Star of david Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⣴⡄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡜⠀⠘⣄⠀⠘⣄
⠀⠀⠀⠰⣗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠀⠘⠒⠒⠒⠚⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡷
⠀⠀⠀⠀⠘⢆⠀⠀⣠⡤⠤⠤⢄⠀⠀⢠⣤⡤⠤⠤⠤⠤⣤⣄⠀⠀⢀⠤⠤⠤⣤⡀⠀⢀⠎
⠀⠀⠀⠀⠀⠈⢆⠀⠈⢧⠀⣠⠃⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⢣⠀⢠⠏⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠈⢣⠀⠈⠳⠃⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠳⠋⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⡴⡄⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡴⣄⠀⠘⣄
⠀⠀⠀⠀⠀⢀⠞⠀⢀⡞⠀⠙⡄⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡜⠀⠘⣆⠀⠘⢆
⠀⠀⠀⠀⢠⠎⠀⠀⠙⠓⠒⠒⠊⠀⠀⠘⠛⠓⠒⠒⠒⠒⠛⠋⠀⠀⠈⠒⠒⠒⠛⠀⠀⠈⢦
⠀⠀⠀⠰⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⠀⠀⢠⠤⠤⠤⢤⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⢧⠀⢠⠋⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠻⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣢⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠄⢹⠋⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠐⢱⠋⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠆⣠⠏⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠠⣀⠃⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢐⠃⠀⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⢄⣀⣀⣤⢤⠄⠤⠤⠤⠤⠤⠰⠲⢒⣤⣀⡀
⢶⣶⣖⣒⣒⣒⣒⣒⣚⣊⣩⠹⠽⠭⠭⠭⠿⠶⠃⠀⠀⠀⠀⡀⠋⠛⠛⠀⠀⠀⠀⠀⠀⠓⠒⠛⠋⠚⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢳⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠳⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠙⡄⠣⠀⠀⠀⠀⠀⠀⣦⢤⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢎⠱⡄⠀⠀⠀⠀⠈⢇⠐⠁
⠀⠀⠀⠀⠘⢆⠑⡄⠀⠀⠀⠀⠈⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡘⢄⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠈⢢⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠳⡄⢧⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢤⣧⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡰⠛⠚⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠁⠜⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⠁⠀⡨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠡⠄⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡼⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠅⠒⡝⠁
⠀⠀⠀⠀⣌⠡⢐⡞⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⡜⠁⠀⠀⠀⠀⣠⣟⣆
⠀⠀⠀⡼⠐⠊⡜⠀⠀⠀⠀⠀⣴⠛⠿⠄⠀⠀⠀⠀⠀⠹⠯⠭⠭⠿⠍⠷⠒⠘⠚⠀⠀⠀⠀⠀⠀⠛⠋⠛⠁
⠀⠀⡜⠀⠀⡔
⠀⠜⠀⢄⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢲⡀⠀⠀⠀⠀⠀⢤⢴⡄⠀⠀⠀⠀⠀⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠑⣄⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡱⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣤⣤⣤⣿⣤⣿⣤⣤⣤⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠿⠟⠱⠿⠎⠻⠿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢧⣠⠏⠀⠀⠀⠹⣆⡼⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡘⠟⣴⣿⣿⣿⣆⠻⢃⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⢿⡀⠀⠀⠀⢀⡿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⡘⢿⣿⣿⣿⡿⢃⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢼⣧⣬⣷⣤⣤⣤⣾⢤⣼⡦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⣘⣛⡈⢛⣛⡛⢁⣛⣃⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣦⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⣷⡀
⣿⣿⣿⣿⣿⣏⢩⣭⢁⣬⣭⣥⡈⣭⡭⣹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⣏⢉⡟⠉⠉⠉⢻⡉⣹⠏
⣿⣿⣿⣿⣿⣿⣆⢡⣾⣿⣿⣿⣿⡌⢴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⣻⡇
⣿⣿⣿⣿⣿⡿⢡⣦⠹⣿⣿⣿⠟⣴⣌⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣏⣈⣧⣀⣀⣀⣼⣁⣹⣄
⣿⣿⣿⣿⣿⣷⣶⣶⣦⡰⣶⢆⣴⣶⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⢯⣉⡽⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡴⢦⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⠏⠀⠼⠥⠬⠧⠀⠹⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢠⡤⠤⣤⠀⢀⡤⠤⠤⠤⠤⢤⡀⠀⣤⠤⢤⡄⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢳⣴⠃⢠⠏⠀⠀⠀⠀⠀⠀⠹⡄⠘⣦⡞⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⡼⠻⡄⠘⣆⠀⠀⠀⠀⠀⠀⣰⠃⢠⠟⢧⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠘⠓⠒⠛⠀⠈⠓⠒⠒⠒⠒⠚⠁⠀⠛⠒⠚⠃⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⣆⠀⢲⡒⢒⡖⠀⣰⠒⠒⠒⠒⠒⠒⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠳⠞⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣼⣿⣿⢁⡀⢀⣈⣿⣿⣧⡀⣀⣀⣀⣀⠀⢀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄⠀⠀⢀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⢠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣁⣀⣀⣀⣻⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣟⣁⣀⣀⣈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠑⠳⠈⢆⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣀⠀⠀⢣⡀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⡔⠙⣦⠀⠀⠱⡀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⣼⣵⣰⣜⣱⠀⠀⠑⣄⣈⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢫⢉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⣰⠛⠱⡄
⠀⠀⠀⠀⠀⠀⢣⠁⠀⢠⣤⣶⢶⣶⠆⠀⠀⣠⠶⣶⢶⢶⢶⢦⢶⢆⢦⡀⠀⠀⢶⣶⡲⡶⡶⡖⣲⠆⠀⠀⢈⠼⠁⢡⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠹⡙⣯⠋⠀⠀⣰⠣⢡⢩⠟⠙⠉⠉⠉⠉⠉⠙⡄⠀⠈⢏⠙⣯⠙⢙⠏⠀⠀⣠⢫⢂⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠑⠂⠀⠀⡜⠏⡄⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⢢⠈⢦⠃⠀⠀⡴⠇⢄⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡞⠘⣀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠳⠁⠀⢀⠞⡊⢤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠘⡿⣦⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⣠⠏⡈⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⠀⠀⠘⡄⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠘⣿⢿⡅
⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢠⠛⣇⠀⠀⠈⢄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⣄⠀⠀⠘⢆⠛⣄
⠀⠀⠀⠀⠀⠀⡜⠀⠀⣠⢃⣃⢡⢧⠀⠀⠈⢆⠘⣄⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⣠⡋⠙⣦⡀⠀⠈⢦⠘⢆
⠀⠀⠀⠀⠀⢌⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠙⠛⠃⠀⠀⠠⠣⡈⢆
⠀⠀⠀⠀⢬⣈⣌⣤⡤⡤⡤⡤⡤⡤⡤⣤⡀⠀⠀⣤⣤⠤⣤⣤⡤⡄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⢀⠬⢅⣑⡈⢦
⠀⠀⠀⠀⠀⠉⠳⠾⠾⠶⠶⠴⠕⠓⠓⠓⢳⠀⠀⠘⡋⢻⡚⢻⠎⠀⠀⣰⠏⠉⣷⠮⠮⠬⠽⠽⢭⣩⣋⣌⣍⣍⣒⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠸⡄⣳⠋⠀⢀⡴⡉⠐⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘⠁⠀⢀⠺⠐⢠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢀⠀⣡⠏⠀⢡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠂⡰⠋⠈⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⣀⡝⠁
⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠑⠨⠒⠡⠊⠔⠡⠊⠔⠡⠊⠔⠡⢊⠔⠡⠊⡐⢁⠐⡠⢊⠄⡂⢊⠔⡢⢂⠕⡡⢊⠀⡡⢊⠔⠡⠀⠜⠡⢪⠊⠔⠱⠳⠂⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠋⠹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠃⠀⠀⠘⢿⣿⣿⣧⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄⠀⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆⠀⠀⠀⢠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣄⠀⣴⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⠀⣠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠀⠙⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃⠀⠀⠀⠹⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⠘⢿⣿⣿⣷⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡄⠀⠀⢠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣄⣰⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏⠀⠀⠹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣧⣤⣤⣤⣤⣼⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄⠀⢀⣼⣿⡿⠁⢀⠀⡀⠀⠀⢀⠀⠀⠈⢿⣿⣧⡀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⣾⣿⡟⠁⡆⢰⠀⡆⢀⡀⢰⠀⡆⢰⠀⢻⣿⣷⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⠀⠀⡇⢸⠀⢇⠘⠃⡸⠀⡇⢸⠀⠀⠻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⠀⠀⠘⣄⠳⢌⣹⣏⡡⠞⣠⠃⠀⠀⣴⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⢿⣿⣧⡀⠀⠈⠓⠤⢸⡇⠤⠚⠁⠀⢀⣼⣿⡿⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃⠀⠈⢻⣿⣷⡀⠀⠀⠀⣸⣇⠀⠀⠀⢀⣾⣿⡟⠁⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣶⣶⣶⣶⣿⣿⣷⣶⣾⣿⣿⣿⣿⣷⣶⣾⣿⣿⣶⣶⣶⣶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣬⣭⣤⡹⣿⡄⣤⣤⣤⣤⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠓⠒⠛⢣⡀⠹⡟⠛⠛⠛⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⣿⡍⢩⣭⠍⠉⠉⠁⠘⣿⣎⢩⣿⠟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡀⠰⡶⠒⣲⣶⣶⣶⣷⡀⠹⣶⠂⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⣱⣿⠃⠀⠀⠀⠀⠀⠘⢣⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡜⠁⣰⣿⣿⣿⣿⣿⣷⡴⠁⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⡿⢱⣆⠀⠀⠀⠀⠀⣰⣿⢣⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⡼⠙⣿⣿⣿⣿⣿⡟⠁⣰⠉⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣼⣿⣥⣌⢿⣧⢠⣤⣤⣤⣭⣥⣬⣿⣷⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⠀⠘⠛⣄⠘⢟⠛⠛⠛⠒⠚⠛⠂⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⢻⣷⡉⣭⡭⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣆⠈⢷⠖⢒⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣠⡄⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣰⠟⠻⣆⠀⠀⠹⣧
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⠃⠀⠀⣼⣋⣀⣀⣙⣧⠀⠀⠘⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⢿⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠙⣿⡉⠉⠉⠉⣿⠏⠀⠀⣰⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣆⠀⠀⠹⣿⠉⠉⠉⢉⣿⠋⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠘⢷⡀⢀⣾⠃⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠘⣷⡀⢀⡾⠃⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠈⢻⡾⠁⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⢷⡟⠁⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠰⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⣼⢷⡀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢀⡾⣧⡀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⢠⡾⠁⠈⢿⡄⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢠⡿⠁⠈⢷⡄⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⣿⣁⣀⣀⣀⣿⣆⠀⠀⠹⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠏⠀⠀⣰⣿⣀⣀⣀⣀⣿⣄⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠉⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⢀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⡀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⡄⠀⠀⢻⣏⠉⠉⣩⡟⠀⠀⢠⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠹⣦⣴⠏⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠘⠃⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠀⠀⠀⠀⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⠉⠉⠉⠉⠉⠉⠉⣩⣿⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣿⣍⠉⠉⠉⠉⠉⠉⠉⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⣀⣀⣀⣀⣀⣀⣀⣙⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⣋⣀⣀⣀⣀⣀⣀⣀⣻⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⠀⠀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢀⣴⠿⢷⣄⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢀⣾⠏⠀⠈⢿⣆⠀⠀⠙⣷⡄
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣟⣀⣀⣠⣿⣃⣀⣀⣀⣀⣻⣧⠀⠀⠘⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⣾⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣧⡀⠀⠈⢿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠹⣷⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣷⡀⠀⠈⢿⣦⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣴⡟
⠀⠘⣿⡄⠀⠀⠀⣾⡛⠛⠛⠛⢻⡿⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⡄⠀⠀⢻⣿⠛⠛⠛⠛⣿⡆⠀⠀⠀⣼⡟
⠀⠀⠘⢿⣆⠀⠀⠹⣷⡀⠀⣰⡿⠁⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠹⣷⡀⠀⣰⡿⠁⠀⢀⣾⠏
⠀⠀⠀⠈⢻⣄⠀⠀⠙⣷⣴⡿⠁⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠹⣷⣴⡟⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⢻⣦⠀⠀⣼⡟⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⢀⣼⠟⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠻⣧⣼⠏⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣾⠏⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣰⣿⣇
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣰⡿⠁⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⣼⡟⠀⢻⣧
⠀⠀⠀⠀⣠⡿⠃⠀⠀⣼⣿⡄⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢀⣼⣿⡄⠀⠀⠻⣧⡀
⠀⠀⠀⣰⡿⠁⠀⢀⣼⠏⠈⢿⣆⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⢀⣾⠏⠘⢿⣄⠀⠀⠹⣷⡀
⠀⠀⣴⡟⠁⠀⠀⣾⡏⠀⠀⠈⢿⣆⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢀⣾⠋⠀⠀⠈⣿⡆⠀⠀⠙⣷⡄
⠀⣼⡟⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⢿⣧⠀⠀⠘⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⠘⣿⡄
⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⡀⠀⠈⢻⣿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠻⣧⠀⠀⠀⣠⡿⠃⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠹⣷⡀⣰⡿⠁⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠙⠛⠛⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⣰⡿⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡼⢧⠀⠈⢧
⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣡⣤⣼⣡⣌⣧⡀⠈⢷⣄⣀⣠⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢳⡄⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⢿⡉⢹⠟⠉⢩⡟⠉⠉⠉⠉⠉⠉⠳⡄⠀⠹⡏⢉⡿⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⣳⠏⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢹⡞⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⠃⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠟⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣴⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢠⠟⢧
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⡼⣇⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣰⢯⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⣾⣁⣸⣆⠀⠘⢧⣀⣀⣀⣀⣀⣀⣴⣃⣀⣴⣇⣈⣷⡀⠀⢳⡀
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠘⢧⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠙⢷⡀⠈⢻⡙⢋⡟⠛⢋⡿⠛⠛⠛⠛⠛⠛⠛⠛⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢳⡞⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠤⣀
⠀⠀⠄⣱⠿⠿⠁⠻⠿⢦⡡⠄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣞⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣤⣤⠿⣤⣬⢣
⠀⢀⡀⣿⡷⢰⣿⡷⠰⣿⡇⡀⠀⠀⠀⠀⠀⠀⠈⢫⠏⠁⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠄⣤⣥⡈⠄⠌⢀⠀⠀⠀⠀⠀⠀⠸⠀⣸⣇⢀⣸⣇⢸⠁
⠀⠀⠤⢻⣶⣶⡁⣴⣶⡿⠣⠀⠀⠀⠀⠀⠀⠀⠰⠿⠷⡤⡴⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⢁⠌⠌⠿⠿⠏⠜⠄⠀⠀⠀⠀⠀⠀⠀⣀⣷⣄⠈⠿⠁⣠⣮⣀
⠀⠀⠀⠁⠭⠛⠛⠛⠫⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢂⢃⠊⠀⠉⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡿⠿⢿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⡐⠄⢀⣠⣤⣤⣀⠀⢆⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣴⡀⠀⠀⠀⢀
⠀⢐⠂⣰⡟⠛⣁⡹⠛⣧⡀⢂⠀⠀⠀⠀⠀⣎⠃⠠⣤⠾⠳⣤⠄⠘⡕⠀⠀⠀⠀⠀⠈⣠⣾⠿⠟⠘⠿⢿⣦⡈⠀⠀⠀⠀⢸⡄⠀⠰⣖⡞⠚⠳⣲⠆⠀⢸⡇
⠀⠤⠀⣿⡟⠘⠿⠗⠘⣿⠇⠠⠀⠀⠀⠀⠀⣾⠀⢠⣿⣄⣠⣿⡄⠀⡷⠀⠀⠀⠠⢀⠀⣿⣿⠆⢾⣿⠆⢾⣿⡇⢀⠠⠀⠀⠘⡱⠀⢠⣟⣄⣀⣠⣿⡄⠀⣎⠇
⠀⠈⠄⠘⡿⠿⠦⠾⠿⡏⠀⠍⠀⠀⠀⠀⠀⠁⢀⣀⠀⠘⠃⠀⣀⠀⠀⠀⠀⠠⢀⣸⠓⠛⠻⠶⠦⡠⠶⠾⠛⠓⢿⣀⠠⠀⠀⠈⢀⣈⣁⣘⣿⣋⣉⣁⣀⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣏⠀⠀⠀⠀⣹⠦⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠙⡯⣿⠿⠿⠶⠿⠿⣿⢿⠋


⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣖⠖⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣤⣄
⠀⠀⠀⠀⣤⣤⠯⢦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⢋⡹⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣜⠗⣁⣠⣦⣀⡑⢟⡄⠀⠀⠀⠀⠀⠀⢠⣿⡏⢋⣤⡙⢹⣿⡅
⠀⠀⠀⠀⣸⣇⠀⢠⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠂⢿⡿⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⡇⠀⠽⡁⢸⡧⠀⠀⠀⠀⠀⠀⣾⣿⣏⣜⢿⣇⣹⣿⣧
⠀⠀⠀⠀⢉⣈⠿⢋⣉⠁⠀⠀⠀⠀⠀⠀⢠⣄⡀⠹⢿⣦⣼⡿⠋⢀⣠⡄⠀⠀⠀⠀⠀⣻⠦⠉⠀⠆⠀⠠⢶⡆⠀⠀⠀⠀⠀⢠⡼⠛⠛⠛⠶⠛⠛⠛⢣⡀
⠀⠀⠀⠸⠋⠰⡄⡇⠁⠻⠊⠀⠀⠀⠀⠐⠛⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠛⠂⠀⠀⠀⢠⣿⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠐⠚⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃⠂
⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠄⢀⣠⣤⣤⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀
⠀⠀⢐⠀⣾⡍⣥⡍⣽⡄⢐⠀⠀⠀⠀⠀⠀⠀⠁⡆⠐⣓⡈⠂⠲⠈⡀⠀⠀⠀⠀⢲⣶⠤⣄⠀⢂⣂⠀⢀⡤⢴⣶⠂⠀⠀⢀⡀⣀⡄⢀⠈⣀⡡⡀⢠⣀⢀⡀
⠀⢠⣤⣤⡿⣶⣍⣵⡾⣧⣤⣄⠀⠀⠀⠀⣀⣈⣀⡜⠐⢿⣿⠂⢧⣀⣀⡀⠀⠀⠀⢸⣾⠂⠀⠁⠻⡿⠃⠂⠄⢺⣾⠀⠀⠀⢨⡯⣿⣿⠊⢾⣿⡷⠠⣿⣿⣿⡅
⠐⣿⣿⡿⠿⠾⠿⠷⠾⠿⣿⣿⡗⠀⠀⠀⢹⠿⠿⠷⠾⢄⡰⠶⠾⠿⠿⡏⠀⠀⠀⣼⣿⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠃⠈⢌⠩⡁⠁⠘
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣾⠁⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⣼⠃⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣼⠃⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⣰⠏⠀⠀⠀⡼⢧⡀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣀
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣴⡷⠶⣼⠏⠀⠀⠀⣸⡧⠤⠷⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹
⢀⠜⢿⡉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠫⡀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⡄⠀⠀⠀⠀⣴⠋
⠀⠡⡀⠈⢻⡄⠀⠀⠀⠰⣎⠋⠉⠉⠉⠉⢉⡟⠀⠀⠀⠀⢀⡤⠤⠤⠴⠶⠶⠖⠒⠒⣄⠀⠀⠀⠀⠀⠹⣇⠀⣀⡴⢣⡟⠀⠀⠀⠀⣼⠇
⠀⠀⠑⠀⠀⢻⣄⠀⠀⠀⠘⣇⠀⠁⡡⢊⡾⠁⠀⠀⠀⢠⡞⡁⢀⣠⠀⡀⡀⠤⡠⠁⠘⢧⡀⠀⠀⠀⠀⠹⣧⠉⣠⠏⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠈⠀⠀⠻⡆⠀⠀⠀⠘⢧⡀⢠⡞⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠈⠳⣄⠀⠀⠀⠀⠘⢷⠏⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠄⠀⠹⣄⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠘⡄⢀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠈⢢⡀⠘⣷⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⣠⠞⠁⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣼⠏⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣰⢷⡀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣰⠇⡈⢷⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⣼⠁⠘⢧⡀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⣰⠃⠳⠈⠈⢳⡄⠀⠀⠹⣆
⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠾⠤⠷⠶⠶⠿⠄⠀⠀⠀⠀⠘⠷⠶⠒⠒⠚⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠚⠓⠚⠚⠛⠛⠛⠛⠀⠀⠀⢹⣆
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⣰⣏⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⡄⠀⠀⠀⠀⣠⡶⠶⠶⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⠟
⠀⠀⢀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡞⢻⡄⠀⠀⠀⠘⣧⡀⣀⠔⣾⠃⠀⠀⠀⣴⣟⣀⢀⡀⣀⣀⡄⠠⠀⠤⠤⠤⠤⠞⠁
⠀⠀⠂⠀⠄⠠⠠⠤⠰⠀⠀⠀⠀⠀⠂⠒⠐⠂⠺⡁⠀⠻⣆⠀⠀⠀⠈⢿⡉⣹⠇⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠙⢷⡀⠀⠀⠀⠻⡟⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠐⡀⠙⣷⡀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⢿⣄⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢠⡤⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡰⠁
⣶⣶⡶⢶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣧⠀⠀⠀⠀⢀⣀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣇⠘⢀⠀⠀⠀⠈⠋⢀⠀⠁⠀⠠⠄⢀⣠⡀⠀⠴⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠃⠀⠽⠟⠀⠛⡿⠛⠻⠖⠛⠒⠂⠛⢀⣀⠉⣁⠀⡀⣉⠉⠁⣰⠖⢠⠀⠀⠈⠀⢀⡀⠉⠉⢡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⠖⠢⠀⠌⠋⠀⠀⠦⢼⡶⠳⡆⠲⣿⠟⠀⢹⣟⣉⠈⠁⢐⢁⡲⢯⣀⡀⢣⠘⠮⠹⠆⢐⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢤⡤⠀⠘⢛⢙⡶⠛⠃⢠⣁⣰⡀⠀⠚⣉⠁⠈⠈⠀⠈⢧⡄⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⡀⠙⡿⠀⠀⣀⣿⣥⣁⣤⣤⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣶⡅⠀⠘⢿⣶⡆⠸⠛⢻⡉⠋⠈⡿⠇⠀⠀⢀⣀⠀⠀⢎⣘⠴⡶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠸⣤⡼⣂⠛⠀⠠⠏⠀⢀⣀⡤⢴⣠⡼⠁⡇⠀⠰⣿⡁⠀⠀⠀⣿⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠞⠃⠀⠀⠀⣦⠐⠋⠁⠀⠀⠈⠳⡆⠁⠘⠀⠈⠉⣾⣦⠾⠿⣶⣾⣇⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⠇⠸⠢⠀⢠⣴⡀⡿⢈⡇⠰⣧⠀⠀⠂⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠙⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡂⣻⠀⠀⠀⢰⠁⢸⣹⣿⢉⠀⠁⠀⠀⠀⠐⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠛⣀⣀⠠⠏⠀⢸⣿⡅⣈⠲⠹⠂⣀⡤⠔⠛⢓⡀⠀⠀⠀⠀⣀⠀⢀⡀⢰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⣻⡍⠉⢁⠀⣀⡀⠀⣩⠽⠯⢔⣒⣀⣀⠀⣀⣠⠿⠏⠋⠁⠀⠀⠀⠈⠉⠁⠘⠃⠈⢁⣀⣀⠀⢁⠉⠢⠤⠾⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣟⣠⡏⠀⠘⠁⠉⠉⢚⣿⡄⠀⠛⠉⠁⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠓⠦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⡟⠀⠀⠀⠀⠀⠈⢻⣿⣖⣶⠳⣦⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⢀⡇⠀⠀⢠⠞⠀⠳⣾⣟⡏⣿⡸⡿⢿⣿⣿⡀⠰⠀⢰⠀⠀⠀⠰⠄⠐⠀⠀⠀⠀⠀⠲⠈⠀⠃⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡄⠀⠀⣿⣤⠀⠀⣻⣿⠀⢋⠀⢴⡾⣾⣿⣷⣶⢀⣼⠁⠀⢰⣶⡀⠴⢆⢠⣶⡀⣴⣦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⡀⠈⠁⣶⢰⣤⣿⠄⠈⠀⠀⢙⣿⣿⣿⣿⣿⣟⠁⠀⢠⡈⢠⣀⡀⠀⢀⣀⣠⠈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠾⠿⠿⠛⠁⠀⠈⠙⠷⣤⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡇⡜⠀⡇⠀⠀⢲⡄⠐⢤⡀⠘⠀⠀⢐⡇⡖⢠⠆⠀⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠁⠘⠀⠁⠀⠈⠀⠉⠀⠀⠀⠈⠀⠀⠈⠁⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢤⢰⡄⠀⠀⠀⡦⠀⢰⠀⣴⢰⡆⠀⠀⢰⠀⡰⢀⡄⠀⡄⠰⠀⠀⢦⡀⠀⠀⠰⠀⠀⠀⣦⠀⠰⣄⠀⠠⠄⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠚⠈⠁⠀⠀⠀⠃⠀⠸⠀⠻⠸⠇⠀⠀⠸⠀⠑⠈⠁⠀⠃⠈⠀⠀⠀⠇⠀⠀⠸⠀⠀⠀⠈⠂⠀⠈⠃⠐⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠠⠀⠤⠤⠤⠀⠤⠄⠠⠀⠀⠀⠠



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠧⠰⠄⠰⠞⠄⠀⠸⠃⠴⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠻⣧
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢀⣾⢏⣀⣈⣛⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣰⣿⣁⣀⣹⣷⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠙⣿⡋⠉⢉⣾⠃⠈⠉⠋⠉⢩⣍⠋⠉⣻⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣟⠉⠉⣻⡟⠉⠉⠉⠋⠙⣿⡋⠉⢙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⣿⠃⠀⠀⠀⠀⠀⠀⠻⣧⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣼⠟⠀⠀⠀⠀⠀⠀⠘⣿⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣈⢿⣆⠀⠀⠀⠀⠀⠀⠀⢀⡹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠈⢻⣆⠀⠀⠀⠀⠀⢀⣾⠃⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠙⣷⡀⠀⠀⠀⠀⠀⣰⡟⠉⢻⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣶⣶⣶⣶⣶⢶⣶⡶⢢⣾⢣⣶⣶⣾⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⣶⣾⣷⣶⣶⣶⣶⣾⣿⣶⣶⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡟⢁⠙⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠋⡈⢻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣰⣿⣿⠿⠿⠿⠿⠟⠠⠾⠦⠸⠿⠿⠿⠿⢿⣿⣷⡀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠿⠿⠿⠃⠰⠷⠀⠻⠿⠿⠿⠿⣿⣿⣆
⠀⠀⠀⠀⢠⣿⣿⣿⣧⡀⢶⠎⢠⣷⣶⣶⣶⠐⢶⠆⢠⣿⣿⣿⣷⠀⠀⠀⠀⣾⣿⣿⣿⡄⠰⡶⠀⣴⣶⣶⣶⡄⠰⡶⢀⣼⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⢰⣿⣿⣿⣿⣿⣷⡈⢲⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⣾⣿⣿⣿⣿⣿⡆⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⣰⡄⢻⣿⣿⣿⣿⠃⣴⡄⢻⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡟⢠⣦⠘⣿⣿⣿⣿⡟⢠⣦⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣅⣀⣉⣉⣉⠉⣉⡭⠁⣬⣉⣁⣀⣽⣿⣿⠇⠀⠀⠀⠀⠸⣿⣿⣯⣀⣈⣉⣀⠈⢉⣭⠉⢀⣈⣉⣀⣨⣿⣿⡿
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣆⠙⢁⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⡈⠋⣰⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⣠⠳⡀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡰⠁⠀⠑⡄⠀⠀⠈⢣
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠋⠀⠀⠀⣜⣁⣀⣀⣀⣘⡄⠀⠀⠀⠣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠱⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠘⣄⠀⠀⠈⢣⡀⠀⠀⢀⡜⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠘⢆⠀⠀⠀⢠⠏⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠳⡀⢀⠎⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠈⢦⠀⣰⠃⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠑⠋⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠳⠁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡰⢆⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢀⡴⡀⠀⠀⠈⢆
⠀⠀⠀⠀⢠⠎⠀⠀⠀⡴⠁⠈⢦⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠎⠀⠙⡄⠀⠀⠈⢣⡀
⠀⠀⠀⣠⠃⠀⠀⢀⡜⠀⠀⠀⠈⢣⡀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢠⠏⠀⠀⠀⠈⣆⠀⠀⠀⠱⡀
⠀⠀⡰⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠹⡄
⠀⣜⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⢫⡀⠀⠀⠀⢠⠃⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠱⡄⠀⡰⠃⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠙⠖⠁⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⢶⣴⡿⠗⠙⡟⠀⠀⢰⡆⡿⢳⠀⠀⢈⠓⢲⣴⡇⣾⣸⣾⡁⣷⣟⢰⠛⡆⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠈⠙⠉⠀⠁⠀⠀⠈⠁⠁⠘⠀⠀⠈⠛⠉⠀⠙⠋⠉⠈⠉⠁⠈⠈⠋⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⢀⣴⠟⡫⠟⢽⣫⡿⢋⣿⡿⣻
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠶⠀⡄⠢⣄⢠⠀⠀⠒⡆⠀⠀⠀⠀⢀⣠⢖⣏⣡⡎⣀⢄⣡⣟⢕⣫⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠼⠸⠇⡴⠋⠸⠀⠀⠀⠃⢀⠀⠐⠀⠈⠙⠉⠋⡋⠋⠙⠛⠛⠙⢿⠫⡞⠹
⠀⠀⠀⠈⠀⠀⠀⠠⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣄⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠂⠅⠠⠀⠈⠀⠀⠀⠁⠀⠨⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠇⠐⠁⠀⢸⢁⣨⡆⠆⣾⠠⠃⠀⢀⠀⡀⠀⠀⠠⢀⠄⠐⠀⠠⠀⠀⠀⡂⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠂⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠤⠊⠀⠅⠨⠔⠀⠀⠐⠒⠒⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⡀⠲⠉⠀⡄⠌⠀⢀⠀⠤⢠⣾⠠⡴⡐
⠀⠀⠀⠀⠀⠀⠀⠤⠀⡤⠀⠀⠀⠀⠀⠀⢐⠉⠁⠀⠀⠀⠀⠀⠀⡀⠁⠀⡀⡉⣐⢀⣀⣈⡁⠀⠀⠆⡟⡤⠢⠁⠀⢻⡈⠀⢄⠐⠊⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠣⡃⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉
⠀⠀⠀⢀⠟⠩⠀⠀⠀⠀⠀⠐⠂⠒⠊⠀⠐⠀⠄⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠠⠋⠉⢲⠤⠄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣀⣀⡀⠤⠤⠄
⠀⠀⠀⡊⢐⠞⠀⠄⡀⣀⡀⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠰⠵⠻⠉⢉⣉⠤⠤⠴⠖⠂
⠀⠀⠀⡲⠃⠀⠀⠀⠐⠘⠲⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡰⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢐⡮⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⢀⠀⢀⠀⠸⣀⣠⡆⢤⠀⣠⡛⠀⠀⠄⢀⠠⠤⡀⠂
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⢈⠩⣼⠀⠀⠠⠀⠄⠀⠈⠃⠀⡉⠈⡇⢈⡠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡂
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠌⢰⡀⠀⠀⠀⠀⠀⠀⢸⣿⣈⣿⠀⠀⠄⠀⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⡃⠀⠀⡀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠠⢰⠁⠀⠀⠀⠀⠀⠀⠸⠏⠋⠘⠀⠀⠀⠀⠀⠀⠐⠒⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⢠⠀⣤⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⣀⣴⣶⣶⣾⣶⣶⣦⣤⣴⠻⣷⣰⣄⣿⠀⣰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣆⣠⡄⢴⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠷⢿⣭⣽⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣧⣄⣔⣠⣶⡶⣦⣤⣀⡀⠀⠀⠘⣹⣿⣿⣟⡿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣿⣽⠿⣿⡿⣿⣷⣶⣦⣜⣻⣏⡉⢻⣇⣴⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠁⠉⠃⠈⣹⣿⣏⣹⣿⡿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣙⡋⢻⣿⠻⠿⣽⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣠⡤⠖⠠⣤⣿⣠⣿⣿⣿⣟⣴⣿⣃⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣻⣿⣿⣟⣶⠀⠀⠘⣿⣿⣃⣒⣶⠆⠋⠉⣿⣿⣿⣿⣿⣿⣯⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⠿⠿⠿⠃⠀⠀⠛⠏⠙⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣯⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⣿⣿⣆⠀⣿⣀⣠⡤⠀⠀⠀⠀⠀⠀⢀⣿⡿⠹⠛⣿⡁⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⣿⣿⣿⣿⣧⣿⣿⣧⣤⠤⠀⠀⠀⠀⠀⣼⣿⣷⡈⣧⢸⣿⡄⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁⢠⣿⣿⣿⣿⣿⣷⣿⣿⡦⠀⠀⠀⠀⣠⣾⣿⣿⣿⡷⣿⠀⢿⣿⡄⢻⡄⠀⠲⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⡼⢨⣿⢏⣿⣿⣿⣿⣿⣦⣀⣤⡶⠛⠋⠀⢹⣿⣿⠁⠹⣦⠈⠻⣿⣼⣿⣔⣤⡈⠓⠀⣤⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⡿⠃⠀⣃⡼⣣⣾⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⣿⣿⣿⣆⠀⠘⢧⠀⠀⠙⠛⠿⠿⣿⣇⠀⠀⠘⠛⠻⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣿⣿⠀⠠⣯⣿⣿⣿⣿⣿⣿⡟⠃⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣤⣿⣿⣿⣶⣶⣶⣆⣉⠁⠢⠀⠀⠰⠦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡏⠀⢰⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⣤⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠃⡀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿⣿⢿⣿⣿⠯⠀⠀⣿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠾⠳⠉⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⡀⠀⠈
⠀⠀⠀⠀⠀⠀⡼⠿⠛⣉⣁⣼⣿⣿⠀⠀⢀⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⣷
⠀⠀⠀⢀⣠⣤⣤⣤⣄⠀⣷⣿⣽⣯⠄⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⣿⣿⣧⡀
⠀⠀⣴⣿⡿⠿⠿⣿⡿⣿⣿⡑⢞⠠⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⢿⣿⣧
⠀⠀⣿⣿⣿⣶⣀⠈⢙⡲⣿⣿⣿⡀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⢻⣿⡆
⠀⠀⠈⢷⠈⠙⢿⣧⣨⡷⠟⠿⢿⣿⣶⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠈⣿⣿
⠀⠀⠀⠀⠈⣤⣾⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠘⠿
⠀⠀⠀⠀⠀⠀⠁⠉⠑⠲⣦⡐⢢⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⠀⠀⠀⠀⠀⢉⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣦⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣀⠀⠀⠀⣀⠈⡿⣻⡄⣀⠀⣺⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠧⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢿⣿⣿⣯⠒
⠀⠀⠀⠀⣷⣤⣀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣥
⠀⠀⠀⠀⢹⣿⣿⣿⣷⣦⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣖⠂⠄
⠀⠀⣰⣤⣼⢾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⡛⡛⠉⣩⡿⠋⢀⡄⠀⠂⠀⢠⡿⠻⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⠷⣄
⠀⠀⣿⠿⣿⣿⣷⣿⣿⣧⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣴⠟⠁⣠⠞⠁⠀⠀⣰⡿⠁⢀⣿⣿⣿⣿⠟⠉⠉⠋⠓⠀⠀⠈
⠀⠀⠉⠀⠸⠟⢟⣿⣿⣿⣿⣽⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣴⠙⠃⠀⣠⡾⠋⢀⣰⠿⣛⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠁⠀⠀⠈⠋⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⣴⣟⢁⢈⠀⠀⡈⣉⣻⡆⠠⣀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⡿⢻⣽⣼⢾⡿⢿⡧⣯⢽⢶⣴⣮⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠛⠃⠸⠯⠇⠐⠁⠀⠄⠰⠾⠀⠀⠚⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠿⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠻⠿⠿⠛⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣧⣤⣤⣼⣿⣷⣤⣤⣤⣤⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⠛⠛⢻⣿⡿⠛⠛⠛⠛⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣤⣤⣼⣿⣷⣤⣤⣤⣤⣤⣤⣾⣿⣧⣤⣤⣽⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠛⠛⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣄
⢠⣷⣾⢀⣤⣄⣀⣀⣀⣤⡄⣷⣿⡇⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⢣⣾⣿⢫⣤⣤⣶⣶⣶⣶
⢸⣿⣿⢰⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡸⣿⠃⠛⠛⣛⣛⡻⣿⠟
⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣤⣀⣀⣀⣤⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⣿⣿⡄⠀⠀⠀⡙⣿⣷⡉
⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠉⠉⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣬⣭⣥⣤⡄⣼⣿⡜⣿⣿⡄
⢘⣿⣿⡈⠻⠿⠿⠿⠿⠟⢃⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠷⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⣛⣋⣼⣿⡟⠼⠿⠿⠿
⠀⠉⠍⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣺⣿⣓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⣶⣶⣶⣶⠶⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣿⣿⢻⣿⣿⣮⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⢠⠀⢀⣏⡀⢠⡀⢠⡀⢀
⠀⠀⣿⣿⡉⣡⣤⡽⢭⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⢗⣿⣿⣿⠿⠿⢿⣿⣿⣿⠸⡶⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡁⣸⡀⢸⣿⠁⢸⡇⢸⡅⣸
⠀⠀⢹⣿⣁⡙⠛⣳⣚⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⡿⢡⡾⠛⢷⡜⣿⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡅⠈⣿⠀⣸⡇⢸⡇⣹
⠀⠀⠀⠙⠻⣿⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣧⣀⣼⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⣧⡈⠛⢿⣿⠟⠋⢀⣾⠃⣿
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⡇⢸⠤⠭⠨⡇⣿⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣈⠛⠷⢾⣿⠶⠞⠛⣁⣼⠏
⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢸⣇⢸⣀⣄⣰⡇⣿⡗⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⣶⣿⡶⠾⠟⠋⠁
⠀⠀⢀⣀⣤⣤⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠓⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠿⠷⠄


⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠰⠉⠂⢠⣄
⠀⠀⠀⣀⣀⣀⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣨⢸⣿⡗⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢛⡛⠛⢶⣀⣶⢟⣛⡛⢶⡄
⠀⢠⣿⡶⣝⡻⣿⠾⣻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠇⠀⠀⣿⠁⢘⣛⠃⠀⣿
⠀⣾⣿⣇⠙⢿⡮⣩⣭⣽⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡿⣿⢸⣟⡇⣿⢿⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟⠀⠀⣿⠀⠠⡽⠇⠀⣿
⢰⡿⠿⣯⡇⣾⢣⣿⣯⡀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣼⣿⣯⣾⣿⣷⣽⣿⣦⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠶⡆⠀⣿⠀⠀⠽⠄⠀⣿
⢨⡳⣤⡈⢸⡏⣾⣿⡿⢱⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣶⡄⠀⣿⠀⠀⢻⠂⠀⣿
⠀⠹⣮⣝⠿⣹⣿⣴⣾⡾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠷⠾⡷⠶⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣭⠀⠀⣿⠀⢐⡶⡆⠀⣿
⠀⠀⢹⣿⢣⣶⣶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣤⣤⣤⣶⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡽⠁⢡⢏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣠⣮⣾⣁⣀⣀⣹⣆⣧⣀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⡿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⢿⠿⠶⠾⠿⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢠⣾⢿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣮⢆⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⣰⡿⡫⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⣆⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡼⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⠋⠀⠀⠘⢞⢗⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣧⣤⣤⣤⣤⣬⣦⣿⣦⣤⣤⣤⣤⣤⣤⣿⣧⣀⣀⣀⣀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢻⣿⣏⠉⠉⠉⢩⣿⠏⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⢦⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣡⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⢀⣸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣤⣤⣤⣄⣾⣿⣿⣀⣁⣀⣌⣙⠻⣿⣿⣿⣋⣰⣇⣤⣾⣿⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠉⠉⠉⣩⣿⡟⠉⠉⠛⠛⠻⠋⠻⢿⣿⣿⣿⠛⠛⠛⠉⣝⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠹⣿⣄⠀⠀⠀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⢀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⢀⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡋⠀⠀⠀⢹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⣴⣶⣶⣾⣿⣿⣾⣶⣶⣶⣶⣶⣶⣦⣴⣾⣿⣷⣶⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣯⣉⠡⠼⢻⣿⡿⠋⠉⠉⠉⠉⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⣠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⢀⡀⡀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷
⠀⠀⠀⠄⣀⣼⣿⣷⠞⠛⠲⢿⣷⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡞⠉⠋⠀⠀⢀⡴⠦⡶⠆
⠀⡀⣰⠋⠀⣠⡄⠐⢲⣷⣦⠀⠀⢳⣆⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⢸⣟⢰⠇⠈⠓⡀
⢀⣷⠃⠀⣄⣿⣝⣷⣶⡁⣿⣽⠀⠀⢻⣀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⠋⢹⣿⣤⣛⣾⡃
⢸⡇⠀⠀⠹⣿⣛⠏⠈⠀⣄⠃⡰⡶⠋⠉⢱⠇⠀⠀⠀⠀⠀⠀⢠⠙⢦⣀⠀⠀⠀⣀⣈⣿⡷⣽⠛⡇
⠀⢷⠆⠙⣦⠈⠙⠛⠄⠠⡤⣶⠏⢸⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⣦⠀⠀⠀⠈⠿⠆
⠀⠈⠀⠛⠁⣀⣀⠐⢄⣨⠇⠉⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⣿⡀⠀⠀⠀⠀⢀⠀⠀⣀⣴⡄
⠀⠀⠀⠠⡈⠛⠠⡄⠉⠀⢠⠀⠀⠈⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⣿⠉⠀⠀⢀⠀⣴⡃⠀⢿⡽⡀⠹⠂
⠀⠀⠀⠀⣹⣦⡴⣿⣄⠀⢁⠀⠀⢠⣾⡽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⠛⠁⠀⠴⣦⠛⢧⡀⠀⠀⠀⠂
⠀⠀⠀⠈⠉⠁⠐⠋⡉⠁⠀⠀⠀⠀⢸⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢋⣠⠀⠀⠀⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣦⣠⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⣿⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢤⡶⠦⣤⣄⣛⣁⣩⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣴⣧⣙⣿⠀⠀⠀⠰⠋⠀⠀⡀
⠀⠀⠀⠀⣹⡾⡇⠀⠋⢹⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠏⣼⣿⣿⣿⡇⠀⠀⠀⠿⠶⣢⣿⣄⠀⡀
⠀⠀⠀⠀⢨⠟⡠⣷⢖⣼⡷⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⣿⣿⡝⠋⠹⣆⢀⣤⣤⡼⠋⠀⠀⠉⠁
⠀⠀⠀⣰⠃⣾⢃⠻⣿⣿⡟⢷⣤⠀⠀⢀⢀⢄⠀⠀⠀⠀⠀⠀⠈⢫⠟⠓⠢⠀⠘⠛⠻⠟
⡀⢀⠠⠋⢝⡻⠈⢇⡈⢻⣷⣤⣿⠀⠀⠀⢿⡙⡇⠀⠀⣠⣤⠀⡟⣿⣠⡀⠀⠀⠀⠘⠁⠄
⠈⠉⠀⠀⠈⠀⢀⡺⠃⢀⢻⣿⣌⢧⠀⠀⢀⢻⡇⠀⠀⢛⠖⣠⡇⢿⡟⠛⣀
⠀⠀⠀⠀⠀⠀⣮⠀⢠⣶⣿⣿⣿⢀⠀⢀⡼⢫⠃⠀⠀⠘⣄⣇⠁⣼⣡⢸⠏⠀⠀⠀⠰⣶⡀
⠀⠀⠀⠀⠀⠀⠻⣷⠷⢿⣿⣿⣿⢆⣀⣤⡿⠋⠀⠀⠀⠀⠈⢺⡤⢿⠗⢉⣦⣤⣄⣠⣶⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣬⣤⣬⣥⣶⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡤⠀⢀⠀⠀⣠⣤⡴⠟⠃⢸⡄⠀⠀⡆⢾⠉⢹⣍⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⣀⠐⢶⣽⠀⠀⠀⢀⡀⣀⢀⡀⣿⣀⠀⠀⢸⠇⠀⠀⣾⡏⠀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠂⠀⣀⣀⣿⣿⣶⣿⣿⠀⢀⠀⠀⡄⠀⠀⠐⣿⣧⣽⣿⣿⢠⣤⡄⣿⣇⣠⣶⣽⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢠⣄⠀⣤⣀⣭⣿⣿⡟⣛⡸⢿⣇⣿⣏⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠴⢄⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠰⡼⠈⣍⠑⠙⣯⣭⠛⠙⠃⠃⠐⠁⠹⣿⣿⡿⣻⣽⣧⣶
⣋⣐⣃⣀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⣿⡶⠿⠶⠶⠤⢴⡶⠴⢶⣷⣾⣿⣿⣿⣿⣿⣿
⣿⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣴⡇⠴⠖⢶⠶⠶⠦⠒⠞⠿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣄⣼⡿⠛⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣶⡄⠀⢠⣴⣶⣿⣿⣟⣙⣿⣿⡿⠛⠛⠋⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣼⣿⣿⣿⣿⣗⠀⠀⠘⢉⣩⣿⣿⣿⣿⣿⣇⣠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢈⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⣿⣧⣴⣾⡆⠀⠀⢀⣠⣬⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢹⣿⢻⣿⡅⠀⠀⠈⠉⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡽⠷⠀⠀⠀⢀⣤⣴⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⠇⠀⠛⠁⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⣿⣿⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣄⠀⠀⠀⠀⠀⠚⠛⣽⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣧⣦⠞⠀⠀⠀⠀⠈⢻⡿⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠃⠀⢀⢀⠠⠂⠀⢸⠁⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⣼⡧⠊⠉⠳⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠛⠉⠀⠀⠀⣹⣷⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠚⢲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠿⡿⠂⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⢦⢿⣧⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠒⡶⡶⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣶⣾⣷⡀⠀⠀⠀⠀⠀⢠⣟⣿⣾⣿⡷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⢻⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⠛⣿⣿⡟⠀⠀⠀⠀⠀⠀⢐⢻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡴⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠙⠻⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣆⠀⠀⠀⢀⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡄⠀⠀⠀⣠⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦⠀⢠⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡄⠀⣰⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠀⠘⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠀⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠏⠀⠀⠀⠘⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣛⣛⣛⡻⠃⠀⡀⠀⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⢠⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣆⢠⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁

⠀⠀⠀⠀⠀⠀⠊⠳⠄⠀⠀⠀⠀⠲⠉⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀⠙⠀⡫⠀⡀⠀⡀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣉⡡⠀⠈⠒⠀⠁⠐⠠⠂⠀⠀⢀⠈⠈⠈⠊⠁⠠⡉⣩⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠒⠒⠚⠈⠉⣀⡀⠀⣀⣀⠀⠈⠉⠀⠇⠒⠐⠁
⠀⠀⠀⠀⠀⢠⠀⠠⠄⠄⠄⠆⠠⠠⠀⢤⠀⠀⠰⠀⠀⠀⠀⠀⢠⠂⢀⠀⠀⠀⠀⠠⠒⠈⠈⠁⠐⠄⠄
⠀⠀⠀⠀⠀⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠌⠐⠀⠀⠒⠀⠀⠈⠁⠀⠀⠈⠈⡁⡀⠀⢀⢈⠈⠁⠀⠈⠈
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢧⡁⢀⠽⠠⠀⠀⢔⠒⠒⠒⡊⢠⢧⢰⢿⡈⡒⠒⠒⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠄⠀⠤⠤⠀⠈⠶⡬⡉⡣⢲⡤⢏⡞⠀⢇⠳⢼⡲⡜⡁⡤⠎⠀⠀⠠⠄⠀⠤
⠀⠀⠀⠄⠠⠃⠂⠀⠀⢄⡀⢀⠠⠀⠀⠀⠣⠐⣀⠀⠀⠑⠒⠈⣡⢔⣀⡀⢀⢠⢭⠴⠀⠀⠀⠀⠰⠮⡭⢀⡀⡀⣐⢠⡈⠐⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢍⠉⠁⠁
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠠⣄⣀⣠⠄⣴⠀⠀⢀⣀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣼⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢉⢼⠩⡧⡘⡡⠀⢀⣸⡈⢦⠀⠀⢸⣻⡇⠀⠀⢸⢟⡇⠀⠀⡰⠃⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢸⠀⡇⡛⠁⠀⢸⡀⠳⡀⠱⡀⠀⣿⣟⣠⣆⣻⣿⠄⠀⠜⠀⡜⢁⡏
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⡵⠀⠀⠀⢣⠙⢆⠙⠢⡀⢠⡙⠙⠃⠘⠛⢙⡠⢈⡄⠊⡠⠊⡼
⠀⠀⠀⠀⠠⣶⣀⠈⢂⠀⠀⠌⠁⣠⡶⠄⠀⠀⠀⠀⠀⠀⢀⠸⢲⠥⢴⡛⠀⠄⣗⠦⠤⣽⣏⠛⡈⠳⡄⢠⡖⢁⠗⣩⣮⠤⠤⢺⠁
⠀⠀⠀⠰⠲⣌⢉⢷⡶⠂⠲⣶⡾⠉⣰⠾⠀⠀⠀⠀⠀⠀⣤⡴⣹⠇⠸⢧⠣⠶⠈⢟⠒⣚⠣⣄⠘⡀⡽⢻⡀⡪⢀⠬⣓⠒⣺⠁
⠀⠀⢀⣀⡀⡘⠻⡖⠁⠀⠀⠘⠾⠋⢁⢀⣠⠀⠀⠀⠀⠠⡤⠖⢹⠂⠀⣧⠙⠲⠂⠀⠉⠣⢖⣁⡍⢲⡁⢈⡗⢸⣈⣒⠄⠋
⠀⠀⢸⠀⠈⠙⠒⡗⠀⠀⠀⠀⢓⠚⠋⠀⠈⡀⠀⠀⠤⠀⣀⡠⠋⠀⠀⠈⠓⣠⠤⣹⡄⠀⠀⠀⠈⠉⡀⠀⠈⠉
⠀⠀⠳⢆⢤⢇⡐⡏⠀⠀⠀⠀⢟⣂⡤⠤⠰⠃⠀⠀⠙⠉⠈⠀⠀⠀⣠⠞⢉⠍⡓⢀⢦⠀⠶⡆⢒⡩⡉⠳⣄
⠀⠀⠠⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⢸⡰⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡜⠁⠐⢅⠒⠤⢏⢨⠀⡑⠁⠥⠒⡈⠃⠈⡑⡄
⠀⠀⠈⢱⡊⣯⠀⠀⠀⠀⠀⠀⠀⢠⣿⢱⠈⠀⠀⠀⠀⠀⠀⠀⠰⡠⢃⠸⡼⠐⠢⣀⠍⠀⠈⣇⡔⠂⢣⠜⢘⢄⡠
⠀⠀⠀⢻⡟⢋⠀⠀⠀⠀⠀⠀⠀⢈⠛⣿⡃⠀⠀⠀⣀⠤⠂⣡⠔⡀⠀⡀⠀⡌⢠⠀⠀⠀⠀⠀⢇⢰⡀⢀⠀⢀⠲⣄⠐⠤⣀⡀
⠀⠀⠀⠈⠓⠬⢧⠀⠀⠀⠀⠀⠀⡾⠴⠋⠁⠀⠺⠩⢤⡶⢛⣡⠞⣡⢾⣁⠜⠑⠋⠀⠀⠀⠀⠀⠈⠊⠑⣄⡷⢄⠳⣬⣙⠶⡤⠌⠵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⣶⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⣾⣿⣿⡄⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠜⢠⣿⣿⣿⣿⣿⣆⠹⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⠓⡌⢠⣿⣿⡿⠉⠹⣿⣿⣆⠘⣜⣳⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⢈⣂⠎⣠⣿⣿⡟⠁⣴⣆⠹⣿⣿⣧⡘⢆⠣⣄⠓⢡⠴⠄
⠀⠀⢀⣀⣀⢀⣀⣀⣀⣀⣠⣠⣀⣠⣀⣓⣚⣓⣂⣲⠎⣰⣿⣿⡟⠀⣼⣁⣜⣦⠘⣿⣿⣷⡈⠧⠆⠡⠥⠠⠠⠹⠡⠤⠠⠤⠤⠤⠤⠤⠤⢦⠂
⠀⠀⠐⣏⠉⢩⣥⣭⣭⣡⣅⣤⣤⣤⣠⣤⣤⣤⣤⣤⣴⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣷⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡶⢀⡞
⠀⠀⠀⠈⢧⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡿⣿⠿⣿⢿⠿⠿⢿⡿⣿⢿⣿⣿⣿⣿⠿⢿⠿⠿⢿⠟⢻⣿⣿⣿⡿⢁⠚⢆
⠀⠀⠀⠀⢘⠁⢘⢻⣿⣿⡄⢰⠁⢨⢶⢶⡆⢀⡆⠀⢸⡏⣰⣶⣎⠀⣷⣶⣖⣶⣶⢀⣰⣆⣻⣿⣿⣯⡀⣸⣖⣲⡏⠀⢠⣿⣿⡿⠡⠮⠀⠸⠄⣀
⠀⠀⠘⡅⠈⠁⠈⠉⢩⣿⡍⠈⠉⠉⠉⢹⢯⠉⠁⠀⠉⠉⣿⠁⠀⠁⠈⠁⣿⡂⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡼⣇⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠐⣄⠀⠀⣠⣎⠹⣷⣦⠀⠀⡤⢢⣾⣦⡄⠀⢠⡾⡭⣱⡆⠀⢠⡞⡡⢹⡆⠀⠠⣾⡳⣌⣗⠀⢀⣲⠳⢞⡖⠀⢀⣺⣱⠉⡗⠀⢂⡖⠁
⠀⠀⠀⠀⠛⢦⠀⢴⠋⢦⠙⣿⣦⠀⢿⣿⣿⡿⠻⣄⠘⠳⡜⠈⢳⣄⠘⠧⠴⢞⣡⠴⢛⠄⡿⠛⢋⠐⣿⣿⣧⣼⠏⣰⠏⡐⣁⢁⠇⡠⠋
⠀⠀⠀⠀⠀⠀⢣⡘⢎⡙⢧⠘⢿⣷⡈⠻⣿⢁⣞⡿⣤⣤⣙⠳⠼⣭⣧⠜⣆⣸⣠⡶⠝⢊⣡⡤⢾⣧⠈⣿⡿⢋⣾⠋⣜⠀⣤⢋⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠟⢦⡹⣴⠁⣼⣿⣿⣦⠈⠉⢻⣉⠩⣆⠭⠙⡒⠢⣼⡀⢸⣀⠤⠶⣎⢹⠘⠓⣵⡼⠁⠈⣶⣿⣿⣄⣳⠟⣡⠾⠁
⠀⠀⠀⠀⠀⠀⠠⠒⠢⡽⠢⡙⢿⣿⣿⣿⣷⡶⢦⣍⡚⠳⢤⣅⣹⡉⠉⠉⠉⠉⢹⢦⣤⠰⠛⢊⣥⢖⣶⣾⣿⣿⡿⠟⣡⠚⡽⣆
⠀⠀⠀⠀⠀⠀⢀⡬⡽⢁⣾⣿⣷⣬⠛⠉⢻⣿⡄⠻⣿⡛⢶⣤⣭⣵⡤⠠⢤⣴⢭⠤⢒⡞⣉⡟⢁⣾⣿⡏⠈⢉⣴⣾⣿⣆⠈⢆
⠀⠀⠀⠀⠀⠀⢨⡝⢀⣾⣿⣿⠟⣸⢦⢤⡴⣍⣛⠢⠼⣿⣞⣄⣤⣤⠇⠀⠐⢷⣈⣀⣭⣭⠾⠥⠚⣋⣥⠞⣶⢻⣆⠹⣿⣿⣧⠈⢖⠄
⠀⠀⠀⠀⠀⢠⠏⢠⣿⣿⣿⠋⠴⠥⠤⠬⠆⠘⣿⣿⣷⠒⠦⠤⠍⣩⠀⠐⠀⢀⣉⡡⠤⠴⢲⣿⣿⡿⠁⠘⠒⠒⠛⠃⢘⣿⣿⣷⡀⢪⠄
⠀⠀⠀⢀⣴⠃⢠⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣾⡇⠀⠔⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢡⡀
⠀⠀⠀⣴⠁⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣟⠛⢛⡇⠀⠀⡟⠛⠛⣿⣿⣿⠟⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠛⠙⠛⠃⠀⣳⡠
⠀⠀⠐⠛⢻⠛⠚⡗⠘⢻⠛⡟⣫⣽⡿⠿⣯⢻⠟⢷⡀⠻⣿⣿⡆⠻⡇⠀⠀⣿⠟⣼⣿⣿⠋⢀⡮⡭⢾⣭⢭⡛⠣⠁⠉⠉⠋⠉⠋⠩⠍⠉⠈
⠀⠀⠀⠰⠒⠛⠋⡉⠒⢮⡟⠙⣁⠂⠀⠠⠴⠀⢱⢈⡽⡄⠹⣿⣿⣦⡁⠀⠀⢫⣼⣿⡿⠃⢠⡇⠚⠀⠲⠒⠀⠁⠐⡀⠙⡂⡂⢄⠖⠃⠒⠲⣦⣄
⠰⠀⣀⠄⢠⠤⠤⡀⠑⢸⡀⠠⠁⠀⢢⠀⠀⢀⠞⠡⠐⠸⣄⠙⣿⣿⠀⠀⠀⢸⣿⡿⠁⣠⠏⠘⠦⠀⠀⠠⡴⠀⠀⠀⠀⢋⠔⠁⡤⠤⡀⠠⠄⡀⠱
⠀⢀⣁⠀⠀⠀⠀⢱⡀⠈⢇⡀⠀⠀⠀⣀⡀⠈⠔⠂⠀⠀⢹⣆⡸⠃⠀⠀⠀⠀⢿⣁⣴⡃⠚⢑⠲⢇⠀⠄⠂⠀⠠⠇⠠⠃⠠⠊⠀⠀⠀⠀⠘⢁⠔
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⢏⠐⠒⠀⠀⠈⠉⠀⠀⡀⠀⣠⣏⢀⡀⠀⣀⣀⠀⡀⢈⠷⣄⠀⠀⠀⠀⠀⠀⢐⡈⠡⠀⡆⠀⠀⠉⠀⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠠⣀⢀⡀⠀⠀⠀⣼⣼⠏⠠⠀⡀⠀⠀⠨⠀⠀⡀⢂⠰⠀⢳⡀⠀⠀⢀⡠⠊⢃⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠤⠴⣛⡛⠋⠤⠂⠀⠀⠁⠶⠃⠆⠔⠀⠈⠥⠴⠤⠽⢯⡿⠭⠒⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠄⠀⠀⠀⠀⠀⠤⠤⠤⠤⠦⠴⠶⠶⠶⠶⠃

⢸⣶⣶⣶⣶⣶⣤⣶⣤⣤⣀⣀⣤
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣦⣾⠖⣶⣶⣤⠀⣀⣀⠀⠀⢀⠀⣀⢀⣀⣀⣀⢀⣠⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣧⣤⢀⢻⣿⣿⣿⠿⠿⣿⣾⣿⣿⣿⣟⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⢠⡿⠋⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡞⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣷⡼⠻⢿⣛⡂⣁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⢿⣿⢯⢁⡱⢦⡆⢈⣲⣖⣄⣙⣽⣾⡀⠀⠀⠀⠀⢀⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⣛
⢸⣿⠛⠻⣫⣿⠛⣿⣼⢷⣊⠊⣿⣉⡽⢛⣩⣿⣿⣇⣀⣀⢸⢦⣾⣿⣿⣿⣿⣿⣿⣿⣧⣶⣯⣉⡟⣿⠿⣿⢿⢿⣿⣿⣿⣛⣿⢿⡿⡿⢿
⢨⣴⣦⡀⡀⠂⣻⣦⡶⠾⣫⠠⠄⠘⠃⠀⠈⠁⠉⠂⠁⠀⠈⠉⠉⠉⠉⣉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠁⠁⠀⠀⠉⠉⠙⠛⠛⠋⢿⣷⣶
⢸⣯⣅⠈⠑⠤⠎⠽⠿⠶⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⠤⠀⠀⠀⠀⠈
⢸⣿⣾⠂⠀⡤⣶⡀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘
⢸⣿⣧⣤⠈⠙⠶⡷⠆⠀⠀⠁⡐⣂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⣤⠀⠀⠁
⢸⣿⡟⣥⣄⠀⢀⡏⠠⠀⠀⠀⠀⠀⠘
⢸⣿⣿⣿⣿⢔⡩⡏⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡼⠿⡄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡼⠃⠀⠹⡄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠏⠀⠀⠨⡥⠤⠤⠤⠭⠀⠀⠀⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⣶⠀⠀⠀⠀⢸⣿⣽
⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⢸⣿⣾
⠀⠀⠀⠀⠀⠈⢆⠀⠀⢠⣶⠶⠶⠶⢆⠀⠀⠀⣰⠶⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⠀⢠⡶⠶⠶⠶⣶⠄⠀⠀⡼⠁⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢱⡄⠀⢠⠏⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢳⡀⠀⣰⠏⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⣤⠟⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢳⡠⠏⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣰⠛⣆⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡼⠱⡄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣰⠃⠀⠘⣆⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡼⠁⠀⠹⡄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠐⠿⠶⠶⠶⠜⠂⠀⠀⠻⠶⠶⠦⠤⠤⠤⠴⠶⠶⠃⠀⠀⠘⠥⠤⠤⠤⠿⠆⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣆⠀⠀⢐⡖⠒⠒⠒⢖⠀⠀⠀⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠀⠀⠀⠀⢸⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢳⡀⠀⣰⠋⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢳⣴⠋⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠉⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠙⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⠀⠘⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣌⢿⣿⣷⡰⣶⣶⣶⣶⣶⣶⣶⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠛⠛⠛⠛⢛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠊⢻⣿⣷⡜⠛⠛⠛⠛⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡷⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢢⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⢾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣤⣤⣤⣤⡜⢿⣿⣷⡠⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣥⣤⣤⣤⣤⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠎⢿⣿⣷⡙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢘⣓⣤⣶⡂⣟⠀⠀⣖⣶⡶⡆⠀⡷⠋⣖⡆⣾⡐⣗⣴⣲⢦⠶⣶⣲⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠐
⠀⠀⠀⠀⠰⠀⠁⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⣀⡤⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠆⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠦⢆⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⢁⠀⠌⠁
⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠐⠈⡉⠀⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⢀⠀⠀⠀⢠⠋⡔⢀⡀⠀⠀⠀⣤⡀
⠀⠀⣸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⠒⠚⠀⠐⠚⠀⠀⣋⡼⣿⣛⠓⠀⠈⡉⢹
⠀⠀⣿⡍⣿⠀⠀⠀⠀⠀⠀⠛⢣⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠓⠂
⠀⠀⣿⣷⡿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⠀⠀⠠⠤⡀
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⣄⠀⠀⡏⠀⠀⢏⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣿⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠦⡀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠐⠆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠈⠀⡀⠀⠀⠀⣀⡤⠄⢀⣰⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠉⠀⠀⢪⠤⢴⠍⠀⢀⣼⡁⠀⠀⣼⠃⢐⣳⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⠀⢠⠂⠂⠀⡋⠀⣈⢻⡿⠖⠛⠛⢳⣦⣾⣿⡇⢳⠀⣠⠆⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠘⢀⣧⢀⡄⠀⠀⠀⠉⠀⠇⠀⠃⠶⣬⣿⢟⣿⡟⠀⠲⠁⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣯⠈⠄⠀⣿⡆⠎⠀⣰⠀⣾⣦⣾⣿⣿⡿⢆⠀⠀⠀⡠⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣥⢼⡄⠈⣻⠄⠀⠀⠀⠀⣿⣿⣿⢾⣿⢨⡿⣢⠄⠘⠑⣠⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠘⠳⠤⠇⠀⠈⢱⠀⡔⠀⠑⠁⣸⣿⣿⣾⡟⠀⠀⠀⢇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢹⠀⠎⠀⠀⠀⡄⠸⣧⠀⠘⢋⠿⣼⣿⠇⠀⠂⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠑⠀⠀⠀⣄⣧⠀⠉⢄⣤⣯⣤⠸⠃⠀⠐⢡⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡆⠄⢄⠆⠀⠀⠘⠙⢧⠀⠈⠠⣸⡿⠁⣀⠀⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢅⠀⠰⠦⠄⢴⣤⣢⣄⣠⢿⠅⣀⠄⠈⠄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⢀⠀⠀⠀⡩⠋⣉⣸⠅⠈⡀⢄⡾⠃⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⣠⡈⣆⠀⠀⡤⠾⠋⠀⠀⢤⡤⠎⡀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢂⡀⠈⠈⠉⠁⠀⣀⣀⡴⠟⠋⣠⡔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⢑⢖⠒⣛⢚⣿⠏⠃⣠⡴⣠⠍⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠚⣇⠀⠄⢈⠤⠟⠉⠁⠀⠀⠀⠀⠁⠔⠐⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠴⠊⠁⠀⠀⠀⠀⢀⠴⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠈⢇⣁⣶⠒⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢑⠁⠀⢄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠣⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠠⠠⣤⣤⣤⣶⣶⣶⣖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠀⠀⠀⠰⣲⣿⣿⣷⣿⣿⣿⡿⡳⣶⢶⢘⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⢛⡀⣴⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⢯⣦⣦⣜⣶⡦⠄⢘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣗⡷⢆⣤⣿⣿⣿⣰⣾⣿⣿⣿⣿⣿⣛⣺⠻⡟⢿⡟⢿⣿⣲⣤⢴⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⢿⠿⣾⣷⣿⣿⢿⣿⡿⣿⣿⣿⢿⡟⠛⠛⠛⠛⠛⠛⠛⢻⠿⣿⠟⠻⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡆⣾⣿⡆⡸⢁⠋⣿⣏⣼⠀⢿⣿⣿⢸⡇⠀⠀⠀⠀⢀⠀⠀⠸⣧⢹⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡅⠿⠛⣧⣏⠁⠌⢀⣠⣿⠀⢼⣿⣛⣼⡗⣴⣿⣧⠀⠉⠳⡀⢀⣿⡄⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡂⣄⢿⡿⢀⣰⡈⣁⣿⠀⣿⣇⣿⠿⣿⣿⣿⣿⣿⣿⣿⣣⣈⣿⡇⡇⠂⠀⠓
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢂⣽⢾⣿⠘⣹⠅⠈⣿⠀⣻⡿⠃⠀⣿⣷⢿⣿⣿⡿⠁⣿⢿⣿⡇⡇⠼⠀⡯
⠀⠀⠀⠀⠀⠀⠀⢐⣿⠆⣽⣿⣿⢸⣿⣷⣶⣿⢠⣿⡗⣂⣂⣿⡇⢼⣿⣿⡷⠀⣿⣿⣿⡇⡇⣽⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠁⠘⠟⠋⠉⠀⠉⠉⠉⠁⠀⢀
⣠⣴⣤⣤⣴⣖⣠⣶⣶⣶⣾⣶⣦⣾⣦⣤⣤⣄⣠⡄⣤⣀⡀⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢦⣤⣀⣤⡄⠠⠔⠀⢀⣤⠂
⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢛⠛⠛⠛⠋⠙⠉⠁⠀⠀⠀⠀⠀⠀⡀⡈⣀⢀⠀⠀⢹⣯⣛⠿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⣤⡀⠀⢀⡀⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠐⣟⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⡄⣤⡄⣶⣮⣤⣤⣄⠀⢀⣀⣀⣀⣀⣀⠘⢹⣿⣀⣠⣤⣤⣤⣤
⠐⣿⣿⣿⣿⣿⣻⣓⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠿⠀⣿⣿⣿⣏⣿⣷⣯⣭⣿⢿⣿⣿⢴⣼⣿⣷⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⡟⣛⣛⣛⢻⣭⣿⣿⣮⠁⡆⢠⠀⠀⠀⠈⠽⣿⣿⣿⠀⣿⢸⣿⣿⣿⣿⣿⣛⣻⣿⣿⣦⡵⠚⠀⠸⣿⡓⣳⣟⣻⣿⣟
⢘⣿⣻⣭⣭⣭⣭⣿⣿⣬⣥⣤⣤⣤⣤⣤⣤⣭⣽⣁⡤⠤⢤⣬⣽⣮⣥⠤⠞⠉⠀⠀⠌⠉⠀⠀⠀⠉⢒⣒⣒⣶⣾⣿⣿⣻⣿⣿⣿
⢸⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣭⣿⣿⡷⣶⣼⣿⣿⣿⡇⠀⣀⠀⠀⠀⠠⢤⣤⣤⣤⣼⣿⣿⣿⣿⢿⣟⣿⣿⣿⠟⠋
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢯⣽⣿⠿⠻⠯⠬⠟⣿⠿⠟⠛⢉⣁⣤⣶⣿⣧⠈⢛⣀⣰⣶⣶⣿⣟⣻⣿⣿⡏⣿⣿⡿⢰⢈⣼⣋⣀⣀⠠
⢨⣿⣿⣿⣿⣶⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆⣭⢩⣭⣽⣿⣿⣷⠘⠁⢀⣽⣿⣿⡛⠛⠛⢹⣷⣿⠏⠳⣟⣻⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⣉⣙⠛⠸⣿⢿⠋⢿⠁⢰⡿⣿⣿⣿⣍⢉⠉⢈⡸⢛⢿⣿⣿⣿⣼⢿⡿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣽⡆⣠⡆⢸⣧⢠⣄⢨⣄⣿⣿⣿⣿⡌⢙⣿⣶⣿⢋⣷⣾⣷⣞⣮⠓⠻⣿⣿
⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣍⣛⣿⣿⣾⡇⣸⡏⢸⣿⢸⣿⡟⢻⣿⣿⠁⣻⣿⣿⣿⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠸⡿⠿⠿⠿⠿⠟⠋⢉⣉⡉⠀⣀⠐⣚⣢⣹⣿⣿⣿⣿⢹⣿⣷⣾⣏⢸⣿⢸⣿⡇⢸⣿⣿⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣶⣶⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⡃⠘⡋⢸⣿⣿⣿⡇⢸⣿⣿⣿⣿⢠⡙⣿⡿⢫⣿⣾⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⠦⢹⣋⣾⣿⡿⣶⣾⣯⣈⣹⠼⣇⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣟⣻⣿⣿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⣠⣽⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⡯⡀
⠀⠀⠀⠀⠈⡻⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠵
⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠈⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠈⢩⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣠⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣸⣿⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣸⣿⣁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠋⠟⠋⠉⠻⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⡿⠟⠉⠻⠏⠳⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣼⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠙⣿⣿⣿⣿⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⣿⣿⣿⣿⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠿⣿⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠠⣿⡇

⠀⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠀⢰⢀⡆⢰⠀⢸⠀⢰⠀⡆⢰
⠀⠀⠀⢀⣸⢿⣻⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡆⢧⠘⣦⣸⣠⡾⢠⡇⣼
⠀⠀⢪⣽⣾⣿⣿⣿⣾⣽⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⣌⠳⠦⣭⡥⠶⢋⡼⠃
⠀⠀⢸⣿⣿⡏⠉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢶⠖⠚⠉
⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣤⣄⡀


⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⢰⣤⣤⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠹⣟⣿⠛⠋⠙⣏⣿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠿⠄⢹⡟⠛⠛⢻⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢩⣿⣿⣿⣯⠙⢶⣄
⠀⠀⢀⡼⣧⠀⠀⠀⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⢻⣿⣄⣼⣿⠃⣠⡿
⠀⠀⠚⠛⠚⢷⣛⡟⠓⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠿⠿⠿⠷⠚⠉
⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣘⣿⡷⢦⣄⡀
⡔⠉⢩⡄⠀⢰⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣾⣷⡀⢀⣀
⠹⣿⣿⣿⣶⣶⣿⣿⣿⣃⣠⡴⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠏
⠀⠈⠙⠛⠿⠿⠛⠋⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣱⠶⣄⠀⠀⠀⠀⠀⠀⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⢻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠙⠱⣦⣹⡇⠀⠀⠀⠀⢠⠰⠆⠓⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣿⣋⡀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⣠⣿⡁⠀⠀⠀⠀⠘⣧⣀⣾⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⣿⣇⣽⣆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠻⣤⣿⠛⣤⠀⠀⠀⠀⠀⣽⡿⠛⠉⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡄⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣦⡀⠀⠈⢻⣴⣿⡄⣠⠀⠀⠸⡟⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠍⠛⢿⡟⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠳⣄⠐⢀⣿⣿⣏⠹⣆⢺⡖⠀⠀⠀⠀⡜⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠀⠀⢸⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠈⠲⣤⣍⣿⣿⣥⠛⣾⣱⣆⣠⣴⢞⣱⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢘⡅⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠲⣦⣴⣿⠿⡏⢻⡏⠀⠉⢻⣿⣿⣷⣿⣿⡗⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡄⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡄⠙⠀⠟⠀⠁⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⢻⡷
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⣷⣀⣠⡶⢘⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⡿⢹⡷⢿⣯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⠁⠀⠈⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡏⠟⣿⡀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠃⣿⠇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢰⡎⣷⡀⠀⡄⠀⢸⡇⠀⢀⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⣤⠹⣷⣀⣀⢀⣸⠇⠀⠀⣿⡄⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣼⣿⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⡀⠙⢿⣿⣿⠛⠀⠀⠀⢹⣿⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣤⣸⣿⠏⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡿⣴⡀⣸⡏⠀⠀⠀⠀⠀⠛⣦⣀⡀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢧⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣁⣾⣿⣤⣀⣡⣀⠀⠀⢀⣰⢿⡏⠚⢿⡄
⠀⠀⠀⠀⢀⣴⣶⣶⡆⢾⡿⠿⢿⣿⣿⣧⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠐⠒⠛⠻⠟⠀⠉⠉⠉⠉⠉⠀⠘⠛⠛⠛⠳⠞⠳⣦
⣶⣖⣀⣈⡁⠀⢈⣉⠀⠀⠀⢀⡀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠶⠶⠶⠖⠶⠶⠶⠶⠶⠖⠒⠲⠲⠶⢶⣶⣶⣶⠶⠶⠿
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉



⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢀⡴⠛⠋⠉⠉⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠈⠙⠓⠀⠀⠀⠀⠀⣠⠤⠤⣀
⠀⠀⠀⢨⣿⠟⠻⣷⣄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢒⣲⣤⢀⣿⡄⠀⠀⣾⣥⠀⣤⣈⣶
⠀⠀⠀⠸⣷⡆⢠⣼⣿⡄⢀⡿⢦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠎⠁⠉⣻⣿⡏⠀⠀⠹⡿⠀⢤⢸⡷
⠀⠀⠀⠀⠈⠹⢆⣆⣸⣧⡾⠁⠘⠉⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⢼⣿⠟⢛⣦⡤⣦⡗⢀⣀⠸
⠀⠀⠀⠀⠀⢀⡾⢿⣿⣋⣠⠆⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⣠⠖⠉⠁⠀⠘⣿⠉⠛⠂⠠⣄
⠀⠀⠀⠀⢠⡟⠀⠀⠈⠙⠉⠶⠀⢀⣀⣀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠶⠆⠈⣷⡄⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⠁⠄
⠀⠀⠀⠀⣼⠷⣤⡀⠀⠀⠀⢈⣉⣋⣉⣋⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢷⡄⠀⢸⠁⢆⠀⠀⠀⠀⠹⣄⠀⠀⠁⠀⠰
⠀⠀⠀⠀⢰⡄⠈⠻⣶⣤⣄⣀⠀⠀⠀⠉⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣶⣾⠀⡌⠁⠀⠀⠀⠀⠹⣆⠐⣦⡴⠶
⢀⣴⡶⢦⠈⡇⣠⣤⣴⣿⠿⠿⢿⡿⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠈⢻⡟⡁⠀⠀⠀⠀⠀⠀⢈⠰⡏⠀⣴
⣬⡅⣰⡆⠀⠀⡉⠁⠀⠈⠢⠀⠀⠀⠀⠘⣿⣾⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠷⢤⣴⠋⣿⣧⡀⠀⠘⣦⣰⣶⣌⣷⣤⠞⠁
⢸⡟⠋⠈⣷⣄⣿⡄⣠⠀⠀⠀⠢⣄⠀⠠⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⡾⠃⠀⢻⣿⠟⠋⠉⠉⠭⠜⣛⢧⣄⡠
⠀⠹⣤⠾⠋⠙⠻⣿⣎⢷⣄⠀⠀⠈⢳⣄⢹⡿⠃⣻⠻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⠟⠁⠀⠀⠸⡇⠀⠀⣀⣤⠄⠚⠉⢸⡏
⠀⢸⡇⠀⠀⣀⠘⣷⠈⢣⣝⢧⡀⠀⠘⣿⣿⡖⡿⠏⠁⡌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠄⠹⠀⠀⣠⣣⡾⠛⠉⠀⠀⠀⠀⣾⠁
⠀⢸⣇⢰⡾⣿⡀⢿⡇⠀⠙⢷⣝⠲⣄⣻⣿⣧⣷⣼⣦⠼⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⣿⡿⠃⠉⠀⢀⣠⣀⣼⠂
⠀⠙⣿⣼⣿⣾⡟⢸⡇⢱⡆⠀⠈⠻⢾⣏⣉⣩⡭⣶⡶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⡇⢤⠀⣿⡇⠀⠀⠐⣋⣵⠟⢉
⠀⠀⣿⣿⡃⢸⡇⢸⡇⢸⡅⠀⠀⠀⠀⠉⠻⣿⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣁⠀⠀⣿⡇⣠⣴⡿⠋⠁⠀⢸
⠀⠀⢸⣿⠁⢸⡇⢸⣷⣟⡓⠀⠀⠀⠈⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢹⠀⠀⣿⡟⠋⠁⠀⠀⠀⣠⡾
⠀⠀⢸⣿⡄⢸⡇⠀⢻⣿⡷⣄⠀⠚⢷⣦⣀⣀⡙⢧⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⣼⠀⡁⣿⢻⣧⣴⣤⣶⡟⠋⠁
⠀⠀⣾⣿⣷⢾⣧⣀⣠⣿⣷⣹⣿⣿⣶⣦⣬⢩⣥⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣶⡇⠀⣿⡇⣿⠁⣿⠁
⠀⠀⣿⣿⠛⣃⣴⣹⣿⣋⣀⣈⣿⣿⣿⠿⠛⠈⠛⠏⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⠟⣲⣿⡁⠋⣰⠃⢿⡀
⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠓⠀⠖⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣿⣿⣿⣥⣴⣿⣿⠟⢛⣛⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣓⣋⣙⣿⣶⣿⣻⣿⣧⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⠿⣿⣿⡿⠿⠿⠿⣿⡟⢿⠿⡿⠿⠿⢿⣿
⣿⣦⣶⣶⣤⣦⣶⣶⣶⣶⣶⣤⣧⣧⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣿⣿⣷⣾⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⢀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⢠⡿⣧⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣰⡟⠁⠙⣷⡀⠀⠘⣷⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣟⣀⣀⣴⣟⣀⣀⣀⣘⣷⡄⠀⠘⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠹⣿⡛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡄⠀⠈⢿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡿⠁
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣌⣿⣆⠀⠈⢻⣦⣤⣤⣤⣤⣤⣤⡀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠈⢿⡍⠉⠉⢩⡿⠋⠉⢉⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣆⠀⠀⠻⣯⠉⠉⠉⣽⠏⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠈⢿⣄⢠⡿⠁⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠹⣧⢀⣼⠏⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⣻⡟⠁⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⢹⣿⠃⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣰⡟⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⡿⠃⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠏⠀⠀⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⢠⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⣼⠏⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣰⡟⠉⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢀⣾⣿⡀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⣴⣿⣆⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⢠⣾⠃⠈⢿⡄⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣼⠏⠀⠻⣦⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⢠⣿⣁⣀⣀⣈⣿⣄⠀⠈⢻⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣏⣀⣀⣼⣏⣀⣀⣀⣹⣧⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢻⣆⠀⠀⢻⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠘⣷⡀
⠀⠀⠀⠀⣴⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⠀⠀⠹⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠙⣧⡀⠀⠀⣰⡟⠁⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠘⣷⡀⣰⡟⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠈⢿⠏⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣾⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⣿⠄⠀⠉⠈⠉⠀⠉⠀⠀⠀⠉⠠⠄⠀⠀⡀⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠘⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⡼⢿⣾⡄⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⠀⠀⢠⣇⠀⢾⣿⣿⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠎⢹⠀⡼⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⠀⠸⠄⠀⠄⠆⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⢄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀
⠠⡔⠈⠀⠀⠀⠀⠀⠀⠘⣆⡠⢄⠀⠀⠐⢦⣄⡀⠀⢀⡀⣤⠀⠀⢀⡤⠀⠀⠀⠀⡀⠀⠰⠟⢁⠊⠁⠚⠀⠀⡀
⠀⠀⠀⢠⡀⠁⢤⠀⠂⠁⡟⠳⠶⠄⠀⠀⠀⠙⠛⠓⠈⠁⠀⠒⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠀⠀⠀⠙
⠀⢸⠲⠤⠷⠔⠈⡓⠀⣴⣿⣤⣄⣤⣤⣤⣄⣀⣀⣄⣀⣠⣀⣀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⡇⠈
⠀⠁⠲⢆⠀⠀⠙⠇⢀⡿⠿⢿⡋⠘⠛⠛⠀⠀⠘⠁⠍⠘⠙⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖
⠀⣀⠀⠀⠁⠀⣬⡅⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀
⠀⠈⠻⣤⣠⢀⣿⠛⣻⡿⣛⣛⠿⠛⣿⣿⣻⣿⠛⠿⣻⣿⢻⡛⠛⡷⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉
⠀⢀⣤⠾⠛⣮⡄⢺⣿⡓⡇⡆⠀⢹⣿⠧⡇⠀⢈⣹⣿⡇⣏⢘⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣧⠀⣴⠃
⠀⠀⠐⠀⠺⡁⢉⣿⣿⢱⡷⡟⠚⣿⣿⣃⡷⠺⠛⢻⣿⡇⡷⠲⠀⠀⠀⠸⠓⠀⠀⠀⠠⠀⠀⣀⠠⠙⣛⣿⣏⢡⡄
⠀⠀⠀⠀⢰⣭⡿⣿⣿⣺⣇⣇⠀⢿⣿⡥⣇⣘⡀⢸⣿⡇⣿⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠐⠙⠂⠉⣶⠂
⠀⠀⠀⠀⠈⠉⠀⣿⣿⠯⣏⡇⠀⢸⣿⡂⣿⢸⠀⢸⣿⠗⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠓
⠀⠀⠶⠂⠀⠀⣤⣿⣿⡿⠏⢇⡃⡀⠀⢠⠦⠀⠀⠀⠀⠀⠀⢀⣤⠖⠇⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⢠⡞⣿⣗⣀
⠀⠀⠀⠀⢰⠀⣼⣿⣿⡷⠿⠿⠻⠛⢧⣸⡴⠿⢀⠱⡶⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠀⠐⢀⡹⠃⠀⠉⠁
⠀⠀⠀⠀⣬⠀⠘⡿⠿⠧⣴⣷⢸⠛⢿⣧⣠⣼⣿⡅⠉⣆⡀⠀⠀⠀⠀⠸⢂⡔⠠⢄⠂⠀⠀⠀⠐⠊⡄
⠀⠂⠀⠸⢷⣾⠀⢠⣂⣶⠛⢣⣤⡎⠁⠻⣧⡄⠈⣁⠐⠈⣁⣤⡁⠀⣁⠀⠚⣁⠀⢈⡀⠅⠀⠇⢀⠀⠹⡄⠀⠁⢀
⠀⠀⢡⠰⣿⡗⡂⠾⠿⢏⠘⡆⢻⠁⢸⣿⣿⠁⠀⠸⠄⠔⣿⣿⣱⡶⣶⣦⢀⣿⡐⣯⠛⠠⠀⠄⡇⠀⢀⠉⠀⠀⣈
⠀⠀⢀⣴⢿⠦⠋⣠⣴⣺⡀⡜⣨⣼⣿⣿⣿⠀⠀⢸⣿⠀⠇⢸⣿⡃⠟⣩⡌⢹⡇⠺⣗⠀⠀⢠⠁⠀⠨⡧⠀⠸⡇
⠀⠀⣿⠇⢢⠀⢀⣿⣿⡿⢓⣮⣿⡿⣿⣿⣿⠀⣀⠼⣿⠀⠀⠈⢹⣷⣴⣿⡇⠀⣿⠰⢿⠛⠻⢿⣿⠿⠶⠀⠀⠸⠇
⠀⢀⣠⣴⡏⣀⣾⡿⣼⡆⢻⡿⠘⠃⡟⣿⣿⣞⠃⠀⣿⠀⠀⠀⠸⡅⢸⣿⣷⠀⣿⠄⠀⠀⠁⠀⣭⠈⠉⢋⠀⠑⣄
⠐⠁⢒⡿⣷⣿⣿⣿⣝⡟⠀⢤⣤⡀⠁⣼⡏⡇⠀⠀⣿⠀⠤⠤⠤⠥⠄⠸⠋⠀⠉⠀⠀⢁⣞⣷⣾⠀⢐⠈⠂⢰⣿
⠀⠀⠺⠇⣻⢿⣿⣿⣧⣷⡀⠀⠈⠉⠑⠾⠂⢷⠀⣛⣳⣤⣤⣄⣀⠀⠀⠋⠉⠀⠰⠀⠀⢿⣵⣿⣿⢃⣄⣀⢀⣈⣁
⠀⠙⠦⢼⣿⡯⠏⡿⣿⠁⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠉⡛⠉⠛⣯
⠀⠀⠀⠀⠘⠃⢀⠀⠛⠒⠒⠛⠛⠛⠳⠚⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠯⣝⡻⡇⠀⡄⠀⠀⢙
⠀⢠⣤⣀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠹⢙⣇⠂⠀⠀⢀⣀⡌
⠀⠈⠈⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠠⣠⠀⣠⠔⣊⠥⠲⠚⠋⠀⠁⠐⠿⢸⣇⣠⣴⣌⣸⢁
⠀⠈⢷⣧⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠀⠀⠀⠀⣿⣿⠆⡀⢶⢹⢴⣿⣿⣸⣷⠈⡀⢀⣠⣀⠀⠀⢀⠉⠉⡉⠉⠡⠏
⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣬⣤⣤⣤⣥⣤⣭⣭⣭⣥⣤⣤⣬⣵⣬⣥⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣄⠀⠀⠀⠀⢸⠟⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠩⠀⠠⣄⡐⠆⠀⢀
⠀⠀⠀⠀⢻⡇⢤⡀⠀⠘⠆⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣀⠀⠀⠐⠀⠀⠀⠀⠀⣠⣾⣿⡆⠀⠁⠐⠚⠦
⠀⢠⡏⠀⢸⡆⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⢀⣼⣿⡿⠃⠘⠁⢀
⠀⠉⠁⢠⣶⡅⠘⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⣾⠃⣀⣀⣀⣿⣿⠏⠐⠛⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠻⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⢿⣿⡎⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⢲⣶⠿⠇⡟⢋⣿⣿⣦⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠼⣿
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⢋⢸⠁⢠⣼⡄⢱⠀⠀⠀⠀⣀⠤⢠⣴⣶⠖⠐⠉⠀⣿
⠀⠀⠀⠀⣘⡳⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠂⠈⠻⠇⠈⠀⠀⠐⠁⠀⠀⠘⢻⣽⠀⠀⠀⠀⣿
⠀⠀⠀⢼⣿⣿⣿⡏⢶⡄⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⠺⡎⠀⠀⠀⠀⠀⠀⠀⠸⣰⡾⠀⠀⠀⠀⣾
⠀⠀⠀⣴⣭⣿⠏⠁⣿⣇⣼⣿⠀⢗⠀⠠⠀⡄⠀⠨⠃⠈⠑⠀⠀⠀⠀⠀⡀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼
⠀⠀⠀⢻⠿⡟⡴⣸⡿⠻⣿⡟⠐⢸⠀⠂⠀⠀⠀⢰⡶⠘⣷⣾⣧⣼⣿⣿⣿⣟⣷⡞⠐⠀⢀⣀⣤⣤⠠⡤⢄⣤⣶⣾⠭
⠒⠖⠶⣿⠇⠀⠁⠸⣏⡀⠿⠀⠰⢸⢁⡀⠀⠀⠀⠀⢠⣶⣈⣿⢿⠉⢻⣿⣿⣏⣿⠷⢿⣶⡀⢩⣥⣤⣾⣤⣤⣼⡍⢻
⠀⡀⠀⣿⡤⡀⠀⠘⣯⡏⢀⠀⢀⠸⠞⠣⢭⠁⠒⠀⣾⣿⣿⡉⢺⢄⠐⣜⣿⣿⢷⣙⣻⣽⣿⣾⣿⣿⣿⡿⣿⡿⣿⠸⡄
⠀⠄⠀⣿⠀⡁⠀⠂⢈⡅⠀⢐⡠⣴⡆⣁⣴⢶⠀⠨⣿⣿⡿⣟⣿⣿⢄⠈⠉⣹⣾⣭⣽⣟⡿⢻⣿⣿⡛⠂⠆⠀⠽⠃⠈
⣄⡄⠀⠈⠐⡃⠀⠀⠈⠀⣸⣿⡷⠃⣠⠀⣷⡖⢀⠖⠚⣿⣗⣿⣯⡋⢸⣶⣴⡿⠍⡸⢺⠿⡟⠉⢿⡛⣿⠀⠂⣤⠀⠘⠄
⣿⠃⣤⢰⡜⣿⣾⡇⠀⣸⣿⣿⣧⣌⣽⡡⣿⣏⡟⠀⢰⢿⢩⣿⣿⡟⠈⣡⣤⣤⣠⡥⢾⠀⢀⣴⣶⠀⡉⠀⠁⢠⣴⣠⣔
⣿⣾⣿⣾⣟⠙⣿⠂⠀⣿⣿⣿⣿⣿⡟⠀⡛⡿⠇⠀⢾⣿⡿⣿⡿⣟⣠⣿⢿⣉⣾⠅⠈⠁⣼⣿⡆⠯⠅⠀⠀⢈⣅⡙⣿
⣿⠀⣽⣷⣴⢴⣸⠀⠀⢿⣿⣿⣿⢈⡠⢬⠁⠀⠀⠀⡄⠿⢙⡋⠑⠿⠟⡻⢿⡿⢻⣧⠀⢠⣼⡿⠃⠀⠀⠄⠀⢸⢽⡧⣿
⣿⠀⢸⣿⣯⣸⣿⠀⠀⣼⣿⣿⣿⠁⠁⠀⠀⠀⠀⠀⠃⠀⠈⠃⠀⠀⠀⠁⠈⡇⢸⠃
⣿⠂⢘⣿⡷⢸⣾⠀⠀⢽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠖⠀⡶
⠿⣀⢳⣿⠟⠚⠟⠀⠀⢽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠘⠁⠀⠀⢁
⠸⣿⠿⠟⠂⠀⠀⠀⠀⠸⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⡀
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⢠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢸⣿⠌⡌⣷
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⢠⣿⣿⢣⡼⣿⢈⣧⠘⣷
⠀⠀⢠⣄⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠂⠀⠀⢠⢳⣜⣧⠀⠀⣠⣿⣿⠟⣸⡇⣿⢸⣷⢲⠈⠃
⠀⠀⠉⡀⠀⠀⠀⠀⠦⡐⠐⠂⡀⠀⠀⠰⢊⣰⢃⠀⠀⠀⠘⣷⠻⠎⣡⣴⣿⡿⡃⣴⡘⢁⠏⢸⣿⢸⡆⢰⡄
⠀⢀⠁⠁⠀⠀⠈⠘⠀⢣⡠⠀⡞⠢⠄⣀⡀⠀⠀⠀⠁⠀⢿⣿⣿⡷⠿⢟⢻⣾⣷⡙⢡⣿⢠⢲⣏⠸⡇⣧⠁
⠀⠟⢐⠀⠀⠀⠈⠘⠀⠀⠀⠔⠁⡜⠀⠄⠈⠎⠒⠀⠀⠀⠀⢫⣶⣾⡏⣿⠿⡷⠉⣴⣼⣿⢸⢻⣿⢦⣌⢹⡀
⠀⡆⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠙⠖⢤⠴⠣⠀⠀⠀⠀⠀⠘⣣⣙⢃⡌⣴⢰⣿⣻⠆⡟⣾⣻⣿⢸⣿⠘⡇
⠐⠁⣤⠀⢨⠀⢠⠀⢡⡰⡀⡄⠠⠀⢧⠈⠂⠣⠀⠀⠀⠀⠀⠀⠫⠏⣿⢣⣿⢸⡇⡿⡐⢰⣿⢹⣿⢸⣿⠀⣧
⠀⢠⠹⠀⠀⠀⢨⠀⠀⠀⠛⠱⡧⠁⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠸⢹⠸⢡⡄⣿⠈⣿⣼⡏⣿⣿⢀⢹
⠀⡣⠈⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣻⠇⣿⢰⡏⣿⠇⣿⡋⣾⡌
⠀⠀⢠⠃⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⠘⢱⡟⠀⢿⠇⢫⡄
⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡄⡾⠁⠆⠘⠀⠈⡃
⠀⠀⠂⠀⡜⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⡇⠀⡾⠀⠀⡼⡀⡁
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢠⣡⠀⡇⢽⡇
⠀⠀⠀⠈⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢏⣾⡏⣴⡆⠻
⠀⠀⠀⡀⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢃⣼⡟⣰⡟⠀⠰
⠀⠀⠘⠀⠀⠀⡀⢀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⣾⠟⢠⠟⣴⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⢀⠠⣾⣿⠘⣿⡇
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⣽⣿⠸⣿⣿⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢿⣿⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠀⠘⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⣴⡄⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⢀⣾⠟⢿⣄⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢀⣾⠏⠀⠈⢿⣆⠀⠀⠘⣿⡄
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣏⣀⣀⣠⣿⡋⣀⣀⣀⣀⣻⣧⠀⠀⠈⢿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠙⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣧⡀⠀⠈⢿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟
⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠈⢿⣆⠀⠀⠙⣿⡟⠛⠛⠛⠛⣿⡿⠛⠛⠛⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⡄⠀⠀⠻⣿⡟⠛⠛⠛⠛⣿⡟⠀⠀⢀⣾⠏
⠀⠀⠀⠈⢿⣦⠀⠀⠘⢿⣆⠀⠀⣰⡿⠁⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠹⣷⡀⠀⢀⣾⠟⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⢻⣦⠀⠀⠈⢿⣆⣼⡟⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠙⣿⣄⣾⠏⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⢈⣿⠟⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⢸⣿⠋⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣀⣾⠏⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣠⣿⠃⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⣰⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠁⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⣰⡿⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⢀⣾⠟⠻⣧⡀
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⣼⣿⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢠⣾⡏⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⣰⡟⠁⠀⢀⣼⠟⢻⣧⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⢠⣿⠛⢿⣄⠀⠀⠙⣷⡀
⠀⠀⠀⠀⣼⡟⠀⠀⢀⣾⠏⠀⠀⠻⣧⡀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⣠⣿⠃⠀⠈⢿⣆⠀⠀⠘⣿⡄
⠀⠀⢀⣾⠟⠀⠀⢠⣾⣧⣤⣤⣤⣤⣽⣷⡀⠀⠀⢻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣴⣿⣥⣤⣤⣤⣬⣿⣦⠀⠀⠘⢿⣆
⠀⢀⣾⠏⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⣷⡄⠀⠀⠹⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⢿⣆
⢠⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡄⠀⠀⠹⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⣆⠀⠀⠘⣿⡍⠉⠉⠉⠉⣽⡟⠉⠉⢉⣿⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠘⢿⣄⠀⢀⣼⠟⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠈⢿⣦⣾⠏⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠈⠻⠋⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏
⣿⣿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢁⣄⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡾⠻⣦⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢠⡿⠁⣤⣿⣧⠘⢷⡄
⠀⠀⠀⠀⠀⠐⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣾⠏⣰⣿⣥⣤⣤⣤⣤⣤⣬⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠘⢷⡄⢲⣶⠶⠶⣶⡶⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠹⣧⠐⠟⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⢠⣄⢠⡿⠁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠹⣧⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠹⡿⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣄⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢁⣾⣧⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢻⣆⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⢠⣿⣁⣙⣁⣈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢁⣴⠄⠹⣧⠘⣷⡀
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠺⠛⠿⠿⠿⠇⠈⢿⡄
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡛⠛⠛⠛⠛⠛⢛⣿⠋⣰⡿⠿⠿⠷⠶⠶⠶⠾⠿⠿⠾⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠻⣿⠛⢠⡾⠃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠹⣷⡿⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠙⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⣿⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⡿⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⣾⣧⣠⣶⣿⠟⠛⠛⠻⣿⣶⣄⣼⣷⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢿⣿⣿⣯⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣧⣠⣤⣤⣄⣼⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠟






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⣻⣷⣦⣽⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⣀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠓⠶⣿⣿⣟⣿⣿⣿⣿⣿⣦⣤⡀⠀⢀⠠⢀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣷⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣦⢸⣷⣤⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠻⢿⠿⣿⣿⣿⣿⣿⣿⠿⠷⠿⠛⠀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠟⢻⣿⣿⣷⣿⣦⣀⠀⠀⣸⣿⣿⡿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⢿⣿⣿⡤⠀⣼⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣨⠏⠀⣿⣿⣅⢀⡘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣰⣿⣿⣿⣿⣇⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣷⡟⠙⠁⠀⢹⣿⣿⣿⣿⣿⠀⠀⠹⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡀⠀⠀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠴⠶⠷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠍⠀⠒⠀⢉⣙⠉⠉⠁

⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠄⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣽⣿⣖⣁⠀⠀⠀⠠⢉⣷⣷⣋⠠⠀⠀⠀⠀⣹⣿⣿⠁⠀⠀⠀⠠⠤⢾⣿⡧⠤⠀⠀⠀⠤⠴⢾⣿⠶⠤⠄
⠀⠀⠀⠀⠜⠉⠇⠀⠀⠀⠀⠀⠡⠀⡇⠌⠁⠀⠀⠀⠈⠀⠹⠁⠈⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣄
⠀⠀⢀⣀⣸⣶⣏⣀⡀⠀⠀⠀⢀⣦⣧⣀⠀⠀⠀⠀⢤⣀⣼⣀⡠⠄⠀⠀⠀⣀⣠⣧⣀⡀⠀⠀⠀⢀⣠⣾⣧⣈⠋
⠀⠀⠀⠀⡸⠛⢧⠀⠀⠀⠀⠀⢀⡿⡿⡅⠀⠀⠀⠀⠀⣨⠟⢯⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠙⢿⠟⣩⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢠⣿⣿⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣰⣿⡟⢻⣿⣆⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣴⣿⠟⢠⡄⠻⣿⣦⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⣼⣿⠏⠀⠛⠛⠀⠙⠛⠃⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⣧
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣦⠘⢿⣿⣿⣿⣿⣿⣿⡟⢁⣾⣿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣴⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠈⢿⣿⡄⠠⣤⡄⢠⣾⡿⠃⣤⣤⣤⣤⣤⣤⣤⣤⠀⢤⣤⡄⢠⣤⠄⢠⣿⡿⠁⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⡈⢻⣿⣆⠉⢠⣿⡿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢿⣿⡄⠉⣰⣿⡟⢁⣾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢻⣿⣆⠘⠿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢻⣿⣆⠹⠟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠹⣿⣧⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⢻⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣴⣆⠹⣿⣧⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣴⡄⠹⣿⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⢁⣼⣿⠏⣀⠘⣿⣷⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣾⣿⠃⣀⠹⣿⣧⡈⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⢁⣾⣿⠃⠐⠛⠃⠈⠛⠓⠀⠛⠛⠛⠛⠛⠛⠛⠛⢠⣾⡿⠃⠘⠛⠂⠘⣿⣷⡀⢻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢠⣿⡿⢁⣼⣿⣿⣿⣿⣿⣿⣷⡄⠻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢠⣤⣄⠀⣤⣤⠀⣰⣿⡟⢁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠻⣿⣦⠘⠃⣴⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⣿⣧⣼⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⣿⣿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⠃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠔⠒⠀⠀⠀⢀⣠⣼⢨⣿⣽⣿⣿⣿⣿⣋⣃⣁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⣠⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣟⠓⡑⣋⣑⡐⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡃⠁⠀⢾⣿⣿⡿⢋⣽⣿⣿⣿⣿⣿⣿⣿⣭⣭⡿⣿⣾⣿⣏⠂⠐⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣻⣻⡣⠶⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣩⣹⣉⣛⣉⣻⣙⣻⣟⣺⣞⣙⣿⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠻⠛⢻⡿⢻⢿⣟⡿⣿⢛⣿⣿⣟⢻⡏⠉⠉⠉⠉⠉⠉⠉⠉⢻⡛⡿⠋⠙⢉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⣿⣿⡇⣼⢁⢊⢸⣿⢰⡇⠘⣿⣿⣿⣸⡇⠀⢀⡀⢀⣀⣀⠀⠀⠸⣧⢹⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠛⠛⡧⣿⠸⠨⠠⣀⣼⡇⠀⣾⣟⣛⣾⡏⣰⣿⣷⡄⠀⠙⢧⠀⢈⣿⡄⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⣔⢿⣿⢰⣰⡄⠂⢸⡇⠀⣿⣆⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡱⡉⣿⡇⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠴⢀⡹⠴⣿⠘⢹⡯⠀⢹⡇⠀⣿⡷⠂⠀⣿⡇⡿⣿⣿⣿⡿⠀⡿⣿⣿⡇⡇⣨⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡌⣽⣟⣿⢰⣿⣷⣴⣼⡇⣠⣿⠧⠐⠂⣿⡟⢤⣿⣿⣿⡤⠀⣷⣿⣿⡇⡗⣲⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠞⠻⠛⠛⠐⠛⠛⠛⠛⠓⠓⠛⠓⠋⠉⠛⠃⠘⠿⠟⠛⠒⠐⠛⠛⠙⠁⠁⠉⠁⠈
⠀⣠⣀⣀⣀⡠⠄⣀⣠⣤⣠⣰⣤⡄⣠⣄⡀⣀⣀⠀⠀⢀⢀⠀⠀⠀⠀⠈⠀⠠⢁⠀⡀⠀⠈⠀⠀⣀⣘⠀⣀⡀⠀⣀⡀⠠⠀⣀⡀

⣿⣟⡛⠻⠿⠿⣿⢿⣟⠻⠿⠿⠌⠉⠉⠉⢉⣠⣽⣿⣿⣿⣿⣯⠛⠙⠿⠿⠿⠿⣿⣿⣿⣿⣿⣤⠼⠛⢛
⣿⣯⣽⣤⡤⠄⢀⣠⣬⣥⣤⣤⣤⣴⣾⣿⣿⣿⣿⣛⠟⠛⠋⠁⠈⣠⣤⣤⣉⡒⢋⣙⣻⣿⣿⣿⣷⣶⡶
⠋⠉⠁⠀⠀⠘⠛⠻⠿⠿⠯⠍⠙⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⢠⢠⠈⠛⠛⠛⠻⠏⠙⠛⠛⠉
⣶⣶⣶⣦⣶⣷⠰⢘⡛⡁⠀⠁⢀⠀⠉⠀⠄⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠀⠉⠉⠀⠀⠠⠤⢤⣤⣜⣋⣉
⣿⣿⣿⣗⣿⣧⠶⠶⠛⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⢶⡙⢹⣿⡏⠻⠍
⡿⠆⠙⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠦⣦⣤⣄⣀⣸⣿⠃⠦⣴
⠁⣀⣤⣄⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡛⠛⢌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⣝⠿⠛⠀⠀⢀
⣴⣿⣟⠟⠙⠷⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠭⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠠⣿⣾⣶⣄⣘
⣤⣶⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢻⡿⠂⠀⠀⠀⠀⠀⠀⠀⡤⣿⡿⣿⣿⣿⣇⣛⡙⢳
⣻⣻⣿⠻⢿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⢠⢦⡉⠁⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠿⠛⡿⣿⣿⣟⣿
⢿⣿⣿⣷⣯⣥⣤⢄⣀⣀⡀⡀⣠⠖⠀⢈⣠⣤⣤⣿⣿⣿⣥⡄⠀⠀⠀⠀⠀⣀⢀⣛⣀⣿⣇⣑⣫⣁⡌
⣽⣿⣿⣿⣋⣁⣾⣿⣿⣥⣤⣤⠀⠄⣼⣿⣿⣿⡋⠙⣩⣿⣿⣿⣿⡀⠀⠀⠀⢿⣿⣮⣿⠉⠉⠉⠉⠉⠠

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠤
⠀⠈⠙⡦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠀⠈⠀⢀⠰⠂
⠀⠀⠀⠈⠀⢠⠉⠓⠶⠠⢄⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠐⠪⠀⠢⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠠⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⣠⡧⣿⡄⠀⠀⠀⡘⠰⠁⠀⠀⠀⠀⢠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠑⠄⢹⡷⠁⢁⠘⢥⣶⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠈⢷⡁⠀⠀⢰⡝⠒⠀⡀⠀⢀⠐⠀⠀⠂⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡠⠀⠀⠠⠀⢈⡖⠀⠀⡏⠀⡈⠁⠠⢀⢳⡨⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⢁⠘⠀⡌⠙⠒⠀⠀⠀⢰⡀⠀⠀⠑⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣿⡌⢸⠀⡇⠰⠀⠀⠀⠀⠀⠓⣼⠀⢰⠒⠲⠅⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠛⢹⠀⠀⣇⠀⠀⠀⠀⣀⣀⠀⠈⣕⢺⡧⠀⠀⠚⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⡦⠴⠖⠀⠉⠁⠀⠀⡀⠤⡀⢄⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⡧⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⠚⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⣧⠀⠀⠐⢄⠀⠀⠀⠀⠀⠃⠀⠐⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠈⠳⡄⠀⠀⣼⣀⡤⠀⢷⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠤⠂⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠤⡀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠴⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢰⡶⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠢⡇⠀⠀⠰⡇⠀⠀⢸⡆⠊⠉⠉⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢰⠋⠉⢑⡄⣿⠀⡰⠃⡀⢣⠀⢸⡇⡎⢉⠉⣂⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⡠⠀⠀⠀⣿⣰⠇⣀⠀⠈⡧⢸⡇⠀⡀⠀⠸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢟⠀⣀⣀⠀⣿⠈⠀⠻⡼⠁⠀⣿⠇⣀⣀⠂⣼⠁⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡔⢁⠄⠀⣿⠀⢄⠲⠴⣡⠂⢸⡇⠑⠄⢡⡌⠐⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⢽⢧⡘⠆⣿⠀⢐⣽⣿⡕⠀⢸⡇⠖⣤⢟⢤⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢊⠙⠧⡙⠀⣿⣰⢏⠀⡀⠙⣦⢸⠃⠜⠵⠟⣁⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢟⠃⢀⠀⠀⣟⣿⡆⠸⢡⢀⣿⣿⡆⠠⡀⠈⢻⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣘⣢⠆⠀⣄⡿⢿⡧⢰⣶⢬⣿⢿⣂⠀⠀⢤⣢⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⠠⠤⣄⡀⠸⣿⡟⡄⠀⠀⠟⡇⠈⡶⢻⠿⠻⡇⠘⠋⡀⠀⠰⡹⣷⡇⠀⣀⠤⠤⠀⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡎⣁⠶⠶⢶⣦⣍⣀⠙⢶⣿⡜⡰⡄⠀⠰⢰⠿⠆⣼⢷⡀⣟⠇⠐⢰⢀⠶⠑⣽⡷⠊⢅⣨⣤⣶⠶⠶⢂⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠿⠥⢤⣖⣠⠰⠈⡑⢍⠳⣄⡟⡴⠁⠘⡇⡀⣴⢁⡼⢃⡈⢻⡄⢦⣠⠈⡏⠀⢳⡘⣇⡼⠋⢋⠁⠂⣠⢐⡤⠤⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡞⠳⣄⠄⣱⡘⣿⠁⠀⠀⢰⠟⣿⠞⠡⠊⠙⠦⠙⢾⡹⠧⠀⠀⠀⢻⢋⣔⠁⢄⡴⠙⣾⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢺⣤⡇⠻⣟⠐⢠⡰⢀⠖⡡⠊⡀⠀⠀⠀⡀⠠⡱⣄⠖⣀⠀⢸⠞⠁⣇⡾⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⡇⠀⡙⢀⠿⠡⠋⠜⠀⠀⢹⠀⡇⢰⠀⠀⠑⢌⢯⠻⣅⠘⠄⠀⣿⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⢇⠁⣼⣇⡱⢀⠀⢀⣼⢶⣿⣿⣿⣿⣿⢦⡄⠈⠆⢀⢺⣿⠀⢁⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢛⢈⡇⠸⢠⡿⠛⢩⠁⡴⢢⠀⢩⠓⠹⡇⠰⠘⡧⢺⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢄⢾⠠⡇⢠⠸⣦⡀⢻⡂⠈⠉⠀⣾⢁⣴⡇⠸⠀⡇⣨⢢⠜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠊⢇⢳⣀⠀⠀⠟⢓⡾⠿⡿⢻⡖⢻⠃⠀⠁⡰⢃⠋⠃⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠑⠀⠙⡷⣀⠀⠐⠈⠳⠀⠇⠸⠉⠘⠀⢀⡴⠋⠀⠀⢤⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡄⠈⠒⢸⡏⠲⣤⡀⠀⠀⠀⠀⠀⣠⠰⠈⡙⠐⠊⠢⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⡎⢆⡜⢠⣌⡇⠀⠀⢠⣏⡄⠑⡐⡼⡄⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠸⣧⡜⣷⢀⡉⠀⠀⡇⠀⠉⡀⢰⢁⣮⡇⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣿⣼⠣⣽⠄⣤⣧⠀⣘⡤⣿⣼⣿⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠾⠿⠿⠿⠿⠿⠖⠛⠉


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

⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡄⢀⡄⠀⠲⠀⠀⡄⢀⡄⢠⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⣤⣤⣤⣤⣤⣤⣤⣶⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣶⠰⣷⠰⣇⠀⣿⠁⢰⡷⢰⡇⢰⡆
⠀⠀⠀⣀⡠⢜⣛⣭⣝⣛⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢻⣄⠙⢦⣿⣶⠞⢁⡾⢃⣾⠁
⠀⠀⠈⣵⣾⣿⣿⣿⣿⣿⣿⣶⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡙⠳⠦⣤⡤⠶⠛⣡⡾⠁
⠀⠀⠀⣾⣿⣿⡟⠉⠙⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣶⡶⠶⠛⠁
⠀⠀⠀⣽⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠛⠛⠛⠃⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠛⠓⠂



⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⠀⠀⢀⣀⣀⣀
⠀⠀⠠⣤⣤⣤⣬⣭⣜⣷⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠘⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣶⣶⣶⣶⢦⣄⡀
⠀⠀⠀⠙⢇⣾⠂⠀⠀⠈⠷⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⢠⣿⡇⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢠⣿⣿⠟⢿⣿⣦⠈⠻⣦⡀
⠀⠀⠀⢠⡾⢧⡀⠀⠀⠀⣼⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠸⣿⣷⣤⣼⣿⡟⠀⣠⡾⠃
⠀⠀⠠⠿⠷⠎⢷⡴⢶⡶⠷⠾⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢾⣿⣿⣿⣿⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠈⢿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢿⣿⣶⣤⣀
⢀⠔⠒⢢⠀⠀⠀⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣾⣿⣿⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣶⣶⣶⣶⣾⣝⢷⡄
⢧⣀⣤⣾⣦⡀⠀⢻⣿⣾⣿⣿⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣮⡻⢿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣠⣴⣶⠦
⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠙⠛⠿⠿⠟⠛⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⣿⣿⡿⠿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⡆⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⢉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠁⠀⠀⠈⠉⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠰⡅⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⡷⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠂
⡀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠐⠒⠐⠀⠒⠲⠒⠒⠀⠀⠦⣴⠖⠒⠒⠂⠒⠒⠠⠔⠀⠀⠐⠒⠂⠀⠂
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠀⠀⠀⣄⣀⢀⡀
⠂⠔⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⢰⠀⠂⠃⠀⠀⠈
⠂⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠃
⢰⣤⡼⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤
⢸⡂⠓⣽⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿
⣸⢆⠀⢀⡧⣿⠀⠠⢒⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠈⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣽
⣉⣿⢻⣾⣿⣿⢧⠀⡌⠎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⢻
⢸⠨⣹⣿⣿⢿⢊⡳⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠄⣰⠆⣴⣦⢢⣶⠀⠀⠀⠀⣠⠚⢁⣠⣧⠶⠚
⠸⡘⣓⠿⢺⣴⣾⣟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣐⢂⡔⢻⡔⠈⠠⠆⢻⠁⠀⣰⠇⢹⠂⢈⢿⡉⠰⣠
⢀⠹⢣⣷⣿⣿⣟⢯⢻⣿⡄⠀⠀⠀⠀⠀⠀⠲⠛⠓⠓⠿⠦⠀⠀⠀⠀⠀⠄⡠⠐⠀⠐⠃⠘⠀⠈⡌⡄⠀⠃⡈⢀⣀⠿⢉⢀⡀⡰⡄⠀⢶⣬
⢸⢳⠠⠿⠿⠧⠹⢟⣪⢷⣿⣦⡀⠈⠈⠀⠁⠀⠈⠄⠁⠀⠒⠀⠀⠀⣈⡴⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣽⠀⠉⣁⡆⡀⠁⠀⠀⣠⢰⠁⣉⡍⢻
⠀⣐⣝⣷⣤⡗⢨⣷⣬⢸⣾⣿⣷⡄⠠⠀⠀⠄⠀⠤⠁⠃⠀⣁⣴⠞⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡴⢉⠀⡀⠊⠀⣀⣧⠘⠀⠅⠠⢸
⠀⣸⠜⡿⠳⡙⣤⡹⣯⡹⢿⣟⣿⣿⡆⡀⠀⠀⠀⣀⣠⣶⠟⠋⠀⠀⠄⠀⠀⠀⠠⠂⢠⠁⠂⠃⠀⠀⣷⠼⠹⠁⠀⠁⠀⡐⢉⡟⡒⠃⡤⠀⢰
⠨⡏⠶⠙⠓⣔⣨⠙⠙⡷⣾⣿⣿⣿⣿⣶⡀⠀⢞⠉⠀⠀⠀⠀⣠⠊⠀⠀⠀⠈⠃⡸⡇⢨⣁⠀⡤⢦⣾⠸⡶⡀⠀⠃⡘⠀⢻⠹⡇⠱⠟⡟⢸
⠐⡧⠁⠳⡳⣌⠣⣀⠈⠋⣿⡤⣿⣿⣿⣿⣿⣦⡀⠐⠀⠀⠠⠠⠀⠠⠀⠠⢀⠄⠀⡀⠸⠆⠿⠀⠠⠮⠉⢚⣾⠁⠐⠀⠀⡴⡿⢀⠂⠀⠀⢇⢸
⠀⠀⠈⢀⢣⣌⣼⢯⣆⠪⡍⢫⡻⣝⣿⣿⢿⣿⣿⡧⣴⠀⠀⠀⠈⠀⠀⠂⡄⠴⠊⠁⠀⠀⠀⠐⠂⠀⠀⠚⡾⠀⠕⢀⡥⡧⠂⠉⠀⠸⠘⡀⠠
⠀⠀⠀⠈⠀⠨⠈⠀⠉⠂⡤⠙⢓⣨⣿⣯⣟⣿⣿⣿⣿⣧⡄⣀⣴⠴⣠⣀⡀⠈⠷⠀⠀⠀⠐⠄⠐⠁⠀⠀⠁⠀⠘⢼⣥⣧⡀⠠⠀⢠⡐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟
⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠘⣷⡀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣤⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇
⠀⠀⠀⠀⠀⠀⣴⡟⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢻⣦
⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠹⣧
⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣼⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠠⣖⣂⣤⢠⡄⠀⢀⣄⣠⡄⠀⢠⣒⢰⣄⢀⣄⣦⣄⣦⡀⣠⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠓⠀⠃⠀⠘⠊⠃⠃⠀⠒⠚⠙⠘⠙⠚⠓⠋⠓⠋⠓⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢄⠠⠔
⠀⠀⠀⠀⠰⠀⠁⠘⡀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠘⠘⠋⠛⠛⠐⠀⠀⠀⠀⠀⠈⡀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠈⠉⠚⠛⠘⠋⠉⠉⠀⠀⢀⠀⠀⠀⠰⣀⡤⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢂⢀⠀⠀⠀⠀⠀⠀⣄⠁⠈⠁⢁⣀⠀⠘⠘⠧⠋⠂⠀⠀⠂⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⢋⡘⠁⢀⡣⡤⠀⠒⢀⣈⡉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠋⠡⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠈⠁
⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠐⠁⠀⠀⠀⠀⠀⠙⠀⠈⠀⠀⠀⠀⣀⠈⢹⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢠⡄⠀⠀⢠⡄⠀⠀⠀⣠⠀⠀⠀⠆⠀⠀⠀⠀⣀
⠀⠙⠃⢀⡀⠀⠉⠀⠀⠤⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠆⠀⠀⣠⡀
⠀⠀⠀⠈⠀⠀⢀⡀⠀⠠⡄⠀⠀⡀⠀⠀⡀⠀⠀⢠⡀⠀⠀⠘⠀⠀⠀⠐⠷⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠙⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠐⠂⢀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⢠⣤
⠘⠃⠀⢠⠀⠀⣤⡄⠀⠀⠀⠀⢰⡆⠀⠀⠴⠆⠀⠛⠀⠀⠖⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠛⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣀⡀⠀⠀⠶⠀⠀⠀⠛⠀⠀⠀⠉⠀⠀⠒⠀⠀⠙⠀⠀⠀⠘⠃
⠀⢠⡄⠀⠀⠀⠀⠘⠃⠀⠀⠀⠆⠀⠀⠀⠀⠘⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠦⠀⠀⠤⠀⠀⠀⠀⠀⠶⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠛⠀⠀⠀⠀⢀⡀⠀⠀⠛⠀⠀⠀⠙⠀⠀⠀⠁⠀⠰⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢤
⠀⠀⠀⠴⠀⠀⢀⡀⠀⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠘⠀⠀⢀⠀⠒⠀⠀⠃⠀⣀⠀⠀⠘⠁⠀⠀⠶⠀⠀⠀⠀⣠⠀⠀⣀
⠀⠆⠀⠀⠀⠀⠈⠃⠀⠀⠀⠠⠄⠀⢠⠀⠀⠘⠀⢀⡄⠀⠙⠁⠀⡀⠀⠀⠀⠀⢀⡀⠀⠠⠄⠀⠀⠰⠂
⠀⠀⠀⠀⠘⠃⠀⠀⠘⠃⠀⠀⠀⡀⠀⠀⠀⢤⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠁⠀⠈⠁⠀⠀⠀⣀⠀⠀⠠⡦⠀⠀⠀⠠⠄
⠀⠀⠶⠆⠀⠀⢠⡄⠀⠀⠦⠀⠀⠁⢀⡀⠀⠀⠀⠐⠂⠀⠀⠀⣦⠀⠈⠀⠀⠛⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠰⠆⠀⢀⡀⠀⠀⠀⠀⠠⠄⠀⢠⡄⠀⠀⠉⠀⠀⠀⡀⠈⠁⢀⠀⠀⠀⠐
⠀⠀⠘⠃⠀⠐⠂⠀⠈⠀⠀⠠⠀⠀⠀⠶⠀⠀⠀⠀⠈⠁⠀⠶⠆⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠈⠀⠸⠇
⠀⠀⠀⢠⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⢀⡀⠀⠘⠃⠀⠀⠆⠀⠀⠀⠀⠀⠉⠀⠀⠐⠛⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⣄⠀⠰
⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠤⠀⠀⣀⠀⠀⢠⠀⠀⠛⠀⠰⠂⠀⠀⠀⠀⠐⠆⠀⠀⠠⠶
⠀⠀⠀⠛⠁⠀⡈⠁⠀⢀⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⡄⠀⠀⡄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉
⠀⠰⠀⠀⠀⠀⠉⡀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠗⠀⠀⡄⠀⢠⡄⠀⠈⠁⠀⢠⡀⠀⠀⠀⠀⠀⠃⠀⠐⠂⢀⡀⠀⠀⠆
⠀⠀⠀⠐⠂⠀⠈⠛⠀⠀⠀⠠⠀⠀⢀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢶⠄⠀⢀⣀⠀⠀⠀⠀⠀⠰
⢀⡀⠀⡄⠀⠀⡤⠀⠀⠠⡄⠀⠀⠀⠘⠃⠀⠀⢠⡄⠀⠀⠐⠀⠀⠈⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡀
⠀⠁⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠤⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⢀⡀⠀⠘⠋⠀⠀⠀⠀⠒⠀⠀⢠⡆⠀⠀⠉⠁⠀⠆
⠀⠀⠀⠀⠶⠀⠀⠀⣀⠈⠁⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠀⠈⠁⠀⠀⠀⢀⡀⠐⠆⠀⠀⠀⠀⠀⠐⠂
⠰⠂⠀⠀⠀⠲⠀⠀⠀⠀⠐⠂⠀⠀⠀⠠⠀⠀⠀⠀⠰⠂⠀⠈⠀⠀⠙⠃⠀⠀⠀⠀⠠⡄⠀⠀⠈⠃⠀⠀⠀⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⢀⢠⢤⢀⠀⠀⠀⣤⣦⠖⠒⣤⣶⣶⣷⣶⡖⣶⣦⣐⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣶⣤⣾⣼⡏⣽⣿⣧⣼⣿⣿⣷⣶⣾⣿⣿⣶⢯⣭⠉⠻⠷⠶⣤⣄⠀⢀⠀⣽⣵⡄⡀⢀⡄⣀⡀⣀
⠀⠀⠀⢀⡤⣞⡟⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣭⣻⡯⠻⢿⣎⣵⣴⣏⣷⣶⣤⢌⡁⠨⡙⣿⣿⣧⣡⣞⡉⢿⣰⡇⠀⢂⣀
⠀⠀⢰⣿⣿⣹⣿⣢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣽⣯⣿⣷⣤⡶⡷⢂⣿⣾⣿⣿⣏⣿⣿⢶⣾⣿⣴⣶⣦⡯⠄
⠀⠀⢈⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡿⠿⠿⢽⠧⣿⣿⡶⡾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢫⡀⢀
⠀⠀⠚⢃⢽⣿⣿⣻⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣖⣠⣬⣷⢬⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡷⢹⣿⢯⣿⣿⡇
⠀⠀⢸⣿⣾⣿⣿⡿⢿⣿⣿⡏⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣠⣾⢧⣻⣿⣿⣿⣿⣾⣿⣿⣿⢷⣟⣻⣿⣼⣿⣿⣛
⠀⠀⢨⣿⣿⣿⣿⣿⣾⣿⡟⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣵⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⣿⣿⣯⠟⣿⣿⡿⣿⠟⣿⡿⠟⣿⣿⠟⣿⣿⢿⠟⠋⠋
⠀⠀⠰⠿⡿⢿⣿⣿⣿⣿⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣽⣷⣿⣿⣠⣿⣿⣇⣻⢿⣖⡀
⠀⠀⠀⢒⣶⣾⣿⣿⣿⣿⡣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢿⣿⣧⠉⠃
⠀⠀⠀⠈⠛⠀⣽⣿⣿⣧⣆⣻⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠿⡇⢿⠊⢿⣷⠆
⠀⠀⠀⣲⣶⣾⣿⣿⣿⣷⢹⣷⣿⣶⣼⣛⡶⠚⢿⠴⠳⣬⣿⣿⣦⣀⣻⣽⣿⣶⣶⠜⠚⠯
⠀⠀⠀⠀⠉⠛⠉⠟⠽⠣⡞⠿⠯⡹⠿⠻⢷⠀⠀⠀⠀⠀⠀⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠
⠀⠀⠀⠀⠀⠀⢦⠀⠀⠸⡇⠈⢠⠂⠄⢀⠀⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⢁⣘⡟⡈⠂⠀⠀⢀⡄⣲⣻⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣢⠄⢀⣠⠞⠀⠀⢠⣄⠁⢀⡇⠀⠀⠀⡀⠀⠠⣠⣤⠀⠀⠀⠀⠀⠎⠄⠀⠀⠀⠑⡄⠀⠀⢠⠡⠋⣃⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠈⠀⠀⠀⡊⢠⣁⡀⠤⡺⠁⠀⠀⠀⠟⠒⠒⠒⠚⠆⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠨⠔⠐⠣⢀⠀⠀⠀⠀⠀⠈⠮⠉⠈⠁⠀⠀⠀⠀⡌⢃⣀⣀⣀⣀⣘⡀⠀⠀⠀⠢⡀⠀⠀⠀⡠⠊⠀⢀⠠⢀⡏
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡇⠄⠀⢒⠂⠀⠀⡇⠀⠀⠀⠀⠀⠉⠈⠁⠀⢀⠎⠀⠀⠀⠀⠁⢆
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠃⠀⠀⠐⠀⠀⠐⠁⠀⠀⠀⢀⣄⠀⠀⠀⠀⢈⠀⠀⠚⠲⠀⠀⠸
⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⠀⠀⠌⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢲⠟⠚⢷⢲⠆⠀⠈⠢⢄⣀⡀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠂⠄⣀⣀⣀⠄⠊⠀⠀⡄⠜⠇⠆⠀⠀⠀⢀⣀⠀⣀⡀⡀⣀⢀⠀⠀⠀⠀⣠⢧⡀⠀⣠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⡎⣷⢀⡇⢹⣃⡟⠀⠀⠀⠀⠉⠉⠻⡹⠋⠉⠁⠀⠀⠀⠀⠀⢀⣴⠗
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠲⣭⢲⠀⠀⠀⠀⠘⠇⠀⠀⠀⡄⠀⣄⡀⠨⣅⡀⢡⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡨
⠀⠀⠀⠀⠀⢠⠊⠐⠀⠀⡀⠢⠢⡀⠀⠀⠀⠘⠄⠀⠀⡇⢸⠆⢿⢠⣤⣧⢸⠸⣄⡿⠀⢀⣸⠲⢐⠀⠀⠸⡷⠀⠀⠀⡐⠁⡜
⠀⠀⠀⠀⠀⡇⠀⢀⡡⠈⠀⡃⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠛⠀⠀⠀⡄⠃⠀⡔⢀⠎⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢇⢀⠈⠀⠈⠅⠀⠂⠌⠀⠀⠀⠀⠀⠀⠀⣄⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠠⠊⠀⢀⣴⣧⡿
⠀⠀⠀⠀⠀⠈⠢⠄⣀⣀⣀⠤⠊⠀⢀⠁⠁⠀⠀⠀⠰⠎⡔⠊⠳⣄⠀⠀⠀⠀⠀⡠⢪⠃⠀⠀⠀⠐⠢⠀⢀⠀⠈⠀⠀⣀⣦⣟⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡐⢪⣀⠮⠦⢒⣀⢞⠕⠁⠀⠀⢀⡀⠀⡓⠀⠈⠀⠀⠀⠀⢑⠯⠶
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⠭⠭⠍⠂⠁⠀⡀⠀⠀⠀⠊⠀⠀⠐⠊⠀⠀⢀⠀⠘
⠀⠀⠀⠀⠀⠤⢠⠂⠤⠤⠤⠾⠓⠒⠂⠀⠘⡓⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠳⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⢱⢠⠂⣹⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⢀⠀⠀⢀⠀⡤⠞⠈⠀⠀⠠⠀⢄⠰⡂⠤⠀⠀⠀⠄⣹⠿⢴⣶⣼⣀⠀⠀⠀⠀⠀⠀⢡⢢⠗⣚⡁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠐⠃⠄⠑⠦⠄⠀⠀⠀⠀⠀⠀⠉⠋⠥⠤⠀⠀⠀⡠⠀⠼⡦⠊
⠀⠀⠀⠀⠀⠀⡠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠜⠀⠀⠀⢀⠄⠀⠠⡤⠄⠄⠀⠀⠀⠀⠀⠀⠎⠐⠈⠀⡈
⠀⠀⠀⠀⠀⠴⠥⠤⠦⠤⠤⠤⠦⣤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠖⠀⠉⠐⢆⠁⠀⢄⠀⠀⠀⢆⠀⠈⠀⡰⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠙⠰⡒⠚⠛⠓⠒⠛⠐⠒⠀⠀⠤⠤⠅⠀⠀⠀⠀⠀⠢⣀⣀⣸⣄⣀⡀⣀⠼⠄⠠⠜⠀⠀⠀⠀⠖⠉⠈⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠁⠀⠠⠡⠀⠁⠀⠀⠀⠀⠀⠠⠘⠀⠀⠐⠡⠀⠀⠀⠀⠀⠐⠄⠀⠃⠠⠄
⠀⠀⠔⠀⠀⡀⠀⠒⠀⠀⠀⠀⠐⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⢀⠄⠀⡀⠢⠄⠀⠀⠀⠀⠀⠐⠀⡀⠈
⠀⠀⢀⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈
⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀
⠀⠀⠀⣰⠛⠙⢳⣀⠀⠀⠀⠀⢀⡀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠓⠀⠀⠊
⠀⠀⠀⡰⡄⠀⠐⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣈⠁⠀⠈⣀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠃⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡄⠰⠄⠀⠀⠀⠀⠀⠠⠂⠐⡀⠀⠲⠄⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⢀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢀⠄⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠠⠌⠡⠂⠲⠁⠤⠀⠀⠀⠠⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⣈⠂⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠲⠀⠀⠂
⠀⠀⠠⠂⠁⠄⢔⠐⠢⠀⠀⠀⠀⠀⠀⡄⠀⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠉⠀⡠⠀⠠⡀⠈⠁
⠀⠀⠀⠐⠁⠸⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠐⠀⠀⠀⠀⠊⠠⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⢰
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠂⠀⠖⠀⡄⢠⢎⡈⣠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠙
⠀⠀⠀⠀⠀⠀⠘⠒⠀⢨⢠⡆⢼⢨⡧⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢆
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠎⠀⠂⢰⡼⠇⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⢠⠈
⠀⠀⠀⠀⠀⠀⠙⢆⡰⠣⠆⠆⡤⠀⢀⡉⡛⠈⠀⠀⢀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠇⠈
⠀⠀⠀⠀⠀⠀⢨⢸⠀⠀⠶⠀⡏⢸⠇⠀⡇⠀⠀⠀⢠⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢠⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠘⠓⠃⠂⠛⠈⠈⠀⠁⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠍⠈
⠀⠀⠀⠀⠀⠀⢰⠴⢸⠑⠂⣋⡗⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢶⡴⢃
⠀⠀⠀⠀⠀⠀⢸⡆⢸⠃⠀⡏⠰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄
⠀⠀⠀⠀⠀⠀⡸⠆⠈⠠⠷⠆⠈
⠀⠀⠀⠀⠀⠀⢠⢻⠠⡄⠉⣻⠈⠁⠄
⠀⠀⠀⠀⠀⠐⠞⢹⢁⡝⣇⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠧⡬⠤⣅⢀⡿⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠊⠈⢃⠄⠀⠀⠀⠀⠄⠀⠀⡀⢀⠀⠄⣠⡀⠀⢀⡄
⠀⠀⠀⠀⠀⢀⢰⣤⡇⣿⣀⠂⠀⠀⠀⠀⠀⢀⢄⢠⡆⡦⢏⠛⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠧⠘⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣸⠀⠇⠧⢸⡄⠀⠀⠀⠀⠀⠈⠲⢌⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡡⢴⢰⠶⠆⠀⠀⠀⠀⠀⠀⠁⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢳⣄⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠰⣌⣹⡿⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⣴⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠀⠀⠀⠀⢤⡀⣀⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠈⢸⢛⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣼⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⠀⣏⠻⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢘⡃⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁
⠀⠀⠀⠀⠀⢈⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠸⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠈⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠌⢰⣤⠁⡀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⢰⣦⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⢸⡄
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣤⣲⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣴⡾⣇⣨⣏⠙⡀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠎⡏⣻⢹⡉⢰⡄⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠸⠏⢘⣟⣰⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣤⢾⣧⣼⣿⡇⣷⣿⠈⢧⡾⠉⠀⠀⠀⠀⠀⢰⡦⣼⠉⣿⡏⣿⠀⠀⠈⠓⢦⣄⠀⠀⣀⡴⠞⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢸⡇⢸⠟⢧⠟⣿⣄⣸⡇⠈⣰⠀⠀⠀⠀⣘⡇⣿⣙⠸⡧⡟⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣟⢀⣼⡧⢸⣦⢠⡇⣶⠾⢁⠀⠀⠀⠀⢹⡷⣻⢰⠚⡱⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠞⠀⣿⣹⠋⣿⢸⡏⢿⡇⢸⡀⠈⡇⠀⠀⠀⡸⢸⡉⢼⠗⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠾⠛⠻⠶⣽⣼⣧⡜⢧⣸⠦⠞⠃⠀⠀⠀⢹⣖⣿⣾⢿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣎⠻⣵⡟⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⢸
⠀⠀⠀⠀⠐⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠚⠋⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠓⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⠃⠀⠀⠀⡀⠀⢀⠀⠀⠐⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣢⣜⢀⢠⠄⠀⠀⠘⠁⢀⠀⠀⢀⠀⡐⠀⢃⠀⠀⠀⠀⠰⠃⠀⠀⠠⢄⡀⠈⡥⢘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠲⠀⠈⠀⠘⡀⠀⠀⠀⠸⠄⠁⠀⠀⠀⠀⠈⠀⡁⠀⢀⠀⡄⣀⡀⠀⠐⠄⠁⠒⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢼⡀⣀⠀⢈⡿⠀⠀⠀⠘⠀⢀⣀⣀⠀⠀⠀⠘⠃⠂⠘⠀⢀⡈⢀⠀⠀⠀⠒⠼⠠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠉⡴⠃⠸⢗⠉⣿⠆⠀⠀⠀⠃⠀⡉⠁⡅⠀⡄⡀⠀⠀⢠⡔⠌⣉⠀⢄⡁⠈⠁⠂⠂
⠀⠀⠀⠀⡀⠠⡆⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠆⠈⠂⠀⡐⡄⠠⢀⢇⠀⠀⡰⠀⠁⠀⠀⣻⠇⠐⡃⣀⠀⠁⠁⠀⠀⠀⠀⠀⣠⢄⠄
⠀⠀⠀⠀⠨⡁⢀⠠⠀⠀⠀⠠⠀⡄⠀⡆⠀⠀⠀⠀⠀⡀⠀⠀⡤⠂⠈⠀⠘⠁⡠⠀⣄⠀⠀⠃⠀⠁⠉⠑⢢⡎⠁⠀⠀⠀⢀⠰⠠⠁
⠀⠀⠀⠀⠀⠐⡈⡆⠰⡀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠠⡡⠈⠒⠰⠀⠰⠀⠀⠀⠐⢸⠀⠈⠀⠀⠀⠠⠐⢤⠄⠈⠃⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠳⠎⠃⠂⠀⠀⠰⠭⠐⠐⠂⠀⠀⢶⠈⠂⠀⠀⠀⠀⠀⡞⠀⠀⠀⡄⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄
⠀⠀⠀⠀⠀⠀⠈⣷⣀⠀⠜⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠂⠀⠐⢠⠀⡖⠀⠂⠀⠀⠀⠀⠀⢰⠦⡐⢦⡤⠥⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠁⠱⠊⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠁⠀⠀⠀⠀⠀⠀⠀⠊⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠃⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⡤⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠨⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠛⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢤⣠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠀⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠄⠀⠰⡆⢠⠀⢀⠄⠀⢂⡠⠀⢠⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠃
⠀⠀⠀⠀⠀⠰⠤⠄⠀⡇⢘⣄⣸⠀⣏⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠂⢀⠗⢉⠁⢸⠀⣏⠉⠀⠁⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠃⠘⠁⡀⠠⢁
⠀⠀⠀⠀⠠⢤⡉⠀⣡⠀⠈⠁⠘⠂⠉⣠⣀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠇⠀⠁⠐
⠀⠀⠀⠀⠀⢸⠱⡆⠋⠨⠉⠉⡏⠑⠄⠂⢠⠇⢀⠀⠀⢀⠄⠂⣼⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠂
⠀⠀⠀⠀⠀⢸⠀⠇⠈⠘⠆⠀⡇⢀⡆⡀⢠⠀⠈⠀⠀⠘⠄⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢡⢄
⠀⠀⠀⠀⠀⢸⡀⠀⠛⠞⠃⠀⠛⠈⠀⡁⢈⠀⠁⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠰⠧⡄⠁⠂
⠀⠀⠀⠀⠀⢤⠤⡄⡖⢲⠈⡏⡣⠐⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⣶⢰⠈⡂
⠀⠀⠀⠀⠀⢸⣀⠀⣷⢌⠀⡗⢣⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠣⠘⠀⠀⠁
⠀⠀⠀⠀⠀⢸⡉⠀⠃⢈⣔⠃⢆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠚⢁⡀⠀⠈⠩⢈⠀⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⢊⡇⢻⣀⠆⣸⡀
⠀⠀⠀⠀⠐⢻⠉⡇⣹⠸⡇⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠨⣠⣅⢉⡀⠀⡾⠏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢂⠀⠁⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡆⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⡐⠛⠃⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠉⠁⠀⠀⠉
⠀⠀⠀⠀⠈⠃⡇⠉⢸⡇⢧⠀⡈⠀⠀⠀⠀⠀⠠⠌⡐⠋⡸⠁⠈
⠀⠀⠀⠀⠀⠖⠀⡉⢀⡁⠈⠃⠀⠀⠀⠀⠀⠀⡀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣆⠙⠃⣿⠈⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⡳⡄⢿⠂⢀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢤⣀⠥⠷⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠠⠄⠀⡀⠀⡀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠒⡗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠰⠈⠁⠁
⠀⠀⠀⠀⠀⢀⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠘⡒⠀⠘⠃⠀⠉⠉⠀⠈⠀⠐⡛
⠀⠀⠀⠀⠐⠟⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠆⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⠐⢍⣠⠀⠀⠀⡇
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⢰⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡴⢿⢱⢠⢸⡁⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠘⠈⠁⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⠸⠩⠸⠅⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣇⠀⣯⢸⠀⡇⢸⠀⠤⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠆⡐⢺⠁⡆⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠉⡏⠈⣿⢸⠀⡇⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡁⠄⠏⢹⠀⠃⠀⠀⠀⠀⠀⠉⠓⢤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣫⡀⠉⣈⡀⣁⠘⠓⠒⠻⠄⠤⠎⠀⠀⠀⠀⠀⠀⢠⡄⠇⢀⡈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢀⡀⢿⡀⢰⢏⡇⢸⠳⡄⣿⠈⣟⠁⠢⡀⠀⠀⠀⠀⠰⢊⡥⠉⡃⢠⡉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠊⠀⢸⠁⠟⠈⣇⢸⡇⠘⣿⠀⣿⠀⠀⡿⠀⠀⠀⠀⠀⠹⣄⠘⠃⢨⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠋⠐⠒⠠⠬⣈⣃⡄⠈⢄⠛⠂⠚⠁⠀⠀⠀⠀⠠⠐⣨⠖⠋⠠⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠆⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠉⠀⠐⠃⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⢿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⢠⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠙⢿⡄⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⡘⠁⠀⠈⠉⠉⠙⢻⠇
⠀⠀⠀⠀⠀⠀⢾⣩⠀⠀⠀⠀⠀⠀⣔⢤⡀⠀⠀⠀⠀⠀⠈⠛⠏⠀⠉⠉⠙⠻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠁⠀⠀⣀⣠⡀⠀⠈⣆⡄
⠀⠀⠀⠀⠀⣖⣩⠈⠠⠀⡀⠀⢀⡤⢨⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠣⠀⢀⠁⠙⠻⣿⣿⣿⣧⠀⠀⠀⠀⠓⠀⠀⡀⠀⠀⠡⠑⡠⣄⡀
⠀⠀⠀⠀⠀⠀⡙⠆⠐⠀⠀⠀⠠⢆⠠⣎⡁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡀⠀⠘⠇⡀⡆⡸⠿⠿⠋⠀⠀⠀⠀⣰⡆⠐⢉⠀⣦⠀⢀⢻⣿⣿⣷⣄
⠀⠀⠀⠀⣀⣤⠁⠀⠁⠀⠀⠀⠀⠀⠀⠘⢁⣀⠀⠀⠀⠀⠀⢨⡀⠁⢤⡘⢆⡇⠀⠐⢹⡷⠄⠀⠀⠀⠀⠀⠀⢈⠦⠀⠁⠈⡸⠀⠈⢯⣿⣿⣿⣿⡇
⠀⠀⠀⣾⣿⣿⡆⡁⠀⠀⠐⠀⠢⠀⠑⢔⡆⣿⣷⡀⠀⠀⢠⠒⢁⠀⢀⠊⡃⠁⠀⡃⢸⠇⠀⠀⠀⠀⠀⠰⡦⠀⢀⠀⠐⠊⠂⢰⠘⣺⣿⣿⣿⣿⠃
⠀⠀⠀⠹⣿⣿⡧⠁⡄⠀⡀⠆⠌⠀⠐⣉⣽⣿⡿⠀⠀⠀⠘⣤⠸⢠⡜⡀⣵⠀⣀⡟⢸⡄⠀⠀⠀⠀⠀⠀⠑⠃⠘⡆⠀⠀⠂⢚⣻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠘⣿⣿⣯⣤⠀⠟⢘⡀⢘⢻⣿⣿⡿⠁⠀⠀⠀⠀⢛⢿⠻⡷⢇⡹⢅⠀⠑⠘⠀⠀⠀⠀⠀⠀⠀⠀⢌⠌⣕⡠⡣⣰⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⢻⣡⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⢸⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡸⠿⢻⣼⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠓⣄⠀⠀⠀⠀⠀⠀⠀⣼⠃⠰⠆⠘⣧⠀⠀⠀⠀⠀⠀⠀⡠⠚⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⢳⡀⠀⡄⠀⠀⣼⠃⠈⢡⡌⠁⠘⣧⠀⠀⣠⠀⢀⠞⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠹⡄⠹⡄⣼⣧⠊⣠⣈⣡⣄⠑⣼⣧⢰⠃⢰⠏⠀⠀⡀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⡀⠀⠀⢃⢀⣿⠃⠈⠈⠂⡸⢇⠐⠁⠁⠘⣿⠀⠌⠀⠀⢀⣇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠩⡑⢤⡀⠘⣼⢃⠤⠖⠛⠉⢉⠉⠉⠓⠒⠤⡘⣧⠃⢀⡤⢋⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣨⣷⡦⠴⠶⡾⠟⠋⠀⢤⡄⠁⠰⠉⠉⠆⠈⢠⡤⠀⠜⠿⡷⠶⠦⢴⣾⣅⣀⣀⣀
⠀⢶⡖⠚⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠈⠑⣄⠀⠀⣠⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠓⢲⡶
⠀⠀⠙⢦⡙⠓⠦⠤⡜⠉⠀⡀⠠⠄⠀⢁⡦⠴⢤⡀⠀⠀⡨⠂⠒⢅⠀⠀⢀⡤⠤⢴⡁⠀⠠⢀⣀⠀⠉⢣⠤⠴⠚⢋⡴⠋
⠀⠀⠀⠀⠙⢦⡀⢀⠴⠄⠊⠀⠀⠀⡰⠋⠀⠀⠀⠘⡆⠀⠄⠲⠆⡠⠀⢰⠃⠀⠀⠀⠙⣆⠀⠀⠀⠑⠢⠦⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⡠⠐⠀⣇⠀⠠⣀⢀⡠⠃⠀⠈⡓⢚⠁⠀⠘⢄⡀⣀⠄⠀⣸⠀⠀⢄⠀⠀⢀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣷⣔⠀⣎⠑⠘⢦⡀⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄⠀⠀⠀⠀⢀⡴⠃⠊⣱⠀⣢⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣙⣷⣄⠀⢀⣀⣉⡷⠶⠶⠏⠁⢠⠀⢀⡄⠈⠹⠶⠶⣾⣉⣀⡀⠀⣠⣾⣋⣠⣜⡀
⠀⠀⠀⠀⠀⠀⡴⠛⠋⡁⢀⠤⡿⠀⢀⠤⡈⢂⠢⢄⣈⠁⠀⠉⠉⠀⠈⣁⡤⠀⡐⢁⠤⡀⠀⢿⡠⡀⠈⠙⠛⢦
⠀⠀⠀⢀⣀⣜⣀⣠⠤⠒⣁⢾⡇⠀⠀⠀⠊⠸⣶⠔⠚⠛⠿⡛⢛⡿⠛⠓⠲⣶⠇⠙⠀⠀⠀⢸⡷⣌⠓⠤⣄⣀⣣⣀⡀
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠈⠉⣼⠃⠒⠲⡈⠉⠱⡏⢰⠉⢳⠀⢹⡎⠀⡞⠉⡂⢸⡎⠉⢁⠖⠒⠘⣷⠈⠁⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⡀⠑⠂⠲⣝⠲⠶⠊⠀⠘⠃⠀⠙⠶⠖⣫⠆⠒⠊⠀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡈⠦⢴⠣⠄⠀⠘⠆⢀⣠⠶⠛⠛⠷⣄⡀⠰⠃⠀⠠⠜⡮⠴⢁⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢁⠒⣈⣑⣁⣤⠴⠚⠛⠐⠂⠒⠐⠂⠒⠙⠓⠦⣤⣈⣊⣁⠒⡈⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢋⣠⡼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧⣄⡙⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢾⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡆⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⣆⢀
⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⢰⣦⣺⣞⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣟⣿⣢⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⡁⠈⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣻⣿⣷⣄⡀⠀⣀⣴⣿⣿⣻⣿⣿⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⣿⣿⣿⠋⠀⠈⠊⠀⠈⢻⣿⣿⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠟⣿⡿⣿⣦⡀⠀⠀⠀⣠⣾⡿⣿⡿⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣦⣠⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣻⣿⣿⠟⠛⠛⠛⠛⣿⣿⣿⣻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⣿⣷⡄⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⢻⣿⣿⣟⣿⣿⣿⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠁