀⠀⠀⠀⠀⣠⡖⠂⠈⠉⠉⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠉⣤⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣀⣤⣤⣾⣿⣷⡄⠀⠀⠀⠀⣠⡄⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⠋⠉⠀⠀⠀⡀⠒⠛⢦⣄⣠⣤⣠⣾⣿⣿⣿⣯⣿⣿⣿⣷⣤⣴⠾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣴⠏⠀⠒⠒⠋⠉⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠛⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣰⡇⠀⠀⠀⠀⠀⠀⠀⢰⡆⠈⢡⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⢠⡌⠁⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠙⠛⠋⠉⢻⡟⠁⠀⠀⠀⠸⠟⠉⠋⠛⠛⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⠶⠶⠎⠙⢇⠀⠀⠀⠀⠀⠀⢰⡏⠁⠶⠶⠶⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠶⠶⠈⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⠈⠑⠶⢄⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠈⠁⠀⠀⣀⣀⣀⣀⣄⣀⡀⣸⡇⠀⠀⠀⠀⠸⣷⣀⣀⣀⣀⣀⣀⣀⠀⢀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⣠⣤⣤⣠⣤⣀⣀⣈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠙⠛⠋⢉⣀⣠⣄⣠⣄⣀⣤⣀⣠⣽⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠛⠉⠻⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣤⣀⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠉⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⠀⢀⠀⠔⠀⠀⠀⠐⠢⣀⠀⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠁⠔⠚⠃⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⡄⠀⠀⠘⣿⣿⣿⣿⡏⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⢿⠀⠦⠀⠀⢢⡄⠀⠀⠀⢀⠒⠀⣿⣿⠿⢟⡳⡆⠀⢰⣿⣿⣿⣿⢀⠂⠀⣀⣤⡀⠀⠀⠀⠀⢠⣤⣀⠀⠁⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠈⠳⡄⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⣿⣿⠇⢰⡏⢀⡅⠄⣴⣿⣿⣿⢿⣌⢀⣀⣀⠄⡀⠀⠀⠀⠀⠀⠤⠄⣀⠀⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠⡆⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⡶⠘⠇⠀⣸⣿⣿⣿⠒⢿⠈⣿⣷⡄⠀⠀⠀⠀⣰⣿⣄⠀⠉⢀⡯⠞⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⢔⠠⠄⢀⡀⠀⠀⠀⠀⡾⠃⠀⠈⡈⡆⠻⣿⣿⣆⣸⣄⠘⠛⠃⠀⠀⠀⠀⠈⠛⠋⠀⠀⢸⣁⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡀⠀⠀⠀⠉⠀⠉⠀⠀⠀⢀⣼⡆⠀⠀⠀⡇⠰⠀⠘⠛⡿⢿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣾⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⠐⣠⣀⣀⠀⢀⣀⣐⠊⣻⣿⣿⠀⠀⠀⢠⠀⢣⠀⠘⠅⢀⠽⣄⠀⠀⠀⠈⠳⠶⠟⠁⠀⠀⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣋⠠⡤⢯⡈⠀⢀⡽⠄⢠⠈⣻⣿⣷⡀⠀⠀⣆⠀⠓⣤⠂⡄⠀⠈⠓⢀⣀⠀⠀⠀⢀⣀⠤⠊
⠀⠀⠀⠀⠸⣾⣿⡇⣿⣿⠿⠛⠃⢹⠀⡇⠀⠈⠉⠁⠀⠀⠈⡀⢋⠙⠻⢿⣿⣘⣿⣿⣿⠁⢠⠉⠒⢶⢉⣁⠸⡏⠐⠈⡹⢅⡀⡒⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣨⠀⠀⠀⠀⣰⠀⡅⠀⠀⠀⠀⠀⠀⠀⢡⠈⡄⠀⠀⠸⠂⠤⠔⠁⣠⢆⡀⠀⠆⢸⠄⠀⠈⠉⠉⠀⢠⠀⡎⠙⠣⠄⡀
⠀⠀⠀⠀⠀⠀⡰⠁⠈⣄⢀⣠⠆⢁⣀⡃⠀⠀⠀⠀⠀⠀⠀⠈⡄⢡⡄⡠⢧⠀⠀⢠⠔⠁⠀⠀⠹⢀⠃⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⡅⠀⠚⠉⠙⠒⠢⡎⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⡈⠀⠀⠀⠀⠀⠀⠀⢸⠀⡎⠰⢤⡀⣰⡀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⣶⠚⡳⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⡚⠓⠓⠂⠉⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠚⡉⢿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠒⠂⠀⠈⠉⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠠⠄⠀⢀⣀⣀⡀⠀⠠⠀⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠾⢯⣽⣿⣓⡒⣒⣒⣒⣻⣿⡭⠿⠟⢻⠄⠀⠀⠀⠀⠀⠀⠀⡿⣿⣷⣶⣦⣤⣤⣤⣤⣤⡤⣴⣶⣺⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠠⠤⡌⠉⣉⣉⣉⡉⢁⡤⠄⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⢉⡙⠛⠓⠚⠛⠛⠚⠛⣛⡉⠡⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡌⠉⠉⠉⠉⠉⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡃⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡏⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠃⠀⠀⠀⠀⠀⠀⠸⠄⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠒⢃⠀⠀⠀⠀⠀⠀⠀⢀⠓⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠐⠳⠀⠀⠀⠀⠀⠀⠀⢠⠛⠚⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠙⣀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⢢⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠂⠙⠄⠀⠀⠀⠀⠀⢉⠈⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠄⠁⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠀⠀⠀⠀⠀⢐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠀⢄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣧⠀⠀⠀⠀⡐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠀⠀⠀⡠⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠤⠤⠬⠀⠀⠀⣾⡄⢀⢀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠻⢿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠃⠀⠀⠀⠀⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠄⠀⠀⠀⠀⡇⠀⠀⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⡄⠀⢀⡀⢠⠀⠀⠀⠀⠀⠠⠖⡲⢦⡀⠀⠀⠀⠠⡊⢳⣿⣿⣿⡏⢑
⠀⠀⠀⠀⣤⣷⣿⠇⠀⠀⠀⠀⠀⠈⢃⠀⣾⡿⠟⠀⠀⠀⠀⣈⠂⠜⢠⠃⢽⠀⠀⠀⠀⠙⠺⣿⣿⣿⠗⠁
⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀⠠⣶⣾⡟⢀⣤⡀⠀⠀⠀⠀⠘⢕⣒⣣⡈⠎⠀⠀⠀⠀⠀⠀⢉⣿⠁
⠀⠀⠀⠀⠘⠿⠛⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛

⠀⠀⠀⠀⢀⠀⢀⣤⡄⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣧⡂⠀⠀⠀⠀⠀⢠⣤⡄⣀⡀
⠀⠀⠀⠀⢨⣿⣿⡿⠋⠀⠀⠘⢟⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣰⣿⡣⣹⣦⡄⠀⠀⠀⢻⣿⣧⣿⡿⣦
⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠻⡟⠉⠉⠉⠀⠀⠀⠀⠀⠰⢟⣱⠿⢛⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⡥
⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠞⠋⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⢀⡈⣶⣷⠆⢳⡀⠀⠀⠀⢀⡤⡴⠗⠳⠤⢤⠄⠀⠀⠀⢠⣿⢃⡼⣭⣤⠀⠀⠀⠀⠀⣼⡙⣎⣣⠶⣀
⠀⠀⠘⢇⣈⠉⢰⣶⠀⠀⠀⠀⠀⠙⣷⠀⠀⣴⠃⠀⠀⠀⠀⠰⣟⣶⣽⣷⡝⠀⠀⠀⠀⡼⠡⠗⢉⣤⠶⣉⡄
⠀⠀⠀⠈⠛⠃⠈⠁⠀⠀⠀⠀⠀⠰⠿⠛⠛⠾⠀⠀⠀⠀⠀⠀⠈⠛⠓⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⡿⣶⠄⠀⠀⠀⠀⠀⠘⣿⠇⣴⡿⠃⠀⠀⠀⢐⣴⣿⠟⢻⢹⣿
⠀⠠⠞⠋⡳⠤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⣶⣿⣿⡾⠋⠀⠀⠀⠀⠀⣾⢏⠘⡐⣼⡼⡋
⠀⠀⠀⠀⠁⠢⢈⠗⢦⡄⠀⠀⠀⠀⠛⠛⠟⠃⠀⠀⠀⠀⠀⢻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠿⠦⠾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣆⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠠⣿⣿⢿⡿⢿⡿⠛⠿⠿⠻⢿⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠄⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⡟⠁⠀⠈⠉⠛⠋⠉⢻⡇⠀⠀⠀⠀⠀⠘⣿⠋⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⠃
⠀⠀⠀⢀⣀⠀⠉⡙⠀⠀⠀⠈⠀⡀⢀⠀⠁⠀⡀⠀⡸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇
⠀⠀⠀⠿⢏⡹⢠⣷⣄⠀⠀⠀⠀⠛⠛⠁⠀⢀⠃⠘⣛⡏⠁⠀⠀⠆⠀⠀⠀⠰⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠃⣀⣀⡀⠁⠀⠀⠀⡫⠉
⠀⠀⠀⠀⠀⠈⠀⢍⠻⢿⣶⣦⡄⠀⣤⣤⣶⠟⡂⠔⠁⢃⠀⠀⠀⠀⠐⠀⠐⠀⠀⣠⠠⠀⢠⡄⠄⡀⢀⠀⠀⠙⠋⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡈⠑⠁⠊⠃⢉⠄⠈⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⡤⠊⠀⡠⠊⠀⠀⠁⠒⠒⠁⠐⠲⡄⠠⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢲⡀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠈⣤⠎⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠢⢔⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠈⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢩⡀⣆⠀⠀⠀⠀⠀⠀⠌⠀⢀⡆⠀⠀⠀⠀⠘⣆⡀
⠀⠀⠀⠀⠀⣀⢩⡏⠀⠀⠀⢀⡈⠉⠁⡅⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠙⢥⡀⠀⠀⠀⡠⠊⠀⡠⠚⠒⠀⠀⠀⠀⠒⠉⠚
⠀⠀⠀⠀⠀⠐⠀⠀⠁⢲⢚⣍⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠒⠂⠈⠀⠀⠉⠒⠒⠂⠀⠀⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢰⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢴⠁⠀⠀⠀⠀⠂⢘⡆⢠⠀⠀⠀⠀⠀⠀⠀⠤⣶⣀⡂⠈⢶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠀⠀⠈⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⠄⠀⠀⠀⣤⡄⠀⢨⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠨⠄⠀⠀⠀⢽⠇⠀⠀⠈⠁⠠⠀⣂⡀⠀⠀⠀⠀⠀⠘⠛⠀⠄⠀⠿⠀⠄
⠀⠀⠀⠀⠀⠀⠐⡂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⡄⠐⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⠠⡬⢤⠄⡤⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠋⠉⠁⠄⠀⠌⠦
⠀⠀⠀⠀⠀⠀⡇⢀⡀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣇⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠀⠀⠀⠀⠀⠀⠀⡀⢀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢹⡏⠙⠉⣿⠉⠁⠀⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⠉⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠚⠒⠒⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⢚⣦⣌⠛⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠻⣿⣦⣝⣻⣿⠇
⠀⠀⠀⠀⣀⠀⡤⠀⠀⣰⣿⣿⣿⣿⣿⡿⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⣬⠚⡿⠻⢋⠀⢋⣄
⠀⠀⠀⠀⢡⣄⡨⢿⣯⣛⠿⣿⣿⣿⡟⠁⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⠉⠁⢂⠰⢆⠀⡤⠊⡸⣇
⠀⠀⠀⠀⠊⠉⠛⠋⠉⠉⠉⠿⢿⣛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠿⠛⠛⠿⣶⣌⡁⠀⠛⣤⡿
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣀⣠⢼⣟⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠁⠄⠙⢻⣟⣻⡯⣄⠀⠠
⠀⠀⠀⣠⣼⣲⣤⣾⣿⡿⠁⠉⠻⣿⣇⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⠐⠀⠀⠀⠀⠒⠀⣿⣟⣷⠴⠨⢷⡄
⠀⣠⣿⡟⣿⣿⣿⣿⢏⠀⢠⡈⢢⣽⣿⠆⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢠⡀⠀⠀⣰⣿⣿⣮⡀⣷⣿⡧
⢠⣿⣿⣷⢿⣿⡟⣿⠿⠁⠀⠘⠋⠁⠀⠙⠣⡀⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠶⠛⢂⣠⠄⠉⠁⣠⡄⠸⠋⠁⡄⠀⣿⣿⣿⣿⣷⢛⣿⡇
⠟⢹⣿⣿⢸⣿⣇⣿⠀⠀⠂⠀⠁⠀⠀⠠⠀⠈⡑⠚⠝⠣⢴⠥⣀⡀⠀⠀⢀⣠⡄⠙⠀⠃⠀⠀⠀⠘⠁⢀⠀⠘⠋⠀⢀⣤⡄⠳⣦⣿⣿⣿⣦⣿⣿⣿⠃⢠⣦
⢀⣾⣿⣿⡿⣿⣯⣿⢆⢠⡀⠀⠀⠀⠐⠆⠀⠘⡗⠀⢀⠀⡀⡀⠀⢠⡇⠐⣿⣿⠄⠐⠀⠒⠁⠀⠀⢄⠘⠌⣠⣤⣾⠋⢑⠘⣿⣆⠛⣿⣿⣿⣼⣿⣿⣿⠀⢿⡟
⠘⣿⣿⣿⣷⢿⣟⣿⡆⣱⡿⠴⣧⣤⣌⡀⠀⠇⠀⠞⣁⣾⡔⠠⠠⣼⣧⠀⠙⠻⢷⠖⠂⠀⠈⠁⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠘⣿⣿⠻⣿⣿⣿⣿⣿⠏⢠⢃⡇
⠈⣿⣿⣿⡷⣿⣿⣿⣿⠟⠠⠇⠀⠉⠉⢛⣛⣦⣾⣶⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⡐⠀⢠⠆⠀⠀⢸⣿⣾⣿⣿⣿⣿⣏⣼⡶⡾
⠀⠘⣿⣿⣷⣿⣿⣿⠁⠀⠀⠃⠀⠀⠀⠙⠛⠈⣛⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⡇⠀⢸⠦⠀⢀⠈⡙⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠉⠻⠿⠿⠛⡠⢀⠀⢠⡄⠖⠀⠚⠉⠉⣡⣄⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⣸⠂⠀⠔⠁⠀⠙⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠃⠀⠀⠀⠀⠀⠀⠋⢀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⣀⣀⠀⠀⠀⠙⠛⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⣀⣀⣼⣶⣦⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣀⠐⡓⠀⣀⣤⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣄⣤⢤⣕⣶⣾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⣿⣿⣿⣿⠟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣧⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣷⣆⢺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⢿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡆⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡿⣿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣻⣿⣿⠇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣿⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣯⣿⡟⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢻⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⢶⡟⢻⠴⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⣽⣿⣷⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣝⣿⣽⡋⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣷⣿⣿⠤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠛⣹⣫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⣿⣿⡿⠿⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣶⣿⣿⣟⣿⣧⠴⣾⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠵⣿⣷⠿⠖⠾⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⠿⣿⣿⣿⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠃

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

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡠⡀⠀⡀⣄⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⣼⠉⡁⠚⠁⢀⠆⠀⡔⠂⢀⡎⣰⡊⢀⠤⣤⣠⠠⣆⢄⣠⣠⣤⡦⢲⡄⣀⠘⢧⣾⡋⣻⠀⠉⠀⣀⣀
⠀⠀⠀⠐⣠⣾⣶⣿⡇⣴⣶⣄⣿⣳⣾⣿⣷⣿⠇⣜⣦⣿⣿⣠⣯⢤⣉⠁⣿⣿⣸⣿⡿⣷⣿⣿⣿⣿⣷⣴⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣿⣿⣿⢻⣴⡿⣿⣿⡏⢽⡿⣘⠿⠿⣸⡿⠿⣯⡏⣸⣿⠻⣿⢒⣿⣿⢿⣯⢡⣿⣿⢻⣿⣿⠉⢹⣇⠉⢿⣿⣧
⠀⠀⠀⠸⠿⠿⠿⢸⣿⣷⣼⣇⣰⡌⢛⠈⡅⢀⡏⢻⢸⡏⣹⣿⠻⡾⣷⠿⣃⣽⠈⢿⠀⠀⠠⣿⣿⣿⢀⣼⡽⠷⢿⣹⣏
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣏⠴⢿⣿⡟⣾⣧⡌⠈⣿⡿⠁⡀⢼⠧⠀⠋⠉⠡⠄⠀⠀⠀⠙⣿⣿⠚⢿⣧⣀⣼⣟⠛
⠀⠀⠀⠀⠀⠀⠀⣥⢀⣿⣿⠟⠋⠀⠘⠻⠣⠋⠀⠀⠈⠻⢳⡄⣿⠞⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣆⣾⡉⠙⣿⡏
⠀⠀⠀⠀⠀⠐⣴⣾⣿⣿⠇⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⣦⠀⠉⠉⢀⠄⠐⣷⠀⠀⠀⢀⠀⠀⠀⢀⠛⠿⣿⣷⣆⠘⣿⣧⡀
⠀⠠⠄⠀⠀⠀⠋⠀⠉⣭⣤⣤⡴⣂⣤⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⠀⠀⠘⡆⠀⣠⡾⢀⠺⣯⣽⣿⣿⣯⣉
⠈⠋⠀⠀⠀⢀⠀⠀⠐⠛⠛⡿⠦⠟⠛⠁⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠰⣶⡄⠀⠀⠈⢳⡏⠀⡙⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠁⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣠⡙⠟⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠛⠁
⣶⣶⣶⣶⢶⢶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⢶⣶⡶⠶⣶

⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠔⠁⠀⢀⠀⠉⢢⡀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⠔⣅⠀⠀⠀⡨⠢⡀⠀⠀⢀⣤⣻⣿⣿⣿⢟⣤⡀
⠀⢸⠀⠀⠀⠀⠀⠠⠄⡃⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡌⠀⢸⡄⠀⢠⡇⠀⠁⠀⠀⢸⣿⣧⣿⣿⣿⣼⣿⡇
⠀⠸⡀⢀⠀⠂⠀⠀⣠⠃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠰⡀⡸⠁⠀⠀⢇⢀⠆⠀⠀⠸⣿⢏⣿⣿⣿⡹⣿⠇
⠀⠀⠈⠲⠤⠠⠤⠞⠁⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⠣⠄⠀⠠⠜⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠗⠁

⠀⠀⢀⠄⠒⠀⠒⠠⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⡀⠀⠀⠀⠀⠠⢔⣂⣀⠐⠢⢄⠀⠀⠀⠀⣠⡴⠶⠿⣶⣦⣄
⠀⢰⠁⠀⠀⠀⠀⠀⠽⠆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣻⡄⠀⠀⣐⡝⠁⠀⠀⠙⡄⠈⠂⠀⠀⣰⢫⣾⣿⣿⣮⢻⣿⡆
⠀⢀⠀⠀⠀⠀⠀⠀⡅⡂⠀⠀⢸⣿⣿⣿⣿⣿⣿⢾⡇⠀⠀⠌⡇⠀⠀⠀⠀⠸⣄⠘⠀⠀⢿⢸⣿⣿⣿⣿⣧⢻⡿
⠀⠈⠢⡀⠀⠀⠀⣀⠜⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⢜⣦⡀⠀⠀⠀⡨⠁⠀⠀⠈⢷⡻⣿⣿⣿⣿⡷
⠀⠀⠀⠈⠁⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠂⠂⠁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠋⠁
⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⢀⢔⣿⣷⡆⠀⠉⠦⡀⠀⠀⢀⣴⠋⠉⢽⣿⣷⣤⡀⠀⠀⢀⠎⠁⠀⠐⠆⣌⠱⡀⠀⠀⢀⣾⣿⣿⣯⣹⠻⣷⡀
⠀⠈⠀⠙⠉⠀⠠⣴⣆⠁⠀⠀⢸⣿⣶⣾⣿⣿⠋⠹⡇⠀⠀⡂⠀⠀⢀⣄⠀⠀⠀⢠⠀⠀⣾⣿⣿⣿⠻⣿⣿⣿⣧
⠀⠸⣤⣶⣄⠀⠘⠻⢋⠇⠀⠀⠸⡟⠙⠿⣿⣯⣄⣴⠇⠀⠀⠡⡀⠀⠀⠉⠀⠀⢀⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠈⠛⠏⠤⠤⠖⠁⠀⠀⠀⠀⠈⠲⠾⠿⠛⠋⠁⠀⠀⠀⠀⠑⠢⠀⣀⠠⠔⠂⠀⠀⠀⠀⠘⠻⢿⣿⡿⠟⠋

⠀⠀⠀⠀⣀⠄⢤⡄⠤⠀⠀⠀⠀⠀⠀⣀⣤⡀⢤⣤⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⢠⠊⠀⢀⣤⠙⢶⠀⠀⠀⠀⢠⣾⣿⡿⢛⣷⡙⠀⠀⠀⠀⠀⡠⠊⠀⢠⠂⢘⠀⠀⠀⠀⠀⣠⣾⣿⣟⣿⣿
⠀⠀⣇⠀⣴⠟⠁⠀⡔⠀⠀⠀⠀⠻⣿⠋⣡⣾⣿⡟⠀⠀⠀⠀⠎⠀⣀⠔⠁⡠⠊⠀⠀⠀⠀⣼⣿⢿⣿⣿⡿⠃
⠀⠀⡉⢳⣄⣀⠤⠊⠀⠀⠀⠀⠈⣷⣝⠻⣿⠿⠋⠀⠀⠀⠀⠀⠳⣈⢀⡠⠊⠁⠀⠀⠀⠀⠀⠹⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠊⠉⠉⣢⣀⣀⢀⡔⠈⠁⠢⠄⢄
⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⢀⡀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⡇⢀⡴⠒⠉⠁⠀⠀⠀⠈⠉⠓⡱⠄⠀⠀⠁
⠀⠀⠀⠀⢠⠊⡠⠤⠤⠞⠁⠘⣏⠡⠤⠤⠭⣱⣄⠀⠀⠀⠀⠠⠘⠃⠀⠀⠀⠀⠀⠀⡔⠉⠠⡀⠀⠘⠆⠀⠉
⠀⠀⠀⠀⡆⠀⢣⠀⢰⠊⠑⠒⠃⠀⠀⠀⠀⠈⠪⣆⡀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣦⡌⠁⠐⠌⡀⢀
⠀⠀⠀⠀⠃⠀⠀⠑⠁⢘⠢⢄⣤⡀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠈⠑⠦⠤⠤⠤⠤⢐⡂⠒⠁⡄⠡⠀⠊⠀⡏⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠠⠄⠈⠑⠒⠒⠒⠆⠘⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠
⠀⠀⠀⡸⠀⠀⠀⠀⢠⠁⠀⠠⠀⠀⠀⢀⡉⠘⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⡀⠀⠀⠀⠀⠐⠒⡨⠒⠁
⠀⠀⠚⢤⠀⠀⠀⣠⣇⡀⠀⠀⠢⠀⠄⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⠁⢀⡀⠀⣤⡐⠁
⠀⠀⠀⠈⠐⢳⡴⠀⢉⠑⠐⢶⠔⣀⡬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠳⠔⠁
⠀⠀⠀⠀⠀⠘⢃⠀⠘⠀⠀⢀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡄⠈⠀⠀⣄⠀⢀⠀⠂⠤⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⡄⠀⠈⠁⠄⠑⢄⠀⠁⠚⠥⠶⡀⢀⠁⢲⠤⣀⠀⠀⢀⠀⠀⠀⠀⠀⠠⢃⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠐⠀⢠⡼⠊⠑⠊⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⢀⢉⡀⠀⢐
⠀⠀⢈⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠐⠃⠀⠀⠀⠁⠀⠈⠀⠀⠁⣀
⠀⠀⠸⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠤⣀⡀⠬⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣤⠀⠀⠀⣀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡠⣄
⠀⠀⠀⠀⠀⠀⣿⠿⠋⠛⠋⠀⠀⠀⠀⠀⠀⠀⠟⡙⡻⢿⠀⠀⠀⠀⠀⠀⠀⠀⡿⡙⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠷⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠦⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⡀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⣰⢿⡟⣦⠡⣦⠀⠀⠀⠀⠀⠀⡤⣥⣠⣤⣽⣤⠀⠀⠀⠀⠀⠀⢠⡶⠆⠼⢳⡄⠀⠀⠀⡀⠀⠀⠀⣠⣄⣁⣠⣇⣄
⠀⠀⠀⠀⠀⠃⢸⠁⠈⢿⠋⠀⠀⠀⠀⠀⢠⣷⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠈⠀⣤⡀⣰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢿⣿⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡜⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣶⣶⣶⣶⡇⠀⠴⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠻⠇
⠀⠀⠀⠐⠂⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⡿⠿⠿⠿⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠒
⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣿
⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠸⠷⠀⠐⠶⠆⠀⠀⠀⠀⠀⠀⠸⠗⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠴⠶⠀⠠⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣷⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠉⠛⠛⠛⠛⠛⠋⠁⠀⠀⢹⣿⡇⠀⠀⣼⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣤⣤⠀⠀⠀⠀⠀⣤⣄⡀⢸⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⢸⡟⠀⠀⠀⣀⠀⠀⠀⠀⡀⢠⣄⠀⢿⣆⡀⠀⢻⣿⡇⠀⠶⠶⠀⠀⠀⠀⠠⠶⣄⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠠⠐⢪⡅⠀⠀⣴⣿⡆⠀⠀⠀⢠⣾⣿⠀⡼⠋⡅⢰⢮⡛⠀⣶⡆⠢⠀⠀⠀⢀⡤⠀⠀⠀⣿⣏⡀⠀⠀⢰⣾⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠳⠄⡇⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⢧⠔⠁⠘⡀⢹⠀⠿⣿⠀⠀⠀⠀⢺⣿⡆⠀⠀⡼⠊⠁⠀⠐⢆⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣰⠈⡄⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⡀⠠⠀⠀⠀⠀⢃⠀⠀⠸⣦⣤⢤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣠⠄⢀⠠⣤⠂⠁⣀⠀⢃⠀⠣⡀⠀⠀⠀⠑⠄⡀⠈⠉⠁⠀⢀⠤⠊⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡅⠲⠞⡢⠀⠀⢘⠶⠒⢠⣆⠈⠢⡠⠊⠈⠐⠴⡀⢀⢸⠁⠐⠀⣟⣀⠠⣀⢀⠤⢤⠒⠁⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠀⠁⠀⠀⠈⠁⠈⠀⠀⠀⡂⠀⠉⠒⢶⠤⣀⠀⡰⢁⠎⠑⠭⠤⠅⠊⣥⠐⠉⠀⠀⠀⡤⠂⠁
⠀⠀⠀⠀⠀⡠⠚⡆⠀⠀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄⠀⠀⠘⠑⢄⠸⠁⠆⠀⠀⠀⠀⠀⠀⠹⡄⢠⡠⠐⠈⠁
⠀⠀⠀⠀⢰⠀⠰⠯⠴⠽⢤⠖⠒⠀⠀⠀⠀⠀⢰⠂⠣⡼⠯⠶⠑⠂⠈⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠲⡤⠘⡤⡤⠂⠀⠀⠀⠀⠘⠤⢴⠲⡶⠀⠀⠀⡋⡌⣀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠊⡀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠲⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢶⠀⠀⠀⡌⠃⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄⠀⢀⣀⡀⠀⢀⣀⠀⠠⠐⡎⠀⠀⠀⠆⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢰⠀⢰⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸⠀⡘⠃⢄⡀⠀⡠⠊⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠠⠒⠀⠀⠀⠂⢤⠀⠒⢺⠁⢠⠁⠀⢠⠊⠈⠀⠀⠀⠀⠐⢢⠄⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⢠⠀⠰⠣⠄⣀⣀⣀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢰⠀⠀⠀⠀⠀⠀⠀⡆⠀⣸⠀⠀⠈⠓⠂⠠⡌⠀⠀⢹⡄⠀⢰⡂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠀⠀⠀⠀⠀⠀⠀⢘⠀⠉⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠇⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡄⠀⠀⠀⠀⠀⠀⠀⢈⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠃⠀⠖⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠢⢀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠳⣤⣀⠤⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⣭⢽⣟⠛⠛⠛⠛⠛⢛⠻⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢓⢂⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡬⠒⠂⠠⠤⠤⠤⠄⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣩⣁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢊⠠⠐⠁⢸⠈⡉⠒⣬⣑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⢉⡠⠖⠁⠀⠀⠠⢸⠠⣂⠥⢒⠁⠉⠒⠬⣑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⢁⡠⡦⣗⠥⡆⠀⠀⠀⠀⢘⡩⢐⡈⢴⡄⠀⠀⠈⢸⡁⢦⢌⡑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⢁⠠⠐⠉⡇⡈⠀⠈⣇⠀⠀⠀⠀⠀⡂⠅⢂⠉⡀⠀⠀⠀⢸⡂⠀⠈⠌⢓⠦⢈⡐⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⢀⣤⠖⠁⠀⠀⠀⣉⠠⠀⠌⠧⠀⠀⠀⠀⠸⡇⠤⢒⢨⡇⠀⠀⠀⢸⠃⠂⠄⠀⠸⠅⠀⠈⠁⠦⢍⡒⠢⣀
⠀⠀⠀⠀⢀⠴⠊⣀⣤⣾⣋⣩⣁⣀⠀⠀⠀⠷⠨⠴⠬⠤⠤⠤⠤⠄⠐⠓⠚⠒⠓⠓⠒⠒⠒⠚⠚⠀⠚⠐⠓⠒⠒⠒⠒⠒⠚⠛⠒⠒⣿⡆
⠀⠀⠀⠀⣿⣻⠉⡉⣥⡤⠤⠤⠤⢶⠲⣲⡖⢒⣲⡒⢒⣔⠚⠚⣋⡙⣭⡙⠋⣏⡙⣋⢍⣙⣉⣍⣉⣉⣭⠛⣻⡝⢻⠉⡛⠛⢛⢰⠀⢐⡯⡇
⠀⠀⠀⠀⠻⣽⠀⢸⠀⠠⠀⠐⠀⡸⠛⠉⠉⠉⢉⠍⠛⠚⠹⢉⠂⠐⠀⠋⠙⠁⠉⠡⠔⠉⠁⡐⢉⠉⠔⠡⠋⠜⢩⡀⢑⠔⠁⠨⠅⢸⠟
⠀⠀⠀⠀⠀⠙⠀⢸⠀⠀⡠⠞⠀⡟⠀⠀⠀⠀⠘⢀⣈⢥⣘⣧⠯⠶⠇⠧⡀⠀⠀⠀⠀⠀⠀⠀⠡⠊⠔⡡⢂⠔⡠⠇⠠⠂⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡆⢸⠁⠀⠉⠲⠘⠁⠀⠀⠀⠀⠸⣩⣴⠾⠿⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢊⠔⣣⠀⡁⢢⠀⡇
⠀⠀⠀⠀⠀⠀⠇⢸⡄⠀⢠⠃⠃⡠⠂⡀⠆⠀⠀⡏⠃⠴⢀⢸⠀⠀⠀⠠⣦⠠⠀⠀⠀⠀⢠⣆⠠⡀⠀⠀⠀⠀⠈⠘⡄⢠⢸⡅⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⢀⠃⠀⠇⠁⣉⠦⡀⠀⢃⠳⠖⠞⡰⠀⢀⠠⡆⠙⢠⠀⠀⠀⠀⠈⠩⢷⡌⠢⡀⠀⠀⠀⠀⠹⡈⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠇⢠⡆⠀⠠⡈⡞⢡⢤⠾⢂⠈⠁⠉⠉⠉⡒⢿⢤⣝⠖⢡⠀⠀⢀⣠⣄⣀⠀⠙⣦⡘⠢⣀⡠⢤⡀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣛⣻⠠⠀⠈⠀⠈⠓⢶⣆⢹⠰⡇⡜⠶⠙⣯⡴⠞⠉⠀⠁⠀⠀⠘⠢⣈⠉⠙⠙⠾⠷⣦⠀⠚⠓⣅⣀⣙⣛
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣯⠀⠄⠀⠀⠀⠀⠀⢀⡿⡗⠉⣦⠒⠫⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⡐⠂⠄⡀⡀⢀⠀⠰⣿⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⠄⡀⢀⢀⡀⠀⠀⣾⣃⡸⢾⢀⢶⣎⣙⡀⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠨⠁⢢⠠⢂⠅⡂⢤⠁⢘⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠀⡃⠀⢻⣂⠔⠐⡡⢂⢐⡦⠛⣀⡼⣇⢀⠽⣿⠊⢔⠕⡡⢊⠔⡡⢊⠔⡡⠊⡠⢂⠌⠥⠸⡀⠐⡠⢄⡟⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⢸⠀⣈⣀⡉⣉⢁⡨⠡⢉⣍⡥⢀⡭⣈⡩⣉⠈⠙⠉⠉⣛⢁⣋⣘⣁⣉⣘⢁⣚⡒⠃⣚⠚⠚⠓⠀⠒⠛⠒⠒⠲
⠀⠀⠀⠀⠀⠀⢸⠀⣩⢁⠬⢃⠉⠀⠀⠀⠆⢐⠡⢂⠡⠂⡇⠀⠀⠈⠅⢰⠈⢂⡡⢚⡇⠀⠀⠄⢸⣚⠭⢚⠭⢘⡫⠁⠀⠐⢒⢸⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡂⠁⣺⠀⠀⠀⠁⡠⢂⡁⠐⡁⡇⠀⠀⠀⠀⡩⢒⡡⢀⠩⠂⠀⠀⠀⢸⠄⣊⠄⢊⠁⡚⠀⠀⠀⢀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠀⣸⠀⡀⠀⢲⠀⠀⠀⠀⣗⡡⢒⡡⢐⡇⠀⠀⠀⠀⡒⡡⢂⠁⣢⠀⠀⠀⠀⢸⠊⠄⡊⠄⡊⡅⠀⠀⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣺⠄⢂⡡⠔⠀⠀⠀⠀⡯⢂⠅⢂⠥⡇⠀⠀⠀⠀⡥⡢⠑⡀⢴⠀⠀⠀⠀⢨⠕⡠⢐⠥⢒⡇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡆⢸⠂⠅⡠⠅⠀⠀⠀⠀⡯⠔⡈⠄⣂⠅⠀⠀⠀⠀⡃⠔⡀⠐⡮⠀⠀⠀⠀⣅⠜⡈⠔⡊⠔⡇⠀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠁⡀⠤⠀⠀⠀⠀⣗⠠⢐⠠⢀⠀⠀⠀⠀⠠⢖⠁⣂⠴⡃⠀⠀⠀⠀⣗⠀⢐⠡⠐⠫⠇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⢐⠠⢐⢹⠀⠀⠀⠀⡗⡈⠔⡨⢘⠀⠀⠀⠀⢸⢇⠈⢀⠨⡇⠀⠀⠀⠀⠃⡊⠐⡈⠄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠐⡨⠕⣪⠀⠀⠀⠀⡯⢀⠅⠂⢹⠀⠀⠀⠀⢸⠅⡊⠔⣀⡇⠀⠀⠀⠀⡯⠀⡁⢀⠡⢒⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠇⢸⠠⢐⣈⠴⠀⠀⠀⠀⠋⠀⢀⢐⢸⠀⠀⠀⠀⢸⠩⠀⠈⣰⡇⠀⠀⠀⠀⡤⠁⠊⠀⢐⢹⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠁⠀⠀⠀⠀⠒⠈⠂⠉⠈⠘⠀⠀⠀⠀⠀⠃⠀⠃

⡅⣩⢩⣉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⢁⠉⠀⢀⡀
⠲⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⠆⠉⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠐⠒⠀⠄⠉⠀⠀⠰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⢄⡊⢡⡅⠨⠁⠈⠀⠀⠀⠘⠀⢀⣄⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠁
⠀⠀⢰⠃⣈⠀⠀⠀⠀⠀⠀⠀⣜⢁⠀⠈⠡⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠠
⠀⠀⠈⠈⠉⢃⡀⠀⠀⠀⠀⠀⠀⠟⠂⢷⢂⡙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠚
⠀⠀⣄⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⠀⡀
⠀⠀⠈⠉⠥⠀⡲⠀⠐⠀⠀⠀⠀⠠⡠⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈
⠀⠀⠀⠀⠈⢍⡀⠠⠐⠊⢰⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠚⡃⠶⠶⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉
⠀⠀⠀⡀⠀⠀⢈⡀⢂⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠤⠀⠀⠂⠀⠐⢂⠀⠠⠄⡀⠀⠀⠀⢀⡀⠴⠄⢂
⠀⠈⠀⠀⠀⢐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⠄⢡⣠⣀⠀⣤⠀⠀⠆⠉⠀⠀⠘⠈⠁⢂⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠁⠀⡠
⠀⠀⠀⠶⣀⣠⢘⠀⡀⢰⠠⡤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢠⠀⠐
⠀⠈⠀⠐⠀⢀⠹⠌⠷⢞⠀⣉⣷⡁⠠⠴⡤⡀⠆⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣦⠀⣤⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠤⠀⡁⠝⠀⠀⠈⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢀⠃⠀⠀⠀⠉⠉⠀⠈⠉⡀⠀⠈
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⡀⠄⠀⠈⠀⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠘⠿⠿⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆⠙⠿⠿⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠿⠿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠻⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠛⠿⣿⣿⣿⡿⢿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠈⢿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⡿⠃⠘⢿⣿⡟⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠸⣿⣿⠏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡄⠠⠄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠐⠓⠘⠁⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⢦⠀⢠⣆⠀⡄⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠇⣮⣼⡄⠸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠋⠘⠃⠀⠃⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠔⠀⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠢⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⠂⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠐⠄⢀⡀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠒⠒⠒⠢⠭⢍⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡈⠀⠌⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⠊⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠡⠐⢄⡀⠈⠀⢠⠂⢠⠀⡇⠀⠀⠀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⣠⠃⢠⠇⠀⠀⢀⡎⠀⠀⠀⠀⠉⠉⠙⠒⠢⠤⣀⡀⠑⣄⠀⠀⠀⠀⠀⠑⢄⠈⠉⠀⠄⠀⢘⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠇⠀⠸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⢆⠘⡆⠀⠀⠀⠰⠀⠀⠉⠢⠄⠂⢀⡀⠀⠉⠉⠉⠉⠉⠉⠍⠠⠤⠁⠍⠉
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⢸⠀⡇⠀⠀⠀⢠⠀⠀⠀⠆⠀⠄⠁⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⡆⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠐⢀⠀⠀⠔⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠀⠀⠀⢸⠇⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⡠⠒⠞⠿⠭⡁⡒⢤⡀
⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠉⠑⠃⠀⠀⢀⣠⠞⠀⡟⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠐⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢱⠀⠀⢇⠀⠀⠀⠀⠈⠂⠤⠤⠤⠐⠒⠒⠊⠉⡉⠁⠠⠀⣠⠃⠀⠀⡆⠀⠀⠀⠀⠀⠐⠀⢰⠀⡇⡀⠀⠀⠀⠈⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠈⢆⠀⠈⢄⠀⠀⠀⠰⠆⠐⠂⠈⠁⠈⢀⣀⣀⣤⠤⠴⠚⠁⠀⠀⢀⠁⠀⠀⠀⠈⠀⠀⠀⢸⠀⠘⢌⠢⠤⣤⠤⠒⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠢⠬⠓⠤⠤⠤⠤⠀⠒⠒⠈⢉⡃⡀⠀⠉⠉⠘⡄⠉⠁⠀⠩⡀⣀⡀⠀⣀⠀⢀⡀⣀⠀⠀⠐⢉⢶⣒⠖⠒⡂⠁⠀⠀⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⡜⠠⠒⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⣕⡀⠀⠀⠀⠈⠀⡀⢉⠀⠉⠉⠈⠁⠀⢀⠴⠟⠀⠀⠀⡇⢰
⠀⠀⠀⣀⠤⣒⠱⣁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠐⠀⠉⠀⠀⠀⢀⡤⡀⠀⠀⣍⠃⠀⠢⠄⢀⡔⠁⡸⠀⠀⠈⠉⠐⠒⠀⠠⢀⡀
⠀⠒⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡂⠁⠀⠀⠨⡀⠐⠂⠁⠨⡬⠀⠀⢀⣀⡀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢁⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠬⠐⠊⠁⠀⠁⠒⠒⠂⠉⠀⢠⠀⡁⠀⠀⠚⠗⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢆⠔⠄⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣴⣿⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢆⡴⠂
⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠺⣿⣿⣿⡿⠿⡿⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡟⠀⢰⣿⣿⡟⠙⠋⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠐⣿⣿⠟⠥⠄⠀⠀⠈⠥⣤⠀⣿⣿⡿⠀⠀⠘⡁⠘⣿⠋⠻⠿⠟⠃⠈⠛⠿⠿⠹⣿⠇⠀⢸⣿⣿⠁⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⣢⡀⠀⠀⠐⣰⡆⢹⡿⠄⠈⢈⠎⠉⡟⢯⢠⣿⡍⠂⠀⠀⣴⡈⠀⠀⣿⣦⠀⣸⣿⢃⣈⢉⠀⠀⠀⣈⠹⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠄⣣⠀⠘⠟⠃⠀⠀⠘⠟⠃⣸⠰⠀⠀⠸⡐⠀⠑⠸⠀⠙⠃⠀⠀⠀⠙⠛⠀⢠⣃⠄⠀⢋⢳⠘⣿⡆⠀⠀⠀⢿⡆⠀⢹⠟⡃
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⣄⢀⣠⡄⠀⢀⠃⠀⠀⢰⡀⠃⠀⠀⠀⠆⠀⠀⠆⠀⠀⠄⠀⠀⢰⠈⡀⠀⠈⢹⡆⠀⠀⠀⠀⢀⡀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⠙⠋⢀⠀⠂⠀⠀⢀⠌⢃⠈⢄⣀⠀⠈⠂⢄⡀⠀⠀⢀⠠⠊⠀⠆⠠⠀⠀⢸⣷⡀⠀⠈⠛⠛⠁⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠸⢃⠀⣸⠇⣐⡆⠊⠉⠣⡄⠉⢰⠊⠀⠈⢲⢬⠄⢏⣁⡈⢧⡔⣀⠀⠘⢄⢈⠂⠺⢿⡿⢷⡶⠠⠀⡶⠾⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⢻⣧⠀⠀⠊⠁⠀⠀⠈⠐⠠⠂⠟⠀⠀⠀⠀⠐⢠⠈⠑⠠⡀⠣⠀⣀⠀⠈⡌⠈⠐⠊⠀⠉⠃⢿⡇
⠀⠀⠀⠀⠀⠀⡎⢰⣿⠃⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢀⢘⢢⡀⢀⠓⢀⠀⢸⡟⠘⠀⠀⠀⠀⠀⠀⠀⠀⢑
⠀⠀⠀⠀⠀⡌⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢌⢈⡖⣛⠋⠁⡠⠂⠰⢠⠁⠀⠀⠀⠀⠀⠀⠃⣤⠈⢆
⠀⠀⠀⠀⢰⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠈⢍⠢⡋⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢁⠠⣸⠃⠈⡄
⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⡇
⠀⠀⠀⠀⠸⠀⠀⣅⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢠⣾⠆⣾⣿⣿⣿⣷⣶⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠀⢀⠁⠒⠀⠤⠀⠤⠤⠀⠂⢹⠠⣶⡀
⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⡿⠃⠀⠻⢿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠐⡤⠀⣀⠠⠁⠈⢀⣀⣀⡠⠂⠀⠀⠀⠀⠟⡛⠿⠿⣿⣶⣾⡿⠿⠟⡛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡐⠀⠐⠀⠀⠏⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠰⡆⡼⠀⠀⠀⠀⠀⡧⢼⠀⠀⠀⠀⠀⠀⠀⠀⡗⢲⠀⠀⠀⠀⠀⢰⢠⡆⠀⠀⠀⠀⠀⠀⣤⣀⡇⠀⠀⠀⠀⢰⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡂⡈⠀⠀⠀⠀⠀⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠌⠁⢡⠀⠀⠀⠀⠀⠌⠈⠢⠀⠀⠀⠀⠀⢀⠜⠁⠱⠀⠀⠀⠀⠀⠘⠉⢢⠀⠀⠀⠀⠀⢠⠊⠘⡀⠀⠀⠀⠀⠠⠃⠣⡀
⠀⠀⠀⠀⠀⠀⠒⢠⡴⠚⠀⠀⠀⠀⠀⠳⢤⡔⠃⠀⠀⠀⠀⡛⢠⡴⠚⠀⠀⠀⠀⠀⠳⣄⡠⢧⠀⠀⠀⠰⠇⣀⡴⠃⠀⠀⠀⠀⠰⣤⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠈⠀⢄⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠀⠒⠀⠐⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠆⠀⠀⣀⠀⢀⡀⠀⠀⢂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠉⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⢧⠀⡀⢠⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠺⠇⠀⠀⠀⡙⠠⠄⠀⢠⠘⠀⠀⠀⠀⠀⠠⠮⣦⠊⠁⢀⠀⠰⠁⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠂⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠉⠀⠑⢆⠘⠄⠀⠀⠀⠐⠤⠀⠂⠉⠃⠤⠀⠀⡴⠉⡀⠛
⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⣀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠢⡈⠀⡠⢊⠈⠔⠁⠀⠀⠀⠀⠘⢌⠘⡁⢰⢄
⠀⠀⠀⣀⠊⡘⢃⠀⠀⠤⠀⠂⠁⠁⠠⠀⡌⠀⠀⠀⡟⠀⠀⠀⢑⣁⠂⠀⠇⠀⠀⠂⠀⠀⠀⠀⠀⠈⠙⠖⠉⠀⠀⠀⠀⠀⡐⠁⠀⠸⠀⠀⠀⠂⠁
⠀⠀⠠⠤⠢⣽⡶⢀⡄⠀⠂⠢⠤⠖⠒⠈⠀⠀⠀⠀⢈⣢⣤⠀⠤⠠⠁⡂⠀⠀⠐⠄⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣀⠀⢤⠤⠒⠂⠉⠉⠉⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠀⠀⠀⠀⠄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⢈⠄⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⣐⠀⠠⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡉⠀⣀⣀⡀⠈⢢⣀⡠
⠀⠠⠀⠀⠀⠀⠀⠀⣀⠀⢐⠀⠀⢈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠤⠤⣀⣀⣀⡠⠴⢏⣹⠂⠀⠀⠈⠙⢿⠒⠭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠈⠀⠀⠠⠶⠀⠀⠀⡀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⡘⠀⠀⠤⠄⠐⠿⠋⠉⠉⠁⠀⡜⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠤⠰⠐⠎⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠠⠭⠥⠴⠖⠒⠒⠀⠀⠁⠀⠀⠀⡄⠶⡉⠐⢄⠀⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠀⠢⠘⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠄⠀⡀⡀⠀⠀⠀⠀⢀⡄⠀⠉⠔⠀⢡⡀⠄⡀⠄⠃
⠀⠀⠀⠀⠀⠀⠂⠁⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠈⠀⠌
⢠⠥⠀⣀
⠐⠰⠀⠀⡤⠤⠄⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⡔⠉⡍⠁⠀⠑⡄⠀⠀⠀⡠⠚⠡⠊⠀⠉⢢⡀⠀⠀⠀⡴⠋⠀⢀⠀⡙⢆⠀⠀⠀⡠⠊⠁⣀⣀⠈⠳⡀
⠀⢸⠲⠀⠀⠐⠐⠐⢽⠀⠀⠀⡇⠈⠦⠀⠀⠐⠀⡇⠀⠀⢸⠀⠀⠀⠈⠠⠄⢸⠀⠀⠀⡃⢰⠋⣶⡆⢱⠀⠁
⠀⠘⣄⠅⠀⠣⡀⣀⡜⠀⠀⠀⠳⣁⡆⠀⠀⠀⣰⠃⠀⠀⠘⣄⠀⠀⠀⠀⢀⡜⠀⠀⠀⢣⡈⠢⠽⠥⠊⣠⠃
⠀⠀⠈⠑⠒⠒⠛⠁⠀⠀⠀⠀⠀⠈⠉⠓⠒⠉⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁

⠀⠀⡠⢚⡽⠋⠉⠲⡄⠀⠀⠀⢀⣴⠉⠉⠉⠑⣄⠀⠀⠀⠀⣠⠖⣩⡙⢉⡲⡀⠀⠀⠀⣀⡠⠤⠤⠤⢤⣀
⠀⢰⢱⠃⠀⠀⠀⠀⣸⠀⠀⠀⡏⢸⠀⠀⠀⠀⡇⡇⠀⠀⢸⠡⠈⢄⠈⠀⠐⢻⠀⣠⢚⠁⠀⠀⠀⠀⠀⢸⡓
⠀⠸⡈⢆⡀⠀⣠⠞⡸⠀⠀⠀⢧⠈⠲⠤⠤⠞⢠⠃⠀⠀⠸⡔⢢⡀⠡⠀⠤⡻⠀⠘⢿⣍⠉⠀⠀⠉⣉⡽⠟
⠀⠀⠙⠢⠬⠭⠕⠊⠀⠀⠀⠀⠀⠑⠦⠤⠤⠒⠁⠀⠀⠀⠀⠈⠳⠤⠦⠥⠊⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⢀⠤⡶⢲⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢤⢒⢢⠛⡿⠾⡆⠀⠀⠀⠀⣠⠴⡶⠖⡤⣀
⠀⢠⠃⠐⠀⠀⠀⠀⢳⠀⠀⢠⡖⠉⠁⠀⠀⠉⠑⢦⠀⠀⠀⢸⣿⢸⢰⢻⣸⠁⠀⠀⠀⡜⢡⠛⢊⡅⠂⡏⡆
⠀⠸⡀⠀⠀⠀⠀⠀⡿⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢻⣾⣾⡾⠃⠀⠀⠀⠀⣗⠰⠐⡆⠐⣩⠒⡇
⠀⠀⠑⠤⣄⣀⡤⠔⠁⠀⠀⠀⠈⠉⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⡿⠿⡇⠀⠀⠀⠀⠀⠘⠲⣥⣔⣷⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⡠⠤⠤⠤⣀⠀⠀⠀⠀⠀⢀⠤⠄⠒⡢⢄⡀⠀⠀⠀⠀⠀⢰⠉⡆⠀⠀⠀⠀⠀⠀⠈⢣⣈⣯⡰⠋
⠀⢠⠊⠀⠀⠀⠀⠈⢳⠀⠀⠐⢭⣁⣠⠀⣟⣭⠆⢱⠀⠀⠀⠀⠀⢸⠉⡇⠀⠀⠀⠀⠀⠀⡠⠚⠛⠙⠣⣄
⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⣲⣷⣖⣧⣜⣋⡇⢸⠀⠀⠀⠀⢀⣎⣀⣧⠀⠀⠀⠀⠀⡜⡐⢡⡀⠀⠈⡘⡄
⠀⠈⠣⣄⡀⢀⣀⠴⠃⠀⠀⢸⣞⣻⢽⣿⣁⠤⣤⠎⠀⠀⠀⠀⢸⠦⠢⢾⠀⠀⠀⠀⠀⢡⡀⠀⠀⢀⠠⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠃⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣷⣄⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⢃⣽⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣏⠰⣿⣿⣧⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣹⣿⣯⠙⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⡿⠟⢣⣷⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢻⡷⠀⢈⡿⠀⠀⠀⠀⠀⠀⠀⣿⡖⠀⠉⠉⠹⠟⠀⠙⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣾⠀⠐⣷⣷⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣠⡄⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣛⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠊⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡄⠀⠀⠀⠀⠉⠑⠢⣄⠀⠀⠀⠀⠀⠰⣊⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⣴⠈⠀⠒⠒⠠⢄⡀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠑⢄⠒⡀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⠁⠀⠙⠄⠀⠠⡄⠀⠈⠐⡄⠀⠀⢣⠀⠀⠀⡼⠉⠉⠫⡱⡀⠀⠀⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⠀⠀⡀⠐⠂⠢⠀⠀⠀⠘⢀⠄⠀⠇⠀⠀⠀⢀⠀⠀⣀⢁⠀⢑⢲⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡒⠓⠬⠩⡦⠀⠌⠀⠀⠀⢆⠚⠀⡀⠀⠀⠀⠀⠼⢆⡀⢛⠧⠀⢨⠀⡕⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⢀⠔⠀⡀⠀⠁⠤⠃⠀⠀⡀⢠⡐⠀⠀⠀⠁⠉⡁⠀⢐⡀⡄⠀⠀⠑
⠀⠀⠀⢀⢀⡀⠀⠀⣀⠈⠓⢄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢯⡃⠈⢦⠀⠀⡠⠾⠁⡠⠂⢭⠇⠀⠈⠁
⠀⠀⠀⢀⠤⢴⠀⣦⡄⠀⠀⠀⠱⠂⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑⠀⠁⣄⠀⢁⠀⠀⠀⠠⢤⠄
⠦⠶⠪⢆⠀⠸⠀⠅⠀⠀⢚⣃⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢉⠇⠀⠀⠀⠉
⠀⠀⠀⠀⠉⠃⠉⠀⠀⠀⡐⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⡂⠀⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⠀⠠⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢖⡂⢲⠀⠤⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠳⡖⡖⠂⠁

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⠿⠛⠛⠛⠛⠛⠛⠛⢿⣛⠛⠛⠛⠛⠛⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣄⣀⡀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡈⠉⠙⠒⠶⢤⣀⣼⠷⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⠀⠺⠧⢤⣀⡀⠉⠙⠒⠦⢤⣀⡀⠀⠀⢸⠁⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⠶⠒⠛⠛⠲⡦⣄⣀⠉⠓⠲⠤⣀⡀⠀⠉⠙⢲⡞⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⢠⣼⡇⠀⠀⠀⣞⠙⡆⠉⠙⣳⣦⣤⣀⡉⠙⠒⠢⢼⡷⢤⣀⡀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢀⡼⠀⠀⠀⠀⠙⠋⠀⠀⠸⣇⡼⠀⠈⠉⠛⠲⠤⣌⣀⡀⠈⠉⠓⢾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠻⠤⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠉⠛⠒⠦⢼
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢀⣀⣀⣤⠤⣴⠶⠶⡶⠶⣶⣒⣺⠉⡽⢤⣤⣀⡀⣶⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⢦⠀⠀⠀⢘⡇⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⢀⡔⠛⢿⠛⢛⣿⠿⣽⣱⢻⢶⡃⠂⢢⠇⢰⣹⡆⡭⡟⢷⣏⣠⠏⠻⣿⠀⣇⠀⢀⠀⠀⠀⠀⣀⠀⠱⢄⡠⠏⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⢸⡉⢈⡞⠀⠑⠛⢐⣶⠈⠛⣄⢻⣞⡁⠀⠀⠀⠈⠣⠟⠀⠈⢣⠀⠀⣿⠀⢹⡀⠀⠑⠢⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣤⡄⠀⠀⠀⢠⣶⡖⢻⣯⣙⣿⢃⠀⠀⠀⠀⢠⠖⠒⡄⡇⠀⣿⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢧⡀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠘⣻⣿⠿⣿⡑⠤⠴⠀⠸⠽⢦⣀⣀⣹⡇⣿⣿⣶⣶⣦⠘⠦⠴⢃⡇⣸⣿⠀⢠⠞⠉⠳⠶⣤⣤⣀⣀⣠⣴⠶⠟⠋⠀⠀⠈⢦⡀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠈⣛⣥⣴⣾⠿⠒⠒⣛⡿⡟⠛⠻⣿⠟⠓⡻⠿⣿⣿⠋⠀⠀⢀⡼⣱⢟⣿⣰⠃⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⢣⠀⠀⢳⡀⠀⠀⢸
⣿⠀⠀⠀⢀⣠⣖⠋⠉⠻⠶⠋⠀⠀⠘⢯⣥⡇⠀⠀⠀⠀⢺⡙⢛⡆⠉⠛⠦⣴⣾⣴⣿⡞⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡄⠀⠀⢧⠀⠀⢸
⣿⠀⣠⠖⠋⠸⣏⢛⡷⠀⣠⣄⣤⠀⠀⠀⠀⠀⣤⣴⢶⠀⠀⠙⠉⠀⣴⠾⢻⡌⣿⠿⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠸⡄⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣄⠀⠀⠀⠀⠀⠀⠤⢾⠿⢡⡾⣿⣿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⢯⠶⢻⡇⠈⠓⠶⠤⣄⠀⠀⠀⢀⣴⣾⣇⣼⣧⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠻⠴⠛⣷⡼⠋⠉⠉⠛⠲⣌⢳⡄⣴⠟⠉⠀⣉⠙⠛⠛⠛⣿⣷⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⡟⠀⠀⠀⠀⣄⠀⠈⢳⣿⡏⠀⠀⠀⠛⠁⢠⣤⠀⣿⣿⡦⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢠⡾⠁⠀⠺⠂⠀⠉⠀⠀⠀⣿⠀⢠⡏⠀⠀⠀⠀⠁⠀⢻⣿⣤⡼⠟⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢾⡁⠀⠀⢠⡄⠀⠀⢀⡇⠀⣿⣧⡀⠻⢦⣤⡴⠏⠀⠀⠈⢻⡟⠀⢰⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠙⢦⣄⠀⠙⠳⠚⠋⢀⡼⣿⠀⠙⠲⢦⣤⣀⣀⣀⣤⡶⠋⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢀⣴⠿⠷⣶⣶⣖⠛⠋⢀⣧⣤⣤⣤⣼⡆⣸⡭⣋⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣯⠀⠀⠀⠈⠙⠛⠷⢾⣧⣄⣀⣤⡾⠟⠋⠉⠳⠵⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡴⠏⠀⠀⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣧⣄⣀⣀⣠⡤⠐⠊⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⣶⠛⠩⣿⠿⣦⡀⠀⠀⠀⠀⢀⣾⣏⣀⣀⣀⣀⣤⣤⢤⣀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣰⠊⠀⠈⠻⣦⣀⠀⣠⣿⠋⠙⢯⡉⢉⣵⠟⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠉⠂⠴⠖⠚⠋⠉⠛⣋⣡⣤⡶⠛⠙⠋⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⣀⡠⠴⠶⠶⠶⠞⠛⠋⢹⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⢦⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⣠⠶⠛⠉⢹⡆⠀⠀⠀⣀⣤⠴⠚⠁⠀⠀⠉⠛⠶⣦⣤⣀⣀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⢀⡴⠋⠁⢠⣤⢤⡴⠻⠦⠤⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠸⡧⠴⠶⠚⠻⠞⠋⠑⠲⢤⣀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣌⢙⣿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠈⢻⡶⣦⣄⡀⠀⠀⠀⢸⡆⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⠉⠛⠳⠶⠾⠇⠀⠈⠛⠿⠶⣶⡶⠶⣶⡶⠶⣿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣷⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⢻⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢦⡿⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⢿⣿⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣄⣀⠁⠿⠻⢄⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠛⠚⣛⣿⣛⣋⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣄⣀⣹⡄⠀⠀⠀⢺⣅⡀⢀⡴⠖⠛⠉⠈⠉⠉⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠷⢤⣤⣤⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⡀⢀⡀⢀⠀⢀⠀⣀⠠⢠⠀⡀⢀⡀⢀⡀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠛⠘⠋⠘⠃⠙⠃⠛⠡⠋⠀⠀⠚⠯⡘⠃⠙⢁⠂⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢉⣄⣄⡀⢠⡦⠤⠀⡐⠀⠠⠀⠀⠀⠀⠠⠁⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠘⠛⠍⠿⠆⠃⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠁⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⣰⣦⣿⠅⠢⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠸
⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡠⠈⠈⢀⠁⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠈⡃⠀⠀⠁⠀⠀⠅⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢦⠰⠀⠀⠀⠀⠀⢀⠀⠴⠤⠦⠆⠀⠀⢀⢄⠀⠀⠠⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠚⣆⡀⠀⠀⠈⠂⠀⠖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠁⠀⠀⠀⠄⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠌⢠⠂⠀⠀⢐⣦⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠠⠀⠀⠀⠀⠐⠀⠂⠀⠀⠊⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠘⠄⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⣏⠁⠀⢿⢠⣾⣽⡿⠞⣿⠀⠰⡏⠥⠀⠀⢾⣿⣿⣹⠷⠀⡷⠺

⠀⠀⠀⠀⠀⠀⣾⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⢴⣤⡓⠃⡀⠀⠃⠀⠀⠀⠀⢀⠐⠒⠶⣃⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠉⠀⠀⢘⣀⣴⣿⠃⠰⠀⠀⠀⠀⠀⠀⡁⠀⢠⣿⠟⠋⢀
⠀⠀⣦⡄⠀⠀⠀⠀⠀⠀⠀⣴⠶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣉⣩⣀⠀⠀⠀⠀⠀⠀⠀⠻⡶⣿⡟⠀⠀⣬⣷
⠀⠀⢹⣷⡀⠀⠤⠔⠋⠓⠿⠮⢕⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠂⠀⠀⣷⡀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠺⡼⠐⠔⠋⣷⠀⡀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⢠⣆⠀⢸⣿⠁⠀⠲⠂⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⣀⣶⠒⠉⠉⠉⠁⠘⠀⠘⠋⡁⢀
⠀⠀⠀⠀⣴⠖⠲⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢺⣁⣀⣀⠀⠀⠀⠀⠀⢢⠈⠀⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⣀⠀⠀⢀⠗⠶⠒
⠐⠶⢶⡌⠁⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠓⠓⠀⢿⣯⠭⡀⠐⠢⣦⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣄⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡅⠀⠀⠀⠀⠀⠀⢳⠾⠿⢭⣯⣶⣷⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠃
⠀⠀⠀⠀⢠⠶⣿⡆⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍
⠀⠀⠀⠀⠀⠀⠈⣓⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣚⡺⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠘⣿⣤⡀
⠀⠀⠀⠀⠀⠈⡱⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣿⣵⢾⡀
⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⠈⠛⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⢀⣾⣿⠟⠋⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⣿⠆⠘⠛
⣄⠠⠂⠀⠀⠀⠀⠀⠸⡇⠀⠀⣼⣿⠃⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⣬⣥
⠫⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠻⡟
⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣼⣯⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿