Satellite Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠈⠁⠀⠀⠀⠀⠀⢀⣩⣶⡦⣄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⢋⡴⠋⠉⠢⡀
⠀⠀⠐⢌⠭⠄⠒⣓⣤⡀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠋⢁⡴⠋⠀⠀⠀⠀⣸⣆
⠀⠀⠀⠀⠙⠮⡉⠵⢦⡛⣢⡀⠀⠀⢀⠎⠀⣀⣀⣤⣴⣶⣿⠿⠛⠉⠀⡠⠖⠁⠀⠀⠀⠀⣠⣾⡿⣫⣦
⠀⠀⠀⠀⠀⠀⠈⠲⣉⠽⠦⣞⢓⣄⣼⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢋⣴⠟⠉⠆
⠀⠀⠀⠀⠀⠀⢀⡠⠤⢵⣎⣉⠳⣤⠝⠀⣀⣀⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⢁⡴⠋⠁⠀⠀⣰
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⢷⣎⠙⠍⣴⣶⡶⠵⠀⢇⡁⠀⠀⣀⣤⣴⣾⡿⠟⢉⡠⠞⠁⠀⠀⠀⣠⡾⢿
⠀⠀⠀⠀⠀⠂⠀⣀⡠⠄⠀⠘⢸⡄⢰⠚⠈⣣⣤⣠⣶⣶⣾⣿⠿⠟⠋⣁⠤⠒⠁⠀⠀⠀⠀⣠⠾⠋⣠⡾
⠀⢀⠤⡤⠴⢯⠉⠒⠂⠉⠀⠀⡿⠃⢺⡿⠿⢏⡩⠷⢶⣛⣥⡠⠔⠒⠉⠀⠀⠀⠀⠀⢀⠤⠚⠁⣠⡾⢛⠃
⠀⠀⠉⠀⠀⠀⠣⡀⠀⠀⢀⡼⠁⠀⠀⢳⡶⠶⠗⠪⣉⠽⠦⡚⢢⣀⠀⠀⢀⡠⠔⠊⠁⣀⣴⠟⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠑⡄⠀⠀⠈⠲⣌⠩⠷⣔⠓⣎⠁⢀⣠⡴⠞⠋⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⠛⠋⠉⢑⣆⣉⣁⡤⠝⠋⠁⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⢋⣉⣀⣀⣀⠀⠤⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠴⣅⠈⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠔⢧⡀⣨⠿⣅⢈⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠙⢦⣠⠟⢧⣀⡼⠻⣄⢈⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡴⠛⢦⣀⡼⠻⣄⣨⠷⣄⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡴⠛⢦⣠⠞⠳⣄⡨⢃⠔⠁⠙⢿⣿⣦⣀⠀⠀⠀⢀⡨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡞⠙⢦⣠⣺⠔⠁⠀⠀⠀⠀⠙⢿⠇⠉⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢊⠔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣠⡪⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⡠⡾⠋⠳⣄⡼⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⡠⢪⡊⠙⢦⡴⠋⠳⣤⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⢶⡋⠙⣦⡞⠉⠳⣤⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡁⠙⣦⡞⠉⢳⣴⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡁⢙⣶⡋⠈⢳⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡀⢙⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⢷⣄⢀⣴⠟⠁⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁⠀⠀⢀⣽⢿⣅⠀⠀⢀⣴⠟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠈⠳⣦⣴⠟⠁⠀⠙⢷⣴⠟⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣧⡀⠀⢀⣴⠟⠻⣤⠀⢀⣴⠛⠙⢷⣄⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢻⣶⡟⠁⠀⠀⢈⣿⣏⠀⠀⠀⢀⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢃⠀⢀⡤⢄⠀⠀⠀⢀⣴⠟⠙⢷⣄⢀⣴⠟⠉⠙⢦⣀⡴⠋⠁⠈⠻⣦⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢈⠔⠃⠀⠀⠑⢄⢰⣟⠁⠀⠀⢀⣽⢿⣅⠀⠀⢀⡼⠛⣧⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⣀⡀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠑⢝⢷⣄⣴⠟⠁⠀⠙⢷⣴⠋⠀⠀⠀⢹⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⣷⣄⠀⢀⡴⠋⠙⢷⣄⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢟⢷⣏⠀⠀⠀⢀⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⠷⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⢷⣌⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁⠀⠀⢀⣽⢷⣼⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠈⠳⣄⣴⠟⠁⠀⠙⢿⣄⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠰⢀⡀⠀⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣧⡀⠀⢀⣴⠟⠳⣄⠀⢀⡴⠋⠙⢷⣄⠑⢄⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠈⢻⣶⡟⠁⠀⠀⢈⡿⣏⠀⠀⠀⢀⣽⠇⠈⠱⢄⠠⠎⣀⢀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⢀⣴⠟⠙⢦⡀⢀⣴⠟⠉⠻⣦⣀⡴⠋⠀⠈⠳⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⢀⣴⣟⠁⠀⠀⢀⣽⢿⣅⠀⠀⢀⡼⠻⣦⡀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁⢠
⢀⣴⠟⠁⠙⢷⣄⡴⠋⠁⠀⠙⢧⣴⠋⠀⠀⠈⢻⣶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⣼⠀⢹⠀⢺
⠻⣧⡀⠀⢀⣴⠟⢷⣄⠀⢀⡴⠋⠙⢷⣄⢀⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢀⠔⠛⠛⠋⢀⡴⠃⢀⡟⠀⣼
⠀⠈⠻⣶⡋⠀⠀⠀⢙⣷⣏⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢀⠔⠁⠀⠐⠛⠛⠉⢀⣠⠞⠁⣰⠏
⠀⠀⠀⠈⠻⣦⣀⡴⠋⠀⠘⢳⣤⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠚⠚⠛⠉⢀⣠⠾⠃
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⣠⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣈⣻⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢉⠶⣁⡠⠓⣄
⠀⠀⠀⠀⠀⠀⠀⢞⢁⠜⢣⡠⠊⡢⡎⢀⠷⢄⠀⠀⠀⠀⢀⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⠊⡨⠮⣀⠜⢣⡄⠊⣱⠄⢀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡔⢉⠶⡋⣨⠞⢉⣶⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⣠⡞⢁⣀⢘⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡅⠉⢁⡼⠋⡱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣿⣿⣿⣿⣿⠿⣁⡴⡋⡈⠲⣀⠜⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣆⠀⠉⠁⠐⢏⡠⠘⢢⡔⠉⡲⡊⡠⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠙⢿⣿⣿⡄⠀⠀⠀⠀⠑⢶⠁⡸⠪⡠⠊⢣⡔⢉⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠎⠁⠀⠀⠀⠙⢿⠃⠀⠀⠀⠀⠀⠀⠙⢤⠊⢉⠶⣁⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠑⠀⡀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠈⠲⣄⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠈⡢⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠁⠀⠀⠜⠃⢁⠀⡀⠀⠀⠀⠀⠀⠁⠠⡀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠀⢀⠌⢀⣇⢀⣁⣢⣐⢄⠀⠀⠀⠀⡀⠌⠂
⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣌⢊⠀⠀⢀⠂⠀⠈⠉⠌⠀⠌⠀⠀⠈⠀⠐⠈
⠀⠀⠀⠀⢀⠔⠁⠀⢀⣮⣤⣤⡄⠀⡑⠠⡀⠀⠌⠀⠌
⠀⠀⡀⠈⠀⠈⠲⠀⠀⠄⠀⠀⠉⠑⠢⣴⡄⠩⠀⠌
⡀⠊⠈⠐⠄⡀⠀⠑⠀⠈⠄⡀⠀⠀⡠⠀⠙⣤⠋
⠀⠑⠠⡀⠀⠈⠐⠄⡀⠑⢨⢦⡔⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠑⠠⠀⠀⠌⠂⠁⠁⠈⠉⠊⠂⠐⠊
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠒⠒⠉⢉⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢸⡇⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⠻⣄⠀⠀⠀⠀⡴⠁⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡹⣆⣠⠤⠚⠋⠹⢯⡕⡄⠀⢀⣀⣠⠤⠤⠒⠂⠉⠉⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠓⠘⡄⠀⠀⠀⠀⢈⠜⡽⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⡠⠶⠖⠒⠒⠒⢲⠖⠉⠀⠀⠀⠀⠀⠀⣠⠔⣩⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⢀⡴⡇⠀⠀⠀⠀⣀⡠⠖⠋⠁⠈⠉⠙⠛⠀⢤⠲⡖⠢⠤⠤⠤⢄⡴⠃
⠀⢀⠴⠋⢀⣀⡠⠤⠔⠒⠋⠀⢳⣒⣒⣮⡭⢥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡱⡀
⠐⠓⠊⠉⠉⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⠞⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢷⣦⣄
⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⢀⡎⠀⠈⢦⡀⠀⠀⠀⢀⣀⠤⠖⠉⠀⠀⠀⢙⡷⡷
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠉⠛⠛⠉⠏⠀⠀⠀⠀⠀⢀⡠⣪⠜⠁
⠀⠘⠓⠒⠒⠒⠒⠒⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣠⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣦⣀⡀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡹⠿⣫⢆⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣎⠹⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠉⠀⡀⠈⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣴⣿⣝⣿⣿⣿⣶⣭⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣿⣿⣯⣿⠟⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠐⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓⠀⠀⠈⠀⠀⢀⣴⣞⢻⣿⣿⣿⣿⣿⡷⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢀⢀⣴⣿⣿⣿⣧⣙⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣗⡘⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣝⢿⣿⣿⣿⣿⡿⠃⠉⠈⢀⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣼⣿⣿⣿⣝⡿⡷⠯⠀⠈⢀⡀⠀⠀⠢⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣎⡻⣿⠎⠛⠋⠀⠀⠀⢀⡀⠀⠀⠤⠀⠀⠈⠢⡀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠿⠏⠁⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠆⠀⠀⠈⠢⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠱⢄⠀⠀⠙⠄⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠓⢄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠈⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄


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

⠀⠀⠀⠀⠀⢨⠁⡆⠠⠀⡤⢄⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠁⠀⢀⠃⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⠘⠀⠆⠸⠀⠃⢰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⡺⠅⡀⠉⢂
⠀⠀⠀⠀⢠⠀⡆⢠⠀⡄⢰⠀⡄⢀⠕⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠀⠇⠀⢀⠴⠀⢠
⠀⠀⠀⠀⡄⢀⠀⠀⢀⠁⠈⢀⠀⠸⠒⠢⡄⢠⣀⣀⣀⡤⠚⠁⠀⠀⢠⠜⠁⢀⢴⠃
⠀⠀⡀⠀⠃⠘⠀⠃⠘⠀⠇⠘⠀⠀⠀⢀⠔⠉⠀⠐⠀⠈⡐⠄⠒⠈⠀⠀⢨⠄
⠀⠀⠀⠉⠀⠓⠐⠀⠆⢸⢀⡆⠰⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⡆⠀⠰⠀⠆⢠⠀⡜⢀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡀⠀⠈⠢⠀⠀⠀⠀⠀⣿⠀⣀⠂⡘⠀⠆⠠⠀⡌⢀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠀⠀⠈⠳⡀⠀⠀⠀⠀⢀⠀⠀⠁⡌⢠⠁⡘⠀⠂⠰⠀⡄⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠀⢰⡏⠒⢄⠀⠀⡠⠃⠀⠀⣀⠖⠁⠀⠀⠠⠀⡄⢠⠁⡈⠀⠃⠰⠀⠆
⢠⠀⠉⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠉⠉⠠⢘⠓⠀⠤⠔⠃⠀⠀⠀⠀⠀⠰⠀⠆⢠⠀⠈⢀⠃⡐
⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠁⠢⡀⠀⠀⠀⠀⠀⠈⠐⠂⠼⢀⡀⠀⠀⡌⢀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠒⢌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠆
⠀⠀⠀⠀⠑⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣹⣂⡶⠄⠤⠄⠀⠀⣀⣀⠀⠀⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟⣬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⣿⢯⣾⣷⣿⡳⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣿⣿⣿⣿⣿⢋⡝⣛⣻⣿⠛⠎⠛⢿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⣿⢿⣿⣿⡿⣷⣿⣿⣿⡾⣿⣿⠏⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡷⠄⠾⠛⠻⣛⣿⢶⣟⣿⣿⡿⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡛⣿⣿⣻⣿⣿⣷⣶⣭⣝⠿⡻⣮⣹⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣿⣿⣿⡻⣿⣷⣻⣿⣿⣿⣿⣿⡷⣦⡸⣯⠋
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣯⣿⢿⣿⣯⡿⣷⡗⣿⣿⣻⣵⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣯⣟⠿⠟⠰⠇⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⡽⢷⣄
⠀⠀⠀⠀⠀⠀⢠⡶⠳⣆⠀⠀⠀⠀⠀⠀⢀⡶⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⢷⣄⡴⠋⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠘⠷⠶⢯⡀⠀⠀⠀⢀⡴⠋⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⡴⠛⢷⣄⠀⢀⡴⠋⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡴⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠳⣄⢀⡴⠋⠀⠀⠀⢙⣷⣏⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢰⡟⠁⠀⢀⡼⠿⣄⠀⠀⢀⡴⠋⠀⠙⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⡾⠛⣷⡴⠛⢦⡀⠀⠈⠳⣤⡴⠋⠀⠀⠈⢳⣶⠋⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⡴⠋⠀⢀⣴⠟⢦⡀⠀⠈⣳⣄⠀⢀⡴⠋⠉⠳⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⣀⣀⣠⣤⣾⡋⢀⡴⠋⠀⢀⣴⠟⠁⣠⡾⠛⢦⡾⠋⠈⠻⣏⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢈⡿⠋⠀⠀⣴⠟⠁⣠⡾⢫⡞⠻⣆⢙⣦⡀⠀⠈⠳⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣰⠞⠁⣠⡾⠋⠀⠈⠳⢞⣷⠟⠁⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣠⡾⠋⠀⠀⠀⢀⣴⠟⠁⣠⡾⠋⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣄⠀⠈⠻⣦⡀⠀⢀⣴⠟⠁⣠⡾⠋⠀⢀⣴⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣽⢷⣄⡶⠋⠻⣶⡟⠁⣠⡾⠋⠀⢀⣴⢟⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⢷⣄⣴⠟⠁⠀⠙⢷⣄⠀⠈⠻⣾⡋⠀⢀⣴⢟⣵⠟⠁
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣴⠟⢷⣄⠀⢀⡴⠋⠙⢷⡄⠀⠈⠻⠶⠿⢷⠟⠁
⠀⠀⠀⢀⣴⠟⠻⣄⣀⣴⠟⠁⠀⠀⢙⣷⣏⠀⠀⣠⡾⠃
⠀⠀⢠⡟⠁⠀⢀⣼⠿⣅⠀⠀⢀⡴⠋⠀⠙⣷⡾⠋
⠀⠀⠈⠻⣦⣴⠟⠁⠀⠈⢳⣶⠛⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠈⠻⣦⡀⢀⣴⠏⠉⠳⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣯⡁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠦⣀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠣⠑⠀⡀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠀⠀⠀⠌⢀⣅⡀⠑⢀⡀⣄⠀⠀⠀⠀⢀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡈⠀⠀⠀⠈⠀⠈⠉⠌⠀⠈⠁⠀⠀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠁⣞⣁⣀⠀⢀⠑⠄⡀⠀⠄⠀⠌
⠀⠀⠀⠀⠀⠀⡀⠀⠁⠁⠠⠂⠀⡄⠀⠈⠉⠓⠤⣀⡈⢊⠀⠄
⠀⠀⠀⠀⡀⠊⠁⠂⢀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠌⠻⣄⠎
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠄⠀⠀⢨⢦⡄⠈⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢀⠄⠂⠀⠈⠉⠁⠀⠂⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠏⡆⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢦⡀⠀⠀⠀⠀⠀⠀⡜⡎⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⢆⠳⡙⢆⠀⠀⢀⡤⡞⡜⠀⠚⠂⣿⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⠀⢀⡱⠬⠖⠛⠉⠈⣻⣰⡥⠤⠖⠊⢫⢳⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠴⠗⠒⠉⠁⣀⡠⠤⠖⠒⠉⠉⠈⢷⡀⠀⠀⠀⢇⢇⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠉⢁⣀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⢀⣀⡬⣧⢀⣀⠤⢾⣿⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡳⣤⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⡺⠯⠗⠓⡏⠀⠀⠀⢸⣼⣿⡿⠟⢡⡾⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⠼⢄⡀⠳⠗⠀⠀⠀⠀⠀⣀⡤⡴⣖⡹⠷⠛⠋⠁⠀⠀⠀⠀⣇⣀⣠⣴⣾⠟⠋⢀⡴⠋⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⣾⣷⡜⠌⠓⢖⣉⡹⠀⢀⣤⢴⣲⣹⠓⠛⠉⠀⠀⠀⠀⣀⡠⠤⠔⠒⠻⡍⠳⡜⢯⠁⣠⠴⠋⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠚⠉⠁⠀⢀⣀⡤⠴⠒⢎⠉⠀⠀⠀⠀⠀⠀⠈⢦⡈⢢⡙⣅⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣽⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⠀⢀⠀⢀⣀⣀⡠⠤⠶⢶⣯⣭⣭⣭⣭⣭⡭⠭⠶⣒⡤⠄⣀⠀⠀⠀⠑⣄⠙⣌⣳⡀⠀⣀⣴⣿⣿⠟⣡⣾⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠁⠒⠫⠵⠶⠶⠮⠍⠚⢉⣤⣾⣿⡿⠋⣁⣾⠟⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠉⣠⡾⠋⠁⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠉⣀⡴⠚⠁⠀⠀⠀⠀⢀⣴⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠋⠁⣠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣤⠿⠋⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠋⠉⢀⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠉⠉⢁⣀⣠⡤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⣠⣶⡿⠋⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⣀⣴⡾⠛⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⢀⣠⣴⠾⠛⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣶⡶⠶⠞⠛⠋⠁⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣀⣀⣀⣤⣴⣶⡾⠟⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠥⠄⣀⣀⣀⣀⠠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠦⠤⠤⠤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⣴⠛⠙⠓⠢⣄⣏⠉⠉⠉⠉⠙⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣠⣀⣀⣀⣀⣀⣹⡀⠀⢰⠟⠛⠲⠤⠤⣠⣏⢻⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠈⣇⢀⡟⠀⠀⠀⠀⠄⠀⠘⣿⡟⠒⠒⠒⠒⠒⠒⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢹⢹⣁⣀⣀⡒⠂⠀⠠⢀⣿⣧⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⠉⠉⠉⠋⠉⠁⠘⡄⠄⣠⠤⠤⡉⠁⠁⢰⡶⢾⡤⠤⠤⢄⠠⣤⡄⣧
⠀⠀⠀⠰⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡄⠀⢸⡄⠀⠀⠀⠀⠀⠀⢱⡝⠀⠀⠀⠈⢂⠐⣼⠁⢸⡇⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠲⢤⡀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠁⣀⣤⠀⠷⢤⣤⣤⣤⠤⣤⣼⠁⠀⠀⠀⠀⣼⠉⠁⠀⠈⣷⣀⠀⠀⠀⠀⠀⠀⣧
⠀⠲⠤⡀⠀⠀⠀⣰⠋⠀⠀⠈⢳⠀⠀⠀⠈⠁⣀⣀⠀⠀⠀⠀⠀⠀⠠⡿⠐⠒⠒⢄⣠⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠋
⠀⢀⣀⡀⠀⠀⠀⢻⠀⠀⠀⠀⣸⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣄⡤⠟
⠀⠀⢀⡀⠀⠀⠀⢨⠗⠒⡶⠞⢷⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠈⠀⠀⠀⠀⡾⢀⠀⠇⣤⠘⣇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠙⠀⠉
⠀⠀⠀⠀⠀⠀⣸⠃⢸⠀⠀⠘⢆⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠚⠛⣦⡀
⠀⠀⠀⠀⠀⢠⠇⠀⠻⠀⠀⠘⠛⠀⢻⡀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀⠀⢠⠞⠓⢦⠀⠀⣀⡴⠋⠁⠀⠀⠀⢀⡏⠹⡄
⠀⠀⠀⠀⢀⡟⠀⡴⢦⠀⠀⢠⠚⠆⠈⣧⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⢻⡀⠀⡸⢦⡞⠁⠀⠀⠀⠀⠀⢠⡞⠀⠀⣷
⠀⠀⠀⠀⡼⠁⢰⠇⢺⠀⠀⠀⠔⢜⡄⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⣩⠷⢄⡉⠢⣀⠀⠀⠀⣠⠏⠀⠀⠀⢸
⠀⠀⠀⣸⠃⠀⣀⠉⠺⠀⠀⠀⠒⢁⡀⠀⢹⡄⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠞⠁⠀⠀⠙⢦⡈⠳⣄⣰⠋⠀⠀⠀⠀⡾
⠀⠀⢠⡏⠀⢸⠕⠻⣆⠀⠀⠀⡔⠡⢸⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠙⣦⠞⠁⠀⠀⠀⠀⢠⠇
⠀⠀⡾⠀⢀⠈⠲⢄⡁⠀⠀⠀⢂⠠⠊⢀⠀⠈⣧⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⢀⣞⠢⣄⡀
⠀⠸⣇⠀⣎⡽⠦⡀⠁⠀⠀⠀⠀⣠⢾⡙⡆⢀⡽⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⢀⣰⠋⠀⡱⢀⡨⢷
⠀⠀⠀⠙⠋⠀⠀⠈⡆⠀⢀⠀⠘⠁⠀⠉⠓⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⣞⣛⣉⡀⠀⠀⠀⠀⢀⣀⡠⠴⣯⣡⠴⣺⠕⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢠⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢹⠯⣄⣀⢀⡤⠚⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣿⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄⣈⣍⢀⣀⡤⠴⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⢺⣿⡛⠛⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢤⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠴⡲⠏⢼⣿⡿⠶⢶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠀⢸⣿⣇⠀⠸⠄⠳⠐⠒
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⢤⣾⣿⣿⢤⣤⣀⣀⣀
⠀⠀⠒⠀⠀⠻⠉⠹⠁⠇⠀⢸⣿⣿⣈⢀⣿⠩⣏⣣⣀⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠑⠒⠁⠐⠉⣻⣿⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢸⣿⠟⢋⣅⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡡⠶⠊⢹⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⣍⡤⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠁
⠀⠉⠁⠈⠓⠀⠒⠂⠄⠲⠶⡆⢀⣈⢀⣦⣀⣀⢀⣀⣀⣀⣀⠠⢄⣀⣀⣀⡀⣠⣀⡠⠠⠒⠒⠤⠄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠠⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣦⣄⣀⣰⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡰⢁⣾⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⣿⣿⣿⡷⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⢼⣻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣟⢳⡞⠛⠛⣷⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⢇⡂⢄⣩⣰⣷⠚⠉⢷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⣿⣛⣶⠖⠛⠚⠏⠉⢻⡭⣹⡝⣹⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠉⠩⣁⠀⠀⠀⠀⣞⡀⠌⢉⠛⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣃⣀⣠⠬⠛⠒⢺⣉⣈⣀⠀⠀⠀⢠⡽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠡⠊⡉⠁⠀⠈⠀⠉⡉⠣⢌⣦⠤⡀⠔⠁⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠆⠀⠈⠀⢀⡀⠀⠀⠍⢉⣀⢿⠊⠀⠈⠐⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡿⠀⠀⡀⢀⡤⠜⠲⠀⠀⠀⠀⠀⠄⡀⢀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣿⠇⣈⠁⠀⠀⠀⠀⠘⠀⠓⠤⡀⠀⢠⣃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠲⢄⠀⠀⠉⠀⠒⠂⠀⠀⠀⠈⠲⠚⠃⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠉⠀⣀⠀⠀⠀⠀⠀⢀⡀⠴⠈⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⣀⠱⠦⠀⠀⠉⠀⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠤⠐⠂⠉⠁⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⠀⠁⠀⠁
⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣍⠀⠁⠀⢂⠲⢀⡀⠀⠀⠐⡡⢂⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢊⠈⠁⠀⠀⢠⡀⣋⠀⠈⠉⠚⠠⠅⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠐⠐⠀⠄⢀
⠀⠀⠀⠀⠀⡀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⣼⢤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⡀⠉⠀⠂⠀⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠀⠈⠈⠐⠘⠀⢰⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡐⠈
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢇⠀⠀⠀⣠⡿⢛⡿⠈⠐⡠⢄⣠⠀⠀⠀⠀⠀⠈⠉⠚⠃⢇⡖⡤⣀⠀⠀⠀⠀⠀⠑⠀⣀⠔⢢⠜⢠⡀
⠀⠀⠀⠀⡠⠸⣽⣿⠿⠗⢛⣴⣵⢄⢘⡶⣶⡇⢀⠄⡀⠀⠀⠉⠀⠀⠒⠠⠀⡀⠀⠀⠀⠀⠁⠙⠽⢃⢂⢤⠀⠀⠄⠠⠃⠀⠂⣴⣿⣦⡀
⠀⠀⠀⢠⠽⡺⢁⡰⣢⣿⡶⢣⡇⠘⡻⢀⣍⡗⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⢄⣀⡀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡆
⠀⠀⠀⡾⠄⢡⣳⠉⠹⡿⠂⢴⡟⢲⠁⠀⢖⠅⢀⡐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠠⠀⠀⠀⠀⠠⠿⠿⠿⠛⠁
⠀⠀⢰⣄⠀⣘⠉⢠⡔⠂⢹⣿⠰⠈⠓⠊⠀⠁⠀⠀⠐⠀⠀⠀⠉
⠀⠀⢸⠀⠩⡇⢐⡺⠋⠸⠃⢐⣤⡟⡄⠀⠀⠈
⠀⠀⠀⠀⠴⠇⢈⡟⢶⣤⠶⢦⣟⣴⠇
⠀⠀⠀⠈⠠⢺⡞⢛⠻⠮⡶⡾⢞⣁⠈
⠀⠀⠀⠀⠘⠿⠗⠀⠀⠒⢃⣍⣄⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠘⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠄⠀⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣡⡀⢠⠴⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠞⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⣿⡾⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⢮⣄⣀
⠀⠉⠉⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢉⣿⣭⡃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⡠⠰⠾⢳⣛⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡟⣯⣘⠀⠀⠀⡤⠀⠀⢰⡲⢬⢬⡀
⠀⠀⠠⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣿⠈⠀⠰⠒⠒⠓⠂⢈⠹⠴⢎⡄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⢩⡽⢿⠉⠉⠉⠉⠛⡉⠀⢈⢻⠒⠯⡁
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠂⠀⢀⣠⠄⠀⠀⣚⡉⠨⡇⠀⠀⠀⣷⣿⠀⣞⣻⡁⣥⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⣀⠀⠀⠀⠀⠀⠀⣀⣶⠄⠀⠋⠀⠀⢸⢸⠀⠀⢠⣉⣁⣄⣻⠀⠀⠀⣇⢹⢀⠌⣿⡀⢸⣸⡆
⠀⠀⠀⡀⠀⠠⡀⢠⠀⡀⠈⠷⣄⢀⣀⣴⣛⣓⣚⣶⣄⣡⣤⣤⣼⠘⢰⣶⣺⣤⣭⣿⣼⣃⢀⣼⠇⠂⣧⢈⣯⡧⢸⣬⣇⣀
⠀⠀⠀⠇⢀⡿⣷⣾⣶⣿⣿⠿⠟⠟⠋⠉⠙⠛⠥⢴⣤⣩⣟⡻⠿⣼⠟⠟⣩⣿⣟⣿⣯⣝⣿⣿⣿⣷⣿⣿⣿⣿⠶⣿⣿⣿
⠀⠀⠶⣗⡚⣒⣧⢼⡿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠦⢤⣬⣉⡛⠛⠛⠛⠿⢿⣟⣨⣿⣿⣿⣿⣛⣛⡇
⠀⠐⢫⡷⢚⡭⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠂⠀⢀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⣷⣶⣬⣍⣉⠛⠻⠿⢿⠿⠏
⠀⣠⠴⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛

⠀⠀⠀⠀⠇⠀⠀⢀⣀
⠀⠀⠰⢶⣿⣤⠠⠴⠪⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡠⢶⣾⡟⠗⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠉⠁⢨⢸⡇⣷⣐⡲⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⣠⣴⣿⣿⡵⠂⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣦⠀⠬⠗⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠛⠉⢸⣼⣯⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣖⡀⣠⣅⡀
⠀⠀⣈⣸⡷⣿⠉⠙⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢐⣼⣿⠚⠋⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⣷⠦⣄
⠶⠊⠻⢿⡿⣬⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⢡⣸⣌⣀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠜⠛⠃⣻⣷⣀
⠀⠁⡣⢸⣏⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡿⠏⠉⠣⠀⠀⢀⣤⡀⢀⣅⠀⠀⠀⠀⢀⣀⣼⡯⠉⠑
⠀⢀⠀⢹⡧⠚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢈⢹⡟⠀⠀⠀⠀⠀⠀⠀⡀⢹⡷⠦⢀⠀⠀⠁⠀⣻⣷⡦⣄
⠀⠰⠁⢼⡟⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢺⡅⠃⠀⠀⠀⠀⠀⠉⠉⢿⣿⣤⡀⠀⠀⠀⠀⡹⡇⠂
⠀⡀⠀⢹⣷⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⡇⢄⠀⠀⠀⠀⠀⠠⠀⢪⣏⡀⠁⠀⠀⠀⢀⠠⣇
⢀⠁⠈⣿⣇⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣼⡇⠠⠀⠀⠀⠀⠀⠀⠀⢚⡏⠉⠁⠀⠀⠀⠀⠰⣏⠐
⣸⠗⢆⣺⣿⢢⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠤⣴⡷⣤⡄⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⣀⣴⣯⣠⡆
⣿⣤⣤⣿⣿⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣤⣿⣷⣤⣧⣤⣤⣤⣤⣤⣤⣴⣧⣤⣤⣤⣤⣤⣧⣼⣿⣤⣮
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⢶⣞⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⡟⣿⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠾⠿⢛⣛⣛⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣟⣟⣟⣿⢯⡇⣿⡧⠽⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣛⣟⣽⣭⣭⢿⣾⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣽⣽⡗⢒⣹⢸⡏⠉⠉⢹⠁⣿⣿⣶⣶⠇⠀⠀⠀⠀⠀⠀⣠⣮⡭⡥⠶⠶⠶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣿⣚⡺⣺⠠⢬⣸⣸⠇⠀⠀⢸⠀⣿⣿⢸⣿⠀⠀⠀⠀⠀⢀⡶⠷⠒⢒⣚⣛⣻⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣭⢍⢭⡭⡭⡏⠀⠀⡉⠁⠀⠀⠀⣾⠘⠉⢛⢻⣿⡌⢠⡄⠀⣰⠛⠛⠛⣛⢋⣩⡏⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢖⢓⢒⢖⠓⠂⠁⣀⣴⣽⣿⣿⢿⣿⣿⣷⣶⣶⡟⠁⠀⠀⢠⠾⢟⠛⢛⣟⣟⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣯⣩⣭⣉⣭⠉⠁⠀⢛⣿⣿⣿⣿⠆⢰⣿⢻⡹⣿⠟⠀⠀⠀⣴⣛⣛⣋⣹⣯⣭⣍⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢤⠭⠭⠦⠤⣦⠖⠀⠀⢀⠔⢩⣏⣟⣹⣯⣴⣾⣥⣮⣿⠏⠀⠀⢠⣾⣭⣭⣩⣭⣬⣬⣭⠋
⠀⠀⠀⠀⠀⠴⠷⠴⠶⠶⠴⠶⠊⠀⢠⠔⠁⢠⣶⣿⡿⣿⢸⡟⢻⠟⣏⠋⠀⢀⣴⣯⣤⣴⡤⢤⠶⢶⠔
⠀⠀⠀⡴⠶⠒⠒⣒⢓⢗⠒⠀⡠⠂⠀⠀⠀⠀⣿⡿⢹⣷⣚⡧⢾⣦⣏⠀⣠⠾⠶⠶⠶⠶⣶⠚⡷⠋
⠀⣠⠶⢛⠛⣋⣋⣛⠙⣁⠔⠉⠀⠀⠀⠀⢀⣬⣶⡏⣿⡟⣽⠛⢻⠟⣿⠒⣿⠻⣿⣿⢿⠛⠛⠋⠁
⠚⠊⠙⠉⠙⠛⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣳⣽⣧⣿⠅⢸⠀⢾⢰⣿⠀⢹⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡹⠋⠟⢳⡿⠤⣾⣾⢾⠦⠺⡟⠉⠇⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡹⢁⣀⣀⣸⣧⣤⣴⣿⣤⣤⣀⣀⣀⡀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠛⠋⢹⠉⢸⣿⡏⠀⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠀⠀⠀⠀⢆⣘⣂⣚⣿⣷⣶⣿⣿⣿⣹⣾⣿⡿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢃⣠⣴⣾⣿⠭⢿⣿⣿⣻⣻⣮⣫⣝⣯⣀⣩⣭⣽⣯⣻⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣯⡿⡿⠷⠿⢿⡿⠩⣿⡄⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⡀⣿⣿⣿⠿⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣄⣾⣿⣦⣣⣾⡏⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠽⠿⠮⢿⣿⣝⣿⣿⣿⠿⠿⠿⢿⡯⢽⣿⣏⣀⠽⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠦⠔⠒⠒⠢⠴⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⣠⢤⣔⣒⣒⣈⣉⣭⣭⣭⣀⣒⣒⡢⠤⣄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠉⠙⢻⣿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣻⣿⣿⡀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⠘⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣀⣥⣭⣭⣅⢉⣾⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⣷⣄⠙⠻⢿⣿⣷⣶⣤⣤⣄⣀⣀⣠⣤⣤⣶⣿⣿⣷⡧⠄⣠⣴⠬⠶⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣄⣀⣿⣿⣷⡄⣀⡉⠙⠿⢛⣻⣿⣿⣿⣿⣛⡛⠿⠛⢉⣀⣴⣾⣿⣿⣀⣠⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠿⡿⣿⣿⣿⣿⣗⡏⠉⠑⠒⠲⠶⠤⠤⠤⠤⠶⠒⠒⠊⠉⢹⣿⣿⣿⣿⣿⢿⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢻⡆⠇⢱⠀⠀⠙⠿⢿⣴⣒⡠⠀⠀⠀⠤⠄⠀⠀⠀⢄⣒⣾⡿⠿⠋⠀⠀⡎⠸⢰⡟⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢸⡇⢰⠈⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠀⡇⢸⡇⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢸⠃⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠸⠀⠀⠀⡰⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⢦⠀⠀⠀⠇⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⣀⣂⣠⣤⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠈⣰⢂⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⢎⣰⣿⣿⣶⣅⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⡀⠔⢈⠃⣀⡛⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⣿⣿⠋⠁⠀⡀⠁⠙⠿⣿⣿⠇⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⢀⠄⠊⠀⠀⠀⠈⠀⠈⡄⠁⠀⠀⠀⠈⠂⠀⠐⠁⠀⠀⢀⠀⢀⠁⠀⠀⠀⠘⠣⠀⠁⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠔⠁⠀⢀⢔⣥⣶⣷⣶⣷⢄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠋⣠⣶⣿⣿⣷⣦⡐⡀⠀⠀⠀⢀⣤⣤⣈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠂⣾⣿⣿⣿⣿⣿⠈⡆⠀⠘⠿⠃⠀⠀⠀⠔⢁⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠘⣿⣿⣿⣷⣌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠆⠀⢀⡀⠀⠀⠀⠻⡿⠟⢛⠏⢁⠀⠘⠘⠿⠈⠁⠀⠀⠈⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠌⣴⣿⣿⡆⢰⠁⠀⠀⠠⠀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡁⢹⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠞⣿⣿⣿⣿⣿⡇⠆
⠀⠀⠀⠀⠀⠀⡈⣼⣿⣿⣿⡇⢈⠀⠀⠀⠂⠀⠀⠀⡠⠔⣀⣠⣤⣤⣀⠀⠀⡂⠀⠀⠀⠀⢀⠙⠿⠿⠿⠿⠟⠛⡡⠃⠀⠀⠀⠀⠄⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡘⢰⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠄⠐⣩⣶⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡟
⠀⠀⠀⠀⠐⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⢂⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡕⡀⠀⢠⣶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⢁⠈⠙⠛⡡⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠡⠀⠀⠀⡠⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢐⠀⢠⣤⡀⢀⠀⠐⠁⢀⡠⠈⡇
⠀⠀⠀⠀⢀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠐⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢰⣿⣿⣿⠸⠀⠀⠐⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠡⡸⣿⡿⠀⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠁⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠠⠸⣿⣿⠏⠂⠀⠀⠀⡐⠀⣠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠠⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠅⠀⠀⠀⠀⢀⠀⣀⣭⣴⣶⣮⡕⠄⠀⠢⡀⡔⠈⠁⠀⢀⢚⣠⣄⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⢈⠛⠿⠛⠉⠁⠀⠒⠈⠀⠀⠀⢢⠀⢔⣵⣿⣿⣿⣿⣿⣿⣿⠰⠀⠀⠆⠀⠀⠀⡀⢁⣿⣿⣿⣷⠘
⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⡀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⡌⠀⠀⠀⠀⠀⢀⢁⣼⣿⣿⣿⠏⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢡⢊⣀⡘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣄⠀⠀⠈⣾⣿⣿⣿⣿⣿⣿⣿⡫⠊⠀⠀⠀⠐⡀⠀⣡⣾⣿⣿⣿⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡑⢂⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⠈⠋⠀⠀⢂⢿⣿⣿⣿⣿⡿⠋⠏⠀⠀⠀⠀⠀⠠⢠⣾⣿⣿⣿⣿⡟⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢿⣿⣿⣿⣌⢂⠀⠀⠀⠀⠀⠀⠙⠿⠟⣋⠀⠀⠀⠀⠀⢜⢻⡿⢟⠿⠊⠉⠀⠀⠀⢄⠀⡠⢡⣿⣿⣿⣿⣿⡟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠻⣿⣿⣿⣧⠉⠤⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⢸⣿⣿⣿⡿⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⢿⣿⣧⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⣀⣤⣤⣤⣥⣌⠐⢄⡠⠷⠀⠉⠋⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠁⠒⠛⠠⣄⡀⠀⠀⠀⠀⠀⢀⠁⠒⣀⣭⣴⣾⣿⣿⣿⣿⣿⡿⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⡤⠄⢧⠀⣿⣿⣿⣿⣿⣿⡿⠟⡫⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⠙⠭⠍⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠉⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣏⢿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡝⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠀⠰⡀⠀⠀⠈⠻⣿⣿⣷⡸⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣾⡜⡄⠀⠀⢀⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣀⠀⣤⣤⣤⠀⠀⠀⠀⢸⣿⣿⠀⠀⢹⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠸⡍⢹⣿⣿⣿⣿⡇⣿⣿⣿⣿⠿⠲⢶⣶⣿⣶⣿⣿⠟⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⣋⠞⠙⠿⣿⣿⣷⣶⣿⣿⣿⠁⣽⡏⠀⣠⣦⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠈⠀⠁⠒⠀⠀⠈⠉⠛⠻⠿⢿⣿⡿⠿⣿⣿⠟⢣⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠊⠁⠰⠟⠛⢉⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣤⣄⣀⣀⣤⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣟⣵⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⠟⣵⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⡤⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⣤⣤⣦⠀⢀⠋⠙⠉⠐⠲⠽⡟⠋⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠹⣿⣿⣰⣃⡀⢀⣀⣠⡞⠸⣍⡉⠿⡀⢰⡶⠆⢀⣤⣄⠀⠀⢀⡀⠀⠀⢠⠄⠀⣀⠀⠀⣀⡠⠖⠂⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⣄⣲⣦⡽⠶⠛⠋⠉⠀⠈⠈⠙⠻⡀⣿⣤⠀⢀⡀⡄⠀⠀⠙⠒⠋⠰⢋⣽⠟⠀⠀⠀⢀⣾⠤⠀⣤⠘⢶⣤⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣀⣼⣿⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠞⠈⠁⠀⠰⣿⠿⡟⠘⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣟⠲⣶⣤⣄⡀⡀⠀⠉⠐⠠⠎⠀⠀⠀⢀⣶⠋⣠⡇⠀⢀⡀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠤⠒⠊⠙⠻⡏⠃⠠⢄⣐⠀⠀⠀⢠⣾⠁⣾⣏⣷⣠⠉⣁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠓⡤⣤⠖⠋⠈⢣⣄⣠⠴⠒⠉⢻⣷⣶⣦⣿⣄⣀⣀⠀⣿⢐⡏⡿⠗⠸⢁⡿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⢀⠀⠼⠂⢞⡉⠀⣈⡱⠶⠒⠉⠁⠀⠀⠀⠀⠀⠘⠉⠛⢿⡿⣶⣬⣽⣟⠈⠁⠘⠒⢳⣿⢁⠆
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡠⠘⠲⢎⠀⠀⣀⣤⠴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⢿⡀⠟⠏⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠁⠀⢀⣀⡤⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠴⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠹⣿⣮⡢⢀⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣧⡨⢻⡵⣫⡤
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠄⡀⠈⠑⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣮⡘⠻⣿⣖⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠁⢄⡀⠀⠁⢀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣴⣿⣮⡻⣿⣿⣿⣷⣤⡭⠂
⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⢢⡀⠀⠀⢠⠀⠀⠑⢤⠀⠀⠀⠤⡀⠀⠀⠀⠀⢀⢿⣿⣿⣿⣮⡻⡿⠋⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢢⠀⠀⠀⢠⠀⠀⠀⠤⠀⠀⠈⠀⠀⢀⣴⣿⣷⣽⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢤⠀⠀⠀⠤⠀⠀⠀⡂⣀⡦⣴⣿⣿⣿⣿⣿⣷⣝⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⢄⠀⠀⠀⠀⣠⣾⣿⣮⡘⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠑⠠⣠⣾⣿⣿⣿⣿⣷⣄⡙⡿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣍⠋⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣽⣿⣿⣿⣿⣿⣿⡿⠏⠉⠁⠑⢄⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡻⣿⣿⣿⣟⢿⣿⢽⣍⠀⠐⢄⠀⠀⠈⠑⢄⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣕⠻⣿⣮⡻⢿⡿⠃⠈⠀⠠⡀⠀⠀⠙⢄⡀⠀⠀⠑⢤⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⣷⣌⡭⠄⠀⠀⠀⠄⠀⠀⠘⠢⡀⠀⠀⠙⢄⡀⠀⠀⠑⠤⡀⠀⠀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠋⠀⠈⠉⠁⠀⠀⠀⠀⠈⠢⡀⠀⠀⠘⢆⠀⠀⠀⠑⢀⡀⠀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠙⢄⠀⠀⠀⠑⢤⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠛⢄⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠧⠶⠶⢚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⡯⠉⠒⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣦⡷⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠛⠉⢩⣿⢽⣷⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡄⢉⣿⠚⣷⠿⠿⠓⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⢾⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢈⣷⠚⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣖⢂⣿⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠉⠙⠛⠺⠭⣿⢿⣿⣾⣵⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠈⣿⠛⢻⠾⢷⠶⠶⠄
⣿⣿⠿⢿⠿⠿⢿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠛⣤⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣜⢦⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣭⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣝⠋⠶⠛⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠐⣮⡻⠟⣕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠘⢣⡿⠂⠀⠀⠀⠀⠀⠘⣧⡜⠀⠀⠀⠀⠀⠀⠈⢠⡁⠀⠀⠳⣉⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠶⣷⣦⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠙⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢘⣿⡷⣔⣄⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣟⣻⣛⣻⡃⣶⣆⣀⡀⠀⠀⠀⠀⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣝⣾⣿⢯⣶⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠸⠟⣽⡝⣿⣟⣿⡟⣽⣯⢛⣤⣤⣀⣀⢧⡜
⠀⠀⠀⠀⠀⠀⠀⠾⠷⠯⠉⠭⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠐⠉⠙⠛⠛⠛⠛⠊⠙⠛⠛⠛⠛⠛⠃⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠀⢠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⢰⡆⣶⢸⣿⡇⣶⢰⡆⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⣠⠀⠀⠸⡀⠈⢧⡀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⢸⣻⡋⠉⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣮⡁⠀⠀⠀⠙⣄⠀⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠏⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⣦⡀⠀⠀⠀⠙⠒⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣧⣤⣭⣽⣯⣥⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⡛⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢐⣶⠶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣛⠧⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣯⠿⢽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠿⣽⠇⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣛⣛⣛⣛⣋⣶⣭⡇⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁⠼⠁⠀⠀⠀⠀⠀⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⣖⣒⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢶⣮⣭⡅⠒⠚⠋⠁⠈⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠰⣾⣿⡶⠉⠲⠄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⣤⣤⣤⣄⠙⠛⠋⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⢻⣶⣝⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠸⣏⠁⣀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢕⣤⣬⡽⠀⠀⠀⠀⠈⠻⢮⡇
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠉⠒⠊⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⠏⠀⣠⠖⢻⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⡃⠀⠀⠀⠀⠀⠀⢰⠲⣶⣄⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠓⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠀⢰⢸⠀⠈⢿⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡎⣠⡀⣾⡇⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣌⢦⠳⣀⢸⡗⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢟⣧⣻⠀⠀⠀⠀⢰⢹⠁⠈⢢⡑⡄⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠈⠉⠀⠀⣴⢦⡸
⠀⠀⠀⠀⠀⢸⠈⠁⠀⠀⠀⠀⠈⢎⢆⠀⠀⡇⠸⠀⠀⠀⠀⠀⣀⠤⠂⠠⠀⠀⠀⣀⣀⠀⢧⡿⡇
⠀⠀⠀⠀⠀⠈⣇⠀⢠⡀⠀⠀⠀⠀⠙⠛⠊⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠓⡄⢰⣿⣿⣿⡃⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⢰⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⣧⣀⣀⡠⣶⠇⠀⠀⠀⠈⠁⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠁⠀⠀⠀⠀⠀⣼⡿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⢤⢄⣀⡀⠀⠀⠀⠀⡠⠶⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣵⣺⣿⡀⠀⠀⠀⠛⠛⠛⠉⠀⣠⠶⢦⠀⠰⣂⠴⠋⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠢⠤⠤⣀⣀⣀⣀⠿⠿⠟⠚⠉


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

⠀⢀⣤⣤⣶⠟⢦⡀
⢀⡹⣶⢟⠁⠀⠀⠙⢦⡀
⢸⠛⠁⠀⠑⢄⠀⠀⠀⠙⢦⡀
⠀⠳⣄⠀⠀⠀⠑⢄⠀⠀⢀⡹⢦⡀
⠀⠀⠈⠳⣄⠀⠀⠀⠳⣤⠋⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠈⠳⣄⡰⠋⠀⠳⣄⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠳⣄⢀⡴⠋⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠼⠻⣄⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣏⠀⠀⠈⠳⣄⠀⠀⠀⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠳⣄⣴⠋⠀⠀⠀⠀⠀⣀⢀⡴⠊⠹⣄⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⡴⠿⣉⠣⡀⠀⠀⣠⢞⡽⠋⠀⠀⠀⠈⠱⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠈⢳⣈⡟⣻⠗⠉⠀⠀⠀⠀⠀⠀⠀⣨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣥⠞⠉⢢⡀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠶⠶⠤⣤⣴⠋⠀⠀⠀⠀⠀⠈⠢⡀⣠⢚⣷⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡩⣶⡾⠿⠿⠷⣶⣤⣈⠙⠢⣄⠀⠀⠀⠀⣠⣏⣴⠿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⠞⠁⠀⠀⠀⠀⠀⠈⠙⠿⣦⡈⠳⣄⠀⣮⡼⣏⣱⡛⢧⡀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠘⢆⣠⠞⠉⠁⠙⢦⡉⢢⡴⠋⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠹⣧⠘⣇⠀⠀⠀⠀⢀⡝⠋⠳⣄⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⢀⠞⠉⣱⠄⠀⠀⠀⠀⠀⢻⡇⢻⠀⠀⠀⠀⠫⡀⠀⠀⠈⠳⡄⠀⠀⢈⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣀⡴⢋⡠⠞⠁⠀⠀⠀⠀⠀⠀⢸⡗⣸⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠈⢢⣤⠋⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢏⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣧⠏⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡴⠋⠈⠳⣄⠀⠀⠈⠳⣄
⠀⠀⢀⡄⠀⡖⠀⡞⠀⠀⢀⣀⣴⠟⠉⠈⠻⢿⣶⣤⣤⣀⣤⣤⣶⣵⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⡄⠀⠀⠈⠳⣄
⠀⠀⣼⠀⢸⡇⠀⣇⠀⠀⣏⣹⠃⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⡀⢀⠔⠉⠳⣄
⠀⠀⣿⠀⠈⣧⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⠜⠣⡀⠀⠀⠈⠳⣄
⠀⠀⠸⡄⠀⠈⢧⡀⠀⠉⠓⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡁⠀⠀⠈⠢⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠙⣄⠀⠀⠉⠓⠶⠤⠤⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠢⡀⠀⣠⣾
⠀⠀⠀⠀⠈⠙⠦⣤⣀⣀⣀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣨⣾⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡾⠟⠉⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⢂⣴⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣴⣾⡟⣼⡇⣾⣿⣿⣿⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠚⢿⠟⠂⠀⠀⠀⠀⠀⣀⡤⣷⣿⣿⣿⣇⣿⣇⢿⣿⣿⣿⣿⣦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢖⣵⣿⣷⣾⣿⣿⣿⣿⣿⡸⣿⡌⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⣠⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣦⡙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⠿⠶⣤⠄
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢿⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣌⠻⣿⣿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣾⣿⠀⠀⢀⣰⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⠀⣶⣿⣿⣿⣿⣿⣿⡿⠟⢩⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⡀⠙⠟⠉⠁⠈⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⣼⣿⡟⠀⠈⠙⠉⠸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣰⣦⣀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣧⠹⣿⣆⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠹⣿⡆⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⢻⣦⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣶⣶⣾⣿⣶⡶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠀⠘⠀⠘⠃⠀⠛⠀⠀⠀⠸⠀⡂⠀⠀⢀⠀⠁⠘⢀⠀⠠⠀⠀⠐⠂⢐⠀⠀⠀⠐⠟⠀⠘⠃⠀⠃⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣿⣽⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠭⣽⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠟⠋⡁⢄⣤⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⡛⢉⡁⢀⣢⣤⢶⣿⣿⣿⣿⣿⠿⠟⠟⣩⣝⢿⣿⣿⣿⠟⠡⠀⠂⣀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣏⠰⠟⣡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠀⠀⠀⡝⢏⢷⣻⣿⠰⠀⢀⠰⡆⠀⣂⣿⡇
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠛⣿⣿⣷⣨⣿⣿⣿⣿⠛⣉⡀⠀⠀⢀⠀⠀⡄⣇⢸⡄⣏⣿⣷⣄⣈⣐⣆⣬⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣶⣿⣿⣿⡟⠋⠉⠁⣌⢰⣉⣽⢀⠀⢸⡄⠀⠸⡜⢾⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⣿⣿⣴⣿⣿⠉⣟⣡⣴⣧⣤⣤⣶⣿⣼⣇⢹⣿⣿⣿⣿⣶⣶⣾⣢⣭⣿⣿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⢏⠉⠛⢿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣽⣿⡿⢻⣿⠟⠿⣿⣿⡻⣿⣿⠛⢻⣿⢍⠚⠀⠈⠉⠑⠈⢻⣿⣿⣿⡄
⠀⠀⠀⢸⡿⣫⡟⠡⠤⠖⢀⠀⠀⠘⢻⣯⣻⣿⡿⣿⣧⣽⣶⡇⢰⣿⣿⣷⣜⠪⠐⢾⣿⣇⠠⠀⠀⠀⠀⠉⢈⣭⡛⢿⠁
⠀⠀⠀⠈⣱⣏⠀⠠⠀⠀⠀⠀⠀⣀⣤⠹⠷⢻⣷⣿⣿⠿⢿⣳⡊⠛⢿⣿⣿⣷⣧⣶⣦⣍⡛⠳⠶⢦⣤⣤⡄⠀⠐⡀⢄
⠀⠀⠀⢠⣿⠒⠖⠀⠀⢀⢀⣀⣠⡟⠈⢘⢸⣿⣿⣷⡝⣅⣀⣭⣭⣁⣟⢡⣿⣟⢻⣿⣿⣄⠙⠛⠳⢶⣦⣬⣤⣀⢀⣠⣾⣫
⠀⠀⠀⠸⣿⣳⣖⡻⢿⣿⡿⠟⠋⠀⠁⠀⣺⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠘⣿⣿⢿⣿⣿⣿⣿⣶⣿⣿⣿⡟⢿⣿⡿
⠀⠀⠀⠀⢻⣅⡀⠉⠀⠉⠀⠀⡀⢀⣢⣞⣿⣿⣿⡇⡌⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣷⠛⠛⠛⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠿⣤⣀⠀⠠⠀⢶⣯⢷⣿⣿⣿⣿⢡⢿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⠿⣛⡀⠀⣸⣿⣿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢬⣭⣭⣭⣶⣿⣿⣿⣿⡏⣌⣾⣿⣿⡿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣿⣿⣿⢶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢦⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡟⣼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣧⢻⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠙⠛⠛⠛⢇⠛⠛⠛⠛⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⣬⡝⢿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣥⣿⢯⣻⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⣾⡷⣿⣷⣿⡳⣽⡿⣿⣿⣿⣿⣿⣿⡷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⣿⣿⣿⢿⣿⣿⣜⣠⣧⣿⡏⠈⠁⠈⠛⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⢀⣸⣼⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⣿⣿⣿⣭⣭⣥⣚⣻⠿⣿⣿⣿⡿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣾⣿⣿⡿⣿⣿⣿⣿⣿⣦⣝⠙⢿⣿⡟
⠀⠀⠀⠀⠀⢤⣾⣿⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠃
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣾⣿⣿⣿⣿⡻⣨⡟⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⠾⠛⠁⠉⠁⠙⠛⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⣀⣤⠴⠶⣶⡛
⠀⠀⠀⠀⠀⣠⠞⠁⣠⡴⣋⡥⠖⠛⠉⣉⣹
⠀⠀⠀⢀⡾⠁⢀⡾⣡⠞⠉⣠⠶⢛⣩⣭⣭
⠀⠀⠀⡾⠁⢠⠏⡼⠃⣰⢟⣵⠞⣩⡤⠶⠾⠁
⠀⠀⢸⠃⢀⡟⣼⠁⣼⢃⡞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢂⡭⢧⡀
⠀⠀⣿⠀⢸⣧⡇⢠⡇⣾⢠⡏⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⡥⢚⣡⠖⣩⠵⡄
⠀⠀⠙⠛⠛⠈⠓⠞⠃⠛⠞⠃⠀⠀⠀⠉⠻⣦⣄⠀⠀⣠⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢂⡵⢞⣡⠖⣫⠔⣋⡴⢊⡽⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣾⣿⣿⡿⣁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢂⡥⢚⣭⠖⣩⠔⣋⠴⢊⡥⢚⠡⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣫⢞⠟⢹⠿⣷⣄⠀⠀⠀⠀⠀⠐⣥⢚⡩⢖⣩⠔⣫⡴⢊⡥⠚⡩⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⣫⢞⠟⢁⠴⣏⡴⠟⠻⣷⣄⠀⠀⠀⠀⠈⣯⡔⣊⠵⢊⡥⢚⡡⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠛⠋⠁⢘⣟⣁⠴⢯⣴⠟⠁⠀⠀⠈⠻⣷⣄⠀⢀⡼⠁⠹⡥⠞⡩⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣴⣿⣷⣄⠀⠀⠀⠀⠀⠈⠻⣷⣟⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣼⡏⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⣀⣴⡿⠋⠀⠀⠀⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣤⡀⠀⣀⡴⠿⠙⢿⣿⣿⣿⡿⠋⠀⠀⠀⣀⡀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⡞⡰⢋⠿⣅⡀⠀⠀⠀⠙⢿⡋⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣡⢋⡼⣱⢏⡜⣡⠋⠆⠀⠀⠀⠀⠙⢦⣀⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡴⢡⠟⡴⣡⠎⡼⢣⠊⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠞⡱⢃⠞⡰⢋⠞⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣡⢋⡼⣡⢊⡜⣡⠋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡴⢡⠞⡴⢡⠎⡴⢣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢇⡜⡰⢋⠞⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢏⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠈⠉⠉⠉⠉⠉⠉⠀⠒⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⢌⡑⠊⠥⠴⠶⠦⢤⣤⣤⣤⣤⡤⠴⠶⠶⠎⠓⣈⠕⠢⢄⡀
⠀⠀⠀⠀⠀⣀⡠⠤⢄⡠⠔⠊⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠂⠠⠀⠒⠒⠒⠒⠂⠉⠉⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠈⠣⡔⠒⣊⣉⠵⠦⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⣒⠏
⠀⠀⠀⠀⠀⠈⠙⢤⠜⠓⣆⣉⠷⣒⠋⠭⠭⠐⠒⣒⣂⣐⣒⣒⣒⣒⣒⡒⣂⣒⡒⠒⠠⠭⠍⠓⠒⠉⣩⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⡤⠽⢶⣚⣓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠬⣶⣉⣉⠳⣤⡙⠒⣂⣤⡤⠤⣀⣀⢀⡀⢀⡀⢠⠤⠄⠒⠒⠉⡡⠊
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠙⡶⣎⠙⡍⣭⣶⣶⠝⡄⢸⣈⣆⣀⣀⣠⡀⠠⠤⠖⠒⠉
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠀⠀⢰⢹⡀⢰⠹⠀⠳⣀⣇⡸⠁
⠀⠀⠀⣀⣀⣤⣜⡒⠬⠥⠐⠃⠀⠀⣾⠥⠜⠒⠈⣕⢛⣻⣭⠵⠦⡀
⠀⠀⠈⠒⠁⠀⠀⠣⡀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠑⢤⠒⠛⣯⣉⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⡤⠘⠲⣌⡱⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢅⠬⠗⠒⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⢸⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠃
⠀⠀⠀⠀⠈⠣⡀⠑⢄⠀⠀⠀⠀⠀⠐⠚⠛⠿⣿⡿⢶⠶⠶⠖⠂⠀⠀⢀⢄⡔⢁⠄
⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠂⠀⠀⠀⠀⠦⡀⠈⣧⠀⢀⠎⠀⠀⠠⠊⢠⠎⠐⠁
⠀⠀⠀⠀⠀⠁⠢⡀⠈⠳⣄⠂⡀⠢⡀⢀⠀⠤⠀⠹⠈⠄⢀⡀⠐⠁⡰⠡⠂⠀⠀⠊⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡈⠓⢄⡠⠓⢈⡀⠴⠒⠀⠀⠂⠤⡜⢁⢞⠀⠀⠀⠠⠂⠀⠠
⠀⠀⠀⠀⠠⠀⠀⠀⢀⡜⢄⠀⠌⠀⡰⠛⠄⠀⠀⠸⠀⠀⠀⠀⠻⣄⢲⠀⠀⠀⢁
⠀⠀⠀⠀⠐⠀⠀⠀⡼⠉⠀⠛⠀⢀⠥⠶⠢⡄⠒⢤⡔⠢⡀⠀⠐⠄⡀
⠀⠀⠀⢀⣀⣀⣀⣄⠃⡀⠀⠐⢈⠀⢀⡀⠀⠐⢆⠀⠙⠆⠈⠲⣀⠀⠉⠓⠤⡀⢀⠀⠀⢀
⠀⠀⠀⠒⢒⠓⠀⠐⡀⡤⠊⠈⠀⠔⠡⠀⡀⠄⠀⠒⠀⠀⣄⡀⠀⠱⠔⠉⠑⠬⢸⠂⠀⢒⠀⠂
⠀⠀⠀⠀⠄⠒⠀⠈⠁⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⡆⠀⠠⠀⠈⠀⠰⢥⣀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⠐⣈⠀⠀⠀⠈⠢⡈⠀⠀⠀⠀⠸⠇⠑⡀⠀⠀⠀⠀⠀⠀⠉⠒⠢⢤⡀
⠀⠀⠀⠂⠀⠀⠆⠠⠃⠀⢀⠄⠀⠀⠈⠁⠀⠄⢀⣼⡀⠤⠐⡄⢀⠄⠀⠀⠌
⠀⠀⠀⠀⠀⠀⢀⣇⡔⣠⠂⠀⣁⠀⡀⠁⠀⠀⠞⠀⢆⠀⠀⠈⣆⠠⠂⠀⠐⠄⠀⡠⢪⠃
⠀⠀⠀⠀⠀⠀⠈⠉⣿⡥⠔⠊⠁⠐⠀⠀⠒⢀⠀⠤⡀⠒⠈⠃⠈⢢⠀⠀⠀⣠⠞⢠⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⢁⠀⠀⢀⡀⢐⠊⠐⢀⠑⢄⠀⠀⠄⠀⠳⣠⠞⠁⡰⠁⡰
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠁⠀⠀⡉⠀⠔⠊⡀⠈⡀⠈⠁⠀⠈⠀⠀⠀⢀⡔⠁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⡀⠌⠠⠊⠀⠀⡍⢦⡀⠀⠀⠈⠄⠀⠈⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠨

⠀⠀⠀⠀⠀⣴⣶⣷⣦⡀⠀⠀⠀⠀⠀⣠⡴⠶⠚⠓⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠰⣝⡻⣿⣿⠇⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠰⠂⠒⠂⠐⠂⢰⡿⠷⠶⢶⡄
⠀⠀⠀⢀⣾⣿⣿⠛⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣶⣶⢲⣶⡇
⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⢸⣾⣷⡀⠀⠀⠀⠀⣴⣷⡾⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⡄⠀⢀⣼⣿⣿⠇⠀⠀⠀⠀⠘⠶⠒⠒⠒⠒⠺⠟⠛⠛⠛⠁
⠀⠸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠂⠸⠿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠿⢿⡿⠀⠺⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣉⣉⣩⣉⣉⣉⣀
⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢿⣿⣿⢿⣿⠛⣩⣍⠙⣿⡆
⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣯⣽⡇⢸⡇
⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⢭⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣍⣩⣤⣿⡇
⠀⠘⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣃⡄⠀⠀⠀⠀⣤⣱⠀⠀⠀⠀⠀⠀⠀⣎⣤⠀⠀⠀⠀⢠⣜⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉⠛⠛⠉
⠀⠀⠙⣩⠙⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠼⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⠁⠀⠀⠈⣆⣀⣀
⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠚⠯⢿⡿⢿⡿⠝⠻⠟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣞⣦⣺⣾⡿⣿⣷⣷⣴⣷⡗
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣦⡀⠀⠀⢸⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⡀⠀⠀⣴⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⠿⠋⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠟⠀⠀⣿⣿⣿⣿⣿⣼⣧⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⠶⠶⠶⢶⡄
⠀⠀⣀⣤⣼⣿⡿⠁⠉⠁⠀⠀⠀⠀⠀⠸⣿⡿⠛⠉⠉⠀⣼⡟⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡗⠀⠀⣴⣶⣷⣾⣶⣶⣶⣾⣷⣿⣶⣆
⠀⠘⠻⠿⠛⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢼⡟⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠛⠛⠛⠻⠟⠛⠋
⠀⠀⠀⠀⠀⣄⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠀⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⠏⠀⠀⠀⠘⣿⣿⣿⣿⣿⣶⣤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠀⢀⣴⣿⣿⡿⠀⠀⠀⠀⠸⣿⣿⣷⣄⠀⠀⡴
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢁⣴⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄
⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠉⠉⠉⢙⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠾⠿⠿⠟⣯⣿⣧⣤⠀⠀⠀⠀⠀⠀⢠⣤⣿⣭⡟⠻⠿⠿⠆
⠀⠀⠀⠀⢤⣶⣿⣭⣵⣬⣵⣶⡄⠀⠀⠀⠀⣶⣾⣭⣴⣭⣽⣽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣉⣄⣄⣉⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⡀⠸⢀⣄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⣀⣴⣾⡯⠤⠾⠛⠁⠀⠀⠀⢠⣿⣥⣿⣿⡷⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠛⠋⠂⠈⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣇⠄⠈⣾⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠏⠈⡄⠀⡠⠔⢏⠉⢹⣇⣇⣦⠖⠁⠀⣴⣿⣿⣿⡆⠀⠀⠀⠤⣿⡀⠃⣼⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⣢⣇⣴⠿⣦⣤⣼⣿⠾⣷⣆⠀⠹⣿⣿⡿⠃⠀⠀⠀⡖⣿⣏⣹⣿⢸
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠄⠄⠀⠀⠈⠁⢻⣄⣽⣿⡿⠧⠤⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⢸⣿⣹
⠀⠀⡤⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⢰⠀⠀⢻⣥⣽⣷⠀⣠⣴⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⡷⡿⡇⢸⢿⢾
⠀⠀⣤⣀⣴⡇⠀⠀⠈⠉⠛⠶⢶⣾⣄⣤⡀⠀⠈⢿⡏⠀⣿⡓⣶⣵⡀⠀⠀⠀⠀⠀⠀⣿⠃⡇⢸⡎⢿⡀
⠀⢾⣿⠋⠙⠳⣄⠀⠙⠓⠶⠟⠋⠀⠈⠉⠉⠀⠀⠈⢿⣤⣴⡝⢿⣟⠁⠀⠀⠀⠀⢀⣾⣹⡿⢹⣾⢹⣿⣷⡄
⠀⢠⣿⣦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠁⠘⣉⠀⠀⠀⠀⠀⠀⠁⣟⣷⢿⣿⢾⣘⠆⠁
⠀⠉⠀⠀⠉⠳⢴⣄⣀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠻⢥⡑⣄⠀⠀⠀⠀⠘⠁⠈⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢂⠀⠀⠉⠁⠀⠀⠀⠀⣠⡖⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠈⠙⠲⢕⢄⠀⠀⠀⠀⠀⠀⠀⢹⣤⡆⠀⠀⣰⠃⠀⠘⠋⠉⠠⠂⡀⠐⠀⣆
⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⢀⢈⠀⠀⠀⣄⠁⠰⠆⠀⡀⣐⣰⣿
⠀⠀⠀⠀⠘⢠⣤⡀⠀⣠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣤⠶⠷⠛⣨⠀⠀⠈⣶⣴⣰⣾⣥⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠛⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡤⠚⠋⠀⣀⣴⣾⣯⣄⣀⠀⠈⠙⠛⠛⠛⠛⠁⠀⡀
⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠚⠋⠀⢀⣤⡼⢟⣋⠥⠭⠯⠥⠭⠵⠖⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠈⠁⠀⠀⢀⠐⠁⠀⣠⢾⠀⣀⣴⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠔⠂⠄⠀⠄⠒⠐⠁⠀⠰⠿⠷⠛⠋⠉⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠠⠴⣶⠖⠢⠄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⣠⠶⠛⠒⠆⡀⠐⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠒⠒⠁⡄⢠⡄⢠⠈⠒⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠌⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⠀⢀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠁⡌⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢰⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠈⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⠔⠁⠀⠀⡠⠂⠀⠀⠋⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠘⢤⡀⠀⠈⠢⡈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀⠔⠁⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠢⡄⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠂⠁⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠈⠐⢄⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠊⠁⡠⠊⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠧⠤⠄⠠⠤⠤⠠⠤⠤⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄⠈⠑⢦⡀
⠀⠀⠀⣰⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠀⠠⠠⡔⢢⠤⠄⠀⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠢⡀⠙⣆
⠀⠀⠀⢹⠀⠃⠒⠀⠒⠒⠒⠀⠒⠂⠀⠒⠒⠒⠒⠒⠒⠂⠖⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠲⠐⠀⠒⠒⠒⠒⠀⠐⠂⠒⠂⠒⠒⠒⠂⠒⠘⠀⡏
⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠚⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠃⢤⣀⣀⡀⣴⢿⡿⢠⣀⣀⣀⠠⡞⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣀⣭⣹⣯⣨⣅⣀⣈⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡿⠋⣍⣀⣀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⢀⣤⣤⡖⢂⠐⣄⠀⡸⡄⡿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣠⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣾⣟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣵⣦⣀
⣟⡿⠗⠛⠛⠉⡀⢶⠞⡙⢉⣴⡿⢿⣿⠿⣿⣿⠀⠀⢀⣈⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠏⠲⡤⣄⠀⠀⠀⢀⡀⠀⠀⠉⠉⠻⣿⣿⣿⣧
⠀⡀⣿⠖⠚⠃⠄⢠⣤⡾⠋⠁⣰⣾⡁⠄⠀⠀⠀⠀⣸⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⡨⠓⣼⣧⣴⣶⣠⣿⣷⣴⣷⣿⣶⣿⣿⣿⣿
⠛⠃⠐⠒⢀⣴⣾⠛⠃⠀⢸⣿⠯⢠⠄⠀⠀⠀⢀⣾⣿⣿⣯⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠋⢑⣻⣧⡘⢻⣿⣿⣿⣿⣿⣿⣿⣟⣻⠟⣻⣯⡟⠉
⠀⢀⣤⣾⠟⠉⢀⠀⢸⣄⡟⠁⠀⠀⠀⠀⢀⢔⣼⢻⣿⣿⢿⠿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢹⣿⣿⣿⠟⣉⣬⣿⣿⣿⣿⣽⡿⣾⣭⣓⡾⠛
⣶⠟⠛⠀⠀⠀⠀⣤⢾⠟⠀⠀⠀⠀⠀⡰⣺⡿⢥⣿⣏⣁⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⢻⣿⣿⡿⣿⣿⣿⣿⣷⣏⡱⢿⣧⣷⣷⣿⣿⣿⢝⣶
⠁⠀⠀⠐⠂⣠⣞⣽⠁⠀⠀⠀⠀⠠⢚⠴⠏⢠⣿⣻⣿⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⢿⣿⠀⢽⣿⣿⣿⣟⡽⢿⣿⡿⠟⠻⣿⣿⣿⣧⠉
⠀⡀⠂⢰⠖⠹⠟⣳⣶⠆⠀⢀⣼⣀⡝⠛⢹⣿⣿⠃⠒⢀⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠻⣿⣆⠀⠀⠀⠿⠀⠀⠙⢿⣿⣿⣿⣻⣧⣶⣶⣿⡟⣽⣿⣿⡿
⠀⠀⡴⠶⠘⠓⠊⠹⠋⢀⡐⠁⣿⠟⣠⣲⣾⣋⠷⠀⣤⢞⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠙
⡠⠞⢱⠋⠀⠀⠀⠀⡀⠊⠁⢸⠟⠀⢩⣿⣧⡟⠀⢀⠽⣿⢽⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠹⣿⡻⣄⠀⠀⠀⠀⠀⠀⠀⠻⣟⣿⣿⣿⢻⡿⠆⠀⠀⠀⢀
⠂⡴⠁⠀⠀⠀⢀⠎⠁⢀⣴⠇⠐⢢⣾⡟⠨⠁⡠⢂⠀⣩⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠹⡇⢨⣦⡀⠀⠀⠀⠀⠀⠀⠹⣷⠻⣿⣷⡀
⠚⠁⡰⠁⢀⠔⠁⠀⢐⡟⠉⠀⢀⣿⡟⠰⠂⠀⠀⠠⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠿⢻⡼⣿⣿⣶⡄⠀⠀⠀⠀⠀⠹⣧⡹⣿⣿⣦⢦⠄⡀
⠀⠈⠀⠠⠋⠀⠤⣢⠏⡀⠀⢀⣿⡿⠀⠂⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣧⡽⢿⢦⣠⣄⠀⠀⠀⠀⠀⠘⢿⡟⣿⣿⣿⡴⠒⠉
⠀⢀⠞⠀⡀⢀⣶⠯⠊⠀⢀⣾⡟⠁⡈⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠸⡆⠿⣯⡛⠛⠀⠀⠀⠀⠀⠀⢌⢿⡏⢻⣿⣷⡄
⠔⠉⠠⠚⢀⡟⠁⠒⠀⢀⣾⣿⡁⡸⠁⠀⠀⢀⢀⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡄⠀⠀⠀⠀⠀⠀⣀⢻⡴⣺⣿⣷⠶⠀⠀⠀⠀⠀⠀⠡⠹⣦⡻⣿⣿⣦
⣠⠀⠀⢡⠞⠄⠀⠀⢀⣾⣿⠄⢰⠃⠀⣀⠀⢀⡏⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢘⣿⡷⡀⠀⠀⠀⢐⢈⢽⠚⣿⠀⣼⣿⣷⠶⠀⠀⠀⠀⠀⠀⠱⣿⣿⠹⣿⣿
⠚⠓⣴⡏⢀⣈⣄⣀⣿⣿⡃⢠⣟⡠⠀⠀⠀⠞⠦⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠩⠀⠀⠀⠀⠐⠀⠾⠅⢹⣷⣾⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠘⢿⣆⠈⢻
⣴⣾⠃⠟⠀⢀⣠⣿⣿⡟⢀⣾⠘⡁⠀⠠⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣰⣄⠀⠀⠀⠀⢠⣴⣿⣿⣿⡆⠘⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠙⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢧⡀⠀⣨⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡀⠀⠀⣹⢾⡁⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠙⣦⡞⠁⠀⠙⢦⡴⠋⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠞⠁⠙⢦⣀⡴⠋⠙⢦⡀⣠⠞⠳⣄⠀⠀⣠⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢀⡴⠛⢦⡀⠀⣠⠿⢧⡀⠀⠈⣳⠞⠁⠙⢿⡄⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡀⠀⠀⣹⣾⡁⠀⠀⠙⣦⠞⠁⠀⢀⠀⠀⠙⢦⣄⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡞⠁⠀⠙⢦⣠⠞⠁⠀⢀⡴⠋⠀⠀⠀⠀⠙⢯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣠⠞⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡴⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡖⠒⠚⠛⠛⠳⣄⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠋⠙⢦⡀⢀⡼⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⡴⠛⢦⡀⠀⢀⡽⢯⡀⠀⠈⣳⣄
⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⣄⢀⡴⢿⡀⠀⠀⢙⣶⡋⠀⠀⠙⣦⡔⠁⠈⠳⣄
⠀⠀⠀⠈⡇⠀⠀⢿⠀⠀⠸⡆⠀⠀⠀⢀⡽⢦⡀⠀⠀⠀⠈⡟⠀⠀⠙⢦⡴⠋⠀⠙⢦⣠⠞⠉⠙⢦⡀⣠⠟⠳⣄
⠀⠀⠀⠀⢻⡀⠀⠘⣇⠀⠀⠹⣄⠀⠀⠉⠀⠀⠙⢦⡀⠀⢸⡇⠀⠀⠀⠀⠙⢦⡀⣠⠞⠛⣦⡀⠀⣠⠟⢧⡀⠀⣨⠇
⠀⠀⠀⠀⠈⢧⠀⠀⠘⢦⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠙⢦⣸⠁⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⣹⢾⡁⠀⠀⣹⠞⠁
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠈⠳⣄⠀⠀⠈⠙⠓⠶⠆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡞⠁⠀⠙⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣤⣄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⣠⠤⠔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⢠⡤⣼⢾⢚⢚⡹⠀⠀⠀⠀⠀⢀⣀⠠⠆⠀⢠⠛⠠⣤⣄⣀⣀⠀⠀⠀⠀⠙⠃⠀⠚⠁
⠀⠀⠀⠀⣆⣏⣥⡦⠶⠒⠒⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠀⢀⣠⠴⣺⠟⠋⠉⠀⠒⠀⠀⠀⠠⠞⢈⣹⠎⠍⣳⡄⠀⠀⠁
⠀⠀⠀⠀⠏⡧⠖⠗⠁⠀⠀⠳⣶⣤⣶⡏⠀⠀⠀⠀⠀⠀⢺⣧⣀⠈⠓⠒⠖⠒⠶⠒⠶⠒⠚⠋⢉⣀⣤⡾⠟⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠔⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⣀⢤⠶⠋⠁⠀⠀⠀⠀⢀⡀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⢀⠠⠀⠫⡽⢀⠘⠋⠀⣠⣀⠀⠀⠠⡀⠀⠀⣖⢻⢻⠚⡗⣦⡄⠓⠋⣼⡏⠇⢹⠸⠸⡇
⠀⠀⠀⠴⠶⠚⠃⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⣵⠀⢸⣸⣸⡤⠷⠋⠀⢀⠘⠀⠙⠛⠚⠒⠗⠃
⠀⠀⠀⠀⠀⠀⣀⣬⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⢀⠝⠀⠈⠈⠀⠀⠀⠀⠀⣸⣣⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠔⠐⠒⠀⠀⣀⡀⢀⣀⣀⡀⠀⣀⣘⠁⠀⠀⠀⠸⡀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠉⠓⢤⣸⣷⣦⠀⠀⠀⢀⣴⣶⣦⣰⡆⠀⣾⣷⠿⣿⡟⠃⣾⣯⣭⡍⣿⣏⣽⠇⠀⠀⠀⠀⡗⠒⠒⠀⡖⠀⠂
⠀⠀⠀⠀⣤⣀⠀⠀⠈⠀⠀⠙⠣⠀⠀⢼⡇⣸⣿⠹⣿⣤⣿⠇⠀⣿⡇⠀⣿⣥⣤⡀⣿⡟⠻⠦⠀⠄⣀⠀⢹⠚⠓⢺⡃
⠀⠀⠀⠀⠀⠈⠙⠡⠀⠀⣠⢤⡀⠀⠀⠘⠛⠛⢁⣀⡈⣩⣥⣄⠀⣬⣤⡀⠀⣠⣶⡄⣠⡴⠶⠆⠀⠀⠀⠀⣼⠦⠤⠼⣇⡈⠄⠤
⠀⠀⠀⡤⢤⣤⡄⡀⠀⢸⡁⢀⡽⠀⠀⠀⡀⠀⠿⣤⣅⢹⣧⣾⠗⣿⡾⣇⢸⣏⠀⡀⢿⡷⠚⠃⠀⠀⠀⠀⠀⠁⠒⠀⠉⠁
⠀⠀⠀⠀⠠⢶⣷⣤⡄⠀⠉⠉⠀⠀⠀⠁⡀⠀⠶⠶⠿⠸⠿⠀⠀⠛⠀⠉⠈⠛⠛⠁⠸⠷⠚⠛⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠘⠈⢀⡄⠂⡂⠠
⠀⠀⠀⠀⠰⠄⣀⡉⠀⠀⠀⠀⠀⣀⠤⠆⠚⠀⠈⠒⠂⠀⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⡁⠀⡀⠧⠖⠀⠀⠤⠀⣀⢀⡼⠀⠀⠆⠂⠀⢀⠀⡐⢶⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠑⠦⠦⣄⣀⡑⠠⠤⠠⠚⠉⢀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⠀⠰⠀⠑⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠈⣶⠆⠀⢤⠒⠈⠳⣤⡀⠀⠀⠀⠀⠐⠒⠘⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠑⠀⠀⠸⣦⣄⡒⠀⠠⠤⠀⠀⠂⣀⡜
⠀⠀⠀⠸⠇⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠳⠶⠖⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡗⠀⠀⠀⠀⠀⠀⠀⠀⠐⣗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠎⢠⢾⠃⠀⣀⣤⣀⠀⢿⢦⠈⣎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⢘⡾⠀⢼⣿⣿⢿⡇⢘⢾⠄⢹⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣆⠘⣜⡄⠈⣻⠿⣏⠀⣜⡼⠀⣏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⢿⣆⠈⠁⢠⢃⣀⠘⡆⠉⢀⣾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⢟⢽⣤⣤⡀⠀⠁⠀⢠⠇⣼⣿⣇⠹⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢰⣿⣿⢟⣵⣷⡿⡳⡷⣻⠀⠀⢀⡞⢰⣿⠛⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⡛⠿⠯⠮⠯⠾⠾⠞⠁⠀⠀⡼⢠⣿⡏⠀⢹⣷⡀⢧⠀⠀⠀⠀⢀⣠⣾⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⡰⠁⣾⡟⠀⠀⠀⢻⣧⠈⣇⠀⢠⠠⣟⣽⣿⣿⢿⢟⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⣿⡤⠤⠤⠤⠬⣿⣇⠘⡄⠈⠂⠙⠛⠛⠓⠓⠛⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⣰⣿⠋⠉⠉⠉⠉⠉⠙⣿⡆⠹⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⣿⣿⢯⢫⢯⠻⡿⢿⢻⢯⢋⢿⣿⡈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣾⡿⠿⠿⠿⢻⠿⠿⠿⡿⠿⠿⠿⢿⣷⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣞⣿⠁⠀⣠⢞⡵⠊⠁⠑⢮⡳⣄⠀⠘⣿⣷⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⣽⠇⡠⢊⠔⠉⠀⠀⠀⠀⠀⠉⠪⡑⢄⠸⣷⢣⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣼⣾⣿⣮⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣷⣿⣷⣦⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠶⠿⠿⠿⠿⠿⠿⠤⠴⠶⠶⠶⠶⠶⠤⠴⠶⠦⠤⠴⠾⠿⠿⠿⠿⠿⠶⠶⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠔⠒⠒⠒⠒⠒⠒⠒⠂⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢣⣉⠑⠒⠭⠭⠵⢒⣒⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠾⠯⠑⠚⣉⠴⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠉⠑⠒⠢⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⠤⠄⠒⠒⠉⠁⠀⠀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠠⡒⢊⣉⡡⠤⠝⠯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽
⠀⠀⠀⠀⠀⠈⠳⣤⠤⠐⠶⣍⢉⣻⣕⣒⠠⠤⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠠⠤⠤⣒⣂⠭⢴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⡊⠉⣻⣤⡤⠟⢭⡉⠑⠒⠒⠠⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠩⠭⠭⠭⠭⠅⠒⠒⠊⠉⠉⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣤⠤⠟⢶⣈⢉⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣓⣖⠊⠉⢳⣄⣨⠵⠢⠤⠄⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⠠⠤⠴⠒⠊⢉⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⠙⢿⢭⡥⠾⢿⠗⣒⣋⡩⢭⢢⠀⢹⠉⢣⠈⠁⠉⠉⠠⠀⠀⢀⣀⣁⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢣⢻⡄⠈⡆⢿⠟⣿⡋⠁⡇⠀⡗⠚⠒⠒⠒⠚⠓⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⠤⢤⠀⠀⠈⡆⣧⠀⢰⠘⠃⣈⣳⣤⣧⣔⣁
⠀⠀⠀⢀⢤⣠⣤⣒⣛⠭⠥⠒⠒⠊⠀⠀⠀⣧⠧⠤⠋⠉⠉⢴⡒⣉⡹⢦⡄⠓⢦⡀
⠀⠀⠀⠘⠒⠁⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠈⠣⣤⠔⠛⠻⣍⣉⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢖⡉⣉⣳⣤⡌⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣤⠔⠛⢶⢉⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢞⠉⠁⣀⣠⠬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣌⣤⠐⠺⠷⣀⣢⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣾⠋⠉⠁⠀⠀⠀⠀⠀⣀⠀⠀⣀⡤⠄⠀⣀⣀⠀⠀⠀⠐⠋⠉⠉⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⠀⠀⠀⠀⣻⡄⠀⠀⠀⠠⠄⠀⠀⠁⠀⠉⢁⡀⠈⠉⠉⠀⠀⣠⠤⠴⠄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡴⠋⢹⡇⠀⠀⠠⠖⠋⠉⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⢠⠎⣀⣴⠞⢁⡈⠁⠀⠠⠶⠖⠂⠀⠀⠘⠃⠀⠀⠉⠉⠉⠉⠁⠠⠤⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠉⠐⠒⠀⠀⠛⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠙⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣙⢷⣄⠙⠀⠉⠀⠀⢤⣄⠀⠀⠤⠀⠀⠘⠛⠛⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢹⡆⠙⠀⢀⡀⠀⠶⣤⡀⠀⠀⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⡾⠃⠀⠠⣄⡉⠃⠀⣀⠀⠀⠀⢤⣀⠀⠀⠉⠉⠀⠀⠀⠉⠁⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠙⠂⠀⠈⠓⠆⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⠐⠦⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠰⠀⠢⣄⠀⠀⠀⣄⡀⠀⠦⣄⣀⠀⠀⠀⠀⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠈⠙⠲⢦⣀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣄⣀⣐⣀⣒⣽⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⢀⡀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠈⠛⠀⠀⠙⠶⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⣤⣤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣂⣯⣭⣭⣭⣭⣭⣥⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣆
⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣯⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣛⣯⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣮⣻⢿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣯⣻⢿⣿⣿⣿⣷⣽⡻⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣭⣿⣷⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣻⢿⣿⣿⣷⣯⣻⣿⣿⣿⠿⠊
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣟⣿⣽⣷⡄⠀⠀⠀⠀⠀⣠⣶⣯⣻⢿⣿⣿⣷⣯⣻⢿⣿⣿⠷⠉⠁
⢻⣿⣿⡶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⣀⢀⣠⣾⣽⡻⣿⣿⣷⣮⣟⢿⣿⣿⡷⠊⠁⠀⠀⣀⣠⠤⠴⢶⡤⢤⡀
⠀⢻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⢶⣶⣶⣤⣬⣿⡉⠀⠀⢈⣿⣿⣿⣿⣿⠿⢿⣿⣷⣯⣻⢿⣿⣿⣎⣥⣤⠴⢶⡾⠋⠙⣧⣀⣴⠋⣀⣠⣬⣦
⠀⠀⠉⠉⠙⠛⠻⠿⢿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠋⠀⠀⠀⠉⠉⠉⠛⠃⠀⠀⠈⢹⣿⣿⠟⠉⠉⠉⠀⠀⢠⠿⠗⠒⣶⣿⣿⣿⣿⣉⡁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠉⣁⣿⣿⡿⠿⢿⣿⣿⠀⢠⠶⠛⠛⠛⠛⠿⢶⣤⣀⠀⠰⣿⣿⣿⣿⣷⣶⣦⣤⣄⣿⣠⡴⠛⢹⡟⠛⢿⠀⠉⠙⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣤⡟⠉⠀⠀⠀⠀⢀⣬⣄⠀⣀⣤⣤⣤⣄⣀⠀⠈⠛⢷⡄⠘⠁⢈⣿⣿⣿⢷⣿⣿⣿⣻⣿⣿⣿⣶⣤⣼⣧⣔⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⣼⣿⣦⡀⠀⠀⣠⣾⣿⣿⣷⣿⡟⠛⠛⠻⣿⣷⣄⠀⠈⠷⢠⡀⠸⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠛⢛⣴⣾⣿⣿⣿⣿⣿⣿⡿⢷⣶⣤⣤⣿⣿⣤⣤⣾⣿⣿⣆⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿⣿⣯⣿⣿⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠉⠉⠉⠁⠈⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠛⠻⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷





⣶⣶⣶⣶⣶⡾⢶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣷⡾⣶⠾⡶⠾⠷⣶⡾⠷⠶⡿⠿⢶⡶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣴⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣠⣤⣴⢿⣿⣿⣿⡟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣤⣿⣿⣿⣿⣿⣿⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠏⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠈⠛⢿⠟⠛⠛⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢻⣿⣿⡦⣀
⠀⠀⠀⣀⣴⣚⣋⣁⣀⣰⣋⣛⣀⣻⣌⣛⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⡏
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⣠⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢠⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢛⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠋⠉⣻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠛⠋⣵⣶⣾⣿⣷⣶⣶
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠡⡀⠀⠀⡜⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠥⠤⠤⠼⠦⠮⠤⠤⠤⠵⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠴⠦⠾⣿⣿⠿⠶⠶⠶⠤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⠀⠤⢀⡲⣿⣿⢶⣒⠂⠤⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⡿⢿⠛⠛⠘⠓⠀⠸⠤⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣟⣺⡄⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢇⠀⠀⠀⠀⠀⢠⡯⢽⡇⠀⠀⠀⠀⠀⡤⠊⡇
⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣀⢀⣸⡗⢺⣇⡀⠀⠀⠀⠉⠀⠀⡇
⠠⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⢀⣔⣤⡀⠰⠖⢿⣏⡡⠰⣮⣸⣟⣻⣯⣿⢿⠅⣒⣆⣰⢤⣥⣀⣀
⠀⠑⢦⡀⠀⠀⠀⠀⠀⠈⠐⠤⠀⣀⡀⠐⠁⣋⠀⠀⠀⢹⡏⠹⡁⢉⠩⣽⢏⡙⡏⠉⠉⠁⠉⠉⡟⠉⢩⠁⠈⠉⢳
⠀⠀⠀⠈⠁⠒⠀⠤⠄⠀⠀⠒⠋⠉⠀⠀⠀⣿⠤⠄⠀⢸⠇⠂⠙⡀⠀⣿⣤⡜⡇⠀⠀⠀⠀⢀⣇⠀⠸⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠘⡄⠀⡠⠋⠀⠀⠀⠈⠤⡿⣀⠤⣿⠀⠀⠀⠴⠉⡿⠀⢰⠄⠀⡜⣧
⠀⠈⢶⠤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⡀⠰⠊⠀⠀⠀⠄⠀⠀⢈⡿⣄⣁⣿⡤⠒⠀⠀⠀⡏⠀⠀⢱⠊⠀⢻
⠀⠀⠈⠙⠛⠿⠓⠒⠒⠀⠀⠉⠀⠀⠒⠚⠉⠀⠀⠀⠀⠄⠀⠀⠐⠀⢸⣷⣛⣿⣿⠀⠀⠀⠀⠀⢛⣠⠔⠈⢣⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢤⡴⢿⣿⠛⠳⣽⢦⣄⡠⠴⠖⡻⣧⠀⠀⠘⣷⢼⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠀⠀⠲⠂⠀⠀⠀⠀⠀⣼⣸⡗⣦⣈⠀⠀⠀⣠⡀⠙⣾⣆⡀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠈⣟⠛⣌⣨⣿⡷⣤⡀⠀⠙⠦⣌⣻⣎⠘⢶⣻⣿⣟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣟⡉⠉⢻⡟⠲⢌⡡⢄⡀⠀⠙⠛⠷⣏⢹⣏⢿⠻⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠴⠟⠛⢯⣿⠀⠀⠀⠀⠉⠢⣤⣀⡴⠞⣿⣿⣺⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢲⣤⣤⡖⢿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣿⣿⠈⣢⣥⡀
⠀⠀⠀⠀⣀⣠⣤⣶⣦⣄⣠⣴⣆⣀⣀⣄⠀⢀⣤⡀⠀⠀⠀⠀⠀⢼⠟⠋⠉⠉⠙⢻⡆⢀⣀⡀⠀⠀⠀⠀⠀⢠⣟⠿
⠀⠀⠀⠀⠙⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠻⠻⠛⠿⠶⠾⠷⠶⣶⣶⣶⣾⣿⣿⣿⣾⣷⣾⣿⣿⣦⡶⢶⣤⣤⣼⣥⣬⣧⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣀⡤⠤⢠⠾⠒⣲⠞⠛⠛⠒⠚⠓⠒⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠈⠉⠉⠋⠀⠒⠂⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡀⠃⠐⠒⠀⠀⠀⠀⠀⠒⠘⢀⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠒⠒⢰⣏⣀⣀⣹⡤⠀⠒⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢀⠀⢘⠠⢨⡿⠭⡓⠆⠃⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠈⠁⠁⠰⠆⠌⠈⠁⢀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⢹⡖⠂⠐⠂⠀⢲⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠰⢀⣾⡇⠀⠀⠀⠀⢸⣷⡀⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⢿⡃⠐⠀⠀⠀⢘⡻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣰⡏⢸⡇⠀⠀⠀⠀⢸⡅⢹⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⠘⢙⡏⠉⠉⠉⠉⢹⡋⠃⢿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⢖⠁⠁⡆⢘⡃⠀⠀⠀⠀⢘⡁⢸⠈⠨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⠁⡠⠊⠀⢨⡇⠉⠉⠈⠉⢸⡇⠀⠑⢄⠈⠪⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠔⢁⠄⠊⠀⠀⠀⢸⣇⢀⡀⢀⡀⢸⡇⠀⠀⠀⠁⠠⡈⠢⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⡡⠂⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⢴⡦⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢳
⠀⠀⠀⠀⠀⠀⠀⣧⣉⣉⣉⣉⣭⣉⣉⣍⣉⣩⡇⠀⠀⢸⡇⠀⠀⢸⣍⣉⣉⣉⣉⣭⣉⣉⣩⣉⣼
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⠀⢀⡤⠀⣀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣀⠀⢠⡀⠀⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⣤⠤⠬⡆⢲⠤⢤⣾⡷⣤⠤⠾⢰⠏⠤⣬⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢁⠈⣿⣿⡿⠿⣿⣿⡇⠘⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣶⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣶⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡾⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣏⠝⣟⣿⡟⠛⠙⠿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣬⣭⣍⣻⠿⣿⣿⣿⢿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣶⣝⠙⣿⡿⠁
⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣌
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⡿⠏⠾⢹⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠠⣀⠂⠄⠈⢹⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠐⠂⠀⠠
⠀⠀⠀⠀⠠⣈⠒⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⢠⡀⢀⣀⠀⡆⡆⠁⢠⡀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡄⣖⣿⢸⡀⢟⣚⠀⡇⡇⡏⢸⡀⣗⡚⠈⣳⠄


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣬⣭⣬⣭⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠲⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠞⠶⠶⣦⡀
⠀⠀⠀⠀⠀⢎⡻⢿⣿⡿⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠖⠒⠒⠒⠒⠂⢺⣿⣶⢶⢶
⠀⠀⠀⠀⣰⣿⣿⡗⠂⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⢸⣶⣶⣶⣾⡇
⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢡⣴⣆⠀⠀⠀⠀⠀⣠⣶⡬⠀⠀⠀⠀⢸⣶⣶⡀⠀⠀⠀⠀⢠⣶⣶⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⠈⣿⣿⣷⡄⠀⠀⣰⣿⣿⡿⠀⠀⠀⠀⠀⠦⠤⠤⠤⠤⠤⠼⠿⠿⠿⠿⠁
⠀⢰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡷⠀⢸⣿⡿⠏⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠄⢰⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠛⣛⠛⠓⠒
⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⢛⠛⢿⣷⡀
⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⣷⣷⣾⣶⣾⠁⣾⡟⣿⡆⢹
⠀⠀⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡈⠛⠋⣠⣿
⠀⠰⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⢨⢀⠀⠀⠀⠀⠀⡀⢇⠀⠀⠀⠀⠀⠀⠀⡼⣀⠀⠀⠀⠀⢀⡘⡄⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠻⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⢺⣿⡆⠀⠀⠀⢀⣿⡶⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⢲
⠀⠀⠒⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣿⢶⣳⣶⣤⣴⡶⣲⣟⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣅⣠⢅⣿⣭⣿⡥⣅⣠⢥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⣼⣿⣧⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠙⠛⠑⠛⠉⠉⠉⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠷⠀⠀⠀⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⣿⢿⣽⣿⣿⢻⡿⣻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠂
⠀⠀⠀⠀⢀⣾⣿⡿⠻⠟⠋⠁⠀⠀⠀⠀⣿⣿⣿⡿⠿⠟⠛⣿⣿⠟⠉⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣠⣀⣟⠉⢉⡉⣹⣀⣅⣀
⠀⠠⣴⣿⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠘⠀⢀⣠⣤⣶⣶⣶⠀⠀⠀⠀⣴⣶⣶⣦⣤⣀⡀⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡇⠀⠀⠀⠀⣿⣷⣄
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⣠⣾⣿⣿⣿⠁⠀⠀⠀⠀⢹⣿⣿⣿⣤⡀⠘⠋
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣤
⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣀⣽⣿⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢹⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠷⠾⠿⡶⡆⠀⠀⠀⠀⠀⠀⠴⠾⠿⠶⠍⠙⠛⠛⠃
⠀⠀⠀⠀⠐⠿⠞⠶⠞⠷⠲⠿⠓⠀⠀⠀⠀⠐⠿⠗⠶⠻⠶⠚⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠤⠴⠶⠤⠴⠆⠀⠀⠀⠀⠤⠤⠤⠴⠤⠤⠄

⣿⠉⠉⡿⠛⠟⡟⢛⠛⢿⠛⣟⠛⠛⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣻⣿⡟⣿⢛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠛⠀⠀⠀⢘⠃⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⡦⠀⠀⠀⢸⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠀⠀⠀⠈⠈⠓⢄
⠀⠀⠀⢄⠀⠠⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠃⠀⣶⠀⠹⡀⠀⠀⠀⡁⠀⠀⡑⡀
⠀⠀⠀⠈⠐⢌⠲⣅⢄⡀⠀⠀⠁⠀⠉⠑⠒⠒⠒⠒⠒⡊⠉⠁⠈⠒⠀⡚⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠐⢬⡓⣎⣢⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⣤⣤⣿⣤⢤⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠹⡶⡞⠀⠄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠛⠛⠛⠿⠿
⠀⠀⢀⡤⠴⠀⠀⠃⠀⠿⠐⠀⢞⠻⣕⠢⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠀⠉⠂⡙⢭⡣⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠈⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣞⣻⣿⣯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⡿⣿⣿⣿⣾⢻⣿⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⡿⢿⣿⣿⣭⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣽⢿⣻⣿⣯⡿⣿⣿⡷⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⣻⣿⣿⣷⡿⢋⣽⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣶⡟⢋⣷⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢛⣶⣿⣿⣿⣿⣿⣿⣿⡿⢏⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣏⣠⡾⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣦⡿⢿⣿⣿⣿⠋⣠⣿⣿⣵⣟⣻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣄⠙⠋⢨⣿⣿⣷⣟⣿⣿⣿⣾⣻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⢿⣿⣿⣿⣿⡄⠀⠀⠙⠿⣿⣿⣧⡾⢿⣿⣿⣤⠟⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⠉⠿⣿⣿⣿⣽⢿⣿⠟⣩⠻⣋⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣹⡿⣿⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣻⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠳⡀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⠾⣗⢚⣻⣿⡈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢶⣾⣴⠶⠒⠛⠛⢫⠉⠉⠘⣷⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠅⣾⠛⡄⠐⠦⢀⠀⢼⣔⢄⠐⠈⠁⠐⢄
⠀⠀⠀⠀⠀⠀⡰⠋⡴⠋⠀⣿⠀⠘⢦⠙⣤⡉⠓⣿⡠⠈⠰⠆⢴⡄⢷⡤
⠀⠀⠀⠀⣠⠞⣡⠞⠀⠀⠀⠛⠀⠀⠀⠳⣐⠙⢦⢸⠀⢱⡶⣿⡁⠙⠢⢷⡄⠁⠀⢀
⠀⠀⠀⠘⢁⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠈⢷⡃⠒⣺⡿⢷⡄⢤⣷⣤⡴⠞⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡴⠾⢻⣖⣯⣿⣿⡌⣻⣆⣹⣿⡟⡉
⠀⠀⠈⠀⠀⠀⠀⢀⡀⢶⡾⠷⠿⠶⠟⠛⠛⠋⠉⠙⣷⣀⢲⡌⣻⣿⡅⠚⢞⠿⢄
⠀⠀⠀⠀⠀⢀⡞⢁⡅⣿⠟⣄⠀⠰⠄⠀⠀⠐⠂⠐⠛⢿⠫⣅⣼⡃⠹⢶⠙⢾⠀⢢⡀
⠀⠀⠀⠀⡴⠋⣰⠏⠀⣿⠀⠈⢦⠀⢢⠀⠀⠐⠤⢽⡗⢀⢳⡜⣻⣿⣧⣈⣿⡷⡇⠀⠻⡀
⠀⠀⣠⠞⣡⠞⠁⠀⠀⣿⠀⠀⠀⠳⡈⡆⠀⠀⠀⠈⠙⡟⠛⢯⣽⡓⠘⠟⠳⣷⣾⠓⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠
⢀⡾⢁⡴⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⣄⠀⠀⠀⢧⠀⠘⣢⡶⢿⣿⣆⠀⢳⡆⠹⡓⠦⣄⣌⢧⡀⠀⠀⢀⣀⡤⢴⣻⠭⠖
⠁⠨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠐⢛⣷⡾⠛⢖⣻⣧⠼⢷⣶⡷⢶⡈⠀⠀⠉⠳⢷⠔⠺⠟⠛⠉⠁
⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⡴⠷⣾⣻⣟⣧⣭⣴⣿⢷⣖⠀⢿⣽⣾⣇⠀⠀⡠⢀
⠀⠀⠀⠀⠀⢴⣶⣾⣭⣭⣭⣭⣥⣶⡶⠶⠖⠲⠿⢿⡯⣽⢿⡆⢿⣞⣘⣆⣀⣽⢆⣽⢣⡼⡵⠋
⠀⠀⣠⠎⢻⢸⡿⢧⠈⠁⠂⣤⢀⠀⠀⠈⠓⢀⠀⠀⢷⡞⠒⣥⣾⣯⣀⣿⢳⣅⢊⣿⣣⢿⡡
⢀⡾⢃⣴⠋⢸⡇⠈⢳⡀⠀⢰⠀⠀⠁⠐⠆⣈⣿⠛⢉⣻⣖⠋⣿⣻⣿⣦⣠⣿⣿⢿⡇⠀⠑⣀
⠏⣠⡞⠁⠀⣿⡇⠀⠀⠹⣄⠀⣇⠀⠀⠀⠂⠛⠉⣿⡿⠏⠻⣏⣻⡷⠴⠻⣿⢷⣿⣶⡇⠀⠆⠘⢇
⡼⠋⠀⠀⠀⢿⠃⠀⠀⠀⠘⢧⡸⠀⠀⠀⢀⣴⠍⠌⢹⣄⢀⣉⣿⣷⣤⠂⠈⣧⣀⡝⢿⡲⠦⣄⠈⢧⠀⠀⠀⣀⣠⣤⣶⡾⠟
⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠈⠻⣄⡄⠀⢉⡱⣆⣀⣀⡱⣞⣡⣿⣿⠙⢶⣄⣈⢻⡄⠸⡃⠀⠀⠉⠛⠷⠐⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣗⠀⡈⢉⡹⢟⠉⢀⡼⣯⣿⣿⣏⠛⢿⠛⢾⡛⢶⡇⠀⢀⠔⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣁⣀⣈⣷⣾⡭⢹⣿⣿⣉⣇⠀⣱⣀⣛⢻⣿⡰⢯⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣹⣷⣤⣿⡟⢣⣤⣊⣿⣿⡇⣠⣤⠤⡿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠁⣳⣟⡹⢧⣤⣻⣶⣿⣿⡿⣿⣶⠾⢋⣽⣿⠟⢷⡼⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠉⠀⠾⢲⡽⠋⢇⣠⡤⠛⢛⣿⣿⠸⣗⡛⣯⣶⣍⣾⣩⢳⣼⣧⡝⣶
⠀⠀⠀⠀⠀⠀⢀⡠⡞⠛⠛⠒⠒⡶⢺⡟⣶⠶⢾⣧⡤⣶⣯⢭⣿⣾⡿⢿⣾⣷⣿⠟⢿⡅⠹⣿⣇⠀⢣
⠀⠀⠀⢀⡤⠚⠁⠀⠉⢠⠀⠀⢴⣵⣷⣯⠴⠒⠁⢙⡖⢂⡠⠦⣿⢿⡟⣦⣄⣽⠻⡶⢾⣭⣷⣌⢻⣄⠀⢷⡀
⢀⡠⠚⠁⠀⠀⠀⠀⡼⣬⠠⣴⡿⢏⡉⠀⠀⠀⣠⡄⢛⠀⠀⢸⣿⣸⡇⠀⣼⠳⢦⡙⢆⡈⡙⠻⢿⣿⣤⣄⣳⡀⠀⣀⣠⣤⣴
⠛⢦⣀⠀⠀⠀⠀⠀⠸⡄⢀⡏⢹⡔⢀⣉⢵⠞⠁⠀⠀⢢⡴⠛⣿⠁⣿⣸⣏⣆⣀⣉⣻⣷⣤⠀⠦⡹⣷⣀⣉⠛⠈⠉⠉⠁
⠀⠀⠈⠑⠢⣄⠀⠀⢸⡇⠸⠀⡀⢷⡊⣰⠋⠀⠀⠀⠀⠉⡟⠂⣿⣤⣿⡧⠴⠛⠋⠉⠁⠞⠹⣦⣄⠁⣼⣿⡏
⠀⠀⠀⠀⠀⠈⠙⠲⢤⣧⠛⠉⠀⢠⣿⠁⠀⠀⠈⠂⠀⠀⠈⢠⣿⡍⣿⢷⡄⠀⠀⠀⢀⠀⢠⣝⣯⣿⣿⣿⡜⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠉⠈⠀⠀⠀⠀⡸⡢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⠀⠀⠀⠀⠠⠠⠄⠤⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠈⠀⠀⠀⠀⠀⢀⢀⡄⣄⢀
⠀⠀⠀⠀⠀⠀⠨⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⢀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠲⠀⠀⠈⠈⠁⠉⠈⠀⠀⠠⠂⠀⠀⠀⠀⠠⠈⡐
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠈⠁⠉⠈⠀⡀
⠀⠀⠠⡄⠊⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠖⠖⠐⡄
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠀⠀⠀⠠⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣠⠀⡀⠀⠀⠀⠀⡀⢠⣀⠁⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢂⡀⠐⢂⠀⠜⠁⡈⠀⠀⠀⠈⡀⠙⠀⢀⠒⠄⣀⠂⠄⡀
⠀⠀⠀⠀⠀⠠⠄⠀⠡⠔⠈⠡⠀⠈⠠⠐⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠀⠠⠉⠐⠤⠁⠀⠤
⠀⠀⡀⠂⢣⠀⠊⣁⠀⠘⣀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⢀⡙⠀⢀⡉⠂⢠⠃⠂
⠀⠀⠐⡈⠀⠲⠈⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠆⠀⠀⠀⠀⠀⠐⠂⠈⠰⠂⠈⡐
⠀⠀⠀⠈⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠍⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡁⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠙⢿⣿⣇⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⠁⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠈⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿
⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿
⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿
⣿⣿⠁⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⡄⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣤⣶⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣀⣀⣀⣀⣸⣿⣿⠉⠉⠉⠉⢹⣿⣇⣀⣸⣿⣿⠉⠉⠉⠉⢹⣿⣇⣀⣀⣾⣿⠟⠉⠉⠙⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠀⠀⠀⠀⢸⣿⡿⠿⢿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠿⠿⣿⣿⡀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⣶⣴⣦⣾⣿⡇⠀⢀⣿⣿⣶⣶⣶⣦⣾⣿⡇⠀⠀⠹⣿⣿⣶⣶⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣿⣿⡟⠛⠛⠁⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⡿⠿⠿⠿⠿⣿⣿⡇⠀⠀⠀⣴⣿⣿⠿⠿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⡇⠀⠀⠀⠀⣿⣿⣷⣶⣶⣾⣿⣿⠁⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⣿⣇⠀⠀⠀⠀⣿⣿⡏⠙⠋⠙⣿⣿⣄⡀⢀⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿
⣿⣿⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⢿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣶⣿⣿⣿⣿⣷⣿⣾⡆⠀⠀⠀⣶⣷⣿⣾⣿⣿⣶⣾⣦⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣿⣿⠉⠉⠉⠉⢹⣿⣧⣀⣀⣀⣿⣿⡏⠉⠉⠉⢹⣿⣿⠀⠀⠀⠀⠸⠿
⣿⣿⡀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⣿⣿⠁⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣤⣤⣼⣿⡇⠀⠀⠀⣿⣿⣧⣤⣤⣤⣼⣿⣿⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⢺⣿
⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿
⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠠⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡐⡀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠆⠢⠀⡄⠀⠎⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠈⢂⠀⠀⠀⠀⠂⢠⠓⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⠬⢀⡀⠀⠀⢀⠀⡀⠀⠀⠈⠀⠀⠊⠀⠀⠚⣄⣀⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡀⠀⠀⠀⣠⠴⢳⡾⡿⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠊⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢀⡜⠋⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣇
⠀⠀⠀⠠⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⠀⠁⡠⠀⠂⠈⠁⠀⠀⠐⠠⣡⠀⠈⢳⣔⡋⠀⠀⠓⠀⠀⠀⡔⣿⡿
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠰⠫⠂⠀⠀⠀⠀⣠⡿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠒⠀⠄⠀⡀⠠⠀⠀⠀⠀⠀⠀⡃⠀⠠⢀⠠⠀⠀⠀⠠⣀⣾⢧⣞⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠠⠀⢂⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢠⠾⢟⣥⢶⣹⣟⡮⣿⣷
⠀⠀⠀⠀⠀⠀⢀⢀⠅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢸⠄⢀⠀⣐⢴⡾⢸⣗⣿⡿⠿⠿⠿⡝⠷⡿
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⠤⣤⣔⣲⢴⣿⣬⣿⣬⣿⣖⣿⣅⣤⣶⣷⣤⣶⣿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠁⠀⣀⣤⣿⡿⢷⣷⣴⣤⣥⣶⣤⣦⣔⢤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⢷⡾⢋⣴⡿⣿⢻⠉⠟⠛⠯⣙⡿⢢⡉⢷⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠮⣵⣷⠗⣡⠟⡴⣤⠀⢻⣶⠶⢋⠥⡒⢹⠟⢦⣼⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡾⣏⢹⣿⠀⠀⠸⢦⡬⡟⠛⡐⠀⠀⢸⡄⣸⡽⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡏⢀⢰⡏⠀⠀⠸⣧⠀⡇⢀⠠⠂⠀⠊⣡⢷⣣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠠⠬⠥⢄⣐⣂⣛⣛⣓⣒⣶⢦⠯⠟⠛⠋⠁

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣾⣿⡶
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣛⣟⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣦⣄⡀⡀⠀⠀⣠⣶⣿⠿⣷⣦⣄⠀⢠⣴⣶⡿⢿⣶⣦
⠀⠀⠀⣼⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⡟⢸⣿⣿⣇⢹⡇⠀⠀⠀⣀⣤⣤⣸⣿⡿⢻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠙⠿⠿⠿⠿⠿⢿⣤⣿⣛⠿⠿⠿⠿⠛
⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠁⣿⡏⢹⣿⠈⠁⠀⠀⠀⠹⣿⡿⠟⣿⣿⣸⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠏
⠀⠛⠛⠿⣿⡿⠛⠛⠃⠀⠀⠀⠀⠀⠘⡿⠁⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⡅⠀⠀⠀⣤⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠛⠂

⠀⠀⠀⠰⣀⣀⢰⣂⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠙⣤⡄⠀⢀⣀⡀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠫⣿⡯⠋⣠⣤⡝⠆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢘⣶⣿⣿⠛⣿⣿⡛⡄
⠀⠀⠀⠈⢻⣿⣅⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⣿⣿⠇⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⡿⠽⣿⣿⣏⣀⠿⣿⠃
⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⠀⠃⠀⠀⠀⠀⠀⠰⣜⠿⠟⢁⣾⣿⣦⣄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢈⣿⣧⣞⣙⠿⣿⣿⠖⠁
⠠⠮⠀⠀⠉⠉⠭⠍⠉⠉⠀⠀⠥⠄⠀⠀⠀⠀⠈⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠈⠛⠿⠿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠈⠛⠴⠿⠟⠂⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗

⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⡀
⠀⠀⠀⣍⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣯⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠘⠳⣦⠄⢠⣴⣶⣶⣦⡀
⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡷⠘⣿⣿⠷⣟⢀⡤⣾⡇⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠈⠑⣿⣿⢿⣿⣿⣿
⠀⣤⣾⣿⣻⣿⣦⡄⠀⠀⠀⠀⣰⣿⣿⣯⡻⠿⠤⠼⢟⣶⣿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠯⣰⣤⣀
⠀⣿⠋⢸⣿⠈⢻⡇⠀⠀⠀⠀⠙⢿⣿⠟⠀⠀⠀⠀⠀⠻⣿⡿⠇⠀⠀⠀⠀⠀⠘⠛⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉
⠀⠃⠀⠀⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣷⣦⣤⡀
⢀⣾⣿⣿⣿⣹⣿⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣿⣿⢿⣶⣤
⠸⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⢿⣟⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⢀⣀⣴⣹⣛⡟⢻⣛⣏⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢧⠀⠈⠉⠙
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⢻⣽⣽⢿⣿⣿⠄⠀⠀⠀⠀⠿⣿⣿⣷⣿⣿⣿⣷⣾⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⡸⠠⡄⢆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣼⣧⣬⡉⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠉⢹⡏⠉⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣴⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣷⣶⣿⣍⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦
⠀⠀⣀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠏⣳⠢⣄⠀⠀⠀⠀⣄⠀⠀⠃⠀⠙⢿⣿⣿
⠀⢀⣽⣿⣿⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣤⡼⠛⣼⣧⣽⣶⣤⣿⣷⣴⣿⣾⣿⣶⣿⣿
⣠⣾⣿⣿⣭⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠈⢻⣻⣧⡘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣙⡿⢛⣿⣽
⡟⣿⣿⡿⢿⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢻⣻⣿⣿⠟⢋⣤⣿⣿⣿⣿⣽⣿⢷⣫⣝⣢
⣴⣿⡏⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠈⣿⣿⣿⢿⣿⣿⣿⣿⣵⣟⢡⠿⣧⣴⣷⣞⣿⣿⣯
⣿⣿⣿⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⡀⠀⠀⠘⣿⣿⢡⣽⣿⣿⣿⣟⣻⣷⣾⣿⡿⠿⣿⣿⣿⣿
⡿⠧⠅⢈⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡼⢿⣧⠀⠀⠀⠘⡟⠀⠈⠹⣿⣿⣿⣷⣾⣯⣥⣶⣶⣿⣿⣿⣿
⡇⠀⣠⣼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⠟
⠀⠘⢿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⣿⡾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣿⣿⣿⣿⣯
⢀⠉⠈⢃⠯⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢿⡈⢣⡀⠀⠀⠀⠀⠀⠀⠀⢹⣯⢿⣾⣧⡉
⠁⠀⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠐⡾⣇⣹⣿⣦⣤⠀⠀⠀⠀⠀⠀⠻⣆⠹⣿⣷⣤⢀⡀
⠀⠨⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣡⣹⡜⣿⣿⡛⠃⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⣾⣷⣍⣩
⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠟⣿⠉⣾⡵⣿⣿⠀⠀⠀⠀⠀⠀⠙⣿⣼⢿⣿⣿⡄
⠰⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠸⣇⢉⣿⣦⣄⣀⠀⠀⠀⠀⠀⠈⢎⢻⡌⣻⣿⣿
⠞⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡀⠀⠀⠀⠀⡄⠰⣆⣿⡯⠚⢿⣿⣉⣀⠀⠀⠀⠀⠀⠀⢠⣹⣿⡽⣿
⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢽⢿⠥⠀⠀⠀⠀⠀⠐⢹⠈⣷⡀⢻⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⡛
⠑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢙⣆⣽⣿⠓⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠘⢿⣦
⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⢷⠄⠀⠀⠀⠀⢸⣾⣿⣿⣿⣆⠈⠻⣿⠿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠆⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⡰⠀⠀⡢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠇⠀⠐⠂⠀⠀⠤⠀⠤⠄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢄⠀⣠⡤⠤⠤⠶⠀⠀⠐⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢆⠀⠀⠈⠓⢤⣀⣠⠤⠔⠒⢺⢦⣠⠜⠀⠀⠉⠑⠒⠠⠄⡀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠡⡀⠀⡠⢊⠁⠴⡀⠁⠐⠁⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠈⠪⡔⠁⠀⣠⠌⠡⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠈⠢⡜⡠⠊⠀⠘⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠈⠣⢀⢠⠂⡌⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⡀⠀⠀⠀⠀⠀⠀⠑⠾⣀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠑⣤⣀⡀⠀⠀⠀⠀⠀⠈⠐⢒⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠈⠉⠓⠒⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠀⠘⠢⢄
⠀⠀⠀⠀⠀⣀⣀⣀⣠⣎⣀⣀⣀⣀⣀⣀⣀⣱⣀⣀⡀
⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠭⢱⠂
⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣽⣆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠈⢷⡟⣶⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⢻⣛⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⢿⣿⣟⣧⡀⠀⠀⠀⠀⠀⣠⣤⣦⣶⣴⣇⣿⣇⣿⣴⣿⡿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠙⢳⡿⢤⡀⠀⠀⠀⠙⣻⣿⣿⢻⡏⣿⡟⣿⠻⠟⠿⠛⠁
⠀⠀⠀⠀⠀⠀⣶⢶⣶⣶⣶⢶⡆⠀⠀⣠⠀⠈⠻⠉⠀⠀⠀⣸⣿⣿⣿⣿⣾⣼⣥⠗⠁
⠀⠀⠀⠀⠀⠀⣿⠼⣿⡹⠿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠙⢼⢣⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢴⣿⢧⣄⠀⠀⠀⠀⠰⡜⢜⡆
⠀⠀⠀⠀⠀⠀⠀⣀⠥⢐⣒⣒⡂⢙⣾⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣛⣶⡄⣠⣴⠦⠈⠓
⠀⠀⠀⠀⠀⠠⣪⣴⣿⣿⣿⣿⣷⣳⠟⡑⢏⡶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⣿⣿⣿⣿⣷⠀⠀⠑⠏⠀⠀⠀⠀⠀⠀⡀⠰⣮⣻⡿⢫⣾⢳⣄⡀
⠀⠀⠀⠀⠀⢘⣝⣿⣿⣿⣯⣿⣿⣿⠇⡆⠀⠀⠀⠀⠀⠀⠀⠒⠉⢀⠐⠈⠛⠀⠻⠙⠻⣿⣿⣦
⠀⠀⠀⠀⠀⢸⣿⣿⣽⢿⣿⣿⣿⢏⠜⠀⠠⠶⠀⠀⠀⠀⠀⠀⠔⠁⢀⠎⠀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠈⠛⠛⠋⠈⠙⠋⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠱⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠰⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠠⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢡⡃⠀⠉⠉⠀⠀⠀⢹⠁⠀⣧⠁⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⣰⠏⠈⢧⢸⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠂⠀⠀⠘⡇⠀⢰⠏⠀⠀⠀⢿⠀⠀⡟⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢀⠃⠀⢀⣀⠀⠀⠆⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⣀⣀⣀⡇⢸⠰⢩⡿⠯⢏⠡⢸⠀⣇⢀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⡀⠁⠱⠦⠦⠄⠁⢹⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡍⠙⡖⠒⠒⠒⠒⠒⣿⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠒⠒⠂⠒⡀⣾⡇⠀⠀⠀⠀⠀⣿⡆⡗⠐⠂⠒⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡀⣠⠀⠖⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡿⠈⡇⠀⠀⠀⠀⠀⣷⠸⣧⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⠃⢤⠀⠒⠀⠈⣀⡈⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠁⠈⢈⡟⢳⠚⡗⠒⠒⠒⠒⠚⣟⢲⠛⣅⠁⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠑⠦⡉⠁⣙⢶⣤⠄⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠴⡫⠀⡀⢈⡇⠀⠀⠀⠀⠀⣿⠈⡈⠩⠳⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠌⠳⣶⡀⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠞⡡⠊⢀⠔⠁⠸⡏⠉⠉⠉⠉⠉⣿⠀⠑⢄⠈⠢⡑⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠴⠈⠛⢷⣀⠈⡡⢄⠀⠀⠀⠀⠀⢀⡴⢊⡵⠋⡠⠒⠁⠀⠀⠐⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠤⡈⠢⣝⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠲⡂⠈⠉⠳⣄⣀⠱⠆⠀⢀⡴⣋⢔⣁⠀⡎⠀⠀⠀⠀⠀⢸⡇⠀⣀⣄⢀⠀⣿⠀⠀⠀⠀⠀⠈⡆⠈⠑⢌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠈⠉⠻⣧⣉⡡⠬⢷⠦⠄⠐⢚⣁⡙⠦⠀⠐⡗⠲⣖⠒⢒⢒⡎⠀⠀⢸⣿⠀⠀⠈⣖⠒⠒⠒⠒⠒⠓⠒⡒⠒⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠐⡉⢷⡀⠀⢂⢻⣿⡷⡾⠟⠃⢃⠀⢡⠤⠋⠉⠉⠉⡇⠀⠀⢸⣿⠀⠀⠀⡏⠉⠉⠉⠉⠉⡏⢉⣉⠉⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣧⠀⠸⠀⠷⠀⠛⣤⡀⡘⢀⡜⠓⠀⠠⠤⡄⣇⣀⣀⣸⣿⣀⣀⣀⡧⡄⠤⠀⠐⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠤⠀⠒⣈⡉⠆⠀⠀⠀⠀⡿⠀⢰⠤⠤⠀⢒⣊⠹⢧⣝⠛⠫⣯⠉⠉⢃⠸⣯⣭⣿⣿⣿⣭⣽⢀⠉⠈⠹⠉⠉⠹
⠀⠀⠀⢀⠀⡶⠖⠙⠒⢂⠈⠉⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠀⠀⠀⠀⠙⢤⡄⠒⠙⢻⣄⡈⠵⢄⣘⠀⠉⠉⠉⠉⠉⠉⠉⢘⣀⣀⣀⡀⣀⣀⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣉⠁⢈⠻⣦⡄⠙⠢⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠙⠋⠛⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⡀⡀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⣻⣌⣀⠀⠙⠶⣄⠋⠓⣄⠀⠀⠀⠀⣃⣀⣀⣀⣀⣀⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡤⠀⠛⠻⠎⢀⣑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢖⠀⠉⠀⣀⣀⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⠀⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢸⣿⣿⠁⠀⠀⠁⠁⠀⠀⠉⠉⠉⠉⢙⣿⣿⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⣿⣿⡏⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⣿⡟⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠀⠀⣸⣿⡇⢹⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⢰⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⣿⠀⠸⣿⣷⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡟⢠⡀⣿⣿⡄⠀⣽⣿⡇
⠀⠀⠀⢀⣀⣀⣀⣀⣸⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⡇⢸⡇⢹⣿⡇⠀⣿⣿⣇⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠟⠀⠸⠿⠿⠀⠀⠀⠀⠀⠀⠿⠿⢀⣼⣧⠸⠿⠿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢀⣤⣤⡀⠀⣰⣶⣾⣿⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⣺⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⣿⣷⣶⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗

⠀⠀⠀⠀⣿⠛⠛⠈⣿⣠⡿⠀⣿⠛⣿⡇⣿⠀⠀⢸⡟⠛⣿⠀⣿⠛⢻⡇⠀⣾⣷⠀⠙⢻⡟⠋⣿⡇⣾⡟⢻⣿⢸⣿⡀⢺⡇
⠀⠀⠀⠀⣿⣤⡤⠀⢸⣿⠃⠀⣿⣀⣿⡇⣿⠀⠀⢸⡇⠀⣿⠀⣿⣤⣼⡇⢀⡿⢻⡄⠀⢸⡇⠀⣿⡇⣿⡇⢸⣿⢸⣿⣷⣸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⣼⢿⣇⠀⣿⠉⠉⠁⣿⠀⠀⢸⡇⠀⣿⠀⣿⠘⣷⠀⢸⣷⣼⣇⠀⢸⡇⠀⣿⡇⣿⡇⢸⣿⢸⣿⠸⣿⡇
⠀⠀⠀⠀⠛⠛⠛⠘⠛⠀⠛⠀⠛⠀⠀⠀⠛⠛⠛⠘⠛⠛⠛⠀⠛⠀⠙⠃⠛⠃⠀⠛⠀⠘⠃⠀⠛⠃⠙⠛⠛⠛⠘⠛⠀⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢄⣸⣶⣷⣾⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠛⠀⠠⣦⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠺⠇⢀⣤⠀⢶⡆
⠀⠠⢼⣻⣿⣿⣿⡷⠄⠀⢀⣤⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠙⠃⠀⠀⠀⠉
⠀⠀⠊⣻⠿⡷⢟⠑⠂⠀⠀⠉⠀⣠⣦⣦⣼⣿⣦⣀⣄⣀⣴⣶⣶⣤⣄⠀⠀⠀⠀⠀⢸⣿⣦⣤⣤⣤⣤⣤⣤⣴⣤⣴⣶⣤
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣀⢀⠀⠿⠂⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⢸⠭⣿⣰⡆⠀⠀⢀⡶⠀⠀⠀⣆⠀⠀⢀⡀⠀⣀⠀⠰⡀⢸⣿⡇⠀⠀⢀⣦⠇⢠⡄⠀⠀⠀⠀⠀⡴⠀⠀⠀⢀⠄⠊
⠀⠀⢸⣴⣿⠈⡟⠀⠀⠸⡏⠁⠀⠙⣿⠀⠀⣾⣴⣶⣼⡆⠒⡇⢸⣿⡇⠀⠀⢸⣿⠀⣾⣿⠀⠀⡠⠤⠬⡷⠤⠐⠊⣁⣤⣾⡆
⠀⠀⠘⠛⠛⠀⠃⠀⠀⠀⠃⠀⠀⠀⠛⠀⠀⠛⠉⠉⠙⠃⠀⠃⠘⠛⠓⠀⠀⠀⠛⠀⠛⠛⠃⠀⠐⠒⠒⠓⠒⠒⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠩⠅⠻⠋⠉⡇
⠀⠀⠀⣀⠀⠀⢀⡀⢠⡀⠀⠀⠀⣤⢸⣿⣿⠀⠀⠀⢠⠍⢤⠀⠀⠀⢀⣀⠀⠀⠀⠀⠤⢠⡄⠀⠀⠀⡇⠀⠀⢀⣶⡀⣶⡇
⠀⠀⣿⡗⠀⠀⠈⠀⣿⡏⠀⠀⠸⣿⢺⣿⣿⠀⠀⠀⣟⠢⢺⠇⠀⠀⢈⣻⠓⠀⠀⢀⡀⣼⡇⠀⠀⠀⡇⠀⠀⢸⣧⠀⠙⡇
⠀⠀⠞⠛⠀⠀⠀⠀⠹⠁⠀⠀⠰⠋⠛⠛⠛⠀⠀⠀⠚⠀⠸⠀⠀⠀⠐⠉⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠧⠤⠤⠾⠼⠤⠤⠇

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿