Warmly Emojis & Symbols

⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠐⠀⣤⡈⢂⠀⠀⠀⠀⡆⡤⠶⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠆⠈⣠⠄⠀⡏⢡⢘⢸⠇⠀⠟⠆⢀⠇⠀⡔⢂⠙⠂⡀⠘⠁⡀⠘⢂⡠⠀⠀⠀⢀⠎
⠀⠀⠃⠀⢡⣄⠀⡇⡌⠘⢸⠀⢠⢟⠦⢸⠀⠀⣥⠈⠠⡀⠉⢢⠀⠉⢂⠌⠡⠀⠀⠀⠌
⠀⠀⠠⠀⡽⠢⡀⣷⠁⠀⠸⣀⣐⠭⠅⢸⡀⠀⠀⢠⠈⠑⣤⠊⠑⡄⠈⢠⢴⠀⠀⠐⠖⠤⠤⢄⣀⡀
⠀⠀⢾⣉⢉⣠⢀⡉⠀⠀⠀⢣⢠⡴⢰⣤⠇⠀⠈⠀⠰⠀⠀⠡⡀⠀⠠⠀⠙⠀⢸⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒
⠀⠀⠘⠿⠟⠛⠚⠃⠀⠀⠀⠈⠙⠋⠉⠉⢀⠔⠚⣧⠀⠀⠃⠀⠈⠃⠀⡱⢷⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢸⠭⢷⡡⠓⠒⠒⠒⠒⠲⢾⣳⣔⣷⠀⠀⢀⡤⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⣸⣒⣚⣒⣒⣒⣒⣒⣒⣂⠤⠿⠤⢆⠀⠀⣼⢹⡑⡾⢡⡆⢀⡀
⠀⠀⠀⠀⠓⠲⠶⠶⠶⠦⠤⠤⠞⡠⢄⠀⢯⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⣿⠀⢰⣇⠎⣴⢃⠶⢠⢍⡐⢡⢆⡆
⠀⢐⠉⣉⣩⣉⣉⣉⣉⠉⠉⠉⠉⠀⠈⡄⠘⣦⣄⣠⠀⠀⠀⠀⠀⠀⡀⣀⣠⠇⠀⡼⣉⠃⠲⣬⣈⠓⡾⢴⢉⡜
⠀⠀⠐⠄⠀⠀⠄⠀⠈⠉⠀⠈⠉⠉⠁⢑⠀⠸⡼⠀⠀⠀⠀⠀⠀⠀⠀⢅⡇⢀⡜⠁⠆⡍⡒⠢⠮⣌⢁⠆⣸
⠀⠀⠀⠀⠀⠀⠀⣠⢴⣶⡆⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠄⣅⡒⠰⢡⠜⡰⠢⢌⢹⠏
⠀⠀⠀⣠⣶⡶⠟⣺⣿⡟⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠤⢒⡔⠀⠂⠄⠙⠒⠰⠤⣅⡂⡌
⠀⠀⠀⠈⠓⠿⠿⠗⠋⣀⣀⡀⠀⠀⠀⣘⡄⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠐⠊⠀⠀⠀⠀⠁⠠⠀⠀⠀⠉
⢀⠀⢀⣤⡶⣄⠀⠖⠰⣏⢝⣻⠗⠲⠀⠨⠽⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⢄⢹⣿⣿⣿⡄⠀⠀⠈⠙⢏⠀⠀⠀⠀⠀⠀⠈⡕⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀
⠀⠈⢺⠉⠉⠀⠰⡀⠢⡀⠀⠀⠉⣉⠖⠌⠁⠤⠊⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠀⠈⠣⠀⠀⠀⣀
⠀⠀⠈⡄⠀⠀⠀⠳⢀⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠢⠊⠀⡠
⠀⠀⠀⠰⡀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⣀⣄⠠⠤⠐⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠑⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠇
⠀⠀⡄
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⢸⡃
⢠⠀⢸⡇⠀⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠣⠄⠀⠐⢀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⡀
⠰⠀⠀⠀⠀⠀⠀⣸⡇⠀⠐⢀⠀⠈⠀⠙⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠇⢰⠀⠀⠈⠁
⠀⡄⠀⠀⠇⠀⠀⡿⢐⠀⠁⠀⠀⠀⠀⠠⠩⢀⣠⣠⡆⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠀⠀⠀⠈⠀⠉⠠⡀⠀⢰⠀⠀⡀
⠈⠁⠀⠀⡀⠀⢈⡇⠀⢄⠀⠀⠀⠀⠀⢀⠠⠈⠛⠛⠀⠀⠀⠀⠀⠑⠀⠀⠀⡖⠀⠀⠉⠀⠀⠈⠁⠀⠔⠂⠀⠀⠀⠀⠀⡌⠀⡈
⠀⠀⠀⠨⣯⠀⠀⡇⠀⠬⠇⠐⣢⠀⠀⠀⠉⠀⢠⣧⣌⣡⠈⠛⠀⠶⠒⡀⠀⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢀⠀⠌
⡴⣶⡀⠀⢹⡆⠀⣇⡤⠀⠀⠈⠛⠀⠀⠀⠀⠀⡀⠙⡟⠁⢐⠂⠀⠀⠻⠄⢀⠀⢠⣀⠀⠄⣤⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⢨⠂⠀⠂
⠳⠸⠃⠀⠈⢻⣶⠟⢡⠎⠀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠁⠀⠨⠄⠂⠛⠁⢁⡁⠀⠚⠀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂
⠛⠁⠀⠀⠀⣈⠁⢠⡋⡠⠂⠀⣀⡞⠉⠉⢩⡉⢉⠉⣉⢑⣖⠂⠀⠀⣤⠖⠒⠤⠤⡄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⢂⠤⠁
⠀⠀⠀⠰⠄⠈⠁⠀⠁⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⣤⠀⠈⠀⠀⠀⣧⣴⣆⢀⡤⢃⡮⣽⠿⣯⠴⢤⡤⢀⣀⣀⢀⠀⠁⡀⠀⠘
⠀⠀⡀⡀⡀⠀⠀⠘⠛⠁⣄⠀⠀⠈⢄⢀⠤⠀⠀⡏⢈⠓⠙⠲⣦⡘⢽⣟⣇⠀⠀⠀⡽⠛⡶⠀⢀⣾⠀⣻⡿⠿⡎⠓⠀⣀⣐⠂
⠞⠀⠈⠀⢱⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠰⣎⣼⡁⠀⠀⠠⢤⡈⢡⣍⠀⠀⠈⠀⠴⠿⠀⠺⡷⠋⡹⠀⠀⠇⡀⠾⠋⠉⢙⡶
⠀⠀⠀⠀⠔⠙⠋⠀⠀⠂⠀⠀⠀⠀⠀⠀⠰⠃⢰⠀⣼⡿⠓⠗⠀⠀⠁⠀⠘⡆⠀⠀⠀⡰⠀⠀⠀⠀⢀⣀⣘⡁⠀⢀⠃⠀⠀⢀⠐
⣠⣶⣿⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠙⠀⠈⠙⠛⠁⣴⣧⣀⠠⡎⠐⠠⢷⣐⡠⠐⠋⠁⢠⣠⣶⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣦
⠻⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⣇⢡⠁⢀⡀⡠⠀⠿⠛⠀⠤⠄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠻⠀⢊⡌⠀⢳⠀⠀⠀⠀⠂⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠙⢿⣿⣿⣿⣅⣀⡀⠀⠀⢄⡄⠀⠀⠰⠂⠀⠀⡀⠀⠁⠀⡈⠉⣀⠈⠀⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢋⠉⠙⠛⠟⠿⣦⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠂⠤⠀⠀⠀⠁⠀⠀⡀⢠⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠂⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡴⠛⠛⠛⠛⠛⠛⠻⣤⣤⣤⣤⣤⣤⣿⣿⣿⡟⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡟⠁⠀⢐⠀⠀⠀⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣄
⠀⠀⠰⠿⠿⠿⠿⠿⢿⣯⣤⣤⣤⣤⣡⣤⣠⣤⣤⣤⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣄⣿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠋⠉⠉⠈⠉⠉⠀⠀⠀⠀⠉⠉⠉⠓⠲⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀⠀⠀⠀⠀⠀⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠐⠊⠐⠀⠐⠂⠀⠀⠀⠀⠀⢸⡀⢆⠀⠀⠀⠀⡀⠀⠀⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⣦⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠹⡀⠘⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣸⠟⢦⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢸⡌⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣦⠮⣈⠑
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠈⣴⠀⠈⠃⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣤⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⡆⠀⢠⡶⠲⣞⠿⠻⠛⠀⠀⠀⠀⠀⠀⠀⢀⡿⢹⠀⠻⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⠀⠀⠉⠢⠹⣄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠂⠀⠀⠀⠀⢀⣠⡴⠒⠁⢀⣀⠤⠴⠞⠋⡁⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠄⠰⠀⢻⣿⣶⠞⠈⠁⠀⢀⠖⠀⠀⠃⠀⠀⠀⠀⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⡀⠀⢀⠄⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣨⠶⠧⢿⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢰⢟⠷⠛⠆⣀⠐⠃⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⣀⠙⠛⠛⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⡥⢷⣀⡔⠊⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠘⣧⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⢴⡿⠶⣞⣉⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣠⠿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⢠⠟⣹⠋⠉⠛⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣰⠏⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢀⠀⠊⠁⠀⠈⠂⠀⠉⠲⢄⡀⠀⠀⠀⠀⢠⡄⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⡼⠋⣰⠟⠀⠀⠀⠀⠀⣤⠀⠀⠀⣾⣧⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣤⣤⠴⡿⢻⡆
⠀⠀⠀⠀⠀⢠⠞⠀⠀⡾⠁⣰⡟⠀⠀⠀⠀⠀⠀⠘⢷⡀⡰⠀⢿⣧⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢿
⠀⠀⠀⠀⠀⠃⠀⠀⢸⡇⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣸⣿⠟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⡇
⠀⠀⠀⠀⢸⠀⠀⠀⡸⣧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡟⠀⠀⠀⣿⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⢹⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠆⣗
⠀⠀⠀⢰⡇⠀⠀⢠⣼⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡣⣶⣿
⠀⠀⠀⠀⡇⢀⣴⠋⠙⠷⣄⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡹⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣔⠄⣂⠀⠀⠀⠀⠀⢤⠀⡀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠌⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⣴⣶⣶⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣛⣋⣹⣿⣷⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⣲⣶⣾
⠀⠀⣰⣤⣿⣿⣿⣿⣿⡆⣤⠀⠀⠐⣿⣿⣿⡟⡉⣻⣿⡟⠿⣿⣿⣿⣿⣦⣀⡂⠠⣤⠀⠛⢿⣿⣽⣿⣿⢏⠁⣿⣤⣶⣶⢨⣍⣿⣿
⢀⣴⣿⣿⡟⣉⢉⣿⣿⠁⠙⠀⡤⠤⠼⠋⠩⣤⣄⡀⠀⠚⠀⠻⣿⣿⣿⣿⣿⠁⠀⠆⠀⣹⡄⠛⣯⣉⣼⣄⣼⣷⣤⣸⡏⢹⣿⣿⣿
⠺⢿⣿⣿⡄⠀⠈⣿⠃⠈⠓⠀⠀⠀⠀⠀⣰⣿⣿⠗⠈⠀⠀⢦⣼⣿⣿⣿⡁⠀⠀⠀⢸⣿⣷⣿⣿⣿⠿⠛⠉⠉⠀⠈⢡⣼⣿⣿⣿
⠀⠀⢀⠙⣿⣶⣿⣿⠀⠀⠀⠀⠀⠁⠀⣸⣿⣿⣽⣄⣦⡀⣰⣾⣿⡿⠻⠿⠃⠀⠀⠀⣾⣿⡿⡏⣿⠃⢀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿
⠀⢰⡌⠇⢻⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣠⢤⠀⠀⠀⠀⠰⣿⣿⣷⣁⡟⠀⣠⠆⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⣏⡟⣠⣾⣤⣟⣿⣿⡄⠀⠀⣀⡀⠀⢹⣿⣿⣿⣯⣸⡏⢣⣤⡘⣃⣸⡇⢸⡛⠀⠀⢿⣿⣿⣿⣧⣾⡇⠀⠀⢠⣇⠀⠀⠀⢿⣿⣿
⠀⢟⣠⣿⠟⢹⣿⣿⣿⣿⣶⠟⢻⠁⠀⢸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠃⠀⠈⢆⠀⣾⣿⣿⣿⡏⠿⠀⠀⠀⣾⣿⠀⠀⠀⠘⣿⣿
⠀⠀⣿⠟⠀⢸⣿⣿⣷⣼⡇⠀⠘⣦⣀⣼⣿⣿⣿⣿⣿⢹⠟⠛⠿⣿⣿⠈⠀⠀⠰⠀⡇⣿⣿⣿⠀⠀⠀⠀⠀⢹⡿⠀⠀⢁⠀⣸⣿
⠀⠀⠋⠀⠀⣽⣿⣿⣿⣿⡆⠀⠉⢯⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⠈⢹⡿⠟⠀⠀⠀⠀⢠⣼⣴⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡏⣿⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡀⢱⣼⣶⡴⡆⠀⢸⣷⡆⠀⠀⠀⠀⠈⠻⠏⠛⠉⠁⢀⠀⠹
⠀⣀⠀⠀⠀⣿⣿⣿⣇⣿⡧⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡷⢰⡀⠀⠀⠑⠈⠛⢿⣿⡀⠀⢾⣿⠄⠀⠀⢠⠀⠀⠀⣤⡆⠀⠀⠀⠹⡇
⢀⣿⣷⣦⣄⣿⣿⣿⣿⡇⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣧⠀⡇⡄⠀⠀⠀⠈⠠⠌⠓⢤⣾⣿⠀⠀⠀⢸⠀⠀⣸⡇⠀⠀⣾⣴⣤⣾
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⢸⡇⠹⣆⠀⠀⠀⠀⠀⠈⠳⠿⢿⡀⠀⠀⢸⡆⠀⠈⣥⣤⣴⣿⣿⣿⣿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⡇⠀⢻⠶⠄⠀⠀⠰⢀⡈⢂⣈⠟⠀⣶⣿⡇⠀⠀⣽⣿⠿⠿⠿⠻⠿
⠀⠀⠀⢹⣿⢿⣿⠟⠉⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡇⠀⠀⠀⠸⢤⣴⣾⣿⣿⣿⣇⠀⠀⢹⣿⡇⠀⠀⢿⣿
⠀⠀⠀⠸⣿⢸⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⡟⢿⣦⣸⣿⡇⠀⢀⣸⣿⣄⣤⣤⣿⠿
⠀⠀⠀⢠⣿⣼⣿⣄⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⢿⣿⡛⠷⠞⣫⣿⠿⠷⠚⠉⠉⠉⠉⠁
⠀⠀⠀⠸⡿⠛⠳⠿⠁⠀⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣤⣤⣿⣿⣆⣈⣉⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢻⣿⣿⣿⣶⣿⣿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣿⣧⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⢿⡿⣿⣿⣽⡗⠿⠯⠿⠛⠓⠚⠛⠛⠛⠛⠛⠋⠙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠶⠶⣶⣴⣶⣶⠀⠐⠂⠶⢶⣶⣶⡶⠒⠲⢦
⠀⠀⠀⠀⣀⣬⣳⠎⠉⠀⠀⠀⠀⠀⠀⡠⢿⠟⠻⠄⠶⠞⠛⢫⡽⠻⡤⢤⡀⠀⠈
⠀⠀⠀⠀⠠⣴⣿⠐⡷⢤⠄⠀⠀⠀⠀⠁⢸⠀⠀⠂⠀⠀⠔⠸⠇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠸⠛⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠅
⠀⠀⠀⠆⠈⠉⠑⠶⣄⡀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣰⣶⣀⡀⠀⠀⠀⠀⠄⠀⠀⠀⣠⣦⢴⡄⠀⢿⣾⠃⠀⡀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⡀⠀⣴⣠⠿⣟⢒⠀⠀⠾⠇⢠⣀⣷⠀⠀⠀⣠⣴⠋⠉⠋⠙⠙⠟⠿
⠀⠀⠀⣿⣿⣿⣏⣤⠾⠉⠀⠀⠀⠀⠀⡇⢸⣴⣿⣿⠀⠀⢠⣶⣾⣿⣿⣦⣠⣾⣿⣿⣧⡀⠀⠀⠀⠀⠠
⠀⠀⠀⣽⣏⢙⡟⠛⠶⣲⣶⣶⠇⢀⣾⠀⠈⣿⣿⣿⣤⣤⣤⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣒⣶⣤⣤
⠀⠀⠸⣿⣿⣾⣿⢿⣗⣚⣿⡯⢠⣼⣇⠀⠀⣟⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⢰⣿⣿⣿⣿⣏⣿⣿⣿⡇⠸⣿⣿⣤⣼⣿⢿⠟⢿⡸⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣟⣍⢿⠃
⠀⠀⠀⣿⣿⣿⡟⣿⣿⣿⣿⣇⠀⣿⣻⡿⢿⣿⣾⣷⡄⢳⣽⣿⠿⣿⣿⣿⢿⣿⠃⠿⡇⢨⣿⣿⣿⣿⣾
⠀⠀⠀⣿⣿⠉⠁⠛⠉⢻⢹⡟⠳⢿⣴⣿⠚⢿⠉⠛⡏⠙⢉⣧⠀⠉⡉⢷⠀⢹⠀⠀⠖⠋⠉⣿⣿⣟⣿
⠀⠀⢀⣿⣷⡖⣾⢡⡴⣾⡾⣄⣀⣙⡏⠻⡆⠀⢆⠀⠃⠀⣻⡶⡇⠘⠀⠀⠀⢻⠀⠀⣤⣀⣴⠎⠈⢿⡙
⠀⠀⢸⣿⣿⡷⣿⣾⣿⡿⠱⣿⣿⣿⠀⠀⠙⣀⠈⡄⠀⠀⠘⠃⠈⡆⠀⠀⠀⣸⠀⠀⣿⣿⡏⣶⠀⣾⣹⡂
⠀⠀⠈⠋⠉⠁⠁⠈⠉⠁⠀⢹⣿⡏⠀⠀⠀⢻⢻⡏⠀⠈⠈⠃⠘⠃⢸⡄⠀⢸⠀⠀⠉⠀⠀⡏⠲⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣼⣧⡀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⣀⡤⢻⣀⣀⣀⣀⣀⡀⠀⢸⡇⠘⣧⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡿⠚⠛⠉⠉⢁⣭⠻⢿⡋⠉⠀⠀⠀⣼⠀⠈⡟⣩⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⣀⣄⢀⢀⣀⣠⣄⣀⣄⣄⣀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠶⠶⠶⢿⠆⣠⡿⠿⠋⠉
⠀⠀⢰⣿⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⣤⣄⡀⠀⠀⠉⠁⠉
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣄⠉⠄⠀⠀⢀⠈
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⢶⠤⣶⡞⠃⠂
⠀⠀⠘⠷⠾⠮⠥⠈⠛⠿⠿⠛⠛⠻⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠿⠟⠛⠒⠔⠪⠟⠀⠰⠤

⣿⣿⠿⣿⡿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣯⣭⣿⣯⣭⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡎⠙⠛⠙⢿⠟⠛⣷⣦⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀⣾⢇⣀⠈⣿⡿⢻⣦⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣵⣶⡄⢀⠀⢴⡄⠀⣻⣿⣿⣇⢾⣿⣧⡀
⠀⠀⠀⠀⣸⣿⣿⣟⣿⣿⣿⡟⣿⣶⣾⣧⣶⣭⠟⢿⣿⢿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠁⠀⢻⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⢀⣤⣼⠗
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢶⣾⠉
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣴⣿⣿⣦⣤⡀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠉⠉⠉⢿⣿⠶⣶
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣤⣄⠀⠀⠀⠀⠀⠹⣶⣿⡀
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠻⠿⠃⠀⢠⣤⡀⠀⠙⠻⣷⡀
⠀⣾⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⡀⠀⠘⠿⠋⠀⠀⠀⠈⣿⣦
⣴⣿⣿⣿⣿⣧⣤⣿⣿⣿⣧⣀⣉⣿⣿⣿⣿⣾⣿⡟⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠛⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣯⣉⡛⠻⣷⣄⢠⣶⣤⣀⣼⣿⠇⠀⠀⠀⠀⠘⢷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⣿⣿⣤⣸⣿⣿⣟⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡄
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⣿⣿⣿⣿⣿⣯⣽⣿⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⢻⣇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣹⣿⣿⡟⠉⠉⠉⠻⠿⠋⠙⠛⠁⠀⠀⠀⠀⠀⢶⡟⢉⣥⠄⠀⠘⠻⣿⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⢿⣷⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠉⠀⠀⠀⠀⢰⡟
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠋⣹⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⢿⣿⡀⠀⠀⠀⠀⠰⠆⢠⡄
⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣦⢠⡄⠈⠁⠀⠈⣿⡆
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠻⠿⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠄⠀⠀⠀⡰⠆⠈⠀⠀⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡀⠐⠿⠿⢱⡤⠀⠀⠀⣨
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠙⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸⣧⣀⠀⢀⡙
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠋⠉⣿⠟⠋⠀⠀⠀⢰⣿⣿⡀⠰⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣧⠀⠉
⠀⠀⠀⠀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⢀⣤⣤⣀⣤⣶⠞⠁
⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⡀⠀⢸⣿⠿⢿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⣷⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠻⣿⣿⣿⠀⠈⢻⣤⠘⠋
⠀⠀⠀⠀⠸⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⣠⣿⡿⠃⣀⣤⣾⠟⠀⠀⠠⡄
⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢀⣼⣿⠋⣴⣾⢿⣿⠛⠂
⠀⠀⠀⠀⠀⠘⣿⣿⠄⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢺⣿⣿⠇⠉⣿⣿⣛⠃⣀⣶⠆
⠀⠀⠀⠀⠀⠀⠹⣿⠐⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠉⠀⠀⠛⠈⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠈⠙⠿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⣦⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣏⣉⠀⠀⠀⠀⣼⣿⠀⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠋⠀⠀⠀⠰⣿⠉⠀⠀⠀⣾⣿⣾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⢀⣾⡆⠀⠀⠀⠀⠀⠟⢻⣿⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣦⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⡄⠻⣧⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⣿⣷⡀⠈⠓⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣸⡟⣹⡇⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⢀⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣶⠉⠀⠀⠀⠀⠀⠀⠈⣿⠟⠂⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣹⡦⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⢀⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⢾⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣿⣿⣶⣄⣀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⢀⢀⣴⣶⣾⠁
⠀⠀⠀⢸⣿⣿⠀⠀⠉⠿⠛⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⠋⠀⠈⠃⡀
⠀⠀⠀⣾⣿⡿⠇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠁
⠀⠀⢸⣿⣿⣷⡄⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄
⠀⠀⣾⣿⣿⣿⠃⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣶⣦⣀⣀⠀⠙⠓
⠀⠀⢿⣿⣿⣿⣦⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠛⠛⠿⠿⠿⠿⠿⣶⣤
⠀⠀⠈⠻⠿⢿⣿⣷⣄⡀⠀⠀⠈⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⡿⠿⠟⣿⣿⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⠿⠛⠻⠿⠿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠒⠲⠤⢤⣄⣀⣀⠀⣠⠔⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⣉⢙⡄⡴⠋⠁⠀⠀⡴⢦⠀⢠⠖⠶⢄⡀⢈⡧⡿⠒⠚⠓⢦⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠖⠋⠉⠙⣾⣅⡤⠊⠉⠉⠀⠈⠓⠋⠀⠀⠀⠈⠉⠙⠧⣄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⣠⠞⢁⣀⡤⣤⣀⡀⠀⠀⢀⡤⠴⠒⠒⠒⠦⢄⣀⠈⠳⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡡⠚⠉⠀⠀⠀⠀⠉⠓⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⢃⡞⠀⠀⠀⣀⣀⣀⠀⠀⠀⣤⣶⢤⠀⠀⠀⣀⣀⣀⠀⠀⠀⢳⠀⠙⣦⣠⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡾⣍⣻⢦⣀⣀⣚⣁⣀⣈⣳⣀⢸⡁⠈⣹⠀⠀⠚⠁⠀⠈⢳⠀⢀⣨⣷⣋⣉⣷⣤⣀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠒⠋⠉⠉⠉⠉⠉⠥⣽⠁⠀⠈⣻⣭⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠦⣈⠁⣠⢚⣉⣉⠉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣠⠄⠀⠀⠀⠀⠀⣀⣋⣉⣹⣄⣀⣠⣯⣍⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⣸⡆⡇⣾⠀⠈⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⠤⠒⠋⠙⢧⡁⠀⣰⠃⢰⠇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⣲⠦⠤⣀⣀⡀⠀⠀⠈⠉⠓⡷⠘⠢⠤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠴⠛⢇⠀⠐⡆⠀⠀⠙⣲⠃⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⢀⠆⠀⢩⠟⠓⠶⢤⣄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁⠀⠀⠈⠑⠚⠁⠀⠀⢠⡏⠀⢰⡏⠓⠒⠒⠒⠒⠒⠊⠉⠁⠀⠀⠀⠀⠀⠻⣤⠴⠃⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢸⢰⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠒⠂⠀⠀⠀⠀⠀⠈⠙⣇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠓⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣀⣀⣀⣠⣤⡄⠀⠀⠀⠈⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣀⣀⣀⣀⣀⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⢿⣿⠿⣿⡿⢿⡿⢿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠰⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠰⢿⡿⠷⢿⠷⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠰⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣷⡄
⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦
⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇
⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣼⣿⣿⣧⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⢹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏
⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟
⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠘⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠹⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣷⣦⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠋⠀⢙⠋⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣥⡀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠐⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⠖⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠁⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠘⣃⠜⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠔⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠸⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠖⠲⠶⠤⠵⡄⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠐⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠺⢿⡀⠀⠡⡀⠀⠀⢀⣼⠥⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣤⣡⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⢈⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀
⡖⠐⠂⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⢀⡀⡀⠄⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⡕
⠘⠲⠤⣤⣤⣤⠤⠤⠭⠶⠄⠀⠀⠀⠵⠶⠏⠀⠀⠺⡁⠙⢂⠀⠠⡀⠀⠀⠀⡀⠤⠒⠁⠀⠀⠀⡠⠃⡄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠊⠑⠀⠈⠉⠢⠁⠀⠴⠲⠦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠⠊⠀⣀⣀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢠⠀⠢⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⡤⢤
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡒⠋⠉⠀⠀⠠
⠀⢞⠁⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠠⡀
⠀⠀⠉⡡⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡌⠁⢒⠒⠒⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠲⣄⠀⠀⠀⠀⠙⠶⠤⠂
⠀⠀⠰⣁⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢞⣉⠁⠀⠀⠀⠀⠒⠋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠙⢦⣀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢫⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣴⠃⠀⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢦⡀
⠀⠀⠀⠀⠓⠔⠊⢠⠀⢀⡇⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣿⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠇⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⢢⡀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠴⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠴⠶⠖⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠉⠀⠀⠀⠀⠠⠔⠒⠚⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠙⣷⣄⡀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⠄⠀⠀⠀⠀⢀⣀⣁⣤⣤⣽⣷
⠀⠀⠀⠀⠀⠀⢸⡇⠉⡛⠷⣤⣀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠚⠉⠁⢀⣀⣀⣠⣤⠤⠶⠖⠚⠛⠋⠉⠉⠀⢀⣀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣹⣦⣈⠓⢬⣙⠻⣦⣄⠀⠀⠀⠀⠀⣠⠶⠚⢛⡉⠭⠥⠄⢒⣂⣀⣤⣤⡴⠶⠶⠟⠛⠋⠉⣛⠛⢶⡄
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠙⠳⢦⣌⠓⠦⣙⠳⣦⣀⢀⡾⠁⠀⠀⠠⠖⣒⣛⣋⣭⡭⠥⠤⠤⠤⠀⢀⣀⣀⣠⣤⠈⠃⠈⣿
⠀⠀⠀⠀⢰⡏⠀⠀⠈⠙⠶⣤⡉⠻⢶⣄⡁⠊⠙⠻⢧⣄⣀⣀⣀⣤⣤⣤⣴⡶⠶⠶⠶⠟⠛⠛⠛⠉⢉⣁⡠⠄⢀⣼⠇
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠙⠳⣤⡈⠛⢶⣤⡀⠀⠈⣩⡭⠭⠤⠄⠀⠀⠐⠒⠒⠒⢒⣒⣊⣉⣭⣭⣤⡶⣾⠋⠁
⠀⠀⠀⠀⠀⢀⡿⠷⣦⣄⠀⠀⠀⠀⠀⠉⠓⠦⠈⠛⢷⣤⣠⣤⣤⠶⠶⠶⠒⣚⣛⣛⣿⣭⡿⠿⠿⠟⠛⠛⠛⣻⡇
⠀⠀⠀⠀⠀⠈⢧⣈⠲⢭⡛⢶⣤⡀⠀⠀⠀⠀⠀⠀⠉⢀⣉⣥⠶⠶⠚⣛⠛⠉⠉⠉⠀⢀⣀⣀⣤⣤⣴⡶⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣉⠳⢮⣙⢷⣤⡀⠀⠀⢠⠞⠉⠀⠒⣈⣉⣥⡤⠤⠶⠖⠛⠛⢛⣉⣉⠉⠀⠀⠀⠲⣎⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣌⠛⠮⣿⢶⣄⣿⠀⠀⠀⠈⠀⠀⠒⢚⣛⣉⣉⣉⣭⣭⣤⡶⠶⠶⠞⠛⢀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠈⠙⠻⠶⠶⠾⠛⠛⠛⠛⠉⢉⣉⣉⣀⣀⣤⠤⠤⠴⢒⣊⣥⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠒⠋⢉⣉⣁⣀⣤⣤⣤⡤⠤⠶⠶⠖⠒⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡀⢀⡀⠀⠀⢀⠂⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠒⠊⠁
⠀⠀⠀⠀⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠈⠑⠀⠁⠀⠀⠘⠁⠈⠀⠛⢩⠋⠉⠆⠀⠀⡜⡿⠐⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠖⠀⠀⠐⠀⠀⠁⠈⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠰⢦⠍⠀⠀⠰⠶⢶⠆
⠀⠀⠀⢀⠀⠀⠠⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠃⠀⠂⠀⠀⢰⡈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⣾⠆⠀⠀⠄⠀⠀⠈⡀⠛⠀⠀⠄
⠀⠀⠐⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠿⠀⠀⠀⠀⠀⠀⢠⠄⠘⠃⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠃⠀⠠⢀⠀⠀⠀⠀⠀⠀⠛⠀⡆⠀⠀⠀⠁⠀⠀⣰⠈⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⡀⠀⢰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡜⠩⣰⢠⡄⠀⠀⠸⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠃⢀⠈⠀⣶
⠀⠀⣀⠀⠀⠀⠀⠀⡀⣀⢀⣀⠀⢀⡀⢀⣍⠀⠀⠀⠀⠀⠈⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⡀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠈⠀⡄
⠀⠀⠈⠙⠒⠀⢰⠀⢹⡇⢸⣇⠀⡾⢷⠀⣶⠁⠀⠀⡀⠀⠀⠰⠈⠀⢀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠈⠀⠐⠂⠀⠐⠒⠙⢂⠀⠀⠀⢀
⠀⠀⠀⢰⡆⠀⠏⠀⠀⠀⠀⠉⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢂⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠉
⠀⠸⢀⡈⠀⠀⠀⠀⠑⠲⠤⢴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠈⠑⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠠
⠀⠀⠈⠁⠀⠀⠐⠒⠒⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣄⡀⡠⠀⠉⠈⠀⠀⠀⠀⢀⡀⠀⠀⠀⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡆⠀⠀⡄⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⡀⠂⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠐⠀⠀⢓⠐⢀⡀⡀⠀⠀⠠⡜⠤⠀⢠⠀⠀⡦⠀⠃⠀⠀⠀⠀⠒⠀⠀⠀⡀⠀⠀⠀⡀⠘
⠀⠀⡅⢀⠀⢀⣀⣰⢀⣀⡅⠀⢠⠁⠀⠀⠨⠀⠀⡤⠐⠀⠒⠂⠀⠠⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠁⠀⠀⠒⠒⠂⠈⠭⠭⠃⠀⠀⢦⡼⠀⠀⠈⠦⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⢀⡠⠴⢂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⢆⠀⣴⡀⠁⠀⢠⡄⢀⠀⠉⢩⠀⠈⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⣉⠈⠀⠀⡇⠐⠤⠀⠀⠑⠉⠉⠁⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⣁⣉⠀⠀⠀⡀⣀⡠⠠⠖⠀⠀⠀⢀⠁⢀⠐⠉⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⠑⠆⠀⡴⣀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠃⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣆⠀⠀⠛⠨⠁⠀⠀⠀⡘⠂⠄⠀⠀⠙⠶⢞⠁⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣍⡠⣁⠶⠂⠀⠉⠀⠠⠂⠀⠀⢄⡠⣔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠎⠀⢙⡗⠒⢲⠶⠚⠛⠛⡖⠒⠓⣿⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠚⠃⢠⡞⠀⠀⣽⠀⠳⢀⣠⢙⠤⠔⠃⠑⠂⡴⢌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣀⢏⠀⠀⡠⠘⠃⠂⢡⠀⡖⠈⠉⠀⠤⠠⡄⠠⡀⠀⠀⡏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢰⠇⡀⠱⠞⠀⣤⢀⡠⠟⠠⠉⡀⠼⠀⡀⢀⢷⠀⢨⠉⠘⠣⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠟⠢⡀⡘⠀⠀⠀⠈⢠⠁⠸⣀⠀⠀⠀⢄⠀⢀⣄⠈⠠⠄⢠⡇⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⡏⠀⠀⠚⠁⠀⢀⠉⠂⢁⠀⠠⠉⠁⡈⠀⢀⠁⠀⢨⠀⢀⠀⠁⠈⡃⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡞⠁⠀⠀⡐⠀⠀⠤⣀⢠⠊⠀⡧⠒⠋⠁⠀⠀⠒⠒⢻⠀⠈⠒⠂⢠⡇⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⢰⠀⠠⠴⡎⠀⠀⢸⠀⢀⡌⠀⠀⠉⠂⢀⠄⠀⠀⠆⠀⠊⠀⠀⡀⠒⠁⢸⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⡼⢆⠀⡰⠁⠀⡀⠀⠈⠉⠀⠀⠀⠀⠂⠉⠀⠀⠸⡀⢀⠾⠀⠀⢑⠦⠀⣼⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡞⠀⠄⠚⠁⠀⠀⠧⠀⠤⠛⠀⠀⠜⠒⠀⡠⠀⠀⢠⠀⠁⢀⠆⠀⢨⠀⠋⠀⡆⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⢳⡀⠀⡠⠀⠀⠀⡄⠀⠀⡄⠀⢘⠀⠐⠊⠑⠀⠀⠘⠂⠒⠡⠀⠀⠀⠃⠒⢂⣯⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠉⠉⠛⠲⠦⣤⣄⣄⣀⣀⠀⠀⠀⢸⠢⠄⠀⢀⠂⠀⢀⡀⠀⢀⡀⢀⢀⣃⣠⠮⠜⠂⠀⠡⠀⠡⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⡇⠀⠀⠉⠹⠉⠛⠛⠛⠒⢓⠚⠛⠛⠛⠋⠉⠉⠉⡍⠀⠀⠀⡀⠀⢀⠀⠀⠀⠆⠀⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⢘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⡀⠀⠀⢠⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠸⠀⠀⡇⠀⠀⢘⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢤⠀⣄⢠⣀⣀⣀⣀⣆⣄⠀⠀⠀⠀⣄⢀⣀⠀⠀⠀⠸⠀⠆⠀⠀⠀⢀⣄⣠⠦⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠈⠀⠀⠈⠀⠁⠈⠛⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣲⣶⣶⣿⣾⣿⣿⣿⣶⣶⡦⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⣧⠾⠷⠺⣇⣿⣻⠻⡷⠛⡭⠳⡋⠻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢛⠀⠁⠘⢧⣷⣽⣄⣛⣏⣘⡢⣳⣵⡿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡿⢻⡧⣩⣿⣼⡵⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣇⢿⣌⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣏⠼⣿⡿⣻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣼⣧⢭⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⠘⣿⠾⠨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠦⠠⣵⡄⢦⠨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣳⠾⢯⣭⣿⢏⠭⣆⡚⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣱⠎⣺⣖⣚⣗⢶⣊⣱⣂⡌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣚⣱⣑⣞⣻⣿⣿⣑⣂⣐⣒⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢹⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣋⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣿⣿⣿⣿⡃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡿⣿⣿⣿⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⠛⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢶⡖⠶⠶⠶⠶⠶⠶⠦⠴⠶⠖⠒⣒⣒⣸⢫⣿⠳⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠛⠉⠙⠻⣄⠙⢿⡛⠛⠛⠛⠋⠹⠿⠉⠙⠏⣀⡴⡡⠇⠘⠀⠀⠈⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣉⠳⠤⣄⣀⣀⣀⠤⠴⠚⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⠤⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⡴⠋⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⣠⠞⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⢾⡏⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠳⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰
⠀⠀⠹⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠹⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠙⣦⡀⠈⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠎
⠀⠀⠀⠀⠀⠈⠻⠾⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣇⠀⠀⠀⠀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡼
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⣾⣦⣴⣶⣮⣭⣭⣤⣤⣬⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠒⢄
⠀⠠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠘⢦⡀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⣀⡀
⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠈⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡯
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢵⡪
⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣖⢽⡪
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡫⢕⢽⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠽⣪⣕⢕⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀⠀⠀⠀⠀⢠⡿⢖⡭⡪⣿⢕⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠭⡛⢗⢦⣄⣀⣟⣻⢕⣿⡪⣚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡺⢕⡪⡕⡜⡼⡜⠄⠈⠉⠺⢕⢽⣪⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⣴⡫⢕⢪⣳⣹⢱⡱⠑⠈⠀⠀⠀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⣠⣖⠯⣓⢬⢢⢧⢇⠣⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣍⡎⡞⣎⠎⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⠜

⠀⠀⠀⠀⠀⠀⢠⣾⢡⣶⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠋⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠿⣿⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⠟⠛⠛⢿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⡛⢠⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠘⡟
⠀⠀⠀⠀⠀⢸⣿⣿⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⣿⡟⠀⠀⠁⢰⣿⣇⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢠⣧
⠀⠀⠀⠀⢀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢿⣿⡆
⠀⠀⠀⠀⣼⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⣿⠛⠛⠋⠀⠘⢿⣿⡄⠀⠀⠀⠀⡀⠋⢀⠀⠀⠀⠋⡁
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⣿⣀⣀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠆⠀⠖⠀⠀⠀⠀⠆⠁⠠
⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⣼⣿⣷⣿⣶⣶⣤⣶⣾⣿⣿⣿⡀⠀⠀⣟⣛⣉⣿⠃⠀⠀⠟⠀⠀⠀⠠⠄⠁⠤⠉⠠⠈⠀⠄⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⢀⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠋⠉⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠐⢀⠐⠂⡀⠃⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣷⣦⣼⡇⢸⣿⣧⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢛⠀⠠⠀⠀⠆⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠉⠸⠇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠈⡋⠁⠀⠀⠀⠁⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠁⠀⠘⠀⠀⠃
⠀⠀⠀⢀⣤⣀⣀⡀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣄⣀⣀⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣿⣏⢰⣿⣹⣿⣿⠁
⠀⠀⠀⠿⠿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠉⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⠿⠟⠛⠉








⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠀⠀⠀⣴⣶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠘⠆⢀⡀⠀⠀⠃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠈⡠
⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡰⠃⠀⠀⠌⡵⠃⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠐⠈⠃⠠⠊⠀⠈⠀⡀⠀⠊
⠀⠘⠓⠂⣤⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢀⡡⠢⠉⠀⠀⠀⠀⠀⠀⢀⠴⠇⠀⠀⠀⡄⠀⢀⣶⣶⡶⣶⣶⣦
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡄⣀⠀⡀⠀⠐⠋⠀⠂⠀⠤⠀⠀⢀⠄⠊⢁⠔⠀⣗⠀⠀⢰⡆⠘⠿⠛⠁⠈⠙⠋
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠀⠀⠈⠐⠀⠀⠤⠀⠀⠀⢀⡠⠒⠁⠀⣼⣿⣶⣶⣿⣧⠀⡀⠀⠀⠀⠠
⠀⣀⠀⢀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠰⡄⠤⠀⠀⠀⣀⠀⠄⠤⠀⠀⢀⡀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⠂
⢀⢀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢸⣿⣿⣿⣿⠀⠈⠀⠀⠂⠂⠀⠈
⠀⠘⠿⠋⠀⠘⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢠⠄⠀⠀⠀⠀⢀
⠀⠐⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠠⢀⠇⠀⠀⠉⠹⣿⡿⠋⠁⠈⠀⠀⠀⠀⠀⠈
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⡤⠐⠂⠀⠐⢢⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡈⠀⠀⠀⢀⠀⢀⣤⣤⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠄⠀⣘
⠀⠀⠀⠀⢀⡁⠀⠺⠿⠇⠀⠙⠋⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠁⠀⠈⠀⠀⠂⠀⠂⠀⠈⠀⠀⢀⠃⠀⠀⠄
⠀⠁⠠⣀⠈⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠄⠐⠀⠀⠀⠂⠀⢠⠀⡠⠠
⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠐⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠆
⠀⠀⠠⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠂
⠀⠀⠀⠈⠀⠀⠀⠠⠀⢰⠂⠀⠀⠀⠉⠂⠀⠀⢀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠃⠀⡈⠀⠀⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠐⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠃⠀⠀⠈⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡷⡢⠀⠀⠀⠀⢴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠁⠀⠀⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⢀⡀⡀⠄⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠋⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣿⣥⡀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣟⠁⢩⣿⣿⣶⣿⣮⣃⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣧⣘⣿⠴⣾⣿⣿⢫⣿⣿⣽⣛⠻⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⠶⣿⡿⣿⢳⣮⣿⣿⣿⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣿⣭⣴⠾⡿⢱⠇⣮⣿⠙⢻⣯⠙⡓⠙⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡜⣿⠁⠀⢠⣡⡿⠀⠷⠁⠀⡀⠛⡇⠘⢌⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣷⠗⣴⢓⠛⣴⡇⠀⡄⣀⠀⠠⠐⡸⡄⠈⢧⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠈⡘⠋⠀⣿⡇⢸⡀⠈⠀⠈⠄⠆⢳⠀⠀⢣⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢸⢿⠁⠈⡀⠀⠀⠒⠀⢄⠀⢇⠀⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸⢺⡇⠀⢥⠀⠁⠀⠀⠀⠀⠘⡆⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⣊⣿⡀⠘⡀⠀⠀⠀⠀⡀⠀⣷⣴⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠈⠀⠀⠀⠸⢿⣿⣷⣴⣿⡦⠀⠂⠀⠈⠀⣸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣠⠴⣦⡏⠀⢀⠈⡠⠀⠀⠀⠀⠀⠀⠀⠛⠻⠋⠀⠀⠀⠀⠀⠀⢣⠁
⠀⠀⠀⠀⣤⡞⣉⢥⡹⠀⠠⠉⠀⠄⠀⠀⠀⠀⠀⢰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣈⣇⠀⠀⠀⠀⠀⢄
⠀⠀⢀⠞⠁⠸⠐⠋⢀⠼⠄⠀⢰⠀⠀⠀⠀⠀⠀⣨⣀⣀⣄⣠⣀⣀⣤⣴⣼⣿⣿⣿⡄⠀⠀⠀⣀⣾⠿⠁
⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⣏⣿⣹⣿⣟⣿⡿⠿⣛⠲⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⡿⢿⠽⡿⠷⠚⠟⠙⠋⠉⠀⠀⠀⠹⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠶⠒⠚⠛⠓⠦⣤⠶⠖⢲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣸⣦⣀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⡒⠊⠉⠉⢉⣽⠉⠉⠉⠙⠛⠒⠻⢷⡀
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢒⡎⠉⠁⣀⣀⠤⠕⠒⠊⠉⠉⠛⠓⠒⠲⠶⠤⢤⣤⣼⣇
⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⣀⡠⠄⢲⡎⠉⢀⣀⠤⠬⠓⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄
⠀⠀⠀⡼⠉⠳⢤⡀⠀⡰⠁⠀⡠⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠶⠒⠛⠉⠙⡇
⠀⠀⠀⣇⠀⠀⠀⠉⠓⣧⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠒⠚⠛⠉⠉⠉⠀⣀⣀⣀⣠⡤⠤⠶⠶⢺⡇
⠀⠀⠀⢘⣦⣀⠀⠀⠸⠀⢨⠓⢦⡀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠀⣀⣀⣠⡤⠴⠶⠒⠚⠛⠉⠉⠁⠀⣀⣀⣤⠴⠖⢻⡄
⠀⠀⠀⣾⠀⠈⠓⢦⣵⠀⢸⠀⠀⠉⠳⣄⡀⢠⡏⠀⠤⠖⠚⠋⠉⠉⠁⠀⢀⣀⣠⡤⠴⠶⠒⠛⠋⢉⣉⣀⣤⡤⠤⢾⡁
⠀⠀⠀⢻⡄⠀⠀⠀⡸⠳⢼⡀⠀⠀⠀⠀⠉⠛⣧⣤⡤⠤⠶⠖⠒⠛⠛⠉⣉⣁⣤⡤⠤⠶⠒⠛⠉⠉⠁⣀⣀⣠⣤⡞⠁
⠀⠀⠀⠀⠙⠳⣄⡀⡇⠀⡘⠉⠓⠦⣄⡀⠀⢰⡇⠀⠀⢠⡤⠴⠖⠒⠛⠉⠉⠀⢀⣀⣠⡤⠶⠖⠚⠋⠉⠉⢀⣀⣀⣽
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠇⠀⠀⠀⠈⠙⠶⣼⣄⠀⢀⣀⣀⣠⣤⠤⠶⠖⠚⠋⠉⠁⣀⣀⣠⡤⠴⠖⠛⠋⠉⠁⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠀⠀⠀⢠⠏⠉⠉⠉⠀⠀⣀⣀⣤⠤⠶⠒⠛⠋⠉⠀⠀⢀⣀⣠⡤⠴⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⢸⠀⠀⠐⠒⠛⠉⠉⠀⣀⣀⣤⡤⠤⠶⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠷⠤⠤⠶⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣰⠀⡄⢀⢀⡔
⠀⠀⠠⠐⠈⠅⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠃⠈⢠⠆⡄⡤⡀
⠀⠀⠐⠒⠒⠒⠂⠉⠉⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢒⠚⠙⠓⠂
⠀⠀⠀⠀⢠⣟⠁⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠄⠀⠘⠣
⠀⠀⠀⠀⣸⠋⠀⠀⠉⣷⠀⠀⠀⠀⠄⠀⡜⠀⢠⢤⠀⡀⠤⢤⠀⠢⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠘⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⢀⡄⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠔
⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠢⡀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⣽⠁⠀⢺⡋
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⠆
⠀⠀⠀⠀⠀⠀⡀⠸⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠉⠉⣹⠐⠀⠀⠀⠀⠀⠀⡀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣀⡀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠰⠏⠀⠀⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠚⠛⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠐⠊⠀⠀⠀⠀⠀⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠛⠋⠉⠂⠀⠀⣀⠀⠀⠲⠒⠂
⠀⠀⠀⠀⠀⢀⣴⣿⣿⡵⠂⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣠⣶⠄
⠀⠀⠀⠀⠐⠛⢫⣿⣿⠃⠀⠀⠀⠀⠀⠀⠿⠻⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣻⣿⣡⡷⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⣼⢿⣿⣣⠆⠀⠀⠀⠀⠛⠂⠀⠀⢤⣿⣿⡷
⠀⠀⠀⠀⠀⠀⢋⣾⣷⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡿⠃⡀
⠀⠀⠀⠀⠀⠀⠈⡾⢽⣿⠁⢘⣆⠀⠀⠀⠀⢠⣶⣿⣧⣄⡀⠘
⠀⠀⠀⠀⠀⠀⣠⣶⣮⣴⣴⣿⣧⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⠈⠛⠁⣀⣠⣿⡿⣿⣿⣿⣿⣿⠻⣶
⠀⠀⠀⠸⣟⠹⢿⣯⣿⣿⣿⠙⠂⠛⠘⠋⠈⡟⢿⣯⡿⣿⠧⡹⣇⢹⣷⣀⡀
⠀⠀⠀⠀⠈⠙⢾⡿⠟⣿⠟⠥⣀⠀⠀⡀⢰⠇⠈⢿⣷⡘⢧⠘⡿⣾⣇⠙⢿⣤⢶⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠱⢠⣟⠀⠀⠉⠖⠚⠁⠋⠀⠀⠈⢿⣷⡌⣀⠛⢻⣿⡄⠀⠇⠀⢉⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡜⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⡆⠈⣿⣿⡂⣠⣤⡁
⠀⠀⠀⠀⠄⠀⢠⣿⠇⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠏⠙⣽⣿⡿⣡⡆
⠀⠀⠀⠀⠀⢴⣿⣿⠆⡇⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⢰⡀⠘⠏⣠⣿⣧⣄
⠀⠀⠀⠀⠀⠹⣿⡿⠈⠀⠀⠐⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣮⣻⠈⠃⠀⠈⠉⠉⣻⣿⣶⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠀⡰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠂⠀⣴⡿⠛⢼⣿⠀⠀⠀⠀⠀⠀⢿⠛⠛⠲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣀⠤⠤⠄⠀⡁⠀⠀⠀⠀⠑⠀⠀⠀⢻⣿⡄⢠⠀⠀⠀⠀⡜⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣿⡇⠈⠀⠀⠀⡼⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠂⠀⠀⠀⢠⣴⣆⠀⠀⠀⠀⠀⠀⢸⡜⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠉⠹⠿⣄⠀⠀⠻⠿⠛⠀⠀⠀⠀⠀⡴⢙⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⣤⡴⢺⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠸⣇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠿⠂⢣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⢀⡀⠄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⠆⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡀⢛⢉⡉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠙⢦⡀⠀⣙⠃⠉⠳⠆
⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠢⡀⠀⢙⡂⠉⠻⠦
⠀⠀⠀⢸⣿⣷⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢮⣄⠈⠛⠶⠄⠻⢶⡤⣮⡀
⠀⠀⠀⠘⣿⠟⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠈⠛⠦⠶⢦⡄⢶⣄⠈⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡦⣤⣀⠴⣤⠰⣇⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡄⠀⠀⠐⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠉⢳⡄⠀⢹⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⣿⢤⣾
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⢳⡆⠀⠘⣿⠁
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣤⣤⣄⣀⡀⠀⢤⣤⣶⣶⣶⣶⣶⣶⣤⠄⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠉
⠀⠀⠀⠙⠻⢿⣿⣿⣷⣦⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠀⠂⠀⠰⠀⠀⠈⠃⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠆⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⢿⠿⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⣆⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⡀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣤⣤⣴⣿⠂⠀⠀⠀⠀⠂⠀⢀⡈⠀⡉⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠿⠿⠿⢿⣿⠿⠿⠿⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠿⢿⣷⣄⣀⡀⠀⢸⣿⠀⠀⢀⣀⣴⣿⠿⠿⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠈⠛⠛⠻⠿⢿⣿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠄
⠀⠀⣠⣿⢿⣦⣄⠀⠀⠀⠀⠀⣰⡿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣿⡄⠀⠀⠀⠀⢀⣠⣾⢿⣷⡀
⠀⠀⠻⣷⣦⡉⠻⢿⣦⣄⣠⣼⡿⠁⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠙⣿⣦⣀⣤⣶⠿⠋⣡⣶⡿⠃
⠀⠀⠀⠀⠙⠻⣷⣤⣿⠟⠛⠉⠀⠀⠀⠀⠈⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠀⠀⠀⠀⠈⠉⠛⢻⣧⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⠄
⠀⠀⣀⣠⣴⣿⣣⣤⡦⠉⠀⢀⢀⣤⣶⣶⠖⣲⣤⡄⠐⠲⢤⣤⠴⠖⠒⠲⢶⣄⠀⢀
⣴⣾⣿⣿⣿⣿⡯⡿⠁⢀⣰⡶⣯⡿⢚⠉⠚⠉⢹⠋⠻⠗⢠⣭⣝⡟⠳⣄⠉⢪⡳⣄⠄
⣿⣿⣿⣿⣿⠟⠑⠀⣠⡾⣫⡾⢁⠔⠁⣠⣴⣶⣿⣿⣿⣦⣦⠀⠉⡳⣆⠘⣧⡀⡟⢾⡤⠈⠀⠀⠀⡠⡀
⣿⣿⣿⣟⠁⠀⠀⣴⢋⢪⠞⡴⣡⣶⣿⣿⣿⣿⣿⣿⣿⣾⣿⣗⠄⠙⣎⢳⡘⣿⣮⡘⡿⡂⠀⡄⠈⢼⡅
⣿⣿⡫⠊⠀⠀⣼⡇⢡⢊⠞⣵⣿⣿⡏⠀⠁⠉⠙⢿⣿⣿⡿⣿⣦⠀⠘⡄⠻⣼⣿⣍⣁⡷⣼⣷⣷⡟
⣿⠟⠀⠀⢐⣼⢏⣴⠇⠈⣼⣯⢽⡿⠀⠀⠀⠀⠀⠀⠆⠈⠛⢻⣿⠁⠀⠸⡄⠹⣿⣿⣿⢏⠬⣹⣷⠆
⠁⠀⢀⣤⣾⣿⡟⠁⠈⣼⡷⠃⣿⢃⡆⠀⠠⠀⠃⠀⣦⠀⠀⠀⢹⣇⠀⠀⠇⠀⢻⣿⣷⡾⢽⣾⣮
⠀⠴⠀⢆⣶⠏⣠⠞⡜⣿⡗⣰⣿⠁⠀⠀⡀⡰⠀⠀⢹⡄⠀⠀⠀⣿⡆⠀⠀⠘⢸⣿⣿⣿⣦⣿⣧⡄
⠀⠰⠶⢿⣿⢿⣃⣾⡇⣿⣷⣉⣻⡆⠀⣠⣷⠃⠀⣠⠏⢳⠀⠀⠀⣼⡇⠀⠀⠠⢸⡏⣿⣻⣷⡿⣿⣿⡀
⠀⠙⠛⠿⣷⣿⣿⡇⢆⣿⣿⢻⣿⣧⣴⡟⡡⠂⢼⠟⣒⠒⠳⡄⢠⣿⡇⠀⠀⡄⣼⢣⣿⣿⣿⠀⠈⠻⣧
⠀⠀⠀⡈⢹⣻⣿⡇⠘⢿⡅⠀⠏⠛⠹⠿⠁⠀⠸⢻⣿⣿⣿⣷⣿⣿⠁⠀⠀⢀⣿⣅⣿⡁⢧⡀
⠀⠐⠁⢠⠈⣿⣿⣿⣄⣿⡧⠀⠀⠄⠀⠀⠀⠀⠀⠀⠛⠈⠿⠛⡹⠃⠀⠠⢃⣼⣿⡯⣿⣷⡜⢷⡄
⠀⠀⠀⠔⠃⠐⠹⣿⣿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⣠⠄⣠⣾⣿⠻⢻⣟⢿⣿⣶⣿⣦⡄
⠀⠀⠀⠀⠤⢀⣬⣿⢸⡔⣿⣷⡀⠠⢄⣀⡀⠀⠀⠀⠘⢋⣹⣷⣛⣥⡾⢿⡏⢠⣼⢸⣿⣦⡹⠿⡗⠂
⠀⠀⠐⠀⣀⣴⢿⣿⡸⣧⢻⣿⣿⣄⠀⠉⠀⠀⠀⠀⣈⠡⣶⣿⠿⠟⢠⣾⢹⣾⣽⢸⣿⣿⣷⣶⣌⠳⢤
⠀⢠⠘⠠⠈⢋⡙⣿⣷⣼⣮⢻⣿⣿⣦⣤⠤⠒⢻⣏⠁⢰⡟⢉⠀⣠⣿⡿⢸⣿⣿⣈⣿⣽⣿⣿⣿⣿⣻
⠀⠸⣷⣌⣷⣤⡀⠻⣿⣯⣿⣧⣷⢹⣿⣧⠀⡀⣾⣿⣷⣿⠁⣧⣴⣿⣿⡇⢸⣿⣿⣏⢿⡿⣿⣿⣿⣿⡟
⠀⠀⢿⣿⣿⣟⠿⣾⣾⢿⡟⣿⡞⣿⣷⣌⠣⣼⣿⠿⣿⡿⣾⣿⣿⠿⣿⣿⠈⢿⣿⣿⣤⣙⣲⣿⣿⣿⠂
⠀⢀⡈⣿⣿⠻⢦⣄⡎⠸⣷⣿⣷⣿⣿⣷⣽⡟⣣⠘⢿⣀⣿⠿⣣⠞⢿⣟⣷⣌⣿⣿⣿⣿⣿⣿⣟⠝⠄
⠀⠀⢿⣿⣿⣷⢼⡿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡷⢾⣿⣆⣾⣿⣿⡿⠁⠈⠉⠉⠉⠙⢿⣦
⠀⠀⠈⢿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⢯⣿⠿⢋⣾⣿⣾⣿⣧⣼⣿⡿⣿⣏⣿⠁⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠈⢿⣿⣿⣿⣿⡛⢵⣿⠟⣱⡟⠁⢀⣿⣫⠦⣿⡏⣿⠟⣿⠇⣿⣾⣿⠀⠀⠀⠀⠀⠀⣠⣤⠀⠁
⠀⠀⠀⠀⠀⡦⠼⣿⣿⡟⡿⢫⡾⠋⠀⠀⢾⡿⣿⡾⠃⠀⣿⢂⠉⡄⣟⡿⣿⣿⣦⣄⣤⣶⠟⠋⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡅⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠇⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⠀⠀⣠⡴⠶⠛⠉⢷⡀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠛⠶⣤⣄⠀⢠⠞⣷
⠀⠀⠀⠀⠀⢀⣀⡘⣇⠸⡆⣰⠏⠀⠀⠀⠀⠈⠳⣤⣀⡀⢀⣀⣤⠞⠁⠀⠀⠀⠀⠹⣆⡿⣠⣯⡴⠶⡄
⠀⠀⠀⠀⠀⠻⣌⣉⡻⣷⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡥⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢰⣟⡀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⣀⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⣰⠏⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⠋⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⢻⡄⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠹⣆⣼⠋
⠀⠀⠀⠀⠀⠀⠺⣅⢀⡽⠃⠀⠀⢿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣾⠁⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣀⣀⣀⣀⣀⣀⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠘⠀⠀⢀⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢀⡤⠼⢶⣄⢰⠀⠘⣿⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡰⠺⣷⣀⣆⠀⣽⣧⣶⡾⣿⠟⣤⣾⣇⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠋⢰⣿⡿⣿⣿⡿⠛⠃⠀⠘⢏⠓⠘⢰⣿⡇⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢡⠰⡜⣿⡗⠩⠤⠁⠀⠀⠀⠀⠀⠀⢀⣿⣻⣧⣧⣼⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠁⣸⣿⣤⡀⠀⠀⡀⠀⠀⠀⠀⢈⣽⣿⣿⣿⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⣧⡁⠀⠀⠀⠀⠀⠀⣠⢾⣿⣿⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⡶⣿⣬⣱⣶⠦⣀⣴⣾⣿⣿⣿⡁⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠈⠻⣷⣈⠙⣯⣿⡏⣴⣿⡟⣻⣵⣴⢶⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣄⢀⣽⣿⣿⣿⣧⣿⣟⠛⠋⢑⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡘⠀⠀⢛⣼⠿⣿⠟⠷⡿⠿⠛⢿⠗⠀⠬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠛⠁⡀⠉⠐⠣⠙⠀⠀⠀⠀⠀⠀⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠃⡀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣉⡀⠀⠀⡀⢀⠃⣀⠴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡅⠀⠀⠀⣄⣄⠀⠀⠀⠀⣷⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⡀⠛⠢⡄⡀⠷⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⣿⣟⣠⣇⡀⢸⣿⣿⣧⣀⣀⣠⡀⠀⣠⡿⢻⣶⠃⠀⠀⠀⠱⣆⠀⠀⠀⠐⠆⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣄⣠⣼⣿⡙⣿⣼⠀⢛⣙⣛⢛⣛⠻⠛⠛⡛⠷⣼⡏⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⡀⠊⠀⠠⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡌⢿⣾⣽⡯⠛⢧⣤⣿⣿⣿⡟⢃⡾⣿⠈⣙⠲⣏⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋⠉⠉⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠻⠁⠸⣿⣻⠇⣜⣷⣿⣿⣿⡟⣻⣿⣿⠃⣾⣹⢣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡄⢠⠘⣯⢹⡇⣾⣿⣿⣿⣿⠁⠀⠈⢟⡰⣿⠏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⢧⠈⢀⣿⠞⠀⢘⣿⡿⠏⢸⠀⢄⡀⠀⠙⠙⡾⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⠟⠀⠀⠀⢸⠉⣥⠀⠘⡆⠀⠀⠀⠂⡐⠇⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡾⢀⠿⠀⠀⡷⠒⠤⠀⣀⣀⠐⣦⠀⡀⢀⠤⢬⡄⠠⣤⠀⠀⠀⢀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠌⣾⠇⠸⠀⠀⠀⢠⡼⠁⠺⠧⠄⠀⢨⣄⠀⠀⠤⠄⠀⠘⠿⣯⣿⡄⠀⣇⣠⣿⠞⠳⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡠⠎⣸⠃⠸⠀⠀⠀⠁⠀⠀⠠⣀⣰⠄⠒⠀⠀⠀⡀⠒⠂⠙⢳⡞⢻⡿⣥⠂⠗⠋⠉⠠⡘⠀⠘⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠐⢋⠴⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡘⠦⣄⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠖⠉⢰⣿⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠈⠁⠒⢪⠿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠄⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⡆⠀⠀⠀⠓⢦⣀⠀⠀⠀⢤⣦⠀⠀⡀⢠⡶⠆⠈⠀⠀⠀⣼⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⢸⣏⣿⡀⠀⠀⠀⠀⠁⠑⣀⠀⢸⡇⠀⠀⢀⣀⠀⠄⠒⠀⠉⠀⠉⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣣⣼⢿⡿⠛⠛⠒⠀⠀⠀⠀⠀⠿⠷⣴⡿⣤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠻⠟⠋⠁⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠀⠀⠀⠀⠀⠀⠀⠒⠀⠰⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠀⠤⠀⠀⠀⠀⡀⠀⠀⠀⠑⠠⠀⠠⠀⠐⠀⠀⠀⠀⠀⠈⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢤⡵⣾⢟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⣶⣄⡀⠀⠀⠀⢀⡉⠀⠂⠀⡰⠆⠀⠀⠀⠀⡀⠚⣴⣓⣿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡍⢻⣷⠂⠀⠀⠀⠁⡄⠀⠂⢀⠀⡀⠀⠶⠉⠀⡺⠇⠉⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠂⠚⠙⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣐⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠔⠛⠉⠹⠶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢲⡉⠠⠤⠀⠒⠒⠀⠀⠀⠀⠐⠀⠄⠉⠀⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠤⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡗⠸⡇⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠨⡗⢀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠳⣟⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⡞⣫⠾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣽⠀⢀⠑⠺⠯⢵⡶⢒⡶⠾⠙⠓⠉⠁⢀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠳⢤⡤⢀⣀⣀⣀⣀⠠⠄⠀⠀⡸⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣎⣁⣢⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⢀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠤⡤⢀⠀⢀⣤⣀⠀⠀⣤⢠⣿⣿⡛⠛⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄⣀⠀⠀⠀⠀⠀⣄⣀⣀⣼⣿⣿⣷⣴⣤⣔⡿⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠑⠛⣩⣯⡿⠻⣿⢿⡿⣿⢿⣿⣿⡻⣿⣷⢿⣭⣡⣤⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢰⠋⢀⠁⠀⠘⡄⠀⡿⠀⠁⠿⡅⠹⠈⢢⠙⠻⣿⣄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⡇⠀⢰⠀⠀⠸⠁⠀⠰⠀⠃⠀⠉⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⡴⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⢲⡀⠀⠠⣑⡀⠀⠠⡱⣄⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⠀⠤⢇⠀⠀⠱⡀⠀⠀⠈⢆⠀⠀⡈⢆⠀⢉⠱⡀
⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⠠⡄⠀⠀⠤⠀⠀⢀⠠⡆⠀⠀⠸⡇⠀⠀⢀⡁⢡⠀⠀⠠⠼⠀⠀⠀⢳⠀⠀⠀⠈⢷⡀⠀⠈⣧⠀⠁⣡
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⣚⠀⠀⠀⢐⠁⠀⠀⢘⠀⠀⠰⠒⡇⠀⠀⠀⠀⠸⠆⠀⠐⠂⡦⠀⠀⠄⢧⠀⠀⠀⡘⡇⠀⠠⠘⣇⠀⠄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⠃⠀⠠⠎⡛⠀⠀⠀⣙⠀⠀⢈⣩⡇⠀⠀⠈⠀⢘⡂⠀⠀⠀⣿⠀⠀⡁⢸⡇⠀⠀⠀⢹⡄⠈⠁⢻⡂⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⡠⠀⡌⠀⠀⠐⢸⡇⠀⠀⠤⠆⠀⠀⠀⠴⠁⠀⠀⢈⡁⣉⡅⠀⠀⠀⢸⠀⠀⠀⠈⣧⠀⠀⢀⠘⡇⠀⠠⢨⣇⠀⠁⣿
⠀⠀⠀⠀⠀⠀⠂⠤⢚⠇⠀⠀⣈⣨⠀⠀⠐⠒⠂⠀⠀⠀⠲⠀⠀⠀⠠⠀⢰⡆⠀⠀⠈⢸⠆⠀⠀⠀⣿⠀⠀⠐⠠⣏⠀⠀⠐⣿⠀⠀⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⡤⣤⠀⠀⠘⠛⠅⠀⠀⣉⣙⡀⠀⠀⠒⠒⢺⡇⠀⠠⠤⢼⡄⠀⢈⠒⠾⠀⠀⠘⢰⣯⣀⣤⡸⠷⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠈⠓⠳⠄⣀⡠⢨⡇⠀⠀⠈⢁⣼⠀⠀⠀⢀⣾⡀⠀⣀⠸⠛⠀⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠂⠠⠀⢀⠢⠀⠀⠀⠠⠀⠀⠄⠂⠐⠄⠠⠐⠁⠀⠐⠔⠠⠂⠄⠀⠊⠐⠁⠂⡠⠀⢀⠆⠈⠀⠀⡐⡁⠀⠀⠌⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⡀⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢺⣡⠴⠂⠒⠒⠦⢄⠀⢠⠀⠀⠑⠀⠀⠀⠀⠀⠀⠢⠘⠃⡁⠐⠒⠃⠒⢤⣣⠐⠀⠞⠘⠋⠉⠉⠁⠐
⠀⠀⢀⠸⣄⣸⡀⠐⠋⠉⠙⢺⣋⣀⡀⢠⣄⡀⢀⠀⠑⠄⢈⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⢤⣾⣀⣀⣻⡿⠔⠚⠳⠏⣴⠃⣴⠦⠀⠀⠹⢄
⠀⠁⠀⡴⣯⡟⠁⠀⠀⢀⣾⣭⣿⠉⢾⣋⣻⠈⣤⢀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣱⡷⣾⠿⢿⠾⡿⣿⣄⡤⠖⠀⠈⠡⠈⠠⠐⢃
⠀⠀⢠⡧⣽⠀⠀⣌⠥⠀⠈⠃⢩⣧⣄⠳⠇⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠗⠉⢁⣄⣨⡞⠁⢻⣨⡧⣴⣂⠀⠀⠀⠀⠀⠐⠔⣣
⠀⠀⠘⠃⠀⠀⠀⠈⠃⠀⠀⢀⡼⢿⠈⡀⣄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⢉⡛⠀⠀⣘⠷⠆⢫⡥⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠈⠛⠛⠓⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠃⠀⠀⠤⠀⠀⠀⣴⠟⡩⠊⠑⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠠⣆⣔⣤⠆⠀⠀⡛⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠡⠀⠊⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠴⢀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠉⢉⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣷⢀⣠⡀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⡀⠀⠀⠀⠀⠀⠠⠤⣤⠠⠤⠄
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠁⠀⠁⠠⠀⠠⠐⠒⢿⣿⡄⠡⣷⣤⡀⠀⠀⠀⠀⠀⠀⠁⡐⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⢐⣤⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⣤⡀⠁⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢄⡀⢄⠀⠈⠓⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠣⣀⡙⠅⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢀⠀⠐⢀⠀⠀⠀⠃⠋⠀⠀⠀⡀⠀⠒⢬⣶⡴⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡤⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠉⢀⡤⠀⠸⡓⣄⢀⣴⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠴⠂⠀⠀⠀⠀⠀⣏⢙⣄⡁⢈⣽⣡⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠾⣍⢩⣍⣸⡿⠁⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠛⠛⣀⡙⢡⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠀⠀⠇⠀⠀⠀⢘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣤⠐⠻⣿⠀⠈⠁⠀⠀⠀⠀⣼⠠⠆⠂⢠⡄⠀⠀⠀⠀⠀⠀⠸⠀⡀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟⠁⠃⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⠀⠀⢠⠁⠀⠀⢠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠈⠿⠀⣼⠈⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠗⠀⣦⠀⡀⠈⠁⠘⠂⡀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠃⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⣴⠃⢀⣾⠉⠀⠈⠦⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠄⢀⣀⢾⠋⠉⠀⠀⠀⡦⠑⠮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠟⠧⠘⠉⢙⣾⡇⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢍
⠀⠀⠀⠀⠀⠀⢀⠴⠀⢹⠀⣼⠀⠀⠘⢻⡇⠀⠀⠀⢂⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⡀⠀⠘⣬⣿⡀⢀⠀⠀⠁⢰⡀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡘⠃⠀⠘⡸⣧⠓⠘⣆⠀⠄⢈⣉⣙⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⣄⠑⠆⣠⠚⠣⠸⠀⡄⣈⠇⠀⠀⣤⣤⡄⠀⠴⠶⠶⣾⠓⠖
⠀⠀⠀⠀⠀⠀⣠⠂⣴⣉⡀⣶⢶⡏⣡⡰⣷⢸⡇⠙⣦⠀⠀⠀⠀⠐⠀⠄⠀⢂⣀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠈⠁⢀⣠⣿⣿⢻⠀⠻⠛⡴⣽⠃⢈⡷⣸⠓⠦⠲⣄⠀⠐⠀⠚⠻⠍⢃⡴⠀⡀⠀⠀⠀⠀⢳⡒⠀⠆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠰⢦⡞⣋⣉⠛⡀⠀⠀⠀⠀⠀⠠⠏⠁⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠲⣅⠐⢄⠀⠀⢠⡬⠐⢻⣯⠂⠐⠉
⠀⠀⠀⠀⠀⠀⠀⣚⣓⣛⠉⠾⠃⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠇⠄⣸⠀⠀⠀⠚⠀⠀⠒
⠀⠀⠀⢀⣠⡴⣺⠙⢋⠻⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣄⢸⡏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠠⠴⠚⠓⠲⠂
⠀⠀⠰⠚⠟⣰⠃⠀⣼⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣙⠓⠈⠈⠀⠿⠀⠉⠁⠲⠆⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠻⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠈⠻⢇⠀⡤⣠⠴⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠁⠴⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⠈⡇⠀⠀⠀⠀⠀⠁⠾⠛⠁
⠀⠀⠀⠀⡀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⠀⠁⠠⢀⠀⠀⠄⡁⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢱⠂⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣰⠂⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠃⡀⠁⠀⠀⠀⠀⠀⡨⠁
⠁⠀⠀⠀⠰⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⠌⠁⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠆⠀⠀⡇⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢰⠄⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡑⠈⠛⢇⠄
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⠿⠃⠐⠀⠀⠀⠀⠀⢀⡶⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠋⣰⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠉⣿⠏⣀⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣯⣌⠀⠀⠀⠟⣋⡴⠛⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠻⣿⠿⠀⢠⣤⠀⠉⣅⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠇⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⣰⡶⢌⠀⠈⡤⠾⠋⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣦⣄⠈⢿⣌⣷⣿⣄⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⣿⣿⠀⠛⣿⡇⠀⠹⣿⡝⢿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠃⠼⣿⣿⣿⡄⢠⣿⣿⠶⢶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⠀⣴⣿⣿⣿⣇⣼⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⢠⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠀⣠⠟⠁⠀⠸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠋⢀⣠⣴⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⣴⠏⠀⠀⠀⠀⠙⠛⠉⠀⠈⠙⠻⠿⠿⠿⣶⣿⣯⣽⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⢀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠰⢿⣶⡶⠚⠳⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⣠⣿⣿⣿⣷⣶⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠔⠀⠀⠀⠀⠀⠀⠀⠀⠂⡂⠀⠰⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠑⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡉⠀⣸⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠑⠕⠙⣬⣇⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠶⡄⠀⠀⠀⠀⠀⠀⠀⠈⢛⣷⣄⠀⠘⣞⡟⠀⡴⠮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⣇⣄⣑⣒⣒⣒⣒⣒⣶⣥⣬⡵⠢⣾⣯⣽⣆⡇⠀⣅⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠅⠘⠒⠣⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠌⠚⠥⠤⠤⠤⠑⠚⠤⠵⡄
⠀⠀⠀⠀⠀⠀⠀⢧⠀⡠⠠⠤⠠⠄⠀⠠⠤⠄⠤⠤⠤⠀⡀⠠⠠⠤⠤⠤⠤⠤⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⢀⣀⡁⠀⡀⠀⢀⠀⡀⣀⠀⠀⣀⠀⠀⣀⢀⠀⢀⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠒⠢⢸⢒⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡇⠀⠀⠰⠒⢢⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠉⠀⠀⠀⠀⠀⢣⢱⠀⠀⠀⣧⡀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠀⠀⠀⠀⢀⡴⠃⣜⡔⠲⠀⡗⠷⠂⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠀⠤⠀⡔⠁⠀⠀⠻⠀⠰⡀⡇⠈⠁⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⢸⠀⠚⠀⢇⠀⠀⠀⠀⠀⢀⠇⡇⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠤⠜⢸⠀⠀⣠⣊⣑⣖⣉⣑⣖⣉⣢⡇⠀⢀⠐⠄⠚⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⠉⠉⠉⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠙⢿⣿⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠐⠛⠁⠐⠉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⣀⡀⣀⡀⠀⢀⣀⣀⡀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⢸⣏⡛⣿⣿⣿⣿⣿⣿⡗⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡿⢿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠀⢸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡘⣿⡀⠀⠀⠀⠀⠀⠀⠲⣄⠲⣮⣿⣿⠀⢸⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢳⣄⠀⠀⠀⢀⣴⠗⠦⣽⣿⣿⣿⣆⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣝⣧⣀⣠⣿⣿⣄⠀⠈⠩⢝⡻⢿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣶⣤⣈⡑⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣽⢯⡟⢹⡯⠙⠿⢿⢿⣿⣷⣬⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢱⣿⣾⠁⡌⠃⠰⠁⠀⡸⠻⢦⡙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⢠⡟⣹⠇⠀⠁⢀⣀⣀⠐⡁⠀⠄⠿⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⡾⢠⠏⢀⣼⣿⣿⣿⣿⣿⣷⣆⠐⠀⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⡈⠀⡛⢀⣾⡏⢹⣿⣿⠉⠙⣿⣿⣷⣣⡈⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣠⠖⠉⠀⠼⠁⢸⣿⣷⠀⢿⣿⠀⢠⣿⣿⣿⣷⠑⣸⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⡼⠁⠀⠀⢰⠄⠀⣼⣿⣿⡆⠈⣿⡆⢸⣿⢸⢿⣿⣆⠋⢿⣿⣿⡻⣿⠛⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠘⠃⠀⠀⢀⡆⠀⢠⣿⣿⣿⣿⡀⢸⡇⣼⡿⡏⠸⣿⣿⡄⠈⠿⣿⣷⡘⣷⡀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⢀⡞⠁⢠⣿⣿⣿⣿⣿⠷⢼⣷⣿⣧⡇⠇⣿⢻⣦⠀⠀⣿⣿⢿⠿⣿⡄⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠘⠃⢀⠎⠀⣰⠏⣾⣿⣿⣿⡇⠀⢸⣷⣿⡗⡇⠐⢻⡼⣿⣌⠂⣿⡇⠈⢆⡘⣿⣦⡈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠈⠀⠀⣿⣾⣿⣿⣿⣿⠇⠀⢸⣿⣿⡇⣧⡆⢸⣇⣯⢹⣧⠘⣿⣀⡀⠱⡌⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⣿⣿⣿⣿⡀⠀⣿⣿⡏⠘⢿⠇⢸⡿⣿⠠⠈⣷⠿⣧⣄⠢⡘⢄⣿⣟⠈⢿⣆
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⣼⠏⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⠀⠀⢼⠀⣸⡇⢿⡀⢂⢹⠀⠈⠻⣧⡈⠈⢿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⣸⡿⢾⣿⣿⣿⣿⣿⣿⣿⠀⢿⡏⠀⠀⡼⠀⣿⡇⠘⣷⠀⠀⠆⠀⠀⠉⢿⡄⠈⣿⣇⢠⣿
⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⣰⡾⢋⣴⣿⣿⢿⣿⣿⣿⣿⣿⣇⢸⡷⠀⣼⡃⠀⣿⠀⠂⢸⡇⠃⠸⠀⠀⠀⠸⣿⠀⢻⣃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣷⣿⣿⣿⣿⣨⣿⡟⣿⣿⡿⣹⠾⣇⠐⠈⠃⠀⢸⡀⣛⣸⣷⠀⠀⣷⣶⣦⣴⠟⠠⣾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠛⣿⣿⣷⣿⣿⢁⣿⠀⢿⣄⣀⣈⣀⣿⣿⣿⣿⣿⣿⣤⣼⡏⠉⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⢹⣶⠏⠁⢸⡟⠛⠛⠛⠋⠁⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠺⠿⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣁⣌⣙⣥⣬⣉⣹⣍⣡⣿⣋⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠀⣐⡲⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⣴⣿⣿⣿⣷⣦⠀⠙⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⠙⠙⠛⠿⢿⣿⣧⠀⠀⣄⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠙⠃⠀⠸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠋⠀⢀⣴⠀⠀⠃⠀⠀⠀⠀⢠⡀⢀⢀⠀⢿⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣈⣠⣰⠏⠍⠀⠀⠀⡀⢀⠄⣀⣀⣙⠘⠸⢠⣷⡌⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⠀⡆⠛⠋⠛⠋⠀⠀⠀⠁⠉⠉⠀⣠⢈⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢸⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣐⠀⢼⣄⠁⠀⠀⠀⠀⠀⠀⣤⡎⠄⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⢠⡟⣧⢐⣉⠁⣸⣿⣿⣷⡀⣿⣦⡀⢠⣾⡟⠁⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣼⣧⡙⢿⣿⣽⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⢹⡟⣾⠋⣿⡽⣿⣿⣿⣿⣿⣿⣿⣟⢿⢹⡿⢿⡇⢸⡇⠈⢷⣿⣿⣶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠈⣀⣬⣸⢿⣿⣿⣿⣿⡿⢿⣿⡶⣴⣼⣿⣷⡸⢷⣄⠀⠙⣿⠃⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣝⢻⣁⠀⣼⣿⣿⡿⠿⠋⣠⣾⣿⣿⡿⠹⣯⠈⠻⣷⣶⣤⠀⢠⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡤⠛⣫⣿⣿⣿⣿⡟⣿⡿⠋⠀⠀⣴⢋⣿⣿⡿⠁⠀⠘⣇⠀⠀⠈⠫⡀⠘⢀⠀⠀⢠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡐⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣾⣧⣼⣿⠋⠀⠀⠀⢀⣸⣆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢟⣿⣿⡉⣛⣻⠃⠀⠀⠀⣾⠟⣻⠏⣀⣀⣀⣀⠀⠛⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡇⣾⡿⠋⣟⣿⣿⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⢸⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠉⠀⢰⠙⢫⣿⣤⣾⣿⢛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⡠⠀⠀⠀⠀⠁⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠉⠀⢨⣿⣯⣿⣿⠭⣠⣞⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⡀⠐⠤⡀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⣤⣿⣿⣿⣿⣯⣴⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⡿⠃⠀⠀⠀⢄⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⣠⣾⣿⣿⢻⣿⡿⣿⢿⣷⠘⠷⡀⣼⣿⣿⣿⣿⣿⣿⣿⡄⠐⠋⠤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⣠⠾⣿⣿⣿⣿⡇⠸⢿⠇⠙⢷⣝⡆⠀⠈⠛⢿⣿⣿⣿⡿⠟⠉⠄⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣏⠀⠘⣿⣿⣿⠀⠸⣿⠀⠀⠀⠉⠛⠀⠀⠀⠀⠈⠛⠉⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠈⠻⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⡀⣀⣀⡤⠤⠤⠤⠤⠤⠴⠁⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠹⡀⠀⠀⠀⠀⠀⠀⣔⠙⡀⠀⡘⡅⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⢀⡤⠟⠒⠈⠉⠁⠒⠲⡄⠂⠐⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠈⢍⠑⠂⠤⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠀⠄⡀⢀⠙⢄⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⡰⠁⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠈⢣⠀⠀⠀⠠⠈⠳⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠈⣹⣄⠀⠀⠀⠀⠀⣱⡄⠀⠈⢄⣁⡇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠒⠒⢄⠀⠸⠛⠒⠀⣠⡾⠃⠈⡗⠓⠊⠁⠸⠦⡄⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠣⡀⢀⠜⠀⠀⠀⣠⡾⠋⠀⠀⢰⠀⠀⠀⠀⠀⣀⠦⠒⠉⠀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠁⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⣠⠗⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⠀⠀⢀⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⣀⣀⣠⣤⠶⠛⠉⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠊⠉⠀⠀⠀⠀⣀⡤⠖⠉⠀⠈⡆
⠀⠀⠀⠀⠀⠀⣠⡴⠚⣩⡿⣿⠉⠣⠤⣀⣀⣀⣀⣀⡤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉⠁⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⢸⣇⣤⠞⡿⠁⣿⡇⠀⠦⠄⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⣀⡠⠔⠚⣇
⠀⠀⠀⠀⠀⠀⠉⠀⢸⡇⣰⢿⡇⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠉⠁⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⢸⡷⠀⠉⠁⠀⠀⠀⠠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⠇⢣⡀⠀⠙⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⣄⠠⣠⠀⠈⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠙⠣⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠰⠖⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣖⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⡶⠖⠀⢠⡤⠀⠐⠖⡖⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢸⠱⡄⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡌⠀⠑⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣄⠀⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡈⠧⡏⢇⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡉⠀⠠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡶⠖⠻⣇⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠃⣀⣴⠖⠋⠉⠩⡄⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣾⣿⣯⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠧⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠛
⣿⣿⣿⣿⣿⣿⣿⡇⣿⣟⣼⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣾⣿⠋⠁⠀⢀
⣿⣿⣿⣿⣿⣿⠠⢻⣿⣤⠀⠀⢨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⢀
⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⠃⠀⠀⠀⠀⠀⠀⠁⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡄⣾⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿⠧⠤⡶⠒⠐⠦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣶⣿⠿⠿⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠲⣼⠀⠀⠀⠀⢀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠃⠙⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⣄⠀⠐⠀⣀⣀⣈⣹⣦⡀⠀⢾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⣄⡘⣏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠁⣄⠆⠈⠯⡉⠹⣿⣧⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠈⠙⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠑⢀⠀⠸⠅⢠⡀⠙⣷⣴⣿⣿⣿⣿
⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠢⡀⠀⠀⠘⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢣⣜⡄⠐⣷⣾⣼⣿⣿⣿⣿⣿
⣷⡾⢿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣆⠀⠀⠙⢷⠭⠟⢻⠿⣆⡀⠀⠀⠀⠀⠀⠀⡾⠁⣀⣿⣧⣼⡹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣛⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣧⡀⢰⣿⣴⣺⣧⣤⣤⣉⡒⠀⠀⠀⠀⣼⡗⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢂⢻⣿⣿⣿⣷⣄⢌⠙⣛⣶⣤⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡶⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣯⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣥⠾⣿⣿⣷⣄⡀⠈⠙⢿⣿⣷⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⣷⣦⣅⣻⠿⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠤⠔⠂⠀⠁⠀⠀⠸⠀⠄⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠘⠀⠀⠠⠠
⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠄⠀⢀⡀⢉⠂⠀⠀⡀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠰⢊⡄⠊⠀⠄⠀⠄
⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠀⠀⢀⡠⠀⠄⠃⠀⠀⢀⡤⠊
⠀⠀⣁⠀⠀⠀⠀⠀⠀⡈⢀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⡠⠦⠆⠠⠀⠀⠀⠀⠀⡀⢀⢁⠂⠀⠐⠉⠀⠀⠀⡰⠄⠀⠁
⠀⠘⠉⠀⠀⡦⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠂⠱⠀⠄⠀⠀⠐⠀⠀⠀⢀⠴⠛⠁⠀⠀⠀⡰⠀⠀⣴⣶⣿⡇⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠀⢀⠀⠀⠀⠛⠁⠀⠒⠀⠠⠄⠀⠀⣠⠤⠈⠁⡠⠊⢸⡇⠀⠀⢐⣷⠀⠙⠿⠛⠁⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠄⠀⢀⠈⠁⠈⠃⠘⠀⠀⠀⠒⠠⠄⠀⣀⠀⠀⠀⠀⠀⣀⠔⠈⠀⢀⣾⣿⣤⣤⣾⣿⡀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄⠀⠀⠀⡀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠈⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠰⠀⡀⠀⠀⠠⠀⠄
⠀⠐⠂⠀⠘⠀⠂⠀⠀⠀⠀⠁⠈⠀⠀⠈⠘⡀⠁⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⢈⠑⢦⠀⠀⠀⠀⢹⣻⣿⣿⣿⡏⠀⠃⠀⠐⡀⡀⠀⠀⠄
⠐⠠⣤⣴⡄⠀⢠⣦⣤⠐⠂⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⡒⠄⠀⠀⢀⠀⢀⠀⠨⠀⠀⠇⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠙⠋⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠀⠊⠂⠈⠀⠀⡀⠀⡀⠀⠀⢼⣿⣿⣿⣿⣿⣿⠀⡖⠂⠀⠀⠀⠀⠀⢠
⠀⠰⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⡀⠀⠁⠀⠀⠀⢀⠀⠅⢠⠁⠀⠀⠀⠀⢹⣿⡿⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠀⠠⡀
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⡂⠀⠀⢠
⠀⢀⠀⠀⠐⠂⠄⠀⢀⣤⣶⡀⠐⣿⣿⠇⠀⠐⠁⠀⠀⠀⠀⠂⠀⠀⠈⠢⡀⠀⠀⠄⠀⠠⠄⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⡆
⠀⡄⠀⠀⠀⢠⡄⠁⠀⠙⠛⠁⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⠁⠀⠀⠰
⠀⠀⠀⠦⡀⠈⠁⠐⠂⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠂⠀⠀⠀⠀⠀⠀⠂
⠀⢆⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠠⠄⠀⠢⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠠⠄⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠈⠃⠀⠀⠀⠈⠀⠀⣠⡄⠀⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠐⠀⠐⢰⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠀⠀⠀⠀⡀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠑⠂⠀⠐⠑⠀⠀⠀⠁⠀⠀⠀⠈⠁⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠦⡀⠀⢂⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⠆⢀⡴⠃⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⠸⠏⠂⡈⠀⠀⠀⢤⡀⠀⠀⠀⠀⡤⠀⠀⠀⠀⡰⠸⠇⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠒⠒⠀⣀⠤⠤⠈⠀⠀⠀⠠⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⡀⠃⠠⡤⢀⡀⠒⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠂⣠⡭⠏⠀⠀⠸⣇⠀⠲⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡴⠖⠂⢸⡇⠀⠀⠙⠭⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠴⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠲⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠘⠃⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠃⠀⠀⠀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⡄⠠⠣⠀⠤⠤⢤⡀⠀⢰⠊⠉⠀⠃⠈⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢹⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠱⡘⡀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⠧⠀⠃⠄⠀⢀⣀⣀⣠⣀⣼⡃⠐⡄⠀⠀⠀⠊⠉⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣤⡀⢸⠀⠠⡀⠀⠀⠘⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⢠⣿⡇⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠉⠻⣿⣷⠈⠣⠤⠩⠠⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠀⠀⢸⣿⣧⡀⠀⣠⣿⡿⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠟⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠙⠉⠉⠉⠛⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠙⠉⠉⠉⠉⠋⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⣯⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠘⣧⠀⠀⠀⠀⡇⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣦⠀⠈⢧⡀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⠚⠁⠀⠀⠈⢳⣄⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣤⣴⠆⠀⠀⠰⣶⣏⢉⣷⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢰⠟⢱⡆⠙⣯⡉⠀⡿⠀⣿⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⢸⠏⣧⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠈⠳⠛⠁⠀⠈⣻⣤⣟⠀⣿⠀⠀⠀⣼⠹⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣻⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡟⢻⣆⣀⣄⠀⠀⠀⠀⢠⣾⠉⣹⣽⠷⣿⠀⠀⠀⣟⠀⣿
⠀⠀⠀⠀⠀⣤⣬⣽⣥⣤⡀⠘⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⣩⣟⣥⡀⠀⠀⠀⠈⠻⣏⠉⣿⠀⣿⠀⠀⣤⣬⣯⣥⣤
⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⠀⢰⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⢯⣀⣿⠀⠀⠀⢠⡴⢬⡻⣯⡀⣿⠀⠀⡇⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⢸⡇⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⢶⣞⣉⣀⣀⣀⣀⣩⣤⠶⠶⣤⣼⣦⣼⣃⣀⣉⣻⡶⠴⡇⠀⠀⠀⣿
⠀⠀⣠⠶⠶⡏⠀⠀⠀⢸⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠶⣇⠀⠀⢸⠷⠾⠿⠿⠿⠿⠿⠶⠶⡇⠀⠀⠀⢹⠶⠶⣄
⣠⡞⣁⡀⣀⣙⠂⣛⣛⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣛⣙⣛⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣛⣓⡋⢀⣀⣈⣳
⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣟
⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹
⠀⠙⡟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠛⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡟⠙⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⣿⠋
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⢛⡟⠋⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⠀⠈⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⡟⠋
⠀⠀⠀⢻⠀⢀⡴⠛⠛⠻⢦⡀⢠⡿⠒⣶⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢀⡴⠛⠛⠳⣦⠀⠘⡇
⠀⠀⠀⣿⠀⢸⠁⠀⠀⠀⠘⡇⢸⣇⣀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠻⣶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢰⡏⠉⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠶⢶⡄⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢠⡟⠀⢀⣀⣀⡀⠀⠀⣿⡀⢸⡇⠀⣛⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣾⢁⡴⠋⢉⡾⠛⠃⠀⣿⠉⢹⡏⠉⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⢰⡇⣾⠁⠀⠈⣷⠀⠀⠀⣿⠀⠸⣧⣤⣿⠀⣿⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⣠⣤⣼⡇⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠈⢷⣏⠀⠀⠀⠸⣇⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⠘⡇
⠀⠀⠀⣼⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢹⡄⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢀⡇
⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸⡇⠀⠙⠛⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⢺⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⣿⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⣿⠀⠸⣆⡀⠀⢀⣼⠃⠸⡇⠀⠀⠀⢸⡇⢀⣴⠖⠛⠻⢦⣄⣠⠶⠛⠛⠶⣄⣠⡴⠞⠛⠲⣦⡀⣿⠀⠀⠀⠀⣿⠀⠹⣄⡀⠀⢀⣼⠃⢸⡇
⠀⠀⠀⣿⠀⠀⠈⠙⠛⠋⠁⠀⢐⡇⠀⠀⠀⢸⣇⣼⣁⣀⣀⣀⣀⣹⣇⣀⣀⣀⣀⣸⣏⣀⣀⣀⣀⣈⣧⣿⠀⠀⠀⠀⣿⠀⠀⠈⠙⠛⠋⠁⠀⢘⡇
⠀⠀⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣤⣤⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣶⣿⣿⣾⣿⣾⣿⣾⣿⣿⣿⣾⣶⣶⣶⣶⣷⣤
⣿⣿⣿⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⡛⠘⠃⡼⣉⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠛⢺⠿⠀⢤⠀⣃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄⠀⣀⠀⠀⢀⡿⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢧⡀⢻⣕⣴⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⣰⣿⣇⠈⣿⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⢀⣼⣿⣿⣼⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⢠⡾⠛⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⣰⠋⠀⠀⠙⠟⠁⠙⠻⡿⠿⣴⣾⣿⣿⣿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⣼⣿⣄⢀⣀⣠⣤⣻⣷⣏⣱⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⢀⠀⠀⠀⠀⠀⠀⡐⠉⠀⠈⠐⠀⠀⠀⠀⠀⠠⠚⠁⠙⠠⠀⠀⠀⠀⠀⠀⠔⡙⢈⠂⠄⠀⠀⠀⠀⠀⠀⠧⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠛⠁⠀⠀⠀⠠⠀⡆⠀⠀⡆⠀⠀⠀⠀⠀⠈⢰⠐⠞⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡄⠀⠀⠀⠀⠘⠛⡀⠀⠀⡘⠋⠀⠀⠀⠀⠯⠳⠀⡇⠢⠬⠂⠀⠀⠀⠀⢠⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠠⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⠲⠶⡀⠀⠀⠀⠀⠀⢠⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⢀⠄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠂⠘⠀⠀⠀⢠⡀⠐⠦⠤⠴⠖⠀⠠⠄⠀⠀⠀⠉⠀⠀⠉⠃⠀⠀⠀⠀⠀⢈⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠐⠂⠀⠀⠀⠀⠀⢠⠀⠀⠀⡤⠀⠀⠀⠀⠉⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢸⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠅⠀⠀⠀⠀⠀⠀⠖⠐⠂⠂⠀⠄⠀⠀⠀⠀⡄⠲⠂⠲⠢⡀⠀⠀⠀⠀⢀⡔⢶⡰⠖⢄⠀⠀⠀⠀⠀⠐⢲⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡀⠀⠀⠀⠀⠀⠘⠶⠀⠀⠀⠶⠂⠀⠀⠀⠀⢀⠀⠻⠃⢀⠁⠀⠀⠀⠀⠂⡰⠰⠀⠶⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡹⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠦⠞⠆⠀⠤⠚⠶⠇⠀⠀⠸⠶⠃⠀⠀⠀⠳⠾⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠂⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠆⠶⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠷⠰⠄⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠉⣄⠀⠀⠀⠀⠀⠈⠑⡇⠉⠓⠈⠀⠀⠀⠀⠀⢘⡡⠤⢜⡆
⠀⠀⠀⠀⠀⠰⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠢⠶⠴⠒⠉⠁⠀⠂⠀⠀⠀⣄⡀⣰⣠⣀⣀⡄⠀⠀⠀⠀⠠⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢠⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⣄⠀⠀⠀⠀⠀⠀⢀⡔⠦⠲⣀
⠀⠀⠀⠀⠀⠸⠏⠀⣀⣀⠘⠉⠆⠀⠀⠀⠠⢤⡮⠀⢼⡤⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠠⠏⠀⠘⡀⠀⠀⠀⠀⠈⡄⠶⠆⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠨⢀⠅⠨⡅⢨⡀⠡⠀⠀⠀⠰⠞⡷⠶⠶⠇⠗
⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠒⠒⠂⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠛⠎⣋⢀⡃⣘⠑⠚⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈


⣿⣿⡿⠿⣿⡿⢿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⣼⣿⣿⡿⠟⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠀⠻⣿⠟⢁⠼⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣼⣿⡿⠋⣻⣿⣿⡟⠻⣿⣿⣿⡿⢁⣤⣶⣶⣤⡈⠀⢠⣴⣶⣶⣄⠙⣿⣿⠏⢻⣿⣿⡏⠙⣿⣿⣧
⡿⠛⠁⠀⣿⠿⠋⠀⠀⢿⠟⠁⠀⣾⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⡆⠸⡿⠀⠀⠙⠿⣿⠀⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⢋⣁⣤⣉⠙⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣄⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⠛⣀⡿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢉⣠⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡈⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣆⠸⣿⣿⣿⣄⡙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡛⠛⢻⣧⠀⠀⠀⠀⠀⠀⢸⣿⣆⠘⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡈⠻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣉⣉⣉⣉⣁⣀⣀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣯⣉⣽⣿⣿⣿⣿⣿⣿⡿⠿⠿⠀⣿⣿⣿⣿⣿⣿⠃⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣈⣉⣁⣀⠛⠛⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂
⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠿⡄⠀⢀⢀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢠⣤⣤⡤⣠⡄⠀⠘⠃⠀⠀⡁⠀⢀⠀⠀⠺⠿⠂⠀⢀⣄⣀⣄⡀⠀⠀⠀⠀⡄⠀⠈⠀⢠⡄
⠀⠀⠸⢿⣿⡽⠿⠓⠀⠀⠀⠀⠐⠛⠉⠙⠋⢢⡀⠀⠀⢀⣾⣶⣶⣾⣿⡆⠀⠀⠜⠱⢀⠀⢰⡵⡰
⠀⠀⠀⠀⠁⢻⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠄⠀⠈⠉⣿⣿⣿⣿⡆⠀⠀⠓⡁⢺⠦⠚⠧⠘
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢠⣤⠤⠀⠀⠁⠀⠀⠀⠀⠙⡿⠀⠘⠇⠀⠀⠀⠈⠀⠀⠀⣤⣄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⢠⣻⣼⣃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⣏⠋⠀⠀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡾⢀⣀⣀⢷⠀⠀⠀⡀⣤⢄⡐⠊⠉⡉⢿⠄⠀⢀⣄⣠⣤⣴⠀⢀
⠀⠀⢰⡧⠛⢡⢤⠘⠿⣧⠀⠘⢻⣯⣿⣯⡼⠃⠀⠰⠡⢷⣓⢀⣀⠀⠓⠋⠀⠀⠉⢻⣿⡿⣯⠔⠉
⠀⠀⠘⡧⣴⠚⢀⣌⣦⣾⠀⠀⢰⠍⠋⢹⠀⠀⠀⠀⠀⠩⠉⠉⠀⠀⠀⠀⡀⠀⠀⠐⠛⢻⠞⠇
⠀⠀⠀⠑⠈⠓⠿⠿⠛⠁⠀⠀⠐⠶⠿⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠴⠶⠶⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⣶⡶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⣦⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⣤⠾⠋⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠈⠻⣿⣍⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⣩⡽⠛⠁⠀⠀⠀⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠻⣦⡀⠀⠀⠀⠙⠻⢶⣤⣠⡤⠶⠦⣤⣠⠖⠲⠶⣤⣴⠞⠉⢀⡄⠀⢀⣠⠶⠛⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠉⠻⣄⠀⠀⠀⠀⢀⣀⣉⡙⣷⡦⠄⠙⢷⡄⠀⠀⠙⢷⣶⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠙⠛⢿⡄⠀⠀⠻⠀⠀⠀⠀⠹⣦⣤⠴⠆⠀⠀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣬⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣠⣤⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣘⣷⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠉⠉⢻⡟⠛⢻⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣾⠃⢀⡿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢰⡏⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠁⠀⣿⠁⠀⠀⣿⠀⠀⠀⠀⣆⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠀⠀⠀⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⠀⠀⣿⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢻⡀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢻⡇⠀⢸⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠃⠀⢹⣇⢀⣤⣤⣶⣦⣤⣤⣤⣤⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣠⣤⣶⠶⠿⠿⠿⠿⠷⢶⣦⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠛⠻⢿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⠗⠈⠀⠀⠀⠿⠓⠂⠀⠘⡿⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠀⠀⢰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⠠⣤⡈⠲⡀⡠⠺⣿⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠈⠉⠑⠎⠀⠀⠈⠁⠀⡝⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠂⠈⠀⠀⠲⣦⠀⠀⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡏⣕⣶⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣮⣹⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠘⣧⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⡟⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠁⠀⠀⠀⠢⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⢤⣮⠤⠀⠀⠀⢀⠄⠀⢄⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⡿⣽⢮⣽⠤⠄⠀⠠⢮⠀⠀⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢻⠇⠉⣷⠀⠀⠀⠚⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠈⢧⡐⠁⠀⠀⠀⠀⠙⢦⣀⢠⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠁⣰⠀⠀⠀⢀⠀⠀⠀⠀⠒⣤⣤⣀⣤⣽⣷⡷⠿⠓
⠀⠀⠀⠀⠀⠀⠀⢤⡻⣚⠾⠁⠀⢠⣤⠈⣐⠀⠀⠀⠚⠉⠀⠀⠀⢰⡇⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠠⠶⡚⣛⠿⢉⠈⠀⠀⠀⠹⡃⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠈⢳⡄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠚⠁⠺⠀⠀⠑⠀⠁⠢⢰⡧⠀⠀⡀⠀⠀⢀⡰⢿⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠃⢄⠀⠀⠀⠀⠀⠀⠁⠄⠈⢧⠠⠀⠠⠈⠣⠈⠀⠈⠃⠀⠀⠀⠀⠀⠀⣀⡠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠄⢄⢀⣈⢀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⢀⠀⠤⠠⠖⠂⠈⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠀⠘⠀⠈⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⢦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠑⡌⠢⡀⠀⣠⠖⠋⢁⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⡄⡷⡺⠁⠢⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠂⣉⣉⢉⡙⠓⠶⣧⡘⣳⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣛⣋⣩⡴⠶⠶⣏⠳⣚⠓⠶⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⣡⠔⠋⣁⣀⣤⢼⠃⢰⡘⢧⡀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⡡⠔⠊⢁⢟⠔⢻⠠⣘⢇⢸⡏⠓⣆⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠕⠉⠀⠀⠀⢸⠎⠀⠰⠀⠹⠈⣼⠀⠀⠈⠳⡀⢟⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⢣⡇⠀⠃⠀⠀⠀⠀⠈⢾⠑⣽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢤⠠⠤⠥⢤⣀⣀⡀⠀⠀⠀⠀⠀⠈⠳⣹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡶⠋⠀⢠⠃⠀⠀⠑⢄⠀⠀⠈⠑⠮⡉⠒⠦⢄⡀⠀⠀⠘⢾⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⡠⠋⠀⠀⠀⠎⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠈⠳⣄⠀⠈⠒⢄⡀⠀⠻⣈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⡜⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢀⣀⡈⢢⠀⣀⠄⠝⢦⡀⠙⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡸⠀⠀⠀⠀⠀⠆⠀⢀⡠⢤⢤⠤⠤⡀⣦⢖⡩⠲⣄⣈⠉⠷⠹⢩⠱⠴⠒⣄⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠒⠢⠤⣀⣃⠤⠒⠂⠒⠤⣐⡰⠊⠀⠈⡜⡀⠀⣈⠵⠉⠩⠤⠀⠈⠻⣏⠱⡀⠚⠒⣿⣿⣓
⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠐⠒⠐⠂⢀⡸⠷⠖⠂⠐⠂⠀⠈⠐⠂⠒⠀⢳⢱⠚⠒⠒⠒⢂⣒⠒⠒⠒⠒⠒⠚⡒⠒⠛⢃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆⠀⠀⣠⣿⢠⢻⢉⠏⡙⢉⡿⠋⠀⠀⠈⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢐⡢⠀⠀⠀⠀⠀⠀⠀⠘⡸⣀⡼⠫⢣⢢⠆⠜⡰⢡⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠆⠒⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⣿⢕⡵⡡⢃⠎⡴⣱⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠠⠤⢜⡫⠔⡩⢊⠔⣡⢎⡼⠃⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣰⡞⠃⠖⢭⣖⣡⠞⢡⡞⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠆⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⣃⣯⣶⠿⠲⠦⣤⣤⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⠦⣤⣄⣀⠀⢁⣼⠃⠀⠀⣀⢤⣌⡉⠙⠋
⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⢻⡿⠁⠀⠀⡞⡽⡺⠯⡙⡄
⠀⠀⠀⠀⠀⠀⠣⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠻⣀⡇⠀⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠒⠒⠆⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⠀⠰⢮⡒⢺⠟⠊⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣫⠳⠛⢉⣿⣁⡀⠀⠿⣓⣲⠖⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⡤⣤⣀⢶⠺⠯⣥⠀⠀⠀⠠⢴⠒⠷⠀⠀⠈⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡤⠛⠆⠈⠘⢉⣙⣷⡆⣰⡞⡛⢲⣿⡚⠦⣤⡶⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢩⢋⡤⢦⣄⢦⢍⡶⢧⡏⡇⡷⠯⣬⠉⠀⠁⠋⠒⣤⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠘⠋⠀⣿⡿⠞⠀⠉⠈⢰⢱⠁⠺⠕⣲⣶⡀⣌⢻⡉⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⢱⠀⠀⠀⢀⡏⡸⠀⢀⣔⡝⠉⣇⠏⠹⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠘⣈⣆⣠⡤⠼⢠⣇⣴⣫⠎⠀⠀⠉⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠽⣤⣤⣤⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣭⠭⠥⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣀⣛⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠭⠭⣭⠭⠤⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠭⢭⡭⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢯⣭⣤⣤⣤⠀⠀⠀⠉⢩⣭⠭⠽⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠚⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠈⠙⠲⡄
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡎⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡇⠀⡀⠀⠀⠀⠋⢀⢀⢀⠀⠀⠈⡗⠋⠉⠙⢆⠀⠀⡴⠋⠀⠉⢺⠁⠀⠀⡀⠀⡀⠽⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢧⠀⠀⠘⠍⣘⡃⢪⣤⠄⠀⠀⠀⣧⠀⠀⢀⡞⠀⠀⢳⡀⠀⠀⡾⠀⠀⠀⠠⣤⠕⢘⣁⠉⠂⠀⠀⡼
⠀⠀⠀⢸⠀⠀⠀⠐⠀⡔⠝⠆⠮⠦⢀⠀⢸⠀⡠⠊⠀⠀⠀⠀⠙⢄⠀⡇⠀⠀⠴⠴⠰⠋⢦⠈⠤⠀⠀⢀⡇
⠀⠀⠀⠈⡇⠀⠈⠀⠀⠀⢝⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⡎⠀⠀⠀⠑⠀⢸
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⡇⠀⠀⣀⣀⣀⣀⠤⠤⠤⠴⠒⠓⢤⠀⠀⠀⠀⠀⢀⠴⠛⠒⠦⠤⠤⢤⣀⣀⣀⡀⠀⠀⣸
⠀⠀⠀⠀⡔⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣄
⠀⠀⠀⠀⢇⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠤⠖⠒⠋⠀⠀⠀⠀⠀⠀⠑⠒⠲⠤⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠚⠷⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠐⠀⠀⠌⠙⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠂⠀⠠⠭⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣫⠇⢈⠠⠄⠀⠈⢻⣤⠚⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠰⠆⠃⠀⠀⠀⠘⠳⡀⣈⠁⣀⢤⠀⠀⠰⠓⠀⢀⠀⠀⠀⠀⢀⡀⠶⢀⠀⠀⠞⠉⠁⡤⢨⡄⠀⠀⡠⠟⣷
⠀⠀⠀⠀⠀⠀⠆⠩⠁⠀⠀⠀⠄⠀⠒⣀⡤⠿⢥⡨⠀⠛⣶⡃⠀⠈⠿⠂⣴⠐⠋⠘⠁⠀⠀⢀⠀⠀⠀⠀⠸⠃⠰⢀⣙⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣼⠛⠀⠀⠀⢹⠄⠀⠀⠀⢀⠀⠐⠀⢄⠀⠀⠀⠐⢀⡀⢀⡀⠤⠀⠀⠠⢉⡐⠂⡀⠃⠂⣠⡀⡠⠾
⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⡀⠮⠛⠗⠀⠰⣲⠦⡬⠖⠀⠀⠀⠀⠀⢀⠀⠀⠆⠀⣀⠤⠈⡐⠉⡀⡀⠀⠀⢀⠐⠁⠀⠙⣤⠬⠠⠁
⠀⢀⣄⣠⢌⣰⠀⠁⠀⢐⣁⣀⣴⠂⠛⣁⠀⠀⣙⡘⠄⠀⠀⠀⠀⠀⠀⢰⢠⠒⠄⠡⢣⠌⠡⠀⠀⠀⠨⠀⠄⠂⢀⠐⠛⠁⠀⠀⠁
⠀⠂⡀⠀⠀⠁⢁⡀⠐⠌⠀⢋⠁⠀⠘⠟⠀⠘⢛⠧⠁⠀⠀⡀⠀⣀⣀⠀⠁⠀⠀⠀⠀⠲⡄⠀⠀⠀⠀⠠⡄⠀⠄⢀
⠠⠀⢐⡀⠐⠒⡘⠁⠀⠀⠀⠐⠑⠀⢀⠀⢁⠀⠁⠀⠀⠀⠐⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠑⢀⣶⣅⡠⢌⣤⣒⣤⠄⣐⣧⡀
⠐⠀⠌⠀⠉⠀⠂⠀⠀⠀⠀⢀⠀⠐⠾⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⠀⠀⠀⠘⠛⠋⠈⢿⣿⣙⣛⣩⢥⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⣀⠀⠈⠀⠹⠄⠲⠤⣄⡀⠀⢀⠀⠀⠀⠀⠠⢀⠁⠀⣀⣠⢰⡞⠒⠀⠀⠀⠀⠀⠀⠘⢇⠐⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠊⠉⣻⡉⠀⠀⠀⠀⠀⠀⠀⠈⢳⡦⣤⡴⣄⠀⢈⠀⠘⠣⡤⠁⠀⢄⠠⠀⠈⠤⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠑⠒⠤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠌⠐⣎⠠⣀⠀⠀⠀⠠⠖⣊⠓⠂⠰⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⢀⠀⠀⠡⠈⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠀⠠⠈⢃⡜⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠐⠀⠀⠒⠬⡑⠢⢀⡀⠀⢀⣀⠠⠤⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡔⠁⠀⠀⠀⠀⠀⠀⠑⠂⠀⠤⠀⠀⠐⠀⠉⠉⠁⢂⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⢀⠔⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⠉⠈⠉⠁⠐⠀⠬⢐⣒⣒⠨⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠄⠄⢀⡀⠀⠀⢀⠔⣊⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠮⡀
⠀⠀⠀⢀⡠⠤⠀⠀⠤⠤⠀⣀⡀⠀⠀⣀⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⠔⡡⠒⠀⠉⠉⠁⠐⠒⠀⠀⠈⠁⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠒⠀⠤⢀⡀⠀⠀⠀⠀⢀⠆
⠀⡌⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠎⠀⠀⠀⠀⠀⠀⠈⠉⠐⡂⠁
⠀⢃⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢠⠊
⠀⠈⢄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⡄
⠀⠀⠀⠑⠬⠁⣒⣒⠀⠠⣀⡠⠐⠈⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠈⠣⡂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠑⠐⢺⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢀⡠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠈⢦⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘⣈⠱⠒⠦⠤⣄⣀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠢⡀⠀⠀⠀⠀⠀⡄⠠⡩⢄⢋⡙⠖⠲⡤⢄⣉⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠈⢢⠀⠀⠀⠀⡇⠀⠀⠈⠁⠘⠈⠳⠒⠦⠄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⢀⠂⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠤⠜⢁⣁⠄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⡰⠠⠋⡉⡰⠒⠃⡤⠴⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠥⠖⡚⣠⠆⠎⠉⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠒⠁⠁⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠈