Cautious Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⣠⣿⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⡟⣤⡄⠀⣤⢻⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⣰⠛⢿⣿⢻⡄⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣰⠏⠀⣼⣿⡟⠃⠀⠹⣆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⣴⣿⠃⠀⢥⣼⣿⣄⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⠏⠀⢸⡇⠹⣿⠖⢷⡀⠹⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⣼⣿⡇⣸⣆⠈⣭⡙⣿⣆⠹⣿⣿⣿⣿
⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠹⣇⠈⠁⠀⠘⣧⡀⠀⠀⢸⣿⣿⡿⢃⣬⣭⣽⣿⣿⣿⣧⢻⣿⣾⣿⣧⡹⣿⣿⣿
⠀⠀⠸⣧⣤⣤⣤⣭⣭⣭⣥⣭⣤⣤⣤⣤⣼⠇⠀⠀⢸⣿⣿⣇⠘⠟⠛⠛⠒⠒⠒⠛⠚⠛⠛⠛⠛⢃⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠔⡉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡑⡄
⠀⠀⠀⢀⠎⡴⠿⣆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⣾⣷⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢰⣿⣆⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⢷⡌⢄
⠀⠀⢠⠊⣼⣇⠀⣿⣆⠣⡀⠀⠀⠀⠀⠀⠀⠀⡰⢡⣿⠟⡰⢷⡌⢄⠀⠀⠀⠀⠀⠀⠀⢀⠌⣴⡶⣤⢤⣤⠱⡀⠀⠀⠀⠀⠀⠀⠀⡠⢃⣾⣿⠸⢿⡌⢦
⠀⢀⢂⣾⣿⣿⣀⣿⣿⣧⠱⡀⠀⠀⠀⠀⠀⡔⣡⣯⣩⡄⣿⣿⣿⡌⢦⠀⠀⠀⠀⠀⢠⠎⣼⣿⣿⠉⣧⠘⣧⡐⡀⠀⠀⠀⠀⠀⡴⢡⣞⣼⡏⠉⠁⠀⢀⢣
⡠⢁⣾⣿⣿⣟⣉⣿⣿⣿⣧⡘⡀⠀⠀⢀⡜⣰⣟⣛⣛⣓⣛⣀⣨⣿⣆⢣⠀⠀⠀⢠⢃⣾⣿⡿⠋⠉⠁⣾⣿⣷⡘⢄⠀⠀⠀⡐⣠⣿⣷⣹⣇⣠⣤⣤⣾⣆⠡
⠣⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠜⠀⠀⠐⠤⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠃⠀⠀⠣⠬⠭⠭⠤⠤⠤⠤⠭⠭⠭⠥⠔⠀⠀⠘⠤⠭⠭⠭⠭⠭⠬⠭⠭⠭⠭⠤


⠀⠀⠀⠀⠀⡔⡉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡑⡄
⠀⠀⠀⢀⠜⣰⠿⣌⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⡾⢷⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣰⣿⣆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣼⣷⡘⢄
⠀⠀⢀⠎⣼⠛⢙⢿⣆⠣⡀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣞⠁⠻⣷⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠌⣰⣦⠁⣺⣦⠡⡀⠀⠀⠀⠀⠀⠀⠀⡠⢃⡚⠿⣿⣿⡌⢆
⠀⢠⢋⣼⣿⠇⡌⣿⣿⣦⠱⡀⠀⠀⠀⠀⠀⡐⢠⣿⡿⢀⢹⣿⣿⡄⢂⠀⠀⠀⠀⠀⢀⠆⣼⣿⣿⣆⢠⣹⣦⠑⡀⠀⠀⠀⠀⠀⡰⢡⣾⣥⡈⠉⣭⣿⣌⢢
⠠⢃⣾⣛⣛⣚⣓⣩⣿⣿⣧⠱⡀⠀⠀⠀⡜⣰⣟⣛⣓⣛⣚⠻⢿⣿⣄⢢⠀⠀⠀⢠⢂⣼⣿⣿⣿⣷⣛⣛⣛⣓⡘⡄⠀⠀⠀⡐⢡⣿⣿⣿⣝⣷⣼⣿⣿⣆⠣
⠣⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠜⠀⠀⠘⠤⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠃⠀⠀⠣⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠜⠀⠀⠘⠤⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤


⠀⠀⠀⠀⠀⡔⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠑⡄
⠀⠀⠀⠀⡔⣰⣿⡌⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣾⣧⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⣿⣆⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡼⢷⡘⡄
⠀⠀⢀⠎⣴⣿⣿⣿⣆⢣⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⡀⠀⠁⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠜⣰⢿⣄⣽⠂⠳⡀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⢁⡾⢷⡌⢆
⠀⢀⠊⠈⠁⠀⢠⣬⣿⣆⠣⡀⠀⠀⠀⠀⠀⡰⠁⠀⣠⢿⣿⣆⠀⠈⢄⠀⠀⠀⠀⠀⢀⠌⣴⠿⢾⡿⠁⠀⠀⠱⡀⠀⠀⠀⠀⠀⡠⢃⣾⣇⣤⠄⣿⣿⡌⢢
⣠⠂⠀⠀⠀⠀⠘⠿⠿⠿⠧⠱⡀⠀⠀⠀⡔⠀⣠⡋⢹⠧⣼⢿⣗⢀⡈⢦⠀⠀⠀⢠⢎⣼⣿⣳⡿⠁⠀⠀⠀⠀⠑⡄⠀⠀⠀⡰⢡⣿⣿⣿⣏⢈⣿⣿⣿⣆⢢
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠸⠄⠈⠉⠉⠉⠈⠁⠈⠉⠉⠉⠠⠃⠀⠀⠣⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠐⠄⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠂⢙⡆⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⠙
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠠⠠⣤⣈⠳⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠖⠒⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣀⡤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣀⠀⠄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⡘⠋⠁⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣴⠲⢦⣄
⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠈⠁⠉⢉⡉⠛⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠶⠶⠾⠃⠀⠀⠀⠀⠀⣿⠠⣦⠈⠛
⠀⠉⣳⡄⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣨⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠇
⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢀⣿⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⠋⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡆⠈⠙⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠻⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣋⣤⡴⠞⠋⠁
⠀⠀⣀⣤⡶⠞⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⠶⠛⠉
⠶⠛⠋⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠
⠀⠶⠶⠤⠤⢠⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠉⠉⠉⢉⣭⠉⡽⠀⠀⠀⠀⠀⠸⣇
⠷⣦⡀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⣤⠤⠴⠶⠒⠚⠛⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡠⠆⠉⢀⡾⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠙⠷⣤⡀⠀⣿⠀⠀⠀⠀⠀⠘⣦⠀⠀⠘⡁⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢀⡞
⠀⠀⠀⠀⠈⠙⠟⠋⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣧⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿⠉⠉⠛⡛⠛⠛⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢦⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡆⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠻
⣻⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣆⣸⣇⣠⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣁⣆⣄⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⣾⡳⡟⣿⣿⣷⡖⣟⣿⣿⣷⣷⣟⣿⣿⣦⡳⣟⣿⣿⣿⣿⣽⣿⣿⣦⣶⡿⣿⣿⣶⣒⠿⣿⣿⣷⣠⢿⣿⣿⣟⣺⡻⣿⣿⣶⣿⣿⣿⣿⣮⣶⣿⣿⣿⣶⣎⡻⣿
⣿⠟⠉⣿⣿⣿⢿⡝⣮⣿⣿⣿⡿⣠⣿⣿⣿⣾⣪⣿⣿⣿⣽⣨⣿⣿⢿⡝⣪⣿⣿⢯⣶⣺⣿⣿⢯⣾⣽⣿⣿⣟⡋⣼⣿⡿⠇⠙⣺⣿⣿⣿⣿⣼⣿⣿⣯⣽⣿
⠃⠀⠸⠿⠿⠛⠙⠻⠿⠟⠛⠋⠻⠿⠟⠋⠋⠺⠿⠟⠛⠛⠻⠿⠟⠃⠛⠺⠿⠟⠋⠚⠿⠿⠟⠛⠛⠻⠿⠟⠓⠋⠻⠿⠟⠃⠁⠺⠿⠟⠛⠛⠿⠿⠟⠛⠛⠿⠿
⣤⣤⡀⠀⠀⠀⠀⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⣤⣤
⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠉⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠈⠉⠀⠈⠀⠀⠀⠀⠀⠈⠀⠉⠉⠈⠀⠀⠀⠀⠀⠈⠉⠈⠀⠉⠀⠀⠀⠀⠀⠈⠀⠁⠀⠉⠀⠀⠀⠀⠀⠈
⡓⢒⣶⣿⣿⣿⢖⣷⣿⣿⣟⢶⣶⣿⣿⣿⢖⣶⣿⣿⠟⢖⣶⣿⡿⠓⢒⣶⣿⣿⠳⢖⣶⣿⣿⠷⢒⣶⣿⣿⣳⣆⣴⣿⡿⢳⣆⣶⣿⡿⡳⣖⣾⣿⣿⣗⣖⣾⣿
⣽⣟⣛⣭⣃⣖⣜⡛⢋⣭⡻⠋⠉⠉⢉⣇⣾⣿⣟⣵⣛⣾⣿⣿⣣⣺⣝⣻⣯⣣⣷⣞⣛⣿⠁⠀⠈⣛⣻⣿⣿⣿⣯⣯⣱⣿⢟⣩⣿⣿⣿⣽⣿⣯⣿⣮⡛⠉⠉
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⠿⣏⣿⣿⣛⣻⣋⣱⣾⣿⣿⣿⣿⣾⣿⣿⣿⣯⣿⣶⣦
⣻⣟⣟⣻⢿⡿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣛⣿⣿⢯⣽⣻⣻⣿⣿⣿⣛⡻⣿⣿⣟⣟⣿⣿⣿⣿⣋⡸⣿⣿⣛⣟⣻⣿⣿⣞⣿⡿⣟⣿⣿⣿⣟⣿⡿⣟⣻⣿
⣣⢛⣿⣿⣷⣟⢟⣿⣿⣿⣣⣟⢿⣿⣿⣿⣽⣿⣿⣧⣶⣙⣿⣿⣷⡱⣟⣿⣿⣿⣿⣿⢿⣿⣷⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⡻⣹⣿⣿⣾⡻⣟⣿⣿⣮⣻⡙⣿⣿
⠉⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠀⠁⠉⠉⠉⠀⠀⠉⠉⠉⠈⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠈⠉⠉⠁⠉⠈⠉⠉⠉⠈⠈⠉⠉⠁⠈⠈⠉⠉⠁⠁⠈⠉⠉⠁⠈⠈⠉
⣀⣀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⡀⠀⠀⣀⣀⡀⠀⢀⣀⣀⡀⠀⠀⣀⣀⠀⠀⢀⣀⣀⠀⠀⢀⣀⣀⠀⠀⢀⣀⣀⠀⠀⢀⣀⣀⠀⠀⢀⣀⣀⠀⠀⢀⣀⣀
⡿⠋⠀⣠⣾⣿⠏⠀⣠⣾⣿⠏⠀⣠⣾⣿⠋⠀⣠⣾⣿⠋⠀⣠⣾⡿⠋⠀⣠⣿⡿⠋⠀⣰⣿⡿⠋⠀⣰⣿⡿⠋⠀⣰⣿⡿⠋⠀⣴⣿⡿⠋⢀⣴⣿⡿⠋⠀⣰
⠀⢀⣼⣿⠟⠁⢀⣼⣿⠟⠁⢀⣾⣿⠟⠁⢀⣾⣿⠟⠁⢀⣾⣿⠟⠁⢠⣾⣿⠟⠁⢠⣾⣿⠟⠀⢠⣾⣿⠟⠀⣠⣾⣿⠏⠀⣠⣾⣿⠏⠀⣠⣾⣿⠏⠀⣠⣾⣿
⣴⣿⡿⠋⠀⣰⣿⡿⠋⠀⣴⣿⡿⠋⢀⣴⣿⡿⠃⢀⣴⣿⡿⠁⢀⣴⣿⡿⠁⢀⣴⣿⡿⠁⢀⣴⣿⡟⠁⢀⣼⣿⡟⠁⢀⣴⣿⠟⠁⢀⣼⣿⠟⠁⢀⣴⣿⡟⠁
⠛⠋⠀⠀⠚⠛⠛⠀⠀⠚⠛⠛⠀⠀⠚⠛⠋⠀⠀⠚⠛⠋⠀⠀⠚⠛⠋⠀⠀⠛⠛⠋⠀⠀⠛⠛⠋⠀⠀⠛⠛⠋⠀⠐⠛⠛⠋⠀⠐⠛⠛⠋⠀⠀⠛⠛⠋

⣿⣷⡀⣜⢿⣿⣷⣔⣿⣿⣿⣷⡀⠘⢿⣿⣷⡿⠚⢿⣿⣧⣲⡿⢿⣿⣶⣻⣜⣿⣿⣷⡾⣿⢿⣿⣧⣶⡛⣿⣿⣷⣂⣿⣿⣿⣦⣠⡻⣿⣿⣾⣷⣿⣿⣿⣷⠁⠘
⣻⣿⣿⣮⣧⡹⣿⣿⣾⣾⣻⣿⣿⣦⣶⣿⣿⣿⣾⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⢿⡿⣿⣿⣯⣦⣹⣿⣿⣿⣾⡿⣿⣿⣾⠚⢻⣿⣿⣄⢟⠿⣿⣿⣯⣮⣻⣿⣷⣄
⣿⣩⢿⣿⣿⣮⣿⡿⣿⣿⣥⠻⣿⣿⣿⣯⣼⣿⣿⣿⣼⢾⢿⣿⣿⣼⣿⢿⣿⣿⣿⣿⣿⣿⣷⣽⣛⢿⣿⣿⡟⣿⢿⣿⣿⣯⣿⣿⣿⣷⡽⣿⣿⣿⣷⡽⣝⢿⣿
⠁⠀⠀⠈⠈⠁⠀⠀⠈⠈⠁⠀⠀⠈⠉⠁⠀⠀⠈⠈⠁⠀⠀⠈⠉⠀⠀⠀⠈⠉⠁⠀⠀⠈⠈⠁⠀⠀⠈⠈⠁⠀⠀⠈⠉⠁⠀⠀⠈⠉⠀⠀⠀⠈⠉⠁⠀⠀⠈
⢀⣾⣿⠟⠀⢀⣼⣿⠟⠁⢀⣾⣿⠟⠀⢠⣾⣿⠟⠀⢀⣾⣿⠟⠀⢠⣾⣿⠟⠀⣠⣾⣿⠟⠀⣠⣾⣿⠟⠀⣠⣾⣿⠏⠀⣠⣾⣿⠏⠀⣠⣾⣿⠋⠀⣠⣾⣿⠋
⣿⡿⠃⠀⣴⣿⡿⠃⢀⣴⣿⡿⠃⢀⣴⣿⡿⠃⢀⣴⣿⡿⠁⢀⣴⣿⡿⠁⢀⣴⣿⡟⠁⢀⣴⣿⠟⠁⢀⣼⣿⡟⠁⢀⣼⣿⡟⠁⢀⣼⣿⠟⠁⢀⣼⣿⠟⠁⢀
⠋⠀⢠⣾⣿⠏⠀⣠⣾⣿⠏⠀⣠⣾⣿⠋⠀⣠⣾⡿⠋⠀⣠⣾⡿⠋⠀⣠⣿⡿⠋⠀⣠⣿⡿⠋⠀⣰⣿⡿⠋⠀⣰⣿⡿⠋⠀⣰⣿⡿⠋⢀⣴⣿⡿⠋⠀⣠⣾
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⢿⡿⠿⠿⠿⢿⣿⠿⢿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠳⠤⢤⣄⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⢧⠀⠀⠈⠙⢆⣀⠜⠃⠈⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠹⡆⠀⠀⠐⢦⣿⡤⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⣊⡇⠀⠀⠀⠐⣿⡖⠂⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡓⠀⠀⠀⢣⡹⣇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠉⣦⠀⠀⠀⢹⠀⣤⢤⣀⠀⣿⠇⠀⠀⠀⠀⢀⣀⢚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢶⣿⡠⢤⡘⠮⠋⠀⢀⡟⢻⡿⠷⣶⢶⠞⠋⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⣴⣃⠀⠀⠘⡆⢀⡴⠋⠀⣾⡿⠒⠟⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⠒⠒⠲⠶⠤⠤⠤⣀⣀⣀⣀⣸⡁⠀⠀⠀⡿⠋⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⠦⠤⠞⠁⠀⠀⣰⣿⣋⡍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⣠⣿⣏⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠠⢀⡀⡀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⣰⣯⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⠴⠒⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡀⢀⣴⡟⠁⠀⠀⠀⠸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣻⡏⠀⠀⠀⠀⠀⡠⠗⢠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⣴⡿⠋⠁⠈⠀⠀⠄⠂⢁⡠⢴⠋⠁⠈⠓⢤⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣀⣀⣠⣚⣻⣟⣁⡴⡒⣛⠁⠀⢀⡤⠚⠉⠀⠘⢆⠀⠀⠀⠀⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠶⠏⢭⣽⣩⡍⣵⣭⠍⠠⣮⣠⠖⠁⢀⣀⣀⠀⠀⠈⢆⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠈⢹⠀⣀⡤⢊⡀⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⢉⢷⡽⠉⠙⣿⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣶⣄⣀⡈⣦⣀⣳⡀⢀⡀⡧⢤⣤⣶⡶⠿⡗⠀⠀⠀⢙⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠠⠛⠙⠛⠓⠓⠓⡖⠛⠛⢻⠉⠁⠀⠀⠀⣧⣄⡠⢀⣾⠀⣘⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣸⠒⠒⠂⠫⢉⠎⠨⠉⣉⡼⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⣴⡇⢄⠹⡮⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⣤⣤⣤⣄⣀⣤⣤⣤⣌⡧⠀⢀⡴⡷⠁⠘⠋⠑⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣽⣹⠥⠽⡧⠬⢉⣉⢱⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⠤⠴⠗⠋⠀⢀⠇⠀⠀⠸⡌⢆⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠿⠝⠛⠓⠒⠒⠚⠢⠤⢀⣀⣈⣙⣛⣛⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠶⠁⠹⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡀⠀⠀⠀⣿⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⣿⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠿⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠿⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠃⠀⠀⠀⠀⠘⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢠⠞⣋⣿⣷⡘⣦⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢠⠞⣋⣿⣷⡜⣦⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⣤⡄⠀⠀⢿⢸⣿⣿⣿⡇⣿⠀⠀⢤⣴⠶⠶⠆⠀⠀⠰⠶⠶⣦⡤⠀⠀⢸⢸⣿⣿⣿⣇⣸⠀⠀⢠⣤⡶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣬⣭⣭⣭⣭⣭⣭⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣭⣭⣭⣭⣭⣭⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣉⣩⡉⣉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⣩⡉⢉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡈⣛⣛⣁⣛⡀⢀⣀⣀⣀⡀⢀⣀⣀⣀⣀⠀⣀⣀⣀⣀⠀⣀⣀⣛⣛⡃⢘⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣌⢻⣿⣿⣿⣌⠻⣿⣿⣿⣦⠹⣿⣿⣿⣧⡙⢿⣿⣿⣷⡌⢿⣿⣿⣿⣌⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣎⠻⣿⣿⣿⣦⡹⣿⣿⣿⣷⡙⢿⣿⣿⣷⡌⢿⣿⣿⣿⣌⠻⣿⣿⣿⣦⠹⣿⣿⣿⣧⡙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡘⢿⣿⣿⣷⣌⢻⣿⣿⣿⣎⠻⣿⣿⣿⣦⡹⣿⣿⣿⣷⡙⣿⣿⣿⣷⣌⢿⣿⣿⣿⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠂⣛⣛⠛⢛⠂⠙⠛⠛⠛⠃⠘⠛⠛⠛⠓⠈⠛⠛⠛⠛⠈⠛⠛⠛⠛⡂⣛⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣙⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠇⣿⣿⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⠀⠉⠛⠻⠷⣶⣤⣀⣀⣀⣤⣴⡶⠿⠛⠉⠁⠀⠀⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⠀⠀⠀⢀⣠⣤⣾⠿⠟⠿⢿⣤⣄⣀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⢠⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣤⡀⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠛⠛⠃⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠟⠛⠉⠉⠉⣻⠷⣶⠶⣶⣤⡀⠀⠀⠀⠀⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢀⣴⠞⢋⣁⠤⠀⠀⠀⠀⠀⠈⠀⠀⠋⠀⠈⠆⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⣠⣿⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣠⡀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⡴⠀⠀⠀⠀⢀⠄⠀⢠⣾⠆⢀⡴⢋⣤⠞⠉⢿⡀⢸⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠞⠁⠀⠀⢀⣠⠞⣁⣤⠾⠋⣿⣴⡿⠞⠋⠁⠀⠀⠘⣷⣼⠀⠀⢸⣿⠀⠀⠀⠐⠛⠛
⠀⠀⠀⠀⠀⠀⢀⡴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣀⣠⣶⣿⠷⠛⠋⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠠⣿
⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⡋⠉⠀⠀⠀⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢸⡏⠁⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡅⠀⠀⠀⢀⣀⠉⠛⠀⠀⡆⠀⠀⠀⢠⡾⠟⠋⠀⠀⢸⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣿⣿⣷⡆⠀⣸⠃⠀⠀⠀⠀⣤⣶⣿⠀⠀⠀⡇⢰⣿⡇⠀⠀⠐⠶⠶⠶
⠀⠀⢀⣠⡤⠖⠛⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠛⠉⠀⢠⡟⠀⠀⠀⠀⠀⠻⠿⠋⠀⠀⠀⣇⣸⡿⠁
⣠⠶⠋⠀⠀⠀⠀⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉
⣧⠀⠀⢴⠀⢸⣿⡉⠉⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠹⣦⣀⣼⠟⠛⠉⠳⠆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢀⡀⠴⠛⠉⠠⠤⠀⠉⠻⢀⣶⠀⠀⣴⠏
⠀⠀⠘⢧⡀⠀⠀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠞⠁
⠀⠀⠀⠀⣟⣿⠙⡇⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣤⣤⣄⣀⣀⣤⣤⣤⢶⡟⠁
⠀⠀⠀⠀⡿⠿⠶⠿⠟⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠈⣿⣀
⠀⠀⠀⢸⡇⠰⠶⠖⠂⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠋⠙⡟⣿⠋⢉⣀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡿⢿⠶⣤⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⢸⢇⡟⢧⠈⠉⠁⠀⠀⠀⠐⠛⠛⠃⣼⡇⡸⠀⠈⠙⢶⣄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⡇⠀⠀⠀⣾⢸⡇⠘⢷⡀⠀⠀⠀⠀⠀⢀⣴⠞⣿⢃⡇⠀⠀⠀⣠⠙⢷⣄
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⡇⠀⠀⠀⡏⣿⠀⠀⠈⢷⡄⠀⢀⣠⠶⠋⠁⠀⣿⢸⠃⠀⠀⣴⠁⠀⠀⢻⡆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣿⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣿⠀⠀⠀⠀⠹⠞⠋⠁⠀⠀⠀⢸⡇⣾⠀⠀⠰⠃⠀⠀⠀⠘⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⢀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠈⣧⣿⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⡰⡆⠀⠀⠀⠀⠀⠀⢀⣼⠏⣿⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⢀⣴⡾⠋⠀⣿⠀⠀⠀⢸⢸⡇⠀⣀⣠⣤⣄⣀⣀⡀⠀⠀⠀⢸⡇⡇⠀⠀⢹⡇⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁⠀⠀⠀⣿⠀⠀⠀⣸⢸⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢸⡇⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⢿⣤⣤⣤⣴⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⡇⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⠀⠀⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⢀⠀⣿⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡀⠀⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⢸⠀⢹⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⡇⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠀⡼⠀⢸⣇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⢰⠃⠀⢸⣿⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠶⣷⣿⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣤⣴⣿⣶⡾⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⣠⣶⣦⡀⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣿⣿⣿⡇⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⢧⣴⠁⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠄⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠀⠒⠤⠄⠤⢼⡦⠤⠐⠒⡍⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣁⣀⡍⠀⢈⠘⣸⠆⠀⠀⠀⠱⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠀⠉⠁⠈⠒⠀⠀⣼⢆⠀⠀⠀⠀⢈⡩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠄⠀⠀⠀⠀⣰⠃⡰⠟⠀⢀⠀⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⣴⠗⠈⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠿⠏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠘⠲⡎⢩⠇⠤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢺⠀⠀⠀⠀⠀⠈⢉⠏⠂⠈⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⢵⡘⢰⠀⠀⠀⠀⠭⢰⠀⠈⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠙⠛⠶⣆⣀⡘⢁⣐⢤⣤⣈⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⢻⠒⠉⠃⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⠀⠀⠀⣀⡈⠀⠀⠀⣏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠒⠣⠅⠀⠀⠀⠀⡣⠤⠩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⡄⣀⡀⠀⠀⠀⣀⢠⣤⠚⡠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⢏⡨⠐⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠈⠉⠐⢒⡃⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⣍⠉⠉⢩⣭⣍⣉⠉⣩⣍⠉⢩⣍⢩⣭⠉⢉⣩⣭⣉⠉⢉⣩⣭⣍⠉⠉⣍⣭⣉⡉⢉⣍⡉⢉⣭⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⢸⣿⠉⣿⣷⢸⣿⠀⣿⡟⢸⣿⠀⣿⡟⢹⡿⠆⣾⣿⠉⣿⣷⠀⣿⡏⢻⣿⡎⣿⡇⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⣧⠀⢸⣿⠀⣿⣿⠈⣿⣆⣿⠇⢸⣿⠄⠻⣿⣦⡀⠀⣿⡇⠀⣿⣿⠀⣿⣧⣼⡿⠃⢸⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣼⣿⡄⢸⣿⠀⣿⣿⠀⢿⣿⣿⠀⢸⣿⠀⣀⡈⢻⣿⡄⣿⣇⠀⣿⣿⠀⣿⡏⣿⣇⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠈⣿⣇⢸⣿⣦⣿⠏⠀⢸⣿⡇⠀⢸⣿⠀⢻⣿⣴⣿⠃⠻⣿⣤⣿⠏⠀⣿⡇⢸⣿⡄⠀⣿⣏
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠀⠀⠀⠘⠉⠉⠓⠀⠀⠧⠤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠉⢿⣿⡏⠉⠉⠛⢻⡏⠙⣿⡏⠉⡟⠉⢻⡟⠋⠀⠙⢿⡿⠛⣉⠙⢻⣿⠉⢉⡙⠛⣿⠉⢻⣿⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⢸⣿⡇⠀⣿⡆⠀⣷⠀⢻⠁⢸⡇⠀⢸⡀⠘⣿⣤⣼⠃⠀⣿⡇⠀⣿⠀⢸⡷⠀⢸⡄⠸⠇⢠⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⣇⠀⣿⡇⠀⣿⡇⠀⣿⡀⠘⠀⣼⡇⠀⢸⣷⣄⠀⠹⣿⠀⠀⣿⡇⠀⣿⠀⢀⠁⢠⣿⣷⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠇⢀⣉⠀⢹⡇⠀⣿⡇⠀⣿⡇⠀⢀⣿⡇⠀⢸⡛⠙⣷⠀⢸⡄⠀⣿⡇⠀⣿⠀⢸⡆⠈⣿⣿⡆⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣀⣸⣿⣄⣸⣇⣀⣀⣤⣼⣿⣷⣀⣸⣿⣧⣀⣼⣧⣄⣀⣠⣾⣷⣄⣀⣠⣼⣿⣀⣸⣷⣀⣹⣿⣇⣠⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡆⢰⢰⠇⡹⠁⡔⡰⠒⡆⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡀⠸⠙⠈⠋⠀⠁⠀⠀⠑⠚⠑⠹⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡜⠇⠃⢠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠂⠐⢣⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡎⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⣆⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠱⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠋⠙⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⣠⠐⠂⠁⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⣤⠃⢤⠄⠀⢖⢈⠷⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠁⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠤⠄⢀⡀⠀⠁⠠⢀⠈⠀⢄⡀⠀⣠⡆⠀⢀⠀⠂⠉⠀⢀⣀⡀⠀⠀⠐⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⣘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⢢⡌⠿⠒⣙⡧⠆⠑⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⣒
⠀⠀⠀⠀⠀⠀⠀⢬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⠾⠛⠋⠁⡀⣀⠈⠉⢛⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡅
⠀⠀⠀⠀⠀⠀⢈⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⠈⠃⢺⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂
⠀⠀⠀⠀⠀⠀⠀⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⣯⡶⣓⣲⡶⣬⣛⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠅
⠀⠀⠀⠀⠀⠀⠀⣙⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⡟⢈⠤⠖⠶⠀⠉⠛⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠂
⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⢀⠴⠒⠓⠂⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠩⠥⠀⠀⠀⠀⠀⠀⠀⠃⠀⣸⠀⠸⡀⠔⠋⣉⡁⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠙⠤⢈⡀⣀⠀⠀⠁⠠⣄⠀⠀⠀⠀⠀⠀⢤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣔⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣂⠘⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢋⠴⡾⢡⢀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣴⠈⢹⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢞⠂⠀⢀⣶⢠⢃⠖⢢⠀⢸⡆⠸⠘⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠈⠓⠋

⣶⣶⣦⡀⠀⠀⠀⠰⣶⣶⣶⣦⡀⠀⠀⠀⠲⣶⣶⣶⣤⡀⠀⠀⠀⢶⣶⣶⣶⣤⡀⠀⠀⠀⢶⣶⣶⣶⣄
⣿⣿⣿⠟⠀⠀⠀⣠⣾⣿⣿⣿⠟⠀⠀⠀⣠⣾⣿⣿⣿⠟⠀⠀⠀⣠⣿⣿⣿⡿⠋⠀⠀⠀⣠⣿⣿⣿⡿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀
⣾⣿⣿⡿⠋⠀⠀⠀⣠⣾⣿⣿⡿⠋⠀⠀⠀⣠⣾⣿⣿⡿⠋⠀⠀⢀⣠⣾⣿⣿⡿⠋⠀⠀⢀⣴⣿⣿⣿
⠿⠿⠿⠤⠤⠤⠤⠾⠿⠿⠿⠿⠤⠤⠤⠤⠾⠿⠿⠿⠯⠤⠤⠤⠴⠿⠿⠿⠿⠯⠤⠤⠤⠴⠿⠿⠿⠿⠯

⠀⠘⣛⣛⣛⢣⣀⣀⣀⣄⣘⠟⠛⣛⠁⣀⣀⣠⢠⣚⣛⣛⣟⣁⠄⠀⣀⠀⢚⣛⢻⣛⣄⣄⣀⣀⣀⡛⠛
⣤⣤⣭⣉⣉⣉⣉⣭⣭⣭⣭⣯⣀⣉⣀⣈⣭⣭⣭⣭⣍⣉⣉⣉⣋⣤⣭⣤⣬⣍⣉⣉⣉⣉⣭⣭⣭⣽⣄

⢀⣤⣤⣤⣤⠄⠀⠀⠀⠀⠤⠤⠤⠤⠄⠀⠀⠀⢀⣤⣤⣤⣤⠄⠀⠀⠀⠀⠤⠤⠤⠤⠄⠀⠀⠀⢀⣤⣤
⣿⣿⣿⡿⠁⠰⠒⠂⠓⠂⠚⠐⠐⠒⠖⠖⢂⣴⣿⣿⣿⠟⠁⠖⠢⠂⠓⠒⠂⠐⠘⠲⠒⠒⣠⣶⣿⣿⣿
⠛⠛⠛⠒⠒⠒⠒⠛⠛⠛⠛⠛⠒⠒⠒⠒⠛⠛⠛⠛⠓⠒⠒⠒⠚⠛⠛⠛⠛⠓⠒⠒⠒⠚⠛⠛⠛⠛⠓
⠀⣀⠀⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⠀⠀⠀⢀⢀⣀⢀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⣀⣀⡀⣀
⠉⡉⣅⡤⢤⣤⣤⡌⠉⡍⠉⢡⢠⣄⡠⢤⣬⣭⠉⢩⡅⠀⡄⣤⣄⢭⣭⣩⣍⠁⢠⠀⢠⢠⣭⡩⡭⣭⣥
⠶⠶⠶⠤⠤⠤⠤⠶⠶⠶⠶⠦⠤⠤⠤⠤⠶⠶⠶⠶⠦⠤⠤⠤⠤⠶⠶⠶⠶⠦⠤⠤⠤⠤⠶⠶⠶⠶⠦

⠀⠀⠀⢀⣠⣖⣿⣦⡀⣀⡀
⠀⠀⣶⣿⣿⣿⣿⠃⠀⠀⢳⡀⢀⠆⠀⠀⠀⠒⠂
⢀⣼⣹⣿⣿⣿⣏⠀⠀⠀⠘⡇⢨⠤⠤⣄⠀⠀⢀⡤⠖⠒⠲⡄
⣎⢿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠙⣿⠀⠀⠈⠳⡴⣋⠀⠀⠀⣠⠇⠀⠀⣀⣀⣤⠤⠤⠤⣀
⠀⠉⢻⣿⣿⡿⣿⣿⢿⣷⡄⠀⢸⡀⠀⠀⢀⣹⣍⣀⡤⠶⠥⠖⠚⠋⠉⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠈⠉⠈⠉⠙⣿⣿⡀⠀⢳⡀⠀⠀⢸⡇⢠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠿⣧⠀⠀⠙⠲⠶⠫⠂⠈⠑⠦⣄⣀⠀⠀⠀⡀⠀⠀⠄⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣸⠡⠶⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⡖⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠸⢤⠤⠖⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⣒⠒⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⡑⠢⢼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠙⣦⠤⠤⡤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀⣀⠈⠉⠓⠒⠢⠤⠤⣀⣀⣀⣀⡀⠈⠁⠀⢸⣽⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠙⣴⣿⠶⣿⣟⡒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠒⠢⠤⣀⣀⠀⠀⠀⠀⠈⠉⠉⡽⠓⢼⣇⢫⡑⠒⠤⣤⡀⠀⢀⡠⣪⠔⢧⣼⡄⢹⡄⢸⣍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⡀⠀⠉⠒⠤⢄⡀⠀⡔⠀⢀⠘⢿⠃⠙⢦⣾⣿⡧⠖⡩⠊⠁⠀⠀⡝⠻⡏⠑⠋⠈⠛⢿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡓⢸⠀⠀⢋⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⠀⠀⠈⠳⣄⠙⠊⠀⠀⠀⠀⠀⠱⠶⠃⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢸⣤⣤⡖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⢴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡒⠒⠒⢏⠹⡆⠀⠀⠀⠀⠒⠒⠒⠂⠉⡿⢹⠏⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠸⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡼⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠰⣄⡀⢀⡀⠀⣀⣀⣀⣠⣤⣧⣴⣗⣚⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣴⠧⣀⣀⡀⠀⠉⠁⠀⠀⠐⠒⣋⣉⣡⠧⣤⣤⣶⣿⣋⡩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠤⠤⠽⠗⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣿⠀⣼⣿⣿⡟⠀⢀⣾⣿⣿⠏⠀⢠⣿⣿⣿⠃⠀⣰⣿⣿⡿⠁⠀⣼⣿⣿⡟⠀⢀⣾⣿⣿⠏⣿
⠀⠀⠀⠀⠀⣿⣾⣿⣿⣟⣀⣠⣾⣿⣿⣏⣀⣰⣿⣿⣿⣃⣀⣼⣿⣿⣿⣁⣀⣾⣿⣿⣟⣀⣠⣿⣿⣿⣏⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡇
⠀⠀⠀⠀⠀⣶⠒⢲⣶⣾⣷⠞⠓⣲⣶⣶⣶⠒⠒⣶⣶⣶⡶⠒⢒⣶⣶⣶⡖⠒⢲⣶⣶⣾⠗⠚⣳⣶⣶⣶⣶
⠀⠀⠀⠀⠀⣿⢠⣿⣿⣿⠋⠀⣰⣿⣿⡿⠁⠀⣼⣿⣿⡟⠁⢀⣾⣿⣿⠏⠀⢠⣿⣿⣿⠋⠀⣰⣿⣿⡿⠁⣿
⠀⠀⠀⠀⠀⠻⠿⠿⠿⢷⡶⢾⠿⠿⠿⠷⠶⠾⠿⠿⠿⠶⠶⠾⠿⠿⠿⠶⠶⠿⠿⠿⠷⠶⡶⢿⡿⠿⠷⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⣀⠤⢄⡀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⡜⠁⠀⠀⠙⣄⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⠞⠀⠀⠀⠀⠀⠘⢆⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠘⣆
⠀⠀⠀⠀⠀⢀⠎⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⢆
⠀⠀⠀⠀⢠⠏⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⢧
⠀⠀⠀⣠⠃⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢣⡀
⠀⠀⡰⠃⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠳⡀
⠀⡴⠁⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠱⡄
⡼⠁⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠹
⡇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠱⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⡰
⣀⣈⣒⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣔⣊⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠒⠒⠒⢤⡀
⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠈⠆
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠠⣇⡀⠀⠀⠀⠀⠀⠀⢀⣰
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⠉⠉⠉⠉⠓⡀⠘⠉⠹⣧⡀
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⣀⣴⢇⠀⠀⠀⠀⢀⡁⠀⠀⠀⡌⠻⣆⣀
⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢸⡔⠁⠀⠑⠤⢤⡴⠂⠈⡒⠒⠊⠀⠀⠈⢦⡇
⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡠⠊⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡜⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠔⠲⠖⠒⠒⠒⠚⠀⠀⠀⠘⠀⠀⠀⠡⢀⠀⠀⠈⠒⠤⠤⠤⠄⠚⠁⠀⠀⡠⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⡰⠢⡄⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠤⠄⠂⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠴⢤⠀⢸⠄⠠⠤⠤⠤⠴⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠦⠔⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢺⡀⠞⠀⠠⠤⢄⡀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁⢀⠀⢀⠠⢄⡀⡱⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠀⠀⠀⠀⠀⢀⢃⠔⠁⠀⠀⠀⠉⠒⠲⣆⠠⠤⠤⠤⠤⠤⠤⠄⣰⠖⠒⠉⠀⠀⠀⠀⠢⡘⡄⠀⢏⠀⢠⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠰⣀⠌⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡉⠑⠒⠒⠠⠤⠤⠄⣀⣀⣀⣀⣀⡰⢆⣀⣀⣀⣀⡀⠠⠠⠤⠄⠒⠒⠈⢉⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠉⠛⡇⠈⠀⠀⠁⠀⠁⠈⠁⢸⠋⠉⢈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠈⠁⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠑⠢⢄⡀
⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣯⣀⣄⣠⢦⣀⣠⣃⣀⣀⣀⣀⣀⣀⣀⡀⣘⣄⣀⡴⣄⣀⣀⣽⡀⣀⣀⡀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢀⣀⡀⢩⣿⣿⠁⠈⣿⣿⣿⣿⣿⣿⠟⠉⢁⣀⡀⠉⠙⢿⣿⡏⠀⢹⣿⣿⡿⠁⠈⢿⣿⣿⡏⠀⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣏⠀⠰⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⠃⠀⣴⣿⣿⣿⣷⡀⠈⢿⣿⡀⠀⣿⣿⠇⠀⠀⢸⣿⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠙⠛⢿⣿⣿⠀⠀⣿⣿⣿⣿⡏⠀⢰⣿⣿⣿⣿⣿⡇⠀⢸⣿⣇⠀⢹⡿⠀⣼⡇⠀⣿⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⢿⣿⠀⠀⣿⣿⣿⣿⣧⠀⠈⣿⣿⣿⣿⣿⠇⠀⣸⣿⣿⠀⠘⠇⢀⣿⣧⠀⢹⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⠿⠟⠁⢀⣾⣿⠀⠀⠻⠿⠿⠟⢿⣧⡀⠈⠻⠿⠿⠋⠀⣰⣿⣿⣿⡇⠀⠀⣸⣿⣿⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣶⣿⣿⣿⣦⣤⣤⣤⣤⣤⣾⣿⣿⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣤⣤⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡿⠿⣿⡿⠿⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡇⠀⣠⣤⣤⣀⠈⠙⢿⣿⣿⠟⠁⢀⣤⣤⣄⠀⠙⢿⣿⡆⠀⢻⣿⣿⠇⠀⠀⣿⣿⣿⠀⢠⣿⡇⠀⠀⠹⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣷⠀⠈⣿⡟⠀⢠⣿⣿⣿⣿⣷⠀⠈⣿⣷⠀⠸⣿⣿⠀⣀⠀⢹⣿⡇⠀⣼⣿⡇⠀⣦⠀⠙⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⠀⠀⣿⣿⡄⠀⣿⡇⠀⣿⡀⠘⣿⠀⢠⣿⣿⡇⠀⣿⣧⠀⠘⣿⠀⠸⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡟⠀⢠⣿⣧⠀⠘⣿⣿⣿⣿⡿⠀⢀⣿⣿⣧⠀⠘⠀⢸⣿⣇⠀⠛⠀⣼⣿⣿⡇⠀⣿⣿⣷⡀⠘⠀⠰⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⠇⠀⠙⠛⠋⠁⣀⣤⣿⣿⣿⣦⡀⠈⠙⠛⠋⠀⣠⣾⣿⣿⣿⡄⠀⠀⣿⣿⣿⡀⠀⢠⣿⣿⣿⡇⠀⢿⣿⣿⣷⡄⠀⠰⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⠀⠀⠀⢀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⡀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⣴⣿⠿⠿⠗⠀⣿⡇⠀⠀⣿⣿⠀⣿⣿⠀⣿⣿⠀⠀⠀⢸⣿⡿⠿⣿⣦⡀⠀⣿⡿⠿⣿⣦⠀⣿⣿⠿⠿⠗⠀⣿⣿⡄⠀⢸⣿⠀⠀⣿⡇
⠀⢸⣿⠀⠀⣸⣿⠃⠀⠀⠀⠀⣿⣇⣀⣀⣿⣿⠀⣿⣿⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠈⣿⣧⠀⣿⡇⠀⣸⣿⠀⣿⣿⣀⣀⡀⠀⣿⡿⣿⡄⢸⣿⠀⠀⣿⡇
⠀⢸⣿⠀⠀⢿⣿⠀⠀⠀⠀⠀⣿⡿⠿⠿⣿⣿⠀⣿⣿⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⣿⡿⢿⣿⡁⠀⣿⣿⠿⠿⠃⠀⣿⡇⢻⣷⣼⣿⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠸⣿⣧⣀⣀⡀⠀⣿⡇⠀⠀⣿⣿⠀⣿⣿⠀⣿⣿⣀⣀⡀⢸⣿⣀⣀⣴⣿⠇⠀⣿⡇⠀⢿⣷⠀⣿⣿⣀⣀⣀⠀⣿⡇⠈⢿⣿⣿⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠈⠛⠿⠿⠛⠀⠿⠃⠀⠀⠻⠟⠀⠻⠟⠀⠻⠿⠿⠿⠃⠘⠿⠿⠟⠛⠁⠀⠀⠻⠃⠀⠘⠟⠀⠻⠛⠿⠿⠟⠀⠿⠃⠀⠈⠻⠛⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⢠⣤⣤⣤⡀⠀⢠⣤⣤⣤⣤⠀⠀⣠⣤⣤⡀⠀⣤⣤⣤⣤⡄⢠⣤⣄⠀⠀⣤⡄⢠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⣿⣟⠛⢻⣿⡀⢸⣿⠋⢻⣿⡆⢸⣿⠟⠛⠛⠀⣾⡿⠛⠛⠃⠀⣿⡟⠛⠛⠃⢸⣿⣿⡄⠀⣿⡇⠘⠛⣿⣿⠛⠛⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⣿⣯⣀⣼⣿⠃⢸⣿⣀⣴⡿⠃⢸⣿⣦⣤⣤⠀⢻⣿⣦⣄⠀⠀⣿⣧⣤⣤⡄⢸⣿⠹⣿⡄⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⠿⠛⠁⠀⢸⣿⠛⢿⣧⠀⢸⣿⡋⠉⠉⠀⠀⠈⠛⣿⣧⠀⣿⡟⠉⠉⠁⢸⣿⠀⢻⣷⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⢸⣿⠀⢸⣿⡆⢸⣿⣦⣤⣤⠀⣶⣤⣴⣿⡟⠀⣿⣷⣤⣤⡄⢸⣿⠀⠈⢿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠈⠉⠉⠉⠉⠀⠉⠙⠋⠉⠀⠀⠉⠉⠉⠉⠁⠈⠉⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠻⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⠟
⠀⠀⠀⣀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⡄⠀⠀⠀⠀⠀⣠⡶⠛⠛⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠫⢋⡼⣻⠄⠀⠀⠀⢰⠏⠀⠀⠄⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⡜⠁⠀⠀⠀⠐⢿⡤⣄⣀⠈⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⣃⡀⠀⠀⣠⡤⠖⠋⠀⠀⠀⠘⠁⠀⠉⠛⠛⠛⠛⢻⡟⠁⠀⣠⠞⠉⢙⡿⠛⠻⡛⠋⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠉⢹⠏⠉⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢰⠇⣠⠞⠁⢀⡴⠋⠀⠀⠀⠙⢦⡀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⣠⡧⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠈⠱⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⠷⠆⠀⠀⠘⣻⣿⣀⡴⢋⣤⠀⣠⠀⠀⢀⣀⠀⠀⠀⠈⢳⣄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢀⣧⣀⣀⣤⠶⢶⡶⠂⠋⠉⠉⣳⢤⡄⠀⠠⠔⣫⡟⣿⠁⠀⠸⣦⡀⣿⠀⢰⡏⢻⠄⢱⣰⣧⡟⠈⠳⣾⡇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣸⡏⠈⠁⠀⠀⢸⠸⠀⡀⠀⢰⣿⠀⠙⢦⣴⠟⡿⢸⣿⠢⣄⠠⣼⠗⢹⣄⠘⢷⡿⠀⠸⡏⢹⡇⣀⡴⣿⡇
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⢈⠀⡇⠀⣎⢿⠀⠀⠀⠀⢰⠇⠸⣧⡀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡶⠶⣷⠀⠀⠀⠀⢾⠀⡀⠃⡆⢹⢸⡇⠀⠀⠀⡿⠀⠀⠈⠹⣦⡀⠈⠻⣦⡀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⠛⠛⠛⠶⣄⢸⣀⣇⠀⣧⡸⠈⣷⠀⠀⣼⠃⠀⡆⢀⠀⠈⢿⣄⠀⠈⠳⣄⡚⠃⣠⠟⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⠋⢙⢙⡇⠀⢹⣆⡼⠁⠀⠀⡇⢸⠀⠀⢲⠈⣷⠲⢶⣼⣷⠚⢷⠆⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢸⢸⡇⠀⠀⣿⠁⠀⠀⠀⡇⢸⠀⡇⢸⠘⣧⠀⠈⢿⣭⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣾⠀⣼⡇⠀⠀⣿⠀⠀⠀⠀⠃⠘⠀⢰⣏⣼⣿⡄⠀⢻⡶⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢤⣶⡄⠾⠂⠀⠀⣿⠀⠀⠀⢸⠀⡇⠀⣼⠟⠒⠒⣿⠦⣾⠙⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣈⢢⣕⠀⠤⢤⠀⠀⣿⠀⠀⠀⢸⣸⠃⠀⣿⡦⡤⠄⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣿⠏⡄⢀⡎⠀⠀⢿⣄⠀⠀⡘⡇⠀⢰⡘⢛⡟⠋⣻⠀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⣇⠐⠀⢇⠀⣀⡻⠷⣤⣤⣧⣀⣸⣇⣼⠃⠀⠘⡄⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣙⣤⣞⠛⠛⠉⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠈⠉⠀⠙⠳⣦⣤⣀⣀⣀⣀⣠⣤⡴⠶⠞⠛⠋⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⡀⠀⢀⣀⣀⣀⣀⣰⡏⠀⠀⠀⢸⡷⠀⣤⣤⣤⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣩⢭⠉⠙⠛⠒⠀⣿⠀⠀⠀⠀⢸⣗⠰⠖⠒⢦⣄⡅⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡀⠠⠤⠤⠤⠀⠀⣿⠀⠀⠀⠀⠘⣿⡆⡀⠢⠤⣤⠤⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠴⠴⠶⠶⠒⠚⠛⠛⡏⠀⠀⠀⠀⠀⣿⡏⠙⠓⠶⠶⠶⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠋⠛⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣰⡆⠀⢀⠀⠀⠀⢀⡇⢻⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⣤⣀⡀
⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⣠⠞⠓⢦⣰⠃⠀⠀⠀⠀⠀⢀⣴⣿⢁⣠⣯⣤⣤⠴⠚⣁⡀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⠤⠒⠒⠒⠒⠒⠬⣉⠲⣄
⢠⠎⠁⠀⠀⠀⠉⠛⠲⢤⣠⢤⡿⠀⠀⠀⠃⠀⠀⠀⠀⣀⡴⠞⠁⢉⣉⠀⠀⠀⢿⣠⡴⠶⢬⣄⠀⠀⠀⠀⢠⠎⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠳⡄
⢸⡀⠀⠀⠀⠀⠀⠀⣀⣀⡇⠀⡇⠀⠀⠀⠀⠙⢻⠏⠉⠁⠀⠀⣈⣤⢤⣄⠀⢠⠟⠁⠀⠀⠀⠙⣧⠀⠀⢰⠃⡜⠀⢠⣴⡦⣶⣶⢦⣶⣿⣦⡶⣶⠆⢣⠹
⠈⢷⡀⠀⠀⢠⡞⠉⠉⠉⠻⢤⡇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡼⠁⠀⠀⠈⠻⡏⣤⠀⠀⠀⠀⠀⢻⠀⠀⡎⢰⠁⠰⠿⢶⡆⢸⡇⣿⠁⣸⣿⠟⠁⠀⠀⡇
⠀⠈⢷⡀⠀⠸⠻⣄⠀⠀⠀⠀⢻⠀⠀⠀⠉⢳⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⣾⠀⠀⡇⢸⠀⠀⠟⠋⠀⠛⠁⠻⠟⠁⠛⠀⠀⠀⠀⡇
⠀⠀⠀⠹⡄⠀⠀⠈⢷⡄⠀⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⢈⣧⣉⠁⠀⠀⢸⠷⣄⡀⠀⢀⣠⠞⢧⡀⠀⢱⡈⢆⠀⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡇⡰⢠
⠀⣠⣤⡤⠗⠀⠀⠀⠀⢹⡄⠀⠀⢹⣄⡀⠀⣼⠀⠀⠀⠀⡀⠀⠉⠉⠓⢶⡋⠀⠈⠉⠉⠉⣁⠤⣄⢳⡄⠀⠱⡈⢦⡙⠛⠉⠉⠉⠉⠉⠉⠉⠉⢉⠔⣡⠏
⢸⡇⣠⠶⠶⠶⠶⠶⠖⠋⠀⣠⠞⠋⣉⠙⢷⡏⠀⠀⢠⠿⡁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠁⠀⣸⠂⣷⠀⠀⠘⠢⡉⠲⠤⣀⣀⣀⣀⣀⠤⢒⣩⠞⠁
⠈⢷⡏⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠛⠙⠳⠀⠁⠀⠀⠀⠀⠈⠓⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⡿⠀⠀⠀⠀⠈⠙⠒⠦⢤⣤⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⢖⣊⣙⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⢀⡤⣼⡷⠓⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣍⠙⢦⠉⠹⣷⣶⠶⠶⠤⠤⣤⠶⣶⣖⣚⣛⡭⠵⣶⡆⠀⠀⠀⠀⠀⣀⣀⣸⠀⡀⡷⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠈⢷⡈⢧⠀⠹⡍⠓⢦⣀⠀⡿⢆⢹⡍⠁⠀⠀⠀⡟⠛⠛⠛⣉⣉⣉⣉⣉⣉⠀⢿⡵⣾⣃⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢻⡈⡇⠀⢹⡄⠀⠈⠻⡇⠸⡄⢣⠀⠀⠀⠀⣟⢹⠉⠉⠁⠀⠀⠀⠀⢹⣤⠼⡇⡇⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⢸⠀⠀⢷⠀⠀⠀⢷⠀⢷⠀⢿⠒⠲⠤⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣇⠈⡂⠀⠸⡆⠀⠀⢸⡄⠸⡆⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⢶⣶⣶⠚⠋⢀⣇⣀⣠⣧⠤⠤⠴⢷⠶⠓⠒⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⢻⡆⣿⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣸⣀⣀⣀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣷⣌⠙⠁⢸⣶⡶⢶⡖⠛⠛⠉⠉⠉⠉⠙⣧⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡶⠷⠀⣀⣤⣶⣲⡞⠃⠀⢀⣀⣀⣀⣠⡼⠁⠳⠶⠚⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠋⢻⡀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⡄⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣁⣤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃⠀⠀⠀⠀⠀⠀⠨⠭⠟⣻⠋⢓⣂⣠⡼⠦⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡁⠀⠉⢙⣛⠷⢒⡶⠒⠒⠚⠋⠉⠛⠯⢍⣃⣠⠤⠤⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠭⠭⠭⠭⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⣀⢀⣤⡀⣈⠉⡉⠁⠉⠉⠉⠉⢉⣭⡁⢈⡉⢉⡉⢉⢩⣩⢉⡉⣩⣍⢉⡁⣈⠉⠉⠉⠉⠉⠉⣩⣍⠉⣀⠀⣀⢀⡀⣌⣍⡉⢉⣭⡉⣉⢉⡉⠉⠀⠈
⣺⢾⠀⡿⣾⢷⡇⠀⠀⠀⠀⠀⢸⠀⡁⣼⣽⡸⡇⣸⠀⡇⢸⡿⡇⢸⢾⠷⣾⠀⠀⠀⠀⠀⠰⡇⢈⢰⣿⡆⣿⢸⡇⢸⠀⡇⣏⠀⡇⡟⣾⠃
⣈⣈⣉⣁⣉⣈⣁⣀⠀⣀⣀⣀⣈⣉⣁⣁⣀⣁⣉⣁⣀⠁⣈⣁⣉⣉⣈⣀⣉⣀⣀⣀⣀⣀⢀⣉⣁⣈⣀⣉⣈⣉⣀⣈⣀⣁⣈⢉⣀⣁⣈⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⣿⡿⠦⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣶⡀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⣿⣿⣦⡀⠀⡈⠻⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠶⣾⣿⣦⡀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⣿⣿⣦⡀⠀⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠙⢿⣿⣿⣦⠀⠀⠙⢿⣿⣿⣦⠀⠀⠙⢿⣿⡿⠆⠀⠐⠉⠉
⢤⣤⣶⣆⠀⠀⠙⢿⣿⣿⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⡙⠿⠟⠛⠂
⠀⠙⢿⣿⣷⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⠙⠛⠉⠁
⣄⠀⠀⠙⣿⣿⣷⡤⠄⠀⣉⣁⣀⡀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⣀
⣿⣷⡆⠁⠈⢿⣿⣷⡀⠀⠘⢿⣿⣧⡀⠀⠙⣿⣿⣦⠀⠀⠹⣿⣿⣆⠀⠀⠻⣿⣿⡄⠀⠈⢿⣿⣷⡀⠀⠘⣿⣿⣧⠀⠀⠹⣿⣿⣆⠀⠀⠹⣿⣿⣆
⠻⣿⣿⣄⠀⠈⢻⣿⣷⡄⠀⠈⢿⣿⣷⡀⠀⠘⣿⣿⣧⠀⠀⠹⣿⣿⣆⠀⠀⢹⣿⣿⡄⠀⠈⢿⣿⣷⡀⠀⠈⢿⣿⣷⡀⠀⠘⣿⣿⣦⠀⠀⠹⣿⣿
⠀⠛⠛⠛⠂⠀⠀⠛⠛⠛⠀⠀⠈⠛⠛⠓⠀⠀⠘⠛⠛⠃⠀⠀⠙⠛⠛⠂⠀⠀⠛⠛⠛⠂⠀⠀⠛⠛⠛⠀⠀⠘⠛⠛⠓⠀⠀⠘⠛⠛⢧⣤⣤⣽⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⢶⣶⣿⣦⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠘⢿⣿⣷⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣤⣶⣆⠁⠀⠙⢿⣿⣷⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⢙⠿⠿⠓⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⢶⣾⣷⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⠙⢿⣿⣷⣄⠀⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⠻⣿⣿⣷⣄⠀⠀⠻⣿⣿⣷⣄⠀⠀⠹⣿⣿⠷⠄⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⣤⣴⣶⡀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠻⠿⠟⠂
⠐⠒⠛⠃⠀⠀⠈⠛⠛⠛⠂⠀⠀⠈⠛⠛⠛⠂⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⡞⠙⠀⡾⡄⢺⠀⡇⢹⠋⣾⣼⠙⣦⣿⣄⡇⠀⠀⠀⠀⠀⣰⠙⠂⣸⢧⢸⡆⢺⠘⡟⢹⣦⠏⢳⣸⣆⣷⠀⠀⠀⠀⠀⢠⡏⠓⢀⣿
⠀⠀⠀⠀⠀⠀⠳⠞⠸⠋⠳⠘⠶⠃⠸⠀⠛⠙⠶⠋⠗⠹⠇⠀⠀⠀⠀⠀⠘⠶⠃⠏⠙⠎⠷⠟⠀⠇⠸⠋⠳⠞⠸⠈⠗⠀⠀⠀⠀⠀⠈⠳⠞⠼⠉
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⣀⠤⠶⠒⠉⠉⠉⠉⠉⠓⠶⢤⡀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠁⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⠈⢇⣀⣤⣞⣡⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⣠⠾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⣸⡇⠀⢠⠀⢸⣧⣠⣿⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⢠⡟⣷⢀⡿⣇⣸⡏⠉⠘⢿⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⢿⣤⣴⠶⠋⣠⣿⠟⠁⠹⠟⠀⠀⠀⠈⣷⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣠⣶⡿⠿⠟⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠈⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠀⠀⠀⠉⠀⢀⣠⡤⢤⣤⣉⡻⣦⡀⠀⠀⠀⠀⢴⣤⣴⣾⢿⣋⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣆⡴⠋⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⠀⠈⠙⢮⡻⠆⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠻⢶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢹⡏⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣠⣶⣿⡆⠀⠈⢷⠀⠀⠀⠀⠀⣰⠁⠀⣰⣶⣿⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⣿⠻⡆
⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⠸⡇⠀⢀⡀⠀⠀⠀⢸⡀⠀⠀⠀⢿⣿⣿⣿⡿⠀⢸⠇⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⡿
⠀⠀⠀⣀⡀⣰⠏⠉⠉⠙⢾⣿⠟⠋⠙⢳⡀⠀⠈⠃⠀⠀⠀⠈⠛⠛⠛⠁⠀⡌⠀⠀⠀⠀⠀⠸⣄⠀⠈⠛⠛⠋⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢈⣃⡼⠃
⣴⠟⠉⠉⠘⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣧⠀⠀⠀⠈⠉⠛⠒⠶⠶⠶⠶⠊⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠒⠒⠊⠙⠉⠀⠀⠀⠀⠀⠀⠀⣾⠋
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡟⠋⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⣼⠋
⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠇⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠻⣯⠀⠀⠀⠀⣠⡾⠁
⣿⠀⠀⠀⠀⢹⡇⠀⠀⠀⠘⠗⠀⠀⠀⠀⢸⠂⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⣿⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟
⢻⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠘⠷⠀⠀⢻⣤⡽⠿⠶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡞⠛⠛⠶⣤⡀
⢸⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣾⠁⠉⠉⠉⠻⣿⣟⣛⣉⣽⠟⠁⠀⢸⡇⠀⠀⠀⠈⠙⢧⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣉⣉⡉⠀⠀⠀⠀⠘⣿⢶⣄⠀⠀⠀⠀⢻⣆
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⠾⢻⡟⠀⠀⠀⠀⣰⣿⣿⣿⣽⣷⡄⠀⠀⠀⣿⠀⠙⣧⡀⠀⠀⠀⢿⡀
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⢸⡇⠀⠀⠀⠀⢻⡟⣿⠿⣿⣿⡇⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⠀⣸⠇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⢠⣤⣤⣤⠌⠁⠀⠀⠀⢸⣄⡴⠟⠀⠀⠀⢠⡟
⠀⠙⠷⢦⣤⣤⣤⣤⣴⠴⠶⠿⠿⠟⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠶⠶⠶⠶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⢘⣧⠀⣴⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⢿⣄⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠀⠀⠀⣿⠛⠛⠛⠶⢶⡀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠛⠛⠒⠂⠴⠿⣤⡀⠀⠀⣨⣷⣤⡴⠶⠒⠒⠛⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⠖⠲⣶⠶⠾⠏⠀⠀⠀⠘⠷⠶⢶⡶⠶⠲⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣤⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⣚⣋⣉⣉⣉⣉⡙⠓⠒⠶⢤⣄⡀
⠀⣀⣤⡴⠶⡏⠁⢹⠶⢦⡀⠀⠀⠀⠀⣠⠔⠚⠉⠁⠀⠀⠀⠻⣭⠄⠐⠒⠒⠦⣽⠟⠀⠀⠀⠈⠙⠳⢦⣄
⢰⠏⢹⠀⠀⡇⠀⢸⠀⠘⡇⠀⠀⠀⠘⠷⢦⣄⠀⠀⠀⣀⡤⠼⣿⡀⠀⠀⠀⠀⣿⠤⠄⣀⠀⠀⠀⠀⠀⣈⣻⡆
⢸⠀⠘⠀⠀⠃⠀⠈⠀⠀⣇⡴⠛⣩⡷⠀⠀⢹⣦⢖⣉⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣉⡳⢦⣠⡾⠋⠁
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⣏⠀⠀⢰⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢰⠋⠀⠀⣿⢠⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣭⣤⠾⠛⢡⡄⣿
⣿⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⢸⡆⠀⠀⠙⢿⠀⠀⠛⢂⠈⠉⢩⡉⠉⠉⠉⢉⡍⠉⠀⠲⠏⠀⠈⣷⠟
⢿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⠀⢀⣶⠵⠚⠛⠛⠻⣦⣤⡴⠟⠛⠓⠶⣵⣤⡀⠀⢻
⠈⠻⣦⣀⠀⠘⣷⣄⣀⣠⡤⠖⠋⠙⠓⠒⢶⡏⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⢸⠋⠉⠉⠓⠒⠶⠦⢤⣤⣀⡀
⠀⠀⠈⢿⡉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣷⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣸⣦⡶⠃⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠶⠶⠿⠿⠿⠶⠶⠒⠚⠉⠁⠀⢠⣿⣿⣿⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣦⠀⠀⠀⠀⢠⣴⣶⣦⡀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠀⠀⠀⠀⠘⢿⣿⠿⠁⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠋⠉⠛⠿⣿⠆⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡆⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠉
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀

⠀⠀⠀⣠⢞⣿⣿⣿⣭⣭⣭⣝⢦⡀
⠀⣠⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢦⡀
⡞⣵⡿⠿⣿⠿⠿⢿⣿⠿⢿⣿⠿⠿⣷⢹⡆
⡇⣿⠠⢶⣿⣶⢰⣾⠃⣶⡄⣿⠰⡆⢸⣾⡇
⡇⣿⣷⡄⢹⣿⢸⣿⡀⢿⠇⣾⠀⣤⣾⣿⡇⠀⠀⠀⢀⣠⠤⠖⠒⠒⠋⠉⠀⠒⠒⠒⠤⢄⡀
⣧⢿⣦⣤⣾⣿⣼⣿⣷⣤⣶⣿⣶⣿⣿⢻⡇⠀⡠⠖⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠓⢤⡀
⠈⠳⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡵⠋⣠⠊⠀⠀⠀⠀⠀⠠⠟⠛⠀⠀⠀⠀⠈⠛⠳⠀⠀⠀⠀⠙⢦
⠀⠀⠈⠳⣝⡻⢿⣿⣿⣿⣟⣛⡵⠋⠀⢰⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠔⠁⠀⠀⠑⡀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠈⠉⠉⢹⢹⠉⠉⠉⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢘⠀⠀⢠⡆⢸⠀⠀⣤⠀⠀⡇⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠘⢄⠀⠈⠁⣸⡀⠀⠛⠀⢠⠃⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢀⢸⠸⠤⡀⠀⠀⠀⠀⠘⣆⠀⢢⠀⠀⠀⠀⠀⠁⠀⠈⠀⠑⠄⠀⠔⠁⠀⠀⢀⠉⠀⢀⠎⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠅⢈⠁⠀⢸⠀⠀⠀⠀⠀⠈⢧⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠟⠉⢻⠁⠀⢹⣤⣀
⠀⠀⠀⠀⠀⠘⡆⠀⢻⢁⡸⠧⣤⣀⠀⠀⠀⠀⢳⡀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⣸⡄⠀⢸⠀⠀⡜⠀⠈
⠀⠀⠀⠀⠀⠀⠉⢸⢹⠁⠀⠀⠀⠉⠙⠛⠲⠶⠤⣷⠀⠀⠀⠀⠀⠀⠉⠰⢁⡉⠁⠀⠀⣶⠴⠾⠧⣇⠀⠘⠀⠀⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⠀⠧⢄⡠⠋⠘⡦⣀⠀⠹⢄⡀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⡇⠀⠀⠑⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⡀⠀⠧⣀⣀⣀⣤⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⣀⣀⡠⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠒⠉⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢩⠀⢀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⡀⡸⠀⠂⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠉⠁⠂⡀
⠀⠀⠀⣰⠋⢹⠀⢠⠞⠉⡆⠀⠀⠀⠀⠀⢸⣀⣀⠤⠤⠒⠒⢊⣀⡚⢥⣒⣄⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⣴⠢⡀⢹⠀⢸⠀⡿⠀⢰⠃⣠⠞⢳⡄⠀⠾⡿⠁⠊⢀⣠⠀⠀⢒⡂⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⢿⠀⢳⢸⠀⢸⣧⡃⠀⡏⣼⠃⢠⡞⠀⠀⠲⢿⡄⠀⢶⢈⠇⠀⠘⠁⠀⠠⣷⠶⣄⠀⠀⠀⠀⢸
⠸⣧⠀⠛⠉⠉⠉⠉⠉⠛⠃⡴⠋⠀⠀⠀⡴⣾⡇⠀⠀⠧⠤⠄⣂⠀⠀⠀⢻⡇⡁⠀⠀⠀⠀⠼
⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠈⢧⡘⣿⠀⠀⢀⣀⣤⣤⣶⡄⠀⢸⡏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣿⡀⠀⠀⢀⠴⠒⠚⠛⠒⠚⠛⢶⡄⠀⠀⠉⢹⠀⠀⢸⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠀⠃⠀⠀⠀⢸⡀⠀⠈⠛⠛⣉⠥⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊
⠀⠀⢻⣇⡀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣤⣄⣀⣤⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡖⡤⠤⠀⣀⡠⠚⠁
⠀⠀⠀⠛⠿⠷⣤⠬⠘⢧⡀⠀⠀⠀⠀⣀⣀⢀⣤⠤⠶⣦⡿⠀⠀⢸⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⠒⠛⡛⠉⠉⢻⡇⠀⠀⠈⠳⡟⣱⠞⠁⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⣇⠀⠀⢸⡇⠀⠀⠀⠠⢼⠇⠀⠀⠀⠀⢸⡏⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⢺⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣸⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡿⢀⠠⢾⠀⠀⠀⠀⠀⣠⠴⠲⠤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠹⡄⠀⠦⠤⣤⠔⠞⠁⢠⡛⠒⠊⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠇⠀⠀⠙⠦⢤⣤⣼⡀⠀⠠⢾⣉⣉⣉⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⠀⠀⢀⣼⠃⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠤⣼⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠙⠛⠉⠈⠓⠚⠛⠋⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢀⡴⠚⠓⠶⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⡄⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠠⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⡤⠖⠲⢦⣯⣄⣀⣀⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣠⠤⠶⢾⠷⠒⠶⣄
⠀⠀⠀⠀⠀⠀⣜⠤⢄⣀⣀⣁⠀⢈⣉⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢂⣤⣀⠀⣀⣀⡠⠼⡂
⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣬⣿⣶⣶⣾⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣧⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣶⣾⣿⣿⣿⣿⣷⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣤⣤⣤⣄⣀⣄⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣠⣄⣀⣠⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠲⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⣀⠠⠤⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢳⠀⢇⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠃⠀⠉⠉⠓⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠘⣇⠀⠿⡄⣰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠾⢦⣀⠠⠟⠀⠸⠄⠠⡤⠀⠀⢧⣀⣀⠊⠀⠀⠹⡆⠀⠹⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠰⣤⡀⠀⣠⡔⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠶⢄⡀⠀⠀⠀⡴⠤⣄⠀⣠⠀⠀⠀⠀⠀⡄⢠⡀⠀⠀⣤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠿⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠙⣦⠀⡆⠀⠀⠈⡇⠘⡄⠀⠀⠀⢠⠁⢸⢳⡄⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⣿⠀⠀⠀⡇⠀⢣⠀⢰⣆⠸⠀⣼⠀⠻⣄⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣤⣧⠠⠖⠁⠀⠻⣤⣤⠞⠀⠀⠘⣦⠁⢹⡇⠀⠉⠀⠀⠹⠟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⢤⣤⠤⠤⠄⠒⠉⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⣄⠀⠀⡀⢀⡄⢀⡀⣀⡠⢀⢠⣄⣀⢀⣄⣤⠀⣀⣤⣤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⠾⡌⢳⠼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⢵⡀⡿⢸⡀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⠠⡴⣾⠁⠈⢷⡀⣀⠄⠀⠀⠀⣼⠋⠻⣇⡸⠷⣦⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡄⢠⠙⠀⡇⠹⡀⢀⣴⠈⠟⠀⠀⠀⠘⠉⠀⠀⠛⢿⣦⠿⡀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⡀⠀⠈⠶⠰⡄
⠁⠈⣼⣀⡷⠶⠶⠂⢸⠀⠀⠀⠠⠁⢴⢂⠀⣀⠆⠀⢸⡆⠹⣄⠀⠀⠙⠁⠉⠈⠁⠉⠉⠉⠉⡇⠀⠀⠸⠋⢻⣉⣋⡀⠀⠁
⠀⠀⠈⠉⡓⠶⠖⠊⠁⣀⠀⠂⠀⠀⠀⡏⠈⠁⡇⠀⠀⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠉⠁
⠀⠀⠀⠀⢇⡀⢀⣥⠘⠉⠀⠀⠀⠀⢠⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠰⠶⠀⠀⠀⢠⠠⠆⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢸⠀⢰⠀⢸⠖⠂⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠸⠀⢸⠀⢼⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠁⠀⠁⠀⢸⠀⠘⠒⠒⠂⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⢃⡀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⢸⠘⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠶⠦⠠⠾⠀⠲⠶⠶⠶⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠠⣖⣀⣲⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣄⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣄⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⣤⢠⢠⣀⡀⡀⡀⡀⢤⠀⠀⣠⡄⠀⣠⡄⠀⣠⣤⠀⢠⣤⠀⢀⣤⠄⠀⣤⡄⠀⣠
⠀⠀⠀⣺⣍⣉⣋⣉⣉⣉⣙⣿⠀⠀⡆⣵⣿⡿⣫⣻⣿⣿⠪⠀⠀⠃⣠⡾⠋⣠⣾⠋⢁⣴⠟⢁⣴⡿⠃⣠⡿⠋⣠⡾⠃
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡿⠋⣠⠀⢠⣤⡤⡤⣤⣤⣤⣤⢤⢤⣤⣄⠀⠋⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣤⣾⠟⠀⢈⡉⣁⣁⣉⣉⣉⡉⢉⣈⡉⡉⠀⣾⠟⠁
⠀⠀⠀⣠⠄⣠⢀⡤⢀⡄⣠⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣬⠀⠀⠏⢁⣴⠀⢉⣉⠉⠉⣉⡉⠉⣉⡈⠉⣉⣁⠀⢁⣴⠟
⠀⠀⠀⣠⢞⡵⢋⡴⣫⠞⡱⢚⠀⠀⣿⣇⣧⣄⣷⣦⣤⣼⡿⠀⠀⣶⠟⢁⣴⠟⠋⣤⡾⠋⣠⡾⠋⢠⣾⠟⢁⣴⠟⢁⡤
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣶⠖⢀⣶⠖⢀⣶⡶⠂⣰⡶⠂⣰⣶⠖⢀⣶⠖⢀⣶⠀⠀⣴⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠁⠴⠟⠁⠴⠿⠋⠠⠾⠋⠠⠾⠟⠁⠶⠟⠁⠴⠟⢩⠀⠀⣿⣿⡟⠛⠛⠛⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿
⠀⠀⠀⠀⠀⢀⣤⢤⡤⣤⢤⡤⣤⣠⢤⡤⣄⡤⡄⣄⡄⠀⢸⠀⠀⣿⣿⡇⢀⣷⣾⣾⢅⣧⢸⣿⣷⣾⣷⡇⣿⠂⢸⣿⣿
⠀⠀⠀⠀⠀⠸⠿⠸⠄⠿⠸⡧⠿⠿⠸⠇⠟⢧⠿⠟⠇⠀⢸⠀⠀⣿⣟⡇⢀⠤⣤⣤⢤⢠⢤⣤⣤⢤⢤⣤⣤⡀⢸⣿⢿
⠀⠀⠀⠀⣠⡤⠀⣠⣤⠀⢀⣤⠄⢀⣤⡄⠀⣠⡤⠀⣠⣤⠈⠀⠀⣿⣿⣇⣀⣁⣉⣈⣈⣀⣈⣈⣉⣉⣀⣉⣈⣀⣸⣿⣿
⠀⠀⠀⣿⠋⣠⣾⠟⢁⣴⡟⢁⣴⡿⢋⣤⣾⠋⣠⣾⠛⣡⣾⠀⠀⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿

⠀⠀⠀⡿⡿⡿⡿⡿⡿⡿⡿⡿⡿⡿⡿⡿⡻⡻⡿⡿⡿⡻⣻⠀⠀⠟⣡⠟⣡⡾⢋⡼⢋⣴⠀⠀⠆⣠⠖⣠⡶⢀⡴⢂⡤
⠀⠀⠀⠀⠀⠀⣴⣶⢲⠒⣶⢴⡒⢲⣶⣆⣦⣶⣆⡆⠀⠀⠀⠀⠀⠛⠁⠚⠋⠐⠋⠐⠋⠰⠀⠀⠟⠁⠚⠋⠐⠋⠐⠋
⠀⠀⠀⣴⣴⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⡤⣰⠀⠀⠄⣠⣀⡄⣀⣀⣄⡄⠀⠀⠀⡀⢀⣀⡀⣀⣀⣀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠃⢛⠛⡃⢛⢛⠃⡃⠀⠀⠀⠀⢘⡛⠃⣙⢛⢛⠃
⠀⠀⠀⠖⡰⢆⡶⣢⢖⡰⢂⡶⣢⢖⡴⢂⡶⣠⢖⡴⢂⡶⣢⠀⠀⡅⢯⠽⠿⣿⢿⡍⡧⠀⠀⠀⠄⠸⠸⠿⠹⢿⠸⠇
⠀⠀⠀⠉⠀⠉⠈⣥⣬⡀⣭⣬⢥⢨⣤⣍⣬⢥⣌⠀⠉⠈⢙⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⢄⡤⢀⡍⣩⢅⡭⢉⡄⣨⢌⡭⢉⡍⣡⢀⡤⢀⡴⠀⠀⣥⠞⣡⡾⢃⡴⢋⡴⢛⠀⠀⣤⠞⣡⡾⢃⡴⢋⡴⠓
⠀⠀⠀⠑⠛⠐⠋⠚⠑⠛⠐⠋⠚⠑⠛⠐⠋⠚⠑⠛⠐⠋⠚⠀⠀⠃⠚⠋⠐⠛⠐⠛⠐⠛⠀⠀⠃⠘⠋⠀⠋⠀⠋⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠉⠉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⢠⣶⠶⣶⡄⢰⣶⠀⠀⠀⣴⡶⢶⣦⠀⣶⡆⢠⣶⡆⢰⣶⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⢸⣿⡀⠙⠃⢸⣿⠀⠀⠀⣿⡇⠀⣿⡇⣿⡇⢸⣿⡇⢸⡏⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⢸⣿⠀⠀⠀⣿⡇⠀⣿⡇⢸⣷⢸⣿⡇⣾⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⢠⣤⠈⣿⡇⢸⣿⠀⠀⠀⣿⡇⠀⣿⡇⢸⣿⣿⠛⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠘⠿⡶⠿⠃⠸⠿⠶⠶⠆⠻⠷⠾⠟⠀⠀⠿⠿⠀⠿⠿⠀⠀⠀⠀⠀⢀⣤⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⣀⣀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠇⠀⣀⣴⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡀⠀⠀⠈⠙⠛⠛⠋⢁⣤⣾⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⢀⣤⣾⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠉⠛⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣄⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠉⠻⡄⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣦⡀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠈⠛⠛⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⡶⠒⢒⡲⠞⠋⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡗⢸⠇⣠⠊⠀⠀⠀⠀⠀⠁⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇⢸⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⡋⠉⠉⠉⠒⠂⠀⠀⠀⠀⠀⢀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣀⡉⠉⢉⣻⠟⠛⠛⢲⠟⠛⠛⣯⣟⡛⠓⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢙⣶⣯⡀⠀⠀⢀⡞⠀⠀⣾⣡⠞⠙⠒⠋
⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠤⠤⠤⠤⠤⠤⣤⣤⣤⣼⣧⣀⣻⣧⡟⠈⠛⠃⠀⣠⠞⠀⢀⣾⠟⠁
⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠁⠉⠉⢳⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⠘⡦⠴⠖⠒⠺⠅⠀⢀⣿⠿⠒⠒⢤
⠀⠀⢠⠞⢡⣤⡀⣤⡀⠀⠀⣀⡀⠀⡀⠀⠀⠹⣄⠀⠀⠈⢧⣀⠀⠀⡄⢀⠄⠀⠀⢀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠶⠛⢃⡼
⠀⠀⣿⠀⣼⡏⢹⣿⡇⠀⣼⣿⣧⣿⡇⣶⢠⡇⠘⡆⠀⠀⠀⠈⠙⠒⠲⢲⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⢰⠶⠶⠖⠋
⠀⠀⡇⠀⢹⣧⢸⣿⠀⠀⣿⠁⣿⢸⣿⣿⡾⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠈⠀⢀⡏
⠀⢀⡇⠀⠀⣿⣿⣿⠀⠀⣿⣤⡟⢸⣿⣿⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣼
⠀⠈⢣⡈⢷⡿⢻⣿⣤⡄⣿⣿⠃⢸⡟⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⢰⡇
⠀⠀⠀⠱⡄⠁⠈⠉⠉⠁⠈⠁⠀⠘⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠙⠦⠤⠤⠤⣤⣠⣀⣀⣀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣸⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣲
⠀⠀⠀⠀⠀⠀⠐⣏⡀⢨⠟⢩⢉⡗⠀⠀⢀⣤⣄⣀⣀⣀⠖⠙⢋⡟⠳⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠉⢉⠕⡶
⠀⠀⠀⠀⠀⠀⠸⣏⡉⢻⣤⠊⠼⣀⣀⣀⠾⡷⠀⠀⣨⠋⠀⢀⡞⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⢛⡹⠛⠁⠀⠀⣀⠔⣧⡜⠁
⠀⠀⠀⠀⢶⡋⠉⢉⣹⢹⡇⢀⡖⠒⠒⠲⠖⡇⣀⡴⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⢀⡜⠁⠀⠸⡀
⠀⠀⠀⠀⠀⠉⠉⠉⡟⢸⠉⠉⠀⠀⠀⠀⠛⢋⡟⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⣠⠋⡆⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠉⣧⣀⣤⡶⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠸⠤⣤⣀⣼⣅⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢣⡠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠉⠓⠲⢶⠤⣤⠤⢴⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⢻⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢸⠀⣿⠀⠀⠉⠛⠒⠒⠒⠒⠚⠛⠋⠉⠀⠀⠀⢠⠎⡇⠱⢤⣠⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⣸⠁⠀⠀⠙⢀⡞⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠋⠠⠬⠉⢿⡶⠷⠶⢖⣂⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠦⣀⠀⠤⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣉⣹⣟⠶⠖⠒⠒⠓⠶⠤⠤⠤⢤⣤⣤⣤⣤⡤⣤⠴⢾⠦⣀⠀⠀⠀⠈⠙⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧⠤⠬⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣄⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⢰⣶⡄⠀⠀⠀⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡈⠁⠀⠀⢸⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⠈⣿⣷⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡆⠀⠀⢸⣿⠇⠀⣀⠀⠀⣿⣿⣍⣿⣿⠀⠀⠀⢻⣿⣴⣷⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣠⣿⡟⠀⠀⢸⣿⣷⣶⣿⡇⠀⠹⣿⣿⣿⡟⠀⠀⠀⠘⣿⣿⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠙⠋⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⢰⣦⠀⠀⠀⠀⣴⣦⠀⠀⠀⣤⣄⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⢠⣿⣿⣿⣷⠀⠀⢸⣿⡇⠀⠀⢀⣿⡟⠀⠀⠀⣿⣿⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣏⣹⣿⡆⠀⠈⣿⣿⣰⣦⣸⣿⠇⠀⠀⠀⣿⣿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⢻⣿⣿⢿⣿⡿⠀⠀⠀⠀⣿⣿⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠈⠻⠋⠘⠛⠃⠀⠀⠀⠀⠘⠊⠀⠙⠛






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⢀⢀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠏⢨⠷⠹⠹⠸⠾⡥⠀⢍⡶⢹⠸⡹⢸⡹⠀⠶⢾⡅⢏⠗⠏⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠿⠿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠛⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⢀⣼⡿⣷⡄⠀⠀⠀⢀⣴⠟⠁⠀⠀⡄⣶⣀⠀⠙⢷⣄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢀⣴⡿⠋⠀⠘⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇⡇⣿⣿⠀⠀⠀⣿⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣴⡿⠋⠀⠀⠀⢀⣿⠃⠀⠀⢸⡇⠀⠀⠀⣷⣿⣿⣿⣴⠆⠀⣿⠀⠀⠀⢿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣆⣴⡿⠋⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠘⣷⡀⠀⠀⢿⣿⣿⣿⠋⠀⣠⡿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣄⣀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⣉⣉⡀⣠⡾⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⢀⡄⠀⢙⣷⣄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⡿⠹⢸⢸⢸⣿⣿⣿⣧⠀⠀⠀⢰⡏⠀⠀⢰⣾⣿⣿⡇⠋⠀⠹⣆
⠀⠀⠀⠀⠀⢸⣿⣧⡒⢿⡆⣞⢰⡎⡇⣒⣽⣿⠀⠀⠀⢸⣿⣿⣿⣧⠘⠘⠘⠸⠟⣻⣿⣿⠀⠀⠀⢾⠀⠀⠀⢸⣿⣿⣯⣇⣤⠄⠀⣿
⠀⠀⠀⠀⠀⢸⣿⣷⣶⣿⣷⣿⣶⣾⣷⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⢸⡆⠀⢀⣸⣿⣿⣿⣿⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠘⢿⣿⣿⣦⣤⣤⣾⣿⣿⡿⠁⠀⠀⠀⠀⠹⣶⡛⠀⠙⠛⠛⠁⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⡾⣷⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠏⠁⠈⠻⣷⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠟⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣤⣤⡀⠀⢀⣤⣄⠀⣤⡀⢠⡄⢠⣤⣤⠀⣤⡄⠀⣠⣤⣀⠀⣠⡄⢠⣤⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⣿⡏⠹⠇⠀⣿⠉⣿⡇⣿⠀⢸⡇⠀⣿⡏⠀⣿⡇⢸⣿⠉⣿⠀⣿⣿⣼⣿⠀⠀⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⣿⣇⣰⡆⠀⣿⠟⣿⡇⣿⣄⣼⡇⠀⣿⡇⠀⣿⡇⢸⣿⣀⣿⠀⣿⡟⣿⣿⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠛⠛⠁⠀⠛⠀⠙⠃⠈⠛⠋⠀⠀⠙⠃⠀⠛⠃⠀⠙⠛⠁⠀⠙⠃⠘⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠐⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠂⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⣦⡀⢀⣴⡿⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⢿⡿⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣠⡾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⢠⣀⡀⠀⢀⣀⡀⠀⠀⣠⣄⢤⣤⣄⣤⡄⠀⠀⣤⠀⠄⣀⣠⠀⢀⣄⡀⠀⢠⠀⠀⠀⡀⣤⢠⡄⢤⡠⣤⠠⡄⢤⡀⣄
⠀⠀⠀⡾⠟⢁⣴⣾⠟⠁⠀⠀⡯⠲⡺⣯⣾⣗⡇⠀⠀⠉⣠⣾⠟⢁⣴⡾⠋⣠⣾⠟⢉⠀⠀⡳⣌⠳⣝⢦⡙⢮⡓⣌⠢⡝⢦
⠀⠀⠀⢃⣰⣾⠟⠁⣠⡞⠀⠀⡯⡪⡪⡺⡻⡧⡇⠀⠀⣾⠟⢁⣴⡿⠋⣠⣾⠟⠁⣠⡞⠀⠀⡙⢮⡳⣌⠳⣝⢦⡙⢮⠳⣌⠲
⠀⠀⠀⠹⠟⠁⣠⣾⠟⠡⠀⠀⣯⣮⡺⠪⡪⡫⡇⠀⠀⢀⣴⡿⠋⣠⣾⠟⠁⣠⡿⠋⢐⠀⠀⣝⢦⡙⢮⡳⣌⠳⡝⢦⡙⢮⡛
⠀⠀⠀⠂⣠⣾⠟⠁⣠⡆⠀⠀⣿⠯⢯⣪⡪⡪⡇⠀⠀⠿⠋⣠⣴⠟⠁⣠⡾⠋⢀⣴⡇⠀⠀⣎⠳⣝⠦⡙⢮⡳⣌⠳⣝⢦⣉
⠀⠀⠀⣼⠟⠁⣠⣾⠟⢁⠀⠀⣗⢟⣞⢯⣪⡪⡇⠀⠀⣠⣾⠟⠁⣠⡾⠋⢀⣴⡟⠁⣠⠀⠀⢦⡑⢌⠳⣝⢦⡙⢮⡳⣌⠳⣍
⠀⠀⠀⠄⣠⣾⠟⠁⣠⡆⠀⠀⣷⢵⢕⢝⣿⣮⡇⠀⠀⠛⠁⠀⠚⠋⠀⠀⠛⠃⠀⠘⠃⠀⠀⠓⠙⠂⠛⠊⠓⠘⠂⠛⠊⠓⠈
⠀⠀⠀⣼⠟⠁⣠⣾⠟⠁⠀⠀⣟⢕⢕⣵⢕⣟⡧⠀⠀⠀⢀⠠⡀⢀⡀⢀⠀⣀⠀⡀⠀⠀⠀⣀⣄⣄⣄⣀⣄⢀⡄⢠⣄⣠⡀
⠀⠀⠀⠅⣠⣾⠟⠁⣠⡆⠀⠀⣷⣵⣕⢝⠔⢕⡇⠀⠀⠖⣫⠞⡡⢋⡵⣫⠞⣡⢞⡵⠂⠀⠀⣷⡪⣉⣸⡮⣮⡻⣾⣱⣟⢷⣿
⠀⠀⠀⣻⠟⠁⣠⣾⠟⠁⠀⠀⣫⣷⣵⣷⣕⢕⡇⠀⠀⠚⡥⢚⡵⣫⠞⣡⢞⡵⢋⡴⢋⠀⠀⣏⡭⣮⣿⣮⡮⡪⣺⡲⣿⣻⣿
⠀⠀⠀⠀⣠⣾⠟⠁⣠⡆⠀⠀⡯⡿⡫⣫⠕⣕⠇⠀⠀⢋⡵⢫⠞⣡⢞⡵⢋⡴⣫⠞⡱⠀⠀⣷⣟⢷⣿⣺⣫⣮⡮⡺⣺⣿⣿
⠀⠀⠀⣼⠟⠁⣠⣾⠟⠁⠀⠀⡯⡪⡪⠢⡎⣴⡇⠀⠀⣉⠞⣡⢞⡵⢋⡴⣡⠞⣡⢎⡰⠀⠀⢳⣝⢷⣝⢿⣻⣮⡫⣢⣮⣮⣾
⠀⠀⠀⠂⣠⣾⠟⠁⣠⡆⠀⠀⡷⡮⡪⡪⡢⡢⡇⠀⠀⣡⢞⡵⢋⡴⣫⠞⡡⢚⡵⣋⡔⠀⠀⣷⣽⢷⣝⢷⡝⢾⢗⣊⡻⣼⣻
⠀⠀⠀⠘⠛⠁⠀⠘⠛⠃⠀⠀⠛⠞⠞⠞⠊⠛⠃⠀⠀⠀⠋⠐⠋⠘⠁⠚⠀⠋⠘⠃⠀⠀⠀⠛⠟⠿⠓⠳⠛⠳⠛⠻⠻⠛⠛
⠀⠀⠀⣀⡀⠀⣀⡀⠀⣀⣀⠀⣀⣀⠀⢀⣀⠀⢀⣀⠀⢀⣀⠀⢀⣀⠀⢀⣀⠀⢀⣀⡀⠀⠀⣀⠀⣀⡀⢀⣀⠀⣀⡀⣀⡀
⠀⠀⠀⠏⣠⡾⠋⣠⣾⠟⢁⣼⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢋⣬⠀⠀⣡⡾⢋⣴⠟⣡⠾⢋⡴⠋⣠⠿
⠀⠀⠀⣾⠋⣡⡼⠛⢁⣶⠛⢁⣴⠏⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠿⢋⣴⡾⠫⠀⠀⣯⡀⢻⣅⠺⣯⡐⢿⣄⠺⣇⠐
⠀⠀⠀⣠⣾⠟⢀⣼⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢃⣴⡿⢋⣴⡾⠋⣤⣿⠀⠀⣦⠙⣦⡙⢳⣌⠻⣦⡙⢷⡌⠛
⠀⠀⠀⠛⢀⣾⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠿⢋⣤⡿⠋⣴⡿⠋⣀
⠀⠀⠀⣾⠟⢁⣰⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⡟⢉⣴⡿⢋⣴⡾⠋⣠⡾⢣⠀⠀⡞⢿⣿⣿⣫⡪⡳⡻⣶⣟⢝⢶
⠀⠀⠀⣁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢃⣴⡿⢋⣴⡾⠋⣠⡾⠋⣠⣾⠀⠀⡕⡕⣝⣿⢻⢪⢪⢪⢪⢩⣷⣕
⠀⠀⠀⡛⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⠟⢁⣴⡿⢁⣴⡿⠋⣠⡾⠋⣠⡾⠋⣠⡾⠋⣀⠀⠀⡫⣮⢗⢕⢕⢕⢕⢐⡷⡯⡪⡺
⠀⠀⠀⠐⠛⠁⠐⠛⠃⠐⠛⠃⠀⠛⠃⠐⠛⠋⠐⠛⠋⠀⠛⠛⠀⠚⠛⠀⠚⠛⠀⠚⠋⠀⠀⠛⠓⠛⠓⠛⠛⠋⠋⠊⠚⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⠾⣿⠙⠻⠇⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠈⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣀⠤⠒⠊⠉⠁⠀⠀⠉⠉⠐⠢⢄⡀⠀⣰⠿⠐⠂⠈⠉⠉⠁⠐⠒⠬⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠛⢦⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⣠⡤⠞⠋⠙⠳⣄⡀⠀⠀⠀⠀⠀⢠⣤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀⡈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⠀⠀⠙⠳⡄⠀⠀⠀⠀⠀⢀⠖⠉⣁⡀⠈⠙⠲⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣴⣿⣿⣄⠀⠀⢹⠀⠀⠀⠀⢠⡏⢀⣾⣿⣧⣀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣿⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠻⣿⣿⠿⠀⠀⢸⠃⠀⠀⠀⠸⡄⠈⠿⣿⡿⠏⠀⠀⢠⠇⠀⠀⠀⠀⠀⣿⠀⡷
⠀⠀⠀⠀⢀⡶⠛⠉⠳⣼⡶⠖⠒⢦⡀⠀⠀⠘⠒⠤⢤⣀⣀⡀⠀⣀⠚⠀⠀⠀⠀⠀⠳⣄⠀⣀⣀⣀⡀⠠⠎⠀⠀⠀⠀⠀⠀⣋⡼⠃
⣠⠞⠋⠉⢿⠁⠀⠀⠀⣿⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⡿⠀⠀⠀⢸⡀⠀⠀⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢻⠀⠀⠀⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠆⠀⠀⠀⣰⠏
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠁⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠞⠋⠉⠁⠙⠃⠀⢀⡼⠋
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡄⠀⠀⠀⢸⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⣿⠀⠀⠀⠘⠃⠀⠀⠀⠈⠀⠀⠀⠀⢸⡀⠀⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋
⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠸⡇⠀⠸⡏⠙⢳⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠾⠿⣥⣀
⢸⠀⠀⠈⠉⠉⠉⠁⠀⠀⣼⠋⠁⠀⠀⠀⠀⠀⣿⠛⠉⠁⢀⡏⠉⠉⠛⠛⢛⡿⣟⠉⠉⠁⣷⠀⠀⠀⠀⠙⠳⣄
⢸⡄⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⡟⠢⡀⢀⠴⢿⣿⣯⠓⢄⡸⢹⠶⠶⣄⠀⠀⠀⠘⣧
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠶⠛⠉⡇⠀⠘⠁⠀⢸⣿⣿⠀⠀⠀⠸⡇⠀⠈⢷⡀⠀⠀⠸⡇
⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢐⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠙⠂⠀⠀⠀⣇⠀⠀⣘⠃⠀⠀⢰⡇
⠀⢿⡀⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⣀⣠⣤⡴⠟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⢸⡴⠞⠁⠀⠀⣠⡟
⠀⠀⠉⠙⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠘⡇⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠿⣿⣿⠿⠁⠀⠀⠀⣿⡀⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡶⠶⠶⠶⠶⣄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢠⡇⠀⠀⠀⠀⠹⣆⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣫⣄⣀⣠⣼⣧⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣄⣌⣷⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⣀⠶⠦⠲⠶⠔⠶⠆⠲⠦⠐⢶⠀⠀⣔⢶⡦⠲⣦⡴⣶⣤⠶⣦⡶⣦⠀⠀⣴⣶⣤⣶⣦⣶⣶⣴⣶⣤⣶⣦
⠀⠀⠀⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡻⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣮⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠀⠀⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⢷⡟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛
⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⣶⡽⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⣬⡀⣀⣀⢀⣀⡀⣀⡀⢀⣀⢻⠀⠀⣮⡻⣧⣟⣿⣟⣻⣿⣛⣷⣟⢿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠈⠉⠀⠉⠁⠉⠉⠈⠉⠀⠉⠁⠀⠀⠉⠉⠈⠉⠀⠉⠁⠈⠉⠈⠉⠁⠀⠀⠉⠁⠈⠉⠀⠉⠁⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⡠⢔⣢⣭⣷⣶⣶⣮⣬⣔⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⡠⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣕⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢌⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢢⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⡌⣾⣿⠿⢿⣿⠿⠿⠿⢿⣿⠿⠿⣿⣿⠿⠿⢿⣇⠆⠀⠀⠀⢸⣿⡿⠿⢿⡿⠿⠿⠿⣿⡿⠿⢿⣿⡿⠿⠿⣿⡇
⠀⠀⢠⣿⣇⠻⠷⣿⣿⡇⣿⣿⢱⣿⣷⡜⡇⠾⠿⢆⣿⢰⠀⠀⠀⢸⣿⡘⠿⢶⣿⣿⢸⣿⡏⣾⣿⣦⢹⡇⠿⠷⣸⡇
⠀⠀⠈⣿⣝⠿⠗⣸⣿⡇⣿⣿⣜⠿⠟⣱⡇⣺⣿⣿⣿⠘⠀⠀⠀⢸⣯⡻⠿⢂⣿⣿⢸⣿⣧⡻⠿⢋⣼⣇⣿⣿⣿⡇
⠀⠀⠡⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠡⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠊⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠪⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠊⠭⠟⠛⠛⠫⠩⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠂⠀⣎⢀
⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠀⠂⠲⠄⠘⠇⠀⡄
⠀⠐⠀⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠓⠀⠄⡲⠖⠀⠀⠠⠀⠀⠂⠀⡀
⠀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠅⡆⠀⠀⠀⡠⠨⡢⠀⠀⢀⠀⠉⠄
⠀⠀⠀⠀⠄⠀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⡙⢀⡑⠀⠋⠀⠀⠠⠉⠁⠀⠀⣷⠀⡀
⠀⢀⠴⠁⠠⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠉⠄⠁⣰⠃⠀⠂⠔⠦⡀⠌⠀⠉⠈⢀
⠀⠀⠀⠀⠠⠈⠀⠀⢀⠀⠀⠀⠈⠣⠄⡀⠀⠀⠀⠠⠄⠀⠀⠀⡠⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⡀⢠⠀⠀⠀⠀⠣⠤⢔⠬⢒⡞⠂⡃⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠑⠜⠤⠀⠄⢠⣁⠀⡆⠀⠀⠀⠀⢄⠋⠁⠀⠈⠑⣤⣤⣇⣀⡀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠡⠔⠐⠀⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠷⠛⠡⡿⠋⣩⣅⠚⢡⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠀⠙⢉⡴⠞⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⢀⠰⠀⠀⢐⠀⠀⢀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠃⠀⠀⠀⠀⠈⡄⢨⠤
⠀⠀⠀⠠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢤⣄⠀⡀⠀⠀⠀⠀⢁⡠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣀⠄⢀⠆
⠀⠀⠀⠀⠀⠨⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠾⠋⠀⠐⠀⠀⠀⠀⠀⢁⠐⠀⠀⠀⠐⠒⠀⡐
⠂⢐⠁⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠠⠂
⠀⠀⠀⠀⠀⠂⠀⠁⠈⠁⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢣⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣱⡯⢷⡹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⡰⣵⣶⣶⡽⣘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⣾⡇⣿⣿⣿⣟⡟⡜⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢏⣾⡟⣏⣿⣿⣿⡇⣿⠿⠎⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢿⠛⣀⣿⢸⣿⣿⢿⡇⠀⠈⢌⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣦⣤⣾⣿⢻⢸⣿⣿⢸⣷⣤⣦⣾⠏⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡳⢻⣿⣿⣿⣿⣧⢸⣿⣿⢸⣻⣿⣿⣿⣾⣧⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣱⣿⣷⣿⣿⠟⣿⣿⠸⣿⡿⢸⣿⣯⣿⡟⢿⣿⢷⡹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⣼⣿⣻⣾⣿⡇⠀⢫⣿⢣⣮⣔⢟⣿⠙⣟⣶⣾⣿⣏⢻⣜⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⣼⣿⣿⣿⣿⣿⣧⣤⣾⠧⡸⣿⡿⢳⣿⢰⣿⣿⣿⣿⣿⣿⣿⡜⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⣬⣭⣭⣭⣿⣿⣭⣭⣿⣭⣥⣽⣷⣮⣿⣯⣤⣭⣭⣭⣭⣭⣭⣵⣿⣬⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⣿⡄⣼⠀⡾⣇⠀⣿⠛⣳⠄⣿⣆⢸⠀⡇⢸⣷⡀⡇⢰⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⣷⠇⣸⠷⢿⡄⣿⠉⢿⠀⡇⠹⣾⠀⡇⢸⠈⢳⡇⢹⣄⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⢄⣾⣿⣿⡟⠁⠀⣰⣿⣿⣿⠃⠈⢀⣾⣿⣿⡟⠀⠀⣰⣿⣿⣿⠃⠠⢀⣾⣿⣿⡟⠀⠀⣰⣿⣿⣿⠃⠀⢀⣾⣿⣿⠟⠀⠀⣸⣿⣿⡿⠁
⢀⣼⣿⣿⡟⠂⠀⣰⣿⣿⡿⠇⠀⢀⣾⣿⣿⡟⠀⠀⣰⣿⣿⣿⠃⠀⢀⣾⣿⣿⠟⠀⠀⣼⣿⣿⡿⠁⠠⢠⣾⣿⣿⠏⠀⠀⣼⣿⣿⡿⠁⠀⢠
⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠟⠄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋
⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠄⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⡟⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠂⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⣱⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⣿⡿⠃⠀⢀⣤⣤⣤⠄⠀⠀⣤⣤⣤⡤⡀⠀⢠⣤⣤⣤⠄⠀⠀⣤⣤⣤⡤⠀⠀⢠⣤⣤⣤⠄⠀⠀⣤⣤⣤⡤⠀⠀⣠⣤⣤⣤⠀⠀⢀⣾⣿⣿
⡿⠁⠀⢠⣿⣿⣿⠏⠀⠀⣼⣿⣿⡿⠁⠀⢠⣿⣿⣿⠏⠀⠈⣼⣿⣿⡿⠡⠀⢠⣿⣿⣿⠏⠀⠀⣾⣿⣿⡟⠁⠀⣰⣿⣿⣿⠃⡀⢀⣾⣿⣿⡟

⡀⢀⣀⣠⡀⣀⣀⣀⣀⣀⢀⣀⣀⣄⣠⡀⣀⣀⣀⣀⣄⢀⣀⣀⣀⣠⡀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣠⡀⣀⢀⣀⣠⣄⢀⡀⣀⣀⣠⡀⣀⣀⣀⣠⡄
⣿⣿⣿⢏⡴⣩⣿⣿⡟⣡⢟⣿⣿⣿⢋⡼⣻⣿⣿⠟⢡⢟⣽⣿⡿⢋⡲⣻⣿⣿⡟⢱⢟⣿⣿⡿⢫⡜⣿⣿⣿⠟⣕⢾⣿⣿⡿⠋⡞⣵⣿⣿⣟⣥⢀
⠛⠛⠁⠀⠈⠛⠛⠋⠀⠉⠙⠛⠋⠁⠉⠉⠛⠛⠋⠀⠉⠙⠛⠋⠁⠉⠙⠛⠛⠉⠈⠁⠛⠛⠋⠀⠉⠘⠛⠛⠁⠈⠁⠛⠛⠋⠀⠉⠙⠛⠛⠉⠉⠁⠛

⣤⣤⢀⣤⣤⣀⣠⣤⠀⣠⣤⡄⣠⣤⡄⣠⣤⣄⢀⣤⡄⢀⣤⣤⣀⣤⣤⢀⣠⣤⢀⣠⣤⢄⣠⣤⡄⣠⣤⡄⣀⣤⣄⢀⣤⣤⢀⣤⣤⠀⣠⣤⠀⣠⣤
⣟⠡⣾⣟⠁⣾⣿⡁⣾⣿⡋⣾⣿⡋⢴⣿⣏⢰⣿⣏⠰⣿⣟⠡⣿⣟⠑⣾⣿⡁⣾⣿⡁⢾⣿⡋⢰⣿⡏⢰⣿⣏⢰⣿⣟⢡⣾⣟⠁⣾⣿⡡⣾⣿⡳
⠛⠗⠚⠛⠗⠈⠛⠛⠊⠛⠛⠚⠙⠟⠒⠙⠻⠂⠙⠛⠗⠛⠛⠗⠊⠛⠓⠚⠛⠓⠊⠛⠻⠒⠛⠛⠂⠙⠻⠒⠛⠛⠗⠙⠛⠓⠙⠛⠓⠈⠛⠓⠚⠛⠛

⣤⣀⣶⣶⣦⣄⣄⣰⣶⣶⣤⣀⣀⣶⣶⣦⣠⣀⣰⣶⣶⣄⣀⣀⣶⣶⣦⣀⣄⣰⣶⣶⣄⣀⣀⣶⣶⣦⣀⣀⣲⣶⣶⣄⣤⣠⣶⣶⣦⣠⣀⣶⣶⣶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠛⠛⢿⣿⣿⠟⠛⠻⣿⣿⡿⠛⠛⢿⣿⣿⠟⠛⠻⣿⣿⡟⠛⠛⢿⣿⣿⠛⠛⠻⣿⣿⡟⠛⠛⢿⣿⣿⠟⠛⠻⣿⣿⡟⠛⠛⢿⣿⣿⠛⠛⢻⣿

⢀⣀⣀⢀⣄⣀⢀⣀⣀⠀⣀⣀⢀⣀⣀⣀⣀⣀⠀⣀⣀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⢀⣀⣀⠀⣀⣀⡀⣀⣀⡀⣀⣀⡀⢀⣀⡀⢀⣀⣀⢀⣀⣀⢀⣄
⡯⢻⣿⡆⢙⣿⣧⢙⣿⣶⠙⣿⣷⠘⣿⣷⠈⣻⣷⡌⢻⣿⡌⢻⣿⡆⢹⣿⡦⢙⣿⣧⠙⣿⣷⠈⣿⣷⠈⣻⣿⡌⢻⣿⡆⢻⣿⣇⢙⣿⣧⣙⣿⣦⠙
⠳⠿⠟⠲⠿⠟⠲⠾⠟⠓⠾⠟⠓⠾⠿⠓⠶⠿⠋⠰⠿⠛⠲⠿⠟⠰⠿⠟⠢⠾⠟⠃⠾⠟⠃⠾⠿⠓⠶⠿⠛⠶⠿⠛⠲⠿⠟⠲⠿⠟⠳⠾⠟⠓⠾

⣤⡤⣤⣤⣤⣠⣤⢠⣤⣤⣄⣤⡀⣤⣤⣤⣠⣄⢤⣤⣤⣄⣤⡠⣤⣤⣤⣠⡄⢤⣤⣤⣄⣄⡠⣤⣤⣤⣠⣄⢤⣤⣤⣄⣤⡠⣤⣤⣤⢠⡀⣤⣤⣤⣄
⣟⣿⡘⣿⣿⣿⣿⣇⢻⣿⣿⣞⣷⡙⣿⣿⣿⡿⣇⢻⣿⣿⣾⣷⡙⣿⣿⣿⣻⣌⢿⣿⣿⣿⣷⠹⣿⣿⣿⢻⡎⢿⣿⣿⡼⣇⠹⣿⣿⣿⢿⡎⢿⣿⣿
⣿⣾⠿⠈⢿⣿⣷⡽⠇⠻⣿⣿⣎⠷⠘⣿⣿⣷⡹⠆⢻⣿⣿⣞⠷⠘⣿⣿⣷⡛⠆⢻⣿⣿⣝⠗⠘⣿⣿⣧⡻⠎⢻⣿⣿⡜⠷⠙⣿⣿⣧⠱⢌⢿⣿

⣀⢠⣤⣤⡄⣀⣀⣤⣤⣤⣀⣀⣠⣤⣤⣄⣀⣀⣤⣤⣤⣀⣀⣠⣤⣤⡄⣀⣀⣤⣤⣤⣀⣀⣠⣤⣤⣄⣀⣀⣤⣤⣤⢀⣀⣤⣤⣤⣄⣀⢀⣤⣤⣤
⣷⣿⣿⡟⣴⣟⣾⣿⣿⢣⣿⣷⣿⣿⡟⣴⢏⣾⣿⡿⢣⡿⣻⣿⣿⢟⣬⢿⣿⣿⡿⣣⡿⣽⣿⣿⠟⢰⢯⣿⣿⡿⣡⣿⣽⣿⣿⢏⣾⣣⣿⣿⡿⣱⡄
⣿⣿⠟⠘⢏⣾⣿⡿⠣⠞⣽⣿⣿⠏⠼⢇⣾⣿⡿⠡⠎⣽⣿⣿⠏⡰⢯⣾⣿⡿⠡⠏⣽⣿⣿⠏⢴⣫⣿⣿⡿⠡⢏⣽⣿⣿⠋⡼⣧⣿⣿⡿⣽⠏⣼
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣤⡴⢶⣤
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⣀⣀⣀⣀⠀⣤⡴⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣠⡴⢶⣬⠀⣄⡀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⣀⣀⡘⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣀⣀⣈⠀⠈⠙⠳⣤⢀⡿⠋
⠀⠀⠀⠀⢰⡟⠉⠉⠉⠀⠻⣿⣿⣿⡿⠟⢡⣿⣿⣿⣾⣿⣿⣿⣿⡿⢂⣤⠀⠀⢠⡞⠀⠛⠛⠲⠶⢤⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⠙⠟⣭⣶⣾⣿⣿⣿⣿⣿⣿⡿⢟⣩⠾⣯⡁⠀⣠⠏⣤⣤⣀⠀⠀⠀⠀⠙⡇
⠀⠀⠀⠀⢸⣇⠀⠀⣰⠟⠁⠀⠀⢻⣿⣿⣿⣿⣿⠋⠉⠁⠀⠈⠁⠀⠈⢙⡛⠁⠀⠀⠀⠙⣧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠉⠙⠛⣿⠀⠀⠀⠀⠀⠻⠿⠛⣩⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⣽⣧⣤⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀⢀⣠⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⢠⡄⢠⡄⣤⡄⣤⠀⢠⡄⡄⢀⠤⣤⣠⣠⠀⡤⣄⡤⣄⣠⣄⢤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠘⠂⠋⠑⠐⠋⠓⠀⠐⠋⠓⠈⠒⠁⠃⠉⠀⠓⠉⠓⠃⠋⠙⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⣋⣤⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠈⢿⣿⣿⣿⣿⣧⣤⣾⣿⣿⡿⠛⠛⠛⠛⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣟⣡⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠛⠉⠉⠁⠀⠘⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣶⣷⠀⠀⠀⣀⣤⣶⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠈⠭⠥⠶⠶⠾⠿⠿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠈⠙⠛⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⣤⣀⣸⣿⠀⡀⠀⣿⣇⠀⢰⡆⢰⡇⣠⠀⢠⡀⣷⠀⢀⣿⠏⠀⣤⠀⠙⣿⡄⠀⠙⡆⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠿⠋⣻⠏⢠⡥⠀⠻⣧⠀⠸⠇⣸⣿⡿⠀⠸⣿⡿⠀⠐⣿⡄⠀⠿⠀⢀⣿⠃⢲⡄⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣶⣾⣷⣶⣶⣿⣷⣶⣶⣿⣿⣷⣶⣶⣿⣷⣶⣶⣿⣿⣶⣶⣶⣿⣿⣶⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⣣⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⡿⠿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣍⡀⠀⠀⠀⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⣿⣷⣶⣤⡀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⡀⠀⠀⠀⠀⢠⡄⠀⢀⣠⡄⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠔⠉⠀⢚⡛⠉⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢰⣿⣛⠿⠀⢿⣿⡟⠀⠀⢻⣿⡟⠸⣿⣿⢛⣿⣷⠸⣿⣿⠛⣿⣷⠸⣿⣿⠻⠿⠆⢿⣿⡟⣻⣿⡞⢿⣿⣃⣿⠟⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢈⣙⢻⣿⡆⣸⣿⣇⣠⡄⣸⣿⡇⢀⣿⣿⠛⠛⠉⢀⣿⣿⠛⠛⠋⢀⣿⣿⣛⣀⡀⣸⣿⡟⢿⣿⡀⠀⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠘⠛⠛⠋⠀⠛⠛⠛⠛⠃⠛⠛⠋⠘⠛⠛⠃⠀⠀⠘⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠁⠛⠛⠃⠈⠛⠃⠀⠛⠛⠃⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣀⣤⣤⣄⠀⠀⠀⣀⣤⣤⣀⠀⠀⣀⣤⣤⣄⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣭⠙⠃⣿⣿⡇⠀⠀⣾⣿⡋⢹⣿⡆⢰⣿⡏⢙⣿⣷⠀⣿⣿⣭⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡅⠀⠀⣾⣿⣧⣾⡇⠹⣿⣧⣼⡿⠃⠘⢿⣷⣼⣿⠟⢠⣿⣿⡌⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠈⠻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⠤⣀
⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⢰⠃⠀⠀⡠⠀⠃⠀⡠⢄⠀⡀⠀⠀⠀⡀⠀⠀⢀⡀⠀⡤⠤⡀⡀⠀⠀⠀⡄⢠⠒⢄⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠈⠢⡄⢸⢸⠀⠀⡇⢡⠀⣄⢀⠇⠀⡔⠁⢹⢸⠀⠀⡇⢁⠀⡄⢠⠁⢸⠀⢸⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠠⠐⠁⠸⠈⠢⠔⠁⠘⠶⠙⠚⠀⠀⠘⠒⠚⠀⠑⠚⠁⠈⠒⠙⠊⠀⠈⠀⠈⠀⠀⡞
⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠙⠲⣄⡤⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢩⣀⡌⠹⡄⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢧⡞⠁⠀⠀⢀⡤⢦⠀⠀⠀⠀⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠒⠓⢄⣀⡎⠉⠛⠛⠉⠁⠀⠀⠑⠦⠴⠚⠉⠒⠦⠤⠚⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠙⠲⠤⠤⠤⠔⠚⢦⣀⡀⢀⣠⠴⠤⢀⣀⣠⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠧⠶⠦⣄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡠⠤⠤⢤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⠀⠀⢀⡴⠛⠲⠤⠤⠖⠒⠓⠒⠢⠤⠤⠼⠧⣀⡀⠀⡀⣍⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠛⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠛⠛⠋⠉⠛⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⣠⡴⠞⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠳⢦⣄⠀⢄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⡾⢁⣴⣶⣶⣶⣤⡀⠀⣤⡀⠀⢀⣀⠀⠀⣀⣤⣤⡌⢷⡀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣼⠃⣾⡏⠀⠀⠀⢈⣿⠀⣿⡇⠀⢸⣿⠀⣼⡟⠉⠉⠁⠈⣷⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⣿⣇⣠⣤⣴⡿⠃⠀⣿⡇⠀⠘⣿⠀⢿⣦⣄⡀⠀⠀⣸⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⡋⠉⠉⠻⣷⠀⠀⣿⠁⠀⠀⣿⠀⠀⠈⠉⢿⡆⠀⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢻⡄⣿⣇⣀⣀⣠⡿⠀⠀⢻⣧⣀⣸⡿⠀⣤⣀⣠⣿⠃⢠⡏⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠻⣌⠙⠛⠛⠋⠀⠀⠀⠀⠉⠛⠋⠀⠀⠉⠉⠉⠁⢀⡾⠁⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⠙⠲⢦⣤⣀⣀⣀⣀⣀⣤⣤⠴⠚⠉⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣐⠀⠀⠈⠉⠉⠉⠁⠀⠀⢀⣀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⠷⢶⣶⣶⠶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣄⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢀⣶⡟⠛⠛⠿⠀⠛⠛⠛⣿⡿⠛⠛⠃⢀⣴⠿⠛⠛⠻⣷⣄⠀⠀⣿⡿⠛⠛⢿⣶⡀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠏⠀⠀⠀⠀⠈⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠉⠛⠿⣶⣄⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣹⣿⠀⣿⣷⣤⣴⡾⠟⠁⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣿⡇⠀⠀⠘⣿⣆⠀⠀⠀⠀⢀⣿⠇⠀⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠰⣶⣤⣤⣶⠿⠁⠀⠀⠀⣿⡇⠀⠀⠀⠈⠻⣷⣤⣤⣴⠿⠋⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣤⣶⣿⣿⣿⡟⢉⡉⢿⢿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣾⣯⠀⣾⣿⠀⣿⠀⣾⡇⢸⡄⢻⣿⣿⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣧⢹⣿⠀⣿⠀⢻⡟⢸⣧⠸⢋⠿⢸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⠿⠿⢿⣭⣼⣿⠀⢿⣧⣀⣤⣿⣿⣆⣾⣆⣼⠋⠹⣿⡏⢙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⣶⡄⢻⣿⣿⡶⠶⣿⡟⠛⣿⣿⣿⣿⠻⣿⠀⠀⢹⡇⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⠀⣿⡇⢸⣿⡏⢠⣆⢹⣿⡀⢹⣿⣿⡇⢸⣿⠀⢀⠀⠇⠸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⠃⣼⣿⡇⢸⣿⢸⣿⣇⠸⢏⡸⠇⣼⣿⠀⢸⣧⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣇⠀⢁⣼⣿⣿⣇⠘⠋⣸⣿⣿⡀⣼⣧⠀⣿⣿⡀⣸⣿⣆⣸⣿⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⢿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠉








⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠉⠈⠀⠀⠈⠁⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⠿⠿⣿⣿⡿⠿⠿⠿⣿⣿⠿⠿⢿⣿⣿⡿⠿⠿⣿⣿⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⠁⣴⣦⠈⣿⣷⡆⠀⣶⣿⠃⢰⣶⡀⢹⣿⡇⢰⣦⡀⢻⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣄⡈⠛⢿⣿⣿⡇⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇⠸⠿⠃⣸⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⠛⣿⣷⠀⢻⣿⡇⠀⣿⣿⠀⢸⣿⠇⢸⣿⡇⢠⣶⣾⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣄⡉⢉⣠⣿⣿⣇⣀⣿⣿⣧⣈⢉⣀⣾⣿⣇⣸⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣜⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⢈⣉⣉⣉⣁⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣶⣒⠒⠒⣒⣲⣤⣤⣤⣄⣀⣀⣈⡃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣿⡭⠽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢀⡴⠄⢰⠀⠀⡠⣄⠀⡄⠀⢀⡆⠀⠀⣿⠲⡄⣰⢳⡀⡄⠀⢀⡆⢀⠀⣶⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⠀⠀⠀⠓⣄⢸⡇⠐⢇⡼⠀⢧⡶⣼⠃⠀⠀⣿⣠⠇⠹⠼⠁⣷⡿⣼⠁⡟⢦⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠐⠦⠞⠈⠉⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠘⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢠⠴⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠶⠦⠤⢤⠀⠀⠘⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⠀⠀⢠⠄⠒⠒⠒⠒⠒⠒⠒⠒⠲⡀⠀⢀⣀⠆⣀⢠⡄⡔⠀⠀⠤⠠⡄⠀⠀⠀⠀⠀⠀⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄⠀⠸⠀⠀⠀⠀⠀⠄⠀⠁⠁⠀⡇⠀⢈⡉⡀⢁⠍⠀⣀⠀⠀⢉⡐⠋⠀⢀⠂⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⡁⠀⠀⠀⠙⠘⠁⠀⠟⠉⠚⠦⠗⠟⠇⠀⠐⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠸⠄⠤⠤⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣴⠶⠶⣶⣶⣶⣶⡶⠶⠶⣶⣶⣶⣶⠶⠶⢶⣶⣶⣶⣶⠶⠶⢶⣶⣶⣶⡶⠶⠶⣶⣶⣶⣶⡶⠶⠶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⣿⠀⣰⣿⣿⣿⠟⠀⢀⣼⣿⣿⣿⠋⠀⢀⣾⣿⣿⡿⠃⠀⣠⣿⣿⣿⡟⠁⠀⣰⣿⣿⣿⠏⠀⠀⣼⣿⣿⣿⠋⣿
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣏⣀⣀⣾⣿⣿⣿⣃⣀⣠⣿⣿⣿⣿⣁⣀⣰⣿⣿⣿⣟⣀⣀⣼⣿⣿⣿⣏⣀⣀⣾⣿⣿⣿⣃⣀⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢻⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⣏⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣼⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣧⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣿⠀⣠⣿⣿⣿⡟⠁⠀⣰⣿⣿⣿⠏⠀⢀⣼⣿⣿⣿⠋⠀⢀⣾⣿⣿⡿⠁⠀⢠⣿⣿⣿⡟⠁⠀⣰⣿⣿⣿⠏⣿
⠀⠀⠀⠀⠀⠀⣿⣰⣿⣿⣿⡟⠀⠀⣼⣿⣿⣿⠏⠀⢀⣾⣿⣿⣿⠃⠀⢠⣿⣿⣿⡿⠁⠀⣰⣿⣿⣿⡟⠀⠀⣼⣿⣿⣿⠏⠀⣿
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⢻⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⡟⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣠⠀⠀⠀⠀⠀⠀⢣
⠀⠀⢀⠀⣤⡀⠠⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⠘⠃⠈⠀⢤⠀⠀⠀⠀⠀⡆
⠆⠁⠀⡁⢙⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠁⠀⠀⠀⠀⢆⣀⡄⠀⠤⠃
⠀⠀⠀⠀⠀⡑⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⢀⡀
⠀⠀⠀⣰⠞⠋⠛⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡄⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡁
⠀⠀⠀⠙⠄⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠒⠈⠁⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢈⢂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠐⡄⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠁⠒⠤⢀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠂⠀⠀⠀⠈⠐⠂⠤⢀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠣⡀⠁⠀⠀⠀⠀⠀⠀⠐⡈⠉⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠂⠀⠀⠀⠀⠀⠐⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⣿⣿⣿⠟⠁⠀⠀⠀⣠⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⣴⣾⣿⣿⣿⡟⠋⠀⢀⠀⢀⣶⣿⣿⣿⣿⠟⠁⠀⠀⠑⣠⣾⣿
⣿⠟⠡⠀⠀⠀⣠⣾⣿⣿⣿⡿⠏⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠐⣡⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣾⣿⣿⣿
⠁⠀⠀⢸⣿⠿⠛⠛⠟⠛⠻⠿⠟⠻⠿⠿⠻⠟⠛⠛⠻⠟⠻⠿⠿⠻⠿⠛⠛⠛⠛⠿⠿⠛⠿⠿⠻⠟⠻⣿⣿⡿⠋
⠀⠀⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⣠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣴⣿
⣿⡿⠋⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠟⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟
⠠⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁
⢀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣠⣾
⣿⣿⡿⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⡿⠋⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠠⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋
⠀⣠⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣴
⣿⣿⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿
⣿⠟⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠁⠐⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠃
⠀⠂⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣱⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⣠
⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣾⣿
⣿⡿⠋⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠋⠀⠂⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋
⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⢀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣴⣿
⣿⣿⠟⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠟⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⡀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉
⠀⣠⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇⠠
⣾⣿⣿⣿⡿⠋⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠀⢀⣴⣾⣿⣿⣿⠿⠋⠀⠠⠀⣠⣾
⣿⣿⣿⢋⣀⣀⣀⣀⣴⣿⣿⣿⣿⡟⣁⣀⣀⣀⣠⣾⣿⣿⣿⣿⣛⣀⣀⣀⣀⣴⣿⣿⣿⣿⡿⣁⣀⣀⡀⣠⣾⣿⣿
⣿⡿⡛⢻⡿⠿⠿⠿⠿⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣥⣼⣷⣶⣶⣶⣶⣶⣾⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⢀⣾⠏⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣧⡀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⣴⡟⠁⠀⠀⢀⣤⣶⣶⣤⡀⢀⣤⣤⣦⣤⣤⠀⢀⣤⣤⣤⡀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠈⣿⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⢸⣿⠉⠉⣿⣿⠀⠉⢹⣿⡏⠉⠀⣿⡏⠈⢻⣿⠀⢸⣿⡏⠉⢿⣷⠀⠀⠀⣿⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⢸⣿⡀⠀⠻⠟⠀⠀⢸⣿⡇⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣸⣿⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠘⠻⣿⣶⣤⡀⠀⠀⢸⣿⡇⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡿⠿⠿⠋⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⢠⣤⠀⠈⣿⣷⠀⠀⢸⣿⡇⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⢸⣿⡀⠀⣿⣿⠀⠀⢸⣿⡇⠀⠀⣿⡇⠀⢸⣿⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢿⣄⠀⠀⠀⠈⠻⢿⣿⠿⠋⠀⠀⠸⠿⠇⠀⠀⠙⠿⢷⠿⠟⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣶⡄⠀⠀⠀⠀⠀⢰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠟⠀⣀⣤⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⣥⣶⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣀⣤⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠉⠛⠿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⢀⣠⠴⠛⢉⣠⣴⣶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣉⣤⠴⠞⠛⠉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠟⠀⠀⠀⢉⣁⣤⡤⠶⠾⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢟⡻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢫⣿⣿⡝⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣣⣿⠛⠛⣿⣜⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⢀⣤⣤⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣱⣿⣧⠀⠀⣸⣿⣎⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠸⣿⣿⠇⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣠⡿⣽⣿⣿⣿⠀⠀⣿⣿⣿⣯⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⣰⣟⣼⣿⣿⣿⣿⡄⢠⣿⣿⣿⣿⣧⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⣴⢏⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⡹⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⣼⢯⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣷⡽⣧⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⣾⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣷⠀⠀⠀⠀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷
⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠉⠀⠀⠀⠀⠉⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⡤⠖⠒⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢦⡀
⠀⠀⠀⡜⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠱⡀
⠀⠀⢸⠁⠀⠀⡞⠀⢸⢀⠞⠑⡆⢠⠀⠀⠀⡔⠀⢀⡤⢾⠀⡎⠉⢢⢰⠀⠀⠀⡆⠀⡇⠙⡆⠀⠀⡇
⠀⠀⢸⠀⠀⠀⢈⡳⢸⠘⡄⢀⡇⠸⡀⣶⣠⠇⠀⠸⣀⢸⠀⢇⣀⡜⠘⣄⡿⣠⠃⠀⡇⠀⠇⠀⢀⠇
⠀⠀⠘⡆⠀⠀⠁⠀⠈⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠔⠋
⠀⠀⠀⠀⠀⠈⠷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⢤⡀⠀⠀⢀⣠⠴⠒⠒⠛⠓⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠷⠃⠀⣇⢀⠔⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣀⡀⠀⢸⠯⣀⣀⡠⠤⠚⠙⢆⡀⠀⣀⠤⣄⠀⠀⢘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠑⢦⢧⣀⠀⠀⠀⠀⣀⡀⠀⠉⠉⠁⠀⠀⠉⠉⠉⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⠉⠙⠉⠉⠁⠉⠲⠤⠤⠔⠋⠙⠒⠒⠒⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⣱⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⡖⠉⠉⠙⠚⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣥⣄⡤⠊⠁⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠒⠦⣄⣢⡽⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠊⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠑⢄⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠊⡠⠊⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠑⢄⠑⢄
⠀⠀⠀⠀⡠⠊⡠⠊⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠑⢄⠑⢄
⠀⠀⡠⠊⡠⠊⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡤⣄⡏⠀⢹⡞⠉⢣⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠑⢄⠑⢄
⡠⠊⡠⠊⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠑⢄⠑
⡇⢸⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⡖⠒⣿⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⢸⡇⠀⢸⡇⠀⢸⠀⢠⠖⢲⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣷⠀⢸⡇⠀⢸⡇⠀⢸⢀⡇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠋⠀⠈⠁⠀⠈⠁⠀⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⠸⡀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⡇
⠙⢦⡙⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⢋⡴
⠀⠀⠙⢦⡙⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠙⢦⡙⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⡀⠉⠦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠔⠃⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣦⣈⣉⣉⣉⣉⣍⣉⣉⣉⣉⣉⣍⣉⣍⣉⣅⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⡿⠃⠁⣼⣿⣿⠏⡘⢡⣿⣿⡟⡁⢋⣾⣿⣿⠃⠀⣰⣿⣿⡟⡤⢠⣾⣿⡿⢡⠂⣼⣿⣿⢏⠴⣰⣿⣿⡟⠑⢃⣾⣿⣿⠃
⣿⡿⢁⠀⣼⣿⣿⢏⠈⣰⣿⣿⡟⡔⢄⣾⣿⡿⠃⠊⣴⣿⣿⠏⡌⢡⣿⣿⡿⡁⢈⣾⣿⣿⠃⠈⣰⣿⣿⡟⢱⢠⣾⣿⡿⠡⠂
⡿⢁⢀⣼⣿⣿⠃⡄⣰⣿⣿⡟⣄⢠⣾⣿⡿⢁⠀⣼⣿⣿⢏⠌⣱⣿⣿⡟⠁⢁⣾⣿⣿⠃⠘⣴⣿⣿⢏⠐⢡⣿⣿⡿⣄⢀⣼
⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠈⠉

















⣿⣿⡿⠠⢄⣴⣿⣿⢃⠀⣰⣾⣿⡗⡠⢀⣶⣿⡿⢠⠄⣴⣿⣿⠇⡄⣠⣾⣿⡟⠠⢄⣶⣿⣿⠢⠄⣰⣿⣿⠗⠠⢠⣾⣿⡿⢀
⣿⡿⠡⢀⣾⣿⣿⠃⡄⣰⣿⣿⠏⡢⢠⣿⣿⡿⣁⠌⣼⣿⣿⠏⠔⣱⣿⣿⡟⠀⢠⣾⣿⡿⠃⠀⣼⣿⣿⠏⠀⣣⣿⣿⡟⠰⢁
⡟⣠⢁⣾⣿⡿⢣⠘⣼⣿⣿⢏⠌⣡⣿⣿⡟⡱⢏⣾⣿⣿⠃⠜⣴⣿⣿⠟⠸⣨⣿⣿⡿⢁⡅⣼⣿⣿⢏⠖⣰⣿⣿⡟⢀⢡⣾

⠀⠀⢀⣠⣤⣴⣷⡾⠋⠐⠄
⠀⢲⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠤⢀
⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⣆⠸⡀
⠀⠀⠁⠻⢿⣿⣿⡄⠀⡔⢀⡴⢡⡼⠀⢀⠀⠀⠀⠀⠒⠒⠂⡀
⠀⠀⠀⠀⠸⡀⡉⡛⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠑⠠⠐⡄⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠋⡔⠒⠒⠂⠀⠁⠀⠀⠀⢀⢘⠉⣤⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢰⣥⠬⠉⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢧⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⣿⡀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠀⡆⠀⢀⡀⠀⢄⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢹⡷⠒⡇⠈⠂⠀⠀⠀⠀⠀⡀⠀⠠⠀⠃⢀⡨⠀⡌⢀⠣⡀⠑⠄⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡢⠤⠁⠀⠀⠀⠀⢀⠄⠠⠀⠀⠀⡠⠰⠁⠀⢰⠀⡟⢄⠈⠂⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠎⠀⠃⠀⢰⠫⡀⠇⠀⠁⠢⠻⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠂⠀⠀⠁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠢⠸⡀⠀⠀⠀⠑⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠠⠀⠐⢆⠀⠀⠀⠈⠁⠒⠀⡸⠒⠀⠂⠚⠀⠀⠁⠰⠀⠀⠀⠀⠈⠢⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠔⠁⠀⠀⠀⠈⠢⡀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠉⡰⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠄⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠌⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⡘⠐⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣄⠈
⠀⠀⠀⠀⠀⢀⠠⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⢄⠀⠀⠀⠀⠀⡘⠀⠄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⢁⠂
⠀⠀⠀⢀⠴⡁⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢳⠀⡀⠀⠈⡠⢊⣴⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠁⠉
⠀⠀⠀⠘⡲⢁⡀⢀⠀⡼⠭⠤⣌⠀⠀⠀⠈⢁⡄⠀⠀⠀⠀⠰⡈⢶⠈⠑⠮⠔⠈⠀⠈⢋⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠊⢀⠄
⠀⠀⠀⠘⠤⠀⠉⠉⠀⠀⠀⠀⠘⡛⠂⠒⠨⠞⡹⠁⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⠑⢄⠀⠀⠀⠀⠔⠁⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠓⠀⠁⠀⠐⠢⠀⣥⡤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⢀⠐⠆⠀⠈
⠀⠀⠀⠀⠰⠶⠶⣶⣶⣶⣶⣶⣶⣶⣠⣤⣤⡄⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⢿⣿⣿⡿⠿⢟⣟⡛⠛⠛⠛⠛⠉⠉⠀⠀⠤⣶⣦⣴⠄⠀⣀⡀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⡻⠿⢿⣿⣦⣤⣖⣂⣀⣉⣉⣁⣒⣛⣉⣡⣶⣿⣿⡉⠙⠓⠛⠿⠿⡅
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶⣛⣉⠀⢀⣀⣀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠳⠶⠶⠶⠶⠦⠴⠖⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣾⠿⣶⡄⠀⠀⠀⠀⠀⣿⣏⠉⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠿⢛⣿⣿⠀⢸⣿⢿⣿⣦⡀⣀⠀⠙⣷⡄⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠀⠈⢿⡇⠀⢸⠏⠈⣽⢹⡿⠛⡿⠟⢿⣿⢀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⣰⣿⠆⠀⠸⡇⠀⡟⠀⢰⡟⡼⠀⢰⠁⠀⣾⠏⣼⣿⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠳⢸⡏⠀⣿⣧⣛⣀⣼⣷⣄⣼⣶⣤⣾⣷⣀⣩⣤⣀⣡⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣿⣄⠙⢻⣿⣿⠋⢻⣿⡿⣶⣤⣠⣬⣝⣫⣽⡿⠿⣷⡾⠿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢛⣿⣿⠿⠇⠀⣼⠃⠈⣩⣿⡟⠉⢻⡟⠉⣧⡁⢿⠁⢠⡏⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣾⡏⠀⣾⠀⠀⡇⠀⣸⣿⣻⠀⠀⣿⠀⢀⣿⠇⡆⠀⢘⡆⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣿⣆⠀⢟⡀⠸⣇⠀⢿⢏⣿⡄⠸⢟⡀⠸⢋⣼⣷⡀⣸⡇⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠻⠿⠿⢿⣶⠿⠷⡶⠿⠛⠻⠿⠟⠻⠿⠟⠋⠙⠿⠟⠿⣶⠿⠁

⠀⠀⠀⣠⠞⠛⠛⠉⠉⠉⠉⠛⢦⡀
⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⣠⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣀⡀
⡞⠁⢀⣀⠀⣀⣀⡀⢀⣀⡀⠀⣀⣀⠀⢹⣀⠔⠋⠀⠀⠀⠀⠀⠙⢦⡀⢀⡴⠋⠁⠀⠀⠀⠀⠙⢦⡀
⡇⠀⣿⣉⠁⠉⡟⠁⣾⠉⢻⠀⣏⣹⡇⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠹⡄
⣷⠀⣈⢻⡇⠀⡇⠀⣿⣀⣼⠀⡟⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠙⠛⠁⠀⠋⠀⠈⠙⠁⠀⠁⠀⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣰⠃
⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠳⢄⣰⠀⠀⠀⠉⠛⠻⠀⠀⠻⠋⠁⠀⠀⠀⣤⡠⠞⠁
⠀⠀⠈⠳⣄⣀⣀⣀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⢹⡄⠀⠈⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠉⠉⠉⢩⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠒⠀⠠⠤⠀⡆⠴⠆⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⡀⠀⠀⠁⢄⠀⠐⠁⠀⢠⠗⢦⡞⠛⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡇⠀⢸⠉⠙
⠀⠀⠀⠀⠀⢀⣀⡰⠘⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠠⠄⠂⠀⠀⢀⣀⣀⣸⠀⠀⡇⠀⠘⠀⢠
⠀⠀⠀⠀⠀⢫⣀⡱⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠂⠁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠙⣆⠀⠀⠂⠀⢠⠃
⠀⠀⠀⠀⠀⠈⣤⣉⢶⣁⡠⠌⢉⣒⠢⠄⠀⢀⣀⡀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠈⠙⠄⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠙⠢⢴⠁⠀⠀⠀⠀⠉⠁⠐⠂⠒⠒⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢀⡀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠦⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⣀⡴⠋⠀⠑⢦⣀⠀⠀⠀⠀⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⣋⡴⠚⠀⠈⠙⣿⣷⣄⣀⣤⣄⣀⡀⠀⠀⠀⢠⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢃⣼⠏⠀⠀⠀⠀⣠⣾⣿⣿⠿⠿⢿⣿⣿⣷⣤⡀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⣾⠇⠀⠀⠀⢀⣼⡿⠋⢹⣿⡇⠀⠀⢬⡙⠿⣿⣿⣷⣦⣀⡀⠉⠉⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⠀⠀⠀⠀⣾⡟⡴⠀⣸⣿⡇⠀⢀⣀⣈⠳⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⣿⡀⠀⠀⢰⣿⠀⠀⠀⣿⣿⠀⣰⣿⡿⣻⠀⠈⠳⣮⡙⠿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠸⣇⠀⠀⠘⣿⠀⠀⣼⣿⠇⣰⣿⡿⢡⠏⠀⣠⡄⠈⠻⣦⣈⠉⠛⠛⠛⣉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣿⣿⣧⠹⣆⠀⠀⠙⠧⣼⠿⠃⢠⣿⣿⣣⠏⠀⢀⡙⠁⠀⠀⠀⠉⠛⠛⠛⠋⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠙⢿⣿⣧⡘⠆⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⢠⣾⠟⠻⣧⠀⣴⡄⠀⣴⡄⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⡛⠂⠀⠻⢿⣷⡄⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⣾⣿⡇⠀⣿⣤⣿⠇⢰⣿⣧⠀⢠⣿⠃⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠰⠶⠀⠈⠙⢿⣦⣀⣴⠾⠋⢀⣿⡿⠀⠀⣿⣿⣅⣠⣿⢹⣿⢠⡏⢸⡏⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣤⡐⣿⡇⠀⢀⣿⡟⠉⣸⡇⢸⣿⡿⠀⢸⣇⣾⡟⠀⠀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⣠⣶⣿⡿⠟⠋⣩⣿⠃⠀⡏⠉⠛⢿⣿⣷⣤⠟⢿⣧⣴⠟⠀⠸⡿⠁⠀⢸⣿⡟⠀⣠⡾⠟⢉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢻⣿⡇⣷⠀⣿⠃⠀⢸⡇⠀⠀⠈⣿⡏⠁⠀⠀⠉⠁⡀⢀⡀⢀⣀⣠⡈⡉⠀⢈⣥⣴⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣫⠀⢸⣿⣷⣿⠀⣿⣄⣠⡿⠃⠀⠀⢰⣿⡇⠀⠻⠶⠛⠞⠛⠛⠛⠋⠉⠉⠛⠁⠀⠀⠠⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠱⠁⠀⢸⣿⣿⣿⠀⠈⠛⠋⠁⠀⠀⠀⣾⣿⡷⢶⡄⢠⡄⢀⣤⠀⣴⢠⡆⢀⣴⣶⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣤⠀⠀⢀⣾⣿⣿⣷⣨⣧⣾⡇⣼⣿⢠⡟⢸⣷⡟⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣿⣿⠀⠀⠀⠀⢀⣴⣿⡿⢻⣿⢀⣽⠃⢿⡿⠉⣿⡾⠃⠸⠟⠀⠸⣿⣄⣠⡟⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⣀⣤⣶⣿⠿⠋⠀⠈⠻⠿⠃⠀⠈⠁⠀⠀⠀⠀⠀⢀⣤⠀⠙⠛⠋⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠛⣋⣡⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⡜⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⣀⣴⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⣛⣛⣛⣉⣭⣭⡤⠶⠚⠉⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢛⣥⣶⠶⠂⠀⠀⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⡻⣿⠿⠷⠝⠿⠝⠿⠿⠿⠮⠿⠮⠻⠿⠿⠷⠭⠿⠽⠿⠿⠿⠦⠽⠯⠙⠿⠿⠷⠭⠻⠮⠻⠿⠿⠷⠽⠷⠝⠿⠿⣿
⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣷⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣮⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣝⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣶⣦⣴⢶⣖⢶⣶⣶⣦⡲⣦⣔⢶⣶⣶⣔⢶⣶⠲⣶⣶⣶⣔⣶⣖⢶⣶⣶⣦⡲⣶⡲⣶⣶⣶⣔⢶⣴⠶⣶⣿

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢸⣿⣿⣿⠏⠉⠉⠉⠿⠿⠿⠿⠋⠉⠉⠉⠿⠿⠿⠿⠋⠉⠉⠉⠿⠿⠿⠿⠉⠉⠉⠹⠿⠿⠿⠟⠉⠉⠉⢹⣿⣿⣿⠟⠉⠉⢹⡇
⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⣰⡿⠻⠷⠀⣰⣿⣆⠀⣿⠀⢸⡇⠚⢻⡿⠛⢸⣷⢠⣾⠛⢷⡄⢺⣷⡀⣿⠀⣿⠀⢸⣿⣿⠏⠀⠀⠀⣼⡇
⠀⠀⢸⡿⠃⠀⠀⢠⣿⠀⢻⣧⣠⣤⢠⣿⣶⣿⡄⣿⣄⣼⡇⠀⢸⡇⠀⢸⣿⠸⣷⣀⣼⡇⣸⡏⢿⣿⠀⣛⠀⢸⣿⠋⠀⠀⢀⣾⣿⡇
⠀⠀⢸⡇⠀⠀⢠⣿⣿⣀⡀⠉⠉⠁⣈⣁⣀⣈⠁⠈⠉⠉⣀⣀⣈⣁⠀⠈⠁⠀⣈⣉⣁⣀⠈⠁⠀⢉⣀⣉⣀⡸⠃⠀⠀⢠⣾⣿⣿⡇
⠀⠀⠸⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⢠⣿⣿⣿⠏⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣿⣿⣿⡟⠘⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⢀⣿⢻⣿⡟⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣾⠻⣿⡟⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣼⠏⠀⢻⣆⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⣼⠏⠀⢻⣇⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣸⡟⠀⠀⠈⢿⡄⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣰⡟⠀⠀⠀⢿⡆⠀⠀⠀⢻⣆
⠀⠀⠀⠀⣸⡟⠀⠀⠀⢰⡿⠀⠀⠀⠀⠘⣿⡀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⣿⡄⠀⠀⠈⢿⡄
⠀⠀⠀⢠⡿⠁⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠹⣷⣀⣀⣀⣸⣷⡀⠀⠀⠀⢠⣿⠁⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠘⣷⣀⣀⣀⣘⣿⡀
⠀⠀⢀⣿⣧⣤⣤⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣾⣧⣤⣤⣤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⣾⣿⣎⠻⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣹⣿⣿⣿⣿⣿⣿⣿⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⣿⣯⣿⣯⣽⣿⣿⣿⣿⣿⣿⣭⠿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⡻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⣿⣦⡄
⠀⠀⠀⠀⠀⢠⣶⣟⣿⣾⡿⠿⢿⣿⣿⣿⣿⣿⣾⣿⣥⣿⣿⣿⣿⣷⣾⣿⣟⣻⣷⣽⣿⣦⣄
⠀⠀⠀⣠⣾⣟⣽⣯⡿⣻⣿⣿⣽⢻⣯⡟⣿⣿⣯⣽⣿⣯⣾⢿⣿⡿⣿⣿⣿⣯⣿⢏⣷⣿⡿⣦⡀
⠀⢠⣼⡿⣿⣿⣿⣿⣧⣿⣿⣭⡹⢺⣿⡇⣻⢳⣿⣯⢭⢻⣿⡞⣿⣇⣿⣿⣇⣿⡟⣿⣿⣟⣻⣿⣿⣦⡀
⠀⠈⠻⣷⣝⢿⣿⡿⣿⣟⣹⣻⣿⣾⣿⣇⣟⡸⣿⣟⣶⣼⣿⢇⣿⣿⣿⢹⣿⣿⢳⣿⣿⣿⣿⢏⣾⠿⠃
⠀⠀⠀⠈⠻⣷⣙⣧⣟⣛⣿⣿⣯⣼⣿⣿⣻⣿⣽⣻⣿⣛⣽⣿⣼⣿⣿⣾⢿⣿⣾⣿⣷⢟⣽⡿⠉
⠀⠀⠀⠀⠀⠈⠻⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣽⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣻⣿⡿⢿⣿⣿⡿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣼⡿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣿⣿⠏⠵⢿⡇⠛⣯⠟⠋⢼⣿⠄⠘⠻⠿⠋⠾⠛⠾⠟⠷⠟⠳⠟⠋⠀⣿⣵⣿⣵⣿⣿⣿⣯⣾⣯⣾⣿
⢾⣽⠗⣤⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣿⠧⠀⢽⡿⠿⡿⡿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣧⡄
⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣶⣀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿
⣿⣷⡄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿
⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿
⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙
⠙⢿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦
⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿
⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿
⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿
⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈
⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷
⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿
⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿
⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀
⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿
⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿
⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿
⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻
⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠈⠈

⠀⠀⢀⡴⠋⠉⠉⠉⠉⠉⠛⢦⡀
⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⡏⠀⣤⣤⢠⣤⣤⢀⣤⣄⢠⣤⣄⠀⡇
⡇⠀⢷⣤⠀⢻⠀⣾⠀⢹⢸⣧⡿⠀⣷
⡇⠀⢦⡾⠀⠸⠀⠸⠦⠟⠸⠇⠀⠀⣿⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⡴⡟⠈⠀⠀⠀⠂⠀⡀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⠄⠀⠀⠐⡀⢀⠁⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠙⠉⠙⠛⠋⠉⠉⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣼⠉⠙⡏⠀⢸⡤⢄
⠀⠀⠀⠀⠀⢀⢠⡄⠢⠄⠀⠀⠀⠁⠀⠆⠀⡄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢹⠀⠀⡇⠀⠨
⠀⠀⠀⠀⠀⡔⠀⢄⡀⢸⡀⠀⠀⠀⠀⠀⠀⠇⠠⠈⠁⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⠂⣴⠚⠓⠺⡀⠀⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠨⣀⡹⡧⠢⡈⠢⢄⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠑⠒⠒⠒⠀⠉⠀⠀⠈⡲⢄⠀⠓⢤⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠌⠀⠀⠀⠄⠀⢀⠀⠂⠰⡀⠀⢠⡀⠀⢀⠀⠰⠁⠉⠀⢼⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠐⠠⢀⠀⠀⠐⠚⠁⠀⠀⡀⢀⠄⠀⠓⠲⠖⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⠀⠀⠃⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠖⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠉⠉⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣿⠀⠀⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠀⠐⠿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣠⠶⢤⡀⠀⣾⡏⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⣿⣄⠀⠿⠆⣿⡏⠀⢠⡤⠀⢠⡦⠀⡀⢶⡄⠀⠀⠀⢀⣼⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣍⡂⣤⠀⠀⠈⠙⠻⣶⡎⣿⡇⠀⣼⡗⢺⢻⡇⣸⡇⢨⡇⠀⣠⡐⣋⣥⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣩⣥⠶⠀⠉⠀⠀⠀⠀⡜⠀⡻⠹⣿⠞⠙⠗⠃⠘⠓⠁⠙⠊⠀⠀⠈⠁⠲⣦⣍⡉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣀⣀⣀⠀⠀⠈⣒⣁⣀⠀⢀⡀⠀⢀⡀⠀⣀⡀⣀⡀⠀⣀⡀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣿⠋⠛⣿⡄⣾⡟⠉⠻⣷⠸⣷⣠⣿⣷⢰⣿⠁⣿⣷⣄⣿⡇⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⡸⠟⠀⠀⠀⣿⣤⣤⣿⠃⢻⣧⣤⣴⡿⠀⢻⣿⡏⢿⣿⠇⠀⣿⠍⢻⣿⡇⠀⠀⠻⠇⢸⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠷⣤⡄⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⣩⡥⣤⣤⣤⣭⣤⣬⠉⠀⠀⠉⠀⠀⠉⠁⠀⠀⢠⣤⠾⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠻⢶⣄⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣿⣳⣻⣦⡙⡷⣿⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠟⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠶⠂⠀⠀⠀⢠⡞⠋⠉⠉⠁⠀⠀⠀⠀⠉⣋⣋⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⠀⠀⠀⠀⠘⠦⣤⣤⣶⡦⢀⡀⣠⣶⡄⠙⣧⡀⠀⠀⢠⣶⣗⢲⣠⣾⡇⠀⣶⠀⠀⠀⠀⠀⠀⠀⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡟⣄⠀⠀⠀⠀⠀⣼⠟⠁⢀⣿⣿⠁⣿⡇⠀⡿⣿⣤⠞⣿⠃⠈⢹⠁⣿⠃⢀⣿⠀⣠⡶⠀⠀⠀⣠⢻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣼⣿⠀⠀⠀⠀⠀⢿⣀⣠⠎⠀⣿⠀⣿⠀⡸⠁⡼⢿⣆⠙⠂⠖⠃⠀⠙⠷⠋⢹⡿⠛⠁⠀⠀⠀⣿⣧⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠚⠁⢸⡇⠘⡿⠀⠀⠀⠀⠀⠀⠀⣰⠋⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡴⢛⡛⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣚⣁⡀⠀⡀⠀⢀⠀⠀⠧⡠⠎⠀⢠⠘⣛⠳⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡾⢫⡍⠐⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⣿⣟⣛⡛⢸⣿⣛⣛⠃⠀⠀⠀⠀⠐⠃⠊⣍⢳⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣧⠀⠐⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⡀⣿⡇⣿⡟⠛⠃⢸⣿⣛⣛⡀⠀⠀⠀⠀⠀⠀⠐⠈⣸⢣⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠙⠃⠙⠃⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⣀⣀⡤⠞⢋⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⠉⠀⠀⠀⠉⣩⣤⣤⣟⣀⠲⠀⣀⠀⠀⢀⡀⠀⣀⠀⠀⢠⠀⠶⣀⣘⣤⣤⣌⡉⠀⠀⠀⠈⠱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣷⠈⠉⠻⢷⣤⣀⠀⠛⡱⢨⠙⠃⢀⣤⣶⠟⠋⠁⣴⡇⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠈⠀⠀⠀⠀⠉⠙⠛⠊⠁⠀⠁⠚⠛⠉⠀⠀⠀⠀⠈⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢓⡜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣤⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢐⠖⢺⠋⠉⠋⡽⠙⠤⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡔⠉⠱⡆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⢇⠀⣰⡷⠊⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡀⠀⠀⠀⠈⢹⠀⠀⣀⠀⠀⠀⠉⠀⠀⣀⡤⠚⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡇⢀⠀⠀⠀⠈⠙⠉⠈⡆⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⡞⢳⠀⠓⢻⡀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⠣⣱⢸⡀⠀⠀⠙⠢⠤⠤⠤⠖⠊⡻⠀⠀⠀⠀⠊⠉⡩⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠽⠆⠀⣟⠉⠙⠳⢆⠀⠀⠀⠀⢀⠇⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠉⠑⢢⣼⠉⠉⢣⡈⢣⠀⣀⣀⠞⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠐⢂⠔⢋⡜⠁⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⡠⠤⠔⢒⣒⣒⣒⣢⡀⠀⠀⠀⠀⠘⢆⡱⠀⠀⣳⡀⠛⠘⠧⢤⣴⠯⡴⠦⠀⠀⣠⠞⡹⠂
⠀⠀⠀⢀⠀⠀⠐⣅⠤⡞⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣊⡥⠔⠊⠉⠁⠀⠀⢸⣻⡇⠀⠀⠀⠀⣠⠤⠤⠔⠋⠉⠃⠀⠀⣠⠤⠔⠒⠻⡀⢀⠞⠱⠮⠐⠊⣉⡗
⠀⠀⢸⠉⡇⠀⠀⠈⠳⢝⢦⡀⠀⣀⡀⠀⡴⢊⡠⠊⣧⠀⢀⣦⡴⣆⡀⣄⠺⡟⠀⢀⠔⠲⡟⠁⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⣯⢁⡀⣘⠭⣉⡈⠁
⠀⠀⠸⡀⠹⣄⡠⠔⣖⣨⢧⡉⣹⠋⢡⡎⡴⠋⠁⢁⡨⢗⣺⡌⠧⡿⢀⠏⠉⡀⠒⠁⠀⠀⠙⠷⣴⣢⡀⠀⠀⡰⠋⠀⠀⠀⠀⠀⣯⡠⠎⠀⠈⠳⠤⠼
⠀⠀⠀⠳⠤⠤⠓⠉⠀⠀⡰⢱⠋⠀⢀⡉⠓⠶⠛⠓⠊⠁⠀⠀⢠⠇⠈⠉⢻⣿⣯⠁⠀⠀⠀⠀⠀⠉⣹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢮⠤⣎⣁⠘⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠉⠁⠀⢈⣵⣦⣤⣤⣤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠔⠊⠁⠈⠑⠋⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⣠⠔⠊⣁⡤⣊⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣞⠃⠈⠉⡽⣿⡿⠿⣿⡛⠛⠩⢍⡙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡛⠓⠦⠤⠤⠤⠤⠤⠤⠴⠯⢈⠱⠦⠤⠼⠒⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠏⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠃⠀⠀⢻⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⣀⣀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⠘⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄
⢀⣼⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣄
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣟⡛⠃
⠀⠀⠀⠀⢀⣬⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣽⠟⢫⠉⢻⡄
⠀⣴⢟⣿⡿⢷⣎⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⡄⠈
⠀⠃⣾⠁⣷⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇
⠀⠀⢿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢷⡀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⢲⣴⡖⢀⣀⣀
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣧⡄⢠⡼⠀⠤⢤⣤⠖⠀⡏⠀⠀⠀⠀⠻⣿⡤⠅⠛
⠀⠀⠀⠀⠀⠀⢸⠷⠄⠀⠀⠉⠉⠀⠀⠀⠘⠉⠀⡶⠷⠄⠀⠀⡀⠀⠙⣿⡷⣿⠒⢶⡄
⠀⠀⠀⠀⠀⠀⢩⠟⠃⠉⠉⠉⢉⡉⠀⠁⠀⠀⠈⠁⠈⠁⠀⢈⠋⠘⢛⣁⣀⣌⣛⠚⣾⣤
⠀⠀⠀⠀⠀⠨⢽⠀⠀⠀⠀⠀⠳⠶⠀⢰⡇⡀⠀⠀⠀⠀⣆⠀⠞⠀⣿⣿⣿⣿⣿⡇⡙⠛
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡏⢉⣉⢹⡂⢰⣶⣶⣶⣶⣶⡀⠀⠀⠀⠀⠈⠉⠁⠈⠁⣠⣷⠞
⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⡇⠘⠋⢸⠀⠸⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠐⣺⠀⠀⠀⡇⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⣿⣵⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠈⢽⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⡟⠀⡈⠻⣷⣄⠀⢀⣠⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⣽⠀⠀⠀⠁⣉⢉⣉⣁⠀⢀⣠⣀⣀⣀⣴⠞⣛⠉⠛⠻⣷⣄⣤⣴⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟
⠀⠀⠀⠀⣠⣾⡿⠶⣯⣤⣶⢟⠛⠛⠛⢿⡿⠛⠉⠉⠉⢿⡄⠁⠁⠆⣰⡿⠋⠉⠙⢿⣿⣿⠟⠋⠁
⠀⠀⠀⠘⢿⣿⣷⣦⣤⣿⣇⡀⠐⠀⢀⣾⡇⠀⠠⠀⢀⣸⣷⠶⠶⢾⣿⣀⣀⣤⡤⢾⡽⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣶⣶⣿⣶⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣏⠙⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠙⠿⣿⠿⠟⠿⠡

⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⡇⠀⡇⠈⠀⡄⡘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢀⡆⠀⠀⠣⣘
⠀⠀⠀⠀⠀⠰⠠⠀⠀⠆⠄⠀⠃⠐⠀⠀⠛⠑⠁⠀⠀⠀⠀⠀⠰⠐⠁⠀⠂⠀⠀⠀⠉⠘⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠃⠐⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈

⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀⠀⠀⢀⣀⣀⠀⠀⣀⣀⣀⠀⠀⣀⣀⡀⠀⠀⣀⣀⡀⠀⢀⣀⣀⡀⠀⢀⣀⣀⠀⠀⢀⣀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡿⠋⢀⣼⣿⡿⠫⣠⣼⣿⠟⠉⣠⣾⣿⠟⠁⣠⣾⣿⠟⠁⣠⣿⣿⠟⢁⣴⣿⡿⠋⢕⣴⣿⡿⠋⢈⣴⣿⡿⠋⣠⣾⣿⠟⠁
⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣠⡆
⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠋⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⡀
⠀⠀⠀⠀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠿⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁
⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡆
⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠁⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢁⡀
⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡇
⠀⠀⠀⠀⠟⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟
⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⡆
⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃
⠀⠀⠀⠀⠑⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡄
⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇
⠀⠀⠀⠀⢛⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋
⠀⠀⠀⠀⣠⣿⣿⠋⢅⣴⣿⡿⠋⢀⣴⣿⡿⠋⣀⣴⣿⠿⠋⣠⣾⣿⠟⠉⣠⣾⣿⠟⠁⣠⣾⣿⠟⢅⣠⣾⡿⠋⢅⣴⣿⡿⠋⢀⣴⡇
⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁⠀⠈⠉⠉⠁⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⠀⠉⠉⠁




⡿⢿⠿⠿⠿⡿⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⡟⡻⢿⣿⣿⣿