Expletive Emojis & Symbols

⠀⠀⠀⣄⣀⣄⠀⠀⣀⣄⠀⠀⢰⠀⣠⠀⠀⠠⡶⠀⠀⡖⣿⣀⠀⠀⣤⣴⢤⡆⠀⢰⢤⠄⢀⣄⢠⡄
⠀⠀⢈⣿⣭⡧⠀⠀⣯⣘⡽⠀⠸⠀⢹⠇⠀⢰⢤⡀⠀⠓⣷⠋⠀⠀⣻⣀⣸⡋⠀⢈⠃⠀⠀⢽⣉⠇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠑⠚⠁⠀⠀⠁⠀⠀⠀⠉⠘⠉⠁⠀⠙⠁⠀⠀⠘
⠀⠀⠀⣴⣶⣦⠀⢀⣠⣦⢆⠀⠀⢰⣿⡦⠀⠀⣠⠖⡊⢒⢦⠀⠀⢦⣤⡤⡄⠀⣤⢠⡖⠀⠀⣀⣠⣀
⠀⠀⠀⠸⣿⠃⠀⣼⢋⡴⡟⠀⠰⢦⣉⡡⠦⠀⣇⣼⣷⣿⣄⠇⠀⣾⠦⠟⡇⠀⢻⠸⠁⠀⢿⣬⢣⣼
⠀⠀⠀⠀⠋⠀⠀⠘⠉⠋⠁⠀⠘⠋⠉⠙⠂⠀⠈⠓⠒⠚⠋⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢉⠁⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⡌⠀⣴⣿⣿⣿⡆⠀⠀⠣⡀
⠀⠀⠀⡇⠀⢻⣿⣟⣿⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀
⠀⠀⠀⢦⠀⠀⠙⣿⣿⡷⠀⠀⠂⠀⠀⠀⠀⠀⠐⠈⢀⠀⠀⢀⠀⣿⠸⣇⠀⡆⠀⠀⠀⠀⠠⢊⣽⣿⣿⣿⣆
⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⡀⡆⢸⣀⣻⣿⣦⢙⢤⠼⠀⠁⠀⠀⠀⠀⡗⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠂⠐⠈⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⢳⢉⠘⣿⠀⠟⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠉
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠊⠰⣾⣷⡯⠘⠞⡂⠋⠀⠀⠀⠠⠐⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣺⣿⢻⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠘⠀⡀⠀⠐⠀⠁⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⡠⠀⠂⠈⠀⠀⣦⢸
⠀⠀⠀⠀⠀⠔⠂⠡⡄⣎⠁⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡀⢀⣤⡀⠀⠀⠀⠀⠐⠀⠀⠀⢸⣀⡀⢿⠘⣳⠀⠁
⠀⠀⠀⠀⠀⢀⣶⣧⠉⠌⠄⠃⠀⠌⠀⠁⠀⠀⠀⠀⠀⠀⠈⣄⢮⡻⣗⠌⠀⠀⠀⢠⢖⣛⢲⡘⠿⡇⠰⠈⠁
⠀⠀⠀⠀⢦⣤⡻⢯⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⢽⣤⣉⠊⢀⠀⠀⠀⣿⢿⣹⠞⠁⠀⠛⠀⠀⠀⢀⠠
⠀⠀⠀⢠⡆⣋⠳⠄⠀⠐⠀⠀⠀⡀⠄⠈⠂⠀⠀⠄⠀⠑⠀⠀⠹⡄⠠⠀⠀⠀⠀⠘⠳⠖⠋⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⢉⣉⡇⢀⠂⠀⠀⠄⠀⣀⡄⣯⡅⣶⣦⠐⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠆⡀⢸⡿⠂⠿⠟⠿⠧⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠉⠀⠉⠁⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⡆⢀
⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣰⢦⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠘⡋⠐
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⣀⠀⢠⡞⣋⣳⡀⣾⣄⢠⣾⣿⣓⡂⣼⣅⣶⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⡿⠇⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⡅⠀⠘⣎⣛⣿⠉⢩⡇⠈⠏⠏⠀⠘⡿⠁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠒⠒⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠓⠒⠒⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀
⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⡠⠊⠁⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠿⣿⣷⠄⢸⣿⡷⠀⠀⠀⣰⣿⡏⠀⠀⠀⣿⣿⠀⠀⠀⠙⢦⡀
⡼⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⡿⠁⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠉⠀⠘⣿⡇⠀⠀⣰⣿⠟⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⢳
⡇⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⢿⣿⡇⠀⠘⠛⣻⢿⡟⠛⠃⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠾⣿⣿⡃⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢸
⠹⡄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠻⠆⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⣀⠀⢀⣿⡇⠀⠈⢿⣷⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⡾
⠀⠘⠦⣀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣶⣿⠿⠃⢸⣿⣷⠀⠀⠈⢿⣿⣆⠀⠀⠀⣶⣶⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟⠀⠀⠀⠀⠀⣠⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠒⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⣀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⣴⣾⡭⠴⠶⠞⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⣠⡴⣺⣿⣿⣷⠁⠀⠀⠀⠂⢠⡄⠒⠀⠀⠀⠀⠀⢸⡃⠄⠀⠀⠀⠀⠀⠀⠀⣶⣂⠂⠀⠀⠀⠀⠀⡠⠂⠈
⠀⠀⠞⣗⣻⡭⢟⣯⣹⡆⠀⠀⠀⠁⠄⠐⠈⠁⠀⠀⠀⠀⠹⠛⠁⠀⠀⢀⠀⠀⠀⠀⢯⠭⠀⠀⠀⠀⠀⠀⠢⣖⣉⣈⢉
⠀⠀⢀⠘⠯⡿⠿⠩⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠊⠒⢇⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⢑⣀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡤⣀⣴⠀⠀⠀⠀⠀⠀⠈⢙⡄⠀⢀⣨⡁⠀⠀⠀⠀⡀⠲⠰⢀⡀⠀⠙⠚
⠀⠀⠀⠀⢀⢠⠐⡴⠀⠀⠀⢀⣠⠝⠉⣘⣲⡂⢹⠋⠁⠀⠀⠀⠀⠁⠈⠕⠁⠀⠀⠀⠀⠀⠀⢉⡀⠛⠠⡄
⠀⠀⠀⠪⡄⣕⠌⠀⠀⠀⠀⠀⠀⢈⠡⢸⣁⡆⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⢀⡄⠀⠀⠀⠘⠈⠂
⠀⠀⠀⠀⢰⡌⠀⡀⠀⠀⠀⠀⠀⠁⢠⠏⠀⠙⠀⠀⢀⠀⠀⡀⠀⠀⠀⠈⢳⣢⢴⡼⣠⡤⠀⠀⠀⠀⠀⢰⠀⢀⡄
⠀⠀⠀⣄⠸⠁⠜⡃⢀⡠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠓⢀⠃⠴⠀⠀⠉⠉⣿⣺⣍⣅⠀⠀⠀⠀⠘⠦⠆⠁⡚⠅⣠⠆
⠲⠦⣤⠊⠠⡞⢤⡎⠘⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⠛⡀⠙⠂⠀⠀⠀⠏⠀⠃⠈⠀⠀⠀⠙⠲⠀⢲⠈⠳⠶⢣⣀
⠀⠀⣀⣇⡐⠓⣔⠉⣸⣄⠀⢠⠄⣒⠶⣦⣴⡀⠀⠀⠁⠐⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⠨⠏⠙⢀⣌⡀
⠀⠀⠀⠀⠈⣧⠤⠼⢄⠀⠀⢡⡿⢿⣿⣗⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠁⢲⠎⠁⠈⠉
⠀⠀⠀⢀⢘⡁⠀⠀⠀⠀⠀⠘⣿⣳⠿⣿⣿⢍⠀⠀⠀⠀⠀⠀⠀⠱⣠⢀⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣎⣉⣟⠁⠀⠀⠀⠀⠰⢿⣽⡿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⣦
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⠁⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⣧⢾⣧⡯⣿⡄⠀⢠⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠄⠠⠀⠀⠈⠇⢩⠜⠠⣄⠇⣀⠀⠀⠀⠀⠀⢠⣶⣕⠀⠀⠀⠀⠰⢿⡿⣻⡿⠆⠀⢳⡸⠃
⠀⠀⢰⠀⠀⠀⠀⠀⢠⠋⣴⡒⠃⠀⠀⢠⠆⣘⢢⠤⠏⡎⠀⠀⠀⠀⠀⠀⠈⢿⠂⠀⠀⠀⠀⠀⠚⠉⠁⠀⠁⠀⠐
⠀⠐⠂⢠⠄⣈⠀⠀⠀⠸⢵⠛⠀⠀⠀⠀⠀⠀⡧⠊⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠡⠆⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡠⣴⢖⠘⣷⣾⠀⡀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠠⠆⣀⠀⠀⠀⠎⣀⠄⠀⠀⠀⣦⢿⠻⣶⣫⡿⠛⠃⠄⠀⢀⠐⠀⡬⢃⠐⡏
⠀⠀⠀⢀⠐⣅⠆⠀⠀⠀⠀⠀⠐⠂⡀⢶⠎⠥⠀⠀⠀⠌⡡⠎⢉⡀⠀⠈⠀⠹⢤⠞⠈⠀⠀⠀⠈⢉⡆⣸⡩⠇⢀⠜⠒
⠀⠂⠲⠄⠳⠈⠀⠀⠀⠀⠀⠀⠀⠋⢡⠆⠲⠀⠀⠀⠀⠀⠁⠺⠥⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢈⣄⣁⡎⠈⠓
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⢈⡉⠛⠛⠿⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⣿⣷⠀⠀⢰⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⣤⣤⣤⣤⣬⣍⣙⠻⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣿⣤⣀⣀⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡀⠀⠀⣀⣯⣉⣉⣉⣉⣉⣉⠉⠉⠉⠁⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⠿⢷⡀⢸⣏⣉⣉⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠸⣏⣉⣉⣉⣉⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠌⢿⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣾⣦
⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠿⠛⣋⣹⣿⡿⠿⢿⣿⡏⠀⠸⢿⣿⡁⠆⣿⢋⣽⣿⠇⣸⣿⣿⣿⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⣾⣿⣿⣿⡏⢰⠲⡆⢻⣷⡤⠀⣸⣿⣿⡿⢣⣾⣿⣿⡀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⢉⣡⣴⣿⣧⠘⢦⡴⠟⣿⡧⠀⣠⣿⣟⣱⣿⣐⣠⡿⢛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⡟⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣶⣶⣾⣿⣇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⡿⠓⠖⠒⠒⠚⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠶⠖⠒⠚⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠒⠒⠲⠦⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⡀
⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⣀⠀⠀⠀⠀⠙⢦⣀
⢠⠞⠁⠀⠀⠀⠀⠀⢱⣄⠀⠀⡠⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠀⢠⡿⠃⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⠀⢠⣦⡄⠀⠙⠛⢃⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠙⢦
⡟⠀⠀⠀⠀⠀⢤⣄⣰⣿⣿⣿⣁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡗⠀⠠⠿⠷⣶⠆⠀⠀⠐⠿⠿⢷⣤⣀⣠⣤⠾⠿⠿⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠘
⣇⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡷⠄⠀⠀⢸⡟⠉⣽⣏⠉⢻⡇⠀⠀⠀⣼⠏⠀⠀⠀⢠⣶⣤⣴⠿⠛⠻⢷⣤⣴⣦⡄⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀
⠘⣆⠀⠀⠀⠀⠖⠛⠋⣿⠛⠛⠻⣄⠀⠀⠈⠷⣾⣧⣸⣷⠛⠀⠀⠀⠾⢷⡶⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢀⣼
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠛⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⡆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡴⠶⠶⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⣠⣴⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢁⣀⣤⣴⠶⠛⠋
⣀⣀⣀⣀⣀⣀⣐⣿⣷⣾⣛⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⡻⠻⣟⠟⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠐⠒⠀⠤⢄⣀⠀⢀⡀⠤⠒⠒⠒⠒⠢⠤⣀⠀⠀⠀⠀⠀⣀⡀⠤⠤⢄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣝⢦
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠠⠄⣀⠐⠃⢰⠆⠀⠀⠈⢦
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⢀⠞⢉⢍⠳⡄⠀⢰⠀⠼⢵⠄⠀⠀⣠⣃⣸⣀⠀⠀⠸⣷⣦⣼⣿⣀⡀⠀⠀⢸⠀⡜⠀⠀⡞⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⠈⠦⣀⠼⢀⡇⠀⣮⢴⢠⠋⠀⠀⢉⡏⣁⣏⠀⠀⠠⢾⣿⣿⣿⣿⣿⡁⠀⠀⠸⠤⠇⡄⣸⠁⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠓⠢⠤⠖⠋⠀⠀⠀⡿⠃⠀⠀⠀⡽⠉⢸⠁⠀⠀⠀⠼⠛⠉⠻⠇⠈⠉⠀⠀⠰⣵⢆⠍⣅⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠠⡙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠨⠲⠍⠒⠦⠤⠤⠤⠤⠤⠤⠤⠀⠒⠒⠂⠤⠤⠄⣀⣀⠀⠀⠀⣀⡠⠤⠒⠉⠉⠉⠒⠢⠤⠤⠔⠊⠉⠱⣄⠀⢖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠉⠀⠀⠀⠀⠀⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠈⠉⠢⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠛⠛⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡇⠀⠀⠀⠀⠀⠀⠠⠤⡔⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⣠⠶⠋⠉⠉⠉⠁⠀⣀⣀⡀⠀⠀⠘⠢⢄⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣰⠟⠉⠀⠈⢧⠀⠀⠀⢀⡀⡀⠀⠀⠀⢸⠂⠀⢀⡤⡖⠀⠀⠀⣾⠁⠀⠀⠀⠠⠀⠀⠀⠀⢳
⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⣿⠀⠀⠠⠤⠟⠀⠀⠀⣸⣷⡧⠶⠀⠀⢀⡄⠈⠁⡜⠀⣀⣀⣠⣯⣠⣄⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⠀⠀⠀⠚⠋⣿⣽⡶⠀⠒⠊⢁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⣀⣠⡤⠞⠁
⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠢⡀⠠⢶⡟⡟⠀⠀⠀⠀⣎⡴⠂⢸⠉⠀⠁⢀⡏⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠉⢹⡆⢀⣀⠀⠀⠀⠀⠀⣿⠀⡾⠸⠇⠀⠀⠀⠘⠋⠀⠈⣼⠀⠀⠀⠈⢧⣀⣀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢧⡀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⣀⣠⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠤⠶⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣿⣿⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣀⢀⣙⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑⠀⠀⠀⠘⠛⠿⠻⣿⡿⠀⡰⢿⡿⠿⣿⠿⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⡛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣁⣀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣤⣤⣤⣾⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣟⣋⣩⣭⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠀⠀⠈⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠁⢀⣶⣦⣾⣦⠀⡀⢀⠞⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠂⠈⠛⠋⢉⣉⡨⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣦⣀⠀⠀⢰⣧⠀⠀⠀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠉⠉⠉⠉⠉⠙⠓⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⣿⣿⣇⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠘⠊⠀⠈⠂⠸⠃⠀⠀⠀⠉⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣀⣀⣀⣀⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣤⣤⣶⣾⣧⠀⠀⠀⠀⠀⠀⣰⡖⠀⠒⠀⣰⡖⠀⢒⠀⠆⠀⠀⢀⡟
⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣶⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠛⠋⠉⠙⠋⠙⠲⢤⣀⣀⠀⠀⠀⠀⠀⢀⣀⠀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⠀⢉⡓⢦⡀⠀⡴⣚⡁⠀⠀⠈⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢩⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡆⠀⠻⠿⠀⢷⠄⠘⠿⠃⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢾⣠⡆⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢋⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣚⡭⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢠⣿⣿⣿⣿⣦⠀⢀⡏⠀⠀⠐⠒⠒⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⣦⡀
⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢹⡇⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⢠⡀⠀⣀⠀⠀⢠⡖⣶⠀⣰⠂⠀⠤⠤⣶⠶⠶⠆⠀⣾⠛⡷⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⣉⠀⠀⠁⠀⢀⣼⣧⡤⣿⠶⠂⠀⠉⠁⣰⠏⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⣰⠇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠉⠉⠛⢳⡄⠀⠈⣷⣀⣸⣇⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣄⣠⣤⡾⠃⠀⠛⢻⡏⠹⠇⠀⠀⢠⡞⠀⢰⣉⡷⠀⠀⠸⠇⠀⠀⢠⣞⣷⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⢦⡀⠀⠀⢰⣿⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣤⠄⠀⠀⠀⠀⠜⠀⠄⠀⠀⠸⠈⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⢻⣿⣿⣿⣽⣿⣿⣿⣼⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠙⢿⣿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠔⠁⣿⡉⠁⠀⠀⣼⣏⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣶⣶⡶⠀⠀⠀⠀⠂⠀⣄⠀⠀⠠⠂
⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⢈⠁⠀⠀⠀⠈⢿⣶⣤⣼⣿⣶⣾⣿⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠁⡀⠀⢀⣤⣾⣿⣿⣿⠄⢈⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⣽⡿⣿⣿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠁
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠿⣿⣿⡿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠁⡀
⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣧⣒⣼⣯⣷⣼⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠛⠿⢷⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⢀⣄⠀⠀⠀⠁⠀⠀⠀⠀⣒⡦⢠⡄⠀⡠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⣿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠁⠐⠋⠀⠘⠁⠀⠑⠀⠀⠀⠀⠸⠀⠀⠐⠀⠀⠀⠀⢴⠆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⣢⠏⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂

⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠂⠒⠒⠑⠀⢠⠤⠄⡀
⠀⠀⠀⢀⠔⠁⠀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⣀⠀⢀⡄⠀⢠⡄⢠⣄⠀⠀⠀⠱
⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠀⠀⠀⡠⠀⠀⡖⠒⢄⠌⠈⠢⠖⠂⠂⠀⠸⡀⠀⠀⢱⠀⠸⠝⠃⠀⠁⠈⠉⠀⠀⠨
⠀⠀⠀⠀⠈⠐⠠⠄⢀⣀⡀⠀⣀⣀⠀⠀⡄⠈⠀⠀⢀⡰⠀⠀⢀⠀⢀⡀⠀⠐⠀⠀⠈⠠⠀⠀⣠⣀⠀⠤⣀⡀⣀⡠⠔⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠠⠄⠀⢎⠁⠀⠀⠀⠿⠃⠘⠃⠀⢠⠄⠈⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⣀⠀⠀⠠⠄⣀⠇
⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠀⠢⡤⠃⠀⠀⣀⡤⠤⠐⠒⠒⠠⠤⡀⣀⠤⠒⠒⠒⠚⠒⠲⢤⡀
⠀⠀⠀⠀⠆⠀⡄⡠⠄⢀⠄⢀⠂⠠⠤⠀⠀⠀⠀⣀⠠⠀⠐⠒⠠⢄⠀⠀⠀⠞⠁⠀⠰⢰⢠⠀⠀⠜⠎⠀⠀⠀⠆⢒⠆⡆⠀⠀⠙⡄
⠀⠀⠀⠀⠈⡀⠁⠒⠀⠁⠁⠀⠀⠁⠁⠀⠄⠀⡔⠀⠀⠈⠍⠆⠀⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢆⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠈⠖⢀⠀⠐⠂⠂⠐⠒⠂⠈⠀⠀⠘⠠⠀⠀⠀⠀⣀⡠⠃⠀⠀⠀⠑⠆⢀⠤⠀⠀⠠⠄⠓⠁⠀⠠⠄⢀⡀⣀⠀⡆⠁
⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐

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

⠀⠀⠀⠀⣤⠀⠀⢠⣦⠀⠀⢠⡄⠀⠀⠀⣠⡶⠶⠲⠶⣄⠀⠀⠀⢠⠶⢶⠆⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⢿⡇⠀⠸⣿⠀⠀⣿⡇⠀⢠⣾⠋⣠⠤⣠⡀⢸⡇⠀⠀⣾⠀⣾⠤⢴⡖⠀⠀⠀⣼⠀⢸⠇
⠀⠀⠀⠀⠘⣧⠀⠀⣿⠀⢀⡿⠀⠀⣼⡇⢸⡏⠀⣿⡇⣸⠇⠀⠀⣿⣤⡄⢠⠏⠀⠀⠙⠛⣿⠛⣿⠛⠒⠂
⠀⠀⠀⠀⠀⢻⠀⠀⡿⠀⢸⠃⠀⠀⠸⣧⡀⠛⠚⠙⠛⠁⡀⠀⠀⠀⢠⣧⠋⠀⠀⠀⠛⠛⡿⠛⣿⠒⠒⠂
⠀⠀⠀⠀⠀⠈⡀⠀⣁⠀⣈⠀⠀⠀⠀⠘⠻⢦⣤⣤⣤⠞⠁⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⡇⠀⡇





⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠰⠂⠒⠒⠑⠀⢠⠤⠄⡀
⠀⠀⠀⢀⠔⠁⠀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⣀⠀⢀⠄⠀⠠⡄⢠⣤⠀⠀⠀⠁
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⡠⠀⠀⡖⠒⢄⠌⠈⠦⠖⠂⠂⠀⠸⡀⠀⠀⢱⠀⠸⠍⠃⠀⠁⠈⠉⠀⠀⠨
⠀⠀⠀⠀⠈⠐⠠⠠⢀⣀⣀⣀⣀⣀⠀⠀⡔⠊⠀⠀⢀⡱⠀⠀⠀⠀⢠⡀⠀⠐⠂⢄⠐⠀⠢⠀⣠⣀⠠⠤⠀⠀⣀⡠⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠐⠄⠀⢎⠁⠀⠀⠀⠟⠂⠘⠃⠀⢠⠔⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⣀⠀⠀⠠⠄⢀⠃
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠒⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠀⠢⡴⠃⠀⠀⣀⡤⠤⠐⠒⠒⠠⠤⡀⣀⠤⠒⠒⠒⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠂⠀⡀⡠⠀⢂⠄⠀⠂⠠⠤⠀⢀⠂⠀⠀⠀⠐⠒⠒⠢⢄⠀⠀⠀⠞⠁⠀⢰⢰⢰⠀⠀⠌⠊⠀⠀⠀⠆⢚⠆⡄⠀⠀⠙⡄
⠀⠀⠀⠀⠈⡀⠁⠂⠀⠀⠁⠀⠀⠁⠁⠀⠄⠀⡌⠀⠀⠈⠍⠆⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⠐⢀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠠⠃⠀⠀⠀⠑⠄⠀⠄⠀⠀⠠⠀⠓⠀⠂⠀⠄⢠⡠⡀⠀⡆
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⣴⣿⣿⠧⠀⠀⠠⢾⣿⣷⡄⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⢸⡇⢰⠰⠰⢠⣆⡀⢀⠢⢀⣤⠀⠀⣿⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⢸⡇⠀⠀⠀⠈⠂⠀⠀⠁⠀⠉⠀⠀⣿⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⢶⣶⡶⠶⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣀⣀⣠⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠠⢤⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠴⢋⡴⠋⠉⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠁⠀⠀⠉⠉⠛⠶⣍⡳⡝
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣸⣇⣀⣾⣀⠀⠀⠀⡾⠀⠀⠀⢰⡏⣹⠆⠀⠀⢺⣼⠇⢠⡞⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟⠉⢹⠏⠉⠀⠀⢰⠇⠀⠀⢀⣼⢿⡋⢀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠛⢻⠛⠛⡟⠓⠂⠀⠀⣈⠀⠀⠀⣟⠁⢈⣷⣏⠀⠀⠀⢠⠟⠀⣔⢲⡄⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠙⠛⠋⠀⠈⠀⠀⠀⠋⠀⠀⠉⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠰⣌⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢌⡳⠭⠳⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠤⣤⣤⣤⣤⠴⠖⠋⠉⠀⠀⠀⠈⠙⠦⣄⣀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠙⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠐⠶⠷⣀⠀⠀⠈⠉⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⣄⣠⣿⢿⣦⣀⡠⠀⠙⠃⠤⠤⢤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⠶⠋⠉⠁⠀⠈⠉⠳⢾⣥⡔⠂⠀⠐⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⠶⣤⣄⣠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⠦⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⡾⢣⡄⠀⠰⠶⠶⠤⠄⠀⢀⣴⣿⣤⣈⠷⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠸⣿⣄⠙⢦⠀⠠⠤⠤⠄⣠⣾⣿⣿⠟⠉⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠙⢷⣄⠑⡄⠀⠀⣰⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⣀⣀⡀⠉⢛⣿⣦⣾⣿⣟⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠻⣟⠛⠛⠛⢻⣿⣟⢛⣿⠟⣟⠛⠛⠛⢻⣿⣿⠛⣿⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠳⣤⣤⣤⣬⣽⡟⠁⠀⠘⠳⣤⣤⣀⣩⣭⠟⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣠⠋⠉⣿⡇⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠛⣦⡀
⠀⠀⠀⠀⢻⣄⣀⣿⠇⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣸⡄
⠀⠀⠀⠀⠀⠉⠉⣿⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣤⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⡀⠤⠴⠶⠦⠴⠶⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠲⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣷⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠟⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣷⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⠶⠶⠛⠛⠛⣿⣷⡀
⠀⠀⠀⠀⠀⢠⡜⢫⠀⠀⠀⣸⠉⠀⠀⠈⢻⠉⠀⣻⡇⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷
⠀⠀⠀⠀⠀⡎⠀⡸⠀⠀⢠⡏⠀⠀⠀⠀⢸⠀⢠⡟⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸⡇⠀⢿⡀
⠀⠀⠀⠀⣶⠁⢰⡅⠀⠀⢸⣧⣴⠖⠀⠀⠈⠁⣸⠇⠀⠀⠐⣶⣤⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⢠⠁⢀⣘⠁⠀⠀⠘⠉⢿⡀⠀⠀⠀⣸⡇⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⢸⡇
⠀⣀⠐⣾⠶⠊⣩⣤⣦⠀⠀⠀⠈⢻⡄⠀⢰⣿⠃⠀⠀⣰⠇⠀⠀⠀⠀⠀⠉⣉⣨⣿⠂⠀⠘⣇
⠀⢳⡶⠃⢀⣴⠋⢁⣼⣇⠀⠀⠀⠀⣹⣆⣀⣿⣀⣠⣴⣷⣤⣤⣤⣤⣤⣜⠛⣉⠉⣿⡆⠀⠀⣿⡇⢐⢀⡀
⣠⠏⠀⣠⠞⠁⢠⠟⠉⢹⡦⠴⠶⠞⠉⠉⠉⠉⠉⠀⠀⠀⢀⣠⣤⠶⠟⠋⠉⠉⠉⢿⠇⠀⠀⢻⡏⣉⠉⠳⠆
⠻⣤⣴⡏⠀⣰⠃⠀⣴⠟⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⡟⣋⣉⣁⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡧⠈⠛⢺⣆
⠀⢸⡇⠀⣶⠋⢀⣴⠿⠶⠶⠶⠶⠶⠛⠛⠛⣿⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⢿⣧⠈⢶⣄⣼
⠀⠘⢷⣤⣽⠟⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠶⠉⠁
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠢⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣿⠃
⠀⠀⠀⠀⠻⠶⢶⣶⣶⣤⣤⣤⣤⣤⣄⣠⣠⣿⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⢶⣿⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⠤⢾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠸⡿⠖⠊⠙⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡞⠋⠀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⣀⠀⠀⠙⠛⢷⣦⣄⣀⡀
⠀⠀⠀⠀⣠⠴⠖⠋⠁⠀⣠⣤⠾⢿⣄⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣤⣝⠙⠶⢦⣤⣄⡀⠀⠈⠉⠙⢶⣶
⣀⣘⣿⣦⣤⣤⣤⣴⣛⣛⣉⣁⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣈⣉⣉⣙⣛⣛⣉⣀⣉
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠇⠀⠈⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠛⠙⡳⢦⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣋⠅⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡠⠀⠀⠀⠀⠀⢀⣤⠀⣠⠎⣠⠞⣧⠀⡆⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⣠⢔⣡⡴⠋⣯⣶⠷⠋⠁⠀⠘⣧⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣤⡴⠿⠛⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀⠘⣷⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡆⠀⠘⠛⠲⣤⣀⠀⡀⠀⢀⣠⠴⠋⠉⠀⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⢸⣷⣶⠀⢸⠁⠀⠈⢡⣶⣶⠀⠀⣿⢰⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠬⣉⠥⢀⡟⠀⠀⠀⠈⣛⣫⠄⠀⢿⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⠓⠶⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⠈⠉⠉⠩⠍⠉⠁⠈⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⠒⡶⠒⠒⠒⠻⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⢺⡏⠙⡇⠀⠑⠀⠀⠀⠀⡷⠈⣿⠲⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢉⠀⠀⠀⠀⢸⠀⠀⠹⡄⠀⠀⠀⠀⣰⠇⠀⢸⠀⠀⠀⢀⠈⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢢⠀⠀⠀⢸⠀⢀⣀⠘⢦⡀⢀⡼⠃⣀⡀⡾⠀⠀⠀⡎⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠃⠀⠀⠈⠛⠉⢻⣀⠀⢹⠋⠀⣠⡏⠉⠁⠀⠀⠐⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⢻⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢱⡖⠛⠛⠛⠛⢦⠀⠀⢸⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠸⣏⡙⠲⠤⣄⣀⡄⠘⡇⠀⠀⠀⢀⡏⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡤⠶⠋⣁⡿
⠀⠀⠀⠀⠀⠀⠀⣸⠛⠳⠶⢤⣭⣷⠀⠈⠓⠲⠚⠉⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⠴⠶⠛⡏
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢀⣀⣸⣧⡤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠙⠛⠛⠶⢦⣤⣤⣶⠶⠶⢤⣤⣼⣧⣇⣤⣤⣴⡶⠛⢋⡉⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀
⠒⠒⠒⠒⠒⠒⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢲⣶⣟⠛⠋⠉⠉⠉⠙⠛⠿⣿⠆⠀⠀⠀⠀⢀⡼⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣟⠀⠀⠀⠀⠀⠀⢤⣓⢮⣷⣤⣤⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠳⠶⣤⣤⣒⣮⣳⣵⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠲⡄⠀⣀⣀⡀
⠀⠀⠀⠀⢀⡀⠻⣦⣀⣠⡾⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠹⠊⠀⠀⠈⡆⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠙⣧⠀⠉⠁⢠⡞⠁⠀⠀⠠⣄⡀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠁⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡜⠁⠈⠖⠊⠓⡆
⠀⠀⠀⠀⠀⣠⡟⠀⢀⠀⠸⣧⡀⠀⠘⢶⣌⠛⠶⠞⢡⡾⠃⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠀⢀⠔⠚⠀⠀⠀⠀⠴⢥⡀
⠀⠀⠀⠀⠘⠋⣴⠞⠉⠛⢦⡈⠋⠀⠀⢀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⢀⡴⣄⢸⠉⠉⠀⠀⠈⠢⢤⡄⣀⠀⠀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣡⡶⠲⢦⣌⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⢦⡇⠀⠀⠀⠀⠀⢸⠖⠙⢄⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⣠⠤⠤⠤⢤⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢄⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⠎⠀⠀⣠⡀⠀⠈⢦⠀⠀⠀⠞⣛⣓⣒⠾⠽⢶⡀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⣤⠀⠀⠙⣄⠀⠀⠠⢟⣛⣓⡲⠯⠽⢦
⠀⠀⠀⡏⢀⡠⠞⠁⠙⠢⣄⡀⡇⠀⠀⣠⡇⢐⡄⠀⣔⠀⢳⡀⠀⠀⠀⠀⠀⢰⠃⣀⠤⠊⠀⠑⢤⣀⠘⡄⠀⢀⣼⠀⢢⠄⠠⡔⠈⣇
⠀⠀⠀⣧⢻⠀⠾⢁⠘⠇⢠⢷⡇⠀⠀⠺⡇⠈⠁⠂⠉⠀⡼⠎⠀⠀⠀⠀⠀⢸⡜⣇⠀⠏⡀⠹⠀⣼⣢⡇⠀⠘⢾⠀⠈⠠⠀⠁⢀⡧⠃
⠀⠀⠀⠹⡉⢣⣀⣔⢠⣠⠎⣩⠃⠀⠀⠀⠙⢄⡢⠀⣁⡴⠁⠀⠀⠀⠀⠀⠀⠀⢧⠙⢆⡀⠀⢀⡴⠋⡼⠀⠀⠀⠀⠳⣄⠐⠂⣠⠞
⠀⠀⠀⠀⠙⢦⡜⢋⠛⣦⠔⠁⠀⠀⠀⠀⠀⣴⠛⡍⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠟⣉⠻⡤⠊⠀⠀⠀⠀⠀⢀⣞⢩⡍⢷⡀
⠀⠀⠀⠀⠀⡾⠑⠊⠒⠹⡆⠀⠀⠀⠀⠀⣸⠁⠉⠀⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠂⠉⠂⢱⡀⠀⠀⠀⠀⠀⡞⠀⠁⠈⠀⢳⡀
⠀⠀⠀⠀⡼⠀⠡⠀⠠⠁⠸⡄⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢂⠀⡐⠀⢡⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣇
⠀⠀⠀⢀⡇⠀⠀⠑⠁⠀⠀⣇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⢱⠀⠀⠒⠀⠐⠘⡆⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡾⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⡄⠀⠀⠀⠀⢸⢸⡆
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡇⡀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠃⠁⠀⠀⠀⠀⠈⠀⠃
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⢀⣿⣇⣤⣴⣶⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁⢀⣙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠛⠛⠉⠁⢀⣀⣤⣤⣶⣾⡿⠿⠟⠛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣴⣴⣦⣴⣦⣴⣴⣶⣾⠿⠿⠛⠛⠉⠁⢀⣀⣤⣤⣶⣶⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠉⢹⣿⠟⠉⠁⢀⣀⣤⣤⣶⣾⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⠄⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⣿⣶⣶⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⣴⣿⢿⣦⠉⠹⣿⡆⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡍⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠰⣿⡌⣿⣿⣩⡀⢻⣿⠈⣥⡄⠀⢻⣿⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣧⠀⣤⣴⣦⣄⢸⣿⣟⡋⠀⠀⠀⢿⣷⢸⣿⣟⣡⣌⢿⣿⡿⠃⠀⠈⠉⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠸⣿⡆⢿⣿⣽⣿⠈⣿⣿⠛⣀⢰⣷⣾⣿⠂⠻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⣀⣀⡀⢻⣷⠘⣿⡿⣿⣦⠹⣿⠿⠟⠃⠉⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⠿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠛⠛⠛⠛⠃⠘⣿⣇⠙⠋⠈⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⡿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⣀⣠⣤⣴⣶⣿⠿⠿⠛⠋⠉⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡤⣴⠶⢍⡇⢠⡎⠉⠃⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⡀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡌⠠⡶⡤⣟⠄⠁⢠⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠀⢘⡄⠀⠀⠀⠀⠀⠀⠆⠀⠡⣀⣙⠁⠎⢈⣒⠮⠐⢐⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⠀⣼⠀⠀⠀⠀⠀⠀⠀⠰⡀⠤⠀⠂⠀⣁⣡⡤⠖⢛⣩⡥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠈⠈⡱⠂⠀⠀⠀⠀⠀⠰⠶⠒⠛⠉⠉⠰⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠒⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⢉⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠚⢉⣁⠤⠴⠒⠚⠉⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⠒⠊⢉⣁⠤⠴⠖⠒⠉⠁⣀⣠⠤⠴⠒⠋⠁⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⡠⠤⠶⠒⠚⠉⠁⣀⡠⠤⠶⠒⠋⠉⢀⣀⡤⠤⠖⠚⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡤⠤⠖⠒⠊⠉⢀⣀⡤⠤⠖⠒⠉⠁⠀⣀⣠⠤⠴⠒⠉⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣀⠤⠴⠒⠒⠉⠁⠀⣀⡠⠤⠖⠒⠋⠉⠀⢀⣀⠤⡤⢖⠿⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⡤⠤⠖⠒⠊⠉⠀⢀⣀⣤⠴⠒⢊⣉⣥⠴⠊⠉⠀⣀⣀⠀⢹⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⡠⠤⠴⠒⠚⠉⠁⢀⣠⡄⠀⠀⠁⢀⣀⣤⡴⠺⣿⣹⣇⠀⢻⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⢀⣠⡤⠔⠚⠉⠁⠀⣤⣦⡞⠻⣏⠙⣿⣫⡹⠿⠟⠀⠀⢻⡆⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠁⠀⠀⣤⢰⣟⣁⢹⡄⢤⡄⠻⠆⠙⠉⠀⣀⣄⠶⠚⠉⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠉⠀⣀⣄⠰⣾⣃⠀⢹⣆⢻⣥⠆⠛⠋⠀⢀⡀⡴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⣿⣽⣅⢻⣯⡼⠛⠛⠀⠀⣀⣤⡴⠜⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠘⠷⠙⠃⠁⠀⣠⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠄⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⣄⠴⠚⠈⠉⠀⡀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠉⠀⠰⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠛⣛⣛⡻⠶⠿⢛⠛⠏⢩⡙⠻⣦⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠛⠋⣉⣩⣤⣽⣿⣿⣿⣷⣼⣿⣶⣌⣿⣷⣌⢻⠟⠛⣙⡛⠻⢶⣤⣄
⠀⠀⠀⠀⠀⠀⣠⡶⠾⠛⣛⣭⣵⡾⠿⢿⣛⣭⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣶⣶⢗⣠⣿
⠀⠀⠀⠀⠀⠀⠹⣷⣬⠛⢛⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⡻⣦⡀
⠀⠀⠀⠀⠀⢀⣴⠟⢡⣾⣿⣿⣿⣿⣿⡿⠿⠯⠙⢳⣶⣶⣿⣿⣿⣿⣋⣘⣿⣿⣿⣿⣿⡛⠛⠿⣿⣿⣿⣮⣛⠛⢷⣦⡀
⠀⠀⠀⠀⠀⣼⠇⣴⣿⣿⣿⣿⣿⣿⣍⣀⣶⣾⣿⡿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⢸⡇
⠀⠀⠀⠀⢰⡟⠼⢻⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⣠⠔⣠⣿⣿⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢠⣿⡃
⠀⠀⠀⠀⠸⣧⡀⣾⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⣿⣿⣿⣿⣿⡿⠿⠆⣸⡇
⠀⠀⠀⠀⠀⢿⡄⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⢿⣿⣿⣿⣿⣿⣿⠷⠦⠘⣿
⠀⠀⠀⠀⠀⠸⣧⠛⠁⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡆⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠙⢻⡇⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⠛⢿⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⠋⠉⠻⣿⠇⢿⡿⢿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⢿⣧⠀⠉⠃⠀⢹⡏⠰⠀⡜⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⡀⠀⠀⠹⠀⣈⣃⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠀⠀⣠⡔⠉⠀⠀⡃⣠⢎⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡱⡄⠀⠠⠀⠀⠉⢛⣶⣶⣤⣤⡤⠀⠀⠀⠠⣤⣤⣶⡿⢯⡄⠀⠀⠀⠣⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢹⠈⢻⠀⠀⠀⠈⠑⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⡸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣜⠤⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢂⣀⠀⠀⠀⠀⠀⠀⠀⢰⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⢣⠀⠀⠀⠀⠀⠀⢰⣦⣄⣀⣉⣠⣴⣿⡄⠀⠀⠀⠀⣠⠣⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⠔⠉⢣⠀⠀⠀⠀⠀⣾⣿⡿⢿⡿⠛⢿⣿⡇⠀⠀⢀⠔⠀⠉⠢⡙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡱⠃⠀⠀⠀⠁⢀⠀⠀⠀⡿⢹⣧⣀⣀⣠⣾⡆⠀⢀⠴⠁⠀⠀⠀⠀⠘⡜⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠡⠁⠀⠀⠀⠀⠀⠀⢙⠲⠤⠔⠈⠙⢛⣛⣛⠩⡰⠿⠉⠀⡄⠀⠀⠀⠀⠀⠘⡸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠠⡈⠑⠂⠤⠄⠊⢁⠃⠂⠀⠁⠀⡀⠀⠀⠀⠀⠁⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢎⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠒⠀⠤⠀⠂⠅⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⣰⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡎⠓⢤⣀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠤⢶⢫⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠉⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠸⢷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠘⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⢸⠉⠢⠭⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⡠⠤⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣋⣉⡁⠀⢠⠞⠙⣇⠈⠑⠒⢹⠉⠉⠀⠀⠀⠠⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡢⣄⠀⢀⡴⠋⠀⢀⣹⣠⠏⠀⡖⠉⠐⣦⢄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢩⣷⣤⠎⠀⠀⠀⠙⡟⠁⠀⣼⡴⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠈⠳⠤⢀⣀⣠⢭⣟⠀⠀⠀⣠⠞⠁⠀⠰⣻⡥⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⡉⠉⠙⠟⡛⠋⠁⠀⠀⠀⠈⠡⠒⠛⠛⢂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣐⣠⣂⣤⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠍⢀⣀⣀⣀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡽⡬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠒⠸⡇⠀⠀⡇⠀⠀⡏⠉⢱⡖⠤⣄⠀⠀⠀⠀⣠⠔⢉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢘⣉⠦⠔⣉⡤⠚⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢏⠉⠑⠒⠳⣤⣤⣧⠠⠼⠒⠈⠉⣠⠔⠊⠁⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢛⢿⡖⠊⠋⢍⠁⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀
⠀⠀⠀⠀⢠⠎⡴⢳⡌⣵⠦⣀⠀⠀⠀⢀⣠⠤⣦⣀⣜⠸⡿⠀⠀⠀⠙⠲⢤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠄⡀
⠀⠀⠀⠀⣾⣸⣃⡷⣼⢣⡟⠊⡗⢰⣹⠹⣇⢉⠔⠋⢁⣠⠒⡄⠀⠀⣀⣠⣄⣀⣀⣀⠤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣴⠃
⠄⠀⠀⠀⠈⠛⠨⢽⣇⣾⣠⠦⠽⢆⢀⠀⠀⢧⢀⡜⠁⣀⠙⠠⡔⠚⢦⠸⣸⠈⣆⠁⡀⠀⣀⠔⠉⠁⠀⣀⡤⠴⠋⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠘⠓⣿⠏⠀⠐⠄⠐⡋⢸⡦⢉⠀⢸⠀⣷⣊⣠⡤⠶⠚⠉⠁⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠤⠜⠛⠛⠯⠽⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡴⠤⢄⣸⡇⢴⠂⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠑⢿⣄⢀⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠄⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⣙⣻⣄⡴⠶⠻⣉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⢉⣳⣶⡀⢤⣤⣤⣤⣤⠤⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠻⣧⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠿⣇⠀⡟⢳⣾⠟⢻
⠀⠀⠀⠀⠀⠀⣠⠀⠀⣿⢀⣇⣸⣿⠀⠘
⠀⠀⠀⠀⠀⠀⣿⠀⠸⠿⠿⠿⠿⠿⢿⡇⠀⢠⣶⣷
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣿⠹⡇⠀⠀⠀⠀⢀⣴⣿⣿⡿⠻⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣦⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠉⠉⠉⠙⠻⢿⣿⣿⣿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠋⠀⠀⣀⣀⠀⠀⠈⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠀⠀⠀⠙⠛⠉⠀⠀⠀⠘⢿⡟⣿⡏
⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡃⠀⠀⠀⢠⣶⣿⣿⣆⠀⠀⣠⣄⠉⢻⡇
⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣀⠶⠛⠻⠿⢿⣿⣿⠇⠀⠀⠀⠈⠛⣉⣴⠟⠀⢸⣿⡟⢡⣿⡇
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⣤⣴⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⠉⣥⣾⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⣧⣄⣸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠉⠉⠛⠛⣋⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠻⣶⣤⣤⣾⠟⠁
⠀⣠⠾⢿⣇⠀⠀⠀⠀⢠⣿⢻⣧⡀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠶⣿⣁⠀⠀⠀⠀⠀⠀⣹⠇
⣴⠋⣠⣾⣿⣦⣄⣀⣴⡿⢿⡄⠹⣿⣆⠀⠀⠀⣠⣄⠀⢸⣿⣿⣆⠀⠀⠀⠀⠈⠙⠿⠷⠶⠶⢀⣼⡏
⣿⡐⠻⠿⣿⣿⣿⠋⠉⢿⡌⢻⣆⠈⢿⣷⣀⣾⠉⠉⢻⣿⣇⠘⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠹⣿⡀⠀⠈⢻⣟⣧⠀⠀⠹⣦⡘⢧⡀⠻⢿⣧⣴⣶⣛⣻⣿⣷⣤⡉⢿⣿⣦⣄⣀⣀⣠⣾⠟⠁
⠀⠘⢿⣦⡀⠀⠻⣯⣷⣄⠀⠈⠻⣮⡷⣄⣸⣿⣥⡶⠛⠋⠀⠈⢻⣿⣾⣿⠛⠛⠿⢿⣿⠃
⠀⠀⠀⠙⢿⣦⣄⠉⢿⣿⣷⣄⠀⠉⠳⠉⠹⣿⡃⠀⣠⣤⣶⣿⣿⡿⢿⣿⣦⣀⠀⠈⣧⣀
⠀⠀⠀⠀⠀⠉⠻⣷⣄⠙⠛⡹⣷⣤⣤⠀⠀⢻⣿⣾⡿⠛⢻⡟⠛⠀⠀⢹⣿⠛⢿⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡏⢻⣿⠁⠀⠀⣾⢻⣿⠀⠀⢸⡟⠀⠀⠀⢸⣿⠀⠀⠈⢻⠁⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣷⣾⣟⠀⠀⠀⣿⠘⡇⠀⠀⠸⣿⠀⠀⠀⣸⣿⡀⠀⠀⣾⠀⣿⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⠈⢻⣿⣄⡀⠀⣿⡄⠀⠀⢠⡀⠙⠀⠀⠀⡿⣿⡇⠀⠀⣿⠀⣿⢠⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⢷⣾⡏⠙⠻⣿⣿⡟⠀⠀⢸⡇⠀⠀⠀⠀⣧⢸⡇⠀⠀⢿⠀⠃⢸⣷⣈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⢨⣿⠇⠀⠀⣿⣿⠃⠀⠀⠘⣷⠀⠀⠀⠀⢿⡼⠇⠀⠀⠈⣦⠀⣸⣿⠟⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⡟⠀⠀⣠⠻⣷⠀⠀⢠⡄⢸⠀⠀⠀⡆⢸⡇⠀⠀⢸⠀⣿⠀⣿⣇⢀⣀⡾⠿
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢠⣿⠇⠀⠀⣿⡆⠁⠀⠀⠘⠃⠘⠀⠀⠀⡟⠀⠇⠀⠀⠸⢀⠈⠀⣿⣿⠉⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⣼⣿⠀⠀⢰⡀⠻⠀⠀⢠⡆⢰⡄⠀⠀⣤⠀⡄⠀⠀⢸⡀⣼⠀⠀⣿⣿⠀⢹⡇
⠀⠀⠀⠀⠀⣰⣿⠃⠀⢠⣿⠇⠀⢀⠘⣷⠀⠀⠀⡘⢃⡘⠁⠀⠀⢋⡀⠃⠀⢀⡘⣧⠈⠀⢠⣿⣿⡆⢸⡇
⠀⠀⠀⠀⠀⣿⣟⠀⢀⣿⡏⠀⠀⠈⣷⡏⠀⠀⢸⡧⠸⡇⠀⠀⢾⠀⠟⠀⠀⠸⠇⠹⠁⠀⢸⣿⡿⠃⢸⣷
⠀⠀⠀⠀⠀⠻⣿⣦⣾⣿⠀⠀⠸⣆⡛⠀⠀⢀⣾⣠⣤⣀⠀⣠⡀⣤⠀⠀⢰⡆⢸⠆⠀⠀⣿⡟⠀⢠⡾⠇
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣤⣴⣷⣿⣧⣤⠴⣾⢻⣿⠉⠀⢀⣹⣷⠁⠀⢀⡜⢳⡄⠀⢀⣼⣿⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣷⣿⣿⡏⠀⢤⣼⣿⡎⠁⣠⡿⠟⠃⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠈⠛⠻⣿⣿⣿⠋⠛⠛⠛⠿⠿⢿⣿⣀⣘⣿⣿⣿⣾⣿⡇⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡟⠉⠀⠀⢹⣧⣾⠏
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡇⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣸⡇⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣴⡿⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣿⡿⠁⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢸⣧⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠸⣇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⠟⣿
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢹⡄⠘⢃⠀⠀⠀⠀⠀⠀⢰⡟⠀⠘⣧
⠀⠀⠀⢠⡜⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⡄⠀⠀⠀⠀⠀⡿⠃⠀⠀⢿
⠀⠀⢰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣷
⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⢿⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢻⣆
⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠐⣿⣯⡙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠨⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⢸⣿⣿⣦⣌⡙⠿⣶⣄⣀⠀⠀⠀⢠⠉⣿⡄⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⡿⠀⠈⠙⠻⠿⣶⣮⣭⣿⡷⠶⣶⡆⠀⠘⣷⠀⢻⣦⣀⣀⣀⡠⠶⠶⠶⠆⠻⣷⡀
⣶⡀⠀⠀⠀⠀⠀⠈⣿⣭⣽⣿⠿⠟⠛⠛⠾⣿⣧⠈⢻⣏⠉⠁⠀⠀⠀⠀⢀⣤⣿⠿⠂⣀⣠⣤⣤⣤⣤⣤⡀
⠿⣿⣦⣤⣤⣤⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⡉⠀⣀⣀⣀⣤⣶⡟⠋⠀⠀⣠⣾⠟⠋⠉⠉⠉⠉⢿
⠀⠈⠉⠠⢿⣿⣷⡶⠄⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡘⠻⣿⠛⠛⠛⠛⠁⠈⠻⠶⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠉⠛⠷⠶⠿⣟⣿⣿⣿⣿⣿⡶⠶⠟⠁⠀⢹⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠿⠿⠟⠋⣉⣥⣤⡴⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠙⠃⠈⠉⠉⠉⢉⠖⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠖⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⡀⠀⣀⣀⡀⠤⠤⠒⠒⠉⠁⠀⠀⢡⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠑⢄⡲⠉⠀⢀⣐⠲⠒⠀⠀⠀⠘⢋⡤⠤⠤⠈⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠘⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⡙⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⣤⣄⣀⠀⠀⠀⣀⣀⣤⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⣈⠒⠀⠐⠁⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣰⣋⣉⣉⣭⣉⣉⣉⣛⣻⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠧⢤⡤⢄⠀⠀⢀⠀⣀⡨⠗⠁⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣄⣸⣀⡾⠃⠀⢷⣂⣠⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠐⠀⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⢹⠉⢩⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣓⣢⡼⣤⣜⣢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠛⠛⠛⠛⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡾⠁⠀⢠⣂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⣸⠇⠀⢈⠏⢹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⣿⠀⠀⠺⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣾⠀⠀⠀⠀⣴⠞⠋⠉⠁⠀⠀⠈⠉⠉⠛⠶⣤⣀
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣧⡴⠇⠀⣀⣀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠳⢼⡇⠀⠀⠺⠤⠜⠘⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠹⣆
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠳⣼⠁⢀⣤⣤⣄⣀⣀⣹⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣦⢄⢻⡄
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠋⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⠸⡇
⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⢀⡏⠓⢾⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢸⠷⢤⡾⠀⠀⢀⡔⣶⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⡾⠀⠀⢀⠀⠀⣸⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠇⠀⠐⠭⠇⠀⣿⠃⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⣯⠤⠶⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣌⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⡀⠀⢀⣀⣀⡀
⠀⠀⠀⣠⠤⠶⠶⠚⠉⠀⠀⠀⠀⠈⠛⠋⠉⠉⠉⠉⠛⢦⡀
⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄
⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟
⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⡴⠦⣤⣀⣀⣀⣀⣀⡼⠖⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡓⣶⡄⠈⠉⠉⠁⠀⠀⢀⣰⣤⣤⣤⣀⣠⣄⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢛⣀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢀⣿⣿⣿⡿⠟⠛⠙⠻⣿⣿⣿⡿⠃⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠀⢀⣀⠀⠀⠈⠉⠁⠀⣀⣀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣻⣿⣷⣦⡀⢀⣴⣿⣿⣿⡂⠀⢸⣧⠀⠀⡒⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡸⠿⡇⠀⠘⢠⢊⣝⡟⠁⠀⠛⠻⣭⡆⢹⠀⠈⡼⢿⠁⢤⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣁⠸⠲⠱⡀⠀⠈⠮⠾⠭⠄⠀⣀⠘⠦⡠⠄⠋⠀⠀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⡤⠟⠀⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠠⠤⠤⠤⠤⠔⠒⠚⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⡖⡄⠀⠀⡠⠒⠈⠁⠀⠂
⠀⠉⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⣠⠈⢂
⠀⠀⠀⠰⠀⠀⠀⢀⠀⠀⢀⡠⠼⠁⡶⠃
⠀⠀⠀⠀⠀⠀⠀⢁⡀⠁⠀⠐⣹⣇⡄
⠀⠀⠀⠀⠀⠀⠀⢫⠸⠀⠀⠀⠉⢈⢣
⠀⠀⠀⢈⡄⣀⠴⠁⢄⡀⠀⠘⣧⠎
⠀⠀⠀⢼⣵⠏⠀⢀⡐⠁⠁⠀⠁
⠀⢰⠁⠃⠃⠀⠀⠈⡇
⠀⠈⠀⠀⢀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢤⠀⠠⢀
⠀⢀⠀⡠⢾⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⡐⠀⡈⢢⠀⠁⠄
⠀⡼⡏⠀⠀⠆⠀⠀⣆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣼⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣵⣖⡆⠀⡟⢠⠀⠀⠀⡆
⠀⢠⠃⠀⠀⠰⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠊⠁⠢⠅⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢂⡉⠀⡠⠄⠀⠀⠀⠀⡁
⠀⠈⠀⠀⠀⠀⠁⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠲⢀⡀⠀⠁⠀⠀⠀⠠⡎⢹⡘⡟⠁⠀⡇⠀⠀⠀⠀⢀⠃
⠀⠀⠀⢃⠀⠀⢸⣇⠀⠀⠱⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⣀⠀⠀⠀⠈⡏⣶⡄⠐⠋⣶⣖⠐⠂⠁
⠀⠀⠀⠈⠂⠀⠘⠘⢀⣀⡤⠦⡴⠆⠲⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠒⠠⣀⡇⡆⠈⠂⠒⢛⣏⢀⡇
⠁⠆⢤⠀⣀⣀⡗⠀⠈⣧⣀⣴⢦⠖⠺⣍⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠜⠁⠀⠀⠀⠸⡟⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⢿⡟⠀⠐⣽⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⢁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣌⡁⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠀⠀⠀⠀⠀⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠑⠒⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⠀⠀⠀⠡⡀⠀⠀⠀⢫
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⡀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠀⠀⠀⠈⠐⠀⠀⠀⠣
⠀⠉⠒⠤⢤⠒⠒⠲⠄⠒⢻⠀⠑⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠈⠀⠀⠈⠢⣤⠚⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⠹⡄⠀⠂⢲⠀⠐⠋
⠀⠀⠀⢀⠘⠀⠀⢸⠀⠀⢦⠀⠀⠀⠀⢡⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⠰⠀⠂⠈⡇
⠀⠀⠀⠨⢀⠀⠀⢸⠀⠀⠘⡀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⡰⡇⠀⠀⠁
⠀⠀⠀⡙⠶⢌⠀⠸⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠈⠲⠊⠀⠀⠀⠀⠐
⠀⠀⠀⠉⠣⠔⢍⠂⠴⢄⠀⢠⠀⠀⠀⡘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⡅⠀⠀⠀⠠⡄⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠸⡀⠀⠀⠙⠂⢬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⢦⣀⠈⣂⠀⠀⠀⢡⠀⠈⠂⠀⠤
⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃⠀⠀⠀⠈⣦⡀⢀⣤⢚

⠀⠀⣠⣤⣤⣤⣀⣸⣷⣦⣴⣾⣿⣦
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠘⠿⠿⢟⠋⡙⢛⡉⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠸⠙⠁⠼⠇⢸⣿⡿⠛⣿⣿⣿⠛⠉
⠀⠀⠀⢱⠄⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠈⠇⡀⠀⢀⣀⣤⣼⣿⣍⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣤⡄
⠀⠀⠀⠀⠀⣰⠟⠛⠉⠉⠁⣨⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⠟⠻⠿⢿⣿
⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⠿⢿⣿⡿⠃⠀⠀⠀⠀⢹
⠀⠐⣠⡗⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣿⣿⣿⠀⠂⣹⡇⠀⠒⣀⠀⠄⠸
⠈⣼⠋⠀⢀⣴⡀⠀⠀⢼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣤⣹⡏⠛⢛⠇⢠⣿⡇⠀⠁⠿⠋⠀⡄
⠰⠛⠄⣠⠏⢹⡷⠀⠀⢺⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠉⠉⠛⠿⠛⣷⡐⠃⠀⠀⠉⠁⠀⠀⠀⠀⢰⠁
⡆⠀⠀⠁⢀⢼⣇⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢠⡟⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜
⠧⠠⠐⠊⠀⣸⠏⠀⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⣿⠃⠀⣸⣿⠟⠀⢀⣀⣠⣀⠔⠁⢂⡀⠃
⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⣼⠗⠀⠀⠀⢠⡿⠀⣴⣿⠁⡘⠂⠚
⠀⠀⠀⢀⣾⠁⡶⢠⣄⣀⣀⡠⣤⢶⢾⡆⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢀⡼⠁⠐⣿⡷⢠⣿
⠀⠀⠀⢸⡏⢰⡇⣾⢹⡇⢻⡇⠹⡎⣇⢻⡀⠀⠀⠀⢀⣿⠁⠀⠀⢀⣾⠁⠀⣠⣤⡄⠠⢿⡀
⠀⠀⠀⢸⡄⣾⠀⣿⢸⣿⠘⣷⠀⣿⢹⢸⡇⠀⠀⠀⣸⡇⠀⠀⠀⢸⣿⠀⠀⣀⣀⡀⣷⢸⣷
⠀⠀⠀⢸⡇⣿⠀⣿⠈⣿⡄⢻⡀⣿⢸⢸⡇⠀⠀⠀⣿⠁⠀⠀⠀⢼⣿⠀⠈⢉⡉⣱⠋⢸⣿⡄
⠀⠀⠀⢸⡇⣿⠀⣿⡆⢻⡇⢸⡇⣿⠸⢸⡁⠀⠀⢸⡏⠀⠀⠀⠀⠈⣿⡀⠀⢸⡇⢿⠀⣸⣿⣿⡄
⠀⠀⠀⢸⠁⣿⠀⢸⡇⠘⡇⢸⡇⢸⡄⢞⡇⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⢸⡆⢿⣿⣿⠇
⠀⠀⠀⠧⢠⣿⡠⠾⡁⢰⡇⢸⡇⢸⡇⢸⠅⠀⢸⡏⠀⠀⠀⠀⠀⢘⣿⠀⠀⢸⣿⠊⣧⠈⢿⡏
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠸⣿⣇⣀⣼⡿⢧⡿⠄⠈
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠸⣅⣀⡀⠀⠀⠀⠀⠀⢹⣿⣷⡖⠒⠒⠠⠄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡄⠀⠀⠆⠀⠀⠀⠸⠳⡀⠉⠒⠢⠤⠄⣀⣈⣙⡃⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢀⣀⡹⢆⣀
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⡸⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠶⣌⣠⠐⠢
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠐⠂⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠲⣎⣸⡒⢠
⠀⠀⠀⠀⠀⠀⠀⠸⢀⣶⣶⣴⡆⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠡⠘⠬⠋⠤⣀⠀⡄
⠀⠀⠀⠀⠀⠀⢠⣤⡿⢻⣿⣿⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⢰⠈⠦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢲⠚⣛⠠
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠋⠉⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡏⠛⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⣴⣀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⡄
⠀⢀⠀⢀⣉⠄⡠⠊⠀⠀⠀⠈⠀⢉⡝⠶⢄⡀⠀⠀⡜⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠐⠸⣍⡵⢃⠀⠀⠀⠀⠐⠠⠁⠀⠀⡀⠈⠐⠶⠁⠀⠀⠀⠀⠀⠇⠀⠀⢀⣀⡤⠐⠸⠉⡁⠀⠢
⠀⠀⠐⠀⠤⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⡀⠀⠣⠀⠀⠀⠀⠀⢛⠉⠱⠒⠒⠂⠒⠒⠊⠉

⠀⣜⣔⣾⣿⠀⡰⣡⣾⣿⣿⣴⡼⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉⠀⠄⣿⣿⣿⣿⣾⣦⠀⣷⠈
⠀⡟⢻⣿⣿⣴⣿⣿⣿⣿⣿⠾⠀⠀⠀⠀⠀⡎⠁⢸⠀⠀⠆⠂⠀⡖⠀⠀⠀⠀⡶⣿⡟⣻⣿⣿⣿⣷⣿⡆
⠀⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⡴⠀⠀⠀⠀⠉⣧⠉⢹⡀⠉⣁⠀⠀⣇⠋⠉⣡⠞⣶⡏⣹⣿⣿⣿⣵⣿⢹⣷⡀
⠀⣨⣿⣿⡗⠋⢿⡟⠛⠿⠿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⣥⣾⣿⣷⠿⠟⠋⢿⠙⣿⡏⢻⡇
⠀⣙⣧⣿⣷⠀⢸⠇⠘⣟⠛⠛⠛⢛⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡛⠛⠙⠙⢻⠆⠀⢸⢠⣿⠇⢼⡇
⠀⢻⡇⢆⢻⡄⣼⠀⠀⠘⠦⣄⡀⠀⠀⢀⠈⣻⣿⡟⠋⠁⢈⣿⡿⠋⠀⠀⠁⠀⣀⡴⠋⠀⠀⢸⣾⣿⢀⣴⡇
⢀⡘⢧⢨⣷⣷⡿⡄⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣤⡄⠀⠀⢘⣯⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⠏⣤⡏
⣽⡷⠚⢿⡿⠋⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡁⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣤⣟⠁⢀
⠛⠀⣀⠈⣻⢶⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡁⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣰⣿⡟⢶⣄
⢀⡤⠀⠀⡏⠀⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⢹⡝⠀⠀⠉
⠋⠀⠀⣼⣥⡆⠀⠸⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣾⢦⡁
⡀⠀⣰⠿⣏⢹⢀⢦⠀⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢸⣿⣷⣯⡀⠀⣀
⡿⣷⣤⡀⠿⣣⣿⣿⣧⢸⣿⣷⡀⠀⠀⠀⠀⢉⣁⠀⢀⣀⣀⡀⢈⠉⠀⠀⠀⠀⢀⣾⢿⣟⣾⣿⣿⣶⣿⣿⣿
⣿⠾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠙⠀⢀⣀⣤⣴⣿⠾⣿⣻⣿⣿⣿⣿⣿⣿⣿
⣏⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡿⣦⡀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠿⠿⣿⣦⠀⢀⠴⠛⠉⠉⠉⠁⡼
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡏⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣀⣀⠀⠀⠀⠹⡤⠴⠒⠋⠱⣄
⠀⠀⠀⠀⠀⠀⢸⣧⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⡿⠟⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⣿⠟⠃⠈⠛⠿⣿⣿⣿⣷⣶⣤⡷⣄⠀⠀⠀⠀⠙⠲⣤
⠀⠀⠀⠀⠀⡤⠖⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠛⠛⠉⠁⠀⢠⣿⣿⡿⠋⢀⣾⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⠗⢤⡉⠓⠒⠒⠒⠋
⠀⠀⠀⠀⡞⠀⠀⣿⣿⣯⠉⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠠⠿⠛⠁⠀⠀⠸⠛⠋⠉⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠹⡄
⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⢈⣩⣭⣿⣿⣷⣤⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣶⣿⣿⣭⣍⣉⠀⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⡇
⠀⠀⠀⠈⣇⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠉⠻⠾⠏⠀⠈⠃⠀⠀⠀⣸⠃⠀⠀⠀⠘⠉⠀⠙⠾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⢠⠇
⠀⠀⠀⠀⠈⠢⢤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠦⠤⣤⣤⣤⣤⣤⣤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠠⡀⠀⠀⠀⠀⡰⠁⠈⠉⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⢀⠀⠀⠀⢸⠁⠀⠀⢀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⠀⠀⠀⠨⢤⠑⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⢢⠊⡃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡗⢄⠀⠀⠀⠰⡄⠀⠀⠀⠈⠠⠌⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⣠⢴⠁⢐⠀⠀⠀⠀⠈⡑⠀⠀⠀⠀⠀⣧⢰⡀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⢠⠀⠀⠀⢠⡎⢨⡴⠚⠁⠈⢦⠰⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⢆⠘⠤⣀⠀⠀⢀⣠⣾⠃⠄⠰⠤⢤⡤⠄⠤⠤⢤⡴⠞
⠞⠀⠀⠀⠉⠰⡁⠀⠀⠀⠀⠀⢳⡐⢄⡀⠀⠀⠀⠀⢀⡴⠀⠈⠑⠤⢄⣌⣭⠅⠖⠁⢀⠛⢒⠒⠒⡒⠒⠒⠚⠉⠀⠀⠀⠀⠀⡀
⠆⢄⠀⠀⠈⠳⢌⣺⣿⠀⠀⠀⠈⡳⣄⡈⠁⠒⠒⣈⡥⠃⠀⠀⠀⢀⠠⡀⠀⢰⠊⠁⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠠
⠠⡀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠉⠉⠀⠉⠁⠀⢀⢦⡀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠤⠤⢄⡠⣤⡄⠀⠀⠆⢀
⠀⠘⢦⡀⠑⠂⠤⠤⠤⠰⠒⣉⠜⢀⠠⠂⠀⠀⠀⠀⠢⡓⢄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡀⠀⠀⠀⠀⠀⠸⠦⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠑⠲⠤⠤⠤⠖⠊⠁⡄⠁⠀⠀⢀⣤⣄⠀⠀⠈⠀⠑⢄⡈⠒⢄⠀⠀⠀⠀⠠⠁⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⠆⢠⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠊⠀⠀⡤⢀⠀⠀⠙⠢⣐⠹⠀⠀⠀⠀⠆⠀⡇⠰⠀⠀⠀⠀⠴⠶⢶⠶⠾⠁⠀⣼⢀⡏⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠈⠄⠃⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠒⠒⠒⠒⠲⠀⢀⡗⠚⠛⠲⠦⣄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⣴⠒⡀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠰⠀⠁⢸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠝⠋⠉⠚⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⠀⠀⠀⠙⠋⠀⠀⠀⢀⡆⠀⠀⠐⠀⡣⡴⠒⡟⠀⣤⠁⠁⠉⠉⠉⠃⠒⠒⠲⠁⢠⢊⠄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⡟⠀⠀⠀⠀⠶⠖⠚⠚⠀⠀⠈⠉⠉⠓⠓⠓⠒⠆⠊⠀⠀⢸⠀⢧⡀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠶⠒⢩⣡⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⢀⡍⠿⠋

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⢉⣉⣉⣹⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣉⣉⣉⢹⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡟⠛⠻⠿⠿⠾⠿⠿⠷⠿⠿⠿⠾⠿⠿⠛⠛⣿⣾⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠰⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠐⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢠⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣿⣾⣿⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⢟⣭⣶⣶⣾⣭⡻⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⢱⡿⣉⣿⣾⣿⠿⠛⠘⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣇⠻⠟⠛⢉⣁⣤⣴⣾⡇⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢀⣀⣙⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⡄⢶⣿⣿⣻⣿⣠⡿⣣⡿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡿⣶⣽⣛⣛⣛⣭⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣻⣿⠉
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣿⢻⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣼⣇⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣁⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⢀⣠⣴⣾⣿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⡿⠿⠶⣤⣀
⠀⠐⠚⠉⠉⠉⣿⠛⠿⠿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⣤⡄⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⡿⠿⠛⠋⠉⢹⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⣿⡿⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣾⣿⣿⣿⣿⣿⣟⠋⠁⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠘⠧⠴⠿⠋⠀⠈⠉⠙⢻⡏⢿⡋⠉⠹⣇⣀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⢻⡏⠉⠉⠀⠀⠀⠀⢸⡏⠉⠉⠙⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠛⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡍⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⣤⣤⠿⠤⠴⠶⣶⠶⠶⠾⠷⠶⠒⠒⠒⣶⠟⠛⠛⠛⠋⠉⠉⣹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢰⣿⠹⣦⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⣀⣀⣀⣀⣀⣿⣄⣀⣀⣀⣀⠀⢰⡿
⠀⠀⠀⠀⣆⠀⠘⣿⠀⠈⢷⣄⣀⣤⠴⠶⠞⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠲⠦⠤
⠀⠀⠀⠀⢻⡄⠀⠙⠳⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠙⠶⢤⣤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣂⣊⣴⣿⣿⣷⣿⣿⣢⣤⣄⠚⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⠈⣿⣿⣿⣿⣿⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣿⣿⡿⣛⣿⣿⣿⣿⣿⣿⣿⡿⢻⡟⢻⡇⠘⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⠏⠀⠀⣿⡀⠸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⢿⡟⢉⣰⠏⠀⣴⠀⢿⡇⠀⠀⠈⣿⣿⣿⡿⣋⣿⣟⣉⣻⣤⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣷⢿⣿⣿⡟⣹⣿⣿⠇⢸⠇⠀⣼⡟⠀⠈⠁⠀⠀⡼⣿⣧⣿⠳⣼⡟⠛⠻⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠠⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣸⠀⠀⣿⡇⠀⠀⠀⠀⠞⢠⣿⣿⣷⣮⣿⣗⣤⣴⢼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⠙⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⣿⣄⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣿⡁⠀⡠⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣼⣿⣷⣮⡀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣼⣿⣿⣿⣿⠆⠀⠀⠀⠀⢻⣿⣿⣿⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿⠉⠛⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠸⢿⣼⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⡿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⣿⡟⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡆⣈⣉⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡦⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠒⠲⠶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⢳⡀
⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣧
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠰⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⡌⠀⠀⣀⡀⡀⠀⠀⠀⠈⡇
⠀⠀⠈⠙⠒⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡇⢀⡜⠀⡄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⡸⠓⠲⢤⣤⣴⠆⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢸⠃⡼⠀⠀⡇⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⣠⣌⡉⠓⠚⠹⣅⡀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠸⠺⠶⣇⠀⠀⢹⡀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣄
⠀⠀⠀⠀⡌⠀⢀⡤⠖⠋⠀⠀⠈⠳⣄⢀⡷⣌⠉⠙⠒⠀⠀⠄⠀⣀⣀⠀⠀⠀⠈⠳⡄⠈⢧⡀⢰⣄⠀⠀⠀⠣⠤⠄⠉
⠀⠀⠀⠰⠀⠈⢃⠀⠀⠀⠀⠀⠀⢀⣨⡏⠀⠈⠳⣄⠀⠀⠀⠀⠀⡿⠂⠀⠀⠀⠀⠀⣿⠀⠀⠉⠛⠈⠳⢄⡀⠀⢀⡠⠔
⠀⠀⠀⡆⠀⠀⠀⠳⣤⣀⠀⠀⠀⠈⢹⡆⠀⠀⣾⣾⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠉⠓⠒⠋⠉⣇⠀⠀⡠⠊⠀⠀⠀⠀⡄⢀⣀⣀⣀⣠⠞⠁
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠀⠀⠀⠀⠀⠀⠉⠠⠀⢄⠈⡁
⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣀⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣤⣄⣀⣐⣀⣀⣤⡤⠤⠶⠒⠋⠉⠀⢳⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣤⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣀⣀⡀⣀⣀⣀⣀⣀⣀⡠⠤⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⣈⣒⡫⠭⠛⠳⠶⢖⣒⣚⣿⣋⣈⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⣀⣀⣤⠶⠶⠞⠛⠛⠛⠛⠛⠻⠷⠶⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣴⡏⠹⣷⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⡠⠶⣄⣠⣶⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢀⣀⠶⢦⣀⣠⡼⠛⢦⣴⠟⠁⠈⣉⣀⣠⣼⡏⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⠛⢶⡶⠉⠉⠙⠛⠁⠀⠀⠘⠋⠀⠀⢠⡇⠀⠀⠀⠀⠹⣿⣿⠇⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢀⣀⣰⣿⣋⣁⡠⠶⠶⢶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⣀⣀⣀⡀⠾⠋⠀⠀⠘⢿⡀
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠈⠛⣿⣅⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⢠
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠋⠑⠒⠒⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡾
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠠⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣴⣦⣀⢠
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⣍⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⢿⣤⡟⠃⠀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠘⠛⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡄⢀⣤⢶⣆⠀⣠⡀⠙⢷⣤⣄⠀⠀⠀⠀⠀⡖⠋⠙⠉⡀⠉⣷⡄⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠻⠿⠉⠻⠿⠟⠉⠙⠷⣦⣀⠀⠀⠘⡇⠀⠻⠟⢀⡘⠀⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠶⣼⣧⣤⣠⣤⠞⠁⠀⢾⡋
⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣀⣀⣠⠾⢷⣄⣠⣴⣶⣶⣶⣤⣀⣀
⣿⣏⠉⠉⠙⠛⠻⠷⢾⡏⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⠾⠿⠷⢶⣷⣶⣶⣶⣿⣧⠀⠉⠉⠁⠀⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⠷⣤⡀
⠀⠛⣷⡶⣴⡞⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⢈⣿⢿⣄⣠⣶⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠈⢿⣤⡿⢃⣽⠋⢿⡶⠶⠶⠶⠶⠶⢶⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢻⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⣼⠇⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣼⠋⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⢂⡀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣈⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠓⠂⠀⠀⠠⠀⠀⠀⠀⠀⠒⠒⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠒⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⣿⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⣿⠆⠘⠓⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠟⠀⠸⠀⠀⠀⠿⠟⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⣇⠀⠀⠀⠀⠀⠀⠀⢻⣇
⢀⣀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⠀⣘⣿⠛⠛⠛⠛⠛⠛⠛⢻⣂⣤⣤⣤⣾⡟⠀⣀⠀⠀⣀⠀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠅⠀⠨⠁⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀⠀⣀⡀⢀⣀⣀⣀⠀⣀⠀⠀⡀⠀⣀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠟⢸⣿⣛⣿⡇⣙⡁⣿⣿⠛⠛⢹⣿⣷⣸⣿⢸⣿⠿⣿⣆⣾⣿⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠗⠀⣿⣿⣿⣏⠠⣿⡇⣿⣿⣛⣛⢹⣿⠿⣿⣿⢸⣿⣀⣿⡟⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠻⠏⠈⠻⠇⠿⠇⠿⠿⠿⠿⠋⠿⠃⠻⠟⠸⠿⠿⠛⠀⠻⠿⠟⠋

⠀⠀⠀⠀⠀⢰⣿⣦⢸⣷⢰⣿⡿⠿⠇⣾⡇⠀⣾⡆⣾⣿⠿⠿⢀⣾⡿⢿⣷⠀⠀⠀⠀⣶⣿⠿⠆⢠⣾⣷⣴⣷⠀⣼⡷
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⢸⣿⡷⠶⠆⢻⣿⢰⣿⡇⣿⡿⠶⠶⢸⣿⣿⣿⠋⠀⠀⠀⠀⠙⢿⣶⡄⣼⣧⣼⣷⢹⣷⡟⠁
⠀⠀⠀⠀⠀⠸⣿⠀⢿⡿⠸⣿⣿⣿⡷⠈⠻⣿⠟⠀⣿⣿⣿⣿⠾⡿⠉⠻⡧⠀⠀⠀⠀⢶⣾⡿⠏⢿⠏⠙⣿⠆⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⢀⣤⣶⣦⡀⢀⣴⣶⣤⠀⣶⣶⣦⣄⢠⣶⡶⣶⣄⣴⡆⢀⣴⣄⣶⣶⣶⣶⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⢀⣀⢼⣿⠉⣿⣧⣾⣟⢹⣿⡇⣿⡏⢹⣿⢸⣿⣷⣿⡋⠘⣿⣼⡟⢸⣿⣿⣤⣤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣾⡿⠸⣿⣶⣿⠇⠹⣿⣾⡿⠃⣿⣷⣿⠟⢸⣿⣭⣿⡿⠀⢸⣿⠀⠘⣿⣷⣶⣶⠀⣩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠀⠀⠉⠉⠉⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡠⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠀⠀⡀⠐⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠤⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠬⣉⠐⠁⠀⠂⠐⠠⢂⠄⡀⡀⢀⠀⠀⠀⠊⠊⠄⠀⢁⠁⠈⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠒⠀⠀⠐⠢⠁⠬⠀⠡⠒⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠠⠄⠄⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠁⠀⣈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢱⠁⠈⢢⢀⠀⢱⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⠈⠀⠀⠐⡇⢁⠈⠀⠀⠀⠁⢠⣀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢘⡠⠀⠀⠀⠀⠀⠁⠙⢢⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⢴⡀⠀⠆⠀⠀⠀⢘⡠⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠣⡄⡎⠀⠀⠀⡬⡀⠄⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⢱⠘⠀⠀⠑⠀⠀⠀⡜⠩⢐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢤⠀⠰⠧⠱⠘⠆⠀⠀⡠⠾⠶⠭⢖⣞⡮⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣟⢀⣤⠀⠌⠠⠶⠚⠄⠀⠀⠀⠀⠈⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡂⠑⢖⣲⠀⠀⠄⡂⠐⠠⢂⠔⡠⢂⠖⠔⠕⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢐⠁⠀⠘⠀⠀⠤⠂⠰⡶⠤⠬⣄⣡⡀⢤⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠅⡪⠀⠀⠀⠀⡨⢐⠀⠇⢨⠀⡀⠀⠐⠄⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠄⠨⠂⠀⡄⠂⠀⠀⠐⠀⠘⡈⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⡀⣀⡀⢀⣜⡽⠀⠀⠁⠠⠀⠃⠃⠀⠀⠠⠪⠂⠀⠐⡆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠲⣁⡀⡀⢀⣠⡤⠚⠝⠒⠒⠂⠚⠉⠛⠑⠋⠀⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠁⠀⠀⠀⠀⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣦⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⣿⣿⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣾⣿⣿⣆⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣼⣿⣿⣿⣦⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣫⣿⣿⣿⡿⠟⠋⠹⣿⣿⣿⣿⣿⣿⡿⠀⠒⠒⡄⠀⠙⢿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⢁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠆⠀⣾⣿⣿⣿⠿⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⠂⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⠀⣿⣿⣿⠃⠀⠀⠈⠙⠻⣿⣿⡷⠶⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⣿⣿⣷⡄⠀⣠⣤⣤⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⡀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠿⠿⠟⠁⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠃⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣾⣿⣷⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⣠⠃⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣍⢀⣤⣇⠀⠆⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡀⠀⠈⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⢀⠘⢿⣿⣷⣄⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠈⠋⠀⠈⢻⣿⠿⣦⡀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠘⣿⣿⣿⣿⣿⡃⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠘⢿⣿⣿⣿⡇⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⠀⠈⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠈⠛⢿⣿⣍⢦⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣤⣄⣀⡙⢠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣇⠀⣾⣿⣿⣿⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⡇
⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡟⠁⠨⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠆⠉⠉⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠄⠘⠋⠉⠑⠻⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⣳⣤⣄⠀⢀⣴⣎⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢻⡿⠇⠀⠸⠿⠟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡄⠀⠀⠀⢠⠁⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠈⠑⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠆⢤⣤⣤⠀⠶⠆⢸⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡇⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⡀⠀⢠⣥⠀⠀⠒⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠎⠁⠀⠀⠈⠑⢂⡀⠀⠀⠀⠀⠀⠀⠀⡸⢿⣿⠟⢣⡀⠂⢀⣤⠰⠂⠠⣀⠈⢢⡀⠀⠀⠀⢀⠘⣹⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢪⡑⠦⡀⠀⠀⠉⠑⢦⡄⠀⠀⢠⠌⠀⢠⣿⠀⠀⢹⢠⡉⠀⠀⠀⠀⠈⠳⣀⠻⡀⢠⠶⠉⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡰⠁⠀⠀⠈⠱⢮⡑⢂⠀⠀⠀⠈⠑⢦⡁⠀⢀⣾⣿⠀⠀⠘⣾⡁⠀⠀⠀⠀⠀⣠⣷⠀⢃⣤⠷⠊⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡄⠒⠢⡀⠀⠀⠈⠓⢬⡑⢠⡀⠀⠀⠘⠳⣬⡿⣿⡇⣠⠈⠀⠀⠀⠀⠀⢀⣄⠸⠄⠀⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠰⢡⡟⢀⡔⢀⠔⡀⠈⡆⢀⡀⠀⠀⠈⠑⢮⡔⢠⠀⠀⠀⠙⠷⣦⡈⠁⠀⠀⠀⣀⡘⠓⠒⠒⠈⠀⠀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⡌⢠⠋⢠⠂⠀⠑⠂⠂⢠⡑⠀⠀⠁⠢⡀⠀⠀⠈⠑⢪⡕⣠⠀⠀⠀⠉⠓⢢⣄⢸⡟⠘⢷⡀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⢀⠘⡰⠃⡰⠂⡄⢻⠀⠴⠊⡑⠁⢴⠉⠒⠠⠂⠀⢀⣄⠀⠀⠈⠑⢮⡑⠂⠀⠀⠀⠈⠙⢦⡀⠀⠙⣦⡀⢀⠆
⠀⢀⠂⠉⢫⡴⠁⠀⠓⠒⠀⠈⠀⡀⠀⢀⢤⡄⢉⡄⠀⠀⡰⠂⣈⠳⠆⣄⠀⠀⠈⠑⢮⡕⠢⡀⠀⠘⠈⠑⢠⣈⠻⠎
⠀⢼⡀⠀⡜⠓⢤⡀⠀⠀⠀⠒⢰⡏⠀⠂⡄⠈⣁⡀⠀⡰⠁⣰⠆⠀⠀⠀⠁⠀⠀⠀⠀⠈⠑⢮⡑⠢⡀⠀⠀⠈⠑⠂⡀
⠧⡀⠉⠁⠀⠀⠀⠈⠐⢦⣄⠀⠀⠈⠒⢤⢀⡿⠁⢸⠆⢀⠲⠉⠀⠐⠀⠀⢀⠄⠂⠀⠰⠀⠀⠀⠈⠑⢮⡑⠢⠄⠀⠀⠀⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣄⠀⠀⠀⠘⢤⡄⠐⠠⠃⡐⠁⠂⣄⠾⠁⡰⠈⢱⠀⣤⠂⠐⢄⠀⠀⠈⠑⢦⡉⠲⣄⠀⠀⠉⠁⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠘⠣⡀⠈⠒⠀⡘⠁⠞⠁⠀⠈⠈⠁⡤⢀⡀⢉⡶⢠⠀⠀⠈⠑⢦⡉⠲⣄⡀⠀⠀⠙⢢⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠙⠲⣤⡀⠀⠈⠑⠒⠀⡁⠘⠄⠐⢂⠀⢀⠜⠀⠎⢀⡛⢠⡄⠈⠒⡄⠀⠀⠈⠑⢦⡉⠂⢄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⡀⠀⠀⠑⠂⠀⠒⠁⢠⠂⠠⠁⢀⠎⠀⠣⡈⠑⢺⠂⠉⠒⢀⠀⠀⠈⠑⡌⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⠀⠐⢦⠔⠀⡀⠀⡀⠒⢤⠀⡐⠀⡄⠁⠄⢸⠀⠀⢠⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠗⢤⡀⠀⠀⠀⠐⢅⠀⠑⠂⣀⠐⢠⠌⠀⠘⢠⠄⠀⡰⢃⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠑⠦⣀⠀⠀⠉⠒⠰⠏⠀⠃⠄⠂⡠⠁⠀⠞⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠈⠒⠀⠐⠊⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠢⣄⡀⠀⠀⠀⣴⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠉⠲⣤⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠬⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⠤⠤⠤⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣤⡀⠀⢠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠛⠻⢿⣿⡿⠛⠉⠙⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠁⢠⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⣿⣿⢁⡀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⣤⡄⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠼⠿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣟⣀⣉⢩⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠰⠿⠶⡾⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⡼⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⣶⠷⠖⢲⣢⡀⠀⢰⠁⠀⠀⠀⠈⡂⠀⠀⠀⠠⡶⠤⠀⠀⠁⠀⠀⠀⠀⢀⡔⠁⠈⠢⠀⠀⠀⠀⡷⢺⣄⠀⠚⢺⡆
⠀⠀⠀⠀⢈⣻⠕⡶⡒⡦⡽⣓⠀⢸⠀⠀⠀⢀⡔⣓⡢⡀⠀⢀⣠⣤⣤⣄⢤⣤⡀⠀⠀⠘⢆⣂⣴⣶⠷⡀⠀⠀⡿⣀⣤⣤⣄⣸⣇
⠀⠀⠀⠀⢈⣭⠘⡙⣉⠀⢥⣭⠀⢸⢀⣴⣿⣏⢃⡥⢠⡇⠀⠈⠻⠉⡒⢈⣠⡠⢃⠀⠀⠀⠀⢹⠷⠦⠟⠁⠀⠀⡿⢈⣿⣿⣟⣻⣭⡇
⠀⠀⠀⠀⠈⠻⢗⡤⣤⣶⠗⠁⠀⢸⠀⠀⠀⠈⠛⡟⠋⠀⠀⡀⠘⣛⣃⠉⡥⠐⠊⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣷⣈⠉⠉⢉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣈⣀⣁⣈⡀⠀⠀⠀⣀⣈⣉⣉⣀⣁⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡀⠈
⠀⠀⠀⠀⠀⢠⠔⣂⢄⠀⠀⠀⠀⠀⡏⢉⣀⣀⠀⣡⡀⠀⠀⠶⠒⣚⡒⠒⠒⠒⠲⠀⠀⠐⠷⠆⠉⠛⠒⡆⠀⠀⠀⠀⣾⠒⠙⡆
⠀⠀⠀⠀⠀⠘⠀⡀⡏⢀⢀⣀⠀⠀⡗⣾⡷⢺⣟⣌⠼⠀⠀⠀⠐⠛⢃⣭⡙⠁⠀⠀⠀⡀⣰⡞⢋⠛⣿⠃⠀⠀⠀⠀⣫⡦⣶⣅
⠀⠀⠀⠀⢀⠀⠾⢿⢛⣷⣛⠛⠇⠀⡟⢿⠿⠞⣋⡃⠀⠀⠀⠀⢯⣸⣟⣽⣭⡿⠀⠀⠀⡇⠚⠷⠤⡶⡣⡀⠀⠀⠀⡮⡽⢖⡠⡇⠇
⠀⠀⠀⠀⠥⠤⠨⠿⠟⠁⠬⠷⠄⠀⠇⠀⠀⠘⠧⠇⠀⠀⠀⠤⠯⠠⠯⠽⠯⠽⠤⠀⠀⠣⠠⠀⠸⠼⠏⠚⠀⠀⠀⠢⠤⠨⠱⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⡀⠀⢀⡀⣀⣀⣀⠀⣀⣄⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢾⡏⢸⡷⠶⠀⡀⢸⡇⣿⡷⠶⢸⣏⠈⣉⠈⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠛⠘⠛⠛⠃⠙⠛⠁⠙⠛⠛⠀⠙⠛⠉⠀⠘⠁

⠀⠀⠀⠀⣒⣒⣲⣒⣒⣒⣒⣒⡄⢀⠤⡠⠤⠤⠤⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠲⢒⠒⠀⠀⠀⠀⢀⣀⣔⣢⣀⣀⣀
⠀⠀⠀⠀⢡⣋⣉⣉⣉⣉⣉⣦⡁⢸⠊⠁⠂⣀⣀⠂⠑⡆⠀⢄⡀⢀⠀⠀⠀⢀⣠⠀⠀⢸⠸⠾⠌⢀⡀⠀⠀⠀⠸⡕⢾⣻⣏⠢⢸
⠀⠀⠀⠀⢸⠩⠛⠍⣩⣩⣉⢹⡀⢸⠀⢴⣾⢿⣛⣥⡀⡇⠀⢁⠄⢄⠀⠤⠤⠤⠈⠀⠀⢸⢩⢭⡿⣟⣫⠄⠀⠀⠀⡀⠢⠉⠫⢁⡍
⠀⠀⠀⠀⢨⠐⠒⠂⣈⣉⣉⢸⡃⠘⠈⠋⠉⠙⠻⣿⡗⠂⠀⠘⠌⠔⠀⠉⢡⠤⡄⠀⠀⢸⢸⣖⣿⢿⠋⠀⠀⠀⠀⢁⠐⠉⠊⢁
⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠀⠈⠉⠈⠀⠀⠈⠉⠈⠁⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⡀⠀⠀⣀⢀⣀⠀⢀⠀⢠⣶⡶⡄⠀⢀⢴⣶⡄⠀⢀⡴⠼⠟⡛⣧⡦⡀⠀⠀⡴⢑⡶⣴⡂⠒⣶⡀⢠⠀⠼⢄⠔⢢
⠀⠀⠀⠀⠂⡤⡉⣭⠍⠉⠠⠐⡇⠀⢿⠀⡈⣤⠁⠀⡸⠀⠀⢲⠁⣼⢀⠮⠻⠗⡔⠀⠘⢫⣣⡏⢩⡶⡴⡎⠀⠸⣀⠉⠩⠉⠩⢈⡍⠉
⠀⠀⠀⠀⠀⠤⠤⣠⣴⣾⣭⠕⡁⢸⠰⠌⠋⠀⡿⠣⠔⠃⠀⠜⢴⡯⠓⠫⠀⢀⡣⠀⠀⢸⣿⡠⠿⠄⡛⠄⠀⠀⠀⢄⣼⣶⣌⡁⣀
⠀⠀⠀⠀⠈⠉⠁⠛⣽⣰⠟⠉⠀⠸⡀⡀⠔⠈⠢⢄⢀⠄⠀⠈⠢⢺⣧⣤⡿⠕⠁⠀⠠⢿⠟⠀⠀⠈⢣⠖⠀⠀⠀⠘⢵⣲⠇⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠖⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⠁⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠉⠙⠓⢦⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡼⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⢳⣴⣾⣯⠶⠶⠶⠶⠶⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢀⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢶⣄⠀⠀⠀⠀⢻⠀⠀⠀⠀⡀⠀⣸⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢀⣿⡀⢀⣆⠠⣿⣶⢿⣆⠙⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠃⠀⣠⡾⢹⣧⡾⠻⣶⡏⠀⠀⠹⣆⠸⣇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠰⠛⠋⠀⠀⠉⠀⠀⠀⠀⢻⡄⢿⡀
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣆⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣤⣶⣶⠶⠾⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠶⠶⠶⣶⣄⠀⠀⠀⠀⠀⣿⠘⣧
⠀⠀⠀⠀⠀⠀⠈⣷⢀⡴⠋⠁⠀⠀⠀⠀⣠⠶⠚⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠒⠓⠒⠶⣄⠉⠀⠀⠀⠀⠀⣿⣠⡟
⠀⠀⠀⠀⠀⣴⠟⣿⠋⠀⠀⠀⠀⠀⠀⡾⠁⠀⣴⣾⣿⡇⠀⠀⢳⡀⠀⠀⠀⠀⢠⠞⢁⣤⣴⡆⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⢸⡿⣦
⠀⠀⠀⠀⢨⣧⡀⣿⣀⡀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣶⡇⠀⡇⠀⠀⠀⠀⣾⠀⣿⣿⣿⣷⣴⡆⠀⠀⣧⠀⠀⠀⠀⠀⢸⡇⢸
⢀⣤⠤⡼⠋⠉⠻⡟⠋⠙⣷⠀⠀⠀⠘⠇⠀⠈⠻⢿⣿⣿⠿⠁⢠⡇⠀⠀⠀⠀⠸⠀⠻⣿⣿⣿⡿⠁⠀⠀⠃⠀⠀⠀⠀⠀⢸⣇⣼
⣿⠁⠀⣿⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⠀⠉⠒⠶⠦⢤⣤⣤⣴⠏⠀⠀⠀⠀⠀⠀⠰⣦⣤⣭⠥⠶⠖⠒⠁⠀⠀⠀⠀⠀⠀⣾⠛⠁
⣿⠀⠀⣿⠀⠀⠸⡇⠀⠀⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⣿⠀⠀⢿⡇⠀⠀⣷⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⣿⠀⠀⠸⠇⠀⠀⠋⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⣿⠀⢠⣤⣤⡶⠶⢶⣤⡶⠘⠇⠈⣷⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⢺⡀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⢹⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⢸⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠋⠁
⠸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢨⡟⣷⠶⢦⣤⣤⣀⣀⣀⣀⣤⣤⣤⠶⢶⡟⣿⠁
⠀⠻⠶⠶⣶⠶⠶⠛⠛⢻⣟⠛⠋⠀⠀⠀⠀⣠⠟⠀⣿⠀⠀⠀⠙⠿⣭⣭⣥⠼⠋⠀⠀⢸⡇⠸⡇
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠙⢦⣄⣀⣤⠶⠟⠁⠀⢸⡟⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⣷⠀⢿⡀
⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠⣼⡇⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⠀⠀⠀⣿⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⠀⠀⣀⢀⣀⠴⠞⠉⢸⠃⠀⠀⠀⠘⡏⠙⠛⠛⡿⠋⠀⠀⠀⢸⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣧⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠶⠶⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠋⢻⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡶⣤⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⣀⣀⣸⣇⣹⣀⢀⠀⠀⠀⣠⣿⣿⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠉⠉⠉⠁⠀⣹⠈⠁⠀⠀⠈⠈⠉⣻⣷⣴⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⢀⣀⣀⣀⣀⣿⣤⣤⣄⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣉⣹⡏⠀⠀⢸⡇⢈⣟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠙⢿⣄⠀⣼⠟⠋⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⣀⢀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣿⣠⣏⣀⣀⣀⣀⣸⣇⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠄⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠼⠛⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⢿⣿⣧⠀⠀⠀⠀⢀⡀⢴⡆⠰⠀⢂⡀⠀⢠⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⣷⣿⣘⣿⢆⠠⢀⠀⠁⠈⠁⢚⣐⠃⡀⣀⠀⠀⠄⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⢁⣥⣿⡗⣤⣃⢦⠘⠳⣦⠠⠖⡀⠰⠊⠁⠀⠀⠀⠀⣠⡀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣸⣿⣿⣧⡞⣼⣫⠏⠳⠠⠾⠨⠁⠚⠰⠗⠀⠊⠀⠄⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢀⣠⣼⣿⣾⣿⡛⣿⣽⡘⠧⠥⢁⣶⡘⠁⢈⡀⢀⢉⡄⠈⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⣻⣿⣼⣿⣿⣿⡿⡏⣿⣿⣫⡿⢊⢓⣈⠁⠄⠙⠡⠠⠀⠄⠀⠐⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣿⣧⣶⣬⣉⠻⠛⠉⠀⢀⣹⣏⣻⢾⡤⠜⠀⣈⢃⡚⠂⠠⠄⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠛⠻⢿⣿⣿⣶⣦⣄⡆⠺⣿⣯⣻⣓⡒⡐⠏⠁⣅⠘⢒⠀⡀⠈⠁⠀⠐⠃
⠀⠀⠀⠀⠀⠳⡄⠀⠀⡀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣦⣴⣴⣶⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁⢀⠈⠲⠀⠀⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⢷⠶⠛⠻⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠿⠿⠿⢿⠻⢿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⣤⣤⣦⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣠⡾⠁⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠓⠶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡏⠀⢹⠀⠀⠀⠀⣸⠃⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣄⠀⠀⠀⡾⠀⠀⠈⢧⡀⠀⠀⣿⡤⠞⠁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⡗⠤⠾⠃⠀⠀⠀⠀⢻⡄⠀⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣰⠟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⢷⡄⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢾⡋⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠈⣷⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠆⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⡞⠀⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠠⠟⠒⠛⠉⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠳⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣤⣤⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡄⢸⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣠⢃⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⣴⡿⠛⢯⡉⠻⠏⢸⡇⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢰⠃⠀⢡⡇⠀⣿⣻⣦⡤⢷⣤⣾⣟⣷⠀⠈⡧⠀⠀⢸⡇⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⡟⠀⠀⠈⠓⠦⢤⡥⠟⠀⠀⠈⠷⠬⣅⣤⠞⠃⠀⠀⢸⡇⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠇⠀⠀⠀⠀⢀⡤⠶⠲⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠻⠶⠾⢷⠀⠀⠀⠀⢀⡏⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⣷⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠚⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⠋⠉⠉⠉⠉⠉⠙⠻⢷⣦⡀
⠀⠀⠀⠀⢀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⣿⣧⣶⣶⣆⠀⠀⢀⣤⣶⢿⣦⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠈⠉⢸⡟⠻⠷⠿⠛⠋⠀⠀⢻⣧⣰⡿⣫⡙⣷⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⣼⡇⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⢿⣆⠻⠛⠛⠿⠀⠀⠀⠀⠈⠁⠀⣴⣿⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⣸⡏⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⠋⢻⣧⠀⠀⠀⣠⣿⡇⠀⠀⠀⠰⣿⠄
⠀⠀⠀⠀⢀⣿⠃⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣴⡾⠋⠹⣧⡀⠀⠀⠀⠏⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠈⠛⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡛⠛⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⢀⣴⡶⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢀⣀⣤⣴⣿⣯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠉⠉⠉⣹⣷⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠍⠩⠉⠉⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠫⣝⠛⢻⣟⣻⣿⠽⠷⢿⣛⠿⢯⠿⠿⠽⠟⠛⠛⠛⠛⠛⠛⠋⠉⠋⠉⠁⠈⢿⣄⠀⠀⠀⠀⠀⠀⠨⢍⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠁⠀⠘⣳⢊⠁⠌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢬⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿
⠀⠀⠀⠀⠀⢤⣦⣤⣤⣠⣝⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣾⣿⣿⣿⣷⣧⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠉⢛⣻⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠐⠛⠁
⠀⠀⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢿⡎⠉⠻⣿⣿⣿⣿⣆
⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣄⢙⣶⠋⠈⠉⠛⠿⣿⣦⡀
⠈⠉⢉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢉⣿⠏⠀⠀⣀⣀⣀⠀⠀⠹⠇
⣀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢿⣿⣿⢰⡟⠁⢠⣶⣿⣿⣿⣿⣿⡷
⢠⠿⢻⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡑⠢⡀⠀⠘⣿⣿⠀⠁⠀⣼⣿⠟⠉⠉⠙⠁
⠁⢀⡾⢋⣿⣿⣿⠃⣹⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣸⠂⢠⡸⣿⣷⣡⣠⣿⡿
⠀⠈⠀⢸⣿⣿⡟⢰⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣾⣿⣷⣿⣿⡇
⠀⠀⠀⡯⠉⡿⠁⡼⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢐⠆
⠀⠀⠀⠁⠀⠀⠘⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣶⡄⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⡿⠀⣸⣿⣿⣿⣿⠟⣿⠋⢛⠊⣿⠁⣿⠿⠻⠀⠛⣿⢻⡷⣿⣷⠊⠈⢹⠇⠈⠉⠀⠀⠀⠐⠋⠹⡧
⠀⠀⠀⠀⠀⠁⢀⣽⡟⢰⡿⠋⠔⠁⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⣷⣾⣏⠀⠀⢻⠀⠀⠀⢰⡖⠀⠀⠀⣰⡷
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠗⠀⢻⡄⠄⣼⢹⣷⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣶⢀⣴⣿⡆⠁⠈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⣿⡇⠀⣬⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣏⣼⣿⠀⣴⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠙⠳⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠿⣫⣷⣶⣶⣯⣽⣛⣛⣟⣛⡋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⢨⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⢸⣿⣿⣿⣷⣶⣶⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣾⣿⣴⡎⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢟⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣱⣿⣿⣿⡇⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣧⣿⣿⣿⡿⣱⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠝⠛⠛⠛⠽⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀



⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣻⣿⣟⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣷⣿⣿⣟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠉⠈⠉⠙⣿⣯⣼⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⣾⣿⣿⣷⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣾⣿⣿⣿⣭⡼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣭⣭⣍⣀⠾⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣾⣿⣿⣿⣷⣶⣮⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⣛⡉⢙⣛⣛⡛⠛⢭⣭⣬⣿⣿⣿⣿⡿⠟⢿⣿⣟⣿⡿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣺⣽⣶⣿⣿⣿⣿⣿⣿⣿⣤⣶⣶⣌⣉⣿⣿⣉⣡⣶⣶⣄⣿⣿⣿⣿⣿⣿⣶⣾⣭⣓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⠟⠿⠾⢻⣿⠏⢹⣿⡿⠳⠿⠿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡆⠀⠻⣄⡀⠀⣸⣛⠁⠈⣛⣇⠀⢀⣠⡟⠀⢸⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣭⡄⠀⠈⠿⠛⠁⠀⠈⠳⠖⠁⠀⠈⠛⣿⡁⠀⢠⣽⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡎⠀⢀⡇⢀⣴⠞⠛⠿⠿⠛⠳⣤⡀⢸⡀⠀⢰⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢖⣿⣿⣿⣿⢠⠀⠀⡇⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⢸⠃⠀⡄⣿⣿⣿⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⡽⣿⣿⣿⣿⣎⠀⢀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡀⠀⢸⣿⣿⣿⣿⣟⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⠨⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢆⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣻⣿⣿⣧⢳⠀⡀⠀⠀⠀⠀⡘⠁⠀⠀⢀⠀⡾⣼⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣻⣿⣮⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⣷⣿⣿⣿⡏⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⢿⣿⡄⠀⡄⠘⠄⠀⠀⢀⢿⣿⢿⣿⠎⠀⠀⣀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣿⠟⠻⣆⢘⣆⢀⡎⢠⡟⠻⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⢹⠏⠡⠁

⠀⠀⠀⢀⣤⠞⠂⠀⠀⠀⠀⠀⠀⠠⠄⢀⢀⠕⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣀⢠
⠀⠀⠀⠞⠁⠂⠖⠀⠀⠀⠀⠀⠀⠘⠈⠇⠢⡀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠋⢠⢛⢱⢤⠀⠀⠀⠀⠀⠀⠀⠀⠁⣏⢸
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠀⠄⠀⠀⠀⠀⠀⠀⢸⠀⣿⣷⣷⡆⡰⠀⠀⠀⠀⠀⠉⡀⠀⠉⡋
⠀⠀⢘⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⡆⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢉⡙⠀⠀⠀⠀⠀⠀⠀⠈⢀⠆⠐⢋⠰⡆
⠀⠀⠘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀
⠀⠀⢰⡿⠟⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢸⣀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠈⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠂⠀⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⠀⠊⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀
⠀⠀⡠⢗⠀⠺⢄⡀⠀⠀⠀⠀⠀⠀⢀⠄⠙⠿⠂⡀⠀⠀⠀⠀⠀⢀⢉⡓⠀⠚⡋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⣹⠁
⡎⢠⡆⠀⣿⢐⠁⠉⢱⠀⠀⠀⢠⣦⡀⠀⠠⠢⠞⢿⣶⠀⠀⠀⠀⠀⠈⠄⠠⠠⢃⣀⠁⠀⠀⠀⠀⢠⣄⠀⠀⠉⠁⠈
⠣⠀⠄⠸⠿⠏⠃⠈⠘⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢸⠀⡆
⠀⠀⠀⡦⣄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢈⠀⠃⠀⡀
⠀⠉⠉⠀⣸⠠⡚⡠⠀⠀⠀⠀⠀⠀⢸⠉⠛⠉⠟⠀⠀⠀⠀⠀⠀⠀⠐⡯⢇⠈⠠⠈⠀⠀⠀⠀⠀⠀⠄⢀⠀⡀
⠀⠁⠀⠇⠈⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠚⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⣰⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢴⣤⡦⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣷⠶⠿⣿⠀⠀⠀⠀⠀⠀⠀⢹⢟⣛⣻⡇⠀⠀⠀⠀⠀⠀⣶⣶⡆
⠀⠀⢰⡖⣒⢶⡆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⢶⣿⠁⠀⠀⠀⠀⠀⠀⢸⣞⣿⣇⣿⠇⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠸⠧⠭⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠋⠿⠀⠀⠀⠀⠀⠀⠀⠀⢷⡶⣾⠋⠀⠀⠀⠀⠀⣘⣻⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠿⠀⠀⠀⠀⠀⠀⠛⠽⠛⠛⠛⠻⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠤⠤⣴⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠚⡓⠲⠚⠁⠠⣤⣴⣶⣿⡷⣸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⡴⠾⢿⡿⠟⣡⠔⣂⡀⠉⠉⠰⠶⠿⠟⣋⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢠⢏⡔⠂⠀⠀⣼⠇⣾⣿⣿⣶⣶⣶⠄⡀⢤⣭⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠁⠀⠀⢿⢸⡄⣿⣿⣇⢻⢸⣿⣿⡿⠿⢿⣿⡀⠙⠲⠾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠰⠖⢦⣀⠘⠇⠙⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⢤⡈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⢀⣼⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢹⠀⠀⣿⣄⣹⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⣧⠀⢮⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣀⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⢙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠙⣿⣿⣶⡗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⣠⣿⡀⠀⢀⣤⡤⢤⣤⣀⣀⠀⠀⠀⠀⢠⡶⢋⣡⣶⠿⠷⠖⠀⢻⣯⣽⣳⣼⣿
⠀⠀⠀⠀⠀⠀⠀⣉⣿⠠⣿⣿⣧⠀⠋⠀⠠⢤⣤⣬⣉⠛⢶⡀⠛⣾⠀⠋⠙⠻⠶⠀⠀⠀⠈⣿⣬⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡿⢻⣷⠙⠿⠛⠀⠀⠸⠿⢛⣿⡧⠌⠁⠀⠁⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠸⡇⠘⢯⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣶⣥
⠀⠀⠀⠀⠀⠀⠀⣽⣦⣼⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣀⣀⢀⡶⠇⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⣾⡿⢹⡿⣠⣽⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⢸⣿⣏⣿⠁
⠀⠀⠀⠀⠀⠀⠹⣷⣜⣧⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠾⣿⡷⠶⠀⠀⠀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡽⣦⠀⠀⠀⠀⠀⠀⠉⠁⠲⠶⠿⠛⠁⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⡌⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⢀⣀⣴⠟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉⠁⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡡⠚⠉⠉⠉⠉⠒⢤⡀⢰⡕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠁⠆⠀⢀⠖⠚⠓⠢⢵⠀⠐⠫⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠸⡀⢠⠎⠐⠃⠀⢉⠀⡇⠠⡄⢒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⠀⡟⢳⠇⣤⠘⠀⠀⠀⠀⡇⢈⢡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠦⠄⠈⣆⡼⢲⢄⣰⢣⣠⠒⢲⡦⡃⡏⢿⢸⢾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⣤⣧⠙⢘⠜⡻⢹⠉⢯⠙⢦⠈⢘⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠦⠀⠸⠀⡧⠊⠀⢸⢦⣘⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠢⠤⢾⠁⠀⠀⡜⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠇⠀⠀⠀⣇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠫⣑⡒⠃⠀⠀⠀⢀⣼⠑⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣒⠯⠤⢭⡭⠭⣗⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⡞⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣇⠀⠀⡇⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⠀⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡄⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣬⣹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⠉⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣶⠄⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣏⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣼⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⠉⠉⣿⠋⠉⠉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠈⣹⠋⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠐⢛⠇⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⡿⠟⠃⠾⠟⣿⣻⣿⡿⠿⢿⣿⣿⣧⢹⠻⠷⣏⣛⣦⣍⠐⢰⣨⡯⣯⣽⣏⣿⣷⣿⠗⠸⣗⣞⣿⣿
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⣿⣷⣾⣿⣷⣏⠻⢿⡿⠇⠘⠙⢃⠈⠻⠿⢞⡧⣾⣿⣶⡉⣿⣿⣿
⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡋⣯⣭⣿⣾⣶⣦⣿⠹⢳⣾⡸⢋⣄⠻⣷⢸⣷⣿⣶⡈⣿⡿
⠃⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣍⢻⢻⣿⣻⣿⣿⣿⣁⢘⢫⣝⣻⣍⣻⣦⠀⠀⣽⣏⣿⡌⢹
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣹⢿⣿⣟⣿⡿⣵⣿⣸⡇⣮⣿⣿⣿⣭⠘⢶⣌⡛⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⣿⣟⠿⣶⣤⣿⠹⣿⣿⣿⣿⣾⡟⢸⣷⣾⣟⢮⡛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⢟⣰⣷⣿⣿⢿⣷⣿⣽⣿⣿⣿⡖⣽⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⣶⠶⡶⠀⢀⣼⣿⡳⢻⣿⣿⣻⡟⢿⣿⣬⣽⡿⣿⢻⣿⣥⣿⣿⣷⣽⠛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢸⣿⡿⢰⣾⡿⣋⣻⡿⣷⣘⠻⢿⣿⡿⣼⣿⠛⢿⣿⣇⣴⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣧⠀⠀⠀⠸⠏⣴⢺⣿⣿⣿⣿⣿⣷⣼⣃⢘⡿⢧⣻⣿⠁⣿⣾⣏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⢿⣿⣿⢿⣿⡛⣿⢿⣧⣸⣷⣾⠀⡽⠒⢻⣿⣯⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡟⣿⣻⣏⠙⣨⢹⣧⣦⡈⢻⣿⣷⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣉⣣⣹⣷⠿⢾⡵⠂⣰⣿⡙⢿⣧⣾⣿⡛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢀⡀⢀⣼⣿⣿⡟⢫⣶⣾⡇⣼⢿⣿⡟⣦⠙⣿⣿⣿⣦⣉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣾⣿⡿⢿⣷⣿⡎⣿⠈⡆⣿⣿⢾⣅⡿⣿⢿⣿⣿⢿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣫⣴⢿⣽⣿⣧⡀⢾⠰⠿⢃⣽⡽⠇⣀⡰⣿⣾⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣊⡿⣿⣷⠷⣾⠿⡛⠋⢠⠠⠀⠸⠋⣥⣾⣏⢻⣿⣽⠏⠙⣛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠁⠼⢛⣚⣉⠁⠀⠸⣀⣟⡀⠀⠀⠶⠆⠾⢨⣯⣏⢨⣭⣼⡄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣻⣋⠃⢠⣞⠿⢻⣩⣤⣴⣶⣯⠘⠇⣀⣹⡁⡆⣀⡘⣻⡿⠘⠿⣧⡶⢡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢈⠹⢷⠔⢻⠟⣾⢻⣽⣿⣾⣦⠟⣿⡆⠉⠸⣣⣼⣿⣦⣷⠆⠀⣾⡿⡇⢨⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠁⡜⢻⠟⢼⣟⣭⡙⣿⡾⣇⣾⣿⣷⡏⣿⣮⣛⢹⣷⡆⢞⢠⣷⣿⠇⢘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣦⣿⡆⠉⠩⣿⣿⣿⣯⡙⠣⢾⣿⡏⢇⣿⣿⠟⠀⠉⣾⠉⢸⣧⣿⣷⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡿⣿⠰⠆⠉⣋⣥⣾⡧⣔⢻⣿⡇⡀⢄⢰⣶⣾⣤⢰⣤⣼⣯⣿⡇⣿⣟⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣙⣠⣭⣤⡝⠶⢿⢿⡽⠛⣡⣦⣼⣿⣿⡟⣡⢸⣿⣿⣿⡧⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠉⣣⡀⠐⠨⠈⣵⡿⣿⠿⣛⣿⣿⣿⠙⡄⠻⣛⣥⡄⣾⡷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣭⣝⣷⡀⠚⢷⡟⠿⢗⣭⠹⠿⠟⡀⠃⡀⢿⣿⠃⣿⣱⣾
⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣯⣽⡲⣿⣿⣿⣿⠿⠿⡗⣈⠂⠛⠀⠉⠐⣒⡛⢃⠀⣼⣌⣿⣅⣿⣾⣿
⣞⣿⣿⡏⣿⢷⣄⡀⠀⠀⠀⢀⢠⣴⣯⣛⣛⣫⣭⡤⣄⠀⠈⡁⠤⠶⣶⣷⣬⣰⣿⣷⣽⣛⣿⣤⣿⣿⣾⣹⣿⣿⣿
⠛⢛⠛⠓⢙⣛⣹⣿⣶⣶⣾⡿⢿⣿⣿⣻⣿⠶⠛⢩⣔⣿⣷⣶⣶⢦⣙⣛⠿⣿⣿⣟⢰⣿⣧⠙⣿⣿⢿⠏⢻⣿⣿
⢠⢈⡆⣠⣻⢸⡿⣿⣿⣿⣿⣼⣿⠿⣛⣭⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢳⣿⣿⣛⣿⣽⣿⡄⣷⣾⡘⢻⣿⣿⣿
⣄⣨⣀⣋⠉⣾⡾⣿⠿⠟⡁⢭⣶⣞⣿⢟⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡛⣿⢳⡾⣿⣿⣿
⣈⣉⢁⣿⠀⣼⡆⣴⡶⣴⣶⣿⣿⣿⣝⣽⣽⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣆⣾⣦⣟⣿⣿
⣿⣒⣞⣿⣧⣿⣷⣜⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡆⣽⣿⣭⣭⣽⣭⣩⣭⣿⣹⣿⣷⠰⣎⣷⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⣀⣀⣤⣤⣯⣉⠢⣤⣤⣀⣤⡔⣩⣴⣷⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣤⣾⣿⠋⣉⢭⣭⡍⡉⠉⣿⣿⣿⢉⡤⣶⣶⠂⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⠤⠬⡙⠛⢿⣿⣿⣀⠀⣈⣛⣁⣈⣀⣿⣿⣿⣅⣀⣈⣡⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠠⠒⢚⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠿⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀⢀⡠⢄⠀⡠⠤⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡀⠀⠀⠀⠀⠐⠁⠀⠈⠋⠀⠀⠉⠀⠀⠈⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠹⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢿⣿⣿⣿⣶⣶⣶⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⣀⣀⣀⣀⣀⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⠀⢀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠋
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣶⣆⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⢀⣀⣀⠀⠀⣀⣀⠀⠊⠁
⠀⠀⠀⠀⣤⡀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⡀⣿⡇⠀⣰⣿⣿⡿⢠⣿⡿⠿⠀⠀⢀⡤⠾
⠀⠀⠀⠀⠿⠃⠘⠿⢷⣶⣦⣄⠀⣿⣿⠸⣿⣿⣿⣿⡿⠁⣿⣿⠃⠹⣿⣧⡀⠈⠻⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣿⣿⡆⣿⣿⠀⣿⣟⢻⣷⡀⠀⣿⣿⣀⣀⠈⢻⣿⣦⢀⣀⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⠿⠛⠀⢿⣿⠀⣿⣿⠈⣿⣷⣄⡛⠛⠛⠛⣀⣸⣿⣿⢺⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠟⠃⠈⠉⠉
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠒⠶
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⣾⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢈⣩⣿⡿⠁⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⣻⣿⠀⠂⠀⠀⠀⠈⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠈⠙⠿⣶⣤⡴⠂⠀⠙⠻⣿⣄⠀⠀⠀⢈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡈⣿⣿⣁⣀⣀⣀⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠿⡿⠿⠿⠿⠿⠟⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣄⣴⠋⠉⠀⠀⠀⠀⠀⠉⠉⠙⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⡛⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢀⡎⠀⠀⠀⣠⠂⠀⠀⢠⠂⠀⢀⠀⠀⡀⠀⠀⡆⠀⡄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣼⠀⠀⠀⢠⠃⠀⠀⢠⠏⠀⠀⡎⠀⠀⡇⠀⠀⡇⠀⣷⡀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣇⠀⠀⢠⡏⠀⠀⢠⠏⠀⠀⡼⠀⠀⢸⠁⠀⢰⠁⠀⡇⢳⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⡎⢿⣀⠔⠹⡇⢀⡴⢻⠀⢀⡴⠁⠀⣠⠏⠀⢀⡞⠤⠞⠀⢸⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡤⢤⡀⢸⠀⠀⠀⢀⠉⠉⠀⠈⠉⠉⠐⠶⠖⠉⠦⠤⠊⠀⡀⠀⠀⠘⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢧⣐⡌⠚⠁⠀⠀⠀⠨⠕⣶⠆⠀⡀⣀⣀⣀⣀⠀⠲⣾⠭⠤⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠈⠙⣄⠀⠀⠀⠀⠐⠉⠀⠀⠈⠉⠁⠉⠉⠉⠁⠀⠈⠑⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⠀⠀⡈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⢿⣄⣴⣧⣀⣤⣈⡑⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣻⠶⠶⣏⠳⣄⠀⢒⡶⠲⣟⣫⣭⠦⠾⠦⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⢢⡴⠛⢧⡤⠖⠉⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣠⢼⡀⠀⠀⠀⠀⠀⣸⢄⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢆⣀⠇⠀⠀⠀⠀⠀⢇⣀⠟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⠀⠈⠉⠉⠁⠉⠙⡇⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⢀⡟⠀⠀⠀⠀⠀⠀⠀⣇⢠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⡛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣾⣿⣿⣷⣿⣾⣦⣾⣿⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡛⠉⠉⠙⢿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠈⠻⣿⣿⣿⠟⠉⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⢀⣤⡀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣀⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠐⣿⣿⣿⣦⣀⠀⠀⠀⣠⣶⣿⣿⣿⡇⠀⠀⠈⣿⡇⠀⠀⣠⠴⠢⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⡰⠛⣛⣿⣿⣿⡷⠀⠸⣿⣿⣿⣟⡋⠳⡄⠀⠠⠟⡳⣄⡠⠕⠀⠀⠀⠚
⠀⣀⣠⠴⠤⡀⠀⢀⠜⣛⠻⡀⠀⠀⡇⢾⢠⣴⣮⢏⠀⠀⠀⢫⠀⢷⣾⠇⠀⣿⠀⠀⠸⠏⣹⠊⠉⠯⣤⣴⠟
⢾⣁⠀⠀⠀⠉⣀⢽⠲⡋⡇⠀⠀⠀⠙⢬⣿⣯⣥⠎⠀⠀⠀⠘⢷⣄⡀⠤⠺⠃⠀⠀⢰⠚⠁
⠀⢧⣀⡀⠀⣹⠆⠈⠓⠶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⣤⣤⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠁⠐⠒⠂⠬⠿⠟⠢⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣠⣤⡤⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣻⣿⣿⠋⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠛⠃⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⢠⣿⣿⣿⣿⣿⣿⣶⣤⣤⡄⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⣿⣿⣿⣿⣿⣟⠛⠋⠉⠹⠿⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣦⠶⠶⠞⠛⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣥⠀⠀⠀⣴⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣿⣶⡾⠛⠄⡈⢻⣿⠛⠛⠿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣿⣿⣿⣿⣶⣄⠙⢺⣿⣀⣀⣀⣠⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⣿⣿⣷⣻⣟⡛⣿⣿⣿⣿⠋⢹⣿⣿⣦⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣅⡢⠀⢿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣤⣹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠏⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡿⠋⠀⣿⣿⡿⠟⠁⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⡈⠛⠋⠀⠀⠀⠈⢉⡀⠀⠀⠙⠛⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠛⠋⢀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠀⠀⢚⡫⠥⢤⡀⠀⠀⠀⠀⠀⠀⡠⠖⠈⠹⣍⠉⠀⠀⠀⠈⢿⣿⡿⠿⠿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣬⣷⠀⣞⣠⠖⠋⠉⠆⠀⠀⠀⠀⠜⠉⠒⠲⢤⣘⡇⠀⠀⠀⠀⠀⠉⢠⣴⣶⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⣯⣹⡇⡏⠀⣔⣐⣦⠀⠀⠀⠀⠐⠀⡰⢒⣦⠀⠈⡇⠀⠀⠀⠀⠀⠀⢿⣍⡹⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⢻⡇⠜⠀⠌⠿⠋⡖⠉⠚⠀⠀⠠⡝⣛⣋⢠⠔⠀⠀⠀⠀⠀⢀⣀⣘⣋⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣰⠟⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣠⣤⣈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠶⢤⣤⣀⣀⣀⣀⣁⣈⣠⣤⣴⠞⠋⠁⠀⢀⠋⠀⠀⠙⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣠⣤⣭⣭⣍⡉⠙⣛⣶⣿⣥⣤⣄⡀⢀⡞⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⣴⡟⢉⣠⣾⣿⣿⣻⣾⡋⢁⣠⣾⣿⣿⣿⠟⠀⠀⠀⠀⢀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⣾⣿⣿⣿⠿⠾⣷⠌⠙⢻⡿⠿⠿⠿⠛⠉⠁⠀⠀⠀⠀⢀⣼⠃⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⡀⠀⢀⣼⠟⣿⣿⣿⣟⠓⠆⠀⠀⠀⢸⠁⡆⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠈⠉⠙⢧⣹⡏⢸⣿⣿⣿⣿⣭⡍⠀⠀⠀⢸⣄⣧⣤⣤⣤⣶⣶⣾⣿⣿⣿⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠄⠀⠀⠀⠀⢸⡟⠁⠐⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⣾⣿⡦⢠⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠛⠛⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣾⣿⣯⣟⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⢹⣿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⣿⣛⣛⣛⣻⣿⣿⡿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⡤⠤⣀⢀⡀⠀⢀⣀⡀⢸⢉⠇
⠀⠀⠀⠀⠀⢀⠎⠀⢀⣷⠋⠈⠹⠉⠙⠇⣀⠘⠖⡖⢲⠊⠀⠒⢼⠃⡈⠉⠍⠰⠏⠻⣸⠉⠒⣄⢀⣀
⠀⠀⣀⡴⠋⠉⢄⠘⠉⠉⡆⠁⠠⢈⡁⠦⠀⠔⠀⠀⠀⠀⠀⢁⡠⠀⣃⡀⢀⠈⡠⠠⠆⠀⠀⠈⠁⠀⠉⠒⣄
⢠⠊⠀⠀⠀⠀⠀⠸⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠑⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠒⠶⠖⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣖⠠⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣛⡿⠅⠀⠀⠀⠀⠀⣈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢴⣷⣤⡤⠄⠀⠰⠀⢠⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠧⡴⠀⠀⠀⠠⣠⠈⣀⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⡶⠛⠁⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄⠚⠒⠂⢄⡰⢫⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⡼⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢃⠀⠀⠀⠠⢄⡇⠈⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠒⠒⠲⠞⠛⣦⣤⠴⣚⠉⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⠀⠀⠀⠀⠙⠫⣈⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡞⢉⢑⠦⣤⣜⣠⡤⡄⠐⠤⣄⣰⠀⠀⢀⣌⡗⠀⠈⢧⠀⢠⠔⣫⡍⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⠒⡏⠁⠈⠛⢑⣞⣢⣀⠀⠉⠉⣉⣩⠑⢒⣏⣧⢀⠓⡃⠚⠉⢁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢈⣤⠼⠯⢍⡉⠁⠛⠃⠤⠀⠁⠈⠯⣁⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡈⡉⠑⠒⠂⠀⠰⡀⠀⢸⠁⠀⠀⠀⠀⠈⠲⠄⢀⠃⠀⠀⢀⣔⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⠉⠉⠓⠒⠂⠐⠲⠦⠷⠀⠘⣦⣀⣀⣀⣀⣀⡠⠤⣾⠀⠀⡠⠋⠀⠀⣠⡔⣉⡳⠶⠖⠋⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⢻⣹⣋⡩⣭⠽⠍⠉⠉⠀⠚⠁⠳⣄⠀⣠⠚⠓⢦⡏⣀⣰⣟⣫⠭⣭⢭⡥⠤⠤⠤⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡰⠇⠀⡀⠀⠀⠀⠀⠀⠀⢨⣿⡿⠀⠀⠀⠉⠀⠀⠀⠑⡖⠾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠓⠲⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⠉⠙⠓⢲⠖⠶⠶⠒⠒⠒⠚⠛⠉⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢠⣀⣀⡀⣼⠀⡇⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠐⠛⣿⡻⠿⣒⠒⠂⢀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣾⣶⣤⣤⣄⣦⣤⣭⣥⣿⣴⣦⣦⡁⠀⠀⢈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠻⠿⠧⠽⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠠⠤⠀⠄⠀⢠⠠⠠⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠆⠀⠄⠄⠆⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⣋⣭⠉⢡⡦⣤⠈⢭⣭⡙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢟⠉⠀⢿⡄⢻⡶⠂⢸⡇⣤⠀⣸⡇⢀⣿⡛⠆⢙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠰⣏⣥⠀⣬⣿⢘⣛⣯⣬⣿⣯⣤⣿⣀⡾⣯⠉⣴⠟⣿⠌⠹⣷⣄
⠀⠀⠀⠀⠀⠀⣴⡟⣱⢶⣆⣙⢷⢞⣴⡾⠛⠉⠀⢀⠀⠀⠀⠉⠉⠛⢶⣮⡓⠾⠃⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⣼⡟⠀⠙⠻⣯⢙⣴⠟⠁⠀⢠⣤⠀⣾⢿⠀⠾⠇⠀⠀⠀⠈⠻⣦⣀⣀⣀⣠⣤⣿⣧⣴⣶⠆
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢠⡿⠃⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⢛⣛⣉⡉⣩⣥⣤⣄
⠀⢀⣀⣠⣿⣧⣤⣴⣶⠶⠿⠿⠟⠛⠛⠋⢋⣉⣉⣉⠀⣠⣶⣶⣄⠰⢶⣾⠿⠃⣿⡟⠛⠁⢸⣿⠉⣿⡇
⠀⠈⠉⣉⣭⣤⣄⠀⣴⣶⡶⠆⠀⠀⣿⡇⢸⣿⡋⣉⠀⣿⡇⠘⠛⠀⠈⣿⡇⠀⢿⣿⠿⠃⢸⣿⡀⣻⡇
⠀⠀⠀⢹⣿⢙⣿⠇⢻⣷⣤⡆⠀⠀⢹⣯⠈⣿⡟⠛⠀⢻⣧⠀⣶⡆⠀⣿⣇⠀⢸⣿⣶⣶⠀⠿⠿⠟⠃
⠀⠀⠀⢸⣿⠛⣿⡆⢸⣿⣀⣀⠰⣶⣸⣿⠀⣿⣷⠶⠆⠘⠻⠿⠛⢀⣀⣈⣉⣀⣤⣤⣤⣤⣶⠶⢶⣾⠟⠛⠃
⠀⠀⠀⠈⠿⠇⠻⠿⠀⠛⣛⣛⣁⣉⣩⣥⣤⣤⣶⡶⠶⠶⠟⠻⠛⠛⠛⠋⠉⢉⣿⠏⠀⠀⠀⠀⣸⡏
⠀⠀⢀⣶⡶⣶⣾⠿⠟⠛⠛⠛⢿⣍⠉⠉⠀⠀⢠⣤⠀⣦⣖⠀⠾⠂⠀⠀⣠⡾⢣⣤⣄⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⣠⡙⢷⣤⣀⠀⠈⠁⠈⠙⠋⠀⢀⣀⣴⡾⢋⡤⠻⣦⣙⡷⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⢀⣴⢯⣽⢆⣬⣝⠛⠷⠶⠶⠶⠶⠿⠛⢫⣥⡖⠻⣶⣃⡀⣉⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣅⡿⢁⣾⠷⠌⠀⣸⠇⣾⣛⠀⣿⠙⠂⠈⣷⡀⢘⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣭⣓⡐⣧⡟⠀⣿⣭⠀⠻⣶⣟⣀⣬⡷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠶⠶⠶⠾⠿⠛⠛⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⡠⢴⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡖⢉⣁⣰⠶⠶⢞⣖⠀⢀⣴⣶⣦⠀⠏⡀⡈⡣⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⢱⠀⠀⠻⣜⡾⢐⠂⠈⠅⠈⣿⢄⠈⣟⣯⡸⢰⠤⠥⠥⢄⣀⠁⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠘⠤⠐⣜⠆⢀⣠⠴⠿⠥⡄⢠⢒⣢⣄⣼⣋⣴⡁⠀⢹⡥⠔⣚⢦⢤⡈⠳⡄
⠀⠀⠀⠀⢠⠒⠒⢦⠀⢀⠔⣉⡤⢄⣀⣀⣩⠞⢉⣄⠀⣀⡀⠀⠉⢇⣸⠁⠐⠊⠀⣐⠙⢦⣽⠀⢠⠔⠲⣄
⠀⠀⠀⠀⠻⠀⠔⢊⣀⡜⢀⠇⢀⠀⠋⠄⢣⡔⠉⠬⠙⠃⠭⠏⢹⣈⠿⡀⢾⡛⢢⠀⢀⡼⡯⠀⠀⠔⡄⠸⠂
⠀⠀⠀⠀⠀⡠⠊⠁⢀⢙⣜⡄⠑⠀⢀⡀⢇⣇⠀⠉⣀⣀⡈⠀⢸⡯⠂⠈⢆⣉⣥⡴⠛⠛⠓⠒⠦⣴⠄
⠀⠀⠀⠀⡼⠀⡞⣉⡛⠽⡘⠿⣦⣀⣉⣠⢴⠺⢄⣸⣍⣩⣽⣀⠾⠲⣄⣾⠋⢣⡼⡧⠖⢊⡹⣤⣄⠈⠳⣄
⠀⠀⠀⠈⢳⣸⠁⠰⠀⢀⣤⡄⣸⣟⣿⠋⠀⠀⠈⢏⠑⠀⢫⠇⠀⠁⠈⢻⣧⡤⢻⠁⢀⣁⡀⠐⡊⢱⡄⡏
⠀⠀⠀⠀⠀⠹⣷⣄⡸⣟⣭⡾⠚⡽⠁⢀⡴⡆⠀⠀⠉⠋⠁⠀⠀⡗⣄⠀⠙⣆⡸⢦⡯⣙⠛⡆⠀⣰⢻⠁
⠀⠀⠀⠀⠀⠀⢀⣴⡉⠻⡃⠀⣠⡇⠀⣎⠀⢹⠀⠀⠀⠀⠀⠀⢰⠃⠈⡇⠀⡾⠀⠀⠧⡤⠿⢥⡞⠉⠁
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠶⠬⢯⣤⣿⣀⣿⠷⠾⠤⠦⠤⠶⠦⠶⠼⠦⢶⣇⣰⢧⣤⡤⠤⠤⠦⣀⡇
⠀⠀⠀⠀⠀⡠⠛⠁⠀⠀⠰⣿⣿⣦⠤⣜⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠭⠥⣜⣿⠷⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠝⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢟⠻⡏⠸⣿⣿⡿⣿⣿⠟⠻⠿⡿⠟⢋⣿⣿⣿⠿⠛⠁
⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠀⠀⠠⠐⢩⠥⠀⠈⣀⠀⠜⠛⠋⠈⠁⢀⣀⣤⣤⣄⣚⣉⡉⠁
⠀⠀⠐⠒⠯⣭⣉⣈⣈⡁⠀⣀⣀⣀⠀⠀⠈⠁⠘⠛⠛⠛⠃⠀⠀⠈⠃⠀⠀⠀⠀⠀⠒⠒⠠⠶⠞⠛⢿⣿⢦⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⣻⠆⠀⠀⠀⠀⠀⢈⣶⡀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢸⣌⠁⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣄⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⠀⠀⣄⣠⣀⡀⠐⠶⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣤⣤⣐⣚⣿⣿⣿⣿⡇⠀⣋⣠⠼⠛⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡆⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣾⠂⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣿⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣰⣿⣿⡄⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠟⠿⠿⠛⠛⠛⠛⠋⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢆⣠⡄⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠿⠟⠛⠻⠷⣶⣤⡀⠀⠀⠀⠀⣠⣴⠾⠟⠻⠿⢿⣽⣿⣾⣷⣀⣠⡀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⣠⠛⠉⠀⠀⠀⠀⠀⠀⠉⠁⠙⢿⣿⣿⣿⣄
⠀⠀⠀⠀⣄⣾⣿⣿⠉⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄
⠀⠀⠀⠰⢻⣿⣿⠇⠀⠀⠀⠀⢰⣀⠴⠈⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃
⠀⠀⠀⢠⡟⣿⣿⡖⠀⠀⠀⠀⠈⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⠀⠀⣩⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⣀⡠⠀⠴⠒⢐⡩⠋⠀⠀⠀⠀⢠⣿⣿⠫⡉
⠀⠀⠀⠀⠈⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠔⠒⠊⠉⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⣰⣿⣿⠾⠃
⠀⠀⠀⠀⠀⠿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠎
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⢀⡀⢀⡀⠀⣶⣿⣿⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣁⡀⠤⠄⠒⠂⠈⠉⠀⡔⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⣿⣶⡄⠀⠀⠀⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣌⣴⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠜⠻⡾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡶⡀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⣠⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠖⠒⠒⠒⠒⠶⠤⢤⣄⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢯⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠐⠺⠥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠡⠖⣀⡴⠛⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣠⠴⢏⣉⠓⠒⠢⠤⠄⣈⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡤⠖⠋⠀⠀⠀⠈⠳⡀⠀⠠⡀⠀⠈⠉⠉⠛⢒⡶⠂⠀⠀⠀⠸⠧⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢧⠀⠀⠀⠀⠀⢠⣀⠌⠀⠀⠈⠳⡀⠀⠀⠀⣹⠋⠀⠀⠀⠀⠀⠹⡓⢄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⢰⣶⣼⠂⠀⢸⠃⢀⡤⠤⢤⣄⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠒⠮⡙⠒⠒⠂⠘⣄⠀⠀⠈⣡⠏⠀⠀⢉⣉⠉⠐⢦⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⠤⠤⠄⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⢀⡀⠻⠽⠆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀⠚⠉⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠠⢄⠀⠀⠀⠀⢀⣠⢤⡎⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⣀⣤⠤⠴⠚⠉⠀⡾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢾⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠤⢤⡤⣀⣤⣤⠤⠤⠤⡃⢸⠤⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⠀⣬⠀⠀⠀⠀⠈⠉⠋⡀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡗⠛⠂⠀⠀⠀⠀⠐⠋⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⣧⣄⣠⠼⣇⣀⣀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣯⠈⣾⡀⣰⠛⡏⢫⢹⡙⠓⣒⣒⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠋⠩⠭⠭⠭⠭⠷⠖⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⡤⠖⠚⠉⢩⣟⣥⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣀⡤⢶⡦⣄⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠇⠀⢠⠀⠀⢀⠀⠀⠀⠤⠤⠒⢀⡾⠋⠀⠀⠘⢦⠀⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠠⡀⢈⣆⣱⢄⣉⠙⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠈⡇⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣬⠽⠛⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢸⡀⢠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⡄⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠈⡇⠘⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⠻⣤⡇⠀⠀⠈⣷⣤⠖⠁⠀⠈⠓⢶⣾⠅⠀⠀⠀⡿⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠈⢢⡀⠈⠀⠀⠀⠀⠉⠁⠀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⣧⠞⢉⡉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢷⢰⡏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡼⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣝⡂⣠⠀⠀⠀⠀⣠⣴⣚⣋⣉⣉⣉⣉⣿⡄⠀⠀⠀⠘⢀⠇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠀⠀⠀⢰⣿⣿⠛⣛⣻⣿⣿⠿⣿⠃⠀⠀⢸⣇⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠘⣯⠭⠽⠿⠒⠒⠒⠒⠾⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⢤⣤⣤⣤⣤⡤⠴⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢄⠀⢸⠀⠀⠀⣖⢦⠀⠀⠀⢈⠀⠀⠀⠀⣏⠳⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠋⠀⠀⠀⡤⡀⠀⠀⠀⡄⠀⠀⠀⢉⡁⠀⠀⠀⢸⠀⠀⠀⠀⠈⠄⠁
⠀⠀⠀⠀⠀⢀⡠⡆⢀⠀⠀⠀⠘⠓⠀⠀⠀⠀⢱⢠⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⡀⠀⠀⢠⠎⡠⠗⠉⠀⠣⡀⠀⠀⠀⠁⠀⠀⠀⡈⢣⡑⠤⢀⣀⡀⢴⠏⢈⠐⠤⠤⠤⠐⢒⣶⠞⠁
⠃⠀⠀⠀⣘⢀⣠⡄⠀⠀⢱⡀⠤⣀⣀⣀⡠⡴⠁⠀⠉⠒⠒⠒⢂⠁⠀⠈⠙⠉⡩⠉⠉⠉⠀⠀⠒⠀⠀⠄⠀⠀⠀⡀
⠀⢠⡀⠀⠈⠑⠛⠁⠀⠀⣰⠉⠒⠤⠤⠴⠚⠀⣀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠣⡈⠂⠤⡀⣀⠀⠤⢒⠟⠀⠠⠀⠀⠀⠀⠰⢌⠳⢄⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⠬⢍⣉⡉
⠀⠀⠈⠑⠒⠠⠄⠐⠒⠁⡄⠀⠀⠀⣤⡀⠀⠀⠂⠑⢄⠑⢤⡀⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⢀⠂⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠰⠀⠀⠀⠑⠦⠘⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⣤⣤⣶⠶⠀⠀⠆⣘⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠉⠉⠉⠉⠀⠘⢯⣭⠽⣳⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠀⢀⠀⠀⠀⠈⣀⡠⢤⠀⡀⠒⠒⠒⠤⠤⢄⣀⡀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡎⠀⠀⠀⠴⠷⠒⠃⠀⠙⠒⠒⠐⠢⠤⠔⠀⠀⢰⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠒⣚⡭⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠭⠗⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠎⠹⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠋⠈⣋⠿⠂⠀⠀⠀⠀⣴⡀⠀⠀⠀⣹⢂⣄⣰⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡰⢦⣹⡄⠀⠀⠰⡆⠴⠉⠧⢄⡀⠀⠟⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠁⠄⠀⠀⢸⣠⣄⡞⠋⠁⡀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⢀⠀⠀⠃⠀⠀⣀⣀⣤⣟⣁⣘⣃⠀⠀⠁⠀⢉⣀⣴⠏⠧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣖⠋⠘⣓⡶⠂⠈⠁⠀⢀⡴⠚⠻⠤⣄⣀⡀⠈⢉⣻⣦⡀⠀⠈⣳⣠⣄⢾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠞⠳⣯⠀⣀⠀⠀⢀⡾⠲⢤⣄⡀⢀⣨⠿⠛⠉⢁⣦⡽⣄⠀⠟⠁⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⣯⣤⣀⣀⡤⠟⠉⣀⣠⠴⠚⠉⠀⠣⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣀⡴⠋⢁⣠⣴⣚⣉⣀⣀⣀⣀⣀⣈⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⢀⢀⢠⠰⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠰⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣬⠼⠖⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣄⠉⠓⠛⠛⠋⣉⡉⠉⠉⠉⠡⣤⣀⠀⠀⠀⠘⠓⣋⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣼⠃⠀⠀⠴⠻⠟⠀⠀⠀⠀⠀⠘⠼⠛⠆⠀⠀⢺⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⣠⠀⠀⠀⢁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣗⠀⢰⠲⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⣄⠀⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣹⠖⣶⠤⣤⠤⣴⢋⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠾⠀⠚⢡⠞⢡⠞⡡⠋⡼⠓⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢾⡉⠈⢧⡙⢦⡀⠀⠀⠐⠃⠞⠡⠎⢚⡵⠋⣰⢋⡿⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠛⢷⡂⣹⠦⣍⡓⠶⠤⠤⠴⠶⠚⣉⡤⢺⡧⠟⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠉⠓⠶⣮⣩⣟⣳⣖⣒⣟⣩⡿⠗⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠴⢃⡴⠂⡀⠀⠀⠀⠀⠈⢀⠉⠉⡀⠀⢀⠀⢠⠄⢀⡴⠁⡴⠃⣠⠆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋⡠⠊⣠⠞⢀⡴⠃⡰⠋⣠⠊⢁⡴⠃⡴⠋⣠⠋⢠⠞⢀⡞⢁⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠠⠞⠠⠞⠁⠠⠋⠀⠚⠡⠞⠁⣰⠋⢠⠞⠀⡼⠁⡴⠃⢠⠏⢠⠋⡴⠂⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⣶⠉⠛⠛⠛⢲⠐⠟⢵⣤⣦⣴⣦⡶⠻⠂⡶⠚⠛⠛⠋⣶⣶⡿⠁
⠀⠀⠀⠀⠀⢀⣠⠤⣤⠶⠶⣿⣦⣤⣴⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣶⣾⣷⣶⣦⣤⣴⣿⠴⠶⣦⠤⢄⡀
⠀⠀⠀⠀⢠⡋⣴⠋⢡⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠙⣦⢙⡄
⠀⠀⠀⠀⣠⢿⡛⠒⢚⣿⢿⣟⣭⣭⠀⠀⠀⣀⣠⣶⠿⠛⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡓⠒⢛⡿⣄
⠀⠀⠀⣞⣿⠀⢷⣿⣿⣿⡿⠿⠟⠉⠀⢠⡿⢟⡩⠀⠀⠀⠀⠠⠐⠂⠨⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⡶⠄⢿⣳
⠀⠀⠀⣯⢧⡻⠚⠉⠀⠀⠀⠀⠀⠀⣴⡟⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⢹⣿⣿⣿⣿⣿⣯⡓⠟⣴⣽
⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⡟⠁
⠀⠀⠀⠀⡿⠐⠂⢀⠁⠀⠀⣰⡿⠏⠀⠀⠀⠀⣤⣤⡤⡀⠀⠀⠀⠀⠀⠀⠒⠒⠂⣿⣿⣿⣻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⣿⡆⠀⠀⠀⢀⡼⠏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⠋⠐⠀⣿⣿⣿⣿⣷⣶⣬⣲⣿⠄
⠀⠀⠀⠀⣿⡇⣸⠀⠀⢈⣜⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢿⣹⢻⣿
⠀⠀⠀⠀⣿⡷⠁⠀⠀⣯⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⡏⢼⣿
⠀⠀⠀⠀⣿⡇⠀⠀⣰⣿⣟⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢹⡇⣸⣿
⠀⠀⠀⠀⣿⡇⠀⣰⣿⣿⣿⡜⣿⣿⣦⠀⠀⠀⠀⠀⠠⠴⢶⣶⣶⡄⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⣨⣮⢿⣿
⠀⠀⠀⠀⣿⡇⢠⣿⣿⣿⣿⣿⣼⡗⠸⡳⡀⠀⠀⠀⠀⠘⠲⠶⠖⠁⠀⢀⣾⣿⣿⣿⣿⣿⣿⣳⣿⡿⢸⣿
⠀⠀⠀⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⡟⠘⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣾⣻
⠀⠀⠀⠀⣿⣟⠟⢿⣿⣿⣿⣿⣿⣿⠀⡠⠀⠀⠙⠒⠦⠤⢤⠄⠀⢠⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢹⣽
⠀⠀⠀⠀⣿⡇⡇⠀⠿⣿⣿⣿⣿⣿⡇⠄⢀⠀⠀⠀⢈⡁⠀⠀⠀⢸⡇⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿
⠀⠀⠀⠀⣿⡧⠀⠈⠀⢻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢧⡀⢢⠀⠹⣿⣿⣿⣿⣿⠛⢌⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠌⠢⠀⢘⣿⣿⡇⣿⢸⠀⡸⠊⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢻⣿⣿⣿⣿⣧⠈⢺⢿
⠀⠀⠀⠀⣿⡇⢀⡀⠀⠀⠈⣿⠋⢁⣿⣠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⠄⠀⣸⣿⣿⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⣿⣗⠟⠁⢀⡀⣰⠇⠀⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣹⣿⣤⣤⡌⣷⠋⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⣿⣟⠛⠻⣿⣿⣷⣿⣏
⠀⠀⠀⠀⣁⢤⣿⣄⣶⣻⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⣾⣈⣾⣭⣀
⠀⠀⠀⠤⣼⣿⣿⣿⣧⣿⣧⣆⣀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣧⠤
⠀⠀⠠⠶⣿⣿⡿⠿⠻⢻⣿⣏⣉⣭⣿⡆⠉⠻⠷⣶⣶⣶⣶⣶⣶⡾⠿⠉⢰⣿⣿⣿⣿⣿⡿⠻⠿⢷⣿⣿⠶⠄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡾⠟⠛⠿⠶⠶⠾⠓⣺⢩⡍⣗⡚⠷⠶⠶⠾⠛⠻⠷⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠇⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⢿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠐⡻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢠⠔⠋⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠬⠥⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠠⢧⣀⠀⠀⠈⢷⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠃⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠧⠤⢴⠄⠀⠈⠳⡄⠀⠀⢧⠀⠀⢀⣋⣀⣀⣤⣄⣼⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣧⢀⡤⣍⡦⡀⠀⠳⣞⠉⠀⠀⠀⠀⠀⡀⣹⢀⣀⣰⣃⣀⣀⣠⡤⠤⠶⠖⠚⠋⠉⠉⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⡟⡾⠀⠈⠳⣿⢆⠀⠈⠓⠒⠦⠤⠴⠚⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠴⠚⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡴⠙⠉⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⠀⠀⠀⠀⠤⠀⢀⣉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⣠⠜⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠉⠉⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⡴⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢌⡓⠠⠧⢤⣀⣀⣀⣀⣀⡀⠀⠊⢀⢀⣀⣀⣀⣸⣓⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢬⣀⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣼⡙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣇⣀⡉⠉⠐⠒⠠⠤⢤⣀⡀⠀⠀⡜⠉⢧⣿⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⣥⣄⣀⡀⠀⣀⡉⠉⠉⠉⠉⠓⠐⠒⠒⠘⠳⠶⠤⠤⢸⣇⣣⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣿⡿⢸⣿⠀⣿⡿⠙⠇⠀⢰⣶⠆⢰⣶⠠⣤⠀⣠⠄⣀⢀⡀⣀⠈⠉⠉⠉⠐⠒⠒⠒⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⡏⣿⣦⢀⣿⡗⠟⠀⠀⢸⣿⠀⣾⣿⣰⠁⣼⣿⠀⠹⠺⢹⣿⠋⠇⣿⡿⠸⡆⢲⣶⢠⣄⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⠛⠃⠻⠟⠸⠿⠇⣼⠺⡟⣿⡿⢀⣿⡇⢈⡄⣿⡇⠀⠀⠀⢸⣿⠀⠀⣿⡇⡎⠀⣿⡿⢸⣿⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠁⠘⠓⠒⠁⠀⠿⠿⠀⢸⣿⣇⣴⢀⣿⡇⣾⡟⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⠉⠁⢛⢲⠒⠒⡶⢤⠤⠤⠤⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⡌⢧⡼⢁⡼⠃⠀⣀⣀⣀⣀⡍⢸⣍⣩⡭⠭⢶⡶⠒⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣛⡉⠉⠉⠁⠀⡼⠛⠃⠉⠉⠉⠁⡏⢰⣷⡌⢱⡄⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣀⡀⠀⣀⠔⠁⠀⠀⠀⠀⠀⡼⢁⣏⠙⢇⡼⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡟⠉⠉⠁⠀⠀⠀⠀⠒⠛⠛⣛⢋⣉⣑⣂⣾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠧⠽⠿⠿⠷⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉⠓⠛⠛⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⠂⠀⠀⠀⠈⠓⠃⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⢀⡞⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡇⠀⢀⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⣼⣿⣿⣿⠏⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⣀⣤⣤⡈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠁⠀⠀⠀⡀⠀⢀⠀⠀⡈⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠙⠒⠛⠀⢹⠀⠈⠛⠛⠁⠀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠐⠛⠙⠃⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣾⣿⣛⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⡁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢿⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⡏⠘⠙⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⣿⣿⣿⣿⡄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠙⢿⣿⣿⡟⢤⠀⡨⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢻⡿⠉⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⠴⠶⠶⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣼⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡏⠔⠉⣠⠖⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠉⠀⠀⣠⠴⠖⠛⠋⠉⠉⠛⠲⢤⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠸⢠⠟⢁⣴⣯⡥⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⢀⣤⠖⠻⣇⣀⣔⣡⡤⠤⠶⠶⠶⠶⢤⣄⡀⠀⠀⠀⠀⢹⠀⠀⠀⢠⡀⢸⡟⢧⡀
⠀⠀⣰⠟⠀⠀⣠⡞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄⠀⠀⢸⣇⠀⣦⠘⡿⠟⢳⣄⢳⡄
⠀⣸⠃⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⠞⢹⡼⠛⢷⠃⠀⠀⠹⡆⠻⡀
⢀⡏⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⢹⡀⣷
⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⢦⠄⠀⠀⠀⠀⠈⣇⢸⡆
⠸⡇⠀⠀⣿⠀⠀⠀⠀⣠⡶⠿⠿⢿⣦⣄⠀⠀⠀⠀⠀⠀⣰⡿⠋⣠⠖⠲⢦⡀⠀⠀⠀⠀⠿⢼⣇
⠀⣧⠀⠀⣹⠀⠀⠀⠀⢀⡴⠒⠙⠦⣌⠛⢷⣄⠀⠀⠀⣰⡟⢁⡴⠃⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⣿⢻⡄
⠀⣹⣤⡞⠁⠀⠀⠀⢠⡟⠀⠀⠀⣠⣬⠳⢄⠉⠃⠀⠀⠋⣰⢫⣾⣿⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⣿⢸⡇
⣾⠉⣇⠀⠀⠀⠀⠀⢨⡅⠀⠀⢰⣿⣿⣶⡄⢠⠀⠀⠀⠀⠻⡈⠷⣾⣿⣀⣠⠤⠟⠒⠋⠀⠀⠀⣿⠈
⢻⡀⢻⡀⠀⠀⠠⢤⠈⣧⣠⣄⣀⣙⣛⣋⣠⡏⠀⠀⠀⠀⠀⠉⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠛⠾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠛⠋⠉⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣤⡤⠀⠀⠐⠶⠶⠾⢷⡄⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⢠⠋⠁⠀⠀⢀⣀⣀⣀⣀⢀⣀⣁⠀⠀⠀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠚⠒⠚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⢀⣤⠞⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠷⠦⣤⣀⣀⣀⠀⠀⠀⢀⣀⣠⣤⠶⠚⢻⣆⠀⠀⠰⠞⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠈⢻⡉⠉⠉⠉⠛⠛⠉⠀⠀⠀⢸⡟⣇⠀⢀⣤⠾⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠙⠲⣤⣤⣻⣿⣿⡦⠀⢠⡞⠋⢻⣤⣤⣄⡴⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠉⠛⠛⠛⠛⠒⠋⠀⠀⠀⣤⣤⣤⣽⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⠙⠓⠲⣶⣶⣶⣶⣶⣦⣤⣤⠥⠞⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠉⠉⠉⠉⣽⠛⠉⠋⠁⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣀⣀⣤⣿⠋⠀⠀⠀⠀⣴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠧⠤⢤⣄⣀⣰⡏⠀⠀⠹⡄⠀⠀⠀⢀⡼⠿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⢸⡷⠶⠶⠊⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡿⠲⡶⠶⠶⠾⠃⠀⠀⠀⠀⠷⠶⠶⣶⠚⠻⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠉⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣶⣾⣿⣛⣷⣿⣿⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣵⣿⠿⠟⠛⠛⠉⠉⠉⠉⣉⠙⠛⠿⠻⠿⣿⣿⣷⣶⣆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣶⠿⠛⢉⠀⠀⠀⢀⠘⣧⡀⠀⠰⢦⣿⠠⠄⠀⣀⣴⠀⠀⠈⠉⠛⠿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠿⠋⠁⠀⢀⣤⡷⣆⠀⠀⠸⠛⠃⠀⠀⠚⠛⠆⠀⠀⠞⢛⠋⠀⢰⣤⡄⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠁⠀⣳⣴⡆⠀⠀⠇⠀⠀⣀⣠⣤⣤⠶⠶⠶⠶⢶⣤⣤⣄⣀⡀⠀⠚⠻⠇⠁⠀⣄⣠⠈⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⢀⣶⣾⠏⠁⠀⡀⠈⠉⡟⠋⠀⣠⣴⠾⠛⠉⠉⠑⠲⠶⠦⠤⠄⣀⣀⠀⠈⠉⠛⢶⣦⣄⠀⠀⠴⠿⠿⠄⠀⠀⠙⣿⣥⠆
⠀⠀⠀⠀⣼⣿⡿⠁⠀⣻⣾⣇⠀⠀⢀⣴⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠲⠶⠭⢿⣦⡀⠐⠀⠀⢰⣀⡤⠈⠻⣿⢃⡀
⠀⠀⠀⣼⡿⠋⠀⢀⠈⠉⠉⠀⣠⣶⠏⠀⠀⠉⠉⠐⠂⠀⠀⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠈⡉⠐⠀⠀⠀⠀⠹⣇⠀⠉⢹⠟⠦⠀⠀⠹⣿⡅
⠀⢀⣼⡿⠉⠐⢶⠿⠦⠄⠀⣰⢟⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⢀⡀⠀⠀⠀⠈⠳⣄⠀⠀⢀⣰⣀⡀⠹⣶⣦
⠀⣼⠟⠃⠀⠀⠉⠙⠂⢠⡼⠃⠀⡦⠈⠉⠀⠒⠐⠀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢉⡿⠿⡄⠀⠸⠿⣀
⠀⢻⡶⠀⡀⢴⡀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠢⠤⠤⢄⣈⡁⠐⠘⣦⠈⠀⠀⣠⠀⠀⢻⣿
⠀⢠⠁⠀⠸⠿⡄⠀⠀⣿⡖⠲⣆⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠀⠤⠤⢄⣀⡀⠀⠀⠀⠠⠤⠄⣉⣁⢻⡆⠀⢻⣿⡗⠂⠈⣿⡇
⠀⣾⠀⠀⡀⠀⠀⠀⢀⣿⣀⡠⠿⠃⢨⡟⠉⠙⠗⠀⠀⠀⡖⢂⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠀⠀⠀⠀⠉⣷⠀⠉⠀⠉⠀⠀⢹⣗
⢀⡛⠀⣤⣷⡶⠂⠀⢸⡏⠰⣧⠀⠀⣼⠃⠐⠶⠀⠀⠀⢘⠃⢰⡟⠀⠉⠏⣠⠖⠙⡷⢆⡦⢠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠐⢦⣾⣄⡀⢨⣿
⢸⡇⠀⠈⠏⠙⠂⠀⠉⠁⠀⠛⠗⠠⣿⣀⣀⣤⠰⣷⣀⣾⠇⣾⠃⠒⠾⢠⣉⠀⠀⠀⠀⠀⢬⠃⠈⠃⣻⠓⠒⢶⠀⣤⠤⣄⠀⠀⠀⠛⠩⠁⠀⢸⣿
⢾⡇⠀⠠⡀⢀⡀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠉⠰⠿⢄⣠⣴⠘⣿⣤⣠⡖⠀⠀⢠⠀⠀⢠⡿⠠⢤⡄⢠⡛⠀⢸⣧⠀⠀⣀⣠⠀⠀⢸⣿
⣸⣿⠀⠴⣿⡿⡄⠀⢿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠼⢬⠀⠀⣼⡇⠀⢀⡀⣾⡇⠀⣾⡿⠀⠠⠿⡟⠢⠀⠺⠋
⢨⣽⡇⠀⠘⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠑⠿⠧⠶⠛⠁⢀⠀⠀⠀⠀⢀⡂
⠈⠉⣿⠀⠀⢠⣴⡆⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣷⣜⠀⠀⣸
⠀⠀⠸⣇⠐⠛⢿⠋⠀⠀⠡⣤⠄⣀⣀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠐⠒⠠⠄⡤⡀⣀⡀⠀⠀⠀⠀⠨⣭⡉⠉⠒⠒⠂⣶⠃⠀⠀⠀⠛⠉⠁⣠⠛
⠀⠀⠀⣿⣆⠀⠀⢤⣠⡆⠀⠻⣦⡄⠤⢀⣉⠀⠐⠀⠀⠀⠄⠀⠉⠁⠒⠒⠳⠤⠤⢄⣉⡁⠒⠂⠀⠀⠀⠀⠀⢀⡾⠃⢀⣸⣶⡖⠀⠀⣰⣿⠃
⠀⠀⠀⢻⣿⣇⠀⠤⠶⡝⠋⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠐⢠⡶⠉⢀⠀⠀⠸⠏⠛⠀⣸⡟⠃
⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⢠⣤⣧⣀⠈⠻⢦⣄⡀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠄⢠⣀⠀⠀⠀⠀⠀⣀⣠⠾⠋⠀⠠⣌⢳⡖⠀⠀⣠⣞⠏⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠠⠟⠻⠀⠀⢠⡄⠈⠛⠷⢦⣤⣀⣉⣀⠀⠀⠀⠀⠀⣀⣈⣉⣴⠶⠟⠉⢁⣀⣶⠀⠀⠚⠉⠃⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⢶⣀⠀⠀⠘⣶⣾⡋⠀⠠⣄⣀⠉⠉⠉⠋⠛⠛⠋⠋⠉⠈⠁⠀⢀⡄⠀⠀⣽⢼⡏⠀⠀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢤⣀⠀⠀⠀⠀⠲⣿⣯⠀⢀⣹⣄⡃⠀⠀⣷⣾⣄⠀⠀⠸⠃⠂⠀⠀⠀⣀⣤⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣀⡈⠁⠀⠀⠈⠹⠏⠀⠀⠈⠉⠋⠁⠀⠀⠀⢁⣀⣤⣶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠤⣤⣤⣤⡤⣤⡤⠴⠶⠶⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⣶⠓⠓⠒⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠚⠛⠒⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⣿⡴⣾⠁⠀⠀⠉⠳⣄⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠈⢳⡄⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢀⡀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⠶⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⣾⡴⠞⠁⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠜⠋⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⢒⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⠴⠖⠛⠋⠉⠁⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣚⣡⣤⠤⠶⠖⠛⠋⠉⣡⠏⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⢈⡭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠛⠁⠈⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⠋⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠛⠁⠀⠀⠀⠀⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⡀⢀⣠⣶⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡄
⠀⠀⠀⠀⠀⢰⣿⣿⠀⣴⣿⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣽⣿⢿⡿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠻⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣾⣥⣶⠎⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠿⡏⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣿⠂⣀⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠟⠉⢡⠀⠀⠁⢀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⣻⠀⠀⡀⣘⣠⣤⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣰⣷⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡋⠁
⠀⠀⠀⠀⣠⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢻⣿⡟⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣧
⠀⠀⠀⣿⣿⡙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⡆⣾⣿⡇⣸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠋⢹⣿⡟
⠀⠀⠀⣿⣿⣡⡾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠉⠀⡁⣿⣿⣿⢻⡟⣼⣿⣿⣦⡀⠀⢀⣠⣾⣿⣿⢻⠋⡀⠸⣿⣿⡇
⠀⠀⠀⢻⣿⣿⣰⣯⣼⣿⣿⣷⡄⠀⠀⣠⣾⣿⠟⠋⢸⡀⣠⣿⣿⣿⣿⣾⡿⢻⢏⣻⣿⣷⣶⣿⣿⡟⠋⡞⢸⢀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠙⣿⣿⣏⣼⠇⣹⣿⣿⣦⣾⣿⣿⠋⠀⢀⣸⣴⣿⣿⠟⠙⢿⣿⣧⣿⡿⣿⠙⣿⣿⢻⠏⣠⠀⡇⠆⣸⣴⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⡾⢻⠇⢻⣿⣿⢛⡏⢀⡆⣸⣿⣿⠟⠁⠀⠀⠀⠻⣿⣿⣤⣿⢰⣿⣿⠀⡀⡇⣰⣷⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣾⠀⢸⣿⣿⠀⡇⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣼⣿⡟⣀⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣸⣿⣿⢠⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠍⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠓
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠐⠒⠂⢄⠀⠠⠀⣀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠈⠁⢢⠀⠀⠣⠀⠀⠀⠀⠀⢀⠀⠀⣀⠔⠒
⠀⠀⠀⠀⢀⠄⠀⠀⠀⢠⠤⢀⣉⢈⠑⠄⠤⠀⠀⠀⠀⡐⢡⣴⣶⣦⡀⠀⠀⢠⣶⣶⣶⠄⢡⠀⠠⠅⠀⠀⢀⠡⠁⠴⠞⠁⠄
⠀⠀⠀⠀⠀⣀⣶⣢⡴⠋⠀⠀⠙⣷⣵⡀⠀⠀⠀⠀⠀⠀⠈⠭⠤⠭⠀⠐⠢⡀⠰⠶⠒⠀⠀⠡⠀⠀⠀⠐⠓⢂⡰⢃⠔⠁
⠀⠀⠀⠀⢀⠡⠉⣡⡤⠀⠠⠶⠦⠈⣿⠁⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠤⠄⠠⠠⠕⠀⠀⠀⠀⢠⡀⠀⢀⡠⠒⠁⠶⠁⠀⢀
⠀⠀⠀⠀⢠⠑⡆⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠘⠅⠀⢀⢤⠤⢴⡖⠒⣶⠒⢲⡖⠦⣆⠄⠀⠲⠀⠈⡀⠁⠋⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠐⠀⠐⠀⢀⣟⠁⠃⠀⠀⠀⠀⠀⡄⠐⠸⣖⣲⣿⠿⣻⢗⡾⢭⣶⡏⠀⠀⠡⠶⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠉⢷⠄⣀⣀⣴⣟⣥⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠙⢏⣳⣠⠼⣭⣶⡿⠋⠀⢀⠂⣄⡀⠙⠋
⠀⠀⠀⠀⠀⡆⠩⡜⠀⠈⠁⠰⢻⠿⠦⣀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠈⠉⠉⠉⠁⢀⣀⡔⠁⠀⠈⢁
⠀⠀⠀⠔⠁⠀⠀⠈⠁⠀⠀⠈⢁⢄⠄⠀⠑⢄⠀⠀⠀⢀⠀⢲⠲⣈⠓⠶⠤⠄⠔⠩⢀⡵⢤⢤⣀⣀
⠀⠀⠂⠀⠠⠉⠠⠪⠀⠪⠢⠪⠀⠀⠠⠂⠆⠀⠂⡠⠀⠋⠠⠘⡇⠀⠈⠑⠢⢤⠐⠂⠁⠀⠈⠠⠙⠌⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⡠⠚⠄⢳⠔⠢⠤⡾⠈⢳⡤⠴⠚⠇⡀⠳⡇⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠈⠀⢆⠀⠁⢸⠄⠀⠀⣿⢛⠻⡅⠀⢀⠀⡆⣐⠁⠀⠄⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⡄⠠⠀⡇⠀⢠⠇⠔⡀⢇⠀⠸⠀⠀⠁⡀⢀⠀⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠁⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢆
⠀⠀⠀⠀⠀⠀⣴⣶⣤⣶⣿⣶⣶⣿⣿⠿⠿⠿⠉⠉⠉⠉⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⡟⠀⠇
⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠄⠈⠄⠠
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣶⣶⣶⣶⣷⣶⣶⣤⡀⠀⢾⣿⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠞⠿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠇⣶⣶
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢹⣿⣿⣿⣿⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣰⣿⣿⡿⢿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣿⣿⣟⣠⡏⠀⠀⣸⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠁⠙⡀⠀⠀⣿⡿⠀⠈⠻⠿⣿⣿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⡿⠀⠀⢹⣧⡀⠀⢦⠇⢹⣿⣷⠀⠠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣧⠀⠀⠃⠀⠀⠘⣿⣿⣆⠨⠀⢸⣿⣿⣆⠀⢙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣴⠀⠀⠀⣯⢸⡿⠀⠀⢸⣿⣿⣿⡇⠈⠛⢻⣿⣿⣿⣿⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡃⠀⠀⠀⣿⣼⡇⠀⠀⣸⠛⢻⡄⢡⣼⣇⣸⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⠐⠀⠀⣰⣿⣿⡇⠀⠀⡀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣐⣿⠈⡇⠀⣿⣿⣿⣷⠂⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⠀⣼⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡇⢰⠿⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠂⠀⡄⠀⢀⣀⣤⡴⠂⠀⠀⠀⠀⠉⠉⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⡀⢄⠐⢌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⡁⢀⠀⣾⣠⡶⣛⣝⢿⡇⠀⠀⠀⣀⣸⣦⣬⢧⣼⣥⣤⣿⣤⠤⠀⠀⠀⠀⣾⣷⡄⢿⠬⣐⣌⣣⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣥⣾⣾⣿⡿⢿⣷⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠹⡏⢹⠉⠿⠀⠹⠇⠀⠀⠀⠀⠐⣿⣿⣿⣾⠀⢣⣻⣿⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢹⣮⣿⣿⣿⣿⣿⣧⣤⣦⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣄⣀⣤⣠⣠⣸⣿⣿⣿⣿⣴⠢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢓⡸⣷⢸⠉⢛⣻⣿⣿⣯⣛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⡿⣿⠛⠛⡟⢙⣹⣛⠓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⣻⣿⣿⢸⠀⠀⢿⡀⡉⠛⡛⣿⣷⠀⠀⠀⢰⡾⡟⠛⠟⣡⡤⠇⠀⠀⣧⢸⢡⡟⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⡶⣿⡙⢸⠀⠀⠀⠉⠉⠉⠉⠅⢩⡄⠀⠀⣽⠃⠈⠉⠉⠁⠀⠀⠀⠀⢹⡎⣹⠇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢛⣛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⡥⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠇⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠡⠿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⠆⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⡴⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣸⣿⣿⣿⢩⣭⠤⢈⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣫⡆⠀⢳⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⣿⡄⣿⣿⣿⣿⡏⣡⣴⢙⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇⠀⠀⢣⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⠫⠀⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠉⠉⠓⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⣀⡤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡞⠉⢳⡞⠁⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠚⠹⡅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⢸⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢹⡄⠀⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡄⠀⡇⠀⢸⠀⠀⡇⠀⠀⠠⠶⠖⠶⠄⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⡇⠀⢸⠀⠀⣇⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⣀⡀⠀⢀⣀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⡠⠆⡆⠀⢸⣈⣉⣉⣉⣁⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⢉⡽⠉⠉⠉⠉⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡿⢤⠀⡀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⣄⠀⠀⠈⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⠓⠦⠤⠤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄



⡏⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏

⠀⠀⠀⠀⠀⠀⠀⢸⠥⢸⠁⢙⠀⠃⢹⠁⡜⡄⢸⠭⡃⡇⠀⡯⠄⠀⠸⢄⠀⡏⠈⡇⡕⡜⠁⢱⢸⢔⠁⡯⠄
⣤⣤⣤⣤⣤⣤⣤⣴⣶⣼⣶⣮⣤⣤⣼⣼⣤⣼⣼⣶⣥⣷⣦⣷⣶⣤⣴⣾⣤⣧⣤⣧⣴⣼⣶⣮⣼⣬⣦⣷⣦⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿