Dialog Emojis & Symbols

⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠛⠛⠛⠛⠛⢹⠀⠀⠀⢿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠓⠒⠒⢄⣶⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⢦⣠⠔⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡤⠤⢤⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡵⠒⠚⢢⠇⠀⠉⠉⡟⣠⠔⠋⠉⠉⠉⠉⠀⠀⡴⢚⠙⠛⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠑⠛⢰⣿⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇⣞⢲⡤⠤⢼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠦⢍⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡷⣟⣥⢀⣟⣸⡄
⠀⠀⠀⠀⠀⠀⣀⠴⠚⠙⢮⣽⣯⡏⠙⠦⡀⠀⠀⢠⠤⠤⠔⠿⢥⠀⠀⠈⣩⣽⠇⢸⢦⣅
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⣿⢿⠀⠀⠀⠹⡄⠀⠘⠀⠀⠀⠀⢸⠀⠀⣰⠃⠻⣷⠟⠉⠙⢦
⠀⠀⠀⠀⢰⠃⠀⣄⠀⠀⠀⣿⢽⡀⠀⢰⠀⢹⡀⠈⠀⠀⠉⠉⠉⠀⢀⡇⡼⠘⠃⠀⠀⣰⡈⡆
⠀⠀⠀⠀⡟⠀⢰⢿⠀⠀⠀⣟⣻⡇⠀⢸⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⢠⠧⡄⠳⠤⢴⡇⡇⡿
⠀⠀⠀⠀⡇⠠⣏⢸⠀⠀⠀⡧⣖⡇⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠸⡆⡇⠀⠀⠘⡿⢡⠇
⠀⠐⠒⠒⠻⡄⠘⣾⠶⠖⠒⠛⠛⠓⠒⢲⠟⢀⡗⠒⠒⠒⠒⠒⠒⠒⠊⣇⢻⠓⣶⣶⢞⡵⠓⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠙⢦⣞⠛⢻⣶⡄⢀⡤⣴⣃⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣽⡷⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠘⠻⠆⠛



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡒⠐⠲⡄⠀⢀⡔⡋⢙⢦
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⡇⠀⠘⣀⣀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠤⠤⠤⠤
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠄⠠⠃⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⣄
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠁⠈⠀⠒⠂
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠀⠠⠴⠖⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠤⠤⠖
⠀⠀⢀⠄⠶⠂⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉
⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⡀⠀⠐⠠⠀⢀⠀⠀⠀⠀⠙⠦⠴⠦⠤⠄⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠰⠀⠀⠀⠀⠀⠀⢀⠑
⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⡰
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠦⠄⠀⠤⠖⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢁⠀⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⡃⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢈⠆⠀⠈⠀⠀⠀⠀⠀⠀⠈⠠⠴⡀⠀⠀⠄⠁⠀⠀⠀⠀⡀⠀⠀⠐⠁⠀⠀⠀⠤⠄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠀⠀⢀⠂⠀⠀⠐
⠀⠁⠀⠤⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠐⠒⠂⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⠎⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠐⢫⠷⡀⠄⠀⠀⠀⡞⠉⠀⠀⠀⠑
⠀⠀⠀⠨⡀⠀⠀⠀⣠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠨⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⢀⡠

⠀⠀⠀⠀⠀⠀⢰⠄⠠⠆⠰⠄⣀⡀⢠⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⢶⠊⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡀⠀⠀⠀⠘⢶⠄⠀⣄⠋⢦⠛⠆⡼⠄⣶⠟
⠀⠀⠀⠀⠐⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡀⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⢉⣷
⠀⠀⠀⢀⠤⠾⢤⠀⠀⠀⠀⠀⠀⠀⢰⡟⠂⠠⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⠉⠓⠦⡄⠀⠀⠀⠀⠀⠀⠀⢾⣿⠆
⠀⠀⠀⠐⠀⠊⠸⠖⠿⣦⣴⣦⣴⡟⠛⠓⠐⠀⠀⠀⠀⠈⠒⠚⢦⠀⠀⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⡀⢰⣤⣄⡙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠟⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠚⠋⠹⠛⡼⠛⣾⠋⠁⠉⠉⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠁⢀⡀⠀⣠⡟⢢⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⠆⠱⠤⠦⢖⡠⡄⠀⠀⠀⠀⠀⠀⢀⣤⣺⠴⠖⠲⢸⢧⡖⣀⡤
⠀⠀⠀⠀⠀⠀⠻⡍⠁⠀⠀⠈⠉⠉⢧⠀⠀⠀⠀⠀⠀⠀⠈⢶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⡲⠒⠀⠀⠠⢄⣿⡞⠀⠀⠀⠀⠀⠈⣵⡋
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢀⣱⠦⠄⠐⠂⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂⠀⠀⢠⠈⠼⡃⠀⠀⠀⠀⠀⠀⣻⣧
⠀⠀⠀⠐⠂⠐⠉⠛⣦⣀⠴⠶⢦⣰⠏⠀⠀⠀⠀⠀⠀⠀⠴⠒⠢⣄⢾⣤⢆⡶⠶⣴⡛⠓⠚⠂⠀⠀⠀⠁⠀⠱⣤⡀⠀⠀⣀⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠙⠓⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢀⠋⠉⠉⠁⠘⠢⢤⡄
⠀⠀⠀⠀⠀⠀⣰⣤⣤⣤⣤⡄⣤⣤⣤⡾⠃⠀⠀⠀⠀⢠⡔⠁⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⣄⠀⠀⡀
⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠃⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⣱⠆⠀⠀⠀⠠⠄⢅⠈⣄⣀⠠⠤⠤⠤⠀⠀⢰⠆
⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠈⠙⠛⣻⠇⠻⠿⠿⠿⠿⠥⠄⢀⡀⠀⠀⠈⠉⠉⢳⠀⠀⠀⠀⠀⢠⡶⠟⠀⠀⠀⢡⣀⣀⣀⣀⠀⠤⠤⠤⠤⣤⠐⠾⢓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠸⠶⠟⠷⡀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⢶⠀⣒⣶⡿⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⣀⣀⡠⠞⠀⠀⠀⠀⠀⠀⠀⢨⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠐⠒⠒⢲⡄⠀⠀⠀⠀⠀⠠⣌⠄⠀⠀⠀⠀⠀⠀⣀⡀⣀
⠀⠀⠀⠀⢀⠀⠀⡸⢦⣤⣄⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⢀⠂⠀⠉⠁⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠢⣴⠞⠁⠀⠀⠀⠀⠈⠳⣶⠊⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⠒⠤⣄⣀⠀⠀⠀⠀⠀⢘⣇⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⢠⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢟⣠⣴⣾⠛⠁⠀⠀⠀⠀⠸⣀⣀⣼⠀⠀⠀⢀⣀⣀⣤⣤⠾⠃
⠀⠁⠀⠀⠀⠀⠀⡷⠛⠻⣧⡿⣿⠿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣍⢉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠴⠦⢄⡀⣿⠉⠩⡔⠲⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠆⠀⠀⠈⠙⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠂⠀⠈⠁⠀⢄⠀⠀⠀⠀⠀⠄⠤⠠⠀⠄
⠀⠀⠀⡀⢠⠄⠀⡄⣄⢰⠁⠀⠀⠀⠀⢀⠠⠠⠀⢀⠀⠃⠁⣇⠀⠀⠀⠐⡂⠀⡄⡀⠀⠂⠂⡄⠀⠀⠋⠁⠀⡆⢠⠀⠄⢹⡀
⠀⠀⠀⣸⢸⠠⠬⠥⠘⢰⡤⠀⠀⠀⠀⡉⢆⣄⠸⠬⠸⠼⣄⣯⠀⠀⠀⠸⡇⠀⠃⠁⠃⠒⣰⠃⠀⠀⠴⣤⠠⠴⠾⠰⢓⣠⡏
⠀⠀⠀⠈⠓⠳⢦⠴⠖⠾⠀⠀⠀⠀⠀⠠⣴⣶⣀⣴⣤⡴⠋⠁⠀⠀⠀⠀⠈⠛⠦⠴⠶⠚⠁⠀⠀⠀⠀⠈⠛⠛⠛⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢤⢀⡀⡀⢠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠠⠆⡘⠀⡀⢀⢁⢰⠂⢲⠀⠀⠀⢀⡀⠀⠈⡀⢈⠀⠀⠀⠀⢀⡀⠀⠇⠘⡁⠌⡐⢢⡆⢀⡀⠀⡀⡎⠆⠄⣄⢠⠈⡄
⠀⠀⠀⠀⣷⠀⣃⡁⢌⣠⣤⠬⠧⠀⠀⠈⡆⢸⠀⡇⢸⢀⣸⢃⡀⠀⠀⢸⡘⢠⣡⣠⡡⣀⣘⠀⠄⠀⠰⣷⡶⠄⣄⣘⠐⠏⢇⡐
⠀⠀⠀⣀⠙⣋⣤⣄⣠⣤⠏⠘⠀⠀⠀⠈⠺⡎⣂⡈⢉⣁⡙⠁⠄⠀⠀⡀⠨⠑⠆⠴⠷⢜⠀⠠⠀⠀⠀⠀⠑⠶⣤⠶⠼⠛⠊
⠀⠀⠀⠀⠀⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠁⠀⢀⠘⢰⠀⠄⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⠀⠈⢀⠤⠀⠀⠀⠀⢀⠄⠔⠒⠢⣠⣀⡀
⠀⠀⠀⠀⠀⡀⠄⠀⠘⡄⣀⠜⣀⠀⠀⠀⢠⠇⢀⠀⡀⠄⠔⠤⠀⠀⠀⠀⠀⠰⢀⠃⠠⢌⢦⠇⢀⠀⠰⠘⠉⠉⡍⢀⠀⠀⣆
⠀⠀⡀⠘⢇⠁⣠⡣⠄⢊⣁⡜⠁⠀⠀⠀⣏⠀⣘⣀⣣⡸⠄⢼⠇⠀⠀⠱⣥⡁⠦⡸⠤⠒⣸⡇⠐⠂⠈⠱⣖⡂⢑⡈⢈⣁⠏
⠀⠀⠀⣤⣶⣩⣥⣠⣤⠛⠉⠀⠀⠀⠀⠀⠈⠓⢤⣀⣠⠶⠒⠀⠀⠀⠀⠀⠀⠣⢤⠴⠶⠾⠋⠁⠀⠀⠀⠀⠈⠙⢩⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠄⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⢀⠄⠀⠀⠀⠠⠁⠀⢄
⠀⠀⠀⠠⠈⠀⢀⡀⡄⠬⠀⠀⠀⠀⠀⠈⠀⠀⡀⠄⡀⠠⡄⠁⠤⠀⠀⠈⢣⠀⠀⠈⠁⡠⡀⠀⠀⠀⠨⠀⢠⠀⠄⡀⠆⠀⡄
⠀⠀⠀⠠⣐⠐⢎⡇⡅⣀⣀⡌⠀⠀⢀⠀⡀⢃⢁⡀⣇⠀⠇⢠⠄⠀⠐⢚⠀⠇⠸⠀⡇⠁⣤⠂⠀⠠⡀⠄⠨⡘⢠⢃⣀⡄⠇⡜
⠀⡀⠀⠀⠘⣇⣤⣄⡀⣠⡴⠃⠀⠀⠐⠳⢮⣈⣭⡉⠀⢀⣤⠞⠀⠀⢀⣺⠧⣍⣉⡉⢩⣽⠃⠀⠀⠀⠙⠶⣄⠀⣀⠀⢀⡰⠊
⠀⠄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉
⠀⠂

⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣧⠀⠀⠀⣰⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣠⣤⣄⣀⣠⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⢀⣀⣴⣿⣷⣴⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⢿⠿⣿⠛⠋
⠀⠀⣴⣿⣶⣾⣿⣶⣿⣿⣷⣶⣶⡄⠀⠀⠀⢀⣀⣴⣿⣿⣷⣤⣤⣄⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠉⠉⠹⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⡿⣿⣿⣿⡿⠿⠃⠀⢀⣸⣿⣿⣿⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠙⠋⠀⠀⠀⠘⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣾⣿⣶⣤⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⡿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣠⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠘⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠃⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣛⣿⡿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢿⣿⣿⡟⠋⢿⡯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⢠⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣴⣶⣦⣀⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡿⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠿⠿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠙⠿⠿⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠐⠂⢄⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠉⠁⠢⠄⢀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠉⠀⠀⡀
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⡀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠈⠂⠀⢀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠁⠂⠐⢄⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⡤⠈⠠⠄⠐⠈⠀⠀⠀⠀⠀⠀⠀⡠⠃⢀⠀⠠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠤⠄
⠀⠀⠀⢀⠠⠀⠔⠈⠀⠀⠁⠂⠀⢄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠄⠠⠂⠉⠑⢀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠁⠈⠑⠄⢀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠒⠈⠈⠀⠁⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠈⠒⠠⠀⠀⠈⠀⠀⠀⠈⠓⠀⠀⠀⠀⠈⡦⠄⠐⠈⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠁⢁⠂⠂⠐⠀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠉⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠐⢄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠑
⠀⠀⠀⠔⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠠⡀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠀⠀⢄⠀⠀⠀⠀⡀⠀⠐⠁⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠂⢂⠧⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⢨⠔⢀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠂⠄⠀⠄⠠⢄⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠈⠢
⠀⠀⠀⡠⠐⠀⠊⠀⠀⠀⠀⠐⠂⠂⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠄⡀⡀⠀⠀⠀⡠⠀⠀⢀⡀⡀⠂⠀⠀⠀⣰⢀⠀⢀⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⢀⡀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠄⠀⠈⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠁⠁

⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠋⠁⠀⠀⠀⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿
⣿⣷⣶⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣄⡀⣤⣀⣀⣠⣿⣿⣿⡇⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠉⠙⠲⣄⣀⣤⣤⣤⣤⣄⡀⠀⠀⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠁⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠛⠋⠉⠙⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍⠁⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣆⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣀⠀⠉⠉⠉⠀⠈⣿⣿⣷⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⢀⡤⠒⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠓⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡂⠉⠁⠀⠀⠀⠀⠀⠉⠝⠣⣆⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢠⠂⠀⢲⡄⠀⠀⢀⠊⠀⠀⠈⠳⡄⠀⢀⠠⠄⡀⠀⣿⠀⢘⠈⠉⣿⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢠⠃⠀⠀⠀⡔⡚⠃⠀⠉⣉⣹⡆⠀⠀⠀⠀⠀⠀⠘⡄⢨⡀⠀⡀⠀⢈⣁⡉⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇⠀⠀⠀⡞⠉⠉⡀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⣃⢀⣿⠁⢨⠀⠄⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⣀⡀⠀⠈⣑⣘⣿⠈⢄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠋⠀⢀⠀⢀⠃
⠀⠀⠀⠀⠐⡄⠀⠀⠀⠈⠙⣇⠀⣹⡏⠉⠉⠀⠀⠑⠄⡀⢀⣲⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠐⣀⡡⠔⠁
⠀⠀⠀⠀⠀⠈⠢⢅⠀⠀⠀⠄⠉⠉⠀⠀⠀⠀⠀⠀⣀⡠⠝⠉⢃⠀⠈⠉⠉⠁⠒⠒⠂⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠉⠉⠂⠀⠀⠇⠀⠀⠀⠈⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣀⠀⠀⠀⡠⠁⢀⠎⠀⠀⠀⠀⠀⠘⢄⢃⠀⡠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣴⠦⠈⠆⣀⣔⡡⠔⠁⠀⠀⠀⠀⠀⠀⡤⢆⢙⠂⠁⠀⠤⣌⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡜⡋⠀⢣⡀⠀⠑⡘
⠀⠀⠀⠀⠀⠀⢀⠤⠰⢿⡖⠈⠀⠀⠀⠀⡠⠙⠄⢀⠀⠀⠀⠀⣀⡕⡖⡷⡄⠀⠉⣲⠾⠛⢦⡀
⠀⠀⠀⠀⠀⡰⠃⠄⠒⠠⠙⠶⣦⠤⣴⣯⢴⠒⠋⠘⠀⠀⠀⠰⢁⡉⠓⢦⣽⠔⠊⢀⠀⢔⠄⠙⡄
⠀⠀⠀⠀⢠⠁⠈⡀⠔⣣⣔⡂⢈⠉⢁⣊⡴⠚⠑⠊⠀⠀⠀⠘⠒⢬⣊⢔⣔⣁⡀⣄⣥⢕⣕⡡⢸⡀
⠀⠀⠀⠀⠇⢠⡡⠔⢁⡏⢠⡾⠷⠛⠛⠀⠄⠀⠀⠠⠤⠤⠤⠤⢤⢀⢨⣭⣿⣿⣿⣏⢕⣷⡫⢞⢽⡇
⠀⠀⠀⠀⠀⢅⠂⢀⠶⠤⣀⣙⠋⠁⠑⠀⠒⠒⠲⡶⢲⡖⠒⠒⠺⠏⠦⠤⠷⠾⠠⡅⡬⣨⢄⢥⣼⡇
⠀⠀⠀⠀⢇⢐⠠⠀⠈⠐⠄⡈⠉⠓⠤⡀⠀⠀⢠⡗⠀⡇⠀⠀⡔⢒⢒⠀⡀⡙⢋⡡⢈⢔⢡⠾⢺⠇
⠀⠀⠀⢰⠚⠓⢻⠶⠦⣦⢥⣦⡃⠄⡺⠀⠀⠀⠠⠉⢺⡇⠀⠀⠇⠐⡡⡢⠢⡰⠋⢔⢕⠅⢁⣔⡝
⠀⠀⠀⠘⣶⢲⠚⠛⢛⣿⡾⠅⡂⢽⠁⠀⠀⠀⠠⠀⢸⠃⠀⠀⢰⠨⠊⡨⢊⣖⢗⢗⢟⢛⠛⠉⠋⠈⡃
⠀⠀⠀⠀⠋⣾⠀⠀⡾⣼⠀⠅⡲⠇⠀⠀⠀⠀⢰⠀⢻⠀⠀⠀⢸⡆⠋⠀⠑⢻⡟⠉⠉⠉⠉⠉⢹⢻⡇
⠀⠀⠀⠀⠄⣿⠀⢸⣻⡇⡀⠅⢺⠁⠀⠀⠀⠀⣨⡀⣽⡆⠀⠀⢸⣿⡀⠕⠀⡨⣧⠀⠀⠀⠀⠀⢈⢼⡇
⠀⠀⠀⠠⠦⠿⠤⠾⢺⢥⣢⣄⣬⣷⣤⢀⣀⡜⠁⠈⢥⣟⢦⣴⣿⣟⣁⣅⣤⣰⣿⡦⡀⢄⠄⡀⣀⣹⣧⣀⢀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠁⠈⠈⠙⠙⠋⠛⠉⠙⠉⠊⠁⠉⠈⠀⠉⠉⠈⠈⠁⠁⠈⠉⠈⠀⠀⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠶⠖⠒⠚⠛⠋⠛⠛⠒⠒⠒⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠉⣀⡠⠤⠶⠖⠛⠛⠉⠉⠉⠉⠉⠉⠐⠒⠲⠤⢤⣈⡑⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⣀⡴⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⣀⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣍⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣄
⠀⠀⠀⠀⠀⣴⠋⣠⠞⠁⠀⠀⠀⠀⢀⣀⡤⠤⠒⠖⠂⠒⠒⠒⠂⠢⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠢⡀
⠀⠀⠀⢀⡞⢁⡴⠃⠀⠀⢀⡴⠶⠛⣉⡤⠶⠒⠚⠛⠛⠙⠛⠛⠛⠓⠒⠢⢤⣀⡁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡑⡀
⠀⠀⢀⡞⢀⠞⠀⠀⣠⠞⢉⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡘
⠀⠀⡜⢀⡞⠀⢠⡞⢁⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑
⠀⣼⠁⡞⠀⣰⠋⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣧⣀⢀
⢠⡇⢸⠃⢠⠇⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠋⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠂⠄
⣸⢃⡟⠀⡎⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋
⣿⢸⡇⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⣿⢨⡃⠘⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢿⢸⡇⠀⢿⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⢸⡌⡇⠀⠈⢷⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠸⡇⣧⠀⠀⠈⢳⡄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢻⢹⡄⠀⠀⠀⠙⢦⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠈⠆⢇⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠰⠁
⠀⠀⠘⣮⣆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⣙⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠠⠃
⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢄⡀⢫⠉⠀⠈⠙⠒⠦⡄⢀⣀⡀⠀⠀⢀⣀⣀⣠⡤⠶⠚⠉⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣧⠀⠀⠀⠀⠀⠀⠀⡹⢁⡴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⣤⣄⣤⣄⣀⣀⣀⣀⣀⣀⠤⠤⠐⠒⠉⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠉⠙⠛⠛⢿⣿⠀⣿⣿⡟⠋⠁⠀⠙⠙⢻⣿⣿⣿⣿⣿⡇⢸⣿⡟⠛⠛⠋⠉⠙⠉⠻⠿⢿⣿⣿⣿
⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⣿⣿⣶⣄⣀⣤⣤⣀⣤⣀⠀⠲⣶⣿⣿⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⢸⣿⣧⣀⣀⠀⣀⠀⢀⡀⢀⣼⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡙⢻⣿⠀⣿⣿⣿⣷⣶⣶⣤⣶⣷⣾⣅⣨⠛⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠛⣿⣿⣿
⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠾⠿⠿
⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢰⣶⣶⡶⠖⠒⠶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⣿⣿⡿⠛⠛⠋⠀⠀⠙⠛⠛⢿⣿⣿⣿⠀⣿⣿⡿⠛⠁⠀⠈⠉⠙⠛⠻⠿⢿⣿⡇⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⣿⣿⣦⣀⠀⠀⠀⠀⢀⣀⡠⠴⣿⣿⣿⠀⣿⣿⣷⣤⣤⣀⣀⣤⣄⣀⡠⠒⢾⣿⡇⢸⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⠹⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣙⡇⢸⣿⣿⣿⣦⣤⣤⣶⣶⣶⣿⣤⡜⢻⣿
⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣾⠻⣦⠀⠀⠀⠀⠀⠀⣼⠋⠸⡇⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⣿⠀⠘⢧⡀⠀⠀⠀⡼⠃⠀⠀⣿⠀⠀⠀⠀⣠⡾⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣦⡀⢀⠈⣿⠀⠀⠈⢳⡌⢀⡾⠁⠀⠀⠀⣿⠀⠀⣠⡾⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠈⠻⢦⡰⣿⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⢸⣀⡾⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣠⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⣸⡇⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⢰⡟⣀⣀⡤⠶⠛⠉⣵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡶⠿⣶⣤⣍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡘⢛⣩⣥⣤⣤⡴⠶⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢈⣳⣄⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⡄⣴⠶⠶⠞⣷⠀⣸⠟⠉⢹⡇⣿⠁⠀⣿⡿⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⣤⠈⠁⢨⡴⠶⠶⣶⣰⠟⠛⢻⣿⠉⠉⠁⠀⠀⣾⣱⠇⠀⡀⠀⢻⡄⣿⠀⠀⣼⣷⡇⠀⢠⣿⡇⠀⢰⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣠⣴⡶⠄⣷⠀⠀⣿⡟⠀⢠⣿⡇⠀⢰⣶⣶⣾⣿⡟⠀⣸⣿⠀⢸⣇⡟⠀⠀⠈⠀⠀⠀⢸⣿⣇⣀⣾⠀⠉⠛⢛⣶⣦⡤
⠀⠀⠀⠀⠀⠐⡺⠿⢿⣿⣭⣄⣀⡀⢻⡄⠀⠘⠀⢀⡾⣿⠃⠀⠈⣁⣀⣾⣿⠁⠀⠛⠛⠀⠈⣿⡇⠀⢠⡶⣶⠀⠀⣿⣾⠛⠉⢻⡄⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠁⢠⠀⠁⣤⠉⠉⣿⠘⣧⠀⠀⠀⣼⠃⣿⠀⠀⠿⠿⠛⣿⡏⠀⣠⣤⣤⠀⠀⣿⣇⣀⣸⠇⣿⣤⣤⡿⠻⢦⠴⠟
⠀⠀⠀⠀⠀⠀⠒⢀⠰⠗⣀⠺⠗⣀⡦⣹⠇⠀⣸⠏⢀⣟⣀⣀⣠⣤⣼⣿⠶⠶⠟⠀⠻⠛⠛⠛⠋⠉⠉⠀⠁⠀⠀⠀⠻⣶⠒
⠀⠀⠀⠀⠈⠀⡀⠉⢡⣄⠙⣠⣤⣿⠡⠿⠖⠛⣛⣤⡌⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌
⠀⠀⠀⠀⠰⠀⠀⠶⠆⠁⠶⠈⣿⣁⣤⣴⣾⣟⣻⣿⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⣤⣤⣀⣀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠃⢀⡘⠃⣠⡾⢟⣭⠛⢃⣤⠿⢿⡇⠀⢀⣴⠿⣧⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⣇⠛⢈⣉⣟⢛⠿⠿⣦
⠀⠀⠀⠀⠀⠀⡤⠈⠡⡆⠉⢠⡆⠉⣴⡎⠋⣶⡿⠀⣠⡟⣻⣷⣿⠀⠀⠀⣼⣷⣄⠀⠀⠀⠀⢸⠛⣷⣄⠀⠀⠀⠹⣦⢨⡟⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠾⢀⡐⠿⢀⡰⠿⣸⣧⣾⣉⡿⢿⣉⢿⡆⠀⣼⣏⣹⡿⣆⠀⠀⠀⣿⠿⣀⠹⠷⣄⠀⠀⢻⠟⢰⣆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⣤⠈⠁⣤⠙⢣⣤⣿⢯⣬⠛⣡⣾⠛⣼⣇⣼⣷⡟⢻⣦⡝⢷⡀⢠⣧⣤⠛⢡⡄⠛⢻⣦⡀⠀⢸⡛⣦⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠆⠉⠰⠆⠉⢷⠎⠁⢶⢎⠱⣶⢈⣿⣷⢉⠱⡿⣉⠰⡾⢻⣼⡇⠁⠶⠆⠀⠶⠀⠀⠿⢦⣿⠀⠈⢧⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡀⠛⢀⡘⠛⣀⡘⠟⣤⠛⠃⣤⠻⢃⣤⠻⢃⣄⠛⢁⣜⠟⢹⡙⠛⣀⠘⠃⣀⠈⠃⠀⠀⠁⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠁⡤⠈⢠⣦⠉⢠⡤⠉⢴⡌⠉⣴⠌⠉⣶⠈⠁⣶⠈⠁⣦⠉⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠂⠀⠲⢀⡀⠟⢀⠐⠿⢀⠐⠗⣀⠐⠂⣀⠐⠂⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⢀⠈⠃⡀⠈⠁⣄⠉⠀⡄⠉⢀⡀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠓⠒⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄⠀⣰⠶⠶⠶⢶⡄⠀⠀⣴⠖⠒⠒⡖⢠⠋⡇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⣆⠀⡏⠁⠀⠀⠹⣤⠟⡆⠀⠀⠀⡇⢠⡎⡘⠀⠀⠀⡇⡏⢸⠃⠀⠀⠀⢀⡎
⠀⢠⣄⣀⠀⠀⠀⠀⣠⠞⠓⠶⣏⡁⠀⠀⠀⠀⠘⣶⠇⠀⠀⠀⠀⠹⡀⡇⠀⠀⠀⢹⣸⠀⡇⠀⠀⠀⣷⡇⢸⠀⠀⠀⠀⣼
⢀⡏⠀⠉⠙⠲⢤⡴⠃⠀⠀⠀⢀⡝⠂⠀⠀⠀⢀⣘⡆⠀⢠⣄⠀⠀⢳⡇⠀⠀⠀⠈⣿⢠⡅⠀⠀⠀⣿⠀⢸⠀⠀⠀⢀⡇
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢾⣿⣿⠀⠀⠘⠋⠀⠀⠀⢣⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⠀⣼⣄⣀⣀⣸⠁
⠈⢯⠉⠒⢢⠀⠀⠀⠀⠀⡴⡟⢆⠀⠀⠀⠀⠀⣠⣿⠀⡀⠀⢀⣄⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠓⣶⣶⣟⡁
⠀⠈⠓⣶⠋⠀⠀⠀⢀⡜⢡⣷⠘⡄⠀⠀⠀⠀⠈⠉⠉⠹⣀⣸⠀⠙⢦⣴⠋⠀⠀⠀⡖⠒⡦⠀⠀⠀⣧⢠⣾⠁⠀⠈⣷
⠀⢀⡼⠁⠀⠀⠀⢀⡞⣠⠏⠈⢧⠹⡀⠀⠀⠀⠀⢀⣀⣀⣫⣀⣰⠷⣴⢷⣇⣀⣀⣀⡷⢲⣇⣀⣀⣀⣿⢿⠘⠦⣤⡴⠏
⠀⠸⣥⣀⡀⠀⠀⡞⢠⠏⠀⠀⠘⣆⠳⠖⠒⠋⠉⠉⠀⢀⡾⠁⠀⠀⠀⠸⣥⣀⣀⣸⠃⢯⣤⠤⠤⣤⡏⠈⠛⠚⠉
⠀⠀⠱⣄⠉⠓⠺⢁⡟⠀⠀⠀⠀⠹⣄⣤⠤⠶⠒⠚⠋⠉⠁
⠀⠀⠀⠈⠙⠦⣤⠏

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿
⣿⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠀⠀⠀⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏
⣿⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣿⠀⠀⠀⢾⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢻⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠂⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣹⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡿⠛⠿⠿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠂⠀⠍⠀⠂⠀⠈⠀⡀⠀⠄⢤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠥⢈⡑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠿⠷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢠⢦⣀⠀⣀⠤⠔⠒⠉⠀⢀⣠⣴⡶⡆⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠙⠻⠶⣶⣤⣀⠠⠜⠋⠉⠀⣀⣤⠀⢰⠟⠛⠉⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡇⢀⣤⡴⠞⠛⠉⢩⠀⢸⡇⠀⠀⠀⠀⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠅⡂⠀⠠⢀⣾⠀⢸⣅⠀⢀⡐⠁⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢢⡇⠀⡤⠂⠸⠀⢼⡗⠀⠝⠔⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⢸⠌⠇⠀⠁⠀⠀⠀⣸⠁⠁⠎⠀⠸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠅⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠁⣧⠀⠎⠀⠀⠀⣾⡆⠀⠀⠀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⡇⢸⣧⡷⠤⠦⠠⠆⠀⠛⠂⠒⠂⢈⠥⢤⡄⢸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⡂⢘⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡄⢠⣤⣤⣤⣤⣄⡄⠀⡴⠴⠶⠶⠾⣐⡾⠇⢸⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠁⢀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⢸⠁⣦⡀⠀⠀⢀⠀⣷⠀⠠⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡀⢈⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⢸⢸⡆⠀⠀⠆⢸⠀⣿⠀⠀⠀⡐⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠈⡂⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸⠇⠰⠃⠄⠘⠀⣿⣈⠀⢬⠂⡀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠨⡀⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⢸⣤⠈⠀⢄⠀⠀⠀⣿⠴⠺⠣⠈⠁⠀⠀⢼⡗⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢾⣿⡲⢀⣠⣴⡇⠀⣿⠁⡐⠀⠂⡆⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠓⣂⡇⠀⣿⠀⠀⢀⡀⡇⠀⠀⣋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢰⢨⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⠼⠾⠦⣬⣀⠀⠀⡀⠀⠁⠀⠛⠳⠖⣀⠤⡇⠀⠀⢵⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠕⠄⠰⢸⠀⠀⡀⢸⠀⠀⠀⠀⡀⠤⡐⠪⢉⢁⣈⠀⠂⠀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠉⠁⠀⢀⢿
⠀⠀⠀⠀⠀⠀⡀⠁⠰⢸⠀⠀⠂⢸⣀⣠⠴⠊⠀⠁⠀⠀⠀⠀⠈⣁⢀⡀⠀⠒⠐⠀⠀⠈⠉⠉⠒⣢⣄⣀⠀⠀⠠⢸⣽⠀⢀⠐⠊⠉
⠀⠀⠀⠀⠀⢀⠈⡀⢸⣸⠀⢐⣩⡼⠿⠟⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠤⠄⠈⠉⠛⠲⢍⣸⣿⠠⣀⣀⠠⢤⣆
⠀⠀⠀⠀⠀⢈⣡⡤⠚⠛⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠈⠄⢐⡀
⠀⠀⠀⠴⠚⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⡀⠀⡀⠀⠠⣼⣗⠂⢠⠀⠐⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠠⠀⣸⣿⣤⠀⠐⠂⣿⣿⠰⠀⡀⣡⣿⡇⡐⠀⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢶⣄⡐⠂⣤⠉⣿⠻⣮⠁⢾⡏⢹⡇⣤⣵⠟⣿⡌⠁⠶⢀⣨⡶⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⢈⣿⣿⢶⣤⡘⢻⡇⠙⣿⣿⠇⠸⣿⣾⠋⢸⡷⢀⣸⡶⣻⣟⠓⢠⡌⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⡈⠀⠆⢀⡻⣧⡉⠻⢿⣷⠀⠈⠿⠀⠀⠿⠁⠀⣼⡷⠛⠁⣴⢫⡉⠠⠆⢀⠉⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠹⠿⣷⣷⠶⣮⣴⣿⣷⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣾⣱⣦⠴⠿⢶⡾⠛⠃⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⣠⠉⠹⣷⠦⠤⠬⣍⣩⠤⣤⣄⣀⠀⣠⠤⠤⡄⢀⡠⠤⠬⢥⡤⠐⣶⠾⠷⣄⠉⠰⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣀⠉⠰⢄⣈⡟⠁⢀⣀⣀⣿⠁⠀⠈⠀⠈⡇⡇⠀⠀⢱⠎⠀⢀⣀⣸⠃⠀⡿⠀⢸⣇⡘⠂⣤⠁⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⡚⠛⣿⢿⣇⠀⠀⠉⠉⠙⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⡄⠀⠈⠉⠉⠀⠀⠃⠀⢸⡇⠁⠶⢀⡈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡌⠁⠲⢀⡜⡿⠻⠛⠃⠀⠀⠀⣸⣴⡆⠀⠈⠀⠘⠀⠀⠛⠿⠛⠀⠀⠀⠀⡇⠀⢸⣁⡙⠠⡄⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⢀⡘⠠⡌⣡⣧⣤⣤⣴⡾⣤⢤⡏⠁⣇⡠⠤⣀⡞⢇⡠⠤⠤⠤⠶⣿⣶⣺⣧⡖⢾⣌⠑⠖⢠⡉⠀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠌⠀⠖⢺⡋⠹⠿⣠⡛⡳⠃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠹⣿⣤⡍⠹⠟⣹⠛⠲⠄⣀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠉⠰⠆⣀⠛⢴⠍⡝⢁⣠⡖⣻⠇⠀⢠⣦⠀⠀⣿⣄⠀⠸⣿⣷⣦⣘⢧⣜⠛⢴⠌⡐⠃⢤⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠃⣤⠉⠰⢆⣼⠒⠷⢎⣘⢿⠀⣰⣻⢿⡄⢰⡿⢿⣆⠀⢹⣏⣘⠛⣿⣿⡼⠂⣄⠉⠰⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⡈⠃⠴⢁⡘⠃⣤⢉⣆⢼⣽⠋⢾⣇⣸⠃⣶⢙⣧⡀⢿⣋⠶⢂⡜⠋⠶⢀⡘⠁⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠤⠁⠘⠀⡌⠁⠶⢀⣸⠋⠶⢁⡛⠣⣿⣟⠻⢢⡜⠛⢷⡈⣿⠡⡦⢁⠛⠀⡌⠁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠦⢀⡙⠠⡮⢁⠛⢠⡌⠁⠶⢸⡟⠱⠦⣁⡛⢨⣿⣼⣷⢠⡌⠁⠆⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⠌⠀⠒⢠⡉⠀⠆⣀⠛⠠⠎⣀⠛⢠⡌⠀⠆⢠⠻⣿⡆⢀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠉⠠⠄⣀⠃⠠⠈⠐⠂⢠⠉⠰⠆⣀⠋⠠⠄⡀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠁⠰⠀⡀⠃⠠⠀⠐⠂⠀⠁⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⣀⣀⣶⡄⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⣾⡟⠉⢩⠀⠲⠁⠀⠀⠀⠀⠀⠠⡚⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠴⠋⢉⣽⠆⠀⠀⠀⠀⠐⠀⠀⠀⣀
⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠀⠀⠀⠐⠄⠣⠄⢠⠀⠀⠀⠀⠀⣼⠏⠁⠀⠀⠀⠀⠀⠀⠀⠐⡂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⣸⣿⣶⡶⡆⠀⣯⡅⠀⠀⠈⡄⠀⠐⠀⠀⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣭⠁⣀⠇⠀⠿⠃⠄⠀⠔⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠻⠏⠀⠈⠂⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⣶⣶⡄⢀⢀⣤⣴⣀⠀⣤⣀⣀⣠⣀⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡔⠶⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢉⠁⠀⠙⠛⠛⠉⣰⢿⣿⣿⣾⠋⠀⡷
⠀⠀⠀⠀⠀⡀⠄⠀⢀⡰⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠒⠀⢀⠀⡀⠀⠀⠀⢠⠀⠈⡀⢁⣅⣀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠇⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠉⢉⠂⣠⠄⠀⢀⣤⣴⣄⢲⠀⠀⠀⣤⣦⣄⢠⡄⢠⠀⠀⠀⠾⣦⡴⢿⣿⢻⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢺⣿⢷⣏⡷⡾⠀⠀⠸⠟⣉⠁⠘⣦⠀⠀⢹⣧⣽⡿⣷⡎⠀⠀⠀⠀⠀⢀⣼⠿⣯⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠆
⠀⠀⠀⠀⠀⠐⣟⣓⠙⢓⠳⠲⠲⠒⠀⠉⠒⠈⠁⠀⠀⠘⠟⠋⠇⠀⠑⠲⠖⠀⠀⠂⠉⠀⠆⠘⣯⢃
⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⣀⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠳⣄⣠⠞⠁⠀⠀⢀⡠⠒⠒⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⢀⠞⠀⠀⠀⠀⣠⡶⠿⠛⠛⠛⠛⠛⢷⣄⠀⠀⠀⠀⡸⠁⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢹⣦
⠀⠀⠀⡎⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⣰⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢿⡆
⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡷⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⡇⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣵⡄⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣷⠀⠀⠀⠀⢱⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡟
⠀⠀⠁⢷⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⣆⠀⠀⠀⠨⣿⣶⠶⣶⠀⠀⠀⠀⠀⠀⣽⡃⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⣸⡇⠀⣀⣀⣤⣄⣿⠇⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠙⢻⢦⡀⠀⠀⠿⠶⠶⠶⠶⠦⠄⠀⠀⠀⠀⣠⣾⠋⠹⣷⣄⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠉⠁⠀⢀⢴⡿⠁
⠀⠀⠀⠀⠀⠀⠑⠿⠷⣦⣄⣀⣀⠀⠀⠀⢀⣀⣠⣴⠾⠀⠀⠀⠀⠈⠪⣷⣦⣤⣀⡀⠀⠀⠀⠀⢀⣀⣴⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⣼⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⠀⢰⡿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣰⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣆⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠄⢤⡄⣤⢀⡄
⠀⡁⠢⡦⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢲⣴⣧⣷⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠙⠁⣿⡀
⠀⠑⡴⢀⡆⠠⠀⠀⠀⡀⠤⠀⠰⠀⠀⠀⠀⢿⡟⠛⠛⠛⠙⢷⠀⠀⠀⠀⢀⡳⢆⠂⠀⠠⢀⡀⠀⠀⠄
⠀⠀⠰⣤⠐⠊⠈⠉⠉⠁⠘⠐⠀⠂⠀⠀⠀⠈⠁⠤⠀⠀⠄⠈⠀⢀⠀⠀⠦⠭⢽⣄⣀⠀⠀⠀⠉⠲⠄
⠀⠀⠘⠛⢳⣦⣀⣀⢀⠀⢀⡄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠿⣦
⠀⠀⠀⠀⠈⠁⠉⠈⠉⠅⠊⠂⠑⠆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⠀⠀⠀⠀⠀⣫⢣⡏⡦⠒
⠀⠀⠐⠀⠀⠘⠄⠀⢀⠀⣾⣿⡿⡖⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠶⠲⠾⢡⠇⠆⠋
⠀⢠⠀⠀⠀⠀⠈⠀⡏⠘⢿⡿⠱⣀⠂⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠐
⠀⠀⠧⣄⡀⢀⣀⣠⠀⠀⠩⠉⠁⢤⠶⠛⢠⡇⡀⠄⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐⠟⣯⣼⠗⠂⠁⢀⣀⣴⠁⠀⠀⠀⠀⠠⠀⡄⠈⢛⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠐⠒⠄⠤⠠⡀⠀⠀⠀⠀⠈⠃⠁⠀⠈⠁⠈⠀⢠⠥⠒⠉⠉⠁⠈⠈⠁⢙⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⡼⠂⠀⠸⠾⠞⠁
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠁⠀⢠⠁⣀⠲⡄⠀⠈⡦⠀⣾⣽⡧⠀⠀⠐
⠀⠀⢸⠦⡄⠂⠀⠀⣀⣀⣠⡌⠀⢸⣿⣅⣀⠄⠀⠀⠈⢩⠄⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢄⠀⠁⢀⠀⠛⠛⠛⠉⡀⢉⣭⣭⠏⣀⠀⠀⡠⠀⠀⠀⠀⣠⠆⣈⣡⣤⣶⣶⣤⣄⡐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠅⣶⣮⣊⡴⡀⠀⠀⠀⢀⠜⣡⣾⠏⡁⠠⠀⠀⢌⠙⠻⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢿⣶⣯⣽⣿⡗⠻⠧⠀⠀⠌⣰⡏⢳⣄⣶⡀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡾⠿⢿⣻⣟⡟⠐⠂⣀⠀⠀⣿⠣⠀⠿⢽⣏⣼⣇⣶⣠⢴⠇⢸⡇
⠀⠀⢀⠀⠄⠀⠀⠀⠀⠄⠀⢸⡿⡇⠄⢸⣿⣿⡇⠀⠀⠟⠀⠀⣿⣇⢀⡀⠀⠀⠀⠉⠁⠛⠚⢀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⣀⠘⠸⠿⣇⠀⠀⠁⠀⠀⠀⠁⠈⠳⣄⣀⣤⠀⢀⣠⣠⡉⠋
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠓⡤⠉⠛⠨⢶⣾⣿⢧⡀
⠀⠀⢀⠠⣬⠀⠀⠀⠀⠁⠀⠄⠀⠀⠖⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠰⣷⣿⣷⣽⡄
⠀⠁⡮⠥⢼⠠⡄⠀⠀⠀⠀⠀⠳⠀⡀⠀⠀⠀⠔⠀⠀⠀⠀⠈⠀⢈⡀⠁⢤⠀⠀⠀⠀⠟⣿⣿⢇⡅
⠀⠀⠀⣆⡁⠀⠈⠉⠒⢤⡀⡴⢃⣆⣆⠀⡀⢀⡄⠀⠀⠀⠀⡀⠠⢰⡅⠀⠁⠀⠀⠀⠀⣴⡿⠿⠋
⠀⠀⠀⢿⣧⣤⣤⡄⠀⠀⢈⢲⣠⣾⡇⢠⡍⣶⡚⠈⠉⠉⠉⠀⠂⣲⣠⡇⠠⣀⠀⠀⠀⠈⠀⣤⠶⠛⠃
⠀⠀⠀⠈⠹⢿⣿⣿⣶⣿⣿⣿⣿⣿⢁⣿⣿⣿⣶⣦⣶⣴⣦⣤⣾⣿⣿⡏⢰⣟⡀⠀⠀⠀⢀⣿⣁⣾⡇

⠀⢰⡶⠶⠶⠶⠶⢦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢸⠃⠀⠀⢰⠶⠶⠶⠶⠖⠒⠒⠒⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⠀⠀⠀⣿
⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⡆
⠀⢸⡀⠀⠀⢸⠀⠀⠀⠀⣟⡽⠯⠭⠿⠿⠛⠳⠙⡇⠀⢸⣏⡭⠭⠭⠭⠽⠭⢍⣶⠀⠀⢸⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⢰⣧⡄⠀⠀⠀⠀⠀⢸⣽⠀⠀⢸⠇⠀⠀⢸⡇
⠀⢸⡅⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡏⡇⠀⠸⣿⡇⠀⠀⠀⠀⠀⢸⣼⠀⠀⢸⡀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⣻⠇⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⠇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡀⠀⠀⠀⡇⡧⠀⠀⠀⠀⠀⠀⣷⡇⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⣿⠂⠀⠀⠀⠀⠀⢸⣾⠀⠀⠘⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢻⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣾⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣿⡆⠀⠐⣿⡁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢼⠀⠀⠀⠀⠀⠀⣿⡁⠀⢰⣿⡇⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⡇⠀⠀⢀⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢷⢸⠀⠀⠀⠀⠀⠀⣿⠇⠀⠈⣧⠆⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⡇⠀⠀⠘⡇
⠀⢸⡇⠀⠀⠘⡇⢀⢤⡀⢸⣜⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⣻⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠘⡇⠀⠀⠀⡇⢼⠀⠇⠸⠯⠭⠿⠿⠷⠿⠟⠿⠆⠀⠀⠫⠿⠶⠶⠾⠶⠿⠿⠿⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⡜⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⠳⠤⠞⢀⣄⣀⢀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⢤⣄⣤⣄⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⢀⠁⡆⢸⢹⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠋⠛⠉⢻⢸⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⠈⠒⠁⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⠀⣧
⠀⠈⡇⠀⠀⠸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠎⡄⠀⠀⡇⠀⠀⠀⡿
⠀⢸⡇⠀⠀⢐⡇⠀⠀⠀⠸⢾⠭⠭⠿⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⠭⠭⠭⠭⢶⠇⠀⠀⣗⠀⠀⠀⣟
⠀⢸⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣟
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢰⣖⣒⣒⣶⣶⣶⢶⣶⣶⣶⣾⣷⡾⣶⠶⠶⠶⠶⢶⣶⠀⠀⠀⡗⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⡇⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠘⡇⠀⠀⢀⡇⠀⠀⠀⠀⡿⣿⠾⣿⣿⣿⣿⡿⠿⠾⠶⠿⠷⢾⣓⠺⠶⣖⠚⠛⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⣠⣧⣤⣤⡤⣷⠶⠦⡶⢦⣴⠴⠦⣤⣶⠶⣦⣶⣶⣶⣶⡶⢶⣶⡦⠶⠶⢶⠿⢿⠿⠗⠒⠧⠀⢤⡤⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢿⣿⣿⣿⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢿⣿⡄⢨⣿⣿⣗⢸⣿⡿⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠛⠛⢻⣿⣦⡀⠀⠀⠀⠘⣿⣧⣼⣿⣿⣿⣼⣿⠃⠀⠀⠀⢀⣴⣿⣿⡿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⢿⣿⡏⣿⣿⣧⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⣼⣿⡏⢹⣽⣿⡟⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⣼⣿⡇⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⢿⣿⡇⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣿⣿⣧⣿⡿⠁⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⣷⣿⣿⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣿⣿⣄
⠀⠀⠀⢀⣤⣶⣶⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⡿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣶⣶⣤⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⠇⣿⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⣇⣿⡄
⠀⠀⣿⣿⣿⣿⣿⣿⡿⣱⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⣿⣿⣿⡿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⡍⣧⣿⣿⣿
⠀⠀⠹⣿⣿⣿⣿⣿⣷⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⡿⣿⣿⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢹⣧⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠻⠿⣿⣿⠿⠛⠁⠀⠀⠰⢅⣐⠑⢤⠁⠙⠀⠀⢿⣿⣿⣿⡏⠀⢆⣰⡔⠀⡆⠀⠀⠰⢁⠀⠀⠈⠻⠿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇⠀⠀⠉⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟⡛⠿⣿⣷⡄⠀⠀⠀⠀⠀⠼⠟⠁⠿⠧⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣴⣿⣿⣷⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡙⠿⠿⢟⣹⣿⡿⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⢿⣿⠛⠛⠁⠈⠛⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣾⣿⣿⠟⠁⠀⠀⠀⢠⣿⢻⣿⣿⠟⣿⠻⣿⡄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢾⣇⠀⣿⣿⠂⣧⠀⣿⡷⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣿⣿⣧⣿⣬⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⣠⣴⣦⡀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣴⡄⠀⠀⢸⣿⣇⣴⣾⠟⠉⣿⣇⣀⠀⣠⣴⣿⢿⣿⠀⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣙⢻⣿⡇⢸⣿⡟⣿⣿⠉⣠⣴⣴⣿⣟⠻⢿⣿⡏⠀⣾⡏⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢈⣉⡛⠻⢿⣧⣼⣿⣿⡟⣿⣶⣿⠋⢘⣿⡛⠿⢶⣿⣷⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣤⣀⣀⣹⣿⠉⣿⠃⠘⣿⣿⣀⣼⣿⣿⣅⠀⢸⣿⣿⣿⣤⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣛⣻⣁⣀⣀⣀⡀⠈⠛⠉⠉⠛⠈⠙⠛⠿⠿⠋⠹⠇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋⠉⢻⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠓⠀⠈⠇



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⢀⣴⡀⢢⣀⣀⡴⢁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠛⠒⠚⠋⠀⠻⢦⣭⡭⠴⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠴⣾⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⣀⠤⠄⢀⡤⠀⢻⡸⣄⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢃⡼⠃⠀⡴⠊⠁⠀⠀⢰⡏⠉⣿⠃⠀⠀⣰⠋⢀⡤⢦⣼⠀⢠⠀⠓⠦⣭⣭⡥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⠖⠋⠀⠀⣼⠁⢠⡇⠀⠀⣾⠀⠀⡟⠀⠀⣰⠇⠀⡎⠀⠀⣿⠀⡘⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⢠⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⡇⠈⢉⣿⢠⠃⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢸⡇⠀⢸⠇⠀⣿⠀⢰⠀⢠⡇⠀⣿⡀⠀⣿⠀⢸⠛⠚⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠾⢥⣀⠀⠈⣷⡀⠈⢀⣼⡇⠀⣾⡇⣼⡇⢀⡝⢷⣄⣀⣀⣸⣁⣠⠀⠀⣼⡯⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⣬⢹⣆⠀⠈⠛⠛⠋⠘⠛⠛⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠁⠉⠀⠀⠀⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⢿⣿⣿⣷⡄⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣟⡟⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡛⣟⣷⣿⣿⣿⣿⣿⣷⠀⣀⣀⠀⠀⠀⢰⣿⣿⡟⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠋⠃⠃⠘⢻⣿⣻⣿⣿⣿⣶⣄⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠘⠘⠚⢻⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣇⠀⠀⠀⠀⢀⡀⠀⠀⣠⣦⠀⠀⠀⣿⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⣸⣄⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣰⠏⠛⠶⣄⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠁⠈⠑⠦⣄⣀⠀⠀⠀⢀⣀⠴⠚⠁⠀⠀⠀⠈⠳⣄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣠⠞⠁⠀⠀⠀⠀⣀⡀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⣰⠁⢀⡜⠁⠀⠸⣄⣠⠏⢸⠉⠁⣴⠉⣇⢸⡄⠀⡇⠀⠀⡄⢤⠐⡠⠤⠀⠈⢳⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢰⠃⠀⡞⠀⠀⠀⠀⢘⡇⠀⢸⠉⠁⣧⣤⣽⢸⡗⠒⡇⠀⠀⡇⢸⠀⠙⢄⠀⠀⠀⢳⠀⠀⠀⢻
⠀⠀⠀⠀⢀⡏⠀⢰⠇⠀⠀⠀⠀⠘⠃⠀⠸⠤⠄⠇⠀⠸⠸⠃⠀⠇⠀⠀⠃⠸⠀⠤⠜⠀⠀⠀⠈⣇⠀⠀⠸⡇
⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠀⣤⠤⠤⠀⣤⠦⣄⠀⢠⡀⢠⠤⣄⡀⠀⠀⣠⢤⡀⢤⠀⠀⠀⡴⠀⠀⢹⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⢸⠇⢸⡇⢸⠀⠈⢳⡄⢸⠁⠀⣷⠈⢧⣀⡞⠁⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⡷⠒⠀⠀⣿⡴⠋⠀⢸⡇⢸⠀⠀⢸⠇⣿⣀⠀⠿⠀⠈⣿⠀⠀⠀⠀⢸⡇⠀⢰⡇
⠀⠀⠀⠀⠸⡆⠀⠈⡇⠀⠀⡇⠀⠀⠀⣿⠳⣄⠀⢸⡇⣿⠀⣠⡟⠀⡿⠉⠀⠶⠀⠀⠾⠀⠀⠀⠀⡿⠀⠀⣾
⠀⠀⠀⠀⠀⢧⠀⠀⠹⡄⠀⠁⠀⠀⠀⠉⠀⠈⠀⠈⠁⣙⠋⠉⠀⠀⠁⠀⠀⠉⠀⠀⠉⠀⠀⢀⡾⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⢰⣛⣏⣉⣻⣋⣳⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠉⠉⠉⠙⢲⡄⠈⡏⡀⠁⢉⠉⡷⢦⢠⡖⠚⠛⠋⠉⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢐⣛⣿⠀⢰⡇⡇⣿⢸⠀⡇⢸⡆⣹⠛⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⣀⣈⣉⡽⠀⠸⡇⠃⠛⠘⢀⡷⠚⠀⠈⣟⣁⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠓⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡞⠉⠉⠉⡟⠉⠉⠙⡆⠀⠀⡇⠀⢰⠖⠒⠒⢲⠒⠒⠒⢲⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡟⠉⠉⠉⡏⠉⠉⠉⡇⠀⠀⡇⠀⢸⠒⠒⠒⢺⠒⠒⠒⢺⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⣀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠛⢻⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⣧⣤⣤⣤⣧⣤⣤⣤⡇⠀⠀⡇⣶⣼⣀⣀⣀⣸⣀⣀⣀⣸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠁⠀⠀⢸⠁⠀⠀⢹⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡗⠒⠒⠒⡗⠒⠒⠒⡇⠀⠀⡇⠀⢸⠦⠤⠤⢼⠤⠤⠤⢼⠀⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠧⠤⠤⠤⠧⠤⠤⠤⠇⠀⠀⡇⠀⢸⣄⣀⣀⣸⣀⣀⣀⣸⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣽⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⠀⢠⡆⣶⠶⠶⠶⠀⢠⡶⢶⣦⠀⣴⠄⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⢠⡿⠀⣿⠀⠀⠀⢀⣿⠁⠀⢿⡆⣿⠃⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⣿⠁⠀⣿⡆⠀⠀⢸⡇⠀⠀⢸⡇⣿⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⣿⣷⡶⠀⢸⡷⠶⠶⢾⡇⣿⡶⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣇⣀⣀⢸⡇⠀⠀⢸⡇⣿⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠉⠙⠛⠋⠈⠁⠀⠀⠘⠁⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣧⠀⠀⢠⡿⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⢿⣿⡿⠿⠿⣿⡇⢸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣿⢰⣶⣶⢸⡇⢸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣩⣭⡍⣿⢸⣿⣿⢸⡇⢸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠛⠛⠃⠛⠘⠛⠛⠘⠃⠘⢻⣷⡼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⣋⣥⣶⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⡂⠠⣶⣿⣿⣿⠁⡦⢙⣛⣛⣓⣦⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠀⢠⡄⠀⠀⣍⡛⠿⣷⣶⣾⠿⢛⣉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⢸⡇⠀⠀⣿⣿⣷⣦⣭⣴⣾⣿⣿⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢸⡇⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⡁⠀⠀⠀⠀⠀⠀⢬⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⣤⣾⣿⣿⡷⠞⢤⡴⣦⣀⡻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⠀⣷⣤⣤⢄⣿⠉⠛⠐⢀⠀⠀⠚⠙⠿⣾⣿⣿⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⠾⣷⠀⠀⠀⠙⠂⠀⠀⢈⡀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠉⢙⣿⣿⠟⢩⡄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⣾⡿⠟⠀⢼⡇⠀⠀⣿⡄⠀⢀⡀⣤⠀⠐
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠐⣭⣾⠏⠀⠀⠀⠀⠀⠀⠀⢿⠛⠀⠠⣾⣿⣄⠀⠉⠛⠳⠾⠃⠉
⠀⠀⠀⠀⠻⣧⡤⠀⠀⢠⡼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⣾⠁⣸⣷⣆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠒⠛⠛⠛⠉⣀⣨⣴⣾⣷⣶⣦⣄⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠷⠀⠠⠆⠹⣿⠿⠛⠋⠛⠛⢏⡁⠀⠛⠻⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣽⣿⣦⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⢤⣈⣻⣿⣿⣴⣦⣀⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢠⡀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣶⣾⣿⡿⠋⠁⠁⠉⠉⠉⠀⠿⢃⣶⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⢀⠳⣄⠀⠀⠀⠀⠀⣤⣄⣀⣀⡀⢀⣠⣾⡿⣟⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⡀⠈⠃⠀⠀⠀⠀⠀⠉⠱⣿⣷⣾⡿⠷⠿⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⡀⠓⠀⠀⣦⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠣⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠴⢿⣿⣿⡄⠀⣀⠶⠀⠤⣴⣿⣿⠀⣀⠀⠈⠃⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⠞⠃⠀⠀⠀⠀⠤⠊⠁⡴⠀⢘⣿⣿⣧⣿⣿⡀⠀⠀⠀⢀⡌⠀⣿⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢤⠈⢛⣛⢶⣦⣄⣤⡄⠀⠴⠟⠁⡀⠋⠉⢁⣿⣿⡿⣿⣿⣶⣤⣴⣾⡇⣸⣿⣷⠄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠩⠶⠂⠀⠀⠘⠀⢀⣴⣿⢏⡼⠁⠀⠙⣿⣷⠙⣿⣴⠏⣿⠋⠀⠀⠀⠠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠘⠃⣰⣦⠀⠀⢀⡀⢀⡾⠿⠟⠋⠀⢠⣴⡾⠛⠁⠈⠉⢀⡀⣿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣄⠠⡑⠂⢦⠀⡰⣤⡄⠈⠁⣿⡇⠀⢟⠁⠀⠀⠀⠀⠀⠐⢛⣿⠀⠀⠀⣀⣴⣾⣿⣿⣤⣴⣶⣶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⣬⡲⢶⣬⡛⢾⣷⣄⣴⣿⣿⣶⡿⠀⠀⠀⢰⣦⡴⣶⡾⢻⣶⣴⣾⢟⣩⣿⣿⣿⣿⣧⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠄⠀⣸⡿⠛⠋⠉⢹⡇⣸⢏⣼⣿⠟⠻⣇⠀⠉⠛⠛⠃⠀⠀⠀⠀⡇⠀⠀⠈⢿⣿⣻⡿⠛⠉⠛⠿⠛⢻⣿⡙⠻⠿⢿⡿
⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠿⢿⣥⣶⣷⣿⣧⡀⠀⠀⠀⠀⢀⣀⣀⣴⠀⠀⠀⠀⡛⢹⡇⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⡾⠃
⠀⠀⠀⠀⠈⣿⣷⣰⣆⠀⣠⣾⣿⣿⡇⠀⠀⢨⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⣏⠙⠿⣷⣤⣶⣿⣿⣦⣝⣦⡀⠀⠀⢠⣶⣿⣿⠿⠋⢀
⠀⠀⠀⠀⠀⠘⢿⣿⣯⣙⣿⣿⣿⣿⣥⣴⠎⣻⣿⣿⣿⣿⣿⠟⣧⠀⠀⠀⠸⣿⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣥⣤⣤⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣟⣛⣛⣿⣧⣬⣿⣿⣿⣿⣿⡟⠀⠙⣧⣄⡀⠀⣿⣶⣦⣴⣿⣿⢿⣿⠛⠛⠛⠻⠿⠛⠛⠻⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⡿⡿⢿⠇⢸⣿⣿⣿⣿⣿⢿⣷⡀⠀⠈⣟⢻⣿⣿⣿⠋⣨⡿⢫⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⣿⣿⣿⣿⡟⠘⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⣻⠛⣿⠃⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡟⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⣿⢻⡿⠛⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢊⡈⠁⢨⡉⠉⠉⢠⡈⠁⢁⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠈⡇⠀⣸⡀⠀⠀⣸⡀⠀⣸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠠⠏⠉⢻⠋⠉⠉⢻⠋⠉⢻⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢠⡆⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢀⡗⠒⢻⠗⠒⠒⢻⠗⠒⢻⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢘⡇⠦⢾⡆⠶⠤⢾⠦⠤⢾⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠸⣇⣀⣸⣀⣀⣀⣸⣀⣀⣸⣷⣶⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⢀⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⠀⢀⣾⣿⣿⡆⢀⣠⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢿⣷⣦⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣤⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠘⢿⡟⠋⠙⢻⡿⠛⠉⢻⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⠋⠀⠀⠀⠀⠀⣹⠛⠀⢲⡖⠂⠒⡖⠂⠐⡆
⠀⠀⠀⣀⣀⣤⣼⠀⠀⠠⣤⣼⡇⠀⠀⠘⠇⠀⠀⢸⣿⡇⠀⠀⢠⠀⠀⠘⣿⣿⠀⠀⠀⠲⠶⣶⣿⠀⠀⠸⠇⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠉⠛⢿⠷⣤⡀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠻⠀⠀⠀⢻⣿⣷⣤⣀⠀⠀⠈⢻⠀⠀⢀⡀⠀⠀⡇⠀⢠⣿⡶⠤
⠀⠀⠀⠀⠀⠀⢾⣀⣀⣀⣀⣼⡇⠀⠀⡄⢀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠉⠉⠀⠀⠀⣸⠀⠀⢸⡇⠀⠀⡿⠖⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣠⣿⣆⣀⣀⣀⣷⣦⣤⣤⣤⣤⣶⣿⣿⣿⣟⠛⠛⠛⢷⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠻⠿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠁⠀⠀⠸⣿⣿⣿⠏⢹⣿⣿⣿⣿⡟⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋⣡⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⡿⢋⣴⣿⣷
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣷⣌⠙⢋⣴⣿⣿⣿⣿⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⣤⣶⣌⠻⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣿⣿⣿⣿⡿⠃⣼⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⢹⣿⣿⣿⠁⣤⣤⣴⣶⣿⣿⣿⣿⣿⣷⣤⡉⠛⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠟⢋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣴⣿⠏⣹⣿⡄⢻⣿⣿⣿⣅⣿⣿⣿⣿⣿⣿⣇⣿⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣼⣿⣿⠀⣿⣿⣧⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⢉⣤⣌⠙⢿⣿⣿⣿⣿⣿⣷
⠀⢰⣿⣿⣿⣧⡘⢿⣿⣧⡘⢿⣿⣿⣿⡿⠿⠟⢻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡅⢺⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣷⣄⠙⣿⣷⣄⠙⠿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣤⡈⢡⣴⣿⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⣿⣿⣿⣷⡈⢻⣿⣿⣦⠀⣉⣉⣉⣉⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⠀⠻⣿⣿⡿⢋⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡿⠋
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⠀⣷⡌⠁⣴⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡇⢸⣿⣿⡟⠉
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠃⣼⣿⠇⣸⠟⢠⣦⠘⢿⣿⡿⠋⣠⣶⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⣴⣄⡙⠿⣿⣿⣿⣿⣿⡟⠀⠋⣰⣿⣿⣷⡈⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠸⠿⠿⠇
⠀⠀⣼⣿⣿⣿⣦⣌⡙⠛⠿⠟⢀⣿⣦⠈⢛⢛⠋⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⡿⢀⣿⣿⣇⠸⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⡿⠇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⢶⡆
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⡟⠡⣾⣷⣤⣴⣶⣿⣿⣿⣷⣦⣤⣴⣿⡦⠀⠀⠀⠰⣶⣶⣶⣶⣶⡾⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡆⢈⣿⣿⣿⣿⠿⠛⠿⢿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⢀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⠀⣿⣿⡿⠛⠛⢀⣿⣿⣿⠏⣠⣶⡆⠀⠀⠈⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⡆⢹⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣷⠈⣿⣧⣤⣤⠈⣿⣿⣿⣆⠙⢿⡇⠀⠀⢀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⠇⢘⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠻⢿⣿⣿⣿⣿⣿⠿⢃⣼⣿⣿⣇⠐⢿⡿⠛⠿⢿⣿⣿⣿⣿⠿⠛⠻⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⠟⢁⣴⣿⣿⣿⣿⣿⣷⣄⣠⣾⣶⡆⢸⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⡀⢀⣀⣀⠀⠀⢀⣀⡀⢀⡀⢀⡀⣀⢀⡀⠀⢀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣼⠋⠈⣷⢸⡧⣽⡃⠀⠻⣧⡁⢸⣷⡏⠀⣿⢸⡇⠀⢸⡇⠀⠸⢧⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣇⣘⣷⣶⣋⣸⣷⣾⣃⣀⣴⣶⣏⣸⣇⣻⣆⣿⣸⣷⣶⣸⣷⣶⣰⣶⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⡰⠀⡀
⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢆⠀⢃⠀⠀⠀⡀⠀⠀⠀⡄⠇⠀⠀⠀⡴⠦⢧⣜⡀
⠀⠀⠈⠢⢄⠀⠀⠀⢀⠔⢋⠀⠀⣠⡘⢧⣀⠠⠤⣤⡿⠤⠄⣀⣸⠶⠤⣀⡰⠁⠀⢀⡤⠋
⠀⠀⠀⠀⢀⡵⠲⠴⣁⡀⢀⡰⣾⣿⣿⣮⡷⠒⠚⠛⢛⡲⠶⢤⣽⡙⠒⠀⢧⣀⡔⠋
⠀⠀⠀⠀⠈⠉⠓⠲⠴⣷⡋⠀⡬⠛⠁⢸⠀⠀⣠⠖⠉⠀⠀⠀⠈⠙⢦⡄⠀⡝⠠⡀⠐⠋⠀⢀⡠⠔⠊
⠀⠀⠀⠀⠀⠈⣂⡴⢺⡀⢩⠟⠲⠤⣴⡃⢀⠞⠁⠀⢀⣠⠤⠤⠤⢤⣀⡙⣿⣧⡀⢈⣢⡔⣏⢉⡇
⠀⠀⠀⠀⣴⠋⠁⠀⠀⢷⠏⠀⣠⠞⠁⢨⠛⠲⣤⡎⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠛⠁⠈⢹⣀
⠀⠀⠀⠀⠉⡷⣄⣀⣴⡾⢠⠞⠁⠀⢠⠃⠀⢠⠇⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⣀⣬⣽⡷⠒⠂
⠀⠀⠀⠀⢨⠁⡏⢻⠛⣿⠁⠀⠀⢠⠇⠀⠀⢸⡀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⢸⣿⣿⣿⠟⠛⢻⠃
⠀⡚⠒⠂⣾⠀⡇⢸⠀⢻⠀⠀⠀⡞⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣸⣿⣿⣿⣿⣿⣿
⠀⣧⠀⣀⣻⣀⣧⣼⣤⣼⣇⠀⢰⡇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠙⣶⣿⣿⣿⣿⠿⢿⡇⠀⠀⠐
⠉⠙⡍⠁⠀⠀⢿⠀⢻⣿⣿⣧⡘⡇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏
⠀⠀⠘⣆⣠⣤⣼⣀⡤⠿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠹⣄⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡁⠀⠀⡐⠠⣀
⠀⠐⠉⣹⢶⠃⣼⠱⡀⠀⡹⡟⠉⠻⣿⣿⣷⣶⣶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣰⠃⠀⠀⠉
⠠⠐⠉⠀⠘⠒⠼⣦⠙⠀⡀⠘⢦⡴⠋⠉⠛⢻⣿⣿⡟⠛⠛⣿⣿⣿⣿⡿⠿⣿⡿⠋⢰⣤⠛⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠞⠉⠒⠤⢤⣿⣿⣿⡇⠀⠀⣿⡇⠻⡉⠀⠀⠉⠀⠀⠈⡎⠳⣌⣆⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣀⠀⡜⠩⠉⠀⢀⣿⡇⠀⢹⣄⡀⠀⠀⠀⠀⢳⠀⣬⠟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠐⠁⠀⠀⠔⠀⢀⡏⠘⠒⠒⢾⠿⡗⠊⠁⠈⢆⠀⠀⠀⠀⠘⠟⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣘⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡖⠒⠚⠉⠉⢹⠀⠀⠀⠀⠀⣦⠞⣿⠉⠉⠉⠉⠁⢈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⠒⠒⠒⢻⠦⡀⢀⣀⣐⠶⠀⠀⣀⣀⣴⣩⣧⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡇⡿⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⢐⡲⠤⠀⠀⢰⠃⠀⠀⢀⡿⠖⠛⠉⠁⢹⠀⢠⢚⠏⠀⠀⠀⠀⢷⠀⠀⠀⢸⠀⢐⣲⣯⣥⡀⡇⡇⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢀⠔⠉⠙⢆⠀⠀⡞⠀⠀⠀⡼⠀⠀⠀⠀⠀⢸⡄⡜⡜⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⡞⣏⠀⠀⠀⡇⠇⡇⠀⠀⠀⢀⡞
⠀⠀⢠⠷⣄⠀⠀⠀⠳⣴⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠸⣷⢳⠃⠀⢰⣦⡀⠀⠀⢹⡄⠀⠸⣿⡿⠀⠀⠀⡇⣶⡇⠀⠀⠀⣼⠁
⠀⠘⡜⠲⣌⡳⣄⠀⠀⠀⠀⠀⢠⠃⠀⠀⣠⠤⠴⠶⣾⢏⡎⠀⠀⠀⠉⠀⠀⠂⠀⣷⠀⢀⣀⣀⠀⠀⠀⣇⡟⣧⣠⣤⣴⠇
⠀⠀⠉⠉⠀⠙⠮⣵⠀⠀⠀⢠⢿⡇⠀⠀⢧⣴⡖⠚⡝⣸⠁⠀⠀⠀⡴⢶⣄⠀⠀⢸⡄⢺⡏⢹⠀⠀⠀⡏⠙⢓⣒⡒⠃
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⡏⡌⣇⠘⠀⢀⡤⣏⣼⡥⠷⠛⢳⡀⡼⠷⡏⠹⡄⠀⠈⣇⢸⡇⢸⠀⠀⠀⣇⣴⡏⠁⢙⡆⡆
⠀⠀⠀⠀⠀⡴⠃⠀⠀⢀⡞⠀⡇⢹⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⢿⠁⠳⠘⡄⣷⠀⢀⣹⡿⡇⢸⠀⠀⠀⡿⠑⠽⠿⠟⠈⠁
⠀⠀⠀⢀⡜⠁⠀⠀⢀⡞⠀⡴⣇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡧⠀⢂⠱⣼⡟⣉⡽⢠⣧⠿⠖⠛⢹⡇
⠀⢠⡇⢺⣄⡀⠐⢀⡞⢀⠞⠀⢹⠘⡇⠀⠀⢀⣀⡤⠶⠒⠋⠉⣰⠃⡀⠀⠀⠈⠁⠀⠀⠀⠁⠛⠛⠻⢿⣟
⠀⠀⠁⠀⢧⡉⠛⠺⣠⠏⠀⠀⢸⡆⡷⠚⢋⣉⣠⠤⠴⠖⠚⠛⠁⠘⠃
⠀⠀⠀⠀⠀⠉⠉⠙⢃⣀⡄⠀⠈⠻⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠈⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⣾⠇
⣀⣄⣴⣶⡟⠃
⠈⣿⠿⠋⠙⠷⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠂⠠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣦⡀
⠀⢀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠑⠤⠄⡀⠀⠀⠀⠛⡋⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘
⠀⢸⠀⠀⠃⠀⠀⢠⡀⠀⠀⠀⢠⠀⢀⠀⠀⠂⠀⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡘⣧⢀⡧⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⡀⠀⠀⠀⠀⠾⠋⠋⠀⠀⠀⠁⠀⠀⠀⢀⢠⡀⢂⡰⣤⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣾
⠀⠘⣧⣀⣦⣄⠀⠀⠂⡀⢠⣤⣶⣴⣦⣶⣤⡉⠈⠉⠈⠁⠀⠹⠂⠀⠀⡀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⢀⠦⣍
⠀⠀⠈⠛⠛⠛⠃⠀⠚⠛⠀⠈⠁⠈⢻⣿⡟⠀⠀⠀⠀⠀⣠⠇⠀⢠⡎⠁⠀⠈⣦⣄⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⡀⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣿⣷⡀⠀⠀⠀⣠⣿⣤⣀⣀⠄⠀⢀⡈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣹⡇⠀⠀⢻⡿⠉⠉⠙⠺⢿⠀⢹⠻⣿⢿⡷⠂⠀⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠀⠀⠀⠠⣤⣤⣀⣀⡀⠈⠣⠉⠀⠁⠀⠀⠀⠀⠀⣤⠔⢠⢵⡄⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠠⣤⠂⣄⠉⠀⠀⠀⠀⠀⣤⡀⠒⠋⠀⠀⠀⠛⠂⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠹⢢⡄⠀⠀⣄⠠⣇⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣄⡐⣧⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠰⠂⠀⠀⠰⠋⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠙⢻⡟⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⣰⠀⠀⠈⡀⠀⠀⢠⣿⡴⠻⡦⣀⠶⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⠄⠂⠉⠀⠀⣠⡤⠓⠀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠉⠁⠀⠀⠀⢡⠀⢠⣾⣿⣿⣿⣶⠟⠀⠀⠻⣷⣄⠈⠀⠀⠀⢈⣹⣮⣩⠋⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣹⡆⠈⠁⠀⠀⠀⠀⠀⢿⣿⣶⡆⠈⠐⠺⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⢳
⠀⠀⠀⠀⣦⣤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠙⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠛⠋⠉⠉⢀⠀⠉
⠀⠀⠀⠀⢸⡿⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣤⡀⠈⠳⢦⣤⣀⣀⣀⣔⣙⣯⣽⣿⣿⣀⠤⠀⣷⡀⠀⠀⠀⡸⠀⠀⠀⢠⣤⡀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠁⠷⣾⣿⣇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⣄⠈⠙⠶⣄⣀⡀⢉⠉⠉⠙⢻⣿⡿⣿⣿⣯⡀⠀⣨⣿⠀⠠⠞⠁⢰⠄⠀⠞⠛⠛⠛⢷⠆⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠖⠀⠀⠀⠀⠀⠀⢀⠀⢿⣿⣶⢄⡀⠀⠐⠛⠀⠀⠀⠊⠉⢠⣿⠟⢿⣧⣬⢿⣧⠀⢀⠀⢰⣴⠀⣀⠈⠀⠀⠀⠀⠰⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢎⡃⠈⠉⠛⣦⠙⠓⠖⠲⣦⣤⣀⣠⠴⠛⠃⠀⠸⠋⠀⠀⠉⠷⠾⣧⣼⣿⡆⠛⠋⢷⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠖⠆⠁⠀⠀⠻⣿⢿⣇⠀⢊⡱⠦⣤⣤⠀⢀⡀⠀⠀⠤⠀⠀⢀⡀⠒⠛⠶⠀⠀⠈⠁⠀⠀⠀⠀⢰⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠋⠿⡀⠈⠀⠀⠀⠁⠀⠉⠉⠀⠀⠠⠆⠀⠐⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⠀⠀⠐⠒⠒⠤⣀⣴⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣷⣄⡀⠀⣀⣤⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡟⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⠿⣿⠿⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣻⡿⠋⠀⠸⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⡿⠋⠀⠀⠀⠀⠈⠹⠛⠓⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡐⠟⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠁⠐⠀⢀⠀⠀⠤⣶⣶⣾⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠩⠔⠀⢀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⢠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠶⣀⣶⣶⡀⠀⣈⣽⣿⠟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⣿⣷⠂⠈⠀⠐⠀⠀⠀⠀⠀⠀⡠⢿⣶⠂
⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠇⠾⣿⠿⠇⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢶⣤⣤⣾⣿⣿⣶⣤⠀⠀⠀⠀⠒⠦⣈⣿⠁⠈⠋
⠀⠀⠀⡉⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⢹⣀⠀⣀⠀⡠⠤⢤⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣠⣤⣭⣹⠟⠁⠃
⠀⠀⠀⠁⠀⠀⠤⣚⣧⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠇⠀⠼⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠿⠗⠚⠛⠛⠉⠁
⠀⠀⠀⠀⠀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣦⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⠀⠀⠆⠈⡀⢹⣮⣷⠀⠀⠀⠀⢠⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠃⠈⠈⢻⠀⠀⠀⠀⠀⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⣆⠀⣤⣾⣴⠏⠀⠀⠀⠀⠀⠀⠘⠋⡕⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠾⠶⠂⠓⠼⠦⠶⠗⠇⠆


⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⡷⠈⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠈⠁⠐⡀⢀⠀⠀⠀⠀⠀⠀⢂⣘⣃⡀⠀⠀⠀⠀⠂⣀⣭⣁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣴⣛⣳⡆⠀⠀⠀⠀⠀⠰⠿⠿⠆⠀⠀⠀⠀⣶⣾⣿⣿⣷⣦⠀⠀⠀⢀⣿⣜⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣿⣧⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠄⠀⠀⢉⡉⢁⡈⢩⠉⠀⠀⠀⠈⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣷⡆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢂⡂⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⣿⣿⣿⡋⠁⠀⠀⠀⠀⣀⣽⣯⣀⠀⠀⠀⠀⠀⣀⣽⢯⣀⠀⠀⠀⠀⠀⣀⣽⣯⣀⠀⠀⠀⢠⣄⣀⠛⢁⣀⣤⠀⠀⡇
⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣥⣦⣿⡆⠀⠀⠀⢰⣿⣧⣬⣿⡆⠀⠀⠀⢰⣿⣄⣨⣿⡆⠀⠀⠀⠻⣿⠂⢾⡿⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡠⠂⠡⠂⠁⠀⠀⠀⠀⠈⠙⢦⡤⢀⠒⠒⠠⢤⡀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡂⠥⠐⠚⠓⠚⠋⠠⠀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠜⢓⠒⠣⠀⠏⠉⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡰⠊⠀⠀⠀⠀⠀⠀⠸⠁⠘⠁⠉⠁⠀⢻⡀⠀⠀⡠⢐⠤⠤⣾⠉⡝⠉⠂⠀⡄⠀⠆⠸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⢷⠀⡜⡐⠁⢀⣀⠀⠣⢠⠀⠘⡀⠠⠀⢀⠀⡄⠈⡸
⠀⠀⠀⠀⠀⠀⠀⠃⠸⠀⠀⠀⠀⡀⢠⡤⠤⣴⠀⡇⢸⠀⠀⠻⠀⠀⠈⢷⠀⢃⠀⠰⡈⣀⣀⡀⡆⠀⠓⠊⠀⠈⡄⠰⠀⢣
⠀⠀⠀⠀⡠⢄⡴⢦⡴⠀⠤⠒⠲⡃⠈⠀⠀⠈⢇⡃⢘⠀⠀⢰⢠⠀⠀⠘⡀⠈⠀⡀⠈⠒⠠⡀⢀⠀⠠⠒⢢⠀⠰⠀⠷⠺⣦
⠀⠀⢀⠎⡰⠁⢀⡀⠈⢂⠂⠀⠀⠐⣀⠀⠀⠀⢚⡇⠀⠀⠀⠸⠀⠆⠀⠀⠰⠀⢀⠬⠑⠢⠀⠈⡌⡄⠀⢃⠀⠄⠀⡄⢀⠀⡀⠆
⠀⠀⠈⠀⠇⠀⢨⠘⡀⠀⠳⠀⠰⡄⠸⠀⢸⠀⠈⡁⠀⠀⠀⠀⡀⣐⠀⠀⠀⠡⠡⡀⠑⠒⠁⢀⠆⠰⡀⠰⠀⠈⠈⠁⣀⠀⠀⠆
⠀⠀⠘⡄⠘⢄⠀⠁⠂⠌⡀⠃⠀⢳⡀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢣⠈⠂⠤⠔⣪⠦⠀⠀⡤⠰⠞⢛⠥⠁
⠀⠀⠀⠈⢢⠀⣉⠒⢄⠀⠐⡌⠀⠘⡱⡀⢨⠘⠀⠘⠀⠀⠀⠀⠐⢈⠉⢣⠀⠀⠀⢻⡂⠒⠈⠀⠀⠀⠀⠖⠀⠈⠁
⠀⠀⠀⢔⠁⠪⡀⠑⠤⠃⠀⠇⢆⠀⣡⠉⡉⠀⠂⠀⠁⠀⠀⠀⠀⠏⢆⠀⠢⠄⠐⠒⠱⡀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠣⡀⠈⠢⠀⠤⢊⡄⣀⢠⠤⠇⠈⠉⠁⠁⠀⠀⢀⣀⡰⠀⠈⠒⠒⢂⠈⠭⠀⢀⣀⡀⠄⡪⠎
⠀⠀⠀⠀⠀⠈⠐⠀⠀⠒⠁⠀⠃⠈⡄⠀⠀⠀⠀⢀⣀⡀⠀⠤⠄⠔⠀⠀⢀⢌⠖⠂⠀⠀⠠⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢂⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠑⠤⠀⣀⠀⠀⠄⣊⠔⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠄⠐⠈




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⢄⣤⢀⡄⢠⡄
⠀⢂⢀⡀⡀⠀⠀⠂⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⣔⣐⣇⣷⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⢀⠹⠇⠀⠀⠀⠈⠛⠱⣟
⠀⠢⣄⠌⢠⠀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠸⢼⡿⠿⠿⠿⠟⢿⡀⠐⠀⠀⠀⠀⢦⡄⠀⢀⠄⡀⠀⠀⠀⠉⡁
⠀⠀⠨⣀⣉⠤⠤⠒⠒⠐⠣⠠⠀⠀⠰⠀⠀⠀⠈⠹⠀⣀⠀⠀⢀⠸⠃⠀⠀⠀⢀⣘⡓⢾⠁⠀⠀⠀⠉⠒⠠⣀
⠀⠀⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠉⠉⢫⡷⣤⣄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠿⠷⠴⠶⡲⠀⡠⢃⠈⠢⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢫⢎⠈⡇⢀⡂
⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠋⣠⣬⣁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠤⡀⠀⠀⢀⡠⣠⢫⢣⠱⠃⠁
⠀⠀⠀⠀⠈⠀⠈⣀⠀⡄⢠⣿⣾⣿⠛⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠛⠰⠇⠘
⠀⠀⡆⠀⠀⠀⠀⠀⢈⠇⠈⢻⣿⢀⢢⠠⢘⡇⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠤⡤⠐⠂⠄
⠀⠀⠿⢤⣄⠀⣀⣀⡤⠀⠀⠏⠉⠁⠢⡶⠟⠁⣼⣇⡐⡀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢀
⠀⠀⠀⠀⠸⠋⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⢺⣻⣯⣼⡏⡅⠀⠀⡀⣰⡌⠄⠠⠀⠀⠀⠠⠀⠀⠈⠋⡁⠈⠈⠒⠈
⠀⠀⠀⠀⠀⠀⢀⠐⠂⠤⡤⠤⡀⠀⠀⠀⠀⠀⠻⠓⠀⠀⠈⠈⠉⠀⢱⣈⡴⠒⠉⠚⠀⠘⠈⠓⠚⣽
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⣀⡲⠄⢀⣰⣴⡰⠂⠁
⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⣆⠉⠉⠒⡀⡀⣰⢦⣦⠀⠀⠀⠀⠈
⠀⠀⠐⣤⡁⢀⠐⠀⠀⠀⠀⠀⠐⠂⠒⣦⡞⠀⠀⠘⠁⠀⠻⠿⠋
⠀⠀⠈⠉⠋⠠⠀⠀⢠⣶⣶⣾⣧⠄⠠⠿⡿⢶⠞⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡀⠤⠖⠒⣒⣀⡀⠀⠀⣀
⠀⠐⠀⠁⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡾⢒⣟⠔⠂⠀⠀⠈⠁⠀⠀⠀⡰⠋⣠⡶⠼⠟⠛⡛⠻⢿⣦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡙⢹⣿⣾⢰⡏⣄⡀⠀⠀⠀⠜⢡⣾⠟⠁⠆⠁⠀⠀⠹⠤⠈⠻⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⣿⣿⣴⣿⣿⡏⠛⠛⠀⠂⠈⢠⣿⡀⢻⣴⣯⣀⠀⡀⠀⠀⠀⠠⠙⢿⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡿⢿⠻⣯⣿⣭⡏⠈⠐⣤⡅⠀⢸⣟⠆⠀⠿⢿⣧⡾⣿⣼⣤⣦⡿⠀⣿⡇
⠀⠀⢀⡀⠀⠀⠂⠀⠀⠀⢄⠀⢸⣿⡏⢨⢄⣻⢻⣾⡇⠀⠀⠟⠃⠀⢸⣿⡀⢀⡀⠀⠀⠀⠈⠁⠘⠘⠃⡐⣿⡇
⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⡀⠀⠿⣿⡇⠀⠀⠐⠀⠀⠀⠉⠀⠙⢧⣀⢀⣠⠀⠀⣀⣄⣌⠙⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠰⠆⣌⠙⠛⢚⣥⣭⣿⣿⡁
⠀⠀⠀⠀⠀⡀⠈⠀⠁⠀⠒⠆⡀⠀⠀⢠⠀⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠠⣿⣿⣷⣝⣦
⠀⠀⢘⡒⢉⡇⢀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠏⠙⠂⠀⠀⠀⠄⠀⢹⣿⣿⣿⡙⠂
⠀⠀⠨⡯⠅⠍⠈⠢⢴⡀⠀⡀⢂⣤⠁⢠⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠂⣲⠀⠀⠙⠀⠀⠀⠀⢈⣼⣿⣯⡼
⠀⠀⠀⢸⡮⠀⠀⠀⠀⠀⠙⠠⡌⡁⢻⣼⠅⠠⢆⠘⣃⡠⠠⠤⢄⣆⣸⢀⡛⢠⠀⠁⠀⠀⠀⠀⠸⠟⠋⠁⣀⣀⡀
⠀⠀⠀⠈⣿⣿⣷⣾⣄⠀⣨⣠⣬⣶⣾⣿⢁⣿⣼⣿⢀⡀⢀⠀⡀⠀⢀⣡⣷⣿⡇⠐⣤⠄⠀⠀⠀⠀⢰⣿⠉⣉⡀
⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣷⣿⣾⣿⣿⣾⣿⣿⣿⡇⢸⠯⢤⣀⣀⢀⣠⣼⣧⣴⣿⡇
⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣟⣿⣯⣙⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣭⣭⣽⣿⣽⣿⣽⣯⣽⣿⣿⣯⣿

⠀⠀⢀⠀⠀⠀⢀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣤⡀⠀⢀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⢠⠤⠀⠀⠀⠀⣠⣤⣴⣶⣤⣤⣄
⠀⠀⠁⠳⠲⠲⠀⠠⠀⣾⡧⣾⠱⢌⠴⠉⢿⣿⠀⡏⢒⣈⢐⣾⠑⠸⢣⣶⡇⢒⣏⠀⠀⠀⠐⠁⢀⠀⢸⣿⣜⣤⡿⡿⣿⢹⡇
⠀⢨⠀⠒⢠⠀⠃⠀⡆⠘⠿⣿⣿⣿⣿⣿⠿⠃⠀⣷⡼⢿⣬⣥⣼⣧⣦⣭⣦⣭⡿⢠⠑⡔⠖⠎⠀⢁⠘⢿⣎⣘⣧⣥⣾⡿⠃
⠀⢸⡂⠤⠴⠀⠄⣴⠀⠀⠀⠟⠃⠀⠀⠀⢀⢢⣀⣈⣀⠄⡀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠶⠆⠚⠃⠀⡇⠀⠼⠋⠉⠉⠁
⠀⠀⠀⠙⠖⡆⠉⠀⠀⠀⢀⡄⠐⠒⠀⠈⡊⣀⣥⠩⣥⡄⢀⠀⠀⠀⠀⡈⢀⠀⠀⠘⣀⡀⠀⠂⠀⠐⠢⠠⢀⣀
⠀⠀⠀⠠⠤⠇⢠⠀⠀⠀⠀⠂⠀⠞⢑⠄⡆⠛⠁⡂⠉⠁⢘⠔⠄⠀⠀⠈⠀⢀⠤⠊⣡⠀⠀⠀⠀⠀⢸⠁⠑⠂⢁⡆
⠀⠘⢸⣿⣆⣿⣿⠀⠀⠀⠀⠢⢀⠈⠠⢤⣶⣦⣦⣴⣶⠒⠏⠚⡈⠄⠀⠀⣐⡁⠔⠁⠀⠆⢀⠀⠀⠄⢐⡆⠀⢠⢀
⠀⠀⢻⣿⡟⢻⣿⡖⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⡛⠉⣻⣦⡘⡅⠢⠀⠀⠄⠠⠁⠀⠀⠀⣸⣶⣖⣺⣿⣿⣷⠀⠣⠬
⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⢀⠂⢀⢊⠇⠀⣿⣿⠷⡶⠿⠿⡇⠀⠀⠊⠔⠙⡡⠆⡆⠀⢰⣿⣿⡿⠿⠿⣿⣿⡆
⠀⠀⢸⣿⡇⠈⣿⡇⠀⠀⢰⡤⠀⠩⠂⠀⡇⠀⠒⠒⠒⠀⠃⠀⠀⡄⠋⣤⠳⢡⠇⠀⠀⠿⢇⠀⠀⠀⠈⣻⣿⡆⢲⠡⠘
⠀⠀⢸⣿⡇⠀⣿⡧⡀⠀⠀⢁⢔⠕⠁⠀⠇⠀⠊⠀⠁⠀⠀⠀⠀⠈⠓⠒⠚⡁⠀⠀⠠⣔⡀⠂⣀⠠⠄⠐⠚⠉⠄
⠀⠀⠉⠄⠀⠀⠐⠒⠀⠀⠘⠕⠁⡀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠾⠀⠁⡀
⠀⣴⠶⣦⢰⣶⠀⣴⡆⣴⠾⣶⠀⣿⡇⠀⠘⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠂⣀⣀⣀⠐⠀⠀⠀⣼⢻⠀⠸⣿⠟⠀⠀⡴⣶
⠰⣿⠀⣿⢻⣿⣿⢿⡇⣿⠰⣶⢸⣿⠃⡐⠈⢉⠀⠀⠀⠓⠓⢢⠀⠀⡀⢀⡁⠉⢡⡀⢀⠀⠀⣯⡞⢀⡶⣦⣾⢸⡟⢿⣡⠆
⠀⠻⠾⠛⠸⠟⠈⠸⠇⠛⠷⠛⢘⣛⠀⠀⠀⠀⡄⠢⠀⠉⠀⠁⠀⠀⠇⠀⠀⠃⠀⠀⠘⠀⠚⢿⣀⡾⣏⡿⠘⠞⠀⢀
⠀⠀⠌⢀⡀⣀⢀⡀⠐⠐⠀⠀⠈⠉⢀⡠⡎⢠⡹⣤⡄⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠃⠀⠀⠀⠁⠀⠀⠀⢀⢰⠒⠁⠢⡀
⢀⠔⡀⢀⡁⢈⠈⢁⣀⣀⡢⡄⢠⠺⠇⠀⢫⠶⡆⡿⠚⣄⠀⢠⠀⠀⣄⡄⠀⠀⠠⡄⡀⠀⠀⠀⠀⡠⠐⢀⠀⠴⠠⣀⠄
⠘⣸⣿⣿⡟⠂⠑⢿⣿⣿⣷⠁⠈⡷⠖⣰⠖⠋⠉⠀⠀⠀⣴⢀⣹⠈⡠⠁⠀⠀⠀⡃⠀⠀⢀⠔⠁⠀⢀⠕⠁⠠
⠀⠋⠈⠉⠁⠀⠀⠈⠉⠋⠈⠀⠀⣉⢉⠁⣀⠀⣀⠀⡀⢑⡀⠊⠀⠀⢸⠈⠉⣉⠀⠁⡃⠀⠀⠀⠐⠀⠁⠀⠌
⠀⠰⠀⠀⠀⠄⡀⠀⠀⢀⠆⠀⠀⣻⣯⢸⣏⣱⣹⣿⡱⣿⣹⠆⠀⠀⢨⠀⢸⠀⠇⠀⡇
⠀⢀⣤⣀⣠⣀⣀⣤⡀⣀⣤⣄⣀⣀⣉⣀⣉⠁⠀⠈⠁⠉⠉⠀⠀⠀⠈⠂⠚⠀⠈⠉⠁⠐⣏⢴⢋⡷⠢⣄⢶⠉⠀⣠⠂⣄⠂
⠀⣾⢀⠹⢃⢸⡏⠴⠹⡏⣴⠶⡇⠸⢡⣤⡼⠂⠀⠀⡤⠔⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⡆⢸⡇⢅⡿⢸⠀⠀⣇⢂⣸⢃
⠀⣿⣈⣇⣸⣀⣀⣿⣄⣅⡛⣃⣄⣸⡘⠟⣲⡆⠀⠀⠒⠢⠊⠀⠀⠀⡠⠇⠰⠀⠸⡒⢓⠀⠠⡷⠎⠻⠬⠴⢻⠤⠥⠿⢪⡥⠆
⠀⠈⠛⠙⠉⠛⠛⠉⠛⠋⠛⠛⠙⠛⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠒⠒⠲⢦⣄⠀⠀⠀⢀⣤⠴⠚⠛⠉⠉⠉⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠚⠛⠛⠓⢶⣾⠛⠛⠒⠶⠶⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⣤⣤⣤⣤⡄⠀⠀⠀⣠⡦⠤⠤⢤⡄⢠⡟⡿⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀⠀⣴⠞⠛⠋⠙⢷⠀⣼⣇⠀⠀⠀⠀⣷⠀⢀⡔⣿⠀⠀⠀⢸⠁⡞⠀⡇⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⣀⣻⡄⠀⠀⠀⣠⣄⡀⣀⣤⠶⠚⠋⠉⠁⠈⣧⠀⡇⠀⠀⠀⠀⠈⢷⠇⢿⠀⠀⠀⠀⢹⡀⣾⠀⡇⠀⠀⠀⢸⢰⡇⢠⠇⠀⠀⠀⠀⢠⡏
⠀⠀⢠⡶⢾⣏⡉⠁⠀⠀⢀⡼⠋⠈⠙⠻⢤⣀⠀⠀⠀⠀⠀⠘⢾⡇⠀⠀⠀⠀⠀⠘⣆⢸⠀⠀⠀⠀⠘⣇⡇⢰⠇⠀⠀⠀⢸⢸⠀⢸⠀⠀⠀⠀⠀⣾
⠀⢠⡾⠀⠀⠀⠉⠓⢦⣤⠏⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⢀⣈⡧⠀⠀⣤⡄⠀⠀⠘⣾⠂⠀⠀⠀⠀⢻⡇⢸⠀⠀⠀⠀⢸⣿⠀⢸⠀⠀⠀⠀⢠⡇
⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⢰⣟⣻⡏⠀⠀⠀⠓⠋⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢸⣿⠀⣸⣀⣀⠀⠀⣸⣅
⣸⠙⣿⠒⠲⠤⣄⡀⠀⠀⠀⠀⠀⢠⣾⢧⠀⠀⠀⠀⠀⠀⠉⣉⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠾⢧⣤⣉⣭⠟⠃⠈⠳⡄
⢿⠀⠈⢷⣤⣀⡼⠁⠀⠀⠀⠀⣠⠏⣸⡈⢇⠀⠀⠀⠀⠀⠀⠿⠿⠖⠲⡄⠀⢸⠙⠦⡀⠀⢠⠼⠂⠀⠀⢀⣠⣤⣄⠀⠀⠀⢸⡇⢀⣤⡾⠋⠉⠻⣆⠀⠀⢹⡆
⠘⢧⡀⠀⣰⠏⠀⠀⠀⠀⠀⣰⠃⣰⠟⣧⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⠞⢀⣀⠈⣻⢻⠀⠀⠀⠀⠸⠁⠀⡏⠀⠀⠀⢸⣇⣾⢿⡁⠀⠀⢀⣿⠀⠀⠀⣿
⠀⠀⠙⣿⠋⠀⠀⠀⠀⠀⣰⠃⣴⠃⠀⠘⣇⢹⡀⠀⠀⠀⢀⣀⣀⣤⡤⢬⣧⣤⠼⠋⠳⠟⣿⢦⠤⠄⠀⡚⢋⡛⠷⠄⠀⠠⠸⠛⢛⠈⢛⣶⣶⠟⠃⠀⠀⠀⣿
⠀⠀⠘⣷⣤⣀⠀⠀⠀⣰⠃⣸⠏⠀⠀⠀⢸⡄⠳⠒⠛⠉⠉⠁⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠿⣤⣤⣠⣠⡟⠸⢧⡴⠶⠶⢦⣼⠃⠈⠛⠛⠋⠀⠀⠀⠀⠀⢰⠟
⠀⠀⢸⠋⢦⡈⠉⠓⠶⠏⢠⠏⠀⠀⠀⠀⠀⢷⣀⣀⣤⡤⠴⠶⠒⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠸⡆⠈⠙⠲⣄⡀⣰⠏⠀⠀⠀⠀⠀⠀⢈⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⢳⡄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⡴⠋⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠉⠛⠶⢤⣤⣀⣀⣀⣀⣀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣀⣀⣤⠴⠞⠉⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣄⣀⣀⣀⣤⡤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⠭⠭⠿⠿⠿⠿⠦⠴⠶⠶⠶⠶⠶⠶⠶⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⠗⠛⠛⠛⠛⠒⠒⠒⠒⠒⣒⣶⣶⠶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠤⠄⠒⠒⠈⠉⠉⣀⠀⢸⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢠⠒⠒⠂⠉⠉⠉⢸⠀⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠉⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢠⠖⠒⡆⠠⠖⠒⢲⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠈⠀⠀⢸⠀⢸⡇⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣇⣸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡛⢻⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣇⠈⠒⠒⠃⠠⢄⣀⣸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠉⠁⠒⠒⠤⠄⣀⣀⠀⢸⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠤⠤⠤⠤⠤⠤⠄⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠸⠃⣴⠋⢀⡠⠔⠒⠉⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⡄
⠀⠀⠀⠀⠀⠁⢀⡞⠀⢀⣴⡿⠋⠙⠳⠀⢠⡶⠳⣦⠀⠀⣰⡾⠛⣷⡄⠀⠀⣶⠒⢦⣄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢸⠇⠀⣾⣿⡇⠀⠀⠀⠀⣼⡇⠀⢻⡇⢠⣿⡇⠀⢸⣿⠀⢸⣿⡀⠀⢻⣧⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⣿⡇⢀⣀⣀⠀⣿⡇⠀⢸⣿⢸⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⠈⣿⡇⠀⢸⣇
⠀⠀⠀⠀⠀⠀⢸⠀⠘⣿⣿⡇⠀⢸⣿⢠⣿⡇⠀⢸⣿⢸⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣸⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢻⣿⣧⢀⣼⡟⠀⣿⣧⢀⣾⡏⠈⢿⣷⣀⣿⡟⠀⠸⣿⣧⣾⠿⠋⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠛⠿⠋⠀⠀⠈⠛⠛⠋⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⣤⠀⠀⣤⠀⠀⣠⡴⠒⠂⠀⠀⣾⡄⠀⣠⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⠀⠀⣿⡄⢰⣿⡇⠀⠀⠀⢀⣿⣧⡾⠋⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢹⣿⠀⠀⠀⠸⣿⡇⠀⣿⠇⠸⣿⡇⠀⠀⠀⠘⣿⣿⠷⣦⡄⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠛⠒⠂⠀⠀⠙⠿⠞⠋⠀⠀⠙⠿⠶⠖⠀⠀⠻⠟⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠳⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡄⠀⣴⠶⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⠀⡿⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢰⣇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⠀⠹⣧⣀⣄⡀⠀⢸⣿⡄⠀⠀⠀⢀⢡⠀⠀⣾⠀⠀⠄⠀⠐⡀⣁⠠⣠⣾⣿⠇⢀⣤⣶⣾⡄⢀⠐
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⡀⡰⣿⣿⣿⣿⣆⣰⣿⣿⣖⠀⢡⠰⢖⣂⢨⣿⣧⠀⢐⡈⡄⢠⣰⣾⣿⣿⣿⠀⣿⣿⣿⣷⡶⠃⡀
⠀⠀⠀⠀⠀⡤⠀⠀⠈⢻⣿⣷⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⡔⠀⣹⢥⣾⣿⣿⣷⣮⣴⣶⣿⣿⣿⣿⣿⡇⢸⣿⣿⡟⠻⢡⡌⡤⠐⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣛⣃⣹⣧⣴⠟⢀⡀⠀⠠⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠠⡄⠤⣴⣶⣷⣾⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠈⠵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣦⣶⣼⣿⣿⠉⠁⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠚⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢻⣿⣿⡿⠛⠛⣿⠉⠉⣿⠀⠀⠀⣾⣿⣿⠃
⠀⠀⠰⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠉⢿⣿⣿⣿⠏⠀⠀⠸⣿⣿⡿⠁⠀⠀⠀⢸⣿⣿⡇⠀⠰⠇⠀⠀⣿⠀⠀⢠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⠋⠉⠉⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⢏⣧⠀⠀⠀⠀⠀⢸⣯⠄⠀⠀⠀⠉⠙⢿⣿⠃⠀⠀⠀⠀⠀⣿⣀⣀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⢀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣋⠀⠀⠀⠀⠀⠀⣿⠄⠿⠄⣤⠀⠀⠀⢻⠀⠀⢰⡆⠀⢐⡟⠁⠀⢹⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠙⠛⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣸⣧⣤⣾⣧⡀⢀⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠹⣿⣿⣶⣤⣀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣀⣀⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣠⣿⡟⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣤⡇⣀⣀⣠⣾⣶⣶⣶⣾⣿⣿⣷⣾⣷⣿⣿⣿⣿⣿⠿⠛⢛⠩⠝⢻⢿⡿⠏
⠀⠀⠀⠠⣾⣿⡋⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠴⠔⡂⢋⠨⠄⠂⢀⠈
⠀⠀⠀⠠⣿⣿⣿⣷⣶⣦⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⣇⠡⠀⠆⠂⠈⠀⠄
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠰⠀⣃⢡⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠉⠋⣼⣿⣿⣿⣿⡿⠟⠛⠻⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣠⣴⣾⣯⣯⡟⠻⣦⡁⠠⠀⠂⠀⠀⣀⡀
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠀⢀⣿⣿⡿⠛⣁⣴⣷⣿⣿⣷⣿⢿⣿⡟⠉⣠⣤⡔⣾⢛⣿⣿⣿⡟⠀⣼⣿⣽⣯⣿⣻⡝⣅⠠⠘⡀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣾⡿⠋⢀⢸⣿⢯⣿⣿⣿⣿⠱⠘⣿⢀⢨⠌⠆⢻⡈⠉⠧⠸⠻⣇⠀⢿⡿⣿⣿⢿⡿⡿⣾⠊⡉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⠀⠁⠀⠄⢾⣿⡿⣏⠿⠒⡀⣁⠩⠰⠆⢃⢙⡈⠧⠀⢚⡈⡅⠱⠀⠻⣿⣽⣞⢷⣻⣜⣌⠧
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡽⠖⠘⠈⡀⠡⠰⠀⣃⢡⠨⠄⢂⠚⠀⠈⠁⠰⠈⡃⠡⠐⠜⢏⢫⠻⡜⢦⠐
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠠⠀⠂⢃⠸⠀⠀⠀⠀⠀⠄⠰⠀⠂⠁⠠⠀⠃⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡔⢄⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡘⠀⠈⠄⠘⠄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⠀⠒⠠⠄⣀⡐⠀⡐⠀⠀⠀⠈⢄⠘⢄⠠⠄⠒⠀⢉⡀⣤⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠘⠔⠂⠄⠀⢀⠀⠀⡔⠀⠀⠀⠀⠀⠈⢆⠀⠄⠀⠒⠈⠁⡐⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠈⢆⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠉⡀⠄⠃⠀⠀⠀⠀⡀⢀⠤⠄⠀⣀⠀⢀⠀⠀⡄⠀⠘⠂⢄⡉⠐⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⢉⡠⠄⠂⠁⠀⠀⠀⠀⠀⠑⡔⠀⢘⠀⠀⠀⡘⡀⠀⡄⢀⡇⠀⠀⠀⠀⠈⠁⠂⢄⡈⠁⠂⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢀⢀⠈⠐⠢⠄⣀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⢠⠀⢁⠀⠀⠀⡇⠀⠀⠀⣀⡠⠄⠂⡁⡀⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⢄⠀⠈⡐⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠰⠉⠀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠜⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠱⡀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⡌⠠⠄⠒⠉⢀⡀⠐⡄⠀⠀⠀⠀⢀⠂⢀⠈⠐⠂⠤⠀⢐⣄⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⡀⠤⠔⠂⠉⠀⠑⡄⠘⡄⠀⠀⢠⠃⠠⠁⠈⠁⠒⠀⠤⢀⡈⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠘⡄⢀⠁⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠘⠃⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⠿⠿⣛⣋⣁⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣿⣿⡟⠉⢠⣶⣾⣿⣿⣿⣿⣷⣶⣄⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡀⠀⢸
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣧⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡰⣆⢰⢠⡆⣤
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠷⣌⡡⠉⠀⠀⠘⠁⠛⣤⡤
⠀⣼⡏⣻⣶⣰⢿⣋⣿⣴⣶⢦⡄⠀⠛⢛⠓⢰⢠⣊⣐⡀⠀⠁⠈⡀⠶⠆⠀⠀⢀⠠⢤⣤⣤⡔
⠀⢈⠛⠉⠀⠀⠀⡀⠀⠉⠉⠋⢈⠀⠛⣋⣁⡖⠸⠿⠍⠉⠩⠙⢱⡛⠶⠶⠀⠀⠀⠑⠦⡬⠤⠚⠈
⠀⠀⣷⣄⣠⣴⣆⣀⠲⣤⣤⣆⣨⣥⡘⠋⣡⡶⢀⡞⡓⣦⢺⡁⢈⡙⠷⡆
⠀⠀⠘⠋⠛⡿⠛⠉⢻⣿⣷⣿⣿⣿⣷⣮⡉⠰⠟⠰⠇⠹⠀⠷⠈⠛⠂⡀⢀⡀⢀⣠⣤⠤⣤⣄⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡞⣰⠟⠁⠀⠀⣀⢻⣡⡌⢷
⠀⠀⠀⡀⠐⢀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣤⣚⡇⣿⠈
⠀⣄⠀⠀⠀⠀⠁⠀⡴⣂⠀⠀⠀⠀⠠⡀⠀⡀⢠⠀⠀⠀⠈⢠⢠⠠⢤⠄⣄⢻⡀⢍⣛⣛⡁⠞⣡⠏⣴
⢀⣷⣶⣠⡄⣀⣀⣶⣿⣿⣇⠀⠀⠀⠀⠙⠀⠀⠈⠓⣷⣶⣦⣿⣿⠿⠋⣤⠙⣦⡙⠶⢬⠭⠴⠶⠋⠾⠃
⠈⠛⠋⠉⠉⠉⠉⠉⠉⠉⡉⢠⡀⢤⢠⡄⡄⡄⡄⡠⢀⡴⠖⢀⣀⣀⣀
⠀⠀⠀⠀⠰⣀⣀⠀⢠⡈⠳⣄⠛⣈⠧⠣⠁⠃⢙⠕⠡⣚⡋⢉⣉⣉⡉⠁⠀⠀⠠⠀⠀⠂
⠀⠸⠐⠀⠤⢼⣇⠀⣀⠙⠶⣬⣛⠊⢁⣍⣠⣀⣠⣠⡌⠑⠒⠶⢤⣌⠁⠀⠀⢀
⠀⠀⠤⠆⢠⠀⠀⠀⠉⠛⠲⠶⠌⠅⢙⠛⣛⢿⣷⠛⡅⣰⠎⡙⠦⡌⠀⠀⠐⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢛⣋⡭⢊⠝⡆⡝⡶⡜⡌⢧⡙⠆⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠻⢷⣄⢶⡰⠆⡆⠖⣴⢂⣴⡆⠀⠀⠀⠀⠀⠀⠁⠁⠉⣈⠁
⠘⠻⠶⣍⡀⠀⢀⣠⣤⣶⡏⡧⠶⠀⠀⠀⠀⢀⠴⠉⣉⠁⠀⠁⠒⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠘⣿⣷⠿⠛⢋⣠⣴⡲⠶⠶⢶⠀⠀⢰⡃⠰⡀⠿⠴⠴⠶⡤⠃⠀⡇⢀⢀⢀⠀⠀⠷⣶⣶⡖⣈⡿
⠀⠀⢠⡶⣫⢤⢯⢫⢁⣲⣍⡛⠶⣦⣄⠀⢾⣿⣶⣗⣖⡦⠒⣀⣴⣆⣴⣷⣶⣶⠞⠀⠀⠀⠰⠌⣾⣿⣿
⣀⣀⣀⣀⣁⣀⣘⣘⣃⣛⣈⣛⣂⣀⣉⣀⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣙⣛⣋
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡌⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⠡⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡁⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠔⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡁⠤⠤⠤⠄⠐⠊⠈⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⠀⠀⠀⠀⠄⣀⠠⠤⠒⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⡤⠔⠒⠉⠁⠒⠀⠀⠐⠂⢀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠢⠄⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⡸⠀⣁⠈⠀⠀⠀⠈⠀⠀⠊⠂
⠀⠀⠀⠀⠐⠃⡀⠀⠀⠀⠠⠁⠄⠐⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠌⠂⠀⠁⠀⠀⠀⠁⠄⣀⣀⣀⣀⡀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠒⠘⠤⢤⣀⣀⣀⣀⣀⣀⣤⡼⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⠈⠁⠁
⠁⠅⠀⠉⠁⠉⠀⠀⠒⠂⠀⠀⠰⠀⠀⠀⠃⣀⠀⠀⠀⠄⠄⠀⠄⠀⠂⠒⠀⠐⠀⠈⠀⠀⢀⡤⠤⠤⠤⠄⣸
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠒⠒⠀⠈⠁⠀⡀⡀⠀⠈⠀⠀⠐⠂⠀⠁⠉⠉⠀⠁⠐⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠀⠂⠰⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣴⣶⣶⣶⡶⠶⠶⠶⠿⠶⠾⠿⢿⠿⠿⠿⠿⠿⠟⠿⠿⠻⠛⠛⠛⠻⢿⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⡀⠀⠀⣀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠘⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠻⡇⠀⢸⣿⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣯⢿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣧⠀⠠⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣆⢺⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢹⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣻⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣛⢽⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⠀⠀⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡹⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢾⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⠀⠀⢻⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠤⢢⡄⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⢀⠀⣿⣸⣧⠀⣀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠐⠀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠑⢿⣆⡬⢤⣤⣥⣭⠂⢀⠀⢰⠀⠀⢸⡇⠀⠀⠰⠂⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣽⠀⢰⣿⣠⡸⣷⣀⣀⣀⣩⣠⣾⠀⠀⢸⣗⣒⣒⣒⣒⡷⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⢻⠰⠞⠛⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣻⠀⠓⠤⠛⢠⣦⣶⣶⣶⡶⣶⣶⠀⠀⢶⣶⠶⠶⠶⠶⡿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⢠⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢨⠀⠀⢸⡗⠀⠀⠀⠀⡃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠂⠄⠀⠀⠸⣿⠀⠀⠀⢀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⡇⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⣵⡂⣀⠀⠀⠀⣿⠀⠀⠀⢀⠀⢸⠀⠀⢸⡇⠀⠀⠀⡄⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠘⣿⡇⠀⠀⠀⠀⣿⣄⣀⣀⣉⣀⣸⠀⠀⢸⣷⣀⡀⠀⠀⣤⡇⠀⠀⠀⡄⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡃⠤⠀⠀⠀⠉⠀⠈⠉⠉⠉⠉⠀⠀⠈⠉⠈⠉⠉⠉⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⡀⢿⣇⣀⢀⠀⡀⡀⢀⠀⠀⡀⣀⣀⡀⢀⠀⠀⡀⣀⣀⢀⠀⠀⠴⠖⢠⡇⠀⠀⣀⣘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣭⡭⠱⠛⠛⠛⠛⠛⠻⠛⠿⠛⠿⠿⠽⠧⠽⠬⡧⢤⠶⠶⠶⠾⠶⠶⠶⠿⠾⠷⠶⠤⠭⠬⠴⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠉⡍⠉⠁⠛⢛⠙⢋⠜⢋⠗⠠⠖⠠⠖⡰⠆⡰⠆⠶⠲⠆⠴⢢⠦⢠⠄⡤⢀⢄⠀⢠⠤⠴⢠⠄⠶⠰⠖⠰⠂⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡔⣠⢂⠐⡀⠐⡡⢊⠥⢊⠠⢊⠄⢊⠤⢊⠔⡢⠀⡠⢚⠔⡡⠒⡡⢊⠔⡨⠊⢔⠤⠂⠀⡡⢊⠔⡡⠂⠀⡠⣚
⠀⠀⠀⠀⠀⡀⠤⠔⠊⠁⠁⠈⠉⠉⠙⠃⠚⠒⠛⠒⠓⠶⠃⠶⠃⠶⠅⠾⠤⡾⠶⠷⠞⡴⠿⠶⠷⠶⠶⠳⠗⠶⠒⠷⠲⠧⠾⠖⠺⠚⠛⠙⠢⣄
⠀⠀⠀⠀⠛⠩⠂⠦⢤⠄⣤⣠⡄⣠⣄⣠⣀⣀⣀⣀⣀⡀⢀⡀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡄
⠀⠀⠀⠀⢸⠑⠡⠒⠤⠢⠔⡡⢒⠤⢂⠤⠊⠄⠈⠠⢊⠡⢋⠝⣉⠝⡩⢋⠉⠉⠝⡉⢝⠉⠋⠝⡋⠋⠙⠋⠜⠁⠟⡡⢋⠉⡩⢉⠍⡩⢋⠍⡩⠉⢋⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠉⠀⠁⠐⠁⠊⠀⠀⠔⠁⠈⠀⠊⠐⠁⠐⠀⠂⠐⠀⠊⠁⠂⠠⠊⠐⠀⠂⠀⠀⠈⠀⠁⠈⠀⠁⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣀⣾⣿⣿⣿⣿⣆⣠⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⠿⠟⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢏⣵⣾⣿⠿⣿⣷⣮⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⣾⣿⣿⢣⣶⡌⢿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⣿⡇⢰⣶⣶⢸⣿⣿⢃⣿⣿⣿⣿⣿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⢸⣿⣿⢸⣿⢋⣼⣿⣿⣿⣿⣿⡿⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡞⣵⣿⣿⠏⠉⠛⠛⠋⠀⠀⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠀⠙⠉⠁⠀⠀⠀⢀⣀⡀⠀⠻⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠦⣭⠴⢆⣤⣤⣀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡘⣿⠇⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡉⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣇⢻⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠙⠛⠛⠛⠛⠘⠛⠛⠛⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡭⠤⠴⠒⠒⢻⡎⠁⠀⠀⠀⢠⢠⠶⡟⠉⠉⠉⠉⠉⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡥⠤⠤⠴⣦⣄⠀⠀⠀⠠⣄⠀⠀⠀⠀⣠⢚⣇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⢻⢰⡇⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⣀⠤⣀⡀⠀⠈⡼⠁⠀⠀⢠⣧⠼⠶⠒⠛⣷⠀⢀⢴⡟⠉⠉⠉⠉⢷⠀⠀⠀⠀⡇⠀⢤⣤⣖⡀⢸⢸⡇⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⢀⠖⠛⣄⠀⠀⢰⠃⠀⠀⠀⡟⠀⠀⠀⠀⠀⢻⠀⡟⡸⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣧⢶⠛⠉⠉⢹⢸⢸⡇⠀⠀⠀⠀⡾
⠀⠀⠀⣴⡁⠀⠀⠈⢢⡀⡎⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢸⣤⢣⠇⠀⢀⣀⠀⠀⠀⢻⡄⠀⠀⣿⣼⠀⠀⠀⢸⡄⣺⡇⠀⠀⠀⣼⠃
⠀⢰⡸⣅⡙⢦⡀⠀⠀⠙⠁⠀⠀⣰⠃⠀⠀⠀⣀⣀⣠⣼⡏⡾⠀⠀⠘⠚⠃⠀⠀⠈⢷⠀⠀⠉⠉⠀⠀⠀⢸⣇⣿⡇⠀⠀⣰⠏
⠀⠀⠳⠤⠙⠢⣝⢦⡀⠀⠀⠀⣰⣧⠀⠀⢰⡏⣉⣡⡵⣿⣱⠃⠀⠀⠀⢀⣀⡀⠀⠀⢸⡆⢰⣾⠷⡆⠃⠀⢸⡻⣾⣋⣉⣿⠏
⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⢠⢏⢿⡀⠀⠀⠛⢻⡇⣰⢃⣯⣤⣄⠀⢰⣏⡽⢻⡀⠀⠈⣧⢸⣿⢀⡇⠀⠀⢸⡇⣠⠶⠒⢦⢠
⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢀⡏⢸⠸⡇⠀⠀⢸⡯⠟⠛⠋⠁⠀⠸⣦⠏⣍⢳⠀⣧⠀⡀⢸⣼⣿⢰⡇⠀⠀⢸⡾⣻⣤⣠⡼⠱⠃
⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⡞⠀⣸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⡂⢧⠸⣄⡤⢼⠟⢸⢸⣇⣠⣤⣼⠃⠈⠉⠉
⠀⠀⣠⢠⠏⠀⢀⠀⢀⡞⢀⡜⠹⡆⢿⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⡏⠀⠐⠎⠳⠧⠤⠞⠀⢿⣋⣁⣤⣤⡞⡀
⠀⠐⠗⠘⡗⠦⣄⣀⡞⢠⠎⠀⠀⣇⢸⣀⣀⣤⠴⠖⠛⠉⢀⣀⣠⡼⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠘⢦⣄⣉⣷⠋⠀⠀⠀⣿⣼⣉⣥⡤⠶⠒⠛⠉⠉⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠠⠤⠄⢀⠀⣀⡀⠀⠀⠀⠀⠔⠒⠔⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠒⠒⠖⠶⠶⠤⠤⢤⣀⣉⣀⣁⠀⠐⠒⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⣠⠆⠠⠀⣂⠀⠀⠀⢀⣈⡉⠉⠙⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢻⠀⠀⠀⠀⠸⣿⠈⠉⠀⢻⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢸⠀⠠⠂⠀⠀⣿⠀⠀⠀⢸⠀⠀⠅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠀⠀⢸⡇⠀⠔⠀⠀⣿⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣄⢄⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡼⢻⢸⡇⠀⠀⠀⡄⣿⠀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠷⢮⠞⠿⢳⣶⣦⠇⣿⣤⢄⣀⣨⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⡇⠁⠁⣄⣀⡀⠀⠀⠉⠈⠉⠛⠚⠏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢸⡇⠀⠀⡏⢈⣐⡉⠓⠶⠤⢤⣤⣀⡄⠀⠀⡇⢠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⢰⡂⢸⡇⠄⠉⠡⠒⠂⢤⠈⠃⠀⠀⠃⠘⠀⢤
⠠⣄⣀⠀⠀⠀⠄⠀⢸⠀⣹⡇⠀⢸⣇⠸⠷⢦⣤⣀⣀⠀⢸⠀⠀⠀⠠⠀⠸⠀⠹⠂
⠀⠀⠈⠙⠓⠢⢤⣄⣸⠀⣿⡇⠀⠈⠉⠛⠻⠶⢶⣤⣭⣉⣛⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠑⠲⡢⠤⢄⣀⠀⠀⠀⠀⠈⠉⠙⠃⠀⣸⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠕⠃⠒⠦⠤⣄⣀⣘⣻⢀⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠂⠜⠖⡲⢄⣀⡀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠫⠲⣶⢤⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠐⣩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⣍⠂⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠐⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠆⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠘⣽⠛⠛⠛⢯⠁⠈⡽⠛⠛⠚⣮⠃⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⡀⠀⡇⠀⠀⠀⣿⢀⣀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠠⠝⠒⠒⠒⠊⠄⠠⠑⠒⠲⠒⠣⠯⠬⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠐⣭⠉⠉⠉⢭⠃⠘⡭⠩⠈⠍⣍⠂⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠰⣝⣀⣀⣀⣚⡄⢠⣓⣀⣀⣀⣫⠄⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠋⠀⣾⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⣽⡇⠀⠀⣠⣶⣿⠇
⠀⠀⠀⠀⣠⣾⣿⡟⠁⠀⢀⣾⣿⡿⠀⠀⣰⣷⡆⠀⠀⠀⣤⣶⣶⣶⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⢀⣠⣾⡟⠁⣠⣾⣿⠋
⠀⠀⠀⣼⣿⣿⠏⠀⠀⢠⣾⣿⣿⠁⠀⣰⣿⣿⣿⠀⠀⠸⣿⡿⠛⠛⠛⠛⠻⠿⠿⣿⣿⣶⣶⣤⣤⣾⣿⣯⣴⣶⡿⠟⠉⢀⣴⣿⡟⠁
⠀⠀⣸⣿⣿⡏⠀⠀⢠⣿⣿⣿⠃⠀⣰⡿⣿⣿⣿⠀⠀⣀⣬⣤⡀⠀⠀⠀⣀⣤⣤⡀⢀⣀⠈⣹⣿⣿⠋⣉⣁⠀⠀⠀⢠⣾⣿⠏
⠀⠀⣿⣿⣿⠀⠀⢀⣿⣿⡿⠁⢀⣼⠟⢀⣿⣿⡏⢠⣾⣿⠏⢉⣿⠀⣠⣾⡿⠋⢹⣷⣿⡟⣰⣿⡿⣡⠟⣹⣿⡇⠀⢠⣿⣿⠏
⠀⠀⣿⣿⣿⡀⠀⠘⣿⣿⣥⣴⠟⠁⠀⢸⣿⣿⢇⣿⣿⠃⢀⣼⠏⣼⣿⠟⠀⢠⣿⣿⠟⣰⣿⣿⡿⠃⣰⣿⣿⠁⢀⣿⣿⠏
⠀⠀⠙⢿⣿⣷⣤⡄⢈⣩⣭⣤⣄⡀⠀⣿⣿⣿⢸⣿⣯⣴⠟⢁⣼⣿⠏⠀⣠⣿⣿⠏⢀⣿⣿⡟⠁⢰⣿⣿⠇⠀⣸⣿⡟
⠀⠀⠀⠀⠈⠉⠁⣴⣿⣿⡿⠋⠉⠁⣸⣿⣿⠇⠘⣿⣏⣠⡴⠟⢻⣿⣀⡴⠋⣿⣏⣠⢾⣿⡟⠀⠀⣿⣿⣏⣀⣀⣈⣟⣁⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⢠⣿⣿⡟⠀⢀⡀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⣉⣤⣤⣾⣿⣿⠟⠛⠉⣭⡍⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⠿⠛⠙⣿⣿⠀⠀⣾⣿⠇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠹⣿⡄⠀⠻⣿⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣤⣤⡴⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡰⠀⡠
⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠠⠀⠀⠀⣤⣠⣃⡴⠁
⠀⠀⠙⠦⡀⠀⠀⠀⠀⢀⡤⡖⠁⠀⠢⡈⢷⠀⠁⠀⣠⣀⣆⣀⠀⠀⠁⣧⣀⡀⠀⡼⠁⠀⠈⢉⠷⠂
⠀⠀⠀⠀⠈⢳⣤⣀⡠⠋⠀⠀⢀⣠⣾⣷⣌⢷⡁⠀⠘⣿⡏⠉⠉⠉⢻⠧⠬⣉⢺⡁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠺⠥⣌⣉⣳⣂⡴⠋⠙⣿⣿⡿⠿⠟⠟⠿⣿⣻⠓⠶⢦⣾⣀⠀⠀⠈⠳⣞⠁⠀⢀⠴⠂⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢄⠀⢈⣽⠛⢦⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠩⢟⣶⣂⠌⠙⢿⣷⣤⣸⠁⠁⠂⠀⢀⣤⠴⡞⠉
⠀⠀⠀⠀⠀⢀⡤⠞⠋⢹⡀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣜⡀⠀⠙⢿⣿⣷⣤⣴⣿⡏⠧⣤⠗⠂
⠀⠀⠀⠀⢺⣍⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⢷⣡⠀⠈⢿⣿⣿⣏⠁⠀⢀⣸⡦⠄
⠀⠀⠀⠀⠀⣘⢳⣄⣴⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢿⣵⣤⡘⣿⣿⣿⣧⣤⣶⣶⡟⠋⠉
⠀⢀⠀⠀⠀⡆⢸⠉⡿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⠛⠉⢹⡇
⠀⠙⠉⠀⢰⡂⢸⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁
⠀⢰⠀⢀⣸⣇⣸⣠⣯⣼⣿⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⢿⣏⣀⡀
⠚⠙⣏⠉⠁⠀⠸⡇⠈⢿⣿⣿⢾⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⠐
⠀⠀⠘⣆⢀⣀⣀⣗⣀⣨⢿⣿⣾⡹⢻⡿⣾⣴⣰⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠁⠀⠀⠰⢀⡀
⠀⠠⠒⠛⣯⡿⠉⡟⣏⠀⠀⢻⡿⠛⢦⣄⡁⠋⠛⠛⠟⠿⣶⣤⡀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⡆⠀⠈⠙
⠀⡠⠔⠊⠉⠧⣌⣣⠈⢦⠂⠁⠱⣄⢀⡽⠛⠷⠶⣶⣤⣤⡴⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⣈⠿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⡱⣆⠀⣨⠿⢄⣀⠀⢠⣿⣿⣿⡇⠀⠀⣿⡟⢻⡿⠛⠁⠙⢟⠁⠀⢸⡙⢦⡙⣄⠓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠈⠉⠛⠛⡿⢿⠤⠀⠀⣿⡇⠀⠳⡀⠀⠀⠀⠀⠀⠀⣇⠀⠙⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡄⠀⠀⠀⡰⠂⢰⠇⣸⣀⣀⣀⣿⣇⣀⠤⠻⡖⠂⠀⠀⠀⠀⢸⣠⠟⠁⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠈⠀⠀⠀⠊⠀⠀⡸⠀⠀⠀⠉⠹⠉⡏⠀⠀⠀⠈⢆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢻⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⡀⡀⣀⣻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣔⣿⣿⠟⠛⠛⠻⠋⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣃⣺⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⣙⣿⣿⣿⡿⢧⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠛⢿⣦⣌⠙⠷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣶⣄⡉⠻⣿⡾⢿⣿⣿⣿⣷⣲⣶⣦⣤⣄⣀⣀⣀⣸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣻⡟⢾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠻⠿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⠛⢿⠟⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⣀⠤⡖⠒⠛⠙⠛⢻⣿⣿⠀⣏⡀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠃
⠀⠀⢧⠀⠀⣀⣤⣶⣾⠿⠛⠃⢸⡇⠀⠀⣿⣇
⠀⠀⠘⢷⣿⡿⠛⠋⠁⠀⠀⠀⢸⣇⣀⣠⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡠⠀⢀
⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢰⠀⠘⠀⠀⠀⠀⠀⢰⠃⢠⠎
⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈⢀⠀⠹⣄⣀⠡⠀⠀⠄⠀⣆⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⢀⡞⠓⠻⠶⢯⣀⡀
⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⣠⠞⢏⠀⠀⠀⢱⡄⠻⣄⢀⣀⡀⢼⣤⣽⣤⣤⣀⣀⡀⡿⠶⣤⣀⢀⡾⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠙⣦⣄⣀⠞⠁⠀⠀⠁⣠⣶⣿⣿⣦⠻⣇⠀⠀⠈⣿⣿⠀⠀⠀⠈⢹⡿⠶⠤⢌⢻⡅⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠾⢥⣄⣈⠙⣲⢆⡴⠊⠉⠻⣿⣿⣿⣷⢿⢿⢿⡿⣿⣛⡛⠳⠶⣤⣼⣇⠀⠀⠀⠈⠻⣾⠋⠀⠀⢀⡤⠂⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠙⣹⠿⣅⡀⢀⣤⠟⠋⠁⠀⠀⠀⠀⠀⠉⠞⣻⢿⣰⠀⡉⠛⢿⣶⣤⡀⢠⠏⠈⠢⡐⠉⠀⢀⣀⣤⡖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣂⡴⠞⢿⠀⠈⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣹⣾⣁⡄⠀⠙⢿⣿⣿⣦⡀⠀⣈⣦⡖⢻⠁⢠⣇
⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠸⣆⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⡼⣠⠀⠀⠹⣿⣿⣿⣿⡿⠟⠋⠉⠙⣯
⠀⠀⠀⠀⠀⠽⢦⡀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⣧⢆⠀⠀⠹⣿⣿⣿⣧⡀⢀⠠⠤⢺⣂⣁⣀⠠
⠀⠀⠀⠀⠀⠀⣤⣙⣦⡤⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣻⣞⡦⡄⣿⣿⣿⣿⣷⣶⣶⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⡇⠀⡇⢩⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠝⢻⢻⣿⣿⣿⣿⠛⠋⠉⢹⠁
⠀⢒⠓⠒⠂⢰⠅⠠⡇⢸⠇⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢸⠀⠀⠀⢸⡇⠰⠇⢸⡗⢻⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠠⢄
⣀⣼⣧⠤⠴⠾⠛⠚⣿⠋⢻⣿⣿⣷⣎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠉⠉⠟⠓⠒⠂
⠀⠀⠹⡄⠀⠀⠀⠀⢿⠀⠀⢻⣿⣷⡟⣷⣇⣆⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠹⣆⣀⣠⣤⣸⣂⣡⡴⢿⣿⣿⣧⡚⢻⠿⣾⢦⣧⣧⣰⣀⡄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡖⠤⣀
⠀⡠⠴⠚⠹⣇⣾⠀⣸⠻⡀⠀⠀⢹⣿⠟⠻⣦⣄⡁⠚⠰⠋⠟⠟⡻⢳⣦⣄⡀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⣼⠀⠀⠈⠙⠓
⠀⠀⣠⠴⠚⠩⣧⣀⠻⡀⠹⣄⠔⠋⠙⣦⠀⠀⣹⠿⢷⣦⣤⣤⣤⣤⣤⣥⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣃⠈⣳⡇⠀⡄
⠐⠉⠀⠀⠀⠀⠀⠉⠛⠿⠆⠙⢤⡄⠀⠈⢳⣾⠁⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⣿⣿⢿⣿⣿⠿⠋⠻⣿⠟⠁⠀⣹⠻⣅⠙⣆⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠓⢦⣤⠋⠈⠙⠲⠦⢤⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠈⢻⡁⠀⠀⠀⠈⠂⠀⠀⠘⡇⠈⠳⣌⣧⡀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢠⠀⢀⡏⢩⠋⠁⠀⢰⣿⣿⠀⠀⠻⡄⠀⢀⣀⠀⠀⠀⠀⢻⠀⢀⣨⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡠⠁⠀⠀⠀⡰⠋⠒⣾⠦⢼⣤⣄⣀⣸⣿⣿⣠⠤⠚⠙⢏⠉⠀⠀⠀⠀⠀⠸⣧⠞⠋⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⡟⠀⠀⠀⠀⠀⠹⠉⢻⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⠳⢄⣀⣀⣀⣀⣀⣀⣀⣀⣠⠞⠉⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢠⠎⢀⣠⠴⠛⠉⠀⠀⠀⠙⢄⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡄⠀⣰⠋⠠⡏⠀⣀⡠⠖⠒⠢⣄⠀⠈⠳⣄⠀⠤⠈⣳⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡼⠁⣀⡀⠉⠋⠁⠀⠀⠀⠀⠀⠙⠢⣄⠈⢦⣠⠞⠁⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠈⢳⡞⠁⢸⠦⣄⠀⠀⠀⠀⠀⠐⠦⣀⠀⣹⠊⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⣄⡴⠃⢀⡼⠋⢳⣄⠀⠒⢦⡀⠈⡿⠁⠀⢦⠉⠲⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠶⣏⣀⡴⠋⣠⠃⣄⣀⡽⠉⣤⣤⣤⣬⣧⣤⣬⣷⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠖⠛⠒⠋⠀⠀⠀⢸⠀⠀⠀⢸⡄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⣀⣀⣿⣀⣀⣀⣸⣀⣀⣀⣠⣇⣀⣀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢿⠀⠀⠀⢸⠀⠀⠀⢀⡇⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣸⣀⣀⣀⣸⣀⣀⣀⣸⢀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⢰⡟⠀⠈⢧⡀⠈⢻⡀⠀⢹⠁⠀⢠⠇⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣸⡇⠀⠀⠀⠉⠲⢤⣱⣄⣸⣀⣴⣣⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⣠⡾⠋⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢹⣀⡴⠋⠀⢠⡟
⢴⡦⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⠛⠀⠀⢀⡞
⠀⠙⢦⣀⠈⠉⠉⠛⠲⢒⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠁⠀⠀⠠⣸⠁
⠀⠀⠀⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀⣰⠃⢀⡤⠞⢆⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⢀⣠⠽⠟⠂⡄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠉⠈⠉⠙⢦⡀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠈⣹⣡⡞⠉⠀⠀⠈⢳⡀⣀⣠⠏
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢠⡞⠁⠀⣀⣤⡤⠤⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢹⡆⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠲⠤⣀⠀⠀⠙⢶⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⠤⢤⣀⣀⣤⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠦⣄⣙⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣈⣀⣀⣀⠀⢠⢿⡇⠀⠀⠀⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⡴⠶⠶⢶⡄⢀⣾⠋⠉⠉⠉⣷⠀⠀⣠⣿⠉⠉⠉⣹⢠⡏⢸⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⣀⠀⠀⢀⣠⠤⠶⠚⠛⠙⣇⢰⡏⠀⠀⠀⠀⢻⣼⢹⠀⠀⠀⠀⢹⡀⣼⠁⡏⠀⠀⠀⣟⢸⠀⡌⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣖⠛⠉⠁⠀⢀⡼⠋⠙⠾⣏⡀⠀⠀⠀⠀⠀⠙⣾⠇⠀⠀⠀⠀⠀⢿⠀⡆⠀⠀⠀⠈⣇⡏⢀⠇⠀⠀⠀⣿⡿⠀⡟⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠉⠛⠶⣤⣠⠏⠀⠀⠀⠀⢀⡽⠃⠀⠀⠀⠀⠀⣘⡆⠀⠀⣤⠀⠀⠈⢷⡇⠀⠀⠀⠀⢻⡇⢸⠀⠀⠀⠀⣿⡇⠀⡇⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⢰⣾⣻⡏⠀⠀⠐⠓⠃⠀⠀⠈⢧⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⣿⡇⠀⣇⡀⠀⠀⢸⣇
⠀⠀⠀⠀⢰⠏⢷⡒⠶⠤⣄⠀⠀⠀⠀⠀⠀⣰⡿⡄⠀⠀⠀⠀⠀⠉⣉⣧⠀⠀⠀⠀⢀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠳⠾⣥⣭⣩⡿⠛⠈⠳⣄
⠀⠀⠀⠀⢸⡀⠀⠳⣤⣀⡞⠁⠀⠀⠀⠀⡼⢁⣇⠸⡀⠀⠀⠀⠀⠀⠻⠿⠒⠲⠀⠀⡟⠑⢤⡀⢀⡼⠃⠀⠀⢀⣤⢤⡄⠀⠀⠀⢸⠀⣠⣴⠛⠉⠙⣦⠀⠀⠸
⠀⠀⠀⠀⠀⠳⣄⡀⣰⠏⠀⠀⠀⠀⢀⡞⢁⡞⠹⡄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠖⠋⣠⣀⢙⠛⡄⠀⠀⠀⢸⣀⣠⠇⠀⠀⠀⢸⢰⡟⣇⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠈⣽⠃⠀⠀⠀⠀⢀⡞⢠⡟⠀⠀⢻⡈⡇⠀⢀⣀⣠⣤⠤⠴⠶⢾⡶⠶⠋⠙⠛⢻⠗⠶⠶⠦⣼⢁⡟⠓⠶⠶⠶⢾⠘⢧⣈⣻⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡶⢤⣀⡀⠀⡾⢀⡿⠀⠀⠀⠀⢷⠘⠋⠉⠁⠀⠀⠀⣀⣀⡾⠀⠀⠀⠀⠀⠘⠶⠶⠶⠤⠏⠈⠳⠖⠒⠒⠲⠟⠀⠀⠉⠁⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠘⡇⠳⣄⡀⠉⠛⢁⡾⠁⠀⠀⠀⠀⠘⣧⡤⠶⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠙⠲⣤⠞⠁⠀⠀⠀⠀⠀⣠⢶⡄⢀⣴⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢶⣄⠀⠀⢶⠦⣄⠀⠀⠀⠀⢀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠁⣠⡟⠋⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠒⠲⠶⡄⠘⠶⣶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠶⠶⠒⠋⠉⠀⢀⡾⠉⠀⣰⠛⢦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⡏⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⢀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠐⢯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡯⠤⠤⠄⠤⢠⠄⠀⠀⠀⠀⠠⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⠀⡀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⣻⡶⠀⠀⠀⠀⡀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣄⣘⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢀⠠⣪⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⣄⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡴⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣫⡄⣿⣿⡇⣍⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣇⣿⡇⣿⣿⡇⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⢻⣷⣶⣶⣶⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⡏⣾⡏⣭⣛⣛⣱⡍⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢛⣛⣵⡿⢟⣛⠿⣷⣍⣭⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢸⣿⣿⣮⢹⣿⢸⣿⣿⣿⢸⣿⣵⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⢸⣿⣿⢫⣾⣿⣜⡻⠿⢟⣼⡿⠷⢹⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡸⣿⣿⡷⣶⡌⣿⡿⠿⢿⡏⣴⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣮⣭⣾⣿⣷⣮⣾⣿⣧⣶⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠻⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢻⡜⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢶⡇⠀⢀⡀⠀⠀⠀⠠⣿⠃⡞
⠀⠀⠀⠀⠀⢢⡀⠀⣿⠙⠦⣄⡀⠀⢣⠀⢀⣠⡴⠻⡇⠀⠀⠀⠀⢸⢀⡴⠛⣇⠀⠀⠀⢠⠇⠀⠀⠀⢀⣤⡶
⠀⠀⠀⠀⠀⠀⢣⢀⡿⠀⠀⠀⠉⠙⠛⡟⠉⠁⠀⠀⠻⣄⠀⠀⣠⡿⠋⠀⠀⠈⠢⣀⠀⠏⠀⣀⡴⠞⠉⣿⠀⢀⡴
⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⠀⢹⡴⠋
⠀⠀⠀⠀⢶⡚⠉⠀⠀⣠⠴⠖⠒⠶⣤⣠⣤⡀⣤⣤⣤⣀⣀⣀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢙⣦⣄
⠀⠀⠀⠀⠀⠙⣆⠀⣼⠁⠰⣟⣙⢲⡏⠀⠈⣷⠇⠀⢸⡏⣁⠀⢹⠋⣠⠦⣬⡏⢹⢳⠋⢹⡇⢸⢣⡟⢠⠞⠉⠁
⠀⠀⠀⠀⠀⠀⢸⡆⠘⠢⠴⠶⢬⡉⢳⡀⠀⠙⢠⡄⠸⡇⠙⠁⢸⡇⠳⠤⢾⡀⠙⠛⠀⣸⠀⡟⡾⣰⠯⣀
⠀⠈⠂⠀⠀⠀⢸⡇⠀⢰⣤⣀⣠⠇⣸⠃⣿⢦⡟⣇⠀⡇⢺⢻⠀⣿⠟⠀⣠⡇⡞⢻⢀⡟⢸⣿⢧⡏⠀⠀⠉
⠀⠀⠉⠁⠀⠀⣼⠁⠀⠸⠤⣤⡤⠾⠻⠒⠛⠈⠁⠙⠛⠛⠋⠈⠓⠛⠛⠋⠉⠉⠁⠈⠉⠁⣴⡦⢸⡇⠀⠐
⠀⠀⠀⠀⢀⣼⣃⣠⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⣀⣀⣀⣰⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣀⣸⡇
⠀⠀⠀⠴⠟⠉⠁⠀⠀⠀⣠⠎⠉⢦⡀⣠⠾⠋⠉⠀⠀⡾⠉⠳⣄⠀⠀⣠⠶⠛⠉⢏⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠈⠿⠏⠀⠀⠀⠀⢠⠇⠀⠀⠘⣦⡾⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠀⡰⠃⠰⡄⠀⠀⠀⠀⠀⠀⠀⣼⢁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⠃⠀⠁⠹⣆⠀⠀⠀⠀⠀⡠⠋⡄⣿⢸⠁⠘⠀⢳⠀⠀⢠⡄⠀⠀⢠⠇⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠦⣾⠀⠀⠀⠀⠘⡄⠀⢀⡴⣶⠁⠀⠀⣿⡇⠀⠀⠐⠈⡇⢀⡇⠱⡄⣠⠎⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡜⢻⠀⠦⠈⠀⠀⠀⠀⣰⢁⡴⠋⡀⠈⠀⠀⠀⠈⠁⠐⠀⠀⠀⣝⡸⠑⠀⠙⠁⠀⠀⠀⢠⠷⢄⢀⡄
⠀⠀⠀⠀⠀⠀⠾⠰⠘⠃⠀⠀⠀⠀⠈⢀⡷⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠟⠁⠀⠀⠀⠀⠈⠀⠀⡠⠖⠸⠛⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⡀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠂⢱
⠀⠀⠀⠀⠀⠀⢷⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢊⡏
⠀⠀⠀⠀⠀⣀⡾⠃⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠄⠰⡏
⠀⠀⠀⢀⠞⢉⠰⡆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⢀⠎⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢦
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠈⢧
⠀⣼⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠉⠙⠛⠒⢺⣆
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠴⠶⠶⣆⢀⣾⠛⠛⠋⢷⠀⢀⣴⠋⠉⠉⡏⢸⢡⠃⠀⠀⠀⠀⡞⢹⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⡤⠖⠚⠋⠉⢹⡀⡇⠀⠀⠀⠘⡾⢹⠀⠀⠀⢸⡄⡟⢘⠀⠀⠀⣇⡇⢸⠀⠀⠀⠀⣼⠁⠀⢧
⡎⠀⠀⣰⠓⠲⢤⣀⠀⣠⠞⠁⠈⠙⠲⡄⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠘⣼⡆⠀⠀⠀⣷⡇⡏⠀⠀⠀⣿⠇⣸⠀⠀⠀⢰⡇⠀⠀⢸
⡇⠀⢠⠇⠀⠀⠀⠈⠙⠁⠀⠀⠀⢠⠎⠀⠀⠀⢠⣴⣶⠁⠀⠸⠗⠀⠀⠘⡇⠀⠀⠀⠘⠷⠃⠀⠀⠀⣿⠀⣟⠀⠀⠀⣼
⡇⠀⠘⣶⠢⠦⣄⠀⠀⠀⠀⠀⣴⢧⠀⠀⠀⠀⠈⢉⣿⠀⠀⠀⢀⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠷⠯⣭⣹⡖⠃
⣷⠀⠀⠈⠳⣤⠞⠀⠀⠀⢀⡞⣸⡆⢣⠀⠀⠀⠀⠘⠛⠈⠹⡀⢸⡉⠢⣄⣴⠛⠀⠀⢠⠤⢤⠄⠀⠀⣷⣠⣶⠋⠉⠙⡆⠀⠀⣤⢰
⢹⡄⠀⠀⣰⠃⠀⠀⠀⢀⠎⣰⠋⠹⡌⣆⠀⠀⠀⠀⠀⢀⣀⣟⣉⣼⢂⣨⣉⣀⣀⣀⣸⢶⣿⣀⣀⣀⣿⣿⠹⣦⣤⡴⠇⠀⠀⡏⣼
⠈⣧⠉⢼⣧⣄⡀⠀⢀⠎⣸⠃⠀⠀⢱⡘⠖⠒⠛⠉⠉⠁⢀⡿⠁⠁⠀⠀⢩⣤⣬⣭⡍⢈⣥⣤⣬⣥⡍⠈⠛⠛⠉⠀⠀⠀⣠⢧⡿
⠀⠸⡆⠠⠙⢦⡈⠙⠛⣰⠃⠀⠀⠀⠈⢧⡤⠶⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⠁
⠀⠀⠹⡄⢀⠀⠉⠳⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠃
⠀⠀⠀⠹⣄⠁⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠔⠁
⠀⠀⠀⠀⠈⠳⢦⣈⠑⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠊⣡⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⣀⠈⠉⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣈⣉⣠⡤⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⣠⡶⠶⠶⠶⠤⠤⠤⠤⠤⠴⠶⠶⠒⠒⠚⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣃⣤⡾⠟⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢴⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠊⡀⠀⢤⡤⠤⠤⣤⡤⠤⢄⠆⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣀⢀⣸⣀⣀⡀⣸⣀⡀⣸⡄⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠛⠛⢻⠛⠛⠛⢻⠛⠛⢻⠁⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠄⠀⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠟⠛⢿⠟⠛⠛⢿⠟⠛⢻⠂⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⢸⠃⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡶⠶⢾⡶⠶⠶⢾⡶⠶⢾⠁⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡀⠀⣸⡀⠀⠀⣸⡀⠀⣸⢡⢶⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣹⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡧⠀⠀⠀⠀⠀⠠⣾⣧⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠻⢿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣤⣈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⣴⣶⣶⢗⣶⣶⣦⡀⠠⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢢⠎⠀⠀⣠⣿⣿⣅⣠⡿⠛⢯⣿⣷⡙⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⢇⡏⣼⣷⣿⣿⣿⣿⢿⣿⣦⣀⣾⣿⣿⣷⠙⡼⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⠉⠁⣠⣿⣿⢸⡇⣿⣿⣿⡿⢿⣿⢺⣿⣿⣧⣿⣿⣿⢿⠌⠁⠀⠀⠀⠿⠟
⠀⠀⠀⠀⠀⢀⣴⡾⠿⣿⣿⡎⣇⣿⣿⣿⣿⣶⣿⣿⣿⠿⠿⣿⣛⣛⣘⣀⣀
⠀⠀⠀⠀⠈⠁⠀⠀⢀⣼⣿⣿⡜⣾⣿⣿⣿⣻⠟⠉⠀⣀⣬⡧⣿⢿⣿⣿⣿⣿⣷⣶⣶⡆⢰⠀⢶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⣹⣿⣎⠛⠿⣿⡇⠀⣴⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠇⠘⠀⠸⠿⠿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣉⠙⠛⠻⠦⠤⠀⠰⣿⣛⣛⣛⣛⠛⠛⠛⠛⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠠⠖⢠⣰⣯⣿⣿⢀⡌⢠⣶⣶⣿⣿⣷⣶⣶⣤⣤⣍⣁⡀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠘⠃⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠿⠇⠩⣭⡭⠍⠡⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠠⢬⣽⣛⣻⣿⣿⣟⠛⢴⣾⣿⡷⢤⣴⣶⣤⢄⣼⣿⠁⣽⣿⣧⣤⠤⣤⣤⣶⣤
⠀⠀⠀⠀⠀⠀⠚⠛⢛⣯⣽⣿⣿⠇⢲⣿⣿⠄⣰⣿⠏⠉⣲⣿⠗⢦⣿⣿⣿⡅⢐⣿⣿⣻⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠙⠶⠒⠒⠦⠔⠒⠂⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⢸⠑⢄⠀⢠⠃⢀⠀⢀⡴⡿
⠀⠀⠀⠀⠀⠀⢠⡾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠦⣼⠀⠈⠲⠁⠀⢸⡠⠊⢀⡇⠀⢀⣠⡄
⠀⠀⠀⠀⠀⣠⣿⡼⠀⠀⠀⠀⠀⢀⠠⢀⡠⠄⠒⣤⠎⢉⡌⡁⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠄⡇⠀⠈⠀⠀⠀⠀⠀⠀⠁⣀⣘⣦⣋⡥⣬
⠀⠀⠀⠀⢸⡯⣿⡁⠀⡔⠈⠑⣤⠃⣠⡈⡇⠸⣲⠃⠾⠛⢠⡠⠇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤⣤⠴⣶⠒⢒⣹⡟⢉⢹⣸⠀⣿⠁⣿⢀⡇⣀
⠀⠀⠀⠀⠸⣷⣝⡖⣾⢀⠎⠂⡏⢸⠀⡁⣇⢰⠉⢳⠞⢃⢿⣊⡟⠁⠀⠀⠀⠀⠀⠀⠶⠿⣯⣉⡆⠛⣰⣿⢈⣫⣿⠁⠛⢸⡟⢠⢶⢰⣿⢻⠄⠒⠋⠁
⠀⠀⠀⠀⠐⠐⢺⡧⣿⠘⠄⢎⠷⣈⣊⠅⠳⠚⠀⠘⠂⠁⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⣩⣿⣠⠇⣯⡬⠽⠿⠞⠙⠚⠛⠋⠉⠉⠸⣅
⠀⠀⠀⠀⠀⠀⠘⢿⣞⡷⠖⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣴⠌⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡧⡴⣾⡏⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⣬⣦⡀
⠀⠀⠀⠀⠀⠀⠠⠻⣟⢮⡯⡵⣾⡶⡤⣰⡦⣤⡴⣿⣿⣙⣈⣵⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣁⢎⣿⣡⣾⢿⠀⣰⣦⠀⠀⣟⢦⡀⠸⡼⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢗⢟⠫⡻⢝⢏⠀⠀⠙⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⣞⢟⡳⣻⣶⢿⡳⣳⣠⠷⡁⢘⠢⣀⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠑⠊⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢘⠄⠣⡃⢜⠄⠹⠀⠀⠀⠀⠈⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢆⠀⠀⠀⡔⠇⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⢰⠀⠱⣄⠜⠀⢸⠀⡰⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠻⠀⠀⠈⠀⠀⠘⠊⠀⢠⡃⠠⠄⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣄⠀⠀⠀⠤⠈
⠀⠀⠀⠀⠀⠙⠍⠁⠛⣀⣀⣀⡠⢤⣠⠤⢤⢠⠒⣶⡏⣾⠋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠀⣀⣀⡤⣀⠤⢄⡤⠒⢦⡼⠋⢹⡄
⠀⠀⠀⢀⣤⡥⠼⡉⣿⢉⣿⠠⡶⣾⠇⣶⢸⣾⠀⣛⠀⣿⣰⡇⣒⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⢉⡔⢻⢰⡟⢰⠶⠿⠁⠿⠻⡄
⠀⠀⠀⠉⡉⢟⠷⣧⠀⡜⡿⠰⢾⣿⣠⢤⣈⣧⡼⠿⠼⢛⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢢⢻⣄⠀⢰⣯⣧⢳⣼⢸⣧⣴⢆⣴⡶⠟⣰⠃
⠀⠀⠀⠀⠄⡢⡞⢛⣺⣤⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⡀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢳⢝⣷⡙⢧⣸⣯⠿⣴⢿⡾⢿⣴⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠂⢜⠯⣫⣟⠇⣠⣾⠀⢀⡀⠀⠀⣤⡀⠀⢙⢩⣟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠤⠄⠁⠠⡻⢟⣆⡙⢷⣴⢶⠟⣬⡽⠻⡅
⠀⠀⠀⠀⠀⠊⠒⡕⣿⣮⣟⣽⣂⣾⣝⣆⠀⣗⠝⢢⡈⠋⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⣴⢛⣟⠻⣟⡫⠼⢛⡅⠀⠀⠈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠊⠓⡕⣫⢪⣟⣝⢪⢚⣿⢉⠂⠐⠈⠲⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠂⠪⠢⠄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⡑⠊⠂⠑⠉⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠊⢄⡤⠿⠀⠀⢤⣤⡿⣹⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⡰⢉⢍⢍⠒⢲⡄
⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠿⠧⣼⣿⡟⠂⠀⠒⠠⠂⠀⣄⡚⢤⣤⣶⣴⠄⡠⠴⢒⠦⣀⡆⠆⢍⢳⡄⡀⠠⠄
⠀⠀⠀⠀⠌⠀⢀⠀⠔⠒⢲⡀⣠⣿⣏⠁⣀⣀⠘⠲⣶⠿⢋⢠⠔⠒⡀⡀⠀⣑⠒⠺⠦⠀⠑⢄⠀⢸⣧⣽⠠⠄⠘
⠀⠀⠀⢄⠛⣄⠀⠀⢄⣀⣼⡟⣻⡷⡷⠘⠲⢶⠾⠾⢶⣶⣾⡛⢻⣟⣛⣿⣿⣥⡴⢦⠤⠉⡀⠂⠑⠛⢻⠤⠀⠀⠄⡀
⠀⠀⠀⠀⠈⠈⠓⠤⣀⣒⣋⣴⣿⡿⣧⣒⣀⣦⡗⠛⠟⡻⣤⡤⠾⢏⠉⢗⢠⣤⣶⣋⣸⡓⠛⠂⠀⠀⠀⠀⢈⣤⠴⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠼⠀⠀⢩⡛⠛⢣⡉⠁⠶⠿⢛⢛⡑⠈⠒⠛⠛⢁⠘⠁⠾⠶⠋⠛⠄⡀⠀⠐⣡⢴⡾⠛⠲⢥⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⠉⠋⢁⡄⠀⢾⢅⡁⠀⣱⡄⠠⢦⡀⣐⡒⠂⢂⠀⡀⠀⠀⢰⠉⠊⢸⣥⣀⠀⠀⠀⠉⠑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠁⠰⠄⠀⠉⠛⠛⢋⡥⠀⠠⡀⠈⠘⠛⢋⡃⠈⠢⠅⢐⡤⠾⣛⠉⠉⠳⢦⣄⠀⠀⠀⠀⠈⠐⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣉⣦⣴⡶⠷⠀⠀⠆⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠓⠦⣀⡆⠌⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣴⢋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⠋⠋⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢰⠶⡆⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣠⠏⠀⠹⡄⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣠⠴⠞⠋⢀⡴⠋⠀⠀⠀⠙⢦⡀⠙⠳⠦⣠⡀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⢠⣴⣦⣄⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⣿⡅⢀⣿⠞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠈⠛⠛⠁⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠀⣟⣹⣯⣹⡾⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸⡇⠀⣟⢻⣟⢣⡿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢸⡇⠀⣿⠋⠉⣿⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⣿⠀⠀⣿⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⡶⡿⢏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣶⠋⣙⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣋⡙⢲⡀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⡇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠆⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠻⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡼⠃⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣼


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠛⠿⠿⠿⠻⡿⢫⡝⠿⢿⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣛⣛⣛⣻⣻⣿⣿⣛⣛⣛⣛⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⢸⠁⠀⠀⠀⠀⠀⢠⠇⠀⡰⠁
⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⣄⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠇⠀⠀⠀⢀⣤⣄⣞⢀⡼⠁
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠀⠈⢆⠀⢹⣦⠔⠚⠀⠀⠘⡀⠠⣦⠀⠀⠀⠀⠈⠁⡇⠀⠀⠀⠀⢀⡾⠀⠀⠉⠙⠛⢲⡦
⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⣠⠞⠉⠣⡀⠀⠀⢀⣷⡄⠙⣦⡠⠤⠶⠒⣷⣶⣿⠶⠶⠦⠤⣤⣸⡟⠛⠶⢦⣀⡾⠁⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⢈⣷⢦⣤⣔⠁⠀⠀⠀⠑⣤⣾⣿⣿⣿⣦⠹⣧⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣽⡛⣓⣢⣄⣘⣧⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠾⢧⣄⣀⠉⠙⡶⢀⣴⠚⠉⠻⢿⣿⣿⣿⣧⣹⣦⣤⣤⣶⣿⣿⣤⣴⣶⠾⢿⡟⠛⠛⠛⠛⠛⠛⠛⣿⣷⣦⡄⢀⣠⠖⠁⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠙⠚⣻⣿⡁⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⢧⠀⠀⠀⠳⣄⠤⠔⠒⠈⠉⠉⠉⠉⢹⣿⡏⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢤⣤⢾⡇⠀⠙⢲⣄⣴⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⢈⣧⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢸⢿⣷⣠⠼⡏⠉⢹
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⢿⡀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⢀⣀⢖⡠⢱⠞⠉⠈⢣⡀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⡿⢸⣿⡧⠘⠳⢶⠞⠋⠁
⠀⠀⠀⠀⠀⠰⣾⡋⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠁⠀⡀⠐⡶⣎⠿⣪⡳⡏⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⠳⣄⠀⠀⣼⠃⢸⣿⠀⠀⠀⠈⣇⣀⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠉⢻⣦⡀⠀⠀⣠⣼⣿⣿⣿⣿⡿⠋⠀⠀⢠⡙⣯⢮⡳⠏⠚⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠈⠳⣴⠃⠀⣼⡧⠐⠒⢈⣉⣿⣤⠤⠖⠂⠉
⠀⠀⠀⠀⠀⠐⠂⡶⠬⣿⣶⣻⣿⣿⣿⣿⣿⠟⠀⠀⠐⡶⡘⢮⡶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠓⢦⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡀⠀⠀⠀⢀⡇⠀⡇⠀⣿⢹⣿⣿⣿⠋⠀⠀⡹⣣⢜⡲⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⠀⠀⠀⣾⣿⣿⡏⠉⠉⠀⡟
⠀⠒⡗⠒⠒⠀⣼⠄⠀⡇⠀⡟⠸⣿⣿⠃⠀⠀⡌⠕⡬⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⠓⢦⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢰⡇⠀⠀⠀⢸⡄⠀⡇⠀⣷⠰⣿⡟⢆⠀⢀⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡀
⠀⠈⣇⣀⣀⣀⣼⣧⣴⣧⡶⢿⣶⣿⠀⠈⢣⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠿⠿⢿⠷⠤⠤⠄⠀⠈⠉
⠛⠋⠹⡍⠉⠀⠀⠀⠀⢿⠀⠀⣿⡏⢧⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡇⠀⠀⡈
⠀⠀⠀⠻⡄⠀⠀⠀⠀⢸⡆⢠⣿⠀⢈⢷⡀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠁⠀⠐⠀⢀⡃
⠀⠀⠀⠀⣻⣦⠤⣶⠾⠭⣧⣴⣿⠀⣼⠀⠙⢦⡀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠀⠀⡸⠉⠳⢤⣀⡀
⠀⠠⠖⠋⠁⢹⣦⡿⠀⢰⠛⢶⣿⠀⡇⠀⠀⠀⠻⣄⡀⠀⠈⠳⣄⢀⣠⠴⠚⠁⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⡄⢰⡇⠀⠀⠀⠈⠉
⠀⠀⣀⡤⠖⠋⢹⣧⣀⠹⣄⠈⢿⣿⡇⠀⠀⠀⠀⠀⣙⣦⡀⠀⠀⠙⠦⣀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣁⠀⢹⣟⠀⠢⡀
⠔⠊⠁⠀⠀⠀⠀⠀⠉⠓⠿⣦⠈⢿⣿⣿⣖⣚⠛⠋⠉⠀⠈⠓⣤⣀⣠⣬⣷⣾⡿⠛⠉⠉⢹⣿⡿⣿⣿⣿⡿⠟⠙⢿⡿⠟⠁⠀⢻⡷⣅⠈⢧⡀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢿⣟⢛⡿⠻⠿⠿⢿⣟⣛⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⡇⠹⣟⠋⠀⠀⠀⠈⠣⡀⠀⠀⠙⡇⠈⠳⣄⠳⡄⠈⠙⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠀⠀⠀⠀⠀⠈⢩⠉⠛⢻⠛⢿⠦⠤⠀⠀⣿⣿⡇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⢳⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡀⠀⠀⠀⠐⢀⡼⣀⢀⡟⠀⠸⠀⠀⠀⠀⣿⣿⡇⠀⠀⣨⣷⡤⠴⠖⠒⠀⠁⠀⠀⠸⡇⢀⣷⠟⠁⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠸⠁⠀⠀⠀⣠⠎⠀⠀⢸⡏⠙⠛⠒⠶⠶⢶⣿⣿⡷⠒⠉⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠸⡇⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⢆