Contemplate Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠉⠉⠉⠓⠦⡀
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣴⠼⢶⢄⠘⠲⢢⢴⣀⢠⡆⠀⠀⢀⡀
⠸⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣑⣌⠁⠈⠀⡇⠀⠀⠈⠛⠋⠧⢶⠐⡹⠁
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⠴⢶⠀⠀⠀⠀⠀⠀⠀⡩⠶⢆⠈⠁⠀⠀⣁⣀⣴⣄⣠⡀⠀⠈⢀⢶⠄
⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⠏⠰⠁⢀⡜⠀⣀⢤⠀⠀⠀⠀⠀⠀⠀⡘⢹⠦⠴⠋⠉⠉⠈⠉⢻⣷⣤⠀⠯⣀
⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠒⠋⠀⠀⠢⠴⠋⠀⠐⠁⡸⠀⠜⡳⠀⢴⠄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣝⣋⠀⢰⣁
⠀⠀⠀⠉⠲⢤⣀⣀⠀⠀⣀⣤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠈⠓⠁⠀⠁⢠⠞⠀⠀⢀⣀⠀⠀⠀⠀⠀⣸⣿⠗⠒⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⢁⢴⣀⡈⠙⠂⠀⠀⢀⡟⠛⠀⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⣮⣉⣉⡿⠀⢠⠞⣛⣁⡴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣊⣁⠀⠀⠀⠀⠀⠀⠀⠀⢷⢋⣭⡍⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡙⠂⠀⡀⠀⠀⢀⣤⣤⠞⠘⠈⠂⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⣄⠀⠀⠘⣆⠀⢀⠀⡄⢀⠖⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠉⠀⠀⠿⠀⢻⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⠛⠒⠒⠒⠆⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠤⠐⠒⠉⠅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠈⢦⡶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⢛⠓⠀⠀⢉⢏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⣎⠀⠀⣰⠋⢲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡐⠀⠉⡏⠁⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠁⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠶⢶⣤⡶⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡓⠀⠈⠀⠑⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡋⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠹⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣴⣇⠀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣧⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢡⠀⠀⠀⠀⢀⣞⠓⠛⠋⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠄⡀⠈⣄⣀⠄⠀⠀⠈⠁⠀⠀⠀⠀⠁⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⣆⠈⣿⢯⡐⠧⠖⠠⡤⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠤⠀⠀⠀⢉⡡⠖⠀⠈⣳⡵⣊⠠⠤⠄⠄⠺⣦⡀⠑⠄⠀⠀⡀⠀⠂
⠀⠠⠀⠈⢠⠀⠀⠀⠀⠀⠀⣀⢀⣠⡦⠀⠀⠀⠀⠀⠀⠨⣷⣦⣠⡀⠂⠠
⠀⠀⠀⠀⢹⣤⣤⣀⣰⣤⢤⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⣀⡀⡀
⠀⠀⠀⢻⡿⠻⣿⣽⠏⣻⣿⣿⡿⠈⠂⢈⣉⣭⣷⣆⠀⠀⠀⠀⠋⣿⣶⣿⠀⠁
⠀⠀⠀⠁⠄⠄⢾⣿⣿⣿⣿⣻⣷⠃⢘⣿⣿⣿⣿⣿⡀⠀⢀⣀⣀⣿⡏⠂⠐
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠈⠉⠛⠛⠛⠻⠃⣰⣿⣿⣤⢿⠅
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣏⡽⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⡏
⠀⠀⠀⠀⠀⣼⣿⣿⣿⢿⡆⠀⠀⠀⠀⠀⠀⠰⢶⣦⣀⠀⠀⠀⡿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⢿⠸⣶⣄⠀⠀⢤⣴⣶⣤⡉⠉⠀⣠⡤
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⠌⢻⣶⣄⠀⠳⣤⣉⠷⣠⠾⠋
⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡖⠻⣽⣿⣷⣆⣀⣠⣚⣥⣀⡠⠐⠄⠀⠀⠈⠁
⣿⠿⣿⣿⣿⣿⣿⣿⣷⡽⣿⣿⣤⣶⣾⣿⣿⡿⠿⠻⠛⠫⠀⠀⠀⠀⣀⣤⣤⣴⡤
⠁⣀⣵⣿⣿⡿⠇⢚⡿⢛⡥⠚⠿⠛⠉⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠺⠿⣒⠄⣀
⠄⠰⠡⠁⠠⠄⠚⠉⣲⣵⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⢀⡀⠀⠀⠀⠀⢀⠞⠋⢩⢃⠈⠂
⠰⠃⠀⠀⠀⡰⠯⣀⢲⡀⢿⣆⣤⣧⣤⣴⣶⣶⣶⡖⠾⠟⠛⠀⠓
⠀⠀⠀⣠⣾⠁⠀⠈⠂⣥⠘⡿⣩⣿⣿⣿⣿⣿⠿⡗⠂
⣤⣤⣾⣿⣻⣩⠀⡀⣉⣽⣿⣿⡿⢿⠟⡉⠁

⠀⠀⠀⠀⠀⠀⠀⣿⣛⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣟⣯⣿⣛⣯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⣀⣀⡀⣀⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠠⣤⠿⠇⢠⡄⠀⢰⣾⣦⢀⢷⠀⠀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⣰⢻⠀⠀⣒⣂⣀⡈⠉⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠹⣿⣿⣡⡄⠀⠲⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⣿⣄⣠⣼⡏⠉⢿⣟⡄⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⣼⣿⣇⣽⣿⣧⡀⢸⣷⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠜⣠⠎⠁⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢏⡽⣶⣶⡯⢸⡟⢿⣿⣿⣿⣿⣿⡿⠛⠙⠻⣯⣧
⠀⠀⠀⠀⠀⠀⢔⣷⡇⠀⠳⡌⢆⢒⠛⢻⠶⠽⢿⣿⣿⣿⣅⣄⣀⣠⡽⣿⣆
⠀⠀⠀⠀⠀⢀⣾⢟⣇⠀⠈⠻⣆⢡⠀⣨⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠁⢿⡆
⠀⠀⠀⠀⢀⣾⡟⣼⣿⣷⣄⡀⠈⢳⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣧⣀⠘⣿⡀⠀⠀⢀⣀⠀⡀⣀
⠀⠀⠀⠀⠘⠛⠀⠛⠛⠋⠋⠛⠓⠚⠁⠀⠁⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠃⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⢐⣒⠒⠒⠒⠒⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠈⠀⡔⠂⠀⠀⠈⣼⣰⡀⠀⠀⠙⣀⠐⠂⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⡀⠠⡄⠀⡹⠀⠀⠀⠠⢄⡀⠈⡤⠚⢫⡑⠄⢀⡀⠀⠈⠢⡀
⠀⠀⠀⠀⢠⠊⠀⢀⡀⢇⠒⠂⠈⡾⠒⠒⡆⢸⠀⠀⠀⠀⠀⠂⠒⢄⠔⢅⡀⠀⠀⠀⠈⢢
⠀⠀⠀⠠⠃⠀⠀⠓⡆⢀⢑⡃⠤⠤⠄⠀⠀⠀⡮⡭⢄⠀⡀⠀⢸⡌⠀⠄⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠆⠀⠀⠈⠒⠁⠇⢠⡒⣒⡨⡀⡄⠘⠚⠉⠀⡛⠂⠥⠀⠀⡟⠀⠒⢄⠀⠀⣀⠀⠀⡠⢣
⠀⠀⠰⠀⠀⠀⠀⡙⢀⡄⠑⢟⡺⡏⠆⠏⠝⠘⠠⠊⠿⢤⠀⡔⠐⠣⠄⠀⡠⠻⠈⠀⠀⢈⠴⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠠
⠀⠀⠀⠀⠀⠉⠉⢛⠀⣀⠀⣈⠛⣁⣀⡼⠀⠂⠔⠆⣂⣖⠣⠂⠠⣴⠢⢥⠀⠀⠃⠀⠀⠀⠀⠀⢃
⠀⠀⠀⡀⠀⠀⠀⢈⡀⠦⠿⠀⢀⠁⣄⠙⡀⢳⣦⠤⣉⣛⡧⠄⠊⡀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠁⠀⠀⢀⡀⣡⡄⣂⠁⠈⠀⠐⣶⢤⠜⡁⠐⡀⢰⣴⣾⣅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠈⡄⠄⠀⠀⠀⠑⠄⣀⣈⠓⠚⠛⢢⠰⠀⠀⠇⠚⣈⡏⢩⠐⠀⠀⠀⠀⠀⠀⠀⠠⣀⠠⠀⡠⠃
⠀⠀⠀⠀⠰⡀⠄⠀⠀⠀⠀⠀⠀⣈⠐⠖⢁⠠⠲⢀⠘⠤⠯⠹⠍⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠱⡐⠢⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠆⠀⠁⠀⠀⡰⠒⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠘⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⠀⢂⢄⠠⢀⠄⠀⠀⡀⠀⠀⢀⡀⡈
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⢀⢀⠔⡠⢀⠡⠊⠔⡡⢊⠔⢁⢔⠅⡡⠪⠊⠐⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠐⡡⢊⠄⢊⠔⡡⢐⠕⣁⠬⠐⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠔⢀⠔⡁⠐⡡⢊⢔⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠠⠀⢐⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⡀⠀⣀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠐⠀⠐⠈⠐⠀⠀⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠶⠶⢷⣶⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠋⠉⢉⣉⣡⣤⣤⣀⣀⣀⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣼⡟⠉⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⡾⠋⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠟⠻⡏⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⢰⠀⠀⢠⣿⣿⣿⣿⣿⠟⠁⣀⣤⣤⡉⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⠸⡇⡧⢸⣿⡏⠈⣿⡁⠀⠀⠀⠀⠙⠉⢆⠀⢠⣿⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠘⡆⠳⠹⠘⣿⣿⡄⠈⢀⡀⠀⠀⠀⠀⠐⠂⢀⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⠀⠘⣿⣦⣀⡀⠀⠀⡄⣸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡆⢰⣿⣿⡿⠉⠉⠉⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠙⠁⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢠⣭⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣭⣭⣿⣏⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣼⡿⠃⠀⠉⠙⠀⠀⣙⠉⠉⠛⠆⣀⣀⣀⡀⠀⠀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠛⠿⣦⡀⣶⠀⠀⠀⢿⣧⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠈⠉⠛⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠣⢀⡀⠀⠀⣰⡿⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡄⣰⣿⣶⣦⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠐⣄
⠀⠀⠀⠀⠀⣼⣿⣿⠟⠋⠁⣼⠟⠋⠉⠛⠀⣿⣿⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡿⠛⠁⣤⠘⣧⡀
⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⣼⣿⣦⣄⠀⠀⠀⠀⠀⠀⢸⡆⠸⣷⡀
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣾⣿⣶⠄⠸⣧⠀⠘⣿⣿⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⣷⡀
⠀⠀⠀⠀⠀⣡⣤⡤⠀⠀⣿⣿⠏⠀⠀⣿⣆⠀⢹⣿⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠸⣿⣧
⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⠏⠀⠀⠀⣿⣿⡄⠈⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⡆
⠀⠀⠀⠀⢰⣿⣿⠁⠀⣸⠏⠀⠀⠀⠀⢹⣿⣧⠀⠸⡇⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠋⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⢻⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠛⠿⠿⠇
⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠘⠀⢀⣤⣶⣿⣿⣿⠷⠶⠂⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠈⢿⣿⣷⣀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⣠⣴⠾⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣤⡀⠀⣴⠿⠛⣉⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠛⠛⠁⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣠⠔⠀⠀⠀⢀⣠⡔⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀⢠⣾⣿⡟⠀⠀⠀⠹⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣦⡿⠿⠿⠷
⠀⣠⣾⣿⣿⠃⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣠⣴⡿⠿⠟⠋⠀⣀⣀⡀⢀⡀
⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣨⣭⣭⣤⣶⣶⣶⣿⣿⣿⣿⣿⡇⢸⣿⣿⣷
⠘⣧⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣄⠀⢹⣷⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿
⠀⢻⣧⡀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣧⡘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⢹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣟⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠈⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⢿⣿⡇⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈⣿⣷⠘⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠘⣿⡀⢹⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢹⡇⠀⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠸⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢣⠀⠘⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⣉⣵⣿⣶⣶⣤⡄⠀⠀⢹⣿⣿⠿⠿⠿⠛⢛⣤⣶⣿⣿⣧⠈⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⢸⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠆⠀⠀⠀⣀⣀⠀⠠⠤⠤⠤⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⡀⠀⠀⠨⠿⠧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠉⠛⢲⡖⢒⠖⠂⠀⠀⠀⣔⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢦⠀⢀⡞⠀⣻⠃⠀⠀⠀⠀⠙⣆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⠀⠀⣿⡤⠖⠲⣦⠈⢀⠜⠁⡐⠁⠔⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⣥⠞⠁⠀⠀⠀⣨⠤⠋⠀⡰⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣠⡏⠛⠀⠀⠀⠔⠁⠀⠀⠀⠧⡃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠊⠉⠉⠉⠉⠙⠛⠛⠓⠀⠀⠒⠒⠒⠒⢾⠧⣀⡀⠔⠊⠀⠀⠀⠀⠐⠋⣁⣀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠚⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠆⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⣈⣀⣥⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⢀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢉⠩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⡭⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠘⠉⢉⣉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠋⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠶⢀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠋⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠉⢂⠈⠀⠀⠀⠀⠀⠐⠀⣀⠠⠐⠉⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠌⢀⠀⠈⠄⠀⠀⣀⡤⠖⠒⠋⠁⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⣠⡖⠂⠈⠉⠀⠈⣶⣿⡆⠀⠈⢢⡀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢹⣿⠀⠀⡴⠀⠚⠉⠁⠀⠀⠀⠀⠙⠋⢀⡔⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⣏⡟⢉⠙⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡜⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠋⠉⠒⠀⠀⡈⠙⠒⠦⢀⡀⢾⣥⣄⡀⠀⠀⠀⢸⠁⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⠀⠀⠀⠀⠀⠈⠑⠢⠤⡀⠀⠈⠉⠙⢻⣓⢦⣠⠏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠉⠑⠒⠢⠤⢤⡀⣀⣀⡉⠒⠠⡴⠃⠀⠉⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡆⢸⠀⠀⠈⠁⠀⠈⠉⠁⠈⠛⠛⠛⠋⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠐⣠⣧⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠒⠲⠷⢤⣄⣀⣘⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠶⠖⠚⠃⠉⠉⠙⠒⠦⠤⠤⠤⠤⠤⢤⣤⠶⣶⡾⠛⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡄⠀⡄⠀⠀⠄⠀⠠⠀⠠⠀⠄⠄⠀⠀⠀⠀⠂⠰⠀⠂⠂⠀⠆⠐⠰

⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣴⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣿⣿⣿⣷⣶⣴⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠿⣿⣷⣦⣤⣤⣤⣤⣴⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠠⢀⡀
⠻⣿⣷⣄⣴⡦⠑⠊⠀⠀⠀⠀⠀⠉⠙⣛⣛⡛⢋⣉⡀⠉⠛⢿⣿⣷⣿⣿⣿⣶⣶⣶⣤⣦⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⣸⣿⣿⣧⣀⣠⣄⣶⣶⣦⣶⣤⣄⣀⣠⡿⠿⣟⠉⠙⠻⢗⣢⣿⠋⢙⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣛⣛⡓⠲⠶⠦⣤⣄⣀⠀⠙⣷
⠞⠋⠙⠻⣿⣿⣿⣿⣿⠿⠛⠛⠻⣿⢿⣿⣷⣾⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠻⠟⠛⠛⠫⠉⠉⠉⠉⠀⢀⡀⠀⢈⣹⡟⠛⠛⠿⣷⣶⣦⣤⣌⣉⡓⠚
⠀⠀⠀⠀⠈⠙⠛⠛⠳⠶⠶⠞⠉⠀⠁⠀⠉⠉⠉⢉⠉⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠖⠂⠊⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⡶⣶⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣚⣿⣿⣷⣾⣿⣷⣽⣿⠻⣻⣿⣽⢿⣽⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣮⣽⣿⡞⠟⣻⣿⡟⢿⠟⠂⠀⠈⠙⠋⠛⠷⣿⣫⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡾⣿⣿⣿⣷⣯⣿⣟⣿⣷⣿⣦⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣚⡿⠿⠿⠿⣟⣽⣽⣒⣲⣶⣾⣽⠷⠀⠀⠀⠀⠻⠷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣭⣽⣿⣿⣿⠷⣛⣯⣭⣽⣿⢿⡍⠀⠀⢀⡴⣶⣦⡶⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣟⣻⣿⣯⣶⡫⣶⣮⢿⣯⣬⣍⠁⠀⠀⠉⣀⣤⡞⠀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⣿⣿⣿⣿⢧⡱⣽⠌⠿⠶⠖⠀⠀⠀⠀⠀⠻⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢿⣯⣾⠿⣆⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣾⡽⠃⠀⠘⢧⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣤⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⡀⠀⠀⠀⠀⠀⠉⣶⣄⡀⠀⠀⠀⠀⠀⠀⡈⢉⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣯⡓⢤⡀⠀⠀⠀⠈⠓⢯⣗⠂⠀⠀⠀⠘⣏⠛⠤⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡻⣿⣿⣿⣦⡈⠓⢦⡀⠀⠀⠀⠈⢙⣳⡶⢦⣤⣼⣷⠦⣄⠙⢮⢳⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡜⣿⣿⣿⣿⣦⡀⠈⠓⢦⣄⣀⣼⣧⣦⣤⣼⠇⣷⠶⠼⣧⡈⢳⡹⣄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⣿⣄⠀⠀⠈⢹⢦⡻⣿⣿⣿⡶⣣⠖⠢⣄⡈⠑⢻⡟
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣷⡀⠀⢸⣄⢳⡘⣿⣿⣿⡁⢴⡤⠤⠈⢀⡞
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⣿⡿⠿⠿⣿⣦⢸⡏⢳⣧⡈⢻⠁⣹⠞⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⡟⢸⣿⣿⣷⣄⠘⡇⠙⣾⣾⠁⠀⠀⢀⣾⣧
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⣿⣿⣿⣿⣿⣦⣷⠀⠘⣿⠀⠀⠀⡞⢻⣟⠉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣄⠀⠸⣷⡀⠀⠀⠀⢻⢷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡻⢿⣿⣿⣿⣦⡀⡏⣻⣄⠀⢠⢏⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠿⣿⣿⣿⣧⠿⡽⠛⣡⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣷⣌⡛⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠦⠙⠻⣿⡛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⠛⣉⣉⣉⣉⣉⠛⠻⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⡉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣦⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣈⠉⣉⣁⣤⣄⣉⠙⣉⣡⣤⣤⣈⣉⣁⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⢻⣿⡇⢹⣿⠀⣿⣿⠈⣿⡟⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⢸⣿⡇⢸⣿⠀⣿⣿⠀⣿⣿⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⣾⣿⡇⢸⣿⠀⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⣿⣿⡇⢸⣿⠀⣿⣿⠀⣿⣿⡄⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⣿⣿⣇⣸⣿⠀⣿⣿⣰⣿⣿⡇⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⡿⣿⣿⣿⣿⡇⢻⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠀⠿⣿⣿⣿⠇⢿⠀⢿⠃⣿⣿⣿⡿⠇⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣛⣿⣻⣿⣟⢯⠿⢻⢛⢙⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

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

⠀⠀⠎⠿⣦⡀
⠀⢀⡀⠐⠛⠃⢠⡄⢶⣦
⠀⠸⡷⠰⣦⡄⠀⠿⣤⡉⠈⢳⣄
⠀⠀⠁⠐⠘⢧⡀⢀⠀⠹⣶⣄⠈⠳⣦⡀
⠀⠐⠀⣄⠀⠈⠓⡀⠁⠀⠘⣿⣿⣄⠀⠻⢶⣄⠀⠀⠀⠀⠀⠀⣀
⠀⡇⣼⡏⢠⣶⣄⠙⢦⡢⡀⠈⠻⣿⣷⣦⡀⠹⣷⣄⠀⠀⠠⠊⢹⡇
⠀⠀⣿⡇⠀⢩⣽⣷⠀⠹⣟⢦⡀⠈⢻⣿⡷⡄⠸⣿⣆⢸⡇⠀⠘⣄
⠀⢸⠁⢀⡀⠘⢿⡽⣷⣄⢈⣷⣝⢶⡀⠹⣿⣆⠀⠉⢿⣿⡗⡗⢄⣻⡄
⠀⠇⠀⣾⣿⡄⠈⢿⣜⢿⡁⠀⢙⣿⣧⠀⠘⢿⣦⡀⠀⠿⠇⠹⡈⠹⣷⡀
⢸⠀⢰⣿⢿⣷⡀⠈⠹⣿⣿⣆⠀⠈⢻⣧⠀⠈⣿⣿⣦⠀⠀⠠⣵⣄⠙⢷⡀
⡇⠀⠈⣿⠘⠿⣷⡀⠀⠘⢻⣞⣿⣄⠈⢿⡆⠀⠘⠉⢦⣄⠀⠀⣿⣿⣕⢾⣿⣄
⡇⠀⠀⣿⠃⠈⣿⣷⡀⠀⠈⢻⡞⢿⡄⠀⠁⠀⠀⠀⠀⠙⠗⢀⠘⣧⠙⣷⠈⠛⢷⡄
⠀⠀⢀⣿⣇⠀⠸⣿⣧⠀⠀⠈⣿⣎⠟⠀⠀⠀⠀⠀⠀⢀⡀⢸⣇⠈⢳⣟⠀⠀⢰⣍
⠐⡄⠀⠀⠻⣧⠀⢻⣿⡆⠀⠀⠸⣿⡆⠀⠀⢀⠀⠀⠀⢸⣷⡀⢿⠀⠀⢹⡜⣆⠈⣿⣆
⠀⠘⢇⠀⠀⠘⣷⣼⣿⣿⡀⠀⠀⣿⡃⠀⠀⢸⣦⣤⡀⠈⢿⣿⡄⠀⢀⡀⠠⣿⣆⢹⣿⡄
⠀⠀⠈⢧⠀⠀⠘⣱⣿⣿⣇⠀⠀⢸⣿⣦⣆⠈⢻⣿⣿⣆⠸⣿⣿⡀⢸⣧⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠈⣷⠀⠀⠀⠹⣿⣿⠀⠀⠈⠉⠻⣿⠃⠀⢻⣿⣿⣦⠹⣿⣷⡌⢿⣷⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠉⠀⠀⠈⢿⣿⣿⡇⠙⣿⡛⠀⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠙⣄⠀⠀⠈⠙⣿⣆⠀⠀⠀⠀⢾⣷⣦⣀⠻⣿⡇⠀⣾⣇⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠉⠙⢻⣧⡈⠀⠀⠿⠛⢃⡀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⣿⣿⣆⢠⣄⠀⠀⠈⠛⠃⠀⠀⢀⣴⣿⡇⠀⠈⠙⠻⠿⣿⣿⣷⠆⣴⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠐⣿⣿⣿⣷⡛⢿⣖⠀⠀⠀⠀⠀⠉⠙⠛⣧⣄⡀⡀⢸⣶⣤⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⠟⣋⣙⡻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣯⣄⣀⠀⠀⠹⢿⣿⣿⣿⣦⣝⢿⣦⣀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⠋⠁⠀⠀⢾⣿⣿⣿⣿⣿⡿⢡⣾⣿⣿⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣸⡟⣿⣿⣿⣷⣀⠀⠉⠹⣿⣿⣿⣿⣷⣌⡛⢿⣶⣶⣿⣛⠀⠀⠀⠀⠁⢀⠀⠀⠀⢨⣿⣿⣿⣿⠟⣱⣿⠟⠉⠀⢀⣴⠶⠃
⠀⠀⠀⠀⠀⠀⡟⠈⢿⣿⣿⠻⣷⠂⠀⣤⣿⠿⠋⢋⣵⣿⡿⣶⣮⠻⣿⣿⣤⡀⠀⠀⠀⠘⠓⠶⠶⣄⢙⣿⣿⢃⣴⣿⠃⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⡇⢀⣿⣿⣿⠀⠈⠓⠀⠀⠠⠴⣾⣿⡿⠏⣰⣿⣿⣷⣮⡛⡿⠃⠀⠀⠀⠀⠀⠀⠀⢻⡌⠀⢡⣿⠟⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠆⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⣰⣿⣿⣿⣿⣿⠃⡀⠀⠀⢠⣆⠀⠀⠀⠀⠈⢷⠀⣾⠇⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢊⣠⡖⠀⢩⣿⣿⣿⣿⠟⢸⡇⠀⠀⣿⣿⣦⡔⢀⡀⠀⠈⢠⣿⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⡘⠻⠿⠿⣿⣟⣛⣤⣴⣾⣿⠏⢀⣴⣿⣿⣿⣿⠏⠀⢸⡇⠀⠀⠻⣿⣿⣿⣿⢖⣠⡀⣸⡇⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣴⣿⣿⣿⣿⣿⣿⡿⠃⠀⠈⢡⣿⡿⠟⠁⠀⠀⠈⢷⣤⣤⣼⠶⠍⠉⠁⠉⠀⢀⣿⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣭⣀⣀⣠⠄⠀⠀⡴⠟⠋⠀⠀⠀⠀⠀⣠⣸⣯⣵⡖⠂⠀⠀⠀⠀⢀⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⣿⣿⣥⠤⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠒⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣤⣤⠤⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⡀⠙⢆⠱⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠑⠀⠀⠑⠄⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠄⠀⢦⡀⠀⡆⢠⠄⢦⡎⡏⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣦⠊⠢⠀⢵⠀⠀⠈⢆⠀⠑⣇⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⡆⠀⡁⠀⠀⢄⠀⠀⠣⠀⡈⠦⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠈⢆⠈⠰⠀⠀⢣⠀⠄⠁⠀⢄⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠱⡄⢺⡀⠀⠁⠀⠇⠀⠀⠀⠀⠈⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠈⡞⠃⠀⠔⠀⠀⠀⠀⠀⠀⡀⠀⢣⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣧⠀⠀⠀⠀⢆⠀⠂⠀⠘⡀⠀⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠀⠀⠈⠢⠀⠀⠀⠐⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠂⠤⠀⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠈⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠚⠛⠑⣄⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠈⠳⣄⡰⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⢻⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⠏⠀⠀⠀⠀⠀⢀⣾⠣⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡄⠀⠀⠀⢀⠔⠃⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢄⠀⡤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠋⠀⠀⡇
⠀⠀⠀⠀⠠⡀⠀⠀⠈⢆⠀⠀⢀⠜⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⣄⣀⢀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠱⣦⢀⠠⠀⠑⣴⠋⠀⠠⢀⢠⡶⢁⠀⠀⠀⠀⠀⡠⠊⠀⠠⠐⠈⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠈⠢⡈⠢⠄⠀⠀⠀⠀⠀⠀⠦⠊⡠⠂⠀⠀⠀⠀⠈⠣⣀⡀⠈⠁⠈⠁⠈⠢⡀⠀⠀⢘⠄
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⡄⡌⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⢃⡀⠀⠀⠀⠀⠈⢢⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠇⠇⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢜⡦⠠⠐⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠈⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠉⠁⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠉⠑⣤⠂⢢⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⡄⠀⠀⠐⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⢀⠌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠚⠃⠄⠠⠘⠒⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠐⠄⣠⠊⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⣿⣷⣾⣿⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢻⣿⣿⡏⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠈⠁⠃⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠉⠉⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣆⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡷⠀⠀⠠⣴⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣤⣀⣤⣾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠿⢿⣿⣿⣿⣿⣿⣦⣠⠤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠛⢿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⢤⣀⠀⠀⠉⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣄⢠⢰⣎⡄⡆⠄⡠⡀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡪⠑⣺⠇⡇⡿⠀⢘⢹⣷⡇⡖⣧⡟⣦⡦⡠⠀⠀⠀⠀⠀⠀⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⣷⢄⣂⡄⠱⠂⡁⠒⠀⢸⠂⠘⠁⠼⠃⠥⠧⠷⠶⠀⠀⠀⠀⠰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠻⢯⢥⡽⠧⡀⠀⠗⡆⣠⢸⡇⠹⡆⢠⡀⢷⠀⢒⣛⠀⠉⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⠰⠂⠌⣁⠬⠉⠀⠀⠂⠁⠐⠀⢱⠀⢻⡄⠁⠘⢆⣨⡥⠢⠄⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣣⠄⠒⠀⠧⢈⠁⠒⡐⠌⠁⠐⠃⠀⠡⠜⠁⠀⠀⠤⠄⠰⠂⠀⠀⠹⢸⠇⠀⠀⠈⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠃⡨⠀⠁⠀⠀⠀⠌⠉⠀⠀⠅⠀⠀⢈⠀⠐⠀⠀⣀⠘⠂⢣⢀⠗⢁⣍⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠰⢻⣼⠊⠀⠉⠀⠀⢠⠀⣤⢀⢠⡌⡤⠱⠀⠆⡈⠰⢈⠐⠀⡞⣭⠤⡀⣀⣠⡿⡿⡄⠀⠀⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠢⢆⠐⣀⢀⡀⠀⠢⠉⠀⠨⠐⠀⠂⢁⠤⠤⣸⠓⢿⠀⠧⣇⣽⣽⣧⢘⡙⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢪⠓⠀⠡⢀⣀⠤⡤⡐⣶⡞⡏⣳⢿⠸⣿⠆⠻⠋⠠⠀⣶⠋⠼⣹⣎⠡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⢾⡖⢣⢸⢸⡇⣟⠀⢾⣻⣳⣝⠺⣓⡶⠳⣆⠀⠛⡭⠌⠶⣔⠈⡥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⢻⡌⢿⡴⠞⣜⣹⣾⠬⢕⣺⣭⣭⢵⡖⢋⣏⠀⣿⡀⡴⣧⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠈⠁⣴⣺⣟⡭⠳⠂⣻⠋⡍⢃⢈⣀⣧⣉⡽⠶⠶⠛⠛⠓⠀⠀⠠⢖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠊⠀⠀⠀⠈⠁⢁⣈⣷⣣⣐⢋⡶⢰⡞⢻⣣⢭⣶⠮⠎⣠⡀⠐⠀⠀⠀⠀⠀⠉⠊⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣥⢆⢀⣙⣀⣀⡁⣗⣭⣥⡖⣾⡿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⢋⡭⣭⡻⠩⡊⢧⣸⣲⣵⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⣝⣓⣝⡃⣀⣁⡁⡤⣵⢛⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣮⣝⣲⣥⣬⠿⠓⡲⠏⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⣀⣠⣤⠴⠟⠂⠉⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢉⣀⣀⣠⣤⠶⠞⠋⢁⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡉⠉⣁⣀⣠⡤⠶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡽⠆⢀⡤⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⢧⣤⣄⠳⣿⠟⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢠⡁⣄⡹⣶⠀⠈⢛⠒⠊⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠧⣉⠁⠈⢷⡀⠀⠙⢆⠀⠈⢦⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠙⡄⡄⡘⡧⠀⠀⢻⡆⠀⠀⠹⣄⠀⠳⣤⠯⣠⠟⠁⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⣷⡈⠁⠘⣦⠀⠀⠻⣦⠀⠁⠘⣄⣀⠈⢷⣿⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢺⣿⣷⡀⣀⠘⣆⠀⠀⠘⣷⠀⠀⠘⢯⠀⠈⠻⡖⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⢾⣿⣷⡹⠀⠙⢦⠀⠀⠘⣧⡄⠀⠈⢳⣄⠀⠹⣆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⢀⢸⡿⣿⣧⠃⢀⠙⣾⡄⠀⠈⢿⡆⠀⠀⠁⠀⠀⠹⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢻⣷⣴⣿⣯⢻⡁⠘⢷⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⢸⣿⣿⣿⣿⣆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠽⠿⠟⠉⡉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⡘⠟⢿⣿⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣜⡖⠁⢁⠀⢺⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣦⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠨⠅⣙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣀⠈⠩⢭⣿⣷⣔⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡾⢁⣿⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢫⠛⣿⣿⣅⢦⡀⠀⠀⠀⠢⠀⣤⢾⠔⠟⠛⠋⢠⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣲⣿⠀⠀⠀⣀⠿⣿⣿⣯⢻⣞⣿⣦⠴⠋⠀⠀⠀⠀⢀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡺⠿⢿⣤⣴⠞⠁⣸⣿⡟⢛⣿⠿⠋⠀⠀⠀⣀⠄⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣄⢠⣴⠟⢁⡴⡋⣾⢃⠀⠀⢀⣀⣀⣤⠶⡛⠁⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣦⡼⠟⠋⠀⠀⠀⠺⣿⣿⠀⣓⠚⠛⠍⢀⣠⠖⠚⠉⠩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢁⡀⠀⠠⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⢀⣀⡰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠮⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠐⠋⠉⠀⠀⠀⠸⡷⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⡥⢀⠀⠀⠀⢠⡞⣹⡆⠀⠀⠀⢻⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠈⢰⣋⣿⠀⠀⠀⠛⠋⠀⠀⠀⠀⠘⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠉⠁⠀⢠⣴⠆⠀⠀⠐⡶⠆⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠉⠉⢻⣤⡤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢷⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣴⠁⠀⠀⡟⠀⠀⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⢠⣾⡀⠀⣀⣀⡀⢹⣿⣳⠄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠙⠾⣧⠀⠀⠈⠛⠶⠦⠤⠶⠞⠋⠁⠀⢀⣾⠤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⡄⡠⠀⢰⠏⠀⢧⣀⡏⠈⢹⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⠿⠋⠀⠀⠀⠈⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠋⠀⠀⠀⠀⠀⢀⡀⡀⢰⠁⣤⣸⡗⠛⠚⠉⠀⠀⠈⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⠶⠤⠶⠶⣶⡜⣋⡤⠏⠀⠀⠀⠀⠀⣤⠀⢃⣧⣼⠶⠋⠁⠀⠀⠀⠀⣠⠶⠖⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⢠⡶⠂⢀⡤⢒⡾⠀⠀⠀⠀⠀⠀⠈⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⠾⡗⠒⣿⠁⠉⢻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⠃⠙⠡⣾⣫⣽⡟⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡆⢸⠀⠘⡆⠷⠶⢿⠇⠀⢸⣳⠀⣾⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⡿⢲⠶⠖⠋⠁⠀⠀⠀⢛⣩⣭⡛⠃⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠰⠀⠓⠀⠳⣀⣠⡞⠓⠒⠛⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣾⠃⢹⡄⢻⠀⠀⠀⣀⣀⣀⣠⠬⠥⠀⠛⠃⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠉⠀⠈⠓⠈⢦⡴⠊⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢿⠀⡇⡄⢀⠀⡀⢀⣰⣚⠡⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠻⢧⣇⢸⡀⣧⢘⣿⠥⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢿⡇⠐⣒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⢀⣤⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣐⠳⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⠀⠀⢀⣿⣿⣿⣿⣿⠟⢇⣤⡀⢹⡿⠿⡿⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣗⣲⠶⣶⠂⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⡄⠀⢸⠿⢿⣿⣿⠟⠓⣾⠃⠨⣟⠀⠐⢧⣤⣿⣿⠟⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⡷⣤⣌⠉⠛⠚⠓⠒⠒⠛⠛⠋⠉⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⡷⢏⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣟⠀⠀⠀⠀⠀⠀⣠⣴⣷⢠⣤⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢹⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠸⣿⣽⣍⠂⠀⠀⠀⢰⡿⠉⠛⠀⣨⣿⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀⠀⠛⠋⢻⡿⢻⠀⠀⠀⠀⠀⠀⠀⣿⢞⣷⡄⠀⠀⠀⢠⣤⡀⠀⣠⠈⠋⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⠛⢻⠀⠀⠀⠀⠀⠀⠀⠈⡆⠘⡆⠀⠀⠀⠀⠀⠀⣿⢞⡽⠶⠀⠀⠀⠈⠛⠿⠛⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣼⠀⠀⠀⠀⠀⠀⠀⢠⡗⠚⣧⡀⠀⠀⠀⠀⠀⢸⣏⣽⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⣽⠀⠀⠀⠀⠀⠀⣴⡋⠓⠾⠟⣻⢧⣄⠀⠀⠀⠀⣿⣾⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠻⡅⠙⠛⢹⡄⠀⠀⠀⠀⠀⣹⣝⣷⡀⠊⠤⠚⠙⢧⡀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢴⣶⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠳⠘⢂⣀⠤⠴⢻⠇⠀⠀⠀⠺⠛⢎⣭⣽⣿⣶⣦⣤⣴⣾⡇⠀⠶⢾⡿⢿⣯⣁⣀⣀⣠⣤⡤⠶⠋⠁⠙⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣀⣤⠴⣶⠶⠖⢰⣷⣤⣶⣌⣭⡤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠋⠶⠋⠀⠉⠉⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⡴⠶⢶⣶⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⣿⣿⣿⣿⣶⣾⣿⣿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠁⢀⣠⣬⣤⣤⣿⡿⠉⠁⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⠀⢀⣫⣥⣤⣶⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⠋⣁⣠⠤⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⠴⠒⢋⣡⣤⣶⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⡶⠟⢛⣯⣷⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⣋⣥⣶⣿⠿⠟⢋⣽⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡝⠟⣿⣿⡿⠿⠿⠇⠀⠀⠀⠀⠀⣠⠶⢛⣽⣽⣿⠇⢄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋⠉⠉⠇⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡹⠟⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣾⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣴⣦⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣟⡿⠋⠁⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠲⢶⣄⠀⠀⢴⣏⠭⠛⢶⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⢍⠢⡈⠀⠈⠻⣦⠀⠀⠀⠀⠈⠻⣶⣄⢀⠄⡠⡘⢶⣄⣀⣈⣿⣷⡄⠹⣧⣀⣤⡿⠀⠀⠸⠟⠶⣆
⠀⠀⠀⠀⠀⣴⣿⣿⢿⡙⠨⠂⠑⠁⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠈⠙⢷⣦⣈⣫⣲⣿⣿⣿⠿⠿⣿⣆⣈⠛⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⡿⣏⠢⡕⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⢘⡯⣻⣿⣿⣿⠁⠀⠀⠀⣿⣿⣃⡄⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⣰⡿⣕⠮⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄⠀⠀⠀⢫⣾⣿⡯⠽⣿⠀⠀⠀⠀⣿⣿⡧⠄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⣿⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⢿⣷⣦⣴⣿⣿⣧⢖⣫⡿⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠄⠉⠛⢿⣿⣿⣿⠿⣇⠀⠀⠀⢀⣠⣽⣿⡿⠀⠀⠀⠀⠀⠀⣼⠃
⠠⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⡿⢋⣡⠴⢿⠀⠀⢠⣾⣿⣿⣿⣿⣷⣤⣄⣀⣀⡼⠃
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠋⠀⠀⠘⣇⢠⣾⡿⠛⠁⠀⠀⠀⠀⠈⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⣴⠾⠷⢶⣤⣴⠟⠁⠀⠀⠀⠀⣠⡾⠛⠛⢷⡄⠀⠙⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⢾⡇⠀⠀⠀⠀⠀⠀⢀⣿⠶⣤⣄⠀⠀⠹⢦⣤⣤⡾⠿⣤⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⣠⠾⠛⠛⠁⢠⣤⠟⠷⣤⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⠷⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⢦⣤⣤⠾⠿⣤⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠙⠛⢶⣤⠶⠛⠋⠛⠷⣦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⣠⣤⣄⣀⡀⠀⠈⠛⠷⠾⠋⠉⠙⠳⠶⠛⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠈⠉⠁⠀⠈⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⡀⠀⠀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠈⠄⠐⠀⠀⠀⠃⠐



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡽⠂⠀⠸⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⢻⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡈⡇⠀⠀⠀⠀⠀⢸⣸⠃⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠙⠁⠀⠀⠀⠀⠀⠘⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣤⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⢁⣤⣄⣀⠠⠶⠶⡄⠀⠀⠄⢰⠂⠈⠀⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣾⠋⠉⠓⣄⡀⠀⠀⠀⠑⢄⡀⢰⡀⡀⠈⢠⠒⠒⣤⣄⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢄⠉⠠⠀⠈⣷⣄⠀⠀⠺⠂⠉⠉⠈⠹⢸⠏⠖⠒⠀⣾⠐⠮⠗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡉⠀⠀⠄⠀⠙⠷⣦⣤⡄⠀⠀⠰⠊⠉⠉⠙⠤⠒⢳⡈⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠙⠻⢄⠀⣄⠀⢀⠠⡤⢬⠤⠶⢾⡦⠤⡲⡄
⠀⠀⠀⠀⠀⠀⠀⣎⡤⠴⢤⡤⠀⠀⠀⠀⠁⠀⠀⡔⠀⠀⠂⢀⣄⠀⠈⢣⣶⣽⣉⠀⢀⡄⣱⠗⠸⢁⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠈⠀⠀⠜⠈⠋⡓⢯⡵⠋⠙⠂⢠⣀⠀⢺
⠀⠀⠀⠀⠀⢠⠞⠛⠳⣶⣤⡄⠀⠀⠀⠀⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢨⢁⡞⠧⡈⠓⣄⣤⠄⢉⢠⢯⡆
⠀⠀⠀⣠⠐⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⠀⠘⠀⢀⡀⢐⣦⣀⣀⠀⢀⠀⠰⣈⡂⠑⡌⠲⣄⡉⢠⠄⠀⠈⠉
⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠖⠋⠀⢹⣤⢈⡆⠢⢼⠳⣄⣸⠞⠈⠃⠀⢰⠂⡀
⠀⠀⠀⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣾⡏⣹⠶⠀⠫⣰⡆⠉⡘⢷⡄⢴⠴⠂⠀⠀⠀⠁
⠀⠀⠀⣠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡟⠀⠈⠀⠐⠏⠀⡀⠀⢸⠄⠈⢰⡃⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡀⣰⠋⡾⠀⠀⠀⠀⡀⢀⠶⣄⠀⠀⠀⠀⠛⠲
⠀⠀⠀⠀⠘⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢫⢃⡜⠁⠀⠀⠀⢠⠉⠛⠀⠀⠉
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠱⠤⣄⣀⣀⣀⣀⣀⣠⣄⣀⠀⠀⠀⠀⡰⠀⠐⠀⠀⠀⠀⠔⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠈⣃⡆⢤⣀⡀⠀⠇⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠴⠿⠶⠆⠲⠶⠶⠶⠷⠶⠄⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠋⠲⣂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⠝⠁⣰⠉⡙⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠊⠀⢠⠂⠀⡔⠀⢉⣴⠇⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡅⡸⠁⠀⡔⠁⢀⠄⠀⠠⢆⡀⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠟⠁⠀⡸⠀⠠⠋⠀⠰⠁⠀⢠⠃⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠔⠀⢠⠁⠀⡴⠁⠀⠀⡿⣄⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠖⠁⠀⠀⠘⠀⣠⠂⠀⠘⠀⠀⠀⠜⠀⡇⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⢀⠷⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠴⡃⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⡠⠊⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢠⠞⠓⠆⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⣧⣀⠄⠈⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠠⣖⠉⣉⣉⡳⣽⣗⠈⠂⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠈⠙⢏⠀⠈⠻⣦⣄⠀⠀⠀⠀⣰⣿⣤⣤⣴⣢⣤⣶⡋⠀⠀⠀⠀⢀⡔⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⠜⣧⠤⠂⠀⠉⠉⣿⡯⠊⢻⣮⡛⠿⣿⠛⠷⠒⠉⡷⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣅⣌⡄⣿⡌⠄⠈⠳⣤⣬⡿⣡⣿⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⢻⣛⡓⠢⣀⣤⡇⢹⣿⣧⠹⣧⠀⠀⠀⠈⠻⠿⠿⠏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⡻⠋⠁⠀⠙⠓⢾⠉⡛⢦⡙⠂⠈⠢⠵⠖⡶⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⢀⣠⠇⠀⠀⠉⠉⠐⠒⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⢀⡀⠀⠀⠀⣨⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⣢⠶⠞⠛⠉⠉⠁⠸⠛⢏⡤⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠘⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣤⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⠿⠛⠛⠋⠁⠀⠘⣧⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⣴⢀⣾⣵⣶⠾⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢿⡄⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠟⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢫⣽⣿⠉⠙⠂⠀⠀⠈⣧⢈⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⢿⠀⠀⢀⡤⢶⡖⢲⠖⡤⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⣫⣤⡙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⡇⠀⠋⠀⠛⠛⠛⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢋⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⡴⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⣷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⢠⡤⠴⠖⠂⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠖⠃⠀⠀⠀⠀⠀⠀⣰⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠙⠷⣤⣀⠀⠀⠀⠀⢀⣀⣴⠞⠁⠀⠀⠀⢠⡿⢸⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⣿⠹⣆⠀⠀⠀⠉⠛⠳⠒⠛⠛⠉⠀⠀⠀⠀⠀⢀⣾⠃⠀⣷⠀⠙⠻⠶⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠛⠛⠛⠛⠉⠀⠀⣟⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢹⡆⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠶⠶⣦⣄⡀
⠀⠀⠀⢀⣠⡴⠶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⢤⣀⡀⠀⠀⠀⠀⠀⠉⠛
⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠈⣹⠗⠶⠾⢥⡀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣴⣇⠀⠀⠀⢀⣿⣆⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢀⣾⠃⠈⢷⡀⠀⢸⡇⠙⢧⡀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣼⠃⠀⠀⠈⢃⢀⣈⠃⠀⠀⠙⠶⣤⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠜⠁⠀⠀⠀⠠⡆⠈⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢦⡀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠈⣷⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣄⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠈⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠈⠋⠀⠰⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠁⠀⠠⢠⣤⠠⠋⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠐⠶⣤⣀⠰⠆⡀⢀⣼⢀⣠⣍⠳⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠈⢧⣻⡿⣛⠇⡺⠀⠀⠘⠛⠙⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣯⢸⡀⡇⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢠⡿⣦⠀⠀⠐⠠⣽⠸⣷⡁⠀⠀⠀⠀⢰⡇⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠸⠃⠘⡆⠀⠀⡿⣿⠠⣿⠗⠀⠀⠀⠤⡿⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠃⠀⠀⣿⢜⣽⣿⣠⠀⠀⠀⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡇⠀⠀⠘⠀⠀⣿⣫⣾⠃⠈⠀⠀⠀⡞⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⢀⡀⠀⡄⠀⣿⣦⣻⠀⠉⡇⠀⢠⡧⠄⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠘⠀⢀⣧⠀⣿⣫⣽⡀⠈⡇⠀⢼⠋⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠛⠀⠀⠀⠀⠈⠀⣿⣿⣿⠃⠈⡇⠀⣿⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⢰⡀⠀⠀⠄⠀⠀⣿⢽⣻⡀⠈⠁⠀⢿⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣥⡤⢼⡇⠀⠀⠀⠀⢸⡿⣽⣿⣧⡶⡎⠀⢸⡇⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠱⠀⠀⢀⣀⣀⣷⣷⣿⣟⣛⣵⡆⣼⠛⠛⠒⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⡀⢰⣶⣶⣤⣬⣵⣿⣿⣿⣿⣿⣿⣿⡃⣸⠀⣀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠿⠿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣰⣿⠄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣦⣴⡿⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠉⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡜⠁⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠂⠀⠀⠴⠀⠀⠀⠘⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠁⠀⢀⡸⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⡘⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⠈⠀⠀⡜⠁⠀⠀⠀⠀⡄⠀⠀⠀⠇⠀⠀⠀⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⢰⠇⠀⠀⠠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣧⠀⠀⠈⠀⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡟⠀⢸⡄⠀⠀⠇⠀⠸⠀⠀⠀⠀⠆⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣿⠀⢸⡃⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠀⢠⠀⠁⡀⠘⠁⠀⠀⢸⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⢇⠀⢺⠀⠀⠸⠇⢰⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠆⠈⢣⠀⠷⡀⠀⢀⡈⠀⠀⣠⠇⠀⠀⠋⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠢⣀⠐⠀⠉⠠⠞⠁⢀⡴⠃⠀⣤⠦⣀⠀⠀⠈⠉⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⢶⠀⠀⠒⠉⠀⠀⠀⠀⠰⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⡀⠀⠁⠀⢰⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡀⠖⠀⢀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠉⠁⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠛⠓⢠⡀⠶⠀⠀⠰⢀⡆⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠶⠁⠀⠀⠀⠀⠁⢀⡇⠸⠀⠘⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣾⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠰⡀⠰⠄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠖⣀⣶⡤⠞⠁⠀⠀⠀⠀⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⠲⠻⢶⣤⣭⣉⣁⠀⠈⠉⠉⠉⠉⠉⠀⠈⢉⠙⣆⡰⠃⣠⠶⠛⠓⠐⠒⠈⣉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠂⠀⠱⠖⠲⠾⠋⢉⣍⣀⣤⣤⣄⣀⣤⣤⡤⣤⡄⠈⣤⢈⣷⠌⢡⣤⠄⠚⠛⠓⢠⡀⠃⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠉⠁⣀⣀⣠⡤⠀⠀⠀⠀⠰⠶⡄⠐⠂⠈⢉⠤⠟⠛⠛⠛⠁⠀⠀⠁⠀⠸⣞⠡⠞⠃⠄⡀⠀⠀⠀⠀⠈⠶⠀⢰⡄
⠀⠀⠀⠀⠀⠀⡰⠊⠀⣠⠖⠊⣉⠛⠉⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⠀⠴⣆⠀⠀⢀⣄⡀⠁⠀⠀⣀⠀⠀⠀⣇⣸⠁
⠀⠀⠀⠀⢀⠞⠀⣠⠞⠁⣠⠛⠀⠀⠀⠀⠀⠀⢀⠈⢠⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢀⡿⣦⡀⠘⠏⠰⠾⢦⣄⠈⠀⠀⣶⠉⣿
⠀⠀⠀⢀⠉⠀⣘⠃⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⡇⠀⠀⠀⠀⠈⠙⠃⣠⡞⣀⠇⡸⠃
⠀⠀⠀⠘⠀⠀⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢴⠀⠈⠓⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⢀⣤⠾⢯⠚⠁⡸⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣤⣄⣀⡀⠛⢣⣶⣶⣶⡄⠀⣶⣴⠾⠟⠁⠀⠘⠋⡰⠛⠂⠀⠀⠀⠠⠾⠁
⠀⠀⠀⢀⠀⠀⠈⠉⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠛⠳⠶⠶⠀⠀⠉⠁⠀⠀⠀⠀⠰⠎⠀⠀⠰⠶⠖⠈⠀⠀⡀
⠀⠀⡄⠈⢀⠀⠀⡆⠀⠀⠘⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠒⠀⣀⠃
⠀⠀⠁⠀⠀⠀⠀⠙⠦⣀⣀⡤⠃⢀⡠⠦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢤⣤⠿⠏⠁⠀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣴⠎⠀⠀⠈⠉⠉⠁⠀⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠀⢀⣉⣉⠲⠂⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠓⣤⠀⠀⠀⠀⠀⠀⠛⠛⠓⠀⠀⠤⣤⣤⣤⣤⣤⣤⣤⠤⣤⣤⣤⣤⣤⠤⠒⠒⠛⠋⠀⠀⢀⣀⣠⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠶⠆⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠤⠶⠶⠆⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣤⣤⣴⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣟⣿⣭⣿⣭⣿⣿⣯⣿⡗⢿⣿⣿⣹⢿⣽⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣛⣿⣿⠿⢿⣿⣿⠿⣿⠿⠈⠀⠉⠙⠛⠿⠷⣿⢿⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣺⣷⣶⣶⣵⣚⣟⢿⣷⣽⡦⣄⠀⠀⠀⠀⠀⠀⠈⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠯⢿⣾⣿⣿⣿⣿⣛⣯⠭⠭⣭⣿⣳⡆⠀⠀⠀⠀⢲⣦⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢟⣛⣷⣽⣷⣲⣯⠷⢿⣟⣿⣿⣿⡓⠀⠀⠀⣠⣤⣀⣀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⢿⣿⣿⣿⡽⣒⣝⣻⣿⡛⠿⠟⠃⠀⠀⠞⠉⣉⣻⠁⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣷⠙⡟⣇⢻⣯⣿⡟⠀⠀⠀⠀⠚⣿⠃⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡯⣻⣿⣞⣿⣯⠛⢼⠏⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣩⡿⣷⠋⠙⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡽⠛⠁⠀⠀⠈⠳⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⣄⡀⠀⠀⠀⠀⠘⢿⣲⢤⣀⠀⠀⠀⠀⢠⣤⣠⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣦⡙⠲⣄⡀⠀⠀⠀⠉⠓⢮⣄⣀⠀⠀⠈⢧⣈⠙⠲⢍⡳⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⠻⣿⣿⣿⣦⡀⠙⠲⣄⡀⠀⠀⠀⣨⠿⠛⠛⢳⡾⣟⠛⢶⡀⠹⡜⢦
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣦⡀⠀⠙⠳⢦⣴⠿⣿⣿⣶⣾⠁⡿⠋⠛⠻⢦⣘⣎⣳
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣆⠀⠀⠀⣿⢳⡹⣿⣿⣿⣟⡴⠋⠓⠶⢤⠈⢹⡟
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣷⡄⠀⢿⣆⠻⠜⢿⣿⡿⣔⢾⡗⠲⠀⢠⡾⠁
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠻⠿⠿⠛⢻⣷⣤⢠⠄⢤⡧⡌⣿⢁⡽⠟⠁⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠃⣾⣿⣿⣶⣄⢸⡇⠰⣶⣾⠀⠀⠀⢀⣶⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢸⣿⣿⣿⣿⣿⣾⡇⠀⠘⣿⡀⠀⠀⡞⠹⣯⡉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣄⠀⠸⣷⡄⠀⠀⠀⢹⠿⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⣿⣿⣿⣿⣦⡀⡇⣹⣆⠀⢀⠞⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡙⢿⣿⣿⣿⣧⠿⡽⠗⣁⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣦⣍⠻⣿⣿⣧⣿⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⢿⣷⣤⠙⢿⣿⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠸⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢄⢀⣤⣴⣯⣶⣿⣿⣿⣧⡄⠂⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠙⣷⣿⣾⢿⣿⣿⣿⣯⣿⣿⣿⣿⣵⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⠶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣙⢵⣶⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⠿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣁⣾⣭⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⣧⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⡿⣶⠒⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⢠⣬⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⡁⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠓⠛⢷⣞⡩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣿⠏⠓⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡀⡼⠧⠏⢻⣿⣿⣿⣿⣿⣿⣿⣷⠴⣿⣿⡉⠿⠃⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠛⠚⠋⠿⣿⡯⣉⠗⠙⠾⣁⠙⠖⠂⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂⢈⠈⠑⠃⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡠⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣶⣦⣴⡁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⣻⣟⣏⠿⣿⣿⣿⡆⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⠻⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⠠⢀⣨⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⢿⠉⣀⡀⢛⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣯⣷⣦⣽⣿⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣷⣴⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⢄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠃⠀⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡰⠳⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡇⠀⠧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⢴⡆⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡒⢲⣞⣛⣿⠟⠿⠿⣿⠿⠿⠿⣿⣛⣳⡖⢒⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢸⡇⣠⠖⠉⠀⠀⠀⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣄⢈⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠈⠙⢿⡄⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢿⡄⣠⠀⣠⠾⠛⢩⠭⣉⠉⣱⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣙⣿⡞⣿⡞⠁⢀⡤⣀⢳⡈⠑⢎⠀⠙⢄⠀⠀⢀⣤⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⣠⠞⠋⠙⠉⢻⣷⣿⣿⡿⣶⣽⡄⠙⣇⠀⠈⢆⠀⢸⡠⠖⠉⠀⢠⠃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⢼⡇⠁⠀⠀⠀⠈⢳⠀⠹⡀⠀⠸⡀⢸⠀⠀⠀⠀⢸⠤⠒⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢷⣄⠙⢦⡀⠙⣾⠀⣀⡠⠖⠁⢀⣼⠖⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠤⠜⠙⡄⠀⠀⠀⠀⠀⠀⡸⠀⠉⠓⢚⣿⣽⠉⠉⠉⠑⠒⠊⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠦⣤⠤⠴⠁⠀⣤⠶⠋⠁⠘⢟⠢⣄⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

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

⠀⠀⠀⠀⠔⠛⠛⠂
⠀⠀⠀⠄⠂⢀⡀⢀⡀
⠀⠀⣠⣴⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⡌⢹⣿⣇⣰⣶⣘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢀⢔⣇⠀⡀
⠀⣷⠄⠙⢿⣾⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣹⣷⣶⣧⣿⡿⡂
⠀⢹⢿⣄⠈⠻⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢟⢛⠁⢋⡲⢠⣴⣶⣶⣄⡀
⠀⢸⠈⢿⣧⡀⠈⢷⣿⢿⠀⠀⠀⠀⠀⣀⠀⣠⠆⠀⠀⢀⣴⣼⣭⣿⡏⠀⢉⣼⡟⣻⣿⣿⣿⣿⡏⠅
⠀⢸⢷⡅⠈⢡⣄⡀⢿⡘⢤⣤⣤⣷⣶⣼⣿⣷⣴⣦⣶⣶⣶⣿⣯⣿⣧⣤⣿⣿⡿⢿⣿⣥⣼⣉⠀⢀⡀
⠀⠀⠂⠃⠀⠚⢻⠫⠴⠇⠈⠉⠉⠉⢉⢉⡩⠉⠉⠉⠉⠉⠉⠉⠉⠻⠉⠉⢻⠟⠁⠀⠀⠀⠀⠀⠀⢸⠋
⠀⢰⠷⠤⠀⠢⣾⡶⠆⡀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢘⡄⠀⠀⢰⡾⢹⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⡇⠄⠀⣼⣿⣈⣠⣾⣷⣆⡤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢰⡇⠀⢰⣧⡟⣇⣀⣸⣿⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⣷⣧⣼⣷⣦⣟⠛⠐⣿⡄⠠⠳⠟⠿⠿⠿⠿⠷⠶⠶⠶⢶⠤⠤⣤⢤⠀⠀⠀⢤⣤⣤⣴⡶⠲⠆
⠀⠸⢿⣿⡟⠛⠻⣿⣛⣇⡛⠇⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉
⠀⠀⠀⢀⣒⠲⠶⠮⠉⠉⠙⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠠⠀⠠⠀⠠⠀⠀⠀⠀⠀⠠⠠⠠⠀⠠⠠⠀⠂
⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠄⡀⠀⠀⠀⡀
⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠁⠁
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⣤⡄⡀⢠⡤⢄⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠃⠀⠐⠀⢀⠀⠀⠀⠀⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡇⠈⠉⠀⠸⠀⠈⠉⠀⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⢢⣀⣀⣀⣤⠚⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠙⢦⠀⢠⠞⠁⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠘⠲⢄⣳⣃⠴⠚⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢰⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣰⡇⠀⠀⡀⠀⠀⢸⠀⠀⠀⣀⠀⠀⣷⡀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⣰⢿⢀⡤⠊⠁⠉⠉⠹⠉⠉⠁⠉⠢⣀⢸⢷⡀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⢄⡀⠀⠀⠀⣰⣃⣾⣼⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣸⣇⣳⡀⠀⠀⠀⣀⠤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠥⣀⠀⠙⣄⡠⠚⠉⠁⠀⠀⠈⠑⠒⠦⠤⠼⠤⠤⠖⠒⠉⠀⠀⠀⠉⠙⠦⣀⡜⠁⢀⡠⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠱⡔⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡑⡴⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣜⢰⢁⡶⢠⠇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠢⣤⠒⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢧⠰⡄⢱⠘⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⣠⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠶⠔⠂⠁⠲⢄⠀⠀⠀⠀⠀⠈⠒⢡⡀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡆⠀⠀⠀⠀⠀⠙⠒⠤⣀⠀⠀⠀⠀⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠷⠤⠤⠴⠶⠶⠶⠦⠤⠴⠷⠶⠖⠒⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣿⠇⠀⡀⣫⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⡿⠈⢿⣿⣿⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣏⣀⣜⣳⣿⣿⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣇⣀⣈⣻⠏
⠀⠀⠀⠀⠘⠿⣿⠿⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤
⣿⣿⣥⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡞⡟⠳⣄⠀⣀⡀
⠀⢰⢻⣌⣖⠋⠻⣄⣹⠶⣄
⠀⠘⣟⢸⡉⣳⡀⠈⠳⣄⠈⠳⣄
⠀⢰⠻⡄⠉⠉⢳⡀⠀⠀⠙⢆⠈⠙⢦⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣿⠛⢦⠀⠈⠲⣄⠀⠀⣱⣦⡀⣙⢷⣄⢀⣠⠞⡭⣷
⠀⢀⣴⣇⠒⠚⢇⠀⠀⠈⢧⡀⠈⡠⢷⡄⠋⠙⣯⡟⠀⠳⢿⡀
⠀⢸⠁⠙⡄⠀⠈⢆⠀⡠⣢⡙⣦⠀⠀⢻⡄⠀⠘⣧⡀⠀⠸⡇
⠀⡆⠀⢀⠿⣄⠀⠈⠣⡀⠀⠘⠉⣧⠀⠀⠙⣆⠀⠈⠳⣤⡀⠙⣆
⢰⠇⠀⢹⠀⢹⡆⠀⡴⡟⢦⠀⠀⠘⣧⠀⠀⠘⢦⠀⠀⠈⠻⣄⠙⣆
⣼⠀⠀⢸⡀⠉⢻⡄⠀⠀⠞⢦⡀⠀⠘⢷⡀⠀⠀⠑⠀⠀⠀⠈⠓⠮⠳⣄
⡇⠀⠀⠜⢧⡀⠈⣧⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠹⡄⠀⠀⠀⠱⣄⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠙⣆⠀⠀⠀⣹⠁⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠈⢦⠀⠀⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⡆⠀⠀⠀⠀⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣧⠈⣧
⠀⠀⠀⠀⢳⡀⠀⠀⠔⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠸⡄⢸⡄
⠀⠀⠀⠀⠀⢹⡗⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣳⡈⢧⣀⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⣨⠷⠋⠁⠀⠀⠈⠛⢳⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⡸⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠙⠳⠦
⠀⠀⠀⠀⠀⠀⡈⠀⠙⣆⠀⠀⠀⠁⠀⣸⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⣇⠀⠀⠀⢀⠼⠁⣠⣿⣿⠳⣄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢸⠀⠀⢀⡏⠈⠓⠦⠀⠀⠒⠾⢡⠞⠁⢀⠈⠻⢦⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⢸⠀⠀⠸⣧⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⣸⠀⠀⡞⡽⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠨⢸⣄⣇⣿⣆⣀⣤⠤⠞⠁⠀⠀⠁⠀⡏⢀⡾⠁⡇⠀⣄⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠳⡉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣧⠀⠙⣦⣀⣀⣠⠤⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢲⠒⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⣀⣼⠷⠒⠊⠉⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣀⡴⠋

⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣤⡄
⣿⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣀⣄⣀⣀⣶⣶⣿⣿⣿⣿⣿⣿⢣⠤⡤
⡅⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣦⣄⠀⣀⣠⣶⡄⣀⡀⣠⣶⣿⠃⣹⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠉⠀⠀⣀
⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣿⣿⣿⣿⣿⣿⣿⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄
⣧⣿⣿⠀⠀⠀⠀⠀⣠⠀⢸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⡿⠀⠀⢀⣤⣤⣿⣷⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡀
⣿⣿⡇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿
⡿⠟⠇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⣤
⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⡿⠿⠛⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣠⣴⣶⣿⣿⣿⡿⠛⠛⠉⠁⠐
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠋⠉⢻⡇
⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⣧⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⣼⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣟⣁⡀⠀⢀⣀⣠⣤⠔⠚⠛⣥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣋⣿⣛⣿⣿⣿⣿⣿⠁⠐⠛⢻⡇⠀⠀⠀⠀⢤⣤⣀⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢷⣿⣿⠏⠁⠤⠾⠋⣿⣷⡇⠀⡄⣾⣿⣦⣄⡀⠀⣾⣿⣿⣿⣿⡇⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡁⠀⠀⠀⣲⠏⠀⢻⠛⢋⠀⢀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣨⣿⣿⣿⢿⣿⣿⣿⣿⠋⠀⣾⣀⣤⣦⣤⣴⡿⠛⠋⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⣼⣿⣿⣿⣿⣄⠀⣟⣹⣿⠟⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣭⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⣶⠀⣿⡿⠇⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠻⣿⣿⣿⣿⡟⣿⣿⠏⢠⣿
⠀⠀⠀⠀⠀⠀⠀⣴⡄⢻⣿⣧⣠⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡄⢀⣿⡇⠀⢺⣷⣿⠏⠀⣸⣿
⠀⠀⠀⠀⣰⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⣿⣿⣿⠇⠀⠘⠿⠏⠀⠀⠛⠋
⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠉⠉⣿⣿⣿⣷⣴⣿⣿⣿⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠏⠉⠉
⠀⠀⢰⣿⣿⣿⠙⠁⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡿⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠿⠟⠀⠙⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠁⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣏⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⣶⣿
⠄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣿⢿⣿⣿⣿⣿⣿⡿⠋⠀⣀⣴⣾⣿⣿⢿⡟⢻⡿⠿⣿⣿⢀⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠸⣿⣿
⠀⣿⣿⣷⠂⠀⠀⠀⠀⢠⣴⠉⠙⣸⣿⣿⣿⣿⣿⣶⣾⣿⢿⣿⣿⣴⣇⣿⣡⣿⣃⣴⣿⣿⣽⡁⠀⢸⠁⠀⠀⣰⣦⡀⢀⣶⠂⠀⠀⠀⠈
⠀⣿⣿⣿⣿⣙⠋⣿⠃⠙⡋⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣶⣰⣿⠀⣿⣿⣿⢸⣿⣦⡀⠀⠀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢸⡟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢸⠇⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢸⠀⠀⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⡇⠀⢀⡇⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⡘⠀⠀⡼⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠐⠁⠀⡰⠁⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠠⡄⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⡔⠁⠀⡼⢹⣿⠇
⠀⠀⠀⠀⢀⡙⠄⠸⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠊⠀⠀⠊⢀⣿⡿
⠀⠀⠀⠀⢀⡙⠆⠰⣳⠂⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠙⠦⠀⢫⠞⣛⡛⠻⣦⡀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢉⡀⠀⠀⠀⠈⢻⣆⠹⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠙⢧⡘⣧⠀⠀⠀⠀⠀⠀⢠⣿⠁⣰⢦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠳⣽⡄⠀⠀⠀⠀⠀⣿⠃⣰⠃⠘⠁⠘⢉⠽⠛⢛⣳⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡞⢁⡰⠁⠀⠀⠀⢀⡠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠒⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢀⣴⠀⠀⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⡀⠀⢸⢹⣈⡃⣀⣷⠀⠨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣜⠷⣟⡬⣃⣼⣿⣿⣿⣻⣽⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡯⣭⢞⢽⢳⣷⣿⣿⣿⣿⣿⡿⣿⣽⡀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⢬⣾⣿⣿⠿⠿⠛⢿⣿⣿⣦⣿⣽⡄⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠉⠻⣿⣿⣿⠃⢀⠀⠀⠀⢈⠛⠿⣿⣿⣿⣦⣇⡐⣀⢡⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢴⣂⡈⠁⠠⣿⣿⠁⠀⠈⠉⠀⠈⠁⠀⠀⠈⣿⣿⣿⣻⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿⠉⠉⠁⠀⢾⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⢉⣹⠈⢠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡧⠂⠾⠿⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⢀⣼⣿⣿⡁⣨⡀⠘⠇⠀⠳⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠹⢟⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣠⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢈⣧⡄⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⢹⡏⠩⠙⢳⡄⡨⢁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠈⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠈⢌⠸⠋⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡛⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠐⠒⠒⠒⠒⠒⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠉⢀⠠⠤⠒⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣢⡄⡠⠋⠀⠀⠀⠀⢨⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡵⠊⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⣬⠞⢧⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⠉⠀⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⣀⣀⣀⣴⣠⣿⣹⣿⣶⣄⢀⣼⡧⠤⠤⣤⠄
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠥⠒⠊⠉⠉⠉⠛⠩⢿⣴⣿⠿⢿⡿⣿⣿⣿⣅⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡠⠒⢦⣠⠒⠒⠉⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠳⣶⠖⢠⣿⠇⠘⢷⡀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣼⡟⢈⣑⠷⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠉⠀⢸⡿⠀⠀⠀⠻⣆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢠⣿⢀⣾⠋⡶⣷⠶⠲⣞⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠰⠶⢛⣣⠄⠀⠀⠀⢙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠹⣦⣿⣿⣆⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⡾⠿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣶⣬⡙⢿⡋⠇⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣼⣿⣿⣆⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⣿⣿⣿⡃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠋⠉⣉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⣿⡟⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣿⠃⢿⣀⢻⢂⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢟⠏⠙⠒⠈⠉⠙⠻⢦⡀⠀⠀⠀⠁⠺⢿⣷⣦⣦⣴⣠⣀⡀⠀⣀⣀⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠉⠀⠀⠀⠀⠉⠙⠲⠶⠦⢤⡤⣈⣙⡋⠟⠿⢿⣿⣿⡿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠲⣿⡶⠂⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠑⠠⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣀⠀⠐⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢄⣀⠀⠀⠀⡏⢇⠈⠙⠒⠤⣶⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡟⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⠉⠀⠲⣥⠘⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣠⠈⢸⠃⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣛⣿⣷⣶⠶⠋⠀⠹⡄
⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠈⠢⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⢀⡴⢂⡀⠀⠉⢂⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠱⡄
⠀⠀⠀⠀⠀⢀⠔⠉⠉⠁⠲⣤⡦⠔⠊⠝⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡄
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢀⡼⠳⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠟⠛⠋⠉⠉⠉⠛⠻⢿⣶⣄⠀⠀⠀⢀⡴⢞⣡⡤
⠀⠀⠀⢀⣠⣤⣶⠶⢶⣤⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣠⣿⠟⠋⠁
⠀⢀⣾⡟⠉⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⠿⠿⠶⠶⠶⢶⣦⣤⣄⣀
⣰⣿⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠋⠉⢁⣠⡀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣦⡀
⣿⣿⠀⠀⠀⢤⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠘⢿⣷⡀⠀⠀⠀⠈⠛⠶⠶⣶⡶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠉⠛⠷⠶⣤⣄⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠠⣄⠈⢲⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⢀⠀⠀⢠⣿
⠀⠀⠀⠀⢙⣿⣏⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣫⠞⠻⣷⣄⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⡞⠀⠀⡾⢃
⠀⠀⠀⠀⢸⣿⠈⠳⢬⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠈⠙⠻⠷⠶⠶⠚⠉⠀⣠⢴⣿⠋⠀⢀⣾⣵⠏
⠀⠀⠀⠀⢾⣿⠀⠀⠀⠙⢮⡳⠄⠀⠀⠀⠀⢠⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠤⠖⣫⡴⠋⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠖⠋⠁⠀⠀⢠⣾⣟⣁⣀
⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠉⢻⣷
⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠤⠴⠚⠁⠀⠀⠀⠀⠐⠉⠘⠃⣠⡶⢷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⡦⠀⣿⣷⠀⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣣⣴⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣄⣀⣀⣀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋
⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⡿⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢻⡏⢻⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⢸⡇⠈⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠴⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⣼⠃⠀⣿⠀⠀⠀⠀⠀⠘⠋⠉⣀⣀⣠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟
⠀⠀⠀⠀⠀⣿⡄⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠙⠷⣦⣤⣤⣤⣴⣶⠶⢶⣦⣀⠀⠀⠀⣀⣤⡴⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣶⠶⠀⠀⢀⣾⠏
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⡏⠩⠿⠿⠿⠿⠄⠀⠀⠀⠠⠶⠒⠒⠶⠄⢸⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣄⠘⡀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢇⣠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣄⠀⠀⠀⠉⠍⠍⠀⠀⠀⣠⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⠈⢉⠛⠛⠛⠷⣤⡀⠀⠀⠀⢀⣤⠞⠛⠛⠉⠁⠀⠀⠀⠀⠀⢀⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠘⡌⠱⡏⢣⠋⢣⠀⠀⠀⠈⠙⠻⠶⠟⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠉⠓⢄⠀⢨⠀⠸⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⢸⣿⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢓⡂⢀⣿⣻⣦⡗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠲⢦⣄⡈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣂⣉⣉⣁⠟⠫⣸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠙⢸⣈⠔⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⠟⠿⡹⠿⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⡆⠀⠀⠀⠢⠀⠀⢀⡄⢯⡌⠉⠐⠐⠈⠉⣭⠇⣤⠀⠀⠠⠒⠀⠀⠀⣶⠶⠂
⠀⠀⢀⣠⣴⣶⣿⣿⣷⣦⠘⠋⠘⠦⣤⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠤⠤⠞⠉⠙⠀⠀⠀⠀⠀⣀⣠⣾⡷
⠀⣰⣿⠿⣫⣽⠄⢀⣽⣿⠃⢀⣴⣶⣿⣶⡄⣠⣶⣿⣿⡆⠀⢠⣶⣤⣾⣷⡆⠀⠀⣠⣶⣿⣷⣄⠀⣠⣴⣾⣷⣦⠘⣿⣿⠿⠟
⠀⠛⢁⣾⣿⣯⣴⣿⠿⠋⢠⣾⣿⣋⣼⣿⢳⣿⣏⣁⡚⠁⢠⣿⡿⠋⣰⣿⡇⢀⣾⣿⣋⣴⣿⠇⣴⣿⡟⠡⣿⠏⣼⣿⠋⠀⡀
⠀⢀⣿⡿⠉⢹⣿⡇⢀⣴⣿⣿⠛⠛⣋⣵⣟⡉⢉⣿⣿⣴⣿⣿⣀⣴⣿⣟⣠⣿⣿⡛⠛⢛⣥⣾⣿⡏⠀⢀⣤⣾⣿⡏⣠⡾⠁
⠀⠘⠿⠁⠀⠘⠿⡿⠿⠋⠘⠿⣿⡿⠟⠉⠿⣿⣿⠿⢻⣿⡟⠿⣿⣿⠿⠟⠁⠘⠿⣿⡿⠿⠋⠀⠻⢿⡿⠿⠋⠘⠿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⣄⣀⡀
⠀⠀⠤⢤⣤⠄⡾⠋⡡⠀⠀⠀⠀⠤⢤⡀⠀⠀⣀⡤⠭⢄⠐⢎⡓⠀⡠⠤⢤⡀⠀⠀⣀⡤⠄⠀⠀⠀⠠⢌⠙⢷⠠⣤⡴⢤
⢀⣤⣶⣶⠆⠈⠁⠀⠀⣀⡴⠃⠀⠈⠀⠛⠋⢲⠛⠗⠀⠀⠀⠊⠓⠈⠀⠀⠾⡻⡆⠙⠛⠀⠁⠀⠘⢦⡀⠀⠀⠈⠁⠰⣶⣶⣤
⠘⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠝⢧⣉⣚⠻⢳⠃⠰⡞⠟⣓⣫⡼⠡⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⣀⠁⠈⣀⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡯⠁⠈⢿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈








⣿⠿⠿⠿⠿⢟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣛⣿⣿⣿⣟⣛⣛⡛⣿

⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣥⣦⣴⣀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦
⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣼⣇
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠉⠈⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣿⣿⣿⣄⠀⠴⠖⣂⣂⡀⠀⣿⣿⣿⣿⣯⠿⠁
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠻⠛⠋⠀⢀⡟⠹⠀⠻⠇
⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡇⣼⡈⣿⠄⠀⠀⣠⣾⣿⣿⠇
⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⢷⣾⣧⣤⠀⠀⠀⠀⣠⣿⡇⣧
⣿⣿⣿⣿⣿⣦⡄⢀⣼⣿⣿⣿⣿⣿⣿⣿⠸⠙⢿⠖⠀⢀⣠⣶⣿⣿⡇⢸⡆
⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠷⣋⣶⡀⠌⠛⠙⣿⣿⣿⣿⣿⡇⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡮⠟⣡⠔⠧⠌⢠⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣾⡟⠀⡀⢠⣾⡿⣻⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠉⠀⠀⢠⣿⡟⠀⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡰⢠⣿⣿⡁⢐⣿⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⡿⠋⠁⠘⢿⡇⠀⠀⠀⠀⠀⠀⡄⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠘⣿⣷⣤⣄⣀⣈⡄⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣷⠀⣼⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣾⣧⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣽⠟⠁⠀⢿⣿⡇⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⠀⢸⡀⠀⣿⡿
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⣤⣤⡌⠙⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢠⣀⣠⠀⡄⠀⣄⣀⡄⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠢⠤⠴⠂⠀⣠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣏⣀⣀⣀⣹⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡆⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢻⣸⠇⠀⠀⠀⠀⠀⠀⠀⢸⣠⠇⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⡀⠀⠀⠙⢦⣀⢀⣀⣀⣀⠀⢀⡼⠋⠀⠀⣠⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠙⢦⣀⠀⠀⢉⣙⠒⣄⠉⠛⠉⠀⠀⣀⡴⠃⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠙⠓⠲⠦⣄⣈⠛⠒⠯⣌⣉⣁⣀⣠⡤⠶⢚⣁⣠⠤⠴⠖⠒⠂⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀⢀⣠⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠷⠶⠤⠤⠤⠶⠶⠖⠚⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠀⠉⠉⠉⠉⠉⠉⠁⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⢠⠎⠉⠙⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⣄
⠀⠀⡇⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠘⠇
⠀⢀⠁⠀⠀⠀⠀⢻⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰
⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⠋⠛⠓⠄⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠋⠛⠓⠄⠀⡈⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸
⢀⡾⠤⠤⢀⠀⠀⣼⠀⢀⠤⢺⣿⣿⣖⠢⣄⠀⠀⠀⠀⠀⠀⣀⠤⣺⣿⣿⡗⠠⡀⠸⠀⠀⣀⠤⠤⣼
⡎⠁⠀⠀⠈⠷⡀⠛⠀⠻⢄⣻⣿⣿⡿⣀⠞⠀⠀⠀⠀⢰⡆⠻⢄⣻⣿⣿⣟⣠⠟⠀⠳⠞⠁⠀⠀⠈⡦
⠃⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠈⠶⢀⣀⡤⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠤⠶⠶⠶⠶⠷⠦⠤⢄⠀⠀⠀⠀⠀⠀⣀⣀⣀⠴⠁
⠀⠀⠀⠀⠸⠄⢣⠀⠀⠀⠀⢠⠞⠁⡠⠦⠀⠀⠙⠛⠛⠛⠛⠛⠰⡄⠈⡇⠀⠀⠀⠀⣤⢁⠟
⠀⠀⠀⠀⠀⠙⢾⣄⠀⠀⠀⣿⠀⣸⠃⠀⠀⠉⠓⠶⠶⠶⠛⠛⠀⡁⠀⠃⠀⠀⠀⢰⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠉⢆⠀⠀⠿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠠⡄⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⢘⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣇⠀⠈⠑⢦⣄⠀⠀⠀⠀⢠⠞⠁⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣤⡀⠀⠀⠉⠉⠉⠉⠁⠀⢠⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⢉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡤⠐⠒⠚⢤⡄⡀⢀⡤⠞⠒⠒⠠⣤⢄⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⢤⣀⡀
⠀⠀⠀⠀⣠⠖⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠲⣄
⠀⠀⢀⡼⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢷⡀
⠀⠀⡜⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢃
⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸
⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀
⢰⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⢸⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬
⠓⠦⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣤⣤⣤⣤⠶
⠀⠀⡇⠀⠀⠀⡄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⢸
⠀⠀⡇⠀⠀⢾⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡷⠀⠀⢸
⠀⠀⡇⠀⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠁⠀⠀⢸
⠀⠀⡇⠀⠀⢰⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⢸
⣀⣀⣤⣤⣤⣾⣀⣀⣰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣇⣀⣀⣷⣤⣤⣼⣀
⣿⣿⣿⡿⠟⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣁⣀⣀⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢿⣏⡉⣶⡆⣉⡽⠗⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠑⢂⣩⣉⣊⠘⡇⡟⢈⣊⣉⣁⡒⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣀⣸⠋⠀⠀⠉⠓⠷⠗⠋⠁⠀⠈⢻⣀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢥⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡘⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠸⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢊⡝⠛⢛⣻⣟⠦⠀⠠⢞⣿⣛⠛⠛⣍⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣧⠀⠛⠛⠃⠀⠀⠀⠈⠛⠛⠃⢠⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠶⠄⠀⠴⠁⠀⠀⠑⠄⠀⠰⠎⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣄⠀⠀⠀⣀⠀⢀⡀⠀⠀⢀⡄⠀⠀⠀⡀
⠀⣠⡄⢤⡤⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠠⣤⠀⣠⡀
⠀⠐⠀⠼⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠑⠀⠀⠀⠀⠀⠀⠸⠄⠐
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⡆
⠰⠁⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠁⠀⠹
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⡠⡀
⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠘⢤⠜⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠈⠐⠠⠇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠧⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠛⠛⠉⠉⠙⠛⠛⠿⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⢿⣷⡄⠀⠀⠀⠀⣴⣯⠙⢆
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⣠⣤⡤⣄⡀⠀⠀⠀⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⣧⠀⠀⠀⠀⠘⠋⢀⠏
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⢹⣦⡀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠘⠤⣀⣀⣸⠇⢀⣀⣀⣠⠤⠖⠋⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⢺⠁⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠶⠚⠉⢉⣭⣭⣥⣤⣤⣤⣄⡀⠀⠀⢰⣿⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣸⣿⠃⠀⠀⢀⣴⣿⡿⠁⠀⣀⣀⡀⢀⣤⢦⠀⠀⠀⠀⠀⠀⠀⣼⢩⣶⠄⠀⠀⠈⠙⠛⠿⢿⣶⣿⣿⣤⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣿⣿⡤⠶⠾⠿⠟⠋⢀⣴⡿⠛⢉⡇⣿⣧⡞⠀⠀⣴⣄⣤⣤⣀⠈⠉⢉⣤⣴⡄⢀⣴⡿⢷⡄⢸⣿⠏⠉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣧⠀⠀⠀⠀⢠⣿⣿⢁⣠⠞⠀⣿⣿⡀⠀⣴⣿⡟⠉⢻⣿⡇⢠⣿⣟⡽⠃⣾⡏⠀⢿⠇⣿⣿⢀⡾⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠈⣿⡄⠀⠀⠀⢸⣿⡟⠉⠀⢀⡴⠻⣿⣷⡞⢹⣿⡇⠀⠘⣿⡷⢻⣿⠁⠀⣸⣿⡀⠀⠀⣠⢿⣿⡾⣁⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢹⣧⠀⠀⢀⡞⣿⣧⣀⣠⡾⠁⠀⣸⣿⣷⢸⣿⠀⢠⢾⣿⠃⠈⢿⡴⠞⠁⠻⣧⣀⡴⠃⠀⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠙⢦⣀⡞⠀⠈⠻⠟⠋⠀⠀⠀⣇⣼⠟⢸⣿⠀⠯⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡴⢲⣦⡀⠀⠀⠀⠀⣠⣴⣶⣶⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠛⠻⡉⠁⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠛⠁⠀⠀⠀⣼⠟⠉⠈⢹⡿⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⠋⠁⠀⠀⠀⠈⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡠⠞⠁⠀⠀⠀⠀⠻⠃⠀⠀⠀⢀⣤⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠤⠴⢷⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⣀⣠⣤⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠉⠉


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

⠀⠀⠀⢀⠞⠙⢦
⠀⠀⠀⠘⡄⠀⠤⠳⣤⠤⢤⡀
⠀⠀⠀⢠⠫⠿⡄⠀⠙⣆⠒⠉⠲⣄
⠀⠀⢠⣷⠘⠤⠽⣦⠀⠈⢳⣄⠀⠈⠳⣄
⠀⠀⢻⠙⣧⠀⠀⠈⢷⡀⠀⠙⢦⡀⠀⠈⠻⣦⡀
⠀⠀⢸⠀⣆⠘⣆⠀⠈⢳⡄⠀⠀⠙⢦⡀⠀⠈⣙⣦⡀⠀⠀⠀⠀⡴⢋⡙
⠀⠀⣼⢶⡏⠛⠹⣆⠀⠀⠹⣄⠀⠀⣤⠻⣄⠈⠵⠊⢳⣄⣠⠶⠻⠁⠈⠳⡆
⠀⢸⠃⠀⢇⠀⠀⢻⡄⠀⠀⠘⢧⡀⠘⠁⠹⣆⠀⠀⠀⠹⣯⠀⣴⠀⠀⠀⡇
⠀⣼⣄⠀⡸⣆⠀⠀⢻⡄⠠⢶⡨⣷⡀⠀⠀⠘⣧⡀⠀⠀⠸⣷⡇⠀⠸⣿⡇
⠘⠇⠘⣇⡇⠹⣆⠀⠀⠙⣦⠀⠁⠀⢱⡀⠀⠀⠹⣷⡄⠀⠀⢻⣷⠀⠀⠸⣇
⢸⠀⠀⣿⠀⠀⠹⣧⠀⢻⠊⣧⠀⠀⠀⢷⣀⠀⠀⠙⠛⠂⠀⠈⢿⣦⠀⠀⢿⡄
⡆⠀⢀⣿⠀⢧⠀⢹⡄⠀⠀⠘⣦⠀⠀⠀⠙⠗⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠈⢷⡀
⢧⠀⢸⠟⢳⡈⣆⠀⣷⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠘⣧
⠈⢧⠀⠀⠈⢧⠸⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠈⣧⠀⠀⠈⢣⡹⠖⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⣇⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠠⢋⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠉⠉⠉⣙⣿⣿⣿⣶⣦⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠰⠹⣿⣿⣿⣶⣶⣤⣤⣶⠖⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⠟⠋⠹⣷⣄⡀
⠀⠀⠀⠀⠀⣴⠟⢿⣦⣀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⢿⣿⡉⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⢴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⢠⡾⠁⠀⠈⣿⠿⠿⠂⠀⠀⠀⠀⠉⠁⠀⠀⢀⣿⣿⣄⠀⠀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢈⢵⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⣰⠟⠀⠀⠀⣼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⣿⣿⣿⣦⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⢸⡏⠀⠀⠀⠀⣿⣿⣄⣀⠀⠀⢀⣠⣤⣶⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⣿⠃⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠃⠀⠀⠀⠀⠀⢀⣴⣿⠟
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢋⣁⣴⡾⠁⠿⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⢀⣴⣿⠟⠁
⢸⡟⠀⠀⠀⣠⣤⣤⣤⣤⣶⣶⣿⣿⡿⠋⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⢀⣴⣿⡿⠋
⣸⡇⢰⣶⣤⣽⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⣠⣿⣿⠟
⣿⣿⡈⠛⢿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠇⢀⣠⣾⣿⡿⠃
⠘⢿⣿⣦⣄⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣶⣿⡿⠟⠋
⠀⠀⠙⠻⣿⣿⠶⠄⠀⠀⢠⣾⣿⣿⣶⣿⣷⣴⠟⠋⠁⠀⠻⣿⣅⡠⠴⣾⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠈⠻⣶⣶⣶⣶⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠀⠂⠀⠄⢀⠀⠀⠀⠀⠠⠄⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢺⣿⠆⠀⠀⣀⠐⠓⠋⠀⢀⣀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣤⣌⠁⠂⠄⢀⠀⠀⠀⠁⣴⠏⣷⠀⣴⡄⢠⠀⢠⡲⢶⡿⢹⡇⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡿⠃⠀⠀⢀⠃⠀⡄⠰⣿⣴⠇⣴⣿⡅⠀⣥⠈⢶⣁⣀⣸⠃⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠆⠀⠀⠀⠒⣤⣄⣈⠛⠾⠀⠀⠁⠆⠀⠉⠛⠁⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⢾⠆⠀⣼⣤⢄⡀⢀⡜⠿⠁⠀⠈⠑⠚⠀⠀⠘⡀⢀⠠⣤⣴⠈⠉⠀⠀⠀⠀⡀⠤⠄⠐⠒⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠀⠤⠴⠀⠉⠋⢥⣬⠞⠒⠂⡀⠀⠠⣄⠀⠀⠀⠁⠀⠀⠘⠹⠀⠀⠀⡆⠀⠀⠀⠠⣤⠒⠲⡄⢡
⠀⠀⣀⡀⣀⠀⠀⠠⠤⡀⠀⠐⢐⠢⡀⠀⠀⠙⣍⠠⢀⡀⠀⠈⠠⡈⠆⡀⠀⠀⠠⠀⠔⠁⠀⠀⠀⢰⠀⠄⠰⠆⣀⣰⠒⠻⢧⢄⣀
⠀⠀⠁⠀⣠⠀⣄⠀⠀⠁⠘⡀⢠⣄⡄⠀⠀⠀⠈⠑⠀⣌⢢⠀⠀⠈⠢⠈⢠⠂⣴⠁⠀⠀⠀⠀⠀⣠⡆⠂⠐⠒⠉⠙⠀⡀⠐⠚⢤⢉⠉⠑⠆
⠀⠀⠀⠀⢻⣿⣿⣻⡷⢸⣿⠹⠦⢌⡃⢄⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠃⡞⠛⠡⢀⠂⠤⣂⠅⠛⠙⣄⣀⡀⠤⡆⠀⡇⠰⠶⠄⣗⠀⠘
⠀⠀⠀⠀⠘⠿⠿⠿⠇⢸⠀⠀⠀⠀⠈⣂⠷⠠⡄⣠⠜⢡⡀⠀⠀⠀⠀⢀⡄⠃⣀⠀⠀⠉⠴⡷⣄⠀⠀⠀⠀⠀⠰⠀⠈⢀⣷⣾⠋
⠀⠀⠀⠦⠄⠀⠀⠀⠐⠋⠀⠀⠀⢀⠔⣡⡶⠆⠀⡌⢠⣟⠋⡼⣣⣴⣤⣬⡙⢎⡠⠤⣙⡞⣀⢰⠃⣄⡀⠀⠀⠀⠦⠀⣀⠈⠙⠃⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣽⠀⠋⢁⠔⡙⠇⢸⠛⣦⣌⠃⣬⡗⠡⣼⣦⣢⡼⢉⠇⡀⡾⠀⠛⠫⠐⠤⣀⠠⠐⡻⠏⠉⠐⠒⠒
⠀⠀⠀⠀⣀⠴⠲⡄⠀⠀⢀⠰⠷⠀⣤⣡⣤⡄⢇⣈⣤⡙⠛⠋⣣⠘⠋⠳⣦⣤⡤⢶⣧⡤⢋⣴⣧⢀⡀⠈⠑⠶⠆⠊⠀⢀⢀⣀⡀⠀⠠⠶⠴⡆
⠀⡠⠐⠉⠠⠀⠀⠀⢦⣾⠄⠂⠁⠒⢄⠙⢮⣻⡊⢹⠋⢉⠟⡟⢴⠀⠀⢠⢏⠳⣚⡺⠷⣝⢯⡹⠁⢆⡆⠂⠀⠀⠀⠀⡏⠀⢀⠒⢄⠀⢀⡀⠀⡇
⠻⡀⠀⢠⠊⠈⢂⠠⠈⢃⠀⠀⠀⠀⠀⠉⠐⢶⣶⣡⢞⣉⠁⡾⢃⣜⠳⢭⣒⡷⢈⣄⢀⣍⣷⠑⣶⣯⣴⣷⠐⠒⠤⠿⠇⠀⠙⠂⠋⡠⢒⢉⠴⡇
⠀⠡⡀⠀⡂⠐⠀⠐⠀⢀⠃⠀⠀⠀⣀⠀⢨⣽⣯⣇⣉⣻⠟⣰⣿⣿⡷⣦⠙⣿⣾⡟⢿⣿⣾⢀⣿⣿⣭⠉⢋⠁⠀⠀⢠⠀⠀⠀⠸⡐⠉⠋⠀⠇
⠀⠀⠁⡀⠀⣁⠤⠒⠉⠀⠀⠀⠀⢠⠁⠉⠒⠩⠙⠛⠛⠿⠾⢧⣼⣏⣴⣿⣰⣿⣵⣿⣤⡿⠧⠾⠟⠛⠩⠐⠉⠀⠀⠀⠸⡀⠀⠰⠶⠧⠤⠄⠀⠚
⠀⠀⠀⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⣀⠣⠤⠤⠤⣈⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠂⠉⣑⠂⢄⠠⡂⢤⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⢾⠀⠀⠾⠁⠀⠀⠀⢠⡀⠙⠃⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠉⠺⠦⠀⣿⢀⣖⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⣦⠠⠀⠀⢠⡾⠛⡢⣼⡗⠻⠳⡀⠀⠀⠀⠀⠀⠀⣰⠾⡖⣶⣶⡦⣶⡄⠀⠅⡻⢬⣶⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠋⠀⠀⠀⠀⠑⠢⠤⠐⠊⠠⠐⠁⠀⣄⠀⠀⣦⠀⠁⠦⡀⢋⣛⣂⠤⠃⠀⠀⢵⠠⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣇⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠴⡎⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡅⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⠀⢦⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⠀⠀⠀⠀⢀⣰⠧⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠶⠶⠟⠛⠐⠃⠀⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⣈⣐⡠⠈⠁⠐⠢⡄⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠾⡯⣭⣄⣤⣀⣀⣈⣉⣉⠐⠚⢷⡄⠀⠀⠀⡀⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠰⣀⠙⠳⢬⣭⣡⣽⡽⠟⠛⠛⠋⠀⠀⠀⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⣀⡄⠈⠀⠀⠀⠀⢐⠑⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢆⣀⡀⠀⠀⠀⠀⠀⠀⠉⠘⠙⠋⠉⠀⠀⠀⠀⠀⠀⣀⢾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠚⠃⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⠣⢄⣀⠀⠀⠀⠀⠀⣠⡀⠄⢠⡟⠀⠄⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠊⠌⡙⠛⠓⠰⠴⠴⠤⠞⠋⠀⠌⠀⠀⢀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠘⠀⠐⠀⠃⠚⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠠⠀⠐⠖⠀⠞⠑⠞⠳⠶⠳⠞⠰⠗⠿⠞⠷⠋⠞⡑⠟⠺⠒⠷⡺⢞⠶⢃⠾⡚⢟⠳⠛⠚⠛⡛⠟⠻⠟⠛⠀⠆⠀⠂⠀⠀⠀⠠⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⢀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⠀⠀⠀⠀⡀⣤⡀⠀⠀⠀⣤⣼⡄⠀⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⢀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠋⠀⠀⠀⠀⠈⢻⡟⠀⠀⠀⢹⣿⡿⠋⠢⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣰⡆⠀⠀⣸⣧⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⣿⣇⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢹⡇⠀⠀⣿⣿⠀⠀⠀⢸⣿⡀⠀⠀⢸⣿⠀⡀⠀⣿⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣧⡄⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⡇⠀⣿⠀⢠⠀⠀⣾⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣀⣀⣸⣿⣀⣀⣿⣿⣀⣀⣀⣿⣿⣇⣀⣀⣿⣿⣀⣀⣀⣿⣁⣸⣀⣀⣿⣇⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣴⣿⠿⠿⠿⣿⡄⠀⠀⠀⠀⢠⣿⡟⠁⠙⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠁⠀⠈⣿⡇⠀⠀⠀⠀⢺⡏⠀⠙⢻⢿⣿⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⠟⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⣷⣄⠀⢀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠃⠀⠛⠿⠏⠀⠀⠀⠀⣠⣤⣶⠂⢰⣦⣤⡀⠀⠀⠀⣀⣤⣴⢲⣤⢲⣦⣤⡀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⢸⣿⣿⡏⠀⠀⠀⣿⣿⣿⣯⣿⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢠⣾⠿⠿⢿⣿⣆⠀⠀⠀⠀⢀⣾⡿⠿⠟⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠈⠉⠛⠿⢿⠀⠀⠀⠀⣹⣿⡿⠿⠛⠹⣿⠄⠀⠀⠀⠀⣾⠃⠀⠀⠈⣿⣿⡀⠀⠀⠀⢸⡏⠠⠀⠠⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠉⡁⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⢷⡄⠀⠀⠄⢩⡿⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠛⠛⠅⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠒⠀⠋⠀⠀⠀⠀⢀⠀⠀⠘⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⣠⠀⠂⣈⢂⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⠀⠒⢀⠀⠒⠀⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢢⡀⠠⠄⢀⡌⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡊⠀⠺⡀⢀⠗⠀⢑⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠐⢰⡦⠂⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⡜⣏⠄⠂⠀⠘⠃⠀⠐⢄⢹⢧⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠢⡀⢀⠼⠔⠓⠢⠄⡀⠀⠀⢀⠠⠔⠚⠢⠧⡀⠀⠐⠘⢧
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠑⠤⢊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠤⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢸⡇⠄⢆⠂⠀⠀⠀⠀⠀⠀⠀⠉⠐⡌⢁⢀⡀⡀⠀⠀⠀⠀⠐⣰⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠑⠠⢀⠀⠀⠀⢙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠐⠒⠒⠒⠒⠚⠒⠚⠂⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠚⠛⠛⠛⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣶⣿⡿⠁⠀⠀⢹⣿⣶⡄⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠉⣿⣿⣄⣀⣠⣾⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠻⠟⢻⣿⡿⠛⠿⢁⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⣶⡟⠛⢿⡖⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣦⣠⣼⢧⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠉⣉⣉⡉⠛⠓⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢉⣠⠴⠂⠀⠀⠀⠀⠀⢀⣴⠛⣧⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⡴⠋⠁⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠘⣆⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⡴⠋⠀⠀⠀⠀⠀⣠⡴⠞⠋⠀⢀⣀⠀⠀⠘⣦⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠉⠉⠛⠲⠎⢳⣄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠴⠶⠚⠋⠉⡙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠷⣄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡏⠀⠀⠀⣀⡻⣤⢷⣶⣤⣀⠀⠀⠀⠀⠀⢀⣤⣶⡾⣤⢟⣀⠀⠀⠈⠙⢿⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⢸⠃⠀⠀⠀⠈⢻⠃⠸⢿⡿⠹⡆⠀⠀⠀⢠⡏⠻⣿⠇⠈⣿⠀⠀⠀⠀⠀⠈⡇⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡴⢻⡟⢿⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⣸⡇⠀⠀⠀⠈⢧⡀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡿⢻⡟⢦⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠷⣼⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⢹⡇⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣯⠼⠁⠀⣸⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⡶⣶⠿⣶⠶⠶⣶⡶⢶⡶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⢀⡤⢷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣠⠖⠛⠉⠀⢸⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠀⣴⣤⠀⠀⡼⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡉⣰⣿⠃⢠⣾⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠉⠉⠉⠉⠉⠉⠁⣿⡇⠰⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠻⣦⣌⠳⢦⣄⡀⠀⠸⣿⠁⠀⠀⠀⠉⠛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢙⡻⠶⣤⣭⣉⣛⣓⡒⠒⠒⠒⠒⠒⣂⣈⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠛⠳⣦⡉⠉⠉⠛⠛⣛⣛⡛⠛⠛⠛⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⢸⣇⡀⣀⣴⠟⠋⠙⢷⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠈⠛⠷⣶⣤⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣴⠾⠟⠛⠁⠀⠀⠀⠀⠈⣿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⠶⠟⠃
⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣩⣿⣟⡉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠛⠛⠶⠶⣶⣶⣤⣤⣤⣶⡀⠈⠉⠉⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠾⠟⠛⠁

⠀⠀⠀⢀⣠⣤⣤⣤⣤⡤⠤⠶⠶⠶⠶⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢸⡇⢀⣤⣤⣤⡀⢠⣴⣶⣦⢀⣶⣶⣶⡀⢸⣿⢿⣷⡀⣶⣶⣶⡆⢠⣶⣦⣄⢠⣤⣤⣄⡀⢸⡇
⠀⠀⠀⢸⡇⢸⣿⠉⢹⡇⢸⣿⠀⠀⢸⣿⠀⣿⡇⢸⡇⠀⣿⡇⣿⡇⠀⠀⣿⡏⢸⣿⠈⢹⣿⠉⠀⢸⠇
⠀⠀⠀⢸⡇⠈⣿⡀⣸⡇⢸⣿⣀⡀⢸⣿⡀⠉⠁⢸⡇⠀⣿⡇⣿⡇⣀⠀⣿⡇⠀⠉⠀⢸⣿⠀⠀⣾
⠀⠀⠀⠈⣇⠀⣿⠛⣿⠃⢘⣿⠛⠃⠀⠛⢿⣦⡀⢸⣧⣠⣿⠁⣿⡿⠟⠀⣿⡇⠀⠀⠀⢸⡿⠀⠀⣿
⠀⠀⠀⠀⣿⠀⣿⠀⢻⡇⠀⣿⠀⠀⢀⣀⠀⣿⡇⢸⡟⠛⠋⠀⣿⡇⠀⠀⣿⡇⢠⣤⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⣿⡀⠙⠁⠘⠷⠀⣿⣦⣤⠸⣿⣀⣿⡇⢸⡇⠀⠀⠀⣿⣇⣀⠀⣿⣷⣾⠇⠀⠘⠃⠀⣀⡟
⠀⠀⠀⠀⠉⠉⠛⠓⠶⠶⣤⣤⣈⣉⠀⠙⠻⠟⠁⢸⡇⠀⠀⠀⠿⠛⠛⠀⣀⣀⣤⡤⠶⠖⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⣤⣤⣀⣀⣤⡤⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⢛⣛⣛⣉⣉⣩⣭⣭⣥⣬⣭⣭⣉⣍⣉⣛⣛⡛⡟⡛⠻⠿⠿⠿⠿⠿⣿⣿⣿
⣿⣿⣿⡿⢲⣶⠿⠿⠿⣿⡿⠛⠛⠻⣿⠛⠉⠛⢿⡏⠉⠉⠙⣿⡏⠛⠙⣻⡿⠛⠻⢿⡿⠿⠿⠿⣷⡆⣼⣿⣿
⣿⣿⣿⣯⢸⣇⠀⣶⡄⢸⡇⠀⣶⣾⡇⠀⣿⠀⢸⡇⢸⣿⠀⣿⡇⢸⣿⣿⡗⣴⣦⠈⣷⡆⠀⣶⣿⡇⣿⣿⣿
⣿⣿⣿⣿⢸⣿⠀⣿⡇⢸⣧⠀⣿⣿⡇⠠⣿⣶⣾⡇⢸⣿⠀⣿⠂⢸⣿⣿⠀⢸⣿⣶⣿⡇⠀⣿⣿⠃⣿⣿⣿
⣿⣿⣿⣿⢸⣿⠀⣉⠀⣼⣿⠀⣄⣼⣷⣄⡈⠙⣿⡇⠸⠿⠀⣿⠀⢀⣀⣿⠁⣿⣿⣿⣿⡇⢠⣿⣿⠀⣿⣿⣿
⣿⣿⣿⣿⠘⣿⠀⣿⡄⢹⣿⠀⣿⣿⣿⣿⣿⠀⢹⡇⢀⣠⣴⣿⠀⢸⣿⣿⠀⣿⡿⢻⣿⡇⢸⣿⣿⠘⣿⣿⣿
⣿⣿⣿⣿⠠⣿⣤⣿⣧⣘⣿⠀⠛⠿⡇⠀⣿⠀⢹⡇⢸⣿⣿⣿⠀⢽⠿⣿⠀⠛⠃⣼⣿⣇⣼⣿⣿⢰⣿⣿⣿
⣿⣿⣿⣿⣶⣤⣬⣉⣙⡛⠻⠿⢶⣶⣿⣦⣀⣀⣾⡇⢸⣿⣿⣿⣀⣀⣤⣿⣷⡶⠿⠟⢛⣋⣩⣥⣴⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣬⣭⣙⡛⠻⠿⢿⠿⠿⠟⢛⣯⣽⣧⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣀⠠⠤⠒⠋⠀⠳⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣠⠚⠙⠓⠂⠀⠀⠐⠒⠂⠘⡶⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠰⣽⠀⠈⠉⠁⣸⠀⠈⠉⠁⢠⣳⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⢧⠀⠀⠀⠁⠀⠀⠀⠀⡜⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣨⠷⣄⠀⠉⠉⢀⣠⠾⣄⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⠋⠀⠐⠾⡉⠑⠂⢉⠗⠂⠀⠙⡆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠚⠂⠀⢄⡀⠘⣆⣰⠃⢀⡀⠀⠐⠓⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⠑⠾⠧⠊⠁⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡞⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢳⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢀⡞⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢳⡀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠒⢤⡀⠀⠀⣠⣮⠤⠶⠓⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠶⠤⣵⣄⠀⠀⢀⡴⠒⠙⢧⡀
⠀⠀⠀⠀⠀⠀⢠⠿⠲⢤⡀⠀⢹⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡏⠀⢀⡤⠖⠿⡄
⠀⠀⠀⠀⠀⣰⢃⠀⠀⠀⠘⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠃⠀⠀⠀⡙⣆
⠀⠀⠀⠀⠀⡏⡎⣸⢁⡶⢠⠇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠲⢤⠖⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠸⡆⢶⡈⣇⢱⢹
⠀⠀⠀⠀⠀⢳⡄⡇⢸⠱⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠛⠓⡦⠴⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠳⠎⡇⢸⢠⡞
⠀⠀⠀⠀⠀⠀⠉⠳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠀⠀⠀⠀⠈⠑⠢⣄⣀⠀⠀⠀⠀⠀⠈⢊⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣵⣤⣤⣤⠤⠖⠋

⠀⠀⣾⠋⠁
⠀⠀⢿⡈⠁⠀⠚⣍⠙⠦⡀
⠀⢠⡾⠁⠘⣶⠋⠉⠳⢄⠘⠷⣄
⠀⠀⢷⡈⠀⠈⢳⡀⠀⠀⠙⠲⣄⠑⠢
⠀⠀⢨⣷⠀⠈⠵⠻⢄⠀⠀⠀⠘⢷⣄⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⡄⢹⣿⠀⠘⡆⠀⠀⠀⣄⠀⠠⡶⣌⢳⡀⢀⣧⣀⠀⠀⠀⠈
⠀⢻⠘⣿⠀⣀⡀⠀⠀⠀⠈⠳⣄⡀⠘⡇⢿⡉⠈⢿⣴⠃
⠀⠈⠀⣹⡜⠃⠀⠰⣄⠀⠘⠉⠣⣿⠄⠀⠈⠻⡄⠀⠙⡆⠐⣦⡴
⠀⠀⢀⣿⣿⣆⠀⠀⠈⠣⡀⠀⠀⠆⢠⠀⠀⠀⠹⣆⠀⠀⠀⡙⠦⣼⡄
⠸⠀⢸⣿⣿⣿⣄⠀⠠⢶⡌⣆⠀⠀⠀⢷⡀⠀⣠⣼⠇⠀⠀⠘⣦⡀
⠄⠀⢸⣿⣿⣿⣿⣆⠀⠀⠁⠘⣧⠀⠀⠈⠃⠀⠋⠙⢷⣦⡀⠀⠈⠻⣖⢤⣀
⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠃⠙⠀⠀⠀⢸⠀⠙⠧⡄
⢇⠀⠈⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡀⠀⠻⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠹⣇⠀⠙⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠹⠀⠀⠈⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠹⣄⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⡀⠇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣷⡾⡀
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⡄⠀⠢⣀⣤⣶⣶⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠸⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠔⣹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢨⣷⡀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣴⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⢠⠃⢸⣿⣄⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⢰⠋⠘⠷⠤⠤⢄⡀⠈⢿⣿⡿⠁⣠⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢀⡔⠁⠀⠀⠀⠀⠀⠀⢿⣆⠀⣿⠃⣴⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣰⣇⣼⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⢛⣛⣛⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣦⡀⠀⠀⠀⢀⣀⡼⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣧⣠⣿⣿⣿⠶⠖⠋⠉⠀⢀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⣠⣾⣿⣿⣿⡇⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠸⣿⣿⣿⣿⠟⠁⠀⠀⢀⣾⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⣀⣴⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠋⠉⠹⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡏⠉⣠⣌⠈⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣅⠸⢿⡿⠀⣼⠿⠿⣿⠟⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠻⠴⣄⣰⡶⠿⠀⢀⣀⣀⠀⠛⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢠⣶⡄⠙⣿⣄⠀⢰⣿⣿⣿⣷⡀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠈⡛⢁⣨⣿⡏⠀⠘⢿⣿⣿⠟⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⢀⣀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠟⠛⠛⠛⠋⠉

⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⣀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⢀⣤⣿⠿⠁⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⣿⣧
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡏⠀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠸⣿⡇
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢀⣴⣶⣦⣤⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣶⣄⠀⠀⣸⡏
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⢠⣿⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⢀⣿⣧⣀⣀⣀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣿⣿⣷⣄⣀⣀⠘⠃⠀⠀⠀⠀⠀⢀⣀⣤⣀
⣀⣼⣿⡿⠿⠿⢿⣧⡀⠀⢸⣿⡆⠀⣴⣾⠟⢿⣿⣿⣿⣿⣿⠛⢷⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⡇⣰⣶⣄⣀⣰⡿⠿⢿⣿⣿⣆
⣿⡏⠀⠀⠀⠀⠀⠙⢷⣤⣼⣿⡇⠀⠻⣿⣤⣼⣿⣿⣿⣿⣿⣤⣼⡿⠏⠀⠀⠀⠀⠀⣠⡾⠟⠋⣠⣾⣿⣿⣿⣿⣿⣦⣤⣄⠈⠙⠻⢿⡿⠏⠀⠀⠀⠀⢹⣿⡆
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⠁⠀⠀⠈⠻⠿⠿⢿⣿⡿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠙⠻⠿⠿⣿⣿⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠛⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⠁
⠀⠀⠀⠀⠻⣿⣿⣷⣶⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣦⣀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⢀⣽⠿⠿⢿⡿⠟⠛⠛⠛⠛⠛⠉⠙⠿⣿⣿⡄⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⠀⠀⠀⠘⠋⠀⠀⣿⠿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠈⢿⡟⠀⢸⣿⡄⠀⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡆⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠟⠛⠉⠀⠀⠀⣠⡄⠀⢸⣿⡇⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣤⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢸⣿⡇⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣇⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢸⣿⡁⠀⣴⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⣠⣿⡇⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠁⠀⠀⢸⣿⣷⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠙⢿⣿⡦⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠟⠀⠀⠀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣤⣀⡀⠀⠀⠀⠸⠿⠿⠿⣿⣿⠿⠿⠋⠀⠀⢀⣠⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠧⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠼⢿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣼⣿⣇⠀⠀⠀⠀⠀⠀⣼⣿⣧⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⡴⠛⠛⠛⠛⠛⠛⠛⢦⣤⣄⠀⠀⠚⠛⠛⠛⠛⠛⠛⠁⠠⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡆⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢰⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠱⣶⡄
⠀⠀⠀⠀⠀⣾⡿⠁⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠻⢿⣄⡀
⠀⠀⠀⢠⣾⡏⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠸⣿⡇
⠀⠀⠀⢸⣿⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠈⠋⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⣧
⠀⠀⢠⡄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠛
⠀⠀⣿⠇⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠛⠿⢿⣶⣤⣀⠀⢀⣀⣀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣀⠀⠀⣀⣠⣴⣾⡿
⠀⠀⠀⢻⣿⠟⠋⠛⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠛⠛⠿⣿⡇
⠀⠀⠀⢨⣧⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡸
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⠇⠀⠀⠀⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠸⡟⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣷⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠻⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⣆⠀⠀⠀⢸⣿
⠀⠀⠀⣶⠀⠀⠀⢀⡄⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠃⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠏⠀⠀⠀⣼⣿⠀⠀⠀⠀⢸⣿⣷⣦⣀⣀⣀⣠⣤⣤⣤⣀⣠⣿⣿⡁⠀⠀⢹⣿⣿⣀⣠⣤⣤⣤⣤⣤⣄⣀⣰⣿⣿⠁⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡿⢿⣿⣿⠿⠿⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢰⣦⡀⠀⠘⠃⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣦
⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⢸⣿⠃⠀⢠⡄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠸⠟⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⢰⣆
⠀⠀⠀⠀⢀⣀⣴⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⣸⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠃
⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟
⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⠟⠋⠁⠰⣿⣿⣷⡀⣀⣸⣿⠀⠀⢸⣷⡀⠀⣶⣿⣿⡆⠈⠙⢿⣿⣷⣦⡀
⠀⠀⠀⠀⣀⣀⡀⠀⢀⣶⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣿⣿⠃⠀⠀⠘⣿⣿⡏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡆⠀⢀⣀⣀
⠀⠀⠀⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠈⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠷
⣿⣿⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⠇⠀⠀⠀⠀⢸⣿⣧⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⡆
⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣠⣤⣤⣀⣀⣀⣤⣤⣀⣤⣤⣤⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣻⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠚⠙⠃⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⡟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⠀⠙⠚⠁⣸⠀⠙⠚⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⣀⣀⠄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣏⠀⢀⡠⢟⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⢹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠈⠉⠙⠋⠉⠀⠀⠉⠻⢿⣿⣿⣾⣿⣦⣛⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⢸⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠃⣇⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢧⡝⠀⠀⠀⠀⠀⠀⠀⢸⣴⠃⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠷⡄⠀⠀⠙⢦⣄⡤⠤⢤⣀⣠⠞⠁⠀⢀⡼⢧⣀⡀
⠀⠀⠀⢀⡴⠋⠉⣀⣀⠀⠙⠲⣄⣀⣠⠤⢍⠵⠀⠀⠀⠀⣀⡴⠋⠀⠀⢈⠙⠳⣄
⠀⠀⢠⠏⠀⠀⠀⠀⠉⠙⠛⠲⢤⣌⡉⠓⠶⠶⠶⠖⢚⣋⡥⠴⠒⠛⠋⠉⠁⠀⠈⣧
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⣀⠀⢀⣀⣀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣾⣿⣿⣿⣴⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡿⣿⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠿⠿⠿⠿⢿⣿⣿⢻⣿⡿⢿⣿⠛⢻⣿⣿⣿⠏⡟
⠀⠀⢀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣋⣁⣨⣿⡛⠛⠈⠛⠷⠗⠾⠋⢠⡾⠇⠐⣶⣿⣿⡷⢴⠇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⣷⣾⣿⣿⣿⣷⣾⣶⣦
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈
⠀⠘⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡦
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠷⣬⣯⣽⣿⣿⣿⣿⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠈⠒⠢⣤⣀
⠀⠀⠀⢀⣄⡆⢠⣦⣮⢽⣦⡀
⠀⠀⠀⣿⠟⠇⡿⣿⢃⣿⡇⠈⠁⠲⡄⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠒⠒⠒⠒⠒⠶⢤⡀
⠀⠀⢄⣿⣾⢾⣿⣿⣸⣿⡇⠀⠀⠀⡏⠁⠑⠒⠒⠋⠉⠉⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠑⠲⠦⣤⣄⡀⠀⠀⠀⣀⣀⢠⣶⣿⣿⣿⠿⣿⣍⠩
⠀⠀⣻⣿⣷⣿⣿⣣⣿⣿⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠃⢀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠉⠙⠶⠤⠴⠿⣆⠀⢹⣿⣿⡾⢻⣷⣿⣷
⠀⣰⣿⣷⣟⣿⣽⣿⣿⡗⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠶⠿⠟⠋⠙⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢹⣿⣿⣧⣍⣸⣿
⠀⣵⣻⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣿⣿⣽⣿⣿⣿
⠸⣻⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢰⣼⣿⣿⣿⣻⣿
⢠⢧⣿⣿⣿⣿⣿⣃⢈⢘⡴⠛⠿⢶⡜⢦⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⠈⠀⠀⢀⠀⠀⠀⡿⠀⢰⣾⣿⣯⣷⣿⣿
⠸⣿⣿⣿⣿⠟⠀⠀⠈⠀⠀⠀⠀⠘⣷⡄⢹⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠹⣦⡀⠀⠀⠙⣆⣀⣤⣮⣦⣰⣤⡆⠀⣸⣿⣿⣯⣿⣿⢿
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠘⣧⣬⣷⣖⡂⠤⣠⣶⣄⠀⠀⠹⡆⠀⠀⠹⣿⣿⣶⣄⡿⠛⠋⠉⠹⣿⣿⠧⣀⣿⣿⣿⣿⣟⢻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⢿⣍⠙⣦⠀⠈⢷⣄⣀⠀⠙⢦⣒⣠⠟⢿⣿⠟⠁⠀⠀⠀⠀⠙⠛⣶⣶⣶⣿⣿⡟⡏⡄⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠙⢶⣐⣻⠿⠿⠴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠿⣃⠛⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⢦⡉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠓⢄⠀⠉⠳⠦⡀⠉⠀⣄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢢⠀⠑⠦⣀⠀⠈⢢⡀⠀⠙⠦⡅⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⣄⠀⠱⣄⠀⠈⣧⡀⠀⠙⢦⡀⠀⠘⣦⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣦⠀⠈⠣⡀⠈⠻⣆⠀⠀⠑⠀⠀⠈⠳⣄⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⣧⡀⠀⠐⣄⠀⠈⠳⡄⠀⠀⠁⠀⠀⠀⠑⠺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⠉⢿⡀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡄⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢚⢦⡈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠹⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢱⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠈⠀⠻⣆⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⣆⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⡿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣄⣻⣤⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣬⣉⠛⠟⡡⠖⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢠⠛⣿⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⢠⡝⢁⡾⠁⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠘⠁⠜⠉⢳⢦⣀⡀⠀⢀⠄⠀⣠⠟⠁⣾⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡟⠳⣄⡀⠀⠀⠀⣸⣿⠞⠁⣠⣾⡏⠀⢸⠏⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠐⠒⠈⠙⠣⢠⣿⡟⣡⣴⣿⠛⠉⢀⣠⠞⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡸⣆⠀⠀⠀⠀⢸⠏⣼⢫⣿⣃⣤⣶⠟⢉⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠳⠄⡁⠀⠈⢩⣏⣼⢣⣿⣿⣿⣿⣥⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣾⣿⣿⣿⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠾⠛⠋⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣻⣫⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣽⣯⣽⣿⣿⣿⣿⣯⣭⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠒⠒⠒⠲⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣀⣤⣤⠴⠶⠶⢒⣺⣿⣀⣀⠀⠀⠀⠀⠈⠙⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⡠⠔⠊⠉⠀⠀⢀⣠⣔⠾⠟⠛⠋⠉⣉⣩⣽⠶⠂⠀⠀⠀⠀⠈⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⡴⠋⠀⠀⣀⣤⣴⣾⠟⠋⠀⢀⣠⡴⠶⠛⠉⠁⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣰⢃⣾⣡⡴⠞⠛⣩⠶⠋⢀⣠⠴⠚⠉⠁⣀⣤⠶⠖⠛⠋⠉⠉⠉⠙⠛⠛⠓⠶⢤⡀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⣴⣇⣾⠟⠁⠀⣠⠞⠁⣠⠖⠋⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠷
⠀⠀⠀⠀⣸⣿⣾⠃⢀⡴⠋⠁⠀⠸⢧⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢰
⠀⠀⠀⠀⣿⣿⣫⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾
⠀⢀⡤⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣀
⠀⡞⠀⢸⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⣧⣙⣦
⢸⠁⠀⢸⣿⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣼⠋⢹
⠸⡆⠀⠟⣿⠀⠀⠀⠀⠀⣀⣤⠴⠶⠒⠒⠒⠺⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⢒⣛⠉⠉⠿⠤⠤⠾⣿⠿⠛⢶⣼
⠀⢻⣆⣀⣿⠀⣰⣶⡶⠿⠛⠃⠈⠉⠉⠉⠁⢉⣙⣛⣛⠓⠶⠶⠦⠤⠶⠶⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢰⣿
⠀⢸⡿⠻⣟⠛⢻⣧⡀⠀⠀⠀⠶⠶⠒⠛⣛⣉⣉⠉⢹⠇⠀⢀⣤⣤⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⠖⠀⠀⣿⡇⠀⢀⣾⣿
⢀⡿⡇⠀⣿⠛⠛⣿⣇⠀⢀⠀⠀⣠⣶⡿⠻⣿⡿⠛⢛⣀⣴⠁⠀⠀⠙⣦⠸⡀⣰⣿⡋⢉⡽⠋⠛⠀⢠⣿⠁⠀⣾⣿⠋
⢸⠁⣷⠀⢸⡆⠀⠸⣿⡄⢸⡇⠀⠰⠟⠷⠾⠋⠀⠀⢠⣿⣿⠀⠀⠀⠀⠸⣆⢱⡄⠀⠉⠉⠀⣀⣀⡤⠞⣁⢀⣾⣿⠁
⠈⢧⣹⡄⠀⣷⠀⠀⡿⣿⠛⠿⢷⣦⣄⣀⣀⣀⣀⣤⣾⠃⠹⠀⠀⠀⠀⠀⠙⠷⠿⠿⠛⠋⠉⠉⠉⠁⠈⢉⣿⣿⡇
⠀⠈⢻⣅⠀⢹⡆⠀⡇⠈⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠙⢶⣼⣧⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠈⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡴⢦⡀⠀⠀⠰⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠀⠀⠀⢶⣤⣶⣷⣦⣀⡀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⠁⣀⣠⣤⢤⠉⠋⠛⠁⡙⣿⣧⣤⣤⠞⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣦⡀⠀⠀⠀⠀⢠⣶⣼⠁⣀⣤⣴⠾⡥⠤⠶⠶⠛⠛⠶⢾⡅⠀⠉⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠻⢄⡀⠀⢀⣿⠉⣿⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠙⢶⣾⠻⣦⡿⣇⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⠀⣸⠇⠀⣠⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⢷⣄⠀⠘⢦⡀⢀⣰⠏⠹⣿⠇⠀⠀⠀⠚⠉⠀⣐⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣍⣛⠶⢤⣤⣤⣤⣤⣤⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢯⣭⣽⡾⠿⠿⢛⣫⡽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠉⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣁⠤⠤⠤⠤⠤⠤⠤⢄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡇⠀⠒⠆⠀⠀⠀⠀⠖⠆⠀⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠒⠒⠀⡆⠀⠐⠒⠂⠀⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣧⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠂⠂⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⣦⢄⣀⣀⢴⡊⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⠀⠀⠈⠙⢄⠀⢀⠜⠉⠀⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠲⢤⡈⢦⠏⣠⠴⠂⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣴⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⡄⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠙⠢⡄⠀⠀⢀⣾⠥⠖⠓⠒⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠓⠶⢽⣆⠀⠀⠀⡤⠚⠉⠹⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠒⠤⣀⠀⢸⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣸⠀⢀⡤⠔⠚⢧⡀
⠀⠀⠀⠀⠀⠀⣼⢃⠀⠀⠀⠈⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠎⠀⠀⠀⢀⢹⡄
⠀⠀⠀⠀⠀⢰⡇⡏⣸⠁⣴⠂⡜⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠦⣤⠖⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠘⡄⢲⡄⢹⡈⡆⣷
⠀⠀⠀⠀⠀⠘⢷⡀⡇⢸⠙⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠶⢤⣄⡤⠴⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠱⠜⢹⠀⡇⣰⠟
⠀⠀⠀⠀⠀⠀⠀⢻⡓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠁⠀⠀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠈⠒⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⠀⠀⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠉⠻⣿⣿⡿⠋⠀⢀⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠈⠋⠀⢀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣄⣀⣴⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠘⢥⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠚⢿⡟⠃
⠀⠀⠀⢠⠀⠀⢠⢖⡄⠀⡇⢸⠀⠀⠀⡅⠀⢀⢰⠀⠀⠀⣄⠀⠀⣴⣷⡄⠀⣿⡇⠀⠀⢸⠀⠀⢀⣼⡀
⠀⠀⡐⠉⢧⠆⠠⠀⠆⠀⠐⠌⠀⠀⠂⠀⠂⠈⠀⠈⠀⡸⠿⡤⠀⠿⡿⠇⠀⠻⠇⠀⠐⠀⠐⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠄⠀⠀⢐⣉⡆⠀⠀⣌⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠀⠀⢀⡇⠀⠀⡇
⠀⠀⠀⣄⠀⠀⠈⠉⡇⠀⠰⢤⠀⠀⡉⠁⠀⡀⠀⠀⠀⠰⡀⠀⠀⢸⣿⡇⠀⢿⣿⠀⢸⣿⠀⢠
⠀⠐⠚⠃⠸⠀⢴⠀⢃⡄⢰⠀⡀⠀⡇⠀⢐⠙⡂⠀⢰⣾⣶⣶⠀⡼⡿⢣⠄⢸⣿⠀⢸⣿⡄⣿⣿
⠀⠘⠂⠀⠚⠀⠀⠀⠀⠀⠈⠀⠁⠀⠉⠀⠀⠈⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠈⠋⠀⠈⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢠⠎⠁⠀⣰⣣⠀⡷⣄⡇⡌⢉⡇⡆⠀⢸⡯⠁⠀⠀⠀⡏⢉⡆⡞⠳⡄⡔⢦⡄⡖⢢⠀⠀⠀⢲⣒
⠀⠈⠳⠶⠲⠃⠈⠗⠃⠘⠇⠷⠚⠑⠓⠶⠘⠶⠂⠀⠀⠰⠟⠉⠀⠓⠚⠁⠳⠚⢡⠷⠋⠸⠦⠦⠸⠤
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠤⠀⠀⠀⠐⠓⠠⠄⠀⠀⠀⠀⠶⠈⠐⠴⠂⠀⠅⠒⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣀
⠀⠀⠀⠀⠀⣤⡶⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠻⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠈⠻⣶⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣴⣶⣦⣤⣄⠀⠀⠀⠀⢰⣿⣿⡿⠏⠑⢠⣸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣸⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⣿⣿⣷⡀⠀⠀⢿⡟⠹⡄⠀⠀⠺⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠁⠀⠀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠉⡁⠀⠀⠀⠀⠈⠁⠀⠟⣀⠀⠀⢻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣤⣴⠉⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⡇⠀⠀⠀⠀⠀⠉⠙⠛⠋⠀⠀⠀⠀⣬⣍⣸⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡇
⠀⠀⠀⠀⢼⣿⣷⠄⠀⠿⠿⠿⠶⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣯⣿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠉⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠠⣄⣤⣄⣠⣄⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠂⣀⠀⠁⠀⠁⢈⣙⠳⡄⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡆⠀⠀⢀⡀⠀⢀⡀⠈⠁⢙⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣀⠠⠀⠈⠉⠀⠀⠀⠀⠉⠙⠂⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠛⠀⠀⠀⢀⡄⠀⠰⠀⠀⠀⠀⣆⠸⠒⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⢲⠀⠀⠀⠀⠈⣇⠀⠀⡇⠀⠀⠀⠀⢀⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⢦⠤⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠲⢦⣄⣀⠀⠀⠀⢀⣀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⢤⡀⠀⠀⠀⣀⠴⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣄⠀⠈⢩⣭⣍⠀⢀⡹⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠙⠂⠻⣿⡿⠛⠁⠀⠀⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡄⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢸⡄⠀⠀⠀⠀⢸⣧⠀⠀⠀⢠⠻⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡌⡇⠀⠀⠀⠀⠘⢸⠀⠀⠀⢸⡆⢳⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⢰⠁⡇⠀⠀⠀⠀⣿⢚⡇⠀⠀⠀⢳⠈⣇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡎⢀⠃⠀⠀⠀⠀⣿⠈⣧⠀⠀⠀⠘⣆⠸⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡸⠀⡸⠀⠀⠀⠀⠀⡗⡉⢼⡀⠀⠀⠀⠸⡄⢱⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣥⡄⠀⢰⠁⢀⠃⠀⠀⠀⠀⢠⡗⡊⠈⡇⠀⠀⠀⠀⢳⠀⢧⠀⠐⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠈⠀⢰⠇⠀⡘⠀⠀⠀⠀⠀⢸⢋⠴⠊⣧⠀⠀⠀⠀⠈⣇⠘⣆⠀⠀⠀⠙⢦⡀
⡠⡔⢳⠎⠵⡄⠀⠀⢀⠴⠊⠀⠀⠀⠀⣠⠋⠀⢠⠁⠀⠀⠀⠀⠀⢸⠠⢂⡡⢼⡀⠀⠀⠀⠀⢸⡆⠈⢷⣄⠀⠀⠀⠙⠲⢄⡀⠀⢀⡒⢠⠉⡎
⡇⡃⠀⠀⢀⢣⡄⠺⢅⠀⠀⠀⢀⡠⠊⠁⠀⠀⡎⠀⠀⠀⠀⠀⠀⡾⢂⠡⠐⣘⠇⠀⠀⠀⠀⠈⣇⠀⠀⠙⠳⣄⠀⠀⠀⠊⠙⣶⢡⠀⠈⡇
⢣⠃⠆⠈⠘⡟⢙⣆⡈⣗⣠⠔⠉⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠻⣇⠔⢋⡼⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⣤⡖⢀⣊⡈⠹⠀⠀⠁⢁
⠈⠳⠤⣄⢠⡔⠉⠉⠉⠻⣄⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢦⡜⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣠⠔⠃⠉⠉⠉⠓⣄⠠⠒⠉
⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⣤⣙⠲⣄⡀⠀⠀⠈⣃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣀⠤⠒⢉⡠⠞⠓⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠘⢯⡀⠀⠀⠀⠀⠈⠓⠦⣍⡓⠦⣤⡿⠿⣿⣶⣶⣶⣶⣶⣶⡖⠲⣶⣶⣾⣿⣿⠿⠻⠗⢊⡡⠔⠊⠁⠀⠀⠀⠀⠀⠀⠰⠠⠁
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢥⣄⣀⠨⠭⢉⣉⡙⠛⢛⢛⠉⠉⠖⢁⣠⠴⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠤⠤⣀⣈⠠⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⢶⣄⣀⠀⠉⠛⢦⣄⣀⣈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⢸⠇⠀⠀⢀⠀⣤⡔⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠆⠼⣛⣻⣿⣿⣿⠿⣏⠁⠙⣦⡀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠉⠀⠀⡠⠛⠷⣶⣶⣭⠑⠆⣈⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⣭⣍⣙⡲⠬⠭⠒⠉⠉⠉⠉⠉⠓⠦⠤⠐⠒⠧⠤⠤⣤⣀⡤⠊⠉⠙⢦⣽⣿⣿⡛⠀⣡⠊
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⡈⣉⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢉⡟⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⣲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠑⢆⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣠⡀⡜⠀⠀⠀⠀⠀⠀⠈⠦⣰⢂⠜⠀⠀⠀⠀⠀⠀⠀⢣⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠢⠤⢩⣯⠤⠤⠤⡄⠀⣠⠖⠒⠒⠿⣠⠤⣄⠀⠀⡰⠒⠒⠒⠴⠧⠤⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢰⠋⠀⡿⠤⠒⡆⠀⠠⠦⣾⠀⠀⢸⡄⢠⠖⣢⠀⣀⡀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢰⠀⣿⠀⠀⢸⠈⠁⠀⡇⠸⠀⡇⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠈⠀⠁⠀⣿⠀⠀⢸⠀⠀⠀⠉⠊⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠀⠀⠈⢢⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠣⠤⠄⣀⣀⡀⠤⠋⢲⠀⠣⣄⣀⠀⠀⢀⡠⠇⠀⢠⡾⢄⡀⠀⠀⠀⢀⡸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠐⣒⠂⠀⠀⠀⠀⢀⡼⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠙⠦⠤⣄⣉⣉⡭⠤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠋⠉⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠢⡀⠀⠀⠀⠀⠀⠀⢠⠋⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠂⡅⠀⠀⠀⠀⠀⠀⠘⡄⢠⡠⠃⠀⠀⠀⠀⠀⠀⠀⢸⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠤⢼⣭⣤⠤⠤⢤⠀⠀⡠⠤⠬⣿⠥⠤⢄⡀⠀⠀⢀⠤⠄⠀⠛⠥⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡀⢠⠖⢆⣠⠲⢤⡞⠁⢸⠓⠒⠐⡄⠀⡔⠒⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢇⣀⠇⠀⡇⠀⠀⢸⢱⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠂⠠⡀⠀⠀⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⡇⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢣⠀⡉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⠀⠀⠀⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⣉⡐⠒⠒⠒⠒⠒⠊⠈⣃⠈⠒⠠⠄⠀⠠⠤⠚⠁⠀⣴⠋⠢⠤⠄⠀⠀⠤⠐⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡑⠒⠒⠀⠀⠈⢀⣀⣀⣈⠉⠬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠐⠒⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠉⠉⠙⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⢿⣿⣿⣿⣿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡴⢋⡙⠳⡄⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠉⢀⣀⠀⢛⡟⢠⡤⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠘⣇⠺⠼⢀⡗⣼⣿⣦⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⠀⠸⠿⠀⢠⣿⡤⣤⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠈⠓⢒⡯⠦⣝⠟⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣠⣤⣟⠁⣤⡀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢸⡸⡽⣸⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⡉⣠⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢧⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠛⠋