Pepper Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠚⠛⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠠⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣙⠛⠀⠀⢂⠀⠀⢄⠀⠀⠀⠈⢧⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢆⡾⠇⠀⠀⠀⠐⣄⣀⠀⠀⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣜⢣⠀⠀⠀⠻⡇⠁⠊⠂⠀⠀⠀⠙⣦⠀⢠⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠛⠻⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠸⣩⣏⣷⠂⠀⠀⣀⡟⢺⣤⣄⢠⡀⢀⡀⠈⢧⣀⡀⠀⠀⠠⣘⠿⠀⠀⠀⠀⠀⠀⠀⠐⢥⣀⡀⠀⢀⣘⡫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢦⠷⣎⡵⣄⠀⢀⣘⠖⣼⠉⡯⢧⠏⠉⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣇⡀⠩⠙⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠦⠛⡤⡇⢺⠄⢉⣈⡗⣇⡴⢽⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠄⠀⠀⠈⠉⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠲⢿⡙⠁⠰⠗⠠⠯⡟⢣⡕⢰⡿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠁⠁
⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠈⣉⣞⢖⠀⠀⠠⠈⢶⣉⡿⡇⠰⢾⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡽⢎⣧⠸⠁⠐⣾⢥⡟⡀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠈⠿⣍⢠⠀⠀⠀⣰⠶⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡙⢺⡆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⣲⣥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣧⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⢙⠝⠀⠀⠀⠀⠀⠠⣴⣠⢞⢉⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣉⠁⠀⠈⠃⠀⠀⠀⠶⡊⣴⠇⠾⣿⠦⢟⡹⣎⢓⣠⠀⠀⠀⠀⠀⠀⢍⠒⠂⠀⠄⠠⠐⠒⢻⡅
⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣷⣿⣿⣿⠿⠒⠀⠀⠀⠀⠀⠀⡙⣧⠭⣝⢳⣛⡗⢊⣵⠛⣼⠃⠀⠀⠀⠀⠀⠀⢸⠟⠲⠲⠦⠤⠖⢺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠿⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠉⠀⠉⠁⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠰⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⠀⠀⠀⠐⠃

⢀⠀⡀⢀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠿⡻⡿⠛⡿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢂⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠛⠊⠁⠀⠀⠀⠀⠀⠀⠀⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⢦⠋⣸⠶⠦⢄⡀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢀⣀⡄⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⢠⠒⠉⠉⠑⠲⢤⣴⣾⡇⣯⣤⠔⠚⠛⠒⢄⠀⠀⠀⠀⠀⠀⠐⠒⢻⣿⡇⠀⠸⣿⠏⣠⣶⣦⡀
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡼⠟⠋⠛⠛⠷⡄⠀⠀⠀⠈⠆⠀⠀⠀⠀⣰⣾⡷⠆⠈⠑⠀⠀⠀⠀⠻⡛⢟⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡆⠀⠀⠀⠀⣿⣛⠤⠀⠀⠀⠀⠀⠀⠀⠘⡇⠐⣿⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣿⣟⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣿⡇
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢹⣿⡆⠈⠢⡄⠀⠀⠀⠀⠀⠀⢨⣿⠃
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠈⢿⣏⠀⠀⠀⠀⣷⣿⡆⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⡀⠤⠀⠉⠀⠁⢂⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠠⣆⠀⠀⠀⠀⠀⣠⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⢀⣠⣤⣀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠔⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠈⠹⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠶⠂⢠⣿⣀⣀⣀⣀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠈⠋⢘⣧⣤⣾⠿⣿⡯⠶⠶⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠐⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠣⡄⠀⠀⠀⠀⠀⣎
⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣘⠁⠀⣤⣤⣤⡀⠈⠑⢄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠊⠉⠉⠉⠉⠉⠛⠶⣤⡆⣹⠀⠀⢹⣿⣿⣿⣷⡄⠈⠓⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣆⠙⠛⣿⡂⠀⠈⠉⠛⠻⣿⣿⣦⡀⠀⠑⢄⡀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⡀⠏⠀⠀⠀⠀⠀⠀⠘⣿⢿⢯⣦⣄⠀⠉⠳⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣀⠷⠁⠀⠀⠀⠀⠀⠀⠀⣰⣗⠀⠁⠙⢿⣿⣶⣄⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⣰⣿⣿⣶⣤⣀⠈⠉⠊⢿⡆⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢤⣾⣿⣿⣿⣷⡶⢿⢿⣿⣿⣿⣿⣯⣻⣿⣷⣦⣼⡇⠀⠀⣿⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠙⠻⠿⠿⣿⣶⣄⡀⠀⠀⠉⢻⣿⣿⣷⣯⣿⠿⠀⢠⣾⣿⣀⢀⠀⣰⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠄⠀⠀⡀⠈⠙⠻⢶⣤⣤⢹⣿⣿⣿⣿⡇⠀⠰⠋⠀⠉⠁⠁⠈⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⡀⠉⠛⠻⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠒⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠊⡀⠤⠤⠀⠀⠀⠐⣌⠑⢄⠀⠀⠀⠀⠀⠀⠀⠋⢀⣀⣐⠒⠤⣑⢄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⠀⠀⠀⠘⠁⠀⢀⡄⢣⠀⠀⠀⢀⠀⠲⠀⠈⠀⢄⠱⡢⢄⡁⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⢀⡀⠀⡈⠀⠀⢀⢌⠃⠠⠤⠀⠀⠀⢱⠘⠀⠈⠂⠘⣆
⠀⠀⠀⠀⠀⠀⠆⡀⠀⠢⠟⠁⢀⣀⡀⣀⢀⠊⢀⠇⡇⠀⢸⠈⠄⠐⠂⠀⠀⢀⠎⠄⠀⠀⠐⠀⡼
⠀⠀⠀⠀⠀⡐⠀⡇⠀⠀⠙⡁⠜⠒⠐⠁⠁⠀⢀⠎⠀⠀⠈⡔⠂⠀⠬⠖⡀⡀⠀⠈⠀⡀⠀⡰⠁
⠀⠀⠀⠀⠀⠃⠀⠁⠠⠒⠸⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠘⠦⣀⠀⣀⣀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠤⠀⣂⠨⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⢰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠐⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠆⢒⠠⢀⡀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠂⠀⠀⠁⠈⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⠀⠀⠠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠿⠻⠿⠿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⢶⣠⣤⣤⣤⣀⠀⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⠆
⠀⠀⠀⠀⠀⢀⣤⡶⣾⣿⣭⣽⣿⣿⡿⠿⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣧⡀⠀⢹⣟⣿⣷⣤⣤⣤⣤⣤⣀⡀⠀⠀⣀⣀⣤⣴⣿⣿⡟
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣶⣶⣶⣮⣝⢿⣿⣿⣎⢻⣿⣿⣿⣧⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢹⡾⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢻⡽⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣌⢿⣿⣿⣿⣿⣿⣿⢣⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠂⠻⠿⣿⣿⡿⠏⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⠉⠉⠉⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⠀⠀⠀⠀⠀⠀⢽⣿⠃
⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠂⠀⠀⠀⠀⣰⣿⣿⣄
⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⢀⣀⡀⠀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣷⣤⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⠿⠋⠁⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠛⠙⠗⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢄⡀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⡴⠋⠉⠉⠲⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⢀⡼⠁⠀⠐⣠⠟
⠀⠀⠀⠀⠀⣀⣠⡤⠶⠶⠒⠒⠶⣞⠋⠉⠉⠀⠀⠀⠀⠀⢨⠟⠀⠀⢀⡞⠁
⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⠢⠤⠒⡆⣀⠴⠋⠀⠀⠀⡟⠛⠛⠓⠲⢤⣀
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⠘⣀⠀⢄⣠⠎⠁⠀⢀⠄⠀⠘⣇⠀⡀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣄⠀⠠⠻⣄⡄⠀⠀⢀⠁⠀⢹⡀⠃⠤⢄⠀⠀⠀⠀⠀⢻
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠉⠟⠛⠛⠶⠶⠿⠙⠒⠀⠣⡀⠀⠚⠛⠓⠻⢤⣀
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠚⠛⠛⠛⠲⢤⡌⠈⠁⠀⠀⠀⠀⠀⠀⠙⢳⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢢
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⣿⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀
⢿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣼
⢸⡀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠘⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⢻⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠻⣦⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⢳⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⣀⣀⣀⣀⣀⣠⣤⠶⠟⠉⠀⠈⠻⣦⣤⣤⣤⣤⡶⠾⠛⠉
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠤⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠒⠆⠀⠀⠀⠀⠀⠀⢀⡤⠶⡆⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠂⢣⣤⡤⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡚⠁⢐⠿⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⣋⣩⡴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠈⠁⢤⠓⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣦⠀⠀⠀⠀⠀⠀⠀⣾⣁⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡽⠛⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⢯⢽⡿⠂
⠀⠀⠀⠀⠀⢤⢰⠾⠛⠀⠀⠀⠀⠀⠀⠀⢀⠔⣃⡳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠳⡹⠌⣟⡏⡿⠐⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀⢠⢸⣀⣀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⠆⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠣⠴⠘⠛⠒⠂⠝⠄⠀⠀⠀⠀⢠⠜⠂⠘⠋⠘⠇⠙⠢⢠⣤⡀
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠤⠤⣠⣴⣖⠃⠀⠀⠀⠰⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⡆
⠀⠀⠀⠀⠀⠘⣆⣄⠀⠀⠀⢹⠆⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⣀⣠⣤⠤⣚⡋⠕⠀⠀⠐⢶⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⣶⣆⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡆⢸⠜⠅⠀⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠸⣵⣦⡘⢻⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⣀⠀⠻⣷⣜⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⢘⠐⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠙⢿⣮⣷⠀⡀⢀⢀⣀⣰⠒⠂⠒⠀⠀⠠⠀⠀⡄⠈⠀⠂⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠺⢃⡦
⠀⠀⠀⠀⠀⠀⡀⠠⠚⠳⢀⣤⡀⠀⠀⠀⠀⠙⣿⣷⡝⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠉⠘⢆
⠀⠀⠀⠀⢰⡬⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠻⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⠀⠀⠀⠀⠀⠸⠂⠀⠉⠬⠥⠤⠄⠐⢺⡀⠀⠀⠀⢀⡎⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠒⣦⠅⢀⣀⣠⣠⢄⣤⢴⠆⠀⠀⠀⠘⠇⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⢸⠂⠀⠀⠀⠀⠈⠀⠐⠛⠓⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⠀⠀⠀⡀⠤⠠⠧⠘⠃
⠀⠀⠀⠀⠀⠀⣡⣀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣤⢤⠀⠂
⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⢨⡵⢫⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠦⠑⠢⣴⠠⣤⣄
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣤⣌⣉⠶⠲⠀⠀⠐⠀⠀⠀⠀⠀⠀⠿⣦⣄⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⢶⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠡⡰⡈⠀⠉⠙⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⢀⠀⠀⠀⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠞⠁⠀⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⠋⠀⠉⠀⠀⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⣠⡴⢾⣿⡷⢂⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⢀⠞⠉⣴⡟⢉⡴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠠⠎⣠⡟⠉⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⡿⠏⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠎⠻⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠈⠛⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡷⠋⠉⠀⠀⠀⢀⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠏⢀⣠⣤⣤⣤⣤⡼⠃⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣠⣶⣿⣿⣿⡷⣤⡄⠀⠂⣠⠋⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⣿⠛⠶⣿⡄⠀⠀⠡⠈⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣀⣤⣀⠹⠇⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡆⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⡆⠀⠀⠀⠀⠘⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠇⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⡏⠸⠋⠈⠁⠀⠀⠀⠀⠀⠈⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣀⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⣿⣿⣏⣶⡇⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠁⠈⠁⢿⡧⠀⠀⠐⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠃⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡋⠋⠀⠀⠀⠀⠀⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠋⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⡆⡀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⢛⡴⢂⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠋⠹⢻⠟⠁⣀⣽⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠠⠖⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣷⡶⠋⠁
⠀⠀⠀⠀⠀⣼⣿⣿⡇⣉⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠹⢿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠄
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⡐⣤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢤⣄⣠⡀
⠀⠀⠀⠀⢀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢂⡀⠀⠀⠀⠨⠴⠛⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣛⠓⠿⡁⠈⠇
⠀⠀⠀⢰⠏⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡮⠅⠀⠀⠀⠀⠐⢚⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢿⠀⠀⠀⡆⢰
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠓⠷⣶⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣆⠀⢀⣿⣼
⠀⠀⠀⢀⠀⠀⠀⡃⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⢠⡤⠴⠆⠀⠀⠀⠀⠀⠈⢧⡀⢸⡄⠈⢿⡟
⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⣠⠾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠈⠉⣴⡏⠀⠀⠐⠂⠀⣀⠀⠀⠀⠘⣇⠚⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⡏⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣿⣧⣶⣿⠿⠋⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠘⢷⡄⠀⣾⠃
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡻⠿⣛⠭⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⡀⢤⡀⠀⠐⠐⠃
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣴⠃⠀⠀⠀⣤⡀⠀⠀⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢒⡶⠈⠀⠈⠻⢾⣷⡸⡄
⠀⠀⠀⠀⢰⡴⠀⠀⠀⠀⢸⡟⠀⠀⠀⢰⣅⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣿⣷⣄⢰⣶⣶⡟⣾⠇⡅
⠀⠀⠠⠀⠞⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠁⡀⢴⣿⣿⡿⢧⡀⠀⡀⠠⠤⣤⡀⠀⠀⠀⠈⠆⢹⡋⣿⡀⢿⡈⠁⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣾⣾⣿⡟⠀⠀⠀⠀⠈⠻⣶⡞⢳⡀⠀⠀⠀⠘⢆⠳⢮⣿⣼⡇⣀⣼⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠀⠠⠚⣯⢾⣧⠀⠀⠀⠀⠀⠀⠀⣀⣴⡇⠀⠀⠀⠀⠀⠑⠦⡙⢿⣷⣬⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⢾⣮⣙⣃⠀⠀⠰⠶⣾⡟⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡺⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠛⠋⠀⠐⢤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣷⣀⣄⣀⣴⡾⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣧⠈⣿⣿⣿⣧⣀⠸⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⢋⣵⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣤⣉⣿⢻⣿⣿⣷⠈⠛⠻⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⢀⣹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣮⣿⣿⣿⣇⠀⡀⠀⠙⠳⠶⢙⣻⣿⣶⣦⢀⣀⣀⣠⣤⣶⠆⢈⣩⣶⣶⣿⣿⡿⠃⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠉⠻⡿⢿⣿⣯⡙⠻⠿⣿⣶⣿⣷⣶⣾⣿⣇⠀⠀⣀⣉⡉⢉⣉⣴⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠿⣦⡀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣿⣧⣀⡘⠛⠻⢿⣿⣿⣶⣼⣿⡿⠿⠿⢿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⢆⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⣿⣷⣶⣤⣮⣤⣸⣿⣿⣿⣷⣿⠿⠋⠁⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠈⢯⠑⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⣀⢴⣠⠔⠛⠻⢿⣷⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠗⠋⢁⡀⣀⣀⣈⣿⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣔⢦⡀⠰⠶⠤⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣾⡿⠋⠀⢀⡴⠟⢛⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⢶⣾⣟⡛⠛⠲⠶⠶⠟⠛⢛⣛⠛⠛⠉⣩⣭⣷⡶⠾⣿⣶⣦⣿⣿⡿⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣶⣶⣿⡍⠛⢷⣤⣤⣄⣀⣨⣤⡶⠶⠿⢛⣫⣍⢀⣼⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣭⣭⣿⣿⣿⣯⣴⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠛⠛⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⣶⣆
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠘⣇⠀⠀⠀⠀⠀⠀⣸⠇⠹⣦
⠀⠀⠀⠀⠀⠀⣸⠋⢻⣄⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠹⣄⠀⠀⠀⠀⣰⠏⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⢷⣄⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠹⣆⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠳⣄⢸⠃⠀⠀⣴⡀⠀⠀⠀⢹⡾⠃⠀⠀⢠⣦⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⡇⠘⢦⣀⠀⠈⠀⣀⣠⠔⠃⠈⣆⠀⠀⠀⠀⠈⢷⡀
⣤⣄⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠙⠓⠛⠋⠉⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣧
⠈⢿⡛⠶⢦⣄⡈⣇⠀⠀⠀⢀⡀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣴⠒⠒⡶⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⢳⡀⠀⠈⠙⠻⠆⠀⠀⠀⡏⠳⢄⣸⠁⠀⠀⠀⠀⢰⠃⠀⡼⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠁⢀⡀⠀⠀⠀⢸⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⠲⣄⡀⣸⠀⡟⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡄⠀⠉⣙⣻⣧⣀⡀⠀⠀⠀⠰⠿⠞⣷⠀⠀⠀⠀⢰⠇
⠀⠀⢰⡇⠀⠀⠀⠸⠓⠦⣤⣸⠀⠀⠀⠀⣠⠖⠛⠶⢤⣄⣠⠟⠉⠙⠳⣄⠀⠀⠀⠀⡏⠀⠀⠀⢠⡏
⠀⠀⢸⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡜⠁⢀⣤⣄⠀⣠⠶⠖⠶⣄⠀⠈⢧⠀⠀⠀⡇⠀⠀⠰⢿⣤⠴⠶⠛
⠀⠀⠈⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰⠁⠀⡏⠠⠎⢻⠁⠀⡶⠀⠈⡇⠀⠘⡆⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠸⡆⠀⠙⠦⠤⠾⣧⡀⠀⢀⣰⠇⠀⠀⣷⠀⠈⣩⠗⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠁⠢⣀⠀⠹⣄⠀⠀⠀⣀⡈⠉⠛⠉⣀⡄⠀⠀⣽⠖⠋⠁⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠙⠒⠾⠷⢦⡀⠈⠉⣟⠛⡟⠉⠀⠀⠀⡇⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠹⣄⠀⢸⡖⣇⠀⠀⠀⣼⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠲⠶⠶⠖⢻⡄⠈⣧⢹⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⠿⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠠⠤⠆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠛⠲⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣶⠓⠟⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢦⣤⣤⣤⣤⡤⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣆⠀⠀⠐⠄⠀⠀⠀⠀⣠⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠠⣄⣀⣀⣤⣿⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡀⠀⠀⠀⠈⠛⠛⠛⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⠦⢄⡀⠀⠀⠀⣠⢊⡈⡑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠰⠦⣀⠈⠐⡄⠀⠘⠦⡅⠰⣔⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⡈⠉⠀⠀⡀⠠⠐⠈⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣄⠀⠇⠀⠀⠄⢠⡄⠀⠘⠀⠀⠀⠙⢢⡀⠂⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⠔⢉⠀⠀⠀⠀⢀⠄⠊⣀⢀⡀⠄⠀⠀⠂⡀⡀⠄⠤⡑⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠴⠷⠤⠤⢄⡤⢠⠀⠊⠀⠀⠀⠤⠤⢒⡉⠀⠂⠀⠀⠀⠀⠐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⢐⣃⠀⠀⡥⡵⠀⣀⠀⠀⠀⠀⢀⣩⠀⠔⢊⠀⠀⠀⢠⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠂⠉⠉⡄⠸⠤⠤⡄⠈⠑⠒⠒⠋⠁⠣⣭⡤⠔⣀⠔⢊⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⠀⢠⡀⠀⢠⠀⠀⠂⠠⡭⠉⠈⠀⠞⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣶⣦⣬⣹⣯⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⣾⣽⠿⠿⠟⠋⠀⢈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠐⣿⠅⠀⠀⢸⠀⠐⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡄⠀⠀⠈⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⣾⡇⠀⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡈⢹⡇⠀⠀⣸⠀⣸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⣁⣼⣇⢀⡤⠿⣶⡇⠀⣤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠄⠀⠠⡆⣈⡹⠉⠉⣉⣹⣿⡟⠂⠀⣄⠀⠀⡀⠸⠟⣻⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⢠⡀⠀⠃⠀⠀⠀⢛⣛⣳⣀⣤⣿⡿⠟⠿⠷⠾⠿⠀⢿⣥⡤⠶⠖⠋⠈⠁⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣠⢀⡀⢀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢛⣿⣯⣥⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡴⠊⠙⠋⠉⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣄⣀⣀⣀⣀⣀⡤⣶⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠠⠐⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠚⠁⠉⠂⢹⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⢉⠳⡀⡇⠀⠀⠀⠀⠀⠤⠤⠭⣕⣆⠀⠀⠀⠀⢀⣴⣂⣀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢣⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠉⠀⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⡴⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠴⡇⠀⠀⠀⠀⠀⠀⢴⣖⠒⠒⠶⠤⢤⣀⣀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⡀⢹⡄⠀⠀⠀⠀⠀⠸⡿⠿⣷⣶⠶⠤⠤⠭⡿⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣋⠀⢀⣱⣄⠀⠀⠀⠀⠀⠹⣄⠈⢻⡀⠀⢀⡼⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠢⣄⠀⠀⠀⠀⠈⠙⠒⠓⠒⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⣀⣀⣠⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦
⠀⣤⠖⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡞
⠘⢧⠾⠻⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢷
⠀⠀⠀⠀⠀⠹⣆⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⢧
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠙⣦⣄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠉⠓⠒⠒⠒⠋⠉⢳⡀
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠤⠀⢀⡴⠃⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⡷⣄⣀⣀⣠⣤⠶⠭⣀⡠⠒⠀⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢋⣉⢳⠀⠀⡸⠓⠒⣲⡞⢑⡒⠐⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡘⠀⣤⠀⠈⡇⠰⠄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⠋⠈⠉⠀⠀⡇⠀⠀⠉⣇⡈⠀⢀⠞⠀⡀⠸⡇⠠
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠱⡀⠀⠀⣠⠗⡤⡀⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡏⠀⠀⠀⠀⠀⠘⢲⣴⣚⣉⣉⣉⠥⠔⠊⡇⢠⡇⠄
⠀⠀⠀⠄⠀⠀⠀⠻⡄⠐⢷⠬⢥⣉⣁⣀⣀⡀⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢸⡇⠀⠀⠀⢀⠄⠀⠸⣀⢈⠀⠀⢀⣀⠠⢾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⣄⣀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢹⠉⠉⢉⠀⠀⠐⠉⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡼⠳⢤⣀⡀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⡟⠉⠉⠉⢻⠈⠉⢻⠀⠀⢀⡤⠒⠒⠺⣅⠀⣀⣈⣉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠐⠀⠀⠸⠋⠀⢀⠈⠱⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆⠀⠀⠀⠀⠀⢸⢀⠀⠘⢄⡀⠀⣀⣧⠀⠀⠀⠀⢠⠃⢠⠏⠀⠑⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠣⣀⠀⠀⠀⡠⠎⠈⠀⠀⠀⠈⣹⡁⠀⠑⠦⠤⠒⣁⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⠀⠀⠀⠀⠀⢉⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣈⠁⠉⠉⢉⣀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢿⡿⠿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⢠⠋⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡿⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠲⠀⠀⠈⠀⠀⠀⠀⠀⠳⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⠄⢀⣤⠀⠀⠀⠀⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠣⣈⡿⠏⠀⠀⢀⠔⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠀⢀⡤⢶⠁⠀⠈⡍⠉⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠊⠀⠔⠊⠀⠠⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⡉⠡⠀⠈⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣇⢀⣤⣤⣛⠁⠢⠡⠄⠠⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣀⠅⠀⣽⣿⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⢀⣀⣀⡀⠀⠤⠐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠐⢐⠒⠒⠒⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡴⡇⠀⠀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⢁⣘⣿⡷⠴⠇⠀⠀⠀⣠⠀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣛⣛⣹⣋⣎⣛⣟⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣿⣭⣭⣽⣿⣭⣭⣭⣭⣽

⠋⠀⠬⠘⠀⠀⠀⠀⠀⠁⠀⠀⢰⠂⡀⠀⠀⠀⠊⠀⠀⠀⠉⠀⠀⠁⠀⠁⠈⠈⠀⢈⠉⠀⠀⢮⣭⡷⠂⠀⠀⠈
⠏⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠨⡀⠆⠀⠀⠀⠀⠤⢀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁
⠀⠀⣀⡔⢒⠙⠛⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⣀⠂⠀⠀⠀⠀⠀⠀⠀⢲⠄⠀⠀⠀⣠⠂⡸⠂⠀⠰
⠁⠀⠈⢁⡉⠀⢬⠀⡀⠀⠀⠀⡐⠀⢀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠘⠁⠀⠀⠁⠀⠀⠸⠁⠐⠂⠀⠀⠀⠀⠐⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⣑⢶⣄⡀⠀⠀⠀⠀⠘⠁⠀⣀⡀⠀⠀⠀⠀⠘⡅
⠃⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠂⠀⠀⠀⠀⠀⢀⠠⠝⠛⠖⠀⣀⠀⠀⠃⢀⠀⢰⠒⢰⡄⠀⠲
⠁⠠⠂⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⠁⠀⠀⠀⠉⣈⠁⠠⡌⠁⠀⠀⠀⠂⠙⠀⠀⠁⠩⠤⠀⠀⠁
⠀⢀⡀⠀⠀⠀⡆⠀⠉⠠⠄⠈⠇⠀⣠⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠠⠓⠀⠀⠀⠀⠠⣤⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰
⠰⠈⠀⠀⠀⠀⣁⠀⠀⠀⠀⢱⠀⠀⠀⣔⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠋⠙⢠⠁⢐⣀⡄⠀⠀⡀
⠀⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⢠⠴⣶⣧⠀⠀⠨⠁⠀⠀⠀⠂⠀⠀⠑⠘⠾⡌⠁⠙⠀⠀⠀⠋⠠⡐⠆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠖⠐⠀⠀⠠⠀⠀⠈⠙⠉⠁⠀⠀⠀⢠⠄⠀⠐⠀⠤⠄⠲⠄⠀⠀⠀⠀⢠⠆⠀⠀⠀⠡
⠲⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣛⡂⡀⠀⠀⠀⣀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠈⡀⠀⠈
⠄⠀⠠⠤⠒⠂⠀⡀⠘⠠⠐⠂⡀⠈⠁⠀⠻⢡⡗⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡅⠀⠀⠈⠀⠀⠀⢃⠀⡅⠀⠀⠀⠀⠶
⣄⠀⠳⠀⠀⠀⠙⠉⠁⠀⠂⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⡁⠀⠀⠢⠀⠀⠀⠀⠖⠆⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⠳⠀⠀⠀⢠⠁⡸⡃⠀⠀⠀⠀⠀⢠⡄⠘⠁⠀⠀⠐⠀⠀⠀⢀⣀
⠷⠃⠀⠀⣀⣤⠊⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢲⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢲⡀⠀⠀⠀⠀⡠⡐⢰
⠀⠀⠀⠯⠽⠞⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠑⢀⠀⠀⢀⠀⠀⠀⠀⠆⠀⣀⠀⠀⠀⠀⠈⢄⠸⠀⠀⢱⠈⠁
⠀⣶⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠈⠩⠡⢤⠄⡀⠀⠀⠲⠊⠁⠀⠀⠀⠉⠁⠀⠀⠠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠁
⢠⠋⠠⠂⠀⠀⠸⠸⢰⠋⠁⠀⠄⠀⠀⠰⠂⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⢠⡄⡀⠀⡄
⡮⢄⠢⠀⣤⡴⢒⢉⣢⠖⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠎⠸⠇⠀⠈⠠⠀⡀⠐⠚⢀⣠⠀⠀⠀⠠⡌⠀⠀⠀⠁⠈⠐⠆
⠁⠁⠀⠀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠂⠛⠠⠁⠀⠈⢃⣘⠆⠀⠦⠄⠀⠀⠀⠁⠀⠀⠄
⢀⠀⠀⠀⠀⢰⠂⣢⡀⠀⠆⠀⠀⠀⠀⠀⡀⠀⡒⡒⡄⢂⠀⠀⠀⠀⠀⠀⠴⡿⠞⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⣀⠁⠀⢀⠀⠀⢈⢉⡅⠁⠀⠁⠀⡇⠀⠀⠟⢋⠄⢁⠪⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣀⣠⠐⠂⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢃⣴⣿⣷⣾⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⡿⠉⡀⠛⢠⡤⠿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠻⣦⣣⣤⠒⠀⠀⠀⠀⠉⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣿⣋⣤⠖⠀⢀⠀⠀⠲⢶⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⢀⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠧⠠⠶⠾⢿⣭⣭⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡛⠓
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠻⠿⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡻⣿⣦⡝⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡿⣯⣍⠉⠉⠀⢠
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠌⠛⠃⠠⣰⡆
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡟
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⡇⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠆⢠⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢸⣿⡇⠀⡰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡸⠋⠁⣆⠀⠈⠀⢰⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣤⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡟
⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠁⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇
⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡇⠒⣿⠇⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⣿⣿⠀⠘
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⠀⠀⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠃⠀⠀⠀⠀⠀⠀⢠⡿⠁⣼⣿⠀⠀⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠂⠀⠀⠀⠀⠀⠀⠀⣾⠇⢠⣿⡇⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣾⣿⠃⠀⠀⢃⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⢠⣿⡏⠀⠀⠀⣼⣿⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⣼⡿⠁⠄⠀⢠⣾⣿
⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⣿⠁⠀⠀⢀⣾⣿⡇⠀⠀⠆
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⣼⢟⣿⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠜⢁⣾⠋⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣰⡿⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⣶⡾⠋⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⣾⡿⠋⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠀⠀⠁⠀⡀⠀⠀⠛⠁
⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠒⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠁⠀⠀⠀⠀⢰⠇
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⣀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⠐⠲⡄
⠞⠃⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡿
⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠙⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⡟⣻⠇⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠟⣿⡿⡿⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡛⠫⠿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣟⡿⠀⠀⠘⠀⠄⠒⢨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⠑⢢⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢿⣦⣠⢄⠀⠀⠀⠴⠶⠾⠿⠟⠛⠿⠖⠀⠀⠋⢉⣿⣿⣿⣿⡙⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡟⢿⣿⣆⠀⠀⠀⢰⣴⣶⣾⣶⣶⠂⠀⠀⠀⠀⠸⣿⣿⣿⣟⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠚⠛⠯⠀⠀⠀⠈⣿⣤⠿⠛⠛⡁⠀⠀⠀⠀⠀⠈⣿⢹⣽⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣠⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣝⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⣇⠀⢀⠐⢦⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢀⣬⡛⣷⣿⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣄⢂⣰⣬⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢐⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣆⣀⣉⠩⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⠀⠸⣿⠿⠛⠻⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡷⡉⠁⠀⡀⠀⠀⣀⣶⡿⠋⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣰⠏⣴⢏⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢲⡶⠛⠉⣼⠃⣇⢙⢲⣶⣦⡐⠺⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠘⣿⣼⣷⣿⠟⠻⣿⣿⣷⣤⣀
⠀⠀⠀⠀⡀⢠⣶⡿⠃⠀⠀⠀⠀⠀⢹⡄⢹⡌⢾⡉⠋⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⣿⣿⡿⠀⠀⠉⣽⣿⣿⣿⣿
⠀⠀⡠⠊⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣿⡘⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⠀⠀⠀⠉⠁⠛⠻⠿
⠀⢀⣄⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⣿⣷⡿
⠀⣤⡛⠷⣤⣀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢏⡇⢷
⠀⢻⠁⠀⠀⠉⠉⠛⠒⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⢀⡄⣧⠘⡄⠀⢠⣦⣄
⠀⠈⠇⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠰⣿⡆⠀⠲⠶⣦⣄⣀⠀⠀⠠⠈⠰⠠⠀⠀⠀⣀⣀⡀⠀⠠⢴⡎⠁⠀⠈⣷⣄⠂⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢈⢀⡃⠀⠀⠀⠀⠸⢤⡀⠀⠐⡄⢿⠂⠀⠀⠞⠭⠇⠀⠀⢐⢠
⠀⠀⠀⠀⠀⠐⠞⣮⢫⢧⡀⠀⠀⠀⢂⠀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠢⢀⡀⠀⠠⣄⠠⠤
⠀⠀⠀⠀⠀⠀⢠⡹⣇⣩⣆⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⡄⠠⣤⠀⠀⠀⠁⠲⠦⠤⠶⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠶⠇⢠⠀⠀⡀⠀⠀⠀⠀⠉⠰⠬⠀⠀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠸⠒⠒⠚⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠀⠀⠀⠀⠆⠐⢢⡀⠀⡀⠈⢲⠑⢲⣀⡐⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠄⡀⣠⠔⠀⠀⢠⠞⠁⠀⠀⠀⣀⠔⠚⠓⢲⣷⡀⡉⠫⢿⣤⡧⠉⣹
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⠇⠀⠀⢠⠟⠘⠁⠀⠀⠆⠀⠀⣰⣦⣆⣿⣿⣿⣠⣴⣿⣤⣤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠴⣴⣻⠖⠁⠀⠀⢟⣃⠈⠀⠀⠀⠀⢶⣶⣿⠿⠛⠋⠉⠙⠛⠛⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠠⢀⣈⣥⡴⠶⠚⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠚⠻⣦⣀⡀⠀⠀⢀⣄⣹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢄⠀⠀⠀⠀⠀⡄⠀⡇⢀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠈⢷⠀⠀⠀⢰⡿⣄⠐⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠜⠀⠀⠀⠈⡧⢈⠑⢄⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⣧⠀⠀⠀⠀⠀⠘⢖⠁⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠈⢸⠀⡼⠀⠀⠀⠀⠀⠑⣤⠂⢀⠙⣆⠀⠀⠀⢀⠞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⢀⠨⣸⠀⢇⠳⡀⢤⣀⠀⠀⠈⢢⠀⢐⡘⡆⠀⢠⢁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠍⡂⠑⠄⢳⠀⠀⠁⠺⠌⡮⠳⡄⠀⢸⠂⡁⠁⢵⠀⠘⣡
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⡇⠀⠘⢄⠈⠢⡀⠀⠀⢠⠇⠂⣻⢠⠏⠀⠇⠠⡍⠀⢰⢻⡀
⠀⠀⠀⠀⠀⢯⠢⣄⡀⠀⠀⢳⠅⠘⡀⠑⢄⠙⣆⣠⠋⣠⠀⠿⠉⡠⠁⠂⢠⠃⠀⡼⠐⣇
⠀⠀⠀⠀⠀⠈⠢⡀⠙⡄⠀⠀⠱⡄⠑⡀⠀⠳⡈⠃⡐⠀⡄⢀⠎⠀⡐⢠⠏⢀⣴⠇⠀⢻⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠰⣤⣅⡚⠀⠀⠀⠘⣦⠐⡀⠀⠱⡐⠀⠀⡖⠁⠀⡐⢠⣮⡴⠛⢁⡀⠀⣼⠀⠀⢀⠜⣹
⠀⠀⠀⠀⠀⠀⠀⠈⠑⡌⠓⢤⠀⠀⢸⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠁⠛⣀⠄⠒⡉⠀⣴⠏⠀⢐⡡⠞⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢠⡀⢳⡀⢸⠁⡀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠊⠀⢀⠎⢀⡤⠁⠀⠀⢈⠔⢫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⡄⢷⠟⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡘⠀⡜⢁⣴⠏⡴⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠐⠀⢠⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠘⣵⠯⢹⢰⠃⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⠃⢦⢘⠟⡌⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⠈⢦⡰⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⠀⠀⠀⣀⣀⠀⠠⠌⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠡⠐⠈⠐⠠⠀⠠⠀⠤⠂⠄⠂⠀⠠⠀⠠⠤⠆⠑⠀⠀⠁⠒⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠛⠉⠉⠉⠉⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠂⠀⠀⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠓⣤⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⢀⡞⠁⠀⠂⠁⠀⣱⠏⡴⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⢸⠁⠎⠀⠀⠀⠊⡏⣸⠃⠀⢀⠤⠂⠈⠁⠀⠀⣀⣈⣐⣠⡀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⠀⢻
⠀⠀⠀⢸⡌⠀⠀⠀⠀⠀⠻⡇⡠⠂⠁⠀⠀⢀⡠⠖⠋⠉⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄
⠀⠀⠀⠀⢻⡀⠀⠀⠀⡆⠠⡏⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⣀⣀⣠⣤⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠂⠀⣇⠀⡠⠐⠂⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠒⠀⠀⢸⡇⢸⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⠀⢹⡎⠀⠀⠀⣇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡾
⢀⣀⠀⠀⠀⢸⠀⠀⠀⠁⠀⢸⢷⡀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⠀⢸⡽⠁
⡞⠻⣄⣠⡴⠃⠀⢀⠈⠀⠀⡞⠀⡷⠋⠙⠛⠂⠀⠀⠀⠀⣀⣠⠆⠀⠀⠀⠠⣄⠀⠀⠈⠉⠀⠀⠘⡇
⢳⡀⠀⠀⠤⠀⠂⠀⠀⢀⡞⠁⠸⡇⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠀⠀⠀⠀⣿
⠀⠙⠦⢤⣤⣤⣤⠤⠖⠋⠀⠀⠀⠙⠲⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣠⣤⣴⣾⡆⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢿⡿⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠇⠖⠒⠋⠀⢀⣤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠴⠦⢄⣀⣀⣀⣀⣤⡤⠤⠖⠛⢻⠉⢹⡗⠒⠒⠒⠒⠊⢉⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⢀⠀⢠⠁⠀⠀⠠⠤⠤⠒⠀⠀⠈⣧⣼⠦⠶⠒⠒⠒⠢⢄⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠒⢦⠏⢀⡴⣄⣀⣄⠀⠠⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠈⠉⠋⠀⠉⠴⡇⠈⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣉⣙⣲⣌⣿⣁⣀⣤⠤⠤⠴⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠉⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣈⣹⣏⣁⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠚⠒⢺⡗⠓⠚⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣸⣇⣀⣰⣇⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡺⠉⠩⢕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠈⣆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠾⠳⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⣀⣠⡤⠶⠖⠛⠉⢁⣀⡤⠤⣄⠀⠘⢧⡀⠘⡜⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡥⠴⠚⠋⠉⢀⡠⠤⠔⠚⠉⠉⠀⠀⠀⠈⢆⠀⠀⠻⣄⠈⠒⠡⠙⠡⢐⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⠋⠁⠀⡤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠙⣆⠀⠀⠀⠀⠀⠙⡜⡄
⠀⠀⠀⠣⡀⠀⠀⠠⠀⠀⠀⠀⠌⠀⠀⣠⠏⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠘⣧⠀⠀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⡆⠀⠀⡆⠀⠀⢀⠎⠀⠀⡴⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡤⠚⠁⣷⠀⠀⠀⠘⣧⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠘⠀⢀⡞⠀⣠⣤⡀⠀⡇⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⣨⣧⠔⠋⠁⣀⡤⠚⠁⠀⠀⠀⣠⠾⡆⠀⠀⠀⢇⢇
⠀⠀⠀⠀⠀⠀⡀⢀⠊⢀⠄⢀⡞⠀⠀⠀⠈⢻⣤⡷⠤⣄⣀⣟⠶⠶⠶⠶⢒⣫⠕⣃⡤⠖⠋⠁⠀⠀⠀⢀⡤⠚⠁⠀⣿⠀⠀⠀⠈⠢⣕⡂⠤
⠀⠀⠀⠀⠀⠔⠠⠁⠰⠁⢀⡞⡴⠾⣯⠞⠉⠙⣿⢧⣀⣀⣀⣈⣩⠭⠭⠿⠛⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠉⠀⠀⢀⡠⠟⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢈⠀⠱⠀⠀⠀⡞⠰⡁⢿⡏⠀⣾⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠉⠉⠳⣤⣬⣿⣡⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠉⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠑⠒⠒⠲⠦⠤⠤⠤⢤⠶⢶⣶⣖⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⡠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⣠⠃⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣸⠃
⠣⣈⠢⢄⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠶⠚⠋⠓⠚⠋⣡⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠛⣉⣠⠴⠊⠁
⠀⠀⠑⠢⠬⣙⡒⠲⠦⠤⢤⣤⣤⣤⣤⣤⣤⣬⣿⣿⠥⠤⠶⠒⠚⠋⣉⣁⡤⠴⠖⠋⠉
⣀⣀⣀⣀⣀⣀⣈⣉⣙⣒⣲⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣖⣒⣚⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⣛⢟⣙⠛⣛⢏⣿⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠖⠒⢲⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠖⠋⠉⠙⠢⣼
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⡠⠒⠈⠁⠀⠀⠈⠙⠲⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡤⠤⢷⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⡄⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣴⣾⣿⣿⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣋⠉⠈⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⡀⠀⠀⣀⣀⣀⠤⠖⠚⡅⠠⡤⠤⠤⠄⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢈⠉⠉⠉⠀⠀⠀⠀⠀⢹⣤⡧⠤⠤⠤⠤⣈⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠲⣞⢀⡴⣄⣀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⡈⣉⠀⢀⢲⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠤⠵⠶⠗⠒⠒⠒⠛⢩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠠⣤⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⠶⠶⣿⠶⠶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⣀⣿⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠢⡤⠖⠉⠉⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠴⢅⡀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣔⠲⣄⠂⠉⠉⠀⠉⠉⠒⢄⡔⠋⠀⠀⠀⢀⠤⠒⠑⠬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠈⢁⡢⠤⠄⣀⠀⠀⡀⢀⡤⠒⠂⠁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⣱⠊⠉⠫⣠⠤⠤⣄⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠠⠤⢰⣍⠉⠉⠁⠀⠀⠀⠀⠀⣼⡻⠭⢯⡉⣏⢩⡆⠀⣀⡴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣽⡋⢭⣭⣺⣀⠀⠀⠀⠀⣀⣀⣤⠯⣤⠶⢖⣛⣻⡄⢉⡩⠥⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢿⣉⣉⣁⣤⠭⣽⠋⠉⠉⢫⡏⠀⠀⠈⢏⠉⠀⢰⡇⠸⣄⠀⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⢮⠽⠖⠒⠛⡇⠀⠀⠀⣀⠀⠀⠤⠐⢺⠀⠀⢸⡇⢠⠤⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣽⠀⠀⠀⠀⡗⠀⠁⠁⢀⢆⠀⠀⢀⠃⠀⠀⠘⡇⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠈⠢⠤⠔⠃⠀⠉⠈⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢘⡧⠤⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠉⠒⠤⠤⠀⠀⠤⠤⠐⠉⠀⠀⠀⢨⡗⠒⠒⢦⡀⠙⢄
⠀⠀⠀⠀⠀⣀⠤⠤⠼⠧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢹⠀⠈⠓⢤⡀
⠀⠀⠀⡠⠊⢀⠔⠒⢰⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⡜⠀⡰⡦⡀⠑
⠀⠀⢰⠁⢰⠃⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠑⠊⠁⠳⠝⠒
⠀⡴⠋⠀⠈⡆⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇
⡜⠀⡴⡦⣀⣙⡆⠀⢸⡇⠀⣿⠀⠀⠀⢀⡀⢀⠀⡄⡆⠀⠄⠪⠀⢒⣂⣠⡼⠇
⠳⠴⠵⠃⠀⠀⠀⠀⠈⢧⠀⣿⠀⠀⠀⠐⢃⣘⣪⡥⠤⠶⠒⠛⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣿⣶⣶⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣠⣀⡤⠀⠀⠀⠀⠀⡀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢐⣰⢴⡿⠛⠀⠉⠀⠀⠀⠀⠀⠀⣍⣉⣩⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⢽⠂⣀
⠀⠀⠀⠀⠀⠀⠟⠛⣷⡄⠈⠁⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⡶⢲⣴⡏⠀⠀⠀⠀⠐⣑⠀⠰⡂⠐⠀⠙⠉⠁
⠀⠀⠀⠀⢀⠄⠀⢀⡈⢷⠌⣹⠀⢰⣄⠀⠹⣧⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠄⠁⠠⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠈⠁⣠⡶⡞⠈⠈⢠⡿⠀⠀⣿⠀⠀⠘⣷⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠂⠀⠀⠀⢀⣠⣄⢠⡄⠈⠁
⠀⠀⠀⠀⢈⠋⠈⠀⠀⠀⠀⣇⠀⠀⠸⣇⠀⠀⠸⡇⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⣠⣔⣯⣽⣿⣿
⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⢻⡀⠀⠀⠁⠀⠀⠀⠛⠛⠛⠃⠀⠀⢀⡴⢹⡷⠃⠀⠀⠊⠁⠀⠀⢂⡀
⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣻⣏⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⣘⢘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢤⣦⣀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⠂⡁⠁
⠀⠀⠀⠀⠀⡤⣤⣤⡇⠀⠀⠀⠴⠂⠀⠀⣐⣖⣷⣻⣾⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⡟⢯⣿⡇⠀⠀⢀⠀⠂⠀⠀⢻⣿⠟⠋⠉⠛⢛⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀
⠀⠀⠀⠀⠀⡧⢤⣾⡇⠀⠀⠫⠀⠠⠈⠀⠀⠻⣦⣀⣀⣀⣤⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠒⠋⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣷
⠀⠀⠀⣠⠴⠋⠀⠀⠀⢀⣀⣤⠴⠖⠒⠛⠋⠉⠉⠉⢙⠓⠲⢤⣹⡀
⠀⢀⣠⠟⠀⠀⠀⣠⠖⠋⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠹⡇
⣠⣟⠁⠀⠀⢠⡞⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢉⡟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡤⠜⠂⠀⡇
⠘⣧⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠹⣆⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠆⠀⠀⠀⠀⠀⠙⣦
⠀⠀⢹⡄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸
⠀⠀⠀⠻⣄⠀⠈⣻⠄⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⣠⣤⣴⣾⡄⠀⠀⡾
⠀⠀⠀⢰⠏⠉⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⡇
⠀⠀⠀⠘⠷⡤⣦⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠋⠙⣋⡡⠟⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠧⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠲⣤⣤⡤⠴⠶⠶⠚⠛⠉⠉⠈⢹⣆⠀⠀⡏⠉⠉⠉⠉⠉⠐⠒⠉⠁⢀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠞⠋⠉⠉⠉⠉⠉⠉⠁⠛⠶⣤⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠈⡷⠤⣤⡞⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣿⣁⣀⡀⡴⠃⠱⠖⢊⡦⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⡀⠀⠀⢀⣉⣙⠆⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠒⠒⠤⣩⠇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⣶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠰⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠛⠛⡟⠛⠛⡿⠛⢻⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡇⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢼⡇⠀⠀⣿⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡇⠀⠀⣿⣿⣿⣷⣤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢃⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣵⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠋⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⢿⠿⢿⣶⣶⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢸⡇⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢤⣬⣭⠤⠶⢴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠀⠀⢠⣤⣀⣤⡄⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣄⣀⣉⣠⣄⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⡇⠀⠀⠀⠀⣈⣉⣩⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠟⠀⠀⠀⠀⠈⠉⣉⡀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠘⠉⠉⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡀⠀⠀⠀⠀⠀⠒⠛⠉⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣼⠃⠀⠀⠀⠀⣘⣋⣉⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠃⠀⠀⠀⢀⣀⣩⣭⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⡏⠀⠀⠀⠀⢈⣉⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡟⠀⠀⠀⠀⠀⢬⠍⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣠⠏⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⠁⠰⠋⠀⠀⠀⠛⠓⠚⢒⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⣀⣘⣛⣛⣓⣺⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠤⣤⣉⡉⣩⣭⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠶⢦⣤⣤⣠⣤⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⢀⣀⣀⠙⠛⠶⠦⣴⣾⡿⠋
⠀⠀⠀⠀⣀⣠⣶⠿⠋⢀⡀⠴⢤⣀⣉⣉⣙⣛⣿⣿⠟⠁
⣀⣤⣾⡿⠟⠋⠁⠠⣤⣀⠙⢻⣶⣤⣽⣿⠿⠛⠋⠁
⠛⠻⠿⣶⣶⣴⣿⣿⣿⣿⡿⠿⠛⠛⠉
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠐⢾⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢛⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⢠⣀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠯⠁⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢘⠉⢹⠇⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⣀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡆⠁⠸⠘⠛⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⠀⠀⠈⠂⠾⠀⢀⠀⠀⠀⠀⠀⠸⢙⣿⡿⣦⡄⠈⠀⠠⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡗⠁⠈⠂⣰⣦⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣦⡴⣶⣾⣿⣿⠀⠀⠀⠀⢠⡜⢻⣧⣜⣿⣧⠶⣶⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣠⡟⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣹⣿⣿⣟⣿⢻⡿⠛⢠⣄⠀⠀⠀⣿⠇⣼⣿⣿⣿⡿⣧⡟⠀⠀⠀⠀⠀⠀⠈⢻⣿⢻⠀⠀⢰⣏⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠩⢹⡏⠀⠀⡼⠁⠀⠀⢻⣦⠀⠀⣿⠀⣿⣿⡿⠏⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠈⠀⠀⢸⠋⡈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⡾⠀⠀⡰⠁⠀⢀⣴⠛⠁⠀⢰⡟⠀⣿⣿⣿⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⣾⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⠜⠀⠤⣴⣤⣤⡴⢛⠁⠀⠀⠀⢘⠃⠀⣿⣿⣿⣀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠃⠀⠉⠉⣿⠉⠁⠀⠀⠀⠀⠀⣈⠉⠀⣰⣿⣿⣿⣿⣷⠀⠀⢸⠇⠀⠀⠀⠀⠀⠠⠀⡇⠀⠀⠇⠀⢦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⡹⠃⠀⣿⣿⣿⣿⣿⣽⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⣴⠾⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⠧⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⢤⡀⠀⠀⠀⠀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠠⡏⠀⠋⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡹⠛⢻⠁⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡄⠀⡇⠀⠀⠀⠀⠀⠀⠛⣸⡇⠀⢸⣀⡁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⡇⢰⠃⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠈⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡖⠀⠤⠄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠈⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⣾⣋⠇⠀⢰⠏⡵⠀⠀⢠⣶⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠘⠷⣤⣄⡀⠀⢀⣠⣴⠏⣴⡿⠀⢠⢿⣇⠀⠀⡸⣿⣀⣠⠖⣿⡏⠀⡴⢻⡇⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⡋⠀⠻⠞⠹⣷⣠⠎⠀⢻⣶⠋⢀⡼⢿⠁⠀⠙⠷⠞⠁⣾⣇⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⣿⡇⠀⠀⠀⠀⠈⠁⠀⡴⠁⣻⠄⠈⠳⠛⠀⠀⢀⣤⠤⢶⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠀⣰⠋⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠴⠞⠁⢰⠀⠈⡏⠠⣠⢆⠋⠁⠀⡷⠴⡇⡇⢠⣶⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠰⣇⠐⡓⠀⠀⡇⠀⡇⠉⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠉⠁⠀⠁⠀⠁⠀⠉⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⢠⣶⣶⣶⡄⠀⣶⡄⠀⣶⡆⢰⡶⢤⣤⠴⡶⠀⣛⡃⠀⣰⢶⣶⡶
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⢸⣿⣿⣿⡇⠀⣿⣿⡄⣿⡇⠀⠀⢸⣿⠀⠀⠀⣿⡧⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⡄⠘⣿⣿⣿⠂⠀⣿⠈⢿⣿⡇⠀⠀⢸⣿⠀⠀⠀⣿⡗⠀⢹⣿⠟⠃
⠀⠀⠀⠀⠀⠘⠿⠃⠀⠻⠃⠀⠛⠿⠟⠀⠀⠻⠀⠈⠛⠃⠀⠀⠘⠟⠀⠀⠀⠻⠃⠀⠸⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢒⢒⠢⡀⠀⢀⢀⠀⠀⠀⡀⡀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢛⣙⡩⠋⠉⠙⠙⠉⠙⠉⠛⠋⠉⠹⠿⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⣶⢻⣷⢻⠀⠀⠀⡼⢷⡀⠀⢀⠎⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡋⠽⣹⠀⠀⡏⡶⡙⡟⡟⡟⣿⢃⢹⠇⠀⡞⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠟⣟⣶⢚⡛⡟⣛⢻⢟⣝⡟⢻⣽⠆⢰⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⣾⣤⣿⡏⣼⢴⢇⣯⣾⣬⠿⣥⣾⡗⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⢛⣲⡮⠭⠝⣓⣚⣞⣻⣽⣿⣾⠟⠋⠀⠀⠘⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠭⠤⢶⠖⢷⣎⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠟⢋⣀⢀⡤⠴⠶⠶⠶⠶⢤⣤⣀⠀⠘⢄⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠀⡌⠀⠀⠀⠀⠀⠀⠀⣼⠋⠙⠳⢦⣵⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⢠⡞⣿⣀⠀⠀⠀⠈⠛⢦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡎⠀⠟⢎⣩⣠⣇⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⣆⠀⡂⠀⡄⡑⢸⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠧⢡⠀⢾⢤⠼⣟⢑⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢏⢸⡀⡇⠘⠲⠻⣍⣱⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢨⣬⡧⡜⠀⠈⢏⣹⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠰⠴⡧⢜⠀⠀⠀⢭⣷⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢈⡶⡷⢎⠀⠀⠀⢘⡾⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡗⣟⣣⠀⠀⠀⠀⢟⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣿⡷⠀⠀⠀⠀⢪⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢟⠀⠀⠀⠀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣠⡾⠛⠉⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⡾⠋⠀⠀⢀⣀⣀⣀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣴⡟⠁⠀⣤⣾⣿⣿⣿⣿⡷⠀⠀⠉⠛⠳⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⠏⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⡛⠿⢶⡿⠛⠛⠛⠛⠻⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⢷⣀⠀⢀⣀⣄⠀⠀⢀⣠⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠈⠙⠛⠛⠋⠁⢀⣴⠟⠁⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠈⠋⢠⡏⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠻⣦⣀⠀⢀⣀⡀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢀⣠⣤⣤⣾⠁
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣴⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⢛⣉⠀⠈⠹⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣷⣄⠀⠀⠀⠀⠀⠀⣴⡟⠉⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠘⣯⡉⠙⠛⠛⠷⠶⠶⠿⠿⠟⠛⠛⣉⣽⣿⠈⠁⠀⠀⢀⡀⠀⢸⣧⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢿⠙⠳⠶⣤⣤⣤⣤⣤⣴⣶⣶⠟⠋⠉⣿⠀⠀⠀⢀⣿⡇⠐⠛⠁⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣰⠏⠀⢠⡟⠁⠀⠀⣸⡟⠀⠀⠀⣾⠿⣷⣄⣀⣠⣴⠟⠃
⠀⠁⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⢻⡄⠀⠀⢰⡏⠀⠀⠈⠀⣀⣤⡾⠟⠀⠀⢀⣾⠟⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠙⠶⢦⣼⣷⡶⠶⠾⠛⠋⠉⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣄⣀⡀⠀⠀⠀⣀⣀⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠠⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⢠⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⡀⡜⠀⠐⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠉⣐⡉⠈⠀⠄⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠂⠀⠀⠀⠀⠀⠂⠀⡉⠃⠀⠐⢀⠀⠴⠁⠀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣯⣤⣬⣉⣁⣂⡐⠀⠀⠤⠿⠽⠛⠂⠴⢂⣄⣀⣀⣐⣷⣃⠟⠀⢀⣀⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠉⠙⠋⠙⠉⠉⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⡏⢹⡅⢰⡊⠁⣀⣠⣄⡉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⣒⣚⣻⣿⣽⡂⠨⡂⣼⡠⠊⠁⠀⠀⠹⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣚⣋⡭⠭⣖⠀⠆⣼⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠈⠥⠀⢒⣒⣛⡭⠅⢘⠁⡇⠀⠀⠀⠀⢀⠜⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠍⠩⠭⠵⣖⣒⠀⠐⠀⡇⢀⠠⠤⠒⠉⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⣨⡭⠭⠽⣒⠈⠥⢠⡗⢀⣠⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡀⣒⣺⠭⠭⠗⢀⠅⡼⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢂⣴⣾⣿⡟⠡⣡⠞
⠀⠀⠀⢀⣀⡠⠤⠤⣶⠒⠒⠒⢒⣈⣹⣦⣅⡀⠀⠀⠐⠄⠶⠀⢶⣾⣿⡟⣇⣡⣾⣿⣿⣿⣖⠒⠶⠶⠦⢤⣄⣀⡀
⠀⠀⣞⢉⢀⡀⠀⠠⡂⠀⠀⠀⠀⢄⠀⠉⠫⢉⣀⣀⠀⠀⠘⡛⢫⣭⣭⣍⣭⠽⠋⣀⠄⠈⡉⠉⠉⢓⡦⠀⣀⠙⡯⣷
⠀⠀⠈⠓⠿⣟⣒⠶⠤⠤⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣅⣉⣀⣀⣴⣶⣶⣦⣤⡤⠬⠴⠖⣚⣫⠽⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠦⠤⣤⣤⣤⣤⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣭⣤⣤⣤⠤⠴⠶⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡟⠻⠛⡛⡛⠛⠛⠛⢻⡟⠛⡟⡟⠟⡛⣿⠻⣿⠛⠛⠛⠛⢻⡟⡟⡟⠛⡛⣛⢿⠛⡟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠙⢿
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠚⠛⠓⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠂⠀⠀⠀⠀⠐⠂⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠍⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⡇
⠀⠀⡇⠀⠀⠀⠈⢂⠀⠀⠀⡠⠁⠀⠂⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⢀⠈⠁⡃⢠⠎⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠑⠄⡀⠀⠂⡴⣶⠐⡀⠀⠀⠔⡭⣅⠢⠀⡤⠁⠀⠠⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠛⠛⠄⠁⠀⠀⠢⡳⢛⡴⠁⠁⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡜⠀⠰⢓⠆⠀⢰⠦⠀⠀⠀⡾⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠑⡀⠂⠋⠀⠀⠋⠀⠀⣀⡞⠁⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡌⡀⠈⠉⣁⡀⠤⠚⠁⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠁⡄⢂⠀⠀⠀⡀⡀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠈⠁⠀⠈⠁⠉⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠰⡄⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠂
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠂
⠈⠀⠀⠠⠀⠀⠄⠨⠀⠀⠘⠀⠈⠀⢀⡀⠀⠘⠘⠨⠀⠃⠀⠆⠁⠈⠀⠀⠇⠈⠀⠃⠡⠠⠀⢀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢘⠀⠀⠀⠀⠀⠣⠈
⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠁⠒⠙⠅⠀⠀⠀⠀⡘⠀⠀⡀⠀⠁⠜⠀⠀⠀⠀⠀⠀⢔⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠠
⠀⠀⠠⠀⠀⠀⠀⢘⠆⠀⠀⠀⠀⢥⠀⠀⠁⠤⠘⠁⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈
⠀⠀⢀⡀⢐⣀⠔⠃⡀⠀⠠⠀⠄⠀⠓⢤⣀⣂⣀⣀⠀⠀⠀⠠⡠⠀⠀⠄
⠀⠀⠉⠫⠉⠉⡁⡀⠘⢀⠀⠀⠠⠐⠀⠐⡈⠉⠉⠁⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠄⠄⠀⠤⠤⠀⠀⠀⠄⠀⠈⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠐⠆⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣖⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠫⣍⡙⠛⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠐⠀⠒⠀⠃⠘⠂⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣬⣉⡉⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠭⠭⠭⠊⠉⠻⠀⠀⠀⠀⣀⣀⣴⠖⠻⣟⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢀⠐⠫⠴⠦⠼⠛⠀⠉⠻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡶⣤⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣷⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠇⠀⠸⠋⠽⣿⠿⣿⡇⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠟⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣭⢭⣽⣿⣏⠀⠀⡆⠀⠀⠀⢸⠀⣻⣿⣿⣿⣯⡈⡙⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣶⡿⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠟⠋⣭⣿⡿⠿⣿⡆⠀⠀⠀⠀⡄⠲⣷⢹⣿⠉⡿⣿⣿⣎⢣⡀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠷⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠉⠒⠊⠁⠈⠣⡀⣜⠀⠀⡿⠸⣿⡆⠘⢿⣿⣿⡄⣱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡇⠀⡇⢠⣿⡇⠀⠀⠈⢻⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢹⣿⣧⣬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣿⣿⣧⣸⡇⠈⠛⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡏⡉⣿⣯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣷⡬⣿⣿⣿⡗⣹⣤⣀⠀⠀⠀⢀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⣿⣿⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣫⡿⢟⠋⠩⡄⠁⠀⢸⣿⣿⡇⠸⣿⣧⡀⣰⢞⣯⣥⣴⣞⣛⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠻⣾⡿⠂
⠀⠀⠀⠀⠀⠀⠀⣰⣟⣥⠞⢁⠴⠣⢆⠌⠀⠀⠀⣾⣿⡿⣷⠀⠽⣿⣾⠟⡛⠛⠭⠭⣉⢉⡉⠛⠓⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⣟⣹⣿⣏⠅
⠀⠀⠀⠀⠀⠀⢀⣿⢞⡥⠖⣫⢔⠕⠁⠀⠀⠀⠀⣿⡿⠀⢸⡄⠀⣽⣤⠤⠬⣍⣙⣒⣶⣤⣤⣤⡴⠆⠂⠙⠓⠀⠀⠀⠀⠀⠀⠚⡏⢹⣿⣌
⠀⠀⠀⠀⠀⠀⠼⠃⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⣰⣮⡅⠀⠀⣧⠀⣿⣿⠇⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡽⠀⠀⢰⣿⠁⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢻⣼⣿⢿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠘⡏⠠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠗⣾⣿⠶⣾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⡇⠰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢷⣶⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠺⡛⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣤⣀⣶⠦⣿⡆⠀⠀⠀⠀⠀⠀⢰⣻⣿⠫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣼⡿⢫⣉⡿⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣛⣛⣯⣝⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣯⣽⣭⣽⣿⣯⣭⣭⣭⣯⣭⣭⣿⣭⣭⣽

⢫⣵⢤⣤⣤⣤⣤⠀⣠⣬⡙⠿⢀⠙⠿⣷⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠⠀⠀⣀⠐⠟⠁⠀⢠⠊⠁⠀⠀⠡⣀⣀⣸⡿⠀⢠⣤⠈⠉⠀⠀⠀⠈⠨⢀⢁
⠀⡝⢟⣿⢵⣿⣿⣿⣿⣷⡶⠀⠀⢰⣄⠘⠃⠍⠑⡀⠈⠇⠀⠀⠀⠀⢤⠀⠁⢀⠇⠀⡩⠤⠄⠄⠀⢀⠠⠀⠸⢦⠀⠈⣉⠉⠀⢠⣶⠿
⢦⣀⣀⠈⠙⠇⢻⣷⠈⣿⣿⡦⠀⠀⢑⡙⠄⠸⡀⠀⠀⠀⠀⢀⠀⠀⡈⠃⢠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⡐⢒⠉⠀⣀
⠻⠿⠟⠁⠀⣀⠀⠈⠀⠙⢋⠀⠶⠀⠈⢴⣤⠄⣀⣈⠀⠐⣤⡀⠀⡼⢇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢠⠘⢇⣴⡀
⢀⡆⠰⡄⠺⠿⢣⠀⠀⠀⠘⠁⠀⠀⢴⡄⠙⣀⣴⠋⠀⠀⠹⣤⣶⣤⣮⠤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠑⠊⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣡⠆⠚⠛⣦⢾⣿⣿⡛⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⢯⠿⣦⣀⡠⠤⠄⠀⠀⠀⠀⠀⠀⣠⣾
⣶⣲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠞⠫⠰⠀⠚⠋⠈⣿⡿⠁⠲⠷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣽⡿⢷⣿⢋⣠⡤⡠⠀⢤⡶⠄⣀⣴⣿⢋
⡧⣷⣞⡷⠀⠀⠀⠀⠀⠠⠀⣸⣿⡆⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠀⠂⠀⠁⠀⠀⡀⡈⠙⠋⠁⠈⠛⠀⠀⠘⠉⠀⠈⠉⠉⠀⢸
⠛⠚⠋⢀⣀⣠⣤⠶⠆⠁⠀⢈⢉⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣶⣾⡇⠀⣤⡀
⠚⠛⠛⠛⠉⠁⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠑⠂⠀⠀⠀⠈⠉⠀⢠⣽⠗
⠀⠰⠂⣀⡴⠟⠉⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠰⠃⠀⠀⠀⠉⠢⠀⠀⠀⠈⠁
⠀⠒⠛⠉⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠈⢻⠓⢄⡛⠁⠐⠒
⠰⠵⠀⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⢀⡐⣷⠄⠒⠲⠈⡆⠀⠉⢀
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣦⠉⠀⠀⠀⠀⠀⠀⠔⠂⢀⣈⣿⣿⣤⣿⣿⢷⣟⢷⢛⠁⠀⢠⠃⡀
⠀⠀⠒⠰⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠚⠀⡀⠀⣄⠀⣠⣽⣿⢋⣽⡿⢾⠏⠋⠁⠘⣠⢀⠠⠂⠀⠀⠁⠀⠰⠆⠐
⠀⠂⠀⠀⠘⠒⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⠀⠀⠀⡈⠆⠙⠻⠟⡘⠋⠁⠀⠀⠐⢾⣶⡷
⠒⠀⠀⠀⠀⠀⠰⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠠⠖⠂⠀⠀⠷⠀⡰⣧⠀⠀⢀⡀⠀⠀⠉⠀⠀⠠
⠀⠀⣉⣀⡀⢀⣤⡄⠈⠁⠀⠀⠀⠲⠶⠤⠰⢂⠀⠀⠠⡤⠀⠀⠠⣤⠆⠀⠀⠀⣤⣀⠀⠀⠀⠐⢶⠶⢤⣇⣤⣔⣨⠟⠀⢆⡀⢧
⣿⣿⢿⣿⣷⣿⣽⣶⡖⠀⠧⡀⠇⢀⡈⠑⠚⠁⠀⠚⠀⠀⠐⠈⠢⡀⠀⠀⠀⠢⠛⠛⠀⡯⢸⡆⢀⠔⠚⣿⣿⣿⣁⣤⣄⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⡠
⠚⠙⠽⣟⢿⣷⡌⣿⣿⡗⠀⠀⠀⠈⠘⠐⣶⣄⣠⠞⠀⠀⡄⠈⢢⢑⡄⠀⢠⡄⠐⠂⠀⢤⡍⠐⠛⠓⠈⣿⣟⠏⢿⡌⠁⠐⡦⠀⠀⠀⠐⠦⢤⣤⡴⠶⠚⣁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣵⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣟⡆⠐⠀⠀⠀⠀⠀⠀⠀⡠⢚⣩⣭⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢽⣿⣷⣤⡀
⠀⠀⠀⠉⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢈⡴⠋⡵⣿⢹⣿⣿⣿⡄⠀⠀⠀⠀⣀⡄⠋⠁⠯⠞⣭⠛⠻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⡤⠋⢠⠎⣴⢃⣾⣟⣿⣿⠃⠀⠀⢠⣾⣯⠀⠀⠀⠀⠞⢵⣾⢗⢙⣿⣿⣿
⠀⠀⠀⠀⠀⡈⢻⣿⣿⣿⣿⣿⣿⡇⠂⠀⠀⠀⠀⣸⢁⡴⠁⠰⠁⠊⢹⣿⣿⣿⠀⠀⠀⠀⠙⠳⢦⣴⣄⣤⣠⣼⣿⣶⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠩⠿⣿⠿⠋⠀⠀⠀⠀⠀⠀⣿⠟⣀⠴⠃⠀⠀⠈⣻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠸⠏

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣰⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠹⣶⣶⣤⣤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡎⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣓⣒⣰⣿⠁⠀⠀⠀⠀⠀⣼⣿⣏⡿⢿⡿⠟⠁⣴⣾⣿⣢⠀⠀⠀⠀⠀⣴⠉⠙⠛⠛⠛⠓⠐⠋⠁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⣹⡿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁⠀⠀⡀⠀⠻⣿⣿⠟⠀⠀⠀⠀⠀⢹⣿⡿⠷⣶⣶⣶⠒⠀⠀⡹
⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠋⠑⠈⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣤⣴⠳⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢸⣟⡷⠾⠿⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠋⠀⠀⠀⠀⠀⠀⢰⡍⠋⠉⠉⠀⠀⠒⣧
⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣥⠷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠛⠣⢿⣿⢲
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡎⠌⢡⠀⢒⠲⠟⣫⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠃⠑⠂⠸⠇⠅⣤⣹⣯⡿⢀⣀⡠⣄⣤⣀⡀⠀⠀⢠⡾⣿⡾⢻⣦⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⣯⣍⣒⣤⡤⠀⠛⣿⣽⢛⠛⠀⠀⠀⠙⠻⢿⣷⣠⠾⠟⠛⠙⠯⢄⠙⡿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⠠⡇⠀⢧⣈⡩⠋⠀⢀⠞⠁⠻⢬⠔⠀⢀⣘⢳⡄⠀⡼⣵⡘⠀⡇⢀⡔⠄⠘⡿⣻⢛⣿⣇
⠀⠀⠀⠀⠀⠀⢧⣀⣱⣀⠀⠀⠀⠀⣠⠃⢸⣒⠢⡈⠀⠀⠻⡀⠀⠁⣿⠁⡈⠱⠈⠀⠘⠓⠁⠀⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠑⣦⡀⢠⠃⢠⠟⢻⣷⣌⣢⣰⣄⠀⠀⣤⡿⠘⣷⣒⣒⣒⣢⠗⠒⠂⠈⢹⠋⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⢸⠀⠘⠤⠤⠔⠋⠉⠀⠀⠀⠀⡬⢷⠀⢸⠿⠯⡽⠃⠀⠀⠀⢀⣾⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠎⢆⠀⠀⠀⠀⠀⠀⠀⠀⣇⣈⡠⢾⣆⠀⠑⠋⠀⠀⠀⠀⣠⢻⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⠣⡀⠀⠀⠀⢀⣠⠞⠉⠁⠀⠰⠋⠑⢦⣀⠀⠀⠀⣰⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢶⣴⠊⢉⡴⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠿⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⣀⣀⡠⠖⠉⠀⠀⠀⠀⠑⠎⠁⠀⠀⠀⠈⠓⢤⣀⣀
⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⢸⠒⠤⢀⣀⠤⠔⠊⠁⠀⠙⠢⠤⢀⣀⠤⠔⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡒⢤⠀⠀⢀⠔⠈⠉⠒⠈⠉⠒⢄⠀⠀⢠⠄⠠⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡎⠮⠜⣀⠔⠁⠀⠀⠀⣠⠀⠀⠀⠀⠑⢄⠐⠼⠌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠀⠉⠀⠀⠀⠀⢀⠔⠁⠳⡄⠀⠀⠀⠀⠉⠒⠊⡰
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⡀⠤⠒⠁⠀⠀⠀⠈⠑⠤⢀⣀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⢦⠔⠒⠢⣄
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⣀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡇⡇⠀⠀⡎⡔⠃⠀⡠⠃⠀⠀⠀⢠⠏⢆⠀⠀⠀⠀⢣⡀⠐⠓⡌⡆
⠀⠀⠀⠀⠀⡇⡇⠀⠀⢇⠉⠒⠊⠀⠀⠀⠀⡰⠃⠀⠈⠣⡀⠀⠀⠀⠈⠒⠊⢁⠃
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠑⠠⠤⠠⠤⠔⠋⠀⠀⠀⠀⠀⠈⠓⠤⠄⠠⠤⠔⠁⠀⠀⣀⠀⡀⡀⢀⡀
⠀⠀⠀⠀⠀⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠⠂⠀⠀⠢⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠈⠄⢸
⠀⠀⠀⠀⠀⣄⣀⣀⡠⠔⠒⠁⠀⠀⠀⠀⠈⠑⠢⠄⣀⣀⣀⠄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠘⠦⣀⡀⠀⢀⣀⠤⠚⠑⠦⣄⣀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠋⠉⠉⠉⠉⠑⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠋⠉⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⢠⡶⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⡄
⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡎⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⣸
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢸⠐⠀⠀⠀⠀⢣⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠩⣅⠂⠀⠀⠀⠀⡀⠀⠀⠀⠘⠒⠒⡆
⠀⠀⠀⠀⢸⠈⠄⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⣾⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢴⡁
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡀⢣⡀⢸⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠛⠖⢇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⡆
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠸⠆⠀⠐⠀⠰⠆⠐⠁
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠉⠑⠠⡀⠀⡄⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢸⠀⠀⠀⢈⣠⡐⠀⠀⠀⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⡌⠀⠠⠖⠉⠀⠙⠆⣠⣖⣒⡰⠃
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⢄⡔⠃⠒⠒⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢄⠀⠀⡇⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠑⣲⣄⣑⢄⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠉⠉⠑⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠐⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⡀⠀⠀⣦⠀⢦⡀⢺⣦⡀⠀⠀⠀⠀⠀⠀⠀⠲⢦⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠓⠊⠉⠉⠁⠈⠑⠂⠀⠀⣤⡰⠬⠽⠏⣴⣴⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠉⠻⠟⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡴⠊⠀⠉⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠓⢀⡀⠀⠀⠀⠀⢀⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⠀⡀⠀⠀⠀⠀⠀⡀⠈⠂⠀⠀⠀⠄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡾⠁⠀⠀⠀⠐⠚⠹⠆⠀⠀⠀⣠⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠙⢃⠀⠀⠀⠀⣶⣄⠒⠂⢠⡀⠁⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢀⠟⡀⠀⠀⠀⠹⡙⢿⡆⠒⠚⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡉⠀⠘⢤⡀⠀⠀⠑⠰⠧⠄⠂⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠊⠀⠀⠠⡁⠀⠀⠀⠽⠶⠦⢤⣤⣤⣤⡐⠋
⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠢⢀⣀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠦⠴⠤⠤⠤⠤⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢸⠁
⠀⠀⠀⢀⠸⣄⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠂⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⡠⠊⠀⢀⠗⠐⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠰⠁⣀⡔⠁⠀⠀⠀⠃⠀⠀⠀⠀⢀⣤⠄⠀⣀⠀⠀⠀⢠
⡌⠀⠀⠈⠀⠀⠀⠙⠛⠋⠁⠀⠀⠜⠁⠉⠻⣿⣿⠆⠀⠀⠈
⠓⠀⠀⠒⠒⠂⠀⠀⡞⠉⠉⠁⠀⠈⠁⠀⠚⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⡆⢰⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⠒⠶⠒⠀⠘⠲⠒⠒⠒⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠘⠀⠀⠀⠀⠀⡆⠘⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠋⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣸⡆⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠑⢄⡀⠀⠀⠀⠀⢠⣌⠙⢩⣶⣶⠆⠀⢴⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⠎⠙⢂⠀⠀⠀⠀⠉⠛⠛⠿⠻⢤⣤⣌⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⣀⡚
⣿⣿⣿⣿⠿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣠⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠀⡒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠆⠵⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢦⡽⠉⠑⠰⠤⡀⠀⠔⠃⣀⡤⠀⠀⠀⡘⢀⠞⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⢠⠀⠠⠃⠉⢠⡶⢀⠀⠀⠠⢲⠌⣠⡁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠛⠒⠏⠠⡘⢀⠀⠀⣀⢶⣡⠧⠄⠎⡡⢡⢰⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠂⠈⣴⣴⣶⣿⣇⡸⣷⠀⠠⢀⣀⠀⠀⠂⡞⣨⠞⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠔⠂⠀⠉⠁⡜⠿⠴⢟⠇⠀⠃⠀⠀⢀⠈⠉⠃⢌⢁
⠀⠀⠀⠀⠀⠀⢠⠀⠇⡴⠗⠴⠶⠀⢀⠀⠁⠀⠈⠁⠀⢀⠖⠀⠀⠈⣗⠀⠖⠀⡈⠑⠠⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠈⠘⢠⠤⠤⠄⢠⡰⠁⠀⠐⡦⢰⡀⠄⠋⠀⠀⠀⢠⡷⠠⠄⠀⢁⠀⣼⠐⣧⠗⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠘⠌⢤⡀⢀⠄⢪⠌⢔⠆⠀⠘⠃⡏⠀⠀⢀⡀⠶⠀⠈⠐⣧⠔⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⡴⢀⠀⣩⠀⠀⠨⣿⢍⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⡒⠋⠑⠒⠪⠷⠈⠀⠀⢀⠃⠀⠀⠀⠀⠀⠦⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠠⠀⠀⠘⢦⠀⠀⠀⢀⡜⠈⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢈⡀⢢⠀⠀⡅⠀⠀⠀⢀⠾⢷⡖⠀⠀⠘⡆⠀⢀⠰⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠰⠂⠀⠀⡰⠃⠀⠀⠘⡇⠀⠀⡀⠀⠈⠁⠀⠀⠀⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⡀⠄⠀⡀⢀⠃⠀⠀⠀⠀⢸⠔⠋⠉⠑⠢⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠚⠝⠂⠉⣱⡄⠀⠀⠀⠸⠤⠄⢀⣄⡀⠀⠠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡈⠀⢀⡴⠊⠉⠁⡈⠀⠀⠀⡀⢀⠀⠀⠀⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠔⠉⠁⠀⠀⠀⠫⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠀⡀⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢀⡄⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣴⠏⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢿⣆⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠈⢿⣦⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⢻⣇⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⠏⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠋⠀⠀⠤⠤⠫⠤⠀⠀⠀⢉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢂⣉⣩⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣌⣉⣁⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⡷⠤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⢸⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡠⠊⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⠐⠉⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠡⠤⠤⣔⣂⠁
⠀⠀⠀⠀⡤⠒⠊⠉⣁⣤⣴⣶⣶⣾⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣶⣶⣶⣦⣤⣈⠉⠑⠒⢤
⠀⠀⠀⠀⠑⠢⠤⣀⣉⠛⠻⠿⠿⢿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⣉⣀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⢖⡐⠒⠀⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠄⠀⠒⢂⡲⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠂⠀⠀⠀⠀⠒⠒⠒⠒⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠛⠉⠻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⣠⣴⣶⣶⣶⣄⣈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶⣾⣿⣷⣼⣿⣿⣿⣿⣿⣿⠿⠿⠛⠟⣯⣿⣽⠻⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠀⠀⠀⠀⠀⣿⣿⣿⡿⠉⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣼⣿⠿⠏
⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡴⠦⠟⠛⠛⠉⠀⠀⠀⠀⢠⣿⣷⣿⡿⠁
⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠘⢻⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⡵⠟⠀⠀⠀⠀⠀⣀⡤⠀⣌⡜⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠈⡆⠈⣄⣀⣀⣤⣴⣶⡿⠋⠀⠀⡟⠀⠹⣿⠿⡄
⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣄⣀⠀⠀⠘⢆⢀⠍⠁⠀⣿⠏⠀⠀⠀⡼⠁⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣦⣌⠉⠂⠤⠀⡿⠀⠀⠀⡜⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⣀⠴⢊⠟
⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣶⣤⣀⣟⠀⠀⠞⠀⠀⠀⡠⠐⠈⠀⠀⢙⣦⠊⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⡜⣿⣿⣿⣿⣿⣿⠿⢻⢿⠀⠀⠀⠀⡠⠊⠀⠀⢀⣤⣾⡿⠁⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⡀⠀⢸⢸⣿⣿⣿⠟⠁⠀⠈⡄⠀⠠⠐⠈⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡌⣿⣿⡏⠀⠀⠀⢀⣄⠀⠀⢀⣠⣤⣤⣄⣿⣿⣿⡿⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⡇⠀⠀⠀⣈⣁⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣇⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⣰⠆⠀⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⢆⣎⣀⠀⠀⠀⠀⠀⠀⠠⠠⠁
⠀⠀⠀⠀⢀⠴⠒⠃⠤⣀⣤⣷⢸⣁⡠⠽⠦⢄⠀⠀⠀⠀⠀⠀⠠⢾⣿⠀⠸⣿⠗⣠⣤⡀
⠀⠀⠀⠀⡜⠀⠀⠀⠀⡸⠟⠛⠛⠿⠄⠀⠀⠀⠁⠀⠀⠀⢀⣴⡶⠌⠙⠀⠀⠈⠀⢿⠿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⣯⡀⢡⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⡧⠀⠈⠀⣤⣤⡀⠀⢰⣿
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠘⢿⠄⠠⡄⠙⠋⠁⠀⣹⡏
⠀⠀⠀⠀⠀⠈⠀⠀⢀⣆⠀⠀⠀⠀⣠⠤⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⣀⣄⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠐⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣯⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢸⣁⣘⠒⠂⠀⠀⠀⣠⡄⠀⡴⣄⠀⢠⡄⠀⡀⠀⠐⠒⣃⣘⡇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⣧⠤⠅⠀⠀⠀⠰⣧⡀⠿⠽⠀⠛⠓⠐⠛⠧⢀⡟⠃⢀⠀⠀⠨⠤⣼⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠸⠤⠒⠀⣠⡄⠙⠦⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣠⡆⡀⠐⠒⠤⠇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠻⢖⠉⠀⣀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣁⣀⠀⠉⡲⠟⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠳⠖⠉⠀⠘⢧⠼⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⣤⣤⣤⣤⣤⣤⠀⠀⠀⢳⠎⠃⠀⠉⠲⠞⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠳⠴⡋⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠛⢿⣿⡀⠀⣿⡿⠿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢙⠦⠞⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠞⡁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⣀⣾⡿⠀⠀⣿⣧⣤⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠿⢿⣿⣆⠀⠻⠟⠋⠉⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⠀⢀⡀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣈⢡⠇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣴⣾⣿⠏⠀⠿⣿⣶⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⡌⣁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⣸⠀⠀⠐⣶⠖⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠰⠳⠄⠀⠈⣇⠉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠋⢸⢀⠀⠁⡠⢶⣀⠀⠀⠀⠀⠃⠸⠅⠾⠄⠿⠀⠧⠀⠀⠀⠀⠀⢞⡧⠄⠀⡀⡇⠙⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠋⢸⠀⡀⠀⠛⢁⣴⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢶⣦⠀⠁⢀⠀⡇⠙⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣀⡇⢠⠀⠈⠀⠋⢀⠇⢰⣀⠆⢠⡆⠀⣖⣦⠈⢳⠆⠈⠀⠀⡄⢸⣀⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣁⣬⠆⢣⡠⡄⣀⠀⠀⠉⠀⠘⠓⠀⠙⠉⠀⡀⢠⢀⡜⠸⣥⣈⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠚⢁⣸⠎⠳⣄⠀⠀⠀⠀⠀⠀⣠⠞⠱⣇⡈⠓⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⠴⠚⠋⠙⣒⡦⢤⡚⠋⠉⠓⠤⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⣏⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⣤⣄⠀⠀⢾⣿⣿⣿⣿⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⣵⣿⣿⠀⠀⠈⠛⠿⣿⣵⣶⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⣄⡀⠀⠀⠀⠀⠘⠿⠿⡿⠇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⣶⡾⢿⣷⣶⣦⡀⠀⠀⠙⠧⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣽⣷⣿⣶⣿⣿⡿⠿⠿⠿⠷⠀⡄⠀⠀⠀⠉⠙⠒⢦⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣆⢿⡟⣠⣄⠀⠀⠀⠀⠀⢹⣷⣄⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠅⣰⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⢋⣾⣿⣿⣦⣾⣐⣛⠳⡀⢀⣠⠀⠀⠀⢹⡏⣻⣾⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣮⣿⣯⣟⣽⡿⣿⡿⢟⣫⡭⡁⢺⣿⡟⣏⢹⡏⠉⢹⣄⣷⣿⣿⡷⢦⣶⣏⡀⣿⣿⣝⣻⠿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣮⠆⢚⣀⣿⣯⡿⣫⣵⣾⣿⡟⣰⡇⡏⠁⢲⠋⠀⢱⡶⠘⢯⣧⣿⣿⠀⠸⠛⠉⠻⣿⣿⣿⣿⣷⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⣯⣿⣹⣿⣿⣿⣿⡿⢱⣿⡧⠁⢉⣸⣷⣤⣾⣅⣀⣼⣿⣗⡿⢦⣦⣤⣤⡂⢯⡣⠠⠙⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣿⣿⣿⣿⠁⢰⣿⣿⠀⠀⠙⠉⠙⣿⡿⠿⣏⠨⣿⣧⣸⠋⠁⢠⣽⡼⣿⣦⡈⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⢿⣿⣿⣿⡏⠀⣿⣿⣿⠀⠀⢀⣀⣴⣟⠀⠀⠙⠸⣿⡿⢻⣦⡀⠛⠻⣧⠘⣿⣿⣄⠀⣔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⢱⣷⣿⣿⣿⣿⠁⠨⣿⣿⣿⠀⣼⣄⢨⡿⠛⠀⡀⠀⠘⠧⢐⣾⣿⠅⠀⠀⣼⣇⠸⣿⣿⣆⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⣿⣿⣿⣿⣿⡇⠀⣯⣿⣿⡿⠀⣳⡼⢿⣄⡀⢸⡟⠀⠀⠀⠘⣿⣧⣼⠃⠈⣿⣿⡀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣷⣿⣿⣿⠃⠀⣽⠿⢿⡇⠀⠁⠐⠈⠿⣧⡀⠁⠀⠀⠀⠀⢀⣽⡿⠀⠠⣿⣿⣇⠀⠿⡯⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⢻⣯⣿⣿⠀⠈⠉⠷⠀⠁⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠺⣿⡏⠀⠀⢿⣿⣿⡆⠈⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣿⣿⣿⡀⠀⢂⡴⠀⠀⠀⠀⠀⠀⠸⣿⠃⠐⠀⠀⠀⠾⠆⠌⠃⠀⠀⠠⣿⣿⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⣿⡄⠠⠈⠀⠀⠀⠀⠀⢴⣦⠀⠉⠳⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠈⣹⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠧⠴⠀⠀⠀⠀⠀⠀⢠⡞⠛⠃⠀⠀⠀⠘⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣆⠀⠠⣶⡄⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡀⠀⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠊⠁⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⣀⣀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠑⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠔⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣸⣇⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⢻⣶⣄⣰⣿⣿⣆⣠⣾⡇⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⢰⡟⠀⠀⠀⢀⣴⠟⠙⡀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⣼⠃⠀⠀⠀⣼⡟⠀⢀⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⢰⠏⠀⠀⠀⢰⣿⠃⠀⡜⠀⠀⠀⠀⠀⠀⣰⣾⢳⠀⢠⣿⠏⡕⠀⣴⡄⠀⠀⢀⣄
⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢠⠞⠀⠀⠀⠀⠀⠀⠀⣿⣧⠋⠀⢸⣿⡞⠀⣸⣿⠀⢀⡴⠋⣿⣆⣀
⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⢠⣴⣶⣤⣶⠀⠀⠹⣿⡴⠞⠉⢻⣷⡟⠉⠻⠶⢛⣡⠶⣿⡏
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⡇⠀⢰⣿⡟⠉⣿⣿⠀⠀⣰⢿⣿⡀⣠⠟⣿⣧⠀⢀⡴⠛⠁⢸⣿⠁
⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⣿⡇⠀⣿⣿⢀⣴⣿⣿⠀⣸⠇⢘⣿⣇⡏⠀⣸⣿⢠⠟⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠻⣿⣦⣤⣤⣤⣶⡿⠟⠁⠀⠀⢿⣧⢀⡿⢿⠟⠁⢻⣿⣠⣿⡀⣸⡿⠙⣧⣠⣿⢻⡏⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⣈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠉⠁⠈⠛⠋⠁⠀⠈⠉⠀⢸⡇⠀⢀⣼⡿⠁⠀⣾⣿⣷⡄⢀⣀
⠀⠀⠀⠀⣀⣀⡛⠻⠶⠄⠀⢠⣾⠓⠀⠀⣠⢾⡆⠀⢠⣾⠋⠂⠀⣴⡟⠠⠀⣸⡇⠀⢸⡇⠀⠀⠙⠛⠛⠉⠀⠀⠀⠈⢿⣿⣷⣿⣿
⠀⠀⠀⠻⠿⠿⠿⠿⠖⠀⠀⠘⣿⡄⠀⢀⡃⢸⣷⠀⠈⢿⣆⠀⠀⢻⣧⠀⠀⣿⠃⠀⢸⡇⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠈⠟⠛⠛⠋
⠀⠀⣀⣠⣤⣤⡄⠀⠀⠀⠀⡀⠈⢿⡄⢸⠁⠈⣿⠀⠀⠈⢻⡆⠀⠀⠹⣧⠀⢻⣄⡀⢿⡇⠀⠘⠻⣿⠟⠃⠀⢠⣤⣤⣀
⠘⣿⣿⣿⠿⣿⣿⣶⣶⣄⠀⠣⣠⠾⠃⠀⠀⠀⣿⠀⠡⢄⡼⠃⠀⢄⡤⠟⠀⠀⠠⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡆
⠀⢹⣿⣿⣦⣀⠉⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⡤⠀⠀⠀⠈⣿⣿⣿⣶⣾⣿⣦
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣶⡆⣤⣴⠀⡴⣾⠃⠀⠀⢠⣶⡆⠀⠀⠀⣟⢤⠀⡤⣸⠃⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠙⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠚⠀⠁⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⢄⠀⠀⣴⣦⢀⣴⣶⡄⢀⣴⣿⣷⠀⠀⠀⢀⣴⣦⡀⠀⠀⢀⣶⡄⣠⣶⣦⠀⠀⣀⣠⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠈⠀⠀⣿⣿⠏⠈⣿⣿⡾⠁⢸⣿⠀⠀⢀⡞⠙⣿⡇⠀⠀⢸⣿⡟⠁⢹⣿⡇⠈⠉⢸⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⢸⣿⡟⠀⠀⣿⣿⠁⠀⢸⣿⠀⠀⣸⠁⠀⣿⣷⠀⠀⣿⣿⠁⠀⣸⣿⠁⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⢸⣿⠇⠀⠀⣿⡟⠀⠀⢸⣿⠀⠀⡟⠉⠀⢹⣿⠀⠀⣿⡇⠀⣰⡿⠃⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢹⣿⠀⠈⣿⠀⠀⠀⣿⠇⠀⠀⢸⡟⠀⢠⠇⠀⠀⢸⣿⠀⠀⣿⡟⢻⣏⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠾⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠈⠀⠀⠀⠸⣿⠀⠀⠈⠁⠀⠻⣷⣄⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⣠⣤⢄⠀⣠⡶⡦⠀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⢀⣀⣀⣀⡙⠛⠷⠆⠀⠀⢠⣾⠟⢻⣿⠃⠀⣿⡷⠃⣘⣿⡖⢁⣠⣿⡁⢀⡴⠋⣿⣤⡤⠀⠐⠛⢋⣉⣤⣤⣤⣀
⠀⠀⠀⠀⠛⠿⠿⠿⠛⠛⠂⠀⠀⠀⣿⣇⡴⢿⣿⢀⡼⢹⣿⡞⠉⣹⣿⡏⠀⠙⢛⡩⠖⢻⣿⠃⠀⠀⣀⣀⡉⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠟⠋⠀⠀⠈⠉⠀⠈⠙⠉⣰⠋⢹⣿⣼⠃⢸⣿⢀⡴⠋⠀⠀⣾⡟⠀⠀⠀⠈⠙⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣼⠟⢿⣤⡾⢏⡏⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⢧⣤⣴⠿⠋

⠀⠀⣿⣿⣿⣿⣿⣿⣟⣏⣋⣍⣏⣋⣋⣻⣻⣿⣛⣿⣏⣿⣋⣋⣉⣿⣟⣟⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⡟⠉⠉

⠀⠈⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠐⠠⢀⡀⡀⠀⢀⣀⡀⠤⠂⠁⠀⠀⠀⠁⠒⠀⠤⠤⠤⠀⢒⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⡹⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣎⠦⠤⡀⠀⠚⠑⡬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣄⠀⠀⠀⠀⢀⡀
⠀⠰⠀⠤⠤⠒⢒⠯⠭⠭⡀⠀⣠⠀⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡄⠀⢀⠁⠐⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠒⡖⠁⠀⡌⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈⠒⠁⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠇⢀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⠀⠀⣧⠀⠴⠀⡀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠐⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡐⠋⠣⣤⠴⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢎⠰⠀⠀⠀⠀⣀⣣⠜
⠀⢀⠀⠀⡀⡀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⡁⠉⠁⠑⠊⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣤⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⢀⣤⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠞⠀⠀⣠⡶⠶⣦⠀⠀⣾⣏⡽⠀⢠⣿⣁⡟⠀⣼⡿⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠹⢿⣶⣄⠀⢸⡿⠀⣼⣿⠀⠀⡿⣿⡄⣀⠼⢿⣿⣤⠴⣿⡇⢀⡴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⠈⠓⠋⢸⡏⢀⡜⢀⠼⣿⠁⢠⠞⢹⣿⠀⠙⠟⠋⢠⣿⠃⠀⡦
⠀⠀⠀⠀⠀⠀⠀⢯⣀⣠⠟⢀⣀⠀⠀⠉⠉⠀⠸⠴⠟⠀⠘⠷⠟⠋⠀⡤⠒⢛⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⣤⣤⡤⠤⠴⠶⠚⠛⢛⡷⠀⠀⠀⠷⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣴⢶⡄⠀⣶⠀⢠⣤⡀⠀⣴⠀⣠⣶⣦⠀⣴⡶⠆⠀⠀⣤⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠁⠀⣿⡇⢸⣿⣷⡀⣿⢠⣿⠃⠉⠀⣿⠀⠀⠀⠀⡇⠛⣫⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⣿⡇⢸⡇⢿⣧⡟⠸⣿⠀⠀⡀⣿⠛⠃⠀⠀⠳⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠋⠀⠿⠁⠸⠇⠈⠻⠃⠀⠹⠷⠞⠁⠻⠷⠆⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣤⢶⠀⢀⣤⣄⠀⠀⢠⣶⡄⣠⡀⠀⢀⡀⠀⢀⣶⠄⠀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠺⢀⣾⢃⡿⠀⠀⢀⡍⠀⣧⣻⢀⣈⡙⠛⢻⣿⠛⢻⣿⡏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢞⡁⠀⣠⣿⡏⠀⠀⡟⢩⣿⡇⠀⢸⣿⠀⢸⣿⣣⠏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣏⠁⣿⡇⢀⡼⠁⣾⡟⠀⢀⣾⣿⠀⢸⣿⡏⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⣠⡏⠀⠈⠛⠛⠁⠀⢿⣇⣠⠟⠈⠻⠷⠋⠙⠁⢺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞


⣏⣈⣯⣭⣯⣭⣿⣽⣽⣽⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢈⣙⠻⢦⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣶⣟⣉⣈⣭⣿⣿⣿⣿⣯⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠟⠉⢭⣽⣿⣿⣿⣿⣿⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃⠀⠀⠐⣺⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣯⠁⠀⠀⢤⣼⣿⣿⣿⣿⡭⠪⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⢒⠺⢿⣿⣿⡋⠐⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⡀⣸⣿⣿⠿⡉⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠈⣙⣿⡿⠓⠚⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⣀⣂⣈⡉⠉⠁⠀⠀⢴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣴⣿⡿⠇⣠⠄⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠉⢀⣤⡾⠃⠀⣠⣴⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣀⣤⣶⡿⠟⣡⣶⣿⣿⠏⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⢀⣾⡿⠛⢉⣐⣿⣿⣿⠿⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠚⠉⠀⠀⢀⣤⠞⠋⠀⣀⣼⣿⣿⣿⣣⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠖⠒⠋⠉⠀⠀⠀⢀⣠⣤⡤⠈⠉⠀⠠⢀⣨⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⣤⣾⡿⠋⠉⣀⣠⣤⣶⣶⣾⣿⣿⣿⣟⢛⣋⣀⣤⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠰⣿⣏⠠⠚⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁

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

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⢀⣤⢖⣻⢿⡿⠂⠀⠀⣀⣤⡀⠀⠀⠀⠀⣠⠿⠛⠉⠉⠁⠀⠀⣠⠔⢶⣄⠀⠀⠉⠉⠓⠦⠤⠀⠀⠀⠀⠀⠀⣤⠀⣀⡀⢀⣤⣶⠟
⠀⠁⠈⠉⡵⠋⠀⢀⣤⣾⣿⣿⣿⣦⡀⠀⠀⠁⠀⠀⠀⠀⠀⣠⣶⠭⣽⢿⣻⢳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣷⣴⡄⢁⡘⣿⠁
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⡮⢝⣮⣵⢂⠛⣽⢚⣻⢷⠀⠤⠀⠀⠀⢨⣿⣟⣿⣾⠛⣿⡇⣠⡖⣿⣄
⠀⠀⠀⠀⠒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠄⠀⠀⠀⠙⠚⠓⠴⠯⠹⠤⠛⠘⠉⠀⠀⠂⠀⠠⠾⡵⠋⠐⠉⡌⣿⡀⠙⣿⣿⣿⠂
⠀⠀⠀⢀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠀⠀⢹⠇⣷⣿⡇⢀⢹⣿⡧
⠀⠀⢀⣞⣇⢀⡴⠂⠀⣀⣀⠀⠀⠀⠀⠀⠀⣽⡽⢢⢆⡄⣀⣠⡇⠐⢩⢭⣆⢀⣀⣀⣀⣀⡀⠀⠀⣤⣀⡀⠘⠃⠀⣿⠀⠸⢹⣿⠇
⠀⠀⠀⢾⠟⠼⠁⠸⡟⠛⠉⠛⠲⡀⠀⠀⠰⠇⠚⠓⠛⠛⠓⠋⠂⠀⠋⠘⢳⠗⡟⠚⠛⠘⠃⠀⠀⠪⠽⠃⢀⣤⡄⠹⠁⠀⠘⠿⠆
⠀⠀⠀⠀⠀⠀⠀⣤⢸⡄⠀⠻⢧⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠈⠀⠁⠀⠀⠀⠀⠀⠀⢠⠢⡄⠈⠽⠟⠀⢠⡶⡄
⠀⠀⠐⠠⣾⢿⣾⣿⣾⡇⣄⠀⠈⡏⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣾⣯⣽⣷⠀⠙⠑⠋⠀⢀⣀⣀⣈⡁⠁⠀⠀⠺⡧
⠀⠀⠀⠀⢤⣾⣿⣿⣿⣷⣿⠀⠀⠀⠀⠸⢀⡀⠀⣤⡒⠿⠛⠋⠀⠀⠾⠁⢲⣆⡄⠀⢈⠀⠉⢁⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠆⠀⠀⢉⣿⠛⢻⡟⢿⠂⠀⠀⠀⢀⡸⠁⠀⢿⡇⠀⠀⠀⠀⠀⠀⠰⠿⢿⡇⠀⠿⠥⠿⠿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠒⠀⠀⢸⡿⠀⣿⡇⣿⠀⠀⠀⠀⢸⠁⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣘⣿⠇⠀⡀⠀⠄⠀⠀⠀⠙⣿⡿⣿⠛⠛⢻⣿⣿⣿⣿⡿⠃
⠀⠀⡄⠀⠀⢸⡇⠀⣿⠁⣿⠀⠀⠀⠀⠀⠄⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠰⣿⡿⠀⠀⣦⣤⣠⣤⡅⠀⠀⠀⠀⠀⠀⠀⠈⢠⠁⣀⡏
⠀⠀⠀⠀⠀⠘⠃⠀⣿⠀⠛⠀⠀⣧⣀⣸⣧⡄⠀⠀⠀⠉⠠⠀⠀⠤⠔⠻⡋⠀⠀⠀⣿⣀⣿⣿⡆⠀⠀⠙⠢⢄⡀⠀⢀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠟⡻⠓⠤⢼⠂⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⢰⠶⢶⡶⡆⣶⢶⣀⡶⠆⣶⣦
⠀⠀⠀⠀⠀⠀⢠⡮⠄⠠⠤⣀⠀⠀⢀⠀⢰⠁⠀⡐⠀⠔⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⠟⣸⠗⢸⣷⠇⣿⡾⢹⡷⠂⣷⣟
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠈⠀⢀⠞⡱⠃⠀⠄⣇⡄⠀⣠⠠⠶⠌⠑⣆⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠼⠶⠸⠇⠀⠿⠀⠸⠷⠆⠇⠹⠆
⠀⠀⠀⠀⢰⠃⠀⠀⠀⣠⠂⠀⠀⠈⠉⠓⠀⠀⠂⠷⠀⠀⠀⠹⡄⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡀⠀⠐⠒⠒⠒⢒⡒⠒⡂⠒⢒⠀⠒⢂⡀
⠀⠀⠀⠀⣸⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸⣡⡶⢦⠖⠀⠠⠾⠳⡜⠁⠠⠿⠶⠶⡟⡵⠴
⠀⠀⠀⠀⢹⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡄⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡿
⠀⠀⠀⠀⠈⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠘⣆⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⢦⣄
⠀⠀⠀⠀⠀⠀⠘⢧⡀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢰⠇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡰⠃⠀⠀⠀⠀⠐⢦⢂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠀⠀⠀⠀⣠⣔⡁⠴⠊⠀⠀⠀⠀⢢⠀⠀⢾⡀⠑⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠘⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡠⠤⢤⡀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⠇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠊⠀⢠⠊⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠀⡎⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢠⠇⠀⢸⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡘⠄⠀⡠⠶⠒⣊⠋⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣆⠀⠈⠀⠀⠀⠀⠀⣀⠠⠄⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⣄⠀⢤⡀⢸⠀⠀⠀⠀⠙⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠄⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠃⠉⢶⠿⠂⠀⠀⠀⠀⠀⠀⠙⠢⣍⡉⠐⠒⠐⠒⠀⠠⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠆⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠋⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠉⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡈⠛⢀⣤⠂⠀⢠⡾⠁⠂⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠉⠰⠛⠖⠦⠀⠶⠞⠁⠀⢀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢦⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠹⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠐⢻⡀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠆⣠⠏⠀⠀⢀⠀⠀⢀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⢻⠆⠐⡸⠃⠀⠀⠀⢀⠀⢀⡇⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⢣⣀⠀⠀⠀⠀⠸⠀⠈⢃⠀⠀⠀⣴⠏⢠⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠉⠀⠀⣀⠀⠀⠀⠀⠈⠳⣀⠜⢁⣴⠛⠒⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠉⣿⣽⣤⣤⡴⠀⠀⡀⠀⠀⣿⣶⣿⡔⠻⣖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢶⠓⣶⠷⠎⢡⡀⠈⣧⠀⠀⠋⠙⢻⣿⡆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⠃⠃⢰⡆⠑⢄⠙⠀⠀⠀⠀⢁⡜⡅⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠏⠀⠏⠹⠃⢠⢠⡗⡀⠀⢀⡀⠉⠓⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣿⣤⡆⠀⠐⠄⠩⠀⠒⠉⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⣿⣬⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢼⡙⢶⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⣿⣇⣸⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠄⠀⠀⢹⣿⣿⢿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⡀⠀⠀⠀⣀⣷⣄⠀⠀⢰⡆⠀⠀⢸⡇⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠸⠇⠀⠀⠀⠿⡿⠹⠃⠀⠀⠛⠀⠀⠸⠇⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄⠀⢀⠀⣴⣆⣴⠀⠀⠀⠀⣦⠀⢀⣤⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠈⠁⠀⠀⠀⢈⠟⢿⠀⠀⠀⠀⠈⠃⠀⢩⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠓⢻⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⡼⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡰⠃⢸⠀⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⣼⡦⠔⠁⠀⠀⠸⢀⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠆⠘⠥⠂⠀⠀⡇⠸⣀⣀⠰⠖⠂⠐⠖⠒⢀⠛⢠⡄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠑⠂⠀⠈⠀⠛⠀⠀⠈⠫⣉⠓⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢄⠄⢀⠆⠀⠀⠀⠀⠀⢣⠀⠌⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⠀⠀⠀⠀⠀⠀⠀⠐⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠴⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠕⠁⠀⠀⡴⠀⠀⠢⡀⠀⠀⢢⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠰⡇⠀⠀⠀⣷⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠳⡀⠀⠀⠏⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⠀⠐⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠆⠠⠟⠁⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⢀⡴⠋⠉⠈⢲⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠀⡔⠀⠄⡀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢀⠠⠠⠤⣀⠀⠈⡁⠀⠀⠀⠀⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠉⠀⠀⠈⠳⡆⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠆⠀⠃⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣆⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⢠⡇⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢿⡄⠀⠀⠀⡄⠀⠀⠀⣠⠂⠀⠀⠈⠂⠀⠀⢀⣼⢃⣼⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡈⠙⠀⠐⠋⠀⣀⣤⠞⠁⠀⠀⠀⠀⠈⠶⣄⡙⠛⠛⠃⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠾⠆⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠧⠴⠖⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢉⣠⣤⣴⠶⠶⣤⣤⣈⠙⢿⣿⣿⣿⣿⣿⠀⣤⣄⣈⣉⢉⣉⣉⣉⣉⣡⡄⢰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣴⣿⣿⡏⢀⣴⣶⣄⠹⣿⣷⡄⢻⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⣿⠈⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣏⠀⣿⣿⣿⣿⣿⣄⡙⠿⠟⢠⣿⣿⠏⣸⣿⣿⣿⣿⠀⡇⢠⣌⣉⣉⣉⣉⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣶⣤⡞⠋⣡⣶⣿⣿⣿⣿⣿⠀⡇⠸⠛⣋⣉⣉⣉⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠙⢿⣿⣿⣿⣿⢰⣧⠔⠛⠛⢿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⢈⡹⠿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢿⣿⣿⣿⡈⢡⣴⣿⣿⡄⢹⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⣾⡿⠁⣤⣄⠙⠻⣿⣿⣿⣿⣿⣇⢸⣿⣿⠋⣠⣤⣌⡙⢿⡇⠈⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⢰⣿⣿⡇⢰⣿⣿⣷⣄⠘⣿⣿⣿⣿⡿⢸⣿⡇⣸⣿⣿⣿⣷⠀⣷⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡄⠘⣿⣿⡇⠘⣿⣿⣿⣿⠀⣿⣿⣿⣿⠃⣼⣿⡇⢻⣿⣿⣿⡿⠀⡇⢀⣿⣿⣿⣿⡏⠀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⡀⠙⢿⣿⡀⠙⠿⠿⠟⣰⣿⣿⡿⠃⣴⣿⣿⣧⠈⢿⣿⠋⠀⠜⠁⣸⣿⣿⡿⠋⢀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠉⠛⠲⠤⠴⠞⠛⠋⣁⣤⣾⣿⣿⣿⣿⣷⣄⡉⠓⠂⠶⠾⠛⠛⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠁⡈⢉⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠄⠔⣀⡠⠤⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠁⣀⡨⢄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠠⠊⠀⢰⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠁⠀⡀⠸⠈⡁⡇⡠⠀⣠⣔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠃⣀⢵⣄⠀⠃⢞⡈⢉⡽⠠⠔⠁⠀⠉⠉⠉⠉⠉⠁⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⣚⠳⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠈⠉⠀⠒⠤⠄⡀⠀⣀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣼⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢊⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢑⠃⡠⠃⢠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⡀⠑⢌⡠⢁⣳⠤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠁⡀⡈⣿⣛⡠⠊⠁⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣉⠀⠐⠲⠖⠀⠉⠀⣰⢿⡵⡂⢠⠆⠀⠀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠑⣔⠀⠀⠀⠀⢀⡷⢛⡗⢞⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⢄⡀⠅⠋⠘⠤⠊⡶⢯⡶⠛⠹⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠈⠦⣙⣀⢠⠛⢄⠊⢃⠸⢷⠆⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⣿⠀⠆⡡⢚⠀⣆⠤⠢⠊⡜⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠢⠀⠠⠔⠂⣡⣂⣤⣤⣾⣽⡾⠷⠔⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠛⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣤⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣯⡍⠉⠉⠉⠙⠂⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⣴⣮⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⣨⠟⠁⠀⣠⠖⢻⠉⠉⠙⠒⠲⠤⠤⠤⠤⢤⡀⠀⣼
⠀⠀⠀⢀⣾⣵⠆⠀⡞⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠃
⠀⠀⠀⢘⡝⠁⠀⠰⡇⠀⠈⣠⡶⠶⠶⠴⡄⠀⠀⢠⣤⣤⠀⣇
⠀⠀⢠⣾⣴⠂⠀⢀⡷⠀⠀⠋⡠⠔⠢⢄⠀⠀⠀⣈⠤⣈⠓⡿
⠀⠀⠘⣱⡇⢀⡀⢸⠀⠀⠀⡞⠀⠀⠀⠀⢣⢀⡞⠀⠀⠈⢷⡇
⠀⠀⡞⡡⠒⠂⣼⡎⣇⠀⠀⣇⠈⣀⡤⢀⡼⢸⣀⡤⣈⠁⡼⠓⠲⢤⣀⡤⠤⢄⡀
⠀⠰⡏⢷⠁⣎⠉⠓⠛⠀⠀⠈⠙⠛⠒⠊⠀⠹⡀⠀⢢⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠦⣄
⠀⠀⢳⣄⣈⣃⠀⠐⣫⠔⣒⣢⠀⠀⠀⡀⠀⠀⡇⢀⣮⠤⢤⡀⠀⢠⠇⠀⠀⠀⢀⠀⠀⠀⠘⠦⠤⢤⡀
⠀⠀⢸⣤⠀⣸⠀⠀⢠⡾⠓⣌⠑⡄⠸⣀⡀⢠⠃⡞⠓⠒⢤⣸⢀⠎⠀⠀⠀⢠⠋⠀⠀⠀⡴⠀⠀⠀⣧
⠀⠀⠈⠹⡄⡏⠀⠀⣾⣤⢼⠋⠉⢻⣦⣤⠬⣭⡤⢷⣤⣀⣀⣥⡟⠀⠀⠀⠀⡏⠀⠀⠀⡸⠁⠀⢀⠀⡟
⠀⠀⠀⠀⢷⡇⠀⠀⢸⠀⠀⠓⠶⢯⡀⢈⣟⠉⠈⣯⠃⠀⠀⡼⠀⠀⠀⠀⡸⠀⠀⠀⢰⠃⠀⢀⣏⣰⠇
⠀⠀⠀⠀⠀⠹⡄⠀⠈⢧⡖⠢⣄⣠⢍⠉⢉⠛⠛⡏⠀⠀⠀⣧⡀⠀⠀⣰⠃⠀⠀⢠⠏⠀⢠⠟⢩⠏
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠑⠶⣿⣄⣠⢿⣉⣿⠷⠳⡄⠀⠀⠈⠛⢍⡉⠉⠒⠒⠲⣅⣀⡴⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠈⠉⠉⠉⢁⣠⠔⠛⢦⠀⠀⠀⠀⣹⢤⣀⠀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠶⠒⠚⠉⠀⠀⠀⠈⠳⣄⣀⣠⠃⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢿⣮⣿⣟⡛⠛⣛⣿⣤⡀⢤⣤⣄⣴⣤⠶⠶⠶⢶⡷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣋⠵⠞⠟⣛⣭⣶⣿⣿⠟⣿⡇⠀⠀⠈⣿⠁⠀⠀⠀⠀⠈⠙⢽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⡋⢀⣠⣾⣿⣿⣿⠏⣀⣴⡿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢘⢻⣦⣤⣴⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣼⡟⢀⣱⣾⣿⠿⠋⠉⢋⡠⢿⡵⠁⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢀⠶⠉⠀⣻⣦⠌⠁⠈⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⣼⣿⠿⡿⣿⠰⣿⣾⣋⡵⠋⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢀⣿⠀⠀⠀⢸⣿⡦⣤⣤⡈⠰⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣳⣾⣿⣿⡇⣠⣽⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢀⡰⠛⠚⠀⠀⠀⢤⢹⣦⠀⠙⠛⣷⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⢀⣀⣤⠀⠐⠐⠄⠀⣔⣿⡔⠋⠀⠀⠀⠀⠑⢄⡠⠈⣿⠆⢶⣤⣄⣻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⣴⣿⡿⠟⠁⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣿⡅⠀⠉⡿⣿⣛⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠛⠦⣷⢨⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⣿⣿⣷⣤⣀⡀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⠛⠿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣭⣿⣿⣿⣷⣣⠿⡵⢿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⢀⣴⡛⣿⣟⣷⣾⣿⠟⢋⣈⣉⣿⣭⣱⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⣰⣿⣿⣻⣶⣿⡟⠋⣤⣶⣿⣿⡿⠿⢫⣟⠛⠛⠛⢿⣷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⣴⣏⣙⣼⣿⡟⢁⣤⣿⣿⡟⢻⣄⠄⠀⣿⠃⠀⠒⢀⣰⢿⠏⠐⠒⠤⠀⣸⣿⣇⣠⢀⣀⣀⡀
⢰⣿⣿⣿⣿⢋⣠⣿⣿⢿⡉⢳⡄⢁⣤⣰⡟⠀⠀⣦⡎⠰⡏⠀⠀⠀⠀⠀⠹⣿⡏⠀⠀⠀⠉⠉⠓⠢⢄⡀⢀⣀⣀
⢾⣿⣿⠟⢁⣿⣿⠟⡙⢧⠙⠦⢸⡏⢋⡿⠁⠀⠀⠃⢈⡿⠁⠀⠀⠀⠀⠀⠀⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡄
⢸⣿⣏⣴⣿⡿⢉⢳⠟⣧⠐⢧⠀⣰⡞⠁⠀⠀⠀⡀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⡟⣠⠌⣿⣾⡯⠁⣀⣼⠟⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣦
⣿⣿⣿⢟⡗⣻⣿⡏⣹⢇⣸⡟⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣁⡀
⣿⣿⣯⣼⣿⣿⡟⢾⣧⣾⠷⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄⠀⠀⠀⠀⠀⠀⠠⠐⡀⠀⢨⠿⣿⣿⣿⣿⠖⠂⣀
⣿⣿⣿⣿⣿⣿⡷⠟⢹⡏⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠤⠄⠒⠚⠓⠒⠒⠀⠤⣄⡀⢻⡄⠀⣀⠠⢤⣾⣇⡘⠧⣤⢘⣿⣿⣿⣿⣏⢿⣏⣉⠷⣄
⣿⣿⣿⣿⠿⠋⠁⠀⠘⠃⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⠟⠋⠀⣤⣾⣷⣌⠙⡶⣁⡈⠿⣿⣿⣿⣷⣾⣏⣵⣆⣉⣿⡀
⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⣈⣿⡙⢿⣎⡴⠏⠷⢚⣿⣿⣿⣿⣿⣮⢽⣿⣶⡼⣇
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⣿⣿⣟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⣴⢈⣥⠚⢿⣦⠀⠀⠉⣸⣿⣿⣿⣿⠟⢚⣽⢿⣿⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠐⠂⢀⣴⡨⢿⣧⡀⠀⠈⢻⣿⣿⣷⣦⠘⠗⣿⣽⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⠉⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠐⠏⠈⢥⠈⠻⣷⡀⠀⢿⣿⣿⣿⣿⡦⣟⣌⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡛⠿⠋⠀⠀⢀⣽⢳⣶⣄⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠷⠀⠹⢿⣆⠀⣤⣿⣿⣿⣛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⣿⣿⣿⣆⠸⢋⣀⣸⣿⣿⣿⡟⠀⠀⢀⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣿⣄⠀⢻⣷⣼⡿⢻⣿⡿⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣹⣿⣿⣿⣿⣥⡈⣤⣾⣿⣿⡿⠋⠀⠉⠢⢠⡌⢰⣧⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠛⠀⠀⠉⠛⢣⣿⣏⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣯⣤⣿⣿⣿⣿⡟⣠⣿⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⣷⣸⡟⠀⠀⠈⠓⢠⠀⢠⣶⣤⡀⠀⠀⠀⠀⠓⢦⡴⣤⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⡞⠙⣿⣿⣿⣿⠏⢠⣽⣿⠿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⢠⣶⣿⣻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠙⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠺⡖⢿⣿⣿⣫⣿⣿⣿⠿⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠸⣿⡿⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡿⠛⠛⠻⢿⣿⣿
⠀⠀⠀⠀⢸⣾⣛⠃⢠⣿⣿⣿⠟⡻⣠⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⠀⠀⠀⠀⠉⢿
⠀⠀⠀⢀⣿⠻⠏⣤⣿⣿⣿⣿⠆⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇
⠀⠀⠀⢸⣿⡿⠶⢶⣿⣿⣿⣿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟
⠀⠀⠀⣿⣿⣯⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿⠃
⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇
⠀⠀⠀⣿⣿⣿⠋⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠁
⠀⠀⠀⣿⡿⠁⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠃⠀⠀⠀⣸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣶⣾⣷⣾⣿⣾⣷⣿⣿⣷⣿⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣾⣿⣷⣶⣿⣿⣿⣿
⣿⣿⣿⡏⠀⠈⠉⠀⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣶⣶⡶⠶⢦⣅⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠚⠛⠚⠋⣢⣤⣬⣉⣉⠉⠉⠒⠻⢼⣃⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠉⢀⣯⣽⠟⠛⠛⠿⢿⣿⣿⣒⣀⣄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣷⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⡿⣿⣰⡟⢁⠞⠉⠉⣹⠟⢿⢷⣶⣦⡄
⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⡟⣿⡿⠀⠀⠀⠠⠊⠀⠀⠀⠈⢿⣿⣿⡦⠀⢰
⠀⠀⠀⠀⠀⣿⣿⣥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⢟⡠⠔⣊⣉⣉⠉⠉⠉⠁⠐⠒⠦⢤⡀⠿⣩⡍⣷⣤⠀⣀⢀⡄⢤⣤⣔⠶⢞⣡⣾⡀⠀⠺
⠀⠀⠀⠀⠀⢿⣿⣿⡈⢻⣿⣿⣦⣤⣄⣀⣤⣤⣤⣤⣂⠴⠋⣴⡻⢍⠉⡉⢝⡟⣶⣄⠀⠀⠀⠀⠈⠑⢿⣧⠸⣿⡇⢽⣿⣿⡌⢿⣿⡆⡟⠀⠹⠃⡁
⠀⠀⠀⠀⠀⣤⡙⠻⣷⡀⢻⣿⡿⠉⢹⣿⣿⣏⢙⡿⠃⠀⣾⣵⣬⣤⠶⠾⠻⠒⠛⠚⠓⠀⠀⠀⠀⠀⠀⠹⡠⠟⠋⠀⠈⢙⠿⣤⡿⠟
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⡉⠃⠀⠀⠟⠿⢿⠎⠀⠀⠴⠛⠉⠁⢀⣀⣤⣤⣤⡤⣤⣤⣤⣤⣀⣀⡀⠀⠀⢰⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀⠞⠁⠐⠀⠐⠈⠄⣌⢴⡊⠠⣤⣤⣶⣟⣽⣿⣿⡿⢼⡵⠿⢿⡹⠿⠿⣿⣿⡿⠆⡀⠈⠓⠀⠀⠛⠿⠶⠂⠀⠘⠳
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠜⢀⣄⠀⠀⠀⠀⠾⠁⣸⣷⡿⠛⠉⠁⠈⠉⠀⠀⠀⠀⠈⠀⠉⠀⠀⣬⠿⢿⣿⣯⣽⡃⠀⠀⠈⠀⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠁⠰⣿⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢰⣃⡃⠈⠿⣿⣇⢈⡆⠠⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠐⠀⠤⠂⠀⡠⢻⣿⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣤⠼⠁⣠⣶⣾⣿⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠒⠊⠀⠼⣿⣟⠲⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣴⣾⣿⣿⣿⡿⢻⣧⡄⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣴⣭⣶⢾⣿⠿⣿⣶⡄⠘⣿⣿⣿⣿⣶⣶⣶⣿⡟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢸⣟⠑⢺⣿⣷⣶⣦⣀⣀⢀
⠀⠀⠀⠀⠀⣠⣛⡿⠋⠑⠂⠀⠀⠨⢐⣿⣿⠿⢣⣿⠿⢿⣿⣿⣿⣿⣿⣦⡀⠀⢙⣿⣿⣿⠿⠛⠁⠀⠀⢸⡄⢜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣴⡵⠋⠀⣀⠀⢠⠠⠶⠲⢦⣽⣿⣾⡦⣞⠆⠀⠈⠙⠛⠛⠛⢻⠛⠋⠙⠟⠉⠀⠀⠀⠀⠀⠀⢸⣧⠟⠛⠛⠿⣿⣛⣯⣽⡷⣏⢙⡿⣆
⠀⠀⠀⢠⠊⠀⢠⣾⣷⠞⠓⠒⠀⠀⠁⠀⠀⠀⠙⠟⢷⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⢕⠀⠀⠀⠀⠀⠀⠀⡜⠹⠀⠀⠀⠀⠈⠉⠉⠙⠻⣿⡎⡟⣿⡆
⠀⠀⠀⢰⢃⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠘⠧⠤⠀⠴⠚⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣧⣿⡇
⠀⠀⠀⠨⡎⠐⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡸⣄⠀⠀⠀⡠⠄⠀⡀⣀⣁⠒⠀⠀⠀⠀⣤⣽⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⢳⣸⣏⠁⢀⣀⣀⣀⣤⣀⣀⡀⢀⣠⣿⡉⠻⣿⣟⢆⠀⠀⠐⠿⠛⠛⠋⠙⠋⠀⠀⠀⣰⠳⣿⣿⡏⢹⣷⣦⣀⣀⣤⡤⣤⣄⡀⠀⣿⢋
⠀⠀⠀⠀⠀⣿⠛⢿⢽⣿⣧⡀⠻⣿⣿⠟⠛⢿⣿⣿⣄⣸⣿⡀⠛⣄⠀⠀⠀⠐⠂⠀⠀⠀⢀⡴⠁⠀⠈⠛⠿⢴⠿⠿⠻⣿⣿⣇⠘⣿⣿⣾⠁⡟
⠀⠀⠀⠀⠀⢁⠘⠂⠙⢿⣿⣷⣄⣽⠇⠀⠀⠨⠉⠋⠉⠁⢸⠁⠀⠀⠓⠢⠤⣀⣀⣠⠴⠞⠋⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠙⠛⠋⠉⠁⢶
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠈⠉⢁⠔⠀⠠⢤⡤⠆⠀⣲⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢠⠃⠀⠀⠀⠀⠈⠅⠐⢀⣳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡆⠀⠀⠒⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣤⣀⣀⣀⣠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⡄⠢⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠒⠉⣁⠀⠀⠀⠀⠉⢢⠃⠀⣀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡲⠊⠉⠲⡄⠀⠀⠀⠈⡀⠀⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⡀⠀⢀⠀⠀⠀⠀⢀⢠⣀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠒⠤⡀⠀⠀⠀⠀⠀⠋⠀⠈⠀⠀⠤⠴⠁⠤⠌⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠈⠑⠢⡤⠐⠀⠀⠀⠀⠀⠀⠀⠇⢁⣠⣞⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠒⠃⠠⢄⡉⠑⠂⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢤⠙⠲⢤⣀⡀⠀⠀⠀⠰⠿⡏⠐⠤⣀⣀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠈⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⢦⠀⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠐⠦⢠⡀⠀⠀⠀⢀⠔⠉⠀⠈⠀⠠⢄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⠀⢀⣀⠄⠀⠀⠀⠡⡑⢀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡈⠁⠛⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡄⠀⣀⠄⠀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠘⡰⠐⡀⠀⠀⠀⠈⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠤⠄⠀⠀⠀⠀⠀⢡⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠂⠀⠠⠀⠀⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠘⠉⠁

⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠈⠈⠠⢽⣦⢬⡬⡖⢓⠅⢤⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠒⠠⣄⡀⠀⠀⠀⠉⠐⠦⠐⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⣂⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⠇⢀⣀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠘⠗⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡜⠁⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣴⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡻⠄
⠀⠀⠀⠀⠀⠠⠄⠀⠤⠭⠛⠁⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡦⣉⠒⠈⠀⠀⠜⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⣀⠉⠀⠀⠉⠸⠓⠁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠸⢀⡐⡀⣄⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠘⠃⠂⢁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠤
⠀⠀⠀⠀⠈⠀⠂⠏⠋⠑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⣰⠫⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣀⣤⣴⡶⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⣟⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠞⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⢀⠾⠛⠀⠀⠀⠀⢀⡀⠀⠐⢀⡀⠀⠀⠀⠀⠋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠐⠶⠠⠘⣆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢰⡏⠉⢻⠄⠀⡾⠉⠉⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⡀⢀⡀⠀⡀⢀⢀⠀⢀⡀⡠⠤⠀⡠⢀⡀⣀⠤⠀⠀⣀⠀⠀⡄⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⢠⠟⠒⢯⡀⠀⠛⠙⢶⡀⠀⠀⠀⣰⣭⡆⢸⠙⢄⡇⢸⠑⣄⡇⢸⠈⢆⡼⠀⡷⠂⠀⡧⣴⠃⠙⠦⡄⢠⡏⣆⠀⡦⡜⠁⠘⡞
⠀⠀⠀⠀⠀⢻⣄⣀⣸⠇⢴⣀⣀⡼⠁⠀⠀⠀⠁⠀⠘⠈⠀⠈⠃⠈⠀⠈⠃⠈⠀⠘⠁⠀⠓⠒⠀⠁⠈⠂⠐⠒⠁⠃⠀⠈⠀⠁⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉


⠀⠀⠀⠀⠀⣠⠴⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⡠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⢠⡿⣵⠞⢀⠀⠀⢠⡄⠘⢮⡹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⣿⠆⠰⣿⠈⠛⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣽⠀⢸⣧⠤⢤⡀
⠀⠀⠀⣿⢸⠇⠀⢈⠳⡄⠘⢦⠀⠈⣿⢹⡆⠀⠀⠀⠀⠀⠀⣀⢤⠲⠏⠈⠀⠀⣁⠹⠖⣤⣀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠉⢦⠈⠁⠀⠀⢹⡀
⠀⠀⠈⣷⢸⡆⠀⠀⠀⢀⠉⠀⠀⢀⡿⢸⡇⠀⠀⠀⠀⠀⠀⡘⡁⠀⠀⡳⢆⠐⢦⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣠⠞⠘⣆⣀⣀⣼⣀⡀
⠀⠀⠀⢹⣯⡻⣦⣁⠀⠀⠀⢀⣠⡾⣡⡟⠁⠀⠀⠀⠀⠀⠀⠃⠧⠀⠀⠁⠀⠈⡀⠀⠀⠼⠘⠀⠀⠀⠀⠀⣀⣀⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠙⠷⣎⣽⡃⠀⠀⣫⣥⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠁⠀⣾⣦⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠛⠛⠋⠉⠉⠁⠀⠀⠀⠘⠛⠻⠧
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠂⠀⠀⠈⠉⠀⠉


⠀⠀⠀⠀⣤⠤⢤⡀⠀⠀⣤⡤⣤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⠸⣇⠀⢀⡗⠀⢠⣇⣀⡀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠁⠀⠀⠈⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⡆⠀⠀⣸⣏⡈⠉
⠀⠀⠀⣠⠟⠉⠛⣦⠀⠈⠉⠉⢹⡆⠐⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⢰⣚⡦⢰⣒⠂⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⢈⣿⠷⠾⣯⡀⠀⠀⠛⡉⠛⣷⡄
⠀⠀⠀⠻⣄⣀⣠⡾⠀⢦⣀⣀⡼⠃⠀⡀⠀⠀⠀⠀⠀⠀⠈⠟⣛⠀⠸⣍⡷⢨⣨⠇⠀⡛⠛⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⣸⡇⠀⣀⡁⠋⢑⣾⠇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢆⡀⠀⠀⠀⠀⢀⠲⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠟⠋⠀⠀⠙⠛⠿⠛⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⢰⠸⠈⠠⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠚⠓⠃⠚⠒⠋⠑⠛⠚⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⢄⣀⠀⠀⠐⢴⡟⠆⠀⠰⣾⠧⠀⢰⢦⠀⡀⢂⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠘⠤⠄⠞⠻⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⢀⠊⡇
⠀⠀⢀⡀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠚⠉⠐⠒⠲⠤⣀⠀⠀⢀⠄⣠⠄
⠀⠀⢸⡅⢸⡁⠀⠀⠀⢸⠀⠒⢒⠢⡀⠀⠀⠀⡠⠺⠷⠶⠀⠀⠀⠀⠀⠀⠐⠛⠺⡗⢦⡀⠐⠁⢀⡀
⢾⠀⢺⠀⢮⢠⠆⠀⠀⠈⠦⠼⠷⠤⡐⠀⢀⠞⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠑⡄⠀⠹⠛⠂
⢻⠀⣻⠄⡿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣷⠶⠸⠀⣰⠀⠀⠀⠸⠿⣆⣸⠀⠀⠀⠸⠀⠠⣤⣄
⣹⠀⢻⠀⠗⣧⠀⠀⠀⠀⠂⢤⣀⠤⠀⢸⠀⠀⠀⠀⡀⠀⢎⡀⢤⠄⣤⡀⠀⠀⠉⠒⠀⠀⠀⡇⠀⠛⠁
⢹⠂⡠⠲⡄⢁⣀⠀⠀⠑⢄⣤⡕⠀⠀⢸⠀⠀⡠⠚⠀⢀⡀⠀⠀⠀⣀⣙⠦⠤⢤⠀⠀⠀⠀⡇⠀⢠⢄
⠈⠁⢈⠟⠓⠚⢤⡠⢦⠀⠀⠀⠀⠀⠀⠈⡆⠀⠁⠈⠣⠤⣀⣀⣀⣀⣀⠤⠜⠃⠄⠀⠀⠀⢠⠃⠀⠲⠄
⠀⢰⡃⠀⢰⡀⠀⢣⠀⠀⠒⠤⣀⣀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠩⢂
⠀⠈⣧⠀⠈⣧⠀⠈⡆⠀⠐⢄⣤⡶⡀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠰⠽⠇
⠀⠀⠸⡆⠀⠺⡏⠀⠸⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⣀⣀⡀⠀⠔⠊⠁⠀⠀⠀⣱⠒⠒⠄
⠀⠀⠀⢻⡀⠀⢻⡆⠀⢣⠀⠀⠀⠀⣠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⠑⠂
⠀⠀⠀⠈⢷⠀⠀⢷⠂⠈⢆⠀⠀⡠⠉⠀⡣⠀⡐⡀⡀⠀⢀⠀⠠⠂⠀⠀⢠⡀⠰⣄⣀⣛⡃⢹⠀⠈
⠀⠀⠀⠀⠘⣇⠀⠘⣧⠀⠘⡄⠂⠀⠀⠉⠀⠀⡧⠁⢣⠀⡷⠷⣧⠀⠸⡿⠂⠃⠀⠀⠀⠀⠙⠈⣄
⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⢱⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠈⠄⡀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠐⠄⠀⠑
⠀⠀⠀⠀⠀⠀⢷⠀⠸⠿⢇⣀⠇
⣀⣀⣀⣀⣀⣀⣘⣧⣴⣚⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠖⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣄⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢷⠿⠿⠛⠻⢿⡉⢵⣄⠀⠀⠀⠘⢳⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢀⡀⠈⢳⠀⠸⣦⠀⠀⠀⠈⠣⡀⠈⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⢀⠀⠀⠑⡀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠙⠢⣀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⢣⠤⠀⠀⢀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠈⠑⠄⢈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡌⠛⢄⠀⠀⠀⠀⠈⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠐⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⢤⡀⠀⠀⠀⢉⠦⣀
⠀⠀⠀⠀⠀⠠⡴⠶⡤⠗⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣄⣀⡈⠉⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠤⢼⡀⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠈⠉⠙⢾⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠆⢁⠂⠄⢀⣀⠤⡀⠀⠀⠀⠀⣠⡼⡃⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠓⠒⠤⠅⠈⠁⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠐⢎⣀⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣤⡄⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⢸⠇⢰⠇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢏⠀⡐⡅⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⢸⠀⢸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⣿⡇⠙⢿⣷⣦⣄⣘⠀⣂⣀⣴⣷⡀⠀⠀⠀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠻⣿⣿⣿⠿⢿⣿⡟⠀⠀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⣠⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠸⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⢿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠖⠛⠉⠉⠉⠓⠶⣤⣿⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⢸⣿⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠁⠹⣾⡣⠀⣠⣴⣶⣤⣀⣰⣟⣹⡆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣀⣀⣤⡾⠟⠓⠒⠶⣄⣹⣯⠝⢦⡀⠀⢀⡇⠀⠀⢻⡗⣾⠋⢰⡖⠛⠉⠉⠉⣿⣷⠀⠀⣧⣀⡀
⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⠀⣀⠀⠀⠀⠙⢿⡯⡈⠻⣄⢸⠃⠀⠀⠀⢻⣻⣄⠸⣇⣤⡄⠀⠀⡿⣿⠖⠒⢻⡀
⠀⠀⢈⡏⠀⡣⠀⠀⠀⠀⠀⢠⣿⣷⢅⣴⠿⠛⠻⢷⡄⠀⠈⣿⣔⠀⠈⠿⠀⠀⣆⠀⠈⢷⡉⠛⠛⠉⠀⢀⣼⢃⣸⣄⣀⣼⡿⠛⠓⠒⠂
⠀⠀⠈⠁⠙⠃⠀⠀⠀⠀⢀⣼⣿⣿⣺⡇⠀⠀⠀⠈⢿⠀⠀⢸⣿⣷⡄⠀⠀⢰⣿⡆⠀⠘⣷⣤⣤⣴⡶⠟⣱⠟⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⡞⠀⠀⣼⣷⣿⠻⣦⣀⣸⠀⢻⡶⠾⠿⠉⢉⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡆⠠⡄
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠉⠉⣹⠿⣿⣿⣾⠉⠛⠛⠁⠀⠀⣰⣿⣿⣿⠄⠘⠛⠋⠀⠀⠀⠀⠀⠐⣤⠷⣖⠒⠒⠛⠦⠀⠀⠀⠈⢠⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⡰⠛⠻⣷⣧⣤⣤⣤⣴⡾⢿⣯⡁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡎⠈⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⠞⠁⠀⠀⠜⠋⠉⠉⠉⠀⠀⠀⠙⣷⡀⠀⠀⣠⠞⠋⠉⠉⠉⠲⣇⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡼⠃⢀⡴⠞⠛⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠒⠈⠦⡄⠀⠀⠀⠀⠀⣿⡼⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣠⣀⡎⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠛⠛⠻⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠁⠀⠀⠀⠀⠀⢀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠐⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⡀⠀⠀⠐⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠄⠢⡀⠀⠈⠦⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⢡⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢂⠹⣆⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠚⡀⠹⣦⣤⡶⠾⠛⠛⠋⠉⠙⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣴⢿⡄⠀⠀⡀⠀⢡⠀⠙⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⢀⣴⠾⠛⠋⠉⠉⠀⠀⠈⠉⠉⠉⠀⠈⣷⠀⠀⢁⠀⠀⠑⢌⢘⡳⢶⣤⡄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⢀⡿⠀⠀⠘⠀⠠⡀⠀⠉⠑⢦⣀⡷⠔⢉⠘⠂⠀⠀⠀⣠⣤⠶⠾⠿⠶⣶⣤⣀
⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⢾⣇⣀⡀⠀⠆⠀⠈⠢⢀⣠⡼⠿⠄⢀⣘⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀
⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣀⣤⣤⣭⣽⣷⣶⣼⣦⣤⡴⠟⢁⣠⠖⠚⠉⠛⠛⠛⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠛⠉⠁⠀⠀⠀⠈⠉⠙⠻⣦⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⣤⣀⠀⠀⠀⠀⠀⠀⢻⣆
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠈⠛⢷⣦⣄⣀⣀⣀⣀⣠⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣤⣤⣤⣴⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢒⣂⠒⠶⠶⠶⢖⡤⣀
⠀⠀⠀⠀⠀⠀⡰⡻⠉⠈⣧⠀⠀⠀⠀⠀⡠⠃⠊⠳⣄
⠀⠀⠀⠀⢀⣎⠜⠀⠀⠀⠙⠢⠠⢔⠶⠛⠐⠪⣆⠀⠈⢦
⠀⠀⠀⠀⣾⠁⠀⠀⠀⣠⡣⠤⠠⠁⠀⠀⠀⠀⢹⠢⡀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⣀⡀
⠀⠀⠀⠀⣯⠓⠦⠴⠛⠉⠀⢠⡝⠀⠀⠀⠀⠀⢸⠔⠘⢦⢻⠀⠀⠀⠀⢀⣴⠮⠓⠀⠈⠉⡕⠐⠪⡛⠲⡢⣀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠋⣣⡤⡀⠀⠀⡰⡡⠁⠀⠈⣼⠀⠀⠀⡔⠝⠁⠀⠀⡠⢐⠭⠂⠉⠑⡴⠀⣀⠨⡁⡀
⠀⠀⠀⠀⢷⡀⠀⠀⠀⣠⠕⠊⠀⠈⢪⠐⠩⠊⠁⠀⠀⣰⠇⠀⠀⢸⠐⠀⢤⡴⢊⠰⠁⠀⠀⠀⢀⢰⠉⠐⢷⣜⡸⡀
⠀⠀⠀⠀⠈⢮⣢⢔⡪⠁⠀⠀⠀⠀⠀⡇⢡⠀⠀⢀⢔⠏⠀⠀⠀⢂⠜⠁⠀⠀⢀⢰⠀⠀⠀⢀⠌⡌⠀⠀⠀⠁⡐⡇
⠀⠀⠀⠀⠀⠀⠙⢦⡝⢄⠀⠀⣠⢖⠺⣉⡝⠛⠻⢗⡣⡀⠀⠀⢰⠀⠀⠀⠀⢀⠌⠀⣓⡐⠦⢅⢘⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢪⠟⡱⠁⠀⠸⢑⠀⠀⠀⠈⠊⠢⠀⢸⠈⢒⠶⠏⠤⠤⢄⢇⠀⠀⠀⠹⣢⣄⡀⠤⢪⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠊⠀⠀⠀⣨⠖⠳⡀⣤⣔⠗⠢⡱⠀⢆⢣⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠁⠀⡄⠆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⢦⡠⠔⠞⠃⠀⠀⠘⣇⠀⠀⠀⡿⡇⠀⠓⢅⡢⣀⠀⠀⡘⢡⠀⠀⠀⠀⢀⠜⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⠀⠀⠀⢀⢜⡬⣦⣀⣴⢥⠇⠀⠀⠀⠉⠓⠫⠬⡀⠈⠁⠀⠀⢊⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⢠⣱⠤⢔⠩⠊⠀⠈⡶⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣒⡏⠀⣾⠁⠀⠀⢀⠜⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠷⠵⠶⠮⠑⠊



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣄⣼⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⣿⢿⣿⣿⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⣠⣶⣿⣇⣀⣿⣷⠀⠋
⠀⠀⠀⠀⠀⠀⢠⡐⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠞⣽⣿⣿⣿⣿⣿⡟⠠⢤⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⣇⠈⠙⠀⠀⠀⠀⠀⠀⠀⠞⢡⣼⣇⣄⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢐⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣯⠁⢩⣿⡆⢰⡖⠀⣀⣴⠟
⠀⠀⠀⠀⠈⢲⠃⠀⠐⡀⣴⣦⡿⠂⠀⠀⠀⠀⠀⠀⠉⠋⠸⢿⣿⣿⣿⣿⠷⠋⠠⠞⠉⣡⡶
⠀⠀⠀⠀⠰⣷⢀⢰⣴⣿⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣼⣷⣮⣤⣶⣾⠏
⠀⠀⠀⠀⠀⢻⣦⣷⣽⣿⣷⡍⠀⢠⡶⠞⠛⣳⠀⠀⠀⠾⠷⢶⣶⡌⢸⣿⣿⣿⠛⣋⣽⠃
⠀⠀⠀⠀⠀⠀⣹⣿⡹⣿⣿⣿⢀⣏⣤⠷⢾⠟⠀⠀⢀⢒⡷⠿⣄⣹⠀⢻⣿⣿⣿⣻⠇
⠀⠀⠀⠀⠀⠀⢻⣿⡟⠻⢿⡏⠈⠉⠓⣾⠾⠇⠂⠀⠀⡜⢺⡗⠊⠉⠂⠸⠿⢛⡛⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠙⠊⢟⣦⡷⠀⠀⠀⠈⡠⠄⠀⠀⠀⢠⡈⠀⠀⠀⢀⢇⣴⡋⠥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠆⠀⡔⠊⠓⠿⠦⡴⠾⠶⠓⠦⠀⠰⠃⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣿⠈⠀⢦⣤⡤⠤⠤⠤⠤⠤⣤⡤⠀⠀⠀⣼⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢿⣿⣷⣶⣶⣶⣿⡟⠁⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠻⣄⡉⢋⣩⠟⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠢⢭⡭⠞⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠛⠉


⠀⡀⣀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣯⢹⣟⡛⣛⣛⡛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠈⠛⠟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡔⠀⣼⣃⠀⢀⠈⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⠛⢅⣺⣤⣴⡿⢿⣿⠭⠭⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⣄⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡏⠀⢠⣤⣤⣄⠀⠑⢆⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠋⠉⠙⠳⣤⡴⠄⣇⠀⠈⣿⣿⣿⣷⣆⠀⠑⠆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢰⡈⠛⢿⣷⠀⠀⠙⠙⠛⢿⣿⣷⣄⠀⠉⠠⣀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⢘⣇⠸⠁⠀⠀⠀⠀⠀⠀⢻⣿⠿⣳⣤⡀⠈⠙⠾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡘⠎⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠈⠈⠻⣿⣷⣤⡀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⣀⣀⠀⠀⠀⠀⢀⣾⣿⣷⣦⣄⡀⠉⠑⠻⣷⠀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⣴⣾⣿⣿⣿⣶⠾⣿⣽⣿⣿⣿⣿⣟⣿⣿⣷⣤⣿⠂⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠛⠿⠿⢿⣿⣦⡀⠀⠀⠸⠙⣿⣿⣿⣾⣽⡿⠏⠀⣴⣿⣇⠀⠀⢀⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠄⠀⢀⡀⠉⠙⠷⣦⣴⣄⣫⣿⣿⣿⣿⠀⠀⠞⠁⠉⠉⠉⠀⠁⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢀⠉⠙⠛⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⣄⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠛⠛⠻⣿⣆⠀⠀⠀⣿⣿⠟⠛⠛⠛⠛⠋⠀⠀⠀⢸⣿⠉⢻⣷⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠏⠸⠀⠏⠸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⣼⡟
⠀⠀⠀⡗⢸⠂⡇⠰⠀⠇⠸⠀⡇⢸⠀⡇⢸⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⣰⡿⠁
⠀⠀⠀⡦⢰⠀⡦⢰⠀⡆⢰⠀⡆⢰⠀⡆⢰⠀⠀⠀⢹⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⢰⣿⠁
⠀⠀⠀⡥⢠⠀⡅⢠⠀⡄⢨⠀⡄⢠⠀⡅⢠⠀⠀⠀⠘⣿⣧⣀⣀⣴⣿⠏⠀⠀⠀⣿⣿⣀⣴⣶⣶⣦⡀⠀⠀⠀⠘⢿⣶⡿⠏⢠⣿⠃
⠀⠀⠀⣅⢨⠀⣅⢨⠀⡅⢨⠀⡅⢨⠀⡅⢨⠀⠀⠀⢀⣴⣿⠿⠿⢿⣤⡀⠀⠀⠀⣿⣿⠏⠀⠀⠙⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⡃⢘⠀⡃⢘⠀⡃⢘⠀⡃⢘⠀⡃⢘⠀⠀⠀⣼⣿⠃⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⠏⢠⣾⠟⢿⣦
⠀⠀⠀⠏⠘⠁⠏⠸⠀⠃⠸⠀⠇⠘⠀⠇⠘⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⣶⣶⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣼⡟⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠗⠸⠀⠗⠸⠀⠇⠸⠀⠇⠸⠀⠇⠸⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣼⡟⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⡖⢰⠀⡗⢰⠀⡆⢸⠀⡆⢰⠀⡇⢰⠀⠀⠀⢿⣿⡀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⣰⡿⠁⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⡆⢰⠄⡦⢰⠀⡆⢰⠀⡆⢰⠀⡆⢰⠀⠀⠀⠘⣿⣷⣤⣤⣾⣿⠏⠀⠀⠀⠹⣿⣷⣤⣤⣾⣿⠃⠀⠀⠀⢰⣿⠁⠀⠀⠀⠸⣿⣄⣼⡿
⠀⠀⠀⠀⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⡶⢆⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠉⢟⣩⣶⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⢸⣸⣿⡇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣄⡀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠐⣿⣷⣸⣿⣿⣷⣿⢟⣩⣭⡾⠿⢿⣿⣿⣿⣾⣼⣷⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷⢻⢷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣶⣾⣿⡿⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣥⣶⣶⠀⠈⠉⠁⠀⠉⠉⠀⠉⠉⠉⠛⠛⣿⣿⡷⣮⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⣹⣾⣿⣿⣿⣿⣿⣁⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣴⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣯⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠟⠋⠉⠁⣠⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣻⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⢻⡇⠀⠚⠶⠶⠖⠁⠀⠀⠀⠀⠀⠈⠓⠿⠿⠟⢿⣿⣿⣿⣿⡏⣷⢹⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠀⣠⢼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠶⠠⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣾⣿⣿⣿⣯⡙⠛⡆⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⣶⣾⡆⠀⡀⢀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠁⣨⡇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣧⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢤⡄⠀⠀⠈⣿⣿⣿⣿⣿⠿⠿⠿⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣸⣿⣿⣇⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⣀⣀⣸⣿⡇
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠐⠶⡶⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⠁⢀⡟⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⢴⠧⣬⠃⢀⣿⣷⣶⡴⢒⣒⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠚⠛⠻⠿⢩⠟⠋⠉⠉⠉⠢⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠊⠉⠉⠉⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣦⣤⣀⠀⠀⡀⣼⠀⠀⢶⣶⣶⣦⣄⠀⠑⢄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠈⠙⠳⣾⣃⣸⠀⠀⢸⣿⣿⣿⣿⣷⣄⠀⠑⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⢨⢅⣁⡛⣿⠃⠀⠀⠈⠉⠙⢿⣿⣿⣦⡀⠀⠁⠦⣀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡿⡐⠇⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⢍⣢⣄⡀⠈⠙⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡈⠷⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⡃⠀⠀⠟⢿⣟⣶⣄⠀⠈⠱⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⢐⣾⣿⣿⣦⣄⡀⠈⠙⠋⢿⣆⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⣠⣶⣿⣿⣿⣶⣴⣾⣿⣻⣿⣿⣿⣿⣟⢿⣿⣷⣦⣼⡿⠀⠀⢸⡟⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠙⠻⢿⣿⣿⣿⣿⣀⠀⠀⠉⢹⠹⣿⣿⣿⣮⣽⣻⢿⠃⠀⣰⣿⣇⣀⠌⠁⣠⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⢀⣀⠀⠀⠈⠉⠛⠾⣦⣄⣠⡈⢿⣿⣿⣿⣿⡏⠀⠀⠞⠉⠛⠛⠓⠃⠄⠁⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⠄⠀⠉⠛⠿⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⠀⣀⡀⠄

⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⢀⣴⠊⠙⠁⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⡴⠆⠀⠐⠃⠀⠀⠐⠚⠂⡀⢀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⣀
⠀⠀⡘⠁⠀⣠⣄⠀⠈⠛⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⣀⠀⠀⠀⠀⠘⡻⠁⠀⢰⠀⠠⠟⠂⠀⠀⢠⣄⠉⠁
⠀⠀⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠊⠁⠀⣀⠠⠁⠀⣾⠦⠐⠉⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⡇⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠰⠦⠀⠀⢀⡤
⠀⠀⠻⠄⠀⢀⡀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠘⠁⠘⠃⠀⢀⣤⢤⡄⠀⣤⣤⣤⣤⣀⣰⣏⡼⢀⠀⠀⣠⢤⠀⠀⠀⠘⠺⢇⠀⠀⣉⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢡⣾⠇⢈⣋⠀⣨⢭⢩⣿⠭⡄⣮⣰⢒⢛⢮⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠸⠯⢠⣿⢸⣾⣧⣸⣾⣿⣞⠹⠧⠻⠅⢲⣆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣠⠿⣏⠟⠙⠳⢋⣈⣡⡿⢷⣋⣉⣱⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠘⠿⠿⠟⠋⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⢤⡀⠀⠀⠀⠀⠀⠀⣠⢇⣈⣁⠠⠤⣄⠀⠂⠀⣀
⠀⠀⠀⠚⢱⡄⠘⠁⠀⠀⣠⡀⠀⠘⠿⠀⠀⠀⠀⢀⠠⠞⠁⠠⡦⠄⠀⠀⣀⠀⠀⠀⠀⠀⠐⣿⠞⣉⠀⠀⠤⠀⠀⣀⠀⠁
⠀⠀⢸⣇⠀⠀⣄⠀⣤⡄⠁⢠⣤⡄⠠⢠⠆⠀⢠⡆⠐⠂⠀⠀⠀⠘⠀⠈⠉⠀⠙⠀⠀⠀⠀⣈⣤⡭⠓⠛⡛⠛⠁⠳⠄⣀⠁
⠀⠀⣾⣿⣿⣦⣦⣤⣠⣤⣄⣄⣀⣤⠤⠏⠀⠀⡰⠀⠈⠉⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⣿⠃⢀⣄⠀⠉⠀⠀⣄⠀⠈
⠀⠀⠙⣻⣦⣤⠆⠈⠛⠉⠀⣀⣰⣿⣧⡾⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⡆⢀
⠀⠀⠈⣛⡛⠛⠉⠙⠛⠛⣻⡏⠉⠁⣀⣀⠤⠀⠀⠀⠀⠲⠀⠀⠀⠀⠘⠀⠈⠁⠠⠒⠂⠀⠀⣷⠉⢀⡀⠀⠀⠙⠀⢀⠀⠆
⠀⠀⢘⡹⠋⢹⣗⠛⠛⡿⢓⡖⠚⠛⠙⣋⡀⠀⠀⠀⠀⣀⠀⠠⠔⠒⠀⠂⠀⠀⢀⡀⠀⠀⠀⠸⡆⠈⠋⠀⠤⠀⠀⠀⠁
⠀⠀⢾⡿⣻⡗⠶⢦⡶⠶⠶⠶⠚⢋⣍⠡⢠⠀⠀⠀⠀⠀⣄⠀⠀⠀⣶⡆⠀⠀⠈⠁⣰⠂⠀⠀⠈⠲⠤⣀⡀⠰⠖⡠⠖
⠀⠀⠀⠛⠻⠷⠶⠍⢁⣀⣠⠤⠤⠤⠿⠚⠁⠀⠀⠀⠀⢄⠀⢤⡄⠀⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠘⢢⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠠⣄⠀⠀⠀⠃⠀⡤⠤⠀⠰⡀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⠀⠘⣿⢷⣄⡘⠀⠰⠀⠀⠑⣄⠙⢄⠀⠀⠀⠀⣿⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⢨⠊⡝⢧⣘⣀⣀⣀⣈⣿⡷⣌⣡⣄⠀⠀⠏⠀⠀⠀⠀⠙⠛⠿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡔⠒⠂⠒⠂⠸⢧⠙⢿⠟⣷⣾⡛⠙⢋⡉⢩⡍⣃⠀⢸⣧⣿⢤⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⠊⠀⡴⠛⠒⠂⠠⢼⣷⠟⠻⠋⡀⠴⠂⢚⠁⠠⠤⠇⠀⠀⠉⠁⠘⡗⠁⢑⠀⠀⠀⠠⠒⢢⡄⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⡄⠀⡄⠀⠀⣴⡟⠁⢄⠴⢊⣠⡴⢿⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢏⠀⢻⠀⠀⢸⡷⠀⠀⠁⠀⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠸⣷⣼⠏⣈⠶⢃⠔⠈⠻⣷⣟⣽⣧⡀⠀⠀⠀⠀⠀⢈⡇⠀⠁⠉⠀⠀⠀⠀⠀⠊⠀⠀⢀⣼⣿⡟⠃⠈⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠿⠩⠤⣍⠰⠁⠀⠀⠀⠈⢿⣦⡙⣷⡄⠀⠀⠀⠀⠈⢃⠀⠀⢰⠶⣄⠀⠀⠀⠀⠀⠀⠈⠁⡌⠓⠀⣠⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⣦⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⣷⢾⣷⠀⠀⠀⠀⠀⠈⢤⠀⠀⠦⢀⠆⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣹⣧⠀⠀⠀⠀⠀⠀⢱⣤⣀⠀⠀⠀⠀⢀⠀⠤⠊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣙⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠛⢻⠏⠁⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⠖⠒⠒⠊⠉⠉⣿⡟⢿⡗⠢⣤⣄⣀⡀⠸⣿⡟⠀⠀⠰⠎⠀⠀⠠⠤⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢫⡝⠣⠶⢶⣤⣄⣀⡀⠀⠀⣿⣧⣼⡇⠀⠀⠈⠉⠙⠲⠿⣥⣀⡀⠀⠀⠉⠁⠀⠀⠀⠃⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢸⡇⠀⠀⠈⠁⠘⠛⠩⠶⢶⣿⣧⣼⡇⢀⣀⣀⣠⣤⡤⠔⠒⠛⠋⠑⢄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⢻⡅⠀⠀⠀⠀⠰⡄⠀⠀⠈⠁⠐⢋⢿⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⢀⠀⠀⢠⣽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠲⡖⠒⠊⠀⢡⡄⠀⠀⠀⠀⣾⠀⠹⣆⠀⠀⠀⢀⣀⣠⣤⣤⣴⠒⠚⠃⠀⠀⠀⠀⠈⢸⡏⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠑⠀⠀⠀⠀⠈⠐⢦⠀⠀⣿⠂⠀⠙⠷⠚⠉⠙⣿⣿⡏⠉⢡⣄⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡏⠀⠀⢰⡀⠀⠀⠀⠠⠊⠁⠀⠀⢸⠇⠀⠀⠀⠀⢀⣬⢻⣿⠟⢻⡁⠉⠙⠚⣤⡉⠉⠉⠁⠘⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠈⠉⠉⠉⠲⣤⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣿⢰⣯⣼⠀⢀⠉⠁⠂⢰⡄⡸⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢦⣄⣀⡀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠿⡌⠛⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢦⣄⣀⣸⡅⠀⠀⠀⠀⢀⣀⣠⠄⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⠖⠈⠉⠁⠀⠀⠈⠦⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠤⢤⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⠠⡤⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⢩⠀⠀⠀⠀⠈⢉⠉⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣸⠀⠀⠀⠀⠀⠀⠀⣇⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠁⠙⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣸⢿⣶⠟⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠛⡈⠉⣿⣿⣦⣤⣼⠀⠀⠀⠀⠀⣤⡊⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠘⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⡈⠀⠈⢺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⣿⡿⣿⣿
⣿⣤⣶⣴⣴⣦⣾⣤⣼⣧⣤⣼⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣷⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣷⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢠⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣼⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠛⠇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠒⠀⠀⠀⢠⠖⠠⠀⠠⠀⠀⠸⡄⠀⠃⠁⠀⡿⠃⢴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠄⠀⠀⠀⢀⠀⢀⠄⠀⠘⠁⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠀⠀⠀⠉⠀⠀⠀⠋⠓⢀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠀⡆⠀⠰⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠐⠀⠀⠀⠒⠁⠀⠤⠀⢈⠀⠀⠄⢀⠄⠀⣴⣶⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⡤⠴⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠑⠀⠀⠀⠀⡀⠸⠇⠀⣠⠀⠌⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠈⠁⠀⠋⠁⠀⠀⠀⠀⠀⠀⠈⢀⣀⠀⠳⠀⠈⠀⠀⣀⠀⠃⠀⣀⠀⠷⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠖⠛⠁⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⠰⠶⠛⠀⠀⠈⢉⣡⡀⠡⠆⢠⠀⡗⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠰⠄⠀⠐⠈⠁⠁⢠⠒⠂⣀⡀⠘⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⠄⢠⡀⠙⠀⠙⢅⠀⠸⣦⠀⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠄⠀⠀⠀⠉⢁⡀⠀⠃⣶⡄⢀⠀⠀⠀⠀⢀⡀⠀⠀⡄⠈⠇⠠⠠⠄⠃⠀⠺⠃⢈⠁⣀⠀⠀⠈⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡤⠀⢀⣧⣀⠉⢁⠘⠻⠟⠀⠁⠀⢀⣀⠀⠄⠀⢀⡎⠀⠓⠂⠀⡀⣨⣁⠀⣦⡄⠙⠂⢹⠄⠐⠷⠄⢿⣷⡆⠂⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠒⠀⠀⠀⠲⠒⠌⠉⠀⠀⠛⠉⠀⠈⠉⠠⠘⣦⡟⠁⠀⠒⠄⠀⡏⠀⠙⢧⠌⠁⢰⡄⠘⠆⢰⡆⠀⠀⡿⢁⠀⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠘⠀⠀⠲⠆⢀⣄⠉⢀⡀⠀⣷⡦⠀⢀⠋⠀⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠳⠀⠲⠆⠛⠉⢀⠈⠁⡀⠁⣠⠤⠟⣁⡀⠀⣤⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠐⢦⠀⠾⠓⢀⠀⠣⠀⠀⠨⠄⠘⠉⣡⣄⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠄⠸⠆⠀⠄⠀⠴⠌⡀⠀⠀⠀⠰⠂⠀⣠⣄⠀⠀⠄⠀⠀⣇
⠀⠀⠀⠠⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠂⠒⠒⠐⠓⠒⠒⠒⠒⠒⠀⠒⠀⠒⠒⠒⠒⠲⠂⠀⠤⠀⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⣤⢰⡾⠛⠃⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠏⢸⣧⣤⡀⠀⠐⡆⠃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟⣠⠀⢀⡝⠻⣶⠀⠀⠈⡇⠀⠘⠁⠀⠀⠀⣀⠀⠄⠀⠒⡀⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠹⠄⢸⣇⢀⠞⠰⣦⠐⠁⠶⣆⣒⠠⡌⣥⠀⣾⠈⣷⠰⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠐⢘⠛⠏⠉⢲⡆⢠⠃⣿⠀⠀⢹⡆⣿⢤⣏⠀⠈⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⡄⡐⢂⣤⢉⡔⠐⣆⢸⡇⢸⠤⠀⠀⠀⢿⡎⠀⣿⠘⢘⡀⠃⣿⠀⠻⣄⡀⠀⠏
⠀⠀⠀⠀⠀⠀⣬⠁⣶⡀⢸⢹⡇⠀⣿⢸⠁⡄⣿⢸⡇⠸⡀⠆⠀⠀⢸⡇⠀⣿⠀⣘⣂⠄⡉⠤⠀⠀⢋⡉
⠀⠀⠀⠀⠀⠀⢹⡆⠻⣇⠇⢸⡏⠻⣿⢸⡄⠀⡟⢸⡇⠺⢀⠀⠀⠀⠈⢥⣄⠒⠈⡍⢡⢆⣬⠆⠀⠈⢀⠡⡀
⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⠀⢸⡇⠀⠿⠈⣓⠈⠤⠄⣀⢉⡉⢠⡆⠲⢸⡇⢸⣧⠀⡇⣿⠀⢀⡄⢀⣀⣠⠤⠤⣦
⠀⠀⠀⠀⠀⠀⠀⠙⡀⠁⠤⠄⡂⣀⠈⢵⠂⠲⠀⠀⣿⢀⡟⢸⡇⡄⢸⡇⢸⠘⣧⡇⢿⠀⢸⡇⠈⣿⡇⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠄⠀⣼⠁⠸⣇⢸⣤⠄⠀⢀⢿⡜⣶⢸⡇⠀⣼⠁⣼⠀⠘⢧⡈⠁⠈⠀⠀⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣿⠸⢁⡏⢸⠀⠀⠀⢇⠘⢃⠟⢘⢓⠠⠤⠞⠐⠂⣨⣄⠈⢀⣾⠃⠀⣿⡧⠴⡇
⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠻⡄⠸⠁⠚⢂⠀⠤⠀⢐⡂⡉⠰⡄⢠⠎⠈⢷⡄⢸⣿⡄⡜⣿⡄⠀⣿⡇⠀⣠⡶⢦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⢀⣒⠈⣡⢄⠀⡖⢹⡇⠀⠑⢷⣄⠀⣿⢀⣄⢸⡏⠙⠙⢻⠁⣿⡇⠀⣿⣿⠀⠻⢗⣾
⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⡄⢹⣷⠀⣿⡆⢰⠁⢸⣇⣰⠀⡀⢹⡟⣿⠀⠇⢸⡇⣈⢀⣹⡇⣹⡁⣠⣿⣷⣶⣖⣿⠅
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣇⠀⣿⡄⣽⣷⡘⠀⢸⡇⠈⢀⡷⠮⡀⢻⡄⢀⡞⠁⠼⢀⠉⠀⠩⠑⠂⠉⠁
⠀⠀⠀⠀⠀⠀⢸⠗⠚⢻⣿⡀⢸⣷⠀⢿⠇⠀⢸⡇⠀⠘⠃⣠⠃⠠⠉⣁⠀⠀⠁⠀⢀⠄⢀⡐⡄
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠈⣿⣧⠀⠏⢀⠘⠀⠤⠉⠈⠉⠙⠋⠑⣲⠀⠀⠳⢀⠄⢀⡴⠃⠀⠘⠮⠔
⠀⠀⠀⠀⠀⠠⠗⢀⠀⠄⠈⠻⠷⠆⠀⠀⡠⠀⠀⠤⢄⣀⡃⠓⢶⣾⠈⠻⠿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⣹⠆⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣋⠁⠀⠀⠀⢢⡀⠀⠀⠀⠀⡀⠂⠐⢀⠀⠀⠀⠠⣠⠀⠀⠀⠀⠀⠉⢠⡈⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠐
⠀⠀⠀⠀⢰⢄⠀⢀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⠀⠑⢤⠀⠈⠃⠐⠄⠀⠀⠀⠀⠀⠑⠠⠐⠐
⠀⠀⠀⠀⣀⠀⠀⠘⣃⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠰⢢⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣷⡾⠛⠀⠀⠀⠈
⠀⠀⢰⢔⣽⠄⢀⣼⡿⣶⡄⠀⠹⠛⠿⣷⣦⠀⠀⠀⠀⠁⠀⠀⠀⠐⠈⠋⠁⠀⠀⠀⠀⠱⠒⠀⠀⠀⠈⢿⡽⠷⠛⠉⠀⠀⠀⠀⠀⠀⢶⣢⠀⠀⠉
⠀⠀⠀⠉⠁⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠁⠀⠘⠀⠢⠄⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉
⠀⡠⠖⠀⠀⠀⠀⣿⣿⣻⣿⡆⠀⠀⢀⣰⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⣤⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠈⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⡄⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠈⠃⠀⠀⠼⠂
⠀⠀⠀⡀⠻⣄⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠬⣤⢁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠁
⠤⢄⠀⠀⠀⠐⠀⠀⠀⣠⣶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠰⠀⠀⠀⠀⡀⡤⢦⡄
⣰⠀⠌⠁⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⣆⠰⠂⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀
⠀⠖⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠉⠀⠀⠀⠀⠤⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡎⠀⠀⠊⠂⠀⠠⠀⢀⡀⠑⠕⠢⢀
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⣲⣿⣾⢁⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⡴⢂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠀⠀⠋⠚⠁⣿⣀⣠⠟⢚⡷⠶⢀⡤⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠐⠋⠤⣀⠐⠀⠀⠀⠀⣀⠠⠖⠨
⠀⠀⡄⠀⠀⠀⢀⡀⢀⠀⠀⠀⢰⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠁⠠⠎⠀⣰⠟⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠠⠀⠀⡁⠀⠂⠀⠀⠀⠀⠐⠆
⠔⠈⠀⠀⠀⠀⢸⣥⡛⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⢂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢀⠈⠀⠀⢀
⢀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣤⣤⣤⣤⡄⠞⠃⠘⠀⠀⠀⠀⠱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣄⡤⠀⠀⢀⣾⣄⣀⠀⠀⠀⠀⠸⠷⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾
⣿⣿⣯⣿⣿⡇⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⣯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⡀⡖⠀⠀⢀⣾⣿⣿
⣿⣿⣿⣿⡿⠁⠀⣀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠁⠄⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⣉⣡⠔⠂⠀⠀⠹⣿⣿
⠛⢉⡉⠉⠀⠀⠀⠜⢀⠂⠀⡄⠀⠤⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⡠⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠉⠀⠐⠇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠲⡄⠐⠀⠀⠄⠀⠀⠀⣠⠁⠀⠀⠠⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⢠⠀⢒⠀⠀⠀⠀⠐⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠈
⠀⠡⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠐⠂⢀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠠⠀⠀⠇⠀⠌⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤
⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠞⢀⠈⡁⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠂⠸⠀⠀⠀⠄⠀⠀⠀⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⢀⡶⣆⣀⠐⠃⠀⢀⢰⣯⡴⢇⡴⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁⠀⠀⠀⠀⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠈⠁⠓⠆⢹⠄⠀⣴⣟⣸⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠠⣤⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢈⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠸⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠐⠂⠀⣉⢀⣄⣁⣐⣲⣦⠤⣴⠶⠶⢤⣶⢟⡻⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂⠀⠉⠀⣀⠤⣶⠋⠉⠁⣠⠊⠀⡀⠔⡈⠵⠋
⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠉⠀⠀⣀⢀⣤⠴⠋⠀⡐⠁⢀⣴⠟⠁⠀⠘⣡⠊
⠀⠀⠀⠀⠀⠀⠎⠃⠀⠀⠀⠉⢀⣠⠞⠁⠀⠀⠴⡠⢀⠟⠁⠀⠀⠶⣄⠇
⠀⠀⠀⠀⠀⠘⡸⠂⠁⠀⡠⠒⣡⠊⠀⠀⣠⠴⢊⡴⠋⣄⠲⠄⠓⠈⡟⡀
⠀⠀⠀⠀⢠⠞⣁⠀⠚⠁⠠⠄⡁⠠⠀⢈⡡⠐⢿⣍⠖⠀⡀⠀⠀⠀⣺⣍⠛⠛⠓⢒⣀
⠀⠀⠀⣴⣇⣀⣀⣠⣶⣟⡅⠠⠤⠒⠚⠋⢳⣤⡼⠇⠀⣀⡀⠀⠀⠠⠉⠀⣷⣀⠀⠀⠉⠓⠦⡀
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡥⠸⢇⠀⡙⠁⠀⠀⠀⠀⠀⢀⡿⢟⠓⢤⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡨⠄⣀⡼⠉⠑⠀⠀⢀⠀⠀⢀⡼⠀⢾⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡈⠉⠙⡶⢾⠀⠠⡜⠀⢀⡼⠁⠀⣾⢷⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢙⣶⠊⠑⡔⠀⢠⡇⠠⠎⠀⠀⢴⣟⠀⠑⢄⣀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠙⡦⠤⠃⠀⠘⢀⠀⠀⠀⣠⠋⢸⡅⠀⠀⠈⠲⣂⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⡌⠁⢲⡀⠀⠀⡟⠀⠀⢀⠇⠀⢻⡿⡇⠀⠀⠦⠘⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢳⣦⡌⣱⠀⠀⣧⠀⠀⠉⠀⠀⠼⠓⡀⠙⢀⡑⠀⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢂⣿⠀⠀⣇⠀⠀⢹⡄⠈⡅⠀⠀⡄⠀⢿⠂⠈⢢⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣽⠂⠚⣀⠅⠀⠀⢣⠐⡃⠀⠀⠀⠀⣼⣇⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠛⢤⡴⠁⠀⠀⠀⠈⠳⣆⠀⠘⠁⠀⣰⡧⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡜⠁⠒⠀⠀⠀⠀⠀⠈⠱⢦⣄⠀⠈⣇⠈⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⢳⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⠛⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣟⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠦⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢸⠀⠀⠹⣯⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣟⣿⡆⠀⠀⠀⠘⢆⣀⡀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⠘⠿⣽⡯⠁⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣹⣷⣯⣛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⢋⣼⢛⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢭⣾⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣄⠀⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⡿⠿⠿⡿⢿⡹⠿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠛⡻⢛⣿⣿⣿⣿⡿⠻⠿⢿