Goofy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠀⠀⠀⠀⣠⠆⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⠛⠻⢷⣶⣾⣋⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⢿⣿⣿⣷⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣿⣿⣷⣶⣿⣟⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣾⣿⣿⡛⠉⢻⣿⡟⠿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣃⠤⠄⠻⡧⠄⢼⡟⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣀⡀⠀⢠⣤⠀⡇⠀⠀⠀⠀⠀⢀⣀⡤⠿⣿⣿⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣄⠸⣿⡇⣧⣠⠶⣢⡴⠋⣹⠏⠀⠀⠀⠉⠉⠁⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⣀⠉⠛⠿⠆⠛⡷⠟⣥⠞⠋⢠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣈⣙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡤⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠹⣿⣷⡌⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⢻⣿⣆⠀⠀⠀⠉⠛⠒⠒⠦⠶⠶⠶⣶⠖⠛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠉⠛⣷⠖⠛⠛⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⣷⣦⣀⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢠⡾⠻⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆⠀⠸⣤⡀⢹⣿⣿⣿⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡗⠲⣶⣞⠻⣦⡿⢿⣿⣷⠀⢸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣼⣶⣾⣿⣿⣿⣿⠀⠦⠀⢻⣿⣿⣆⠀⠛⠛⠛⠛⣿⢻⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⡟⠉⠉⠀⢿⣿⣿⣿⠇⠀⠀⠀⠛⠛⠿⣷⣦⣤⣤⣞⡋⢾⡝⢿⣿⣄
⠀⠀⠀⢀⣠⡴⠾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠈⠀⠈⠁⠀⠻⢿⣷⡀
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣇⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠘⢿⣷⡄
⠀⠀⠀⠀⠀⢀⢸⣿⣧⣾⣿⠿⠿⠟⠛⠛⠛⢻⣿⣿⣷⣦⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡆⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡷⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠘⠻⣷⡄
⠀⠀⠀⠀⠀⡈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣴⣤⣄⣀⢀⣴⣾⣿⡿⣿⣿⣶⡀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠛⢻⣯⣿⣿⡇⠀⠀⠀⠀⠀⢹⡆
⢀⢀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⣿⣿⣿⣿⣾⣍⣘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⢆⠀⠀⢀⠐⡇
⠀⢆⣡⣄⣀⣀⣼⠷⠆⠀⠀⠀⠀⠀⢀⣾⣿⣤⢿⣿⣛⠀⠀⠉⠛⠻⠿⣾⣿⣿⣿⣿⣿⡟⠻⣿⡯⣿⣝⠛⠛⠋⠉⢀⡰⠋⠐⠃
⢀⠈⠛⠛⠉⠉⠀⠀⠀⠀⠀⣀⣀⣤⣾⣏⠀⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡟⢿⣦⢸⣿⡏⠀⠀⠃⢀⣤⠿⠋
⠘⢷⣄⣀⣀⠀⢀⣠⣴⡇⣻⣟⣉⣯⡉⠛⠷⣦⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⠀⠙⠃⠈⣿⣀⣴⠿⠋⠁
⠀⠀⠀⣉⣉⠛⠋⣡⡟⢻⣿⣿⣿⠿⣿⡿⣦⠈⠙⠿⣦⣀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣎⠀⣿⣶⣿⠋⠁
⠐⠳⢶⣌⠿⠻⠿⠿⠀⠸⣿⣿⣿⣷⣿⢿⢷⣆⠀⠀⠈⠙⠷⣶⣤⣶⣿⣿⣿⣿⡿⣿⢿⣧⡴⠾⠋⠁
⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣉⠃⠀⠉⠀⠀⠀⠀⠀⢤⣤⣤⣾⣿⠋⢻⡤⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠉⣤⣿⠁⠀⢷⡟⠋⠛⠶⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣤⠿⠿⠀⠀⠈⢳⠉⠉⠉⠙⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⣠⣊⠁⠀⠀⠀⠀⠉⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠑⠀⠀⡤⠔⢒⠟⠛⢿⣿⣦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠁⢸⠃⠐⡡⠂⠀⠀⠙⣿⠷⠐⠢⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢠⡀⠀⠈⠀⢢⡀⠀⠀⣸⡰⣿⣁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠛⡄⡠⢶⠀⠙⢲⡾⠛⠻⡟⠉⢳⡀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⡄⠀⡀⠀⢀⠝⠀⠀⣾⠁⠀⠀⠀⠀⠀⣯⡀
⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠔⠑⠎⠀⢠⠊⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⢠⣦⣀⠀⣀⣶⠉⠀⠀⠀⠈⠀⠁⠀⢀⡾⣿⣧⠀⠀⣿⠀⡇⠀⢸⡿⠠⣄
⠀⠀⠔⠋⠀⢀⣾⣿⣿⣿⡃⠀⡀⠀⠀⠀⠀⠀⠀⣴⠟⡄⠉⠛⠀⢀⠸⠄⢃⠀⠈⠀⢠⣼
⠀⢨⠀⠈⠛⠛⠛⠛⠛⠉⠀⣠⠇⠀⠀⠀⠀⣠⣾⠏⠀⣤⠀⠂⠈⢠⣾⣿⣿⣿⡇⠀⠀⠀⠈⠓⠄
⠀⢈⠷⠀⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⠀⢀⣼⣿⠇⠀⠊⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⢀⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢠⣿⣿⡟⠀⠀⢄⠀⠀⠀⠀⢀⣤⣶⣿⣿⡏⠀⠂⣶⡆⠁
⠀⠉⠀⠀⠠⠀⠀⠐⠚⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⠀⠀⠁⢰⣄⣠⡔⠉⠻⠛⠻⠏⠀⣿⣿⣷
⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠠⠊⠀⢠⣇⠀⠀⠀⠀⢊⠀⢠⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⣀⡀⠀⠙⠋⠃⠀⢀⠀⠀⠀⢸⣿⣿⡿
⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠈⠉⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠹⡀⠀⠀⠀⣀⣴⣄⡀⠀⠀⠀⠈⠛⠋
⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠉⢉⣽⣿⠟⠁⠀⠁⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠀⠀⠀⠀⠀⣀⣠⣤⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠉⢀⠀⢈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠦⠘⠓⢾⡿⠁⠀⠀⢀⠃⠀⡨⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢄⠀⢀⣸⣀⣬⣴⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠘⣷⣿⠿⣿⣿⠟⠛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠈⠏⠀⠀⠀⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠐⠀⠀⠀⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⡇⠀⠀⢀⣄⢠⣦⠀⠀⣿⣿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡽⢁⣨⣿⣷⠀⠀⢸⣿⣾⣿⠀⠀⣿⠟⠣⡌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡼⢁⡞⠀⢹⡿⣆⠀⠈⣿⡯⠿⠤⠴⠋⢰⢰⣇⣀⡙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣼⢁⡞⣀⠤⠞⠳⠈⠓⡩⠥⣒⣂⣁⡀⠀⠉⠁⠀⠀⠈⡏⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢇⡞⢰⠁⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⡴⠃⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡅⠈⠧⣄⣀⠀⠀⠀⠉⠉⢉⣉⣁⣀⣠⡤⠴⠒⡏⠑⣄⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⡢⠀⠀⢹⡉⠛⣛⣹⠟⠿⠿⠻⢿⣷⡶⡶⠚⠁⠀⠘⡆⠀⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠛⠶⣄⠀⠈⠙⡍⢣⠀⠀⠀⠀⢀⡼⣸⣷⡀⠀⠀⠀⢹⡀⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠿⣻⠁⠀⠀⠈⣿⣦⡀⠘⢦⡉⠒⠒⠒⠋⣴⠛⣿⣇⠀⠀⠀⠀⢧⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣰⠃⠀⠀⠀⠀⣿⣿⡍⠳⣄⣙⠓⢦⡤⠒⠁⠀⢻⣿⠀⠀⠀⠀⠸⡆⠀⠀⠈⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡤⡿⣿⣿⣦⣀⣉⣩⠟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣼⠀⡇⢹⣿⣿⡿⢫⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣠⣠⠴⠊⣿⠀⠀⠀⠙⠿⠀⢸⠃⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⡀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⠤⣚⣩⠤⠤⠤⠼⢦⣤⣄⣀⡀⣠⠞⠀⠀⠀⠀⠀⠀⣿⣿⣨⣿⠛⠛⠂⠉⠉⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⠴⢋⠭⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠦⣖⠤⣴⡛⠉⠛⠓⠤⡉⠙⠳⣄⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠞⣡⠞⢁⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣿⣿⣦⡀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡞⢁⡴⠋⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢀⡿⠿⠿⢿⠀⠀⠀⠀⢸⣇⠀⣀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠼⠤⠤⠤⠖⠒⠉⠉⠈⠉⠁⠒⠒⠶⠶⠶⠚⠛⠓⠋⠑⠦⣤⣤⠤⠮⠶⠒⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠂⠘⠀⠛⠸⠄⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠂⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠠
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⣤⠐⡀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⣾⠁⠀⠀⣴⣿⣿⡟⠛⠳⡀⠀⠀⠀⢸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⡘⠀⠀⢸⡇⠀⠀⢸⣿⡃⠀⠑⠀⠀⢡⠀⠀⠐⠹⣿⡿⠁⠑
⠀⠀⠀⠀⠀⠀⢄⡀⠌⠀⠐⠤⠓⣾⣿⣧⣀⠀⠀⣿⡄⠀⡀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⡘⢋⠋⠁⠀⠀⠀⢀⠖⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢣⡐⠪⢋⠀⠀⠀⠀⠀⡀⠴⠋⠈⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⢸⡷⣄⠀⠀⠀⠁⠃⠄⠊⠠⠁⠀⣆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⢀⣿⡇⠈⢿⣦⡀⠀⠀⠀⠀⢀⠤⠒⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⣸⣿⡃⣀⣘⢿⣿⣧⠠⣀⡀⠃⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⢻⠟⠁⠹⣧⠢⠉⠛⠂⢠⡈⠄⡁⠀⢀⡀⠀⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⡠⠊⠀⠀⠀⢻⣇⠡⠀⣀⠰⠻⣦⠒⠀⠲⡀⠀⢀⢝⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⡠⠊⠀⠀⠀⣠⠀⢸⣿⢀⠠⠄⠐⠀⠙⠓⠂⠄⡘⢭⣥⣾⠇⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠤⠤⠴⠠⢚⣠⢚⠋⠁⠀⠀⠀⠀⠂⠄⡀⠢⡈⠂⢝⠋⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⢠⣿⣷⡘⠄⣀⠀⠀⠀⣤⡀⠀⠈⠂⠌⢸⠄⢑⣂⣁⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠤⡀⣄⠈⡩⠁⠈⠉⠀⠀⠀⠀⠀⠀⠁⠐⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠈⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠛⠳⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣶⣦⣄⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢛⡇⠀⠀⠀⠉⠻⢿⣦⡀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⢸⡇⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠻⣄⣀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢠⣼⣇⡄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⣴⣿⣥⣴⣶⡾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⣘⣿⣿⣴⣃⣀⠀⠀⠀⠀⠀⠀⠘⣿⣏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠛⠛⠿⠟⠛⠛⢿⣶⣄⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠃⠀⣴⡿⠛⠳⣄⡴⠛⠻⣷⡄⠀⢻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡏⠀⣼⡟⠀⠀⠀⠘⠁⠀⠀⠘⣿⡄⠈⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢻⣿⠁⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢹⣿⢻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠁⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠹⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠀⠀⢸⣿⠀⢸⣿⠀⢰⣿⣷⠀⢰⣿⣷⠀⢸⣿⠀⢸⣿⠀⠀⠘⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⠀⢸⣿⠀⢸⣿⣿⡇⣿⣿⣿⠀⢸⣿⠀⢸⣿⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣯⣴⡶⠶⣶⣤⣿⡇⠀⢿⣆⣈⣿⣿⣧⣿⣿⣏⣀⣾⠇⠀⣿⣇⣴⡶⠶⣶⣬⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠁⠀⢀⣀⠈⠻⣧⣴⠾⠛⢋⣩⣭⣤⣭⣭⣉⠛⠻⢶⣤⡿⠋⠁⣀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣼⣿⣿⣿⣿⣿⢻⣿⠀⠾⢻⡉⠀⠐⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠓⠀⠀⣹⠻⠆⢸⣿⢻⣿⣿⣿⣿⣿⡄
⠀⣰⣿⣿⣿⣿⣿⡟⠀⠙⣦⣴⠾⠋⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠳⢶⣤⡞⠁⠘⣿⣿⣿⣿⣿⣿⡀
⢠⣿⣿⣿⣿⣿⣿⡇⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠻⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⣿⣿⣿⣿⣿⣿⣇
⣸⣿⣿⣿⣿⣿⣿⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠻⣷⣤⣀⠀⠀⠀⢀⣀⣤⣶⡿⢿⣿⠁⠀⠀⢹⣿⠿⣷⣦⣄⣀⠀⠀⠀⢀⣀⣴⡿⠃⠈⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⡇⠀⠀⠈⠙⠻⣿⣿⠿⠟⠛⠉⠁⣀⣼⣿⣤⣤⣤⣼⣿⣆⡀⠉⠙⠛⠿⢿⣿⡿⠛⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣶⣿⡟⠋⠀⠀⠈⠙⠄⠈⠛⣿⣷⣶⣶⣶⡿⠟⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠃
⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠈⢿⣄⠀⠀⠀⠀⠀⢀⣼⠏⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠙⢷⣦⣤⣤⣶⠟⠁⢠⣾⡋⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠈⠉⢀⣠⣿⠇⠀⢈⣀⡁⢈⣀⣃⣈⣃⣄⣘⣀⣀⣀⣀⣘⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣠⣶⡀⠀⣸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈
⠀⠀⠀⠀⢠⣾⣿⡿⠛⠁⠀⢸⣈⠃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⣀⣄
⠀⠠⠀⣰⣿⡿⠋⠀⠀⠀⠀⠈⢿⣾⣶⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠈⠀⠒⢺⣿⣿⠿⠂
⠀⠀⠀⠙⠟⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⠏⠀⠉⠣⠀⠀⠀⠀⠀⠀⢰⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠁⠈⠙⢆⠀⣀⠀⠐⠀⠀⠀⠀⠀⠺⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠈⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣷⡄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⣤⣤⣄⠀⠘⣿⠋⠘⠁⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣿⣿⣿⣧⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠠⠄⠀⢂⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⡏⠀⣀⣀⠀⠀⠀⠀⠀⣀⡠⡄⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⢀⣀⠉⠻⣿⠿⠿⠛⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⠀⠀⠐⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣼⣿⡟⠘⣧⠀⠀⠀⠀⠀⠀⢰⢁
⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⠀⠀⠸⢿⠟⠀⠀⠸⣷⣤⣀⣀⠐⠂⠀⠁
⠀⠀⠀⢀⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠀⠀⢾⣦⣄
⠀⠀⢠⠾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡅⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣦⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⢿⣧⣄⣀⡀⣀⣀⡤⠖⠁⠈⠛⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⣠⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⠛⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡞⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠀⠀⠀⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⢈⠙⠲⢤⡤⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣈⠳⣄⢢⡎⠉⣷⣶⣶⣶⣶⣿⣿⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢼⣆⣿⠦⠈⠛⠿⠿⣿⠿⠧⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⣿⣧⣤⣤⠀⠀⠀⠀⠀⣴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢿⣿⣿⣿⠃⠘⣆⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⣠⠿⠛⠉⣹⣿⣀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⠀⠈⠿⠿⠋⠀⠀⢹⣄⡀⠀⠀⠀⣠⠋⢀⣠⠚⠁⠀⠀⢠⣿⣧⣿⣦⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⣤⣄⣠⠃⣴⡟⠁⠀⢀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⡀⠉⢾⡏⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠱⢤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠳⣠⡿⠟⠛⠛⠛⠛⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡰⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⡖⠒⠂⠀⡸⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡟⠀⣠⣶⡏⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⢹⠖⠒⠲⢶⣶⣾⣿⣿⣿⢿⡟⠀⠀⣿⣿⡇⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠋⠀⠀⠌⣼⠟⠉⡩⠋⠀⡼⠁⠀⢸⣿⣿⡗⠲⣄⠸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⠃⠀⠀⠀⠀⣰⠃⠀⠀⢸⣿⣿⣧⠀⠘⡇⠹⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⠀⠀⠀⠀⡰⠃⠀⠀⠀⣸⣿⣿⠏⠀⢠⡇⠀⠙⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠧⣤⡤⠞⣁⠖⠀⣀⡴⠋⠉⠁⠀⠀⠈⢹⠀⠀⠈⠻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⠤⠞⠉⠊⠉⠳⣄⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⠛⠿⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠤⠒⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡄⢀⠀⢀⣀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣿⠟⢣⡀⠀⠀⠀⠀⠴⠚⢛⣿⣿⣿⡎⠀⠀⢻⣆⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠇⠀⠀⡴⠁⢀⠤⠋⠁⢸⣿⠃⠀⠀⠈⣿⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠹⢆⡈⢠⡴⠁⠀⠀⣠⣿⣿⣇⣀⡀⠀⡿⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠐⡄⠀⠀⠀⠀⠈⠙⢿⠁⢀⣴⣶⣿⣿⣿⠇⠀⠑⠒⠀⠀⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⢆⠀⠀⠀⠀⠀⠀⠀⢛⣛⠛⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠉⠒⠂⠀⠀⠂⣀⡈⣤⡔⠉⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⣠⣴⡿⠋⠀⠘⢿⣄⠀⠀⠹⢿⣶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣠⠟⢳⣶⣿⣿⠛⢤⠀⠀⠀⠈⠻⣿⣦⣄⠀⠙⠻⢿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠈⠑⢦⡔⠋⠁⢀⣼⠏⠀⠙⠋⣭⠀⠀⠉⠒⠢⣄⡀⠀⠈⠻⣿⣿⣶⣤⡀⠈⠙⠻⠿⣿⣿⣿⣿⣦
⠀⠀⢀⠠⠤⠤⠤⠤⠤⠄⡠⠎⠀⠀⠀⠀⠀⠄⠀⢱⠀⢀⡾⠁⠈⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⡀⠉⠙⠛⠛⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠔⢁⣀⣀⣀⣀⡀⢀⠜⠁⠀⠀⠀⠠⡀⠀⠀⢡⠀⣃⣾⠃⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠲⠶⠦⢠⣄
⢀⠊⠁⢀⣤⣤⡿⠙⠉⠀⠀⠀⠀⠀⠀⠈⠆⢀⡼⠊⢹⡇⠀⠀⠀⠀⠀⢸⣇⠀⢠⣤⣤⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠈⢦⡀⠿⣿⣿⡇⠂⠀⠀⠀⠔⠃⠀⠀⣠⠾⡉⣀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠻⣦⠸⣿⣿⣆⠀⠈⠉⠘⠓⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠙⠢⠤⠉⠙⠒⠓⠒⠒⠈⠉⠉⠉⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠈⠁⠀⠲⠆⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠉⠻⣿⡇⠀⠀⠀⠀⣀⠤⠀⢩⠒⣤⣤⣴⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠛⠢⡀⠙⠃⠀⠀⠀⡞⠁⠀⠀⠀⢄⠙⠛⠛⢁⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠘⣃⢀⡀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⢈⠳⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠈⠉⠉⠀⠀⠀⠀⢀⠴⠊⠁⠀⠘⢇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⣀⣀⡀⠤⠚⠁⠀⠀⠀⠀⠀⠈⢣⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠈⠲⠄⠰⠤⣀⠠⠄⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠐⡀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡇⠀⢠⠁⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡋⠈⠣⡄⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠀⠀⠸⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠳⣄⠘⢦⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠀⠰⠃⡰⠉⠀⠒⠢⣄⠀⠀⠀⢠⣞⠁⠀⠀⠀⠀⠀⠈⠳⣄⠱⡄⠀⠀⠒⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⢠⣾⣿⣶⣦⣄⠀⠀⠑⣄⠔⠁⠈⢣⡀⠀⠀⠀⠉⠁⠀⠈⠀⡱⣤⣄⣤⣤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⢠⠀⡁⠘⣿⣿⠋⠀⠈⢢⡄⠁⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⢦⡘⠻⠿⠃⢈⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠤⢀⠈⡄⠱⡄⢹⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣄⡀⠀⠀⠀⠉⠀⠂⠈⠁⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⢟⡄⠸⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠂⠶⢦⣤⡤⠤⠤⣄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⣱⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠙⠋⠙⠓⢦⣄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠄⣀⠀⣀⠀⡀⠀⣀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠌⡠⠤⡈⢋⠀⠀⠈⠃⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡌⠀⠀⠈⠆⠀⠀⠑⡠⠀⠀⡄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢡⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢐⠈⣇⠀⠀⠸⠀⠆⠃⠀⡄⡼⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠔⢁⡀⠸⠀⢀⠀⠄⠤⠀⠰⠬⠸⣅⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠔⠉⢀⡀⠔⠐⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠎⢠⠂⠀⠀⠀⠀⠀⠐⠢⠤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⡎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⣀⡀⠀⠀⠀⢀⡠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢸⠀⠀⠈⠢⠀⠀⠀⠀⠤⡖⠉⠀⠀⠀⠀⡦⢨⡝⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣑⢴⠘⡄⠈⠂⠀⢢⠔⢀⢾⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⡇⠀⠀⠀⣠⠖⠉⢀⠜⡄⠘⠄⠄⠀⠁⡠⠃⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡠⠃⠀⠀⠀⢹⡀⠀⠈⠂⠙⠦⣀⢀⠠⠊⠀⢰⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡜⠁⠘⠀⠀⠀⣀⢀⡤⠊⠀⠀⠀⢸⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠈⠦⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠒⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣀⠀⣀⣠⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣃⣘⣻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⢈⠉⠉⠉⣩⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣼⣾⣿⣿⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠉⠻⡿⠟⠁⠀⠀⠀⠀⠉⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⣴⣿⣿⣷⡀⣠⣤⡀⢠⣾⣿⣿⣆⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⠀⠀⠀⢿⣿⣿⣿⠣⣿⣿⣿⠸⣿⣿⣿⠟⠀⠀⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠙⢿⠀⠀⠀⠀⠉⠉⠀⠀⠙⠿⠋⠀⠈⠉⠁⠀⠀⠀⠠⠟⠉⢹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⠀⠀⢀⣠⣴⣶⣷⣶⣤⣤⣤⣤⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣴⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠟⠁⠀⠀⣾⣿⣿⠏⠉⢻⣿⣿⣿⣿⣿⠾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢵⣶⣿⣿⣿⣿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠓⠀⠀⠀⠀⠀⢸⣿⣿⣯⣾⣿⣯⣿⣿⣿⠀⠀⠀⠀⢀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠈⣽⣧⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠛⠟⠻⣿⣿⣿⣆⠀⠀⠀⠀⢿⢿⡵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⠘⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠙⠻⢿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠁⠈⣠⠠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠁⠀⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣷⣶⣧⡶⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⡿⢛⠋⠉⢙⣿⣿⡿⠁⠀⠀⠀⠀⠹⣿⠇⠀⢸⣿⣿⡆⠈⢿⠇
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠁⠛⠀⠀⠘⢹⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⡆⠙⢿⣿⣧⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠂⠁⠀⠀⠀⠀⠉⣹⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠙⠃⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⢰⣿⡇⠀⠀⠀⠀⠀⢰⡿⠟⠀⠀⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠈⠘⠁⠀⠘⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠄⠀⢀

⠀⠀⠀⠀⠀⢠⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠀⠉⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣿⠀⠀⠀⠀⠀⢤⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠁⠀⠀⠀⠀⣠⠾⢛⣷⠶⠻⢶⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢠⡞⢡⡾⠋⠀⠀⠀⢬⡟⣠⢞⣥⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣴⠷⢯⣴⠏⠀⠀⠀⠀⠀⠀⢹⣿⣯⠵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡏⠀⠀⠀⣠⠖⠳⣦⣿⣿⠷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⡀⣰⠏⣠⣤⡌⠿⢁⣤⣄⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡹⠏⢰⠃⠀⠘⣴⠋⠀⠘⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡎⠀⠀⠀⡏⠀⠀⠀⣿⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⣿⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠧⣸⡇⠀⠀⠀⠀⠀⠀⢀⡟⣀⣴⣮⡻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡒⠙⢧⠤⠤⠶⠤⠤⣤⣼⠟⣩⠌⣿⠙⣦⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠁⠀⠐⣦⡴⢿⣿⣥⠲⠄⠀⠘⠶⢿⣤⣈⠻⣄⠳⣄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢾⣁⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠙⣧⠙⣧⠘⣧⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠈⢷⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠒⠶⡶⠶⠚⠋⠙⠶⢤⣤⣤⣤⣤⣤⡶⠊⠁⠀⠈⣇⠀⠘⣧⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⠟⡃⢀⡀⠤⣀⡤⠤⢤⣳⡤⣴⠾⠃⠀⠀⠀⢠⡏⠀⠀⢻⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⣾⢳⡋⠀⠀⠉⠀⠀⠈⣿⠁⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⢣⡙⢦⠀⠀⣀⣠⢾⠁⡼⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⢀⡟⢢⡬⠭⠭⢴⣿⠛⣧⠀⠀⠀⠀⠀⠀⠘⢧⣀⡼⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⣾⠀⢸⡇⠀⠀⠀⣿⡀⢸⣀⣠⣄⣀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠿⠋⢻⡄⠹⣧⣀⣠⣤⠟⣡⡏⢹⠁⠉⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⣰⠏⡻⣦⡀⠈⠉⠀⠰⢛⡇⣸⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢠⡏⢀⢷⡈⠁⠀⠀⠀⠀⣸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⣿⠀⣾⠀⠙⠶⣤⣤⡴⠞⢻⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢸⡇⢰⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠴⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠸⢷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣶⣶⣾⣷⣦⣾⣿⣷⣶⣶⣶⣶⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠂⢄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⣼⡷⡟⢡⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣠⢆⣴⠋⠀⣷⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠊⠋⠛⣁⣴⣶⣿⡿⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⣿⠉⠹⠇⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣄⣿⣆⠀⣶⣾⡇⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢊⠔⢉⠆⠈⢟⠢⠽⠛⠛⠐⠁⣼⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡰⠃⡴⠋⢀⣠⠷⢀⣴⣶⣶⣦⡀⠀⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡼⠁⠀⡏⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⢀⢾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⢀⠀⠑⠢⣤⠀⢲⣾⡷⡶⠖⢶⣈⢹⣤⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠈⠰⣄⠀⠀⠨⡑⡀⠀⡠⠂⠀⣾⡿⠀⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⢠⠸⣷⠦⣀⠈⠂⢭⣄⡀⢼⢹⡇⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⠂⡈⢹⣶⣌⣉⣁⣹⠀⣀⠆⠘⣿⠀⠀⠘⣿⠟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⠁⠀⣿⣿⡿⠋⠉⠉⠉⠁⠀⠀⣿⡀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠤⠐⠉⠀⠀⠀⠀⠏⠀⣤⢀⣀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣲⣿⣆⠈⣧⠀⠀⠀⠀⠈⣿⡄⡀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠤⠋⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢹⣷⣜⡄⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⠔⠒⣤⠤⠐⠤⡄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢄⡀⠴⠃⠀⠀⣆⠀⠀⢠⡦⣄
⠀⠀⡴⠦⣀⣀⣀⣀⡠⣴⠊⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣆⠀⠀⠀⠀⢀⡞⠁⣀⡀⠴⠛⡇⠀⢀⠞⠀⠀⢹
⠀⢸⠁⠀⠀⠀⠁⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⣿⣿⣿⡿⠋⣤⠔⠒⠀⠁⠐⣿⣿⣿⣿⣆⠀⠀⢠⠊⢀⡴⠁⠀⠀⠀⢃⢀⡌⠀⣀⡴⠚⠠⢄
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣿⠿⠋⠀⢰⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢣⠀⠀⣠⠏⠀⠀⠀⠀⠀⠸⣁⣤⠊⣁⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠳⣄⡀⠀⠀⠀⣩⠔⠂⠀⠐⠒⠢⢼⣿⣿⡿⠁⠀⢑⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⡆⠀⣼
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⠦⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⢋⡻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠈⠐⠋
⠀⠀⠀⢀⠋⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠁
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠉⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡳⠃
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⡠⠚⠀⠉⠳⣄⠀⠀⠀⠀⠀⢀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠈⢦⣄⡎⠀⠀⠀⣠⣆⡙⠇⠀⠀⢀⠔⠊⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠲⢤⣾⣿⣿⣿⣾⣍⢳⣾⣷⣦⠀⠀⠀⢳⡀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⢙⠋⠀⣼⣿⣿⣿⡿⠳⡀⠀⠈⢅⠀⠀⣠⣞⣁⣠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣉⠛⠛⣫⠀⠀⢣⠀⠀⢸⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⢡⠈⢦⡈⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⢤⣤⣀⣀⣀⣀⣀⡠⣾⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠈⠀⠶⠄⣤⣤⣄⣠⣤⣤⡤⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣀⣤⣤⣴⡶⠶⣶⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣠⣴⠿⠋⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⣀⣴⡿⠃⠀⠀⠀⢀⣤⠶⠶⠚⠉⠁
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠻⠶⣶⣴⣞⠋
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⢷⣤⣠⣤⠔⠒⠛⠛⠛⠛⠓⠂⠤⠄
⠘⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣁⣀⣠⡶⠛⠻⣿⡀
⠀⠈⠻⣦⣄⡀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⡏⢉⣽⠟⢻⣦⣤⠶⣿⡷⠶⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠉⠛⠻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣀⠀⡟⢀⡾⢁⣴⣿⣿⣿⣶⣿⡁⠀⠀⠀⠀⠀⠀⠈⠑⢦⣄
⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣤⡟⢠⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡿⠟⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠋⠁⠀⠀⠉⠻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠙⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠹⣧⣤⣀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣉⣿⣿⠀⠀⠀⠀⠀⢰⡏⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠿⠿⣿⡆⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿⡿⠀⠀⠀⠀⢠⣿⡿⠟⠛⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⣰⡟⠁⣀⣀⠀⠈⢻⣆⠀⠀⠀⠀⢻⣿⣿⣿⣀⣀⣀⣀⣀⣸⣿⣿⣿⠃⠀⠀⠀⣠⡿⠉⠀⢴⣶⣦⡀⢸⣿
⠀⠀⠀⠀⠀⢰⣿⢀⡾⢿⣍⠁⠀⠀⠻⣄⣀⣤⣤⣤⡿⠛⢋⣉⣭⣭⣭⣭⣉⣛⠛⠿⣶⣟⠛⠛⠏⠀⠀⢠⠞⠁⠘⠇⣸⡟
⠀⠀⠀⠀⠀⠈⢿⡀⠑⠀⢈⣣⠀⠀⠀⠛⢉⣴⠟⢁⣴⣾⡿⠛⠉⣻⣿⣿⣿⣿⣷⣦⡀⠉⠻⣆⠀⠀⠀⠹⣤⣀⣀⣠⣿⣀
⠀⠀⠀⠀⢀⣠⣼⣷⡶⠶⠟⠁⠀⠀⠀⠀⠈⠀⢠⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉⠉⠉⠛⠻⣶⣄
⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠈⢻⣦
⠀⠀⣼⡏⠀⠀⠘⠛⠛⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠈⠁⠀⠀⠀⠀⣿⣤
⢀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⠄⠀⢠⣿⠁
⠛⠉⠻⣧⠀⠀⠀⠐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣤
⠀⠀⠾⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡅
⠀⠀⠀⠀⠈⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⠀⢀⣀⣀⣀⣤⣴⣾⡟⠁
⠀⠀⠀⠀⠀⣿⠀⠉⠛⠻⠿⠶⠶⠶⠶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠋⠁⢸⡇
⠀⠀⠀⠀⠀⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠉⠙⣿⣶⣶⣤⣤⣤⣾⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⣷⣦⣤⣤⣤⣶⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠹⣿⠁⠀⠀⠀⠉⠻⣏⠁⠀⠀⢀⣿⠋⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠹⣧⡀⠀⠙⢷⣤⣀⣀⣀⣤⡾⠋⠀⢀⣼⠏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢿⣦⡀⠀⠈⠉⠉⠉⠁⠀⢀⣤⠟⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠙⠻⣶⣤⣄⣀⣤⣴⡾⠛⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠈⡇⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠈⣿⢹⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠟⠛⠛⠛⠛⠛⠿⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⡀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⠛⠋⠉⠉⠛⠛⠻⢶⣦⣀
⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣤⡀⠀⠀⣿⡆⢀⠄⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⢤⣿⣿⠀⠀⢸⡷⣿⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢼⠿⠋⠀⠀⢸⡟⠈⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣄⣀⣀⣀⣀⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⣤⣴⣦⣤⣀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠻⠿⣷⣤⣀⡀⢀⣀⣤⡾⠟⠋⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⠉⠉⠛⢿⣷⣦⣄⣠⣤⣾⡿⠃⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠈⠛⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣽⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣾⣿⣷⣾⣶⣷⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⢤⡀⠀⠀⠀⡸⠀⠀⠀⠀⠈⣆⠀⠀⠀⢀⠤⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠙⢦⡀⠀⡇⠀⠀⠀⠀⠀⢸⡀⢀⠔⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠱⣼⠁⠀⠀⠀⠀⠀⢸⣧⠋⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠑⠲⢌⣆⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣼⠧⠒⠋⠉⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⣈⣧⠤⠤⠤⠼⣄⣀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠔⠋⣁⠤⢦⠀⢠⠦⠤⣈⠑⢶⡋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠀⠀⢸⠀⢸⠀⠀⠈⠑⢄⠙⢦⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⢀⡤⠖⠒⠉⠉⠉⠉⠉⠑⠲⠤⠤⡼⠀⡞⠸⡀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠈⢆⠈⣷⠒⠊⠉⠉⠉⠉⠉⠑⠒⠢⢄⡀
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⠀⢳⡀⠀⡸⠀⠸⣄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠸⡀⠀⠀⡹⠊⢀⠀⢀⠈⠳⣄⠀⠀⢸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠈⠉⠒⠲⠶⢤⣤⠴⠖⠒⠊⠉⢳⠀⠳⡤⠊⢀⡴⢻⠀⢸⠷⢄⠈⠳⣶⠏⢠⠏⠉⠑⠒⠦⣤⠤⠶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢰⡋⠀⢸⠀⢸⠀⠀⣱⠄⠀⣠⠋⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠲⢄⡈⠙⠚⠀⠈⠒⠉⣁⡤⠊⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣀⣤⠚⠁⠀⠀⠀⠀⢨⠋⠙⠒⠒⢻⠉⠁⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠑⠤⢄⣠⣤⠤⠶⠛⢻⠃⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠹⡌⠑⠲⠤⠤⣀⡠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢀⣾⠻⡄⠀⠀⠀⠀⠀⢸⡏⢢⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢀⡴⠟⠁⠀⣇⠀⠀⠀⠀⠀⣼⠁⠀⠙⢦⡀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠉⠀⠀⠀⠀⠸⡄⠀⠀⠀⢰⠏⠀⠀⠀⠀⠈⠑⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣿⠀⡰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡆⠀⡠⠞⡟⣡⠋⠈⠷⡄⠈⢧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡴⠁⠈⣧⠊⠀⠀⢯⠁⠀⠀⠀⠀⠀⢸⠃⠘⣦⠶⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣠⠏⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢦⣼⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⠀⠀⢀⡠⠶⠒⠒⠒⠒⠦⣄⠀⠀⣇⠀⠀⠀⢀⡠⠔⠒⠢⣄⠀⠀⠀⠀⠀⠀⢸⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⡀⡆⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢽⡶⠤⠒⠁⠀⠀⠀⠀⠈⢦⡀⠀⠀⢰⠀⠈⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣀⡘⡄⠀⢣⡙⢤⡎⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⢠⡀⠀⠈⠙⢲⣼⡇⠀⠀⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠙⣏⠁⠀⡱⢭⣻⡇⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣀⠀⠳⠄⠀⠀⠀⣟⡵⠀⡴⠓⣹
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠍⣲⣶⡇⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⢿⣿⣶⣤⠀⠀⠀⡿⣊⠀⢀⣴⣃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢑⣒⡶⠋⠀⠀⠀⠀⣀⣤⣶⣿⣿⡇⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠙⢿⠇⠀⠀⡿⠁⠴⠋⢁⠎
⠀⠀⠀⠀⠀⢀⣀⣸⠄⠀⠀⠀⡸⠁⠠⠶⠾⠟⠛⠛⠛⠻⣟⠋⠁⠀⠀⠀⠀⣠⣤⣤⣤⡤⣤⡤⠤⢤⡤⠀⢰⡇⠀⡰⢠⠏
⠀⠀⠀⠀⠀⠀⠉⢢⣄⣆⠀⢀⡇⠀⠰⡗⠒⢶⣖⣲⡶⠒⢻⡟⠛⠀⠀⠀⠀⢸⡄⠀⠈⠛⠛⠁⢀⣾⠁⠀⣸⠁⡼⠓⠻⣄
⠀⠀⠀⠀⠀⠀⢰⠋⣁⢹⡄⠘⡇⠀⠀⠱⣄⣀⠉⠉⣁⣠⣾⠁⠀⠀⠀⠀⠀⠸⣯⡓⠦⣤⣤⠶⢻⠏⠀⢀⡏⢸⠃⡤⡄⢹
⠀⠀⠀⠀⠀⠀⣿⢰⠁⠀⣧⠀⢷⠀⠀⠀⠘⠛⠿⠛⢛⡽⠃⢀⣤⠤⠶⠶⣤⡀⠙⠍⠂⠀⠀⠀⠁⠀⠀⢸⠃⣾⠰⠛⡷⣸
⠀⠀⠀⠀⠀⠀⢸⣼⡩⠗⣾⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣟⠀⠀⠀⠀⠀⣹⡄⠀⠀⠀⠀⠀⣰⡄⠀⢸⠀⡿⡀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢳⡝⢦⠜⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠚⠋⠀⠀⠀⠀⣠⠞⠁⠀⠀⢿⠼⣳⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣻⠁⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠴⠖⢒⣋⣡⣴⣾⡟⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠳⣶⣿⡿⠛⠛⢋⣠⠞⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠈⠛⠶⠶⢒⣋⡥⢠⣄⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠿⣄⡀⠀⠀⠉⢉⣡⡾⠋⠉⠁
⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠛⠛⠛⠋⠁
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⠤⣒⣒⠂⠀⠠⠖⠚⣉⣉⣉⠉⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⣡⠖⠉⢁⣀⡀⠠⠖⠋⣉⣉⣀⣉⣉⠙⠒⢤⡈⠑⢦⡰⡄
⠀⠀⠀⠀⠀⠀⡔⡽⢡⠞⢀⡴⠊⠁⣀⣤⠤⠖⠋⠉⠀⠀⠀⠉⠙⠲⣄⠉⢦⡀⠳⡜⣄
⠀⠀⠀⠀⠀⡜⡸⢡⢏⡴⠋⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠳⡄⠹⡜⣆
⠀⠀⠀⠈⣿⣶⣷⣟⣭⢶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⡄⢳⡘⡄
⠀⠀⠀⠀⡏⡏⣿⣿⡄⡾⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣄⠀⠈⣧⡹⣄⢳⡹⡄
⠀⠀⠀⢸⡇⡇⡟⣿⣿⡇⠀⠾⠋⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠁⠀⢸⣟⠮⣀⣿⣷⣀⡀
⠀⠀⠀⢸⢣⡇⡇⣿⣿⡇⠀⠀⡴⡖⡛⢳⣆⠀⠘⡆⠀⠀⠞⡏⠈⢹⢳⠀⠀⠀⡿⣿⡄⡀⣏⠙⣏⠁
⠀⠀⠀⢸⢸⡇⡇⣿⡏⣧⠀⠀⠀⠉⠀⠈⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢯⠁⢇⢹⠀⢸⡄
⠀⠀⠀⢸⢸⡇⡇⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡀⠸⡈⣇⠀⣧
⠀⠀⠀⢸⢸⡇⡇⡿⣧⣏⠀⢀⡀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣿⠀⣇⠀⢧⠸⡄⢹⡀
⠀⠀⠀⣸⢸⡇⡇⡇⣿⣇⠳⣿⠙⣄⡀⠀⠀⠈⠉⠂⠉⠀⠀⠀⠀⣀⡞⢸⡷⢁⣿⣇⠸⡄⠈⢆⠹⡄⢷
⠀⠀⢀⡟⢸⢣⣇⡇⣿⡸⡄⢹⡄⢿⠹⡄⢀⡴⠒⠉⠉⠢⡀⠀⡾⢹⠀⣼⠀⡼⡸⠹⡄⢱⡀⠈⢧⠙⣌
⠀⠀⣼⠁⡼⣸⢸⠁⡏⢣⣧⠘⡇⢸⡄⢻⣎⣴⣭⣭⣭⣶⡹⡼⠁⣾⠀⡏⢰⣧⠃⠀⢳⠀⠳⡀⠀⠳⡌
⠀⣰⠃⣰⢣⡏⡟⠀⡇⢈⡿⡄⣧⠈⣇⠀⠻⣿⣿⣿⣿⣿⡿⠁⢀⡇⢰⠇⡼⡏⠀⠀⠈⢧⠀⠁
⡼⢁⡴⢁⡞⣸⠁⢰⠃⡸⢻⡹⡜⣆⢹⡀⠀⢹⡄⠀⢀⡼⠁⠀⣼⢁⡟⡼⣼⠁⠀⠀⠀⠈
⠰⠋⣠⠏⡴⠃⢀⠏⣰⠃⣸⣧⠀⠀⠀⠁⠀⠀⢻⡉⣹⠃⠀⠀⠁⠈⠀⢰⡇
⠀⠀⠀⠚⠁⣠⠋⡴⠁⡴⠃⣿⠀⠀⠀⠀⠀⠀⠘⣧⡏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⡡⠊⣠⠞⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠛⠛⠓⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣠⡴⠞⠋⠉⣿⠀⠀⠀⠀⣿⠉⠙⠳⢦⣄⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢻⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⡟⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠸⣇⣴⠞⠁⠀⠀⠀⠀⣠⡴⣿⠀⠀⠀⠀⣿⢦⣄⠀⠀⠀⠀⠈⠳⣦⣸⠇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⠞⠉⠀⣿⠀⠀⠀⠀⣿⠀⠉⠳⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣿⠀⠀⠀⠀⣿⣀⣠⡴⠞⠋⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠦⢤⣤⣤⡤⠴⠶⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢠⠖⠢⠀⣠⣦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠂⠤⠀⢀⣀⠀⠀⠀⠈⣦⠈⠂⣀⣂⣀⠨⡦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢉⣉⣬⠟⡏⠁⠀⠐⢈⠉⣹⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⡧⡄⠀⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠓⢀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⣣⡀⠘⡀⠀⠇⠀⠀⠀⠄⠀⣀⡴⠁⠀⠀⡄⠀⡨⡡⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠤⡘⢼⠀⠇⠒⠂⢀⠐⠀⠀⠀⠀⣦⠖⠐⠀⢰⢘⡔⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠚⠀⠠⠓⢂⠑⠀⠀⠀⠀⠀⠀⡆⠬⣑⠲⠉⠀⠀⣓⠈⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠡⠀⢀⢠⡆⠀⠀⠀⠀⠀⠀⢀⠇⠙⢄⡁⠀⠀⠈⡀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⠀⠀⠀⠀⣠⠌⠀⡍⠀⡏⠁⡖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⢐⠀⠉⡀⠀⠰⠀⡴⠁⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠷⠂⠀⠐⠡⠊⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⡠⠆⠀⡄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡀⠀⢠⠁⠠⠂⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠛⢏⣀⠟⠒⠊⢨⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠈⠂⠦⠀⠀⠀⠀⠀⠀⠀⢘⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢮⠀⠑⠄⠀⠀⠀⠉⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠈⡉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⠰⠊⢃⡤⠀⠀⠀⠀⠙⠒⠀⣀⡀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠠⠙⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣉⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣮⡳⣄⢀⣟⠢⢄⣀⣆⡃⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠑⢌⠪⣳⠎⠀⠀⢙⠈⠉⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⣀⣀⡀⣀⣑⣁⣀⣀⠀⢸⣤⣤⣄⣠⣠⣤⠬⢷⡶⠶⢶⠀⠐⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠀⠈⠑⠂⠀⠈⠉⠉⠁⠀⠀⠒⠊⠉⠩⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠢⠐⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠐⠰⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠛⠛⠳⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⣠⡴⠟⠋⠉⠀⠀⠀⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢶⣤⠀⠀⠀⠀⠀⣿⢀⡔⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠲⣾⣿⡇⠀⠀⠀⠀⣿⢸⡗⢺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⣴⣶⣶⣤⡀
⠀⠀⣠⣴⣶⡶⣶⣤⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠟⠋⠀⠀⠀⠀⢀⡿⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢠⡏⠙⠻⠄
⠀⠰⠛⠉⣿⡇⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⡀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠶⠶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠣⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡎⠞⠛⠛⠳⠆⠀⠀⠀⠘⠛⠋⠉⠙⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠀⠀⠸⡄⢀⠴⣶⡦⠀⠀⠀⠀⠀⠔⣶⡲⡄⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢣⠈⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⣠⠔⠚⠁⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠘⢄⣀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡆⠀⠀⣶⠆⣰⡶⡏⠀⠀⠀⠀⠑⠀⣀⡀⢿⠆⠀⢠⣶⡌⢱⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⢸⠀⢰⠉⠀⡏⠁⣿⢦⠀⠀⣠⡎⢰⠹⡏⠀⠀⠀⡌⠉⡇⡸⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⢸⠀⢰⡇⠀⡏⢸⠟⢻⣏⠀⡎⠀⡇⠀⠀⠀⠇⢠⠇⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⢸⠀⢸⠀⠠⠁⢸⣴⣮⢹⠀⡇⠀⠇⠀⠘⠸⠀⣸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⡀⠀⠀⠀⡇⠀⠈⠀⠀⠀⢸⠃⢸⠀⡇⠃⠀⠀⠀⢠⠇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠘⣆⠀⢠⠇⠀⠀⠀⠀⠀⠞⠀⡿⠀⠘⠄⠀⠀⠀⠈⠀⠀⡇⠀⢀⠔⠋⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⠢⠋⠀⠀⠀⠀⠀⠀⠀⢰⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⢠⠏⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣠⣤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⠿⢡⡝⢇⣀⠿⠟
⠀⠀⠀⠀⠀⡀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⡿⠓⣬⠛⢭⡄
⠀⠀⠈⣿⢘⣿⢀⣻⣈⣝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⡤⠟⣢⣤
⠀⠀⠀⠀⠀⠁⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠶⠾⠧⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣀⡀⠀⠀⢸⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠉⠛⠲⠾⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⣤⣤⡤⠀⣀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀⣰⣿⣿⣿⠃⠀⠀⠈⢿⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠀⠀⠀⢀⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⢛⣡⣴⣶⡟⠉⣍⠙⠿⠀⠀⠀⣺⡟⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠃⠀⡀⠀⠀⠀⠀⠸⠿⢁⣿⡇⠀⠀⠀⣀⣀⣠⡴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⣄⠸⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⣿⡷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⣿⠉⠛⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⣒⣘⣿⡐⠒⣤⣤⣤⣶⣿⣿⢻⠀⣿⠀⢠⣾⣿⠈⠐⢤⣀⠀⠀⠀⣀⠀⠀⠀⢡
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣤⣶⣿⣿⣿⣿⡇⢸⣿⣿⣿⠟⠋⠁⠀⠂⠹⣷⣾⠿⠃⠀⠀⠀⠌⠋⠉⠉⠀⠁⠀⠂⠁
⣿⣷⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⣤⡄⠀⢀⢸⣷⣿⣿⣿⣿⣿⣿⡿⠁⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠠⡀
⢿⣿⣧⡀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠘⢿⣿⢿⣿⣿⡿⠟⠁⠀⠀⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡄
⠀⠛⠿⣿⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁⣾⣿⣿⢀⠤⠤⡄⠀⣰⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠉⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢠⣿⣿⡿⠃⠀⢐⠠⠊⠙⠻⢿⣿⣿⣿⣶⣦⣤⣀⣀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⣸⣿⣿⡧⠴⠚⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣸⡀⠀⠀⢀⡐⠉⠉⠑⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣀⠁⠀⠀⠠⣄⣀⠜⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣦⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⢰⠋⠀⠀⣈⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⢠⠃⠀⠀⠂⠉⠀⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠀⠘⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣀⢀⠀⣠⡞⠁⢠⡾⠀⢠⣇⣴⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢋⠽⠒⠸⠋⠀⠐⠏⠀⣠⠋⠀⠀⠀⠈⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢲⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠢⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⠤⠤⢤⣬⣭⣥⣤⣤⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⡿⠿⠟

⠀⡀
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠆⠰
⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢲⢀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢲⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣰⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⠉⠈
⠀⠸⣿⠿⢿⡿⣷⣄⡀⠹⣤⢤⣀⡀⠀⠀⠀⠀⡝⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⢀⣄⣀⠀⠀⠀⡈⠁⢩⣿⠁⡀⠈⠣⠀⠐⠂⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂⠐⠂⠠⠀⠀⠀⠠⢀
⠘⠿⠿⠀⣀⣀⡀⠀⢸⣿⣿⡏⠀⠀⠀⢂⣀⣀⣤⣤⣤⣤⣖⣨⢣⣤⡀
⠀⣀⠀⢀⠀⣠⣿⣦⣾⡏⢿⡇⠀⠀⠈⢿⠽⠿⠟⣿⣿⣿⣽⣷⣾⣿⣿⣧⣶⣤⡾⣲⣤⣄⡀
⠢⣦⡀⠋⠀⠿⣿⣿⣿⣷⣾⣿⣧⣴⡁⢼⣷⠗⣋⣭⡭⢍⠭⠿⣿⣿⣿⡿⣿⣿⣿⣶⣿⡄⠉⠓⠶⠖⠒⠃
⠠⣼⣷⠆⠄⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⢘⣷⣶⣉⣡⣾⡀⢀⡸⠀⣀⣀⣀⣨⠿⠀⠙⠿⠷⠤
⠠⡿⣿⡄⠀⠀⠈⠈⠘⢻⣿⣿⣿⣿⣿⣼⣿⣻⡛⠿⣷⣾⣘⣉⠋⠉⠛⠛⠛⠛⠒⠒⠒⠒
⠀⠀⣨⣏⠀⣀⠀⠀⠀⢚⡉⠛⢿⣏⣻⣯⣍⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠁⠀⠀⠀⣆⡀⢂⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⡀⠂⠀⣷⣄⡄⠴⠆⡄⢀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣥⠶⠶⠚⠛⠛⠲⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠉⠉⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣄⡀⠀⠀⠀⠀⠀⣿⠀⢀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣠⣿⣷⠀⠀⠀⠀⠀⣸⠀⣿⠤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⡿⠏⠀⠀⠀⠀⠀⣿⠀⠙⠶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⢀⣤⣶⣶⣶⢷⣶⣤⡀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠉⠛⠷⢶⣤⣤⣤⡶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠘⠟⠉⠙⣿⣄⠀⠀⠉⠃⠀⠀⠀⠀⠉⠛⠷⠶⠶⠶⠶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠈⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢀⡟⠈⠉⠛⠻⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠾⠛⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⣾⠁⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⡤⢤⣤⡤⠴⠾⠷⠶⠶⣶⠶⠶⠶⠶⠶⠷⠶⣶⠶⠶⠶⠶⠶⠿⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣿⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣿⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣿⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠾⢷⣤⣤⣤⣤⣤⣤⡾⠷⠞⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣟⣻⣛⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣭⣭⣽⣿⣿⣯⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠿⠛⠛⠉⠉⠉⠁⠀⠀⠀⠉⠉⠙⠛⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡄⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⡇⠀⠀⠀⣿⣿⠋⠈⢻⣿⡇⠀⠀⠀⣸⣿⡏⠀⠘⣿⣿⠀⠀⢸⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠿⠇⠀⠀⠀⣿⣿⡄⠀⣸⣿⡇⠀⠀⠀⠸⣿⣧⣀⣠⣿⡿⠀⠀⠈⠛⣫⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⡀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠙⠿⣿⡿⠟⠁⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣡⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠿⠿⣿⣿⣦⣄⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠙⢻⣿⣦⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣸⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⡆⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁

⣿⡟⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⠀⣠⣶⡾⣷⣄⡹⣶⣿⣻⡄⢀⠀⠄⠀⢸⣿⣿⣺⡒⢏⣖⢺⢙⣃⢖⠻⢿⠿⡿⠟
⣿⣿⣿⣿⣠⣿⣿⡟⠧⢈⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢸⢹⡇⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣷⡄⢰⡀⠀⠀⠀⠂⢸⢸⣧⣤⣤⣤⣠⣼⣿
⣿⣿⣿⣿⣿⣿⡇⢠⣍⣻⠸⣿⣿⣿⣾⣿⣿⣿⠂⠀⠘⠀⠀⠀⠀⠀⣀⠀⠙
⡋⣿⣿⣿⣿⡟⠈⢘⣿⣿⣤⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⠀⠰⣎⣻⣽⣶
⠀⣿⣿⣿⣿⡇⠀⠈⠻⣶⡟⣠⣾⢿⡏⠉⠀⡟⠀⠀⢺⣶⣿⣷⣶⡌⠻⣿⢿⣶⡞
⢠⣿⣿⣿⣿⡏⠵⢶⣄⣵⣿⠟⠁⠄⢻⣾⣧⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠺⣿⣿⣿⡿⣇⠀⠠⣄⡀⠀⠘⠒⠀⠈⡿⢷⠶⠶⠶⢶⣶⣶⣶⣀⠐⠀⣀⠈⢙⠀⢹⣿⣀⠀⡀
⢀⣬⣭⡉⢉⣿⣦⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣷⣷⣿⣯⣄⣠⡀
⣶⣶⣶⣾⣿⣿⡿⣷⣄⠀⠀⠀⠀⠀⠺⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣉⡿⠙⠛⠋
⠛⠛⢻⡟⠛⣿⣧⠙⠛⠁⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⢀⣀⡈⠨⠿⢿⣿⠁
⠀⠀⢸⡇⡁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠙⠿⠿⠆⠀⢸⣷⢿⣶⡀
⣷⠶⣼⡇⣧⡴⠶⠶⠶⠞⠛⠛⠛⠛⠛⠛⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀
⠋⠉⠉⠛⠃⠀⠈⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣠⣤⣤⣤⣠⠱⣦⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠃⠀⠀⠀⠀⠀⢀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⢀⠜⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⣀⣀⡀
⠀⠀⠀⠀⠀⠉⠉⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠈⠉⠒⠢⡀
⠀⠰⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢀⡀⠤⠤⠤⠤⠤⠤⠤⠀⠀⠠⠠⠤⢀⣀⡞⠉⢳⡠⢦⡠⢆⠀⠀⢀⡄⠀⠀⡔⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡄⡀⠉⢨⠁⡜⠁⡷⠊⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣧⠀⠸⡃⠸⢡⠞⠢⢤⡞⠀⠀⠐⠀⠀⠀⠆⠀⠀⢸⠃
⠀⠰⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⢇⠱⠄⠘⠀⠧⠀⣧⡴⠶⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⠈⠓⠐⠁⠀⢠⠆⢄⣸⠃⠀⠀⠆⠀⠀⠐⠀⠀⢀⡏
⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⢧⠐⠈⠉⠀⡄⠀⠀⠀⠁⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠀⢀⡴⠀⠀⠀⡝⠀⠀⡘⠀⠀⠀⠁⠀⠀⡸⠁⠉
⠣⠀⢶⠀⠈⠀⠀⠀⢁⠀⠀⡀⠀⠀⠀⢓⣀⣤⣴⠒⠈⠀⠀⠀⠀⠀⡀⠤⠤⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⡇⠀⠀⢰⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀⠷⢤⡤
⠀⠀⠘⠀⠀⠁⠀⠀⠈⠀⠀⢷⠀⠀⠀⠈⠈⠁⠰⠀⠀⠀⠀⠀⠀⢈⣠⠄⠀⢠⡀⡈⠢⠀⠀⠀⠀⢀⠆⠀⠀⠀⠇⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⠠⠀⠀⠀⡇⠀⠀⠀⣰⠃⠙⢁⠄⠒⢢⠀⠈⠀⣧⡀⠀⠀⠸⠀⠀⠀⢠⠀⠀⠀⠘⠲⢤⣀⣀⣀⠀⠀⠀⣠⠇
⠀⠀⠀⠈⠒⠒⠒⠒⠐⠋⠁⠀⠀⠀⠀⢰⠀⠀⠀⠰⣤⠊⠃⠀⠀⠀⣸⣄⣠⠾⣤⢄⠀⠀⣼⠂⢀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠹⠉⠇⢀⡤⠾⠛⠠⠧⣾⣁⡜⣱⠈⢷⠀⠘⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢼⡄⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⢀⡾⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠈⢻⠒⣤⣤⠀⠀⠀⠀⠠⢶⠒⣉⠴⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠸⠠⠀⠀⠀⠀⠀⠀⠀⢁⢡⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠮⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠌⠐⢤⡀⠀⠀⠀⣀⠜⠁⠘⢆⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠖⠋⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠁⠒⠀⠺⠅⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢲⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⣤⡀⠀⢀⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠯⠍⣉⣟⠀⠈⣿⣁⣀⠤⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠈⠓⠉⠁⠀⠀⠀⠀⠉⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⣀⡤⠤⠒⠚⠂⠤⠄⢀⣀⡀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⡎⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠉⠐⢠⡄⠀⠀⠀⡠⣾⣍⣀⡀⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠵⠿⠶⠏⠉⠐⢄⠼⠁⠎⠉⠛⠒⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠶⠆⠐⠒⠶⠤⠤⠾⠁⠤⠤⠀⠀⠤⠤⠤⠞⠉⠂⠤⠤⠤⠤⠶⠾⠃⣠⣤⡄⣀⣀⣀⣀⢀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣶⣯⣥⣤⣤⣤⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣁⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣦⣀⣀⡀⠀⠀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠛⠉⠙⠉⠉⠋⠛⠛⠛⠛⠛⠛⠛

⣟⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⣀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠐⡀⠀⠄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠥⠀⠀⠀⢴⡈⠀⠀⠁⢁⣂⡴⠶⠤⠼⡂⠀⣀⠀⠀⠀⠀⠀⠀⣶⡆⠀⣤⠄
⠀⠠⠀⠀⠄⣀⠀⣶⠖⠀⡄⠀⠀⠀⠀⠈⢷⠊⠙⢿⠇⠸⡿⠀⠈⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⠒⠃
⠀⠀⠁⠀⠀⠙⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠠⠖

⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⠁⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠘⢠⣴⠀⣤⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣨⠀⠀⢸⠄⠠⠞⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠈⡍
⠀⢀⠀⠀⢠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⢀⡀⠰⠖⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⡠
⠀⠃⠀⠀⠈⠄⠀⠀⠠⠀⠀⠰⠄⠀⠀⠀⠀⠀⢨⠟⠀⠈⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⣀⠂⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠀⠒⡄⠀⠀⠀⠀⠀⠀⠐⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠂⠀⠀⠈⠃⠀⠈⠄
⠀⠀⠀⠀⡀⠀⠀⢅⢀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣶⡌⠁⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠴⠘
⠀⠀⠠⠂⠁⠸⢌⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣀⣀⣀⠠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡇
⠀⠀⠀⢸⠉⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⡆
⠀⠀⠀⠈⠓⠒⠒⠒⠓⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠚⠁
⠀⠠⠒⠀⣀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠻⠿⠛⠛⠛⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠿⠿⢿⣟⠷⢦⡄⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⠒⠄
⠀⠈⠉⠉⣀⣤⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠉⠉⠁
⠀⠀⢠⡾⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠈⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠉⠛⠶⠦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠤⢤⣀⣀⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣠⠴⠾⠷⢦⡀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⢀⡔⠉⠀⠀⠀⠀⠉⠳⠀⠀⠀⣠⡿⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣤⣄⣀⣀⣀⣤⡤⠶⠖⠛⠉⠀⠀⠙⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠙⠛⠓⠶⢦⣴⠦⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⣀⠀⠀⠀⣀⣀⡀⠈⠳⢆⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⢀⡴⠋⠁⣠⣶⣿⣿⣿⣿⣿⣦⡄⢀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣠⠏⠀⠀⣼⠿⠛⠉⠁⠀⠀⠀⠀⠁⠀⠉⠓⢦⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢠⡟⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠀⠀⢸⡇⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⡞⠀⠀⠘⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡼⠁⠀⠀⠀⢻⡄⣸⠇⢀⣴⣶⣶⣶⣶⣶⣦⡄⠀⠀⠀⢀⣴⣿⠟⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣠⡞⠁⠀⠀⠀⠀⠘⣷⠋⠀⠋⠀⠀⠀⠉⠉⠉⠙⠁⠀⠀⠀⠈⢉⣠⣤⣄⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠾⠋⠀⠀⢀⡎⠀⠀⠀⢹⡆⠀⠀⢀⡶⠋⠉⣯⢦⠀⠀⠀⠀⢰⠀⠉⠀⢿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⠟⠀⠀⠀⠀⠈⣷⠀⠈⠉⠓⠰⠖⠚⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⣄⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡇⠀⠀⢀⣿⡋
⠀⠀⠀⠀⢸⡏⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⢸⡇⠈⠳⡄
⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢀⣤⣶⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⠀⠀⢸⠁⠀⠀⡇
⠀⠀⠀⠀⠘⣷⡀⠀⠸⣧⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⣼⠿⠛⠉⠹⣿⠀⢀⡿⢲⡦⠘⢧⡀
⠀⠀⠀⠀⠀⠈⠻⣄⡀⠹⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣶⡖⢄⠀⠀⠀⢸⠇⢠⠀⢾⣁⡀⠀⣀⣴⡏⠀⠈⣠⠞⣣⣄⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⣿⡀⠘⢿⣿⣿⣿⣿⣿⣿⣧⠘⡇⠀⠇⠀⠀⡇⠀⢸⣰⡿⡿⢿⣿⣿⡿⢃⣠⣞⠁⢰⠟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡄⡇⠀⢸⡀⢸⠁⢀⡾⠋⢠⠇⠀⠈⢁⠔⠁⣸⠏⠠⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣣⠀⠀⣃⠃⣠⠎⠀⣠⠏⠀⠀⢠⠎⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠏⡴⠁⠀⡴⢃⣤⡶⣠⠋⠀⣰⠃⠠
⣴⣞⣉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠜⡜⠀⠀⣾⠵⠋⢀⡜⠁⠀⡰⠁⡴⠁⠀⠀⠀⠀⠀⠀⣠⡴
⠉⠉⠁⠀⠀⠀⠀⠀⠐⠛⣻⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡸⠀⣠⠞⠁⢀⡴⠋⠀⢀⡞⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡰⠁⠐⠁⠀⣰⠋⠀⠀⠀⠘⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⠉⠻⠿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⢰⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⠈⠁⠀⣀⠎⠑⠀⠀⠈⠀⠀⠀⠀⠈⠶⠀⠀⠦⡞⣣⠄⠀⢠⣏⣹⠄⣿⣙⠀⣾⣿⡄⣴⠟⠀⣿⣙⠀⠀⢶⣦⣰⣿⠀⢀⡄⠀⠓⠃⠑⠂⠀⡆
⠀⢒⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⣠⣴⡄⠀⠀⠀⠀⠀⠋⠑⠀⠀⠈⠯⠁⠀⠿⠶⠀⠿⢿⠇⠻⠶⠂⠿⠶⠀⠠⠀⠙⣿⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸
⠀⠈⠂⡀⠀⠀⠀⠀⠀⡀⠀⢀⡾⠍⠁⠀⣠⠞⠉⡏⠙⠲⡄⠀⠦⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⣀⠀⠀⢀⣿⡀⠀⠀⠀⢠⡄⠀⠀⠀⠁⠀⠒⡀
⠀⠀⠀⠑⢄⡀⠀⠀⠂⠀⠀⠈⠁⠀⠀⢰⠇⠒⢀⣧⡒⠃⢹⠀⠀⠀⠀⠀⢀⢿⢑⣤⠀⠁⠀⠀⠀⠁⡀⠀⠈⡟⠁
⠀⢀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠶⠀⠈⢧⡴⠋⣿⠙⢲⡼⠀⠀⠀⠐⠉⠙⠈⠀⠁⠀⢀⡀⠀⠱⠇⠀⠀⠀⠂⠀⠀⠀⠵⠎⠀⠈
⠀⢒⡋⡈⠈⠄⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⡈⠓⠶⠿⠖⠋⠀⠀⠀⣀⣤⣴⣶⣦⣤⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠺⠇⠀⠀⠀⠀⠄⡀⢺⣿⣿⣷⣿⣿⡿⠀⡀⠀⠀⠤⠀⣠⣾⣿⣿⣿⣿⣿⣿⢿⣷⣄⠀⠘⢻⡟⠃⠀⠀⡀⠀⠀⠘⢟⡿
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠈⠀⠙⢿⣿⡿⠋⠀⠈⠀⠀⠀⠀⢰⣿⣿⡏⠋⣯⠋⣍⠏⢸⣟⣿⣇⠀⠀⠀⠀⠀⠸⠿⠄⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⡀
⠀⠀⠢⠤⠄⠀⠀⠀⠃⡀⠀⠀⠀⠛⠀⠀⠀⢀⠘⠀⣤⡂⣿⣟⣿⣷⣿⣷⣷⣿⣶⣾⣿⣿⣟⠀⠋⠀⠉⠉⠉⠉⠉⠉⠃⢠⠀⠂⠀⠀⠀⠀⢐⠀⠀⠖
⠀⠀⢀⣠⡤⣤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⡔⠀⢻⣿⣿⣿⢦⡰⢃⡓⣇⣾⣿⣿⡿⢀⣤⣤⡀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⣠⠏⠁⠀⣼⠀⠀⠙⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠁⢸⡇⠈⠻⣦⠶⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⡏⠀⠀⢀⣿⡀⠀⠀⢹⠀⡼⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡿⣿⠿⡿⠟⠉⠀⠀⠀⢻⣄⣴⣿⣄⢀⡼⠃⠀⠀⠁⠀⠀⠀⠐⠂
⠀⢻⡤⠞⠋⣿⠙⠳⣄⡾⠀⠀⠀⠀⠀⠀⠂⠀⠘⠄⠀⠂⠀⠘⠂⢀⣤⣶⡖⠰⢦⡀⠀⠈⠢⡄⠀⠹⣾⣧⠿⠋⠀⠀⠖⠀⠀⠠⠐
⠀⠀⠙⠦⣤⣿⣠⡴⠟⠁⠀⠒⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢠⣿⣿⢫⠁⠀⣽⡿⡆⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⢀⠀⠀⠀⠁⠠⣤⣴⡶⠶⠦⡔⠁
⠀⠀⠀⠀⡀⠀⠀⢀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠒⠠⠥⠀⠀⣤⠀⠈⣭⣥⡄⠀⣴⣾⣶⡿⠀⣤⠀⠀⠀⣀⡄⠀⠀⠈⠀⠀⠤⣤⣴⣻⣛⣯⡽⢯⡾⠁
⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠸⣧⠀⢮⢿⢟⠀⢻⣿⠏⠀⡴⠇⠀⠀⠉⠟⠣⠀⠀⠀⠀⠀⠈⠉⠚⠑⠋⠉⠀⠋⠀⠘⠁
⠀⠀⠀⠄⢀⢀⢀⠀⠈⠀⠀⠠⠄⠄⠀⠀⠀⠀⠀⢀⠀⠀⠁⣉⣴⢀⠀⠀⠐⠊⠁⠐⣦⡉⠁⠀⠀⠀⠐⠀⠐⣦⡶⣰⢦⠀⣲⡀⢰⡆⡰⣦⠀⣶⣦⡄
⠀⠀⠀⠀⠀⠈⠉⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠘⠛⠟⠘⠃⠀⠸⠟⠻⠃⠙⠛⠄⠻⠻⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠿⠿⠿⣶⣦⣀⠀⢀⣠⣶⡾⠿⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢁⡴⠚⠛⠛⠛⠲⣝⠻⠃⠻⢋⡵⠞⠛⠛⠛⠲⣤⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⡀⠀⠀⠈⢷⣠⣠⠏⠀⠀⠀⡀⠀⠀⠈⢳⡀
⠀⠀⢀⣀⣤⠤⠶⠶⠒⠚⠛⣿⠀⠀⠀⣿⣿⡇⠀⠀⢸⡏⣿⠀⠀⠀⣿⣿⡇⠀⠀⢈⡟⠛⠒⠲⠶⠦⢤⣄⣀
⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠉⠀⠀⠀⡼⠁⠸⣆⠀⠀⠈⠉⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⡤⠞⠁⠀⠀⠙⠶⣄⣀⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣀⣀⣀⣀⣹⣶⣄⣀⣀⣀⣀⣴⣾⣁⣀⣀⣀⣠⣤⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⠿⡿⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⢏⢹⠉⢹⠀⢸⠀⠀⠀⡇⠀⡇⠉⡏⢙⠟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠉⠒⠼⠤⣼⣀⣤⣀⣇⡠⠧⠔⠋⣁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣄⣀⣀⠀⢀⣀⣀⣤⠴⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡤⠖⣋⣩⠤⠤⠄⠀⠐⢋⣁⣠⠤⠤⠤⠤⢤⣀⡀⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⢋⡴⠚⠁⢀⣀⣤⣤⠀⠞⠉⢁⣀⣠⣤⣤⣤⣀⡀⠉⠓⠦⣄⠈⠑⣄⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣱⠏⣠⠏⠀⣠⠞⠉⠀⢀⣀⣤⠤⠞⠋⠁⠀⠀⠀⠀⠈⠉⠛⢦⣄⠈⠳⡄⠈⢳⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏⣰⠃⣠⠞⠁⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠙⣆⠀⢳⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣡⣏⣼⡧⠚⣁⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠘⣆⠀⢷⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⡿⢿⢷⠞⠏⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣆⠈⢧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡇⣾⢸⣆⢡⡇⠀⠀⣠⣶⣶⣶⠶⣤⣄⠀⠀⠀⠀⠀⢀⣤⠶⠾⠿⠿⢿⣦⡀⠀⠘⣷⡜⢧⡈⢧⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⢸⣿⣾⡇⠀⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣏⠲⣅⡘⣷⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⣿⢸⢹⣿⡇⠀⠀⢀⢤⠶⡒⢲⣄⠀⠀⠸⡄⠀⠀⠀⡴⡟⠩⠉⣳⢦⠀⠀⠀⠀⣿⣿⡄⣄⢹⡟⠛⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⣿⢸⢸⣿⡇⠀⠀⠉⠘⠦⠁⠘⠉⠃⠀⠀⣷⠀⠀⠀⠀⠉⠀⠀⠁⠈⠀⠀⠀⢸⡇⡏⠧⢸⠀⣇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⢸⢸⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⠀⠘⡇⢻⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⢳⠈⣧⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⢸⢸⣿⣇⠀⢀⣀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣸⣿⠀⢷⠀⠈⣇⠸⡆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡟⣼⢸⣿⡌⢳⣾⡉⢷⠀⠀⠀⠀⠀⠘⠘⠂⠘⠉⠀⠀⠀⠀⠀⢀⡞⢩⣧⠞⢹⣿⡇⠘⣇⠀⠘⣆⠹⡄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣸⠃⣸⢠⡇⣿⢸⡇⢷⠀⢹⡇⠈⣿⢶⡄⠀⠀⣀⡤⠤⠦⢤⡀⠀⠀⢠⡾⣿⠀⣸⠏⠀⡞⣼⢻⡀⠹⡄⠀⠘⣆⠙⣆⠹⠆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⡿⢸⢇⡏⢸⡿⡜⡇⠀⣷⠀⣿⠈⢷⣀⠞⢁⡠⠤⠤⢄⠙⣆⢀⡿⠀⡿⠀⣿⠀⣸⢡⠇⠈⢧⠀⢹⡄⠀⠈⢦⡈⢧⡀
⠀⠀⠀⠀⠀⠀⣼⠁⢰⠇⡾⢸⠃⢸⡇⢳⣻⠀⣿⠀⢸⡄⠈⢯⣴⣿⣶⣶⣾⣿⣷⣜⡿⠁⢰⡇⢀⡇⢀⣏⡞⠀⠀⠘⣆⠀⠹⡄⠀⠀⠳⠄⠙
⠀⠀⠀⠀⠀⣼⠃⢠⠏⣸⠃⡟⠀⢸⠀⢸⡟⣇⢸⡆⠘⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣼⠀⣸⠇⡼⣹⠁⠀⠀⠀⠹⡄⠀⠉
⠀⠀⠀⢀⡾⠁⣠⠏⣰⠏⣸⠃⠀⡾⠀⡼⢻⡜⣆⢻⡀⢻⡄⠀⠈⢷⡈⠉⠉⢀⡿⠁⠀⢠⡏⢠⡏⣸⢣⡏⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣠⠞⠁⣴⠋⣰⠃⠀⣸⠃⣰⠃⢠⣷⠈⠈⠃⠈⠳⠀⠀⠘⣷⠒⠒⣿⠁⠀⠀⠞⠀⠘⠀⠁⡟
⠀⠀⠀⠀⠀⠀⠚⢁⡼⠁⠀⣴⠃⣰⠃⢠⡟⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠞⢁⡼⠁⣰⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠋⣠⠞⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠞⠂⠀⠀⠀⠀⠀⣴⣦⠀⢿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠋⣀⠀⠀⠀⠀⠈⢿⢏⡄⢀⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣈⣩⡉⠀⠀⠀⠀⠐⠙⠟⠈⠙⠻⢤⣀⠀⠀⠀⠀⠀⢀⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⣍⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠀⠀⣰⣖⣿⠟⢽⣿⡖⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣨⢷⡀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢶⡟⠋⠀⣸⣣⣿⣟⡀⣾⣿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡙⠂⡤⠀⠀⠀⠀⠀⢠⣴⡞⠀⢸⡇⠀⢀⡏⣿⣿⠅⣿⣿⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⠀⠀⠀⠀⠀⢸⣿⣇⡀⢰⡇⠀⡼⣶⣿⠃⣴⣿⠟⢠⢤⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠈⠉⢷⢸⡇⣼⣷⡿⣁⣾⡿⢃⣤⡿⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣱⣿⠟⣠⣿⠈⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠻⣆⠀⠀⠀⠀⣠⣾⣿⡿⣋⣼⡟⠃⢸⠙⢿⣆⣿⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠹⣧⠀⠀⣼⣫⠟⢋⣴⣿⠟⠀⠀⠸⣆⠸⣏⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢀⡀⠘⢷⣄⣼⡃⢠⣾⡿⠋⠀⠀⢀⡴⢋⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠸⠃⠀⠀⠙⠛⠿⠮⠿⠁⠀⠀⠴⠋⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠈⠉⠛⢶⣄⠀⠀⠀⠀⠀⣠⣤⠶⠶⠒⠒⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣠⣤⣄⡀⠀⢀⣤⣤⣤⣄⠀⠀⠹⣿⠁⢀⣠⣤⣤⡀⢀⣤⣤⣤⣄⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣰⡿⠋⢹⣿⣿⣷⣿⣿⣿⣿⣿⣷⠀⠀⢻⣄⣿⠋⣹⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣷⠖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣇⣽⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢸⡇⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣿⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⢠⣴⣶⡶⣶⣄⠀⠀⠀⠀⢷⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⢀⣿⠀⠀⠈⠙⠿⣿⣿⡿⠟⠁⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠉⠀⢀⡏⠙⠿⡄
⠀⠰⠛⠉⣿⡆⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠉⠛⠿⢿⡿⠟⠋⠀⠀⠀⣠⡾⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠈⠙⠷⣦⣤⣀⣀⣀⣀⣤⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣤⣤⣤⣤⣴⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⡉⠉⠉
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⠐⡀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠤⡡⠀⠀⠘⠀⠀⠀⠀⠀⢡⡀⡀⠀⠠⢄
⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡀⠀⣀⠘⠁⠀⠶⡃⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⡀⠔⠀⠀⠀⠀⠈⠅⠀⢨⠀⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠒⠂⠀⠀⡜⠁⠆⡀⢀⠐⠀⠀⠰⠅⠀⠀⠀⠀⢢⠈⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢐⡐⠟⠀⠠⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⢀⠀⠀⠀⠰⠄⠀⢀⠑⠠⠩⠡⣄⠤⠀⠀⢀⠀⠌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⡀⠀⠀⡠⠀⠁⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⠄⠊⠉⠀⠈⠁⠒⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠨⣀⠈⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰
⡄⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡠
⡇⠀⠀⠠⠀⠀⠀⠀⠀⠄⢀⠔⠁⠀⠀⠀⠀⠀⢀⡤⠄⠒⠂⠀⠀⠤⠔⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⡈
⡇⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⡀
⠃⠀⠀⢰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡁⠁⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠠⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠄⠀⣠⠀⠀⠄⠀⠂⠀⠀⢉⠀⠴
⠀⠂⠐⠀⠀⠀⢠⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡀⠤⠀⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⣀⡀⠤⠤⠐⡖⠒⠛⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁
⠀⠀⠐⠂⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠈⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶⣼⢠⡟⣽⣿⣡⣾⣵⣶⣾⣶⣶⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦⡻⢿⣿⣾⠳⡇⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣫⠵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠋⣩⣽⣶⣶⣶⣾⡿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣮⣍⡳⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣭⡓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣌⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⡶⠟⠛⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠉⠀⣀⣀⡀⠙⢿⡂⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⢸⣷⢸⡿⣮⡛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣩⢿⡿⠿⣷⣄⢹⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠸⣿⡏⢿⣆⡙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠉⠸⠧⠂⠟⠙⡿⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣄⡀⠀⠀⠙⠀⠀⠀⠀⠀⣿⢿⡌⠻⣟⡂
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠫⣴⣏⠄⢸⠿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣧⠻⣄⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢠⡟⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣌⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣙⣦
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢧⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡻⣍⣠⣤⣄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢉⣠⠄⠘⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣾⠿⠷⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠋⢸⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣼⣽⠟⠙⠛⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣸⠇⢉⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣰⢿⡇⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢶⢉⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⡯⣷⡀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣄⣀⣀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣿⡇⡏⠛⠦⠤⠤⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢹⣷⣟⡒⠒⠶⢶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⢿⣿⣿⣷⣶⡿⠏⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣰⠶⠀⠀⠀⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠀⠀⠀⠀⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠈⠀⠁⠢⡠⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡠⠐⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣠⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠁⢰⠀⠀⢰⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣬⠈⠁⠀⠀⡆
⠀⢀⣤⠀⠠⢀⠀⢀⠠⠓⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠔⠃⠄⠀⠤⠂⠀⠀⠀⠀⠀⢴⠀⠁⠀⠀⠈⠙⠂⠀⠀⠐
⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠀⠆⣄⡀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⣈⠓⠂⠄⠀⠠⠐⠒⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠈⠢⡒⠁⠒⠀⠒⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠉⠈⠁⠁⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠁⠀⠀⠀⠚⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠃⠀⠀⠀⠀⠻⢦
⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠈⠢⡤⠂⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⢄⡀⠀⠂
⠀⠀⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠖⢸⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⡆⢷⣿⣿⠆⠀⠀⢀
⠀⠀⠀⠐⡄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⠠⠓⠄⣀⡠⠔⠀⠀⠀⠀⠀⢻⠀⠀⠀⠙⠛⢋⠔⠢⢉⡁⠔⠃⠀⠈
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⢈⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⣠⠔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⡀⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⢠⣦⡀
⠀⠀⠀⠴⠶⢶⣤⣄⣀⢀⡀⣀⣤⣴⠶⠿⠒⠀⠀⠀⠀⢀⠤⠖⠋⠉⠐⡔⠉⠓⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⢀⠠⠐⠀⠁
⠀⠀⠀⠘⡀⠀⠉⠛⠉⡛⢋⠋⠋⠀⠀⡸⠀⠀⠀⠀⠀⠘⡀⠐⠘⠋⠀⡅⠉⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣤⡄⠈⣤⡄⠀⠀⢡
⠀⠀⠀⠀⠈⠐⠀⠀⠊⠀⠀⠁⠂⠐⠈⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠈⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠰⢄⡀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⣀⠀⢐⠂⠀⢐⠒⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⣠⠮⠧⢤⠘⠂⠠⠞⠒⢲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢏⡢⠤⠘⠒⠒⠘⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠆
⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⢉⡛⠳⣄
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣀⠔⡃⢸⠄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠿⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡠⠜⡁⢸⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣽⡇⣿⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠠⠴⠊⠁⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢈⣿⡁⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠖⠋⠀⡟
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣸⣿⣇⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠙⠛⠇⢸⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠲⣾⣿⣿⣏⢹⡄⠀⠀⠀⠀⠀⢰⡇⠀⠀⠒⠒⢋⠁⣼
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⢾⣿⣿⢸⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢙⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⣟⡟⠘⠆⠀⠀⠀⠀⢠⠏⠀⠀⠀⢉⣀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣍⣛⠂⣦⠀⠀⠀⠀⢸⠀⠄⠀⣀⣠⡤⠤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣩⣭⣦⠟⡀⠀⠀⠀⢸⠀⠀⢀⣴⣶⣶⡗⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠛⠷⠟⢻⣽⢸⡇⠀⠀⠀⣮⠀⠀⠉⣡⣤⠴⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢠⣽⡎⢷⠀⠀⠀⣿⠀⠀⢀⣀⣠⡴⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢻⣽⠿⠀⠀⠀⡦⠀⠀⠀⢀⣀⣠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⠤⢙⢦⡇⠀⠀⡇⠀⠀⠘⠛⢉⡁⠀⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣤⣴⣿⣶⡰⣀⢀⠇⠀⢀⣛⢉⣟⣻⡿⠿⠿⢿⣍⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠛⠿⠉⣉⡨⣿⣤⡾⠛⢋⡽⠯⡁⡄⠀⠀⠀⢻⡆⠀⢿⣦⡴⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠷⣾⠿⠛⠉⠀⠀⠀⢹⣆⠀⠀⡿⠀⠀⠈⢸⡇⠀⣾⣿⠁⠀⢀⣀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⣿⣿⠹⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⡤⠰⢂⣾⠁⢰⣿⡿⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣶⣿⠃⠈⠄⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢔⣶⡿⠏⠀⣾⣿⠃⠀⢰⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⠃⠀⠀⠀⠀⠀⠢⡀⠀⠀⣻⣿⡿⣛⣵⠟⠋⠀⣰⢧⣼⠇⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠏⠀⠀⠀⠀⠀⢀⢠⣬⣥⣿⣿⣿⣾⡟⠀⠀⠀⢶⢿⣿⠃⠀⠀⣾⣿⣿⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⠀⢀⡐⣷⣿⣟⣹⣿⣿⣿⣿⡇⠀⠀⢤⣤⣸⣿⣠⣦⡾⢋⣿⠇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠐⠈⠙⢿⣿⣿⣿⣿⣿⣿⣯⡧⢀⣘⣻⣟⣿⣁⣡⣽⣷⣿⣿⡷⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⠉⠙⠟⠋⣿⣿⣿⣿⣿⣥⣤⣿⣿⣿⣿⣿⠿⠟⠋⠉⠋⠠⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡿⢹⡟⣿⣿⣿⣿⣿⣿⣿⡛⠆⠀⠀⠀⠀⠀⢀⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠸⠃⠟⢻⣿⣏⠀⠀⡱⠀⠀⠀⣠⠖⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⢸⢠⣿⣆⠉⠀⠀⡀⣼⠯⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠲⡋⠿⢻⣄⣤⣸⠏⠀⢈⠂⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠃⠀⣿⢹⠆⠈⠀⠀⠀⢸⣷⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠙⠘⠆⠀⠀⠠⣴⣿⣿⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣉⠀⠠⢘⣧⠞⣤⣶⣰⡆⢸⣿⣿⢏⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠠⣄⡀⠀⠀⠀⠀⠀⠀⡾⠛⠾⠃⣺⣿⢿⡛⣿⣿⣿⣿⣿⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⠔⣁⡀⠀⠀⠀⠤⠤⢤⡽⠿⠟⠗⠉⠈⠛⠋⡹⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠺⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠋⠉⢹⣿⡏⠉⠙⠛⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⣠⣾⣿⡿⢻⣿⡟⢿⣿⣷⣄⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⣠⣾⣿⡿⠋⠀⢸⣿⡇⠀⠙⢿⣿⣷⣄⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣾⣿⡿⠋⠀⠀⠀⢸⣿⡇⠀⠀⠀⠙⢿⣿⣷⣤⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣏⡀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣹⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣦⣤⣀⣼⣿⣇⣀⣤⣴⣶⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄

⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣶⣾⣿⣶⣶⣄⡀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠁⠉⠀⡀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⡈⠀⠊⠀⠀⠀⠐⠀⠀⠨⠀⠀⠀⠄⢼⣯⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠿⠿⣿⡿⠿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠄⠀⠀⢀⠀⠀⠠⠙⠿⠟⠿⠻⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡀⠙⣇⠀⢹⡄⡐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠋
⠀⠀⠰⠶⠀⣠⠀⠀⠀⠀⠀⠀⠀⠂⣠⣏⣨⢀⡼⢀⣠⡇⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠐⠉
⠀⠀⠀⠀⢀⣤⣤⣀⣤⡏⠀⠀⢀⣴⡄⠀⠘⠃⠀⣀⠀⠘⠁⠀⠀⠀⠀⠘⠻⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⡁⠀⠀⢼⣿⠀⠀⠀⠀⠘⠛⠀⠀⢀⣤⢄⣠⣀⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⠀⠀⢾⡷⠀⠀⠀⠀⠀⠀⠠⣤⣾⣷⣿⣿⣿⣷⣶⣤⠀⠀⠀⢻⡿⢿⡿⡶⠀⠀⠀⠀⢀⠤⠔⢮⠒⠄⠀⠀⠀⢸⣿⣦
⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⣀⣿⣟⣿⣿⣿⣿⣿⣿⣿⣟⣧⣄⠀⠈⠃⠘⠀⢧⠀⠀⠀⠀⢻⠀⠀⠀⠀⠸⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⢀⣠⣄⠀⠀⠉⠁⠉⠉⠈⠉⠉⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⢤⣀⡀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠜⠁⠁⠐⠒⢀⡀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⢀⠠⠀⠄⢈⠀⠀⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠐⡀⢀⠀⠀⠀⠀⠀⢠⠆⠀⠀⣤⡀⠈⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠛⠛⠛
⠀⠀⠀⠀⠘⠀⠃⠀⠐⠀⠀⠀⢠⠛⠃⠀⣴⣿⣿⣿⣶⡀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣙⠚⠀⠸⣿⣿⣿⣿⣿⡇⠀⠘⠄⠀⠀⠀⠈⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠊⠉⠀⠀⠀⠙⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠉⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠒⠒⠶⢤⣌⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡴⢋⣁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣴⠏⣼⢃⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠳⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣹⣿⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣿⣿⡿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⣠⣴⡾⠿⠗⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⡼⠋⢹⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⠀⠙⠳⠦⢤⣤⣤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠁⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠙⠻⠷⢶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⠶⣾⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⢀⣀⣤⣄⣀⣀⣀⣀⡀⠀⢀⣠⠴⠛⠛⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡶⠛⠉⠀⠀⠀⠀⠉⠉⠙⠛⠯⣀⠀⠀⠀⠀⠀⠈⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣄⡀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣄⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠿⠿⢿⠿⠿⢿⣦⠄⠀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⢿⡇⠀⠀⠈⠻⢿⠿⠿⠿⣿⣿⣿⣿⣿⣯⣭⣹⡇⣶⠦⠶⣦⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢧⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⡔⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⠀⠀⢠⣤⡄⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡇⡀⢸⠀⠀⠀⡀
⢀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⢰⣾⡟⠁⠀⠀⠚⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠼⠾⣠⡼⠁
⣾⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⣿⣀⡸⠿⣤⣄⢀⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡧⣰⣤⡴⠓⠂
⠉⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠻⢤⣼⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣹⣿⣿⣽
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡵⢲⣶⣦⣤⡤⠄⠀⢶⡿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡙⠛⠻⠏⠹⠈⡇⠀⣀⠀⡀⣀
⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡏⠀⠉⠻⢧⣶⣴⡟⠙⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣧⣤⣽⣄⣨⠔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢘⣉⡙⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡽⢿⣽⣉⣹⠟⣡⣄⣰⠟
⠀⠀⠀⠀⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣴⡾⠟⠁⣠⣃⢘⣿⣿⣷⣿⣿⣿⣿⣿⣿⡉⣿⣿⡁⢸⣿⣿⣯⣀⣴⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⡍⠁⡀⠀⠀⠠⢀⡿⣿⣿⣿⡿⠛⣉⡉⠉⣉⣉⣉⣈⣉⣉⣀⣀⣉⣁⣈⣓⣒⣃⣀⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣴⣿⣿⡄⠉⠁⢀⣾⣿⣿⣦⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣋⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠒⠒⠾⠛⠛⠛⠓⠻⠄⠀⠸⣿⣿⣿⣿⡧⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣉⣡⣤
⠄⠀⠀⢀⠀⣀⢀⣤⣤⣴⠶⠶⠿⠿⠿⠇⠈⠉⠉⠉⠉⠉⠀⠀⠀⠘⠓⠲⠲⠶⠶⠆⠀⢀⣀⣴⠀⠀⣰⣿⣿⣿⡏⠀⠾⡖⠒⠛⣻⣿⣼⣟⣹⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣿⠈⠁⠂⠈⡁⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠤⠤⠤⠤⠒⠚⠁⢀⡠⠀⠀⢀⣶⢾⡿⠛⠁⠀⠀⣿⣿⣿⣿⣶⡄⢀⣻⢿⣿⣿⣟⠛⡿⠋⣿⣿⣿⣿⣿⣿
⠀⣉⣠⠜⠁⠀⠀⠀⣷⣦⣤⣤⣤⣄⣀⣀⢀⠀⠀⠀⠀⢠⢤⣶⣿⣷⡋⠁⢰⠀⠌⠁⣹⣿⣄⡀⢀⣾⣿⣿⣿⣿⣿⣿⠘⠻⣿⣿⣿⣿⣶⣿⣤⣿⣿⣿⣿⣿⣿
⣭⣿⣾⣦⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⠸⣌⢷⣬⣥⠟⠀⣰⡏⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣦⣏⣡⣴⣿⡟⣠⣆⠀⠀⣾⣿⣿⣿⣿⣤⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⠟⠛⢋⣴⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣋⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣇⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠉⠻⣆⠀⣀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡾⠶⣄⣀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠃⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣦⢷⡀⠸⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠄⡸⢟⣣⣀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠐⠿⠟⠋⣀⡠⠔⡹⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⠠⠤⢰⡈⣀⢈⡟⠁⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣲⡐⢏⠀⠀⠀⠷⠂⠀⢄⠀⠈⢻⣿⡄⠀⠀⠀⠀⠀⠀⡠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⣿⣿⡟⢿⣀⠀⢊⣼⡄⠀⠀⠑⢄⠀⠙⠋⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⣿⣿⣇⠈⠛⠉⣠⢿⣷⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣿⣿⣿⠂⠄⠄⠂⢸⣿⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣠⠎⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠠⣄⣀⣤⡄⠴⠁⠀⠀⢀⡄⢸⣿⡿⠀⠀⠀⠀⢸⣿⡅⠢⡀⠀⠀⠈⠓⢰⠊⣽⡏⠀⠀⠀⠀⠂⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠄⠀⠈⠈⣷⡌⠂⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⡇⠀⠈⢂⠀⠀⠀⠀⠀⠋⠉⠳⠆⡀⠀⠀⠀⠐⠀⢄
⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢠⡔⠉⠀⡀⠀⠠⠀⠈⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠙⠂⠠⠸⠄⠰⠊⠀⠀⠈⠐⢄⠀⠈⢂⢊⡆
⠀⠀⠀⠀⠀⠌⠀⢀⠄⠀⡠⠐⠁⠀⠀⠉⠀⠀⠀⡀⠞⢻⣿⣿⠀⠀⠀⠀⠀⣿⠿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⡀⠇
⠀⠀⠀⠀⢰⡈⡔⠁⠠⠊⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠡⣻⣿⠀⠀⠀⠀⠈⠋⣰⠁⠉⠛⠒⠂⠤⢀⡀
⠀⠀⠀⠀⠈⠁⠥⠐⠁⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠈⠣⠄⣀⣀⠠⠔⢂⡌⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠐⠒⢒⠾⢹⢯⣿⡛⠫⡉⠉⠉⠉⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢀⣀⣤⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⢸⡄⠘⠬⠻⠿⠲⠶⠿⠓⠂⠈⠉⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⣀⣀⣀⣯⣩⡾⡩⠁⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡀⠀⠐⠀⠔⠀⠀⠀⠀⠀⠀⠀⠉⡆
⠀⠀⢀⠤⠐⠒⠀⠀⠉⠉⠍⠉⠁⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⠀⣀⣀⣀⢀⣠⠴⠁
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠈⠉⠀⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢮⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⣇⣠⣤⣴⣶⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠉⢹⠟⠁⠀⠙⣯⠉⠉⠙⠓⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⢀⣸⣄⠀⠰⢆⣼⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⠙⠿⡞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⢴⡶⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠳⠄⠀⣈⢧⡀⠀⠀⠀⠀⢠⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟
⢿⡉⠛⢦⡀⠀⠀⠀⠸⣧⣀⣀⡀⢰⠁⠚⠛⣦⣀⠀⠀⠀⠀⠉⠉⣿⠛⠛⠛⠛⠛⠋⠉
⠈⢷⡀⠀⠙⢦⡀⠀⠀⠈⣿⠋⠁⠈⠳⠤⠞⠋⠉⠙⠒⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠈⠻⣄⠀⠈⠳⣄⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠱
⠀⠀⠀⠙⢧⡀⠀⠹⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠠⣿⣷⣦⡈⢷⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡜
⠀⠀⠀⠀⠀⣨⡿⠟⢻⡏⠙⢿⡛⠲⠶⠶⠶⢶⠶⠶⣶⠶⠾⡛⠻⡏⢷⣄⠀⠀⡇
⠀⠀⠀⠠⢾⠿⠶⠾⠛⠙⣦⣤⣷⢾⣶⣦⡀⠈⢆⡰⠁⠀⠀⠈⠶⠁⠸⣟⠷⣤⣹⣄⠀⠀⠀⠀⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻⡟⠱⢪⡿⠉⠃⠈⠁⠀⠀⠄⠚⠀⠀⠀⢿⠀⠀⠈⠉⢷⣴⣶⣷⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⠻⣗⠋⠀⠀⠰⣄⠀⠀⠀⠀⠑⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⠉⠑⠒⡛⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⢶⠶⠶⣞⢹⠶⠶⠶⡶⠶⠶⠶⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡛⠛⠓⠲⣤⢀⡼⣛⠚⠛⠛⠒⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣾⣭⣿⣶⣶⣯⣬⣷⣯⣽⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠁⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣭⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋⣱⣶⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⣿⠂
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⡘⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢀⣾⣿⣿⣆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠘⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⡿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠛⠋⠉⠉⠉⠉⠙⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⢁⣈⡓⢤⠖⡀⠀⠀⣀⠀⠀⣀⡀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠯⠬⠞⠃⠛⠛⠛⠇⡾⠁⠙⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠴⠒⠒⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠶⠤⢤⣄⣀⡀
⠀⠀⣀⣤⠤⠖⠚⠋⠉⠁⠀⠀⣀⠀⢠⡀⠀⡄⠐⢰⡎⠀⡏⠙⢸⡇⢳⡀⠀⢷⣄⠀⢹⡇⢡⡎⠘⡆⢸⠒⣆⢲⡀⠠⢐⡦⣤⠀⠀⠀⠈⠉⠉⠓⠲⠤⣤⣀
⢾⠉⠁⠀⡀⠀⠠⣄⠐⠀⢻⠀⠹⣆⠸⡇⠀⢻⡄⠘⡇⠀⣿⠃⢸⡇⢸⠇⠀⡈⢻⡆⢸⡇⢸⡇⢀⡇⣼⢶⡃⠸⡇⠀⢸⠧⡟⣰⠋⢹⡄⣠⠠⡄⢀⡀⠀⠀⠉
⠘⡆⠀⠀⢿⡄⠀⠉⢷⡄⠸⡇⢀⠹⠂⣷⠠⠀⠻⠀⠓⠀⠛⠚⠈⠁⠉⠀⠀⠉⠈⠀⠈⠁⠀⠑⠊⠀⠛⠘⠃⠸⠇⠀⣿⣠⠇⣿⠀⣼⢹⠇⢀⡇⣾⡠⠈⠀⢰
⠀⢹⡀⠀⠆⠻⣆⠠⠀⠙⠀⠉⠈⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⢤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠈⠚⠁⠘⠤⠞⣱⠋⡇⠀⠀⡞
⠀⠈⣇⠀⠁⠀⠀⠀⢀⣀⣤⠤⠶⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠶⠤⣤⣀⣀⠀⠀⠀⠉⠀⣸⠁
⠀⠀⠸⣤⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠆⢀⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠤⠂⡩⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠐⠋⠀⠀⡉⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠡⠤⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⡀⠀⠀⠀⡜⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡇⠀⠀⠘⡄⢳⣎⠔⠈⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠀⠀⠀⠁⠘⠏⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢪⢒⢀⠀⢀⠁⡈⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢦⠤⠤⠃⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⢨⡀⢡⢸⢐⡰⠀⡀
⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠝⠌⠁⠀⣐⠁
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠉⡙⣀⠁⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠠⢈⠀⢄⡨⠀⠌
⠈⠓⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣀⠀⠠⠀⠀⠀⠘⠁⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠴⡁⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠊⠢⡐⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⡂⢀⡀⠀⠀⠀⠄⠀⠂⡐⠀⠀⢀⡀⠐⠢⢀⠈⠺⠁⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⡔⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⢠⠂⠄⠐⠀⠈⠁⠀⠀⠽⠉⢹⢩⢘⠶⢤⡥⠀⢀⢴⡁⠀⠐⠁⠈⢀⠔⠁⢲⠁⠁⡌⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠕⢁⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⣘⠇⢘⡆⠀⠀⠀⠀⠀⠈⠍⠐⠢⠀⠈⠀⠀⠀⠲⡲⢁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠈⠀⠩⠇⠀⠀⢀⣀⠀⠀⠒⠉⠀⠀⠀⡎⠀⠀⢠⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠈⠁⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠁⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⣀⡀⠀⠀⠒⠀⠀⠈⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡀⠬⠚⠛⠋⢇⠀⠀⠀⠀⠀⠀⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⢤⣤⣞⣆⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠂⠀⠈⠙⠓⠒⠖⠐⠄⠐⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠢⠐⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠐⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣔⣒⣀⠒⠶⠶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣺⣽⠟⢉⣩⣤⣤⣤⣤⣤⣬⣿⣒⠂⠀⠀⠀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠯⠛⠁⠢⢴⡿⠟⠙⣓⡚⢓⠛⡛⠳⠮⢤⣀⣄⡐⠻⡢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⡫⠂⣰⡆⢴⣆⣤⣤⣈⠛⠿⢛⠉⠁⢀⣅⢀⡀⠀⠈⠙⠻⣦⣌⠐⠕⢄
⠀⠀⠀⠀⠀⠀⣠⣿⢿⠏⠠⣺⠟⣡⠀⠸⠟⠉⣤⢶⣳⣬⡛⢿⠿⣶⣦⣬⣉⠒⢦⡀⠈⠻⣷⣄⠲⡡⢀
⠀⠀⠀⠀⠀⣰⣳⠋⠁⢀⡼⢁⠞⡱⠃⡨⢤⣼⡾⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣙⢾⡀⡈⢿⣷⣔⡹⣷⡀
⠀⠀⠀⠀⢰⢳⡏⢀⢠⡿⢠⢃⠠⠀⠐⣁⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⣄⢩⣍⡈⠛⠻⠎⢻⢳⡀
⠀⠀⠀⠀⣿⢿⠇⠆⡟⠁⠋⢁⡄⡸⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⣆⢻⡟⣦⣶⠀⠈⣏⣧
⠀⠀⠀⠀⣿⡞⢠⠁⠀⠁⠠⣿⢣⠃⣿⡇⣀⡤⡄⠀⠀⠀⠀⡠⣄⣀⣀⣃⡀⠄⠀⠀⢸⡘⡈⠗⢇⡀⣴⢻⢸⣿⡀
⠀⠀⠀⣶⣿⡇⡀⢸⣠⡆⣄⣈⠘⢢⠟⠋⠁⢀⠇⠀⠀⠀⡜⠀⢿⣿⣶⠏⡘⠀⠀⠀⠈⠓⠐⣿⢹⠆⣿⠸⢸⣿⡇
⠀⠀⠀⣿⣿⡇⣷⠈⢿⡇⡇⢻⠞⠁⠀⠀⢀⠎⠀⠀⠀⠀⣇⠀⠀⠀⣠⠜⡤⠊⠙⠀⡘⢠⠰⣿⠈⡄⡿⢀⢹⣿⡇
⠀⠀⠀⢻⢿⠀⢸⣇⠸⣟⢰⣸⡄⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⣜⣠⣄⡸⡰⢃⠃⣦⠁⠘⠄⠁⡈⣸⣻⠃
⠀⠀⠀⠸⣾⡇⠈⢏⡆⢿⣖⠗⣷⡀⠀⠀⠈⠷⢄⠀⠀⠶⠮⣥⣦⠀⠀⠈⠉⣩⡞⠁⢃⣼⣿⠃⠈⠀⢠⢡⠇⡿
⠀⠀⠀⠀⢿⣿⣆⠘⣺⡜⢿⢆⠻⡷⠀⠀⠀⠀⠀⠉⠒⠦⠤⢤⡤⠤⠤⢶⠻⠗⢀⡀⠾⠙⠁⠞⠀⠐⣿⠏⡼⠃
⠀⠀⠀⠀⠈⢻⣎⢆⠱⡹⣎⠻⣧⣿⠀⠀⠀⠀⠀⠀⢀⡠⠤⢶⡲⢤⣴⣋⣤⠴⣪⡵⡣⢦⠄⠀⠀⠠⢊⡾⠁
⠀⠀⠀⠀⠀⠀⠛⢯⣧⡙⢿⢧⡈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⣤⣤⠴⠾⣋⠤⠖⢁⠔⡹⢋⡴⡫⠂
⠀⠀⠀⠀⠀⠀⠀⠪⡻⣟⠎⠳⣝⡻⡆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠷⠾⠋⣀⠤⣚⡵⢊⠕⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⣤⡀⠙⢳⠀⠀⠀⠀⢸⠍⢉⣳⣄⣀⣴⣛⡤⢄⣠⣶⠟⢉⣪⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡻⣶⣧⠀⠀⠀⠈⣗⣒⡲⠞⠒⠒⢒⣉⣉⣥⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⡀⠀⠀⢸⣽⡿⠿⠟⠛⠛⠛⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣵⣄⡀⠀⢀⣥⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢤⣶⠒⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢵⢞⣡⠿⣹⡏⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠤⣸⡟⠓⠛⣿⣇⠭⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⣲⣛⡫⠥⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠿⢿⣿⣿⠿⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠁⣴⣶⣄⠉⢠⣾⣷⣄⢹⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡇⠈⠿⣿⣿⡆⠘⣿⣿⣿⡄⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠛⠁⣻⠿⢧⠀⢿⣿⣿⠇⠸⣿⣿⣿⣧⡿⢿⠉⠙⠻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠰⠛⡿⠋⠁⠀⠀⠀⠁⠐⡦⠃⠀⠉⠁⠠⠤⠄⠉⠙⠷⡶⠀⠃⠀⠀⠈⠙⠿⠟⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⡖⠃⠀⠀⠀⣾⡛⠛⢛⣆⠀⠀⠧⠶⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣃⠀⠀⠀⠀⠉⠻⠿⠿⠛⠀⠀⠀⠐⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠦⣄⣀⣀⣀⣠⣤⡀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠈⣿⣿⣿⣿⡗⠒⠒⠚⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠟⠛⠛⢿⠛⠻⢶⣶⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡓⢤⣀⣀⣀⡤⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠛⣶⣦⣤⣤⠴⠮⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡄⠀⠸⣿⣿⡿⠀⠀⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⢡⠀⠀⠀⠘⠟⠁⠀⠀⡛⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠃⠀⡼⠀⠉⠑⡶⠒⡄⠀⠀⠇⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠄⠀⠸⠆⢠⠇⠀⠉⠒⠓⠒⠃⠀⠀⢀⠀⠘⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⢀⡐⢶⠃⣜⠀⠀⠀⣠⣾⣆⠀⠀⠀⢸⠀⢸⡍⠀⠀⠹⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⣴⣿⣷⣶⣾⣿⣿⣿⣧⣤⣤⣴⡀⠘⡤⠄⠀⡀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠐⠂⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠁⠀⣿⣿⡿⠿⠿⠿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⡀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠀⠈⠁⠳⢤⣉⣥⡜⠀⢪⣉⣛⣋⣱⣀⢠⡤⠠⠤⡀⠀⠀⠸⡀⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠡⠀⢰⠀⠃⠈⠽⠋⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣤⣀⣀⣀⣀⣤⠴⠃⠀⠰⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠛⠉⠁

⠀⠀⠀⢀⡴⠒⠋⡟⠙⠲⢄⠀⠀⠀⠀⠀⢀⣤⠖⠛⣿⠛⠳⢦⡀⠀⠀⠀⠀⠀⣠⠶⠛⢻⡟⠛⢶⣄
⠀⠀⢠⠏⠀⠀⠀⡇⠀⠀⠀⢳⡀⠀⠀⢀⡞⠁⠀⠀⣿⠀⠀⠀⠹⡆⠀⠀⠀⣾⠁⠀⠀⢸⡇⠀⠀⠙⣧
⠀⠀⢾⠀⠀⠀⢀⣧⡀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⢀⣿⣄⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢀⣼⣇⠀⠀⠀⣻
⠀⠀⠸⡄⢀⠔⠁⡇⠙⢦⡀⣰⠃⠀⠀⠘⣇⢀⡴⠋⣿⠙⢷⡀⢠⠏⠀⠀⠈⢷⣀⣴⠟⢹⡏⠳⣄⢠⡿
⠀⠀⠀⠘⠣⣄⣀⣇⣀⣠⠟⠁⠀⠀⠀⠀⠈⠻⣤⣀⣿⣀⣠⠽⠋⠀⠀⠀⠀⠈⠻⣧⣄⣸⣇⣠⡼⠟
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣀
⠀⠀⢀⣴⠟⠉⠀⣿⠀⠉⠻⣦⠀⠀⠀⠀⣴⡿⠋⠁⣿⡇⠉⠻⣷⡀⠀⠀⠀⣴⡿⠛⠉⣿⡏⠉⠻⣷⣄
⠀⠀⣾⠃⠀⠀⠀⣿⠀⠀⠀⢹⡇⠀⠀⣸⡏⠀⠀⠀⣿⡇⠀⠀⠸⣷⠀⠀⢸⣿⠁⠀⠀⣿⡇⠀⠀⠘⣿⡄
⠀⠀⣿⡀⠀⣠⣾⣿⣦⡀⠀⢸⡟⠀⠀⢻⡇⠀⢀⣴⣿⣷⣄⠀⢠⣿⠀⠀⢸⣿⠀⢀⣴⣿⣿⣦⡀⢀⣿⡇
⠀⠀⠘⢷⣾⠋⠀⣿⠈⠻⣦⡿⠁⠀⠀⠈⢿⣶⡟⠁⣿⡏⠻⣷⣾⠏⠀⠀⠈⢿⣶⡿⠋⣿⡏⠙⣿⣾⠟
⠀⠀⠀⠈⠙⠷⠶⠿⠶⠟⠋⠀⠀⠀⠀⠀⠀⠙⠻⠶⣿⠷⠿⠛⠁⠀⠀⠀⠀⠀⠙⠻⠷⣿⣷⠿⠟⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢀
⠀⠀⠀⠀⠟⠁⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠒⠄
⠀⠀⠀⠀⠀⣀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠓
⠀⢠⡄⠀⠀⠀⠀⠃⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠈⠑
⠀⠀⠑⠄⡀⠀⠀⠀⢀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠱⣤⠨⠄⠨⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠴⠒⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠠⠖⠂⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⡴⠆⠀⠀⠀⠶⡄
⠀⠀⠀⠀⠤⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⣀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡉⠀⠀⡀⠀⠀⠀⢀⠀⠀⠺⣿⡿⢘⠿⠿⠁⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠠⡤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠠⣤⣤⣀⡀⢀⢀⣀⣤⠴⡦⠀⠀⠀⢀⡠⠴⠛⢡⠐⠲⠤⢀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠉
⠀⠀⠀⢄⠀⠙⢉⠛⢉⠋⠀⠠⠃⠀⠀⠀⠘⡀⠀⠃⢀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠡⠄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠴⢀⡠⠀⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠦⣀⡀
⠀⠀⠀⠃⠄⠀⠂⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢼⣿⣿⣿⣿⣷⣦⣤⡀⠀⣶⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠙⠛⠻⠿⢿⣿⣿⣿⣦⡈⠙⠛⠿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣦⣀⠀⠀⠀⠉⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣄⣀⣤⣤⣿⡟⣃⣀⣀⢄⣄⠀⠀⠀⠀⠀⣀⢴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠳⠳⡿⠏⠼⠏⠀⠀⠀⠀⠐⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠠⣉⠛⠛⢻⢿⣶⣶⣦⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠾⠛⠛⠿⡆⣀⠈⠉⠀⠈⠉⠁⠈⠉⠱⡄⠀⠀⠀⢀⣠⠎⣾
⠀⠀⠀⠂⠀⠂⣀⡀⢿⡀⠀⠀⠀⠀⠐⠃⠀⠀⠀⢠⠒⠻⢹⣧⡀⣀⠁
⠀⠀⢠⠀⡂⢀⠌⠱⣼⡇⠀⠀⡄⠀⠀⠀⠀⢰⣶⡉⠅⠀⠈⠛⠛⢦⡀
⠀⠀⡜⠀⠠⢋⣴⢶⢍⣿⣄⠀⣿⠀⠀⠀⠃⠀⣏⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⡇⡘⢀⡈⠀⣹⡿⠿⢯⠀⢀⠀⠀⠀⠆⠀⢸⡄⠀⠀⡀⢀⡄⡀
⠀⠀⣏⠁⠞⠁⣜⡁⠈⢂⡄⢀⡄⣴⠀⠀⣰⣇⡾⠁⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠉⠴⢁⡾⢠⣾⣿⣿⣶⡾⠣⢟⣵⠾⠷⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⣠⣼⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⢻⣿⠗⡄⠀⠀⡠⠚⣻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⡀⠀⠀⢱⢄⣰⢃⡈⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠒⠶⠤⣤⣀⡑⠛⢆⠀⠀⠸⠈⡉⢾⠁⡴⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢀⡾⡁⠀⡄⠹⣯⠁⢐⡊⠩⢉⡫⣕⣦⣤⣤⣤⣤⡞⣩⡞⢹⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀
⠀⠀⠀⠀⠀⠀⣾⠀⠖⠅⠘⣠⠿⠾⣉⠉⠳⣄⡰⢄⣿⣿⣿⣿⡟⠀⡼⠀⠸⣧⠀⠀⠀⣴⣶⠊⠩⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣏⣠⣴⣶⣶⣧⣄⠀⡈⠆⣠⠋⠹⡆⢹⣝⣟⣯⣶⡾⢤⡀⠀⢻⣷⡀⠀⠹⣇⠀⠀⢻⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠁⡔⢳⣄⣿⣿⣿⣿⡇⠠⠙⢦⡀⠻⣿⣆⠀⣿⣦⣀⣠⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠛⢢⡴⢆⣸⣿⣿⣿⣿⣿⣅⠆⡖⢀⢙⣦⣌⣻⡥⢿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣼⡀⣼⠋⣿⣿⣿⣿⣿⠈⠳⢤⡒⠂⢈⠋⠁⣤⠻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⡻⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⡄⠀⠀⠙⠳⢥⣑⣉⣒⣀⣨⠟
⠀⠀⠀⠀⠀⠀⢀⣾⣆⢨⡔⠈⠉⠛⠛⠉⡅⣿⠀⠀⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣾⣿⡿⣷⣄⡐⠠⠠⠀⠀⣡⠟⠀⠀⠸⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣿⣿⣿⣿⡏⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠉⠙⠉⠛⠀⠀⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠞⠋⣹⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠘⠿⢿⣿⠿⠿⠛⢛⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣾⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⣀⣀⡀⢀⣀⣀⣀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠠⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠸⣿⣿⣶⣭⡻⣿⢿⠿⢿⣿⡋⡀⠀⠰⣇⡀⠀⣿⢀⠀⣹⡟⠏⠁⣀⣴⡤⠲⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢶⣾⠿⠟⢿⣶⢦⡹⢷⠀⠁⠖⠼⣉⣴⣶⣋⣯⣀⣿⣶⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⠋⠀⠰⢶⣾⣿⣿⣶⣬⣤⣄⣠⣽⣯⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⣿⣿⠟⠁⠀⢀⣀⡓⢲⣿⣿⣿⣯⣹⣿⣿⣿⣻⣿⣿⣿⣿⡏⢿⣿⡟⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⣠⣿⣿⣿⣿⡏⠿⠿⢿⡿⣿⡿⢿⡿⠿⢻⣧⢸⣷⠸⣿⠁⣿⢏⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⢻⣿⣟⠟⠀⠀⣠⣿⡄⠿⠂⠛⠘⠏⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣈⡀⢀⣰⣿⣿⣧⠀⢀⣀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣇⠛⠀⠈⠍⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⢛⠋⠉⠀⠀⠈⠀⠈⠁⠀⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠬⠛⠿⢿⣿⡄
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠹⠋⠓⠚⠻⢷⣤⣀⠀⠀⣀⣠⣴⡶⢶⠤⢤⡀⠐⠲⣅⠀⠀⠀⢠⣤⣤⣤⣤
⠀⠈⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⣰⣿⡇⠀⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⣿⣇⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⣶⣤⡤⠀⣠⣄⣀⣀⣀⣀⢀⣠⣴⣶⣾⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣿⣍⡉⢋⣴⣿⣿⣿⣿⡿⣯⣿⣭⣿⣿⣿⣮⠿⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡟⣰⢿⣿⣷⣿⢯⣭⣭⠝⠟⢿⣿⠟⡿⢿⣿⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠙⡞⠻⣟⣿⠠⡉⠛⣿⣄⠈⢹⡶⠃⠘⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣀⠀⠈⡙⠈⣃⣛⣁⠟⠀⡀⠃⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠄⠀⠉⠉⠋⠀⢠⠄⠀⣶⡆⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣧⡄⢀⣀⣀⣀⠈⢠⣴⡿⠁⠀⠀⣄⣀⣀⡤⠤⣄⢶⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⢛⣿⠁⢀⣰⠟⠀⠀⠀⠀⠈⠛⠿⣦⢸⡿⣙⡾⠶⠀⠁⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢘⡇⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⡠⠖⠚⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠾⢤⡀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⣀⣉⡉⠁⡖⣒⠣⠄⠀⠀⠀⢳⠀⠀⠀⠀⠀⢸⡆
⠀⠀⢀⣤⣶⣋⡁⠀⠉⠓⣼⢁⠈⠐⠢⢄⡀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠹⡅⠀⠈⠉⢉⡓⡏⠻⠂⠀⠀⠀⠈⡾⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠱⣄⠀⠀⠿⠿⡇⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠉⠓⡖⠒⢻⣄⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠉⠛⠡⠄⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠃
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠈⢯⠉⢹⠻⡟⠓⢲⢶⠒⠒⣶⠠⠄⠄⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢧⠙⠒⠋⠀⠑⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⡇⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⣄⡈⠓⠞⢁⣇⠀⠙⢦⡀⠀⢄⠁⢢⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠉⠉⠉⠉⠀⠀⠀⠀⠉⠒⠦⠎⠸⡄⠠⣀⡙⣢⣄⠑⠀⠁⢨⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠒⠒⠒⠶⠷⠄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢄⣉⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠲⠞⠅⠤⠀⠀⠠⠀⠠⠤⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⢰⡶⣶⣴⡶⣶⣶⡶⢶⣆⢰⡦⠀⢠⣶⢶⣤⣶⣶⡆⢴⣶⠀⣶⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣾⡇⣿⣿⣇⣿⣿⣿⠊⢿⣿⡇⠶⢾⣿⢿⣿⣿⣿⡇⣾⣿⡇⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⢟⢿⠛⡿⠻⠿⠿⠹⡟⢿⠐⠶⠙⡟⡿⠛⠛⣛⣓⠛⠛⠛⠛⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣷⣶⣾⣷⣗⣠⡴⠖⠚⠛⠿⠶⣤⣶⣶⣦⣦⡾⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣶⠒⠒⣲⣶⣾⣥⣶⣦⣀⣤⠀⠀⠈⣻⡟⣙⣿⣿⣾⣿⠿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢝⣳⣾⠿⣿⣿⣿⣿⣿⡟⠿⠤⠤⠶⢛⣿⣿⣿⣾⣷⣶⣿⣷⣿⣿⠗⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡸⣿⠋⢻⣿⣩⣭⣿⣇⣼⣾⣷⠆⠀⠀⠻⣿⣿⣭⣹⣿⣿⡞⠛⣿⣿⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⢂⢐⢹⣷⣿⠁⠀⠀⠈⠁⠀⢀⡆⣰⣿⣿⣿⠟⢻⣿⣷⠾⣁⣸⠛⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣦⣸⠋⠑⠢⡀⠀⣠⣾⣶⣿⠧⣾⡿⢋⡤⠶⠿⣿⡿⠀⠈⣿⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡷⠸⣷⢎⡄⣿⠹⣻⡶⠿⣛⣁⣀⡀⠀⣀⣸⣿⡗⠀⠸⡉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠶⣻⣖⣉⣴⣿⣿⣿⣽⣿⠟⠛⣻⡿⣷⣤⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢟⣶⣿⣿⣿⣿⣿⣍⣿⣿⠿⠁⢸⡿⢏⠀⢀⠀⣼⠛⠉⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣿⠁⠀⠀⡼⠱⠋⠀⠈⢰⣷⠘⠛⠙⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣾⣿⣿⡇⢸⡟⠋⠉⠁⠀⠀⣰⣷⣠⠂⠀⢀⡾⢻⠄⢀⣴⢴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣻⣿⣼⣧⣤⣤⣴⣦⣿⣼⣿⣃⣠⡴⠋⣠⣿⢨⢊⡀⢌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢃⣹⣿⣿⠋⢁⠿⢻⣿⣿⣿⣿⣿⣿⣿⣴⣞⣯⠿⡣⠋⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⢁⡺⠛⣿⣿⣤⣿⣿⣱⣿⣯⣿⣯⣽⣷⡶⠞⣻⢀⣿⡟⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢹⡿⢧⣶⡁⣰⣿⣻⢷⣿⣿⡟⣉⡿⠀⡿⢷⣿⣿⢿⣿⡟⠣⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣇⣀⡜⠉⣿⣿⣿⣿⣿⣿⠿⠛⠉⣴⣿⣿⠟⢿⣷⣈⠻⠿⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢸⠿⣷⣾⣄⡼⣶⠿⣿⣿⠿⢻⣾⣿⣿⣿⣶⣞⣿⣿⣷⣦⡿⡇⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣧⡤⠞⠋⠉⣤⣿⠀⢸⣿⠿⣻⣿⣿⣿⣿⠻⣿⡟⣋⣴⠏⣓⣖⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⢠⣾⣶⣿⣿⣿⣯⣿⣾⣿⣿⡿⠟⠋⣡⣤⣾⣻⢇⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠟⣫⣽⠿⠋⣁⣤⠶⠋⠉⣁⣤⠿⣋⡥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⡷⠛⠉⣠⠴⠚⠉⣀⣤⡶⠟⣫⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠼⠿⠿⣤⡴⢾⡏⠀⠀⢴⣾⣿⣿⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡌⠙⠉⣀⣤⡾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠴⠞⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣀⣀⣀⣹⡆⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠉⠉⠉⠉⣷⠀⠀⠀⠀⠀⠀⣿⠿⠒⠀⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠛⠒⠦⠤⢴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣿⡄⠀⠀⣸⠇⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⢀⣠⡟⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣤⡶⠾⠛⠛⠛⠙⠛⠶⠦⣤⣀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣱⣿⠟⠿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⡀⠀⢸⣿⠛⠳⠶⣤⣀⣀⣀⡠⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⡀⠙⠛⠛⠁⠀⠀⢠⡿⢿⣯⡁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠘⢿⣤⡀⠀⠀⠀⠀⠘⠁⠀⠈⠻⣶⣤⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⣿⡿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢷⣄⠀⠀⢀⣼⣇⠀⠉⣻⣷⣤⡀⠀⠀⠀⠀⠀⠙⢿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠺⢿⣦⠀⠈⠉⣡⣄⣰⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢷⡀⣴⣿⣿⣿⣿⣿⠛⢿⣿⡇⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠙⠿⢿⡟⢉⣿⠏⠀⠀⠙⠇⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⡿⠿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠟⠛⠻⣿⠀⠉⠁⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⡀⠀⠀⠀⢸⣇⡗⠀⠀⢀⣀⣴⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠻⠶⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣅⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢀⠀⣄⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠠⡾⠶⣶⡷⢬⣻⡿⢛⡂⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⡴
⠀⠀⠀⠀⠀⠤⣤⣈⠉⣽⣿⣣⣾⣿⠿⢿⡇⠀⠀⠁⠞⣿⣿⣿⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠁⠉⠙⠛⠛⠛⣷⣾⣿⣿⣤⡀⠀⠸⡏⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠙⠷⠼⣿⣿⣾⢿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢀⣶⣻⣿⣿⣿⡿⠿⣿⡟⢧⣦⣤⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣦⣀
⡀⢀⣠⣼⡿⢟⣿⠿⡛⢻⣿⣒⣒⠓⠂⠸⣿⣆⠀⠁⠨⣭⣹⣿⣿⣧⣤⣼⣿⣿⣿⣿⠿⣧⡀⠀⠀⠀⢀⡀
⣾⣏⣨⣠⣾⣿⣿⣿⡷⠴⠶⢤⣬⣅⢀⠀⠉⣿⣿⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⡿⢄⣖⡀⠾⣿⡟⣂⡀
⠈⠹⣯⣽⡟⣿⣿⣿⣧⡀⢀⣼⡿⣿⡿⠀⠐⣿⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣛⣛⣿⣟⣯⣸⣿⣷⣶⣦⠀⠤⣀
⠀⠀⢹⣿⠀⠀⠹⠛⠛⠏⠉⠉⠈⠁⠀⠀⢰⡟⣾⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣶⣿⣭⣽⣻⣿⠻⠷⠾⠇⠉
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠁⠀⢈⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣷⡝⣷⡍⠩⣜⣻⣿⠟⣀⡀⠤
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⢸⣿⣿⣿⣿⣯⣶⣿⣿⣿⣯⣛⣻⣽⣶⣿⡀⡍⣤⠀⣽⣌⣛⣛⢛⣂⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠈⣿⣿⣏⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡻⣷⢠⣼⣶⣾⣿⣿⣿⣷⠿⣻⠋⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅⠈⢹⣿⣿⣿⣿⣿⣿⣿⣽⣯⠙⣻⣿⣿⣿⡟⠛⠛⣇⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡠⠿⣶⣦⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣷⣀⣨⣿⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠻⣿⠿⣿⠉⠀⠀⠀⠀⠀⠀⢰⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⠀⣶⣴⠀⠿⠳⣦⣀⡀⠀⢀⣾⣽⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠁⠀⠀⢀⣠⣤⡀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⡀⢈⣛⡛⢋⡀⠀⠀⢀⣾⣽⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣄⣀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠠⣿⣿⣿⣯⠄⢀⣴⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣟⣷⣶⣦⣤⣀⣀⣿⣿⣄⣀⣀⠀⠀⢻⣦⣨⣯⣷⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢁⣠⣤⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣀⣤⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠏⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣾⣿⣿⣿⣿⣏⣴⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⢿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠋⠉⠉⠉⠉⠛⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⣴⠟⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⠏⠀⣴⡏⠉⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣿⠓⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠹⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⢿⣶⣄⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣠⣼⣿⣿⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⣿⣿⣿⡟⢀⣾⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⣠⣴⣾⡿⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠿⠋⢠⡾⠃⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⢠⣾⠿⣿⡄⠀⠀⠀⠀⠀⠀⠹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠈⠻⢦⣄⣀⣀⣀⣀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⢰⠟⠁⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣤⣤⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠁⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠈⠉⢻⠿⢷⣶⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢸⠀⠀⠀⠈⠉⣻⠛⠛⠛⠛⠛⢿⠿⠿⠿⠿⠟⢻⠟⠛⠉⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠒⠢⠿⠤⣀⣀⡀⠀⣿⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⢸⣇⣀⣀⣠⣤⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠈⠉⠋⠁⠙⠋⠉⠉⠉⠉⢉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⣠⣴⠶⠛⠛⠛⠛⠛⠛⠻⠶⣦⣶⡛⠁⠀⠀⠀⠀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠘⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠿⠿⣶⣄⠀⠀⠀⠀⠀⣴⣾⠿⠿⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠸⣿⡀⠀⠀⠀⣸⡿⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⣷⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠸⣿⡀⠀⣼⡟⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠿⠿⣶⣄⠀⠀⠀⢹⣿⢸⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠿⢷⣶⣿⠁⠀⠀⠀⠈⣿⡆⠀⠀⠘⣿⡄⠁⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⣠⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⣶⣤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣀⣀⣀⣼⠿⣷⣤⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠙⠛⠛⠛⠁⠀⠀⠁⠀⠘⢿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠷⢶⣦⡀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⠶⣶⣶⣶⣶⣶⣶⡶⠾⠿⠛⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⠻⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⡙⠀⠀⡀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠁⠛⢲⠆⢶⠋⢀⢄⣴⠶⢦⣄⣴⠖⠒⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢬⠐⠖⢠⠃⡾⢁⠂⠀⠙⡇⠀⠈⡅⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⣀⣀⣬⠴⢾⢺⡇⣒⠀⠀⢀⡇⠀⠀⠺⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣠⡴⡞⢁⢄⡁⠀⠀⠪⣧⣀⣀⡴⠟⠳⠦⣤⠞⠁⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠙⠃⢡⣀⢀⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣰⣤⣍⣙⠁⠀⣰⠀⠀⠤⠤⡤⠄⣀⣀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⣸⠚⠉⠁⠀⠀⡠⠊⠉⠊⠈⠂⠚⢗⡢⠵⠂⠀⠀⠒⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠸⢄⡑⠌⠀⠈⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠢⡴⡋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠑⠀⠀⢇⡹⢀⠁⠀⠀⠀⠀⠀⠀⢘⠁⠑⠋⠀⣘
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠖⢂⣉⡁⣀⠀⢀⠎⢣⡀⢆⠀⠀⠀⢀⡀⠔⠛⠰⢤⠋⠈⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣈⣄⢃⣘⠀⠀⠀⡏⠒⡿⢄⠀⠙⠉⡀⠒⠄⠺⡀⠀⡈⠀⠤⡁⢈⡡
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⢤⠀⠀⠀⠆⠀⡠⠄⠀⣷⠐⢻⠲⠀⠘⣍⠐⠂⠑⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠒⡤⠀⠒⠉⣈⣀⡜⠀⣼⠀⠈⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠀⠀⠀⠀⢇⠀⢧⢤⣿⡄⠀⠀⣧⣀⣀⠀⠠⠵⠀⠀⠀⠀⠀⢠⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡰⢆⡰⣄⣸⠉⠛⠛⠛⠒⠒⠉⠁⠀⠀⠀⠀⠀⠐⠤⡄⣀⣀⣇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢸⠒⠛⠀⠀⠀⠀⠀⢠⠔⠚⠉⠉⠠⡀⠀⠀⠀⠀⠈⢡⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢂⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡦⢀⡆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡠⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡉⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡔⠋⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠀⠀⠀⠉⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⡙⠒⠦⣤⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⣠⣶⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡈⢦⡐⣖⠁⢽⣶⣶⣶⣶⣿⣿⠓⢤
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣵⣹⠳⠄⠙⠻⠿⠿⠿⠿⠂⣠⠃
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢻⣿⣿⣴⣦⡄⠀⠀⠀⠀⢰⠋⠁
⠀⠀⠀⣠⠏⢿⣿⣿⡿⠁⠘⡆⠀⠀⠀⠀⠀⠀⡤⠋⠀⣀⠾⠋⠉⢹⣿⢇⡀⠀⠀⡰⠃
⠀⣰⠋⢹⠀⠈⠛⠛⠁⠀⠀⢹⣤⡀⠀⠀⢀⡞⢀⣠⠞⠁⠀⠀⢀⣾⣿⣿⣿⡀⠞⠁
⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠹⡤⢤⡜⢰⡿⠁⠀⣀⣄⣠⣿⣿⣿⣿⣿⠃
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠱⠀⠈⢿⡁⢀⣼⣿⣿⣿⣿⣿⣿⣿⡟
⠀⡁⠈⢲⢼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠾⠛⠉⠉⠉⠉⠙⣿⡟
⠀⠙⠤⠎⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠉⠉⣀⡴⠋⣷
⠀⠀⠀⠀⠀⠀⠊⠓⢤⣀⡀⠀⠀⠀⢀⣀⣠⣴⣾⣿⡿⠀⢰⣾⡟⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⢣⡀⠀⡼⠉⠉⡙⣹⣿⠿⢿⡿⠛⡿⠁⠀⣾⣿⣇⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠊⠁⠀⠠⢰⠟⠁⠀⠊⠀⣰⠃⠀⠀⣿⣿⣯⠉⠳⡄⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡌⠀⠀⠀⠀⢠⠇⠀⠀⠀⣿⣿⣿⠆⠀⣿⠀⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢱⡀⠀⢀⡰⠃⡀⠀⠀⣰⣿⠿⠋⠀⢀⠷⡀⠀⠹⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⠚⢋⣠⠎⠤⢴⡚⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠙⢶⠤⠄⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠈⠉⠉⠉⠍⠉⠀⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠘⠀⠐⠂⠀⠐⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠤⠄⠀⠀⠀⠀⠀⠚⠒⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠉⠁⣤⠠⠤⠄⠀⠀⠄⠠⠀⠒⠂⠀⢺⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠⡄⠈⠀⡄⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢸⠀⠀⠀⢀⠈⠁⠐⠀⠁⠀⠀⠀⠀⢸⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠸⡀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡰⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢹⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠄⠁⠒⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠇⠀⠈⠀⠄⠠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣀⣀⣀⣀⣀⣧⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣼⣀⣀⣀⣀⣀⣀⣀
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼
⣎⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⢈
⡧⠀⢀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⠀⠠
⠑⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⢀⠔
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⠓⠒⠒⠒⠒⠒⣶⠒⠒⠒⠒⠒⠒⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡖⠒⠒⠀⠀⠀⢲⠂⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠠⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠷⣄⠀⠀⠀⠀⠀⠀⠀⢀⠴⠥⠤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣦⠠⠀⣶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣿⠀⣀⡿⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠶⠖⠃⠀⢀⠀⡃⠀⠐⢒⠖⣦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠉⠄⢀⣠⠾⠁⠱⢤⣀⠠⠌⠀⠀⠀⢳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⠉⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠛⠿⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⢈⣷⡖⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠿⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⣠⣄⣴⣶⠿⣄⣀⡴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠐⠁⣿⡇⠀⢀⣬⡟⡿⠟⢡⡄⠘⣤⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠈⢁⢴⣾⣯⢤⡀⠀⠉⠀⠀⣸⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠀⠀⠀⠀⠀⢁⠀⠿⢻⣿⡇⠀⠀⠀⠰⠿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⢻⡆⠀⠀⠀⠀⣬⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⢿⠃⠀⠀⣀⣀⣸⡀⠀⠀⠀⢀⡆⢀⡀⠀⣼⡿⠋⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠠⠰⠿⣿⣿⡟⠃⠀⠀⣠⠟⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣆⠀⠀⣀⠆⠀⢻⣿⣷⣤⣤⣤⣤⣴⣶⣆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⠀⠰⣟⡗⠻⠈⣿⣿⣿⣿⠿⠿⠛⠋⠉⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⣜⡟⠱⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣶⣶⡤⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠒⢶⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣿⣿⣿
⠀⠀⠀⠉⠙⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⡁⠤⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠡⠂⣡⠔⢉⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⠀⠈⢸⠃⡀⠀⠈⠁⠀⣀⣕⣊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢣⠀⠣⡀⠀⠑⡄⠀⢈⡡⠄⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈⠖⠒⠺⡀⢀⠀⣠⠊⠉
⠀⠀⠀⠀⠀⠀⡠⠔⠁⠐⠀⠀⠸⠀⠀⠀⠱⠀⡿⡥⠊⡉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠡⡀⠀⣆⠀⠀⠀⡇⠘⠃⠞⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⡐⠱⣈⠀⡀⢸⠀⠄⠔⠈⠂
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠊⠀⠀⠀⠖⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢃⢁⠣⡀⢇⡸⠈⠔⢀⠠
⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢘⡖⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⠊⠀⢀⠘⡀⠀⠀⡁
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠄⠐⠄⠀⠀⠡⣰⡠⠧⠒⠂⠀⠀⠀⠒⢄⠀⠀⠀⠀⠐⡨⠠⠤⠜⠀⠂⠭⠤⠀⡀
⠈⠒⠀⠀⠀⠐⠂⠀⠉⠀⠀⣀⠠⡀⠀⢀⠀⠀⠀⢹⣊⣉⣀⣀⣀⣀⠀⠀⣨⠀⠀⠀⣀⠚⡁⠀⠀⠒⠂⠠⠁⠂⠊
⠀⠀⠀⠀⠀⢀⢠⠒⠠⣀⠘⡁⠠⠁⠄⠈⠆⠀⠀⢠⠀⠀⠡⠀⠀⠀⠀⠈⠁⢴⠀⠐⠙⠳⡄⢄⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⣭⠚⠈⣎⠈⡀⠈⠦⠔⠣⣀⠀⠀⠣⡔⢉⢄⣀⠠⠊⠻⢂⠀⠁⠈⠆⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢐⣉⠉⠅⢐⠄⡠⠊⡼⢡⡞⠠⠀⠀⣀⢁⠢⣄⡈⢺⠀⠡⠀⠀⠀⠡⣂⢢⠀⡀⢀⠌⠀⢉⢄⠀⠠⢀⠔⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡌⡔⠀⠆⠈⠀⠀⠀⠠⠄⡏⠉⠉⡨⣹⠒⠦⠇⠀⡠⢶⡁⠀⠀⠀⠀⢀⠌⠀⢳⠁⠁⡎⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠊⡰⠃⠀⠀⠀⠀⠀⠀⠀⣇⣀⣘⠄⣸⠒⠂⠀⠀⠀⠀⠈⠈⢁⠦⡠⠃⠀⠀⠀⠰⡳⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠁⢀⠴⠁⠀⠧⠃⠄⠀⣀⡠⠤⠐⠊⠁⠀⠀⠀⠆⡀⠀⢠⢇⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠒⠈⠉⢹⠹⠁⠀⠀⠀⠀⠀⠀⠀⢀⣨⠞⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⡄⢇⢀⣀⡤⠀⠐⠂⠉⠀⠉⡆⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠬⡚⠙⠉⠆⠀⠀⠀⠀⠀⠀⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⣤⣰⣾⡀⠀⠂⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠐⠒⠀⠈⠓⠓⠖⠖⠢⠂⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠴⠊⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⠤⢒⡩⠥⠐⠛⠛⠓⠢⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢀⡔⡡⠒⠁⠀⠀⠀⠀⢀⣐⢀⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⡾⠋⠀⠀⠀⢀⡠⠄⣀⡨⢠⢊⡵⠿⢬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⡐⠁⠀⠀⠈⠳⡱⠃⣀⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡜⢀⠔⠉⠉⠲⡀⢁⠜⠁⠀⠑⡄⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⠋⡰⠁⠀⠀⠀⠀⢘⡌⠀⠀⠀⠀⠘⡄⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⡜⢻⡄⠀⠀⡐⢳⡄⠀⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⣿⣿⣷⠀⠀⣿⣿⣷⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠺⡄⠀⠀⣿⣿⡿⠀⠀⢿⣿⡟⠀⠀⣬⠔⢊⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡴⢆⠀⠘⡤⠤⠼⣛⣯⣥⣤⣝⣛⠷⠂⠚⠀⢠⠋⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠥⠤⠜⠁⠀⠀⢠⣾⣏⣀⣼⣿⣿⣿⣷⠀⠀⠀⠈⠂⠚⠆⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠴⠄⠀⡀⠀⠀⠀⠻⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⠀⠐⠂⠀⢙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠼⡃⠀⢀⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠀⠀⢤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠐⠒⠂⠀⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠀⣤⣤⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⠙⣤⡄⠠⠤⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢱⡈⢣⡀⠀⠈⠁⠀⡠⠃⣼⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠑⢄⠉⠒⠤⠔⠊⢠⠎⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⢰⠀⠀⠑⠦⠤⠤⠚⠁⠀⡄⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⡶⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⢤⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠉⠁⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠔⠋⠁⠀⠀⠀⠛⠶⡌⢣
⠀⢀⠌⠀⠀⢐⠈⠲⣀⠁⠈⡆⠀⠀⠀⠀⣠⠞⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠃⠒⠤⡀⠀⠀⠀⠀⠀⠀⡀⠤⡀⠀⠀⠈⠐⢄⠀⢂
⠂⠘⠀⡠⠤⢘⣄⠀⢹⠄⠀⣧⠂⠰⠄⡼⠁⠀⠀⡇⠀⠀⠰⠇⠀⠀⠶⠀⠀⠀⡅⠀⠀⠀⠘⢦⠀⠀⠀⢀⢰⠁⠀⣡⠄⠀⢱⠀⠀⢱
⠀⠀⠀⠣⡀⠀⠙⠁⠀⠀⠔⠁⢠⠊⢀⡇⠀⠀⠀⡇⣤⠀⠀⠀⠀⠀⡄⠀⢠⡄⢧⠀⠀⠀⠀⠈⢇⠀⢇⠀⢽⡀⠀⠟⠀⣠⠂⠀⢄⠀⡇
⠀⠀⠀⠀⠈⠑⡄⠀⠀⠀⠀⢀⡌⠀⠈⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠘⡆⠈⡆⠀⠑⠀⠀⠈⠃⠀⢀⠜⢠⠁
⠀⠀⠀⠀⠀⠀⠈⠒⠠⠄⠀⠈⡑⠀⠀⡇⠀⠀⠀⡇⣤⠀⠁⠀⠀⠀⠀⠀⠀⣠⠀⢱⠀⠀⠀⠀⠀⣧⠀⠱⣄⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⠛⢄⠀⠀⣧⠀⠀⠀⡇⢸⣿⣶⣤⣤⣤⣤⣴⣿⡇⠀⠘⡄⠀⠀⠀⠀⢹⠀⢢⣀⠁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠑⠂⣿⠀⠀⠀⡇⠀⢿⣿⡟⠉⠉⠋⠙⠻⠁⠀⠀⡇⠀⢀⠤⣄⢸⠄⢊⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⣿⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢈⠀⣀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⡐⠉⢈⠀⠀⠀⠀⠀⠀⠇⠀⠀⠑⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢰⠇⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠸⠀⠀⠀⠀⠑⡄⠈⡄⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡇⠀⠀⠀⠀⠔⠁⡠⠁⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠃⠀⠀⠀⠈⡀⠀⠃⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢸⠀⠀⠀⠀⢀⠜⢀⠂⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⡆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢣⡀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡻⠦⣀⠀⠀⠀⠀⢀⡀⠰⡂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡈⠀⠀⠀⠀⠈⠀⠈⠁⠉⢂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠴⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢄⠁⠀⠆⢄⠀⠀⠀⠀⠀⡀⢠⠈⠈⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠈⠀⠈⠀⠀⣡⠀⠀⠀⡌⠀⠀⠁⠀⠃⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⡀⠤⢒⠭⠊⠀⠀⠀⠫⢕⠢⢄⠀⠀⠀⠀⠀⠁⢄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣓⣂⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣂⣈⣁⣐⣂⣐⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠶⢶⣚⣿⣿⣿⣿⣿⣿⣶⣤⣈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⡓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣭⣿⣿⣿⣿⠿⣿⣿⣿⣯⣭⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣿⣋⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣶⣶⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣠⣾⣿⣿⣷⣄⣤⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣷⣽⣿⣿⣯⣾⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠏⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


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

⠀⠀⠀⢰⣊⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⣀⠀⠀⠐⣠⣤⣤⣀
⠀⠀⠀⢈⣿⣭⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⠂⢀⢴⠖⠸⠛⠡⣴⣿⣿⣿⡄⠀⠀⠀⠀⢀⡤⡤⠀⠀⠀⢀⣤⡀
⠀⠀⠀⢨⣽⠴⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢰⠋⢀⡄⣠⣾⣿⣿⢉⡀⠀⠀⠀⠀⠀⠀⢰⢫⣿⣧⠀⠀⣼⣿⣿⡦
⠀⠀⠀⠰⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⠘⠂⣠⣾⣿⣿⣿⣿⣷⣿⡾⠀⠰⡶⠀⠀⢠⢥⡜⠋⠀⣸⣿⣿⠋⡀
⠀⠀⠀⠚⣿⣛⠀⠀⠀⠄⠀⠀⠀⢠⣭⣬⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢰⣳⠀⠀⠀⠀⠀⠉⣰⣿⠟⣡⣾⠟
⠀⠀⠀⣙⣟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⢀⣾⣄⣰⠟⢁⡼⠏⠁
⠀⠀⠀⣭⣯⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⣿⣿⣿⡏⠸⣿⡿⠀⠀⠈⡑⠀⢀⣾⣿⣿⣧⣴⢋⣀⣤⡄
⠀⠀⠀⢴⣷⠶⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣻⠇⠁⡈⠉⠁⠀⢻⡇⠀⠀⠀⠃⢀⣾⣿⣿⣿⣿⣿⣯⡉⠉
⠀⠀⠀⠶⡷⠶⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣴⣿⣤⡔⠋⠀⠀⠀⠀⢱⡅⢀⡜⠁⣼⣿⣿⣿⠏⠹⠿⠿⠷
⠀⠀⠀⠻⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄
⠀⠀⠀⣙⣛⣋⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⠿⠿⢿⡏⠁⠀⠀⠀⣤⠂⣿⣿⡿⡿⠿⠿⠿
⠀⠀⠀⣉⣋⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠄⢴⣾⣿⣷⣿⣶⡆
⠀⠀⢠⣬⣥⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠛⠋
⠀⠀⢠⣤⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣾⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠛⠋⠁⠀⠀⢸⣿⣿⡇⠀⠀⠈⠉⠻⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⢸⣿⣿⡇⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⢠⣶⣿⣿⡿⠋⠀⠀⢸⣿⣿⡇⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣴⣿⣿⡿⠋⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣦⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⣀⡀⠀⠀⢸⣿⣿⡇⠀⠀⢀⣀⣤⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⡀⢀⣀⠀⣀⠀⣀
⠀⠀⠀⠀⠘⡇⢺⣿⣏⣻⣿⣋⡇
⠀⠀⠀⠀⠀⠁⠈⠉⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠠⠈⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣤⣤⣌⡄⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠠⠟⠛⠉⠉⠉⠛⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠁⠠⠴⣶⣶⣶⣾⠀⢀⣖⠀⢀⣠⣴⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣠⣤⣤⡀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⠛⠛⠉⠁⠸⣿⡆⠀⠀⠿⠛⠉⠉⠻⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠈⠙⢦⠀⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠾⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠛⠃⠀⡇⠀⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠙⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠔⢁⠜⠀⠀⠀⢹⣿⣿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⣀⣀⠤⠖⠉⠓⠒⠈⠀⢀⣃⣠⡄⠀⢀⣼⣿⣿⠀⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠐⠒⠒⠒⢦⣤⣶⣿⣿⣿⠃⠀⠈⡛⠛⢣⢡⠀⣼⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣄⣀⢀⣤⣤⣴⣤⠛⠿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⢻⣿⣿⣿⣏⢹⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢳⡀⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⢻⣿⠟⠛⠋⠉⢹⣿⣿⣿⣿⣿⡀⠀⢀⣀⠀⠀⣻⠀⠈⠉⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⢀⡞⠀⠀⠀⠀⠀⠁⢨⠉⠉⠙⠙⢃⡐⠁⠀⠀⣄⡹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠻⡄⠀⠘⢧⣀⠀⣀⠀⠀⠀⠚⠀⠂⠈⠉⠉⠛⢴⠛⠉⠉⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢳⡀⠀⠀⠀⠀⡏⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⠀⠀⠉⠲⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠱⡄⠀⠀⠀⠉⠉⠀⠱⠆⠀⢰⠆⠀⡀⢀⡧⠁⡰⠃⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠙⠂⠠⠶⠣⠎⡙⡆⠠⠴⠾⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠄⢧⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⡸⢧⠀⠀⠠⠐⢄⠀⠀⢀⡀⠀⠒⠺⠂⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠑⠀⠈⡇⠀⠀⢸⠓⠄⠈⡇
⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢸⠄⠀⠀⢇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠑⠤⢤⣤⡤⠶⠖⢲⣤⡀⠀⣴⠀⠀⠀⢸⡀⣠⠶⠌⠇⠀⣀
⠀⠀⠀⠀⢠⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⠊⠁⠀⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⢀⣤⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢈⣯⣼⣿⣿⠃⠀⠀⡰⠋⣤⣤⠀⠀⠀⡀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⣿⣿⣿⡳⡀⠀⠀⠀⠀⣿⠿⠀⠀⠀⠅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠹⣿⣿⣷⢹⡄⠀⠘⠃⡠⢶⣾⣿⡑⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡈⠛⠿⠋⡅⠀⠀⢤⣀⢀⡛⠛⠃⢇⠀⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⡀⠀⠉⠛⠷⠰⠮⠯⠳⢤⣀⣀⣙⠶
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠂⠰⡆⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢠⠦⠀⠀⠀⠉⠀⠠⡄⠀⠀⠐⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠸⠂⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⣤⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠘⠃⠀⠀⠀⠀⢾⠆⠀⠙⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⢠⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠈⠁⠦⠀⠂⠀⣼⣿⣿⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣄⡀⠀⠀⠀⣺⣿⣿⣿⣷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣶⡄⠀⠀⢸⣿⣿⣿⣿⡟⢉⡻⣿⣿⣿⠡⠝⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⠀⢸⣿⣿⣿⣿⣷⣬⣴⣿⣿⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣧⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣯⠿⠛⠛⠉⠉⠁⣿⣿⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠘⣿⡿⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⣀⠀⠀⠀⠀⢀⠀⠈⠢⠄⠀⣀⣀⡀⠜
⠀⠀⠀⠀⠀⡀⠂⠩⠓⠈⠥⢀⠇⠀⠀⢻⣿⠶⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠂⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⠃
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⣀⣤⣤⣾⣷⣴⣦⣠⣦⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠙⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠄⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠛⠋⠛⠀⠀⠀⠀⠀⢿⠗⠀⠀⠀⠈⠁⠂⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠘⠻⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⡰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠈⠀⢀⠁⠀⢠⣆⣀⣀⡀⢀⣀⡀⠀⡁⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⢂⠀⠀⢀⠀⠆⠀⠀⠀⠙⠋⠉⠛⠛⢩⠃⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠴⣄⠴⠶⠒⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠜⠉⠉⠙⠒⠒⠒⠒⠚⠉⠀⠀⠈⢢⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠉⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣀⣠⡤⢶⡋⠀⠙⠲⠤⠤⠷⢦⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⢉⡟⠀⠀⠀⢻⡀⢧⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠋⠀⠀⠀⠀⠈⢷⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⠀⢀⣠⡼⠋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠎⠁⠀⠈⠉⠉⠉⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠉⠈⠉⠉⠁⠈⠱⠀⠀⠀⠀⠀⠠⣄⣠⣤⣤⠴⠶⠶⠶⠶⠞⠛⠛⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠶⢤⣄⡀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠈⣏⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠈⢈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠂⠉⠙⡛⠛⡉⣡⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠒⠒⠚⠓⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣜⠃⠀⠀⠀⠀⢀⣊⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠉⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠛⠃⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠋⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣤⣼⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠃⠀⠀⠀⠈⠉⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⠐⠟⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠄⠀⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠁⠀⠀⠀⠀⠀⠀⣩⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢺⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠳⡄⠐⠂⠀⢀⣤⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⠀⠈⠁⢀⠀⠀⢀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠿⠿⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡋⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠀⠀⠀⠀⠀⠈⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡁⠀⠀⢀⠀⣀⣠⣿⣿⣧⠀⠀⢀⣴⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⢀⣀⣀⠈⠉⠙⠿⠗⠒⠛⠛⠃⠀⠀⠀⣀⣀⣀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⡿⠃⠀⠀⠙⠛⠿⢿⣿⡷⠶⢶⣿⣿⣀⣀⣀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⢀⡴⠞⠉⠙⠛⣿⠃⠀⠀⠀⠀⠀⠀⢰⠉⢡⠀⠀⠉⠉⠛⠛⠛⠉⠉⠀⠉⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⢰⣿⠁⢀⠀⠀⠀⢻⣆⠘⣤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠀⠀⠀⣿⡏⢠⠛⠀⠀⠀⠀⢻⣦⡈⠙⠀⠀⠀⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿
⠀⠀⠀⢀⡞⠀⣼⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣀⠀⠀⠀⠁⠀⠀⠀⠰⣿⡇
⠀⠀⠀⣸⣧⠘⣿⡄⠀⠀⠀⠀⠀⠙⠛⢿⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⣿⡇⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣷⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡇⠀⠀⢈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠘⣷
⠀⠀⠀⢿⣿⠀⠀⠈⢹⣿⣿⡄⠀⠀⢀⠀⠀⠀⠘⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣷⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⠀⠀⠀⢿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠈⢻⣿⣷⡀⠀⣼⠄⠀⠀⠐⢊⡿⣇⠺⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠏⠀⠀⠀⢀⣷
⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢻⣿⣷⡀⠙⡀⠀⠀⠀⠈⢻⣿⠃⠠⡟⢻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠛⢻⣿⣿⣄⠀⠄⠀⠀⠀⠀⢿⣇⠀⠐⣿⣩⡿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⣀⣀⣀⣠⣾⡏⠀⠀⢸⡟⢠⣾⣿⣿⣿⣿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏
⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⢿⣿⣿⣿⡿⠋⠀⠀⠀⠛⠀⣾⣿⣿⡿⠋⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠛⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⢸⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢸⣿⣿⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⠇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣶⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣤⣴⣦⣾⣿⣿⢇⣀⠀⠀⣰⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠿⠿⠿⣿⣶⣶⣤⣶⣾⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠍⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⢿⣷⣆⠀⠀⠀⠀⠀⢠⣾⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢾⡿⠿⠛⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣖⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⣴⠋⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠉⢳⡄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⢻⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⡀⠀⠀⠀⢀⣠⡴⢻⡟⣿⠀⣿⣿⠷⣤⣀⠀⠀⠀⣀⣤⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢯⣉⠙⠻⣿⡏⠁⣰⠟⠀⣿⠀⣷⠘⣧⡈⣽⣿⠛⢋⣩⠟⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠙⠳⠶⠤⠶⠟⠁⠀⢀⡏⠀⣿⠀⠈⠻⠦⠤⠶⠛⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠸⣇⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⣠⠾⠁⠀⠀⠀⠙⢧⣄⣀⣀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢋⣡⡤⡴⠶⠒⠒⠶⢦⢬⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠉⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠉⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⣤⣄⣀⣇⣀⣀⣸⣤⣤⣤⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠋⠁⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⢀⣄⠀⢠⠋⠀⣀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢦⡀⠀⠀⠀⠀⠀⠀⣾⣿⣋⣀⣸⣿⣧⠈⢆⠘⢿⡇⠛⠛⠋⡞⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣶⣶⣿⡄⠀⠀⠀⠀⠸⠻⣟⠿⣿⡿⡷⠚⠇⠀⠑⢸⡇⠀⡠⠴⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠟⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡦⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⢀⣴⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠀⣼⠟⣡⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢘⠀⠀⠀⢀⣀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⣧⣾⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⠀⠈⡙⠑⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠐⡟⢿⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⣈⡐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⠀⢀⠞⠇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⡀⣿⣿⣿⣿⠿⠿⠿⣿⣿⢿⣿⠿⠿⠂⠀⠸⠿⠇⠋⢠⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠻⣿⣷⣦⣀⣴⡷⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡴⠁⠀⣠⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⢀⠔⠊⠀⣠⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣀⣀⡤⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠽⠄⠠⠚⠗⠐⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⣀⠀⠀⢀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠙⡦⠀⠌⠙⠁⠓⠁⠀⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠐⢲⠀⣀⠀⢀⡄⡄⠤⠀⠀⢀⡴⠋⠀⠤⢐⣶⠁⢸⣇⡀⠀⡄⠀⠀⠀⠳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⡁⠠⠄⢀⣅⡄⣀⠀⠀⠀⠀⠀⢸⣄⢺⢤⡈⠁⠀⠀⠉⠀⠀⠙⢂⣄⡀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⣉⠑⠊⢀⣈⡁⣀⡀⢀⡀⠀⠀⠀⠀⡟⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⢳⠮⠄
⠀⠀⠀⠀⠀⠀⠰⠙⠃⠮⠍⠀⠑⠾⠍⠑⠛⠁⠈⠀⠀⠀⠀⠀⠚⠃⠀⢤⡴⣾⣦⠀⠀⠀⠀⠀⠀⣀⡬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⠀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠴⠋⠁⣀⣀⣀⠀⠀⢶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣲⣿⡯⢟⣥⣶⣶⣿⣿⣿⣿⣿⣷⣶⡘⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡫⡹⣯⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠷⠘⢫⠀⣠⣴⣦⠀⠀⠀⠀⢤⢀⣄⢠⣶⣷⣶⣿⣿⣟⣿⣿⡟⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⢸⡟⣿⣿⣿⣿⠀⠀⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣟⣿⡟⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠃⠀⠀⣾⡇⣿⣿⣿⣿⡀⠀⢿⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣭⣭⠑⢃⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣿⠁⣹⣿⣹⣿⣧⡆⢸⡇⠸⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣿⣿⣿⣽⣭⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠠⠀⠀⠀⠀⠀⠀⠈⠛⠀⣿⣿⡿⣌⣿⣿⣸⠷⠴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡾⢿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⠿⣿⣿⣿⡀⠰⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⣧⠀⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡛⠻⢷⡾⠁
⠀⠀⠀⠂⠈⠀⠀⠠⠀⠀⠄⢀⣀⣠⡤⢶⡙⠯⠒⣄⠀⠀⠀⠀⠀⠀⠀⣼⠇⠶⠃⠎⣼⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣍⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⢀⣼⣿⣿⣦⡀⠀⠀⠀⠀⣿⣾⡇⠀⠀⣿⡩⢏⣿⡯⠉⢉⣽⣾⣷⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⡇⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠇⠀⠀⢀⣼⣿⢧⡀⠀⢸⣿⣠⠟⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣶⠁⣼⡄⢰⣦⣤⣠⠤⠒⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠈⠈⠻⢆⠀⠀⠈⣶⡇⣠⣾⡿⠟⠛⠍⠙⠛⠛⠛⢻⣿⣿⣿
⠀⠀⠀⣼⡟⠀⠉⠀⢦⠙⠀⠠⠀⣼⣿⣿⣻⣾⣷⣶⣯⣄⣀⡀⠀⠀⠀⠀⢀⣦⣤⣤⣤⡿⣹⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⣿⣿⣿⣾⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⣿⣟⣯⣿⣿⣿⣷⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⠉⠉⠉⠉⠉⠉⠛⠿⣿⣿⣾⣿⣿⣿⡿⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠟⠓⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⣦⠀⠀⠀⠀⠀⠀⠀⠒⠠⢧⣼⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠒⠂⠀⠀⠀⢀⡀⠀⠀⠠⠔⠂⠁⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠺⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡏⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣧⣿⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠠⠄⠸⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⣭⣿⣿⣿⣿⣿⣿⣷⣿⣿⢿⣱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡾⣷⣷⣄⠀⠀⠀⠀⠀⠀⢀⠄⠀⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡃⠀⠀⢀⡴⣻⣿⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣦⣤⣾⣿⣷⣿⠀⣰⣿⣾⣿⠃⠀⢀⣠⣴⣾⣭⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣛⣓⣶⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣧⣴⣾⣿⣿⣿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠋⠙⠛⠛⠛⠻⠿⣿⣟⠿⣿⣖⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠋⢁⣀⣀⣀⡀⠀⠀⣠⡴⠒⠋⠉⠉⠛⠲⢦⣈⠓⢤⡉⠉⠉
⠀⠀⠀⠀⠀⠀⣰⡿⣫⣾⣿⡟⢁⡴⠋⠉⠉⠀⠉⠉⠙⠺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠙⢄⠀⠀⠀⣠⡶⠾⠿⠷⣶⣄
⠀⠀⠀⠀⠀⣴⢫⣾⣿⣿⡏⢰⠋⠀⠀⠀⠀⠀⠀⣀⣀⠀⠹⡄⢀⡀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠈⠳⣠⡞⠁⢀⣤⣤⣄⡀⠙⢷⡀
⠀⠀⠀⠀⢰⣿⡿⢻⣿⡿⢀⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⣷⣿⣿⡆⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠙⠀⠀⠸⠟⠁⠙⢿⡄⠘⡇
⠀⠀⠀⠀⢼⠋⠀⣾⣿⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣾⡈⠋⠁⠀⠀⣀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡟⠀⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠉⣛⣻⣿⡿⢿⡿⠷⠶⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣧⠀⠀⠀⠀⠀⠀⢀⣠⣾⣯⠶⠚⡏⠉⠁⠀⠀⠈⣇⠀⠀⠀⠀⡇⠙⢦⠀⠀⠀⢶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠻⣦⣄⣀⣀⣤⢶⡿⣟⠁⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⣄⠀⣀⣠⣶⣤⣼⣷⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠇⠀⠀⠉⠉⠁⣠⠋⠀⢸⡀⠀⠀⠀⠹⣦⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢻
⠀⠀⠀⠀⠀⣼⠋⠀⣀⣀⠀⠀⠀⠀⠀⡼⢹⠀⠀⠈⢇⡀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⡇
⠀⠀⠀⠀⢰⡇⠀⣼⡟⠁⠀⠀⠀⠀⢠⡇⠘⣆⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠸⣧⠀⢹⣇⠀⠀⠀⠀⠀⠀⣷⣀⣸⣿⣿⣿⣿⣿⣿⣿⠟⢫⠟⠁⠀⠀⠀⠉⢻⣿⢿⣿⡟⠙⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⢻⣦⠀⠙⠷⠦⠀⢀⣀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠘⠀⠀⢀⡤⠶⣄⣟⠀⠀⠹⣷⠀⢸⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣤⠶⠋⠙⣆⠀⠹⣿⣿⣿⣿⣿⣿⠿⣤⡶⠞⠛⢷⣹⡄⠀⠈⢿⡄⠀⠀⢻⣆⠜⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠹⣿⣿⣿⢿⡇⠀⠘⡇⠀⠀⠀⢻⣧⠀⠀⠈⢷⡀⢀⡴⠋⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠙⢿⣇⠀⠙⣆⠀⠘⣄⠀⠀⠀⠻⣆⠀⣀⠬⠓⢁⣠⠞⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠳⣄⣈⣷⣄⣈⣧⡤⠤⠖⠛⢉⣤⠴⠞⣋⣤⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠉⠁⠀⠀⠀⠀⣀⣀⣤⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠲⠶⠿⠿⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶⡄⣿⡆⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢯⡟⠚⠁⢱⠖⢊⠕⠊⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢆⠉⠀⢀⡤⠊⣠⣺⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⢰⣋⡷⢏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡊⢀⣾⠿⠞⠉⠉⠁⠈⠐⢲⣤⡄⠀⠀⠀⠀⠉⢣⡑⢧⠉⠛⣷⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣴⣾⣿⡞⢁⡀⠀⢠⠞⠋⠀⠀⡉⢉⡱⣄⠀⠀⠀⠀⠘⡆⠀⠁⣰⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⡿⠁⠀⠀⠀⢶⠀⠀⠑⠺⢅⠈⢦⠀⠀⢀⣴⡝⢼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠁⠀⠀⠀⢀⣠⠀⠀⠛⠒⠉⠀⢸⡆⢀⣾⣿⣿⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⠀⠰⠣⠏⣤⠤⠄⠐⣒⣻⠁⢸⣷⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄⠀⠀⠘⢿⣿⣿⣿⠏⠀⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⡐⢄⡀⠉⠉⣠⢊⣾⣿⣿⣿⣿⠟⢀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣶⡮⣍⠭⣴⣿⣿⣿⣿⣿⡗⡾⣿⣱⣷⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⠇⠈⠑⠪⢏⠈⠏⠉⠛⠛⠛⠛⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠉⠚⡀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⠿⠿⠿⠿⣿⣿⣿⣿⣧⡤⡤⣄⣀⠀⠀⠀⠀⠒⠒⠀⡤⠐⠤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠟⢸⢡⠁⠃⢩⢛⢶⣄⡀⠀⠀⣀⣒⡀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⣆⡆⠀⠀⠌⡟⠈⠈⠿⣤⢾⠉⠈⡹⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡟⣻⠀⢰⠰⠰⠀⢀⣰⠌⠆⠀⣃⣵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⢣⢇⢇⡇⠇⠀⣧⠾⠁⢸⣼⡾⡏⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠻⠀⠘⠸⡘⡸⣧⠞⢁⣤⠶⢏⠻⢱⢹⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠘⡀⠃⢇⠁⣧⣗⡽⠞⠙⣧⡀⠀⡀⠌⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣷⢃⠘⣧⡾⠋⠁⠀⠀⠀⠈⠻⣦⡳⠀⢀⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣾⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⣿⣶⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠈⠉⠛⠛⠛⠁

⠀⠀⠀⠀⢠⠾⠗⠀⠀⠀⢰⣤⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⠺⠦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡄⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠐⠀⡀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡄⠀⠀⠐⡄⠀⠀⠀⠀⠀⢙⠛⠋⠁⠈⠑⣤⠂⠈⠉⠛⡂
⠀⠀⠀⠀⠀⠀⣤⠀⢡⡤⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠆⠀⡾⠂⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠰⠰⠆
⠀⠠⠆⠀⠀⠀⢀⠠⠓⠤⠀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠⠛⠦⣀⣀⠀⠁
⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⢀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⠉⠀⠐⠒⠀⠀⠉⠶
⠀⠀⠀⠀⠈⠢⡀⠀⠀⢈⠴⡄⠀⠀⠀⠀⠀⠀⠀⢁⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠴⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡧⠀⠀⠀⠀⣤⡀
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡁⠀⡀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠉⠻⠀⡀⠉⠙⠄⠀⠀⠀⠀⠀⢴⢶⣤⣄⣀⠀⡀⢀⣠⣤⡶⠶⠒⠀⠀⠀⠀⠀⠐⠁⣶⡶⣤⠱⢪⡼⠦⠖
⠀⠀⡆⠀⠀⠀⠀⢊⢀⡀⠀⠘⡀⠀⠀⠀⢀⠘⡄⠀⠙⠋⢙⠛⡙⠛⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⡦⢀⡀⣽⣿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠔⠀⠀⠀⠊⢇⠀⠈⠀⠀⠀⠁⠀⠈⠀⠀⠈⠀⢀⠄⠀⠀⠩⡁⠀⠐⠨⠭⠐⠁⠁⠂⠀⠊⠀⡠
⠈⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠈⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠑⣄⡀⠀⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⢉⡐⠀⠠⠀⠀⠄⠐⡂⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠹⠀⠀⠈⠁
⠀⠀⠀⠳⣰⠐⠀⠀⠰⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠠⠔⢂⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣬⠄⣀⠤
⠀⠀⠀⠀⠈⠁⠒⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠞⠛⡁⠀⠀⠀⠘⠛⠆⠀⠀⠀⠀⠀⠀⠰⠛⠋⠀⠀⠀⠀⣤⣤⣄
⠀⢩⠉⠙⠿⠷⢶⡶⠾⠿⠉⠉⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠠⠆⠀⣄⠀⠤⠖⠀⠀⡄
⠀⠀⠃⠄⠀⠤⠊⠈⠂⠀⠄⠐⠁⠀⠀⠀⠤⠀⠀⠀⠀⠘⠃⢸⠘⠀⠀⠠⠀⠒⠂⠀⠀⠈⠂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠂⠁⠁⠀⠀⠁
⠀⠀⠀⢀⡠⠔⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⣀⠴⠒⠉⠁
⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿